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quantum degenerate ultracold atoms with spin - degree of freedom exhibit both magnetic order and superfluidity , offering a rich system in which to explore quantum coherence , long - range order , magnetism and symmetry breaking . many aspects of spinor atoms in a trap have been investigated with spin @xmath3 atoms , such as @xmath0na and @xmath1rb @xcite . spin-2 @xcite and spin-3 @xcite spinor gases have been studied to a lesser extent . spinor condensates are described by a vector order parameter @xcite . the distinctive feature is its spin - dependent interaction which organizes spins giving rise to ferromagnetic and antiferromagnetic ( polar ) order . it can also coherently convert a spin @xmath4 and a @xmath5 atom to two @xmath6 atoms and vice versa @xcite , while conserving magnetization and energy . in parallel to ultracold spinor physics , optical lattices have become a powerful tool to create strongly correlated many - body states of bosons and fermions @xcite . lattice systems offer flexibility as the lattice parameters and particle interactions can be controlled easily , thereby facilitating progress towards the creation of quantum emulators @xcite . since the seminal observation of the superfluid to mott insulator transition with spinless bosons @xcite , steady progress is being made towards the understanding of spinor atoms in an optical lattice @xcite . issues of temperature and entropy @xcite are among the challenges that need to be overcome to create a many - body correlated state of spin-1 atoms . theoretical studies of lattice - trapped spinor condensates have mainly explored the phase diagram and the nature of the superfluid - mott insulator transition @xcite . due to the tunability of cold atom and optical lattice parameters , it is also possible to study non - equilibrium dynamics . dynamics of many - body quantum systems is still an emerging field , and only a number of issues have so far been investigated @xcite . an early experiment @xcite studied the dynamics of spinless bosons in a suddenly - raised optical lattice , observing the collapse and revival of the matter wave field in the momentum distribution . in a more - recent experiment @xcite , tens of oscillations in the momentum distribution or visibility were observed , and the predicted @xcite signature of effective higher - body interactions confirmed . as for spin-1 atoms , dynamical studies have mainly focused on large atom continuum or trapped systems in the mean field regime exploring spin - mixing dynamics @xcite , quantum quench dynamics @xcite , and various instabilities @xcite . the goal of this paper is to study the dynamics of spin-1 bosonic atoms in a three - dimensional ( 3d ) optical lattice and probe its many - body state and system properties . starting with a ferromagnetic ( @xmath1rb ) or antiferromagnetic ( @xmath0na ) superfluid ground state in a shallow lattice , suddenly raising the lattice depth creates a non - equilibrium state , which can be followed in various scenarios with and without a magnetic field and with and without effective three - body interactions . the evolution shows collapse and revival of matter - wave coherence measured by visibility oscillations , in a more complex pattern than for spinless bosons @xcite . it also shows oscillations in spin populations due to the combined effect of the spin - mixing collisions of the @xmath6 and @xmath7 components and differential level shifts proportional to the square of the magnetic field strength , the quadratic zeeman shift . linear zeeman shifts do not affect the behavior of spinor condensates . both spin - mixing and visibility oscillations reveal details about the system such as the composition of the initial many - body state , and thereby its superfluid and magnetic properties . by analyzing the frequency spectrum of the visibility , we show that the ratio @xmath8 of spin - dependent and spin - independent atom - atom interactions can be deduced . combined with spectra of spin - mixing dynamics at various magnetic field strengths , this gives us a method to measure the interaction couplings for spin-1 atoms . finally , we find that the presence of quadratic zeeman shift enhances spin mixing oscillations for ferromagnets and shows collapse and partial revivals in the transverse magnetization . the hamiltonian that quantitatively describes the physics of the final deep lattice comprises of two - body as well as effective multi - body interactions , which arise due to virtual excitations to higher bands . we derive the induced three - body interaction parameters for spin-1 atoms in a deep harmonic well , approximating the minimum of a single lattice site as such , and find the existence of _ spin - dependent _ three - body interactions . we show how to detect the signature of the three - body interactions and argue that they are directly exhibited in the _ in situ _ density as opposed to the time of flight visibility measurements as is the case for spinless bosons @xcite . the article is organized as follows . in sec . [ sec : model ] we setup the spin-1 bose - hubbard model , sketch the mean - field theory to determine the initial ground state , describe the exact hamiltonian after the quench , and discuss observables and computational aspects . we present our main results in sec . [ sec : polar ] , [ sec : ferro ] , and [ sec : three ] . section [ sec : polar ] explores the non - equilibrium dynamics of antiferromagnetic spin-1 atoms , with and without a magnetic field . section [ sec : ferro ] describes the dynamics for a ferromagnetic spinor . section [ sec : three ] shows the effects of effective three - body interactions in the dynamics . we summarize our results in sec . [ sec : conclusion ] . a derivation of the effective three - body interaction is given in the appendix . ultracold spin-1 bosons in the lowest band of a 3d cubic optical lattice and an external magnetic field @xmath9 along the @xmath10 axis are modeled by the free energy @xmath11 here @xmath12 is the creation operator of a boson in magnetic sublevel @xmath13 , or @xmath14 in the energetically - lowest wannier function or orbital of lattice site @xmath15 . the first term in eq . ( [ hamil ] ) represents the hopping of atoms between nearest neighbor sites and is proportional to the hopping energy @xmath16 . the second term describes the on - site atom - atom repulsion with strength @xmath17 , @xmath18 , and @xmath19 . the third term is a spin - dependent atom - atom interaction with a strength @xmath20 that can be either positive or negative . the three operators @xmath21 on site @xmath15 satisfy angular momentum commutation rules and are defined by @xmath22 for @xmath23 , @xmath24 , or @xmath10 and @xmath25 are matrix elements @xmath26 , @xmath27 of component @xmath28 of the spin-1 angular momentum @xmath29 . the fourth term corresponds to the quadratic zeeman energy of the magnetic sublevels with strength @xmath30 . finally , the terms containing the lagrange multipliers @xmath31 and @xmath32 control the total atom number , @xmath33 , and total magnetization , @xmath34 , respectively . here @xmath35 is the on - site magnetization . both @xmath36 and @xmath37 commute with @xmath38 . ( the large linear zeeman hamiltonian of the atoms is `` absorbed '' in the term @xmath39 and seen not to affect the physics of the spinor condensate . ) the interaction strengths are given by @xmath40 and @xmath41 @xcite , where @xmath42 with @xmath43 or 2 are scattering lengths for the two allowed values of the total angular momentum of @xmath44-wave collisions of two spin-1 particles at zero collision energy and zero magnetic field . @xmath45-wave scattering with total angular momentum @xmath3 is prohibited due to bosonic wave function symmetry . the mean density of the local orbital @xmath46 is determined by the laser parameters and polarizability of the atom . finally , @xmath47 is the mass of the atom , @xmath48 , and @xmath49 is planck s constant . the ratio of the two interaction strengths is independent of lattice parameters as the @xmath50 dependence cancels . for @xmath0na @xmath51 @xcite and the system is antiferromagnetic . for @xmath1rb @xmath52 @xcite and the system is ferromagnetic . the quoted uncertainty in @xmath8 for @xmath0na and @xmath1rb are one - standard deviation and obtained from the corresponding references . for @xmath2k we find @xmath53 @xcite . for our investigation we use @xmath54 for sodium and @xmath55 for rubidium . the quadratic zeeman strength @xmath56 with @xmath57 hz/(mt)@xmath58 for @xmath0na and @xmath59 hz/(mt)@xmath58 for @xmath1rb . the phase diagram for the spin-1 bose - hubbard model has been calculated with numerical methods such as quantum monte carlo @xcite and density - matrix renormalization group @xcite for a one - dimensional lattice . mean - field approaches for spin-1 bosons , which give predictions for any dimension , have also been performed @xcite and are extensions of those for scalar bosons @xcite . as mean - field models are most predictive in three dimensions and we focus on such a lattice , we use the decoupling mean - field theory @xcite to find the initial many - body ground state . in a mean - field approximation , the hopping term in eq . ( [ hamil ] ) can be decoupled as @xmath60 when fluctuations around the equilibrium value are negligible . we can define @xmath61 , for @xmath62 , as the site - independent superfluid order parameter . using eq . ( [ mfa ] ) we can rewrite eq . ( [ hamil ] ) as a sum of independent single site hamiltonians , @xmath63 where @xmath64 and @xmath10 is the number of nearest neighbors , e.g. @xmath65 in 3d . for a given @xmath31 and @xmath32 the superfluid order parameters and ground state wavefunction are obtained by finding those values of @xmath66 for which the energetically - lowest eigenstate of @xmath67 is smallest . the character of the ground state depends on whether the spin - dependent term @xmath20 is positive or negative @xcite . for the antiferromagnetic @xmath68 superfluid ground states , the order parameters can be written as @xmath69 , while for the ferromagnetic @xmath70 superfluid we have @xmath71 . here the functions @xmath72 are wigner rotation matrices @xcite with euler angles @xmath28 , @xmath73 , and @xmath74 determined by minimizing @xmath75 . the real valued @xmath76 and angle @xmath77 are the spin - independent superfluid density and a global phase , respectively . we have @xmath78 . within mean - field theory the many - body superfluid wavefunction is given by the product wavefunction @xmath79 over sites @xmath15 , where @xmath80 and kets @xmath81 are elements of the _ occupation - number _ basis of fock states of the three @xmath26 projections . the single - site wavefunction is a superposition of fock states with amplitudes @xmath82 . in fact , it is also a superposition of fock states with different magnetization . we only present results for ground states with zero magnetization @xmath83 at every site , ensured by setting @xmath84 . results for other magnetizations show similar physics . for @xmath68 the bosons condense into a state with @xmath85 . this is called a polar ( antiferromagnetic ) superfluid . there are two kinds of polar order @xcite : the ground state is a transverse polar state with @xmath86 when @xmath87 , and a longitudinal polar state with @xmath88 when @xmath89 . for ferromagnetic atoms with @xmath70 the magnetic order maximizes the total angular momentum with @xmath90 @xcite , leading to a partially - magnetized superfluid with order parameters @xmath91 for @xmath92 , and a longitudinal superfluid with order parameters @xmath93 for @xmath94 . our numerical simulations are performed in the occupation number basis . only basis functions with @xmath95 are included . we use @xmath96 leading to 84 basis functions in a site and negligible truncation errors when the mean atom number per site is less than three . all current optical - lattice experiments use mean atom numbers of this order of magnitude . after preparing the initial superfluid the depth of the optical lattice is suddenly increased so that tunneling is turned off . this lattice ramp - up is assumed to be slow enough to prevent excitations to a higher band yet fast enough compared to interactions . we can then treat subsequent time evolution due to the single - site hamiltonian @xmath97 exactly . as each site evolves under the same hamiltonian , we have suppressed the site index . for induced three - body interactions , additional terms appear in @xmath98 as discussed in sec . [ sec : three ] . recent observation of multi - body effects for lattice - trapped spinless bosons @xcite , where a similar quench was used , a mean - field treatment @xcite of the initial state followed by exact on - site evolution was found to agree well with the experiment . if lattice sites are not completely decoupled ( @xmath99 ) during the evolution , a correlated multi - site treatment is necessary @xcite . we do not study that scenario in this paper . following ref . @xcite we realize that , in addition to the occupation number basis , eigenfunctions of the operators @xmath100 , @xmath101 , and @xmath102 also form a complete basis for the on - site hilbert space of @xmath98 . in fact , these _ angular - momentum _ basis states @xmath103 diagonalize @xmath98 when @xmath104 with energy spectrum @xmath105 where @xmath106 is the local atom number and the quantum number @xmath107 is restricted to @xmath108 and even / odd @xmath107 for even / odd @xmath106 . the integer @xmath109 is the magnetization quantum number with @xmath110 . for @xmath111 the quadratic zeeman interaction couples the angular - momentum basis states . finding its matrix elements is involved , leading us to perform all simulations in the occupation number basis . to analyze the non - equilibrium dynamics of our system , we follow several observables . the first is the atom number per lattice site in each spin component , @xmath112 , which can be detected either in situ @xcite or , after release of the atoms from the lattice , by the stern - gerlach separation method where the spin states are first spatially separated and then detected @xcite . the second observable is the visibility , which is a measure of coherence of the wavefunction and equals the number of atoms with zero momentum in the spin - dependent momentum distribution . this is the standard quantity measured after releasing atoms from the lattice and a time of flight expansion @xcite . within our simulation it is given by @xmath113 for any site @xmath15 . finally , we study the square of the in - situ transverse magnetization @xmath114 . transverse magnetization can be measured by faraday rotation spectroscopy which allows for continuous observation of spin population in a bec @xcite . in this section we analyze the dynamics of a longitudinal polar superfluid ground state after the optical lattice strength is rapidly raised . the condensate evolves under @xmath98 with @xmath104 for hold time @xmath115 , after which one or more of the observables is measured . figure [ fig : mixingaf](a ) shows typical evolution of the in - situ population of spin components @xmath7 , @xmath116 as a function of hold time . here we use typical numbers for @xmath0na atoms in the initial lattice with @xmath117 , @xmath118 , and a mean atom - number per site of @xmath119 , and in the final lattice @xmath120 , where @xmath121 is the harmonic frequency near the minima of the lattice potential . ( an inifinitesimal @xmath89 is applied to ensure formation of the longitudinal polar state . ) ( color online ) spin - mixing dynamics after a sudden increase of lattice depth starting with the longitudinal antiferromagnetic ground state of spin-1 @xmath0na in a zero magnetic field . ( a ) the on - site atom number of the spin components , @xmath122 , as a function of hold time showing spin - mixing . time is in units of @xmath123 and system parameters are described in the text . ( b ) fourier spectrum of the time trace in ( a ) showing the frequencies involved in the spin - mixing dynamics . the inset shows the initial ground state fock state number probabilities @xmath124 as a function of atom number @xmath125 in spin component @xmath6 . there are no atoms with @xmath7 . ( c ) contribution @xmath126 to @xmath127 of having @xmath125 atoms in the @xmath6 state , where @xmath128 . the curves are for @xmath129 , and @xmath130 . it shows that the @xmath131 periodicity is due to @xmath132 and @xmath133 periodicity due to @xmath134 and 3 , respectively.,scaledwidth=40.0% ] initially , all atoms are in the @xmath6 state and as time evolves atoms begin to appear in states @xmath7 because of spin changing collisions , and a pattern of periodic modulation emerges with a period of @xmath123 , while conserving zero magnetization . this spin - mixing time trace also contains information about the composition of the initial many - body state . to explore this , we further analyze the dynamics in fig . [ fig : mixingaf](b ) and ( c ) . figure [ fig : mixingaf](b ) shows the fourier analysis of the time trace in panel ( a ) , and the inset shows on - site fock state probabilities @xmath135 for the initial state . the shape of the initial state number distribution is characteristic of a nearly coherent or a slightly - squeezed state . in the frequency spectrum , peaks are observed at frequencies that are integer multiples of @xmath136 . in fact , they occur at @xmath137 , @xmath138 , and a small contribution at @xmath139 . these features can be understood from an analysis of the eigenenergies in eq . ( [ energy ] ) of @xmath98 at @xmath104 . similar to number fock state composition of the initial state , it is also a superposition of angular momentum states @xmath140 . the observables @xmath141 commute with total atom number @xmath100 and thus only measure the coherence between states with different @xmath107 but the same @xmath106 . for states with @xmath142 the two allowed @xmath107 are @xmath116 and @xmath143 with energy difference @xmath144 . this leads to a peak at @xmath145 in fig . [ fig : mixingaf](b ) . for @xmath146 states , @xmath3 and @xmath147 exist leading to the frequency at @xmath148 . the small feature at @xmath149 indicates the presence and mixing of @xmath150 and @xmath151 states . the above analysis of the eigen energies is confirmed in fig . [ fig : mixingaf](c ) . it depicts the time evolution of the contribution @xmath126 to @xmath127 of having @xmath125 atoms in the @xmath6 state , where @xmath128 . atom number @xmath132 has a period of @xmath152 as we oscillate between states @xmath153 and @xmath154 with a total of two atoms , while that for @xmath134 and @xmath155 has a period of @xmath156 . here , we oscillate between the three atom states @xmath157 and @xmath158 . the spin - mixing dynamics can be compared and contrasted with other spin-1 experiments . in ref . @xcite , a pair of @xmath3 @xmath1rb atoms were prepared in a single site of a deep optical lattice in the fock state @xmath159 , and allowed to spin - mix with @xmath160 . spin mixing oscillations between two levels analogous to rabi oscillations were observed with a single frequency . on the other extreme , spin - mixing for a spinor bec with a large number of atoms has been discussed in theory and observed in experiments @xcite . they are in a regime where a classical pendulum phase - space analysis is appropriate @xcite , and although there can be spin - mixing oscillations for specific initial states , quantum recurrences due to the discrete energy spectrum is absent . our analysis here explores the regime which is between these two the single fock state regime and the regime of large atom number condensate . as such , we are exploring a regime which can shed light on the semi - classical transition to large condensate dynamics , a topic for future investigation . in our analysis here , we can analyze the multiple frequencies of the dynamics time trace to probe the composition and atom number statistics of the initial many - body state . figure [ fig : visaf ] shows the dynamics of visibility of the @xmath6 state the occupation of the zero momentum state for @xmath6 component , for the same initial state and parameters as in fig . [ fig : mixingaf ] . the visibility @xmath161 measures the phase coherence in this spinor superfluid system . we show the relative visibility @xmath161/@xmath162 in fig . [ fig : visaf](a ) . we see that the atoms oscillate between being completely coherent to completely incoherent ( @xmath163 ) . the pattern is more complex than the spinless boson visibility @xcite . there is a fast oscillation with time scale @xmath164 , which is modified by a slower envelope with a time scale @xmath165 . the exact nature of the complex oscillations is revealed in the frequency spectrum shown in fig . [ fig : visaf](b ) . similar to fig . [ fig : mixingaf ] features appear at small integer multiples of @xmath136 . here they are located at @xmath166 , @xmath137 , and @xmath138 . peaks also occur at much larger frequencies with a dominant frequency at @xmath167 , which is @xmath168 in this example . twenty six is also the number of fast oscillations in a full period @xmath123 as can be seen in panel ( a ) . this indicates that for an unknown ratio @xmath8 , one full revival oscillations of the visibility can help determine this ratio by counting the number of fast oscillations or equivalently performing a frequency analysis . combined with the realization that this ratio is independent of lattice parameters , this method of determining spinor interactions is one of the key findings of this paper . the visibility spectrum frequencies appear because the expectation value of the annihilation operator is sensitive to the overlap between fock states of different atom number @xcite . for example @xmath169 connects to @xmath170 giving rise to the dominant frequency @xmath171 . other frequencies can similarly be explained by performing an expansion of the initial state in the angular momentum basis , and applying the evolution operator for the final hamiltonian eq . ( [ final ] ) at @xmath104 . for higher occupation numbers not shown here we find that the visibility patterns become more complex . in fact , by controlling the initial squeezing @xcite ( i.e. by controlling the initial tunneling energy @xmath16 ) as well as the total occupation , we can change the complexity of the frequency spectrum and thereby make it amenable for analysis . ( color online ) ( a ) visibility @xmath172/@xmath173 of the @xmath6 state as a function of hold time @xmath115 after a sudden increase of the lattice depth , for the same parameters as in fig . [ fig : mixingaf ] . the pattern of collapse and revival of coherence is more complex than the spinless boson case due to the competition between spin - dependent and spin - independent interactions . there are @xmath174 fast oscillations in one full collapse and revival period of @xmath123 , yielding a method to determine the ratio @xmath8 from visibility oscillations . ( b ) spectrum of the visibility oscillations showing the range of contributing frequencies , the most dominant one being at @xmath175.,scaledwidth=45.0% ] the energies of three component spin-1 atoms are sensitive to external magnetic fields . such external fields have been exploited in manipulating spinor atoms in an optical trap to access ground state properties @xcite , in detection such as in stern gerlach separation method @xcite , or to influence the dynamics in quench experiments @xcite . for spinor atoms in an optical lattice external magnetic fields can not be ignored and , in fact , lead to unique physics . for example , the quadratic zeeman shift affects the phase diagram @xcite . the ratio of quadratic shift to the spin - dependent interaction strength , @xmath176 , controls the physics of this system . two different regimes emerge the zeeman regime for @xmath177 and the interaction regime for @xmath178 @xcite . ( color online ) spin - mixing oscillations in the presence of a magnetic field in the form of a quadratic zeeman shift . panel ( a ) shows the population in @xmath6 as a function of hold time for several values of the quadratic zeeman shift . panel ( b ) shows the frequency analysis of panel ( a ) with two dominant frequencies in the time evolution . at @xmath104 these frequencies are @xmath137 and @xmath138 . the inset shows that the two dominant frequencies as a function of @xmath30 first dips before slowly increasing due to the competing nature of @xmath30 and @xmath20 . ] figure [ fig : afzeeman ] shows an analysis of spin - mixing oscillations @xmath179 in the presence of a quadratic zeeman shift @xmath30 , during the initial state preparation and during the evolution . the other parameters are as in fig . [ fig : mixingaf ] . to highlight the effects of a magnetic field , we show a comparison of the dynamics for @xmath180 , and @xmath181 in fig . [ fig : afzeeman](a ) . we see that the oscillations become faster while simultaneously the amplitudes get smaller for increasing @xmath30 in the zeeman regime @xmath177 . the spin - mixing dynamics vanishes for a large enough @xmath9-field . in panel ( b ) we plot their frequency spectra comparing the frequencies and the amplitudes . the inset shows two dominant frequencies as a function of the zeeman strength . a closer look at panel ( b ) reveals that for a @xmath30 in the interaction regime @xmath182 , the dominant frequency initially decreases before starting to increase , due to a competition between the spin - dependent interaction and the zeeman term . much of the @xmath9-field effects can be understood from the eigenvalues of @xmath98 calculated in the fock state basis @xcite using conservation of atom number and magnetization . for up to three atoms per lattice site this involves diagonalizing at most a @xmath183 matrix . the energy splitting of the eigenvalues of the @xmath183 matrices is observed . in fact , from top to bottom the two curves in the inset of panel ( b ) are due to contributions from two - atom and three - atom fock states . in a condensate with large particle numbers the competition between interaction and zeeman energy gives rise to a sharp phase boundary at @xmath184 which can be manifested through magnetic field induced resonances @xcite . ( color online ) ( a ) spin - mixing dynamics @xmath185 for a ferromagnetic @xmath186 rubidium condensate as a function of hold time . ( b ) fourier spectrum showing the relevant frequencies of 3@xmath136 , 5@xmath136 and 7@xmath136 , and their relative amplitudes . ( c ) visibility @xmath187/@xmath188 as a function of hold time . ( d ) frequency spectrum of the visibility showing features at @xmath189 and @xmath190 , pointing a way to measure interaction ratios @xmath8 from the visibility spectrum.,scaledwidth=50.0% ] figure [ fig : ferro ] shows the dynamics when our initial state is a ferromagnetic superfluid state of @xmath1rb with @xmath70 created in a shallow lattice . for a ferromagnetic state , the collective spin configuration is such that the spin - dependent interaction energy is maximized . this means that the ground state is a superposition of magnetization states in the angular momentum basis @xmath140 with variance @xmath191 although it still has magnetization @xmath192 . the parameters are @xmath193 , @xmath117 and @xmath194 in the shallow lattice and @xmath195 and @xmath194 in the final deep lattice . the population dynamics @xmath185 is shown in fig . [ fig : ferro](a ) . the oscillation amplitude is not as large as in the polar case in the previous section . there are two reasons for this : first , the ground state has comparable populations in all three components and therefore , the population difference between the spin components is smaller to begin with , unlike for @xmath0na where initially only the @xmath6 state is occupied . second , the initial state is much closer to the ground state in the deep lattice . we would like to point out that the smallness of the ferromagnetic interaction is not responsible for the oscillation amplitudes being small . a fourier analysis in panel ( b ) of the @xmath6 population shows that the frequencies present are @xmath145 , @xmath148 and @xmath149 , as in the polar state . again , energy differences obtained from eq . ( [ energy ] ) give us those frequencies . these frequencies and their spectral weight determine the composition of different fock components in the initial many - body state , and therefore , can be used as an experimental probe . we will show in the next subsection that adding magnetic - fields can enhance the amplitude of spin - mixing dynamics . the visibility @xmath187/@xmath188 is shown in fig . [ fig : ferro](c ) . it has a simpler pattern than for the polar case in fig . [ fig : visaf ] . the coherence of the initial matter - wave exhibits collapse and revival modulations with a fast timescale of @xmath196 . its frequency spectrum in panel ( d ) shows two dominant frequencies , @xmath189 and @xmath190 , although other frequencies with extremely small amplitudes do exist . for @xmath1rb the dominant frequencies are @xmath197 and @xmath198 in units of @xmath136 . as explained for @xmath0na , these frequencies appear from the overlap of fock states connected by the annihilation operator @xmath199 @xcite . here , the peak at @xmath189 appears due to the overlap of number states @xmath170 and @xmath159 . similarly , the peak at @xmath190 appears due to number states @xmath159 and @xmath200 . as with @xmath0na , finding these frequencies yields a method to experimentally determine @xmath8 . nevertheless , since we need to observe many oscillations , @xmath201 for @xmath1rb and @xmath202 for @xmath2k , this will be a challenging application of quantum phase revival spectroscopy . ( color online ) effect of quadratic zeeman interactions for @xmath1rb dynamics for @xmath184 and @xmath203 for the top and bottom row respectively . ( a ) and ( c ) shows spin mixing dynamics of the @xmath6 state , ( b ) and ( d ) shows in - situ transverse magnetization @xmath204 oscillations . spin - mixing and @xmath204 oscillations depend on a combination of @xmath30 , @xmath205 , @xmath20 and the initial state . we see that the spin - mixing amplitudes increase as we increase the field . on the other hand transverse magnetization oscillations get weaker.,scaledwidth=50.0% ] in this subsection we study the effects of the quadratic zeeman interaction on a ferromagnetic spinor . in fig . [ fig : bferro ] we show the dynamics when the quadratic zeeman shift is @xmath184 and @xmath203 for the top and bottom row respectively . we plot spin population @xmath127 and in - situ transverse magnetization @xmath206 for @xmath1rb . the parameters are @xmath207 , @xmath117 and @xmath194 in the shallow lattice and @xmath195 and @xmath194 in the final deep lattice . in all the cases , the oscillations are no longer periodic in @xmath123 , but the values of @xmath208 and @xmath20 combined with the initial state composition influence the dynamics . we find that the spin - mixing modulation amplitudes become large in the presence of a b - field unlike the antiferromagnetic case in the previous section . the oscillations also get faster , and the fast spin oscillations are modified by an envelope of a slower modulation pattern which is manifested in a beat - like pattern involving the two dominant frequencies as clearly evident in fig . [ fig : bferro](c ) . as we increase @xmath30 , the population differences in the @xmath6 and @xmath7 states become larger , and therefore a large spin - mixing modulations can occur . transverse magnetization @xmath209 shown in ( b ) and ( d ) goes through partial revivals and complete collapse . faraday rotation spectroscopy can be used to detect transverse magnetizations @xcite . this could give us a direct probe of the magnetic properties in our quench set up , in addition to the coherence properties which we can obtain through population and visibility revivals . we show in panels ( b ) and ( d ) that the initial amplitude for @xmath209 decreases for increasing b - field , but the number of collapse sequences increases . when @xmath94 the transverse ferromagnetic superfluid turns into a longitudinal superfluid for which @xmath210=0 . in that regime transverse magnetization is zero throughout the evolution , but large amplitude spin - mixing modulations still occur . in our dynamical scheme , the evolution of the many - body state takes place in a deep optical lattice with no tunnelling to the neighbors . in such a setting , even within the single - band bose - hubbard model , there are effective three and higher - body interactions due to collision - induced virtual excitations to higher bands or vibrational levels . for the spinless bosonic case , such effective multi - body interactions have been predicted in theory @xcite . a recent experiment @xcite monitored matter wave collapse and revival dynamics for tens of oscillations and observed the signature of higher - body effects in the visibility time trace . a more accurate treatment of a multi - component system in a deep lattice should therefore also incorporate higher - band induced multi - body interaction terms . three - body interactions can be important for efimov physics @xcite and for many - particle systems giving rise to novel and exotic phenomena @xcite . in a deep lattice , the minimum of the potential at a single site can be approximated as a harmonic potential . for spin-1 bosons in a single isotropic harmonic trap , the derivation of the effective three - body interactions is given in the appendix . the effect can be concisely represented by adding @xmath211 to @xmath98 in eq . ( [ final ] ) . here , as previously , @xmath212 is the on - site atom number and @xmath213 is the on - site total angular momentum . the hamiltonian term with strength @xmath214 is similar to the spin-0 effective three - body interaction of refs . @xcite and only depends on total particle number . the new hamiltonian term with strength @xmath215 depends intricately on the atomic spin . for a harmonic potential with frequency @xmath121 , @xmath214 is attractive and equal to @xmath216 @xcite . the strength @xmath215 satisfies @xmath217 . there is no magnetic field dependence in the effective three - body terms . finally , we note that the perturbative effective three - body hamiltonian is only valid when the three - body interaction strengths ( @xmath218 ) are much smaller than the corresponding two - body strengths ( @xmath219 ) . the on - site effective hamiltonian @xmath220 is diagonal in the angular momentum basis states @xmath103 with diagonal matrix elements @xmath221 with the same restrictions as in eq . ( [ energy ] ) on allowed values of @xmath107 . equation ( [ spectrum3b ] ) is one of the key results in this paper . this spectrum extends and generalizes the spectrum of spin - mixing spinor hamiltonian as presented in ref . @xcite to the effective three - body case . as discussed next , this helps us quantify and understand the effects of three - body interactions on our dynamics scenario . effects of effective three - body interactions on the spin - mixing dynamics for a polar ( @xmath0na ) initial state . ( a ) the population @xmath222 as a function of hold time for occupation @xmath223 and @xmath104 with ( solid line ) and without ( dashed line ) the effective three - body interaction , ( b ) frequency analysis of the time trace with effective three body interactions reveals the presence of additional frequencies , which could be used to determine the three - body interaction strength.,scaledwidth=45.0% ] for spinless bosons in an optical lattice , a time of flight measurement of the visibility dynamics determined the strength of the effective multi - body interactions @xcite . here for spin-1 bosons in an optical lattice , we show that the effective three - body interaction effects can be observed directly in the on - site population density in the spin oscillation dynamics . this opens up the possibility that in situ measurements such as quantum microscopes @xcite and other techniques @xcite in lattices could be used to detect effective multi - body interactions . because of the more complex nature of visibility patterns , we only analyze spin - mixing population dynamics . in fig . [ fig : af3b ] we show the effects of the effective three - body interactions in the spin - mixing dynamics of an initial polar superfluid state of @xmath0na at @xmath117 , @xmath224 and @xmath104 . the occupation is @xmath225 , which highlights the three - body effects , and @xmath226 . the interaction strengths for the deep lattice are @xmath227 and @xmath228 , so that @xmath229 and @xmath230 . in an earlier section we have seen that spin - mixing dynamics is controlled by @xmath20 . here , we find that this spin - mixing scaling is also influenced by the three - body interaction @xmath215 . figure [ fig : af3b](a ) shows a comparison of the dynamics with and without the three - body term . we see that the time traces start to differ after the first oscillation , and the periodicity in @xmath123 is destroyed . a frequency analysis of the oscillations in panel ( b ) elucidates the exact nature of the modulations . without a three - body term , strong features appear at @xmath231 , and @xmath149 . with a three - body term , additional frequencies appear at @xmath232 , and @xmath233 , which follow from the energy differences of the three - body spectrum in eq . ( [ spectrum3b ] ) . identification of any of the frequencies gives us the @xmath215 coupling strength . for example , the peak at @xmath234 arises as the initial state contains contributions of angular momentum states @xmath235 and @xmath236 containing three atoms . these two states have an energy difference of @xmath237 whose signature is the @xmath234 frequency . in experiments an unknown effective three - body strength @xmath215 can be deduced by assigning several frequencies in the time trace . the presence of all the other frequencies can be used to reduce error in the measurement and verify the spectrum eq . ( [ spectrum3b ] ) . the spectral weights and the values of the frequencies also give us clue about the initial superfluid state . control of initial squeezing @xcite , by varying @xmath238 in the shallow lattice , can be used in such a way that some of the frequencies are more dominant so to make detection easier . effects of effective three - body interactions on the spin - mixing dynamics for a ferromagnetic ( @xmath1rb ) initial state . ( a ) the population @xmath179 as a function of hold time with ( solid line ) and without ( dashed line ) the three - body interaction . for @xmath1rb the oscillation amplitude is small . ( b ) frequency analysis reveals the presence of several frequencies , which are a signature of induced three - body interactions ; compared to @xmath0na in fig . [ fig : af3b ] , one frequency is missing for @xmath1rb due to its ferromagnetic nature.,scaledwidth=45.0% ] in fig . [ fig : ferro3b ] we show the spin dynamics @xmath239 for a ferromagnetic ( @xmath1rb ) initial state at @xmath117 , @xmath104 , @xmath226 , and @xmath240 in the shallow lattice . in the deep lattice we use @xmath195 and @xmath194 . the spin - mixing amplitude is not that prominent for @xmath1rb . nevertheless , the influence of the three - body interaction is discernible in our simulation . the frequency spectrum of the time trace shown in ( b ) makes it clear that new frequencies emerge at @xmath234 and @xmath233 proving the existence of three - body interactions and yielding a method to measure its strength . the appearance of additional frequencies is similar to the @xmath0na case , except that we do not observe a feature at @xmath241 , which is due to a coherence between four - atom angular momentum states @xmath242 and @xmath151 . for the ferromagnetic initial superfluid the angular momentum state @xmath242 is absent . it is conceivable that initial squeezing control of a ground state @xcite or other specific initial state preparations @xcite can be used to see larger amplitude spin - mixing oscillations to make the experimental detection of three - body effects easier for ferromagnetic coupling . in this paper we performed a theoretical study of the dynamics of spin-1 bosons in an optical lattice in a quench scenario where we start from a ground state in a shallow lattice and suddenly raise the lattice depth . we have shown that the ensuing spin - mixing and visibility oscillations can be used as a probe of the initial superfluid ground state . the spectral analysis of time evolution reveals the fock state composition of the initial state and thereby its superfluid and magnetic properties . analysis of visibility oscillations , i.e. quantum phase revival spectroscopy , further yields a method to determine the spin - dependent and spin - independent interaction ratio @xmath8 , which is an important quantity for spinor gases . we treat both antiferromagnetic ( e.g. @xmath0na atoms ) and ferromagnetic ( e.g. @xmath1rb ) condensates . for ferromagnetic interactions the spin - mixing oscillation amplitudes are small . when external magnetic field can not be ignored , the inclusion of a quadratic zeeman field is necessary , and we have quantified such dynamics . we have shown that the presence of a magnetic field increases the spin - mixing amplitude for a ferromagnetic condensate . the hamiltonian that more accurately describes the physics of the final deep lattice comprises of two - body as well as effective multi - body interactions , which arise due to virtual excitations to higher bands . we derive the induced three - body interaction parameters for spin-1 atoms in a deep harmonic well and show its effect on the spin - mixing dynamics . we demonstrate that a frequency analysis of the oscillations can detect the signature and strength of the spin - dependent three - body interactions . we stress our finding that the three - body interactions for spinor atoms can be observed directly in the _ in - situ _ number densities in addition to the time of flight visibility as observed for spinless bosons @xcite . although there have been many theoretical studies for spin-1 bosons in an optical lattice , a many - body correlated ground state has not yet been achieved experimentally . there are many unexplored questions in that regard . here , we have combined the study of dynamics with optical lattice spinors to show how non - equilibrium dynamics can be used as a probe for revealing ground state properties and spinor interactions . there are other dynamic scenarios that can give different perspectives on spinor lattice physics , such as a quench from mott insulator to superfluid , evolution in a tunnel coupled lattice , to name a few . interplay of superfluidity , magnetism and strong correlations makes this a rich system where study of quantum dynamics may lead to a better understanding of collective phenomena . in this appendix we derive the effective three - body spinor interaction due to virtual excitations to excited trap levels of the isolated sites of the deep optical lattice . the derivation closely follows refs . atoms are held in the ground state of a site , which we approximate by an isotropic 3d harmonic - oscillator potential with frequency @xmath121 . then the hamiltonian for spin-1 bosons is @xmath243 with @xmath244 where operators @xmath245 annihilate an atom in spin projection @xmath246 and 3d harmonic oscillator state @xmath28 . we use the convention that roman and greek subscripts represent spin projections and harmonic oscillator states , respectively . repeated indices are summed over . the single - particle energies @xmath247 and @xmath248 are the harmonic oscillator and quadratic zeeman energies , respectively . for the ground state @xmath249 . we choose @xmath250 and will require that @xmath251 for @xmath252 . the spin - independent and spin - dependent atom - atom interactions @xmath214 and @xmath215 have `` bare - coupling '' strengths @xmath253 and @xmath254 , respectively . the vector @xmath255 are spin-1 matrices . each symmetric real coefficient @xmath256 is a 3d integral over the product of the four oscillator wavefunctions @xmath28 , @xmath73 , @xmath74 , and @xmath30 . in this appendix the two atom - atom interactions are explicitly normal - ordered in order to facilitate the derivation . we can now derive the effective hamiltonian with atoms in the lowest oscillator state using degenerate perturbation theory with zeroth - order hamiltonian @xmath257 and perturbation @xmath258 . first , we define for every spatial mode @xmath28 the spin wavefunction @xmath259 with @xmath260 atoms in spin state @xmath26 . the ground states @xmath261 are formed by the orthonormal basis functions @xmath262 for any value of @xmath263 and where @xmath264 indicates that there are no atoms in excited spatial modes . their energy is @xmath265 . excited states @xmath266 with energy @xmath267 are states where at least one atom occupies a @xmath268 spatial mode . we reproduce the hamiltonian in eq . ( [ final ] ) in first - order perturbation theory once we make the assignment @xmath269 and @xmath270 . to second - order in degenerate perturbation theory the matrix element for ground states @xmath271 and @xmath272 is @xmath273 the sum over excited states can be evaluated by realizing that only states @xmath274 with @xmath275 and @xmath276 contribute as both @xmath214 and @xmath215 conserve atom number and magnetization , and can only change the state of two atoms at the same time , i.e. states @xmath266 are those with at most two atoms in the higher trap levels . by inspection , we then realize that the energy differences @xmath277 are independent of the total number of atoms in the ground states and to good approximation are also independent of the quadratic zeeman energy , as @xmath278 for @xmath268 . similar expressions hold for @xmath279 . inserting the ( normalized ) expression for @xmath266 and performing the sums over @xmath266 as well as those appearing in the potentials @xmath214 and @xmath215 we first find @xmath280 where the remaining sums over trap levels have been made explicit , repeated roman indices are still summed over , and we have omitted the contribution proportional to @xmath215 times @xmath215 as the spin - dependent interaction strength is an order of magnitude smaller than the spin - independent one . by normal ordering the creation and annihilation operators in eq . ( [ first ] ) and using that ground states @xmath261 contain no atoms in excited trap levels we find @xmath281 where @xmath282 is a correction to the pair - wise two - body interaction and @xmath283 is an effective three - body interaction . here @xmath284 and @xmath285 the sums in @xmath286 diverge and must be regularized and renormalized @xcite . that is we require that the bare coupling constant @xmath253 is defined such that , by combining the first- and second - order contributions , @xmath287 is finite and equal to @xmath205 . similarly , we require that @xmath288 is finite and equal to @xmath20 . the sums in coefficient @xmath289 of the effective three - body hamiltonian do converge . hence , we redefine the effective three - body interaction as @xmath290 where , consistent within our second - order perturbative calculation , @xmath291 are the spin - independent and spin - dependent three - body interaction strength , respectively . for a spherically symmetric harmonic oscillator @xmath292 @xcite . since @xmath68 for na atoms , @xmath293 , while for @xmath1rb @xmath70 and thus @xmath294 . the two- and three - body interaction can be rewritten in terms of the angular momentum operators @xmath295 defined in eq . ( [ hamil ] ) following ref . @xcite . by combining the quadratic zeeman interaction as well as the two - body and effective three - body interaction we find our final result @xmath296 where we suppressed the ground - state oscillator index , and @xmath297 and the number operator @xmath100 commute . kwm would like to thank richard scalettar and hulikal krishnamurthy for discussions during early stages of this work . we acknowledge support from the us army research office under contract no . 60661ph and the national science foundation physics frontier center located at the joint quantum institute . y. liu , e. gomez , s. e. maxwell , l. d. turner , e. tiesinga , and p. d. lett , phys . lett . * 102 * , 225301 ( 2009 ) ; y. liu , s. jung , s. e. maxwell , l. d. turner , e. tiesinga , and p. d. lett , phys . lett . * 102 * , 125301 ( 2009 ) ; a. widera , f. gerbier , s. flling , t. gericke , o. mandel , and i. bloch , phys . rev . lett . * 95 * , 190405 ( 2005 ) ; a. widera , f. gerbier , s. flling , t. gericke , o. mandel , and i. bloch , new j. phys , * 8 * , 152 ( 2006 ) .

we study the dynamics of spin-1 atoms in a periodic optical - lattice potential and an external magnetic field in a quantum quench scenario where we start from a superfluid ground state in a shallow lattice potential and suddenly raise the lattice depth . the time evolution of the non - equilibrium state , thus created , shows collective collapse - and - revival oscillations of matter - wave coherence as well as oscillations in the spin populations . we show that the complex pattern of these two types of oscillations reveals details about the superfluid and magnetic properties of the initial many - body ground state . furthermore , we show that the strengths of the spin - dependent and spin - independent atom - atom interactions can be deduced from the observations . the hamiltonian that describes the physics of the final deep lattice not only contains two - body interactions but also effective multi - body interactions , which arise due to virtual excitations to higher bands . we derive these effective spin - dependent three - body interaction parameters for spin-1 atoms and describe how spin - mixing is affected . spinor atoms are unique in the sense that multi - body interactions are directly evident in the _ in - situ _ number densities in addition to the momentum distributions . we treat both antiferromagnetic ( e.g. @xmath0na atoms ) and ferromagnetic ( e.g. @xmath1rb and @xmath2k ) condensates .

1304.7565

video compression is a major requirement in many of the recent applications like medical imaging , studio applications and broadcasting applications . compression ratio of the encoder completely depends on the underlying compression algorithms . the goal of compression techniques is to reduce the immense amount of visual information to a manageable size so that it can be efficiently stored , transmitted , and displayed . 3-d dwt based compressing system enables the compression in spatial as well as temporal direction which is more suitable for video compression . moreover , wavelet based compression provide the scalability with the levels of decomposition . due to continuous increase in size of the video frames ( hd to uhd ) , video processing through software coding tools is more complex . dedicated hardware only can give higher performance for high resolution video processing . in this scenario there is a strong requirement to implement a vlsi architecture for efficient 3-d dwt processor , which consumes less power , area efficient , memory efficient and should operate with a higher frequency to use in real - time applications . + from the last two decades , several hardware designs have been noted for implementation of 2-d dwt and 3-d dwt for different applications . majority of the designs are developed based on three categories , viz . ( i ) convolution based ( ii ) lifting - based and ( iii ) b - spline based . most of the existing architectures are facing the difficulty with larger memory requirement , lower throughput , and complex control circuit . in general the circuit complexity is denoted by two major components viz , arithmetic and memory component . arithmetic component includes adders and multipliers , whereas memory component consists of temporal memory and transpose memory . complexity of the arithmetic components is fully depends on the dwt filter length . in contrast size of the memory component is depends on dimensions of the image . as image resolutions are continuously increasing ( hd to uhd ) , image dimensions are very high compared to filter length of the dwt , as a result complexity of the memory component occupied major share in the overall complexity of dwt architecture . + convolution based implementations @xcite-@xcite provides the outputs within less time but require high amount of arithmetic resources , memory intensive and occupy larger area to implement . lifting based a implementations requires less memory , less arithmetic complex and possibility to implement in parallel . however it require long critical path , recently huge number of contributions are noted to reduce the critical path in lifting based implementations . for a general lifting based structure @xcite provides critical path of @xmath0 , by introducing 4 stage pipeline it cut down to @xmath1 . in @xcite huang et al . , introduced a flipping structure it further reduced the critical path to @xmath2 . though , it reduced the critical path delay in lifting based implementation , it requires to improve the memory efficiency . majority of the designs implement the 2-d dwt , first by applying 1-d dwt in row - wise and then apply 1-d dwt in column wise . it require huge amount of memory to store these intermediate coefficients . to reduce this memory requirements , several dwt architecture have been proposed by using line based scanning methods @xcite-@xcite . huang et al . , @xcite-@xcite give brief details of b - spline based 2-d idwt implementation and discussed the memory requirements for different scan techniques and also proposed a efficient overlapped strip - based scanning to reduce the internal memory size . several parallel architectures were proposed for lifting - based 2-d dwt @xcite-@xcite . y. hu et al . @xcite , proposed a modified strip based scanning and parallel architecture for 2-d dwt is the best memory - efficient design among the existing 2-d dwt architectures , it requires only 3n + 24p of on chip memory for a n@xmath3n image with @xmath4 parallel processing units ( pu ) . several lifting based 3-d dwt architectures are noted in the literature @xcite-@xcite to reduce the critical path of the 1-d dwt architecture and to decrease the memory requirement of the 3-d architecture . among the best existing designs of 3-d dwt , darji et al . @xcite produced best results by reducing the memory requirements and gives the throughput of 4 results / cycle . still it requires the large on - chip memory ( @xmath5 ) . in this paper , we propose a new parallel and memory efficient lifting based 3-d dwt architecture , requires only @xmath6 words of on - chip memory and produce 8 results / cycle . the proposed 3-d dwt architecture is built with two spatial 2-d dwt ( cdf 9/7 ) processors and four temporal 1-d dwt ( haar ) processors . proposed architecture for 3-d dwt replaced the multiplication operations by shift and add , it reduce the cpd from @xmath7 to @xmath8 . further reduction of cpd to @xmath9 is done by introducing pipeline in the processing elements . to eliminate the temporal memory and to reduce the latency , haar wavelet is incorporated in temporal processor . the resultant architecture has reduce the latency , on chip memory and to increase the speed of operation compared to existing 3-d dwt designs . the following sections provide the architectural details of proposed 3-d dwt through spatial and temporal processors . organization of the paper as follows . theoretical background for dwt is given in section ii . detailed description of the proposed architecture for 3-d dwt is provided in section iii . implementation results and performance comparison is given in section iv . finally , concluding remarks are given in section v. + lifting based wavelet transform designed by using a series of matrix decomposition specified by the daubechies and sweledens in @xcite . by applying the flipping @xcite to the lifting scheme , the multipliers in the longest delay path are eliminated , resulting in a shorter critical path . the original data on which dwt is applied is denoted by @xmath10 $ ] , and the 1-d dwt outputs are the detail coefficients @xmath11 $ ] and approximation coefficients @xmath12 $ ] . for the image ( 2-d ) above process is performed in rows and columns as well . eqns.(1)-(6 ) are the design equations for flipping based lifting ( 9/7 ) 1-d dwt @xcite and the same equations are used to implement the proposed row processor ( 1-d dwt ) and column processor ( 1-d dwt ) . [ lift_2d_eq ] @xmath13 & \leftarrow a'*x[2n-1]+\{x[2n]+x[2n-2]\ } \ldots p1\\ { l_1}[n]&\leftarrow b'*x[2n]+\{{h_1}[n]+{h_1}[n-1]\ } \ldots u1\\ { h_2}[n ] & \leftarrow c'*{h_1}[n]+\{{l_1}[n]+{l_1}[n-1]\}\ldots p2\\ { l_2}[n ] & \leftarrow d'*{l_1}[n]+\{{h_2}[n]+{h_2}[n-1]\}\ldots u2\\ h[n ] & \leftarrow k0 * \{{h_2}[n]\}\\ l[n ] & \leftarrow k1 * \{{l_2}[n]\}\end{aligned}\ ] ] where @xmath14 , @xmath15 , @xmath16 , @xmath17 , @xmath18 , and @xmath19 @xcite . the lifting step coefficients @xmath20 , @xmath21 , @xmath22 , @xmath23 and scaling coefficient @xmath24 are constants and its values @xmath25 , @xmath26 , @xmath27 , and @xmath28 , and @xmath29 lifting based wavelets are always memory efficient and easy to implement in hardware . the lifting scheme consists of three steps to decompose the samples , namely , splitting , predicting ( eqn . ( 1 ) and ( 3 ) ) , and updating ( eqn . ( 2 ) and ( 4 ) ) . haar wavelet transform is orthogonal and simple to construct and provide fast output . by considering the advantages of the haar wavelets , the proposed architecture uses the haar wavelet to perform the 1-d dwt in temporal direction ( between two adjacent frames ) . @xcite developed a lifting based haar wavelet . the equations of the lifting scheme for the haar wavelet transform is as shown in eqn.([eq2 ] ) @xmath30 = \left ( { \begin{array}{*{20}{c } } { \sqrt 2 } & 0\\ 0&{\frac{1}{{\sqrt 2 } } } \end{array } } \right)\left ( { \begin{array}{*{20}{c } } 1&{s(z)}\\ 0&1 \end{array } } \right)\left ( { \begin{array}{*{20}{c } } 1&0\\ { - p(z)}&1 \end{array } } \right)\left ( \begin{array}{l } { x_0}(z)\\ { x_1}(z ) \end{array } \right)\ ] ] @xmath31 eqn.([eq3 ] ) is extracted by substituting predict value @xmath32 as 1 and update step @xmath33 value as 1/2 in eqn.([eq2 ] ) , which is used to develop the temporal processor to apply 1-d dwt in temporal direction ( @xmath34 dimension ) . where l and h are the low and high frequency coefficients respectively . the proposed architecture for 3-d dwt comprising of two parallel spatial processors ( 2-d dwt ) and four temporal processors ( 1-d dwt ) , is depicted in fig . [ blockdia_1 ] . after applying 2-d dwt on two consecutive frames , each spatial processor ( sp ) produces 4 sub - bands , viz . ll , hl , lh and hh and are fed to the inputs of four temporal processors ( tps ) to perform the temporal transform . output of these tps is a low frequency frame ( l - frame ) and a high frequency frame ( h - frame ) . architectural details of the spatial processor and temporal processors are discussed in the following sections . in this section , we propose a new parallel and memory efficient lifting based 2-d dwt architecture denoted by spatial processor ( sp ) and it consists of row and column processors . the proposed sp is a revised version of the architecture developed by the y. hu et al.@xcite . the proposed architecture utilizes the strip based scanning @xcite to enable the trade - off between external memory and internal memory . to reduce the critical path in each stage flipping model @xcite-@xcite is used to develop the processing element ( pe ) . each pe has been developed with shift and add techniques in place of multiplier . lifting based ( 9/7 ) 1-d dwt process has been performed by the processing unit ( pu ) in the proposed architecture . as shown in fig . [ 3d_2 ] , the proposed pu is designed with five pes , and each pe ( except first pe ( shift@xmath35pe ) ) has been constructed with two pipeline stages for further reduction of cpd . this modified pu , reduces the cpd to @xmath36 ( adder delay ) . fig . [ blockdia_1 ] shows that the number of inputs to the spatial processor is equal to 2p+1 , which is also equal to the width of the strip . where p is the number of parallel processing units ( pus ) in the row processor as well as column processor . we have designed the proposed architecture with two parallel processing units ( p = 2 ) . the same structure can be extended to p = 4 , 8 , 16 or 32 depending on external bandwidth . whenever row processor produces the intermediate results , immediately column processor start to process on those intermediate results . row processor takes 9 clocks to produce the temporary results then after column processor takes 9 more clocks to to give the 2-d dwt output ; finally , temporal processor takes 3 more clocks after 2-d dwt results are available to produce 3-d dwt output . as a summary , proposed 2-d dwt and 3-d dwt architectures have constant latency of 18 and 21 clock cycles respectively , regardless of image size n and number of parallel pus ( p ) . details of the row processor and column processor are given in the following sub - sections . let @xmath37 be the image of size @xmath38 , extend this image by one column by using symmetric extension . now image size is @xmath39 . refer @xcite for the structure of strip based scanning method . the proposed architecture initiates the dwt process in row wise through row processor ( rp ) then process the column dwt by column processor ( cp ) . [ 3d_1](a ) . shows the generalized structure for a row processor with @xmath40 number of pus . @xmath41 has been considered for our proposed design . for the first clock cycle , rp get the pixels from @xmath42 to @xmath43 simultaneously . for the second clock rp gets the pixels from next row i.e. @xmath44 to @xmath45 , the same procedure continues for each clock till it reaches the bottom row i.e. , @xmath46 to @xmath47 . then it goes to the next strip and rp get the pixels from @xmath43 to @xmath48 and it continues this procedure for entire image . each pu consists of five pipeline stages and each pipeline stage is processed by one processing element ( pe ) as depicted in fig . [ 3d_2](b ) . first stage ( shift@xmath35pe ) provide the partial results which is required at @xmath49 stage ( pe@xmath35alpha ) , likewise processing elements pe@xmath35alpha to pe@xmath35delta ( @xmath49 stage to @xmath50 stage ) gives the partial results along with their original outputs . ( e.g. , consider the pe@xmath35alpha of pu-1 , it needs to provide output corresponding to eqn.(1 ) ( @xmath51 $ ] ) , along with @xmath52 $ ] , it also provides the partial output @xmath53 $ ] which is required for the pe@xmath35beta ) . structure of the pes are given in the fig . [ 3d_2](b ) , it shows that multiplication is replaced with the shift and add technique . the original multiplication factor and the value through the shift and add circuit are noted in table.[tab1 ] , it shows that variation between original and adopted one is extremely small . as shown in fig . [ 3d_2](b ) , time delay of shift@xmath35pe is one @xmath9 and remaining all pes are having delay of @xmath54 . to reduce the cpd of pu , pes from pe@xmath35alpha to pe@xmath35delta are divided in to two pipeline stages , and each pipeline stage has a delay of @xmath9 , as a result cpd of pu is reduced to @xmath9 and pipeline stages are increased to nine and is shown in fig . [ 3d_2](c ) . the outputs @xmath55 $ ] , @xmath56 $ ] , and @xmath57 $ ] corresponding to pe@xmath35alpha and pe@xmath35beta of last pu and pe@xmath35gama of last pu is saved in the memories memory@xmath35alpha , memory@xmath35beta and memory@xmath35gama respectively , shown in fig . [ 3d_1](a ) . those stored outputs are inputted for next subsequent columns of the same row . for a @xmath58 image rows is equivalent to @xmath59 . so the size of the each memory is @xmath60 words and total row memory to store these outputs is equals to @xmath61 . output of each pu are under gone through a process of scaling before it producing the outputs h and l. these outputs are fed to the transposing unit . the transpose unit has @xmath4 number of transpose registers ( one for each pu ) . [ 3d_3](a ) shows the structure of transpose register , and it gives the two h and two l data alternatively to the column processor . the structure of the column processor ( cp ) is shown in fig . [ 3d_1](b ) . to match with the throughput of rp , cp is also designed with two number of pus in our architecture . each transpose register produces a pair of h and l in an alternative order and are fed to the inputs of one pu of the cp . the partial results produced are consumed by the next pe after two clock cycles . as such , shift registers of length two are needed within the cp between each pipeline stages for caching the partial results ( except between @xmath62 and @xmath63 pipeline stages ) . at the output of the cp , four sub - bands are generated in an interleaved pattern , @xmath64 and so on . outputs of the cp are fed to the re - arrange unit . [ 3d_3](b ) shows the architecture for re - arrange unit , and it provides the outputs in sub - band order @xmath65 and @xmath66 simultaneously , by using @xmath4 registers and @xmath67 multiplexers . for multilevel decomposition , the same dwt core can be used in a folded architecture with an external frame buffer for the ll sub - band coefficients . .original and adopted values for multiplication [ cols= " < , < , < " , ] the proposed 3-d dwt architecture has been described in verilog hdl . a uniform word length of 14 bits has been maintained throughout the design . simulation results have been verified by using xilinx ise simulator . we have simulated the matlab model which is similar to the proposed 3-d dwt hardware architecture and verified the 3-d dwt coefficients . rtl simulation results have been found to exactly match the matlab simulation results . the verilog rtl code is synthesized using xilinx ise 14.2 tool and mapped to a xilinx programmable device ( fpga ) 7z020clg484 ( zynq board ) with speed grade of -3 . table [ fpga_results ] shows the device utilization summary of the proposed architecture and it operates with a maximum frequency of 265 mhz . the proposed architecture has also been synthesized using synopsys design compiler with 90-nm technology cmos standard cell library . it consumes 43.42 mw power and occupies an area equivalent to 231.45 k equivalent gate at frequency of 200 mhz . the performance comparison of the proposed 2-d and 3-d dwt architectures with other existing architectures is figure out in tables [ 2dcompare ] and [ 3dcompare ] respectively . the proposed 2-d processor requires zero multipliers , 34p ( pis number of parallel pus ) adders , 60p+3n internal memory . it has a critical path delay of @xmath9 with a throughput of four outputs per cycle with @xmath68/2p computation cycles to process an image with size @xmath38 . when compared to recent 2-d dwt architecture developed by the y.hu et al . @xcite , cpd reduced to @xmath9 from @xmath69 with the cost of small increase in hardware resources . table [ 3dcompare ] shows the comparison of proposed 3-d dwt architecture with existing 3-d dwt architecture . it is found that , the proposed design has less memory requirement , high throughput , less computation time and minimal latency compared to @xcite , @xcite , @xcite , and @xcite . though the proposed 3-d dwt architecture has small disadvantage in area and frequency , when compared to @xcite , the proposed one has a great advantage in remaining all aspects . table [ 3d_asic ] gives the comparison of synthesis results between the proposed 3-d dwt architecture and @xcite . it seems to be proposed one occupying more cell area , but it included total on chip memory also , where as in @xcite on chip memory is not included . power consumption of the proposed 3-d architecture is very less compared to @xcite . in this paper , we have proposed memory efficient and high throughput architecture for lifting based 3-d dwt . the proposed architecture is implemented on 7z020clg484 fpga target of zynq family , also synthesized on synopsys design vision for asic implementation . an efficient design of 2-d spatial processor and 1-d temporal processor reduces the internal memory , latency , cpd and complexity of a control unit , and increases the throughput . when compared with the existing architectures the proposed scheme shows higher performance at the cost of slight increase in area . the proposed 3-d dwt architecture is capable of computing 60 uhd ( 3840@xmath702160 ) frames in a second . 30 q. dai , x. chen , and c. lin,``a novel vlsi architecture for multidimensional discrete wavelet transform,''__ieee transactions on circuits and systems for video technology _ _ , vol . 14 , no . 8 , pp . 1105 - 1110 , aug . 2004 . c. cheng and k. k. parhi , `` high - speed vlsi implementation of 2-d discrete wavelet transform , '' _ ieee trans . signal process . 393 - 403 , jan . 2008 . b. k. mohanty and p. k. meher , `` memory - efficient high - speed convolution - based generic structure for multilevel 2-d dwt.''__ieee transactions on circuits and systems for video technology , _ _ vol . 353 - 363 , feb . 2013 . i. daubechies and w. sweledens , `` factoring wavelet transforms into lifting schemes , '' _ j. fourier anal . 247 - 269 , 1998 . huang , p.c . tseng , and l .- g . chen , `` flipping structure : an efficient vlsi architecture for lifting - based discrete wavelet transform , '' _ ieee trans . signal process . 1080 - 1089 , apr . 2004 . xiong , j .- w . tian , and j. liu , `` a note on flipping structure : an efficient vlsi architecture for lifting - based discrete wavelet transform , '' _ ieee transactions on signal processing , _ vol . 54 , no . 5,pp . 1910 - 1916 , may 2006 c .- t . huang , p .- c . tseng , and l .- g . chen , `` analysis and vlsi architecture for 1-d and 2-d discrete wavelet transform , '' _ ieee trans . signal process . 1575 - 1586 , apr . cheng , c .- t . huang , c .- y . ching , c .- j . chung , and l .- g . chen , `` on - chip memory optimization scheme for vlsi implementation of line based two - dimentional discrete wavelet transform , '' _ ieee transactions on circuits and systems for video technology , _ vol . 17 , no . 7 , pp . 814 - 822 , jul . liao , m. k. mandal , and b. f. cockburn , `` efficient architectures for 1-d and 2-d lifting - based wavelet transforms , '' _ ieee transactions on signal processing , _ vol . 5 , pp . 1315 - 1326 , may 2004 . b.f . wu and c.f . chung , `` a high - performance and memory - efficient pipeline architecture for the 5/3 and 9/7 discrete wavelet transform of jpeg2000 codec , '' _ ieee trans . circuits syst . video technol . , _ vol . 15 , no . 12 , pp . 1615 - 1628 , dec . xiong , j. tian , and j. liu , `` efficient architectures for two - dimensional discrete wavelet transform using lifting scheme , '' _ ieee transactions on image processing , _ vol . 607 - 614 , mar . w. zhang , z. jiang , z. gao , and y. liu , `` an efficient vlsi architecture for lifting - based discrete wavelet transform , '' _ ieee transactions on circuits and systems - ii : express briefs , _ vol . 158 - 162 , mar . b. k. mohanty and p. k. meher , `` memory efficient modular vlsi architecture for high throughput and low - latency implementation of multilevel lifting 2-d dwt , '' _ ieee transactions on signal processing , _ vol . 5 , pp . 2072 - 2084 , may 2011 . a.darji , s. agrawal , ankit oza , v. sinha , a.verma , s. n. merchant and a. n. chandorkar , `` dual - scan parallel flipping architecture for a lifting - based 2-d discrete wavelet transform,''__ieee transactions on circuits and systems - ii : express briefs , _ _ vol . 61 , no . 6 , pp . 433 - 437 , jun . 2014 . b. k. mohanty , a. mahajan , and p. k. meher , `` area and power efficient architecture for high - throughput implementation of lifting 2-d dwt , '' _ ieee trans . circuits syst . briefs , _ vol . 59 , no . 7 , pp . 434 - 438 , jul . 2012 . y. hu and c. c. jong,``a memory - efficient high - throughput architecture for lifting - based multi - level 2-d dwt,''__ieee transactions on signal processing , _ _ vol . 20 , pp.4975 - 4987 , oct . 15 , 2013 . y. hu and c. c. jong , `` a memory - efficient scalable architecture for lifting - based discrete wavelet transform,''__ieee transactions on circuits and systems - ii : express briefs , _ _ vol . 60 , no . 8 , pp . 502 - 506 , aug . j. xu , z.xiong , s. li , and ya - qin zhang , `` memory - constrained 3-d wavelet transform for video coding without boundary effects , '' _ ieee transactions on circuits and systems for video technology , _ vol . 812 - 818 , sep . 2002 . m. weeks and m. a. bayoumi , `` three - dimensional discrete wavelet transform architectures,''__ieee transactions on signal processing , _ _ vol . 50 , no . 8 , pp.2050 - 2063 , aug . z. taghavi and s. kasaei , `` a memory efficient algorithm for multidimensional wavelet transform based on lifting , '' _ in proc . acoust speech signal process . ( icassp ) _ , vol . 401 - 404 , 2003 . q. dai , x. chen , and c. lin , `` novel vlsi architecture for multidimensional discrete wavelet transform , '' _ ieee transactions on circuits and systems for video technology , _ vol . 14 , no . 8 , pp . 1105 - 1110 , a. das , a. hazra , and s. banerjee,``an efficient architecture for 3-d discrete wavelet transform,''__ieee transactions on circuits and systems for video technology , _ _ vol . 286 - 296 , feb . 2010 . b. k. mohanty and p. k. meher , `` memory - efficient architecture for 3-d dwt using overlapped grouping of frames,''__ieee transactions on signal processing _ _ , vol . 11 , pp.5605 - 5616 , nov . 2011 . a. darji , s. shukla , s. n. merchant and a. n. chandorkar , `` hardware efficient vlsi architecture for 3-d discrete wavelet transform , '' _ proc . of @xmath71 int . conf . on vlsi design and @xmath72 int . conf . on embedded systems _ pp . 348 - 352 , 5 - 9 jan . 2014 . w.sweldens , `` the lifting scheme : a construction of second generation of wavelets , '' _ siam journal on mathematical analysis , _ vol.29 no.2 , pp . 511 - 546 , 1998 .

this paper presents a memory efficient , high throughput parallel lifting based running three dimensional discrete wavelet transform ( 3-d dwt ) architecture . 3-d dwt is constructed by combining the two spatial and four temporal processors . spatial processor ( sp ) apply the two dimensional dwt on a frame , using lifting based 9/7 filter bank through the row rocessor ( rp ) in row direction and then apply in the colum direction through column processor ( cp ) . to reduce the temporal memory and the latency , the temporal processor ( tp ) has been designed with lifting based 1-d haar wavelet filter . the proposed architecture replaced the multiplications by pipeline shift - add operations to reduce the cpd . two spatial processors works simultaneously on two adjacent frames and provide 2-d dwt coefficients as inputs to the temporal processors . tps apply the one dimensional dwt in temporal direction and provide eight 3-d dwt coefficients per clock ( throughput ) . higher throughput reduces the computing cycles per frame and enable the lower power consumption . implementation results shows that the proposed architecture has the advantage in reduced memory , low power consumption , low latency , and high throughput over the existing designs . the rtl of the proposed architecture is described using verilog and synthesized using 90-nm technology cmos standard cell library and results show that it consumes 43.42 mw power and occupies an area equivalent to 231.45 k equivalent gate at frequency of 200 mhz . the proposed architecture has also been synthesised for the xilinx zynq 7020 series field programmable gate array ( fpga ) . index terms : discrete wavelet transform , 3-d dwt , lifting based dwt , vlsi architecture , flipping structure , strip - based scanning .

1509.04268

in the past three decades the cooling history of neutron stars was investigated by several authors ( e.g. @xcite ) . recent numerical simulations account for non - isothermal interior , as well as for general relativistic effects . nevertheless , as far as we know , all investigations assumed spherical symmetry of geometry and temperature distribution . as it was pointed out by miralles et al . ( 1993 ) , the effect of rapid rotation on the cooling of neutron stars can be as important as general relativistic effects , whereas the effect of slow rotation should be negligible . although the assumption of slow rotation holds for most of the known pulsars , there exist a couple of millisecond pulsars ( @xcite ) , for which rotation should yield a rather different cooling behaviour . these millisecond pulsars are generally located in binary systems . it may however be expected that young , isolated millisecond pulsars can be detected in the near future , too . a candidate might be the supernova remnant of sn1987a . although the observed neutrino burst lasting for ten seconds indicates that a neutron star was formed in the supernova , there is no evidence for the continued existence of it ( s. @xcite for a recent review ) . however , the neutron star can still be hidden by the surrounding matter , and the continued observations might reveal a rapidly rotating neutron star . the aim of this letter is to study the effect of non - spherical geometry on the cooling of neutron stars . as far as we know , this is the first investigation of rotational effects beyond the isothermal core ansatz ( @xcite ) and also the first completely two dimensional simulation of neutron star cooling . the letter is organized as follows : we first derive the general relativistic equations of thermal evolution and describe the numerical method in sect . [ sec : eq ] . in sect . [ sec : res ] , we apply the two dimensional cooling code to static and rotating neutron star models based on a relativistic equation of state including hyperonic degrees of freedom . finally , we summarize our conclusions and discuss further improvements and applications of the current work in sect . [ sec : concl ] . already a few seconds after the formation of a neutron star in a supernova , its interior settles down into catalysed , degenerate matter . the subsequent cooling involves only thermal processes and does not change the space time geometry . however , the structure of a neutron star depends on the rotational velocity , which generally decreases as the star loose angular momentum , e.g. due to emission of magnetic dipole radiation . since the time scale for reaching hydrostatic equilibrium is much smaller than the time scale for the variation of angular velocity ( s. @xcite , p. 239 ) , one can treat the evolution of a neutron star in quasi stationary approximation . though the partial transformation of rotational energy into thermal energy may considerably change the cooling behaviour of a neutron star ( see , e.g. , @xcite ) , and also the variation of space time geometry might have an effect on it , we study here , as a first step , the simplest case of constant angular velocity . the stationary , axisymmetric , and asymptotic flat metric in quasi isotropic coordinates reads @xmath2 where the metric coefficients @xmath3 are functions of @xmath4 and @xmath5 only . the metric coefficients are determined by the einstein equation ( @xmath6 ) @xmath7 , and the energy - momentum conservation @xmath8 . the obtained elliptic differential equations ( @xcite ) are solved via a finite difference scheme ( @xcite ) once , before the cooling simulation starts . in the case of uniform rotation , @xmath9 , considered here , the equations for thermal evolution are ( @xcite ) @xmath10 where @xmath11 @xmath12 denote the heat flux 3-vector in the comoving frame , @xmath13 the heat capacity , @xmath14 the neutrino emissivity , and @xmath15 the heat conductivity . the partial radial and angular differentials are abbreviated by @xmath16 and @xmath17 , respectively . thermal equilibrium is described by @xmath18 . at the surface of the neutron star the heat flux @xmath19 and @xmath20 is determined by the normal heat flux @xmath21 @xmath22 where @xmath23 is the @xmath4-coordinate of the surface . @xmath21 is taken from a non - magnetic photosphere model which describes the temperature gradient in the region between @xmath24 and the star s surface ( e.g. @xcite ) . in these models @xmath25 depends on the temperature at the density @xmath24 and on the surface gravity @xmath26 the parabolic differential equations obtained after inserting eqs . and into eq . are solved via an implicit finite difference scheme by using a alternating direction implicit method . this yields a non - linear equation system which can be solved iteratively . the obtained linear equation systems have tridiagonal coefficient matrices which can be inverted rather fast . the correctness of the two dimensional code was checked by comparing the outcome of it with simple , analytically solvable models and with the results of the one dimensional code described by schaab et al . we consider a superfluid neutron star model basing on the relativistic hartree - fock equation of state labelled rhf8 in huber et al . ( 1997 ) , which accounts for hyperonic degrees of freedom . the global properties of uniformly rotating models with fixed gravitational mass @xmath27 and angular velocity @xmath28 , @xmath29 , and @xmath30 are summarized in tab . [ tab : models ] . @xmath31 denotes the maximum possible kepler angular velocity , above which mass shedding sets in . all models allow for both the direct nucleon urca and for the direct hyperon urca processes ( cf . all direct urca processes are suppressed by nucleon and lambda pairing below the respective critical temperature ( cf . the ingredients to the cooling simulations are similar to those discussed by schaab et al . ( 1996 ) in detail and are published on the web ( http://www.physik.uni-muenchen.de/sektion/suessmann/astro/cool/schaab.0198/input.html ) . .models of uniformly rotating neutron stars with fixed gravitational mass @xmath32 and angular velocity @xmath28 , @xmath29 , and @xmath30 [ tab : models ] [ cols="<,^,^,^",options="header " , ] entries are the gravitational mass @xmath33 , the baryonic mass @xmath34 , the circumferential radius as measured by a distant observer @xmath35 , the angular velocity @xmath36 , the central mass density @xmath37 , the central baryon density @xmath38 , and the surface gravity @xmath39 . figure [ fig : cool ] shows the evolution of the surface temperature as measured by a distant observer . it is possible to distinguish three epochs of evolution . in the first epoch @xmath40 yr , large temperature gradients occur in the interior of the neutron star . radial temperature gradients were already found in one dimensional simulations ( see , for example , @xcite ) . in rotating neutron stars , azimuthal temperature gradients , which cause transverse heat flow , @xmath41 , exist , too . the polar temperature is by 16% or 63% higher than the equatorial temperature for the models rotating with @xmath42 and @xmath30 , respectively ( see also fig . [ fig : deviation ] ) . after about 100 yr the cooling wave reaches the surface and the interior becomes isothermal , @xmath18 the temperature deviation between pole and equator is now mainly caused by the smaller surface gravity @xmath39 at the equator , since @xmath43 for fixed internal temperature . whereas the temperature deviation is still high in the case of nearly kepler rotation ( @xmath44% ) , it almost disappears for the model with @xmath42 ( @xmath45% ) . after about @xmath46 yr , the photon surface radiation dominates the cooling . the temperature distribution tends to a new equilibrium with @xmath47 , on a timescale of @xmath1 yr . @xmath48 denotes the temperature at the inner boundary of the photosphere at @xmath24 . since the heat capacity is only weakly temperature dependent for @xmath49 k in the outer crust , the surface temperature tends to an isothermal state , whereas the temperature in the outer crust varies with the surface gravity . this behaviour is accompanied by the breakdown of the scaling @xmath50 of the surface temperature with surface gravity for small surface temperatures . however , as one can see by comparing the thin dashed curve , which assumes @xmath51 over the whole temperature range , with the thick curve in fig . [ fig : deviation ] , this is only a small effect and can not explain the tendency of the surface temperature fraction to unity . besides these effects on the angular dependency of the temperature , rotation has also a net effect on cooling in the intermediate epoch . this effect is caused by the reduction of the neutrino luminosity , which sensitively depends on the central density . a further effect of rotation on cooling of neutron stars is the lengthening of the thermal diffusion time by several hundred years . additionally the drop of the surface temperature is smoother in the case of rotation ( see also @xcite ) , since the cooling wave reaches the polar region earlier than the equator . during the cooling wave is reaching the different surface regions , the fraction @xmath52 falls below unity ( s. fig . [ fig : deviation ] ) . in this letter , we have studied the effect of non - spherical geometry on the cooling of neutron stars . our general finding is that rapid rotation plays a significant role in the thermal evolution . this role is particularly important in the early epoch , when the star s interior is not yet isothermal . it is precisely this epoch , when even isolated pulsars might rotate very rapidly . the angular dependent radial and transverse heat flow cause azimuthal temperature gradients in the interior of the star and thus also at the surface . the polar surface temperature is by up to 63% higher than the equatorial temperature . the simulation of this non - isothermal epoch can not be performed with isothermal core approaches or one dimensional codes . in the intermediate epoch , heat conduction is unimportant , since the interior is nearly isothermal . nevertheless two significant , though smaller , effects remain . the first effect is the reduction of the central density with respect to the static , non - rotating case , if one fixes the gravitational mass or the rest mass . since the neutrino emissivity depends sensitively on the density , especially in the vicinity of threshold densities for fast neutrino emission processes , the neutrino luminosity of the star is reduced as well . the second effect is the angular dependency of the surface temperature caused by the angular dependent surface gravity . the polar surface temperature is still by up to 31% higher than the equatorial temperature . both effects could also be studied with one dimensional codes or even within the isothermal core ansatz ( see @xcite ) . whereas miralles et al . ( 1993 ) assumed that the interior is still isothermal in the late photon cooling epoch , we find that the surface temperature distribution itself tends to an isothermal equilibrium , which implies that the temperature distribution in the outer crust becomes angular dependent . the transition between the early and the intermediate epochs turns out to be delayed by rotation . because of the enlargement of the crust in the equatorial plane , the cooling wave formed in the inner region of the star needs several hundred years longer to reach the surface furthermore , the transition becomes smoother , since the radial diffusion time it not isotropic any more ( see also @xcite ) . the surface temperature , especially in the intermediate epoch , is rather sensitive to micro physical parameters ( superfluid energy gap , neutrino emissivity , equation of state , etc . ; see @xcite for a recent review ) . the comparison of the net effect with observation might therefore be difficult . nevertheless , the comparison of the observed soft x - ray spectra with the theoretical obtained spectra may reveal the angular dependency of the surface temperature due to rotational effects . the deduction of the spectrum from the surface temperature distribution is , unfortunately , not at all trivial in non - spherical geometry ( @xcite ) . so far , only the thermal spectra of slowly rotating neutron stars ( @xmath53 for the vela pulsar 0833 - 45 ) are known . it may however be expected that the thermal spectra of a young , isolated millisecond pulsar can be detected in the near future . for example , the period @xmath54 ms of the fastest pulsar b1937 + 21 ( @xcite ) known today corresponds to a ratio @xmath55 , which lies between the respective ratios of our two rotating models . since we neglected some important effects , like differential rotation and the loss of angular momentum , our work can only be understood as a first step towards two dimensional simulation of cooling of rotating neutron stars . of course , the two dimensional code presented here may also serve to investigate other non - spherically symmetric effects , such as the effect of strong magnetic fields on heat conductivity and neutrino emissivity . the inclusion of differential rotation and strong magnetic fields , as well as the mentioned deduction of the photon spectra will be addressed to future work . we would like to thank the referee k. a. van riper for his valuable suggestions . one of us ( ch . s. ) gratefully acknowledges the bavarian state for financial support . tables with detailed references to the used ingredients and the obtained cooling tracks can be found on the web : http://www.physik.uni-muenchen.de/sektion/suessmann/astro/cool/schaab.0198 . page d. , 1997 , thermal evolution of isolated neutron stars , to be published in the proceedings of the nato asi ` the many faces of neutron stars ' ( kluwer ) , eds a. alpar , r. buccheri & j. van paradijs , preprint astro - ph/9706259

the effect of rotation on the cooling of neutron stars is investigated . the thermal evolution equations are solved in two dimensions with full account of general relativistic effects . it is found that rotation is particularly important in the early epoch when the neutron star s interior is not yet isothermal . the polar surface temperature is up to 63% higher than the equatorial temperature . this temperature difference might be observable if the thermal radiation of a young , rapidly rotating neutron star is detected . in the intermediate epoch ( @xmath0 yr ) , when the interior becomes isothermal , the polar temperature is still up to 31% higher than the equatorial temperature . afterwards photon surface radiation dominates the cooling , and the surface becomes isothermal on a timescale of @xmath1 yr . furthermore , the transition between the early and the intermediate epochs is delayed by several hundred years . an additional effect of rotation is the reduction of the neutrino luminosity due to the reduction of the central density with respect to the non - rotating case .

astro-ph9806180

there is a number of new physics ( np ) models beyond the standard model ( sm ) of particle physics . motivated by the hierachy and/or the fine - tuning problem in the sm , most np models propose new states with tev - scale masses . a few examples are the susy models , models with extra gauge bosons ( @xmath6 models ) , and the models with extra dimensions . when the masses of the np states are heavier than the center of mass ( cm ) energy of the collider , the effects of the np can be measured indirectly in terms of the deviations of the sm observables such as the total cross section and various asymmetries . the deviations from the sm in the scattering processes are determined by the mass , spin , and coupling strength of the new states being exchanged by the initial state particles . the question of how to distinguish new states with different spins and couplings at the low energies arises at the sub - tev @xmath7 collider . while the cern large hadron collider ( lhc ) will probe np models with the tev - scale masses , we certainly need the precision measurements to distinguish signature of one model from the others . the precision measurements of the four - fermion scattering at the @xmath7 collider are expected to efficiently reveal the nature of the intermediate states being exchanged by the fermions . the angular distributions and the asymmetries induced by various new states provide information of the spin and coupling of the interactions . at the international linear collider ( ilc ) with the center of mass energy @xmath8 gev , the tev masses could not be observed directly as the resonances since they are heavier than @xmath9 . low energy taylor expansion is a good approximation for the signals induced from the np models and the corrections will be characterized by the higher dimensional operators . for the 4-fermion scattering , the leading - order np signals from the states with spin-0 and spin-1 such as leptoquarks , sneutrino with @xmath10-parity violating interactions @xcite and @xmath6 will appear as the dimension-6 contact interaction at low energies . as a candidate for the np state with spin-2 , the interaction induced by the ( massive ) gravitons , @xmath11 , can be characterized at the low energies by the effective interaction of the form @xmath12 @xcite , a dimension-8 operator . in the viewpoint of effective field theory , this effective interaction does not need to be originated from the exhanges of massive graviton states and it is not the most generic form of the interaction containing dimension-8 operators . however , it certainly has the gravitational interpretation due to the use of the symmetric energy - momentum tensor @xmath13 . it can be thought of as the low - energy effective interaction induced by exchange of the kk gravitons ( in add @xcite and rs @xcite scenario ) interacting with matter fields in the non - chiral fashion . in the braneworld scenario where the sm particles are identified with the open - string states confined to the stack of d - branes subspace , and gravitons are the closed - string states propagating freely in the bulk spacetime @xcite , table - top experiments @xcite and astrophysical observations allow the quantum gravity scale to be as low as tevs @xcite . since the string scale , @xmath14 , in this scenario is of the same order of magnitude as the quantum gravity scale , it is possible to have the string scale to be as low as a tev . the tev - scale stringy excitations would appear as the string resonances ( sr ) in the @xmath15 processes at the lhc @xcite . the most distinguished signals would be the resonances in the dilepton invariant mass distribution appearing at @xmath16 each resonance would contain various spin states degenerate at the same mass . these srs can be understood as the stringy spin excitations of the zeroth modes which are identified with the gauge bosons in the sm . they naturally inherit the chiral couplings of the gauge bosons . in this article , it will be shown that the leading - order stringy excitations of the exchanging modes identified with the gauge bosons in the four - fermion interactions will contain both spin-1 and 2 . their couplings will be chiral , inherited from the chiral coupling of the zeroth mode ( identified with the gauge boson ) . namely , we construct the tree - level stringy amplitudes with chiral spin-2 interactions ( in addition to a stringy dimension-8 spin-1 contribution contrasting to the dimension-6 contributions from other @xmath17 model at low energies ) which can not be described by the non - chiral effective interaction of the form @xmath12 as stated above . this article is organized as the following . in section ii , we discuss briefly the construction of the stringy amplitudes in the four - fermion scattering as is introduced in the previous work @xcite , the comments on the chiral interaction are stated and emphasized . in section iii , the low - energy stringy corrections are approximated . the angular momentum decomposition reveals the contribution of each spin state induced at the leading order . in section iv , we calculate the angular left - right , forward - backward , and center - edge asymmetries . the extensions to partially polarized beams are demonstrated in section v. in section vi , we make concluding remarks and discussions . the low - energy ( @xmath18 ) expressions for the asymmetries @xmath19 induced by the sm and the np models ( kk gravitons and sr ) up to the order of @xmath20 are given in the appendix . the 4-fermion processes that we will consider are the scattering of the initial electron and positron into the final states with one fermion and one antifermion , @xmath21 . we will ignore the masses of the initial and final - states particles and therefore consider only the processes with @xmath22 where @xmath23 . the physical process will be identified as @xmath24 . the @xmath25 and @xmath1-channel are labeled @xmath26 and @xmath27 respectively . to make a relatively model - independent phenomenological study of the tev - scale stringy kinematics , we adopt the parametrised bottom - up " approach for the construction of the tree - level string amplitudes @xcite . in this approach , the gauge structure and the assignment of chan - paton matrices to the particles are not explicitly specified . the main requirement is that the relevant string amplitudes reproduce the sm amplitudes in the low energy limit ( @xmath28 , field theory limit ) . this implies that we identify the zeroth - mode string states as the gauge bosons of the sm . the expression for the open - string 4-fermion tree - level helicity amplitude for the process @xmath29 with @xmath30 is @xcite @xmath31 where @xmath32 and the regge slope @xmath33 . the interaction factors from the exchange of photon and @xmath34 ( chiral ) are given by @xmath35 here @xmath36 and the @xmath37 coupling @xmath38 . the neutral current couplings are @xmath39 . the chan - paton parameter @xmath40 represents the tree - level stringy interaction which can not be determined in the field theory limit @xmath41 since @xmath42 . the amplitude for other helicity combination @xmath43 is given by @xmath44 and an index exchange in the @xmath45 factor @xmath46 since the veneziano - like function @xmath47 and @xmath48 are appearing with the chiral couplings ( the factor @xmath45 ) in the amplitudes , the piece of the stringy states induced by this function will have the similar chiral couplings to those of the sm . on the other hand , the purely stringy interaction piece , proportional to @xmath40 , are assumed to be non - chiral and the values of @xmath40 are set to the same value for all helicity combinations . for @xmath49 , we can use taylor expansion to approximate the leading order stringy corrections to the amplitudes . they are in the form of the dimension-8 operator @xmath50 the amplitude in eq . ( [ eq:3 ] ) becomes @xmath51 again , for @xmath52 , the amplitude is @xmath44 and @xmath53 of the above . the stringy correction can be decomposed into the contribution from the angular momentum states , @xmath54 . using the wigner functions @xmath55 and @xmath56 , @xmath57 where @xmath58 is the angle between the incoming electron and the outgoing antifermion in the c.m . frame . from eq . ( [ eq:12]-[eq:13 ] ) , we can see that the couplings of the @xmath54 states , proportional to @xmath59 , inherit the chirality from the coupling @xmath45 of the zeroth mode gauge boson exhange in the sm . this is the distinctive feature of the couplings of the string states in this bottom - up " approach . because of the chirality of the coupling , the @xmath60 interaction in these stringy amplitudes can not be described by the effective interaction of the form @xmath12 . as long as we couple the spin-2 state @xmath61 to the energy - momentum tensor @xmath62 which does not contain information of the chirality of the fermions , the interaction will always be non - chiral . therefore , the effective interaction of this kind can never describe the _ chiral _ stringy interaction induced by the worldsheet stringy spin excitations in the string models under consideration ( i.e. the models which address chiral weak interaction ) . as a comparison , we give the sm - kk amplitudes where the kk part is induced by the effective interaction of the form @xmath12 as @xmath63 for @xmath64 and @xmath65 . the kk - gravitons contributions can be represented by the wigner functions as @xmath66 the chiral spin-1 and spin-2 stringy interaction will lead to remarkable and unique phenomenological signatures in the 4-fermion scattering at the electron - positron collider even when compared with the contributions from kk gravitons as we will see in the following . the left - right ( @xmath67 ) and forward - backward ( @xmath68 ) asymmetries quantify the degrees of the _ chirality _ of the interaction under consideration regardless of the spin of the intermediate states . on the other hand , the center - edge asymmetry ( @xmath69 ) does not contain any information on the chirality of the spin-1 interaction ( at least in the massless limit ) @xcite . for spin-2 , @xmath69 shows dependence on the chirality of the couplings of the intermediate states in the scattering as we will present in section v. therefore , we can use @xmath69 to distinguish np models with spin-2 mediator from the models mediated by the spin-1 state . among the class of models with spin-2 interactions , we can use the @xmath70 to distinguish one model from another as demonstrated in fig . [ .1-fig ] and fig . [ 10-fig]-[12-fig ] for the case of sr versus kk - gravitons . the angular left - right asymmetry is defined as a function of @xmath71 as @xmath72 where @xmath73 is the cross section of the scattering of the 100% left(right)-handedly polarized electron beam with the 100% right(left)-handedly polarized positron beam . the angular left - right asymmetries induced by the stringy corrections are plotted as in fig . [ .1-fig ] in comparison to the kk - graviton model . it is interesting to comment that the angular left - right asymmetries induced by the stringy corrections differ significantly from the sm distributions only in the quark ( @xmath1 and @xmath2 ) final states . the difference in @xmath74 is hardly visible . this feature is similar to the asymmetries induced by the kk gravitons @xcite . = 5.5 in the forward - backward asymmetry is defined as @xmath75 where @xmath76 is the number of events in the forward(backward ) direction . the numerical values of the unpolarised forward - backward asymmetries for the sm and the stringy amplitudes with @xmath77 gev , @xmath78 tev , and @xmath79 are @xmath80 the deviations of the values with the stringy corrections from the sm values are linearly dependent on @xmath40 and @xmath81 at the leading order as we can see from the expression in the polarized beams section with @xmath82 . this is true only for the scattering at low energies comparing to the string scale . at higher energies @xmath83 or around the srs at @xmath84 , the forward - backward asymmetries become very small due to the non - chiral choice of the chan - paton parameters @xmath40 which efficiently dilutes the chirality of the interaction @xcite . the center - edge asymmetry is defined as a function of the cut of the central region @xmath85 as @xmath86 . \label{eq:2.10}\end{aligned}\ ] ] the deviations of the center - edge asymmetry from the sm values of the unpolarised beams , @xmath87 , are plotted with respect to @xmath85 in fig . [ 4-fig ] . a few interesting remarks are worthwhile making . there is distinctive feature between @xmath69 of the lepton @xmath74 and the quarks , @xmath1 and @xmath2 . the effect of the non - chiral purely stringy interaction represented by the value of @xmath40 is opposite for @xmath74 and @xmath1 . on the other hand , the features of @xmath74 and @xmath2-type quark are , for the small value of @xmath88 , roughly the same . for large value of @xmath89 , the deviation from the sm of the @xmath2-type becomes negative and appears very distinctive from the corresponding value of the @xmath74 . this strange behaviour is originated from the competing contributions between @xmath90 appearing in the @xmath91 terms and the terms of order @xmath92 . it should be emphasized that the numerical results as in fig . [ 4-fig ] are calculated using full expression upto the order of @xmath91 . they are different from the results obtained from the approximation using the leading order upto @xmath92 as is given in section v. the difference becomes very obvious in the @xmath2-type quark with high value of @xmath89 . let @xmath93 be the degrees of the longitudinal polarization of the @xmath94 beam defined as ( @xmath95 ) @xmath96 where @xmath97 is the number of particle @xmath98 with the left(right)-handed helicity in the beams . the polarized differential cross section can be expressed as @xmath99 \label{eq:3.1}\end{aligned}\ ] ] where @xmath100 upper(lower ) signs are for @xmath101 . @xmath102 represents the scattering involving the left(right)-handed electron . in practice , the observable left - right asymmetry is defined with respect to the partially polarized beams of electron and positron by taking difference of the total number of events when the polarizations of the beams are inverted . it is therefore given by @xmath103 where @xmath104 . for @xmath105 , @xmath106 is @xmath107 , while @xmath106 becomes @xmath108 when @xmath109 . the full expressions of @xmath110 for the kk - gravitons and sr models are given in the appendix . with the partially polarised beams of electron and positron , we can calculate @xmath68 using eq . ( [ eq:3.1 ] ) . the full expressions for the asymmetries induced by kk gravitons and sr are given in the appendix . up to the leading order of @xmath92 in the energy taylor expansion , the polarized forward - backward asymmetry induced by the stringy corrections is @xmath111 @xmath112 \label{eq:3.3 } \end{aligned}\ ] ] where the einstein summation convention is implied for @xmath113 . @xmath114 and @xmath115 . the other combination is obtained by exchanging @xmath116 . the sm value is given by @xmath117 with @xmath118 and @xmath119 . [ 10-fig]-[12-fig ] show @xmath68 induced by kk gravitons and sr in comparison to the sm values with respect to the cm energy . assuming the string scale @xmath78 tev and the effective quantum gravity scale @xmath120 tev , the differences between the asymmetries induced by the two models appear at higher cm energies even in the case when they are indistinguishable at the low cm energies . this is due to the fact that while the _ chirality _ of the stringy interaction keeps the terms of both order @xmath92 and @xmath91 in the expression of @xmath68 , the non - chiral graviton interaction , on the other hand , keeps only the term of order @xmath121 in the numerator @xmath122 , and only term of order @xmath123 in the denominator @xmath124 ( see appendix ) . this leads to distinguishable aspects of the two models . with the partially polarised beams of electron and positron , we can calculate @xmath69 using eq . ( [ eq:3.1 ] ) . the full expressions for the asymmetries induced by kk gravitons and sr are given in the appendix . the deviations from the sm values for various polarizations of the beams for the stringy model are plotted as in fig . [ 7-fig]-[9-fig ] . up to the leading order of @xmath92 in the energy taylor expansion , the polarized center - edge asymmetry induced by the stringy corrections is @xmath125 @xmath126 where the sm value is given by @xmath127 eq . ( [ eq : c3 ] ) is unique for the spin-1 contribution in the 4-fermion scattering . any np particles with spin-1 will not change this @xmath85-dependence . remarkably , the spin-2 contributions , either in the form of kk @xcite or the string states ( eq . ( [ eq:3.5 ] ) ) , will induce the deviations from this sm value proportional to @xmath128 . similar to the @xmath68 case , the _ chirality _ of the stringy interaction keeps terms of both order @xmath92 and @xmath91 in the expression of @xmath129 while in the case of kk gravitons , there is no terms of order @xmath121 in the constant term @xmath130 and the denominator @xmath131 . the differences become obvious when @xmath132 is relatively large ( @xmath133 ) as shown in fig . [ 4-fig ] . = 5.5 in = 4.0 in = 4.0 in = 4.0 in = 4.0 in = 4.0 in = 4.0 in we have constructed and approximated the tree - level stringy amplitudes of the scattering @xmath134 . the low - energy stringy corrections for the 4-fermion processes contain both spin-1 and spin-2 contributions with the chiral couplings inherited from the zeroth mode states identified with the gauge bosons in the sm . the chirality of the couplings is diluted by the non - chiral choice of the purely - stringy piece of the stringy interaction , characterized by @xmath40 . the contributions from both stringy spin-1 and spin-2 are of dimension-8 in nature . the low - energy dimension-8 spin-1 contribution is remarkably distinctive from the dimension-6 contributions in other @xmath17 models . the chirality of stringy spin-2 interaction also leads to unique phenomenological features distinguishable from the non - chiral spin-2 interaction induced by the kk - gravitons ( or massive gravitons ) . then we investigated the signatures of the tev - scale string model at the @xmath7 collider in comparison to the kk - gravitons using angular left - right , forward - backward , and center - edge asymmetries . the deviations of the asymmetries from the sm values are investigated separately for each model . all asymmetries show significant differences between the low - energy corrections of the two models . for the @xmath7 collider with variable center - of - mass energies from 500 to 1000 gev and assuming @xmath135 tev , the forward - backward asymmetries show drastic differences between stringy signals and the kk - graviton ones . the center - edge asymmetries also show significant differences between the two models if the chan - paton parameter @xmath132 ( representing purely stringy piece of the interaction ) in the sr model is sufficiently large ( @xmath133 ) . the origin of the differences between the two models is mainly ( another reason is the fact that sr also has spin-1 contribution in addition to spin-2 ) the _ chirality _ of the interactions . while the string interaction is chiral , the interaction induced by the kk gravitons , couple to the energy - momentum tensor @xmath62 , is non - chiral . as we can see in the appendix from the full espressions of @xmath19 , the chirality of the stringy interaction keeps almost all of the terms of order @xmath92 and @xmath91 while the non - chirality of the interaction of the kk gravitons eliminates certain terms of order @xmath121 and @xmath123 in the asymmetries . specifically , the @xmath136 term in @xmath110 contains only the sm term in the case of kk gravitons , in contrast to the stringy case . in @xmath68 , as discussed above , @xmath137 in the case of kk gravitons does not contain the term of order @xmath138 . for @xmath129 , @xmath139 does not contain terms of order of @xmath121 in the case of kk gravitons . this is in contrast to the processes where the stringy interaction is non - chiral as well as having only the spin-2 contributions such as in the scattering @xmath140 ; in which case the two models give exactly the same low - energy signatures @xcite . in the intermediate energy range ( @xmath141 ) with @xmath142 , since the deviations induced by kk gravitons and sr depend only on @xmath143 , the results in this article therefore are also valid for the clic with center - of - mass energies 3 to 6 tev with @xmath144 tev . i would like to thank tao han for helpful discussions . this work was supported in part by the u.s . department of energy under contract number de - fg02 - 01er41155 . @xmath145 [ [ section ] ] @xmath146 @xmath147 [ [ section-1 ] ] @xmath148 @xmath149 @xmath150 [ [ section-2 ] ] @xmath151 [ [ section-3 ] ] @xmath152 @xmath153 a. antoniadis , n. arkani - hamed , s. dimopoulos , and g. dvali , phys . lett . * b436 * , 257 ( 1998)[hep - ph/9804398 ] ; n. arkani - hamed , s. dimopoulos and g.r . dvali , phys . * b429 * , 263 ( 1998 ) [ hep - ph/9803315 ] . l. randall and r. sundrum , phys . lett . * 83 * , 3370 ( 1999 ) [ hep - ph/9905221 ] . a. antoniadis , n. arkani - hamed , s. dimopoulos , and g. dvali , phys . lett . * b436 * , 257 ( 1998)[hep - ph/9804398 ] . g. shiu and s.h . tye , phys . rev . * d58 * , 106007 ( 1998)[hep - th/9805157 ] ; l.e . ibanez , r. rabadan , and a.m. uranga , nucl . phys . * b542 * , 112 ( 1999)[hep - th/9808139 ] ; i. antoniadis , c. bachas , and e. dudas , nucl . phys . * b560 * , 93 ( 1999)[hep - th/9906039 ] ; k. benakli , phys . rev . * d60 * , 104002 ( 1999)[hep - ph/9809582 ] ; e. dudas and j. mourad , nucl . phys . * b575 * , 3 ( 2000)[hep - th/9911019 ] ; e. accomando , i. antoniadis , and k. benakli , nucl . phys . * b579 * , 3 ( 2000)[hep - ph/9912287 ] ; l.e . ibanez , f. marchesano , and r. rabadan , jhep * 0111 * , 002 ( 2001)[hep - th/0105155 ] ; g. aldazabal , s. franco , l.e . ibanez , r. rabadan , and a.m. uranga , jhep * 0102 * , 047 ( 2001)[hep - ph/0011132 ] ; m. cvetic , g. shiu , and a.m. uranga , phys . rev . lett . * 87 * , 201801 ( 2001)[hep - th/0107143 ] ; d. cremades , l.e . ibanez , and f. marchesano , nucl . * b643 * , 93 ( 2002)[hep - th/0205074 ] ; c. kokorelis , nucl . phys . * b677 * , 115 ( 2004)[hep - th/0207234 ] ; c. kokorelis , jhep * 0208 * , 036 ( 2002)[hep - th/0206108 ] ; i. klebanov and e. witten , nucl . b664 * , 3 ( 2003)[hep - th/0304079 ] ; m. axenides , e. floratos , c. kokorelis , jhep * 0310 * , 006 ( 2003)[hep - th/0307255 ] . smullin , a.a . geraci , d.m . weld , j. chiaverini , s. holmes , and a. kapitulnik , phys . rev . * d72 * , 122001 ( 2005 ) [ hep - ph/0508204 ] ; e.g. adelberger , b.r . heckel , and a.e . nelson , ann . partl . sco . * 53 * , 77 ( 2003 ) [ hep - ph/0307284 ] .

we investigate the tev - scale stringy signals of the four - fermion scattering at the electron - positron collider with the center of mass energy @xmath0 gev . the nature of the stringy couplings leads to distinguishable asymmetries comparing to the other new physics models . specifically , the stringy states in the four - fermion scattering at the leading - order corrections are of spin-1 and 2 with the chiral couplings inherited from the gauge bosons identified as the zeroth - mode string states . the angular left - right , forward - backward , center - edge asymmetries and the corresponding polarized - beam asymmetries are investigated . the low - energy stringy corrections are compared to the ones induced by the kaluza - klein ( kk ) gravitons . the angular left - right asymmetry of the scattering with the final states of @xmath1 and @xmath2-type quarks , namely @xmath3 and @xmath4 , shows significant deviations from the standard model values . the center - edge and forward - backward asymmetries for all final - states fermions also show significant deviations from the corresponding standard model values . the differences between the signatures induced by the stringy corrections and the kk gravitons are appreciable in both angular left - right and forward - backward asymmetries . @xmath5 0.4 cm

hep-ph0601142

in newtonian theory a self - gravitating incompressible fluid body rotating at a moderate velocity around a fixed axis with respect to some inertial frame takes the shape of a maclaurin ellipsoid , which is axisymmetric with respect to the rotation axis . for a higher rotation rate , namely when the ratio of kinetic to gravitational potential energy @xmath6 is larger than @xmath7 , another figure of equilibrium exists : that of a jacobi ellipsoid , which is triaxial and rotates around its smallest axis @xcite . actually the jacobi ellipsoid is a preferred figure of equilibrium , since at fixed mass and angular momentum , it has a lower total energy @xmath8 than a maclaurin ellipsoid , due to its greater moment of inertia @xmath9 with respect to the rotation axis . indeed , at fixed angular momentum @xmath10 , the kinetic energy @xmath11 is a decreasing function of @xmath9 , and for large values of @xmath10 , this decrease overcomes the effect of the gravitational potential energy @xmath12 , which increases with @xmath9 . therefore , provided some mechanism acts for dissipating energy while preserving angular momentum ( for instance viscosity ) , a maclaurin ellipsoid with @xmath13 will break its axial symmetry and migrate toward a jacobi ellipsoid @xcite . this is the secular `` bar mode '' instability of rigidly rapidly rotating bodies . the qualifier _ secular _ reflects the necessity of some dissipative mechanism to lower the energy , the instability growth rate being controlled by the dissipation time scale , the maclaurin spheroids are subject to another instability , which is on the contrary _ dynamical _ , i.e. it develops independently of any dissipative mechanism and on a dynamical time scale ( one rotation period ) . ] . as shown by christodoulou et al . @xcite , the jacobi - like bar mode instability appears only if the fluid circulation is not conserved . if on the contrary , the circulation is conserved ( as in inviscid fluids submitted only to potential forces ) , but not the angular momentum , it is the dedekind - like instability which develops instead . the famous chandrasekhar - friedman - schutz ( cfs ) instability ( see @xcite for a review ) belongs to this category . the jacobi - like bar mode instability , applied to neutron stars , is particularly relevant to gravitational wave astrophysics . indeed a jacobi ellipsoid has a time varying mass quadrupole moment with respect to any inertial frame , and therefore emits gravitational radiation , unlike a maclaurin spheroid . for a rapidly rotating neutron star , the typical frequency of gravitational waves ( twice the rotation frequency ) falls in the bandwidth of the interferometric detectors ligo and virgo currently under construction . neutron stars being highly relativistic objects , the classical critical value @xmath14 , established for incompressible newtonian bodies , can not a priori be applied to them . the aim of the present article is thus to investigate the effect of general relativity on the secular bar mode instability of homogeneous incompressible bodies . we do not discuss compressible fluids here . it has been shown that compressibility has little effect on the triaxial instability @xcite . chandrasekhar @xcite has examined the first order post - newtonian ( pn ) corrections to the maclaurin and jacobi ellipsoids , by means of the tensor virial formalism . this work has been revisited recently by taniguchi @xcite . however , these authors have not computed the location of the maclaurin - jacobi bifurcation point at the 1-pn level . this has been done only recently by shapiro & zane @xcite and di girolamo & vietri @xcite . on the numerical side , bonazzola , frieben & gourgoulhon @xcite have investigated the secular bar mode instability of rigidly rotating compressible stars in general relativity . in the newtonian limit , they recover the classical result of james @xcite ( see also @xcite ) , namely that , for a polytropic equation of state , the adiabatic index must be larger than @xmath15 for the bifurcation point to occur before the mass shedding limit ( keplerian frequency ) . in the relativistic regime , they have shown that general relativistic effects stabilize rotating stars against the viscosity driven triaxial instability . in particular , they have found that @xmath16 is an increasing function of the stellar compactness , reaching @xmath17 for a typical neutron star compaction parameter . this stabilizing tendency of general relativity has been confirmed by the pn study of shapiro & zane @xcite and di girolamo & vietri @xcite mentioned above . note that this behavior contrasts with the cfs instability , which is strengthened by general relativity @xcite . in this paper , we improve the numerical technique over that used by bonazzola et al . @xcite by introducing surface fitted coordinates , which enable us to treat the density discontinuity at the surface of incompressible bodies . indeed the technique used in refs . @xcite did not permit to compute any incompressible model . in particular , it was not possible to compare the numerical results in the newtonian limit with the classical maclaurin - jacobi bifurcation point . we shall perform such a comparison here . the very good agreement obtained ( relative discrepancy @xmath18 ) provides very strong support for the method we use for locating the bifurcation point and which is essentially the same as that presented in ref . @xcite . the plan of the paper is as follows . the analytical formulation of the problem , including the approximations we introduce , is presented in sec . [ s : basic ] . section [ s : num ] then describes the numerical technique we employ , as well as the various tests passed by the numerical code . the numerical results are presented in sec . [ s : cal ] , as well as a detailed comparison with the pn studies @xcite and @xcite . finally , sec . [ s : summ ] provides some summary of our work . let us consider a rotating star that is steadily increasing its rotation rate , e.g. by accretion in a binary system . before the triaxial instability sets in , the spacetime generated by the rotating star can be considered as _ stationary _ and _ axisymmetric _ , which means that there exist two killing vector fields , @xmath19 and @xmath20 , such that @xmath19 is timelike ( at least far from the star ) and @xmath20 is spacelike and its orbits are closed curves . when the axisymmetry of the star is broken , the stationarity of spacetime is also broken . in newtonian theory , there is no inertial frame in which a rotating triaxial object appears stationary , i.e. does not depend upon the time . it can be stationary only in a corotating frame , which is not inertial , so that the stationarity is broken in this sense . the stationarity in the corotating frame can be expressed geometrically by stating that the newtonian spacetime possesses a one - parameter symmetry group , whose integral curves are helices . a generator of this symmetry group thus has the form @xmath21 where @xmath22 and @xmath23 are respectively the time and azimuthal coordinates associated with an inertial observer , and @xmath24 is the angular velocity with respect to the inertial frame . in general relativity , a rotating triaxial system can not be stationary , even in the corotating frame , as it radiates away gravitational waves and therefore loses energy and angular momentum . however , at the very point of the symmetry breaking , no gravitational wave has yet been emitted . for sufficiently small deviations from axisymmetry , we may neglect the gravitational radiation . therefore we shall assume that the spacetime has a helical symmetry , as in the newtonian case . the ( suitably normalized ) associated symmetry generator @xmath25 is then a killing vector , which can be written in the form ( [ e : helical ] ) in weak - field regions ( spacelike infinity ) . note that spacetimes with helical symmetry have been also used for describing binary systems with circular orbits @xcite . we model the stellar matter by a perfect fluid , for which the stress - energy tensor takes the form @xmath26 , where @xmath27 is the fluid 4-velocity , @xmath28 the fluid proper energy density , @xmath29 the fluid pressure and @xmath30 the spacetime metric tensor . a rigid motion is defined in relativity by the vanishing of the expansion tensor @xmath31 of the 4-velocity @xmath27 . in presence of the killing vector @xmath25 , this can be realized by requiring the colinearity of @xmath27 and @xmath25 ( supposing that the fluid occupies only the region where @xmath25 is timelike ) : @xmath32 where @xmath33 is a scalar field related to the norm of @xmath25 by the normalization of the 4-velocity @xmath34 . for a perfect fluid at zero temperature , the momentum - energy conservation equation @xmath35 can be recast as @xcite @xmath36 @xmath37 where @xmath38 is the proper baryon number density and @xmath39 is the momentum 1-form @xmath40 , @xmath41 being the fluid specific enthalpy : @xmath42 , where @xmath43 is some mean baryon mass . in eq . ( [ e : canon ] ) , @xmath44 denotes the exterior derivative of @xmath39 , the so - called _ vorticity 2-form _ @xcite . for the rigid motion we are considering , the baryon number conservation equation ( [ e : conserv ] ) is automatically satisfied , thanks to the colinearity of @xmath27 with the symmetry generator @xmath25 . moreover , the equation of motion ( [ e : canon ] ) can be reduced to a first integral . indeed cartan s identity applied to the lie derivative of the 1-form @xmath39 along the vector field @xmath25 leads to @xmath45 where the second equality holds thanks to the helical symmetry : @xmath46 . inserting relation ( [ e : rigid ] ) into the equation of fluid motion ( [ e : canon ] ) shows that the first term in eq . ( [ e : cartan ] ) vanishes identically , so that one gets the first integral of motion @xcite @xmath47 in the axisymmetric and stationary case , where @xmath25 is a linear combination of the two killing vectors [ eq . ( [ e : ell_axi ] ) below ] , one recovers the classical expression @xcite . the first integral ( [ e : int_prem_rigid ] ) can be re - expressed in terms of the 3 + 1 formalism of general relativity . let us introduce the spacetime foliation by the @xmath48 hypersurfaces @xmath49 and the associated future - directed unit vector field @xmath50 everywhere normal to @xmath49 . @xmath50 is the 4-velocity of the so - called _ eulerian observer _ ( also called _ zamo _ or _ locally non - rotating observer _ ) . we have the following orthogonal split of the fluid 4-velocity : @xmath51 with the lorentz factor @xmath52 the spacelike vector @xmath53 is the fluid 3-velocity as measured by the eulerian observer and the second equality in the above equation results from the normalization of the 4-velocity @xmath27 . similarly , we have an orthogonal split of the helical killing vector : @xmath54 where @xmath55 is the lapse function , governing the proper time evolution between two neighboring hypersurfaces @xmath49 . from relation ( [ e : rigid ] ) and the two orthogonal decompositions ( [ e : u_ortho ] ) and ( [ e : ell_ortho ] ) , we get @xmath56 , so that the ( logarithm of the ) first integral ( [ e : int_prem_rigid ] ) can be written @xmath57 with @xmath58 in the non - relativistic limit , @xmath59 tends toward the classical specific enthalpy ( excluding the rest - mass energy ) of the fluid , whereas @xmath60 tends toward the newtonian gravitational potential . therefore , in the newtonian limit , where @xmath61 and @xmath62 , eq . ( [ e : int_prem ] ) reduces to the classical first integral of motion @xmath63 when the rigidly rotating star is still stationary and axisymmetric , it is well known that a coordinate system @xmath64 can be chosen so that the metric takes the papapetrou form ( see e.g. @xcite and references therein ) @xmath65 where @xmath55 , @xmath66 , @xmath67 and @xmath68 are four functions of @xmath69 . the coordinate vectors @xmath70 and @xmath71 are then the two killing vectors @xmath19 and @xmath20 mentioned in sec . [ s : helic ] . the helical killing vector @xmath25 is expressible as @xmath72 with the form ( [ e : metric_axi ] ) , the einstein equations reduce to a set of four coupled elliptic equations ( see e.g. @xcite for the precise form ) . when the triaxial instability sets in , we shall consider that the metric is a perturbation of ( [ e : metric_axi ] ) , which we write as @xcite @xmath73 where @xmath55 , @xmath74 , @xmath75 , @xmath66 , @xmath67 and @xmath68 are six functions of @xmath76 , with @xmath77 note that @xmath70 and @xmath71 are no longer killing vectors , only @xmath25 remains killing . the coordinate system @xmath78 is adapted to the killing vector @xmath25 and @xmath22 is an ignorable coordinate in this system . we call @xmath79 the _ corotating _ coordinate system , and @xmath80 the _ nonrotating _ one . @xmath55 is the lapse function already introduced in eq . ( [ e : ell_ortho ] ) . @xmath81 is ( minus . ] ) the _ shift vector _ of the nonrotating coordinate system . in the triaxial case , there is no equivalent of the papapetrou theorem @xcite which , in absence of convective motions ( in the meridional planes ) , allows one to set to zero all the off - diagonal components of the metric tensor of axisymmetric and stationary spacetimes , except for @xmath82 . so , in principle , all the metric components should be non - vanishing in eq . ( [ e : metric_3d ] ) . however , we retained only @xmath83 and @xmath84 as the extra non zero components with respect to the axisymmetric case . we did so as an approximation in order to simplify the writing of einstein equations . this approximation can be justified by the following remarks : ( i ) the metric element ( [ e : metric_3d ] ) is exact in the stationary axisymmetric case , ( ii ) it encompasses the first order pn metric . this means that the neglected terms are of second order pn and moreover vanish for stationary axisymmetric rotating stars . we consider these terms to be negligible in our study of the verge of the non - axisymmetric instability . the einstein equation leads to the following set of partial differential equations : @xmath85 \ ; = \ ; 16 \pi na^2 b p \ , r \sin\theta \label{e : einstein3 } \\ & & \delta_2 \ , \zeta \ ; = \ ; 8\pi a^2 \ , [ p + ( e+p)u_i u^i ] + \frac{3}{2 } a^2 k_{ij } k^{ij } - \overline\nabla_i \nu \overline\nabla^i \nu \ , \label{e : einstein4}\end{aligned}\ ] ] where the following notations have been introduced [ see also eq . ( [ e : hnu_def ] ) ] @xmath86 and @xmath87 denotes the covariant derivative with respect to the flat 3-metric @xmath88 , @xmath89 the corresponding laplacian and @xmath90 is the laplacian in the 2-dimensional space spanned by @xmath69 : @xmath91 in eqs . ( [ e : einstein1])-([e : einstein4 ] ) , @xmath92 denotes the spatial components of the fluid 3-velocity @xmath53 [ cf . ( [ e : u_ortho ] ) ] . they can be expressed as @xmath93 and @xmath94 is deduced from @xmath53 by means of eq . ( [ e : gamma ] ) : @xmath95 - b^2 r^2 \sin^2\theta ( u^\varphi)^2 \right ) ^{-1/2 } \ .\ ] ] we have also introduced the energy density @xmath96 as measured by the eulerian observer : @xmath97 another quantity not yet defined and which appears in eqs . ( [ e : einstein1])-([e : einstein4 ] ) is the extrinsic curvature tensor @xmath98 of the hypersurface @xmath49 . the above equations must be supplemented by an equation of state ( eos ) , i.e. a relation between @xmath59 , which appears in the first integral ( [ e : int_prem ] ) , and @xmath28 and @xmath29 which appear in the source terms of the einstein equations ( [ e : einstein1])-([e : einstein4 ] ) , via eq . ( [ e : ener_euler ] ) . for the incompressible matter we are considering , we choose this eos to be @xmath99 where @xmath100 is a constant , representing the constant proper energy density of the fluid . the basic idea is to solve the equations presented in sec . [ s : basic ] by means of an iterative scheme to get an axisymmetric stationary solution , and then to introduce some triaxial perturbation and resume the iterative scheme . if the perturbation is damped ( resp . grows ) as long as the iteration proceeds , the equilibrium configuration will be declared stable ( resp . unstable ) . this procedure has been used already in the works @xcite . we shall prove here , by comparison with the analytical result , that it correctly locates the secular instability point along the maclaurin sequence . we solve the system of non - linear elliptic equations ( [ e : einstein1])-([e : einstein4 ] ) by means of the multi - domain spectral method presented in ref . the nice feature of this method is that it introduces surface - fitted coordinates @xmath101 , so that the density discontinuity at the stellar surface is exactly located at the boundary between two domains . in this way , all the fields are @xmath102 functions in each domain . this avoids any spurious oscillations ( gibbs phenomenon ) and results in a very high precision . as discussed in sec . [ s : present ] , this technique constitutes the major improvement with respect to the numerical method used in ref . another difference with ref . @xcite is that we employ the technique presented in ref . @xcite to solve the vector elliptic equation ( [ e : einstein2 ] ) for the shift vector . in particular , we solve for the cartesian components of the shift vector , whereas @xcite solved for the spherical components . a weak point of the surface - fitted coordinate technique of ref . @xcite is that the transformation from the computational coordinates @xmath101 to the physical ones @xmath103 becomes singular when the ratio of polar to equatorial radius @xmath104 is lower than @xmath105 . this prevents us from computing very flattened configurations , but fortunately this causes no trouble in reaching the maclaurin - jacobi bifurcation point , which is at @xmath106 . the iterative procedure is as follows . first one must set a value of the central enthalpy @xmath107 and the rotation frequency @xmath24 , in order to pick out a unique rotating axisymmetric configuration . the iteration is then started from very crude values : flat spacetime and spherical stellar shape . we solve the einstein equations ( [ e : einstein1])-([e : einstein4 ] ) with the spherical energy density and pressure distribution in their right - hand side . we then plug the obtained value of @xmath60 in the first integral of motion ( [ e : int_prem ] ) , along with @xmath107 , to get the enthalpy field @xmath59 . the zero of this field defines the new stellar surface to which we adapt the computational coordinates @xmath101 ( see @xcite for details ) . from @xmath59 we compute new values of the energy density and pressure according to the eos ( [ e : eos_e])-([e : eos_p ] ) . these values are then put on the right - hand side of the einstein equations ( [ e : einstein1])-([e : einstein4 ] ) and a new iteration begins . usually the rotation velocity is set to zero for the ten first steps . it is then switched on , either integrally or ( for very rapidly rotating configurations ) gradually . the convergence of the procedure is monitored by computing the relative difference @xmath108 between the enthalpy fields at two successive steps ( long - dashed line in fig . [ f : evol_q ] ) . the iterative procedure is stopped when @xmath108 goes below a certain threshold , typically @xmath109 , or for high precision computations @xmath110 . evolution during the iterative procedure of the convergence indicator @xmath108 ( long - dashed line ) , the triaxial perturbation @xmath111 ( short - dashed line ) and its growth rate @xmath112 ( solid line ) . the left figure corresponds to a stable configuration with respect to nonaxisymmetric perturbation , the right one an unstable one . @xmath108 and @xmath111 are depicted in logarithmic units , while @xmath112 is multiplied by 10 for better readability . the discontinuity at step 10 is due to the switch of the rotation and that at step 25 to the switch of the triaxial perturbation.,title="fig:",height=226 ] evolution during the iterative procedure of the convergence indicator @xmath108 ( long - dashed line ) , the triaxial perturbation @xmath111 ( short - dashed line ) and its growth rate @xmath112 ( solid line ) . the left figure corresponds to a stable configuration with respect to nonaxisymmetric perturbation , the right one an unstable one . @xmath108 and @xmath111 are depicted in logarithmic units , while @xmath112 is multiplied by 10 for better readability . the discontinuity at step 10 is due to the switch of the rotation and that at step 25 to the switch of the triaxial perturbation.,title="fig:",height=226 ] isocontours of the enthalpy @xmath59 in the equatorial plane at steps 20 , 150 , 180 and 200 for the unstable model presented in fig . [ f : evol_q ] . dashed lines denote negative values of @xmath59 , which correspond to the exterior of the star . the thick solid line denotes the stellar surface.,title="fig:",height=170 ] isocontours of the enthalpy @xmath59 in the equatorial plane at steps 20 , 150 , 180 and 200 for the unstable model presented in fig . [ f : evol_q ] . dashed lines denote negative values of @xmath59 , which correspond to the exterior of the star . the thick solid line denotes the stellar surface.,title="fig:",height=170 ] isocontours of the enthalpy @xmath59 in the equatorial plane at steps 20 , 150 , 180 and 200 for the unstable model presented in fig . [ f : evol_q ] . dashed lines denote negative values of @xmath59 , which correspond to the exterior of the star . the thick solid line denotes the stellar surface.,title="fig:",height=170 ] isocontours of the enthalpy @xmath59 in the equatorial plane at steps 20 , 150 , 180 and 200 for the unstable model presented in fig . [ f : evol_q ] . dashed lines denote negative values of @xmath59 , which correspond to the exterior of the star . the thick solid line denotes the stellar surface.,title="fig:",height=170 ] after this convergence has been achieved , we switch on a triaxial @xmath113 perturbation by modifying the metric potential @xmath60 according to @xmath114 where @xmath115 is a small constant , typically @xmath116 to @xmath117 . then we continue the iteration procedure as described above , without any further modification of the equations . at each step , we evaluate the quantity @xmath118 where @xmath119 and @xmath120 are the @xmath113 coefficients of the fourier expansion of the @xmath23 part of @xmath60 : @xmath121 and the @xmath122 in eq . ( [ e : max_triax ] ) is taken over all the @xmath123 and @xmath124 coefficients . @xmath111 is used to monitor the evolution of the triaxial perturbation : if @xmath125 as the iteration proceeds , we conclude that the perturbation decays and the axisymmetric configuration is a stable one . in the vicinity of the marginally stable configuration , the decay or growth of the perturbation turns out to be pretty small . in order to facilitate the diagnostic , we monitor instead the _ relative _ growth rate of the @xmath113 part , defined as @xmath126 after some transitory regime , @xmath112 turned out to be constant . if @xmath127 ( resp . @xmath128 ) we conclude that the configuration is unstable ( resp . the behaviors of @xmath108 , @xmath111 and @xmath112 in two typical computations are shown in fig . [ f : evol_q ] , whereas fig . [ f : bar ] depicts the development of the triaxial instability . note that the above method does not require to specify some value of viscosity . whatever this value , the effect of viscosity is simulated by the rigid rotation profile that we impose at each step in the iterative procedure via eq . ( [ e : u_i ] ) . if we consider the iteration as mimicking some time evolution , this means that the time elapsed between two successive steps has been long enough for the actual viscosity to have rigidified the fluid flow . consequently , a quantity that we can not get by our method is the instability time scale . as recalled in the introduction , this time scale depends upon the actual value of viscosity , but not the instability itself . the numerical code implementing the above procedure has been constructed upon the c++ library lorene @xcite . numerical computations have been performed on sgi origin200 as well as linux pc workstations . three domains have been used , the innermost one corresponding to the interior of the star , and the outermost one extending to spacelike infinity by means of a suitable compactification ( see @xcite for details ) . the number of spectral coefficients used in each domain is @xmath129 . the corresponding memory requirement is 40 mb and a typical cpu time ( e.g. corresponding to the first 60 steps of fig . [ f : evol_q ] , which are sufficient to conclude about the stability of the configuration ) is 3 min on an intel pentium iv 1.5 ghz processor . numerous tests have been performed to assess the validity of the method and the accuracy of the numerical code . we present here successively tests for axisymmetric configurations in general relativity and tests about the determination of the triaxial instability point in the newtonian limit . tests regarding the triaxial instability in the relativistic case are deferred to sec . [ s : resu_inst ] , where we present a detailed comparison with analytical ( post - newtonian ) results . the multi - domain spectral technique with surface - fitted coordinates has already been tested in the newtonian regime , giving a rapidly rotating maclaurin ellipsoid with a relative error of the order of @xmath110 @xcite . it has been also shown in ref . @xcite that the error is evanescent , i.e. that it decays exponentially with the number of spectral coefficients , which is typical of spectral methods . the accuracy of the computed relativistic axisymmetric models is estimated using two general relativistic virial identities grv2 @xcite and grv3 @xcite . these two virial error indicators are integral identities which must be satisfied by any solution of the einstein equations and which are not imposed during the numerical procedure ( see ref . @xcite for the computation of grv2 and grv3 ) . grv3 is a generalization to general relativity of the classical virial theorem . we have obtained values of grv2 and grv3 of the order of @xmath110 in newtonian regime and @xmath130 for high rotation and high compaction parameter . the virial errors are at least one order of magnitude better than obtained in the previous code used for calculating rapidly rotating strange stars . .[t : ansorg ] comparison with the numerical results of ansorg et al . @xcite for a highly relativistic stationary axisymmetric model @xmath131 ( @xmath132 ) and @xmath133 , with compaction parameter @xmath134 . meaning of the symbols [ in geometrized units ( @xmath135 ) ] are as follows : dimensionless angular velocity @xmath136 , gravitational mass @xmath137 , baryon mass @xmath138 , circumferential equatorial radius @xmath139 , angular momentum @xmath140 . @xmath141 , @xmath142 , and @xmath143 are the redshifts in the polar direction , in the equatorial plane along the direction of motion and opposite to that direction respectively . [ cols="<,<,<",options="header " , ] some important quantities at the viscosity driven instability points for different values of compaction parameter are reported in table [ t : instab ] . we are using two kinds of compaction parameters : @xmath144 already defined and the _ proper compaction parameter _ @xmath145 , where @xmath146 is the circumferential radius , i.e. the length of the equator ( as given by the metric ) divided by @xmath147 of the actual configuration ( unlike @xmath148 which is the circumferential radius of the non rotating star having the same baryon mass ) . our results are shown as thick lines in figs . [ f : omeg_ecc_ours ] , [ f : omeg_tsw_ours ] and [ f : ecc_comp ] . according to our analysis the critical value of the eccentricity very weakly depends on the compaction parameter and for @xmath149 is by only @xmath5 larger than newtonian value of the onset of the secular bar mode instability . the critical value of the ratio of the kinetic energy to the absolute value of the gravitational potential energy @xmath150 for compaction parameter as high as 0.275 is by @xmath151 ( table [ t : instab ] ) higher than the newtonian value . the dependence of @xmath1 on the compactness can be very well approximated by the function @xmath152 the comparison between our relativistic calculations of the viscosity driven instability and the corresponding two different pn calculations derived by @xcite and by @xcite is shown in fig [ f : omeg_ecc ] , [ f : omeg_tsw ] and [ f : tsw_comp ] . shapiro and zane @xcite employ an energy variational principle to determine equilibrium shape and stability of homogeneous triaxial ellipsoids . the method used by them is valid for arbitrary rotation rate , but only for constant density bodies . di girolamo and vietri @xcite determined the value of the eccentricity and @xmath153 at the instability onset point using landau s theory of second - order phase transitions . this method is the extension to pn regime of that used by bertin and radicati @xcite for the newtonian treatment of bar mode instability and valid for any equation of state . considering the dependence of @xmath153 with respect to the eccentricity ( fig . [ f : omeg_ecc ] ) we found good agreement between our results and the pn ones of @xcite . as can be seen , both calculations show much weaker influence of relativistic effects on location of the instability onset point than shapiro & zane pn calculations . figure [ f : tsw_comp ] shows the dependence of the critical eccentricity and the critical value of @xmath6 on the compaction parameter @xmath144 . the authors of @xcite use a proper `` conformal compaction parameter '' @xmath154 , where @xmath155 , @xmath156 , @xmath157 , @xmath158 being the ellipsoid semiaxes . in order to make a comparison between our relativistic calculations and those of @xcite we find the @xmath144 corresponding to their @xmath159 using the formula @xmath160 ( see @xcite , expression [ 4 ] , p. 422 ) . this formula is valid in the spherical limit , but the difference between the proper `` conformal compaction parameter '' and the `` spherical conformal compaction parameter '' is at most 3 % . according to our analysis the critical value of the eccentricity depends very weakly on the compaction parameter ( solid line ) ; this is in agreement with the pn calculations by di girolamo & vietri @xcite . the relative differences between our and their pn calculations are less than @xmath161 . a much stronger weakening of the bar mode instability by general relativity is suggested by the pn study of shapiro & zane @xcite , who find that @xmath162 ( dashed line in the left panel of fig . [ f : tsw_comp ] ) could be as large as 0.94 at @xmath163 . this discrepancy may be ascribed mainly to the ellipsoidal approximation for the deformation and equilibrium shape . we refer to the article @xcite for a detailed explanation of discrepancies between the two pn calculations of instability points . the right panel of fig . [ f : tsw_comp ] presents the comparison between @xcite and our calculation of the critical @xmath6 as a function of the compaction parameter @xmath144 . we found that for the compaction parameter as high as 0.05 ; 0.15 ; 0.25 @xmath1 is only by @xmath164 @xmath165 higher than newtonian value , while according to ref . @xcite , the increase is @xmath166 respectively . according to our study relativistic effects weaken the jacobi - like bar mode instability ( solid line in figs . [ f : omeg_ecc_ours ] , [ f : omeg_tsw_ours ] , [ f : ecc_comp ] and [ f : tsw_comp ] ) , but the stabilizing effect is not very strong . this is in agreement with the pn calculations @xcite . eccentricity ( left ) and ratio of the kinetic energy to the absolute value of the gravitational potential energy ( right ) at the onset of the secular bar mode instability , versus the compaction parameter @xmath145.,title="fig:",height=226 ] eccentricity ( left ) and ratio of the kinetic energy to the absolute value of the gravitational potential energy ( right ) at the onset of the secular bar mode instability , versus the compaction parameter @xmath145.,title="fig:",height=226 ] eccentricity ( left ) and ratio of the kinetic energy to the absolute value of the gravitational potential energy ( right ) at the onset of the secular bar mode instability , versus the compaction parameter @xmath144 . our results are denoted by the solid line . the dashed line and the dash - dotted line correspond to the pn calculations by @xcite and @xcite respectively.,title="fig:",height=226 ] eccentricity ( left ) and ratio of the kinetic energy to the absolute value of the gravitational potential energy ( right ) at the onset of the secular bar mode instability , versus the compaction parameter @xmath144 . our results are denoted by the solid line . the dashed line and the dash - dotted line correspond to the pn calculations by @xcite and @xcite respectively.,title="fig:",height=226 ] triaxial instabilities of rotating compact stars can play an important role as emission mechanisms of gravitational waves in the frequency range of the forthcoming interferometric detectors . a rapidly rotating neutron star can spontaneously break its axial symmetry if the ratio of the rotational kinetic energy to the absolute value of the gravitational potential energy @xmath6 exceeds some critical value . we have investigated the effects of general relativity upon the nonaxisymmetric viscosity - driven bar mode instability of incompressible , uniformly rotating stars . this is the relativistic analog of the newtonian maclaurin - jacobi bifurcation point . our method of finding the instability point is similar to that used in ref . @xcite for compressible fluid stars . the main improvement with respect to this work regards the numerical technique . we have indeed introduced surface - fitted coordinates , which enable us to treat the strongly discontinuous density profile at the surface of incompressible bodies . this avoids any gibbs - like phenomenon and results in a very high precision , as demonstrated by comparison with the analytical result for the newtonian maclaurin - jacobi bifurcation point , that our code has retrieved with a relative error of @xmath167 . according to our results , general relativity weaken the jacobi - like bar mode instability : the values of @xmath168 , eccentricity and @xmath153 increase at the onset of instability above the newtonian values . this general tendency is in agreement with pn analytical results @xcite for rigidly rotating incompressible bodies and with the numerical calculations of @xcite for relativistic polytropes . however we found that the stabilizing effect of general relativity is much weaker than that obtained in the pn treatment of @xcite . the critical value of the ratio of the kinetic energy to the absolute value of the gravitational potential energy @xmath1 for a compaction parameter as high as 0.275 is only @xmath151 larger than the newtonian value , whereas it has been found @xmath169 larger by @xcite . according to our analysis the critical value of the eccentricity very weakly depends on the compaction parameter and for a compaction parameter as high as 0.275 is only @xmath5 ( @xmath170 according to @xcite ) larger than the newtonian value of the onset of the secular bar mode instability . regarding the dependence of @xmath171 and @xmath162 with respect to compactness , we found very good agreement between our result and the recent pn ones of @xcite , the relative differences being lower than @xmath161 . we are grateful to nick stergioulas and tristano di girolamo for helpful discussions and to brandon carter for reading the manuscript . we also thank stuart shapiro and silvia zane for providing tables of their results and tristano di girolamo and mario vietri for providing their results prior to publication . this work has been funded by the following grants : kbn grants 5p03d01721 ; the greek - polish joint research and technology program epan - m.43/2013555 and the eu program `` improving the human research potential and the socio - economic knowledge base '' ( research training network contract hprn - ct-2000 - 00137 ) .

we perform some numerical study of the secular triaxial instability of rigidly rotating homogeneous fluid bodies in general relativity . in the newtonian limit , this instability arises at the bifurcation point between the maclaurin and jacobi sequences . it can be driven in astrophysical systems by viscous dissipation . we locate the onset of instability along several constant baryon mass sequences of uniformly rotating axisymmetric bodies for compaction parameter @xmath0 . we find that general relativity weakens the jacobi like bar mode instability , but the stabilizing effect is not very strong . according to our analysis the critical value of the ratio of the kinetic energy to the absolute value of the gravitational potential energy @xmath1 for compaction parameter as high as @xmath2 is only @xmath3 higher than the newtonian value . the critical value of the eccentricity depends very weakly on the degree of relativity and for @xmath4 is only @xmath5 larger than the newtonian value at the onset for the secular bar mode instability . we compare our numerical results with recent analytical investigations based on the post - newtonian expansion .

gr-qc0205102

a large number of cosmological and astrophysical observations have evidenced that 85% of the matter content of the universe is in the form of dark matter ( dm ) . a generic weakly interacting massive particle ( wimp ) is a well - motivated candidate for this new kind of matter , since its thermal production in the early universe would match the observed dm abundance . in addition , wimps can be easily accommodated in theories beyond the standard model , such as supersymmetry ( susy ) . wimps can be searched for indirectly , through the particles produced when they annihilate in the dm halo ( photons , neutrinos and antiparticles ) . among the different annihilation products , gamma - rays provide an appealing detection possibility because the signal can be traced back to the source . the large area telescope ( lat ) aboard the fermi gamma - ray space telescope has produced the most detailed maps of the gamma - ray sky for a wide range of energies , with unprecedented angular and energy resolutions . using data from the fermi - lat , various studies have revealed the presence of an excess from an extended gamma - ray source in the inner region of the galaxy @xcite , a signal that is robust when known uncertainties are taken into account @xcite . although the explanation of this galactic centre excess ( gce ) is still under debate , if it were interpreted in terms of dm annihilations @xcite it would correspond to a particle with a mass in the range @xmath2 gev for a @xmath3 final state ( @xmath4 gev for a @xmath5 final state ) and with an annihilation cross section in the dm halo , @xmath6 @xmath7/s , remarkably close to that expected from a thermal relic . several attempts have been made to explain the gce in terms of simplified models for dm @xcite , considering dm annihilation into pure channels . however , as pointed out in ref . @xcite , it is crucial to investigate if this excess can be obtained within a complete theoretical framework . for example , it has been recently shown that the neutralino could reproduce the gce for dm masses up to hundreds of gev depending on the primary annihilation channel within the context of the mssm @xcite and the next - to - mssm ( nmssm ) @xcite . in this article , we carry out a complete analysis of the right - handed ( rh ) sneutrino in the nmssm @xcite and demonstrate that it can successfully account for the gce while fulfilling all the experimental constraints from direct and indirect dm searches as well as collider physics . we apply the lux and supercdms limits on the spin - independent elastic scattering cross section of dm off protons , which are currently the most stringent bounds from direct detection experiments . we also consider the latest results from the lhc on the higgs boson mass and couplings to the sm particles , which are known to be specially constraining for light dm scenarios through the upper bound on the invisible and non standard model higgs decays . besides , the latest bounds from the measurement of the rare decays @xmath8 , @xmath9 and @xmath10 are also applied . finally , we incorporate the fermi - lat constraints on dwarf spheroidal galaxies ( dsphs ) and spectral feature searches in the gamma - ray spectrum , including an estimation of the effect that the most recent results derived from the pass 8 data impose on our results . this model has been extensively described in refs . it is an extended version of the nmssm , in which a new gauge singlet superfield , @xmath11 , is introduced in order to account for rh neutrino and sneutrino states as in @xcite . the superpotential of this construction is given by @xmath12 where flavour indices are omitted and the dot denotes the antisymmetric @xmath13 product . @xmath14 is the nmssm superpotential , @xmath15 is a new dimensionless coupling , @xmath16 is the neutrino yukawa coupling , and @xmath17 are the down and up type doublet higgs components , respectively . as in the nmssm , a global @xmath18 symmetry is imposed so that there are no supersymmetric mass terms in the superpotential . since we assume @xmath19-parity conservation in order to guarantee the stability of the lsp , the terms @xmath20 and @xmath21 are forbidden . furthermore , we do not consider cp violation in the higgs sector . after radiative electroweak symmetry breaking the higgs fields get non - vanishing vacuum expectation values ( vevs ) and the physical higgs states correspond to a superposition of the @xmath22 , @xmath23 and @xmath24 fields . the rh sneutrino interacts with the sm particles through the mixing in the higgs sector thanks to the coupling @xmath25 , thereby behaving as a wimp . interestingly , light rh sneutrinos with masses in the range of @xmath26 gev are viable dm particles @xcite and constitute ideal candidates to account for the gce , as we already pointed out in ref.@xcite . their phenomenology is very rich , as they can annihilate into a variety of final states , some of which include scalar and pseudoscalar higgses . in particular , if @xmath27 , the annihilation final state of rh sneutrinos is dominated by a @xmath28 pair in vast regions of the parameter space . the subsequent decay of each scalar and pseudoscalar higgs into pairs of fermions or gauge bosons gives rise to non - standard final states , which often display spectral features coming from the @xmath29 final states . given that the rh sneutrino annihilation contains a mixture of final states , often including exotic configurations , the gce model - independent approach generally found in the literature is not applicable . to fit the gce we have followed the approach of ref . @xcite , where the authors take into account theoretical model systematics by exploring a large range of galactic diffuse emission models final state with a mass of 49@xmath30 gev and a velocity averaged annihilation cross section of @xmath31 @xmath7/s . other analyses of the gce employ different assumptions on the galactic diffuse and point source components , and the reconstructed dm mass and annihilation cross section differ slightly . ] . regarding the dm distribution , we have considered a generalised nfw profile with slope @xmath32 in the region of interest ( roi ) @xmath33 and @xmath34 , as in ref . @xcite . to implement the aforementioned analysis , we have performed a series of scans over the parameter space of the model in the rh sneutrino mass range @xmath35 gev , computing the gamma - ray spectrum as well as the rh sneutrino relic abundance with micromegas 3.6.9 @xcite . the number of free parameters and the corresponding ranges of variation coincide with those used in ref . @xcite ( only the range in the rh sneutrino mass has been enlarged to accommodate new solutions ) . in our analysis , we have also incorporated the most recent constraints from direct detection experiments using the latest results of lux and supercdms @xcite . lhc constraints on the masses of supersymmetric particles , the mass and couplings of the sm higgs boson , as well as bounds from the rare decays @xmath8 , @xmath9 and @xmath10 , have also been implemented ( for more details , see ref . @xcite ) . we have set an upper bound on the rh sneutrino relic abundance , @xmath36 , consistent with the latest planck results @xcite . besides , we have considered the possibility that rh sneutrinos only contribute to a fraction of the total relic density and set for concreteness a lower bound on the relic abundance , @xmath37 . to deal with these cases , the fractional density , @xmath38 $ ] , has been introduced to account for the reduction in the rates for direct and indirect searches ( assuming that the rh sneutrino is present in the dm halo in the same proportion as in the universe ) . then , we have convoluted the differential photon spectrum of the scan points fulfilling all these constraints with the energy resolution of the lat instrument . we have used the p7rep - source - v15 total ( front and back ) resolution of the reconstructed incoming photon energy as a function of the energy for normally incident photons . afterwards , we have calculated the @xmath0 function as follows @xcite : @xmath39 where @xmath40 is the covariance matrix containing the statistical errors and the diffuse model and residual systematics @xcite . @xmath41 ( @xmath42 ) stands for the measured ( predicted ) flux in the @xmath43th energy bin . the vector @xmath44 refers to all parameters of our model which determine the predicted flux . the fermi - lat satellite has also provided bounds on the dm annihilation cross section in the galactic halo derived from the study of the gamma - ray spectrum from dsphs and the search for spectral features in the galactic centre . these limits play an important role in the current analysis . let us review them in more detail . + @xmath45 dwarf spheroidal galaxies the mass of these objects is dominated by dm ; hence , they constitute ideal targets for indirect searches . the fermi - lat collaboration has performed an analysis of the gamma - ray emission from 25 dsphs using four years of data @xcite . the absence of a signal can be interpreted as constraints on the annihilation cross section of dm particles . it is customary to assume annihilation into pure sm channels in the calculation of these bounds . the occurrence of non standard annihilation final states in our model prevents us from using these results directly . instead , we have extracted independent upper bounds on @xmath46 for each of the six more constraining dsphs ( coma berenices , draco , segue i , ursa major ii , ursa minor and willman i ) , using the dm flux predicted by our model and the mean values for the @xmath47 factors from ref . then , we have applied the most restrictive of these limits to our data . we have checked that this method leads to slightly less stringent bounds than the combined limit from the fermi - lat collaboration ( by a factor smaller than @xmath48 ) , when applied to the region of dm masses from 10 to 100 gev with pure annihilation channels . lastly , we have also estimated the impact of the preliminary results derived from the latest data ( pass 8) presented by the fermi - lat collaboration @xcite . in general for any final state , the limit on @xmath46 improves by approximately a factor @xmath49 for a dm mass in the range @xmath50 gev . conservatively , we have used a factor 4 to assess the dsph bounds derived from the newest fermi data . + @xmath45 gamma - ray spectral features the fermi - lat collaboration has performed a search for spectral lines in the energy range @xmath51 gev @xcite . not having found any globally significant spectral feature , this analysis has been translated into 95% c.l . upper limits on the dm annihilation cross section into a pair of photons , @xmath52 . the rh sneutrino in the nmssm can give rise to a complex spectrum , displaying lines and box - shaped spectral features , which arise from the diagrams shown in figure [ fig : diagram ] , and involve annihilation into pairs of light scalar and pseudoscalar higgs bosons . the first diagram shows the usual contribution to the primary production of a pair of photons through a loop of charginos , sfermions , top quarks , @xmath53 and charged higgses , and would produce a line with @xmath54 . the second and third diagrams would produce a line with energy @xmath55 or @xmath56 if any of the higgs bosons were produced nearly at rest , when @xmath57 or @xmath58 . otherwise , the decay in flight of the boosted higgs bosons gives rise to box - shaped features with a maximum energy @xmath59 and widths @xmath60 . the published bounds @xcite on @xmath52 do not include the specific dm halo used in this paper for the analysis of the gce . in order to recalculate this limit , we have computed the @xmath47 factor for our halo in the region of interest ( roi ) r41 circular region centred on the galactic centre with a mask applied to @xmath61 and @xmath62 , and has been optimised for a regular ( @xmath63 ) nfw profile . ] , @xmath64 gev@xmath65 @xmath66 , and compared it with the one used in ref.@xcite , which yields @xmath67 gev@xmath65 @xmath66 . the ratio @xmath68 is then applied to the fermi - lat bounds on @xmath52 from ref . @xcite . then , we have applied the bounds on the annihilation cross section into two photons to the @xmath69 predicted by our model for monochromatic gamma - ray lines . concerning the box - shaped contributions , first we have derived the corresponding limits on the annihilation cross section from the fermi - lat gamma - ray line bounds and afterwards we have applied them to our prediction weighted by the fractional dm density squared , @xmath70 , along the box width . finally , note that stronger bounds could be obtained if we used a roi which is optimised for the profile used here , but this is out the scope of this article . in figure [ fig : sv95p7 ] , we show @xmath46 vs. @xmath71 for the points of the parameter space that fit the gce at 95% c.l . the different colours indicate the main annihilation channel ( remember that the whole annihilation spectrum is considered when calculating the gamma - ray flux ) . the stars represent the best fit point for each of the dominant annihilation channels and their properties are summarised in table[tab : chi ] , where we distinguish `` pure final states '' ( if the main annihilation channel contributes to more than 90% to @xmath46 ) and `` mixed final states '' . lastly , black circles correspond to the points that would be allowed by our estimation of the pass 8 constraints on dsphs . final state & @xmath71 ( gev ) & @xmath72 ( @xmath7/s ) & @xmath73 & @xmath0 + @xmath74 ( @xmath75% ) & 119.8 & @xmath76 & 0.094 & 21.9 + @xmath77 ( @xmath78% ) & 65.0 & @xmath79 & 0.109 & 22.3 + @xmath3 ( @xmath80% ) & 46.1 & @xmath81 & 0.038 & 22.6 + + final state & @xmath71 ( gev ) & @xmath72 ( @xmath7/s ) & @xmath73&@xmath0 + @xmath77 ( @xmath82% ) & 63.8 & @xmath83 & 0.061 & 20.8 + @xmath3 ( @xmath84% ) & 63.2 & @xmath83 & 0.042 & 21.0 + @xmath74 ( @xmath85% ) & 121.4 & @xmath86 & 0.075 & 21.6 + @xmath87 ( @xmath88% ) & 39.6 & @xmath89 & 0.071 & 23.7 + @xmath90 ( @xmath91% ) & 39.0 & @xmath92 & 0.099 & 25.4 + @xmath93 ( @xmath94% ) & 127.4 & @xmath95 & 0.054 & 25.9 + @xmath77 @xmath96 ( @xmath97% ) & 25.5 & @xmath98 & 0.068 & 27.4 + @xmath99 ( @xmath100% ) & 72.4 & @xmath101 & 0.104 & 29.2 + as we can observe , there are solutions that fit the gce for rh sneutrino masses in the range @xmath102 gev , while fulfilling all other experimental constraints ( from direct and indirect dark matter searches as well as from the lhc ) . the best fit points for pure annihilation channels are in good agreement with model - independent studies @xcite , but we have also obtained new non - standard annihilation channels ( into light scalar and pseudoscalar singlet - like higgs bosons ) and examples with mixed final states which provide a slightly better fit to the gce . notice also that the points are separated in two regions in the rh sneutrino mass . let us comment in more detail these two regions . + @xmath45 @xmath103 gev . we have found solutions where the rh sneutrino annihilates mainly into a pair of very light , singlet - like , cp - odd higgs bosons ( cyan ) . since @xmath104 , these pseudoscalars can not decay into a pair of @xmath105 quarks and instead they do it predominantly into a pair of @xmath106 leptons . the resulting process , @xmath107 , leads to a leptonic final state ( with best fit around @xmath108 gev ) , which differs from the usual @xmath109 final state ( whose best fit is around @xmath110 gev @xcite ) . we have also found @xmath109 final states ; however , these appear only for @xmath111 gev @xcite and therefore fall out of the 95% c.l . + @xmath45 @xmath112 gev . this region is populated by points which present annihilation mainly into @xmath3 ( grey ) , @xmath90 ( green ) , @xmath87 ( violet ) , @xmath113 ( cyan ) , @xmath114 ( blue ) and @xmath115 ( dark blue ) . the best fit for a pure annihilation into a @xmath3 pair is obtained for @xmath116 gev ( see table [ tab : chi ] ) , in good agreement with ref . @xcite , but it shifts to larger masses @xmath117 gev if mixed final states are considered . very few solutions with dominant @xmath90 and @xmath87 final states are found . these channels dominate when the up component of the lightest higgs is larger than the down component , which enhances the higgs coupling to up - type fermions and top loop contributions to @xmath87 final states . however , these loop contributions also enhance the @xmath29 line production and most of the points are excluded for this reason . besides , these final states are always related to the resonant annihilation of rh sneutrinos through a light singlet - like @xmath118 @xcite and typically have a smaller relic abundance than the lower bound considered in this article . this also happens for other channels when @xmath119 gev and explains the gap in the plot . the annihilation into a pair of cp even higgs bosons takes place mostly for @xmath120 gev . these subsequently decay mainly into @xmath121 ( if the down component of @xmath118 is large ) , giving rise to a four quark final state @xmath122 . it can also decay into gluons and @xmath123 ( if the up component of @xmath118 is large ) . it is worth noting that this light @xmath118 is singlet - like , typically has a mass of @xmath124 gev , very typical of the nmssm . for this reason the best fit point ( which in our analysis is obtained for @xmath125 gev with @xmath126 ) differs from other solutions found for the sm higgs @xcite . finally , regarding the annihilation into @xmath113 , the pseudoscalar in this region satisfies @xmath127 and thus decays predominantly into @xmath3 . the annihilation into four @xmath105 quarks , @xmath128 , produces a best fit of the gce for @xmath129 gev when this channel is almost pure . mixed scenarios are in fact responsible for the best fit point found in our analysis with 44% of annihilation into @xmath113 , a rh sneutrino mass @xmath130 gev and a @xmath1 . + the spectra for the best fit points in table [ tab : chi ] are shown in figure [ fig : spec ] . as we can see the pseudoscalar and scalar channels provide the best fits . this is mostly due to the presence of spectral features ( mainly lines in the case of @xmath113 and box - shaped emissions for @xmath114 ) that improves the fit at large energies . on top of this , the energy distribution of the spectra of four - body final states ( fermions , gauge bosons or combinations between these two , as shown in fig.1 ) is also known to provide better fits to the gce @xcite . other two - body final states , such as @xmath3 , @xmath90 , @xmath87 and @xmath99 , also benefit from the presence of lines in the spectrum which might be probed by fermi - lat using the latest pass 8 reprocessed data . in figure [ fig : sv95p7 ] , we also show all the points that are not ruled out by our estimation of the pass 8 dsph bounds and provide a 95% c.l . fit to the gce . it is remarkable that all the solutions found that fit the gce at 68% c.l . would be now ruled out , highlighting the tension between the pass 8 data and the gce . moreover , the points that survive the pass 8 constraints are mainly related to dominant annihilation into non standard final states , such as @xmath77 and @xmath74 . this is due to the fact that these solutions produce four particles in the final state , which makes their spectrum different from that of the usual two - body final states . furthermore , at the same time they produce spectral features at high energies . it is worth noting that the points which are still alive and fit the gce at 95% c.l . might show up in the results of the fermi - lat collaboration in the very near future . the rh sneutrino elastic scattering off quarks is mediated by the exchange of a cp - even higgs boson on a @xmath131 channel , and the dominant contribution arises from the lightest of these particles , @xmath132 . as it was shown in ref . @xcite , the annihilation cross section and the spin - independent scattering cross section off protons , @xmath133 , are correlated by crossing symmetry when the main annihilation final state in the early universe is any quark pair . this generically leads to a large @xmath133 , already excluded by current direct detection limits . however , this correlation can be broken in the presence of resonant annihilations . because of this as well as the different annihilation final states , the predictions for @xmath133 span many orders of magnitude @xcite . as already emphasised in the previous section , there is a wide variety of final states that can fit the gce at 95% c.l . ; therefore the prospects for direct detection can be very different . in figure [ fig : si95p7 ] , we show the theoretical predictions for @xmath134 as a function of the rh sneutrino mass for all the points that fit the gce at 95% c.l . we have also included the most stringent bounds from direct detection experiments @xcite and some of the predicted sensitivities of next - generation detectors . it is noteworthy that many of the points have a relatively large scattering cross section and might be within the reach of future experiments , such supercdms and lz . in particular , this is the case of most of the examples with @xmath77 final states and @xmath135 gev that also satisfy our estimation of the pass 8 constraints . on the contrary , especially in the points with resonances , the predicted @xmath136 can be extremely small . the resonance with the sm - like higgs @xmath137 is narrow and occurs around @xmath138 gev . the resonance with the lightest singlet - like higgs , @xmath118 , seems broader due to the variation of @xmath139 throughout the scan and generically occurs for @xmath140 gev . fortunately , the presence of such light scalar and/or pseudoscalar higgs bosons in these cases might give rise to interesting signals at the lhc . as already pointed out , most of the cases that provide good fits to the gce also contain box - shaped features and lines in the gamma - ray spectrum . although we have included in our analysis the constraints from the fermi - lat collaboration on these contributions , future searches for these kind of features might provide new insight on the dm properties . in this subsection , we will determine which of the points that fit the gce at 95% c.l . are also within the reach of future searches for gamma - ray lines . to this aim , we have defined the following ratio : @xmath141 which quantifies how far our predictions are from the current experimental limits . here @xmath142 is the contribution from our model to spectral features ( box or lines ) on the gamma - ray spectrum and @xmath143 is the expected limit from ref . @xcite for r41 . the factor @xmath144 has been introduced in the previous section to convert the bounds of ref . @xcite to the dm halo considered in this article . @xmath145 is evaluated at the energy @xmath146 , for which the ratio is maximised . for gamma - ray lines @xmath146 coincides with the energy of the line . on the other hand , for box - shaped features @xmath146 represents the mean value of the fermi - lat energy bin for which @xmath142 is closer to the current expected limit . notice that @xmath147 , since the current bound on spectral features has already been applied to our data . in figure [ fig : r ] , we represent @xmath145 vs. @xmath146 for all the points that fit the gce at 95% c.l . we also indicate with a dashed line the expected improvement on these kind of searches with the pass 8 data . we can observe that many of these scenarios have @xmath148 , which means that an improvement of a factor 2 in the fermi - lat sensitivity to the search for spectral features would be enough to probe these solutions . as has already been emphasised , the points in which the rh sneutrino annihilates mainly into scalar or pseudoscalar higgs bosons typically present box - shaped features and/or lines in their spectrum . the energy @xmath146 at which we evaluate @xmath145 ( and at which we expect them to be detected ) is systematically shifted towards low values , since the sensitivity of the fermi - lat instrument is better is typically achieved close to the lower - end of the box . ] . in particular , all the points with @xmath114 final states have @xmath149 gev ( in spite of the rh sneutrino mass being @xmath150 gev ) . regarding the points with @xmath113 final states , only a portion of them with @xmath151 gev might be observable in the near future . unfortunately , all of the points that satisfy our expectation of the pass 8 bound on dsphs have very small values of @xmath145 and therefore would not present any observable feature in the gamma - ray spectrum . finally , in this section we investigate the implications of new physics searches at the lhc . in general , the nmssm introduces profound changes in the structure of the higgs sector . most notably , the addition of new states opens the door to the occurrence of light singlet - like scalar and pseudoscalar higgs bosons with a mass that can be considerably smaller than that of the sm higgs . as we have seen in the previous sections , these scenarios are very common when we try to fit the gce with rh sneutrino dm at 95% c.l . these light particles provide new decay channels for the sm higgs boson , @xmath137 . some of them contribute to the invisible branching fraction , as is the case of the rh sneutrino , the rh neutrino and the lightest neutralino ( @xmath152 , @xmath153 and @xmath154 ) , whereas others would only appear as non - standard decay channels(@xmath155 and @xmath156 ) . current lhc data leave considerable room for these contributions [ e.g. , br@xmath157 . nevertheless , the future high luminosity lhc run ( 3000 fb@xmath158 at @xmath159 tev ) will significantly narrow down these ranges ( e.g. , the invisible branching ratio in @xmath160 associated production will be probed at 95%c.l . down to 13% and the higgs couplings to sm particles will be measured individually with 10% accuracy @xcite ) . it is therefore conceivable that some of the scenarios considered in this article can be probed in this way . we have defined the invisible branching ratio of the sm higgs and the branching ratio of the sm higgs into non sm particles as follows , @xmath161 mostly only the rh sneutrinos contribute to the invisible branching fraction , since the rh neutrino mass is generally too large . neutralinos in this scenario would decay into the rh sneutrino through @xmath162 or @xmath163 @xcite , contributing to the invisible branching ratio of the sm higgs boson . nevertheless , this contribution is negligible in most of the points of the parameter space since either the lightest neutralino is too heavy or the branching ratio of the @xmath154 process is very small . in figure [ fig : higgs ] , we plot @xmath164 as a function of @xmath165 for all the points that fit the gce at 95% c.l . we denote with dashed and dot - dashed lines the expected sensitivity in the future high luminosity lhc run to these two observables , respectively . many solutions found here would lead to observable signals , mainly in searches of non sm decays of @xmath137 . interestingly , this is the case of most of the solutions to the gce that survive our estimation of the pass 8 constraints on dsphs , which have @xmath166 . furthermore , the @xmath77 final states that fulfil the pass 8 bounds from dsphs might be also probed at the lhc , through the direct production of @xmath167 and its subsequent decay into @xmath106 leptons producing a multilepton signal @xcite . as seen previously , some of the best fits to the gce in this model require the presence of either a light cp - even or a cp - odd higgs boson ( see table [ tab : chi ] ) . in fact , after the inclusion of the dsph pass 8 bounds , only the regions with a very light singlet - like cp - odd higgs boson survive . in figure [ fig : masses ] ( left panel ) , we represent the solutions that fulfil all the experimental constraints and fit the gce at 95% c.l . in the plane @xmath168 . in general , points with light cp - even higgs bosons below the mass threshold @xmath169 are more difficult to obtain than those with a light cp - odd higgs ( the sm - like higgs ) . our choice of signs for @xmath170 and @xmath171 makes the coupling of @xmath132 to @xmath172 higher than the @xmath167 one ; thereby the existing lhc constraints on the sm - like higgs couplings to sm particles are more stringent for the @xmath132 final states @xcite . this does not apply to the limit @xmath173 , where the relative sign between @xmath170 and @xmath171 does not play any role . ] . the mass of the pseudoscalar higgs can even be smaller than @xmath174 ( thus favouring its decay into leptons ) , whereas the mass of the lightest scalar higgs boson remains above 20 gev . it is noteworthy that the solutions allowed by the estimated pass 8 bounds from dsphs always contain a pseudoscalar higgs below 50 gev , and some of them also include a scalar higgs below 60 gev . in figure [ fig : masses ] ( right panel ) , we show the mass ranges of the different supersymmetric particles , as well as the higgs sector for the aforementioned solutions . in general , these solutions have a mass spectrum that contains several non sm particles below the frontier of 100 gev . namely , we can see that neutralinos in these scenarios can be as light as 20 gev while in some cases also the rh neutrinos can reach a mass around 60 gev . notice that the masses of sleptons and squarks do not vary much , since the soft masses and trilinear parameters for these particles were fixed in our scan . figure [ fig : inputs ] displays the solutions fitting the gce at 95% c.l . in the planes @xmath171 versus @xmath170 ( left panel ) and @xmath175 versus @xmath176 ( right panel ) , where the main constraint on these parameters is the bound on the @xmath172 mass @xcite . for @xmath74 final states ( blue ) we observe a slight preference for small values of @xmath170 , while for @xmath77 ( cyan ) and @xmath3 ( grey ) final states the solutions can be found for a wide range of values . very light pseudoscalar higgses can be found in the peccei - quinn ( pq ) limit @xmath177 and @xmath178 , for which the @xmath179 symmetry is restored , and it is precisely in these regions where we find most of the points that are in agreement with the expected dsph bounds using pass 8 data . in the right panel of fig . [ fig : inputs ] , we can observe that these solutions are grouped in the region of @xmath180 gev while the values of @xmath175 do not show any preferred region . the pq limit is not needed for points that only fulfil pass 7 data as the pseudoscalar mass can be larger . the nmssm can also accommodate low - mass neutralino dm @xcite . several studies have recently pointed out that these neutralinos can also explain the gce @xcite when they are either singlino / higgsino or bino / higgsino admixtures @xcite . the former situation can occur when @xmath181 and requires some degree of fine - tuning in the higgs mass matrix parameters in order to avoid large deviations from the sm - like higgs composition . moreover , to account for the observed relic abundance the mass of the pseudoscalar higgs must be very close to the resonant condition , @xmath182 . the bino / higgsino scenario requires @xmath183 and is however less fine - tuned since no resonant condition is required to fulfil the relic density constraint . the region of the parameter space in which rh sneutrinos can explain the gce is similar to that with singlino / higgsino neutralinos , as @xmath184 ( in fact , light neutralinos are also frequent as we showed in fig.[fig : masses ] ) . it is worth noting that , especially when pass 8 constraints are imposed , both models feature a similar higgs spectrum with light singlet - like scalar and pseudoscalar higgs bosons . nevertheless , in the nmssm with rh sneutrinos the amount of fine - tuning required is largely reduced as resonant annihilation is no longer necessary in order to obtain the correct relic abundance . in this model , the annihilation cross section increases when the annihilation channels into light scalar and pseudoscalar higgs bosons ( @xmath185 and @xmath186 ) are open @xcite . this is illustrated in figure [ fig : mh1-msn ] , where @xmath187 is represented as a function of @xmath188 for the set of points for which the gce is fit at 95% c.l . the solid line denotes @xmath189 ( threshold for the annihilation in @xmath74 ) and the dashed line corresponds to @xmath190 ( condition for resonant annihilation ) . only the points with @xmath121 final states require resonant annihilation . it should finally be emphasized that , contrary to the neutralino case , the parameters that determine the rh sneutrino properties ( the soft mass , @xmath71 , the trilinear term , @xmath191 , and the coupling , @xmath15 ) do not affect the masses of the higgs sector of the nmssm . therefore , it is much easier to obtain the correct relic abundance without violating the lhc bounds . in this article , we have demonstrated that rh sneutrino dark matter in the nmssm can account for the observed low - energy excess in the fermi - lat gamma - ray spectrum from the galactic centre . since we are working with a complete theoretical framework , we have also explored the complementarity with other dm search strategies , such as indirect searches for gamma - ray lines , direct dm detection , and the implications for a future lhc run . more specifically , we have performed a scan in the parameter space of the model , incorporating all the constraints from the lhc as well as direct and indirect dark matter searches . for the latter , we have included the fermi - lat bounds on dsphs and gamma - ray spectral features . in addition , we have estimated the effect of the latest fermi - lat reprocessed data ( pass 8) . we have computed the gamma - ray spectrum for each point in the parameter space of the model , and then we have performed a @xmath0 fit to the excess . it should be emphasised that , contrary to usual model - independent approaches , we have taken into account the full annihilation products , without assuming pure annihilation channels . we have obtained good fits to the gce for a wide range of the rh sneutrino mass , @xmath192 gev , with annihilation cross section in the dm halo @xmath193@xmath7/s at 95% c.l . there is a large variety of final states for the rh sneutrino annihilation . in general , we observe that the fit to the gce is good when the rh sneutrino annihilates mainly into pairs of light singlet - like scalar or pseudoscalar higgs bosons , @xmath114 or @xmath113 , that subsequently decay in flight . these produce spectral features , such as gamma - ray lines and/or box - shaped emissions that improve the goodness of the fit at large energies . the best fit of our analysis ( @xmath1 ) corresponds to a rh sneutrino with a mass of @xmath194 gev which annihilates preferentially into a pair of light singlet - like pseudoscalar higgs bosons ( with masses of order 60 gev ) . good fits are also obtained for a variety of mixed final states and have been summarised in table[tab : chi ] . our estimation of the constraints from the new pass 8 data from fermi - lat rules out all the points that fit the gce at 68% c.l . and leaves only a few of them within the 95% c.l . interestingly , these correspond mainly to points with predominant annihilation into light pseudoscalar higgs bosons . it is remarkable that a region with @xmath108 gev remains valid.in this region , the light pseudoscalar has a small mass , @xmath104 , and therefore leads to a leptonic final state with 4@xmath106 , @xmath107 , which differs from the usual result for @xmath109 . we have then studied the implications for direct detection of the points that fit the gce at 95% c.l . because of the presence of light non sm particles in the mass spectrum of the model , and the fact that many of our solutions involve resonant annihilation through the lightest scalar higgs , the prospects for the rh sneutrino scattering cross section off quarks span several orders of magnitude . it is remarkable that many of the points found here are within the reach of next - generation experiments such as supercdms and lz . interestingly , this is the case of the few points which also survive our estimation of the pass 8 constraints . we have also investigated the prospects for the detection of these scenarios in future searches for gamma - ray spectral features , using the estimated future sensitivity of fermi - lat for this kind of studies . we have found that only a small fraction of the points that fit the gce at 95% c.l . will be observable in this way , mainly those associated with resonant annihilation through the lightest or second - lightest scalar higgs boson . unfortunately , none of the scenarios allowed by our estimation of the pass 8 constraints would be detectable by spectral feature searches . finally , we have computed the contribution of these scenarios to the invisible and non - sm branching ratios of the sm higgs and compared them with the predicted reach of the future high - luminosity lhc run at 14 tev . we have determined that a substantial amount of viable scenarios ( many of which also satisfy the pass 8 constraint ) can be probed in such a way . it is remarkable that many of the scenarios that can explain the gce with rh sneutrino dm can be subject to a complementary detection through other methods in the near future , such as direct dm searches or collider tests of the higgs sector . the authors are grateful to v. gammaldi , g. a. gomez - vargas , d. hooper , t. linden , r. lineros , and c. weniger for useful comments . is partially supported by the stfc , and s.r . by the campus of excellence uam+csic . we thank the support of the consolider - ingenio 2010 program under grant no . multidark csd2009 - 00064 , the spanish micinn under grants no . fpa2012 - 34694 and fpa2013 - 44773-p , the spanish mineco centro de excelencia severo ochoa program under grant no . sev-2012 - 0249 , and the european union under the erc advanced grant sple under contract erc-2012-adg-20120216 - 320421 . v. vitale _ et al . _ [ fermi - lat collaboration ] , arxiv:0912.3828 . d. hooper and l. goodenough , phys . b * 697 * ( 2011 ) 412 [ arxiv:1010.2752 ] . a. morselli _ et al . _ [ fermi - lat collaboration ] , nuovo cim . c * 034n3 * ( 2011 ) 311 [ arxiv:1012.2292 ] . d. hooper and t. linden , phys . d * 84 * ( 2011 ) 123005 [ arxiv:1110.0006 ] . k. n. abazajian and m. kaplinghat , phys . d * 86 * ( 2012 ) 083511 [ arxiv:1207.6047 ] . t. daylan , d. p. finkbeiner , d. hooper , t. linden , s. k. n. portillo , n. l. rodd and t. r. slatyer , arxiv:1402.6703 . c. gordon and o. macias , phys . d * 88 * ( 2013 ) 083521 [ erratum - ibid . d * 89 * ( 2014 ) 4 , 049901 ] [ arxiv:1306.5725 ] . k. n. abazajian , n. canac , s. horiuchi and m. kaplinghat , phys . d * 90 * ( 2014 ) 023526 [ arxiv:1402.4090 ] . b. zhou , y. f. liang , x. huang , x. li , y. z. fan , l. feng and j. chang , arxiv:1406.6948 . f. calore , i. cholis and c. weniger , arxiv:1409.0042 . p. agrawal , b. batell , p. j. fox and r. harnik , arxiv:1411.2592 . f. calore , i. cholis , c. mccabe and c. weniger , arxiv:1411.4647 . j. petrovic , p. d. serpico and g. zaharijas , arxiv:1411.2980 . a. v. belikov , j. f. gunion , d. hooper , and t. m. tait , phys . lett . * b705 * ( 2011 ) , 8286 , [ arxiv:1009.0549 ] . m. r. buckley , d. hooper , and t. m. tait , phys . b702 * ( 2011 ) , 216219 , [ arxiv:1011.1499 ] . g. zhu , phys.rev . * d83 * ( 2011 ) , 076011 , [ arxiv:1101.4387 ] . g. marshall and r. primulando , jhep * 1105 * ( 2011 ) , 026 , [ arxiv:1102.0492 ] . m. buckley , p. fileviez perez , d. hooper , and e. neil , phys . b * 702 * ( 2011 ) , 256259 , [ arxiv:1104.3145 ] . m. boucenna and s. profumo , phys . d * 84 * ( 2011 ) , 055011 , [ arxiv:1106.3368 ] . m. r. buckley , d. hooper , and j. l. rosner , phys . b * 703 * ( 2011 ) , 343347 , [ arxiv:1106.3583 ] . d. hooper , n. weiner , and w. xue , phys . d * 86 * ( 2012 ) , 056009 , [ arxiv:1206.2929 ] . r. c. cotta , a. rajaraman , t. m. p. tait and a. m. wijangco , phys . d * 90 * ( 2014 ) 013020 [ arxiv:1305.6609 ] . m. r. buckley , d. hooper , and j. kumar , phys . d * 88 * ( 2013 ) , 063532 , [ arxiv:1307.3561 ] . k. hagiwara , s. mukhopadhyay and j. nakamura , phys . d * 89 * ( 2014 ) 015023 [ arxiv:1308.6738 ] . n. okada and o. seto , phys . d * 89 * ( 2014 ) 043525 [ arxiv:1310.5991 ] . e. c. f. s. fortes and m. tonasse , phys . d * 89 * ( 2014 ) , 016015 , [ arxiv:1312.2837 ] . a. alves , s. profumo and f. s. queiroz , jhep * 1404 * , 063 ( 2014 ) [ arxiv:1312.5281 ] . k. p. modak , d. majumdar , and s. rakshit , arxiv:1312.7488 . j. d. ruiz - alvarez , c. a. de s.pires , f. s. queiroz , d. restrepo and p. s. rodrigues da silva , phys . d * 86 * ( 2012 ) 075011 [ arxiv:1206.5779 ] . c. boehm , m. j. dolan , c. mccabe , m. spannowsky and c. j. wallace , jcap * 1405 * ( 2014 ) 009 [ arxiv:1401.6458 ] . a. alves , s. profumo , f. s. queiroz and w. shepherd , phys . d * 90 * ( 2014 ) 11 , 115003 [ arxiv:1403.5027 ] . c. boehm , m. j. dolan and c. mccabe , phys . d * 90 * ( 2014 ) 023531 [ arxiv:1404.4977 ] . e. izaguirre , g. krnjaic and b. shuve , phys . d * 90 * ( 2014 ) 055002 [ arxiv:1404.2018 ] . m. abdullah , a. difranzo , a. rajaraman , t. m. p. tait , p. tanedo and a. m. wijangco , phys . d * 90 * ( 2014 ) 035004 [ arxiv:1404.6528 ] . a. berlin , d. hooper and s. d. mcdermott , phys . d * 89 * ( 2014 ) 115022 [ arxiv:1404.0022 ] . d. k. ghosh , s. mondal and i. saha , arxiv:1405.0206 . a. martin , j. shelton and j. unwin , phys . d * 90 * ( 2014 ) 10 , 103513 [ arxiv:1405.0272 ] . a. berlin , p. gratia , d. hooper and s. d. mcdermott , phys . d * 90 * ( 2014 ) 015032 [ arxiv:1405.5204 ] . l. wang and x. f. han , phys . b * 739 * ( 2014 ) 416 [ arxiv:1406.3598 ] . c. arina , e. del nobile and p. panci , arxiv:1406.5542 . n. okada and o. seto , phys . d * 90 * ( 2014 ) 8 , 083523 [ arxiv:1408.2583 ] . n. f. bell , s. horiuchi and i. m. shoemaker , arxiv:1408.5142 . a. d. banik and d. majumdar , arxiv:1408.5795 . k. cheung , w. c. huang and y. l. s. tsai , arxiv:1411.2619 . d. hooper , arxiv:1411.4079 . j. liu , n. weiner and w. xue , arxiv:1412.1485 . m. j. dolan , c. mccabe , f. kahlhoefer and k. schmidt - hoberg , arxiv:1412.5174 . a. biswas , arxiv:1412.1663 . k. ghorbani and h. ghorbani , arxiv:1501.00206 . m. cahill - rowley , j. gainer , j. hewett and t. rizzo , arxiv:1409.1573 . c. cheung , m. papucci , d. sanford , n. r. shah and k. m. zurek , phys . d * 90 * ( 2014 ) 7 , 075011 [ arxiv:1406.6372 ] . j. huang , t. liu , l. t. wang and f. yu , phys . d * 90 * ( 2014 ) 11 , 115006 [ arxiv:1407.0038 ] . t. gherghetta , b. von harling , a. d. medina , m. a. schmidt and t. trott , arxiv:1502.07173 . d. g. cerdeo , m. peir and s. robles , jcap * 1408 * ( 2014 ) 005 [ arxiv:1404.2572 ] . d. g. cerdeo , c. muoz and o. seto , phys . d * 79 * ( 2009 ) 023510 [ arxiv:0807.3029 ] . d. g. cerdeo and o. seto , jcap * 0908 * ( 2009 ) 032 [ arxiv:0903.4677 ] . r. kitano and k. y. oda , phys . d * 61 * ( 2000 ) 113001 . [ hep - ph/9911327 ] . b. garbrecht , c. pallis and a. pilaftsis , jhep * 0612 * ( 2006 ) 038 . [ hep - ph/0605264 ] . d. g. cerdeo , j. -h . huh , m. peir and o. seto , jcap * 1111 * ( 2011 ) 027 [ arxiv:1108.0978 ] . g. belanger , f. boudjema , a. pukhov and a. semenov , comput . phys . commun . * 185 * ( 2014 ) 960 [ arxiv:1305.0237 ] . d. s. akerib _ et al . _ [ lux collaboration ] , phys . * 112 * ( 2014 ) 9 , 091303 [ arxiv:1310.8214 ] . r. agnese _ et al . _ [ supercdms collaboration ] , phys . * 112 * ( 2014 ) 041302 [ arxiv:1309.3259 ] . r. agnese _ et al . _ [ supercdms collaboration ] , phys . * 112 * ( 2014 ) 241302 [ arxiv:1402.7137 ] . p. a. r. ade _ et al . _ [ planck collaboration ] , astron . astrophys . * 571 * ( 2014 ) a16 [ arxiv:1303.5076 ] . m. ackermann _ et al . _ [ fermi - lat collaboration ] , phys . rev . d * 89 * ( 2014 ) 4 , 042001 [ arxiv:1310.0828 ] . m. ackermann _ et al . _ [ fermi - lat collaboration ] , arxiv:1503.02641 . m. ackermann _ et al . _ [ fermi - lat collaboration ] , phys . rev . d * 88 * ( 2013 ) 082002 [ arxiv:1305.5597 ] . a. ibarra , s. lopez gehler and m. pato , jcap * 1207 * ( 2012 ) 043 [ arxiv:1205.0007 ] . a. ibarra , h. m. lee , s. lpez gehler , w. i. park and m. pato , jcap * 1305 * ( 2013 ) 016 [ arxiv:1303.6632 ] . d. g. cerdeo , v. martn - lozano and o. seto , jhep * 1405 * , 035 ( 2014 ) [ arxiv:1311.7260 ] . d. g. cerdeo , p. ghosh and c. b. park , jhep * 1306 * ( 2013 ) 031 [ arxiv:1301.1325 ] . d. g. cerdeo , p. ghosh , c. b. park and m. peir , jhep * 1402 * ( 2014 ) 048 [ arxiv:1307.7601 ] . j. f. gunion , d. hooper and b. mcelrath , phys . d * 73 * ( 2006 ) 015011 [ hep - ph/0509024 ] . f. ferrer , l. m. krauss and s. profumo , phys . d * 74 * ( 2006 ) 115007 [ hep - ph/0609257 ] . c. e. aalseth _ [ cogent collaboration ] , phys . * 101 * ( 2008 ) 251301 [ erratum - ibid . * 102 * ( 2009 ) 109903 ] [ arxiv:0807.0879 ] . d. a. vasquez , g. belanger , c. boehm , a. pukhov and j. silk , phys . d * 82 * ( 2010 ) 115027 [ arxiv:1009.4380 ] . j. j. cao , k. i. hikasa , w. wang , j. m. yang , k. i. hikasa , w. y. wang and j. m. yang , phys . b * 703 * ( 2011 ) 292 [ arxiv:1104.1754 ] . d. albornoz vasquez , g. belanger and c. boehm , phys . d * 84 * ( 2011 ) 095008 [ arxiv:1107.1614 ] . j. kozaczuk and s. profumo , phys . d * 89 * ( 2014 ) 9 , 095012 [ arxiv:1308.5705 ] .

we show that the right - handed ( rh ) sneutrino in the nmssm can account for the observed excess in the fermi - lat spectrum of gamma - rays from the galactic centre , while fulfilling all the current experimental constraints from the lhc as well as from direct and indirect dark matter searches . we have explored the parameter space of this scenario , computed the gamma - ray spectrum for each phenomenologically viable solution and then performed a @xmath0 fit to the excess . unlike previous studies based on model - independent interpretations , we have taken into account the full annihilation spectrum , without assuming pure annihilation channels . furthermore , we have incorporated limits from direct detection experiments , lhc bounds and also the constraints from fermi - lat on dwarf spheroidal galaxies ( dsphs ) and gamma - ray spectral lines . in addition , we have estimated the effect of the most recent fermi - lat reprocessed data ( pass 8) . in general , we obtain good fits to the gce when the rh sneutrino annihilates mainly into pairs of light singlet - like scalar or pseudoscalar higgs bosons that subsequently decay in flight , producing four - body final states and spectral features that improve the goodness of the fit at large energies . the best fit ( @xmath1 ) corresponds to a rh sneutrino with a mass of 64 gev which annihilates preferentially into a pair of light singlet - like pseudoscalar higgs bosons ( with masses of order 60 gev ) . besides , we have analysed other channels that also provide good fits to the excess . finally , we discuss the implications for direct and indirect detection searches paying special attention to the possible appearance of gamma - ray spectral features in near future fermi - lat analyses , as well as deviations from the sm - like higgs properties at the lhc . remarkably , many of the scenarios that fit the gce can also be probed by these other complementary techniques .

1501.01296

since the end of the eighties and by this time works on designing and development of physical programs for @xmath0 and @xmath1 - colliders are under development in different countries . now there are projects on their creation in usa [ 1 ] , germany [ 2 ] , japan [ 3 ] . physical programs for these colliders created as a result of long - term cooperation of the representatives of many high - energy physics centres are stated in `` conceptual design reports '' [ 1 - 3 ] . in these projects electron - photon and photon - photon beams are supposed to be obtained on the basis of linear accelerators with @xmath2 beams . one of the best methods of obtaining intensive @xmath3 beams is the use of the compton backscattering of laser light on an electron beam of the linear collider . for the first time in works [ 4 ] it was shown , that on the basis of linear colliders with @xmath2 beams it is possible to realize @xmath4 and @xmath5 - beams with approximately the same energies and luminosities , as for initial electron beams . the necessary intensive bunches of @xmath6 quantums were offered for receiving at scattering of powerful laser flash on electron bunches of these accelerators . the small sizes of linear colliders beams make it possible to obtain conversion coefficient ( the attitude of number of high - energy photons to number of electrons in a bunch ) @xmath7 at energy of laser flash in some joules , i.e. it is possible to convert the most part of electrons to photons . the detailed description of the scheme of an electron beam conversion in @xmath6 beam , the basic characteristics of @xmath8 and @xmath9 collisions , problems of a background and calibration of luminosity were considered in detail in [ 5 ] . the region of laser conversion @xmath10 is unique by its physical properties . it is the region of an intensive electromagnetic field ( the focused laser bunch ) . this fact allows one to investigate such processes of nonlinear quantum electrodynamics as radiation of a photon by electron in a field of an intensive electromagnetic wave , and also `` subthreshold '' pairs production @xcite . at sufficient power of laser flash in the field of conversion the processes are essential due to absorption from a wave more than one of laser photons simultaneously @xmath11 @xmath12 processes ( [ 1]),([2 ] ) represent nonlinear by intensity of a field processes of interaction electrons and photons with a field of an electromagnetic wave . the first of these nonlinear processes results in expansion of spectra of high - energy photons and occurrence of additional peaks in spectra of scattered radiation due to absorption of several photons from a wave , and the second one effectively reduces a threshold of @xmath2 pairs creation . the interaction of electrons and positrons with a field of an electromagnetic wave results in effective increase of their masses ] : @xmath13 which is characterized by parameter of intensity of a laser wave @xmath14 : @xmath15 where @xmath16- density of photons in a laser wave , @xmath17 - their energy , @xmath18 - amplitude of classical 4-potential of electromagnetic wave , @xmath19 - a charge of electron . regular research of nonlinear breit - wheeler ( [ 2 ] ) and compton ( [ 1 ] ) processes was carried out in @xcite,@xcite . now the area of nonlinear effects is rather actual and is of great interest because here essential are the processes of radiation due to absorption from a wave of a few of photons , and their probabilities are essentially nonlinear functions of intensity of a field . recently on accelerator slac @xcite a series of experiments e-144 with check of predictions of nonlinear qed was finished in the field of parameter @xmath20 that became possible due to use of the supershort and rigidly focused laser pulses . thus for the first time the experiment was set up in which the process of @xmath2 - pair production at participation of only real , instead of virtual photons was carried out . the main features of the conversion are described by a quantity @xmath21 which is determined via the initial electron beam energy @xmath22 and the laser photon energy @xmath23 as @xmath24 the differential probability of process of radiation of a photon by electron performed by a summation over polarizations of final electron and a photon has the following form @xcite : @xmath25 @xmath26,\ ] ] @xmath27,\ ] ] @xmath28 @xmath29 @xmath30 is the bessel functions of nth order , @xmath31 is energy of high - energy photon . the expression in the sum ( [ 5 ] ) , determines probability of radiation of n- harmonics by electron in a field of circular -polarized electromagnetic wave ( from a wave n laser photons can be absorbed ) . the change of a variable @xmath32 corresponds to the change of a variable @xmath33 : @xmath34 @xmath35 the influence of nonlinear effects results in the fact that the maximum energy of high - energy photons of the first harmonic @xmath36 decreases in comparison with the maximum energy of photons in usual compton effect and the energy of the @xmath6 - quanta formed at absorption from a wave of several photons exceeds energy , achievable in usual copmton effect . and various values of parameter @xmath37 , @xmath38,title="fig : " ] and various values of parameter @xmath37 , @xmath38,title="fig : " ] results of numerical calculations of power spectra of photons in a nonlinear case at @xmath39 are given in fig 1 . apparently from these figures , the account of nonlinear effects results in essential change of spectra in comparison with spectra of usual copmton scattering . first , simultaneous absorption from a wave of several laser photons results in expansion of spectra of rigid @xmath40 quanta and occurrence of the additional peaks , appropriate to radiation of harmonics of higher order . this expansion at the same parameter x increases with the intensity of a wave . second , effective electron weighting results in compression of spectra , i.e.in shift of the first harmonic aside smaller values y. with increase of parameter x compression of the first harmonic decreases . the process discussed in this section , is the good test on check of nonlinear quantum electrodynamics . the `` hard '' photons propagated in the region of laser conversion can produce @xmath2 pairs in collision with s laser photons simultaneously @xmath41 threshold value for energy of a photon is defined from the relation @xmath42 here @xmath43 , @xmath44 - 4- momentum of photons @xmath40 and @xmath45 . the maximum energy of compton photon emitted at interaction of electron with n laser photons is @xmath46 thus the corresponding value of electron energy for @xmath47 pairs creation at absorption from a laser wave s of photons is @xmath48 in particular if the `` hard '' photon is emitted as a result of interaction with one laser photon@xmath49 @xmath50 when @xmath51 electron - positron pairs will be created only due to nonlinear processes qed . thus at the first stage of realization of projects nlc ( in particular for tesla ) at x=4.5 occurrence of the big number of positrons will be a consequence of nonlinear effects qed . calculated in @xcite the probability of @xmath2 pair production at interaction with circle - polarized laser photons can be submitted by the high - energy non - polarized photon as @xmath52 @xmath53 \biggl]\ ] ] @xmath54 the total number of the produced positrons can be obtained by averaging on an energy spectrum of compton photons ( [ 5 ] ) @xmath55 here @xmath56 . expression ( [ 7 ] ) contains two factors 1/2 . the first one arises at the account of relative movement of bunches of photons @xmath40 and @xmath45 . the same multiplier appears because of the fact that photons interact in the average with the half of a laser bunch ( the length of a laser pulse does not exceed the sizes of area of conversion and distribution of density of bunches in a direction of movement is homogeneous ) . in fig . 2 the number of the created positrons per one electron is shown depending on x at various values of parameter @xmath57 , value of factor of conversion @xmath58 , the length of a laser bunch @xmath59 . at the first stage of project tesla will be observed @xmath60 pairs per one collision . these pairs will form the essential background , therefore the produced positrons should be removed from the region of interaction with the help of a magnetic field . to observe the process of creation of pairs it is necessary to extract the positrons moving practically along the direction of initial electrons . we shall consider distribution on energy of the created positrons . energies of positrons ( electrons ) are distributed in the interval @xmath61 from here it is easy to get restricts of distribution of positrons on energy @xmath62 @xmath63 @xmath64 table [ cols="^,^,^,^,^,^ " , ] this table represents borders of positrons energies for various number s of the absorbed laser photons , parameter @xmath65 . differential distribution of the produced pairs on energy of positrons looks like @xmath66 \biggl ] , \label{9}\ ] ] @xmath67 at small @xmath57 distribution on energy ( [ 9 ] ) becomes @xmath68 in fig . 3a differential distributions of the created positrons by a `` hard '' photon of the maximal energy for various number s of the absorbed laser photons are submitted . electron energy @xmath69 . in fig.3b we show the same distributions , but averaged in the spectrum of high - energy photons . in fig . 4a and 4b distributions for energy of electrons @xmath70 are shown . for different number of absorbed laser photons , ( a ) - without averaging on a spectrum , ( b ) - with averaging on a spectrum . , title="fig : " ] for different number of absorbed laser photons , ( a ) - without averaging on a spectrum , ( b ) - with averaging on a spectrum . , title="fig : " ] for different number of absorbed laser photons , ( a ) - without averaging on a spectrum , ( b ) - with averaging on a spectrum . , title="fig : " ] for different number of absorbed laser photons , ( a ) - without averaging on a spectrum , ( b ) - with averaging on a spectrum . , title="fig : " ] the total distribution of positrons is given in fig.5 from figures it is possible to see that the maximum in distribution of positrons after averaging on a spectrum of high - energy photons is shifted to the bottom border of a spectrum . it enables precise identification of `` subthreshold '' @xmath2 pairs creation as effect of nonlinear quantum electrodynamics . at @xmath71 it is possible to observe `` subthreshold '' creation of positrons in a positron spectrum near the bottom border . for energy of initial electron @xmath72 positrons with energy in an interval @xmath73 are created by absorption of 5 laser photons simultaneously . for energy of initial electron @xmath70 , positrons with energies @xmath74 are created only due to nonlinear effects qed . let s note that the contribution of other mechanisms to creation of @xmath75 pairs is much less . for the bethe - heitler process @xmath76 threshold values of energy are much greater @xmath77 except for it , the probability of creation contains the additional factor @xmath78 . the authors would like to thank i.f.ginzburg , g.l.kotkin and v.g.serbo for useful discussions .

in this paper we consider the process of `` subthreshold '' electron - positron pairs creation in the region of laser conversion . the total number of positrons and their distribution are obtained . this phenomena is offered for use as a good test to examine nonlinear effects of quantum electrodynamics on tesla .

hep-ph0211214

turbulent transport and mixing play an essential role in many natural and industrial processes @xcite , in cloud formation @xcite , in chemical reactors and combustion systems @xcite , and atmospheric and oceanographic turbulence determines the spread of pollutants and biological agents in geophysical flows @xcite , @xcite , @xcite , @xcite . concentration fluctuations are often of great importance in such systems and this is related to the separation of nearby fluid particles ; turbulent pair diffusion therefore plays a critical role in such systems . the idea of locality has been fundamental to the theory of turbulent particle pair diffusion since richardson s pioneering paper , @xcite , which established turbulent pair diffusion as an important scientific discipline and laid the foundations for a theory of how ensembles of pairs of fluid particles ( tracers ) initially close together move apart due to the effects of atmospheric winds and turbulence . richardson argued that as particle pairs separate the rate at which they move apart is affected mostly by eddies of the same scale as the separation distance itself this is the basis of the locality hypothesis . richardson was also motivated by a desire to bring molecular and turbulent pair diffusional processes into a unified picture through the use of a single non - fickian diffusion equation with scale dependent diffusivity , @xmath13 , where @xmath14 is the pair separation variable . assuming homogeneous isotropic turbulence , richardson posed the problem in 3d in terms of the probability density function ( pdf ) of the pair separation , @xmath15 , and assuming the normalization , @xmath16 , he suggested the following diffusion equation to describe @xmath17 , @xmath18 the scaling of the pair diffusivity @xmath19 with the pair separation and what it means for the pair diffusion process is of paramount importance . from observational data of turbulent pair diffusivities collected from different sources , richardson assumed an approximate fit to the data namely , @xmath20 . this is equivalent to @xmath21 , @xcite , @xcite , often referred to as the richardson - obukov @xmath22-regime . @xmath23 is the pair separation , @xmath24 is the time , and the angled brackets is the ensemble average over particle pairs . it is no longer believed that it is possible to unify molecular and turbulent diffusional processes because their physics are fundamentally different ; brownian motion characterizes molecular diffusion , while convective gusts of winds that increase the pair separation in surges characterizes turbulent diffusion @xcite , @xcite , @xcite . nevertheless , the idea of a scale dependent turbulent diffusivity has survived . richardson s assumed 4/3-scaling law is equivalent to a locality hypothesis , according to which for asymptotically large reynolds number only the energy in eddies whose size is of a similar scale to the pair separation inside the inertial subrange is effective in further increasing the pair separation . furthermore , richardson s scaling for the pair diffusion is consistent with @xcite , ( k41 ) ; this can be seen from the form of the turbulence energy spectrum in the inertial subrange which is , @xmath25 , from which it follows that the pair diffusivity depends only upon @xmath26 and @xmath27 ( the rate of kinetic energy dissipation per unit mass ) . this leads directly to the locality scaling , @xmath28 . it is usual to evaluate @xmath19 at typical values of , @xmath26 , namely at @xmath29 , so this scaling is replaced by , @xmath30 . the requirement of large reynolds number , @xmath31 , implies that this scaling is true only inside an asymptotically infinite inertial subrange , @xmath32 , where @xmath33 , and @xmath34 is the kolmogorov micro - scale and @xmath35 is a scale characteristic of energy containing eddies , typically the integral length scale , or the taylor length scale . in this limit , the pair separation is initially zero , @xmath36 , as @xmath37 . for finite inertial subrange , there are infra - red and ultra - violet boundary corrections so the inertial subrange still has to be very large in order to avoid these effects and to observe inertial subrange scaling in the pair diffusion . with the 4/3-scaling for @xmath19 , an explicit solution for equation ( 1.1 ) for diffusion from a point source with boundary conditions @xmath38 , can be derived , @xmath39 turbulence is both scale dependent and time correlated ( non - markovian ) , and it is not clear whether the pdf in equation ( 1.2 ) , which describes a local and markovian process , can accurately represent the turbulent pair diffusion process . attempts have been made to derive alternative non - markovian models for pair diffusion , @xcite , @xcite , and this remains a subject of ongoing scientific research . however , this does not affect richardson s hypothesis of scale dependent diffusivity , which is the main focus of this study . it is possible to generalize the scaling for the pair diffusivity to be time dependent and still be consistent with k41 and with locality , @xcite , @xcite , such that , @xmath40 for some @xmath41,and @xmath42 . dimensional consistency then gives , @xmath43 and @xmath44 , which leads to @xmath45 , and @xmath46 . thus we obtain , @xmath47 if the further constraint @xmath48 is satisfied , then this yields , @xmath49 . thus , a @xmath22-regime is not a unique signature for richardson s 4/3-scaling for the pair diffusivity . however , a time dependent pair diffusivity is hard to justify physically if we assume steady state equilibrium turbulence , because a time dependent diffusivity implies , for @xmath50 that the pair diffusivity at the same separation is ever increasing in time without limit , or for @xmath51 that the pair diffusivity approaches zero with time and the separation process effectively stops . both cases seem unlikely , and in the ensuing we will restrict the discussion to steady state equilibrium turbulence and consider only the case , @xmath52 . however , it is worth remarking that time - dependent diffusivities like equation ( 1.3 ) may exist in the context of non - equilibrium turbulence . the main aims of this research are two - fold . firstly , to re - examine the body of evidence available on turbulent pair diffusion especially , if any , from large scale turbulence containing large inertial subranges , section 2 . particular attention is focussed upon a reappraisal of richardson s original 1926 dataset , section 3 . secondly , based upon these findings , a new theory that does not _ a priori _ make the assumption of locality is constructed from which new scaling laws for the pair diffusivity is derived through a novel mathematical method , and compared to the reappraised 1926 dataset , in section 4 . finally , a lagrangian diffusion model is used to examine the new theory and the results from the simulations are found to agree closely with the 1926 dataset and with the predictions of the new theory , section 5 . we discuss the significance of these findings and draw conclusions in section 6 . although the general consensus among scientists in the field at the current time is that the collection of observational data , experimental data , and direct numerical simulation , suggests a convergence towards a richard - obukov locality scaling , the relatively low reynolds numbers in the experiments and dns , and the high error levels , and other assumptions made in collecting the data means that this is by no means a forgone conclusion . as noted by @xcite , `` .. there has not been an experiment that has unequivocally confirmed r - o scaling over a broad - enough range of time and with sufficient accuracy . '' it is not known precisely what size of the inertial subrange is required to observe unequivocally the pair diffusion scaling , but it is widely assumed to be about at least four orders of magnitude or more . only geophysical turbulent flows , such as in the atmosphere and in the oceans , can produce such extended inertial subranges . a number of claims of observing locality scaling , including the existence of an approximate @xmath22-regime in geophysical flows , have been made by @xcite , @xcite , @xcite , and @xcite . more recent observations include @xcite in the atmosphere , and @xcite , and @xcite in the oceans . but high error levels and other assumptions ( such as two - dimensionality ) made in these observations means that decisive conclusions can not be drawn from them regarding pair diffusion theories . direct numerical simulations ( dns ) is inconclusive at the current time because it does not produce a big enough inertial subrange in order to be able to test pair diffusion laws reliably . for example , @xcite perform a dns with @xmath53 , at taylor scale based reynolds numbers @xmath54 showing an approximate inertial subrange energy spectrum over a very short range of just 40 . other dns of particle pair studies at low reynolds numbers are , @xcite at @xmath55 , @xcite at @xmath56 , @xcite at @xmath57 , @xcite at @xmath58 , @xcite at @xmath59 . @xcite at @xmath60 . see also @xcite , and @xcite . the maximum separation of time scales between the integral time scale and the kolmogorov times scale observed to date in dns is about a factor of , 100 . this is still about two orders of magnitude smaller than the minimum size required to adequately test inertial subrange pair diffusion scalings . for 2d turbulence , see @xcite , @xcite . particle tracking velocimetry ( ptv ) laboratory experiments @xcite , @xcite have been providing pair diffusion statistics at low to moderate reynolds numbers . like dns , these reynolds numbers are too small to reliably test pair diffusion laws . @xcite obtain pair diffusion measurements from ptv . more recently , @xcite obtained measurements in a water tank at @xmath61 , and @xcite and @xcite tracked hundreds of particles at high temporal resolution at @xmath62 . although higher resolution tracking experiments using high - energy physics methods have been performed for single particle trajectories @xcite , they have not yet been applied to particle pair studies . when intermittency is accounted for , the scaling in fully developed turbulence should in fact be slightly greater than the r - o scaling . this will be discussed further in discussions and conclusions , section 6 . @xcite reported in the proceedings of the royal society of london data on turbulent diffusivities collected from different sources , which is reproduced here with a brief description in table 1 . he plotted the turbulent diffusivity against the pair separation in log - log scale , shown as the red and black filled circles in figure 1 . motivated by an attempt to unify pair diffusional processes across all possible scales , he assumed the scaling @xmath20 as a reasonable fit ( fig . 1 , dotted blue line ) . this is not the least squares line of best fit to the data , it is just a theoretical approximation to the data . the actual line of best fit is , @xmath63 ( fig . 1 , solid red line ) . @xcite and @xcite , commenting on diffusion of floats in the sea noted that their new measurements were roughly consistent with the 1926 data , i.e. @xmath20 . at the end of their paper they wrote , `` note added in proof . after this manuscript was submitted the writers have read two unpublished manuscripts by c. l. von weisaecker and w. heisenberg in which the problem of turbulence for large reynolds number is treated deductively with the result that they arrive at the 4/3 law . the agreement between von weisaecker and heisenberg s deduction and our quite independent induction is a confirmation of both '' , see @xcite . but they also observed that , `` any power between @xmath64 and @xmath65 would be a tolerable fit to the data '' . thus , as early as 1948 , it was noted that @xmath20 was only a rough fit , which makes it all the more surprising that an alternative theory for pair diffusion has not been developed till now . .datum number , source , the turbulent diffusivity , @xmath0 , and the length scale , @xmath66 . references : [ 1 ] @xcite , [ 2 ] @xcite , [ 3 ] @xcite , [ 4 ] @xcite , [ 5 ] @xcite , [ 6 ] @xcite , [ 7 ] @xcite , [ 8 ] @xcite [ cols="<,<,^,^ " , ] the results obtained from ks are summarized in table 2 . this table shows @xmath7 from the simulations in column2 ; @xmath67 in cloumn 3 . the equivalent mean square separation scaling laws , @xmath68 , are also shown : @xmath69 is the non - local scaling , column 4 ; @xmath70 is the local scaling , column 5 . the ratio of scaling powers , @xmath71 is shown in column 6 . log - log plots of the turbulent pair diffusivity @xmath72 against the separation @xmath73 for different energy spectra @xmath74 display clear power law scalings of the form , @xmath75 over wide ranges of the separation inside the inertial subrange ( fig . the @xmath76 s are the slopes of the plots in figure 5 ; these are more easily observed by plotting the compensated relative diffusivity @xmath77 against the separation ( fig . the @xmath7 s are the powers that give near - horizontal lines over the longest range of separation , a procedure that determines @xmath7 to within @xmath78 error for most @xmath74 , except near the singular limit , @xmath79 , where the errors are around @xmath80 . the plots of @xmath7 and @xmath67 against @xmath74 show that the new scaling powers , @xmath7 , are in the range , @xmath8 , ( fig . 7 , black filled circles and dotted blue line respectively , left hand scale ) . it is observed that as @xmath81 , then @xmath82 ; and as @xmath83 , then @xmath84 . furthermore , the , @xmath76 s , display a smooth transition between the asymptotic limits at @xmath79 and @xmath85 . the range of separation over which the scalings , @xmath86 , are observed to progressively reduce from both ends of the range as @xmath83 . it reduces from below due to the diminishing energies contained in the smallest scales so that the pair separation penetrates further into the inertial subrange before it forgets the initial separation . it reduces from above because the long range correlations are increasingly stronger as @xmath83 causing the particles in a pair to become independent at earlier times and at smaller separation . the plot of @xmath87 against @xmath74 show a peak at @xmath88 , where @xmath89 , ( fig . 7 , right hand scale ) . in fact , @xmath87 remains relatively close to this peak value in a neighbourhood of @xmath90 , in the range @xmath91 . plots of the scaling powers , @xmath92 , obtained from the simulations ( filled black circles ) , and from @xmath93 ( dotted blue line ) , against @xmath74 are shown in figure 8 . the same is shown in figure 9 , but focussing in the range @xmath91 which covers the intermittent turbulence range which is indicated by the two red vertical lines . for kolmogorov turbulence , @xmath94 , ks produces , @xmath95 , so that @xmath96 . for real turbulence with intermittency corrections , @xmath97 , such that @xmath98 , three extra cases , @xmath99 , where simulated . the currently accepted value for the intermittency lies in the range , @xmath100 , @xcite , @xcite , @xcite , @xcite . therefore , three cases that cover most of this range were simulated , @xmath101,and @xmath102 . for these spectra , ks produced,(fig . 7 and table 2 ) , @xmath103 the mid - point in the above intermittency range is close to , @xmath104 , which gives the scaling law @xmath105 , which is an error of less than @xmath106 in the scaling power compared to the data in figure 1 . for , @xmath107 , we obtain @xmath108 , and the error in the scaling power is @xmath109 ; for @xmath110 , we obtain @xmath111 , and the error in the scaling power is @xmath112 . it is reasonable to infer that the error will be almost zero close to @xmath113 . the kolmogorov spectrum @xmath94 produces , @xmath114 , and the error in the scaling power is @xmath115 . the richardson s locality law , @xmath2 , is @xmath116 in error in the scaling power compared to the reappraised geophysical data in figure 1 . the simulations produce scalings for @xmath117 , in the range @xmath118 to @xmath119 for the same intermittent spectra considered above , fig . 9 , table 2 . it is interesting to note that even under the assumption of locality intermittency spectra should produce scalings in the range @xmath120 to @xmath121 thus , a true r - o @xmath22-regime can not exist in reality . in general ks predicts the correct scaling for the turbulent pair diffusivity to within @xmath78 of the geophysical data in figure 1 in most of the accepted intermittency range centred in the neighbourhood of , @xmath122 . finally , ks produces a non - richardson 4/3-scaling , @xmath123 , where @xmath124 at @xmath125 , ( fig . 7 and table 1 ) . with this spectrum , @xmath126 , this is equivalent to a new @xmath127 regime . the numerical results presented here are the most comprehensive obtained to date from ks due to the very large ensemble of particle pairs and the small time steps used . the statistical fluctuations in the results are therefore small . the @xmath128s , which are the slopes of the plots in figure 5 , can be determined to within @xmath78 error . an exception is close to the singular limit @xmath79 where the numerical errors can be large . an accurate estimate of this error can be obtained as follows , noting first that the error level in @xmath7 is identical to the error level in @xmath87 . as @xmath81 , then @xmath129 ; but very close to this limit , @xmath130 is still a good approximation . for @xmath131 , ks produces , @xmath132 ( fig . 7 and table 1 ) . this is an error of @xmath80 which is small considering that it is so close to the singular limit . an error of around @xmath78 away from @xmath79 is therefore reasonable . in the limit @xmath133 there is no detectible error in @xmath87 from ks to three decimal places ( table 2 ) . ks is an established method used by many researchers in turbulent diffusion studies , as noted in the earlier references in this paper . however , some researchers , @xcite , @xcite , @xcite , have expressed reservations regarding ks which they believe produces incorrect scalings for pair diffusion statistics because of the lack of true dynamical sweeping of the inertial scales eddies by the larger scales . these authors strongly support locality which they believe to be true in reality , and their inference that ks _ mus_t be incorrect is a direct consequence of this thinking . as we are examining the hypothesis of locality itself here , such a presumption can not be taken as a given . furthermore , @xcite has shown through an analysis of pairs of trajectories that the quantitative errors in ks due to the sweeping effect is in fact negligible provided that the simulations are carried out in a frame of reference moving with the large scale sweeping flow . the close agreement of ks with the geophysical data in figure 1 obtained here provides further support for this conclusion . richardson pioneered the scientific discipline of turbulent particle pair diffusion , and assumed a locality scaling for the pair diffusivity due to the inclusion of one data - point from a non - turbulent context . here , the reappraised 1926 data shows an unequivocal non - local scaling for the turbulent pair diffusivity , @xmath134 , fig . 1 . consequently , the foundations of turbulent pair diffusion theory have been re - examined here in an effort to resolve one of the most important and enduring problems in turbulence . the main contribution of this investigation is to propose a new non - local theory of turbulent particle pair diffusion based upon the principle that the turbulent pair diffusion process in homogeneous turbulence is governed by both local and non - local diffusional processes . the theory preditcs that in turbulence with generalized energy spectra , @xmath4 , over extended inertial subranges , the pair diffusivity scales like , @xmath6 , with @xmath8 , in the range @xmath5 , which is intermediate between the purely local and purely non - local scaling power laws . the reappraised 1926 geophysical data , fig . 1 , table 1 , provides strong support for the new theory . additional support comes from a lagrangian simulation method , ks , whose results for turbulence with intermittency is in remarkably close agreement with the geophysical data in figure 1 , to within @xmath78 error in the scaling power for energy spectra in the accepted range of intermittency ; for @xmath135 , ks produces , @xmath136 , which is an error of just @xmath137 in the scaling power . all other predictions of the new theory have also been confirmed using ks . thus , an important corollary of the present work is that ks is a more accurate lagrangian simulation method than previously thought . a detailed mathematical approach has been developed by expressing the pair diffusivity through a fourier integral decomposition . _ a priori _ assumptions regarding locality was not made , and this has led to an expression for @xmath19 as the sum of local and non - local contributions , equation ( 4.29 ) . a feature of this approach is that it illustrates how various scalings and closure assumptions are inherent in developing any theory for pair diffusion . such assumptions are always present even in apparently simpler scaling rules , but are often hidden and unstated . this work also raises questions about previous works . some dns in particular have reported pair separation scaling that are close to , @xmath138 . how convincing is this in support of a locality hypothesis ? in the first place , richardson s 1926 data still remains the most comprehensive data on large geophysical scales till now , and it shows unequivocal non - local scaling . it is difficult to reconcile this with the apparent locality scaling reported in some studies , especially in view of the very short inertial subranges that they contain . furthermore , fully developed turbulence containing intermittency in a large inertial subrange should not in fact produce a r - o scaling of , @xmath2 , even under the locality hypothesis as we have seen in section 5.4 . the scaling in @xmath139 in particular is very sensitive to the power in the energy spectrum through equation ( 41 ) . with the intermittency observed in real turbulence and under the assumption of locality this would produce a scaling of , @xmath140 to @xmath141 , or equivalently @xmath142 to @xmath121 , which should be detectable if present ; but except for richardson s revised 1926 dataset presented here , no study has reported a scaling greater than @xmath138 strongly indicating that asymptotically large inertial subranges have not yet been achieved in experiments and dns . it is expected that attention will now turn towards the implications of this new theory for the general theory of turbulence , and for turbulence diffusion modeling strategies . + * acknowledgements * the author would like to thank sabic for funding this work through project number sb101011 , and the itc department at kfupm for making available the high performance computing facility for this project . the author would also like to thank mr . k. a. k. k. daoud for producing the parallel version of the ks code . the author has benefitted from useful discussions on this topic with professor a. umran dogan of king fahd university of petroleum & minerals and the university of iowa . 1987 a stochastic analysis of the displacements of fluid element in inhomogeneous turbulence using kraichnan s method of random modes . _ adv . in turb . g . comte - bellot & j. mathieu , pp . 191203 . springer .

richardson s theory of turbulent particle pair diffusion [ richardson , l. f. proc . roy . soc . lond . a 100 , 709737 , 1926 ] , based upon observational data , is equivalent to a locality hypothesis in which the turbulent pair diffusivity @xmath0 scales with the pair separation @xmath1 with a 4/3-power law , @xmath2 . here , a reappraisal of the 1926 dataset reveals that one of the data - points is from a molecular diffusion context ; the remaining data from geophysical turbulence display an unequivocal non - local scaling , @xmath3 . consequently , the foundations of pair diffusion theory have been re - examined , leading to a new theory based upon the principle that both local and non - local diffusional processes govern pair diffusion in homogeneous turbulence . through a novel mathematical approach the theory is developed in the context of generalised power law energy spectra , @xmath4 for @xmath5 , over extended inertial subranges . the theory predicts the scaling , @xmath6 , with @xmath7 intermediate between the purely local and the purely non - local scalings , i.e. @xmath8 . a lagrangian diffusion model , kinematic simulations [ kraichnan , r. h. , phys . fluids 13 , 22 - 31 , 1970 ; fung et al . , j. fluid mech . 236 , 281 - 318 , 1992 ] , is used to examine the predictions of the new theory all of which are confirmed . the simulations produce the scalings , @xmath9 to @xmath10 , in the accepted range of intermittent turbulence spectra , @xmath11 to @xmath12 , in close agreement with the revised 1926 dataset . turbulence , pair diffusion , richardson , non - local , mathematical foundations , simulation

1405.3625

wave dynamics can be tailored by symmetries and topologies imprinted by dint of underlying periodic potentials . in turn , the symmetries and topologies of the periodic potentials can be probed by excitations in the system into which the potential is embedded . in particular , flatband ( fb ) lattices , existing due to specific local symmetries , provide the framework supporting completely dispersionless bands in the system s spectrum @xcite . fb lattices have been realized in photonic waveguide arrays @xcite , exciton - polariton condensates @xcite , and atomic bose - einstein condensates ( becs ) @xcite . fb lattices are characterized by the existence of compact localized states ( clss ) , which , being fb eigenstates , have nonzero amplitudes only on a finite number of sites @xcite . the clss are natural states for the consideration of their perturbed evolution . they feature different local symmetry and topology properties , and can be classified according to the number @xmath0 of unit cells which they occupy @xcite . perturbations may hybridize clss with dispersive states through a spatially local resonant scenario @xcite , similar to fano resonances @xcite . the cls existence has been experimentally probed in the same settings where fb lattices may be realized , as mentioned above : waveguiding arrays vicencio15,mukherjee15,weimann16 , exciton - polariton condensates baboux16 , and atomic becs @xcite . the impact of various perturbations , such as disorder @xcite , correlated potentials @xcite , and external magnetic and electric fields @xcite , on fb lattices and the corresponding clss was studied too . a particularly complex situation arises in the case of much less studied nonlinear perturbations , which can preserve or destroy clss , and detune their frequency @xcite . here we study the existence of nonlinear localized modes in a pseudospinor ( two - component ) diamond chain , whose components are linearly mixed due to spin - orbit - coupling ( soc ) . the system can be implemented using a binary bose - einstein condensate ( bec ) trapped in an optically imprinted potential emulating , e.g. , the `` diamond chain '' @xcite . the two components represent different atomic states , and the soc interaction between them can be induced by means of a recently elaborated technique , making use of properly applied external magnetic and optical fields @xcite . the possibility to model these settings by discrete dynamics in a deep optical - lattice potential was demonstrated , in a general form , in refs . we consider two types of nonlinearities produced by interactions between atoms in the bec , _ viz_. , intra- and inter - component ones . the main objective of the analysis is to analyze the impact of the soc on the linear and nonlinear cls modes , as well as on exponentially localized discrete solitons . we demonstrate the possibility to create diverse stable localized modes at and close to the fb frequency , and inside gaps opened by the soc . in a previous work @xcite , we studied the effect of the soc on the dynamics of discrete solitons in a binary bec trapped in a deep one - dimensional ( 1d ) optical lattice . among new findings related to the soc were the tunability of the transition between different types of localized complexes , provided by the soc strength , and the opening of a minigap in the spectrum induced by the soc . inside the minigap , miscible stable on - site soliton complexes were found @xcite . in the opposite , quasi - continuum limit , one- and two - dimensional discrete solitons supported by the soc were studied too @xcite . the paper is structured as follows . the model is introduced in section ii . following a brief recapitulation of the spectral properties of the single - component linear quasi-1d diamond - chain lattice , the two - component system is considered . it is shown that the soc opens gaps between the fbs and dbs in the spectrum . in section iii , exact solutions for cls modes are constructed in the linear system with the soc terms . effects of the soc on nonlinear cls modes , and a possibility to create other types of the localized ones , in gaps between the fb and db is considered in section iv . in particular , the nonlinear clss are found in an exact analytical form too . in that section , localized modes in the semi - infinite gap ( sig ) are briefly considered too . the paper is concluded by section v. ) . circles and solid lines designate lattice sites , and hoppings , respectively . the dashed rectangle defines the unit cell , consisting of a ( upper ) , b ( middle ) and c ( bottom ) sites . ( b ) the dispersion relation for the linear case @xmath1 ( see details in the text).,width=453 ] we consider the one - dimensional `` diamond - chain '' lattice shown in fig . [ fig : diamond](a ) . its bandgap structure , shown in fig . [ fig : diamond](b ) , consists of two dbs which merge with the fb at conical intersection point located at the edge of the brillouin zone @xcite . the tight - binding ( discrete ) model governing the propagation of waves through this system is based on the following equations : @xmath2where @xmath3 is the nearest - neighbor coupling strength and @xmath4 the nonlinearity coefficient . these discrete gross - pitaevskii equations ( gpes ) describe a bec trapped in the deep optical lattice . the same system can be realized in optics , as an array of transversely coupled waveguides . in that case , time @xmath5 is replaced by the propagation distance @xmath6 . the evolution equations ( [ first ] ) can be derived from the hamiltonian @xmath7 \right\ } , \end{gathered}\]]which is conserved , along with the norm , @xmath8 . in the linear limit , @xmath1 , the modal profiles , @xmath9 are looked for as @xmath10 using the bloch basis , @xmath11 , with wavenumber @xmath12 and the polarization eigenvectors @xmath13 we obtain the band structure which consists of three branches @xmath14 two dbs are @xmath12-dependent branches , and the @xmath12-independent one is the fb , see fig . [ fig : diamond](b ) . henceforth , we set @xmath15 , by means of rescaling . the fb states with @xmath16 in eq . ( [ eqnosocmode ] ) are independent of @xmath12 , and may be perfectly localized in a single cell in the form of the linear cls @xcite , @xmath17where @xmath18 is the kronecker s symbol , and @xmath19 determines the location of the cell . diffractive decay of the cls is prevented by the opposite signs of the field amplitudes at sites @xmath20 and @xmath21 and the corresponding destructive interference . the clss are , strictly speaking , infinitely degenerate states , as they can be positioned in any unit cell . any superposition of clss eigenvectors is also an eigenvector . the present cls belongs to class @xmath22 , as it occupies only one unit cell . the basic model in our study is a two - component ( pseudospinor ) system carried by the diamond chain , with the ( pseudo- ) soc of strength @xmath23 which induces the linear mixing between the two components , following the lines of ref . @xcite ( where the soc of the rashba type @xcite was adopted ) . the components represent two hyperfine states of the same atomic species in the binary bec . one can think of the diamond chain as a stripe embedded into a two - dimensional network , with the wave functions vanishing at all other sites . the anisotropy of the soc defines a direction which forms some angle with the diamond - chain s axis connecting all @xmath24 sites . below we set this angle to be @xmath25 , although the cases of @xmath26 or @xmath27 , as well as the general case of an arbitrary angle can be readily considered as well . thus , the positive @xmath28-axis is directed from site @xmath24 to @xmath21 in a unit cell , and the positive @xmath29-axis from @xmath24 to @xmath20 in the same cell . the two different components of the discrete wave functions are denoted @xmath30 . we can also add a ( pseudo ) magnetic field @xmath31 which induces a zeeman splitting , in terms of soc . the corresponding system of discrete gpes is @xmath32 + ( \gamma & & i\frac{dc_{n}^{+}}{dt}+bc_{n}^{+}+b_{n}^{+}+b_{n+1}^{+}-\lambda \left ( b_{n}^{-}+ib_{n+1}^{-}\right ) + ( \gamma |c_{n}^{+}|^{2}+\zeta & & i\frac{da_{n}^{-}}{dt}-ba_{n}^{-}+b_{n}^{-}+b_{n+1}^{-}-\lambda \left ( b_{n+1}^{+}-ib_{n}^{+}\right ) + ( \gamma _ { 1}|a_{n}^{-}|^{2}+\zeta & & i\frac{db_{n}^{-}}{dt}-bb_{n}+a_{n}^{-}+a_{n-1}^{-}+c_{n}^{-}+c_{n-1}^{-}-% \lambda \left [ c_{n}^{+}-a_{n-1}^{+}+i(a_{n}^{+}-c_{n-1}^{+})\right ] + ( \gamma _ { 1}|b_{n}^{-}|^{2}+\zeta |b_{n}^{+}|^{2})b_{n}^{-}=0 , \notag \\ & & i\frac{dc_{n}^{-}}{dt}-bc_{n}^{-}+b_{n}^{-}+b_{n+1}^{-}+\lambda \left ( b_{n}^{+}-ib_{n+1}^{+}\right ) + ( \gamma _ { 1}|c_{n}^{-}|^{2}+\zeta types of the local nonlinear terms are included : collisions between atoms belonging to the same component generate the cubic self - interaction with coefficients @xmath33 and @xmath34 in eq . ( [ socsveeq ] ) , while collisions between atoms from different components give rise to the cross - interaction accounted for by coefficient @xmath35 . the above set of equations is invariant under the symmetry operation , which involves the permutation of the soc components , simultaneous sign change of the magnetic field , the complex conjugation , and time reversal : @xmath36note that in the nonlinear case this symmetry holds only if @xmath37 . the linear system with @xmath38 and in the absence of the soc and magnetic fields , @xmath39 , is characterized by a double - degenerate single - component spectrum of the diamond chain from the previous section . therefore , the fb exists and is double degenerate too , due to the presence of two pseudospin components . for @xmath40 but @xmath41 , the spectrum of each component is shifted by @xmath42 relative to the single - component model outlined in the previous section , see fig . [ fig3 ] ( a ) . thus , both fbs survive , but their degeneracy is lifted . when the soc is present , @xmath43 , together with the zeeman terms , the fbs are deformed ( made dispersive ) , losing the flatness , see fig . [ fig3](b - d ) . and ( a ) @xmath44 , ( b ) @xmath45 , ( c ) @xmath46 , ( d ) @xmath47 . the zeeman term @xmath48 acts against the soc term @xmath49 , making the former fbs dispersive ( curved).,width=377 ] being interested in effects related to the presence of the fb with nonvanishing soc , in what follows below we focus on the case when the soc is present ( @xmath43 ) , without the zeeman splitting , @xmath50 . besides the analytically found cls solutions , other types of stationary solutions of the nonlinear system are found below by dint of a numerical procedure based on the powell method @xcite . the linear stability analysis of all found localized solutions was performed numerically , solving linearized equations for small perturbations which determine the stability eigenvalues ( evs ) . the linearized equations are derived following the well - known procedure @xcite . namely , small perturbations are added to the localized solutions whose stability is under consideration : @xmath51 , @xmath52 denote , respectively , the multi - component discrete wave function of the perturbed solution , and of its stationary ( unperturbed ) counterpart . perturbation eigenmodes are looked for as @xmath53 , where @xmath54 is the ev . generally , @xmath54 is a complex number whose positive real part , if any , implies the exponential instability of the solution . next , the perturbed solution is substituted into eq . ( [ socsveeq ] ) , which is linearized with respect to the small perturbations . with the help of a straightforward but cumbersome algebraic procedure , the resulting linear system may be reduced to the ev problem of the corresponding evolution matrix @xcite . finally , the latter problem is solved in a numerical form . after that , results predicted by the linear - stability analysis are verified against direct simulations of eq . ( [ socsveeq ] ) , using the runge - kutta algorithm of the sixth order . for the case @xmath50 and @xmath43 , the following branches of the linear dispersion for eigenmodes taken as in eq . ( [ e ] ) can be obtained from eq . ( [ socsveeq]):@xmath55these dispersion curves are plotted in fig . [ fig1 ] , which demonstrate that , in accordance with eq . ( [ bands ] ) , the double - degenerate fbs survive keeping their eigenvalues @xmath56 , unaffected by the soc . this is by no means trivial , since the soc is lowering the system symmetry . further , the fbs are separated from the dbs by minigaps , whose widths , @xmath57 , increase with @xmath23 , as seen in fig . [ fig1](a ) : @xmath58 on dispersion relations ( [ bands ] ) . ( b ) the dispersion relations explicitly shown for @xmath45.,width=377 ] the two mutually symmetric sigs , with @xmath59 and @xmath60 , are located , respectively , above and beneath the dbs namely , at @xmath61values @xmath62 correspond to maximum values of dispersion branches @xmath63 given by eq . ( [ bands ] ) , which are attained at @xmath64 . two analytical solutions for the clss inside the fbs of the linear system ( @xmath65 ) are @xmath66where @xmath67 is an arbitrary constant which determines the norm of the wave function , i.e. , the number of atoms , in terms of the bec model . both solutions are clss with @xmath68 , as they occupy two unit cells . in the limit of @xmath69 , they split into two obvious cls states ( [ nosoc ] ) set in adjacent cells . thus , the soc induces a change of the cls class from @xmath22 at @xmath70 to @xmath68 at @xmath43 . the application of transformation ( [ symmetry ] ) produces two additional symmetry - related solutions . the total norm of each of these solutions is given by @xmath71 , as follows from eq . ( [ soclin ] ) . localized solutions are characterized by their _ participation number _ , which indicates the number of strongly excited sites : @xmath72and by the spatial decay rate of the localized - mode s tail vs. @xmath73 . for the linear - cls solutions ( [ soclin ] ) obtained above , the participation number is @xmath74 + ( 1/2)(\lambda ^{4}+1)^{2}+6\lambda ^{4}}\ ; , \label{soclinpt}\ ] ] which yields the correct limit value @xmath75 in the limit of vanishing soc . in the limit of strong soc , it returns to the same limit value , @xmath76 ( practically , it is attained for @xmath77 ) . at two values of the soc strength , @xmath78 , expression ( [ soclinpt ] ) attains a maximum , @xmath79 . at an intermediate point , @xmath80 , there is a local minimum , @xmath81 . the localized modes in the linear system are restricted to clss , sitting solely on the fbs , with @xmath16 , see eq . ( [ bands ] ) . in the absence of the soc , there is no coupling between the equations for the two components , hence clss are generated in each component independently , the respective general solutions being linear superpositions of the simple stationary modes given by eq . ( [ nosoc ] ) . the soc terms alter the shape of the cls according to eq . ( [ soclin ] ) . in that case , the two components are coupled , and the structure of the compact modes changes . they feature complex field amplitudes , and , as said above , occupy ( at least ) two unit cells , see fig . [ figls1 ] . ( left ) and @xmath82 ( right ) , as given by eq . ( [ soclin ] ) . values of the phase , marked in the figure near the respective lattice sites , are @xmath83 and @xmath84 . the color code displays the squared absolute value of the wave function on each site and for each component . note that the amplitudes strictly vanish on all @xmath24-sites . here , @xmath46.,width=340 ] the next , obviously important , objective is to explore the impact of nonlinearity in eq . ( [ socsveeq ] ) . we find that the clss survive solely if both the self- and cross - interactions are present , and the corresponding nonlinearity parameters are _ exactly equal _ , @xmath85 . this is clearly related to the above - mentioned validity condition for symmetry operation ( [ symmetry ] ) in the nonlinear system . being interested in self - trapped modes , we consider the case of attractive nonlinearity , with @xmath86 for the opposite sign of the nonlinearity , essentially the same modes are obtained , with the opposite sign of the frequency , @xmath59 . the clss have precisely the same structure as given by eq . ( [ soclin ] ) for the linear system , while the frequency of the nonlinear cls families is @xmath87[recall that @xmath67 is the amplitude of solution ( [ soclin ] ) ] . thus , the nonlinear cls modes are found in the _ exact analytical form _ , although they exist solely under the strict condition ( [ > 0 ] ) . equation ( [ detuning ] ) demonstrates that , varying @xmath88 , one can tune the nonlinear cls frequency to any positive value across the whole minigap , through the dispersive branches , and into the sig . in particular , the existence of the clss in the spectral areas occupied by the dispersive bands implies that the clss may be identified as _ embedded solitons _ @xcite , i.e. , ones which are embedded into the bands of dispersive waves . straightforward algebraic manipulations show that , for the nonlinear cls modes , which can be created only if @xmath89 , frequency @xmath90 and the total norm are related as @xmath91 this relationship does not depend on @xmath23 and is not valid for discrete solitons ( with exponentially decaying tails ) considered below . both the linear stability analysis and direct simulations show that the nonlinear clss are stable for frequencies @xmath90 staying in a vicinity of the corresponding fb , _ viz_. , @xmath92 . according to eqs . ( [ detuning ] ) and ( [ n ] ) , this implies that stable are clss with sufficiently small amplitudes and norms . the dependence of the stability threshold on the nonlinear - cls frequency on @xmath23 is shown in fig . [ lsacls ] in units of the minigap s width , @xmath57 ( [ shirina ] ) . in the limit of weak soc , almost the whole minigap is populated by stable nonlinear clss ( but note that the minigap itself becomes narrow in this case ) . on the contrary , the increase of the soc strength @xmath23 leads to a quick reduction of the relative stability threshold , which falls to @xmath93 for @xmath94 . to test the results of the linear - stability analysis , we simulated perturbed evolution of the nonlinear clss in the framework of eq . ( [ socsveeq ] ) . to this end , the nonlinear clss , belonging to the predicted stability and instability regions , with added a small amplitude random perturbation @xcite , were used as initial conditions . in the course of the evolution , stable nonlinear clss keep constant phase differences between adjacent lattice sites , see fig . [ phasedif ] . on the other hand , unstable clss feature irregular variations of the phase differences , which is accompanied by emission of energy into the lattice background . the latter feature is illustrated in fig . [ partcomp ] by plots for the participation ratio ( [ part - ratio ] ) of the respective modes . in addition to the nonlinear clss which persist in the linear limit , the inclusion of the attractive nonlinearity gives rise to exponentially localized discrete solitons ( dss ) , which are similar to usual dss in nonlinear lattices @xcite , see examples in figs . [ figmg0 ] and [ figmg1 ] . the structure of the ds tails , and the comparison to the cls shape [ taken from fig . [ figmg0](d ) ] , are shown on the logarithmic scale in fig . [ diverse ] . a drastic difference of the discrete solitons from the clss is that the dss exist as well when the cross - interaction coefficient , @xmath35 , is different from its self - interaction counterparts , @xmath33 and @xmath95 ( for instance , in fig . [ figmg0 ] @xmath96 but @xmath97 ) . different ds modes are numerically obtained by changing the respective input , i.e. , its shape and norm , and ratios between the self- and cross - interaction parameters , @xmath33 , @xmath95 , and @xmath35 . similar to the clss , the dss have narrow stability regions inside the minigaps . exact location of boundaries of the regions is a challenging issue , as they are sensitive to variations of all the parameters . on the other hand , the existence and stability of similar exponentially localized dss in minigaps was studied in other lattice models ( see , e.g. , ref . therefore , we here focus on accurate delineation of the stability area for the clss in the minigaps of the nonlinear lattice , which is a new problem . the simulations demonstrate that perturbed stable states give rise to robust breathers periodically oscillating in time . outside of the narrow stability regions found in the neighborhood of the fb , unstable perturbed clss and dss spread over the entire lattice in the course of simulations . the evolution of participation number for perturbed stable and unstable dss of type shown in fig . [ figmg0 ] ( d ) is displayed in fig . [ partcomp ] , confirming the predictions of the linear - stability analysis . the spreading rate of unstable dss is indeed determined by unstable evs for these modes . , which is plotted , in the log scale , in units of the minigap width @xmath57 [ see eq . ( [ shirina ] ) ] . nonlinearity parameters are @xmath98.,width=226 ] and @xmath99 ( black circles ) , @xmath100 and @xmath101 ( red empty circles ) , @xmath102 and @xmath103 ( blue squares ) , vs. time . ( b ) the same for an unstable nonlinear clss . the soc strength is @xmath46 . the stable and unstable clss have , respectively , @xmath104 , and @xmath105 . the nonlinearity parameters are @xmath106 . , title="fig:",width=226 ] and @xmath99 ( black circles ) , @xmath100 and @xmath101 ( red empty circles ) , @xmath102 and @xmath103 ( blue squares ) , vs. time . ( b ) the same for an unstable nonlinear clss . the soc strength is @xmath46 . the stable and unstable clss have , respectively , @xmath104 , and @xmath105 . the nonlinearity parameters are @xmath106 . , title="fig:",width=226 ] ( the black dashed line ) and of an unstable one , with @xmath107 ( the blue solid line ) versus time . nonlinearity parameters are @xmath108 . also shown are the time dependence of the participation ratio for a stable ( @xmath109 , the red dashed line ) and unstable ( @xmath110 , the green solid line ) discrete solitons ( dss ) . the soc strength is @xmath111 . , width=377 ] and @xmath112.,width=529 ] and @xmath112.,width=529 ] ( d ) ( the blue dashed line with empty squares ) , and the discrete soliton found in the semi - infinite gap ( the red line with empty circles ) . for comparison , the line with constant slope @xmath113 is shown as the olive dot - dashed one . parameters are @xmath46 , @xmath114 for the modes denoted by solid lines , and @xmath115 , @xmath116 , @xmath117 for those denoted by dashed lines . the values of @xmath90 and @xmath118 are the same as in fig . [ partcomp].,width=302 ] finally , it has been checked that stable clss and dss may coexist in a vicinity of the fb . they all are characterized by mutually close ( and small ) values of the norm . in fact , their shapes are close too , apart from the presence of the exponentially decaying tails in the dss , which are absent in the clss . the ds solutions existence area covers the entire minigaps , but their stability region is a narrow strip adjacent to the fb . unlike the clss , ds states do not exists at values of @xmath90 belonging to the dbs , i.e. , the dss can not be modes of the embedded type . it is natural to explore the existence and stability of localized states in the sig . first , as mentioned above , the clss can be found in the sig . note that the simple relation between the cls norm and frequency , given by eq . ( [ n ] ) , applies to the sig as well . the stability area of the clss inside of sig has been identified too , through the linear - stability analysis and direct simulations alike . the stability area in the plane of @xmath119 is shown in fig . [ clsinsi ] , cf . the stability region for the clss in the minigaps , shown in fig . [ lsacls ] . similar to the fact that the clss are stable only a small part of the minigap , the stability area occupies only a small portion of the sig . . the nonlinearity parameters are fixed as @xmath120 for both solution types ( recall that the cls does not exist if the these coefficients are not equal ) . the dashed line shows the sig boundary , as given by the inversion of eq . ( [ sig ] ) : @xmath121 . , width=377 ] in the absence of the soc ( @xmath40 ) , dss , which are characterized by the largest amplitude at site @xmath24 ( on the contrary to the clss , which have zero amplitudes at @xmath24 ) , also exist inside the sig . they are found to be stable at @xmath122 for @xmath123 ( or at @xmath124 for @xmath125 ) . a typical example of such a mode is displayed in fig . [ fig13](a ) . , @xmath126 , and ( b ) @xmath45 , @xmath127.,width=529 ] in the presence of the soc , single - peak dss are found in the sig too . their symmetry with respect to the central @xmath128 site is broken by the soc . the stability region of the single - peak dss almost overlaps with the sig area , as shown in fig . [ clsinsi ] ( the gray area bounded by the blue solid curve ) . an example of the stable ds is shown for @xmath129 , @xmath130 in fig . [ fig13](b ) . the affinity of these states to the usual discrete solitons is confirmed by plotting tails of the corresponding solutions in fig . [ diverse ] ( the red line with empty circles ) . for the sake of comparison , an exponentially decaying function is shown by the green line in the same plot . we have introduced a dynamical lattice system , which may be realized for the two - component bec with the soc ( spin - orbit coupling ) , loaded into the optically imprinted ribbon with the structure of the diamond chain , that serves as a paradigm for realizing fb ( flatband ) spectra . the system includes interactions between its components , both nonlinear and linear , the latter mediated by the soc . the linear version of the system , in the absence of the soc , is characterized by the spectrum which consists of a pair of dbs ( dispersive bands ) which touch the fb at the edge of the corresponding brillouin zone , for each component . such a linear decoupled system creates cls ( compact localized states ) in each component which extend over one unit cell and belong to the class @xmath22 , at a single value of the frequency which exactly corresponds to the fb . the introduction of the soc changes the spectrum , keeping the double - degenerate fbs and opening narrow minigaps between them and the dbs . the clss remain available in the exact analytical form . they increase their size to @xmath68 , and can be viewed as a bound state of two clss of class @xmath22 , with an appropriate phase and amplitude modulation , depending on the strength of soc . in the presence of the onsite self - attractive cubic nonlinearity , we have shown that the clss persist in the form of exact analytical solutions . their frequency can be tuned throughout the minigap , as well as into the sig ( semi - infinite gap ) . parallel to that , usual dss ( discrete solitons ) with exponentially decaying tails have been found too , in a numerical form , in the minigap and sig alike . both the cls and ds modes are stable inside of the minigap in narrow stripes abutting on the fb . stability areas for the cls and ds families have been found in the sig too , also being small in comparison with the sig size . a main motivation for the interest in clss and fbs is due to the macroscopic degeneracy they enforce . such macroscopic degeneracies are usually removed due to external perturbations , leading to new eigenstates , whose properties are sensitively depending on the type of perturbation added . in this work we show that spin - orbit coupling with cold atoms in optical lattices acts as a degeneracy - preserving perturbation . we also find that properly tuned interactions destroy the notion of a band structure , yet preserve the clss which simply detune their frequencies . a detuning of these interactions will finally destroy the clss . in addition to that , in this work we have demonstrated that the presence of the fb strongly affects not only the clss but also regular dss . it is expected that these dss will keep their properties even if a perturbation of the system will lend the fb a weak curvature . as a development of the analysis , it may be interesting to analyze a fully two - dimensional version of the system . in particular , modes in the form of stable localized vortices have not been found in the present system , both with and without the soc terms . it is plausible that vortex modes may exist in the fully two - dimensional lattice . g.g . , a.m. , and lj.h . acknowledge support from the ministry of education and science of serbia ( project iii45010 ) . the work of b.a.m . is supported , in part , by the joint program in physics between the national science foundation ( us ) and binational science foundation ( us - israel ) , through grant no . this author appreciates hospitality of the vina institute of nuclear sciences at university of belgrade ( serbia ) . this work was supported by the institute for basic science , south korea ( project code ibs - r024-d1 ) . d. l. campbell , g. juzelinas , and i. b. spielman , phys . a * 84 * , 025602 ( 2011 ) ; y. j. lin , k. jimenez - garcia , and i. b. spielman , nature * 471 * , 83 ( 2011 ) ; v. galitski i. b. and spielman , nature * 494 * , 49 ( 2013 ) ; h. zhai , rep . . phys . * 78 * , 026001 ( 2015 ) .

we report the coexistence and properties of stable compact localized states ( clss ) and discrete solitons ( dss ) for nonlinear spinor waves on a flatband network with spin - orbit coupling ( soc ) . the system can be implemented by means of a binary bose - einstein condensate loaded in the corresponding optical lattice . in the linear limit , the soc opens a minigap between flat and dispersive bands in the system s bandgap structure , and preserves the existence of clss at the flatband frequency , simultaneously lowering their symmetry . adding onsite cubic nonlinearity , the clss persist and remain available in an exact analytical form , with frequencies which are smoothly tuned into the minigap . inside of the minigap , the cls and ds families are stable in narrow areas adjacent to the fb . deep inside the semi - infinite gap , both the clss and dss are stable too .

1609.09640

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 .

1612.04066

light plays an essential role in quantum communication and is indispensable in most practical applications , for example quantum cryptography . photons are attractive carriers of quantum information because the interactions of light with the surroundings are normally weak , but for the same reason it is generally difficult to prepare , manipulate , and measure quantum states of light in a nondestructive way . repeated interactions provide a method to increase the effective coupling strength between light and matter , and the backreflection of light in a cavity thus constitutes an interesting tool , in particular , because experiments are currently moving into the strong coupling regime @xcite , where coherent dynamics takes place on a faster time scale than dissipative dynamics . in this paper we propose a versatile setup consisting of an array of cavities and passive optical elements ( beam splitters and phase shifters ) , which allows for quantum state engineering , quantum state purification , and non - destructive number resolving photon detection . the setup builds on two basic ingredients : the hong - ou - mandel interference effect @xcite generalized to input pulses containing an arbitrary number of photons and the possibility of projection onto the subspace of even or the subspace of odd photon - number states by use of cavity quantum electrodynamics in the strong coupling regime . regarding quantum state engineering , the basic setup provides a possibility to conditionally generate photon - number correlated states . more specifically , the setup allows us to project an arbitrary photonic two - mode input state onto the subspace spanned by the state vectors @xmath0 with @xmath1 . we denote this subspace by @xmath2 . the scheme is probabilistic as it is conditioned on a specific measurement outcome . the success probability equals the norm of the projection of the input state onto @xmath2 and is thus unity if the input state already lies in @xmath2 . in other words , the setup may be viewed as a filter @xcite , which removes all undesired components of the quantum state but leaves the desired components unchanged . we may , for example , use two independent coherent states as input and obtain a photon - number correlated state as output . photon - number correlated states , for example einstein - podolsky - rosen ( epr ) entangled states @xcite , are an important resource for quantum teleportation @xcite , entanglement swapping @xcite , quantum key distribution @xcite , and bell tests @xcite . in practice , however , the applicability of these states is hampered by noise effects such as photon losses . real - world applications require therefore entanglement purification . the proposed setup is very attractive for detection of losses and can in particular be used to purify photon - number entangled states on site . if a photon - number correlated state , for example an epr state , is used as input , the desired state passes the setup with a certificate , while states which suffered from photon losses are detected and can be rejected . photon losses are an especially serious problem in quantum communication over long distances . it is not only a very common source of decoherence which is hard to avoid , but also typically hard to overcome . the on - site purification protocol mentioned above can easily be adopted to a communication scenario such that it allows for the purification of a photon - number correlated state after transmission to two distant parties . purification of two mode entangled states has been shown experimentally for qubits @xcite and in the continuous variable ( cv ) regime @xcite . ( cv - entanglement purification is especially challenging @xcite . nevertheless , several proposals have been made to accomplish this task @xcite , and very recently takahashi _ et al . _ succeeded in an experimental demonstration @xcite . ) a special advantage of our scheme lies in the fact that it does not only allow for detection of arbitrary photon losses , but is also applicable to many modes such that entanglement can be distributed and purified in a network . with a small modification , the basic setup can be used for number resolved photon detection . the ability to detect photons in a number resolved fashion is highly desirable in the fields of quantum computing and quantum communication . for example , linear optics quantum computation relies crucially on photon number resolving detectors @xcite . moreover , the possibility to distinguish different photon - number states allows for conditional state preparation of nonclassical quantum states @xcite , and plays a role in bell experiments @xcite and the security in quantum cryptographic schemes @xcite . other applications include interferometry @xcite and the characterization of quantum light sources @xcite . existing technologies for photon counting @xcite such as avalanche photodiodes , cryogenic devices , and quantum dots typically have scalability problems and can not reliably distinguish high photon numbers , destroy the quantum state of light in the detection process , or do not work for optical photons . here , we present a non - destructive number resolving photo detection scheme in the optical regime . this quantum - non - demolition measurement of the photon number allows for subsequent use of the measured quantum state of light . an advantage of the counting device put forward in this work compared to other theoretical proposals for qnd measurements of photon numbers @xcite is the ability to detect arbitrarily high photon numbers with arbitrary resolution . the scheme is based on testing successively all possible prime factors and powers of primes and the resources needed therefore scale moderately with ( width and mean of the ) photon number distribution . in particular , a very precise photon number measurement can be made even for very high photon numbers by testing only few factors if the approximate photon number is known . the paper is structured as follows . we start with a brief overview of the main results in sec . [ overview ] . in sec . [ filter ] , we explain how the conditional projection onto @xmath2 can be achieved and discuss some properties of the proposed setup in the ideal limit , where the atom - cavity coupling is infinitely strong and the input pulses are infinitely long . in sec . [ detector ] , we show that a modified version of the setup can act as a non - destructive photon number resolving detector , and in sec . [ purification ] , we investigate the possibility to use the setup to detect , and thereby filter out , losses . in sec . [ nonideal ] , we consider the significance of finite input pulse length and finite coupling strength , and we obtain a simple analytical expression for the optimal choice of input mode function for coherent state input fields . section [ conclusion ] concludes the paper . the most important ingredient of the proposed setup is the possibility to use the internal state of a single atom to control whether the phase of a light field is changed by @xmath3 or not @xcite . the basic mechanism , which is explained in fig . [ cavity ] , has several possible applications , including preparation of superpositions of coherent states @xcite , continuous two - qubit parity measurements in a cavity quantum electrodynamics network @xcite , and low energy switches @xcite . concerning the experimental realization , basic ingredients of the scheme such as trapping of a single atom in a strongly coupled cavity and preparing of the initial atomic state have been demonstrated experimentally in @xcite , where the decrease in cavity field intensity for an atom in the state @xmath4 compared to the case of an atom in the state @xmath5 is used to subsequently measure the state of the atom . state preparation and readout for a single atom in a cavity have also been demonstrated in @xcite . another promising candidate for an experimental realization is circuit quantum electrodynamics . see , for instance , @xcite for a review . ( color online ) a single atom with level structure as shown in ( a ) is placed in a cavity in the strong coupling regime . the light field , which is on resonance with the cavity , couples the ground state level @xmath4 resonantly to the excited state @xmath6 , and the state @xmath6 decays to the state @xmath4 through spontaneous emission at a rate @xmath7 . ( b ) if the atom is in the state @xmath5 , the incident field is not affected by the presence of the atom , and for a sufficiently slowly varying input pulse , the interaction with the resonant cavity changes the phase of the light field by @xmath3 . ( c ) if the atom is initially in the state @xmath4 , the possibility of spontaneous emission prevents the light field from building up inside the cavity ( provided the photon flux of the input beam is not too high ) , and the incoming field is reflected from the input mirror without acquiring a phase shift . this transformation is insensitive to the precise values of @xmath8 , @xmath7 , and the cavity decay rate as long as the system is in the strong coupling and weak driving regime . ] the generation of quantum superposition states can be achieved as follows . the atom is initially prepared in the state @xmath9 , and the input field is chosen to be a coherent state @xmath10 . after the interaction , the combined state of the atom and the light field is proportional to @xmath11 , and a measurement of the atomic state in the basis @xmath12 projects the state of the light field onto the even @xmath13 or the odd @xmath14 superposition state . more generally , the input state @xmath15 , where @xmath16 is an @xmath17-photon fock state , is transformed into the output state @xmath18 , i.e. , the input state is conditionally projected onto either the subspace spanned by all even photon - number states or the subspace spanned by all odd photon - number states without destroying the state . with this tool at hand , we can project an arbitrary two - mode input state onto the subspace @xmath19 , @xmath1 . if two modes interfere at a 50:50 beam splitter , a state of form @xmath0 is transformed into a superposition of products of even photon - number states . if we apply a 50:50 beam splitter operation to the input state , project both of the resulting modes conditionally on the subspace of even photon - number states , and apply a second 50:50 beam splitter operation , the input state is thus unchanged if it already lies in @xmath2 , but most other states will not pass the measurement test . to remove the final unwanted components , we apply opposite phase shifts to the two modes ( which again leaves @xmath20 unchanged ) and repeat the procedure ( as shown in fig . [ setup ] ) . for an appropriate choice of phase shifts , the desired state is obtained after infinitely many repetitions . in practice , however , a quite small number is typically sufficient . if , for instance , the input state is a product of two coherent states @xmath21 with @xmath22 , the fidelity of the projection is @xmath23 for one unit , @xmath24 for two units , and @xmath25 for three units . the scheme is easily generalized to an @xmath26 mode input state . in this case , we first project modes 1 and 2 on @xmath2 , modes 3 and 4 on @xmath2 , etc , and then project modes 2 and 3 on @xmath2 , modes 4 and 5 on @xmath2 , etc . the setup can also be used as a device for photon number resolving measurements if the phases applied between the light - cavity interactions are chosen according to the new task . each photon - number state @xmath16 sent through the array leads to a characteristic pattern of atomic states . as explained in section [ destructivescheme ] , one can determine the photon number of an unknown state by testing the prime factors and powers of primes in the range of interest in subsequent parts of the array . the scheme scales thereby moderately in the resources . three cavity pairs suffice for example for detecting any state which is not a multiple of three with a probability of @xmath27 . however , in this basic version of the counting scheme , the tested photon state may leave each port of the last beam splitter with equal probability . deterministic emission of the unchanged quantum state of light into a single spatial mode is rendered possible if we allow atoms in different cavities to be entangled before the interaction with the field ( see section [ qndscheme ] ) . more generally , the proposed scheme allows to determine the difference in photon numbers of two input beams without changing the photonic state . the correlations in photon number between the two modes of states in @xmath2 facilitate an interesting possibility to detect photon losses . to this end the state is projected onto @xmath2 a second time . if photon loss has occurred , the state is most likely orthogonal to @xmath2 , in which case we obtain a measurement outcome , which is not the one we require in order to accept the projection as successful . on the other hand , if photon loss has not occurred , we are sure to get the desired measurement outcome . we note that loss of a single photon can always be detected by this method , and the state can thus be conditionally recovered with almost perfect fidelity if the loss is sufficiently small . we can improve the robustness even further , if we use an @xmath26-mode state . it is then possible to detect all losses of up to @xmath28 photons , and even though it is @xmath26 times more likely to lose one photon , the probability to lose one photon from each mode is approximately @xmath29 , where @xmath30 is the probability to lose one photon from one mode and we assume @xmath31 . in a situation where many photon losses are to be expected , this procedure allows one to obtain photon - number correlated states with high fidelity , although with small probability . we can also distribute the modes of a photon - number correlated state to distant parties , while still checking for loss , provided we send at least two modes to each party . as the proposed scheme can be used as a filter prior to the actual protocol it has an important advantage compared to postselective schemes . if the tested entangled state is for example intended to be used for teleportation , the state to be teleported is not destroyed in the course of testing the photon - number correlated resource state . the dynamics in fig . [ cavity ] requires strong coupling , a sufficiently slowly varying mode function of the input field , and a sufficiently low flux of photons . to quantify these requirements , we provide a full multi - mode description of the interaction of the light with the cavity for the case of a coherent state input field in the last part of the paper . we find that the single atom cooperativity parameter should be much larger than unity , the mode function of the input field should be long compared to the inverse of the decay rate of the cavity , and the flux of photons in the input beam should not significantly exceed the rate of spontaneous emission events from an atom having an average probability of one half to be in the excited state . we also derive the optimal shape of the mode function of the input field ( eq . ) , when the mode function is only allowed to be nonzero in a finite time interval . the proposed setup for projection of an arbitrary two - mode input state onto @xmath2 is sketched in fig . [ setup ] . we denote the field annihilation operators of the two input modes by @xmath32 and @xmath33 , respectively . the total transformation corresponding to one of the units consisting of a beam splitter , a set of cavities , and a second beam splitter , conditioned on both atoms being measured in the state @xmath34 after the interaction , is given by the operator @xmath35 , where @xmath36\ ] ] and @xmath37 as explained above , the hong - ou - mandel effect ensures that @xmath38 , while most other possible components of the input state are removed through the conditioning , for instance all components @xmath39 with @xmath40 odd . there are , however , a few exceptions , since all states of the form @xmath41 , @xmath1 , @xmath42 , are accepted . the phase shifts between the @xmath35 units are represented by the operator @xmath43,\ ] ] which leaves states of the form @xmath0 unchanged , while states of the form @xmath39 with @xmath44 acquire a phase shift . for a setup containing @xmath45 units , the complete conditional transformation is thus represented by the operator @xmath46\label{on},\end{aligned}\ ] ] where @xmath35 in the last line commutes with the product of cosines . for @xmath47 , the product of cosines vanishes for all components of the input state with different numbers of photons in the two modes if , for instance , all the @xmath48 s are chosen as an irrational number times @xmath3 . we note that even though we here apply the two - mode operators one after the other to the input state corresponding to successive interactions of the light with the different components of the setup , the result is exactly the same if the input pulses are longer than the distance between the components such that different parts of the pulses interact with different components at the same time . the only important point is that the state of the atoms is not measured before the interaction with the light field is completed . the setup using an array of cavities as in fig . [ setup ] may thus be very compact even though the pulses are required to be long . ( note that it would also be possible to use a single pair of cavities and atoms repeatedly in a fold - on type of experiment ; however in that case the compactness would be lost due to the need for long delay lines necessary to measure and re - prepare the atoms before they are reused . ) a natural question is how one should optimally choose the angles @xmath48 to approximately achieve the projection with a small number of units . to this end we define the fidelity of the projection @xmath49 as the overlap between the unnormalized output state @xmath50 after @xmath45 units and the projection @xmath51 of the input state @xmath52 onto the subspace @xmath2 . the last equality follows from the fact that @xmath53 , where @xmath54 lies in the orthogonal complement of @xmath2 . maximizing @xmath55 for a given @xmath56 thus corresponds to minimizing @xmath57\\ \times\langle\psi_{\textrm{in}}|u^\dag pu|n\rangle|m\rangle,\end{gathered}\ ] ] i.e. , we would like to find the optimal solution of @xmath58(n - m)\\ \times\prod_{\substack{i=1\\i\neq j}}^n\cos^2[\phi_i(n - m ) ] \langle\psi_{\textrm{in}}|u^\dag pu|n\rangle|m\rangle=0.\end{gathered}\ ] ] a set of solutions valid for any input state can be obtained by requiring @xmath59\prod_{i\neq j}\cos^2[\phi_i(n - m)]=0 $ ] for all even values of @xmath60 ( note that @xmath61 for @xmath40 odd ) . within this set the optimal solution is @xmath62 . it is interesting to note that choosing one of the angles to be @xmath63 , @xmath64 , all terms with @xmath65 are removed from the input state according to . when all angles are chosen according to @xmath62 , it follows that @xmath66 only contains terms with @xmath67 , @xmath68 , which may be a useful property in practical applications of the scheme . even though this is not necessarily optimal with respect to maximizing @xmath69 for a particular choice of input state , we thus use the angles @xmath62 in the following , except for one important point : if the input state satisfies the symmetry relations @xmath70 , it turns out that the operator @xmath35 by itself removes all terms with @xmath71 , i.e. , we can choose the angles as @xmath72 , and @xmath66 only contains terms with @xmath73 , @xmath68 . for @xmath74 , for instance , only terms with @xmath75 contribute . in fig . [ fidelity ] , we have chosen the input state to be a product of two coherent states with amplitude @xmath76 and plotted the fidelity as a function of @xmath77 for different numbers of units of the setup . even for @xmath77 as large as 10 , the fidelity is still as high as 0.9961 for @xmath74 , and the required number of units is thus quite small in practice . the figure also shows the success probability @xmath78 for @xmath79 . for @xmath80 , for instance , one should repeat the experiment about 11 times on average before the desired measurement outcome is observed . ( color online ) fidelity ( eq . ) as a function of the expectation value of the number of photons in one of the input modes for @xmath81 and setups with one , two , three , and infinitely many units . the angles are chosen as @xmath72 . the dotted line labeled @xmath82 is the probability in the limit of infinitely many units to actually obtain the required measurement outcome , i.e. , all atoms in @xmath9 after the interaction with the field . ] in this section , we show how a photon number measurement can be implemented using a modified version of the setup introduced in the previous section . the key idea is explained in subsection [ destructivescheme ] where we describe the basic photo - counting scheme . in subsection [ qndscheme ] , this protocol is extended to allow for a qnd measurement of photon numbers . in the following , we analyze the setup shown in fig . [ setup ] when the input is a product of an @xmath17-photon fock state in the lower input beam and a vacuum state in the upper input beam . since the setup contains a series of beam splitters , it will be useful to define @xmath83 and @xmath84 , such that @xmath85 and @xmath86 at beam splitters bs@xmath87 , bs@xmath88 , bs@xmath89 , @xmath90 , and @xmath91 and @xmath92 at beam splitters bs@xmath93 , bs@xmath94 , bs@xmath95 , @xmath90 . as before , all atoms are initially prepared in the state @xmath96 and will after the interaction with the field be measured in the @xmath97 basis . when we start with an @xmath17 photon state , there are only two possible outcomes of the measurement of the atoms in the cavities labeled c@xmath87 and c@xmath93 in fig . [ setup ] dependent on whether @xmath17 is even or odd . in the even case , the two atoms can only be in @xmath98 or @xmath99 and in the odd case @xmath100 or @xmath101 . to handle the odd and even case at the same time we denote @xmath102 as @xmath103 and @xmath104 as @xmath105 . a measurement of @xmath103 indicates an even number of photons in the @xmath33-beam , resp . , @xmath105 an odd number of photons in the @xmath33-beam . we start with the state @xmath106 as input . after the beam splitter bs@xmath87 , the state has changed into @xmath107 and interacts with the two atoms in the cavities c@xmath87 and c@xmath93 . by measuring the atoms in the @xmath97 basis , the state is projected into the subspace of an even or odd number of photons in the @xmath33 path . the photon state after the measurement can be written as @xmath108\ket{0}\ket{0}=\frac{1}{\sqrt{2\ n!}}(\hat{a}^n \pm\hat{b}^n)\ket{0}\ket{0}$ ] , where @xmath109(@xmath110 ) is the state with an even ( odd ) number of @xmath33 photons and corresponds to the measurement result @xmath103 ( @xmath105 ) . note that this first measurement result is completely random . after bs@xmath93 the state simplifies to @xmath111\ket{0}\ket{0}$ ] . now due to the phase shifters the two modes pick up a relative phase of @xmath112 so that the state is given by @xmath113\ket{0}\ket{0}$ ] . finally , after bs@xmath88 we have the state @xmath114 , which is equal to @xmath115 this can also be rewritten as @xmath116 . so the result of measuring the state of the atoms in cavities c@xmath88 and c@xmath94 will be @xmath117 with probability @xmath118 and @xmath119 with probability @xmath120 . since the state is again projected into one of the two states @xmath121 we can repeat exactly the same calculations for all following steps . whereas the first measurement result was completely random , all following measurement results depend on @xmath17 and the previous measurement outcome , i.e. , with probability @xmath122 the @xmath123-th measurement result is the same as the @xmath124-th result , and with probability @xmath125 the measurement result changes and the state changes from @xmath121 to @xmath126 . if the number of units is infinite ( or sufficiently large ) and we have chosen all phases equal as @xmath127 , then we can guess from the relative frequency with which the measurement result has switched between @xmath103 and @xmath105 the number of photons with arbitrary precision for all photon numbers @xmath128 . @xmath129 , where @xmath130 and @xmath131 ( @xmath132 ) is the number of cases , where the measurement outcome is the same ( not the same ) as the previous measurement outcome . measuring this relative frequency with a fixed small phase is not the optimal way to get the photon number . we propose instead the following . let us use a setup with a total of @xmath45 units and choose the phases to be @xmath133 , @xmath134 , for an arbitrarily chosen value of @xmath135 and let us calculate the probability @xmath136 that the measurement results are all the same , @xmath137 this probability is equal to one for all photon numbers that are a multiple of @xmath138 and goes to zero otherwise in the limit of infinitely many units of the setup . this way we can measure whether the photon number is a multiple of @xmath138 . for example , for @xmath139 and @xmath140 we detect any state which is not a multiple of three with a probability of at least @xmath141 , resp . , @xmath142 for @xmath143 , where @xmath144 for @xmath145 , already @xmath140 is sufficient to achieve @xmath146 . for @xmath147 and @xmath140 we have @xmath141 , which increases to @xmath142 for @xmath143 . in fig . [ 20photo ] and [ 100photo ] we have shown @xmath136 for @xmath148 and @xmath149 . the number @xmath150 needed to get a good result typically scales logarithmical with @xmath138 , e.g. for @xmath151 already @xmath152 is enough to reach @xmath153 . ( color online ) probabilities @xmath136 ( eq . ) for @xmath148 and @xmath143 resulting in @xmath154 ( eq . ) . for @xmath155 , @xmath156 . ] ( color online ) probabilities @xmath136 for @xmath149 and @xmath157 resulting in @xmath158 . for @xmath159 , @xmath153 . ] given an unknown state we can test all possible prime factors and powers of primes to identify the exact photon number . if , e.g. , we have a state consisting of @xmath160 to @xmath161 photons the following factors have to be tested @xmath162 ( where 2 does not need to be checked separately ) . 24 measurement results are sufficient to test all factors with a probability over @xmath163 . if we require reliable photon number counting for @xmath17 ranging from @xmath160 to @xmath164 , for large @xmath164 , all primes and power of primes that are smaller than @xmath164 need to be tested . this number can be bounded from above by @xmath164 . all @xmath164 tests are required to work with high probability . to this end each single test needs to succeed with a probability better than @xmath165 . it can be checked numerically that this is the case if @xmath166 , leading to a photon counting device with reliable photon detection up to @xmath164 using an array consisting of less than @xmath167 basic units . note that this setup does not destroy the photonic input state but changes @xmath168 randomly into @xmath169 $ ] , i.e. , the photons leave the setup in a superposition of all photons taking either the @xmath32-beam or the @xmath33-beam . this can not be changed back into @xmath168 by means of passive optical elements . the output state - a so - called @xmath170 state - is , however , a very valuable resource for applications in quantum information protocols and quantum metrology @xcite . for a non - demolition version of the photon number measurement we use the basic building block depicted in fig . [ block ] . the atoms in the two upper cavities are initially prepared in an entangled state @xmath171 , and the atoms in the lower cavities are also prepared in the state @xmath172 . this can , for instance , be achieved via the parity measurement scheme suggested in @xcite . during the interaction the upper and the lower atoms will stay in the subspace spanned by @xmath173 . the state changes between @xmath172 and @xmath174 each time one of the two entangled cavities interacts with an odd number of photons . as in the previous subsection we define @xmath117 to handle the even and the odd case at the same time . for @xmath17 even , @xmath175 and @xmath176 , while in the odd case we define @xmath177 and @xmath178 . and c@xmath88 and between the atoms in cavities c@xmath93 and c@xmath94 . the gray box is either a cavity or a mirror . ( in the latter case , we may as well send the light directly from the cavities c@xmath88 and c@xmath94 to the first two cavities of the next block . ) ] using the same notation as before , the state is transformed as follows , when we go through the setup from left to right . the initial state is @xmath179 and after the first beam splitter we have the state @xmath180 the interaction with the first two cavities leads to the state @xmath181 the second beam splitter transforms the state into @xmath182\ket{0}\ket{0}\ket{b_+}\\ + [ ( \hat{a}^\dag)^n-(\hat{b}^\dag)^n]\ket{0}\ket{0}\ket{b_-}\big\}.\end{gathered}\ ] ] the two modes pick up a relative phase shift of @xmath183 at the phase shifters so that @xmath184\ket{0}\ket{0}\ket{b_+}\\ + [ e^{i\phi_i n}(\hat{a}^\dag)^n -e^{-i\phi_i n}(\hat{b}^\dag)^n]\ket{0}\ket{0}\ket{b_-}\big\}.\end{gathered}\ ] ] after the third beam splitter we get @xmath185\\ = \frac{1}{\sqrt{2}}\cos(\phi_i n)\ket{b_+}\ket{b_+ } + \frac{i}{\sqrt{2}}\sin(\phi_i n)\ket{b_-}\ket{b_+}\\ + \frac{i}{\sqrt{2}}\sin(\phi_i n)\ket{b_+}\ket{b_- } + \frac{1}{\sqrt{2}}\cos(\phi_i n)\ket{b_-}\ket{b_-}.\end{gathered}\ ] ] note now that independent of whether @xmath17 is even or odd , the interaction with the last two cavities turns the states @xmath186 into @xmath187 and the states @xmath188 into @xmath189 . the state is thus changed to @xmath190,\end{gathered}\ ] ] and after the last beam splitter we have @xmath191.\end{gathered}\ ] ] note that the photonic modes are now decoupled from the state of the atoms . the photons will continue after the final beam splitter unchanged in @xmath168 while the atoms still contain some information about the photon number . note that @xmath192 and @xmath193 . by measuring all atoms in the @xmath97 basis we can easily distinguish between @xmath194 by the parity of the measurements . the probability to obtain @xmath195 is @xmath196 , and the probability to get @xmath197 is @xmath198 . after the measurement , all photons are found in the @xmath32 beam for both outcomes . the probability for measuring @xmath195 is the same as the changing probability in the previous setup such that we can do the same with a chain of the demolition free block . if we prefer also to get the parity information in each step , we can add an additional cavity at the end of each block as shown in fig . [ block ] . more generally , the demolition free element leaves photonic input states @xmath199 , where @xmath17 photons enter through the lower and @xmath200 photons enter through the upper port , unchanged . a calculation analogous to the previous one shows that one obtains the atomic states @xmath195 and @xmath197 with probabilities @xmath201 and @xmath202 respectively . this way , we can test for photon number differences @xmath203 in two input states in the same fashion as for photon numbers in a single input beam described above . similarly , one can project two coherent input states @xmath204 onto generalized photon - number correlated states @xmath205 with fixed photon number difference @xmath206 .... in a realistic scenario we may be faced with photon losses . both setups have a built - in possibility to detect loss of one photon . in the first case we get the parity of the total number of photons in every single measurement of a pair of atoms . if this parity changes at some place in the chain then we know that we lost at least one photon . in the demolition free setup the valid measurement results are restricted to @xmath195 and @xmath197 . if we measure @xmath207 or @xmath208 we know that we lost a photon in between the two pairs of cavities . in addition , the optional cavity at the end of each block provides an extra check for photon loss . we next investigate the possibility to use a comparison of the number of photons in the two modes to detect a loss . as in sec . [ filter ] , we start with the input state @xmath81 and use the proposed setup to project it onto the subspace @xmath2 . we then use two beam splitters with reflectivity @xmath209 to model a fractional loss in both modes . after tracing out the reflected field , we finally use the proposed setup once more to project the state onto @xmath2 . in fig . [ recover ] , we plot the fidelity between the state obtained after the first projection onto @xmath2 and the state obtained after the second projection onto @xmath2 , the probability that the second projection is successful given that the first is successful , and the purity of the state after the second projection . the second projection is seen to recover the state obtained after the first projection with a fidelity close to unity even for losses of a few percent . this is the case because a loss of only one photon will always lead to a failure of the second conditional projection . the main contribution to the fidelity decrease for small @xmath209 is thus a simultaneous loss of one photon from both modes . it is also interesting to note that the final state is actually pure for all values of @xmath209 , which is a consequence of the particular choice of input state . finally , we note that a single unit is sufficient to detect loss of a single photon , and for small @xmath209 we thus only need to use one unit for the second projection in practice . ( color online ) projecting the input state @xmath81 onto the subspace @xmath2 followed by a fractional loss @xmath209 in both modes and a second projection onto @xmath2 , the figure shows the fidelity between the states obtained after the first and the second projection onto @xmath2 , the probability that the second projection is successful given that the first is successful , and the purity of the state after the second projection for two different values of @xmath77 . ] let us also consider a four mode input state @xmath210 . as before we use the setup to project this state onto the subspace spanned by the state vectors @xmath211 , @xmath1 . we then imagine a fractional loss of @xmath209 to take place in all modes . if two of the modes are on their way to alice and the two other modes are on their way to bob , we can only try to recover the original projection by projecting the former two modes onto @xmath2 and the latter two modes onto @xmath2 . the results are shown in fig . [ loss ] , and again the curves showing the fidelity and the purity are seen to be very flat and close to unity for small losses . this scheme allows one to distribute entanglement with high fidelity , but reduced success probability . ( color online ) projecting the input state @xmath210 onto the subspace spanned by the vectors @xmath211 , @xmath1 , followed by a fractional loss @xmath209 in all four modes and a projection of modes 1 and 2 onto @xmath2 and of modes 3 and 4 onto @xmath2 , the figure shows the fidelity between the states obtained after the first and the second set of projections , the probability that the projections onto @xmath2 are successful given that the first projection is successful , and the purity of the state after the projections onto @xmath2 for two different values of @xmath77 . ] so far we have considered the ideal limit of infinitely long pulses and an infinitely strong coupling . in this section , we use a more detailed model of the interaction of the light field with an atom in a cavity to investigate how long the input pulses need to be and how large the single atom cooperativity parameter should be to approximately achieve this limit . the backreflection of light in a cavity leads to a state dependent distortion of the shape of the mode function of the input field , and to study this effect in more detail we concentrate on a single cavity as shown in fig . [ cavity ] in the following . for simplicity , we assume the input field to be a continuous coherent state @xcite @xmath212|0\rangle\ ] ] with mode function @xmath213 , where @xmath214 is the expectation value of the total number of photons in the input beam . the light - atom interaction is governed by the hamiltonian @xmath215 and the decay term @xmath216 where @xmath8 is the light - atom coupling strength , @xmath217 is the annihilation operator of the cavity field , @xmath7 is the decay rate of the excited state of the atom due to spontaneous emission , and @xmath218 is the density operator representing the state of the atom and the cavity field . for @xmath219 , where @xmath220 denotes the trace , the population in the excited state of the atom is very small , and we may eliminate this state adiabatically . this reduces the effective light - atom interaction dynamics to a single decay term @xmath221 i.e. , the atom is equivalent to a beam splitter which reflects photons out of the cavity at the rate @xmath222 if the atom is in the state @xmath4 and does not affect the light field if the atom is in the state @xmath5 . assume the atom to be in the state @xmath223 , @xmath224 . since the input mirror of the cavity may be regarded as a beam splitter with high reflectivity , all components of the setup transform field operators linearly . for a coherent state input field , the cavity field and the output field are hence also coherent states . we may divide the time axis into small segments of width @xmath225 and approximate the integrals in by sums . the input state is then a direct product of single mode coherent states with amplitudes @xmath226 and annihilation operators @xmath227 , where @xmath228 , @xmath229 . in the following , we choose @xmath225 to be equal to the round trip time of light in the cavity , which requires @xmath230 to vary slowly on that time scale . denoting the coherent state amplitude of the cavity field at time @xmath231 by @xmath232 , we use the beam splitter transformation @xmath233=\bigg[\begin{array}{cc}t_c&-r_c\\r_c&t_c\end{array}\bigg ] \bigg[\textrm{in}\bigg]\end{aligned}\ ] ] for the input mirror of the cavity to derive @xmath234 where @xmath235 is the reflectivity of the input mirror of the cavity , @xmath236 is the transmissivity of the beam splitter , which models the loss due spontaneous emission , i.e. , @xmath237 , and @xmath238 denotes the output field from the cavity . we have here included an additional phase shift of @xmath3 per round trip in the cavity to ensure the input field to be on resonance with the cavity , i.e. , the second element of the input vector in is @xmath239 . taking the limit @xmath240 and @xmath241 for fixed cavity decay rate @xmath242 , and reduce to @xmath243 where @xmath244 is the single atom cooperativity parameter . according to the steady state solution of @xmath245 we need @xmath246 to efficiently expel the light field from the cavity for @xmath247 . we should also remember the criterion for the validity of the adiabatic elimination , which by use of takes the form @xmath248 i.e. , for @xmath246 the average flux of photons in the input beam should not significantly exceed the average number of photons emitted spontaneously per unit time from an atom , which has an average probability of one half to be in the excited state . solving , @xmath249 we obtain the output field @xmath250 with @xmath251 note that @xmath252 is not necessarily normalized due to the possibility of spontaneous emission . the ideal situation is @xmath253 and @xmath254 , and we would thus like the norm of @xmath255 to be as small as possible . for @xmath247 and @xmath246 , the exponential function in is practically zero unless @xmath256 , and as long as @xmath257 changes slowly on the time scale @xmath258 , we may take @xmath257 outside the integral to obtain @xmath259 and @xmath260 . as this result is independent of @xmath257 , a natural criterion for the optimal choice of input mode function is to minimize @xmath261 under the constraint @xmath262 . we also restrict @xmath257 to be zero everywhere outside the time interval @xmath263 $ ] . since we would like the double integral to be as close to 2 as possible , we should choose @xmath264 . a variational calculation then provides the optimal solution @xmath265\\ 0&\textrm{otherwise } \end{array } \right.,\ ] ] where @xmath266 @xmath267 and @xmath268 . deviation @xmath269 of the overlap between the output mode function and the input mode function from the ideal value when the atom is in the state @xmath223 . note that @xmath252 is defined such that the norm is less than unity if a loss occurs during the interaction . the single atom cooperativity parameter is assumed to be @xmath270 . solid lines are for the optimal input mode function given in , and dashed lines are for an input mode function , which is constant in the interval @xmath271 $ ] and zero otherwise . the inset illustrates these functions for @xmath272 . for @xmath273 , the shape of the output mode function is almost the same as the shape of the input mode function as long as @xmath274 , but the norm is slightly decreased due to spontaneous emission from the atom such that @xmath275 . this result is independent of the actual shape of the input mode function , and the solid and dashed lines for @xmath276 are hence indistinguishable in the figure . the results for @xmath277 are independent of @xmath278 , because the light field does not interact with the atom in this case . nonzero values of @xmath277 only occur due to distortion of the shape of the mode function . for @xmath279 , @xmath280 for the optimal input mode function and @xmath281 for the constant mode function . ] this solution gives @xmath282 for fixed @xmath283 , @xmath284 is a decreasing function of @xmath285 , and for @xmath286 , @xmath287 and @xmath280 . distortion of the input mode function can thus be avoided by choosing a sufficiently long input pulse . this may be understood by considering the fourier transform of the input mode function . for a very short pulse the frequency distribution is very broad and only a small part of the field is actually on resonance with the cavity , i.e. , most of the field is reflected without entering into the cavity . for a very long pulse , on the other hand , the frequency distribution is very narrow and centered at the resonance frequency of the cavity . @xmath288 is plotted in fig . [ optmode ] as a function of @xmath285 both for the optimal input mode function and an input mode function , which is constant in the interval @xmath263 $ ] . the optimal input mode function is seen to provide a significant improvement . in fact , for the constant input mode function @xmath289/(\kappa t)$ ] , and hence @xmath277 scales only as @xmath290 for large @xmath291 . finally , we note that @xmath292 , and @xmath279 thus also ensures @xmath293 , which justifies the approximation in eqs . [ gammaeq ] and [ inout ] . we would like to determine how a single unit of the setup transforms the state of the field , when we use the full multi - mode description . for this purpose it is simpler to work in frequency space , and we thus use the definition of the fourier transform @xmath294 on eqs . and to obtain @xmath295 where @xmath296 assume now that the density operator of the two - beam input field to the unit may be written on the form @xmath297 where @xmath17 and @xmath298 are summed over the same set of numbers and @xmath299 and @xmath300 are continuous coherent states in frequency space , i.e. , @xmath301|0\rangle\end{gathered}\ ] ] and similarly for @xmath300 . note that and are consistent when @xmath302 and @xmath303 as well as @xmath304 and @xmath305 are related through a fourier transform @xcite . after the first 50:50 beam splitter , the input state is transformed into @xmath306 and this is the input state to the two cavities . the initial state of the two atoms is @xmath307 . we saw in the last subsection that a cavity is equivalent to an infinite number of beam splitter operations applied to the cavity mode and the input field modes . to take the possibility of spontaneous emission into account , we also apply a beam splitter operation to the cavity mode and a vacuum mode in each time step and subsequently trace out the vacuum mode . as the beam splitters are unitary operators acting from the left and the right on the density operator , the field amplitudes are transformed according to as before , but when the ket and the bra are different , the trace operations lead to a scalar factor . since the reflectivity of the beam splitter modeling the loss is @xmath308 for an atom in the state @xmath223 , this factor is @xmath309\\ = \prod_{k=-\infty}^\infty\langle\sqrt{2c\kappa\tau}\gamma_{m , p}(k\tau)\delta_{p\uparrow}| \sqrt{2c\kappa\tau}\gamma_{n , i}(\kappa\tau)\delta_{i\uparrow}\rangle\\ = \exp\bigg\{-\int_{-\infty}^{\infty}\frac{c\kappa^2}{(1 + 2c)^2(\kappa/2)^2+\omega^2 } \big[|\alpha_{\textrm{in},n}(\omega)|^2\delta_{i\uparrow}\\ + |\alpha_{\textrm{in},m}(\omega)|^2\delta_{p\uparrow } -2\alpha_{\textrm{in},n}(\omega)\alpha^*_{\textrm{in},m } ( \omega)\delta_{i\uparrow}\delta_{p\uparrow}\big]d\omega\bigg\},\end{gathered}\ ] ] where @xmath310 is the amplitude of the cavity field corresponding to the input field @xmath311 and the atomic state @xmath312 as given in eq . . altogether , @xmath313 is thus transformed into @xmath314 \langle k_p(\omega)\alpha_{\textrm{in},m}(\omega)| \otimes|i\rangle\langle p|$ ] . projecting both atoms onto @xmath315 and taking the final 50:50 beam splitter into account , we obtain the output state after one unit @xmath316\\ \times d_{jq}[(\beta_n(\omega)-\alpha_n(\omega))/\sqrt{2},(\beta_m(\omega)-\alpha_m(\omega))/\sqrt{2}]\\ \times\left|\left\{\frac{1}{2}(k_i+k_j)\alpha_n(\omega ) + \frac{1}{2}(k_i - k_j)\beta_n(\omega)\right\}\right\rangle \left\langle\left\{\frac{1}{2}(k_p+k_q)\alpha_m(\omega ) + \frac{1}{2}(k_p - k_q)\beta_m(\omega)\right\}\right|\\ \otimes\left|\left\{\frac{1}{2}(k_i+k_j)\beta_n(\omega ) + \frac{1}{2}(k_i - k_j)\alpha_n(\omega)\right\}\right\rangle \left\langle\left\{\frac{1}{2}(k_p+k_q)\beta_m(\omega ) + \frac{1}{2}(k_p - k_q)\alpha_m(\omega)\right\}\right|,\end{gathered}\ ] ] where we have written @xmath317 for brevity . a subsequent phase shifter is easily taken into account by multiplying the coherent state amplitudes by the appropriate phase factors . we note that the result has the same form as the input state if we collect @xmath17 , @xmath124 , and @xmath318 into one index and @xmath298 , @xmath30 , and @xmath200 into another index . we can thus apply the transformation repeatedly to obtain the output state after @xmath45 units of the setup . ( color online ) fidelity as defined in as a function of the single atom cooperativity parameter @xmath278 for different values of the expectation value of the number of photons in one of the input modes and different numbers of units of the setup , assuming @xmath279 and @xmath319 . the dotted lines are the asymptotes for @xmath320 . ] if we assume @xmath302 and @xmath321 to have the same shape for all @xmath17 and @xmath298 , the output field simplifies to a two - mode state for @xmath279 as expected . this may be seen as follows . when @xmath285 is much larger than @xmath322 , the width of the distribution @xmath302 in frequency space is much smaller than @xmath283 . for the relevant frequencies we thus have @xmath323 , and in this case the right hand side of reduces to @xmath324 , i.e. , @xmath325 is independent of frequency , and the amplitudes of all the continuous coherent states of the output density operator are proportional to @xmath302 . in fig . [ finitec ] , we have used in the limit @xmath279 to compute the fidelity as a function of the single atom cooperativity parameter for @xmath319 . the ideal values for @xmath320 are also shown , and the deviations are seen to be very small for @xmath278 larger than about @xmath326 . current experiments have demonstrated single atom cooperativity parameters on the order of @xmath327 @xcite , and high fidelities can also be achieved for this value . when @xmath278 is finite , there is a relative photon loss of @xmath328 due to spontaneous emission each time the field interacts with a cavity containing an atom in the state @xmath4 . we note , however , that the setup is partially robust against such losses because the next unit of the setup removes all components of the state for which a single photon has been lost . the setup put forward and studied in this article acts in many respects like a photon number filter and has several attractive applications for quantum technologies . based on the ability to distinguish even and odd photon numbers using the interaction of the light field with a high finesse optical cavity , photonic two mode input states can be projected onto photon - number correlated states . naturally , this protocol is very well suited to detect losses and can in particular be adapted to purify photon - number entangled states in quantum communication . we studied deviations from ideal behavior such as finite length of the input pulses and limited coupling to estimate for which parameters the idealized description is valid . the setup can be modified such that it is capable to perform a quantum - non - demolition measurement of photon numbers in the optical regime . the non - destructive photon counting device completes the versatile toolbox provided by the proposed scheme . we acknowledge valuable discussions with ignacio cirac and stefan kuhr and support from the danish ministry of science , technology , and innovation , the elite network of bavaria ( enb ) project qccc , the eu projects scala , qap , compas , the dfg - forschungsgruppe 635 , and the dfg excellence cluster map . d. i. schuster , a. a. houck , j. a. schreier , a. wallraff , j. m. gambetta , a. blais , l. frunzio , j. majer , b. johnson , m. h. devoret , s. m. girvin , and r. j. schoelkopf , nature ( london ) * 445 * , 515 ( 2007 ) .

we propose and analyze a multi - functional setup consisting of high finesse optical cavities , beam splitters , and phase shifters . the basic scheme projects arbitrary photonic two - mode input states onto the subspace spanned by the product of fock states @xmath0 with @xmath1 . this protocol does not only provide the possibility to conditionally generate highly entangled photon number states as resource for quantum information protocols but also allows one to test and hence purify this type of quantum states in a communication scenario , which is of great practical importance . the scheme is especially attractive as a generalization to many modes allows for distribution and purification of entanglement in networks . in an alternative working mode , the setup allows of quantum non demolition number resolved photodetection in the optical domain .

1002.0127

the ads / cft correspondence @xcite has revealed the deep relations between gauge theories and string theories and has provided powerful tools for understanding the dynamics of strongly coupled field theories in the dual gravity side . in recent years , this paradigm has been applied to investigate the properties of certain condensed matter systems @xcite . the correspondence between gravity theories and condensed matter physics(sometimes is also named as ads / cmt correspondence ) has shed light on studying physics in the real world in the context of holography . it is well known that in realistic condensed matter systems , the presence of a finite density of charge carriers is of great importance . according to the ads / cft correspondence , the dual bulk gravitational background should be charged black holes in asymptotically ads spacetimes . the simplest example of such charged ads black holes is reissner - nordstrm - ads(rn - ads ) black hole , which has proven to be an efficient laboratory for studying the ads / cmt correspondence . for instance , investigations of the fermionic two - point functions in this background indicated the existence of fermionic quasi - particles with non - fermi liquid behavior @xcite , while the @xmath5 symmetry of the extremal rn - ads black hole is crucial to the emergent scaling symmetry at zero temperature @xcite . moreover , adding a charged scalar in such background leads to superconductivity @xcite . a further step towards a holographic model - building of strongly - coupled systems at finite charge density is to consider the leading relevant ( scalar ) operator in the field theory side , whose bulk gravity theory is an einstein - maxwell - dilaton system with a scalar potential . such theories at zero charge density were analyzed in detail in recent years as they mimic certain essential properties of qcd @xcite . solutions at finite charge density have been considered in @xcite in the context of ads / cmt correspondence . recently a general framework for the discussion of the holographic dynamics of einstein - maxwell - dilaton systems with a scalar potential was proposed in @xcite , which was a phenomenological approach based on the concept of effective holographic theory ( eht ) . the minimal set of bulk fields contains the metric @xmath6 , the gauge field @xmath7 and the scalar @xmath8 ( dual to the relevant operator ) . @xmath8 appears in two scalar functions that enter the effective action : the scalar potential and the non - minimal maxwell coupling . they studied thermodynamics of certain exact solutions and computed the dc and ac conductivity . the main advantage of this eht approach is that it permits a parametrization of large classes of ir dynamics and allows investigations on important observarables . however , it is not clear whether concrete ehts can be embedded into string theories . for subsequent generalizations see @xcite . on the other hand , strongly coupled quantum liquids play an important role in condensed matter physics , where quantum liquids mean translationally invariant systems at zero ( or low ) temperature and at finite density . by now there are two successful phenomenological theories of quantum liquids : landau s fermi - liquid theory and the theory of quantum bose liquids , describing two different behaviors of a quantum liquid at low momenta and temperatures . in particular , the specific heat of a bose liquid at low temperature is proportional to @xmath9 in @xmath10 spatial dimensions , while the specific heat of a fermi liquid scales as @xmath11 at low @xmath11 , irrespective of the spatial dimensions . one may wonder if the newly developed techniques in ads / cft correspondence can help us understand the behavior of quantum liquids . in @xcite the authors considered a class of gauge theories with fundamental fields whose holographic dual in the appropriate limit was given in terms of the dirac - born - infeld ( dbi ) action in ads space . they found that the specific heat@xmath12 in @xmath13 spatial dimensions at low temperature and the system supported a sound mode at zero temperature , which was called `` zero - temperature sound '' . one interesting feature was that the `` holographic zero sound '' mode was almost identical to the zero sound in fermi liquids : the real part of the dispersion relation was linear in momentum ( @xmath14 ) and the imaginary part had the same @xmath15 dependence predicted by landau . the crucial difference was that the zero - temperature sound velocity coincided with the first - sound velocity , while generically the two velocities are not equal for a fermi liquid . such analysis was performed in the case of massive charge carriers in @xcite and in the case of sakai - sugimoto model in @xcite . the specific heat of general @xmath16 systems was calculated in @xcite and the specific heat of lifshitz black holes was discussed in @xcite and @xcite , while the zero sound was also investigated in @xcite . in this paper we will study the low - temperature specific heat and the holographic zero sound in effective holographic theories . here the bulk effective theory is @xmath0-dimensional einstein gravity coupled to a maxwell term with non - minimal coupling and a scalar . it was found in @xcite that the theory admitted both extremal and near - extremal solutions with anisotropic scaling symmetry . we consider dynamics of probe d - branes in the above mentioned backgrounds and find that by appropriately fixing the parameters in the effective theory , the specific heat can be proportional to @xmath11 , resembling a fermi liquid . we also compute the current - current retarded green functions at low frequency and low momentum , and clarify the conditions when a quasi - particle excitation exists . moreover , we also explore the possibility of observing the existence of fermi surfaces in such a system by numerical methods . we find that although the system possesses some features of fermi liquids , such as linear specific heat and zero sound excitation , we do not observe any characteristic structure in the wide range of @xmath17 . in addition , the ac conductivity is also obtained as a by - product . the rest of the paper is organized as follows : the exact solutions of the effective bulk theory will be reviewed in section 2 and the thermodynamics of massless charge carriers will be discussed in section 3 . we shall calculate the correlation functions in section 4 and identify the quasi - particle behavior , while the existence of fermi surfaces will be explored in section 5 via numerics . we will calculate the ac conductivity in section 6 , including the zero density limit . finally we will give a summary and discuss future directions . in this section we will review the solutions obtained in @xcite , which can be seen as generalizations of the four - dimensional near - extremal scaling solution discussed in @xcite . in the beginning we consider the following action in @xmath0-dimensions , without any reference to string theory or m / theory origin nor specifying the forms of the gauge coupling @xmath18 and the scalar potential @xmath19 explicitly , @xmath20.\ ] ] the resulting solutions are charged dilaton black holes , which have been investigated in the literature for a long period @xcite . let us focus on solutions carrying electric charge only . the general configuration with planar symmetry can be written as follows @xmath21 after plugging in the scaling ansatz @xmath22 into the equations of motion , we can arrive at several constraints on the parameters and the scalar functions : * we require that @xmath23 , so that the extremal solutions have smooth connections to the finite temperature solutions ; * the field equations indicate that @xmath24 and @xmath25 and @xmath26 when the scale invariance is restored . * the equations of motion determine that the scalar field must take the form @xmath27 and both @xmath18 and @xmath19 are constrained to be exponential in @xmath8 , power law in @xmath28 . * once we have fixed @xmath23 , the metric must have @xmath29 , where saturation occurs for vanishing flux , e.g. in @xmath30 with @xmath31 . subsequently , according to the constraints discussed above , we take the following forms of @xmath18 and @xmath19 . @xmath32 then we will consider charged dilaton black holes with a liouville potential @xcite . now the scaling ansatz turns out to be @xmath33 since @xmath34 and @xmath35 can be eliminated by rescaling @xmath28 and @xmath36 , we shall set @xmath37 and @xmath38 . the remaining parameters can be explicitly given in terms of @xmath39 , @xmath40^{2}}{(d-2)(2+\alpha^{2}+\alpha\eta ) [ 2(d-2)+(d-1)\alpha^{2}-\eta^{2}(d-3)+2\alpha\eta]}.\end{aligned}\ ] ] it can be easily seen that there is no scale invariance in such backgrounds . we will restrict to @xmath41 and @xmath42 without loss of generality , which implies that @xmath8 must diverge to positive infinity at the horizon . in the specific limit @xmath43 , the scalar potential becomes constant and the scale invariance of the solutions can be restored : * when we set @xmath44 with vanishing flux , the scalar @xmath8 becomes constant and the resulting solution is @xmath30 ; * when we set @xmath45 with flux through @xmath46 , @xmath8 becomes constant and the resulting solution is @xmath47 ; * when we set @xmath48 arbitrary , @xmath25 with flux through @xmath46 , the scalar @xmath49 and the resulting solution is the modified lifshitz solution , whose dynamical exponent @xmath50 before coming to practical calculations we should determine the range of parameters . firstly , by requiring that @xmath51 for small @xmath28 and the flux to be real , one can impose the bound on @xmath48 in terms of fixed @xmath52 , @xmath53 secondly , when the flux is zero , @xmath54 we should require @xmath55 to ensure a well - defined boundary in the sense of ads / cft , @xmath56 combining constraint derived in the general background in the beginning of this section , we can obtain the following complete restrictions @xmath57 we will impose such constraints in the subsequent calculations . the near extremal solution can be obtained in a similar way , @xmath58 where @xmath59 and the other parameters and fields remain the same as the extremal solutions . one can easily get the temperature @xmath60 and the entropy density @xmath61 in this section we will investigate thermodynamics of massless charge carriers in the backgrounds reviewed in previous section . according to ads / cft , @xmath62 probe d - branes correspond to @xmath62 fields in the fundamental representation of the gauge group in the probe limit @xmath63 @xcite . an efficient method for evaluating the dc conductivity and dc hall conductivity of probe d - branes was proposed in @xcite and @xcite . moreover , a holographic model building approach to `` strange metallic '' phenomenology was initiated in @xcite , where the bulk spacetime was a lifshitz black hole and the charge carriers were described by d - branes . here we will consider probe d - branes as massless charge carriers and explore the thermodynamics in the near - extremal background . the dynamics of probe d - branes is described by the dirac - born - infeld ( dbi ) action @xmath64 where @xmath65 denotes the tension of d - branes , @xmath66 is the induced metric and @xmath67 is the @xmath68 field strength on the worldvolume . in the second line we set @xmath69 , where @xmath70 denotes volume of the internal space that the d - branes may be wrapping . furthermore , we assume that the d - branes are extended along @xmath71 spatial dimensions of the black hole solution . if @xmath72 , the fundamental fields are propagating along certain @xmath10-dimensional defect . we will introduce a nontrivial worldvolume gauge field @xmath73 and absorb the factor @xmath74 into @xmath67 . since we are not studying realistic string theories , the wess - zumino terms will be omitted in the following discussions . before proceeding we should make sure that the backreaction of the probe branes onto the background can be neglected . our discussion is along the line of @xcite . expanding the dbi action to quadric order of @xmath75 in the background , we can obtain @xmath76 to avoid backreaction of the probes on the background , the stress energy of the probes must be smaller than that generating the bulk spacetime . it can be easily seen that the stress energy of the original background@xmath77 , where @xmath78 denotes the planck length in @xmath0-dimensional spacetime and @xmath79 is the corresponding cosmological constant . therefore by varying the quadric action of the probes with respect to @xmath80 , we can arrive at the following condition @xmath81 one can see that as long as the effective tension @xmath82 is sufficiently small , the backreaction can be neglected . in this background configuration , after performing the trivial integrations on @xmath83 and dividing out the infinite volume of @xmath84 , we can obtain the action density . @xmath85 where @xmath86 and the prime denotes derivative with respect to @xmath28 . then the charge density is given by @xmath87 we can also solve for @xmath88 @xmath89 by plugging in the solution for @xmath88 , we can find the on - shell action density @xmath90 following the methods used in @xcite , some interesting physical quantities like the chemical potential and the free energy , can be evaluated analytically . we will see that this is still the case for our background . the chemical potential is given by @xmath91 where @xmath92 notice that in order to obtain the above results , we have made use of the following useful formulae for beta function and incomplete beta function @xmath93 as well as for hypergeometric function @xmath94 after choosing the grand - canonical ensemble , the free energy density is given by @xmath95 where @xmath96 moreover , other thermodynamic quantities can also be calculated from the thermodynamic relations . the charge density can be written as @xmath97 which is consistent with previous result . the entropy density is given by @xmath98 notice that when @xmath26 , there exists a nontrivial contribution to the entropy density at @xmath99 like those observed in @xcite and @xcite . on the other hand , the entropy density is vanishing at extremality as long as @xmath100 . the specific heat is @xmath101 it is well known that for a gas of free bosons in @xmath10 spatial dimensions , the specific heat at low temperature is proportional to @xmath102 , while for a gas of fermions the low temperature specific heat is proportional to @xmath11 , irrespective of @xmath13 . when @xmath100 and @xmath82 is sufficiently small , the first term dominates . one can easily obtain @xmath103 when the specific heat is proportional to @xmath11 . then the parameter @xmath48 can be expressed in terms of @xmath52 @xmath104},\ ] ] combining with ( [ res ] ) , we can arrive at the following conclusions : * when @xmath105 both @xmath106 and @xmath107 are permitted solutions ; * when @xmath108 only @xmath107 is a permitted solution ; * when @xmath109 there is no solution , which means that we can not realize @xmath110 in this regime . when @xmath111 , the second term provides the only contribution . the linear dependence on @xmath11 fixes @xmath112 . note that in this limit , @xmath113 . so @xmath114 by taking into account of ( [ res ] ) , we can find that only @xmath115 is permitted . we can also formally evaluate the `` speed of sound '' . in grand - canonical ensemble , the pressure is given by @xmath116 while the energy density is @xmath117 therefore , @xmath118 however , it was emphasized in @xcite that this quantity is only the speed of normal / first sound in the relativistic case @xmath119 . actually the speed of normal sound is dimensionful in a system with @xmath120 . we will calculate the holographic zero sound in the next section . in this section we will calculate the holographic zero sound in the anisotropic background at extremality . the basic strategy is to consider fluctuations of the worldvolume gauge field on the probe d - branes in the background with nontrivial @xmath121 . such analysis was performed for @xmath122 background in @xcite and for lifshitz background in @xcite . we will calculate the holographic zero sound in a similar way and classify the behavior of the zero sound in different parameter ranges . zero sound should appear as a pole in the density - density retarded two - point function @xmath123 at extremality @xcite . in @xcite the authors provided a general framework for evaluating the corresponding retarded green s functions with background metric @xmath124 here we will take the nontrivial dilaton into account . the symmetries in the spatial directions allow us to consider fluctuations of the gauge fields with the following form @xmath125 where @xmath126 denotes one of the spatial directions . the quadratic action for the fluctuations is given by @xmath127,\ ] ] where @xmath128 . note that we are working in the gauge of @xmath129 . after performing the fourier transform @xmath130 the linearized equations of motion can be written as @xmath131-\frac{e^{-\phi}g^{q/2 - 1}_{xx}g_{rr } } { \sqrt{|g_{tt}|g_{rr}-a^{\prime2}_{t}}}(k^{2}a_{t}+\omega ka_{x})=0,\ ] ] @xmath132+\frac{e^{-\phi}g^{q/2 - 1}_{xx}g_{rr } } { \sqrt{|g_{tt}|g_{rr}-a^{\prime2}_{t}}}(\omega^{2}a_{x}+\omega ka_{t})=0.\ ] ] in addition , the following constraint can be obtained by writing @xmath133 s equation of motion in @xmath129 gauge @xmath134 the above equations are not independent , as we can obtain the equation of motion for @xmath135 by combining the constraint equation and the equation of motion for @xmath136 . therefore it is sufficient to solve the constraint and the equation for @xmath135 only . by introducing the gauge - invariant electric field @xmath137 we can obtain the equation of motion for @xmath138 @xmath139e^{\prime}-\frac{g_{rr}}{|g_{tt}|}(u^{2}k^{2}-\omega^{2})e=0,\ ] ] where @xmath140 moreover , the quadratic action can also be expressed in terms of @xmath138 , @xmath141,\ ] ] for our specific background , the metric can be rewritten in terms of the new radial coordinate @xmath142 as follows @xmath143 in the new coordinate system , the solution of the worldvolume gauge field and the function @xmath144 are given by @xmath145 where dot denotes derivative with respect to @xmath146 . integrating the quadratic action by parts @xmath147 introducing a cutoff at @xmath148 and taking the limit @xmath149 , the quadratic action turns out to be @xmath150 after imposing the incoming boundary condition at the `` horizon '' @xmath151 and plugging in the solutions of @xmath152 , the retarded correlation function reads @xcite @xmath153 by defining @xmath154 the retarded correlation functions can be written in terms of @xmath155 @xmath156 in order to evaluate the retarded correlation functions , we should try to solve ( [ 4eq6 ] ) , whose analytic solutions are always difficult to find . we will leave the numerical work to section 5 , while here we will obtain the low - frequency behavior of @xmath157 by solving ( [ 4eq6 ] ) in different limits and matching the two solutions in an overlapped regime , following the spirit of @xcite and @xcite . to be concrete , we will solve ( [ 4eq6 ] ) in the limit of large @xmath146 and then expand the solution in the small frequency and momentum limit . next we will take the small frequency and momentum limit first and then perform the large @xmath146 expansion . the integration constants can be fixed by matching the two solutions . first let us take @xmath158 , which leads to the following equation for @xmath138 @xmath159 the solution can be given in terms of a hankel function of the first kind , @xmath160 in the limit of small frequency with @xmath161 , the asymptotic expansion reads @xmath162 it should be pointed out that the case of @xmath163 must be treated separately . in this case the corresponding parameter is given by @xmath164 now the expansion contains a logarithmic term @xmath165 where @xmath166 is the euler constant . next we take @xmath167 with @xmath168 being fixed . then the last term in ( [ 4eq6 ] ) can be neglected and the equation of @xmath138 becomes @xmath169\dot{e}=0.\ ] ] when @xmath170 and @xmath171 , the solution is given by @xmath172,\end{aligned}\ ] ] where @xmath173 . for either @xmath170 or @xmath171 , the powers of @xmath146 do not match , which will be displayed in appendix a. we will make use of the following useful formulae for the asymptotic expansion @xmath174 @xmath175 therefore the large @xmath146 limit is given as follows when @xmath176 , @xmath177.\ ] ] when @xmath178 the expansion also contains a logarithmic term @xmath179\nonumber\\ & \simeq&d_{1}+d_{2}[\frac{c_{1}\mu_{0}k^{2}}{md}-\frac{\omega^{2}}{md}\log 2d + \frac{\omega^{2}}{(\beta-1)d}\log\omega-\frac{\omega^{2}}{(\beta-1)d}\log(\omega z^{\beta-1})].\end{aligned}\ ] ] to evaluate the retarded correlation functions , we need small @xmath146 expansion of the solution . it can be seen that the second term in the expansion always tends to zero more rapidly , being irrespective of @xmath178 or not . therefore we have @xmath180 assuming that @xmath181 can be dealt with in a similar fashion , see footnote 7 of @xcite . ] , the leading order behavior of @xmath138 reads @xmath182 , so the quadratic action turns out to be @xmath183 thus @xmath184 the relation between the integration constants @xmath185 and @xmath186 can be obtained by matching the expansions of the solutions in different limits and eliminating the other integration constant @xmath187 . finally we summarize our result for @xmath155 @xmath188 when @xmath176 the parameters are given by @xmath189 when @xmath178 the parameters are given by @xmath190 the dispersion relation of the holographic sound mode is given by setting the denominator of @xmath155 to vanish . similar to the situations discussed in @xcite , the value of @xmath191 determines which term in the denominator dominates . it can be seen that when @xmath192 , the @xmath193 term dominates . then we can expand @xmath194 @xmath195,\ ] ] and then invert to find @xmath196 notice that when @xmath192 , @xmath197 , so at low momentum @xmath198 , that is , the real part is bigger than the imaginary part . therefore this mode describes a quasi - particle excitation . as a check of consistency , we can take the specific limit @xmath199 , which just gives the @xmath30 background . one can obtain @xmath200 which agrees with @xcite . the speed of the holographic zero sound is given by @xmath201 it can be seen that in the relativistic case @xmath31 , the speed of zero sound coincides with the speed of normal / first sound . one special case is @xmath202 , where all the @xmath203 functions cancel and the speed of zero sound reads @xmath204 which turns out to be finite as @xmath205 . when @xmath206 as well as @xmath207 has a pole as @xmath205 , so @xmath208 goes to zero from above . when @xmath209 dominates , then @xmath210 inverting the relation above , @xmath211 notice that since @xmath212 is real and @xmath213 is complex , the leading term has a complex coefficient . furthermore , the real and imaginary parts are of the same order , hence this mode is not a quasi - particle . finally when @xmath178 , @xmath214 expanding for small @xmath215 , @xmath216 it can be seen that the dispersion relation differs from the holographic zero sound mode by logarithmic terms . in previous section we observed a sound - like excitation in the regime of @xmath192 . it is known that such zero sound mode is associated with the deformation of the fermi surface away from the spherical shape . the theory of normal fermi liquids tells us that the jump in the distribution function can be observed as a singularity in the retarded current - current green s function in the @xmath217 limit . to see if we can observe such fermi surface we need the complete solution to the equation for the gauge fluctuations ( [ 4eq6 ] ) with @xmath217 , at least numerically . we investigated thermodynamics of probe d - branes in section 3 , where we found that when @xmath100 , the specific heat was proportional to the temperature @xmath11 under certain conditions , which is just the behavior of fermi liquids . as we have many groups of parameters @xmath218 which lead to specific heat linear in @xmath11 , we choose the following parameters @xmath219,@xmath220 in @xmath1 dimensional spacetime as an example . for simplicity we also fix @xmath221 and @xmath222 . then the equation for the gauge fluctuations ( [ 4eq6 ] ) in the @xmath217 limit is reduced to @xmath223 in the near horizon region @xmath224 , the perturbative solution of the above equation is given by @xmath225 here , we choose the boundary condition @xmath226 at the horizon , which is compatible to impose the incoming boundary condition for @xmath227 . so we set @xmath228 . at the boundary @xmath229 , the asymptotic solution of ( [ eq : red ] ) becomes @xmath230 notice that since the first term is finite at the boundary , it corresponds to the source term and the coefficient of the second term implies the vev of the dual gauge operator . then , the green function is proportional to @xmath231 . in figure 1 , we plot the dependence of the green s function on the momentum @xmath17 for @xmath232 . .,scaledwidth=50.0% ] from the figure it can be easily seen that the characteristic structure does not appear in a wide range of @xmath17 , while the specific heat and zero sound exhibit typical features of fermi liquids . such phenomenon was also observed in @xcite . it was pointed out in @xcite that one difficulty in applying landau s theory of fermi liquids was the assumption that the particle number should be conserved as the strength of the interaction was varied , which was not obvious for the case they discussed . their conclusion seems to be still applicable to the present case . in this section we calculate the ac conductivity by making use of the correlation functions , which can be seen as a by - product of section 4 . furthermore , we will take the limit of zero density and compare the results with those appeared in previous literatures . it can be easily seen that the current - current correlation function is given by @xmath233 therefore we can obtain the following expressions in the small frequency limit , @xmath234 @xmath235 @xmath236 recalling that the definition of ac conductivity is given by @xmath237 we can arrive at the following results @xmath238 @xmath239 @xmath240 as a check of consistency , we consider the specific limit , @xmath241 . when @xmath242 , @xmath243 , which agrees with the result obtained in @xcite . the behavior of the ac conductivity is qualitatively similar to that investigated in @xcite . to be concrete , when @xmath176 @xmath244 is real while @xmath213 is complex . the conductivity is purely imaginary when @xmath192 and has a simple pole at zero frequency . then according to kramers - kronig relation , one can conclude that the real part of the conductivity , and hence the spectral function , consists of a delta function at zero frequency . when @xmath245 the conductivity , and hence the spectral function has a power - law dependence , and no explicit quasi - particle excitation exists . on the other hand , it was observed in @xcite that after transforming the equation of @xmath136 into a schrdinger - like form , the ac conductivity was directly related to the reflection amplitude for scattering off the potential . for our system , it can be observed that the equation for @xmath136 at @xmath246 becomes @xmath132+\frac{e^{-\phi}g^{q/2 - 1}_{xx}g_{rr } } { \sqrt{|g_{tt}|g_{rr}-a^{\prime2}_{t}}}\omega^{2}a_{x}=0\ ] ] in the original coordinate , @xmath247 the asymptotic behavior of @xmath136 is given by @xmath248 where we have introduced a background electric field @xmath249 , the above equation can be put in a schrdinger - like form , @xmath250 where @xmath251 and @xmath252.\ ] ] generally speaking , the ac conductivity can be obtained by numerical methods . however , in the zero density limit @xmath253 one can find @xmath254 so the potential possesses the same form as those investigated in @xcite and @xcite . assuming that the solution asymptotes to @xmath30 , we can obtain @xmath255 following their approach . in particular , when @xmath256 , the potential vanishes and the conductivity is constant at all temperatures @xmath257 which agrees with the analysis performed in @xcite . in this paper we explore the zero sound in @xmath0-dimensional effective holographic theories , whose bulk fields include the graviton @xmath6 , the @xmath68 gauge field @xmath7 and the scalar field @xmath8 . the solutions possess anisotropic scaling symmetry and they reduce to previously known examples , such as @xmath30 , @xmath47 and `` modified '' lifshitz solutions under certain conditions . we consider thermodynamics of massless probe d - branes in the near - extremal background and clarify the conditions under which the specific heat is linear in the temperature , which is a characteristic feature of fermi liquids . subsequently we study the zero sound mode by considering the fluctuations of the worldvolume gauge fields on the probe d - branes . rather than analytically solving the equations of motion , we obtain the low - frequency behavior by solving the equation in two different limits and then matching the two solutions in a regime where the limits overlap , following @xcite and @xcite . the resulting behavior of the zero sound looks similar to that investigated in @xcite , that is , when the parameter @xmath258 , the dispersion relation reveals a quasi - particle excitation ; while the zero sound is not a well - defined quasi - particle when @xmath259 . furthermore , we plot the correlation function in @xmath1 at @xmath217 with fixed parameters which lead to linear specific heat . the result is that we can not observe any characteristic structure of fermi liquids in a wide range of @xmath17 , which is similar to what was found in @xcite . as a by - product , we also evaluate the ac conductivity via the current - current correlation function , which reduces to previously known results at specific limits . by now there are mainly two approaches for studying condensed matter physics in the context of holography . one approach can be thought of as `` top - down '' , that is , we consider certain exact solutions or brane configurations in string / m theory which possess the desired properties of condensed matter systems . the main advantage is that we have clear understanding about the dual field theories , while it is difficult to find such solutions in string / m theory . a complementary approach can be seen as `` bottom - up '' , which means that we consider certain toy models of gravity which possess solutions with the desired properties . it allows a parametrization of large classes of ir dynamics and provides useful information in the dual field theory side . however , the main disadvantage is that the embeddings of such toy models into string / m theory are not obvious , thus many things in the field theory side remain unknown . however , the `` bottom - up '' approach is an efficient tool for investigating the ads / cmt correspondence . the background solutions we studied in this paper are exact solutions of theories with domain wall vacua . in @xcite the author also constructed interpolates between two exact solutions of the single- exponential , domain wall gravity theory , which lead to the argument that the domain wall / qft correspondence @xcite can be considered as an effective holographic tool which is applicable in settings beyond the regime of domain wall supergravities . therefore the domain wall / qft correspondence can also be taken as one specific class of effective holographic theories . moreover , the author of @xcite argued that even when the uv completion of some bulk theory was unknown , if the theory admitted an approximate domain wall solution at some intermediate value of @xmath28 then one could use domain wall / qft correspondence to develop a holographic map . thus it is interesting to develop the ads / cmt holography by making use of this domain wall / qft correspondence . in particular , we can establish precision holography in this anisotropic background along the line of @xcite and study the fermionic correlation functions following @xcite . we leave such fascinating projects in the future . * acknowledgments * this work was supported by the national research foundation of korea(nrf ) grant funded by the korea government(mest ) through the center for quantum spacetime(cquest ) of sogang university with grant number 2005 - 0049409 . in this appendix we show that when the parameter @xmath261 takes some specific values , the large @xmath146 expansions of the solutions @xmath260 to equation ( [ small ] ) can not match that of ( [ daz ] ) . firstly , when @xmath262 but @xmath263 , the solution to ( [ small ] ) is given by @xmath264.\end{aligned}\ ] ] when performing the expansion , we just focus on the powers of @xmath146 . the hypergeometric function gives @xmath265 which does not match that of ( [ daz ] ) . secondly , when @xmath170 but @xmath266 , the solution is @xmath267.\end{aligned}\ ] ] the @xmath193 term gives @xmath268 which does not match that of ( [ small ] ) either . therefore we just considered the cases with @xmath170 and @xmath171 in the main text . j. m. maldacena , `` the large n limit of superconformal field theories and supergravity , '' adv . * 2 * , 231 ( 1998 ) [ int . j. theor . * 38 * , 1113 ( 1999 ) ] [ arxiv : hep - th/9711200 ] . + s. s. gubser , i. r. klebanov and a. m. polyakov , `` gauge theory correlators from non - critical string theory , '' phys . b * 428 * , 105 ( 1998 ) [ arxiv : hep - th/9802109 ] . + e. witten , `` anti - de sitter space and holography , '' adv . theor . math . * 2 * , 253 ( 1998 ) [ arxiv : hep - th/9802150 ] . s. a. hartnoll , `` lectures on holographic methods for condensed matter physics , '' class . * 26 * , 224002 ( 2009 ) [ arxiv:0903.3246 [ hep - th ] ] . + c. p. herzog , `` lectures on holographic superfluidity and superconductivity , '' j. phys . a * 42 * , 343001 ( 2009 ) [ arxiv:0904.1975 [ hep - th ] ] . + j. mcgreevy , `` holographic duality with a view toward many - body physics , '' arxiv:0909.0518 [ hep - th ] . + g. t. horowitz , `` introduction to holographic superconductors , '' arxiv:1002.1722 [ hep - th ] . + s. sachdev , `` condensed matter and ads / cft , '' arxiv:1002.2947 [ hep - th ] . s. s. gubser , `` breaking an abelian gauge symmetry near a black hole horizon , '' phys . d * 78 * , 065034 ( 2008 ) [ arxiv:0801.2977 [ hep - th ] ] . s. a. hartnoll , c. p. herzog and g. t. horowitz , `` building a holographic superconductor , '' phys . * 101 * , 031601 ( 2008 ) [ arxiv:0803.3295 [ hep - th ] ] . s. a. hartnoll , c. p. herzog and g. t. horowitz , `` holographic superconductors , '' jhep * 0812 * , 015 ( 2008 ) [ arxiv:0810.1563 [ hep - th ] ] . s. s. gubser , a. nellore , s. s. pufu and f. d. rocha , `` thermodynamics and bulk viscosity of approximate black hole duals to finite temperature quantum chromodynamics , '' phys . rev . lett . * 101 * , 131601 ( 2008 ) [ arxiv:0804.1950 [ hep - th ] ] . k. goldstein , s. kachru , s. prakash and s. p. trivedi , `` holography of charged dilaton black holes , '' jhep * 1008 * , 078 ( 2010 ) arxiv:0911.3586 [ hep - th ] . j. gauntlett , j. sonner and t. wiseman , `` quantum criticality and holographic superconductors in m - theory , '' jhep * 1002 * , 060 ( 2010 ) [ arxiv:0912.0512 [ hep - th ] ] . m. cadoni , g. dappollonio and p. pani , `` phase transitions between reissner - nordstrom and dilatonic black holes in 4d ads spacetime , '' jhep * 1003 * , 100 ( 2010 ) arxiv:0912.3520 [ hep - th ] . c. m. chen and d. w. pang , `` holography of charged dilaton black holes in general dimensions , '' jhep * 1006 * , 093 ( 2010 ) arxiv:1003.5064 [ hep - th ] . k. goldstein , n. iizuka , s. kachru , s. prakash , s. p. trivedi and a. westphal , `` holography of dyonic dilaton black branes , '' arxiv:1007.2490 [ hep - th ] . b. h. lee , d. w. pang and c. park , `` strange metallic behavior in anisotropic background , '' jhep * 1007 * , 057 ( 2010 ) [ arxiv:1006.1719 [ hep - th ] ] . y. liu and y. w. sun , `` holographic superconductors from einstein - maxwell - dilaton gravity , '' jhep * 1007 * , 099 ( 2010 ) [ arxiv:1006.2726 [ hep - th ] ] . u. gursoy , `` gravity / spin - model correspondence and holographic superfluids , '' arxiv:1007.4854 [ hep - th ] . t. faulkner , g. t. horowitz and m. m. roberts , `` holographic quantum criticality from multi - trace deformations , '' arxiv:1008.1581 [ hep - th ] . a. bayntun , c. p. burgess , b. p. dolan and s. s. lee , `` ads / qhe : towards a holographic description of quantum hall experiments , '' arxiv:1008.1917 [ hep - th ] . m. ali - akbari and k. b. fadafan , `` conductivity at finite t hooft coupling from ads / cft , '' arxiv:1008.2430 [ hep - th ] . d. astefanesei , n. banerjee and s. dutta , `` moduli and electromagnetic black brane holography , '' arxiv:1008.3852 [ hep - th ] . a. karch , d. t. son and a. o. starinets , `` zero sound from holography , '' arxiv:0806.3796 [ hep - th ] . m. kulaxizi and a. parnachev , `` comments on fermi liquid from holography , '' phys . d * 78 * , 086004 ( 2008 ) [ arxiv:0808.3953 [ hep - th ] ] . m. kulaxizi and a. parnachev , `` holographic responses of fermion matter , '' nucl . b * 815 * , 125 ( 2009 ) [ arxiv:0811.2262 [ hep - th ] ] . a. karch , m. kulaxizi and a. parnachev , `` notes on properties of holographic matter , '' jhep * 0911 * , 017 ( 2009 ) [ arxiv:0908.3493 [ hep - th ] ] . b. h. lee and d. w. pang , `` notes on properties of holographic strange metals , '' arxiv:1006.4915 [ hep - th ] . c. hoyos - badajoz , a. obannon and j. m. s. wu , `` zero sound in strange metallic holography , '' arxiv:1007.0590 [ hep - th ] . k. c. k. chan , j. h. horne and r. b. mann , `` charged dilaton black holes with unusual asymptotics , '' nucl . b * 447 * , 441 ( 1995 ) [ arxiv : gr - qc/9502042 ] . + r. g. cai and y. z. zhang , `` black plane solutions in four - dimensional spacetimes , '' phys . d * 54 * , 4891 ( 1996 ) [ arxiv : gr - qc/9609065 ] . + r. g. cai , j. y. ji and k. s. soh , `` topological dilaton black holes , '' phys . d * 57 * , 6547 ( 1998 ) [ arxiv : gr - qc/9708063 ] . + c. charmousis , b. gouteraux and j. soda , `` einstein - maxwell - dilaton theories with a liouville potential , '' phys . rev . d * 80 * , 024028 ( 2009 ) [ arxiv:0905.3337 [ gr - qc ] ] . s. kachru , x. liu and m. mulligan , `` gravity duals of lifshitz - like fixed points , '' phys . d * 78 * , 106005 ( 2008 ) [ arxiv:0808.1725 [ hep - th ] ] . m. taylor , `` non - relativistic holography , '' arxiv:0812.0530 [ hep - th ] . a. karch and e. katz , `` adding flavor to ads / cft , '' jhep * 0206 * , 043 ( 2002 ) [ arxiv : hep - th/0205236 ] . s. kobayashi , d. mateos , s. matsuura , r. c. myers and r. m. thomson , `` holographic phase transitions at finite baryon density , '' jhep * 0702 * , 016 ( 2007 ) [ arxiv : hep - th/0611099 ] . a. karch and a. obannon , `` metallic ads / cft , '' jhep * 0709 * , 024 ( 2007 ) [ arxiv:0705.3870 [ hep - th ] ] . s. a. hartnoll , j. polchinski , e. silverstein and d. tong , `` towards strange metallic holography , '' jhep * 1004 * , 120 ( 2010 ) [ arxiv:0912.1061 [ hep - th ] ] . a. karch and a. obannon , `` holographic thermodynamics at finite baryon density : some exact results , '' jhep * 0711 * , 074 ( 2007 ) [ arxiv:0709.0570 [ hep - th ] ] . d. t. son and a. o. starinets , `` minkowski - space correlators in ads / cft correspondence : recipe and applications , '' jhep * 0209 * , 042 ( 2002 ) [ arxiv : hep - th/0205051 ] . g. t. horowitz and m. m. roberts , `` zero temperature limit of holographic superconductors , '' jhep * 0911 * , 015 ( 2009 ) [ arxiv:0908.3677 [ hep - th ] ] . c. p. herzog , p. kovtun , s. sachdev and d. t. son , `` quantum critical transport , duality , and m - theory , '' phys . d * 75 * , 085020 ( 2007 ) [ arxiv : hep - th/0701036 ] . h. j. boonstra , k. skenderis and p. k. townsend , `` the domain wall / qft correspondence , '' jhep * 9901 * , 003 ( 1999 ) [ arxiv : hep - th/9807137 ] . i. kanitscheider , k. skenderis and m. taylor , `` precision holography for non - conformal branes , '' jhep * 0809 * , 094 ( 2008 ) [ arxiv:0807.3324 [ hep - th ] ] . s. s. gubser , f. d. rocha and p. talavera , `` normalizable fermion modes in a holographic superconductor , '' arxiv:0911.3632 [ hep - th ] . t. faulkner , g. t. horowitz , j. mcgreevy , m. m. roberts and d. vegh , `` photoemission experiments on holographic superconductors , '' jhep * 1003 * , 121 ( 2010 ) [ arxiv:0911.3402 [ hep - th ] ] . t. faulkner , n. iqbal , h. liu , j. mcgreevy and d. vegh , `` from black holes to strange metals , '' arxiv:1003.1728 [ hep - th ] .

we investigate zero sound in @xmath0-dimensional effective holographic theories , whose action is given by einstein - maxwell - dilaton terms . the bulk spacetimes include both zero temperature backgrounds with anisotropic scaling symmetry and their near - extremal counterparts obtained in 1006.2124 [ hep - th ] , while the massless charge carriers are described by probe d - branes . we discuss thermodynamics of the probe d - branes analytically . in particular , we clarify the conditions under which the specific heat is linear in the temperature , which is a characteristic feature of fermi liquids . we also compute the retarded green s functions in the limit of low frequency and low momentum and find quasi - particle excitations in certain regime of the parameters . the retarded green s functions are plotted at specific values of parameters in @xmath1 , where the specific heat is linear in the temperature and the quasi - particle excitation exists . we also calculate the ac conductivity in @xmath0-dimensions as a by - product . cquest-2010 - 0395 * zero sound in effective holographic theories * bum - hoon lee@xmath2 , da - wei pang@xmath3 and chanyong park@xmath3 + _ seoul 121 - 742 , korea + _ _ @xmath4 center for quantum spacetime , sogang university _ + _ seoul 121 - 742 , korea + _ [email protected] , [email protected] , [email protected]

1009.3966

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 ) ; m. alford , k. rajagopal , and f. wilczek , phys . b * 422 * , 247 ( 1998 ) . m. g. alford , k. rajagopal and f. wilczek , nucl . * b537 * , 443 ( 1999 ) . d. t. son , phys . rev . d * 59 * , 094019 ( 1999 ) ; d. k. hong , v. a. miransky , i. a. shovkovy , and l. c. r. wijewardhana , phys . d * 61 * , 056001 ( 2000 ) ; d. k. hong , nucl . * b582 * , 451 ( 2000 ) ; t. schfer and f. wilczek , phys . d * 60 * , 114033 ( 1999 ) ; r.d . pisarski and d.h . rischke , phys . d * 61 * , 051501 ( 2000 ) ; s. d. h. hsu and m. schwetz , nucl . phys . * b572 * , 211 ( 2000 ) ; w. e. brown , j. t. liu , and h .- c . ren , phys . d * 61 * , 114012 ( 2000 ) . t. schfer , nucl . * b575 * , 269 ( 2000 ) ; i. a. shovkovy and l. c. r. wijewardhana , phys . b * 470 * , 189 ( 1999 ) . k. rajagopal and f. wilczek , hep - ph/0011333 ; d. k. hong , acta phys . b * 32 * , 1253 ( 2001 ) ; m. alford , ann . nucl . part . sci . * 51 * , 131 ( 2001 ) ; t. schfer , hep - ph/0304281 ; d. h. rischke , prog . part . phys . * 52 * , 197 ( 2004 ) ; m. buballa , phys . rept . * 407 * , 205 ( 2005 ) ; h .- c . ren , hep - ph/0404074 ; m. huang , int . j. mod . e * 14 * , 675 ( 2005 ) ; i. a. shovkovy , found . phys . * 35 * , 1309 ( 2005 ) . m. alford and k. rajagopal , jhep * 0206 * , 031 ( 2002 ) . a.w . steiner , s. reddy and m. prakash , phys . d * 66 * , 094007 ( 2002 ) . m. huang , p. f. zhuang and w. q. chao , phys . d * 67 * , 065015 ( 2003 ) . f. bedaque , nucl . phys . a * 697 * , 569 ( 2002 ) ; o. kiriyama , s. yasui and h. toki , int . j. mod . phys . e * 10 * , 501 ( 2001 ) . a. i. larkin and yu . n. ovchinnikov , sov . jetp * 20 * , 762 ( 1965 ) ; p. fulde and r. a. ferrell , phys . rev . * 135 * , a550 ( 1964 ) . m. g. alford , j. a. bowers and k. rajagopal , phys . d * 63 * , 074016 ( 2001 ) ; d. k. hong and y. j. sohn , hep - ph/0107003 ; i. giannakis , j. t. liu and h. c. ren , phys . d * 66 * , 031501 ( 2002 ) ; r. casalbuoni and g. nardulli , rev . phys . * 76 * , 263 ( 2004 ) ; j. a. bowers , hep - ph/0305301 . m. g. alford , k. rajagopal , s. reddy and f. wilczek , phys . d * 64 * , 074017 ( 2001 ) f. neumann , m. buballa , and m. oertel , nucl . * a714 * , 481 ( 2003 ) ; i. shovkovy , m. hanauske and m. huang , phys . d * 67 * , 103004 ( 2003 ) ; h. grigorian , d. blaschke and d. n. aguilera , phys . c * 69 * , 065802 ( 2004 ) ; s. reddy and g. rupak , phys . c * 71 * , 025201 ( 2005 ) . i. shovkovy and m. huang , phys . b * 564 * , 205 ( 2003 ) . m. alford , c. kouvaris and k. rajagopal , phys . lett . * 92 * , 222001 ( 2004 ) ; phys . d * 71 * , 054009 ( 2005 ) . g. sarma , j. phys . solids * 24 * , 1029 ( 1963 ) . m. g. alford , j. berges and k. rajagopal , phys . lett . * 84 * , 598 ( 2000 ) ; e. gubankova , w. v. liu and f. wilczek , phys . lett . * 91 * , 032001 ( 2003 ) ; a. mishra and h. mishra , phys . d * 69 * , 014014 ( 2004 ) ; s. b. rster and d. h. rischke , phys . rev . d * 69 * , 045011 ( 2004 ) . m. huang and i. shovkovy , nucl . * a729 * , 835 ( 2003 ) ; j. f. liao and p. f. zhuang , phys . d * 68 * , 114016 ( 2003 ) . k. fukushima , c. kouvaris and k. rajagopal , phys . d * 71 * , 034002 ( 2005 ) . s. b. rster , i. a. shovkovy and d. h. rischke , nucl . * a743 * , 127 ( 2004 ) ; h. abuki , m. kitazawa and t. kunihiro , phys . b * 615 * , 102 ( 2005 ) ; s. b. ruster , v. werth , m. buballa , i. a. shovkovy and d. h. rischke , phys . d * 72 * , 034004 ( 2005 ) ; d. blaschke , s. fredriksson , h. grigorian , a. m. oztas and f. sandin , phys . d * 72 * , 065020 ( 2005 ) ; h. abuki and t. kunihiro , hep - ph/0509172 . w. meissner and r. ochsenfeld , naturwiss . * 21 * , 787 ( 1933 ) . d. h. rischke , phys . d * 62 * , 034007 ( 2000 ) ; d. h. rischke , phys . d * 62 * , 054017 ( 2000 ) . m. huang and i. a. shovkovy , phys . d * 70 * , 051501 ( 2004 ) . m. huang and i. a. shovkovy , phys . d * 70 * , 094030 ( 2004 ) . r. casalbuoni , r. gatto , m. mannarelli , g. nardulli and m. ruggieri , phys . b * 605 * , 362 ( 2005 ) ; m. alford and q. h. wang , hep - ph/0501078 . s .- wu and s. yip , phys . a * 67 * , 053603 ( 2003 ) . w. v. liu and f. wilczek , phys . lett . * 90 * , 047002 ( 2003 ) . k. iida , t. matsuura , m. tachibana and t. hatsuda , phys . d * 71 * , 054003 ( 2005 ) . pao , s - t . wu , and s .- k . yip , cond - mat/0506437 ; d. t. son and m. a. stephanov , cond - mat/0507586 ; m. kitazawa , d. h. rischke and i. a. shovkovy , hep - ph/0602065 . i. giannakis and h. c. ren , nucl . b * 723 * , 255 ( 2005 ) . i. giannakis , d. f. hou and h. c. ren , phys . b * 631 * , 16 ( 2005 ) . e. v. gorbar , m. hashimoto and v. a. miransky , phys . lett . * 96 * , 022005 ( 2006 ) . e. v. gorbar , m. hashimoto and v. a. miransky , phys . b * 632 * , 305 ( 2006 ) . m. huang , hep - ph/0504235 , to appear in phys . d. r. casalbuoni , z. y. duan and f. sannino , phys . d * 62 * , 094004 ( 2000 ) ; d. h. rischke and i. a. shovkovy , phys . d * 66 * , 054019 ( 2002 ) . m.franz , z. tesanovic , phys.rev.lett.87 , 257003(2001 ) ; m.franz , z. tesanovic , o.vafek , phys.rev.*b66*(2002 ) , 054535 ; a. melikyan , z. tesanovic , cond - mat/0408344 . d. k. hong , v. a. miransky , i. a. shovkovy and l. c. r. wijewardhana , phys . d * 61 * , 056001 ( 2000 ) , [ erratum - ibid . d * 62 * , 059903 ( 2000 ) ] ; d. d. dietrich and d. h. rischke , prog . part . phys . * 53 * , 305 ( 2004 ) .

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 .

hep-ph0602092

the star - product operation introduced by groenewold for phase - space functions @xcite permits formulation of quantum mechanics in phase space . it uses the weyl s association rule @xcite to establish one - to - one correspondence between phase - space functions and operators in the hilbert space . the wigner function @xcite appears as the weyl s symbol of the density matrix . the skew - symmetric part of the star - product , known as the moyal bracket @xcite , governs the evolution of symbols of heisenberg operators . refined formulation of the weyl s association rule is proposed by stratonovich @xcite . the weyl s association rule , star - product technique , star - functions , and some applications are reviewed in refs . @xcite . a one - parameter group of unitary transformations in the hilbert space @xmath0 with @xmath1 being hamiltonian , corresponds to a one - parameter group of canonical transformations in the classical theory @xcite , although canonical transformations provide a broader framework @xcite . weyl s symbols of time dependent heisenberg operators of canonical coordinates and momenta induce quantum phase flow . osborn and molzahn @xcite construct quantum hamilton s equations which determine quantum phase flow and analyze the semiclassical expansion for unconstrained quantum - mechanical systems . an earlier attempt to approach these problems is undertaken in ref . @xcite . the infinitesimal transformations induced by the evolution operator ( [ u7 ] ) in phase space coincide with the infinitesimal canonical transformations induced by the corresponding hamiltonian function @xcite . the quantum and classical finite transformations are , however , distinct in general , since the star- and dot - products as multiplication operations of group elements in quantum and classical theories do not coincide . the quantum phase flow curves are distinct from the classical phase - space trajectories . this fact is not well understood ( see e.g. refs . @xcite ) . osborn and molzahn @xcite made important observation that quantum trajectories in unconstrained systems can be viewed as a `` basis '' to represent the evolution of quantum observables . such a property is usually assigned to characteristics appearing in a standard technique for solving first - order partial differential equations ( pde ) . the well known example is the classical liouville equation @xmath2 this equation is solved in terms of characteristic lines which are solutions of classical hamilton s equations @xmath3 with initial conditions @xmath4 . equations ( [ classham3 ] ) are characteristic equations . they represent a system of first - order ordinary differential equations ( ode ) for canonical variables . physical observables @xmath5 evolve according to @xmath6 it is remarkable that despite quantum liouville equation is an infinite - order pde its solutions are expressed in terms of solutions of the quantum hamilton s equations which are infinite - order pde also . a technical advantage in using the method of characteristics in quantum mechanics stems from the fact that to any fixed order of the semiclassical expansion the quantum hamilton s equations can be viewed as a coupled system of first - order ode for quantum trajectories and generalized jacobi fields obeying certain initial conditions . the evolution can be considered , respectively , as going along a trajectory in an extended phase space endowed with auxiliary degrees of freedom ascribed to generalized jacobi fields . the evolution problem can be solved e.g. numerically applying efficient ode integrators . quantum characteristics can be useful , in particular , for solving numerically many - body potential scattering problems by semiclassical expansion of star - functions around their classical values with subsequent integration over the initial - state wigner function . among possible applications are transport models in quantum chemistry and heavy - ion collisions @xcite where particle trajectories remain striking but an intuitive feature . a covariant extensions of quantum molecular dynamics ( qmd ) transport models @xcite is based on the poincar invariant constrained hamiltonian dynamics @xcite . we show , in particular , that quantum trajectories exist and make physical sense in the constraint quantum systems also and play an important role similar to that in the quantum unconstrained systems . the paper is organized as follows : in sects . ii and iii , characteristics of unconstraint classical and quantum systems are discussed . sects . iv and v are devoted to properties of characteristics of constraint classical and quantum systems . quantum phase flows are analyzed using the star - product technique which we believe to be the most adequate tool for studying the subject . we give definitions and recall basic features of the method of characteristics in sect . ii . in sect . iii , fundamental properties of quantum characteristics are derived . the weyl s association rule , the star - product technique , and the star - functions are reviewed based on the method proposed by stratonovich @xcite . we show , firstly , that quantum phase flow preserves the moyal bracket and does not preserve the poisson bracket in general . secondly , we show that the star - product is invariant with respect to transformations of the coordinate system , which preserve the moyal bracket . thirdly , non - local laws of composition for quantum trajectories and the energy conservation along quantum trajectories are found in sect . iii - d . applying the invariance of the star - product with respect to change of the coordinate system ( [ brinva ] ) and the energy conservation , we derive new equivalent representations of the quantum hamilton s equations eq.([qf2 ] ) - ( [ qf4 ] ) . in sect . iii - e , we derive using the star - product technique the semiclassical reduction of the quantum hamilton s equations to a system of first - order ode involving along with quantum trajectories their partial derivatives with respect to initial canonical variables . finally , we express the phase - space green function @xcite in terms of quantum characteristics and reformulate relation between quantum and classical time - dependent observables @xcite using the method of characteristics . the possibility of finding quantum trajectories and generalized jacobi fields by solving a system of ode gives practical advantages because of the existence of efficient numerical ode integrators . it would be tempting to extend method of characteristics to constraint systems such as gauge theories , relativistic qmd transport models , etc . the skew - gradient projection method is found to be useful to formulate classical and quantum constraint dynamics @xcite . in sect . iv , we show that in classical constraint systems characteristic lines exist and the method of characteristics is efficient . the proof we provide does not presuppose that constraint equations can be solved . the phase flow is commutative with the phase flows generated by constraint functions . characteristic lines , if belong to the constraint submanifold at @xmath7 , belong to the constraint submanifold at @xmath8 also . v gives description of quantum characteristics in constraint systems . although the formalism is complete , we encounter unexpected difficulty to formulate simple geometric idea that quantum trajectory belongs to a constraint submanifold . using tools of the analytic geometry , any idea like that requires the use of composition of functions . in quantum mechanics , one has to use the star - composition . this calls for a modification of usual geometric relations `` belong '' , `` intersect '' , and others . in a specific quantum - mechanical sense , the hamiltonian and constraint functions can be said to remain constant along quantum trajectories , while in the usual geometric sense they obviously do nt . the problem of visualization of relations among quantum objects in phase space is discussed in sects . iii - d and v - b . conclusion summarizes results . the phase space of system with @xmath9 degrees of freedom is parameterized by @xmath10 canonical coordinates and momenta @xmath11 which satisfy the poisson bracket relations @xmath12 with @xmath13 where @xmath14 is the identity @xmath15 matrix . the phase space appears as the cotangent bundle @xmath16 of @xmath9-dimensional configuration space @xmath17 . the matrix @xmath14 imparts to @xmath18 a skew - symmetric bilinear form . the phase space acquires thereby structure of symplectic space . in what follows , physical observables are time dependent , whereas density distributions remain constant . such a picture constitutes the classical analogue of the quantum - mechanical heisenberg picture . in the classical unconstrained systems , phase flow : @xmath19 , is canonical and preserves the poisson bracket . the classical hamilton s equations ( [ classham3 ] ) are first - order ode . the energy is conserved along classical trajectories @xmath20 the classical hamilton s equations ( [ classham3 ] ) can be rewritten as first - order pde : @xmath21 the phase - space trajectories can be used to solve the liouville equation ( [ class ] ) which is the first - order pde . any observable @xmath5 is expressed in terms of @xmath22 , as indicated in eq.([charac ] ) . classical trajectories obey the dot - composition law : @xmath23 the stratonovich version of the weyl s quantization and dequantization @xcite is discussed in the next subsection and in more details in refs . @xcite . the phase - space variables @xmath24 correspond to operators @xmath25 acting in the hilbert space , which obey commutation rules @xmath26 = -i\hbar i^{kl}. \label{987897}\ ] ] operators @xmath27 acting in the hilbert space admit multiplications by @xmath28-numbers and summations . the set of all operators constitutes a vector space . the basis of such a space can be labelled by @xmath24 . the weyl s basis looks like @xmath29 the association rule for a function @xmath30 and an operator @xmath27 has the form @xcite @xmath31 , \;\;\ ; \mathfrak{f } = \int \frac{d^{2n}\xi } { ( 2\pi \hbar ) ^{n}}f(\xi ) \mathfrak{b}(\xi ) . \label{inv}\ ] ] the value of @xmath32 can be treated as the @xmath33-coordinate of @xmath27 in the basis @xmath34 , while @xmath35 $ ] as the scalar product of @xmath34 and @xmath27 . using eqs.([inv ] ) one gets an equivalent association rule @xmath36 the half - fourier transform , @xmath37 provides the inverse relation . the weyl - symmetrized functions of operators of canonical variables have representation @xcite @xmath38 where the subscripts indicate the order in which the operators act on the right . given two functions @xmath39 $ ] and @xmath40 $ ] , one can construct a third function , @xmath41,\ ] ] called star - product . in terms of the poisson operator @xmath42 one has @xmath43 the star - product splits into symmetric and skew - symmetric parts , @xmath44 the skew - symmetric part is known under the name of moyal bracket . the wigner function is the weyl s symbol of the density matrix . in the heisenberg picture , the wigner function remains constant @xmath45 , whereas functions representing physical observables evolve with time in agreement with equation @xmath46 this equation is the weyl s transform of equation of motion for operators in the heisenberg picture . it is the infinite - order partial differential equation ( pde ) . the series expansions of @xmath47 over @xmath48 is given by @xmath49 where @xmath50 is the initial data function . using the @xmath51-adjoint notations of ref . @xcite , equation ( [ evol ] ) can be represented in the form @xmath52(\xi ) . \label{haki } \end{aligned}\ ] ] its formal solution , @xmath53 is equivalent to eq.([expa ] ) . if the target symbol @xmath54 is semiclassically admissible , the evolution operator has asymptotic expansion @xcite @xmath55 the power series expansion in @xmath56 is valid for semiclassically admissible symbols @xmath57 and @xmath58 . if , however , @xmath58 is a rapidly oscillating symbol , then ( [ haki3 ] ) fails and the solution of the evolution equation becomes of the wkb type whose exponential phase is a symplectic area ( see for details ref . @xcite ) . active transformations modify operators @xmath27 and commute with @xmath59 . passive transformations change the basis and keep operators fixed . these views are equivalent . we choose the former . consider transformations depicted by the diagram @xmath60 where @xmath61 is given by eq.([u7 ] ) . the operators of canonical variables are transformed as @xmath62 the coordinates @xmath63 of new operators @xmath64 in the old basis @xmath59 are given by @xmath65 . \label{uxit}\ ] ] since @xmath61 is the evolution operator , functions @xmath66 can be treated as the weyl s symbols of operators of canonical coordinates and momenta in the heisenberg picture . for @xmath7 , we have @xmath67 the set of operators of canonical variables is complete in the sense that any operator acting in the hilbert space can be represented as a function of operators @xmath68 . one can indicate it as follows : @xmath69 . the taylor expansion of @xmath70 permits the equivalent formulation of the weyl s association rule . transformations @xmath71 generate transformations of the associated phase - space functions : @xmath72 \nonumber \\ & = & \sum_{s=0}^{\infty } \frac{1}{s!}\frac{\partial ^{s}f(0)}{\partial \xi^{i_{1}} ... \partial \xi ^{i_{s } } } tr[\mathfrak{b}(\xi ) \mathfrak{u}^{+}\mathfrak{x}^{i_{1}} ... \mathfrak{x}^{i_{s}}\mathfrak{u } ] \nonumber \\ & = & \sum_{s=0}^{\infty } \frac{1}{s!}\frac{\partial ^{s}f(0)}{\partial \xi ^{i_{1}} ... \partial \xi ^{i_{s}}}tr[\mathfrak{b}(\xi ) \acute{\mathfrak{x}}^{i_{1 } } ... \acute{\mathfrak{x}}^{i_{s } } ] \nonumber \\ & = & \sum_{s=0}^{\infty } \frac{1}{s!}\frac{\partial ^{s}f(0)}{\partial \xi ^{i_{1}} ... \partial \xi ^{i_{s}}}u^{i_{1}}(\xi , \tau)\star ... \star u^{i_{s}}(\xi,\tau ) \nonumber \\ & = & \sum_{s=0}^{\infty } \frac{1}{s!}\frac{\partial ^{s}f(0)}{\partial \xi ^{i_{1}} ... \partial \xi ^{i_{s}}}u^{i_{1}}(\xi , \tau)\circ ... \circ u^{i_{s}}(\xi,\tau ) \nonumber \\ & \equiv & f(\star u(\xi,\tau ) ) . \label{tf}\end{aligned}\ ] ] last two lines define the star - composition . the star - function @xmath73 is a functional of @xmath74 . the @xmath75-product is not associative in general . however , the indices @xmath76 for @xmath77 are symmetrized , so the order in which the @xmath75-product is calculated is not important . the antisymmetrized products @xmath78}$ ] of even number of operators of canonical variables are @xmath28-numbers as a consequence of the commutation relations . these products are left invariant by unitary transformations : @xmath79}\mathfrak{u}=\mathfrak{x}^{[i_{1}} ... \mathfrak{x}^{i_{2s}]}$ ] . in phase space , we get @xmath80}(\xi,\tau ) = \xi ^{[i_{1}}\star ... \star \xi ^{i_{2s } ] } $ ] and , in particular , @xmath81 phase - space transformations induced by @xmath61 preserve the moyal bracket and do not preserve the poisson bracket , so the evolution map @xmath82 , is not canonical . using eq.([teu ] ) , one can check e.g. that for @xmath83 where @xmath84 is the kronecker symbol functions @xmath85 do not satisfy the poisson bracket condition for canonicity to order @xmath86 . for real functions @xmath66 satisfying eqs.([area ] ) one may associate hermitian operators @xmath64 which obey commutation rules for operators of canonical coordinates and momenta . as a result , functions @xmath66 appear in the coincidence with a unitary transformation relating @xmath68 and @xmath64 . the conservation of the moyal bracket for a one - parameter set of continuous phase - space transformations is the necessary and sufficient condition for unitary character of the associated continuous transformations in the hilbert space . applying eq.([tf ] ) to product @xmath87 of two operators , we obtain function @xmath88 associated to operator @xmath89 and function @xmath90 associated to operator @xmath91 . these operators coincide , so do their symbols : @xmath92 the star - product is calculated with respect to @xmath93 and @xmath33 in the left- and right - hand sides , respectively . equation ( [ brinva ] ) is valid separately for symmetric and skew - symmetric parts of the star - product of the functions . the substantial content of eq.([brinva ] ) is that one can compute the star - product in the initial coordinate system and change variables @xmath94 , or equivalently , change variables @xmath94 and compute the star - product , provided eq.([area ] ) is fulfilled . the functions @xmath95 define quantum phase flow which represents quantum deformation of classical phase flow . ) . the solid line stands for a quantum trajectory @xmath96 at @xmath97 . the dashed line is assigned to a trajectory @xmath98 which we would have at @xmath99 for the classical dot - composition law . the distance between the solid and dashed trajectories is of order of @xmath100 . , width=2 ] in the usual geometric sense , quantum characteristics @xmath101 can not be considered as trajectories along which physical particles move . the reason lies , in particular , in the star - composition law @xmath102 which is distinct from @xmath103 , see fig . [ fig10 ] . in classical mechanics , the composition law has the form of eq.([compcl ] ) . the energy conservation in the course of quantum evolution implies @xmath104 where @xmath105 $ ] is hamiltonian function . @xmath106 is , however , not conserved along quantum trajectories in the usual geometric sense , so @xmath107 . in classical mechanics , the conservation law has the form ( [ clco ] ) . to express the idea that a point particle moves continuously along a phase - space trajectory , one has to use the star - composition ( [ comp ] ) . the dot - composition is not defined in quantum mechanics . similarly , @xmath108 does not make any quantum - mechanical sense . one has to work with @xmath109 . if so , the only way to express quantitatively the fact of the energy conservation along a phase - space trajectory is to use eq.([ec ] ) . the similar problem arises in constraint systems when we want to decide if quantum trajectories belong to a constraint submanifold . the analytic geometry provides tools to formulate relations among geometric objects . those relations which are expressed through composition of functions are modified . we discuss if possible to assign a geometric sense to formulas involving the star - composition in sect . v - b . quantum hamilton s equations can be obtained applying the weyl s transform to evolution equations for heisenberg operators of canonical coordinates and momenta @xmath110 to reach the step 2 , the energy conservation ( [ ec ] ) is used . going from ( [ qf2 ] ) to ( [ qf3 ] ) , the change of variables ( [ brinva ] ) is performed . to achieve ( [ qf4 ] ) , we exploit @xmath111 . the time derivative of @xmath112 can be computed classically using the poisson bracket . the substitution @xmath113 leads , however , to deformation of classical trajectories . equations ( [ qf3 ] ) and ( [ qf4 ] ) are the quantum analogues of eq.([classham3 ] ) , eq.([qf ] ) is the quantum analogue of eq.([classham2 ] ) , and eq.([qf2 ] ) is the quantum analogue of eq.([classham ] ) . as distinct from the de broglie - bohm trajectories ( see e.g. @xcite ) , @xmath112 are not related to specific states in the hilbert space . the functional form of quantum hamilton s equations ( [ qf ] ) is left invariant by the change of variables @xmath114 provided the map @xmath115 : @xmath116 , preserves the moyal bracket . equations ( [ qf ] ) are not invariant under canonical transformations . consider e.g. canonical map : @xmath117 , with generating function @xmath118 such that @xmath119 and @xmath120 . one can compare @xmath121 and @xmath122 in the coordinate systems @xmath123 and @xmath124 . for functions @xmath125 and @xmath126 , one gets , respectively , @xmath127 and @xmath128 . the symmetric and skew - symmetric parts of the star - product are both not invariant under canonical transformations . coordinate systems in phase space if related by a canonical transformation provide non - equivalent quantum dynamics . this ambiguity is better known as the operator ordering problem . the quantum deformation of classical phase flow can be found by expanding @xmath129 the right - hand side of eqs.([qf ] ) @xmath130 is a function of @xmath131 ( i.e. functional of @xmath74 ) , so we have to expand @xmath132\ ] ] using e.g. the cluster - graph method @xcite . classical trajectories @xmath133 satisfy classical hamilton s equations @xmath134 and initial conditions @xmath135 . given @xmath133 , the lowest - order quantum correction @xmath136 can be found by solving first - order ordinary differential equations ( ode ) @xmath137 with initial conditions @xmath138 . the functions @xmath139 and @xmath140 entering eq.([qhe ] ) is a particular case of generalized jacobi fields @xmath141 given @xmath142 and @xmath143 for @xmath144 , the next corrections @xmath145 can be found from first - order ode involving generalized jacobi fields ( [ gj ] ) with @xmath144 . for a harmonic oscillator , @xmath146 for @xmath147 , in which case quantum phase flow is both canonical and unitary . the generalized jacobi fields ( [ gj ] ) satisfy ode also . the lowest order equations have the form : @xmath148 the first of these equations describes the evolution of small perturbations along the classical trajectories . being projected onto a submanifold of constant energy it becomes the jacobi - levi - civita equation @xcite . in stochastic systems , @xmath139 grow exponentially with time . at any fixed level of accuracy of the semiclassical expansion , we have a coupled system of ode for @xmath149 and @xmath143 subjected to initial conditions @xmath150 where @xmath151 and @xmath151 or @xmath152 , respectively . the evolution problem can be solved e.g. numerically applying efficient ode integrators . a numerical computation of the semiclassical expansion of the quantum phase flow in the elastic scattering of atomic systems is performed in ref . @xcite . an alternative approach allowing to reduce the semiclassical quantum dynamics to a closed system of ode is proposed by bagrov with co - workers @xcite . the phase - space trajectories appearing in @xcite are connected to specific quantum states like in the de broglie - bohm theory . properties of quantum paths , localization of quantum systems , and a coherent - type representation of the quantum flow are discussed in ref . @xcite . the series expansions of @xmath112 and @xmath153 over @xmath48 are given by @xmath154 in general , quantum phase flow is distinct from classical phase flow . this feature holds in integrable systems also , as discussed in appendix a. the lowest order operators @xmath155 entering eq.([haki3 ] ) can be found to be @xcite @xmath156 here , the derivatives of @xmath157 are calculated with respect to @xmath158 : @xmath159 the jacobi fields with the upper indices are defined by @xmath160 according to eq.([fina2 ] ) , time dependence of the zero order term @xmath161 is determined by time dependence of the classical phase - space trajectory and form of the function @xmath30 . the similar conclusion holds for @xmath162 : eq.([fina3 ] ) tells that time dependence enters through the classical trajectory , the first quantum correction to the classical trajectory , and the classical jacobi fields . the problem of convergence of a formal power series expansion is always a difficult subject . the convergence rate of the time series depends obviously on the system . generally , the series expansion in @xmath163 has a finite convergence radius . however , in the itegrable systems one has a truncated series expansion ( cf . eq.(iii.27 ) and eq.(a.3 ) ) . using orthogonality condition @xmath164 = ( 2\pi \hbar)^{n}\delta^{2n}(\xi - \zeta)\ ] ] and eq.([tf ] ) , we express green function for the weyl s symbols @xcite in terms of the quantum characteristics : @xmath165 \nonumber \\ & = & ( 2\pi \hbar)^{n}\delta^{2n}(\star u(\xi,\tau ) - \zeta ) \nonumber \\ & = & ( 2\pi \hbar)^{n}\delta^{2n}(\xi - \star u(\zeta,-\tau ) ) . \label{gf}\end{aligned}\ ] ] a compact operator relation between the classical and quantum time - dependent observables is established in ref . solutions of the quantum and classical liouville equations , @xmath5 and @xmath166 , with initial conditions @xmath167 are related through the product @xmath168 where @xmath169 is the classical green function @xmath170 in terms of the characteristics , we obtain @xmath171 it is assumed that classical and quantum hamiltonian functions coincide i.e. @xmath172 . given the green function is known , the quantum trajectories can be found from equation @xmath173 for @xmath174 where @xmath175 is an infinitesimal parameter , the associated transformations of canonical variables and phase - space functions are given by @xmath176 and @xmath177 . the transformations of canonical variables are canonical to order @xmath178 only . the infinitesimal transformations of symbols of operators are not canonical . any function @xmath179 can be used to generate classical phase flow or quantum phase flow , according as the dot - product or the star - product stands for multiplication operation in the set of phase - space functions . the analogue between unitary and canonical transformations is illustrated by dirac @xcite in terms of the generating function @xmath180 defined by @xmath181 . the evolution map @xmath182 , is canonical for @xmath183 and @xmath184 . the parallelism of the transformations is manifest , but trajectories are complex . the generating function defined by the phase of @xmath185 yields real trajectories . it is not clear , however , if time - dependent symbols of operators are entirely determined by such trajectories . the weyl s symbols of operators of canonical variables @xmath186 are the genuine characteristics in the sense that they allow by equation @xmath187 the entire determination of the evolution of observables . the quantum dynamics is totally contained in @xmath186 , whereas the deformation of symbols of the operators calculated at @xmath188 has a kinematical meaning . we give first description of second - class constraints systems and of the skew - gradient projection formalism . the details are found elsewhere @xcite . second - class constraints @xmath189 with @xmath190 and @xmath191 have the poisson bracket relations which form a non - degenerate @xmath192 matrix @xmath193 if this would not be the case , it could mean that gauge degrees of freedom appear in the system . after imposing gauge - fixing conditions , we could arrive at the inequality ( [ nongen ] ) . alternatively , breaking the condition ( [ nongen ] ) could mean that constraint functions are dependent . after removing redundant constraints , we arrive at the inequality ( [ nongen ] ) . constraint functions are equivalent if they describe the same constraint submanifold . within this class one can make transformations without changing dynamics . for arbitrary point @xmath33 of the constraint submanifold @xmath194 , there is a neighbourhood where one may find equivalent constraint functions in terms of which the poisson bracket relations look like @xmath195 where @xmath196 here , @xmath197 is the identity @xmath198 matrix , @xmath199 . the matrix @xmath200 is used to lift indices @xmath201 up . the basis ( [ sb ] ) always exists locally , i.e. , in a finite neighbourhood of any point of the constraint submanifold . this is on the line with the darboux s theorem ( see e.g. @xcite ) . all symplectic spaces are locally indistinguishable . the concept of the skew - gradient projection @xmath202 of canonical variables @xmath33 onto a constraint submanifold plays important role in the moyal quantization of constraint systems . geometrically , the skew - gradient projection acts along phase flows @xmath203 generated by constraint functions . these flows are commutative in virtue of eqs.([sb ] ) : using eqs.([sb ] ) and the jacobi identity , one gets @xmath204 for any function @xmath58 , so the intersection point with @xmath205 is unique . to construct the skew - gradient projections , we start from equations @xmath206 which say that point @xmath207 is left invariant by phase flows generated by @xmath208 . using the symplectic basis ( [ sb ] ) for the constraints and expanding @xmath209 in the power series of @xmath210 , one gets @xmath211 similar projection can be made for function @xmath30 : @xmath212 one has @xmath213 the projected functions are in involution with the constraint functions : @xmath214 consequently , @xmath215 does not vary along @xmath216 , since @xmath217 the skew - gradient projection is depicted schematically in fig . [ fig1 ] . in the classical second - class constraints systems , one has to start from constructing @xmath218 from @xmath219 . the evolution equation for phase - space functions can be converted then to the classical liouville equation : @xmath220 similarly , the canonical variables obey the classical hamilton s equations : @xmath221 with initial conditions @xmath222 equation @xmath223 tells that @xmath224 remain constant along @xmath225 : @xmath226 equations ( [ invo-2 ] ) show that trajectories do not leave level sets @xmath227 and therefore do not leave the constraint submanifold @xmath228 given @xmath218 is constructed , it becomes possible to extend standard theorems of the hamiltonian formalism to second - class constraints systems without modifications . the novel element is the interplay between the evolution and the skew - gradient projection . let the coordinate system @xmath229 is obtained from the coordinate system @xmath230 by the canonical transformation @xmath231 . is commutative with classical projection @xmath232 onto constraint submanifold @xmath205 . , width=7 ] eq.([fsg ] ) may be applied for @xmath225 . using eq.([invo-2 ] ) , we replace the arguments of the constraint functions to @xmath225 and replace everywhere @xmath225 with @xmath233 , as long as the poisson brackets are invariant and the constraint functions are scalars . we arrive at @xmath234 the first line is a consequence of eq.([fssf ] ) . the evolution is commutative with the skew - gradient projection . equation ( [ abba ] ) is illustrated on fig . [ fig4 ] . the liouville equation can be solved provided phase - space trajectories @xmath22 are known . in general , @xmath235 applying projection ( [ fsg ] ) , one gets @xmath236 the first line follows from eq.([fssf ] ) . equation ( [ 23456 ] ) shows how to use characteristics in order to solve evolution equations in the classical second - class constraint systems . the evolution depends on choice of the constraint functions up to a canonical transformation . suppose we found two sets of the constraint functions @xmath208 and @xmath237 describing the same constraint submanifold . each set can be transformed to the standard basis ( [ sb ] ) . such bases are related by canonical transformations , so one can find a canonical map : @xmath238 , such that @xmath239 . the inverse transform is @xmath240 . the skew - gradient projections @xmath232 and @xmath241 are related by : @xmath242 the skew - gradient projection depends on choice of the constraint functions up to a canonical transformation . the same is true for projected hamiltonian functions : @xmath243 where @xmath244 . two sets of the constraint functions @xmath208 and @xmath237 lead to the canonically equivalent hamiltonian phase flows . the constraint systems represent high interest since all fundamental interactions in the elementary particle physics are based on the principles of gauge invariance . gauge fixing turns gauge - invariant systems into constraint systems . in the classical mechanics , the constraint systems can be treated as a limiting case @xmath245 of systems in a potential @xmath246 which rapidly increases when the coordinates @xmath247 go away from the constraint submanifold . in the limit of @xmath245 , @xmath248 if @xmath247 belongs to the constraint submanifold and @xmath249 when @xmath247 does not belong to the constraint submanifold . the classical systems obtained by imposing the constraints and by the limiting procedure have equivalent dynamic properties @xcite . in the quantum mechanics , this is not the case . the limiting procedure applied to a particular system of ref . @xcite to model holonomic constraints , results to the quantum dynamics which depends on the way the constraint submanifold is embedded into the configuration space . from other hand , the quantization of constraint holonomic systems leads to the conclusion that the dynamics is determined by the induced metric tensor only @xcite . the limiting procedure and imposing the constraints are not equivalent schemes of the quantization . in what follows , we discuss the constraint dynamics as it appears in the gauge theories . the groenewold - moyal constraint dynamics has many features in common with the classical constraint dynamics . the projection formalism developed for constraint systems allows , from other hand , to treat unconstrained and constraint systems essentially on the same footing . we recall that classical hamiltonian function @xmath250 and constraint functions @xmath224 are distinct in general from their quantum analogues @xmath251 and @xmath252 . these dissimilarities are connected to ambiguities in quantization of classical systems . it is required only @xmath253 in what follows @xmath254 the quantum constraint functions @xmath252 satisfy @xmath255 the quantum - mechanical version of the skew - gradient projections is defined with the use of the moyal bracket @xmath256 the projected canonical variables have the form @xmath257 the quantum analogue of eq.([fsg ] ) is @xmath258 the function @xmath259 obeys equation @xmath260 the evolution equation which is the analogue of eq.([evol ] ) takes the form @xmath261 where @xmath262 is the hamiltonian function projected onto the constraint submanifold as prescribed by eq.([sgrad4 ] ) . any function projected quantum - mechanically onto the constraint submanifold can be represented in the form @xcite @xmath263 in the space of projected functions , the set of projected canonical variables @xmath264 is therefore complete . defined by eq.(4.21 ) . the submanifold @xmath265 does not coincide with the constraint submanifold @xmath266 . the variance is of order @xmath267 . the constraint submanifold @xmath205 can be parameterized by classical projection @xmath268 constructed with the use of the quantum constraint functions @xmath269 . , width=7 ] the evolution equation in the quantum constraint systems has the form of eq.([pev2 ] ) which is essentially the same as in the quantum unconstrained systems . replacing @xmath106 by @xmath270 , one can work further with solutions @xmath271 of quantum hamilton s equations ( [ qf ] ) . it is not required for points @xmath272 to belong to the constraint submanifold , so phase - space trajectories @xmath271 occupy the whole phase space . the quantum phase flow preserves the constraint functions in the following sense : @xmath273 the alternative equation @xmath274 which would carry the conventional geometric meaning uses pre - conditionally the dot - composition law which is not allowed quantum - mechanically . it is obviously violated , so in the usual sense @xmath275 for @xmath8 even if @xmath276 ( see fig . [ fig12 ] ) . any attempt to decide if @xmath277 involves the dot - composition e.g. @xmath278 statements involving the dot - composition are , however , forbidden . surprisingly , expressive means of the star - product formalism are not enough to formulate the simple geometric idea that a trajectory belongs to a submanifold . we wish to find statements admissible quantum - mechanically and from other hand which would support relations of belonging and intersection inherent for geometric objects . it is tempting to interpret eqs.([concon ] ) as the evidence that quantum trajectories @xmath101 do not leave , in a specific quantum - mechanical sense , level sets of constraint functions @xmath279 . such a statement has the invariant meaning with respect to unitary transformations : suppose the map @xmath280 : @xmath281 , corresponds to a unitary transformation in the hilbert space . the inverse unitary transformation generates the inverse map @xmath115 : @xmath282 , such that @xmath283 and , by virtue of eq.([brinva ] ) , @xmath284 . in the coordinate system @xmath285 , the constraint functions become @xmath286 equation ( [ brinva ] ) allows to change the variables @xmath287 in eq.([concon ] ) to give @xmath288 where @xmath289 represents the quantum phase flow in the coordinate system @xmath285 . equations ( [ concon ] ) and ( [ concon-2 ] ) are therefore equivalent . they show that `` do not leave '' represents a predicate invariant under unitary transformations . the non - local character of relations between the quantum phase flows is displayed in eq.([nonloc ] ) explicitly . one can conclude that quantum trajectories do not transform under unitary transformations as geometric objects . and @xmath290 ( solid lines ) and quantum trajectories @xmath74 and @xmath291 ( dashed lines ) in unitary equivalent coordinate systems @xmath230 and @xmath285 , respectively . as shown , @xmath74 crosses @xmath205 twice , whereas its image @xmath291 crosses @xmath290 once . any counting of the intersections rests on an implicit use of the dot - composition , an operation which is forbidden quantum - mechanically . the property of the statements @xmath292 and @xmath293 be true or false depends on unitary transformations . from the viewpoints of eqs.([concon ] ) and ( [ concon-2 ] ) , @xmath74 and @xmath291 belong to the level sets of @xmath294 and @xmath295 , respectively . however , from condition @xmath296 it does not follow that @xmath297 and _ vice versa_. geometric relations among quantum objects , which use the dot - composition , do not have objective meaning . , width=2 ] the coordinate transformation @xmath115 : @xmath282 does not superpose @xmath298 and @xmath299 assuming @xmath300 , we obtain @xmath301 and therefore @xmath302 in general . the constraint submanifold does not transform under unitary transformations as a geometric object also . we see that points of @xmath205 transform differently from @xmath205 . they are `` not attached to @xmath205 '' . in new coordinate system , @xmath205 represents a set of new points . to put it precisely , @xmath303 unitary transformations affect the visualization of trajectories and submanifolds . the relation `` do not leave '' supports , however , some features inherent to the usual geometric relations `` belong '' and `` intersect '' . one can show e.g. that if quantum trajectories do not leave the level sets of @xmath294 and each level set of @xmath294 is a subset of one of the level sets of @xmath304 then quantum trajectories do not leave the level sets of @xmath304 . one can not assign to quantum trajectories definite values of energy and constraint functions . in the coordinate system @xmath230 one has @xmath305 , whereas in the coordinate system @xmath285 one has @xmath306 where @xmath307 is defined by eq.([hprime ] ) . the constants @xmath308 and @xmath309 do not depend on time . however , @xmath310 in general even if trajectories are related by a unitary transformation . the same conclusion holds for constraint functions , as shown on fig . [ fig12 ] . finally , the syntax of the star - product formalism is not rich enough to express the simple geometric idea that trajectory belongs to a submanifold . the star - product geometry admits the statement that quantum trajectories do not leave level sets of the constraint functions . the validity of this statement is not affected by unitary transformations and has the objective meaning . the quantum - mechanical relation `` do not leave '' is the remnant of usual relations of belonging and intersection inherent to geometric objects . it can not be completely visualized , however . the classical phase flow commutes with the classical skew - gradient projection , as discussed in sect . iv . we want to clarify if such a property holds for quantum systems . given the quantum trajectories @xmath311 are constructed , the evolution of arbitrary function can be found with the help of eq.([tf ] ) and its projection can be computed using eq.([sgrad4 ] ) . the quantum projection applied to arbitrary function can not be expressed in terms of the same function of the projected arguments eq.([great ] ) , basically because the classical relation @xmath312 turns to the quantum inequality @xmath313 . in terms of a function @xmath314 defined for @xmath315 in eq.([great ] ) , the quantum analogue for eqs.([23456 ] ) reads @xmath316 the construction of @xmath314 from @xmath32 is a complicated task , so practical advantages of this equation are not seen immediately . equation ( [ 0000 ] ) accomplishes solution of the evolution problem for observable @xmath5 in terms of quantum characteristics . it remains to prove @xmath317 the first line is a consequence of the fact that the constraint functions @xmath294 are moyal commutative with the projected hamiltonian function @xmath318 . to arrive at the second line , it is sufficient to use eq.([concon ] ) to replace arguments of the constraint functions entering the skew - gradient projection . the quantum phase flow commutes with the quantum projection , as illustrated on fig . [ fig6 ] . the composition law ( [ comp ] ) for quantum phase flow holds for the constraint systems . it holds for projected quantum trajectories also : @xmath319 as a consequence of eqs.([qabba ] ) . . the phase - space trajectory @xmath320 does not belong to the submanifold @xmath265 except for @xmath7 , so the white planes on figs . [ fig5 ] and [ fig6 ] are distinct.,width=8 ] the method of characteristics for solving evolution equations in classical and quantum , unconstrained and constrained systems has been discussed . the analysis rests on the groenewold - moyal star - product technique . the classical method of characteristics applies to first - order pde and consists in finding characteristics which are solutions of first - order ode . for the classical liouville equation , the corresponding first - order ode are the hamilton s equations and the characteristics of interest are the classical phase - space trajectories . the quantum liouville equation is the infinite - order pde . nevertheless , it can be solved in terms of quantum characteristics which are solutions of the quantum hamilton s equations . these equations represent infinite - order pde also . using the star - product formalism , we showed that to any fixed order in the planck s constant , quantum characteristics can be constructed by solving a closed system of ode for quantum trajectories and generalized jacobi fields . the quantum evolution becomes local in an extended phase space with new dimensions ascribed to generalized jacobi fields . this statement holds for constraint systems also . one - parameter continuous groups of unitary transformations in quantum theory represent the quantum deformation of one - parameter continuous groups of canonical transformations in classical theory . quantum phase flow , induced by the evolution in the hilbert space , does not satisfy the condition for canonicity and preserves the moyal bracket rather than the poisson bracket . the knowledge of quantum phase flow allows to reconstruct quantum dynamics . .solutions of evolution equations for functions ( second column ) and projected functions ( third column ) of classical systems ( first row ) and quantum systems ( second row ) in terms of characteristics . @xmath321 are solutions of classical hamilton s equations with hamiltonian function @xmath250 ( second column ) and projected hamiltonian function @xmath322 ( third column ) . @xmath323 are classical projections of @xmath321 . @xmath74 are solutions of quantum hamilton s equations with hamiltonian function @xmath251 ( second column ) and projected hamiltonian function @xmath262 ( third column ) . @xmath324 are quantum projections of @xmath74 . @xmath325 is defined in terms of @xmath326 by eq.([0000 ] ) . classical and quantum projections are defined by eqs.([fsg ] ) and ( [ sgrad4 ] ) , respectively . [ cols="<,^,^",options="header " , ] the results reported in this work are valid for semiclassically admissible functions , i.e. for functions regular in @xmath56 at @xmath327 . physical observables are normally associated with classical devices and expressed as classical functions of classical variables . the quantum evolution turns , however , the set of classical functions into the set of semiclassically admissible functions . the use of the skew - gradient projection formalism allows to treat unconstrained and constraint systems essentially on the same footing . we showed that the skew - gradient projections of solutions of the quantum hamilton s equations onto the constraint submanifold comprise the complete information on quantum dynamics of constraint systems . the formalism we developed applies in particular to the dynamics of gauge - invariant systems which become second class upon gauge fixing . the quantum dynamics of charged particles in external gauge fields on flat and curved manifolds is discussed within the star - product formalism in a gauge - invariant manner in refs . @xcite . the evolution equations for semiclassically admissible functions admit solutions in terms of characteristics in all physical systems , as summarized in table [ lab3 ] . the analytic geometry uses the dot - product and rests on classical ideas how to arrange composition of functions . it is well known that all theorems of geometry can be reformulated using tools of the analytic geometry . given the dot - product is replaced with the star - product , we arrive at the star - product geometry with well defined coordinate systems , transformations of the coordinates and equations for functions of the coordinates . however , objects of the star - product geometry , defined algebraically , can hardly be visualized : we found that quantum trajectories and constraint submanifolds do not transform as geometric objects . the statement `` quantum trajectory belongs to a constraint submanifold '' can be changed to the opposite by a unitary transformation . the star - composition law ( [ comp ] ) shows also that the quantum evolution can not be treated literally as moving along a quantum trajectory . we attempted to find statements whose validity can not be reverted by transformations of the coordinate system and which , from other hand , express relations similar to `` belong '' , `` intersect '' , etc . a weak but consistent geometric meaning can be attributed to the statement `` quantum trajectories do not leave level sets of constraint functions '' . the dot - product composition of linear functions coincides with the star - product composition of linear functions , so under linear transformations straight lines and hyperplanes turn to straight lines and hyperplanes . relations of the linear algebra , imbedded into the star - product geometry , preserve the consistent geometric meaning . finally , this work extended the method of characteristics to quantum unconstrained and constraint systems . from the point of view of applications , it is motivated by the fact of using classical phase - space trajectories in transport models and by the appearance of constraints in relativistic versions of qmd transport models . the method of quantum characteristics represents the promising tool for solving numerically many - body potential scattering problems . this work is supported by dfg grant no . 436 rus 113/721/0 - 2 , rfbr grant no . 06 - 02 - 04004 , and european graduiertenkolleg gr683 . a completely integrable classical system admits a canonical transformation which makes the hamiltonian function depending on half of the canonical variables only ( see e.g. @xcite ) . such variables if exist can be taken to be canonical momenta which usually referred to as actions . the canonically conjugate coordinates are referred to as angles . in quantum mechanics , we search for a unitary transformation ( or a half - unitary transformation @xcite ) allowing to express the hamiltonian as an operator function of operators of canonical momenta . if such a transformation exists , the hamiltonian commutes with the operators of canonical momenta , so that the canonical momenta are integrals of motion , whereas the operators of canonical coordinates depend linearly on time . the quantum integrable systems admit an equivalent treatment in the framework of the groenewold - moyal dynamics @xcite . given the hamiltonian function @xmath179 is known , one has to search for a map @xmath280 : @xmath328 preserving the moyal bracket , for which the system admits a hamiltonian function @xmath329 depending on actions @xmath330 , i.e. , canonical momenta only . we restrict the discussion by unitary transformations ( [ unita ] ) , leaving aside more involved cases described in refs . @xcite . let @xmath66 and @xmath331 be solutions of eq.([qf ] ) with hamiltonian functions @xmath179 and @xmath307 , respectively . in the coordinate system @xmath285 , the series expansion ( [ teu ] ) is truncated at @xmath332 . the quantum hamilton s equations give a motion by inertia : @xmath333 the actions ( @xmath334 ) remain constant , whereas the angles ( @xmath335 ) evolve linearly with time . equations ( [ iner ] ) can be derived as the weyl s transform of the equations of motion for the heisenberg operators of the canonical coordinates and momenta obtained by a unitary transformation from the initial set of operators the canonical coordinates and momenta . the poisson bracket @xmath336 depends for any @xmath337 on the actions only , so one has @xmath338 the map @xmath339 : @xmath340 showing the evolution in the coordinate system @xmath285 is both canonical and unitary , as the left - hand side of eq.([iner ] ) is a first - order polynomial with respect to the angles . the actions @xmath330 poisson and moyal commute with @xmath307 . composite functions @xmath341 , where @xmath115 is the inverse unitary map : @xmath342 such that @xmath343 , obey eqs.([qf ] ) and proper initial conditions and coincide with @xmath331 . it can be expressed as follows : @xmath344 the functions @xmath345 are defined using the star - product and depend on @xmath56 accordingly . we thus conclude that quantum phase flow is distinct from classical phase flow for integrable systems also . for a one - dimensional system @xmath346 , which is a classical integrable sysem for any potential @xmath347 , the first quantum correction to the phase - space trajectories appears to order @xmath348 . 99 h. groenewold , physica * 12 * , 405 ( 1946 ) . h. weyl , z. phys . * 46 * , 1 ( 1927 ) . h. weyl , _ the theory of groups and quantum mechanics _ , dover publications , new york inc . e. p. wigner , phys . rev . * 40 * , 749 ( 1932 ) . j. e. moyal , proc . . soc . * 45 * , 99 ( 1949 ) . m. s. bartlett and j. e. moyal , proc . * 45 * , 545 ( 1949 ) . r. l. stratonovich , sov . phys . jetp * 4 * , 891 ( 1957 ) . a. voros , ann . henri poincar * 24 * , 31 ( 1976 ) ; * 26 * , 343 ( 1977 ) . f. bayen , m. flato , c. fronsdal , a. lichnerowicz , d. sternheimer , ann . * 111 * , 61 ( 1978 ) ; ann . phys . * 111 * , 111 ( 1978 ) . p. carruthers , f. zachariasen , rev . * 55 * , 245 ( 1983 ) . m. hillery , r.f . oconnell , m.o . scully , e. p. wigner , phys . rep . * 106 * , 121 ( 1984 ) . n. l. balazs and b. k. jennings , phys . rep . * 104 * , 347 ( 1989 ) . b. fedosov , _ deformation quantization _ , akademie verlag , berlin ( 1996 ) . m. i. krivoruchenko , a. a. raduta , a. faessler , phys . * d73 * , 025008 ( 2006 ) . m. i. krivoruchenko , talk given at the xiii annual seminar `` nonlinear phenomena in complex systems : chaos , fractals , phase transitions , self - organization '' , minsk , belarus , may 16 - 19 , 2006 ; e - print archive : hep - th/0610074 . p. a. m. dirac , _ the principles of quantum mechanics _ , 3-rd ed . , claredon press , oxford ( 1947 ) . a. m. dirac , phys . sowjetunion * 3 * , 64 ( 1933 ) . b. leaf , j. math . * 10 * , 1971 ( 1969 ) ; * 10 * , 1980 ( 1969 ) . a. anderson , phys . b305 * , 67 ( 1993 ) ; phys . lett . * b319 * , 157 ( 1993 ) ; ann . phys . * 232 * , 292 ( 1994 ) . j. aichelin , phys . rep . * 202 * , 233 ( 1991 ) . s. w. huang , a. faessler , guo - qiang li _ et al . _ , prog . part . . phys . * 30 * , 105 ( 1993 ) . e. lehmann , r. k. puri , a. faessler _ et al . * 30 * , 219 ( 1993 ) . h. sorge , h. stcker and w. greiner , ann . phys . * 192 * , 266 ( 1989 ) . t. maruyama _ _ , nucl . phys . * a534 * , 720 ( 1991 ) . a. m. dirac , rev . * 21 * , 392 ( 1949 ) ; proc . a246 * , 326 ( 1958 ) . b. r. mcquarrie , t. a. osborn , and g. c. tabisz , phys . rev . * a58 * , 2944 ( 1998 ) . v. g. bagrov , v. v. belov and m. f. kondrateva , teor . mat . fiz . * 98 * , 48 ( 1994 ) [ theor . . phys . * 98 * , 34 ( 1994 ) ] . v. g. bagrov , v. v. belov , m. f. kondrateva , a. m. rogova , and a. yu . trifonov , j. moscow phys . soc.*3 * , 1 ( 1993 ) . v. g. bagrov , v. v. belov , a. m. rogova , and a. yu . trifonov , mod . * b7 * , 1667 ( 1993 ) . v. g. bagrov , v. v. belov , a. yu . trifonov , ann . phys . * 246 * , 231 ( 1996 ) . m. karasev , russ . phys . * 11 * , 157 ( 2004 ) . m. s. marinov , phys . a153 , 5 ( 1991 ) . g. braunss and d. rompf , j. phys . * a26 * , 4107 ( 1993 ) . m. nakamura and n. mishima , nuovo cim . * 79b * , 287 ( 1984 ) ; prog . . phys . * 81 * , 451 ( 1989 ) . m. nakamura and h. minowa , j. math . phys . * 34 * , 50 ( 1993 ) . m. nakamura and k. kojima , nuovo cim . * 116b * , 287 ( 2001 ) . m. i. krivoruchenko , a. faessler , a. a. raduta and c. fuchs , int . j. mod . phys . * a * 22 , 787 ( 2007 ) . v. i. arnold , _ mathematical methods of classical mechanics _ , 2-nd ed . , springer - verlag , new york inc . 1989 r. c. t. da costa , phys . rev . * a * 23 , 1982 ( 1981 ) . m. i. krivoruchenko , amand faessler , a. a. raduta , c. fuchs , phys . lett . * b608 * , 164 ( 2005 ) . t. hakioglu and a. j. dragt , j. phys . * a34 * , 6603 ( 2001 ) . t. a. osborn and m. f. kondratieva , j. phys . * a35 * , 5279 ( 2002 ) . m. i. krivoruchenko , c. fuchs , amand faessler , nucl - th/0605015 . m. v. karasev and v. p maslov , _ nonlinear poisson brackets _ , nauka , moscow ( 1991 ) . m. karasev , lett . * 59 * , 229 ( 2001 ) . m. v. karasev and t. a. osborn , j. phys . * a * 37 , 2345 ( 2004 ) . m. v. karasev and t. a. osborn , j. phys . * a * 38 , 8549 ( 2005 ) . m. v. karasev and v. p. maslov , rus . math . surv . * 39 * , 5279 ( 1984 ) .

the knowledge of quantum phase flow induced under the weyl s association rule by the evolution of heisenberg operators of canonical coordinates and momenta allows to find the evolution of symbols of generic heisenberg operators . the quantum phase flow curves obey the quantum hamilton s equations and play the role of characteristics . at any fixed level of accuracy of semiclassical expansion , quantum characteristics can be constructed by solving a coupled system of first - order ordinary differential equations for quantum trajectories and generalized jacobi fields . classical and quantum constraint systems are discussed . the phase - space analytic geometry based on the star - product operation can hardly be visualized . the statement `` quantum trajectory belongs to a constraint submanifold '' can be changed e.g. to the opposite by a unitary transformation . some of relations among quantum objects in phase space are , however , left invariant by unitary transformations and support partly geometric relations of belonging and intersection . quantum phase flow satisfies the star - composition law and preserves hamiltonian and constraint star - functions .

quant-ph0604075

opportunistic beamforming ( obf ) is a well known adaptive signaling scheme that has received a great deal of attention in the literature as it attains the sum - rate capacity with full channel state information ( csi ) to a first order for large numbers of mobile users in the network , while operating on partial csi feedback from the users . in this paper , we consider a cellular network which operates according to the obf framework in a multi - cell environment with variable number of transmit beams at each bs . the number of transmit beams is also referred to as the transmission rank ( tr ) in the paper , and we focus on optimally setting the transmission rank at each bs in the network . the earliest work of obf appeared in the landmark paper @xcite , where the authors have introduced a single - beam obf scheme for the single - cell multiple - input single - output ( miso ) broadcast channel . the concept was extended to @xmath0 random orthogonal beams in @xcite , where @xmath0 is the number of transmit antennas . the downlink sum - rate of this scheme scales as @xmath1 , where @xmath2 is the number of users in the system @xcite . recently , the authors in @xcite have considered using variable trs at the bs , and they have showed that the downlink sum - rate scales as @xmath3 in interference - limited networks , where @xmath4 is the tr employed by the bs . the gains of adapting variable tr compared to a fixed one is clearly demonstrated in @xcite , however , how to select the tr for obf is still an open question which has only been characterized in the asymptotic sense for the single - cell system in @xcite-@xcite , and a two - transmit antenna single - cell system in @xcite . in all of the above works , the users are assumed to be homogeneous with the large - scale fading gain ( alternatively referred to as the path loss in this paper ) equal to unity . obf in heterogenous networks has been considered in @xcite-@xcite . in @xcite , the authors focused on the fairness of the network and obtained an expression for the ergodic capacity of this fair network . in @xcite , the authors modeled the user locations using a spatial poisson point process , and studied the outage capacity of the system . in @xcite , the authors considered an interference - limited network and derived the ergodic downlink sum - rate scaling law of this interference - limited network . the trs in @xcite-@xcite are considered to be fixed . in this paper , we are interested in the quality of service ( qos ) delivered to the users . more precisely , we focus on a set of qos constraints that will ensure a guaranteed minimum rate per beam with a certain probability at each bs . previous studies have shown that user s satisfaction is a non - decreasing , concave function of the service rate @xcite ; this suggests that the user s satisfaction is insignificantly increased by a service rate higher than what the user demands , but drastically decreased if the provided rate is below the requirement @xcite . the network operator can promise a certain level of qos to a subscribed user . to this end , the qos is closely related to the tr of the bs . increasing the tr will increase the number of co - scheduled users . however , increasing the tr will also increase interference levels in the network , which will decrease the rate of communication per beam . a practical question arises ; what is a suitable tr to employ at each bs while achieving a certain level of qos in multi - cell heterogeneous networks ? the authors in @xcite have performed a preliminary study of this problem for a single - cell system consisting of homogeneous users with identical path loss values of unity . the main contributions of this paper are summarized as follows . we focus on finding the achievable trs without violating the above mentioned set of qos constraints . this can be formulated into a feasibility problem . for some specific cases , we derive analytical expressions of the achievable tr region , and for the more general cases , we derive expressions that can be easily used to find the achievable tr region . the achievable tr region consists of all the achievable tr tuples that satisfy the qos constraints . numerical results are presented for a two - cells scenario to provide further insights on the feasibility problem ; our results show that the achievable tr region expands when the qos constraints are relaxed , the snr and the number of users in a cell are increased , and the size of the cells are decreased . we consider a multi - cell multi - user miso broadcast channel . the system consists of @xmath5 bss ( or cells ) , each equipped with @xmath0 transmit antennas . each cell consists of @xmath2 users , each equipped with a single receive antenna . a bs will only communicate with users in its own cell . let @xmath6 denote the channel gain vector between bs @xmath7 and user @xmath8 in cell @xmath9 . the elements in @xmath6 are independent and identically distributed ( i.i.d . ) random variables , each of which is drawn from a zero mean and unit variance _ circularly - symmetric complex gaussian _ distribution @xmath10 . the large - scale fading gain ( may alternatively referred to as path loss in this paper ) between bs @xmath7 and user @xmath8 in cell @xmath9 is denoted by @xmath11 . the path loss ( pl ) values of all the users are governed by the pl model @xmath12 for @xmath13 , where @xmath14 represents the distance between the user and the bs of interest . therefore , the random pl values are also i.i.d . among the users , where the randomness stems from the fact that users locations are random . moreover , we assume a quasi - static block fading model over time @xcite . the bss operate according to the obf scheduling and transmission scheme as follows . the bss will first pre - determine the number of beams to be transmitted . bs @xmath9 generates @xmath15 random orthogonal beamforming vectors and transmits @xmath15 different symbols along the direction of these beams ( @xmath15 is the tr employed by bs @xmath9 ) . this process is simultaneously carried out at all bss . for bs @xmath9 , let @xmath16 and @xmath17 denote the beamforming vector and the transmitted symbol on beam @xmath18 , respectively . the received signal at user @xmath8 in cell @xmath9 can be written as where @xmath19 is the additive complex gaussian noise . we assume that @xmath20 = \rho_i$ ] , where @xmath21 is a scaling parameter to satisfy the total power constraint @xmath22 at each bs . for conciseness , we assume @xmath23 . each user will measure the sinr values on the beams from its associated bs , and feed them back . for the beam generated using @xmath16 , the received sinr at user @xmath8 located in cell @xmath9 is given by @xmath24 \left\{\sigma_n^2 l_i + g_{i , i , k } \sum_{\substack{l\neq m \\ l=1}}^{l_i } |\mathbf{h}_{i , i , k}^\top \mathbf{w}_{i , l}|^2 \right . \nonumber \\ & \hspace{1.5 cm } \left . + \sum_{j \neq i}^m g_{j , i , k } \frac{l_i}{l_j } \sum_{\substack{t=1}}^{l_j } |\mathbf{h}_{j , i , k}^\top \mathbf{w}_{j , t}|^2 \right\}^{-1}.\end{aligned}\ ] ] once the bss have received the feedback from the users , each bs will select a set of users for communication by assigning each beam to the in - cell user having the highest sinr on it , _ i.e. _ , the user with sinr value @xmath25 . for cell @xmath9 , let @xmath26 and @xmath27 denote the distributions of the sinr on a beam at user @xmath8 and the maximum sinr on a beam , respectively . since the maximum number of co - scheduled users in cell @xmath9 is equal to @xmath15 , increasing @xmath15 will have the effect of increasing the number of co - scheduled users . however , increasing @xmath15 will also increase the amount of intra - cell and inter - cell interference , which will decrease the rate of communication per beam . therefore , we focus on finding an achievable @xmath28 tr m - tuple with a set of qos constraints at all the bss that will ensure a guaranteed minimum rate per beam with a certain probability . to this end , we consider that an outage probability of @xmath29 can be tolerated at each bs , where the outage event refers to the received sinr of the scheduled user on a beam being below a target sinr threshold value @xmath30 , _ i.e. _ , @xmath31 for all @xmath9 . there is also a natural constraint on @xmath15 due to the orthogonality requirement among the beams , _ i.e. _ , @xmath32 for all @xmath9 . we focus on finding the achievable @xmath15s such that these constraints are not violated . this is a non - trivial problem for the system of interest due to the presence of intra - cell and inter - cell interferences , and the sinr values on a beam being not identically distributed among the users due to their different locations . we note that there is an implicit constraint that @xmath15 must be an integer . for the analysis , we will relax the integer constraint and assume that @xmath0 is sufficiently large such that the constraints @xmath33 is always satisfied for all @xmath9 . denote @xmath34 as an achievable tr m - tuple with the relaxed constraints ; the corresponding achievable @xmath28 m - tuple is given by @xmath35 for all @xmath9 , where @xmath36 represents the floor function . since the sinr on a beam is a strictly decreasing function of the tr , we have the following property ; given an achievable m - tuple @xmath37 , another m - tuple @xmath38 is achievable if @xmath39 for all @xmath40 . in the remaining parts of the paper , we will focus on finding the achievable trs and the achievable tr region , where the achievable tr region is defined to consist all the achievable @xmath34 m - tuples . we will call the constraints on @xmath41 the qos constraints . we will start our analysis with a simple single - cell scenario . we drop the cell index @xmath9 for brevity . for a single cell , the sinr expression in ( [ eq : sinr_expression ] ) reduces to @xmath42for a given pl value @xmath43 , by using techniques similar to those used in @xcite , it is not hard to show that @xmath44 is given by @xmath45 therefore , by conditioning on @xmath46 , the cdf of @xmath47 is given by @xmath48.\end{aligned}\ ] ] first we consider the simplest case where the user s are located equidistant to the bs , _ i.e. _ , the user s pl values are identical and deterministic , and given by @xmath49 . for this simplest case , a closed - form expression for the achievable tr can be obtained , and it is formally presented through the following theorem . [ thm : single_homo_iid ] for the system in consideration with @xmath50 and @xmath51 , the achievable trs are given by @xmath52where @xmath30 is the target sinr threshold value . with equal pl values @xmath49 , the qos constraint is given by @xmath53^k \leq p.\end{aligned}\]]solving for @xmath4 completes the proof . setting @xmath54 makes the result in theorem [ thm : single_homo_iid ] consistent with @xcite . now , we will consider the users to be heterogenous as in section [ sec : sys_model ] . we model the cell as a disk with radius @xmath55 . given the non - identical pl values , @xmath56 is given by ( [ eq : cdf_sc_noniid ] ) for this setup , and the qos constraint can be written as @xmath57 \leq p.\end{aligned}\ ] ] since the user locations are random in our setup , removing the conditioning of @xmath58 by averaging over the pl values gives us the qos constraint of interest . this idea is formally presented in the following lemma . [ lem : unbounded_path_model ] for the system in consideration with @xmath50 and the random pl values governed by the pl model @xmath12 for @xmath13 , the qos constraint is given by @xmath59^k \leq p,\end{aligned}\]]where @xmath30 is the target sinr threshold value and @xmath60 is the lower incomplete gamma function . since the users are located uniformly over the plane , the cdf of the distance from the user to its associated bs is given by @xmath61 . let @xmath62 denote the cdf of the pl value , which is given by @xmath63 . since the pl values are i.i.d . among the users , we have @xmath64^k.\end{aligned}\]]substituting for the cdfs and setting @xmath65 gives us @xmath66^k.\end{aligned}\]]evaluating the integral completes the proof @xcite . the achievable tr region consists of all the achievable @xmath67s that satisfy ( [ eq : qos_constraint_unbounded_pathloss_model ] ) . next , we will focus on the general multi - cell scenario . similar to what we have done in the previous section , we will start the analysis by obtaining an expression for the conditional distribution of the sinr on a beam at a user . this result is formally presented in the following lemma . [ lem : sinr_cdf ] consider user @xmath8 in cell @xmath9 . given the pl values from all the bss to user @xmath8 , _ i.e. _ , given @xmath68 , the conditional distribution of the sinr on a beam is given by @xmath69 the conditional distribution of the sinr can be obtained using a result in @xcite , which is summarized as follows . suppose @xmath70 , @xmath71 are independent exponentially distributed random variables ( rvs ) with parameters @xmath72 . then @xmath73 where @xmath74 is a constant . given all the pl values @xmath75 , @xmath26 can be re - written as @xmath76where @xmath77 is a constant , and @xmath78 and @xmath79 are independent exponentially distributed rvs with parameters @xmath80 and @xmath81 , respectively . therefore , directly using the result in @xcite completes the proof . using the above lemma , given all pl values , the conditional cdf of the maximum sinr on a beam can be written as @xmath82next , we will use this expression to find the achievable tr region considering different scenarios , similar to what we have done in section [ sec : l ] . for the clarity of presentation and the ease of explanation , we present the analysis for the two - cells scenario ; the analysis of the @xmath5-cells scenario can be easily extended using the same techniques . first we consider the classical wyner model @xcite for the two - cells scenario . the users pl values are deterministic as follows ; the pl value between all the users to their associated bs is unity , and the pl value between all the users to the interfering bs is @xmath83 . for this setup , the qos constraint for cell one is given by @xmath84^k \leq p,\end{aligned}\]]where @xmath30 is the target sinr threshold . the qos constraint for cell two can be easily obtained by interchanging @xmath85 and @xmath86 in the indices . analytical expressions that characterize the achievable tr region for this setup are formally presented through the following theorem . [ thm : wyner_model ] for the wyner model , given a fixed @xmath87 , the achievable trs for cell one is given by @xmath88where @xmath89 is the lambert - w function given by the defining equation @xmath90 , @xmath91 , @xmath92 , @xmath93 , @xmath94 , and @xmath30 is the target sinr threshold . with some simple manipulations , we can re - write the qos constraint in ( [ eq : qos_mc_iid_unequal ] ) @xmath95the following chain of inequalities holds which completes the proof . @xmath96 given a fixed @xmath97 , the achievable trs for cell two can be easily obtained by interchanging @xmath85 and @xmath86 in the indices . the achievable tr region consists of all the achievable @xmath98 tuples . when @xmath99 , the result in theorem [ thm : wyner_model ] can be further simplified , and the result is presented in the following corollary . [ cor : wyner_model ] for the wyner model , if @xmath99 , the achievable trs are given by @xmath100 next , we consider the users to be heterogeneous as in section [ sec : sys_model ] . for this scenario , if all the path loss values are given , the qos constraint for cell one can be written using as @xmath101 \leq p.\end{aligned}\]]the qos constraint for cell two can be easily obtained by interchanging @xmath85 and @xmath86 in the indices . since the user locations are random , we need to remove the conditioning on @xmath102 by averaging over the pl values . with multiple bss , the pl values between the user and each bs are correlated . hence , it is difficult to average over the pl values directly as in the single cell case because it is difficult to obtain the cdf of the path loss value . nonetheless , since the pl values are directly related to the distance between the user and each of the bss , we can perform a change of variables by writing each pl as a function of the user and bs locations , and then average over the location process by making use of the fact that the users are located uniformly over the plane . for the purpose of illustrating the idea , consider a user @xmath8 in cell one and let @xmath103 denote its exact location coordinate on the two dimensional plane . for convenience , we assume that a user is always connected to the closest bs geographically , _ i.e. _ , the two cells are arranged in a rectangular grid on the two dimensional plane . hence @xmath104 and @xmath105 are independent and uniformly distributed within the cell for all @xmath8 . let @xmath106 denote the location coordinate of bs @xmath9 . figure [ fig : two_cell_model ] illustrates the setup . the distance from the user to bs one and two is therefore @xmath107 and @xmath108 , respectively . thus the pl values are given by @xmath109 and @xmath110 , respectively . the following lemma presents the qos constraints for a two - cells scenario consisting of heterogeneous users with random pl values . [ lem : multi_cell_hetero ] for the system in consideration with bs @xmath9 being located at @xmath106 , given a fixed @xmath87 , the qos constraint of cell one is @xmath111 where @xmath30 is the target sinr , @xmath112 is the area of cell one , and @xmath113 is defined by the following integral @xmath114^{\alpha/2}\right)}{\left [ \left(\frac{(x - x_1)^2+y^2}{(x - x_2)^2+y^2}\right)^{\frac{\alpha}{2 } } \frac{l_1}{l_2}\eta+1\right]^{l_2 } } dx dy,\end{aligned}\]]and the integration is over the area of cell one . first we substitute @xmath115 and @xmath116 to ( [ eq : sinr_distribution ] ) to get @xmath117 . given a user s location coordinate @xmath103 , @xmath117 is given by @xmath118^{-\alpha/2 } } \right ) \\ & \left[(s+1)^{l_1 - 1 } \left ( \left(\frac{(x_{1,k}-x_1)^2+(y_{1,k})^2}{(x_{1,k}-x_2)^2+(y_{1,k})^2}\right)^{\frac{\alpha}{2 } } \frac{l_1}{l_2}s+1\right)^{l_2}\right]^{-1}.\end{aligned}\]]averaging ( [ eq : cdf_mc_noniid ] ) over cell one gives us @xmath119where @xmath120 is the joint pdf of @xmath104 and @xmath105 . since the location coordinates are i.i.d . among the users , we have @xmath121^k.\end{aligned}\]]substituting for @xmath122 completes the proof . given a fixed @xmath97 , the qos constraint for cell two can be easily obtained by interchanging @xmath85 and @xmath86 in the indices . the achievable tr region is given by all the @xmath98 tuples that satisfy the qos constraints for both cells . in this section , we present our numerical results for the single cell and two - cell scenarios . in all the simulations , the cell is modeled as a disk with radius @xmath55 for the single cell scenario , and each cell is modeled as a square with cell area @xmath123 for the two - cell scenario . in figures [ fig : region_homo1]-[fig : region_hetero ] , we show the achievable tr regions for the two - cells scenario . figures [ fig : region_homo1 ] and [ fig : region_homo2 ] show the achievable tr regions for the wyner model with @xmath124 and @xmath54 , respectively . the dotted line connecting the origin and the corner point in each region represents the achievable tr set given in corollary [ cor : wyner_model ] . figure [ fig : region_hetero ] shows the achievable tr regions for heterogeneous users , using the result of lemma [ lem : multi_cell_hetero ] . for a given @xmath125 , any @xmath126 below the boundary can be achieved , whereas any @xmath126 above the boundary will violate the qos constraints . moreover , if the system wants to maximize the multiplexing gain at each bs , operating at @xmath127 strictly below the boundary is sub - optimal in a sense that we can further increase the trs without violating the qos constraints . therefore , the boundary curve can be considered as the pareto optimal boundary between the achievable and un - achievable tr pairs . as can be observed from the figures , tr region expands when the qos constraints are relaxed , _ i.e. _ , @xmath29 is increased and/or @xmath30 is decreased . relaxing the qos constraints allows more interference in the network , thus expanding the achievable tr region . moreover , the achievable tr region also expands when @xmath2 is increased . the achievable rate on a beam increases due to multi - user diversity , therefore , more beams / interference can be tolerated without violating the constraints . the achievable tr region will also change with @xmath55 and @xmath128 and will be discussed further in figures [ fig : vsk]-[fig : vssnr ] . .,width=259 ] .,width=259 ] . , width=259 ] let @xmath129 denote the maximum achievable tr with the relaxed constraints on @xmath4 . figure [ fig : vsk ] shows @xmath129 vs. @xmath2 , for the single - cell scenario and two - cells scenario with equal trs . as can be observed from the figure , for a fixed @xmath2 , @xmath129 decreases as the cell size increases . this is because the users are uniformly located in the cell and as the cell size increases , the users locations will be more spread out . as a consequence , the sinr on each beam will decrease and we must compensate this by decreasing the tr ( to decrease the interferences ) . figure [ fig : vssnr ] shows @xmath129 vs. snr for fixed number of users , where snr is defined as @xmath130 . as can be observed from the figure , @xmath129 increases with snr . intuitively , when @xmath128 decreases , we can increase the tr ( effectively introduces additional interferences ) while still satisfying the qos constraints . therefore , the achievable tr region will expand with decreased cell size or increased snr . finally , @xmath129 decreases as @xmath5 , the number of cells , increases . this is because the sinr on each beam decreases with @xmath5 . for single cell and two - cell scenarios with @xmath131.,width=288 ] in this paper , we considered a multi - cell multi - user miso broadcast channel . each cell employs the obf scheme with variable trs . we focused on finding the achievable trs for the bss to employ with a set of qos constraints that ensures a guaranteed minimum rate per beam with a certain probability at each bs . we formulated this into a feasibility problem for the single - cell and multi - cell scenarios consisting of homogeneous users and heterogeneous users . analytical expressions of the achievable trs were derived for systems consisting of homogeneous users and for systems consisting of heterogeneous users , expressions were derived which can be easily used to find the achievable trs . an achievable tr region was obtained , which consists of all the achievable tr tuples for all the cells to satisfy the qos constraints . numerical results showed that the achievable tr region expands when the qos constraints are relaxed , the snr and the number of users in a cell are increased , and the size of the cells are decreased . 99 p. viswanath , d. n. c. tse and r. laroia , `` opportunistic beamforming using dumb antennas , '' _ ieee trans . inf . theory _ , 1277 - 1294 , jun . 2002 . m. sharif and b. hassibi , `` on the capacity of mimo broadcast channels with partial side information , '' _ ieee trans . inf . theory _ 506 - 522 , feb . 2005 . m. wang , f. li and j. s. evans , `` opportunistic beamforming with precoder diversity in multi - user mimo systems , '' in _ proc . ieee vehicular technol . dresden , germany _ , jun . 2 - 5 , 2013 . j. wagner , y .- c . liang and r. zhang , `` random beamforming with systematic beam selection , '' in _ proc . symp . on personal , indoor and mobile radio commun . , helsinki , finland _ , pp . 1 - 5 , sept . 11 - 14 , 2006 . j. l. vicario _ et al . _ , `` beam selection strategies for orthogonal random beamforming in sparse networks , '' _ ieee trans . wireless commun . _ , vol . 7 , no . 9 , pp . 3385 - 3396 , sept . et al . _ , `` transmission mode selection in a heterogeneous network using opportunistic beamforming , '' in proc . _ ieee global commun . atlanta , ga _ , 9 - 13 , 2013 . y. huang and b. rao , `` performance analysis of random beamforming with heterogeneous users , '' in _ proc . annual conf . on inf . sciences and systems , princeton , nj _ , pp . 1 - 5 , mar . 21 - 23 , 2012 . t. samarasinghe , h. inaltekin and j. s. evans , `` outage capacity of opportunistic beamforming with random user locations , '' in _ proc . ieee global commun . atlanta , ga _ , 9 - 13 , 2013 . m. wang , t. samarasinghe and j. s. evans , `` multi - cell opportunistic beamforming in interference - limited networks , '' in _ proc . australian commun . theory workshop , sydney , australia _ , feb . 3 - 5 , 2014 . n. enderl and x. lagrange , `` user satisfaction models and scheduling algorithms for packet - switched services in umts , '' in _ proc . ieee vehicular technol . conf . , jeju , korea _ , pp . 1704 - 1709 , apr . 22 - 25 , 2003 . n. zorba and a. i. prez - niera , `` robust power allocation scheme for multibeam opportunistic transmission strategies under quality of service constraints , '' _ ieee j. on sel . areas commun . 26 , no . 8 , pp . 1025 - 1034 , aug . n. zorba and a. i. prez - neira , `` optimum number of beams in multiuser opportunistic scheme under qos constraints , '' in _ proc . ieee workshop on smart antennas , vienna , austria _ , feb . 26 - 27 , 2007 . d. n. c. tse and p. viswanath , _ fundamentals of wireless communications_. cambridge , u.k . : cambridge univ . press , 2005 . t. samarasinghe , h. inaltekin and j. s. evans , `` the feedback - capacity tradeoff for opportunistic beamforming under optimal user selection , '' _ performance evaluation _ , 70 , issues 7 - 8 , pp . 472 - 492 , jul . y. huang and b. rao , `` multicell random beamforming with cdf - based scheduling : exact rate and scaling laws , '' in _ proc . ieee vehicular technol . conf . , las vegas , nv _ , sept . 2 - 5 , 2013 . i. gradshteyn and i. ryzhik , _ table of integrals , series , and products seventh edition . _ academic press , 2007 . s. kandukuri and s. boyd , `` optimal power control in interference - limited fading wireless channels with outage - probability specifications , '' _ ieee trans . wireless commun . _ , vol . 1 , , jan . 2002 . a. d. wyner , `` shannon - theoretic approach to a gaussian cellular multiple - access channel , '' _ ieee trans . inf . theory _ , 1713 - 1727 , nov .

in this paper , we consider a multi - cell multi - user miso broadcast channel . the system operates according to the opportunistic beamforming framework in a multi - cell environment with variable number of transmit beams ( may alternatively be referred as the transmission rank ) at each base station . the maximum number of co - scheduled users in a cell is equal to its transmission rank , thus increasing it will have the effect of increasing the multiplexing gain . however , this will simultaneously increase the amount of interference in the network , which will decrease the rate of communication . this paper focuses on optimally setting the transmission rank at each base station such that a set of quality of service ( qos ) constraints , that will ensure a guaranteed minimum rate per beam at each base station , is not violated . expressions representing the achievable region of transmission ranks are obtained considering different network settings . the achievable transmission rank region consists of all achievable transmission rank tuples that satisfy the qos constraints . numerical results are also presented to provide further insights on the feasibility problem .

1405.2387

sciboone @xcite is a muon neutrino scattering experiment located at the boone neutrino beam at fermilab . the 0.8 gev mean energy neutrino beam is produced with a 8 gev proton beam . protons hit a beryllium target producing charged pions that are selected and focused using a magnetic horn . the ability to switch the horn polarity allows to select @xmath1 to produce neutrino beam or @xmath2 to produce anti - neutrino beam . only neutrino beam is currently used in this analysis . sciboone detector consists in three sub - detectors : the main detector scibar , the electromagnetic calorimeter ec , and the muon range detector mrd. * scibar@xcite is a fully active and fine grained scintillator detector that consists in 14,336 bars arranged in vertical and horizontal planes . scibar is capable to detect all charged particles and perform de / dx based particle identification . * the electron catcher ( ec)@xcite , is a lead - scintillator calorimeter consisting in two planes , one vertical and one horizontal , with a width corresponding to 11 @xmath3 . * the mrd@xcite , consists in 12 steel plates sandwiched between vertical and horizontal planes of scintillator . the mrd has the capability to stop muons with momentum up to 1.2 gev . the mrd detector is used in this analysis to define charged current events by tagging the outgoing muon . the current analysis is covering scibar contained events , which means that events with particles other than muons escaping from scibar detector are not being considered . ec detector will be introduced in the analysis in the near future allowing us to use events with particles escaping from scibar in the forward direction and reaching the ec . neut @xcite event generator is used in this analysis . the rein - sehgal model is implemented to simulate charged current resonant pion production with an axial mass @xmath4 gev/@xmath5 . all resonances up to 2 gev are taken into account . however @xmath6 is the resonance that more largely contributes to the @xmath0 production . a cc-@xmath0 event is defined in this analysis as such event that contains at least a muon and a neutral pion coming out from the interaction vertex . this definition includes neutral pions generated by secondary interactions inside the target nucleus as , for instance , charge exchanges . though the @xmath0 decays almost immediately to two photons , and those produce em cascades with an average flight distance of 25 cm , topologically a cc-@xmath0 scibar contained event contains a muon reaching the mrd and two or more tracks contained in scibar ( see fig . [ fig : event ] ) . the non - muon tracks are considered gamma candidates and are used to , at the end , reconstruct the neutral pion . event . muon track in green , reconstructed em showers in yellow and blue . ] given the signal definition we can use some event topology and track property based cuts in order to reduce the background events in the sample ( see table [ tab : summary ] for summary ) . the chosen filters are applied sequentially as follows : * scibar uses a cc event definition based on the muon tagging using the mrd . then , the first applied selection is over events that contains a track reaching the mrd tagged as a muon . because we do nt expect any other particle to reach the mrd , we also require only one tagged muon in the event . * given that we are selecting scibar contained events , we use a veto filter to dismiss events with outgoing tracks . the veto filter applies to events with outgoing tracks either from the upstream or the sides of the detectors . the veto filter does not apply on tracks pointing to the ec because those tracks will be fully reconstructed once the ec information will be used . the veto filter is also useful in order to remove events with in - going tracks originated in interactions outside the detector ( called dirt interactions ) . * as discussed before , we expect events with 3 tracks in scibar , the muon and the 2 electromagnetic cascades from the pion decay . we thus use a filter to meet this topology . * we also use a time based filter in order to avoid cosmic rays and dirt generated tracks in our selected events . this filter requires that the photon candidates should match the muon time with a difference of 20 ns or less . * as commented before , we use the scibar de / dx capability in order to separate minimum ionizing particles as muons or photons from protons . most protons are rejected using this filter . * finally , a cut is placed requiring that the photon tracks should be disconnected from the event vertex taking advantage of the larger photon flight distance . this cut is particularly useful to reject protons and charged pions , which track starts always from the event vertex . .event selection summary . [ cols= " < , > , > , > , > " , ] after the above commented cuts , we get reconstructed photons with a typical energy between 50 and 200 mev ( see fig . [ fig : photone ] ) . also , for correctly associated photon candidates , the energy is reconstructed with 100 mev resolution and small bias . the photons are reconstructed at all angles . once we have the 2 reconstructed gammas , we are able to reconstruct also the @xmath0 observables . in particular we reconstructed the invariant mass and also the momentum and angle . as you can see in fig . [ fig : angle ] neutral pions are produced at all angles with a momentum in 50 - 300 mev / c range . it is also visible a peak in the invariant mass plot near the @xmath0 mass ( fig . [ fig : mass ] ) . from the plots we can see that our neut - based mc reproduces well the @xmath0 observables . reconstructed mass . mc background broken in events with neutral pion and events without neutral pion . mc normalized to cc - inc events . ] [ fig : angle ] since the poster was presented , some reconstruction improvements have been performed , in this section we are going to discuss them . the track reconstruction in scibar is performed by a cellular automaton which essentially travels among the beam direction connecting hits to create tracks . the first reconstruction improvement was to implement in the code a second run of the automaton but this time traveling in the transversal direction , that is perpendicular to the beam , and using the hits that are not associated to any track from the first processing . in this way we found abut a 10% more events containing 3 or more tracks and also we got the ability to reconstruct tracks at larger angle , close to 90 degrees like in fig . [ fig : sbtcat_event ] . this has been an important upgrade given the low statistics of the analysis mainly due to the lack of events with 3 or more reconstructed tracks . a second improvement is to use a new algorithm that improves the reconstruction performance of the em cascades . the em cascades in scibar are characterized by disconnected track segments and isolated hits , making difficult to recover and correctly associate all the photon visible energy . the new algorithm seeks for those disconnected track segments and merge them into a single extended track via an energy - flow algorithm . it also seeks for hits around the gamma candidate track in order to add the energy coming from those hits to the gamma track . i this way , the photon energy reconstruction is improved and so is the @xmath0 observables . by using the new algorithm we find a narrower invariant mass peak with less low mass @xmath0 than in the fig . [ fig : mass ] . also , the bias in the @xmath0 momentum is reduced and it can be observed an increment of the high momentum pions . the sciboone collaboration gratefully acknowledges support from various grants and contracts from the department of energy ( u.s . ) , the national science foundation ( u.s . ) , the mext ( japan ) , the infn ( italy ) and the spanish ministry of education and science .

sciboone , located in the booster neutrino beam at fermilab , collected data from june 2007 to august 2008 to accurately measure muon neutrino and anti - neutrino cross sections on carbon below 1 gev neutrino energy . sciboone is studying charged current interactions . among them , neutral pion production interactions will be the focus of this poster . the experimental signature of neutrino - induced neutral pion production is constituted by two electromagnetic cascades initiated by the conversion of the @xmath0 decay photons , with an additional muon in the final state for cc processes . in this poster , i will present how we reconstruct and select charged - current muon neutrino interactions producing @xmath0 s in sciboone address = ific ( u. valencia / csic )

0910.5627

legendrian contact homology has been an effective tool in studying legendrian submanifolds in @xmath7 in @xmath8 , chekanov @xcite and eliashberg and hofer ( unpublished but see @xcite ) used contact homology to show that legendrian knots are not determined up to legendrian isotopy by the so - called classical invariants ( topological isotopy class , thurston - bennequin number , and maslov class ) . subsequently , contact homology has been used to greatly illuminate the nature of legendrian knots in @xmath9 the contact homology of legendrian submanifolds in @xmath10 ( for @xmath11 ) was given a rigorous foundation in @xcite and its efficacy was demonstrated in @xcite . very roughly speaking contact homology is the homology of a differential graded algebra ( dga ) associated to a legendrian submanifold @xmath12 . the algebra is generated by double points in the ( lagrangian ) projection of @xmath13 into @xmath5 and the differential counts rigid holomorphic disk with corners at these double points and boundary on the projected legendrian submanifold . in the initial definition of contact homology the disks were counted modulo 2 since in that version of the theory orientations and orientability of spaces of holomorphic disks need not be considered . a @xmath4-lift of contact homology of legendrian knots in @xmath8 have been introduced in a purely combinatorial fashion in @xcite . it is however still not known if the oriented version of the theory in this case is any stronger that the unoriented version of the theory . orientations for the moduli space of certain riemann surfaces without boundary has been discussed in @xcite . in this paper we show how to lift the dga of legendrian submanifolds , of @xmath14 , which are spin to @xmath4 . we demonstrate that this lift gives a more refined invariant of legendrian isotopy than does the theory over @xmath2 in dimensions @xmath15 . for legendrian knots in @xmath8 , our analytical approach to orientations recovers the combinatorial sign rule of @xcite and furthermore gives rise to another combinatorial sign rule not mentioned there . we also use legendrian contact homology to produce lower bounds on the double points of exact lagrangian immersions into @xmath16 ( a lagrangian immersion @xmath17 is _ exact _ if the closed form @xmath18 , where @xmath19 are standard coordinates on @xmath20 is exact . ) generically an exact lagrangian immersion can be lifted to a legendrian embedding . a dga is called good if it is ( tame ) isomorphic to a dga without constant terms in its differential @xcite . we show that if @xmath17 is an exact self - transverse lagrangian immersion of a closed manifold such that the dga associated to a legendrian lift of @xmath21 is good then the number @xmath22 of double points of @xmath21 satisfies @xmath23 where @xmath24 or @xmath25 for any prime @xmath26 if @xmath6 is spin and where @xmath27 otherwise . it is easy to construct exact lagrangian immersions of spheres and tori of arbitrary dimensions which shows that the estimate is the best possible . while the hypothesis on the exact lagrangian immersion seems somewhat unnatural it is frequently satisfied and from anecdotal evidence one would expect exact lagrangian immersions with non - good dga s to have more double points than ones with good dga s . despite this evidence it does not seem straightforward to use contact homology for estimates when the algebra is not good . however , we prove that if one can establish an estimate like with any fixed constant subtracted from the right hand side then is true too . the paper is organized as follows . in section [ sec : basnot ] we introduce basic notions which will be used throughout the paper . in section [ sec : orimdli ] we show how to orient moduli spaces of holomorphic disks relevant to contact homology . to accomplish this we discuss orientations of determinant bundles over spaces of ( stabilized ) @xmath28-operators associated to legendrian submanifolds and their interplay with orientions of spaces of conformal structures on punctured disks . similar constructions are carried out in @xcite but some of the details differ . in section [ sec : legch ] we define the dga associated to a legendrian spin submanifold @xmath13 as an algebra over @xmath29 $ ] with differential @xmath30 and prove that @xmath31 . furthermore we prove the invariance of contact homology under legendrian isotopy by a mixture of a homotopy method and the more direct bifurcation analysis , making use of the stabilization mentioned above . ( over @xmath2 this invariance proof gives an alternative to the invariance proof given in @xcite . ) we also describe how the contact homology depends on the choice of spin structure of the legendrian submanifold and we derive diagrammatic sign rules for legendrian knots in @xmath8 . in section [ sec : lmt ] , we adapt a theorem of floer @xcite to our situation so that , in special cases , the differential in contact homology can be computed . we also apply these results to construct examples which demonstrates that contact homology over @xmath4 is more refined than contact homology over @xmath2 . in section [ sec : dbpt ] we prove the results related to the double point estimate for exact lagrangian immersion mentioned above . acknowledgments : the authors are grateful to lenny ng for many useful discussions concerning the sign rules in dimension three . we also thank aim who provided some support during a workshop where part of this work was completed . part of this work was done while te was a research fellow of the swedish royal academy of sciences sponsored by the knut and alice wallenberg foundation . ms was partially supported by an nsf vigre grant as well as nsf grant dms-0305825 . he also thanks the university of michigan and msri for hosting him while working on this paper . je was partially supported by nsf grant dms-0203941 , an nsf career award ( dms0239600 ) and frg-0244663 . in this section we introduce notation and briefly describe the construction of the contact homology of a legendrian submanifold in @xmath14 . for more details on this construction , see @xcite . let @xmath32 be standard coordinates on @xmath7 throughout this paper we consider the standard contact structure @xmath33 on @xmath14 and we make use of the following two projections : the _ front projection _ @xmath34 which forgets the @xmath35-coordinates and the _ lagrangian projection _ @xmath36 which forgets the @xmath37-coordinate . note that if @xmath38 is a legendrian submanifold then @xmath39 is a lagrangian immersion . maslov class _ of a lagrangian immersion @xmath40 is the homomorphism @xmath41 such that @xmath42 is the maslov index of the loop of lagrangian planes which consists of tangent planes to @xmath43 along some loop @xmath44 representing @xmath45 . the _ maslov number _ @xmath46 of @xmath47 is the positive integer which is a generator of the image of @xmath48 if @xmath49 and @xmath50 if @xmath51 . if @xmath52 is a legendrian immersion we sometimes write @xmath53 for the maslov number of the lagrangian immersion @xmath54 we define the contact homology of a spin legendrian submanifold @xmath13 in @xmath55 equipped with a spin structure as an algebra over the group ring @xmath29 $ ] if @xmath13 is connected and as an algebra over @xmath4 otherwise in the following way . consider first the connected case . assume that @xmath13 is generic with respect to the lagrangian projection and let @xmath56 be the free associative ( non - commutative ) algebra over @xmath29 $ ] generated by the ( transverse ) double points of @xmath57 there is a @xmath4 grading on this algebra defined as follows . for a double point @xmath58 denote the two points in @xmath13 mapping to @xmath58 by @xmath59 and @xmath60 where @xmath59 has the larger @xmath37-coordinate . choose , for each double point @xmath61 a path @xmath62 in @xmath13 that runs from @xmath59 to @xmath63 the conley - zehnder index @xmath64 of @xmath58 is the maslov index of the loop of lagrangian subspaces in @xmath5 which is the path of planes tangent to @xmath65 along @xmath66 closed up in a specific way , see @xcite . the grading of @xmath58 is defined as @xmath67 and the grading of a homology class @xmath68 is defined as the negative of the maslov index of the loop of lagrangian subspaces in @xmath5 which are tangent to @xmath65 along some loop @xmath69 representing @xmath45 . if one defines the algebra over @xmath4 instead of @xmath29 $ ] the grading is only in @xmath70 . in the disconnected case @xmath56 is graded over @xmath2 , see subsection [ defofal ] . ( there is also a relative @xmath70-grading which we will not discuss in this paper . ) the differential of the algebra @xmath56 , @xmath71 lowers grading by @xmath72 and is defined by counting holomorphic disks in @xmath5 with boundary on @xmath65 . more precisely , for @xmath73 , a double point of @xmath65 , @xmath74 , a word in the double points of @xmath65 , and @xmath68 , we consider the moduli space @xmath75 . this space consists of holomorphic maps from the punctured unit disk to @xmath5 which maps the boundary to @xmath65 , which are asymptotic , see @xcite , to the double points specified at the punctures , and which when restricted to the boundary satisfies a certain homology condition specified by @xmath45 . if @xmath13 is connected then the differential on @xmath56 is @xmath76 where the sum runs over all @xmath74 such that @xmath77 is @xmath78-dimensional . the moduli spaces which are @xmath78-dimensional are compact manifolds . in section [ oms ] we orient @xmath77 . this means in particular that each component of a @xmath78-dimensional moduli space comes equipped with a sign and @xmath79 in is the algebraic number of points in the moduli space . if @xmath13 is not connected then the differential is defined as in except that the homology class @xmath45 there should be deleted . the _ contact homology _ of @xmath13 is @xmath80 . finally , we note that if @xmath12 is a legendrian submanifold then double points of @xmath81 correspond to segments in the @xmath82-direction of @xmath83 with its endpoints on @xmath13 . the vector field @xmath84 is the reeb field of the contact form @xmath85 and thus such a segment is called a reeb chord . we will therefore use the words _ reeb chord _ and _ double point _ interchangeably below . in this section we orient moduli spaces of holomorphic disks in @xmath5 with boundary on @xmath65 , where @xmath12 is a legendrian submanifold which is spin . to orient the moduli spaces we find an orientation of the determinant bundle of a stabilized version of the linearization of the defining @xmath28-equation . the source space of the linearization splits into an infinite dimensional space and a finite dimensional space arising from automorphisms or variations of the conformal structure of the source disk . the restriction of the linearization to the infinite dimensional space is a fredholm operator . we orient the determinant bundles over spaces of such fredholm operators and orient spaces of automorphisms and conformal structures separately . the orientations we define depend on several choices . we make these choices so that it is possible to endow the graded algebra discussed in section [ sec : basnot ] with a differential . in particular , this differential must respect the multiplication of the algebra in the sense that it satisfies the graded leibniz rule , see equation . let @xmath86 denote the unit disk @xmath87 in @xmath88 with @xmath89 distinct punctures @xmath90 on the boundary , @xmath91 . the orientation on @xmath87 induces an orientation on its boundary . if one puncture , @xmath92 say , is distinguished then the orientation on @xmath93 induces an order of the punctures . the punctures subdivide @xmath93 into @xmath89 disjoint oriented open arcs . we denote their closures by @xmath94 , with notation such that @xmath95 ( where @xmath96 and @xmath97 ) . for convenience we write @xmath98 and @xmath99 for @xmath100 thought of as a point in @xmath101 and @xmath102 , respectively . ( we let @xmath103 , @xmath104 , and in the special case when @xmath105 , @xmath106 . ) a _ lagrangian boundary condition _ on @xmath86 is a collection of maps @xmath107 where @xmath108 , where @xmath109 denotes the space of lagrangian subspaces of @xmath5 . for non - punctures @xmath110 we write @xmath111 to denote @xmath112 where @xmath113 is the unique subscript such that @xmath114 . if @xmath26 is a puncture then @xmath115 and @xmath116 . we write @xmath117 and @xmath118 . an _ oriented lagrangian boundary condition _ on @xmath86 is a collection of maps @xmath119 where @xmath120 , where @xmath121 denotes the space of oriented lagrangian subspaces of @xmath5 . note that any lagrangian boundary condition on a disk with at least one puncture lifts ( non - uniquely ) to an oriented boundary condition , and that there are boundary conditions on the @xmath78-punctured disk which do not lift . a _ trivialized lagrangian boundary condition _ for the @xmath28-operator on @xmath86 is a collection of maps @xmath122 where @xmath123 . a trivialized lagrangian boundary condition induces a lagrangian boundary condition @xmath124 via @xmath125 and a trivialization @xmath126 of a lagrangian boundary condition ( i.e. an on - basis in @xmath124 ) gives a trivialized lagrangian boundary condition by defining @xmath127 as the matrix with column vectors @xmath126 . fixing an orientation on @xmath128 , a trivialized boundary condition can be considered as an oriented boundary condition . neighborhoods of the punctures of @xmath86 will be thought of as infinite half strips @xmath129 $ ] or @xmath130\times[0,1]$ ] with coordinates @xmath131 , and we consider the @xmath28-operator @xmath132(d_m,{{\mbox{\bbb c}}}^n)\to { { \mathcal{h}}}_{1,\epsilon}[0](d_m , { t^\ast}^{0,1}d_m\otimes{{\mbox{\bbb c}}}^n),\ ] ] where @xmath133 , and @xmath134 $ ] is the closed subspace of the sobolev space @xmath135 , with weight @xmath136 in the neighborhood of the @xmath137 puncture , which consists of elements @xmath138 with @xmath139 for @xmath114 , and @xmath140 on @xmath93 and where @xmath141 $ ] is the closed subspace of the sobolev space @xmath142 which consist of elements with vanishing trace ( restriction to the boundary ) . see @xcite , section 5 . we denote the operator in by @xmath143 . often in our applications , the weight will be clear from the context and we drop it from the notation writing simply @xmath144 . we recall the following notion from @xcite . let @xmath145 and @xmath146 be lagrangian subspaces of @xmath5 . define the _ complex angle _ @xmath147 inductively as follows . if @xmath148 let @xmath149 and let @xmath150 denote the hermitian complement of @xmath151 and let @xmath152 for @xmath153 . if @xmath154 then let @xmath155 , @xmath153 and let @xmath156 . then @xmath157 and @xmath158 are lagrangian subspaces . let @xmath159 be smallest angle such that @xmath160 . let @xmath161 . repeat the construction until @xmath162 has been defined . let @xmath163 be a lagrangian boundary condition . let @xmath164 denote the complex angle of @xmath165 and @xmath166 . as shown in @xcite , the operator @xmath143 is fredholm when @xmath167 the _ determinant line _ @xmath168 of a fredholm operator @xmath169 between banach spaces is defined as the tensor product of the highest exterior power of its kernel and the highest exterior power of the dual of its cokernel , respectively . that is , @xmath170 let @xmath171 denote the space of all lagrangian boundary conditions and weights for which is satisfied . it is a standard result that there exists a determinant line bundle @xmath172 over @xmath173 with fiber over @xmath174 equal to @xmath175 . the bundle @xmath172 is not orientable . in fact , the corresponding determinant bundle over oriented lagrangian boundary conditions on the @xmath78-punctured disk is non - orientable . what is needed for orientability in that case are trivialized boundary conditions , see lemma [ lmacanor ] . we say a boundary condition @xmath176 , @xmath177 is _ transverse _ if @xmath178 and @xmath179 are transverse for each puncture @xmath26 . we will subdivide the punctures on a disk with such boundary conditions into four classes . the first subdivision seems rather ad hoc at this point but in our applications to legendrian submanifolds it has a clear geometric meaning : each puncture has a sign . that is , there are _ positive _ and _ negative _ punctures . ( in the applications , disks have one positive puncture and all other negative . ) the second subdivision comes from the _ parity _ of a puncture defined in the following way . let @xmath145 and @xmath146 be two _ oriented _ transverse lagrangian subspaces of @xmath5 . let @xmath180 be the complex angle of @xmath181 . then there exists complex coordinates @xmath182 on @xmath5 so that in these coordinates @xmath183 and @xmath184 we will call such coordinates _ canonical coordinates of @xmath181_. note that the canonical coordinates are not unique . however , in the constructions below the choice of specific canonical coordinates will be irrelevant . for example , the unitary linear map of @xmath5 with matrix @xmath185 in canonical coordinates is well - defined and maps @xmath145 isomorphically to @xmath146 . the ordered pair @xmath181 is _ even ( odd ) _ if is an orientation reversing ( preserving ) map from @xmath145 to @xmath186 consider a transverse oriented boundary condition @xmath187 . if @xmath26 is a negative puncture then @xmath26 is _ even _ if the pair @xmath188 is even and @xmath26 is _ odd _ if @xmath188 is odd . if @xmath189 is a positive puncture then @xmath189 is _ even _ ( _ odd _ ) if the pair @xmath190 is even ( odd ) . we define gluing operations for the boundary conditions discussed in subsection [ ssec : boundcond ] and explain how orientations of the determinant lines over the pieces relate to an orientation of the determinant line of the resulting glued boundary condition . more precisely , the gluing operations give rise to exact sequences involving kernels and cokernels of operators and the orientations of the determinant lines are related via these . in our applications , the exact form of the relations is important . we therefore start out with explaining our conventions for exact sequences of oriented vector spaces . all our gluing operations give rise to exact sequences with at most four non - zero terms so we discuss only this case . let @xmath191 be an exact sequence of finite dimensional vector spaces . this sequence induces an isomorphism @xmath192 our interest in this isomorphism is its effect on orientations . note that there is a natural correspondence between orientations on a finite dimensional vector space @xmath193 and on its dual @xmath194 . to see this , think of an orientation of @xmath193 as a non - zero element in @xmath195 up to multiplication by a positive number . the correspondence can then be obtained as follows . pick any basis @xmath196 in @xmath193 and let @xmath197 be the dual basis in @xmath194 . now identify the orientation in @xmath193 given by @xmath198 with the orientation in @xmath194 given by @xmath199 it is easy to see that this identification is independent of the choice of basis . if @xmath193 and @xmath200 are finite dimensional vector spaces then the correspondence just described gives rise to a correspondence between orientations of the @xmath72-dimensional vector space @xmath201 and pairs @xmath202 of orientations of @xmath193 and @xmath200 respectively , modulo the following equivalence relation : a pair represented by @xmath203 is identified with the pair represented by @xmath204 for any non - zero @xmath205 . we call an equivalence class an _ orientation pair_. using this terminology , we interpret the isomorphism on the orientation level as follows : the sequence gives a correspondence between orientation pairs of @xmath181 with orientation pairs of @xmath206 . we next give a concrete description of this correspondence . let @xmath207 be an orientation pair on @xmath181 . pick bases @xmath208 in @xmath145 and @xmath209 in @xmath146 such that @xmath210 represents @xmath207 . pick vectors @xmath211 such that @xmath212 is a basis of @xmath213 . by exactness , the vectors @xmath214 are linearly independent in @xmath215 . let @xmath216 be any vectors in @xmath215 such that @xmath217 for all @xmath113 . define @xmath218 to be the orientation pair represented by @xmath219 ( note the _ reverse order _ of the vectors @xmath220 . ) it is easy to see that @xmath218 is well defined . the inverse @xmath221 of @xmath47 can be described as follows . let @xmath222 be any orientation pair of @xmath206 . pick bases @xmath223 in @xmath213 and @xmath224 in @xmath215 such that @xmath225 represents @xmath222 . pick any basis @xmath208 in @xmath145 then there are vectors @xmath226 in @xmath213 so that @xmath227 represent the same orientation on @xmath213 . furthermore , there are vectors @xmath228 such that @xmath229 represent the same orientation on @xmath215 . define @xmath230 to be the orientation pair on @xmath181 represented by @xmath231 it is easy to verify that @xmath221 is well defined and that @xmath221 and @xmath47 are inverses of each other . let @xmath176 be a transverse boundary condition on the punctured disk . let @xmath144 be the operator with this boundary condition and with trivial weights . then , by proposition 5.14 in @xcite , @xmath232 where @xmath233 is the maslov index and where @xmath234 is the loop in @xmath109 obtained from the map @xmath163 as follows . let @xmath235 denote the complex angle between @xmath165 and @xmath166 then there exists coordinates @xmath19 in @xmath5 such that @xmath236 and @xmath237 let @xmath238 , @xmath239 , be the path of lagrangian subspaces given in these coordinates as @xmath240 then @xmath234 is the loop obtained by concatenation of the paths in @xmath124 with the paths @xmath241 . that is , @xmath242 let @xmath243 and @xmath244 be trivialized lagrangian boundary conditions . let @xmath245 and @xmath246 be points which are not punctures and assume that @xmath247 . we assume also that @xmath45 and @xmath248 are constant in neighborhoods of @xmath26 and @xmath189 , respectively . ( if this is not the case , we may homotope the boundary conditions so that they become constant around @xmath26 and @xmath189 . note that such homotopies may change the dimensions of the kernels and cokernels of the operators involved . however , if the homotopy can be chosen sufficiently @xmath249-small then the dimensions can be kept constant . ) we glue @xmath86 and @xmath250 to one disk @xmath251 with a trivialized lagrangian boundary condition which is the concatenation of @xmath45 and @xmath248 . to define this operation more accurately we proceed as follows . puncture the disks @xmath86 at @xmath26 and @xmath250 at @xmath189 . use conformal coordinates @xmath129 $ ] in a neighborhood of @xmath26 in @xmath86 and conformal coordinates @xmath130\times [ 0,1]$ ] in a neighborhood of @xmath189 in @xmath250 . since the boundary conditions @xmath45 and @xmath248 are constant in neighborhoods of @xmath26 and @xmath189 , respectively , it follows that for all sufficiently large @xmath252 the boundary condition @xmath45 is constant in @xmath253)$ ] and the boundary condition @xmath248 is constant in @xmath254\times({\partial}[0,1])$ ] . for such @xmath255 we define a new disk @xmath256 with @xmath257 punctures as follows . let @xmath258)$ ] , let @xmath259\times[0,1])$ ] , and let @xmath256 be the disk which is obtained from gluing @xmath260 to @xmath261 by identifying @xmath262 $ ] with @xmath263 $ ] . note that the boundary conditions glue in a natural way to a boundary condition @xmath264 on @xmath256 . since the maslov index is additive under this gluing we find that @xmath265 we call this type of gluing _ gluing at a boundary point_. [ lmaglue1 ] for all sufficiently large @xmath255 there exists an exact sequence @xmath266 where @xmath128 is naturally identified with @xmath267 . ( we describe the maps @xmath268 , @xmath269 , and @xmath270 and the identification in remark [ rmkmapsinseq ] below . ) in particular , together with an orientation on @xmath128 the above sequence induces an isomorphism between @xmath271 and @xmath272 [ fo^3 ] the special case @xmath273 of this lemma ( together with lemmas [ lmacanor ] and [ lmagluecanor ] below ) fixes a misstatement in lemma 23.5 of @xcite ( on page 206 ) . consider two bundles of index @xmath78 with no kernel and no cokernel then lemma 23.5 implies that the kernel and cokernel of the glued problem are trivial as well . however this can not be correct since the problem has index @xmath274 by . using sobolev spaces with a small negative exponential weight @xmath275 , @xmath276 at the punctures @xmath245 and @xmath277 ( i.e. in coordinates @xmath278 $ ] the weight functions @xmath279 and @xmath280 equal @xmath281 in neighborhoods of the punctures and equal @xmath72 elsewhere ) , we get a canonical identification of the kernels and cokernels on the punctured and the non - punctured disks , see lemma 5.2 in @xcite . consider the operator @xmath282(d(\rho),{{\mbox{\bbb c}}}^n)\to , { { \mathcal{h}}}_{1,\rho}[0](d(\rho),{t^\ast}^{0,1}d(\rho)\otimes{{\mbox{\bbb c}}}^n),\ ] ] where the subscript @xmath255 indicates that we use sobolev norms with weight functions which are the gluing of the weight functions on the two disks . ( since the support of the glued weight function in @xmath256 is compact this weight function is not very important but it will be convenient to use that norm in the estimates below . ) more precisely , this weight function is a smoothing of the function which equals @xmath279 on @xmath283)$ ] and @xmath280 on @xmath284\times[0,1])$ ] . pick bases @xmath285 and @xmath286 in @xmath287 and @xmath288 , respectively . also , pick bases @xmath289 and @xmath290 in @xmath291 and @xmath292 , respectively . ( a priori , elements in the cokernels are elements in the dual of @xmath293 $ ] . however , they can be represented by @xmath294-pairing with smooth functions , see lemma 5.1 in @xcite , and we will think of them as such . ) let @xmath295 be a cut - off function which equals @xmath72 on @xmath296)$ ] , which equals @xmath78 on @xmath297)$ ] and with @xmath298 for @xmath299 . let @xmath300 be a similar cut - off function on with support in the part of @xmath256 corresponding to @xmath250 . define @xmath301 to be the @xmath294-complement of the @xmath302 functions @xmath303 we claim that there exists a constant @xmath304 such that for all sufficiently large @xmath255 @xmath305 assume not , then there exists a sequence of functions @xmath306 , @xmath307 as @xmath308 such that @xmath309 let @xmath310 denote the subset @xmath311\times[0,1]\cup [ -\rho ,- a]\times[0,1],\ ] ] where the first part is a subset of @xmath86 and the second of @xmath250 . let @xmath312 be a cut - off which equals @xmath72 in @xmath313 , which equals @xmath78 in the complement of @xmath314 and which has @xmath315 , @xmath299 . then using the elliptic estimate for the @xmath28-operator on @xmath316 $ ] with small positive exponential weight @xmath317 at both ends we find using that @xmath318 let @xmath319 be a cut off function which is @xmath72 on @xmath320)$ ] , which is @xmath78 outside @xmath283)$ ] , and with @xmath321 . using the elliptic estimate on the @xmath294-complement of the kernel of @xmath28 on @xmath86 we find @xmath322 ( note since the support of the cut - off functions @xmath295 are contained in the support of the cut - off functions @xmath319 we know @xmath323 is orthogonal to the kernel of @xmath28 on @xmath324 ) with @xmath325 similarly defined but with support on @xmath250 instead we find @xmath326 however , , , and contradicts and we find the estimate on @xmath327 holds as claimed . the restriction of @xmath28 to @xmath327 is a fredholm operator of index @xmath328 consider the elements @xmath329 and @xmath330 in the dual of @xmath331 $ ] . furthermore , we pick in this dual constant functions @xmath332 on the gluing region with values in the orthogonal complement of @xmath333 and consider @xmath334 as elements in the dual . moreover , choose them so that their @xmath294-norms ( in the dual weight , which is large in the gluing region ) are @xmath72 . now pick elements ( in @xmath331 $ ] ) dual to these elements and all of norm one . then they stay a uniform positive distance away from the intersection @xmath335 and thus give a direct sum decomposition @xmath336=w\oplus{{\mbox{\bbb r}}}^{s_1+s_2+n}.\ ] ] let @xmath337 be the projection induced by the above direct sum decomposition . we claim that @xmath338 is an isomorphism . assume not , then there is a sequence @xmath339 with @xmath340 we need only note that also @xmath341 for @xmath207 one of the chosen elements in the dual to see that this contradicts the previous estimate . but this is clear since @xmath342 where @xmath343 is the function which was cut - off to get @xmath207 . we thus find the exact sequence @xmath344 where @xmath268 is the inclusion followed by orthogonal projection to the subspace spanned by @xmath345 , where @xmath269 is the @xmath28-operator followed by second projection in the direct sum decomposition of @xmath331 $ ] , and where @xmath270 is the inclusion followed by the projection to the quotient @xmath331/{\operatorname{im}}(\bar{\partial})$ ] . [ rmkmapsinseq ] we note that the natural identification of @xmath128 in the above sequence is obtained by identifying the cokernel of the @xmath28-problem on the gluing strip with the constant functions taking values in @xmath346 . the map @xmath268 can be described as follows . first we map the space @xmath347 into @xmath348 by cutting off kernel elements with cut - off functions as in the proof . the obtained map is clearly injective . the map @xmath268 is then simply the inclusion of @xmath349 into @xmath350 followed by the @xmath294-orthogonal to the image of @xmath347 . the map @xmath269 can be described as follows . first we map the space @xmath351 into the space @xmath352 using cut - off functions as in the proof above . the map is injective . we then consider @xmath347 as a subspace of @xmath348 and map it to the subspace of @xmath352 with the @xmath28-operator followed by @xmath294-orthogonal projection . the map @xmath270 is simply mapping @xmath351 , included into @xmath352 as above , to @xmath353 by projection to the quotient . we next consider another type of gluing . let @xmath243 and @xmath244 be as above . let @xmath26 be a puncture of @xmath86 and let @xmath189 be a puncture on @xmath354 . assume that @xmath355 and that @xmath356 and that @xmath45 and @xmath248 are constant in neighborhoods of @xmath357 and @xmath358 , respectively . ( in general , our assumption that boundary conditions are @xmath359-small in standard coordinates around the punctures implies that we can actually homotope them to constant without changing the kernel / cokernel dimension . ) choose conformal coordinates @xmath129 $ ] in a neighborhood of @xmath26 and conformal coordinates @xmath130\times[0,1]$ ] in a neighborhood of @xmath189 . for @xmath252 large enough we can glue @xmath283)$ ] to @xmath284\times[0,1])$ ] to obtain a disk @xmath256 with @xmath360 punctures . also the boundary conditions glue in a natural way to a boundary condition @xmath264 on @xmath256 . the index of the corresponding operator satisfies @xmath361 ( here the equality follows from the fact that to obtain @xmath362 from @xmath363 and @xmath364 , @xmath0 negative @xmath337-rotations are removed at their common puncture . ) we call this type of gluing , _ gluing at a puncture_. [ lmaglue2 ] for all sufficiently large @xmath252 there exist an exact sequence @xmath365 where @xmath268 , @xmath269 , and @xmath270 are defined similarly to the maps in remark [ rmkmapsinseq ] . in particular , orientations of the determinants spaces @xmath366 induces , via , an orientation on @xmath272 the proof of this result is similar to the proof of lemma [ lmaglue1 ] . however , in the present situation the proof is simpler since the operator in the gluing region is fredholm of index @xmath78 with trivial kernel and cokernel . this is also the reason for the absence of the @xmath128 summand in the third term of the gluing sequence . we show that orientations behave associatively with respect to both gluings at punctures and at boundary points . we will often deal with many vector spaces . for convenience we employ the following notation : if @xmath367 are ordered vector spaces then we denote their direct sum by a column vector containing the vector spaces . that is , @xmath368.\ ] ] ( note that , when dealing with _ oriented _ vector spaces the ordering of the terms in a direct sum is important . ) we first consider the case of two gluings at punctures . let @xmath369 , @xmath370 , and @xmath371 be trivialized transverse boundary conditions for the @xmath28-operator on three punctured disks . assume that @xmath45 can be glued to @xmath248 at a puncture and that @xmath248 can be glued to @xmath304 at another puncture . performing these two gluings we obtain a boundary condition @xmath372 . fix orientations @xmath373 , @xmath374 , and @xmath375 on @xmath376 , @xmath377 , and @xmath378 , respectively . via lemma [ lmaglue2 ] these orientations induce orientations @xmath379 and @xmath380 on @xmath381 and @xmath382 . [ lmaassocpp ] let @xmath383 be the orientation on @xmath384 induced via gluing ( as in lemma [ lmaglue2 ] ) from the orientations @xmath379 and @xmath375 on @xmath381 and @xmath378 , respectively . let @xmath385 be the orientation on @xmath384 induced via gluing ( as in lemma [ lmaglue2 ] ) from the orientations of @xmath386 and @xmath380 on @xmath387 and @xmath382 , respectively . then @xmath388 where @xmath389 is the orientation induced from the exact sequence @xmath390 @>{\beta_\rho}>>\end{cd}\\ & \begin{cd } \left[\begin{matrix } { \operatorname{coker}}(\bar{\partial}_a)\\ { \operatorname{coker}}(\bar{\partial}_b)\\ { \operatorname{coker}}(\bar{\partial}_c ) \end{matrix}\right ] @>{\gamma_\rho } > > { \operatorname{coker}}(\bar{\partial}_{e } ) \end{cd } \longrightarrow 0 , \end{split}\ ] ] @xmath268 , @xmath269 , and @xmath270 are defined similarly to the maps in remark [ rmkmapsinseq ] . to see this consider the diagram @xmath391 @>{{\operatorname{id } } } > > \left[\begin{matrix } { \operatorname{ker}}(\bar{\partial}_{a\sharp b})\\ { \operatorname{ker}}(\bar{\partial}_c ) \end{matrix}\right ] @ > > > \left[\begin{matrix } { \operatorname{coker}}(\bar{\partial}_{a\sharp b})\\ { \operatorname{coker}}(\bar{\partial}_c ) \end{matrix}\right ] @>{{\operatorname{id } } } > > \left[\begin{matrix } { \operatorname{ker}}(\bar{\partial}_{a\sharp b})\\ { \operatorname{ker}}(\bar{\partial}_c ) \end{matrix}\right ] \\ @aaa @vvv @aaa @vvv\\ { \operatorname{ker}}(\bar{\partial}_{e } ) @>{\alpha_\rho } > > \left[\begin{matrix } { \operatorname{ker}}(\bar{\partial}_a)\\ { \operatorname{ker}}(\bar{\partial}_b)\\ { \operatorname{ker}}(\bar{\partial}_c ) \end{matrix}\right ] @>{\beta_\rho } > > \left[\begin{matrix } { \operatorname{coker}}(\bar{\partial}_a)\\ { \operatorname{coker}}(\bar{\partial}_b)\\ { \operatorname{coker}}(\bar{\partial}_c ) \end{matrix}\right ] @>{\gamma_\rho } > > { \operatorname{coker}}(\bar{\partial}_{e } ) \end{cd},\ ] ] where the @xmath78 s at the ends of the horizontal rows are dropped as are the @xmath78 s above the two middle terms in the upper horizontal row . completing the diagram with an arrow from the left middle entry in the lower row to the right middle entry in the upper row we find the direct sum of the gluing sequence of @xmath392 and the trivial sequence for @xmath393 . we check orientations . let @xmath394 represent an orientation pair for @xmath384 . pick a wedge of vectors @xmath395 on the complement of its image in @xmath396 where @xmath397 is a wedge of vectors in @xmath349 and @xmath66 a wedge of vectors in @xmath398 . then the induced orientation on the pair @xmath399 , \left[\begin{matrix } { \operatorname{coker}}(\bar{\partial}_{a\sharp b})\\ { \operatorname{coker}}(\bar{\partial}_{c } ) \end{matrix}\right ] \right)\ ] ] is represented by @xmath400 where @xmath401 is the image of the form @xmath159 under the appropriate map in remark [ rmkmapsinseq ] with the vectors in the opposite order , see subsection [ ssseclinalg ] . this latter orientation pair induces the orientation pair @xmath402 on the pair @xmath403 , \left[\begin{matrix } { \operatorname{coker}}(\bar{\partial}_{a})\\ { \operatorname{coker}}(\bar{\partial}_{b})\\ { \operatorname{coker}}(\bar{\partial}_{c } ) \end{matrix}\right ] \right),\ ] ] where @xmath222 is any wedge of vectors in @xmath404 on the complement of the image of @xmath397 . on the other hand the orientation pair @xmath405 induces the orientation pair @xmath406 on the same pair of spaces . since @xmath47 and @xmath407 represents the same orientation on the complement of @xmath408 if and only if @xmath409 represents the same orientation as @xmath410 on the complement of @xmath411 we see that the orientation pairs agree and the two gluing sequences give the same orientation on @xmath384 . an identical argument shows that also the gluing sequence for @xmath412 and @xmath413 induces this orientation on @xmath384 . second we consider the mixed case . let @xmath369 , @xmath370 , and @xmath371 be trivialized transverse boundary conditions for the @xmath28-operator on three punctured disks . assume that @xmath45 can be glued to @xmath248 at a boundary point and that @xmath248 can be glued to @xmath304 at puncture . performing these two gluings we obtain a boundary condition @xmath414 . fix orientations @xmath373 , @xmath374 , and @xmath375 on @xmath376 , @xmath377 , and @xmath378 , respectively . via lemmas [ lmaglue1 ] and [ lmaglue2 ] these orientations induce orientations @xmath379 and @xmath380 on @xmath381 and @xmath382 , respectively . [ lmaassocbp ] let @xmath383 be the orientation on @xmath384 induced via gluing ( as in lemma [ lmaglue2 ] ) from the orientations of @xmath379 and @xmath375 on @xmath381 and @xmath378 , respectively . let @xmath385 be the orientation on @xmath384 induced via gluing ( as in lemma [ lmaglue1 ] ) from the orientations of @xmath386 and @xmath380 on @xmath387 and @xmath382 , respectively . then @xmath388 where @xmath389 is the orientation induced from the exact sequence @xmath390 @>{\beta_\rho } > > \end{cd}\\ & \begin{cd } \left[\begin{matrix } { \operatorname{coker}}(\bar{\partial}_a)\\ { { \mbox{\bbb r}}}^n\\ { \operatorname{coker}}(\bar{\partial}_b)\\ { \operatorname{coker}}(\bar{\partial}_c ) \end{matrix}\right ] @>{\gamma_\rho } > > { \operatorname{coker}}(\bar{\partial}_{e } ) \end{cd } \longrightarrow 0 , \end{split}\ ] ] where @xmath268 , @xmath269 , and @xmath270 are the naturally defined maps in a simultaneous gluing process ( see remark [ rmkmapsinseq ] ) . the proof is similar to the proof of lemma [ lmaassocpp ] . finally we consider the case of two gluings at boundary points . let @xmath369 , @xmath370 , and @xmath371 be trivialized transverse boundary conditions for the @xmath28-operator on three punctured disks . assume that @xmath45 can be glued to @xmath248 at a boundary point and that @xmath248 can be glued to @xmath304 at a boundary point . performing these two gluings we obtain a boundary condition @xmath415 . fix orientations @xmath373 , @xmath374 , and @xmath375 on @xmath376 , @xmath377 , and @xmath378 , respectively . via lemma [ lmaglue1 ] these orientations induce orientations @xmath379 and @xmath380 on @xmath381 and @xmath382 , respectively . [ lmaassocbb ] let @xmath383 be the orientation on @xmath384 induced via gluing ( as in lemma [ lmaglue1 ] ) from the orientations of @xmath379 and @xmath375 on @xmath381 and @xmath378 , respectively . let @xmath385 be the orientation on @xmath384 induced via gluing ( as in lemma [ lmaglue1 ] ) from the orientations of @xmath386 and @xmath380 on @xmath387 and @xmath382 , respectively . then @xmath388 where @xmath389 is the orientation induced from the exact sequence @xmath390 @>{\beta_\rho}>>\end{cd}\\ & \begin{cd}\left[\begin{matrix } { \operatorname{coker}}(\bar{\partial}_a)\\ { { \mbox{\bbb r}}}^n\\ { \operatorname{coker}}(\bar{\partial}_b)\\ { { \mbox{\bbb r}}}^n\\ { \operatorname{coker}}(\bar{\partial}_c ) \end{matrix}\right ] @>{\gamma_\rho } > > { \operatorname{coker}}(\bar{\partial}_{e } ) \end{cd } \longrightarrow 0 , \end{split}\ ] ] where @xmath268 , @xmath269 , and @xmath270 are naturally defined maps in a simultaneous gluing process ( see remark [ rmkmapsinseq ] ) . the proof is similar to the proof of lemma [ lmaassocpp ] . we describe , following @xcite , the canonical orientation of the determinant bundle over the space of trivialized lagrangian boundary conditions on the closed ( @xmath78-punctured ) disk . we also relate trivialized lagrangian boundary conditions on a punctured disk with trivialized boundary conditions on the closed disk using capping disks . the space of trivialized lagrangian boundary conditions over the @xmath78-punctured disk is the space @xmath416 of ( free ) loops in @xmath417 . [ lmacanor ] the determinant bundle over @xmath416 is orientable . in fact , an orientation of @xmath128 induces an orientation of this bundle . consider the fibration @xmath418 which is evaluation at @xmath419 . we find that if @xmath420 is any component of @xmath416 then @xmath421 since @xmath422 and @xmath423 . in fact , a generator of @xmath424 can be described as follows . fix any element @xmath425 in @xmath420 and a loop @xmath426 in @xmath417 which starts and ends at @xmath427 and which generates @xmath428 . then the loop @xmath429 generates @xmath430 . let @xmath431 be the determinant bundle . we need to check that @xmath432 , where @xmath433 is the first stiefel - whitney class . to see this note that the monodromy of the orientation bundle of @xmath172 along the loop described is the following . if @xmath434 is a basis in the kernel and @xmath435 one in the cokernel of @xmath412 then the monodromy is given by the orientation pairs @xmath436 hence it preserves orientation and @xmath432 . to prove the second statement , we follow the argument in @xcite , 21 . the trivialization @xmath45 on the boundary allow us to choose a trivialization of the trivial bundle @xmath5 near the boundary of @xmath437 such that the real plane field is constantly equal to @xmath438 . extending this trivialization inwards towards the center of the disk and stretching the neck , we eventually split off a @xmath439 with a complex vector bundle at @xmath440 . using gluing arguments similar to those above we find that orientations of the determinant line over the disk with constant boundary condition together with the complex orientation of the determinant line of the complex bundle over @xmath439 induces an orientation of @xmath376 . since the determinant line of the @xmath28-operator is canonically ( by evaluation at any point in the boundary ) identified with @xmath441 we get an induced orientation as claimed . as shown in @xcite the orientations on the index space induced by different trivializations ( different @xmath442 if @xmath443 ) are different . fix an orientation on @xmath128 . the orientation of the determinant bundle over @xmath416 induced from the fixed orientation on @xmath128 will be called the _ canonical orientation_. consider two trivialized boundary conditions @xmath45 and @xmath248 on the @xmath78-punctured disk as in lemma [ lmaglue1 ] and construct the glued boundary condition @xmath264 . [ lmagluecanor ] the gluing sequence in lemma [ lmaglue1 ] induces from the canonical orientations on @xmath376 and on @xmath377 , and the orientation @xmath444 times the standard orientation on @xmath128 , the canonical orientation on @xmath381 . as in the proof of lemma [ lmacanor ] we pick special trivializations of the bundles near the boundary . note that these trivializations glue in a natural way . when pushing inwards we obtain two complex bundles over two copies of @xmath439 sitting over the origins of the two disks glued . using a gluing argument similar to the one in lemma [ lmaglue1 ] , we see that the lemma will follow as soon as it is proved for the disk with constant boundary conditions @xmath438 . in this case @xmath287 , @xmath288 , and @xmath349 are all isomorphic to @xmath128 and the cokernels of all three operators are trivial . moreover , the fixed orientation on @xmath128 gives the canonical orientation on the determinants . the gluing sequence is @xmath445 where , as is easily seen , @xmath446 . to determine @xmath269 , let @xmath447 be thought of as an element in the kernel on the first disk . we then cut this constant function off by a cut - off function @xmath47 which equals @xmath72 on the first disk and equals @xmath78 on the second . the cokernel of the @xmath28-operator on the strip with positive weights and constant boundary conditions is a complement of the intersections of the kernel of the @xmath294-pairing with constant functions . we can thus represent it as the subspace spanned by constant functions in the strip which are cut - off _ outside _ the supports of the cut - off functions @xmath47 . now , @xmath448 , and taking the @xmath294-inner product with the cut - off constant functions we find that the sign of @xmath449 essentially determines the map . since the orientation of the @xmath450-axis is from the first disk to the second we find that @xmath451 . it follows from our conventions in section [ ssseclinalg ] that the standard orientation on @xmath452 and the orientation @xmath444 times the standard orientation the second @xmath128 induces the standard orientation on the first @xmath128 . the lemma follows . note that it follows from the above lemma [ lmagluecanor ] that picking two disks with constant boundary conditions and gluing these we get a disk with constant boundary conditions and the orientation induced on the determinant of the later from orientations on the determinants of the former two is independent of the ordering of the former two . the reason for this is that the change of order of the summands is accompanied by a change of the map @xmath269 to @xmath453 . if the dimension @xmath0 is even then both these changes preserve orientation and , if @xmath0 is odd both reverse orientation . thus in either case the induced orientation is not affected by the change of order . let @xmath145 and @xmath146 be two transverse lagrangian subspaces of @xmath5 . let @xmath454 be the complex angle of @xmath181 and recall that there exists canonical complex coordinates @xmath19 in @xmath5 such that @xmath455 and @xmath456 . let @xmath457(s)$ ] be the @xmath72-parameter family of unitary transformations of @xmath5 given by the matrix @xmath458 in canonical coordinates . then @xmath459(s)$ ] defines a map of the space of pairs of transverse lagrangian subspaces in @xmath5 into the path space of @xmath417 . ( note that @xmath457 $ ] is independent of the choice of canonical coordinates . ) the next lemma shows that this map is continuous . the @xmath72-parameter family of unitary transformations @xmath457(s)$ ] depends continuously on @xmath181 . the space of lagrangian subspaces of @xmath5 transverse to @xmath128 is identified with the space of symmetric linear matrices @xmath13 via @xmath460 if @xmath461 is the inverse of @xmath462 then it is easily checked that @xmath463(s)$ ] is given by the matrix @xmath464 in the standard basis of @xmath5 . the lemma is a straightforward consequence of . recall , see subsection [ typesofpath ] , that we subdivided the set of pairs of transverse oriented lagrangian subspaces into two subsets : even pairs and odd pairs . we associate to such pairs , with complex angle meeting certain conditions , @xmath72-parameter families of unitary transformations . we define these unitary transformations by writing their matrices in canonical coordinates . note that the extra conditions on the complex angle ensures that the transformations are independent of the choice of canonical coordinates . let @xmath465 denote the complex angle of the ordered pair @xmath181 . * if @xmath181 is even and if @xmath466 for all @xmath467 then define @xmath468(s)= { \operatorname{diag}}\left(e^{i\theta_1s},e^{-i(\pi-\theta_2)s } , \dots , e^{-i(\pi-\theta_n)s}\right).\ ] ] * if @xmath181 is odd and if @xmath469 , @xmath470 then define @xmath471(s)= { \operatorname{diag}}\left(e^{i\theta_1 s } , e^{-i(2\pi-\theta_2)s } , e^{-i(\pi-\theta_3)s},\dots , e^{-i(\pi-\theta_n)s}\right).\ ] ] * if @xmath472 is even and if @xmath473 , @xmath474 then define @xmath475(s)= { \operatorname{diag}}\left(e^{-i(2\pi-\theta_1)s},\dots , e^{-i(2\pi-\theta_{n-1})s } , e^{-i(\pi-\theta_n)s}\right).\ ] ] * if @xmath472 is odd and if @xmath476 , @xmath477 then define @xmath478(s)= { \operatorname{diag}}\left(e^{-i(2\pi-\theta_1)s},\dots , e^{-i(2\pi-\theta_{n-2})s } , e^{-i(\pi-\theta_{n-1})s},e^{-i(\pi-\theta_n)s}\right).\ ] ] let @xmath479 and note that if the complex angle of @xmath181 equals @xmath480 then the complex angle of @xmath472 equals @xmath481 . it is then easily seen that if @xmath181 satisfies * ne * and @xmath472 satisfies * pe * then @xmath482(1)\circ r_{ne}[v_1,v_2](1)={\operatorname{id}}$ ] . similarly , if @xmath181 satisfies * no * and @xmath472 satisfies * po * then @xmath483(1)\circ r_{no}[v_1,v_2](1)={\operatorname{id}}$ ] . let @xmath193 be an oriented lagrangian subspace of @xmath5 and let @xmath484 . let @xmath485 and @xmath486 be the oriented lagrangian subspaces of @xmath487 , with standard coordinates @xmath488 , given by the orienting basis @xmath489 define the _ upper @xmath490-stabilization _ of @xmath193 to be the oriented lagrangian subspace @xmath491 of @xmath492 given by @xmath493 define the _ lower @xmath490-stabilization _ of @xmath193 to be the oriented lagrangian subspace @xmath494 of @xmath492 given by @xmath495 let @xmath496 , @xmath497 be a continuous family of transverse lagrangian subspaces parameterized by a compact space @xmath498 . let @xmath499 be the complex angle of @xmath496 . by compactness of @xmath498 there exists @xmath500 such that @xmath501 and @xmath502 for all @xmath497 and @xmath503 . fix such a @xmath500 and let @xmath504^u(\beta)$ ] and @xmath505^l(\beta)$ ] . note that @xmath506 is even ( odd ) if and only if @xmath496 is even ( odd ) . moreover , by the choice of @xmath490 , @xmath506 satisfies the condition * ne * ( * no * ) if @xmath507 is even ( odd ) and @xmath508 satisfies the condition * pe * ( * po * ) if @xmath496 is even ( odd ) , for all @xmath497 . thus we can construct the corresponding @xmath498-families of unitary operators . let @xmath496 and @xmath500 be as above . if @xmath496 is even then the families of unitary operators @xmath509(s)$ ] and @xmath510(s)$ ] depend continuously on @xmath497 . if @xmath496 is odd then the families of unitary operators @xmath511(s)$ ] and @xmath512(s)$ ] depend continuously on @xmath497 . this is a straightforward consequence of . let @xmath496 , @xmath497 be a continuous family of transverse lagrangian subspaces of @xmath5 parameterized by a compact simply connected space @xmath498 . fix @xmath500 small enough and consider the stabilized family @xmath506 of transverse lagrangian subspaces in @xmath513 . assume that @xmath514 and @xmath515 are equipped with positively oriented frames @xmath516 and @xmath517 which vary continuously with @xmath497 . we associate to this family two families of trivialized lagrangian boundary conditions on the @xmath72-punctured disk . to simplify notation , if @xmath496 is even then let @xmath518 and if @xmath496 is odd then let @xmath519 . note that @xmath520(1)x_1(\lambda)\ ] ] is a framing of @xmath515 . hence there exists @xmath521 such that @xmath520(1)x_1(\lambda ) = x_2(\lambda)\cdot\alpha(\lambda).\ ] ] since @xmath498 is simply connected the map @xmath522 lifts to a map @xmath523 , where @xmath524 is the space of paths in @xmath525 with initial endpoint at the identity matrix and which projects to @xmath525 by evaluation at the final endpoint . pick such a lift . identify @xmath526 with @xmath527 $ ] . define two families of trivialized boundary conditions for the @xmath28-operator on @xmath528 as follows @xmath529(s ) & = r_{n\ast}[\tilde v_1(\lambda),\tilde v_2(\lambda ) ] x_1(\lambda)\cdot\tilde\alpha[\lambda](s),\\ a_{p\ast}[\lambda](s ) & = r_{p\ast}[\tilde v_2(\lambda),\tilde v_1(\lambda ) ] x_2(\lambda)\cdot\tilde\alpha^{-1}[\lambda](s),\end{aligned}\ ] ] where @xmath530(s)$ ] is the inverse of the matrix @xmath531(s)$ ] . let @xmath26 be the puncture on @xmath528 and note that @xmath532(p^\pm)=a_{p\ast}[\lambda](p^\mp)$ ] so that these boundary conditions can be glued . let @xmath533 and @xmath534 denote the @xmath28-operators with boundary conditions @xmath532 $ ] and @xmath535 $ ] , respectively . then the bundles @xmath536 and @xmath537 are orientable since @xmath498 is simply connected . we orient them as follows . let @xmath538 . pick an orientation of @xmath539 . together with the canonical orientation of the @xmath28-operator on the @xmath78-punctured disk and lemma [ lmaglue2 ] this orientation determines an orientation on @xmath540 . since the bundles @xmath541 and @xmath542 are orientable the orientations of @xmath539 and @xmath540 induce orientations on @xmath543 and @xmath544 , respectively , for any @xmath497 . the orientations on @xmath543 and @xmath544 as defined above glue to the canonical orientation on the @xmath78-punctured disk . after adding a finite dimensional vector space we may assume that all operators are surjective . the lemma then follows from properties of the direct sum operation on vector bundles . we call the operators @xmath545 , @xmath546 , @xmath547 , and @xmath548 which arises as above _ capping operators _ and we call an orientation pair on @xmath541 , @xmath542 with the properties above a pair of capping orientations . let @xmath498 be any compact simply connected space and let @xmath549 , @xmath550 be @xmath498-families of transverse lagrangian subspaces . construct as in the previous section the stabilizations @xmath551 and assume that these families are equipped with positively oriented frames @xmath516 and @xmath517 , respectively . construct the families @xmath552 , @xmath553 , @xmath554 , and @xmath555 of operators on the @xmath72-punctured disks with oriented determinant bundles as there . let @xmath556 be any topological space and consider an @xmath557-family of oriented lagrangian boundary conditions @xmath558\colon{\partial}d_{m+1}\to{\operatorname{lag}}(n)$ ] , @xmath559 on the @xmath560-punctured disk with the following properties . for all @xmath561 , if @xmath562 then @xmath558(p_j^+)=v^1_j(\lambda)$ ] , @xmath558(p_j^-)=v_2^j(\lambda)$ ] , @xmath558(p_0^+)=v_2 ^ 0(\lambda)$ ] , and @xmath558(p_0 ^ -)=v_1 ^ 0(\lambda)$ ] . let @xmath563 denote the determinant bundle of @xmath564}$ ] . we construct , under some additional trivialization conditions , the _ capping orientation _ of @xmath172 in the following way . first stabilize the family @xmath558\colon{\partial}d_m\to{\operatorname{lag}}(n)$ ] to @xmath565\colon{\partial}d_m\to{\operatorname{lag}}(n+2)$ ] by letting @xmath566(\zeta)= a[x,\lambda](\zeta)\times { \operatorname{span}}\left(e^{i\tfrac{\gamma}{2}(\zeta)}{\partial}_{x_1 } , e^{i\gamma(\zeta)}{\partial}_{x_2}\right),\quad\zeta\in{\partial}d_m,\ ] ] where @xmath492 , @xmath567 , and where @xmath568 $ ] satisfies @xmath569 in neighborhoods of @xmath98 , @xmath562 and @xmath570 , and @xmath571 in neighborhoods of @xmath99 and @xmath572 . noting that the @xmath28-operator on @xmath86 with boundary conditions @xmath573 has both trivial kernel and trivial cokernel we find that the kernels and cokernels of @xmath564}$ ] and @xmath574}$ ] are canonically isomorphic . thus to orient the determinant bundle of @xmath564}$ ] it suffices to orient the determinant bundle of @xmath574}$ ] . we find such an orientation in the case when the boundary conditions @xmath565 $ ] are equipped with a certain type of trivialization . thus , assume that the boundary conditions @xmath565\colon { \partial}d_m\to{\operatorname{lag}}(n+2)$ ] are trivialized , i.e. they are represented by @xmath575\colon { \partial}d_m\to u(n+2)$ ] . assume moreover that this trivialization satisfies the following conditions for all @xmath561 , if @xmath562 then @xmath575(p_j^+)=x^j_1(\lambda)$ ] , @xmath575(p_j^-)=x_2^j(\lambda)$ ] , @xmath575(p_0^+)=x_2 ^ 0(\lambda)$ ] , and @xmath575(p_0 ^ -)=x_1 ^ 0(\lambda)$ ] . glue to @xmath576}$ ] first the capping operator @xmath577 ( @xmath578 ) at @xmath579 if the puncture @xmath579 is even ( odd ) . then glue to @xmath576}$ ] the operators @xmath580 ( @xmath580 ) to @xmath575 $ ] at its even ( odd ) negative puncture @xmath100 , @xmath562 , _ in the order opposite _ to that induced by the boundary orientation of @xmath581 . ( we glue in the opposite order so that the leibniz rule works out appropriately , see section [ sec : legch ] . ) we obtain in this way a trivialized boundary condition @xmath582\colon { \partial}d\to u(n+2)$ ] on the closed disk . lemma [ lmaglue2 ] and repeated application of lemma [ lmaassocpp ] gives the gluing sequence , where we write @xmath583 for the capping operator at the puncture @xmath189 at @xmath497 and where the sign indicates the sign of the puncture , @xmath584 } ) @ > > > \left[\begin{matrix } { \operatorname{ker}}(\bar{\partial}_{p_1 ^ -,\lambda})\\ \vdots\\ { \operatorname{ker}}(\bar{\partial}_{p_m^-,\lambda})\\ { \operatorname{ker}}(\bar{\partial}_{p_0^+,\lambda})\\ { \operatorname{ker}}(\bar{\partial}_{\tilde a[x,\lambda ] } ) \end{matrix}\right ] @ > > > \end{cd}\\ & \begin{cd } \left[\begin{matrix } { \operatorname{coker}}(\bar{\partial}_{p_1 ^ -,\lambda})\\ \vdots\\ { \operatorname{coker}}(\bar{\partial}_{p_m^-,\lambda})\\ { \operatorname{coker}}(\bar{\partial}_{p_0^+,\lambda})\\ { \operatorname{coker}}(\bar{\partial}_{\tilde a[x,\lambda ] } ) \end{matrix}\right ] @ > > > { \operatorname{coker}}(\bar{\partial}_{\hat a[x,\lambda ] } ) \end{cd } \longrightarrow 0 . \end{split}\ ] ] we give the determinant of @xmath576}$ ] the unique orientation @xmath585 which together with the chosen orientations for the capping disks , via the the gluing sequence , give the canonical orientation on @xmath586})$ ] . it is clear that the orientation so defined gives an orientation of the determinant bundle over @xmath587})\to x\times\lambda$ ] and thus by the above mentioned isomorphism also the bundle @xmath588})$ ] gets oriented . [ dfncap ] we call the orientation of the determinant bundle @xmath563 the _ capping orientation_. note that the same construction can be applied when @xmath589 is replaced by a locally trivial fibration @xmath590 . let @xmath181 be a pair of transverse lagrangian subspaces in @xmath591 and let @xmath592 in @xmath5 be its stabilization . note that in canonical coordinates of @xmath592 the boundary conditions of the capping operators constructed from @xmath592 are split and we may determine the dimensions of the kernel and cokernel from properties of the classical riemann - hilbert problem . to this end let @xmath593 , @xmath594 , be the complex angle of @xmath592 . we think of the @xmath72-punctured disk @xmath528 as of the unit disk in @xmath88 punctured at @xmath72 boundary parameterized by @xmath595 , @xmath596 . * the _ negative even _ boundary condition is , in canonical coordinates , given by @xmath597 thus , @xmath598 the kernel is spanned by the function @xmath599 , where @xmath600 * the _ negative odd _ boundary condition is , in canonical coordinates , given by @xmath601 thus , @xmath602 the kernel spanned by the function @xmath599 , where @xmath600 the cokernel is spanned by the function @xmath603 , where @xmath604 where we view this function as a linear functional on the target space of @xmath28 via the @xmath294-pairing and where the cokernel is a one dimensional subspace complementary to the kernel of this functional . * the _ positive even _ boundary condition is , in canonical coordinates , given by @xmath605 thus , @xmath606 the cokernel is spanned by the functions @xmath603 , where for @xmath607 @xmath608 * the _ positive odd _ boundary condition is , in canonical coordinates , given by @xmath609 thus , @xmath610 the cokernel is spanned by the functions @xmath603 , where for @xmath611 @xmath608 we orient moduli spaces of holomorphic disks with boundary on a generic legendrian submanifold equipped with a spin structure . the orientation is obtained by comparing capping orientations with fixed orientations of spaces of automorphisms and of conformal structures . let @xmath612 denote the unit disk @xmath613 with @xmath72 positive , and @xmath89 negative punctures on the boundary , @xmath614 . let @xmath579 be the positive puncture and let @xmath90 be the negative punctures . as mentioned above the positive puncture @xmath579 and the orientation of @xmath437 induces an ordering of the negative punctures @xmath615 . let @xmath616 denote the space of conformal structures on @xmath86 . if @xmath617 then @xmath616 is a one - point space . for @xmath618 let @xmath619 denote the group of conformal automorphisms . we orient @xmath620 , @xmath621 in the following way . let @xmath86 with conformal structure @xmath622 be represented by a disk with positive puncture @xmath579 and ordered negative punctures @xmath623 . then any conformal structure @xmath624 in a neighborhood of @xmath622 can be represented uniquely by a disk with positive puncture at @xmath579 its first two negative punctures at @xmath92 and @xmath625 and with the rest of its negative punctures at @xmath626 . thus , the tangent space of @xmath620 at @xmath622 is identified with the direct sum of the tangent spaces of @xmath437 at @xmath627 . we orient @xmath616 by declaring the oriented basis @xmath628 where @xmath629 is the positive unit tangent to @xmath437 at @xmath100 , to be positive basis in @xmath630 . we orient the one - point space @xmath631 by declaring it positively oriented . next consider @xmath632 and @xmath633 . to orient @xmath632 , consider @xmath528 punctured at @xmath26 . pick two points @xmath634 and @xmath635 in @xmath526 a small distance from @xmath26 . the tangent space of @xmath632 at the identity is the @xmath636-dimensional space of holomorphic vector fields on @xmath87 tangent to @xmath437 along the boundary and vanishing at @xmath26 . evaluation of such a vector fields at @xmath634 and @xmath635 gives a map @xmath637 we use this map to orient @xmath638 and the group structure of @xmath632 to orient @xmath632 . to orient @xmath633 , consider @xmath639 as @xmath87 with positive puncture at @xmath72 and negative at @xmath640 . pick a point @xmath189 in the _ lower _ hemisphere of the two into which @xmath437 is subdivided by @xmath640 and @xmath72 and orient @xmath638 by evaluation at @xmath189 as above . let @xmath6 be an orientable manifold of dimension @xmath0 . let @xmath641 denote the stabilized tangent bundle of @xmath6 , where @xmath642 denotes the trivial bundle over @xmath6 . fix some triangulation of @xmath6 . then a spin - structure on @xmath6 can be viewed as a trivialization of @xmath643 restricted to the 1-skeleton that extends to the 2-skeleton . note that the extension to the 2-skeleton is homotopically unique if it exists since @xmath644 . for the same reason , any trivialization over the 2-skeleton automatically extends over the 3-skeleton but this extension is in general not homotopically unique . let @xmath13 be an oriented manifold equipped with a spin structure and let @xmath498 be a compact simply connected space . ( in fact , for our applications @xmath645 and @xmath646 $ ] are sufficient . ) let @xmath647 , @xmath497 be a family of _ chord generic _ legendrian embeddings . the chord genericity implies that we get a continuous family of reeb chords @xmath648 of @xmath649 and that their endpoints vary continuously with @xmath124 in @xmath13 . fix @xmath538 and choose a family of diffeomorphisms @xmath650 such that @xmath651 for all @xmath652 . for each reeb chord @xmath653 of @xmath654 , fix a capping path @xmath655 connecting its upper end point to the lower one , see @xcite , section 2.3 . fix a triangulation @xmath656 of @xmath13 , where @xmath657 denotes the @xmath113-skeleton of @xmath658 , such that each reeb chord endpoint @xmath659 lies in @xmath660 and each capping path lies in @xmath661 . as mentioned above the spin structure gives a trivialization of the restriction of @xmath662 to @xmath663 . fix such a trivialization . as @xmath497 varies we move the triangulation and capping paths by @xmath664 and the trivialization by the bundle isomorphism @xmath665 covering @xmath664 given by @xmath666 using the notation from @xcite , section 4 , let @xmath667 denote the space of candidate maps . such a trivialization enables us to construct the capping orientation of the determinant bundle over the space of linearized boundary conditions over @xmath668 in the following way . consider first the trivial @xmath5-bundle over @xmath13 which is the the pull - back @xmath669 and its stabilization @xmath670 . associate to each point @xmath671 the lagrangian subspace @xmath672 where @xmath673 $ ] is a family of smooth functions such that @xmath674 for all reeb chords @xmath675 and where @xmath490 is such that @xmath676 is smaller than any component of any complex angle at any reeb chord and @xmath677 is larger than such components . let @xmath678 . then the restriction of @xmath679 to the part of the boundary of @xmath612 lying between @xmath100 and @xmath680 is a path in @xmath13 connecting two reeb chord endpoints and we obtain from the field of lagrangian subspaces just defined a family of lagrangian subspaces over @xmath668 which is the stabilization of the tangent plane family and which satisfies the conditions in section [ caportrivbdcond ] . pick a homotopy , fixing endpoints , of this path to a path which lies in @xmath661 . then the trivialization @xmath222 of @xmath662 along @xmath661 induces a trivialization of @xmath681 on the corresponding parts . this in turn induces a trivialization of the stabilized lagrangian boundary condition and we define the capping orientation of the determinant bundle of the linearized @xmath28-operator over @xmath667 as in definition [ dfncap ] . we must check that this orientation is well - defined . choosing a different homotopy to some path in @xmath661 , the two end paths in @xmath661 can be connected with a homotopy in @xmath682 . using the capping orientation over this homotopy proves that the orientation is well - defined . we call also the orientation of @xmath683 over @xmath684 the _ capping orientation_. using the following lemma we orient all moduli spaces of holomorphic disks . as in @xcite we let @xmath685 denote the full @xmath28-operator . [ lmadgamma ] let @xmath686 denote the full linearization of the @xmath28-operator at some holomorphic @xmath687 . the determinant bundle over @xmath684 with fiber @xmath688 is canonically isomorphic to the tensor product of the determinant bundle of @xmath689 with the highest exterior power of the tangent bundle to @xmath620 . let @xmath690 . we must show that ( at @xmath691 ) @xmath692 note that @xmath693 and that @xmath694 for some map @xmath695 let @xmath696 be the map induced by projection . then the following sequence is exact @xmath697 and induces a canonical isomorphism @xmath698 on the other hand @xmath699 and @xmath700 . hence @xmath701 let @xmath687 be a transversely cut out holomorphic disk . using lemma [ lmadgamma ] we orient the moduli space to which @xmath691 belongs as follows . * if @xmath702 then @xmath686 simply agrees with the ordinary @xmath28-operator and the transversality condition implies that this operator is surjective . moreover , @xmath703 for @xmath704 gives an injection @xmath705 . the quotient @xmath706 can be identified with the tangent space to the moduli space @xmath707 . the capping orientation on @xmath708 together with the orientation on @xmath709 thus gives an orientation of the moduli space . * if @xmath710 then an orientation of @xmath683 and @xmath711 together give an orientation of @xmath688 . the surjectivity assumption implies that @xmath712 is trivial hence we get an orientation on @xmath713 which is the tangent space of the moduli space . an oriented connected @xmath78-dimensional manifold is a point with a sign . the above definition says that , to get the sign of a rigid disk we compare the capping orientation of the kernel / cokernel of the @xmath689-operator with the orientation on @xmath709 or @xmath620 depending on the number of punctures @xmath714 in this section we associate to any legendrian submanifold @xmath12 which is equipped with a spin structure a graded algebra @xmath715 over the group ring @xmath29 $ ] if @xmath13 is connected and over @xmath4 otherwise . we define a map @xmath716 and prove that it is a differential . with this established we prove that the stable tame isomorphism class of the differential graded algebra @xmath717 remains invariant under legendrian isotopies . this implies in particular that the contact homology @xmath718 is a legendrian isotopy invariant . we then show how the differential @xmath30 depends on the particular spin structure on @xmath13 and in the final subsection discuss the relation of our approach to contact homology over @xmath719={{\mbox{\bbb z}}}[t , t^{-1}]$ ] for legendrian @xmath72-knots with the completely combinatorial approach taken in @xcite . let @xmath12 be ( an admissible chord generic ) oriented connected legendrian submanifold . define @xmath715 as the free associative algebra over @xmath29 $ ] generated by the reeb chords of @xmath13 . that is @xmath720{\langle}c_1,\dots , c_m{\rangle}.\ ] ] recall from @xcite ( also see section [ sec : basnot ] above ) that each generator @xmath721 comes equipped with a grading @xmath722 and a capping path @xmath723 elements @xmath68 have gradings @xmath724 . so , @xmath715 is a @xmath4-graded algebra . we note that if @xmath13 is an orientable manifold then @xmath724 is even for any @xmath68 . if the legendrian submanifold @xmath13 is not connected ( see also @xcite ) then we will use a simpler version of the theory : we let @xmath725 in this case the algebra @xmath715 has a natural @xmath2-grading . there is also a relative @xmath70-grading which we will not discuss here . ( note that the orientability implies that the maslov number is even , see also remark [ rmkdcgr ] ) . in the connected case this simpler version corresponds to setting all @xmath68 to @xmath72 and reducing the grading modulo @xmath636 . assume that @xmath13 is equipped with a spin structure . define the _ differential _ @xmath726 by requiring that it is linear over @xmath29 $ ] ( over @xmath4 in the disconnected case ) , that it satisfies the graded leibniz rule on products of monomials @xmath727 and define it on generators as @xmath728 where @xmath729 is the algebraic number of rigid disks in the moduli space ( where the sign of rigid disk is defined as in the previous section ) . it follows from @xcite lemma 1.5 , that @xmath30 decreases grading by @xmath72 in the connected case , see remark [ rmkdclower1 ] for the disconnected case . the purpose of the next subsection is to complete the proof of the following theorem . [ thmd^2=0 ] the map @xmath716 is a differential . that is , @xmath730 it is a straightforward consequence of the signed leibniz rule that the lemma follows once has been established for generators . the fact that it holds for generators will be established below . [ rmkcoeff ] note that the above definition also make sense for other coefficient rings . for example one could replace @xmath4 above with @xmath731 , for any @xmath732 or by @xmath733 . in order to prove theorem [ thmd^2=0 ] , we will determine the relations between orientations on @xmath72-dimensional moduli spaces and the signs of the pairs of rigid disks which are their boundaries . we connect our abstract definitions of even and odd to the geometrical situation under study . let @xmath721 be a reeb chord of an oriented connected chord generic legendrian submanifold @xmath12 . then the two tangent spaces of @xmath734 the projections of which intersect at @xmath735 are oriented . we order these tangent spaces by taking the one with the _ largest _ @xmath37-coordinate _ first_. the holomorphic disks we study have punctures mapping to reeb chords . translating the definition of even and odd punctures to the present situation , we say that a reeb chord @xmath721 is _ even _ if the even negative boundary condition @xmath736 takes the orientation of the upper tangent space to that of the lower . otherwise we say that @xmath721 is _ odd _ and , as is easy to see , in this case the odd negative boundary condition @xmath737 takes the orientation of the upper tangent space to the orientation of the lower . [ eochords ] a reeb chord @xmath721 is even ( odd ) if and only if its grading @xmath722 is even ( odd ) . recall that @xmath738 where @xmath66 is a capping path of @xmath721 and where @xmath124 is the @xmath739-trick path connecting the lower to the upper tangent space at @xmath740 . consider instead the inverse of the path @xmath741 . this path is a negative rotation @xmath481 where @xmath480 is the complex angle of the upper and lower tangent spaces followed by @xmath66 backwards ( i.e. from bottom to top ) . the maslov index of this path is even ( odd ) if and only if this path preserves ( reverses ) the orientation of the upper tangent plane . now , closing up with the negative even boundary condition path instead of the inverse @xmath739-trick path changes the maslov index by one . thus the negative even boundary condition path preserves orientation if and only if @xmath722 is even . [ rmkdcgr ] in the case of an oriented disconnected legendrian submanifold we use this notion of even and odd punctures to define the @xmath2-grading discussed above . [ rmkdclower1 ] to see that the differential decreases grading by @xmath72 in the disconnected case note that ( see proposition 5.14 in @xcite ) the formal dimension of a component of a moduli space @xmath742 with boundary mapping to the collection of paths @xmath743 on the legendrian submanifold @xmath12 equals @xmath744 where @xmath745 is the maslov index of the closed path @xmath234 obtained by closing up the paths @xmath124 at the corners by rotating the lagrangian subspace of the incoming edge to the lagrangian subspace at the outgoing edge in the negative direction . note that at a negative corner the incoming lagrangian subspace is the upper one . thus , the negative close up preserves orientation if and only if the puncture is odd . at the positive puncture the incoming lagrangian subspace is the lower one and the negative close up preserves the orientation if and only if @xmath0 is even and the puncture is even or @xmath0 is odd and the puncture is odd . since the maslov index of a loop of lagrangian subspaces is even if and only if it is orientation preserving we find that @xmath746 where @xmath747 denotes the modulo @xmath636 grading . we conclude that the differential changes grading . we shall determine the relation between orientations on spaces of conformal structures on a disk which is close to splitting up into two disks and the conformal structures on the two disks into which it splits . to this end we first describe the standard orientation on @xmath620 in various local coordinates . consider a disk @xmath612 with @xmath748 punctures @xmath749 on the boundary representing the conformal structure @xmath622 and where @xmath579 is the positive puncture . then the standard orientation of @xmath620 at @xmath622 is given by the @xmath750-tuple of vectors @xmath751 where @xmath752 denotes the vector tangent to @xmath581 at @xmath100 and points in the positive direction along @xmath581 . on the other hand we can coordinatize a neighborhood of @xmath622 in @xmath620 by fixing any three punctures on @xmath581 and letting the remaining punctures move . we will however restrict attention to coordinates in which the positive puncture remains fixed . for such coordinates the positive orientation is given by the following lemma , the proof of which is straightforward . [ lmashuffle ] the positive orientation of @xmath620 at @xmath622 in coordinates obtained by fixing the ordered points @xmath753 is given by the @xmath750-tuple of vectors @xmath754 secondly , we look at the outward normal to the space of conformal structures at a disk which is obtained by gluing two disks . let @xmath755 , @xmath153 . consider @xmath756 and @xmath757 with punctures @xmath758 and @xmath759 , respectively . let @xmath86 be the disk obtained from gluing @xmath756 to @xmath757 by identifying @xmath760 and @xmath100 , @xmath761 . to see the outward normal of the conformal structure of the glued disk we use coordinates on @xmath757 which fixes @xmath579 , @xmath100 , and one of the punctures next to @xmath100 , and coordinates on @xmath756 which fixes @xmath760 , @xmath634 , and @xmath762 . note that @xmath763 has punctures corresponding to all @xmath764 except @xmath100 and all @xmath765 except @xmath760 . we use coordinates on @xmath766 fixing @xmath579 , @xmath634 , and @xmath762 . the outward normal is then represented by the tangent vector to the circle at the puncture formerly next to @xmath100 directed away from @xmath100 . we next note that there are natural inclusions @xmath767 , @xmath153 . these induce the decomposition @xmath768 , where the @xmath82-direction is spanned by the normal direction discussed above . let @xmath769 be the standard orientation on @xmath770 , @xmath153 and let @xmath585 be the orientation of the outward normal then [ lmaglueconfstr ] the direct sum map above induces the orientation @xmath771 . in other words , @xmath772 we must consider two cases separately according to whether the fixed puncture in @xmath773 is @xmath774 or @xmath680 . let @xmath775 and @xmath776 denote the positive tangent vector of the boundary of the glued disk at @xmath764 and @xmath765 , respectively . assume first that @xmath774 is fixed . then by lemma [ lmashuffle ] @xmath777 the computation for @xmath680 fixed is similar . when proving the gluing theorem , section 7 in @xcite , holomorphic disks with @xmath778 punctures and holomorphic disks with @xmath779 punctures were treated simultaneously by adding marked points to disks with few punctures . we will take the same approach here and thus we need to discuss orientations on determinants in the presence of marked points . recall that we introduced marked points on a holomorphic disk @xmath780 with boundary on a legendrian submanifold @xmath12 by picking a codimension one submanifold @xmath781 of @xmath13 such that @xmath782 and such that @xmath783 is transverse to @xmath781 for some @xmath784 . we then consider the @xmath28-problem for maps @xmath785 in a neighborhood of @xmath138 , which takes the additional marked point @xmath26 into @xmath781 and we use a neighborhood of @xmath138 in that sobolev space to find local coordinates on the moduli space . in our study of orientations below , we will take @xmath781 as codimension one spheres in @xmath13 around one of the endpoints of some reeb chord to which some puncture of @xmath138 maps . we start by describing the corresponding situation on the level of the linearized equation . let @xmath786 be a trivialized boundary condition for the @xmath28-operator . pick @xmath787 points @xmath788 in @xmath93 which are not punctures . let @xmath789(d_m,{{\mbox{\bbb c}}}^n)$ ] and let @xmath790 $ ] then @xmath791 . for each @xmath792 , pick a linear form @xmath793 and consider the linear functionals @xmath794 let @xmath795 and define the operator @xmath796 as the restriction of @xmath412 . then the index of @xmath797 satisfies @xmath798 pick @xmath787 elements @xmath799 such that @xmath800 . this gives a direct sum decomposition @xmath801 which induces the exact sequence @xmath802 where @xmath159 is the direct sum decomposition followed by the @xmath294-projection to @xmath803 in the first summand , where @xmath490 is @xmath412 followed by projection to @xmath804 , and where @xmath66 is the natural projection induced form the inclusion @xmath805 . this sequence induces an isomorphism @xmath806 to facilitate our sign discussions we will assume that @xmath787 is _ even_. in that case the position of @xmath807 in the direct sum decomposition is of no importance for orientations and we have a canonical isomorphism @xmath808 thus , for @xmath787 even , an orientation on @xmath807 gives a canonical isomorphism between orientations on @xmath809 and orientations on @xmath376 . we next consider gluing isomorphisms in the presence of marked points . let @xmath810 and @xmath811 be trivialized boundary conditions such that the positive puncture of @xmath45 can be glued to some negative puncture of @xmath248 . assume that there are @xmath812 marked points near the positive puncture on @xmath86 and @xmath813 marked points near the negative puncture of @xmath248 to which @xmath45 is glued . consider the glued boundary condition @xmath814 and note that @xmath815 inherits the marked points from @xmath86 and @xmath250 , and thus comes equipped with @xmath816 marked points . consider the gluing sequence in lemma [ lmaglue2 ] . note that the cut - off of a function in @xmath287 ( @xmath288 ) vanishes on the part of @xmath815 which correspond to @xmath250 ( @xmath86 ) . this observation implies that the gluing sequence obtained by replacing the operators @xmath412 , @xmath817 , and @xmath392 in lemma [ lmaglue2 ] with @xmath818 , @xmath819 , and @xmath820 , respectively , is also exact . let @xmath373 and @xmath374 be orientations on @xmath376 and @xmath377 . orient @xmath821 and @xmath822 . let @xmath823 and @xmath824 be the induced orientations on @xmath809 and @xmath825 and endow @xmath826 with the orientation induced from the orientations of its summands . let @xmath379 be the orientation induced on @xmath381 from the gluing sequence of @xmath412 and @xmath817 . then @xmath379 and the orientation on @xmath826 induces an orientation @xmath827 on @xmath828 . the orientation on @xmath828 induced from the gluing sequence of @xmath818 and @xmath819 and the orientations @xmath823 and @xmath824 equals @xmath829 . to show this we consider the following diagram . @xmath830 @ > > > \left[\begin{matrix } { \operatorname{ker}}\bar{\partial}_a'\\ { \mathbb r}^{2a}\\ { \operatorname{ker}}\bar{\partial}_b'\\ { \mathbb r}^{2b } \end{matrix}\right ] @ > > > \left[\begin{matrix } { \operatorname{coker}}\bar{\partial}_a'\\ { \operatorname{coker}}\bar{\partial}_b ' \end{matrix}\right ] @ > > > \left[\begin{matrix } { \operatorname{coker}}\bar{\partial}_a\\ { \operatorname{coker}}\bar{\partial}_b \end{matrix}\right]\\ @aaa @aaa @vvv @vvv \\ { \operatorname{ker}}\bar{\partial}_{a\sharp b } @ > > > \left[\begin{matrix } { \operatorname{ker}}\bar{\partial}_{a\sharp b}'\\ { \mathbb r}^{2a}\\ { \mathbb r}^{2b } \end{matrix}\right ] @ > > > { \operatorname{coker}}\bar{\partial}_{a\sharp b } ' @ > > > { \operatorname{coker}}\bar{\partial}_{a\sharp b}\\ @aaa @aaa @vvv @vvv \end{cd}\ ] ] where the @xmath78 s on the left and right in the first and second horizontal rows have been dropped and where the @xmath78 s below the lower row of arrows have been dropped as well . the upper horizontal row is the direct sum of the sequences inducing @xmath831 and @xmath832 from @xmath373 and @xmath374 , respectively . the lower horizontal row is the sequence inducing @xmath829 from @xmath379 . the gluing sequence for @xmath392 is obtained from the above diagram by adding an arrow from the top left entry to the top right entry . ( thinking of it as a half - circular arc the gluing sequence is then the outer boundary of the diagram . ) the gluing sequence for @xmath820 is obtained from the above diagram by adding an arrow from the middle left entry in top row to the middle right entry in the same row and forgetting the @xmath821 and @xmath822 summands . ( thinking also of this arrow as a half - circular arc the gluing sequence is the inner boundary of the diagram . ) let @xmath405 be wedges of vectors representing the orientation pair on the pair @xmath833 the induced orientation on the pair @xmath834 , \left[\begin{matrix } { \operatorname{coker}}(\bar{\partial}_{a})\\ { \operatorname{coker}}(\bar{\partial}_{b } ) \end{matrix}\right ] \right)\ ] ] is then represented by @xmath835 where @xmath397 is a wedge of vectors on the complement of the image of @xmath408 . this pair in turn induces the orientation pair @xmath836 on the pair @xmath837 , \left[\begin{matrix } { \operatorname{coker}}(\bar{\partial}_{a}')\\ { \operatorname{coker}}(\bar{\partial}_{b } ' ) \end{matrix}\right ] \right)\ ] ] where @xmath47 is a wedge of vectors on the complement of the image of @xmath838 . on the other hand the orientation pair @xmath405 induces the orientation pair @xmath839 on the pair @xmath840 , { \operatorname{coker}}(\bar{\partial}_{a\sharp b}')\right).\ ] ] this in turn induces an orientation @xmath841 on the pair in . noting that the orientations of @xmath842 and @xmath843 agree if and only if those of @xmath844 and @xmath845 do and that the vertical maps on the @xmath826 summand is the identity we find that the lemma holds . we now turn to induced orientations on moduli spaces . we show that there is a natural way to pick marked points and an orientation on the corresponding copies of @xmath82 so that the orientation of a moduli space of a disk with marked points is determined exactly as for disks with many punctures . in our applications we use only moduli spaces of dimensions @xmath78 and @xmath72 so we consider these two cases . we also assume we are in a generic situation where all moduli spaces are transversely cut out . let @xmath780 be a holomorphic disk in a moduli space of dimension @xmath78 or @xmath72 . note then that the corresponding @xmath689-operator which is a part of the linearization of the @xmath28-map at @xmath138 ( @xmath622 denotes the conformal structure ) has either only kernel or only cokernel . consider the case of only cokernel first . in that case we choose an even number of marked points @xmath788 near a puncture of @xmath138 . we obtain in that way a new source disk @xmath846 with @xmath89 punctures and @xmath787 marked points . pick local @xmath72-parameter families of diffeomorphisms @xmath847 around the marked points which move them in the positive direction along the boundary and set @xmath848 . in this case the orientation inducing sequence reduces to @xmath849 viewing @xmath850 as a subspace of @xmath141 $ ] , we obtain a splitting @xmath851 and a corresponding direct sum decomposition @xmath852 . we obtain the diagram @xmath853 where the top arrow is an isomorphism with inverse which maps the tangent vector of a moving marked point to the corresponding factor in @xmath807 . note that the marked points in @xmath93 lies in an arc which contains no punctures . thus , this map is orientation preserving since @xmath787 is even . likewise the leftmost vertical map is the identity on the @xmath807-component . it follows that the orientation on the moduli space induced by @xmath854 is the same as the one induced by @xmath855 . consider next the case when @xmath856 . in this case , if @xmath787 is sufficiently large then @xmath857 and the orientation inducing sequence reduces to @xmath858 thus we get @xmath859 . we use the convention that marked points are added near the positive puncture in the negative direction of it . we then get the diagram @xmath860 where the top horizontal map is the quotient of the ( oriented ) automorphism group acting on the positively oriented tangent vectors to the marked points moving on the boundary which is then orientation preserving , and where the left vertical arrow is the map induced by @xmath861 . it follows again that the orientation on the moduli space induced by @xmath862 is the same as that induced by @xmath863 . we now determine how capping orientations behave under gluing as in lemma [ lmaglue2 ] . in fact , we concentrate on the case important for our applications : when two _ rigid _ disks @xmath138 and @xmath864 with boundary on a legendrian submanifold @xmath865 are being glued . as we have seen gluings of disks with few punctures ( when kernels of @xmath28-operators are present ) can be treated as gluings of disks with many punctures . we concentrate on the case of many punctures first and explain in remark [ rmkfewpunct ] how to modify the arguments in the presence of marked points . let @xmath45 be a trivialized stabilized boundary condition on the @xmath89-punctured disk ( corresponding to @xmath138 above ) , with positive puncture at a reeb chord @xmath866 and negative punctures at reeb chords @xmath867 . let @xmath248 be a trivialized stabilized boundary condition on the @xmath868-punctured disk ( corresponding to @xmath864 above ) , with positive puncture at a reeb chord @xmath73 and negative punctures at reeb chords @xmath869 . let @xmath392 be the operator which is obtained by gluing @xmath412 and @xmath817 at @xmath866 . since we model the case when the disks @xmath138 and @xmath864 are rigid and have many punctures we assume that @xmath870 and that @xmath871 in particular , it follows from the dimension formula that @xmath872 where @xmath873 are the homology classes of the boundary paths of @xmath138 and @xmath864 capped off with the capping paths at each puncture . since @xmath13 is orientable , @xmath874 and @xmath875 are even . let @xmath373 , @xmath374 and @xmath379 denote the capping orientations of the determinant lines of @xmath412 , @xmath817 , and @xmath392 . [ lmaglueridgop ] let @xmath876 , @xmath153 . the gluing map in lemma [ lmaglue2 ] induces from orientations @xmath877 and @xmath878 the orientation @xmath879 consider the boundary condition @xmath880 obtained by gluing the two boundary conditions @xmath363 and @xmath364 . this boundary condition can be obtained in two ways . * glue all capping disks to @xmath45 and @xmath248 , respectively and then join @xmath363 and @xmath364 . * glue the operators @xmath881 and @xmath882 to obtain an operator @xmath883 on the closed disk . glue all relevant capping disks to @xmath264 giving the boundary condition @xmath884 on the closed disk . then glue @xmath883 to @xmath885 . repeated application of lemmas [ lmaassocpp ] and [ lmaassocbp ] gives the following gluing sequence corresponding to the gluing ( i ) : @xmath886 @ > > > \end{cd}\\ & \begin{cd } \left [ \begin{matrix } \bigoplus_{j=1}^{m-1}{\operatorname{coker}}(\bar{\partial}_{f_j-})\\ { \operatorname{coker}}(\bar{\partial}_{b_k+})\\ { \operatorname{coker}}(\bar{\partial}_a)\\ { { \mbox{\bbb r}}}^n\\ \bigoplus_{j=1}^{r}{\operatorname{coker}}(\bar{\partial}_{b_j-})\\ { \operatorname{coker}}(\bar{\partial}_{a+})\\ { \operatorname{coker}}(\bar{\partial}_b ) \end{matrix } \right ] @ > > > { \operatorname{coker}}(\bar{\partial}_{\hat a\sharp \hat b } ) \end{cd } \longrightarrow 0 . \end{split}\ ] ] note that the capping orientation @xmath373 together with the orientations on the capping disks of @xmath45 induce the canonical orientation on @xmath887 , by definition . similarly , the capping orientation @xmath374 together with the orientations on the capping disks of @xmath248 induce the canonical orientation on @xmath888 . thus lemma [ lmagluecanor ] implies that that these orientations in the above sequence induce the canonical orientation on @xmath889 . to find the corresponding gluing sequence for ( ii ) we first look at the gluing sequence for @xmath392 this sequence is the following . @xmath890 it allows us to identify the two spaces involved . second the gluing sequence for @xmath883 is @xmath891 @ > > > { \operatorname{coker}}(\bar{\partial}_e ) @ > > > 0 , \end{cd}\ ] ] where both the leftmost and the rightmost non - trivial maps are isomorphisms . moreover , by definition , the chosen orientation pair on @xmath892 corresponds to the canonical orientation on @xmath883 . with these two identifications made we apply lemmas [ lmaassocpp ] and [ lmaassocbp ] and find that the gluing sequence in case ( ii ) is @xmath893 @ > > > \end{cd}\\ & \begin{cd } \left [ \begin{matrix } { \operatorname{coker}}(\bar{\partial}_{b_k+})\\ { \operatorname{coker}}(\bar{\partial}_{b_k-})\\ { { \mbox{\bbb r}}}^n\\ \bigoplus_{j=1}^{k-1}{\operatorname{coker}}(\bar{\partial}_{b_j-})\\ \bigoplus_{j=1}^{m-1}{\operatorname{coker}}(\bar{\partial}_{f_j-})\\ \bigoplus_{j = k+1}^{r}{\operatorname{coker}}(\bar{\partial}_{b_j-})\\ { \operatorname{coker}}(\bar{\partial}_{a+})\\ { \operatorname{coker}}(\bar{\partial}_a)\\ { \operatorname{coker}}(\bar{\partial}_b ) \end{matrix } \right ] @ > > > { \operatorname{ker}}(\bar{\partial}_{\hat a\sharp\hat b } ) \end{cd } \longrightarrow 0 . \end{split}\ ] ] lemma [ lmagluecanor ] implies that the chosen orientations on all capping disks together with the capping orientation @xmath379 on @xmath894 induce the canonical orientation on @xmath889 . we are now in position to determine the gluing sign . note first that in both sequences the middle map takes the complement of @xmath895 to @xmath78 and maps this vector non - trivially into the @xmath128 summand in the next term in the sequence . to compare the signs we first move @xmath128 so that it becomes the first summand in its respective sum in both sequences . note that because of , the dimension of the space @xmath896\ ] ] equals @xmath897 if @xmath866 is odd and @xmath898 if @xmath866 is even . thus , the dimension of this space is congruent to @xmath0 modulo @xmath636 in either case . in sequence ( i ) , moving @xmath128 to the first position thus gives an orientation change of sign @xmath899 to find the corresponding change in the sequence ( ii ) note that the dimension of the space @xmath900\ ] ] equals @xmath901 and we find that the orientation change is @xmath902 the orientation difference is thus @xmath903 with @xmath128 moved to the first position we compute the sign change by changing the order of summands in sequence ( i ) to agree with those in sequence ( ii ) : the orientation difference in the two sequences arising from their second terms are @xmath904 where the first factor comes from moving @xmath895 to the left and the second from moving @xmath905 to its position . the orientation difference arising in the third term in the sequences ( with @xmath128 already moved to the first position ) is calculated as follows . first we move the @xmath906-term in sequence ( i ) to the left . this subspace has dimension @xmath901 if @xmath907 is even and @xmath908 if @xmath907 is odd . since @xmath909 modulo 2 , by , we see that if @xmath907 is odd then there is no sign arising from this move since @xmath910 is even . on the other hand if @xmath907 is even we find the sign @xmath911 . in general the sign is thus @xmath912 second we move the subspace @xmath913 over @xmath291 . the latter space has dimension @xmath914 . we must calculate the dimension modulo @xmath636 of the former . since @xmath915 modulo @xmath636 , we find that the dimension of this space is @xmath916 if @xmath73 is even and @xmath917 if @xmath73 is odd . thus in any case we get a sign @xmath918 from this motion . finally we must move out @xmath919 if this term is non - zero and move @xmath920 to the right position . we consider two cases . first if @xmath866 is even then @xmath919 is zero dimensional and @xmath920 is odd dimensional giving a sign @xmath921 . on the other hand if @xmath866 is odd then @xmath920 is even dimensional and the sign is still @xmath921 since we must move the @xmath72-dimensional space @xmath919 . thus in any case we get @xmath922 multiplying out all the signs gives the over all sign @xmath923 as claimed . [ rmkfewpunct ] in the presence of marked points we simply need to replace the operators @xmath412 , @xmath817 , and @xmath392 above with @xmath818 , @xmath819 , and @xmath820 , respectively . in particular by adding sufficiently large even numbers of marked points we can assure that none of the latter operators has non - trivial kernel . of course adding marked points effects the dimensions of the cokernels of the operators but the final expression for the sign is independent of these dimensions so the lemma holds in the presence of marked points as well . let @xmath12 be a generic legendrian submanifold oriented and with a fixed spin structure . then as shown above , all moduli spaces of holomorphic disks with boundary on @xmath13 come equipped with orientations . we show the following result . let @xmath924 be a component of a @xmath72-dimensional moduli space with boundary @xmath925 . for @xmath926 , @xmath927 let @xmath928 if the orientation of @xmath924 is outwards at @xmath926 and let @xmath929 otherwise . a boundary component @xmath926 is a broken holomorphic disk . that is , two rigid holomorphic disks @xmath138 and @xmath864 such that the positive puncture of @xmath138 is identified with some negative puncture of @xmath864 . assume that the positive puncture of @xmath138 is @xmath866 and that its negative punctures are @xmath867 . assume also that the positive puncture of @xmath864 is @xmath73 and that its negative punctures are @xmath869 . [ lmamorient ] let @xmath930 and @xmath931 be the signs of @xmath138 and @xmath864 , respectively . then @xmath932 it is enough to consider the case of many punctures , see remark [ rmkfewpunct ] , so assume @xmath933 and @xmath934 . let @xmath45 and @xmath248 be the boundary conditions corresponding to @xmath138 and @xmath935 respectively . consider the commutative diagram @xmath936 here @xmath766 , @xmath937 , and @xmath938 are endowed with their standard orientations and @xmath82 is endowed with the orientation corresponding to the _ outward _ normal @xmath939 . moreover , the spaces appearing in the lower horizontal row are endowed with their capping orientations . we compute the orientations . the leftmost vertical map restricted to the complement of @xmath82 is an isomorphism between oriented spaces of sign @xmath940 . the lower horizontal arrow is an isomorphism of oriented vector spaces with sign @xmath941 by lemma [ lmaglueridgop ] . the upper horizontal arrow is an isomorphism of sign @xmath942 by lemma [ lmaglueconfstr ] . now , the oriented tangent space to @xmath924 is identified with the oriented kernel of the rightmost vertical map . chasing the diagram we find that this oriented kernel is identified with @xmath82 oriented by @xmath943 as claimed . finally we are in position to complete the proof of theorem [ thmd^2=0 ] . let @xmath73 be a reeb chord of @xmath13 and let @xmath944 be a word appearing in @xmath945 . consider a component of a @xmath72-dimensional moduli space @xmath946 with oriented boundary @xmath947 . assume that @xmath948 consists of two broken disks in @xmath949 respectively , where @xmath950 , and that @xmath951 consists of two broken disks in @xmath952 respectively , where @xmath953 . by the definition of the differential the disks in contribute @xmath954 where @xmath930 and @xmath931 are the signs of the two disks and where we use the fact that @xmath724 is even . on the other hand the disks in contribute @xmath955 where @xmath956 and @xmath957 are the signs of the two disks and where we use the fact that @xmath958 is even . by lemma [ lmamorient ] @xmath959 thus these two terms cancel in @xmath945 . since all terms contributing to @xmath945 arise in this way we conclude @xmath960 . we show that legendrian isotopies of a legendrian submanifold @xmath12 with a fixed spin structure does not change the stable tame isomorphism class of its associated algebra @xmath715 . more precisely [ thminv ] let @xmath961 , @xmath962 be a legendrian isotopy then the dga @xmath963 of @xmath964 and the dga @xmath965 of @xmath966 are stable tame isomorphic . ( in particular the contact homology of @xmath964 is isomorphic to that of @xmath966 . ) the theorem follows from lemma [ lmaeasyinv ] and corollaries [ corhsinv ] and lemma [ ststi ] which are proved below . our approach is to study the bifurcations in moduli spaces under generic isotopies . during such deformations three events effect the moduli - spaces ( and therefore possibly the differential ) : they undergo morse modifications , there appear disks of formal dimension @xmath640 ( so called handle slide disks because of analogous phenomena in morse theory ) , and there appear self tangency instances . to prove invariance it is sufficient to study these events separately . to show invariance in the first case we use a parameterized moduli space . to show invariance in the second we use an auxiliary legendrian embedding of @xmath967 $ ] with certain boundary conditions at @xmath968 and @xmath969 , and its differential graded algebra . we mention these proofs have an advantage over the more straight forward proofs of invariance given in @xcite . in particular , we avoid the need for `` degenerate gluing '' , which is technically much more difficult than the gluing results needed to prove @xmath970 let @xmath961 , be a generic @xmath72-parameter family of legendrian submanifolds such that @xmath964 and @xmath966 are generic and such that there are neither handle slide moments nor self tangency moments during the isotopy . that there are neither handle slide moments nor self tangency moments during the isotopy . [ lmaeasyinv ] @xmath971 note that since the complex angles in the added directions are taken to be smaller than any complex angle of a double point of @xmath972 , the capping orientations are continuous in @xmath973 . therefore , the parameterized moduli spaces are naturally oriented . in particular , the rigid disks on @xmath964 and @xmath966 equals the oriented boundary of the one - dimensional parameterized moduli space . this oriented cobordism immediately gives @xmath974 . we associate to a @xmath72-parameter family of legendrian embeddings @xmath975 , @xmath962 , legendrian embeddings @xmath976 , depending on @xmath276 and a positive morse function @xmath977 . let @xmath978 , @xmath979 $ ] be a legendrian isotopy . for small @xmath276 , fix smooth non - decreasing functions @xmath980\to[-\delta,\delta]\ ] ] such that @xmath981 for @xmath982 , and such that @xmath983 for @xmath984 . note that @xmath985 as @xmath986 . fix standard coordinates @xmath987 on @xmath83 . define @xmath988 , @xmath989 as @xmath990$,}\\ \phi_{\alpha^\delta(t ) } & \text{for $ t\in[-1,1]$,}\\ \phi_{\delta } & \text{for $ t\in[1,\infty)$. } \end{cases}\ ] ] write @xmath991 fix a _ positive _ morse function @xmath977 and @xmath276 . let @xmath992 denote the derivative of @xmath21 . define @xmath993 , @xmath994 where @xmath995 it is straightforward to check that @xmath996 is a legendrian embedding . assume that the morse function @xmath977 above has local minima at @xmath997 and no critical points in the region @xmath998 . then the @xmath999-coordinate @xmath1000 of each reeb chord @xmath721 of @xmath1001 satisfies @xmath1002 . let @xmath138 be a holomorphic disk with boundary on @xmath996 and one positive puncture . [ lmamin ] if the positive puncture of @xmath138 maps to a reeb chord @xmath721 of @xmath1001 with @xmath1003 . then the image of @xmath138 lies in @xmath1004 . for definiteness assume the positive puncture of @xmath138 maps to @xmath721 with @xmath1005 . project @xmath1006 to the @xmath1007-plane . the image of this projection is contained in the region @xmath1008 for some @xmath159 . if the projection @xmath1009 of @xmath138 to the @xmath1007-plane is non - constant then it covers at least one of the regions @xmath1010 for some ball @xmath1011 . since @xmath138 has boundary on @xmath1012 , @xmath1009 takes no boundary point to the line @xmath1013 . this and the above covering property contradicts @xmath1009 being bounded in the @xmath1014-direction . the lemma follows . [ lmaalg ] the image of every holomorphic disk with boundary on @xmath996 is contained in the region @xmath1015 . arguing as in the proof of lemma [ lmamin ] we find that the projection to the @xmath1007-plane of a holomorphic disk with boundary on @xmath996 can not intersect the lines @xmath1016 in interior points . it follows that the image of any disk lies entirely in one of the regions @xmath1017 , @xmath1015 , or @xmath1018 . however , a disk with image in @xmath1019 ( @xmath1017 ) must have its positive puncture at a reeb chord @xmath721 with @xmath1005 ( @xmath1020 ) . the lemma follows from lemma [ lmamin ] . assume now that @xmath1021 is generic for each @xmath1022 . then it is a consequence of lemma 6.25 in @xcite that any rigid disk with boundary on @xmath996 and positive corner at some reeb chord @xmath721 with @xmath1003 is transversely cut out . moreover by lemma 6.12 in @xcite transversality of the @xmath28-equation can be achieved by perturbation near the positive puncture of a disk and it follows that there exists ( arbitrarily small ) perturbations of @xmath996 which are supported in the region @xmath1023 and which makes every moduli space ( of formal dimension @xmath1024 ) transversely cut out . we fix such a perturbation of @xmath996 but keep the notation @xmath996 for the perturbed legendrian embedding . let @xmath1025 denote the algebra over @xmath1026={{\mbox{\bbb z}}}[h_1(l)]$ ] generated by the reeb chords of @xmath996 as in subsection [ defofal ] and define the map ( differential ) @xmath30 of @xmath1025 as there . [ lmaalg ] the map @xmath1027 satisfies @xmath1028 . in the light of lemma [ lmaalg ] , a word by word repetition of the proof of theorem [ thmd^2=0 ] establishes the lemma . let @xmath975 , @xmath1029 be a legendrian isotopy such that @xmath964 is a generic handle slide moment . that is , there exists one handle slide disk in some @xmath75 , which is the only non - empty moduli space of formal negative dimension , that all moduli spaces of holomorphic disk with boundary on @xmath1030 , @xmath1031 of negative formal dimension are empty , and that all moduli spaces of rigid disks are transversally cut out . we choose notation so that @xmath1032 are the reeb chords of @xmath964 and so that @xmath1033 recall for a reeb chord @xmath1034 @xmath1035 is the difference in @xmath37-coordinates of its endpoints in @xmath7 let @xmath977 be a positive morse function with local minima at @xmath997 , no critical points in the region @xmath998 , and one local maximum at @xmath78 . [ lmarchords ] for all sufficiently small @xmath276 the reeb chords of @xmath996 are @xmath1036 , b_j[1 ] , b_j[0]\bigr\}_{j=1}^r\cup \bigl\{a[-1 ] , a[1 ] , a[0]\bigr\}\cup \bigl\{c_j[-1 ] , c_j[1 ] , c_j[0]\bigr\}_{j=1}^s,\ ] ] where for any reeb chord @xmath721 of @xmath964 , @xmath1037|=|c[1]|=|c[0]|-1=|c|$ ] . it is easy to see that for @xmath1038 the reeb chords are as described above and that the corresponding double points in @xmath1039 are transverse . this shows that the reeb chords are as claimed for all sufficiently small @xmath1040 . the second statement in the lemma is a straightforward consequence of the following grading formula from @xcite . let @xmath1041 be the @xmath37-coordinates of the upper and lower points in the front projection corresponding to the reeb chord @xmath1042 assume @xmath1043 is above @xmath1044 . near @xmath1043 we can represent the front as the graph of a function @xmath1045 @xmath1046 with @xmath1047 for some @xmath1048 we can similarly find a function @xmath1049 for the front near @xmath1050 let @xmath1051 since @xmath1041 correspond to a double point in the lagrangian projection , @xmath1052 has a critical point at @xmath1053 if @xmath721 is a transverse double point @xmath189 is a non - degenerate critical point . from @xcite we have latexmath:[\[\label{eq : frontform } @xmath66 is a path in the front connecting @xmath1043 to @xmath1044 and @xmath1055 and @xmath1056 is the number of down- and up - cusps of @xmath1057 we call @xmath1058 $ ] , @xmath1059 $ ] , and @xmath1060 $ ] , @xmath1061$]-reeb chords , and @xmath1062 $ ] , @xmath1063 $ ] , and @xmath1064 $ ] , @xmath1065$]-reeb chords . as above we perturb @xmath996 slightly in the region @xmath1066 to make it generic with respect to holomorphic disks . note that the @xmath999-coordinate of @xmath1065$]-reeb chord equals @xmath997 and that the @xmath999-coordinate of a @xmath1061$]-reeb chord is very close to @xmath78 for small @xmath276 . consider a sequence of functions @xmath1067 as above with @xmath1068 as @xmath1069 ( i.e. each @xmath1067 has a non - degenerate maximum at @xmath78 and non - degenerate local minima at @xmath997 ) . fix @xmath1070 and pick @xmath276 sufficiently small so that @xmath1071 satisfies lemma [ lmarchords ] . let @xmath1072 . we next note that as @xmath986 , @xmath1073 where @xmath1074 with @xmath1075 . [ lmaconnstrip ] there exists @xmath1076 such that for all @xmath1077 there exists a @xmath1078 such that for all @xmath1079 and any reeb chord @xmath721 the following holds . the moduli spaces @xmath1080,c[1])$ ] and @xmath1080,c[-1])$ ] of holomorphic disks with boundary on @xmath1012 consists of exactly one point which is a transversely cut out rigid disk . moreover the sign of the rigid disk in @xmath1080,c[1])$ ] and that of the disk in @xmath1080,c[-1])$ ] are opposite . first consider the case @xmath1038 . it is easy to find rigid disks in the @xmath1007-plane with positive puncture at @xmath1081 $ ] and negative puncture at @xmath1082 $ ] . moreover , by @xcite lemma 6.25 these disks are transversely cut out . to see that these are the only disks , let @xmath1083 and @xmath193 be neighborhoods of the endpoints of the reeb chord @xmath721 in @xmath964 and consider the projections of @xmath1084\times u)$ ] and @xmath1084\times v)$ ] to @xmath5 . for sufficiently large @xmath1070 , these projections intersect only at @xmath78 and it follows that there exists a positive @xmath1085 such that the area of the projection of any disk with boundary on @xmath1086 , positive puncture at @xmath1081 $ ] , and negative at @xmath1082 $ ] is either equal to zero or larger than @xmath1052 . since @xmath1087)\to{{\mathcal{z}}}(c[\pm 1])$ ] as @xmath1069 it follows that for @xmath1070 large enough the disks in the @xmath1007-plane are the only ones . we also check the statement about signs in the case @xmath1038 . to this end , note that the trivialized boundary conditions of the two disks in the @xmath1007-plane are identical and that multiplication by @xmath640 is a holomorphic automorphism relating them . since multiplication by @xmath640 reverses the orientation of the kernel of the linearized problem , it follows that their signs are opposite . finally , we note that the fact that the moduli space @xmath1080,c[\pm 1])$ ] corresponding to @xmath1086 is transversely cut out implies that the statement of the lemma holds also for @xmath1012 for all sufficiently small @xmath1040 ( where the smallness depends on @xmath1070 ) . we next note that as @xmath1069 , @xmath1086 approaches the legendrian submanifold @xmath1088 the projection of this legendrian submanifold to @xmath88 is simply the @xmath999-axis and its projection to @xmath5 agrees with that of @xmath964 . [ lmacn = a ] there exists @xmath1076 such that for all @xmath1077 there exists a @xmath1078 such that for all @xmath1079 and any reeb chord @xmath1089 the following holds . if the moduli space @xmath1090;{\mathbf e})$ ] , where @xmath1091 is a word constant in the @xmath1061$]-generators and @xmath1092 $ ] , has formal dimension @xmath78 then it is empty . again we start with the case @xmath1038 . consider a disk @xmath138 as above with boundary on @xmath1086 . as @xmath1069 , @xmath1093 and the projection of @xmath138 converges to a broken disk @xmath1094 with boundary on @xmath964 . the components @xmath1095 of such a broken disk either have formal dimension at least @xmath78 , or equals the handle slide disk . also , any reeb chord @xmath1096 appearing as a puncture of some @xmath1095 has @xmath1097 and exactly one component of the broken disk must have its positive puncture at @xmath1089 . this component has formal dimension at least @xmath78 ( since it is not the handle slide disk ) . for a disk @xmath1095 let @xmath1098 is the grading of its positive puncture and @xmath1099 the sum of gradings of its negative punctures and the negative of the grading of the homology data . then the formal dimension of ( the moduli space of ) @xmath1095 is @xmath1100 . the above implies that @xmath1101 since the positive puncture of @xmath138 is its only @xmath1061$]-puncture it follows that the formal dimension of @xmath138 equals @xmath1102 . the statement of the lemma follows for @xmath1038 . since emptyness of a moduli space is an open condition the lemma follows in general . let @xmath1103 be the map from @xmath1104 to @xmath1105 which maps @xmath1082 $ ] to @xmath721 and @xmath1081 $ ] to @xmath78 for any reeb chord @xmath721 of @xmath13 . [ lmac = a ] there exists @xmath1076 such that for all @xmath1077 there exists a @xmath1078 such that for all @xmath1079 the following holds . if @xmath138 is a holomorphic disk with boundary on @xmath1012 in @xmath1106,{\mathbf e})$ ] , where @xmath1091 is a word constant in the @xmath1061$]-generators and @xmath1107 $ ] , and if this moduli space has formal dimension @xmath78 then @xmath1108 and @xmath1109 . consider first the case @xmath1038 . taking the limit as @xmath1069 and arguing as in the proof of lemma [ lmacn = a ] we see that the projection of @xmath138 converges to a broken disk @xmath1094 , that all reeb chords @xmath1096 appearing as a puncture of some @xmath1095 satisfies @xmath1110 , and that there is a unique component with its positive puncture at @xmath73 . if this component is not the handle slide disk then the argument in the proof of lemma [ lmacn = a ] shows that the formal dimension of @xmath138 is at least @xmath72 . if , on the other hand , this component is the handle slide disk then the formal dimension of @xmath138 equals @xmath78 only if the broken disk has no other components . this shows the lemma for @xmath1038 . again since the condition that a moduli space is empty is open the lemma follows in general . fix @xmath1070 sufficiently large and @xmath276 sufficiently small so that lemmas [ lmaconnstrip ] , [ lmacn = a ] , and [ lmac = a ] holds for @xmath1012 . we also assume that @xmath1012 is generic with respect to holomorphic disks . let @xmath1111 let @xmath1112 . we denote the differential of @xmath1113 by @xmath658 , see lemma [ lmaalg ] . there are natural inclusions @xmath1114 . lemma [ lmamin ] implies that this is an inclusion of dga s in other words , @xmath1115= \gamma_{\pm}({\partial}_{\pm } c),\ ] ] where @xmath1116 is the map defined on generators by @xmath1117 $ ] , and where @xmath1118 is the differential on @xmath1119 . for generators @xmath1058 $ ] we have by lemmas [ lmaconnstrip ] and [ lmacn = a ] @xmath1120 } \delta b_j[0]= b_j[1]- b_j[-1 ] + \beta_1^j + { { \mathcal{o}}}(2),\ ] ] where @xmath1121 is linear in the @xmath1081$]-generators and @xmath1122 denotes a linear combination of monomials which are at least quadratic in the @xmath1061$]-generators . for the generator @xmath1059 $ ] we have by lemmas [ lmaconnstrip ] and [ lmac = a ] @xmath1123 } \delta a[0]= a[1]-a[-1 ] + \epsilon + \alpha_1 + { { \mathcal{o}}}(2),\ ] ] where @xmath1124 , where @xmath1125 and where @xmath1126 is linear in the @xmath1061$]-generators . for generators @xmath1060 $ ] we have by lemmas [ lmaconnstrip ] and [ lmacn = a ] @xmath1127 } \delta c_j[0]= c_j[1]-c_j[-1 ] + \gamma_1^j + \delta_1^j(a[0])+ { { \mathcal{o}}}(2),\ ] ] where @xmath1128)$ ] is linear in the @xmath1061$]-generators , where @xmath1129)$ ] lies in the ideal generated by @xmath1059 $ ] , and where @xmath1130 is constant in the @xmath1059 $ ] generator . below we will consider @xmath1131 and @xmath1132 as different differentials on the algebra @xmath56 . let @xmath1133 be as in and write @xmath1134 . consider the tame isomorphism @xmath221 of @xmath56 defined on generators as @xmath1135 if @xmath1136 and @xmath721 is a generator of @xmath56 then let @xmath1137 be the map defined on monomials by replacing each occurrence of @xmath721 by @xmath864 . then with @xmath1138 denoting the induced differential , a straightforward calculation gives @xmath1139 [ lmahsinv ] the algebra @xmath1140 is isomorphic to the algebra @xmath1141 . we prove that the two differentials agree on generators . by lemma [ lmaalg ] , @xmath1142 . thus , summing the terms constant in the @xmath1061$]-generators after acting by @xmath658 in we find @xmath1143 } 0={\partial}_+b_j[1 ] -{\partial}_-b_j[-1 ] + \left(\delta\beta_1\right)_0,\ ] ] where @xmath1144 denotes the part of @xmath1145 which is constant in the @xmath1061$]-generators . since the constant part of @xmath1146 $ ] equals @xmath1147-b_k[-1]$ ] it follows that @xmath1148 therefore , applying @xmath1103 in , we conclude @xmath1149 since no monomial in @xmath1150 contains @xmath73 . applying @xmath658 to we find similarly @xmath1151 } { \partial}_- a[-1 ] = { \partial}_+ a[1 ] + \delta\epsilon + \left(\delta\alpha_1\right)_0.\ ] ] the first equality in implies that @xmath1152 since every @xmath1061$]-generator in @xmath1126 is for the form @xmath1058 $ ] we find , as with @xmath1153 above , that @xmath1154 . we conclude @xmath1155 applying @xmath658 to gives @xmath1156 } { \partial}_-c_j[-1]={\partial}_+c_j[1 ] + \left(\delta\gamma_1^j\right)_0 + \left(\delta\delta_1^j(a[0])\right)_0.\ ] ] applying @xmath1157-\epsilon}{a[1]}\right)\bullet$ ] to both sides in and noting that no monomial in @xmath1158 $ ] contains an @xmath1159 $ ] generator we get @xmath1160 = & \left(\frac{a[-1]-\epsilon}{a[1]}\right)\bullet({\partial}_+c_j[1])+ \left(\frac{a[-1]-\epsilon}{a[1]}\right)\bullet \left(\delta\gamma_1^j\right)_0\\ & + \left(\frac{a[-1]-\epsilon}{a[1]}\right)\bullet \left(\delta\delta_1^j(a[0])\right)_0 . \end{split}\ ] ] each term in @xmath1161)\right)_0 $ ] arises by replacing @xmath1059 $ ] in every monomial @xmath1162\eta$ ] of @xmath1129)$ ] with @xmath1163-a[-1]+\epsilon)$ ] yielding @xmath1164-a[-1]+\epsilon)\eta$ ] . when @xmath1157-\epsilon}{a[1]}\right)\bullet$ ] is a applied to @xmath1164-a[-1]+\epsilon)\eta$ ] the result is @xmath1165-\epsilon - a[-1]+\epsilon)\eta=0.\ ] ] thus , the last term in vanishes . since the @xmath1061$]-generator of any monomial in @xmath1130 equals either @xmath1166 $ ] for some @xmath1070 , or @xmath1167 $ ] for some @xmath1168 and since the constant part of @xmath1169 $ ] equals @xmath1170-c_k[-1]$ ] , and the constant part of @xmath1146 $ ] equals @xmath1147-b_k[-1]$ ] , we conclude that @xmath1171-\epsilon}{a[1]}\right)\bullet \left(\delta\gamma_1^j\right)_0=0.\ ] ] thus , applying @xmath1103 to we arrive at @xmath1172 the lemma follows from , , . [ corhsinv ] if @xmath961 , @xmath1173 , is a legendrian isotopy with a generic handle slide at @xmath1174 as above then the stable tame isomorphism classes of @xmath1175 and @xmath965 are the same . we now turn our attention to self tangency instances . as shown in @xcite there is no loss of generality in assuming that our isotopy near the self tangency instant has standard from . consider a @xmath72-parameter family of legendrian submanifolds @xmath961 , @xmath979 $ ] . we assume that @xmath972 is constant in @xmath973 outside @xmath1176 for some ball @xmath1177 or radius @xmath1178 around the origin . the intersection of @xmath972 with @xmath1179 consists of two sheets @xmath1180 and @xmath1181 . let @xmath1182 , @xmath1183 , and @xmath1184 be coordinates on @xmath83 . assume that the sheet @xmath1181 is constant in @xmath973 and that it satisfies @xmath1185 the second sheet is moving with @xmath973 . let @xmath1186 , @xmath1187 , and let @xmath1188 denote the standard inner product on @xmath1189 . then @xmath1190 is given by the map @xmath1191\times{{\mbox{\bbb r}}}\times{{\mbox{\bbb r}}}^{n-1}\to{{\mbox{\bbb c}}}^n\times{{\mbox{\bbb r}}}$ ] , @xmath1192 where @xmath1193 where @xmath1194 is a constant . in the region @xmath1195 the isotopy interpolates by the one above and the constant isotopy . we denote this @xmath72-parameter family of legendrian embeddings @xmath975 , @xmath979 $ ] . let @xmath276 . note that @xmath1196 has two reeb chords more than @xmath1197 and that the extra reeb chords of @xmath1196 converges to the self tangency reeb chord @xmath585 of @xmath1198 . we choose notation so that the reeb chords of @xmath1197 are @xmath1199 those of @xmath1198 are @xmath1200 and those of @xmath1196 are @xmath1201 and so that @xmath1202 fix a positive morse function @xmath21 with local minima at @xmath997 and no critical points in @xmath998 and with one local maximum at @xmath1203 , for some small @xmath490 . assume that @xmath1204 is a generic self tangency moment which means that all moduli spaces of negative formal dimension are empty and that all rigid disks with boundary on @xmath1204 are transversely cut out . [ lmastreebchords ] fix @xmath21 as above . then there exists @xmath1205 such that for all @xmath1206 the reeb chords of @xmath1207 are @xmath1208,b_j[1],b_j[0]\}_{j=1}^r\cup \{b[-1],a[-1]\}\cup \{a_j[-1],a_j[1],a_j[0]\}_{j=1}^s,\ ] ] where the @xmath999-coordinates of the @xmath1065$]-reeb chords are @xmath997 , and where the @xmath999-coordinates of the @xmath1061$]-reeb chords are close to @xmath78 . consider first the case @xmath1038 . the reeb chords of @xmath1209 are easily seen to be @xmath1208,b_j[1],b_j[0]\}_{j=1}^r\cup \{o[-1],o[1],o[0]\}\cup \{a_j[-1],a_j[+1],a_j[0]\}_{j=1}^s,\ ] ] and all except @xmath1210 $ ] and @xmath1211 $ ] have transverse tangent planes at their ends . we conclude that for @xmath276 sufficiently small @xmath1207 has the reeb chords @xmath1208,b_j[1],b_j[0]\}_{j=1}^r\cup \{a_j[-1],a_j[+1],a_j[0]\}_{j=1}^s,\ ] ] and possibly some reeb chords in a neighborhood of @xmath1212 $ ] and of @xmath1211 $ ] . to find these we take a closer look at @xmath1213 if @xmath1214 denotes the lower endpoint of @xmath585 then in a neighborhood of @xmath1215 , @xmath1207 is simply the embedding ( with notation as above ) @xmath1216 in neighborhoods of @xmath1217 , @xmath1207 is given by @xmath1218 and we find that there are two reeb chords @xmath1219 $ ] and @xmath1220 $ ] in a neighborhood of @xmath1221 $ ] and no reeb chords in a neighborhood of @xmath1222 $ ] . in a neighborhood of @xmath1223 it is given by @xmath1224 to have a reeb chord we note first that the equation @xmath1225 must hold . since @xmath276 , the @xmath973-coordinate of any reeb chord thus satisfies @xmath1226 . moreover , @xmath1227 implies that at a reeb chord @xmath1228 and from the expression for @xmath1229 , @xmath1230 . the final condition is @xmath1231 , which implies @xmath1232 by the choice of @xmath21 , the local maximum of @xmath21 lies at @xmath1233 thus @xmath1234 for @xmath1235 and letting @xmath986 we see that @xmath996 does not have any reeb chords near @xmath1211 $ ] . this finishes the proof . consider a sequence of functions @xmath1067 as above with @xmath1068 as @xmath1069 ( i.e. each @xmath1067 has a non - degenerate maximum at @xmath1236 and non - degenerate local minima at @xmath997 ) . fix @xmath1070 and pick @xmath276 sufficiently small so that @xmath1071 satisfies lemma [ lmastreebchords ] . let @xmath1072 . [ lmastconnstrip ] there exists @xmath1076 such that for all @xmath1077 there exists a @xmath1078 such that for all @xmath1079 and a reeb chord @xmath1237 the following holds . the moduli spaces @xmath1080,c[1])$ ] and @xmath1080,c[-1])$ ] of holomorphic disks with boundary on @xmath1012 consists of exactly one point which is a transversely cut out rigid disk . moreover the sign of the rigid disk in @xmath1080,c[1])$ ] and that of the disk in @xmath1080,c[-1])$ ] are opposite . the proof is a word by word repetition of the proof of lemma [ lmaconnstrip ] . we next note that as @xmath1069 , @xmath1086 approaches the legendrian submanifold @xmath1088 the projection of this legendrian submanifold to @xmath88 is simply the @xmath999-axis and its projection to @xmath5 agrees with that of @xmath964 . [ lmastc = b ] there exists @xmath1076 such that for all @xmath1077 there exists a @xmath1078 such that for all @xmath1079 and any reeb chord @xmath1238 , @xmath1239 , the following holds . if the moduli space @xmath1240;{\mathbf e})$ ] , where @xmath1091 is a word constant in the @xmath1061$]-generators and @xmath1241 $ ] , has formal dimension @xmath78 then it is empty . for @xmath1070 sufficiently large , @xmath1242)<{{\mathcal{z}}}(a[-1])<{{\mathcal{z}}}(b[-1])$ ] . therefore , for such @xmath1070 , a disk with its positive puncture at @xmath1058 $ ] must have negative punctures mapping to reeb chords in the set @xmath1243,b_i[\pm 1]\}_{i=1}^s$ ] . after this observation the lemma follows from the proof of lemma [ lmacn = a ] . [ lmastc = a ] there exists @xmath1076 such that for all @xmath1077 there exists a @xmath1078 such that for all @xmath1079 and any reeb chord @xmath1244 , @xmath1245 the following holds . if the moduli space @xmath1246;{\mathbf e})$ ] , where @xmath1091 is a word constant in the @xmath1061$]-generators and @xmath1247 $ ] , has formal dimension @xmath78 then @xmath1219 $ ] appears at least once as a letter in the word @xmath1091 . let @xmath138 be a disk in @xmath1246;{\mathbf e})$ ] . as in the proof of lemma [ lmac = a ] its projection to @xmath5 converges to a broken disk with boundary on @xmath964 as @xmath1069 . let @xmath1094 be the components of this broken disk . as above we let @xmath585 denote the degenerate reeb chord of @xmath964 . let @xmath1248|=|a[-1]|-1 $ ] . then the formal dimension of a disk @xmath1249 satisfies the following . * if the positive puncture of @xmath1249 does not equal @xmath585 then the formal dimension of @xmath1249 equals @xmath1250 , where @xmath1251 is the grading of the reeb chord at its positive puncture , and where @xmath1252 is the sum of the gradings at its negative corners and the negative of the grading of its homology data . * if the positive puncture of @xmath1249 equals @xmath585 then the formal dimension of @xmath1249 equals @xmath1253 since each of the disks @xmath1249 have non - negative formal dimension and since at least one of them does not have its positive puncture at @xmath585 , we find @xmath1254 thus , assuming that none of the negative punctures of @xmath138 map to @xmath1219 $ ] we find that the formal dimension of @xmath138 is at least @xmath72 which contradicts it being rigid . it follows that at least one of the negative punctures of @xmath138 map to @xmath1219 $ ] . fix @xmath1070 sufficiently large and @xmath276 sufficiently small so that lemmas [ lmastc = b ] and [ lmastc = a ] hold for @xmath1255 . let @xmath1256 . consider the algebra @xmath1112 and let its differential be @xmath658 . note that @xmath1257 and @xmath1258 can be considered as subalgebras of @xmath1113 via the map which takes @xmath1244 and @xmath1238 to @xmath1259 $ ] and @xmath1062 $ ] and @xmath73 and @xmath1096 to @xmath1219 $ ] and @xmath1220 $ ] , respectively . it is a consequence of lemma [ lmamin ] that @xmath1119 are differential graded subalgebras of @xmath363 . in other words , denoting their respective differentials @xmath1131 and @xmath1132 we have @xmath1260 ) & = \gamma_\pm({\partial}_\pm b_j),\quad\text { for } j=1,\dots , r,\\ \delta(a[-1 ] ) & = \gamma_-({\partial}_-a ) , & \\ \delta(b[-1 ] ) & = \gamma_-({\partial}_-b),&\\ \delta(a_j[\pm 1 ] ) & = \gamma_\pm({\partial}_\pm a_j),\quad\text { for } j=1,\dots , s,\end{aligned}\ ] ] where @xmath1261 $ ] , @xmath1262 $ ] , @xmath1263 $ ] , and @xmath1264 $ ] . it follows from lemma [ lmastc = b ] that @xmath1265=b_j[1]-b_j[-1]+\beta_1^j+{{\mathcal{o}}}(2),\ ] ] where @xmath1121 denotes term which is linear in the @xmath1061$]-generators and @xmath1122 the term which is quadratic and higher . it follows from lemma [ lmastc = a ] that @xmath1266=a_j[1]-a_j[-1]+\gamma(a[-1])+\alpha_1^j+{{\mathcal{o}}}(2),\ ] ] where @xmath1267)$ ] lies in the ideal generated by @xmath1219 $ ] and is constant in the @xmath1061$]-generators and where @xmath1268 is the linear in the @xmath1061$]-generators . consider the stabilized algebra @xmath1269 with extra generators @xmath1270 and @xmath1271 , @xmath1272 and @xmath1273 and define the algebra homomorphism @xmath1274 @xmath1275 where @xmath864 is the unique element in @xmath1276 such that @xmath1277 ( in @xmath1278 ) . for the existence of such an element see @xcite , lemma 1.16 . let @xmath1279 be the natural projection and let @xmath1280 be the subalgebra generated by @xmath1281 . then [ lmaalg1 ] @xmath1282 for all @xmath1283 and @xmath1284 let @xmath1103 be the map which takes @xmath1285 $ ] and @xmath1286 $ ] to @xmath1238 and which takes @xmath1287 $ ] and @xmath1288 $ ] to @xmath1244 . then , for generators @xmath1238 , follows by applying @xmath1289 to . we also have @xmath1290 and @xmath1291 thus , holds and it is sufficient to prove for @xmath1244-generators to conclude it holds in general . applying @xmath658 to and considering the constant term we find @xmath1292={\partial}_- a_j[-1 ] -\delta(\gamma(a[-1 ] ) ) - ( \delta\alpha^j_1)_0.\ ] ] letting @xmath1293 be the map which takes @xmath1219 $ ] to @xmath1270 and @xmath1220 $ ] to @xmath1294 we find that @xmath1295=\tilde\phi_0{\partial}_-a_j[-1 ] -\tilde\phi_0(\delta(\gamma ( a[-1 ] ) ) -\tilde\phi_0(\delta\alpha^j_1)_0.\ ] ] since the constant part of @xmath1296 $ ] equals @xmath1297-b_j[-1])$ ] we note that @xmath1103 annihilates each polynomial in @xmath1298 which originates from a monomial in @xmath1268 in the ideal generated by @xmath1058 $ ] . moreover , the constant part of @xmath1299 $ ] equals @xmath1287-a_j[-1]+\gamma(a[-1])$ ] . since @xmath450 annihilates @xmath1300))$ ] and since @xmath1103 annihilates @xmath1287-a_j[-1]$ ] and since @xmath450 and @xmath1103 commutes we find that @xmath1301 annihilates the last term . finally , each term in @xmath1302))$ ] contains either @xmath1219 $ ] or ( when @xmath1219 $ ] is differentiated ) @xmath1220+\gamma_-v$ ] . hence , @xmath450 annihilates @xmath1303))$ ] and we conclude @xmath1304 as claimed . [ ststi ] @xmath1305 is stable tame isomorphic to @xmath1306 . given lemma [ lmaalg1 ] the proof of this lemma is standard , see @xcite . in our construction of legendrian contact homology in subsection [ defofal ] we assume that our legendrian manifolds @xmath12 are spin and we fix a spin structure on @xmath13 . in general the dga @xmath717 of @xmath13 depends on the fixed spin structure . we explain here the exact form of this dependence . let @xmath6 be an oriented @xmath0-manifold which is spin . in subsection [ oms ] we viewed spin structures as trivializations of @xmath1307 over the @xmath72-skeleton that extends over the @xmath636-skeleton . given a spin structure @xmath1308 on @xmath6 we can identify the set of spin structures @xmath1309 with @xmath1310 specifically , let @xmath1311 be another spin structure . denote the trivialization associated to @xmath1311 by @xmath1312 isotop @xmath1313 and @xmath1314 to be the same on the 0-skeleton @xmath660 of @xmath1315 now along each edge in the 1-skeleton @xmath661 of @xmath6 we can compare @xmath1313 to @xmath1312 this gives us a loop in @xmath1316 thus we have a map @xmath1317 one may easily check that this gives a well defined cohomology class @xmath1318\in h^1(m;{{\mbox{\bbb z}}}_2).$ ] it is a standard fact that @xmath1319 is a one to one correspondence . we denote this map @xmath1320 let @xmath12 be a legendrian submanifold with a fixed spin structure @xmath1308 . let @xmath1311 be another spin structure on @xmath13 . let @xmath1321 and @xmath1322 denote the differentials on @xmath715 induced from the spin structures @xmath1308 and @xmath1311 respectively . let @xmath68 . [ changeos ] let @xmath73 be any reeb chord of @xmath13 . assume that @xmath1323 where @xmath1324 , @xmath1325 , and where @xmath1326 is a reeb chord word . then @xmath1327 where @xmath1328 is the non - trivial homomorphism . take a triangulation of @xmath13 containing all the double points in the complex projection in the 0-skeleton and containing the capping paths for the double points in the 1-skeleton . ( we can assume the capping paths are all disjoint arcs in @xmath1329 ) note that we may find trivializations @xmath1313 and @xmath1314 corresponding to @xmath1308 and @xmath1311 , respectively , of @xmath662 over @xmath682 such that the trivializations agree over all capping paths . let @xmath138 be any rigid holomorphic disk . consider the trivializations of the lagrangian boundary conditions on the closed disk which arises when @xmath138 is capped off and which is induced from @xmath1313 and @xmath1314 respectively . the difference of these trivializations arises in differences of the framings along the boundary paths of @xmath138 . since @xmath1313 and @xmath1314 agrees on the capping paths it follows that this framing difference is exactly @xmath1330 , where @xmath45 is the homology class encoding the boundary of @xmath138 . by @xcite the canonical orientation on the determinant line of homotopically different trivializations of a lagrangian boundary condition on the closed disk are opposite . the lemma follows . while it seems crucial that a legendrian submanifold be spin to define an oriented version of contact homology , there is no real dependence on the spin structure , when we define contact homology over @xmath29.$ ] let @xmath13 be a legendrian submanifold of @xmath7 the dga s of @xmath13 over @xmath29 $ ] associated to any two spin structures are tame isomorphic . given two spin structures @xmath1311 and @xmath1331 let @xmath1332 and @xmath1333 be the dga s associated to @xmath13 using the two spin structures . using the notation from theorem [ changeos ] define @xmath1334 to be the identity of the generators of the dga but for @xmath68 let @xmath1335 ( thus the isomorphism @xmath47 arises from an isomorphism of the base ring @xmath29 $ ] ) . one may easily check that @xmath47 is a chain map and hence a tame isomorphism from @xmath1332 to @xmath1336 it is quite interesting to note that if one merely uses orientations to define the dga for @xmath13 over @xmath4 then there is a dependence on spin structures . the dga s of a legendrian submanifold @xmath13 defined over @xmath4 using two different spin structures are not necessarily stable tame isomorphic . if the dga s are stable tame isomorphic then the contact homology associated to @xmath13 with the two spin structures would be isomorphic . let @xmath13 be the legendrian unknot in @xmath8 whose lagrangian projection has one double point . thus @xmath1337 where @xmath73 is the double point in the projection . if we use the spin structure on @xmath13 defined in subsection [ 3drel ] then @xmath1338 so the contact homology is @xmath1339 by theorem [ changeos ] we see that using the other spin structure on @xmath13 will give a differential @xmath1340 so the contact homology with this spin structure is a copy of @xmath4 in gradings 0 and 1 . thus the dga s associated to @xmath13 using the two spin structures are not stable tame isomorphic . to get examples in higher dimensions one can use the spinning construction , see @xcite . in this section we show that with the proper choice of spin structure on @xmath1341 the dga we associate to a legendrian knot in @xmath8 is the same as the combinatorially defined one given in @xcite . we also deduce an alternative combinatorial description and demonstrate that these two in a certain sense constitute a complete list of possible combinatorial definitions . recall that our construction of orientations on the moduli spaces relevant to contact homology depend on choices . specifically ( see section [ cocd ] ) we chose an orientation on @xmath1342 on the capping operators , on spaces of conformal structures @xmath1343 and automorphisms @xmath1344 and on @xmath88 . this last choice was largely hidden in previous sections : we were simply using the natural complex orientation @xmath88 . however , there is no real need to choose this orientation and , as we shall see , the choice matters . we will call the choices listed above _ the choice of basic orientations_. let @xmath1345 be an oriented legendrian knot and consider a double point of its lagrangian projection . near this double point the lagrangian projection subdivides the plane into four quadrants . we describe two shading rules . * a quadrant of a double point with even grading ( see section [ subsec : orion1d ] ) is _ a - shaded _ if it is adjacent to the incoming edge of the overcrossing , the other two quadrants are _ a - unshaded_. all quadrants of odd double points are _ a - unshaded_. see the left hand side of figure [ signrule ] . * a quadrant of a double point with even grading ( see section [ subsec : orion1d ] ) is _ b - shaded _ if it is adjacent to the incoming edge of the overcrossing and to the outgoing edge of the undercrossing , the other three quadrants are _ b - unshaded_. a quadrant of an odd double points is b - shaded if it adjacent to the incoming edges of both the over- and the undercrossing , the other three are b - unshaded . see the right hand side of figure [ signrule ] . the main goal of this section is to prove the following theorem . [ thm:3dsigns ] let @xmath13 be an oriented legendrian knot in @xmath8 equipped with the lie group spin structure of @xmath1346 . then there exists a choice of basic orientations such that for any reeb chord @xmath73 of @xmath13 , @xmath1347 where the first sum is over words @xmath1348 in the double points of the lagrangian projection of @xmath13 with @xmath1349 and @xmath1350 is the number of a - shaded corners in the image of @xmath1351 moreover , there exists another choice of basic orientations ( where the orientation of @xmath88 is opposite , for more detail see lemma [ lmacombor ] ) such that @xmath1352 where the first sum is over words @xmath1348 in the double points of the lagrangian projection of @xmath13 with @xmath1349 and @xmath1353 is the number of b - shaded corners in the image of @xmath1351 the sign rule presented in @xcite is the one corresponding to the a - shading . [ rmk:3dsigns ] note that we have not explicitly identified which orientation on @xmath88 gives which orientation convention . we also note that other changes in the basic orientations give differentials closely related to those given here . ( for reasonable definitions of basic orientations they differ by an over - all sign . ) to compute the differential with respect to the other spin structure on @xmath1341 ( the null - cobordant one ) one can appeal to theorem [ changeos ] . however , there is also a simple way to compute the differential in this case . start with the trivialization of the stabilized tangent bundle to @xmath1341 corresponding to the lie group spin structure and add a @xmath337-rotation in a small neighborhood of a point @xmath1354 we may assume that no capping path used in computing the algebra and differential contains @xmath1355 if @xmath1356 then define @xmath1357 to be the number of times @xmath1358 intersects @xmath1355 as in the proof of theorem [ changeos ] we see that the differential @xmath1359 corresponding to the new spin structure is @xmath1360 where @xmath1361 or @xmath1362 . recall from subsection [ oms ] that when given a chord generic legendrian knot @xmath1345 we first consider the lagrangian projection @xmath1363 of @xmath13 to @xmath88 then the inclusion of @xmath88 into @xmath1364 as the first coordinate . in the @xmath72-dimensional case the complex angle has only one component and therefore a @xmath72-dimensional stabilization is sufficient . we thus consider a @xmath487 bundle over @xmath13 and a field of lagrangian subspaces of the bundle by assigning to each point @xmath26 in @xmath13 the lagrangian subspace @xmath1365 where @xmath1366 for the unit tangent vector @xmath1367 to @xmath13 at @xmath26 and where @xmath1368,$ ] for some small @xmath1369 and @xmath1370 for @xmath26 in a neighborhood of each upper end of a reeb chord and @xmath1371 in a neighborhood of each lower end point of a reeb chord . the auxiliary linearized problem for a holomorphic disk @xmath1372 is then the @xmath28-problem with boundary condition given by the above plane field along @xmath1373 . we note that this auxiliary linearized problem is split : @xmath1374 where @xmath1375 acts on the @xmath137 coordinate of a section . moreover , @xmath1376 and @xmath1377 is an isomorphism . to fit the above into the orientation scheme presented in previous sections , we need a trivialization of the lagrangian plane field which meet the conditions presented in subsection [ cappingrulegeq3 ] at the double points of @xmath13 . ( note that gives a trivialization which does not necessarily meet the conditions at reeb chords . ) to achieve this we change the trivialization in in a neighborhood of the upper end of each _ odd _ reeb chord as follows . following the orientation of the knot we add a @xmath337-rotation to the trivialization right before we come to the upper reeb chord end and a @xmath1378-rotation right after ( here we think of the lagrangian plane field as oriented by the trivialization presented above ) . with this modification the trivialization does meet the necessary conditions and can be capped off with @xmath1379 , @xmath1380 , @xmath1381 , and @xmath1382 . consider the space @xmath1383 of holomorphic immersions @xmath1384 with @xmath560 convex corners on the boundary . one called positive and the rest called negative . the weak homotopy type of @xmath1383 equals that of @xmath1385 . ( that is @xmath1383 is a @xmath1386-space . ) each disk immersion with a marked point on its boundary can be contracted through immersed disks to a small standard immersed disk near its marked point . in the present set up we use the positive corner as the marked point and note that there is no problem keeping control of the rest of the corners during such a deformation ( parameterized by a compact space ) . we associate to each element @xmath1387 a lagrangian boundary condition @xmath1388 for the @xmath28-operator on @xmath487-valued functions on @xmath612 by defining , with @xmath1389 denoting the positive tangent vector of @xmath581 at @xmath1390 @xmath1391 where @xmath1392 and where @xmath1393 ( @xmath1394 ) near the negative punctures along the incoming ( outgoing ) part of @xmath581 . along the corresponding parts of @xmath581 near a positive puncture we let @xmath1395 have the opposite signs . note that the @xmath28-operator just mentioned has index @xmath1396 and that the dimension of its kernel ( cokernel if @xmath1397 ) equals @xmath78 over each @xmath1387 . let @xmath1398 be the vector bundle with fiber over @xmath138 equal to the cokernel ( kernel ) of this operator . since the generator of @xmath1399 can be represented by a rigid rotation of a convex disk around its positive puncture it is easy to see that @xmath87 is orientable . we consider also the bundle @xmath1400 , the fiber of which over @xmath138 is the tangent space of conformal structures ( automorphisms if @xmath1397 ) of the source @xmath612 of @xmath138 . this is a fiber bundle of fiber dimension @xmath1401 . moreover , the natural linearization map @xmath1402 , where @xmath1403 is a variation of the conformal structure gives a fiberwise isomorphism @xmath1404 ( this is a consequence of the general transversality properties of the @xmath28-equation in dimension @xmath72 : rigid disk are automatically transversely cut out ) . the natural orientation of spaces of conformal structures induces an orientation on @xmath172 . assume that @xmath87 is oriented . then orientations of @xmath87 and @xmath172 either agrees or disagrees over every @xmath1405 . we consider special types of trivializations of @xmath1406 these trivializations model those coming from a knot diagram as mentioned above . let @xmath1407 , be a subdivision into connected components where @xmath1408 ( @xmath1409 ) has its negative ( positive ) end at the positive puncture . for @xmath1387 we define a _ diagram orientation _ of @xmath1410 over @xmath581 to be a trivialization of this pull back bundle which has the form @xmath1411 let @xmath26 be the positive corner of @xmath138 then we call @xmath26 _ even ( odd ) _ if a positive rotation of the incoming trivialization vector with magnitude equal to the exterior angle of @xmath138 at @xmath26 gives the negative ( positive ) outgoing trivialization vector . let @xmath189 be a negative corner of @xmath138 then we call @xmath189 _ even ( odd ) _ if a positive rotation of the incoming trivialization vector with magnitude equal to the exterior angle of @xmath138 at @xmath189 gives the positive ( negative ) outgoing trivialization vector . note that a corner is even or odd according to whether or not the corresponding reeb chord has even or odd grading ( see lemma [ eochords ] ) , thus the number of odd corners of any diagram orientation of a rigid disk in @xmath1383 is odd . we associate a _ diagram trivialization _ of the lagrangian boundary condition @xmath1412 to a diagram orientation as follows . fix a small neighborhood @xmath1413 of the positive ( negative ) end point of each @xmath102 which has its positive ( negative ) end point at a negative ( positive ) _ odd _ puncture . the restriction of the diagram trivialization to the complement of the union of these fixed neighborhoods is simply @xmath1414 to complete the definition of the diagram trivialization we proceed as follows . on @xmath1415 corresponding to a negative odd puncture we first make a positive @xmath337-rotation inside @xmath498 begining at @xmath1416 and ending at @xmath1417 and then continue like that to the puncture . on @xmath1418 corresponding to a positive odd puncture we start out at the puncture with the framing @xmath1419 make a negative @xmath337-rotation inside @xmath498 ending up at @xmath1420 and then continue like that until we get into the region where the trivialization was already defined . recall from subsection [ cocd ] the capping disk @xmath1380 , @xmath1382 , @xmath1379 , @xmath1381 . the determinant bundles over these types of boundary conditions are in general _ not _ orientable . consider for example the split boundary condition @xmath1379 with one - dimensional kernel and zero - dimensional cokernel . the kernel is spanned by a function with second coordinate equal to @xmath78 . now , applying a uniform @xmath337-rotation to the first or second line in the split lagrangian boundary condition brings us back to the original boundary condition . transporting the orientation along such a path changes it when the first coordinate is rotated and does not change it when the second coordinate is rotated . thus the bundle is non - orientable . however , the determinant bundles over subspaces of the spaces of capping disks are orientable . let @xmath1421 , @xmath1422 , @xmath1423 , and @xmath1424 be the spaces of capping disks such that the second component of the trivialization at the puncture equals @xmath1425 . the determinant bundles of the @xmath28-operator over @xmath1421 , @xmath1422 , @xmath1423 , and @xmath1424 are orientable . note that all spaces above are homotopy equivalent to @xmath1385 and that the monodromy of a generating loop in @xmath1426 preserves orientation . fix orientations on the bundles over @xmath1423 and @xmath1424 . this determines orientations over @xmath1421 and @xmath1422 , respectively by requiring that the orientations ( of two glueable representatives ) glue to the canonical orientation of the determinant of the resulting trivialized boundary condition over the closed disk . diagram orientaions of @xmath1412 which vary continuously with @xmath1387 induce an orientation on @xmath87 by gluing capping disks ( with orientations as above ) to the corners of @xmath138 . this construction has already been discussed , see subsection [ cappingrulegeq3 ] . the result of gluing capping disks @xmath1380 , @xmath1379 , @xmath1382 , and @xmath1381 to the corresponding punctures is a trivialized boundary condition on the closed disk . in the gluing sequence the operators of the capping disk have oriented determinant bundles which together with the canonical orientation on the closed disk induce an orientation on @xmath1427 . this construction is clearly continuous in @xmath138 . to compute signs in contact homology we need to compare the orientation on @xmath87 induced from the orientation on @xmath172 and the orientation on @xmath87 induced by a diagram trivialization . to this end we look at the details of basic orientations to derive the shading rules . we first list the basic orientation choices . * first we choose an orientation of @xmath642 . this orientation will be fixed throughout the discussion . * second we choose an orientation on @xmath88 . * third choose orientations on @xmath1423 and @xmath1424 . recall that the first two choices determines canonical orientations on the determinant bundles over trivialized boundary conditions over the closed disk . then the third choices induced orientations on @xmath1421 and @xmath1422 respectively via gluing . with all capping operators oriented we orient all the bundles @xmath87 over @xmath1428 we next describe how the choices of orientation of @xmath88 changes canonical orientations . recall that the canonical orientation on the determinant bundle over trivialized boundary conditions on the @xmath78-punctured disk was defined by , after deformation , expressing the operator as an operator over @xmath1429 with constant @xmath642 boundary conditions and an operator over @xmath439 with complex kernel or cokernel attached at the origin of @xmath1429 . the kernel and cokernel of the latter gets oriented by viewing them as complex vector spaces . now if the orientation of @xmath88 is switched then the orientation of each odd dimensional complex vector space changes and the orientation of each even dimensional complex vector space remains the same . since the dimension of the kernel and cokernel of the operator over @xmath1429 with constant @xmath642 boundary conditions is 2 and 0 , respectively , the effect on the canonical orientations is as follows : changing the orientation of @xmath88 changes the canonical orientation for each operator of index @xmath1430 and keeps the orientation of each operator of index @xmath1431 . recall the index of the operators @xmath1432 are @xmath1433 respectively . fix an orientation of @xmath88 . consider @xmath1434 with one positive odd corner and with a diagram orientation which agrees with that of the boundary of the disk . pick the orientation of the determinant bundle over @xmath1423 such that when gluing @xmath1382 to the trivialized boundary condition just described the orientation induced on @xmath87 agrees with the one induced on @xmath87 by the bundle @xmath172 of linearized conformal structures . note that the operator obtained by gluing @xmath1382 has index @xmath636 and hence the same choice of orientation works also for the other orientation of @xmath88 . denote this orientation @xmath1435 [ lma1punct ] let @xmath1434 be any disk with any diagram orientation . the orientation induced on the fiber of @xmath87 over @xmath138 obtained by gluing @xmath1382 with orientation @xmath1435 agrees with the orientation induced by @xmath1436 there are two possible diagram orientations for such a disk . for diagram orientations which can be obtained continuously from the diagram orientation given above the lemma is clear . to see that the lemma holds also for other diagram trivializations we need only check it for one . to this end we rotate the first coordinate of the glued boundary condition by @xmath337 . this transports the canonical orientation to the canonical orientation and the orientation of the capping disk to the right orientation . moreover , it preserves the orientation on the @xmath636-dimensional kernel of the @xmath28-operator on the one punctured disk . thus the induced orientations on @xmath138 with the two different diagram trivializations must be the same . our main theorem ( theorem [ thm:3dsigns ] ) clearly follows from the following lemma . [ lmacombor ] let @xmath1437 be the orientation induced on @xmath87 from the diagram trivialization corresponding to a diagram orientation and let @xmath389 be the orientation induced by @xmath172 . for a choice of basic orientations @xmath1438 on @xmath88 , @xmath1435 on @xmath1423 , and @xmath1439 on @xmath1424 we have @xmath1440 where @xmath1441 is the number of a - shaded punctures in the diagram orientation . for the choice of basic orientations @xmath1442 , @xmath1435 , and @xmath1443 we have @xmath1444 where @xmath1445 is the number of b - shaded punctures in the diagram orientation . we will prove this lemma by induction on the number of punctures in the disk . as a `` base case '' we must understand the orientations on one- and two - punctured disks . the lemma for the disk with one positive odd puncture and no other punctures is simply a restatement of lemma [ lma1punct ] . fix an orientation @xmath1446 on @xmath88 . to determine the sign of an arbitrary diagram orientation we will use orientations of @xmath72-dimensional moduli spaces and in particular lemma [ lmamorient ] . more precisely , we consider holomorphic disks with several convex and one concave corner . exactly as above one may use capping disks to orient the one - parameter family of disks in which it lives . for such model families lemma [ lmamorient ] relating various orientations hold . to determine the sign of the disk with one even positive corner and one odd negative corner , consider the disk with one concave ( even ) corner shown in figure [ fig : signs1 ] . it is easy to arrange a one - parameter family which splits as follows . one splitting gives a one punctured disk @xmath87 and a two punctured disk @xmath1447 the other splitting gives a one punctured disk @xmath87 and a two punctured disk @xmath1448 from the above we know the signs on @xmath87 and @xmath1449 are both @xmath1450 now lemma [ lmamorient ] says we must have @xmath1451 note that @xmath248 and @xmath1452 are the two possible twice punctured disks with a positive even puncture and a negative odd puncture . thus we know these disks have opposite signs . one choice of orientation @xmath1439 yields the a - shading rule and the other one @xmath1453 the b - shading rule . note that the operator obtained by capping off @xmath248 has index @xmath78 and hence its canonical orientation changes with @xmath1446 . since the gluing of @xmath1382 and @xmath1380 also gives an operator of index @xmath78 also @xmath1454 changes with the orientation of @xmath88 and thus @xmath1439 and @xmath1453 are independent of @xmath1446 . we have chosen orientations on the determinate line bundles over @xmath1424 and @xmath1423 so that the lemma is true for the cases considered so far . to finish the `` base case '' of our induction we are left to check that the lemma holds for disks @xmath1455 and @xmath1456 with one positive odd puncture and one negative even puncture . arguing as in the above paragraph we see that the two configurations of such disks must have opposite signs . moreover , with respect to one choice of signs on these disks the a - shading rule in the lemma is correct with respect to the other the b - shading rule is . we finish the base case by checking that a change of @xmath1446 changes the signs of @xmath1455 and @xmath1456 . note that the operator obtained by capping off @xmath1455 has index index @xmath636 . thus its canonical orientation is not affected by @xmath1446 . however , the operator obtained by gluing @xmath1381 and @xmath1379 has index @xmath78 so the orientation @xmath1457 on @xmath1422 induced from @xmath1439 changes with @xmath1446 . it follows that there is a choice @xmath1458 so that the a - shading rule holds in all base cases . moreover , with @xmath1442 we find since @xmath1459 that the induced orientation @xmath1460 on @xmath1422 satisfies @xmath1461 and that the b - shading rule applies in all base cases for @xmath1462 . to finish the proof we use the same argument in both cases . for simplicity we give it only in the case of a - shading . assume by induction that the lemma is true for all disks with @xmath0 punctures . let @xmath138 be a disk with @xmath1463 punctures . if the puncture counterclockwise of the positive puncture is odd then we can glue a once punctured disk @xmath87 to it . one of the four possible cases if shown in figure [ fig : signs2 ] . we will finish the argument in this case . the remaining cases are similar . the one dimensional moduli space formed by this gluing also splits into an @xmath0 punctured disk @xmath1464 and a twice punctured disk @xmath1465 by induction we know the sign of @xmath1464 is @xmath1466 where @xmath1467 is the number of @xmath45-shaded regions in @xmath1468 moreover , the sign of @xmath1449 is @xmath1450 thus lemma [ lmamorient ] implies @xmath1469 note the number of @xmath45-shaded regions of @xmath138 is @xmath1470 thus the lemma holds for @xmath1351 an entirely analogous argument works if the puncture counterclockwise of the positive puncture of @xmath138 is even . the only difference is one must glue a twice punctured disk to @xmath1351 ( note in this case one must actually have determined the signs on thrice punctured disks , but this may be done as in the previous paragraph by noting that at least one of the punctures must be odd . ) in this section we describe how to perturb highly degenerate legendrian submanifolds into generic submanifolds . this will be our main tool in constructing examples in the next section . in subsection [ pdls ] we perturb a legendrian submanifold , whose complex projection has double points along the interior of a compact codimension zero submanifold with boundary and a certain behavior at this boundary using a morse function on the submanifold , into a generic legendrian submanifold . for the perturbed legendrian there will be a double point in the complex projection for each critical point of the morse function . adapting a construction of floer @xcite and pozniak @xcite , we show that the contact homology boundary map for these double points is related to the morse `` gradient flow '' boundary map . in subsection [ sec : signsfordegen ] we compare the signs in these two boundary maps . consider a legendrian submanifold @xmath13 in @xmath14 . let @xmath1083 be an @xmath0-manifold with boundary @xmath1471 and let @xmath1472 be a contact embedding ( also respecting contact forms ) such that @xmath1473 is an embedding . let @xmath1474 $ ] be a collar neighborhood of @xmath1471 in @xmath1083 , with @xmath1471 corresponding to @xmath1475 . let @xmath1476),$ ] @xmath1477),$ ] and @xmath1478 $ ] . assume that @xmath1479 consists of two sheets @xmath966 and @xmath1480 represented in coordinates of @xmath1481 as the @xmath72-jet extensions @xmath1482 and @xmath1483 of two functions @xmath1484 and @xmath1485 respectively . assume further that @xmath1486 a positive constant , for @xmath1487 , that @xmath1488 for points in @xmath1489 and that @xmath1490 is monotone in @xmath973 for @xmath1491 $ ] . we will frequently think of @xmath1083 and @xmath193 as subsets in @xmath1492 we will consider functions @xmath1493 satisfying 1 . @xmath1052 is a morse - smale function on @xmath200 2 . the support of @xmath1052 contains @xmath193 and is contained in @xmath200 3 . all critical points of @xmath1494 are critical points of @xmath1052 and occur in @xmath1495 4 . @xmath1052 is real analytic near its critical points . define the legendrian submanifold @xmath1496 to be the one obtained from @xmath13 by replacing @xmath1483 in the front projection of @xmath13 with @xmath1497 the double points of @xmath1498 which lie inside @xmath1499 correspond to the critical points of @xmath1052 and double which lie outside @xmath1499 corresponds to double points of @xmath1500 denote the part of @xmath1498 corresponding to @xmath1501 by @xmath1502 ( and similarly for @xmath1503 etc . ) . having identified the double points of @xmath1504 we know the generators of its algebra @xmath1505 . we next consider holomorphic disks . let @xmath1506 and @xmath1507 be distinct critical points of @xmath1508 thought of as double points of @xmath1509 let @xmath1510 $ ] be conformal coordinates for @xmath639 and @xmath1102 a small tubular neighborhood of @xmath1502 in @xmath5 that contains @xmath1511 define @xmath1512 where 1 . @xmath1513 2 . @xmath1514 and @xmath1515 and 3 . @xmath1516 and @xmath1517 . to define the operator @xmath1518 we use the standard complex structure on @xmath16 [ thm : lmt ] with the notation established above , there exists an @xmath1519 such that if the @xmath359 norm of @xmath1052 is less than @xmath1133 , and @xmath1102 is contained in an @xmath1133 neighborhood of @xmath1502 , for positive @xmath124 sufficiently small there is a one to one correspondence between @xmath1520 and gradient flow lines of @xmath1521 connecting @xmath1506 to @xmath1522 our strategy for proving theorem [ thm : lmt ] is to transplant our problem into the cotangent bundle of some manifold and then use a slightly modified version of a construction of floer . to this end we recall floer s construction @xcite . given a manifold @xmath1523 consider its cotangent bundle @xmath1524 thought of as a symplectic manifold . we will denote the zero section of @xmath556 by @xmath1525 fix any almost complex structure @xmath739 on @xmath556 that agrees with the canonical one along the zero section . let @xmath1052 be a morse function on @xmath1523 and define @xmath1526 to be the graph of @xmath1527 let @xmath1528 be the projection map and set @xmath1529 the hamiltonian @xmath781 generates a flow @xmath1530 on @xmath556 and @xmath1531 define the time dependent almost complex structure @xmath1532 and @xmath1533 where 1 . @xmath1534 2 . @xmath1514 and @xmath1515 and 3 . @xmath1535 and @xmath1536 . in @xcite it was shown that if the @xmath359 norm of @xmath1052 is sufficiently small then the map @xmath1537 from @xmath1538 to @xmath1539 is a bijection onto the set of bounded trajectories of the gradient flow of @xmath1052 connecting @xmath1506 and @xmath1522 moreover if @xmath1052 is a morse - smale function then the moduli space @xmath1538 is transversely cut out by its defining equation . define @xmath1540 just as @xmath1541 except instead of the equation in condition ( 1 ) use the equation @xmath1542 the modification of floer s construction we need is given in the following theorem . [ lem : interp ] if the @xmath359 norm of @xmath1052 is sufficiently small then there is a one to one correspondence between @xmath1543 and @xmath1544 for any @xmath124 sufficiently small . let @xmath1545 be the flow of the hamiltonian @xmath1546 and @xmath1547 in local darboux coordinates one may compute that @xmath1548 where @xmath1549 is the hessian of @xmath1550 now consider the two parameter family of complex structures @xmath1551 note @xmath1552 ( independent of @xmath124 ) . so for a fixed @xmath1553 @xmath1554 interpolates between @xmath1555 and @xmath1556 define @xmath1557 just as @xmath1558 ( including boundary conditions ) except use the complex structure @xmath1554 in condition ( 1 ) . for @xmath124 sufficiently small the spaces @xmath1559 are transversely cut out for all @xmath1560.$ ] this claim establishes the lemma since it will set up a one to one correspondence between @xmath1561 and @xmath1562 to prove the claim note that since @xmath1052 is morse - smale we know @xmath1563 is transversely cut out by its defining equation @xcite . said another way , @xmath1564 is a regular operator . if the claim is not true then we will find a sequence of complex structures converging to @xmath1565 that are not regular . but this is a contradiction since by gromov compactness ( and the upper semi - continuity of the dimension of the kernel ) we know there is an @xmath1566 neighborhood of @xmath1565 in the space of almost complex structures that contain only regular complex structures . assume the claim is false . so for all @xmath1567 there is some @xmath1568 so that @xmath1569 is not transversely cut out . thus there is always a solution @xmath1570 for which @xmath1571 is not surjective . since @xmath1572 satisfies @xmath1573 the map @xmath1574 has boundary conditions @xmath1575 and @xmath1576 and satisfies @xmath1577 in addition , since @xmath1578 intertwines the complex structures @xmath1554 and @xmath1579 we see that @xmath1579 is not regular . one may compute that @xmath1580 is of order @xmath124 so for @xmath124 small enough @xmath1579 must be regular . thus for @xmath124 small enough @xmath1581 is regular for all @xmath1560.$ ] to finish the proof of theorem [ thm : lmt ] consider the manifold @xmath1582 and its cotangent space @xmath1583 we can find a symplectomorphism @xmath221 from a neighborhood @xmath1102 of @xmath1502 in @xmath5 to a neighborhood @xmath1584 of @xmath1585 thought of as part of the zero section , that takes @xmath1502 to @xmath1083 and sends the standard complex structure along @xmath1586 to the canonical complex structure along the zero section . ( to ensure that @xmath1587 still sits in @xmath1102 one merely needs to make sure the @xmath249 norm of @xmath1052 is sufficiently small . ) now let @xmath739 be the complex structure on @xmath556 that extends the one pushed forward from @xmath1102 and is standard along the entire zero section . there is a function @xmath1588 that is constant on @xmath1589 and satisfies @xmath1590 we can choose @xmath1083 to be a small enough neighborhood of @xmath1591 so that the function @xmath1592 has @xmath359 norm small enough for floer s results to hold . we can further find a function @xmath1593 such that @xmath1594 in addition there is some @xmath1519 so that if @xmath1595 then @xmath1596 will also have @xmath359 norm small enough for floer s results to hold . according to lemma [ lem : interp ] we can now find a @xmath124 so that @xmath1597 is in one to one correspondence with @xmath1598 it is easy to see that all the holomorphic curves in @xmath1599 lie in @xmath1584 and correspond , via @xmath1600 to holomorphic curves in @xmath1601 this establishes theorem [ thm : lmt ] . we briefly recall how to compute the homology of a manifold via a morse function . given a morse - smale function @xmath1602 we get a chain complex generated by the critical points of @xmath21 . the boundary map of this chain complex comes from counting isolated gradient flow lines between critical points . we assign a sign to each such flow line as follows . let @xmath1506 and @xmath1507 be two critical points . the morse index of @xmath1506 being one larger than the morse index of @xmath1522 let @xmath1603 be the unstable manifolds of @xmath1506 ( which is the set of points in @xmath6 that under the gradient flow , flow to @xmath1506 as time goes to @xmath1604 ) . and @xmath1605 be the stable manifold of @xmath1507 ( which is the set of points that flow to @xmath1507 as time goes to @xmath1606 ) . the gradient flow lines connecting @xmath1506 to @xmath1507 are exactly @xmath1607 if we chose orientations on @xmath1603 for all critical points @xmath1506 and assume @xmath6 is oriented then we get induced orientations on @xmath1608 for all critical points @xmath1609 if such a choice has been made then @xmath1610 is an oriented manifolds . if it is a one dimensional manifold then it also gets an orientation by @xmath1611 thus for isolated flow lines connecting @xmath1506 and @xmath1507 we get a sign by comparing these orientations . the boundary map in the chain complex comes from the signed count of these flow lines . the homology of this complex agrees with the homology of the manifold . for more details see @xcite . consider the set up in section [ sec : lmt ] : a legendrian with an open set @xmath193 of double points in its lagrangian projection and a morse - smale function @xmath1052 on @xmath193 with which we perturb the legendrian . we now wish to compare the signed count of flow lines above with the signs of holomorphic disks associated to the flow lines by theorem [ thm : lmt ] . to this end choose a tree of gradient flow lines of @xmath1052 that connect all the critical points of @xmath1052 ( we are assuming @xmath193 is connected so this can be done ) . each of the moduli spaces of flow lines can be assigned an orientation using the coherent orientations from section [ oms ] , by identifying it with the space of holomorphic curves using theorem [ thm : lmt ] . we may now choose orientations on the unstable manifolds of the critical points so that the orientations induced by intersecting the stable and unstable manifolds agrees with the coherent orientations on the flow lines in the chosen tree . this is possible since we chose a tree of flow lines . note that the orientations on all other moduli spaces of flow lines are determined by the orientations on the spaces of flow lines in the chosen tree and gluing constructions . since the orientations on the moduli space of flow lines coming from section [ oms ] and the procedure outlined in the previous paragraph are both `` coherent '' they are both determined by the orientations on the flow lines in the tree and gluing constructions . we have proved [ thm : sfh ] under the correspondence between holomorphic disks and gradient flow lines set up in theorem [ thm : lmt ] there is a choice on orientation on the unstable manifolds of @xmath1052 so that the morse - smale orientations and orientations from section [ oms ] agree on all moduli space of flow lines . in this section we give examples showing that oriented contact homology is a finer invariant than contact homology over @xmath1339 to keep computations simple , we give examples showing that the linearized contact homology over @xmath4 will distinguish legendrian links in the 1-jet space of @xmath1612 not distinguished by the full contact over @xmath1339 similar constructions can be applied to construct legendrian submanifolds in @xmath1613 however , the computation of the full contact homology is more difficult . ( it relies on a morse theoretic description of all holomorphic disks relevant to contact homology , involving gradient flow trees with cusps . ) we therefore defer discussions of this to a forthcoming paper . recall @xmath1614 has a natural contact structure @xmath1615 where @xmath124 is the liouville 1-from on @xmath1616 pulled back to @xmath1617 and @xmath973 is the coordinate in the @xmath82 factor . projecting out the @xmath82 factor is called the lagrangian projection and is analogous to projecting out the @xmath37-coordinate in @xmath7 note that since @xmath1616 admits a complex structure coming from @xmath5 by quotienting by a holomorphic action the analytic set up of contact homology in @xcite for @xmath1618 also works for @xmath1619 moreover , the proofs in @xcite carry over word - for - work to this setting . ( note , in general , setting up contact homology in jet spaces requires more work , see @xcite . ) with the above understood we can take the contact homology of legendrian links in @xmath1617 to be well - defined . our main theorem of this subsection is for any @xmath1620 there are infinitely many legendrian links , topologically isotopic to two copies of the zero section , in @xmath1617 that are distinguished by their oriented linearized contact homology , but have the same classical invariants and tame isomorphic contact homology dga s over @xmath1339 let @xmath964 be the zero section of @xmath1617 and @xmath21 a non - negative self - indexing morse function on @xmath1612 with minimal number of critical points . we will now alter @xmath21 near the unique index 0 critical point . for @xmath26 odd figure [ fig : b4 ] gives a handle decomposition of @xmath1621 there are clearly analogous decompositions for all @xmath1620 . > from this we get a morse function @xmath1052 on @xmath1622 with critical points @xmath1623 the critical points are labeled by their morse index . we can assume that @xmath1624 and @xmath1625 moreover , the boundary map for the morse - witten complex is @xmath1626 let @xmath1627 be the morse function on @xmath1612 equal to @xmath21 on @xmath1628),$ ] that is , outside a neighborhood of the index 0 critical point of @xmath1629 and in the neighborhood equal to @xmath1550 we can assume @xmath1627 is smooth . let @xmath1630 be the 1-jet of @xmath1631 in @xmath1632 @xmath1633 we now fix the index 0 critical point and measure all gradings relative to this . according to theorem [ thm : lmt ] and the discussion following it we know , for @xmath1133 small enough , all the holomorphic disks coming in to the computation of the boundary map come form gradient flow lines . thus one may easily compute the boundary map of @xmath1634 on the generators associated to @xmath1052 are @xmath1635 moreover , the boundary map on the other generators comes from the morse boundary map for @xmath1636 thus when everything is reduced mod 2 the boundary map is independent of @xmath26 but the contact homology over @xmath4 ( with relative grading chosen so that @xmath1000 has grading @xmath0 ) of the link @xmath1637 when @xmath1638 is @xmath1639 one obtains a similar answer when @xmath1640 note these gradings are only relative gradings , but this is irrelevant as the @xmath26 torsion distinguishes the legendrian links independent of grading . the gradings were computed using the morse index , see @xcite . we describe how to use contact homology to derive a lower bound on the number of double points of an exact lagrangian immersion . contact homology has also been used to study intersections of a pair of immersions @xcite . let @xmath1641 be an exact lagrangian immersion . then after small perturbation we may assume that the legendrian lift of @xmath13 is an embedding which is chord generic . let @xmath1642 denote its dga . recall a dga is augmented if the differential of no generator contains a constant . an _ augmentation _ of an algebra is a graded map @xmath1643 such that @xmath1644 and @xmath1645 given an augmentation @xmath1133 the graded algebra tame isomorphism @xmath1646 will conjugate @xmath1642 to an augmented algebra . a dga is called _ good _ if it admits an augmentation , and is hence tame isomorphic to an augmented dga . we show the following theorem where we use the algebra @xmath715 with coefficients other than @xmath4 , see remark [ rmkcoeff ] . [ dpegood ] let @xmath1647 be an exact lagrangian immersions and let @xmath1642 be the dga associated to an embedded chord generic legendrian lift of @xmath21 . if @xmath1642 is good then @xmath21 has at least @xmath1648 double points , where @xmath24 or @xmath25 for any prime @xmath26 if @xmath13 is spin and @xmath27 if @xmath13 is not spin . to simplify notation we identify @xmath13 with its image under the embedding which is the lift of @xmath21 and write @xmath12 let @xmath1649 be a copy of @xmath13 shifted a large distance in the @xmath37-direction , where as usual @xmath37 is a coordinate in the @xmath82-factor . then @xmath1650 is a legendrian link . moreover , assuming that the shifting distance in the @xmath37-direction is sufficiently large , shifting @xmath1649 @xmath1568 units in the @xmath1651-direction gives a legendrian isotopy of @xmath1652 . after a large such shift @xmath1652 projects to two distant copies of @xmath65 and it is evident that an augmentation for @xmath13 gives an augmentation for @xmath1653 . moreover , the linearized contact homology of @xmath1654 equals the set of sums of two vector spaces from the linearized contact homology of @xmath13 . we will next compute the linearized contact homology of @xmath1650 in a different manner . let @xmath1655 be a morse function on @xmath13 and use @xmath1592 to perturb @xmath13 in @xmath1656 , where @xmath1083 is a small neighborhood of the @xmath78-section . after identification of @xmath1083 with a neighborhood of @xmath1649 in @xmath1657 we use this isotopy to move @xmath1649 to @xmath1658 . the projection of @xmath1658 into @xmath5 then agrees ( locally ) with an exact deformation of @xmath13 in its cotangent bundle and there is a symplectic map from that cotangent bundle to a neighborhood of @xmath65 in @xmath5 . pulling back the complex structure from @xmath5 we get an almost complex structure on @xmath1659 . the intersection points of @xmath13 and @xmath1658 are of three types . * critical points of @xmath1592 . * pairs of intersection points between @xmath13 and @xmath1658 near the self - intersections of @xmath13 . * self intersection points of @xmath13 and of @xmath1658 near self intersections of @xmath13 . fix augmentations of @xmath715 and of @xmath1660 . if @xmath30 is the differential of @xmath1661 it is easy to see that any monomial in @xmath1662 , where @xmath721 is a reeb chord of type ( 1 ) or ( 2 ) must contain an odd number of reeb chords of type ( 1 ) and ( 2 ) . therefore the augmentations of @xmath715 and @xmath1660 give an augmentation for @xmath1661 that is trivial on double points of type ( 1 ) and ( 2 ) . denote by @xmath787 the linearized differential induced by the augmentations chosen and by @xmath1663 the span of the double points of type ( @xmath1664 ) , @xmath1665 suppose @xmath73 is a type ( 3 ) double point , then @xmath1666 has no constant part and its linear part has no double points of type ( 1 ) or ( 2 ) , since each holomorphic disk with a positive puncture at @xmath73 must have an even number of negative corners of type ( 1 ) or ( 2 ) . thus @xmath1667 if @xmath1096 is of type ( 1 ) or ( 2 ) then the linear part of @xmath1668 involves only double points of type ( 1 ) and ( 2 ) . denote by @xmath1669 the projection onto @xmath1670 and @xmath1671 then @xmath1672 on @xmath1673 consider @xmath1674 . we claim that for sufficiently small perturbation @xmath1592 , @xmath1675 . to show this we consider gluing of two ( two - punctured ) disks contributing to @xmath1676 . this gives a @xmath72-parameter family two - punctured disks . now , for sufficiently small perturbation , no reeb chord of type ( 2 ) has length lying between the lengths of two reeb chords of type ( 1 ) . moreover , every reeb - chord of type ( 3 ) has length bigger than the difference of the lengths of two reeb chords of type ( 1 ) . this shows that the @xmath72-parameter family must end at another pair of broken disks with corners of type ( 1 ) . it follows that @xmath1677 . it follows that @xmath1678 agrees with the floer differential of @xmath1679 , where @xmath1680 is the @xmath78-section and where @xmath1681 is the graph of @xmath1682 . hence , @xmath1683 write @xmath1684 , where @xmath1685 and let @xmath1495 be a direct complement of @xmath1686 . then @xmath1687 . fix the augmentations for @xmath13 and @xmath1649 which gives the element of the linearized contact homology of @xmath13 which has the largest dimension . by the above discussion we find that @xmath1688 equals a direct sum of two copies of this maximal dimension vector space . it follows that the contribution to the linearized contact homology involving double points between @xmath1658 and @xmath13 must vanish . we check how double points of type ( 2 ) kill off the double points of type ( 1 ) that exist in the homology of @xmath1689 we compute @xmath1690 where the third equality is due to the fact that @xmath1691 is in @xmath1692 it follows that @xmath1693 moreover , notice an element @xmath1694 in @xmath1495 is a non zero element in the linearized contact homology if and only if @xmath1695 and @xmath1696 thus if @xmath1695 then @xmath1694 is in @xmath1697 showing that @xmath1698 we find @xmath1699 and conclude that @xmath1700 [ improve ] suppose there is a constant @xmath1523 such that for any exact lagrangian immersion @xmath1701 @xmath21 has at least @xmath1702 double points , where @xmath24 or @xmath25 for any prime @xmath26 if @xmath13 is spin and @xmath27 if @xmath13 is not spin . then @xmath21 has at least @xmath1648 double points . [ lmamultsph ] let @xmath1703 be a chord generic legendrian embedding with @xmath22 reeb chords and maslov number @xmath1704 . then , for any @xmath1705 there exists a legendrian embedding @xmath1706 with @xmath1707 reeb chords and @xmath1708 . for @xmath671 , let @xmath1711 . note that translations in the @xmath1712-direction , @xmath503 and that the scalings @xmath1713 , @xmath1714 , @xmath1715 are legendrian isotopies which preserves the number of reeb chords . we may thus assume that @xmath1716 is contained in @xmath1717 , where @xmath1133 is very small . for convenience , we write @xmath1718 with coordinates @xmath1719 and corresponding coordinates @xmath1720 in @xmath1721 . consider the embedding @xmath1722 , where @xmath1723 is the unit sphere in @xmath1724 . let @xmath1725 @xmath1726 be polar coordinates on @xmath1727 . fix a morse function @xmath47 , with one maximum and one minimum on @xmath1723 which is an approximation of the constant function with value @xmath72 . define @xmath1728 , @xmath1729 as follows @xmath1730 where we think of the gradient @xmath1731 as a vector in @xmath1724 tangent to @xmath1723 at @xmath408 . it is then easily verified that @xmath1732 is a legendrian embedding . moreover , the reeb chords of @xmath1732 occur between points @xmath1733 and @xmath1734 such that @xmath1735 , @xmath1736 , @xmath1737 , @xmath1738 , and either @xmath1739 or @xmath1740 . these conditions are incompatible with @xmath21 being an embedding unless @xmath1740 and we conclude that the number of double points of @xmath1732 are as claimed . the statement about the maslov number is straightforward . given @xmath1523 in the statement of the theorem , choose @xmath1741 so that @xmath1742 for any immersed exact lagrangian @xmath1647 lift @xmath21 to an embedded legendrian in @xmath1743 and @xmath1070-spin this legendrian @xmath1741 times . the lagrangian projection of the resulting legendrian gives a new exact lagrangian immersion @xmath1744 since reeb chords correspond to double points @xmath1745 has @xmath1746 time as many double points as @xmath1636 we have @xmath1747 thus @xmath1748 m. akaho , _ lagrangian intersections via contact homology _ , preprint 2003 . f. bourgeois and k. monkhe , _ coherent orientations in symplectic field theory _ , preprint 2001 , ` arxiv : math.sg/0102095 . ` y. chekanov , _ differential algebras of legendrian links _ , invent . * 150 * ( 2002 ) , no . 3 , 441483 . t. ekholm , j. etnyre and m. sullivan _ the contact homology of legendrian submanifolds of jet spaces _ , in preparation . t. ekholm , j. etnyre and m. sullivan _ non - isotopic legendrian submanifolds in @xmath14 _ , preprint 2002 . t. ekholm , j. etnyre and m. sullivan _ the contact homology of legendrian submanifolds in @xmath14 _ , preprint 2002 . y. eliashberg , _ invariants in contact topology _ , proceedings of the international congress of mathematicians , vol . ii ( berlin , 1998 ) . doc . math . 1998 , extra vol . ii , 327338 . y. eliashberg , a. givental and h. hofer , _ introduction to symplectic field theory _ , preprint 2000 . ` arxiv : math.sg/0010059 . ` j. etnyre , l. ng and j. sabloff , _ invariants of legendrian knots and coherent orientations _ , j. symplectic geom . * 1 * ( 2002 ) , no . 2 , 321367 . a. floer , _ witten s complex and infinite dimensional morse theory _ , j. differential geom . 30 ( 1989 ) , 207221 . a. floer and h. hofer , _ coherent orientation for periodic orbit problems in symplectic geometry _ , math . 212 , ( 1993 ) 1338 . k. fukaya , y. oh , h. ohta , k. ono , _ lagrangian intersection floer theory -anomaly and obstruction- _ , preprint . k.mishachev , _ the n - copy of a topologically trivial legendrian knot _ , j. symplectic geom.*1 * ( 2003 ) , no . 4 , 659682 . l. ng , _ computable legendrian invariants _ , topology * 42 * ( 2003 ) , no . 1 , 5582 . m. pozniak , ph . d. thesis , 1994 . d. salamon , _ lectures on floer homology _ , in synplectic geometry and topology , edited by y. eliashberg and l. traynor , ias / park city mathematics series , vol 7 , 1999 , pp . 143230 . , _ contact surgery and symplectic handlebodies _ , hokkiado math . journal * 20 * ( 1991 ) , 241251 . e. c. zeeman , _ twisting spun knots _ , trans . * 115 * ( 1965 ) , 471495 .

we show how to orient moduli spaces of holomorphic disks with boundary on an exact lagrangian immersion of a spin manifold into complex @xmath0-space in a coherent manner . this allows us to lift the coefficients of the contact homology of legendrian spin submanifolds of standard contact @xmath1-space from @xmath2 to @xmath3 we demonstrate how the @xmath4-lift provides a more refined invariant of legendrian isotopy . we also apply contact homology to produce lower bounds on double points of certain exact lagrangian immersions into @xmath5 and again including orientations strengthens the results . more precisely , we prove that the number of double points of an exact lagrangian immersion of a closed manifold @xmath6 whose associated legendrian embedding has good dga is at least half of the dimension of the homology of @xmath6 with coefficients in an arbitrary field if @xmath6 is spin and in @xmath2 otherwise .

math0408411

the cherenkov telescope array ( cta , acharya et al . 2013@xcite ) is the project of a new array of several imaging atmospheric cherenkov telescopes ( iacts ) for very high - energy ( vhe ) astronomy . the array shall be composed by three different types of telescopes , in order to maximize the performance in three different energy ranges : the large size telescope ( lst ) for the low energy range ( e @xmath1 20 gev 1 tev ) , the medium size telescope ( mst ) for the core energy range ( e @xmath1 0.110 tev ) , and the small size telescope ( sst ) for the high energy range ( e @xmath2 1 tev ) . the astri project ( ` astrofisica con specchi a tecnologia replicante italiana ' ) is included in this framework : it is a ` flagship project ' of the italian ministry of education , university and research , which , under the leadership of the italian national institute of astrophysics ( inaf ) , aims to realize and test an end - to - end prototype of the sst . the astri sst-2 m prototype is characterized by two special features which will be adopted for the first time on a cherenkov telescope ( pareschi et al . 2013@xcite ) : a dual - mirror schwarzschild couder ( sc ) optical design ( vassiliev et al . 2007@xcite ) , which is characterized by a wide field of view ( fov ) and a compact optical configuration , and a light and compact camera based on silicon photo - multipliers , which offer high photon detection sensitivity and fast temporal response . figure [ fig1 ] ( left panel ) shows the telescope layout , whose mount exploits the classical altazimuthal configuration . the proposed layout ( canestrari et al . 2013@xcite ) is characterized by a wide - field aplanatic optical configuration : it is composed by a segmented primary mirror made of three different types of segments , a concave secondary mirror , and a convex focal surface . the design has been optimized in order to ensure , over the entire fov , a light concentration higher than 80 % within the angular size of the pixels . the telescope design is compact , since the primary mirror ( m1 ) and the secondary mirror ( m2 ) have a diameter of 4.3 m and 1.8 m , respectively , and the primary - to - secondary distance is 3 m. the sc optical design has an f - number f/0.5 , a plate scale of 37.5 mm/@xmath3 , a logical pixel size of approximately 0.17@xmath3 , an equivalent focal length of 2150 mm and a fov of 9.6@xmath3 in diameter ; the mean value of the active area is @xmath0 6.5 m@xmath4 . the primary mirror is composed by 18 hexagonal segments , with an aperture of 849 mm face - to - face ; the central segment is not used because it is completely obstructed by the secondary mirror . according to their distance from the optical axis , there are three different types of segments , each having a specific surface profile . in order to perform the correction of the tilt misplacements , each segment will be equipped with a triangular frame with two actuators and one fixed point . the secondary mirror is monolithic and has a curvature radius of 2200 mm and a diameter of 1800 mm . it will be equipped with three actuators , where the third actuator will provide the piston / focus adjustment for the entire optical system . for both the segments of the primary mirror and the secondary mirror the reflecting surface is obtained with a vapor deposition of a multilayer of pure dielectric material ( bonnoli et al . 2013@xcite ) . the sc optical configuration allows us designing a compact and light camera . in fact , the camera of the astri sst-2 m prototype has a dimension of about 56 cm @xmath5 56 cm @xmath5 56 cm , including the mechanics and the interface with the telescope structure , for a total weight of @xmath0 50 kg ( catalano et al . 2013@xcite ) . such small detection surface , in turn , requires a spatial segmentation of a few square millimeters to be compliant with the imaging resolving angular size . in addition , the light sensor shall offer a high photon detection sensitivity in the wavelength range between 300 and 700 nm and a fast temporal response . in order to be compliant with these requirements , we selected the hamamatsu silicon photomultiplier ( sipm ) s11828 - 3344 m . the ` unit ' provided by the manufacturer is the physical aggregation of 4 @xmath5 4 pixels ( 3 mm @xmath5 3 mm each pixel ) , while the logical aggregation of 2 @xmath5 2 pixels is a ` logical pixel ' ( figure [ fig1 ] , lower right ) ; its size of 6.2 mm @xmath5 6.2 mm corresponds to 0.17@xmath3 . in order to cover the full fov , we adopt a modular approach : we aggregate 4 @xmath5 4 units in a photon detection module ( pdm ) and , then , use 37 pdms to cover the full fov . the advantage of this design is that each pdm is physically independent of the others , allowing maintenance of small portions of the camera . to fit the curvature of the focal surface , each pdm is appropriately tilted with respect to the optical axis . the camera is also equipped with a light - tight two - petal lid ( figure [ fig1 ] , upper right ) in order to prevent accidental sunlight exposure of its sipm detectors . the astri sst-2 m prototype will be placed at the ` m. g. fracastoro mountain station ' , the observing site of the inaf catania astrophysical observatory ; it is at serra la nave , on the etna mountain , at an altitude of 1735 m a.s.l . ( maccarone et al . 2013@xcite ) . the prototype is currently under construction and it will be tested on field : it is scheduled to start data acquisition in 2014 . although the astri sst-2 m prototype will mainly be a technological demonstrator , it should be able to perform also scientific observations . based on the foreseen sensitivity ( @xmath1 0.2 crab unit at 0.8 tev ) , a source flux of 1 crab at e @xmath2 2 tev should be detectable at 5 @xmath6 confidence level in some hours , while a few tens of hours should be necessary to obtain a comparable detection at e @xmath2 10 tev ( bigongiari et al . 2013@xcite ) . in this way we would obtain the first crab observations with a cherenkov telescope adopting a schwarschild couder optical design and a sipm camera ; in addition , also the brightest agns ( mkn 421 and mkn 501 ) could be detected . beside the prototype , the astri project aims to realize , in collaboration with cta international partners , a mini - array of a few sst dual - mirror ( sst-2 m ) telescopes . thanks to the array approach , it will be possible to check the trigger algorithms and the wide fov performance , to compare the mini - array performance with the monte carlo expectations and to validate the performance predictions for the full sst array . the astri / cta mini - array shall constitute the first seed of the cta observatory at its southern site and shall perform the first cta science , starting operation in 2016 . considering 7 telescopes at an optimized distance of 250 - 300 m , preliminary monte carlo simulations yield a minimal improvement in sensitivity compared to the current iacts ( di pierro et al . 2013@xcite ) : starting from e = 10 tev , the mini - array expected sensitivity is slightly better at few tens of tev ; moreover , the astri / cta mini - array sensitivity is still competitive up to 100 tev , where the performance of the present generation of iacts drops dramatically . in this energy regime the mini - array will operate as the most sensitive iact array . the astri / cta mini - array will be able to study in great detail sources with a flux of a few 10@xmath7 erg @xmath8 s@xmath9 at 10 tev , with an angular resolution of a few arcmin and an energy resolution of about 10 - 15 % . the astri / cta mini - array will observe prominent sources such as extreme blazars ( 1es 0229 + 200 ) , nearby well - known bl lac objects ( mkn 421 and mkn 501 ) and radio - galaxies , galactic pulsar wind nebulae ( crab nebula , vela - x , hess 1825 - 137 ) , supernovae remnants ( vela - junior , rx j1713.7 - 3946 , kepler ) and microquasars ( ls 5039 ) , as well as the galactic center . in this way it will be possible to investigate the electron acceleration and cooling , to study the relativistic and non relativistic shocks , to search for cosmic - ray ( cr ) pevatrons , to study the cr propagation and the impact of the extragalactic background light on the spectra of the nearby sources ( vercellone et al . 2013@xcite ) . this work was partially supported by the astri flagship project financed by the italian ministry of education , university , and research ( miur ) and lead by the italian national institute of astrophysics ( inaf ) . we also acknowledge partial support by the miur bando prin 2009 .

the astri project aims to develop , in the framework of the cherenkov telescope array , an end - to - end prototype of the small - size telescope , devoted to the investigation of the energy range @xmath0 1100 tev . the proposed design is characterized by two challenging but innovative technological solutions which will be adopted for the first time on a cherenkov telescope : a dual - mirror schwarzschild couder configuration and a modular , light and compact camera based on silicon photo - multipliers . here we describe the prototype design , the expected performance and the possibility to realize a mini array composed by a few such telescopes , which shall be placed at the final cta southern site .

1405.4187

in the context of novel materials with unusual physical properties , the researchers are interested in the fe - based double perovskite multiferroics with the general formula pbfe@xmath0m@xmath0o@xmath1 ( m = nb , ta , sb ) and their solid solutions with substitution of a or b type ions in the abo@xmath1 perovskite structure , see , e.g. @xcite and references therein . recent studies @xcite of these substances reveal a lot of interesting properties like large magnetoelectric coupling and high dielectric permittivity . it had been shown in the above papers that these properties occur in substantial range of temperatures and chemical compositions , revealing the existence of ferroelectric ( fe ) , antiferromagnetic ( afm ) and spin glass ( sg ) phases on the corresponding phase diagrams . in the above compounds , fe@xmath3 and m@xmath5 cation positions may be ordered or disordered within the simple cubic b sublattice of the perovskite abo@xmath1 structure . the degree of chemical ordering depends on the relative strengths of electrostatic and elastic energies and on the ionic radii of these cations in particular . it is commonly accepted that pb(fe@xmath0nb@xmath0)o@xmath1 ( pfn ) and pb(fe@xmath0ta@xmath0)o@xmath1 ( pft ) are chemically disordered compounds due to almost equal ionic radii of fe@xmath3 and nb@xmath5 or ta@xmath5 @xcite , while sb - contained compounds can be chemically ordered up to 90% as sb@xmath5 is much larger than fe@xmath3 @xcite . the magnetism of the compounds is due to fe@xmath3 , s = 5/2 ions that occupy half of octahedral sites of the perovskite lattice . the magnetic moments of the fe@xmath3 ions interact with each other via various superexchange paths , considered in ref . in details . the majority of papers consider the spin glass state as the magnetic ground state of both pfn and pft at @xmath6 k. there are several ambiguous statements about sg nature of the magnetic ground state in pfn at @xmath7 k , see @xcite and references therein . the statement about glasslike state , starting at @xmath8 k for low magnetic fields @xmath9 oe or at @xmath10 k at @xmath11 oe @xcite along with reference to some superparamagnetic ( spm ) behavior with blocking temperature @xmath12 increase the confusion in understanding of the above magnetic ground state nature . the light was poured in the paper @xcite with the help of @xmath13sr spectroscopy and neutron scattering . the authors @xcite have shown that magnetic ground state of pfn is a spin glass like state , that coexists with the long - range afm order below @xmath14 k in the time scale of their experiment . the sg state has also been identified from @xmath15o nmr as distinct anomalies in the spin - lattice and spin - spin nuclear magnetic relaxation @xcite . however , the microscopic nature of the above sg state as well as essential increase of magnetic susceptibility in pfn and pft below the neel temperature remain unclear till now . it has been proposed in refs . and that along with infinite - range percolation cluster responsible for the long - range ordered afm phase , superantiferromagnetic fe@xmath3 clusters are created also . the latter are responsible for the spin - glass like ( so - called cluster glass ) behavior of magnetic properties . in principle , this fact agrees with nmr and esr results @xcite . @xmath16nb nmr spectra in pfn @xcite show the existence of two different nb sites with different local magnetic fields : fe - rich , nb - poor and fe - poor , nb - rich nanoregions . these data suggest that a spin - glass state of pfn below 11 k might arise from the latter regions and a phase separation exists , at least , at nanometric scale . the second model , recently proposed in ref . , is based on coexistence of the long - range order and sg on the microscopic scale . it assumes that all fe@xmath3 spins in the system form afm order below the neel temperature , but there are additional long - range spin - spin correlations along @xmath2 direction , while the transversal @xmath4 spin components undergo random thermal reorientations between energetically equivalent ( or nearly equivalent ) orientations . it has been suggested that such system of heisenberg spins gradually froze into a sg state , known as @xmath17 reentrant sg phase @xcite . however , the theoretical description of such reentrant phase is absent so far for pfn or pft so that the microscopic origin of this state still remains unknown . the detailed magnetoelectric studies of pfn single crystals have been performed in refs . . in particular , it had been found @xcite that below @xmath18 k the lattice point group symmetry changes from @xmath19 to @xmath20 . it was concluded therefore that a weak ferromagnetism , usually observed in pfn , is induced in an originally antiferromagnetic spin structure by lowering the crystal symmetry . this finding increase the confusion in understanding of magnetic ground state nature of both pfn and pft . the aim of this paper is to make regimentation of the diverse ( and sometimes controversial ) facts about the coexistence of long - range magnetic order and spin glass phase in the above double perovskite multiferroics . for that , based on so - called random local field method ( see @xcite and references therein ) we are going to present the theoretical description of the mixed afm - sg phase in the perovskite multiferroics . besides we present strong experimental evidence of such coexistence . the main peculiarities of above perovskites , making them different from ordinary antiferromagnets are the sharp increase of magnetic susceptibility in the antiferromagnetic phase @xmath21 with its subsequent diminishing at low temperatures @xmath22 , where @xmath23 and @xmath24 are , respectively , neel and glassy transition temperature . in this section we are going to show that these anomalies can be well described within our model of mixed afm - sg phase , which is realized in the pfn and pft . it has been demonstrated experimentally in ref . @xcite that sg and afm phases coexist in pfn on the microscopic scale . the crux of the matter is that in ground and low - lying excited states of any magnet the length of its magnetization vector ( or the lenghts of sublattice magnetizations in afm ) is conserved , i.e. that vector can only rotate , keeping its length constant , see fig . [ fig1a ] ( a ) . at the same time if we assume that the interplay between disorder ( random positions of magnetic fe@xmath3 ions and nonmagnetic nb@xmath5 ions ) and anisotropic spin - spin interaction makes @xmath25 spin components fluctuate so that ( by virtue of conservation of spin vector length ) @xmath2 spin components , which are antiferromagnetically aligned , have different lengths . this means that while fe@xmath3 spins in pfn form afm order along @xmath2 -axis ( fig . [ fig1a ] ( b ) ) , their @xmath4 components contribute to sg phase . for this scenario to realize , the interaction between spins should have several contributions . namely , although there are short - range exchange and superexchange interactions , generating long - range ( afm in our case of pfn ) magnetic order , there is one more type of interaction , inevitably present in any magnetic system and fixing the direction of its magnetization . this is so - called relativistic spin - spin interaction , forming magnetic anisotropy energy and magnetic dipole interaction . although the amplitudes of latter interactions are usually ( much ) smaller then that of exchange interaction , they play an important role , being responsible both for spontaneous magnetization ( and/or antiferromagnetic vector ) direction as well as for features like magnetic domain wall width . as the magnetic dipole interaction depends on the angle between spins , its action in the disordered spins system leads to their gradual freezing in random orientations , yielding spin glass state moreover , the freezing temperature , @xmath26 k is much smaller then neel temperature , @xmath27 k as the resulting exchange interaction is accordingly weaker then initially afm one . in other words , @xmath23 is determined by the afm exchange interaction amplitude , while @xmath24 - by the synergy of afm and fm interactions . here we argue that the main reason for above behaviour is the synergy between disorder and the presence of several types of interactions between fe@xmath3 spins in pfn . namely , below we assume that there is short - range ( of range @xmath28 ) exchange interaction of afm sign along @xmath2 axis , while in the @xmath4 plane there is so - called frustrating exchange interaction of fm sign ( as opposed to afm one in @xmath2 - direction ) , which contributes to sg behavior , rising @xmath24 and lowering @xmath23 . note that the consideration of the sole chemical fe - nb disorder in pfn does not explain the values of @xmath23 and @xmath24 temperatures in it . really , consider perovskite abo@xmath1 ferrites without above disorder , i.e. the systems like yfeo@xmath1 , where all b lattice sites are occupied by fe . if we imagine the dilution of such systems by nonmagnetic ions like in pfn and extrapolate ( linearly with fe concentration ) their @xmath23 , which is around 600 k ( see , e.g. , ref . ) to , say , 50% dilution , we obtain @xmath29 300 k , i.e. much larger value then @xmath29 150 k for pfn . our theory explains this and other experimental facts by considering the reciprocity between the chemical disorder and frustrating fm - afm exchange interactions , leading to glassy type of spin - spin correlations at low temperatures . in principle there is also long - range magnetic dipole interaction of relativistic nature , but its amplitude is negligibly small in pfn @xcite . according to estimations of ref . , the afm exchange interaction between nearest fe@xmath3 is around 42 k , while the next - nearest neighbour fm interaction constant is @xmath30 . at the same time , the constant of magnetic dipole interaction is around 0.077 k , which is more then one order of magnitude less then above smaller exchange constant . below we use these values of exchange constants to fit the experiment in pfn . as the length of spin vector is usually conserved , in our picture ( see fig.[fig1a ] ) the fluctuations of the spin is due to those of its @xmath4 projections , which means that @xmath2 - projection is always directed along @xmath2 axis but its length fluctuates due to those of @xmath4 projections . note that in this case the average balance between @xmath31 and @xmath32 directions of @xmath33 is conserved so that the overall structure is antiferromagnetic . component formation ( a ) and sketch of fe@xmath3 spins @xmath2 components afm alignment in pbfe@xmath0nb@xmath0o@xmath1 ( b ) . black and red arrows show the magnetic sublattices in afm alignment and blue circles on panel ( b ) show the spinless nb@xmath5 sites . the lengths of spin vectors @xmath34 and @xmath35 are the conserved quantites , their @xmath36 and @xmath37 components @xmath38 fluctuate from site to site so that the lengths @xmath39 are different.,scaledwidth=47.0% ] to describe the above discussed effects quantitatively , we introduce so - called random field model , which includes naturally the explicit form of the interaction between the fe@xmath3 spins in pfn and pft . the hamiltonian of the ensemble of spins 5/2 reads @xmath40 where @xmath41 is heisenberg spin in @xmath42 - th host lattice site , @xmath43 is an external magnetic field , directed along @xmath2 axis , @xmath44 @xmath45 ( @xmath46 ) is a spin - spin interaction , which , in the spirit of above discussion , we choose in the form @xmath47 here summation is running over host lattice sites , where fe@xmath3 ions ( i.e. spins ) are present . we consider the strengths of the interactions @xmath48 and @xmath49 to be close to those from ref . , but they may be adjusted to achieve the better fit to experiment . for simplicity , the hamiltonian does not contain so - called zero field splitting terms , related to single ion anisotropy . hamiltonian incorporates two sources of randomness . the first is the spatial disorder , which means that spin can be randomly present or absent in the specific @xmath50-th cite of a host lattice . the second is the thermal disorder , i.e. random spin projection in the @xmath50-th cite . having this randomness in mind , we consider every spin @xmath51 as a source of a random field @xmath52 affecting other spins at the sites @xmath53 . in other words , every spin in our approach is subjected to some random , fluctuating field , created by the rest of the spin ensemble . thus all thermodynamic properties of the system are determined by the distribution function @xmath54 of the random field @xmath55 . that is , any spin - dependent macroscopic quantity @xmath56 ( like magnetization ) , reads @xmath57 , where @xmath58 and @xmath59 is auxiliary single particle thermal average with effective hamiltonian @xmath60 . the explicit form of distribution function @xmath61 reads @xmath62 where bar denotes the averaging over random spatial positions of spins ( spatial averaging ) and angular brackets denote the thermal averaging over possible spin orientations . to actually perform the above averagings , we use the spectral representation of @xmath63 - function . @xcite as we can not do these averagings exactly @xcite , we do them self - consistently in the framework of statistical theory of magnetic resonance lineshape . @xcite for the general form of interaction @xmath64 , incorporating not only @xmath48 and @xmath49 , eq . , this procedure yields [ r4 ] @xmath65d^3r\right\ } , \label{rf4a } \\ & & \gamma=\biggl\{\bigl[\phi^{-1}(m)\bigr]^2-(q_x^2+q_y^2+q_z^2)+ 2i\biggl[q_xm_x+q_ym_y+q_zm_z\biggr]\biggr\}^{1/2},\ m=\sqrt{m_x^2+m_y^2+m_z^2 } , \nonumber \\ & & q_x = j_{xx}\rho_x+j_{xy}\rho_y+j_{xz}\rho_z,\ q_y = j_{xy}\rho_x+j_{yy}\rho_y+j_{yz}\rho_z , \ q_z = j_{xz}\rho_x+j_{yz}\rho_y+j_{zz}\rho_z , \nonumber \\ & & m_{x , y , z}=\int \frac{h_{x , y , z}}{h}\ b_{5/2}\left(\frac 52 g\beta \mu_b h\right)f({\bf h},{\bf m})d^3h . \label{rf4b}\end{aligned}\ ] ] the equations and constitute a self - consistent set for determination of the dimensionless magnetization components @xmath66 ( @xmath67 is saturation magnetization ) within random local field model . we pay attention that as we are dealing with antiferromagnet , consisting of two sublattices with opposite directions of magnetizations , there are two types of magnetization in afm . one is above magnetization @xmath68 and the other is staggered magnetization ( or so - called afm vector ) @xmath69 , where @xmath70 is ordering wavevector ( see , e.g. ref . and references therein ) . the equations for @xmath71 then will be almost similar to and except that the interaction is now renormalized by the phase factors @xmath72 , i.e. @xmath73 @xmath74 . in the equations and @xmath75/5 $ ] is brillouin function for spin 5/2 and @xmath76 is its inverse , @xmath77 is spins concentration i.e. number of spins per unit volume , @xmath78 is reciprocal temperature . the equations and have a solution only for sufficiently large ratio @xmath79 , where @xmath48 and @xmath49 are the amplitudes of afm and fm exchange interactions . as exchange interaction is short - range , to effectively create the long - range ordered afm state , the concentration @xmath80 of fe@xmath3 spins should be sufficiently large . this means that the concentration @xmath80 can also be regarded as parameter of transition between spin glass and magnetically ordered phase . if there is no dilution of fe@xmath3 spin cites in above perovskites , the distribution function becomes @xmath63 - function @xmath81 @xcite and we have ordinary mean field approximation , where mean field @xmath82 @xmath83 . it had been shown earlier @xcite that regular procedure of transition from mean field to random field model is to expand the integrand of eq . in power series in @xmath84 . the second approximation , proportional to @xmath85 , generates gaussian distribution function of random fields . we note that our random field model in gaussian approximation for ising spins 1/2 gives ordinary replica - symmetric solution @xcite for spin glass . our aim is to calculate experimentally observed @xmath86 component of dynamic magnetic susceptibility @xmath87 @xmath88 where @xmath89 is @xmath2 component of dimensional magnetization ( which at nonzero frequency @xmath90 becomes complex ) and @xmath91 is above external magnetic field . it can be shown ( see @xcite and references therein ) that within above random field model the expression for dimensionless dynamic magnetization @xmath92 has the form @xmath93 where @xmath94 is defined by eq . and @xmath95 is a single spin relaxation time , averaged self - consistently over above random fields . namely , assuming the arrenius law for single spin relaxation , we obtain @xmath96 where @xmath97 is a barrier between single spin orientation and @xmath98 sec is inverse attempt frequency or initial relaxation time . the parameters @xmath97 and @xmath99 are experimentally adjustable . we note that while single spin relaxation in our model obeys arrhenius law , the relaxation of the whole spin ensemble at low temperatures @xmath100 obeys vogel - fulcher law @xcite , inherent for glassy systems . latter law can be extracted from the resulting @xmath101 curves only numerically . explicit form of @xmath95 obtained with the help of eq . for spin 5/2 reads @xmath102 the expression with respect to , and permits to determine theoretically the dynamic magnetic susceptibility of considered disordered perovskite multiferroics . figure [ fig2 ] reports the fit of the _ ac _ @xmath103 to experimental curves measured in @xcite for pfn at different frequencies . the excellent coincidence between theory and experiment is seen . we fit the experimental curves by their maximum temperatures @xmath104 , which varies from 13.8 to 14.4 k with frequency increase from 1.08 to 29.16 hz . corresponding mean relaxation times of spin fluctuations as a function of the @xmath104 temperature are reported in fig . [ fig3 ] . to extract the experimental symbols shown in fig . [ fig3 ] , we use both relaxation time data from the _ ac _ magnetic susceptibility and those from nuclear magnetic resonance ( nmr ) of @xmath15o isotope . it has been shown previously @xcite that both spin - lattice ( @xmath105 ) and spin - spin ( @xmath106 ) nuclear magnetization relaxation times show distinct minimum around the sg transition temperature . in particular , the @xmath105 relaxation time depends on electron spin fluctuations at the nuclear larmor frequency @xmath107 mhz , while the @xmath106 relaxation time feels spin fluctuations at much lower frequency , defined by the spin - echo delay time @xmath108 100 @xmath13s . the nmr data thus extend the measured relaxation times of the spin fluctuations up to nanosecond range . the data in fig . [ fig3 ] are described well by the vogel - fulcher law @xcite at the time scale from 1 down to @xmath109 s : @xmath110,\ ] ] where @xmath104 corresponds to the peak temperature either of _ ac _ magnetic susceptibility or nuclear magnetic relaxation at frequency @xmath111 . [ fig4 ] shows the fit of the dc magnetic susceptibility ( measured at field cooling ( fc ) and zero - field cooling ( zfc ) protocols ) as a function of temperature for different strength of the field . the excellent coincidence between theory and experiment is seen again . note that with increase of magnetic field strength the @xmath104 temperature at the zfc curves shifts to lower temperature in accordance with our theory prediction , see also ref . . the above shift is related to the interplay between spins polarization by the applied magnetic field and transversal spin components freezing , giving rise to the sg phase . namely , at higher magnetic fields the polarized ( i.e. aligned along field direction ) spins form effectively paramagnetic phase so that lower temperatures are required for transversal spin components to `` feel '' the glassy correlations i.e. to freeze into sg phase . our fitting of experiment in figs . [ fig2 ] and [ fig4 ] has been done for the following set of parameters . considering the magnetic moment of fe@xmath3 ion to be 5.2 @xmath112 ( @xmath112 is bohr magneton ) and taking the best fit value @xmath113 , we obtain that concentration of fe@xmath3 ions is approximately 7.8@xmath114 @xmath115 . the barrier @xmath97 between single spin orientation in arrhenius law for our fit turns out to be @xmath116 k. likewise , the parameter @xmath117 sec . we point here to the difference between parameters of single - spin relaxation arrhenius law and those for the vogel - fulcher law , which results from the average relaxation of the whole spins ensemble . also , the best fit is achieved for the exchange constants ratio @xmath118 , see eq . . the discrepancy between theoretical and experimental curves at low temperatures are due to the fact that additional defects contribute experimental susceptibility , making it to decay slower . the pretty good coincidence between theoretical and experimental magnetic susceptibility curves at different frequences and magnetic fields suggests that the observed behavior can be well attributed to the joint action of site disorder , anisotropy ( when @xmath4 components of spin vector fluctuate making the length of @xmath2 component to vary ) and hierarchy of exchange interactions in pfn . namely , although the short - range exchange interaction between spins dominates , the smaller frustrating exchange interaction in the @xmath4 plane forms the observable sg anomalies owing to transversal spin fluctuations . to conclude , here we present the experimental explanation and its quantitative theoretical description of the coexistence of long - range ordered ( afm ) and spin - glassy phases in the crystalline double perovskite fe - contained multiferroics like pbfe@xmath0nb@xmath0o@xmath1 . the main physical mechanism of such coexistence is the interplay between chemical disorder ( half of corresponding lattice cites are occupied by spinless nb@xmath5 ions ) and the anisotropy and frustration of exchange interactions between fe@xmath3 spins in above multiferroics and pfn in particular . namely , if the afm long - range order is related to the strong exchange interaction of @xmath2 - components of spins , the glassy effects are due to the much weaker exchange interaction of ferromagnetic sign , which couples transversal @xmath4 spin components . our theory , explicitly considering randomness in the spin space along with anisotropy of exchange interactions , is able to describe our experimental data quantitatively thus showing that considered physical mechanism of afm and glassy phases coexistence in above fe - contained multiferroics is quite accurate . v. v. laguta , v. a. stephanovich , m. savinov , m. marysko , r. o. kuzian , i. v. kondakova , n. m. olekhnovich , a. v. pushkarev , yu . v. radyush , i. p. raevski , s. i. raevskaya and s. a. prosandeev , new j. phys . 16 , 113041 ( 2014 ) . vogel , h. the law of the relationship between viscosity of liquids and the temperature . z. _ * 22 * , 645 - 646 ( 1921 ) ; fulcher , g. s. analysis of recent measurements of the viscosity of glasses . * 8 * , 339 - 355 ( 1925 ) .

we propose experimental verification and theoretical explanation of magnetic anomalies in the complex fe - contained double perovskite multiferroics like pbfe@xmath0nb@xmath0o@xmath1 . the theoretical part is based on our model of coexistence of long - range magnetic order and spin glass in the above substances . in our model , the exchange interaction is anisotropic , coupling antiferromagnetically @xmath2 spin components of fe@xmath3 ions . at the same time , the @xmath4 components are coupled by much weaker exchange interaction of ferromagnetic sign . in the system with spatial disorder ( half of corresponding lattice cites are occupied by spinless nb@xmath5 ions ) such frustrating interaction results in the fact that antiferromagnetic order is formed by @xmath2 projection of the spins , while their @xmath4 components contribute to spin glass behaviour . our theoretical findings are supported by the experimental evidence of coexistence of antiferromagnetic and spin glass phases in chemically disordered fe - contained double perovskite multiferroics .

1512.08725

organic semiconductors are envisioned to revolutionize display and lighting technology . the remaining engineering - related challenges are being tackled and the first products are commercially available already . to guarantee a sustainable market entry , however , it is important to further deepen the understanding of organic semiconductors and organic semiconductor devices . electronic trap states in organic semiconductors severely affect the performance of such devices . for organic thin - film transistors ( tft s ) , for example , key device parameters such as the effective charge mobility , the threshold voltage , the subthreshold swing as well as the electrical and environmental stability are severely affected by trap states at the interface between the gate dielectric and the semiconductor . trap states in organic semiconductors have been studied for several decades.@xcite although the first organic field - effect transistors emerged in the 1980 s , ( polymeric semiconductors : ref . , small molecule organic semiconductors : ref . ) it is only recently , that trap states in organic field - effect transistors are a subject of intense scientific investigation ( refs . @xcite and references therein ) . the present study is focused on trap densities in small molecule organic semiconductors . these solids consist of molecules with loosely bound @xmath2-electrons . the @xmath2-electrons are transferred from molecule to molecule and , therefore , are the source of charge conduction . small molecule organic semiconductors tend to be crystalline and can be obtained in high purity . typical materials are oligomers such as pentacene , tetracene or sexithiophene but this class of materials also includes e.g. rubrene , c@xmath3 or the soluble material tips pentacene ( ref . ) . trap densities are often given as a volume density thus averaging over various trapping depths . the spectral density of localized states in the band gap , i.e. the trap densities as a function of energy ( trap dos ) , gives a much deeper insight into the charge transport and device performance . in this paper we compare , for the first time , the trap dos in various samples of small molecule organic semiconductors including thin - film transistors ( tft s ) where the active layer generally is polycrstalline and organic single crystal field - effect transistors ( sc - fet s ) . these data are also compared with the trap dos in the bulk of single crystals made of small molecule semiconductors . it turns out that it is this comparison of trap densities in tft s , sc - fet s and in the bulk of single crystals that is particularly rewarding . the trap dos in organic semiconductors can be derived from several different experimental techniques , including measurements of field - effect transistors , space - charge - limited current ( sclc ) measurements , thermally stimulated currents ( tsc ) , kelvin - probe , time - of - flight ( tof ) or capacitance measurements . for the present study , we focus on the trap dos as derived from electrical characteristics of organic field - effect transistors or from sclc measurements of single crystals . we begin with a brief discussion of charge transport in small molecule semiconductors followed by a summary of the current view of the origin of trap states in these materials . after a comparison of different methods to calculate the trap dos from electrical characteristics of organic field - effect transistors we are eventually in a position to compile , compare and discuss trap dos data . even in ultrapure single crystals made of small molecule semiconductors , the charge transport mechanism is still controversial . the measured mobility in ultrapure crystals increases as the temperature is decreased according to a power law @xmath4.@xcite this trend alone would be consistent with band transport . however , the mobilities @xmath5 at room temperature are only around 1@xmath6/vs and the estimated mean free path thus is comparable to the lattice constants . it has often been noticed that this is inconsistent with band transport.@xcite since the molecules in the crystal have highly polarizable @xmath2-orbitals , polarization effects are not negligible in a suitable description of charge transport in organic semiconductors . _ holstein s _ polaron band model considers electron - electron interactions and the model has recently been extended.@xcite with increasing temperature , the polaron mass increases . this effect is accompanied by a bandwidth narrowing and inevitably results in a localization of the charge carrier . consequently , this model predicts a transition from band transport at low temperature to phonon - assisted hopping transport at higher temperatures ( e.g. room temperature ) . the model may explain the experimentally observed increase in mobility with decreasing temperature and seems to be consistent with the magnitude of the measured mobilities at room temperature . on the other hand , thermal motion of the weakly bound molecules in the solid is large compared to inorganic crystals . such thermal motions most likely affect the intermolecular transfer integral . troisi et al . _ have shown that , at least for temperatures above 100k , the fluctuation of the intermolecular transfer integral is of the same order of magnitude as the transfer integral itself in materials such as pentacene , anthracene or rubrene.@xcite as a consequence , the fluctuations do not only introduce a small correction , but determine the transport mechanism and limit the charge carrier mobility.@xcite clearly , the thermal fluctuations are less severe at a reduced temperature and the calculations predict a mobility that increases with decreasing temperature , according to a power law . this is in excellent agreement with the measured temperature - dependence in ultrapure crystals . moreover , the model predicts mobilities at room temperature between 0.1@xmath6/vs and 50@xmath6/vs , which also is in good agreement with experiment.@xcite interestingly , the importance of thermal disorder is supported by recent tetrahertz transient conductivity measurements on pentacene crystals.@xcite in essence , the band broadening due to the thermal motion of the molecules is expected to result in electronic trap states which would be related to the intrinsic nature of small molecule semiconductors.@xcite clearly , trap states can also be due to extrinsic defects and these traps can completely dominate the charge transport resulting in an effective mobility @xmath7.@xcite for amorphous inorganic semiconductors such as amorphous silicon , the mobility edge picture has been developed ( fig . [ figure - dossketch]).@xcite the mobility edge separates extended from localized states . the existence of extended states in amorphous silicon is attributed to the similarity of the short - range configuration of the atoms in the amorphous phase which is similar to the configuration in the crystalline phase.@xcite hopping in localized states is expected to be negligible if transport in extended states exists , i.e. we have an abrupt increase in mobility at the mobility edge . only the charge carriers that are thermally activated to states above the mobility edge contribute to the transport of charge . schematic representation of the mobility edge separating localized states ( traps ) from extended states . at the mobility edge , the mobility as a function of energy abruptly rises and only the charge carriers that are thermally activated to states above the mobility edge contribute to the charge transport . ] in the following we assume that charge transport in small molecule semiconductors can be described by an effective transport level and a distribution of trap states below this transport level . the mobility edge model is a specific realization of this very general assumption . in a _ completely _ disordered material ( no short - range order ) all electronic states are localized.@xcite the charge carriers are highly localized and hop from one molecule to the next . however , even this situation can be described by introducing an effective transport level and a broad distribution of trap states below the transport level.@xcite in the following , we use the term valence band edge . this term may generally be interpreted as the effective transport level and denotes the mobility edge in the mobility edge picture . we proceed by summarizing the current view of the microscopic origin of traps in small molecule semiconductors . charge carrier traps within the semiconductor are caused by structural defects or chemical impurities . chemical impurities may also cause a surrounding of structural defects by distorting the host lattice.@xcite on the other hand , chemical impurities tend to accumulate in regions with increased structural disorder ( ref . ) as well as at the surface of a crystal ( ref . ) . trap states caused by the gate dielectric can become very important in organic field - effect transistors . finally , as mentioned already , also the thermal fluctuations of the molecules are expected to result in shallow trap states within the band gap . in the bulk of ultrapure anthracene or naphthalene crystals , typical densities of vacancies ( a dominant point defect ) are of the order of @xmath8@xmath9 ( ref . @xcite , p. 222 ) . vacancies are expected to be concentrated close to other structural defects due to a reduced formation energy.@xcite extended structural defects ( e.g. edge dislocations , screw dislocations or low - angle grain boundaries ) can be present in significant densities in organic crystals , e.g. 10@xmath10@xmath9 ( ref . @xcite , p.226 ) . therefore , extended structural defects are thought to be the main source of traps in ultrapure organic crystals.@xcite thin films of small molecule semiconductors are expected to have a higher density of structural defects than single crystals . thin films of small molecule semiconductors are often polycrystalline and grain boundaries can limit the charge transport in such films . for example , measurements of sexithiophene - based transistors with sio@xmath11 gate dielectric and an active channel consisting of only two grains and one grain boundary show , that the transport is in fact limited by the grain boundary.@xcite at the grain boundary , a high density of traps exists and the density of these traps per unit area of the active accumulation layer is of the order of @xmath12@xmath13.@xcite in the following , we focus on structural defects in vacuum evaporated pentacene films which are of particular relevance for this work . since pentacene films are often polycrystalline , large angle grain boundaries are expected to produce additional structural defects also in this material . the effect of grain boundaries on charge transport in pentacene films is still controversial . atomic force measurements ( afm ) of ultrathin pentacene films have clearly shown , that the field - effect mobility in pentacene - based transistors can be higher in films with smaller grains.@xcite in addition , some experimental evidence indicates that there is no correlation between charge trapping and topographical features in pentacene thin films.@xcite on the contrary , it has recently been shown that long - lived ( energetically deep ) traps that cause gate bias stress effects in pentacene - based tft s are mainly concentrated at grain boundaries.@xcite another important cause of structural disorder in pentacene films is polymorphism since pentacene can crystallize in at least four different structures ( phases ) . it is quite common that at least two of these phases coexist in pentacene thin films.@xcite a theoretically study deals with in - grain defects in vacuum evaporated pentacene films.@xcite structural defects are formed during the film growth . upon addition of more and more `` defective '' molecules at a given site , the ideal crystal structure becomes energetically more and more favourable . the system eventually relaxes into the ideal crystal structure during the continuation of the film growth . the relaxation happens , provided that the evaporation rate is low enough and that there is enough time for relaxation.@xcite in this study it is suggested that structural defects within the grains of a pentacene film that resist relaxation can not exceed densities of 10@xmath14@xmath9 , at typical growth conditions . a structural defect can , however , influence the electronic levels of 10 surrounding molecules even if these molecules are in the perfect crystal configuration . it is concluded that grain boundaries ( and not in - grain defects ) are the most prominent cause of structural defects in pentacene thin films.@xcite on the other hand , an experimental study identifies pentacene molecules that are displaced slightly out of the molecular layers that make up the crystals.@xcite by means of high impedance scanning tunneling microscopy ( stm ) , specific defect islands were detected in pentacene films with monolayer coverage . within the defect islands , the pentacene molecules are displaced up to 2.5 along the long molecular axis out of the pentacene layer with a broad distribution in the magnitude of the displacements . electronic structural calculations show that the displaced molecules lead to traps for both electrons and holes . the maximum displacement of the pentacene molecules as seen by stm is 2.5 and this corresponds to a maximum trap depth of 0.1ev.@xcite the best method to produce crystals of small molecule semiconductors includes a zone refinement step in the purification procedure ( ref . @xcite , p. 224 ) . even such crystals still have a considerable impurity content . anthracene , for example , still has an impurity content of 0.1ppm in the best crystals , which corresponds to a volume density of @xmath15@xmath9 ( ref . @xcite , p. 224 ) . zone refinement produces organic materials of much higher purity as compared to purification by sublimation.@xcite however , zone refinement can only be applied if the material can be molten without a chemical reaction or a decomposition to occur . this is not possible for many materials including tetracene or pentacene . thus , much higher impurity concentrations are expected e.g. in tetracene or pentacene.@xcite an experimental study indicates that in tetracene single crystals the charge carrier mobility is limited by chemical impurities rather than by structural defects.@xcite the ability of a chemical impurity to act as trap depends on its accessible energy levels . in a simplistic view a hole trap forms if the ionization energy of the impurity is smaller than the ionization energy of the host material.@xcite we focus on pentacene , and the center ring of the pentacene molecule is expected to be the most reactive.@xcite an important impurity is thus thought to be the oxidized pentacene species 6,13-pentacenequinone , where two oxygen atoms form double bonds with the carbon atoms at the 6,13-positions . according to theoretical studies , pentacenequinone is expected to lead to states in the band gap of pentacene ( ref . and ) and may predominantly act as scattering center ( ref . ) . repeated purification of pentacene by sublimation can result in very high mobilities in pentacene single crystals.@xcite another common impurity in pentacene is thought to be 6,13-dihydropentacene , where additional hydrogen atoms are bound both at the 6- and at the 13-position.@xcite properties of the gate dielectric s surface such as surface roughness , surface free energy and the presence of heterogenous nucleation sites are expected to play a key role in the growth of small molecule semiconductor films from the vapour phase thus influencing the quality of the films . apart from growth - related effects , the sole presence of the gate dielectric can influence the charge transport in a field - effect transistor especially because the charge is transported in the first few molecular layers within the semiconductor at the interface between the gate dielectric and the semiconductor . thus , also fet s based on single crystals are affected , even laminated ( flip - crystal - type ) sc - fet s . the surface of the gate dielectric contains chemical groups that act as charge carrier traps . the trapping mechanism may be as simple as the one discussed above for chemical impurities . this means that the trapping depends on the specific surface chemistry of the gate dielectric but the ability of certain chemical groups on the surface of the gate dielectric to cause traps will also depend on the nature of the small molecule semiconductor . the trapping mechanism can also be seen as a reversible or irreversible electrochemical reaction driven by the application of a gate voltage.@xcite chemical groups on the surface of the gate dielectric certainly affect the transport of electrons in n - type field - effect transistors.@xcite water adsorbed on the gate dielectric may dissociate and react with pentacene . one possible reaction product is 6,13-dihydropentacene . the number of impurities that are formed can depend on the electrochemical potential and would thus increase as the gate voltage is ramped up in a field - effect transistor.@xcite it has also been suggested that water causes traps by reacting with the surface of the gate dielectric . water on a sio@xmath11 gate dielectric with a large number of silanol groups ( -si - oh ) causes the formation of sio@xmath16-groups and the latter groups can act as hole traps.@xcite in addition to chemical reactions involving water , water molecules may act as traps themselves just like any other chemical impurity . a polar impurity molecule leads to an electric field dependent trap depth though.@xcite even if a polar impurity does not lead to a positive trap depth , its dipole moment modifies the local value of the polarization energy since we have highly polarizable @xmath2-orbitals in organic semiconductors . this results in traps in the vicinity of the water molecules.@xcite the net effect is a significant broadening of the trap dos at the insulator - semiconductor interface.@xcite it has been suggested that the polarity of the gate dielectric surface impedes the charge transport as described in the following.@xcite a more polar surface has randomly oriented dipoles which lead to a modification of the local polarization energy within the semiconductor and thus to a change of the site energies . as in the case of polar water molecules , this brings a broadening of the trap dos . the dependence of the mobility on the dielectric constant of the gate dielectric has been observed with conjugated polymers ( refs . and ) and with rubrene single crystal field - effect transistors.@xcite more recently , a model has been put forward to quantitatively study the effect of randomly oriented static dipole moments within the gate dielectric.@xcite the model predicts a significant broadening of the trap dos within the first 1 nm at the insulator - semiconductor interface and can explain the dependence of the mobility on the dielectric constant of the gate dielectric quantitatively.@xcite in this context , it is important to realize that surfaces with a low polarity have a low surface free energy and are thus expected to have a high water repellency as well . clearly , the high water repellency leads to a a reduced amount of water at the critical insulator - semiconductor interface.@xcite as already mentioned in sec . [ section - chargetransport ] , the thermal fluctuations of the intermolecular transfer integral may be of the same order of magnitude as the transfer integral itself in small molecule semiconductors such as pentacene , anthracene or rubrene.@xcite a theoretical study has pointed out that the large fluctuations in the transfer integral result in a tail of trap states extending from the valence band edge into the gap.@xcite moreover , the band tail is temperature - dependent . the extension of the band tail increases with temperature due to an increase in the thermal motion of the molecules.@xcite for pentacene the theoretical study predicts exponential band tails @xmath17 with @xmath18mev at @xmath19k and @xmath20mev at @xmath21k . some experimental evidence suggests , that trap states due to the thermal motion of the molecules play a role in samples with a low trap density.@xcite field - effect transistors are often used to measure the trap dos . the trap dos can be calculated from the measured transfer characteristics with various analytical methods or by simulating the transistor characteristics with a suitable computer program . in sec . [ section - comparison ] we quantitatively compare the trap dos from various studies in the literature with our data . since in these studies different methods were used to derive the trap dos , it is necessary to ensure that all these methods lead to comparable results . analytical methods that are relevant for the comparison in sec . [ section - comparison ] were developed by _ lang et al . _ ( ref . ) , _ horowitz et al . _ ( ref . ) , _ fortunato et al . _ ( ref . ) , _ grnewald et al . _ ( ref . ) and _ kalb et al . _ ( method i : ref . , method ii : ref . ) . the trap dos as calculated with the different methods from the same set of measured data is shown in fig . [ figure - compmethods].@xcite clearly , the choice of the method to calculate the trap dos has a considerable effect on the final result . the graph also contains the trap dos obtained by simulating the transistor characteristics with a computer program developed by _ oberhoff et al . _ and this may be seen as the most accurate trap dos.@xcite the analytical results agree to a varying degree with the simulation . method i by _ kalb et al . _ gives a good estimate of the slope of the trap dos but overestimates the magnitude of the trap densities which can be attributed to a neglect of the temperature - dependence of the band mobility @xmath5.@xcite for the method by _ lang et al . _ , the effective accumulation layer thickness @xmath22 is assumed to be constant ( gate - voltage independent ) . an effective accumulation layer thickness of @xmath23 nm is generally used . the method by _ lang et al . _ leads to a significant underestimation of the slope of the trap dos and , with an effective accumulation layer thickness of @xmath23 nm , to a significant underestimation of the trap densities very close to the valence band edge ( vb ) . these deviations need to be considered in the following analysis . ( color online ) spectral density of localized states in the band gap ( trap dos ) of pentacene as calculated with several methods from the same set of transistor characteristics . the transistor characteristics were measured with a pentacene - based tft employing a polycrystalline pentacene film and a sio@xmath11 gate dielectric . the energy is relative to the valence band edge ( vb ) . the choice of the method to calculate the trap dos has a considerable effect on the final result . adapted from ref . . on the one hand , trap dos data were taken from publications by various groups that are active in the field . the data were extracted by using the dagra software which allows to convert plotted data e.g. in the figures of pdf files into data columns . on the other hand , we also add to the following compilation unpublished data from experiments in our laboratory . we focus on the trap dos in small molecule semiconductors . since almost no data exists in the literature on the trap dos in solution - processed small molecule semiconductors , we almost exclusively deal with the trap dos in vapour - deposited small molecules . more specifically , the data are from tft s which were made by evaporating the small molecule semiconductors in high vacuum . the single crystals for the sc - fet s and for the measurements of the bulk trap dos were grown by physical vapour transport ( sublimation and recrystallization in a stream of an inert carrier gas).@xcite moreover , the electron trap dos close to the conduction band edge ( cb ) has rarely been studied so far in small molecule semiconductors and , with one exception , we are dealing with the hole trap dos in small molecule semiconductors in the following . ( color online ) trap dos from thin - film transistors ( tft s ) made with small molecule organic semiconductors . several different semiconductors , gate dielectrics and methods to calculate the trap dos were used . some details of the tft fabrication are listed in table [ table - tfts ] along with the method that was used to calculate the trap dos and the reference of the data . small molecule semiconductors tend to be crystalline and can be obtained in high purity . typical materials are oligomers such as pentacene or sexithiophene but this class of materials also includes e.g. rubrene or c@xmath3 . the molecules interact by weak van der waals - type forces and have loosely bound @xmath2-electrons which are the source of charge conduction . ] [ cols= " < , < , < , < , < , < , < , < , < , < " , ] it is interesting to compare the trap dos in small molecule organic semiconductors with the trap dos in hydrogenated amorphous silicon ( a - si : h ) and polycrystalline silicon ( poly - si ) . for a - si : h , the mobility edge picture is used to describe the charge transport and trap states have been studied extensively.@xcite the distribution of bond angles and interatomic distances in amorphous silicon ( a - si ) around a mean value leads to a blurred band edge , i.e. to band tails extending into the gap . the trap densities at a given energy reflect the volume density of certain bond angles and interatomic distances . for example , a rather large deviation from the atomic configuration in the crystalline phase ( from the mean value in the amorphous phase ) leads to traps with energies far from the band edge . these traps are present with rather low densities since small deviations are much more likely to occur . in addition , we may have dangling bonds in a - si acting as traps . it is well known , that hydrogenation of a - si leads to a reduction in the trap dos due to a passivation of dangling bonds with hydrogen.@xcite for fig . [ figure - asi ] we have selected typical trap dos data from samples with small molecule semiconductors ( data from fig . [ figure - together ] ) . the data are compared with a typical hole trap dos in a - si : h ( dash - dotted green lines ) and with a typical electron trap dos in a - si : h ( full green line ) . details of the data are given in table [ table - si ] . in fig . [ figure - asi ] we see that the hole trap dos in tft s with small molecule semiconductors such as pentacene is surprisingly similar to the hole trap dos in a - si : h . both the magnitude of the trap densities and the slope of the distribution are very similar . finally , in fig . [ figure - polysi ] we similarly compare data from small molecule semiconductors with a typical hole trap dos in poly - si ( dash - dotted blue line ) and an electron trap dos in poly - si ( full blue line ) . the trap distribution is less steep in poly - si as compared to the trap dos in organic thin films such that we have higher trap densities far from the transport band edge . we compared the hole trap dos ( trap densities as a function of energy relative to the valence band edge ) in various samples of small molecule organic semiconductors as derived from electrical characteristics of organic field - effect transistors and space - charge - limited current measurements . in particular , we distinguish between the trap dos in thin - film transistors with vacuum - evaporated small molecules , the trap dos in organic single crystal field - effect transistors and the trap dos in the bulk of single crystals grown by physical vapour transport . a comparison of all data strongly suggests that structural defects at grain boundaries tend to be the main cause of `` fast '' traps in tft s made with vacuum - evaporated pentacene and supposedly also in related materials . moreover , we argue that dipolar disorder due to the presence of the gate dielectric and , more specifically , water adsorbed on the gate dielectric surface is the main cause of traps in sc - fet s made with a semiconductor such as rubrene . one of the most important findings is that bulk trap densities can be reached in organic field - effect transistors if the organic semiconductor has few structural defects ( e.g. single crystals ) and if a highly hydrophobic gate dielectric is used . the highly hydrophobic cytop@xmath24 fluoropolymer gate dielectric essentially is a gate dielectric that does not cause traps at the insulator - semiconductor interface and thus leads to organic field - effect transistors with outstanding performance . the trap dos in tft s with small molecule semiconductors is very similar to the trap dos in hydrogenated amorphous silicon . this is surprising due to the very different nature of polycrystalline thin films made of small molecule semiconductors with van der waals - type interaction on the one hand and covalently bound amorphous silicon on the other hand . although several important conclusions can be drawn from the extensive data it is clear that the present picture is not complete . more systematic studies are necessary to consolidate and complete the understanding of the trap dos in organic semiconductors and organic semiconductor devices . the present compilation may serve as a guide for future studies .

we show that it is possible to reach one of the ultimate goals of organic electronics : producing organic field - effect transistors with trap densities as low as in the bulk of single crystals . we studied the spectral density of localized states in the band gap ( trap dos ) of small molecule organic semiconductors as derived from electrical characteristics of organic field - effect transistors or from space - charge - limited - current measurements . this was done by comparing data from a large number of samples including thin - film transistors ( tft s ) , single crystal field - effect transistors ( sc - fet s ) and bulk samples . the compilation of all data strongly suggests that structural defects associated with grain boundaries are the main cause of `` fast '' hole traps in tft s made with vacuum - evaporated pentacene . for high - performance transistors made with small molecule semiconductors such as rubrene it is essential to reduce the dipolar disorder caused by water adsorbed on the gate dielectric surface . in samples with very low trap densities , we sometimes observe a steep increase of the trap dos very close ( @xmath0ev ) to the mobility edge with a characteristic slope of @xmath1mev . it is discussed to what degree band broadening due to the thermal fluctuation of the intermolecular transfer integral is reflected in this steep increase of the trap dos . moreover , we show that the trap dos in tft s with small molecule semiconductors is very similar to the trap dos in hydrogenated amorphous silicon even though polycrystalline films of small molecules with van der waals - type interaction on the one hand are compared with covalently bound amorphous silicon on the other hand . although important conclusions can already be drawn from the existing data , more experiments are needed to complete the understanding of the trap dos near the band edge in small molecule organic semiconductors .

1002.1611

recent experiments@xcite on conductance fluctuations and weak - localization effects in quantum dots have stimulated theoretical work@xcite on phase - coherent conduction through cavities in which the classical electron motion can be regarded as chaotic . if the capacitance of the quantum dot is large enough , a description in terms of non - interacting electrons is appropriate ( otherwise the coulomb blockade becomes important@xcite ) . for an isolated chaotic cavity , it has been conjectured and confirmed by many examples that the statistics of the hamiltonian @xmath2 agrees with that of the gaussian ensemble of random - matrix theory.@xcite if the chaotic behavior is caused by impurity scattering , the agreement has been established by microscopic theory : both the gaussian ensemble and the ensemble of hamiltonians with randomly placed impurities are equivalent to a certain non - linear @xmath12-model.@xcite transport properties can be computed by coupling @xmath13 eigenstates of @xmath2 to @xmath1 scattering channels.@xcite since @xmath14 this construction introduces a great number of coupling parameters , whereas only a few independent parameters determine the statistics of the scattering matrix @xmath0 of the system.@xcite for transport properties at zero temperature and infinitesimal applied voltage , one only needs to know @xmath0 at the fermi energy @xmath15 , and an approach which starts directly from the ensemble of scattering matrices at a given energy is favorable . following up on earlier work on chaotic scattering in billiards,@xcite two recent papers@xcite have studied the transport properties of a quantum dot under the assumption that @xmath0 is distributed according to dyson s circular ensemble.@xcite in refs . [ barangerm ] and [ jpb ] the coupling of the quantum dot to the external reservoirs was assumed to occur via ballistic point contacts ( or `` ideal leads '' ) . the extension to coupling via tunnel barriers ( non - ideal leads ) was considered in ref . [ brouwerb ] . in all cases complete agreement was obtained with results which were obtained from the hamiltonian approach.@xcite this agreement calls for a general demonstration of the equivalence of the scattering matrix and the hamiltonian approach , for arbitrary coupling of the quantum dot to the external reservoirs . it is the purpose of this paper to provide such a demonstration . a proof of the equivalence of the gaussian and circular ensembles has been published by lewenkopf and weidenmller,@xcite for the special case of ideal leads . the present proof applies to non - ideal leads as well , and corrects a subtle flaw in the proof of ref . [ lewenkopfweidenmueller ] for the ideal case . the circular ensemble of scattering matrices is characterized by a probability distribution @xmath16 which is constant , that is to say , each unitary matrix @xmath0 is equally probable . as a consequence , the ensemble average @xmath17 is zero . this is appropriate for ideal leads . a generalization of the circular ensemble which allows for non - zero @xmath17 ( and can therefore be applied to non - ideal leads ) has been derived by mello , pereyra , and seligman,@xcite using a maximum entropy principle . the distribution function in this generalized circular ensemble is known in the mathematical literature@xcite as the poisson kernel , @xmath18 here @xmath19 is the symmetry index of the ensemble of scattering matrices : @xmath20 or @xmath21 in the absence or presence of a time - reversal - symmetry breaking magnetic field ; @xmath22 in zero magnetic field with strong spin - orbit scattering . ( in refs . [ mellopereyraseligman ] and [ melloleshouches ] only the case @xmath20 was considered . ) one verifies that @xmath23 for @xmath24 . ( [ mainres ] ) was first recognized as a possible generalization of the circular ensemble by krieger,@xcite for the special case that @xmath17 is proportional to the unit matrix . in this paper we present a microscopic justification of the poisson kernel , by deriving it from an ensemble of random hamiltonians which is equivalent to an ensemble of disordered metal grains . for the hamiltonian ensemble we can use the gaussian ensemble , or any other ensemble to which it is equivalent in the limit @xmath8.@xcite ( the microscopic justification of the gaussian ensemble only holds for @xmath8 . ) for technical reasons , we use a lorentzian distribution for the hamiltonian ensemble , which in the limit @xmath8 can be shown to be equivalent to the usual gaussian distribution . the technical advantage of the lorentzian ensemble over the gaussian ensemble is that the equivalence to the poisson kernel holds for arbitrary @xmath9 , and does not require taking the limit @xmath8 . the outline of this paper is as follows : in sec . [ sec3 ] the usual hamiltonian approach is summarized , following ref . [ vwz ] . in sec.[sec2 ] , the lorentzian ensemble is introduced . the eigenvalue and eigenvector statistics of the lorentzian ensemble are shown to agree with the gaussian ensemble in the limit @xmath8 . in sec . [ sec4 ] we then compute the entire distribution function @xmath16 of the scattering matrix from the lorentzian ensemble of hamiltonians , and show that it agrees with the poisson kernel ( [ mainres ] ) for arbitrary @xmath9 . in sec . [ sec5 ] the poisson kernel is shown to describe a quantum dot which is coupled to the reservoirs by means of tunnel barriers . we conclude in sec . the hamiltonian approach@xcite starts with a formal division of the system into two parts , the leads and the cavity ( see fig . [ fig1]a ) . the hamiltonian of the total system is represented in the following way : let the set @xmath25 represent a basis of scattering states in the lead at the fermi energy @xmath15 ( @xmath26 ) , with @xmath1 the number of propagating modes at @xmath15 . the set of bound states in the cavity is denoted by @xmath27 ( @xmath28 ) . we assume @xmath9 . the hamiltonian @xmath29 is then given by@xcite @xmath30 form a hermitian @xmath3 matrix @xmath2 , with real ( @xmath20 ) , complex ( @xmath31 ) , or real quaternion ( @xmath22 ) elements . the coupling constants @xmath32 form a real ( complex , real quaternion ) @xmath33 matrix @xmath34 . the @xmath35 scattering matrix @xmath36 associated with this hamiltonian is given by @xmath37 for @xmath7 the matrix @xmath0 is respectively unitary symmetric , unitary , and unitary self - dual . usually one assumes that @xmath2 is distributed according to the gaussian ensemble , @xmath38 with @xmath39 a normalization constant and @xmath5 an arbitrary coefficient which determines the density of states at @xmath15 . the coupling matrix @xmath34 is fixed . notice that @xmath40 is invariant under transformations @xmath41 where @xmath42 is orthogonal ( @xmath20 ) , unitary ( @xmath31 ) , or symplectic ( @xmath22 ) . this implies that @xmath16 is invariant under transformations @xmath43 , so that it can only depend on the invariant @xmath44 . the ensemble - averaged scattering matrix @xmath17 can be calculated analytically in the limit @xmath8 , at fixed @xmath1 , @xmath15 , and fixed mean level spacing @xmath45 . the result is@xcite @xmath46 it is possible to extend the hamiltonian ( [ hamham ] ) to include a `` background '' scattering matrix @xmath47 which does not couple to the cavity.@xcite the matrix @xmath47 is symmetric for @xmath48 and can be decomposed as @xmath49 , where the matrix @xmath50 is orthogonal and @xmath51 is real and diagonal . in the limit @xmath52 , the average scattering matrix @xmath17 is now given by@xcite @xmath53 lewenkopf and weidenmller@xcite used this extended version of the theory to relate the gaussian and circular ensembles , for @xmath54 and @xmath24 . their argument is based on the assumption that eq.([sbarsupsymext ] ) can be inverted , to yield @xmath44 and @xmath47 as a function of @xmath17 . then @xmath55 is fully determined by @xmath56 ( and does not require separate knowledge of @xmath44 and @xmath47 ) . under the transformation @xmath57 ( with @xmath42 an arbitrary unitary matrix ) , @xmath17 is mapped to @xmath58 , which implies @xmath59 for @xmath24 one finds that @xmath16 is invariant under transformations @xmath60 , so that @xmath16 must be constant ( circular ensemble ) . there is , however , a weak spot in this argument : equation ( [ sbarsupsymext ] ) can _ not _ be inverted for the crucial case @xmath24 . it is only possible to determine @xmath44 , not @xmath47 . this is a serious objection , since @xmath47 is not invariant under the transformation @xmath57 , and one can not conclude that @xmath23 for @xmath24 . we have not succeeded in repairing the proof of ref . [ lewenkopfweidenmueller ] for @xmath24 , and instead present in the following sections a different proof ( which moreover can be extended to non - zero @xmath17 ) . a situation in which the cavity is coupled to @xmath61 reservoirs by @xmath61 leads , having @xmath62 scattering channels ( @xmath63 ) each , can be described in the framework presented above by combining the @xmath61 leads formally into a single lead with @xmath64 scattering channels . scattering matrix elements between channels in the same lead correspond to reflection from the cavity , elements between channels in different leads correspond to transmission . in this notation , the landauer formula for the conductance @xmath65 of a cavity with two leads ( fig . [ fig1]b ) takes the form @xmath66 [ sec2 ] for technical reasons we wish to replace the gaussian distribution ( [ gaussens ] ) of the hamiltonians by a lorentzian distribution , @xmath67 where @xmath5 and @xmath6 are parameters describing the width and center of the distribution , and @xmath39 is a normalization constant independent of @xmath5 and @xmath6 . the symmetry parameter @xmath19 indicates whether the matrix elements of @xmath2 are real [ @xmath20 , lorentzian orthogonal ensemble ( @xmath68oe ) ] , complex [ @xmath31 , lorentzian unitary ensemble ( @xmath68ue ) ] , or real quaternion [ @xmath22 , lorentzian symplectic ensemble ( @xmath68se ) ] . ( we abbreviate `` lorentzian '' by a capital lambda , because the letter @xmath69 is commonly used to denote the laguerre ensemble . ) the replacement of ( [ gaussens ] ) by ( [ lorens ] ) is allowed because the eigenvector and eigenvalue distributions of the gaussian and the lorentzian ensemble are equal on a fixed energy scale , in the limit @xmath8 at a fixed mean level spacing @xmath45 . the equivalence of the eigenvector distributions is obvious : the distribution of @xmath2 depends solely on the eigenvalues for both the lorentzian and the gaussian ensemble , so that the eigenvector distribution is uniform for both ensembles . in order to prove the equivalence of the distribution of the eigenvalues @xmath70 ( energy levels ) , we compare the @xmath61-level cluster functions @xmath71 for both ensembles . the general definition of the @xmath72 s is given in ref . [ mehta ] . the first two @xmath72 s are defined by @xmath73 the brackets @xmath74 denote an average over the ensemble . the cluster functions in the gaussian ensemble are known for arbitrary @xmath61,@xcite for the lorentzian ensemble we compute them below . from eq . ( [ lorens ] ) one obtains the joint probability distribution function of the eigenvalues , @xmath75 we first consider the case @xmath76 , @xmath77 . we make the transformation @xmath78 the eigenvalues @xmath79 of the unitary matrix @xmath0 are related to the energy levels @xmath80 by @xmath81 the probability distribution of the eigenphases follows from eqs.([lorense ] ) and ( [ eigenphaseigenval1 ] ) , @xmath82 this is precisely the distribution of the eigenphases in the circular ensemble . the cluster functions in the circular ensemble are known.@xcite the @xmath61-level cluster functions @xmath83 in the lorentzian ensemble are thus related to the @xmath61-level cluster functions @xmath84 in the circular ensemble by @xmath85 for @xmath86 one finds the level density @xmath87 independent of @xmath88 . for @xmath89 one finds the pair - correlation function @xmath90 eq . ( [ pairlor ] ) holds for @xmath31 . the expressions for @xmath91 are more complicated . the @xmath61-level cluster functions for arbitrary @xmath5 and @xmath6 can be found after a proper rescaling of the energies . ( [ kappae1 ] ) generalizes to @xmath92 the large-@xmath13 limit of the @xmath72 s is defined as @xmath93 for both the gaussian and the lorentzian ensembles , the mean level spacing @xmath45 at the center of the spectrum in the limit @xmath8 is given by @xmath94 . therefore , the relevant limit @xmath95 at fixed level spacing is given by @xmath8 , @xmath96 , @xmath94 fixed for both ensembles . equation ( [ lorclufunc ] ) allows us to relate the @xmath97 s in the lorentzian and circular ensembles , @xmath98 it is known that the cluster functions @xmath99 in the circular ensemble are equal to the cluster functions @xmath100 in the gaussian ensemble.@xcite equation ( [ clusterlc ] ) therefore shows that the lorentzian and the gaussian ensembles have the same cluster functions in the large-@xmath13 limit . the technical reason for working with the lorentzian ensemble instead of with the gaussian ensemble is that the lorentzian ensemble has two properties which make it particularly easy to compute the distribution of the scattering matrix . the two properties are : + * property 1 : * if @xmath2 is distributed according to a lorentzian ensemble with width @xmath5 and center @xmath6 , then @xmath101 is again distributed according to a lorentzian ensemble , with width @xmath102 and center @xmath103 . + * property 2 : * if the @xmath3 matrix @xmath2 is distributed according to a lorentzian ensemble , then every @xmath35 submatrix of @xmath2 obtained by omitting @xmath104 rows and the corresponding columns is again distributed according to a lorentzian ensemble , with the same width and center . + the proofs of both properties are essentially contained in ref . [ hua ] . in order to make this paper self - contained , we briefly give the proofs in the appendix . the general relation between the hamiltonian @xmath2 and the scattering matrix @xmath0 is given by eq . ( [ sheq ] ) . after some matrix manipulations , it can be written as @xmath105 we can write the coupling matrix @xmath34 as @xmath106 where @xmath42 is an @xmath3 orthogonal ( @xmath20 ) , unitary ( @xmath31 ) , or symplectic ( @xmath22 ) matrix , @xmath107 is an @xmath35 matrix , and @xmath108 is an @xmath33 matrix with all elements zero except @xmath109 , @xmath110 . substitution into eq . ( [ sheq ] ) gives @xmath111 where we have defined @xmath112 . we assume that @xmath2 is a member of the lorentzian ensemble , with width @xmath5 and center @xmath113 . then the matrix @xmath114 is also a member of the lorentzian ensemble , with width @xmath5 and center @xmath15 . property 1 implies that @xmath115 is distributed according to a lorentzian ensemble with width @xmath116 and center @xmath117 . orthogonal ( unitary , symplectic ) invariance of the lorentzian ensemble implies that @xmath118 has the same distribution as @xmath119 . using property 2 we then find that @xmath120 [ being an @xmath35 submatrix of @xmath121 is distributed according to the same lorentzian ensemble ( width @xmath122 and center @xmath123 ) . we now compute the distribution of the scattering matrix , first for a special coupling , then for the general case . first we will consider the special case that @xmath124 is proportional to the unit matrix . the relation ( [ snn ] ) between the @xmath0 and @xmath120 is then @xmath125 thus the eigenvalues @xmath126 of @xmath120 and @xmath79 of @xmath0 are related via @xmath127 since transformations @xmath128 ( with arbitrary orthogonal , unitary , or symplectic @xmath35 matrix @xmath42 ) leave @xmath129 invariant , @xmath16 is also invariant under @xmath130 . so @xmath16 can only depend on the eigenvalues @xmath79 of @xmath0 . the distribution of the @xmath131 s is [ cf . ( [ lorense ] ) ] @xmath132 from eqs . ( [ eigenphaseigenval ] ) and ( [ eigenvaltildeh ] ) we obtain the probability distribution of the @xmath133 s , [ ke ] @xmath134 eq . ( [ ke ] ) implies that @xmath16 has the form of a poisson kernel , @xmath135 the average scattering matrix @xmath17 being given by @xmath136 now we turn to the case of arbitrary coupling matrix @xmath137 . we denote the scattering matrix at coupling @xmath137 by @xmath0 , and denote the scattering matrix at the special coupling ( [ wspecial ] ) by @xmath47 . the relation between @xmath0 and @xmath47 is @xmath138 where we abbreviated [ stransformw ] @xmath139 the symmetry of the coupling matrix @xmath137 is reflected in the symmetry of the @xmath140 matrix @xmath141 which is unitary symmetric ( @xmath20 ) , unitary ( @xmath31 ) or unitary self - dual ( @xmath22 ) . the probability distribution @xmath142 of @xmath47 is given by eq . ( [ p0distr ] ) . the distribution @xmath143 of @xmath0 follows from [ probrel ] @xmath144 where the jacobian @xmath145 is the ratio of infinitesimal volume elements around @xmath47 and @xmath0 . this jacobian is known,@xcite @xmath146 after expressing @xmath47 in terms of @xmath0 by means of eq . ( [ s0froms ] ) , we find that @xmath16 is given by the same poisson kernel as eq . ( [ p0distr ] ) , but with a different @xmath17 , @xmath147 in the limit @xmath8 at fixed level spacing @xmath148 , eq . ( [ sbarlorensarbw ] ) simplifies to @xmath149 the extended version of the hamiltonian approach which includes a background scattering matrix @xmath47 can be mapped to the case without background scattering matrix by a transformation @xmath150 ( @xmath20 ) , @xmath151 ( @xmath31 ) , or @xmath152 ( @xmath153 ) , where @xmath42 and @xmath39 are unitary matrices.@xcite ( @xmath154 is the transposed of @xmath42 , @xmath155 is the dual of @xmath42 . ) the poisson kernel is covariant under such transformations,@xcite i.e. it maps to a poisson kernel with @xmath156 ( @xmath20 ) , @xmath157 ( @xmath31 ) , or @xmath158 ( @xmath22 ) . as a consequence , the distribution of @xmath0 is given by the poisson kernel for arbitrary coupling matrix @xmath34 and background scattering matrix @xmath47 . this proves the general equivalence of the poisson kernel and the lorentzian ensemble of hamiltonians . the circular ensemble of scattering matrices is appropriate for a chaotic cavity which is coupled to the leads by means of ballistic point contacts ( `` ideal '' leads ) . in this section we will demonstrate that the generalized circular ensemble described by the poisson kernel is the appropriate ensemble for a chaotic cavity which is coupled to the leads by means of tunnel barriers ( `` non - ideal '' leads ) . the system considered is shown schematically in fig . we assume that the segment of the lead between the tunnel barrier and the cavity is long enough , so that both the @xmath35 scattering matrix @xmath47 of the cavity and the @xmath159 scattering matrix @xmath160 of the tunnel barrier are well - defined . the scattering matrix @xmath47 has probability distribution @xmath161 of the circular ensemble , whereas the scattering matrix @xmath160 is kept fixed . we decompose @xmath160 in terms of @xmath35 reflection and transmission matrices , @xmath162 the @xmath35 scattering matrix @xmath0 of the total system is related to @xmath47 and @xmath160 by @xmath163 this relation has the same form as eq . ( [ sfroms0 ] ) . we can therefore directly apply eq . ( [ probrel ] ) , which yields @xmath164 hence @xmath0 is distributed according to a poisson kernel , with @xmath165 . in conclusion we have established by explicit computation the equivalence for @xmath9 of a generalized circular ensemble of scattering matrices ( described by a poisson kernel ) and an ensemble of @xmath3 hamiltonians with a lorentzian distribution . the lorentzian and gaussian distributions are equivalent in the large-@xmath13 limit . moreover , the gaussian hamiltonian ensemble and the microscopic theory of a metal particle with randomly placed impurities give rise to the same non - linear @xmath12-model.@xcite altogether , this provides a microscopic justification of the poisson kernel in the case that the chaotic motion in the cavity is caused by impurity scattering . for the case of a ballistic chaotic cavity , a microscopic justification is still lacking . the equivalence of the poisson kernel and an arbitrary hamiltonian ensemble can be reformulated in terms of a central limit theorem : the distribution of a submatrix of @xmath101 of fixed size @xmath1 tends to a lorentzian distribution when @xmath8 , independent of the details of the distribution of @xmath2 . a central limit theorem of this kind for @xmath166 has previously been formulated and proved by mello.@xcite this work was motivated by a series of lectures by p. a. mello at the summerschool in les houches on `` mesoscopic quantum physics '' . discussions with c. w. j. beenakker , k. frahm , p. a. mello , and h. a. weidenmller are gratefully acknowledged . this research was supported by the `` stichting voor fundamenteel onderzoek der materie '' ( fom ) and by the `` nederlandse organisatie voor wetenschappelijk onderzoek '' ( nwo ) . the two proofs given below are adapted from ref . [ hua ] . the matrix @xmath2 and its inverse @xmath101 have the same eigenvectors , but reciprocal eigenvalues . therefore , property 1 of the lorentzian ensemble is proved by showing that the distribution of the eigenvalues of @xmath101 is given by eq . ( [ lorense ] ) , with the substitutions @xmath167 and @xmath168 . this is easily done , @xmath169 \nonumber \\ & = & { 1 \over v } \lambda^{m(\beta m + 2 - \beta)/2 } \prod_{i < j } \left|e_i e_j ( e_i^{-1 } - e_j^{-1 } ) \right|^{\beta } \prod_i \left [ \left({\lambda^2 + ( e_i - \varepsilon)^2}\right)^{-(\beta m + 2 - \beta)/2 } e_i^{2 } \right ] \nonumber \\ & = & { 1 \over v } \lambda^{m(\beta m + 2 - \beta)/2 } \prod_{i < j } \left|e_i^{-1 } - e_j^{-1}\right|^{\beta } \prod_i \left({\lambda^2 e_i^{-2 } + ( 1 - \varepsilon e_i^{-1})^2}\right)^{-(\beta m + 2 - \beta)/2 } \nonumber \\ & = & { 1 \over v } \tilde \lambda^{m(\beta m + 2 - \beta)/2}\prod_{i < j } \left|e_i^{-1 } - e_j^{-1}\right|^{\beta } \prod_i \left({\tilde \lambda^2 + ( e_i^{-1 } - \tilde \varepsilon)^2}\right)^{-(\beta m + 2 - \beta)/2}.\end{aligned}\ ] ] in order to prove property 2 , we may assume that after rescaling of @xmath2 we have @xmath76 , @xmath77 . first consider @xmath170 . in this case , one can write @xmath171 where @xmath65 is the @xmath35 submatrix of @xmath2 whose distribution we want to compute , @xmath172 is a vector , with real ( @xmath20 ) , complex ( @xmath31 ) , or real quaternion elements ( @xmath22 ) , and @xmath173 is a real number . for the successive integrations over @xmath173 and @xmath172 we need two auxiliary results . first , for real numbers @xmath174 , @xmath175 , @xmath176 such that @xmath177 and @xmath178 , and for real @xmath179 we have @xmath180 second , if @xmath181 is a @xmath182-dimensional vector with real components , and if @xmath183 , then @xmath184 since @xmath185 is a quadratic function of @xmath173 , the integral over @xmath173 can now be carried out using eq . ( [ lemma1eq ] ) . the result is : @xmath186 next , we integrate over @xmath172 . we may choose the basis for the @xmath172-vectors so that @xmath187 is diagonal , with diagonal elements @xmath188 . after rescaling of the @xmath172-vectors to @xmath189 one obtains an integral similar to eq . ( [ lemma2eq ] ) , with @xmath190 . the final result is @xmath191 property 2 now follows by induction . notice that eq . ( [ finproof2 ] ) allows us to determine the normalization constant @xmath39 , @xmath192 c. m. marcus , a. j. rimberg , r. m. westervelt , p. f. hopkins , and a. c. gossard , phys . 69 * , 506 ( 1992 ) ; c. m. marcus , r. m. westervelt , p. f. hopkins , and a. c. gossard , phys . b * 48 * , 2460 ( 1993 ) ; chaos * 3 * , 643 ( 1993 ) . m. w. keller , o. millo , a. mittal , d. e. prober , and r. n. sacks , surf . sci . * 305 * , 501 ( 1994 ) . a. m. chang , h. u. baranger , l. n. pfeiffer , and k. w. west , phys . lett . * 73 * , 2111 ( 1994 ) . j. p. bird , k. ishibashi , y. aoyagi , t. sugano , and y. ochiai , preprint . r. a. jalabert , a. d. stone , and y. alhassid , phys . lett . , 3468 ( 1992 ) . h. u. baranger , r. a. jalabert , and a. d. stone , phys . rev.lett . * 70 * , 3876 ( 1993 ) ; chaos * 3 * , 665 ( 1993 ) . v. n. prigodin , k. b. efetov and s. iida , phys . lett . * 71 * , 1230 ( 1993 ) . [ pei ] [ barangerm ] h. u. baranger and p. a. mello , phys.rev . lett . * 73 * , 142 ( 1994 ) . [ jpb ] r. a. jalabert , j .- l . pichard , and c. w. j. beenakker , europhys . * 27 * , 255 ( 1994 ) . z. pluhr , h. a. weidenmller , j. a. zuk , and c. h. lewenkopf , phys . lett . * 73 * , 2115 ( 1994 ) . p. w. brouwer and c. w. j. beenakker , phys . b * 50 * , 11263 ( 1994 ) . [ brouwerb ] e. r. mucciolo , v. n. prigodin , and b. l. altshuler , preprint ( cond - mat/9406048 ) . o. bohigas , m. j. giannoni , and c. schmit , phys . rev.lett . * 52 * , 1 ( 1984 ) ; o. bohigas in _ chaos and quantum physics _ , edited by m .- giannoni , a. voros , and j. zinn - justin ( north - holland , amsterdam , 1991 ) . m. v. berry , proc . r. soc . london a * 400 * , 229 ( 1985 ) . k. b. efetov , adv . phys . * 32 * , 53 ( 1983 ) . j. j. m. verbaarschot , h. a. weidenmller , and m. r. zirnbauer , phys . rep . * 129 * , 367 ( 1985 ) . [ vwz ] s. iida , h. a. weidenmller , and j. zuk , phys . rev . lett . , 583 ( 1990 ) ; ann . * 200 * , 219 ( 1990 ) . a. altland , z. phys . b. * 82 * , 105 ( 1991 ) . c. h. lewenkopf and h. a. weidenmller , ann . * 212 * , 53 ( 1991 ) . [ lewenkopfweidenmueller ] r. blmel and u. smilansky , phys . , 477 ( 1988 ) ; _ ibid . _ * 64 * , 241 ( 1990 ) ; u. smilansky , in _ chaos and quantum physics _ , edited by m .- giannoni , a. voros , and j. zinn - justin ( north - holland , amsterdam , 1991 ) . f. j. dyson , j. math . phys . * 3 * , 140 ( 1962 ) . [ mehta ] m. l. mehta , _ random matrices _ ( academic , new york , 1991 ) . p. a. mello , p. pereyra , and t. h. seligman , ann . ( n.y . ) * 161 * , 254 ( 1985 ) . [ mellopereyraseligman ] p. a. mello , in _ mesoscopic quantum physics _ , edited by e. akkermans , g. montambaux , j .- l . pichard , and j. zinn - justin ( north - holland , amsterdam , to be published ) . [ melloleshouches ] [ hua ] l. k. hua , _ harmonic analysis of functions of several complex variables in the classical domains _ ( amer . math . soc.,providence , 1963 ) . t. j. krieger , ann . * 42 * ( 1967 ) , 375 . g. hackenbroich and h. a. weidenmller , preprint ( cond - mat/9412088 ) . c. mahaux and h. a. weidenmller , _ shell - model approach to nuclear reactions _ ( north - holland , amsterdam , 1969 ) . h. nishioka and h. a. weidenmller , phys . b * 157 * , 101 ( 1985 ) . [ nishiokaweidenmueller ] [ friedmanmello ] w. a. friedman and p. a. mello , ann . phys . * 161 * , 276 ( 1985 ) .

we consider the problem of the statistics of the scattering matrix @xmath0 of a chaotic cavity ( quantum dot ) , which is coupled to the outside world by non - ideal leads containing @xmath1 scattering channels . the hamiltonian @xmath2 of the quantum dot is assumed to be an @xmath3 hermitian matrix with probability distribution @xmath4^{-(\beta m + 2 - \beta)/2}$ ] , where @xmath5 and @xmath6 are arbitrary coefficients and @xmath7 depending on the presence or absence of time - reversal and spin - rotation symmetry . we show that this `` lorentzian ensemble '' agrees with microscopic theory for an ensemble of disordered metal particles in the limit @xmath8 , and that for any @xmath9 it implies @xmath10 is the ensemble average of @xmath0 . this `` poisson kernel '' generalizes dyson s circular ensemble to the case @xmath11 and was previously obtained from a maximum entropy approach . the present work gives a microscopic justification for the case that the chaotic motion in the quantum dot is due to impurity scattering .

cond-mat9501025

in quantum information theory entanglement between parts of a system has been identified as the key resource that can possibly make quantum information processing more powerful than classical information processing . entanglement can also be a resource for long - distance quantum communication or distributed quantum computation , and it is at the heart of some quantum communication protocols . but entanglement is fragile under the influence of environment induced decoherence . all engineering hence thrives to better control and manipulate the quantum information stored in the system while keeping the detrimental effects of decoherence low . in nature , on the other hand , we mostly find less controllable systems , especially if the system size becomes macroscopic as in gases , fluids , solids or even biological systems . since these systems are usually open , noisy systems at possibly high temperatures one expects that environment induced decoherence will erase all entanglement between system degrees of freedom . this reasoning is true except for three cases . first , the environment and its coupling to the system could be special in a way that it creates rather than destroys entanglement . however , it is unlikely to find such an environment in nature where usually thermalization dominates , and we will only briefly touch upon the subject of such environments in this paper . second , if the system has an entangled ground state , as many solid state systems do , its thermal state will be entangled in a certain temperature range above zero by a continuity argument . coupling to a heat bath drives a system into its thermal state . but there is a temperature threshold for the bath above which the thermal state will be unentangled . third , the system might have a built - in entropy drain , meaning that the correlations with the environment are , by one way or another , erased such that the system can re - build entanglement through its quantum mechanical interactions . this entropy drain may even be local to exclude the trivial cases where entanglement is simply pumped " into the system , e.g. , by injecting fresh , entangled bell pairs . in @xcite we proposed such a local entropy drain in form of a reset mechanism , where system particles are randomly replaced by particles in some standard , mixed state of sufficiently low entropy . note that such a mechanism can not create entanglement , on the contrary , it erases any entanglement that might still be present between the particle that is reset and the rest of the system . only the interplay with the system hamiltonian can lead to entanglement in a steady state that is possibly far from thermodynamic equilibrium . this reset mechanism was studied for a toy model with two qubits , where analytic solutions could be obtained . also a multipartite scenario for a ( simplified ) gas model was discussed , and further generalizations were suggested . by gas - type systems we mean systems in which the decoherence processes act locally on the system particles , by strongly coupled systems we mean those where the decoherence processes act globally . to be more precise , local decoherence processes are those , which induce transitions between the eigenstates of the local , free hamiltonian alone , while global decoherence processes induce transitions between eigenstates of the total hamiltonian . in this paper we review the key idea of a reset mechanism but provide more in - depth material than in @xcite . we elaborate on the generalizations suggested in @xcite , namely on the influence of local entropy drains on the dynamics of entanglement and on the steady - state entanglement in gas - type systems as well as in strongly coupled systems . we prove that the master equation describing the evolution of the system coupled to a heat bath and subject to a reset mechanism is of lindblad form and hence generates a completely positive , i.e. , physical map . we analytically solve the master equation for small systems of two spins with special interaction hamiltonians , which enables us to illustrate the main features of the reset mechanism . in particular we show the following . 1 . steady - state entanglement in systems with reset mechanism is different from the entanglement in thermal states . 2 . in strongly coupled systems with constant coupling steady - state entanglement with reset can exist for higher temperatures than the entanglement in the thermal state , which is the steady state without reset . 3 . in gas - type systems steady - state entanglement with reset can exist even for arbitrary temperatures . these features are not due to the specially chosen interaction hamiltonians and decoherence processes . we demonstrate that the above properties are almost independent of both . one can also relax the conditions on the reset states and take mixed states with sufficiently low entropy instead of pure states . finally , a generalization to larger system sizes , possibly even with fluctuating particle numbers , still leads to similar results . hence , the reset mechanism is at the same time simple and generic . we remark that in cavity qed an incoherent generation of entanglement has been proposed , which bears resemblance to our work @xcite . there , an atom couples to two leaky optical cavities and is driven by a white noise field . this incoherent driving can , when the atom is finally traced out , result in entanglement between the cavity modes . entanglement is generated for intermediate cavity damping rates and intensities of the noise field , an effect labeled `` stochastic resonance '' in @xcite . we believe that this effect is more correctly interpreted as an example for a reset mechanism . in a subsequent work @xcite strongly related to @xcite , one single cavity entangles two atoms , giving yet another example for a reset mechanism even closer to the setups of this paper . the reset mechanism is certainly not a preferred way to actively protect entanglement and mostly can not even be compared to such strategies , but , because of its simplicity and generality , there is hope that such a mechanism may ultimately be identified in natural processes leading to an increased understanding whether entanglement can play a role in systems at high temperatures . the paper is organized as follows . we first concentrate on simple models with only two particles ( qubits ) . in section [ gasmodel ] we motivate the description by a master equation , explain in which cases the model is valid , and study several specific hamiltonians and noise channels analytically and others numerically . we also compare entangled steady states resulting from a reset mechanism to entangled steady states resulting from special choices of interaction hamiltonian and decoherence process . we show in section [ stronglycoupledmodel ] that we can find the same features in strongly coupled systems , and we give the conditions to be met by the reset mechanism such that entangled steady states can exist . then , in section [ multipartite ] , we extend the model to include more qubits and discuss the meaning of different kinds of entanglement that we use . finally we give a summary of the results in section [ summary ] . in this chapter we discuss a toy model with only two particles , which we take as spin-@xmath0 systems or qubits for simplicity . the toy model shows all the features that we will later find in larger systems and it has the advantage that we can show many results analytically leading to an increased understanding of the involved processes . we will formulate the equations for an arbitrary number @xmath1 of qubits , so that we can refer to them later in section [ multipartite ] . in a gas particles are weakly coupled in the sense that most of the time they do not considerably interact with each other unless they collide . in the meantime they only feel their local , free hamiltonian and are subject to individual , local decoherence processes , e.g. through interactions with thermal photons ( radiative damping ) . if we pick a subset consisting of @xmath1 gas particles and consider these as the system , collisions with the remaining gas particles are another source of decoherence ( non - radiative damping or dephasing ) . in a master equation that models this gas - type system we replace the original , time - dependent collision hamiltonian by an averaged , time - independent interaction hamiltonian . since the interaction hamiltonian does not modify the energy landscape in this model , the local , radiative decoherence processes tend to drive the system to the thermal state of the free hamiltonian , for which we choose the form @xmath2 we leave the interaction hamiltonian @xmath3 unspecified for the moment . for two qubits , we will often use the ising hamiltonian @xmath4 for analytic discussions , whereas more complicated hamiltonians will be treated numerically . we write the total hamiltonian as @xmath5 such that the master equation is @xmath6+{\cal l}_{\rm{noise}}\rho,\ ] ] where @xmath7 is a liouville operator representing the noise channels . we describe the noise channels by the lindblad operator @xcite @xmath8\nonumber\\ -\frac{b}{2 } s[\sigma_-^{(i)}\sigma_+^{(i)}\rho+\rho\sigma_-^{(i)}\sigma_+^{(i)}-2\sigma_+^{(i)}\rho\sigma_-^{(i)}]-\frac{2c - b}{4}[\rho-\sigma_z^{(i)}\rho\sigma_z^{(i)}]\end{aligned}\ ] ] where @xmath9 and the @xmath10s are pauli operators . parameters @xmath11 and @xmath12 give the decay rate of inversion @xmath13 and polarization @xmath14 under the action of @xmath7 , and @xmath15 $ ] depends on temperature , where @xmath16 corresponds to @xmath17 ( we set the boltzmann and planck constant equal to one ) . the definition of @xmath18 stems from laser physics where inversion occurs corresponding to negative temperatures " . many authors use @xmath19 instead of @xmath18 and @xmath20 instead of @xmath21 . then , no negative temperatures are possible . the noise channel is derived assuming certain approximations , e.g. the markov approximation . note , however , that this is not an essential assumption as we will demonstrate later in an example ( see figure [ spingas ] and related text ) . an important special case of ( [ localnoise ] ) , obtained by setting @xmath22 and @xmath23 , is the dephasing channel @xmath24,\ ] ] well - known especially in its integrated form as a completely positive map @xmath25 with @xmath26 . as with the ising hamiltonian ( [ isinghamiltonian ] ) , we will often use the dephasing channel for analytic discussions because of its simplicity . we are interested in the steady state of this master equation for @xmath27 qubits and the question whether there is entanglement in this state . at this point we simply state the following results since we will later solve a more general master equation that contains equation ( [ gasme ] ) as a special case . as an easy example we start with the ising interaction hamiltonian ( [ isinghamiltonian ] ) . the steady state will be the tensor product of the thermal states of each free hamiltonian @xmath28 since this state commutes with @xmath29 and since dephasing noise does not change the diagonal elements of the density matrix . in conclusion , we have the unentangled steady state @xmath30 when can we hope to find an entangled steady state ? as we know the radiative decoherence processes drive the system into the thermal states of the free hamiltonian . if the interaction hamiltonian can entangle these states we may find an entangled steady state at least for low temperatures . as an example , consider an interaction hamiltonian @xmath31 while the other terms stay the same as in the example above . we set @xmath32 ( no non - dissipative processes ) . the steady - state density matrix is then @xmath33 this density matrix can be entangled . we measure the entanglement between two sets of qubits by the negativity @xcite , which is given with respect to a bipartition @xmath34-@xmath35 as @xmath36 , where @xmath37 means the partial transpose with respect to @xmath34 . for two qubits , we omit the label @xmath34 since there is only one bipartition , and the negativity can assume values between zero ( separable state ) and @xmath0 ( maximally entangled state ) . the reason why we choose the negativity as a measure throughout the paper is that we will use a generalization thereof in the multipartite case where the generalization of other entanglement measures might be hard to compute . the negativity of the state above is @xmath38\left(b^2 + 4 \left(g^2+\omega ^2\right)\right)^{-2}\big\},\end{aligned}\ ] ] which can be larger than zero but will always vanish for high temperatures of the bath , @xmath39 ( see figure [ sxsx ] ) . entanglement as measured by the negativity in the steady state of equation ( [ gasme ] ) with interaction hamiltonian @xmath31 and free hamiltonian @xmath40 . the decoherence processes are given by equation ( [ localnoise ] ) . we choose the decay rate of inversion , @xmath11 , as inverse unit timescale and the other parameters as @xmath41 ( no dephasing noise ) and @xmath42 . the parameter @xmath43 on one of the axes is also measured in units of @xmath11 , whereas @xmath18 is dimensionless.,scaledwidth=70.0% ] we see that the steady state of the master equation ( [ gasme ] ) can be entangled for specially chosen interaction hamiltonians , but only below a certain temperature threshold . furthermore , if the free , local hamiltonian is too strong , i.e. , @xmath44 dominates by far all other parameters , there is also no entanglement . this statement applies to other models involving the hamiltonian ( [ atomichamiltonian ] ) as well , and , accordingly , @xmath44 should have the same order of magnitude as the other parameters . spin gases @xcite are an example for such gas - type systems . a spin gas is a system of quantum spins with stochastic , time - dependent interactions . a physical model of a spin gas is a system of @xmath1 classically moving particles with additional , internal spin degrees of freedom . upon collision , these quantum degrees of freedom interact according to some specified hamiltonian . in @xcite the interaction hamiltonians were chosen locally unitarily equivalent to the ising interaction leading to a description in terms of weighted graph states . hence , in such spin gases , classical kinematics drives the evolution of the quantum state , and also the decoherence of arbitrary probe systems put into the gas and subjected to interactions with it . in general , multiple non - consecutive collisions of particles are possible . the spin gas remembers its whole interaction history , and it provides a microscopic model with _ non - markovian decoherence_. assume that we have two selected gas particles ( e.g. another species ) that we consider as the system , while the other gas particles act as the environment that induces decoherence when interacting with the system particles . the rare interactions between the two system particles happen only during the short times when they collide . in the longer times in between they are not coupled and subject to local decoherence processes , i.e. , interactions with the environment gas particles . the induced decoherence processes are equivalent to dephasing channels ( corresponding to @xmath45 in equation ( [ localnoise ] ) ) . in such a situation , any entanglement between the two qubits that may either have been present initially or have built up on a short time scale will eventually be destroyed by the interactions with the other gas particles @xcite . + _ the spin gas with reset mechanism _ + for the moment , we stick to the toy - model with only to selected gas particles . imagine now that the two particles can , at a certain rate , leave the box in which the gas is contained and are instantly replaced by fresh qubits that are in a standard mixed state with sufficiently low entropy . instead of a replacement of system particles one can equivalently picture a measurement of the particle and a subsequent preparation in this standard state . note that the last step need not be an active procedure but can , e.g. , result from a spontaneous decay to this state . we call both procedures a reset mechanism . certainly , by a reset , we did not introduce entanglement into the system always consisting of two qubits . on the contrary , any entanglement that might have been present between the particle that has left the box and the one that is still inside leads to a description of the latter by a more mixed density matrix ( closer to the identity ) . but the advantage is that we have lowered the local entropy of the system since the new particle has no correlations with the environment . this new particle can then become entangled with the other one on short time scales . we said above that in a spin gas with zero rate of qubit exchange the steady state will not be entangled . for infinite exchange rate the system would always be in a pure ( or a standard separable ) state and there is also no entanglement . if , however , the rate at which the qubits leave the spin gas is in a certain intermediate parameter regime one can hope that there is entanglement in the system on average . here , averaging means taking the mean density matrix of many simulation runs . later , the solutions to master equations are assumed to resemble the evolution of such a mean density matrix and , for explanations of certain ( entanglement ) features , this picture will sometimes be invoked . note that there are also other ansatzes . in @xcite the solution of the master equation represents a smoothed version of a single simulation run , where smoothing is achieved by a time - integration kernel . the solution of the master equation does then not follow the rapid changes of the single density matrix , but sees only the slower changes resolved by a so - called coarse - grained timescale , which is related to the support of the integration kernel . figure [ spingas ] indeed shows entanglement in steady states in a simulation of a spin lattice gas with an ising - type interaction @xcite . steady - state entanglement between two selected qubits as measured by the negativity in a spin lattice gas ( @xmath46 lattice , @xmath47 qubits ) . the probability that a particle is exchanged for a fresh one in one time step of the simulation is plotted on the horizontal axes . hence , the value @xmath48 corresponds to an infinite exchange rate . the special qubits interact @xmath49 times stronger with each other than with the @xmath50 qubits that form the environment , i.e. , as physical particles , they are e.g. of a different kind than the environment particles . the density matrices from which the negativity is derived were averaged over @xmath51 simulation runs . other parameters : initial distance between special qubits : @xmath48 , interaction phase picked up during a collision between them : @xmath52 ; interaction phase for interactions with environment spins : @xmath53 ; for details see @xcite.,scaledwidth=70.0% ] the above scenario with two qubits might seem a little artificial . however , if we extend the setup to systems with more qubits and allow fluctuating particle numbers we can drop the requirement that selected particles must be instantaneously replaced . we can regard the spin - gas particles that do not belong to the system as a hot bath " introducing decoherence and destroying entanglement in the system , and the source of qubits in standard states as a cold bath " that can counteract the effect of the hot bath and preserve entanglement in the steady state . the analogy with a cold bath has some limitations as we will point out in section [ stronglycoupledmodel ] . we will deal with multipartite qubit systems in section [ multipartite ] . for the moment , we will stick to the two qubit system , for which we can find a master equation that reflects the properties of the cold bath " and that we can solve analytically in certain cases . observe that the master equation will again incorporate the markov - assumption , whereas the spin gases are non - markovian systems @xcite and also partly have non - local decoherence processes . the essential features on the other hand will be qualitatively the same in some parameter regimes of the spin gas , where these effects play a minor role . in the next subsection , our goal is to transfer the idea of a reset mechanism from the specific example of a spin gas to a description in form of a master equation , suitable for any spin system . compared to ( [ gasme ] ) , the master equation that models a gas - type system with reset mechanism has an additional term @xmath54 , which we describe as follows . with some probability @xmath55 particle @xmath56 , @xmath57 , is reset during the time interval @xmath58 to some specific state @xmath59 . the other qubits are left in the state @xmath60 . the change in the density matrix during the time @xmath58 due to @xmath54 is @xmath61 . observe that the time interval must be longer than the timescale of any of the involved processes but short enough so that we can replace it by the time differential to obtain , for the rate of change @xmath62 , the following master equation : @xmath63+{\cal l}_{noise}\rho+\sum_{i=1}^n r(|\chi_i\rangle\langle\chi_i|\mbox{tr}_i\rho-\rho)\ ] ] before we proceed to discuss the solution of ( [ generalme ] ) let us establish that the problem is well - defined , i.e. , that the master equation leads to a completely positive map , which is true when the master equation is of lindblad form . for the noise part this is known , so we have to bother only about the reset part . since @xmath64 is local we have to show that each summand is of the form @xmath65+[\sigma_m^{(j)},\rho\sigma_n^{(j)}])$ ] where the @xmath66s are pauli operators and @xmath67 must be positive ( semidefinite ) matrices . we expand @xmath68 and @xmath69 in the @xmath10-basis as @xmath70 and @xmath71 . we insert these expressions into @xmath54 and also into the lindblad - expression , collect the coefficients that belong to each @xmath10-matrix of @xmath68 using the scalar product , and compare the coefficients @xmath72 in each expression , which leads us to a simple linear system of equations for the @xmath73 . solving this system of linear equations we obtain @xmath74 the eigenvalues of @xmath67 are @xmath75 . since we assumed @xmath69 to be pure , we have @xmath76 and eigenvalues @xmath77 . we note that also a mixed reset state would be fine to ensure that the @xmath67 are positive semidefinite . because the sum of positive matrices is a positive matrix , and because we know that the noise terms also have positive @xmath78-matrices , we have shown that the master equation is of lindblad form and preserves the positivity of the density matrix . up to this point , we have modeled interacting , gas - type systems coupled to a noisy environment . we have described a toy model consisting of only two particles by a master equation and compared predictions about the entanglement properties of steady states to simulations with a spin gas as an example for such gas - type systems . we have seen that in general there will be no entanglement in the steady state . we have extended the example of the spin gas by allowing particle exchange with a cold bath " of particles in standard states ( or an equivalent reset mechanism ) , and we have found that steady states of such systems can be entangled . we have derived a master equation that models systems with reset mechanism and have proved that the master equation is of lindblad form . in the following subsection we study the solutions of ( [ generalme ] ) . in principle , the solution to the master equation ( [ generalme ] ) with noise channels as in equation ( [ localnoise ] ) is simple . the equation is of the form @xmath79 with solution @xmath80 mapping @xmath68 to a column vector @xmath12 containing the @xmath81 coefficients @xmath82 of the density matrix and accordingly mapping the liouville operator @xmath83 to a @xmath84-matrix @xmath85 we get the equivalent @xmath81 coupled linear differential equations @xmath86 with solution @xmath87 . to compute the matrix exponential we need the spectral representation of @xmath85 , i.e. , we must solve the eigenproblem @xmath88 . observe that the steady state ( if it exists ) is given by the eigenvector @xmath89 corresponding to the eigenvalue @xmath90 . in the following we will first analyze the solutions for an ising interaction hamiltonian and later generalize to generic cases . we specialize to the ising hamiltonian ( [ isinghamiltonian ] ) as ( effective ) interaction hamiltonian , @xmath91 , and to a specific reset state , namely @xmath92 for both qubits . the free hamiltonian is @xmath93 as before . we can solve the problem through spectral decomposition of the liouville operator @xmath94+{\cal l}_{\rm noise}\rho+{\cal l}_{\rm reset}\rho$ ] , but the expression for the corresponding matrix @xmath95 is very lengthy . one obtains shorter expressions if one does not solve all @xmath81 differential equations at once through the matrix exponential , but step by step , since not all differential equations are coupled . still , we have chosen to move the solution derived in this way to [ appendix ] not to overburden the text with technical details . for illustration , we will restrict the noise to the special case of a dephasing channel ( [ dephasingchannel ] ) in the following . + _ dephasing channel _ + as pointed out above , the solution is given by the spectrum of the total liouville super - operator defined by @xmath96 and its corresponding eigenvectors . in the case of a dephasing channel one obtains the eigenvalues @xmath97 with multiplicities @xmath98 , respectively . the eigenvector belonging to the eigenvalue @xmath99 represents the density matrix in the steady state , and we will come back to this matrix in the next subsection . the full solution , derived by solving the differential equations in a step - by - step manner as explained above , is given in [ appendix ] . to demonstrate the time - evolution of an initial density matrix governed by the master equation we plot the entanglement between the two qubits as a function of time . note that figure [ timedependence ] is based on the analytic solution given in [ appendix ] , and the plot shows how the negativities of different initial states approach the final negativity of the steady state . we choose @xmath100 as unit timescale ( setting @xmath101 ) . the parameters @xmath102 , @xmath43 , @xmath44 here have the fixed values @xmath103 , @xmath104 , @xmath104 , respectively . the initial states are weighted graph states @xcite with density matrix @xmath105 where @xmath106 . through the parameter @xmath107 we can continuously tune the entanglement in the initial state from the product state @xmath92 for @xmath108 to the maximally entangled , bell - equivalent state for @xmath109 . states that are initially highly entangled are first driven into separable states before the steady - state entanglement value is approached from below . vice versa , an initial product state gets highly entangled first , before the steady - state value for the entanglement is reached from above . ( 430,200 ) ( 0,0 ) ( a ) time development of entanglement for different initial states . the @xmath110-axis displays the entanglement as measured by the negativity . the initial states are weighted graph states , characterized by a parameter @xmath107 ( see text ) . the unit timescale is @xmath100 , @xmath111 is measured on this timescale , and the parameters of the master equation are @xmath112 , @xmath113 , @xmath114 . ( b ) cut through the @xmath115d plot at @xmath108 ( orange curve ) and @xmath109 ( blue curve ) . , title="fig:",scaledwidth=60.0% ] ( 182,65 ) ( a ) time development of entanglement for different initial states . the @xmath110-axis displays the entanglement as measured by the negativity . the initial states are weighted graph states , characterized by a parameter @xmath107 ( see text ) . the unit timescale is @xmath100 , @xmath111 is measured on this timescale , and the parameters of the master equation are @xmath112 , @xmath113 , @xmath114 . ( b ) cut through the @xmath115d plot at @xmath108 ( orange curve ) and @xmath109 ( blue curve ) . , title="fig:",scaledwidth=50.0% ] as we said earlier , to display the full analytic solution of ( [ generalme ] ) for more general noise channels would be quite space - consuming . we will not present it since we are primarily interested in the entanglement properties of the steady state . in the following we will discuss these properties for the master equation with general local noise channels . as we have seen , any initial state of the density matrix evolves exponentially fast into a steady state on a characteristic timescale given by the largest non - zero characteristic exponent ( or the smallest in absolute values since they are negative ) . the characteristic timescale thus depends on the parameters of the master equation , too . the steady state is also a function of these parameters . to smooth the presentation , we have again transferred the steady - state solution of ( [ generalme ] ) with ising hamiltonian , local noise channels as in ( [ localnoise ] ) and reset states @xmath116 to [ appendix2 ] . here , we illustrate the solution with a plot . in figure [ qomeequstate ] we choose @xmath117 as unit timescale and fix the values @xmath118 , @xmath119 and the dimensionless parameter @xmath120 . then , for certain values of @xmath102 and @xmath43 , measured in units of @xmath11 , we see entanglement as measured by the negativity in the steady state . entanglement in the steady state of the master equation ( [ generalme ] ) . the entanglement is measured by the negativity , the unit timescale is @xmath117 , the parameters @xmath102 and @xmath43 ( in units of @xmath11 ) are on the axes , while the other parameters are @xmath118 , @xmath119 , and @xmath120.,scaledwidth=70.0% ] to get entangled steady states when @xmath43 becomes small , we have to go to higher reset rates @xmath102 . however , @xmath43 can not be arbitrarily small . there is a weak coupling threshold below which no reset rate can ensure entanglement in the steady state . this threshold depends on the parameters of the decoherence processes , and its existence is intuitively clear . if the decoherence processes simply dominate the entangling processes , then no entanglement can be created by any means . to see this better , we momentarily put @xmath121 for simplification . let us also turn our attention once more to the dephasing channel as a special case of the noise terms of the quantum - optical master equation . then , in the steady state , the anti - diagonal coefficients are all the same , namely @xmath122 , the other off - diagonal elements are @xmath123 , and the diagonal elements all have the values @xmath124 . all other matrix elements are given by the hermiticity of the density matrix . we compute from the above expressions for the density matrix the following analytic expression for the negativity in terms of the parameters @xmath43 ( hamiltonian interaction ) , @xmath125 ( strength of the dephasing channel ) , and @xmath102 ( reset rate ) : @xmath126}\}\ ] ] equation ( [ negativity ] ) contains the full information about the entanglement properties of the two qubits . note that @xmath127 depends , in fact , only on two parameters , @xmath128 and @xmath129 . in fig . [ toymodel](a ) we see a plot of the negativity function n. the key feature is the color - coded region in the @xmath102-@xmath43-plane with steady - state entanglement , where a darker color indicates higher entanglement . the entangled region is bounded by the red line given by one of the roots of the non - trivial part of equation ( [ negativity ] ) . outside of this region , the state is separable ( white area ) . the entangled region approaches asymptotically the straights @xmath130 and @xmath131 plotted black in fig . [ toymodel](a ) . the asymptotic line @xmath130 is independent of @xmath102 and simply tells us that , in the weak coupling regime , decoherence / noise will always triumph over the hamiltonian part that tries to create entanglement as pointed out before . that is , as a necessary condition , we need to be above this threshold to observe entanglement . three lines are marked in the colored , entangled region : 1 . the upper , white line is the maximum in @xmath43-direction ( at constant @xmath102 ) . the lower , white line is the maximum in @xmath102-direction ( at constant @xmath43 ) . the middle line in black is the straight @xmath132 . to this middle line the upper and lower white curves go asymptotically for large @xmath43 , @xmath102 . the global maximum of the negativity is on this middle line at infinity with a value of approximately @xmath133 , about 20% of the maximally possible value . the darkest , most entangled area in our plot has negativity approximately @xmath134 . [ toymodel](b ) shows a cut at @xmath113 through the color - plot . most notable is the existence of a threshold value for @xmath135 above which entanglement is present in the steady state . ( 230,160 ) ( 0,0 ) ( a ) separable states ( white area ) and entangled states ( colored area ) in the @xmath102-@xmath43-plane , where @xmath102 is the rate of the reset process , and @xmath43 is the coupling strength in the ising hamiltonian , and we use @xmath136 as unit timescale . the color encodes the amount of entanglement measured by the negativity : the darker the area , the more entanglement is present . for a discussion of the other lines , please see the text . ( b ) the second plot shows a cut for constant @xmath113.,title="fig:",scaledwidth=60.0% ] ( 220,47 ) ( a ) separable states ( white area ) and entangled states ( colored area ) in the @xmath102-@xmath43-plane , where @xmath102 is the rate of the reset process , and @xmath43 is the coupling strength in the ising hamiltonian , and we use @xmath136 as unit timescale . the color encodes the amount of entanglement measured by the negativity : the darker the area , the more entanglement is present . for a discussion of the other lines , please see the text . ( b ) the second plot shows a cut for constant @xmath113.,title="fig:",scaledwidth=40.0% ] in this part we want to compare steady - state entanglement that is due to the reset to entanglement that is due to a special combination of interaction hamiltonian and decoherence process as in subsection [ mewithoutreset ] . there , the interaction hamiltonian is @xmath137 while the free hamiltonian are given by ( [ atomichamiltonian ] ) . the decoherence process ( [ localnoise ] ) is determined by the parameter choice @xmath41 . in [ mewithoutreset ] , it has been established that the steady state is entangled , but only in a finite temperature range . if we add the reset mechanism to the master equation of this example we can show that the steady state is entangled for arbitrary temperatures . as reset states we choose the eigenstate @xmath138 of the pauli operator @xmath139 . the steady - state density matrix of the master equation has now the matrix elements @xmath140\right\}\nonumber\\ \fl c_{0011}=\frac{g ( 2 s b - b - r ) ( i ( b+2 r)+\omega ) } { ( b+r ) \omega ^2+(b+2 r ) \left(4 g^2+(b+r ) ( b+2 r)\right)}\end{aligned}\ ] ] while all other coefficients are zero or are given by hermiticity . the negativity of this density matrix is @xmath141 we plot the function ( [ negwithandwithoutr ] ) in figure [ variationofsandr ] ; see figure caption for details . we observe two features : 1 . at @xmath142 we are back to the situation of subsection [ mewithoutreset ] , where entanglement vanishes above some temperature threshold ( remember : @xmath143 as @xmath144 ) . there is a threshold reset rate @xmath102 , above which an entangled steady state exists for arbitrary temperature . from the coefficients of the steady - state density matrix ( [ eswithandwithoutr ] ) we also see that the entanglement created by the reset stems from a different density matrix than the entanglement present without reset . in figure [ variationofsandr ] this is visible in the region of small @xmath102 , where the reset tends to destroy this latter entanglement . then , for larger @xmath102 , the effect of the reset mechanism kicks in . ( 430,200 ) ( 0,0)(a ) negativity for @xmath145-qubit system with @xmath146 as a function of the reset rate @xmath102 and temperature - dependent parameter @xmath18 . the noise is described by the quantum - optical master equation ( [ localnoise ] ) with @xmath147 . with @xmath117 as unit timescale the other parameters are given by @xmath148 and @xmath149 . the entanglement due to the reset mechanism exists for all temperatures , while the entanglement without reset mechanism ( @xmath142 ) vanishes above a certain temperature threshold . ( b ) cut at constant @xmath150 ( orange curve , corresponding to zero temperature ) and @xmath151 ( blue curve , corresponding to infinite temperature).,title="fig:",scaledwidth=55.0% ] ( 204,38)(a ) negativity for @xmath145-qubit system with @xmath146 as a function of the reset rate @xmath102 and temperature - dependent parameter @xmath18 . the noise is described by the quantum - optical master equation ( [ localnoise ] ) with @xmath147 . with @xmath117 as unit timescale the other parameters are given by @xmath148 and @xmath149 . the entanglement due to the reset mechanism exists for all temperatures , while the entanglement without reset mechanism ( @xmath142 ) vanishes above a certain temperature threshold . ( b ) cut at constant @xmath150 ( orange curve , corresponding to zero temperature ) and @xmath151 ( blue curve , corresponding to infinite temperature).,title="fig:",scaledwidth=45.0% ] up to now we studied rather special interaction hamiltonians . in the following we demonstrate the genericity of entanglement that is present in a steady state due to a reset mechanism . in the previous example we have pointed out that entanglement , if present at all without reset , stems from a special combination of hamiltonian and decoherence process . one may ask whether adding a reset mechanism with special reset states is not just as artificial as the choice of special combinations of hamiltonians and decoherence processes . we are now going to show that one and the same reset with fixed reset states can lead to steady - state entanglement for many combinations of hamiltonians and decoherence processes . the reasoning was the opposite without reset , where only very few combinations of hamiltonians and decoherence processes lead to steady - state entanglement . we can also relax the condition that the reset states are pure states to a certain extent . hence , we show that we have found generic features by generalizing the system in various directions . \(i ) the qualitative behavior of the two - qubit model does not depend on the particular choice of the interaction hamiltonian or details of the decoherence model other than its local action on individual qubits . figure [ variationofs ] shows e.g. steady - state entanglement for an xyz hamiltonian as function of reset rate @xmath102 , and decoherence described by the noise operator @xmath152 . the qualitative behavior is similar to figure [ toymodel](b ) or figure [ variationofsandr ] , and we observe steady - state entanglement even for infinite temperature of the bath . ( 430,200 ) ( 0,0)(a ) negativity for two - qubit system with @xmath153 interaction and magnetic field , @xmath154 , as a function of the reset rate @xmath102 and temperature - related parameter @xmath18 . the noise is described by the quantum - optical master equation channel ( [ localnoise ] ) with @xmath147 ( @xmath117 as unit timescale ) . the hamiltonian parameters are @xmath155 , @xmath156 . ( b ) cut through the plot . the upper curve corresponds to zero temperature @xmath157 , the lower one to infinite temperature @xmath158 of the bath . curves for any finite temperature lie in between.,title="fig:",scaledwidth=55.0% ] ( 204,38)(a ) negativity for two - qubit system with @xmath153 interaction and magnetic field , @xmath154 , as a function of the reset rate @xmath102 and temperature - related parameter @xmath18 . the noise is described by the quantum - optical master equation channel ( [ localnoise ] ) with @xmath147 ( @xmath117 as unit timescale ) . the hamiltonian parameters are @xmath155 , @xmath156 . ( b ) cut through the plot . the upper curve corresponds to zero temperature @xmath157 , the lower one to infinite temperature @xmath158 of the bath . curves for any finite temperature lie in between.,title="fig:",scaledwidth=45.0% ] \(ii ) the idealized reset mechanism we consider can be replaced by a more realistic imperfect reset mechanism . in this case , fresh particles are in mixed states with sufficiently low entropy rather than in pure states ( with entropy 0 ) . still , the steady state turns out to be entangled . when we vary @xmath102 there is a new , third threshold . first , for very small @xmath102 , there can be an entangled steady state , which is not due to the reset and which is present only for a finite temperature range above zero . second , there is one threshold value for @xmath102 above which the steady state is entangled due to the reset mechanism ( for arbitrary temperature ) . third , whereas for pure reset states the entanglement goes down to zero again only in the limit @xmath159 ( permanent projection to the product of the reset states ) , the entanglement goes down to zero for some finite @xmath102 if the reset states are mixed states . this behavior is easy to understand , since it is more difficult for the interaction hamiltonian to create entangled states from mixed reset states . the higher the entropies of the reset states are , the smaller is the range of the reset rate @xmath102 for which there is entanglement in the steady state . this range can also become zero , so we must demand reset states of sufficiently low entropy . figure ( [ variationofpimperfectprojection ] ) clearly shows this new threshold appearing for large reset rates . negativity for two - qubit system with @xmath153 interaction and parameters as in figure ( [ variationofs ] ) . the temperature related parameter has the fixed value @xmath120 . now , the reset state is not the pure state @xmath160 but the mixed state @xmath161 . the three curves , from top to bottom , correspond to @xmath162 , @xmath163 , and @xmath164 . with a mixed reset state , a third threshold for large but finite reset rate @xmath102 appears above which the state is not entangled in contrast to the case of pure reset states where the entanglement vanished only for @xmath159 ( topmost curve).,scaledwidth=60.0% ] the picture that emerges from all these results is the following . entanglement can prevail in dissipative , open quantum systems that are far away from thermodynamic equilibrium . for gas - type systems treated in this section , a reset mechanism can evoke steady - state entanglement even for infinite temperature of the environment generically , i.e. , independently of the specific form of the interaction hamiltonian or decoherence channel . in the next section we show that steady - state entanglement appears also in strongly coupled systems with an appropriate reset mechanism . in gas - type systems , we can treat the local noise channels separately for each qubit as explained above . if these local channels correspond to a heat bath , they drive each qubit individually to the thermal state of the local , free hamiltonian , i.e. , they populate the eigenstates of the free hamiltonian according to the boltzmann factor . although the effective interaction hamiltonian in the master equations is represented as continuously acting , the physical interaction process in gas - type scenarios is viewed as a short collision event . hence , the interaction hamiltonian does not influence the energy spectrum considerably and does not modify this thermal state . in strongly coupled systems , on the other hand , interactions of quanta of the heat bath with the system qubits affect the system as a whole . in this sense , the decoherence process acts globally on the system , inducing transitions between joint eigenstates . in this section we will shortly discuss the master equation describing a strongly coupled spin system in contact with a thermal , photonic bath . we will see that the resulting equilibrium state , the thermal state , can be entangled below a certain temperature threshold if the ground state of the hamiltonian is entangled . when we add a reset mechanism we find that , in contrast to the gas - type scenario , entanglement in the steady state can exist only below a certain temperature threshold . however , the novel feature is that this threshold is typically much higher than for the thermal state . finally , we describe the influence of the master equation parameters on the respective steady state , and , with this insight , formulate a general condition under which a reset mechanism can lead to an entangled steady state . + let @xmath165 ( @xmath166 ) be momentary eigenstates of some non - degenerate system hamiltonian @xmath167 with eigenenergies @xmath168 ( @xmath169 ) throughout the paper . ] . we define @xmath170 with @xmath171 being the inverse temperature . often @xmath172 is written as @xmath173 , and we explained the connection to the parameter @xmath18 in the last section . the master equation for a spin system , coupled with strength @xmath125 to a heat bath consisting of photons , is @xcite @xmath174-\gamma\sum_{j , a , b}[n_{ba}|g_{ba}|^2|\langle a|\sigma_-^{(j)}|b\rangle|^2+(n_{ab}+1)|g_{ab}|^2|\langle b|\sigma_-^{(j)}|a\rangle|^2]\nonumber\\ \qquad\times\left\{|a\rangle\langle a|\rho+\rho|a\rangle\langle a|-2\langle a|\rho|a\rangle |b\rangle\langle b|\right\},\end{aligned}\ ] ] here , @xmath175 is the spectral density , for which @xmath176 if @xmath177 and @xmath178 else . for small system one can justify to treat the spectral density as constant ( @xmath179 if @xmath177 ) and merely tune the overall coupling constant @xmath125 @xcite . observe that we did not include non - radiative contributions as opposed to the master equation ( [ gasme ] ) with noise terms ( [ localnoise ] ) . the master equation ( [ mesolidstate ] ) drives any initial density matrix to the thermal state of inverse temperature @xmath180 . that means , the ground state and also the excited states are populated according to the canonical distribution . we will study the master equation ( [ mesolidstate ] ) for an ising hamiltonian with transverse magnetic field , briefly discuss the solution without reset mechanism ( thermal state ) , and then turn to an analysis of the full master equation with reset mechanism . an ising hamiltonian with transverse magnetic field has the form @xmath181,\ ] ] and the eigenvalues are @xmath182,@xmath183,@xmath43,@xmath184 with corresponding eigenvectors , expressed in the standard basis , @xmath185 where @xmath186 provides normalization . we exclude the case @xmath187 where the ground state would be degenerate , a case not properly described by the master equation ( [ mesolidstate ] ) . the ground state of this system is the first eigenvector in equation ( [ eigenvectorhising+b ] ) . since this state is entangled , so is the thermal state below a certain temperature threshold . we see this directly from the negativity of the thermal state @xmath188 , which is @xmath189 for any fixed @xmath43 and @xmath190 , the non - trivial part in this formula goes to the value @xmath191 when @xmath192 , while it goes to @xmath193 for @xmath194 . the threshold value for @xmath180 , where the non - trivial part becomes exactly zero , can be easily computed numerically for any given parameters @xmath43 and @xmath190 . in terms of @xmath180 , the thermal mixture of the eigenstates is separable below this threshold , which , in terms of @xmath195 , means that the mixture is separable above that critical temperature . from [ strongly_coupled_neg_thermal_state ] one can see that the critical temperature grows linearly with @xmath43 and monotonously , but sub - linearly with @xmath190 . we keep the ising hamiltonian with transverse magnetic field as above , but extend the master equation ( [ mesolidstate ] ) by the reset term @xmath196 with @xmath27 and reset state @xmath197 for both qubits . we solve the resulting master equation numerically . in figure [ timedepsolidstate ] we see how the entanglement for different reset rates @xmath102 develops over time @xmath111 from the value zero in the initial product state @xmath198 to its final value in the steady state while all other parameters are kept fixed ( see figure caption ) . we notice that small , non - zero reset rates decrease the entanglement in the steady state until it is gone , while larger rates can bring entanglement back . to explain this effect , recall that the master equation mimics the averaged density matrices that would be obtained from ( infinitely many ) simulation runs of the system . the reset rates of the master equation are related to probabilities that in a simulation a reset took place during a certain time interval . although our reset processes are strictly speaking local , let us assume for the sake of argumentation that the reset happens on both qubits simultaneously , thus effectively restarting the process again from the beginning whenever a reset occurs in a simulation . for small rates @xmath102 , i.e. , for small probabilities that a reset takes place in the simulation , the system can come close to its thermal equilibrium state before it is reset . when we average the density matrices over many simulation runs , we average matrices that are mostly close to the unique thermal equilibrium state , and hence also the mixture will still retain entanglement . when the rates get larger , the density matrices over which we average become more and more diverse since they will be far from equilibrium and fluctuations occur . as a consequence the average density matrix will have no entanglement . when the reset rate is above a certain threshold , we will find entanglement in the system again ( as we did in the gas - type systems ) because now the density matrices over which we average become similar again . now , they are close to the state that has unitarily evolved for a time of order @xmath199 from the initial reset state . in the limit @xmath159 the state is constantly kept in the initial product state with zero entanglement . in this way we can understand how the two entangled regions arise . the first is an artifact of the entangled thermal state that is more and more destroyed by the reset mechanism . this first region could also be present in a gas - type model . the second region is the one that is really created by the reset mechanism just as in the gas - type model . we can directly see in figure [ timedepsolidstate ] that entanglement in a thermal state is a truly different effect from entanglement that is created by the reset mechanism . ( a ) solution of the master equation ( [ mesolidstate ] ) with reset term ( [ reset_strongly_coupled_systems ] ) for a strongly coupled system . the unit timescale is @xmath100 and the time @xmath111 is plotted in these units . the temperature parameter was chosen as @xmath200 , and , at this low temperature , the steady state of ( [ mesolidstate ] ) is entangled even without reset , the other parameters being @xmath201 and @xmath202 . when increasing the parameter @xmath102 of the reset mechanism , the steady - state density matrix changes , as explained in the text , fist becoming separable and then entangled again . ( b ) cut through the same plot for different @xmath102.,scaledwidth=95.0% ] although the two effects are truly different , this does not mean that for certain parameter regimes , the two regions can not overlap , see figure [ couplingtobath ] . imagine the coupling to the photon bath , @xmath125 , is increased . this means that the system is driven towards thermal equilibrium faster than before . hence , following the arguments from above , the system can tolerate higher reset rates before the entanglement in the first region is destroyed . the stronger coupling to the photon bath suppresses the entanglement in the second region , and as an overall effect we see that the two regions need not be separate . note that the entanglement in the thermal state for @xmath142 is independent of @xmath125 because then it does not matter how fast equilibrium was approached . ( a ) influence of the parameter @xmath125 , which describes the strength of the coupling to the photon bath . here , we choose @xmath203 as unit time . the other parameters are @xmath204 and @xmath202 . ( b ) cut through the same plot for different values of @xmath125.,scaledwidth=95.0% ] the transverse magnetic field with relative strength @xmath190 splits up the energy levels of the hamiltonian ( we exclude the degenerate point @xmath187 ) . the ground state will contain less and less entanglement as @xmath190 increases ( and will approach the product state @xmath205 for @xmath206 ) . hence , at zero temperature , the entanglement in the equilibrium state for @xmath142 will go down for increasing @xmath190 ( see red line in figure [ nvsbr](a ) ) . for a thermal state with @xmath207 , i.e. , @xmath208 , the ground and exited states get mixed . when @xmath190 is small , the splitting between ground and first exited state is small , and the mixture will be close to a separable state . for increasing @xmath190 , the larger energy split leads to an increased population of the ground state relative to the exited states at the same temperature @xmath209 , and the entanglement will increase . when @xmath190 gets even larger , the thermal state will be close to the ground state , but we know , that the ground state for large @xmath190 is only weakly entangled . hence , there will be some @xmath190 for which the entanglement in the thermal state is maximal ( see red line in figure [ nvsbr](b ) , and the identical line in figure [ nvsbr](c ) ) . when we switch on the reset mechanism , we see that an increasing @xmath102 destroys the entanglement in the thermal state as before , but for increasing @xmath190 the regions one and two ( artifact of thermal state vs. true reset state entanglement ) quickly overlap . that is because the magnetic field tends to drive the state towards @xmath205 , whereas an increasing reset mechanism drives the state towards @xmath210 . if we choose the reset state as @xmath211 , the two effects do not compete and both drive the state towards an unentangled product state ( see light grey areas in figure [ nvsbr](c ) and compare to the light grey areas in figure [ nvsbr](b ) ) . how fast an increasing @xmath102 destroys the entanglement in the thermal state is almost independent of the entanglement in the thermal state . for larger @xmath102 the influence of @xmath190 plays less and less a role and there is almost no difference between figures [ nvsbr](b),(c ) for large @xmath102 . influence of the relative magnetic field @xmath190 . the unit timescale is given by @xmath100 and @xmath212 . plots ( a ) and ( b ) show the difference between ( almost ) zero temperature ( @xmath213 ) and some finite temperature ( @xmath214 ) . the behavior of the red curves , for @xmath142 , are explained in the text . plots ( b ) and ( c ) demonstrate the influence of different reset states in connection with the magnetic field @xmath190 . plot ( b ) has reset state @xmath215 , ( c ) has reset state @xmath211.,scaledwidth=98.0% ] the coupling strength of the hamiltonian @xmath43 also splits the energy levels . hence with increasing @xmath43 there will be more entanglement in the thermal state at some finite temperature . again , the speed with which an increasing reset rate destroys the entanglement in the thermal state is almost independent of @xmath43 ( see figure [ essolidstate](a ) ) . most interesting is the influence of the temperature @xmath209 . figures [ essolidstate](a)-(c ) show plots which contain information about the equilibrium - state entanglement for different temperatures ( @xmath142 ) . we see how the thermal states get less and less entangled for increasing temperatures , so that region one vanishes quickly as expected . but , for the same temperatures , the reset rate @xmath102 can still produce entanglement in a steady state ! on the other hand , for fixed @xmath43 , there is some temperature threshold above which no reset rate can produce entanglement . this threshold is in contrast to the case of gas - type systems . negativities for increasing temperature ( decreasing @xmath171 ) . the unit timescale is given by @xmath100 , the relative magnetic field by @xmath202 . the parameter region in the @xmath216 plane , where steady - state entanglement occurs , becomes smaller for increasing temperatures . however , the temperatures , for which entanglement can exist with reset mechanism , are much higher than the temperatures , for which the thermal state is entangled . the red line in ( c ) corresponds to the red line in figure [ entcondition ] ( see that figure caption and the text ) and is drawn for comparison only.,scaledwidth=98.0% ] this discrepancy between gas - type and strongly interacting systems raises the deeper question : what are the conditions under which the reset mechanism can create entanglement in the steady state ? we already see that the reset mechanism is different from a cold bath equivalently the replacing of system particles by fresh , standard ones because we would expect that some cold bath can always counteract the influence of the hot photon bath . the condition that the solution of the master equation ( a completely positive ( cp ) map ) at @xmath142 is entangling at some point in time is certainly necessary , since , as pointed out before , the reset mechanism does not introduce entanglement itself . the question is : is a solution of the master equation at @xmath142 that creates entanglement on some short time scale sufficient such that the solution for some @xmath217 is a cp - map with entangled steady state ? unfortunately this is not true . the reset rate @xmath102 itself can influence the solution of the master equation in such a way that although the solution for @xmath142 was entangling on some short time scale , the solution for larger @xmath102 need not be . the condition for steady - state entanglement is the following . if the solution of the master equation for some @xmath217 is entangling on a time scale of order @xmath199 then the steady state of this solution will also be entangled . for illustration , we think once more in terms of many hypothetical simulation runs . as stated above , the states over which we have to average will be close to the state that has unitarily evolved for a time @xmath199 from the initial reset state . when this state has a certain amount of entanglement , then so does the mixture of states close to it ( see figure [ entcondition ] ) . illustration of the condition under which a reset mechanism can create entanglement in the steady state . at time @xmath218 the state is already close to the true steady state , so the red line corresponds to the red line in figure [ essolidstate](c ) since @xmath219 in this plot , and the other parameters are the same as in figure [ essolidstate](c ) . the reset can create entanglement in the steady state if , for given @xmath102 , there is entanglement in the time - evolved state at time @xmath220 ( see text for details ) . the green curve is the curve @xmath221 illustrating this result . , scaledwidth=70.0% ] the condition does not only hold for the specific hamiltonian or the specific heat bath chosen here . it is valid for a large class of hamiltonians ( with appropriately chosen reset states ) and baths . in the next section we treat the multipartite case . for multipartite spin systems , we will show that the reset mechanism can create steady - state entanglement in a similar way . the parameter regions where this happens are comparable to the @xmath145-qubit case . the values of the negativity , or rather its generalization , the average negativity , stay almost constant with increasing system size . note , however , that larger systems could have larger negativities , so if we divide the actual ( constant ) negativity by the maximal possible negativity , then this quantity would go down for growing system size . we will also consider entanglement in reduced density matrices . since in a reduction from @xmath1 to , say , @xmath145 qubits the traced out @xmath222 qubits act as an additional noise source , it is not surprising that the parameter region where we find steady - state entanglement in the reduced systems shrinks with growing @xmath1 . if the number of particles fluctuates according to some distribution , our best description of a reduced density matrix is a mixture of reductions originating from different system sizes . it is remarkable that even in this case there is some parameter region , where steady - state entanglement is found . in the following , we motivate and explain the entanglement measures we are using and then demonstrate the above features in both gas - type systems and strongly interacting systems . it is straightforward to generalize the master equation for the gas - type system , equation ( [ generalme ] ) to @xmath223 qubits . the relevance of the entanglement quantities we are going to use needs to be motivated , though . we turn once more to the example of a spin gas . imagine that the spin gas is in a box of volume @xmath224 . this box has one semi - permeable wall through which particles can leave the box , and another through which particles from a cold reservoir " can enter . by cold reservoir " we mean that the quantum state of the particles in this reservoir is in some sufficiently pure standard state . the motional degrees of freedom of theses particles , on the other hand , are in thermal equilibrium with the outside environment just like the system particles in the box . assume that the density of the gas in the box is @xmath225 . then there are on average @xmath226 particles in the box . the distribution of the number of particles that are in the box is a poissonian @xmath227 . when we observe the spin gas after certain time intervals , which should be long enough such that the gas always reaches its equilibrium state , we sample the distribution and get information about the density matrices with a corresponding number @xmath228 of qubits . the density matrices we can reconstruct after we collected a certain , sufficient amount of information is close to the steady - state density matrix of the master equation for @xmath228 qubits . we are interested in the entanglement properties of the gas and we will look at different aspects of entanglement in the following . all these aspects of entanglement are quantified by measures that are based on the negativity or the average negativity . the average negativity @xmath229 is the negativity averaged over all possible bipartitions of the system @xcite . non - zero average negativity ensures the presence of some form of entanglement in the system . specifically , we study three types of entanglement : 1 . the average negativity of the density matrices with @xmath228 qubits , averaged over the poissonian distribution of the number of particles in the system : @xmath230 . the negativity of reduced @xmath145-qubit density matrices averaged over a renormalized , truncated poissonian distribution : @xmath231 . the negativity of averaged , reduced @xmath145-qubit density matrices : @xmath232 . now , we lay out , what these quantities mean , and which aspect of entanglement they describe . + ( i ) if we ask how much entanglement we find in the system on average we are led to the quantity @xmath230 , which is the expectation value of the average negativity of density matrices with different @xmath228 , where @xmath233 is the poissonian probability distribution for the number @xmath228 of particles in the system . observe that if we disallowed the fluctuation of the particle number in the system and introduced the reset mechanism by other means ( e.g. measurement and decay to standard state inside the box ) , the quantity of interest would simply be @xmath234 for fixed system size @xmath1 . \(ii ) when we look at subsystems we are led to slightly different quantities . let us fix the subsystem size to @xmath145 qubits . we call the reduced density matrices of originally @xmath228 qubits @xmath235 where we assume @xmath236 . in gas - type systems there can also be zero or one particle in the box ( especially if @xmath237 is small ) , and the entanglement is simply zero in these cases . since a reduction " of a one or zero - qubit density matrix to a @xmath145-qubit density matrix makes no sense , we simply exclude these cases and rescale the truncated poissonian @xmath238 to the distribution @xmath239 . the quantity @xmath231 therefore tells us how much entanglement a subsystem of two qubits contains on average ( for @xmath145 qubits @xmath240 ) . \(iii ) when we look only at a subsystem of two qubits disregarding the number of particles @xmath228 in the system , then our best description of the @xmath145-qubit density matrix is the average density matrix @xmath241 with entanglement @xmath232 . + when we compare the three kinds of entanglement defined above we see that the conditions that one of these quantities be non - zero are increasingly stringent . to find entanglement in the reduced system is a more stringent condition since tracing out the other particles has the same effect as an additional noise source . also , to find entanglement in the averaged density matrix @xmath242 is a stricter condition since the averaging increases entropy , i.e. , tends to make the matrix more mixed . if we keep the particle number fixed , @xmath243 , there is just the quantity @xmath244 to describe the entanglement in the reduced state . to compute the average negativity is a hard task . the system size , and hence the number of differential equations we must solve , and also the number of bipartitions scale exponentially . to simplify the computation , we consider a symmetric situation , where all qubits interact pairwise via ising interactions and are subject to dephasing noise ( [ dephasingchannel ] ) . in figure [ morequbits ] we plot the three entanglement measures @xmath230 ( blue , dashed curve ) , @xmath231 ( orange , solid curve ) , and @xmath232 ( black , dashed - dotted curve ) for @xmath245 and for a qubit number that fluctuates between two and five . since the meaning of these measures is different , one can not compare the absolute values represented by the curves directly with one exception . the points , where the curves become non - zero , must , from left to right , appear in the order explained in the previous paragraph , i.e. , blue first , representing measure ( i ) , orange next , representing ( ii ) , and black last , representing ( iii ) . measures @xmath230 ( blue , dashed ) , @xmath231 ( orange , solid ) , and @xmath232 ( black , dashed - dotted ) as functions of the reset rate @xmath135 . the particle number fluctuates between two and five , and the fluctuations are accounted for by a poissonian or truncated poissonian weighting with @xmath245 as explained in the text . the interaction strength of the ising interactions between all particles is @xmath246 , while the strength of the local , free hamiltonian ( [ atomichamiltonian ] ) is @xmath247 . kinks in the orange curve stem from the averaging over negativities whith different supports ( see ( ii ) in the text).,scaledwidth=70.0% ] eventually , we study equations ( [ mesolidstate ] ) and ( [ reset_strongly_coupled_systems ] ) in the multipartite case . we obtain similar results as in the case of gas - type systems underlining again how generic the reset mechanism is . although a fluctuation of particles may seem less natural in strongly coupled systems as compared to gas - type systems , we will look at the exact same entanglement quantities , so that the corresponding plots are directly related to each other . as hamiltonian , we choose a sum of pairwise ising interactions and magnetic fields in @xmath248 and @xmath110 direction according to @xmath249\ ] ] where the small gradient magnetic field in @xmath110 direction is introduced for technical reasons to lift degeneracies in the hamiltonian . figure [ 5qubitsstronglycoupled ] shows a plot of the same entanglement measures as in the previous subsection . as in the gas - type scenario , the feature that entanglement can be created by a reset mechanism holds also in the multipartite case . while we have shown the genericity of the reset mechanism with respect to the hamiltonian and noise process already in previous sections , here we demonstrate that the reset mechanism is also generic with respect to system size . measures @xmath230 ( blue , dashed ) , @xmath231 ( orange , solid ) , and @xmath232 ( black , dashed - dotted ) as functions of the reset rate @xmath135 . the particle number fluctuates between two and five , and the fluctuations are accounted for by a poissonian or truncated poissonian weighting with @xmath245 . the other parameters of the hamiltonian ( [ hisinggradientmagnfield ] ) are @xmath250 and @xmath202 . the inverse temperature is @xmath214 . kinks in the orange curve stem from the averaging over negativities whith different supports ( see ( ii ) in the text).,scaledwidth=70.0% ] we have shown that entanglement can be present in dissipative , open quantum systems far from thermodynamic equilibrium if we assume the existence of an additional mechanism that resets " the particles , at a certain rate , into a single - particle , low - entropy state . for a @xmath145-qubit toy model of a gas - type system , we have analytically solved the master equation consisting of a hamiltonian part , a noise channel , and the proposed reset mechanism . for special cases we have been able to give closed expressions for the entanglement as a function of the parameters of the master equation . we have extended the analysis to similar models with other interaction hamiltonians , decoherence models , and imperfect reset mechanisms . we have treated the situation of strongly correlated systems by the same means and we have given conditions under which steady - state entanglement arises in this case . finally , we have shown that in systems consisting of more qubits , and even in systems with fluctuating particle number , steady - state entanglement can prevail . many systems are conceivable for an experimental realization of such dissipative , open quantum systems . for instance , one may consider ions in microtraps that interact via an induced dipole moment @xcite leading effectively to a continuously operating ising interaction . decoherence , dominated by dephasing noise , appears naturally in such systems , and the reset mechanism may , e.g. , be achieved by periodically applying a @xmath251-pulse that couples the internal level @xmath252 to a metastable auxiliary level @xmath165 that decays rapidly to @xmath253 . the state afterwards is always @xmath253 , which can be mapped to @xmath215 by a subsequent hadamard operation . for charge manipulated quantum dots , the exchange interaction leads to a continuously operating heisenberg interaction between neighboring electron spins by lowering the potential barrier @xcite . the effect of surrounding nuclear spins may be described by dephasing noise , while the reset mechanism can consist in replacing an electron by a fresh one from the surrounding fermi sea , prepared in a suitable state ( e.g. , @xmath253 ) . atomic beams interacting via a cavity mode @xcite may also serve as a toy example of such systems far from thermodynamic equilibrium . note that a reset mechanisms could be realized in many physical ways , including a measurement with subsequent preparation of the state , coupling or decay to metastable auxiliary states , as well as replacing a qubit by a fresh one . while these suggestions for implementations aim at demonstrating how a reset mechanism would be realized in experiments , the effect itself is generic and other realizations are conceivable . in particular , one might try to find such a reset mechanism in less controlled , maybe even biomolecular systems consisting of many particles . this work was supported by the austrian science foundation ( fwf ) , the european union ( olaqui , scala , qics ) and the aw through project apart ( w.d . ) . [ [ appendix ] ] in this appendix we derive the solution of the master equation ( [ gasme ] ) for the ising hamiltonian ( [ isinghamiltonian ] ) explicitly . in the final formulae we restrict ourselves again to the dephasing channel ( [ dephasingchannel ] ) . instead of solving the master equation by spectral decomposition of the matrix @xmath85 associated with the liouville operator @xmath83 ( see section [ solutionmegastype ] ) , we solve the system of linear differential equations step by step . since not all of the differential equations are coupled , this leads to simpler expressions . the disadvantage is , however , that the set of solutions contains @xmath254 integration constants that must be determined afterwards for given initial conditions . we expand the @xmath145-qubit density matrix @xmath68 in the standard basis @xmath255 with @xmath256 and @xmath257 . the expression for the density matrix becomes @xmath258 . inserting this expansion into the master equation , and defining the two functions @xmath259\\ & -&\textstyle\frac{b}{2}s[(1-(x^\prime\oplus1))+(1-(x\oplus1))]-\textstyle\frac{2c - b}{4}[1-(-1)^{x^\prime+x}],\\ f_2(x^\prime , x)&:=&bs(1-x^\prime)(1-x)+b(1-s)(1-(x^\prime\oplus1))(1-(x\oplus1)),\end{aligned}\ ] ] we get the following linear system of coupled differential equations @xmath260\nonumber\\ & -&i\omega/2 \left((-1)^{s_1^\prime}-(-1)^{s_1}+(-1)^{s_2^\prime}-(-1)^{s_2}\right)\nonumber\\ & + & f_1(s_1^\prime , s_1)+f_1(s_2^\prime , s_2)-2r\}c_{s_1^\prime s_2^\prime s_1s_2}\nonumber\\ & + & f_2(s_1^\prime , s_1)c_{(s_1^\prime\oplus1 ) s_2^\prime ( s_1\oplus1 ) s_2}+f_2(s_2^\prime , s_2)c_{s_1^\prime ( s_2^\prime\oplus1 ) s_1 ( s_2\oplus1)}\nonumber\\ & + & r/2\{c_{0 s_2^\prime 0 s_2}+c_{1 s_2^\prime 1 s_2}+c_{s_1^\prime 0 s_1 0}+c_{s_1^\prime 1 s_1 1}\}.\end{aligned}\ ] ] here , the operation @xmath261 means addition modulo @xmath145 . fortunately , these @xmath81 differential equations are not fully coupled . the coefficients @xmath262 , @xmath263 , @xmath264 , and @xmath265 on the diagonal of the density matrix are coupled only to themselves . once we have solved these equations , we can treat the diagonal coefficients as known inhomogeneities in the other equations . the off - diagonal coefficients @xmath266 and @xmath267 are coupled among themselves and to the diagonal , so we can solve them next . the same is true for the pair @xmath268 and @xmath269 . finally , the anti - diagonal coefficients @xmath270 and @xmath271 are coupled to @xmath266,@xmath267,@xmath268 , and @xmath269 , or to their complex conjugates . we solve these as a last step , and all other coefficients are given by the hermiticity of the density matrix . the solution is now straightforward in principle . however , the expressions for the matrix coefficients are still space - consuming , so we will give them only for the special case of the dephasing channel . + for the dephasing channel ( [ dephasingchannel ] ) , the structure of the differential equations is still the same and not simplified , we save space only because we have fewer parameters and a symmetric situation . the solution for the diagonal elements then reads : @xmath272 the integration constants @xmath273 , @xmath274 , and @xmath275 accommodate the initial conditions . since @xmath276 is a constraint , there is no free constant @xmath277 . the off - diagonal elements are : @xmath278 @xmath279 the coefficients @xmath268 , @xmath269 are very similar to @xmath266 , @xmath267 , except that @xmath280 must be replaced by @xmath281 , and the integration constants are @xmath282 , @xmath283 instead of @xmath284 , @xmath285 . finally , the elements on the anti - diagonal are @xmath286 and @xmath287 all other coefficients follow from the hermiticity of the density matrix . the form of all matrix coefficients is similar . first , there are the parts with the integration constants that fall off exponentially with time . the characteristic exponents are the eigenvalues of the homogeneous parts of each linear differential equation or system of equations ( multiplied with time ) . as pointed out in [ gastype_solutionising ] , these exponents are the spectrum of the total liouville super - operator defined by @xmath96 with values @xmath97 and multiplicities @xmath98 respectively . second , there always is a part independent of @xmath111 ( belonging to the eigenvalue @xmath99 ) that represents the value in the steady state . [ [ appendix2 ] ] in this appendix we present the steady - state solution for the master equation ( [ generalme ] ) with ising hamiltonian , local noise channels as in equation ( [ localnoise ] ) and reset states @xmath116 to [ appendix2 ] . the solution in form of the matrix coefficients in the computational basis is given by : @xmath288 furthermore , @xmath289 , and @xmath290 are obtained from @xmath266 by replacing @xmath291 , @xmath292 in the numerator . the other coefficients are given by the hermiticity of the density matrix . observe that the dephasing and the depolarizing channels are included as special instances of the parameters @xmath293 in this analytic expression . the dephasing channel results from putting @xmath294 , @xmath18 drops out , and renaming @xmath23 . the depolarizing channel is given by @xmath16 , @xmath295 , and renaming @xmath296 . the plot in figure [ qomeequstate ] is based on this solution . 10 [ 1]`#1 ` [ 1 ] [ 2 ]

quantum mechanical entanglement can exist in noisy open quantum systems at high temperature . a simple mechanism , where system particles are randomly reset to some standard initial state , can counteract the deteriorating effect of decoherence , resulting in an entangled steady state far from thermodynamical equilibrium . we present models for both gas - type systems and for strongly coupled systems . we point out in which way the entanglement resulting from such a reset mechanism is different from the entanglement that one can find in thermal states . we develop master equations to describe the system and its interaction with an environment , study toy models with two particles ( qubits ) , where the master equation can often be solved analytically , and finally examine larger systems with possibly fluctuating particle numbers . we find that in gas - type systems , the reset mechanism can produce an entangled steady state for an arbitrary temperature of the environment , while this is not true in strongly coupled systems . but even then , the temperature range where one can find entangled steady states is typically much higher with the reset mechanism .

quant-ph0703138

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 .

1305.7022

registration , the task of establishing correspondences between multiple instances of objects such as images , landmarks , curves , and surfaces , plays a fundamental role in a range of computer vision applications including shape modeling @xcite , motion compensation and optical flow @xcite , remote sension @xcite , and medical imaging @xcite . in the subfield of computational anatomy @xcite , establishing inter - subject correspondences between organs allows the statistical study of organ shape and shape variability . examples of the fundamental role of registration include quantifying developing alzheimer s disease by establishing correspondences between brain tissue at different stages of the disease @xcite ; measuring the effect of copd on lung tissue after removing the variability caused by the respiratory process @xcite ; and correlating the shape of the hippocampus to schizophrenia after inter - subject registration @xcite . in this paper , we survey the role of symmetry in diffeomorphic registration and deformation modeling and link symmetry as seen from the field of geometric mechanics with the image registration problem . we focus on large deformations modeled in subgroups of the group of diffeomorphic mappings on the spatial domain , the approach contained in the large deformation diffeomorphic metric mapping ( lddmm , @xcite ) framework . connections with geometric mechanics @xcite have highlighted the role of symmetry and resulted in previously known properties connected with the registration of specific data types being described in a common theoretical framework @xcite . we wish to describe these connections in a form that highlights the role of symmetry and points towards future applications of the ideas . it is the aim that the paper will make the role of symmetry in registration and deformation modeling clear to the reader that has no previous familiarity with symmetry in geometric mechanics and symmetry groups in mathematics . one of the main reasons symmetry is useful in numerics is in it s ability to reduce how much information one must carry . as a toy example , consider the a top spinning in space . upon choosing some reference configuraiton , the orientation of the top is given by a rotation matrix , i.e. an element @xmath0 . if i ask for you to give me the direction of the pointy tip of the top , ( which is pointing opposite @xmath1 in the reference ) it suffices to give me @xmath2 . however , @xmath2 is contained in space of dimension @xmath3 , while the space of possible directions is the @xmath4-sphere , @xmath5 , which is only of dimension @xmath4 . therefore , providing the full matrix @xmath2 is excessive in terms of data . it suffices to just provide the vector @xmath6 . note that if @xmath7 , then @xmath8 . therefore , given only the direction @xmath9 , we can only reconstruct @xmath2 up to an element @xmath10 which preserves @xmath1 . the group of element which preserve @xmath1 is identifiable with @xmath11 . this insight allows us to express the space of directions @xmath5 as a homogenous space @xmath12 . in terms of infomation we can cartoonishly express this by the expression @xmath13 this example is typically of all group quotients . if @xmath14 is some universe of objects and @xmath15 is a group which acts freely upon @xmath14 , then the orbit space @xmath16 hueristically contains the data of @xmath14 minus the data which @xmath15 transforms . thus @xmath17 reduction by symmetry can be implemented when a problem posed on @xmath14 has @xmath15 symmetry , and can be rewritten as a problem posed on @xmath18 . the later space containing less data , and is therefore more efficient in terms of memory . registration of objects contained in a spatial domain , e.g. the volume to be imaged by a scanner , can be formulated as the search for a deformation that transforms both domain and objects to establish an inter - object match . the data available when solving a registration problem generally is incomplete for encoding the deformation of every point of the domain . this is for example the case when images to be matched have areas of constant intensity and no derivative information can guide the registration . similarly , when 3d shapes are matched based on similarity of their surfaces , the deformation of the interior can not be derived from the available information . the deformation model is in these cases over - complete , and a range of deformations can provide equally good matches for the data . here arises _ symmetry _ : the subspaces of deformations for which the registration problem is symmetric with respect to the available information . when quotienting out symmetry subgroups , a vastly more compact representation is obtained . in the image case , only displacement orthogonal to the level lines of the image is needed ; in the shape case , the information left in the quotient is supported on the surface of the shape only . we start with background on the registration problem and the large deformation approach from a variational viewpoint . following this , we describe how reduction by symmetry leads to an eulerian formulation of the equations of motion when reducing to the lie algebra . symmetry of the dissimilarity measure allows additional reductions , and we use isotropy subgroups to reduce the complexity of the registration problem further . lastly , we survey the effect of symmetry in a range of concrete registration problems and end the paper with concluding remarks . the registration problem consists in finding correspondences between objects that are typically point sets ( landmarks ) , curves , surfaces , images or more complicated spatially dependent data such as diffusion weighted images ( dwi ) . the problem can be approached by letting @xmath19 be a spatial domain containing the objects to be registered . @xmath19 can be a differentiable manifold or , as is often the case in applications , the closure of an open subset of @xmath20 , @xmath21 , e.g. the unit square . a map @xmath22 can deform or warp the domain by mapping each @xmath23 to @xmath24 . acts on an image @xmath25 by composition with the inverse warp , @xmath26 . given two images @xmath27 , image registration involves finding a warp @xmath28 such that @xmath29 is close to @xmath30 as measured by a dissimilarity measure @xmath31 . ] the deformation encoded in the warp will apply to the objects in @xmath19 as well as the domain itself . for example , if the objects to be registered consist of points sets @xmath32 , @xmath33 , the set will be mapped to @xmath34 . for surfaces @xmath35 , @xmath28 similarly results in the warped surface @xmath36 . because those operations are associative , the mapping @xmath28 acts on @xmath37 or @xmath38 and we write @xmath39 and @xmath40 for the warped objects . an image is a function @xmath25 , and @xmath28 acts on @xmath41 as well , in this case by composition with its inverse @xmath42 , see figure [ fig : registration ] . for this @xmath28 must be is invertible , and commonly we restrict to the set of invertible and differentiable mappings @xmath43 . for various other types of data objects , the action of a warp on the objects can be defined in a way similar to the case for point sets , surfaces and images . this fact relates a range registration problems to the common case of finding appropriate warps @xmath28 that trough the action brings the objects into correspondence . trough the action , different instances of a shape can be realized by letting warps act on a base instance of the shape , and a class of shape models can therefor be obtained by using deformations to represent shapes @xcite . the search for appropriate warps can be formulated in a variational formulation with an energy @xmath44 where @xmath45 is a dissimilarity measure of the difference between the deformed objects , and @xmath2 is a regularization term that penalizes unwanted properties of @xmath28 such as irregularity . if two objects @xmath46 and @xmath47 is to be matched , @xmath45 can take the form @xmath48 using the action of @xmath28 on @xmath49 ; for image matching , an often used dissimilarity measure is the @xmath50-difference or sum of square differences ( ssd ) that has the form @xmath51 . the regularization term can take various forms often modeling physical properties such as elasticity @xcite and penalizing derivatives of @xmath28 in order to make it smooth . the free - form - deformation ( ffd , @xcite ) and related approaches penalize @xmath28 directly . for some choices of @xmath2 , existence and analytical properties of minimizers of have been derived @xcite , however it is in general difficult to ensure solutions are diffeomorphic by penalizing @xmath28 in itself . instead , flow based approaches model one - parameter families or paths of mappings @xmath52 , @xmath53 $ ] where @xmath54 is the identity mapping @xmath55 and the dissimilarity is measured at the endpoint @xmath56 . the time evolution of @xmath52 can be described by the differential equation @xmath57 with the flow field @xmath58 being a vector field on @xmath19 . the space of such fields is denoted @xmath59 . in the large deformation diffeomorphic metric mapping ( lddmm , @xcite ) framework , the regularization is applied to the flow field @xmath58 and integrated over time giving the energy @xmath60 if the norm @xmath61 that measures the irregularity of @xmath58 is sufficiently strong , @xmath52 will be a diffeomorphism for all @xmath62 . this approach thus gives a direct way of enforcing properties of the generated warp : instead of regularizing @xmath28 directly , the analysis is lifted to a normed space @xmath59 that is much easier control . the energy has the same minimizers as the geometric formulation of lddmm used in the next section . direct approaches to solving the optimization problem must handle the fact that the problem of finding a warp is now transfered to finding a time - dependent family of warps implying a huge increase in dimensionality . this problem is therefore vary hard to represent numerically and to optimize . for several data types , it has been shown how optimal paths for have specific properties that reduces the dimensionality of the problem and therefor makes practical solutions feasible . in the next section , we describe the geometric framework and the reduction theory that allows the data dependent results to be formulated as specific examples of reduction by symmetry . we survey these examples in the following section . we here describe a geometric formulation of the registration problem @xcite and how symmetry can be used to reduce the optimization over time - dependent paths of warps to vector fields in the lie algebra resulting in an eulerian version of the equations of motion . secondly , we describe how symmetry of the dissimilarity measure allows further reduction to lower dimensional quotients . we here introduce a number of notions from differential geometry in a fairly informal manner . for formal definitions we refer to @xcite . while it is neccessary for the purpose of rigour to learn formal definitions , independent of cartoonish sketches , when learning differential geometry , one can still get quite far with cartoonish sketches . for example , by picturing a manifold , @xmath19 , as a surface embedded in @xmath63 . this is the approach we will take . the tangent bundle of @xmath19 , denoted @xmath64 , is the set of pairs @xmath65 where @xmath66 and @xmath67 is a vector tangential to @xmath19 at the point @xmath68 ( see figure [ fig : tm ] ) . a vector - field is a map @xmath69 such that @xmath70 is a vector above @xmath68 for all @xmath66 . in summary , a vector on @xmath19 is an `` admissible velocity '' on @xmath19 , and @xmath64 is the set of all possible admissible velocities . of the manifold @xmath19 consists of pairs @xmath65 of points @xmath23 and tangent vectors @xmath71 . it s a fiber bundle over @xmath19 with fibers @xmath72 for each @xmath73 m . ] given a vector - field @xmath74 we may consider the initial value problem @xmath75 for @xmath76 $ ] . given an intial condition @xmath77 , the point @xmath78 given by solving this initial value problem is uniquely determined ( if it exists ) . under reasonable conditions @xmath79 exists for each @xmath77 , and there is a map @xmath80 which we call the flow of @xmath74 . if @xmath74 is time - dependent we can consider the initial value problem with @xmath81 . under certain conditions , this will also yield a flow map , @xmath82 which is the flow from time @xmath83 to @xmath84 . if @xmath74 is smooth , the flow map is smooth as well , in particular a diffeophism . we denote the set of diffeomorphisms by @xmath43 . conversely , let @xmath85 be a time - dependent diffeomorphism . thus , for any @xmath66 , we observe that @xmath86 is a curve in @xmath19 . if this curve is differentiable we may consider its time - derivative , @xmath87 , which is a vector above the point @xmath86 . from these observations it imediately follows that @xmath88 is a vector above @xmath68 . therefore the map @xmath89 , given by @xmath90 is a vector - field which we call the _ eulerian velocity field of @xmath52_. the eulerian velocity field contains less data than @xmath91 , and this reduction in data can be viewed from the perspective of symmetry . given any @xmath92 , the curve @xmath52 can be transformed to the curve @xmath93 . we observe that @xmath94 thus @xmath52 and @xmath93 both have the same eulerian velocity fields . in other words , the eulerian velocity field , @xmath95 , is invariant under particle relablings . schematically , the following holds @xmath96 finally , we will denote some linear operators on the space of vector - fields . let @xmath97 and let @xmath98 . the _ push - forward _ of @xmath74 by @xmath99 , denoted @xmath100 , is the vector - field given by @xmath101(x ) = \left . d\phi \right|_{\phi^{-1}(x ) } \cdot u ( \phi^{-1}(x)).\end{aligned}\ ] ] by inspection we see that @xmath102 is a linear operator on the vector - space of vector - fields . one can view @xmath100 as `` @xmath74 in a new coordinate system '' because any differential geometric property of @xmath74 is also inherited by @xmath100 . for example , if @xmath103 then @xmath104 ( y ) = 0 $ ] with @xmath105 . if @xmath38 is an invariant under @xmath74 then @xmath106 is invariant under @xmath107 . as @xmath102 is linear , we can transpose it . let @xmath108 denote the dual space to the space of vector - fields , i.e. the set of linear maps @xmath109 , and let @xmath110 . we define @xmath111 by the equality @xmath112 for all @xmath113 where @xmath114 denotes evaluation of @xmath115 on @xmath67 . finally , we define the lie - derivative as the linear operator @xmath116 defined by @xmath117 = \left . \frac{d}{d \epsilon}\right|_{\epsilon=0 } ( \phi^w_{\epsilon } ) _ * u.\end{aligned}\ ] ] as @xmath118 is linear , we can take its transpose . if @xmath110 , then we can define @xmath119 \in \mathfrak{x}(m)^*$ ] by the equation @xmath120 , w \rangle + \langle m , \pounds_u[w ] \rangle = 0\end{aligned}\ ] ] for all @xmath121 . this is a satisfying because for a fixed @xmath115 and @xmath122 we observe @xmath123 , w \rangle + \langle m,\pounds_u[w ] \rangle = 0 = \frac{d}{dt } \langle m , w \rangle = \frac{d}{dt}\langle ( \phi_t^u ) _ * m , ( \phi_t^u ) _ * \rangle \label{eq : product_rule}\end{aligned}\ ] ] this is nothing but a coordinate free version of the product rule . the variational formulation of lddmm is equivalent to minimizing the energy @xmath124 where @xmath125 is a distance metric on @xmath43 , @xmath126 is the identity diffeomorphism , and @xmath127 is a function which measures the disparity between the deformed template and the target image . given images @xmath128 , we consider the dissimilarity measure @xmath129 [ ex : imssd ] in this article we will consider the distance metric @xmath130 , \mathfrak{x}(m ) ) \\ \phi^v_{0,1 } \circ \varphi_0 = \varphi_1 } } \left ( \int_0 ^ 1 \| v(t ) \| dt \right),\end{aligned}\ ] ] where @xmath131 is some norm on @xmath132 . if @xmath131 is induced by an inner - product , then this distance metric is ( formally ) a riemannian distance metric on @xmath43 . note that the distance metric , @xmath133 , is written in terms of a norm @xmath131 , defined on @xmath132 . in fact , the norm on @xmath132 induces a riemannian metric on @xmath43 given by @xmath134 and @xmath133 is the reimannian distance with respect to this metric . if the norm @xmath131 imposes a hilbert space structure on the vector - fields it can be written in terms of a psuedo - differential operator @xmath135 as @xmath136 , u \rangle$ ] @xcite . given @xmath137 , minimizers of @xmath138 must neccessarily satisfy @xmath139\\ \langle m(1 ) , w \rangle = \left . \frac{d}{d\epsilon } \right|_{\epsilon=0 } f ( \phi^w_\epsilon \circ \phi^u_{0,1 } ) \quad,\quad \forall w \in \mathfrak{x}(m ) . \end{cases } \label{eq : extreme1}\end{aligned}\ ] ] this is a vector - calculus statement of proposition 11.6 of @xcite . that this equation of motion is even well - posed is nontrivial , since @xmath137 is merely an injective map , and there is no guarantee that it can be inverted to obtain a vector - field @xmath74 to integrate into a diffeomorphism . fortunately , safety guards for well - posedness are studied in @xcite . if the reproducing kernel of @xmath137 is @xmath140 , then the equations of motion are well - posed for all time . there is something unsatisfying about using for the purpose of computation . doing any sort of computation on @xmath43 is difficult , as it is a nonlinear infinite dimensional space . moreover , the dissimilarity measure @xmath45 only comes into play at time @xmath141 and the distance function is an integral over the vector - space @xmath132 . it would be nice if we could rewrite the extremizers purely in terms of the eulerian velocity field , @xmath74 and the flow at @xmath141 . in fact this is often the case . given @xmath137 , the minimizer of @xmath138 must neccessarily satisfy the boundary value problem @xmath142 = 0 , m = p[u ] \quad \forall t \in [ 0,1 ] \\ \frac{d}{d \epsilon}|_{\epsilon = 0 } \left [ f ( \phi^w_\epsilon \circ \phi^u_{0,1 } ) \right ] + \langle p[u(1 ) ] , w \rangle = 0 \quad , \forall w \in \mathfrak{x}(m ) . \end{cases } \label{eq : extreme2}\end{aligned}\ ] ] this is an alternative formulation of , obtained simply by taking a time - derivative . in the language of fluid - dynamics , is an eulerian version of . the advantage of this formulation , is that the bulk of the computation occurs on the vector - space @xmath132 , and this observation is the starting point for the algorithm given in @xcite . this reduction of the problem to the space of vector - fields is a first instance of reduction by symmetry . in particular , this corresponds to the fact that the space of vector - fields @xmath132 , is identifiable as a quotient space @xmath143 and the map @xmath144 . is the quotient projection . the reduction to dynamics on @xmath43 to dynamics on @xmath132 occurs primarly because the distance function is @xmath43 invariant . however , one can not completely abandon @xmath43 because the solution requires one to compute the time @xmath145 flow , @xmath146 . fortunately , there is a second reduction which allows us to avoid computing @xmath146 in its entirety . this second reduction corresponds to the invariance properties of the dissimilarity measure @xmath45 . let @xmath147 denote the set of diffeomorphisms which leave @xmath45 invariant , i.e. : @xmath148 one can readily verify that @xmath149 is a subgroup of @xmath43 , and so we call @xmath149 the _ isotropy subgroup of @xmath45_. having defined @xmath149 we can now consider the homogenous space @xmath150 , which is the _ quotient space _ induced by the action of right composition of @xmath149 on @xmath43 . this quotient space is `` smaller '' in the sense of data . in terms of maps , this can be seen by defining the map @xmath151_{/g_f } \in q$ ] , where @xmath152_{/g_f}$ ] denotes the equivalence class of @xmath28 . we call this mapping the _ quotient projection _ because it sends @xmath43 to @xmath153 surjectively . while these notions are theoretically quite complicated , they often manifest more simply in practice . [ ex : two_particles ] let @xmath154 be the closure of some open set . let @xmath155 with @xmath156 and consider the dissimilarity measure @xmath157 we see that @xmath158 and @xmath159 where @xmath160 denotes the diagonal of @xmath161 . the quotient projection is @xmath162 . note that @xmath43 is infinite dimensional while @xmath153 is of dimension @xmath163 . this is massive reduction . for a one - point matching problem with dissimilarity measure @xmath164 visualized by their effect on an initially square grid . the isotropy subgroup leaves @xmath45 invariant by not moving @xmath68 . , title="fig : " ] for a one - point matching problem with dissimilarity measure @xmath164 visualized by their effect on an initially square grid . the isotropy subgroup leaves @xmath45 invariant by not moving @xmath68 . , title="fig : " ] for a one - point matching problem with dissimilarity measure @xmath164 visualized by their effect on an initially square grid . the isotropy subgroup leaves @xmath45 invariant by not moving @xmath68 . , title="fig : " ] if one is able to understand @xmath153 then one can use this insight to reformulate the dissimilarity measure @xmath45 as function on @xmath153 rather than @xmath43 . in particular , there neccessarily exists a unique function @xmath165 defined by the property @xmath166_{/g_f } ) = f(\varphi)$ ] . again , this is useful in the sense of data , as is illustrated in the following example . consider the dissimilarity measure @xmath45 of example [ ex : two_particles ] . the function , @xmath167 is @xmath168 finally , note that @xmath43 acts upon @xmath153 by the left action @xmath169_{g_f } \in \operatorname{diff}(m ) / g_f \stackrel{\psi \in \operatorname{diff}(m ) } { \longmapsto } [ \psi \circ \varphi]_{/g_f } \in \operatorname{diff}(m ) / g_f.\end{aligned}\ ] ] usually we will simply write @xmath170 for the action of @xmath92 on a given @xmath171 . this means that @xmath132 acts upon @xmath153 infinitesimally , as it is the lie algebra of @xmath43 . consider the setup of example [ ex : two_particles ] . here @xmath172 and the left action of @xmath43 is given by @xmath173 for @xmath92 and @xmath174 . the infinitesimal action of @xmath98 on @xmath153 is @xmath175 these constructions allow us to rephrase the initial optimization problem using a reduced curve energy . minimization of @xmath138 is equivalent to minimization of @xmath176 where @xmath177 is obtained by integrating the ode , @xmath178 with the intial condition @xmath179_{/g_f}$ ] where @xmath180 is the identity transformation . we see that this curve energy only depends on the eulerian velocity field and the equivalence class @xmath177 . minimizers of @xmath181 must neccessarily satisfy @xmath142 \quad , \quad m = p[u ] \\ \langle u(1 ) , w \rangle = - df(q ) \cdot ( w \cdot q ) \quad,\quad \forall w \in \mathfrak{x}(m ) . \end{cases } \label{eq : extreme3}\end{aligned}\ ] ] again , the solution only depends on the eulerian velocity and @xmath177 . for this reason , we see that the @xmath149 symmetry of @xmath45 provides a second reduction in the data needed to solve our original problem . in addition to reducing the amount of data we must keep track of there is an additional consequence to the @xmath149-symmetry of @xmath45 . in particular , there is a potentially massive constraint satisfied by the eulerian velocity @xmath74 . to describe this we must introduce an isotropy algebra . given @xmath182_{/g_f}$ ] we can define the ( time - dependent ) isotropy algebra @xmath183 this is nothing but the lie - algebra associated to the isotropy group @xmath184 . it turns out that the velocity field @xmath95 which minimizes @xmath138 ( or @xmath181 ) is orthogonal to @xmath185 with respect to the chosen inner - product . intuitively this is quite sensible because velocities which do not change @xmath186 do not alter the data , and simply waste control effort . this intuitive statement is roughly the content of the following proof . let @xmath74 satisfy or . then @xmath187 $ ] anihillates @xmath185 . let @xmath74 be the solution to . we will first prove that @xmath188 ( this is @xmath74 at time @xmath141 ) is orthogonal to @xmath189 . let @xmath190 . we observe @xmath191 , w(1 ) \rangle \stackrel{\text{by \eqref{eq : extreme3}}}{= } - \left.\frac{d}{d\epsilon}\right|_{\epsilon=0 } f_{q } ( \phi^{w(1)}_\epsilon z(1 ) ) . \end{aligned}\ ] ] however , @xmath192 leaves @xmath193 fixed , so @xmath194 . therefore @xmath195 , w(1 ) \rangle = 0 $ ] . let @xmath196^*w(1)$ ] in coordinates this means @xmath197^{-1}(x ) } [ \phi^u_{t,1}]^i w^j \left(1 , [ \phi^u_{t,1}]^{-1}(x ) \right ) \end{aligned}\ ] ] one can directly verify that @xmath198 for all @xmath76 $ ] . denoting @xmath199 $ ] , as in , we find @xmath200 , w(t ) \rangle & = \frac{d}{dt } \langle m(t ) , w(t ) \rangle \langle \partial_t m , \rangle + \langle m , \partial_t w \rangle \\ & = \langle - \pounds_u [ m ] , w \rangle + \langle m , -\pounds_u[w ] \rangle = 0 . \end{aligned}\ ] ] where the last equality follows from . thus @xmath201 , w(t ) \rangle$ ] is constant . we ve already verified that at @xmath141 , this inner - product is zero , thus @xmath201 , w(t ) \rangle = 0 $ ] for all time . that @xmath192 is an arbitrary element of @xmath202 makes @xmath203 an arbitrary element of @xmath185 at each time . thus @xmath95 is orthogonal to @xmath185 for all time . at this point , we should return to our example to illustrate this idea . [ ex : ex5 ] again consider the setup of example [ ex : two_particles ] . in this case the space @xmath205 is the space of vector - fields which vanish at @xmath206 and @xmath207 . therefore , @xmath95 is orthogonal to @xmath186 if and only if @xmath187 $ ] satisfies @xmath208 for some covectors @xmath209 and for any @xmath113 . in other words @xmath210 where @xmath211 denotes the dirac delta functional cetnered at @xmath68 . this orthogonality constrain allows one to reduce the evolution equation on @xmath132 to an evolution equation on @xmath153 ( which might be finite dimensional if @xmath149 is large enough ) . in particular there is a map @xmath212 uniquely defined by the conditions @xmath213 and @xmath214 with respect to the chosen inner - product on vector - fields . consider the setup of example [ ex : two_particles ] with @xmath215 . then @xmath216 . let @xmath217 be the matrix - valued reproducing kernel of @xmath137 ( see @xcite ) . then @xmath218 is given by @xmath219 where @xmath220 are such that @xmath221 and @xmath222 . one can immediately observe that @xmath59 is injective and linear in @xmath223 . in other words @xmath224 is an injective linear map for fixed @xmath171 . because the optimal @xmath95 is orthogonal to @xmath185 we may invert @xmath225 on @xmath95 . in particular , we may often write the equation of motion on @xmath226 rather than on @xmath132 . this is a massive reduction if @xmath153 is finite dimensional . in particular , the inner - product structure on @xmath132 induces a riemannian metric on @xmath153 given by @xmath227 , v(q , v_2 ) \rangle.\end{aligned}\ ] ] the equations of motion in and map to the geodesic equations on @xmath153 . let @xmath74 extremize @xmath138 or @xmath228 . then there exists a unique trajectory @xmath229 such that @xmath230 . moreover , @xmath186 is a geodesic with respect to the metric @xmath231 . let @xmath74 minimize @xmath138 . thus @xmath74 satisfies . by the previous proposition @xmath95 is orthogonal to @xmath185 . as @xmath232 is injective on @xmath233 , there exists a unique @xmath234 such that @xmath235 . note that @xmath138 can be written as @xmath236 thus , minimizers of @xmath138 correspond to geodesics in @xmath153 with respect to the metric @xmath231 . if we let @xmath237 be the hamiltonian induced by the metric on @xmath153 we obtain the most data - efficient form or and . minimizers of @xmath138 ( or @xmath228 ) are : @xmath238_{/g_f}. \end{cases } \label{eq : extreme4}\end{aligned}\ ] ] we see that this is a boundary value problem posed entirely on @xmath153 . if @xmath153 is finite dimensional , this is a massive reduction in terms of data requirements . consider the setup of example [ ex : two_particles ] with @xmath215 . the metric on @xmath239 is most easily expressed on the cotangent bundle @xmath240 . if @xmath241 is the matrix valued kernel of @xmath137 , the metric on @xmath240 takes the form @xmath242 a related approach to defining distances on a space of objects to be registered consists of defining an object space @xmath243 upon which @xmath43 acts transitively there exists a @xmath244 such that @xmath245 with distance @xmath246 here the distance on @xmath243 is defined directly from the distance in the group that acts on the objects , see for example @xcite . with this approach , the riemannian metric descends from @xmath43 to a riemannian metric on @xmath243 and geodesics on @xmath243 lift by horizontality to geodesics on @xmath43 . the quotient spaces @xmath153 obtained by reduction by symmetry and their geometric structure corresponds to the object spaces and geometries defined with this approach . intuitively , reduction by symmetry can be considered a removal of redundant information to obtain compact representations while letting the metric descend to the object space @xmath243 constitutes an approach to defining a geometric structure on an already known space of objects . the solutions which result are equivalent to the ones presented in this article because @xmath247 where @xmath248 for some fixed reference object @xmath249 . we here give a number of concrete examples of how symmetry reduce the infinite dimensional registration problem over @xmath43 to lower , in some cases finite , dimensional problems . in all examples , the symmetry of the dissimilarity measure with respect to a subgroup of @xmath43 gives a reduced space by quotienting out the symmetry subgroup . the space @xmath153 used in the examples in section [ sec : red ] constitutes a special case of the landmark matching problem where sets of landmarks @xmath250 , are placed into spatial correspondence trough the left action @xmath251 of @xmath43 by minimizing the dissimilarity measure @xmath252 . the landmark space @xmath153 arises as a quotient of @xmath43 from the symmetry group @xmath149 as in in example [ ex : two_particles ] . reduction from @xmath43 to @xmath153 in the landmark case has been used in a series of papers starting with @xcite . landmark matching is a special case of jet matching as discussed below . hamilton s equations take the form @xmath253 on @xmath240 where @xmath254 denotes the spatial derivative of the reproducing kernel @xmath241 . generalizing the situation in example [ ex : ex5 ] , the momentum field is a finite sum of dirac measures @xmath255 that through the map @xmath59 gives an eulerian velocity field as a finite linear combination of the kernel evaluated at @xmath256 : @xmath257 . registration of landmarks is often in practice done by optimizing over the initial value of the momentum @xmath258 in the ode to minimize @xmath138 , a strategy called shooting @xcite . using symmetry , the optimization problem is thus reduced from an infinite dimensional time - dependent problem to an @xmath259 dimensional optimization problem involving integration of a @xmath260 dimensional ode on @xmath240 . the space of smooth non - intersecting closed parametrized curves in @xmath261 is also known as the space of embeddings , denoted @xmath262 . the parametrization can be removed by considering the right action of @xmath263 on @xmath264 given by @xmath265 then the quotient space @xmath266 is the space of _ unparametrized curves_. the space @xmath267 is a special case of a nonlinear grassmannian @xcite . it is not immediately clear if this space is a manifold , although it is certainly an orbifold . in fact the same question can be asked of @xmath268 and @xmath269 . a few conditions must be enforced on the space of embeddings and the space of diffeormophisms in order to impose a manifold structure on these spaces , and these conditions along with the metric determine whether or not the quotient @xmath267 can inherit a manifold structure . we will not dwell upon these matters here , but instead we refer the reader to the survey article @xcite . when the parametrization is not removed , embedded curves and surfaces can be matched with the current dissimilarity measure @xcite . the objects are considered elements of the dual of the space @xmath270 of differential @xmath271-forms on @xmath19 . in the surface case , the surface @xmath38 can be evaluated on a @xmath4-form @xmath122 by @xmath272 where @xmath273 is an orthonormal basis for @xmath274 and @xmath275 the surface element . the dual space @xmath276 is linear an can be equipped with a norm thereby enabling surfaces to be compared with the @xmath276 norm . note that the evaluation does not depend on the parametrization of @xmath38 . the isotropy groups for curves and surfaces generalize the isotropy groups of landmarks by consisting of warps that keeps the objects fixed , i.e. @xmath277 the momentum field will be supported on the transported curves / surfaces @xmath278 for optimal paths for @xmath138 in @xmath43 . images can be registered using either the @xmath50-difference defined in example [ ex : imssd ] or with other dissimilarity measures such as mutual information or correlation ratio @xcite . the similarity will be invariant to any infinitesimal deformation orthogonal to the gradient of dissimilarity measure . in the @xmath50 case , this is equivalent to any infinitesimal deformation orthogonal to the level lines of the moving image @xcite . the momentum field thus has the form @xmath279 for a smooth function @xmath280 on @xmath19 and the registration problem can be reduced to a search over the scalar field @xmath280 instead of vector field @xmath281 . minimizers for @xmath138 follow the pde @xcite @xmath282 with @xmath283 representing the deformed image at time @xmath62 . -difference will be orthogonal to level lines of the image and symmetry implies that the momentum field will be orthogonal to the level lines so that @xmath279 for a time - dependent scalar field @xmath284 . ] in @xcite an extension of the landmark case has been developed where higher - order information is advected with the landmarks . these higher - order particles or _ jet - particles _ have simultaneously been considered in fluid dynamics @xcite , the spaces of jet particles arise as extensions of the reduced landmark space @xmath153 by quotienting out smaller isotropy subgroups . let @xmath285 be the isotropy subgroup for a single landmark @xmath286 let know @xmath271 be a positive integer . for any @xmath271-differentiable map @xmath287 from a neighborhood of @xmath288 , the @xmath271-jet of @xmath287 is denoted @xmath289 . in coordinates , @xmath289 consists of the coefficients of the @xmath271th order taylor expansions of @xmath287 about at @xmath68 . the higher - order isotropy subgroups are then given by @xmath290 that is , the elements of @xmath291 fix the taylor expansion of the deformation @xmath28 up to order @xmath271 . the definition naturally extends to finite number of landmarks , and the quotients @xmath292 can be identified as the sets @xmath293 with @xmath294 being the space of rank @xmath295 tensors . intuitively , the space @xmath296 is the regular landmark space with information about the position of the points ; the 1-jet space @xmath297 carry for each jet information about the position and the jacobian matrix of the warp at the jet position ; and the 2-jet space @xmath298 carry in addition the hessian matrix of the warp at the jet position . the momentum for @xmath296 in coordinates consists of @xmath299 vectors representing the local displacement of the points . with the 1-jet space @xmath296 , the momentum in addition contains @xmath300 matrices that can be interpreted as locally linear deformations at the jet positions @xcite . in combination with the displacement , the 1-jet momenta can thus be regarded locally affine transformations . the momentum fields for @xmath298 add symmetric tensors encoding local second order deformation . the local effect effect of the jet particles is sketched in figure [ fig : discimage ] . when the dissimilarity measure @xmath45 is dependent not just on positions but also on higher - order information around the points , reduction by symmetry implies that optimal solutions for @xmath138 will be parametrized by @xmath271-jets in the same way as @xmath296 parametrize optimal paths for @xmath138 in the landmark case . the higher - order jets can thus be used for landmark matching when the dissimilarity measure is dependent on the local geometry around the landmarks . for example , matching of first order structure such as image gradients lead to 1-order jets , and matching of local curvature leads to @xmath4-order jets . the image matching problem can be discretized by evaluating the @xmath50-difference at a finite number of points . in practice , this alway happens when the integral @xmath301 is evaluated at finitely many pixels of the image . in @xcite , it is shown how this reduces the image matching pde to a finite dimensional system on @xmath153 when the integral is approximated by pointwise evaluation at a grid @xmath302 @xmath303 where @xmath304 denotes the grid spacing . @xmath305 approximates @xmath45 to order @xmath306 , @xmath307 . the reduced space @xmath153 encodes the position of the points @xmath308 , @xmath309 , and the lifted eulerian momentum field is a finite sum of point measures @xmath310 . for each grid point , the momentum encodes the local displacement of the point , see figure [ fig : discimage ] . in @xcite , a discretization scheme with higher - order accuracy is in addition introduced with an @xmath311 approximation @xmath312 of @xmath45 . the increased accuracy results in the entire energy @xmath138 being approximated to order @xmath311 . the solution space in this cases become the jet - space @xmath298 . for a given order of approximation , a corresponding reduction in the number of required discretization points is obtained . the reduction in the number of discretization points is countered by the increased information encoded in each 2-jet . the momentum field thus encodes both local displacement , local linear deformation , and second order deformation , see figure [ fig : discimage ] . the discrete solutions will converge to solutions of the non - discretized problem as @xmath313 . , @xmath304 and the image matching pde is reduced to an ode on a finite dimensional reduced space @xmath153 . with the approximation @xmath305 , the momentum field will encode local displacement as indicated by the horizontal arrows ( top row ) . with a first order expansion , the solution space will be jet space @xmath297 and locally affine motion is encoded around each grid point ( middle row ) . the @xmath311 approximation @xmath312 includes second order information and the system reduces to the jet space @xmath298 with second order motion encoded at each grid point ( lower row ) . ] image matching is symmetric with respect to variations parallel to the level lines of the images . with diffusion weighted images ( dwi ) and the variety of models for the diffusion information ( e.g. diffusion tensor imaging dti cite , gaussian mixture fields cite ) , first or higher - order information can be reintroduced into the matching problem . in essence , by letting the dissimilarity measure depend on the diffusion information , the full @xmath43 symmetry of the image matching problem is reduced to an isotropy subgroup of @xmath43 . the exact form of the of dwi matching problem depends on the diffusion model and how @xmath43 acts on the diffusion image . in @xcite , the diffusion is represented by the principal direction of the diffusion tensor , and the data objects to be match are thus vector fields . the action by elements of @xmath43 is defined by @xmath314 the action rotates the diffusion vector by the jacobian of the warp keeping its length fixed . similar models can be applied to dti with the preservation of principle direction scheme ( ppd , @xcite ) and to gmf based models @xcite . the dependency on the jacobian matrix implies that a reduced model must carry first order information in a similar fashion to the 1-jet space @xmath297 , however , any irrotational part of the jacobian can be removed by symmetry . the full effect of this has yet to be explored . incidentally , the equation of motion @xmath315 = 0 \\ u = k * m\end{aligned}\ ] ] is an eccentric way of writing euler s equation for an invicid incompressible fluid if we assume @xmath74 is initially in the space of divergence free vector - fields and @xmath241 is a dirac - delta distribution ( which impies @xmath316 . ) this fact was exploited in @xcite to create a sequece of regularized models to euler s equations by considering a sequence of kernels which converge to a dirac - delta distribution . moreover , if one replaces @xmath43 by the subgroup of volume preserving diffeomorphisms @xmath317 , then ( formally ) one can produce incompressible particle methods using the same reduction arguments presented here . in fact , jet - particles were independently discoverd in this context as a means of simulating fluids in @xcite . it is notable that @xcite is a mechanics paper , and the particle methods which were produced were approached from the perspective of reduction by symmetry . in @xcite one of the kernel parameters in @xcite which controls the compressibility of the @xmath74 was taken to the incompressible limit . this allowed a realization of the particle methods described in @xcite . the constructions of @xcite is the same as presented in this survey article , but with @xmath43 replaced by the group of volume preserving diffeomorphisms of @xmath318 , denoted @xmath319 . the information available for solving the registration problem is in practice not sufficient for uniquely encoding the deformation between the objects to be registered . symmetry thus arises in both particle relabeling symmetry that gives the eulerian formulation of the equations of motion and in symmetry groups for specific dissimilarity measures . for landmark matching , reduction by symmetry reduces the infinite dimensional registration problem to a finite dimensional problem on the reduced landmark space @xmath153 . for matching curves and surfaces , symmetry implies that the momentum stays concentrated at the curve and surfaces allowing a reduction by the isotropy groups of warps that leave the objects fixed . in image matching , symmetry allows reduction by the group of warps that do not change the level sets of the image . jet particles have smaller symmetry groups and hence larger reduced spaces @xmath297 and @xmath298 that encode locally affine and second order information . reduction by symmetry allow these cases to be handled in one theoretical framework . we have surveyed the mathematical construction behind the reduction approach and its relation to the above mentioned examples . as data complexity rises both in term of resolution an structure , symmetry will continue to be an important tool for removing redundant information and achieving compact data representations . hoj would like to thank darryl holm for providing a bridge from geometric mechanics into the wonderful world of image registration algorithms . hoj is supported by the european research council advanced grant 267382 fcca . ss is supported by the danish council for independent research with the project `` image based quantification of anatomical change '' . d. c. alexander , j. c. gee , and r. bajcsy , _ strategies for data reorientation during non - rigid warps of diffusion tensor images _ , medical image computing and computer - assisted intervention miccai99 ( chris taylor and alain colchester , eds . ) , lecture notes in computer science , no . 1679 , springer berlin heidelberg , january 1999 , pp . 463472 ( en ) . r abraham and j e marsden , _ foundations of mechanics _ , benjamin / cummings publishing co. inc . advanced book program , reading , mass . , 1978 , second edition , revised and enlarged , with the assistance of tudor ratiu and richard cushman . reprinted by ams chelsea , 2008 . d. c. alexander , c. pierpaoli , p. j. basser , and j. c. gee , _ spatial transformations of diffusion tensor magnetic resonance images _ , ieee transactions on medical imaging * 20 * ( 2001 ) , no . 11 , 11311139 ( eng ) . thomas brox , andrs bruhn , nils papenberg , and joachim weickert , _ high accuracy optical flow estimation based on a theory for warping _ , computer vision - eccv 2004 ( toms pajdla and ji matas , eds . ) , lecture notes in computer science , no . 3024 , springer berlin heidelberg , january 2004 , pp . 2536 ( en ) . richard g. boyes , daniel rueckert , paul aljabar , jennifer whitwell , jonathan m. schott , derek l. g. hill , and nicholas c. fox , _ cerebral atrophy measurements using jacobian integration : comparison with the boundary shift integral _ , neuroimage * 32 * ( 2006 ) , no . 1 , 159169 ( eng ) . guang cheng , baba c. vemuri , paul r. carney , and thomas h. mareci , _ non - rigid registration of high angular resolution diffusion images represented by gaussian mixture fields _ , proceedings of the 12th international conference on medical image computing and computer - assisted intervention : part i ( berlin , heidelberg ) , miccai 09 , springer - verlag , 2009 , pp . 190197 . r. derfoul and c. le guyader , _ a relaxed problem of registration based on the saint venant kirchhoff material stored energy for the mapping of mouse brain gene expression data to a neuroanatomical mouse atlas _ , siam journal on imaging sciences ( 2014 ) , 21752195 . suma dawn , vikas saxena , and bhudev sharma , _ remote sensing image registration techniques : a survey _ , image and signal processing ( abderrahim elmoataz , olivier lezoray , fathallah nouboud , driss mammass , and jean meunier , eds . ) , lecture notes in computer science , no . 6134 , springer berlin heidelberg , january 2010 , pp . 103112 ( en ) . vladlena gorbunova , sander s. a. m. jacobs , pechin lo , asger dirksen , mads nielsen , alireza bab - hadiashar , and marleen de bruijne , _ early detection of emphysema progression _ , medical image computing and computer - assisted intervention : miccai ... international conference on medical image computing and computer - assisted intervention * 13 * ( 2010 ) , no . pt 2 , 193200 ( eng ) . joan glauns , _ transport par diffomorphismes de points , de mesures et de courants pour la comparaison de formes et lanatomie numrique _ , ph.d . thesis , universit paris 13 , villetaneuse , france , 2005 . sarang c. joshi , michael i. miller , and ulf grenander , _ on the geometry and shape of brain sub - manifolds _ , international journal of pattern recognition and artificial intelligence * 11 * ( 1997 ) , no . 08 , 13171343 . alexis roche , grgoire malandain , xavier pennec , and nicholas ayache , _ the correlation ratio as a new similarity measure for multimodal image registration _ , proceedings of the first international conference on medical image computing and computer - assisted intervention , miccai 98 , springer - verlag , 1998 , acm i d : 709612 , pp . 11151124 . d rueckert , l i sonoda , c hayes , d l hill , m o leach , and d j hawkes , _ nonrigid registration using free - form deformations : application to breast mr images _ , ieee transactions on medical imaging * 18 * ( 1999 ) , no . 8 , 712721 .

we survey the role of symmetry in diffeomorphic registration of landmarks , curves , surfaces , images and higher - order data . the infinite dimensional problem of finding correspondences between objects can for a range of concrete data types be reduced resulting in compact representations of shape and spatial structure . this reduction is possible because the available data is incomplete in encoding the full deformation model . using reduction by symmetry , we describe the reduced models in a common theoretical framework that draws on links between the registration problem and geometric mechanics . symmetry also arises in reduction to the lie algebra using particle relabeling symmetry allowing the equations of motion to be written purely in terms of eulerian velocity field . reduction by symmetry has recently been applied for jet - matching and higher - order discrete approximations of the image matching problem . we outline these constructions and further cases where reduction by symmetry promises new approaches to registration of complex data types .

1412.7513

in 1968 , mack , wilson and gell - mann recognized that scale invariance is a broken symmetry of strong interactions@xcite . in 1969 , salam and strathedee showed that to formulate a broken chiral as well as scale symmetry within an effective lagrangian approach one has to assume the existence of a chirally invariant scalar field @xmath16 of dimension 1@xcite . in 1970 , ellis proposed to identify this scalar field with the @xmath17 meson@xcite the existence of which was suggested by earlier measurements of forward - backward asymmetry in @xmath18@xcite . the scalar meson dominance of the trace of the energy - momentum tensor ( also referred to as a partially conserved dilatation current ) has been used to study the couplings of the @xmath17 meson@xcite . with the advent of qcd in 1970 s it has been recognized that the quantization of qcd lagrangian leads to breaking of scale invariance in qcd . the anomalous breaking of scale symmetry results in qcd scale anomaly which was shown@xcite to have the form @xmath19 here @xmath20 and @xmath21 are the gluon field strength and the quark field with running mass @xmath22 . @xmath23 and @xmath24 are the gell - mann @xmath23-function and quark anomalous dimension@xcite . the summation over colour is understood . @xmath25 is the trace of the energy - momentum tensor . in the absence of a technology to solve the fundamental qcd theory and find the hadron spectrum and the interactions of the composite states , we use the effective lagrangian method to study the hadron dynamics at low energies@xcite . the basic ingredient in constructing effective lagrangians is anomaly matching . the effective lagrangian must posses not only the symmetries but also the anomalies of the original qcd theory@xcite . in 1981 , schechter suggested that a scalar gluonium field @xmath16 plays a major role in effective qcd lagrangian through its connection to the qcd trace anomaly@xcite . effective lagrangians with such dilaton - gluonium field @xmath16 were subsequently examined from various aspects in a series of studies@xcite . in 1985 , ellis and lanik@xcite constructed an effective qcd lagrangian with broken scale and chiral symmetry in which the dilaton - gluonium scalar field @xmath26 is related to the scalar gluonic current @xmath27 by a relation @xmath28 in eq . ( 1.2 ) @xmath10 is the @xmath16 meson mass and @xmath29 is related to gluon condensate @xmath30 @xmath31 by an approximate relation@xcite @xmath32 the gluon condensate @xmath30 parametrizes the nonperturbative effects of qcd and is related to the energy density of qcd vacuum . the relation ( 1.2 ) is unique to ellis - lanik lagrangian . starting with the salam - strathedee chirally invariant field @xmath33 , it is the result of matching of the qcd trace anomaly in gluonic sector with the trace of the energy - momentum tensor of the @xmath16 field@xcite and low - energy theorems for scalar gluonic current @xmath27@xcite . from their lagrangian ellis and lanik derived the following relations for @xmath16 decay widths @xmath34 @xmath35 where @xmath36 . the appearance of the gell - mann function @xmath23 in the scale anomaly ( 1.1 ) reflects the qcd confinement . in the ellis - lanik lagrangian the @xmath16 field codes the qcd confinement which is often a missing feature in other effective qcd lagrangians . the cern measurements of @xmath18 and @xmath37 on polarized targets reopened the question of existence of the @xmath38 meson . these measurements allow a model independent determination of normalized production amplitudes , including the two @xmath1-wave transversity amplitudes . evidence for a narrow @xmath38 resonance was found in amplitude analyses of cern data on @xmath18 at 17.2 gev / c in the mass range 600 - 900 mev and on @xmath37 at 5.98 and 11.85 gev / c in the mass range 580 - 980 mev@xcite . further evidence was found recently in amplitude analysis of measurements @xmath18 on polarized target at 1.78 gev / c at itep@xcite . our new amplitude analysis@xcite of the cern measurements of @xmath18 on polarized targets at 17.2 gev / c and momentum transfer @xmath39 = 0.005 - 0.20 @xmath40 extends the mass range to 580 - 1080 mev and allows to study the effects of @xmath41 interference . there are two solutions for the unnormalized moduli @xmath42 and @xmath43 of the two @xmath1-wave transversity amplitudes @xmath44 and @xmath1 corresponding to recoil nucleon transversity `` up '' and `` down '' relative to the scattering plane . here @xmath45 is the integrated cross - section . both moduli in both solutions exhibit a resonant behaviour around 750 - 780 mev . in our analysis@xcite we supplement the cern data with an assumption of analyticity of production amplitudes in dipion mass . analyticity allows to parametrize the transversity amplitudes @xmath1 and @xmath44 as a sum of breit - wigner amplitudes for @xmath38 and @xmath46 with complex coefficients and a complex background . next we performed simultaneous fits to the moduli @xmath43 and @xmath42 in the four solution combinations @xmath6 , @xmath7 , @xmath8 and @xmath9 . in each solution combination we obtained two fits , a and b , with the same resonance parameters for @xmath38 and @xmath46 and the same @xmath47 . the average values of @xmath16 mass and width are @xmath48 mev and @xmath49 mev . the transversity amplitudes @xmath1 and @xmath44 are linear combinations of nucleon helicity nonflip amplitude @xmath3 and nucleon helicity flip amplitude @xmath4 corresponding to @xmath50 and @xmath51 exchange in the @xmath52-channel , respectively . these amplitudes are physically interesting since the residue of the pion pole in @xmath4 is related to the @xmath1-wave partial wave in @xmath5 scattering . the residue of the @xmath50 pole in @xmath3 is related to the @xmath1-wave partial wave in @xmath53 scattering . analyticity imparts the fitted transversity amplitudes with an absolute phase . this allows to determine the helicity amplitudes from the fitted transversity amplitudes . due to a sign ambiguity there are two solutions , `` up '' and `` down '' in each solution combination and in each fit a and b. in the `` down '' solution the @xmath2 is suppressed in the flip amplitude @xmath4 while it dominates the nonflip amplitude @xmath3 . in the `` up '' solution the situation is reversed . in ref . @xcite we show that unitarity in @xmath5 scattering excludes the `` up '' solution . furthermore , the `` down '' solution - and thus the evidence for the narrow @xmath2 - is in agreement with unitarity in both @xmath5 and @xmath53 scattering . in the `` down '' solution the narrow @xmath2 manifests itself as a broad resonance centered at @xmath54 720 mev in the helicity flip mass spectra @xmath55 in solution combinations @xmath6 and @xmath7 . in contrast , the contribution of @xmath2 is small in @xmath55 in combinatins @xmath8 and @xmath9 . the dual manifestation of the @xmath2 as a narrow resonance in nonflip amplitude @xmath3 and as a broad resonance in the flip amplitude @xmath4 is connected to symmetries of qcd . in this paper we show that ellis - lanik relation ( 1.5 ) selects the solution combinations @xmath6 and @xmath8 . weinberg s mended symmetry of spontaneously broken chiral symmetry@xcite selects solutions @xmath6 and @xmath7 . the combined preferred solution is thus @xmath6 for which @xmath12 mev and @xmath13 mev . the agreement with ellis - lanik relation imparts the @xmath2 resonance with a dilaton - gluonium interpretation and relates its existence to breaking of scale symmetry in qcd . at the same time , the degeneracy of @xmath2 mass with @xmath56 and the metamorphosis of the narrow @xmath2 into a broad @xmath1-wave structure in @xmath5 scattering is in agreement with weinberg s mended symmetry which predicts @xmath57 with a broad width@xcite . we suggest that the broad resonant @xmath1-wave structure in @xmath5 scattering associated with the spontaneous breaking of chiral symmetry is due to the interaction of the chirally invariant and narrow @xmath2 with the chirally noninvariant qcd vacuum in @xmath5 scattering . the paper is organized as follows . in section ii . we use the most recent values of gluon condensate to show that ellis - lanik relation ( 1.5 ) selects the solutions @xmath6 and @xmath8 . in section iii . we show how weinberg s mended symmetry selects the solutions @xmath58 and @xmath7 and comment on the role of qcd vacuum in hadron reactions . the observed degeneracy of @xmath2 and @xmath59 is used in section iv . to derive new relations between gluon condensate and pion decay constant , and between chiral condensate and gluon condensate . we find that these relatons are well satisfied . the paper closes with a summary and remarks in section v. the moduli of the @xmath1-wave transversity amplitudes @xmath44 and @xmath1 measured in @xmath18 at 17.2 gev / c on polarized target are shown in figure 1 . the figure shows the two solutions for @xmath42 and @xmath43 and the results of simultaneous fits to the four solution combinations @xmath6 , @xmath7 , @xmath8 and @xmath9 . two fits , a and b , give essentially identical curves , the same resonance parameters for @xmath38 and @xmath46 , and the same @xmath47@xcite . the transversity `` up '' amplitude @xmath42 clearly resonates below 800 mev while the transversity `` down '' amplitude @xmath43 shows a broader structure around 800 mev . in table i we present the mass @xmath10 and the width @xmath61 of the @xmath16 resonance for the best fits in each solution combination . the measured width @xmath61 is the total width of the @xmath16 decays . to compare our results with the ellis - lanik relation ( 1.5 ) we need partial width @xmath62 where @xmath63 is the fraction of charged @xmath64 decays . as we shall see below , the partial width @xmath65 is small . we also assume that @xmath66 decays of @xmath16 are suppressed . then isospin conservation requires that @xmath67 . with this value of @xmath63 we then calculate the value @xmath30 of the gluon condensate from ellis - lanik relation for each solution combination . the numerical values of @xmath30 were estimated originally by the itep group@xcite to be @xmath68 ( gev)@xmath69 or up to @xmath70 ( gev)@xmath69 in later calculations@xcite . more recent estimates are @xmath71 ( gev)@xmath69 @xcite and @xmath72 ( gev)@xmath69@xcite . the latest estimate@xcite based on analysis of the @xmath73 violating ratio @xmath74 is @xmath75 ( gev)@xmath69 . the most recent estimate based on charmonium sum rules@xcite gives a similar result @xmath76 ( gev)@xmath69 . this new itep value includes a contribution from @xmath77 condensate in dilute instanton model@xcite . in this paper we will use the average value of the two last results @xmath78 the comparison of @xmath30 in ( 2.1 ) with the results for @xmath30 from ellis -lanik relation in table i shows the best agreement is for solution combinations @xmath6 and @xmath8 . the ellis - lanik relation thus selects the solutions @xmath6 and @xmath8 and imparts the @xmath2 resonance with a dilaton - gluonium interpretation . we can now use the relation ( 1.6 ) to calculate the decay width @xmath79 . assuming @xmath80 and @xmath67 we obtain results shown in table i. we see that @xmath79 is too small for @xmath2 to be observed in @xmath81 decays . this result is in full agreement with the measurements of reactions @xmath82@xcite and @xmath83@xcite which found no evidence of resonance structure below @xmath46 . the fact that @xmath2 is not observed in these reactions strongly suggests a gluonium interpretation of this state since gluons do not couple directly to photons . these experiments support the ellis - lanik identification of the scalar dilaton field @xmath16 with a scalar gluonium which they based solely on the dominance of the scalar gluonic current @xmath27 by the field @xmath16@xcite . a theoretical support for the gluonium interpretation of @xmath2 comes from recent studies of qcd sum rules . it was shown by elias and collaborators in @xcite that qcd sum rules require that a scalar state with a mass below 800 mev has a narrow width . several analyses using qcd sum rules concluded that the scalar states with a mass below @xmath46 are non-@xmath84 states@xcite . the qcd sum rules are thus consistent with the identification of the narrow @xmath2 resonance with the lowest mass scalar gluonium . in 1990 , weinberg showed that even where a symmetry is spontaneously broken it can still be used to classify hadron states@xcite . such mended symmetry leads to a quartet of particles with definite mass relations and c - parity . he showed that @xmath51 , @xmath16 , @xmath85 and @xmath50 mesons corresponding to spontaneously broken chiral symmetry of strong interactions occupy the quartet . weinberg s mended symmetry predicts @xmath57 in agreement with our results for @xmath10 shown in table i. central to weinberg s mended symmetry are current algebra and superconvergence sum rules for helicity amplitudes . in 1968 , gilman and harari studied saturation of such sum rules in a framework of a linear sigma model@xcite . they found that @xmath57 and @xmath86 which gives @xmath87 mev . a broad resonant structure below 900 mev in the @xmath1-wave in @xmath5 is required to saturate the adler - weisberger sum rule for @xmath88 scattering and to produce the equality @xmath57 . a narrow resonance like @xmath2 will not do it . but then , what does saturate the adler - weisberger sum rule for @xmath5 scattering?@xcite we suggest that the solutions @xmath6 and @xmath7 provide the answer . in these two solutions the narrow @xmath2 manifests itself as a broad resonant structure at @xmath54 720 mev in the flip helicity amplitude @xmath4 . this broad resonant structure will appear also in the @xmath1-wave in @xmath5 and we conjecture that it will saturate the adler s sum rule as required by weinberg s mended symmetry . weinberg s mended symmetry then selects the solution combinatins @xmath6 and @xmath7 . when combined with the ellis - lanik relation , the preferred solution is @xmath6 . from table i we see that the corresponding @xmath12 mev compares well with particle data group value @xmath89 mev@xcite . in the following we thus refer to the @xmath16 resonance as @xmath2 . a question now arises how a chirally invariant and narrow resonance in the nonflip amplitude @xmath3 can also manifest itself as a broad resonant state breaking chiral symmetry in the flip amplitude @xmath4 , i.e. in @xmath5 scattering . we suggest that the answer may lie in the interaction of the chirally invariant @xmath2 with the chirally noninvariant qcd vacuum in the @xmath5 scattering . it is well - known that qed vacuum participates in electromagnetic processes . for instance , the photon propagator is modified by the vacuum polarization@xcite . in ref . @xcite it is shown that the scalar gluonic current interacts strongly with qcd vacuum . recently kharzeev and collaborators demonstrated that qcd vacuum participates in hadron interactions at low as well as at high energies with involvement of the qcd trace anomaly@xcite . in general , the involvement of the qcd vacuum may depend on the hadronic process and in @xmath5 scattering it could result in the metamorphosis of the narrow and chirally invariant @xmath2 into a broad resonant structure associated with breaking of the chiral symmetry . chiral symmetry allows for two phases of qcd vacuum : asymmetric and degenerate nambu - goldstone phase and a symmetric wigner - weyl phase@xcite . the interaction of the chirally invariant @xmath2 with qcd vacuum in @xmath5 scattering could also modify the vacuum itself@xcite . brown and rho used the dilaton - gluonium scalar field @xmath16 and ellis - lanik effective lagrangian to describe the change of qcd vacuum in a dense medium@xcite . superconvergence sum rules assumed in weinberg s mended symmetry are based on analyticity of helicity amplitdes . analyticity connects hadron dynamics at low and high energies and is thus used also in amplitude analyses of hadronic reactions . qcd vacuum participates in hadron scattering also at low and high enegies@xcite . the reason why analyticity seems to reflect important aspects of hadron dynamics could be that it originates in some way from the structure of qcd vacuum . finally we note that in 1968 when gilman and harari published their paper@xcite , only @xmath16 and @xmath85 were assumed to contribute to @xmath88 scattering . today we know of many more resonances the contribution of which should be taken into account in the saturation of the sum rules each of which becomes more complicated . also , the presence of these resonances enlarges the set of sum rules equations . we suggest that the sum rules equations factorize into a chain of solvable sets with the gilman - harari set of @xmath90 and @xmath50 forming the first ( lowest mass ) set . these sets could in fact be the representations ( multiplets ) of the weinberg s mended symmetry . if such is the case , then mended symmetry could provide a new organizing principle of the hadron spectra and effective qcd dynamics . the mass and width of the @xmath16 resonance in the preferred solution combination @xmath6 are from table i @xmath92 a comparison with particle data group@xcite values for mass and width of the @xmath93 meson @xmath94 indicates a strong @xmath95 degeneracy . this experimentally observed @xmath96 degeneracy leads to some interesting consequences . in the following we set @xmath57 and @xmath97 . using the usual @xmath98 coupling , the @xmath99 decay width is given by@xcite @xmath100 where @xmath101 is the pion decay constant and @xmath102 is the pion momentum in the rest frame of @xmath93 . with @xmath62 , the ellis - lanik relation ( 1.5 ) then relates the gluon condensate and the pion decay constant @xmath103 the @xmath15 chiral condensate @xmath104 is related to the pion decay constant by the gell - mann , oakes and renner ( gmor ) relation@xcite @xmath105 the new measurements of @xmath106 decay by e865 collaboration at bnl@xcite have been used recently to verify the validity of the gmor relation@xcite . combining ( 4.4 ) and ( 4.5 ) we get a new relation for chiral condensate in terms of the gluon condensate @xmath107 the new relations ( 4.4 ) and ( 4.6 ) are a nontrivial consequence of ellis - lanik effective lagrangian which is coding the breaking of scale symmetry and confinement in qcd and the cern measurements of pion production on polarized target . with pdg value @xmath101 = 92.4 mev , the relation ( 4.4 ) gives the following value for the gluon condensate @xmath108 with pdg average value @xmath109 = 8 mev and using @xmath30 from ( 4.7 ) we obtain from relation ( 4.6 ) a value of chiral condensate @xmath110 the value of @xmath30 is in agreement with the value from ellis - lanik relation for the solution combination @xmath6 shown in table i and in a reasonable agreement with the accepted value ( 2.1 ) . the value of the qcd scale @xmath111 from ( 4.8 ) is in agreement with recent expectations . it is interesting to calculate the gluon and chiral condensates for @xmath112 @xmath113 @xmath114 comparing the values ( 4.7 ) and ( 4.9 ) for the gluon condensate @xmath30 we notice that the value ( 4.9 ) for @xmath63 = 1 is in a better agreement with the accepted value @xmath30 = 0.0117 @xmath115 0.0051 gev@xmath69 in ( 2.1 ) . we note that the values of @xmath30 calculated from ellis - lanik relation assuming @xmath112 are also closer to the accepted value of @xmath30 in ( 2.1 ) but still selecting solutions @xmath6 and @xmath8 with @xmath30 = 0.0116 @xmath115 0.0028 gev@xmath69 and @xmath30 = 0.0122 @xmath115 0.0056 gev@xmath69 , respectively . it appears that these improved agreements with the accepted value of gluon condensate require an additional factor of @xmath116 in the ellis - lanik relation ( 1.5 ) and in relations ( 4.4 ) and ( 4.6 ) . however we note that @xmath2 is suppressed in @xmath1-wave intensity in @xmath117 at 18.3 gev / c@xcite . this suggests that @xmath118 so that @xmath119 and improved agrrements follow . an additional term in the ellis - lanik effective lagrangian may thus be required by the @xmath120 production data to suppress or cancel the @xmath121 coupling . cern measurements of @xmath18 on polarized target enable experimental determination of partial wave production spin amplitudes . the measured @xmath1-wave transversity amplitudes provide evidence for existence of a narrow scalar resonance @xmath2 . the assumption of analyticity of production amplitudes in dipion mass allows the deterimation of the @xmath1-wave helicity amplitudes @xmath3 and @xmath4 . the unitarity in @xmath5 selects `` down '' solutions @xmath6 , @xmath7 , @xmath8 and @xmath9 . ellis - lanik relation selects solutions @xmath6 and @xmath8 while weinberg s mended symmetry selects solutions @xmath6 and @xmath7 . the combined preferred solution @xmath6 has the @xmath16 meson mass and width degenerate with @xmath59 . the agreement with ellis - lanik relation imparts the @xmath2 resonance with a dilaton - gluonium interpretation and relates its existence to breaking of scale symmetry in qcd . the qcd vacuum participates in hadron reactions and the interaction of chirally invariant and narrow @xmath2 with chirally noninvarinat qcd vacuum in @xmath5 scattering is proposed as the mechanism for its metamorphosis into a broad resonant @xmath1-wave structure in @xmath5 related to spontaneous breaking of chiral symmetry . the observed @xmath122 degeneracy leads to two new relations for pion decay constant and @xmath15 chiral condensate which are both well satisfied experimentally . scalar mesons are probes of qcd vacuum@xcite and the complexity of scalar states reflects the complex structure of the qcd vacuum . there is an emerging evidence for a @xmath123 resonance from tau decays @xmath124@xcite , from @xmath125 decays@xcite and from measurements of @xmath117 at 18.3 gev / c@xcite . the cern measurement of @xmath126 on polarized target at 5.98 gev / c suggests existence of a strange scalar @xmath127@xcite . several recent investigations examined the possibility of a scalar nonet below 1 gev@xcite . however a non-@xmath128 structure of @xmath123 and @xmath127 has been also proposed@xcite . high statistics measurements of @xmath117 and @xmath129 on polarized targets@xcite would help clarify the evidence for these scalar states . due to its dilaton - gluonium nature , the @xmath2 may play a special and a broader role . estimate of the cosmological constant from qcd scale anomaly yields a value in a remarkable agreement with the latest astrophysical data@xcite . a dilatonic nature of the quintessence has been recently proposed by damour , piazza and veneziano@xcite . bulk dilaton field in ads brane world models can be stabilized on the brane and allow for an interaction of the dilaton field with the standard matter through its fluctuation corresponding to an observable massive dilaton@xcite . qcd / string duality stemming from the ads / cft correspondence@xcite in ads brane world model predicts the lowest mass of the scalar gluonium @xmath130@xcite where @xmath132 is the string coupling constant , @xmath133 is the string energy scale , @xmath134 is the number of branes and @xmath135 is the ads radius . it is inviting to identify the predicted @xmath136 with the mass of @xmath2 . the resonance parameters of @xmath2 may then be related to large scale structure and evolution of the universe . lattice qcd calculations in pure - gauge theory predict large masses for @xmath130 gluonium . recent calculations predict @xmath137 mev@xcite . lower masses are obtained in lattice calculations with scalar glueball - scalar meson mixing in a quenched approximation@xcite . implementing chiral symmetry in lattice calculations can also modify the predicted mass of the scalar gluonium@xcite . conformal symmetry may be another additional ingredient in lattice calculations that may bring the predicted gluonium mass closer to the 770 mev inferred from the cern measurements of pion production on polarized target . the @xmath14 degeneracy leads to observable effects in hadron reactions . the degeneracy implies that the @xmath2 regge trajectory @xmath138 is a daugther trajectory of the @xmath56 regge trajectory @xmath139 . such @xmath16 exchange regge amplitude is required to describe the observed anomalous energy dependence of polarizations in @xmath140 and @xmath141 elastic scattering and the deviation from the mirror symmetry in polarizations in @xmath142 and @xmath143 elastic scattering at intermediate energies@xcite . this indicates that polarization measurements of hadron reactions even at intermediate energies may provide a testing ground for the renewed efforts to develop regge amplitudes from ads / cft and qcd / string duality@xcite . the regge behaviour of helicity amplitudes is essential to superconvergence sum rules assumed in weinberg s mended symmetry@xcite . the relationship between nonlinearly realized conformal invariance developed by salam and strathdee@xcite and the breaking of chiral symmetry and its connection to mended symmetry was first discussed by beane and van kolck in 1994@xcite . a remarkable feature of hadron collisions is the conversion of the kinetic energy of colliding hadrons into the matter of produced particles . the pion production is the simplest of these matter creation processes . the measurements of pion production on polarized targets access the production process on the fundamental level of spin amplitudes rather than spin averaged cross - sections . our results emphasize the need for a dedicated and systematic study of various production processes on the level of spin amplitudes measured in experiments with polarized targets . such `` amplitude spectroscopy '' will be feasible at high intensity hadron facilities@xcite . the first high intensity hadron facility will be the japan hadron facility at kek . it will become operational in 2007 and the formation of its experimental program is now in progress@xcite . i wish to thank v. elias for his interest and helpful e - mail correspondence concerning qcd sum rules , scale invariance and @xmath16 meson . i thank m. rho for bringing to my attention the role of @xmath16 meson in the change of qcd vacuum in a dense medium and to references@xcite . my special thanks go to s. weinberg for stimulating and encouraging e - mail correspondence concerning @xmath16 meson and mended symmetry . .mass @xmath10 and width @xmath61 from simultaneous fits to amplitudes @xmath144 and @xmath145 in four combinations of solutions of @xmath144 and @xmath145 . the gluon condensate @xmath30 and the width @xmath146 are calculated from ellis -lanik relations ( 1.5 ) and ( 1.6 ) with @xmath67 . [ cols="^,^,^,^,^ " , ]

cern measurements of @xmath0 on polarized target at 17.2 gev / c enable experimental determination of partial wave production amplitudes below 1080 mev . the measured @xmath1-wave transversity amplitudes provide evidence for a narrow scalar resonance @xmath2 . the assumption of analyticity of production amplitudes in dipion mass allows to determine @xmath1-wave helicity amplitudes @xmath3 and @xmath4 . the helicity flip amplitude @xmath4 is related to @xmath5 scattering . there are four `` down '' solutions @xmath6 , @xmath7 , @xmath8 and @xmath9 selected by the unitarity in @xmath5 scattering . ellis - lanik relation between the mass @xmath10 and partial width @xmath11 derived from an effective qcd theory with broken scale and chiral symmetry selects solutions @xmath6 and @xmath8 and imparts the @xmath2 resonance with a dilaton - gluonium interpretation . weinberg s mended symmetry selects solutions @xmath6 and @xmath7 . the combined solution @xmath6 has @xmath12 mev and @xmath13 mev . the observed @xmath14 degeneracy leads to new relations between gluon condensate and pion decay constant and @xmath15 chiral condensate . the two relations are well satisfied . ellis - lanik relation relates the existence of @xmath2 to breaking of scale symmetry in qcd . interaction of chirally invariant and narrow @xmath2 with chirally noninvariant qcd vacuum in @xmath5 scattering is proposed as a possible mechanism for the metamorphosis of @xmath2 into a broad resonant @xmath1-wave structure in @xmath5 related to spontaneous breaking of chiral symmetry in qcd . we comment on possible connections of @xmath2 to cosmological contstant and qcd / string duality in ads / cft brane world model .

hep-ph0209323

since the first radio pulsar , cp 1919 , was discovered in november 1967 ( hewish , bell , et al . 1968 ) , more and more radio pulsars are found , the number of which reaches about 750 ( becker & trumper 1997 ) . these objects are now almost universally believed to be rotating magnetized neutron stars . however , it was suggested that there might be no neutron star , only strange stars ( e.g. alock et al.1986 ) . hence , one of the very interesting and most fundamental questions is ` what is the nature of pulsars?'(i.e . do the signals of pulsars come from neutron stars or from strange stars ? ) . unfortunately , the question could still not be answered with certainty even now ( broderick 1998 ) . soon after the discovery of pulsars , by removing the possibilities of the pulse periods due to white dwarf pulsation or duo to rapid orbital rotation ( see , e.g. a review by smith 1977 ) , many people widely accept the concept that the pulsars are the neutron stars , which were conceived as theoretically possible stable structures in astrophysics ( landau 1932 , oppenheimer & volkoff 1939 ) . following this , many authors discussed the inner structure of neutron stars , especially the properties of possible quark phase in the neutron star core ( e.g. wang & lu 1984 ) . as the hypothesis that strange matter may be the absolute ground state of the strong interaction confined state has been raised ( bodner 1971 ; witten 1984 ) , farhi & jaffe ( 1984 ) point out that the energy of strange quark matter is lower than that of matter composed by nucleus for quantum chromodynamical parameters within rather wide range . hence , strange stars , that might be considered as the ground state of neutron stars , could exist , and the observed pulsars might be strange stars ( alcock et al . 1986 , kettner et al . therefore , the question about the nature of pulsars , which seems to have been answered , rises again . now , there are two kinds of candidates for the pulsar s correspondence object : one is the classical neutron star , and another is the strange star which was proposed about a decade ago . the key point on this question is to find the difference of the behaviors of strange stars and neutron stars , both observationally and theoretically . as both the strange - star models and neutron - star models for pulsars predict the observed pulsars mass ( @xmath4 , from double - pulsars systems ) and radius ( @xmath5 cm , from x - ray bursters ) , it is hard to differentiate these two type models in observations . the dynamically damping effects ( wang & lu 1984 ; dai & lu 1996 ) , the minimum rotation periods ( friedman & olinto 1989 ) , the cooling curves ( benvenuto & vucetich 1991 ) and the vibratory mode ( broderick et al . 1998 ) have been discussed in detail in the literature . however , no direct observational clue has yet shown that the pulsars are neutron stars or strange stars . while , in the terrestrial physics , to search the new state of strong interaction matter , the so - called quark - gluon plasma ( qgp ) , is the primary goal of relativistic heavy - ion laboratory ( mclerran 1986 , muller 1995 ) . many proposed qgp signatures have been put forward in theory and analyzed in experimental data , but the conclusion about the discovery of qgp are ambiguities . more likely , it is suggested in theory that there is a possibility of existing strange hardron cluster ( schaffner et al . 1993 ) and strangelet ( benvenuto & lugones 1995 ) , however , no experiment has affirmed or disaffirmed the suggestion . this laboratory physics researches should be inspired if the pulsars are distinguished as strange stars rather than neutron stars . also , the rudimental strangelet in the early universe might have implications of fundamental importance for cosmology ( e.g. the dark matters , witten 1984 ) . almost all of the proposed strange star models for pulsars have addressed the case generally contemplated by most authors that the strange star core is surrounded by a normal matter crust ( alcock et . al . 1986 ; kettner et al . 1995 ) . the essential features of this core - crust structure is that the normal hadron crust with @xmath6 and the strange quark matter core with mass of @xmath4 and radius of @xmath5 cm are divided by a @xmath7 fm electric gap . it is believed that a crust can be formed during a supernova explosion ( alcock et al.1986 ) by accretion . however , as discussed below , after the strange star was born , a magnetosphere could be established soon , and the radiation from the open field lines region would prevent the accretion . therefore , a crust is difficult to form beyond a newborn strange star . it is accepted that a strange star without crust will not supply charge particles to develop a rotating space charge separated magnetosphere ( alcock et al . 1986 ) . the reason for this is that the maximum electric field induced by a rotating magnetized dipole , @xmath8 v @xmath9 , is negligible when being compared with the electric field at the strange matter surface , @xmath10 v @xmath9 . pulsar emission mechanisms , which depend on the stellar surface as a source of plasma , will not work if there is a bare quark surface ( alcock et al . hence the bare strange stars will not be the observed pulsars . however , there are two points pointed out here : * the electric field due to electron distribution near the surface decreases quickly outward , from @xmath10 v @xmath9 at the surface to @xmath11 v @xmath9 at a very small height of 10@xmath12 cm above the surface . therefore the induced electric field will control most area of the magnetosphere . * the magnetosphere can be established through pair production in @xmath0 or two photon processes . so , it is proposed that the bare strange stars could act as the observed radio pulsars . five conclusions are obtained in this paper : 1 . the bare strange stars may be not bare in fact ; a magnetosphere would be settled around the strange stars if the pair production process were taken into account . the magnetosphere of a strange star is very similar to that of a neutron star ; the radio pulsars might be the ` bare ' strange stars rather than the neutron stars if the strange matter hypothesis is correct . 3.both pulsars with parallel and anti - parallel magnetic axes relative to rotational axes can be observed . 4 . the idea , that the radio pulsars are the strange stars , is supported by some observations . it is suspected that the strange stars with normal matter crusts are formed in binary systems ; and strange stars with crusts would act as x - ray pulsars or x - ray bursters . the structure of this paper is as follows . in section 2 the formation of the magnetosphere of the bare strange star is discussed , including the pair production processes , which is very important for the exist of magnetosphere . a comparison of properties between magnetospheres of the trange stars and of the neutron stars is presented in section 3 . in section 4 , the emission of strange star with a magnetosphere is discussed . conclusion and discussion are shown in the section 5 . why the bare strange stars are bare out ? the main reason for this is that the bare surface will not supply charged particles to form a rotating space charge separated magnetosphere ( alcock 1986 ) . however , it is suggested as follows that the magnetosphere can be formed if the pair production process is to be taken into account . if the strange stars can be the candidates of pulsars , it should be strongly magnetized and rapidly rotating . the directed radiation pencil and the cyclotron absorption lines in pulsar observations show that there might be @xmath13 gauss magnetic field in the pulsar polar caps . for a rotating magnetized strange star , the unipolar induction effect should be included . maxswell equations in the frame , which corotates with the star , are ( fawley et al . 1977 ) @xmath14 where @xmath15 is the space - charge density , @xmath16 is the current density , @xmath17 is a complicated combination of the fields and their derivatives ( see the appendix a in fawley et al . 1977 ) , and @xmath18 where @xmath19 is the angular velocity of the rotating star . if we treat a time independent problem , and simply let @xmath20 and @xmath21 , we come to the space charge separated density ( goldreich & julian 1969 ) @xmath22^{-1 } , \eqno(1)\ ] ] which is required for the electric field in the inertial frame to be entirely given by the corotation electric field . in the vicinity of a strange star , @xmath23 , near the cap , @xmath24 where @xmath25 ( @xmath26 is the polar cap magnetic field ) , and @xmath27 in unit of second . in fig.4 ( right ) , near @xmath28 cm , the distributed electron ( bounded to the strange quark matter ) charge density is comparable to the induced charge separated density , hence @xmath29 should be dominant when @xmath30 cm if there is an electric force equilibrium . as @xmath31 is very small compared with the quark charge density @xmath32 ( @xmath33 , see the appendix ) in the strange star interior , it is a good approximation to think the quarks and electrons are in chemical and thermal equilibrium although charged particles have been slightly separated to balance the unipolar induced electric force in the star interior . we can discuss the electrons distribution in the corotation frame by thomas - fermi model . in the star and near the surface , @xmath34 could be negligible , and the boundary problem could be approximated in one dimension as [ see equ.(a5 ) ] @xmath35 and the equilibrium state ( @xmath36 , and @xmath37 ) is not the solution of the above problem when the charge separated density @xmath38 ( in nature unit ) @xmath39 . hence , the electrons could not in equilibrium dynamically in the corotation frame , and also in the observers frame . if we have vacuum outside the strange star , the induced electric field along the magnetic field , @xmath40 , would be given by ( goldreich & julian 1969 ) @xmath41 where a dipole magnetic field is assumed , @xmath42 , @xmath43 and @xmath44 are the usual polar coordinates with @xmath44 measured from the rotation axis , @xmath45 is the strange star radius , @xmath46 , @xmath47 is the direction of magnetic field . in fig.4 ( left ) , near @xmath48 cm , the electric field to bound the electrons to the quark matter , @xmath49 , is comparable to the unipolar induced electric field along @xmath47 , hence , when @xmath50 cm , the motion and distribution of electron should be mainly controlled by @xmath40 , as all of the other forces ( e.g. gravitation and centrifugal acceleration ) can be negligible ( goldreich & julian 1969 ) . thus , the distributed electrons near and above @xmath51 cm could not be mechanically or quantum mechanically equilibrium ( detailed discussion below ) , and a magnetosphere around strange star could be established . from equ . ( 3 ) and equ.(a4 ) , the critical height where the two electric fields are equal , @xmath52 , can be obtained as a function of @xmath44 , and the solution to the electric equilibrium height with @xmath53 mev is shown in fig.1 . almost at all of the latitude degree the induced electric force can exceed that caused by strange quark matter attraction . @xmath54 as discussed above , the unipolar induced electric field can have considerable contribution to the distribution of electrons almost at the strange star surface ( @xmath55 or @xmath56 cm is a very small number in the astronomical view point ) , which can pull or push the out part electrons near the strange star surface . the potential difference between @xmath57 and @xmath58 is given by ( goldreich 1972 ) @xmath59 if there are no charged particles outside the star . an electron in this electric field can not be continually accelerated , as the pair creation processes could stop the acceleration . electrons or positrons with large lorentz factors should produce @xmath60 rays by , for example , inverse compton scattering , curvature radiation , synchrotron radiation and , perhaps , pair annihilation . also most of the produced high energy @xmath60 rays in such strong magnetic field could convert to electron - positron pairs through @xmath61 or two photon processes , such as @xmath62 . hence , if a strange star have a vacuum outside , this cascade of pair creation should bring about the appearance of a large enough pair plasma to construct a charge separated magnetosphere around the strange star , both a corotation part and a open field lines part ( michel 1991 ) . according to energy conservation law , the energy of the above cascade process is from the strange star s rotation . as long as the magnetosphere has been established , the detailed discussed pulsar inner as well as outer accelerators , such as the polar gap model ( ruderman & sutherland 1975 ) , the slot gap model ( arons 1983 ) , and the out gap model ( cheng ho & ruderman 1984 ) , would work for the radio as well as higher energy photons emission , because of the electromotive force caused by the potential difference between the center and the edge of the polar cap region . let s come to some details . for @xmath63 , the induced electric field would pull the electrons out , accelerates them ultra - relativistically . as electrons lost from the quark matter , the strange star could be positively charged , the global electric circuit ( shibata 1991 ) might be set up . however , this global circuit could be in quasi - equilibrium , and a small vacuum gap similar to that of rs model ( ruderman & sutherland 1975 ) could be possible near the polar cap . for @xmath64 , the induced electric field would push the electrons inward , and a large vacuum region should be above the polar cap . some physical processes , such as cosmic @xmath60 rays interaction with strong magnetic fields , or electrons ( scattered by neutrinos from strange star ) synchrotron radiation ( jump between two landau levels ) , could trigger the pair creation cascade . hence , a space charged limited flow ( sturock 1969 ) would take place , the outward accelerated particles ( electrons or positions ) might coherently radiate radio waves ( melrose 1995 ) and incoherently emit high energy photons , and the inward accelerated particles could interact with electrons and quarks electrically , which may result in the observed hot spot ( wang & halpern 1997 ) . therefore , a bare strange star could act as a radio pulsar or a @xmath60 ray pulsar . if a strange star forms soon after a supernova , a magnetosphere composed by ions would not be possible since a lot of very energetic outward particles and photons are near the star . however , the time scale t to form an @xmath65 pair plasma magnetosphere is very small . the total number of @xmath65 pairs in the magnetosphere might be estimated as @xmath66 , @xmath67 where the radius of light cylinder @xmath68 ( the radius of strange star ) . the mean free path @xmath69 of a photon with energy greater than @xmath70 mev moving through a region of magnetic b can be estimated as ( erber 1966 ) @xmath71 where @xmath72 could be approximated as 1/15 ( ruderman & sutherland 1975 ) , @xmath73 is the angle between the direction of propagation photon and the magnetic field . @xmath74 cm for @xmath75 ( @xmath60 is the lorentz factor of electron ) . also , the mean free length @xmath76 of electron to produce photon by curvature radiation etc . could be assumed in order of @xmath69 . for the cascade processes discussed , the time scale @xmath77 to build up a magnetosphere might be @xmath78 which is in order of @xmath79 seconds for typical pulsar parameters . considering the photon escaping and the global magnetic field structure , this time scale should not change very much . hence , ions would have less possibility in the strange radio pulsar s magnetosphere that might be mainly consist of @xmath65 pairs quickly , especially in the open field lines region where large outward pressure of @xmath65 pairs and electromagnetic waves exists . while strange stars are in binaries , as the accretion pressure of wind or matter could be greater than the outward pressure from the polar cap , the accretion process should be involved , and a strange star could be accretion powered @xmath1ray source . for an accreting strange star , there could be two envelope crusts shielding the two polar capes . as strange matter does not react with ions because of the coulomb barrier ( the height of which is @xmath80 mev ) , there could be an electrostatic gap of thickness hundreds fermi above the surface ( alcock et al . if the magnetic field is very strong ( @xmath81 gauss ) , and in case of high accretion rate ( high massive x - ray binaries ) , those accreted crusts should be small , where some violent processes , such as the huge release of gravitational energy and the thermal nuclear reactions , could take place . in this case , the ion penetration probability might be large enough to keep a quasi - equilibrium accretion process , and the accretion strange star could be an x - ray pulsar ( bhattacharya et el . 1991 , nagase 1989 ) . if the magnetic field is less strong ( @xmath82 gauss ) , and in case of low accretion rate ( low massive x - ray binaries ) , those two polar accreted crusts could be large enough to form a united crust , where the accretion process is mild , and the electric gap could prevent strong interactions between the crust and the strange matter . however , as the accreted matter piled up , the crust could be enough hot and dense to trigger the thermonuclear flash . in this case , the strange star could act as an x - ray burster ( lewin et al . 1993 ) , and a lot of ions might also be pushed to the strange matter through the coulomb barrier . if the magnetosphere of a bare strange star could be formed , a question is raised : what is the difference between the magnetospheres of bare strange stars and neutron stars ? our calculations show that , just outside of the surface ( beyond @xmath83 cm ) , the induced electric field can be large enough to control the magnetosphere . this situation is very similar to that for the neutron stars . hence , goldreich and julian model ( goldreich & julian 1969 ) can also describe the magnetosphere of strange stars . there are two differences between the magnetospheres properties of strange stars and of neutron stars . * only electrons and positrons ( no ions ) are charged particles in the magnetosphere of a strange star . for the magnetosphere of a neutron star , the iron ions would be exist because the binding energy of a neutron star surface is too low to stop the irons flow out . * in the magnetosphere of strange stars the rs inner gap ( ruderman & sutherlad 1975 ) can be formed easily . while in case of neutron stars , the rs inner gap is difficult to form , and the free - flow models ( see e.g. arons , 1983 ; harding & muslimov 1998 ) will work . polar gap , as well as gap sparking , should be on the scene in the strange - star model for pulsars . gil & cheng ( 1998 ) have noted the importance of polar gap sparking near the pulsars surface . the short time scale sparking could be essential for the observed micro - pulses as well as ( drifting ) sub - pulses . a wealth of observations has been collected for pulsars since the discovery of pulsars thirty years ago . some important gaps still remain in our understanding of the emission process . general agreement begins and ends the statement that the very strong magnetic fields expected for neutron stars must play a prominent role ( sutherland 1979 ) . goldreich and julian ( 1969 ) proposed a model to show that a rotating magnetic neutron star is surrounded by a charge - separated magnetosphere . sturrock ( 1971 ) was the first to develop a comprehensive model for pulsar radiation , which suggests that an acceleration , immediately above the polar cap , will take place due to space - charge limited flow . ruderman and sutherland ( 1975 ) proposed an ` inner gap ' model . the model starts with the assumption that the binding of ions within the neutron star surface restrict the out free - flow of ions and a so - called ` inner gap ' will be formed . in the inner gap @xmath84 , @xmath65 particles can be produced through @xmath85b process , and can be accelerated relativisticly to produce @xmath60-rays through curvature radiation ( cr ) . eventually , a cascade of pair production results in a discharge of the gap . besides rs model , there are some models such as ` slot gap ' model ( see e.g. arans 1983 ) , beskin , gurevich and istomin ( 1988 ) model . but the ` user friendly ' nature of rs model is a virtue not shared by others ( shukre 1992 ) . it is not uncommon to read contemporary observational papers in which the sole theoretical reference is to rs ( 1975 ! ) ( michel 1992 ) . unfortunately , the rs model still faces great difficulties . from the theoretical point of view , some calculations ( hillebrandt and muller 1976 ; neuhauser et al . 1986 , 1987 ; kossl et al . 1988 ) show that the ion binding energy is at least one order less than what is required in the rs model , which means that the inner gap can not be formed . this is the so - called ` binding energy problem ' . in another hand from observational point of view , rankin s phenomenological work provides a much firmer basis for emission model than earlier models with only a hollow cone ( taylor and stinebring 1986 ) . the emission beams of radio pulsars can be divided into two ( core and conal ) emission components ( lyne and manchester 1988 ) or three ( core , inner conal and outer conal ) emission components ( rankin 1983,1990,1993 ) . but in rs model only one hollow cone emission component can be obtained . this is to say that there is serious conflict between the theory of rs model and the observations . under an assumption of the inner gap sparking qiao and lin ( 1998 ) proposed an inverse compton scattering model for radio emission of pulsars . in the model all of the core , inner cone and the outer cone emission components can be obtained . so most important problem faced by inner gap model is the ` binding energy problem ' . now the ` bare ' strange star is not bare in fact , a magnetosphere can be formed due to the very strong electric field at the strange star surface , and the ` binding energy problem ' will be easy retrievable . hence , the user - friendly nature of rs model can be survived . for rs inner gap model , it is assumed that the magnetospheric charge density above the polar cap is positive ( in the goldreich - julian model this means that the rotational angle velocity @xmath19 and the magnetic momentum @xmath86 are anti - parallel ) . for neutron stars at which @xmath19 and @xmath86 are parallel , the inner gap can not be formed and the inner gap model does not work . if the angle between @xmath87 and @xmath88 of the neutron stars is distributed uniformly , the virtue above means that the rs inner gap model does not apply to half of the neutron stars . this is a strange virtue . if a magnetosphere can be formed around the ` bare ' strange star , this limitation does not exist again . let s go to some details as follows for @xmath89 . for neutron stars , all of the electrons of irons in the crust are free as the electron fermi energy is so high that the electron sea is only slightly perturbed by the nuclei coulomb fields . the thickness @xmath90 of the crust for typical neutron star models is about several hundred meters , and the density of the crust @xmath91 g @xmath92 ( smith 1977 ) . the number density of electron in the crust would be @xmath93 as electrons flow out from the polar cap surface , the radius of which is @xmath94 , @xmath95 where @xmath45 and @xmath27 are the radius and period of a neutron star , the matter below the cap should be positively charged , and the electrons out of this region will drift in across magnetic field lines at a velocity of @xmath96 ( jackson 1975 ) , @xmath97 where @xmath98 is the electric field due to positive charging up , @xmath99 is the magnetic field . to consider the equilibrium case above , assuming the @xmath100 is the positive charged number - density , we come to @xmath101 which means only a very slight departure of electron number density from @xmath102 can support a equilibrium free flow , here @xmath103 , @xmath104e.u.s is the electron electricity . so , an equilibrium of electron flows could be established because @xmath105 is very small . in fact , there is a space charge separation in the neutron star , while , this effect is negligible in the above estimates . similarly , we consider the case for a bare strange star . from fig . 4 ( left ) , only electron above @xmath106 cm could be flow out by induced electric field . the out flow rate @xmath107 could be @xmath108 while , if the total electrons being available to flow are pulled out , the maximum discharging rate @xmath109 should be @xmath110 in fact , a discharging rate in a real case should be much smaller than @xmath109 because the total electrons being available to flow are not pulled out at all . so , @xmath111 and @xmath112 are comparable , which result in the instability of the polar cap electron flow , and the inner gap similar to that of rs model could be possible . the total numbers of electrons available to flow out from neutron star and from strange star are @xmath113 respectively , here @xmath114 . if the charge density of relativisticly flowing electrons at the surface is @xmath115 , then the times scales to pull all of this electrons out from neutron star and from strange star are @xmath116 therefore , the instability processes in the strange stars are more acute than that in the neutron stars , which could result in the depletions of charge above the polar caps . it is shown that there might be a magnetosphere surrounding a bare strange star or namely isolated strange star . ` bare ' strange stars might not be bare at all , which could be observed as rotation - powered pulsars . it is difficult to form an accretion crust around an isolated strange star . strange stars with accretion crusts could be formed in binary systems , which can act as x - ray pulsars and x - ray bursters . it is easy to understand why the k - shell lines of iron have never be observed in x - ray emission bands of radio pulsars . the iron emission lines at @xmath117 kev and absorption edges at @xmath118 kev have been observed in many accretion - powered x - ray pulsars ( nagase 1989 ) . absorption lines at 5.7 kev or 4.1 kev have also been detected in several x - ray bursters ( lewin et al.1993 ) , which could be considered as the iron element origin . while , there is not any observational signature of iron lines in the x - ray emission of rotation - powered pulsars ( cheng et al . 1998 , becker & trumper 1997 , thompson 1996 ) , although the @xmath65 annihilation line in carb have been observed and well explained ( zhu & ruderman 1998 ; agrinier et al . 1990 ; massaro et al . 1991 ) . if pulsars are rotating magnetized neutron stars as universally believed , the binding energy per ion in the neutron star surface is too low to support the inner gap scenario , hence the ions will free - flow from the surface ( neuhauser et al . 1986,1987 ; harding and muslimov 1998 ) . therefore the composition of iron in the magnetosphere should be un - negligible , and it is hard to explain why the emission line has never been observed . but for strange star , the open field line regions of a strange star magnetosphere consists of @xmath65 pairs only , no iron can result in the radative processes . so , one of the discrimination criteria for strange star and neutron star in observation is to seek the iron lines of rotation - powered pulsars in x - ray bands . it is easy to resolve the so - called ` binding energy problem ' . the rs model(ruderman & sutherland 1975 ) has come closest to enabling comparison of observations and theory ( radhakrishnan 1992 ) . a fatal point of the model is that there is an inner gap above the polar cap , which inquires the binding energy of the positive ions large enough ( larger than 10 kev ) to restrict them from free - flowing out . however , the iron bounding energy can not be large enough to support the rs gap of neutron star . the calculations by hillebrandt and muller ( 1976 ) , muller ( 1984 ) , jones ( 1985,1986 ) , neubauser et al.(1986,1987 ) , kossl et al.(1988 ) show that the ion binding energy at least one order less than what is required in the rs model . the irons have its lowest energy state as unbound atoms rather than the chains . this is so - called ` binding energy problem ' , which is faced if the signal of pulsars is coming from neutron stars . in case of strange star , as the positive charged quark matter near the surface is held by strong interaction , the binding energy should be approximately infinity when @xmath119 . because of the attraction of the quark matter and the quasi - equilibrium of electric current , the electrons could not be easily pulled out when @xmath120 . \3 . the strange star model for pulsars could be employed for @xmath119 as well as @xmath120 . is there any striking difference in observation and theory between these two situations of @xmath119 and @xmath120 ? this is an open question . what s more , it is easier to collapse during the last stage of the supernova explosion if a strange quark matter star , rather than a neutron star , leaves over . during the last stages of the collapse of a supernova core , a shock wave will form and move outward due to the very stiff nuclear equation of states when the central density exceeds the nuclear - matter density . as the collapse continues , the phase transitions from nuclear matter to two - flavor quark matter and from two - flavor quark matter to three - quark matter may occur ( dai et al 1995 ) . after the conversions , the neutrino energy in the whole collapse core increases obviously , which could result in the enhancement of both the probability of success for supernova explosion and the energy of the revived shock wave . a critical point to distinguish neutron star and strange star is that the strange star have very high coulomb barrier which can support a large body of matter , while , the neutron stars do not have . as the bare strange stars acting as the radio or @xmath85ray pulsars are very similar to the neutron stars ( the differences of rotation periods and cooling curves between them are hard to be found ) , we suggest to search the differences between strange star and neutron star in accretion binaries , especially for the bursting x ray pulsar gro j1744 - 28 ( strickman et al . 1996 ) . one of the most serious difficulties is the possibility of strange star glitch . an important element of the theory ( pines & alpar 1985 ) to explain the observed glitches by neutron star is the ability of superfluid neutrons to move freely across magnetic field lines , while , there seems no neutral particle in a strange star . nevertheless , perhaps , a strange star model on glitch might be developed by further research . there might be no neutron stars in the galaxy ( alcock et al . 1986 ) , all of the observed pulsars are the strange stars . if this is true , the strange matter hypothesis could be right , which can help us to interpret the properties of the six flavor quarks and the composition of the universe . for a static and nonmagnetized strange star , the properties of strange quark matter are determined by the thermodynamic potentials @xmath121 ( i = u , d , s , e ) which are functions of chemical potential @xmath122 as well as the strange quark mass , @xmath123 , and the strong interaction coupling constant @xmath124 ( alcock et al . 1986 ) . by assuming weak interaction chemical equilibrium and overall charge neutrality , we come to @xmath125 and the total energy density @xmath126 reads @xmath127 where @xmath99 is the bag constant , and @xmath121 referred to the appendix in the paper by alock et al.(1986 ) . the above equations ( a1a - a1b ) have only one free independent parameter , @xmath88 , and establish the relations between \{@xmath128 ; i = 1,2,3,4 for u , d , s , e , respectively } ( 9 equations for 9 quantities ) . the calculation results from equ.(a1 ) are shown in fig.2 , and fig.3 , where the number densities of u , d , s quarks , and the quark charge density @xmath32 ( in unit of coulomb per @xmath131 ) are varied as a function of total energy density @xmath126 . in the computation , we choose @xmath132 , @xmath133 mev , and the renormalization point @xmath134 mev , both for @xmath135 and @xmath136 . as @xmath126 has a mild rise variation from the outer part to the interior of a strange star ( alcock et al . 1986 ) , the number density of u , d , and s quarks increase almost in a same degree . however , the equilibrium quark charge density @xmath32 changes significantly , as @xmath126 increases ( fig.3 ) , which means the number of equilibrium electrons becomes smaller as one goes to a deeper region of a strange star . for a strange star with a typical pulsar mass 1.4 m@xmath137 , the total energy @xmath126 has a very modest variation with radial distance of strange star ( alcock et al 1986 ) , from @xmath138 g @xmath92 ( near surface ) to @xmath139 g @xmath92 ( near center ) , therefore the quark charge density @xmath32 would be order of @xmath140 ( @xmath136 ) to @xmath141 ( @xmath135 ) coulomb @xmath92 . physically , as the fermi energy of quarks becomes higher ( for lager @xmath126 ) , the effect due to @xmath142 would be less important , hence , the charge density should be smaller . since the quark matter are bounded up through strong interaction ( the thickness of the quark surface will be of order 1 fm ) , and the electrons are held to the quark matter electrically , hence the electrons distribution would extend beyond the quark matter surface . a simple thomas - fermi model has been employed to solve for this distribution ( alcock el al . 1986 ) , and the local charge distribution can be obtained by poisson s equation @xmath143 where z is a measured height above the quark surface , @xmath73 is the fine - structure constant , @xmath144 is the quark charge density , @xmath145 is the electrostatic potential , and the number density of electrons is given by @xmath146 physically , the boundary condition for equ.(a2 ) are @xmath147 by a straightforward integration of equ.(a2 ) , we get @xmath148 from equ.(a4 ) , the continuity of @xmath149 at @xmath150 requires @xmath151(alcock et al . 1986 ) , and we can consider the solution of equ.(a4 ) for @xmath152 by @xmath153 hence , @xmath154 it is interesting that , although the electric field near the surface is order of @xmath155 v @xmath9 , the electric field decrease very quickly above the quark surface . we choose @xmath53 mev ( hence @xmath156 mev@xmath157 coulomb @xmath92 ) , and the calculated electric field as a function of @xmath158 is shown in figure 4 ( left ) , where the electric field is @xmath159 v @xmath9 when @xmath150 . also the electron charge density curve is drawn in fig.4 ( right ) , which decreases from @xmath160 coulomb @xmath92 at the surface to @xmath161 coulomb @xmath92 when @xmath162 cm . * acknowledgments : * we are very grateful to prof . t. lu and q. h. peng for their valuable discussion and encouragement . we would like to thank b.zhang , b.h.hong and j.f.liu for helpful discussions . this work is partly supported by nsf of china , the climbing project - national key project for fundamental research of china , the doctoral program foundation of institution of higher education in china , and the youth scientific foundation of peking university . : : agrinier , b . , et al . 1990 , apj , 355,645 : : alcock , c . , farhi , e . , & olinto , a . 1986 , apj , 310 , 261 : : arons , j . 1983 , apj , 276 , 215 : : becker , w . , & trumper , j . 1997 , a&a , 326 , 682 : : benvenuto , o.g . , lugones , g . 1995 , phys . d51 , 1989 : : benvenuto , o.g . , & vucetich , h . 1991 , nucl . ( proc . suppl . ) b24 , 160 : : beskin , v.s . , gurevich , a.v . , & istomin , y.n . 1986 , apss , 146 , 205 : : bhattacharya , d . , van den heuvel , e.p.j . 1991 , phys.rep . , 203 , 1 : : broderick , j.j . , et al . 1998 , apj , 192 , l71 : : bodmer , a.r . 1971 , phys . d4 , 1601 : : cheng , k.s . , ho , c . , & ruderman , m . 1984 , apj , 300 , 500 : : cheng , k.s . , , & zhang , l . 1998 , apj , 493 , l35 : : dai , z.g . , & lu , t . 1986 , zeit . f. physik a , 355 , 415 : : dai , z.g . , peng , q.h . , 1995 , apj , 440 , 815 : : erber , t . 1966 , rev . 38 , 626 : : farhi , e . , & jaffe , r.l 1984 , phys . d30 , 2379 : : fawley , m . , arons , j . , & scharlemann , e.t 1977 , apj , 217 , 227 : : flower , e.g . , 1977 , apj , 215 , 291 : : frieman , j.a . , & olinto , a . 1989 , nature , 341 , 633 : : gil , j.a . , & cheng , k.s . 1998 , a&a , submitted : : goldreich , p . 1972 , in the physics of pulsars , ed . a.m.lenchek , gordon and breach , science publisher , 151 : : goldreich , p . , & jullian , w.h . 1969 , apj , 157 , 869 : : harding , a.k . , & muslimov , a.g . 1998 , submitted to apj : : hewish , a . , bell , s.j . 1968 , nature , 217 , 709 : : hillebrands , w . , muller , e . 1976 , apj , 207 , 589 : : jackson , j.d . 1975 , _ classical electrodynamics _ , john wiley & sons : : jones , p.b . 1985 , phys . , 55 , 1338 : : jones , p.b . 1986 , m.n.r.a.s . , 218 , 477 : : kettner , ch . , weber , f . , & weigel , m.k . 1995 , phys . , d51 , 1440 : : kossl , d . , wolff , r.g . , muller , e . , hillebrands , w . 1988 , a&a , 205 , 34 : : landau , l . 1932 , phys.z.sowjetunion , 1 , 285 , : : lewin , w.h.g . , van paradijs , j . , taam , r.e . 1993 , space sci . , 62 , 223 : : lyne , a.g . , & manchester , r.n . 1988 , m.n.r.a.s . , 134 , 477 : : massaro , e . , 1991 , apj , 376 , l11 : : mclerran , l . 1986 , rev . phys . , 58 , 1021 : : melrose , d.b . 1995 , j.astrophy.astr . , 16 , 138 : : michel , f.c . 1991 , _ theory of neutron star magnetospheres _ , univ . chicago press : : michel , f.c . 1992 , in the magnetospheric structure and emission mechanisms of radio pulsars , eds . hankins , h . , rankin , m . , & gil , a . , pedagogical univ . press , 405 : : muller , b . 1995 , rep . phys . , 58 , 611 : : muller , e . 1984 , a&a , 130 , 415 : : nagase , f . , 1989 , japan , 41 , 1 : : neuhauser , d . , langanke , k . , & koonin , s.e . 1986 , phys . a33 , 2084 : : neuhauser , d . , koonin , s.e . , & langanke k. 1987 , phys . rev . , a36,4163 : : oppenheimer , j.r . , & volkoff , g.m . 1939 , phys . , 55 , 374 : : pines , d . , & alpar , m.a . 1985 , nature , 316 , 27 : : radhakrishnan , v . 1992 , in the magnetospheric structure and emission mechanisms of radio pulsars , eds . hankins , h . , rankin , m. , and gil , a. , pedagogical univ . press , 367 : : qiao , g.j . , & lin , w.p . 1998 , a&a , 333 , 172 : : rankin , j.m . 1983 , apj , 274 , 333 : : rankin , j.m . 1990 , apj , 352 , 247 : : rankin , j.m . 1993 , apj , 405 , 285 : : ruderman , m.a . , & sutherland , p.g . 1975 , apj , 196 , 51 : : schaffner , j . 1993 , phys . lett . , 71 , 1328 : : shibata , s . 1991 , apj , 378 , 239 : : shukre , c.s . 1992 , in the magnetospheric structure and emission mechanisms of radio pulsars , eds . hankins , h . , rankin , m . , & gil , a . pedagogical univ . press , 412 : : smith , f.g . _ pulsars _ , 1977 , cambridge univ . press : : strickman , m.s . 1996 , apj , 464 , l131 : : sturock , p.a . 1971 , apj , 164 , 529 : : thompson , d.j . 1996 , in pulsars : problem & progress , ed . s.johnston et al . , 307 : : taylor , j.r . , & stinebring , d.r . 1986 , ann . 24 , 285 : : wang , f.y .- h . , & halpern , j.p . 1997 , apj , 482 , l159 : : wang , q.d . , & lu , t . 1984 , phys . b148 , 211 : : witten , e . 1984 , phy . rev . , d30 , 272 : : zhu , t . , & ruderman , m . 1998 , apj , 478 , 701

it is suggested in this paper that the ` bare ' strange star might be not bare , and there could be a magnetosphere around it . as a strange star might be an intensely magnetized rotator , the induced unipolar electric field would be large enough to construct a magnetosphere around the strange matter core . this kind of magnetosphere is very similar to that of the rotating magnetized neutron stars discussed by many authors . a magnetosphere will be established very soon through pair production by @xmath0 or two photon processes after a strange star was born in a supernova explosion . it is emphasized that the fact that the strange star surface can not supply charged particles does not stop the formation of a space charge separated magnetosphere around the bare strange star . an accretion crust is quite difficult to come into being around an isolated strange star . therefore the observed radio signals of an rotation - powered pulsar may come from a bare strange stars rather than a neutron stars or a strange star with an accretion crust . the idea , that the radio pulsars are the strange stars without crusts , is supported by some observations . for example , the electron - positron annihilation line in the spectrum of the crab pulsar has been reported ( agrinier et al . 1990 ; massaro et al . 1991 ) ; the iron emission lines have been observed in many x - ray pulsars but never been reported in @xmath1ray emission of radio pulsars . this fact is difficult to be understood if the radio pulsars are the neutron stars where the surface binding energy of iron ions is too low to avoid a ion free - flow from the surface ( neuhauseret et al.1986 , 1987 ) . @xmath2department of geophysics , peking university , beijing 100871 , china + @xmath3chinese academy of sciences - peking university joint beijing + astrophysical center + email : [email protected] , [email protected] + _ sent to astrophysics journal ( part 1 ) on april 11 , 1998 _ elementary particles - pulsars : general - stars : neutron - hydromagnetics

astro-ph9804278

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% .

1503.03516

optical properties of low - dimensional semiconductor nanostructures originate from excitons ( coulomb - bound electron - hole pairs ) and exciton complexes such as biexcitons ( coupled states of two excitons ) and trions ( charged excitons ) . these have pronounced binding energies in nanostructures due to the quantum confinement effect.@xcite the advantage of optoelectronic device applications with low - dimensional semiconductor nanostructures lies in the ability to tune their properties in a controllable way . optical properties of semiconducting carbon nanotubes ( cns ) , in particular , are largely determined by excitons,@xcite and can be tuned by electrostatic doping,@xcite or by means of the quantum confined stark effect.@xcite carbon nanotubes are graphene sheets rolled - up into cylinders of one to a few nanometers in diameter and up to hundreds of microns in length , which can be both metals and semiconductors depending on their diameters and chirality.@xcite over the past decade , optical nanomaterials research has uncovered intriguing optical attributes of their physical properties , lending themselves to a variety of new optoelectronic device applications.@xcite formation of biexcitons and trions , though not detectable in bulk materials at room temperature , play a significant role in quantum confined systems of reduced dimensionality such as quantum wells,@xcite nanowires,@xcite nanotubes,@xcite and quantum dots.@xcite biexciton and trion excitations open up routes for controllable nonlinear optics and spinoptronics applications , respectively . the trion , in particular , has both net charge and spin , and therefore can be controlled by electrical gates while being used for optical spin manipulation , or to investigate correlated carrier dynamics in low - dimensional materials . for conventional semiconductor quantum wells , wires , and dots , the binding energies of negatively or positively charged trions are known to be typically lower than those of biexcitons in the same nanostructure , although the specific trion to biexciton binding energy ratios are strongly sample fabrication dependent.@xcite first experimental evidence for the trion formation in carbon nanotubes was reported by matsunaga et al.@xcite and by santos et al.@xcite on @xmath0-doped ( 7,5 ) and undoped ( 6,5 ) cns , respectively . theoretically , rnnow et al.@xcite have predicted that the lowest energy trion states in all semiconducting cns with diameters of the order of or less than 1 nm should be stable at room temperature . they have later developed the fractional dimension approach to simulate binding energies of trions and biexcitons in quasi-1d/2d semiconductors , including nanotubes as a particular case.@xcite binding energies of @xmath1 mev and @xmath2 mev are reported for the lowest energy trions@xcite and biexcitons,@xcite respectively , in the ( 7,5 ) nanotube . however , the recent nonlinear optics experiments were able to resolve both trions and biexcitons in the same cn sample,@xcite to report on the opposite tendency where the binding energy of the trion _ exceeds _ that of the biexciton rather significantly in small diameter ( @xmath3 nm ) cns . figure [ fig0 ] shows typical experimental data for conventional low - dimension semiconductors ( left panel ) and small diameter semicondicting cns ( right panel ) . in the left panel , the biexciton resonance is seen to appear at lower photon energy than the trion one , in contrast with the right panel where the biexciton resonance manifests itself at greater photon energy than the trion resonance does . this clearly indicates greater trion binding energies than those of biexcitons in small diameter semiconducting cns as opposed to conventional low - dimension semiconductors . = 17.5 cm more specifically , colombier et al.@xcite reported on the observation of the binding energies @xmath4 mev and @xmath5 mev for the trion and biexciton , respectively , in the ( 9,7 ) cn . yuma et al.@xcite reported even greater binding energies of @xmath6 mev for the trion versus @xmath7 mev for the biexciton in the smaller diameter ( 6,5 ) cn . ( their spectra are reproduced in fig . [ fig0 ] , right panel . ) in both cases , the trion - to - biexciton binding energy ratio is greater than unity , decreasing as the cn diameter increases [ 1.46 for the 0.75 nm diameter ( 6,5 ) cn versus 1.42 for the 1.09 nm diameter ( 9,7 ) cn ] . trion binding energies greater than those of biexcitons are theoretically reported by watanabe and asano,@xcite due to the energy band nonparabolicity and the coulomb screening effect that reduces the biexciton binding energy more than that of the trion . watanabe and asano have extended the first order ( @xmath8)-perturbation series expansion model originally developed by ando for excitons ( see ref.@xcite for review ) to the case of electron - hole complexes such as trions and biexcitons . figure [ fig00 ] compares the differences between the trion and biexciton binding energies delivered by `` phenomenological '' and `` unscreened '' models termed as such to refer to the cases where the energy band nonparabolicity , electron - hole complex form - factors , self - energies and the screening effect are all neglected , and where all of them but screening are taken into account , respectively , with the difference given by the `` screened '' model . the latter is the watanabe asano model which includes _ all _ of the factors mentioned within the first order ( @xmath8)-perturbation theory . one can see that the `` screened '' model does predict greater trion binding energies than those of biexcitons as opposed to the phenomenological and unscreened models . however , the most the trion binding energy can exceed that of the biexciton within this model is @xmath9 equal to @xmath10 and @xmath11 mev for the ( 6,5 ) and ( 9,7 ) cns , respectively , which is obviously not enough to explain the experimental observations . = 10.0 cm this article reviews the capabilities of the configuration space ( landau - herring ) method for the binding energy calculations of the lowest energy exciton complexes in quasi-1d/2d semiconductors . the approach was originally pioneered by landau,@xcite gorkov and pitaevski,@xcite holstein and herring@xcite in the studies of molecular binding and magnetism . the method was recently shown to be especially advantageous in the case of quasi-1d semiconductors,@xcite allowing for easily tractable , complete analytical solutions to reveal universal asymptotic relations between the binding energy of the exciton complex of interest and the binding energy of the exciton in the same nanostructure . the landau - herring method of the complex bound state binding energy calculation is different from commonly used quantum mechanical approaches reviewed above . these either use advanced simulation techniques to solve the coordinate - space schrdinger equation numerically,@xcite or convert it into the reciprocal ( momentum ) space to follow up with the ( @xmath8)-perturbation series expansion calculations.@xcite obviously , this latter one , in particular , requires for perturbations to be small . if they are not , then the method brings up an underestimated binding energy value , especially for molecular complexes such as biexciton and trion where the kinematics of complex formation depends largely on the asymptotic behavior of the wave functions of the constituents . this is likely the cause for watanabe asano theory of excitonic complexes@xcite to significantly underestimate the measurements by colombier et al.@xcite and yuma et al.@xcite on semiconducting cns . = 11.5 cm the landau - herring configuration space approach does not have this shortcoming . it works in the _ configuration space _ of the _ two _ relative electron - hole motion coordinates of the _ two _ non - interacting quasi-1d excitons that are modeled by the effective one - dimensional cusp - type coulomb potential as proposed by ogawa and takagahara for 1d semiconductors.@xcite since the configuration space is different from the ordinary coordinate ( or its reciprocal momentum ) space , the approach does not belong to any of the models summarized in fig . [ fig00 ] . in this approach , the biexciton or trion bound state forms due to the exchange under - barrier tunneling between the equivalent configurations of the electron - hole system in the configuration space . the strength of the binding is controlled by the exchange tunneling rate . the corresponding binding energy is given by the tunnel exchange integral determined through an appropriate variational procedure . as any variational approach , the method gives an upper bound for the _ ground _ state binding energy of the exciton complex of interest . as an example , fig . [ fig000 ] compares the biexciton binding energies calculated within several different models , including those coordinate - space formulated that are referred to as phenomenological in fig . [ fig00 ] , as well as the configuration space model . it is quite remarkable that with obvious overall correspondence to the other methods as seen in fig . [ fig000 ] , the landau - herring configuration space approach is the only to have been able consistently explain the experimental observations discussed above and shown in fig . [ fig0 ] , both for conventional low - dimension semiconductors and for semiconducting cns . whether the trion or biexciton is more stable ( has greater binding energy ) in a particular quasi-1d system turns out to depend on the reduced electron - hole mass and on the characteristic transverse size of the system.@xcite trions are generally more stable than biexcitons in strongly confined quasi-1d structures with small reduced electron - hole masses , while biexcitons are more stable than trions in less confined quasi-1d structures with large reduced electron - hole masses . as such , a crossover behavior is predicted,@xcite whereby trions get less stable than biexcitons as the transverse size of the quasi-1d nanostructure increases quite a general effect which could likely be observed through comparative measurements on semiconducting cns of increasing diameter . the method captures the essential kinematics of exciton complex formation , thus helping understand in simple terms the general physical principles that underlie experimental observations on biexcitons and trions in a variety of quasi-1d semiconductor nanostructures . for semiconducting cns with diameters @xmath12 nm , the model predicts the trion binding energy greater than that of the biexciton by a factor @xmath13 that decreases with the cn diameter increase , in reasonable agreement with the measurements by colombier et al.@xcite and yuma et al.@xcite the article is structured as follows . section 2 formulates the general hamiltonian for the biexciton complex of two electrons and two holes in quasi-1d semiconductor . carbon nanotubes of varying diameter are used as a model example for definiteness . the theory and conclusions are valid for any quasi-1d semiconductor system in general . the exchange integral and the binding energy of the biexciton complex are derived and analyzed . section 3 further develops the theory to include the trion case . in section 4 , the trion binding energy derived is compared to the biexciton binding energy for semiconducting quasi-1d nanostructures of varying transverse size and reduced exciton effective mass . section 5 generalizes the method to include trion and biexciton complexes formed by indirect excitons in layered quasi-2d semiconductor structures such as coupled quantum wells ( cqws ) and bilayer self - assembled transition metal dichalchogenide heterostructures . section 6 summarizes and concludes the article . the problem is initially formulated for two interacting ground - state 1d excitons in a semiconducting cn . the cn is taken as a model for definiteness . the theory and conclusions are valid for any quasi-1d semiconductor system in general . the excitons are modeled by effective one - dimensional cusp - type coulomb potentials , shown in fig . [ fig1 ] ( a ) , as proposed by ogawa and takagahara for 1d semiconductors.@xcite the intra - exciton motion can be legitimately treated as being much faster than the inter - exciton center - of - mass relative motion since the exciton itself is normally more stable than any of its compound complexes . therefore , the adiabatic approximation can be employed to simplify the formulation of the problem . with this in mind , using the cylindrical coordinate system [ _ z_-axis along the cn as in fig . [ fig1 ] ( a ) ] and separating out circumferential and longitudinal degrees of freedom for each of the excitons by transforming their longitudinal motion into their respective center - of - mass coordinates,@xcite one arrives at the hamiltonian of the form@xcite @xmath14 here , @xmath15 are the relative electron - hole motion coordinates of the two 1d excitons separated by the center - of - mass - to - center - of - mass distance @xmath16 , @xmath17 is the cut - off parameter of the effective ( cusp - type ) longitudinal electron - hole coulomb potential , @xmath18 , @xmath19 with @xmath20 ( @xmath21 ) representing the electron ( hole ) effective mass . the `` atomic units''are used,@xcite whereby distance and energy are measured in units of the exciton bohr radius @xmath22 and the rydberg energy @xmath23 , respectively , @xmath24 is the exciton reduced mass ( in units of the free electron mass @xmath25 ) and @xmath26 is the static dielectric constant of the electron - hole coulomb potential . = 11.0 cm the first two lines in eq . ( [ biexcham ] ) represent two non - interacting 1d excitons . their individual potentials are symmetrized to account for the presence of the neighbor a distance @xmath27 away , as seen from the @xmath28- and @xmath29-coordinate systems treated independently [ fig . [ fig1 ] ( a ) ] . the last two lines are the inter - exciton exchange coulomb interactions electron - hole ( line next to last ) and hole - hole + electron - electron ( last line ) , respectively . the binding energy @xmath30 of the biexciton is given by the difference @xmath31 , where @xmath32 is the lowest eigenvalue of the hamiltonian ( [ biexcham ] ) and @xmath33 is the single - exciton binding energy with @xmath34 being the lowest - bound - state quantum number of the 1d exciton.@xcite negative @xmath30 indicates that the biexciton is stable with respect to the dissociation into two isolated excitons . the strong transverse confinement in reduced dimensionality semiconductors is known to result in the mass reversal effect,@xcite whereby the bulk heavy hole state , that forming the _ lowest _ excitation energy exciton , acquires a longitudinal mass comparable to the bulk _ light _ hole mass ( @xmath35 ) . therefore , @xmath36 in our case of interest here , which is also true for graphitic systems such as cns , in particular,@xcite and so @xmath37 is assumed in eq . ( [ biexcham ] ) in what follows with no substantial loss of generality . the hamiltonian ( [ biexcham ] ) is effectively two dimensional in the configuration space of the two _ independent _ relative motion coordinates , @xmath28 and @xmath29 . figure [ fig1 ] ( b ) , bottom , shows schematically the potential energy surface of the two closely spaced non - interacting 1d excitons [ second line of eq . ( [ biexcham ] ) ] in the @xmath38 space . the surface has four symmetrical minima representing isolated two - exciton states shown in fig . [ fig1 ] ( b ) , top . these minima are separated by the potential barriers responsible for the tunnel exchange coupling between the two - exciton states in the configuration space . the coordinate transformation @xmath39 places the origin of the new coordinate system into the intersection of the two tunnel channels between the respective potential minima [ fig . [ fig1 ] ( b ) ] , whereby the exchange splitting formula of refs.@xcite takes the form @xmath40 here @xmath41 are the ground - state and excited - state energies , eigenvalues of the hamiltonian ( [ biexcham ] ) , of the two coupled excitons as functions of their center - of - mass - to - center - of - mass separation , and @xmath42 is the tunnel exchange coupling integral responsible for the bound state formation of two excitons . for biexciton , this takes the form @xmath43 where @xmath44 is the solution to the schrdinger equation with the hamiltonian ( [ biexcham ] ) transformed to the @xmath45 coordinates . the factor @xmath46 comes from the fact that there are two equivalent tunnel channels in the biexciton problem , mixing three equivalent indistinguishable two - exciton states in the configuration space one state is given by the two minima on the @xmath47-axis and two more are represented by each of the minima on the @xmath48-axis [ cf . [ fig1 ] ( a ) and fig . [ fig1 ] ( b ) ] . the function @xmath44 in eq . ( [ jxx ] ) is sought in the form @xmath49\ , , \label{psixxxy}\ ] ] where @xmath50 \label{psi0xy}\ ] ] is the product of two single - exciton wave functions ( ground state ) representing the isolated two - exciton state centered at the minimum @xmath51 ( or @xmath52 , @xmath53 ) of the configuration space potential [ fig . [ fig1 ] ( b ) ] . this is the approximate solution to the shrdinger equation with the hamiltonin given by the first two lines in eq . ( [ biexcham ] ) , where the cut - off parameter @xmath17 is neglected.@xcite this approximation greatly simplifies problem solving , while still remaining adequate as only the long - distance tail of @xmath54 is important for the tunnel exchange coupling . the function @xmath55 , on the other hand , is a slowly varying function to account for the major deviation of @xmath56 from @xmath54 in its `` tail area '' due to the tunnel exchange coupling to another equivalent isolated two - exciton state centered at @xmath57 , @xmath58 ( or @xmath59 , @xmath53 ) . substituting eq . ( [ psixxxy ] ) into the schrdinger equation with the hamiltonian ( [ biexcham ] ) pre - transformed to the @xmath45 coordinates , one obtains in the region of interest ( @xmath17 dropped for the reason above ) @xmath60 up to negligible terms of the order of the inter - exciton van der waals energy and up to the second order derivatives of @xmath61 . this equation is to be solved with the boundary condition @xmath62 originating from the natural requirement @xmath63 , to result in @xmath64 after plugging this into eq . ( [ psixxxy ] ) one can calculate the tunnel exchange coupling integral ( [ jxx ] ) . retaining only the leading term of the integral series expansion in powers of @xmath34 subject to @xmath65 , one obtains @xmath66 the ground state energy @xmath67 of the two coupled 1d excitons in eq . ( [ egu ] ) is now seen to go through the negative minimum ( biexcitonic state ) as @xmath27 increases . the minimum occurs at @xmath68 , whereby the biexciton binding energy takes the form @xmath69 in atomic units . expressing @xmath34 in terms of @xmath70 , one obtains in absolute units the equation as follows @xmath71 = 11.0 cm the trion binding energy can be found in the same way using a modification of the hamiltonian ( [ biexcham ] ) , in which two same - sign particles share the third particle of an opposite sign to form the two equivalent 1d excitons as fig . [ fig2 ] shows for the negative trion complex consisting of the hole shared by the two electrons . the hamiltonian modified to reflect this fact has the first two lines exactly the same as in eq . ( [ biexcham ] ) , no line next to last , and one of the two terms in the last line either the first or the second one for the positive ( with @xmath72 ) and negative ( with @xmath73 ) trion , respectively . obviously , due to the additional mass factor @xmath74 ( typically less than one for bulk semiconductors ) in the hole - hole interaction term in the last line , the positive trion might be expected to have a greater binding energy in this model , in agreement with the results reported earlier.@xcite however , as was already mentioned in sec . [ sec2 ] , the mass reversal effect in _ strongly _ confined reduced dimensionality semiconductors is to result in @xmath37 in the trion hamiltonian . the positive - negative trion binding energy difference disappears then . the negative trion case illustrated in fig . [ fig2 ] , is addressed below . just like in the case of the biexciton , the treatment of the trion problem starts with the coordinate transformation ( [ transformation ] ) to bring the trion hamiltonian from the original ( configuration space ) coordinate system @xmath38 into the new coordinate system @xmath45 with the origin positioned as shown in fig . [ fig1 ] ( b ) . the tunnel exchange splitting integral in eq . ( [ egu ] ) now takes the form @xmath75 where @xmath76 is the ground - state wave function of the schrdinger equation with the hamiltonian ( [ biexcham ] ) modified to the negative trion case , as discussed above , and then transformed to the @xmath45 coordinates . the tunnel exchange current integral @xmath77 is due to the electron position exchange relative to the hole ( see fig . [ fig2 ] ) . this corresponds to the tunneling of the entire three particle system between the two equivalent indistinguishable configurations of the two excitons sharing the same hole in the configuration space @xmath38 , given by the pair of minima at @xmath51 and @xmath78 in fig . [ fig1 ] ( b ) . such a tunnel exchange interaction is responsible for the coupling of the three particle system to form a stable trion state . like in the case of the biexciton , one seeks the function @xmath79 in the form @xmath80\ , , \label{psixy}\ ] ] with @xmath81 given by eq . ( [ psi0xy ] ) , where @xmath82 is assumed to be a _ slowly _ varying function to take into account the deviation of @xmath83 from @xmath54 in the `` tail area '' of @xmath54 due to the tunnel exchange coupling to another equivalent isolated two - exciton state centered at @xmath57 , @xmath58 ( or @xmath59 , @xmath53 ) . substituting eq . ( [ psixy ] ) into the schrdinger equation with the negative trion hamiltonian pre - transformed to the @xmath45 coordinates , one obtains in the region of interest @xmath84 ( @xmath85 , cut - off @xmath17 dropped ) up to terms of the order of the second derivatives of @xmath86 . this is to be solved with the boundary condition @xmath87 coming from the requirement @xmath88 , to result in @xmath89 after plugging eqs . ( [ sxy ] ) and ( [ psixy ] ) into eq . ( [ jxast ] ) , and retaining only the leading term of the integral series expansion in powers of @xmath34 subject to @xmath65 , one obtains @xmath90 inserting this into the right - hand side of eq . ( [ egu ] ) , one sees that the ground state energy @xmath32 of the three particle system goes through the negative minimum ( the trion state ) as @xmath27 increases . the minimum occurs at @xmath68 , whereby the trion binding energy in atomic units takes the form @xmath91 in absolute units , expressing @xmath34 in terms of @xmath70 , one obtains @xmath92 = 12.0 cm from eqs . ( [ exx ] ) and ( [ exstar ] ) , one has the trion - to - biexciton binding energy ratio as follows @xmath94 if one now assumes @xmath95 ( @xmath96 is the dimensionless cn radius , or transverse confinement size for quasi-1d nanostructure in general ) as was demonstrated earlier by variational calculations@xcite to be consistent with many quasi-1d models,@xcite then one obtains the @xmath96-dependences of @xmath97 , @xmath98 and @xmath99 shown in fig . [ fig3 ] . the trion and biexciton binding energies both decrease with increasing @xmath96 in such a way that their ratio remains greater than unity for small enough @xmath96 in full agreement with the experiments by colombier et al.@xcite and yuma et al.@xcite however , since the factor @xmath100 in eq . ( [ exstarexx ] ) is less than one , the ratio can also be less than unity for @xmath96 large enough ( but not too large , so that our configuration space method still works ) . as @xmath96 goes down , on the other hand , the biexciton - to - exciton binding energy ratio @xmath101 in eq . ( [ exx ] ) slowly grows , approaching the pure 1d limit @xmath102 . similar tendency can also be traced in the monte - carlo simulation data of ref.@xcite the equilibrium inter - exciton center - of - mass distance in the biexciton complex goes down with decreasing @xmath96 as well , @xmath103 ( atomic units ) . this supports experimental evidence for enhanced exciton - exciton annihilation in small diameter cns.@xcite the trion - to - exciton binding energy ratio @xmath104 of eq . ( [ exstar ] ) increases with decreasing @xmath96 faster than @xmath101 ( fig . [ fig3 ] ) , to yield @xmath105 as the pure 1d limit for the trion - to - biexciton binding energy ratio . = 11.5 cm when @xmath99 is known , one can use eq . ( [ exstarexx ] ) to estimate the effective bohr radii @xmath106 for the excitons in the cns of known radii . for example , substituting @xmath107 for the 0.75 nm diameter ( 6,5 ) cn and @xmath108 for the 1.09 nm diameter ( 9,7 ) cn , as reported by yuma et al.@xcite and colombier et al.,@xcite respectively , into the left hand side of the transcendental equation ( [ exstarexx ] ) and solving it for @xmath106 , one obtains the effective exciton bohr radius @xmath109 nm and @xmath110 nm for the ( 6,5 ) cn and ( 9,7 ) cn , respectively . this agrees reasonably with previous estimates.@xcite in general , the binding energies in eqs . ( [ exstar ] ) and ( [ exx ] ) are functions of the cn radius ( transverse confinement size for a quasi-1d semiconductor nanowire ) , @xmath111 and @xmath26 . figures [ fig4 ] ( a ) and [ fig4 ] ( b ) show their 3d plots at fixed @xmath112 and at fixed @xmath113 , respectively , as functions of two remaining variables . the reduced effective mass @xmath111 chosen is typical of large radius excitons in small - diameter cns.@xcite the unit dielectric constant @xmath26 assumes the cn placed in air and the fact that there is no screening in quasi-1d semiconductor systems both at short and at large electron - hole separations.@xcite this latter assumption of the unit background dielectric constant remains legitimate for _ small _ diameter ( @xmath3 nm ) semiconducting cns in dielectric screening environment , too , for the lowest excitation energy exciton in its ground state of interest here ( not for its excited states though ) , in which case the environment screening effect is shown by ando to be negligible,@xcite diminishing quickly with the increase of the effective distance between the cn and dielectric medium relative to the cn diameter . figure [ fig4 ] ( a ) can be used to evaluate the relative stability of the trion and biexciton complexes in quasi-1d semiconductors . one sees that whether the trion or the biexciton is more stable ( has the greater binding energy ) in a particular quasi-1d system depends on @xmath111 and on the characteristic transverse size of the nanostructure . in strongly confined quasi-1d systems with relatively small @xmath111 , such as small - diameter cns , the trion is generally more stable than the biexciton . in less confined quasi-1d structures with greater @xmath111 typical of semiconductors,@xcite the biexciton is more stable than the trion . this is a generic peculiarity in the sense that it comes from the tunnel exchange in the quasi-1d electron - hole system in the configuration space . greater @xmath111 , while not affecting significantly the single charge tunnel exchange in the trion complex , makes the neutral biexciton complex generally more compact , facilitating the mixed charge tunnel exchange in it and thus increasing the stability of the complex . from fig . [ fig4 ] ( b ) one sees that this generic feature is not affected by the variation of @xmath26 , although the increase of @xmath26 decreases the binding energies of both excitonic complexes in agreement both with theoretical studies@xcite and with experimental observations of lower binding energies ( compared to those in cns ) of these complexes in conventional semiconductor nanowires.@xcite the latter are self - assembled nanostructures of one ( transversely confined ) semiconductor embedded in another ( bulk ) semiconductor with the characteristic transverse confinement size typically greater than that of small diameter cns , and so both inside and outside material dielectric properties matter there . = 12.0 cm figure [ fig5 ] shows the cross - section of fig . [ fig4 ] ( a ) taken at @xmath114 to present the relative behavior of @xmath97 and @xmath98 in semiconducting cns of increasing radius . both @xmath97 and @xmath98 decrease , and so does their ratio , as the cn radius increases . from the graph , @xmath115 and @xmath116 mev , @xmath117 and @xmath118 mev , for the ( 6,5 ) and ( 9,7 ) cns , respectively . this is to be compared with @xmath6 and @xmath7 mev for the ( 6,5 ) cn versus @xmath4 and @xmath5 mev for the ( 9,7 ) cn reported experimentally.@xcite one sees that , as opposed to perturbative theories,@xcite the present configuration space theory underestimates experimental data just slightly , most likely due to the standard variational treatment limitations . it does explain well the trends observed , and so the graph in fig . [ fig5 ] can be used as a guide for trion and biexciton binding energy estimates in small diameter ( @xmath3 nm ) nanotubes . recently , there has been a considerable interest in studies of optical properties of coupled quantum wells ( cqws).@xcite the cqw semiconductor nanostructure ( fig . [ fig6 ] ) consists of two identical semiconductor quantum wells separated by a thin barrier layer of another semiconductor . the tunneling of carriers through the barrier makes two wells electronically coupled to each other . as a result , an electron ( a hole ) can either reside in one of the wells , or its wave function can be distributed between both wells . a coulomb bound electron - hole pair residing in the same well forms a direct exciton [ fig . [ fig6 ] ( a ) ] . if the electron and hole of a pair are located in different wells , then an indirect exciton is formed [ fig . [ fig6 ] ( b ) ] . = 12.0cm-0.5 cm physical properties of cqws can be controlled by using external electro- and magnetostatic fields . ( see , e.g. , refs.@xcite and refs . therein . ) for example , applying the electrostatic field perpendicular to the layers increases the exciton radiative lifetime due to a reduction in the spatial overlap ( contact density ) between the electron and hole wave functions . as this takes place , the exciton binding energy reduces due to an increased electron - hole separation to make the exciton less stable against ionization , in contrast with the exciton magnetostatic stabilization effect under the same geometry.@xcite the tunneling effect is also enhanced as the electric field allows the carriers to leak out of the system , resulting in a considerable shortening of the photoluminescence decay time . cqws embedded into bragg - mirror microcavities show a special type of voltage - tuned exciton polaritons , which can be used for low - threshold power polariton lasing.@xcite new non - linear phenomena are also reported for these cqw systems both theoretically and experimentally , such as bose condensation@xcite and parametric oscillations@xcite of exciton polaritons . for laterally confined cqw structures , experimental evidence for controllable formation of multiexciton wigner - like molecular complexes of indirect excitons ( single exciton , biexciton , triexciton , etc . ) was reported recently.@xcite trion complexes formed both by direct and by indirect excitons , as sketched in figs . [ fig6 ] ( a ) and ( b ) , were observed in cqws as well.@xcite all these findings make cqws a much richer system capable of new developments in fundamental quantum physics and nanotechnology as compared to single quantum wells.@xcite they open up new routes for non - linear coherent optical control and spinoptronics applications with quasi-2d semiconductor cqw nanostructures . very recently , the problem of the trion complex formation in cqws was studied theoretically in great detail for trions composed of a _ direct _ exciton and an electron ( or a hole ) located in the neighboring quantum well@xcite [ as sketched in fig . [ fig6 ] ( a ) ] . significant binding energies are predicted on the order of @xmath119 mev at interwell separations @xmath120 nm for the lowest energy positive and negative trion states , to allow one suggest a possibility for trion wigner crystallization . figure [ fig6 ] ( b ) shows another possible trion complex that can also be realized in cqws . here , the trion is composed of an _ indirect _ exciton and an electron ( or a hole ) in such a manner as to keep two same - sign particles in the same quantum well with the opposite - sign particle being located in the neighboring well . this can be viewed as the two equivalent configurations of the three - particle system in the configuration space ( @xmath121 , @xmath122 ) of the two _ independent _ in - plane projections of the relative electron - hole distances @xmath123 and @xmath124 in the two _ equivalent _ indirect excitons sharing the same electron , or the same hole as shown in fig . [ fig6 ] ( b ) . such a three - particle system in the quasi-2d semiconductor cqw nanostructure is quite analogous to the quasi-1d trion presented in sec . 3 above [ cf . [ fig6 ] ( b ) and fig . [ fig2 ] ] . therefore , the landau - herring configuration space approach can be used here as well to evaluate the binding energy for this special case of the quasi-2d trion state . following is a brief outline of how one could proceed with the configuration space method to obtain the ground state binding energy for the quasi-2d trion complex sketched in fig [ fig6 ] ( b ) . a complete analysis of the problem will be presented elsewhere . the method requires knowledge of the ground state characteristics of the indirect exciton ( abbreviated as `` @xmath125 '' in what follows ) . specifically , one needs to know the quasi-2d ground state energy @xmath126 and corresponding _ in - plane _ relative electron - hole motion wave function @xmath127 for the indirect exciton in the cqw system with the interwell distance @xmath128 . these can be found by solving the radial scrdinger equation that is obtained by decoupling radial relative electron - hole motion in the cylindrical coordinate system with the @xmath129-axis being perpendicular to the qw layers [ see fig [ fig6 ] ( b ) ] . such an equation was derived and analyzed previously by leavitt and little.@xcite the energy and the wave function of interest are as follows @xmath130 \label{indirect}\\[-0.2 cm ] \psi_{i\!x}(\rho , d)=n\exp[-\lambda(\sqrt{\rho^2+d^2}-d)]\,,\hskip0.5cm\nonumber\end{aligned}\ ] ] where @xmath131 is the exponential integral , @xmath132 , the normalization constant @xmath133 is determined from the condition @xmath134 and all quantities are measured in atomic units as defined in sec . 2 . = 12.0 cm with eq . ( [ indirect ] ) in place , one can work out strategies for tunneling current calculations in the configuration space ( @xmath121 , @xmath122 ) of the two _ independent _ in - plane projections of the relative electron - hole distances @xmath123 and @xmath124 in the two _ equivalent _ indirect excitons as shown in fig . [ fig6 ] ( b ) . both tunneling current responsible for the trion complex formation and that responsible for the biexciton complex formation can be obtained in full analogy with how it was done above for the respective quasi-1d complexes . the only formal difference now is the change in the phase integration volume from @xmath135 to @xmath136 . minimizing the tunneling current with respect to the center - of - mass - to - center - of - mass distance of the two equivalent indirect excitons results in the binding energy of a few - particle complex of interest . note that the method applies to the complexes formed by _ excitons only as they allow equivalent configurations for a few - particle system to tunnel throughout in the configuration space ( @xmath121 , @xmath122 ) , thereby forming a respective ( tunnel ) coupled few - particle complex . binding energy calculations for indirect exciton complexes in semiconductor cqw nanostructures are important to understand the principles of the more complicated electron - hole structure formation such as that shown in fig . this is a coupled charge - neutral spin - aligned wigner - like structure formed by two trions , one positively charged and another one negatively charged . the entire structure is electrically neutral , and it has an interesting electron - hole spin alignment pattern . this structure can also be viewed as a triexciton , a coupled state of three indirect singlet excitons . one could also imagine a wigner - like crystal structure formed by unequal number of electrons and holes , as opposed to that in fig . [ fig7 ] , whereby the entire coupled structure could possess net charge and spin at the same time to allow precise electro- and magnetostatic control and manipulation by its optical and spin properties . such wigner - like electron - hole crystal structures in cqws might be of great interest for spinoptronics applications . all in all , indirect excitons , biexcitons and trions formed by indirect excitons are those building blocks that control the formation of more complicated wigner - like electron - hole crystal structures in cqws . the configuration space method presented here allows one to study the binding energies for these building blocks as functions of cqw system parameters , and thus to understand how stable electron - hole wigner crystallization could possibly be in these quasi-2d nanostructures . the method should also work well for biexciton and trion complexes in quasi-2d self - assembled transition metal dichalcogenide heterostructures , where electrons and holes accumulated in the opposite neighboring monolayers are recently reported to form indirect excitons with new exciting properties such as increased recombination time@xcite and vanishing high - temperature viscosity.@xcite presented herein is a universal configuration space method for binding energy calculations of the lowest energy neutral ( biexciton ) and charged ( trion ) exciton complexes in spatially confined quasi-1d semiconductor nanostructures . the method works in the effective two - dimensional configuration space of the two relative electron - hole motion coordinates of the two non - interacting quasi-1d excitons . the biexciton or trion bound state forms due to under - barrier tunneling between equivalent configurations of the electron - hole system in the configuration space . tunneling rate controls the binding strength and can be turned into the binding energy by means of an appropriate variational procedure . quite generally , trions are shown to be more stable ( have greater binding energy ) than biexcitons in strongly confined quasi-1d structures with small reduced electron - hole masses . biexcitons are more stable than trions in less confined structures with large reduced electron - hole masses . a universal crossover behavior is predicted whereby trions become less stable than biexcitons as the transverse size of the quasi-1d nanostructure increases . an outline is given of how the method can be used for electron - hole complexes of indirect excitons in quasi-2d semiconductor systems such as coupled quantum wells and van der waals bound transition metal dichalcogenide heterostructures . here , indirect excitons , biexcitons , and trions formed by indirect excitons control the formation of more complicated wigner - like electron - hole crystal structures . the configuration space method can help develop understanding of how stable wigner crystallization could be in these quasi-2d nanostructures . wigner - like electron - hole crystal structures are of great interest for future spinoptronics applications . this work is supported by the us department of energy ( de - sc0007117 ) . discussions with david tomanek ( michigan state u. ) , roman kezerashvili ( ny@xmath137citytech ) , and masha vladimirova ( u. montpellier , france ) are acknowledged . thanks tony heinz ( stanford u. ) for pointing out ref.@xcite of relevance to this work . [ 99 ] h.haug and s.w.koch , _ quantum theory of the optical and electronic properties of semiconductors _ , 5th edn . ( world scientific , london , 2005 ) . p.y.yu and m.cardona , _ fundamentals of semiconductors _ , 4th edn . ( springer - verlag , berlin 2010 ) . e.g.barbagiovanni , d.j.lockwood , p.j.simpson , and l.v.goncharova , _ appl . * 1 * ( 2014 ) 011302 . m.s.dresselhaus , g.dresselhaus , r.saito , and a.jorio , _ annu . * 58 * ( 2007 ) 719 . j.deslippe , m.dipoppa , d.prendergast , m.v.o.moutinho , r.b.capaz , and s.g.louie , _ nanolett . _ * 9 * ( 2009 ) 1330 . m.steiner , m.freitag , v.perebeinos , a.naumov , j.p . small , a.a.bol , and ph.avouris , _ nanolett . _ * 9 * ( 2009 ) 3477 . c.d.spataru and f.lonard , _ phys . lett . _ * 104 * ( 2010 ) 177402 . t.mueller , m.kinoshita , m.steiner , v.perebeinos , a.a.bol , d.b.farmer , and ph.avouris , _ nature nanotech . _ * 5 * ( 2010 ) 27 . m.yoshida , y.kumamoto , a.ishii , a.yokoyama , and y.k.kato , _ appl . * 105 * ( 2014 ) 161104 . i.v.bondarev , l.m.woods , and k.tatur , _ phys . _ * b80 * ( 2009 ) 085407 ; _ opt . * 282 * ( 2009 ) 661 . i.v.bondarev , _ phys . * b85 * ( 2012 ) 035448 . i.v.bondarev and a.v.meliksetyan , _ phys . _ * b89 * ( 2014 ) 045414 . m.l.trolle and t.g.pedersen , _ phys . * b92 * ( 2015 ) 085431 . r.saito , g.dresselhaus , and m.s.dresselhaus , _ science of fullerenes and carbon nanotubes _ ( imperial college press , london , 1998 ) . m.dresselhaus , g.dresselhaus , and ph.avouris , eds . , _ carbon nanotubes : synthesis , structure , properties , and applications _ ( springer , berlin , 2001 ) . d.j.styers-barnett , s.p.ellison , b.p.mehl , b.c.westlake , r.l.house , c.park , k.e.wise , and j.m.papanikolas , _ j. phys . chem . _ * c112 * ( 2008 ) 4507 . a.hgele , ch.galland , m.winger , and a.imamolu , _ phys . * 100 * ( 2008 ) 217401 . ph.avouris , m.freitag , and v.perebeinos , _ nature photonics _ * 2 * ( 2008 ) 341 . n.m.gabor , zh.zhong , k.bosnick , j.park , and p.l.mceuen , _ science _ * 325 * ( 2009 ) 1367 . t.hertel , _ nature photonics _ * 4 * ( 2010 ) 77 . i.v.bondarev , _ j. comp . nanosci . _ * 7 * ( 2010 ) 1673 . x.dang , h.yi , m .- h.ham , j.qi , d.s.yun , r.ladewski , m.s.strano , p.t.hammond , and a.m.belcher , _ nature nanotech . _ * 6 * ( 2011 ) 377 . i.v.bondarev , l.m.woods , and a.popescu , _ plasmons : theory and applications _ ( nova science , new york , 2011 ) , chap . 16 , p. 381 . s.nanot , e.h.haroz , j .- h.kim , r.h.hauge , and j.kono , _ adv . * 24 * ( 2012 ) 4977 . volder , s.h.tawfick , r.h.baughman , and a.j.hart , _ science _ * 339 * ( 2013 ) 535 . t.hertel and i.v.bondarev , eds . , _ photophysics of carbon nanotubes and nanotube composites ( special issue ) , chem . phys . _ * 413 * , 1 ( 2013 ) . i.v.bondarev , _ optics express _ * 23 * ( 2015 ) 3971 . d.birkedal , j.singh , v.g.lyssenko , j.erland , and j.m.hvam , _ phys . * 76 * ( 1996 ) 672 . j.singh , d.birkedal , v.g.lyssenko , and j.m.hvam , _ phys . rev . _ * b53 * ( 1996 ) 15909 . a.thilagam , _ phys . rev . _ * b56 * , 4665 ( 1997 ) . a.v.filinov , c.riva , f.m.peeters , yu.e.lozovik , and m.bonitz , _ phys . * b70 * ( 2004 ) 035323 . a.s.bracker,@xmath138e.a.stinaff,@xmath138d.gammon,@xmath138m.e.ware , j.g.tischler , d.park , d.gershoni , a.v.filinov , m.bonitz , f.m . peeters , and c.riva , _ phys . _ * b72 * ( 2005 ) 035332 . t.baars , w.braun , m.bayer , and a.forchel , _ phys . rev . _ * b58 * ( 1998 ) r1750 . a.crottini , j.l.staehli , b.deveaud , x.l.wang , m.ogura , _ solid state commun . _ * 121 * ( 2002 ) 401 . y.sidor , b.partoens , and f.m.peeters , _ phys . * b77 * ( 2008 ) 205413 . b.aln , d.fuster , g. muoz - matutano , j.martnez-pastor , y.gonzlez , j.canet-ferrer , and l.gonzlez , _ phys . lett . _ * 101 * ( 2008 ) 067405 . m.j.a.schuetz , m.g.moore , and c.piermarocchi , _ nature physics _ * 6 * ( 2010 ) 919 . t.g.pedersen , _ phys . rev . _ * b67 * ( 2003 ) 073401 . t.g.pedersen , k.pedersen , h.d.cornean , and p.duclos , _ nanolett . _ * 5 * ( 2005 ) 291 . d.kammerlander , d.prezzi , g.goldoni , e.molinari , and u.hohenester , _ phys . * 99 * ( 2007 ) 126806 ; _ physica e _ * 40 * ( 2008 ) 1997 . t.f.rnnow , t.g.pedersen , and h.d.cornean , _ phys . rev . _ * b81 * ( 2010 ) 205446 . t.f.rnnow , t.g.pedersen , b.partoens , and k.k.berthelsen , _ phys . rev . _ * b84 * ( 2011 ) 035316 . r.matsunaga , k.matsuda , and y.kanemitsu , _ phys . * 106 * ( 2011 ) 037404 . s.m.santos , b.yuma , s.berciaud , j.shaver , m.gallart , p.gilliot , l.cognet , and b.lounis , _ phys . * 107 * ( 2011 ) 187401 . i.v.bondarev , _ phys . * b83 * ( 2011 ) 153409 . i.v.bondarev , _ phys . * b90 * ( 2014 ) 245430 . k.watanabe and k.asano , _ phys . _ * b85 * ( 2012 ) 035416 ; _ ibid . _ * 83 * ( 2011 ) 115406 . t.f.rnnow , t.g.pedersen , and b.partoens , _ phys . * b85 * ( 2012 ) 045412 . l.colombier , j.selles , e.rousseau , j.s.lauret , f.vialla , c.voisin , and g.cassabois , _ phys . * 109 * ( 2012 ) 197402 . b.yuma , s.berciaud , j.besbas , j.shaver , s.santos , s.ghosh , r.b.weisman , l.cognet , m.gallart , m.ziegler , b.hnerlage , b.lounis , and p.gilliot , _ phys . _ * b87 * ( 2013 ) 205412 . b.patton , w.langbein , and u.woggon , _ phys . _ * b68 * ( 2003 ) 125316 . m.kaniber , m.f.huck , k.mller , e.c.clark , f.troiani , m.bichler , h.j.krenner , and j.j.finley , _ nanotechnology _ * 22 * ( 2011 ) 325202 . v.jovanov , s.kapfinger , m.bichler , g.abstreiter , and j.j.finley , _ phys . * b84 * ( 2011 ) 235321 . t.ando , _ j. phys . * 74 * ( 2005 ) 777 . l.d.landau and e.m.lifshitz , _ quantum mechanics . non - relativistic theory _ ( pergamon , oxford , 1991 ) . l.p.gorkov and l.p.pitaevski , _ dokl . nauk sssr _ * 151 * ( 1963 ) 822 [ english transl . : _ soviet phys.dokl._ * 8 * ( 1964 ) 788 ] . c.herring , _ rev . * 34 * ( 1962 ) 631 ; c.herring and m.flicker , _ phys . * 134 * ( 1964 ) a362 . t.ogawa and t.takagahara , _ phys . _ * b44 * ( 1991 ) 8138 . a.jorio , c.fantini , m.a.pimenta , r.b.capaz , ge.g.samsonidze , g.dresselhaus , m.s.dresselhaus , j.jiang , n.kobayashi , a.grneis , and r.saito , _ phys . * b71 * ( 2005 ) 075401 . r.loudon , _ am . _ * 27 * ( 1959 ) 649 . f.wang , g.dukovic , e.knoesel , l.e.brus , and t.f.heinz , _ phys . rev . _ * b70 * ( 2004 ) 241403(r ) . d.abramavicius , y .- z.ma , m.w.graham , l.valkunas , and g.r.fleming , _ phys . * b79 * ( 2009 ) 195445 . a.srivastava and j.kono , _ phys . _ * b79 * ( 2009 ) 205407 . t.ando , _ j. phys _ * 79 * ( 2010 ) 024706 . t.byrnes , g.v.kolmakov , r.ya.kezerashvili , and y.yamamoto , _ phys . * b90 * , ( 2014 ) 125314 . o.l.berman , r.ya.kezerashvili , and sh.m.tsiklauri , _ int . j. mod * b28 * ( 2014 ) 1450064 . g.j.schinner , j.repp , e.schubert , a.k.rai , d.reuter , a.d.wieck , a.o.govorov , a.w.holleitner , and j.p.kotthaus , _ phys . rev . _ * 110 * ( 2013 ) 127403 . k.sivalertporn , l.mouchliadis , a.l.ivanov , r.philp , and e.a.muljarov , _ phys . _ * b85 * ( 2012 ) 045207 . g.christmann , a.askitopoulos , g.deligeorgis , z.hatzopoulos , s.i.tsintzos , p.g.savvidis , and j.j.baumberg , _ appl . phys . _ * 98 * , 081111 ( 2011 ) . a.g.winbow , j.r.leonard , m.remeika , y.y.kuznetsova , a.a.high , a.t.hammack , l.v.butov , j.wilkes , a.a.guenther , a.l.ivanov , m.hanson , and a.c.gossard , _ phys . rev . lett . _ * 106 * , 196806 ( 2011 ) . g.j.schinner , e.schubert , m.p . stallhofer , j.p.kotthaus , d.schuh , a.k.rai , d.reuter , a.d.wieck , and a.o.govorov , _ phys . rev . _ * b83 * ( 2011 ) 165308 . g.grosso , j.graves , a.t.hammack , a.a.high , l.v.butov , m.hanson , and a.c.gossard , _ nature photonics _ * 3 * ( 2009 ) 577 . a.t.hammack , l.v.butov , j.wilkes , l.mouchliadis , e.a.muljarov , a.l.ivanov , and a.c.gossard , _ phys . _ * b80 * ( 2009 ) 155331 . z.vrs , d.w.snoke , l.pfeiffer , and k.west , _ phys . lett . _ * 103 * ( 2009 ) 016403 . c.diederichs , j.tignon , g.dasbach , c.ciuti , a.lematre , j.bloch , ph.roussignol , and c.delalande , _ nature _ * 440 * ( 2006 ) 904 . z.vrs , d.w.snoke , l.pfeiffer , and k.west , _ phys . _ * 97 * ( 2006 ) 016803 . a.j.shields , j.l.osborne , d.m.whittaker , m.y.simmons , m.pepper , and d.a.ritchie , _ phys . _ * b55 * ( 1997 ) 1318 . r.p.leavitt and j.w.little , _ phys . * b42 * ( 1990 ) 11774 . f.ceballos , m.z.bellus , h .- y.chiu , and h.zhao , _ acs nano _ * 8 * ( 2014 ) 12717 . m.m.fogler , l.v.butov , and k.s.novoselov , _ nature commun . _ * 5 * ( 2014 ) 4555 .

a configuration space method is developed for binding energy calculations of the lowest energy exciton complexes ( trion , biexciton ) in spatially confined quasi-1d semiconductor nanostructures such as nanowires and nanotubes . quite generally , trions are shown to have greater binding energy in strongly confined structures with small reduced electron - hole masses . biexcitons have greater binding energy in less confined structures with large reduced electron - hole masses . this results in a universal crossover behavior , whereby trions become less stable than biexcitons as the transverse size of the quasi-1d nanostructure increases . the method is also capable of evaluating binding energies for electron - hole complexes in quasi-2d semiconductors such as coupled quantum wells and bilayer van der walls bound heterostructures with advanced optoelectronic properties .

1605.02348

the enhancement of heat transfer by embedded nano - particles in a fluid subjected to a temperature gradient is an important issue that is expected to help in technological application like solar heating , and various cooling devices , including miniaturized computer processors . accordingly , many experiments were conducted in the last couple of decades @xcite , with a breakthrough announcement from a group at argonne national laboratory , who studied water and oil - based nanofluids containing copper oxide nanopaticles , and found an amazing 60% enhancement in thermal conductivity for only a 5% volume fraction of nanoparticles @xcite . subsequent research has however generated what ref . @xcite referred to as `` an astonishing spectrum of results '' . results in the literature show sometime enhancement in the thermal conductivity compared to the prediction of the maxwell - garnett effective medium theory , and sometime values that are less than the same prediction @xcite . confusingly enough , these discrepancies occur even for the same fluid and the same size and composition of the nano - particles . some of these conflicting results were explained by either the formation of percolated clusters of particles ( for the case of enhancement with respect to the maxwell - garret prediction ) or by surface ( kapitza ) resistance ( for the case of reduction with respect to the same prediction ) @xcite . interestingly enough , enhancement is typically seen in _ quiescent _ fluids , and one expects that the agglomeration of clusters will become impossible in flows , be them laminar or turbulent . of course , in technological application flowing nano - fluids may be the rule rather than the exception . thus our aim in this article is to develop models of heat flux in flowing nano - fluids , where the flow can be either laminar or turbulent . in such systems we can not expect an enhancement of heat flux due to the agglomeration of particles ( at least at low volume fractions ) , and for the sake of simplicity we will assume that there is no kapitza resistance . since it was reported in the literature that elongated nano - particles are superior to spheres in quiescent nanofluids @xcite , we will study both spheres and spheroids in flowing nanofluids . we will argue that generically elongation may not be advantageous at all , and will explain why . for completeness , we will begin by studying quiescent nanofluids under a temperature gradient . we will present extensive numerical simulation that will demonstrate a very good agreement with the maxwell - garnett theory for spherical nano - particles and with the generalization of nan et al @xcite for elongated particles . in the case of flowing nanofluid we will offer a model that will provide expressions for the heat transfer for different values of the aspect ratio of the particles , for different volume fractions and for laminar and turbulent flows . the structure of the paper is as follows . in the next section ( [ s : model ] ) we formulate the problem , describe the equations of motion and the numerical procedure , and discuss the dilute suspension approximation . in section [ s:3body ] we describe the results of numerical simulations of spherical and spheroidal particles in a fluid at rest and in a shear flow . in section [ s:4model ] we develop an analytical model of heat transfer characteristic , i.e. the nusselt number , and analyze the model predictions of heat flux enhancement in two limiting cases of very strong brownian diffusion and a weak one . in this section we formulate the model for dilute nanofluids laden with elongated spheroidal nanoparticles . this includes the basic equation of motion for the velocity and temperature fields and the boundary conditions ( with constant velocity and temperature gradients far away from particles ) . we also present details of the numerical simulations and their validation . when the nanofluid is very dilute one can disregard the effect of one particle on the other and consider the nanofluid as an ensemble of noninteracting particles . one of these is shown in fig . [ geometry ] . the center of the spheroidal particle is at the position @xmath0 in the middle of a plane couette flow between two @xmath1-separated horizontal parallel walls that move in the @xmath2-direction with opposite velocities @xmath3 . due to the symmetry , the forces are balanced , and the particle neither migrates nor collides with the walls ; the particle s center remains at @xmath4 . this allows us to eliminate in the numerics any effect of the particle sweeping parallel to the walls . note that this effect is anyway absent in homogeneous cases . we employ periodic boundary conditions along the @xmath5 and the @xmath6 direction . this results in a periodic replication of the computational box ( together with the particle ) in the @xmath7-plane , leading to a `` monolayer '' of an infinite number of periodically distributed particles in the @xmath7-plane . remarkably enough ( and by reasons that will be explained below ) , this allows us to reproduce in numerics with a _ single particle _ effects of _ a finite volume fraction _ @xmath8 at least up to @xmath9 . the walls are kept at fixed temperatures @xmath10 . in the absence of particles these boundary conditions give rise to a vertical temperature gradient @xmath11 and shear @xmath12 ( see fig . [ geometry ] ) . the spheroidal particle s semi - axes are @xmath13 ( the longest ) and @xmath14 ( the shortest ) , @xmath15 ( fig . [ geometry ] ) . the thermal diffusivity inside the particle is @xmath16 . the carrier fluid has a kinematic viscosity @xmath17 and a thermal diffusivity @xmath18 . for simplicity , at this stage we neglect the effect of gravity and the particle mass . the co - ordinate system and polar angles are given by : [ eq : anglesdefs ] @xmath19 where @xmath20 is the projection of the unit - vector @xmath21 onto the axis @xmath22 , @xmath23 , and @xmath24 is the angle between the particle s largest axis and the @xmath25-axis . the particle in a plane couette flow . the largest particle axis is shown as a ( red ) rod . the velocity field ( far from the particle ) is @xmath26 where the shear @xmath27 in the particle - free flow with @xmath1 being the distance between the walls . for the theoretical development we assume the creeping flow conditions as in @xcite.,width=321 ] in the absence of particles , changing the intensity of the laminar shear flow does not change the heat flux since there is no velocity orthogonal to the wall and the system is homogeneous in the direction parallel to the wall . in the presence of a particle , the shear induces it to rotate , the simplest case being a spherical particle which rotates at a constant angular velocity @xmath28 ( provided the particle radius is much smaller than the distance to the wall ) . this rotation induces a vertical flow motion near the particle . the modification of the velocity profile has three consequences . the first directly affects the heat convection in the fluid through a convective contribution @xmath29 . the second comes from the particle rotation which brings up the hotter side of the particle during its rotation . the third changes the heat conduction due to a modification of the temperature profiles at the surface of the particle . in the case of non spherical particles the presence of a shear has also a strong influence of the statistical distribution of particle orientations . to fix the notation we list here the most important dimensionless parameters involved in the problem : [ dimnumdefs ] @xmath30 note that the particle s largest axis defines our length - scale also , @xmath31 the dynamical effects previously mentioned can be quantitatively studied by solving the following system of equations for the temperature in the fluid , @xmath32 , and in the particle , @xmath33 , [ basiceqs ] @xmath34 together with the boundary conditions , @xmath35 the fluid velocity in the whole domain can be found by solving the incompressible navier - stokes equations : @xmath36 together with non - slip boundary conditions at the surface of the particle and at the walls : @xmath37 here @xmath38 is the carrier fluid density and @xmath39 is the pressure . clearly , on nano - scales the temperature difference ( across a particle ) is sufficiently small to allow us to neglect the dependence of the fluid s and particle s material parameters on the temperature , i.e. we take @xmath17 , @xmath18 and @xmath16 as constants . the dynamics of a neutrally buoyant particle is governed by the equations of the solid body rotation @xcite : @xmath40 where @xmath41 is the angular velocity in the body - fixed frame , @xmath42 is the torque in the body - fixed frame and @xmath43 is the moment of inertia tensor . the numerical simulation of the conjugated heat transfer problem given by eqs is performed by means of two coupled d3q19 lattice boltzmann ( lb ) equations under the so - called bgk approximation @xcite . details of the simulations are presented in appendix [ ss : numerics ] . besides purely numerical means , the control of the simulations was done by its comparison with known analytical solutions . as an example , fig [ f:2 ] shows the comparison between the analytical temperature profile ( [ eq : lnl ] ) ( solid red line ) and numerical profile ( black dots ) for a moderately big spherical particle [ @xmath44 . temperature profile along a line passing through the center of a spherical particle ( of radius @xmath45 ) in a quiescent fluid . dashed ( green ) line conductive temperature profile in the pure fluid , solid ( red ) curve eq . ( [ eq : lnl ] ) @xcite for an infinite domain with a temperature gradient , dots numerical results ( at lattice points ) with @xmath46 ( full circles ) and @xmath47 ( squares ) . the present test was performed with pr @xmath48 , @xmath49 and @xmath50 lattice units.,width=321 ] the second test was the comparison in fig . [ jeffery_test ] of the particle s angular velocity theoretically obtained by jeffery @xcite and simulated by our lb - realization . for more details , again see appendix [ ss : numerics ] . here we discuss how to relate the contribution of a single particle to the heat flux with the total contribution of many weakly - interacting particles randomly distributed in the flow occupying a finite volume fraction @xmath8 . the first step is to consider _ fully dilute limit _ , @xmath51 , in which we can exploit the fact that the particle contribution is additive and proportional to @xmath8 . in this way we can introduce the nusselt number as the ratio between the total heat flux ( inside the computational cell ) and the conductive heat flux in the basic cell : @xmath52 where @xmath53 is the total heat flux , @xmath54 is the conductive part and @xmath55 is the effective heat diffusivity of the composite fluid . also , @xmath56 . without particles @xmath57 and nu @xmath48 . for _ dilute suspensions _ , in which @xmath58 , nu can be expanded in powers of @xmath8 @xcite : [ 53 ] @xmath59 where @xmath60 are dimensionless constants dependent on the peclet number , on the particle s aspect ratio , @xmath61 , on the relative heat diffusivity , @xmath62 , and maybe other parameters like reynolds or prandtl numbers [ see eqs . ( [ dimnumdefs ] ) ] , i.e. @xmath63 . as expected , in our simulations the quantity @xmath64nu@xmath65 is indeed proportional to @xmath8 for @xmath66 . this allows us to find @xmath67 as a function of other parameters of the problem . the next order term in the expansion , i.e. @xmath68 , contains very important information about how nu depends on @xmath8 for small but finite values of @xmath8 . generally speaking , to get this information from numerics one needs to solve eqs . for many particles , randomly distributed in space . we demonstrate that this problem can be _ tremendously simplified _ by reducing it to a _ one - particle _ case , in which this particle of volume @xmath69 is put in the center of computational box of volume @xmath70 with _ periodic boundary conditions _ in the horizontal ( orthogonal to the temperature gradient ) @xmath5- and @xmath6-directions . in this way the total system can be considered as constructed from a periodic repetition of the basic elementary cell in the @xmath5- and @xmath6-directions . comparing in subsec . [ sss : sphere - rest ] our numerical results with analytical findings obtained under the assumption of random particle distributions , we see that the precise particle distribution is not essential what really matters is the actual volume fraction . the reason for that is quite simple : as one sees in fig . [ f:2 ] , the deviation from the linear temperature profile becomes important at distances@xmath71 ) that the distance should be smaller than @xmath72 , where @xmath73 is the deviation criterion , i.e. @xmath74 and @xmath75 . ] smaller than the particle diameter @xmath76 for both @xmath77 and @xmath47 . thus any nonlinear dependence of nu on @xmath8 can appear only if the particles are sufficiently close to overlap the deviation from linear temperature profile . we see from our numerics that the this interaction causes 10% deviation from the straight line in nu vs. @xmath8 dependence for volume fractions of about 0.1 and more for @xmath78 . for @xmath79 there is some nonlinear dependence of nu on @xmath8 but it practically coincides for the random and the periodic distributions of particles . in this section we begin with collecting the required information about the dependence of @xmath80 on the various parameters ( [ dimnumdefs ] ) . to do this we consider simple limiting cases , for which some of these parameters are put to zero . we start with the case of spherical particles . ( color online ) . quiescent fluid , spherical particle in the center . dashed curves theory , eq . ( [ a - nlf ] ) and dots simulations for different @xmath62 ( color - coded from `` cold''-blue , @xmath82 , to `` hot''-red , @xmath47 ) . insert : @xmath83 . dashed curve theory , eq . ( [ a - nlf ] ) and dots simulations for different @xmath62 ( same color - code ) . notice , that for @xmath84 runs the accuracy is better then @xmath85% , while for @xmath86 runs it is about @xmath87% , and only for really small @xmath88 runs it reaches the worst value of @xmath89% . these means that for our purposes , usually the runs with @xmath86 box is fully satisfactory.,width=328 ] + here we consider the simplest case of a spherical particle ( @xmath90 ) in a fluid at rest ( re = pe = 0 ) . chiew and glandt @xcite suggested the following formula for this case : [ a - nlf ] @xmath91 we notice that the expression for @xmath83 which controls the very dilute limit ( @xmath51 ) is the same as in the maxwell model @xcite . obviously , for @xmath82 one has zero effect , i.e. nu @xmath48 . for @xmath92 , @xmath93 and one has maximal possible enhancement ( at fixed @xmath58 ) . for @xmath94 one has 25% of this value , @xmath95 gives already 50% of the effect , @xmath96 , @xmath97 and @xmath98 . clearly , the larger @xmath62 the better , but increasing @xmath62 above 100 is not effective . actually , eq . ( [ a - nlf ] ) includes also information about the nonlinear dependence @xmath99 . however this dependence is very weak in the relevant range of parameters : e.g. for @xmath100 and @xmath101 the difference between @xmath99 and its linear approximation is only 15% . for smaller @xmath62 this difference is even smaller . the physical reason for that was explained at the end of previous section . we tested the prediction of eq . ( [ a - nlf ] ) by simulating the heat flux in a periodic box as explained above in sec . [ s : model ] and in appendix [ ss : numerics ] . for the fluid at rest the ( volume averaged ) nusselt number ( as a function of @xmath8 at different values of @xmath62 ) was measured and is shown in fig . [ fig : nu64 ] , where the inset shows @xmath83 . the overall conclusion is that 0.1 cm _ eq . ( [ a - nlf ] ) agrees extremely well with all our simulations and thus can be used in modeling the effect of spherical particles in a fluid at rest . _ the next important question is how the particle rotation affects the heat flux . as already mentioned , this rotation induces fluid motions around the particles thus causing convective contribution to the heat flux . this changes the temperature profile around the particles and , in turn , affects the heat flux inside the particles . to study these issues analytically is extremely difficult and thus numerical simulations can play a crucial role . due to the fact that we have to deal with a number of parameters , we will carefully examine them starting with the simplest case of spherical particles . results for spheroidal particles are discussed in sec . [ sss : el - rot ] . we begin by considering the case @xmath82 ; such a particle at rest does not affect the heat flux , thus nu @xmath48 . in fig . [ f:4 ] , upper left panel , we present nu(pe)/nu@xmath102 as a function of pe for @xmath82 at various volume fractions ( from 9.3% to 19.4% ) , while the inset is showing nu _ vs_. pe . observing the data , we suggest for pe@xmath103 the following ansatz [ pe^2 ] @xmath104 \,,\ ] ] shown in this panel as a straight dashed ( red ) line . one sees that eq . ( [ nu - pe ] ) is approximately valid up to pe @xmath105 , where the deviation is about 1% . we note also that for @xmath82 the value of @xmath106 is very small , @xmath107 . a similar analysis for @xmath77 , ( cf . [ f:4 ] , upper right panel ) , shows that eq . ( [ nu - pe ] ) fits the data even better with a similar value @xmath108 . we see that @xmath106 depends weakly on @xmath8 but more strongly on @xmath62 . thus , to determine the leading @xmath62-dependence we put a sphere of radius 18 in a @xmath86 computational volume ( all in lattice units ) ; this is equivalent to @xmath109 , see fig . [ f:4 ] , lower panels . the @xmath62-dependence of @xmath110 in eq . ( [ nu - pe ] ) in the range of @xmath111 can be fitted by : [ eq : bvsk ] @xmath112 , \\ b_1 & \simeq & 9 \!\times\ ! 10^{-5 } , \ b_2 \simeq 0.15\ . \end{aligned}\ ] ] for larger pe@xmath113 , the nu _ vs. _ pe dependence deviates down , as expected , because at pe @xmath114 this dependence should saturate . our analysis shows that this dependence can be fitted with a good accuracy by the formula that generalizes eq . : ^ 2}\ .\ ] ] the upper left panel in fig . [ f:4 ] shows by a solid ( black ) curve how this model works for @xmath82 ( with pe@xmath116 and pe@xmath117 ) . having in mind that in many applications related to nanofluids the value of pe is smaller than 0.01 and in any case rarely exceeds unity , we reach the conclusion that the convective heat flux around spherical particles and variations of the heat flux inside spherical particles due to their rotation can be neglected . equation ( [ a - nlf ] ) can be used to model the heat flux enhanced by spherical nano - particles with finite volume fraction up to 25% and any actual value of @xmath62 in fluids at rest and in shear flows . the next question to consider is the effect of the particle shape . we will begin with the case of spheroidal particles in fluids at rest . ( color online ) spheroid in a quiescent fluid at different angles @xmath118 . both panels : @xmath77 , pr = 1 , @xmath86 . symbols simulations , solid ( red or blue ) curves eq . ( [ spheroid ] ) with appropriate @xmath61 and @xmath8 , dashed ( black ) curves fits by eq . ( [ nuspheroisboth ] ) to the simulations . * upper panel : * @xmath119 . data : @xmath120 ( upper ) , @xmath121 ( middle ) and @xmath122 ( lower ) . the largest mismatch between the simulations and eq . ( [ spheroid ] ) is 2.8% at @xmath123 . * lower panel : * lower ( red ) dots : @xmath124 , @xmath125 , and upper ( blue ) squares : @xmath126 , @xmath127 . the dotted ( magenta ) curve is eq . ( [ spheroid ] ) with @xmath124 and @xmath127 . the effect of changing @xmath61 is larger than that of @xmath8 . , title="fig:",width=8 ] ( color online ) spheroid in a quiescent fluid at different angles @xmath118 . both panels : @xmath77 , pr = 1 , @xmath86 . symbols simulations , solid ( red or blue ) curves eq . ( [ spheroid ] ) with appropriate @xmath61 and @xmath8 , dashed ( black ) curves fits by eq . ( [ nuspheroisboth ] ) to the simulations . * upper panel : * @xmath119 . data : @xmath120 ( upper ) , @xmath121 ( middle ) and @xmath122 ( lower ) . the largest mismatch between the simulations and eq . ( [ spheroid ] ) is 2.8% at @xmath123 . * lower panel : * lower ( red ) dots : @xmath124 , @xmath125 , and upper ( blue ) squares : @xmath126 , @xmath127 . the dotted ( magenta ) curve is eq . ( [ spheroid ] ) with @xmath124 and @xmath127 . the effect of changing @xmath61 is larger than that of @xmath8 . , title="fig:",width=8 ] the general expectation is that spheroidal nano - particles should be able to enhance the heat flux much more effectively then spherical particles . to achieve this they have to be oriented in the `` right '' direction , i.e. along with the temperature gradient . therefore , logically , the study of the heat flux in nanofluids laden with spheroids should begin with the clarification of the effect of the spheroid orientation on the heat flux . in this section we consider this effect for a given and fixed orientation of the spheroids . for this purpose we perform numerical simulations of the heat flux with spheroidal nano - particles in a quiescent fluid ( pe@xmath128 ) with a temperature gradient , see fig . [ geometry ] , taking for concreteness @xmath77 . for different orientations of the spheroid ( i.e. different @xmath118 the angle between the temperature gradient vector and the largest spheroid s axis ) , we measure different nusselt numbers : when @xmath129 spheroids with higher conductivities tend to forms a thermal shortcut , thus nu is larger than for @xmath130 , where this shortcut effect is reduced ( fig . [ nuvstheta ] ) . our numerics shows that the dependence of the nusselt number on the spheroid orientation , i.e. nu@xmath131 , for small @xmath8 is well described by a simple formula [ k ] [ nuspheroisboth ] @xmath132 the range of validity of equation ( [ nan - form - factor ] ) was tested by means of a series of simulations at varying angles and volume fractions . the results are reported in figure [ nuvstheta ] . we can conclude that the model in eqs . ( [ spheroid ] ) well represents the heat flux in the presence of resting spheroidal particles in the relevant range of the parameters @xmath62 , @xmath61 and @xmath8 . to apply eqs . ( [ spheroid ] ) for spheroids in a simple shear flow we have to find how the ensemble average @xmath143 depends on @xmath61 . for this purpose we first need to know the orientational distribution function of spheroids in the shear flow . this is a subject of the following subsection . the effect of rotation of elongated spheroidal particles ( @xmath144 ) is very similar to that of spherical ones ( @xmath90 ) , just the parameters @xmath106 , pe@xmath145 and pe@xmath146 of the advanced fit depend on the aspect ratio @xmath61 . as an example , we presented in fig . [ f:9 ] a preliminary result of the computed and fitted nu(pe ) dependence for @xmath147 and @xmath77 . in this case @xmath148 , pe@xmath149 , pe@xmath150 . this is interesting to compare with the respective parameters for @xmath90 and @xmath77 : @xmath151 , pe@xmath152 , pe@xmath153 . one sees that the pe - enhancement of the heat flux with elongated particles is smaller then for spherical ones [ @xmath154 and it saturates at smaller pe : pe@xmath155pe@xmath156 and pe@xmath157 . moreover , the overall conclusion that nu(pe ) is almost pe - independent for pe@xmath158 remains valid , thus , the model developed above in eq . ( [ spheroid ] ) with eq . ( [ cos-2 ] ) may be safely used for nano - particles laden flows at pe@xmath158 . -0.24 cm ( color online ) . example of advanced fit for a spheroid @xmath147 , @xmath77 . ( blue ) dots numerical data , solid ( red ) curve fit by eq . fit with @xmath148 , pe@xmath149 and pe@xmath150 . dashed ( red ) line is @xmath159 . here @xmath160 is the time - averaged nu@xmath161 over the period of rotation ( determined in the simulations ) , and @xmath162 is the time - averaged nu@xmath161 for smallest available @xmath163 , e.g. @xmath164 . , title="fig:",width=8 ] in this section we develop an analytical model for @xmath165 . in order to study the orientational statistics of elongated nano - particles we consider the probability distribution function ( pdf ) , @xmath166 , which is the probability @xmath167 of finding any particular spheroid with its axis of revolution in the interval @xmath168\times [ \phi,\phi+d\phi]$ ] on the unit sphere . the pdf is then defined by @xmath169 it was shown by burgers @xcite that @xmath170 satisfies a generalized fokker - planck equation in the presence of a shear : @xmath171 where @xmath172 is the relative velocity on the unit sphere of the axis of revolution for a particle with instantaneous orientation @xmath173 ignoring all brownian effects . the explicit form of this equation in spherical coordinates is : burgers in ref . derived the rotational diffusion coefficient , @xmath182 , of elongated rigid spheroids of revolution : this asymptotics is formally valid for @xmath191 , but give better than 10% accuracy already for @xmath192 , allowing us to suggest the approximate `` practical '' formula @xmath193 that works with an accuracy of about @xmath194 for any @xmath195 and better than with @xmath196-accuracy for @xmath197 . the complete analysis of eqs . ( [ pdfeq ] ) is very involved , see e.g. refs . @xcite and @xcite . we consider only the small - diffusion limit , see the next subsection . jeffery @xcite has shown that if inertial and brownian motion affects are completely neglected , then the motion of the axis of revolution of a spheroidal particle is described by [ jefforbitseqs ] @xmath198 where @xmath199 with @xmath200 , and the constant of integration @xmath201 is called the ( jeffery ) orbit constant . to analyze the small - diffusion limit we introduce two time - scales . the first one defines the periodic motion that a nano - particle with a finite @xmath61 exhibits : @xmath202 , where @xmath203 is the jeffery s period . the second time scale is determined by the inverse shear , @xmath204 . clearly , for large @xmath61 this is a much shorter time scale , so for @xmath205 , @xmath206 it means that the particles spend most of their time near pre- and post - aligned states . if the brownian diffusion is small enough such that @xmath207 one can neglect the effect of the brownian motion on the dynamic motions of particles along jeffery orbits @xcite even during their slow time evolution . in this case the stationary fokker - plank eq . ( [ fp ] ) takes the simple form @xmath208 this equation can be solved @xcite , giving [ pdf1 ] @xmath209 where @xmath210 is the pdf along a particular jeffery orbit with a given integration constant @xmath201 , and @xmath211 is the probability to occupy this orbit . this function is normalized as follows : @xmath212 @xmath211 was found in @xcite for the limiting cases : [ f ] @xmath213 note that eq . ( [ r=1 ] ) is exact , providing a consistency check of the present approach by comparison with spherical particles . the asymptotic , eq . ( [ r=100 ] ) , is very accurate for @xmath214 , but already for @xmath215 it provides reasonable accuracy ( better then @xmath216 ) . the analysis of eqs . ( [ f ] ) together with the available numerical solutions for @xmath217 and @xmath218 @xcite allowed us to suggest the following approximation @xmath219 ^{3/2 } } } } \ .\ ] ] a comparison of this approximation with the exact numerical solution provided in ref . @xcite is presented in fig . [ fig : pdfs ] , upper panel . one sees that eq . ( [ approx - f ] ) fits the numerical data with an accuracy better than @xmath196 . this is more then enough for our purpose to offer an approximate formula for @xmath220 as a function of @xmath61 , see below . ( color online ) . * upper panel : * comparison of exact numerical " , solutions , obtained from eq . ( [ fp-1a ] ) ( solid lines ) and approximate solutions eq . ( [ approx - f ] ) for different @xmath61 ( dashed lines ) . there is visible difference ( about 5% in the value of maximum ) only for @xmath147 . * lower panel : * comparison of dependence of @xmath221 vs. @xmath61 obtained by a ) applying exact numerical " pdf ( black solid line ) , b ) approximate analytical pdf , eq . ( [ approx - f ] ) , ( red dashed line ) and c ) analytical approximation for this dependence eq . ( [ cos-2 ] ) ( blue dot - dashed line).,title="fig:",width=317 ] + ( color online ) . * upper panel : * comparison of exact numerical " , solutions , obtained from eq . ( [ fp-1a ] ) ( solid lines ) and approximate solutions eq . ( [ approx - f ] ) for different @xmath61 ( dashed lines ) . there is visible difference ( about 5% in the value of maximum ) only for @xmath147 . * lower panel : * comparison of dependence of @xmath221 vs. @xmath61 obtained by a ) applying exact numerical " pdf ( black solid line ) , b ) approximate analytical pdf , eq . ( [ approx - f ] ) , ( red dashed line ) and c ) analytical approximation for this dependence eq . ( [ cos-2 ] ) ( blue dot - dashed line).,title="fig:",width=317 ] next we use the fact that jeffery orbits do not intersect on the unit sphere . in other words , fixing @xmath201 results in a relation between @xmath177 and @xmath178 along an orbit . this relationship is obtained by inverting eq . ( [ jefforbitseqstheta ] ) : @xmath222 substituting this function into any one of eqs . ( [ f ] ) we get @xmath223 for a given regime of @xmath61 . this is then substituted in eqs . ( [ pdf1 ] ) , leading finally to a solution of the orientational pdf @xmath224 , which can be used for averaging eqs . ( [ spheroid ] ) in the case of weak brownian motions . as a consistency check of the approach one can consider the trivial case @xmath90 . from eq . ( [ cona-1 ] ) one finds @xmath225 , then from rq . ( [ r=1 ] ) @xmath226 and , finally from ( [ fp-2a ] ) @xmath227 . as the result one has for sphere @xmath228 , as expected . for moderate and strong brownian rotational motion the notion of separate jeffery orbits becomes irrelevant . in this case we need to solve eq . without approximations . once this equation is solved we can compute @xmath229 and substitute the answer in eqs . . to achieve this in the most general case is not a simple task , and here we satisfy ourselves with the two limiting cases of very large and very small rotational diffusion . the second case was discussed above . for the case of very strong rotational diffusion we can use the same eqs . ( [ spheroid ] ) , but averaged with a uniform pdf @xmath230 . this is because the very strong rotational diffusion tends to distribute particles motions around the jeffery orbits uniformly . the corrections up to @xmath231 to such a uniform distribution may be found in ref . @xcite with very strong brownian diffusion , the particles are oriented completely randomly , and @xmath232 . in this case eqs . and ( [ spheroid ] ) give the results reported in fig . [ f:7 ] , upper panels . the upper left panel shows nu vs. @xmath61 dependence for various @xmath62 from @xmath233 to @xmath234 and @xmath235 with volume fraction @xmath236 . these results are rather obvious : for @xmath82 one obtains nu@xmath48 , i.e. no enhancement ; the larger the @xmath62 , the larger the heat flux enhancement ; for any finite @xmath62 there is a saturation of nu@xmath237 for @xmath238 . the value of @xmath64nu@xmath65 may be huge for spheroids ( essentially , rod - like particles at @xmath239 , e.g. for @xmath240 ( diamond in water ) , nu@xmath241 , while for spherical particles @xmath242 , nu@xmath243 at the same volume fraction @xmath244 . the values of nu@xmath245 are bounded , i.e. @xmath246 . the upper right panel shows nu vs. @xmath62 dependence at @xmath236 for three values of @xmath61 : @xmath233 , @xmath247 and @xmath248 . the values of nu@xmath245 are bounded , too : @xmath249 . here , the more elongated particle is ( larger @xmath61 ) , the better the enhancement . and last , but not least : elongated particles may touch each other much easier . as we show in the appendix [ aa : rnphi ] , the basic geometrical requirement that the mean inter - particle distance is less than the largest particle size means that @xmath250 , which is the basic criteria of the dilute limit . the nu@xmath237 dependence for @xmath251 is shown by solid curves , while the region @xmath252 is shown by dashed curves in figs . [ f:7 ] , left panels . provided , @xmath253 at @xmath236 , the enhancement may be still considered as large but not huge : for @xmath77 , the saturation level of ( nu-1 ) is about @xmath254 while for spheres nu@xmath255 . the conclusion in the case of very strong brownian diffusion is that the particles with larger @xmath62 and @xmath61 bring larger heat flux enhancement in the laminar shear flow . to complete the calculations of nu@xmath237 dependence in a weak brownian diffusion limit in the framework of model ( [ spheroid ] ) , we have to find @xmath256 vs. @xmath61 . to make a long story short , we compared in fig . [ fig : pdfs ] , lower panel , the dependence of @xmath257 vs. @xmath61 obtained by numerically solved pdf , eq . ( [ fp-1a ] ) ( see refs . @xcite for more details ) , shown by ( black ) solid curve , and by approximate analytical pdf , eq . ( [ approx - f ] ) , shown by ( red ) dashed curve . as one sees these two dependence coincide within line width . moreover , by careful analysis of various limiting cases , we suggested the following simple model dependence @xmath258 shown in fig . [ fig : pdfs ] , lower panel , as a ( blue ) dash - dotted curve . ( [ cos-2 ] ) fits the exact dependence with an accuracy of 5% . therefore , eqs . ( [ spheroid ] ) and ( [ cos-2 ] ) can be used in our analysis to make predictions on the thermal properties in the limit of small brownian diffusion of fluids laden with spheroidal nano - particles of different aspect ratios and different thermal conductivity ratios with peclet number up to unity . corresponding results are shown in fig . [ fig : nu_nan_jeffery ] , lower panels . + the lower left panel in fig . [ fig : nu_nan_jeffery ] shows nu@xmath237 for different @xmath62 and @xmath244 , at which the maxwell - garnett limit of nu@xmath259 for spherical particles ( @xmath90 ) is @xmath260 . notice , the nu@xmath237 dependence is not monotonic and has a maximum at some @xmath261 , which depends on @xmath62 . the reason for this is the competition of two effects : more elongated nano - particles give larger contribution to the heat flux when their longer axis is aligned with the temperature gradient , which is orthogonal to the velocity gradient ( shear ) in our case . however longer nano - particles are affected more readily by the shear , which tends to orient them in the unfavorable direction orthogonal to the temperature gradient ; then their contribution to the heat flux is even less than the one of the spherical particles ( cf . [ nuvstheta ] ) . again , the values of nu@xmath245 are bounded , i.e. @xmath246 . the lower right panel in fig . [ fig : nu_nan_jeffery ] shows nu@xmath262 for different aspect ratios , @xmath61 at @xmath263 . this is again a consequence of the above described competition . the values of nu are bounded by @xmath264 $ ] . moreover , for @xmath265 the optimal nano - particle shape is spherical . as seen in fig . [ fig : nu_nan_jeffery ] , lower left panel , there exists a maximum of nu@xmath266 for a given @xmath62 . this maximal nusselt number at its maximizing ( optimal ) @xmath267 is shown in fig . [ fig : optimalnu ] , left , as a function of @xmath62 for @xmath268 . for @xmath269 , the maximal nu behaves like @xmath270 , \ \ \varphi = 0.01\ .\end{aligned}\ ] ] since here @xmath58 , the part in parenthesis may be considered as @xmath8-independent , thus , the fit ( [ maxnufit ] ) may be used at any @xmath271 . the right panel of fig . [ fig : optimalnu ] exhibits a contour - density plot of nu@xmath272 at @xmath268 . thick ( black ) solid and dashed curves show @xmath273 dependence ( solid curve is for @xmath274 , and the dashed one is for @xmath275 ) . the inverse dependence @xmath276 is easy to obtain in a symbolic computation software by solving @xmath277nu@xmath278 at @xmath279 , though , the answer appears to be very cumbersome to be shown here . however , our analysis reveals that @xmath280 , and for @xmath281 and @xmath282 , . @xmath283 this dependence is shown in fig . [ fig : optimalnu ] , right , as a wide - dashed ( white ) curve , which deviates from the analytical solution @xmath284 for @xmath285 . again , since @xmath286 , the fit ( [ roptfit ] ) may be used ) and ( [ spheroid ] ) for @xmath58 , @xmath287 /\partial r$ ] , which is a function of @xmath245 only . ] at any @xmath271 . in conclusion , we should notice that the rich information about heat flux enhancement , shown in the four panels of fig . [ fig : nu_nan_jeffery ] , is just an illustration of the analytical dependence nu@xmath288 given by the analytical expressions ( [ spheroid ] ) and ( [ cos-2 ] ) . this is an important result of our modeling . we presented a study of the physics of the heat flux in a fluid laden with nanoparticles of different physical properties ( shape , thermal conductivity , etc ) . we developed a new analytical model for the effective thermal properties of dilute nanofluid suspensions . our model accounts for nanoparticle rotation dynamics including the fluid motion around the nanoparticles . we note that our model reproduces the classical maxwell - garnet model in the appropriate static limits . we used a combination of theoretical models and numerical experiments in order to make progress from the simplest case of spherical nanoparticles in a quiescent fluid to the most general case of rotating spheroidal particles in shear flows . the new physical ingredient that we consider is the exact dynamics of particles in shear flows . this constitute a novelty as most of the models introduced so far , to explain the thermal properties of thermal colloids , have focused only on the static properties of the nanoparticle suspension . our model starts from the realization that particles ( spherical or spheroidal ) in the presence of a gradient of the velocity field are induced to rotate . the dynamics of rotation is absolutely non trivial , but it has been studied at length with correspondece ( for the case of a laminar and stationary shear flow ) to the jeffery orbits . the particles rotation dynamics has a double influence on the thermal properties of the nanofluid . first , particles rotation induces fluid motions in the proximity of the particles , this in turn can enhance the thermal fluxes by means of advective motions along the direction of the temperature gradient . second , the jeffery dynamics of particles leads to a statistical distribution of particles orientation that depends on a multitude of parameters , e.g. the particle aspect ratio , the shear intensity as well as on the intensity of thermal fluctuations . the statistical distribution of particle orientation has a dramatic influence on the heat flux : an elongated particle oriented along the temperature gradient increases the thermal flux , while a particle with perpendicular orientation reduces it . the statistical orientation of particles can thus produce a mixed effects with a non - trivial dependence on the particle aspect ratio . more elongated particles can enhance the heat flux because of the stronger contribution when properly aligned to the temperature gradient but , because of shear , more elongated particles are also spending more time in the unfavorable direction ( i.e. perpendicular to the temperature flux ) thus reducing the thermal conductivity of the fluid . by means of numerical approximations we are able to provide closed expressions for the effective conductivity of the fluid under several flow regimes and for several physical parameters our model considerably extends classical models for nanofluid heat transfer , like e.g. the one of maxwell - garnet , and may help to rationalize some of the recent experimental findings . in particular , we suggest that experiments should consider more carefully measurements performed in quiescent and under flowing conditions : the particles dynamics may lead to very different thermal properties in the two cases . finally , the next steps toward a robust predictive models for the heat transfer in nanofluids should include the effect of surface ( kapitza ) resistance and the effect of nanoparticle aggregation . further it would also be extremely important to extend the model to the case of heat flux in turbulent nanofluids as this case is very relevant to many applications . in the presence of turbulence a particular attention should also be paid to the effect on the drag induced by the presence of spherical , rod - like or maybe even deformable nanoparticle inclusions . we acknowledge financial support from the eu fp7 project `` enhanced nano - fluid heat exchange '' ( henix ) contract number 228882 . the numerical simulation of the conjugated heat transfer problem , equations ( [ eq1]-[eq3 ] ) , is performed by means of two coupled d3q19 lattice boltzmann ( lb ) equations under the bgk approximation @xcite ( for velocity and temperature fields ) and molecular - dynamics simulations ( for particles motion ) : [ lbeq ] @xmath289 where @xmath290 is the lattice boltzmann distribution function for particles at @xmath291 with velocity @xmath292 ( with @xmath293 for d3q19 ) , and @xmath294 is its equilibrium distribution ; @xmath295 is the distribution functions associated with the temperature and @xmath296 is its equilibrium distribution . the first lb , eq . ( [ lbeq1 ] ) , evolves the fluid flow outside of the rigid particle and its momentum is coupled with the particle boundaries by means of a standard scheme , as proposed by ladd @xcite . the second lb , eq . ( [ lbeq2 ] ) , evolves the temperature field , treated as a passive scalar as proposed in @xcite , solving the conjugated heat transfer problem simply by means of adjusting the thermal conductivity to the correct values in the fluid and inside the particle [ eqn . ( [ eq1 ] ) and ( [ eq2 ] ) ] . thermal and velocity boundary conditions , at the top and bottom walls , impose the lb populations to equal the equilibrium populations ( corresponding to the desired velocity and temperature ) . this approach can produce small temperature and velocity slip which are kept into account by measuring the effective temperature and velocity profiles , thus increasing the accuracy . the code employed is fully parallelized by means of mpi libraries @xcite thus allowing large system sizes , important to study the influence of finite size effects . density , momentum and temperature are defined locally at @xmath291 as coarse - grained ( in velocity space ) fields of the distribution functions @xmath297 a chapman - enskog expansion @xcite around the local equilibria @xmath298 and @xmath299 @xcite leads to the equations for temperature and momentum ( [ eq1])-([eq2 ] ) : the streaming step on the left hand side of ( [ lbeq1 ] ) reproduces the inertial terms in the hydrodynamical equations , whereas the diffusive terms ( dissipation and thermal diffusion ) are closely connected to the relaxation ( towards equilibrium ) properties in the right hand side , with @xmath17 and @xmath300 related to the relaxation times @xmath301 , @xmath302 @xcite . consider a spherical particle of radius @xmath45 immersed in a quiescent fluid , in which a constant temperature gradient , @xmath303 , is maintained . the temperature distribution is @xcite : @xmath304\bm g \cdot \bm \rho\ , , \quad r \geq r\ , , \nonumber\end{aligned}\ ] ] where @xmath305 is the distance from the particle s center . in our couette flow simulations , @xmath306 , but the temperature boundary conditions are different from @xcite . one should expect then deviations close to the walls especially for large particles . the results are presented in fig . [ f:2 ] . angular velocity _ vs. _ time . solid ( red ) curve theory of jeffery @xcite , eqs . ( [ eqs : jefferytheory ] ) , for an infinite domain with a constant simple shear at infinity and the creeping ( re@xmath307 ) flow around the particle , dots numerical results for small but finite re@xmath308 . configuration : @xmath86 , pr @xmath309 , @xmath310 , @xmath119 ( @xmath311 , @xmath312 ) , pe @xmath313.,width=321 ] a spheroidal particle of aspect ratio @xmath61 in a simple shear flow undergoes a spinning motion ( in the @xmath314-plane ) . the angular velocity and the period of such a motion were predicted by jeffery @xcite for a case of a creeping flow around the particle , i.e. when re@xmath315 : [ eqs : jefferytheory ] @xmath316^{-1}\ , , \\ \label{eq;jefferyperiod } t_{\rm j } & = & { 2\pi}\left(r+{1}/{r}\right)/{s}\ .\end{aligned}\ ] ] for non - vanishing re , the actual period of rotation of the particle deviates from that predicted by jeffery . in fig . [ jeffery_test ] we compare the results of our lbm simulations with eq . ( [ eq;jefferyangularvel ] ) for the case of @xmath119 and re@xmath317 . there is a 6% difference in periods due to a finite re , but also due to influence of the other particles present due to the periodic b.c . , and also the influence of the walls ( @xmath318 ) . the approximation of non - interacting spheroids is comming from simple geometrical considerations , and it is valid provided the particle aspect ratio @xmath319 , as confirmed by the following derivation : @xmath320{\frac{4\pi\,a^3\,r^{-2}/3}{\varphi } } = a\sqrt[3]{\frac{4\pi}{3\,r^2 \varphi}}\ , , \nonumber \\ 2a & < & \mathcal{l } \quad \rightarrow \quad r < \sqrt{\frac{\pi}{3 \varphi } } \quad \rightarrow \quad r < \varphi^{-1/2}.\end{aligned}\ ] ] here @xmath321 is the number of particles in the total volume ( of fluid and particles together ) , @xmath322 , @xmath323 is an `` effective '' volume per particle , and @xmath324 is a characteristic length / dimension of the effective box / volume embedding the particle . lee , s.h . lee , c.j . choi , s.p . jang , and s.u.s . choi , `` a review of thermal conductivity data , mechanisms and models for nanofluids '' , _ international journal of micro - nano scale transport _ * 1 * , 269 ( 2010 ) . jacopo buongiorno , david c. venerus , naveen prabhat , thomas mckrell , jessica townsend , rebecca christianson , yuriy v. tolmachev , pawel keblinski , lin - wen hu , jorge l. alvarado , in cheol bang , sandra w. bishnoi , marco bonetti , frank botz , anselmo cecere , yun chang , gang chen , haisheng chen , sung jae chung , minking k. chyu , sarit k. das , roberto di paola , yulong ding , frank dubois , grzegorz dzido , jacob eapen , werner escher , denis funfschilling , quentin galand , jinwei gao , patricia e. gharagozloo , kenneth e. goodson , jorge gustavo gutierrez , haiping hong , mark horton , kyo sik hwang , carlo s. iorio , seok pil jang , andrzej b. jarzebski , yiran jiang , liwen jin , stephan kabelac , aravind kamath , mark a. kedzierski , lim geok kieng , chongyoup kim , ji - hyun kim , seokwon kim , seung hyun lee , kai choong leong , indranil manna , bruno michel , rui ni , hrishikesh e. patel , john philip , dimos poulikakos , cecile reynaud , raffaele savino , pawan k. singh , pengxiang song , thirumalachari sundararajan , elena timofeeva , todd tritcak , aleksandr n. turanov , stefan van vaerenbergh , dongsheng wen , sanjeeva witharana , chun yang , wei - hsun yeh , xiao - zheng zhao , and sheng - qi zhou , `` a benchmark study on the thermal conductivity of nanofluids '' , _ journal of applied physics _ * 106 * , 094312 ( 2009 ) . j. a. eastman , u. s. choi , s. li , l. j. thompson , and s. lee , `` enhanced thermal conductivity of through the development of nanofluids '' , _ 1996 fall meeting of the materials research society _ , boston , 26 dec . 1996 ( mrs , pittsburgh , 1996 ) , p.1 . ce - wen nan , r. birringer , david r. clarke , and h. gleiter , `` effective thermal conductivity of particulate composites with interfacial thermal resistance '' , _ journal of applied physics _ * 81 * , 6692 ( 1997 ) . thomas joe mcmillen , _ the thermal constitutive behavior of suspensions _ , phd thesis , caltech , pasadena , ca ( 1977 ) . direct calculation of the velocity field around nanoparticle in terms of the spherical harmonics , eq.(16 ) , p. 108

we developed a model for the enhancement of the heat flux by spherical and elongated nano - particles in sheared laminar flows of nano - fluids . besides the heat flux carried by the nanoparticles the model accounts for the contribution of their rotation to the heat flux inside and outside the particles . the rotation of the nanoparticles has a twofold effect , it induces a fluid advection around the particle and it strongly influences the statistical distribution of particle orientations . these dynamical effects , which were not included in existing thermal models , are responsible for changing the thermal properties of flowing fluids as compared to quiescent fluids . the proposed model is strongly supported by extensive numerical simulations , demonstrating a potential increase of the heat flux far beyond the maxwell - garnet limit for the spherical nanoparticles . the road ahead which should lead towards robust predictive models of heat flux enhancement is discussed .

1204.2694

in the five decades since antineutrinos were first detected using a nuclear reactor as the source @xcite , these facilities have played host to a large number of neutrino physics experiments . during this time our understanding of neutrino physics and the technology used to detect antineutrinos have matured to the extent that it seems feasible to use these particles for nuclear reactor safeguards , as first proposed at this conference three decades ago @xcite . safeguards agencies , such as the iaea , use an ensemble of procedures and technologies to detect diversion of fissile materials from civil nuclear fuel cycle facilities into weapons programs . nuclear reactors are the step in the fuel cycle at which plutonium is produced , so effective reactor safeguards are especially important . current reactor safeguards practice is focused upon tracking fuel assemblies through item accountancy and surveillance , and does not include direct measurements of fissile inventory . while containment and surveillance practices are effective , they are also costly and time consuming for both the agency and the reactor operator . therefore the prospect of using antineutrino detectors to non - intrusively _ measure _ the operation of reactors and the evolution of their fuel is especially attractive . the most likely scenario for antineutrino based cooperative monitoring ( e.g. iaea safeguards ) will be the deployment of relatively small ( cubic meter scale ) detectors within a few tens of meters of a reactor core . neutrino oscillation searches conducted at these distances at rovno @xcite and bugey @xcite in the 1990 s were in many ways prototypes that demonstrated much of the physics required . once the neutrino oscillation picture became clear at the start of this decade , all the pieces were in place to begin development of detectors specifically tailored to the needs of the safeguards community @xcite . longer range monitoring , e.g. that described in @xcite , would also be attactive , but will likely require significant advances before becoming feasible . a more detailed treatment of this topic can be found in a recent review of reactor antineutrino experiments @xcite . antineutrino emission by nuclear reactors arises from the beta decay of neutron - rich fragments produced in heavy element fissions . these reactor antineutrinos are typically detected via the inverse beta decay process on quasi - free protons in a hydrogenous medium ( usually scintillator ) : @xmath0 . time correlated detection of both final state particles provides powerful background rejection . for the inverse beta process , the measured antineutrino energy spectrum , and thus the average number of detectable antineutrinos produced per fission , differ significantly between the two major fissile elements , @xmath1u and @xmath2pu ( 1.92 and 1.45 average detectable antineutrinos per fission , respectively ) . hence , as the reactor core evolves and the relative mass fractions and fission rates of @xmath1u and @xmath2pu change ( fig . [ fig : fisrates]a ) , the number of detected antineutrinos will also change . this relation between the mass fractions of fissile isotopes and the detectable antineutrino flux is known as the burnup effect . following the formulation of @xcite , it is instructive to write : @xmath3 where @xmath4 is the antineutrino detection rate , @xmath5 is the reactor thermal power , @xmath6 is a constant encompassing all non varying terms ( e.g. detector size , detector / core geometry ) , and @xmath7 describes the change in the antineutrino flux due to changes in the reactor fuel composition . typically , commercial pressurized water reactors ( pwrs ) are operated at constant thermal power , independent of the ratio of fission rates from each fissile isotope . pwrs operate for 1 - 2 years between refuelings , at which time about one third of the core is replaced . between refuelings fissile @xmath2pu is produced by neutron capture on @xmath8u . operating in this mode , the factor @xmath9 and therefore the antineutrino detection rate @xmath4 decreases by about @xmath10 over the course of a reactor fuel cycle ( fig . [ fig : fisrates]b ) , depending on the initial fuel loading and operating history . therefore , one can see from eq . [ eq : nu_d_rate2 ] that measurements of the antineutrino detection rate can provide information about both the thermal power of a reactor and the evolution of its fuel composition . these two parameters can not , however , be determined independently by this method , e.g. to track fuel evolution one would need independent knowledge of the reactor power history . measurement of the antineutrino energy spectrum may allow for the independent determination of the fuel evolution , and relative measurements over day to month time scales where @xmath9 varies little allow for tracking of short term changes in the thermal power . this last point may be of safeguards interest , since it is only when the reactor is off that the fissile fuel can be accessed at a pwr , and the integral of the power history constrains the amount of fissile pu that can be produced . there are many efforts underway around the world to explore the potential of antineutrino based reactor safegaurds . the evolution of these efforts is summarized in the agenda of the now regular applied antineutrino physics ( aap ) workshops @xcite . at present , these efforts are funded by a variety of national agencies acting independently , but there is frequent communication between the physicists involved at the aap meetings . this nascent aap community is hopeful that recent iaea interest ( sec . [ sec : iaea ] ) will result in a formal request from the iaea to the national support programs ( the programs within member states that conduct research and development requested by the iaea ) . such a request that clearly laid out the the needs of the agency with respect to detector size , sensitivity , etc , would allow for better coordination between these respective national programs , and would be an important step towards the development of device for use under iaea auspices . as mentioned above , the concept of using antineutrinos to monitor reactor was first proposed by mikaelian , and the rovno experiment @xcite was amongst the first to demonstrate the correlation between the reactor antineutrino flux , thermal power , and fuel burnup . several members of the original rovno group continue to develop antineutrino detection technology , e.g. developing new gd liquid scintillator using a linear alkylbenzene ( lab ) solvent . they propose to build a cubic meter scale detector specifically for reactor safeguards @xcite , and to deploy it at a reactor in russia . a collaboration between the sandia national laboratories ( snl ) and the lawrence livermore national laboratory ( llnl ) has been developing antineutrino detectors for reactor safeguards since about 2000 . our particular focus is on demonstrating to both the physics and safeguards communities that antineutrino based monitoring is feasible . this involves developing detectors that are simple to construct , operate , and maintain , and that are sufficiently robust and utilize materials suitable for a commercial reactor environment , all while maintaining a useful sensitivity to reactor operating parameters . the songs1 detector @xcite was operated at the san onofre nuclear generating station ( songs ) between 2003 and 2006 . the active volume comprised @xmath11 tons of gd doped liquid scintillator contained in stainless steel cells ( stainless was used to avoid any chance of acrylic degradation and liquid leakage ) . this was surrounded by a water / polyethylene neutron - gamma shield and plastic scintillator muon veto ( fig . [ fig : songs1 ] ) . the detector was located in the tendon gallery of one of the two pwrs at songs , about @xmath12 m from the reactor core and under about @xmath13 m.w.e . galleries of this type , which are part of a system for post - tensioning the containment concrete , are a feature of many , but not all , reactor designs . it may therefore be important to consider detector designs that can operate with little or no overburden . the songs1 detector was operated in a completely automatic fashion . automatic calibration and analysis procedures were implemented and antineutrino detection rate data was transmitted to snl / llnl in near real time . an example of the ability to track changes in reactor thermal power is given in fig . [ fig : power ] . a reactor scram ( emergency shutdown ) could be observed within 5 hours of its occurrence at @xmath14 confidence . integrating the antineutrino detection rate data over a @xmath15 hour period yielded a relative power monitoring precision of about @xmath16 , while increasing the averaging period to @xmath17 days yielded a precision of about @xmath18 @xcite . increasing the averaging time to @xmath13 days allowed observation of the fuel burnup @xcite ( fig . [ fig : burnup ] ) . the relatively simple calibration procedure was able to maintain constant detector efficiency to better than 1% over the @xmath19 month observation period . the decrease in rate due to fuel evolution ( burnup ) and the step increase in rate expected after refueling ( exchange of pu laden fuel for fresh fuel containing only u ) were both clearly observed . songs1 was very successful , demonstrating that a relatively simple and compact detector could be operated non - intrusively at a commercial reactor for a long period . however , much of the feedback received from the safeguards community focussed upon the use of a flammable liquid scintillator . as used in the songs1 deployment this scintillator presented no safety hazard to the operation of the reactor - all relevant safety codes and procedures were checked and strictly adhered to . however , deployment of that flammable material did require some compliance effort from the reactor operator . in a safeguards context such situations should be avoided if at all possible . therefore , we decided to develop and deploy two detectors based upon non - flammable materials . the first of these was based upon a plastic scintillator material . our goal was retain as much similarity between this device and songs1 as possible . therefore we wished to use gd as the neutron capture agent ; this was achieved by using @xmath15 @xmath20 m x @xmath21 m x @xmath22 cm slabs of bc-408 plastic scintillator and interleaving them with a gd loaded layer . the total active volume of this detector was @xmath23 m@xmath24 . this device was deployed at songs in 2007 , and was clearly able to observe reactor outages like that shown in fig . [ fig : power ] . we continue to analyze the data from this detector and expect to be able to observe fuel burnup also . inspired by the gadzooks concept presented at this conference in 2004 @xcite , we are also investigating the use of gd - doped water as an antineutrino detector . this detection medium should have the advantage of being insensitive to the correlated background produced by cosmogenic fast neutrons that recoil from a proton and then capture . a @xmath25 liter tank of purified water containing 0.1% gd by weight was been deployed in the songs tendon gallery . initially , this detector was deployed with little passive shielding - the large uncorrelated background rate that resulted has made it difficult to identify a reactor antineutrino signal much beyond a @xmath26 level . in this configuration gd - doped water also appears promising for use in special nuclear material search applications @xcite . the coherent scattering of an ( anti)neutrino from a nucleus is a long predicted but as yet unobserved standard model process . it is difficult to observe since the signal is a recoiling nucleus with just a few kev of energy . nonetheless , this process holds great promise for reactor monitoring since it has cross - section several orders of magnitude higher than that for inverse beta decay , which might eventually yield significantly smaller monitoring detectors . to explore the prospects for this process , we are currently collaborating with the collar group of the university of chicago @xcite in deploying an ultra - low threshold ge crystal at songs . we are also investigating the potential of dual phase argon detectors @xcite . the double chooz collaboration @xcite plans to use the double chooz near detector ( @xmath27 m from the two chooz reactors ) for a precision non - proliferation measurement @xcite . the double chooz detectors will represent the state - of - the - art in antineutrino detection , and will be able to make a benchmark measurement of the antineutrino energy spectrum emitted by a commercial pwr . there is also a significant effort underway within double chooz to improve the reactor simulations used to predict reactor fission rates and the measurements of the antineutrino energy spectrum emitted by the important fissioning isotopes . this work is necessary for the physics goals of double chooz , but it will also greatly improve the precision with which the fuel evolution of a reactor can be measured . the double chooz near detector design is too complex and costly for widespread safeguards use . therefore , the double chooz groups in cea / saclay , in2p3-subatech , and apc plan to apply the technology developed for double chooz , in particular detector simulation capabilities and high flash - point liquid scintillator , to the development of a compact antineutrino detector for safeguards : nucifer @xcite . the emphasis of this design will be on maintaining high detection efficiency ( @xmath28 ) and good energy resolution and background rejection . nucifer will be commissioned against research reactors in france during 2009 - 2010 . following the commissioning phase nucifer will be deployed against a commercial pwr , where it is planned to measure reactor fuel evolution using the antineutrino energy spectrum . an effort to develop a compact antineutrino detector for reactor safeguards is also underway in brazil , at the angra dos rios nuclear power plant @xcite . several deployment sites near the larger of the two reactors at angra have been negotiated with the plant operator and detector design is well underway . this effort is particularly interesting , as a third reactor is soon to be built at the angra site , within which space may be specially reserved for a detector , and because in addition to the iaea , there is a regional safeguards organization ( abbac ) monitoring operations . such regional agencies often pioneer the use of new safeguards technologies , and the detector deployment at angra may occur with abbac involvement . a prototype detector for the kaska theta-13 experiment has been deployed at the joyo fast research reactor in japan @xcite . this effort is notable , since it is an attempt to observe antineutrinos with a compact detector at a small research reactor in a deployment location with little overburden . this may be typical of the challenging environment in which the iaea has recently expressed interest in applying this technique ( sec . [ sec : iaea ] ) . not surprisingly , this effort has encountered large background rates , and to date has identified no clear reactor antineutrino signal . the iaea is aware of the developments occurring in this field . a representative from the iaea novel technologies group attended the most recent aap meeting , and expressed an interest in using this monitoring technique at research reactors . the iaea currently uses a device at these facilities that requires access to the primary coolant loop @xcite . needless to say , this is quite invasive and an antineutrino monitoring technique would clearly be superior in this respect . an experts meeting at iaea headquarters will occur in october of 2008 to discuss the capabilities of current and projected antineutrino detection techniques and the needs of the iaea . _ applications _ of neutrino physics must have seemed somewhat fanciful when first discussed at this conference several decades ago . but even with currently available technologies , useful reactor monitoring appears feasible , as demonstrated by the songs1 results . the iaea has expressed interest in this technique and the applied antineutrino physics community eagerly awaits their guidance as to the steps required to add antineutrino based reactor monitoring to the safeguards toolbox . llnl - proc-406953 . this work was performed under the auspices of the u.s . department of energy by lawrence livermore national laboratory in part under contract w-7405-eng-48 and in part under contract de - ac52 - 07na27344 . 00 a. porta , et . al . , `` reactor antineutrino detection for thermal power measurement and non - proliferation purposes '' in proc . physics of reactors : `` nuclear power : a sustainable resource '' ( 2008 )

nuclear reactors have served as the antineutrino source for many fundamental physics experiments . the techniques developed by these experiments make it possible to use these very weakly interacting particles for a practical purpose . the large flux of antineutrinos that leaves a reactor carries information about two quantities of interest for safeguards : the reactor power and fissile inventory . measurements made with antineutrino detectors could therefore offer an alternative means for verifying the power history and fissile inventory of a reactors , as part of international atomic energy agency ( iaea ) and other reactor safeguards regimes . several efforts to develop this monitoring technique are underway across the globe .

0809.2128

dielectric resonators have applications in microwave and optical frequency ranges , including antennas @xcite and as building blocks of impedance - matched huygen s metasurfaces @xcite . approximate methods for finding the modes of dielectric resonators are known @xcite , which usually assume that @xmath0 . these methods are inaccurate for the moderate values of permittivity available at optical frequencies , and more sophisticated methods are needed to account for radiation effects . open nanophotonic resonators such as meta - atoms , nano - antennas and oligomers are typically strongly radiative systems , where loss can not be treated as a perturbation . furthermore , in many nanophotonic systems , material dispersion and losses can not be neglected , further complicating the problem of finding their modes . in radiating and dissipative systems the modes have complex frequencies @xmath1 , corresponding to damped oscillations of the form @xmath2 , with @xmath3 . the corresponding modal fields @xmath4 do not possess the orthogonality usually found in the modes of closed systems , and they are commonly referred to as quasi - normal modes @xcite . they are particularly useful for solving dipole emission problems @xcite , since they allow a mode volume to be defined for open cavities @xcite . a significant practical difficulty is the requirement to normalise a mode with diverging far - fields @xcite . a different perspective on this problem can be found within the microwave engineering literature @xcite , originally motivated by time - domain radar scattering problems . by using integral methods to solve maxwell s equation , only currents on the scatterer need to be solved for , avoiding having to explicitly handle the diverging far - fields . as it is based on finding the singularities of a scattering operator , this approach is referred to as the singularity expansion method ( sem ) . the field distributions corresponding to these singularities are identical to the quasi - normal modes at the complex frequencies of the singularities @xmath1 . when solving scattering problems on the @xmath5 axis , the fields in the sem approach are reconstructed from the dyadic green s function , which remains finite in the far - field . thus the sem avoids the most significant practical disadvantage of quasi - normal modes based on fields . recently it has been shown that the singularity expansion method can be applied to meta - atoms and plasmonic resonators @xcite , clearly identifying the modes which contribute to scattering and coupling problems . however , finding all modes within a region of the complex frequency plane requires an iterative procedure with multiple contour integrations@xcite . this greatly increases the computational burden , and it remains unclear how robust this procedure is . in addition , it has not yet been demonstrated whether all spectral features can be explained by such a model , particularly the interference between non - orthogonal modes in the extinction spectrum and suppression of back - scattering corresponding to the huygens condition@xcite . in this work , a robust integral approach to finding modes of open resonators is demonstrated for an all - dielectric meta - atom , based on the singularity expansion method . it is shown how this leads to a clear decomposition of the extinction spectrum of a silicon disk , automatically accounting for interference between the non - orthogonal modes . by performing a vector spherical harmonic decomposition of each mode , the unidirectional scattering behaviour of the disk is explained . it is shown that higher - order modes can also interfere to supress back - scattering , corresponding to the previously reported generalized huygens condition @xcite . a brief overview of integral equation methods to solve maxwell s equations is given , followed by the robust approach to find the modes . here dielectric objects are considered , and treated through a surface equivalent problem , with surface equivalent electric and magnetic currents , @xmath6 and @xmath7 , where @xmath8 is the surface normal . these surface currents can be excited by the incident electric or magnetic field , yielding the electric field integral equation and magnetic field integral equation respectively . to yield a stable solution , both of these equations must be combined using some chosen weighting coefficients@xcite . in this work the combined - tangential form is used , as detailed in ref . . this gives us an operator equation relating equivalent surface currents to the tangential components of the incident fields @xmath9 in this work the time convention @xmath10 is used , with @xmath11 . note that in contrast to other conventions , the imaginary part gives the oscillation rate , and the real part gives the decay rate . we could find the corresponding time - domain function @xmath12 of a frequency - domain function @xmath13 through the inverse laplace transform @xmath14 . physically observable quantities must be represented by a real function in the time domain , thus they must satisfy the constraint @xmath15 in the frequency domain . equation is solved using the boundary element method ( also known as the method of moments@xcite ) . after choosing sets of basis functions @xmath16 and testing functions @xmath17 ( both are loop - star functions @xcite in this work ) , the operator equation becomes a finite - dimensional matrix equation @xmath18 where @xmath19 is the vector containing the weighted equivalent surface currents @xmath20^t,\ ] ] and @xmath21 is the source vector containing the projected incident fields @xmath22^t.\ ] ] the impedance matrix @xmath23 is dense and frequency dependent , and contains all information regarding the response of the scatterer to arbitrary incident fields . the unknown current vector @xmath19 is solved for a given incident field vector @xmath21 , with the solution is given by @xmath24 . the singularities of @xmath25 will dominate the spectrum of the response , and by mittag - leffler s theorem the response may be expanded in terms of these singularities @xcite . they correspond to solutions which can exist in the absence of a source , and hence they can be used to model the response to an arbitrary incident field . the most important singularities of the impedance matrix are its poles , corresponding to the quasi - normal modes of the system . in practice it may usually be assumed that all poles are of first order @xcite . the poles correspond to frequencies @xmath26 , satisfying @xmath27 and @xmath28 for non - zero @xmath29 and @xmath30 . these are the left and right eigenvectors of the system respectively , with @xmath29 being the surface current distribution of the mode . the singularities of the impedance matrix are found by the contour integration procedure outlined in ref . . first a pair of matrix integrals @xmath31 and @xmath32 is evaluated about a contour containing all modes of interest , as shown schematically in fig . [ fig : contour ] . the contour is offset slightly from the @xmath5 axis to eliminate any modes which do not couple to incident radiation , hence have @xmath33 . the desired radiating modes are shown by green crosses , and have @xmath3 . since currents must be real functions in the time domain , for each pole there is a corresponding complex conjugate pole at @xmath34 , shown in orange . as the poles and residues are just complex conjugates of those with positive @xmath35 , they can be found by symmetry , and do not need to be included within the contour . note that some poles are over - damped , with @xmath36 , and these poles do not appear in conjugate pairs . the contour incorporates the @xmath37 axis in order to capture these poles . an arc is used to eliminate spurious numerical poles which cluster near the origin when using integral operators of the first kind @xcite . and their residues with only a single integration . green crosses : physical modes with finite radiation damping . red crosses : spurious internal solutions with no damping . orange crosses : conjugate modes which can be found by symmetry.[fig : contour ] ] the mode frequencies and currents are eigenvalues and eigenvectors of @xmath38 . a singular value decomposition is used to determine the number of valid solutions to this equation @xcite and solving for the corresponding left eigenvalue problem yields the projectors @xmath30 . this procedure can yield solutions lying both inside and outside the contour , and those falling outside the contour are discarded . the poles and currents are further improved by newton iteration , then normalised so that @xmath39 . this ensures that the dyadic product of the eigenvectors matches the pole residue , i.e. @xmath40 in the vicinity of @xmath41 , simplifying the pole expansion . note that in general no orthogonality relation exists between these mode currents . as is discussed in appendix [ sec : orthogonality ] , orthogonality is not required for this approach . it will be shown how this non - orthogonality leads to physically meaningful interference effects . due to geometric symmetry , many modes are degenerate , with several different eigenvectors having the same pole location @xmath41 . when solving the structure numerically , the imperfect symmetry of the mesh usually results in some frequency splitting of these degenerate modes , so a thresholding procedure is used to group closely spaced poles . the contour integration and iterative search procedure were found to cope with these nearly degenerate poles without requiring any special handling . note that it is not necessary to orthogonalise degenerate modes , since the method is intrinsically able to account for non - orthogonality , as long as the modes span the full eigenspace . once the modes have been found , currents can be solved for arbitrary incident fields , @xmath42 where we consider excitation at physically realisable frequencies on the @xmath5 axis . the projector @xmath30 operates on the incident field @xmath21 to give its overlap with the mode . the bracketed term accounts for close the excitation frequency is to the mode s resonant frequency . note that this polynomial has the correct asymptotic behaviour , thus improving the convergence and removing the need to include an entire function contribution@xcite . the important result obtained from eq . is a scalar weighting of each mode s current vector @xmath29 . regardless of whether it is calculated directly from eq . or as a superposition of modes from eq . , the current vector @xmath19 can be used to find the currents at any point on the surface . the dyadic green s function can be then be used to find the fields anywhere in space . in practice this is not usually necessary , since many quantities of physical interest such as scattering , radiation forces and torques can be calculated directly from the currents @xcite . the quantity of most interest is the extinction cross - section @xmath43 giving the total work done by the incident fields on the currents . here @xmath44 is the electric field of the incident plane - wave . this quantity can be defined for each mode by substituting the mode s current and its weighting from eq . . , summed over all values of azimuthal index @xmath45.[fig : extinction_multipole ] ] the techniques outlined in section [ sec : modelling ] are now applied to study the scattering behaviour of a silicon disk meta - atom . as a first step , the structure is modelled directly without considering the modes , using eq . . the radius is taken as 242 nm , height 220 nm and edges are rounded with radius 50 nm . the material properties of silicon were obtained by fitting an 8 pole model to the experimental data from ref . . in fig . [ fig : extinction_multipole ] the extinction cross - section of the disk is plotted . the incident wave - vector is parallel to the axis of the disk . as a first attempt to explain the spectral features , a multipole expansion is also shown in fig . [ fig : extinction_multipole ] . details of the expansion are given in appendix [ sec : multipole ] . solid lines show the electric multipole moments @xmath46 , and dashed curves show the magnetic moments @xmath47 . although the multipoles accurately reproduces the total extinction , there is no direct correspondence between modes and multipoles , with each peak exhibiting contributions from many multipole moments . furthermore , several multipole moments show peaks and dips at similar locations , but it is unclear if these moments are linked to each other . therefore _ the multipole decomposition is unable to resolve the internal dynamics _ which are observed in the extinction spectrum . it will be demonstrated that the model based on eq . can resolve these internal dynamics , showing which modes correspond to each of the spectral features . ] the modes of the silicon disk are found by the procedure outlined in section [ sec : finding - modes ] . figure [ fig : extinction_modes](a ) shows the location of the poles in the complex frequency plane , with many of them being doubly degenerate . since currents decay in time as @xmath48 , more highly damped modes have more negative values of @xmath49 . the schematic of the incident field orientation is shown in the inset . the modes which most strongly couple to this incident field are marked with coloured dots . the equivalent surface current @xmath50 of the first 5 of these modes is shown in fig . [ fig : mode - currents ] . note that these currents are complex , hence the plotted vectors give a snapshot of the oscillating current distribution . the divergence @xmath51 is proportional to the equivalent surface charge ( and hence to the normal component of the electric field ) and is indicated by the shading of the surface . the colors of the markers next to each current distribution correspond to the poles shown in fig . [ fig : extinction_modes](a ) . each mode is also given an arbitrary label in roman numerals for reference purposes . . right : spherical multipoles of each mode , scaled normalised according to the total scattered power.[fig : mode - currents],title="fig : " ] + ' '' '' . right : spherical multipoles of each mode , scaled normalised according to the total scattered power.[fig : mode - currents],title="fig : " ] + ' '' '' . right : spherical multipoles of each mode , scaled normalised according to the total scattered power.[fig : mode - currents],title="fig : " ] + ' '' '' . right : spherical multipoles of each mode , scaled normalised according to the total scattered power.[fig : mode - currents],title="fig : " ] + ' '' '' . right : spherical multipoles of each mode , scaled normalised according to the total scattered power.[fig : mode - currents ] ] to understand the nature of these modes , we compare the disk with rounded edges to the sphere , since the two are topologically equivalent . in ref . it is shown that the poles of a sphere can be found from the roots of the denominators of the coefficients from mie theory , involving spherical bessel and spherical hankel functions . the field corresponding to each of these poles is a pure vector spherical harmonic , with poles corresponding to radial index @xmath52 having degeneracy of @xmath53 . note that for each vector spherical harmonic there is an infinite number of poles , corresponding to different number of radial oscillations inside the sphere @xcite . we can consider the dielectric disk to be a sphere which has been transformed in a continuous manner , breaking the spherical symmetry . this means that the total number of poles is the same for both geometries , but the degeneracy is reduced by poles splitting to different locations . as a result , the corresponding current for each pole of the disk is not a pure vector spherical harmonic . by performing a multipole decomposition of the current for each mode of the disk , we can see which mode of the sphere it is most closely related to . this is shown in the right column of fig . [ fig : mode - currents ] , where each mode s multipole moments are normalised to the total scattered power , as outlined in appendix [ sec : multipole ] . in all cases there is a single dominant multipole moment , although for higher order modes the influence of higher moments becomes more significant . in the following sections this multipole expansion of the modes will be used to explain their contributions to extinction and scattering . figure [ fig : extinction_modes](b ) shows the extinction contribution from each of the modes , obtained from each term in eq . . the extinction from degenerate pairs of modes has been combined , along with the contribution of their conjugate modes at @xmath34 . it can be seen that all features in the extinction spectrum can be clearly attributed to the modal contributions . the extinction spectrum for each mode exhibits only a single feature , being a peak and/or dip in the vicinity of its pole frequency @xmath54 . there is a very clear correspondence between the damping rate @xmath49 and the sharpness of the features in the corresponding extinction curve . note that for more highly damped modes , there is some shift between the peak and pole frequencies . this is because such modes couple strongly to the incident field , and therefore the overlap term in eq . can shift the spectral features away from the natural frequency @xmath35 . the accuracy and convergence of this model of extinction is shown in appendix [ sec : accuracy ] . one of the most striking features of fig . [ fig : extinction_modes](b ) is that several modes show negative contributions to extinction . this is due to the non - orthogonality of the modes , which means that even if the incident field matches the profile of one mode , it may still excite others . it can be seen that the dip in extinction at around 260thz can be attributed to a strong negative contribution from mode iii , emitting radiation in the forward direction that is in - phase with the incident field . in ref . it was shown how extinction can be decomposed into direct terms from each mode , plus inteference terms between every pair of modes . the current obtained from eq . does not explicitly show separate direct and interference terms . energy conservation dictates that the total extinction must be positive , so a sufficient set of modes must be included to have a physically meaningful result . to quantify the interference between modes , their overlap @xmath55 is plotted in fig . [ fig : mode - overlap ] , normalised such that @xmath56 . this indicates how strongly an incident field having the shape of mode @xmath45 excites mode @xmath57 . for example , we see strong overlap between modes i and iii , showing that the field which excites mode iii also strongly excites mode i. this leads to the strong negative extinction observed for mode iii . the strong overlap of modes i and iii is is consistent with their multipole decomposition shown in fig . [ fig : mode - currents ] , where both are dominated by the @xmath58 electric dipole term . the strongest mode overlap observable in fig . [ fig : mode - overlap ] is between modes ii and iv , consistent with both having strong magnetic dipole moment @xmath59 . however this does not lead to strong interference in fig . [ fig : extinction_modes](b ) . the low damping rates of these modes seen in fig . [ fig : extinction_modes](a ) results in narrow resonant peaks which have little overlap . a notable feature of fig . [ fig : mode - overlap ] is that mutual overlap terms can be greater than self terms , a consequence of the unconjugated inner product which appears in the formalism . it should also be noted that similarities in the multipole decomposition of modes is not always a good predictor of their overlap . for example , modes ii and v have both have dominant magnetic dipole moments @xmath59 , but nonetheless have relatively weak overlap observable in fig . [ fig : mode - overlap ] . ] ] to calculate the total scattering cross section , vector spherical harmonics are used , since the total scattering is the incoherent sum of all multipole contributions , given by eq . . figure [ fig : multipole - scattering ] shows the contribution of each multipole coefficient to the scattering cross - section . as with the multipole extinction spectrum shown in fig . [ fig : extinction_multipole ] , the features of the multipole scattering spectra are rather complex , but can be explained by considering the contributions of different modes . in the wavelength range above 1000 nm , corresponding to measured range in ref . , it can be seen that the scattering is dominated by the electric dipole and magnetic dipole moments @xmath60 @xmath61 . the magnetic dipole moment can be attributed to the resonance of mode ii , which has negligible contributions from other moments . the electric dipole moment @xmath58 appears to have two distinct maxima in fig . [ fig : multipole - scattering ] . from the coefficients shown in fig . [ fig : mode - currents ] , it is clear that only modes i and iii contribute to this dipolar scattering . from fig . [ fig : extinction_modes](b ) , we can see that mode i has a very broad resonance , while mode iii has a much narrower resonance , with a negative contribution to extinction . this results in cancellation of electric dipole radiation , corresponding to an anapole distribution@xcite . this effect is typically explained in terms of a quasi - static electric dipole ( a linear current distribution ) interfering with a toroidal dipole ( a poloidal current distribution ) . the surface currents shown in fig . [ fig : mode - currents ] are consistent with this explanation , however the explanation in terms of modes is more general , and does rely on any low frequency approximation . indeed , in ref . it was shown that for spheres , this condition occurs when the contributions from the first and second @xmath58 modes cancel . the situation for the disk is similar , the difference being that the interfering modes i and iii have additional contributions from other multipole moments . ] for applications in huygens metasurfaces , the most important attribute of a meta - atom is to have suppressed back scattering and strong forward scattering . this is typically achieved by overlapping electric and magnetic dipole type resonances . [ fig : directional - scattering ] shows the forward and backward scattering amplitudes , with peaks labelled according to the corresponding resonant modes . the first peak of forward scattering corresponds to the overlap of modes i and iii , with almost purely electric dipole radiation , and mode ii , with almost purely magnetic dipole radiation . it can also be seen that at the resonances of modes iv and v there are additional highly directional scattering features , as these modes also overlap with the electric - dipole type modes i and iii . examining the multipole decompositions in fig . [ fig : mode - currents ] , it can be seen that mode iv is dominated by its electric quadrupole response , with a significant contribution from its magnetic dipole response . in contrast , mode v is dominated by its magnetic dipole response , with lesser contributions from electric quadrupole and magnetic octupole moments . it is significant that all of these multipole moments radiate anti - symmetric electric fields into the forward and backward directions . thus all of these moments are able to cancel the electric dipole and magnetic quadrupole moments of modes i and iii , which radiate with symmetric electric fields in the forward and backward direction . considering the contribution of modes to this directional scattering process , the generalised huygens condition introduced in ref . can be re - interpreted as interference between modes of different symmetry . this suggests that to optimise this generalised huygens effect , the meta - atoms should be placed within a homogeneous dielectric environment , as has been done for all - dielectric huygens metasurface @xcite . a dielectric substrate without a compensating superstrate introduces coupling between modes of opposite symmetry@xcite , greatly complicating the design process and degrading the directionality of scattering a robust technique based on the singularity expansion method was presented to find the modes of a meta - atom , fully accounting for radiative losses . by solving maxwell s equations using integral techniques , the normalisation of diverging fields typically required when using quasi - normal modes is avoided . the technique was applied a silicon disk , a building block which enables optical metasurfaces having low loss , and full manipulation of the transmitted phase . it was demonstrated that the complicated features of the extinction spectrum can be readily explained in terms of contributions from the modes . interference between non - orthogonal modes was shown to play a key role . when considering far - field scattering properties , a vector spherical harmonic expansion yields an accurate , if somewhat opaque , description . by combining it with the modal analysis , the nature and origin of all scattering features can be elucidated . in the case of the silicon disk , there are several bands of strong forward scattering and suppressed backscattering , corresponding to the generalised huygens condition . it was shown that each band corresponds to the overlap of modes with odd and even radiation symmetry . the techniques used to find modes and construct models of scatterers are implemented in an open - source code openmodes@xcite , along with notebooks to reproduce all results in this paper@xcite . the author acknowledges useful discussions with andrey miroshnichenko , sarah kostinski , mingkai liu , and yuri kivshar . this research was funded by the australian research council . the electric multipole coefficients @xmath62 and magnetic multipoles coefficients @xmath63 were computed directly from the surface currents using the formulas from ref . . duality allows these formulas to be generalised to include the equivalent magnetic currents through the substitution @xmath64 . the normalisation of multipole coefficients from ref . is used , as this simplifies the expression for scattering cross - section , which is given by @xmath65 where the coefficients include contributions from all values of azimuthal index @xmath45 : @xmath66 in fig . [ fig : mode - currents ] @xmath67 and @xmath68 are normalised to their sum , and their square root is plotted since it more clearly shows the smaller contributions . in fig . [ fig : multipole - scattering ] these terms are plotted including the pre - factor from eq . to give them dimensions of scattering cross - section . for a plane wave propagating in the @xmath69 direction , with incident electric field along the @xmath70 direction , the extinction cross - section is given by @xcite @xmath71\right.\nonumber\\ * + \left.\left[\sum_{m=-1,1}m\mathrm{im}\{b_{lm}\}\right]\right ) . \label{eq : multipole_extinction}\end{aligned}\ ] ] the quantities in square brackets are plotted in fig . [ fig : extinction_multipole ] , including all common pre - factors in eq . . for 3 terms of the multipole expansion , the extinction plotted in fig . [ fig : extinction_multipole ] agrees with the direct calculation to a relative error below 2% for frequencies below 350thz . by adapting the formulas from mie theory @xcite , forward scattering can be found as @xmath72 while back - scattering is given by @xmath73 as losses are low in this system , the total extinction and scattering are approximately equal , due to the optical theorem . however , this still allows each multipole s contribution to extinction shown in fig . [ fig : extinction_multipole ] to be different from its contribution to scattering shown in fig . [ fig : multipole - scattering ] . as discussed in ref . , the electric fields of quasi - normal modes do not obey the usual orthogonality relationship based on a conjugated inner product , i.e. @xmath74 . however , they do obey an unconjugated orthgonality relationship , which is required for normalisation of modes@xcite , and projection of external fields onto modal fields . in contrast , the current vectors on the scatterer obtained from the singularity expansion method do not exhibit any form of orthogonality . however , such orthogonality is not required when working with modal currents , since they are normalised by weighting them to match the residue of the pole , as shown in eq . . in addition to providing the current vector @xmath29 , this approach also yields the correctly normalised projector @xmath30 , which gives the projection of an arbitrary field onto each mode by a simple scalar product , as used in eq . . it is noted that in the literature a number of orthogonal decompositions of the impedance matrix @xmath75 have been presented , most prominently the characteristic mode analysis @xcite . as these mode vectors are real , they exhibit the conventional conjugated orthogonality . however , such decompositions suffer from a number of problems which make them unsuited for physically modelling open resonators . first , the eigenvalue problem must be solved at each frequency , yielding a different set of current vectors at each frequency . this requires some algorithm to track modes with frequency @xcite , and effectively prevents their use in time - domain problems . more significantly , the enforcement of mode orthogonality on an inherently non - hermitian system results in an artificial set of basis vectors which contain a complex mixture of underlying eigenvectors . this manifests itself in unphysical avoided crossings , whereby the nature of a pair of modes is swapped in some frequency region @xcite . the author has observed similar behaviour when utilising other orthogonal decompositions of the impedance matrix , such as the singular value decomposition . in order to reproduce the interference phenomena observed in fig . [ fig : extinction_modes ] , it is essential to use the non - orthgonal modes obtained from singularity expansion method , or quasi - normal modes approaches . ] to confirm the accuracy of the mode expansion , the directly calculated extinction curve is plotted in fig . [ fig : extinction - accuracy ] ( solid line ) , as well as the sum of all contributions plotted in fig . [ fig : extinction_modes](b ) ( red dashed line ) . it can be seen that the agreement is good for frequencies below 250thz , however at high frequencies it becomes poorer . by increasing the number of poles considered from 28 ( i.e. 7 modes , each doubly degenerate and with conjugate poles ) to 145 , much better agreement is achieved , as shown by the blue dashed curve . clearly a model involving so many parameters is less useful as a design tool , thus there is an inevitable trade - off between accuracy and the level of insight provided . however , in contrast to simpler approaches based on point dipole or equivalent circuit models , it is possible to control the level of detail which is included within the model by choosing to include or exclude poles . as this work includes materials with dispersion and dissipative losses , the impedance matrix @xmath23 may exhibit branch point singularities , in addition to poles . the green s function used to calculate elements of the impedance matrix has terms proportional to @xmath76 . the complex wave - number @xmath77 has branch points at the poles and zeros of the permittivity , connected by branch cuts@xcite . for the material data used in this work , all such branch points occur at frequencies above 800thz , thus their contribution is neglected in eq . . the accuracy of the results shown in fig . [ fig : extinction - accuracy ] confirms that no significant contribution from branch points is missing from the result . the lack of branch points in the frequency range of interest also ensures that the integration contour illustrated in fig . [ fig : contour ] does not intersect any of the branch cuts . applying the contour integration in a frequency range of high dispersion would require choosing the contour so that it encircles branch points in pairs to avoid crossing branch - cuts . 39ifxundefined [ 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 , ed . , @noop _ _ , no . ( , , ) link:\doibase 10.1002/adom.201400584 [ * * , ( ) ] link:\doibase 10.1109/tmtt.1975.1128528 [ * * , ( ) ] link:\doibase 10.1103/revmodphys.70.1545 [ * * , ( ) ] link:\doibase 10.1021/ph400114e [ * * , ( ) ] link:\doibase 10.1103/physrevlett.110.237401 [ * * , ( ) ] link:\doibase 10.1103/physreva.92.053810 [ * * , ( ) ] link:\doibase 10.1109/proc.1976.10379 [ * * , ( ) ] link:\doibase 10.1109/jstqe.2012.2227684 [ * * , ( ) ] link:\doibase 10.1103/physrevb.90.075108 [ * * , ( ) ] link:\doibase 10.1103/physrevb.89.165429 [ * * , ( ) ] link:\doibase 10.1109/jphot.2014.2331236 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.110.197401 [ * * , ( ) ] link:\doibase 10.1063/1.4949007 [ * * , ( ) ] @noop * * , ( ) link:\doibase 10.1029/2004rs003169 [ * * , ( ) ] @noop _ _ ( , , ) link:\doibase 10.1109/8.761074 [ * * , ( ) ] link:\doibase 10.1080/02726348108915144 [ * * , ( ) ] link:\doibase 10.1080/02726348108915141 [ * * , ( ) ] link:\doibase 10.1109/jlt.2012.2234723 [ * * , ( ) ] @noop _ _ ( , , ) link:\doibase 10.1109/tap.2015.2438393 [ * * , ( ) ] link:\doibase 10.1002/pip.4670030303 [ * * , ( ) ] @noop _ _ ( , , ) link:\doibase 10.1103/physreva.93.053837 [ * * , ( ) ] link:\doibase 10.1103/physreva.88.053819 [ * * , ( ) ] link:\doibase 10.1038/ncomms9069 [ * * ( ) , 10.1038/ncomms9069 ] link:\doibase 10.1063/1.3486480 [ * * , ( ) ] @noop `` , '' @noop , link:\doibase 10.1088/1367 - 2630/14/9/093033 [ * * , ( ) ] @noop _ _ , ed . ( , ) @noop _ _ ( , , ) link:\doibase 10.1103/physreva.49.3057 [ * * , ( ) ] link:\doibase 10.1109/tap.2016.2579668 [ * * , ( ) ] link:\doibase 10.1109/tap.2016.2556698 [ * * , ( ) ] link:\doibase 10.1109/tap.2016.2550098 [ * * , ( ) ] ( ) pp .

the modes of silicon disk meta - atoms are investigated , motivated by their use as a building block of huygens metasurfaces . a model based on these modes gives a clear physical explanation of all features in the extinction spectrum , in particular due to the interference between non - orthogonal modes . by performing a vector spherical harmonic expansion of each mode , the complex features of the far - field scattering spectrum are also readily explained . it is shown that in general each mode has contributions from many multipole moments . higher order modes with appropriate symmetry are also able to satisfy the huygens condition , leading to multiple bands of strong forward scattering and suppressed back scattering . these results demonstrate a robust approach to find the modes of nano - photonic scatterers , commonly referred to as quasi - normal modes . by utilising an integral formulation of maxwell s equations , the problem of normalising diverging far - fields is avoided . the approach is implemented in an open - source code .

1610.04980

the inner ( @xmath8 1kpc ) parts of some spiral galaxies show star formation complexes frequently arranged in an annular pattern around their nuclei . these complexes are sometimes called `` hot spots '' and we will refer to these as circumnuclear starforming regions ( cnsfrs ) . their sizes go from a few tens to a few hundreds of pc ( e.g. * ? ? ? * ) and they seem to be made of an ensamble of hii regions ionised by luminous compact stellar clusters whose sizes , as measured from high spatial resolution hst images , are seen to be of only a few pc . the luminosities of cnsfrs are rather large with absolute visual magnitudes ( m@xmath9 ) between -12 and -17 and h@xmath10 luminosities which are comparable to those shown by 30 dor , the largest hii region in the lmc , and overlap with those shown by hii galaxies ( * ? ? ? * ; * ? ? ? * ; * ? ? ? * and references therein ) . in the ultraviolet ( uv ) , massive stars dominate the observed circumnuclear emission even in the presence of an active nucleus @xcite . in many cases , cnsfr show emission line spectra similar to those of disc hii regions . however , they show a higher continuum from background stellar populations as expected from their circumnuclear location . the analysis of these spectra gives us the oportunity to measure the gas abundances close to galactic nuclei which , in the case of early type spirals , are expected to be amongst the highest metallicity regions . the importance of an accurate determination of the abundances of high metallicity hii regions can not be overestimated since they constitute most of the hii regions in early spiral galaxies ( sa to sbc ) and the inner regions of most late type ones ( sc to sd ) @xcite without which our description of the metallicity distribution in galaxies can not be complete . in particular , the effects of the choice of different calibrations on the derivation of abundance gradients can be very important since any abundance profile fit will be strongly biased towards data points at the ends of the distribution . it should be kept in mind that abundance gradients are widely used to constrain chemical evolution models , histories of star formation over galactic discs or galaxy formation scenarios . also , the question of how high is the highest oxygen abundance in the gaseous phase of galaxies is still standing and extrapolation of known radial abundance gradients would point to cnsfr as the most probable sites for these high metallicities . accurate measures of elemental abundances of high metallicity regions are also crucial to obtain reliable calibrations of empirical abundance estimators , widely used but poorly constrained , whose choice can severely bias results obtained for quantities of the highest relevance for the study of galactic evolution like the luminosity - metallicity ( l - z ) relation for galaxies . as part of the programme to study the properties of cnsfr we obtained two sets of data with the intermediate dispersion spectrograph and imaging system ( isis ) attached to the 4.2 m william herschel telescope ( wht ) . the first set of observations consisted of high resolution blue and far - red long - slit spectra covering the emission lines of h@xmath1 and [ oiii ] and the caii triplet lines respectively . at the attained resolution of 0.4 and 0.7 respectively , these data allowed the measurement of radial velocities and velocity dispersions of the gas and stars in the regions down to 25 km s@xmath11 . the second set , also obtained with the same instrumentation had a lower resolution of 3.4 and 1.7 in the blue and red spectral regions respectively and a wider total coverage from 3600 to 9650 . these data were used to derive the physical conditions and the abundances of the emitting gas . several cnsfr were observed in the spiral galaxies : ngc 2903 , ngc 3351 and ngc 3504 . archive images obtained with the wide field and planetary camera 2 and the nicmos camera 3 both on - board the hst were also downloaded in order to complement the study . figure [ images ] shows the optical image of ngc 3351 with the slit positions used for the two sets of observations . figure [ profiles ] shows the spatial distribution of the h@xmath1 and continuum emission along one of the slit positions observed . as can be seen from these profiles , some clusters are dominated by continuum emission while in other cases gaseous line emission is clearly important . gas velocity dispersions were measured by performing gaussian fits to the h@xmath1@xmath24861 and [ oiii ] @xmath2 5007 lines on the high dispersion spectra ( figure [ spectra - kin ] ) . stellar velocity dispersions dispersions were measured using the cat lines at @xmath128494 , 8542 , 8662using the cross - correlation technique described in detail by @xcite . late type giant and supergiant stars that have strong cat absorption lines were used as stellar velocity templates . + for the 5 cnsfr observed in ngc 3351 , stellar velocity dispersions are found to be between 39 and 67kms@xmath11 , about 20kms@xmath11 larger than those measured for the gas , if a single gaussian fit is used . however , the best fits obtained involved two different components for the gas : a broad component " with a velocity dispersion similar to that measured for the stars , and a narrow component " with a dispersion lower than the stellar one by about 30kms@xmath11 . these two components were found both in the hydrogen recombination lines ( balmer and paschen ) and in the [ oiii ] @xmath2 5007 line . the narrow component is dominant in the h recombination lines , while the broad component dominates the [ oiii ] one . figure [ velocities ] shows this effect . cnsfr are seen to consist of several individual star clusters , although some of them seem to have an only knot , at the hst resolution . the derived masses for the individual clusters as derived using the sizes measured on the hst images are between 1.8 and 8.7@xmath1310@xmath14m@xmath15 . these values are between 5.5 and 26 times the mass derived for the ssc a in ngc1569 by @xcite and larger than other kinematically derived ssc masses . values for the dynamical masses of the cnsfrs are in the range between 4.9@xmath1310@xmath14 and 4.3@xmath1310@xmath3m@xmath15 . masses derived from the h@xmath1 velocity dispersion under the assumption of a single component for the gas would have been underestimated by factors between approximately 2 to 4 . the masses of the ionising stellar clusters of the cnsfrs have been derived from their h@xmath10 luminosities under the assumption that the regions are ionisation bound and without taking into account any photon absorption by dust . their values are between 8 - 0@xmath1310@xmath16 and 2.5@xmath1310@xmath14m@xmath15 . therefore , the ratio of the ionising stellar population to the total dynamical mass is between 0.02 and 0.16 . the ssc in the observed cnsfrs seem to contain composite stellar populations . although the youngest one dominates the uv light and is responsible for the gas ionisation , it constitutes only about 10% of the total . this can explain the low equivalent widths of emission lines measured in these regions . this may well apply to the case of other ssc and therefore conclusions drawn from fits of ssp ( single stellar population ) models should be taken with caution ( e.g. * ? ? ? also the composite nature of the cnsfrs means that star formation in the rings is a process that has taken place over time periods much longer than those implied by the properties of the ionised gas . for 12 cnsfrs in the galaxies ngc 2903 , ngc 3351 and ngc 3504 chemical abundances of o , n and s were derived from the lower resolution spectra . all the spectra are characterized by very weak [ oiii ] @xmath12 4959 , 5007 lines . figure [ weaklines ] shows the spectrum of a typical region . this low excitation is typical of high abundance regions in which the cooling is dominated by [ oii ] and [ oiii ] emission lines . in these cases the detection of the auroral [ oiii ] @xmath2 4363 line which woukd allow the derivation of the electron temperature is virtually impossible and empirical methods based on the calibration of strong emission lines have to be used . the most commonly used abundance indicator , o@xmath7 , involves the sum of the [ oii ] and [ oiii ] emission lines , supposed to be almost independent of geometrical effects in the nebulae . however , this indicator is not well calibrated at the high metallicity end where the oxygen lines are so weak that the value of the r@xmath7 practically saturates . the sulphur lines , much stronger than the oxygen lines in high metallicity regions , provide an alternative way to derive their abundances . we have developed a new method for the derivation of sulphur abundances based on the calibration of the [ siii ] electron temperature vs the empirical parameter so@xmath7 defined as the quotient of the oxygen and sulphur abundance parameters o@xmath7 and s@xmath7 and the further assumption that t([siii ] ) @xmath17 t([sii ] ) . then the oxygen abundances and the n / o and s / o ratios can also be derived . figure [ scalib ] shows this calibration for which data from different sources as listed in @xcite have been used . the actual fit to the data gives : @xmath18 ) = 0.596 - 0.283 log so_{23 } + 0.199 ( log so_{23})^2\ ] ] only for one of the regions , the [ siii ] @xmath2 6312 line was detected providing , together with the nebular [ siii ] lines at @xmath12 9069 , 9532 , a value of the electron temperature of t@xmath19([siii])= 8400@xmath20k . this value is slightly higher than predicted by the proposed fit and is represented as a solid upside down triangle in figure [ scalib ] . the derived oxygen abundances are comparable to those found in high metallicity disc hii regions from direct measurements of electron temperatures and are consistent with solar values within the errors . the highest oxygen abundance derived is 12+log(o / h ) = 8.85 , about 1.5 solar . = [email protected] ] the lowest oxygen abundance derived is about 0.6 times solar . in all the observed cnsfr the o / h abundance is dominated by the o@xmath21/h@xmath21 contribution , as is also the case for high metallicity disc hii regions . for our observed regions , however also the s@xmath21/s@xmath22 ratio is larger than one . this is not the case for the high metallicity disc hii regions for which , in general , the sulphur abundances are dominated by s@xmath22/h@xmath21 . the derived n / o ratios are in average larger than those found in high metallicity disc hii regions and they do not seem to follow the trend of n / o vs o / h which marks the secondary behaviour of nitrogen . on the other hand , the s / o ratios span a very narrow range between 0.6 and 0.8 of the solar value . from the calculated values of the number of lyman @xmath10 photons , q(h@xmath23 ) , ionisation parameter and electron density , it is possible to derive the size of the emitting regions as well as the filling factor ( see * ? ? ? * ) .the derived sizes are between 1.5 arcsec and 5.7 arcsec which correspond to linear dimensions between 74 and 234 pc . the derived filling factors , between 6 @xmath13 10@xmath24 and 1 @xmath13 10@xmath25 , are lower than commonly found in giant hii regions ( @xmath8 0.01 ) . luminosities are larger than the typical ones found for disc hii regions and overlap with those measured in hii galaxies . the region with the largest h@xmath10 luminosity is r3+r4 in ngc 3504 , for which a value of 2.02 @xmath13 10@xmath26 is measured . ionising cluster masses range between 1.1 @xmath13 10@xmath27 and 4.7 @xmath13 10@xmath28 m@xmath5 but could be lower by factors between 1.5 and 15 if the contribution by the underlying stellar population is taken into account . these values are consistent with those found from the kinematical data set . when compared to high metallicity disc hii regions , cnsfr show values of the o@xmath7 and the n2 parameters whose distributions are shifted to lower and higher values respectively , hence , even though their derived oxygen and sulphur abundances are similar , higher values would in principle be obtained for the cnsfr if pure empirical methods were used to estimate abundances . this can be seen in figure [ abun ] where the distributions of the derived o / h abundances and the o@xmath7 pararameter for cnsfr and high metallicity disc hii regions are shown . cnsfr also show lower ionisation parameters than their disc counterparts , as derived from the [ sii]/[siii ] ratio . their ionisation structure also seems to be different with cnsfr showing radiation field properties more similar to hii galaxies than to disc high metallicity hii regions . this can be seen in figure [ etaplot ] that shows [ oii]/[oiii ] ratio vs [ sii]/[siii ] . since the parameter @xmath29 , which is the quotient of these two ratios , is a good indicator of ionising temperature @xcite , diagonal lines in this plot should run along objects with similar temperatures . cnsfr in this diagram also segregate from the high metallicity disc hii regions pointing to higher values of the ionising temperature . the possible contamination of their spectra from hidden low luminosity agn and/or shocks , as well as the probable presence of more than one velocity component in the ionised gas corresponding to kinematically distinct systems ( see section 3 above ) , should be further investigated . asplund , m. , grevesse , n. , sauval , a. j. 2005 , asp conference series , 336 , 25 castellanos , m. , daz , a. i. , terlevich , e. 2002 , , 329 , 315 colina , l. , gonzlez delgado , r. , mas - hesse , j. m. , leitherer , c. 2002 , , 579 , 545 daz , a. i. 1989 , evolutionary phenomena in galaxies . cambridge and new york , cambridge university press , p. 377 daz , a. i. , lvarez lvarez , m. , terlevich , e. , terlevich , r , snchez portal , miguel , aretxaga , i. 2000 , , 311 , 120 daz , a. i. , terlevich , e. , castellanos , m. , hgele , g. f. 2007 , , 382 , 251 gonzlez delgado , r m. , heckman , t. , leitherer , c. , meurer , g. , krolik , j. , wilson , a. s. , kinney , a. , koratkar , a. 1998 , , 505 , 174 hoyos , c. , daz , a. i. 2006 , , 365 , 454 ho , l. c. , filippenko , a. v. 1996 , , 466 , l83 larsen , s. s. , brodie , j. p. , hunter , d. a. 2004 , 128 , 2295 mccrady , n. , gilbert , a. m. , graham , j. r. 2003 , , 596 , 240 melnick , j. , terlevich , r. , moles , m. 1988 , , 235 , 297 tonry , j. , davis , m. 1979 , , 84 , 1511 vila - costas , b. , edmunds , m. g. 1992 , , 259 , 121 vlchez , j. m. , pagel , b. e. j. 1988 , , 231 , 257

a study of cicumnuclear star - forming regions ( cnsfrs ) in several early type spirals has been made in order to investigate their main properties : stellar and gas kinematics , dynamical masses , ionising stellar masses , chemical abundances and other properties of the ionised gas . both high resolution ( r@xmath020000 ) and moderate resolution ( r @xmath0 5000 ) have been used . in some cases these regions , about 100 to 150pc in size , are seen to be composed of several individual star clusters with sizes between 1.5 and 4.9pc estimated from hubble space telescope ( hst ) images . stellar and gas velocity dispersions are found to differ by about 20 to 30km / s with the h@xmath1 emission lines being narrower than both the stellar lines and the [ oiii]@xmath25007lines . the twice ionized oxygen , on the other hand , shows velocity dispersions comparable to those shown by stars . we have applied the virial theorem to estimate dynamical masses of the clusters , assuming that systems are gravitationally bounded and spherically symmetric , and using previously measured sizes . the measured values of the stellar velocity dispersions yield dynamical masses of the order of 10@xmath3 to 10@xmath4 solar masses for the whole cnsfrs . we obtain oxygen abundances which are comparable to those found in high metallicity disc hii regions from direct measurements of electron temperatures and consistent with solar values within the errors . the region with the highest oxygen abundance is r3+r4 in ngc 3504 , 12+log(o / h ) = 8.85 , about 1.5 solar if the solar oxygen abundance is set at the value derived by @xcite , 12+log(o / h)@xmath5 = [email protected] . the derived n / o ratios are in average larger than those found in high metallicity disc hii regions and they do not seem to follow the trend of n / o vs o / h which marks the secondary behaviour of nitrogen . on the other hand , the s / o ratios span a very narrow range between 0.6 and 0.8 of the solar value . as compared to high metallicity disc hii regions , cnsfr show values of the o@xmath7 and the n2 parameters whose distributions are shifted to lower and higher values respectively , hence , even though their derived oxygen and sulphur abundances are similar , higher values would in principle be obtained for the cnsfr if pure empirical methods were used to estimate abundances . cnsfr also show lower ionisation parameters than their disc counterparts , as derived from the [ sii]/[siii ] . their ionisation structure also seems to be different with cnsfr showing radiation field properties more similar to hii galaxies than to disc high metallicity hii regions .

0801.3078

activity in galactic nuclei is fuelled by a reservoir of low angular momentum gas , but it is unclear how such reservoirs build up . fast outflows from ob stars , supernovae or agn activity can clear out nuclear ism @xcite . gas in the nucleus can also be quickly consumed by star formation @xcite or driven outwards by positive gravitational torques @xcite . if the gas in the reservoir originates outside the nucleus , multiple mechanisms operating on different distance- and time - scales , such as bars feeding nuclear rings @xcite , radiation drag @xcite or interactions @xcite are required . from observations , around @xmath3 of all galactic nuclei in the local universe exhibit low luminosity nuclear activity @xcite , so the mechanism promoting nuclear gas build up is likely to be simple and on - going . in this letter we propose a new mechanism for delivery of gas to the galactic nucleus . we show that the impact of a warm halo cloud ( whc ) , containing @xmath4 , on the central regions of a galaxy will fuel nuclear activity . we develop a simplistic model of this phenomenon to demonstrate its likely importance . while there remains considerable uncertainty in whc parameters , we show that for plausible input parameters , a direct hit by a single whc on the center of a galaxy will supply fuel for star formation and radiatively inefficient accretion onto the central black hole . since whc bombardment of galaxies must occur , some fraction of the low luminosity activity observed in galactic nuclei must be due to whc impacts on galactic nuclei . the assembly of gaseous halos around galaxies naturally produces a multiphase medium with warm halo clouds ( whcs ) embedded in a low density hot gas halo @xcite . whcs could be the local analogue of the high redshift lyman limit systems @xcite . in the halo of our own galaxy , numerous warm clouds are observed with velocities that are inconsistent with galactic rotation ( e.g. * ? ? ? around several other galaxies , hi clouds of @xmath5 have been detected ( e.g. * ? ? ? * ; * ? ? ? extended hi structures ( up to @xmath6 ) are observed out to @xmath7kpc around early - type galaxies @xcite . so , a population of whcs containing @xmath4 per cloud , may be common around most galaxies ( see e.g. * ? ? ? * ) . around our galaxy , two basic models can account for the properties of observed high velocity clouds ( hvcs ) : an accretion model based on whcs ( e.g. * ? ? ? * ; * ? ? ? * ) or a galactic fountain model @xcite . accreting whcs should dominate the mass of clouds in the galactic halo so in this work we shall concentrate on the effects of an accreting whc impact on galactic nuclei . much of the following discussion also applies to galactic fountain clouds , although that population will have higher metallicity , less mass , smaller radius and lower velocity on average , and will be less numerous . large uncertainties exist concerning cloud trajectories around our own galaxy ( e.g. * ? ? ? while we might naively expect radial trajectories for whcs , cloud trajectories may become randomized close to a galaxy . clouds close to the disk become tidally disrupted @xcite , or deflected by magnetic fields @xcite , or disrupted by the kelvin - helmholz instability ( khi ) @xcite and the fragments dispersed about the disk , whereupon the previous cloud trajectory may be irrelevant . since we have no way of predicting actual cloud trajectories , in the discussion below we simply assume that clouds impact the disk randomly . we start by assuming a initial population of @xmath8 clouds on random trajectories raining down on a galaxy of radius @xmath9 . as long as the whc radius is larger than the nuclear region under consideration , the rate of impact ( @xmath10 ) of infalling whc of radii @xmath11 on the galactic center is @xmath12 where @xmath13 is the typical cloud infall time and the cloud material is assumed to arrive within @xmath14 of the galactic center . we calculate the cloud impact rate for multiphase cooling around a @xmath1cdm distribution of halos based on the model of @xcite . in this model the total mass in whcs is based on the mean free path of the clouds and the cloud properties we use are the average of some distribution in the halo . first we calculate the average nuclear impact rate , using @xmath15 , based on conservation of angular momentum @xcite , where @xmath16 is the cooling radius @xcite . we simplify our calculation by assuming that the whc does not fragment and that infall time is @xmath17 where @xmath18 is the maximum circular velocity in the halo . the results are shown in fig . [ fig : results](a ) , where we plot impacts / gyr versus @xmath18 for impacting cloud masses of @xmath19 . evidently the impact rate is relatively flat with @xmath18 ( or equivalently black hole mass ) . note that a nuclear impact rate of @xmath20/gyr from fig . [ fig : results](a ) corresponds to a galactic impact rate of @xmath21/gyr or a mass inflow rate of @xmath22/yr if the typical impactor mass is @xmath23 . this is approximately the low metallicity inflow rate required to explain the so - called g - dwarf problem ( e.g. * ? ? ? * and references therein ) . although observational constraints of cloud impacts are not strong @xcite , we can consider the consequences of whc impact on a galactic nucleus , guided by simulations of hvc impacts with the disk ( see e.g. * ? ? ? * ; * ? ? ? * and references therein ) . the same basic sources of cloud fragmentation in the halo and disk will apply to a whc falling into a galactic bulge . if galactic nuclei have relatively strong magnetic field strengths in general ( e.g. * ? ? ? * ) , the magnetic fields will act as a strong brake on infalling whcs and the cloud may fragment . instabilities in the infalling cloud have growth times of the order of the timescale on which shocks cross the cloud . this cloud crushing timescale , @xmath24 , is given by @xcite @xmath25 where @xmath26 and @xmath27 are the densities of the whc and the surrounding medium respectively and @xmath28 is the relative velocity of the whc and the surrounding medium . if the whc is accreting from a very large distance , @xmath24 can be less than the infall time ( @xmath29 gyrs ) and the cloud can fragment in the halo . indeed many of the hvcs presently observed may be fragments of initially much larger accreting whcs . galactic haloes are hot and diffuse ( e.g. * ? ? ? * ; * ? ? ? * and references therein ) and the disruption of a whc in the halo could create an infalling stringy association of clouds like the hvc a and c complexes . if the trajectory of such a stringy cloud complex intercepted the galactic nucleus then most of the whc mass could be delivered to the region around the central supermassive black hole . sufficiently dense and fast clouds falling through the halo and then the bulge ism will first experience deceleration and compression ( pancaking ) , followed by an expansion of the shocked cloud downstream , then lateral expansion and finally cloud destruction when instabilities and differential forces fragment the cloud . this entire process takes place over a few @xmath24 ( see e.g. * ? ? ? * ; * ? ? ? compression will increase the whc density as the cloud pancakes @xcite . small , fast , dense fragments of cloud can also sweep up large amounts of gas and be shocked @xcite . the transit time for a whc through a bulge of radius @xmath30 to the galactic center is @xmath31 where @xmath32 $ ] is a coefficient incorporating slowdown due to drag and magnetic fields . for a galaxy the size of our own(@xmath33kpc ) , a cloud with @xmath34 km @xmath35 will take @xmath36myr to cross the bulge and reach the center . but will the cloud survive this long ? outside of the inner @xmath290.5kpc , the bulge ism in our own galaxy has very low density ( @xmath37 ) @xcite . at these densities , the cloud crushing timescale is @xmath38myrs , far larger than the transit time through this part of the bulge . within @xmath39kpc of the galactic center the bulge ism density increases to @xmath40 @xcite . here the crushing timescale is @xmath36myr , short enough that the whc will fragment in this region . so , whcs could survive until they reach the central regions , whereupon fragmentation is likely . evidently a whc cloud impact could deliver a large quantity of moderate density gas to a galactic nucleus in a single cloud infall . but what happens once fragments of a cloud of mass @xmath41 arrives in the central few tens of pc of a galaxy ? low angular momentum fragments of the whc will be captured by the central black hole out to a radius ( @xmath42 ) where their velocity is below the escape speed . thus the mass captured is @xmath43 we calculate the average @xmath44 for clouds of mass @xmath45 impacting galactic nuclei . we estimate black hole mass using the @xmath46 relation from @xcite and we assume that the bulge velocity dispersion and maximum circular velocity are related as in @xcite so that @xmath47 which should be valid over the range @xmath48 km @xmath35 . we allowed different values of braking ( @xmath49 ) , ranging from none ( @xmath50 ) to strong ( @xmath51 ) which are shown in fig . [ fig : results](b ) . clearly the braking parameter determines when the average mass captured reaches the total cloud mass . our model predicts that the eddington ratio will be greatest near a critical value of @xmath18 , where the average mass captured reaches the total cloud mass . evidently a nuclear strike on a black hole with mass @xmath52 yields much less mass available for accretion onto the black hole , even for large braking ( @xmath51 ) . however , a much larger fraction of @xmath53 could be captured if the infalling cloud fragments into a stringy cloud complex in the galactic halo , effectively reducing @xmath11 for a given @xmath18 . cloud material not captured by the black hole will instead be gravitationally captured by the bulge . in this case , the material will either persist as part of the bulge ism or will go down the route of star formation ( with some unknown level of mixing ) . this is consistent with observations of nuclear activity due to @xmath54 regions predominating in galaxies with smaller bulges and therefore smaller mass central black holes @xcite . what sort of nuclear activity will result from a whc impact ? once moderately dense , low angular momentum fragments of the whc arrive in the galactic nucleus , star formation is likely , although dependent on poorly constrained variables such as cooling rate and mixing with nuclear gas . the arrival of a large quantity of infalling gas may also initiate activity via shocks or gradual acceleration of gmcs @xcite . in our model , shocked nuclear gmcs or the highest density , shocked fragments resulting from a whc impact could generate a nuclear ob association within @xmath55myr after impact @xcite . a direct hit by a whc fragment on the central black hole may result in a phase of low luminosity bondi accretion . @xcite find two distributions of eddington ratios among active galactic nuclei in the local universe . their first , a lognormal distribution , centered around a few percent of eddington , is associated with nuclear star formation , dominates among black holes @xmath56 and depends critically on feedback from luminous accretion . the second distribution is a powerlaw which occurs in galaxy bulges where there is little or no ongoing star formation . we will only model the second mode since the first mode has a complex association with star formation , stellar evolution and agn feedback . although the distribution of dm / dt in this second accretion mode is a powerlaw , we will use the average value from our @xmath1cdm calculations to get an average activity fraction . we calculate the average fraction of galaxies of a given @xmath18 that are active for impactors of mass @xmath45 and a range of braking and cloud densities . the lifetime of bondi accretion is @xmath57 where @xmath58 here @xmath59 is the average density of the accreting gas , @xmath60 is the black hole mass in units of @xmath61 and @xmath62 is an average combination of gas speed and gas sound speed in units of 200 km / s @xcite . the average gas density around the black hole is fairly unconstrained in our model and clearly there is trade off between more massive clouds that take longer to consume and higher gas densities which are consumed more quickly . nevertheless , in this model the lowest average dm / dt is @xmath63/yr around a @xmath61 black hole , assuming @xmath64 and @xmath65 . in fig . [ fig : results](c ) we show some average values of braking and gas density that yield reasonable models where on average 1/3 of galactic nuclei are active . for @xmath66 clouds this requires @xmath67 , while for @xmath68 clouds with some braking this requires @xmath69 , and with no braking , @xmath70 . recall we are modelling the powerlaw accretion mode of @xcite , which is significant among black holes of mass @xmath71 . therefore , from fig . [ fig : results](b),(c ) , our model requires @xmath72 for impactor clouds of masses @xmath66 and @xmath68 respectively . note that in all cases the plot shows a characteristic shape where the average activity first increases as the amount of the cloud captured increases and then decreases as the accretion rate increases for the same amount of captured cloud . our model predicts that radiatively inefficient bondi accretion should be more commonly detected around larger mass black holes , which is consistent with observations of liner activity in the local universe ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? future observations of low luminosity activity in galactic nuclei as a function of @xmath18 will be able to rule out certain parameters . of course if @xmath49 or @xmath59 depend on @xmath18 then the shapes of the curves in fig . [ fig : results](c ) might differ . mhd simulations of cloud fragmentation in the nucleus are required to establish whether the distribution of densities and braking can in fact generate a powerlaw distribution of dm / dt @xcite and we intend to do this in future work . note that our model of bondi accretion is unlikely to apply for @xmath73 since the dominant activity in that case would be nuclear star formation , which is consistent with observations of smaller mass central black holes @xcite . @xcite argue that luminous accretion can occur in the aftermath of a nuclear starburst . this may yield the lognormal distribution of eddington ratio observed in a fraction of galactic nuclei @xcite , but establishing this is far beyond the scope of the present work . we expect the number of whcs to increase with redshift as the gas accretion rate onto galaxies increases . we naively expect the number of whc clouds and therefore the average cloud impact rate to scale with star formation rate or @xmath74 from @xmath75 to @xmath76 @xcite , corresponding to a rate of @xmath77myr for milky - way sized galaxies at @xmath78 . on these timescales , luminous nuclear activity could result if radiatively efficient accretion occurs onto the central supermassive black hole . a strongly decreasing eddington ratio over cosmic time ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) could naturally produce a high duty cycle at @xmath79 , consistent with fig . [ fig : results](c ) . at even earlier epochs ( @xmath80 ) , the bulk of the growth of supermassive black holes occurs via mergers in massive protogalactic halos ( @xmath81 ) @xcite . at that time , gas accretion is primarily on random trajectories , so nuclear fueling by whc impact would be at its maximum . a natural bridge can then be made to the time before hot halos are established @xcite , when gas accretion is all in the form of infalling gas . in the present epoch , our model predicts that clustering of activity should be random for similar sized galaxies ( comparable @xmath18 ) , since cloud impacts are random . this is consistent with observations of clustering in liners @xcite . in this letter we introduce a new mechanism that delivers large quantities of gas to galactic nuclei on astrophysically interesting timescales . we show that a single warm halo cloud ( whc ) impact on a galactic bulge could potentially deliver a large mass ( @xmath4 ) of gas to the central regions of a galactic nucleus , in a singular event . although there are considerable uncertainties in the parameters of warm halo clouds , some representative numbers suggest that the impacts occur on astrophysically interesting timescales and at a rate that must account for some or even all of the low luminosity activity observed in galactic nuclei in the local universe . based on analytic @xmath1cdm calculations of cooling halos , our model predicts an impact rate relatively independent of galaxy mass . our model also predicts that larger mass black holes ( @xmath82 ) can capture significant fractions of the impacting cloud mass at some critical value depending on the nuclear braking ( @xmath49 ) . at this critical value , our model predicts the highest eddington ratios . below this critical mass , our model predicts that most of the cloud mass will not accrete onto the black hole . instead , this material will mix with the ism in the nuclear bulge or induce star formation , which may lead to delayed episodes of high eddington ratio accretion . finally , our model predicts that , for a reasonable range of cloud masses , densities and braking , the fraction of supermassive black holes accreting at very low eddington ratios in the local universe could be around @xmath83 , which agrees with observations @xcite . bm & kesf gratefully acknowledge the support of the department of astrophysics of the american museum of natural history , psc grant psc - cuny-40 - 397 and cuny grant ccri-06 - 22 . am acknowledges support from an roa supplement to nsf award ast-0904059 . we acknowledge useful discussions with mordecai - mark mac low and a very helpful report from the anonymous referee . blitz l. , spergel d.n . , tueben p.j . , hartmann d. & burton w.b . , 1999 , apj , 514 , 818 boomsma r. , oosterloo t.a . , fraternali f. , van der hulst j.m . & sancisi r. , 2008 , a&a , 490 , 555 cavaliere a. & vittorini v. , 2002 , apj , 570 , 114 chynoweth k.m . , langston g.i . , yun m.s . , lockman f.j . , rubin k.h.r . & scholes s.a . , 2008 , apj , 135 , 1983 ciotti l. & ostriker j.p . , 1997 , apj , 487 , l105 ciotti l. & ostriker j.p . , 2007 , apj , 665 , 1038 clark p.c . , bonnell i.a . , zinnecker h. & bate m.r . , 2005 , mnras , 359 , 809 constantin a. & vogeley m.s . , 2006 , apj , 650 , 727 dekel a. , sari r. & ceverino d. , 2009 , apj , 703 , 785 doherty m. , bunker a. , sharp r. , dalton g. , parry i. & lewis i. , 2006 , mnras , 370 , 331 edmunds m.g . , 1990 , mnras , 246 , 678 fall s.m . & efstathiou g. , 1980 , mnras , 193 , 189 ferrarese l. , 2002 , apj , 578 , 90 garcia - burillo s. , combes f. , schinnerer e. , boone f. & hunt l.k . , 2005 , a&a , 441 , 1011 gultekin k. et al . , 2009 , apj , 698 , 198 heitsch f. & putman m.e . , 2009 , apj , 598 , 1485 ho l. , 2008 , ara&a , 46 , 475 hopkins p.f . & hernquist l. , 2006 , apjs , 166 , 1 hopkins p.f . , hernquist l. , cox t.j . , robertson b. & krause e. , 2007 , apj , 669 , 67 kauffmann g. & heckman t.m . , 2009 , mnras , 397 , 135 kawakatu n. , umemura m. & mori m. , 2003 , apj , 583 , 85 kers d. , katz n. , weinberg d.h . & dave r. , 1997 , apj , 487 , 591 klein r.i . , mckee c.f . & colella p. , 1994 , apj , 420 , 213 kormendy j. & richstone d. , 1995 , ara&a , 33 , 581 krolik j.h . , 2001 , active galactic nuclei ( princeton university press , princeton , new jersey ) kwak k. , shelton r.l . & raley e.a . , 2009 , apj , 699 , 1775 lockman f.j . , benjamin r.a . , heroux a.j . & langston g.i . , 2008 , apj , 679 , 21 maller a. & bullock j. s. , 2004 , mnras , 355 , 694 metzger p.g . , dushcl w.j . & zylka r. , 1996 , a&ar , 7 , 289 miller e.d . , bregman j.n . & waaker b.p . , 2009 , apj , 692 , 470 mckernan b. , yaqoob t. & reynolds c. s. , 2004 , apj , 617 , 232 mckernan b. , yaqoob t. & reynolds c. s. , 2007 , mnras , 379 , 1359 nayakshin s. , cuadra j. & springel v. , 2007 , mnras , 379 , 21 oosterloo t.a . , morganti r. , sadler e.m . , van der hulst t. & serra p. , 2007 , a&a , 465 , 787 ostriker j.p . & rees m.j . , 1977 , mnras , 179 , 541 peek j.e.g . , putman m.e . , mckee c.f . , heiles c. & stanimirovi , 2007 , apj , 656 , 907 peek j.e.g . , putman m.e . & sommer - larsen , 2008 , apj , 674 , 227 putman m.e . et al . , 2002 , aj , 123 , 873 schartmann m. , burkert a. , krause m. , camenzind m. , meisenheimer k. & davies r.i . , 2010 , mnras , astro - ph/0912.4677 schawinski k. , thomas d. , sarzi m. , maraston c. , kaviraj s. , joo s .- j . , silk j. , 2007 , mnras , 382 , 1415 shapiro p.r . & field , g.b . , 1976 , apj , 205 , 762 shankar f. , weinberg d.h . & miralda - escude , j. , 2009 , apj , 690 , 20 shen y. , 2009 , apj , 704 , 89 strickland d. k. , heckman t.m . , colbert , e.j.m . hoopes c.g . & weaver k.a . , 2004 , apjs , 151 , 193 tang s. , wang q.d . , mac low m .- & joung , m. r. , 2009 , mnras , 398 , 1468 thom c. , peek j.e.g . , putman m.e . , heiles c. , peek k.m.g . & wilhelm , r. , 2008 , apj , 684 , 364 vittorini v. , shankar f. & cavaliere , a. , 2005 , mnras , 363 , 1376 wakker b.p . & van woerden , h. , 1997 , ara&a , 35 , 217 white s.d.m . & frenk c.s . , 1991 , apj , 379 , 52 xu y .- d . , narayan r. , quataert e. , yuan f. & baganoff f. k. , 2006 , apj , 640 , 319

we propose a new mechanism for the delivery of gas to the heart of galactic nuclei . we show that warm halo clouds must periodically impact galactic centers and potentially deliver a large ( @xmath0 ) mass of gas to the galactic nucleus in a singular event . the impact of an accreting warm halo cloud originating far in the galactic halo can , depending on mixing , produce a nuclear starburst of low metallicity stars as well as low luminosity accretion onto the central black hole . based on multiphase cooling around a @xmath1cdm distribution of halos we calculate the nuclear impact rate , the mass captured by the central black hole and the fraction of active nuclei for impacting cloud masses in the range @xmath2 . if there is moderate braking during cloud infall , our model predicts an average fraction of low luminosity active nuclei consistent with observations .

1006.0169

attempts for a general classification of halo states were started early in the development of the field , see e.g. @xcite , and have recently led to the suggested definition of halo states @xcite as having more than 50% probability of being in a cluster configuration where more than 50% of this probability should be in a classically forbidden region . this definition is straightforward to apply for a two - body system , where one basically has to find the outer classical turning point in the radial motion . the three - body systems are more challenging @xcite and it is the purpose of this paper to discuss their possible modes of behaviour . one obvious point that needs clarification is how to generalize the outer classical turning point , that also can be used @xcite to scale different halo systems so that e.g. nuclear and molecular halos can be compared in dimensionless units . for the interesting special case of efimov states a universal scaling property predicting one efimov state from the previous one has been developed , see @xcite . another point that has only been discussed briefly in the literature so far is how the transition from a three - body to a two - body state takes place as the binding potentials are changed @xcite . connected to this is the classification of possible three - body configurations into borromean @xcite , tango @xcite or other bound states . to clarify the principles we shall mainly consider systems where all particles are in relative @xmath0-waves , which dominate at large distances . results of more realistic calculations will also be given for @xmath1li and the hypertriton . there is naturally more variability in three - body systems than in two - body systems . the three two - body subsystems might all play a role in the asymptotic region , so even in the weak - binding limit we should expect several types of behaviour to be possible , even for the simplest case of only relative @xmath0-waves . systems with zero and one bound subsystem are called borromean and tango systems , respectively . in a two - body system the classical turning points are found by equating the total energy and the potential energy . in principle we can generalize this to three particles in a specified quantum state described by the wave function @xmath2 and find the probability for being in the non - classical region as @xmath3 , where the integration is confined to regions with potential energy larger than the total energy . this will be much harder to calculate than for a two - body system and could therefore be a rather impractical condition . furthermore , we are not assured that the wavefunction will behave in a simple way in the non - classical region ; there might be configurations where one pair is in a forbidden region whereas the third particle is in an allowed region . we shall therefore here rather explore the possibility of generalizing the classical turning point into a three - body scaling radius and discuss two different types of `` derivation '' of it . we use hyperspherical coordinates to describe the relative motion of three particles with masses @xmath4 , where @xmath5 . the total mass is @xmath6 , the individual momenta and coordinates are @xmath7 and @xmath8 and the hyperradius @xmath9 is defined by @xmath10 where @xmath11 is the center of mass coordinate and @xmath12 is a mass unit chosen for convenience . the hyperradius is an average radius coordinate , applicable to all three - body systems and useful for all angular momenta and for non - spherical systems . the total mean square radius @xmath13 is then via the particle sizes @xmath14 given by @xmath15 it is natural to choose a three - body scaling radius @xmath16 so that the arbitrary mass @xmath12 enters in the same way in @xmath9 and @xmath16 and all measures of size , that typically rely on their ratio , become independent of @xmath12 . the two - body scaling property relating size and binding energy can then be generalized if @xmath17 is an almost single - valued function of another dimensionless quantities @xmath18 , where @xmath19 is the three - body binding energy @xcite . such a scaling property is clearly an advantage when searching for a general definition of the scaling radius @xmath16 . the relation should apply for systems consisting of particles with widely different masses and ranges of interactions . a tempting definition of @xmath16 is to maintain the complete analogy to @xmath9 , i.e. @xmath20 where @xmath21 is interpreted as the equivalent square well radius of the system consisting of particle @xmath22 and @xmath23 . as argued in @xcite this definition is convenient in descriptions of three - body systems intermediate between two and three - body scaling . the choice of hyperspherical coordinates leads to effective radial potentials obtained by adiabatic expansion or by averaging in other ways over the remaining set of angular coordinates . the classically allowed regions for such one - dimensional potentials are easily defined . however , they could be completely different from the regions where the three pairs of particles are located in their classically allowed regions defined by the corresponding two - body potentials . in fact , it is entirely possible to have classical motion in the hyperradial coordinate while the system is in non - classical regions in real space . hyperradial turning points are therefore useless as definitions of quantum halos and often without any resemblance to the length unit @xmath16 defined above in terms of two - body properties . it is instructive to consider the general behaviour of the hyperradial potential . for zero - range two - body potentials the only energy available through combination of parameters is @xmath24 , where @xmath25 is a combination of reduced masses . with this large distance behaviour any number of solutions is possible , ranging from zero to the infinitely many efimov states . for finite range interactions the ranges or alternatively the scattering lengths provide additional length parameters and the hyperradial potentials could approach zero faster than @xmath26 . from @xcite we obtain the large distance behaviour of the dominating lowest @xmath0-wave adiabatic potential as @xmath27 where the reduced mass is @xmath28 , the @xmath0-wave scattering length is @xmath29 and the average scattering length is defined as @xmath30 the chosen normalization reduces @xmath31 to the common @xmath29 when all three particles are identical . thus the large distance behavior of @xmath32 in eq.([e66 ] ) is @xmath33 which is reached when @xmath9 is comparable to @xmath34 , see @xcite . scaling can be shown analytically to occur for the special case of a @xmath35 wavefunction ( @xmath36 being the hypermoment ) for square well two - body potentials of depth @xmath37 and radius @xmath21 . here the effective hyperradial potential @xmath38 can be obtained @xcite as @xmath39 which is valid when @xmath9 is several times larger than any of the square well radii . the square well two - body @xmath0-wave scattering length is given by @xmath40 , where @xmath41 is the zero energy wave number inside the square well , i.e @xmath42 . then @xmath43 is a specific function of @xmath44 , which approaches the constant @xmath45 when @xmath29 becomes much larger than @xmath21 . thus the effective radial potential in eq.([e36 ] ) approaches the form @xmath46 where the definition of @xmath16 , @xmath47 reduces to that of eq.([e22 ] ) for identical masses and radii . note that eq.([e56 ] ) , in contrast to eq.([e22 ] ) , employs a linear ( not squared ) summation . for systems where the two - body scattering lengths are large , most of the wavefunction will reside in the region where eq.([e46 ] ) holds for @xmath35 and one clearly has scaling . a desirable property of the scaling radius @xmath16 would be that classically forbidden regions on average are given roughly by @xmath48 . we can consider this question briefly for the specific @xmath35 solutions , where the probability @xmath49 for particles @xmath22 and @xmath23 being inside their square well radius @xmath21 for large @xmath9 is @xmath50 this probability is 1/2 for @xmath51 which indicates that systems reside mainly in classically forbidden regions if their mean square radii are somewhat larger than @xmath52 . we do not believe that the @xmath35 wave functions are realistic solutions for the weakly bound systems since they do not allow for any form of correlations between the particles . for very loosely bound systems where @xmath31 is much larger than @xmath16 the potential will fall off as @xmath26 between @xmath16 and @xmath31 , i.e. slower than @xmath33 as for a pure @xmath35 solution . the scaling radius will be used to look for scaling properties of three - body systems in the weak binding limit . we are aiming for as universal properties as possible , but the rather different types of structure that occur in three - body systems means we first have to look at how they can be classified . in fig.[fig1 ] we illustrate the various stability regions as function of scattering lengths . at the border between tango and borromean regions one subsystem has a bound state with zero energy . on the thick horizontal line two identical subsystems have bound states of zero energy and the infinitely many efimov states arise . the figure only shows the region where @xmath53 , appropriate e.g. when particles 1 and 2 are identical , but already indicates that two distinct types of transitions can occur as a function of @xmath54 , namely moving from the tango region ( the 23 and 13 subsystems are unbound ) from left to right either directly into unbound systems or through the borromean region . to see this in detail we now turn to the numerical results and show in fig . [ fig2 ] the region of weak binding for a number of both schematic and realistic examples . through eq.([e21 ] ) this scaling plot displays the mean square radius of the system in units of @xmath52 versus the dimensionless binding energy @xmath55 @xcite , where @xmath19 is the three - body binding energy . zero binding energy corresponds to three non - interacting particles at rest . we shall in this letter use the scaling radius defined in eq.([e56 ] ) , where @xmath21 is the radius of the square well with the same scattering length and effective range as the actual two - body potential . changing to the other definition in eq.([e22 ] ) has essentially no effect on the hypertriton examples , whereas the different @xmath1li points move by about 30% . the resulting figure is practically indistinguishable from the present one , except that the perfect scaling for @xmath35 is violated slightly . there is no strong preference for any of these two definitions of @xmath16 . the first analysis in @xcite assumed that simple systems at least aymptotically would correspond to a single value of the hypermoment @xmath36 . we have artificially restricted the wavefunctions for different systems to contain only @xmath36=0 terms and find that these systems , as argued above , indeed do scale and lie on a single curve . however , the points corresponding to more realistic calculations lie above this @xmath36=0 curve and can be grouped and understood as follows . in the realistic @xmath56h - structure the neutron and proton are almost in a deuteron configuration ( close circles ) whereas the @xmath57-particle is far outside but still bound by about 0.14 mev . thus this is a tango system and the large size is almost entirely due to the @xmath57-deuteron extension , i.e. of two - body character . by using a simplified form for the neutron - proton interaction with only @xmath0-waves the position is sligthly higher than the realistic point . by decreasing this neutron - proton attraction the binding decreases moderately while the radius drastically increases ( open circles ) . this off hand surprising property can be understood from fig . [ fig1 ] by moving horizontally from the tango region to the right , increasing @xmath58 . for sufficiently weak initial binding , i.e. large values of @xmath59 , the threshold for @xmath57-deuteron binding is reached instead of the borromean region as we decrease the attraction of the only bound subsystem . thus the diverging radius is due to @xmath57-deuteron two - body threshold and not related to the neutron - proton threshold . decreasing the @xmath0-wave attraction between neutron and proton while maintaining all other parts of the realistic interaction ( filled circles ) , now leads from tango into the borromean region of fig.[fig1 ] , i.e. the three - body system remains bound even after the neutron - proton potential is too weak to form a bound two - body state . the result is a completely different curve in fig . [ fig2 ] much more in agreement with the logarithmic divergence expected from @xcite , although still significantly above the @xmath35 curve . we stress that nothing drastically happens at the arrow where the deuteron becomes unbound ( scaled binding energy about 0.08 ) . the realistic point for the borromean nucleus @xmath1li is at a scaled binding energy of 0.13 in fig . [ fig2 ] . by decreasing the @xmath0-wave attraction in the neutron-@xmath60li systems , i.e. going vertically upwards in fig . [ fig1 ] approaching the threshold , the binding decreases and the radius increases corresponding to a logarithmically diverging curve in fig . [ fig2 ] ( filled triangles ) eventually approaching that of the borromean hypertriton example . a similar behaviour is seen for a hypothetical hypertriton with only @xmath0-waves included ( filled squares ) but with a slightly increased nucleon-@xmath61 attraction . then we move horizontally from the tango to the borromean region . for a borromean system of three particles ( stars ) where the three masses and scattering lengths are equal or differ substantially , we again follow the same trajectory by changing one scattering length . by increasing the @xmath0-wave attraction in the neutron-@xmath62li systems , i.e. vertically approaching the efimov limit in fig . [ fig1 ] , larger binding and smaller radius result as expected . the resulting non - borromean @xmath1li ground state resembles more and more an ordinary nuclear state . however , at some point an excited state appears , i.e. the first efimov state ( open triangles ) . at first it is very weakly bound and close to the efimov line . as the attraction and the three - body binding energy increases the size at first decrease and then `` turn around '' and increase again as the binding energy of the two bound two - body subsystems approach and finally overtake the three - body binding energy @xcite . as the two - body threshold is approached the third excited state ( second efimov state ) should appear on the dashed line . unfortunately the mass ratio ( neutron to @xmath60li ) is relatively small and the next state is many orders of magnitude outside the scale of the plot , i.e. outside the reach of experimental as well as most numerical techniques . one remaining question is the approach of the two sets of points , related to hypertriton and @xmath1li , at very small binding energy . both these sets arise by approach of ( different ) thresholds for borromean binding in fig . [ fig1 ] . when the majority of the radial wave function is located in the tail of the adiabatic potential in eq.([e66 ] ) then @xmath31 is a decisive length parameter . one could then erroneously be led to conclude that @xmath31 determines the size - binding relations and the threshold for borromean binding in fig . however , the absolute scale of the energy can not be determined from one parameter alone , the short distance behaviour is also indispensably necessary @xcite . to understand this we imagine that we reduce the ranges of the potentials while increasing the strengths to maintain the scattering lengths . this zero - range limit results in the thomas collapse and infinitely many bound states at small distances @xcite . to avoid this unphysical behaviour some kind of renormalization is needed . the simplest is to maintain the large distance behavior while only using the two - body potentials down to a distance below which the structure is uninteresting . the two - body results can then be expressed in units of such a rather arbitrary length parameter . the effect on a three - body system can then be anticipated mimicked by a similar renormalization by use of a hyperspherical length unit . however , this is precisely the content of fig . [ fig2 ] where we used @xmath16 as the scale parameter . the curves in fig . [ fig2 ] coinciding at small binding are therefore an almost universal curve as indicated by the convergence in the figure . it would perhaps be rather fortuitous if this simple renormalization procedure in hyperradius results in a universal rescaled curve . the average scattering lengths for the four cases in the weak binding limit vary from 4 fm to 20 fm whereas the ratio @xmath63 varies between 1.7 and 4.2 . however , @xmath31 only determines the adiabatic potential for distances smaller than the scattering lengths and larger than the potential ranges . these conditions are not well fulfilled for the examples in fig . thus constant @xmath63 should not necessarily arise , since @xmath31 should be replaced by a complicated function of all three @xmath29 . this is not contradicting the numerical results in fig . [ fig2 ] which is obtained without use of @xmath31 . thus the emerging numerical curve is almost universal providing rather well defined scaling properties . three - body quantum halos were previously believed to appear along the @xmath35 curve @xcite . this conclusion was reached by omitting constant terms compared to the leading order logarithmically diverging term . however , the present more refined analysis reveals the correct higher lying universal curve approached in the weak binding limit . in fact asymptotically these curves differ by a constant factor . in any case three - body quantum halos can appear above the @xmath35 curve , i.e. allowed in a larger window and being a factor of 23 larger than expected . one implication is that @xmath35 wavefunctions can not be used in the weak binding limit due to the coherent large - distance contributions from more than one subsystem . for stronger binding a @xmath35 basis is simply incomplete . in fig . [ fig1 ] ground state three - body halos appear just below the line separating the borromean from the unbound region . on top of this the efimov effect can also give `` super - halos '' in excited states . ground state two - body halos can appear just below the line separating the tango from the unbound region . in fig . [ fig2 ] , three - body halos where the three - body dynamics dominate will appear on or below the ( almost ) universal curve . all systems falling above are influenced by an effective two - body threshold and correspond either to the efimov line ( caused by the threshold in two subsystems ) or the tango - unbound line in fig . we stress that all quantities used to place a system in fig . [ fig2 ] in principle can be measured experimentally and that our results therefore can be used to classify the three - body nature of realistic systems .

the different kinds of behaviour of three - body systems in the weak binding limit are classified with specific attention to the transition from a true three - body system to an effective two - body system . for weakly bound borromean systems approaching the limit of binding we show that the size - binding energy relation is an almost universal function of the three @xmath0-wave scattering lengths measured in units of a hyperradial scaling parameter defined as a mass weighted average of two - body equivalent square well radii . we explain why three - body halos follow this curve and why systems appearing above reveal two - body substructures . three - body quantum halos 2 - 3 times larger than the limit set by zero hypermoment are possible .

nucl-th0212088

in reference @xcite , barrett and crane have introduced a model for quantum general relativity ( gr ) . the model is based on the topological quantum @xmath1 bf theory , and is obtained by adding a quantum implementation of the constraint that reduces classical bf theory to euclidean gr @xcite . to make use of the barrett - crane construction in quantum gravity , two issues need to be addressed . first , in order to control the divergences in the sum defining the model , the barrett - crane model is defined in terms of the @xmath2-deformation of @xmath1 . in a realistic model for quantum euclidean gr , one would like the limit @xmath3 to be well defined . second , the barrett - crane model is defined over a fixed triangulation . this is appropriate for a topological field theory such as bf theory , which does not have local excitations . but the implementation of the bf - to - gr constraint breaks topological invariance and frees local degrees of freedom . the restriction of the model to a fixed discretization of the manifold can therefore be seen only as an approximation . in order to capture all the degrees of freedom of classical gr , and restore general covariance , an appropriate notion of sum over triangulations should be implemented ( see for instance @xcite ) . a novel proposal to tackle this problem is provided by the field theory formulation of spin foam models @xcite . in this formulation , a full sum over arbitrary spin foams ( and thus , in particular , over arbitrary triangulations ) is naturally generated as the feynman diagrams expansion of a scalar field theory over a group . the sum over spinfoams admits a compelling interpretation as a sum over 4-geometries . the approach represents also a powerful tool for formal manipulations and for model building : examples of this are ooguri s proof of topological invariance of the amplitudes of quantum bf theory in @xcite and the definition of a spinfoam model for lorentzian gr in @xcite . using such framework of field theories over a group , a spinfoam model for euclidean quantum gr was defined in @xcite . this model modifies the barrett - crane model in two respects . first , it is not restricted to a fixed triangulation , but it naturally includes the full sum over arbitrary spinfoams . second , the natural implementation of the bf - to - gr constraint in the field theory context fixes the prescription for assigning amplitudes to lower dimensional simplices , an issue not completely addressed in the original barrett - crane model . this same prescription for lower dimensional simplices amplitudes ( but in the context of a fixed triangulation ) was later re - derived by oriti and williams in @xcite , without using the field theory . the model introduced in @xcite appeared to be naturally regulated by those lower dimensional amplitudes . in particular , certain potentially divergent amplitudes were shown to be bounded in @xcite . these results motivated the conjecture that the model could be finite . that is , that all feynman diagrams might converge . in this letter we prove this conjecture . this paper is not self - contained : familiarity with the formalism defined in @xcite is assumed . the definition of the model is summarized in the section ii ; for a detailed description of the model we refer to @xcite . in section iii , a series of auxiliary results is derived . the proof of finiteness is given in section iv . consider the fundamental representation of @xmath1 , defined on @xmath4 , and pick a fixed direction @xmath5 in @xmath4 . let @xmath6 be the @xmath7 subgroup of @xmath1 that leaves @xmath5 invariant . the model is defined in terms of a field @xmath8 over @xmath9 , invariant under arbitrary permutations of its arguments . we define the projectors @xmath10 and @xmath11 as @xmath12 where @xmath13 , and @xmath14 . the model introduced in @xcite is defined by the action @xmath15=\int dg_i \left [ p_{g } \phi(g_i ) \right]^2 + \frac { 1}{5 ! } \int dg_i \left [ p_{g}p_{h}\phi(g_i ) \right]^5,\ ] ] where @xmath16 , and the fifth power in the interaction term is notation for @xmath17 ^ 5:=\phi(g_1,g_2,g_3,g_4)\ \phi(g_4,g_5,g_6,g_7)\ \phi(g_7,g_3,g_8,g_9)\ \phi(g_9,g_6,g_2,g_{10 } ) \ \phi(g_{10},g_8,g_5,g_1).\ ] ] @xmath11 projects the field into the space of gauge invariant fields , namely , those such that @xmath18 for all @xmath14 . the projector @xmath10 projects the field over the linear subspace of the fields that are constants on the orbits of @xmath6 in @xmath1 . when expanding the field in modes , that is , on the irreducible representations of @xmath1 , this projection is equivalent to restricting the expansion to the representations in which there is a vector invariant under the subgroup @xmath6 ( because the projection projects on such invariant vectors ) . the representations in which such invariant vectors exist are the simple " , or balanced " , representations namely the ones in which the spin of the self dual sector is equal to the spin of the antiselfdual sector can be labeled by two integers @xmath19 . in terms of those integers , the dimension of the representation is given by @xmath20 . simple representations are those for which @xmath21 . ] . in turn , the simple representations are the ones whose generators have equal selfdual and antiself dual components , and this equality , under identification of the @xmath1 generator with the @xmath22 field of @xmath23 theory is precisely the constraint that reduces @xmath23 theory to gr . alternatively , this constraint allows one to identify the generators as bivectors defining elementary surfaces in 4d , and thus to interpret the coloring of a two - simplex as the choice of a ( discretized ) 4d geometry @xcite . using the peter - weyl theorem one can write the partition function of the theory @xmath24}\ ] ] as a perturbative sum over the amplitudes @xmath25 of feynman diagrams @xmath26 . this computation is performed in great detail in @xcite , yielding @xmath27 the first summation is over pentavalent 2-complexes @xmath26 , defined combinatorially as a set of faces @xmath28 , edges @xmath29 and vertices @xmath30 , and their boundary relations . the second sum is over simple @xmath1 representations @xmath31 coloring the faces of @xmath26 . @xmath32 is the dimension of the simple representation @xmath31 . the amplitude @xmath33 is a function of the four colors that label the corresponding faces bounded by the edge . it is explicitly given by @xmath34 where @xmath35 is the dimension of the space of the intertwiners between the four representations @xmath36@xcite . the vertex amplitude @xmath37 is the barrett - crane vertex amplitude , which is a function of the ten colors of the faces adjacent to the 5-valent vertex of @xmath26 . the barrett - crane vertex amplitude can be written as a combination of @xmath38 symbols . however , as it was shown by barrett in @xcite , it can also be express as an integral over five copies of the 3-sphere a representation with a nice geometrical interpretation . this representation is better suited for our purposes so we give it here explicitly @xmath39 where @xmath40 denotes the invariant normalized measure on the sphere . if we represent the points in the 3-sphere as unit - norm vectors @xmath41 in @xmath4 , and we define the angle @xmath42 by @xmath43 , then the kernel @xmath44 is given by @xmath45 this is a smooth bounded functions on @xmath46 , with maximum value @xmath47 . our task is to prove convergence of the feynman integrals of the theory . in the mode expansion , potential divergences appear in the sum over representations @xmath31 . therefore we need to analyze the behavior of vertex and edge amplitudes for large values of @xmath31 . an arbitrary point @xmath48 can be written in spherical coordinates as @xmath49 where @xmath50 , @xmath51 , and @xmath52 . the invariant normalized measure in this coordinates is @xmath53 using the gauge invariance of the vertex , the invariance of the three sphere under the action of @xmath1 and the normalization of the invariant measure , we can compute ( [ vv ] ) by dropping one of the integrals and fixing one point on @xmath54 , say @xmath55 . thus equation ( [ vv ] ) becomes @xmath56 where @xmath57 is the normalized measure on the 2-sphere @xmath58constant . now we bound the barrett - crane amplitude using that @xmath59 , namely @xmath60 the argument is obviously independent of the choice of the six colors in ( [ depe ] ) . weaker versions of the bound in which more colors are included also hold . in particular if we directly bound the @xmath61 s in ( [ vv ] ) we obtain that the absolute value of the amplitude is bounded by the square root of the product of the ten dimensions . more in general , let @xmath62 , with @xmath63 taking the values @xmath64 , be an arbitrary subset of @xmath65 with @xmath66 elements . then the following bound holds for any @xmath62 @xmath67 for @xmath68 we recover ( [ depe ] ) . in @xcite , barrett and williams have analyzed the asymptotic behavior of the oscillatory part of the amplitude , in connection to the classical limit of the theory . we add here some information on the asymptotic behavior of the magnitude of ( [ vv ] ) . the results of this paragraph will not be used in the rest of the paper ; we present them since they follow naturally from our previous considerations . equation ( [ sisi ] ) can be rewritten as @xmath69 where @xmath70 is a smooth bounded function in @xmath71\in \re $ ] , since @xmath72 is given by an integral of smooth and bounded functions on a compact space . therefore , the following limit holds @xmath73 the same argument can be used to prove the following stronger result @xmath74 where , in the limit , the four colors are taken simultaneously to infinity . clearly , these equations are valid for any choice of the @xmath75 s . finally , it is easy to verify the following inequalities for the dimension of the space of intertwiners between four representations converging at an edge ( see @xcite ) @xmath76 which holds for any @xmath77 . using this expression , we can construct weaker bounds in the spirit of ( [ cota ] ) . we define the set @xmath78 ( @xmath79 ) as an arbitrary subset of @xmath80 with @xmath81 elements . then the following inequalities hold for any @xmath82 @xmath83 we have now all the tools for proving the finiteness of the model . we will construct a finite bound for the amplitudes @xmath25 , defined in ( [ z ] ) , of arbitrary feynman diagrams of the model . inserting the inequalities ( [ dd ] ) and ( [ cota ] ) , both with @xmath84 , in the definition of @xmath25 , we obtain a bound for the amplitude of an arbitrary pentavalent 2-complex @xmath26 . namely , @xmath85 where @xmath86 denotes the number of edges of the face @xmath28 . the term @xmath87 in ( [ lala ] ) comes from various contributions . first , we have @xmath32 from the face amplitude . second , we have @xmath88 and @xmath89 from the denominators of the @xmath86 edge amplitudes ( [ xii ] ) and the bounds for the corresponding numerators ( [ dd ] ) respectively . finally , we have @xmath89 from the bounds for the @xmath86 vertex amplitudes ( [ cota ] ) . if the 2-complex contains only faces with more than one edge , then the previous bound for the amplitude is finite . more precisely , if all the @xmath86 s are such that @xmath90 then @xmath91 , and using the fact that @xmath92 the sum on the rhs of the last equation converges . on the other hand , if some of the @xmath86 are equal to 1 , then the right hand side of ( [ lala ] ) diverges , and therefore for this case we need a stricter bound , involving stronger inequalities . this can be done as follows . notice that every time a 2-complex contains a face whose boundary is given by a single edge , the same edge must be part of the boundary of another face , bounded by more than one edge . in fig . 1 an elementary vertex of a 2-complex containing such a face is shown . the thick lines represent edges converging at the vertex , each of them is part of the boundary of four faces . to visualize those faces we have drawn in thin lines the intersection of the vertex diagram with a 3-sphere . there is a face bounded by a single edge . its intersection with the sphere is denoted by @xmath93 . notice that the surface intersecting the sphere in @xmath94 will have a boundary defined by at least two edges . notice also that a single vertex can have a maximum of two such peculiar faces . therefore , using ( [ dd ] ) for @xmath95 we can choose to bound the numerator in ( [ xii ] ) with the colors corresponding to the three adjacent faces like @xmath94 in fig . 1 . if @xmath96 denotes the color of the face @xmath93 then we construct the bound for the amplitude using @xmath97 one of the square roots would be sufficient to bound the edge amplitude , but the symmetry in the previous expression simplifies the construction of the bound for the amplitude of an arbitrary 2-complex . then use ( [ cota ] ) for @xmath95 ( or @xmath98 ) to bound the vertex amplitude corresponding to a vertex containing one ( respectively two ) face(s ) whose boundary is given by only one edge . in this way , we can exclude the color corresponding to these `` singular '' faces from the bounds corresponding to the vertex and the denominator of the edge amplitude . thus these faces contribute to the bound as @xmath32 ( face amplitude ) times @xmath99 ( from the denominator of the single edge amplitude ) , i.e. , as @xmath100 . we keep using ( [ dd ] ) , and ( [ cota ] ) for @xmath84 for faces with @xmath101 and vertices containing no faces with @xmath102 . if we denote by @xmath103 and @xmath104 the set of faces with more than one edge and one edge respectively , the general bound is finally given by @xmath105 where @xmath106 denotes the number of faces in the 2-complex @xmath26 , and @xmath107 denotes the riemann zeta function . this concludes the proof of the finiteness of the amplitude for any 2-complex @xmath26 @xmath108 equation ( [ oui ] ) proves that there are no divergent amplitudes in the field theory defined by ( [ tope ] ) . this field theory was defined in @xcite as a model for euclidean quantum gravity based on the implementations of the constraints that reduce @xmath1 bf theory to euclidean gr . the corresponding bf topological theory is divergent after quantization : regularization is done ad - hoc by introducing a cut - off in the colors or , more elegantly , by means of the quantum deformation of the group ( @xmath109 ) . remarkably , the implementation of the constrains that give the theory the status of a quantum gravity model automatically regularizes the amplitudes . another encouraging result comes from the fact that according to equation ( [ oui ] ) contributions of feynman diagrams decay exponentially with the number of faces . this might be useful for studying the convergence of the full sum over 2-complexes . to carlo rovelli . l crane , _ topological field theory as the key to quantum gravity , _ proceedings of the conference on knot theory and quantum gravity , riverside , j baez ed 1992 . j barret _ quantum gravity as topological quantum field theory _ j math phys 36 ( 1995 ) 6161 - 6179 . l smolin , _ linking topological quantum field theory and nonperturbative quantum gravity _ , j math phys 36 ( 1995 ) 6417 . j baez , _ spin foam models _ , class quant grav 15 ( 1998 ) 1827 - 1858 ; gr - qc/9709052 . _ an introduction to spin foam models of quantum gravity and bf theory _ , in `` geometry and quantum physics '' , eds helmut gausterer and harald grosse , lecture notes in physics ( springer - verlag , berlin ) ( 2000 ) ; gr - qc/9905087 .

we prove that a certain spinfoam model for euclidean quantum general relativity , recently defined , is finite : all its all feynman diagrams converge . the model is a variant of the barrett - crane model , and is defined in terms of a field theory over @xmath0 . -.25 in

gr-qc0011058

soft glassy systems such as foams , colloidal suspensions , emulsions , polymers , glasses @xcite , and granular materials @xcite have a strongly non - linear response to an external perturbation . in such systems , the relation between the stress @xmath0 and the strain rate @xmath1 characterizes the system behavior . although it is known that the relations are diverse and specific to individual systems , a universal law for a certain class of systems may exist . in particular , in sheared granular materials under constant pressure @xmath2 , one of the authors ( hatano ) has found a relation @xcite @xmath3 with @xmath4 by a numerical experiment using the discrete element method . here , @xmath5 is the maximum diameter of the particles ( their diameters are uniformly distributed in the range @xmath6 $ ] ) and @xmath7 is the mass of the particles @xcite . as demonstrated in fig . [ fig : hatano ] , the exponent @xmath8 is not inconsistent with @xmath9 in the range @xmath10 . surprisingly , the power - law behavior given in eq . ( [ hatano : power ] ) is observed in the cases that @xmath11 and @xmath12 , where @xmath13 represents the young modulus of the particle . for example , one can experimentally obtain the power - law behavior under the constant pressure @xmath14mpa by using polystyrene with @xmath15gpa . since @xmath16 corresponds to the shear rate @xmath17/sec in this example , the shear condition leading to eq . ( [ hatano : power ] ) is experimentally possible . as a function of @xmath18 . this result was obtained for a model similar to that explained in the text . the main differences are as follows : ( i ) the top boundary in the @xmath19 direction is modified so as to maintain a constant pressure and ( ii ) the shear is applied directly from the moving layer at the top and the bottom . ( see the inset . ) the parameter values are as follows : @xmath20 , @xmath21 , and @xmath22 . @xmath23 ( data 1 ) , @xmath24 ( data 2 ) , and @xmath25 with @xmath26 ( data 3 ) . furthermore , the square and circle symbols represent the constant pressure data obtained from figs . [ fig : sg ] and [ fig : pg ] , where @xmath27 ( square symbol ) and @xmath28 ( circle symbol ) . ] stimulated by this result , in the present paper , we consider the power - law behavior of stress - strain rate relations in sheared granular materials by investigating a model granular system with the lees - edwards boundary conditions . in this idealized system , we demonstrate that there is a critical volume fraction at which the shear stress and the pressure ( normal stress ) behave as power - law functions of the shear strain rate in the limit @xmath29 . from these power - law behaviors , we derive the scaling relation @xmath30 in the limit @xmath31 at the critical volume fraction . note that this critical condition does _ not _ correspond to a constant pressure . we then present a simple interpretation of eq . ( [ hatano : power ] ) for the system under constant pressure . here , we describe our computational model . the system consists of @xmath32 spheres of mass @xmath7 in a three - dimensional rectangle box whose lengths are @xmath33 , @xmath34 , and @xmath35 along the @xmath36 , @xmath37 , and @xmath19 directions , respectively . in order to realize an average velocity gradient @xmath1 in the @xmath19 direction and average velocity in the @xmath36 direction , we impose the lees - edwards boundary conditions @xcite . the particle diameters are @xmath38 , @xmath39 , @xmath40 and @xmath5 each of which is assigned to @xmath41 particles . when the distance between two particles is less than the sum of their radii , @xmath42 and @xmath43 , an interaction force acts on each of them . this force comprises an elastic repulsion force @xmath44 and the viscous dissipation force @xmath45 , where @xmath46 and @xmath47 represent the relative distance and velocity difference of the interacting particles , respectively . for simplicity , we do not consider the tangential force between the interacting particles . we study the specific case where @xmath48 , @xmath49 and @xmath22 . the control parameters in this system are the volume fraction @xmath50 with the @xmath51th particle diameter @xmath52 , and the dimensionless shear rate @xmath53 . we then calculate the dimensionless shear stress @xmath54 and the dimensionless pressure ( in the @xmath19 direction ) @xmath55 . @xcite as the calculation method for @xmath56 and @xmath57 . note that @xmath58 provides an approximate value of the young modulus of particles . we express the dependence of @xmath59 and @xmath57 on @xmath60 as @xmath61 and @xmath62 , respectively . figures [ fig : sg ] and [ fig : pg ] display these functions with respect to @xmath63 for several values of @xmath64 @xcite . these graphs clearly show that there exists a critical volume fraction @xmath65 at which the power law behaviors are observed as follows : @xmath66 in the limit @xmath67 @xcite . the values of the exponents will be discussed later . here , it is worthwhile noting that similar graphs were obtained in ref . @xcite with the argument on the effect of finite elastic modulus . indeed , these graphs in this reference suggest the existence of the critical state , although the power - law behavior was not mentioned explicitly . upon numerical verification , we found that the critical volume fraction corresponds to the jamming transition point defined as the volume fraction beyond which a finite yield stress appears @xcite . in this paper , we do not argue the nature of the jamming transition , but focus on the power - law behaviors given in eqs . ( [ scaling:1 ] ) and ( [ scaling:2 ] ) . note that a similar critical state was obtained for a sheared glassy system @xcite . as a function of @xmath63 for several values of @xmath64 . the thick solid line represents @xmath68 that is estimated from our theoretical argument . note that the bagnold scaling @xcite is observed for the case in which @xmath69 and @xmath70 . ] as a function of @xmath63 for several values of @xmath64 . the thick solid line represents @xmath71 that is estimated from our theoretical argument . furthermore , the lines @xmath72 const . are drawn to help us to understand the functional form of @xmath73 over @xmath18 . ] the main idea in our theoretical argument is to consider dimensional analysis with kinematic temperature @xmath74 defined as @xmath75 where @xmath76 denotes the velocity of the @xmath51-th particle . although @xmath74 is not a parameter of the system but is determined by @xmath63 and @xmath64 , it is considered that physical processes in granular systems are described in terms of the kinematic temperature @xcite . in particular , the time scale of energy dissipation is assumed to be determined as @xmath77 . one can verify the validity of this assumption by investigating the energy balance equation in the steady state @xcite : @xmath78 which is rewritten as @xmath79 figure [ fig : haff ] indicates that eq . ( [ eb : dless ] ) is plausible as the first theoretical attempt , although a slight deviation is observed . based on this result , hereafter , we assume that the time scale of the energy dissipation is given by @xmath77 . versus @xmath80 for two cases in which @xmath64 is close to the critical volume fraction @xmath65 . the solid line corresponds to eq . ( [ eb : dless ] ) . since we find the best fitting to be @xmath81 , there is a slight deviation from eq . ( [ eb : dless ] ) . ] now , we consider a set of dimensionless constants using the energy dissipation ratio . let us recall that we have three independent dimensionless parameters @xmath82 . we want to use dimensionless parameters each of which represents the ratio of a time scale of a physical process to that of energy dissipation . it is then reasonable that the following two time scales are important : the inverse of the shear rate @xmath83 and the relaxation time @xmath84 of the elastic displacement during the interaction between two particles . thus , instead of @xmath63 and @xmath85 , we introduce @xmath86 then , we wish to determine the functional forms of @xmath87 and @xmath88 in the limit @xmath67 at the critical volume fraction . in order to restrict the possible forms of the functions , we further assume that @xmath0 does not depend on @xmath89 . noting that @xmath90 for @xmath67 , the function becomes @xmath91 where @xmath92 represents the direct contribution of @xmath63 that is not expressed through the dependence of @xmath74 on @xmath63 . here , we consider @xmath92 on the basis of the theory determining the behavior of shear stress near the critical state for a dense colloidal suspension @xcite . according to this theory , the critical behavior is described by an order parameter equation that is similar to the ginzburg - landau equation for magnetization under a magnetic field . assuming that this description is valid for the present problem , we write @xmath93 with numerical constants @xmath94 and @xmath95 . note that , in cases of magnetic materials , @xmath92 and @xmath96 correspond to magnetization and a magnetic field , respectively . because the first term vanishes at the critical volume fraction , we obtain @xmath97 following these assumptions , we can calculate the exponents @xmath98 . concretely , eq . ( [ case1 ] ) with eq . ( [ amp ] ) becomes @xmath99 combining eq . ( [ amp2 ] ) with eq . ( [ eb : dless ] ) , we derive @xmath100 the substitution of this into eq . ( [ amp2 ] ) with eqs . ( [ xi1 ] ) and ( [ xi2 ] ) yields @xmath101 thus , we obtain @xmath102 in eq . ( [ scaling:1 ] ) , which is consistent with the numerical experiment as shown in fig . [ fig : sg ] . in a similar manner , we obtain @xmath103 under the assumption that @xmath104 . this assumption implies that @xmath57 does not depend on @xmath96 because the normal stress is not directly influenced by the shear rate . as shown in fig . [ fig : pg ] , @xmath105 in eq . ( [ scaling:2 ] ) is consistent with the numerical experiment . we next study the power - law behavior observed in the system under constant pressure on the basis of the results obtained above . first , from eqs . ( [ scaling:1 ] ) and ( [ scaling:2 ] ) , which are valid at the critical volume fraction @xmath65 , we derive eq . ( [ hatano : power ] ) with @xmath106 using the values @xmath102 and @xmath105 , we obtain @xmath107 . this value of @xmath8 is consistent with the numerical experiment . however , it should be noted that the scaling relation is obtained at the critical volume fraction , not for systems under constant pressure . in order to discuss quantitatively the behavior of the system under constant pressure , we denote the volume fraction and the shear stress measured on this system as @xmath108 and @xmath109 . we then wish to determine these functions from figs . [ fig : sg ] and [ fig : pg ] . first , the point @xmath110 in fig . [ fig : pg ] determines the volume fraction uniquely . we assume here that this volume fraction is realized in the system under constant pressure @xmath57 with shear rate @xmath63 and that the shear stress in the system is determined by using the volume fraction in fig . [ fig : sg ] . note that this assumption was confirmed directly by a numerical experiment for a constant pressure system whose size is close to that of the system with the lees - edwards boundary conditions . based on this assumption , we determine the volume fraction as a function of @xmath63 at a constant pressure . in addition , using this , we can obtain the dependence of the shear stress on @xmath63 at a constant pressure . hence , we obtain @xmath73 as a function of @xmath18 under constant @xmath57 . for reference , we plot the lines @xmath111 in fig [ fig : pg ] . as an example in fig . [ fig : pg ] , let us consider the case that @xmath112 in which the volume fraction is larger than the critical one for a sufficiently small @xmath63 . this case corresponds to the regime @xmath113 and @xmath114 . then , from fig . [ fig : sg ] , we find that the shear stress remains almost constant . next , in the interval @xmath115 , the states with @xmath112 are close to the critical line . thus , in this regime , it is expected that @xmath73 behaves as that in the critical line , and the scaling behavior given in eq . ( [ hatano : power ] ) is observed approximately . generalizing the above discussion , we expect the typical dependence of @xmath73 on @xmath18 as follows : @xmath116 when spatially homogeneous shear flow is realized . note that @xmath117 and @xmath118 are dependent on the pressure . for example , for states with extreme pressures , such relations would not be observed . since we wish to know the extent to which this approximate power - law relation holds , in fig . [ fig : hatano ] , we include the constant pressure data @xmath119 obtained from figs . [ fig : sg ] and [ fig : pg ] . as expected from the above consideration , the power - law behavior is observed for the case in which @xmath120 . furthermore , the system obeys the power - law regime even for the case @xmath121 in which the line is located below the critical states in fig . [ fig : pg ] . we do not understand the reason why the power law regime is so wide . we have presented numerically and theoretically the scaling relations given in eqs . ( [ scaling:1 ] ) and ( [ scaling:2 ] ) for the system with the lees - edwards boundary conditions . from these new scaling relations , we also have an interpretation of the result observed in the system under constant pressure . the result is summarized in eqs . ( [ sp : re0 ] ) and ( [ sp : re1 ] ) . as far as we know , few experimental results exist in this regard . we expect that the power - law behaviors given in eqs . ( [ scaling:1 ] ) and ( [ scaling:2 ] ) are observed in systems with constant volume , e.g. , by operating a rotating coutte - flow system @xcite . with regard to eq . ( [ sp : re0 ] ) that is valid in the very low shear rate regime , we conjecture that a thermal activation process , which might lead to the logarithmic dependence of @xmath73 on @xmath18 , occurs in this regime . note that such behavior is ubiquitous in shear flow and sliding friction @xcite . furthermore , the result reported in ref . @xcite might be related to eq . ( [ sp : re1 ] ) . we hope that more intensive experimental studies will be performed in this regard . as a function of @xmath18 for the model described in the caption of fig . [ fig : hatano ] with a herzian type interaction @xmath122 . the parameter values are as follows : @xmath20 , @xmath21 , and @xmath123 . @xmath124 ( data 1 ) , @xmath125 ( data 2 ) . furthermore , the square symbols represent the constant pressure data obtained from figs . [ fig : sg : hertz ] and [ fig : pg : hertz ] , where the interaction force is @xmath126 and @xmath127 . ] related to experimental studies , one may be interested in the dependence of our result on the choice of the model we investigate . for example , one may choose the herzian type as an alternative for the interaction force . indeed , such a model dependence has been discussed in the case of zero - temperature and zero applied stress @xcite . in our problem , it is highly expected that there is a critical state at which rheological properties exhibit power - laws . however , it is not evident that the exponents remain the same values for the model with a herzian type interaction . as a function of the dimensionless shear rate @xmath128 for several values of @xmath64 with the interaction force @xmath129 . the dimensionless shear stress and the dimensionless shear rate are defined as @xmath130 and @xmath131 , respectively . the parameter values are as follows : @xmath49 , and @xmath132 . ] in order to consider the model dependence explicitly , we demonstrate the result of numerical experiments in figs . [ fig : hatano : hertz ] , [ fig : sg : hertz ] , and [ fig : pg : hertz ] . it is seen that the exponent of the system under constant pressure does not deviate so much from 1/5 , while the exponents @xmath98 and @xmath133 seem to change slightly . we do not have a theoretical understanding for these values yet , because the time scale related to the particle collision is not directly determined from model parameters . it might be important to develop a theory by using a more physical time scale such as the collision interval . finally , in our theoretical argument , eq . ( [ gl ] ) plays an essential role . as in the case of dense colloidal suspensions @xcite , one may investigate the pair - distribution function of granular systems in order to derive the order parameter equation . the establishment of a complete theory in which the scaling relations are derived from a microscopic model is an important topic for future studies . as a function of the dimensionless shear rate @xmath128 for several values of @xmath64 with the interaction force @xmath129 . the dimensionless shear stress and the dimensionless shear rate are defined as @xmath134 and @xmath131 , respectively . the parameter values are as follows : @xmath49 , and @xmath132 . ] the authors thank h. hayakawa , n. mitarai , and s. tatsumi for their useful comments on this work . this work was supported by a grant from the ministry of education , science , sports and culture of japan ( no . 16540337 ) . ( [ hatano : power ] ) is obtained when a tangential force between interacting particles is ignored . for general cases where the tangential force is taken into account , the left - hand side of eq . ( [ hatano : power ] ) is modified as @xmath135 , where @xmath136 corresponds to the maximum static friction constant . the value of the exponent @xmath8 is identical to that in the case of eq . ( [ hatano : power ] ) . the details will be reported elsewhere .

we investigate a rheological property of a dense granular material under shear . by a numerical experiment of the system with constant volume , we find a critical volume fraction at which the shear stress and the pressure behave as power - law functions of the shear strain rate . we also present a simple scaling argument that determines the power - law exponents . using these results , we interpret a power - law behavior observed in the system under constant pressure .

cond-mat0607511

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

astro-ph0508232

self - gravity is one of the most essential physical processes in the universe , and plays important roles in almost all categories of astronomical objects such as globular clusters , galaxies , galaxy clusters , etc . in order to follow the evolution of such systems , gravitational @xmath0-body solvers have been widely used in numerical astrophysics . due to prohibitively expensive computational cost in directly solving @xmath0-body problems , many efforts have been made to reduce it in various ways . for example , several sophisticated algorithms to compute gravitational forces among many particles with reduced computational cost have been developed , such as tree method @xcite , pppm method @xcite , treepm method @xcite , etc . another approach is to improve the computational performance with the aid of additional hardware , such as grape ( gravity pipe ) systems , special - purpose accelerators for gravitational @xmath0-body simulations @xcite , and general - purpose computing on graphics processing units ( gpgpus ) . grape systems have been used for further improvement of existing @xmath0-body solvers such as tree method @xcite , pppm method @xcite , treepm method @xcite , p@xmath15m@xmath15 tree method @xcite , and pppt method @xcite . they have also adapted to simulation codes for dense stellar systems based on fourth - order hermite scheme , such as nbody4 @xcite , nbody1 @xcite , kira @xcite , and gorilla @xcite . recently , @xcite , @xcite , @xcite , and @xcite explored the capability of commodity graphics processing units ( gpus ) as hardware accelerators for @xmath0-body simulations and achieved similar to or even higher performance than the grape-6a and grape - dr board . a different approach to improve the performance of @xmath0-body calculations is to utilize streaming simd extensions ( hereafter sse ) , a simd ( single instruction , multiple data ) instruction set implemented on x86 and x86_64 processors . @xcite exploited the sse and sse2 instruction sets , and achieved speeding up of the hermite scheme @xcite in mixed precision for collisional self - gravitating systems . although unpublished in literature , nitadori , yoshikawa , & makino have also developed a numerical library for @xmath0-body calculations in single - precision for collisionless self - gravitating systems in which two - body relaxation is not physically important and therefore single - precision floating - point arithmetic suffices for the required numerical accuracy . furthermore , along this approach , they have also improved the performance in computing arbitrarily - shaped forces with a cutoff distance , defined by a user - specified function of inter - particle separation . such capability to compute force shapes other than newton s inverse - square gravity is necessary in pppm , treepm , and ewald methods . it should be noted that grape-5 and the later families of grape systems have similar capability to compute the newton s force multiplied by a user - specified cutoff function @xcite , and can be used to accelerate pppm and treepm methods for cosmological @xmath0-body simulations @xcite . based on these achievements , a publicly available software package to improve the performance of both collisional and collisionless @xmath0-body simulations has been developed , which was named `` phantom - grape '' after the conventional grape system . a set of application programming interfaces of phantom - grape for collisionless simulations is compatible to that of grape-5 . phantom - grape is widely used in various numerical simulations for galaxy formation @xcite and the cosmological large - scale structures @xcite . recently , a new processor family with `` sandy bridge '' micro - architecture by intel corporation and that with `` bulldozer '' micro - architecture by amd corporation have been released . both of the processors support a new set of instructions known as advanced vector extensions ( avx ) , an enhanced version of the sse instructions . in the avx instruction set , the width of the simd registers is extended from 128-bit to 256-bit . we can perform simd operations on two times larger data than before . therefore , the performance of a calculation with the avx instructions should be two times higher than that with the sse instructions if the execution unit is also extended to 256-bit . @xcite ( hereafter , paper i ) developed a software library for _ collisional _ @xmath0-body simulations using the avx instruction set in the mixed precision , and achieved a fairly high performance . in this paper , we present a similar library implemented with the avx instruction set but for _ collisionless _ @xmath0-body simulations in single - precision . the structure of this paper is as follows . in section [ sec : avx ] , we overview the avx instruction set . in section [ sec : implementation ] , we describe the implementation of phantom - grape . in section [ sec : accuracy ] and [ sec : performance ] , we show the accuracy and performance , respectively . in section [ sec : summary ] , we summarize this paper . in this section , we present a brief review of the advanced vector extensions ( avx ) instruction set . details of the difference between sse and avx is described in section 3.1 of paper i. avx is a simd instruction set as well as sse , and supports many operations , such as addition , subtraction , multiplication , division , square - root , approximate inverse - square - root , several bitwise operations , etc . in such operations , dedicated registers with 256-bit length called `` ymm registers '' are used to store the eight single - precision floating - point numbers or four double - precision floating - point numbers . note that the lower 128-bit of the ymm registers have alias name `` xmm registers '' , and can be used as the dedicated registers for the sse instructions for a backward compatibility . an important feature of avx and sse instruction sets is the fact that they have a special instruction for a very fast approximation of inverse - square - root with an accuracy of about 12-bit . actually , this instruction is quite essential to improve the performance of the gravitational force calculations , since the most expensive part in the force calculation is an execution of inverse - square - root of squared distances of the particle pairs . as already discussed in @xcite , the approximate values can be adopted as initial values of the newton - raphson iteration to improve the accuracy , and we can obtain 24-bit accuracy after one newton - raphson iteration . for collisionless self - gravitating systems , however , the accuracy of @xmath16 12 bits is sufficient because the accuracy of inverse - square - root does not affect the resultant force accuracy if one adopts an approximate @xmath0-body solver such as tree , pppm and treepm methods . therefore , we use the raw approximate instruction throughout this study . since the present - day compilers can not always detect concurrency of the loops effectively , and can not fully resolve the mutual dependency among data in the code , it is quite rare that compilers generate codes with simd instructions in effective manners from codes expressed in high - level languages . for an efficient use of the avx instructions , we need to program with assembly - languages explicitly or compiler - dependent intrinsic functions and data type extensions . in assembly - languages , we can manually control the assignment of ymm registers to computational data , and minimize the access to the main memory by optimizing the assignment of each register . in this work , we adopt an implementation of the avx instructions using inline - assembly language with c expression operands , embedded in c - language , which is a part of language extensions of gcc ( gnu compiler collection ) . here , we describe the detailed implementation to accelerate @xmath0-body calculation using the avx instructions . for a given set of positions @xmath17 of @xmath0 particles , we try to accelerate the calculations of a gravitational force given as follows : @xmath18 where @xmath19 is the gravitational constant , @xmath20 the mass of the @xmath21-th particle , and @xmath22 the gravitational softening length . in addition to that , we also try to accelerate the computations of central forces among particles with an arbitrary force shape @xmath1 given by @xmath23 where @xmath1 specifies the shape of the force as a function of inter - particle separation @xmath24 with a cutoff distance @xmath2 ( i.e. @xmath3 at @xmath4 ) . in the above expressions , particles with subscript `` @xmath21 '' exert forces on those with subscript `` @xmath25 '' . in the rest of this paper , the former are referred to as `` @xmath21-particles '' , and the latter as `` @xmath25-particles '' just for convenience . since individual forces exerted by @xmath21-particles on @xmath25-particles can be computed independently , we can calculate forces exerted by multiple @xmath21-particles on multiple @xmath25-particles in parallel . as described in the previous section , the avx instructions can execute operations of eight single - precision floating - point numbers on ymm registers in parallel . by utilizing this feature of the avx instructions , the forces on four @xmath25-particles from two @xmath21-particles can be computed simultaneously in a simd manner . in computing the forces on four @xmath25-particles from two @xmath21-particles , we assign the data of @xmath25- and @xmath21-particles to ymm registers in the way shown in figure [ fig : assignment ] . suppose that @xmath26 and @xmath27 in figure [ fig : assignment ] are @xmath28-components of @xmath25- and @xmath21-particles , respectively . subtracting data in the ymm register ( 1 ) of figure [ fig : assignment ] from data in the ymm register ( 2 ) of figure [ fig : assignment ] , we simultaneously obtain @xmath28-components of eight relative positions @xmath29 in the ymm register ( 3 ) of figure [ fig : assignment ] . in order to effectively realize such simd computations with the avx instructions , we define the structures for @xmath25-particles , @xmath21-particles and the resulting forces and potentials shown in list [ list : structures ] . before computing the forces on @xmath25-particles , the positions and softening lengths of @xmath25-particles are stored in the structure ` ipdata ` , and the positions and masses of @xmath21-particles are in the structure ` jpdata ` . the resulting forces are stored in the structure ` fodata ` . note that the structures ` ipdata ` and ` fodata ` contain the data of four @xmath25-particles , while the structure ` jpdata ` has the data for a single @xmath21-particle . note that the positions , softening lengths , and forces of @xmath25-particles in the structures ` ipdata ` and ` fodata ` are declared as arrays of four single - precision floating - point numbers . thus , the data on each array can be suitably loaded onto , or stored from the lower 128-bit of one ymm register . the assignment of the @xmath25-particles data shown in ( 1 ) of figure [ fig : assignment ] can be realized by loading the data of four @xmath25-particles onto the lower 128-bit of one ymm register , and copying the data to its upper 128-bit . as for @xmath21-particles , since the structure ` jpdata ` consists of four single - precision floating - point numbers , we can load the positions and the masses of two @xmath21-particles in one ymm - register at one time if they are aligned on the 32-byte boundaries . by broadcasting the @xmath30-th element ( @xmath31 and 3 ) in each of the lower and upper 128-bit to all the other elements , we can realize the assignment of the @xmath21-particle data as depicted in ( 2 ) of figure [ fig : assignment ] . after executing the gravitational force loop over @xmath21-particles , the partial forces on four @xmath25-particles exerted by different sets of @xmath21-particles are stored in the upper and lower 128-bit of a ymm register . operating sum reduction on the upper and lower 128-bit of the ymm register , and storing the results into its lower 128-bit , we can smoothly store the results into the structure ` fodata ` . .... // structure for i - particles typedef struct ipdata { float x[4 ] ; float y[4 ] ; float z[4 ] ; float eps2[4 ] ; } ipdata , * pipdata ; // structure for j - particles typedef struct jpdata { float x , y , z , m ; } jpdata , * pjpdata ; // structure for the resulting forces // and potentials of i - particles typedef struct fodata { float ax[4 ] ; float ay[4 ] ; float az[4 ] ; float phi[4 ] ; } fodata , * pfodata ; .... for the readability of the source codes shown below , let us introduce some preprocessor macros which are expanded into inline assembly codes . the definitions of the macros used in this paper are given in list [ list : macros ] . for macros with two and three operands , the results are stored in the second and third one , respectively , and the other operands are source operands . in these macros , operands named ` src ` , ` src1 ` , ` src2 ` , and ` dst ` designate the data in xmm or ymm registers , and those named ` mem ` , ` mem64 ` , ` mem128 ` , and ` mem256 ` are data in the main memory or the cache memory , where numbers after ` mem ` indicate their size and alignment in bits . brief descriptions of these macros are summarized in table [ tab : macros ] . more detailed explanation of the avx instructions can be found in intel s website . .... # define vzeroall asm("vzeroall " ) ; # define vloadps(mem256 , dst ) \ asm("vmovaps % 0 , % " dst::"m"(mem256 ) ) ; # define vstorps(reg , mem256 ) \ asm("vmovaps % " reg " , % 0 " : : " m"(mem256 ) ) ; # define vloadps(mem128 , dst ) \ asm("vmovaps % 0 , % " dst::"m"(mem128 ) ) ; # define vstorps(reg , mem128 ) \ asm("vmovaps % " reg " , % 0 " : : " m"(mem128 ) ) ; # define vloadlps(mem64 , dst ) \ asm("vmovlps % 0 , % " dst " , % " dst::"m"(mem64 ) ) ; # define vloadhps(mem64 , dst ) \ asm("vmovhps % 0 , % " dst " , % " dst::"m"(mem64 ) ) ; # define vbcastl128(src , dst ) \ asm("vperm2f128 % 0 , % " src " , % " src \ " , % " dst " " : : " g"(0x00 ) ) ; # define vcopyu128tol128(src , dst ) \ asm("vextractf128 % 0 , % " src " , % " dst \ " " : : " g"(0x01 ) ) ; # define vgatherl128(src1,src2,dst ) \ asm("vperm2f128 % 0 , % " src2 " , % " src1 \ " , % " dst " " : : " g"(0x02 ) ) ; # define vcopyall(src , dst ) \ asm("vmovaps % 0 , % " src " , % " dst ) ; # define vbcast0(src , dst ) \ asm("vshufps % 0 , % " src " , % " src \ " , % " dst " " : : " g"(0x00 ) ) ; # define vbcast1(src , dst ) \ asm("vshufps % 0 , % " src " , % " src \ " , % " dst " " : : " g"(0x55 ) ) ; # define vbcast2(src , dst ) \ asm("vshufps % 0 , % " src " , % " src \ " , % " dst " " : : " g"(0xaa ) ) ; # define vbcast3(src , dst ) \ asm("vshufps % 0 , % " src " , % " src \ " , % " dst " " : : " g"(0xff ) ) ; # define vmix0(src1,src2,dst ) \ asm("vshufps % 0 , % " src2 " , % " src1 \ " , % " dst " " : : " g"(0x88 ) ) ; # define vmix1(src1,src2,dst ) \ asm("vshufps % 0 , % " src2 " , % " src1 \ " , % " dst " " : : " g"(0xdd ) ) ; # define vaddps(src1 , src2 , dst ) \ asm("vaddps " src1 " , " src2 " , " dst ) ; # define vsubps(src1 , src2 , dst ) \ asm("vsubps " src1 " , " src2 " , " dst ) ; # define vsubps_m(mem256 , src , dst ) \ asm("vsubps % 0 , % " src " , % " dst \ " " : : " m"(mem256 ) ) ; # define vmulps(src1 , src2 , dst ) \ asm("vmulps " src1 " , " src2 " , " dst ) ; # define vrsqrtps(src , dst ) \ asm("vrsqrtps " src " , " dst ) ; # define vminps(src1 , src2 , dst ) \ asm("vminps " src1 " , " src2 " , " dst ) ; # define vpsrld(imm , src1 , src2 ) \ asm("vpsrld % 0 , % " src1 " , % " src2::"i"(imm ) ) ; # define vpslld(imm , src1 , src2 ) \ asm("vpslld % 0 , % " src1 " , % " src2::"i"(imm ) ) ; # define prefetch(mem ) \ asm("prefetcht0 % 0"::"m"(mem ) ) ; .... [ cols="<,<",options="header " , ] & 1000 & 00000000000000000 & 1023 + an affine - transformed squared distance at a sampling point with an index specified by a lower @xmath32 exponent bits @xmath33 and an upper @xmath34 fraction bits @xmath35 is expressed as @xmath36 the ratio between inter - particle distances whose affine - transformed squared distances are @xmath37 and @xmath38 is given by @xmath39 where @xmath40 is assumed for the last approximation . the interval between inter - particle distances whose affine - transformed distances are @xmath41 and @xmath42 is calculated as @xmath43 where we also assume @xmath40 and @xmath44 for the last approximation . therefore , the sampling points with the same fraction bits are distributed uniformly in logarithmic scale , and those with the same exponent bits are aligned uniformly in linear scale unless the fraction bit is small . as an example , we illustrate how the sampling points of the look - up table depend on the pre - defined integers @xmath32 and @xmath34 in figure [ fig : comp_ef ] . we first see the cases in which either of @xmath32 and @xmath34 is zero , in order to see the roles of the integers @xmath32 and @xmath34 . as seen in figure [ fig : comp_ef ] , the intervals of sampling points are roughly uniform in linear scale for the case @xmath45 ( the bottom line in the top panel ) , and uniform in logarithmic scale for the case @xmath46 ( the middle line in the bottom panel ) , unless @xmath47 is small . as expected above , the integers @xmath32 and @xmath34 control the number of sampling points in logarithmic and linear scales , respectively . by comparing the sampling points with @xmath48 and those with @xmath49 ( see the top panel of figure [ fig : comp_ef ] ) , it can be seen that all intervals of the sampling points with @xmath48 ( indicated by the vertical dashed lines and double - headed arrows ) are divided nearly equally into @xmath50 regions by the sampling points with @xmath51 . thus , our binning scheme is a hybrid of the linear and logarithmic binning schemes . and @xmath34 . the top and bottom panels take horizontal axes in linear and logarithmic scales , respectively . ] figure [ fig : decide_ef ] shows the comparison of the several binning in which the number of sampling points is fixed to @xmath52 . one can see that the binning with @xmath51 has sufficient sampling points in the range of @xmath53 , whereas the binning with the other sets of @xmath54 only samples the region of @xmath55 . the number of the extracted exponent bit @xmath32 should be large enough so that the scale of the softening length should be sufficiently resolved . for example , if @xmath56 , @xmath32 should be set to at least equal to or larger than @xmath57 . and @xmath34 . ] in list [ list : bit_binning ] , we present routines for constructing the look - up table . in our implementation , the look - up table contains two values : one is the force at a sampling point @xmath58 , @xmath59 and the other is its difference from the next sampling point @xmath60 divided by the interval of the affine - transformed squared distance @xmath61 where subscript @xmath62 indicates indices of the look - up table , and is expressed as @xmath63 . using these two values , we can compute the linear interpolation of @xmath64 at a radius @xmath24 with @xmath65 by @xmath66 . the @xmath67 and @xmath68 , are stored in a two - dimensional array declared as ` force_table[tbl_size][2 ] ` , where ` tbl_size ` is the number of the sampling points ( @xmath69 ) and the values of the @xmath67 and @xmath68 are stored in the ` force_table[k][0 ] ` and ` force_table[k][1 ] ` , respectively . since the values of @xmath67 and @xmath68 are stored in the adjacent memory address , we can avoid the cache misses in computing the linearly interpolated values of @xmath64 . .... # define exp_bit ( 4 ) # define frc_bit ( 6 ) # define tbl_size ( 1 < < ( exp_bit+frc_bit ) ) // 1024 extern float force_table[tbl_size][2 ] ; // 8 kb union pack32 { float f ; unsinged int u ; } ; void generate_force_table(float rcut ) { unsigned int tick ; float fmax , r2scale , r2max ; union pack32 m32 ; float force_func(float ) ; tick = ( 1 < < ( 23-frc_bit ) ) ; fmax = ( 1 < < ( 1<<exp_bit))*(2.0 - 1.0/(1<<frc_bit ) ) ; r2max = rcut*rcut ; r2scale = ( fmax-2.0f)/r2max ; for(i=0,m32.f=2.0f;i < tbl_size;i++,m32.u+=tick ) { float f , r2 , r ; f = m32.f ; r2 = ( f-2.0)/r2scale ; float r = sqrtf(r2 ) ; force_table[i][0 ] = force_func(r ) ; } for(i=0,m32.f=2.0f;i < tbl_size-1;i++ ) { float x0 = m32.f ; m32.u + = tick ; float x1 = m32.f ; float y0 = force_table[i][0 ] ; float y1 = ( i==tbl_size-1 ) ? 0.0 : force_table[i+1][0 ] ; force_table[i][1 ] = ( y1-y0)/(x1-x0 ) ; } force_table[i][1 ] = 0.0f ; } .... in figure [ fig : binning ] , we compare the conventional binning with equal intervals in squared distances to our binning with @xmath70 and @xmath71 ( i.e. 64 sampling points ) , for the @xmath72-force shape @xcite used in the pppm scheme . although we adopt @xmath73 in the rest of this paper , we set @xmath71 here just for good visibility of the difference of the two binning schemes . it should be noted that the number of sampling points is the same ( 64 ) in both schemes . compared with the conventional binning scheme in the top panel , our binning scheme can faithfully reproduce the given functional form even at distances smaller than the gravitational softening length . in the conventional scheme with 64 constant intervals in @xmath74 ( top panel ) and in our scheme with @xmath70 and @xmath71 ( bottom panel ) between @xmath75 $ ] . although we adopt @xmath73 elsewhere in this paper , we set @xmath71 here for viewability . @xmath76 is assumed as a functional form of @xmath1 , in which @xmath77 is the @xmath72-profile @xcite ( see equation ( [ eq : s2 ] ) ) . solid lines indicate the shape of @xmath64 . vertical dashed lines in both panels are the locations of the gravitational softening length @xmath22 . ] in calculating the arbitrary central forces , the data of @xmath25- and @xmath21-particles are stored in the structures ` ipdata ` and ` jpdata ` , respectively , in the same manner as described in the case for calculating the newton s force , except that the coordinates of @xmath25- and @xmath21-particles are scaled as @xmath78 so that we can quickly compute the affine - transformed squared inter - particle distances between @xmath25- and @xmath21-particles . as in the case of the newton s force , we compute the forces of four @xmath25-particles exerted by two @xmath21-particles using the avx instructions . using the scaled positions of the particles , the calculation of the forces is performed in the force loop as follows ; 1 . calculate an affine - transformed distance between @xmath25- and @xmath21-particles , @xmath79 , as @xmath80 where the function `` @xmath81 '' returns the minimum value among arguments . derive an index @xmath62 of the look - up table from the affine - transformed squared distance , @xmath79 , computed in the previous step by applying a bitwise - logical right shift by @xmath82 bits and reinterpreting the result as an integer . 3 . refer to the look - up table to obtain @xmath67 and @xmath68 . note that the address of the pointer to ` fcut ` is decremented by 1<<(30-(23-f ) ) in advance ( see line 24 in list [ list : arbitraryforce ] ) to correct the effect of the most significant exponential bit of @xmath79 . 4 . derive an affine - transformed distance @xmath83 that corresponds to the @xmath62-th sampling point @xmath58 by applying a bitwise - logical left shift by @xmath82 bits to @xmath62 and reinterpreting the result as a single - precision floating - point number . compute the value of @xmath84 by the linear interpolation of @xmath67 and @xmath85 . using the values of @xmath67 and @xmath68 , the interpolation can be performed as @xmath86 6 . accumulate scaled `` forces '' on @xmath25-particles as @xmath87 after the force loop , the scaled `` forces '' are rescaled back as @xmath88 the actual code of the force loop for the calculation of the central force with an arbitrary force shape is shown in list [ list : arbitraryforce ] . note that bitwise - logical shift instructions such as ` vpsrld ` and ` vpslld ` can be operated only to xmm registers or the lower 128-bit of ymm registers . in order to operate bitwise - logical shift instructions to data in the upper 128-bit of a ymm register , we have to copy the data to the lower 128-bit of another ymm register . bitwise - logical shift operations to the upper 128-bit of ymm registers are supposed to be implemented in the future avx2 instruction set . also note that we can not refer to the look - up table in a simd manner and have to issue the ` vloadlps ` and ` vloadhps ` instructions one by one ( see lines 8992 and 9497 in list [ list : arbitraryforce ] ) . except for those operations , all the other calculations are performed in a simd manner using the avx instructions . .... # define frc_bit ( 6 ) # define align32 _ _ attribute _ _ ( ( aligned(32 ) ) ) # define align64 _ _ attribute _ _ ( ( aligned(64 ) ) ) typedef float v4sf _ _ attribute _ _ ( ( vector_size(16 ) ) ) ; typedef struct ipdata_reg { float x[8 ] ; float y[8 ] ; } ipdata_reg , * pipdata_reg ; void gravitykernel(pipdata ipdata , pjpdata jp , pfodata fodata , int nj , float fcut[][2 ] , v4sf r2cut , v4sf accscale ) { int j ; unsigned long int align64 idx[8 ] = { 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 } ; ipdata_reg align32 ipdata_reg ; static v4sf two = { 2.0f , 2.0f , 2.0f , 2.0f } ; fcut -= ( 1<<(30-(23-frc_bit ) ) ) ; vzeroall ; vloadps(ipdata->x[0 ] , x2_x ) ; vloadps(ipdata->y[0 ] , y2_x ) ; vloadps(ipdata->z[0 ] , z2_x ) ; vloadps(r2cut , r2cut_x ) ; vloadps(two , two_x ) ; vbcastl128(x2 , x2 ) ; vstorps(x2 , ipdata_reg.x[0 ] ) ; vbcastl128(y2 , y2 ) ; vstorps(y2 , ipdata_reg.y[0 ] ) ; vbcastl128(z2 , zi ) ; vbcastl128(r2cut , r2cut ) ; vbcastl128(two , two ) ; vloadps(*jp , mj ) ; jp + = 2 ; vbcast0(mj , x2 ) ; vbcast1(mj , y2 ) ; vbcast2(mj , z2 ) ; vsubps_m(*ipdata_reg.x , x2 , dx ) ; vmulps(dx , dx , x2 ) ; vaddps(two , x2 , x2 ) ; vsubps_m(*ipdata_reg.y , y2 , dy ) ; vmulps(dy , dy , y2 ) ; vaddps(x2 , y2 , y2 ) ; vsubps(zi , z2 , dz ) ; vmulps(dz , dz , z2 ) ; vaddps(y2 , z2 , y2 ) ; vbcast3(mj , mj ) ; vmulps(mj , dx , dx ) ; vmulps(mj , dy , dy ) ; vmulps(mj , dz , dz ) ; vminps(r2cut , y2 , z2 ) ; for(j = 0 ; j < nj ; j + = 2 ) { vloadps(*jp , mj ) ; jp + = 2 ; vcopyu128tol128(z2 , y2_x ) ; vpsrld(23-frc_bit , y2_x , y2_x ) ; vpsrld(23-frc_bit , z2_x , x2_x ) ; vstorps(x2_x , idx[0 ] ) ; vstorps(y2_x , idx[4 ] ) ; vpslld(23-frc_bit , y2_x , y2_x ) ; vpslld(23-frc_bit , x2_x , x2_x ) ; vgatherl128(y2 , x2 , y2 ) ; vsubps(y2 , z2 , z2 ) ; vbcast0(mj , x2 ) ; vbcast1(mj , y2 ) ; vsubps_m(*ipdata_reg.x , x2 , x2 ) ; vsubps_m(*ipdata_reg.y , y2 , x2 ) ; vloadlps(*fcut[idx[4 ] ] , buf0_x ) ; vloadhps(*fcut[idx[5 ] ] , buf0_x ) ; vloadlps(*fcut[idx[0 ] ] , buf1_x ) ; vloadhps(*fcut[idx[1 ] ] , buf1_x ) ; vgatherl128(buf0 , buf1 , buf1 ) ; vloadlps(*fcut[idx[6 ] ] , buf2_x ) ; vloadhps(*fcut[idx[7 ] ] , buf2_x ) ; vloadlps(*fcut[idx[2 ] ] , buf0_x ) ; vloadhps(*fcut[idx[3 ] ] , buf0_x ) ; vgatherl128(buf2 , buf0 , buf2 ) ; vmix1(buf1 , buf2 , buf0 ) ; vmix0(buf1 , buf2 , buf2 ) ; vmulps(z2 , buf0 , buf0 ) ; vbcast2(mj , z2 ) ; vbcast3(mj , mj ) ; vsubps(zi , z2 , z2 ) ; vaddps(buf0 , buf2 , buf2 ) ; vmulps(buf2 , dx , dx ) ; vmulps(buf2 , dy , dy ) ; vmulps(buf2 , dz , dz ) ; vaddps(dx , ax , ax ) ; vaddps(dy , ay , ay ) ; vaddps(dz , az , az ) ; vcopyall(x2 , dx ) ; vcopyall(y2 , dy ) ; vcopyall(z2 , dz ) ; vmulps(x2 , x2 , x2 ) ; vmulps(y2 , y2 , y2 ) ; vmulps(z2 , z2 , z2 ) ; vaddps(two , x2 , x2 ) ; vaddps(z2 , y2 , y2 ) ; vaddps(x2 , y2 , y2 ) ; vmulps(mj , dx , dx ) ; vmulps(mj , dy , dy ) ; vmulps(mj , dz , dz ) ; vminps(r2cut , y2 , z2 ) ; } vcopyu128tol128(ax , x2_x ) ; vaddps(ax , x2 , ax ) ; vcopyu128tol128(ay , y2_x ) ; vaddps(ay , y2 , ay ) ; vcopyu128tol128(az , z2_x ) ; vaddps(az , z2 , az ) ; vmulps_m(accscale , ax_x , ax_x ) ; vmulps_m(accscale , ay_x , ay_x ) ; vmulps_m(accscale , az_x , az_x ) ; vstorps(ax_x , * fodata->ax ) ; vstorps(ay_x , * fodata->ay ) ; vstorps(az_x , * fodata->az ) ; } .... although the avx instruction set takes the non - destructive 3-operand form , the copy instruction between registers appeared in the code above , which was intended to avoid the inter - register dependencies . on multi - core processors , we can parallelize the calculations of the forces of @xmath25-particles for both of the newton s force and arbitrary central forces using the openmp programming interface by assigning a different set of four @xmath25-particles onto each processor core . list [ list : parallel ] shows a code fragment for the parallelization of the computations of the newton s force . the calculation of an arbitrary force can be parallelized similarly to that of newton s force . .... # define isimd 4 extern ipdata ipos[ni_memmax / isimd ] ; extern jpdata jpos[nj_memmax ] ; extern fodata iacc[ni_memmax / isimd ] ; int nig = ni / isimd + ( ni % isimd ? 1 : 0 ) # pragma omp parallel for for(i = 0 ; i < nig ; i++ ) gravitykernel(&ipos[i ] , & iacc[i ] , jpos , nj ) ; .... with the implementations of the force calculation accelerated with the avx instructions described above , we develop a set of application programming interfaces ( apis ) for @xmath0-body simulations , which is compatible to grape-5 library , except that our library do not support functions to search for neighbours of a given particle . the apis are shown in list [ list : api ] . ` g5_set_xmj ` sends the data of @xmath21-particles to the array of the structure ` jpdata ` . ` g5_calculate_force_on_x ` sends the data of @xmath25-particles to the array of the structure ` ipdata ` , and computes the forces and potentials of @xmath25-particles and returns them into the arrays ` ai ` and ` pi ` , respectively . in the function ` g5_open ` , we derive statistical bias of the fast approximation of inverse - square - root , ` vrsqrtps ` instruction . as @xcite reported , the results of this instruction contains a bias which is implementation - dependent . we statistically correct this bias in the same way as @xcite . softening length and the number of @xmath21-particles are set by the functions ` g5_set_eps_to_all ` and ` g5_set_n ` , respectively . ` g5_close ` does nothing and is just for compatibility with the grape-5 library . list [ list : sample ] shows a code fragment to perform an @xmath0-body simulation , using this apis . .... void g5_open(void ) ; void g5_close(void ) ; void g5_set_eps_to_all(double eps ) ; void g5_set_n(int nj ) ; void g5_set_xmj(int adr , int nj , double ( * xj)[3 ] , double * mj ) ; void g5_calculate_force_on_x(double ( * xi)[3 ] , double ( * ai)[3 ] , double * pi , int ni ) ; .... .... int n ; // the number of particles double m[nmax ] ; // mass double x[nmax][3 ] ; // position double v[nmax][3 ] ; // velocity double a[nmax][3 ] ; // force double p[nmax ] ; // potential double t ; // time double tend ; // time at the finish time double dt ; // timestep void time_integrator(int , double ( * ) [ 3 ] , double ( * ) [ 3 ] , double ( * ) [ 3 ] double ) ; // function for time integration g5_open ( ) ; g5_set_eps_to_all(eps ) ; g5_set_n(n ) ; while(t < tend ) { g5_set_xmj(0,n , x , m ) ; g5_calculate_force_on_x(x , a , p , n ) ; time_integrator(n , x , v , a , dt ) ; t + = dt ; } g5_close ( ) ; .... for the version of arbitrary force shape , we provide a new api call to set the force - table through a function pointer , which is not compatible to the grape-5 api . we investigate accuracy of forces and potentials obtained by our implementation for newton s force . for this purpose , we compute the forces and potentials of particles in the plummer models using our implementations and compare them with those computed fully in double - precision floating - point numbers without any explicit use of the avx instructions . for the calculations of the forces and the potentials , we adopt the direct particle - particle method and the softening length of @xmath89 , where @xmath90 is a virial radius of the plummer model and @xmath0 is the number of particles . figure [ fig : newton_error ] shows the cumulative distribution of relative errors in the forces and the potentials of particles , @xmath91 and @xmath92 where @xmath93 and @xmath94 are the force and the potential calculated using our implementation , and @xmath95 and @xmath96 are those computed fully in double - precision . it can be seen that most of the particles have errors less than @xmath97 . these errors primarily come from the approximate inverse - square - root instruction ` vrsqrtps ` , whose accuracy is about 12-bit , and consistent with the typical errors of @xmath98 . while the errors of the forces are distributed down to less than @xmath99 , the errors of the potentials are mostly larger than @xmath100 . it can be ascribed to the way of excluding the contribution of self - interaction to the potentials . in computing a potential of the @xmath25-th particle , we accumulate the contribution from particle pairs between the @xmath25-th particle and all the particles including itself , and then subtract the contribution of the potential between the @xmath25-th particle and itself , @xmath101 to finally obtain the correct potential of the @xmath25-th particle . note that the potential between the @xmath25-th particle and itself is largest among the potentials between the @xmath25-th particle and all the particles , since the separation between @xmath25-particle and itself is zero . thus , the subtraction of the `` potential '' due to the self - interaction causes the cancellation of the significant digits , and consequently degrades the accuracy of the potentials . a remedy for such degradation of the accuracy is to avoid the self - interaction in the force loop . in fact , we do so in calculating the potentials in double - precision ( @xmath96 ) in figure [ fig : newton_error ] . however , such treatment requires conditional bifurcation inside the force loop , and significantly reduces the computational performance . the potentials of particles are usually necessary only for checking the total energy conservation , and the accuracy obtained in our implementation is sufficient for that purpose . for these reasons , we choose the original way to compute the potentials of particles in our implementation . in order to see accuracies of central forces with an arbitrary shape obtained in our implementation , we choose a force shape which is frequently adopted in cosmological @xmath0-body simulations using pppm or treepm methods . such methods are comprised of the particle mesh ( pm ) and the particle particle ( pp ) parts which compute long- and short - range components of inter - particle forces , respectively . our implementation of the calculation of arbitrarily - shaped central forces can accelerate the calculation of the pp part , in which the force shape is different from the newton s force and is expressed as @xmath102 where @xmath103 is the so - called @xmath72-profile with a softening length of @xmath26 @xcite given by @xmath104 we calculate forces exerted between @xmath57k particle pairs with various separations uniformly distributed in @xmath105 in the range of @xmath106 using our implementation described in section [ sec : methodarbitrary ] , where @xmath107k is equal to @xmath108 . we set @xmath22 and @xmath2 to @xmath109 and @xmath110 , and masses to unity . in creating the look - up table of the force shape , we set @xmath70 and @xmath73 . figure [ fig : s2_error ] shows a functional form of @xmath111 ( solid curve ) and @xmath1 ( dashed curve ) in the top panel and relative errors of forces including both pp and pm parts , i.e. @xmath111 , in the bottom panel as a function of @xmath112 . in figure [ fig : s2_error ] , we can see that the relative errors are less than @xmath113 , which are sufficiently accurate for cosmological @xmath0-body simulations . and @xmath1 ( upper panel ) and the relative errors of forces of particle pairs with a separation @xmath24 ( bottom panel ) as a function of @xmath112 , where the forces include both pp and pm parts . here , @xmath114 and @xmath115 are , respectively , an absolute force calculated with our implementation and that obtained by performing all the calculations in double - precision without referring to the look - up table . the separations of particle pairs are distributed uniformly in @xmath105 in the range of @xmath116 . ] in this section , we present the performance of our implementation of the collisionless @xmath0-body simulation using the avx instructions ( hereafter avx - accelerated implementation ) . for the measurement of the performance , we use an intel core i72600 processor with 8 mb cache memory and a frequency of @xmath117 ghz , which contains four processor cores . in measuring the performance , intel turbo boost technology is disabled , and intel hyper - threading technology ( htt ) is enabled . a compiler which we adopt is gcc 4.4.5 , with options -o3 -ffast - math -funroll - loops . to see the advantage of the avx instructions relative to the sse instructions , we also develop the implementations with the sse instructions rather than the avx instructions both for newton s force and arbitrarily - shaped force ( sse - accelerated implementation ) . first , we show the performance of our implementation for newton s force . the performance is measured by executing the direct particle - particle calculation of the plummer model with the number of particles from 0.5k to 32k . the left panel of figure [ fig : newton_pfm ] depicts the performances of the avx- and sse - accelerated implementations . for comparison , we also show the performance of an implementation without any explicit use of simd instructions ( labeled as `` w / o simd '' in the left panel of figure [ fig : newton_pfm ] ) . the numbers of interactions per second are @xmath5 in the case of the avx - accelerated implementation with a single thread , which corresponds to @xmath6 gflops , where the number of floating - point operations for the computation of force and potential for one pair of particles is counted to be @xmath7 . the performances of the sse- and avx - accelerated implementations with a single thread are higher than those without simd instructions by @xmath118 and @xmath8 times , respectively , and higher than those expected from the degree of concurrency of the sse and avx instructions for single - precision floating - point number ( 4 and 8 , respectively ) . this is because a very fast instruction of approximate inverse - square - root is not used in the `` w / o simd '' implementation . on the other hand , the performance with the avx - accelerated implementation is higher than that of the sse - accelerated implementation roughly by a factor of two as expected . furthermore , in the left panel of figure [ fig : newton_pfm ] , we show the performance of a gpu - accelerated @xmath0-body code based on the direct particle - particle method implemented using the cuda language , where the gpu board is nvidia geforce gtx 580 connected through the pci - express gen2 x16 link . the gpu - accelerated @xmath0-body code computes the forces and potentials of the particles using gpus , and integrate the equations of the motion of the particles on a cpu . thus , the communication of the particle data between the main memory of the host machine and the device memory on the gpu boards is required , and can hamper the total efficiency of the code . of course , if all the calculations are performed on gpus , we might not suffer from such overhead . however , the performance of such implementation can not be fairly compared with those of the avx- and sse - accelerated implementations , because the communication of the particle data is inevitable when we perform @xmath0-body simulations with multiple gpus or with multiple nodes equipped with gpus , regardless of the @xmath0-body solvers such as tree and treepm methods . the performances of the avx- and sse - accelerated implementations are almost independent of the total number of particles , @xmath0 . on the other hand , the performance of the gpu - accelerated implementation strongly depends on the number of particles @xmath0 , due to the non - negligible overhead caused by the particle data communication . for @xmath119k , the performance of the gpu - accelerated implementation is only 5% of that for @xmath120k . thus , for small @xmath0 ( @xmath121k and @xmath107k ) , the performance of the avx - accelerated implementation with four threads is higher than that with gpu - accelerated implementation , although , for large @xmath0 ( @xmath122@xmath123 ) , the performance of the gpu - accelerated implementation is higher than that of the avx - accelerated implementation . these features can be explained by the communication overhead in the gpu - accelerated implementation . so far , we see the performance of our code in the case that both the numbers of @xmath25- and @xmath21-particles ( @xmath124 and @xmath125 , respectively ) are the same and equal to @xmath0 . however , in actual computations of forces in collisionless @xmath0-body simulations based on various @xmath0-body solvers such as pppm , tree , and treepm methods , the numbers of @xmath25- and @xmath21-particles @xmath124 and @xmath125 are much smaller than the total number of particles @xmath0 . in the tree method modified for the effective force with external hardwares or softwares as described in @xcite , for example , @xmath124 is the number of particles , for which a tree traverse is performed simultaneously and the resultant interaction list ( size @xmath125 ) is shared , and typically around @xmath118@xmath126 . furthermore , if one adopts the individual timestep algorithm , the number of @xmath25-particles @xmath124 gets even smaller . the number of @xmath21-particles @xmath125 is also decreased in tree and treepm methods . therefore , we show the performance for typical @xmath124 and @xmath125 in the realistic situations of typical collisionless @xmath0-body simulations . the right panel of figure [ fig : newton_pfm ] shows the performance of the avx - accelerated implementation using four threads with four processor cores ( black lines ) and that of the gpu - accelerated one ( red lines ) for various set of @xmath124 and @xmath125 . it can be seen that the obtained performance gets lower for the smaller @xmath124 and @xmath125 , regardless of the implementations . for the avx- and sse - accelerated implementations , this feature is due to the overhead of storing the particle data into the structures ` ipdata ` and ` jpdata ` shown in list [ list : structures ] . the amount of the overhead of storing @xmath25- and @xmath21-particles are proportional to @xmath124 and @xmath125 , respectively , and the computational cost is proportional to @xmath127 . keeping this in mind the low performance with @xmath128 compared with those with @xmath129 can be ascribed to the overhead of storing @xmath21-particles to the structure ` jpdata ` . for the gpu - accelerated implementation , the overhead originates from the transfer of the particle data to the memory on gpus . it can be seen that the performance of the avx - accelerated implementation has rather mild dependence on @xmath124 and @xmath125 , while that of the gpu - accelerated one relatively strongly depends on @xmath125 . such difference reflects the fact that the bandwidths and latency of the communication between gpus and cpus are rather poor compared with those of memory access between cpus and main memory . thus , the performance of the gpu - accelerated implementation is apparently superior to the avx - accelerated one only when both of @xmath124 and @xmath125 are sufficiently large ( say , @xmath130k and @xmath131k ) . at the end of this section , we apply our avx - accelerated implementation to barnes - hut tree method @xcite , and measure its performance . our tree code is based on the pp part of treepm code implemented by @xcite and @xcite , in which they accelerated the calculations of the gravitational forces of the @xmath72-profile using grape-5 and grape-6a systems under the periodic boundary condition . we modify the tree code such that it can compute the newton s force under the vacuum boundary condition . since both of grape-6a systems and phantom - grape library support the same apis , we can easily utilize the capability of phantom - grape by simply exchanging the software library . using the tree code described above , we calculate gravitational forces and potentials of all the particles in a plummer model and a king model with the dimensionless central potential depth @xmath132 . we measure the performance on an intel core i72600 processor . for the comparison with other codes , we also measure the performance of the same code but without any explicit use of simd instructions , and the publicly available code bonsai @xcite , which is a gpu - accelerated @xmath0-body code based on the tree method . the performance of the bonsai code is measured on a system with nvidia geforce gtx 580 . since the bonsai code utilizes the quadrupole moments of the particle distribution in each tree node as well as the monopole moments in the force calculations , for a fair comparison of the performance with the bonsai code , we give our tree code a capability to use the quadrupole moments in each tree node , although the original code uses only the monopole moments . we represent these multipole moments as pseudo - particles , using pseudo - particle multipole method @xcite . figure [ fig : treenw ] shows the wall clock time to compute gravitational forces and potentials for each tree code . we show the both results with the code which uses the quadrupole moments ( lower panels ) and the one which uses only the monopole moments ( upper panels ) . note that the wall clock time includes the time for tree construction , tree traverse and calculations of forces and potentials but we exclude the time to integrate orbital motion of particles . as expected , the wall clock time with the avx - accelerated implementation is roughly 10 times shorter than those without any explicit use of simd instructions , owing to parallelism to calculate forces and potentials . the wall clock time with the avx - accelerated implementation is about only three times longer than those with bonsai , despite that theoretical peak performance of intel core i72600 ( @xmath133 gflops ) is lower than that of nvidia geforce gtx 580 ( @xmath134 gflops ) by a factor of 7.3 in single - precision . we expect that the performance of our avx - accelerated implementation could be close to that of the bonsai in the following situations . when we adopt individual timestep algorithm , the number of @xmath25-particles is effectively decreased , and a part of gpu cores becomes inactive . thus , the performance of gpu - accelerated implementation would be degraded more rapidly than that of our avx - accelerated implementation . furthermore , when we use gpu - accelerated implementation on massively parallel environments , the communication between cpus and gpus is inevitable , which also degrades the performance of gpu - accelerated implementation . the left panel of figure [ fig : s2_pfm ] shows the performance of our implementation to calculate forces with an arbitrary force shape accelerated with the avx and sse instructions . for the comparison , we also plot the performance of an implementation without any explicit use of the simd instructions . the numbers of exponent and fraction bits used to referring the look - up table are set to @xmath70 and @xmath135 , respectively . the performance of the avx - accelerated implementation with a single thread is @xmath9 and @xmath14 times higher than that of the sse - accelerated one and the one without any simd instructions , respectively . these forces with the use of the avx instructions are lower than those expected from the degree of concurrency of their simd operations , @xmath136 , mainly because the reference of a look - up table is not carried out in a simd manner . the performance with multi - thread parallelization is almost proportional to the number of threads up to four threads . if the htt is activated , the performance with eight threads is higher than that with four threads by a few percent . the right panel of figure [ fig : s2_pfm ] shows the performance of the avx - accelerated implementation with eight threads for a various set of @xmath124 and @xmath125 . for @xmath129 , the performance is almost independent of @xmath124 and @xmath125 , and for @xmath137 it is about half the performance with @xmath129 . this is again due to the overhead of copying @xmath21-particle data to the structure ` jpdata ` , as is the case in the calculation of newton s force . such weak dependence of the performance on @xmath124 and @xmath125 are also preferable for the calculations of the forces in the pppm and treepm methods especially with the individual timestep scheme . using the avx instructions , the new simd instructions of x86 processors , we develop a numerical library to accelerate the calculations of newton s forces and arbitrarily shaped forces for @xmath0-body simulations . we implement the library by means of inline - assembly embedded in c - language with gcc extensions , which enables us to manually control the assignment of the ymm registers to computational data , and extract the full capability of a cpu core . in computing arbitrarily shaped forces , we refer to a look - up table , which is constructed with a novel scheme so that the binning is optimized to ensure good numerical accuracy of the computed forces while its size is kept small enough to avoid cache misses . the performance of the version for newton s forces reaches @xmath138 interactions per second with a single thread , which is about @xmath9 times and @xmath8 times higher than those of the implementation with the sse instructions and without any explicit use of simd instructions , respectively . the use of the fast inverse - square - root instruction is a key ingredient of the improvement of the performance in the implementation with the sse and avx instructions . the performance of the version for arbitrarily shaped forces is @xmath9 and @xmath14 times higher than those implemented with the sse instructions and without any explicit use of the simd instructions . furthermore , our implementation supports the thread parallelization on a multi - core processor with the openmp programming interface , and has a good scalability regardless of the number of particles . while the performance of our implementation using the avx instructions is moderate compared with that of the gpu - accelerated implementation , the most remarkable advantage of our implementation is the fact that the performance has much weaker dependence on the numbers of @xmath25- and @xmath21-particles than that of the gpu - accelerated implementation . this feature is also the case for the calculation of the arbitrarily shaped force , and can be explained by the relatively large overhead of the data transfer between gpus and main memory of their host computers . in actual calculations of forces with popular @xmath0-body solvers such as the tree - method and the treepm - method combined with the individual timestep scheme , the numbers of @xmath25- and @xmath21-particles can not be always large enough to extract the full capability of gpus . in that sense , our implementation is more suitable in accelerating the calculations of forces using the tree- and treepm - methods . another advantage of our implementation is its portability . with this library , we can carry out collisionless @xmath0-body simulations with a good performance even on supercomputer systems without any gpu - based accelerators . note that massively parallel systems with gpu - based accelerators , at least currently , are not ubiquitous . even on processors other than the x86 architecture , most of them supports similar simd instruction sets ( e.g. vector multimedia extension on ibm power series , and hpc - ace on sparc64 viiifx , etc . ) our library can be ported to these processors with some acceptable efforts . finally let us mention the possible future improvement of our implementation . fused multiply - add ( fma ) instructions which have already been implemented in the `` bulldozer '' cpu family by amd corporation , and is scheduled to be introduced in the `` haswell '' processor by intel corporation in 2013 . the use of the fma instructions will improve the performance and accuracy of the calculations of forces to some extent . the numerical library `` phantom - grape '' developed in this work is publicly available at http://code.google.com / p / phantom - grape/. we thank dr . takayuki saitoh for valuable comments on this work . a. tanikawa thanks yohei miki and go ogiya for fruitful discussion on gpu . numerical simulations have been performed with computational facilities at the center for computational sciences in university of tsukuba . this work was supported by scientific research for challenging exploratory research ( 21654026 ) , grant - in - aid for young scientists ( start - up : 21840015 ) , the first project based on the grants - in - aid for specially promoted research by mext ( 16002003 ) , and grant - in - aid for scientific research ( s ) by jsps ( 20224002 ) . k. nitadori and t. okamoto acknowledge financial support by mext hpci strategic program . barnes , j . , & hut , p . 1986 , nature , 324 , 446 bdorf , j . , gaburov , e . , & portegies zwart , s . 2012 , journal of computational physics , 231 , 2825 brieu , p.p . , summers , f.j . , ostriker , j.p . 1995 , apj , 453 , 566 fukushige , t. , makino , j. , kawai , a. 2005 , pasj , 57 , 1009 gaburov , e. , harfst , s. , & portegies zwart , s. 2009 , newa , 14 , 630 hamada , t. , iitaka , t. 2007 , submitted , arxiv : astro - ph/0703100 harfst , s. , gualandris , a. , merritt , d. , spurzem , r. , portegies zwart , s. , & berczik , p. 2007 , new astronomy , 12 , 357 hockney , r.w . , eastwood , j.w . 1981 , computer simulation using particles ( new york : mcgraw - hill ) . ishiyama , t. , fukushige , t. , & makino , j. 2008 , pasj , 60 , 13 ishiyama , t. , fukushige , t. , & makino , j. 2009 , apj , 696 , 2115 ishiyama , t. , fukushige , t. , & makino , j. 2009 , pasj , 61 , 1319 ishiyama , t. , makino , j. , & ebisuzaki , t. 2010 , apj , 723 , 195 ishiyama , t. , makino , j. , portegies zwart , s. , groen , d. , nitadori , k. , rieder , s. , de laat , c. , mcmillan , s. , hiraki , k. , & harfst , s. 2011 , submitted , arxiv1101.2020 johnson , v. , & aarseth , s. 2006 , aspc , 351 , 165 kawai , a. , fukushige , t. , makino , j. , & taiji , m. 2000 , pasj , 52 , 659 kawai , a. , & makino 2001 , apjl , 550 , 143 kawai , a. , makino , j. , ebisuzaki , t. 2004 , apjs , 151 , 13 . makino , j. 1991 , pasj , 43 , 621 . makino , j. , & aarseth , s. 1992 , pasj , 44 , 141 makino , j. , fukushige , t. , koga , m. , namura , k. 2003 , pasj , 55 , 1163 nitadori , k. , makino , j. , & hut , p. 2006 , newa , 12 , 169 oshino , s. , funato , y. , & makino , j. 2011 , pasj , 63 , 881 portegies zwart , s. , belleman , r. & geldof , p. 2007 , new astronomy , 12 , 641 portegies zwart , s. , mcmillan , s. , groen , d. , gualandris , a. , sipior , m. , & vermin , w. 2008 , new astronomy , 13 , 285 saitoh , t. r. , daisaka , h. , kokubo , e. , makino , j. , okamoto , t. , tomisaka , k. , wada , k. & yoshida , n. 2008 , pasj , 60 , 667 saitoh , t. r. , daisaka , h. , kokubo , e. , makino , j. , okamoto , t. , tomisaka , k. , wada , k. , & yoshida , n. 2009 , pasj , 61 , 481 sugimoto , d. , chikada , y. , makino , j. , ito , t. , ebisuzaki , t. , umemura , m. 1990 . nature 345 , 33 . tanikawa , a. , & fukushige , t. 2009 , pasj , 61 , 721 tanikawa , a. , yoshikawa , k. , okamoto , t. , & nitadori , k. 2012 , new astronomy , 17 , 82 ( paper i ) xu , g. , 1995 . apjs , 98 355 yoshikawa , k. , fukushige , t. 2005 . pasj 57 , 849 .

we have developed a numerical software library for collisionless @xmath0-body simulations named `` phantom - grape '' which highly accelerates force calculations among particles by use of a new simd instruction set extension to the x86 architecture , advanced vector extensions ( avx ) , an enhanced version of the streaming simd extensions ( sse ) . in our library , not only the newton s forces , but also central forces with an arbitrary shape @xmath1 , which has a finite cutoff radius @xmath2 ( i.e. @xmath3 at @xmath4 ) , can be quickly computed . in computing such central forces with an arbitrary force shape @xmath1 , we refer to a pre - calculated look - up table . we also present a new scheme to create the look - up table whose binning is optimal to keep good accuracy in computing forces and whose size is small enough to avoid cache misses . using an intel core i72600 processor , we measure the performance of our library for both of the newton s forces and the arbitrarily shaped central forces . in the case of newton s forces , we achieve @xmath5 interactions per second with one processor core ( or @xmath6 gflops if we count @xmath7 operations per interaction ) , which is @xmath8 times higher than the performance of an implementation without any explicit use of simd instructions , and @xmath9 times than that with the sse instructions . with four processor cores , we obtain the performance of @xmath10 interactions per second ( or @xmath11 gflops ) . in the case of the arbitrarily shaped central forces , we can calculate @xmath12 and @xmath13 interactions per second with one and four processor cores , respectively . the performance with one processor core is @xmath14 times and @xmath9 times higher than those of the implementations without any use of simd instructions and with the sse instructions . these performances depend only weakly on the number of particles , irrespective of the force shape . it is good contrast with the fact that the performance of force calculations accelerated by graphics processing units ( gpus ) depends strongly on the number of particles . substantially weak dependence of the performance on the number of particles is suitable to collisionless @xmath0-body simulations , since these simulations are usually performed with sophisticated @xmath0-body solvers such as tree- and treepm - methods combined with an individual timestep scheme . we conclude that collisionless @xmath0-body simulations accelerated with our library have significant advantage over those accelerated by gpus , especially on massively parallel environments . , , & stellar dynamics , method : @xmath0-body simulations

1203.4037

in recent years , fifth generation ( 5 g ) wireless networks have attracted extensive research interest . according to the 3rd generation partnership project ( 3gpp ) @xcite , 5 g networks should support three major families of applications , including enhanced mobile broadband ( embb ) @xcite ; massive machine type communications ( mmtc ) @xcite ; and ultra - reliable and low - latency communications ( urllc ) @xcite . on top of this , enhanced vehicle - to - everything ( ev2x ) communications are also considered as an important service that should be supported by 5 g networks @xcite . these scenarios require massive connectivity with high system throughput and improved spectral efficiency ( se ) and impose significant challenges to the design of general 5 g networks . in order to meet these new requirements , new modulation and multiple access ( ma ) schemes are being explored . orthogonal frequency division multiplexing ( ofdm ) @xcite has been adopted in fourth generation ( 4 g ) networks . with an appropriate cyclic prefix ( cp ) , ofdm is able to combat the delay spread of wireless channels with simple detection methods , which makes it a popular solution for current broadband transmission . however , traditional ofdm is unable to meet many new demands required for 5 g networks . for example , in the mmtc scenario @xcite , sensor nodes usually transmit different types of data asynchronously in narrow bands while ofdm requires different users to be highly synchronized , otherwise there will be large interference among adjacent subbands . to address the new challenges that 5 g networks are expected to solve , various types of modulation have been proposed , such as filtering , pulse shaping , and precoding to reduce the out - of - band ( oob ) leakage of ofdm signals . filtering @xcite is the most straightforward approach to reduce the oob leakage and with a properly designed filter , the leakage over the stop - band can be greatly suppressed . pulse shaping @xcite can be regarded as a type of subcarrier - based filtering that reduces overlaps between subcarriers even inside the band of a single user , however , it usually has a long tail in time domain according to the heisenberg - gabor uncertainty principle @xcite . introducing precoding @xcite to transmit data before ofdm modulation is also an effective approach to reduce leakage . in addition to the aforementioned approaches to reduce the leakage of ofdm signals , some new types of modulations have also been proposed specifically for 5 g networks . for example , to deal with high doppler spread in ev2x scenarios , transmit data can be modulated in the delay - doppler domain @xcite . the above modulations can be used with orthogonal multiple access ( oma ) in 5 g networks . oma is core to all previous and current wireless networks ; time - division multiple access ( tdma ) and frequency - division multiple access ( fdma ) are used in the second generation ( 2 g ) systems , code - division multiple access ( cdma ) in the third generation ( 3 g ) systems , and orthogonal frequency division multiple access ( ofdma ) in the 4 g systems . for these systems , resource blocks are orthogonally divided in time , frequency , or code domains , and therefore there is minimal interference among adjacent blocks and makes signal detection relatively simple . however , oma can only support limited numbers of users due to limitations in the numbers of orthogonal resources blocks , which limits the se and the capacity of current networks . to support a massive number of and dramatically different classes of users and applications in 5 g networks , various noma schemes have been proposed . as an alternative to oma , noma introduces a new dimension by perform multiplexing within one of the classic time / frequency / code domains . in other words , noma can be regarded as an `` add - on '' , which has the potential to be harmoniously integrated with existing ma techniques . the core of noma is to utilize power and code domains in multiplexing to support more users in the same resource block . there are three major types of noma : power - domain noma , code - domain noma , and noma multiplexing in multiple domains . with noma , the limited spectrum resources can be fully utilized to support more users , therefore the capacity of 5 g networks can be improved significantly even though extra interference and additional complexity will be introduced at the receiver . to address the various challenges of 5 g networks , we can either develop novel modulation techniques to reduce multiple user interference for oma or directly use noma . the rest of this article is organized as follows . in section [ sec : waveform ] , novel modulation candidates for oma in 5 g networks are compared . in section [ sec : ma ] , various noma schemes are discussed . section [ sec : conclusion ] concludes the article . in this section , we will discuss new modulation techniques for 5 g networks . since ofdm is widely used in current wireless systems and standards , many potential modulation schemes for 5 g networks are delivered from ofdm for backward compatibility reasons . therefore , we will first introduce traditional ofdm . denote @xmath0 , for @xmath1 , to be the transmit complex symbols . then the baseband ofdm signal can be expressed as @xmath2 for @xmath3 , where @xmath4 , @xmath5 is the subcarrier bandwidth and @xmath6 is the symbol duration . to ensure that transmit symbols can be recovered without distortion , @xmath7 , which is also called the orthogonal condition . it can be easily shown that @xmath8 if the orthogonal condition holds . denote @xmath9 to be the sampled version of @xmath10 , where @xmath11 . it can be easily seen @xcite that @xmath12 is the inverse discrete fourier transform ( idft ) of @xmath13 , which can be implemented by fast fourier transform ( fft ) and significantly simplifies ofdm modulation and demodulation . to address the delay spread of wireless channels , a cp is usually used in ofdm . if the length of the cp is larger than the delay span ( the duration between the first and the last taps / paths of a channel ) , then the demodulated ofdm signal can be expressed as @xmath14 where @xmath15 is the frequency response of the wireless channel at @xmath16 and @xmath17 is the impact of additive channel noise . therefore , the channel distortion becomes a multiplication of channel frequency response in ofdm systems while it is convolution in single - carrier systems , which makes the detection of ofdm signal much easier . from the above discussion , ofdm can effectively deal with the delay spread of broadband wireless channels and fft can be used to significantly simplify its complexity , therefore it has been widely used in the current wireless communication systems and standards . however , as we can see from ( [ eq : ofdmsig ] ) , the ofdm signal is time - limited . therefore , its oob leakage is pretty high , especially when users are asynchronized as typical of 5 g networks . to address this issue , a guard band is usually inserted between the signals of two adjacent users in the frequency domain in addition to a cp or a guard interval in the time domain , which reduces the se of ofdm . this is even more severe for the users using a narrow frequency band . 5 g networks have to support not only a massive number of users but also dramatically different types of users that have different demands . traditional ofdm can no longer satisfy these requirements , and therefore novel modulation techniques with much lower oob leakage are required . the new modulation techniques for 5 g networks currently need to consider backward compatibility with traditional ofdm systems but should also have the following key features to address the new challenges . 1 . high se : new modulation techniques should be able to mitigate oob leakage among adjacent users so that the system se can be improved significantly by reducing the guard band / time resources . 2 . loose synchronization requirements : massive number of users are expected to be supported , especially for the internet of things ( iot ) , which makes synchronization difficult . therefore , new modulation techniques are expected to accept asynchronous scenarios . 3 . flexibility : the modulation parameters ( e.g. , subcarrier width and symbol period ) for each user should be configured independently and flexibly to support users with different data rate requirements . the modulation techniques for oma mainly include pulse shaping , subband filtering , precoding design , guard interval ( gi ) shortening , and modulation in the delay - doppler domain . in this section , we introduce those promising modulation techniques subsequently . pulse shaping , which is also regarded as subcarrier - based filtering , can effectively reduce oob leakage . according to the heisenberg - gabor uncertainty principle @xcite , the time and frequency widths of the pulses can not be reduced at the same time . therefore , the waveforms based on pulse shaping is usually non - orthogonal in both time and frequency domains to maintain high se . compared with traditional ofdm , the transceiver structure supporting pulse shaped modulation is more complex . here , we introduce two typical modulations based on pulse shaping , i.e. , filter bank multicarrier ( fbmc ) @xcite and generalized frequency division multiplexing ( gfdm ) @xcite . as shown in fig . [ fig : fbmc ] , fbmc @xcite consists of idft and dft , synthesis and analysis polyphase filter banks . the prototype filter in fbmc performs the pulse shaping . there are two types of typical pulses : the pulse based on the isotropic orthogonal transform algorithm ( iota ) @xcite and the pulse adopted in the phydyas project @xcite . the length of the pulse in the time domain is determined by the required performance and is usually several times the length of the symbol period . the bandwidth of the pulse , which is different from the pulse in the traditional ofdm that has a long tail , is limited within a few subbands . to achieve the best se , offset quadrature amplitude modulation ( oqam ) is usually applied to make fbmc real - domain orthogonal in time and frequency domains @xcite . therefore , the transmit signal over @xmath18 consecutive block periods can be expressed as @xmath19 where @xmath20 and @xmath21 are the numbers of subcarriers and symbols , respectively , @xmath22 is the transmit symbol at subcarrier @xmath23 and symbol @xmath24 , and @xmath25 is the prototype filter coefficient at the @xmath26-th time - domain sample . it is worth noting that the transmit symbols here refer to the pulse amplitude modulation ( pam ) symbols that are derived from the staggering of quadrature amplitude modulation ( qam ) symbols . thus the interval between two adjacent blocks is only half of the block period due to the offset in oqam . the parameter , @xmath27 in ( [ eq : fbmcsig ] ) , is defined as @xmath28 which is used to form the oqam structure . with a properly designed prototype filter such as iota and the oqam structure , the interference from the nearby overlapped symbols caused by a matched filter ( mf ) receiver becomes pure imaginary , which can be easily cancelled . [ fig : gfdm ] demonstrates the block diagram of gfdm . ofdm and single - carrier frequency division multiplexing ( sc - fdm ) can be regarded as two special cases of gfdm @xcite . the unique feature of gfdm is to use circular shifted filters , rather than linear filters that are used in fbmc , to perform pulse shaping . by carefully choosing the circular filter , the out - of - block leakage can be reduced even if the orthogonality is completely given up . we can flexibly adjust @xmath21 frequency samples and @xmath20 time samples for a gfdm block according to the application environment . the transmit signal for each gfdm block can be expressed as @xmath29 for @xmath30 , where @xmath22 is the transmit symbol on subcarrier @xmath23 at subsymbol @xmath24 and @xmath31 is the circular time and frequency shifted version of the prototype pulse shaping filter . in ( [ eq : gfdmsig ] ) , @xmath32 where @xmath33 denotes the @xmath34 modulo operation and @xmath25 is the prototype pulse shaping filter . similar to the traditional ofdm , the modulation process and demodulation process can be expressed by matrix operations . the idft and dft matrices in the traditional ofdm are substituted by some specific matrices corresponding to the modulation and demodulation for gfdm . but , the transceiver structure of gfdm is significantly different from the traditional ofdm . besides fbmc and gfdm , other modulations based on pulse shaping , such as pulse - shaped ofdm @xcite and qam - fbmc @xcite , have also been proposed for 5 g networks . generally , modulations based on pulse shaping try to restrict transmit signals within a narrow bandwidth and thus mitigate the oob leakage so that they can work in asynchronous scenarios with a narrow guard band . fbmc also uses oqam to achieve real - domain orthogonality , which saves the cost of the gi and interference cancellation . in addition , the circular shifted filters in gfdm avoid the long tail of the linear filters in the time domain , which makes gfdm fit for sporadic transmission . furthermore , gfdm is easily compatible to mimo technologies @xcite . subband filtering is another technique to reduce the oob leakage . universal filtered multicarrier ( ufmc ) @xcite and filtered ofdm ( f - ofdm ) @xcite are two typical modulations based on subband filtering , which will be introduced next . fig . [ fig : ufmc ] shows the transmitter and the receiver structures of ufmc @xcite . in ufmc , the subbands are with equal size , and each filter is a shifted version of the same prototype filter . ofdm is applied within a subband for this modulation as shown in the figure . since the bandwidth of the filter in ufmc is much wider than that of the modulations based on the pulse shaping , the length in time domain is much shorter . therefore , interference caused by the tail of the filter can be easily eliminated by adopting a zero - padding ( zp ) prefix with a reasonable length . assuming that @xmath35 subcarriers are divided into @xmath20 subbands , each with @xmath36 consecutive subcarriers , the transmit signal in ufmc can be expressed as @xmath37 where @xmath38 is the filter coefficient of subband @xmath23 , and @xmath39 is the ofdm modulated signal over subband @xmath23 that can be expressed as @xmath40 with @xmath41 denoting the length of the zp , @xmath21 denoting the number of symbol blocks and @xmath42 denoting the signal at subcarrier @xmath23 and symbol @xmath24 . in ( [ eq : ufmcsubsig ] ) , @xmath42 can be expressed as @xmath43 where @xmath44 denotes the @xmath45-th transmit symbol at the @xmath24-th symbol block . at the receiver , the signal at each symbol interval is with the length of @xmath46 and is zero - padded to have a length of @xmath47 so that a @xmath47-point fft can be performed . please note that only the even subcarriers are considered for signal detection after the @xmath47-point fft . f - ofdm has a similar transmitter structure as ufmc @xcite . the main difference is that f - ofdm employs a cp and usually allows residual inter - symbol interference ( isi ) @xcite . therefore , at the receiver , the mf is applied instead of the zp and decimation . besides , downsampling can be applied before the dft operation , which can reduce complexity significantly since the cp can mitigate most of interference caused by the tail of the filter ; the residual interference is with much lower power and can be treated as noise @xcite . thus , the filter in f - ofdm can be longer than that in ufmc and has better attenuation outside the band . with the aid of effective channel coding , the performance degradation caused by residual interference in f - ofdm can be negligible . another difference from ufmc is that the subcarrier spacing and the cp length do not have to be the same for different users in f - ofdm . the most widely used filter in f - ofdm is the soft - truncated sinc filter @xcite , which can be easily used in various applications with different parameters . therefore , f - ofdm is very flexible in the frequency multiplexing . besides ufmc and f - ofdm , other modulations based on subband filtering have also been proposed . for example , resource block f - ofdm ( rb - f - ofdm ) @xcite utilizes filters based on resource block instead of the whole band of users in f - ofdm . in general , modulations based on subband filtering can effectively reduce oob leakage and achieve better performance in comparison with the traditional ofdm . apart from pulse shaping and subband filtering , there are also some other techniques to suppress the oob leakage and meet the requirements of 5 g networks . in the following , we mainly introduce three other modulations , including guard interval discrete fourier transform spread ofdm ( gi dft - s - ofdm ) @xcite , spectrally - precoded ofdm ( sp - ofdm ) @xcite , and orthogonal time frequency and space ( otfs ) @xcite . in gi dft - s - ofdm @xcite , the known sequence is used as the gi instead of a cp . several types of the known sequences , such as the zero sequence @xcite and a well - designed unique word @xcite , can be used . by a fixed known sequence with constant amplitude in gi dft - s - ofdm , the peak - to - average power ratio ( papr ) of the modulated signal can be reduced . moreover , the known sequence can also be utilized to estimate the parameters , such as the carrier frequency offset ( cfo ) in the synchronization process . by utilizing a proper sequence as the gi , the discontinuity between the adjacent time blocks in the traditional ofdm / dft - s - ofdm can be avoided . as a result , the oob leakage is reduced . for gi dft - s - ofdm , the overall length of the gi and useful signal for different users is same . thus , the dft windows for different users at the receiver can still be aligned even if the lengths of the gis are different . therefore , the mutual interference due to asynchronization of users can be mitigated @xcite . [ fig : spofdm ] shows the diagram of sp - ofdm @xcite . from the figure , it consists idft and dft , spectral precoder , and iterative detector . generally , the data symbols mapped on subcarriers are precoded by a rank - deficient matrix in order to project the signal into a properly selected lower dimensional subspace so that the precoded signal can be high - order continuous , and results in much lower leakage compared with the traditional ofdm @xcite . even if precoded by a rank - deficient matrix can reduce the capacity of the channel , the oob leakage of the ofdm signals can be significantly suppressed at the cost of only few reduced dimensions . compared to the modulations based on filtering , sp - ofdm has the following three advantages : * the isi caused by the tail of the filters can be removed without filtering . therefore , the cp applied to combat the multipath of the wireless channels can be shorter , and se is improved . * when fragmented bands are used , sp - ofdm can easily notch specific well chosen frequencies without requiring multiple narrow subband filters @xcite . * furthermore , precoding and filtering can be combined to further improve the performance . the structure of otfs is similar to sp - ofdm , as can be seen in fig . [ fig : spofdm ] . the main difference is that the spectral precoder and the iterative detector are substituted by the two - dimensional ( 2d ) symplectic fourier transform and the corresponding inverse transform modules . otfs maps the symbols in the delay - doppler domain @xcite . through a 2d symplectic fourier transform , the corresponding data in the time - frequency domain can be calculated . then , the calculated data can be transmitted via a time - frequency - domain modulation method as in ofdm . since the 2d symplectic fourier transform is relatively independent of the time - frequency - domain modulation method , pulse shaping and subband filtering can also be applied together to further reduce the leakage in otfs . when a mobile is with a high speed , the channel experiences fast fading . channel parameters need to be estimated and tracked very often therefore , which significantly increases resource costs . moreover , most of the modulations are designed assuming that channels are constant within a symbol block . with a high mobility speeds , extra interference is introduced , which degrades the performance . however , in the delay - doppler domain , the high doppler channel can be expressed in a stable model , which saves the cost of tracking the time - varying fading and improves performance therefore . otfs can be also applied to estimate channel state information ( csi ) of different antennas in mimo systems @xcite . generally , the delay and doppler dispersions are still relatively small compared to the system scale . in this case , the channel can be expressed in a compact and stable form in the delay - doppler domain . as a result , the spread of the pilots caused by the channel are local , which enables to estimate the csi of different antennas in mimo systems by different pilots within a small area of the delay - doppler plane . in addition , a number of modulations based on other techniques have been also proposed , such as windowed ofdm ( w - ofdm ) @xcite , which utilizes windowing to deal with the discontinuity between adjacent ofdm symbols . we compare the power spectral density ( psd ) and bit - error rate ( ber ) of different modulations . suppressing the oob leakage is a key purpose for most of the modulation candidates for 5 g networks . the psds of the some modulations are shown in fig . [ fig : psd ] . from the figure , all modulations achieve much lower leakage compared to the traditional ofdm . among them , ufmc applies subband filtering and also has low leakage , and fbmc and f - ofdm have the lowest leakage . gfdm , gi dft - s - ofdm , and sp - ofdm , although do not reduce the leakage as much as fbmc and f - ofdm , can still achieve much better performance than the traditional ofdm . in order to reduce the oob leakage , many modulations utilize techniques , such as pulse shaping and subband filtering , which may introduce isi and ici . hence , the ber performance of different modulations is compared here . [ fig : ber ] shows the ber performance versus signal - to - noise ratio ( snr ) when the doppler spread @xmath48 and @xmath49 hz . from fig . [ fig : ber ] ( a ) , the traditional ofdm has the best performance when the doppler spread is zero ( @xmath48 ) since the isi caused by the multipath has been completely canceled by the cp . since the bandwidth of each subcarrier is small enough to make the corresponding channel approximately flat , the isi introduced by pulse shaping in fbmc is nearly pure imaginary . therefore , fbmc is approximately orthogonal in the real domain and achieves good ber performance . the performance of ufmc , gfdm , and sp - ofdm is similar to that of fbmc , which is degraded slightly due to noise enhancement and low - projection precoding . however , f - ofdm introduces extra isi that can not be completely canceled , and as a result , it has slightly worse performance , especially in the high snr region . gi dft - s - ofdm and otfs , which are different from the modulation schemes that directly map the symbols on subcarriers , apply spreading before mapping so that their performance does not approach that of ofdm . since the fast - fading channel is difficult to be estimated and tracked accurately , the performance of the most modulation schemes degrades significantly as we can see from fig . [ fig : ber ] ( b ) . while otfs can still achieve good performance due to its specific channel estimation method . moreover , its performance in the high - mobility scenario is even better than that in the zero doppler shift scenario because of doppler diversity . in this section , modulation techniques for 5 g networks will be be discussed . these techniques can be used with oma to effectively deal with the oob leakage in 5 g networks . however , there are still many open issues in the area . a potential application of f - ofdm is iot . in this scenario , the subbands are narrow and therefore , interference caused by a short cp can significantly degrade the performance and should be considered in the detection . to improve the detection performance , additional processing , such as filtering or successive interference cancellation ( sic ) , is needed . residual isi cancellation ( risic ) @xcite could be helpful . the existing designs for subband filtering , such as dolph - chebyshev filter in ufmc and soft - truncated filter in f - ofdm , are with fixed length . however , different users and different application scenarios will have different requirements on the leakage levels , filter lengths , etc . according to the heisenberg - gabor uncertainty principle @xcite , the time and frequency dispersions are dual variables that can not be reduced at the same time . therefore , how to balance the time and frequency dispersions and design an efficient prototype filter according to application scenarios is interesting . similar to the traditional ofdm , multi - carrier based new modulation candidates , such as fbmc and ufmc also have a large papr . in order to improve the efficiency of the power amplifier , the papr should be reduced . the traditional papr reduction methods @xcite applied in the traditional ofdm usually introduce distortions that degrade the performance . therefore , how to properly extend the papr reduction methods in the traditional ofdm to the new modulations is an interesting and meaningful issue . in order to support higher throughput and massive and heterogeneous connectivity for 5 g networks , we can adopt novel modulations discussed in section ii for oma , or directly use noma with effective interference mitigation and signal detection methods . the key features of noma can be summarized as follows : 1 . improved se : noma exhibits a high se , which is attributed to the fact that it allows each resource block ( e.g. , time / frequency / code ) to be exploited by multiple users . 2 . ultra high connectivity : with the capability to support multiple users within one resource block , noma can potentially support massive connectivity for billions of smart devices . this feature is quite essential for iot scenarios with users that only require very low data rates but with massive number of users . 3 . relaxed channel feedback : in noma , perfect uplink csi is not required at the base station ( bs ) . instead , only the received signal strength needs to be included in the channel feedback . low transmission latency : in the uplink of noma , there is no need to schedule requests from users to the bs , which is normally required in oma schemes . as a result , a grant - free uplink transmission can be established in noma , which reduces the transmission latency drastically . existing noma schemes can be classified into three categories : power - domain noma , code - domain noma , and noma multiplexing in multiple domains . we will introduce them subsequently with emphasis on power - domain noma . power - domain noma is considered as a promising ma scheme for 5 g networks @xcite . specifically , a downlink version of noma , named multiuser superposition transmission ( must ) , has been proposed for the 3gpp long - term evolution advanced ( 3gpp - lte - a ) networks @xcite . it has been shown that system capacity and user experiences can be improved by noma . more recently , a new work item ( wi ) outlining downlink multiuser superposition transmission for lte has been approved by 3gpp lte release 14 @xcite , which aims to identify the necessary techniques to enable lte to support the downlink intra - cell multiuser superposition transmission . here , we will expand upon the basic principles of various power - domain noma related techniques , including multiple antenna based noma , power allocation in noma , and cooperative noma . power - domain noma , as illustrated in fig . [ fig : noma ] for the two user case , deviates from conventional oma that uses tdma / fdma / cdma / ofdma allocating orthogonal resource blocks for different users to avoid the multiple access interference ( mai ) . instead power - domain noma can support multiple users within the same resource block by distinguishing them with different power levels . as a result , noma is able to support more connectivity and provide higher throughput with limited resources . the downlink transmission of noma for the two user case is shown in fig . [ fig : noma ] where the users are served at the same time / frequency / code resource block with a total power constraint . specifically , the bs sends a superimposed signal containing the two signals for the two users . this differs from conventional power allocation strategies , such as water filling , as noma allocates less power for the users with better downlink csi , to guarantee overall fairness and to utilize diversity in the time / frequency / code domains . sic is used for signal detection at the receiver . the user with more transmit power , that is , the one with smaller downlink channel gain , is first to be decoded while treating the other user s signal as noise . once the signal corresponding to the user with the larger transmit power is detected and decoded , its signal component will be subtracted from the received signal to facilitate the detection of subsequent users . it should be noted that the first detected user is with the largest inter - user interference and also the detection error in the first user will pass to the other user , which is why we have to allocate sufficient power to the first user to be detected . the extension of noma from two to multiple user cases is straightforward . for the uplink transmission of noma , the transmit power is limited by each individual user . different from the downlink , the transmit powers of the users using the same resource block are carefully controlled so that the received signal components at the bs corresponding to the users with the better csi , have more powers . at the receiver ( the bs ) , the user with the best csi is decoded first . after that , the corresponding component is removed from the received signal . the sic receiver works in a descending order of the csi , which is the opposite to the downlink case . [ fig : noma_oma ] compares noma and oma where two users are served by the same bs if noma is adopted . from the figure , the noma scheme achieves a lower outage probability . however , by adopting noma , a more complex transmitter and receiver are required to mitigate the interference . furthermore , power - domain noma usually works well when only two or a few users share the same resource block . as the number of users multiplexing in power domain increases , the mai becomes severe and the performance of noma degrades . multiple antenna techniques can provide an additional degree of freedom on the spatial domain , and bring further performance improvements to noma . recently , multiple antenna based noma has attracted lots of attention @xcite . different from single - input - single - output ( siso ) based noma , where the channels are normally represented by scalars , one of the research challenges in multiple antenna based noma comes from user ordering ; as the channels are generally in form of vectors or matrices . currently , the possible designs of multiple antenna based noma fall into two categories where one or multiple users are served by a single beamforming vector . by allocating different users with different beams in the same resource block , the quality of service ( qos ) of each user can be guaranteed in multiple antenna based noma systems forcing the beams to satisfy a predefined order . this type of multiple antenna based noma scheme has been first proposed by sun _ et al . _ in @xcite to investigate power optimization to maximize the ergodic capacity . this proposed multiple antenna based noma scheme has proved to be able to achieve significant performance improvement compared with conventional oma schemes . a cluster of users can share the same beam . the spatial channels of different users within the same cluster are considered to be highly correlated . therefore , beams for different clusters should be carefully designed to guarantee that the channels for different clusters are orthogonal to each other in order to suppress the inter - cluster interference . for multiple - input - single - output ( miso ) based noma , a two - stage multicast beamforming scheme has been proposed by choi in @xcite , where zf beamforming has been employed to mitigate interference from adjacent clusters first and then the optimal beamforming vectors have been designed to minimize the total transmit power within each cluster . for mimo based noma , a scheme to simultaneously apply open - loop random beamforming and intra - beam sic , has been proposed by higuchi and kishiyama in @xcite . however , here the system performance is considerably degraded as the random beamforming can bring uncertainties at the user side . more recently , a precoding and detection framework with fixed power allocation has been proposed by ding _ et al . _ @xcite to solve these problems caused by random beamforming , and demonstrated that mimo based noma can achieve better outage performance than mimo based oma even for users who experience strong co - channel interference . a comprehensive summary for the state - of - the - art work on multiple antenna based noma is given in table [ table : mimo noma ] , where `` bf '' , `` op '' , `` su '' and `` mu '' are used to represent beamforming , outage probability , two - user and multi - users cases , respectively . [ cols="<,<,<,<,<",options="header " , ] & no need for user clustering & specific channel coding + [ table : comparison ] several noma schemes have been discussed in this section . even if using different techniques , these schemes share the same spirit to utilize non - orthogonality to increase the system capacity and support more users by the limited resource blocks . beyond the existing work , more research is necessary to improve the performance of these noma schemes from the following aspects . the mpa - sic detection method is usually applied in scma and pdma , in which the user clustering mechanism affects the performance of the method significantly . when users are asynchronous , those with similar time delays should be divided into the same cluster for better performance . if the delays vary a lot among the users within the same cluster , interference among different users becomes large and may break the sparse structure . multi - branch technique @xcite can be applied to improve the performance by regarding each cluster as a branch . by calculating each branch in parallel and selecting the best result as the final one , the performance could be improved compared to the single clustering approach . the joint design of new modulation and noma schemes is an important direction to be explored in 5 g networks . some of the noma schemes , especially the lds based code - domain noma , are based on ofdm , where the output of the sparse spreading matrix is mapped into orthogonal subcarriers . in general , how to properly combine the modulation and noma scheme is under research . for example , for the combination of scma and f - ofdm , the short cp of f - ofdm could introduce isi and ici when the subband is narrow and degrade the detection performance of scma . if the risic algorithm is adopted to cancel the interference introduced by f - ofdm , the multiuser detection of scma should be included in the iteration of cp reconstruction , which poses a requirement of joint design approaches for the receivers . the design of modulation and ma schemes for high frequency bands ( above 40 ghz ) is beginning to receive increased iterest . the millimeter - wave ( mmwave ) and terahertz ( thz ) bands appear to be good candidates to decrease spectrum sacristy due to the availabilities in current circuit design @xcite . however , the propagation properties of mmwave and thz bands have shown to be quite poor , which brings new challenges on system designs . for example , noise is the major limitation of mmwave and thz bands , which makes the transmit power levels extremely important and ultimately impacts the classes of applications that can use them ( e.g. iot ) . moreover , high level impairments including carrier frequency offset ( cfo ) and phase noise also need to be considered in mmwave and thz bands as they are noise - limited . nevertheless , there is already a study on noma based mmwave communications @xcite , we may further see analyses of such systems based on practical scenarios in the future . in this article , we provide a comprehensive survey covering the major promising candidates for modulation and multiple access ( ma ) in fifth generation ( 5 g ) networks . from our discussion , we can see that new modulations for orthogonal ma can be adopted to reduce out - of - band leakage while meeting the diverse demands of 5 g networks . non - orthogonal ma is another promising approach that marks a deviation from the previous generations of wireless networks . by utilizing non - orthogonality , we have convincingly shown that 5 g networks will be able to provide enhanced throughput and massive connectivity with improved spectral efficiency . h. sampath , s. talwar , j. tellado , v. erceg , and a. paulraj , `` a fourth - generation mimo - ofdm broadband wireless system : design , performance , and field trial results , '' _ ieee commun . mag . _ , vol . 40 , no . 9 , pp . 143149 , sep . 2002 . v. vakilian , t. wild , f. schaich , s.brink , and j. f. frigon , `` universal - filtered multi - carrier technique for wireless systems beyond lte , '' in _ proc . ieee globecom workshops ( gc wkshps ) _ , atlanta , ga , usa , dec . 2013 , pp . 223228 . f. schaich , t. wild , and y. chen , `` waveform contenders for 5 g - suitability for short packet and low latency transmissions , '' in _ proc . ( vtc spring ) _ , seoul , korea , may 2014 , pp . j. abdoli , m. jia , and j. ma , `` filtered ofdm : a new waveform for future wireless systems , '' in _ proc . ieee 16th int . workshop signal process . adv . wireless commun . ( spawc ) _ , stockholm , sweden , jun . 2015 , pp . 6670 . x. zhang , m. jia , l. chen , j. ma , and j. qiu , `` filtered - ofdm - enabler for flexible waveform in the 5th generation cellular networks , '' in _ proc . ieee global commun . ( globecom ) _ , san diego , ca , usa , dec . 2015 , pp . 16 . m. bellanger , m. renfors , t. ihalainen , and c.a.f . da rocha , `` ofdm and fbmc transmission techniques : a compatible high performance proposal for broadband power line communications , '' in _ int . . power line commun . and its applications ( isplc ) _ , rio de janeiro , brazil , mar . 2010 , pp . 154159 . n. michailow , m. matth , i.s . gaspar , a. n. caldevilla , l. l. mendes , a. festag , and g. fettweis , `` generalized frequency division multiplexing for 5th generation cellular networks , '' _ ieee trans . _ , vol . 62 , no . 9 , pp . 30453061 , sep . 2014 . a. sahin , i. guvenc , and h. arslan , `` a survey on multicarrier communications : prototype filters , lattice structures , and implementation aspects , '' _ ieee commun . surveys tu . _ , vol . 16 , no . 3 , pp . 13121338 , third quarter , 2014 . z. zhao , m. schellmann , q. wang , x. gong , r. boehnke , and w. xu , `` pulse shaped ofdm for asynchronous uplink access , '' in _ asilomar conf . signals , systems and computers _ , monterey , usa , nov . 2015 , pp . 37 . c. kim , k. kim , y. h. yun , z. ho , b. lee , and j. y. seol , `` qam - fbmc : a new multi - carrier system for post - ofdm wireless communications , '' in _ proc . ieee global commun . conf . ( globecom ) _ , san diego , ca , usa , dec . 2015 , pp . 16 . g. berardinelli , f. m. l. tavares , t. b. sorensen , p. mogensen , and k. pajukoski , `` zero - tail dft - spread - ofdm signals , '' in _ proc . ieee globecom workshops ( gc wkshps ) _ , atlanta , ga , usa , dec . 2013 , pp . 229234 . a. sahin , r. yang , m. ghosh , and r. l. olesen , `` an improved unique word dft - spread ofdm scheme for 5 g systems , '' in _ proc . ieee globecom workshops ( gc wkshps ) _ , san diego , ca , usa , dec . 2015 , pp . p. achaichia , m. l. bot , and p. siohan , `` windowed ofdm versus ofdm / oqam : a transmission capacity comparison in the homeplug av context , '' in _ int . . power line commun . and its applications ( isplc ) _ , udine , italy , apr . 2011 , pp . 405410 x. li and l. j. cimini , `` effects of clipping and filtering on the performance of ofdm , '' _ ieee commun . lett . _ , vol . 2 , no . 5 , pp . 131133 , may 1998 . k. higuchi and y. kishiyama , `` non - orthogonal access with random beamforming and intra - beam sic for cellular mimo downlink , '' in _ proc . ( vtc fall ) _ , las vegas , nv , usa , sep . 2013 , pp . 15 . k. higuchi and a. benjebbour , `` non - orthogonal multiple access ( noma ) with successive interference cancellation for future radio access , '' _ ieice trans . _ , vol . 98 , no . 3 , pp . 403414 , 2015 . z. ding , r. schober , and h. v. poor , `` a general mimo framework for noma downlink and uplink transmission based on signal alignment , '' _ ieee trans . wireless commun . 15 , no . 6 , pp . 4438 - 4454 , jun . 2016 . z. qin , y. liu , z. ding , y. gao and m. elkashlan , `` physical layer security for 5 g non - orthogonal multiple access in large - scale networks , '' in _ proc . conf . on commun . ( icc ) _ , kuala lumpur , malaysia , may 2016 , pp . 1 - 6 . y. liu , z. qin , m. elkashlan , y. gao , and l. hanzo , `` enhancing the physical layer security of non - orthogonal multiple access in large - scale networks , '' _ ieee trans . wireless commun . _ , to appear in 2017 . y. saito , y. kishiyama , a. benjebbour , t. nakamura , a. li , and k. higuchi , `` non - orthogonal multiple access ( noma ) for cellular future radio access , '' in _ proc . ( vtc spring ) _ , dresden , germany , jun . 2013 , pp.15 . y. saito , a. benjebbour , y. kishiyama , and t. nakamura , `` system - level performance evaluation of downlink non - orthogonal multiple access ( noma ) , '' in _ proc . personal , indoor , and mobile radio commun . ( pimrc ) _ , london , uk , sep . 2013 , pp . 611 - 615 . z. ding , z. yang , p. fan , and h. v. poor , `` on the performance of non - orthogonal multiple access in 5 g systems with randomly deployed users , '' _ ieee signal process . _ , vol . 21 , no . 12 , pp . 15011505 , dec . 2014 . y. liu , z. ding , m. elkashlan , and h. v. poor , `` cooperative non - orthogonal multiple access with simultaneous wireless information and power transfer , '' _ ieee j. sel . areas commun . _ , vol . 34 , no . 4 , apr . 2016 . f. liu , p. mahonen , and m. petrova , `` proportional fairness - based user pairing and power allocation for non - orthogonal multiple access , '' in _ proc . personal , indoor , and mobile radio commun . ( pimrc ) _ , hong kong , p.r . china , aug . 2015 , pp . 11271131 . f. liu , p. mahonen , and m. petrova , `` proportional fairness - based power allocation and user set selection for downlink noma systems , '' in _ proc . on commun . ( icc ) _ , kuala lumpur , malaysia , may 2016 , pp . j. mei , l. yao , h. long , and k. zheng , `` joint user pairing and power allocation for downlink non - orthogonal multiple access systems , '' in _ proc . conf . on commun . ( icc ) _ , kuala lumpur , malaysia , may 2016 , pp . n. otao , y. kishiyama , and k. higuchi , `` performance of non - orthogonal access with sic in cellular downlink using proportional fair - based resource allocation , '' in _ proc . wireless commun . ( iswcs ) _ , paris , france , aug . 2012 , pp . 476480 . y. sun , d. w. k. ng , z. ding , and r. schober , `` optimal joint power and subcarrier allocation for mc - noma systems , '' in _ proc . ieee global commun . ( globecom ) _ , washington , dc , usa , 2016 , pp . 1 - 6 . m. al - imari , p. xiao , m. a. imran , and r. tafazolli , `` uplink non - orthogonal multiple access for 5 g wireless networks , '' in _ proc . int . wireless commun . ( iswcs ) _ , barcelona , spain , aug . 2014 , pp . 781785 . l. lei , d. yuan , c. k. ho , and s. sun , `` power and channel allocation for non - orthogonal multiple access in 5 g systems : tractability and computation , '' _ ieee trans . wireless commun . 8580 - 8594 , dec . 2016 . d. diamantoulakis , k. n. pappi , z. ding , and g. k. karagiannidis , `` wireless powered communications with non - orthogonal multiple access , '' _ ieee trans . wireless commun . 8422 - 8436 , dec . 2016 d. lee , h. seo , b. clerckx , e. hardouin , d. mazzarese , s. nagata and k. sayana , `` coordinated multipoint transmission and reception in lte - advanced : deployment scenarios and operational challenges , '' _ ieee commun . mag . 148 - 155 , feb . 2012 . y. liu , z. ding , m. elkashlan and j. yuan , `` non - orthogonal multiple access in large - scale underlay cognitive radio networks , '' _ ieee trans . veh . 10152 - 10157 , dec . 2016 y. liu , z. qin , m. elkashlan , y. gao , and n. arumugam , `` non - orthogonal multiple access in massive mimo aided heterogeneous networks , '' in _ proc . of global commun . ( globecom ) _ , washington d.c , usa , dec . 2016 . f. fang , h. zhang , j. cheng , and v. c. m. leung , `` energy - efficient resource allocation for downlink non - orthogonal multiple access network , '' _ ieee trans . commun . _ , vol . 64 , no . 9 , pp . 37223732 , sep . 2016 . s. m. r. islam , n. avazov , o. a. dobre , k. s. kwak , `` power - domain non - orthogonal multiple access ( noma ) in 5 g systems : potentials and challenges , '' in ieee commun . surveys tutorials , vol.pp , no.99 , pp.1 - 1 , nov . 2016 . m. al - imari , p. xiao , m. a. imran , and r. tafazolli , `` uplink non - orthogonal multiple access for 5 g wireless networks , '' in _ proc . wireless commun . ( iswcs ) _ , barcelona , spain , aug . 2014 , pp . 781785 . d. cai , p. fan , x. lei , y. liu , and d. chen , `` multi - dimensional scma codebook design based on constellation rotation and interleaving , '' in _ proc . technol . conf . ( vtc spring ) _ , nanjing , china , may 2016 , pp . 15 . s. chen , b. ren , q. gao , s. kang , s. sun , and k. niu , `` pattern division multiple access ( pdma ) - a novel non - orthogonal multiple access for 5 g radio networks , '' _ ieee trans . _ , vol . pp , no . 99 , pp . 11 , 2016 . d. fang , y. huang , z. ding , g. geraci , s. l. shieh , and h. claussen , `` lattice partition multiple access : a new method of downlink non - orthogonal multiuser transmissions , '' in _ proc . of global commun . ( globecom ) _ , washington d.c , usa , dec . j. zeng , b. li , x. su , l. rong , and r. xing , `` pattern division multiple access ( pdma ) for cellular future radio access , '' in _ proc . wireless commun . signal process . ( wcsp ) _ , nanjing , jiangsu , china , oct . 2015 , pp . 15 . b. ren , x. yue , w. tang , y. wang , s. kang , x. dai , and s. sun , `` advanced idd receiver for pdma uplink system , '' in _ proc . ieee / cic int . conf . commun . in china ( iccc ) _ , chengdu , china , jul . 2016 , pp . 1 - 6 . y. huang and k. r. narayanan , `` construction @xmath50 and @xmath51 lattices : construction , goodness , and decoding algorithms , '' _ arxiv preprint _ , [ online ] . available : http://arxiv.org/abs/1506.08269 , aug . 2016 . y. cai and r. c. de lamare , `` multistage mimo receivers based on multi - branch interference cancellation for mimo - cdma systems , '' in _ proc . wireless commun . ( iswcs ) _ , tuscany , italy , sep . 2009 , pp .

fifth generation ( 5 g ) wireless networks face various challenges in order to support large - scale heterogeneous traffic and users , therefore new modulation and multiple access ( ma ) schemes are being developed to meet the changing demands . as this research space is ever increasing , it becomes more important to analyze the various approaches , therefore in this article we present a comprehensive overview of the most promising modulation and ma schemes for 5 g networks . we first introduce the different types of modulation that indicate their potential for orthogonal multiple access ( oma ) schemes and compare their performance in terms of spectral efficiency , out - of - band leakage , and bit - error rate . we then pay close attention to various types of non - orthogonal multiple access ( noma ) candidates , including power - domain noma , code - domain noma , and noma multiplexing in multiple domains . from this exploration we can identify the opportunities and challenges that will have significant impact on the design of modulation and ma for 5 g networks . 5 g , modulation , non - orthogonal multiple access .

1702.07673

galaxy mergers ( toomre 1977 ) , galaxy harassment ( moore et al . 1996 ) , ram pressure ( gunn & gott 1972 ) , viscous stripping ( nulsen 1982 ) and dynamical friction ( lecar 1975 , chandrasekhar 1943 ) are believed to play a role in the evolution of galaxies in a cluster enviroment and have explained numerous observational results . different physical mechanisms dominate in different environments . while ram pressure effects are believed to be dominant in the dense inner regions of clusters , tidal interactions are believed to dominate the evolution in the cluster outskirts and in groups of galaxies . however , this picture has been undergoing a gradual paradigm shift as demonstrated by observations , especially at radio frequencies , and simulations of groups and outskirts of clusters ( e.g. virgo ) conducted by various authors . the virgo cluster galaxies have been extensively studied in radio and other wavebands e.g. in hi 21 cm by davies , et al . ( 2004 ) , cayatte et al . ( 1990 ) , warmels ( 1988 ) , huchtmeier & richter ( 1986 ) and in radio continuum by vollmer et al . chung et al . ( 2007 ) have recently reported hi tails for several virgo galaxies which they conclude is ram pressure stripped gas from the galaxy . vollmer et al . ( 2007 ) have studied the polarisation properties of several virgo spirals and inferred that the interaction of the galaxy with the intracluster medium ( icm ) has resulted in peculiar polarisation properties of the cluster spirals . wezgoweic et al . ( 2007 ) find distorted magnetic fields in several virgo cluster members which they attribute to ram pressure of the icm . ngc 4254 is an interesting almost face - on sa(s)c spiral , located in the outskirts of the virgo cluster , with one dominant spiral arm . this galaxy has been studied extensively in the hi line ( cayatte et al . 1990 , 1994 , phookun et al . 1993 ) , in high frequency radio continuum and polarization ( urbanik et al . 1986 , soida et al . 1996 , urbanik 2004 chyzy et al . 2006 ) , in x - rays ( chyzy et al . 2006 , soria & wong 2006 ) and via simulations involving tidal interaction and ram pressure stripping ( vollmer et al 2005 ) . ngc 4254 is both optically and radio bright and lies about @xmath5 ( @xmath6 mpc using a distance of 17 mpc to virgo cluster ) to the north - west of m87 . phookun et al . ( 1993 ) , from their deep hi observations , separate the disk and non - disk emission from the galaxy and find that the latter forms a tail of clouds that extend to @xmath7 from the galaxy . they conclude that there is an infall of a disintegrating cloud of gas onto ngc 4254 and that the one - armed structure is a result of the tidal interaction with the infalling gas . vollmer et al ( 2005 ) have modelled ngc 4254 after including a tidal encounter and ram pressure stripping . they find that the one - armed structure can be explained by a rapid and close encounter @xmath8 years ago with another massive galaxy ( @xmath9 ) . they note that most of the observed hi morphology is reproduced by including ongoing ram pressure effects in the simulation and only two main hi features remain unexplained namely 1 ) the shape of the extended tail of hi emission in the north - west , 2 ) a low surface density hi blob observed to the south of the main disk . davies et al . ( 2004 ) and minchin et al . ( 2005 ) have reported the detection of a massive ( @xmath10 ) hi cloud virgohi 21 in the virgo cluster . this cloud lies to the north of ngc 4254 and connects both spatially and kinematically to ngc 4254 ( minchin et al . recently haynes et al . ( 2007 ) have presented the hi map of this entire region made using the arecibo telescope . they detect a long hi tail which starts from ngc 4254 , includes virgohi 21 and extends northwards to a total distance of 250 kpc . earlier observations had only detected parts of this long hi tail . haynes et al . ( 2007 ) attribute this long hi tail to galaxy harrassment ( moore et al . 1996 ) which the galaxy is undergoing as it enters the virgo cluster . soida et al . ( 1996 ) , from their radio continuum observations near 5 and 10 ghz , have reported enhanced polarisation in the southern ridge which they attribute to interaction of the disk gas with the icm . chyzy et al . ( 2006,2007 ) detect an extended polarized envelope at 1.4 ghz . excess blue emission is observed along the southern ridge ( knapen et al . 2004 ) which indicates the presence of a large population of young blue stars ( wray 1988 ) . the trigger for the enhanced star formation in this region is possibly compression by the ram pressure of the icm . from high resolution co data , sofue et al . ( 2003 ) note that the inner spiral arms of ngc 4254 are asymmetric . they reason that the ram pressure of the icm has distorted the inter - arm low density regions leading to the asymmetric spiral arms . this galaxy has also been observed in the near and far ultraviolet by galex ( gil de paz et al . the uv morphology of the galaxy is similar to the dss optical with several young star forming regions visible in the galex images . in light of all these interesting results , we present new low frequency radio continuum and hi 21 cm observations of ngc 4254 using the giant metrewave radio telescope ( gmrt ) near pune in india . the continuum observations were conducted at four frequencies between 240 and 1280 mhz . we detected extended continuum emission surrounding the optical disk of ngc 4254 . we discuss the integrated spectrum of the galaxy and the environmental effects on this galaxy located at the periphery of the cluster . we adopt a distance of 17 mpc to the virgo cluster following vollmer et al . ( 2005 ) . in table [ tab1 ] , we summarize the physical properties of ngc 4254 and the empirical data from wavebands ranging from x - rays to radio . lcc & * value / property * & * reference * + @xmath11 & @xmath12 & rc3 + @xmath13 & @xmath14 & rc3 + morphological type & sc & rc3 + optical diameter d@xmath15 & @xmath16 & rc3 + b@xmath17 & 10.44 & rc3 + heliocentric velocity & @xmath18 kms@xmath3 & rc3 + distance & @xmath19 mpc & vollmer et al . ( 2005 ) + distance to cluster centre & @xmath6 mpc & + + morphology & distorted & cayatte et al . ( 1990 ) + v ( rotation ) & 150 kms@xmath3 & phookun et al . ( 1993 ) + inclination angle & @xmath20 & phookun et al . ( 1993 ) + non - disk emission & @xmath21 & phookun et al . ( 1993 ) + hi deficiency & @xmath22 & cayatte et al . ( 1994 ) + icm ram pressure & @xmath23 dyn-@xmath24 & cayatte et al . ( 1994 ) + grav pull & @xmath25 dyn-@xmath24 & cayatte et al . ( 1994 ) + galaxy harrassment & 250 kpc tail & haynes et al . ( 2007 ) + + radio spectrum ( 0.4 - 10 ghz ) & @xmath26 & soida et al . ( 1996 ) + polarised emission & excess in south & soida et al . ( 1996 ) + polarised envelope & yes & chyzy et al . ( 2007 ) + + structure & one - arm morphology & ... + star formation & vigorous & ... + blue stars & forms a ridge in south & wray ( 1988 ) + b band & excess in south & knapen et al . ( 2004 ) + + inner arms & asymmetric & sofue et al . ( 2003 ) + possible reason & icm ram pressure & sofue et al . ( 2003 ) + + galaxy & intense emitter & fabbiano et al . ( 1992 ) + discrete source & ulx in south & soria & wong ( 2006 ) + + one arm & tidal & vollmer et al . ( 2005 ) + obs . hi structure & tidal+ram press & vollmer et al . ( 2005 ) + stripping angle & @xmath27 & vollmer et al . ( 2005 ) + [ tab1 ] ngc 4254 was observed at 240 mhz , 325 mhz , 610 mhz and 1280 mhz in the radio continuum and in the 21 cm emission line of hi using the gmrt ( swarup et al . 1991 , ananthakrishnan & rao 2002 ) . gmrt consists of thirty 45-m diameter parabolic dish antennas spread over a 25 km region and operates at five frequency bands below 1.5 ghz . details of our observations are listed in table [ tab2 ] . [ cols="<,^,^,^,^,^ " , ] @xmath28 128 channels over the 8 mhz bandwidth resulted in this channel width ( i.e. @xmath29 kms@xmath3 ) . @xmath30 using data on baselines upto @xmath31 . @xmath32 rms is for a channel of width 12.5 kms@xmath3 . [ tab2 ] the visibility data obtained at gmrt in ` lta ' format were converted to fits and imported to aips . the analysis was carried out using standard tasks in aips . since the field - of - view at 610 , 325 and 240 mhz is large , the images were generated using multiple facets . the final images were generated using robust weighting ( briggs 1995 ) of the visibilities . we made the images using robust=0 ( between pure uniform weighting i.e. robust@xmath33 and pure natural weighting i.e. robust@xmath34 ) and robust=5 . in fig . [ fig1 ] , we show the images made using robust=0 . a sharp cutoff to the radio emission is observed to the south / south - east ( see the 325 and 1280 mhz images in fig . [ fig1 ] ) . a radio continuum minimum is detected in the north - west at all frequencies although the extent and shape varies . part of this difference is likely due to the different spatial frequency coverage of the multi - frequency observations . this minimum is accentuated in the naturally weighted maps with it being most prominent at 240 mhz ( see fig . [ fig3]b ) . the 1280 mhz image ( robust=0 ) was made using a maximum baseline of 14 k@xmath35 to highlight the extended features . in fig . [ fig2 ] , the 610 mhz image has been superposed on the dss optical image to show the envelope of radio emission around the optical disk . we note that our images at 240 mhz and 610 mhz made using robust=5 are by far the most sensitive low frequency images of this galaxy with rms noise of 0.9 mjy / beam and 0.2 mjy / beam respectively , for the beamsizes listed in table [ tab2 ] . the signal - to - noise ratio on the robust=5 image is higher than on the robust=0 image by a factor of about 3 . in fig . [ fig3 ] , the robust=0 and robust=5 images at 240 mhz are shown for comparison . the correlation between the images is good as seen from the radio emission which extends outwards of the optical disk in the north - west , south - east and south - west and is present in both the maps . for rest of the discussion , we use the naturally weighted images at 240 and 610 mhz unless stated . the integrated flux density of ngc 4254 at different frequencies are listed in table [ tab2 ] . visibility data corresponding to the three confusing sources ( marked in fig . [ fig3 ] ) which are engulfed by the radio emission from ngc 4254 were removed before generating the images used for constructing the spectral index maps . an 8 mhz bandwidth spread over 128 channels and centred at 2407 kms@xmath3 resulted in a channel width of 62.5 khz @xmath36 12.5 kms@xmath3 for our hi observations . 3c147 and 1120 + 143 were used as the flux and phase calibrators respectively . the hi data were analysed in aips++ ver 1.9 ( build 1460 ) . the continuum was estimated by fitting a first order baseline to the line - free channels and subtracted from the data . a spectral line cube was then generated from this database . the moment maps shown in fig . [ fig4 ] were obtained using a rms noise cutoff of @xmath37 in the channel maps . the full resolution maps had an angular resolution of about @xmath38 with a position angle of @xmath39 . the hi distribution ( fig . [ fig4 ] ) across ngc 4254 shows emission detected from the disk but little hi is coincident with the radio continuum extensions in the south - east , the north - west and the south - west . the lowest column density is @xmath40 @xmath24 . the velocity field ( fig . [ fig4]b ) shows that the north - east is the receding side of the disk . a slight lopsidedness is visible in the velocity field with the north - east showing lower rotation ( @xmath41 kms@xmath3 ) velocities compared to the south - west ( @xmath42 kms@xmath3 ) wrt to the systemic velocity ( @xmath43 kms@xmath3 ) of the galaxy . [ fig4]c shows the velocity dispersion across the galaxy . widest spectral lines are seen near the centre whereas rest of the disk shows line widths of @xmath44 kms@xmath3 . the disk has a smooth boundary to the south , south - east whereas the northern boundary is jagged as if this part of hi is being ` blown ' off . interestingly , this is also the region where phookun et al . ( 1993 ) detect the largest non - disk hi cloud . we do not detect the non - disk hi clouds ( phookun et al . 1993 ) or the long hi tail ( haynes et al . 2007 ) associated with ngc 4254 likely due to lack of short spatial frequencies resulting in lower sensitivity to such extended faint features . we enumerate below the interesting morphological features detected in the radio continuum emission ( figs . [ fig1 ] , [ fig2 ] , [ fig3 ] ) and the hi ( fig . [ fig4 ] ) from ngc 4254 . 1 . radio continuum emission and hi are detected from the optical disk . bright abrupt cutoff in the radio emission to the south of the galaxy which outlines the optical disk . low surface brightness emission enveloping the disk emission is detected at all frequencies . these features are prominent at 240 mhz ( see fig . [ fig3 ] ) . most notable features are the extension to the north - west , south - east and the finger - like extensions in the south - west which is likely the gas blown out by the icm pressure . no r band or h@xmath45 emission is detected from these regions . a minimum in the radio continuum is detected in the north - west at all the frequencies . this minimum is most extended and prominent at 240 mhz . chyzy et al . ( 2006,2007 ) report low polarization in the region which appears to be partially coincident with the minimum at 240 mhz . thus the minimum could likely be due to a decrease in the magnetic field in that region . about 15% of the total emission at 240 mhz arises in the emission which surrounds the optical disk . the hi disk is extended in the north - east ( see fig . [ fig4 ] ) . the hi velocity field is lopsided . lower recessional velocities are recorded in the north - east and the same region also shows lower column densities . moreover , the velocity field in the north appears to lag . the hi has a jagged border in the north , north - west ( see fig . [ fig4 ] ) suggesting that the atomic gas is being progressively ` blown ' off . a smooth boundary is seen in the south , south - east . moreover the hi shows a vertical edge in the north - east and the west as if gas has been stripped off . hi is not found to be coincident with the radio continuum envelope around ngc 4254 ( see fig . [ fig4 ] ) . we compare the radio continuum envelope to the hi low resolution map in phookun et al . ( 1993 ) who detected disk and non - disk hi components . phookun et al . ( 1993 ) inferred that the non - disk hi clouds around ngc 4254 were an infalling population . a similar conclusion can be drawn from the hi velocity field of the non - disk clouds which are likely to be located between us and the galaxy . the hi blob ( fig 5 in phookun et al . 1993 ) to the south of ngc 4254 is coincident with part of southern extension we detect at 240 mhz . moreover , some of the northern hi clouds are coincident with the northern radio continuum extension ( see fig [ fig3 ] ) . however , it is not clear if these are physically associated or coincident in projection . the hi velocity field is distorted in the north of the galaxy ( fig 4 ( b ) in phookun et al . we estimate a global spectral index between 240 and 610 mhz of @xmath46 . however , fitting a single power law using the least squares method to all the available data ranging from 408 mhz to 10 ghz ( soida et al . 1996 , chyzy et al . 2007 ) and including our data ( excluding the 1280 mhz point ) resulted in the best fit spectrum with an index of @xmath47 ( see fig [ fig5 ] ) . this is in good agreement with the value ( @xmath48 ) that soida et al . ( 1996 ) quote . however @xmath49 obtained from our low frequency data suggests that the spectrum might be flattening at the lower frequencies . since the galaxy is face - on , absorption by foreground thermal gas is expected to be negligible . thus , the break could possibly be due to propagation effects of relativistic electrons and subsequent energy losses as explained by pohl et al . ( 1991 ) & hummel ( 1991 ) . to check this , we fitted the observed integrated spectrum of ngc 4254 using the heuristic model given by hummel ( 1991 ) which describes the radio spectrum as @xmath50 the spectral index of the injection spectrum , @xmath51 and the break frequency , @xmath52 are estimated by fitting the observed spectrum with the above function . the best fit which has @xmath53 is shown in fig . this spectrum gives a lower @xmath54 compared to the single power law fit ( also in fig . [ fig5 ] ) and hence , we believe , is a better fit to the observed data between 240 mhz and 10 ghz . however the fit fails to constrain the break frequency since we do not have data points below 240 mhz . from the above , it is reasonable to conclude that the integrated spectrum of ngc 4254 has a break at lower frequencies which is typical of spiral galaxies ( pohl et al . 1991 , hummel 1991 ) . we also note that lower frequency data ( @xmath55 mhz ) are required to confirm the break . a spectral index map of ngc 4254 was constructed between 240 and 610 mhz ( fig . [ fig6 ] ) . images at both the frequencies were made with a maximum baseline of 15 k@xmath35 and then convolved to an angular resolution of @xmath56 at a position angle of @xmath57 . a cutoff of @xmath58 was used for both the images . while the spectral index across the disk is typically between @xmath59 and @xmath60 with few regions showing @xmath61 , the emission outside the optical disk typically has a spectral index steeper than @xmath62 ( see fig . [ fig6 ] ) . the spectral index of the envelope is reminiscent of the halo emission in edge - on galaxies . chyzy et al ( 2006,2007 ) report the detection of a sharp bright magnetized ridge at 1.43 ghz in the south and present their image at 4.8 ghz . they do not detect the southern feature at 4.8 ghz which implies a spectral index steeper than @xmath63 when combined with our 240 mhz data . this is consistent with our data at 610 mhz . the spectral index near the centre of the galaxy is comparable to the disk ruling out the presence of an agn at the centre . from the above discussion , we conclude that there are two components to the radio continuum emission from ngc 4254 , namely the intense disk component which has @xmath64 and the extended component which has @xmath65 . the two components are difficult to distinguish morphologically . the radio power at 1.4 ghz of most normal spiral galaxies lie between @xmath66 watts - hz@xmath3 and @xmath67 watts - hz@xmath3 ( hummel 1981 ) and absolute blue magnitudes lie between @xmath68 to @xmath69 ( hummel 1981 ) . ngc 4254 has a power of @xmath70 watts - hz@xmath3 at 1.4 ghz . the rc3 ( de vaucouleurs et al . 1991 ) 25 magnitude size of ngc 4254 is @xmath71 and its total blue apparent magnitude is 10.44 yielding an absolute magnitude of @xmath72 . ngc 4254 is thus , both optically and radio bright . high star formation rates ( schweizer 1976 ) of @xmath73yr@xmath3 ( soria & wong 2006 ) have been reported for ngc 4254 . cayatte et al . ( 1990 ) suggested that the sharp cutoff in hi in the south - east could be due to compression by the icm which could then explain the bright disk . simulations by vollmer et al . ( 2005 ) suggest that ram pressure needs to be invoked to explain the observed hi morphology . more recently , chyzy et al . ( 2006,2007 ) have detected a large polarized envelope of emission at 1.43 ghz around the optical disk and also a bright ridge of polarised emission in the south which they suggest is due to field compression by ram pressure . this , they suggest , could be enhanced by the non - disk hi clouds detected by phookun et al . ( 1993 ) falling back on the disk . all the above indicate that ram pressure due to interstellar medium ( ism)-icm interaction is active in this system . on the other hand , ample evidence that this galaxy was involved in a tidal encounter has been found - the single dominant spiral arm , the presence of non - disk hi clouds ( phookun et al . 1993 ) and the presence of a large hi tail which extends northwards from ngc 4254 to a distance of 250 kpc ( haynes et al . haynes et al . ( 2007 ) attribute this long tail to gas stripped from ngc 4254 by the process of galaxy harrassment ( moore et al . 1996 ) as it enters the virgo cluster at high speed . the presence of the large hi cloud virgohi21 ( davis et al . 2004 ) which connects spatially and kinematically ( haynes et al . , minchin et al . 2005 ) to ngc 4254 resembles the results of the simulations by bekki et al . the x - ray emission detected from this galaxy ( chyzy et al . 2007 , soria & wong 2006 ) is confined to the optical disk . this indicates the absence of hot gas around ngc 4254 which is as expected since it is located on the periphery of the virgo cluster at a projected distance of 1.2 mpc from m87 . soria & wong ( 2006 ) suggest that the ultraluminous x - ray ( ulx ) source they detect due south of the optical disk of ngc 4254 has been triggered by the impact of the infalling hi cloud ( phookun et al . 1993 ) on the disk . edge - on galaxies commonly show the presence of halo emission extending along the minor axis which is widely studied at low radio frequencies owing to their steep spectra . we examined the dss maps of a few low inclination galaxies ( e.g. ngc 5236 , ngc 2997 , ngc 6946 ) taken from the atlas by condon et al . ( 1987 ) and find that the radio and optical disks have similar extents . hence we find the radio continunum envelope detected around ngc 4254 interesting and in this section we try to understand the origin of the extended component ( fig . [ fig3 ] ) . we believe one of the following scenarios can account for this extended radio continuum component . if we examine fig [ fig3]b closely , it appears that the radio emission extends further along the spiral arms of the galaxy especially in the north - west and south - east . thus , one possibility is that the spiral arms extend further out in the radio than in the optical . we examine the plausibility of this as follows . the typical synchrotron lifetimes of relativistic electrons diffusing out from supernovae in the galaxy would be @xmath74 years at 240 mhz if the magnetic field was 3 @xmath75 g . since we observe radio continuum emission from the envelope , this suggests that the relativistic electron population was generated less than @xmath74 years ago . if this was the case , then at least the low mass stars ( lifetimes @xmath76 years ) from the progenitor stellar population should be surviving even if the ob associations have disappeared ( lifetimes @xmath77 years ) during the synchrotron lifetime of the electrons . however no optical emission in the dss r band is observed to be coincident with the extended features . we also examined the nir emission in the j , h and k bands from the 2mass maps which trace emission from low mass stars . the nir morphology of the galaxy is similar to the dss optical and no emission coincident with the extended features is detected . this indicates that the extended features are unlikely to be extensions of the galactic spiral arms but are a result of some other physical processes . the extended emission could be from the magnetized electron gas which has been removed from the disk due to external influences like tidal , ram pressure or turbulent viscous stripping . this would lead to the icm being enriched in magnetic field and matter . our multi - frequency radio study of ngc 4254 gives interesting results . we detect radio continuum emission from regions surrounding the optical disk ( see fig [ fig3 ] ) . the spectrum of this emission is steeper ( @xmath65 ) compared to the disk emission ( @xmath64 ) . moreover , the extended emission does not form a uniform halo around the optical disk ( see fig . [ fig1 ] ) . while it could be partially attributed to sensitivity and spatial frequency coverage , most of this feature indicates real asymmetry in the extended emission . at 240 mhz , about 15% of the total emission arises in this extended component . in our hi velocity map , we notice a lagging rotation field in the atomic gas located to the north of the galaxy . it is likely that this part is being stripped and the gas is slowly losing rotation . from the above , we suggest that the extended radio continuum emission which has a steeper spectral index and the atomic gas in the north of the galaxy are likely not part of the disk emission and probably do not lie in the plane of ngc 4254 . this might be gas which has been pushed out of the galaxy by ism - icm interaction . below we summarize the morphological features that we believe can be explained by ram pressure caused by the ism - icm interaction : * a sharp cutoff is present in both the radio continuum and hi to the south / south - east . the galaxy is ploughing into the icm at an angle of @xmath27 ( vollmer et al . 2005 ) which is close to face - on and the south is the leading edge encountering the icm wind . the swept - up gas would , thus , give rise to such a smooth boundary . * the enhanced star formation rates across the entire galaxy . the wind encounters the entire disk and hence star formation would have been triggered throughout the disk explaining the enhanced star formation rates ( @xmath73yr@xmath3 , soria & wong 2006 ) * intense radio emission . this would be a direct result of enhanced star formation and presence of young massive stars in this galaxy . * extended radio continuum around the optical disk . this emission is from the relativistic electron gas which has been influenced by ram pressure and which is subsequently diffusing beyond the galaxy as the latter rushes towards the cluster centre ( see fig . [ fig7 ] ) through the icm . the extended emission arises in gas which is not co - planar with the disk of the galaxy but follows the galaxy . the hi tully - fisher relation gives a distance of 16.8 mpc to ngc 4254 ( schoniger & sofue 1997 ) whereas the mean distance to the virgo cluster is @xmath78 mpc ( federspiel et al . thus , ngc 4254 is closer to us compared to m87 and hence it is reasonable to consider it to be receding and falling in towards the centre of the cluster . the matter giving rise to the extended radio continuum , then , lies between us and the galaxy . * steep spectrum of the extended continuum . the extended emission is similar to the halo emission observed in edge - on disk galaxies which is also found to have a steeper spectrum compared to disk emission . we believe a similar explanation holds for the emission around ngc 4254 . * extended polarized envelope at 1.4 ghz detected around the optical disk by chyzy et al . ( 2006,2007 ) . * minor hi deficiency of 0.17 quoted by cayatte et al . ( 1994 ) . based on the above , we believe that the extended radio continuum emission detected outside the optical disk is mainly because of a ram pressure event as the galaxy moves in the icm . the point we would like to stress is that ram pressure is effective even though the galaxy is located in the low icm density environs in the outskirts of the virgo cluster . using equipartition and minimum energy arguments , we find that the equipartition magnetic field in ngc 4254 is about 3@xmath75 g and the magnetic pressure is @xmath79 ergs-@xmath80 . the ram pressure acting on the galaxy , which lies about @xmath81 from the centre of the cluster was estimated to be @xmath82 ergs-@xmath80 ( cayatte et al . 1994 ) . they estimate the gravitational pull for this galaxy to be @xmath83 ergs-@xmath80 . thus , the ram pressure acting on the galaxy leading to gas stripping is slightly larger than the magnetic pressure holding back the relativistic electrons from escaping and about half of the gravitational pressure of the galaxy . since this galaxy has undergone a tidal encounter , the tidal debris could increase the icm densities or the tidal interaction could have reduced the surface density of gas in the outer parts of the galaxy leading to an enhancement in the ability of ram pressure to influence the ism of ngc 4254 . similar mechanism has been suggested for other virgo galaxies ( e.g. vollmer 2003 , chung et al . 2007 ) and poor groups ( e.g. kantharia et al . 2005 ) . moreover the high star formation rate of the galaxy would lead to intense stellar winds and supernova explosions which would further facilitate the ism - icm interaction . thus , we believe that our radio observations support the earlier results of vollmer et al . ( 2005 ) and chyzy et al . ( 2006 ) where both tidal interactions and ram pressure were put forward to explain the observed morphology of the hi and the radio continuum at 1.4 and 4.8 ghz of ngc 4254 . we also suggest a possible way to determine whether a low inclination galaxy has undergone a ram pressure event by examining its low radio frequency morphology and the distribution of the spectral index across it . several authors have studied ram pressure stripping in a variety of environs and with a variety of initial conditions in clusters and groups ( e.g. schulz & struck 2001 , vollmer 2001 , roediger & hensler 2005 , roediger & bruggen 2006 , hester 2006 ) . we examine these in light of our new low frequency radio observations of ngc 4254 . the angle between the disk of ngc 4254 and its orbital plane in virgo cluster is @xmath27 ( vollmer et al . 2005 ) and the stripping is close to , but not entirely , face - on as the galaxy moves within the cluster . roediger & bruggen ( 2006 ) , in their simulations of ram pressure stripping in disk galaxies , have shown that for face - on stripping , the stripped gas will expand and form a tail behind the galaxy . for a low inclination galaxy , the stripped gas will form a halo around it over a few hundred million years and would be detectable depending on its emissivity . vollmer et al . ( 2005 ) performed simulations to explain the observed hi morphology ( phookun et al . 1993 ) of ngc 4254 . they could explain most of the morphological features only if ram pressure is invoked alongwith a tidal encounter which would have occurred about 300 myrs ago . their results indicated that ram pressure would only succceed in distorting gas with hydrogen column densities @xmath84 @xmath24 . roediger & hensler ( 2005 ) and roediger & bruggen ( 2006 ) , from their simulations on the influence of the icm on the cluster members , infer that galaxies undergo ram pressure stripping in three stages . the first stage , they describe as the instantaneous stripping phase wherein the outer gas disk expands without becoming unbound from the galaxy . it can lead to truncation of the gas disk without much mass loss . this phase is likely to affect galaxies on the outskirts of clusters and in groups and is short - lived , lasting only for 20 to 200 myr . in the intermediate , much longer - lived phase , some of the expanded gas actually evaporates while some of it falls back on the disk . in the final third phase , the galaxy loses mass at a stable rate of 1 m@xmath85yr@xmath3 due to turbulent viscous stripping ( nulsen 1982 ) . comparing this simulation result with our radio continuum data on ngc 4254 , it appears that gas stripping in ngc 4254 is in the first phase of evolution . the extended features which surround the optical disk of ngc 4254 ( see fig . [ fig3 ] ) are probably due to the relativistic gas which has diffused out from the disk but has not yet evaporated . the same argument also probably holds for the hi gas . the gas is being stripped as seen from the jagged morphology of hi that we detect especially in the north and along the vertical edge in the east . the smooth boundary to the south is the leading edge and is caused by the wind blowing across it . some of the gas in the north which shows a lagging rotation field might be due to stripped gas which is slowly losing rotation . we also tried to understand the non - disk hi clouds that phookun et al . ( 1993 ) have detected surrounding the galaxy and not following galaxy rotation . phookun et al . ( 1993 ) conclude that these clouds form an infalling population . while the tidal origin put forward by phookun et al . ( 1993 ) is the most plausible especially since these form part of the long hi tail which haynes et al . ( 2007 ) attribute to galaxy harrassment , we examined it in relation to the three stages of ram pressure stripping inferred by roediger & hensler ( 2005 ) and explained in brief above . if we hypothesize that the infalling clouds were stripped from the galaxy due to ram pressure then the hi gas in this galaxy might have just entered into the second phase of evolution ( roediger & hensler 2005 ) . in this phase , part of the hi falls back whereas part of the gas is lost from the system . some of the far non - disk hi clouds that phookun et al . ( 1993 ) detect could also possibly be part of this population . since the galaxy is receding towards south / south - east , the tail of clouds extending northwards appears to be in the expected direction . however , we also note that roediger & bruggen ( 2006 ) demonstrate in their simulations that the direction of motion of the galaxy can not always be surmised from the tail of stripped gas . it may be mentioned , however , that our explanation of the infalling clouds ( phookun et al . 1993 ) is more speculative and requires further evidence from simulations . since ngc 4254 is located in a relatively low density environment in the outskirts of the virgo cluster , the ram pressure stripping event is relatively young and was begun at most 200 myrs ago ( roediger & hensler 2005 ) . recall that the simulation by vollmer et al . ( 2005 ) indicated that a tidal interaction involving ngc 4254 occurred @xmath86 myrs ago . these timescales support the hypothesis that the tidal encounter boosted the potential of ism - icm interaction making it more effective then it would have otherwise been . the schematic in fig . [ fig7 ] summarizes the scenario for ngc 4254 as discussed above . in this paper , we have presented multifrequency radio continuum observations made with the gmrt of the spiral galaxy ngc 4254 located in the north - west outskirts of the virgo cluster . ngc 4254 is a luminous galaxy with a radio power of @xmath2 watts - hz@xmath3 at 240 mhz . we report the detection of extended , structured , low surface brightness radio continuum emission at 240 mhz surrounding the optical disk . we note that chyzy et al . ( 2006,2007 ) have reported detection of an extended polarized envelope around the optical disk at 1.4 ghz . combining our data with existing spectral data ( soida et al . 1996 , chyzy et al . 2007 ) , we find that the best fit to the global spectrum is given by a spectral index of @xmath47 which is close to the value soida et al . ( 1996 ) quote . however from our data at 240 , 325 and 610 mhz , we deduce that the spectrum flattens at the lower frequencies . a heuristic model which includes electron propogation effects ( hummel 1991 ) gives a better fit to the data . the injection spectrum has an index @xmath4 . from a detailed spectral index mapping between 240 and 610 mhz , we find that the spectral index of the extended emission is steeper with @xmath65 compared to that of the radio emission coincident with the optical disk ( @xmath64 ) . we hypothesize that the extended emission is not associated with the optical disk but has been pushed out due to ram pressure of the icm acting on the ism of the galaxy . the atomic material in the north lags in rotation wrt rest of the disk and this we believe is part of the gas mass which is being influenced by ram pressure . ngc 4254 is moving away from us and the stripped material lies between us and the galaxy as shown in the schematic of fig . [ fig7 ] . from the above , we conclude the following : * we believe that we have detected a clear signature of ram pressure stripping in ngc 4254 , a spiral galaxy in the outskirts of the virgo cluster , in the form of an extended envelope of emission surrounding the optical disk at 240 mhz . this emission exhibits a steep spectral index as compared to the disk emission . we suggest that these might be common signatures of ism - icm interaction which have not been noticed due to a dearth of such low frequency high sensitivity data on low inclination galaxies in cluster / group environments . * we suggest that low frequency ( @xmath87 mhz ) high sensitivity survey of a sample of face - on galaxies in low density groups and clusters will help generate significant information on the radio frequency signatures of ism - icm interactions . signatures such as a steep spectrum and extended emission around low inclination galaxies can be studied for a larger sample . * ism - icm interactions are likely to be more common in lower density , lower velocity dispersion environments than is presently believed . there are already several examples ( e.g solanes et al . 2001 , kantharia et al . 2005 , levy et al . 2007 ) which suggest that ram pressure events are active in x - ray poor groups and in the outskirts of clusters . simulations by several groups ( e.g. roediger & bruggen 2006 , vollmer et al . 2005 , hester 2006 ) show that this is possible . ram pressure in many of these cases is aided by other physical phenomena such as tidal interaction and supernova explosions . the stripped gas would help enhance the icm densities further . systematic low frequency observations of several galaxies in the outskirts of clusters and in poor groups are required for further understanding . we thank the anonymous referee for a detailed report and for giving several helpful suggestions on the manuscript . we thank the staff of the gmrt that made these observations possible . gmrt is run by the national centre for radio astrophysics of the tata institute of fundamental research . this research has made use of the nasa / ipac extragalactic database ( ned ) and nasa/ ipac infrared science archive which is operated by the jet propulsion laboratory , california institute of technology , under contract with the national aeronautics and space administration . this research has made use of nasa s astrophysics data system . ngk thanks prof . t. p. prabhu for useful discussions .

we report the detection of extended low radio frequency continuum emission beyond the optical disk of the spiral galaxy ngc 4254 using the giant metrewave radio telescope . ngc 4254 , which has an almost face - on orientation , is located in the outskirts of the virgo cluster . since such extended emission is uncommon in low inclination galaxies , we believe it is a signature of magnetised plasma pushed out of the disk by ram pressure of the intracluster medium as ngc 4254 falls into the virgo cluster . the detailed spectral index distribution across ngc 4254 shows that the steepest spectrum @xmath0 ( @xmath1 ) arises in the gas beyond the optical disk . this lends support to the ram pressure scenario by indicating that the extended emission is not from the disk gas but from matter which has been stripped by ram pressure . the steeper spectrum of the extended emission is reminiscent of haloes in edge - on galaxies . the sharp fall in intensity and enhanced polarization in the south of the galaxy , in addition to enhanced star formation reported by others provide evidence towards the efficacy of ram pressure on this galaxy . hi 21 cm observations show that the gas in the north lags in rotation and hence is likely the atomic gas which is carried along with the wind . ngc 4254 is a particularly strong radio emitter with a power of @xmath2 watts - hz@xmath3 at 240 mhz . we find that the integrated spectrum of the galaxy flattens at lower frequencies and is well explained by an injection spectrum with @xmath4 . we end by comparing published simulation results with our data and conclude that ram pressure stripping is likely to be a significant contributor to evolution of galaxies residing in x - ray poor groups and cluster outskirts .

0709.4532

the search for the origin of cosmic rays ( cr ) is a continuing one . although many incline to the view that supernova remnants ( snr ) are responsible , it is also possible that pulsars contribute and other possibilities include distributed acceleration . to distinguish between them is not a trivial task but is attempted here , by way of studies of the detailed shape , or ` fine structure ' , of the cr spectrum . our claim that a ` single source ' is largely responsible for the characteristic knee in the spectrum ( erlykin and wolfendale , 1997 , 2001 ) has received support from later , more precise , measurements . very recently ( erlykin and wolfendale , 2011a , b ) , we have put forward the case for further fine structure in the energy spectrum in the knee region which appears to be due to the main nuclear ` groups ' : p , he , cno and fe , but this awaits confirmation . what is apparent , however , is that there should be such fine structure in the spectrum and that this structure should give strong clues as to the origin of cr . it is appreciated that there is the danger of over - interpreting the data but , in our view , this is the only way that the subject will advance . for many years , new measurements merely confirmed that there is a knee in the cr energy spectrum without pushing the interpretation forward . our own hypothesis met with considerable scepticism initially , and , indeed , there are still some doubters but we consider that the new data are sufficiently accurate for the next step to be taken . only further , more refined data will allow a decision to be made as to whether or not the present claims are valid . we start by exploring the properties of snr and pulsars from the cr standpoint and go on to examine the present status of the search for snr - related structure by way of studying the world s data on the primary spectra and the relationship of various aspects to our single source model . particular attention is given to the contribution to the single source peak from the second strongest source and further structure , in the form of curvature in the energy spectra of the various components . the situation in the region of hundreds of gev / nucleon is considered . returning to pulsars , the result of a search for pulsar - peaks is described . type ii sn , which are generally regarded as the progenitors of cr , have a galactic frequency of @xmath1y@xmath2 and a typical total energy @xmath3erg for each one . most models ( eg berezhko et al . 1996 ) yield @xmath4erg in cr up to a maximum rigidity of @xmath5pv . the differential energy spectrum on injection is of the form @xmath6 where @xmath7 ( or a little smaller ) . at higher rigidities , not of concern here , models involving magnetic field compression via cosmic ray pressure can achieve much higher energies ( eg bell , 2004 ) . an alternative origin of cr above the knee is by way of pulsars which , although being thought to be barely significant below 1 - 10 pev , may well predominate as cr sources at higher energies ( eg bednarek and bartosik , 2005 ) . if there are small contributions below 1 pev , however , they might give rise to small peaks ( see 4.2 later ) . unlike snr there are no ` typical ' pulsars in that their initial periods differ , depending as they do on the rotation rate of the progenitor star . with the conventional value for the moment of inertia of @xmath8 , the rotational energy @xmath9i@xmath10 is @xmath11erg , where @xmath12 is the period in ms . sn and pulsars thus have similar total energies only if the pulsar has an initial period less than about 4.5ms . the fraction of energy going into cr is not clear but may be in the range ( 0.1 - 1)@xmath13 ( bhadra , 2005 ) . the maximum rigidity is given by @xmath14 v ( giller and lipski , 2002 ) where @xmath15 is the effective magnetic field , in @xmath16 gauss and , as before , @xmath12 is the period in ms . insofar as there is , presumably , a significant fraction of pulsars with birth periods less than 10ms there appears to be no problem in achieving the necessary pev energies . indeed , the case has been made for the rest of the cr energy range from pev to eev being due to fast pulsars , as already remarked . the differential energy spectrum of all cr emitted by the pulsar during its age will probably be of the form @xmath17 due to there being a delta function in energy at a particular instant and the energy falling with time as the pulsar loses rotational energy . the result of the above is that pulsars can not easily be involved as the source of particles of energy below the spectral knee at @xmath183 pev ; however , they can not be ruled out as being responsible for some of the fine structure there . our first publication claiming evidence for a single source ( erlykin and wolfendale , 1997 ) used measurements from 9 eas arrays . four of the arrays included data above loge @xmath19 7.5 with e in gev . a summary of the sharpness s vs log@xmath20 from that work is given in figure 1a . the sharpness is given by s=@xmath21 where @xmath22 is the shower size and @xmath23 is the shower size at the knee position ( the sharpness for the energy spectrum has @xmath22 replaced by e ) . prominent features in the initial plot are represented by a , b , c and d. a was the concavity first referred to by kempa et al . b was the main peak ( knee ) and c a subsidiary peak ; initially , we identified b with cno and c with fe but latterly , following direct measurements to higher energies than before , and other indications , b is identified with he and c with cno ( erlykin and wolfendale , 2006 ) . d is an important minimum identified now as the dip between cno and fe . the small peak ` e ' was not identified in the 1997 work but it is now realised to be coincident with the fe peak . \(a ) spectral sharpness values , s , for size spectra from the initial work of erlykin and wolfendale ( 1997 ) . the letters a to d were in the initial paper and represented particular features . e is added in view of the most recent work ( see ( d ) ) showing a peak identified by us as due to iron . the letters are carried forward into the plots given below . smooth lines in panels ( a ) and ( d ) below are cubic splines , which pass precisely through the points to guide the eye . \(b ) fine structure in the cherenkov light spectra from the work of erlykin and wolfendale ( 2001 ) . the results are independent of those in ( a ) . the excess , @xmath24(in log(e@xmath25i ) ) is from a 5-point running mean . the dashed line was the previous prediction , based on the fit to figure 1(a ) and the full - line the prediction from a more comprehensive analysis . the latter fit ( @xmath26 ) is better than for the case with no structure ( @xmath27 ) . \(c ) as for ( b ) but for electron size spectra . there is modest duplication of data used also in ( a ) . as in the previous panel ( b ) the full line fit , although not good enough ( @xmath28 ) , is certainly better than that for the case with no structure ( @xmath29 ) . \(d ) fine structure in the primary cosmic ray energy spectrum shown as the difference , @xmath30(in log(e@xmath25i ) ) between the observations and expectation for the galactic diffusion model , with its slowly steepening spectrum . emphasis is placed here on data at energies above those of the knee . the measurements are from 10 arrays and refer to new data published since those reported in ( a ) , ( b ) and ( c ) . ( see erlykin and wolfendale , 2011a , b ) . the non - observation of a cno peak ( and associated minima between b and c , and d ) does not invalidate its presence , as discussed in the text ; there is clearly an excess intensity to be occupied by some nuclei . [ fig : fig1 ] an ` update ' to the structure problem was given in 2001 ( erlykin and wolfendale , 2001 ) where we analysed data from 16 arrays . in many cases , spectra were given for different zenith angles ; the result was that 40 spectra contributed . the measurements superseded the 1997 results but only to a limited extent because the later measurements did not always cover a much longer period . in that work the difference in intensity from the running mean was determined from the cherenkov data and from the 40 eas electron size spectra with the results shown in figures 1(b ) and 1(c ) . we have adopted the same nomenclature for the various features , a , b e . finally , and most recently , figure 1d gives the results from an analysis of 10 more spectra published after 2001 ( and studied by erlykin and wolfendale , 2011a , b ) , the virtue of these most recent spectra is that they extend to higher energies than most of the previous ones and give good statistical precision for region e ( the iron peak ) . it will be noted that yet another ordinate is used , @xmath30(loge@xmath25i ) , the difference of the intensity from that expected from a ` conventional model ' - the galactic diffusion model - in which no structure is present , the datum being a smooth gradually steepening spectrum . the lesson to be learned from figure 1 is that the succeeding analyses give consistent results for the presence of fine structure in the spectra ( size spectra and energy spectra ) ; the difference between features in the various plots is attributed to the non - linear dependence of shower size ( @xmath22 ) on energy ( @xmath31 ) and experimental and model ` errors ' . the lack of observation of the peak c(cno ) in the latest data presumably arises from both statistics and the fact that the arrays were designed to give higher accuracy at high energies ( i.e. regions d and e ) . indeed , very recent observations from tunka-133 ( kuzmichev et al . , 2011 ) and gamma ( martirosov et al . , 2011 ) , not shown in figure 1 , give further strong evidence for the iron peak ( ie e in figure 1a ) . to be consistent with the analyses of figures 1a to 1c we can examine the @xmath32 values for figure 1d , too . as remarked above , the new arrays contributing to figure 1d were designed to go beyond the knee and , accordingly , the resolution of a , b , c and d were not as good . taking this region first , ie , @xmath33 from -1 to + 1 , a smooth line can be drawn through the points and the reduced @xmath32 value determined . it is 1.0 , i.e. a reasonable fit , with no evidence for peaks b or c. however , the datum is @xmath34 , corresponding to the conventionally expected value , so that there is strong evidence for the excess caused by b plus c ( at the many standard deviations level ) . the problem is the lack of the usual dip between b and c ; bearing in mind the fact that the zero datumdiffers between 1d and the rest , the deficit is at the 2 standard deviation level , only - a not very serious problem . turning to the region above @xmath35 , there is clear evidence from figure 1d for the peak e. with respect to the datum it is present at the 4.6 standard deviation level , based on a single point , and about 7 standard deviations if all the points are used . in only figure 1a can a comparison be made ; here the significance is at the 2.6 standard deviation level . the conclusion is that all the peaks , b , c and e are very significant . specifically , for the single points at the appropriate energies / shower sizes , alone , we have , in standard deviations : + b 7.6 , 4.2 , 9.4 and 4.4 + c 2.4 , 2.6 , 3.3 and 3.3 + e 2.6 and 4.0 ( 7 ) ` structure ' can be divided into two broad and overlapping categories : sharp discontinuities ( fine structure ) and slow trends . anything other than a simple power spectrum with energy - independent exponent can be termed ` structure ' . the main fine structure is the very well - known knee at @xmath36pev , and this will be examined here in more detail from the standpoint of the contribution from not just the single source ( snr ) but from the very few most recent / nearest sources . these will have a possible blurring effect on the sharpness of the knee . a related matter for snr is the expected ` curvature ' of the energy spectrum in the energy region where direct measurements for various nuclei have been made ( up to about @xmath37gev for protons and @xmath38gev / nucleus for iron ) . such curvature has been seen in our calculations ( erlykin and wolfendale , 2005 , see also figure 5 below ) for the snr model in which particles from a random collection of snr , in space and time , were followed . of particular relevance to the pulsar case is the search for sharp discontinuities , such as would arise from specific pulsars . we start with the possibility of more than one source contributing to the knee . at any one energy there must be , in principle , contributions from very many sources . by dividing the intensity up into contributions from a background formed by the very many sources and the ` single ' source ( the ` nearest and more recent ' - allowing for propagation effects ) we have , hitherto , ignored those few other sources which are not too far away nor too old ( or too young ) to contribute . the situation is similar to the scattering of charged particles passing through an absorber , where there is multiple scattering , single scattering and plural scattering ; here , we examine the likely contribution from ` plural ' sources . the plural sources have relevance to the sharpness of the knee because their own knees will be at somewhat different energies due to differences in the values of the parameters which determine @xmath39 ( sn energy , ism density , magnetic field and degree of field compression , etc . ) in what follows we give the results of running the programmes of erlykin and wolfendale , ( 2005 ) again to give the strengths of both the strongest peak ( s1 ) and the second strongest ( s2 ) with the result shown in figure 2 , where log(s2/s1 ) is plotted against logs1 . the values of s1 were chosen to be at the energy corresponding to the peak in the observed spectrum . the logic is that if the contribution from s2 is small it will be unnecessary to consider the third and so on . it is evident , and understandable , that the mean value of s2/s1 falls as s1 increases , the point being that the larger the s1 value the less frequent it becomes ; the 2nd largest is , essentially , a typical source. figure 3 shows the frequency distribution of s2/s1 the mean value is about 7% . the probability of s2/s1 being above 20% - a value where a broadening of the single source peak might be expected to be just significant - is only @xmath198% . in an independent approach an analysis has been made of the 24 nearest likely sources identified by sveshnikova et al . ( 2011 ) . of the snr , only 3 are old enough for the cr to have any chance of arriving at earth by now ; in our model the rest are so young that the cr will still be trapped inside the remnant . figure 4 shows the results . it will be noted that monogem ring ( our source ) is also identified as giving the biggest contribution . we added loop 1 , although it was not selected by sveshnikova et al , as the second , but its spectral shape at high energies is seen to be too steep for it to be relevant . if it is assumed that pulsars generate cr of a unique energy at each moment in their lives then the cr energy spectrum should show spikes and if their magnitude can be determined , an estimate could be made of the total pulsar contribution . starting with the experimental data , the precise measurements with the pamela instrument ( adriani et al . , 2011 ) , can be considered . for protons , the intensities are given with statistical ( and other ) errors of 3% . inspection of the fluctuations shows none to be significantly in excess of this value , although there is a small excess at about 50 gev , which may relate to our earlier claim ( erlykin et al . , 2000 ) of a small peak in the ams data . for the range 10 to 300 gev , beyond which the indicated errors become too big , we can take 3% as the upper limit to a spike , the width being the inter - point separation of @xmath40 = 0.05 . although modelling of the cr spectrum above loge = 8.0 due to pulsars has been made by us ( erlykin et al . , 2001 ) and the frequency distribution there can be used to estimate from the few strongest peaks what the total would be , such an analysis has not yet been made in the 10 - 300 gev region . all that can be said at this stage is that the total pulsar contribution is very unlikely to be greater than about 10% . in recent years it has become clear that the fine structure of the cr energy spectrum exists not only in the knee region , but also at lower energies . very precise observations with the pamela , cream and atic instruments ( adriani et al . , 2011 ; ahn et al . , 2011 ; panov et al . , 2009 ) for protons , he and other nuclei support the contention of a concavity in the hundreds of gev region . specifically , the spectra plotted as @xmath41 , where @xmath42 is the rigidity , have local minima at about 220 gv . since the results of these three experiments are very close to each other for the most abundant proton and helium nuclei , we can analyse them together . figure 5 shows the actual intensities . we fitted these spectra using the formula of ter - antonyan and haroyan , ( 2000 ) . it assumes that spectra below and above the minima can be described by power laws with slope indices @xmath43 and @xmath44 , respectively and with a smooth transition between these two slopes . the fit comprises 5 free parameters . the obtained values of the slope indices are @xmath45 and @xmath46 for protons and @xmath47 and @xmath48 for helium nuclei , which is in agreement with values found in the constituent experiments of pamela , cream and atic . before continuing it is necessary to consider figure 5 in some detail . there is the standard problem of the undoubted presence of systematic errors in some or all of the sets of data . for example , the cream and atic intensities are inconsistent for energies above the ankle . in the event this is not too important in that we need only the value of @xmath44 here , for which the mean line seems appropriate ( ie we assume that both are equally accurate ) . more important is the presence of the ankle . it will be noted that the two sets of data ( pamela and atic ) which cover the ankle energy range have have similar slope changes , although that for pamela is sharper . presumably , the systematic errors have little effect on the change of slope at the ankle , rather they give a rotation in the whole spectrum from each set of observations . the joint analysis of these spectra confirms the conclusions made separately by these collaborations that + ( i ) the spectrum of helium nuclei is flatter than the spectrum of protons ; + ( ii ) both proton and helium spectra have a concavity at an energy of hundreds of gev and change their slope by about @xmath49 . + the negative sharpness @xmath50 of the concavity is very high : for both spectra it is @xmath51 , so that it is a real ankle in both cases . interestingly an extrapolation of the best fit proton and helium spectra found by the joint analysis of pamela , cream and atic data up to the knee energy of 3 - 4 pev , shows that helium becomes the dominant component in the all - particle spectrum , with the fraction reaching a value of about 2/3 . measurements of the energy spectra of other nuclei also show a consistent concave shape over the common range 30 - 1000 gev / nucleon ( virtually the same rigidity ) from the cno group to iron ( using the spectra summarised by biermann et al . , 2010 ) and those considered here , although the degree of concavity is variable . we have quantified the concavity using a sagitta for an assumed circular shape on the plot of log(@xmath52 ) vs loge , between the limits above . the frequency distribution of @xmath30 from our model involving randomly distributed snr has been derived from the plots given in erlykin and wolfendale , ( 2005 ) ; these are for the energy spectra of cr protons - but are applicable for any nucleus , replacing energy by rigidity ; the result came from 50 trials , each involving 50,000 snr ( figure 6 ) . the concave or convex shape of simulated spectra in the 30 - 1000 gev interval depends on the particular pattern of the non - uniform snr distribution in space and time . however , in all 50 cases variations of spectral shape are very smooth with the maximum ( negative ) sharpness @xmath53 and the distribution of @xmath30 values shown in figure 7 . this distribution is symmetric around @xmath54 with about equal numbers of positive and negative sagittas . the new experimental data are sufficiently precise to enable @xmath30-values to be determined for individual elements and there are shown in figure 7 . the experimental values for all nuclei including protons and helium are seen to be on the negative side . it is clear that the observed curvature can be hardly explained by the random geometric configuration of cr sources . indeed , the probablity of such or even higher negative curvature for cr nuclei does not exceed 16 - 20% , particularly when it is realised ( see figure 6 ) that many of the random source model spectra with large negative sagittas have shapes which would not fit the measured spectra at energies above 1000 gev . the strongest argument against our random source model being the cause of actual sagittas is the detailed structure ( and sharpness ) of the ankle . in the model , the changes of slope are all smooth ( figure 6 ) , but in the observations they are not as can be seen in figure 5 . the measured spectra for nuclei show similar sharp ankles ( eg the cream data of ahn et al . , 2010 ) . whatever the reason for the ankles , they are a further example of a fine structure of the cr energy spectrum and its origin , which is unlikely to be due solely to the non - uniformity of the snr space - time distribution , needs a more detailed analysis . for the cr energy spectrum between 30 and 1000 gev / nucleon the values derived from our analysis of data from pamela , cream and atic experiments are indicated by the stars.,width=453,height=302 ] there has been progress concerning the abundances of cr nuclei in the knee region as well as at lower energies and this can be reported . in our recent paper ( erlykin and wolfendale , 2011b ) we showed that the relative abundances for the single source ( monogem ring ) at pev energies were essentially the same as those of the ambient cr at production , as measured at 1000 gev / nucleon . this suggests that there is probably only a modest variation from one source ( snr ) to another in terms of the relative abundances , although a complication is that the magnitude of the sagitta in the curvature of the spectra in the much lower range 30 - 1000 gev / nucleon ( see previous section ) varies from one nuclear mass to another . doubtless there are some ( modest ) differences between different sources with regard to relative abundances just as there are differences in energetics , maximum rigidities , etc . and these remain to be elucidated by more precise observations . such abundance variations are expected from a number of causes : interstellar medium changes from place to place , as already remarked , the possible role of pulsars in injecting energetic iron nuclei into the snr shocks , and variable injection of different nuclei from the pre - sn stellar winds and from the sn ejecta . in parenthesis , it can be remarked that the observed spectra of different elements could be different because the different types of snr contributing may well have different spatial and temporal distributions . in the limit it could be argued that different nuclei correspond to different spectra in figure 6 . in this case the difference between the proton and helium spectra in figure 5 can be explained , and , in figure 7 , the average sagitta for c - fe should be compared with a much narrower frequency distribution . the well known flattening of the spectra of the heavier nuclei above the ankle , with respect to @xmath55 and @xmath56 may find the explanation in these terms , too , although injection differences give an alternative explanation . starting with the problem of ` plural ' sources in the knee region , ie the extent to which the single source has a few accompanying contributors , we conclude that probably less than about 10@xmath13 of the ` signal ' represented by the knee is due to other , comparatively nearby , sources . the reduction in sharpness due to this cause is thus expected to be very small . continuing with pulsars , as distinct from snr as the progenitors of cr up to and including the knee , the likelihood of a significant contribution appears to be very small . concerning ` the search for pulsar peaks ' at other energies which could be present even if pulsars are not the main sources of cr , the situation is still open . none has been detected so far , although , as we have pointed out ( erlykin and wolfendale , 2011a ) it is not impossible that the ` iron peak ' at @xmath1970pev could be due to a pulsar , rather than an snr . it is in the region beyond the knee ( where an examination of published spectra suggests another ` bump ' in the spectrum above loge=7.8 ) where pulsar peaks should be visible if , following bednarek and bartosik ( 2005 ) , the bulk of cr in this region come from pulsars . the argument favouring discernable peaks about 10 pv or so is not only that pulsars of unique energy produce sharply peaked energy spectra ( delta - functions in simple models ) and that there should only be a few such sources ( unusually short pulsar periods are required ) but that the inevitably short galactic trapping times also favour fine structure , whatever the sources may be . precise measurements are not yet available to substantiate this argument . turning to curvature in the energy spectrum just before the knee , its presence in the total particle spectrum has been known for a long time ( see , for example , kempa et al . , 1974 ) and it is due to the imminent appearance of the peak due to the single source . there is evidence that the curvature is finite at lower energies too , and occurs at similar rigidity for all nuclei . the magnitude of the sagitta is greater than the average of the values found in our earlier monte carlo calculations . the pamela and cream-2 observations of fine structure at 250 gv are very interesting in their own right and suggest the existence of a new , and significant , extra cr component below 200 gv ; see the work of zatsepin and sokolskaya , ( 2006 ) . ( this suggestion is the subject of contemporary work ) . approaching 10@xmath57 gv rigidities the model of berezhko et al.(1996 ) predicts curvature , see , for example , berezhko and vlk ( 2007 ) for a comparison of the measured proton spectrum with that predicted . the model has maximum curvature at about 2000 gev . at higher rigidities , say @xmath58gv , curvature is observed and is expected because of the onset of nearby sources , with their flatter ( r@xmath59 ) injection spectra - which persist in the absense of serious diffusive losses for particles from nearby sources . the search for fine structure in the cr spectrum will continue to be a profitable one and we regard the search at higher energies than the knee , coupled with very precise measurements below it , as the next step . 1 . adriani , o. , et al . , 2011 , science , * 332 * , 69 . ahn , h.s . , et al . , 2010 , astrophys . j. lett . , 714 , l89 . 3 . bednarek , w. and bartosik , m. , 2005 , 29th icrc ( pune ) * 6 * , 349 . 4 . bell , a.r . , 2004 , mon . not . soc . , * 353 * , 550 . 5 . berezhko , e.g. , elshin , v.k . and ksenofontov , l.t . , 1996 , j.exp.theor.phys . , * 82 * , 1 . 6 . berezhko , e.g. and vlk , h.j . , 2007 , astrophys , j. 661 , l175 . 7 . bhadra , a. , 2006 , astropart . phys . , * 25 * , 226 . 8 . biermann , p.l . , et al . , 2010 , astrophys . j. , * 725 * , 184 erlykin , a.d . and wolfendale , a.w . , 1997 , j.phys.g . , * 23 * , 979 . erlykin , a.d . and wolfendale , a.w . , 2001 , j.phys.g . , * 27 * , 1005 . 11 . erlykin , a.d . and wolfendale , a.w . , 2002 , j.phys.g . , * 28 * , 359 . 12 . erlykin , a.d . and wolfendale , a.w . , 2005 , j.phys.g . , * 31 * , 1475 . 13 . erlykin , a.d . and wolfendale , a.w . , 2006 , j.phys.g . , erlykin , a.d . and wolfendale , a.w . , 2011a , astrophys . space sci . trans , 2011 , 7 , 145 . 15 . erlykin , a.d . and wolfendale , a.w . , 2011b , j.phys.g . ( submitted ) . erlykin , a.d . , fatemi , s.j . and wolfendale , a.w . , 2000 , phys . b , * 482 * , 337 . 17 . erlykin , a.d . , wibig , t. and wolfendale , a.w . , 2001 , new journ . phys.,*3 * , 18.1 . giller , m. , and lipski , m. , 2002 , j.phys.g . , * 28 * , 1275 . kempa , j. , wdowczyk , j. and wolfendale , a.w . , 1974 , j.phys.a,*7 * , 1213 . 20 . kuzmichev , l.a . et al . , 2011 , http://tunka.sinp.msu.ru/en/presentation/kuzmichev.pdf 21 . martirosov , r.m . et al . , 2011 , http://tunka.sinp.msu.ru/en/presentation/martirosov.pdf 22 . panov , a.d . , at al . , 2009 , bull . sci . : physics , * 73 * , 564 . sveshnikova , l.g . , ptuskin , v.s . and strelnikova , o.n . , 2011 , : physics , * 75 * , 334 . 24 . ter - antonyan , s.v . and haroyan , l.s . , 2000 , arxiv : hep - ex 0003006 zatsepin , v.i . and sokolskaya , n.v . , 2006 , astron . and astrophys . , * 458 * , 1

an analysis is made of the fine structure in the cosmic ray energy spectrum : new facets of present observations and their interpretation and the next step . it is argued that less than about 10% of the intensity of the helium ` peak ' at the knee at @xmath0 is due to just a few sources ( snr ) other than the single source . the apparent concavity in the rigidity spectra of protons and helium nuclei which have maximum curvature at about 200 gv is confirmed by a joint analysis of the pamela , cream and atic experiments . the spectra of heavier nuclei also show remarkable structure in the form of ` ankles ' at several hundred gev / nucleon . possible mechanisms are discussed . the search for ` pulsar peaks ' has not yet proved successful .

1111.3191

current research on human dynamics is limited to data collected under normal and stationary circumstances @xcite , capturing the regular daily activity of individuals @xcite . yet , there is exceptional need to understand how people change their behavior when exposed to rapidly changing or unfamiliar conditions @xcite , such as life - threatening epidemic outbreaks @xcite , emergencies and traffic anomalies , as models based on stationary events are expected to break down under these circumstances . such rapid changes in conditions are often caused by natural , technological or societal disasters , from hurricanes to violent conflicts @xcite . the possibility to study such real time changes has emerged recently thanks to the widespread use of mobile phones , which track both user mobility @xcite and real - time communications along the links of the underlying social network @xcite . here we take advantage of the fact that mobile phones act as _ in situ _ sensors at the site of an emergency , to study the real - time behavioral patterns of the local population under external perturbations caused by emergencies . advances in this direction not only help redefine our understanding of information propagation @xcite and cooperative human actions under externally induced perturbations , which is the main motivation of our work , but also offer a new perspective on panic @xcite and emergency protocols in a data - rich environment @xcite . our starting point is a country - wide mobile communications dataset , culled from the anonymized billing records of approximately ten million mobile phone subscribers of a mobile company which covers about one - fourth of subscribers in a country with close to full mobile penetration . it provides the time and duration of each mobile phone call @xcite , together with information on the tower that handled the call , thus capturing the real - time locations of the users @xcite ( methods , supporting information s1 , fig . a ) . to identify potential societal perturbations , we scanned media reports pertaining to the coverage area between january 2007 and january 2009 and developed a corpus of times and locations for eight societal , technological , and natural emergencies , ranging from bombings to a plane crash , earthquakes , floods and storms ( table 1 ) . approximately 30% of the events mentioned in the media occurred in locations with sparse cellular coverage or during times when few users are active ( like very early in the morning ) . the remaining events do offer , however , a sufficiently diverse corpus to explore the generic vs. unique changes in the activity patterns in response to an emergency . here we discuss four events , chosen for their diversity : ( 1 ) a bombing , resulting in several injuries ( no fatalities ) ; ( 2 ) a plane crash resulting in a significant number of fatalities ; ( 3 ) an earthquake whose epicenter was outside our observation area but affected the observed population , causing mild damage but no casualties ; and ( 4 ) a power outage ( blackout ) affecting a major metropolitan area ( supporting information s1 , fig . b ) . to distinguish emergencies from other events that cause collective changes in human activity , we also explored eight planned events , such as sports games and a popular local sports race and several rock concerts . we discuss here in detail a cultural festival and a large pop music concert as non - emergency references ( table 1 , see also supporting information s1 , sec . the characteristics of the events not discussed here due to length limitations are provided in supporting information s1 , sec . i for completeness and comparison . as shown in fig . [ fig : combinedtimeseries : rawtimeseries ] , emergencies trigger a sharp spike in call activity ( number of outgoing calls and text messages ) in the physical proximity of the event , confirming that mobile phones act as sensitive local `` sociometers '' to external societal perturbations . the call volume starts decaying immediately after the emergency , suggesting that the urge to communicate is strongest right at the onset of the event . we see virtually no delay between the onset of the event and the jump in call volume for events that were directly witnessed by the local population , such as the bombing , the earthquake and the blackout . brief delay is observed only for the plane crash , which took place in an unpopulated area and thus lacked eyewitnesses . in contrast , non - emergency events , like the festival and the concert in fig . [ fig : combinedtimeseries : rawtimeseries ] , display a gradual increase in call activity , a noticeably different pattern from the `` jump - decay '' pattern observed for emergencies . see also supporting information s1 , figs . i and j. to compare the magnitude and duration of the observed call anomalies , in fig . [ fig : combinedtimeseries : normedtimes ] we show the temporal evolution of the relative call volume @xmath0 as a function of time , where @xmath1 , @xmath2 is the call activity during the event and @xmath3 is the average call activity during the same time period of the week . as fig . [ fig : combinedtimeseries : normedtimes ] indicates , the magnitude of @xmath0 correlates with our relative ( and somewhat subjective ) sense of the event s potential severity and unexpectedness : the bombing induces the largest change in call activity , followed by the plane crash ; whereas the collective reaction to the earthquake and the blackout are somewhat weaker and comparable to each other . while the relative change was also significant for non - emergencies , the emergence of the call anomaly is rather gradual and spans seven or more hours , in contrast with the jump - decay pattern lasting only three to five hours for emergencies ( figs . [ fig : combinedtimeseries : normedtimes ] , supporting information s1 , figs . i and j ) . as we show in fig . [ fig : combinedtimeseries : changenrho ] ( see also supporting information s1 , sec . c ) the primary source of the observed call anomaly is a sudden increase of calls by individuals who would normally not use their phone during the emergency period , rather than increased call volume by those that are normally active in the area . the temporally localized spike in call activity ( fig . [ fig : combinedtimeseries : rawtimeseries ] , ) raises an important question : is information about the events limited to the immediate vicinity of the emergency or do emergencies , often immediately covered by national media , lead to spatially extended changes in call activity @xcite ? we therefore inspected the change in call activity in the vicinity of the epicenter , finding that for the bombing , for example , the magnitude of the call anomaly is strongest near the event , and drops rapidly with the distance @xmath4 from the epicenter ( fig . [ fig : spatialprops : bombmaps ] ) . to quantify this effect across all emergencies , we integrated the call volume over time in concentric shells of radius @xmath4 centered on the epicenter ( fig . [ fig : spatialprops : dvforsomers ] ) . the decay is approximately exponential , @xmath5 , allowing us to characterize the spatial extent of the reaction with a decay rate @xmath6 ( fig . [ fig : spatialprops : dvintegratedvsr ] ) . the observed decay rates range from @xmath7 km ( bombing ) to 10 km ( plane crash ) , indicating that the anomalous call activity is limited to the event s vicinity . an extended spatial range ( @xmath8 km ) is seen only for the earthquake , lacking a narrowly defined epicenter . meanwhile , a distinguishing pattern of non - emergencies is their highly localized nature : they are characterized by a decay rate of less than @xmath7 km , implying that the call anomaly was narrowly confined to the venue of the event . this systematic split in @xmath6 between the spatially extended emergencies and well - localized non - emergencies persists for all explored events ( see table 1 , supporting information s1 , fig . k ) . despite the clear temporal and spatial localization of anomalous call activity during emergencies , one expects some degree of information propagation beyond the eyewitness population @xcite . we therefore identified the individuals located within the event region @xmath9 , as well as a @xmath10 group consisting of individuals outside the event region but who receive calls from the @xmath11 group during the event , a @xmath12 group that receive calls from @xmath10 , and so on . we see that the @xmath11 individuals engage their social network within minutes , and that the @xmath10 , @xmath12 , and occasionally even the @xmath13 group show an anomalous call pattern immediately after the anomaly ( fig . [ fig : incsocialdist : dvforsomegi ] ) . this effect is quantified in fig . [ fig : incsocialdist : dvintegratedvsgi ] , where we show the increase in call volume for each group as a function of their social network based distance from the epicenter ( for example , the social distance of the @xmath12 group is 2 , being two links away from the @xmath11 group ) , indicating that the bombing and plane crash show strong , immediate social propagation up to the third and second neighbors of the eyewitness @xmath11 population , respectively . the earthquake and blackout , less threatening emergencies , show little propagation beyond the immediate social links of @xmath11 and social propagation is virtually absent in non - emergencies . the nature of the information cascade behind the results shown in fig . [ fig : incsocialdist : dvforsomegi ] , is illustrated in fig . [ fig : incsocialdist : bombcascnet ] , where we show the individual calls between users active during the bombing . in contrast with the information cascade triggered by the emergencies witnessed by the @xmath11 users , there are practically no calls between the same individuals during the previous week . to quantify the magnitude of the information cascade we measured the length of the paths emanating from the @xmath11 users , finding them to be considerably longer during the emergency ( fig . [ fig : incsocialdist : bombpaths ] ) , compared to five non - emergency periods , demonstrating that the information cascade penetrates deep into the social network , a pattern that is absent during normal activity @xcite . see also supporting information s1 , figs . e , f , g , h , l , m , n , and o , and table a. the existence of such prominent information cascades raises tantalizing questions about who contributes to information propagation about the emergency . using self - reported gender information available for most users ( see supporting information s1 ) , we find that during emergencies female users are more likely to make a call than expected based on their normal call patterns . this gender discrepancy holds for the @xmath11 ( eyewitness ) and @xmath10 groups , but is absent for non - emergency events ( see supporting information s1 , sec . e , fig . we also separated the total call activity of @xmath11 and @xmath10 individuals into voice and text messages ( including sms and mms ) . for most events ( the earthquake and blackout being the only exceptions ) , the voice / text ratios follow the normal patterns ( supporting information s1 , fig . d ) , indicating that users continue to rely on their preferred means of communication during an emergency . the patterns identified discussed above allow us to dissect complex events , such as an explosion in an urban area preceded by an evacuation starting approximately one hour before the blast . while a call volume anomaly emerges right at the start of the evacuation , it levels off and the jump - decay pattern characteristic of an emergency does not appear until the real explosion ( fig . [ fig : bomb2:temp ] ) . the spatial extent of the evacuation response is significantly smaller than the one observed during the event ( @xmath14 for the evacuation compared with @xmath15 for the explosion , see fig . [ fig : bomb2:sptl ] ) . during the evacuation , social propagation is limited to the @xmath11 and @xmath10 groups only ( fig . [ fig : bomb2:soct ] , ) while after the explosion we observe a communication cascade that activates the @xmath12 users as well . the lack of strong propagation during evacuation indicates that individuals tend to be reactive rather than proactive and that a real emergency is necessary to initiate a communication cascade that effectively spreads emergency information . the results of figs . [ fig : combinedtimeseries]-[fig : bomb2 ] not only indicate that the collective response of the population to an emergency follows reproducible patterns common across diverse events , but they also document subtle differences between emergencies and non - emergencies . we therefore identified four variables that take different characteristic values for emergencies and non - emergencies : ( i ) the midpoint fraction @xmath16 , where @xmath17 and @xmath18 are the times when the anomalous activity begins and ends , respectively , and @xmath19 is the time when half of the total anomalous call volume has occurred ; ( ii ) the spatial decay rate @xmath6 capturing the extent of the event ; ( iii ) the relative size @xmath20 of each information cascade , representing the ratio between the number of users in the event cascade and the cascade tracked during normal periods ; ( iv ) the probability for users to contact existing friends ( instead of placing calls to strangers ) . in fig . [ fig : summaryprops ] we show these variables for all 16 events , finding systematic differences between emergencies and non - emergencies . as the figure indicates , a multidimensional variable , relying on the documented changes in human activity , can be used to automatically distinguish emergency situations from non - emergency induced anomalies . such a variable could also help real - time monitoring of emergencies @xcite , from information about the size of the affected population , to the timeline of the events , and could help identify mobile phone users capable of offering immediate , actionable information , potentially aiding search and rescue . rapidly - evolving events such as those studied throughout this work require dynamical data with ultra - high temporal and spatial resolution and high coverage . although the populations affected by emergencies are quite large , occasionally reaching thousands of users , due to the demonstrated localized nature of the anomaly , this size is still small in comparison to other proxy studies of human dynamics , which can exploit the activity patterns of millions of internet users or webpages @xcite . meanwhile , emergencies occur over very short timespans , a few hours at most , whereas much current work on human dynamics relies on longitudinal datasets covering months or even years of activity for the same users ( e.g. @xcite ) , integrating out transient events and noise . but in the case of emergencies , such transient events are precisely what we wish to quantify . given the short duration and spatially localized nature of these events , it is vital to have extremely high coverage of the entire system , to maximize the availability of critical information during an event . to push human dynamics research into such fast - moving events requires new tools and datasets capable of extracting signals from limited data . we believe that our research offers a first step in this direction . in summary , similar to how biologists use drugs to perturb the state of a cell to better understand the collective behavior of living systems , we used emergencies as external societal perturbations , helping us uncover generic changes in the spatial , temporal and social activity patterns of the human population . starting from a large - scale , country - wide mobile phone dataset , we used news reports to gather a corpus of sixteen major events , eight unplanned emergencies and eight scheduled activities . studying the call activity patterns of users in the vicinity of these events , we found that unusual activity rapidly spikes for emergencies in contrast with non - emergencies induced anomalies that build up gradually before the event ; that the call patterns during emergencies are exponentially localized regardless of event details ; and that affected users will only invoke the social network to propagate information under the most extreme circumstances . when this social propagation does occur , however , it takes place in a very rapid and efficient manner , so that users three or even four degrees from eyewitnesses can learn of the emergency within minutes . these results not only deepen our fundamental understanding of human dynamics , but could also improve emergency response . indeed , while aid organizations increasingly use the distributed , real - time communication tools of the 21st century , much disaster research continues to rely on low - throughput , post - event data , such as questionnaires , eyewitness reports @xcite , and communication records between first responders or relief organizations @xcite . the emergency situations explored here indicate that , thanks to the pervasive use of mobile phones , collective changes in human activity patterns can be captured in an objective manner , even at surprisingly short time - scales , opening a new window on this neglected chapter of human dynamics . we use a set of anonymized billing records from a western european mobile phone service provider @xcite . the records cover approximately 10 m subscribers within a single country over 3 years of activity . each billing record , for voice and text services , contains the unique identifiers of the caller placing the call and the callee receiving the call ; an identifier for the cellular antenna ( tower ) that handled the call ; and the date and time when the call was placed . coupled with a dataset describing the locations ( latitude and longitude ) of cellular towers , we have the approximate location of the caller when placing the call . for full details , see supporting information s1 , sec . a. to find an event in the mobile phone data , we need to determine its time and location . we have used online news aggregators , particularly the local ` news.google.com ` service to search for news stories covering the country and time frame of the dataset . keywords such as ` storm ' , ` emergency ' , ` concert ' , etc . were used to find potential news stories . important events such as bombings and earthquakes are prominently covered in the media and are easy to find . study of these reports , which often included photographs of the affected area , typically yields precise times and locations for the events . reports would occasionally conflict about specific details , but this was rare . we take the _ reported _ start time of the event as @xmath21 . to identify the beginning and ending of an event , @xmath17 and @xmath18 , we adopt the following procedure . first , identify the event region ( a rough estimate is sufficient ) and scan all its calls during a large time period covering the event ( e.g. , a full day ) , giving @xmath22 . then , scan calls for a number of `` normal '' periods , those modulo one week from the event period , exploiting the weekly periodicity of @xmath23 . these normal periods time series are averaged to give @xmath3 . ( to smooth time series , we typically bin them into 510 minute intervals . ) the standard deviation @xmath24 as a function of time is then used to compute @xmath25 . finally , we define the interval @xmath26 as the longest contiguous run of time intervals where @xmath27 , for some fixed cutoff @xmath28 . we chose @xmath29 for all events . for full details , see supporting information s1 , sec . the authors thank a. pawling , f. simini , m. c. gonzlez , s. lehmann , r. menezes , n. blumm , c. song , j. p. huang , y .- y . ahn , p. wang , r. crane , d. sornette , and d. lazer for many useful discussions . 10 [ 1]`#1 ` urlstyle [ 1]doi:#1 [ 1 ] [ 2 ] _ _ _ _ _ _ _ _ _ _ _ _ _ _ key : # 1 + annotation : # 2 _ _ _ _ _ _ _ _ _ _ _ _ _ _ vespignani a ( 2009 ) predicting the behavior of techno - social systems . science 325 : 425 - 428 . brockmann d , hufnagel l , geisel t ( 2006 ) the scaling laws of human travel . nature 439 : 462465 . gonzlez mc , hidalgo ca , barabsi al ( 2008 ) understanding individual human mobility patterns . nature 453 : 779782 . eubank s , guclu h , kumar vsa , marathe mv , srinivasan a , et al . ( 2004 ) modelling disease outbreaks in realistic urban social networks . nature 429 : 180184 . lazer d , pentland a , adamic l , aral s , barabsi al , et al . ( 2009 ) computational social science . science 323 : 721 - 723 . song c , qu z , blumm n , barabsi al ( 2010 ) limits of predictability in human mobility . science 327 : 1018 - 1021 . onnela jp , saramki j , hyvnen j , szab g , lazer d , et al . ( 2007 ) structure and tie strengths in mobile communication networks . proceedings of the national academy of sciences 104 : 7332 - 7336 . gabrielli a , caldarelli g ( 2009 ) invasion percolation and the time scaling behavior of a queuing model of human dynamics . journal of statistical mechanics : theory and experiment 2009 : p02046 . rybski d , buldyrev sv , havlin s , liljeros f , makse ha ( 2009 ) scaling laws of human interaction activity . proceedings of the national academy of sciences 106 : 12640 - 12645 . singer hm , singer i , herrmann hj ( 2009 ) agent - based model for friendship in social networks . phys rev e 80 : 026113 . hufnagel l , brockmann d , geisel t ( 2004 ) forecast and control of epidemics in a globalized world . proceedings of the national academy of sciences of the united states of america 101 : 15124 - 15129 . colizza v , barrat a , barthlemy m , vespignani a ( 2006 ) the role of the airline transportation network in the prediction and predictability of global epidemics . proceedings of the national academy of sciences of the united states of america 103 : 2015 - 2020 . wu f , huberman b ( 2007 ) novelty and collective attention proceedings of the national academy of sciences of the united states of america 104 : 17599 - 17601 . gonalves b , ramasco j ( 2008 ) human dynamics revealed through web analytics phys rev e 78 : 026123 . ratkiewicz j , fortunato s , flammini a , menczer f , and vespignani a ( 2010 ) characterizing and modeling the dynamics of online popularity . phys rev lett 105 : 158701 . bohorquez j , gourley s , dixon a , spagat m , johnson n ( 2009 ) common ecology quantifies human insurgency . nature 462 : 911914 . balcan d , colizza v , gonalves b , hu h , ramasco jj , et al . ( 2009 ) multiscale mobility networks and the spatial spreading of infectious diseases . proceedings of the national academy of sciences 106 : 21484 - 21489 . caldarelli g ( 2007 ) scale - free networks . oxford university press . centola d ( 2010 ) the spread of behavior in an online social network experiment . science 329 : 1194 - 1197 . helbing d , farkas i , vicsek t ( 2000 ) simulating dynamical features of escape panic . nature 407 : 487490 . kaplan eh , craft dl , wein lm ( 2002 ) emergency response to a smallpox attack : the case for mass vaccination . proceedings of the national academy of sciences of the united states of america 99 : 10935 - 10940 . helbing d , johansson a , mathiesen j , jensen mh , hansen a ( 2006 ) analytical approach to continuous and intermittent bottleneck flows . phys rev lett 97 : 168001 . petrescu - prahova m , butts ct ( 2008 ) emergent coordinators in the world trade center disaster . international journal of mass emergencies and disasters 28 : 133 - 168 . kapoor a , eagle n , horvitz e ( 2010 ) people , quakes , and communications : inferences from call dynamics about a seismic event and its influences on a population . in : proceedings of aaai artificial intelligence for development ( ai - d10 ) . lambiotte r , blondel vd , de kerchove c , huens e , prieur c , et al . ( 2008 ) geographical dispersal of mobile communication networks . physica a 387 : 5317 - 5325 . crane r , sornette d ( 2008 ) robust dynamic classes revealed by measuring the response function of a social system . proceedings of the national academy of sciences 105 : 15649 - 15653 . onnela jp , reed - tsochas f ( 2010 ) spontaneous emergence of social influence in online systems . proceedings of the national academy of sciences . sheetz s , kavanaugh al , quek f , kim bj , lu sc ( 2010 ) the expectation of connectedness and cell phone use in crises . journal of emergency management . rodriguez h , quarantelli el , dynes r , editors ( 2006 ) handbook of disaster research . springer , 1st edition . lind be , tirado m , butts ct , petrescu - prahova m ( 2008 ) brokerage roles in disaster response : organizational mediation in the wake of hurricane katrina . international journal of emergency management 5 : 75 - 99 . .*summary of the studied emergencies and non - emergencies . * the columns provide the duration of the anomalous call activity ( fig . 1 ) , the spatial decay rate @xmath6 ( fig . 2 ) , the number of users in the event population @xmath30 , and the total size of the information cascade @xmath31 ( fig . 3 ) . events discussed in the main text are italicized , the rest are discussed in the supplementary material . ` jet scare ' refers to a sonic boom interpreted by the local population and initial media reports as an explosion . [ cols="<,<,<,<,<,<,<,<",options="header " , ] sixteen events were identified for this work ( see main text table 1 ) , but six events were focused upon in the main text . here we report the results for all events . in figs . [ fig : timeseries_events ] and [ fig : timeseries_controls ] we provide the call activities @xmath23 for all sixteen events used in this study ( compare to fig . 1 ) . in fig . [ fig : spatialpropssi ] we show @xmath32 for the ten events not shown in main text fig . [ mfig : spatialprops : dvintegratedvsr ] . finally in figs . [ fig : socprop_4mainemergs_si ] , [ fig : socprop_4otheremergs_si ] , [ fig : socprop_4concerts_si ] , and [ fig : socprop_4festivals_si ] we present activity levels @xmath33 for @xmath11 through @xmath13 for all 16 events . for the eight non - emergencies . concert 3 takes place at an otherwise unpopulated location and the normal activity is not visible on a scale showing the event activity . [ fig : timeseries_controls ] , scaledwidth=85.0% ] through @xmath13 during the event ( black curve ) and normally ( shaded regions indicate @xmath34 s.d . ) . normal activity levels were rescaled to account for population and selection bias ( see sec . [ sec : calcsocprop ] ) the bombing and plane crash show increased activities for multiple @xmath35 while the earthquake and blackout do not . [ fig : socprop_4mainemergs_si],scaledwidth=100.0% ] for the concerts . all concerts show extra activity only for @xmath11 except concert 4 , which shows a small increase in activity for @xmath10 several hours after the concert started . [ fig : socprop_4concerts_si ] , scaledwidth=100.0% ] for the festivals . interestingly , festival 2 shows no extra activity , even for @xmath11 , indicating that the call anomaly for those events was caused only by a greater - than - expected number of users all making an expected number of calls . [ fig : socprop_4festivals_si],scaledwidth=100.0% ]

despite recent advances in uncovering the quantitative features of stationary human activity patterns , many applications , from pandemic prediction to emergency response , require an understanding of how these patterns change when the population encounters unfamiliar conditions . to explore societal response to external perturbations we identified real - time changes in communication and mobility patterns in the vicinity of eight emergencies , such as bomb attacks and earthquakes , comparing these with eight non - emergencies , like concerts and sporting events . we find that communication spikes accompanying emergencies are both spatially and temporally localized , but information about emergencies spreads globally , resulting in communication avalanches that engage in a significant manner the social network of eyewitnesses . these results offer a quantitative view of behavioral changes in human activity under extreme conditions , with potential long - term impact on emergency detection and response .

1106.0560

intervening damped ly@xmath0 absorption - line systems ( dlas ) in quasar spectra are very rare , with an incidence of @xmath19 per unit redshift at @xmath1 ( rao , turnshek , & nestor 2004 , hereafter rtn2004 ) . consequently , unless dlas are correlated , the appearance of two dlas along any single quasar sightline ( `` double - damped '' ) represents a very unlikely event . as such , the discovery of any double - damped absorption warrants a closer investigation . here we report the discovery of double - damped absorption near @xmath1 in the sloan digital sky survey ( sdss ) quasar q1727 + 5302 during our most recent _ hubble space telescope _ uv spectroscopic survey for dlas ( rtn2004 ) . the purpose of the present paper is to report initial results pertaining to this discovery , and thereby encourage future studies of this region of the sky . the velocity separation of the absorption is 13,000 km s@xmath5 , which corresponds to a proper radial distance of @xmath20 mpc if interpreted as due to the hubble flow . , @xmath8 , and @xmath21 km s@xmath5 mpc@xmath5 ( @xmath22 ) . ] we speculate that this configuration may represent a neutral hydrogen gas filament with a large cosmological extent along our sightline . in fact , the comoving size of this putative filament would be larger than anything previously reported . as discussed in earlier contributions ( e.g. , rao & turnshek 2000 , hereafter rt2000 , and references therein ) , dlas are excellent tracers of the bulk of the neutral hydrogen gas in the universe , and the aim of our most recent dla uv survey has been to improve our knowledge of the incidence and cosmological mass density of dlas at redshifts @xmath23 . the new sample which led to the discovery of the double - damped absorption was derived from the sdss early data release ( edr ) ( schneider et al . we applied a strong - rest equivalent width ( rew ) selection criterion ( rt2000 ) to optical spectra in order to identify candidate dla absorption lines ( @xmath24 atoms @xmath4 ) , and then we obtained hst stis uv spectra to confirm or refute their presence . the current overall success rate for identifying dlas with this method is @xmath17% . since the @xmath252796,2803 absorption lines are saturated , the rew of the absorption is most closely tied to kinematic spread , not column density . recently , nestor et al . ( 2003 ) have discussed evidence for a correlation between kinematic spread and metallicity . the paper is organized as follows . in 2 we present the hst discovery spectrum for the double - damped absorption , a follow - up mmt spectrum used to determine neutral - gas - phase metal abundances , and irtf imaging data used to search for galaxies associated with the double - damped absorbers along the quasar sightline . in 3 we summarize evidence that exists for strong absorption systems near @xmath1 in other sdss quasars in the same region of the sky . a brief summary and discussion of the results is presented in 4 . q1727 + 5302 ( sdss j172739.03 + 530229.16 ) is the j2000 coordinate designation of the quasar which exhibits double - damped absorption near @xmath1 . the quasar has an emission redshift of @xmath26 and an sdss g - band magnitude of 18.3 . a search for the @xmath252796,2803 absorption doublet in bright ( @xmath27 ) sdss edr quasars resulted in the identification of two systems along this sightline at @xmath28 and @xmath29 . table 1 gives rews of some of the identified metal absorption lines . the spectrum shown in figure 1 was obtained during a 73 minute exposure with the hst on 1 january 2003 in the stis nuv g230l mode . see rtn2004 for details of the observing program . the dla lines at @xmath30 and @xmath31 are visible . the insert to the figure shows two voigt damping profiles that have been fitted simultaneously to the two ly@xmath0 lines . these have neutral hydrogen column densities of @xmath32 atoms @xmath4 and @xmath33 atoms @xmath4 , respectively . the @xmath34 errors were determined by assessing uncertainties in the continuum fit as described in rt2000 . we used the 6.5-m mmt on 2 july 2003 to obtain spectroscopic observations of both systems making up the double - damped absorption in order to determine their neutral gas phase metal abundances . we used the and lines ( figure 2 ) , as well as , , and lines ( not shown ) . the method employed to measure the metal abundances is the same as that used by nestor et al . relative to the solar measurements of grevesse & sauval ( 1998 ) , we find zn abundances of [ zn / h ] = @xmath35 ( 26.5% solar ) and @xmath36 ( 4.7% solar ) for the @xmath28 and @xmath29 systems , respectively . the corresponding cr abundances are [ cr / h ] = @xmath37 ( 5.5% solar ) and @xmath38 ( 1.7% solar ) , so there is evidence for depletion onto grains . these determinations fall within the range of dla metallicities that have been reported near @xmath1 ( e.g. , see prochaska et al . also , for the @xmath28 system we find [ si / h ] = @xmath39 ( 15.6% solar ) , [ fe / h ] = @xmath40 ( 4.3% solar)hiresdla by jason prochaska . ] , and [ mn / h ] = @xmath41 ( 6.6% solar ) ; for the @xmath29 system we find [ si / h ] = @xmath42 ( 3.5% solar ) , [ fe / h ] = @xmath43 ( 0.8% solar)@xmath44 , and [ mn / h ] = @xmath45 ( 1.3% solar ) . some of our recent imaging results on low - redshift dlas have been reported by rao et al . ( 2003 ) . on 2 april 2003 we made similar ir ( jhk bands ) observations of the double - damped sightline with the 3.0-m irtf . figure 3 shows a 14 x 14section of the h - band image centered on the quasar . the limiting 1@xmath46 surface brightness is 22.0 h magnitudes per square arcsec . objects labeled g1 and g2 are reasonable candidates for the dla galaxies based on their proximity to the sightline . if at @xmath1 , their impact parameters of @xmath47 and @xmath48 correspond to proper transverse distances of @xmath49 kpc and @xmath50 kpc , respectively . g1 and g2 have h - band magnitudes of @xmath51 and @xmath52 . they are also visible in our j - band and k - band images ( not shown ) , with magnitudes @xmath53 , @xmath54 , @xmath55 , and @xmath56 . adopting @xmath57 from local galaxies studies ( e.g. , bell et al . 2003 ) , a @xmath58-correction appropriate for an sa type galaxy from poggianti ( 1997 ) , and assuming no evolution , a local @xmath59 galaxy redshifted to @xmath1 would have @xmath60 for our adopted cosmology . we note that at @xmath1 @xmath58-corrections for all galaxy types are relatively small in the k - band , and the magnitudes of g1 and g2 would correspond to luminosities of @xmath61 and @xmath62 , respectively , if they were at @xmath1 . however , recently ellis & jones ( 2004 ) have presented k - band observations of the galaxy population residing in three x - ray selected massive clusters at @xmath63 . fits to their data indicate that at redshifts near @xmath1 the galaxy population in massive clusters has @xmath64 lying in the range @xmath65 mag . thus , @xmath66 is @xmath67 mag brighter than @xmath68 , which is what might be expected from passive evolution . therefore , g1 and g2 are @xmath69 and @xmath70 , respectively , and it should be emphasized that there are no good candidate dla galaxies with luminosities near @xmath66 . consequently , if g1 and g2 are passively evolving dla galaxies , they can simply be interpreted as the progenitors of local dwarfs . we note that if g1 was at the quasar redshift ( @xmath26 ) , it would be @xmath71 . however , g1 is offset from the quasar , so if it were related to the quasar it would likely be associated with an interaction . low - redshift quasar host galaxies are generally several times @xmath72 ( e.g. , hamilton , casertano , & turnshek 2002 ; jahnke & wisotzki 2003 ) . one way to test the hypothesis that the double - damped absorption represents an extended filament along our sightline is to search for absorption near @xmath1 in other quasars with @xmath73 that lie approximately in the same direction . this possibility can be considered since the sdss has searched this region of the sky for quasars . five other @xmath73 quasars are known that meet the criteria of having spectra of high enough quality to detect strong - systems and lying within 30 arcmin of the double - damped sightline . an angular size of 30 arcmin ( sdss data presently cut off beyond this on one edge ) is considerably smaller than the putative @xmath74 mpc ( proper distance ) long filament at @xmath1 , but this is reasonable if we are viewing cosmic structure along a filament . all five of these other quasars fall below the brightness limits used for inclusion in our current hst uv survey for low - redshift dlas ( rtn2004 ) . three of the quasars ( sdss q1727 + 5301 at z@xmath75 , sdss q1727 + 5306 at z@xmath76 , and sdss q1729 + 5312 at z@xmath77 ) do not show evidence for strong absorption near @xmath1 , so there is little chance that they have dla absorption near @xmath1 . however , the remaining two do show strong - absorption which meet the criterion for selecting dla candidates ( rt2000 ) . the sightline locations and presence of absorption may constrain the geometry of any filament . the two new absorption systems ( rews are reported in table 1 ) are at @xmath78 ( in sdss q1725 + 5254 at @xmath79 ) and @xmath80 ( in sdss q1727 + 5311 at @xmath81 ) , separated from the double - damped sightline by @xmath82 arcmin and @xmath83 arcmin , respectively . thus , there are four strong - absorption systems that lie along similar sightlines , within a redshift spread of @xmath84 , corresponding to a velocity separation of @xmath85 km s@xmath5 . given our strong - selection criterion , if these new systems are not dlas , they are almost certainly sub - dlas with column densities @xmath86 atoms @xmath4 . figure 4 shows continuum - normalized regions of the sdss spectra for all three quasars with strong - absorption . investigations of the possibility of clustering between dlas and lyman break galaxies ( lbg ) have been made at high redshift ( e.g. , gawiser et al . 2001 , aldelberger et al . 2003 , bouch & lowenthal 2003 ) . gawiser et al . ( 2001 ) and aldelberger et al . ( 2003 ) find no significant evidence for clustering at @xmath87 and @xmath88 , respectively , while bouch & lowenthal ( 2003 ) present weak evidence ( @xmath89 significance ) for clustering at @xmath90 on a size scale up to @xmath91 mpc ( @xmath92 mpc comoving ) . here we have reported the discovery of an apparently non - random ( see below ) structure on the sky near redshift @xmath1 ( 2.1 ) which is much larger than the scales explored in high - redshift dla - lbg clustering studies and much larger than normal galaxy clustering . the structure consists of two extremely high-@xmath15 ( even by dla standards ) dla absorbers separated by 13,000 km s@xmath5 along a single quasar sightline . if interpreted as a cosmological redshift separation , this double - damped structure has a radial proper distance of @xmath74 mpc . the incidence of dlas above a survey threshold of @xmath24 atoms @xmath4 is @xmath19 per unit redshift at @xmath1 ( rtn2004 ) . given this incidence , the probability of observing a second dla within @xmath93 of another dla is @xmath943% . we quote this probability as an upper limit because the incidence of dlas is significantly smaller for higher column density systems , and the column densities of the two systems included in the double - damped absorption are factors of @xmath95 and @xmath96 times larger than the survey threshold.@xmath97 therefore , the double - damped absorption may be the result of correlated dlas and be caused by a cosmologically extended filament of neutral gas along our sightline . in addition to the identification of this remarkable structure , we have made metal abundance determinations for the two systems which make up the double - damped absorption . we find them to have [ zn / h ] = @xmath9 ( 26.5% solar ) and @xmath98 ( 4.7% solar ) , with evidence for some depletion onto grains in both cases ( 2.1 ) . infrared imaging indicates that the two most likely dla candidate galaxies are relatively faint in relation to the galaxy population at @xmath1 , with @xmath99-band luminosities that are @xmath69 and @xmath70 ( 2.2 ) . these add to the list of underluminous galaxies that have been identified as being responsible for dla absorption ( rao et al . the results indicate that the presence of luminous galaxies relative to the local population evidently is not a requirement for the presence of large concentrations of neutral gas . we have also identified two new candidate dla systems in this same region of the sky , separated from the original sightline by 10 arcmin and 25 arcmin ( 3 ) . the discovery of these two additional systems increases the probability that this is a non - random structure . if the two new redshift systems are included , the structure stretches @xmath85 km s@xmath5 along the sightline , corresponding to a radial proper distance of @xmath100 mpc . filaments nearly as large as that implied by the double - damped absorption have been seen in mock redshift surveys and cold dark matter simulations of structure formation . for example , faltenbacher et al . ( 2002 ) report correlations in cluster orientations with respect to one another and find alignments of galaxy clusters major axes on comoving scales of @xmath101 mpc , corresponding to a proper distance of @xmath102 mpc at @xmath1 . the large scale structure simulations of eisenstein , loeb , & turner ( 1997 ) show similar alignments . recently , palunas et al . ( 2004 ) have reported evidence for a structure with a proper size of @xmath103 mpc ( @xmath104 mpc comoving ) , which they found during a search for ly@xmath0-emitting galaxies at @xmath105 . miller et al . ( 2004 ) have reported evidence for structures on even larger scales based on an analysis of the qso distribution in the 2df redshift survey . for this case of double - damped absorption , the size of the putative neutral hydrogen gas filament would be larger than any claimed so far . therefore , our interpretation should be considered speculative pending future studies . nevertheless , the properties of the double - damped absorption are of course relevant to studies of dlas in general . the present - day cosmological model ( approximately 73% dark energy , 24% dark matter and 5% ordinary matter , e.g. , spergel et al . 2003 ) is one in which large - scale structures can form early in time . however , a specific set of cosmological parameters may also indicate that it is highly improbable for certain structures to grow from initial gaussian perturbations . thus , surveys to find evidence for extreme large - scale structure at high redshift have the potential to result in important cosmological constraints . based on the numbers of dlas discovered at low redshift so far and the dla column density distribution , the existence of this double - damped absorption along one sightline represents evidence for a non - random distribution of dlas which should be further investigated . we thank members of the sdss collaboration who made the sdss project a success and who made the edr spectra available . we acknowledge support from nasa - stsci , nasa - ltsa , and nsf . hst - uv spectroscopy made the @xmath15 determinations possible , while follow - up metal abundance measurements ( mmt ) and imaging ( irtf ) were among the aims of our ltsa and nsf programs . funding for creation and distribution of the sdss archive has been provided by the alfred p. sloan foundation , participating institutions , nasa , nsf , doe , the japanese monbukagakusho , and the max planck society . the sdss web site is www.sdss.org . the sdss is managed by the astrophysical research consortium for the participating institutions : university of chicago , fermilab , institute for advanced study , the japan participation group , 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 university of washington . lcccccc q1727 + 5302 & 1.44 & 0.9448 & 2.19(0.13 ) & 2.83(0.07 ) & 2.51(0.07 ) & 0.99(0.07 ) & & 1.0312 & 0.76(0.11 ) & 0.92(0.06 ) & 1.18(0.08 ) & 0.33(0.10 ) q1725 + 5254 & 1.36 & 0.9706 & 0.76(0.13 ) & 1.20(0.12 ) & 1.15(0.12 ) & q1727 + 5311 & 1.81 & 1.1536 & 1.26(0.39 ) & 2.32(0.45 ) & 2.38(0.41 ) &

we report the discovery of two damped ly@xmath0 absorption - line systems ( dlas ) near redshift @xmath1 along a single quasar sightline ( q1727 + 5302 ) with neutral hydrogen column densities of @xmath2 and @xmath3 atoms @xmath4 . their sightline velocity difference of 13,000 km s@xmath5 corresponds to a proper separation of 106@xmath6 mpc if interpreted as the hubble flow ( @xmath7 , @xmath8 ) . the random probability of such an occurrence is significantly less than 3% . follow - up spectroscopy reveals neutral gas - phase zn abundances of [ zn / h ] = @xmath9 ( 26.5% solar ) and @xmath10 ( 4.7% solar ) , respectively . the corresponding cr abundances are [ cr / h ] = @xmath11 ( 5.5% solar ) and @xmath12 ( 1.7% solar ) , respectively , which is evidence for depletion onto grains . follow - up ir images show the two most likely dla galaxy candidates to have impact parameters of @xmath13 kpc and @xmath14 kpc if near @xmath1 . they are significantly underluminous relative to the galaxy population at @xmath1 . to investigate the possibility of additional high-@xmath15 absorbers we have searched the sdss database for @xmath16 quasars within 30 arcmin of the original sightline . five were found , and two show strong - absorption near @xmath1 , consistent with classical dla absorption @xmath17% of the time , but almost always @xmath18 atoms @xmath4 . consequently , this rare configuration of four high-@xmath15 absorbers with a total sightline velocity extent of 30,600 km s@xmath5 may represent a large filament - like structure stretching over a proper distance of 241@xmath6 mpc along our sightline , and a region in space capable of harboring excessive amounts of neutral gas . future studies of this region of the sky are encouraged .

astro-ph0404609

this paper is the second in a series investigating the evolved - star populations in nearby globular clusters . with the large - field ccd imagers now available it is possible to measure nearly complete samples of giant stars in clusters , and at the same time measure stars faint enough that we can normalize the luminosity functions ( lfs ) to the unevolved main sequence . because the lfs for evolved stars directly probe the timescales and fuel consumed in the different phases of stellar evolution , they provide a stringent test of the models for the evolution of low - mass stars . these models are the basis for our use of globular clusters to set a lower limit to the age of the universe and are a fundamental tool in the interpretation of the integrated spectra and colors of elliptical galaxies . the subject of this study is m30 ( ngc 7099 = c2137 - 174 ) , a relatively nearby cluster ( @xmath8 kpc ; peterson 1993 ) at high galactic latitude ( @xmath9 = @xmath10468 ) . m30 has a high central density ( @xmath11 ) , a moderate total mass ( @xmath12 ; pryor & meylan 1993 ) , and is at the metal - poor end of the cluster [ fe / h ] distribution . it is one of approximately 10% of clusters that have cusps at the core of their surface brightness profiles , and it also has one of the largest radial color gradients of any cluster ( stetson 1991b ) . previous studies of the lf for stars in metal - poor clusters have uncovered unexpected features . in a lf formed from the combination of ccd - based observations of the clusters m68 ( ngc 4590 = c1236 - 264 ) , ngc 6397 ( c1736 - 536 ) , and m92 ( ngc 6341 = c1715 + 432 ) , stetson ( 1991a ) found an excess of stars on the subgiant branch ( sgb ) just above the main - sequence turnoff ( msto ) . bolte ( 1994 ) and bergbusch ( 1996 ) both observed m30 using a mosaic of small - field ccd images and found an excess of sgb stars . ( the sgb is defined here as the transitional region between the main - sequence turnoff and the base of the red giant branch at the point of maximum curvature . ) another unexpected observation involving lfs is a mismatch between theoretical predictions and the observed size of the `` jump '' dividing the main sequence ( ms ) and the red giant branch ( rgb ) . when normalized to the ms , there is an excess of observed rgb stars compared to models ( stetson 1991a , bergbusch & vandenberg 1992 , bolte 1994 , bergbusch 1996 ) , although this has been disputed by deglinnocenti , weiss , & leone ( 1997 ) . these results might be explained by the action of core rotation ( vandenberg , larson , & depropris 1998 ) , or perhaps ( as discussed later ) we are witnessing the results of deep mixing and the delivery of fresh fuel into the hydrogen shell - burning regions . langer & hoffman ( 1995 ) suggested that , if the abundance patterns of light elements seen in bright cluster giants ( e.g. kraft 1994 ) are due to deep mixing , hydrogen - rich envelope material is almost certainly mixed into the hydrogen burning shell ( prolonging the giant phase of evolution ) , and some of the helium produced is returned to the envelope . because of the potential importance of such non - standard physics in stars , and because of the caveats associated with earlier lf studies , the most productive next step is to derive better lfs in a number of galactic globular clusters ( ggcs ) . in the next section , we describe our observations of the cluster . in 3 , we discuss the features observed in the color - magnitude diagram , describe the method of computing the luminosity functions , and present the results of artificial star experiments . in 4 , we discuss the constraints that can be put on the global parameters of the cluster metallicity , distance , and age . the method of data reduction is described in appendix a. the data used in deriving the @xmath13- and @xmath14-band lfs of m30 were taken on july 7/8 , 1994 at the cerro tololo inter - american observatory ( ctio ) 4 m telescope . in all , six exposures of 120 s , one exposure of 60 s and two exposures of 10 s were made in @xmath13 , and six exposures of 120 s , one exposure of 60 s , and one exposure of 10 s were made in @xmath14 . all frames were taken using the 2048 @xmath15 2048 pixel `` tek # 4 '' ccd chip , which has a sampling of about 044 per pixel , and a field 15@xmath16 on a side . these exposures were reduced individually for the purpose of constructing the color - magnitude diagram . in performing artificial star experiments and deriving the lf , the three best - seeing images in both @xmath13- and @xmath14-bands were combined into master long - exposure images . the frames were centered approximately 2@xmath16 east of the cluster center , in order to avoid a bright field star nearby . the night of the 4 m observations was not photometric . in order to set the observations on a standard photometric system , we used observations made at the ctio 1.5 m telescope on one photometric night ( october 18/19 , 1996 ) . the detector used was the `` tek # 5 '' 2048 @xmath15 2048 ccd , having a field of about 148 on a side . landolt ( 1992 ) standard star observations were used to calibrate a secondary field that overlapped the 4 m field . on that night , 10 s and 120 s exposures were taken in each band , along with exposures of 27 standards in 7 landolt fields . a sample of 118 stars having @xmath17 and @xmath18 was calibrated as secondary standards in this way . the field was centered approximately @xmath19 south of the cluster center . during the same run on the 1.5 m telescope , frames were taken of m30 on the non - photometric night of october 16/17 . five additional exposures were taken in each band ( 20 s , 200s , and 3@xmath15600s in v , and 15 s , 180 s , and 3@xmath15600s in @xmath14 ) . the details of the data reduction and calibration are described in appendix a. in figure [ m30cmd ] , we plot the total @xmath0 sample of 25279 stars ( upper panel ) and a sample that has been restricted in projected radius to @xmath20 from the cluster center ( lower panel ) . the inner radius was chosen in accord with the restriction placed on stars to be used later in the lfs , while the outer radius restriction was chosen to exclude regions that were affected by field star contamination . fiducial points for the ms and lower rgb of the clusters were determined by finding the mode of the color distribution of the points in magnitude bins . the fiducial line on the upper rgb was determined by finding the mean color of the stars in magnitude bins . once a mean was determined , stars falling more than @xmath21 from the fiducial point were discarded ( so as to eliminate agb and hb stars , as well as blends and poorly measured stars ) , and the mean redetermined . this procedure was iterated until the star list did not change between iterations . at the tip of the rgb and on the agb , the positions of individual stars were included as fiducial points if they appeared to be continuations of the mean fiducial line . the fiducial line for the hb was obtained by determining mean points in magnitude bins for the blue tail , and in color bins for the horizontal part of the branch . no smoothing has been applied . table [ fidtab ] lists the fiducial lines for our samples , as well as the number of stars used in computing each point . the procedure used to correct the `` observed '' lf back to the `` true '' lf is described in detail in sandquist et al . ( 1996 ) . as in that paper , we carried out artificial star tests on only four frames : a long exposure frame ( composed of the average of the three best - seeing images ) and a short exposure frame in both @xmath13 and @xmath14 band . in 11 runs , 20965 artificial stars were processed . allframe s coordinate transformations were found to be unable to follow the nonuniform spatial distortions introduced by the 4 m field corrector . to avoid this problem , we reduced all of the frames through allstar as usual , derived a master detected star list for each filter , and rereduced the frames in allstar with the improved positions . this procedure improved the overall quality of the photometry ( as judged by the scatter around the fiducial lines of the cluster ) , as allframe normally does . in figures [ delta ] [ fsfig ] we plot our computed values for median magnitude biases @xmath22 , median external error @xmath23 , and completeness probability @xmath24 as a function of magnitude and radius . there is little variation in most of the quantities until the innermost radial region ( @xmath25 20 , where crowding of stellar images is worst ) is reached . one change we have made since our first study was in the error estimation . the uncertainty in the incompleteness factor @xmath26 was previously found by simultaneously varying @xmath27 , and @xmath28 in such a way as to cause the maximum change in @xmath26 away from our best value . the magnitude of this change was used as the error estimate . we have improved this , following a suggestion by bergbusch ( 1996 ) , by estimating the error by varying @xmath29 and @xmath28 individually and adding the resulting error estimates in quadrature . using this information , we eliminated stars from consideration for the lf if they fell far enough away from the nearest point on the fiducial line of the cluster . the `` distance '' was defined in terms of difference in magnitude and color divided by their respective external errors , and then added in quadrature . so , stars were eliminated if @xmath30 . on the upper rgb , where contamination by the agb could be a factor , we adjusted the error cutoff by hand until we were sure the agb stars were being eliminated , but not at the expense of the rgb stars . the luminosity functions are listed in tables [ vlftab ] and [ ilftab ] . totals of 14772 and 14507 stars were used in creating the @xmath13- and @xmath14-band lfs . stars on the ms and sgb were only included if they fell more than 20 from the center of the cluster . this reduced the significance of crowding effects on the photometry . stars with magnitudes @xmath31 ( or @xmath32 ) were included in the determination of the lf to within @xmath33 of the center of the cluster . this was done to get a better indication of the `` global '' lf for the red giants since mass segregation is expected to occur in m30 , which would affect the rgb star counts taken from a small range of radii . most previous studies have made estimates of e(@xmath34 ) . we will assume e@xmath35 ( cardelli , clayton , & mathis 1989 ) for the rest of the discussion and refer to the reddening in e@xmath36 . m30 is situated well out of the galactic plane ( @xmath37 ) and the reddening is likely small ( e@xmath38 ) . reddening maps of burstein & heiles ( 1982 ) indicate that e@xmath39 . reed , hesser & shawl ( 1988 ) derive a _ negative _ reddening based on the comparison of m30 s integrated color and spectral type . zinn ( 1980 ) gets a value of 0.01 from integrated - light measurements . reed , hesser , & shawl s data indicates that m30 appears to have the bluest intrinsic colors of all of the globular clusters they examined . however , m30 is the cluster that shows the strongest color gradient of any ggc and reddening values measured from integrated colors or spectra will depend on the range of radii over which the observations are made . given the sense of the gradient , bluer towards the center , it seems likely that these reddening measurements will be biased to lower values . dickens ( 1972 ) and richer , fahlman , & vandenberg ( 1988 ; hereafter rfv ) independently derived significantly higher values , e(@xmath40 , based on the ubv colors of blue hb stars in m30 . on the other hand , @xmath41-band photometry is notoriously difficult to calibrate and the precision of reddening estimates from color - color plots is @xmath42 mag even in the best cases . neither the dickens nor the rfv study appears to have had a photometric calibration good enough to warrant error bars less than this . differential cmd comparisons with m92 ( vandenberg , bolte , & stetson 1990 ; hereafter vbs ) and m68 ( figure [ m30vsm68 ] ) imply e@xmath43 to 0.06 . the recent reddening maps based on iras and cobe measurements of far - ir flux from infrared cirrus suggest e(@xmath44 . we can make our own estimate using using sarajedini s ( 1994 ) simultaneous reddening and metallicity method ( for @xmath45 and @xmath46 , where the quoted errors allow some room for calibration errors ) . we find e@xmath47 . the errors were derived from monte carlo tests with the quoted errors on @xmath48 and @xmath49 . most of the reddening estimates we have thus far indicate a relatively high value e@xmath50 . the compilation of zinn & west ( 1984 ) has [ fe / h ] @xmath51 and numerous studies since have determined values between @xmath52 and @xmath53 ( geisler , minniti , & clari 1992 ; minniti et al . 1993 ; carretta & gratton 1996 ) . from the simultaneous reddening and metallicity method above , we find [ fe / h]@xmath54 . anticipating later discussion of the level of the hb , we examine the effects of an anomalously high on the simultaneous reddening and metallicity method . a high value can come about due to a high helium abundance , whether primordial or the result of a `` deep mixing '' scenario ( langer & hoffman 1995 ) . if the `` true '' value ( in the absence of helium enrichment ) is fainter by 0.10 mag , then we would have @xmath55 , and calculate [ fe / h ] @xmath56 and e@xmath57 . recently reid ( 1997 ) , gratton et al . ( 1997 ) and pont et al . ( 1998 ) have used subdwarf parallaxes measured with the _ hipparcos _ satellite ( esa 1997 ) to redetermine the distance moduli to several of the best observed clusters . the general result of these studies is to increase cluster distance moduli by 0.2 to 0.4 mag , implying high luminosities for the horizontal branches ( as bright as m@xmath58 mag for the [ fe / h]@xmath59 clusters ) . for m30 specifically , gratton et al . find ( m - m)@xmath60 ( for e@xmath61 ) although this is based on only three subdwarfs . we repeat this exercise for m30 with our data and a larger set of subdwarfs . this method is sensitive to uncertainties in the color of the unevolved main - sequence , which result from zero - point errors in the photometric calibration , the reddening uncertainties already mentioned , and uncertainties in the placement of the main - sequence fiducial line . our data for m30 are not optimum for using subdwarf fitting to measure the distance , primarily because of the uncertainty in the reddening , but also because we suffer from each of the other problems to a small degree . nevertheless , we selected subdwarfs that satisfy the following criteria : parallaxes from the _ hipparcos _ mission having relative errors @xmath62 , metal abundances from the study of gratton , carretta , & castelli ( 1996 ) , and @xmath0 photometry . the restriction on parallax error was chosen to minimize the effect of bias corrections . ( as a result , lutz - kelker corrections only change our derived distance moduli by 0.01 mag . ) the gratton et al . metallicity scale was chosen for its homogeneity and because it minimizes the possibility of systematic abundance errors with respect to carretta & gratton globular cluster metallicities . when available , we used @xmath0 photometry for the subdwarfs tabulated in mandushev et al . ( 1996 ) . in the remaining cases , we followed their procedure of combining literature values from the following sources : carney & aaronson ( 1979 ) , carney ( 1980 , 1983b ) , and ryan ( 1989 , 1992 ) . the studies involving carney all used the johnson @xmath14 filter , so we applied the transformation from carney ( 1983a ) to convert them to the cousins system . known spectroscopic binaries were excluded , and , for the cases where it has been measured , any reddening of the subdwarfs ( at most a very small amount ) has been subtracted . our sample of subdwarfs and metal - poor subgiants is shown in table [ subdtab ] . we used the subdwarfs to estimate the distance to m30 in two different ways . first , to simultaneously estimate the distance and e@xmath36 of m30 we created a grid of chi - square - like sums . the m30 main sequence between @xmath63 was represented by a third - order polynomial . for ranges of m - m and e@xmath64 ) , the minimum distance of each subdwarf from the main - sequence polynomial was calculated . this distance was normalized by the combined errors in the subdwarfs colors and magnitudes and the main - sequence fiducial uncertainties in color and magnitude ( assumed to be 0.04 mag in color and 0.05 mag in @xmath13 magnitude ) . our `` @xmath65 '' sum is : @xmath66 we will refer to this as the chi - square sum although it does not match the usual definition of chi - square and it is not normalized in a way to give true confidence intervals . because the slope of the main - sequence changes with @xmath13 , there is a different weighting given to @xmath67 and @xmath68 ) for each subdwarf . the minimum chi - square values are for m@xmath10m@xmath69 and e@xmath70 , although the reddening in particular is poorly constrained . our second approach was to fit the m30 main sequence to the subdwarfs using only the distance modulus as a variable for two e(@xmath71 ) values : 0.02 and 0.06 . table [ dmtab ] shows the changes in this value for different subsets of our sample and for different input data . ( the quoted errors include contributions from the scatter of values in the subdwarf fit , a cluster reddening error of 0.02 mag , and the absolute cluster metallicity error of 0.2 dex . ) clearly , reddening uncertainty is dominant in the total uncertainty . two fits at different reddening are shown in figure [ subdvred ] . if we use the value of metal content given by zinn & west ( [ fe / h ] @xmath72 ) along with the carney et al . ( 1994 ) abundance scale for the subdwarfs ( which has roughly the same zero point as zinn & west ) , the distance modulus is increased by only a few hundredths of a magnitude . restricting the sample to only metal - poor ( [ fe / h]@xmath73 ) subdwarfs also does not significantly change the distance modulus . considering the number of distance modulus measurements for m30 in the literature , it is best to try to compare using a common reddening of e@xmath74 ( and using @xmath75 to correct values ) . from the two methods we presented here we have 14.6 and 14.65 . from the pre - hipparcos ground - based measurements of bolte ( 1987 ) and rfv , we have 14.62 and 14.68 ( the rfv value must also be corrected for their lower assumed metallicity for m30 ) . for the hipparcos - based distances of reid ( 1997 ) and gratton et al . ( 1997 ) , we find 14.75 and 14.87 ( although gratton et al . use only three subdwarfs in their fit ) . it is clear that our distance moduli are consistent with the ground - based measurements , and over 0.1 mag smaller than the other hipparcos - based measurements . neverthless , the different distance measures are in good general agreement , at least for a fixed reddening value . although it is not crucial for the conclusions that follow , we will adopt ( m - m)@xmath76 for the rest of the paper . from figure [ subdvred ] , it is clear that the smaller reddening and distance modulus values are more consistent with the model isochrones and , for our preferred larger reddening value of e@xmath3 , the models do not match the shape of the m30 fiducial above the turnoff . the ratio @xmath77 , where @xmath78 is the number of rgb stars brighter than the luminosity level of the hb , is the traditional quantity used to estimate the helium abundance of stars in globular clusters . dickens ( 1972 ) first noted that m30 s value for `` r '' was unusually large and alcaino & wamsteker ( 1982 ) claimed a significant gradient in this ratio in the sense of a small value at the center of the cluster increasing to one of the largest measured in any cluster at large radii . we have sufficient numbers of stars on the red and blue sides of the instability strip to define the level of the hb . from five stars near the blue edge of the instability strip ( with colors @xmath79 ) , we calculate an average magnitude @xmath80 . from six stars on the red side of the instability strip , we find @xmath81 . all stars used in these averages were found at radii greater than 10 from the center of the cluster , so that the photometry should be quite accurate . because the estimate of the hb magnitude is crucial in later arguments , we examine the topic further . our blue side estimate is consistent with those of bolte ( 1987 ) and bergbusch ( 1996 ) , even with the different calibrations of our respective datasets . because it is possible that the red hb stars in m30 are evolved , it is wise to check this possibility before simply interpolating between the two sides . the number of stars ( 8 with projected radius @xmath82 10 , compared to 93 on the blue hb ) at @xmath83 is roughly consistent with timescales for stars evolving from the blue hb , as the evolutionary tracks tend to parallel each other closely through this part of the cmd as they move toward the agb . thus , we have chosen to look at other clusters of similar metallicity with large , well - studied rr lyrae populations to compute a magnitude correction to go from the red edge of the blue hb into the middle of the instability strip . for the clusters m15 ( bingham et al . 1984 ) , m68 ( walker 1994 ) , and m92 ( carney et al . 1992 ) we find agreement on a correction of @xmath84 mag . thus , we have @xmath85 . to define the rgb star sample for the @xmath5 method , we need to establish the relative bolometric magnitudes of the hb and rgb stars . we have used the hb models of dorman ( 1992 ) in conjunction with the isochrones of bergbusch & vandenberg ( 1992 ; hereafter bv92 ) . the stellar models involved in these studies were computed with a consistent set of physics and compositions . although the composition used is somewhat out - of - date , the _ differential _ bolometric corrections should be fine . the corrections as a function of [ fe / h ] can be approximated by : @xmath86 } + 0.229 \mbox{[m / h]}^{2 } + 0.034 \mbox{[m / h]}^{3 } .\ ] ] because @xmath87-element enhancements influence the position of the hb and rgb in the cmd like a change in [ fe / h ] ( salaris , chieffi , & straniero 1993 ) , they must be taken into account when computing [ m / h ] . for m30 , we find that @xmath88 . we have used [ m / h ] @xmath89 ( correcting [ fe / h ] by 0.21 dex for [ @xmath87/fe]@xmath90 ) , although for this range of metallicities , changes in metallicity have a very small effect on @xmath91 . because of contamination and blending problems toward the center of the cluster , we restrict our samples of rgb and hb stars to @xmath92 . with this choice , we find @xmath93 . this makes m30 s @xmath5 the highest of any of the clusters examined . in table [ clusttab ] , we present @xmath5 values that have been derived from published photometry for the clusters similar in [ fe / h ] to m30 m68 ( walker 1994 ) , m53 ( cuffey 1965 ) , ngc 5053 ( sarajedini & milone 1995 ) , ngc 5466 ( buonanno et al . 1984 ) , and m15 ( buonanno et al . 1983 ) along with several more metal - rich clusters . photometry exists for the central @xmath94 of m30 from the hubble space telescope ( yanny et al . 1994 ; hereafter ygsb ) . by merging their list with ours and eliminating common stars , we have created a master list of hb and rgb stars that completely covers the cluster out to 70 , with portions included out to about @xmath95 . the data for the full sample is presented in table [ poptab ] . using this sample , we find a global value for @xmath5 of @xmath96 . ( even if our value of @xmath48 is too bright , and if we use the magnitude of the red edge of the blue hb , we find @xmath97 still a high value in a relative sense . ) this global @xmath5 value for m30 is on firm ground , because there is no place for the bright stars to hide . the photometry is easily good enough to distinguish between the agb and rgb star samples , so this is not a source of uncertainty . ( in fact , m30 may be deficient in agb stars as well as rgb stars . ) the use of the lower buzzoni et al . ( 1983 ) value @xmath98 for the differential bolometric correction would increase the high @xmath5 value . approximately 30 additional rgb stars would have to be included to bring m30 s value in line with that of other clusters a 24% change in the sample size . there are a few potential explanations for a _ global _ depletion of bright giants in this cluster . first , because the ratio @xmath5 is a helium abundance indicator , the abnormally high value could indicate a higher - than - average helium abundance in m30 stars more luminous than the hb . this does not necessarily imply high y for lower luminosity stars , since a deep - mixing mechanism would also be expected to dredge up freshly produced helium ( sweigart 1997 ) . second , the environment within the cluster may affect the stellar populations by a mechanism that truncates rgb evolution and/or produces additional hb stars . we now consider the arguments for the two sides . using the buzzoni et al . ( 1983 ) calibration and our m30 @xmath5 value , we find @xmath99 . the value derived using @xmath48 of the blue edge of the instability strip gives @xmath100 . that @xmath48 is definitely a faint limit , making @xmath101 a lower limit . in any case , the @xmath5 value for m30 is significantly higher than those for other clusters when the revised differential bolometric corrections are used . [ the low value of @xmath102 for the other clusters is discussed in sandquist ( 1998 ) . ] one check we can make is to examine other helium indicators to see if they also indicate a high abundance relative to other clusters . caputo , cayrel , & cayrel de strobel ( 1983 ) introduced two indicators : @xmath103 ( the mass - luminosity relation for rr lyrae stars of ab type ) and @xmath6 ( the magnitude difference between the hb and the point on the ms where the dereddened @xmath34 color is 0.7 ) . while m30 s rrab variables have not been studied to the extent necessary to compute @xmath103 , we can compute a @xmath6 value from the @xmath104 photometry of bolte ( 1987 ) and rfv . assuming for both that e@xmath105 , we find @xmath106 and @xmath107 respectively , where the primary contributions to the error are the uncertainty in the reddening and the small number of stars used to define the hb magnitude . ( the lower rfv value can be traced to a fainter hb magnitude relative to bolte s data . ) we have computed comparison values for the clusters m68 ( mcclure et al . 1987 , walker 1994 ) , and m15 ( durrell & harris 1993 ) , as summarized in table [ clusttab ] . the @xmath6 values for the other clusters agree with the theoretical value of 6.30 for [ m / h ] @xmath108 ( [ fe / h]@xmath109 ) . we can directly compare our data with @xmath0 for other clusters if we redefine @xmath6 ( hereafter , @xmath110 ) by choosing the ms point to have @xmath111 . from isochrones this is approximately equivalent to @xmath112 . from our fiducial line , we find @xmath113 for e@xmath105 . @xmath0 data exist for m92 ( johnson & bolte 1998 ) and m68 ( walker 1994 ) . we find that @xmath114 ( e@xmath115 ) and @xmath116 ( e@xmath117 ) , respectively . the m92 value relies essentially on one star for the hb magnitude , so the @xmath118 value is uncertain . for @xmath110 and @xmath118 we see evidence ( though not overwhelming ) that the m30 value is high compared to other clusters . the values above indicate that m30 s helium abundance is high by about 0.02 for [ m / h]@xmath119 if the helium is primordial , or 0.03 if the helium enrichment only affects the level of the hb , as in the deep mixing scenario . ( note that a lower value for the reddening would bring the @xmath6 values into consistency with the other clusters . ) a high helium abundance would tend to make the hb distribution bluer on the whole . according to fusi pecci et al . ( 1993 ) , the color of the peak of the hb star distribution in m30 is one of the bluest , but other clusters are rather close ( m53 is bluer , m15 has approximately the same peak color , and m92 and ngc 5466 are slightly redder ) . because m30 has a core of high stellar density , we consider the possibility that this environment has influenced the populations of evolved stars in the cluster . m30 has one of the most robustly determined color gradients ( approximately linear in @xmath120 : @xmath121 mag dex@xmath122 ; piotto , king , & djorgovski 1988 ) of all the globular clusters in the galaxy . the sense of the color gradient is such that the integrated colors become bluer towards the cluster center . in m30 and other post - core - collapse clusters it has been suggested that the color gradient is due to a decrease in the ratio of rgb - to - bhb stars resulting from stellar interactions in the dense cluster cores . although djorgovski & piotto ( 1993 ) claim the color gradient measured in m30 is due to a deficit of rgb stars in the inner few tens of arcseconds , burgarella & buat ( 1996 ) show that the gradient ( which extends to radii @xmath123 ) is not due to differences in the spatial distribution in the evolved - star populations or in the blue stragglers . our results are consistent with this latter claim . buonanno et al . ( 1988 ) claimed to have detected a radial variation in the ratio @xmath124 on the basis of a smaller sample of stars . from our sample , we find @xmath5 ranges from about @xmath125 in the inner @xmath33 to @xmath126 for @xmath127 to @xmath128 for stars with projected radius @xmath129 . there may be marginal evidence for a difference between the outer annulus , and the inner two , but over the range of radii for which the bluer - inward color gradient has been observed , there is no evidence of a trend in the bright populations . the sense of the difference between the outer and inner populations is in any case opposite to that required to make the color gradient . the cumulative radial distribution ( figure [ crd ] ) also shows no strong trends in radius , contrary to claims in other studies with smaller samples ( buonanno et al . 1988 , piotto et al . 1988 ) , but in agreement with studies of the core ( yanny et al . 1994 , burgarella & buat 1996 ) . a kolmogorov - smirnoff test indicates a 33% chance that the two samples are drawn from the same distribution . the inhomogeneity of the rgb sample seems to be responsible for this noncommittal probability . we conclude that despite the color gradient in m30 , and the apparently ripe conditions for interactions to alter the stellar populations , environment - based processes are not responsible for the high r value we measure . based on the color - difference method , vbs claimed that the most metal - poor clusters , including m30 , are coeval at the level of 1 gyr . our comparison ( figure [ m30vsm68 ] ) of the fiducial lines of m30 and m68 ( walker 1994 ) in the neighborhood of the sgb indicates that the ages of m30 and m68 ( assuming similar main - sequence y and [ @xmath87/fe ] ) are nearly identical . with our uniform calibration of msto and evolved stars in m30 , we can determine with good precision the other commonly applied age estimator @xmath7 . in computing the values presented in table [ clusttab ] , we have attempted to use the studies with the largest samples having uniform photometry from the level of the hb to below the to . we were able to derive values for the clusters m68 ( walker 1994 ) , m53 and ngc 5053 ( heasley & christian 1991 ) , m92 ( bolte & roman 1998 ) , and m15 ( durrell & harris 1993 ) . although the clusters all have blue hb morphologies , it has not been necessary to make corrections to find the `` true '' hb level : either the hb is populated on both sides of the instability strip ( m53 , ngc 5053 ) or there are a number of well - measured rr lyrae stars ( m15 , m68 , m92 ) . from our photometry of m30 , we find @xmath132 , and @xmath133 . while m15 , m53 , m68 , m92 , and ngc 5053 have @xmath130 values that agree to within the errors ( and also agree with the values derived for clusters of higher metallicity ) , m30 has a value about 0.15 mag higher . this is in disagreement with values given in the extensive tabulation of chaboyer , demarque , & sarajedini ( 1996 ) . the values for m30 and m92 in particular have been put on firmer ground here since consistent photometry exists from the hb to the to . using the more robust v(bto ) ( the apparent magnitude of a point 0.05 mag redder than the turnoff ; chaboyer et al 1996 ) , we can compare @xmath134 values for m30 and m68 , which also has @xmath0 data . we find @xmath135 for m30 , and @xmath136 for m68 . there are two plausible ways to explain the apparent 0.15 mag excess in @xmath130 for m30 relative to other clusters . first , m30 could be older by @xmath137 gyr . this conclusion would be in conflict with that inferred by vbs based on the color - difference method . ( this is perhaps the first case for which the relative age indicators @xmath130 and the subgiant - branch color extent give significantly different answers . ) alternatively , m30 stars could have a larger initial helium abundance by approximately 0.027 . if the higher helium abundance is restricted to the cluster hb stars , as would be the case in a deep - mixing scenario , the increase required is approximately 0.045 . ( the agreement between the m30 and m68 cmds everywhere but on the hb would argue against a difference in the initial helium abundances in the two clusters , as would beliefs about galactic chemical evolution . ) the size of the potential helium enhancement is close to what was inferred earlier from the helium indicators @xmath6 and @xmath5 for m30 . in the following , we will be using a combination of oxygen - enhanced ( bv92 ) and @xmath87-element enhanced ( vandenberg 1997 ) theoretical lfs to interpret the data . the current state of knowledge indicates that all of the @xmath87 elements have enhancements ( pilachowski , olszewski , & odell 1983 ; gratton , quarta , & ortolani 1986 ; sneden et al . the available evidence also suggests that the oxygen enhancement remains constant , at least for [ fe / h ] @xmath138 ( e.g. suntzeff 1993 , carney 1996 ) . on the rgb , stellar evolution is insensitive to the oxygen abundance because the luminosity evolution is driven almost entirely by the helium core mass ( refsdal & weigert 1970 ) , while the color is primarily determined by h@xmath139 opacity . oxygen has a relatively high ionization potential , and hence does not contribute electrons to the opacity . the fainter one goes on the ms , the more insensitive the evolution is to the oxygen abundance because of the same opacity effect , and because @xmath140 chain reactions are dominant over cno cycle reactions in influencing the luminosity . as a result , the different distribution of heavy elements causes negligible differences in the theoretical lfs on the rgb and lower ms ( see figure 17 of sandquist et al . 1996 ) . it is primarily the turnoff region that is affected by changes in the oxygen abundance , because cno cycle reactions begin to become important , and because oxygen ionization regions are close enough to the surface to influence surface temperatures . increased oxygen abundance increases the envelope opacity , creating redder models . increased cno cycle activity causes a star to adjust to accommodate the increased luminosity by reducing the temperature and density of the hydrogen burning regions , which results in a net _ decrease _ in the luminosity of the turnoff and sgb relative to solar - ratio models . thus , the sgb `` jump '' moves in magnitude in the lf . in the cmd , it also changes slightly in slope , but for metal - poor clusters like m30 , this does not cause a significant change in the shape of the sgb jump in the lf . an examination of bv92 models indicates that the @xmath13-band lf is not very age - sensitive for this range of metallicities . it is most sensitive on the sgb and then , as found in sandquist et al . ( 1996 ) , only when the sgb is nearly horizontal in the cmd . in @xmath13 band for a cluster as metal - poor as m30 , the sgb has a relatively large slope , and so only a large systematic age error will influence the fit . in light of the _ hipparcos _ parallax data , this possibility should be considered , since derived distance moduli indicate brighter to magnitudes ( and thus , younger ages ) for metal - poor clusters . figure [ vlfages ] shows a comparison of the @xmath13-band lf with theoretical lfs for different ages , using an apparent distance modulus of @xmath141 , as derived from one fit to the _ hipparcos _ subdwarf sample . previous studies of m30 s @xmath13-band lf ( piotto et al . 1987 , bolte 1994 , bergbusch 1996 ) uncovered two unusual features in comparisons with theoretical models : an excess of faint red giants relative to main - sequence stars , and an excess of subgiant stars . our photometry goes fainter on the ms , allowing us to verify that the normalization of the theoretical models has not been made in an `` abnormal '' section , while our wide field allows us to measure the largest sample of red giants in the cluster to date . figure [ vlfcomp ] shows a comparison of the studies , with magnitude shifts according to measured zero - point differences . in large part there is excellent agreement . our lf is significantly below bergbusch s at his faint end , most likely due to underestimated incompleteness corrections in his study . at the bright end of the rgb ( @xmath142 ) , our lf points are also below most of bergbusch s . however , we observed a larger number of giants , and our bins are larger , making our points more significant statistically . in the following subsections , we discuss the main features in the lfs . there is a apparently a considerable excess of stars on the rgb for @xmath143 ( we will refer to this as the `` lower rgb '' ) when compared to the models normalized to the unevolved main sequence . to judge the reality of the excess , we need to accurately normalize the theoretical lfs in the horizontal and vertical directions , and choose the photometry subsample to maximize the statistical significance . the horizontal normalization can be accomplished by shifting the theoretical lf in magnitude so that the to matches that of the observational lf ( stetson 1991a ) . in the vertical direction , we have normalized to the ms in a range of magnitudes where there are large numbers of measured stars , and where incompleteness is a relatively small consideration . the mass function controls how well the normalized theoretical lf is fit to the ms portion in the present example . as shown in figure [ vlffig ] , fits using small values for the power - law mass function exponent @xmath144 indicate that the relative numbers of stars on the rgb and lower ms can be matched by canonical stellar evolution models . with such a choice though , bins with @xmath145 are not well - modeled . we find that the lf can be modeled from near the faint limit of our survey to the base of the rgb if we use a higher value for @xmath144 . this alleviates the depression in the star counts in this magnitude range seen by bolte ( 1994 ) . however , we are still left with an excess of rgb stars relative to ms stars in the range @xmath146 . the effects of mass segregation have been previously observed within m30 in the form of a variation of the local mass function exponent @xmath147 with radius ( rfv , bolte 1989 , piotto et al . 1990 , sosin 1997 ) . as a result , the best comparison that can be made would be between theory and a faint sample restricted to the outskirts of the cluster . the models of pryor , smith , & mcclure ( 1986 ) , as well as observational studies , indicate that restricting the sample to stars more than 20 core radii from the center should minimize the effects of mass segregation on faint end of the lf . figure [ vlfrgbms ] shows the lf we computed for this purpose . the presence of the rgb - ms discrepancy in this case suggests that the problem is not related to the dynamical effects on the mass function , at least in the outskirts . we can get good overall agreement with the shape of the lf on the ms , but there is a relative excess of rgb stars , and the sgb region is not well fit . the sgb comparison is insensitive to age and metallicity using this method of matching the msto . helium abundance , however , has a larger effect ( stetson 1991a ) . because the rgb stars in m30 become more populous relative to hb stars ( and presumably ms stars ) towards the center , we expect that the cluster core would show a larger discrepancy . as with the sgb excess , this effect has only been observed in metal - poor clusters ( in other words , not in the lfs of ngc 288 or m5 ) . to add stars to the canonical number at a point in the red giant lf , one must either increase the hydrogen content of the mass being fed into the hydrogen - burning shell , or reduce the density or temperature of the burning shell . one possibility for the excess stars on the lower rgb is that we are seeing the effects of deep mixing , which brings hydrogen - rich envelope material into the energy generating shell . if this kind of mixing occurred on the lower rgb , it could eliminate the rgb bump by erasing the chemical discontinuity left by a surface convection zone . alternately , vandenberg , larson , & depropris ( 1998 ) have examined the effects of rotation on rgb evolution . they found that core rotation can expand the outer portions of the stellar core enough to cause a reduction of the shell temperature . this results in a decrease in the rate of evolution for rgb stars , and hence leads to an increase in the number of stars per luminosity bin . this is in the correct direction to explain the rgb excess . this rotation could be related to deep mixing scenarios that are required to explain abundance anomalies in rgb stars ( e.g. shetrone 1996 ) most notably a decline in the @xmath148c/@xmath149c ratio relative to theoretical predictions , and na - o and al - o anti - correlations ( as surface material is mixed into regions where o is being converted to n in the cno cycle ) . [ note that rotation _ can not _ explain subgiant branch excesses because the burning region in core - burning stages is too small to contain a significant amount of angular momentum ( vandenberg 1995 ) . even if rotation does affect the structure of the star outside the core , and thereby changes the core temperature , this would not produce isothermalization that would lead to sgb excesses . ] the rotation and mixing pictures ( with the assumption that mixing is somehow based on internal rotation ) receive some support from observations of rotation in hb stars of some clusters . there is definite evidence of stellar rotation in blue hb stars in ngc 288 , m3 , and m13 ( peterson , rood , & crocker 1995 ) . m13 has the fastest rotators , with stars falling into two groups : some with @xmath150 km s@xmath122 , and some with @xmath151 km s@xmath122 . m3 has a @xmath152 distribution consistent with @xmath153 km s@xmath122 , while ngc 288 s stars are consistent with @xmath154 km s@xmath122 . cohen & mccarthy ( 1997 ) also found projected rotation rates between 15 and 40 km s@xmath122 for five blue hb stars in m92 . the presence of stellar rotation on the hb implies that angular momentum may have been stored during the rgb phase in a rapidly rotating core , avoiding loss of angular momentum through the stellar wind . ( such mass loss is needed to be able to create hb stars of appropriate masses to match observed cluster hb morphologies . ) if this is true , it would be particularly interesting to compare lfs for m3 and m13 to look for the effects of rotation , and perhaps even different levels of rotation . further stellar rotation measurements for m30 , m68 , and m92 would also be helpful in examining rotation as a cause of ms - rgb discrepancy in the combined lf . figure [ clfhbs ] presents the cumulative lf ( clf ) for the cluster . in this graph we have included rgb stars from @xmath155 to @xmath156 from the center of the cluster . the rgb bump is typically identified from a break in slope in the cumulative lf . at this point , the shell - burning source begins consuming material of constant , lower helium content ( in other words , the shell reaches what was formerly the base of the convection zone at its maximum extent fusi pecci et al . fusi pecci et al . examined clusters over a range of metallicities , and found a linear relation between @xmath157 and [ fe / h ] , as predicted by theory . by combining cmds for three of the most metal - poor clusters ( m15 , m92 , and ngc 5466 ) , they found @xmath158 . in addition , for ngc 6397 , the most metal - poor cluster for which they were able to find the bump , they found @xmath159 . as shown in figure [ clfhbs ] , we have examined data for m68 ( walker 1994 ) , a cluster of nearly the same metallicity as m30 , in order to get a better idea of where the bump should be . there is a clear indication of a slope break for m68 : @xmath160 , or @xmath161 , for m68 . that result shows that the continuation of the fusi pecci et al . relation to lower metallicity appears correct . we have chosen to shift m30 and m68 so that their mstos align because of the evidence that m30 s hb may be anomalously bright ( see [ dist ] ) . the comparison reveals that there may be a feature at the same position as in m68 , although we do not see significant signs of slope change in the clf at the position of the feature . we find that a few bins on the sgb ( @xmath162 ) show an excess of stars relative to the theoretical predictions for the best fitting models , confirming the result of bolte ( 1994 ) . in figure [ vlfsgb ] , we plot the lf with a radius cut closer to the cluster center so as to get better statistics on the sgb . as figure [ vkept ] shows , there is little scatter in the vicinity of the sgb in the cmd that would tend to wash out or contribute to the observed excess at @xmath163 . the excess is based on a single point having a significance of @xmath164 , where the error in almost entirely due to poisson statistics . bolte ( 1994 ) states the significance of the bump as @xmath165 , and it appears to occupy two lf bins in his figure 7 . the significance of his result is probably smaller than that because of the difficulty in determining the position of the `` jump '' ( @xmath166 mag brighter than the msto in @xmath13 ) in his lf . an examination of the @xmath14-band lf in figure [ ilf ] shows the presence of a deviation at the same position in the cmd . this is important because the slope of the sgb is steeper in an @xmath167 cmd than in a @xmath168 cmd . as a result , the bump can no longer be ascribed to a feature caused by the exact slope : it must be the result of an increase in the number of stars congregating near a point on the cluster s fiducial line . at the analogous position in the @xmath14-band lf , there are two bins with excesses of @xmath169 and @xmath170 compared to theory , for a combined significance of @xmath171 . the appearance of the subgiant branch excess in both the @xmath13- and @xmath14-band lfs indicates that the cause must be due to an excess of stars ( rather than being caused by the exact slope of the sgb thus eliminating the exact metallicity , helium content , and oxygen abundance as causes ) . so , the sgb excess has marginal significance in our lfs . in order to more definitively determine the reality of the feature , photometry reaching into the center of this cluster will be needed . the observation of this feature in the combined lf of m68 , m92 , and ngc 6397 ( stetson 1991a ) lends more credence to the phenomenon , but more investigation is necessary . sandquist et al . ( 1996 ) found that there was no evidence for an sgb excess in the lf of the more metal rich cluster m5 ( which has a good @xmath14-band lf for easy comparison with figure [ ilf ] ) . bergbusch ( 1993 ) saw no evidence of an excess in his @xmath13-band lf of ngc 288 . these pieces of evidence seem to indicate that any cause must only be effective at low metallicities . there is , however , a general lack of useful lf data covering the sgb for globular clusters with metallicities between m5 and m30 , or more metal rich than m5 . if the feature is real , there are at least two potential means of creating such an excess : a fluctuation in the initial mass function , and an unknown physical process isothermalizing the stellar core of turnoff mass stars . the excess in stetson s m68-m92-ngc 6397 lf makes mass function fluctuations less likely . a star can be forced to pause on the sgb , but still burning hydrogen in its core , if isothermality is imposed on a large portion of the core ( faulkner & swenson 1993 ) . if such a process occurred in a large enough fraction of the stars in a cluster , a sgb excess could be created in the lf . a way to create such an excess is to invoke a process that increases the efficiency of energy transfer over a large portion of the core . for this to happen , the mean free path of the transporting particle must be large . no such particle has been identified to date . \1 . determinations of the reddening for m30 disagree at a @xmath172 mag level and cases can be made for values ranging from 0.03 to 0.07 in e@xmath64 ) . this uncertainty is the main factor preventing a more accurate determination of the distance modulus . by fitting subdwarfs with _ parallax data to the @xmath0 fiducial line , we find satisfactory fits for @xmath173 ; e(@xmath71 ) pairs ranging from 14.87 ; 0.06 to 14.65 ; 0.02 with the statistical errors of around 0.12 mag ( all for the case [ fe / h ] @xmath174 ) . when shifted to a common reddening , we find our distance modulus is consistent with ground - based estimates , and at least 0.1 mag smaller than other hipparcos - based estimates . m30 has a larger @xmath5 value [ @xmath175 than any of the other metal - poor clusters for which this quantity has been measured . this quantity is usually used as a helium indicator and our measured @xmath5 value suggests a helium abundance @xmath176 larger than the mean of the other metal - poor clusters . m30 s value for the helium indicator @xmath6 is also relatively high although for the case of e@xmath177 , it is consistent with the other metal - poor clusters . if there is a helium abundance enhancement in m30 , it is probably not an initial abundance difference since galactic chemical evolution and the similarity of the m30 and m68 fiducial lines ( see next point ) argue against it . the @xmath130 value for m30 is demonstrably large relative to clusters of similar metallicity . the m30 fiducial line ( except for the hb ) overlies that of m68 ( see figure [ m30vsm68 ] ) and m92 ( vbs ) very closely , indicating that m30 probably has the same age as these two clusters . we suggest that the hb luminosity in m30 is high due to a larger - than - average y for the m30 hb stars . the lfs of the cluster show definite evidence for an excess of rgb stars relative to ms stars , and marginally significant ( @xmath178 ) evidence for an excess of sgb stars , as compared with theory . the sgb feature has slightly higher significance in the @xmath14 band . the possibility remains that these anomalies are only present in low - metallicity clusters . stellar rotation is a possible explanation for the excess number of rgb stars relative to ms stars . alternatively , the excess giants could be a signpost for mixing events on the lower rgb in which fresh hydrogen is mixing into the energy generation region . this could also be identified as the source of the envelope y - enrichment we infer from the hb and brighter rgb stars . we do not find an obvious rgb bump in m30 , in spite of the size of our rgb sample . using the cumulative lf , we have detected the bump in the metal - poor cluster m68 with a @xmath179 value that agrees with the linear trend with [ fe / h ] found by fusi pecci et al . ( 1990 ) . it is possible that points 2 5 are all related to the deep - mixing events inferred for some globulars based on the surface abundances of elements that participate in the energy generation cycles . the hypothesis that we are seeing the effects of the mixing of hydrogen - rich material into the energy - generation regions and helium - rich material out into the stellar envelope can qualitatively explain all of these ( 2 through 5 ) observations . if this hypothesis is correct then we predict that detailed abundance studies of the bright giants in m30 should show the characteristic patterns of deep mixing low oxygen and carbon abundances accompanied by high nitrogen , aluminum , and sodium . we would especially like to thank d. vandenberg for providing us with theoretical @xmath87-enhanced isochrones and luminosity functions prior to publication and p. stetson for the use of his excellent software . it is a pleasure to thank p. guhathakurta , z. webster , and r. rood for useful conversations . this research has made use of the simbad database , operated at cds , strasbourg , france . m.b . is happy to acknowledge support from nsf grant ast 94 - 20204 . electronic copies of the listing of the photometry are available on request to the first author . aperture photometry was performed using the program daophot ii ( stetson 1987 ) . using these data , growth curves were constructed for each frame using daogrow ( stetson 1990 ) in order to extrapolate from the flux measurements over a circular area of finite radius to the total flux observable for the star . the aperture magnitudes and the known standard system magnitudes of landolt ( 1992 ) were then used to derive coefficients for the transformation equations : @xmath180 @xmath181 where @xmath182 and @xmath183 are observed aperture photometry magnitudes , @xmath13 and @xmath14 are the standard system magnitudes , and @xmath184 is the airmass . the primary standard stars covered a color range @xmath185 , completely encompassing the color range of the cluster sample . the coefficients for the transformation equations are given in table [ coeftab ] . the residuals for the sample of 25 stars are shown in figure [ prime ] , and the average residuals are given in table [ restab ] . ( in this and all subsequent comparisons , the residuals are calculated in the sense of ours theirs . ) we chose 118 relatively bright and isolated stars in the m30 field observed with the 1.5 m telescope during the photometric night to be `` secondary standards '' . the stars selected were required to be unsaturated , brighter than the turnoff , in relatively uncrowded regions of the images , and close to the apparent fiducial line of the cluster ( since this acts as an additional check on the accuracy of the photometry ) . once the list was finalized , all other stars were subtracted from the frames and aperture photometry was obtained . the colors for these standards cover the range @xmath186 and @xmath187 both the ctio 4 m and 1.5 m data for m30 were reduced using the standard suite of programs developed by peter stetson ( daophot / allstar ; stetson 1987 , 1989 ) , and following the procedures in sandquist et al . ( 1996 ) . we used the secondary standards established with the 1.5 m observations on the single photometric night to determine the coefficients in the transformation equations for all of the 1.5 m profile fitting photometry : @xmath188 @xmath189 where @xmath182 and @xmath183 are the instrumental magnitudes from the profile fitting , @xmath13 and @xmath14 are the standard values from the aperture photometry , and @xmath190 is an index referring to individual frames . the coefficients of the color terms are given in table [ coeftab ] , the average residuals and standard deviation of the residuals for the comparison of the profile fitting and aperture photometry are given in table [ restab ] , and individual star residuals are shown in figure [ aptopsf ] . in the next step , we chose to calibrate the 4 m profile fitting photometry to the 1.5 m profile fitting photometry rather than the aperture photometry of the secondary standards . this was done primarily to ensure that all of our profile fitting was on the same system over as large a range of magnitudes as possible . the m30 frames taken at the 1.5 m telescope on the one photometric night did not go particularly deep , while the 4 m photometry had few unsaturated observations of the brighter stars in the cluster . the data taken on the non - photometric nights at the 1.5 m telescope did , however , cover a range of magnitudes similar to that of the 4 m data . we selected a sample of 248 stars found in both fields at least 300 pixels away from the cluster center . these stars were used to determine the transformation coefficients for the equations : @xmath191 @xmath192 where @xmath182 and @xmath183 are the instrumental magnitudes from the 4 m observations , and @xmath13 and @xmath14 are the standard values from the 1.5 m observations . the coefficients of the color terms are given in table [ coeftab ] , while the residuals of the comparison of the photometry for the 4 m and 1.5 m measurements of the secondary standards are shown in figure [ secres ] . for the final calibration , we used the transformation equations for the 1.5 m and 4 m profile - fitting data . all of the profile - fitting photometry from both telescopes was combined with weights equal to the inverse square of the internal measurement errors in order to determine our standard - system magnitude and color values . because of the large sky coverage of the ctio frames , most other surveys of m30 overlap the program area at least partially . table [ restab ] provides a summary of the zero - point offsets for comparisons with these studies . we would particularly like to point out that there is considerable difference among them , highlighting the importance of the calibration . the fields used by bolte ( 1987 ) , rfv , and samus et al . ( 1995 ) are completely included on all frames . a comparison with the photometry of bolte is given in figure [ boltecomp ] . in figure [ alccomp ] , we show the comparison with the study of m30 by samus et al . ( we do this partly because it involves the same filter bands , and partly because the residuals are the lowest on average ( although the scatter in star - to - star residuals is large ) . as a note , comparisons with the most recent study ( bergbusch 1996 ) show no signs of color trends in the residuals except within about a magnitude of the tip of the giant branch . cccccc & 17.275 & 0.820 & 29 23.000 & 1.293 & 303 & 17.216 & 0.824 & 28 22.800 & 1.293 & 399 & 17.077 & 0.833 & 20 22.600 & 1.209 & 556 & 16.925 & 0.840 & 23 22.400 & 1.147 & 771 & 16.781 & 0.847 & 14 22.200 & 1.115 & 918 & 16.618 & 0.854 & 38 22.000 & 1.052 & 1048 & 16.391 & 0.863 & 35 21.925 & 1.035 & 827 & 16.140 & 0.880 & 40 21.775 & 1.002 & 811 & 15.868 & 0.901 & 25 21.625 & 0.967 & 850 & 15.621 & 0.913 & 37 21.475 & 0.928 & 896 & 15.367 & 0.927 & 18 21.325 & 0.890 & 934 & 15.127 & 0.953 & 14 21.175 & 0.869 & 907 & 14.877 & 0.969 & 14 21.025 & 0.837 & 910 & 14.662 & 0.990 & 11 20.875 & 0.810 & 965 & 14.395 & 1.013 & 6 20.725 & 0.779 & 908 & 14.117 & 1.052 & 8 20.575 & 0.748 & 876 & 13.924 & 1.068 & 6 20.425 & 0.719 & 856 & 13.604 & 1.115 & 3 20.275 & 0.699 & 843 & 13.319 & 1.143 & 3 20.125 & 0.675 & 740 & 13.125 & 1.178 & 7 19.975 & 0.672 & 765 & 12.830 & 1.223 & 3 19.825 & 0.646 & 694 & 12.634 & 1.260 & 3 19.675 & 0.634 & 620 & 12.384 & 1.329 & 1 19.525 & 0.622 & 566 & 12.016 & 1.481 & 3 19.375 & 0.605 & 519 & 19.225 & 0.588 & 444 & 15.821 & -0.016 & 6 19.000 & 0.576 & 527 & 15.584 & 0.022 & 6 18.800 & 0.568 & 466 & 15.386 & 0.084 & 4 18.600 & 0.558 & 371 & 15.302 & 0.124 & 3 18.400 & 0.575 & 305 & 15.187 & 0.169 & 2 18.240 & 0.595 & 272 & 15.083 & 0.306 & 5 18.094 & 0.645 & 95 & 14.900 & 0.701 & 6 17.968 & 0.695 & 53 & 17.884 & 0.745 & 56 & 14.184 & 0.970 & 4 17.800 & 0.764 & 101 & 14.354 & 0.924 & 1 17.575 & 0.793 & 57 & 14.483 & 0.914 & 1 17.425 & 0.809 & 44 & 14.564 & 0.894 & 1 [ fidtab ] cccccccccc 12.543 & 0.766 & 0.212 & 13 & 1.000 & 18.855 & 191.524 & 14.548 & 184 & 0.960 13.522 & 1.028 & 0.265 & 15 & 1.000 & 19.005 & 240.516 & 16.396 & 229 & 0.952 14.275 & 1.957 & 0.449 & 19 & 1.000 & 19.155 & 276.068 & 17.539 & 261 & 0.945 14.727 & 2.885 & 0.771 & 14 & 1.001 & 19.305 & 285.390 & 17.865 & 267 & 0.934 15.027 & 4.539 & 0.968 & 22 & 1.001 & 19.455 & 332.817 & 19.475 & 306 & 0.918 15.328 & 6.614 & 1.169 & 32 & 0.998 & 19.605 & 361.638 & 20.207 & 331 & 0.915 15.554 & 7.433 & 1.752 & 18 & 1.001 & 19.756 & 386.727 & 20.903 & 351 & 0.907 15.704 & 5.785 & 1.546 & 14 & 1.000 & 19.906 & 451.675 & 22.748 & 403 & 0.892 15.854 & 8.263 & 1.848 & 20 & 1.000 & 20.056 & 475.933 & 23.433 & 419 & 0.880 16.004 & 9.912 & 2.023 & 24 & 1.000 & 20.206 & 556.464 & 25.362 & 488 & 0.877 16.154 & 8.259 & 1.847 & 20 & 1.000 & 20.356 & 574.396 & 25.920 & 498 & 0.867 16.304 & 7.433 & 1.752 & 18 & 1.001 & 20.506 & 632.285 & 27.495 & 537 & 0.850 16.455 & 11.985 & 2.226 & 29 & 1.000 & 20.656 & 664.107 & 28.509 & 555 & 0.837 16.605 & 11.161 & 2.148 & 27 & 1.000 & 20.805 & 698.896 & 29.602 & 575 & 0.824 16.755 & 12.813 & 2.302 & 31 & 1.000 & 20.955 & 733.417 & 30.831 & 592 & 0.809 16.905 & 16.977 & 4.117 & 17 & 1.001 & 21.105 & 781.575 & 32.377 & 622 & 0.798 17.055 & 12.982 & 3.601 & 13 & 1.001 & 21.254 & 751.962 & 32.336 & 588 & 0.784 17.205 & 17.971 & 4.236 & 18 & 1.001 & 21.404 & 805.608 & 33.886 & 620 & 0.774 17.355 & 25.019 & 5.004 & 25 & 0.998 & 21.553 & 779.031 & 34.422 & 580 & 0.749 17.505 & 16.997 & 4.122 & 17 & 0.999 & 21.702 & 903.370 & 38.618 & 648 & 0.722 17.655 & 34.968 & 5.912 & 35 & 1.000 & 21.851 & 922.898 & 40.324 & 643 & 0.702 17.806 & 50.947 & 7.140 & 51 & 1.000 & 22.000 & 986.270 & 44.897 & 643 & 0.658 17.956 & 63.013 & 7.972 & 63 & 1.000 & 22.148 & 998.885 & 46.770 & 622 & 0.631 18.105 & 102.573 & 10.276 & 102 & 0.997 & 22.296 & 1036.098 & 48.374 & 601 & 0.584 18.255 & 117.435 & 11.112 & 116 & 0.990 & 22.445 & 1134.865 & 55.105 & 567 & 0.505 18.405 & 123.294 & 11.517 & 121 & 0.983 & 22.593 & 1350.132 & 71.038 & 501 & 0.378 18.555 & 159.431 & 13.317 & 155 & 0.973 & 22.740 & 3956.208 & 404.959 & 500 & 0.129 18.705 & 191.698 & 14.697 & 185 & 0.964 & & & & & [ vlftab ] cccccccccc 11.149 & 0.590 & 0.170 & 12 & 1.000 & 17.467 & 76.966 & 8.715 & 78 & 1.000 12.225 & 0.850 & 0.236 & 13 & 1.000 & 17.619 & 111.136 & 10.508 & 112 & 0.999 12.993 & 1.281 & 0.355 & 13 & 1.000 & 17.770 & 104.511 & 10.283 & 104 & 0.992 13.452 & 2.373 & 0.685 & 12 & 1.000 & 17.921 & 131.605 & 11.657 & 130 & 0.985 13.756 & 2.779 & 0.743 & 14 & 1.000 & 18.071 & 188.224 & 14.172 & 183 & 0.972 14.060 & 3.586 & 0.845 & 18 & 1.000 & 18.221 & 185.096 & 14.125 & 178 & 0.964 14.287 & 4.359 & 1.315 & 11 & 1.001 & 18.370 & 253.828 & 16.588 & 243 & 0.959 14.439 & 6.772 & 1.643 & 17 & 0.997 & 18.520 & 290.213 & 17.909 & 273 & 0.943 14.591 & 7.986 & 1.786 & 20 & 1.001 & 18.670 & 302.430 & 18.308 & 282 & 0.936 14.742 & 5.598 & 1.496 & 14 & 1.000 & 18.819 & 345.452 & 19.861 & 313 & 0.911 14.893 & 5.998 & 1.549 & 15 & 1.000 & 18.968 & 421.574 & 21.974 & 380 & 0.905 15.044 & 6.780 & 1.645 & 17 & 1.000 & 19.117 & 465.390 & 23.205 & 414 & 0.894 15.195 & 9.954 & 1.991 & 25 & 1.000 & 19.266 & 506.015 & 24.377 & 441 & 0.881 15.347 & 5.985 & 1.546 & 15 & 1.000 & 19.415 & 613.201 & 27.068 & 526 & 0.860 15.498 & 9.583 & 1.956 & 24 & 1.000 & 19.564 & 693.000 & 29.054 & 583 & 0.853 15.649 & 8.810 & 1.879 & 22 & 1.000 & 19.712 & 777.236 & 31.180 & 637 & 0.832 15.799 & 8.811 & 1.879 & 22 & 1.000 & 19.859 & 820.648 & 32.433 & 657 & 0.819 15.950 & 12.407 & 2.229 & 31 & 1.000 & 20.006 & 833.922 & 33.018 & 656 & 0.807 16.101 & 11.995 & 2.190 & 30 & 1.001 & 20.152 & 961.583 & 36.406 & 730 & 0.779 16.252 & 14.400 & 2.400 & 36 & 1.000 & 20.298 & 1024.388 & 38.171 & 758 & 0.762 16.403 & 14.773 & 2.429 & 37 & 1.000 & 20.443 & 1042.008 & 39.215 & 755 & 0.748 16.554 & 21.175 & 2.909 & 53 & 0.999 & 20.587 & 1104.607 & 42.537 & 743 & 0.708 16.705 & 21.109 & 2.900 & 53 & 1.000 & 20.730 & 1145.276 & 43.849 & 759 & 0.695 16.857 & 25.699 & 5.040 & 26 & 1.001 & 20.873 & 1314.175 & 49.073 & 833 & 0.666 17.008 & 26.650 & 5.129 & 27 & 1.000 & 21.013 & 1385.530 & 55.311 & 777 & 0.610 17.161 & 41.972 & 6.400 & 43 & 1.000 & 21.155 & 1525.717 & 70.014 & 738 & 0.501 17.315 & 46.912 & 6.771 & 48 & 1.000 & & & & & [ ilftab ] lcccccccl 14594 & 8.04 & 0.66 & 0.02585 & 0.044 & @xmath193 & @xmath194 & 0.66 & hd19445 18915 & 8.51 & 1.01 & 0.05414 & 0.020 & @xmath195 & @xmath196 & 0.99 & hd25329 24316 & 9.43 & 0.65 & 0.01455 & 0.069 & @xmath197 & @xmath198 & 0.61 & hd34328 38541 & 8.27 & 0.77 & 0.03529 & 0.029 & @xmath199 & @xmath200 & 0.75 & hd64090 40778 & 9.73 & 0.60 & 0.01036 & 0.142 & @xmath201 & @xmath202 & 0.57 & bd+54 1216 53070 & 8.22 & 0.63 & 0.01923 & 0.059 & @xmath203 & @xmath204 & 0.59 & hd94028 57939 & 6.44 & 0.89 & 0.10921 & 0.007 & @xmath205 & @xmath206 & 0.84 & hd103095 60632 & 9.66 & 0.63 & 0.01095 & 0.118 & @xmath207 & @xmath208 & 0.60 & hd108177 74234 & 9.46 & 1.01 & 0.03368 & 0.050 & @xmath209 & @xmath210 & 0.99 & hd134440 74235 & 9.08 & 0.92 & 0.03414 & 0.040 & @xmath209 & @xmath211 & 0.90 & hd134439 98020 & 8.83 & 0.75 & 0.02532 & 0.046 & @xmath212 & @xmath213 & 0.70 & hd188510 100568 & 8.66 & 0.67 & 0.02288 & 0.054 & @xmath214 & @xmath215 & 0.60 & hd193901 100792 & 8.35 & 0.63 & 0.01794 & 0.069 & @xmath216 & @xmath217 & 0.55 & hd194598 104659 & 7.37 & 0.66 & 0.02826 & 0.036 & @xmath218 & @xmath219 & 0.56 & hd201891 3026 & 9.25 & 0.64 & 0.00957 & 0.144 & @xmath220 & @xmath221 & 0.54 & hd3567 33221 & 9.07 & 0.63 & 0.00911 & 0.111 & @xmath222 & @xmath223 & 0.53 & cpd-33 3337 48152 & 8.33 & 0.55 & 0.01244 & 0.085 & @xmath224 & @xmath225 & 0.55 & hd84937 55790 & 9.07 & 0.63 & 0.01099 & 0.135 & @xmath226 & @xmath227 & 0.55 & hd99383 68464 & 8.73 & 0.64 & 0.00977 & 0.135 & @xmath228 & @xmath229 & 0.60 & hd122196 76976 & 7.22 & 0.69 & 0.01744 & 0.056 & @xmath230 & @xmath231 & 0.77 & hd140283 [ subdtab ] cccccc @xmath232 & 0.06 & @xmath233 & * @xmath234 * & @xmath235 & @xmath236 @xmath232 & 0.02 & @xmath237 & * @xmath2 * & @xmath238 & @xmath239 @xmath240 & 0.06 & @xmath233 & @xmath241 & @xmath242 & @xmath243 @xmath240 & 0.02 & @xmath244 & @xmath245 & @xmath246 & @xmath247 [ dmtab ] lccccc ngc 104 ( 47 tuc ) & @xmath248 & @xmath249 & @xmath250 & @xmath251 & @xmath214 ngc 5904 ( m5 ) & @xmath252 & @xmath253 & @xmath254 & @xmath255 & @xmath256 ngc 5272 ( m3 ) & @xmath257 & @xmath258 & @xmath259 & @xmath260 & @xmath261 ngc 4590 ( m68 ) & @xmath262 & @xmath263 & @xmath264 & @xmath265 & @xmath266 ngc 5024 ( m53 ) & @xmath267 & @xmath268 & & @xmath269 & @xmath270 ngc 5053 & @xmath271 & @xmath272 & & @xmath273 & @xmath274 ngc 5466 & @xmath275 & @xmath276 & & & @xmath277 ngc 6341 ( m92 ) & @xmath278 & @xmath279 & & @xmath280 & @xmath281 ngc 6397 & @xmath232 & @xmath282 & & @xmath283 & @xmath284 ( 0.69 ) ngc 7078 ( m15 ) & @xmath285 & @xmath286 & @xmath287 & @xmath288 & @xmath289 ngc 7099 ( m30 ) & @xmath240 & @xmath96 & @xmath290 & @xmath291 & @xmath292 [ clusttab ] lcccccc @xmath293 & 57 & 2 : & 1 & 60 & 3 & 46 & @xmath125 & @xmath294 & @xmath295 & @xmath296 & @xmath297 & @xmath298 & 59 & 4 & 2 & 65 & 4 & 48 & @xmath126 & @xmath299 & @xmath300 & @xmath301 & @xmath302 & @xmath303 & 65 & 3 & 9 & 77 & 4 & 42 & @xmath128 & @xmath304 & @xmath305 & @xmath296 & @xmath306 & total & 181 & 9 & 12 & 202 & 11 & 136 & @xmath96 & @xmath307 & @xmath308 & @xmath309 & @xmath310 & ygsb & 51 & 1 : & 1 & 53 & 1 & 34 & @xmath311 & & & & & [ poptab ] ccccc @xmath13 & @xmath312 & @xmath313 & @xmath314 & 1 & & & @xmath315 & 2 & & & @xmath316 & 3 @xmath14 & @xmath317 & @xmath318 & @xmath319 & 1 @xmath13 & & & @xmath320 & 1 @xmath14 & & & @xmath321 & 1 @xmath13 & & & @xmath322 & 1 & & & @xmath323 & 2 @xmath14 & & & @xmath324 & 1 & & & @xmath325 & 2 [ coeftab ] ccccccccc 1.5 m & landolt & 0.0005 & 0.0006 & @xmath326 & 0.0119 & @xmath327 & 0.0010 & 27 psf & aperture & @xmath327 & 0.0204 & @xmath328 & 0.0198 & @xmath329 & 0.0252 & 118 4 m & 1.5 m & @xmath330 & 0.0447 & @xmath331 & 0.0635 & @xmath332 & 0.0515 & 248 4 m + 1.5 m & bolte 1987 ( s ) & @xmath333 & 0.1067 & & & 59 4 m + 1.5 m & bolte 1987 ( l ) & @xmath334 & 0.0928 & & & 401 4 m + 1.5 m & rfv & @xmath335 & 0.1419 & & & 1374 4 m + 1.5 m & samus et al . 1995 & @xmath336 & 0.1449 & @xmath337 & 0.1771 & @xmath338 & 0.1139 & 255 4 m + 1.5 m & bergbusch 1996 & @xmath339 & 0.0611 & & & 316 [ restab ]

we present new @xmath0 photometry for the halo globular cluster m30 ( ngc 7099 = c2137 - 174 ) , and compute luminosity functions ( lfs ) in both bands for samples of about 15,000 hydrogen - burning stars from near the tip of the red giant branch ( rgb ) to over four magnitudes below the main - sequence ( ms ) turnoff . we confirm previously observed features of the lf that are at odds with canonical theoretical predictions : an excess of stars on subgiant branch ( sgb ) approximately 0.4 mag above the turnoff and an excess number of rgb stars relative to ms stars . based on subdwarfs with _ hipparcos_-measured parallaxes , we compute apparent distance moduli of @xmath1 and @xmath2 for reddenings of e@xmath3 and 0.02 respectively . the implied luminosity for the horizontal branch ( hb ) at these distances is @xmath4 and 0.37 mag . the two helium indicators we have been able to measure ( @xmath5 and @xmath6 ) both indicate that m30 s helium content is high relative to other clusters of similar metallicity . m30 has a larger value for the parameter @xmath7 than any of the other similarly metal - poor clusters for which this quantity can be reliably measured . this suggests that m30 has either a larger age or higher helium content than all of the other metal - poor clusters examined . the color - difference method for measuring relative ages indicates that m30 is coeval with the metal - poor clusters m68 and m92 . = -0.5 in epsf

astro-ph9810259

there are currently several exciting proposals to use the ( 001 ) surface of silicon for the construction of atomic - scale electronic devices , including single electron transistors @xcite , ultra - dense memories @xcite and quantum computers @xcite . however , since any random charge or spin defects in the vicinity of these devices could potentially destroy their operation , a thorough understanding of the nature of crystalline defects on this surface is essential . the si(001 ) surface was first observed in real space at atomic resolution using scanning tunneling microscopy ( stm ) by tromp _ _ et . al.__@xcite in 1985 . in this study they observed the surface consisted of rows of `` bean - shaped '' protrusions which were interpreted as tunneling from the @xmath1-bonds of surface si dimers , thereby establishing the dimer model as the correct model for this surface . since then , stm has been instrumental in further elucidating the characteristics of this surface , and in particular atomic - scale defects present on the surface@xcite . the simplest defect of the si(001 ) surface is the single dimer vacancy defect ( 1-dv ) , shown schematically in figs . [ def1](a ) and [ def1](b ) . this defect consists of the absence of a single dimer from the surface and can either expose four second - layer atoms ( fig . [ def1](a ) ) or form a more stable structure where rebonding of the second - layer atoms occurs @xcite as shown in fig . [ def1](b ) . while the rebonded 1-dv strains the bonds of its neighboring dimers it also results in a lowering of the number of surface dangling bonds and has been found to be more stable than the nonbonded structure . @xcite single dimer vacancy defects can also cluster to form larger defects such as the double dimer vacancy defect ( 2-dv ) and the triple dimer vacancy defect ( 3-dv ) . more complex clusters also form , the most commonly observed@xcite example is the 1 + 2-dv consisting of a 1-dv and a 2-dv separated by a single surface dimer , the so - called `` split - off dimer '' . the accepted structure of the 1 + 2-dv , as proposed by wang _ et . based on total energy calculations,@xcite is shown in fig . [ def1](c ) and consists of a rebonded 1-dv ( left ) , a split - off dimer , and a 2-dv with a rebonding atom ( right ) . recently we have observed another dv complex that contains a split - off dimer , called the 1 + 1-dv , which consists of a rebonded 1-dv and a nonbonded 1-dv separated by a split - off dimer , as shown in fig . [ def1](d ) . here we present a detailed investigation of dv defect complexes that contain split - off dimers . using high - resolution , low - bias stm we observe that split - off dimers appear as well - resolved pairs of protrusions under imaging conditions where normal si dimers appear as single `` bean - shaped '' protrusions . we show that this difference arises from an absence of the expected @xmath1-bonding between the two atoms of the split - off dimer but instead the formation of @xmath1-bonds between the split - off dimer atoms and second layer atoms . electron charge density plots obtained using first principles calculations support this interpretation . we observe an intensity enhancement surrounding some split - off dimer defect complexes in our stm images and thereby discuss the local strain induced in the formation of these defects . finally , we present a model for a previously unreported triangular - shaped split - off dimer defect complex that exists at s@xmath2-type step edges . experiments were performed in two separate but identical variable temperature stm systems ( omicron vt - stm ) . the base pressure of the ultra - high vacuum ( uhv ) chamber was @xmath3 mbar . phosphorus doped @xmath4 and @xmath5 @xmath6 wafers , orientated towards the [ 001 ] direction were used . these wafers were cleaved into @xmath7 mm@xmath8 sized samples , mounted in sample holders , and then transferred into the uhv chamber . wafers and samples were handled using ceramic tweezers and mounted in tantalum / molybdenum / ceramic sample holders to avoid contamination from metals such as ni and w. sample preparation@xcite was performed in vacuum without prior _ ex - situ _ treatment by outgassing overnight at 850 k using a resistive heater element , followed by flashing to 1400 k by passing a direct current through the sample . after flashing , the samples were cooled slowly ( @xmath9 k / s ) from 1150 k to room temperature . the sample preparation procedure outlined above routinely produced samples with very low surface defect densities . however , the density of defects , including split - off dimer defects , was found to increase over time with repeated sample preparation and stm imaging , as reported previously.@xcite it is known that split - off dimer defects are induced on the si(001 ) surface by the presence of metal contamination such as ni , @xcite and w @xcite . the appearance of these defects in our samples therefore points to a build up of metal contamination , either ni from in - vacuum stainless steel parts , or more likely w contamination from the stm tip . after using an old w stm tip to scratch a @xmath10 1 mm line on a si(001 ) sample in vacuum and then reflashing , the concentration of split - off dimer defects on the surface was found to have dramatically increased , confirming the stm tip as the source of the metal contamination . figure [ sods ] shows an stm image of a si(001 ) surface containing a @xmath10 10% coverage of split - off dimer defects . the majority of the defects in this image can be identified as 1 + 2-dvs , however , two 1 + 1-dvs are also present , as indicated . the most striking feature of this image is the difference in appearance of the split - off dimers in contrast to the surrounding normal surface dimers . each split - off dimer in this image appears as a double - lobed protrusion , while the surrounding normal si dimers each appear as a single `` bean - shaped '' protrusion , as expected at this tunneling bias . @xcite line profiles taken across a 1 + 2-dv both parallel and perpendicular to the dimer row direction are shown in fig . [ sods](b ) . the line profile parallel to the dimer row direction agrees with previously reported profiles over 1 + 2-dvs and fits well with the accepted structure , @xcite as shown by the overlayed ball and stick model . the line profile taken perpendicular to the dimer row direction , however , clearly shows that the split - off dimer of this defect is separated into two protrusions while the neighboring si dimers are single protrusions . this is the first recognition and explanation of split - off dimers appearing as double - lobed protrusions . 1 surface with split - off dimer defects is shown in ( a ) . tunneling conditions for this image were @xmath11 v sample bias and 0.8 na tunnel current . line profiles are taken across a single 1 + 2-dv both parallel , x x@xmath12 ( b ) , and perpendicular , y y@xmath12 ( c ) , to the dimer row direction , as indicated in ( a ) . the schematic ( d ) is a top view ball and stick model of a 1 + 2-dv with the approximate positions of @xmath1-bonds indicated by shaded ellipses . ] to understand why split - off dimers appear as double - lobed protrusions we must consider the structure of these defects shown in figs . [ def1](c ) and [ def1](d ) . normally si(001 ) surface dimers appear as `` bean - shaped '' protrusions in stm images because the dangling bonds of each si dimer atom mix to form a @xmath1-bond between the two dimer atoms . however , if we examine the split - off dimer structure closely ( figs . [ def1](c ) and [ def1](d ) ) we see that unlike normal surface dimers , the split - off dimer has two nearest neighbor second layer atoms that each have a dangling bond . the separation distance between the split - off dimer atoms and these second layer atoms is sufficiently close to allow the formation of @xmath1-bonds . the resulting four - atom structure can therefore be referred to as a _ tetramer_. we propose that the four dangling bonds of the split - off dimer tetramer interact primarily along the backbonds between the split - off dimer atoms and the second layer atoms to form @xmath1-bonds down the backbonds , as drawn schematically in fig . [ sods](c ) . these two spatially separated @xmath1-bonds therefore lead to the double - lobed appearance of the split - off dimers under low bias filled - state tunneling conditions , which we confirm in section [ theory1 ] with charge density calculations . in an attempt to fully characterize the appearance of these split - off dimers in stm images , we have performed a series of experiments observing split - off dimers with changing stm sample bias . figure [ sodv ] summarizes our results , showing images where a 1 + 2-dv and a 1 + 1-dv located next to each other are observed at four different sample biases two filled - state images and two empty - state images . in the filled - state image of fig . [ sodv](a ) we see that at @xmath13 v the split - off dimer of both the 1 + 2-dv and the 1 + 1-dv appear as double - lobed protrusions similar to those in fig . [ sods](a ) . however , when the filled - state bias is increased in magnitude to @xmath14 v , fig . [ sodv](b ) , the split - off dimers become single - protrusions and appear very similar to the surrounding normal si surface dimers . this is because as the bias magnitude is increased towards @xmath14 v , the dimer @xmath15-bond and bulk states contribute increasingly to the tunneling current @xcite and the image of the split - off dimer reverts to the bean - shaped protrusion in the same manner as normal surface si dimers . in both of the empty - state images , figs . [ sodv](c ) and [ sodv](d ) , acquired at + 0.8 v and + 2 v , respectively , the appearance of the split - off dimers is very similar to that of the surrounding normal surface dimers . this is because under empty - state tunneling conditions electrons tunnel into the @xmath16-antibonding orbitals of the dimers , resulting in the normal si dimers appearing as double - lobed protrusions . @xcite it is therefore only under low bias magnitude filled - state tunneling conditions that split - off dimers appear significantly different to the surrounding normal si surface dimers . v , ( b ) @xmath14 v , ( c ) @xmath17 v , ( d ) @xmath18 v. ] another noticeable feature of figs . [ sods](a ) and [ sodv](a ) is the enhanced brightness of the 1 + 1-dv compared to the 1 + 2-dv . this is a reproducible effect that we attribute to an increased amount of surface strain induced by the 1 + 1-dv . figure [ strain ] shows a series of adjacent defects forming a short vacancy line channel in the surface . this channel is composed of individual 1-dv , 3-dv , 1 + 2-dv , and 1 + 1-dv defects ( see figure caption ) . in the filled - state image , fig . [ strain](a ) , there is a clear brightening of the dimers on one end of the 1 + 1-dvs and the dimers on both ends of the 1-dv , which is not present for the 1 + 2-dvs . in the empty - state image of the line of defect complexes , fig . [ strain](b ) , we notice that there is a darkening of the same dimers that are enhanced in the filled - state image . v , @xmath19 v , 0.15 na ) of a short chain of dvs in a si(001 ) surface . the individual defects are ( from top left to bottom right ) : 1 + 1-dv , 1 + 1-dv , 1 + 2-dv , 3-dv , 1 + 2-dv , 1 + 2-dv , 1 + 1-dv , 1-dv , and 1 + 2-dv . note the strain - induced brightening of the 1-dv and 1 + 1-dvs in the filled - state ( a ) and the corresponding darkening in the empty - state ( b ) ] owen _ et . _ , @xcite have shown using low bias stm and first principles calculations , that the dimers neighboring a rebonded 1-dv are enhanced in low bias filled - state stm images due to the strain induced by the defect shifting the surface states upwards in energy toward the fermi energy . this effect can be seen for the 1-dv in fig . [ strain](a ) , where the neighboring dimers in the same row as the 1-dv are enhanced in intensity , with the magnitude of the enhancement decaying with distance from the 1-dv . a very similar enhancement can be seen around the 1 + 1-dv sites in this image , with the split - off dimer in particular appearing much brighter than the surrounding normal surface dimers . however , for the 1 + 1-dv only the dimers on one end of the defect are enhanced in intensity while the dimers on the other end of the defect are not . this observation can be readily explained since the 1 + 1-dv is composed of a rebonded 1-dv adjacent to a nonbonded 1-dv ( fig . [ def1](d ) ) and owen _ et . @xcite have shown that while the rebonded 1-dv results in strain - induced image enhancement , the nonbonded 1-dv does not . the observation of an asymmetric strain - induced enhancement of the 1 + 1-dv in fig . [ strain](a ) can therefore be taken as an experimental confirmation of the structure of this defect ( fig . [ def1](d ) ) and the first application of the method of owen _ et . @xcite for identifying strain in more complex surface defect structures . the fact that the 1 + 2-dv causes no enhancement of its neighboring dimers over the surrounding normal surface dimers suggests that the 1 + 2-dv , unlike the 1-dv and 1 + 1-dvs , does not increase the strain of the surface . this at first seems strange , since the 1 + 2-dv involves a rebonded 1-dv similar to the 1 + 1-dv structure . however , wang _ et . _ @xcite have shown , using total energy calculations , that the junction formed between the 1-dv and the 2-dv to create the 1 + 2-dv releases the surface strain that is present when these two defects exist separately from one another . the stm data that we have presented here is therefore the first experimental verification of this calculation . the fact that both the 1-dv and the 1 + 1-dv show local enhancement due to strain , while the 1 + 2-dv does not , indicates that the 1 + 2-dv structure induces less local strain than the 1-dv . in their paper , owen _ et . do not present empty state stm images , nor do they consider empty states in their tight binding calculations . in fig . [ strain](b ) , we show an empty state image of the same line of defects shown in fig . [ strain](a ) . interestingly , in this empty state image the dimers that were enhanced in brightness surrounding the 1-dv and 1 + 1-dvs in the filled - state image are less bright than the surrounding si dimers in the empty - state image . this suggests that the strain associated with these defects causes the lowest unoccupied molecular orbital ( lumo ) of the adjacent dimers to also shift higher in energy , away from the fermi energy . to confirm the interpretation of our stm images , we have performed first principles electronic structure calculations of both the 1 + 2-dv and 1 + 1-dv complexes using the car - parrinello molecular dynamics program . @xcite valence electrons were described using goedecker pseudopotentials @xcite expanded in a basis set of plane waves with an energy cutoff of 18 rydbergs and the exchange - correlation functional was of the blyp form . @xcite slab calculations contained between 124 and 128 si atoms in a @xmath20 @xmath21 supercell , corresponding to six layers of vacuum in the @xmath22-direction , and all calculations were performed with gamma point sampling of the brillouin zone only . a reference calculation was performed with no surface vacancies and assuming the @xmath23 structure in which the dimers buckled alternately along the row . a single 256 atom calculation with a duplication along the y - axis confirmed that the effect of dispersion across the rows is minor as has been noted elsewhere . @xcite both zero temperature geometry optimization and high temperature molecular dynamics calculations were used to explore a variety of surface and second - layer bonding configurations for the 1 + 2-dv and 1 + 1-dv . the results confirm the configurations in figs . [ def1](c ) and [ def1](d ) are the lowest energy geometries of both defect complexes . the dimers are drawn symmetric in these schematics , however , the true minimum energy structure at zero temperature involved charge - transfer buckling of the si dimers . it is well known that at room temperature the barrier is sufficiently small for the dimers to flip - flop between the two equivalent configurations . @xcite our calculations show that the split - off dimer tetramer also has two symmetrically equivalent buckling configurations , with charge transfer between the atoms of the tetramer buckling adjacent atoms in alternate directions . by analogy with the normal dimers we can expect room - temperature stm measurements of the tetramer to image the average of the two configurations . the chemical potential was determined from a 512-atom bulk calculation , which yielded a formation energy of 0.85 ev for the 1 + 2-dv , similar to the value of 0.65 ev computed by wang _ at . al_. @xcite the 1 + 1-dv formation energy has not been previously reported , and we found it to be 1.13 ev . we note that this value is high , but this is consistent with the rarity of observation of the 1 + 1-dv in stm experiments . in fig . [ 1 + 2-dv ] we present a series of calculated electron density slices through various regions of the 1 + 2-dv marked by ( a ) , ( b ) , ( c ) , and ( d ) in the ball and stick schematic . the charge density shown in the figure is the sum of the occupied kohn - sham orbitals within 0.25 ev of the highest occupied molecular orbital ( homo ) . taking into account the @xmath24 ev surface band gap of si(001 ) and the n - type doping of the experimental samples , these states correspond approximately to the accessible states for a @xmath25 v sample bias and can therefore be directly compared to the experimental data in fig . [ sodv](a ) , which was acquired with a @xmath13 v sample bias . -bonding as inferred from the electron density ( see text ) . each electron density plot is an average of both buckling configurations , and the atomic positions and bonds are shown as black balls and sticks . the slices are ( a ) rebonded 1-dv edge dimer , ( b ) split - off dimer , ( c ) split - off dimer backbonds , ( d ) 2-dv edge dimer . ] the four charge density slices in fig . [ 1 + 2-dv ] show : fig . [ 1 + 2-dv](a ) the 1-dv edge dimer , fig . [ 1 + 2-dv](b ) the split - off dimer , fig . [ 1 + 2-dv](c ) the backbond of the split - off dimer , and fig . [ 1 + 2-dv](d ) the 2-dv edge dimer , as indicated schematically in fig . [ 1 + 2-dv](e ) . the charge densities of both buckling configurations of the dimers and backbond atoms are averaged , and the positions of the dimer and tetramer atoms are shown superimposed in both buckling configurations . in the case of the backbonds , the two configurations are not coincident , and so the atoms and bonds are shown in projection onto the plane in fig . [ 1 + 2-dv](c ) . the 1-dv edge dimer in fig . [ 1 + 2-dv](a ) shows a clear three - lobed character with significant overlap between the up - atom charge density of the two buckling orientations , and a single lobe beneath the plane of the surface at the mid - point of the dimer . density functional calculations by hata _ et . al . _ @xcite and tight - binding green s function calculations by pollman _ et @xcite have separately identified this three - lobed feature as being characteristic of @xmath1-bonding in flip - flop dimers on the silicon surface , and we can therefore take this three - lobed feature as a signature of @xmath1-bonding in this work . the backbond of the split - off dimer in fig . [ 1 + 2-dv](c ) connects a first - layer atom to a second - layer atom and also shows a three - lobed structure . by analogy with the surface dimer in fig . [ 1 + 2-dv](a ) we characterize this bond as having @xmath1-character and have indicated this by the shaded ellipse ( c ) shown in fig . [ 1 + 2-dv](e ) . the split - off dimer itself in fig . [ 1 + 2-dv](b ) , however , does not exhibit three - lobed character . instead , the split - off dimer has four lobes ; two located above the up - atoms of the dimer in each buckling configuration , and a second pair of spatially separated lobes beneath the bond . the calculations thus show that @xmath1-bonding occurs down the backbonds of the split - off dimer , but not across the dimer itself . the absence of the @xmath1-bond across the split - off dimer correlates with the double - protrusions observed in the stm images . finally , we also consider the charge density of the 2-dv edge dimer , fig . [ 1 + 2-dv](d ) , and note that it also exhibits three - lobed character , indicative of @xmath1-bonding . this gives the 2-dv edge dimer a bean - shaped appearance in the stm image , as for the 1-dv dimer in fig . [ 1 + 2-dv](a ) . a similar situation exists for the 1 + 1-dv charge density slices shown in fig . [ 1 + 1-dv ] the first three charge density slices , figs . [ 1 + 1-dv](a ) [ 1 + 1-dv](c ) , are analogous to the slices for the 1 + 2-dv as was the case for the 1 + 2-dv , the rebonded 1-dv edge dimer , fig . [ 1 + 1-dv](a ) and the split - off dimer backbonds , fig . [ 1 + 1-dv](c ) exhibit three - lobed @xmath1-like character , while the split - off dimer , fig . [ 1 + 1-dv](b ) exhibits four - lobed character , consistent with an end - on view of @xmath1-bonding down the backbonds . finally , another slice is presented in fig . [ 1 + 1-dv](d ) , which is through the nonbonded 1-dv edge dimer as indicated schematically in fig . [ 1 + 1-dv](e ) . it can be seen that the nonbonded 1-dv edge dimer appears quite different to the charge density slices discussed so far . in particular , we notice that the nonbonded 1-dv edge dimer has a much reduced charge density compared to the other slices , fig . [ 1 + 1-dv](a ) [ 1 + 1-dv](c ) . examination of the structure identifies strain as the characteristic that differentiates the dimer in fig . [ 1 + 1-dv](d ) from the other dimers . since the dimer in fig . [ 1 + 1-dv](d ) is part of a tetramer , one might expect its appearance to resemble the split - off dimer which is also part of the tetramer shown figs . [ 1 + 1-dv](b ) and [ 1 + 1-dv](c ) . however , a detailed examination of the simulated structure reveals that the nonbonded 1-dv tetramer is relaxed , since there is one adjacent dimer present , while the split - off tetramer is highly strained because of the rebonding in the second - layer . since the nonbonded 1-dv tetramer is much less strained , its occupied states lie further from the fermi level , explaining the charge reduction observed in calculations in fig . [ 1 + 1-dv](d ) . as discussed in ref . , the minimum energy arrangement of the electrons in a tetramer is one where the @xmath1-states are delocalized across the four atoms , to form three bonding segments , as indicated by the ellipses in fig . [ 1 + 1-dv](e ) . the charge density slice of fig . [ 1 + 1-dv](d ) is consistent with such an arrangement where the charge density is shared between @xmath1-like bonds on both backbonds and across the dimer atoms . we conclude that this charge density arrangement forms for the nonbonded 1-dv tetramer because it is allowed to relax . in the case of the split - off dimer , the tetramer is constrained by the rebonding and instead forms a higher energy configuration in which the @xmath1-bonds conjugate to form two @xmath1-bonds down its backbonds . -bonding as inferred from the electron density ( see text ) . each electron density plot is an average of both buckling configurations , and the atomic positions and bonds are shown as black balls and sticks . the slices are ( a ) rebonded 1-dv edge dimer , ( b ) split - off dimer , ( c ) split - off dimer backbonds , ( d ) nonbonded 1-dv edge dimer . ] having presented a detailed understanding of the electronic structure of previously observed split - off dimer defects in the si(001 ) surface using both stm and first - principles calculations , we now turn our attention to elucidating the structure of a previously unreported split - off dimer defect . in figs . [ triangular](a ) and [ triangular](b ) we show filled- and empty - state stm images of dv defects at a single - layer s@xmath2-type step edge . at the top of these images white arrows indicate are three defects known as s@xmath2-dvs , which are rebonded 1-dvs at the step edge , which leave a single split - off dimer as the last dimer before the lower terrace begins . @xcite as was the case for the 1 + 1-dv and 1 + 2-dv , the split - off dimers in s@xmath2-dvs appear as double - lobed protrusions under low - bias filled - state imaging conditions , fig . [ triangular](a ) . at the bottom of fig . [ triangular](a ) two similar dv complexes can be observed , as indicated by black arrows , however these defects have a third protrusion giving them a triangular appearance . in empty - state imaging , fig . [ triangular](b ) , however , the additional third feature is not present . these triangular - shaped defects have not been reported on the si(001 ) surface before and most likely arise due to the presence of w contamination . 1.2 v ) of dv defects at an s@xmath2-type step edge . white arrows indicate s@xmath2-dvs , @xcite while black arrows point to a previously unreported defect that exhibits a third protrusion in the filled - state giving it a triangular appearance . we propose the structure ( c ) as a model for this defect . calculated charge density slices at a constant @xmath22-height for the dashed region of ( c ) are shown in ( d ) and ( e ) ( for kohn - sham orbitals summed over 0.45 ev below the homo and 0.45 ev above the lumo , respectively ) . these contour slices are in good agreement with the stm images in ( a ) and ( b ) , in particular predicting the correct spacing of 6.4 between the split - off dimer and third protrusion and also the disappearance of the third protrusion in the empty - state . the horizontal tic - marks in ( d ) and ( e ) indicate the dimer positions on the defect - free surface . ] our proposed structural model of the triangular - shaped defects in fig . [ triangular](a ) is shown in fig . [ triangular](c ) . this model consists of a nonbonded 1-dv defect at an s@xmath2-type step edge , followed by a rebonded split - off dimer and a bound si monomer . swartzentruber has previously observed si monomers on the si(001 ) surface using high - resolution stm after depositing a few percent of a monolayer of si atoms to the surface . @xcite these monomers were bound at rebonded s@xmath2-type step edges , confirming the minimum energy binding position predicted by first principles calculations . the binding position of the monomer in our proposed structure , fig . [ triangular](c ) , is essentially the same position observed by swartzentruber , with the difference being the presence of the dv defect adjacent to the step edge . swartzentruber also observed that the si monomers bound at s@xmath2-type step edges were visible in one bias polarity ( empty - state ) but invisible in the other ( filled - state ) . our images reveal a similar effect , however the feature we observe appears in filled - state images while being invisible in empty - state images . we have performed first - principles calculations to produce charge density contours for our proposed structure . figure 7(d ) shows a constant @xmath22-height contour slice taken 1.2 above the monomer for occupied kohn - sham orbitals within 0.45 ev of the homo . we see in this charge density contour slice the two lobes expected for the split - off dimer as well as a third lobe due to the bound monomer . moreover , the distance between the split - off dimer lobes and the monomer lobe is 6.4 in agreement with the separation seen in the stm image . in fig . [ triangular](e ) we show an empty - state slice taken at the same @xmath22-height and summed over kohn - sham orbitals up to 0.45 ev above the lumo . in this contour the double lobe of the split - off dimer is still present but the monomer lobe is significantly lessened in intensity . the results of our first - principles calculations therefore give good agreement between our proposed structure and the observed defect . the presence of the split - off dimer must therefore be responsible for the reversal of the filled- and empty - state monomer characteristics when compared to those observed for monomers bound to rebonded s@xmath2-type step edges . we have investigated split - off dimers on the si(001)2@xmath261 surface using high resolution stm and first principles calculations . we find that split - off dimers form @xmath1-bonds with second layer atoms which gives them a double - lobed appearance in low bias filled - state stm images . we apply the method of owen _ et . @xcite for identifying local areas of increased surface strain to dimer vacancy defect complexes and thereby present the first experimental confirmation of the predicted strain relief offered by the 1 + 2-dv . finally , we have presented a previously unreported triangular - shaped defect on the si(001 ) surface and a proposed model for this structure involving a bound si monomer .

dimer vacancy ( dv ) defect complexes in the si(001)@xmath0 surface were investigated using high - resolution scanning tunneling microscopy and first principles calculations . we find that under low bias filled - state tunneling conditions , isolated ` split - off ' dimers in these defect complexes are imaged as pairs of protrusions while the surrounding si surface dimers appear as the usual `` bean - shaped '' protrusions . we attribute this to the formation of @xmath1-bonds between the two atoms of the split - off dimer and second layer atoms , and present charge density plots to support this assignment . we observe a local brightness enhancement due to strain for different dv complexes and provide the first experimental confirmation of an earlier prediction that the 1 + 2-dv induces less surface strain than other dv complexes . finally , we present a previously unreported triangular shaped split - off dimer defect complex that exists at s@xmath2-type step edges , and propose a structure for this defect involving a bound si monomer .

cond-mat0305065

the cores of clusters contain great dynamical complexity . the presence of gas cooling ( fabian 1994 ) , relativistic plasma ejection from the central galaxy ( mcnamara et al . 2000 , fabian et al . 2000 ) , the action of magnetic fields ( taylor , fabian & allen , 2002 ) , possible thermal conduction ( voigt et al . 2002 ) , local star - formation ( allen et al . 1995 ) and substantial masses of cold ( @xmath7 k ) molecular gas ( edge 2001 ) all complicate the simple hydrostatic picture . the launch of _ chandra _ and _ xmm - newton _ have allowed the cores of clusters to be studied in unprecedented detail spatially and spectrally . these advances have led to a number of important results ( e.g. , peterson et al . 2001 ; forman et al . 2002 , mcnamara 2002 ) . the very restrictive limits on low - temperature gas from _ xmm - newton _ rgs spectra are forcing us to re - examine some of the basics paradigms of the cooling flows ( e.g. kaastra et al . 2001 ; peterson et al . 2001 , tamura et al . 2001 , molendi & pizzolato 2001 ) . the spatial resolution of _ chandra _ can be used to study the brightest , nearby cooling flows to advance greatly our understanding of the processes occurring in these regions . in this paper we present the _ chandra _ observation of the zwicky cluster zw 0335.1 + 0956 whose properties may help us to address some of the above issues the zw 0335.1 + 0956 was first detected as a strong x - ray source by _ ariel - v _ ( cooke et al . 1978 ) and we therefore use the traditional identification of 2a 0335 + 096 in this paper . 2a 0335 + 096 is among the brightest 25 clusters in the x - ray sky ( edge et al . the presence of a cooling flow was noted by schwartz , schwarz , & tucker ( 1980 ) and its x - ray properties have been studied extensively over the past two decades ( singh , westergaard & schnopper 1986 , 1988 ; white et al . 1991 ; sarazin oconnell & mcnamara 1992 ; irwin & sarazin 1995 , kikuchi et al . the optical properties of the central galaxy in 2a 0335 + 096 have been studied in detail ( romanishin & hintzen 1988 ; mcnamara , oconnell & bregman 1990 ) and the strong , extended optical line emission marks this system out as an atypical elliptical galaxy but a prototypical central galaxy in a cooling flow . a deep , multi - frequency radio study of 2a 0335 + 096 ( sarazin , baum & odea 1995 ) shows a weak , flat - spectrum radio source coincident with the dominant galaxy which is surrounded by an amorphous steep - spectrum ` mini - halo ' . the tentative detection of hi absorption ( mcnamara , bregman & oconnell 1990 ) and firm detection of co emission ( implying 2@xmath8 m@xmath9 of molecular gas ) and iras 60 @xmath10 m continuum ( edge 2001 ) further highlight this cluster as one for detailed study . the implied mass deposition rate from the co detection is low ( @xmath115 m@xmath9 yr@xmath12 ) if the cold molecular gas found is deposited in the cooling flow . we use @xmath13 km s@xmath12 kpc@xmath12 , @xmath14 , and @xmath15 , which imply a linear scale of 0.7 kpc per arcsec at the distance of 2a 0335 + 096 ( @xmath16 ) . unless specified otherwise , all the errors are at @xmath17 confidence level for one interesting parameter . a was observed on 06 sept 2000 with the advanced ccd imaging spectrometer ( acis ) using the back - illuminated chip s3 . the total exposure time is @xmath18 ksec . in this paper we concentrate on the bright central @xmath19 kpc region of the cluster which lies fully within the acis - s3 chip . hot pixels , bad columns , chip node boundaries , and events with grades 1 , 5 , and 7 were excluded from the analysis . we cleaned the observation of background flares following the prescription given in markevitch et al . ( 2003 ) . because chip s3 contains the bright cluster region , we extracted a light curve for the other backside - illuminated chip s1 in the 2.56 kev band , which showed that most of the observation is affected by a mild flare . because the region of the cluster under study is very bright , we chose a less conservative flare cleaning than is recommended , excluding the time intervals with rates above a factor of 2 of the quiescent rate ( taken from the corresponding blank - sky background dataset , markevitch 2001 ) instead of the nominal factor 1.2 . this resulted in the final exposure of 13.8 ks . during the accepted exposure , the frontside - illuminated chips did not exhibit any background rate increases , therefore the residual background flare is of the `` soft '' species and can be approximately modeled as described by markevitch et al.(2003 ) . we fitted the excess background in chip s1 above 2.5 kev with the flare model and included it , rescaled by the respective solid angle , to the cluster fits . the main component of the background was modeled using the above - mentioned quiescent blank - sky dataset , normalized by the rate in the 1012 kev band ( which can be done because the residual flare in our observation is `` soft '' and does not affect the high - energy rate ) . the addition of the flare component has a very small effect on the results , because the cluster core is so bright . we therefore ignored the possible ( not yet adequately studied ) spatial nonuniformity of the flare component . [ fig : x - rayimage ] shows a background subtracted , vignetting corrected image of the central region of the cluster , extracted in the 0.3 - 9 kev energy band and binned to @xmath20@xmath21 pixels . the cluster x - ray surface brightness appears to be regular and slightly elliptical at @xmath22@xmath21 ( @xmath23 kpc ) . to the south , at about 60@xmath21 ( @xmath24 kpc ) from the x - ray peak , we notice the presence of a sharp surface brightness edge similar , to those observed in other clusters of galaxies ( e.g. , markevitch et al . 2000 , vikhlinin , markevitch , & murray 2002a ; mazzotta et al . finally , the x - ray image shows complex structure in the innermost @xmath25@xmath21 ( @xmath26 kpc ) region . below we study these features in detail . l c c c c c c & & & & & & + component & @xmath27 & @xmath28 & @xmath29 & @xmath30 & @xmath31 & @xmath32 + & ( j2000 ) & ( cnt arcsec@xmath33 ) & ( arcsec ) & & & ( deg ) + & & & & & & + narrow & ( 03:38:40.9 ; + 09:57:57.5 ) & @xmath34 & @xmath35 & @xmath36 & @xmath37 & @xmath38 + & & & & & & + extended & ( 03:38:40.4 ; + 09:58:31.7 ) & @xmath39 & @xmath40 & @xmath41 & @xmath42 & @xmath43 + + -model . _ continuous lines _ : isointensity contour levels of the cluster image shown in fig . [ fig : x - rayimage ] after adaptive smoothing . the image was smoothed using a gaussian kernel with a s / n ratio of the signal under the kernel @xmath44 . the levels are spaced by a factor @xmath45 . _ dashed lines _ : isointensity contour levels of the best fit double @xmath30-model . the levels are the same as for the x - ray image . ] -model . to extract the profiles we used concentric ellipses whose center , ellipticity and the orientation where equal to the best fit values of the narrow component of the double @xmath30-model . points with errorbars and lines correspond to the profiles extracted from the x - ray image and the double @xmath30-model , respectively . ] to investigate the cluster spatial structure , and to compare it with previous studies , we start with the classic approach of fitting the surface brightness with a @xmath30-model ( cavaliere , & fusco - femiano , 1976 ) . consistent with previous results , we find that this model does not provide an acceptable fit . hence , we use a double 2-dimensional @xmath30-model defined by : @xmath46 the @xmath47 and @xmath48 components of eq . [ eq : beta ] are given by : @xmath49^{-3\beta + 1/2},\ ] ] where @xmath50 and @xmath51 to perform the fit we use the sherpa fitting program . the cluster region with @xmath52@xmath21 and all the strong point sources , together with the regions defined later in [ par : core_small ] , are masked out . because the image contains a large number of pixels with few number counts ( @xmath53 cnt ) we use the cash ( 1979 ) statistics which , unlike the @xmath54 one , can be applied regardless of the number of counts per bin . the best fit values with their @xmath17 errors are reported in table 1 . unlike the @xmath55 statistics , the magnitude of the cash statistics depends upon the number of bins included in the fit and the values of the data themselves . this means that there is not an easy way to assess from the cash magnitude how well the model fits the data . to have a qualitative idea of the goodness of the fit , we generate a mock x - ray image using the best fit values of the double @xmath30-model and we apply the poisson scatter to it . consistent with previous studies , we find that globally the best fit double @xmath30-model describes very well the overall cluster structure . this is evident in fig . [ fig : mock_image ] where we show the isocontour levels of the model image overlaid to the same isocontour levels of the cluster image after adaptive smoothing . to better illustrate the quality of the fit we also extract the x - ray surface brightness radial profiles in concentric elliptical annuli from both the cluster and the mock images as shown in fig . [ fig : mock_radial ] . the center , the ellipticity and the orientation of each ellipse are set to the best fit values of the narrow component of the double @xmath30-model . the figure clearly confirms the good quality of the fit . the results of the fit reported in table 1 show that the two spatial components have very different slopes , in particular the narrow component is much steeper than the extended one . we also notice that the profile of the narrow component derived from the _ chandra _ data is significantly steeper that the one obtained from the rosat pspc ( kikuchi et al . the large @xmath30 value associated with the narrow component indicates that the cluster contains a very compact and dense gas component with a quite sharp boundary ( this is also clearly visible in fig . [ fig : x - rayimage ] ) . on the other hand , the slope of the extended component derived from the _ chandra _ data is significantly lower that the value derived from rosat ( see e.g. kikuchi et al . 1999 , vikhlinin , forman , & jones 1999 ) . because the _ chandra _ result is obtained fitting the cluster surface brightness in a much smaller radial range ( @xmath56@xmath21 ) than the rosat one ( @xmath57@xmath21 ) , this indicates that a contains one or more radial breaks at @xmath52@xmath21 , similar to the ones observed in other clusters like e.g. hydra - a ( david et al . 2001 ) and perseus ( fabian et al . 2000 ) . finally we note that , although the ellipticity and the orientation are similar , the centroid of the two spatial components are offset by @xmath58@xmath21 with the narrow component centroid being further south than the extended one . although spatial structure of the cluster is globally well described by a simple double @xmath30-model , there are two regions whose structure departs significantly from the best fit model above : i ) a 60 wide sector to the south ; ii ) the innermost @xmath59@xmath21 cluster region . below we study in details the spatial structure of these regions . a closer look at fig . [ fig : x - rayimage ] indicates that a hosts a surface brightness drop at @xmath60@xmath21 from the x - ray peak to the south . this x - ray drop spans a sector from 150 to 210 deg and most closely resembles similar features in other clusters with dense cores and regular morphology on larger linear scales , such as rxj1720 + 26 and a1795 ( mazzotta et al . 2001 ; markevitch , vikhlinin , & mazzotta 2001 ) . to better visualize this feature and to study its nature , we extract the cluster x - ray surface brightness profile within the above sector , using elliptical annuli concentric to the brightness edge . the center , the ellipticity and the orientation of each ellipses are set to the best fit values of the narrow component of the double @xmath30-model . the profile is shown in the upper panel of fig . [ fig : cold_front ] . as extensively shown in previous analysis of cold fronts in clusters of galaxies , the particular shape of the observed surface brightness profile indicates that the density distribution is discontinuous at some radius @xmath61 . l c c c c & & & & + quantity ( @xmath62 ) & @xmath63 & @xmath64 & a & @xmath61 ( arcsec ) + density ( @xmath65 ) & @xmath66 & @xmath67 & @xmath68 & @xmath69 + temperature ( @xmath70 ) & @xmath71 & @xmath72 & @xmath73 & ... + pressure ( @xmath74 ) & @xmath75 & @xmath76 & @xmath77 & ... + + + to estimate the amplitude and the position of the density jump , we fit the profile using a gas density model consisting of two power laws with index @xmath78 and @xmath79 and a jump by a factor @xmath80 at the radius @xmath61 : @xmath81 the fit is restricted to radii @xmath82@xmath21 to exclude the central region with irregular structure . we assume spherical symmetry . the best fit values together with the 90% confidence level errors are shown in table 2 . the best fit model ( which provides a good fit to the data ) is shown with a dashed line in the upper panel of fig . [ fig : cold_front ] . .spatial and spectral properties of the x - ray blobs [ cols="^,^,^,^,^,^,^ " , ] after adaptive smoothing made using a gaussian kernel with a s / n ratio of the signal under the kernel @xmath44 . the green ellipses labeled from 1 to 8 identify 8 x - ray bright extended regions or blobs . the two blue ellipses , labeled 9 and 10 , identify the western and the eastern x - ray holes or cavities . the white square and cross , indicate the position of the cluster d galaxy and its companion , respectively . the white x symbol indicate the position of the other cluster central galaxy see ( fig . [ fig : hst_image ] ) . ] l c c c c c c & & & & & & + model & @xmath83 @xmath84 @xmath85 & @xmath86 ( kev ) & @xmath87 ( kev ) & @xmath88 & @xmath89 @xmath90 & @xmath91 + & & & & & & + tbabs@xmath92mekal & @xmath93 & @xmath94 & ... & @xmath95 & ... & 843/352 + & @xmath96 & @xmath97 & ... & @xmath98 & ... & ( 484/297 ) + & & & & & & + tbabs@xmath92[mekal+mekal ] & @xmath99 & @xmath100 & @xmath101 & @xmath102 & ... & 643/350 + & @xmath103 & @xmath104 & @xmath105 & @xmath106 & ... & ( 393/295 ) + & & & & & & + tbabs@xmath92[mekal+mkcflow]&@xmath107&@xmath108 & ... & @xmath109&@xmath110&658/351 + & @xmath111&@xmath112 & ... & @xmath113&@xmath114&(387/296 ) + & & & & & & + tbabs@xmath92[mekal & @xmath115 & @xmath116& ... &@xmath117 & @xmath118 & 658/351 + + ztbabs(mkcflow ) ] & @xmath119 & @xmath120& ... &@xmath121 & @xmath122 & ( 393/296 ) + + + to study the cluster morphology on smaller scales we adaptively smooth the cluster image as shown in fig . [ fig : im_zoomed ] . the image was smoothed using the ciao tool csmooth with a gaussian kernel and a s / n ratio of the signal under the kernel @xmath44 . the image clearly shows a very complex structure with a number of blobs of x - ray excess emission . the unsmoothed image shows that these blobs are not necessarily distinct but may form a filamentary structure . nevertheless , for the purpose of our analysis we assume that the x - ray blobs are ellipsoid . we select 8 regions as shown in fig . [ fig : im_zoomed ] . each ellipse indicates the region where the surface brightness drops by almost a factor two with respect to the peak of its respective x - ray blob . to measure the volume occupied by each blob we make the simple assumption that the radius of the ellipsoid along the line of sight is equal to the minor radius of the ellipse @xmath123 . in table 3 we report the major and the minor radii corresponding to each region shown in fig . [ fig : im_zoomed ] . the dimensions of the x - ray blobs span from @xmath124@xmath21 to @xmath125@xmath21 ( from 2.8 to 6.3 kpc ) . for each blob we calculate the mean electron density using the following simple procedure : using the method described later in [ par : spectral ] , we extract the spectra from both the blob region and from a cluster region just outside it . using the latter spectrum as background , we fit the blob spectrum with a thermal model , hence deriving the emission measure . finally , the gas density is calculated from the emission measure under the simple assumption that it is constant within the blob . the net number of counts and the mean gas densities obtained using the above procedure are reported table 3 . for completeness in the same table we add the unabsorbed luminosity in the ( 0.6 - 9 ) kev band in the source rest frame and the cooling time ( see [ par : cavities ] below ) . beside the x - ray blobs , we also note a prominent x - ray hole about @xmath126@xmath21 ( @xmath127 kpc ) north - west from the x - ray peak , and a less prominent one at @xmath128@xmath21 ( @xmath129 kpc ) to the east . to check the statistical significance of the eastern hole , we extract the count rates from a circular region in the hole ( @xmath130 ; see fig . [ fig : im_zoomed ] ) and the count rates in two similar regions to the east and to the west of the depression along the isocontour level defined by the best - fit double @xmath30-model . the absorbed flux from the hole is @xmath131 lower than the flux from the neighboring regions at a @xmath132 confidence level . as shown later in [ par : other ] , the cluster image may also present other x - ray holes at larger radii . the previous identification of the eastern hole from rosat hri observations with absorption by a cluster member galaxy by sarazin et al . ( 1992 , 1995 ) is not supported with these higher resolution images ( see fig . [ fig : im_zoomed ] ) . we use the ciao software to extract spectra . the spectra are grouped to have a minimum of 20 net counts per bin and fitted using the xspec package ( arnaud 1996 ) . the position - dependent rmfs and arfs are computed and weighted by the x - ray brightness over the corresponding image region using `` calcrmf '' and `` calcarf '' tools . to account for the absorption caused by molecular contamination of the acis optical blocking filters , all the spectral model we use are multiplied by the acisabs absorption model that models the above absorption as a function of time since launch . the parameter tdays is set to 410 , while the others are left to the default values , namely norm@xmath133 , tauinf@xmath134 , tefold@xmath135 , nc@xmath136 , nh@xmath137 , no@xmath138 , nn@xmath139 . spectra are extracted in the 0.6 - 9.0 kev band in pi channels . at present , the acis response is poorly calibrated around the mirror ir edge , which results in frequently observed significant residuals in the 1.42.2 kev energy interval and high @xmath55 values . we tried excluding this energy interval and found that the best - fit parameter values and confidence intervals do not change considerably while the @xmath55 values become acceptable , as will be seen below . we therefore elected to use the full spectral range . to determine the average spectral properties of this observation , we extract the overall spectrum from the innermost circular region of @xmath140 ( @xmath141 kpc ) centered on the x - ray peak and fit it with different models as listed in table 4 . in the following the absorption of x - rays due to the interstellar medium has been parametrized using the tbingen - boulder model ( tbabs in xspec v. 11.1 ; wilms , allen & mc - cray 2000 ) . furthermore the metallicity refers to the solar photospheric values in anders & grevesse ( 1989 ) . as a first attempt , we use an absorbed single temperature thermal model ( tbabs@xmath92mekal ; see e.g. kaastra , 1992 ; liedahl , osterheld , & goldstein , 1995 ; and references therein ) . we find that this model does not reproduce the spectrum . we also use an absorbed two - temperature model ( tbabs@xmath92[mekal+mekal ] ) with the metallicity of the two components linked together . the hydrogen equivalent column density and the temperature , the metallicity , and the normalization of each thermal component are left free to vary . this model provides a much better fit . in particular , if we exclude the energy band near the ir edge from the spectral analysis , we find that the model is statistically acceptable ( see table 4 ) . we find that the equivalent column density is consistent with the galactic value ( @xmath142 cm@xmath33 ; dickey & lockman 1990 ) . the temperatures of both the hot and the cool components , as well as the metallicity , are consistent with the values obtained from the combined analysis of asca gis and sis spectra extracted from the innermost @xmath144 region ( kikuchy et al . we finally fit the spectrum using two models with cooling flows . the first is an absorbed thermal model plus a cooling flow component ( tbabs@xmath92[mekal+mkcflow ] ) . in the latter we assumed that the cooling flow component itself is intrinsically absorbed by uniformly distributed amount of hydrogen at the cluster redshift ( tbabs@xmath92[mekal+ztbabs(mkcflow ) ] , in xspec v. 11.1 ) . the higher temperature and metallicity of the cooling flow component are fixed to be equal to the ones of the thermal component . in the first cooling flow we leave the absorption free to vary . in the intrinsically absorbed cooling flow model we fix the overall absorption to the galactic value and let the intrinsic @xmath83 free to vary . we find that both models give temperatures , metallicities and the mass deposition rates consistent within the errors . it is worth noting that the cooling flow models give a minimum @xmath55 value similar to the two - temperature model which , as said before , is quite high compared to the degree of freedom ( d.o.f ) . to determine the temperature map we used the same technique used for a3667 ( vikhlinin , et al . 2001a ; mazzotta , fusco - femiano , vikhlinin 2002 ) . we extracted images in 10 energy bands , @xmath145 kev , subtracted background , masked out point sources , divided by the vignetting maps and smoothed the final images . we determined the temperature by fitting the 10 energy points to a single - temperature absorbed mekal plasma model with metal element abundance fixed at @xmath146 . we left as free parameters the equivalent absorption column density , the temperature , and the normalization . to study the cluster temperature structure at larger radii ( where the cluster is less bright ) , we derive first a relatively low spatial resolution temperature map . this is obtained by smoothing the 10 images above using a gaussian filter with @xmath147 . the `` low - resolution '' temperate map , overlaid on the 0.3 - 9 kev acis brightness contours , is shown in the left panel of fig . [ fig : temp_map ] . like the surface brightness , the projected temperature map is not azimuthally symmetric . the gas is substantially cooler in the cluster center and shows a strong positive temperature gradient which varies with position angle . the temperature gradient seems to be higher and lower to the southern and northern sectors , respectively . it is interesting to note that the gradient seems to be maximum somewhere in the south sector , where we observe the surface brightness edge . to study the thermal properties of the x - ray blobs and cavities discussed in [ par : core_small ] , we produce a second temperature map of the innermost central region using a much higher spatial resolution . this region is identified by the green ellipse in left panel of fig . [ fig : temp_map ] . the `` high - resolution '' temperature map , overlaid on the 0.3 - 9 kev acis brightness contours , is shown in the right panel of fig . [ fig : temp_map ] . the map is obtained by smoothing the 10 images above using a very narrow gaussian filter with @xmath148 . for clarity we plot ellipses numbered from 1 to 10 corresponding to the 8 x - ray blobs and 2 cavities defined in the [ par : core_small ] , above ( see also fig . [ fig : im_zoomed ] ) . the high - resolution temperature map shows a very complex structure . in particular the cavities and the x - ray blobs seem to have quite different projected temperatures . we discuss this aspect in [ par : cavities ] below . -model as shown in fig . [ fig : mock_radial ] . the dotted crosses in the upper panel indicate the results obtained by deprojecting the temperature profiles in the assumption of spherical symmetry . the dotted line in the bottom panel is @xmath149 value obtained by fitting a constant to the @xmath83 profile . error bars are @xmath150 confidence level . ] we first measure the overall spectral radial properties extracting spectra in concentric elliptical annuli . center , ellipticity and orientation of each ellipse are fixed to the best fit values of the narrow component of the double @xmath30-model as shown in fig . [ fig : mock_radial ] . the radii are choose in order to have @xmath151 net counts per spatial bin . each spectrum is fitted with an absorbed one - temperature mekal model . equivalent hydrogen column density , temperature , metallicity , and normalization are left free to vary . we get an acceptable fit for all the annuli but the innermost one . the resulting temperature , metallicity , and @xmath83 profiles , with error bars at @xmath150 level , are shown in the upper , middle , and lower panel of fig . [ fig : profile_0_360 ] , respectively ( errors bars are @xmath150 level ) . the temperature profile shows a positive gradient within the innermost @xmath152 kpc with the temperature that goes from @xmath153 kev to @xmath154 kev . at larger radii the temperature profile shows a much flatter profile consistent with being constant . only the last bin at @xmath155 kpc indicates a possible temperature increment at larger radii . although the cluster is clearly not spherical symmetric we try to deproject the overall temperature profiles assuming it actually is . the deprojected profile is reported as dotted crosses in the upper panel of fig . [ fig : profile_0_360 ] . we notice that the deprojected temperature of the innermost bin is @xmath156 kev . the projected metallicity is consistent with the constant value @xmath157 at radii @xmath158 kpc . at lower radii it shows strong positive gradient with a maximum of @xmath159 at @xmath160 kpc . similar metallicity gradients have been observed in other clusters , e.g. m87 ( molendi & gastaldello , 2001 ) , persus ( schmidt , fabian and sanders , 2002 ) , abell 2052 ( blanton , sarazin , mcnamara , 2002 ) , a2199 ( johnstone et al . 2002 ) and centaurus ( sanders & fabian 2002 ) . for some of them , however , it has been shown that the gradient is an artifact induced by the inadequate modeling of the spectra with a single temperature component where a multi - temperature one is actually required . this effect is known as metallicity bias ( see buote 2000 ) . it is interesting to note , however , that if we deprojected the metallicity profile assuming spherical symmetry we find that the profile is consistent with the projected one . this either indicates that the spherical symmetry assumption is inadequate or that the gradient is real . recently morris & fabian ( 2003 ) suggested that apparent off - center peaked metallicity profiles may form as result of the thermal evolution of intracluster medium with a non - uniform metal distribution . finally , as shown in fig . [ fig : profile_0_360 ] , panel c ) , we find that the equivalent column density profile is consistent with a constant value of @xmath161 cm@xmath33 which seems to be 10% higher than the galactic value . as discussed in [ par : tmap ] , the clusters temperature distribution is strongly asymmetric . to verify the statistical significance of this asymmetry , we extract the spectra using the photons from the northern or the southern sectors . spectra are extracted in the elliptical annuli , with the same binning as before . we find that , although both the abundance and metallicity profiles from the north and the south sectors are consistent within the errors , the temperatures profiles are significantly different . [ fig : profile_0_180 ] clearly shows that the temperature gradient in the north sector is shallower than the gradient in the south sector . moreover , the region of constant temperature in the south sector is a factor 1.3 hotter than that in the north sector . in [ par : front ] we show that the cluster has a surface brightness edge similar to the cold fronts observed in other clusters of galaxies . one peculiar characteristic of these cold fronts is that they have a temperature discontinuity ( jump ) over the front itself . in this paragraph we study the thermal properties of the x - ray `` front '' in a . to do this we estimate the 3d temperature profile of the front by deprojecting the temperature profile from the cluster sector whose angles are from 150 to 210 . the 3d temperature profile is shown in the lower panel of fig . [ fig : cold_front ] . we notice that the temperature profile has a quite steep gradient . moreover , unlike the cold fronts observed in other clusters of galaxies , it does not show a clear temperature discontinuity at the front . at first we fit the temperature profile in the range from 30@xmath21 to 150@xmath21 with a single power - law model @xmath162 . the model provides a good fit with @xmath163 , as shown in fig . [ fig : cold_front ] . we notice that the data may also be consistent with a small temperature jump at the front . to constrain the amplitude of this possible jump , we fit the data using a two - power law model with a jump @xmath164 at the position of the front ( see eq [ eq:5 ] ) and we report the result in table 2 . from the table we see that @xmath165 . using the pressure and the density models above , we calculate the pressure profile near the front region . the best fit model together with the 90% errors are reported in table 2 . the observed density and temperature discontinuity at the front result in corresponding pressure discontinuity of a factor @xmath166 . . the two filled triangles give the projected temperature from the western and eastern holes , respectively . error bars are @xmath17 confidence level . ] . the dashed lines are the 90% error measurement on the deprojected temperature of the ambient gas . _ middle panel _ ) thermal pressure associated with the same blobs as in the upper panel . the dashed lines are the 90% error measurement of the thermal pressure of the ambient gas . _ lower panel _ ) cooling time associated with the same blobs as in the upper panel . ] as discussed in [ par : tmap ] , the innermost region of a has a very complex temperature structure . in particular , the temperature map shows that the cavities and the x - ray blobs have quite different projected temperatures ( see fig . [ fig : temp_map ] ) . to better study the thermal properties of these structures , we extract spectra for all the elliptical regions shown in the figure and fit them with an absorbed thermal model . we verify if there is any excess absorption by fixing the metallicity to z=0.6 solar and letting both @xmath83 and the temperature to vary freely . as we find that the resulting @xmath83 values are all consistent with the cluster mean value , we fix @xmath83 to this value and re - fit the model to measure the temperature . the projected temperatures together with the 90% error bars , are shown in fig . [ fig : hardness ] . consistently with the right panel of fig . [ fig : temp_map ] , we find that the central x - ray structures have different projected temperatures . in particular we notice that region 1 and 2 are significantly cooler than region 6 and 7 . furthermore , we notice that the holes are significantly hotter than any other x - ray blobs . it is worth noting that the observed higher temperature from the holes is likely to be the result of a projection effect . projection could also be responsible for the observed difference in the projected temperature of the x - ray blobs . to verify this hypothesis one should find a way to disentangle the contribution to the spectrum due to the blob itself from that due to the gas along the line of sight . to perform this task we assume that the contribution to the spectra due to the gas along the line of sight is almost the same within the innermost 25@xmath21 region ( where all the blobs are ) . this is justified by the fact that the surface brightness profile within this region does not vary more than a factor @xmath167 . we then extract a cluster spectrum using a region just outside the blobs that will be used as background for the spectrum of each blob . the first important result of this exercise is that the final spectra clearly shows the presence of fe - l emission line complex . this proves , without doubt , that the emission from these blobs is indeed mostly , if not all , thermal . we fit these spectra using a single - temperature mekal model leaving as a free parameter only the temperature and the normalization . the result is shown in the upper panel of fig . [ fig : pressure ] . in the same figure we add , as dashed lines , the 90% error temperture estimate of the ambient gas . this estimate cames from the fit of the blob - free cluster spectrum , used before as background , with a two - temperature mekal model with the metallicity of the two thermal components linked together . to compensate our ignorance of the actual 3d cluster structure , we leave the normalization of both thermal components free to vary . we assume that the temperature of the ambient gas coincides with the one of lower temperature component . we believe that this represents a good assumption as the value we find is consistent with the temperature of the innermost bin obtained using a spherical symmetric deprojection model ( see upper panel of fig . [ fig : profile_0_360 ] ) . unlike the emission weighted temperatures , the deprojected ones have a much lower scatter . furthermore , all regions , but the 6th , have temperatures consistent with the 3d temperature of the ambient gas . using the blob density measurements reported in table 3 , it is now possible to estimate the thermal pressure associated with each blob . this is shown in the central panel of fig . [ fig : pressure ] . in the same figure we add , as dashed lines , the 90% error ambient pressure . the blobs , except region 5 , have thermal pressure values significantly higher than the ambient thermal pressure . this unexpected finding will be discussed below . using the temperature and the density of each blob we also estimate their cooling times @xmath168 , where @xmath74 and @xmath31 are the blob s pressure and the emissivity , respectively . these , together with the 90% error bar , are shown in the lower panel of fig . [ fig : pressure ] . we notice that all blobs have similar cooling times which are consistent within the errors ( but see region 6 ) . for practical reasons we report also the blob cooling times in column 7 of table 3 . for completeness we estimate the x - ray blob unabsorbed luminosities in the ( 0.6 - 9.0 ) kev in the source rest frame , as shown in in column 6 of table 3 . the northern and the southern symbols indicate the position of the extended and narrow component , respectively . the thick ellipses indicate the position of the western and eastern holes , respectively . the x - ray excess regions labeled 1 to 8 in fig . [ fig : im_zoomed ] can be easily identified by the small quasi - elliptical isocontour regions . notice that none of these regions have an optical counterpart . ] -model superposed on contours of the 1.515 ghz radio image ( from sarazin , baum , & odea , 1995 ) . the darker and lighter regions indicate negative and positive variation @xmath169 , respectively . the arrows indicate the position of the western and eastern holes and the position of the front . the square and the cross symbols indicate the position of the central d galaxy and its companion while the x symbol indicates the position of the third cluster galaxy in the hst field ( see fig . [ fig : hst_image ] ) . finally the circles in squares symbols indicate the best fit central position of each of the components of the double @xmath30-model . ] we conclude the analysis of a by comparing the cluster properties in the x - rays to those at other wavelengths . in fig . [ fig : hst_image ] we show an hst image of the central region of a superposed on contours of the x - ray image after adaptive smoothing . the optical image shows one central d galaxy ( pgc 013424 , paturel et al . 1989 ) with a companion smaller galaxy 7@xmath21 to the north - west and another elliptical galaxy at 41@xmath21 to the east . in fig . [ fig : hst_image ] the x - ray blobs , defined previously in [ par : core_small ] ( labeled 1 to 8 in fig . [ fig : im_zoomed ] ) , can be easily identified matching the small quasi - elliptical isocontour regions . apart from one x - ray blob coincident with the cluster d galaxy none of the other blobs have optical counterparts . this strongly suggests that the x - ray blobs are produced by some kind of hydrodynamic process . for completeness in fig . [ fig : hst_image ] we report the centroid position of the narrow and extended @xmath30-model components . it is worth noting that the d galaxy position does not coincide with either of these centroids but rather lies in between . the central galaxy of 2a 0335@xmath170096 has been found to have a number of peculiar properties over the past 2 decades . initially it was classified as a seyfert 1.5 or liner by huchra , wyatt & davis ( 1982 ) but more detailed optical observations by romanishin & hintzen ( 1988 ) found a very extended optical emission line region ( h@xmath6[nii ] ) to the north - east of the galaxy ( extended over 30 kpc ) and the central regions of the galaxy are anomalously blue indicating recent star - formation ( see fig . [ fig : halpha ] ) . recently a number of authors have shown that some of the extended h@xmath171 features observed in the central region of a number of clusters correlate with similar structures of x - ray excess ( e.g. a1795 , fabian et al . 2001 ; rx j0820.9 + 0752 , bayer - kim et al . 2002 ; virgo , young , wilson , & mundell 2002 ) . to study the correlation between the x - ray blobs and the line emission in fig . [ fig : halpha ] we show the narrow band h@xmath6[nii ] image from romanishin & hintzen ( 1988 ) with the x - ray isocontours superimposed . the accuracy of the alignment is better than @xmath172 . there is no direct correlation between the x - ray and h@xmath6[nii ] structures but there is a tendency for the two to lie close to each other . however , the statistical significance of this tendency is difficult to judge . note the x - ray blobs labeled 1 and 4 lie too close to the saturated star to the south east of the h@xmath6[nii ] image to detect the fainter h@xmath6[nii ] emission . the radio properties of 2a 0335@xmath170096 are also of note . sarazin et al . ( 1995 ) find an amorphous , steep spectrum ( @xmath173 ) ` mini - halo ' around pgc 013424 which has relatively low radio power compared to other similar systems . [ fig : percentile ] shows the image of the percentile variation ( @xmath174 ) of the cluster x - ray surface brightness @xmath175 with respect to the best fit double @xmath30-model @xmath176 . the isocontours of the vla 1.5 ghz c@xmath170d - array image ( resolution 16@xmath21 ) from sarazin et al . ( 1995 ) are superimposed . the image clearly shows the position of the x - ray front together with the x - ray blobs and the western and eastern holes earlier studied in [ par : front ] and [ par : core_small ] , respectively . we notice that the mini - radio halo is well within the radius defined by the position of the x - ray front . unfortunately the spatial resolution of the radio map is insufficient to correlate it with the x - ray blobs . however , there is some indication that , as observed in other clusters ( see mcnamara 2002 for a review ) , both x - ray holes may contain some steep spectrum radio emission which may be ` relic ' emission from previous radio lobes . future high resolution , low frequency radio observations will be crucial to linking the radio mini - halo and x - ray structure . it is worth noting that fig . [ fig : percentile ] also suggests the presence in a ghost x - ray bubbles at larger radii similar the ones observed in other clusters ( mcnamara 2002 ) . unfortunately , because of the relatively short exposure , these bubbles are not statistically significant . a deeper x - ray observation is needed to confirm their existence . the _ chandra _ observation of a reveals that the core region of this otherwise relaxed cluster is very dynamic . the cluster image shows an edge - like feature at @xmath177 kpc from the x - ray peak to the south . the edge spans a sector from 150 to 210 from the north . on smaller scales ( @xmath178 kpc ) the cluster image shows a very complex structure with x - ray filaments or blobs . moreover it shows the presence of two x - ray cavities ( see fig . [ fig : im_zoomed ] ) . the spectral analysis clearly shows that on large scales the cluster temperature distribution is not azimuthally symmetric ( see [ par : tmap ] and [ par : tprofile ] ) . furthermore , the cluster gas temperature reaches its maximum in a region at @xmath179 kpc from the x - ray peak to the south , just beyond the surface brightness edge ( fig . [ fig : temp_map ] ) . on smaller scales the x - ray blobs have similar temperatures but different pressures . below we discuss the results of our analysis and propose possible dynamical models that may explain the observed properties of the core of a . in [ par : front ] we analyze the spatial structure of the x - ray front . we show that the front is well fit by a discontinuous density profile with a density jump at the front of a factor @xmath180 . similar density jumps have been observed by _ chandra _ in many clusters , including a2142 , a3667 , rx j1720 + 26 , a1795 . in those clusters , they are interpreted as sharp boundaries of a dense , cooler gas cloud moving with a speed @xmath181 through the hotter and more rarefied ambient gas . the surface brightness edge in a appears to be a similar cold front . as discussed by vikhlinin , et al . 2002a , the pressure profile across the front can be used to determine the mach number @xmath182 of the moving cooler gas cloud with respect to the speed of sound in the gas upstream , @xmath183 . in table 2 we show that the pressure jump across the front is @xmath184 . this corresponds to a mach number for the dense central gas cloud of @xmath185 ( see e.g. fig . 6 of vikhlinin , et al . 2001a ) . such subsonic motion of the cool gas cores appears to be very common among the otherwise relaxed clusters with cooling flows ( markevitch , vikhlinin , & forman 2002 ) . in the analysis above , we identified a number of x - ray blobs most of which are likely to be part of a filamentary structure . as can been seen from table 3 , the blobs are quite dense : their densities range from @xmath186@xmath187 @xmath188 . although the projected spectral properties of the blobs are significantly different ( i.e. they have different emission weighted temperatures ; see right panel of fig . [ fig : temp_map ] and fig . [ fig : hardness ] ) , in [ par : cavities ] we show that the temperature of their immediately surrounding ambient gas and the deprojected blob temperatures are consistent within the errors for all but one of the blobs ( see upper panel fig . [ fig : pressure ] ) . this is particularly interesting as it implies that the thermal pressure of the blobs is higher than the thermal pressure of the surrounding ambient gas . this finding , however , does not necessarily mean that the blobs are out of pressure equilibrium with the ambient gas as the latter may have extra non - thermal pressure ( for example , magnetic fields and/or relativistic electrons ejected from the central agn ) . although at the moment it is not clear which physical processes may be responsible for the formation of the observed substructures , we can safely state that the system is dynamically unstable and the presence of blobs is likely to be a transient phenomenon . we notice that the evolution of the system strongly depends on the total pressure supporting the ambient gas . below we discuss two extreme possibilities : \1 ) the ambient gas surrounding the blobs is only supported by thermal pressure and thus the blobs are actually not in pressure equilibrium ; \2 ) the ambient gas , and possibly to some extent the blobs too , are supported by an extra non thermal pressure component such that the blobs and the ambient gas are actually in pressure equilibrium . in the first case , the blobs are overpressured so we expect them to expand . the expansion time is of the order of the blob s sound crossing time : @xmath189 where @xmath190 is the radius of the blob , and @xmath10 , @xmath191 , and @xmath192 are the mean particle weight , the proton mass and the adiabatic index , respectively . using the linear dimension of the blobs reported in table 3 we find @xmath193 yr , which appears to be very short compared with the likely cluster age . furthermore , the blobs cooling time is @xmath194 times longer than the dynamical time ( see lower panel of fig . [ fig : pressure ] ) . thus , in this first case in which the external gas is not supported by an extra pressure component , the blobs should expand before their excess pressure is radiated away . consequently we do not expect to have a significant mass deposition into a cold phase from the blobs . conversely , if the blobs are both in thermal and pressure equilibrium with the ambient gas , as in the second case considered above , the blobs are likely to be more stable . nevertheless , as the blobs are heavier than the ambient gas ( they have higher densities ) , they are expected to sink eventually toward the minimum of the potential well . the time scale @xmath195 needed by the blobs to reach the cluster center is obviously @xmath196 just in the extreme case in which we assume that both the gas is collisionless and the blob have zero angular momentum . ] , where @xmath197 is the free fall time . if we assume the cluster potential well is described by a nfw profile ( navarro , frenk & white , 1996 ) the free fall time for a point mass at rest at a radius @xmath198 can be well approximated by : @xmath199 where @xmath200 , is the scale radius , @xmath201 , @xmath202 is the newton s constant , @xmath203 is the critical density for closure , and @xmath204 is the characteristic density contrast . using the best fit nfw parameters for our cluster and @xmath205 kpc we find : @xmath206 from table 3 we see that the free fall and the cooling times are similar , thus the actual sinking time is much longer than the cooling time . therefore , unlike the first case considered above , if the ambient gas is supported by an extra non thermal pressure component such that the gas and the blobs are in pressure equilibrium , we expect the blobs to cool down before they can actually sink into cluster center . hence , some mass deposition into a cold phase is possible . we conclude this section by discussing possible dynamical models that may explain the origin of the observed complex structure in the core of a . our starting point is the evidence for the presence of a dense gas core that moves from north to south , with a mach number @xmath207 ( see [ par : nature ] ) . we propose two models : + i ) the motion of the cool core induces instabilities that penetrate inside the core and disturb the gas , + ii ) the central galaxy has an intermittent agn which makes the cool gas `` bubbling '' . + the features observed in the cluster result from the development of hydrodynamic instabilities induced by these two phenomena . below we discuss in detail these two models . we conclude that both models are consistent with the available data and that further observations are required to discriminate between the two cases . we first show that small - scale structure can be the result of hydrodynamic instabilities induced by the observed motion of the core gas ( revealed by the presence of the cold front . this motion may be due to a merger ( as in a2142 or a3667 , markevitch et al . 2000 ; vikhlinin et al . 2001 ) , or due to the gas sloshing as in many other cooling flow clusters ( markevitch et al.2001 , 2002 ) . the precise nature of the core motion is not important . the orientation of the front ( see e.g. fig.[fig : percentile ] ) , together with the asymmetry in the temperature distribution ( see fig.[fig : temp_map ] ) , suggests that the core is moving along a projected direction that goes from north to south . if the core is in fact a merging subcluster , it is clear that in projection the position of the subcluster is very close to the cluster center ( see [ par : core ] ) , from the available data , however , we can not measure the merger impact parameter . hence , the subcluster may in fact be passing through the cluster at any distance along the line of sight from the center . when a dense gas cloud is moving with respect to a rarefied one , the interface between the two gasses is subject to hydrodynamical rayleigh - taylor ( r - t ) and kelvin - helmholtz ( k - h ) instabilities . in the following we estimate the effect of both instabilities on the moving core using the same approach as vikhlinin & markevitch ( 2002 ) . we assume that the gas cloud is a sphere of radius equal to the distance of the observed front , @xmath208@xmath21 ( @xmath209 kpc ) . as the cloud moves , the external gas flows around its border . the inflowing gas slows down at the leading edge of the sphere but reaccelerates at larger angles as it is squeezed to the sides by new portion of inflowing gas . in the case of an incompressible fluid , the velocity at the surface of the sphere is purely tangential and is given by @xmath210 , where @xmath211 is the velocity of the inflowing gas at infinity and @xmath212 is the angle of the considered point on the cloud with respect to its direction of motion ( see e.g. lamb 1945 ) . unfortunately there is no analytic solution for the flow of a compressible fluid around a sphere . nevertheless , the qualitative picture is similar . as we are only interested in order of magnitude estimates , in the following we parametrize the actual fluid speed introducing the parameter @xmath213 so that : @xmath214 the parameter @xmath213 should range between 1 ( fluid velocity at the surface of the sphere equal to the inflowing gas at infinity ) and 1.5 ( incompressible fluid ) . first we consider the r - t instability . this instability develops only if the acceleration induced by the drag force @xmath215 is greater than the gravitational potential acceleration near the front due to the cluster mass inside the moving cloud . if we assume hydrostatic equilibrium , then the gravitational acceleration is given by : @xmath216 where , @xmath217 is the mean particle mass , and @xmath65 and @xmath70 are the gas electron density and temperature profiles , respectively . in our case this gives @xmath218 cm s@xmath33 ( see [ par : front ] , [ par : tprofile_ns ] , and fig . [ fig : cold_front ] ) . on the other hand , the cloud acceleration due to the drag force is given @xmath219 ( where @xmath220,@xmath221 the gas electron density inside and outside the edge ) which in our case gives @xmath222 cm s@xmath33 . as the drag acceleration is less than the gravitational one , then the r - t instabilities should be suppressed . now we consider the k - h instability . this instability may develop whenever two fluids in contact have a non - zero tangential velocity . as for the r - t instability , the presence of a gravitational field may suppress the development of the instability . the stability condition is such that a perturbation with wavenumber less than @xmath223 is stable if : @xmath224 where @xmath225 is the gravitational acceleration at the interface between the two regions and @xmath226 is the density contrast . using the eq . [ eq : v ] for the tangential velocity , eq . [ eq : stab ] becomes : @xmath227 eq . [ eq : lambda ] shows that gravity suppresses instabilities only on scales larger than @xmath228 kpc . hence , in absence of other stabilizing mechanism , we expect the formation of the k - h instabilities on smaller scales . in the case of a , we see the instabilities on scales of the order of the radius of the sphere @xmath229 may develop already at @xmath230 . the condition that the instability on the gas cloud scale can develop , although necessary is not sufficient . in fact , in order to effectively disrupt the gas of the moving gas cloud , the growth time @xmath231 of the instability must be much shorter than the dynamical time of the system ( the cluster crossing time , @xmath232 ) . we thus require ( see vikhlinin , markevitch , & murray 2001b , and algebraic correction of mazzotta et al . 2002 ) : @xmath233 if we assume that the cluster radius is @xmath234 mpc , at the angle at which the instability starts to develop of @xmath235 , we find that @xmath236 . furthermore , at the much larger angle of @xmath237 , where the instabilities are most effective , we find @xmath238 . therefore , they have sufficient time to grow to the non - linear regime . the effect of k - h instabilities is to turbulently mix the gas in the dense moving cool cloud with the more diffuse hotter cluster gas . in particular some less dense gas from the cluster could be deposited into the denser gas cloud and vice - versa . such a process may be responsible for the formation of the observed x - ray clumpy structure . similarly , any pre - existing cold ( @xmath239 k ) gas deposited in the cluster and/or sub - cluster core by a cooling flow will also be mixed and re - distributed on these scales . this would explain the observed h@xmath6[nii ] emission morphology . on the other hand , these sub - sonic k - h instabilities will not produce any significant radio signature . hence , in this scenario the radio properties of a ( the mini - halo and the x - ray holes ) must be related to activity of the central galaxy . in conclusion , the observed x - ray properties of the a core may be produced by k - h instabilities induced by the motion of the core . the observed structure of a can also be accounted for if the central cluster galaxy hosts an agn that undergoes intervals of strong activity followed by periods of relative quiescence ( e.g. binney & tabor 1995 , soker et al . 2000 ) . during each active period , the agn inflates two bubbles by filling them with relativistic electrons . as observed in other clusters ( e.g. perseus , fabian et al . 2000 ; hydra a , mcnamara et al . 2000 ; abell 2052 , blanton et al . 2001 ) , this expansion creates an external shell where the gas is denser . the bubbles rise buoyantly , entrain the dense gas underneath them and induce convection ( e.g. , churazov et al . these bubbles are also subjected to hydrodynamic instabilities that destroy them , fragmenting the denser shell into what we observe as denser x - ray blobs . the observed structure in the core may be created by numerous bursts of agn activity during which the jets point in different directions . we can visualize this process as `` gas bubbling '' induced by the energy injection from a central agn . the difference from other clusters with similar x - ray cavities but lacking such filamentary or blobby structure might be caused by a shorter duty cycle of the central agn in a , of the order of @xmath240 yr ( see [ par : dis_blobs ] ) . the relativistic electrons produced by the agn that are likely to be present inside the non - thermal bubbles ( see e.g. mcnamara 2002 and references therein ) should eventually distribute in the central cluster region . however , they may avoid the densest gas blobs due to , for example , an enhanced magnetic field created during the shell compression induced by the bubble expansion . these electrons would add a non - thermal pressure component in the ambient ( inter - blob ) gas that may account for the apparent pressure nonequilibrium of the cluster core . as discussed in [ par : dis_blobs ] , in this case the dense x - ray blobs have time to cool , depositing some mass in a cold phase . this cool gas may explain the presence of the observed co and h@xmath6[nii ] extended emission in the cluster center ( see fig . [ fig : halpha ] ) . the relativistic electrons produced by the agn may also account for the presence of the mini radio halo observed in the cluster core ( see fig . [ fig : percentile ] ) . we finally mention that there may be a connection between the agn activity and the core sloshing , hinted at by the observed correlation between the presence of the x - ray bubbles and cold fronts in the central regions of many cooling flow clusters ( markevitch et al . reynolds , heinz & begelman ( 2002 ) showed that during an agn jet formation , a spherical pulse forms and propagates at the sound speed into the intracluster medium . repeated agn activity may inject sufficient kinetic energy to produce sloshing of the gas , although it is unclear how the bulk motion of the core as a whole could be produced . sloshing of the dense core may also be induced by a merger , even if the merging subcluster did not reach the center of the main cluster ( e.g. , churazov et al . 2003 ) . to conclude this section , `` gas bubbling '' induced by central agn provides a plausible explanation of the observed features in the cluster core the poor cluster a has several remarkable features that are only detectable with the superb resolution of _ chandra_. first , there is a cold front indicating a mildly sub - sonic gas motion . second , two x - ray cavities that may be associated with steep spectrum radio emission indicating previous radio activity in this system . third , a number of small dense gas blobs in the cluster core that may be the shreds of a cooling core disturbed either by k - h instabilities or `` bubbling '' induced by intermittent agn activity . all of these properties relate to processes that may act to disrupt or destroy any cooling flow ( sub - cluster merger , injection of non - thermal electrons into the intracluster medium or direct radiation from an agn ) . from this relatively short x - ray observation it is not possible to determine which process ( if any ) dominates . future deeper observations in the x - ray , radio and optical of a may allow the energetics of each to be assessed . when combined with observations of other similar clusters , it will be possible draw concrete conclusions about the astrophysical nature of cooling flows . we thank david gilbank for useful comments and suggestions and for assistance with the narrow - band optical image , craig sarazin for providing the vla images , and alexey vikhlinin for helpful discussions . this work was supported by an rtn fellowship , cxc grants go2 - 3177x and go2 - 3164x , the royal society and the smithsonian institution . anders , e. & grevesse , n. 1989 , , 53 , 197 allen , s. w. , fabian , a. c. , edge , a. c. , bohringer , h. , & white , d. a. 1995 , , 275 , 741 arnaud , k. a. 1996 , asp conf ser . 101 , astronomical data analysis software and systems v , 5 , 17 bayer - kim , c. m. , crawford , c. s. , allen , s. w. , edge , a. c. , & fabian , a. c. 2002 , , 337 , 938 blanton , e. l. , sarazin , c. l. , mcnamara , b. r. , & wise , m. w. 2001 , , 558 , l15 blanton , e. l. , sarazin , c. l. , mcnamara , b. r. 2002 , , in press , ( astro - ph/0211027 ) buote , d.a . 2000 , apj , 539 , 172 cash , w. 1979 , , 228 , 939 cavaliere , a. & fusco - femiano , r. 1976 , , 49 , 137 churazov , e. , brggen , m. , kaiser , c. r. , bhringer , h. , & forman , w. 2001 , apj , 554 , 261 churazov , e. , forman , w. , jones , c. , & bhringer , h. 2003 , apj , submitted ( astro - ph/0301482 ) cooke , b. a. et al . 1978 , , 182 , 489 david , l. p. , nulsen , p. e. j. , mcnamara , b. r. , forman , w. , jones , c. , ponman , t. , robertson , b. , wise , m. 2000 , apj , 557 , 546 dickey , j. m. & lockman , f. j. 1990 , , 28 , 215 edge , a. c. , stewart , g. c. , fabian , a. c. , & arnaud , k. a. 1990 , , 245 , 559 edge , a. c. 2001 , , 328 , 762 fabian , a. c. 1994 , , 32 , 277 fabian , a. c. et al . 2000 , , 318 , l65 fabian , a. c. , sanders , j. s. , ettori , s. , taylor , g. b. , allen , s. w. , crawford , c. s. , iwasawa , k. , & johnstone , r. m. 2001 , , 321 , l33 forman , w. , jones , c. , markevitch , m. , vikhlinin , a. , & churazov , e. 2002 , assl vol . 272 : merging processes in galaxy clusters , 109 glenn morris , r. & fabian , a. c. 2003 , , 338 , 824 huchra , j. p. , wyatt , w. f. , & davis , m. 1982 , , 87 , 1628 irwin , j. a. & sarazin , c. l. 1995 , , 455 , 497 johnstone , r. m. , allen , s. w. , fabian , a. c. , & sanders , j. s. 2002 , , 336 , 299 kaastra , j.s . 1992 , an x - ray spectral code for optically thin plasmas ( internal sron - leiden report , updated version 2.0 ) kaastra , j. s. , ferrigno , c. , tamura , t. , paerels , f. b. s. , peterson , j. r. , & mittaz , j. p. d. 2001 , , 365 , l99 kikuchi , k. , furusho , t. , ezawa , h. , yamasaki , n. y. , ohashi , t. , fukazawa , y. , & ikebe , y. 1999 , , 51 , 301 lamb , h. 1945 , hydrodynamics ( new york : dover ) liedahl , d.a . , osterheld , a.l . , and goldstein , w.h . 1995 , , 438 , l115 markevitch , m. , ponman , t. j. , nulsen , p. e. j. , bautz , m. w. , burke , d. j. , david , l. p. , davis , d. , donnelly , r. h. , forman , w. r. , jones , c. , kaastra , j. , kellogg , e. , kim , d .- w . , kolodziejczak , j. , mazzotta , p. , pagliaro , a. , patel , s. , vanspeybroeck , l. , vikhlinin , a. , vrtilek , j. , wise , m. , zhao , p. 2000b , apj , 541 , 542 markevitch , m. , vikhlinin , a. , & mazzotta , p. 2001 , , 562 , l153 markevitch , m. , vikhlinin , a , forman , w. r. 2002 : matter and energy in clusters of galaxies , taiwan , in press ( astro - ph/0208208 ) markevitch , m. , et al . 2003 , apj , 583 , 70 mazzotta , p. , markevitch , m. , vikhlinin , a. , forman , w. r. , david , l. p. , & vanspeybroeck , l. 2001 , , 555 , 205 mazzotta , p. , fusco - femiano , r. , & vikhlinin , a. 2002 , , 569 , l31 mcnamara , b. r. , oconnell , r. w. , & bregman , j. n. 1990 , , 360 , 20 mcnamara , b. r. & oconnell , r. w. 1992 , , 393 , 579 mcnamara , b. r. et al . 2000 , , 534 , l135 mcnamara , b. r. 2002 , asp conf . ser . 262 : the high energy universe at sharp focus : chandra science , 351 , ( astro - ph/0202199 ) molendi , s. & gastaldello , f. 2001 , , 375 , l14 molendi , s. & pizzolato , f. 2001 , , 560 , 194 navarro , j. f. , frenk , c. s. , & white , s. d. m. 1996 , , 462 , 563 paturel , g. , fouque , p. , bottinelli , l. , & gouguenheim , l. 1989 , , 80 , 299 peterson , j. r. et al . 2001 , , 365 , l104 reynolds , c. s. , heinz , s. , & begelman , m. c. 2002 , , 332 , 271 romanishin , w. & hintzen , p. 1988 , , 324 , l17 sanders , j. s. & fabian , a. c. 2002 , , 331 , 273 sarazin , c. l. , oconnell , r. w. , & mcnamara , b. r. 1992 , , 397 , l31 sarazin , c. l. , baum , s. a. , & odea , c. p. 1995 , , 451 , 125 schmidt , r. w. , fabian , a. c. , & sanders , j. s. 2002 , , 337 , 71 schwartz , d. a. , schwarz , j. , & tucker , w. 1980 , , 238 , l59 singh , k. p. , westergaard , n. j. , & schnopper , h. w. 1986 , , l308 , l51 singh , k. p. , westergaard , n. j. , & schnopper , h. w. 1988 , , 331 , 672 taylor , g. b. , fabian , a. c. , & allen , s. w. 2002 , , 334 , 769 tamura , t. et al . 2001 , , 365 , l87 vikhlinin , a. , forman , w. , & jones , c. 1999 , , 525 , 47 vikhlinin , a. , markevitch , m. , murray , s.s . 2001a , apj , 551 , 160 vikhlinin , a. , markevitch , m. , murray , s.s . 2001b , apjl , 549 , l47 vikhlinin , a. a. & markevitch , m. l. 2002 , astronomy letters , 28 , 495 voigt , l. m. , schmidt , r. w. , fabian , a. c. , allen , s. w. , & johnstone , r. m. 2002 , , 335 , l7 white , d. a. , fabian , a. c. , johnstone , r. m. , mushotzky , r. f. , & arnaud , k. a. 1991 , , 252 , 72 wilms , j. , allen , a. , & mccray , r. 2000 , , 542 , 914 young , a. j. , wilson , a. s. , & mundell , c. g. 2002 , , 579 , 560

we present a _ chandra _ observation of the central ( @xmath0 kpc ) region of the cluster of galaxies a , rich in interesting phenomena . on large scales ( @xmath1 kpc ) , the x - ray surface brightness is symmetric and slightly elliptical . the cluster has a cool dense core ; the radial temperature gradient varies with position angle . the radial metallicity profile shows a pronounced central drop and an off - center peak . similarly to many clusters with dense cores , a hosts a cold front at @xmath2 kpc south of the center . the gas pressure across the front is discontinuous by a factor @xmath3 , indicating that the cool core is moving with respect to the ambient gas with a mach number @xmath4 . the central dense region inside the cold front shows an unusual x - ray morphology , which consists of a number of x - ray blobs and/or filaments on scales @xmath5 kpc , along with two prominent x - ray cavities . the x - ray blobs are not correlated with either the optical line emission ( h@xmath6[nii ] ) , member galaxies or radio emission . deprojected temperature of the dense blobs is consistent with that of the less dense ambient gas , so these gas phases do not appear to be in thermal pressure equilibrium . an interesting possibility is a significant , unseen non - thermal pressure component in the inter - blob gas , possibly arising from the activity of the central agn . we discuss two models for the origin of the gas blobs hydrodynamic instabilities caused by the observed motion of the gas core , or `` bubbling '' of the core caused by multiple outbursts of the central agn .

astro-ph0303314

since the successful launch of the _ fermi _ gamma - ray space telescope in 2008 june , we now have a new opportunity to study gamma - ray emission from different types of high energy sources with much improved sensitivity and localization capabilities than with the egret instrument onboard the _ compton gamma - ray observatory _ ( cgro ) . with the field of view covering 20% of the sky at every moment ( five times larger than egret ) , and its improved sensitivity ( by more than an order of magnitude with respect to egret ) , the large area telescope ( lat ; * ? ? ? * ) aboard _ fermi _ surveys the entire sky each day down to photon flux levels of @xmath0 @xmath1 few @xmath2 . the number of detected gamma - ray sources has increased , with the 2nd catalog ( 2fgl ; * ? ? ? * ) containing 1873 gamma - ray sources in the 100 mev to 100 gev range , while 271 objects were previously listed in the 3rd egret catalog ( 3eg ; * ? ? ? more than 1,000 gamma - ray sources included in the 2fgl are proposed to be associated with active galactic nuclei ( agns ) and 87 sources with pulsars ( psrs ; * ? ? ? * ) , including 21 millisecond pulsars ( msps ) . other associations included supernova remnants ( snrs ; * ? ? ? * ) , low - mass / high - mass x - ray binaries @xcite , pulsar wind nebulae @xcite , normal and starburst galaxies @xcite , and the giant lobes of a radio galaxy @xcite . however , no obvious counterparts at longer wavelength have been found for as much as 31% of the 2fgl objects so that several hundreds of gev sources currently remain unassociated with any known astrophysical systems . in other words the nature of unassociated gamma - ray sources are still one of the major puzzles , and the mystery has never been solved yet . fortunately , an improved localization capabilities of the ( typical 95% confidence radii @xmath3 , and even [email protected] - [email protected] for the brightest sources ; @xcite ) , when compared to that of egret ( typical @xmath5 ) , enables more effective follow - up studies at radio , optical , and x - ray frequencies , which can help to unravel the nature of the unid gamma - ray emitters . indeed for example , a lot of _ fermi _ sources were identified using wise ir data @xcite . in this context , we started a new project to investigate the nature of unid objects through x - ray follow - up observations with the xis sensor onboard the x - ray satellite ( see section 2 ) . for example , the results of the first - year campaign conducted in ao-4 ( 2009 ) were presented in @xcite . in this campaign , the x - ray counterpart for one of the brightest unassociated objects , 1fglj1231.1 - 1410 ( also detected by egret as 3egj1234 - 1318 and egrj1231 - 1412 ) , was discovered for the first time . the x - ray spectrum was well fitted by a blackbody with an additional power - law component , supporting the recent identification of this source with a msp . in the second - year campaign ( ao-5 ) , another seven unid sources were subsequently observed with @xcite . in particular , this paper presented a convenient method to classify the objects into `` agn - like '' and `` psr - like '' by comparing their multiwavelength properties with those of known agns and pulsars . in the third - year ( ao-6 ) , 1fglj2339.7 - 0531 ( yatsu et al . 2013 in prep ; also romani & shaw 2011 ) and 1fglj1311.7 - 3429 @xcite were intensively monitored with a total exposure time of 200 ksec . both sources are now suggested to be `` black widow '' msp systems and newly categorized as `` radio - quiet '' msps . as these projects show , the x - ray follow - up observations especially using provided various fruitful results to clarify the nature of unassociated gamma - ray sources , and was able to find a new type of gamma - ray emitter . to complete a series of x - ray follow - up programs described above , we further carried out the analysis of all the archival x - ray data of 134 unid gamma - ray sources in the 1fgl catalog of point sources ( 1fgl ; * ? ? ? _ swift_/xrt . note that , all these 134 sources have been detected in the 2fgl catalog , hence the updated data on there lat position and spectra were available from the 2nd @xmath6/lat catalog so are used throughout this work . this allowed us to construct the seds ( seds ) of each objects from radio to gamma - rays ( see section2 and appendix ) for the first time . note that we target all the 1fgl unid sources that satisfy our selections ( see section2 ) , using updated / improved information from the 2fgl catalog on their lat positions and spectra in this paper . moreover , three sources that displayed potentially interesting seds , 1fglj0022.2 - 1850 ( or 2fgl j0022.2 - 1853 ) , 1fglj0038.0 + 1236 ( or 2fgl j0037.8 + 1238 ) and 1fglj0157.0 - 5259 ( or 2fgl j0157.2 - 5259 ) , were deeply observed with as a part of ao7 campaign in 2012 . in the 2fgl catalog , both 1fglj0022.2 - 1850 and 1fglj0157.0 - 5259 are categorized as active galaxies of uncertain type ( agu ) , while 1fglj0038.0 + 1236 was classified as a bl lac type of blazar ( bzb ) based on the positional coincidences to sources observed at another wavelength . but as we see below , the unique seds of these three objects are not well understood as conventional blazars . in section2 we first describe the analysis of the 134 1fgl unid sources with _ swift_. subsequently in section3 , deep follow - up observations of the selected three sources , 1fglj0022.2 - 1850 , 1fglj0038.0 + 1236 and 1fglj0157.0 - 5259 are shown . the results of the analysis are given in section4 . the discussion and summary are presented in section5 and 6 , respectively . @xcite is a gamma - ray observatory launched on 2004 november 20 . the primary goal of this mission is to explore and follow - up the gamma - ray burst , but high mobility and sensitivity to localize sources especially using the xrt @xcite and uvot @xcite , makes it also viable to follow - up unid gamma - ray objects discovered by . in fact , _ swift _ follow - up observations helped study many unid sources @xcite . here we tried to perform a systematic and uniform analysis of unid gamma - ray sources observed thus far with _ swift _ using the archival data . the selection criteria is given as follows ; ( 1 ) categorized as unid sources in 1fgl catalog , ( 2 ) localized at high galactic latitude @xmath7 @xmath8 10@xmath9 , ( 3 ) observational data are in public at the time of october 2011 , ( 4 ) the positional center of _ swift _ field - of - view is within 12 arcmin from the 1fgl sources . among 630 unid sources listed in the 1fgl catalog , this selection yields 134 sources which we analyzed here . in the reduction and the analysis of the _ swift_/xrt and uvot data , headas software version 6.11 and the most recent calibration databases ( caldb ) as of 2011 october 20 were used . we did not use bat data because most sources are not bright enough to be detected within short exposure of typically 10 ksec or less . in the xrt analysis , we only use the pc mode data , while only image data taken from photometry observation is used in the uvot analysis . two types of xrt archival data can be obtained from the _ swift _ data center , level 1 and 2 . level 2 cleaned data have gone through the standard pipeline process ; however , we calibrated the level 1 data ourselves in a way recommended by _ swift _ team . particularly , we selected the good time interval ( gti ) from the level 1 data using ` xrtpipeline ` . in this process , we only changed the default ` xrtpipeline ` selection in the temperature of ccds from `` @xmath10 '' to `` @xmath11 '' , where the former is the default value . in the xrt image analysis , we tried to detect the x - ray counterparts of 1fgl unid sources , and localize each source . first , we extracted x - ray images in the energy range of @xmath12 kev using ` xselect ` . next by using ` ximage ` , we searched for `` possible '' x - ray sources which were @xmath8 3 @xmath13 confidence level in photon statistics against background . the position of these sources are determined with a typical accuracy of @xmath14 using ` xrtcentroid ` . the results of x - ray source detection is listed in table 6 ( see , appendix ) . in the appendix , we also show the xrt images corresponding to each of the 134 1fgl unid gamma - ray sources indicating also the 2fgl error ellipses ( figure11 ) . in these images , bright radio sources and x - ray sources listed in the rosat all - sky survey bright source catalogue ( 1rxs ; * ) corresponding to the xrt sources are also plotted as magenta crosses . finally , if the x - ray sources were detected were within the 2fgl 95@xmath15 position error ellipse , we performed x - ray spectral analysis for those sources . we note that very bright x - ray sources with more than 0.6 c / s should cause serious pile - up effects in the xrt ccd , however , there were no such bright sources in our analyzed sample . in the xrt spectral analysis , pha files were extracted from event files with ` xselect ` and exposure maps were made using ` xrtexpomap ` . we made auxiliary response files ( arfs ) using ` xrtmkarf ` , while we used the current redistribution matrix files ( rmfs ) in caldb . we extracted photons from circles with 30 arcsec radii around the source positions as the source regions , and we set the concentric rings centered at the source positions with radius 30 - 180 arcsec as the background regions . in the case when some background sources appear in the field , or overall regions can not be fitted in the ccd , we simply remove the region surrounding these background sources , or the region outside the ccd chip . if there is no source detected above @xmath16 inside the 2fgl error ellipse , we derived the upper limit assuming the 2fgl error ellipses as the source region of corresponding x - ray flux . the results of spectral analysis are included in the seds given in the appendix . we note that x - ray spectral data were binned at two different ways according to the source brightness ; ( a)binned with ` grppha ` so that at least 20 photons are included in each bin , ( b)divide x - ray spectral data ( 0.5 - 10.0 kev ) logarithmically into 5 bin . when the source have more than 40 counts , we used ( a ) , but otherwise we used ( b ) . we performed the analysis of uvot data only when the x - ray counterpart of 1fgl source was found in the _ swift_/xrt field of view . _ swift_/uvot archival data has six types of filters ( @xmath17 , @xmath18 , @xmath19 , @xmath20 , @xmath21 and @xmath22 ) , with each filter providing different wavelength data . when each filter included more than one observation , the images and exposure maps were summed using ` uvotimsum ` . using ` uvotdetect ` , we detected the sources which have high signal - to - noise ratio ( @xmath23 ) . we set the circles around those sources with radius 5 arcsec as the sources regions if any source was found in the 90% error ellipse of the _ swift_/xrt source . then , circles with radius 30 as the background regions were taken from the area where no sources are found . finally , we obtained the magnitude of each filter using ` uvotsource ` . the correction of galactic extinction were performed following the way described in cardelli et al . ( 1998 ) . to construct seds of each 1fgl sources , we used not only the _ swift_/xrt and _ swift_/uvot flux data analyzed in this paper , but gamma - ray fluxes listed in the 2fgl catalog , and radio fluxes mostly from the ned database and w3browse based on a variety of catalogs . we searched for radio counterparts associated with the xrt or uvot objects which were mentioned above in the heasarc / master radio catalog , ( includes the nrao vla sky survey ( nvss ; * ? ? ? * ghz ) , the first survey catalog of 1.4-ghz radio sources ( first ; * ? ? ? * ghz ) , sydney university molonglo sky survey source catalog ( sumss ; * ? ? ? * mhz ) , vla low - frequency sky survey discrete source catalog ( vlss ; * ? ? ? * mhz ) , the westerbork in the southern hemisphere survey ( wish ; * ? ? ? * ) , australia telescope 20-ghz survey catalog ( at20 g ; * ? ? ? * ) , the green bank 6-cm catalog of radio sources ( gb6 ; * ? ? ? * ) , and parkes - mit - nrao southern , tropical , equatorial and zenith survey ( pmn ; * ? ? ? * ) ) and those radio fluxes are added in the seds . note that if no corresponding radio sources were found in the 2fgl 95% error ellipses , we obtained the upper limit from the most bright radio source in the error ellipse . likewise , if no source was found in the error ellipse but some sources were found outside of the region , we used the sensitivity limits of those sources as an upper limit . finally resultant seds and flux values of each wavelength are given in the appendix ( figure 12 , table 5 and 6 ) . we observed three objects that exhibited potentially interesting or anomalous seds ( which is difficult to be explained by standard emission models of blazars , i.e. , ssc or external compton models as explained in @xcite ) with the x - ray astronomy satellite @xcite . these were denoted in the 1fgl catalog as 1fglj0022.2 - 1850 , 1fglj0038.0 + 1236 , and 1fglj0157.0 - 5259 , and in the 2fgl catalog as 2fglj0022.2 - 1853 , 2fglj0037.8 + 1238 , and 2fglj0157.2 - 5259 , respectively . the observation logs are summarized in table1 . the observations were performed with xis which consists of four ccd cameras each placed in the focal plane of the x - ray telescope ( xrt ; * ? ? ? * ) , and with the hard x - ray detector ( hxd ) which consists of si pin photo - diodes ( hxd - pin ) and gso scintillation counters ( hxd - gso ) ( hxd ; * ? ? ? * ; * ? ? ? one of the xis sensors ( xis 1 ) has a back - illuminated ( bi ) ccd , while the other three ( xiss 0 , 2 , and 3 ) utilize front - illuminated ( fi ) ccds . however , because of an anomaly in 2006 november , the operation of xis2 was terminated . hence , here we use only the three remaining ccds . the xis was operated in the normal full - frame clocking mode with the @xmath24 or @xmath25 editing mode . we analyzed the screened xis data , reduced using the software version 1.2 . the screening was based on the following criteria : ( 1 ) only _ asca_-grade 0 , 2 , 3 , 4 , 6 events were accumulated , while hot and flickering pixels were removed from the xis image using the ` sisclean ` script @xcite , ( 2 ) the time interval after the passage of south atlantic anomaly was greater than 60s , ( 3 ) the object was at least @xmath26 and @xmath27 above the rim of the earth ( elv ) during night and day , respectively . in addition , we also selected the data with a cutoff rigidity ( cor ) larger than 6gv . in the reduction and the analysis of the xis data , headas software version 6.12 and a calibration databases ( caldb ) released on 2009 september 25 were used . the xis cleaned event data - set was obtained in the combined @xmath24 and @xmath25 edit modes using ` xselect ` . the hxd data were also processed in a standard way as follows . first , we obtained the appropriate version 2.0 `` tuned '' non - x - ray background file ( nxb ) for this observation . because the hxd background file has a time variation , we made a new gti file to match the good time interval ( gti ) between observation data and nxb data using ` mgtime ` . next , using this new gti file , we generated time - averaged hxd spectra with ` xselect ` . these were then dead time corrected using ` hxddtcor ` script . epoch appropriate response files for xis - nominal pointing were downloaded from the caldb website . the contribution from the cosmic x - ray background ( cxb ) was simulated following a recipe provided by the hxd team . we extracted the xis images within the photon energy range of @xmath28kev from only the two fi ccds ( xis 0 , xis 3 ) . in the image analysis , we excluded calibration sources at the corner of the ccd chips . the images of the nxb were obtained from the night earth data using ` xisnxbgen ` @xcite . since the exposure times for the original data were different from that of nxb , we calculated the appropriate exposure - corrected original and nxb maps using ` xisexpmapgen ` @xcite . the corrected nxb images were next subtracted from the corrected original images . in addition , we simulated flat sky images using ` xissim ` @xcite , and applied a vignetting correction . all the images obtained with xis0 and xis3 were combined and re - binned by a factor of 4 ( ccd pixel size @xmath29@xmath30m@xmath31@xmath30 m , so that @xmath32 pixels cover an @xmath33 region on the sky ; * ? ? ? * ) . throughout these processes , we performed vignetting correction for all the images . finally , the images were smoothed with a gaussian function with @xmath34 . note that the apparent features at the edge of these exposure corrected images are undoubtedly spurious due to low exposure in those regions . in the spectral analysis of xis , we analyzed the three target sources as point sources , based on the result of our image analysis(see section 5 ) . source regions for spectral analysis indicated by inner green circles were selected around each detected x - ray sources within the error ellipse of a gamma - ray emitters . the corresponding background regions were indicated by outer green ellipse after the removal of source regions . moreover , if the x - ray sources other than target source was found , we excluded the region around those sources from the region for spectral analysis . we extracted the spectra from each source regions with ` xselect ` for each ccd ( xis 0 , xis 1 , xis 3 ) . next , we made redistribution matrix files ( rmfs ) and auxiliary response files ( arfs ) using ` xisrmfgen ` and ` xissimarfgen ` @xcite , respectively . in addition , we used the new contamination files ae_xi0_contami_20120711.fits , ae_xi1_contami_20120711.fits and ae_xi3_contami_20120711.fits , because response function of xis0 is imperfect for recent observations . using these rmfs and arfs , the corrected spectrum about energy response and the effective area of xis were obtained . finally , spectral analysis and model fitting were performed with ` xspec ` version 12.7.0 . in the spectral analysis of hxd , we also subtracted the nxb and cxb to obtain the hxd - pin spectrum , then we performed model fitting together with xis spectrum . we show in this section the results of x - ray image and spectral analysis for each object observed with . since we did nt detect any time - variability for each source in exposures , the results of timing analysis are not shown in this paper . we detected successfully significant signals from the x - ray counterpart of 1fglj0157.0 - 5259 with hxd / pin , but below the sensitivity limit of the hxd / gso , while other two sources were too faint to be detected either with hxd / pin or gso . therefore , as for hxd analysis about 1fglj0157.0 - 5259 in this paper , we only use the data from hxd / pin . in our observations , we detected one x - ray point source ( ra , dec)=([email protected] , [email protected] ) within the updated 2fgl error ellipse corresponding to 1fglj0022.2 - 1850 . figure 1 shows the corresponding x - ray image of 1fglj0022.2 - 1850 , as described in section 4.1 . the radio source nvssj003750 + 123818 appears to be the counterpart of 1fglj0022.2 - 1850 , as indicated by the magenta cross at the center of this x - ray source ( see section 5 ) . moreover , one unknown point source is detected within the background region for spectral analysis ( because the central source is very bright , it is difficult to see this source in fig 1 ) . in figure 4 , the x - ray spectrum of the source , which we argue is the most likely counterpart of 1fglj0022.2 - 1850 , is shown . the xis spectra is given for the energy range @xmath35kev . in the spectral analysis , the target x - ray source is so bright that we selected a source region assuming radii 3@xmath36 ( a typical half - power diameter of the xrt is 2@xmath36:@xcite ) . meanwhile , we excluded one contaminating field x - ray point source detected by , assuming the source region radii 2@xmath36 from the source and background regions for spectral analysis described in section 4.2 . the spectrum is well fitted by a single power - law continuum with a photon index , @xmath37 , moderated by the galactic absorption only . the galactic hydrogen column density was fixed as @xmath38@xmath39 @xcite . the value of @xmath40/d.o.f@xmath41 indicated that this is a satisfactory model for 1fglj0022.2 - 1850 . the details of the fitting results are summarized in table2 . one x - ray point source ( ra , dec)=([email protected] , [email protected] ) was found with _ suzaku _ within the improved 2fgl error ellipse corresponding to 1fglj0038.0 + 1236 . the corresponding x - ray image made through the way described in section 4.1 is shown in figure2 . moreover , one unknown point source is detected within the background region for spectral analysis . the radio source , nvssj00209 - 185332 ( shown by magenta cross ; see section 6 ) , is coincident with the x - ray position and we propose this to be the most likely counterpart of 1fglj0038.0 + 1236 . figure2 shows the x - ray spectrum of the point source detected by near the center of the 2fgl error ellipse of 1fglj0038.0 + 1236 ( @xmath35kev ) . the source region was selected with radii 3@xmath36 , because the target source is too bright not to cover entire region of the emission from target source with radii 2@xmath36 . when we selected the background region , the region from x - ray contaminant source was excluded with radii 2@xmath36 in order to not subtract too much as background . the spectrum could be well fitted by a single power - law continuum with @xmath42 , moderated by the galactic absorption only . the galactic hydrogen column density was fixed as @xmath43@xmath39 @xcite . the value of @xmath40/d.o.f@xmath44 indicates that this is a satisfactory model for 1fglj0038.0 + 1236 . the details of the fitting results are summarized in table3 . we succeeded in detecting a bright x - ray point source with _ suzaku _ within the 2fgl error ellispe corresponding to 1fglj0157.0 - 5259 . figure 3 shows the corresponding x - ray image ( see section 4.1 ) . the x - ray source is located at ( ra , dec)= ( [email protected] , [email protected] ) , as shown in figure3 . the position of the radio source , sumssj015657 - 530157 , is shown by a magenta cross ( see section 5 ) . in this observation , we did not find any other contamination source like in the above two observations . the x - ray spectrum ( xis + hxd ) of the source , which we propose to be the most likely counterpart of 1fglj0157.0 - 5259 , is shown in figure 6 within the energy range @xmath45kev ( xis @xmath46kev , hxd @xmath47kev ) . in this spectral analysis , we set the extraction region to be a radius of 4@xmath36 to encircle this bright source . on the other hand , we set the background region with the more larger radius , and the location of the center of background region displaced from the center of target source not to be over the region covering by ccd . the spectrum could be well fitted by a single power - law continuum with @xmath48 , moderated by the galactic absorption only . the galactic hydrogen column density was fixed as @xmath49@xmath39 @xcite . the value of @xmath40/d.o.f@xmath50 indicates that this is a satisfactory model for 1fglj0157.0 - 5259 . the details of the fitting results are summarized in table4 . in the uniform analysis of archival _ swift _ data , we found several objects which seemed to display anomalous seds that are not typical of agns or psrs . then we performed x - ray follow - up observations of three such sources to more accurately determine the seds of each object ( figures 7 , 8 , and 9 ) . different fluxes between _ swift_/xrt and /_xis _ seen in these seds indicate that these objects should have temporal variability that was not seen within the individual shorter exposures . moreover , thanks to the good sensitivity and long exposure of the data , we have additional hints to reveal the nature of each source as discussed below . an x - ray source found within the updated 2fgl error ellipse of 1fglj0022.2 - 1850 is positionally consistent with the radio source nvssj00209 - 185332 found in the nvss catalog @xcite ( see table2 ) . moreover , infrared counterpart source wisej002209.25 - 185334.7 located at ( ra , dec)=([email protected] , [email protected] ) was found in the wide field infrared survey explorer ( wise ) all - sky release @xcite . the sed of 1fglj0022.2 - 1850/wisej002209.25 - 185334.7/nvssj00209 - 185332 , including our _ suzaku_/xis data and derived xrt and uvot fluxes from _ swift _ , are shown in figure7 . from the relatively high radio flux and flat x - ray spectrum obtained with _ swift_/xrt , this object is likely to be a low - frequency peaked bl lac ( lbl ) . however , during our observation , the x - ray spectrum was observed to be substantially steeper , more typical of a high - frequency peaked bl lac ( hbl ) . moreover , the flat gev gamma - ray spectrum is typical of hbls like mrk 421 and mrk 501 , rather than a lbl . considering the cherenkov telescope array ( cta ) which is an initiative to built a next generation observatory for very - high energy gamma - rays will have an improved sensitivity by an order of magnitude with respect to current instruments ( @xmath51 above a few tev ) , the upward shape of the spectrum suggests the source could be detected also in tev energy in the near future . in the case of 1fglj0038.0 + 1236 , the location of a x - ray counterpart discovered in our observations is consistent with nvssj003750 + 123818 described in table2 , and each optical counterpart sdssj003750.88 + 123819.9 ( classified as galaxy ) located at ( ra , dec)=([email protected] , [email protected] ) and infrared counterpart wisej003750.87 + 123819.9 located at ( ra , dec)=([email protected] , [email protected] ) , were also found in the sloan digital sky survey ( sdss ) catalog @xcite and wise catalog respectively . the constructed radio to x - ray sed together with the _ swift _ xrt / uvot and the lat spectrum is shown in figure8 . since the x - ray spectrum appears very steep , this source seems to be associated with a hbl , while the steep gamma - ray spectrum observed with favors a fsrq origin of this source . while optical and ultraviolet fluxes are extremely bright , this could be due to a contribution of soft photons from the host galaxy as seen in some blazar spectra ( see e.g. , the sed of mrk 501 ; @xcite ) . these results show that this source is difficult to be explained by conventional leptonic models of blazars ( i.e. , ssc or external compton models ) . in the case of 1fglj0157.0 - 5259 , our _ suzaku_/xis observations revealed the presence of a quite bright x - ray counterpart in the lat error circle , and since the hard x - ray fluxes of this source are very high , we could also obtain data from _ suzaku_/hxd . at the position of this x - ray source , the radio counterpart sumss j015657 - 530157 was found in the sumss catalog @xcite(table2 ) . the broad - band sed of 1fglj0157.0 - 5259/sumssj015657 - 530157 with our _ suzaku_/xis , hxd and derived _ swift_/xrt , uvot data is presented in figure9 . since the x - ray fluxes are connected by a straight line with the radio fluxes , these fluxes are seemed to be explained by synchrotron radiation . the peak frequency of the synchrotron spectrum is very high ( @xmath810 kev ) suggesting that the source could be one a rare type of `` extreme '' blazar like mrk 501 in the historical high state @xcite . if this source is an `` extreme '' blazar , observations at tev energies or more deep observations with may detect significant signal in the future . finally , we made the similar figure described in takahashi et al . ( 2012 ) for the 1fgl 134 unid objects in which we performed follow - up observations with _ swift_/xrt ( figure10 ) . this figure presents a comparison of the agns ( aqua ) , psrs ( green ) , and unassociated sources ( red ) which classified in the 2fgl catalog in the x - ray to gamma - ray flux ratios versus radio to gamma - ray ratios plane . the three sources we observed with in this paper are shown in black stars . apparently , that these sources are situated in the typical agn region of this diagnostic plane . it is noteworthy that , 1fglj0157.0 - 5259 is at the right edge of this typical agn region , and this means the x - ray and gamma - ray flux ratios of this source is quite high , such that @xmath52 2 , which is consistent with our speculation that the source is an extreme hbl - type blazar . in this paper we reported on the results of x - ray follow - up observations of three unid gamma - ray sources detected by instrument which indicate anomalous seds . we have successfully detected x - ray counterparts of 1fglj0022.2 - 1850 , 1fglj0038.0 + 1236 and 1fglj0157.0 - 5259 using . the characteristics of each object are summarized below . we also note that these objects display temporal variability in x - rays , as indicated by the different x - ray fluxes measured by _ suzaku_/xis and _ swift_/xrt ( see , figure7 , 8 , 9 ) . the x - ray spectrum of the discovered counterpart of 1fglj0022.2 - 1850 is well fitted by single power - law model with @xmath37 . the spectral shape obtained with ( in x - ray ) and ( in gamma ray ) suggest the source is typical of a hbl - type blazar , but previous @xmath53 observations rather show it was similar to the lbl - type blazar . the source is potentially a tev emitter that could be detected in the near future . in the case of 1fglj0038.0 + 1236 , the x - ray spectrum is well fitted by single power - law model with a photon index , @xmath54 . at first glance , this source also seems to be classified as a hbl because the x - ray spectrum seen in figure8 appear very steep . however , its steep gamma - ray spectrum observed with favors a fsrq origin for this source . these results show that this source is difficult to be explained by standard emission models of blazars , i.e. , ssc or external compton models . in the case of 1fglj0157.0 - 5259 , the x - ray spectrum obtained from xis and hxd are well fitted by single power - law model with a photon index , @xmath55 . from the multiwavelength analysis shown in figure9 , the peak frequency of synchrotron spectrum is very high ( @xmath5210 kev ) suggesting that source could be one of `` extreme '' blazar like mrk 501 in the historical high state . we would like to thank c. c. cheung for useful comments that helped to improve the organization of the manuscript . abdo , a. a. , ackermann , m. , ajello , m. , et al . 2009 , apj , 701 , 123 abdo , a. a. , ackermann , m. , ajello , m. , et al . 2010a , apjs , 187 , 460 abdo , a. a. , ackermann , m. , ajello , m. , et al . 2010b , , 188 , 405 abdo , a. a. , ackermann , m. , ajello , m. , et al . 2010c , apj , 709 , 152 abdo , a. a. , ackermann , m. , ajello , m. , et al . 2010d , apj , 710 , 92 abdo , a. a. , ackermann , m. , ajello , m. , et al . 2010e , apj , 714 , 927 abdo , a. a. , ackermann , m. , ajello , m. , et al . 2010f , apj , 720 , 912 ahn , c. p. , alexandroff , r. , et al . 2012 , apjs , 203 , 21 atwood , w. b. , abdo , a. a. , ackermann , m. , et al . 2009 , apj , 697 , 1071 burrows , d. n. , hill , j. e. , nousek , j. a. , 2005 , space sci . , 120 , 165 cardelli , j. a. , et al . 1998 , apj , 345 , 245 cheung , c. c. , donato , d. , et al . 2012 , apj , 756 , 33 cougnard , i. , guillemot , l. , et al . 2011 , apj , 732 , 47 cohen , a. s. , lane , w. m. , et al . 2007 , aj , 134 , 1245 condon , j. j. , cotton , w. d. , greisen , e. w. , et al . 1998 , aj , 115 , 1693 dabrusco , r. , massaro , f. , paggi , a. , apjs , 206 , 12 day , c. , et al . , 1998 , the asca data reduction guide , tech . , ( greenbelt : nasa gsfc ) , v.2.0 de breuck , c. , tang , y. , et al . 2002 , a&a , 394 , 59 dickey , j. m. , & lockman , f. j. , 1990 , ara&a , 28 , 215 fossati , g. , maraschi , l. , celotti , a. , comastri , a. , & ghisellini , g. 1998 , mnras , 299 , 433 gehrels , n. , et al . 2004 , apj , 611 , 1005 ghisellini , g. 2004 , new astronomy review , 48 , 375 gregory , p. c. , scott , w. k. , douglas , k. , condon , j. j. , et al . 1996 , apjs , 103 , 427 hartman , r. c. , bertsch , d. l. , bloom , s. d. , et al . 1999 , apjs , 123 , 79 ishisaki , y. , maeda , y. , fujimoto , r. , et al . 2007 , , 59 , 113 kataoka , j. , mattox , j. r. , quinn , j. , et al . 1999 , , 514 , 138 kataoka , j. , yatsu , y. , kawai , n. , et al . 2012 , apj , 757 , 176 keith , m. j. , johnston , s. , et al . 2011 , mnras , 414 , 1292 kokubun , m. , makishima , k. , takahashi , t. , et al . 2007 , , 59 , 53 kong , a. k. h. , huang , r. h. h. , et al . 2012 , apj , 747 , 3 koyama , k. , tsunemi , h. , dotani , t. , et al . 2007 , , 59 , 23 maeda , k. , kataoka , j. , nakamori , t. , et al . 2011 , apj , 729 , 103 massaro , f. , dabrusco , r. , paggi , a. , et al . 2013 , apjs , 206 , 13 mauch , t. , et al . 2003 , mnras , 342 , 1117 mitsuda , k. , bautz , m. , inoue , h. , et al . 2007 , pasj , 59 , 1 murphy , t. , sadler , e. m. , et al . 2010 , mnras , 402 , 2403 nolan , p. , abdo , a. a. , ackermann , m. , et al . 2011 , apjs , 199 , 31 ransom , s. m. , ray , p. s. , et al . 2011 , apj , 727 , 16 romani , r. w. , & shaw , m. s. 2011 , , 743 , 26 romani , r. w. 2012 , , 754 , 25 roming , p. w. a. , kennedy , t. e. , mason , k. o. , et al . 2005 , space sci . , 120 , 95 serlemitsos , p. j. , soong , y. , chan , k .- w . , et al . 2007 , , 59 , 9 takahashi , t. , abe , k. , endo , m. , et al . 2007 , , 59 , 35 takahashi , y. , kataoka , j. , nakamori , t. , et al . 2012 , apj , 747 , 64 tawa , n. , hayashida , k. , nagai , m. , et al . 2008 , , 60 , 11 theureau , g. , parent , d. , et al . 2011 , a&a , 525 , 94 voges , w. , aschenbach , b. , et al . 1999 , a&a , 349 , 389 white , r. l. , becker , r. h. , helfand , d. j. , & gregg , m. d. 1997 , apj , 475 , 479 wright , a. e. , griffith , m. r. , et al . 1994 , apj , 422 , 642 wright , e. l. , et al . 2010 , aj , 140 , 1868 ccccc target name & r.a . [ deg ] & dec . [ deg ] & exposure [ ks ] & obs . start ( ut ) + 1fglj0022.2 - 1850 & 5.5540 & -18.9060 & 34.2 & 2012-may.30 12:57:00 + 1fglj0038.0 + 1236 & 9.4627 & 12.6391 & 18.8 & 2012-jun.29 23:56:00 + 1fglj0157.0 - 5259 & 29.3640 & -53.0280 & 12.1 & 2012-may.28 16:19:00 + ccccc target name & r.a . [ deg ] & dec . [ deg ] & @xmath56 [ mjy ] & @xmath57 [ mjy ] + nvss j00209 - 185332 & 5.5381667 & -18.892444 & @xmath58 & - + nvss j003750 + 123818 & 9.461875 & 12.638556 & @xmath59 & - + sumss j015657 - 530157 & 29.240833 & -53.032778 & - & @xmath60 + kev energy band . the magenta cross denote the position of radio counterpart source nvssj00209 - 185332 , the yellow dotted ellipse denotes the 95@xmath15 position error of 2fglj0022.2 - 1853 , and the inner green ellipse denote the source extraction region , the outer green ellipse without inner source region denote the background extraction region.,title="fig:",width=566 ] + + + + + + , 4.6@xmath61 , 12@xmath61 and 22@xmath61 ) are taken from the wise catalog . the optical / uv fluxes were derived from the _ swift_/uvot observations ( this work ) . the x - ray are fluxes taken from the _ suzaku_/xis and _ swift_/xrt observations ( this work ) . finally , the gamma - ray data points are taken from the 2fgl catalog @xcite.,title="fig : " ] + , 4.6@xmath61 , 12@xmath61 and 22@xmath61 ) are taken from the wise catalog . the optical / uv fluxes were derived from the _ swift_/uvot observations ( this work ) and the sdss catalog . the x - ray fluxes are taken from the _ suzaku_/xis and _ swift_/xrt observations ( this work ) . finally , the gamma - ray data points are taken from the 2fgl catalog @xcite.,title="fig : " ] + + + \(1 ) 1fglj0001.94158 + \(133 ) 1fglj2350.13005 + fov . one or more sources are detected within the _ swift_/xrt fov . unfortunately , 2fgl error regions of some sources run off the edge of _ swift_/xrt fov . the signal - to - noise acceptance threshold is set to @xmath63 . the yellow ellipses show @xmath64 error regions of 2fgl catalog gamma - ray sources , and red ellipses show that of 1fgl catalog sources . if there are the radio and bright x - ray sources associated with gamma - ray source , we show those sources as magenta crosses.,title="fig:",width=196 ] \(134 ) 1fglj2352.1@xmath621752 + fov . one or more sources are detected within the _ swift_/xrt fov . unfortunately , 2fgl error regions of some sources run off the edge of _ swift_/xrt fov . the signal - to - noise acceptance threshold is set to @xmath63 . the yellow ellipses show @xmath64 error regions of 2fgl catalog gamma - ray sources , and red ellipses show that of 1fgl catalog sources . if there are the radio and bright x - ray sources associated with gamma - ray source , we show those sources as magenta crosses.,title="fig:",width=196 ]

we have analyzed all the archival x - ray data of 134 unidentified ( unid ) gamma - ray sources listed in the first _ fermi / lat _ ( 1fgl ) catalog and subsequently followed up by _ swift / xrt_. we constructed the spectral energy distributions ( seds ) from radio to gamma - rays for each x - ray source detected , and tried to pick up unique objects that display anomalous spectral signatures . in these analysis , we target all the 1fgl unid sources , using updated data from the second _ fermi / lat _ ( 2fgl ) catalog on their lat position and spectra . we found several potentially interesting objects , particularly three sources , 1fglj0022.2 - 1850 , 1fglj0038.0 + 1236 and 1fglj0157.0 - 5259 , which were then more deeply observed with as a part of an ao7 program in 2012 . we successfully detected an x - ray counterpart for each source whose x - ray spectra were well fitted by a single power - law function . the positional coincidence with a bright radio counterpart ( currently identified as agn ) in the 2fgl error circles suggests these are definitely the x - ray emission from the same agn , but their seds show a wide variety of behavior . in particular , the sed of 1fglj0038.0 + 1236 is difficult to be explained by conventional emission models of blazars . the source 1fglj0022.2 - 1850 may be in a transition state between a low - frequency peaked bl lac and a high - frequency peaked bl lac object , and 1fglj0157.0 - 5259 could be a rare kind of extreme blazar . we discuss the possible nature of these three sources observed with , together with the x - ray identification results and seds of all 134 sources observed with _ swift_/xrt .

1307.5581

in the past decades atom - based metrology has had an enormous impact on science , technology and everyday life . seminal advances include microwave and optical atomic clocks @xcite , the global positioning system , and highly sensitive , position - resolved magnetometers @xcite . atom - based field measurement has clear advantages over other field measurement methods because it is calibrating - free , due to the invariance of atomic properties . atom - based metrology has recently expanded into electric - field measurement . an all - optical sensing approach employed by numerous groups is electromagnetically induced transparency ( eit ) of atomic vapors , utilizing rydberg levels @xcite to measure the properties of the electric field . rydberg atoms are well - suited for this purpose owing to their extreme sensitivities to dc and ac electric fields , which manifest in large dc polarizabilities and microwave - transition dipole moments @xcite . developments include measurements of microwave fields and polarizations @xcite , millimeter waves @xcite , static electric fields @xcite , and sub - wavelength imaging of microwave electric - field distributions @xcite . in the frequency range from 10 s to 100 s of mhz , rydberg - eit rf modulation spectroscopy is a promising method to accomplish atom - based , calibration - free rf electric - field measurement @xcite . rydberg - eit in vapor cells offers significant potential for miniaturization @xcite of the rf sensor . accurate calibration of the electric field is important , for instance , for antenna calibration , characterization of electronic components , etc . conventional calibration with field sensors that involve dipole antennas ( that need to be calibrated first ) obviously leads into a chicken - and - egg dilemma @xcite . in the present work we provide an atom - based calibration method for vector electrometry of rf - fields using rydberg - eit in cesium vapor cells . the basic idea is that the rf generates a series of intersections between levels in the rydberg floquet map ( a map in which field - perturbed floquet level energies are plotted versus rf electric field ) . the ( anti-)crossings occur between floquet states originating in fine - structure components of @xmath4 states with equal principal and angular - momentum quantum numbers @xmath5 and @xmath6 . the crossings present excellent field markers that we use as calibration points for the electric - field strength . specifically , we measure the rf - dressed cs 60@xmath0 states via rydberg - eit spectroscopy at a test frequency of 100 mhz . the dependence of the floquet spectrum on the strength and the polarization of the rf field is investigated . there are exact crossings between states of different @xmath7 , @xmath8 and @xmath9 , which are not coupled in a linearly polarized rf field . the crossings provide field markers , which we use to calibrate the field strength in a test rf transmission system . we also analyze narrow , spectroscopically resolved anti - crossings between floquet states of equal @xmath7 ( or @xmath8 ) , and different @xmath10 and @xmath9 . transitions between those states are allowed via an rf raman process . further , the eit line - strength ratios of intersecting floquet states with unlike @xmath2 yield the field polarization . at various stages of the work , the measured spectroscopic data are matched with the results of floquet calculations to accomplish the calibration tasks . -axis . two electrode plates are located on the both sides of the vapor cell , where the rf field is applied . the polarization of the rf field @xmath11 ( blue arrow ) points along the @xmath12-axis . the polarization of the beams ( red and green arrows ) can be adjusted with the @xmath13/2 wave plates to form an angle @xmath14 with the rf electric field @xmath11 . the probe beam is passed through a dichroic mirror ( dm ) and is detected with a photodiode ( pd ) . polarizing beam splitters ( pbs ) are used to produce beams with pure linear polarizations . ( b ) energy level scheme of cesium rydberg - eit transitions . the probe laser , @xmath15 , is resonant with the lower transition , @xmath16 @xmath17 @xmath18 , and the coupling laser , @xmath19 , is scanned through the rydberg transitions @xmath20 @xmath17 @xmath21 . the applied rf electric field ( frequency @xmath22 = 2@xmath23100 mhz ) produces ac stark shifts and rf modulation sidebands that are separated in energy by even multiples of @xmath24.,scaledwidth=45.0% ] a schematic of the experimental setup and the relevant rydberg three - level ladder diagram are shown in figs . 1 ( a ) and ( b ) . the experiments are performed in a cylindrical room - temperature cesium vapor cell that is 50 mm long and has a 20-mm diameter . the cell is suspended between two parallel aluminum plate electrodes that are separated by @xmath25 mm . the eit coupling - laser and probe - laser beams are overlapped and counter - propagated along the centerline of the cell ( propagation direction along the @xmath26-axis ) . the coupling and probe lasers have the same linear polarization in the @xmath27 plane . the angle @xmath14 between the laser polarizations and the rf field ( which points along the @xmath12-axis ) is varied by rotating the polarization of the laser beams with @xmath13/2 plates , as seen in fig . 1 ( a ) . the weak eit probe beam ( central rabi frequency @xmath28 = [email protected] mhz and @xmath30 waist @xmath31 m ) has a wavelength @xmath15 = 852 nm , and is frequency - locked to the transition @xmath16 @xmath17 @xmath32 , as shown in fig . 1 ( b ) . the coupling beam ( central rabi frequency @xmath33 = [email protected] mhz for 60@xmath34 and @xmath30 waist @xmath35 m ) is provided by a commercial laser ( toptica ta - shg110 ) , has a wavelength of 510 nm and a linewidth of 1 mhz , and is scanned over a range of 1.5 ghz through the @xmath36 rydberg transition . the eit signal is observed by measuring the transmission of the probe laser using a photodiode ( pd ) after a dichroic mirror ( dm ) . an auxiliary rf - free eit reference setup ( not shown , but similar to the one sketched in fig . 1 ( a ) ) is operated with the same lasers as the main setup . the auxiliary eit signal is employed to locate the 0-detuning frequency reference point for all eit spectra we show ; it allows us to correct for small frequency drifts of the coupling laser . the rf voltage amplitude , @xmath37 , provided by a function generator ( tektronix afg3102 ) , is applied to the electrodes as shown in fig . 1 ( a ) , and the rf electric - field vector , @xmath11 , points along @xmath12 ( blue arrow in fig . 1 ( a ) ) . the rf frequency is fixed , @xmath22 = 2@xmath29100 mhz , and the rf field amplitude , @xmath38 , is varied by changing @xmath37 . the rf - field ac - shifts the rydberg levels and generates even - order modulation sidebands ( see fig . 1 ( b ) ) . the rf field amplitude @xmath38 is approximately uniform within the atom - field interaction volume . using a finite - element calculation , we have determined that the average electric field in the atom - field interaction region is @xmath39 of the field that would be present under absence of the dielectric glass cell ( _ i.e. _ the glass shields @xmath40 of the field ) . the glass cell further gives rise to an @xmath41 field inhomogeneity along the beam paths within the cell . the rf transmission line between the source and the cell has unavoidable standing - wave effects . while the standing - wave effect is hard to model due to the details of the experimental setup , which are fairly complex from the viewpoint of rf field modeling , the setup still constitutes a linear transmission system . therefore , for any given frequency and fixed arrangement of the wiring and the electromagnetic boundary conditions , the magnitude of the voltage amplitude that occurs on the rf field plates follows @xmath42 , where @xmath43 is a frequency - dependent transmission factor that is specific to the details of the rf transmission line . as discussed in detail in section [ sec : measurements ] , we use the atom - based field measurement method to determine the transmission factor to be @xmath44 . the average rf electric - field amplitude , @xmath38 , averaged over the atom - field interaction zone inside the cell , is then related to the known voltage amplitude , @xmath37 , generated by the source via @xmath45 ( @xmath46 is the distance between the field plates ) . in this relation , the only factor that is difficult to determine is the transmission factor @xmath43 @xcite . the experiment described in this paper represents a good example of how the atom - based field measurement method allows one to measure @xmath43 and to thereby calibrate rf electric fields . = 2@xmath47100 mhz and the indicated amplitudes @xmath48 v / cm ( bottom ) , 0.252 v / cm ( middle ) and 0.504 v / cm ( top ) , respectively . the main peak at 0 detuning in the field - free spectrum corresponds to the @xmath49 @xmath50 @xmath51 @xmath50 @xmath52 cascade eit . the small peak at -168 mhz ( small arrow ) originates in the intermediate - state hyperfine level @xmath53 @xcite . the peak at -318 mhz is the @xmath54 eit line . the peaks within the magenta circles are rf - induced sidebands of the indicated orders.,scaledwidth=45.0% ] in fig . 2 , we show rydberg - eit spectra for the @xmath55 states for @xmath56 without rf field ( bottom curve ) and with the indicated rf fields ( upper pair of curves ) . the bottom eit spectrum is obtained with the rf - free reference setup . the @xmath57 main peak in the reference spectrum defines the 0-detuning position . since the value of the @xmath55 fine structure splitting ( 318-mhz arrow in fig . 2 ) is well known , the spacing between the zero - field @xmath55 fine structure components is used to calibrate the detuning axis . the top two curves show eit spectra for applied rf field strengths @xmath58 v / cm and 0.504 v / cm . the @xmath58 v / cm plot illustrates the rf - induced ac stark shifts in weak rf - fields . the degeneracy between the @xmath2 = 1/2 , 3/2 , and 5/2 magnetic substates of the @xmath55 levels becomes lifted . ( the quantization axis for @xmath2 is the direction of @xmath59 in fig . 1 ( a ) . ) since the rf field frequency is much lower than the kepler frequency ( 35 ghz for cs 60@xmath60 ) , the ac shifts in weak rf fields are near - identical with @xmath61 , where @xmath62 are the dc polarizabilities of the @xmath63 states , and @xmath64 is the rf root - mean - square field . this has been verified with a dc stark shift calculation ( not shown ) . at higher fields , rf - induced even - harmonic sidebands for @xmath65 appear , which are marked with magenta circles in the top curve of fig . the sidebands come in pairs , the lower - frequency component has @xmath66 , the higher - frequency one @xmath7 . the lines that do not shift much throughout fig . 2 are the @xmath67 states ; these have near - zero polarizability . the ac shifts and sideband separations are on the same order as the fine structure splitting of 60@xmath0 . this similarity in energy scales is important because it gives rise to the level crossings in the floquet maps discussed below . we have performed a series of measurements such as in fig . 2 over an @xmath38-field range of 0 to 0.76 v / cm , in steps of 0.006 v / cm . we have assembled the rf - eit spectra in a floquet map , shown fig . 3 ( a ) . at fields @xmath68v / cm the @xmath2-sublevels shift and split due to the @xmath69-dependent quadratic ac stark effect . the even - harmonic level modulation sidebands , labeled @xmath65 , begin to appear when the rf field is increased further ( also see previous work @xcite ) . to match the measured eit spectra with theory , we numerically calculate rydberg eit spectra using floquet theory , with results shown in fig . 3 ( b ) . for details of the floquet calculation see @xcite . a central point of the present work is that the @xmath70 level , which has near - zero polarizability and ac stark shift , undergoes a series of crossings with the @xmath71 = 1/2 , 3/2 modulation sidebands . the crossings are exact because the linearly polarized rf field does not mix quantum states of different @xmath2 . the crossings can be measured with about @xmath72 precision . as an example , in fig . 3 ( a ) we show a zoom - in of the first level crossing . the crossing is centered at @xmath73 v / cm , with an estimated uncertainty of @xmath74 v / cm , corresponding to a relative uncertainty of @xmath75 . the uncertainty is mostly attributed to the intrinsic eit linewidth , which increases with increasing coupling and probe rabi frequencies . laser linewidths and interaction - time broadening also contribute to the observed linewidths . .the columns show , in that order , the crossing number , the calculated electric field @xmath38 for the crossing , the experimental electric field @xmath76 the atoms would be exposed to for an rf amplitude transmission factor @xmath77 , and the transmission factor @xmath78 . [ cols="^,^,^,^",options="header " , ] in fig . 3 six such crossings are visible within the rectangular boxes . with the rf - source voltage amplitudes @xmath79 at which the crossings are observed , and recalling that the glass cell shields @xmath40 of the electric field from the atoms , the electric field the atoms would experience for an amplitude transmission factor of 1 would be @xmath80 . the ratios between the known ( theoretical ) electric fields where the crossings actually occur , @xmath81 , and the @xmath82 yield six readings for the amplitude transmission factor , @xmath83 . in table 1 it is seen that the @xmath84 have a very small spread and do not exhibit a systematic trend from low to high field . the average , @xmath85 , is the desired calibration factor for the experimental electric - field axis . the @xmath86axis in fig . 3 ( a ) shows the calibrated experimental electric field , @xmath87 , with voltage amplitude @xmath37 at the source . the overall relative uncertainty of the atom - based rf - field calibration performed in this experiment is @xmath3 , similar to what has been obtained in @xcite and about an order of magnitude better than in traditional rf field calibration @xcite . the use of narrow - band coupling and probe lasers , lower rabi frequencies , and larger - diameter laser beams is expected to reduce the uncertainty to considerably smaller values . we note that the calibration uncertainty achieved in this work is based on matching experimental and calculated spectroscopic data at the locations of a series of six isolated level crossings that all occur within a narrow spectral range of less than 50 mhz width ( see rectangular boxes in fig . 3 ) . hence , a fairly small amount of spectroscopic data suffices for the presented atom - based rf field calibration . from fig . 3 it is obvious that this advantage traces back to a specific feature of cesium @xmath88 states , namely that these states offer a mix of magnetic sublevels with near - zero and large ac polarizabilities . rf - dressed rydberg - eit spectra of rubidium atoms do not present a similar advantage @xcite . in fig . 3 we further observe three series of avoided crossings , which are due to an rf - sideband of the @xmath10 level intersecting with an rf - sideband of the @xmath89 level . the first number in the avoided - crossing labels in fig . 3 ( b ) shows the number of rf photon pairs taken from the rf field to access the @xmath10 band , while the second shows the number of rf photon pairs taken from the rf field to access the @xmath89 band . negative rf photon numbers , indicated by underbars , correspond to stimulated rf - photon emission . the coupling between the intersecting @xmath10 and @xmath89 bands is a two - rf - photon raman process in which the atom absorbs and re - emits an rf photon while changing @xmath1 from @xmath8 to @xmath9 , or vice versa . this is a second - order electric - dipole transition , which , for the given polarization , has selection rules @xmath90 and @xmath91 . in fig . 3 , three series of avoided crossings that satisfy these selection rules are visible , one for @xmath92 and two for @xmath93 . each series has a fixed @xmath38-value and consists of copies of the same avoided crossing along the @xmath94-axis , in steps of 200 mhz . the @xmath93 series are particularly easy to spot because one of the two intersecting floquet states has near - zero polarizability . the raman coupling causing the avoided crossings equals the minimal avoided - crossing gap size . for fixed floquet - state wavefunction , the raman coupling strength should scale as @xmath95 . for the @xmath93 avoided crossings at 0.319 v / cm we observe a coupling strength of 8.6 mhz , while those at 0.579 v / cm have a coupling strength of 19.3 mhz . the coupling - strength ratio , which is 2.2 , is somewhat smaller than the @xmath95-ratio , which is 3.3 . the deviation indicates a moderate variation of the floquet - state wavefunctions between 0.319 v / cm and 0.579 v / cm ( which is expected ) . from a field - calibration point of view , the avoided crossings and other details in the spectra could be used to further reduce the uncertainty in the atom - based rf - field calibration factor @xmath43 , which is planned in future work . comparing the cesium and rubidium level structures , it is again noteworthy that cesium offers a combination of @xmath2-dependent polarizabilities that is particularly favorable for this purpose . in the top curve in fig . 2 it is noted that the @xmath96 and @xmath97 floquet states are narrow and symmetric , whereas the other floquet lines are much wider and are asymmetrically broadened . further , the @xmath7-lines exhibit a shoulder on the high - frequency side ( see @xmath98mhz marker in fig . 2 ) , while the @xmath66-lines have no shoulder . the scan in the top curve of fig 2 also corresponds to the vertical dashed line in fig . 3 ( b ) . close inspection of fig . 3 ( b ) reveals that the shoulders of the @xmath7-lines are due to the series of narrow avoided crossings between floquet states in the @xmath7-manifold . the shoulders correspond to the weaker , higher - frequency component of the crossing . the asymmetric line broadening of the wide lines is due to the @xmath99 full - width variation of the rf field within the atom - field interaction zone . for instance , for the @xmath66-lines we estimate for the inhomogeneous linewidth @xmath100mhz , which is close to the observed width of @xmath101 mhz . ( the @xmath7-lines are also inhomogeneously broadened , but we do not give a broadening estimate for those lines because of the interference with the mentioned avoided crossing . ) [ thb ] defined in the text at an rf field of @xmath102=0.415 v / cm , as a function of polarization angle @xmath14 . the data are for the indicated values of the probe rabi frequency . the inset shows sample eit spectra for @xmath103 and @xmath104.,title="fig:",scaledwidth=40.0% ] rydberg - eit spectra generally depend on the laser polarizations @xcite . this also applies to rf - modulated rydberg eit spectra . here , we study the dependence of line - strength ratios on the angle @xmath14 between the rf - field and the polarization of the laser beams ( both laser beams are linearly polarized , and the polarizations are parallel to each other ; see fig . 1 ) . for a line - strength comparison of floquet levels of different @xmath2 it is advantageous to choose an electric field close to one of the exact crossings discussed above , because the two lines of interest will then appear in close proximity to each other , allowing for a rapid measurement . additionally , since states with different @xmath2 do not mix , the line - strength measurements are robust against small variations of the rf electric field . as an example , the inset in fig . 4 we show the rf eit spectra for @xmath105 ( lower curve ) and @xmath104 ( upper curve ) at an rf field @xmath102= 0.415 v / cm , marked with a dashed line in the left panel in fig . 3 ( a ) . the two peaks labeled a1 and a2 within the blue square in the inset in fig . 4 correspond to the floquet levels marked with red circles in fig . 3 ( a ) . the peak a1 , which corresponds to the @xmath106 rf band of the @xmath107 floquet state , increases with the angle @xmath14 , whereas the peak a2 , which corresponds to the @xmath108 rf band of the @xmath109 state , decreases with @xmath14 . to quantify this polarization - angle dependence , we introduce the parameter @xmath110 , where @xmath111 represent the respective areas of gaussian peaks obtained from double - gaussian fits to the spectra at angle @xmath14 . since the intersecting lines have different differential dipole moments , it is important to use the areas and not the peak heights ( see discussion in the last paragraph in sec . [ subsec : spectroscopy ] ) . figure 4 shows @xmath112 as a function of the @xmath14 at @xmath102=0.415 v / cm for the indicated probe laser rabi frequencies , together with the corresponding line strength ratio obtained from floquet calculations ( the floquet calculation yields line strengths valid for the case of low saturation , @xmath113 ) . we find excellent agreement between the measurements and calculations for @xmath28 = @xmath114 9.2 mhz . curves such as in fig . 4 can be used to measure the polarization of an rf - field with unknown linear polarization . the angle uncertainty can be estimated as @xmath115 , where @xmath116 is the difference between experimental and calculated values of @xmath117 , and @xmath118 is the derivative of the calculated curve . for the lowest - power case in fig . 4 , straightforward analysis shows angle uncertainties below @xmath119 for @xmath120 . in the domain @xmath121 the angle uncertainty gradually increases from @xmath119 to @xmath122 because the derivative becomes small . we note that this method of polarization measurement has the advantage of being both simple and very fast , since the areas of only two lines need to be measured . at the expense of reduced acquisition speed , the uncertainty could be improved by measuring line - strength ratios of multiple line pairs and by averaging over a number of spectra . the data for higher probe rabi frequencies in fig . 4 show a more significant deviation from the calculated curve . this is not unexpected , because the calculation is for negligible saturation of the probe transition , whereas the data in fig . 4 vary between moderate and strong saturation of the probe transition . in addition to saturation broadening effects , there may also be optical - pumping effects @xcite that could affect the line strength ratio . this is beyond the scope of the present work . we have demonstrated a rapid and robust atom - based method to calibrate the electric field and to measure the polarization of a 100 mhz rf field , using rydberg eit in a room - temperature cesium vapor cell as an all - optical field probe . the eit spectra exhibit rf - field - induced ac stark shifts , splittings and even - order level modulation sidebands . a series of exact floquet level intersections that are specific to cesium rydberg atoms have been used for calibrating the rf electric field with an uncertainty of @xmath3 . the dependence of the rydberg - eit spectra on the polarization angle of the rf field has been studied . our analysis of certain line - strength ratios has led into a convenient method to determine the polarization of the rf electric field . the rydberg - eit spectroscopy presented here could be applied to atom - based , antenna - free calibration of rf electric fields and polarization measurement . it is anticipated that an extended analysis of all exact and avoided crossings as well as other spectroscopic features will significantly lower the calibration uncertainty . future work involving narrow - band laser sources , miniature spectroscopic cells as well as improved spectroscopic methods ( lower rabi frequencies , wider probe and coupler beams ) are expected to further reduce the calibration uncertainty . the work was supported by nnsf of china ( grants nos . 11274209 , 61475090 , 61475123 ) , changjiang scholars and innovative research team in university of ministry of education of china ( grant no . irt13076 ) , the state key program of national natural science of china ( grant no . 11434007 ) , and research project supported by shanxi scholarship council of china ( 2014 - 009 ) . gr acknowledges support by the nsf ( phy-1506093 ) and bairen plan of shanxi province . 27ifxundefined [ 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 http://dx.doi.org/10.1088/0026-1394/51/3/174 [ * * , ( ) ] http://dx.doi.org/10.1103/revmodphys.87.637 [ * * , ( ) ] http://dx.doi.org/10.1103/physrevlett.95.063004 [ * * , ( ) ] http://dx.doi.org/10.1063/1.4747206 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.98.113003 [ * * , ( ) ] @noop _ _ ( , , ) link:\doibase 10.1038/nphys2423 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.111.063001 [ * * , ( ) ] http://dx.doi.org/10.1088/0953-4075/48/20/202001 [ * * , ( ) ] http://dx.doi.org/10.1063/1.4890094 [ * * , ( ) ] http://dx.doi.org/10.1103/physrevlett.110.123002 [ * * , ( ) ] link:\doibase 10.1364/ol.39.003030 [ * * , ( ) ] \doibase http://dx.doi.org/10.1063/1.4883635 [ * * , ( ) ] http://dx.doi.org/10.1088/1367-2630/12/6/065015 [ * * , ( ) ] http://dx.doi.org/10.1103/physreva.94.023832 [ * * , ( ) ] link:\doibase 10.1038/nphys566 [ * * , ( ) ] http://dx.doi.org/10.1063/1.4891534 [ * * , ( ) ] link:\doibase 10.1109/tap.2014.2360208 [ * * , ( ) ] @noop link:\doibase 10.1103/physreva.90.043419 [ * * , ( ) ] http://dx.doi.org/10.1103/physrevapplied.5.034003 [ * * , ( ) ] http://dx.doi.org/10.1088/1367-2630/18/5/053017 [ * * , ( ) ] https://archive.org/details/generatingstanda1335hill [ ( ) ] in link:\doibase 10.1109/imtc.1993.382655 [ _ _ ] ( ) pp . http://dx.doi.org/10.1103/physreva.62.053802 [ * * , ( ) ] http://dx.doi.org/10.1103/physreva.94.043822 [ * * , ( ) ] @noop ( ) ,

we investigate atom - based electric - field calibration and polarization measurement of a 100-mhz linearly polarized radio - frequency ( rf ) field using cesium rydberg - atom electromagnetically induced transparency ( eit ) in a room - temperature vapor cell . the calibration method is based on matching experimental data with the results of a theoretical floquet model . the utilized 60@xmath0 fine structure floquet levels exhibit @xmath1- and @xmath2-dependent ac stark shifts and splittings , and develop even - order rf - modulation sidebands . the floquet map of cesium 60@xmath0 fine structure states exhibits a series of exact crossings between states of different @xmath2 , which are not rf - coupled . these exact level crossings are employed to perform a rapid and precise ( @xmath3 ) calibration of the rf electric field . we also map out three series of narrow avoided crossings between fine structure floquet levels of equal @xmath2 and different @xmath1 , which are weakly coupled by the rf field via a raman process . the coupling leads to narrow avoided crossings that can also be applied as spectroscopic markers for rf field calibration . we further find that the line - strength ratio of intersecting floquet levels with different @xmath2 provides a fast and robust measurement of the rf field s polarization .

1703.02286

the small dark molecular clouds known as starless cores are are significant in the interstellar medium as the potential birthplaces of stars ( review by * ? ? ? as their name implies , the starless cores do not yet contain stars , but their properties are nearly the same as similar small clouds that do @xcite . furthermore , there is a compelling similarity in the mass function of the starless cores @xcite and the initial mass function ( imf ) of stars . observations of bok globules ( bok 1948 ) and starless cores @xcite suggest that many of these small ( m @xmath0 m@xmath1 ) clouds are well described as quasi - equilibrium structures supported mostly by thermal pressure , approximately bonnor - ebert ( be ) spheres @xcite . observed molecular spectral lines @xcite show complex profiles that further suggest velocity and density perturbations within these cores . we have previously shown that in at least one case , b68 , these profiles can be produced by the non - radial oscillations of an isothermal sphere @xcite . to match the spectral line profiles , we have previously found that the oscillations have to be large enough ( 25 % ) that the amplitudes are non - linear @xcite . if such large amplitude oscillations are to be a viable explanation for the observed complex velocity patterns then their decay rate must be no faster than the sound - crossing or free - fall times of the globules , @xmath2 yrs . numerical simulations in which the external medium is substantially less dense than the gas inside the core have found that mode damping is sufficiently slow that modes will persist for many periods @xcite . however , it is not clear that this description is appropriate for many starless cores . b68 is one of only a handful of cores surrounded by hot , rarefied gas in the pipe nebula , where the vast majority of cores , like those in the taurus & perseus molecular clouds , appear to be embedded in cold , dense molecular gas @xcite . embedded oscillating clouds will generally act as sonic transducers , exciting sound waves in the external medium and thereby loosing energy . a simple one - dimensional analysis , discussed in detail in the appendices , suggests that when the density contrast between the core interior and its bounding medium is close to unity this process can be very efficient , potentially limiting the lifetime of oscillations . in this paper we investigate the damping rate of oscillations of isothermal spheres as a function of the density contrast with the external bounding medium . we do this using numerical simulations of a large amplitude oscillation superimposed upon an isothermal sphere . to evaluate our results we also derive analytical and semi - analytical estimates of the damping rate associated with the excitation of sound waves in the exterior medium . a brief discussion of the numerical simulation is discussed in section [ sec : nhm ] , the presentation and discussion of the numerical results are in section [ sec : rad ] , and conclusions can be found in section [ sec : c ] . the details of the analytical and semi - analytical computation of the damping rates of small amplitude oscillations are relegated to the appendices . we find that a reduced density contrast between the core and the exterior bounding medium does result in increased dissipation of oscillations as momentum and energy are transferred out of the core through the boundary . however , if the oscillations are non - radial the dissipation through the boundary is always less than the dissipation rate due to mode - mode coupling . cccc black & @xmath3 & 0.017 & 8.5 + blue & @xmath4 & 0.037 & 6.5 + green & @xmath5 & 0.087 & 4.8 + red & @xmath6 & 0.22 & 3.44 + because the thermal gas heating and cooling time , via collisional coupling to dust at high densities @xcite and molecular line radiation at low densities @xcite , is short in comparison to the typical oscillation period ( on the order of @xmath7 ) , the oscillations of dark - starless cores are well modeled with an isothermal equation of state @xcite . in our simulations , we use a barotropic equation of state with an adiabatic index of unity ( as opposed to @xmath8 , for example ) , in which case the isothermal evolution is also adiabatic . this allows us to replace the energy equation with a considerably simpler adiabatic constraint . since the unbounded isothermal gas sphere is unstable , we truncate the solution at a given center - to - edge density contrast ( @xmath9 , not to be confused with the interior / exterior density contrast ) , producing a stable bonnor - ebert sphere solution with radius ( @xmath10 ) and mass determined by the central density ( @xmath11 ) and temperature ( @xmath12 ) . this is done by inducing a phase change in the equation of state , i.e. , @xmath13^\epsilon \rho(r ) \frac{k t}{\mu m_p } & \mbox{if } \rho(r ) > \rho \ge \beta\rho(r)\frac{t}{t_{\rm ext}}\,,\\ \rho \frac{kt_{\rm ext}}{\mu m_p } & \mbox{otherwise}\ , , \end{array } \right.\ ] ] where the intermediate state , with @xmath14 and @xmath15 chosen to make @xmath16 continuous , is introduced to avoid numerical artifacts at the surface , and @xmath17 . the specific values of @xmath18 that we chose , their corresponding density contrasts and decay timescales , are presented in table [ tab : t ] . the associated radial density profiles are shown in figure [ fig : ds ] . note that there is a transition region which distributes the sudden drop in density over many grid zones , though the pressure is continuous ( necessarily ) and nearly constant outside the surface of the bonnor - ebert sphere . we employ the same three - dimensional , self - gravitating hydrodynamics code as in @xcite to follow the long term mode evolution . details regarding the numerical algorithm and specific validation for oscillating gas spheres ( in that case a white dwarf ) can be found in @xcite , and thus will only be briefly summarized here . the code is a second - order accurate ( in space and time ) eulerian finite - difference code , and has been demonstrated to have a low diffusivity . because the equation of state is barotropic , we use the gradient of the enthalpy instead of the pressure in the euler equation since this provides better stability in the unperturbed configuration @xcite . the poisson equation is solved via spectral methods , with boundary conditions set by a multipole expansion of the matter on the computational domain . as described in @xcite , the code was validated for this particular problem finding convergence by a resolution of @xmath19 grid zones . the initial perturbed states are computed in the linear approximation using the standard formalism of adiabatic ( which in this case is also isothermal , * ? ? ? * ) stellar oscillations @xcite . the initial conditions of the perturbed gas sphere are then given by the sum of the equilibrium state and the linear perturbation . we employ the same definition for the dimensionless mode amplitude as given in the appendix of @xcite . throughout the evolution , the amplitude of a given oscillation mode , denoted by its radial ( @xmath20 ) and angular ( @xmath21 & @xmath22 ) quantum numbers , may be estimated by the explicit integral : @xmath23 where @xmath24 is the gas velocity , @xmath25 is the mode displacement eigenfunction , @xmath26 is the mode frequency and the dagger denotes hermitian conjugation . in principle , a number of hydrodynamic mechanisms exist by which the oscillations can be damped . in the absence of a coupling to an external medium , these are dominated by nonlinear mode mode coupling , in which large scale motions excite smaller scale perturbations @xcite . however , when embedded in a dense external medium it is possible for the mode to damp by exciting motions in the exterior gas . that is , it is possible for the oscillating sphere to act as a transducer , generating sound waves in the external gas which then propagate outward resulting in a net radial energy flux and damping the oscillations . generally , this will be a strong function of the density contrast between the interior of the core and the exterior medium . if the external density has a low density , and thus little inertia , outwardly moving sound waves will contain little energy density despite their large amplitude and thus inefficiently damp the mode ( as is the case for b68 ) . similarly , if the exterior medium has a very high density the excited outgoing waves will have small amplitudes , which despite the large gas inertia will again carry away only a small amount of energy . conversely , when the density contrast is nearly unity , there will be an efficient coupling between the interior and exterior waves , resulting in a rapid flow of energy out of the cloud . this may be made explicit via the standard three - wave analysis , in which propagating wave solutions for the incident , reflected and transmitted waves on either side of the density continuity are inserted into the continuity and euler equations , which can subsequently be solved for the reflected and transmitted energy flux . in this idealization the ratio of the reflected to transmitted flux is @xmath27 , where @xmath28 is the ratio of the sonic impedances on either side of the discontinuity @xcite . as anticipated , when the density of the external gas is comparable to the surface density of the isothermal sphere we may expect efficient transmission of sound waves from the interior to the exterior ( i.e. , conversion of the oscillation into traveling waves in the exterior ) , and therefore efficient damping of the pulsations . this conclusion is also supported by a 1d analysis of the decay of a standing wave confined to a high density region ( section [ sec:1dt ] ) , for which the damping rate , @xmath29 is given by @xmath30 where @xmath31 is the sound speed in the interior and @xmath10 is the radius of the high - density region . thus , as expected lower external densities ( corresponding to smaller @xmath32 ) have lower damping rates and larger damping timescales . note that when @xmath32 is of order unity the wave decays on the sound crossing time of the dense region , as would be expected if , e.g. , we were to decompose the standing wave into traveling waves . more directly applicable to the damping of oscillating gas spheres is the case of the decay of a standing multipolar sound wave in a uniform density sphere , discussed in section [ sec : sdt ] . this is distinct from the 1d case in an important way : the oscillation wave vectors are no longer only normal to the boundary . as a consequence , the damping rate is a strong function of not only @xmath32 but also the multipole structure of the underlying oscillation . in particular , the damping rate for the @xmath33 multipole mode is proportional to @xmath34 for small @xmath32 ( see , eqs . [ eq : uniform_sphere_damping ] & [ eq : full_linear_damping ] ) , which is a very strong function of @xmath32 for even low @xmath21 ! this limiting behavior is also found in semi - analytic calculations ( section [ sec : flp ] ) in which the linear oscillation of an isothermal sphere is properly matched to outgoing sound waves in the exterior . the damping timescales ( inverse of the damping rates ) from both the uniform - sphere approximation and the full linear mode analysis are shown in figure [ fig : ta ] for the first few multipoles . all of this implies that the damping rates of oscillations of cores embedded in cold molecular regions , where the density contrast between the core and the cloud is near unity , will be much more rapid than those in cores isolated in hot , lower - density regions . , as a function of the ratio of the exterior and surface densities . for reference the corresponding damping times in @xmath35 is shown on the right - hand axis for a cloud similar to barnard 68 . the solid lines show the damping timescale predicted by a linear mode analysis of the isothermal sphere for monopole ( 011 ) , dipole ( 111 ) and quadrupole ( 122,thick ) modes ( see the appendices ) . the asymptotic behavior of the linear mode analysis is shown by the short dashed lines for each , which may be obtained by treating the oscillations of the isothermal sphere as sound waves in a uniform density sphere ( see the appendices ) . the solid circles show the damping timescales measured via numerical simulations , and are colored to match the curves in figures [ fig : ds ] & [ fig : sims ] . finally , we show a power - law fit to the measured damping timescales , which have a power - law index of @xmath36 , and is considerably flatter than we might have expected ( though these represent lower limits upon the decay timescale ) . for comparison the density contrasts observed in some well known examples are also shown , including those for barnard 68 ( @xmath37 ) , l1544 ( @xmath38 ) and the cores in the pipe nebula ( @xmath39@xmath40 ) . ] & [ fig : ta ] . the decay time for the red curve is 0.43 myr , compared to 1.05 myr for the black curve . quadratic fits for each are shown by the dashed lines . matching these up to specific models of the mode decay is complicated by ( i ) the non - linear mode dynamics , and ( ii ) understanding the energy losses due to traveling waves in the external medium ( see the appendices ) . ] however , this is not borne out by numerical simulations of high amplitude oscillations . this can readily be seen by the damping timescales measured via the numerical simulations . figure [ fig : sims ] shows the evolution of the @xmath41 , @xmath42 , @xmath43 mode discussed in @xcite . in all cases the initial amplitude was 0.25 , and the evolution of the mode well fit by a decaying exponential . oscillations on isothermal cores embedded in higher density regions did indeed damp more rapidly , as is apparent in the figure . explicit values of the decay timescale are presented in table [ tab : t ] , though as in @xcite these should be seen as lower limits only due to coupling in the artificial atmosphere . these are plotted as a function of the surface density contrast in figure [ fig : ta ] , shown by the colored filled circles . for reference , we show the approximate surface density contrasts of barnard 68 @xcite , l1544 @xcite and typical for cores in the pipe nebula @xcite . while there is a clear power - law dependence of the damping timescale upon @xmath32 , with @xmath44 this dependence is considerably weaker than that predicted by the linear analysis ( @xmath45 ) . this disparity may be understood in terms of the relative importance of the excitation of sound waves in the exterior medium to the damping of the oscillations . in particular , it is notable that the numerically measured damping timescales are always less , and for most surface density contrasts considerably so , than those implied by the linearized analysis of external sound wave excitation . this is true even with low amplitude oscillations ( e.g. , @xmath46 , for which the damping timescale is roughly a factor of 3 larger than for @xmath47 ) and thus is likely due to mode coupling in the transition region immediately outside the cloud ( seen in fig . [ fig : ds ] immediately outside @xmath48 as the region of rapidly decreasing density ) . within this region the densities are all within a single order of magnitude despite their very different asymptotic densities at infinity ( which differ by many orders of magnitude ) . as a consequence , the damping rates inferred from the isolated isothermal spheres are only a weak function of the properties of the external medium . thus , even for surface density contrasts on the order of @xmath49 , the decay timescale for the quadrupolar oscillation is still larger than @xmath50 , corresponding to many oscillations and comparable to the inferred lifetimes of globules . despite the expectations of the linear mode analysis of the embedded isothermal sphere , the damping timescale of large - amplitude oscillations of embedded bonnor - ebert spheres is only a weak function of the density of the external medium . this is a result of the dominance of nonlinear mode mode coupling in the damping of large oscillations . even in cold molecular environments , the quadrupolar oscillation discussed in @xcite and @xcite has a lifetime of at least @xmath51 , and is thus comparable to the inferred lifetimes of globules @xcite . this suggests that globules supporting large - amplitude oscillations may be common even in these environments . the presence of large - amplitude oscillations on starless cores has a number of consequences for the physical interpretation of observations of starless cores @xcite . necessarily they would imply that starless cores are stable objects , existing for many sound - crossing times . in addition , the signatures of collapse , expansion and rotation in observations of self - absorbed molecular lines ( e.g. , cs ) are degenerate with molecular line profiles produced by oscillating globules ( see , e.g. , figure 6 of * ? ? ? * ) . consequently , it may be difficult to determine the true dynamical nature of motions in a given globule or dense core from self - absorbed , molecular - line profiles alone . this suggests that studies of such profiles in dense cores may be of only statistical value in determining the general status of motions in dense core populations . we would like to thank mark birkinshaw for bringing the problem of damping starless core pulsations in dense media to our attention . we will first review some basic facts about sound waves in uniform media . beginning in subsection [ sec:1dt ] we will discuss the evolution of a one - dimensional standing sound wave as a result of the excitation of waves in the external medium . in subsection [ sec : sdt ] we discuss the application to oscillating uniform density spheres , and treat the full linearized mode analysis of isothermal spheres in section [ sec : flp ] . the governing equations are the linearized continuity and the euler equations : @xmath52 where @xmath53 and @xmath54 are the unperturbed density and sound speed , respectively , @xmath55 is the eulerian perturbation in the density and @xmath56 is the lagrangian perturbation in the velocity ( which is identical to the eulerian perturbation since the initial velocity field is assumed to vanish ) . these may be combined in the normal way to produce the wave equation @xmath57 where we have assumed that the background density is uniform . solutions will generally obey the dispersion relation @xmath58 , and may then be inserted into the euler equation to obtain the associated velocity perturbation . we will consider the decay of a sound wave traveling inside of a uniform high density region ( called the _ interior _ and denoted by sub - script @xmath59 s ) surrounded by a uniform low density region ( called the _ exterior _ and denoted by sub - script @xmath60 s ) . the interior wave oscillation is given by @xmath61 the exterior waves are outgoing , and we will focus upon the boundary condition at the @xmath62 boundary . at this point the exterior sound wave is given by @xmath63 at the interface we require ( i ) pressure equilibrium and ( ii ) continuity of the displacement and hence velocity . since we have assumed the @xmath64 and @xmath65 are constant , the first condition is simply @xmath66 . the second is trivially @xmath67 . thus @xmath68 give two complex equations from which we may determine @xmath69 and @xmath70 as functions of @xmath71 and the background fluid quantities ( note that @xmath72 and @xmath73 by the dispersion relations in each region ) . in particular , note that by taking the ratio of the two equations we find @xmath74 where @xmath75 is the ratio of the exterior and interior sonic impedances . generally this may be solved to find that @xmath76 $ ] , however we will solve this explicitly in the @xmath77 limit to illustrate how we will do this for spherical geometries later . let us begin by assuming that the damping rate is small . specifically , let us assume that @xmath78 is much larger than @xmath79 . then we may taylor expand the left - hand side of eq . ( [ eq:1dmatching ] ) around @xmath80 : @xmath81 and thus , equating real and imaginary parts @xmath82 and @xmath83 \frac{\gamma r}{c_{s , i } } = \zeta \quad\rightarrow\quad \gamma = \frac{c_{s , i}}{r } \zeta\,.\ ] ] we may in principle then insert this into eq . ( [ eq:1dmatching ] ) to obtain the relationship between @xmath71 and @xmath69 . however , we will be concerned with only the damping rate here . in the previous section we discussed the simple problem of the damping of a one - dimensional wave due to the excitation of outgoing exterior sound waves . in this section we will address the more relevant problem of the damping timescale of multipolar oscillations on a sphere . while generally we would use the linearized analysis of the oscillating isothermal sphere , here we will limit ourselves to the spherical analog of the previous section : a sound wave in a uniform density sphere surrounded by a uniform density exterior . before we can discuss the excitation of sound waves outside of an oscillating sphere we must first determine the explicit form of these waves . we do this by separating the radial and angular dependencies using spherical harmonics ( primarily because these were used in determining the mode spectrum of the isothermal sphere ) . that is , we let @xmath84 , and insert this into equation ( [ eq : rho_wave_eq ] ) , producing @xmath85 where @xmath86 . the general solutions of this equation are simply the spherical bessel functions : @xmath87 however , we must still separate the inward and outward traveling waves . this naturally occurs if we consider spherical bessel functions of the 3rd kind , which are necessarily simply linear combinations of spherical bessel functions of the 1st and 2nd kind ( @xmath88 and @xmath89 , respectively ) : @xmath90 outwardly directed waves are given by @xmath91 and inwardly directed waves are given by @xmath92 . in terms of this the density perturbation of the exterior sound waves associated with the @xmath93 multipole is @xmath94 where @xmath69 is a dimensionless wave amplitude . for @xmath95 this results in the standard spherical wave solutions , @xmath96 . the velocity perturbation may be determined via the linearized euler equation : @xmath97 where here and henceforth we have defined @xmath98 for convenience . in contrast , the interior wave must be regular at the origin , and is given by @xmath99 where @xmath100 . at the interface we again require pressure equilibrium and continuity in the radial velocity : @xmath101 and @xmath102 as before , together with @xmath103 and @xmath104 , these are sufficient to determine @xmath69 and @xmath70 . in particular , these imply @xmath105 the right - hand side may be simplified in the @xmath106 limit . since @xmath107 is typically of order unity , this implies that @xmath108 . in this regime , to leading order in small @xmath109 , @xmath110 ^ 2}\,,\ ] ] where @xmath111 . again let us begin with the ansatz that the damping rate is small for small @xmath32 . in which case @xmath112 ^ 2}\,,\ ] ] and thus @xmath113_{\omega_0 } = -\frac{l+1}{\zeta^2 z_i}\nonumber\\ & & \displaystyle \qquad\qquad\qquad\qquad\qquad\quad\rightarrow\quad \frac{j_{l-1}(z_i)}{j_{l}(z_i ) } \simeq -\frac{l+1}{\zeta^2 z_i } \quad{\rm and}\quad \omega_0 \simeq \frac{c_{s , i}}{r } \mathcal{z}_{nl}\end{aligned}\ ] ] where @xmath114 is the @xmath115 root of @xmath116 . with @xmath117 ^ 2 \simeq - \frac{(l+1)^2}{\zeta^4 \mathcal{z}_{nl}^2}\,,\ ] ] ( where we used the fact that @xmath108 ) we find @xmath118 ^ 2}\ , . \label{eq : uniform_sphere_damping}\ ] ] we note that this is quite different than the one - dimensional case . in particular , the decay rate decreases rapidly with decreasing @xmath32 , and is a strong function of @xmath21 , with higher multipoles decaying considerably more rapidly than lower multipoles . this is a consequence of our assumption that not just @xmath77 , but @xmath119 ( justified in our case ) , which implies that @xmath120 . that is , the exterior propagating sound wave necessarily knows that the geometry is converging , with higher multipoles converging more rapidly . we should also emphasize that the scaling of the damping rate , @xmath29 , with @xmath32 is dependent primarily upon the structure of the traveling sound waves in the exterior . rather , the properties of the interior perturbation are encoded in the definition of the @xmath114 , and thus the normalization . therefore , the damping rates associated with the linearized perturbations of the pressure - supported isothermal sphere ( bonnor - ebert sphere ) should be quite similar , differing only in the particular values of @xmath121 . indeed , in the next section we find this to be the case . the rather surprisingly strong scaling of the damping timescale with the density contrast motivates a more careful analysis of the damping rates of a self - gravitating , pressure supported isothermal sphere . the mode analysis of an isothermal sphere has been treated in considerable detail elsewhere ( e.g. , * ? ? ? * ; * ? ? ? * ) and thus we summarize it only briefly here . the perturbed quantities are most conveniently described by the dziembowski variables @xcite @xmath122 which may be separated into radial and angular parts , @xmath123 . in terms of these , the linearized hydrodynamic equations are given by @xmath124 \eta_2 + v \eta_3 \label{eq : dz_a}\\ & & r \frac{\partial \eta_2}{\partial r } = \sigma^2 c \eta_1 + ( 1-u)\eta_2\\ & & r \frac{\partial \eta_3}{\partial r } = ( 1-u ) \eta_3 + \eta_4\\ & & r \frac{\partial \eta_4}{\partial r } = u v \eta_2 + \left [ l(l+1 ) - uv \right ] \eta_3 - u \eta_4\ , , \label{eq : dz_eqs}\end{aligned}\ ] ] where @xmath125 are functions of the background equilibrium configuration . these are solved subject to a normalization condition @xmath126 and the inner boundary conditions : @xmath127 arising from regularity at the origin . a third boundary condition is a result from the continuity of the gravitational potential at the sphere s surface , @xmath128 if the exterior density vanishes , as was assumed in @xcite , the final boundary condition is given by requiring that the lagrangian pressure perturbation , @xmath129 however , we now properly match this onto the outgoing multipolar wave solutions described in section [ sec : sdt ] . namely , @xmath130 subject to @xmath131 therefore , @xmath132 as before we assume that the damping rate is small , set @xmath133 and employ the boundary condition @xmath134_{r = r , \omega=\omega_0}\,.\ ] ] the damping rate is then estimated by @xmath135_{r = r , \omega=\omega_0 } \bigg/ \left . \frac{\partial(\eta_2-\eta_3)}{\partial\omega } \right|_{r = r}\,,\ ] ] where we used the fact that @xmath136 by the normalization condition . in practice @xmath137 is evaluated by solving the eigenvalue problem for the @xmath138 and @xmath139 using @xmath140_{r = r , \omega=\omega_0 } \bigg/ \left . \frac{\partial(\eta_2-\eta_3)}{\partial\omega } \right|_{r = r}\,,\ ] ] and setting @xmath141 this amounts to finding @xmath137 holding the other boundary conditions constant . the results of this procedure are explicitly shown in figure [ fig : ta ] . of particular note is that the intuition obtained from the analysis of sound waves in a uniform density sphere is borne out in the low @xmath142 limit , for which @xmath143

in a previous paper we demonstrated that non - radial hydrodynamic oscillations of a thermally - supported ( bonnor - ebert ) sphere embedded in a low - density , high - temperature medium persist for many periods . the predicted column density variations and molecular spectral line profiles are similar to those observed in the bok globule b68 suggesting that the motions in some starless cores may be oscillating perturbations on a thermally supported equilibrium structure . such oscillations can produce molecular line maps which mimic rotation , collapse or expansion , and thus could make determining the dynamical state from such observations alone difficult . however , while b68 is embedded in a very hot , low - density medium , many starless cores are not , having interior / exterior density contrasts closer to unity . in this paper we investigate the oscillation damping rate as a function of the exterior density . for concreteness we use the same interior model employed in broderick et al . ( 2007 ) , with varying models for the exterior gas . we also develop a simple analytical formalism , based upon the linear perturbation analysis of the oscillations , which predicts the contribution to the damping rates due to the excitation of sound waves in the external medium . we find that the damping rate of oscillations on globules in dense molecular environments is always many periods , corresponding to hundreds of thousands of years , and persisting over the inferred lifetimes of the globules .

0804.1790

one of the surprising properties of extrasolar planets is their distributions around their host stars . since many jovian planets have been found in the vicinity ( far inside the snow line ) of their host stars , numbers of theoretical models have been studied to explain inward planetary migration . recently understanding of planetary migration mechanisms has rapidly progressed through observations of the rossiter - mclaughlin effect ( hereafter the rm effect : @xcite , @xcite ) in transiting exoplanetary systems . the rm effect is an apparent radial velocity anomaly during planetary transits . by measuring this effect , one can learn the sky - projected angle between the stellar spin axis and the planetary orbital axis , denoted by @xmath2 ( see @xcite for theoretical discussion ) . so far , spin - orbit alignment angles of about 15 transiting planets have been measured ( @xcite , and references therein ) . among those rm targets , significant spin - orbit misalignments have been reported for 3 transiting planets : xo-3b , hd80606b (; ) , and wasp-14b @xcite . these misaligned planets are considered to have migrated through planet - planet scattering processes ( e.g. , @xcite ) or kozai cycles with tidal evolution ( e.g. , @xcite ) , rather than the standard type ii migration ( e.g. , @xcite ) . the existence of such misaligned planets has demonstrated validity of the planetary migration models considering planet - planet scattering or the kozai migration . on the other hand , such planetary migration models also predict significant populations of `` retrograde '' planets . thus discoveries of retrograde planets would be an important milestone for confirming the predictions of recent planetary migration models , and intrinsically interesting . in this letter , we report the first evidence of such a retrograde planet in the transiting exoplanetary system hat - p-7 . section 2 summarizes the target and our subaru observations , and section 3 describes the analysis procedures for the rm effect . section 4 presents results and discussion for the derived system parameters . finally , section 5 summarizes the main findings of this letter . hat - p-7 is an f6 star at a distance of 320 pc hosting a very hot jupiter ( @xcite ; hereafter p08 ) . among transiting - planet host stars , f type stars are interesting rm targets because these stars often have a large stellar rotational velocity , which facilitates measurements of the rm effect . however , the rotational velocity of hat - p-7 is @xmath3 km s@xmath4 ( p08 ) , which is unusually slower than expected for an f6 type star . nevertheless , this system is favorable for the rm observations , since the star is relatively bright ( @xmath5 ) and the expected amplitude of the rm effect ( @xmath6 m s@xmath4 ) is sufficiently detactable with the subaru telescope . we observed a full transit of hat - p-7b with the high dispersion spectrograph ( hds : @xcite ) on the subaru 8.2 m telescope on ut 2008 may 30 . we employed the standard i2a set - up of the hds , covering the wavelength range 4940 @xmath7 6180 and used the iodine gas absorption cell for radial velocity measurements . the slit width of @xmath8 yielded a spectral resolution of @xmath960000 . the seeing on that night was around @xmath8 . the exposure time for radial velocity measurements was 6 - 8 minutes , yielding a typical signal - to - noise ratio ( snr ) of approximately 120 per pixel . we processed the observed frames with standard iraf procedures and extracted one - dimensional spectra . we computed relative radial velocities following the algorithm of @xcite and @xcite , as described in @xcite . we estimated the internal error of each radial velocity as the scatter in the radial - velocity solutions among @xmath94 segments of the spectrum . the typical internal error was @xmath95 m s@xmath4 . the radial velocities and uncertainties are summarized in table 1 . we model the rm effect of hat - p-7 following the procedure of @xcite , as described in @xcite and hirano et al . in prep . we start with a synthetic template spectrum , which matches for the stellar property of hat - p-7 described in p08 , using a synthetic model by . to model the disk - integrated spectrum of hat - p-7 , we apply a rotational broadening kernel of @xmath10 km s@xmath4 and assume limb - darkening parameters for the spectroscopic band as @xmath11 and @xmath12 , based on a model by . we then subtract a scaled copy of the original unbroadened spectrum with a velocity shift to simulate spectra during a transit . we create numbers of such simulated spectra using different values of the scaling factor @xmath13 and the velocity shift @xmath14 , and compute the apparent radial velocity of each spectrum . we thereby determine an empirical formula that describes the radial velocity anomaly @xmath15 in hat - p-7 due to the rm effect , and find @xmath16.\ ] ] for radial velocity fitting , including the keplerian motion and the rm effect , we adopt stellar and planetary parameters based on p08 as follows ; the stellar mass @xmath17 [ @xmath18 , the stellar radius @xmath19 [ @xmath20 , the radius ratio @xmath21 , the orbital inclination @xmath22 , and the semi - major axis in units of the stellar radius @xmath23 . we assess possible systematic errors due to uncertainties in the fixed parameters in section 4 . we also include a stellar jitter of @xmath24 m s@xmath4 for the p08 keck data as systematic errors of radial velocities by quadrature sum . it enforces the ratio of @xmath25 contribution and the degree of freedom for the keck data to be unity . we do not include additional radial velocity errors for the subaru data , because we find the ratio for the subaru dataset is already smaller than unity ( as described in section 4 ) . in addition , we adopt the transit ephemeris @xmath26 [ hjd ] and the orbital period @xmath27 days based on p08 . note that this ephemeris has an uncertainty of 3 minutes for the observed transit ; however the uncertainty is well within our time - resolution ( exposure time of 6 - 8 minutes and readout time of 1 minute ) and is negligible for our purpose . the adopted parameters above are summarized in table 2 . our model has 3 free parameters describing the hat - p-7 system : the radial velocity semi - amplitude @xmath28 , the sky - projected stellar rotational velocity @xmath29 , and the sky - projected angle between the stellar spin axis and the planetary orbital axis @xmath2 . we fix the eccentricity @xmath30 to zero , and the argument of periastron @xmath31 is not considered . finally we add two offset velocity parameters for respective radial velocity datasets ( @xmath32 : our subaru dataset , @xmath33 : the keck dataset in p08 ) . we then calculate the @xmath25 statistic ( hereafter `` main - case '' ) @xmath34 ^ 2,\end{aligned}\ ] ] where @xmath35 and @xmath36 are observed radial velocities and uncertainties , and @xmath37 are radial velocity values calculated based on a keplerian motion and on the empirical rm formula given above . we determine optimal orbital parameters by minimizing the @xmath25 statistic using the amoeba algorithm @xcite . we estimate 1@xmath38 uncertainty of each free parameter based on the criterion @xmath39 when a parameter is stepped away from the best - fit value and the other parameters are re - optimized . we also fit the radial velocities using another statistic function for reference ( hereafter `` test - case '' ) , @xmath34 ^ 2 + \left [ \frac{v \sin i_s - 3.8}{0.5 } \right]^2.\end{aligned}\ ] ] the last term is _ a priori _ constraint for @xmath29 to match the independent spectroscopic analysis by p08 . figure 1 shows observed radial velocities and the best - fit model curve for the main - case . figure 2 illustrates the rm effect of hat - p-7b with the best - fit model and also shows a comparison with the case of @xmath40 and @xmath3 km s@xmath4 model . the upper panel of figure 3 plots a @xmath25 contour in ( @xmath41 ) space . as a result , we find the key parameter @xmath42 , implying a retrograde orbit of hat - p-7b . the stellar rotational velocity is @xmath43 km s@xmath4 , which is marginally consistent with the p08 spectroscopic result ( @xmath44 km s@xmath4 ) . residuals from the best - fit model indicate rms of 4.14 m s@xmath4 for the subaru dataset and 4.09 m s@xmath4 for the p08 keck dataset . the rms of the subaru residuals is well within our internal radial velocity errors , and that of the keck residuals is in good agreement with the assumed jitter level of 3.8 m s@xmath4 . one may wonder that a smaller @xmath29 allowed in the main - case would weaken the detection - significance of the rm effect . however , since @xmath45 km s@xmath4 is excluded by @xmath46 , there is very little chance that a true @xmath29 is actually nearly zero and a spin - orbit alignment angle @xmath2 is very small . the lower panel of figure 3 plots a similar @xmath25 contour but for the test - case . in this case , we find @xmath47 and @xmath48 km s@xmath4 . thus our two results ( main and test cases ) are well consistent with each other . in addition , we test the fitting with the eccentricity @xmath30 and the argument of periastron @xmath31 as free parameters . as a result , we do not find any significant ( nonzero ) eccentricity for this planet . the best - fit parameters and uncertainties are summarized in table 3 . in the above analyses , we fixed several parameters as summarized in table 2 , which were based on p08 and . in order to estimate the level of possible systematic errors , we retry the fitting for following four cases ; ( 1 ) @xmath49 ( corresponding to 1@xmath38 lower limit of the impact parameter in p08 ) ; ( 2 ) @xmath50 ( corresponding to 1@xmath38 upper limit of the impact parameter in p08 ) ; ( 3 ) @xmath51 ( a greater limb - darkening case ) ; and ( 4 ) @xmath52 ( a smaller limb - darkening case ) . consequently , we find that respective results for @xmath2 and @xmath29 are ; ( 1 ) @xmath53 and @xmath54 km s@xmath4 ; ( 2 ) @xmath55 and @xmath56 km s@xmath4 ; ( 3 ) @xmath57 and @xmath58 km s@xmath4 ; and ( 4 ) @xmath59 and @xmath60 km s@xmath4 . thus @xmath61 @xmath62 and @xmath63 @xmath64 km s@xmath4 can be still probable if the uncertainties for fixed parameters ( especially for the impact parameter ) are taken into account . these systematic errors would be significantly reduced when the kepler photometric data for hat - p-7 are available @xcite . the derived value of @xmath2 seems to indicate a retrograde orbit by itself . however , since the true spin - orbit angle @xmath65 also depends on the inclination of the stellar spin axis @xmath66 , @xmath65 is not necessarily greater than @xmath67 ( corresponding to a retrograde orbit ) even if @xmath0 . thus one might wonder whether the planet hat - p-7b is statistically in a retrograde orbit . we can roughly estimate the probability using the relation of spherical geometry , @xmath68 note that @xmath66 ranges from @xmath69 to @xmath70 . we compute the true spin - orbit angle @xmath65 of hat - p-7b by substituting the observed values of @xmath71 and @xmath2 into the relation . we adopt @xmath22 ( p08 ) and test three representative cases for @xmath2 ( @xmath72 ) . assuming an uniform distribution for @xmath73 within the range of value , the probabilities of a retrograde orbit ( @xmath74 ) are 99.85% ( @xmath75 ) , 99.70% ( @xmath76 ) , and 91.65% ( @xmath77 ) , respectively . we note that @xmath65 is always larger than @xmath78 ( the adopted value of @xmath71 , in the case of @xmath79 ) . those estimates favor a retrograde orbit of hat - p-7b . on the other hand , it is important to point out that the stellar rotational velocity @xmath3 km s@xmath4 determined by the spectroscopic analysis ( p08 ) is exceptionally slow for an f6 star . for example , hat - p-2 and tres-4 , which are other known planetary host stars with similar spectral type , have larger stellar rotational velocities : @xmath80 km s@xmath4 ( hat - p-2 , @xcite and also confirmed from the rm effect by @xcite ) and @xmath81 km s@xmath4 ( tres-4 , @xcite and also confirmed from the rm effect by narita et al . in prep . ) . the small @xmath29 may suggest a smaller inclination angle of the stellar spin axis . in that case , it is highly possible that the planet is in a nearly polar retrograde orbit . we note that too small @xmath66 can be constrained by the facts that a faster stellar rotation of hat - p-7 than 500 km s@xmath4 would be physically unlikely due to stellar break - up , and that a faster rotation than 100 km s@xmath4 would be empirically unlikely for a f6 star @xcite . translating the constraints into @xmath73 , we find a probability of such unrealistic cases is only @xmath90.03% , which has very little impact on the probability estimations for a retrograde orbit . in any case , it would be important to directly constrain @xmath66 by other observational methods ( e.g. , @xcite or asteroseismology with the kepler ) in order to estimate a true spin - orbit angle of hat - p-7b . we previously experienced a false positive of a spin - orbit misalignment in hd17156b due to lower precision radial velocity data ( @xcite ) . the problem for the hd17156 case @xcite was that radial velocity uncertainties were comparable with predicted rm amplitude and also due to a poor number of radial velocity samples . based on the lesson , we estimate the significance of our rm detection using the equation ( 26 ) in @xcite . as a result , the snr of our rm detection is over 10 , and thus we conclude to have obtained radial velocities of a sufficient number and precision to model the rm effect of hat - p-7b . nevertheless , we finally note that we should care about possible systematic errors in @xmath2 . since the rm amplitude of hat - p-7b is only @xmath82 m s@xmath4 , any systematic shift as much as several m s@xmath4 due to stellar jitter or other reasons at ingress or egress phase would cause a large systematic difference in @xmath2 . thus further radial velocity measurements for this interesting system are desired to confirm a retrograde orbit of hat - p-7b more decisively . we observed a full transit of hat - p-7b with the subaru 8.2 m telescope on ut 2008 may 30 , and measured the rm effect of this planet . based on the rm modeling , we discovered the first evidence of a retrograde orbit of hat - p-7b . this is the first discovery of a retrograde extrasolar planet . the existence of such planets have been indeed predicted in some recent planetary migration models considering planet - planet scattering and/or the kozai migration ( e.g. , @xcite ; @xcite ) . in addition , it is interesting to point out that hat - p-7b is the first spin - orbit misaligned planet having no significant eccentricity . this discovery may suggest that other hot jupiters in circular orbits also have significant spin - orbit misalignments or even retrograde orbits . thus further rm observations for transiting planets , including not only eccentric or binary system planets but also close - in circular planets , would be encouraged in order to understand populations of aligned / misaligned / retrograde planets . we acknowledge the invaluable support for our subaru observations by akito tajitsu , a support scientist for the subaru hds . we are grateful to yasushi suto , ed turner , wako aoki , and toru yamada for helpful discussions on this study ; josh winn and his colleagues for sharing their information in advance of publication . this paper is based on data collected at subaru telescope , which is operated by the national astronomical observatory of japan . the data analysis was in part carried out on common use data analysis computer system at the astronomy data center , adc , of the national astronomical observatory of japan . is supported by a japan society for promotion of science ( jsps ) fellowship for research ( pd : 20 - 8141 ) . we wish to acknowledge the very significant cultural role and reverence that the summit of mauna kea has always had within the indigenous hawaiian community . l|cc parameter & value & source + @xmath83 [ @xmath18 & @xmath84 & p08 + @xmath85 [ @xmath20 & @xmath86 & p08 + @xmath87 & @xmath88 & p08 + @xmath71 [ @xmath89 & @xmath90 & p08 + @xmath91 & @xmath92 & p08 + @xmath93 & @xmath94 & + @xmath95 & @xmath96 & + @xmath97 [ hjd ] & @xmath98 & p08 + @xmath99 [ days ] & @xmath100 & p08 + l|cc|cc & & + parameter & value & uncertainty & value & uncertainty + @xmath28 [ m s@xmath4 ] & 212.6 & @xmath102 & 213.3 & @xmath102 + @xmath29 [ km s@xmath4]@xmath103 & 2.3 & @xmath104 & 3.1 & @xmath105 + @xmath2 [ @xmath89@xmath103 & -132.6 & @xmath106 & -122.5 & @xmath107 + @xmath32 [ m s@xmath4 ] & -14.7 & @xmath108 & -16.6 & @xmath109 + rms ( subaru ) [ m s@xmath4 ] & 4.14 & & 4.32 & + @xmath33 [ m s@xmath4 ] & -37.4 & @xmath108 & -37.5 & @xmath108 + rms ( keck ) [ m s@xmath4 ] & 4.09 & & 4.09 & +

we present the first evidence of a retrograde orbit of the transiting exoplanet hat - p-7b . the discovery is based on a measurement of the rossiter - mclaughlin effect with the subaru hds during a transit of hat - p-7b , which occurred on ut 2008 may 30 . our best - fit model shows that the spin - orbit alignment angle of this planet is @xmath0 @xmath1 . the existence of such a retrograde planet have been predicted by recent planetary migration models considering planet - planet scattering processes or the kozai migration . our finding provides an important milestone that supports such dynamic migration theories .

0908.1673

mixing of states ( fields ) is a well - known phenome- non existing in the systems of neutrinos @xcite , quarks @xcite and hadrons . in hadron systems the mixing effects are essential not only for @xmath8- and @xmath9-mesons but also for the broad overlapping resonances . as for theoretical description of mixing phenomena , a general tendency with time and development of experiment consists in transition from a simplified quantum - mechanical description to the quantum field theory methods ( see e.g. review @xcite , more recent papers @xcite and references therein ) . mixing of fermion fields has some specifics as compared with boson case . firstly , there exists @xmath10-matrix structure in a propagator . secondly , fermion and antifermion have the opposite @xmath11-parity , so fermion propagator contains contributions of different parities . as a result , besides a standard mixing of fields with the same quantum numbers , for fermions there exists a mixing of fields with opposite parities ( opf - mixing ) , even if the parity is conserved in lagrangian . such a possibility for fermion mixing has been noted in @xcite . in this paper we study this effect in detail and apply it to the baryon resonances production in @xmath3 reaction . in section 2 we consider a standard mixing of fermion fields of the same parity . following to @xcite we use the off - shell projection basis to solve the dyson schwinger equation , it simplilies all manipulations with @xmath10-matrices and , moreover , clarifies the meaning of formulas . the use of this basis leads to separation of @xmath10-matrix structure , so in standard case we come to studying of a mixing matrix , which is very similar to boson mixing matrix . in section 3 we derive a general form of matrix dressed propagator with accounting of the opf - mixing . in contrast to standard case the obtained propagator contains @xmath12 terms , even if parity is conserved in vertexes . section 4 is devoted to more detailed studying of considered opf - mixing in application to production of resonances @xmath13 in @xmath3 scattering . first estimates demonstrates that the considered mixing generates marked effects in @xmath3 partial waves , changing a typical resonance curve . comparison of the obtained multichannel hadron amplitudes with @xmath14-matrix parameterization shows that our amplitudes may be considered as a specific variant of analytical @xmath14-matrix . in section 5 we consider opf - mixing for case of two vector - spinor rarita - schwinger fields @xmath15 , describing spin-@xmath16 particles , and apply the obtained hadron amplitudes for descriptions of @xmath3 partial waves @xmath4 and @xmath5 . conclusion contains discussion of results . in application there are collected some details of calculations , concerning the production of spin-@xmath16 resonances . let us start from the standard picture when the mixing fermions have the same quantum numbers . to obtain the dressed fermion propagator @xmath17 one should perform the dyson summation or , equivalently , to solve the dyson schwinger equation : @xmath18 where @xmath19 is a free propagator and @xmath20 is a self - energy : @xmath21 we will use the off - shell projection operators @xmath22 : @xmath23 where @xmath24 is energy in the rest frame . main properties of projection operators are : @xmath25 @xmath26 let us rewrite the equation expanding all elements in the basis of projection operators : @xmath27 where we have introduced the notations : @xmath28 in this basis the dyson schwinger equation is reduced to equations on scalar functions : @xmath29 or @xmath30 the solution of for dressed propagator looks like : @xmath31 where @xmath32 , @xmath33 are commonly used components of the self - energy . the coefficients in the projection basis have the obvious property : @xmath34 when we have two fermion fields @xmath35 , the including of interaction leads also to mixing of these fields . in this case the dyson schwinger equation acquire matrix indices : @xmath36 therefore one can use the same equation assuming all coefficients to be matrices . the simplest variant is when the fermion fields @xmath35 have the same quantum numbers and the parity is conserved in the lagrangian . in this case the inverse propagator following has the form : @xmath37 & = \mathcal{p}_{1 } \begin{pmatrix } w - m_1-\sigma^{1}_{11 } & -\sigma^{1}_{12}\\ -\sigma^{1}_{21 } & w - m_2-\sigma^{1}_{22 } \end{pmatrix } + \\[3 mm ] & + \mathcal{p}_{2 } \begin{pmatrix } -w - m_1-\sigma^{2}_{11 } & -\sigma^{2}_{12}\\ -\sigma^{2}_{21 } & -w - m_2-\sigma^{2}_{22 } \end{pmatrix}. \end{split}\ ] ] the matrix coefficients as before have the symmetry property @xmath38 . to obtain the matrix dressed propagator @xmath17 one should reverse the matrix coefficients in projection basis : @xmath39 & = \mathcal{p}_{1 } \begin{pmatrix } \dfrac{w - m_2-\sigma^{1}_{22}}{\delta_{1 } } & -\dfrac{\sigma^{1}_{12}}{\delta_{1}}\\[4 mm ] -\dfrac{\sigma^{1}_{21}}{\delta_{1 } } & \dfrac{w - m_1-\sigma^{1}_{11}}{\delta_{1 } } \end{pmatrix } + \\[3 mm ] & + \mathcal{p}_2 \begin{pmatrix } \dfrac{-w - m_2-\sigma^{2}_{22}}{\delta_{2 } } & -\dfrac{\sigma^{2}_{12}}{\delta_{2}}\\[4 mm ] -\dfrac{\sigma^{2}_{21}}{\delta_{2 } } & \dfrac{-w - m_1-\sigma^{2}_{11}}{\delta_{2 } } \end{pmatrix } , \end{split}\ ] ] where @xmath40 we see that with use of projection basis the problem of fermion mixing is reduced to studying of the same mixing matrix as for bosons besides the obvious replacement @xmath41 . let us consider the joint dressing of two fermion fields of opposite parities provided that the parity is conserved in a vertex . in this case the diagonal transition loops @xmath42 contain only @xmath43 and @xmath44 matrices , while the off - diagonal ones @xmath45 must contain @xmath46 . projection basis should be supplemented by elements containing @xmath46 , it is convenient to choose the @xmath10-matrix basis as : @xmath47 in this case the @xmath10-matrix decomposition has four terms : @xmath48 where the coefficients @xmath49 are matrices and have the obvious symmetry properties : @xmath50 inverse propagator in this basis looks as : @xmath51 & + \mathcal{p}_2 \begin{pmatrix } -w - m_1-\sigma^{2}_{11 } & 0\\ 0 & -w - m_2-\sigma^{2}_{22 } \end{pmatrix } + \\[3 mm ] & + \mathcal{p}_3 \begin{pmatrix } 0 & -\sigma^{3}_{12}\\ -\sigma^{3}_{21 } & 0 \end{pmatrix } + \mathcal{p}_4 \begin{pmatrix } 0 & -\sigma^{4}_{12}\\ -\sigma^{4}_{21 } & 0 \end{pmatrix } , \end{split}\ ] ] where the indexes @xmath52 in the self - energy @xmath53 numerate dressing fermion fields and the indexes @xmath54 are refered to the @xmath10-matrix decomposition . elements of the basis have simple multiplicative properties ( see table [ multiplic ] ) , so reversing of present no special problems @xcite . .[multiplic ] multiplicative properties of elements of basis . [ cols="^,^,^,^,^",options="header " , ] reversing of gives the matrix dressed propagator of the form : @xmath55 + & \mathcal{p}_2 \begin{pmatrix } \dfrac{w - m_2-\sigma^{1}_{22}}{\delta_2 } & 0\\ 0 & \dfrac{w - m_1-\sigma^{1}_{11}}{\delta_1 } \end{pmatrix } + \\[3 mm ] + & \mathcal{p}_3 \begin{pmatrix } 0 & \dfrac{\sigma^{3}_{12}}{\delta_1}\\ \dfrac{\sigma^{3}_{21}}{\delta_2 } & 0 \end{pmatrix}+ \mathcal{p}_4 \begin{pmatrix } 0 & \dfrac{\sigma^{4}_{12}}{\delta_2}\\ \dfrac{\sigma^{4}_{21}}{\delta_1 } & 0 \end{pmatrix}. \end{split}\ ] ] here @xmath56 the propagator can be compared with the standard case of mixing ( fermion fields of the same parity ) . as for possible applications of considered effect to description of baryon resonances , this is , first of all , @xmath3 scattering , where the high accuracy data exist and detailed partial wave analysis has been performed @xcite . let us consider an effect of opf - mixing on the production of baryon resonances of spin - parity @xmath58 and isospin @xmath59 in @xmath3-collisions . simplest effective lagrangians have the form : @xmath60 in @xmath61-channel case , the scattering amplitude is a matrix of dimension @xmath61 : @xmath62 where @xmath63 and @xmath64 are four - component spinors , corresponding to final and initial nucleon , and @xmath65 is matrix of the same dimension @xmath61 consisting of the propagator and coupling constants . in the two - channel approximation ( @xmath3 and @xmath66 channel ) matrix @xmath65 is of the form : @xmath67 and generalization for @xmath61 channels and @xmath68 mixed states is obvious . here @xmath69 is dressed propagator and we have introduced the short notations for coupling constants : @xmath70 , @xmath71 . after some algebra the matrix @xmath65 turns into into the standard form @xmath72 where @xmath73 and @xmath74 are dimension 2 matrices . note that the @xmath12 matrix has been disappeared after multiplication in , since parity is not violated . after it we obtain from the two - channel @xmath75- and @xmath76- partial waves . @xmath75-waves amplitudes ( produced resonances have @xmath77 ) in standard notations have the form : @xmath78,\\ f_{s,+}(\pi n \rightarrow \eta n ) & = \frac{\sqrt{(e_{1}+m_n)(e_{2}+m_n)}}{8\pi w \delta_{2 } } \big [ g_{1,\pi}g_{1,\eta } ( w - m_{2}- \sigma^{1}_{22 } ) -\\ & - g_{2,\pi}g_{2,\eta}(-w - m_{1}-\sigma^{2}_{11 } ) -\imath g_{2,\eta } g_{1,\pi}\sigma^{3}_{21}- \imath g_{1,\eta } g_{2,\pi}\sigma^{4}_{12}\big ] , \\ f_{s,+}(\eta n \rightarrow \eta n ) & = \frac{(e_{2}+m_n)}{8\pi w \delta_{2 } } \big [ g^{2}_{1,\eta}(w - m_{2}-\sigma^{1}_{22 } ) -g^{2}_{2,\eta}(-w - m_{1}-\sigma^{2}_{11})-\\ & - \imath g_{1,\eta}g_{2,\eta}(\sigma^{3}_{21}+\sigma^{4}_{12 } ) \big],\\ \delta_{2}&=\big(-w - m_1-\sigma^{2}_{11}\big ) \big(w - m_2-\sigma^{1}_{22}\big)-\sigma^{4}_{12}\sigma^{3}_{21 } , \end{split}\ ] ] where @xmath79 and @xmath80 are nucleon energy in the c.m.s . for @xmath3 and @xmath81 respectively . for comparison , we write down the amplitude @xmath82 in a tree approximation : @xmath83.\ ] ] simultaneous calculation of @xmath76-wave amplitudes ( @xmath84 ) gives : @xmath85,\\ f_{p,-}(\pi n \rightarrow \eta n ) & = -\frac{\sqrt{(e_{1}-m_n)(e_{2}-m_n)}}{8\pi w \delta_{1 } } \big [ g_{1,\pi}g_{1,\eta}(-w - m_{2}- \sigma^{2}_{22 } ) -\\ & - g_{2,\pi}g_{2,\eta}(w - m_{1}-\sigma^{1}_{11 } ) - \imath g_{2,\eta } g_{1,\pi}\sigma^{4}_{21 } -\imath g_{1,\eta } g_{2,\pi}\sigma^{3}_{12 } \big ] , \\ f_{p,-}(\eta n \rightarrow \eta n ) & = -\frac{(e_{2}-m_n)}{8\pi w \delta_{1 } } \big [ g^{2}_{1,\eta}(-w - m_{2}w-\sigma^{2}_{22 } ) -g^{2}_{2,\eta}(w - m_{1}-\sigma^{1}_{11})- \\ & - \imath g_{1,\eta}g_{2,\eta}(\sigma^{4}_{21}+\sigma^{3}_{12 } ) \big],\\ \delta_{1}&=\big(w - m_1-\sigma^{1}_{11}\big)\big(-w+m_2-\sigma^{2}_{22}\big)-\sigma^{3}_{12}\sigma^{4}_{21}. \end{split}\ ] ] in tree approximation : @xmath86.\ ] ] one could convince oneself that the constructed partial amplitudes satisfy the multi - channel unitary condition : @xmath87 where @xmath88 is the c.m.s . spatial momentum of particles in @xmath89-th intermediate states . the self - energy ( before renormalization ) is expressed through the components of the standard loop functions @xmath90 and @xmath91 : @xmath92 where function @xmath93 corresponding , for example , @xmath3 intermediate state has the form : @xmath94 it is convenient to calculate first @xmath32 and @xmath33 and then pass to the projections @xmath95 . so , we calculate discontinuities using landau cutkosky rule : @xmath96 then restore functions @xmath97 and @xmath98 through dispersion relation , and finally calculate @xmath95 : @xmath99 let us write down the imaginary parts of @xmath95 : @xmath100 where @xmath101 is momentum of pion in the c.m.s . recall that decomposition coefficients in the projection basis are related with each other by the substitution @xmath102 . so , to renormalize the self - energy , it is sufficient to define an exact form of @xmath103 and @xmath104 , then the components @xmath105 , @xmath106 are fixed by symmetry . we will use the on - mass - subtraction method of renormalization of resonance contribution @xcite . subtraction conditions for the self - energy included in the @xmath75-wave amplitudes have the form see . ] : @xmath107 after it the @xmath76-wave amplitudes are determined by replacing @xmath102 , as it was mention above . , is a consequence of the symmetry properties of coefficients in the projection basis : @xmath108 , @xmath109 . ] recall also the relationships between coupling constants and decay widths in the absence of mixing : @xmath110 the usual definition of the @xmath14-matrix is : @xmath111 where @xmath112 is matrix of partial amplitudes , @xmath76 is diagonal matrix consisting of c.m.s . momenta : @xmath113 @xmath14-matrix representation by construction satisfies the unitary condition . usually , the @xmath14-matrix represents a set of poles and , possibly , some smooth contributions . another variant is the analytical @xmath14-matrix ( for example , @xcite ) @xmath114 the presentation differs from the standard @xmath14-matrix by the presence of a matrix @xmath115 consisting of loops , whose imaginary part is equal to the matrix @xmath76 . it is convenient to rewrite in terms of inverse matrix : @xmath116 it turns out that the our partial amplitudes , can be represented in the form , . as an example consider the two - channel @xmath75-wave amplitudes @xmath117 and use the self - energy in form of , without subtraction polynomials . calculating the inverse matrix of the amplitudes we find that , in accordance with , it consists of a loop matrix and pole matrix @xmath118 our amplitudes , lead to a pole contributions of the form : @xmath119 in resonance phenomenology @xmath14-matrix contains a set of poles , corresponding to bare states . the main feature of our @xmath14-matrix is the presence of poles both with positive and negative energy . if the self - energy in addition to contains the subtraction polynomials , it leads only to redefinition of the poles positions in @xmath14-matrix ( i.e. @xmath14-matrix masses @xmath120 ) . we can see that our multi - channel amplitudes , can be reduced to some specific version of the analytical @xmath14-matrix parametrization . let us use our amplitudes , to calculate @xmath3 partial @xmath75- and @xmath76-waves , where baryons @xmath121 can be produced . we are interested here only in estimates of the observed effects , so we restrict ourselves by the single - channel approach and fix the parameters ( masses and coupling constants ) from rough correspondence to parameters of the observed baryon resonances @xmath59 @xmath122 for estimates we used the relations of the widths and coupling constants in the absence of mixing . the results of calculations of @xmath3 partial waves are shown at figs . [ ests ] , [ estp ] . the results of calculations of @xmath123 @xmath75-wave partial wave . solid lines correspond to the real and imaginary parts of our partial amplitude , in the single - channel approach with the parameters . dashed lines correspond to our amplitudes , neglecting the mixing effect : @xmath124 . all variants of amplitudes satisfy the single - channel unitary condition @xmath125.,width=302 ] the real and imaginary parts of @xmath123 partial @xmath76-wave . notations are the same as in fig . [ ests ] . for @xmath76-wave the solid and dashed lines coincide with each other.,width=302 ] it turns out that the discussed opf - mixing leads to noticeable effects only in @xmath75-wave , while its influence in @xmath76-wave is much less and does not seen at graphics . this feature is explained by the values of the coupling constants in @xmath126 and may be seen at qualitative level from the tree amplitudes , . since we have normalized the coupling constants on the resonance width , inequality between the coupling constants is a consequence of the inequality between the @xmath75- and @xmath76-wave phase volumes . we see that the discussed mixing effect generate the ( unitary ) interference picture `` resonance + background '' in the @xmath75-wave . in this case the @xmath75-wave background contribution originates from the @xmath76-wave resonance and gives the negative contribution to @xmath75-wave phase shift . this fact can be seen from fig . [ ests ] and from eq . . [ gwu ] demonstrates the results of partial wave analysis @xcite for lowest @xmath3 amplitudes with isospin @xmath59 . the discussed effect leads to hard correlation between pair of partial waves . from physical point of view the most interesting is the pair of waves @xmath127 ; recall that in the @xmath128 sector there exist up to now the problems of physical interpretation of the observed states and their correspondence with quark models , see e.g. discussions in @xcite . but this pair of partial waves is not the simplest place for identification of the discussed opf - mixing effect . the reasons are the old problem with roper resonance ( non - standard form of @xmath129 state ) and the existence of several states in @xmath130 channel . but if to look at the partial waves @xmath131 , where resonances @xmath132 are produced , here we observe the more evident situation , which is qualitatively consistent with our expectations , shown at figs . [ ests ] , [ estp ] . namely : in the @xmath133-wave we see a single resonance , whereas in the @xmath76-wave there is a visible interference of resonance with a background . moreover , in accordance with our expectations for interference picture , the background in the @xmath76-wave is evidently negative see fig . so this pair of partial waves @xmath131 looks as a most suitable place for identification of the discussed mixing effect . the results of partial wave analysis @xcite for @xmath123 scattering amplitudes with isospin @xmath59 ( current solution ) . partial waves satisfy the unitary condition @xmath134.,width=491 ] the above discussion was devoted to mixing of two dirac fields of opposite parities , the same effect arises for vector - spinor fields @xmath15 , which describe the spin-3/2 particles . we want to obtain the hadron partial amplitudes , which take into account the discussed effect , and to use them for description of results of @xmath3 partial wave analysis . the details of calculations of the spin-3/2 baryons production are given in the appendix [ ap2 ] . here we present only the results of calculations : the hadron partial amplitudes in two - channel ( @xmath3 , @xmath66 ) approach ( compare them with spin-1/2 case , ) . @xmath76-wave amplitudes ( @xmath135 ) have the form : @xmath136,\\ f_{p,+}(\pi n \rightarrow \eta n ) & = \abs{\mathbf{p}_\pi}\abs{\mathbf{p}_\eta } \frac{\sqrt{(e_{1}+m)(e_{2}+m)}}{24\pi w \delta_{2 } } \big[g_{1,\pi}g_{1,\eta}(w - m_{2}- \sigma^{1}_{22})-\\ & -g_{2,\pi}g_{2,\eta}(-w - m_{1}-\sigma^{2}_{11 } ) + \imath g_{1,\pi}g_{2,\eta}\sigma^{4}_{12 } + \imath g_{2,\pi}g_{1,\eta}\sigma^{3}_{21 } \big],\\ f_{p,+}(\eta n \rightarrow \eta n ) & = \abs{\mathbf{p}_\eta}^{2}\frac{(e_{2}+m)}{24\pi w \delta_{2 } } \big[g^{2}_{1,\eta}(w - m_{2}-\sigma^{1}_{22 } ) -g^{2}_{2,\eta}(-w - m_{1}-\sigma^{2}_{11})+ \\ & + \imath g_{1,\eta}g_{2,\eta}(\sigma^{3}_{21}+\sigma^{4}_{12 } ) \big],\\ \delta_{2}&=\big(-w - m_{1}-\sigma^2_{11}\big)\big(w - m_{2}-\sigma^{1}_{22}\big)- \sigma^{4}_{12}\sigma^{3}_{21}. \end{split}\ ] ] @xmath133-wave amplitudes ( @xmath137 ) : @xmath138,\\ f_{d,-}(\pi n \rightarrow \eta n ) & = \abs{\mathbf{p}_\pi}\abs{\mathbf{p}_\eta } \frac{\sqrt{(e_{1}-m)(e_{2}-m)}}{24\pi w \delta_{1 } } \big [ -g_{1,\pi}g_{1,\eta}(-w - m_{2 } - \sigma^{2}_{22 } ) + \\ & + g_{2,\pi } g_{2,\eta}(w - m_{1}-\sigma^{1}_{11 } ) - \imath g_{1,\pi } g_{2,\eta}\sigma^{3}_{12 } - \imath g_{2,\pi } g_{1,\eta}\sigma^{4}_{21 } \big],\\ f_{d,-}(\eta n \rightarrow \eta n ) & = \abs{\mathbf{p}_\eta}^{2}\frac{(e_{2}-m)}{24\pi w \delta_{1 } } \big[-g^{2}_{1,\eta}(-w - m_{2}-\sigma^{2}_{22 } ) + g^{2}_{2,\eta}(w - m_{1}-\sigma^{1}_{11 } ) -\\ & - \imath g_{1,\eta}g_{2,\eta}(\sigma^{4}_{21}+\sigma^{3}_{12 } ) \big],\\ \delta_{1}&=\big(w - m_{1}-\sigma^1_{11}\big)\big(-w - m_{2}-\sigma^{2}_{22}\big)-\sigma^{3}_{12}\sigma^{4}_{21}. \end{split}\ ] ] where @xmath79 and @xmath80 are nucleon energies for @xmath3 and @xmath66 states respectively . the obtained @xmath76 and @xmath133 partial amplitudes satisfy the two - channel unitary condition . besides , we should take into account the @xmath139-dependent form - factor in a vertex ( the so called centrifugal barrier factor ) . there is no common opinion in literature concerning its form , we take it in two - parameter form : @xmath140 the partial amplitudes , , which take into account the opf - mixing , are written in two - channel approach . but in fact in considered region of energy @xmath141 gev there exist at least five open channels , the most essential are the @xmath142 and @xmath143 channels . in this situation we follow the way suggested in @xcite : we restrict ourselves by the three - channel approach ( @xmath3 , @xmath66 and @xmath144 ) . as for third channel ( @xmath145 ) , it is considered as some `` effective '' channel and its threshold may be a free parameter in a fit . three - channel amplitudes may be obtained from the formulae , in appendix [ ap2 ] , but they are rather cumbersome so we did not write down them . for our local purpose of the description of @xmath146 amplitudes , it is sufficient to use formulae , . the only difference will appear in the self - energy , where we should add the third channel in the similar manner . we use the same procedure of loop renormalization as for spin @xmath147 , see . first of all let s try to describe the @xmath148 , @xmath149 separately . we found that , in accordance with our estimates for spin-1/2 case , the opf - mixing is more essential for lowest @xmath150 wave @xmath148 . results of @xmath5 fitting by formulae in two - channel ( @xmath3 , @xmath144 ) approach are shown at fig . [ d13_s ] . we restricted the energy interval by @xmath151 gev since at higher energy there appears some additional smooth contribution it is seen well from @xmath152 behaviour . as for mass of `` effective '' @xmath153-meson , fit leads to rather low value @xmath154 gev . from other side , the d - wave threshold generates rather smooth contribution in amplitude and is defined badly from data . so we fix it by @xmath155 mev in the following . left : @xmath5 partial wave of @xmath3 scattering @xcite and results of fit by our formulae with @xmath3 and @xmath144 channels ( @xmath156 gev ) . right : inelasticity from pwa @xcite and our curve , corresponding to left panel.,title="fig:",width=302 ] left : @xmath5 partial wave of @xmath3 scattering @xcite and results of fit by our formulae with @xmath3 and @xmath144 channels ( @xmath156 gev ) . right : inelasticity from pwa @xcite and our curve , corresponding to left panel.,title="fig:",width=302 ] fit of real and imaginary parts of @xmath5 gives : @xmath157 parameters of form - factor from @xmath5 wave : @xmath158 now we can describe @xmath4 at fixed parameters of @xmath5 resonance . results are shown at fig . [ p13_single ] . @xmath4 partial wave of @xmath3 scattering @xcite and results of fit by our formulae with @xmath3 and @xmath144 channels ( @xmath159 gev ) . parameters of @xmath5 resonance are fixed by . curves 1 and 2 show the real part of background contribution from @xmath5 resonance ( @xmath160 ) with form - factors and . right : inelasticity from pwa @xcite and our curve , corresponding to left panel.,title="fig:",width=302 ] @xmath4 partial wave of @xmath3 scattering @xcite and results of fit by our formulae with @xmath3 and @xmath144 channels ( @xmath159 gev ) . parameters of @xmath5 resonance are fixed by . curves 1 and 2 show the real part of background contribution from @xmath5 resonance ( @xmath160 ) with form - factors and . right : inelasticity from pwa @xcite and our curve , corresponding to left panel.,title="fig:",width=302 ] @xmath161 parameters of form - factor from @xmath4 wave : @xmath162 we observe that both fits are consistent with each other in parameters of resonances , except for the vertex form - factor . the obtained parameters do not contradict to values of masses and branching ratios of @xmath163 , @xmath164 in rpp tables @xcite . as for @xmath66 channel : pwa results for @xmath4 wave does not require this coupling . for @xmath5 situation is unstable : inclusion of this coupling leads to unphysical big coupling constants . but close inspection shows that this is effect of another threshold with higher mass . so we will restrict ourselves by the two - channel approach . [ d13_s ] , [ p13_single ] demonstrate that fit of @xmath5 and @xmath5 separately leads to rather good quality of description . as for joint fit it gives only qualitative description , as it seen from fig.[pdjoint ] . for better quality it needs `` fine tuning '' , first of all it should include : * more accurate description of @xmath165 channel ; * account of smooth contribution in @xmath5 wave see fig . [ d13_s ] ; * better understanding of role and properties of the vertex form - factor . the observed disagreement may be related with above items . example of joint description of @xmath4 ( w < 2.0 gev ) and @xmath5 ( @xmath166 gev ) partial waves by our formulae with opf - mixing in two - channel approach . in this case @xmath167 . , title="fig:",width=302 ] example of joint description of @xmath4 ( w < 2.0 gev ) and @xmath5 ( @xmath166 gev ) partial waves by our formulae with opf - mixing in two - channel approach . in this case @xmath167 . , title="fig:",width=302 ] thus we can see that the considered mixing of the opposite parities fermion fields leads to the sizeable effects for baryon production and may be identified in production of baryon resonances @xmath132 in @xmath3 scattering . in present paper we have analyzed the mixing effect , specific for fermions , when two fermion fields of opposite parities are mixed at loop level . for fermions it is possibly even if the parity is conserved in a vertex . as a result we have a matrix propagator of unusual form , which contains @xmath12 contributions . but since parity is conserved in vertexes , the @xmath12 matrix disappears after multiplication by the vertexes , and we get the amplitudes containing the resonance and background contributions . note that as a result of solving the dyson schwinger equations we automatically obtain the unitary amplitudes . the derived amplitudes resemble in structure the analytical @xmath14-matrix . the most significant difference is the presence of poles both of positive and negative energies in our amplitudes . if to say about resonance phenomenology , we have a pair of partial waves with strongly correlated parameters , namely , the resonance in one partial wave is connected with background contribution in another wave . the discussed effect is most essential for partial wave with smaller orbital momentum @xmath150 , thit is a consequence of inequality of phase volumes for different @xmath150 . as for manifestation of this effect in @xmath3 scattering , the most simple physical example is connected with production of spin-@xmath16 resonances of opposite parities and isospin @xmath59 . we used the obtained amplitudes for description of two @xmath3 partial waves @xmath168 and @xmath169 . we can conclude that the discussed effect reproduces naturally all the observed features of these partial waves but the joint description of these partial waves needs fine tuning of their properties . we suppose that the most interesting application of this effect is related with the problem of roper resonance @xmath170 , @xmath129 . recall that for these quantum numbers there are still problems of physical interpretation of the baryon states and their comparison with quark models . the effect of opf - mixing in this sector takes a more complicated form because of presence of several states @xmath130 ( see fig . [ gwu ] ) and non - standard form of the roper resonance @xmath129 . but the above mentioned strong correlation between two partial waves gives new possibilities for studying the properties of @xmath170 . * acknowledgements * this work was supported in part by the program `` development of scientific potential in higher schools '' ( project 2.2.1.1/1483 , 2.1.1/1539 ) and by the russian foundation for basic research ( project no . 09 - 02 - 00749 ) . 99 b. pontecorvo . jetp * 6 * ( 1958 ) 429 . . phys.rev.lett . * 10 * ( 1963 ) 531 ; m.kobayashi and t.maskawa . * 49 * ( 1973 ) 652 . . phys.rep . * 375 * ( 2003 ) 105 . d.espriu , j.manzano and p.talavera . phys.rev . * d66 * ( 2002 ) 076002 . m.blasone and j.palmer . phys.rev . * d69 * ( 2004 ) 057301 . b.machet , v.a.novikov and m.vysotsky . * a20 * ( 2005 ) 5399 . b.a.kniehl and a.sirlin . phys.rev . * d74 * ( 2006 ) 116003 . q.duret , b.machet and m.vysotsky . eur.phys.j . * c61 * ( 2009 ) 247 . a.e.kaloshin and v.p.lomov . * 69 * ( 2006 ) 563 . a.e.kaloshin and v.p.lomov . * 19 * ( 2004 ) 135 . m.o.gonchar , a.e.kaloshin and v.p.lomov . * v22 * ( 2007 ) 24 . r.e.cutkosky et al . phys.rev . * d20 * ( 1979 ) 2839 . r.koch . z.phys . * c29 * ( 1985 ) 597 . @xmath3 newsletters * 9 * ( 1993 ) 1 . r.a.arndt et al . phys.rev . * c52 * ( 1995 ) 2120 . r.a.arndt et al . * c74 * ( 2006 ) 045205 ; http://gwdac.phys.gwu.edu k.i.aoki * 73 * ( 1982 ) 1 . a.denner . fortschr.phys . * 41 * ( 1993 ) 307 . s.w.macdowell . phys.rev.*116 * ( 1959 ) 774 . o.babelon et al . * 113 * ( 1976 ) 445 . r.a.arndt , j.m.ford and l.d.roper . phys.rev . * d32 * ( 1985 ) 1085 . s.capstick and w.roberts . prog.part.nucl.phys * 45 * ( 2000 ) 241 . o.krehl et al . * c62 * ( 2000 ) 025207 . a.v.sarantsev et al . phys.lett . * b659 * ( 2008 ) 94 . s.ceci , a.vars and b.zauner . * c58 * ( 2008 ) 47 . m.batinic et al . phys.rev . * c51 * ( 1995 ) 2310 . s.ceci at al . phys.rev . * d77 * ( 2008 ) 116007 . k. nakamura et al . ( particle data group ) j. phys . * g 37 * ( 2010 ) 075021 . p.van nieuwenhuizen . phys.rep.*68 * ( 1981 ) 189 . let us write down the phenomenological lagrangians of interaction of spin @xmath16 particles with @xmath3 system . propagator of rarita - schwinger field has the form ( see more in @xcite ) : @xmath177 where the basis elements are @xmath178 the operator @xmath179 looks like @xcite : @xmath180 where we have introduced the unit `` vectors '' orthogonal to each other : @xmath181 suppose we have two fields @xmath15 of opposite parities . when taking into account opf - mixing the dressed propagator has the following decomposition : @xmath183 where @xmath184 being dimension 2 matrices are solutions of the matrix dyson schwinger equation . since the multiplicative properties of the operators @xmath185 are completely consistent with the properties of the spin-@xmath147 operators ( see table [ multiplic ] ) , the further calculations repeat @xmath186 ones . as a result the matrix propagator looks similar to spin-@xmath147 case . matrix amplitude has the form : @xmath187 where the matrix @xmath65 is constructed from the matrix of the propagator and vertex matrices : @xmath188 the vertex matrix in two - channel approximation looks like @xmath189 the self - energy @xmath190 is expressed through the standard loop function corresponding to one of the channels . for @xmath3 channel this standard function has form : @xmath191 and similarly for @xmath66 the channel . an alternative decomposition of the loop is @xmath192 so that @xmath193

we consider a loop mixing of two fermion fields of opposite parities whereas the parity is conserved in a lagrangian . such kind of mixing is specific for fermions and has no analogy in boson case . possible applications of this effect may be related with physics of baryon resonances . the obtained matrix propagator defines a pair of unitary partial amplitudes which describe the production of resonances of spin @xmath0 and different parity @xmath1 or @xmath2 . + the use of our amplitudes for joint description of @xmath3 partial waves @xmath4 and @xmath5 shows that the discussed effect is clearly seen in these partial waves as the specific form of interference between resonance and background . another interesting application of this effect may be a pair of partial waves @xmath6 and @xmath7 where the picture is more complicated due to presence of several resonance states .

1009.2845

the continuing development and utilization of microwave applications today make electromagnetic interference a serious problem that needs to be solved . although high conductivity metals are very effective for high frequency electromagnetic wave shielding , in many cases they are not suitable when weak or zero reflection is required ( such as for radar stealth technology ) . while metals shield the object by reflecting the incident radiation away , microwave absorbing materials ( mam ) are designed to absorb the radiation and therefore effectively reduce the reflection . strong absorption and weak reflection will lead to a large negative value of reflection loss ( @xmath6 ) and are therefore identified as two strict requirements for high loss mams . minimum @xmath6 values as low as down to less than @xmath12 db have been reported for some materials , most of them are ferri / ferro - magnetic based nanoparticles or composites , _ e.g. _ carbonyl iron@xmath13batio@xmath14 composite ( @xmath15 db ) @xcite , zno@xmath13carbonyl - iron composite ( @xmath16 db ) @xcite , la@xmath17sr@xmath18mno@xmath14@xmath13 polyaniline composite ( @xmath19 db ) @xcite , etc , indicating the dominant role of magnetic losses over the others such as dielectric and conduction losses . dielectrics usually have small permeability and , visa versa , most magnetic materials have small permittivity . to maximize the absorption capability by combining dielectric and magnetic losses , and since zero reflection can be achieved in a mam that has equal permittivity and permeability ( @xmath20 ) to satisfy the impedance matching condition @xmath21 ( @xmath22 is the impedance of the free space ) , much attention has been paid to multiferroic and magneto - dielectric materials . la@xmath0sr@xmath1nio@xmath2 is known as a dielectric compound that has a colossal dielectric constant of up to more than @xmath23 at room temperature @xcite . while la@xmath24nio@xmath2 is an antiferromagnet , the substitution of sr for la introduces holes into the system and suppresses the antiferromagnetic order @xcite . experimental magnetic data show that la@xmath0sr@xmath1nio@xmath2 is a paramagnet at room temperature @xcite , suggesting that the magnetic loss may be negligibly small . with such a large imbalance between permittivity and permeability , @xmath25 , and insignificant magnetic loss , the material is therefore not expected to have a low @xmath6 . in this letter , we show that la@xmath0sr@xmath1nio@xmath2 in fact exhibits a strong microwave absorption capability at the resonant frequencies ; for a layer of 3.0 mm , the minimum @xmath6 reaches down to @xmath26 db at @xmath279.7 ghz . interestingly , the resonance mechanism is found to be impedance matching with @xmath28 @xmath9 . , of the la@xmath0sr@xmath1nio@xmath2 nanoparticle powder . the peaks in the xrd patterns are marked by miller indices . the measurements were carried out at 300 k.,width=283 ] the la@xmath0sr@xmath1nio@xmath2 nanoparticle powder was synthesized using a conventional solid state reaction route combined with high - energy ball milling processes . a pertinent post - milling heat treatment was performed to reduce the surface and structural damages caused by the high - energy milling . to prepare the samples for microwave measurements , the nanoparticle powder was mixed with paraffin in @xmath29 vol . percentage , respectively , and finally coated ( with different coating thicknesses @xmath30 , and 3.5 mm ) on thin plates that are almost transparent to microwave radiation . the free - space microwave measurement method in the frequency range of @xmath31 ghz was utilized using a vector network analyzer . an aluminum plate was used as reference material with 0% of attenuation or 100% of reflection . the permittivity and permeability are calculated according to analyses proposed by nicolson and ross @xcite , and weir @xcite ( hence called the nrw method ) . the impedance and the reflection loss are then calculated according to the transmission line theory @xcite : @xmath32 \label{eqn1}\ ] ] @xmath33 .summary of the microwave absorption characteristics for the paraffin - mixed la@xmath0sr@xmath1nio@xmath2 nanoparticle layers with different thicknesses . here , @xmath5 is in mm ; @xmath34 , @xmath35 , @xmath36 , @xmath37 are in ghz ; and @xmath38 is in @xmath9 . see text for details . [ cols="^,^,^,^,^,^",options="header " , ] [ table1 ] x - ray diffraction ( xrd , fig . [ fig.1 ] ) data indicate that the material is single phase of a tetragonal structure ( f@xmath2k@xmath24ni - perovskite - type , @xmath39 space group ) @xcite ; no impurity or secondary phase could be distinguished . an average particle size of @xmath2750 nm was calculated using the scherrer s equation , @xmath40 ( where @xmath41 is the shape factor , @xmath42 is the x - ray wavelength , @xmath43 is the line broadening at half the maximum intensity , and @xmath44 is the bragg angle ) . the magnetization loop , @xmath45(@xmath46 ) , shows very small magnetic moments with no hysteresis ( fig . [ fig.1 ] inset ) , verifying the paramagnetic characteristic of the material at room temperature . the initial relative permeability , @xmath47 , calculated from the magnetization curve is of @xmath271.005 , which is only slightly higher than that of the air ( 1.00000037 ) @xcite . ( squares ) and @xmath48 ( circles ) curves of the paraffin - mixed la@xmath0sr@xmath1nio@xmath2 nanoparticle layers with different thicknesses : ( a ) @xmath49 mm , ( b ) @xmath50 mm , ( c ) @xmath7 mm , and ( d ) @xmath51 mm . @xmath35 and @xmath36 are the upper and lower frequencies , respectively , where @xmath8 @xmath9.,width=245 ] all of the high - frequency characteristic parameters of the samples are summarized in table [ table1 ] . the @xmath48 and @xmath52 curves for the samples with @xmath53 and 3.5 mm are plotted in fig . [ fig.2 ] . for @xmath54 mm ( not shown ) , no significant absorption or distinguishable resonance could be observed . the @xmath6 value is large ( @xmath55 db ) and has a tendency to decrease when approaching 4 ghz ( from above ) and 18 ghz ( from under ) . it is possible that a resonance peak for this sample would occur at a frequency very close to ( but higher than ) 18 ghz , considering the variation of the resonance frequency @xmath34 on the thickness , as presented below . the @xmath52 curve for @xmath56 mm in fig . [ fig.2]a exhibits a deep minimum of @xmath57 db at @xmath58 ghz , which is very close to the frequency @xmath35 ( @xmath27 14.0 - 14.3 ghz ) where @xmath59 @xmath9 . the close value of @xmath34 to @xmath35 would suggest that the strong microwave absorption would be attributed to the resonance caused by impedance matching . however , the resonance could also be caused by a phase matching if the phases of the reflected waves from the two sample s surfaces differ by @xmath60 . in this case , the resonance frequency and its harmonics are given by @xmath61 , where @xmath62 is the speed of light in the incident medium and @xmath63 , 1 , 2 , ... nevertheless , since the closest @xmath37 value ( 13.9 ghz , obtained for @xmath64 ) is also quite close to @xmath34 , it is difficult to determine conclusively which mechanism is responsible for the deep negative @xmath6 at @xmath34 for this @xmath56 mm sample . the phase - matching calculation for the @xmath65 mm sample predicts @xmath66 ghz for @xmath67 and @xmath2718 ghz for @xmath68 ; both are the lower and upper frequency limits of our measurement system . figs . [ fig.2]b and [ fig.2]c display the @xmath48 and @xmath52 curves for the @xmath69 mm and 3.0 mm samples . with increasing thickness from 1.5 mm to 3.0 mm , the resonance shifts to lower frequencies while the notch in @xmath6 becomes deeper . for @xmath69 mm , the minimum of @xmath6 appears almost at the same frequency as that of @xmath70-matching while the phase matching frequency is a little higher , i.e. , @xmath71 ghz and @xmath72 ghz for @xmath64 . similar scenario is also obtained for @xmath73 mm : @xmath74 ghz whereas @xmath75 ghz . it is quite clear that , although the shift of the resonance to lower frequencies is qualitatively in agreement with the phase matching model , there is still a considerable difference between the calculated values of @xmath37 and the measured @xmath34 that seems to even develop with increasing the sample s thickness . hence , both of the increasing deviation of @xmath37 from @xmath34 and the coincidence of @xmath34 and @xmath35 indicate that the resonance observed in these samples belongs to the @xmath70-matching mechanism . dielectrics absorb microwave s energy and convert it to heat via the rotation of polar molecules at high frequencies and the ion - drag at low frequencies . the @xmath70-matching resonance itself does not necessarily cause any energy dissipation of the electromagnetic wave , but favors the wave s propagation into the sample and hence promotes the absorption . the @xmath48 curves in figs . 2a - c show that there are at least two frequencies ( @xmath35 and @xmath36 ) where the @xmath76 condition is satisfied . nevertheless , a strong absorption is obtained only at @xmath35 while there is no observable anomaly ( except for a shoulder for the @xmath56 mm sample ) in the @xmath52 curve at @xmath36 . this implies that , although @xmath70-matching occurs at both @xmath35 and @xmath36 , the energy dissipation is not promoted at @xmath36 . according to eq . [ eqn2 ] , perfect energy absorption , @xmath77 , occurs if @xmath78 @xmath9 , i.e. @xmath79 @xmath9 and the imaginary part @xmath80 . a deviation of @xmath81 from zero will reduce @xmath6 to a finite value ; the larger the relative value of @xmath38 is , the larger the minimum @xmath6 will be at the resonance . our data show that the @xmath56 mm sample has @xmath82 @xmath9 and 317.2 @xmath9 at @xmath35 and @xmath36 , while those for @xmath73 mm are @xmath83 @xmath9 and 242 @xmath9 , respectively . the larger values of @xmath38 may explain the absence of resonant absorption at the @xmath36 frequencies . this seems to be similar to the analysis reported by pang _ @xcite where the authors introduced an entity of imaginary thickness component that becomes zero at the absorption resonance . in addition , the variation of @xmath38 at @xmath35 ( see table [ table1 ] ) is also in agreement with the decrease of the minimum @xmath6 values ( from @xmath84 db to @xmath26 db ) as @xmath5 increases from 1.5 mm to 3.0 mm . @xmath38 is therefore could be considered as the mismatch at the @xmath70-matching condition . it is also noticeable that the @xmath52 curves do not show any deep minimum at the phase matching frequencies . the reason may lie into the use of the transparent backing plate for the samples . the electromagnetic wave reflects at all the boundaries between two different impedance media . however , without a metal backing plate , the internal reflection at the back side of the sample would be much weaker than the reflection at the front side . moreover , the internal reflection wave is also absorbed again by the sample . so even the phase matching resonance does occur , no significant cancelation of the reflected signals would be detected . the shoulder appearing in the @xmath85 curves in fig . [ fig.2]a may belong to the phase matching effect because of the sample s small thickness ( @xmath49 mm ) . the @xmath70-matching solution for this shoulder is ruled out due to the large value of @xmath38 at @xmath86 . considering the proximity of @xmath87 and @xmath37 , we expect that using metal backing plates for these la@xmath0sr@xmath1nio@xmath2 absorbers would either further decrease the minimum of @xmath6 or widen the absorption band by combining the two matching effects . with further increasing the thickness to 3.5 mm , as displayed in fig . [ fig.2]c , the microwave absorption is strongly suppressed . no @xmath70-matching condition could be observed as the whole @xmath48 curve lies well above @xmath22 . though , the @xmath52 curve still exhibits a notch at @xmath88 ghz . a calculation according to the phase matching model gives @xmath89 ghz ( with @xmath64 ) , which is far above @xmath34 . apparently , none of the mentioned matching phenomena would be the mechanism for the absorption peak at @xmath88 ghz . however , since @xmath90 reaches its minimum of 718 @xmath9 at 8.4 ghz that is closely equal to @xmath34 , this minimum in @xmath90 could be responsible for the deep of @xmath6 at @xmath34 . in summary , we have observed very low microwave @xmath6 values for the powders of la@xmath0sr@xmath1nio@xmath2 nanoparticles despite the large imbalance between permittivity and permeability . for @xmath91 mm , the resonance takes place according to the @xmath70-matching mechanism , where @xmath38 could be considered as a mismatch parameter . the smallest minimum @xmath6 is observed for the absorber with the matching thickness of @xmath7 mm and in the radar x - band . the @xmath70-matching condition is not attained in thick samples ( @xmath92 mm ) that have @xmath11 at all frequencies but the peak absorption occurs where the impedance reaches its minimum . we suggest that ( i ) using a metal backing plate to combine the @xmath70- and phase - matching resonances and ( ii ) mixing la@xmath0sr@xmath1nio@xmath2 with magnetic fillers to balance out @xmath93 and @xmath94 would further improve the material s microwave absorption performance . this research is funded by vietnam national foundation for science and technology development ( nafosted ) under grant number `` 103.02 - 2012.58 '' and by the state key lab at the ims ( vast ) . yuchang q. , wancheng z. , fa l. , dongmei z. , j. magn . magn . mater . 2011;323:600 - 606 . ma z. , zhang y. , cao c. , yuan j. , liu q. , wang j. , physica b 2011;406:4620 - 4624 . cui k. , cheng y. , dai j. , liu j. , mater . chem . phys . 2013;138:810 - 816 . rivas j. , rivas - murias b. , fondado a. , mira j. , sears - rodrguez ma . phys . lett . 2004;85:6224 - 6226 . lunkenheimer p. , krohns s. , riegg s. , ebbinghaus sg . , reller a. , loidl a. , eur . . j. special topics 2010;180:61 - 89 . szpunar b. , smith jr . vh . , spaek j. , physica c : superconductivity 1989;161:503 - 511 . wada s. , furukawa y. , kaburagi m. , kajitan b. , hosoya s. , yamada y. , j. phys : condens . matter 1993;5:765 - 780 . freeman p. , magnetism and the magnetic excitations of charge ordered la@xmath95sr@xmath96nio@xmath97 , ph.d . thesis , trinity , university of oxford , 2005 . tran dt . , , le vh . , , yu sc . , adv . . nanotechnol . 2013;4:025010 - 4 . cullity bd . , graham cd . , introduction to magnetic materials , 2nd edition , new jersey , wiley - ieee press , 2008 , p.16 . nicolson am . , ross gf . , ieee trans . instr . meas . 1970;im-19:377 - 382 . ieee 1974;62:33 - 36 . naito y. , suetake k. , microw . theory tech . 1971;mitt-19:65 - 72 . pang h. , fan m. , he z. , j. magn . magn . mater . 2012;324:2492 - 2495 .

la@xmath0sr@xmath1nio@xmath2 is well known to have a colossal dielectric constant ( @xmath3 ) . the la@xmath0sr@xmath1nio@xmath2 nanoparticle powder was prepared by a combinational method of solid state reaction and high - energy ball milling . magnetic measurements show that the material has a very small magnetic moment and paramagnetic characteristic at room temperature . the mixture of the nanoparticle powder ( 40@xmath4 vol . ) and paraffin ( 60@xmath4 vol . ) coated in the form of flat layers of different thicknesses ( @xmath5 ) exhibits strong microwave absorption resonances in the 4 - 18 ghz range . the reflection loss ( @xmath6 ) decreases with @xmath5 and reaches down to -36.7 db for @xmath7 mm . the impedance matching ( @xmath8 @xmath9 ) , rather than the phase matching mechanism , is found responsible for the resonance observed in the samples with @xmath10 mm . further increase of the thickness leads to @xmath11 at all frequencies and a reduced absorption . the influence of non - metal backing is also discussed . our observation suggests that la@xmath0sr@xmath1nio@xmath2 nanoparticles could be used as good fillers for high performance radar absorbing material . dielectrics ; electronic materials ; energy storage and conversion ; magnetic materials ; nanoparticles ; powder technology .

1308.2711

the concept of gpds @xcite has led to completely new methods of `` spatial imaging '' of the nucleon . the mapping of the nucleon gpds , and a detailed understanding of the spatial quark distribution of the nucleon , have been widely recognized are a key objectives of nuclear physics of the next decade , and is a key justification for the jlab energy upgrade to 12 gev . gpds also allow to quantify how the orbital motion of quarks in the nucleon contributes to the nucleon spin a question of crucial importance for our understanding of the `` mechanics '' underlying nucleon structure . this requires a comprehensive program , combining results of measurements of a variety of processes in electron nucleon scattering with structural information obtained from theoretical studies , as well as with expected results from future lattice qcd simulations . it is well recognized @xcite that exclusive processes can be used to probe the gpds and construct 2-dimensional and 3-dimensional images of the quark content of the nucleon . deeply virtual compton scattering and deeply virtual meson production are identified as the processes most suitable to map out the twist-2 vector gpds @xmath1 and the axial gpds @xmath2 in @xmath3 , where @xmath4 is the momentum fraction of the struck quark , @xmath5 the longitudinal momentum transfer to the quark , and @xmath6 the momentum transfer to the nucleon . having access to a 3-dimensional image of the nucleon ( two dimensions in transverse space , one dimension in longitudinal momentum ) opens up completely new insights into the complex structure of the nucleon . for example , the nucleon matrix element of the energy - momentum tensor contains 3 form factors that encode information on the angular momentum distribution @xmath7 of the quarks with flavor @xmath8 in transverse space , their mass - energy distribution @xmath9 , and their pressure and force distribution @xmath10 . these form factors also appear as moments of the vector gpds @xcite , thus offering prospects of accessing these quantities through detailed mapping of gpds . the quark angular momentum in the nucleon is given by @xmath11~,\ ] ] and the mass - energy and pressure distribution @xmath12 the mass - energy and force - pressure distribution of the quarks are given by the second moment of gpd @xmath13 , and their relative contribution is controlled by @xmath5 . the separation of @xmath14 and @xmath10 requires measurement of these moments in a large range of @xmath5 . dvcs has been shown @xcite to be the cleanest process to access gpds at the kinematics accessible today . it is also a relatively rare process and requires high luminosities for the required high statistics measurements . the beam helicity - dependent cross section asymmetry is given in leading twist as @xmath15d\phi~,\]]where @xmath16 and @xmath17 are the dirac and pauli form factors , @xmath18 is the azimuthal angle between the electron scattering plane and the hadronic plane . the kinematically suppressed term with gpd @xmath19 is omitted . for not too large @xmath5 the asymmetry is mostly sensitive to the gpd @xmath20 . the asymmmetry with a longitudinally polarized target is given by @xmath21~.\ ] ] the combination of @xmath22 and @xmath23 allows a separation of gpd @xmath20 and @xmath24 . using a transversely polarized target the asymmetry @xmath25\ ] ] can be measured , which depends in leading order on gpd @xmath19 and is highly sensitive to orbital angular momentum contributions of quarks . clearly , determining moments of gpds for different @xmath6 will require measurement in a large range of @xmath4 , in particular at large @xmath4 . the reconstruction of the transverse spatial quark distribution requires measurement in a large range in @xmath6 , and the separation of the @xmath10 and @xmath14 form factors requires a large span in @xmath5 . to meet the requirements of high statistics measurements of relatively rare exclusive processes such as dvcs at high photon virtuality @xmath26 , large @xmath6 and @xmath5 , the clas detector will be upgraded and modified to clas12 @xcite . the main new features of clas12 over the current clas detector include a high operational luminosity of @xmath27@xmath28sec@xmath29 , an order of magnitude increase over clas @xcite . improved particle identification will be achieved with additional threshold gas cerenkov counter , improved timing resolution of the forward time - of - flight system , and a finer granularity electromagnetic preshower calorimeter that , in conjunction with th existing clas calorimeter will provide much improved @xmath30 separation for momenta up to 10 gev . in addition , a new central detector will be built that uses a high - field solenoid magnet for particle tracking and allows the operation of dynamically polarized solid state targets . with these upgrades clas12 will be the workhorse for exclusive electroproduction experiments in the deep inelastic kinematics . the 12 gev upgrade offers much improved capabilities to access gpds . figure [ fig : dvcs_alu_12gev ] shows the expected statistical precision of the beam dvcs asymmetry for some sample kinematics . at the expected luminosity of @xmath27@xmath28sec@xmath29 and for a run time of 2000 hours , high statistics measurements in a very large kinematics range are possible . using a dynamically polarized @xmath31 target we can also measure the longitudinal target spin asymmetry @xmath23 with high precision . the projected results are shown in fig . [ fig : aul ] . the statistical accuracy of this measurement will be less than for the @xmath22 asymmetry due to the large dilution factor in the target material , but it will still be a very significant measurement . polarizing the target transverse to the beam direction will access a different combination of gpds , and provide different sensitivity for the y- and x - components of the target polarization . the expected accuracy for one of the polarization projections is shown in fig . [ fig : dvcs_aut_12gev ] . here the target is assumed to be a frozen hd - ice target , which has different characteristics from the @xmath31 target . a measurement of all 3 asymmetries will allow a separate determination of gpds @xmath32 and @xmath19 at the above specified kinematics . through a fourier transformation the t - dependence of gpd @xmath33 can be used to determine the @xmath34quark distribution in transverse impact parameter space . figure [ fig : gpd_h ] shows projected results for such a transformation assuming a model parameterization for the kinematical dependences of gpd @xmath33 . knowledge of gpd @xmath19 will be particularly interesting as it is directly related to the orbital angular momentum distribution of quarks in transverse space . we thank the members of the clas collaboration who contributed to the development of the exciting physics program for the jlab upgrade to 12 gev , and the clas12 detector . much of the material in this report is taken from the clas12 technical design report version 3 , october 2007 @xcite . this work was supported in part by the u.s . department of energy and the national science foundation , the french commisariat lenergie atomique , the italian instituto nazionale di fisica nucleare , the korea research foundation , and a research grant of the russian federation . the jefferson science associates , llc , operates jefferson lab under contract de - ac05 - 060r23177 . 99 d. mller , d. robaschik , b. geyer , f.m . dittes , j. horejsi , fortschr . 42 , 101 , 1994 , hep - ph/9812448 . x. ji , phys . d55 , 7114 , 1997 . a. radyshkin , phys . b380 , 417 , 1996 . m. burkardt , int . j. mod a18 , 173 , 2003 . a. belitsky , d. mueller , a. kirchner , nucl . b*629 * , 323 , 2002 . belitsky , x. ji , f. yuan , phys . rev . d*69 * , 074014 , 2004 . k. goeke _ et al . _ , c**75 * * , 094021 , 2007 . s. stepanyan _ et al._(clas collaboration ) , phys . rev . lett.*87 * , 182002 , 2001 . et al._(clas collaboration ) , phys . rev . lett.*97 * , 072002 , 2006 . et al._(clas collaboration ) , arxiv:0711.4805 , subm . to phys . lett .. c. munoz - camacho _ et al . rev . lett.*97 * , 262002 , 2006 . r. de masi , m. garcon , b. zhao , _ et al . _ ( clas collaboration ) , arxiv:0711.4736 , submitted to phys . jlab experiment e12 - 06 - 119 , f. sabatie _ m. vanderhaeghen , p. guichon , m. guidal , phys . rev . d*60 * , 094017 , 1999 . b. mecking et al . inst . meth . a*503*,513 , 2003 . the complete technical design report document may be obtained from the authors .

an overview is given about the capabilities provided by the jlab 12 gev upgrade to measure deeply virtual exclusive processes with high statistics and covering a large kinematics range in the parameters that are needed to allow reconstruction of a spatial image of the nucleon s quark structure . the measurements planned with clas12 will cross section asymmetries with polarized beams and with longitudinally and transversely polarized proton targets in the constrained kinematics @xmath0 . in addition , unpolarized dvcs cross sections , and doubly polarized beam target asymmetries will be measured as well . in this talk only the beam and target asymmetries will be discussed .

0801.4168

g359.230.82 ( `` the mouse '' ) , with its long axisymmetric nonthermal nebula extending for 12 arcminutes , was first discovered as part of a very large array ( vla ) radio continuum survey of the galactic center at 20 cm ( yusef - zadeh and bally 1987 ) . a bow - shock structure was noted along the eastern edge of the nebula ( yusef - zadeh and bally 1989 ) . in addition , radio continuum data show linearly polarized emission from the full extent of the nebula and the spectral index distribution between 2 , 6 and 20 cm remains flat at the head of the nebula but steepens in the direction away from the head of the mouse ( yusef - zadeh and bally 1989 ) . the detection of x - ray emission from this source and the identification of a young radio pulsar g359.230.82 at the head of the nebula resulted in a breakthrough in our understanding of what powers this source ( predehl and kulkarni 1995 ; sidoli et al . 1999 ; camilo et al . more recently , _ chandra _ observations show detailed structural and spectral characteristics of this bow - shock pulsar wind nebula ( pwn ) associated with the mouse ( gaensler et al . modeling of the x - ray emission suggests a high space velocity @xmath0600 km s@xmath1 in a relatively warm phase of the ism in order to explain the cometary morphology of this source . the region where the mouse is found contains a number of other nonthermal radio continuum sources . figure 1 shows a large - scale 20 cm view of this region where two shell - type supernova remnants ( snrs ) g359.10.5 and g359.10.2 , and a southern segment of a nonthermal radio filament g359.10.2 , known as the `` snake '' , are distributed ( reich & frst 1984 ; gray et al . 1991 ; yusef - zadeh , hewitt and cotton 2004 ) . the shell - like snr g359.1 - 0.5 has a diameter of about 20@xmath2 and the extension of the tail of the mouse appears to be pointed toward the center of this remnant . here , we present high - resolution radio images showing new structural details of the mouse . using polarization data , we present the distribution of the magnetic field and the rotation measure of the bow - shock region of the nebula . we also argue that the mouse and snr g359.10.5 are potentially associated with each other . using the vla in its bna array configuration , we reduced the archival data taken in 1999 october at 3.6 cm wavelength . standard calibrations were carried out using 3c 286 and iers b1741312 as amplitude and phase calibrators , respectively . we also calibrated and combined the 6 and 2 cm data taken in 1987 november and 1988 february in the bna and cnb array configurations , respectively . 3c 286 and nrao 530 were selected as the phase and amplitude calibrators for the 6 and 2 cm data . using different robust weighting to the _ uv _ data , the final images were constructed after phase self - calibration was applied to all 6 , 3.6 cm and 2 cm data . the spectral index distribution of the mouse based on these measurements will be given elsewhere . we believe that the scale lengths of the features that are described here are well - sampled in the @xmath3 plane . in particular , the best well - sampled baselines range between 10 and 550 k@xmath4 at 3.6 cm and and 5 and 250 k@xmath4 at 6 cm , respectively . figures 2a , b shows the combined polarized and total intensity images of the mouse with a resolution of 2.1@xmath5 ( pa @xmath6 ) at 6 cm . the total intensity image of the head of the mouse shows a cone - shape structure within which a bright linear feature with an extent of @xmath7 appears to run along the axis of the symmetry . with the exception of depolarizing features , which are displayed as dark patches in figure 2a , similar morphology is seen in both the total and polarized intensity images . the overall degree of polarization at 6 cm ranges between 10 and 25% . detailed 6 cm images of the head of the nebula show that the peak polarized and total intensity images are offset from each other suggesting that the emission is depolarized at the outermost portion of the bow shock . this offset is best shown in figure 3a , b where a slice is cut along the axis of the symmetry of the distribution of the polarized and total intensity , respectively . a sharp rise in the total intensity of the emission having a 67% degree of polarization is followed by a peak emission which coincides with a drop in the polarized emission ; the degree of polarized emission is less than 1% at this position . in the region where the total intensity falls gradually , the degree of the polarization rises to values ranging between 510% . it is clear that either the magnetic field is highly tangled - up or that there is a high degree of thermal plasma mixed in with nonthermal gas at the head of the nebula where no polarized emission is detected . a more dramatic picture of this depolarization feature at 3.6 cm is shown in figure 6a , as discussed below . figure 4a , b show grayscale and contour images of the head of the mouse immediately to the west of the bright bowshock . the bright compact source that is suspected to be coincident with the pulsar is clearly resolved in these images . the sharp distribution of the emission ahead of the pulsar when compared with a more gradual distribution of emission behind the pulsar supports for the compression of gas and magnetic field in the bowshock region resulting from the supersonic motion of the pulsar . we note a 4@xmath8hole@xmath9 in the 3.6 cm brightness distribution within which the emission is three times fainter than the features surrounding it . the center of the hole@xmath9 lies near @xmath10 . the non - uniform distribution of emission behind the bow - shock can be seen throughout the mouse at 6 and 3.6 cm . additional 2 and 6 cm images presented in figures 5a , b also support the evidence for the modulation of the total intensity along the axis of the symmetry . the morphology of the emission is complicated but figures 2 5 show that the overall brightness temperature along the axis of the symmetry decreases at about 5@xmath11 and 45@xmath12 west of the bowshock . the grayscale distribution of the polarized emission superimposed on contours of total intensity at 6 and 3.6 cm are shown in figures 6a , b , respectively . the length and the position angle of the line segments represent the distributions of the polarized emission and the polarization vectors rotated by 90 degrees . the distribution of the polarization vectors at these wavelengths , including our 2 cm data that are not shown here , are similar to each other . the striking feature in these images is the recognition of a depolarizing feature separating the distribution of the polarized emission in the bowshock region from the region behind the pulsar . the position angle of the polarization angle vectors changes by 90@xmath13 between these two regions . the slice representations of the total and polarized intensity at 6 cm , as shown in figure 3a , b , are consistent with a picture that the scale of the magnetic field within the beam must have changed in order to produce the depolarizing feature . the distributions of the polarization vectors at 3.6 and 6 cm are used to determine the rotation measure ( rm ) distribution of the polarized emission ahead of the mouse . figure 7a , b show slice representations of the distribution of the rm and its error between 3.6 and 6 cm cut along the symmetry axis of the mouse . the rm value is on order of 500@xmath1440 rad m@xmath15 near the bow shock to 400@xmath1420 rad m@xmath15 at the position of the depolarizing feature before it increases to a value 0@xmath1460 rad m@xmath15 away from the bow shock . the low value of the rm and its error in the region behind the pulsar suggests the intrinsic magnetic field traces the direction of the inferred motion . we believe that the faraday rotation toward the mouse due to interstellar material along the line of sight is well represented in the rm distribution behind the pulsar and along the tail of the mouse . however , the high rm values as well as the evidence in field reversal and the low degree of polarization ahead of the pulsar are distinctly different than the region behind the pulsar . these effects , in particular , the increase in rm toward the head of the mouse by two orders of magnitude imply that faraday rotation along the line of sight can not explain the unusual characteristics of polarization ahead of the pulsar . we suggest a picture in which there is a mixture of thermal and nonthermal plasma coexisting in the bowshock region . although , it is expected that the magnetic field gets compressed and becomes more uniform in the bowshock region of a pulsar wind nebula , internal thermal plasma can produce a high degree of faraday rotation and depolarization ( burn 1966 ) . future detailed polarization observations using multiple bands in order to determine the true distribution of faraday rotation should be able to test this interpretation . using the dispersion measure of @xmath0100 @xmath16 pc ( camilo et al . 2002 ) and an upper limit of rm@xmath0 100 rad m@xmath15 , the parallel component of the magnetic field along the line of sight is estimated to be @xmath17 g . this unusually low value of the magnetic field along the line of sight is likely to be underestimated since the sign of the rm could change between the head and the tail of the mouse . the fluctuation of electron density and magnetic field of a turbulent medium along the line of sight , as well as from the region within the nebula , can severely affect the correct estimate of the magnetic field along the line of sight ( beck et al . 2003 ) . in spite of this difficulty , the rm estimate made here is at least an order of magnitude less than the estimate made toward objects lying within a few hundred pcs of the galactic center ( e.g. , yusef - zadeh , wardle and parastaran 1997 ) . the rm value estimated from this study is consistent with a source lying less than 5 kpc from us . one of the most puzzling aspects of radio observations reported here is the distribution of synchrotron emission near the apex of the pulsar wind where the intensity is modulated at 3 cm . the drop in synchrotron emissivity may be accounted for in terms of the unshocked wind arising from the pulsar or an inhomogeneous distribution of synchrotron emission . the numerical simulations carried out by gaensler et al . ( 2004 ) have in fact predicted a closed elongated structure associated with the termination shock behind a supersonically moving pulsar . a more detailed comparison of radio and x - ray data is needed to determine the nature of the intensity distribution and how it may be related to the termination shock behind the pulsar . radio and x - ray observations suggest the the mouse could lie at distance ranging between 2 and 5 kpc and that this object is not associated with snr g359.10.5 . uchida et al . ( 1992 ) considered that g359.10.5 is associated with a molecular ring lying near the galactic center requiring multiple supernova explosions to account for an energy of 6@xmath18 ergs . however , a subsequent detection of maser emission at 5 km s@xmath1 surrounding this large - scale source ( yusef - zadeh et al . 1995 ) , as well as the detection of low value of rotation measure toward the remnant ( yusef - zadeh et al . 2005 , in preparation ) , suggest that snr g359.10.5 is unlikely to be located within a few parsec of the galactic center at the distance of 8.5 kpc . one possibility is that the mouse and snr g359.10.5 are associated with each other . the support for this suggestion may come from the absorption column measured ranging between @xmath19 cm@xmath15 and @xmath21 cm@xmath15 toward the remnant ( bamba et al . 2000 ; egger and sun 1998 ) and @xmath22 cm@xmath15 toward the mouse ( gaensler et al . depending on how one fits the x - ray spectrum , these estimates could be consistent with each other . if snr g359.10.5 and the mouse lie at the distance of 5 kpc , the time that it takes for the pulsar to travel 34 pc , the distance between the center of the remnant and the pulsar , is @xmath23 years . this is more than twice the characteristic age of the pulsar @xmath24 yr . recent studies suggest that the true age of young pulsars can be longer than their characteristic age , so it is possible that the mouse and snr g359.10.5 could have the same origin . in conclusion , we have presented new high - resolution radio continuum images of the mouse , showing a modulation of the intensity behind the pulsar . a polarization image of the mouse at 3.6 cm shows that the overall distribution of the inferred magnetic field is parallel to the axis of symmetry away from the bow shock . however , a high degree of depolarization , field reversal and high rm are detected ahead of the pulsar near the interface where the polarization angle changes by @xmath25 . these images indicate that the direction of the magnetic field in the bow shock is likely to be tangent to the shock normal , though more detailed radio observations are required to correct for faraday effects . we have also argued that the mouse and an adjacent snr g359.10.5 , may be associated . future radio and x - ray observations should be able to shed light on the nature of the mouse and snr g359.10.5 as well as their physical relationship . the national radio astronomy observatory is a facility of the national science foundation operated under cooperative agreement by associated universities , inc . bmg acknowledges the support of nasa through sao grant go2 - 3075x and ltsa grant nag5 - 13032 . fyz was funded by nsf ast-03074234 and nasa nag-9205 . we also thank m. wardle for useful discussion . egger , r. & sun , x. 1998 , `` x - ray emission from g359.1 - 0.5 '' , proceedings of the iau colluquium no . 166 held in garching , germany , 21 - 25 april , 1997 , xxvii , 603pp . springer - verlag berlin heidelberg new york ( isbn 3 - 540 - 64306 - 0 ) , edited by d. breitschwerdt , m. j. freyberg , and j. truemper , 417 - 420

the recent detection of a young pulsar powering `` the mouse '' , g359.230.82 , as well as detailed imaging of surrounding nebular x - ray emission , have motivated us to investigate the structural details and polarization characteristics of the radio emission from this axisymmetric source with a supersonic bow shock . using polarization data at 3.6 and 6 cm , we find that the magnetic field wraps around the bow shock structure near the apex of the system , but downnstream runs parallel to the inferred direction of the pulsar s motion . the rotation measure ( rm ) distribution of the mouse also suggests that the low degree of polarization combined with a high rm ahead of the pulsar result from internal plasma within the bowshock region . in addition , using sub - arcsecond radio image of the mouse , we identify modulations in the brightness distribution of the mouse that may be associated with the unshocked pulsar wind behind the pulsar . lastly , we discuss the relationship between the mouse and its neighboring shell - type supernova remnant g359.10.5 and argue that these two sources could potentially have the same origin . ism : individual : ( g359.230.82 ) , pulsars : individual ( j17472958 ) , stars : neutron , stars : winds , outflows

astro-ph0503031

semiconductor devices have been continuously downscaled ever since the invention of the first transistor @xcite , such that the size of the single building component of modern electronic devices has already reached to a few nanometers ( nm ) . in such a @xmath6 regime , two conceptual changes are required in the device modeling methodology . one aspect is widely accepted where carriers must be treated as quantum mechanical rather than classical objects . the second change is the need to embrace the multi - band models which can describe atomic features of materials , reproducing experimentally verified bulk bandstuructures . while the single - band effective mass approximation ( ema ) predicts bandstructures reasonably well near the conduction band minimum ( cbm ) , the subband quantization loses accuracy if devices are in a sub - nm regime @xcite . the ema also fails to predict indirect gaps , inter - band coupling and non - parabolicity in bulk bandstructures @xcite . the nearest - neighbor empirical tight - binding ( tb ) and next nearest - neighbor @xmath0@xmath1@xmath2 ( kp ) approach are most widely used band models of multiple bases @xcite . the most sophisticated tb model uses a set of 10 localized orbital bases ( s , s * , 3@xmath7p , and 5@xmath7d ) on real atomic grids ( 20 with spin interactions ) , where the parameter set is fit to reproduce experimentally verified bandgaps , masses , non - parabolic dispersions , hydrostatic and biaxial strain behaviors of bulk materials using a global minimization procedure based on a genetic algorithm and analytical insights @xcite . this @xmath8 tb approach can easily incorporate atomic effects such as surface roughness and random alloy compositions as the model is based on a set of atomic grids . these physical effects have been shown to be critical to the quantitative modeling of resonance tunneling diodes ( rtds ) , quantum dots , disordered sige / si quantum wells , and a single impurity device in si bulk @xcite . the kp approach typically uses four bases on a set of cubic grids with no spin interactions @xcite . while it still fails to predict the indirect gap of bulk dispersions since it assumes that all the subband minima are placed on the @xmath9 point , the credibility is better than the ema since the kp model can still explain the inter - band physics of direct gap iii - v devices , and valence band physics of indirect gap materials such as silicon ( si ) @xcite . one of the important issues in modeling of nanoscale devices , is to solve the quantum transport problem with a consideration of real 3-d device geometries . although the non - equilibrium green s function ( negf ) and wavefunction ( wf ) formalism have been widely used to simulate the carrier transport @xcite , the computational burden has been always a critical problem in solving 3-d open systems as the negf formalism needs to invert a system matrix of a degree - of - freedom ( dof ) equal to the hamiltonian matrix @xcite . the recursive green s function ( rgf ) method saves the computing load by selectively targeting elements needed for the matrix inversion @xcite . however , the cost can be still huge depending on the area of the transport - orthogonal plane ( cross - section ) and the length along the transport direction of target devices @xcite . the wf algorithm also saves the computing load if the transport is ballistic as it does nt have to invert the system matrix and finding a few solutions of the linear system is enough to predict the transport behaviors . but , the load still depends on the size of the system matrix and the number of solution vectors ( modes ) needed to describe the carrier - injection from external leads @xcite . in fact , rgf and wf calculations for atomically resolved nanowire field effect transistors ( fets ) have demonstrated the need to consume over 200,000 parallel cores on large supercomputing clusters @xcite . developed by mamaluy _ @xcite , the contact block reduction ( cbr ) method has received much attention due to the utility to save computing expense required to evaluate the retarded green s function of 3-d open systems . the cbr method is thus expected to be a good candidate for transport simulations since the method does nt have to solve the linear system yet reducing the computing load needed for matrix inversion @xcite . the method indeed has been extensively used such that it successfully modeled electron quantum transport in experimentally realized si finfets @xcite , and predicted optimal design points and process variations in design of 10-nm si finfets @xcite . however , all the successful applications for 3-d systems so far , have been demonstrated only for the systems represented by the ema . while the use of multi - band approaches can increase the accuracy of simulation results , it requires more computing load as a dof of the hamiltonian matrix is directly proportional to the number of bases required to represent a single atomic ( or grid ) spot in the device geometry . to suggest a solution to this _ trade - off _ issue , we examine the numerical utilities of the cbr method in multi - band ballistic quantum transport simulations , focusing on multi - band 3-d systems represented by either of the tb or kp band model . the objective of this work is to provide detail answers to the following questions through simulations of small two - contact ballistic systems focusing on a proof of principles : ( 1 ) can the original cbr method be extended to simulate ballistic quantum transport of multi - band systems ? ( 2 ) if the answer to the question ( 1 ) is @xmath10 , what is the condition under which the multi - band cbr method becomes particularly useful ? , and ( 3 ) how is the numerical practicality of the multi - band cbr method compared to the rgf and wf algorithms , in terms of the accuracy , speed and scalability on high performance computing ( hpc ) clusters ? in real transport problems , a device needs to be coupled with external contacts that allow the carrier - in - and - out flow . with the negf formalism , this can be done by creating an open system that is described with a non - hermitian system matrix @xcite . representing this system matrix as a function of energy , we compute the transmission coefficient and density of states , to predict the current flow and charge profile in non - equilibrium . this energy - dependent system matrix is called the retarded green s function @xmath11 for an open system ( eq . ( 1 ) ) . @xmath12^{-1 } , \indent \eta \rightarrow 0^{+}\ ] ] where @xmath13 is is the hamiltonian representing the device and @xmath14 is the self - energy term that couples the device to external leads . as already mentioned in the previous section , the evaluation of @xmath11 is quite computationally expensive since it involves intensive matrix inversions . the cbr method , however , reduces matrix inversions with the mathematical process based on the dyson equation . we start the discussion revisiting the cbr method that has been so far utilized for ema systems . the cbr method starts decomposing the device domain into two regions : ( 1 ) the boundary region @xmath15 that couples with the contacts , and ( 2 ) the inner region @xmath16 that does nt couple to the contacts . as the self - energy term @xmath14 is non - zero only in the boundary region , @xmath13 and @xmath14 are decomposed as shown in eq . ( 2 ) , where subscripts ( @xmath15 , @xmath16 ) denote above - mentioned regions , respectively . @xmath17 then , @xmath18 can be evaluated with the dyson equation defined in eq . ( 3 ) and eq . ( 4 ) , where @xmath19 and @xmath20 are conditioned with a hermitian matrix @xmath21 to minimize matrix inversions by solving the eigenvalue problem ( eqs . ( 5 ) ) . @xmath22 @xmath23^{-1 } \nonumber \\ & = & \begin{bmatrix } g_c^x & g_{cd}^x \\ g_{dc}^x & g_d^x \end{bmatrix } = \sum_\alpha\frac{|\psi_\alpha\rangle\langle\psi_\alpha|}{e-\epsilon_\alpha+i\eta } \nonumber \end{aligned}\ ] ] where @xmath24 and @xmath25 are the @xmath26 eigenvalue and eigenvector of the modified hamiltonian ( @xmath13+@xmath21 ) . here , we note that the matrix inversion is performed only to evaluate the boundary block @xmath27 ( contact - block ) for one time while the rgf needs to perform the block - inversion many times depending on the device channel length . the computing load for matrix inversion is thus significantly reduced , and the method is also free from solving a linear - system problem . instead , the major numerical issue now becomes a normal eigenvalue problem for a hermitian matrix ( @xmath13+@xmath21 ) . for the numerical practicality , it is thus critical to reduce a number of required eigenvalues , and for ema hamiltonian matrices , a huge reduction in the number of required eigenvalues can be achieved via a smart choice of the _ prescription matrix _ @xmath21 . to find the matrix @xmath21 and see if it can be extended to multi - band systems , we first need to understand how to couple external contacts to the device . fig . 1 illustrates the common approach which treats the contact as a semi - infinite nanowire of a finite cross - section . here , @xmath28 is a block matrix that represents the unit - slab along the transport direction , and @xmath29 is another block matrix which represent the inter - slab coupling . the eigenfunction of the plane wave at the @xmath30 mode in the @xmath31 slab , @xmath32 should then obey the schrdinger equation and the bloch condition ( eqs . ( 6 ) ) . @xmath33 where @xmath34 is the plane - wave vector at the @xmath30 mode , @xmath35 is the length of a slab along the transport direction , and m is the maximum number of plane - wave modes that can exist in a single slab and is equal to the dof of @xmath28 . then , the surface green s function @xmath36 and self - energy term @xmath14 can be evaluated by converting eqs . ( 6 ) to the generalized eigenvalue problem for a complex and non - hermitian matrix @xcite . the solution for @xmath36 and @xmath14 are provided in eqs . ( 7 ) , where @xmath37 and @xmath38 are shown in eqs . @xmath39^{-1}k^{-1 } , \nonumber \\ \sigma & = & w^+g_{surf } w\end{aligned}\ ] ] @xmath40 , \nonumber \\ \lambda & = & diag[exp(ik_1l)\text { } exp(ik_2l)\ldots\text { } exp(ik_ml)]\end{aligned}\ ] ] in systems described by the nearest - neighbor ema , each slab becomes a layer of common cubic grids such that each grid on one layer is coupled to the same grid on the nearest layer . the inter - slab coupling matrix @xmath29 thus becomes a @xmath41 , with which the general solution for @xmath36 and @xmath14 in eqs . ( 7 ) can be simplified using a process described in eq . ( 9 ) and eq . we note that previous literatures have shown only the simplified solution for @xmath36 and @xmath14 @xcite . @xmath42^{-1}k^{-1 } \nonumber \\ & = & k[k^{-1}(h_{b}-ei)k+w^+\lambda]^{-1}k^{-1 } \nonumber \\ & = & k[-k^{-1}(w^+k\lambda+wk\lambda^{-1})+w^+\lambda]^{-1}k^{-1 } \nonumber \\ & & [ \because ( ei - h_{b})k = w^+k\lambda+wk\lambda^{-1 } ] \nonumber \\ & = & -k[w\lambda^{-1}]^{-1}k^{-1 } = -k\lambda w^{-1}k^{-1 } \\ \nonumber \\ \sigma & = & w^+g_{surf}w \nonumber \\ & = & w^+(-k\lambda w^{-1}k^{-1})w \nonumber \\ & = & -w^+k\lambda k^{-1}\text { } ( \because w^+ = w ) \nonumber \\ & = & -wk\lambda k^{-1}\end{aligned}\ ] ] the original cbr method coupled to the ema prescribes the hermitian matrix @xmath21 as @xmath43 or its hermitian component ( if @xmath29 is complex ) . the new self - energy term @xmath44 in eqs . ( 5 ) then becomes ( eq . ( 11 ) ) : @xmath45 where the matrix ( @xmath46 ) becomes zero at @xmath9 point , where ema subband minima are always placed on . the resulting new hamitonian ( @xmath13-@xmath29 ) , becomes the hamiltonian with the generalized @xmath47-@xmath48 boundary condition at contact boundaries . the spectra of the matrix ( @xmath13-@xmath29 ) , therefore become approximate solutions of the open boundary problem , and the retarded green s function @xmath49 in eq . ( 4 ) can be thus @xmath50 with an incomplete set of energy spectra of the hermitian matrix near subband minima @xcite . regardless of the band model , the @xmath49 in eq . ( 4 ) can be accurately calculated with a complete set of spectra since it then becomes the dyson equation ( eq . ( 3 ) ) itself . the important question here is then whether we can make the cbr method be still numerically practical for multi - band systems such that the transport can be simulated with a narrow energy spectrum . to study this issue , we focus on the inter - slab coupling matrix @xmath29 of multi - band systems . a toy si device that consists of two slabs along the [ 100 ] direction , is used as an example for our discussion . fig . 2 shows the device geometry and corresponding hamiltonian matrix built with the ema , kp and tb model , respectively . here , we note that the simplifying process in eq . ( 9 ) and eq . ( 10 ) is not strictly correct if the inter - slab coupling matrix @xmath29 is not an identity matrix , since , for any square matrix @xmath37 and @xmath29 , @xmath51 can not be simplified to @xmath29 if @xmath29 is neither an identity matrix nor a scaled identity matrix . when a system is represented with kp model , a single slab is still a layer of common cubic grids as the kp approach also uses a set of cubic grids . but , the non - zero coupling is extended up to next - nearest neighbors such that the inter - slab coupling matrix @xmath29 is no more an identity matrix . the simplified solution for @xmath52 and @xmath14 , however , can be still used to @xmath53 the general solutions in eqs . ( 7 ) since the coupling matrix @xmath29 is @xmath54 and invertible . but , the situation becomes tricky for tb systems that are represented on a set of real zincblende ( zb ) grids . and @xmath14 are even mathematically invalid . ] in the zb crystal structure , a si unit - slab has a total of four unique atomic layers along the [ 100 ] direction . because the tb approach assumes the nearest - neighbor coupling , only the last layer in one slab is coupled to the first layer in the nearest slab while all the other coupling blocks among layers in different slabs become zero - matrices . as described in fig . 2 , this makes the inter - slab coupling matrix @xmath29 be @xmath55 such that matrix inversions become impossible . the simplified solution for @xmath52 and @xmath14 in eq . ( 9 ) and eq . ( 10 ) are therefore mathematically invalid , and they can not be even used to approximate the full solution ( eqs . ( 7 ) ) . a new prescription for @xmath21 is thus needed to make the cbr method be still practical for zb - tb systems , and we propose an alternative in eq . ( 12 ) . @xmath56 where @xmath57 is the energetic position of the cbm ( valence band maximum ( vbm ) ) of the bandstructure of the semi - infinite contact . if only a few subbands near the cbm ( or vbm ) of the contact bandstructure are enough to describe the external contact , the prescription suggested in eq ( 12 ) works quite well as @xmath21 is the hermitian part of the self - energy term , such that ( @xmath58 ) approximates the open system near the edge of the contact bandstructure , the approximation , however , becomes less accurate if more subbands in higher energy ( in lower energy for valence band ) are involved to the open boundaries . away from the band edge , subband placement becomes denser and inter - subband coupling becomes stronger . the prescription @xmath21 in eq . ( 12 ) then would not be a good choice as it only approximate the open boundary solution near band edges , and the cbr method thus needs more eigenspectrums to solve open boundary transport problems . so , for example , the multi - band cbr method would not be numerically practical to simulate fets at a high source - drain bias , since a broad energy spectrum is then needed to get an accurate solution . before closing this section , we note that , if the inter - slab coupling matrix @xmath29 is either an identity matrix or a scaled identity matrix , the prescription matrix @xmath21 in eq . ( 12 ) becomes @xmath59 to the one utilized to simulate 3-d systems in the previous literatures @xcite , where ( @xmath58 ) approximates the open system well near @xmath60 @xmath61 @xmath62 if the system is represented by the ema @xcite . once @xmath52 and @xmath14 are determined from the prescription matrix @xmath21 , evaluation of the transmission coefficient ( tr ) and the density of states ( dos ) can be easily done @xcite . further detailed mathematics regarding derivation of tr and dos will not be thus discussed here . the results are discussed in two subsections . first , we validate the cbr method for multi - band systems with the new prescription for @xmath21 in eq . focusing on a proof of principles , we compute the tr and dos profiles for a toy tb and kp system , compare the result to the references obtained with the rgf algorithm , and suggest the device category where the multi - band cbr method could be particularly practical . second , we examine the numerical practicality of the multi - band cbr method by computing tr and dos profiles of a resonant tunneling device and a nanowire fet . the accuracy , the speed of calculations in a serial mode , and the scalability on hpc clusters , are compared to those obtained with the rgf and wf algorithm . we assume a two - contact ballistic transport for all the numerical problems . to validate the multi - band cbr method that has been discussed in the previous section , we consider two multi - band toy si systems represented by the 10-band @xmath8 tb and 3-band kp approach . here , we intentionally choose extremely small systems to calculate a complete set of energy spectra of the hamiltonian , with which the cbr method should produce results identical to the ones obtained by the rgf algorithm . for the tb system , the electron - transport is simulated while we calculate the hole - transport for the kp system due to a limitation of the kp approach in representing the si material @xcite . _ tb system _ : fig . 3 illustrates the tr and dos profile calculated for the tb si toy device which consists of ( 2@xmath72@xmath72 ) ( 100 ) unit - cells ( @xmath631.1(nm ) ) . the device involves a complex hermitian hamiltonian matrix of 640 dof , and electrons are assumed to transport along the [ 100 ] direction . the tr and dos profiles are calculated using the cbr method for a total of three cases - with 6 , 60 and full ( 640 ) energy spectra that correspond to 1@xmath64 , 10@xmath64 , and 100@xmath64 of the hamiltonian dof , respectively . the transport happens at the energy above 2.32(ev ) which is the cbm of the contact bandstructure . we note that this energetic position is higher than the si bulk cbm ( 1.13(ev ) ) , due to the structural confinement stemming from the finite cross - section of the nanowire device @xcite . with the new prescription matrix @xmath21 suggested in eq . ( 12 ) , the tr and dos profile obtained by the cbr method become closer to the reference result as more spectrums are used , and eventually reproduce the reference result with a full set of spectrums , as shown in the left column of fig . 3 . here , the cbr result turns out to be quite accurate near the cbm even with 1@xmath64 of the total spectrums , indicating that the tb - cbr method could be a practical approach if most of the carriers are injected from the first one or two subbands of the contact bandstructure . this condition can be satisfied when ( 1 ) only the first one or two subbands in the contact bandstructure are occupied with electrons , and ( 2 ) the energy difference between the source and drain contact fermi - level ( the source - drain fermi - window ) becomes extremely narrow . so , the simulation of fets at a high source - drain bias would not be an appropriate target of the tb - cbr simulations since the source - drain fermi - window may include many subbands , and many spectra may be thus needed for accurate solutions @xcite . instead , we propose that rtds could be one of device categories for which the tb - cbr method is particularly practical , since the fermi - window for transport becomes extremely small in rtds in some cases @xcite . the same calculation is performed again but using the old prescription @xmath21 suggested for the ema , and corresponding tr and dos profiles are shown in the right column of fig . 3 . the cbr method still reproduce the reference result with a full set of energy spectra since the dyson equation ( eq . ( 4 ) ) should always work for any @xmath21 s . the accuracy of the results near the cbm , however , turns out to be worse than the one with the new prescription . the results furthermore reveal that the accuracy with 10@xmath64 of the total spectra does not necessarily becomes better than the one with 1@xmath64 , indicating that the old prescription for @xmath21 can not even approximate the solution near the cbm of open tb systems . _ kp system _ : the tr and dos profile of the kp si 2.0(nm ) ( 100 ) cube , are depicted in fig . the structure is discretized with a 0.2(nm ) grid and involves a complex hermitian hamiltonian of 3,000 dof . here , the dof of the real - space kp hamiltonian can be effectively reduced with the @xmath65-@xmath66 approach @xcite . the effective dof of the hamiltonian therefore becomes 500 , where we consider 50 modes per each slab along the transport direction . again , we note that the vbm of the contact bandstructure is placed at -0.4(ev ) , and lower in energy than the vbm of si bulk ( 0(ev ) ) due to the confinement created by the finite cross - section . we claim that the cbr method works quite well for the kp system , since the tr and dos profiles not only become closer to the reference results as more of the energy spectrums are used , but also exhibit excellent accuracy near the vbm of the contact bandstructure as shown in fig . we , however , observe a remarkable feature that is not found in the cbr method coupled to tb systems : the kp - cbr method shows a good accuracy with both the old and new prescription matrix @xmath21 , which supports that the simplified solution for @xmath36 and @xmath14 ( eq . ( 9 ) and eq . ( 10 ) ) are still useful to approximate the full solution ( eqs . ( 7 ) ) as discussed in the previous section . we also claim that the utility of the kp - cbr method could be extended to nanowire fets because the mode - space approach reduces the dof of the hamiltonian such that we save more computing cost needed to calculate energy spectra . in the next subsection , we will come back to this issue again . ) , with respect to the si bulk , is superposed to the channel potential profile to consider the sharp structural confinement stemming from the single donor . ] in this subsection , we provide a detailed analysis of the numerical utility of the multi - band cbr method in terms of the accuracy and speed . based on discussions in the previous subsection with a focus on a proof of principles on small systems , a rtd is considered as a simulation example of tb systems , while a nanowire fet is again used as an example of kp systems to discuss the numerical practicality of the method . the tr and dos profiles obtained by the rgf and wf algorithm are used as reference results . we note that the wf case is added in this subsection to provide a complete and competitive analysis on the speed and scalability on hpc clusters . _ tb system _ : a single phosphorous donor in host si material ( si : p ) creates a 3-d structural confinement around itself . such si : p @xmath67 @xmath68 have gained scientific interest due to their potential utility for qubit - based logic applications @xcite . especially , the stark effect in si : p quantum dots is one of the important physical problems , and was quantitatively explained by previous tb studies @xcite . the electron - transport in such si : p systems should be therefore another important problem that needs to be studied . the geometry of the example si : p device is illustrated in fig . , we consider a [ 100 ] si nanowire that is 14.0(nm ) long and has a 1.7(nm ) rectangular cross - section . the first and last 3.0(nm ) along the transport direction , are considered as densely n - type doped source - drain region assuming a 0.25(ev ) band - offset in equilibrium @xcite . then , a single phosphorous atom is placed at the channel center with a superposition of the impurity coulombic potential that has been calibrated for a single donor in si bulk by by rahman _ the electronic structure has a total of 1872 atoms and involves a complex hamiltonian matrix of 18,720 dof . 6 shows the tr and dos profiles in four cases , where the first three cases are the cbr results with 10 , 20 and 40 spectra that correspond to 0.05@xmath64 , 0.1@xmath64 , and 0.2@xmath64 of the hamiltonian dof , and the last one is used as a reference . due to the donor coulombic potential , the channel forms a double - barrier system such that the electron transport should experience a resonance tunneling . as shown in fig . 5 , the cbr method produces a nice approximation of the reference result such that the first resonance is observed with just 10 energy spectra . it also turns out that 40 spectra are enough to capture all the resonances that show up in the range of energy of interest . the accuracy of the solutions approximated by the cbr method , is examined in a more quantitative manner by @xmath69 the tr and dos profile over energy . fig . 7 illustrates this @xmath70 tr ( ctr ) and dos ( cdos ) profile , which are @xmath71 equivalent to the current and charge profile , respectively . in spite of a slight deviation in absolute values , the ctr profiles still confirm that the cbr method captures resonances quite precisely such that the energetic positions where the tr sharply increases , are almost on top of the reference result . the cdos profile exhibits much better accuracy such that the result with 40 spectra almost reproduces the reference result even in terms of absolute values . we claim that the accuracy in the cdos profile is particularly critical , since it is directly connected to charge profiles that are essential for charge - potential self - consistent simulations . , turn out to be enough to almost reproduce the reference solutions in the entire range of energy of interest ( 0.8(ev ) beyond the vbm of the wire bandsturcutrue ) . ] _ kp system _ : si nanowire fets obtained through top - down etching or bottom - up growth have attracted attention due to their enhanced electrostatic control over the channel , and thus become an important target of various modeling works @xcite . for kp systems , the cbr method could become a practical approach to solve transport behaviors of fet devices since the computing load for solving eigenvalue problems can be reduced with the mode - space approach . a [ 100 ] si nanowire fet of a 15.0(nm ) long channel and a 3.0(nm ) rectangular cross - section , is therefore considered as a simulation example to test the performance of the kp - cbr method . the hole - transport is simulated with the 3-band kp approach , where the simulation domain is discretized with a set of 0.2(nm ) mesh cubic grids and involves a real - space hamiltonian matrix of 50,625 dof . as the device has a total of 75 slabs along the transport direction , the mode - space hamiltonian has 9,000 dof with a consideration of 120 modes per slab . it has been reported that the wire bandstructure obtained with 120 modes per slab , becomes quite close to the full solution for a cross - section smaller than [email protected](nm@xmath72 ) @xcite . the wire is assumed to be purely @xmath73 such that neither the doping nor band - offset are considered . to see if the cbr method can be reasonably practical in simulating the hole - transport at a relatively large source - drain bias , we plan to cover the energy range at least larger than 0.4(ev ) beyond the vbm of the wire bandstucture . for this purpose , we compute 50 , 100 , and 200 energy spectra that correspond to 0.5@xmath64 , 1.1@xmath64 , and 2.2@xmath64 of the dof of the mode - space hamiltonian , respectively . fig . 8(a ) shows the corresponding tr and dos profiles . here , the cbr solution not only become closer to the reference result with more spectra considered , but also demonstrate fairly excellent accuracy near the vbm of the wire bandstructure . the ctr and cdos profiles provided in fig . 8(b ) further support the preciseness of the cbr solutions near the vbm . the cumulative profiles also support that the cbr solution covers a relatively wide range of energy , such that 50 energy spectra are already enough to cover @xmath630.4(ev ) below the vbm quite well . we note that the solution obtained with 200 spectra almost replicates the reference result in the entire range of energy that is considered for the simulation ( @xmath630.8(ev ) below the vbm ) . . the time required to evaluate the tr and dos per single energy point in a serial mode , for the rtd and nanowire fet considered as simulation examples . [ cols="<,<,<,<",options="header " , ] _ speed and scalability on hpcs _ : so far , we have discussed the practicality of the multi - band cbr method focusing on the accuracy of the solutions for two - contact , ballistic - transport problems . another important criterion to determine the numerical utility should be the speed of calculations . we therefore measure the time needed to evaluate the tr and dos per single energy point for the tb si : p rtd and the kp si nanowire fet represented that are utilized as simulation examples . to examine the practicality of the multi - band cbr method on hpc clusters , we also benchmark the scalability of the simulation time on the @xmath74 cluster under the support of the rosen center for advanced computing ( rcac ) at purdue university . the cbr , rgf , and wf methods are parallelized with mpi / c++ , the multifrontal massively parallel sparse direct linear solver ( mumps ) @xcite , and a self - developed eigensolver based on the shift - and - invert arnoldi algorithm @xcite . all the measurements are performed on a 64-bit , 8-core hp proliant dl585 g5 system of 16 gb sdram and 10-gigabit ethernet local to each node . table i summarizes the wall - times measured for various methods in a serial mode . generally , the simulation of the kp si nanowire fet needs less computing loads , such that the wall - times are reduced by a factor of two with respect to the computing time taken for the tb si : p rtd . this is because the kp approach can represent the electronic structure with the mode - space approach such that the hamiltonian matrix has a smaller dof ( 9,000 ) , compared to the one used to describe the tb si : p rtd ( 18,720 ) . compared to the rgf algorithm in a serial mode , the cbr method demonstrates a comparable ( kp ) , or better ( tb ) performance . since a single slab of the kp si nanowire is represented with a block matrix @xmath75 ( fig . 1 ) of 120 dof , the matrix inversion is not a critical problem any more in the rgf algorithm such that the cbr method does nt necessarily show better performances than the rgf algorithm . the tb example device , however , needs a @xmath75 of 720 dof to represent a single slab ( a total of 26 slabs ) so the burden for matrix inversions become bigger compared to the kp example . as a result , the cbr method generally shows better performances . the cbr method , however , does nt beat the wf method in both the tb and kp case since , in a serial mode , the cbr method consumes time to allocate a huge memory space that is needed to store `` full '' complex matrices via vector - products ( eqs . ( 5 ) ) . the strength of the cbr method emerges in a @xmath76 mode ( on multiple cpus ) , where the vector - products are performed via mpi - communication among distributed systems and each node thus saves only a fraction of the full matrix . the scalability of the various methods is compared up to a total of 16 cpus in fig . the common rgf calculation can be effectively parallelized only up to a factor of two , due to its recursive nature @xcite , and the scalability of the wf method becomes worse in many cpus because it uses a direct - solver - based lu factorization to solve the linear system . as a result , the cbr method starts to show the best speed when more than 8 cpus are used . in this work , we discuss numerical utilities of the cbr method in simulating ballistic transport of multi - band systems described by the the atomic 10-band @xmath8 tb and 3-band kp approach . although the original cbr method developed for single - band ema systems achieves an excellent numerical efficiency by approximating solutions of open systems , we show that the same approach ca nt be used to approximate tb systems as the inter - slab coupling matrix becomes singular . we therefore develop an alternate method to approximate open system solutions . focusing on a proof of principles on small systems , we validate the idea by comparing the tr and dos profile to the reference result obtained by the rgf algorithm , where the alternative also works well with the kp approach . since the major numerical issue in the cbr method is to solve a normal eigenvalue problem , the numerical practicality of the method becomes better as the transport can be solved with a less number of energy spectra . generally , the practicality would be thus limited in multi - band systems , since multi - band approaches need a larger number of spectra to cover a certain range of energy than the single band ema does . we , however , claim that the rtds could be one category of tb devices , for which the multi - band cbr method becomes particularly practical in simulating transport , and the numerical utility can be even extended to fets when the cbr method is coupled to the kp band model . to support this argument , we simulate the electron resonance tunneling in a 3-d tb rtd , which is basically a si nanowire but has a single phosphorous donor in the channel center , and the hole - transport of a 3-d kp si nanowire fet . we examine numerical practicalities of the multi - band cbr method in terms of the accuracy and speed , with respect to the reference results obtained by the rgf and wf algorithm , and observe that the cbr method gives fairly accurate tr and dos profile near band edges of contact bandstructures . in terms of the speed in a serial mode , the strength of the cbr method over the rgf algorithm depends on the size of the hamiltonian such that the cbr shows a better performance than the rgf as a larger block - matrix is required to represent the unit - slab of devices . but , the speed of the wf method is still better than the cbr method as the cbr method consumes time to store a full complex matrix during the process of calculations . in a parallel mode , however , the cbr method starts to beat both the rgf and wf algorithm since the full matrix can be stored into multiple clusters in a distributive manner , while the scalability of both the rgf and wf algorithm are limited due to the nature of recursive and direct - solver - based calculation , respectively . h. ryu , h .- h . park and g. klimeck acknowledge the financial support from the national science foundation ( nsf ) under the contract no . 0701612 and the semiconductor research corporation . m. shin acknowledges the financial support from basic science research program through the national research foundation of republic of korea , funded by the ministry of education , science and technology under the contract no . 2010 - 0012452 . authors acknowledge the extensive use of computing resources in the rosen center for advanced computing at purdue university , and nsf - supported computing resources on nanohub.org . 100 g. e. moore , _ electronics _ * 38 * , 114 , ( 1965 ) . m. luisier , a. schenk and w. fichtner , _ phys . b _ * 74 * 205323 , ( 2006 ) . g. klimeck , s. s. ahmed , h. bae , n. kharche , r. rahman , s. clark , b. haley , s. lee , m. naumov , h. ryu , f. saied , m. prada , m. korkusinski and t. b. boykin , _ ieee trans . _ * 54 * , 2079 , ( 2007 ) . y. x. liu , d. z. -y . ting and t. c. mcgill , _ phys . b _ * 54 * , 5675 , ( 1996 ) . g. klimeck , r. lake , r. c. bowen , c. fernando and w. frensley , _ vlsi design _ * 8 * , 79 , ( 1998 ) . t. b. boykin , g. klimeck and f. oyafuso , _ phys . b _ , * 69 * 115201 , ( 2004 ) . r. c. bowen , g. klimeck , w. r. frensley , r. k. lake , _ j. appl _ * 81 * , 3207 , ( 1997 ) n. kharche , m. prada , t. b. boykin and g. klimeck , _ appl * 90 * , 9 , ( 2007 ) . r. rahman , c. j. wellard , f. r. bradbury , m. prada , j. h. cole , g. klimeck and l. c. l. hollenberg , _ phys . lett . _ * 99 * , 036403 , ( 2007 ) . g. p. lansbergen , r. rahman , c. j. wellard , i. woo , j. caro , n. collaert , s. biesemans , g. klimeck , l. c. l. hollenberg and s. rogge , _ nature physics _ * 4 * , 656 , ( 2008 ) . m. shin , _ j. appl . phys . _ * 106 * , 054505 , ( 2009 ) . c. pryor , _ phys . b _ * 57 * , 7190 , ( 1998 ) s. datta , _ superlatt . microstruct . _ * 28 * , 253 , ( 2000 ) . m. stdele , b. r. tuttle and k. hess , _ j. appl . phys . _ * 89 * , 348 , ( 2001 ) . n. kharche , g. klimeck , d. -h . kim , j. a. del alamo and m. luisier , _ proceedings of ieee international electron devices meeting _ , ( 2009 ) . r. lake , g. klimeck , r. c. bowen and d. jovanovic , _ j. appl . * 81 * , 7845 , ( 1996 ) . c. rivas and r. lake , _ phys . ( b ) _ * 239 * , 94 , ( 2003 ) . s. cauley , j. jain , c. -k . koh and v. balakrishnan , _ j. appl . phys . _ * 101 * , 123715 , ( 2007 ) . d. mamaluy , m. sabathil , t. zibold , p. vogl and d. vasileska , _ phys . b _ * 71 * , 245321 , ( 2005 ) . g. klimeck and m. luisier , _ computing in science and engineering _ * 12 * , 28 , ( 2010 ) . d. mamaluy , d. vasileska , m. sabathil and p. vogl , _ semicond . sci . tech _ * 19 * , 118 , ( 2004 ) . h. r. khan , d. mamaluy and d. vasileska , _ ieee trans . dev . _ * 54 * , 784 , ( 2007 ) . h. r. khan , d. mamaluy and d. vasileska , _ ieee trans . dev . _ * 55 * , 743 , ( 2008 ) . h. r. khan , d. mamaluy and d. vasileska , _ ieee trans . elec . dev . _ * 55 * , 2134 , ( 2008 ) . d. mamaluy , m. sabathil and p. vogl , _ j. appl . phys . _ * 93 * , 4628 , ( 2003 ) . in principle , the 4-band kp model uses a total of four bases to model direct bandgap materials such as gaas and inas , where one basis is used to model the conduction band and the remaining three bases are used to model the valence band . the valence band of indirect bandgap materials such si , however , can be still modeled with three bases if vbm is at the @xmath9 point ( ref . both the tb and kp model considered in this work place the vbm of si bulk at 0 ( ev ) . assuming that the source contact is grounded , the fermi - window at @xmath77 = v , becomes [ @xmath78-@xmath79 , @xmath80+@xmath79 ] = [ @xmath82-@xmath83-@xmath79 , @xmath82+@xmath79 ] , where @xmath84 is the single electron charge and @xmath82 is the fermi - level of the system in equilibrium . the maximum and minimum of the window are determined at the source and drain side , respectively . e. lind , b. gustafson , i. pietzonka and l. -e . wernersson , _ phys . b _ * 68 * , 033312 , ( 2003 ) . l. c. l. hollenberg , a. s. dzurak , c. j. wellard , a. r. hamilton , d. j. reilly , g. j. milburn and r. g. clark , _ phys . b _ * 69 * , 113301 , ( 2004 ) . the band - offset between the intrinsic channel and densely doped source - drain leads , is taken from the work of martinez _ et al . _ , where the equilibrium potential profile has been self - consistently obtained for a 14.0(nm ) long [ 100 ] si nanowire that has a 2.0(nm ) rectangular cross - section and 4.0(nm ) long source - drain regions ( ref . [ 32 ] ) . a. martinez , n. seoane , a. r. brown , j. r. barker and a. asenov , _ ieee trans . * 8 * , 603 , ( 2009 ) . n. neophytou , a. paul , m. s. lundstrom and g. klimeck , _ ieee trans . dev . _ * 55 * , 1286 , ( 2008 ) . v. mehrmann and d. watkins , _ siam j. sci . * 22 * , 1905 , ( 2001 ) .

numerical utilities of the contact block reduction ( cbr ) method in evaluating the retarded green s function , are discussed for 3-d multi - band open systems that are represented by the atomic tight - binding ( tb ) and continuum @xmath0@xmath1@xmath2 ( kp ) band model . it is shown that the methodology to approximate solutions of open systems which has been already reported for the single - band effective mass model , can not be directly used for atomic tb systems , since the use of a set of zincblende crystal grids makes the inter - coupling matrix be non - invertible . we derive and test an alternative with which the cbr method can be still practical in solving tb systems . this @xmath3-@xmath4 @xmath5 method is validated by a proof of principles on small systems , and also shown to work excellent with the kp approach . further detailed analysis on the accuracy , speed , and scalability on high performance computing clusters , is performed with respect to the reference results obtained by the state - of - the - art recursive green s function and wavefunction algorithm . this work shows that the cbr method could be particularly useful in calculating resonant tunneling features , but show a limited practicality in simulating field effect transistors ( fets ) when the system is described with the atomic tb model . coupled to the kp model , however , the utility of the cbr method can be extended to simulations of nanowire fets .

1112.3124

weak decays of charmed and beautiful hadrons are quite favorable in particle physics because of their usage in determining fundamental parameters of the standard model and testing various theories and models . among these heavy hadron decays the semileptonic decays @xmath4 and @xmath5 have been observed experimentally . these exclusive decays provide one of the main channels to determine the important ckm matrix element @xmath1 . the difficulty in studying @xmath6 and @xmath5 decays mainly concerns the calculation of the relevant hadronic matrix elements of weak operators , or , equivalently , the corresponding form factors which contain nonperturbative contributions as well as perturbative ones and are beyond the power of pure qcd perturbation theory . up to present these form factors are usually evaluated from lattice calculations , qcd sum rules and some hadronic models . sum rule method has been applied to @xmath7 decay in the full qcd and provided reasonable results@xcite . since the meson b contains a single heavy quark , it is expected that its exclusive decays into light mesons may also be understood well in the effective theory of heavy quark , which explicitly demonstrates the heavy quark spin - flavor symmetry and its breaking effects can systematically be evaluated via the power of inverse heavy quark mass @xmath8 . the effective theory of heavy quark has been widely applied to heavy hadron systems , such as b decays into heavy hadrons via both exclusive and inclusive decay modes . there are two different versions of effective theory of heavy quark . one is the heavy quark effective theory ( hqet ) , which generally decouples the `` quark fields '' and `` antiquark fields '' and treats one of them independently . this treatment is only valid when taking the heavy quark mass to be infinite . in the real world , mass of quark must be finite , thus one should keep in the effective lagrangian both the effective quark and effective antiquark fields . based on this consideration , a heavy quark effective field theory ( hqeft ) @xcite has been established and investigated with including the effects of the mixing terms between quark and antiquark fields . its applications to the pair annihilation and creation have also been studied in the literature@xcite . though the hqeft explicitly deviate from hqet from the next - to - leading order , these two formulations of effective theory trivially coincide with each other at the infinite heavy quark mass limit . in our knowledge the exclusive heavy to light ( pseudoscalar ) decay channels have been discussed in @xcite , where the matrix elements in the effective theory have been formulated , but the two leading order wave functions have not been calculated . in this paper we focus on the calculation of the leading order wave functions of @xmath6 decay by using the light cone sum rule in the effective theory of heavy quark . as an important application , @xmath1 is extracted . in section 2 , the heavy to light matrix element is represented by two heavy quark independent wave functions a and b. in section 3 , we derive the light cone sum rules for the calculation of a and b. in section 4 , we present the numerical results and extract @xmath1 . our short summary is drawed in the last section . the matrix elements responsible for @xmath6 decay is @xmath10 , where b is the beautiful quark field in full qcd . it is generally parametrized by two form factors as follows , @xmath11 in the effective theory of heavy quark , matrix elements can be analyzed order by order in powers of the inverse of the heavy quark mass @xmath8 and also be conveniently expressed by some heavy spin - flavor indenpendent universal wave functions @xcite . here we adopt the following normalization of the matrix elements in full qcd and in the effective theory @xcite : @xmath12 where @xmath13 , and @xmath14 is the heavy flavor independent binding energy reflecting the effects of the light degrees of freedom in the heavy hadron . @xmath15 is the effective heavy quark field in effective theory . associate the heavy meson state with the spin wave function @xmath16 we can analyze the matrix element in effective theory by carrying out the trace formula : @xmath17\end{aligned}\ ] ] with @xmath18 , \nonumber\\ \hat{p}^\mu&=&\frac{p^\mu}{v\cdot p } \;\;.\end{aligned}\ ] ] a and b are the leading order wave functions characterizing the heavy - to - light - pseudoscalar transition matrix elements in the effective theory . they are heavy quark mass independent , but are functions of the variable @xmath19 and the energy scale @xmath20 as well . nevertheless , since the discussion in the present paper is rrestricted within the tree level , we neglect the @xmath20 dependence from now on . combining eqs . ( [ fdef])-([abdef ] ) , one gets @xmath21 where the dots denote higher order @xmath8 contributions which will not be taken into account in the present paper . note that we have used different variables for @xmath22 , @xmath23 and @xmath24 , @xmath25 . the relation between the variables @xmath19 and @xmath26 is @xmath27 the qcd sum rule based on short distance expansion has been proved to be quite fruitful in solving a variety of hadron problems . nevertheless , it is also well known that this method meets difficulties in the case of heavy to light transition because the coefficients of the subleading quark and quark - gluon condensate with the heavy quark mass terms grow faster than the perturbative contribution , which implies the breakdown of the short distance operator product expansion ( ope ) in the heavy mass limit . alternatively , it has been found that heavy to light decays can be well studied by light cone sum rule approach , in which the corresponding correlators are expanded near the light cone in terms of meson wave functions . in this way the nonperturbative contributions are embeded in the meson wave functions instead of the vacuum condensates in the short distance ope sum rule . though there are some differences in the techniques of calculation , the two sum rule methods are based on the same idea of quark - hadron duality and dispersion relation , and furthermore , they follow the same procedure in deriving form factors . for @xmath6 decay , one may consider the vacuum - pion correlation function @xmath28 here @xmath29 and @xmath30 are momenta carried by the pion and leptons . the b meson has momentum @xmath31 . inserting a complete set of states with b meson quantum numbers , we obtain the phenomenological representation @xmath32 with the normalization relation in ( [ normalization ] ) , the matrix elements in ( [ cor ] ) can be expanded into the ones in effective theory of heavy quark in powers of @xmath33 . when all higher @xmath33 order contributions are neglected , ( [ cor ] ) reduces straightforwardly into @xmath34 where @xmath35 is the heavy hadron s residual momentum , @xmath36 . the first term in ( [ phen ] ) is a pole contribution obtained by using ( [ parinhqet ] ) together with the parametrization @xmath37\end{aligned}\ ] ] with f being the leading order decay constant of b meson in effective theory @xcite . the second term in ( [ phen ] ) is the higher resonance contributions given in the form of an integral over the physical spectral density @xmath38 . note that the lorentz indices of the second term in ( [ phen ] ) are not written explicitly but embeded in @xmath38 . on the other hand , the correlator can be calculated and expressed as the form of an integration over the theoretic spectral density @xmath39 , which equals to @xmath38 under the assumption of quark - hadron duality . namely , the correlator ( [ correlator ] ) can be written as @xmath40 equating ( [ phen ] ) and ( [ theo ] ) yields @xmath41 so the next step involves the calculation of ( [ correlator ] ) in the framework of effective theory of heavy quark . substituting the heavy hadron states and heavy quark fields into the effective ones in the effective theory , and then performing the corresponding momentum shift @xmath42 , we obtain when neglecting higher @xmath8 order corrections @xmath43 in the light cone sum rule approach , one should contract the heavy quark fields and expand the correlator into a series in powers of the twist of light cone pion wave functions . these light cone wave functions provide an alternative treatment besides the vacuum condensates . they have been discussed in detail in many references @xcite . up to twist 4 , the pion wave functions relevant to @xmath6 decay are defined as follows , @xmath44\nonumber\\ & + & f_\pi ( x^\mu-\frac{x^2 p^\mu}{x\cdot p } ) \int^1_0 du e^{iup\cdot x } g_2(u ) , \nonumber\\ < \pi(p)|\bar{u}(x)i \gamma^5 d(0)|0>&=&\frac{f_\pi m^2_\pi}{m_u+m_d } \int^1_0 du e^{iup\cdot x } \phi_p(u ) , \nonumber\\ < \pi(p)|\bar{u}(x ) \sigma_{\mu\nu } \gamma^5 d(0)|0>&= & i(p_\mu x_\nu - p_\nu x_\mu ) \frac{f_\pi m^2_\pi}{6 ( m_u+m_d ) } \int^1_0 du e^{iup\cdot x } \phi_\sigma(u).\end{aligned}\ ] ] @xmath45 is the leading twist 2 wave function . @xmath46 and @xmath47 are twist 3 wave functions , while @xmath48 and @xmath49 are wave functions of twist 4 . using the propagator @xmath50 for the contraction of the effective heavy quark fields , we get @xmath51 \nonumber\\ & + & \hat{p}^\mu y[-i\phi_\pi - it^2 g_1(u)-\frac{t}{y}g_2(u)+\frac{t}{6}\mu_\pi \phi_\sigma(u ) ] \}\end{aligned}\ ] ] with @xmath52 and @xmath53 . in order to proceed , we perform a wick rotation of the t axis and then apply the borel transformation @xmath54 to ( [ corresult ] ) . the result is @xmath55 \nonumber\\ + \hat{p}^\mu y[-\phi_\pi(u)+\frac{4}{t^2}g_1(u)+\frac{2}{yt}g_2(u)-\frac{1}{3t}\mu_\pi \phi_\sigma(u ) ] \}.\end{aligned}\ ] ] in deriving this equation we have used the feature of borel transformation : @xmath56 it is found that the spectral function used in sum rule can be obtained by performing a continuous double borel transformation on the amplitude itself @xcite . in order to get the spectral function @xmath57 , we now carry out the continuous borel transformations as follows @xmath58 the result is @xmath59 + \hat{p}^\mu y[-\phi_{\pi } ( u)+\frac{1}{y^2}\frac{\partial^2}{\partial u^2}g_1(u ) \nonumber\\ & -&\frac{1}{y^2 } \frac{\partial}{\partial u } g_2(u ) + \frac{\mu_\pi}{6y}\frac{\partial}{\partial u } \phi_\sigma(u ) ] \}_{u=1-\frac{s}{2y}}.\end{aligned}\ ] ] in the derivation of ( [ spectralfun ] ) , @xmath60 has been first expressed as a derivative of the exponent in ( [ btcor ] ) over u , and then the method of integration by parts over u has been used . ( [ phentheo ] ) and ( [ spectralfun ] ) immediately yield : @xmath61_{u=1-\frac{s}{2y}},\nonumber\\ b(y)&=&-\frac{f_\pi}{4 f } \int^{s_0}_{0 } ds e^{\frac { 2\bar\lambda_b - s}{t } } [ -\phi_{\pi } ( u)+\frac{1}{y^2}\frac{\partial^2}{\partial u^2}g_1(u ) -\frac{1}{y^2 } \frac{\partial}{\partial u } g_2(u ) + \frac{\mu_\pi}{6y}\frac{\partial}{\partial u } \phi_\sigma(u ) ] _ { u=1-\frac{s}{2y}},\end{aligned}\ ] ] for the light cone wave functions appearing in the sum rules ( [ sr ] ) , we take @xcite @xmath62+\frac{15}{8}a_4 [ 21 ( 2u-1)^4 - 14 ( 2u-1)^2 + 1]\ } , \nonumber\\ \phi_p(u)&= & 1+\frac{1}{2}b_2 [ 3(2u-1)^2 - 1]+\frac{1}{8}b_4 [ 35 ( 2u-1)^4 -30 ( 2u-1)^2 + 3 ] , \nonumber\\ \phi_\sigma(u)&= & 6u(1-u)\ { 1+\frac{3}{2}c_2 [ 5(2u-1)^2 - 1]+\frac{15}{8}c_4 [ 21 ( 2u-1)^4 - 14(2u-1)^2 + 1]\ } , \nonumber\\ g_1(u)&= & \frac{5}{2}\delta^2 u^2 ( 1-u)^2+\frac{1}{2}\epsilon \delta^2 [ u(1-u)(2 + 13u ( 1-u)+10u^3 \log u(2 - 3u+\frac{6}{5}u^2 ) \nonumber\\ & + & 10(1-u)^3 \log((1-u)(2 - 3(1-u)+\frac{6}{5}(1-u)^2 ) ) ] , \nonumber\\ g_2(u)&=&\frac{10}{3 } \delta^2 u(1-u)(2u-1).\end{aligned}\ ] ] the asymptotic form of these functions and the scale dependence are given by perturbative qcd @xcite . for the convenience of comparison , we use the same values for the parameters as in @xcite , @xmath63 @xmath64 is the appropriate scale set by the typical virtuality of the beautiful quark , @xmath65 besides all these parameters the numerical analysis of the sum rules ( [ sr ] ) needs also the hadron quantities @xmath66 , @xmath67 , @xmath68 and @xmath69 . these quantities have been studied via sum rules and other approaches by several groups . with the values @xmath70gev , @xmath71gev@xcite , @xmath72gev and @xmath73 @xcite , we get from eqs.([sr ] ) the results for a and b given in the figures fig.1 - 4 . in these figures a and b are shown as functions of t and @xmath74 . we are mainly interested in the range of @xmath75gev , where both the twist 4 corrections and the contributions from excited and continuum states do not exceed 30% . it is seen that the curves in fig.1 and fig.2 are quite stable in this range for the threshold energy @xmath76gev . in fig.3 and fig.4 , a and b become rather stable with respect to the variation of @xmath74 when @xmath77gev . however , they become unstable at small @xmath78 , which corresponds to large momentum transfer @xmath26 . this is in expectation because the light cone expansion and the sum rule method would break down as @xmath26 approaches near @xmath79 @xcite . we also derive @xmath80 and @xmath81 from @xmath82 and @xmath83 by using the relations in ( [ relation ] ) and the beautiful quark mass @xmath84gev . the results are shown in fig.5 - 6 . it is readly seen that when the momentum transfer @xmath26 grows large ( e.g. over @xmath85 for the curve of @xmath86gev in fig.5 - 6 ) , the values of @xmath22 and @xmath23 derived from sum rules become rather unstable and should not be trusted . in order to predict the decay width and @xmath1 , one should have knowledge on the behavior of form factors in the whole kinematically accessible region . now for large momentum transfer we have the single pole approximation @xcite @xmath87 the couplings @xmath88 and @xmath89 have been studied in previous papers . here we would use @xmath90gev , @xmath91gev and @xmath92@xcite . next we write @xmath80 as @xmath93 and fit the parameters @xmath94 and @xmath95 by using the sum rules and eq . ( [ sinpole ] ) . for the threshold @xmath86gev , we choose proper @xmath94 and @xmath95 to make ( [ fitform ] ) approach the sum rule results at @xmath96 but compatible with eq . ( [ sinpole ] ) at @xmath97 . our favorable parameters are @xmath98 the values of @xmath80 at @xmath99gev calculated from ( [ sinpole ] ) , ( [ fitform ] ) and the light cone sum rules are shown in fig.7 . it is found that the single pole model extrapolation matches quite well with the direct estimation from our light cone sum rules ( [ sr ] ) at intermediate momentum transfer around @xmath100 . this implies that our discription of @xmath80 by ( [ sinpole ] ) together with the sum rules ( [ sr ] ) ( but in different applicable regions ) is self - consistent . for the lepton @xmath101 , the lepton mass @xmath102 may be safely neglected , and the decay width of @xmath4 has the distribution on momentum transfer @xmath26 as follows @xmath103 ^ 2.\end{aligned}\ ] ] here @xmath104 is the pion energy in the b meson rest frame . with the pion mass @xmath105gev and the parametrizations of ( [ fitform ] ) , we obtain the integrated width @xmath106 the error in eq.([width ] ) results from the variation of the threshold energy in @xmath107gev . from the branching fraction measured by cleo collaboration @xcite , @xmath108 and the world average of the @xmath109 lifetime @xcite , @xmath110 , one has @xcite @xmath111 comparison of ( [ width ] ) and ( [ cleo ] ) yields @xmath112 at the same time . these effects evidently worsen the accuracy of our extraction of @xmath1 . by taking into account this uncertainty , we arrive at the following result latexmath:[\[\begin{aligned } \label{vub } this estimate is in good agreement with that derived from full qcd calculation @xcite : latexmath:[\[\begin{aligned } \label{vubqcd } furthermore , the value of @xmath1 obtained in eq.([vub ] ) is also close to the one given by cleo @xcite , @xmath114 transitions . in this paper we have studied @xmath4 decay by using the light cone sum rule approach within the framework of effective theory for heavy quark . two leading order wave functions in the effective theory with infinite mass limit have been calculated . the important ckm matrix element @xmath1 has been extracted and its value has been found to be latexmath:[\[\begin{aligned } seen that the value of @xmath1 extracted from the leading order heavy quark expansion coincides well with that extracted from the full qcd calculation , which shows the reliability of the heavy quark expansion and the power of light cone sum rule approach in studying heavy to light exclusive decays . working out @xmath8 contributions should be interesting , and it is expected to cast more light on the treatment of heavy to light decays by applying for the effective theory of heavy quark . v. m. belyaev , a. khodjamirian and r. rckl , z. phys . c * 60 * , 349 ( 1993 ) . a. khodjamirian and r. rckl , wue - itp-97 - 049 , mpi - pht/97 - 85 , hep - ph/9801443 . a. khodjamirian , r. rckl , s. weinzierl , c. w. winhart and o. yakovlev , phys . d * 62 * , 114002 ( 2000 ) . y. l. wu , mod . lett . a * 8 * , 819 ( 1993 ) . w. y. wang , y. l. wu and y. a. yan , int . a * 15 * , 1817 ( 2000 ) . hep - ph/9906529 , 1999 . y. a. yan , y. l. wu and w. y. wang , int a * 15 * , 2735 ( 2000 ) . y. l. wu and y.a . yan , int . a * 16 * , 285 ( 2001 ) . w. y. wang and y. l. wu , int . phys . a * 16 * , 377 ( 2001 ) . w. y. wang and y. l. wu , to be published in int . a ; hep - ph/0012240 . sanchis - lozano , nuov . * a110 * , 295 ( 1997 ) ; hep - ph/9710408 . f. berto and m.a . sanchis - lozano , nuov . * a112 * , 1181 ( 1999 ) . f. berto and m.a . sanchis - lozano , nucl . . suppl . * 93 * , 212 ( 2001 ) , hep - ph/0010091 . g. burdman , z. ligeti , m. neubert and y. nir , phys . d * 49 * , 2331 ( 1994 ) . v. m. belyaev , v. m. braun , a. khodjamirian and r. rckl , phys . d * 51 * , 6177 ( 1995 ) . v. a. nesterenko and a. v. radyushkin , phys . b * 115 * , 410 ( 1982 ) ; sov . jetp lett . * 35 * , 488 ( 1982 ) ; p. ball and v. m. braun , phys . rev . d * 49 * , 2472 ( 1994 ) . v. m. braun and i. b. filyanov , z. phys , c * 44 * , 157 ( 1989 ) . v. l. chernyak and a. r. zhitnitsky , phys . rep . * 112 * , 173 ( 1984 ) . v. m. braun and i. b. filyanov , z. phys . c * 48 * , 239 ( 1990 ) . j. p. alexander et . al.(cleo collab . ) , phys . lett . * 77 * , 5000 ( 1996 ) . particle data group , phys . d * 54 * , 1 ( 1996 ) . b. h. behrens et . al.(cleo collab . ) , phys . d * 61 * , 052001 ( 2000 ) .

@xmath0 decay is studied in the effective theory of heavy quark with infinite mass limit . the leading order heavy flavor - spin independent universal wave functions which parametrize the relevant matrix elements are evaluated via light cone sum rule method in the effective theory . the important quark mixing matrix element @xmath1 is then extracted via @xmath2 decay mode . # 1@xmath3#1

hep-ph0105154

in classical mechanics , ensembles , such as the microcanonical and canonical ensembles , are represented by probability distributions on the phase space . in quantum mechanics , ensembles are usually represented by density matrices . it is natural to regard these density matrices as arising from probability distributions on the ( normalized ) wave functions associated with the thermodynamical ensembles , so that members of the ensemble are represented by a random state vector . there are , however , as is well known , many probability distributions which give rise to the same density matrix , and thus to the same predictions for experimental outcomes @xcite .. the measure that gives equal weight to these two states corresponds to the same density matrix as the one giving equal weight to @xmath5 and @xmath6 . however the physical situation corresponding to the former measure , a mixture of two grotesque superpositions , seems dramatically different from the one corresponding to the latter , a routine mixture . it is thus not easy to regard these two measures as physically equivalent . ] moreover , as emphasized by landau and lifshitz @xcite , the energy levels for macroscopic systems are so closely spaced ( exponentially small in the number of particles in the system ) that `` the concept of stationary states [ energy eigenfunctions ] becomes in a certain sense unrealistic '' because of the difficulty of preparing a system with such a sharp energy and keeping it isolated . landau and lifshitz are therefore wary of , and warn against , regarding the density matrix for such a system as arising solely from our lack of knowledge about the wave function of the system . we shall argue , however , that despite these caveats such distributions can be both useful and physically meaningful . in particular we describe here a novel probability distribution , to be associated with any thermal ensemble such as the canonical ensemble . while probability distributions on wave functions are natural objects of study in many contexts , from quantum chaos @xcite to open quantum systems @xcite , our main motivation for considering them is to exploit the analogy between classical and quantum statistical mechanics @xcite . this analogy suggests that some relevant classical reasonings can be transferred to quantum mechanics by formally replacing the classical phase space by the unit sphere @xmath7 of the quantum system s hilbert space @xmath2 . in particular , with a natural measure @xmath8 on @xmath7 one can utilize the notion of typicality , i.e. , consider properties of a system common to `` almost all '' members of an ensemble . this is a notion frequently used in equilibrium statistical mechanics , as in , e.g. , boltzmann s recognition that typical phase points on the energy surface of a macroscopic system are such that the empirical distribution of velocities is approximately maxwellian . once one has such a measure for quantum systems , one could attempt an analysis of the second law of thermodynamics in quantum mechanics along the lines of boltzmann s analysis of the second law in classical mechanics , involving an argument to the effect that the behavior described in the second law ( such as entropy increase ) occurs for typical states of an isolated macroscopic system , i.e. for the overwhelming majority of points on @xmath7 with respect to @xmath9 . probability distributions on wave functions of a composite system , with hilbert space @xmath2 , have in fact been used to establish the typical properties of the reduced density matrix of a subsystem arising from the wave function of the composite . for example , page @xcite considers the uniform distribution on @xmath7 for a finite - dimensional hilbert space @xmath2 , in terms of which he shows that the von neumann entropy of the reduced density matrix is typically nearly maximal under appropriate conditions on the dimensions of the relevant hilbert spaces . given a probability distribution @xmath1 on the unit sphere @xmath7 of the hilbert space @xmath2 there is always an associated density matrix @xmath10 @xcite : it is the density matrix of the mixture , or the statistical ensemble of systems , defined by the distribution @xmath1 , given by @xmath11 for any projection operator @xmath12 , @xmath13 is the probability of obtaining in an experiment a result corresponding to @xmath12 for a system with a @xmath1-distributed wave function . it is evident from that @xmath10 is the second moment , or covariance matrix , of @xmath1 , provided @xmath1 has mean 0 ( which may , and will , be assumed without loss of generality since @xmath14 and @xmath15 are equivalent physically ) . while a probability measure @xmath1 on @xmath7 determines a unique density matrix @xmath0 on @xmath2 via , the converse is not true : the association @xmath16 given by is many - to - one .- dimensional hilbert space the uniform probability distribution @xmath17 over the unit sphere has density matrix @xmath18 with @xmath19 the identity operator on @xmath2 ; at the same time , for every orthonormal basis of @xmath2 the uniform distribution over the basis ( which is a measure concentrated on just @xmath20 points ) has the same density matrix , @xmath21 . an exceptional case is the density matrix corresponding to a pure state , @xmath22 , as the measure @xmath1 with this density matrix is almost unique : it must be concentrated on the ray through @xmath14 , and thus the only non - uniqueness corresponds to the distribution of the phase . ] there is furthermore no unique `` physically correct '' choice of @xmath1 for a given @xmath0 since for any @xmath1 corresponding to @xmath0 one could , in principle , prepare an ensemble of systems with wave functions distributed according to this @xmath1 . however , while @xmath0 itself need not determine a unique probability measure , additional facts about a system , such as that it has come to thermal equilibrium , might . it is thus not unreasonable to ask : which measure on @xmath7 corresponds to a given thermodynamic ensemble ? let us start with the _ microcanonical _ ensemble , corresponding to the energy interval @xmath23 $ ] , where @xmath24 is small on the macroscopic scale but large enough for the interval to contain many eigenvalues . to this there is associated the spectral subspace @xmath25 , the span of the eigenstates @xmath26 of the hamiltonian @xmath27 corresponding to eigenvalues @xmath28 between @xmath29 and @xmath30 . since @xmath25 is finite dimensional , one can form the _ microcanonical density matrix _ @xmath31 with @xmath32}(h)$ ] the projection to @xmath25 . this density matrix is diagonal in the energy representation and gives equal weight to all energy eigenstates in the interval @xmath23 $ ] . but what is the corresponding _ microcanonical measure _ ? the most plausible answer , given long ago by schrdinger @xcite and bloch @xcite , is the ( normalized ) uniform measure @xmath33 on the unit sphere in this subspace . @xmath34 is associated with @xmath35 via . note that a wave function @xmath36 chosen at random from this distribution is almost certainly a nontrivial superposition of the eigenstates @xmath37 with random coefficients @xmath38 that are identically distributed , but not independent . the measure @xmath35 is clearly stationary , i.e. , invariant under the unitary time evolution generated by @xmath27 , and it is as spread out as it could be over the set @xmath39 of allowed wave functions . this measure provides us with a notion of a `` typical wave function '' from @xmath25 which is very different from the one arising from the measure @xmath40 that , when @xmath27 is nondegenerate , gives equal probability @xmath41 to every eigenstate @xmath42 with eigenvalue @xmath43 $ ] . the measure @xmath40 , which is concentrated on these eigenstates , is , however , less robust to small perturbations in @xmath27 than is the smoother measure @xmath35 . our proposal for the canonical ensemble is in the spirit of the uniform microcanonical measure @xmath35 and reduces to it in the appropriate cases . it is based on a mathematically natural family of probability measures @xmath1 on @xmath7 . for every density matrix @xmath0 on @xmath2 , there is a unique member @xmath1 of this family , satisfying for @xmath44 , namely the _ gaussian adjusted projected measure _ @xmath3 , constructed roughly as follows : eq . ( i.e. , the fact that @xmath10 is the covariance of @xmath1 ) suggests that we start by considering the gaussian measure @xmath45 with covariance @xmath0 ( and mean 0 ) , which could , in finitely many dimensions , be expressed by @xmath46 ( where @xmath47 is the obvious lebesgue measure on @xmath2 ) . this is not adequate , however , since the measure that we seek must live on the sphere @xmath7 whereas @xmath45 is spread out over all of @xmath2 . we thus adjust and then project @xmath45 to @xmath7 , in the manner described in section [ sec : defmeasure ] , in order to obtain the measure @xmath3 , having the prescribed covariance @xmath0 as well as other desirable properties . it is our contention that _ a quantum system in thermal equilibrium at inverse temperature @xmath48 should be described by a random state vector whose distribution is given by the measure @xmath49 associated with the density matrix for the canonical ensemble , _ @xmath50 in order to convey the significance of @xmath3 as well as the plausibility of our proposal that @xmath49 describes thermal equilibrium , we recall that a system described by a canonical ensemble is usually regarded as a subsystem of a larger system . it is therefore important to consider the notion of the distribution of the wave function of a subsystem . consider a composite system in a pure state @xmath51 , and ask what might be meant by the wave function of the subsystem with hilbert space @xmath52 . for this we propose the following . let @xmath53 be a ( generalized ) orthonormal basis of @xmath54 ( playing the role , say , of the eigenbasis of the position representation ) . for each choice of @xmath55 , the ( partial ) scalar product @xmath56 , taken in @xmath54 , is a vector belonging to @xmath52 . regarding @xmath55 as random , we are led to consider the random vector @xmath57 given by @xmath58 where @xmath59 is the normalizing factor and @xmath60 is a random element of the basis @xmath53 , chosen with the quantum distribution @xmath61 we refer to @xmath62 as the _ conditional wave function _ @xcite of system 1 . note that @xmath62 becomes doubly random when we start with a random wave function in @xmath63 instead of a fixed one . the distribution of @xmath62 corresponding to ( [ psi1def ] ) and ( [ marg ] ) is given by the probability measure on @xmath64 , @xmath65 where @xmath66 denotes the `` delta '' measure concentrated at @xmath67 . while the density matrix @xmath68 associated with @xmath69 always equals the reduced density matrix @xmath70 of system 1 , given by @xmath71 the measure @xmath72 itself usually depends on the choice of the basis @xmath73 . it turns out , nevertheless , as we point out in section [ sec : typicality ] , that @xmath74 is a universal function of @xmath70 in the special case that system 2 is large and @xmath14 is typical ( with respect to the uniform distribution on all wave functions with the same reduced density matrix ) , namely @xmath75 . thus @xmath3 has a distinguished , universal status among all probability measures on @xmath7 with density matrix @xmath0 . to further support our claim that @xmath49 is the right measure for @xmath76 , we shall regard , as is usually done , the system described by @xmath76 as coupled to a ( very large ) heat bath . the interaction between the heat bath and the system is assumed to be ( in some suitable sense ) negligible . we will argue that if the wave function @xmath14 of the combined `` system plus bath '' has microcanonical distribution @xmath35 , then the distribution of the conditional wave function of the ( small ) system is approximately @xmath77 ; see section [ sec : hb1 ] . indeed , a stronger statement is true . as we argue in section [ sec : hb2 ] , even for a typical _ fixed _ microcanonical wave function @xmath14 of the composite , i.e. , one typical for @xmath35 , the conditional wave function of the system , defined in , is then approximately @xmath49-distributed , for a typical basis @xmath53 . this is related to the fact that for a typical microcanonical wave function @xmath14 of the composite the reduced density matrix for the system is approximately @xmath76 @xcite . note that the analogous statement in classical mechanics would be wrong : for a fixed phase point @xmath78 of the composite , be it typical or atypical , the phase point of the system could never be random , but rather would merely be the part of @xmath78 belonging to the system . the remainder of this paper is organized as follows . in section [ sec : defmeasure ] we define the measure @xmath3 and obtain several ways of writing it . in section [ sec : prop ] we describe some natural mathematical properties of these measures , and suggest that these properties uniquely characterize the measures . in section [ sec : hb1 ] we argue that @xmath49 represents the canonical ensemble . in section [ sec : typicality ] we outline the proof that @xmath3 is the distribution of the conditional wave function for _ most _ wave functions in @xmath63 with reduced density matrix @xmath0 if system 2 is large , and show that @xmath49 is the typical distribution of the conditional wave function arising from a fixed microcanonical wave function of a system in contact with a heat bath . in section [ sec : rem ] we discuss other measures that have been or might be considered as the thermal equilibrium distribution of the wave function . finally , in section [ sec : two ] we compute explicitly the distribution of the coefficients of a @xmath49-distributed state vector in the simplest possible example , the two - level system . in this section , we define , for any given density matrix @xmath0 on a ( separable ) hilbert space @xmath79 the gaussian adjusted projected measure @xmath3 on @xmath7 . this definition makes use of two auxiliary measures , @xmath45 and @xmath80 , defined as follows . @xmath45 is the gaussian measure on @xmath2 with covariance matrix @xmath0 ( and mean 0 ) . more explicitly , let @xmath81 be an orthonormal basis of eigenvectors of @xmath0 and @xmath82 the corresponding eigenvalues , @xmath83 such a basis exists because @xmath0 has finite trace . let @xmath84 be a sequence of independent complex - valued random variables having a ( rotationally symmetric ) gaussian distribution in @xmath85 with mean @xmath86 and variance @xmath87 ( where @xmath88 means expectation ) , i.e. , @xmath89 and @xmath90 are independent real gaussian variables with mean zero and variance @xmath91 . we define @xmath45 to be the distribution of the random vector @xmath92 note that @xmath93 is not normalized , i.e. , it does not lie in @xmath7 . in order that @xmath93 lie in @xmath2 at all , we need that the sequence @xmath84 be square - summable , @xmath94 is finite . in fact , @xmath95 more generally , we observe that for any measure @xmath1 on @xmath2 with ( mean 0 and ) covariance given by the trace class operator @xmath96 , @xmath97 we have that , for a random vector @xmath36 with distribution @xmath1 , @xmath98 . it also follows that @xmath93 almost surely lies in the positive spectral subspace of @xmath0 , the closed subspace spanned by those @xmath26 with @xmath99 , or , equivalently , the orthogonal complement of the kernel of @xmath0 ; we shall call this subspace @xmath100 . note further that , since @xmath45 is the gaussian measure with covariance @xmath0 , it does not depend ( in the case of degenerate @xmath0 ) on the choice of the basis @xmath101 among the eigenbases of @xmath0 , but only on @xmath0 . since we want a measure on @xmath7 while @xmath45 is not concentrated on @xmath7 but rather is spread out , it would be natural to project @xmath45 to @xmath7 . however , since projecting to @xmath7 changes the covariance of a measure , as we will point out in detail in section [ sec : densitymatrix ] , we introduce an adjustment factor that exactly compensates for the change of covariance due to projection . we thus define the adjusted gaussian measure @xmath80 on @xmath2 by @xmath102 since @xmath103 by , @xmath80 is a probability measure . let @xmath104 be a @xmath80-distributed random vector . we define @xmath3 to be the distribution of @xmath105 with @xmath106 the projection to the unit sphere ( i.e. , the normalization of a vector ) , @xmath107 putting differently , for a subset @xmath108 , @xmath109 where @xmath110 denotes the cone through @xmath111 . more succinctly , @xmath112 where @xmath113 denotes the action of @xmath106 on measures . more generally , one can define for any measure @xmath1 on @xmath2 the `` adjust - and - project '' procedure : let @xmath114 be the adjusted measure @xmath115 ; then the adjusted - and - projected measure is @xmath116 , thus defining a mapping @xmath117 from the measures on @xmath2 with @xmath118 to the probability measures on @xmath7 . we then have that @xmath119 . we remark that @xmath120 , too , lies in @xmath100 almost surely , and that @xmath121 does _ not _ have distribution @xmath3nor covariance @xmath0 ( see sect . [ sec : densitymatrix ] ) . we can be more explicit in the case that @xmath0 has finite rank @xmath122 , e.g. for finite - dimensional @xmath2 : then there exists a lebesgue volume measure @xmath123 on @xmath124 , and we can specify the densities of @xmath45 and @xmath80 , [ mundensity ] @xmath125 with @xmath126 the restriction of @xmath0 to @xmath100 . similarly , we can express @xmath3 relative to the @xmath127dimensional surface measure @xmath128 on @xmath129 , [ mudensity ] @xmath130 we note that @xmath131 where @xmath132 is the microcanonical density matrix given in and @xmath133 is the microcanonical measure . in this section we prove the following properties of @xmath3 : * property 1 * _ the density matrix associated with @xmath3 in the sense of is @xmath0 , i.e. , @xmath134 . _ * property 2 * _ the association @xmath135 is _ covariant _ : for any unitary operator @xmath136 on @xmath2 , @xmath137 where @xmath138 is the adjoint of @xmath136 and @xmath139 is the action of @xmath136 on measures , @xmath140 . in particular , @xmath3 is stationary under any unitary evolution that preserves @xmath0 . _ * property 3 * _ if @xmath141 has distribution @xmath142 then , for any basis @xmath53 of @xmath54 , the conditional wave function @xmath62 has distribution @xmath143 . ( `` gap of a product density matrix has gap marginal . '' ) _ we will refer to the property expressed in property 3 by saying that the family of gap measures is _ we note that when @xmath141 has distribution @xmath3 and @xmath0 is not a tensor product , the distribution of @xmath62 need not be @xmath75 ( as we will show after the proof of property 3 ) . before establishing these properties let us formulate what they say about our candidate @xmath49 for the canonical distribution . as a consequence of property 1 , the density matrix arising from @xmath144 in the sense of is the density matrix @xmath145 . as a consequence of property 2 , @xmath49 is stationary , i.e. , invariant under the unitary time evolution generated by @xmath27 . as a consequence of property 3 , if @xmath146 has distribution @xmath147 and systems 1 and 2 are decoupled , @xmath148 , where @xmath149 is the identity on @xmath150 , then the conditional wave function @xmath62 of system @xmath151 has a distribution ( in @xmath52 ) of the same kind with the same inverse temperature @xmath48 , namely @xmath152 . this fits well with our claim that @xmath49 is the thermal equilibrium distribution since one would expect that if a system is in thermal equilibrium at inverse temperature @xmath48 then so are its subsystems . we conjecture that the family of gap measures is the only family of measures satisfying properties 13 . this conjecture is formulated in detail , and established for suitably continuous families of measures , in section [ sec : uniqueness ] . the following lemma , proven in section [ gmgm ] , is convenient for showing that a random wave function is gap - distributed : [ gaussgap ] let @xmath153 be a measurable space , @xmath1 a probability measure on @xmath153 , and @xmath154 a hilbert - space - valued function . if @xmath155 is @xmath45-distributed with respect to @xmath156 , then @xmath157 is @xmath3-distributed with respect to @xmath158 . in this subsection we establish property 1 . we then add a remark on the covariance matrix . of property 1 from we find that @xmath159 because @xmath160 is the covariance matrix of @xmath45 , which is @xmath0 . ( a number above an equal sign refers to the equation used to obtain the equality . ) * remark on the covariance matrix . * the equation @xmath134 can be understood as expressing that @xmath3 and @xmath45 have the same covariance . for a probability measure @xmath1 on @xmath2 with mean 0 that need not be concentrated on @xmath7 , the covariance matrix @xmath161 is given by @xmath162 suppose we want to obtain from @xmath1 a probability measure on @xmath7 having the same covariance . the projection @xmath163 of @xmath1 to @xmath7 , defined by @xmath164 for @xmath108 , is not what we want , as it has covariance @xmath165 however , @xmath166 does the job : it has the same covariance . as a consequence , a naturally distinguished measure on @xmath7 with given covariance is the gaussian adjusted projected measure , the gap measure , with the given covariance . we establish property 2 and then discuss in more general terms under which conditions a measure on @xmath7 is stationary . of property 2 under a unitary transformation @xmath136 , a gaussian measure with covariance matrix @xmath96 transforms into one with covariance matrix @xmath167 . since @xmath168 , @xmath169 transforms into @xmath170 ; that is , @xmath171 and @xmath172 are equal in distribution , and since @xmath173 , we have that @xmath174 and @xmath175 are equal in distribution . in other words , @xmath176 transforms into @xmath177 , which is what we claimed in . in this subsection we discuss a criterion for stationarity under the evolution generated by @xmath178 . consider the following property of a sequence of complex random variables @xmath84 : @xmath179 ( the phase @xmath180 exists when @xmath181 . ) condition implies that the distribution of the random vector @xmath182 is stationary , since @xmath183 . note also that implies that the distribution has mean 0 . we show that the @xmath184 have property . to begin with , the @xmath185 obviously have this property since they are independent gaussian variables . since the density of @xmath80 relative to @xmath45 is a function of the moduli alone , also the @xmath186 satisfy . finally , since the @xmath187 are functions of the @xmath188 while the phases of the @xmath189 equal the phases of the @xmath190 , also the @xmath184 satisfy . we would like to add that is not merely a sufficient , but also almost a necessary condition ( and _ morally _ a necessary condition ) for stationarity . since for any @xmath36 , the moduli latexmath:[$|z_n| = of @xmath36 takes place in the ( possibly infinite - dimensional ) torus @xmath192 contained in @xmath7 . independent uniform phases correspond to the uniform measure @xmath123 on @xmath193 . @xmath123 is the only stationary measure if the motion on @xmath194 is uniquely ergodic , and this is the case whenever the spectrum @xmath195 of @xmath27 is linearly independent over the rationals @xmath196 , i.e. , when every finite linear combination @xmath197 of eigenvalues with rational coefficients @xmath198 , not all of which vanish , is nonzero , see @xcite . this is true of generic hamiltonians , so that @xmath123 is generically the unique stationary distribution on the torus . but even when the spectrum of @xmath27 is linearly dependent , e.g. when there are degenerate eigenvalues , and thus further stationary measures on the torus exist , these further measures should not be relevant to thermal equilibrium measures , because of their instability against perturbations of @xmath27 @xcite . the stationary measure @xmath123 on @xmath193 corresponds , for given moduli @xmath199 or , equivalently , by setting @xmath200 for a given probability measure @xmath201 on the spectrum of @xmath27 , to a stationary measure @xmath202 on @xmath7 that is concentrated on the embedded torus . the measures @xmath202 are ( for generic @xmath27 ) the extremal stationary measures , i.e. , the extremal elements of the convex set of stationary measures , of which all other stationary measures are mixtures . lemma [ gaussgap ] is more or less immediate from the definition of @xmath3 . a more detailed proof looks like this : of lemma [ gaussgap ] by assumption the distribution @xmath203 of @xmath36 with respect to @xmath1 is @xmath45 . thus for the distribution of @xmath36 with respect to @xmath204 , we have @xmath205 . thus , @xmath206 has distribution @xmath207 . we have already remarked in the introduction that the orthonormal basis @xmath53 of @xmath54 , used in the definition of the conditional wave function , could be a _ generalized _ basis , such as a `` continuous '' basis , for which it is appropriate to write @xmath208 instead of the `` discrete '' notation @xmath209 we used in . we wish to elucidate this further . a generalized orthonormal basis @xmath210 indexed by the set @xmath211 is mathematically defined by a unitary isomorphism @xmath212 , where @xmath213 denotes a measure on @xmath211 . we can think of @xmath211 as the configuration space of system 2 ; as a typical example , system 2 may consist of @xmath214 particles in a box @xmath215 , so that its configuration space is @xmath216 with @xmath213 the lebesgue measure ( which can be regarded as obtained by combining @xmath214 copies of the volume measure on @xmath217 ) . was supposed to be the _ configuration _ , corresponding to the positions of the particles belonging to system 2 . for our purposes here , however , the physical meaning of the @xmath218 is irrelevant , so that any generalized orthonormal basis of @xmath219 can be used . ] the formal ket @xmath220 then means the delta function centered at @xmath221 ; it is to be treated as ( though strictly speaking it is not ) an element of @xmath54 . the definition of the conditional wave function @xmath62 then reads as follows : the vector @xmath222 can be regarded , using the isomorphism @xmath223 , as a function @xmath224 . is to be understood as meaning @xmath225 where @xmath226 is the normalizing factor and @xmath227 is a random point in @xmath211 , chosen with the quantum distribution @xmath228 which is how is to be understood in this setting . as @xmath14 is defined only up to changes on a null set in @xmath211 , @xmath62 may not be defined for a particular @xmath227 . its distribution in @xmath52 , however , is defined unambiguously by . in the most familiar setting with @xmath229 , we have that @xmath230 . in the following , we will allow generalized bases and use continuous instead of discrete notation , and set @xmath231 . of property 3 the proof is divided into four steps . _ we can assume that @xmath232 where @xmath104 is a @xmath80-distributed random vector in @xmath233 . we then have that @xmath234 where @xmath235 is the normalization in @xmath52 , and where the distribution of @xmath227 , given @xmath104 , is @xmath236 @xmath104 and @xmath227 have a joint distribution given by the following measure @xmath237 on @xmath238 : @xmath239 thus , what needs to be shown is that with respect to @xmath237 , @xmath240 is @xmath143-distributed . _ _ if @xmath141 is @xmath241-distributed and @xmath242 is fixed , then the random vector @xmath243 with @xmath244 is @xmath245-distributed . _ this follows , more or less , from the fact that a subset of a set of jointly gaussian random variables is also jointly gaussian , together with the observation that the covariance of @xmath246 is @xmath247 more explicitly , pick an orthonormal basis @xmath248 of @xmath150 consisting of eigenvectors of @xmath249 with eigenvalues @xmath250 , and note that the vectors @xmath251 form an orthonormal basis of @xmath233 consisting of eigenvectors of @xmath252 with eigenvalues @xmath253 . since the random variables @xmath254 are independent gaussian random variables with mean zero and variances @xmath255 , so are their linear combinations @xmath256 with variances ( because variances add when adding independent gaussian random variables ) @xmath257 thus @xmath258 is @xmath245-distributed , which completes step 2 . _ _ if @xmath141 is @xmath241-distributed and @xmath259 is random with any distribution , then the random vector @xmath260 is @xmath245-distributed . _ this is a trivial consequence of step 2 . _ step 4 . _ apply lemma [ gaussgap ] as follows . let @xmath261 , @xmath262 , and @xmath263 ( which means that @xmath221 and @xmath14 are independent ) . by step 3 , the hypothesis of lemma [ gaussgap ] ( for @xmath264 ) is satisfied , and thus @xmath265 is @xmath143-distributed with respect to @xmath266 where we have used that @xmath267 . but this is , according to step 1 , what we needed to show . to verify the statement after property 3 , consider the density matrix @xmath268 for a pure state @xmath269 of the form @xmath270 , where @xmath271 and @xmath272 are respectively orthonormal bases for @xmath52 and @xmath54 and the @xmath82 are nonnegative with @xmath273 . then a @xmath3-distributed random vector @xmath36 coincides with @xmath269 up to a random phase , and so @xmath274 . choosing for @xmath275 the basis @xmath272 , the distribution of @xmath62 is not @xmath75 but rather is concentrated on the eigenvectors of @xmath70 . when the @xmath82 are pairwise - distinct this measure is the measure @xmath276 we define in section [ sec : nu ] . in this section we use property 3 , i.e. , the fact that gap measures are hereditary , to show that @xmath49 is the distribution of the conditional wave function of a system coupled to a heat bath when the wave function of the composite is distributed microcanonically , i.e. , according to @xmath35 . consider a system with hilbert space @xmath52 coupled to a heat bath with hilbert space @xmath54 . suppose the composite system has a random wave function @xmath277 whose distribution is microcanonical , @xmath35 . assume further that the coupling is negligibly small , so that we can write for the hamiltonian @xmath278 and that the heat bath is large ( so that the energy levels of @xmath279 are very close ) . it is a well known fact that for macroscopic systems different equilibrium ensembles , for example the microcanonical and the canonical , give approximately the same answer for appropriate quantities . by this equivalence of ensembles @xcite , we should have that @xmath280 for suitable @xmath281 . then , since @xmath3 depends continuously on @xmath0 , we have that @xmath282 . thus we should have that the distribution of the conditional wave function @xmath62 of the system is approximately the same as would be obtained when @xmath36 is @xmath49-distributed . but since , by , the canonical density matrix is then of the form @xmath283 we have by property 3 that @xmath62 is approximately @xmath152-distributed , which is what we wanted to show . the previous section concerns the distribution of the conditional wave function @xmath62 arising from the microcanonical distribution of the wave function of the composite . it concerns , in other words , a _ random _ wave function of the composite . the result there is the analogue , on the level of measures on hilbert space , of the well known result that if a microcanonical density matrix is assumed for the composite , the reduced density matrix @xmath70 of the system , defined as the partial trace @xmath284 , is canonical if the heat bath is large @xcite . as indicated in the introduction , a stronger statement about the canonical density matrix is in fact true , namely that for a _ fixed _ ( nonrandom ) wave function @xmath14 of the composite which is typical with respect to @xmath35 , @xmath285 when the heat bath is large ( see @xcite ; for a rigorous study of special cases of a similar question , see @xcite ) . , the reduced density matrix becomes proportional to the identity on @xmath52 for typical wave functions relative to the uniform distribution on @xmath7 ( corresponding to @xmath35 for @xmath286 and @xmath287 ) . ] this stronger statement will be used in section [ sec : hb2 ] to show that a similar statement holds for the distribution of @xmath62 as well , namely that it is approximately @xmath152-distributed for a typical fixed @xmath288 and basis @xmath289 of @xmath54 . but we must first consider the distribution of @xmath62 for a typical @xmath290 . in this section we argue that for a typical wave function of a big system the conditional wave function of a small subsystem is approximately gap - distributed , first giving a precise formulation of this result and then sketching its proof . we give the detailed proof in @xcite . let @xmath291 , where @xmath52 and @xmath54 have respective dimensions @xmath20 and @xmath292 , with @xmath293 . for any given density matrix @xmath294 on @xmath52 , consider the set @xmath295 where @xmath296 is the reduced density matrix for the wave function @xmath14 . there is a natural notion of ( normalized ) uniform measure @xmath297 on @xmath298 ; we give its precise definition in section [ sec : typoutline ] . we claim that for fixed @xmath20 and large @xmath292 , the distribution @xmath299 of the conditional wave function @xmath62 of system 1 , defined by and for a basis @xmath289 of @xmath54 , is close to @xmath143 for the overwhelming majority , relative to @xmath297 , of vectors @xmath300 with reduced density matrix @xmath294 . more precisely : _ for every @xmath301 and every bounded continuous function @xmath302 , @xmath303 regardless of how the basis @xmath289 is chosen . _ here we use the notation @xmath304 it is important to resist the temptation to translate @xmath297 into a density matrix in @xmath2 . as mentioned in the introduction , to every probability measure @xmath1 on @xmath7 there corresponds a density matrix @xmath10 in @xmath2 , given by , which contains all the empirically accessible information about an ensemble with distribution @xmath1 . it may therefore seem a natural step to consider , instead of the measure @xmath305 , directly its density matrix @xmath306 , where @xmath307 is the identity on @xmath54 . but since our result concerns properties of most wave functions relative to @xmath1 , it can not be formulated in terms of the density matrix @xmath10 . in particular , the corresponding statement relative to another measure @xmath308 on @xmath7 with the same density matrix @xmath309 could be false . noting that @xmath10 has a basis of eigenstates that are product vectors , we could , for example , take @xmath310 to be a measure concentrated on these eigenstates . for any such state @xmath14 , @xmath299 is a delta - measure . the result follows , by , lemma [ gaussgap ] , and the continuity of @xmath117 , from the corresponding statement about the gaussian measure @xmath245 on @xmath52 with covariance @xmath294 : _ for every @xmath301 and every bounded continuous @xmath311 , _ @xmath312 _ where @xmath313 is the distribution of @xmath314 ( not normalized ) with respect to the uniform distribution of @xmath315 . _ we sketch the proof of and give the definition of @xmath297 . according to the schmidt decomposition , every @xmath300 can be written in the form @xmath316 where @xmath317 is an orthonormal basis of @xmath318 , @xmath319 an orthonormal system in @xmath54 , and the @xmath320 are coefficients which can be assumed real and nonnegative . from one reads off the reduced density matrix of system 1 , @xmath321 as the reduced density matrix is given , @xmath322 , the orthonormal basis @xmath323 and the coefficients @xmath320 are determined ( when @xmath294 is nondegenerate ) as the eigenvectors and the square - roots of the eigenvalues of @xmath294 , and , @xmath298 is in a natural one - to - one correspondence with the set @xmath324 of all orthonormal systems @xmath325 in @xmath54 of cardinality @xmath20 . ( if some of the eigenvalues of @xmath294 vanish , the one - to - one correspondence is with @xmath326 where @xmath327 . ) the haar measure on the unitary group of @xmath54 defines the uniform distribution on the set of orthonormal bases of @xmath54 , of which the uniform distribution on @xmath324 is a marginal , and thus defines the uniform distribution @xmath297 on @xmath298 . ( when @xmath294 is degenerate , @xmath297 does not depend upon how the eigenvectors @xmath328 of @xmath294 are chosen . ) the key idea for establishing from the schmidt decomposition is this : @xmath329 is the average of @xmath292 delta measures with equal weights , @xmath330 , located at the points @xmath331 now regard @xmath14 as random with distribution @xmath297 ; then the @xmath332 are @xmath292 random vectors , and @xmath333 is their empirical distribution . if the @xmath334 coefficients @xmath335 were _ independent gaussian ( complex ) random variables with ( mean zero and ) variance @xmath336 , then the @xmath332 would be @xmath292 independent drawings of a @xmath245-distributed random vector ; by the weak law of large numbers , their empirical distribution would usually be close to @xmath245 ; in fact , the probability that @xmath337 would converge to 1 , as @xmath338 . however , when @xmath339 is a random orthonormal system with uniform distribution as described above , the expansion coefficients @xmath335 in the decomposition of the @xmath340 s @xmath341 will not be independent since the @xmath340 s must be orthogonal and since @xmath342 . nonetheless , replacing the coefficients @xmath335 in by independent gaussian coefficients @xmath343 as described above , we obtain a system of vectors @xmath344 that , in the limit @xmath345 , form a uniformly distributed orthonormal system : @xmath346 ( by the law of large numbers ) and @xmath347 for @xmath348 ( since a pair of randomly chosen vectors in a high - dimensional hilbert space will typically be almost orthogonal ) . this completes the proof . while this result _ suggests _ that @xmath49 is the distribution of the conditional wave function of a system coupled to a heat bath when the wave function of the composite is a typical _ fixed _ microcanonical wave function , belonging to @xmath25 , it does not quite _ imply _ it . the reason for this is that @xmath25 has measure 0 with respect to the uniform distribution on @xmath79 even when the latter is finite - dimensional . nonetheless , there is a simple corollary , or reformulation , of the result that will allow us to cope with microcanonical wave functions . we have indicated that for our result the choice of basis @xmath289 of @xmath54 does not matter . in fact , while @xmath299 , the distribution of the conditional wave function @xmath62 of system 1 , depends upon both @xmath349 and the choice of basis @xmath289 of @xmath54 , the distribution of @xmath299 itself , when @xmath14 is @xmath297-distributed , does not depend upon the choice of basis . this follows from the fact that for any unitary @xmath136 on @xmath54 @xmath350 ( and the invariance of the haar measure of the unitary group of @xmath54 under left multiplication ) . it similarly follows from that for fixed @xmath349 , the distribution of @xmath299 arising from the uniform distribution @xmath237 of the basis @xmath289 , in the set @xmath351 of all orthonormal bases of @xmath54 , is the same as the distribution of @xmath299 arising from the uniform distribution @xmath297 of @xmath14 with a fixed basis ( and the fact that the haar measure is invariant under @xmath352 ) . we thus have the following corollary : _ let @xmath349 and let @xmath353 be the corresponding reduced density matrix for system 1 . then for a typical basis @xmath289 of @xmath54 , the conditional wave function @xmath62 of system 1 is approximately @xmath143-distributed when @xmath292 is large : for every @xmath301 and every bounded continuous function @xmath302 , @xmath354 _ it is an immediate consequence of the result of section [ sec : ref ] that for any fixed microcanonical wave function @xmath14 for a system coupled to a ( large ) heat bath , the conditional wave function @xmath62 of the system will be approximately gap - distributed . when this is combined with the `` canonical typicality '' described near the beginning of section [ sec : typicality ] , we obtain the following result : _ consider a system with finite - dimensional hilbert space @xmath52 coupled to a heat bath with finite - dimensional hilbert space @xmath54 . suppose that the coupling is weak , so that we can write @xmath355 on @xmath356 , and that the heat bath is large , so that the eigenvalues of @xmath279 are close . then for any wave function @xmath14 that is typical relative to the microcanonical measure @xmath35 , the distribution @xmath299 of the conditional wave function @xmath62 , defined by and for a typical basis @xmath289 of the heat bath , is close to @xmath49 for suitable @xmath357 , where @xmath358 . in other words , in the thermodynamic limit , in which the volume @xmath359 of the heat bath and @xmath360 go to infinity and @xmath361 is constant , we have that for all @xmath362 , and for all bounded continuous functions @xmath363 , @xmath364 where @xmath365 . _ we note that if @xmath289 were an energy eigenbasis rather than a typical basis , the result would be false . we review in this section other distributions that have been , or may be , considered as possible candidates for the distribution of the wave function of a system from a canonical ensemble . one possibility , which goes back to von neumann @xcite , is to consider @xmath9 as concentrated on the eigenvectors of @xmath0 ; we denote this distribution @xmath366 after the first letters of `` eigenvector '' ; it is defined as follows . suppose first that @xmath0 is nondegenerate . to select an @xmath366-distributed vector , pick a unit eigenvector @xmath42 , so that @xmath367 , with probability @xmath82 and randomize its phase . this definition can be extended in a natural way to degenerate @xmath0 : @xmath368 where @xmath369 denotes the eigenspace of @xmath0 associated with eigenvalue @xmath201 . the measure @xmath366 is concentrated on the set @xmath370 of eigenvectors of @xmath0 , which for the canonical @xmath371 coincides with the set of eigenvectors of @xmath27 ; it is a mixture of the microcanonical distributions @xmath372 on the eigenspaces of @xmath27 in the same way as in classical mechanics the canonical distribution on phase space is a mixture of the microcanonical distributions . note that @xmath373 , and that in particular @xmath374 is not , when @xmath27 is nondegenerate , the uniform distribution @xmath40 on the energy eigenstates with energies in @xmath375 $ ] , against which we have argued in the introduction . the distribution @xmath366 has the same properties as those of @xmath3 described in properties 13 , except when @xmath0 is degenerate : _ the measures @xmath366 are such that ( a ) they have the right density matrix : @xmath376 ; ( b ) they are covariant : @xmath377 ; ( c ) they are hereditary at nondegenerate @xmath0 : when @xmath378 and @xmath0 is nondegenerate and uncorrelated , @xmath379 , then @xmath366 has marginal ( i.e. , distribution of the conditional wave function ) @xmath380 . _ \(a ) and ( b ) are obvious . for ( c ) let , for @xmath381 , @xmath382 be a basis consisting of eigenvectors of @xmath249 with eigenvalues @xmath250 . note that the tensor products @xmath383 are eigenvectors of @xmath0 with eigenvalues @xmath384 , and by nondegeneracy all eigenvectors of @xmath0 are of this form up to a phase factor . since an @xmath366-distributed random vector @xmath36 is almost surely an eigenvector of @xmath0 , we have latexmath:[$\psi = e^{i\theta } @xmath214 , and @xmath386 . the conditional wave function @xmath62 is , up to the phase , the eigenvector @xmath387 of @xmath294 occurring as the first factor in @xmath36 . the probability of obtaining @xmath388 is @xmath389 . and @xmath390 are multiplicatively independent , in the sense that @xmath391 can occur only trivially , i.e. , when @xmath392 and @xmath393 . in particular , the nondegeneracy of @xmath294 and @xmath390 is irrelevant . ] in contrast , for a _ degenerate _ @xmath379 the conditional wave function need not be @xmath380-distributed , as the following example shows . suppose @xmath294 and @xmath390 are nondegenerate but @xmath394 for some @xmath395 ; then an @xmath366-distributed @xmath36 , whenever it happens to be an eigenvector associated with eigenvalue @xmath384 , is of the form @xmath396 , almost surely with nonvanishing coefficients @xmath397 and @xmath398 ; as a consequence , the conditional wave function is a multiple of @xmath399 , which is , for typical @xmath227 and unless @xmath400 and @xmath401 have disjoint supports , a nontrivial superposition of eigenvectors @xmath402 , @xmath403 with different eigenvalues and thus can not arise from the @xmath380 distribution . also in the case of the degeneracy of @xmath404 : if the orthonormal basis @xmath289 used in the definition of conditional wave function consists of eigenvectors of @xmath390 , then the distribution of the conditional wave function is @xmath380 . ] note also that @xmath366 is discontinuous as a function of @xmath0 at every degenerate @xmath0 ; in other words , @xmath405 is , like @xmath40 , unstable against small perturbations of the hamiltonian . ( and , as with @xmath40 , this fact , quite independently of the considerations on behalf of gap - measures in sections [ sec : hb1 ] and [ sec : typicality ] , suggests against using @xmath406 as a thermal equilibrium distribution . ) moreover , @xmath366 is highly concentrated , generically on a one - dimensional subset of @xmath7 , and in the case of a finite - dimensional hilbert space @xmath2 fails to be absolutely continuous relative to the uniform distribution @xmath407 on the unit sphere . for further discussion of families @xmath408 of measures satisfying the analogues of properties 13 , see section [ sec : uniqueness ] . here is another distribution on @xmath2 associated with the density matrix @xmath0 . let the random vector @xmath36 be @xmath409 the @xmath410 being independent random vectors with distributions @xmath372 . in case all eigenvalues are nondegenerate , this means the coefficients @xmath84 of @xmath36 , @xmath411in sharp contrast with the moduli when @xmath36 is @xmath3-distributed . and in contrast to the measure @xmath366 considered in the previous subsection , the weights @xmath82 in the density matrix now come from the fixed size of the coefficients of @xmath36 when it is decomposed into the eigenvectors of @xmath0 , rather than from the probability with which these eigenvectors are chosen . this measure , too , is stationary under any unitary evolution that leaves @xmath0 invariant . in particular , it is stationary in the thermal case @xmath371 , and for generic @xmath27 it is an extremal stationary measure as characterized in section [ sec : stationarity ] ; in fact it is , in the notation of the last paragraph of section [ sec : stationarity ] , @xmath202 with @xmath412 . this measure , too , is highly concentrated : for a hilbert space @xmath2 of finite dimension @xmath20 , it is supported by a submanifold of real dimension @xmath413 where @xmath292 is the number of distinct eigenvalues of @xmath27 , hence generically it is supported by a submanifold of just half the dimension of @xmath2 . in @xcite , guerra and loffredo consider the canonical density matrix @xmath76 for the one - dimensional harmonic oscillator and want to associate with it a diffusion process on the real line , using stochastic mechanics @xcite . since stochastic mechanics associates a process with every wave function , they achieve this by finding a measure @xmath414 on @xmath415 whose density matrix is @xmath76 . they propose the following measure @xmath414 , supported by coherent states . with every point @xmath416 in the classical phase space @xmath417 of the harmonic oscillator there is associated a coherent state @xmath418 with @xmath419 , thus defining a mapping @xmath420 , @xmath421 . let @xmath422 be the classical hamiltonian function , and consider the classical canonical distribution at inverse temperature @xmath423 , @xmath424 let @xmath425 then @xmath426 is the distribution on coherent states arising from @xmath427 . the density matrix of @xmath414 is @xmath76 @xcite . this measure is concentrated on a 2-dimensional submanifold of @xmath415 , namely on the set of coherent states ( the image of @xmath96 ) . note also that not every density matrix @xmath0 on @xmath428 can arise as the density matrix of a distribution on the set of coherent states ; for example , a pure state @xmath429 can arise in this way if and only if @xmath14 is a coherent state . in a similar spirit , one may consider , on a finite - dimensional hilbert space @xmath2 , the distribution @xmath430 that maximizes the gibbs entropy functional @xmath431 = - \int\limits_{{\mathscr{s}}({\mathscr{h } } ) } \!\!\ ! u(d\psi ) \:\ : f(\psi ) \ , \log f(\psi)\ ] ] under the constraints that @xmath432 be a probability distribution with mean 0 and covariance @xmath433 : @xmath434 a standard calculation using lagrange multipliers leads to @xmath435 with @xmath436 a self - adjoint matrix determined by and ; comparison with shows that @xmath432 is not a gap measure . ( we remark , however , that another gibbs entropy functional , @xmath437 = -\int_{\mathscr{h}}\lambda(d\psi ) \ : f(\psi ) \ , \log f(\psi)$ ] , based on the lebesgue measure @xmath123 on @xmath2 instead of @xmath407 , is maximized , under the constraints that the mean be 0 and the covariance be @xmath0 , by the gaussian measure , @xmath438 . ) there is no apparent reason why the family of @xmath432 measures should be hereditary . the situation is different for the microcanonical ensemble : here , the distribution @xmath439 that we propose is in fact the maximizer of the appropriate gibbs entropy functional @xmath440 . which functional is that ? since any measure @xmath441 on @xmath7 whose covariance matrix is the projection @xmath442}(h)$ ] must be concentrated on the subspace @xmath25 and thus can not be absolutely continuous ( possess a density ) relative to @xmath407 , we consider instead its density relative to @xmath443 , that is , we consider @xmath444 and set @xmath445 = - \int\limits_{{\mathscr{s}}({\mathscr{h}}_{e , \delta } ) } u_{e , \delta } ( d\psi ) \ , f(\psi ) \ , \log f(\psi ) \,.\ ] ] under the constraints that the probability measure @xmath432 have mean 0 and covariance @xmath446 , @xmath447 $ ] is maximized by @xmath448 , or @xmath449 ; in fact even without the constraints on @xmath432 , @xmath447 $ ] is maximized by @xmath450 . brody and hughston @xcite have proposed the following distribution @xmath1 to describe thermal equilibrium . they observe that the projective space arising from a finite - dimensional hilbert space , endowed with the dynamics arising from the unitary dynamics on hilbert space , can be regarded as a classical hamiltonian system with hamiltonian function @xmath451 ( and symplectic form arising from the hilbert space structure ) . they then define @xmath1 to be the classical canonical distribution of this hamiltonian system , i.e. , to have density proportional to @xmath452 relative to the uniform volume measure on the projective space ( which can be obtained from the symplectic form or , alternatively , from @xmath407 by projection from the sphere to the projective space ) . however , this distribution leads to a density matrix , different from the usual one @xmath76 given by , that does not describe the canonical ensemble . as @xmath366 is a family of measures satisfying properties 13 for _ most _ density matrices @xmath0 , the question arises whether there is any family of measures , besides @xmath3 , satisfying these properties for _ all _ density matrices . we expect that the answer is no , and formulate the following uniqueness conjecture : _ given , for every hilbert space @xmath2 and every density matrix @xmath0 on @xmath2 , a probability measure @xmath408 on @xmath7 such that properties 13 remain true when @xmath3 is replaced by @xmath408 , then @xmath453 . _ in other words , we conjecture that @xmath454 is the only hereditary covariant inverse of . this is in fact true when we assume in addition that the mapping @xmath455 is suitably continuous . here is the argument : when @xmath0 is a multiple of a projection , @xmath456 for a subspace @xmath457 , then @xmath408 must be , by covariance @xmath458 , the uniform distribution on @xmath459 , and thus @xmath460 in this case . consider now a composite of a system ( system 1 ) and a large heat bath ( system 2 ) with hilbert space @xmath461 and hamiltonian @xmath462 , and consider the microcanonical density matrix @xmath34 for this system . by equivalence of ensembles , we have for suitable @xmath463 that @xmath464 where @xmath465 . by the continuity of @xmath1 and @xmath466 , @xmath467 now consider , for a wave function @xmath36 with distribution @xmath468 respectively @xmath469 , the distribution of the conditional wave function @xmath62 : by heredity , this is @xmath470 respectively @xmath471 . since the distribution of @xmath62 is a continuous function of the distribution of @xmath36 , we thus have that @xmath472 . since we can make the degree of approximation arbitrarily good by making the heat bath sufficiently large , we must have that @xmath473 . for any density matrix @xmath0 on @xmath52 that does not have zero among its eigenvalues , there is an @xmath474 such that @xmath475 for @xmath476 , and thus we have that @xmath453 for such a @xmath0 ; since these are dense , we have that @xmath453 for all density matrices @xmath0 on @xmath52 . since @xmath52 is arbitrary we are done . markov processes in hilbert space have long been considered ( see @xcite for an overview ) , particularly diffusion processes and piecewise deterministic ( jump ) processes . this is often done for the purpose of numerical simulation of a master equation for the density matrix , or as a model of continuous measurement or of spontaneous wave function collapse . such processes could arise as follows . since the conditional wave function @xmath62 arises from the wave function @xmath56 by inserting a random coordinate @xmath227 for the second variable ( and normalizing ) , any dynamics ( i.e. , time evolution ) for @xmath227 , described by a curve @xmath477 and preserving the quantum probability distribution of @xmath227 , for example , as given by bohmian mechanics @xcite , gives rise to a dynamics for the conditional wave function , @xmath478 , where @xmath479 evolves according to schrdinger s equation and @xmath480 is the normalizing factor . in this way one obtains a stochastic process ( a random path ) in @xmath64 . in the case considered in section [ sec : hb1 ] , in which @xmath54 corresponds to a large heat bath , this process must have @xmath481 as an invariant measure . it would be interesting to know whether this process is approximately a simple process in @xmath64 , perhaps a diffusion process , perhaps one of the markov processes on hilbert space considered already in the literature . in this last section , we consider a two - level system , with @xmath482 and @xmath483 and calculate the joint distribution of the energy coefficients @xmath484 and @xmath485 for a @xmath486-distributed @xmath36 as explicitly as possible . we begin with a general finite - dimensional system , @xmath487 , and specialize to @xmath488 later . one way of describing the distribution of @xmath36 is to give its density relative to the hypersurface area measure @xmath128 on @xmath489 ; this we did in . another way of describing the joint distribution of the @xmath84 is to describe the joint distribution of their moduli @xmath199 , or of @xmath490 , as the phases of the @xmath84 are independent ( of each other and of the moduli ) and uniformly distributed , see . for greater clarity , from now on we write @xmath495 instead of @xmath84 . a relation similar to that between @xmath3 , @xmath80 , and @xmath45 holds between the joint distributions of the @xmath496 , of the @xmath497 , and of the @xmath498 . the joint distribution of the @xmath498 is very simple : they are independent and exponentially distributed with means @xmath499 . since the density of @xmath500 relative to @xmath501 , @xmath502 , is a function of the moduli alone , and since , according to , @xmath503 , we have that @xmath504 thus , @xmath505 where each @xmath506 . finally , the @xmath496 arise by normalization , @xmath507 we now specialize to the two - level system , @xmath488 . since @xmath508 , it suffices to determine the distribution of @xmath509 , for which we give an explicit formula in below . we want to obtain the marginal distribution of from the joint distribution of the @xmath497 in @xmath510 , the first quadrant of the plane , as given by . to this end , we introduce new coordinates in the first quadrant : @xmath511where @xmath512 and @xmath513 . conversely , we have @xmath514 and @xmath515 , and the area element transforms according to @xmath516 with @xmath519 and @xmath520 for @xmath521 . the density @xmath522 of the distribution of @xmath509 is depicted in figure 1 for various values of @xmath24 . for @xmath523 , @xmath522 is identically 1 . for @xmath524 , we have @xmath525 , so that @xmath526 is decreasing monotonically from @xmath527 at @xmath528 to @xmath529 at @xmath530 ; hence , @xmath522 is increasing monotonically from @xmath531 to @xmath532 . for @xmath533 , we have @xmath534 , and hence @xmath522 is decreasing monotonically from @xmath531 to @xmath532 . in all cases @xmath522 is convex since @xmath535 . _ acknowledgments . _ we thank andrea viale ( universit di genova , italy ) for preparing the figure , eugene speer ( rutgers university , usa ) for comments on an earlier version , james hartle ( uc santa barbara , usa ) and hal tasaki ( gakushuin university , tokyo , japan ) for helpful comments and suggestions , eric carlen ( georgia institute of technology , usa ) , detlef drr ( lmu mnchen , germany ) , raffaele esposito ( universit di laquila , italy ) , rossana marra ( universit di roma `` tor vergata '' , italy ) , and herbert spohn ( tu mnchen , germany ) for suggesting references , and juan diego urbina ( the weizmann institute , rehovot , israel ) for bringing the connection between gaussian random wave models and quantum chaos to our attention . we are grateful for the hospitality of the institut des hautes tudes scientifiques ( bures - sur - yvette , france ) , where part of the work on this paper was done . the work of s. goldstein was supported in part by nsf grant dms-0504504 , and that of j. lebowitz by nsf grant dmr 01 - 279 - 26 and afosr grant af 49620 - 01 - 1 - 0154 . the work of r. tumulka was supported by infn and by the european commission through its 6th framework programme `` structuring the european research area '' and the contract nr . rita - ct-2004 - 505493 for the provision of transnational access implemented as specific support action . the work of n. zangh was supported by infn . lebowitz , j. l. : microscopic reversibility and macroscopic behavior : physical explanations and mathematical derivations . in _ 25 years of non - equilibrium statistical mechanics _ , proceedings , sitges conference , barcelona , spain , 1994 , in _ lecture notes in physics _ , j.j . brey , j. marro , j.m . rub , and m. san miguel ( eds . ) ( springer - verlag , berlin , 1995 ) . tasaki , h. : from quantum dynamics to the canonical distribution : general picture and a rigorous example . lett . _ * 80 * , 13731376 ( 1998 ) . urbina , j. d. and richter , k. : random wave models . _ elsevier encyclopedia of mathematical physics _ ( 2006 ) . to appear . von neumann , j. : _ mathematical foundations of quantum mechanics _ ( princeton university press , princeton , 1955 ) . translation of _ mathematische grundlagen der quantenmechanik _ ( springer - verlag , berlin , 1932 ) .

for a quantum system , a density matrix @xmath0 that is not pure can arise , via averaging , from a distribution @xmath1 of its wave function , a normalized vector belonging to its hilbert space @xmath2 . while @xmath0 itself does not determine a unique @xmath1 , additional facts , such as that the system has come to thermal equilibrium , might . it is thus not unreasonable to ask , which @xmath1 , if any , corresponds to a given thermodynamic ensemble ? to answer this question we construct , for any given density matrix @xmath0 , a natural measure on the unit sphere in @xmath2 , denoted @xmath3 . we do this using a suitable projection of the gaussian measure on @xmath2 with covariance @xmath0 . we establish some nice properties of @xmath3 and show that this measure arises naturally when considering macroscopic systems . in particular , we argue that it is the most appropriate choice for systems in thermal equilibrium , described by the canonical ensemble density matrix @xmath4 . @xmath3 may also be relevant to quantum chaos and to the stochastic evolution of open quantum systems , where distributions on @xmath2 are often used . key words : canonical ensemble in quantum theory ; probability measures on hilbert space ; gaussian measures ; density matrices .

quant-ph0309021

attempts to geometrical unification of gravity with other interactions , using higher dimensions other than our conventional @xmath1 space time , began shortly after invention of the special relativity ( * sr * ) . nordstrm was the first who built a unified theory on the base of extra dimensions @xcite . tight connection between sr and electrodynamics , namely the lorentz transformation , led kaluza @xcite and klein @xcite to establish @xmath0 versions of general relativity ( * gr * ) in which electrodynamics rises from the extra fifth dimension . since then , considerable amount of works have been focused on this idea either using different mechanism for compactification of extra dimension or generalizing it to non compact scenarios ( see e.g. ref . @xcite ) such as brane world theories @xcite , space time matter or induced matter ( * i m * ) theories @xcite and references therein . the latter theories are based on the campbell magaard theorem which asserts that any analytical @xmath6dimensional riemannian manifold can locally be embedded in an @xmath7dimensional ricci flat riemannian manifold @xcite . this theorem is of great importance for establishing @xmath1 field equations with matter sources locally to be embedded in @xmath0 field equations without _ priori _ introducing matter sources . indeed , the matter sources of @xmath1 space times can be viewed as a manifestation of extra dimensions . this is actually the core of i m theory which employs gr as the underlying theory . on the other hand , jordan @xcite attempted to embed a curved @xmath1 space time in a flat @xmath0 space time and introduced a new kind of gravitational theory , known as the scalar tensor theory . following his idea , brans and dicke @xcite invented an attractive version of the scalar tensor theory , an alternative to gr , in which the weak equivalence principle is saved and a non minimally scalar field couples to curvature . the advantage of this theory is that it is more machian than gr , though mismatching with the solar system observations is claimed as its weakness @xcite . however , the solar system constraint is a generic difficulty in the context of the scalar tensor theories @xcite , and it does not necessarily denote that the evolution of the universe , at all scales , should be close to gr , in which there are some debates on its tests on cosmic scales @xcite . although it is sometimes desirable to have a higher dimensional energy momentum tensor or a scalar field , for example in compactification of extra curved dimensions @xcite , but the most preference of higher dimensional theories is to obtain macroscopic @xmath1 matter from pure geometry . in this approach , some features of a @xmath0 vacuum brans dicke ( * bd * ) theory based on the idea of i m theory have recently been demonstrated @xcite , in where the role of gr as fundamental underlying theory has been replaced by the bd theory of gravitation . actually , it has been shown that @xmath0 vacuum bd equations , when reduced to four dimensions , lead to a modified version of the @xmath1 brans dicke theory which includes an induced potential . whereas in the literature , in order to obtain accelerating universes , inclusion of such potentials has been considered in _ priori _ by hand . a few applications and a @xmath8dimensional version of this approach have been performed @xcite . though , in refs . @xcite , it has also been claimed that their procedure provides explicit definitions for the effective matter and induced potential . besides , some misleading statements and equations have been asserted in ref . @xcite , and hence we have re derived the procedure in section @xmath9 . actually , the reduction procedure of a @xmath0 analogue of the bd theory , with matter content , on every hypersurface orthogonal to an extra cyclic dimension ( recovering a modified bd theory described by a 4metric coupled to two scalar fields ) has previously been performed in the literature @xcite . however , the key point of i m theories are based on not introducing matter sources in @xmath0 space times . in addition , recent measurements of anisotropies in the microwave background suggest that our ordinary @xmath1 universe should be spatially flat @xcite , and the observations of type ia supernovas indicate that the universe is in an accelerating expansion phase @xcite . hence , the universe should mainly be filled with a dark energy or a quintessence which makes it to expand with acceleration @xcite . then after an intensive amount of work has been performed in the literature to explain the acceleration of the universe . in this work , we explore the friedmann robertson walker ( * frw * ) type cosmology of a @xmath0 vacuum bd theory and obtain solutions and related conditions . this model has extra terms , such as a scalar field and scale factor of fifth dimension , which make it capable to present accelerated universes beside decelerated ones . in the next section , we give a brief review of the induced modified bd theory from a @xmath0 vacuum space time to rederive the induced energy momentum tensor , as has been introduced in ref . @xcite , for our purpose to employ the energy density and pressure . in section @xmath10 , we consider a generalized frw metric in the @xmath0 space time and specify frw cosmological equations and employ the weak energy condition ( * wec * ) to obtain the energy density and pressure conditions . then , we probe two special cases of a constant scale factor of the fifth dimension and a constant scalar field . in section @xmath11 , we proceed to exhibit that @xmath0 vacuum bd equations , employing the generalized frw metric , are equivalent , in general , to the corresponding vacuum @xmath1 ones . this equivalency can be viewed as the main point within this work which distinguishes it from refs . @xcite . in section @xmath12 , we find exact solutions for flat geometries and proceed to get solutions fulfilling the wec while being compatible with the recent observational measurements . we also provide a few tables and figures for a better view of acceptable range of parameters . finally , conclusions are presented in the last section . following the idea of i m theories @xcite , one can replace gr by the bd theory of gravitation as the underlying theory @xcite . for this purpose , the action of @xmath0 brans dicke theory can analogously be written in the jordan frame as @xmath13=\int\sqrt{|{}^{_{(5)}}g| } \left ( \phi \ ^{^{(5)}}\!r-\frac{\omega}{\phi}g^{_{ab}}\phi_{,_{a}}\phi_{,_{b}}+ 16\pi l_{m } \right ) d^{5}x\ , , \ ] ] where @xmath14 , the capital latin indices run from zero to four , @xmath15 is a positive scalar field that describes gravitational coupling in five dimensions , @xmath16 is @xmath0 ricci scalar , @xmath17 is the determinant of @xmath0 metric @xmath18 , @xmath19 represents the matter lagrangian and @xmath20 is a dimensionless coupling constant . the field equations obtained from action ( 1 ) are @xmath21 and @xmath22 where @xmath23 , @xmath24 is @xmath0 einstein tensor , @xmath25 is @xmath0 energy momentum tensor , @xmath26 . also , in order to have a non ghost scalar field in the conformally related einstein frame , i.e. a field with a positive kinetic energy term in that frame , the bd coupling constant must be @xmath27 @xcite . as explained in the introduction , we propose to consider a @xmath0 vacuum state , i.e. @xmath28 , where equations ( 2 ) and ( 3 ) read @xmath29 and ) for later on convenient . ] @xmath30 for cosmological purposes one usually restricts attention to @xmath0 metrics of the form , in local coordinates @xmath31 , @xmath32 where @xmath33 represents the fifth coordinate , the greek indices run from zero to three and @xmath34 . it should be noted that this ansatz is restrictive , but one limits oneself to it for reasons of simplicity . assuming the @xmath0 space time is foliated by a family of hypersurfaces , @xmath35 , defined by fixed values of the fifth coordinate , then the metric intrinsic to every generic hypersurface , e.g. @xmath36 , can be obtained when restricting the line element ( [ 6 ] ) to displacements confined to it . thus , the induced metric on the hypersurface @xmath37 can have the form @xmath38 in such a way that the usual @xmath1 space time metric , @xmath39 , can be recovered . hence , equation ( [ 4 ] ) on the hypersurface @xmath37 can be written as @xmath40,\ ] ] where @xmath41 is an induced energy momentum tensor of the effective @xmath1 modified bd theory , which is defined as @xmath42 with @xmath43 \bigg \}\end{aligned}\ ] ] and @xmath44-\frac{1}{2}g'_{\alpha\beta}\phi ' \bigg \}.\ ] ] also , the induced potential has been defined in the formal identification as @xcite @xmath45\equiv -\epsilon \frac{\omega}{b^{2}}\frac{\phi^{'2}}{\phi } \big |_{_{\sigma_{0 } } } , \ ] ] where the prime denotes derivative with respect to the fifth coordinate . such an identification has been claimed @xcite to be valid depending on metric background and considering separable scalar fields . however , this definition is different from what has been used in ref . @xcite . reduction of equation ( [ 5 ] ) on the hypersurface @xmath37 gives @xmath46-\frac{b_{,\mu}}{b}\phi^{,\mu}\ , , \ ] ] which after manipulation resembles the other field equation of a modified bd theory in four dimensions with induced potential . the definition @xmath41 and equation ( [ freduction ] ) are all we need for our purpose in this work and an interested reader can consult refs . @xcite for further details . in the next section we assume a generalized frw metric in a vacuum @xmath0 universe to find its cosmological implications . for a @xmath0 universe with an extra space like dimension in addition to the three usual spatially homogenous and isotropic ones , metric ( [ 6 ] ) can be written as @xmath47+b^2(t , y)dy^2\,,\ ] ] that can be considered as a generalized frw solution . the scalar field @xmath15 and the scale factors @xmath48 and @xmath49 , in general , are functions of @xmath50 and @xmath33 . however , for simplicity and physical plausibility , we assume the extra dimension is cyclic , i.e. the hypersurface orthogonal space like is a killing vector field in the underlying @xmath0 space time @xcite . hence , all fields are functions of the cosmic time only , and definition ( [ 6.3 ] ) makes the induced potential vanishes . in this case , we will show that such a universe can have accelerating and decelerating solutions . note that , the functionality of the scale factor @xmath49 on @xmath33 , either can be eliminated by transforming to a new extra coordinate if @xmath49 is a separable function , and or makes no changes in the following equations if @xmath49 is the only field that depends on @xmath33 . besides , in the compactified extra dimension scenarios , all fields are fourier expanded around @xmath51 , and henceforth one can have terms independent of @xmath33 to be observable , i.e. physics would thus be effectively independent of compactified fifth dimension @xcite . considering metric ( [ 7 ] ) , equations ( [ 4 ] ) and ( [ 5 ] ) result in cosmological equations @xmath52 @xmath53 @xmath54 and @xmath55 which are not independent equations and where @xmath56 , @xmath57 and @xmath58 . by employing relation ( [ 6.4 ] ) , one can interpret the right hand side of equations ( [ 8 ] ) and ( [ 9 ] ) as energy density and pressure of the induced effective perfect fluid , i.e. @xmath59 and @xmath60 where @xmath61 or @xmath9 or @xmath10 without summation on it . the latter equality in ( [ 11.2 ] ) comes from equation ( [ 22 ] ) which will be derived in the next section . therefor , the equation of state is @xmath62 the usual matter in our universe has a positive energy density , this basically has been demanded by the wec , in which time like observers must obtain positive energy densities . actually , the complete wec is @xcite @xmath63 now , let us consider that the scale factor of the fifth dimension and the scalar field are not constant values , i.e. @xmath64 and @xmath65 . then , by applying conditions ( [ 11.3 ] ) into relations ( [ 11.1 ] ) and ( [ 11.2 ] ) , one gets @xmath66 or @xmath67 where we also have assumed expanding universes , i.e. @xmath68 . using conditions ( [ 11.4 ] ) and ( [ 11.5 ] ) in relation ( [ 11.6 ] ) gives @xmath69 or @xmath70 in where the effective dust matter can be achieved when @xmath71 goes to negative or positive infinity , respectively . in section @xmath12 , we explore characteristic of the corresponding universes for the above results . meanwhile , in the following , we consider two special cases of a constant scale factor of the fifth dimension and a constant scalar field . + + * constant scale factor of fifth dimension * + when @xmath49 is a constant , equations ( [ 8])([11 ] ) reduce to @xmath72 these are exactly the ordinary vacuum bd equations in @xmath1 space time , with @xmath73 , as expected . + + * constant scalar field * + when @xmath15 is a constant , action ( [ 1 ] ) reduces to a @xmath0 einstein gravitational theory that has been considered in ref . @xcite in general situation ( i.e. the extra dimension is not cyclic ) . in this case , equations ( [ 8])([11 ] ) become @xmath74 and , the usual frw equations are equipped with @xmath75 , which refers to a radiation like dominated universe for any kind of geometry without a _ priori _ assumption that the scale factor of the fifth dimension is proportional to the inverse of the usual scale factor , i.e. @xmath76 . actually , the radiation like result is expected . for where there is no dependency on the extra dimension , the usual four dimensional part of metric ( [ 7 ] ) and the third equation ( [ 15 ] ) give a wave equation for the scale factor of fifth dimension . hence , definitions ( [ 6.5 ] ) and ( [ 6.6 ] ) yield a traceless induced energy momentum tensor , as mentioned in ref.@xcite . exact solution of the second equation of ( [ 15 ] ) is @xmath77 substituting solution ( [ 19 ] ) into the first or third equation of ( [ 15 ] ) gives @xmath78 where @xmath79 and @xmath80 are constants of integration , and we have assumed that @xmath1 space time has originated from a big bang . for a closed geometry , solution ( [ 19 ] ) admits @xmath81 and predicts a big crunch at @xmath82 for the usual spatial coordinates while the fifth dimension tends to infinite size and is always real , for the maximum value of the usual scale factor is @xmath83 . but , a flat geometry expands for ever and accepts @xmath81 . an open geometry also expands for ever and admits @xmath84 . in this case , @xmath85 results in @xmath86 and @xmath87 . time evolution of scale factors correspond to closed , flat and open geometries have been illustrated in fig . @xmath88 with constant values of @xmath89 and @xmath90 as an example . in the next two sections , we again consider a more general situation in which the scale factor of the fifth dimension and the scalar field are not constants . [ cols="^,^,^ " , ] analogous to the approach of i m theories , one can consider the bd gravity as the underlying theory . hence , extra geometrical terms , coming from the fifth dimension , are regarded as an induced matter and induced potential . we have followed , with some corrections , the procedure of ref . @xcite for introducing the induced potential and have employed a generalized frw type solution for a @xmath0 vacuum bd theory . hence , the scalar field and scale factors of the @xmath0 metric can , in general , be functions of the cosmic time and the extra dimension . however , for simplicity , we have assumed the scalar field and scale factors to be only functions of the cosmic time , where this makes the induced potential , by its definition , vanishes . we then have revealed that in general situations , in which the scale factor of the fifth dimension and scalar field are not constants , the @xmath0 equations , for any kind of geometry , admit a power law relation between the scalar field and scale factor of the fifth dimension . hence , the procedure exhibits that @xmath0 vacuum frw like equations are equivalent , in general , to the corresponding @xmath1 vacuum ones with the same spatial scale factor but a new ( or modified ) scalar field and a new coupling constant . this equivalency can be viewed as the distinguished point of this work from refs . indeed , through investigating the @xmath0 vacuum frw like equations , we have shown that its equivalent @xmath1 vacuum equations admit accelerated scale factors , contrary to what one may have expected from a vacuum space time . conclusions of the complete investigation of the induced @xmath1 equations are as follows . following our investigations for cosmological implications , we have shown that for the special case of a constant scale factor of the fifth dimension , the @xmath0 vacuum frw like equations reduce to the corresponding equations of the usual @xmath1 vacuum bd theory , as expected . in the special case of a constant scalar field , the action reduces to a @xmath0 einstein gravitational theory and the equations reduce to the usual frw equations with a typical radiation dominated universe . for this situation , we also have obtained dynamics of scale factors of the ordinary and extra dimensions for any kind of geometry without any _ priori _ assumption among them . solutions predict a limited life time for closed geometries and unlimited one for flat and open geometries . a typical time evolutions of scale factors correspond to closed , flat and open geometries have been illustrated in fig . @xmath88 . then , we have focused on spatially flat geometries and have obtained exact solutions of scale factors and scalar field . solutions are found to be in the form of power law and exponential ones in the cosmic time . we also have employed the wec for the induced matter of the @xmath1 modified bd gravity , that gives two conditions ( [ 49.1 ] ) and ( [ 49.2 ] ) . we then have pursued properties of these solutions and have indicated mathematically and physically acceptable ranges of them , and the results have been presented in a few tables and figures . all types of solutions fulfill the wecs in different ranges , where the exponential solutions are more restricted . the solutions fulfilling the wec ( [ 49.1 ] ) have negative pressures , but the figures illustrate that for the power law results there are decelerating solutions beside accelerating ones . for this condition , both @xmath91 and @xmath92 decrease with the cosmic time , but the extra dimension grows . on the other hand , the solutions satisfying the wec ( [ 49.2 ] ) have positive pressures , where the power law results accept accelerating solutions in addition to decelerating ones . for this condition , again decreasing energy density and pressure with the time can occur for some solutions , however all with shrinking extra dimension . the homogeneity between the extra dimension and the usual spatial dimensions , i.e. @xmath93 , can take place in the solutions , but for the power law ones the wecs exclude it . by considering non ghost scalar fields and appealing the recent observational measurements , the solutions have been more restricted . actually , we have illustrated that the accelerating power law solutions , which satisfy the wec and have non ghost scalar fields , are compatible with the recent observations in ranges @xmath3 for the bd coupling constant and @xmath4 for dependence of the fifth dimension scale factor with the usual scale factor . these ranges also fulfill the condition @xmath5 which prevents ghost scalar fields in the equivalent @xmath1 vacuum bd equations . incidentally , this range is more restricted than the one obtained in ref . @xcite , i.e. @xmath94 , where the difference may have been caused by the distinct definition of the induced potential in two approaches of ref . @xcite and ref . however , we should remind that it has also been shown @xcite that the wec , for @xmath0 space times , requires @xmath95 , in which no other experimental evidences have been considered . 1 g. nordstrm , _ phys . z. _ * 15 * , 504 ( 1914 ) . t. kaluza , _ sitz . wiss . _ * 33 * , 966 ( 1921 ) . o. klein , _ z. phys . _ * 37 * , 895 ( 1926 ) . overduin and p.s . wesson , _ phys . * 283 * , 303 ( 1997 ) . m. pavi , `` the landscape of theoretical physics : a global view from point particles to the brane world and beyond , in search of a unifying principle '' _ gr qc/0610061_. p.s . wesson , _ space time matter , modern kaluza klein theory _ ( world scientific , singapore , 1999 ) ; + p.s . wesson , _ five dimensional physics _ ( world scientific , singapore , 2006 ) . campbell , _ a course of differentioal geometry _ ( claredon press , oxford , 1926 ) ; + l. magaard , _ zur einbettung riemannscher raume in einstein raume und konformeuclidische raume _ ( ph.d . thesis , kiel , 1963 ) ; + c. romero , r. tavakol and r. zalaletdinov , _ gen . * 28 * , 365 ( 1995 ) ; + j. lidsey , c. romero , r. tavakol and s. rippl , _ class . * 14 * , 865 ( 1997 ) ; + s.s . seahra and p.s . wesson , _ class . grav . _ * 20 * , 1321 ( 2003 ) . p. jordan , _ projective relativity _ ( friedrich vieweg und sohn , braunschweig , 1955 ) . c. brans and r.h . dicke , _ phys . rev . _ * 124 * , 925 ( 1961 ) ; + r.h . dicke , _ phys . rev . _ * 125 * , 2163 ( 1962 ) . b. bertotti , l. iess and p. tortora , _ nature _ * 425 * , 374 ( 2003 ) ; + y. fujii and k. maeda , _ the scalar tensor theory of gravitation _ ( cambridge university press , cambridge , 2004 ) . n. banerjee and d. pavon , _ phys . d _ * 63 * , 043504 ( 2001 ) ; + a.a . sen , s. sen and s. sethi , _ phys . d _ * 63 * , 107501 ( 2001 ) . r. bean , `` a weak lensing detection of a deviation from general relativity on cosmic scales '' , _ astro - ph/0909.3853 _ ; + s.f . et al _ , `` testing general relativity with current cosmological data '' , _ astro ph/1002.1962 _ ; + r. bean and m. tangmatitham , `` current constraints on the cosmic growth history '' , _ astro ph/1002.4197_. t. appelquist , a. chodos and p.g.o . freund , _ modern kaluza klein theories _ ( addison wesley , menlo park , 1987 ) . aguilar , c. romero and a. barros , _ gen . _ * 40 * , 117 ( 2008 ) . j. ponce de leon , _ class . grav . _ * 27 * , 095002 ( 2010 ) ; + j. ponce de leon , _ jcap _ * 03 * , 030 ( 2010 ) . s.m . m.rasouli and m. farhoudi , `` @xmath8dimensional brans dicke theory from ( @xmath96 ) dimensions '' , _ work in progress _ ; + s.m . m.rasouli , m. farhoudi and h.r . sepangi , `` anisotropic cosmological models in modified brans dicke theory '' , _ work in progress_. l. qiang , y. ma , m. han and d. yu , _ phys . rev . d _ * 71 * , 061501 ( 2005 ) ; + l. qiang , y. gong , y. ma and x. chen , _ astro - ph/0910.1885_. n.a . et al _ , _ science _ * 284 * , 1481 ( 1999 ) ; + p. de bernardis _ et al _ , _ nature _ * 404 * , 955 ( 2000 ) ; + s. hanany _ et al _ , _ astrophys . j. _ * 545 * , l5 ( 2000 ) ; + a. balbi _ et al _ , _ astrophys . j. _ * 545 * , l1 ( 2000 ) . s. perlmutter _ et al _ , _ nature _ * 391 * , 51 ( 1998 ) ; + a.g . et al _ , _ astron . j. _ * 116 * , 1009 ( 1998 ) ; + s. perlmutter _ et al _ , _ astrophys . j. _ * 517 * , 565 ( 1999 ) ; + p. de bernardis _ et al _ , _ astrophys . j. _ * 564 * , 559 ( 2002 ) ; + s.m . carroll , `` why is the universe accelerating ? '' , _ astro ph_/0310342 . peebles and b. ratra , _ astrophys . j. _ * 325 * , l17 ( 1988 ) ; + g. efstathiou _ et al _ , _ nature _ * 348 * , 705 ( 1990 ) ; + j.p . ostriker and p.j . steinhardt , _ nature _ * 377 * , 600 ( 1995 ) . freund , _ nucl . b _ * 209 * , 146 ( 1982 ) . y.m . cho , _ phys . lett . _ * 68 * , 3133 ( 1992 ) ; + j.d . barrow , d. kimberly and j. magueijo , _ class . * 21 * , 4289 ( 2004 ) . aguilar and m. bellini , _ phys . b _ * 619 * , 208 ( 2005 ) ; + m. anabitarte , j.e.m . aguilar and m. bellini , _ eur . j. _ * 45 * , 249 ( 2006 ) . s.w . hawking and g.f.r . ellis , _ the large scale structure of space time _ ( cambridge university press , cambridge , 1973 ) . p.s . wesson and j. ponce de leon , _ j. math . _ * 33 * 3883 ( 1992 ) . w.l . freedman and m.s . turner , _ rev . * 75 * , 1433 ( 2003 ) .

we follow the approach of induced matter theory for a five dimensional ( @xmath0 ) vacuum brans dicke theory and introduce induced matter and induced potential in four dimensional ( @xmath1 ) hypersurfaces , and then employ a generalized frw type solution . we confine ourselves to the scalar field and scale factors be functions of the cosmic time . this makes the induced potential , by its definition , vanishes , but the model is capable to expose variety of states for the universe . in general situations , in which the scale factor of the fifth dimension and scalar field are not constants , the @xmath0 equations , for any kind of geometry , admit a power law relation between the scalar field and scale factor of the fifth dimension . hence , the procedure exhibits that @xmath0 vacuum frw like equations are equivalent , in general , to the corresponding @xmath1 vacuum ones with the same spatial scale factor but a new scalar field and a new coupling constant , @xmath2 . we show that the @xmath0 vacuum frw like equations , or its equivalent @xmath1 vacuum ones , admit accelerated solutions . for a constant scalar field , the equations reduce to the usual frw equations with a typical radiation dominated universe . for this situation , we obtain dynamics of scale factors of the ordinary and extra dimensions for any kind of geometry without any _ priori _ assumption among them . for non constant scalar fields and spatially flat geometries , solutions are found to be in the form of power law and exponential ones . we also employ the weak energy condition for the induced matter , that gives two constraints with negative or positive pressures . all types of solutions fulfill the weak energy condition in different ranges . the power law solutions with either negative or positive pressures admit both decelerating and accelerating ones . some solutions accept a shrinking extra dimension . by considering non ghost scalar fields and appealing the recent observational measurements , the solutions are more restricted . we illustrate that the accelerating power law solutions , which satisfy the weak energy condition and have non ghost scalar fields , are compatible with the recent observations in ranges @xmath3 for the coupling constant and @xmath4 for dependence of the fifth dimension scale factor with the usual scale factor . these ranges also fulfill the condition @xmath5 which prevents ghost scalar fields in the equivalent @xmath1 vacuum brans dicke equations . the results are presented in a few tables and figures . -2.7 cm keywords : brans dicke theory ; induced matter theory ; frw cosmology .

1005.2501

the understanding of the strong and weak interactions of a heavy quark system is an important topic , and the purely leptonic decays @xmath7 seems to be the useful tools for this purpose . in particular , these processes are very simple in that no hadrons and photons appear in the final states . however , the rates of these purely leptonic decays are helicity suppressed with the factor of @xmath8 for @xmath9 and @xmath10 ( the @xmath11 channel , in spite of no suppression , is hard to observe the decay because of the low efficiency ) . therefore it is natural to extend the purely leptonic @xmath12 decay searches to the corresponding radiative modes @xmath5 . these radiative leptonic decays receive two types of contributions : inner bremsstrahlung ( ib ) and structure - dependent ( sd ) @xcite . as is known , the ib contributions are still helicity suppressed , while the sd ones are reduced by the fine structure constant @xmath13 but they are not suppressed by the lepton mass . accordingly , the radiative leptonic @xmath12 decay rates could have an enhancement with respect to the purely leptonic ones , and would offer useful information about the cabibbo - kobayashi - maskawa matrix element @xmath14 and the decay constant @xmath15 @xcite . recently , there has been a great deal of theoretical attention @xcite to these radiative leptonic @xmath12 decays . experimentally , the current upper limits for these modes are @xmath16 and @xmath17 at the @xmath18 confidence level @xcite . with the better statistic expected from the @xmath12 factories , the observations of these decays could become soon feasible . the hadronic matrix elements responsible for the above decays can be calculated in various quark models . however , the relativistic effects must be considered seriously in calculations as the recoil momentum is large . this problem is taken into account by the light - front quark model ( lfqm ) @xcite which has been considered as one of the best effective relativistic quark models in the description of the exclusive heavy hadron decays @xcite . its simple expression , relativistic structure and predictive power have made wide applications of the lfqm in exploring and predicting the intrinsic heavy hadron dynamics . however , almost all the previous investigations have not covariantly extracted the form factors from the relevant matrix elements and paid enough attention to the consistency with heavy quark symmetry ( hqs ) and heavy quark effective theory ( hqet ) . the covariant light - front model @xcite has resolved the above - mentioned shortcomings in the lfqm and has improved the current understanding of the qcd analysis of heavy hadrons . this model consists of a heavy meson bound state in the heavy quark limit ( namely @xmath19 ) , which is fully consistent with hqs , plus a reliable approach from this bound state to systematically calculate the @xmath20 corrections within hqet in terms of the @xmath20 expansion of the fundamental qcd theory . in this paper , we will use the covariant light - front model to investigate the radiative leptonic @xmath12 decays in the heavy quark limit . the paper is organized as follows . in sec . 2 a general construction of covariant light - front bound states is provided , the diagrammatic rules within this model are also listed . in sec . 3 we evaluate the decay constant of heavy meson @xmath2 and the form factors of the radiative leptonic heavy meson decay @xmath21 in a complete covariant way . in sec . 4 the relation between @xmath2 and @xmath0 is obtained . a few numerical calculations made with the help of the gaussian - type wave function are presented . finally , a summary is given in sec . the light - front bound states of heavy meson that are written in a form of exhibiting explicitly the boost covariance have been shown in the literature @xcite . in this paper , we focus on the bound states of heavy mesons in the heavy quark limit : @xmath22[d^3p_q ] 2(2\pi)^3v^+ \delta^3 ( \overline{\lambda}v - k - p_q ) \nonumber \\ & \times & \sum_{\lambda_q,\lambda_q } r^{ss_z } ( x , \kappa_{\bot } , \lambda_q , \lambda_q ) \phi^{ss_z } ( x,\kappa^2_{\bot } ) b_v^\dagger(k , \lambda_q ) d_q^\dagger ( p_q , \lambda_q)|0\rangle,\end{aligned}\ ] ] where @xmath23 is the velocity of the heavy meson , @xmath24 is the residual momentum of heavy quark , @xmath25 is the momentum of light antiquark , @xmath26 = { dk^+ d^2 k_\bot \over { 2(2\pi)^3 v^+}},~~[d^3p_q ] = { dp_q^+ d^2 p_{q\bot } \over { 2(2\pi)^3 p_q^+}},\ ] ] and @xmath27 is the residual center mass of heavy mesons . the relative momentum @xmath28 was first introduced in ref . @xcite as the product of longitudinal momentum fraction @xmath29 of the valence antiquark and the mass of heavy meson @xmath30 , namely @xmath31 . the relative transverse and longitudinal momenta , @xmath32 and @xmath33 , are obtained by @xmath34 in eq . ( [ hqslfb ] ) , @xmath35 and @xmath36 are helicities of heavy quark and light antiquark , respectively . in phenomenological calculations , one usually ignores the dynamical dependence of the light - front spin so that the function @xmath37 can be approximately expressed by taking the covariant form for the so - called melosh matrix @xcite in the heavy quark limit , @xmath38 where @xmath39 and @xmath40 are spinors for the heavy quark and light antiquark , @xmath41 the operators @xmath42 and @xmath43 ) create a heavy quark and a light antiquark with @xmath44 the normalization of the heavy meson bound states in the heavy quark limit is then given by @xmath45 which leads to two things : first , the heavy meson bound state @xmath46 in this model rescales the one @xmath47 in the lfqm by @xmath48 and , second , the space part @xmath49 ( called the light - front wave function ) in eq.([hqslfb ] ) has the following wave function normalization condition : @xmath50 in principle , the heavy quark dynamics is completely described by hqet , which is given by the @xmath20 expansion of the heavy quark qcd lagrangian @xmath51 therefore , @xmath46 and @xmath52 are then determined by the leading lagrangian @xmath53 . the authors of ref . @xcite have shown from the light - front bound state equation that @xmath52 must be degenerate for @xmath54 and @xmath55 . as a result , we can simply write @xmath56 in the heavy quark limit . ( [ hqslfb ] ) together with eqs . ( [ spin ] ) and ( [ msiwf ] ) is then the heavy meson light - front bound states in the heavy quark limit that obey hqs . furthermore , eq . ( [ hqslfb ] ) can be rewritten in a fully covariant form if @xmath57 is a function of @xmath58 : @xmath59 where the antiquark @xmath60 in bound states is on - mass - shell , @xmath61 . hence , @xmath62 as to the normalization condition of @xmath63 , eq . ( [ nwf ] ) can also be rewritten in a covariant form : @xmath64 the left - hand side of eq . ( [ cn ] ) can be easily obtained in a diagrammatic way as shown in fig . 1 : @xmath65 \over 4(v \cdot p_q + m_q)}\nonumber \\ & = & { \rm eq.~(\ref{cn } ) } \ , .\end{aligned}\ ] ] in general , the on - shell feynman rules within this model is given as follows @xcite : \(i ) the heavy meson bound state in the heavy quark limit gives a vertex as follows : @xmath66 \(ii ) the internal line attached to the bound state gives an on - shell propagator : @xmath67 \(iii ) for the internal antiquark line attached to the bound state , sum over helicity and integrate the internal momentum using @xmath68 \(iv ) for all other lines and vertices that do not attach to the bound states , the diagrammatic rules are the same as the feynman rules in the conventional field theory . now , we shall present the evaluations of the decay constants and the form factors for heavy mesons within the covariant light - front model . first , the decay constants of pseudoscalar and vector mesons are defined by @xmath69 and @xmath70 , where @xmath60 and @xmath71 the light and heavy quark field operators , respectively . in the heavy quark limit , the decay constants have the expressions @xmath72 so that @xmath73 using the above bound states and feynman rules , it is very simply to evaluate the relevant matrix elements ( diagrammatically shown in fig . 2 ) : @xmath74 where @xmath75 and @xmath76 here @xmath77 is the number of colors . thus , it is easily found : @xmath78 as expected from hqs . ( [ vponshell ] ) , the integral in eq . ( [ decc ] ) gives @xmath79 next , the form factors for the radiative leptonic decays , which come from vector and axial vector currents are defined by @xcite @xmath80 , \label{fa}\end{aligned}\ ] ] respectively , where @xmath81 . if one ignores the lepton mass ( namely ignores the ib contributions ) , the leading contributions to @xmath82 come from pole diagrams with one vector intermediate state ( @xmath83 ) , and those to @xmath84 from two axial vector states ( @xmath85 ) @xcite . in the non - relativistic quark model ( nrqm ) , the dominant contribution comes from the former state , @xmath86 where @xmath87 is the charge of light quark in units of @xmath88 , @xmath89 , and the hadronic parameter @xmath90 gev@xmath91 @xcite . the contributions of the latter states are also proportional to @xmath92 , the axial vector meson decay constants . but these are zero because the wave function at the origin for an orbitally excited state vanishes . in the heavy quark limit , the above form factors have the expressions @xmath93,\end{aligned}\ ] ] where @xmath94 . so that @xmath95 the contributions to these form factors coming from the coupling of the photon to the heavy and light quark can be diagrammatically shown in fig . 3 ( a ) and ( b ) , respectively . using the above bound states and feynman rules , the former contributions to the form factor , for example , @xmath0 are given by @xmath96 where @xmath97 is the charge of heavy quark in units of @xmath88 and @xmath98 was obtained in eq . ( [ covint1 ] ) . performing the trace and comparing with eq . ( [ decc ] ) , the heavy quark contribution to the form factor @xmath0 is obtained by @xmath99 as obtained from perturbative qcd @xcite . in the case of the light quark contribution , it can be written as @xmath100 where @xmath101 and @xmath102 it is easily found @xmath103 using the light - front relative momentum , the integral eq . ( [ fvq ] ) gives @xmath104 with this way , we also calculate the form factors @xmath105 which come from the coupling to the axial vector current and find that @xmath106 . these results are consistent with those in ref . @xcite , but contrary to those in nrqm @xcite . in addition , it has been emphasized in the literature that various hadronic form factors calculated in the lfqm should be extracted only from the plus component of the corresponding currents . also , the lfqm calculations for decay processes are restricted only with a specific lorentz frame ( namely zero momentum transfer ) . now we see that the covariant light - front model removes the above restrictions , straightforwardly extracts the hadronic form factors @xmath21 , and obtains the result @xmath107 which is consistent with hqs . at first glance , the decay constant @xmath2 in eq . ( [ decc1 ] ) does nt seems to connect with @xmath108 in eq . ( [ fv3 ] ) . however , since the wave function @xmath109 is the function of @xmath110 ( see eq . ( [ cwf ] ) ) for a fully covariant bound state , and @xmath111 , thus @xmath109 is even in @xmath33 . it follows that @xmath112 from eqs . ( [ kzh ] ) and ( [ odd ] ) , @xmath2 and @xmath108 are easily rewritten in the simpler forms : @xmath113 it is known that @xmath114 , and therefore the ratio ( @xmath115 ) in eq . ( [ fvfh ] ) will be equal to unity when the antiquark is at rest . thus , in the non - relativistic case , eq . ( [ fvfh ] ) can be reduced to @xmath116 from eq . ( [ nrqm ] ) , one can extract the hadronic parameter @xmath117 which is consistent with the result of the nrqm @xcite . in the relativistic case , we can evaluate eq . ( [ fvfh ] ) directly to extract @xmath118 . there are serval popular phenomenological wave function that have been employed to describe various hadronic structure in the literature . we choose the gaussian - type wave function @xmath119 @xcite which have been widely used in the study of heavy mesons : @xmath120\bigg\ } \\ ] ] the parameters appearing in this wave function are @xmath4 and @xmath121 . from the nrqm to the lfqm , the value of light quark mass varies from @xmath122 to @xmath123 gev . thus we take several different values of @xmath4 and @xmath15 to evaluate the parameter @xmath118 . these results are listed in table i. we find that , in general , the values of @xmath118 are not only quite smaller than the ones of @xmath124 , but also insensitive to the values of @xmath4 . these results mean that the typical values of @xmath28 in the integrations are much larger than @xmath4 , that is to say , the relativistic effects are very important in these evaluations . 0.2 cm table i. parameters @xmath4 and @xmath121 fitted to the decay constant @xmath15 . also shown are the results of hadronic parameter @xmath118 and @xmath124 . [ cols="^,^,^,^,^,^,^ " , ] this hadronic parameter @xmath118 can be used to calculate the branching ratio of the @xmath12 meson radiative decay . the decay rate for @xmath5 differential in the photon energy is given by @xmath125y^3(1-y ) , \label{br}\ ] ] where @xmath126 . in the heavy quark limit , there is only one contribution @xmath127 to @xmath21 , thus @xmath128 in eqs . ( [ fv ] ) and ( [ fa ] ) can be obtained by @xmath129 integrating the photon energy in eq . ( [ br ] ) , we can obtain the decay rate @xmath130 taking @xmath131 gev , @xmath132 @xcite , and @xmath133 sec @xcite , we obtain @xmath134 . this result agrees well with that in refs . @xcite and @xcite where the light cone qcd sum rules and the lfqm were used in their calculations , respectively . however , this result is about a factor of 2 smaller and larger than that in refs . @xcite and @xcite , respectively . the parameter @xmath118 also relates to a ratio @xmath135 . the pure leptonic decay rate is given by @xmath136 taking the relevant values , we obtain the ratio @xmath137 which is within the range of @xmath138 as expected in ref . @xcite . in this work we have calculated the decay constants @xmath139 for heavy mesons and the form factors @xmath140 for the radiative leptonic decays @xmath141 within the covariant light - front approach . in accordance with hqs and the results of ref . @xcite , @xmath107 and @xmath142 to the tree level , respectively . in addition , the form factor @xmath143 can be related to the decay constant @xmath144 by considering an @xmath145 intermediate state contribution . the relevant hadronic parameter @xmath118 , in contrast to @xmath117 in the nrqm , has been determined by the parameters @xmath4 and @xmath121 in this covariant model . the comparisons between @xmath118 and @xmath124 was listed quantitatively in table i. we conclude that the relativistic effects are quite obvious in these modes . we have also obtained @xmath146 and @xmath147 , in agreement with the general estimates in the literature . the covariant model removes some restrictions often occurred in the usual lfqm calculation , and , as we have shown in this paper , allows one to perform some extremely simple evaluation of various heavy meson properties . further applications to other properties of heavy mesons and physical processes will be presented in subsequent papers . i am grateful to chuang - hung chen for valuable discussions . i also wish to thank the national center for theoretical sciences ( south ) for its hospitality during my summer visit where this work started . this work was supported in part by the national science council of r.o.c . under grant nsc93 - 2112-m-017 - 001 .

we study the radiative leptonic decays of heavy mesons within the covariant light - front model . using this model , both the form factors @xmath0 and @xmath1 have the same form when the heavy quark limit is taken . in addition , the relation between the form factor @xmath0 and the decay constant of heavy meson @xmath2 is obtained . the hadronic parameter @xmath3 can be determine by the parameters appearing in the wave function of heavy meson . we find that the value of @xmath3 is not only quite smaller than the one in the non - relativistic case , but also insensitive to the value of light quark mass @xmath4 . these results mean that the relativistic effects are very important in this work . we also obtain the branching ratio of @xmath5 is about @xmath6 , in agreement with the general estimates in the literature . 16 true pt 10 mm

hep-ph0512006

the hall effect has been continuously playing an important role in experimental condensed - matter research , mostly because the interpretation of hall measurements is rather simple in classical fermi systems @xcite . in such materials the hall coefficient is a remarkably robust property , which is unaffected by interactions and only depends upon the shape of the fermi surface and the sign of the charge carriers . deviations from this simple behavior are generally taken as evidence for the onset of strong correlations and a failure of the fermi - liquid ( fl ) paradigm @xcite . several authors have investigated the hall effect in threeand two - dimensional fl @xcite , but the question of the role of correlations in the hall effect for low - dimensional systems remains largely unexplored . in most three - dimensional systems the interactions play a secondary role and the fl picture is appropriate @xcite . however , the prominence of interactions increases as the dimensionality of the systems decreases and the fl theory is believed to break down for many two - dimensional systems like , _ e.g. _ , the high-@xmath1 cuprate superconductors @xcite . in one - dimensional ( 1d ) systems interactions are dominant , and the fl description must be replaced by the luttinger liquid ( ll ) theory @xcite . this theory predicts a rich variety of physical phenomena , such as spin - charge separation or non - universal temperature dependence of the transport properties @xcite , many of which have been observed experimentally . therefore large deviations from the classical hall effect are expected to occur in _ quasi_-one dimensional systems . among the various experimental realizations of low - dimensional systems ( organic conductors @xcite , carbon nanotubes @xcite , ultra cold atomic gases @xcite , etc . ) the organic conductors are good realizations of quasi-1d materials . studies of the longitudinal transport have successfully revealed signatures of ll properties @xcite . transport transverse to the chains has given access to the dimensional crossover between a pure 1d behavior and a more conventional high - dimensional one @xcite . to probe further the consequences of correlations in these compounds , several groups have undertaken the challenging measurement of the hall coefficient @xmath2 @xcite . the results , different depending on the direction of the applied magnetic field , proved difficult to interpret due to a lack of theoretical understanding of this problem . this prompted for a detailed theoretical analysis of the hall effect in quasi-1d systems . a first move in this direction was reported in ref . where the hall coefficient of dissipationless weakly - coupled 1d interacting chains was computed and found to be @xmath3-independent and equal to the band value . this surprising result shows that in this case @xmath4 , unlike other transport properties , is insensitive to interactions . however the assumption of dissipationless chains is clearly too crude to be compared with realistic systems for which a finite resistivity is induced by the umklapp interactions @xcite . in this work we examine the effect of umklapp scattering on the @xmath3-dependence of the hall coefficient in quasi-1d conductors . we consider @xmath5-filled 1d chains and compute @xmath2 to leading order in the umklapp scattering using the memory function approach @xcite . we find that the umklapp processes induce a @xmath3-dependent correction to the free - fermions value , and this correction decreases with increasing temperature as a power - law with an exponent depending on interactions ( fig . [ fig : graph ] ) . we discuss the implications for quasi-1d compounds . schematics of the model . the chains and the current @xmath6 go along the @xmath7-axis , the magnetic field @xmath8 is applied along the @xmath9-axis , and the hall voltage is measured along the @xmath10-axis . , width=325 ] our model is sketched in fig . [ fig : model ] . we consider 1d chains coupled by a hopping amplitude @xmath11 supposedly small compared to the in - chain kinetic energy . the usual ll model of the 1d chains assumes that the electrons have a linear dispersion with a velocity @xmath12 . for a strictly linear band , however , the hall coefficient vanishes identically owing to particle - hole symmetry . a band curvature close to the fermi momenta @xmath13 is thus necessary to get a finite @xmath4 . we therefore take for the 1d chains of fig . [ fig : model ] the dispersion @xmath14 the upper ( lower ) sign corresponds to right ( left ) moving electrons . ( [ dispersion ] ) can be regarded as the minimal model which gives rise to a hall effect , while retaining most of the formal simplicity of the original ll theory , and its wide domain of validity . in particular , this model is clearly sufficient at low temperatures ( compared to the electron bandwidth ) since then only electrons close to the fermi points contribute to the conductivities . our purpose is to treat the umklapp term perturbatively . we express the hamiltonian as @xmath15 where @xmath16 is the umklapp scattering term and @xmath17 reads @xmath18 . \end{gathered}\ ] ] in eq . ( [ hamiltonian ] ) @xmath19 is the chain index , @xmath20 is a pauli matrix , and @xmath21 . we choose the landau gauge @xmath22 , such that @xmath23 with @xmath24 the interchain spacing . @xmath25 is a two - component vector composed of right- and left - moving electrons . the second term in eq . ( [ hamiltonian ] ) is the band curvature , the third term is the forward scattering and the last term corresponds to the coupling between the chains . in eq . ( [ hamiltonian ] ) we have omitted the backscattering terms ( @xmath26 processes ) which are , for spin rotationally invariant systems , marginally irrelevant @xcite . we therefore take @xmath27 . at @xmath5 filling the umklapp term reads @xmath28 it corresponds to a process where two electrons with opposite spins change direction by absorbing a momentum @xmath29 from the lattice . the hall resistivity @xmath30 relates to the conductivity tensor @xmath31 through @xmath32 here we calculate @xmath30 using a memory function approach @xcite . one rewrites the conductivity tensor in terms of a memory matrix @xmath33 as @xmath34\bm{\chi}^{-1}(0 ) \right\}^{-1}\bm{\chi}(0)\ ] ] where @xmath35 denotes the transpose of @xmath36 . the advantage provided by the memory function is the possibility to make finite - order perturbation expansions which are singular for the conductivities due to their resonance structure @xcite . this formalism is especially useful for ll ( ref . ) and was also used to estimate the hall coefficient in the 2d hubbard model @xcite . @xmath37 is a diagonal matrix composed of the diamagnetic susceptibilities in each direction , @xmath38 , with @xmath39 the thermodynamic average @xmath40 is taken with respect to @xmath17 and @xmath41 is the hamiltonian of eq . ( [ hamiltonian ] ) in the presence of electric and magnetic fields , @xmath42 . the frequency matrix @xmath43 in eq . ( [ sigma_m ] ) is defined in terms of the equal - time current - current correlators as @xcite @xmath44\right\rangle.\ ] ] from eq . ( [ sigma_m ] ) one can directly express the memory matrix @xmath45 in terms of the conductivity tensor . in the following we will only need the off - diagonal term @xmath46 given by @xmath47 it is then straightforward to rewrite the hall coefficient @xmath48 as @xmath49 from eqs ( [ hamiltonian ] ) and ( [ chi0 ] ) we obtain the longitudinal and transverse diamagnetic terms as @xmath50 for evaluating the frequency matrix we write down the current operators : [ currents]@xmath51 \psi_{j\sigma}^{\phantom{\dagger}}(x)\\ & & \\ \nonumber j_y&=&-iet_{\perp}a_y\int dx\sum_{j\sigma } \left(\psi_{j\sigma}^{\dagger}\psi_{j+1,\sigma}^{\phantom{\dagger } } e^{-iea_{j , j+1}}-\text{h.c.}\right)\\ \end{aligned}\ ] ] the expression resulting from eq . ( [ omega ] ) for the frequency matrix is then @xmath52 at this stage we can already evaluate the high - frequency limit of @xmath4 , because the memory matrix vanishes as @xmath53 ( refs and ) and thus @xmath46 drops from eq . ( [ rh ] ) if @xmath54 . the effects of the umklapp disappear at high frequency , and in this limit one recovers from eqs ( [ rh][omegaxy ] ) the result obtained for dissipationless chains @xcite , namely that the hall coefficient equals the band value @xmath55 : @xmath56 in the definition of the memory matrix , eq . ( [ m ] ) , we can ignore the terms of order @xmath57 which do not contribute to @xmath4 in eq . ( [ rh ] ) . furthermore we express the conductivities in terms of current susceptibilities as @xmath58 $ ] , which leads to @xmath59 \left[\chi_y(0)-\chi_{yy}(\omega)\right]}-\omega_{xy}.\ ] ] we rewrite this formula at intermediate frequencies , such that @xmath60 is small . in this expansion we use the equation of motion of the susceptibilitites @xcite , as well as the relation @xmath61=-\omega_{\nu\mu}j_{\nu}. $ ] for @xmath62 the latter equation is exactly satisfied in our model , while for @xmath63 it is only verified in the continuum limit @xmath64 . the resulting expression of the memory matrix is @xmath65 where @xmath66 are the _ residual forces _ operators defined as the part of the hamiltonian which in the absence of magnetic field does not commute with the currents , _ i.e. _ @xmath67 $ ] , and @xmath68 stands for the retarded correlation function of the operators @xmath66 . the terms omitted in eq . ( [ mofk ] ) are either of second order in @xmath60 , of second order in @xmath8 , or vanish in the continuum limit @xmath64 . using eqs ( [ hu ] ) and ( [ currents ] ) we find [ k]@xmath69.\qquad \end{aligned}\ ] ] note that each of the @xmath70 s is of first order in @xmath71 , hence @xmath46 is of order @xmath72 . the quantity @xmath73 entering eq . ( [ mofk ] ) is the real - frequency , long - wavelength limit of the correlator , which we evaluate as @xmath74 it is easy to prove that @xmath73 vanishes for @xmath75 or @xmath76 , by applying spatial inversion and particle - hole symmetry , respectively . retaining only leading - order terms in @xmath11 and @xmath77 , the first nonvanishing contribution in eq . ( [ kk ] ) is of order @xmath78 , and involves three spatial and three time integrations , which we were not able to perform analytically . based on a scaling analysis , we can nevertheless extract the temperature ( or frequency ) dependence of this contribution ( see appendix [ app_scaling ] ) , which yields : @xmath79 where @xmath0 is the ll parameter in the charge sector . in the absence of interactions we have @xmath80 , while @xmath81 ( @xmath82 ) for repulsive ( attractive ) interactions . if the interactions are strong and repulsive ( @xmath83 ) the exponent in eq . ( [ kk - t ] ) changes due to the contraction @xcite of the operators in @xmath84 and @xmath85 , which gives the relevant power - law in this case . correction of the high - temperature / high - frequency hall coefficient @xmath4 by the umklapp scattering in weakly - coupled luttinger liquids . @xmath55 is the value of the hall coefficient in the absence of umklapp scattering , eq . ( [ rh0 ] ) , and @xmath86 is the electron bandwidth . our approach breaks down below some crossover scale ( dashed line , see text ) . in this figure we have assumed that @xmath87 in eq . ( [ rh_result ] ) is negative.,width=283 ] together with eqs ( [ mofk ] ) and ( [ rh ] ) , eq . ( [ kk - t ] ) leads to our final expression for the hall coefficient : @xmath88\ ] ] with @xmath86 the electron bandwidth . ( [ rh_result ] ) shows that in @xmath5-filled quasi-1d systems the umklapp scattering changes the absolute value of the hall coefficient with respect to the band value , which is only recovered at high temperature or frequency . note that eq . ( [ rh_result ] ) also describes the frequency dependence of @xmath4 provided @xmath3 is replaced by @xmath89 . the backscattering term @xmath26 ( neglected here ) could possibly give rise to multiplicative logarithmic corrections to the power law in eq . ( [ rh_result ] ) @xcite . the sign of the dimensionless prefactor @xmath87 can only be determined through a complete evaluation of @xmath73 in eq . ( [ kk ] ) , and is for the time being unknown . the available experimental data are consistent with eq . ( [ rh_result ] ) if one assumes that @xmath87 is negative ( see below ) , as illustrated in fig . [ fig : graph ] . ( [ rh_result ] ) would imply that in the non - interacting limit @xmath90 ( @xmath91 ) the correction to the hall coefficient behaves as @xmath92 . in order to check this prediction we have evaluated the correlator in eq . ( [ kk ] ) for @xmath93 . the corresponding diagram sketched in fig . [ fig : diagram ] involves three frequency - momentum integrations , which in this case can be fully worked out analytically ( see appendix [ app_free ] ) . the resulting expression of @xmath4 at zero frequency is ( @xmath94 ) @xmath95,\ ] ] consistently with eq . ( [ rh_result ] ) . for non - interacting electrons , though , we see that the relative correction induced by the @xmath5-filling umklapp is positive at @xmath96 . since all properties are analytic in @xmath0 , we can also deduce from eqs ( [ rh_result ] ) and ( [ no - int ] ) that @xmath87 tends to @xmath97^{-1}$ ] in the limit @xmath90 . note that eq . ( [ no - int ] ) would also apply to models in which @xmath98 , such as the hubbard model , while eq . ( [ rh_result ] ) is valid only when @xmath99 . example of a diagram appearing in eq . ( [ kk ] ) at first order in @xmath11 and for @xmath93 . the full ( dashed ) lines correspond to free right ( left ) moving fermions , @xmath19 is the chain index , and the arrows represent up and down spins . the magnetic field increases the momentum of the electron by @xmath100 . , width=283 ] the result of eq . ( [ rh_result ] ) shows that in @xmath5-filled quasi 1d systems the umklapp processes induce a correction to the free - fermion value ( band value ) of the hall coefficient @xmath4 , which depends on temperature as a power - law with an exponent depending on interactions . at high temperatures or frequencies , @xmath4 approaches the band value as shown in fig . [ fig : graph ] , implying that any fitting of experimental data must be done with repect to the value of @xmath4 at high temperature or frequency . to study the range of validity of our result , one must consider that at low temperature the quasi-1d systems generally enter either an insulating state characterized by a mott gap @xmath101 , or a coherent two- or three - dimensional phase below a temperature @xmath102 controlled by @xmath11;@xcite in either case our model of weakly - coupled ll is no longer valid , as illustrated in fig . [ fig : graph ] . the variations of @xmath4 below @xmath103 can be very pronounced , and depend strongly on the details of the materials . when the ground state is insulating , for instance , @xmath2 is expected to go through a minimum and diverge like @xmath104 as @xmath105 , reflecting the exponentially small carrier density . other behaviors , such as a change of sign due to the formation of an ordered state or nesting in the fl regime @xcite , can also occur . the validity of eq . ( [ rh_result ] ) is therefore limited to the ll domain : @xmath106 . for the case @xmath107 , we estimate the change of @xmath4 with respect to @xmath55 at the crossover scale @xmath101 , for a system with @xmath108 , where @xmath109 is the coulomb repulsion . the umklapp - induced mott gap in @xmath5-filled systems is given @xcite by @xmath110^x$ ] with @xmath111^{-1}$ ] . we thus find that the largest correction is @xmath112^{\frac{1}{2}}$ ] and has a universal exponent . on the other hand , @xmath4 approaches the asymptotic value @xmath55 quite slowly , and according to eq . ( [ rh_result ] ) a correction of @xmath113 ^ 2 $ ] still exists at temperatures comparable to the bandwidth . the available hall data in the tm family and in the geometry of the present analysis @xcite show a weak correction to the free fermion value which depends on temperature . some attempts to fit this behavior to a power law have been reported @xcite . however the analysis was performed by fitting @xmath2 to a power law starting at zero temperature . as explained above , the proper way to analyze the hall effect in such quasi-1d systems is to fit the _ deviations _ from the band value starting from the high temperature limit . it would be interesting to check whether a new analysis of the data would provide good agreement with our results . however in these compounds both @xmath114-filling and @xmath5-filling umklapp processes are present . for the longitudinal transport , the @xmath114-filling contribution dominates @xcite . for the hall effect , the analysis in the presence of @xmath114-filling umklapp is considerably more involved , but a crude evaluation of the scaling properties of the corresponding memory matrix gives also a weak power - law correction with an exponent @xmath115 , and thus similar effects , regardless of the dominant umklapp . the observed data is thus consistent with the expected corrections coming from ll behavior . however more work , both experimental and theoretical , is needed for the tm family because of this additional complication , and to understand the data in a different geometry where no temperature dependence is observed @xcite . our result eq . ( [ rh_result ] ) is however directly relevant for @xmath5-filled organic conductors such as ( ttm - ttp)i@xmath116 and ( dmtsa)bf@xmath117 @xcite . hall measurements for these compounds still remain to be performed . comparison of the hall effect in these compounds with the one in @xmath114-filled non - dimerized systems @xcite for which only @xmath114-filling umklapp is present , could also help in understanding the dominant processes for the tm family . this work was supported by the swiss national science foundation through division ii and manep . in order to establish eq . ( [ kk - t ] ) we evaluate the correlator @xmath68 to first order in @xmath77 and @xmath11 . let s denote by @xmath118 the curvature [ second term in eq . ( [ hamiltonian ] ) ] , by @xmath119 the inter - chain hopping [ fourth term in eq . ( [ hamiltonian ] ) ] , and by @xmath120 the remaining part of the hamiltonian , @xmath121 . standard perturbation theory yields @xmath122 where the average has to be taken with respect to @xmath120 . the latter corresponds to a 1d chain and can be easily bosonized,@xcite @xmath123 ^ 2+\frac{u_{\nu}}{k_{\nu } } \left[\nabla\phi_{\nu}(x)\right]^2\right\}\qquad \end{aligned}\ ] ] where @xmath124 denotes the charge ( spin ) degrees of freedom , @xmath125 is a velocity , @xmath126 a dimensionless parameter , and @xmath127 and @xmath128 are bosonic fields . in our case we have @xmath81 and @xmath129 . the fields @xmath127 and @xmath128 and the fermionic fields @xmath130 are related by @xmath131\right\}}\ ] ] with @xmath132 corresponding to right ( left ) moving fermions , and @xmath133 a cutoff of the order of the in - chain lattice spacing . with the help of eq . ( [ fermionic ] ) we bosonize each operator in eq . ( [ kkhcbc ] ) : @xmath134\right\}_j}\\ \nonumber & & e^{\frac{i}{\sqrt{2}}\left\{b\phi_{\rho}(x)+\epsilon_{jj'}\theta_{\rho}(x ) + \sigma\left[b\phi_{\sigma}(x)+\epsilon_{jj'}\theta_{\sigma}(x)\right]\right\}_{j'}}\\ & & + \text{h.c.}\big ) \end{aligned}\ ] ] where @xmath19 and @xmath135 are neighboring chains and @xmath136 . for the hopping term we have @xmath137\right\}_j}\\ & & e^{-\frac{i}{\sqrt{2}}\left\{b\phi_{\rho}(x)-\theta_{\rho}(x ) + \sigma\left[b\phi_{\sigma}(x)-\theta_{\sigma}(x)\right]\right\}_{j+1}}\\ \nonumber & & + \text{h.c.}\big ) \end{aligned}\ ] ] and for the band curvature term we take @xcite @xmath138 next we use the identity @xcite @xmath139}\rangle= \exp\big\{-\frac{1}{2}{\sum_{n < m}}'\\ -(a_na_mk+b_nb_mk^{-1})f_1(r_n - r_m)\\ + ( a_nb_m+b_na_m)f_2(r_n - r_m)\big\ } \end{gathered}\ ] ] where @xmath140 , the notation @xmath141 means that the sum is restricted to those terms for which @xmath142 , and @xmath143 are universal functions . the resulting expression for the correlator in eq . ( [ kkhcbc ] ) is @xmath144 the factor @xmath145 results from the linearization in @xmath8 , and we have discarded all factors involving the @xmath146 function , since they correspond to angular integrals of the @xmath147 variables and therefore do not contribute to the scaling dimension . at distances much larger than the cutoff @xmath133 we have @xmath148 , and therefore we find the high temperature , high frequency behavior as @xmath149 we follow the same procedure for the diamagnetic term @xmath150however at zeroth order in @xmath77 and @xmath8and find @xmath151 combining these expressions and collecting the relevant prefactors we deduce eq . ( [ kk - t ] ) . here we provide the derivation of eq . ( [ no - int ] ) , which gives @xmath4 to leading order in @xmath71 but in the absence of forward scattering ( @xmath93 ) . using eqs ( [ rh ] ) and ( [ rh0 ] ) we can express the zero - frequency hall coefficient in terms of @xmath55 and @xmath152 . we then perform a kramers - kronig transform , insert the free - fermion values of the diamagnetic susceptibilities , @xmath153 and @xmath154 , and use eq . ( [ mofk ] ) to arrive at @xmath155 \end{gathered}\ ] ] where @xmath156 is to be evaluated to first order in @xmath8 . the @xmath157 term in eq . ( [ mofk ] ) disappears due to the principal part in the @xmath89 integral in eq . ( [ free_1 ] ) . from eq . ( [ k ] ) one sees that @xmath156 involves 8 fermion fields and can be represented by diagrams like the one displayed in fig . [ fig : diagram ] . there are 32 different diagrams , but all of them can be expressed in terms of only one function @xmath158 , whose expression is given by the diagram in fig . [ fig : diagram ] . we thus obtain @xmath159\right\ } \end{gathered}\ ] ] where @xmath160 and we have pulled all prefactors from eq . ( [ k ] ) , as well as a factor @xmath11 from the diagram , out of the definition of function @xmath87 . the explicit expression of @xmath161 is the frequency summations in eq . ( [ eq : aprime ] ) are elementary , and the various momentum integrals can also be evaluated analytically to first order in @xmath77 , yielding @xmath163 . \end{gathered}\ ] ] the remaining energy integral is divergent and must be regularized . cutting the integral at the bandwidth @xmath86 and assuming @xmath94 we obtain the asymptotic behavior given in eq . ( [ no - int ] ) .

we investigate the hall effect in a quasi one - dimensional system made of weakly coupled luttinger liquids at half filling . using a memory function approach , we compute the hall coefficient as a function of temperature and frequency in the presence of umklapp scattering . we find a power - law correction to the free - fermion value ( band value ) , with an exponent depending on the luttinger parameter @xmath0 . at sufficiently high temperature or frequency the hall coefficient approaches the band value .

cond-mat0608427

nonleptonic @xmath7-meson decays are of crucial importance to deepen our insights into the flavor structure of the standard model ( sm ) , the origin of cp violation , and the dynamics of hadronic decays , as well as to search for any signals of new physics beyond the sm . however , due to the non - perturbative strong interactions involved in these decays , the task is hampered by the computation of matrix elements between the initial and the final hadron states . in order to deal with these complicated matrix elements reliably , several novel methods based on the naive factorization approach ( fa ) @xcite , such as the qcd factorization approach ( qcdf ) @xcite , the perturbation qcd method ( pqcd ) @xcite , and the soft - collinear effective theory ( scet ) @xcite , have been developed in the past few years . these methods have been used widely to analyze the hadronic @xmath7-meson decays , while they have very different understandings for the mechanism of those decays , especially for the case of heavy - light final states , such as the @xmath0 decays . presently , all these methods can give good predictions for the color allowed @xmath8 mode , but for the color suppressed @xmath9 mode , the qcdf and the scet methods could not work well , and the pqcd approach seems leading to a reasonable result in comparison with the experimental data . in this situation , it is interesting to study various approaches and find out a reliable approach . as the mesons are regarded as quark and anti - quark bound states , the nonleptonic two body meson decays concern three quark - antiquark pairs . it is then natural to investigate the nonleptonic two body meson decays within the qcd framework by considering all feynman diagrams which lead to three effective currents of two quarks . in our considerations , beyond these sophisticated pqcd , qcdf and scet , we shall try to find out another simple reliable qcd approach to understand the nonleptonic two body decays . in this note , we are focusing on evaluating the @xmath0 decays . the paper is organized as follows . in sect . ii , we first analyze the relevant feynman diagrams and then outline the necessary ingredients for evaluating the branching ratios and @xmath3 asymmetries of @xmath10 decays . in sect . iii , we list amplitudes of @xmath0 decays . the approaches for dealing with the physical - region singularities of gluon and quark propagators are given in sect . finally , we discuss the branching ratios and the @xmath3 asymmetries for those decay modes and give conclusions in sects . v and vi , respectively . the detail calculations of amplitudes for these decay modes are given in the appendix . we start from the four - quark effective operators in the effective weak hamiltonian , and then calculate all the feynman diagrams which lead to effective six - quark interactions . the effective hamiltonian for @xmath12 decays can be expressed as @xmath13+{\rm h.c.},\ ] ] where @xmath14 and @xmath15 are the wilson coefficients which have been evaluated at next - to - leading order @xcite , @xmath16 and @xmath17 are the tree operators arising from the @xmath18-boson exchanges with @xmath19 where @xmath20 and @xmath21 are the @xmath22 color indices . based on the effective hamiltonian in eq . ( [ heff ] ) , we can then calculate the decay amplitudes for @xmath23 , @xmath24 , and @xmath25 decays , which are the color - allowed , the color - suppressed , and the color - allowed plus color - suppressed modes , respectively . all the six - quark feynman diagrams that contribute to @xmath26 and @xmath27 decays are shown in figs . [ tree]-[annihilation ] via one gluon exchange . as for the process @xmath28 , it does nt involve the annihilation diagrams and the related feynman diagrams are the sum of figs . [ tree ] and [ tree2 ] . based on the isospin symmetry argument , the decay amplitude of this mode can be written as @xmath29 . the explicit expressions for the amplitudes of these decay modes are given in detail in next section . the decay amplitudes of @xmath11 decay modes are quite different . for the color - allowed @xmath8 mode , it is expected that the decay amplitude is dominated by the factorizable contribution @xmath30 ( from the diagrams ( a ) and ( b ) in fig . [ tree ] ) , while the nonfactorizable contribution @xmath31 ( from the diagrams ( c ) and ( d ) in fig . [ tree ] ) has only a marginal impact . this is due to the fact that the former is proportional to the large coefficient @xmath32 , while the latter is proportional to the quite small coefficient @xmath33 . in addition , there is an addition color - suppressed factor @xmath34 in the nonfactorizable contribution @xmath31 . in contrast with the @xmath8 mode , the nonfactorizable contribution @xmath31 ( from ( c ) and ( d ) diagrams in fig . [ tree2 ] ) in the @xmath9 mode is proportional to the large coefficient @xmath32 , and even if with an additional color - suppressed factor @xmath34 , its contribution is still larger than the factorizable one @xmath30 ( from ( a ) and ( b ) diagrams in fig . [ tree2 ] ) which is proportional to the quite small coefficient @xmath33 . thus , it is predicted that the decay amplitude of this mode is dominated by the nonfactorizable contribution @xmath31 . as for the @xmath35 mode , since its amplitude can be written as the sum of the ones of the above two modes , it is not easy to see which one should dominate the total amplitude . the branching ratio for @xmath0 decays can be expressed as follows in terms of the total decay amplitudes @xmath36 where @xmath37 is the lifetime of the @xmath7 meson , and @xmath38 is the magnitude of the momentum of the final - state particles @xmath39 and @xmath40 in the @xmath7-meson rest frame and given by @xmath41\ , \left[m_b^2-(m_{d}-m_{\pi})^2\,\right]}\,.\end{aligned}\ ] ] as is well - known , the direct @xmath3 violation in meson decays is non - zero only if there are two contributing amplitudes with non - zero relative weak and strong phases . the weak - phase difference usually arises from the interference between two different topological diagrams . for three @xmath0 decays , it is seen from the feynman diagrams in figs . [ tree]-[annihilation ] that there are no weak - phase differences , and hence no direct @xmath3 violation in all these three modes , we shall then consider the mixing - induced @xmath3 violation . as the final states @xmath42 can be produced both in the decays of @xmath43 meson via the cabibbo - favored ( @xmath44 ) and in the decays of @xmath45 meson via the doubly cabibbo - suppressed ( @xmath46 ) tree amplitudes . the relative weak - phase difference between these two amplitudes is @xmath47 and , when combining with the @xmath48 mixing phase , the total weak - phase difference is @xmath49 to all orders in the small ckm parameter @xmath50 . thus , the @xmath51 decays can in principle be used to measure the weak phase @xmath52 , since the weak phase @xmath53 has been measured with high precision . the time - dependent @xmath3 asymmetry of such decay modes is defined as : @xmath54 where @xmath55 is the mass difference of the two eigenstates of @xmath56 mesons , and @xmath57 and @xmath58 are given as @xmath59 with @xmath60 where the rephase - invariant quantities @xmath61 , @xmath62 and @xmath63 @xcite characterize the indirect , direct and mixing - induced cp violations respectively . as @xmath64 for neutral @xmath7 system , we have @xmath65 which characterizes direct cp violation . defining @xmath66 and @xmath67 as the amplitudes of @xmath68 and @xmath69 decay modes , respectively , we can further express these two cp asymmetries as @xmath70 where @xmath71 , and @xmath72 represents the relative strong - phase difference between the two amplitudes @xmath66 and @xmath67 . similarly , we can define another two @xmath3-violating parameters @xmath73 and @xmath74 for the @xmath75 decays @xmath76 with the parameter @xmath77 defined as @xmath78 are the charge conjugations of the amplitudes @xmath67 and @xmath66 . since the magnitude of the cabibbo - suppressed decay amplitude @xmath79 is much smaller than that of the cabibbo - favored decay amplitude @xmath80 , the ratio @xmath81 should be quite small and is found to be about @xmath82 in our framework . thus , to a very good approximation , @xmath83 , and the coefficients of the sine terms are given by @xmath84 to compare with the current experimental data , one usually define the following two quantities , which are given by the combination of two @xmath3-violating parameters @xmath57 and @xmath74 , @xmath85 which can provide constraints on the weak phase @xmath6 and the strong phase @xmath86 . using the methods given in the appendix , we can get the @xmath0 decay amplitudes , which are composed of three parts : the factorizable contribution @xmath30 , the nonfactorizable contribution @xmath31 , and the annihilation contribution @xmath87 . the amplitude of @xmath8 mode is found to be @xmath88 with @xmath89 where @xmath90 , @xmath91 and @xmath92 . @xmath93 are the wave functions of mesons . for the @xmath7-meson wave function , we shall take the form given in @xcite @xmath94,\ ] ] with @xmath95 , and @xmath96 being a normalization constant . the @xmath39 meson distribution amplitude is given by @xmath97,\ ] ] with the shape parameter @xmath98 . for the @xmath40 meson light cone wave functions , we use the asymptotic form as given in refs . @xcite : @xmath99 with @xmath100 . @xmath101 where @xmath102 and @xmath103 . @xmath104\nonumber\\ & + & ( c_1+\frac{c_2}{n_c})\phi_b(x)\biggl[\biggl(\big((\bar x - y)m_b+m_b \big)m^2_b\phi(z)+ \mu_\pi m_d \big((2x-\bar y - z)m_b \nonumber\\&-&4m_b \big)\phi_\pi(z ) + \mu_\pi m_dm_b(y-\bar z)\frac{\phi_\sigma'(z)}{6}\biggl ) \frac{m_b}{d_{ba } k_a^2}\nonumber\\ & + & \biggl((x-\bar z)m^2_b\phi(z)-\mu_\pi m_d\big((y-\bar z)\frac{\phi_\sigma'(z)}{6}+ ( 2x - y-\bar z)\phi_\pi(z)\big)\biggl)\frac{m^2_b}{d_{da } k_a^2}\biggl]\biggl\},\label{anni1}\end{aligned}\ ] ] where @xmath105 , @xmath106 , @xmath107 , @xmath108 , @xmath109 and @xmath110 . the annihilation contribution is found to be much smaller than the ones from the factorizable and the nonfactorizable diagrams . numerically , it is negligible . for the color - suppressed @xmath111 decay , its amplitude can be written as @xmath112 with @xmath113 , \label{ckms1}\end{aligned}\ ] ] here @xmath114 , @xmath115 and @xmath116 . @xmath117\,,\label{ckms2}\end{aligned}\ ] ] where @xmath118 and @xmath103 . for the annihilation amplitude @xmath87 , it is the same as the one in eq . ( [ anni1 ] ) since the two modes @xmath119 and @xmath42 have the same annihilation topological diagrams . for the doubly cabibbo - suppressed decay mode @xmath120 , its decay amplitude can be written as @xmath121 here , @xmath122 and @xmath87 can be obtained from the ones of decay mode @xmath123 by simply exchanging the wilson coefficients @xmath14 and @xmath15 . for the @xmath124 decay , its amplitude can be yielded by using the isospin relation @xmath125 . to perform a numerical calculation of the decay amplitudes of @xmath126 decays , the light - cone projectors of mesons are found to be very useful , and the details of these quantities are presented in the appendix . where one encounters the endpoint divergences stemming from the convolution integrals of the meson distribution amplitudes with the hard kernels , which is caused by the collinear approximation . to regulate such an infrared divergence , we may introduce an intrinsic mass scale realized in the symmetry - preserving loop regularization@xcite . at the tree level , it is equivalent to adopt an effective dynamical gluon mass in the propagator . practically , such a gluon mass scale has been used to regulate the infrared divergences in the soft endpoint region @xcite @xmath127^{-\frac{12}{11}},\ ] ] the use of this effective gluon propagator is supported by the lattice @xcite and the field theoretical studies @xcite , which have shown that the gluon propagator is not divergent as fast as @xmath128 . taking the hadronic scale @xmath129 , the dynamical gluon mass scale can be determined from one of the well measured decay mode . numerically , we will see that taking @xmath130 mev , the dynamical gluon mass scale is around @xmath131 . another physical - region singularity arises from the on mass - shell quark propagators . it can be easily checked that each feynman diagram contributing to a given matrix element is entirely real unless some denominators vanish with a physical - region singularity , so that the @xmath132 prescription for treating the poles becomes relevant . in other words , a feynman diagram will yield an imaginary part for the decay amplitudes only when the virtual particles in the diagram become on mass - shell , thus the diagram may be considered as a genuine physical process . the cutkosky rules @xcite give a compact expression for the discontinuity across the cut arising from a physical - region singularity . when applying the cutkosky rules to deal with a physical - region singularity of quark propagators , the following formula holds @xmath133-i\pi\delta[(k_1-k_2-k_3)^2],\label{quarkd}\\ \frac{1}{(p_b - k_2-k_3)^2-m_b^2+i\epsilon}&=&p\biggl[\frac{1 } { ( p_b - k_2-k_3)^2-m_b^2}\biggl]-i\pi\delta[(p_b - k_2-k_3)^2-m_b^2 ] , \label{quarkb}\end{aligned}\ ] ] where @xmath134 denotes the principle - value prescription . the role of the @xmath86 function is to put the particles corresponding to the intermediate state on their positive energy mass - shell , so that in the physical region , the individual feynman diagram satisfies the unitarity condition . equations ( [ quarkd ] ) and ( [ quarkb ] ) will be applied to the quark propagators @xmath135 and @xmath136 in equation ( [ anni1 ] ) , respectively . it is then seen that the possible large imaginary parts arise from the virtual quarks across their mass shells as physical - region singularities . it is seen that for theoretical predictions it depends on many input parameters , such as the wilson coefficient functions , the ckm matrix elements , the hadronic parameters , and so on . to carry out a numerical calculation , we take the following input parameters @xcite @xmath137 the wolfenstein parameters of the ckm matrix elements are taken as @xcite : @xmath138 , with @xmath139 . the coefficient of the twist-3 distribution amplitude of the pseudoscalar @xmath40 meson is chosen as @xmath140 @xcite . with the above values for the input parameters , we are able to calculate the contributions of different amplitudes for each decay mode . our final results at @xmath141 scale are presented in table [ amplitude ] . 0.8pt 0.15 in .[amplitude ] numerical results at @xmath141 scale of the amplitudes for different diagrams in @xmath11 decays . amplitudes @xmath122 , and @xmath87 represent the factorizable ( ( a)and ( b ) diagrams in figs . [ tree ] or [ tree2 ] ) , the non - factorizable ( ( c ) and ( d ) diagrams in figs . [ tree ] or [ tree2 ] ) , and the annihilation ( diagrams in fig . [ annihilation ] ) contributions , respectively . [ cols="<,^,^,^,^",options="header " , ] in summary , we have calculated the decay amplitudes , strong phases , branching ratios , and @xmath3 asymmetries for the @xmath0 decays , including both the color - allowed and the color - suppressed modes . it has been shown that these decay modes are theoretically clean as there are no penguin contributions . as a consequence , direct @xmath3 violations are absent . the contributions from the factorizable diagrams dominate all the decay amplitudes except for the @xmath142 process . all our predictions for branching ratios are consistent with the existing measurements . for the @xmath143 mode , our predictions will be faced with the future experiments as no data are available at present . due to small interference effects between the cabibbo - suppressed and the cabibbo - favored amplitudes , the non - zero @xmath3-violating parameters @xmath57 and @xmath74 have been predicted in the @xmath144 decay modes . it has been shown that the @xmath3-violating parameters have a strong dependence on the weak phase @xmath6 , but they are not sensitive to the dynamical gluon mass scale . with the angle @xmath6 varying within the range @xmath145 , almost all of the values for the cp - violating parameters @xmath146 and @xmath147 are within the range of the current experimental data . thus no constraints on the weak phase @xmath6 could be obtained through those parameters based on the current experiment data , and more precise measurements are needed in future experiments . in this paper , we have further shown that the divergence treatments used in our previous work @xcite are reliable . namely , the endpoint divergence caused by the soft collinear approximation in gluon propagator could be simply avoided by adopting the cornwall prescription for the gluon propagator with a dynamical mass scale . note that when the intrinsic mass is appropriately introduced , it may not spoil the gauge symmetry as shown recently in the symmetry - preserving loop regularization @xcite . meanwhile , for the physical - region singularity of the on - mass - shell quark propagators , it can well be treated by applying for the cutkosky rules . the combination of these two treatments for the endpoint divergences of gluon propagator and the physical - region singularity of the quark propagators enables us to obtain reasonable results , which are consistent with the existing experimental data and also in agreement with the ones @xcite obtained by using the pqcd approach . however , this is different from the treatment of the latter , where @xmath148 and sudakov factors have been used to avoid the endpoint divergence . it is noted that the resulting predictions for the branching ratios are in general scale dependence on the dynamical gluon mass which plays the role of the ir cut - off . this dependence should in principle be compensated from the possible scale in the wave functions which characterizes the nonperturbative effects . in our approach , the dynamical gluon mass may be regarded as a universal scale to be fixed from one of the decay modes . for instance , in our present considerations , if the decay mode @xmath149 is taken to extract the dynamical gluon mass scale , we have @xmath150 mev , and the resulting predictions for other decay modes can serve as a consistent check . within the current experimental errors and theoretical uncertainties for some relevant parameters , it is seen that our treatment is reliable . in order to further check the validity of the gluon - mass regulator method adopted to deal with the endpoint divergence , it is useful to extend this method to more decay modes . anyway , the treatments presented in this paper may enhance its predictive power for analyzing non - leptonic @xmath7-meson decays . this work was supported in part by the national science foundation of china ( nsfc ) under the grant 10475105 , 10491306 , 10675039 and the project of knowledge innovation program ( pkip ) of chinese academy of sciences . to evaluate the hadronic matrix elements of @xmath0 decays , the meson light - cone distribution amplitudes play an important role . in the heavy quark limit , the light - cone projectors for @xmath7 , @xmath39 and @xmath40 mesons in momentum space can be expressed , respectively , as @xcite @xmath151_{\alpha\beta } , \nonumber\\ { \cal m}_{\alpha\beta}^d & = & \frac{i f_d\,}{4}\ , [ ( { \not\ ! p_2 \,}+ m_d ) \,\gamma_5 \,\phi_d(y)]_{\alpha\beta } , \nonumber\\ { \cal m}_{\delta\alpha}^\pi & = & \frac{i f_p}{4}\,\biggl\{{\not\ ! p_3 \,}\gamma_5 \,\phi(u)-\mu_p\gamma_5 \biggl(\phi_p(u)-i\sigma_{\mu \nu}n_-^\mu v^\nu \frac{\phi^{\prime } _ \sigma ( u)}{6 } + i \sigma _ { \mu \nu } p_3^{\mu } \frac{\phi_\sigma(u)}{6 } \frac{\partial}{\partial k_{\bot\nu}}\biggl)\biggl\}_{\delta\alpha},\label{projector } \end{aligned}\ ] ] from the feynman diagrams shown in figs . [ tree]-[annihilation ] , we can get the amplitudes for each decay mode using the relevant feynman rules and the light - cone projectors listed in eqs . ( [ projector ] ) . for the tree diagrams of @xmath8 mode shown in fig . [ tree ] , the amplitudes of each diagrams can be written as @xmath152\frac{-ig_{\alpha\beta}\delta_{ab}}{k^2},\nonumber\\ & = & -i f_\pi g^2_s\frac{c_f}{n_c}\frac{1}{d_b k^2}\mathrm{tr}\big[{\cal m}^b\gamma^\alpha{\cal m}^d{\not\ ! p_3 \ , } ( 1-\gamma_5)({\not\ ! k_b \,}+m_b)\gamma_\alpha\big]\nonumber\\ a^{\ref{tree}b}&=&if_\pi p_3^\mu\mathrm{tr}\big[{\cal m}^b(-ig_s\gamma^\alpha t^a_{ij}){\cal m}^d(-ig_s\gamma^\beta t^b_{kl})\frac{i}{{\not\ ! k_c \,}-m_c}\gamma_\mu ( 1-\gamma_5)\big]\frac{-ig_{\alpha\beta}\delta_{ab}}{k^2},\nonumber\\ & = & -if_\pi g^2_s\frac{c_f}{n_c}\frac{1}{d_c k^2}\mathrm{tr}\big[{\cal m}^b\gamma^\alpha{\cal m}^d\gamma_\alpha({\not\ ! k_c \,}+m_c){\not\ ! p_3 \ , } ( 1-\gamma_5)\big]\nonumber\\ a^{\ref{tree}c}&=&\mathrm{tr}\big[{\cal m}^\pi(-ig_s\gamma^\alpha t^a_{ij})\frac{i}{{\not\ ! k_d \,}}\gamma^\mu ( 1-\gamma_5)\big]\mathrm{tr}\big[{\cal m}^b(-ig_s\gamma^\beta t^b_{kl}){\cal m}^d\gamma_\mu ( 1-\gamma_5)\big]\frac{-ig_{\alpha\beta}\delta_{ab}}{k^2},\nonumber\\ & = & -g^2_s\frac{c_f}{n_c}\frac{1}{d_d k^2}\mathrm{tr}\big[{\cal m}^\pi\gamma^\alpha{\not\ ! k_d \,}\gamma^\mu ( 1-\gamma_5)\big]\mathrm{tr}\big[{\cal m}^b\gamma_\alpha{\cal m}^d\gamma_\mu ( 1-\gamma_5)\big]\nonumber\\ a^{\ref{tree}d}&=&\mathrm{tr}\big[{\cal m}^\pi\gamma^\mu ( 1-\gamma_5)\frac{i}{{\not\ ! k_u \,}}(-ig_s\gamma^\alpha t^a_{ij})\big]\mathrm{tr}\big[{\cal m}^b(-ig_s\gamma^\beta t^b_{kl}){\cal m}^d\gamma_\mu ( 1-\gamma_5)\big]\frac{-ig_{\alpha\beta}\delta_{ab}}{k^2}\nonumber\\ & = & -g^2_s\frac{c_f}{n_c}\frac{1}{d_u k^2}\mathrm{tr}\big[{\cal m}^\pi\gamma^\mu ( 1-\gamma_5){\not\ ! k_u \,}\gamma^\alpha \big]\mathrm{tr}\big[{\cal m}^b\gamma_\alpha{\cal m}^d\gamma_\mu ( 1-\gamma_5)\big],\label{amplitude1}\end{aligned}\ ] ] where @xmath153 stands for the @xmath21th(@xmath154 ) diagrams in fig.[tree ] , @xmath155 and @xmath156 the momentum of @xmath157 quark propagator and gluon propagator , respectively . furthermore , @xmath158 and @xmath159 represent for the @xmath157 quark propagator and gluon propagator , respectively . in fig . [ tree](a ) , the @xmath40 meson can be written as a decay constant since it originates from the vacuum . inversing the fermi lines and writing down the @xmath7 meson projector @xmath160 , gluon vertex @xmath161 , @xmath39 meson projector @xmath162 , the four quark vertex @xmath163 , b quark propagator @xmath164 and another gluon vertex @xmath165 in a trace one by one , and finally the gluon propagator @xmath166 , we can get the amplitude @xmath167 . @xmath168 can be calculated in a similar way . in fig . [ tree](c ) , the @xmath40 meson can no longer be written as a decay constant any more since it exchanges a gluon with the spectator quark . writing down the @xmath40 meson projector @xmath169 , gluon vertex @xmath170 , @xmath171 quark propagator @xmath172 and the four quark vertex @xmath173 in turn in one trace , and writing down the @xmath7 meson projector @xmath160 , gluon vertex @xmath165 , @xmath39 meson projector @xmath174 and the four quark vertex @xmath163 in the other trace one by one , and finally the gluon propagator @xmath166 , we can get the amplitude @xmath175 . similarly , we can get the amplitude @xmath176 . summing up the former and the latter two quantities in eq . ( [ amplitude1 ] ) , we can get the factorizable part @xmath30 ( eq . ( [ favor2 ] ) ) and the nonfactorizable @xmath31 ( eq . ( [ favor3 ] ) ) , respectively . as for the annihilation diagrams for @xmath177 in fig . [ annihilation ] , the amplitudes can be written as @xmath178\frac{-ig_{\alpha\beta}\delta_{ab } } { k^2_a},\nonumber\\ & = & -i f_b g^2_s\frac{c_f}{n_c}\frac{1}{d_{ca } k^2_a}\mathrm{tr}\big[{\cal m}^d\gamma^\alpha ( { \not\ ! k_{ca } \,}+m_c){\not\ ! p_1 \,}(1-\gamma_5){\cal m}^\pi\gamma_\alpha\big]\nonumber\\ a^{\ref{annihilation}b}&=&if_b p_1^\mu \mathrm{tr}\big[{\cal m}^d\gamma_\mu ( 1-\gamma_5)\frac{i}{{\not\ ! k_{ua } \,}}(-ig_s\gamma^\alpha t^a_{ij}){\cal m}^\pi(-ig_s\gamma^\beta t^b_{kl})\big]\frac{-ig_{\alpha\beta}\delta_{ab}}{k^2_a},\nonumber\\ & = & -i f_b g^2_s\frac{c_f}{n_c}\frac{1}{d_{ua } k^2_a}\mathrm{tr}\big[{\cal m}^d{\not\ ! p_1 \ , } ( 1-\gamma_5){\not\ ! k_{ua } \,}\gamma^\alpha{\cal m}^\pi\gamma_\alpha\big],\nonumber\\ a^{\ref{annihilation}c}&=&\mathrm{tr}\big[{\cal m}^b(-ig_s\gamma^\alpha t^a_{ij})\frac{i}{{\not\ ! k_{da } \,}}\gamma^\mu ( 1-\gamma_5)\big]\mathrm{tr}\big[{\cal m}^d\gamma_\mu ( 1-\gamma_5){\cal m}^\pi(-ig_s\gamma^\beta t^b_{kl})\big]\frac{-ig_{\alpha\beta}\delta_{ab}}{k^2_a}\nonumber\\ & = & - g^2_s\frac{c_f}{n_c}\frac{1}{d_{da } k^2_a}\mathrm{tr}\big[{\cal m}^b\gamma^\alpha { \not\ ! k_{da } \,}\gamma^\mu ( 1-\gamma_5)\big]\mathrm{tr}\big[{\cal m}^d\gamma_\mu ( 1-\gamma_5){\cal m}^\pi\gamma_\alpha \big],\nonumber\\ a^{\ref{annihilation}d}&=&\mathrm{tr}\big[{\cal m}^b\gamma^\mu ( 1-\gamma_5)\frac{i}{{\not\ ! k_{ba}-m_b \,}}(-ig_s\gamma^\alpha t^a_{ij})\big]\mathrm{tr}\big[{\cal m}^d\gamma_\mu ( 1-\gamma_5){\cal m}^\pi(-ig_s\gamma^\beta t^b_{kl})\big]\frac{-ig_{\alpha\beta}\delta_{ab } } { k^2_a},\nonumber\\ & = & - g^2_s\frac{c_f}{n_c}\frac{1}{d_{ba } k^2_a}\mathrm{tr}\big[{\cal m}^b\gamma^\mu ( 1-\gamma_5)({\not\ ! k_{ba } \,}+m_b)\gamma^\alpha \big]\mathrm{tr}\big[{\cal m}^d\gamma_\mu ( 1-\gamma_5){\cal m}^\pi\gamma_\alpha \big],\label{amplitude1a}\end{aligned}\ ] ] where @xmath179 and @xmath180 stand for the momentum of @xmath157 quark propagator and gluon propagator , and @xmath181 and @xmath159 represent for the @xmath157 quark propagator and gluon propagator in these annihilation diagrams , respectively . summing up the four quantities in eq . ( [ amplitude1a ] ) , we can get the annihilation contribution @xmath87 ( eq . ( [ anni1 ] ) ) of this decay mode . similarly , as for the tree diagrams of @xmath111 decay mode in fig [ tree2 ] , its amplitudes can be written as follows @xmath182\frac{-ig_{\alpha\beta}\delta_{ab}}{k^2},\nonumber\\ & = & -if_d g^2_s\frac{c_f}{n_c}\frac{1}{d_{b } k^2}\mathrm{tr}\big[{\cal m}^b\gamma^\alpha{\cal m}^\pi{\not\ ! p_2 \ , } ( 1-\gamma_5)({\not\ ! k_b \,}+m_b)\gamma_\alpha\big],\nonumber\\ a^{\ref{tree2}b}&=&if_d p_2^\mu \mathrm{tr}\big[{\cal m}^b(-ig_s\gamma^\alpha t^a_{ij}){\cal m}^\pi(-ig_s\gamma^\beta t^b_{kl})\frac{i}{{\not\ ! k_d \,}}\gamma_\mu ( 1-\gamma_5)\big]\frac{-ig_{\alpha\beta}\delta_{ab}}{k^2},\nonumber\\ & = & -if_d g^2_s\frac{c_f}{n_c}\frac{1}{d_{d } k^2}\mathrm{tr}\big[{\cal m}^b\gamma^\alpha{\cal m}^\pi\gamma_\alpha { \not\ ! k_d \,}{\not\ ! p_2 \,}(1-\gamma_5)\big],\nonumber\\ a^{\ref{tree2}c}&=&\mathrm{tr}\big[{\cal m}^d(-ig_s\gamma^\alpha t^a_{ij})\frac{i}{({\not\ ! k_c \,}-m_c)}\gamma^\mu ( 1-\gamma_5)\big]\mathrm{tr}\big[{\cal m}^b(-ig_s\gamma^\beta t^b_{kl}){\cal m}^\pi\gamma_\mu ( 1-\gamma_5)\big ] \frac{-ig_{\alpha\beta}\delta_{ab}}{k^2},\nonumber\\ & = & -ig^2_s\frac{c_f}{n_c}\frac{1}{d_{c } k^2}\mathrm{tr}\big[{\cal m}^d\gamma^\alpha({\not\ ! k_c \,}+m_c)\gamma^\mu ( 1-\gamma_5)\big]\mathrm{tr}\big[{\cal m}^b\gamma_\alpha{\cal m}^\pi\gamma_\mu ( 1-\gamma_5)\big],\nonumber\\ a^{\ref{tree2}d}&=&\mathrm{tr}\big[{\cal m}^d\gamma^\mu ( 1-\gamma_5)\frac{i}{{\not\ ! k_u \,}}(-ig_s\gamma^\alpha t^a_{ij})\big]\mathrm{tr}\big[{\cal m}^b(-ig_s\gamma^\beta t^b_{kl}){\cal m}^\pi\gamma_\mu ( 1-\gamma_5)\big]\frac{-ig_{\alpha\beta}\delta_{ab}}{k^2}\nonumber\\ & = & -ig^2_s\frac{c_f}{n_c}\frac{1}{d_{u } k^2}\mathrm{tr}\big[{\cal m}^d\gamma^\mu ( 1-\gamma_5){\not\ ! k_u \,}\gamma^\alpha \big]\mathrm{tr}\big[{\cal m}^b\gamma_\alpha{\cal m}^\pi\gamma_\mu ( 1-\gamma_5)\big].\label{amplitude2}\end{aligned}\ ] ] we can get the factorizable contribution @xmath30 ( eq . ( [ ckms1 ] ) ) and the nonfactorizable part @xmath31 ( eq . ( [ ckms2 ] ) ) by summing up the formerand the latter two quantities in eq . ( [ amplitude2 ] ) . as for the annihilation diagrams for @xmath9 decay , its amplitude is the same as the one in eq . ( [ amplitude1a ] ) since the two modes @xmath119 and @xmath42 have the same annihilation topological diagrams , which are shown in fig . [ annihilation ] . 99 m. wirbel , b. stech , and m. bauer , z. phys . c * 29 * , 637 ( 1985 ) ; m. bauer , b. stech , and m. wirbel , z. phys . c * 34 * , 103 ( 1987 ) . m. beneke , g. buchalla , m. neubert , and c. t. sachrajda , phys.rev . * 83 * , 1914 ( 1999 ) ; nucl . phys . * b591 * , 313 ( 2000 ) ; * 606 * , 245 ( 2001 ) . h. n. li and h. l. yu , phys . rev . lett . * 74 * , 4388 ( 1995 ) ; phys . b * 353 * , 301 ( 1995 ) ; y. y. keum , h. n. li , and a. i. sanda , phys . d * 63 * , 054008 ( 2001 ) . c. w. bauer , s. fleming , d. pirjol and i. w. stewart , phys . d * 63 * , 114020 ( 2001 ) ; c. w. bauer , d. pirjol and i. w. stewart , phys . lett . * 87 * , 201806 ( 2001 ) ; phys . d * 65 * , 054022 ( 2002 ) ; * 65 * , 054022 ( 2002 ) . g. buchalla , a. j. buras , and m. e. lautenbacher , rev . , 1125 ( 1996 ) . palmer and y.l . wu , phys . * b350 * 245 ( 1995 ) . h. y. cheng and k. c. yang , phys . b * 511 * , 40 ( 2001 ) ; h. y. cheng and k. c. yang , phys . d * 64 * , 074004 ( 2001 ) . m. beneke and t. feldmann , nucl . * b592 * , 3 ( 2001 ) . p. ball and v. m. braun , nucl . * b543 * , 201 ( 1999 ) . s. descotes - genon and c. t. sachrajda , nucl . b625 * , 239 ( 2002 ) . y. l. wu , int . j. mod . a18 , 5363 ( 2003 ) ; y. l. wu , mod.phys . a19 , 2191 ( 2004 ) . y. l. ma and y. l. wu , phys . b647 , 427 ( 2007 ) ; y. l. ma , y. l.wu , int . j. mod a21 , 6383 ( 2006 ) . j. m. cornwall , phys . d * 26 * , 1453 ( 1982 ) ; j. m. cornwall and j. papavasiliou , phys . d * 40 * , 3474 ( 1989 ) ; * 44 * , 1285 ( 1991 ) . s. b. shalom , g. eilam , and y. d. yang , phys . d * 67 * , 014007 ( 2003 ) ; y. d. yang , f. su , g. r. lu and h. j. hao , eur . j. c * 44 * , 243 ( 2005 ) . f. su , y. l. wu , y. d. yang and c. zhuang , eur . j. c * 48 * , 401 ( 2006 ) . a. g. williams _ et al . _ , hep - ph/0107029 and references therein . r. alkofer and l. von smekal , phys . rept . * 353 * , 281 ( 2001 ) ; l. von smekal , r. alkofer , and a. hauck , phys . lett . * 79 * , 3591 ( 1997 ) ; d. zwanziger , phys . rev . d * 69 * , 016002 ( 2004 ) ; d. m. howe and c. j. maxwell , phys . lett . b * 541 * , 129 ( 2002 ) ; phys . d * 70 * , 014002 ( 2004 ) ; a. c. aguilar , a. a. natale , and p. s. rodrigues da silva , phys . 90 * , 152001 ( 2003 ) ; s. furui and h. nakajima , aip conf . proc . * 717 * , 685 ( 2004 ) . cutkosky , j. math . phys . 1 , 429 ( 1960 ) . w. m. yao _ et al . _ [ particle data group ] , j. phys . g * 33 * ( 2006 ) 1 . j. charles _ et al . _ [ ckmfitter group ] , eur . j. c * 41 * , 1 ( 2005 ) , updated results and plots available at : http://ckmfitter.in2p3.fr . m. beneke and m. neubert , nucl . * b675 * , 333 ( 2003 ) . e. von toerne [ cleo collaboration ] , hep - ex/0301016 . b. aubert [ babar collaboration ] , hep - ex/0610027 . y. y. keum , t. kurimoto , h. n. li , c. d. lu and a. i. sanda , phys.rev . d * 69 * , 094018 ( 2004 ) . b. aubert _ et al . _ [ babar collaboration ] , hep - ex/0207092 . s. blyth _ et al . _ [ belle collaboration ] , phys . d * 74 * , 092002 ( 2006 ) . t. kurimoto , h. n. li and a. i. sanda , phys . d * 67 * , 054028 ( 2003 ) . m. neubert and b. stech , adv . high energy phys . , 294 ( 1998 ) . e. barberio _ et al . _ [ heavy flavor averaging group ( hfag ) ] , hep - ex/0603003 , and online update at http://www.slac.stanford.edu/xorg/hfag .

the branching ratios and cp violations of the @xmath0 decays , including both the color - allowed and the color - suppressed modes , are investigated in detail within qcd framework by considering all diagrams which lead to three effective currents of two quarks . an intrinsic mass scale as a dynamical gluon mass is introduced to treat the infrared divergence caused by the soft collinear approximation in the endpoint regions , and the cutkosky rule is adopted to deal with a physical - region singularity of the on mass - shell quark propagators . when the dynamical gluon mass @xmath1 is regarded as a universal scale , it is extracted to be around @xmath2 mev from one of the well - measured @xmath0 decay modes . the resulting predictions for all branching ratios are in agreement with the current experimental measurements . as these decays have no penguin contributions , there are no direct @xmath3 asymmetries . due to interference between the cabibbo - suppressed and the cabibbo - favored amplitudes , mixing - induced @xmath3 violations are predicted in the @xmath4 decays to be consistent with the experimental data at 1-@xmath5 level . more precise measurements will be helpful to extract weak angle @xmath6 .

0705.1575

whenever the energies of two discrete quantum states cross when plotted against some parameter , e.g. time , the transition probability is traditionally estimated by the famous landau - zener ( lz ) formula @xcite . although the lz model involves the simplest nontrivial time dependence linearly changing energies and a constant interaction of infinite duration , when applied to real physical systems with more sophisticated time dependences the lz model often provides more accurate results than expected . this feature ( which has not been fully understood yet ) , and the extreme simplicity of the lz transition probability , have determined the vast popularity of the lz model , despite the availability of more sophisticated exactly soluble level - crossing models , e.g. the demkov - kunike model @xcite and its special case , the allen - eberly - hioe model @xcite . numerous extensions of the lz model to multiple levels have been proposed . the exactly soluble multistate lz models belong to two main types : single - crossing bow - tie models and multiple - crossings grid models . in the _ bow - tie models _ , where all energies cross at the same instant of time , analytic solutions have been found for three @xcite and @xmath1 states @xcite , and when one of the levels is split into two parallel levels @xcite . in the _ grid models _ , a set of @xmath2 parallel equidistant linear energies cross another set of @xmath3 such energies ( demkov - ostrovsky model ) @xcite . for @xmath4 ( or @xmath5 ) the demkov - ostrovsky model reduces to the demkov - osherov model @xcite . the cases of one @xcite or two @xcite degenerate levels have also been solved . in the most general case of linear energies of arbitrary slopes , the general solution is not known , but exact results for some survival probabilities have been derived @xcite . a variety of physical systems provide examples of multiple level crossings . among them we mention ladder climbing of atomic and molecular states by chirped laser pulses @xcite , harpoon model for reactive scattering @xcite , and optical shielding in cold atomic collisions @xcite . examples of bow - tie linkages occur , for instance , in a rf - pulse controlled bose - einstein condensate output coupler @xcite and in the coupling pattern of rydberg sublevels in a magnetic field @xcite . a degenerate lz model emerges when the transition between two atomic levels of angular momenta @xmath6 and @xmath7 or @xmath8 is driven by linearly chirped laser fields of arbitrary polarizations @xcite . a general feature of all soluble nondegenerate multilevel crossing models is that each transition probability @xmath9 between states @xmath10 and @xmath11 is given by a very simple expression , as in the original lz model , although the derivations are not trivial . in the grid models , in particular , the exact probabilities @xmath9 have the same form ( products of lz probabilities for transition or no - transition applied at the relevant crossings ) as what would be obtained by naive multiplication of lz probabilities while moving across the grid of crossings from @xmath10 to @xmath11 , without accounting for phases and interferences . quite surprisingly , interferences between different paths to the same final state , a multitude of which exist in the grid models , are not visible in the final probabilities . in this paper we develop an analytic description of a three - state model wherein the three energies change linearly in time , with distinct slopes , thus creating three separate level crossings . this system is particularly convenient for it presents the opportunity to investigate quantum interference through different evolution paths to the same final state , and in the same time , it is sufficiently simple to allow for an ( approximate ) analytic treatment ; for the latter we use sequential two - state lz and adiabatic - following propagators . this system is also of practical significance for it occurs in various physical situations , for instance , in transitions between magnetic sublevels of a @xmath0 level @xcite , in chirped - pulse ladder climbing of alkali atoms @xcite , in rotational ladder climbing in molecules @xcite , and in entanglement of a pair of spin-1/2 particles @xcite . the results provide analytic estimates of all nine transition probabilities in this system . we do establish quantum interferences and estimate the amplitude and the frequency of the ensuing oscillation fringes , as well as the conditions for their appearance . the analytic results also allow us to prescribe explicit recipes for quantum state engineering , for example , to create an equal , maximally coherent superposition of the three states . this paper is organized as follows . in sec . [ definition of the problem ] we provide the basic equations and definitions and define the problem . in sec . [ evolution matrix ] we derive the propagator , the transition probabilities and the validity conditions . in sec . [ numerical computation vs analytical approximation ] we compare our analytical approximation to numerical simulations . then in sec . [ applications of analytics ] we demonstrate various applications of the analytics . in sec . [ comparison with the exactly soluble carroll - hioe model for ] we compare our model with the exactly soluble carroll - hioe bowtie model in the limit of vanishing static detuning . finally , we discuss the conclusions in sec . [ sec - conclusions ] . we consider a three - state system driven coherently by a pulsed external field , with the rotating - wave approximation ( rwa ) hamiltonian ( in units @xmath12 ) @xmath13.\ ] ] the diagonal elements are the ( diabatic ) energies ( in units @xmath14 ) of the three states , the second of which is taken as the zero reference point without loss of generality . @xmath15 is a static detuning , and @xmath16 are the linearly changing terms . to be specific , we shall use the language of laser - atom interactions , where the difference between each pair of diagonal elements is the detuning for the respective transition : the offset of the laser carrier frequency from the bohr transition frequency . the pulse - shaped functions @xmath17 and @xmath18 are the rabi frequencies , which quantify the field - induced interactions between each pair of adjacent states , @xmath19 and @xmath20 , respectively . each of the rabi frequencies is proportional to the respective transition dipole moment and the laser electric - field envelope . as evident from the zeroes in the corners of the hamiltonian we assume that the direct transition @xmath21 is forbidden , as it occurs in free atoms when @xmath19 and @xmath20 are electric - dipole transitions . the probability amplitudes of our system @xmath22^t$ ] satisfy the schrdinger equation @xmath23 where the overdot denotes a time derivative . without loss of generality , the couplings @xmath17 and @xmath18 are assumed real and positive and , for the sake of simplicity , with the same time dependence . for the time being the detuning @xmath15 and the slope @xmath24 are assumed to be also positive , @xmath25 we shall consider the cases of negative @xmath15 and @xmath24 later on . with the assumptions above , the crossing between the diabatic energies of states @xmath26 and @xmath27 occurs at time @xmath28 , where @xmath29 , between @xmath27 and @xmath30 at time @xmath31 , and the one between @xmath26 and @xmath30 at time @xmath32 . [ fig1 ] plots diabatic and adiabatic energies vs time for a gaussian - shaped laser pulse . we use @xmath33 and @xmath34 to denote diabatic and adiabatic states , respectively . the objective of this paper is to find analytical expressions for the evolution matrix and for the transition probabilities between different diabatic states . the hamiltonian appears naturally in a number of specific problems of interest in time - dependent quantum dynamics of simple systems . the first example is ladder climbing of electronic energy states in some alkali atoms , for instance , in rubidium @xcite . a linearly chirped laser pulse couples simultaneously both transitions 5s-5p and 5p-6s . if the carrier frequency of the pulse is tuned on two - photon resonance with the 5s-6s transition , then the intermediate state 5p remains off resonance , by a detuning @xmath35 , which leads to the trianglelinkage pattern in fig . the couplings @xmath17 and @xmath18 are the rabi frequencies of the two transitions , which may be different ( because of the different transition dipole moments ) but have the same time dependence since they are induced by the same laser pulse . a second example is found in rf transitions between the three magnetic sublevels @xmath36 of a level with an angular momentum @xmath0 in an atom trapped in a magnetooptical trap . the rf pulse provides the pulsed coupling between the @xmath37 and @xmath38 sublevels , and also between the @xmath38 and @xmath39 sublevels . the trapping magnetic field causes zeeman shifts in the magnetic sublevels @xmath37 and 1 in different directions but it does not affect the @xmath38 level @xcite . this linkage pattern is an example of a bowtie level crossing @xcite . if a quadratic zeeman shift is taken into account , then the sublevels @xmath37 and 1 will be shifted in the same direction , which will break the symmetry of the bowtie linkage and will create the trianglepattern of fig . [ 1 ] . a third example is found in quantum rotors , for instance , in rotational ladder climbing in molecules by using a pair of chirped ultrashort laser pulses @xcite . the energy slope is due to the laser chirp , and the static detuning @xmath15 arises due to the rotational energy splitting . if the laser pulse duration is chosen appropriately then only three rotational states will be coupled , with their energies forming the `` triangle '' pattern of fig . the fourth example is the entanglement between two spin-1/2 particles interacting with two crossed magnetic fields , a linear field along one axis and a pulsed field along another axis @xcite . the role of the static detuning @xmath15 is played by the spin - spin coupling constant . three of the four collective states form a chain , which has exactly the triangle linkage pattern of fig . [ 1 ] . in this system , states @xmath26 and @xmath30 correspond to the product states @xmath40 and @xmath41 , whereas state @xmath27 is the entangled state @xmath42 . an exact solution of the schrdinger equation ( [ 2 ] ) for the hamiltonian is not known . we shall derive an approximation , which is most conveniently obtained in the adiabatic basis . the adiabatic states are defined as the eigenvectors @xmath43 ( @xmath44 ) of the instantaneous hamiltonian @xmath45 . the corresponding adiabatic amplitudes @xmath46^t$ ] and the diabatic ones @xmath47 are related as @xmath48 where @xmath49 is an orthogonal ( because @xmath45 is real ) transformation matrix , @xmath50 , whose columns are the adiabatic states @xmath34 ( @xmath44 ) , with @xmath51 having the lowest energy and @xmath52 the highest energy . as we are only interested in the populations at infinite times , we need only @xmath53 , rather than the explicit function @xmath49 . @xmath53 can be easily obtained using the asymptotic behavior of @xmath45 at infinite times , @xmath54,\quad \r(+\infty)=\left [ \begin{array}{ccc } 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \\ \end{array } \right].\ ] ] the schrdinger equation in adiabatic basis reads @xmath55 with @xmath56 , or @xmath57,\ ] ] where the nonadiabatic coupling between the adiabatic states @xmath43 and @xmath58 is @xmath59 our approach is based on two simplifying assumptions . first , we assume that appreciable transitions take place only between neighboring adiabatic states , @xmath60 and @xmath61 , but not between states @xmath62 and @xmath63 , because the energies of the latter pair are split by the largest gap ( cf . [ fig1 ] ) . second , we assume that the nonadiabatic transitions occur instantly at the corresponding avoided crossings and the evolution is adiabatic elsewhere . this allows us to obtain the evolution matrix in the adiabatic basis by multiplying seven evolution matrices describing either lz nonadiabatic transitions or adiabatic evolution . the adiabatic evolution matrix @xmath64 is most conveniently determined in the adiabatic interaction representation , where the diagonal elements of @xmath65 are nullified . the transformation to this basis reads @xmath66 where [ 8 ] & ( t , t_0)= , & + [ 9a ] & _ k(t , t_0)=_t_0^t _ k(t^)d t^ , & and @xmath67 is an arbitrary fixed time . the schrdinger equation in this basis reads @xmath68 with @xmath69,\ ] ] where @xmath70 . in this basis the propagator for adiabatic evolution is the identity matrix . the lz transitions at the crossings at times @xmath71 are described by the transition matrices @xcite [ 12a ] & _ lz(-)= , & + [ 12c ] & _ lz(0)= , & + [ 12b ] & _ lz()= , & where @xmath72 ( @xmath73 ) is the lz probability of nonadiabatic transition and @xmath74 is the no - transition probability at the crossings at times @xmath71 , @xmath75 here [ 14 ] & & a_-=_12(-)/(2a)^1/2 , + & & a_0=_eff(0)/2a^1/2 , + & & a_+=_23()/(2a)^1/2 , + & & _ = ( 1-i a_^2)++a_^2(a_^2 - 1 ) , where @xmath76 is the effective coupling between states @xmath26 and @xmath30 at crossing time @xmath77 ; it is determined by the splitting between the adiabatic curves @xmath78 and @xmath79 , @xmath80 the propagator in the adiabatic basis reads [ adiabatic propagator ] ^a(,-)=(,)_lz()(,0)_lz(0 ) + ( 0,-)_lz(-)(-,- ) . below we present the diabatic propagator in an explicit form . for simplicity , we assume equal couplings @xmath81 although our approach is valid in the general case . this constraint is not applicable for the ladder climbing system , considered in sec . [ implementation ] , where the couplings are naturally different due to the different transition dipole moments , but is still valid for the other systems discussed . then @xmath82 , @xmath83 , @xmath84 , @xmath85 , and @xmath86 . we find the propagator in the original diabatic basis by using eqs . , and as @xmath87 , or explicitly , @xmath88,\ ] ] with @xmath89 , @xmath90 , @xmath91 . the transition probability matrix , i.e. the matrix of the absolute squares of the elements of the propagator , reads @xmath92,\ ] ] where @xmath93 the element at the @xmath94-th row and the @xmath95-th column of the matrix is the transition probability @xmath96 , that is the population of state @xmath94 at infinite time , when the system starts in state @xmath95 in the infinite past . the survival probabilities @xmath97 and @xmath98 coincide with the exact expressions conjectured @xcite and derived exactly for constant couplings @xcite earlier . in eq . ( [ probabilities ] ) we recognize interference terms , which arise because of the availability of two alternative propagating paths in the hilbert space . there is also a symmetry with respect to the skew diagonal due to the equal couplings between neighboring states and the equal ( in magnitude ) slopes of the energies of states @xmath26 and @xmath30 . as already stressed , our approach presumes that the nonadiabatic transitions occur in well - separated confined time intervals . this means that the characteristic transition times are shorter than the times between the crossings , or @xmath99 . the transition times for diabatic ( @xmath100 ) and adiabatic ( @xmath101 ) regimes are @xcite [ 23 ] & & t_transition , , + & & t_transition2/a , . this leads to the following conditions for validity : [ conditions ] & & _ 0 , , + & & _ 02 , . we shall demonstrate that the lz - based approximation outperforms its formal conditions of validity and is valid beyond the respective ranges . above we assumed that @xmath104 and @xmath105 . now we consider the cases @xmath102 and @xmath103 . we assume that the couplings are even functions , @xmath106 . _ negative static detuning _ ( @xmath102 ) . the schrdinger equation for the propagator @xmath107 is [ 19a ] i ( _ 0;t , t_i ) = ( _ 0,t ) ( _ 0;t , t_i ) . by changing the signs of @xmath15 , @xmath108 and @xmath109 in eq , we obtain the same equation , but with the @xmath110 replaced by @xmath111 [ see eq . ] . it is easy to see that the change of sign of @xmath110 is equivalent to the transformation @xmath112 where @xmath113 is the diagonal matrix @xmath114 . hence we find [ 19b ] i ^(-_0;-t ,- t_i ) = ( _ 0,t ) ^(-_0;-t ,- t_i ) . because the initial condition at @xmath115 for @xmath107 and @xmath116 at @xmath115 is the same , @xmath117 we conclude that @xmath118 ; hence [ 20b ] ( -_0;,- ) & = & ( _ 0;- , ) + & = & ( _ 0;,- ) ^. therefore @xmath119 _ negative chirp rate _ ( @xmath103 ) . we notice that @xmath120 , i.e. the change of sign of @xmath24 is equivalent to exchanging the indices 1 and 3 . hence the probabilities for @xmath103 are obtained from these for @xmath105 using the relation @xmath121 below we compare our analytical approximation with numerical simulations . we take for definiteness the couplings in eq . ( [ 1 ] ) to be gaussians , @xmath122 . for the transition @xmath123 vs the detuning @xmath15 for @xmath124 . each frame compares the numerical ( dashed red ) and analytical ( solid blue ) results.,width=302 ] figure [ fig2 ] shows the nine transition probabilities vs the static detuning @xmath15 . an excellent agreement is observed between analytics and numerics , which are barely discernible . this agreement indicates that the dynamics is indeed driven by separated level - crossing transitions of lz type . the analytic approximation is clearly valid beyond its formal range of validity , defined by conditions , which suggest @xmath125 for the parameters in this figure . the figure also demonstrates that the detuning can be used as a control parameter for the probabilities in wide ranges . for @xmath104 the five probabilities on the first row and the last column vary smoothly , in agreement with the analytic prediction . the two - photon probability @xmath126 vanishes rapidly with @xmath15 , as expected , at a much faster pace than the other probabilities . the other four probabilities @xmath127 , @xmath128 , @xmath129 and @xmath130 exhibit oscillations , in agreement with the analytic prediction , due to the existence of two alternative paths of different length from the initial to the final state ( see fig . [ fig1 ] ) , with an ensuing interference . it is noteworthy that these oscillations , due to path interference , are not particularly pronounced , which might be a little surprising at first glance . however , a more careful analysis reveals that when a control parameter is varied , such as the static detuning @xmath15 here , it changes not only the relative phase along the two paths ( which causes the oscillations ) , but also the lz probabilities @xmath131 and @xmath132 ( @xmath73 ) . indeed , as @xmath15 increases , we have @xmath133 because the crossings at times @xmath134 move away from the center of the pulses and @xmath135 . these probabilities affect both the average value of @xmath9 and the oscillation amplitude , with @xmath9 tending eventually to either 0 or 1 for large @xmath15 , while the oscillation amplitude ( which is proportional to @xmath136 ) is damped . similar conclusions apply to the case of @xmath102 because of the symmetry property . it is easy to see from here that the survival probabilities @xmath137 ( @xmath138 ) are symmetric vs @xmath15 , as indeed seen in fig . [ fig2 ] . for the transition @xmath123 vs the energy slope @xmath24 for @xmath139 . each frame compares the numerical ( dashed red ) and analytical ( solid blue ) results.,width=302 ] figure [ fig3 ] displays the transition probabilities vs the chirp rate @xmath24 . an excellent agreement is again observed between analytics and numerics . we have verified that the analytic approximation is valid well beyond its formal range of validity conditions , which suggest @xmath140 for this figure ; this is not shown because our intention here is to show the small-@xmath24 range that exhibits interference patterns . as with the static detuning in fig . [ fig2 ] , this figure demonstrates the symmetry with respect to the sign inversion of @xmath24 , derived in eq . : the change @xmath141 is equivalent to the exchange of the indices 1 and 3 . the observed additional symmetry , @xmath142 and @xmath143 , is a consequence from the assumptions of equal rabi frequencies and equal ( in magnitude ) slopes of the energies of states @xmath26 and @xmath30 . the figure also shows that , with the exception of the survival probabilities @xmath137 @xmath144 , all other probabilities are asymmetric vs the chirp rate @xmath24 , unlike the two - state level - crossing case . for @xmath105 , as for @xmath104 in fig . [ fig2 ] , oscillations are observed in the four probabilities in the lower left corner but not for the probabilities in the top row and the right column . on the contrary , for @xmath103 , oscillations are observed only in the four probabilities in the top right corner . as discussed in regard to fig . [ fig2 ] , the observation of these oscillations is in full agreement with their interpretation as resulting from interference between two different evolution paths to the relevant final state . like the static detuning @xmath15 , the energy slope @xmath24 can be used as a control parameter because it affects the probabilities considerably . around the origin ( @xmath145 ) the system is in adiabatic regime , while for large @xmath146 it is in diabatic regime . for instance , when the system is initially in @xmath26 , around the origin ( @xmath145 ) the population flows mostly into state @xmath30 , following the adiabatic state @xmath62 . on the contrary , for large @xmath24 it eventually returns to @xmath26 ( not visible for the chirp range in fig . [ fig3 ] ) . for the transition @xmath123 vs the rabi frequency @xmath147 for @xmath148 . each frame compares the numerical ( dashed red ) and analytical ( solid blue ) results . the vertical dashed lines for @xmath97 , @xmath98 and @xmath126 show the values @xmath149 of the rabi frequency for half population in the relevant states , predicted by our model , eqs . and .,width=302 ] diabatic and adiabatic regimes are easy to identify also in fig . [ fig4 ] , where the nine probabilities are plotted vs the peak rabi frequency @xmath147 , which is another control parameter . consider our system initially prepared in state @xmath26 . for weak couplings the system evolves diabatically and therefore it is most likely to end up in the same state @xmath26 . as the couplings increase , the system switches gradually from diabatic to adiabatic evolution ; for strong couplings the evolution proceeds along the adiabatic state @xmath62 , and we observe nearly complete population transfer to state @xmath30 . returning to the issue of oscillations , such are barely seen in fig . [ fig4 ] . as discussed in relation to fig . [ fig2 ] , a varying control parameter changes , besides the relative phase of the interfering paths , also the probabilities @xmath131 and @xmath132 , which eventually acquire their asymptotic values of 0 or 1 ; in these limits the oscillations vanish . the probabilities depend on the peak rabi frequency @xmath147 much more sensitively than on the static detuning @xmath15 and the energy slope @xmath24 ; consequently , clear oscillations are seen vs @xmath15 and @xmath24 , but not vs @xmath147 , because the dependence of @xmath131 on @xmath147 is strongest ( essentially gaussian ) , and hence the approach to the asymptotic values of the probabilities is fastest . in this section we shall use our analytic approximation for the transition probabilities to derive several useful properties of the triple - crossing system . we begin by deriving approximate expressions for the rabi frequency required to reach @xmath150 population in the @xmath95-th state for the transition @xmath123 . simple expressions are found for the transition @xmath151 , @xmath152 and for the transitions @xmath153 and @xmath154 , [ 25c ] _ 1/2 & = & ^ , where @xmath155 . these values are indicated by vertical lines in fig . [ fig4 ] and are seen to be in excellent agreement with the exact values . , @xmath27 and @xmath30 vs the chirp rate @xmath24 for fixed @xmath156 and @xmath157 , provided the system is initially in state @xmath30 . the three curves cross at about @xmath158 , indicating the creation of a maximally coherent superposition with populations @xmath159 , which is very close to the solution of eqs . and , @xmath160 , shown with a vertical dashed line.,width=245 ] if we prepare our system initially in state @xmath26 and use @xmath103 , or in state @xmath30 and use @xmath105 , it is possible to determine by means of our analytical model values of @xmath15 , @xmath24 and @xmath147 , so that we achieve arbitrary preselected populations at the end . for example , for a maximally coherent superposition state , i.e. @xmath161 , we need @xmath162 and @xmath163 . this yields the following set of equations for @xmath15 , @xmath147 , and @xmath24 : [ coha ] & e^2_0 ^ 2/a^22-_0 - 23/2=0 , & + [ cohb ] & _ 0= e^_0 ^ 2/a^2 . & an example is shown in fig . [ fig5 ] where the three final probabilities @xmath126 , @xmath164 and @xmath98 are plotted versus the chirp rate @xmath24 . the three probabilities cross ( indicating the creation of a maximally coherent superposition state ) approximately at the value predicted by eqs . and , shown by the vertical line . for @xmath165 and constant couplings , the hamiltonian allows for an exact solution this is the carroll - hioe ( ch ) bowtie model @xcite . the transition probability matrix for the ch model reads [ ch ] _ch = , where p_c = e^-a^2/2,a=/. we use this exact result as a reference for the @xmath165 limit of our approximate method , applied for constant coupling @xmath166 . we emphasize that taking this limit is an abuse of the method because in the derivation we have assumed that the crossings are _ separated _ , which has justified the multiplication of propagators . nonetheless , it is curious and instructive to push our approximation to this limit . for @xmath165 the lz parameters are @xmath167 and @xmath168 . therefore we have @xmath169 . as functions of the chirp rate @xmath24 . here @xmath170.,width=302 ] figure [ fig6 ] presents a comparison between the exact carroll - hioe solution and our approximate solution . quite astonishingly , our approximate solution is not only qualitatively correct but it is even in a very good quantitative agreement with the exact solution ; we witness here yet another lz surprise where our lz - based model outperforms expectations in a limit where it should not be adequate . the observed feature of our approximate solution can be explained by examining the asymptotics of the approximate probabilities and the exact ch values for @xmath171 and @xmath172 . for @xmath171 the approximation and the ch solution read , up to @xmath173 , respectively [ small a ] [ small a approx ] & & ~ , + & & _ch ~. [ small a ch ] for @xmath172 they read , up to @xmath174 , respectively [ large a ] [ large a approx ] & & ~ , + & & _ch ~. [ large a ch ] equations and demonstrate that our approximate solution reproduces well , for some probabilities even exactly , the correct small-@xmath175 asymptotics , which corresponds to the large-@xmath24 ranges in fig . the reason is that the small-@xmath175 ( diabatic ) regime corresponds to weak coupling ; in the perturbative regime the presence of level crossings , let alone their distribution in time , is less significant . in the large-@xmath175 ( adiabatic ) regime the crossings become very important and definitive for the dynamics . . deviates from the correct asymptotics , but still has the correct asymptotic values for @xmath176 . the correct , or nearly correct , small-@xmath175 and large-@xmath175 asymptotics of our approximate solution explain its surprising overall accuracy in fig . we have developed an approximate analytical model that describes the time - dependent dynamics of a quantum system with three states , which have linearly changing energies of different slopes and are coupled with pulse - shaped interactions . our approach is based upon the two - state lz model , i.e. we assume independent pairwise transitions between neighboring states , described by the lz model . we have performed detailed comparison of our analytic approximation with numerical simulations , versus all possible interaction parameters and for all nine transition probabilities , which has revealed a remarkable accuracy , not only in smooth features , but also in describing detailed interference features . this accuracy shows that indeed , the physical mechanism of the three - state dynamics is dominated by separated pairwise lz transitions , even when the crossings are too close to each other . we have derived the _ formal _ conditions of validity of our lz approach , eqs . , using the concept of transition time . however , a comparison with numeric simulations has revealed that our approximation is valid well beyond the formal ranges of validity . one of the reasons is that for two of the survival probabilities , @xmath97 and @xmath98 , our lz approximation produces the exact results . we have found that even in the extreme case of vanishing static detuning , where our approach _ should not be valid _ because the three crossings coalesce into a triply degenerate bowtie single crossing , it still produces remarkably accurate results because of nearly correct asymptotic behaviors of the transition probabilities . one of the useful and interesting features of the `` triangle '' linkage pattern ( fig . [ fig1 ] ) is the presence of intrinsic interference effects . our `` sandwich '' approach , with its implementation in the adiabatic interaction representation , allows for an easy incorporation of different evolution paths in hilbert space between a particular pair of states . path interferences _ are identified in only four of the nine probabilities . another source of interferences could be nonadiabatic transitions in the wings of the gaussian pulses , where the nonadiabatic couplings possess local maxima ; these interferences would be visible in all nine probabilities . we have found , however , that only the path interferences are clearly identified . a substantial contribution to the path interferences is played by the lz phases @xmath177 . the lz phase is often neglected in applications of the lz model to multiple crossings , in the so - called `` independent crossing '' approximation , where only probabilities are accounted for . although such an approach occasionally works , miraculously , as in the exactly soluble demkov - osherov @xcite and demkov - ostrovsky @xcite models , the present simple , but very instructive model , demonstrates that in general , the lz phase , as well the dynamical adiabatic phases , has to be properly accounted for , which is achieved best in an evolution - matrix approach , preferably in the adiabatic - interaction representation @xcite . in order to be closer to experimental reality , in the examples we have assumed pulsed interactions , specifically of gaussian time dependence . this proved to be no hindrance for the accuracy of the model , which is remarkable because we have applied the lz model ( which presumes constant couplings ) at crossings ( the first and the last ones ) situated at the wings of the gaussian - shaped couplings where the latter change rapidly . this robustness of the approach can be traced to the use of the adiabatic basis where the pulse - shape details are accounted for in the adiabatic phases . we have used the analytic results to derive some useful features of the dynamics , for instance , we have found explicitly the parameter values for which certain probabilities reach the 50% level , and for which a maximally coherent superposition is created of all three states @xmath159 . in the specific derivations we have assumed for the sake of simplicity equal couplings for the two transitions and slopes of different signs but equal magnitudes for two of the energies . these assumptions simplify considerably the ensuing expressions for the probabilities ; moreover , they are actually present in some important applications ( quantum rotors , zeeman sublevels in magnetic field and spin - spin entanglement ) . the formalism is readily extended to the general case , of unequal couplings and different slopes , and we have verified that the resulting lz - based approximation is very accurate again . to conclude , the present work demonstrates that , once again , the lz model outperforms expectations when applied to multistate dynamics , with multiple level crossings and a multitude of evolution paths . this work has been supported by the eu tok project camel ( grant no . mtkd - ct-2004 - 014427 ) , the eu rtn project emali ( grant no . mrtn - ct-2006 - 035369 ) , and bulgarian national science fund grants no . wu-205/06 and no . wu-2517/07 . ostrovsky and h. nakamura , j. phys . a * 30 * , 6939 ( 1997 ) . harmin , phys . rev . a * 44 * , 433 ( 1991 ) . s. brundobler and v. elser , j. phys . a * 26 * , 1211 ( 1993 ) . y.n . demkov and v.n . ostrovsky , phys . a * 61 * , 032705 ( 2000 ) ; y.n . demkov and v.n . ostrovsky , j. phys . b * 34 * , 2419 ( 2001 ) . y.n . demkov and v.n . ostrovsky , j. phys . b * 28 * , 403 ( 1995 ) . demkov , p.b . kurasov , and v.n . ostrovsky , j. phys . a * 28 * , 4361 ( 1995 ) . t. usuki , phys . b * 56 * , 13360 ( 1997 ) . v.n . ostrovsky and h. nakamura , phys . a * 58 * , 4293 ( 1998 ) . y.n . demkov and v. i. osherov , zh . 53 * , 1589 ( 1967 ) [ sov . jetp * 26 * , 916 ( 1968 ) ] . shytov , phys . a * 70 * , 052708 ( 2004 ) . sinitsyn , j. phys . a : math . gen . * 37 * , 10691 ( 2004 ) . m.v . volkov and v.n . ostrovsky , j. phys . b : at . mol . . phys . * 37 * , 4069 ( 2004 ) . m.v . volkov and v.n . ostrovsky , j. phys . b : at . mol . opt . phys . * 38 * , 907 ( 2005 ) . melinger , s.r . gandhi , a. hariharan , j.x . tull , and w.s . warren , phys . lett . * 68 * , 2000 ( 1992 ) . suominen , k. burnett , p.s . julienne , m. walhout , u. sterr , c. orzel , m. hoogerland , and s.l . rolston , phys . a * 53 * , 1678 ( 1996 ) ; k .- a . suominen , j. phys . b * 29 * , 5981 ( 1996 ) ; j. weiner , v.s . bagnato , s. zilio , and p.s . julienne , rev . 71 * , 1 ( 1999 ) ; b. broers , h. b. van linden van den heuvell , and l. d. noordam , phys . . lett . * 69 * , 2062 ( 1992 ) ; b. broers , l. d. noordam , and h. b. van linden van den heuvell , phys . rev . a * 46 * , 2749 ( 1992 ) ; r. b. vrijen , g. m. lankhuijzen , d. j. maas , and l. d. noordam , comments at . * 33 * , 67 ( 1996 ) . n. v. vitanov and b. girard , phys . rev . a * 69 * , 033409 ( 2004 ) . r. g. unanyan , n. v. vitanov , and k. bergmann , phys . lett . * 87 * , 137902 ( 2001 ) ; r. g. unanyan , m. fleischhauer , n. v. vitanov , and k. bergmann , phys . a * 66 * , 042101 ( 2002 ) . a. a. rangelov , j. piilo , and n. v. vitanov , phys . a * 72 * , 053404 ( 2005 ) .

we calculate the propagator and the transition probabilities for a coherently driven three - state quantum system . the energies of the three states change linearly in time , whereas the interactions between them are pulse - shaped . we derive a highly accurate analytic approximation by assuming independent pairwise landau - zener transitions occurring instantly at the relevant avoided crossings , and adiabatic evolution elsewhere . quantum interferences are identified , which occur due to different possible evolution paths in hilbert space between an initial and a final state . a detailed comparison with numerical results for gaussian - shaped pulses demonstrates a remarkable accuracy of the analytic approximation . we use the analytic results to derive estimates for the half - width of the excitation profile , and for the parameters required for creation of a maximally coherent superposition of the three states . these results are of potential interest in ladder climbing in alkali atoms by chirped laser pulses , in quantum rotors , in transitions between zeeman sublevels of a @xmath0 level in a magnetic field , and in control of entanglement of a pair of spin-1/2 particles . the results for the three - state system can be generalized , without essential difficulties , to higher dimensions .

0711.4298

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

astro-ph0602465

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 .

cond-mat0508483

relativistic heavy - ion collisions are the experiments of choice to generate hot and dense matter in the laboratory . whereas in low energy collisions one produces dense nuclear matter with moderate temperature and large baryon chemical potential @xmath11 , ultra - relativistic collisions at relativistic heavy ion collider ( rhic ) or large hadron collider ( lhc ) energies produce extremely hot matter at small baryon chemical potential . in order to explore the phase diagram of strongly interacting matter as a function of @xmath12 and @xmath11 both type of collisions are mandatory . according to lattice calculations of quantum chromodynamics ( lqcd ) @xcite , the phase transition from hadronic to partonic degrees of freedom ( at vanishing baryon chemical potential @xmath11=0 ) is a crossover . this phase transition is expected to turn into a first order transition at a critical point @xmath13 in the phase diagram with increasing baryon chemical potential @xmath11 . since this critical point can not be determined theoretically in a reliable way the beam energy scan ( bes ) program performed at the rhic by the star collaboration aims to find the critical point and the phase boundary by gradually decreasing the collision energy @xcite . since the hot and dense matter produced in relativistic heavy - ion collisions appears only for a couple of fm / c , it is a big challenge for experiments to investigate its properties . the heavy flavor mesons are considered to be promising probes in this search since the production of heavy flavor requires a large energy - momentum transfer . thus it takes place early in the heavy - ion collisions , and - due to the large energy - momentum transfer - should be described by perturbative quantum chromodynamics ( pqcd ) . the produced heavy flavor then interacts with the hot dense matter ( of partonic or hadronic nature ) by exchanging energy and momentum . as a result , the ratio of the measured number of heavy flavors in heavy - ion collisions to the expected number in the absence of nuclear or partonic matter , which is the definition of @xmath6 ( cf . section vii ) , is suppressed at high transverse momentum , and the elliptic flow of heavy flavor is generated by the interactions in noncentral heavy - ion collisions . although it had been expected that the @xmath6 of heavy flavor is less suppressed and its elliptic flow is smaller as compared to the corresponding quantities for light hadrons , the experimental data show that the suppression of heavy - flavor hadrons at high transverse momentum and its elliptic flow @xmath7 are comparable to those of light hadrons @xcite . this is a puzzle for heavy - flavor production and dynamics in relativistic heavy - ion collisions as pointed out by many groups @xcite . for recent reviews we refer the reader to refs . @xcite . since the heavy - flavor interactions are closely related to the dynamics of the partonic or hadronic degrees - of - freedom due to their mutual interactions , a proper description of the relativistic heavy - ion collisions and their bulk dynamics is necessary . in this study we employ the parton - hadron - string dynamics ( phsd ) approach , which differs from the conventional boltzmann - type models in the aspect @xcite that the degrees - of - freedom for the qgp phase are off - shell massive strongly - interacting quasi - particles that generate their own mean - field potential . the masses of the dynamical quarks and gluons in the qgp are distributed according to spectral functions whose pole positions and widths , respectively , are defined by the real and imaginary parts of their self - energies @xcite . the partonic propagators and self - energies , furthermore , are defined in the dynamical quasiparticle model ( dqpm ) in which the strong coupling and the self - energies are fitted to lattice qcd results . we recall that the phsd approach has successfully described numerous experimental data in relativistic heavy - ion collisions from the super proton synchrotron ( sps ) to lhc energies @xcite . more recently , the charm production and propagation has been explicitly implemented in the phsd and detailed studies on the charm dynamics and hadronization / fragmention have been performed at top rhic and lhc energies in comparison to the available data @xcite . in the phsd approach the initial charm and anticharm quarks are produced by using the pythia event generator @xcite which is tuned to the transverse momentum and rapidity distributions of charm and anticharm quarks from the fixed - order next - to - leading logarithm ( fonll ) calculations @xcite . the produced charm and anticharm quarks interact in the qgp with off - shell partons and are hadronized into @xmath0mesons close to the critical energy density for the crossover transition either through fragmentation or coalescence . we stress that the coalescence is a genuine feature of heavy - ion collisions and does not show up in p+p interactions . the hadronized @xmath0mesons then interact with light hadrons in the hadronic phase until freeze out and final semileptonic decay . we have found that the phsd approach , which has been applied for charm production in au+au collisions at @xmath2200 gev @xcite and in pb+pb collisions at @xmath82.76 tev @xcite , describes the @xmath6 as well as the @xmath7 of @xmath0mesons in reasonable agreement with the experimental data from the star collaboration @xcite and from the alice collaboration @xcite when including the initial shadowing effect in the latter case . in this work we , furthermore , extend the phsd approach to bottom production in relativistic heavy - ion collisions . as in case of charm , the initial bottom pair is produced by using the pythia event generator , and the transverse momentum and rapidity distributions are adjusted to those from the fonll calculations . also the scattering cross sections of bottom quarks with off - shell partons are calculated in the dqpm on the same basis as the @xmath14quarks . the bottom quarks are hadronized into @xmath1mesons near the critical energy density in the same way as charm quarks . furthermore , the scattering cross sections of @xmath1mesons with light hadrons ( in the hadronic phase ) are calculated from a similar effective lagrangian as used for @xmath0mesons . presently , there are no exclusive experimental data for @xmath1meson production from relativistic heavy - ion collisions . the phenix collaboration instead measured the single electrons which are produced through the semileptonic decay of @xmath0 and @xmath1mesons in au+au collisions at @xmath2 200 and 62.4 gev @xcite . in this work we will study the bottom production through the single electrons in au+au collisions at @xmath2 200 and 62.4 gev by using the extended phsd . additionally , we make predictions for @xmath0meson and single electron production at the much lower energy @xmath8 19.2 gev while we compare with the experimental results available at @xmath2 200 and 62.4 gev . finally , we study the medium modification of the azimuthal angle @xmath9 of a heavy - flavor pair in relativistic heavy - ion collisions by the interactions with the partonic or hadronic medium . this paper is organized as follows : the production of heavy mesons and their semileptonic decay in p+p collisions is described in detail and compared with experimental data in sec . the initial production of heavy quarks is explained in sec . [ shadowing ] including the shadowing effect in relativistic heavy - ion collisions . we then present the heavy quark interactions in the qgp , their hadronization and hadronic interactions , respectively , in sec . [ qgp ] , [ hadronization ] and [ hg ] . finally , we show our results in sec . [ results ] in comparison with the experimental data while a summary closes this study in sec . [ summary ] . as pointed out in the introduction the charm and bottom quark pairs are produced through initial hard nucleon - nucleon scattering in relativistic heavy - ion collisions . we employ the pythia event generator to produce the heavy - quark pairs and modify their transverse momentum and rapidity such that they are similar to those from the fonll calculations . in the case of heavy - ion collisions at the top rhic energy of @xmath15 gev , the transverse momentum and the rapidity of the charm quark are reduced by 10 % and 16 % @xcite , respectively , and those of the bottom quark are unmodified . the transverse momentum of charm quarks at the invariant energy of @xmath16 gev is modified to @xmath17 , where @xmath18 and @xmath19 , respectively , are the original and modified transverse momenta , while the rapidity is reduced by 15 % . the transverse momentum and the rapidity of a bottom quark at the same energy are , respectively , reduced by 5 % and enhanced by 15 % . the msel code in pythia is taken to be the default value 1 for charm production , and 5 for bottom production . [ h ] spectra ( a ) and rapidity distributions ( b ) of charm and bottom quarks in p+p collisions at @xmath15 gev as generated by the tuned pythia event generator ( dashed ) in comparison to those from fonll ( solid).,width=9 ] spectra ( a ) and rapidity distributions ( b ) of charm and bottom quarks in p+p collisions at @xmath15 gev as generated by the tuned pythia event generator ( dashed ) in comparison to those from fonll ( solid).,width=9 ] [ h ] spectra ( a ) and rapidity distributions ( b ) of charm and bottom quarks in p+p collisions at @xmath16 gev as generated by the tuned pythia event generator ( dashed ) in comparison to those from fonll ( solid).,width=9 ] spectra ( a ) and rapidity distributions ( b ) of charm and bottom quarks in p+p collisions at @xmath16 gev as generated by the tuned pythia event generator ( dashed ) in comparison to those from fonll ( solid).,width=9 ] figures [ pp200q ] and [ pp62q ] show the @xmath20 spectra and rapidity distributions of charm and bottom quarks in p+p collisions at @xmath15 gev and 62.4 gev , where the dashed and solid lines are from the tuned pythia and fonll calculations , respectively . the fonll calculations are rescaled such that the total cross sections for charm production are 0.8 and 0.14 mb at @xmath15 gev and 62.4 gev , respectively . the ratios of the bottom cross section to the charm cross section are taken to be 0.75 % and 0.145 % at the same energies as in the fonll calculations . we find that our tuned pythia generator gives very similar charm and bottom distributions as those from fonll calculations , which fixes the input from pqcd . [ h ] .,width=9 ] furthermore , the produced charm and bottom quarks in hard nucleon - nucleon collisions are hadronized by emitting soft gluons , which is denoted by fragmentation. we use the fragmentation function of peterson which reads as @xcite @xmath21 ^ 2},\end{aligned}\ ] ] where @xmath22 is the momentum fraction of the hadron @xmath23 fragmented from the heavy quark @xmath24 while @xmath25 is a fitting parameter which is taken to be @xmath25 = 0.01 for charm @xcite and 0.004 for bottom @xcite . figure [ peterson ] shows the fragmentation functions of charm and bottom quarks as a function of the hadron momentum fraction @xmath22 . since the charm quark is much heavier than the soft emitted gluons , it takes a large momentum fraction in the fragmentation into a @xmath0meson . it is even more pronounced in the case of a bottom quark . the chemical fractions of the charm quark decay into @xmath26 , and @xmath27 are taken to be 14.9 , 15.3 , 23.8 , 24.3 , 10.1 , and 8.7 % @xcite , respectively , and those of the bottom quark decay into @xmath28 , and @xmath29 are 39.9 , 39.9 , 11 , and 9.2 % @xcite . after the momentum and the species of the fragmented particle are decided by monte carlo , the energy of the fragmented particle is adjusted to be on - shell . furthermore , the @xmath30 mesons first decay into @xmath31 or @xmath32 , and then the @xmath0 and @xmath1mesons produce single electrons through the semileptonic decay @xcite . for simplicity , it is assumed that the transition amplitude for the semileptonic decay is constant and does not depend on particle momentum . denoting the energy , momentum , and mass of a particle @xmath33 by @xmath34 , @xmath35 , and @xmath36 in the decay @xmath37 , the phase space for the final states ( @xmath38 ) is then proportional to @xmath39 in the center - of - mass frame of the leptons @xmath40 and @xmath41 , it is simplified to @xmath42 because @xmath43 , assuming the leptons @xmath40 and @xmath41 to be massless . finally , we perform a lorentz boost to the rest frame of the @xmath0meson , where the solid angle of @xmath44 is assumed to be isotropic : @xmath45 the momentum @xmath46 itself is decided by a random number as follows : @xmath47 where @xmath48 is fixed by energy conservation . accordingly , once @xmath46 is fixed , the invariant mass of the lepton pair ( @xmath40 and @xmath41 ) , and then @xmath49 in the center - of - mass frame of @xmath40 and @xmath41 are fixed . the solid angle of each particle is determined by monte carlo and its energy - momentum boosted back to the p+p collision frame . [ h ] ( dotted ) , @xmath50 ( dashed ) , and @xmath51 ( solid ) in the rest frame of the heavy meson.,width=9 ] figure [ momentum - e ] shows the momentum distribution of single electrons from the semileptonic decay of heavy mesons in the rest frame of the heavy meson . it shows that the single electron from the decay @xmath52 has the largest momentum and that from @xmath50 the lowest momentum according to the mass difference between the mother meson and the daughter meson . we also take into account the decay @xmath53 , @xmath54 , and @xmath55 @xcite . the branching ratio of each decay channel is obtained from the particle data group ( pdg ) @xcite . this completes the description of the semileptonic decays . [ h ] spectrum of single electrons from charm and bottom mesons in p+p collisions at @xmath15 gev ( a ) and @xmath56=62.4 gev ( b ) in comparison to the experimental data from refs . @xcite.,width=9 ] spectrum of single electrons from charm and bottom mesons in p+p collisions at @xmath15 gev ( a ) and @xmath56=62.4 gev ( b ) in comparison to the experimental data from refs . @xcite.,width=9 ] in figure [ ppe ] we show the @xmath18 spectrum of single electrons in p+p collisions at @xmath15 gev ( a ) and @xmath57 62.4 gev ( b ) . the figure shows that our results reproduce the experimental data at @xmath15 gev from the phenix collaboration @xcite as well as at @xmath16 gev from the intersecting storage rings ( isr ) @xcite . we note , furthermore , that the contribution from @xmath1meson decay is compatible to that from @xmath0meson decay around @xmath58 gev at @xmath15 gev , and around @xmath59 gev at @xmath16 gev , respectively . the scattering cross section for heavy - quark pair production in a nucleon - nucleon collision is calculated by convoluting the partonic cross section with the parton distribution functions of the nucleon : @xmath60 where @xmath61 is the distribution function of the parton @xmath33 with the energy - momentum fraction @xmath62 in the nucleon at scale @xmath63 . the momentum fractions @xmath64 and @xmath65 are calculated from the transverse mass ( @xmath66 ) and the rapidity ( @xmath67 ) of the final - state particles by @xmath68 where @xmath69 is the nucleon - nucleon collision energy in the center - of - mass frame . as it is well known the parton distribution function ( pdf ) is modified in a nucleus to @xmath70 where @xmath71 indicates a nucleon in nucleus @xmath72 , and @xmath73 is the ratio of the pdf of @xmath71 to that of a free nucleon . the ratio @xmath73 for a heavy nucleus @xmath72 is lower than 1 at small momentum fraction @xmath62 , and becomes larger than 1 with increasing @xmath62 . the former phenomenon is called shadowing and the latter antishadowing. when increasing @xmath62 further , the ratio reaches a maximum and then decreases again , which is called the european muon collaboration ( emc ) effect @xcite . finally , the ratio increases again close to @xmath74 due to fermi motion . the eps09 package @xcite parameterizes this behavior of @xmath73 and fits the heights and positions of the local extrema of the ratio to the experimental data from deep inelastic @xmath75a scattering , drell - yan dilepton production in p+a collisions , and inclusive pion production in d+au and p+p collisions at rhic . we also employ the eps09 package @xcite in our phsd calculations @xcite . furthermore , the ( anti-)shadowing effect is supposed to depend on the impact parameter in heavy - ion collisions such that it is strong in central collisions and weak in peripheral collisions . therefore , we modify the ratio to @xcite @xmath76 where @xmath77 and @xmath78 are , respectively , the radius of the nucleus @xmath72 and the transverse distance of the heavy - quark pair production from the nucleus center , while @xmath73 is given by eps09 . [ h ] gev and in 0 - 20 % central au+au collisions at @xmath16 gev as a function of @xmath18 ( a ) and of y ( b).,width=9 ] gev and in 0 - 20 % central au+au collisions at @xmath16 gev as a function of @xmath18 ( a ) and of y ( b).,width=9 ] substituting @xmath79 in eq . ( [ factorize ] ) by eq . ( [ shadow ] ) , the cross section for heavy quark production is modified to @xmath80 figure [ shadowingf ] shows the ratio of the cross sections with ( anti)shadowing to without ( anti)shadowing . the scale @xmath63 is taken to be the average of the transverse mass of the heavy quark and that of the heavy antiquark . the total cross sections for charm and bottom production , respectively , decreases by 8 % and increases by 21 % in 0 - 10 % central au+au collisions at @xmath15 gev , and increases by 18 % and 21 % in 0 - 20 % central au+au collisions at @xmath81 gev . the ratio of the charm cross sections increases with increasing transverse momentum in 0 - 10 % central au+au collisions at @xmath15 gev , because @xmath73 increases with increasing @xmath62 in the ( anti)shadowing region . in the case of bottom production , the corresponding @xmath62 is larger and located close to the maximum of the antishadowing region . therefore , the dependence of the ( anti)shadowing effect on @xmath18 or on @xmath62 is monotonous . since the momentum fraction @xmath62 corresponding to bottom production at @xmath57 200 gev is similar to that corresponding to charm production at @xmath57 62.4 gev , the ( anti)shadowing effect on both are similar in @xmath18 as well as in rapidity @xmath67 . finally , the ratio of bottom production at @xmath57 62.4 gev decreases with increasing @xmath18 , because the corresponding momentum fraction @xmath62 moves towards the emc effect region , where @xmath73 decreases with increasing @xmath62 . we note that the cronin effect , which suppresses low @xmath18 particles and enhances high @xmath18 particles , is effectively included in eps09 as shadowing and antishadowing effects . in phsd the baryon - baryon and baryon - meson collisions at high - energy produce strings . if the local energy density is above the critical energy density ( @xmath82 0.5 gev/@xmath5 ) , the strings melt into quarks and antiquarks with masses determined by the temperature - dependent spectral functions from the dqpm @xcite . massive gluons are formed through flavor - neutral quark and antiquark fusion in line with the dqpm . in contrast to normal elastic scattering , off - shell partons may change their mass after the elastic scattering according to the local temperature @xmath12 in the cell ( or local space - time volume ) where the scattering happens . this automatically updates the parton masses as the hot and dense matter expands , i.e. the local temperature decreases with time . the same holds true for the reaction chain from gluon decay to quark+antiquark ( @xmath83 ) and the inverse reaction ( @xmath84 ) following detailed balance . due to the finite spectral width of the partonic degrees - of - freedom , the parton spectral function has time - like as well as space - like parts . the time - like partons propagate in space - time within the light - cone while the space - like components are attributed to a scalar potential energy density @xcite . the gradient of the potential energy density with respect to the scalar density generates a repulsive force in relativistic heavy - ion collisions and plays an essential role in reproducing experimental flow data and transverse momentum spectra ( see ref . @xcite for a review ) . however , the spectral function of a heavy quark or heavy antiquark can not be fitted from lattice qcd data on thermodynamical properties because the contribution from a heavy quark or heavy antiquark to the lattice entropy is small . our recent study shows that the scattering cross sections of heavy quark moderately depend on the spectral function of heavy quark , and the repulsive force for charm quarks as originating from the scalar potential energy density is disfavored by experimental data @xcite . this is expected since the width of the spectral function for a charm quark is very small compared to the pole mass such that space - like contributions to the ( potential ) energy density are practically vanishing . therefore , we assume in this study that the heavy quark has a constant ( on - shell ) mass : the charm quark mass is taken to be 1.5 gev and the bottom quark mass as 4.8 gev , but the light quarks / antiquarks as well as gluons are treated fully off - shell . the heavy quarks and antiquarks produced in early hard collisions - as described above - interact with the dressed lighter off - shell partons in the qgp . the cross sections for the heavy - quark scattering with massive off - shell partons have been calculated by considering explicitly the mass spectra of the final state particles in refs . the elastic scattering of heavy quarks in the qgp is treated by including the non - perturbative effects of the strongly interacting quark - gluon plasma ( sqgp ) constituents , i.e. the temperature - dependent coupling @xmath85 which rises close to @xmath3 , the multiple scattering etc . the multiple strong interactions of quarks and gluons in the sqgp are encoded in their effective propagators with broad spectral functions ( imaginary parts ) . as pointed out above , the effective propagators , which can be interpreted as resummed propagators in a hot and dense qcd environment , have been extracted from lattice data in the scope of the dqpm @xcite . we recall that the divergence encountered in the @xmath86-channel scattering is cured self - consistently , since the infrared regulator is given by the finite dqpm gluon mass and width . for further details we refer the reader to refs . @xcite . [ h ] and @xmath87 from the dqpm calculations described in refs . @xcite.,width=9 ] and @xmath87 from the dqpm calculations described in refs . @xcite.,width=9 ] figure [ sigma ] compares the total ( a ) and differential ( b ) scattering cross sections of charm and bottom quarks with a light quark at @xmath88 . it shows that the total cross section of a charm quark is similar to that of the bottom quark apart from different threshold energies . however , the differential scattering cross section of a bottom quark is more peaked in forward direction , compared to that of a charm quark . this is expected because it is harder to change the direction of motion of a bottom quark in elastic scattering due to the larger mass . we note that charm interactions in the qgp as described by the dqpm charm scattering cross sections differ substantially form the pqcd scenario @xcite , however , the spacial diffusion constant for charm quarks @xmath89 is consistent with the lqcd data @xcite . the heavy - quark hadronization in heavy - ion collisions is realized via dynamical coalescence and fragmentation . here dynamical coalescence means that the probability to find a coalescence partner is defined by monte carlo in the vicinity of the critical energy density @xmath90 gev/@xmath5 as explained below . we note that such a dynamical realization of heavy - quark coalescence is in line with the dynamical hadronization of light quarks in the phsd and differs from the spontaneous coalescence used in our early work @xcite when heavy - quarks are forced to hadronize at a critical energy density @xmath91 gev/@xmath5 via coalescence or fragmentation by monte carlo . indeed , the dynamical realization gives some window in energy density to find the proper light partner and leads to an enhancement of the heavy - quark fraction that hadronizes via coalescence . in phsd all antiquarks neighboring in phase space are candidates for the coalescence partner of a heavy quark . from the distances in coordinate and momentum spaces between the heavy quark and light antiquark ( or vice versa ) , the coalescence probability is given by @xcite @xmath92 , \label{meson}\end{aligned}\ ] ] where @xmath93 is the degeneracy of the heavy meson , and @xmath94 with @xmath36 , @xmath95 and @xmath96 denoting the mass , position and momentum of the quark or antiquark @xmath33 in the center - of - mass frame , respectively . the width parameter @xmath97 is related to the root - mean - square radius of the produced heavy meson through @xmath98 where @xmath99 and @xmath100 are respectively the masses of quark and antiquark . since this prescription gives a larger coalescence probability at low transverse momentum , the radius is taken to be 0.9 fm for a charm quark as well as for a bottom quark @xcite . we also include the coalescence of charm quarks into highly excited states , @xmath101 , @xmath102 , and @xmath103 and the coalescence of bottom quarks into @xmath104 , @xmath105 , and @xmath106 , which are respectively assumed to immediately decay to @xmath107 ( or @xmath30 ) and @xmath108 and to @xmath109 ( or @xmath110 ) and @xmath108 after hadronization as described in ref . @xcite . summing up the coalescence probabilities from all candidates , whether the heavy quark or heavy antiquark hadronizes by coalescence or not , and which quark or antiquark among the candidates will be the coalescence partner , is decided by monte carlo . if a random number is above the sum of the coalescence probabilities , it is tried again in the next time step till the local energy density is lower than 0.4 @xmath111 . the heavy quark or heavy antiquark , which does not succeed to hadronize by coalescence , then hadronizes through fragmentation as in p+p collisions . [ h ] ) as functions of transverse momentum ( a ) and of transverse velocity ( b ) in 0 - 10 % central au+au collisions at @xmath2200 gev taking into account the shadowing effect.,width=9 ] ) as functions of transverse momentum ( a ) and of transverse velocity ( b ) in 0 - 10 % central au+au collisions at @xmath2200 gev taking into account the shadowing effect.,width=9 ] figure [ coalpro ] shows the coalescence probabilities of charm and bottom quarks at midrapidity ( @xmath112 ) as functions of transverse momentum ( a ) and of transverse velocity ( b ) in 0 - 10 % central au+au collisions at @xmath2200 gev . since a heavy quark with a large transverse momentum has a smaller chance to find a coalescence partner close by in phase space , the coalescence probability decreases with increasing transverse momentum . it appears from the upper figure ( a ) that the coalescence probability of a bottom quark is larger than that of a charm quark . it emerges , however , because the bottom quark is much heavier than the charm quark . the lower figure ( b ) clearly shows that the coalescence probability is similar for a bottom or charm quark , when it is expressed as a function of the transverse velocity @xmath113 . after the hadronization of heavy quarks and their subsequent decay into @xmath114 and @xmath110 mesons , the final stage of the evolution concerns the interaction of these states with the hadrons conforming the expanding bulk medium . a realistic description of the hadron - hadron scattering potentially affected by resonant interactions includes collisions with the states @xmath115 . a description of their interactions has been developed in refs . @xcite using effective field theory . moreover , after the application of an effective theory , one should implement to the scattering amplitudes a unitarization method to better control the behavior of the cross sections at moderates energies . the details of the interaction for the four heavy states follows quite in parallel by virtue of the `` heavy - quark spin - flavor symmetry '' . it accounts for the fact that if the heavy masses are much larger than any other typical scale in the system , like @xmath116 , temperature and the light hadron masses , then the physics of the heavy subsystem is decoupled from the light sector , and the former is not dependent on the mass nor on the spin of the heavy particle . this symmetry is exact in the ideal limit @xmath117 , with @xmath118 being the mass of the heavy quark confined in the heavy hadron . in the opposite limit @xmath119 , one can exploit the chiral symmetry of the qcd lagrangian to develop an effective realization for the light particles . this applies to the pseudoscalar meson octet ( @xmath120 ) . although both symmetries are broken in nature ( as in our approach , when implementing physical masses ) , the construction of the effective field theories incorporates the breaking of these symmetries in a controlled way . in particular , it provides a systematic expansion in powers of @xmath121 ( inverse heavy - meson mass ) and powers of @xmath122 ( typical momentum and mass of the light meson ) . following these ideas , we use two effective lagrangians for the interaction of a heavy meson with light mesons and with baryons , respectively . in the scattering with light mesons , the scalar ( @xmath107,@xmath109 ) and vector ( @xmath123 ) mesons are much heavier than the pseudoscalar meson octet ( @xmath120 ) . the latter have , in addition , masses smaller than the chiral scale @xmath124 , where @xmath125 is the pion decay constant . in this case one can exploit standard chiral perturbation theory for the dynamics of the ( pseudo ) goldstone bosons , and add the heavy - quark mass expansion up to the desired order to account for the interactions with heavy mesons . in our case the effective lagrangian is kept to next - to - leading order in the chiral expansion , but to leading order in the heavy - quark expansion @xcite . from this effective lagrangian one can compute the tree - level amplitude ( or potential ) , which describes the scattering of a heavy meson off a light meson as worked out in refs . @xcite . at leading order in the heavy - quark expansion one gets a common result for all heavy mesons due to the exact heavy - flavor symmetry and heavy - quark spin symmetry ( hqss ) . the potential reads explicitly @xmath126 \ , \nonumber\end{aligned}\ ] ] where @xmath127 are numerical coefficients ( fixed by chiral symmetry ) which depend on the incoming @xmath33 and outgoing @xmath128 channels and also on the quantum numbers @xmath129 ( isospin , total angular momentum , strangeness and charm / bottom ) . in ( [ eqq ] ) @xmath125 is the pion decay constant in the chiral limit , and @xmath130 are the low - energy constants at nlo in the chiral expansion ( see ref . @xcite for details ) . finally , @xmath131 denote the mandelstam variables and @xmath132 the four - momentum of the @xmath133particle in the scattering ( @xmath134 ) . for the heavy meson baryon interaction we use an effective lagrangian based on a low - energy realization of a @xmath135channel vector meson exchange between mesons and baryons . in the low - energy limit the interaction provides a generalized weinberg - tomozawa contact interaction as worked out in refs . the effective lagrangian obeys su(6 ) spin - flavor symmetry in the light sector , plus hqss in the heavy sector ( which is preserved either the heavy quark is contained in the meson or in the baryon ) . the tree - level amplitude reads @xmath136 where @xmath137 are numerical coefficients which depend on the initial and final channels ( @xmath138 ) , as well as all the quantum numbers @xmath129 . in ( [ ell ] ) @xmath139 is the meson decay constant in the @xmath33 channel , and @xmath140 are the baryon mass and energy in the c.m . frame . from the form of this potential it is evident that hqss is again maintained . we note again that in both @xmath141 and @xmath142 , hqss is eventually broken when using physical values for the heavy masses . the tree - level amplitudes for meson - meson and meson - baryon scattering have strong limitations in the energy range in which they should be applied . it is limited for those processes in which the typical momentum transfer is low , and below any possible resonance . to increase the applicability of the scattering amplitudes and restore exact unitarity for the scattering - matrix elements , we apply a unitarization method , which consists in solving a coupled - channel bethe - salpeter equation for the unitarized scattering amplitude @xmath143 using the potential as a kernel , @xmath144 where @xmath145 is the diagonal meson - meson ( or meson - baryon ) propagator which is regularized by dimensional regularization in the meson - meson ( or meson - baryon ) channel . we adopt the `` on - shell '' approximation to the kernel of the bethe - salpeter equation to reduce it into a set of algebraic equations . we refer the reader to refs . @xcite for technical details . the unitarization procedure allows for the possibility of generating resonant states as poles of the scattering amplitude @xmath143 . even when these resonances are not explicit degrees - of - freedom , and we do not propagate them in our phsd simulations , they are automatically incorporated into the two - body interaction . this is an important fact , because such ( intermediate ) resonant states will strongly affect the scattering cross section of heavy mesons due to the presence of resonances , subthreshold states ( bound states ) , and other effects like the opening of a new channel when a resonance is forming ( flatt effect ) . to mention some particular examples , in the interaction of @xmath146 mesons with light mesons we generate broad resonances in the @xmath147 channels . in the charm sector we identify them with the experimentally - observed states @xmath148 and @xmath149 that can decay in an @xmath150-wave into @xmath151 and @xmath152 , respectively . in the bottom sector we obtain analogous states @xmath153 and @xmath154 , not yet identified by experiment . we also find a series of bound states in the channel @xmath155 which are identified with the @xmath156 and the @xmath157 states . their bottom relatives @xmath158 and @xmath159 are again predictions . in the meson - baryon channel , we find the experimental @xmath160 and @xmath161 charm resonances in the @xmath162 sector . in our model , the @xmath160 couples dominantly to @xmath163 and @xmath164 , while the @xmath161 to @xmath164 . their bottom homologues are associated with the experimental @xmath165 and @xmath166 baryons seen by the lhcb collaboration @xcite . we finally mention the subthreshold states in the @xmath167 channel , the @xmath168 and @xmath169 ( that strongly couple to the @xmath170 and @xmath171 channels , respectively ) . these states are the counterparts of the experimental @xmath172 in the strange sector , but have not been yet observed , so they can be taken as predictions for future measurements . many other resonant states ( especially in the meson - baryon sector ) are found in the remaining scattering channels . the resulting cross sections for the binary scattering of @xmath173 ( with any possible charged states ) with @xmath115 are implemented in the phsd code considering both elastic and inelastic channels . around 200 different channels are taken into account . although the unitarization method helps to extend the validity of the tree - level amplitudes into the resonant region , one can not trust the final cross sections for higher energies . beyond the resonant region we adopt constant cross sections inspired by the results of the regge analysis in the energy domain of several gev @xcite , where one expects an almost flat energy dependence of the cross sections . [ h ] meson scattering cross sections with pion ( a ) and nucleon ( b).,width=9 ] meson scattering cross sections with pion ( a ) and nucleon ( b).,width=9 ] in fig . [ bh ] we present several examples of @xmath174 meson scattering cross sections with pions and nucleons . the cross sections show a non - smooth behaviour with energy , due to the presence of several mesonic and baryonic beauty states generated dynamically , as described above . as an example , in the scattering with pions we observed the very broad resonant peak of @xmath153 . the clear dip of some of the cross sections around 5830 mev is due to the opening of the coupled - channel @xmath175 at @xmath176 ( flatt effect ) . in the @xmath177 and @xmath178 cross sections , we observe the presence of baryonic states around 6360 mev in both @xmath179 and @xmath180 channels . the position and the width of these states as well as the coupling of these states to the main channels have been carefully analyzed in ref . @xcite in connection with several transport coefficients in heavy - ion collisions . so far we have described the interactions of the heavy flavor produced in relativistic heavy - ion collisions with partonic and hadronic degrees - of - freedom . since the matter produced in heavy - ion collisions is extremely dense , the interactions with the bulk matter suppresses heavy flavors at high-@xmath18 . on the other hand , the partonic or nuclear matter is accelerated outward ( exploding ) , and a strong flow is generated via the interactions of the bulk particles and the repulsive scalar interaction for partons . since the heavy flavor strongly interacts with the expanding matter , it is also accelerated outwards . such effects of the medium on the heavy - flavor dynamics are expressed in terms of the nuclear modification factor defined as @xmath181 where @xmath182 and @xmath183 are , respectively , the number of particles produced in heavy - ion collisions and that in p+p collisions , and @xmath184 is the number of binary nucleon - nucleon collisions in the heavy - ion collision for the centrality class considered . note that if the heavy flavor does not interact with the medium in heavy - ion collisions , the numerator of eq . ( [ raa ] ) will be similar to the denominator . for the same reason , an @xmath6 smaller ( larger ) than one in a specific @xmath18 region implies that the nuclear matter suppresses ( enhances ) the production of heavy flavors in that transverse momentum region . in noncentral heavy - ion collisions the produced matter expands anisotropically due to the different pressure gradients between in plane and out - of plane . if the heavy flavor interacts strongly with the nuclear matter , then it also follows this anisotropic motion to some extend . the anisotropic flow is expressed in terms of the elliptic flow @xmath7 which reads @xmath185 where @xmath9 is the azimuthal angle of a particle in momentum space . in the following subsections , we will present our results on the production of heavy flavors and single electrons in au+au collisions at @xmath2200 gev , 62.4 gev , make predictions for even lower energies , and discuss the azimuthal angular correlations between a heavy - flavor meson and its anti - flavor meson . [ h ] of @xmath186mesons ( a ) and of @xmath1mesons ( b ) with ( solid ) and without ( dashed ) shadowing effect in 0 - 10 % central au+au collisions at @xmath2200 gev in comparison to the experimental data from the star collaboration @xcite.,width=9 ] of @xmath186mesons ( a ) and of @xmath1mesons ( b ) with ( solid ) and without ( dashed ) shadowing effect in 0 - 10 % central au+au collisions at @xmath2200 gev in comparison to the experimental data from the star collaboration @xcite.,width=9 ] the upper ( a ) and lower ( b ) figure [ raa200db ] are , respectively , the @xmath6 of @xmath0mesons and of @xmath1mesons in 0 - 10 % central au+au collisions at @xmath2200 gev from the phsd calculations . the shadowing effect is excluded in the dashed lines and is included in the solid lines . furthermore , in figure [ raa200db ] ( a ) the @xmath6 of @xmath0mesons are compared with the experimental data from the star collaboration @xcite . we note that the @xmath6 of @xmath0mesons without shadowing effect is slightly different from our previous results in ref . @xcite , because the elastic backward scattering has been improved and the coalescence of charm quark takes place continuously as in the later work @xcite . as shown in fig . [ shadowingf ] , the shadowing effect decreases the charm production by @xmath828 % and increases the bottom production by @xmath8220 % . apparently the @xmath6 of @xmath1mesons is much larger than that of @xmath0mesons at the same transverse momentum . however , this is attributed to the larger bottom mass than charm mass as demonstrated in fig . [ coalpro ] before . [ h ] of single electrons from the semi - leptonic decay of @xmath0mesons ( dashed ) and of @xmath1mesons ( dot - dashed ) and the sum of them ( solid ) with ( b ) and without ( a ) shadowing effect in 0 - 10 % central au+au collisions at @xmath2200 gev in comparison to the experimental data from the phenix collaboration @xcite.,width=9 ] of single electrons from the semi - leptonic decay of @xmath0mesons ( dashed ) and of @xmath1mesons ( dot - dashed ) and the sum of them ( solid ) with ( b ) and without ( a ) shadowing effect in 0 - 10 % central au+au collisions at @xmath2200 gev in comparison to the experimental data from the phenix collaboration @xcite.,width=9 ] fig . [ raa200e ] shows the @xmath6 of single electrons from @xmath0meson and @xmath1meson semileptonic decays , which correspond to the dashed and dot - dashed lines , respectively , while the solid lines are sum of them in 0 - 10 % central au+au collisions at @xmath8200 gev . the upper figure ( a ) is the @xmath6 without shadowing effect , and the lower one ( b ) includes the shadowing effect , which enhances the bottom production and suppresses the charm production at low transverse momentum in line with the discussion above . we find that the single electrons from @xmath109 decay have a larger contribution than that from @xmath107 decay above @xmath187 gev . in p+p collisions , the contribution from @xmath109 decay starts to be larger than that from @xmath107 decay at about @xmath188 gev as shown in fig . the reason for the dominance of @xmath109 decay at lower transverse momentum in au+au collisions is that the @xmath6 of @xmath1mesons is larger than that of @xmath0mesons at high transverse momentum as shown in fig . [ raa200db ] . [ h ] of single electrons from the semi - leptonic decay of @xmath0mesons ( dashed ) and of @xmath1mesons ( dot - dashed ) and of both of them ( solid ) with ( b ) and without ( a ) shadowing effect in minimum - bias au+au collisions at @xmath2200 gev in comparison to the experimental data from the phenix collaboration @xcite.,width=9 ] of single electrons from the semi - leptonic decay of @xmath0mesons ( dashed ) and of @xmath1mesons ( dot - dashed ) and of both of them ( solid ) with ( b ) and without ( a ) shadowing effect in minimum - bias au+au collisions at @xmath2200 gev in comparison to the experimental data from the phenix collaboration @xcite.,width=9 ] we present in fig . [ v2e200 ] the elliptic flow @xmath7 of single electrons without ( a ) and with ( b ) shadowing effect in minimum - bias au+au collisions at @xmath2200 gev . the dashed and dot - dashed lines are , respectively , the @xmath7 of single electrons from @xmath0meson and @xmath1meson decays . since the @xmath1 meson is much more massive , the elliptic flow from @xmath1meson decay starts to grow from much higher transverse momentum . the red lines are the elliptic flow @xmath7 of all single electrons . [ raa200e ] and [ v2e200 ] show that phsd can approximately reproduce the experimental data on the @xmath6 and @xmath7 of single electrons from @xmath0 mesons and @xmath1mesons at @xmath2200 gev . they also show that the shadowing effect is not so critical in reproducing experimental data , which is different from the lhc energies @xcite . the beam energy scan ( bes ) program at rhic has been carried out by colliding au nuclei at various energies down to @xmath87.7 gev . the aim of the program is to find information on the phase boundary and hopefully the critical point in the qcd phase diagram as pointed out in refs . it is expected that if the trajectories of the produced nuclear matter in the qcd phase diagram pass close to the critical point , some drastic changes of observables could be measured in experiments . since the phenix collaboration recently measured the single electrons from heavy flavor decay at @xmath262.4 gev @xcite , which is much lower than the maximum energy at rhic , we first address this sytem in the present subsection . [ h ] of single electrons from the semi - leptonic decay of @xmath0mesons ( dashed ) and of @xmath1mesons ( dot - dashed ) and the sum of them ( solid ) with ( b ) and without ( a ) shadowing effect in 0 - 20 % central au+au collisions at @xmath262.4 gev in comparison to the experimental data from the phenix collaboration @xcite.,width=9 ] of single electrons from the semi - leptonic decay of @xmath0mesons ( dashed ) and of @xmath1mesons ( dot - dashed ) and the sum of them ( solid ) with ( b ) and without ( a ) shadowing effect in 0 - 20 % central au+au collisions at @xmath262.4 gev in comparison to the experimental data from the phenix collaboration @xcite.,width=9 ] [ h ] of single electrons from the semi - leptonic decay of @xmath0mesons ( dashed ) and of @xmath1mesons ( dot - dashed ) and of both of them ( solid ) with ( b ) and without ( a ) shadowing effect in 20 - 40 % central au+au collisions at @xmath262.4 gev in comparison to the experimental data from the phenix collaboration @xcite.,width=9 ] of single electrons from the semi - leptonic decay of @xmath0mesons ( dashed ) and of @xmath1mesons ( dot - dashed ) and of both of them ( solid ) with ( b ) and without ( a ) shadowing effect in 20 - 40 % central au+au collisions at @xmath262.4 gev in comparison to the experimental data from the phenix collaboration @xcite.,width=9 ] figs . [ raa62e ] and [ v2e62 ] show , respectively , the @xmath6 and elliptic flow @xmath7 of single electrons from the semileptonic decay of heavy flavors in 0 - 20 % and 20 - 40 % central au+au collisions at @xmath262.4 gev . the upper figures ( a ) are the results without shadowing effect and the lower ones ( b ) are with the shadowing effect . as at @xmath2200 gev , the contribution from @xmath0meson decay is important at low transverse momentum and superseeded by the contribution from @xmath109 decay above 3 gev . the contribution from @xmath109 decay becomes dominant at higher transverse momentum than at @xmath2200 gev , because the ratio of the scattering cross section for bottom production to that for charm production is much lower at @xmath262.4 gev . the latter ratio is 0.75 % at @xmath2200 gev and 0.145 % at @xmath262.4 gev according to the fonll calculations @xcite . our phsd results in fig . [ raa62e ] underestimate @xmath6 besides touching the lower error bars of the experimental data at low and high @xmath18 . although the shadowing effect enhances @xmath6 at low @xmath18 , there is still a large discrepancy between the experimental data and our results in the range of @xmath189 between 2.5 and 4 gev , which clearly lacks an explanation . we mention that a similar pattern of results has been shown in ref . @xcite and partially been attributed to the cronin effect . we recall that the cronin effect for the charm quarks is included in the phsd via the eps09 package used for the shadowing effect as explained in section iii . in spite of the difficulty in reproducing @xmath6 , the elliptic flow @xmath7 of single electrons at @xmath262.4 gev is well described by the phsd approach up to @xmath190 2 gev / c irrespective whether the shadowing effect is included or not , as shown in fig . [ v2e62 ] . the @xmath7 of single electrons from @xmath1meson decay is small at low transverse momentum for the same reason as at @xmath2200 gev . presently there are no available experimental data on open heavy flavors below @xmath262.4 gev from the bes program . accordingly , we will make a prediction on the production of @xmath0mesons and of single electrons at @xmath219.2 gev and compare again with the results at @xmath2200 and @xmath262.4 gev in order to obtain some excitation function . [ h ] of @xmath0mesons ( b ) and of single electrons ( a ) without shadowing effect in 0 - 10 % central au+au collisions at @xmath2200 gev ( solid ) , 62.4 gev ( dashed ) and 19.2 gev ( dot - dashed ) from the phsd approach.,width=9 ] of @xmath0mesons ( b ) and of single electrons ( a ) without shadowing effect in 0 - 10 % central au+au collisions at @xmath2200 gev ( solid ) , 62.4 gev ( dashed ) and 19.2 gev ( dot - dashed ) from the phsd approach.,width=9 ] the upper ( a ) and lower ( b ) panels of fig . [ raa3e ] , respectively , show the @xmath6 of @xmath0mesons and that of single electrons in 0 - 10 % central au+au collisions at @xmath8200 , 62.4 , and 19.2 gev . for simplicity , the shadowing effect is not taken into account in these phsd calculations . comparing the @xmath6 of @xmath0mesons at @xmath2200 and 62.4 gev , the peak of the @xmath6 at 200 gev is slightly shifted to higher @xmath18 than at 62.4 gev . on the other hand , the @xmath191 of @xmath0mesons is more highly peaked at @xmath262.4 gev and the same higher peak is seen in the @xmath6 of single electrons in the lower panel ( b ) of fig . [ raa3e ] . as for the @xmath6 at @xmath219.2 gev , it is peaked at lower @xmath18 for @xmath0mesons as well as for single electrons because of the smaller transverse flow . with a focus to high @xmath18 , the @xmath6 for single electrons increases slightly with increasing collision energy . it is partly attributed to the stronger flow and the harder spectrum of initial heavy flavors at higher collision energies . additionally , in the case of the single electron @xmath6 the contribution from @xmath1meson decay , which dominates at high @xmath18 , increases with collision energy . [ h ] if a heavy quark is produced in the corona region , it will escape from the interaction zone produced in heavy - ion collisions . we show in fig . [ interaction ] ( by the solid line ) the percentages of heavy quarks with no interactions with other partons in 0 - 10 % central au+au collisions as a function of collision energy . the percentages are 3.7 , 6.4 , and 14.2 % at @xmath2200 , 62.4 , and 19.2 gev , respectively , demonstrating the decreasing role of the corona with bombarding energy that goes along with an increasing partonic fraction of the fireball. since parton coalescence is some kind of medium effect not existing in p+p collisions we should interpret it as being generated by interactions with the medium . the dashed line in the figure is the percentage of heavy flavors with no interactions in the qgp and which are additionally hadronized by fragmentation . the percentages drop down from 9.7 to 4.1 and 2.6 % , respectively . this again demonstrates the effects from a larger ( and longer ) qgp phase of the interaction zone with increasing bombarding energy . finally we analyze the azimuthal angle between the transverse momentum of a heavy - flavor meson and that of an antiheavy - flavor meson for each heavy flavor pair before and after the interactions with the medium in relativistic heavy - ion collisions . it is suggested that the analysis of the azimuthal angular correlation might provide information on the energy loss mechanism of heavy quarks in the qgp @xcite because stronger interactions should result in less pronounced angular correlations . since in the phsd we can follow up the fate of an initial heavy quark - antiquark pair throughout the partonic scatterings , the hadronization and final hadronic rescatterings , the microscopic calculations allow to shed some light on the correlation between the in - medium interactions and the final angular correlations . [ h ] and @xmath192 quarks ( upper dashed ) and final @xmath107 and @xmath193 mesons ( upper solid ) and that of initial @xmath194 and @xmath195 quarks ( lower dashed ) and final @xmath109 and @xmath196 mesons ( lower solid ) at midrapidity ( @xmath197 ) in 0 - 10 % central au+au collisions at @xmath2200 gev.,width=9 ] and @xmath192 quarks ( upper dashed ) and final @xmath107 and @xmath193 mesons ( upper solid ) and that of initial @xmath194 and @xmath195 quarks ( lower dashed ) and final @xmath109 and @xmath196 mesons ( lower solid ) at midrapidity ( @xmath197 ) in 0 - 10 % central au+au collisions at @xmath2200 gev.,width=9 ] fig . [ corr ] shows the azimuthal angular distributions of charm and bottom pairs , respectively , in the upper ( a ) and lower ( b ) panels at midrapidity ( @xmath197 ) in 0 - 10 % central au+au collisions at @xmath2200 gev . the dashed lines are the correlations before the interactions with the medium in heavy - ion collisions and the solid lines are those after freeze out of the final heavy mesons . the initial azimuthal correlation of charm pairs produced by the pythia event generator is far from back - to - back due to the associated production of further quark - antiquark pairs . the distribution in the azimuthal angles spreads widely from @xmath198 to @xmath108 although slightly more populated close to @xmath199 ( back - to - back ) . we recall that if a heavy - quark pair is produced to the leading order in pqcd , the heavy quark and heavy antiquark are back - to - back ( or close to it ) in the transverse plane , assuming the transverse momentum of partons producing the pair to be small . however , the pythia event generator also takes into account the gluon splitting @xmath200 which is populated near @xmath199 and the heavy quark excitation @xmath201 where a heavy quark or heavy antiquark is produced from the parton distribution of the colliding nucleon @xcite . as a result , the initial charm pairs from pythia have a mild dependence on @xmath9 at the energy @xmath2200 gev . in the case of bottom quarks , however , the initial pairs from pythia are manifestly peaked near @xmath199 , as shown in the lower panel ( b ) of fig . [ corr ] , because the contribution from gluon splitting and heavy quark excitation is small compared to charm at @xmath2200 gev . [ h ] and @xmath192 quarks ( upper dashed ) and final @xmath107 and @xmath193 mesons ( upper solid ) and that of initial back - to - back @xmath194 and @xmath195 quarks ( lower dashed ) and final @xmath109 and @xmath196 mesons ( lower solid ) at midrapidity ( @xmath197 ) in 0 - 10 % central au+au collisions at @xmath2200 gev.,width=9 ] and @xmath192 quarks ( upper dashed ) and final @xmath107 and @xmath193 mesons ( upper solid ) and that of initial back - to - back @xmath194 and @xmath195 quarks ( lower dashed ) and final @xmath109 and @xmath196 mesons ( lower solid ) at midrapidity ( @xmath197 ) in 0 - 10 % central au+au collisions at @xmath2200 gev.,width=9 ] after the initial production the heavy flavor partons strongly interact with the medium in relativistic heavy - ion collisions . [ interaction ] shows that more than 97 % of the heavy quarks interact by scattering or coalescence with other partons at @xmath2200 gev . accordingly , the azimuthal angular correlation between the initial heavy quark and heavy antiquark is washed out to a large extend . in order to investigate the effect of the in - medium interactions on the initial heavy quark - antiquark correlations we have performed a model study where the initial heavy - quark pairs are always produced back - to - back ( @xmath199 ) . although the initial correlations are all located at @xmath199 , the correlations for charm pairs disappear after the interactions with the medium in heavy - ion collisions as seen from the upper panel ( a ) of fig . [ corrbb ] . however , the lower panels ( b ) of fig . [ corr ] and [ corrbb ] show that the initial angular correlation of bottom pairs survives to some extend , because the bottom quark is too heavy to change the direction of motion in elastic scattering . in the upper panel of fig . [ corr ] ( a ) , we can see that the azimuthal angular correlation is remarkably enhanced near @xmath202 , which implies that the @xmath107 and @xmath193 mesons , which are produced as one pair in the initial stage , move in a similar transverse directions at freeze out . a possible reason for this behavior is the transverse flow : a pair of charm and anticharm quarks are produced at the same point through a nucleon - nucleon binary collision . in case the scattering cross sections of charm and anticharm quarks are large such that they are stuck in the medium and not separated far from each other , they will be affected by similar transverse flows , because the flow will depend on the position of the particles . only in case the charm and anticharm quarks are separated far enough to be located at completely different transverse positions until the transverse flow is generated in heavy - ion collisions , the collective flows of charm and anticharm quarks , respectively , will be independent . [ h ] and @xmath192 quarks ( upper dashed ) and final @xmath107 and @xmath193 mesons ( upper solid ) and that of initial @xmath194 and @xmath195 quarks ( lower dashed ) and final @xmath109 and @xmath196 mesons ( lower solid ) at midrapidity ( @xmath197 ) in 0 - 10 % central au+au collisions at @xmath2200 gev where the initial transverse position of the heavy antiquark is opposite to that of the heavy quark in order to investigate the flow effect on the angular correlation.,width=9 ] and @xmath192 quarks ( upper dashed ) and final @xmath107 and @xmath193 mesons ( upper solid ) and that of initial @xmath194 and @xmath195 quarks ( lower dashed ) and final @xmath109 and @xmath196 mesons ( lower solid ) at midrapidity ( @xmath197 ) in 0 - 10 % central au+au collisions at @xmath2200 gev where the initial transverse position of the heavy antiquark is opposite to that of the heavy quark in order to investigate the flow effect on the angular correlation.,width=9 ] in order to investigate in particular the flow effect on the angular correlation , we reflect the transverse position of the initial heavy antiquark with respect to the origin in fig . [ corrxy ] . since it is a reflection of the transverse position , the initial azimuthal angular correlation in momentum space does not change . however , we find that the distribution of azimuthal angles between the final @xmath107 and @xmath193 mesons from the same pair is completely opposite to that without the reflection , which are shown , respectively , in the upper panels of fig . [ corr ] and [ corrxy ] . if the charm and anticharm quarks from one pair can move a considerable distance and be separated far enough from each other before the transverse flow is developed , the results with and without the reflection should be similar . but the results in fig . [ corrxy ] indicate that the interaction of charm and anticharm quarks with the medium is strong and they get stuck in the nuclear matter and flow together . accordingly , the results in the upper panel of fig . [ corr ] indicating that the charm and anticharm quarks from one pair exhibit similar flows depending on position are naturally explained due to common collective flow . we have studied single electron production through the semileptonic decay of heavy mesons in relativistic heavy - ion collisions at @xmath2200 , 62.4 , and 19.2 gev within the phsd transport approach . the ratio of the initial scattering cross section for bottom production to that for charm production at these collision energies is less than 1 % . however , since the @xmath18 spectrum of bottom quarks is harder than that of charm quarks and the single electrons from @xmath1meson decay is much more energetic than that from @xmath0meson decay , it is essential to take @xmath1meson production into account in order to study the single electron production , especially at high @xmath18 . the parton - hadron - string dynamics ( phsd ) approach has been employed since it successfully describes @xmath0meson production in relativistic heavy - ion collisions at rhic and lhc energies @xcite . in this work , we have extended the phsd to @xmath1meson production and compared single electron production from heavy - meson decays with the experimental data from the phenix collaboration , because there are no experimental data exclusively for @xmath1mesons at the rhic energies . in analogy to the charm quark pairs , the bottom pairs are produced by using the pythia event generator which is tuned to reproduce the @xmath18 spectrum and rapidity distribution of bottom quark pairs from the fonll calculations . the ( anti)shadowing effect , which is the modification of the nucleon parton distributions in a nucleus , is implemented by means of the eps09 package . we have found that the ( anti)shadowing effect is not so strong at rhic energies as compared to lhc energies @xcite . the charm and bottom partons - produced by the initial hard nucleon - nucleon scattering - interact with the massive quarks and gluons in the qgp by using the scattering cross sections calculated in the dynamical quasi - particle model ( dqpm ) which reproduces heavy - quark diffusion coefficients from lattice qcd calculations at temperatures above the deconfinement transition . when approaching the critical energy density for the phase transition from above , the charm and bottom ( anti)quarks are hadronized into @xmath0 and @xmath1mesons through the coalescence with light ( anti)quarks . those heavy quarks , which fail in coalescence until the local energy density is below 0.4 @xmath111 , hadronize by fragmentation as in p+p collisions . the hadronized @xmath0 and @xmath1mesons then interact with light hadrons in the hadronic phase with cross sections that have been calculated in an effective lagrangian approach with heavy - quark spin symmetry . finally , after freeze - out of the @xmath0 and @xmath1mesons they produce single electrons through semileptonic decays with the branching ratios given by the particle data group ( pdg ) . we have found that the coalescence probability for bottom quarks is still large at high @xmath18 compared to charm quarks , and the @xmath6 of @xmath1mesons is larger than that of @xmath0mesons at the same ( high ) @xmath18 . however , this can dominantly be attributed to the much larger mass of the bottom quark . if the coalescence probability and the @xmath6 are expressed as a function of the transverse velocity of the heavy quark , both charm and bottom coalescence become similar since both are comoving with the neighbouring light antiquarks . furthermore , we found that the phsd approach can roughly reproduce the experimental data on single electron production in au+au collisions at @xmath2200 and the elliptic flow of electrons at @xmath262.4 gev from the phenix collaboration . however , the @xmath6 at @xmath262.4 gev is clearly underestimated which presently remains as an open puzzzle . we have additionally made predictions for @xmath0meson and single electron production in au+au collisions at @xmath819.2 gev which can be controlled by experiment in future . finally , we have studied the medium modifications of the azimuthal angular correlation of heavy - flavor pairs in central au+au collisions at @xmath2200 gev . here it has been found that the initial azimuthal angular correlation of charm pairs is completely washed out during the evolution of the heavy - ion collision , even in case they are assumed to be initially produced back - to - back . this decoherence could be traced back to the transverse flow which drives charm pairs ( close in space ) into the same direction such that the azimuthal angular correlation is enhanced around @xmath202 . by considering that the direction of the transverse flow essentially depends on position , the charm and anticharm quarks from each pair apparently are not sufficiently separated from each other before the transverse flow is developed . this decorrelation thus can be attributed to the strong interactions of charm with the medium produced in relativistic heavy - ion collisions . on the other hand , the focussing of pairs around @xmath202 is not observed for bottom pairs at rhic energies due to their significantly higher mass which prevents the bottom quarks ( and mesons ) to change their momentum substantially in the scattering processes . the authors acknowledge inspiring discussions with j. aichelin , s. brodsky , p. b. gossiaux , and p. moreau . this work was supported by dfg under contract br 4000/3 - 1 and by the loewe center `` hic for fair '' . the computational resources have been provided by the loewe - csc . lt acknowledges support from the ramn y cajal research programme and fpa2013 - 43425-p grant from ministerio de economia y competitividad . jmtr acknowledges the financial support from a helmholtz young investigator group vh - ng-822 from the helmholtz association and gsi . dc acknowledges support from grant nr . fis2014 - 51948-c2 - 1-p from ministerio de economia y competitividad , spain . c. bernard _ et al . _ [ milc collaboration ] , phys . d * 71 * , 034504 ( 2005 ) . y. aoki , g. endrodi , z. fodor , s. d. katz , k. k. szabo , nature * 443 * , 675 ( 2006 ) . a. bazavov , t. bhattacharya , m. cheng , c. detar , h. t. ding , s. gottlieb , r. gupta , p. hegde _ et al . _ , phys . d * 85 * , 054503 ( 2012 ) . b. abelev _ et al . _ [ alice collaboration ] , jhep * 1209 * , 112 ( 2012 ) . b. abelev _ et al . _ [ alice collaboration ] , phys . lett . * 111 * , 102301 ( 2013 ) . g. d. moore and d. teaney , phys . rev . c * 71 * , 064904 ( 2005 ) . b. zhang , l. w. chen and c. m. ko , phys . c * 72 * , 024906 ( 2005 ) . d. molnar , eur . j. c * 49 * , 181 ( 2007 ) . o. linnyk , e. l. bratkovskaya and w. cassing , int . j. mod . e * 17 * , 1367 ( 2008 ) . b. gossiaux , j. aichelin , t. gousset and v. guiho , j. phys . g * 37 * , 094019 ( 2010 ) . m. nahrgang , j. aichelin , p. b. gossiaux and k. werner , phys . rev . c * 90 * , 024907 ( 2014 ) . m. he , r. j. fries and r. rapp , phys . c * 86 * , 014903 ( 2012 ) . m. he , r. j. fries and r. rapp , phys . lett . * 110 * , 112301 ( 2013 ) . m. he , r. j. fries and r. rapp , phys . c * 91 * , 024904 ( 2015 ) . j. uphoff , o. fochler , z. xu and c. greiner , phys . c * 84 * , 024908 ( 2011 ) . j. uphoff , o. fochler , z. xu and c. greiner , phys . b * 717 * , 430 ( 2012 ) . s. cao and s. a. bass , phys . c * 84 * , 064902 ( 2011 ) . s. cao , arxiv:1511.04477 [ nucl - th ] . s. k. das , f. scardina , s. plumari , and v. greco , phys . b * 747 * , 260 ( 2015 ) . e. l. bratkovskaya , w. cassing , v. p. konchakovski and o. linnyk , nucl . phys . a * 856 * , 162 ( 2011 ) . v. p. konchakovski _ et al . _ , j. phys . g * 42 * , 055106 ( 2015 ) ; phys . c * 85 * , 011902 ( 2012 ) , phys . rev . c * 85 * , 044922 ( 2012 ) phys . c * 90 * , 014903 ( 2014 ) . t. song , h. berrehrah , d. cabrera , j. m. torres - rincon , l. tolos , w. cassing and e. bratkovskaya , phys . c * 92 * , 014910 ( 2015 ) . t. song , h. berrehrah , d. cabrera , w. cassing and e. bratkovskaya , phys . c * 93 * , 034906 ( 2016 ) . t. sjostrand , s. mrenna and p. z. skands , jhep * 0605 * , 026 ( 2006 ) . m. cacciari , s. frixione , n. houdeau , m. l. mangano , p. nason and g. ridolfi , jhep * 1210 * , 137 ( 2012 ) . l. adamczyk _ et al . _ [ star collaboration ] , phys . * 113 * , 142301 ( 2014 ) . d. tlusty [ star collaboration ] , nucl . a * 904 - 905 * , 639c ( 2013 ) . j. adam _ et al . _ [ alice collaboration ] , jhep * 1603 * , 081 ( 2016 ) . b. b. abelev _ et al . _ [ alice collaboration ] , phys . c * 90 * , 034904 ( 2014 ) . a. adare _ et al . _ [ phenix collaboration ] , phys . lett . * 98 * , 172301 ( 2007 ) . a. adare _ et al . _ [ phenix collaboration ] , phys . c * 91 * , 044907 ( 2015 ) . c. peterson , d. schlatter , i. schmitt and p. m. zerwas , phys . d * 27 * , 105 ( 1983 ) . a. bodek _ et al . _ , phys . lett . * 50 * , 1431 ( 1983 ) . k. j. eskola , h. paukkunen and c. a. salgado , jhep * 0904 * , 065 ( 2009 ) . w. cassing , eur . j. st * 168 * , 3 ( 2009 ) ; nucl . a * 795 * , 70 ( 2007 ) . h. berrehrah , e. bratkovskaya , w. cassing , p. b. gossiaux , j. aichelin and m. bleicher , phys . c * 89 * , 054901 ( 2014 ) . h. berrehrah , p. b. gossiaux , j. aichelin , w. cassing and e. bratkovskaya , phys . c * 90 * , 064906 ( 2014 ) . h. berrehrah , e. bratkovskaya , t. steinert and w. cassing , int . journal of mod . e * 25 * , 1642003 ( 2016 ) . h. berrehrah , m. nahrgang , t. song , v. ozvenchuck , p. b. gossiaux , k. werner , e. bratkovskaya and j. aichelin , arxiv:1604.02343 [ hep - ph ] . t. song and h. berrehrah , arxiv:1601.04449 [ nucl - th ] . c. garcia - recio , v. k. magas , t. mizutani , j. nieves , a. ramos , l. l. salcedo and l. tolos , phys . d * 79 * , 054004 ( 2009 ) . l. m. abreu , d. cabrera , f. j. llanes - estrada and j. m. torres - rincon , annals phys . * 326 * , 2737 ( 2011 ) . o. romanets , l. tolos , c. garcia - recio , j. nieves , l. l. salcedo and r. g. e. timmermans , phys . d * 85 * , 114032 ( 2012 ) . l. m. abreu , d. cabrera and j. m. torres - rincon , phys . d * 87 * , 034019 ( 2013 ) . c. garcia - recio , j. nieves , o. romanets , l. l. salcedo and l. tolos , phys . d * 87 * , 034032 ( 2013 ) . c. garcia - recio , j. nieves , o. romanets , l. l. salcedo and l. tolos , phys . d * 87 * , 074034 ( 2013 ) . l. tolos and j. m. torres - rincon , phys . d * 88 * , 074019 ( 2013 ) . j. m. torres - rincon , l. tolos and o. romanets , phys . d * 89 * , no . 7 , 074042 ( 2014 ) . l. tolos , int . j. mod . e * 22 * , 1330027 ( 2013 ) .

we study the single electron spectra from @xmath0 and @xmath1meson semileptonic decays in au+au collisions at @xmath2200 , 62.4 , and 19.2 gev by employing the parton - hadron - string dynamics ( phsd ) transport approach that has been shown to reasonably describe the charm dynamics at relativistic - heavy - ion - collider ( rhic ) and large - hadron - collider ( lhc ) energies on a microscopic level . in this approach the initial charm and bottom quarks are produced by using the pythia event generator which is tuned to reproduce the fixed - order next - to - leading logarithm ( fonll ) calculations for charm and bottom production . the produced charm and bottom quarks interact with off - shell ( massive ) partons in the quark - gluon plasma with scattering cross sections which are calculated in the dynamical quasi - particle model ( dqpm ) that is matched to reproduce the equation of state of the partonic system above the deconfinement temperature @xmath3 . at energy densities close to the critical energy density ( @xmath4 0.5 gev/@xmath5 ) the charm and bottom quarks are hadronized into @xmath0 and @xmath1mesons through either coalescence or fragmentation . after hadronization the @xmath0 and @xmath1 mesons interact with the light hadrons by employing the scattering cross sections from an effective lagrangian . the final @xmath0 and @xmath1 mesons then produce single electrons through semileptonic decay . we find that the phsd approach well describes the @xmath6 and elliptic flow @xmath7 of single electrons in au+au collisions at @xmath8 200 and the elliptic flow at @xmath2 62.4 gev from the phenix collaboration , however , the large @xmath6 at @xmath2 62.4 gev is not described at all . furthermore , we make predictions for the @xmath6 of @xmath0mesons and of single electrons at the lower energy of @xmath2 19.2 gev . additionally , the medium modification of the azimuthal angle @xmath9 between a heavy quark and a heavy antiquark is studied . we find that the transverse flow enhances the azimuthal angular distributions close to @xmath10 0 because the heavy flavors strongly interact with nuclear medium in relativistic heavy - ion collisions and almost flow with the bulk matter .

1605.07887

the shape of the stellar velocity ellipsoid , defined by @xmath5 , @xmath6 , and @xmath7 , provides key insights into the dynamical state of a galactic disk : @xmath7:@xmath5 provides a measure of disk heating and @xmath6:@xmath5 yields a check on the validity of the epicycle approximation ( ea ) . additionally , @xmath5 is a key component in measuring the stability criterion and in correcting rotation curves for asymmetric drift ( ad ) , while @xmath7 is required for measuring the disk mass - to - light ratio . the latter is where the diskmass survey focuses ( verheijen et al . 2004 , 2005 ) ; however , in anything but face - on systems , @xmath7 must be extracted via decomposition of the line - of - sight ( los ) velocity dispersion . below , we present such a decomposition for two galaxies in the diskmass sample : ngc 3949 and ngc 3982 . previous long - slit studies ( e.g. , shapiro et al . 2003 and references therein ) acquired observations along the major and minor axes and performed the decomposition via the ea and ad equations ; using both dynamical equations overspecifies the problem such that ad is often used as a consistency check . here , use of the sparsepak ( bershady et al . 2004 , 2005 ) integral field unit ( ifu ) automatically provides multiple position angles , thereby increasing observing efficiency and ensuring signal extraction along the desired kinematic axes . long - slit studies have also used functional forms to reduce the sensitivity of the above decomposition method to noise . here , only measures of the los velocity dispersions within a 40@xmath0 wedge about the major axis are used to perform the decomposition by incorporating both the ea and ad equations under some simplifying assumptions . velocities and radii within the wedge are projected onto the major axis according to derived disk inclinations , @xmath8 , and assuming near circular motion . in the end , our method requires neither fitted forms nor error - prone interpolation between the major and minor axes . future work will compare this decomposition method with the multi - axis long - slit method and investigate effects due to use of points off the kinematic axes . following derivations in binney & tremaine ( 1987 ) and assuming ( 1 ) ea holds , ( 2 ) the velocity ellipsoid shape and orientation is independent of @xmath9 ( @xmath10 ) , ( 3 ) both the space density , @xmath11 , and @xmath5 have an exponential fall off radially with scale lengths of @xmath12 and @xmath13 , respectively , and ( 4 ) the circular velocity is well - represented by the gaseous velocity , @xmath14 , the equation for the ad of the stars becomes @xmath15 , where @xmath16 is the mean stellar rotation velocity ; hence , @xmath5 is the only unknown . the third assumption requires mass to follow light , @xmath17 , and constant velocity ellipsoid axis ratios with radius ; @xmath18 is the surface density . the major - axis dispersion is geometrically given by @xmath19 , where @xmath20 is constant with radius . finally , ea , @xmath21 , completes a full set of equations for decomposition of the velocity ellipsoid . data for testing of the above formalism was obtained during sparsepak commissioning ( bershady et al . 2005 , see table 1 ) . both ngc 3949 and ngc 3982 were observed for @xmath22s at one ifu position with @xmath23 nm and @xmath24 or 26 km s@xmath25 . the velocity distribution function ( vdf ) of both gas and stars is parameterized by a gaussian function . in each fiber with sufficient signal - to - noise , the gaseous vdf is extracted using fits to the [ oiii ] emission line ; the stellar vdf is extracted using a modified cross - correlation method ( tonry & davis 1979 ; statler 1995 ) with hr 7615 ( k0iii ) as the template ( westfall et al . 2005 ) . the pointing of the ifu on the galaxy is determined _ post factum _ to better than 1 " by minimizing the @xmath26 difference between the fiber continuum flux and the surface brightness profile . subsequent galactic coordinates have been deprojected according to the kinematic @xmath8 and position angle . figure 1 shows los dispersions for both the gas , @xmath27 , and stars , @xmath28 ; data points are given across the full field of the ifu with points along the major and minor axes and in between having different symbols ( see caption ) . from this figure note ( 1 ) there is no significant difference in @xmath29 along the major and minor axes for ngc 3949 and ( 2 ) the large gas dispersion within @xmath30 " for ngc 3982 is a result of poor single gaussian fits to the multiple dynamical components of its liner nucleus . figure 2 gives the folded gaseous and stellar rotation curves and compares the measured @xmath1 from figure 1 with that calculated using the formalism from 1 . the value of @xmath31 used in figure 2 provides the minimum difference between the two sets of data ( as measured by @xmath32 ; see figure 3a ) . we find @xmath33 and @xmath34 for ngc 3949 and ngc 3982 , respectively ; errors are given by 68% confidence limits . a comparison of these values to the summary in shapiro et al . ( 2003 ) is shown in figure 3b . the disk of ngc 3982 is similar to other sb types studied ; however , ngc 3949 seems to have an inordinately hot disk . the latter , while peculiar , is also supported by the indifference between its major and minor axis @xmath29 from figure 1 and the same indifference seen in the caii data presented by bershady et al . our streamlined velocity - ellipsoid decomposition method appears accurate , as seen by comparison with ( 1 ) galaxies of a similar type for ngc 3982 , and ( 2 ) data in a different spectral region for ngc 3949 . 1 bershady , m. , verheijen , m. , andersen , d. 2002 , in disks of galaxies : kinematics , dynamics and perturbations , eds . e. athanassoula & a. bosma , asp conference series , 275 , 43 bershady , m. a. et al . 2004 , pasp , 116 , 565 bershady , m. a. et al . 2005 , apjs , 156 , 311 binney , j. & tremaine , s. 1987 , _ galactic dynamics _ ( princeton university press : princeton , nj ) shapiro , k. l. , gerssen , j. , & van der marel , r. p. 2003 , aj , 126 , 2707 staler , t. 1995 , aj , 109 , 1371 tonry , j. & davis , m. 1979 , aj , 84 , 1511 verheijen , m. a. w. et al . 2004 , an , 325 , 151 verheijen , m. et al . 2005 , these proceedings westfall , k. b. et al . 2005 , in prep .

we present the decomposition of the stellar velocity ellipsoid using stellar velocity dispersions within a 40@xmath0 wedge about the major - axis ( @xmath1 ) , the epicycle approximation , and the asymmetric drift equation . thus , we employ no fitted forms for @xmath1 and escape interpolation errors resulting from comparisons of the major and minor axes . we apply the theoretical construction of the method to integral field data taken for ngc 3949 and ngc 3982 . we derive the vertical - to - radial velocity dispersion ratio ( @xmath2 ) and find ( 1 ) our decomposition method is accurate and reasonable , ( 2 ) ngc 3982 appears to be rather typical of an sb type galaxy with @xmath3 despite its high surface brightness and small size , and ( 3 ) ngc 3949 has a hot disk with @xmath4 .

astro-ph0508552

because neptune s atmosphere has a relatively low burden of aerosols , its reflected spectrum is strongly influenced by both rayleigh scattering and raman scattering by molecular hydrogen . rayleigh scattering induces polarization that can significantly modify the reflected intensity ( mishchenko 1994 ) , accurate computation of which presents the very large burden of solving the vector radiation transfer equation . sromovsky ( 2004 ) discusses that problem and a new approximation method applicable to low phase angles . accurate treatment of raman scattering is also a computational burden because photons incident at one wavelength lose some energy to rotating and/or vibrating the hydrogen molecule and reappear at longer wavelengths . computation of reflectivity at one wavelength thus requires accounting for contributions from raman scattering at shorter wavelengths . in addition , because the raman source function varies continuously with optical depth , otherwise homogeneous layers become inhomogeneous , requiring many more layers to achieve an accurate characterization of the atmosphere . to avoid the computational burden of rigorous raman scattering calculations , several different approximations have been employed . baines and smith ( 1994 ) used an approximation suggested by wallace ( 1972 ) , in which the raman cross sections for rotational transitions are treated as conservative scattering because the wavelength shifts are relatively small , while the cross section for the vibrational transition , which involves a much larger wavelength shift , is treated as an absorption . this approximation does not produce the sharp spectral features characteristic of raman scattering and is of uncertain accuracy . pollack ( 1986 ) used an alternate approximation in which the rayleigh scattering cross section at a given wavelength is scaled by the solar irradiance ratio at the shifted and unshifted wavelengths . this approximation does produce raman spectral features , but the accuracy is not well known , and can create conservation problems by allowing single - scattering albedo values exceeding unity . karkoschka ( 1994 ) presented a method for correcting observations to remove raman scattering and for converting calculations that ignored raman scattering to spectra that approximately matched spectra that included raman scattering . that method can add or remove raman spectral features and was applied to observations of saturn , jupiter , uranus , and neptune , but was never tested for accuracy or generality . a number of calculations of neptune s geometric albedo have been made that do account for the basic physics of raman scattering . cochran and trafton ( 1978 ) implemented an iterative algorithm in which raman scattering is first treated as an absorption . after solving the scalar radiation transfer equation at each frequency grid point , the photon loss is computed from the radiation field . on the next iteration the lost photons are added back as source terms at the shifted wavelengths appropriate to each raman transition . they achieved convergence after three iterations . their mainly low resolution results are of limited utility however , because the atmospheric structure they assumed is so different from our current understanding . their claim that the residual intensity in the cores of the strong methane bands could be entirely explained by raman scattering will be shown to be invalid because of their assumed ch@xmath4 mixing ratio profile . the first model calculations displaying extensively detailed raman spectral features in neptune s atmosphere are those of courtin ( 1999 ) , who made use of the two - stream code of toon ( 1989 ) to speed the solution of the radiative transfer equation . but this method can deviate from exact solutions by 10 - 15% , does not account for polarization , and is not usable for studying center - to - limb variations . a more rigorous method was used by btremieux and yelle ( 1999 ) , based on the disort radiative transfer code ( stamnes 1988 ) . but they presented only results for jupiter and omitted polarization effects . this paper presents a new method for accurate computation of raman scattering that includes polarization in the context of neptune s atmosphere and makes preliminary applications of that method to resolve several significant issues . the next section reviews the basic physics of raman scattering . that is followed by a discussion of methods for accurate computation of raman scattering . sample computations are then presented to characterize the basic features of raman scattering on neptune . comparisons are made with various prior calculations , and with hst and groundbased observations , to assess the degree of haze absorption required in neptune s atmosphere . the final section evaluates past approximations , discusses how they can be generalized and improved , and presents the new approximation and its performance . the hydrogen molecule can exist in two nuclear spin states . ortho states have parallel nuclear spins and odd total angular momentum quantum numbers ( @xmath5 ... ) with a degeneracy of @xmath6 . the para states have antiparallel nuclear spins and even angular momentum quantum numbers ( @xmath7 .. ) with a degeneracy of @xmath8 . the equilibrium population of these states follows the boltzmann distribution , so that the fraction of molecules with angular momentum @xmath9 is given by @xmath10 where @xmath11 is the degeneracy , @xmath12 is the energy above the ground state , @xmath13 is the boltzmann constant , and @xmath14 is absolute temperature . using published expressions for the energy levels ( farkas 1935 ; massie and hunten 1982 ) we obtain the fractional populations given in table 1 , where @xmath15 is the total fraction of molecules with even @xmath9 and @xmath16 is the total fraction of molecules with odd @xmath9 . this assumes that ortho and para states can exchange energy . however , the time scale for equilibration by means of bimolecular collisions is of the order of years ( massie and hunten 1982 ) and radiative energy exchange is forbidden by selection rules . thus it is also meaningful to consider a different kind of equilibrium condition in which ortho states equilibrate separately from para states . this is relevant when hydrogen equilibrates at high temperature , then is lifted by convection to higher altitude where the fraction of para and ortho molecules remain fixed ( over short time scales ) but equilibration of energy levels within each sub - population does take place . for `` normal '' h@xmath0 , which is defined by the high temperature equilibrium value of @xmath17 , the sub - population distributions are given by @xmath18 where the partition functions @xmath19 and @xmath20 are summations of @xmath21 carried out over even @xmath9 and odd @xmath9 respectively . this distribution is given in table 2 as a function of temperature . c c c c c c c c ' '' '' t(k ) & j=0 & j=1 & j=2 & j=3 & j=4 & f@xmath22 & f@xmath23 + ' '' '' 50 & 0.7704 & 0.2294 & 0.0001 & 0.0000 & 0.0000 & 0.771 & 0.229 + 75 & 0.5173 & 0.4798 & 0.0029 & 0.0000 & 0.0000 & 0.520 & 0.480 + 100 & 0.3747 & 0.6135 & 0.0115 & 0.0003 & 0.0000 & 0.386 & 0.614 + 125 & 0.2947 & 0.6784 & 0.0250 & 0.0018 & 0.0000 & 0.320 & 0.680 + 150 & 0.2450 & 0.7080 & 0.0410 & 0.0059 & 0.0000 & 0.286 & 0.714 + 175 & 0.2112 & 0.7178 & 0.0574 & 0.0135 & 0.0001 & 0.269 & 0.731 + 200 & 0.1865 & 0.7157 & 0.0729 & 0.0245 & 0.0004 & 0.260 & 0.740 + 225 & 0.1673 & 0.7061 & 0.0869 & 0.0387 & 0.0009 & 0.255 & 0.745 + 250 & 0.1520 & 0.6919 & 0.0990 & 0.0552 & 0.0017 & 0.253 & 0.747 + 275 & 0.1394 & 0.6749 & 0.1092 & 0.0732 & 0.0028 & 0.251 & 0.749 + 300 & 0.1287 & 0.6564 & 0.1177 & 0.0919 & 0.0043 & 0.251 & 0.749 + note : f@xmath24 and f@xmath25 denote f@xmath26 and @xmath16 respectively . [ tbl : eqpop ] c c c c c c c c ' '' '' t(k ) & j=0 & j=1 & j=2 & j=3 & j=4 & f@xmath22 & f@xmath23 + ' '' '' 50 & 0.2500 & 0.7500 & 0.0000 & 0.0000 & 0.0000 & 0.25 & 0.75 + 75 & 0.2486 & 0.7500 & 0.0014 & 0.0000 & 0.0000 & 0.25 & 0.75 + 100 & 0.2426 & 0.7496 & 0.0074 & 0.0004 & 0.0000 & 0.25 & 0.75 + 125 & 0.2305 & 0.7480 & 0.0195 & 0.0020 & 0.0000 & 0.25 & 0.75 + 150 & 0.2142 & 0.7438 & 0.0358 & 0.0062 & 0.0000 & 0.25 & 0.75 + 175 & 0.1965 & 0.7362 & 0.0534 & 0.0138 & 0.0001 & 0.25 & 0.75 + 200 & 0.1795 & 0.7251 & 0.0702 & 0.0249 & 0.0004 & 0.25 & 0.75 + 225 & 0.1640 & 0.7109 & 0.0851 & 0.0390 & 0.0008 & 0.25 & 0.75 + 250 & 0.1504 & 0.6944 & 0.0979 & 0.0554 & 0.0016 & 0.25 & 0.75 + 275 & 0.1386 & 0.6762 & 0.1086 & 0.0733 & 0.0028 & 0.25 & 0.75 + 300 & 0.1283 & 0.6570 & 0.1174 & 0.0920 & 0.0043 & 0.25 & 0.75 + note : f@xmath24 and f@xmath25 denote f@xmath26 and @xmath16 respectively . [ tbl : normpop ] the populations for both equilibrium and normal hydrogen are plotted in fig . [ fig : parapop ] , both as a function of temperature and as a function of pressure in neptune s atmosphere . at t @xmath27 300 k , @xmath15 approaches 0.25 and the ortho / para ratio approaches the 3/1 ratio expected from the nuclear spin degeneracy . but at the low temperatures in the upper troposphere and stratosphere of neptune @xmath15 can be much larger , and for pressures less than a few bars , only @xmath28 and @xmath29 ground states need to be considered . ( thick lines ) and for normal h@xmath0 ( thin lines ) . bottom : population fractions vs pressure in neptune s atmosphere.,width=355 ] radiative transitions must satisfy selection rules @xmath30 ( hollas 1992 ) ; the corresponding transitions are named @xmath31 , @xmath32 , and @xmath33 , where @xmath9 is the angular momentum quantum number of the initial state . the @xmath34 and @xmath35 branches can involve changes in rotational energy and vibrational energy , while the @xmath36 branch involves changes in only vibrational energy . for the @xmath34 branch interactions the scattered photons have more energy than the incident photons . but for the @xmath34 branch to play a significant role , there needs to be a significant population in rotational states with @xmath37 , which is not the case for the upper troposphere of neptune . thus , we here ignore the @xmath34 branch . the wavelength - dependent raman and rayleigh cross sections per molecule ( fig . [ fig : xcvswlen ] ) are computed using fits given by ford and browne ( 1973 ) . cross sections at 0.4 @xmath2 m are given in table 3 , along with transition energy energies expressed as corresponding wavenumber shifts . these cross sections for the @xmath35 and @xmath36 raman transitions are 46 - 68% larger than those given by cochran and trafton ( 1978 ) . ground states ( thick lines ) and for @xmath29 ground states ( thin lines ) , where @xmath38 refers to the @xmath39 , @xmath40 vibrational transition . bottom : ratio of raman to rayleigh cross sections.,width=355 ] c c c c c ' '' '' & & & & 0.4-@xmath2 m cross + transition(@xmath9 ) & @xmath41 & @xmath42 & @xmath43 , @xmath44 & section , @xmath45 + @xmath46 & 0&0 & 354.69 & 3.575@xmath4710@xmath48 + @xmath49 & 0&0 & 587.07 & 3.635@xmath4710@xmath48 + @xmath50 & + 2&0 & 354.69 & 1.104@xmath4710@xmath51 + @xmath52 & + 2&0 & 587.07 & 0.642@xmath4710@xmath51 + @xmath53 & 0&1 & 4162.06 & 0.344@xmath4710@xmath51 + @xmath54 & 0&1 & 4156.15 & 0.369@xmath4710@xmath51 + @xmath55 & + 2&1 & 4498.75 & 0.070@xmath4710@xmath51 + @xmath56 & + 2&1 & 4713.83 & 0.041@xmath4710@xmath51 + @xmath36 & & & 4161.00 & 0.412@xmath4710@xmath51 + [ tbl : xctable ] note in table 3 that the @xmath57 cross section sum is only 1% larger than @xmath58 sum at 0.4 @xmath2 m . fig.[fig : xcvswlen ] shows that this close match holds true at all wavelengths of interest . these transitions also all have similar wavenumber shifts . in fact , the shifts of @xmath53 and @xmath54 , which have the largest cross sections , differ by only 6 wavenumbers , which is several times smaller than the spectral resolution that we need to model existing observations . for these transitions we can thus follow courtin ( 1999 ) by ignoring @xmath15 and simply taking the effective cross section as the average of @xmath28 and @xmath29 transitions and using a single wavenumber shift of @xmath59 @xmath44 . the inclusion of the @xmath38 terms correctly accounts for the absorption of photons at the incident wavelength , but does not transfer the scattered photons to exactly the correct longer wavelength . however , this results in a negligible error . the effective scattering cross section per molecule for each transition depends on the fractional population of the initial state as well as the transition cross section . for neptune s the upper troposphere we can make the approximation that there are only two ground states : @xmath28 and @xmath29 . this means that @xmath15 can be taken as the fraction of @xmath28 molecules , and @xmath16 as the fraction of @xmath29 molecules . we can then write the total raman scattering cross section as @xmath60 \label{eq : sigtot}\end{aligned}\ ] ] where we will hereafter refer to the last term as the @xmath36 cross section , as if it were due to a single transition , and its four individual contributions will be assigned the same wavenumber shift of 4161 @xmath61 . the generalized scalar phase function for scattering of unpolarized incident radiation by anisotropic molecules for all types of molecular scattering ( including raman and rayleigh scattering ) can be written as ( placzek 1959 ; soris and evans 1999 ) : @xmath62\label{eq : phase1 } , \ ] ] where @xmath63 is the depolarization ratio , which is the ratio of perpendicular to parallel intensities scattered at 90 . the depolarization ratio is zero for isotropic particles ( the pure rayleigh case ) and unity for isotropic scatterers . for pure rotational raman scattering , @xmath64 ( soris and evans 1999 ) , yielding @xmath65 $ ] , which has a peak deviation of only 7.7% from isotropy . depolarization ratios for vibrational transitions for gases seem to be generally below about 0.38 ( bhagavantam 1942 ) . for the 4156 @xmath44 @xmath36-branch transition , measured values range from 0.045 ( cabannes and rousset 1936 ) to 0.13 ( bhagavantam 1931 ) . either value implies a phase function that is closer to pure rayleigh than to isotropic . however , the deviation from isotropic is greatest for the first scattering , for which the direct beam plays the largest role . backscattered @xmath36-branch light from the first scattering would be reduced by 30 - 33% using the assumption of isotropic scattering . but in sample calculations the reflected flux of raman scattered photons attributable to this transition is only 17% of the total ( see fig . [ fig : ramline1plus2 ] ) and about half of that comes from incident light that already been diffusely scattered . thus , the net effect of assuming isotropic scattering for the vibrational transition will probably never exceed 15% of the @xmath36-branch contribution to single scattering , and will generally be insignificant . thus we will assume isotropic scattering for both rotational and vibrational transitions to benefit from the simplifications that it permits . we follow evans and stephens ( 1991 ) and others in defining downward as positive , so that the direction of light propagation is defined by the vector ( @xmath66 , @xmath67 ) or ( @xmath2,@xmath67 ) , where @xmath66 is the angle from the inward normal at the top of the atmosphere and @xmath68 . the angle @xmath67 is the azimuth angle measured clockwise looking upward . we assume an unpolarized collimated incident solar beam in the direction ( @xmath69 , @xmath70 ) . light that is scattered backward toward the source then has the direction ( @xmath71 , @xmath72 ) . the stokes vector @xmath73 is defined as a 4-element column vector given by @xmath74\label{eq : stokesvec}\ ] ] where @xmath75 is the total intensity and @xmath76 is the intensity of polarized light . separate definitions for @xmath36 , @xmath77 , and @xmath78 are given by hansen and travis ( 1974 ) . with these definitions , the vector radiation transfer equation can be written as @xmath79 where @xmath80 denotes distance downward into the atmosphere , @xmath81 and @xmath82 are linear extinction and scattering coefficients , and where @xmath83 is the source vector due to raman scattering . equation [ eq : rtez ] is often written in terms of optical depth @xmath84 , where @xmath85 , and single - scattering albedo @xmath86 . however , because raman scattering transfers photons from one wavelength to another at a specific physical location in the atmosphere , the wavelength - independent distance @xmath80 is a more convenient vertical coordinate than @xmath84 . in eq.[eq : rtez ] the scattering matrix @xmath87 is a rotated form of the scattering phase matrix @xmath88 , given by @xmath89 where @xmath90 is the scattering angle , @xmath91 is the angle between the scattering plane and the meridional plane of the incoming ray , and @xmath92 is the angle between the scattering plane and the meridional plane of the outgoing ray . formulas for the scattering and rotation angles , and the form of the rotation matrix , are given by hansen and travis ( 1974 ) . in this formulation , the stokes vector @xmath73 includes direct beam as well as diffuse components of the radiation field , and thus there is no solar pseudo source as used by evans and stephens ( 1991 ) . there is an implicit wavelength dependence in eq . [ eq : rtez ] , but the wavelength dependence must be made explicit in the equation for the source vector:@xmath93 where we use wavenumber @xmath94 instead of wavelength because each raman transition is associated with a fixed shift in wavenumber but a variable shift in wavelength . here the quantity @xmath95 is the volume number density of h@xmath0 molecules , and @xmath96 is the cross section for photons at wavenumber @xmath97 to excite transition @xmath98 and exit at wavenumber @xmath99 , where @xmath100 is the energy change associated with the transition . the associated scattering matrix is @xmath101 . note that the vector intensity @xmath102 in the integrand is evaluated at the wavenumber of the incident photon , as are the cross section and scattering matrix in eq . [ eq : source ] . if @xmath73 is measured in photons , then the factor @xmath103 . if @xmath73 is measured in energy units , then @xmath104 . although the radiation transfer equation we use will retain polarization in general , we will ultimately not include polarization in computation of raman source contributions . there is also an extinction contribution from raman scattering , which is given by @xmath105 this is the sum over @xmath106 transitions of extinctions involving photons of incident wavenumber @xmath99 that experience raman scattering and emerge as photons of wavenumber @xmath107 . the actual energy loss at wavenumber @xmath99 due to extinction is proportional to the incident energy at wavenumber @xmath99 , but the source contribution at wavenumber @xmath99 depends on the incident energy at other higher wavenumbers . to solve the vector transfer equation we make use of fortran vector radiation code developed and documented by evans and stephens ( 1991 ) and further validated by sromovsky ( 2004 ) by comparisons with independent solutions of sweigart ( 1970 ) , dlugach and yanovitskij ( 1974 ) , and kattawar and adams ( 1971 ) . using an order-@xmath108 fourier expansion of the intensity and phase matrix , eq . [ eq : rtez ] is separated into @xmath109 uncoupled equations that can be solved independently . the azimuthal components are separately solved and then combined to obtain intensity fields . this separation is valid for randomly oriented particles with a plane of symmetry , so that the 16-element phase matrix has only six unique components ( hovenier 1969 ) . we also assume that the incident radiation is symmetric in @xmath67 and unpolarized . the zenith angle variation is discretized using a double - gauss numerical quadrature . the radiance at any location in the atmosphere is thus represented as a vector involving three components : stokes parameters , quadrature zenith angles , and azimuthal expansion mode . the intensity and source radiance vectors thus become vectors of @xmath110 elements , where @xmath111 is the number of zenith angle quadrature points:@xmath112 , \qquad \qquad \mathsf{j}= \left [ \begin{array}{c } \vec{\mathsf{j}}(\mu_1)\\ \vec{\mathsf{j}}(\mu_2 ) \\ \cdot \\ \cdot \\ \vec{\mathsf{j}}(\mu_{n_\mu } ) \end{array}\right ] , \label{eq : intvector}\ ] ] where each element @xmath113 or @xmath114 is a vector of the form given in eq.[eq : stokesvec ] . the radiance field is separated into upward and downward hemispheres : @xmath115 represents downward radiance ( @xmath116 ) and @xmath117 representing upward radiance ( @xmath118 ) . a model atmosphere is constructed by dividing the atmosphere into uniform sublayers , each of which is characterized by an optical depth , a single scattering albedo , and scattering phase function . the code uses differential generators derived from eq . [ eq : rtez ] and makes use of doubling to compute reflection and transmission matrices for homogeneous layers , which are then combined using standard adding equations . the key equations are those of the interaction principle ( goody and yung 1989 ) : @xmath119 where subscripts refer to the top ( @xmath120 ) and bottom ( @xmath121 ) boundaries of a layer , @xmath122 and @xmath123 are the reflection and transmission matrices , and @xmath124 is the source vector for a layer , which equals @xmath125 for a differential layer . using eq.[eq : interact ] , it is straightforward to derive the properties of a combined layer ( @xmath126 ) from the individual properties of the top ( @xmath14 ) and bottom ( @xmath127 ) layers ( evans and stephens 1991 ) : @xmath128^{-1}}\\ \mathbf{\gamma^- = [ 1-r_b^-r_t^+]^{-1}}\\ \end{array}\label{eq : adding}\ ] ] here the source adding equations refer to vector sources . the same equations can be shown to apply to the matrix sources that will be defined in a subsequent section . these can be converted to doubling equations by making the top and bottom properties equal . the first step in the solution is to convert eq . [ eq : rtez ] into a form that allows computation of the reflection and transmission matrices and the source vector . after azimuthal expansion and zenith angle discretization , the vector transfer equation for azimuthal order @xmath108 can be written in the form@xmath129 where the matrices and source vector here refer to a differential layer of thickness @xmath130 . the single - scattering approximation for the generator equations for reflection and transmission matrices are then@xmath131 where @xmath132 refers to stokes vector index , @xmath133 to zenith angle quadrature index , @xmath134 to the gaussian quadrature weight value , and where @xmath130 is is usually chosen so that @xmath135 , which evans and stephens ( 1991 ) have found to yield about 5 digits accuracy when using double precision . these expressions are different in form but equivalent to those given by evans and stephens . the differential generator for raman source terms can be derived from the single - scattering form of eq . [ eq : rtez ] and discretization of eq . [ eq : source ] , which results in the expression@xmath136 \label{eq : diffram},\ ] ] in which the individual matrix elements are given by @xmath137 where the first @xmath138 superscript indicates the sign of @xmath139 and the second indicates the sign of @xmath140 , and where the raman scattering coefficient per unit distance for transition @xmath98 is given by @xmath141 note that the source expressions depend on the wavenumbers of the incident photons ( @xmath142 ) that are shifted to the current wavenumber , while everything else in eq . [ eq : azeq ] depends on the current wavenumber @xmath99 . this is not a convenient form to work with because once the source vector is computed it applies to one specific incident irradiance direction , meaning that a center - to - limb scan would require many repetitions of the solution algorithm . it is also unnecessarily complex because it retains the full generality of the phase function , which is not needed to accurately approximate the effect of raman scattering . a simpler and more useful matrix formulation is developed in section[sec : smatrix ] . the solution is expressed in terms of reflection and source matrices for the entire atmosphere , from which intensities at desired irradiance and view angles are computed as described in section[sec : compint ] . a uniformly spaced wavenumber grid is used so that the number of photons that are raman scattered out of one bin will be transferred to exactly one other bin . this requires that the wavenumber shift for each transition to be close to an integral multiple of the grid spacing , as illustrated in table 4 . phase functions are introduced as lagrange function expansions of an order that depends on the number of quadrature angles . for sharply peaked forward scattering phase functions we avoid impractically large expansion orders , otherwise needed to avoid angular oscillations in the reflected intensity , by employing the @xmath63-fit method of hu ( 2000 ) . r c c c c c transitions : & @xmath50 & @xmath52 & @xmath143 + @xmath43 : & 354.39 @xmath44 & 587.07 @xmath44 & 4161.00 @xmath44 + ' '' '' grid spacing @xmath144 & & rms error & step sum + ' '' '' 12.240 @xmath44 & 28.953 ( 29 ) & 47.963 ( 48 ) & 339.951 ( 340 ) & 0.044 steps & 417 + 13.640 @xmath44 & 25.982 ( 26 ) & 43.040 ( 43 ) & 305.059 ( 305 ) & 0.042 steps & 374 + 17.780 @xmath44 & 19.932 ( 20 ) & 33.019 ( 33 ) & 234.027 ( 234 ) & 0.044 steps & 287 + 39.260 @xmath44 & 9.027 ( 9 ) & 14.953 ( 15 ) & 105.986 ( 106 ) & 0.032 steps & 130 + 58.620 @xmath44 & 6.046 ( 6 ) & 10.015 ( 10 ) & 70.983 ( 71 ) & 0.029 steps & 87 + 118.860 @xmath44 & 2.982 ( 3 ) & 4.939 ( 5 ) & 35.008 ( 35 ) & 0.037 steps & 43 + [ tbl : wngrid ] while the full polarization machinery of the evans and stephens ( 1991 ) code is retained in our modification , the raman scattering , as we represent it , does not itself introduce any polarization . further , because in most cases rayleigh scattering is the dominant process in creating polarization effects , we usually make use of an approximation that attributes all of the polarized intensity , given by @xmath76 , to the rayleigh scattering , and then use the algorithm described by sromovsky ( 2004 ) to correct the scalar calculation for polarization , rather than carry out the full vector calculation . the approximation has been shown to be accurate to generally better than 1% , and entails no significant additional computational burden , which otherwise would take about 40 times longer . a matrix formulation of the source function is used to facilitate the computation of center - to - limb variations in reflectivity . making the reasonable approximation that raman scattering is isotropic ( see sec . [ sec : ramphase ] ) , only the @xmath145 azimuthal component needs to include a raman contribution and we can write @xmath146 and simplify eq . [ eq : diffram ] to the form @xmath147 where the matrix @xmath148 produces the sum of the upward and downward intensities at @xmath80 by multiplication by the incident intensity at the top of the atmosphere ( @xmath149 ) at wavenumber @xmath142 . there is an @xmath98 subscript on this matrix because it depends on transmission , reflection , and source matrices at the shifted raman source wavenumber @xmath142 . expressing the source in terms of the top - of - atmosphere incident irradiance vector allows us the computational convenience of expressing the source term as a matrix multiplier that has the same structure as the reflection matrix , which will be demonstrated in what follows . the @xmath150 matrix can be derived from the interaction principle ( eq . [ eq : interact ] ) . the radiance at a given level in the atmosphere is proportional to the incident irradiance and can be expressed in terms of the properties of the entire atmosphere above that level ( the top layer ) and the properties below that level ( the bottom layer ) . we assume here that the transmission out of the bottom layer is zero , i.e. if the atmosphere is thin , the bottom layer includes any surface reflection and/or absorption . the relevant source vector for the top layer @xmath151 has multiple terms contributing from different raman transitions that are each linearly related to the irradiance at any single wavelength ( the source direction is the same at all wavelengths ) . this can be shown by using the source addition equations given in eq . [ eq : adding ] . if the sources for each layer to be combined are separately proportional to the top of atmosphere irradiance @xmath152 ( @xmath153 ) , then the source for the combined layer will also be proportional to @xmath152 . thus we can define source matrices @xmath154 and @xmath155 that satisfy @xmath156 using these definitions , it is easy to show that @xmath157 + \mathbf{s_b^-}\label{eq : gdef},\ ] ] which defines the matrix @xmath150 at level @xmath80 in terms of reflection , transmission , and source matrices for the entire atmosphere above @xmath80 ( subscript t for top layer ) and that below @xmath80 ( subscript b for bottom layer ) , all at wavenumber @xmath142 . the source vector at @xmath80 for the unshifted wavenumber @xmath99 can then be written as @xmath158 in which the final two forms define a new matrix @xmath159 and express the source contribution to the current wavenumber @xmath99 in terms of the solar irradiance at @xmath99 , using a scaling that is the ratio of solar spectral functions @xmath160 and @xmath161 , such that @xmath162\mathsf{i_0^+}(\nu)$ ] . if the radiance field is measured in terms of energy per unit wavenumber , then @xmath160 is the solar energy spectrum . if the radiance field is measured in photons per unit wavenumber , then @xmath160 is the solar photon spectrum . substituting the expression given by eq . [ eq : sl ] yields the computationally useful form @xmath163 \big[\frac{\nu}{\mu_j f_\odot(\nu)}\big ] \label{eq : srcmult}\end{aligned}\ ] ] in which we have here made a specific choice that @xmath160 is a solar spectrum in energy per unit wavenumber , and where @xmath164 is the only relevant value of the third index for the assumed unpolarized incident irradiance , and @xmath165 is the only relevant value of the first index for the assumed isotropic and raman scattering phase function , according to eq . [ eq : sl ] . the @xmath138 superscript on @xmath124 was dropped , and never used on @xmath159 , because both values are the same for a differential layer ( see eq . [ eq : simplesrc ] ) . note that the first bracketed factor depends on @xmath142 while the second bracketed factor depends only on wavenumber @xmath99 and @xmath80 , and is independent of @xmath98 . this form suggests the computational procedure described below . atmospheric layers that have uniform extinction coefficients and phase functions become nonuniform due to the vertical variation of the raman source contribution . this requires us to use a much larger number of layers than would be necessary were raman scattering not considered . we typically use 30 - 80 logarithmically spaced pressure levels between 0.0003 and 100 bars , with 80 levels required for accuracy of a few tenths of 1% . the accuracy achieved in any particular problem needs to be assessed by running test calculations using a larger number of levels . each raman solution requires a spectral calculation . following btremieux and yelle ( 1999 ) , we start at the highest wavenumber and work downward so that multiple raman scattering is automatically included in a single pass . at the highest wavenumber , the radiation transfer problem is solved with no source contribution , but with raman extinction included . at each wavenumber including the first , the @xmath166 matrix is computed at each layer bottom boundary @xmath167 , using eq.[eq : gdef ] . the algorithm then computes for each transition @xmath98 , each layer @xmath167 , and each possible incident quadrature index @xmath168 , the function @xmath169 which is essentially the same as the left bracketed factor in eq.[eq : srcmult ] divided by @xmath130 . this is essentially the photon loss term per unit distance for each layer and each transition . here everything is evaluated at the same wavelength and thus the @xmath166 matrix needs no @xmath98 dependence . this form uses the average of @xmath166 at the top and bottom of each layer to represent the radiation field everywhere within the layer . as the computation proceeds to longer wavelengths , the source contributions at the current wavenumber are then obtained by reading the stored values of @xmath170 and computing the source matrix as follows:@xmath171 \frac{\nu}{4\mu_j f_\odot(\nu ) } \delta z \label{eq : finalsrc}.\ ] ] for each wavenumber at which values of @xmath170 are read , the values for transition @xmath98 will have been stored during calculations made at wavenumber @xmath172 . once @xmath170 is read from storage and the matrix @xmath173 is computed , the computation of the radiation field proceeds with the assumption that the source is uniformly distributed over the layer @xmath167 . this is necessary to make use of the doubling equations for a uniform source , but is only a good approximation for layers that are relatively optically thin at the wavenumber of the incident photons . while the input source is assumed to be uniform throughout each layer , and is input as the differential source after dividing by the physical thickness of each layer , the doubling process does modify the source function in accord with the absorption and scattering processes that take place at the wavelength of the scattered raman photon . after computing the source matrix for the entire atmosphere using the same adding equation that is given for the source vector in eq.[eq : adding ] , we then compute the outward intensity at the top of the atmosphere . for arbitrary directions of incidence and reflection this is formally expressed as @xmath174 \nonumber \\ & & \times \left[\begin{array}{c } \mu_0 f_0/\pi \\ 0 \\ 0 \\ 0\end{array}\right ] = \mathbf{\bar{r}^-}\left[\begin{array}{c } \mu_0 f_0/\pi \\ 0 \\ 0 \\ 0\end{array}\right ] \label{eq : toaint}\end{aligned}\ ] ] where the incident flux is @xmath175 in direction ( @xmath176 ) and the combined matrix @xmath177 uses the minus superscript because direction does make a difference for the inhomogeneous layer consisting of the entire atmosphere . by converting the source function into a matrix we have thus put the raman contributions in the same form as the reflection contributions , allowing us to compute a pseudo reflection matrix @xmath178 and use the same mathematical machinery to compute spatially resolved intensity profiles . the azimuthal expansion of @xmath178 is given by @xmath179 where sine components of the expansion are zero by the assumed symmetry of the incident radiation field . since our solution only provides @xmath178 at quadrature points , we carry out a legendre polynomial interpolation to obtain intensities at other angles . how do we know that our computations are correct ? the basic radiation transfer algorithm without raman scattering has been compared with independent calculations for selected cases and shown to be in excellent agreement . this was done by evans and stevens ( 1991 ) for their original code and by sromovsky ( 2004 ) for the modification used here ( as noted in sec . [ sec : solmeth ] ) , but neither of these sets of comparisons deals with the raman scattering additions , which are harder to validate . a few comparisons with past calculations can be made , but most of these lack the resolution , relevance , or rigor that are needed to serve as appropriate standards of accuracy . they do provide a reasonable sanity check , however , and some useful comparisons are made in section [ sec : compcal ] . another validation is to show by means of test cases that the algorithm satisfies conservation of photons . this is done for the case of a monochromatic incident flux in section[sec : monoscat ] . conservation is unlikely to be achieved if the raman transfers are improperly computed . a last validation , though an indirect one , is to demonstrate that the algorithm can reproduce features in the observed spectra with reasonable atmospheric structure models ( sec . [ sec : obscomp ] ) . neptune s thermal structure is obtained from voyager radio occultation observations at ingress ( 45s ) and egress at 61n ) . we used the hinson and magalhes ( 1993 ) analysis for @xmath180 bar and the lindal ( 1992 ) results for 1 bar @xmath181 bars , with an offset of 1.0 k added to match the the hinson and magalhes profile at 1 bar . for @xmath182 bars , we extrapolated using the nearly adiabatic lapse rate at the 6-bar level ( @xmath183 -0.94 k / km ) . the two temperature profiles differ insignificantly except in the stratosphere . the lower egress temperatures in the tropopause region reduce the ch@xmath4 mixing ratio and opacity to a small degree that is only noticed near 0.89 @xmath2 m , where a 10% increase in geometric albedo is produced . we assume a fixed profile at all latitudes for radiation transfer analysis . our altitude scale is computed using gravitational acceleration at 45s ( @xmath184 m / s@xmath185 ) . the hinson and magalhes profile is derived assuming a gas composition of 81% h@xmath0 and 19% he ( conrath 1991 ) . it is possible that n@xmath0 may also be present at a mixing ratio as high as 0.3% ( conrath 1993 ) , in which case our altitude scale would be modified somewhat . the lindal profile assumed 2% ch@xmath4 in the troposphere , which is close to the currently accepted value of 2.2% ( baines 1995 ) that we use for computing ch@xmath4 opacity . the tropospheric mixing ratio is consistent with ch@xmath4 condensation at 1.4 bars , above which we assume the saturated vapor pressure until we reach the stratosphere , at which point the mixing ratio is the smaller of the saturated mixing ratio and the stratospheric limit of 3.5 @xmath186 ( baines and hammel 1994 ) . ( dashed ) , he ( dotted ) , and ch@xmath4 ( dot dash ) . the ch@xmath4 mixing ratio is 3.5 @xmath186 in the stratosphere , 0.022 in the deep troposphere , and otherwise follows the saturation vapor pressure curve . the horizontal dotted line at 1.43 bars indicates the point at which methane reaches the saturation vapor pressure in the troposphere.,width=355 ] the pressure at which the one - way vertical optical depth reaches unity for each opacity contributor , and for the total of all contributors is illustrated in the top panel of fig . [ fig : pendepth ] . the levels at which the total optical depth reaches values from 0.3 to 100 are displayed in the bottom panel . the penetration depth of sunlight into neptune s atmosphere is limited by rayleigh scattering at short wavelengths and by ch@xmath4 and h@xmath0 collision - induced absorption ( cia ) at long wavelengths . the deepest penetration is at 0.935 @xmath2 m , where there is a relatively clean ch@xmath4 window , a window in the cia spectrum , and a low rayleigh cross section . we used ch@xmath4 absorption coefficients derived by karkoschka ( 1994 ) from planetary geometric albedo observations using a technique described by karkoschka and tomasko ( 1992 ) . karkoschka estimates a 5% uncertainty in his 1994 absorption coefficients , plus an additional uncertainty in the continuum baseline : 0.0003 km - am@xmath187 at 400 nm and 0.02 km - am@xmath187 at 1000 nm ( a factor of 2 every 100 nm ) . this continuum uncertainty leads to uncertainties in the ch@xmath4 window regions that are important to determination of the scattering properties and pressure levels of clouds in the several bar range . there is also a likely bias error for weak ch@xmath4 bands superimposed on raman scattering effects due to the nonlinear relationship between i / f and single - scattering albedo ( this is discussed in sec . [ sec : kcorrect ] ) . the cia values are obtained by interpolating tables of pressure and temperature dependencies provided by alexandra borysow , and available at the atmospheres node of nasa s planetary data system . the average rayleigh scattering cross section per molecule was computed using the equation @xmath188 from hansen and travis ( 1974 ) , where @xmath189 is the volume mixing ratio of the @xmath132th gas , @xmath190 is its refractive index , and @xmath191 is its depolarization factor . we used depolarization ratios of 0.0221 for h@xmath0 and 0.025 for he ( penndorf 1957 ; parthasarathy 1951 ) and refractive index values from allen ( 1964 ) . because no depolarization values were give for ch@xmath4 , we used co@xmath0 values . we also used the wavelength dependence of ammonia s refractive index to approximate that of ch@xmath4 . the error in these latter approximations is not significant because of the small ch@xmath4 mixing ratio . absorption ( triple dot - dash ) , h@xmath0 cia ( dot - dash ) , and the sum of all opacities ( solid ) accumulate a one - way vertical optical depth of unity , plotted versus wavelength . bottom : pressure at which one - way total vertical optical depth reaches 0.3 , 1 , 3 , 10 , 30 , and 100 , plotted versus wavelength.,width=355 ] two model structures with aerosols were chosen to illustrate the effects of stratospheric haze and lower altitude cloud aerosols on raman scattering features and to test approximations under more realistic conditions than provided by a clear atmosphere . both models contain a high - altitude absorbing haze of mie - scattering spherical particles with a hansen ( 1971 ) gamma size distribution of effective radius @xmath192 = 0.2 @xmath2 m and variance @xmath121 = 0.02 , where the relative number per unit radius interval at radius @xmath193 is proportional to @xmath194 . the particle size is from pryor ( 1992 ) . the refractive index of the haze is assumed to have a real value of 1.44 and @xmath195-dependent imaginary values given in fig.[fig : nimag ] , which differ from the variation inferred by courtin ( 1999 ) . other index variations could also have been used , with compensating changes in other aerosol components . finding a tightly constrained fit is left for future work . while courtin s @xmath84 = 0.1 haze extends downward only to 20 mb , our haze has twice the optical depth and extends much more deeply ; it has a uniform mixing ratio between 100 and 800 mb , which is more similar to the distribution inferred by moses ( 1995 ) . both of our haze models also contain a 3.8-bar cloud of isotropic scatterers , with a single - scattering albedo 0.99 . this cloud is at the level that is often considered to be the top of a semi - infinite cloud ( baines 1995 ) . haze model i has this deep cloud set to optical depth 0.5 . for haze model ii this cloud is set to unit optical depth and a second cloud is placed at 1.3 bars , the approximate level expected for the base of a ch@xmath4 ice cloud assuming a ch@xmath4 mixing ratio of 2.2% . for this cloud we assumed @xmath84 = 2 , a single - scattering albedo of 0.99 , and a double henyey - greenstein phase function of the form @xmath196 = @xmath197 , with @xmath198 , @xmath199 , and @xmath200 , where @xmath201 . the phase function parameters are due to pryor ( 1992 ) , based on high - phase angle voyager images . our haze and cloud structures are not meant to be optimum fits to neptune s geometric albedo spectrum , but rather a sampling of possible structures that might be encountered . nevertheless , the part of the spectrum below 0.5 @xmath2 m is a fairly good match to the observed geometric albedo spectrum , as demonstrated in sec . [ sec : obscomp ] . the haze ii model mainly serves as a test case for approximations . a key input to modeling of raman scattering is an accurate solar spectrum of the appropriate spectral resolution . for @xmath202 @xmath2 m we used a model spectrum ( kurucz 1993 ) normalized to the 0.41 to 0.87 @xmath2 m results of neckel and labs ( 1984 ) and for 0.12 @xmath2 m @xmath203 0.41 @xmath2 m we used measurements by the upper atmospheric research satellite using the solstice instrument ( woods 1993 ) . the uars spectrum was obtained from the uars web site ( ftp://rescha.colorado.edu/pub/solstice/sol_hires_200.dat ) . it has a nominal resolution of 0.2 nm , but is actually closer to a resolution of 0.5 nm , based on comparisons with convolutions of the very high resolution kurucz model spectrum . we created our standard solar reference spectrum ( fig . [ fig : solarspec ] ) by convolving the combined spectrum with a triangular sampling function to obtain a nominal fwhm resolution of 0.35 @xmath44 sampled at 0.1725 @xmath44 . the nominal wavelength resolution varies from 0.14 nm at 0.2 @xmath2 m to 3.5 nm at 1 @xmath2 m ( the resolution is 1 nm at 534 nm ) . although this has nominally a uniform wavenumber resolution , it is actually limited by the uars observations to about 0.5 nm at the shortest wavelength , where it is oversampled by a factor of 2 or more . the spectrum does nt actually reach the nominal resolution until the transition to the kurucz spectrum at 0.41 @xmath2 m . the solar spectrum matches the karkoschka ground based observing resolution of 1 nm at 0.534 @xmath2 m , but is 3.5 times worse near 1 @xmath2 m . thus careful comparisons between observation and model calculations need to adapt to differing resolutions . following courtin ( 1999 ) , we selected a uars spectrum from 29 march 1992 to provide optimum compatibility with the fos observations from 19 august 1992 . courtin found a peak - to - peak difference of only 2% between two low - resolution uars daily spectra obtained on 2 june 1992 and 19 august 1992 . thus , solar variability is unlikely to be a major error source in comparing calculations with fos observations . most of the solar variability is restricted to @xmath1 0.26 @xmath2 m , so that comparison with the groundbased observations made in 1993 ( karkoschka , 1994 ) are even less influenced by solar variability . the behavior of monochromatic incident photons provides useful insight into the workings of raman scattering in neptune s atmosphere and a useful validation of our basic computational algorithm . for this example we use the three raman transitions given in table 4 , a wavenumber spacing of 58.62 @xmath44 , an atmosphere with ch@xmath4 absorption and h@xmath0 cia , an equilibrium distribution of h@xmath0 , but no aerosols . the propagation of photons from the injection wavelength of 0.228247 @xmath2 m ( 43812.2 @xmath44 ) is illustrated in the bottom panel of fig.[fig : ramline1plus2 ] . the vertical scale in this plot is the ratio of the total reflected flux at normal incidence to the incident photon flux . the first peak shows that 54.6% of the incident photons are reflected by the atmosphere at the incident wavelength . the remaining 45.4% of the photons are shifted to longer wavelengths by raman scattering . the second peak , containing about 8.1% of the photons is due to the @xmath50 transition . this second peak is the fraction of single - scattered @xmath50 photons that exit the atmosphere ; but many @xmath50 photons are further shifted to even longer wavelengths . the third peak , containing 2.2% of the incident photons , is due to the @xmath52 transition . many additional peaks are due to multiple raman scattering with various combinations of @xmath52 and @xmath50 . the relatively large peak near 0.252 @xmath2 m , containing 8.1% of the incident photons , is the first due to the @xmath36 transition . this is the same size as the @xmath50 peak and thus seems surprisingly large given that the @xmath36 transition has about half cross section of the @xmath50 transition ( the ratio is 0.541 at 0.228 @xmath2 m ) . however , the @xmath50 contribution is multiplied by @xmath15 , which is below 0.75 in the pressure range likely to contribute most at this wavelength ( fig.[fig : parapop ] ) . one might thus expect the @xmath36 contribution to be at least 72% , of the @xmath50 contribution on this basis alone . however , there are other factors also at work , such as multiple scattering and the vertical distribution of raman photons . in fact , if the rayleigh scattering cross section is made wavelength - independent , then the @xmath36 peak becomes only about 57% of the @xmath50 peak , which is less than expected from cross section and @xmath15 considerations . apparently , in the real atmosphere the decreasing opacity of the atmosphere with increasing wavelength allows more of the raman photons to make it out of the atmosphere than would otherwise be the case . because the @xmath36 photons undergo the largest wavelength shift , they exhibit the largest effect . the conservation of photons illustrated in the top panel of fig . [ fig : ramline1plus2 ] is computed as follows . for an incident unpolarized flux @xmath204 at zenith cosine @xmath205 , the quadrature value closest to unity , the total reflected flux at @xmath99 is given by by the discrete summation @xmath206\mu_n f_0(\nu ) , & \label{eq : reftot}\end{aligned}\ ] ] where , for a monochromatic incident flux at @xmath207 , only the reflection term contributes at @xmath207 and only the source term contributes at @xmath208 . because of the matrix formulation of the source term that is used here , we must introduce a pseudo - monochromatic flux that has a negligibly small constant offset for @xmath209 , to avoid dividing by zero in eq.[eq : finalsrc ] . using discretization of wavelength as well , in which the incident wavenumber is @xmath210 and @xmath211 , we can write the pseudo - monochromatic incident flux as @xmath212 implying a photon line flux ( flux at wavenumber @xmath210 ) of @xmath213 . the cumulative fractional photon flux can then be written as @xmath214 where @xmath215 and where we ignored @xmath216 compared to unity in the first term and made use of the fact that the source term is zero at @xmath207 . although the second term has an explicit multiplication by @xmath216 , it is not negligibly small because the source term itself contains a division by @xmath216 . for a conservative atmosphere that is either semi - infinite or bounded by a unit - albedo surface , @xmath217 should evaluate to unity if the sum is extended to the last wavenumber . the cumulative flux shown in fig . [ fig : ramline1plus2 ] falls somewhat short of unity because of ch@xmath218 absorption and h@xmath0 cia . the middle panel of fig . [ fig : ramline1plus2 ] shows that raman photons can undergo a surprisingly large number of scatterings before they leak out at the top of the atmosphere . in fact , as shown by the plot of cumulative flux in the upper panel , even by 0.45 @xmath2 m , 1.6% of the incident photons are still inside the atmosphere . this occurs because the atmosphere is conservative at short wavelengths and because photons scattered deep within the atmosphere are more easily lost to another raman scattering than lost to transmission out of the atmosphere ( to space ) . after each additional raman scattering , the optical depth to space gets smaller , as does the cross section for further raman scattering ( fig . [ fig : xcvswlen ] ) , so that photons are more easily lost to space at longer wavelengths . the vertical distribution of raman scattered photons is illustrated in fig . [ fig : linesrcperkm ] for the @xmath36 transition , which produces peaks spaced by 4161 @xmath44 ( fig . [ fig : ramline1plus2 ] ) . the curve of highest magnitude is due to one scattering , the second due to two scatterings , and the last is due to eight successive scatterings involving the @xmath36 transition . because of the large wavenumber shift of this transition , these peaks decline more slowly than those related to the @xmath50 and @xmath52 transitions . the peak at the shortest wavelength is at a location where the incident radiation field times atmospheric density is at a local maximum , which occurs near 300 mb . after each additional scattering , the raman photons diffuse both upward , where they leak out of the atmosphere , and downward , where they provide a reservoir for later contributions . it is worth noting that some of the raman scattered photons that are transferred to a longer wavelength can leave the atmosphere directly without any further scattering ( those at high altitude and directed upward ) . the result is a movement of the peak source intensity further downward after each successive scattering . per km versus geometric mean pressure of the layer boundaries , for selected wavelengths ( labeled by nanometers ) , for a clear neptune atmosphere.,width=355 ] because of vertical variation in the source function , it was necessary to use 72 logarithmically spaced gas layers to avoid cumulative photon fluxes exceeding unity . when cross sections were made wavelength independent , but evaluated at 0.4 @xmath2 m , the accuracy for a given number of levels was much higher . monochromatic calculations were run with three different choices for the number of logarithmically spaced atmospheric levels between 0.0003 and 100 bars : 18 , 36 , and 72 . the ratios to the 72-level fluxes , were 0.991 - 0.998 for the 18-level result , and 0.998 - 0.9996 for the 36-level result . thus it seems likely that errors for 72 levels probably do not exceed 0.1% , and 36 levels are probably within 0.2% . the ratios to the 72-level cumulative flux at 1 @xmath2 m were 0.9976 for 18-levels and 0.9995 for 36 levels . for an irradiance spectrum with a @xmath195-independent photon flux , one might expect the apparent geometric albedo with raman scattering to be the same as without it . one could argue that for every photon lost from a given wavenumber , there is one gained from a higher wavenumber . this argument works well if atmospheric properties are also independent of wavelength . but in a real atmosphere , raman and rayleigh scattering cross sections vary with wavelength , so that the vertical distribution and fraction of photons lost will be somewhat different from the vertical distribution and fraction of photons gained , even with a @xmath195-independent incident photon flux . in spite of this complication , the results shown in fig.[fig : flatphotabs ] are roughly consistent with this argument . the calculation shown here is for a clear neptune atmosphere with ch@xmath4 absorption and cia turned off , and exposed to an irradiance spectrum that has a constant photon flux except for a pseudo - solar absorption feature at 0.311641 @xmath2 m ( 32088 @xmath44 ) , where the photon flux was dropped to 20% of its value elsewere . the irradiance plotted here is the spectrum of energy per unit wavenumber , normalized to unity at 0.2 @xmath2 m , which corresponds to a flat photon spectrum . note the considerable distance from the starting wavelength to the wavelength at which the geometric albedo approaches the raman - free semi - infinite h@xmath0 rayleigh value of 0.7908 ( sromovsky 2004 ) . that is not surprising because of the importance of multiple raman scattering evident from fig.[fig : ramline1plus2 ] . calculation were run with surface albedos of 0 and 1 , to show that for @xmath219 0.6 @xmath2 m even 100 bars of atmosphere is not enough to obscure the lower boundary when only rayleigh and raman scattering are considered . with ch@xmath4 absorption turned on , the effects of the bottom boundary are no longer apparent . among the sharp spectral features present in the reflection spectrum , the largest peak is at exactly the wavelength of the absorption spike in the irradiance spectrum . this peak arises because in addition to photons reflected at the same wavelength , there are also photons raman shifted from shorter wavelengths . because more are shifted in than shifted out , sharp absorption features like this get partially filled in the reflected flux . when the reflected flux is then divided by the irradiance to obtain the albedo , the feature stands out as a positive spike . solar absorption features like this are the most prominent features in the raman spectra of the outer planets . the other , much smaller downward spikes are ghosts of the irradiance spectrum , displaced by the wavenumber shift of the raman transition . in the detailed view shown in the lower part of fig . [ fig : flatphotabs ] the ghost features are labeled by the transitions that produced them . note the relatively small size of the ghost features . the @xmath50 ghost feature has an amplitude of about 5.8% of the geometric albedo , while the corresponding absorption feature in the irradiance spectrum has an amplitude of 80% ; the @xmath52 and @xmath36 features have amplitudes of about 2.7% and 4.8% respectively . the near equality of @xmath36 to @xmath50 ghost amplitudes in spite of their very different cross sections has the same explanation as the relatively large @xmath36 contribution peak in the monochromatic spectrum , discussed in the previous section . m ( 32088 @xmath44 ) and corresponding geometric albedo spectra of neptune for unit ( dashed ) and zero ( solid ) surface albedos . the dotted line indicates the geometric albedo for a raman - free semi - infinite rayleigh atmosphere ( p=0.7908 ) . lower : geometric albedo vs wavenumber , showing raman transition assignments for the main ghost features.,width=355 ] the effect of raman scattering on neptune s aerosol - free geometric albedo is illustrated in fig . [ fig : ramanonoff ] , where spectra including polarization are shown for three calculations of increasing complexity , first with only rayleigh scattering , then including also molecular absorption , and finally including raman scattering . the calculations were started at 0.2 @xmath2 m , and are accurate to better than 1% at wavelengths beyond 0.22 @xmath2 m . experiments with different starting wavelengths show that an accuracy of 1% or better can be achieved at a given short wavelength by starting the calculation at a corresponding wavenumber that is higher by twice the wavenumber shift of the @xmath36-branch transition . this relatively small starting offset is effective because of the steep gradient in the solar spectrum at short wavelengths ( see fig . [ fig : solarspec ] ) . the calculation was carried out with a step size of 17.78 @xmath44 , using a solar spectrum with a nominal resolution of 35 @xmath44 . with only conservative rayleigh scattering , neptune s geometric albedo would be a wavelength - independent 0.791 ( sromovsky 2004 ) . adding methane absorption and collision - induced hydrogen absorption ( cia ) only reduces the albedo at wavelengths beyond 0.4 @xmath2 m . raman scattering produces profound effects throughout the uv and most of the visible spectrum . besides the introduction of sharp spectral features , raman scattering also reduce neptune s baseline geometric albedo by nearly 25% in a clear atmosphere . in fig.[fig : ramanonoff ] the albedo decreases ( shown as red ) appear mostly at shorter wavelengths , but also appear in the vicinity of most of the local peaks in the near - ir reflectivity . the albedo gains ( shown as black ) occur not just at the deep minima in the solar spectrum , but also at the absorption maxima in the ch@xmath4 spectrum , which correspond to minima in the reflectivity spectrum . we find that the fill - in of the reflectivity minima in the near ir is only about 4% of the geometric albedo . this is in sharp contrast to the cochran and trafton ( 1978 ) conclusion that in the cores of the strong ch@xmath4 bands at 0.86 @xmath2 m , 0.89 @xmath2 m and 1.0 @xmath2 m , nearly all of the photons that leave the atmosphere have been raman scattered . however , their conclusion is a result of what we now know to be inappropriate assumptions concerning the amount and vertical distribution of methane . their standard model had a constant mixing ratio of 0.005 relative to h@xmath0 throughout the atmosphere . this resulted in ch@xmath4 supersaturation for @xmath180 bar , by factors of @xmath3100 at the tropopause and more than 10 in the stratosphere . this resulted in a very low single scattering albedo of @xmath220 at 0.89 @xmath2 m , which even for a semi - infinite atmosphere would yield a very low geometric albedo of only 0.00036 , using the single - scattering approximation @xmath221 for @xmath222 ( sromovsky 2004 ) . that is about 20 times smaller than the geometric albedo we computed for our assumed ch@xmath4 profile , which never exceeds saturation levels and is limited to 3.5@xmath4710@xmath223 in the stratosphere . our distribution leads to much more scattering at the center of the disk and significant limb brightening ( sromovsky 2004 ) , while the uniform mixing ratio of cochran and trafton makes the atmosphere dark at all view angles . the geometric albedo spectrum ( green).,width=355 ] the earliest accurate calculation of a raman spectrum relevant to neptune was by cochran and trafton ( 1978 ) for a pure h@xmath0 atmosphere assuming an equilibrium population of ortho and para states . their most relevant case is for 500 km - amagats of h@xmath0 over an opaque cloud layer ( of unspecified but apparently high albedo ) . that amount of h@xmath0 corresponds to an atmospheric depth of approximately 6 bars , which is deep enough to approach semi - infinite behavior only for @xmath1 0.34 @xmath2 m . at 0.5 @xmath2 m the surface albedo can make a difference of 0.25 in geometric albedo . using 2/3 of the ford and browne ( 1973 ) raman cross sections to approximate the values used by cochran and trafton , we computed geometric albedo values for a he - free atmosphere with a unit - albedo lambertian surface placed at the 6-bar level , and ignored polarization . this spectrum , smoothed to a resolution of 5-nm , is displayed in fig . [ fig : savage ] where it compares well with the even lower resolution spectrum of cochran and trafton . the other prior calculation displayed in fig . [ fig : savage ] is due to savage ( 1980 ) , who used the same iterative computational scheme first presented by cochran and trafton ( 1978 ) . according to savage , they used the ford and browne ( 1973 ) raman cross sections that we used , but ignored polarization and assumed a pure h@xmath0 atmosphere , which was deep but of unspecified depth . in spite of their use of different raman cross sections , the savage and cochran and trafton results agree with each other and with our calculations using 2/3 of of the ford and browne cross sections , raising questions about what cross sections were actually used by savage ( 1980 ) . our calculation with the ford and browne cross sections , and other conditions being the same , are shown as the thinner dot - dash curve in fig . [ fig : savage ] . clearly , this spectrum is not compatible with the savage results . all these calculations have a stronger upward slope than we obtained for a truly deep atmosphere ( fig . [ fig : ramanonoff ] ) because the surface reflection becomes more visible at longer wavelengths . this can be seen from the difference between calculations for zero and unit surface albedos , which are shown in the figure . other effects occur when we carry out more realistic calculations . the presence of helium dilutes the raman absorption somewhat , producing about a 2% increase in the baseline value ; using the correct raman cross sections causes an albedo decrease of about 0.02 - 0.03 ( a 3 - 5% decrease ) ; and polarization increases the geometric albedo by about 0.04 ( sromovsky 2004 ) ( a 6.7% increase ) . the approximate net result is a very small difference between the accurate calculation with polarization and the earlier h@xmath0-only calculations without polarization . , without he , excluding polarization , and assuming ford and browne ( 1973 ) raman cross sections ( dot - dash for a@xmath224=1 , dashed for a@xmath224=0 ) and 2/3 ford cross sections ( solid ) , compared to earlier calculations by cochran and trafton ( 1978)(heavy dot - dash ) and by savage ( 1980 ) ( heavy dashed ) . our spectra have been smoothed to an effective resolution of 5 nm to facilitate comparisons . see text for discussion.,width=364 ] the international ultraviolet explorer ( iue ) made full - disk observations of neptune during 2 - 5 october 1985 ( days 275 - 277 ) at a phase angle of 1.87and a sub - earth latitude of -19.95planetographic ( -19.325centric ) . wagener ( 1986 ) converted these observations to geometric albedo assuming an equatorial radius of 25240 km and an oblateness of 0.021 , which are average stellar occultation results referring to a pressure of 1 microbar ( french 1984 ) . using measurements of 24764 km and 0.01708 ( davies 1992 ) , which refer to the 1-bar level , we obtain a conversion factor of 1.035 for adjusting the wagener albedo values ( this disagrees with the factor of 1.047 used by courtin ( 1999 ) ) . we did not make a correction for phase angle , though that might increase their values by as much as 1% if the phase curve at uv wavelengths is similar to that observed at longer wavelengths ( sromovsky 2003 ) . the adjusted iue results are compared with our clear - sky and haze i neptune spectra in fig.[fig : iue ] , where error bars indicate the absolute uncertainty estimates due to wagener ( 1986 ) . the iue results are 5% to 10% below the theoretical clear - sky calculated spectrum , generally showing the smaller difference at shorter wavelengths ( 0.21 - 0.23 @xmath2 m ) and the best defined and larger differences at longer wavelengths ( 0.29 - 0.33 @xmath2 m ) , which is likely due to an absorbing high altitude haze . our haze i model calculations match the iue observations generally well within the iue error bars . if we had neglected polarization , the model calculations would drop about 5% , making the clear - atmosphere model the best fit near 0.25 @xmath2 m . in this spectral range , polarization makes the difference between needing haze absorption and not needing it . the faint object spectrograph ( fos ) observed neptune at a phase angle of 1.22and a sub - earth latitude of -24.78planetographic on 19 august 1992 using apertures of 0.3@xmath225 , covering 21 - 36s , and 1.0@xmath225 , covering 3 - 35s . because these apertures are smaller than neptune s 2.4@xmath225 disk and placed near the middle of neptune s disk , the observed i / f and the relative amplitudes of raman spectral features are increased relative to what would be observed for a full - disk average ( shown in sec . [ sec : angle ] ) . courtin ( 1999 ) obtained uncalibrated geometric albedo spectra by dividing fos flux spectra by a solar flux spectrum measured by the uars solstice instrument ( rottman 1993 ; woods 1993 ) . the fos results we use are the 1-arcsec spectra degraded to a resolution of 1 nm to improve the signal / noise ratio ( shown in courtin s fig . 2 ) . courtin performed a radiometric calibration by matching his 0.22 - 0.33 @xmath2 m fos spectra to full - disk 0.3 - 1.0 @xmath2 m groundbased spectrum of karkoschka ( 1994 ) , obtained during 23 - 26 july 1993 at a phase angle of 0.4 . the fos observations thus have the same 4% absolute uncertainty as the karkoschka ( 1994 ) reference spectrum . the fos noise level varies from 0.4% at 0.33 @xmath2 m to 1% at 0.26 @xmath2 m . courtin s neptune spectrum is compared to our clear - sky and haze i calculations in fig . [ fig : fos ] . our calculations as a function of view angle were integrated over approximately the same field of view as the fos observations to account for the increased raman effect near normal view angles . we then convolved them to a final resolution of 1 nm and scaled this average spectrum to match our calculated geometric albedo spectrum in the wavelength range from 0.31 @xmath2 m to 0.33 @xmath2 m , to simulate the calibration procedure used by courtin . note the excellent agreement between the shapes and amplitudes of the calculated spectral features for the haze i model and those measured by the fos . the most glaring exceptions ( a and b in the figure ) appear to be artifacts in the fos spectrum . these anomalies are not present in the overlapping portion of the karkoschka spectrum , which is in good agreement with the calculated spectrum . the spikes c and d in the ratio spectrum are at points where the i / f spectrum has very sharp gradients , and could be removed by a slight wavelength shift . the fos / haze i ratio spectrum generally agrees with the iue / haze i ratio shown in fig.[fig : iue ] , indicating that haze absorption may be depressing the reflectivity by about 6%-10% below the theoretical value for a clear neptune atmosphere in the 0.23 - 0.27 @xmath2 m range , with increasing absorption indicated at longer wavelengths . figure [ fig : karkcompto1 ] compares clear - atmosphere and haze i model calculations with the disk - integrated groundbased spectra of karkoschka ( 1994 ) . between 0.35 @xmath2 m and 0.45 @xmath2 m the observations are about 13% below the clear - atmosphere calculations , indicating a presence of an absorbing haze that is even more influential than at shorter wavelengths . this behavior suggests that the haze extends deep enough into the atmosphere that it becomes more influential as the molecular scattering optical depth above it decreases with increasing wavelength . this is qualitatively consistent with microphysical models of moses ( 1995 ) , which suggest that haze opacity increases down to the 870 mb level . it is also roughly the character of our haze i model , which is shown to provide good agreement with karkoschka s observations , although the computed amplitudes of the raman spectral features are somewhat larger than the observations in several cases . this discrepancy might mean that more aerosol opacity is needed in the 1 - 3 bar level , or that there is a slight difference in the effective spectral resolution of the observations and the calculations . at longer wavelengths , the high observed albedo values in the deep ch@xmath4 bands at 0.8 @xmath2 m and 0.9 @xmath2 m , indicate that there is also particulate scattering at high altitudes , which is not included in either of the haze models . much of that contribution probably comes from discrete bright features seen in voyager , hst , and groundbased images . effective pressure estimates for such features range from 23 mb to 60 mb at 30 - 40n , 100 - 230 mb at 30 - 50s , and 170 mb to 270 mb near 70s ( sromovsky 2001b ; gibbard 2003 ) . the effective fractional coverage of bright high altitude clouds required to explain the albedo in deep ch@xmath4 bands appears to be about 0.5% to 1% for an upper cloud at 150 mb ( sromovsky 2001a ) . the effective fractional coverage of a coexisting 1.3-bar cloud is @xmath31% assuming a unit - albedo lambertian reflector . the observed albedo in the window regions at 0.825 @xmath2 m and 0.935 @xmath2 m is also significantly above the clear - sky calculation , suggesting a significant aerosol contribution that could be at mid to deep levels , which is largely satisfied by the 1.3-bar and 3.8-bar clouds in the haze i model . the excess i / f calculated in the window regions at 0.59 @xmath2 m , 0.63 @xmath2 m , 0.68 @xmath2 m , and 0.75 @xmath2 m , can be substantially reduced by increasing the ch@xmath4 continuum absorption by amounts comparable to karkoschka s stated continuum uncertainty . it could also be reduced by decreasing the single - scattering albedo of the cloud aerosols . the observations seem to show much less of the collision - induced absorption in the 0.8 - 0.83 @xmath2 m region than is evident in the calculations . another way to describe this difference is to say that there is an imbalance in the 0.83 and 0.94 @xmath2 m window regions that is evident in the calculation , but not in the observations . that imbalance may be due to an error in the ch@xmath4 absorption coefficients . if we add to karkoschka s standard coefficients the continuum absorption uncertainty , we actually obtain very similar albedo values in these two windows . because the @xmath28 raman cross sections are larger than the @xmath29 cross sections ( fig . [ fig : xcvswlen ] ) , the ortho / para ratio will have an effect on the raman features observed in neptune s reflection spectrum . the largest difference in ortho / para ratios for equilibrium and normal h@xmath0 occur near 100 mb ( fig . [ fig : parapop ] ) , and thus associated spectral variations are more likely at short wavelengths where this region is near the @xmath226 level ( fig.[fig : pendepth ] ) . calculations ( fig.[fig : feqcomp ] ) show that the largest difference is for the peaks at 0.280 @xmath2 m and 0.2852 @xmath2 m , which are about 5% larger for equilibrium h@xmath0 . the overlay of fos observations suggests better agreement with equilibrium h@xmath0 , as concluded by courtin ( 1999 ) , although the relative size of the smaller peak is very dependent on the effective resolution of the observations and both peaks have amplitudes that depend on view angle . and since the fos observations cover only the central disk , it is not strictly valid to compare them to geometric albedo observations . uncertainties in the wavelength scale and effective resolution of the fos observations arise from comparisons with overlapping groundbased observations of karkoschka ( 1994 ) , which show significantly less spectral modulation in the 0.3 - 0.33 @xmath2 m overlap region , as noted earlier . the presence of an absorbing haze and high - altitude discrete clouds also affects these peaks . thus it seems premature to make a strong conclusion based on these observations . however , courtin s ( 1999 ) result that @[email protected] is based on higher resolution observations than those shown in fig . [ fig : feqcomp ] , and was derived after special processing to minimize wavelength errors . courtin s results are also compatible with independent conclusions by baines and smith ( 1990 ) and conrath ( 1991 ) that the ortho / para ratio for neptune is near thermal equilibrium . ( solid ) and normal h@xmath0 ( dot - dash ) compared to the observed fos spectrum ( plus signs ) ( courtin 1999 ) scaled by the factor 1.08 . the ratio of equilibrium to normal albedos is in the lower panel . , width=355 ] karkoschka ( 1994 ) presented a method for transforming spectra between raman and non - raman forms based on the assumption that the measured geometric albedo spectrum @xmath228 can be well approximated by a linear combination of terms involving offset versions of the `` physical '' spectrum @xmath229 , which is the reflectivity spectrum without raman scattering . the mathematical expression of this method is @xmath230 where @xmath231 is here the solar photon spectrum , @xmath232 is the fraction of photons not raman scattered , @xmath233 is the fraction undergoing transition @xmath43 , and where @xmath234 equation [ eq : karkcorr ] can be inverted to obtain the raman - free `` physical '' spectrum from the observed spectrum using @xmath235 as a first guess , and then improving the guess using the iterative equation @xmath236 \label{eq : karkinv},\end{aligned}\ ] ] for which four iterations are usually sufficient to converge on @xmath237 for a given set of fractions . karkoschka assumed a power law @xmath195-dependence for @xmath238 , then used minimum roughness of the fitted spectrum for 0.31 @xmath2 m @xmath203 0.405 @xmath2 m as the criterion for picking the optimum fractions and exponent . c c c c c c c c ' '' '' & @xmath195-range & & & & & & rms + spectrum & ( @xmath2 m ) & @xmath239 & @xmath240 & @xmath241 & @xmath242 & @xmath243 & dev + ' '' '' clear sky & 0.31-.405 & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & 0.93% + clear sky & 0.23-.405 & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & 1.47% + haze ii & 0.31-.405 & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & 0.44% + haze ii & 0.23-.405 & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & 0.90% + haze ii ( inv ) & 0.23-.405 & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & 0.58% + k94 coeff . & 0.31-.405 & [email protected] & [email protected] & [email protected] & 0.050@xmath244 & -1@xmath1381 & + [ tbl : fracfits ] we first investigated how well the proposed model could fit raman spectral calculations and how the fitted constants related to raman scattering cross sections . we considered both a clear - atmosphere model and a model with haze and cloud contributions . we first fit the fractional parameters and wavelength dependence exponent by minimizing the rms deviation between the karkoschka model and the calculated spectrum , first over the 0.31 - 0.405 @xmath2 m range used by karkoschka and also over the wider 0.23 - 0.405 @xmath2 m range . the results are summarized in table 5 . note that our fit results are comparable to those of karkoschka , especially when applied to a hazy atmosphere . although we were able to obtain good fits over the limited spectral ranges ( rms deviations of 0.44% and 0.9% for the haze / cloud model , and 0.93% and 1.47% for clear sky model ) , the fits did not accurately match the observed spectrum at slightly longer wavelengths , especially where ch@xmath4 absorption was present . the weak ch@xmath4 bands in the karkoschka conversion of the physical spectrum were much stronger than the corresponding bands in the correct raman calculation . this is understandable as a consequence of the nonlinear relationship between i / f and single - scattering albedo . a small absorption added to a conservative atmosphere can have a much larger fractional effect than it does when added to an absorbing atmosphere . for example , adding @xmath245 to a rayleigh atmosphere with @xmath246 changes @xmath247 from 1.0 to 0.997 , which decreases the i / f from 0.791 to 0.699 , a 9% drop . but if the i / f is already at 0.559 due to raman absorption , adding the same ch@xmath4 absorption as before would change @xmath247 from 0.975 to 0.972 , producing only a 2% drop in i / f . confirming karkoschka s fit results , we find that the vibrational coefficient ( @xmath242 ) is about twice the size of the coefficient for the @xmath29 rotational contribution ( @xmath241 ) , even though its raman cross section is somewhat smaller ( table 3 ) . since we use these cross sections to compute the spectra , the fit results do not suggest that there is something wrong with the cross sections . a large part of this result comes from the fact that the vibrational coefficient represents the average of ortho and para contributions , while the individual @xmath50 and @xmath52 contributions are each multiplied by fractions that decrease their relative contributions by roughly a factor of two ( for @xmath248 , see eq . [ eq : sigtot ] and fig . [ fig : parapop ] ) . additional factors are discussed in sec.[sec : monoscat ] . we next consider how well karkoschka s method is able to remove raman scattering effects from a spectrum that includes raman scattering . we inverted our model raman calculations using the inversion method based on eq . [ eq : karkinv ] . the retrieved parameter values for the haze ii test case are given in table 5 . as illustrated in fig . [ fig : ramremove ] , we found pretty fair agreement between the true raman - free spectrum and inverted spectrum for the haze / cloud case when we used the 0.23 - 0.405 @xmath2 m range for the fitting , although our inverted spectrum is offset about 0.01 albedo units above the true spectrum , and the inverted parameters differ somewhat from the best - fit parameters . using the 0.23 - 0.405 @xmath2 m spectral range yielded a smoother inverted spectrum but a larger offset albedo of 0.03 . much worse results were obtained with the clear sky model , which has larger raman features that are not as well fit by karkoschka s empirical model . as expected , we found that the features due to weak ch@xmath4 bands were much smaller in the transformed spectra than in the true physical spectra . thus , estimates of ch@xmath4 absorption coefficients using spectra corrected in this way will be significantly below the true absorption level . karkoschka s method has several physical flaws that limit its applicability . multiple scattering is one effect that is not accounted for but can be significant ( see sec . [ sec : monoscat ] ) . another problem with the physics assumed in eq . [ eq : karkcorr ] is that the contribution at the present wavenumber is not necessarily proportional to the geometric albedo at the upward shifted wavenumber . in fact , the proportionality constant also depends on the vertical distribution of the scattered photons in relation to the vertical distribution of opacity at the scattered wavelength . the karkoschka forward method also is not generally applicable to spatially resolved observations because the amplitude of raman effects depends on view angle as well as local aerosol structure . the useful spectral range for the karkoschka correction is also rather limited , probably to wavelengths less than about 0.405 @xmath2 m , partly because it does not properly modify weak ch@xmath4 absorption bands . the approximation suggested by wallace ( 1972 ) is a modification of the molecular single scattering albedo of the atmosphere by treating rotational transitions as extra sources of scattering and the vibrational transition as a pure absorption . this can be expressed in terms of cross sections as follows : @xmath249 where @xmath250 is the rayleigh scattering cross section , @xmath251 is the total cross section for absorption by ch@xmath4 and cia by h@xmath0 , @xmath252 is the combined cross section for rotational raman transitions , @xmath253 is the cross section for vibrational raman transitions , and where @xmath254 for the original wallace approximation . for a clear neptune atmosphere , we find that the original wallace approximation provides a crude match to the low resolution baseline of the spectrum , but is generally 5% to 10% low and of course does not produce characteristic raman spectral features . a few detailed calculations by wallace ( 1972 ) at 0.2 @xmath2 m and 0.4 @xmath2 m also indicated that his approximation was about 5% low . the wallace approximation error is nearly zero in the near - ir ch@xmath4 windows , but is about 4% low in the ch@xmath4 absorption bands , which is about what would be expected if raman scattering were ignored ( see lower part of fig.[fig : approx3in1 ] ) . figure [ fig : approx3in1 ] displays a modified form of the wallace approximation that boosts the i / f value by setting @xmath255 , which treats the vibrational cross section as 56.7% absorption and 43.3% scattering . this improves the overall agreement in the blue to orange part of the spectrum , but has little effect elsewhere . the wallace approximation approaches the exact result when high clouds obscure most of the molecular scattering . approximation ( red ) . lower : ratios of the approximate geometric albedo spectra to the accurate raman calculation.,width=355 ] pollack ( 1986 ) approximated raman scattering contributions from shorter wavelengths by scaling the raman scattering cross section at the reflected wavelength by the ratio of solar irradiance at the source wavelength to that at the reflected wavelength . generalizing this approach to include more than one transition , the molecular single - scattering albedo in this approximation can be written as : @xmath256 where @xmath257 , @xmath258 is a function of @xmath259 that can be empirically adjusted to compensate for some of the complexities that are glossed over by this concept , and the upper bound of 1 is used to avoid energy conservation problems . setting @xmath260 , and omitting the rotational transitions completely , would reduce this to pollack s original suggestion , while setting @xmath261 and @xmath262 would reduce this to the wallace approximation . a problem with pollack s original formulation was that it resulted in @xmath263 at some wavelengths , as noted by courtin ( 1999 ) , which leads to instability in the solution of the radiation transfer equation . another problem is that it considered only the @xmath264 vibrational transition , which alone would be unable to generate the ghost features arising from the rotational transitions . the form given in eq . [ eq : pollack ] has the capability to fit all the main ghost features and has the potential , with tuning of the form of @xmath259 , to yield the correct baseline value as well . it also has an advantage over karkoschka s correction , in that it is introduced at a more fundamental physical level that is naturally modified by the presence of aerosols . however , wherever the geometric albedo exceeds the value for a conservative rayleigh atmosphere of 0.7908 ( sromovsky 2004 ) , even @xmath265 will not be able to reproduce the albedo value . this problem can be reduced by working at lower spectral resolutions for which these extreme peak amplitudes become reduced below 0.7908 . preliminary calculations with this approximation used the three transitions given in table 4 , with @xmath266 and @xmath267 the same for all transitions . we found that making the constant @xmath267 = 0.875 for all @xmath98 produced a reasonably good match to the baseline level of geometric albedo , but generated spectral modulations that were too large . to limit these modulations , we used an exponential function to limit the effect of the spectral irradiance ratio , i.e. we used @xmath268 , which is @xmath269 for small @xmath259 and approaches @xmath270 for large values of @xmath259 . we found that @xmath271 provided a reasonable fit to the raman spectral features and to the baseline level , as shown in fig.[fig : approx3in1 ] . while this is certainly an improvement over the wallace approximation , and does reproduce many of the spectral features associated with raman scattering , it does not do so consistently and accurately enough to use for interpreting those features . like the wallace approximation , it also fails to improve geometric albedo accuracy for @xmath219 0.47 @xmath2 m . both wallace and pollack modifications to the single - scattering albedo seem to be far less effective in regions where ch@xmath4 absorption is important . since the actual source function depends on the light level in the atmosphere , which may have a spectral shape that is very different from the incident solar spectrum , it is not appropriate to use the solar spectral ratio as a scale factor in spectral regions with strong ch@xmath4 absorption . the following method goes part of the way in solving this problem . the following describes a new approximation of raman scattering that uses a different kind of modification of the molecular single scattering albedo . instead of using a constant factor or the solar spectral ratio to modify the molecular single - scattering albedo , this approximation uses a @xmath195-dependent multiplier that is tuned to minimize the difference between simulated and exact raman calculations of geometric albedo for an appropriate model atmosphere . the fundamental relation used is from sromovsky ( 2004 ) : @xmath272 , \label{eq : pofomega}\ ] ] where @xmath273 is the geometric albedo including polarization contributions and @xmath247 is the single scattering albedo of a semi - infinite rayleigh atmosphere . this equation is inverted numerically to provide a function @xmath274 so that a given geometric albedo can be related to an equivalent single - scattering albedo of a semi - infinite rayleigh atmosphere . starting with a raman - free calculation producing a spectrum @xmath275 , and a true raman calculation producing a spectrum @xmath276 , we use the inverse transform to compute equivalent single - scattering albedos @xmath277 and @xmath278 . noting that @xmath279 , we then attribute the difference between the equivalent inverse single scattering albedos as due to a difference in the effective absorption due to raman scattering . thus we are led to the approximation for modifying the single scattering albedo of the molecular scatterings that simulate true raman scattering , namely , @xmath280^{-1 } \label{eq : tuneit1}\ ] ] where @xmath281 is the single - scattering albedo simulation of raman scattering on the prior iteration , @xmath282 is the effective semi - infinite single scattering albedo obtained from the spectrum generated using @xmath281 for the molecular scattering . because neptune s atmosphere has a vertically inhomogeneous ch@xmath4 distribution , and thus will not satisfy eq . [ eq : pofomega ] , it is necessary to iterate a few times to achieve convergence . on the first iteration , we have @xmath283 and @xmath284 . the molecular single - scattering albedo spectrum that results is @xmath285 , which reduces to @xmath286 when @xmath287 . the nearly perfect agreement between the geometric albedo computed with raman scattering and the approximation using a tuned single - scattering albedo is shown in fig . [ fig : approx3in1 ] . unlike the previously discussed approximations , it does provide useful accuracy for @xmath219 0.45 @xmath2 m . like the other approximations , it can not reproduce the peaks that exceed the i / f for a unit geometric albedo . while the modified single - scattering albedo can exceed unity , or even go negative , that does not lead to problems as long as the other contributors to the scattering process lead to a combined single - scattering albedo @xmath288 . the important question is whether this approximation or any of the others that are similarly constructed , are useful with clouds present , and for center - to - limb scans . because all of these approximations that modify single - scattering albedos really lack the essential physics of the raman scattering process , it is doubtful whether they can have much utility in situations for which they have not been tuned . a partial test of this utility is the degree to which they can simulate the angular variation of raman scattering , which is described in the next section . section [ sec : aereffects ] compares their performances when haze and cloud aerosols are present . raman scattering effects on neptune s i / f spectrum at zero phase angle are strongest near the center of the disk ( zero zenith angle ) and weakest near the limb ( fig . [ fig : angfix2in1clr ] ) . the errors obtained by ignoring raman scattering ( not shown ) are quite substantial at moderate view angles , especially at short wavelengths and at high spectral resolution , ranging from about 20 to 60% in the 0.25 - 0.3 @xmath2 m range at a zenith angle of 8.1 . at view angles of @xmath360 , the fractional errors are comparable to those seen for the geometric albedo . as noted for the geometric albedo , errors @xmath34% are seen in the deep ch@xmath4 bands , with very little error seen in the near - ir window regions . in the longer wavelength bands , the fractional error does not improve much near the limb , unlike the error at shorter wavelengths , which decreases substantially . the modified pollack approximation helps considerably in reducing errors at short wavelengths ( fig . [ fig : angfix2in1clr ] ) . the extreme error near the central disk drops from 62% to 14% and the rms error from 19% to 4% . at 63the rms error drops from 8.8% to 2.2% . but at long wavelengths , the pollack approximation actually makes things slightly worse in the middle of strong absorption bands , probably because the vertical location of raman photons becomes more critical to the resulting i / f level . overall , the modified pollack approximation has value in modeling a variety of observations . it is far better than ignoring raman scattering and is roughly twice as accurate as the modified wallace approximation . however , using the modified pollack approximation can lead to substantial errors in limb - darkening profiles , depending on the wavelength that is considered . sample profiles are shown in fig.[fig : ctl2in1 ] . under most conditions , the tuned-@xmath247 approximation is even more accurate than the modified pollack approximation , as evident from comparison of spectra at the three sample angles ( fig.[fig : angfix2in1clr ] ) and from a comparison of center - to - limb scans at sample wavelengths ( fig . [ fig : ctl2in1 ] ) . the extreme error near the central disk drops to 10.3% and the mean and rms deviations to only -1.1% and 1.2% , with comparable results at 63view angles . the worst errors are seen at large view angles . at 85.7 , the extreme error increases to 24% and the mean and rms errors increase to 2.6% and 3.3% . but most of these errors occur in the ch@xmath4 bands beyond about 0.53 @xmath2 m . if restricted to 70view angles and @xmath1 0.85 @xmath2 m , this approximation is remarkably effective , with errors rarely exceeding a few percent . approximation ( red ) , for three observing angles at zero phase . the ratios of the two spectral versions are shown at the upper part of each panel ; in the top panel the ratio is offset by 0.4 to prevent overlap . the spectra are convolved to a resolution of 36 @xmath44.,width=345 ] approximation ( red ) . the ratios of the two spectral versions are shown in the lower panel . the calculations were convolved to a resolution of 36 @xmath44 before sampling the selected wavelengths.,width=336 ] calculations for the haze ii atmospheric model containing the cloud and haze aerosols described in sec . [ sec : cloudandhaze ] are presented in fig . [ fig : approxcomp ] , where the calculation that ignores raman scattering is compared to the raman calculation and to the modified pollack and tuned-@xmath247 approximations . at uv - visible wavelengths we see that the size of the raman features is substantially reduced by the addition of a high altitude absorbing haze , which also depresses the baseline reflectivity as needed to approximately match observations . the net effect of raman scattering on the baseline reflectivity is also reduced substantially compared to its effect in a clear atmosphere . this is another example of the nonlinear effects of adding absorption ( see discussion in sec . [ sec : kcorrect ] ) . the effects of the cloud layers are most easily seen at longer wavelengths where the i / f in window regions and regions of intermediate methane absorption are increased substantially compared to that computed for a clear atmosphere . the tuned-@xmath247 approximation is seen to perform well for @xmath289 0.6 @xmath2 m , where haze effects dominate , but is biased about 5% too high in the strongly absorbing regions at longer wavelengths , although remaining quite accurate in window regions . this is not too surprising , because the tuning in these regions is involves a delicate balance between large molecular absorption and large corrections of that absorption to simulate the raman - scattered contributions . as vertical structure is changed from that used for tuning , the tuning can be easily upset . the modified pollack approximation performs somewhat worse at uv - visible wavelengths than it did for the clear atmosphere case ( fig . [ fig : approx3in1 ] ) . the overall negative bias seen here could probably be largely eliminated by adjusting the parameters of the @xmath258 factors in eq . [ eq : pollack ] . that would do much to reduce the positive errors in the window regions beyond 0.45 @xmath2 m or the negative errors in the absorbing regions . these are inherent in the physical model on which the pollack approximation is based . overall , this approximation is less upset by the presence of the cloud layers than is the tuned-@xmath247 approximation . a comparison of these two approximations at three different view angles is presented in fig . [ fig : approxang ] . we see that the tuned-@xmath247 approximation is much better near the central disk , and effective at all wavelengths . it is also effective at all angles for wavelengths less than 0.47 @xmath2 m . but at longer wavelengths , errors in the regions of strong methane absorption grow with increasing view angle , making the modified pollack approximation better for view angles beyond about 50 . the karkoschka approximation was not included in this particular comparison because is not useful for spatially resolved observations , especially those with variable cloud structure , unless the model coefficients are changed for each view angle and structure . even then , the serious problems at @xmath219 0.4 @xmath2 m ( see fig.[fig : ramremove ] ) also reduce its utility , even for disk - average analyses . ) , computed with raman scattering ignored ( green ) , using the pollack approximation ( blue ) , the tuned-@xmath247 approximation ( red ) , and the exact method of calculation ( black ) . the ratios to the true calculation are displayed in the lower panel.,width=355 ] approximation ( red ) , and the exact method of calculation ( black ) . the ratio spectral are also shown in each panel.,width=355 ] major results of this investigation can be summarized as follows : : : procedures for modifying the evans and stevens ( 1991 ) vector radiation transfer code to handle raman scattering are described and justified . equations are provided for the raman differential generators . the use of a raman source matrix facilitates the computation of the first spatially resolved raman scattering spectra for neptune . validation of the code is obtained by demonstration of photon conservation , comparison with prior low - resolution calculations , and comparison with hst and groundbased observations . : : raman scattering reduces the baseline geometric albedo of a semi - infinite conservative rayleigh scattering atmosphere by about 25% from the value of 0.791 it would otherwise have . in the presence of a high - altitude absorbing haze , the effect of raman scattering is reduced by about a factor of two . the well known sharp positive spectral peaks from fill - in of absorption lines in the solar spectrum can greatly exceed the rayleigh conservative limit . for photons introduced at 228 nm , the @xmath36 transition source function for the first scattering has a broad peak near 300 mb ; after further scattering photons move deeper into the atmosphere and leak out the top , moving the peak of the source function further downward after each scattering . : : for a clear atmosphere about 4% of the light in the deep methane bands in the near - ir is due to raman scattering , while the near ir window regions are almost unaffected by raman scattering . the conclusion of cochran and trafton ( 1978 ) that the light in the deep methane bands is almost entirely raman scattered is a result of their assumption of what are now known to be excessively high methane mixing ratios in the stratosphere . : : the calculated history of monochromatic incident photons shows that multiple raman scattering is quite significant in a conservative rayleigh scattering atmosphere , where the raman absorption process can compete with the loss to space . in one example calculation , only 40% of the raman scattered photons exiting the atmosphere experienced only one raman scattering . : : at low phase angles spatially resolved raman spectra show that raman spectral features are enhanced at near the center of the planetary disk and suppressed at the limb . we also found that approximations that work reasonably well in matching accurate disk - integrated calculations do not do as well in matching observations in specific directions , especially at large view angles . : : aerosol - free models of neptune s spectrum correlate well with observed spectral features , but confirm a need for haze absorption to reduce the baseline geometric albedo and to reduce the amplitude of spectral modulations induced by raman scattering . iue full - disk observations , adjusted to account for current knowledge of of neptune s size , have a baseline offset of @xmath36% in the 0.24 - 0.26 @xmath2 m range , increasing to about 10% in the 0.30 - 0.32 @xmath2 m range , both of which are fit well by our haze i model , containing 0.2 optical depths ( at 0.5 @xmath2 m ) of a uv absorbing haze of 0.2-@xmath2 m radius particles uniformly mixed between 0.1 and 0.8 bars . fos observations between 0.22 and 0.29 @xmath2 m , have a similar offset and high - resolution spectral structure that is also well fit by our haze i model . groundbased disk - integrated observations between 0.35 and 0.45 @xmath2 m have a baseline offset that is 13% below the clear - atmosphere calculation which is well matched by our haze i calculation , although several of the spectral features are seen to have greater amplitude in the calculations than in the observations . we consider the haze i model to be a sample calculation , not a tightly constrained fit . : : the karkoschka ( 1994 ) method of applying raman corrections to calculated spectra and removing raman effects from observed spectra is shown to be relatively accurate under restricted conditions , but has practical limitations for problems involving spatially resolved observations or for @xmath290 @xmath2 m . his raman removal algorithm undercorrects the depths of weak ch@xmath4 absorption bands , and thus corresponding absorption coefficients derived from such spectra ( karkoschka 1994 ; 1998 ) will need some revision . the relatively large amplitude of the q - branch contributions found by karkoschka is shown to be consistent with current estimates of raman cross sections , not an indication that they are in error . : : we agree with wallace ( 1972 ) that his suggested approximation yields geometric albedos that are typically @xmath35% lower than an accurate calculation . a generalization of this approximation that counts part of the vibrational cross section as contributing to scattering can remove this bias . this approximation does not help reduce errors in the strong ch@xmath4 bands and is mainly useful at low resolution and for @xmath1 0.45 @xmath2 m . : : the initial form of the approximation suggested by pollack ( 1986 ) generates serious errors . but generalizing it to include rotational and vibrational transitions , enforcing energy conservation , and limiting the solar spectral ratio enhancement , can produce a relatively good approximation , except for @xmath219 0.45 @xmath2 m , where ignoring raman scattering is just as accurate . : : by tuning the absorption attributed to raman scattering it is possible to make a non - raman calculation match a true raman calculation with virtually no error . if that matching is done with respect to geometric albedo , it will not match nearly as well for spatially resolved calculations . it is nevertheless the best approximation to use under cloud - free conditions for zenith angles less than 70and wavelengths less than 0.85 @xmath2 m . it is the only one that provides improvements for @xmath219 0.45 @xmath2 m , although is not very effective for @xmath219 0.47 @xmath2 m when cloud layers of intermediate opacity are present . most of these conclusions apply with equal force to raman scattering in the atmosphere of uranus because of its similarity to neptune in vertical temperature structure , gas composition , and relatively low aerosol loading . conclusion ( 6 ) would need the most revision to accommodate differences in detailed observational results . this research was supported by a grant from nasa s planetary atmospheres program and in part by an archive research grant from the space telescope science institute . i thank pat fry who helped with the installation of the original evans and stevens code and advised on fortran debugging of modifications to it . i also thank evans and stevens for making their well written code available to the community , r. courtin for providing tabular data files of his processed fos observations , and two anonymous reviewers . : : allen , c. w. 1964 . astrophysical quantities , 2nd edition . 291 pages . athlone press , london . : : baines , k. h. , and w. h. smith 1990 . the atmospheric structure and dynamical properties of neptune derived from ground - based and iue spectrophotometry . _ icarus * 85 * _ , 65 - 108 . : : baines , k. h. , h. b. hammel , k. a. rages , p. n. romani , and r. e. samuelson ( 1995 ) clouds and hazes in the atmosphere of neptune . in _ neptune and triton _ , d.p cruikshank , ed . , univ . of arizona press . : : baines , k. h. , and h. b. hammel 1994 . clouds , hazes , and the stratospheric methane abundance in neptune . _ icarus * 109 * _ , 20 - 39 . : : btremieux , y. , and r. v. yelle 1999 . hst detection of h@xmath0 raman scattering in the jovian atmosphere . _ icarus * 142 * _ , 324 - 341 . : : bhagavantam , s. , 1931 . raman effect in gases i. some experimental results _ ind . * 6 * _ , 319 - 330 . : : bhagavantam , s. , 1942 . _ scattering of light and the raman effect . _ , 333 pages , chemical publishing company inc . , brooklyn ny : : cabbanes , j. and a. rousset 1936 . diffusion molculaire : mesure du facteur de dpolarization des raies raman dans les gaz : azote , oxygne , gaz carbonique . _ comptes rendus * 202 * _ , 1825 - 1828 . : : cochran , w. d. , and l. m. trafton 1978 . raman scattering in the atmospheres of the major planets . _ the astrophys . j. * 219 * _ , 756 - 762 . : : conrath , b.j . owen , and r.e . samuelson 1993 . constraints on n@xmath0 in neptune s atmosphere from voyager measurements . _ icarus * 101 * _ , 168 - 171 . : : conrath , b. j. , d. gautier , g. f. lindal , r. e. samuelson , and w. a. shaffer 1991 . the helium abundance of neptune from voyager measurements . _ j. geophys 96 * _ , 18907 - 18919 . : : courtin , r. 1999 . the raman signature of h@xmath0 in the uv spectra of uranus and neptune : constraints on the haze optical properties and on the para - h@xmath0 fraction . _ planet . and space sci . * 47 * _ , 1077 - 1100 . : : davies , m. , v. k. abalakin , a. brahic , m. bursa , b. h. chovitz , jk . h. lieske , p. k. seidelmann , a. t. sinclair , and i. s. tiuflin 1992 . report of the iau / iag / cospar working group on cartographic coordinates and rotational elements of the planets and satellites - 1991 , _ celestial mechanics and dynamical astronomy ( issn 0923 - 2958 ) , * 53 * _ , no . 4 , p. 377 - 397 . : : dlugach , e. g. , and e. g. yanovitskij 1974 . the optical properites of venus and the jovian planets . ii . methods and results of calculations of the intensity of radiation diffusely reflected from semi - infinite homogeneous atmospheres . _ icarus * 22 * _ , 66 - 81 . : : evans , k. f. , and g. l. stephens 1991 . a new polarized atmospheric radiative transfer model . _ j. quant . spectr . and rad . transfer * 46 * _ , 413 - 423 . : : farkas , a. , 1935 . orthohydrogen , parahydrogen and heavy hydrogen . cambridge univ . press , : : ford , a. l. , and j. c. browne 1973 . rayleigh and raman cross sections for the hydrogen molecule . _ atomic data * 5 * _ , 305 - 313 . : : french , r. g. 1984 . oblatenesses of uranus and neptune . in _ uranus and neptune _ , nasa conf . pub . 2330 , 349 - 355 . : : gibbard , s. g. , i. de pater , h. g. roe , s. martin , b. a. macintosh , and c. e. max 2003 . the altitude of neptune cloud features from high - spatial - resolution near - infrared spectra . _ icarus * 2 * _ , 359 - 374 . : : goody , r. m. , y. l. yung 1989 . _ atmospheric radiation _ , oxford university press , oxford , new york . : : hansen , j. e. 1971 . multiple scattering of polarized light in planetary atmospheres . sunlight reflected by terrestrial water clouds . _ j. atmos . * 28 * _ , 1400 - 1426 . : : hansen , j. e. , and l. d. travis 1974 . light scattering in planetary atmospheres . _ space science reviews * 16 * _ , 527 - 610 . : : hinson , d. p. , and j. a. magalhes 1993 . inertio - gravity waves in the atmosphere of neptune . _ icarus * 99 * _ , 142 - 161 . : : hollas , m. 1992 . _ modern spectroscopy _ , john wiley & sons , new york . : : hovenier , j. w. 1969 . symmetry relationships for scattering of polarized light in a slab of randomly oriented particles . _ j. atmos . * 26 * _ , 488 - 494 . : : hu , y .- x . , b. wielicki , b. lin , g. gibson , s.l - c . tsay , k. stamnes , and t. wong 2000 . @xmath63-fit : a fast and accurate treatment of particle scattering phase functions with weighted singular - value decomposition least - squares fitting . _ j. quant . spectr . and rad . transfer * 65 * _ 681 - 690 . : : karkoschka , e. 1994 . spectrophotometry of the jovian planets and titan at 300- to 1000-nm wavelength : the methane spectrum . _ icarus * 111 * _ , 174 - 192 . : : karksochka , e. 1998 . methane , ammonia , and temperature measurements of the jovian planets and titan from ccd - spectroscopy . _ icarus * 133 * _ , 134 - 146 . : : karkoschka , e. and m. tomasko 1992 . saturn s upper troposphere 1986 - 1989 . _ icarus * 97 * _ , 161 - 181 . : : kattawar , g. w. , and c. n. adams 1971 . flux and polarization reflected from a rayleigh - scattering planetary atmosphere . _ astrophys . j. * 167 * _ , 183 - 192 . : : kurucz , r. l. 1993 . smithsonian astrophysical obsevatory cd rom no . 13 . : : lindal , g.f . the atmosphere of neptune : an analysis of radio occultation data acquired with voyager 2 . j. * 103 * _ , 967 - 982 . : : massie , s. t. , and d. m. hunten 1982 . conversion of para and ortho hydrogen in the jovian planets . _ icarus * 49 * _ , 213 - 226 . : : mishchenko th , m. i. , a. a. lacis , and l. d. travis 1994 . errors induced by the neglect of polarization in radiance calculations for rayleigh - scattering atmospheres . _ j. quant . spectr . and rad . transfer * 51 * _ , 491 - 510 . : : moses , j. i. , k. rages , and j. b. pollack 1995 . an analysis of neptune s stratospheric haze using high - phase - angle voyager images . _ icarus * 113 * _ , 232 - 266 . : : neckel , h. and d. labs 1984 . the solar radiation between 3300 and 12500 . _ solar phys . * 90 * _ , 205 - 258 . : : parthasarathy , s. 1951 . light scattering in gases . _ indian j. phys . * 25 * _ , 21 - 24 . : : penndorf , r. 1957 . tables of the refractive index for standard air and the rayleigh scattering coefficient for the spectral region between 0.2 and 20 @xmath2 m and their application to atmospheric optics . * 47 * _ , 176 - 182 . : : placzek , g. , 1959 . _ rayleigh and raman scattering _ , pp . 206 , university of california , lawrence radiation laboratory ( livermore , ca ) ; [ translated from handbuch der radiologie , ed . e. marx , vol . 6 , part ii , end ed . , ( akademische verlagsgesellschaft , leipzig , 1934 ) , pp . 205 - 374 ] : : pollack , j. b. , k. rages , k. h. baines , j. t. bergstralh , d. wenkert , and g. e. danielson 1986 . estimates of the bolometric albedos and radiation balance of uranus and neptune . _ icarus * 65 * _ , 442 - 466 . : : pryor , w.r . west , k.e . simmons , and m. delitsky 1992 . high - phase angle observations of neptune at 2650 and 7500 : haze structure and particle properties . _ icarus * 99 * _ , 302 - 316 . : : rottman , g. j. , t. n. woods , t. p. sparn 1993 . solar - stellar irradiance comparison experiment i : 1 . instrument design and operation . _ j. geophys 98 * _ , 10667 - 10677 . : : savage , b. d. , w. d. cochran , and p. r. wesselius 1980 . ultraviolet albedos of uranus and neptune _ the astrophys . j. * 237 * _ , 627 - 632 . : : soris , c. e. , and f. j. evans 1999 . filling in of fraunhofer and gas - absorption lines in sky spectra as caused by rotational raman scattering . _ applied optics * 38 * _ , 2706 - 2713 . : : sromovsky , l. a. , 2004 . effects of rayleigh - scattering polarization on reflected intensity : a fast and accurate approximation for atmospheres with aerosols . _ icarus * in press . * _ : : sromovsky , l. a. , p. m. fry , s. s. limaye , and k. h. baines 2003 . the nature of neptune s increasing brightness : evidence for a seasonal response . _ icarus * 163 * _ , 256 - 261 . : : sromovsky , l.a . , p.m. fry , k.h . baines , and t. dowling 2001a . coordinated 1996 hst and irtf observations of neptune and triton ii : implications of disk - integrated photometry , _ icarus * 149 * _ , 435 - 458 . : : sromovsky , l.a . , p.m. fry , t. dowling , k.h . baines , and s.s . limaye 2001b . coordinated 1996 hst and irtf observations of neptune and triton iii : neptune s atmospheric circulation and cloud structure , _ icarus * 149 * _ , 459 - 488 . : : stamnes , k. , s .- c . tsay , w. wiscombe , and k. jayaweera 1988 . numerically stable algorithm for discrete - ordinate - method radiative transfer in multiple scattering and emitting layered media . * 27 * _ , 2502 - 2509 . : : sweigart , a. v. 1970 . radiative transfer in atmospheres scattering according to the rayleigh phase function with absorption . _ astrophys . j. * 182 * _ , 1 - 80 . : : toon , o. b. , c. p. mckay , t. p. ackerman , k. santhanam 1989 . rapid calculations of radiative transfer heating rates and photodissociation rates in inhomogeneous multiple scattering atmospheres . _ j. geophys . res * 94 * _ , 16287 - 16301 . : : wagener , r. , j. caldwell , and k - h . fricke 1986 . the geometric albedos of uranus and neptune between 2100 and 3350 . _ icarus * 67 * _ , 281 - 288 . : : wallace , l. , 1972 . rayleigh and raman scattering by h@xmath0 in a planetary atmosphere . _ astrophys . j. * 176 * _ , 249 - 257 . : : woods , t. n. , g. j. ucker , g. j. rottman 1993 . solar - stellar irradiance comparison experiment i : 2 . instrument calibration . _ j. geophys . 98 * _ , 10679 - 10694 .

raman scattering by h@xmath0 in neptune s atmosphere has significant effects on its reflectivity for @xmath1 0.5 @xmath2 m , producing baseline decreases of @xmath3 20% in a clear atmosphere and @xmath3 10% in a hazy atmosphere . however , few accurate raman calculations are carried out because of their complexity and computational costs . here we present the first radiation transfer algorithm that includes both polarization and raman scattering and facilitates computation of spatially resolved spectra . new calculations show that cochran and trafton s ( 1978 , _ astrophys . j. * 219 * _ , 756 - 762 ) suggestion that light reflected in the deep ch@xmath4 bands is mainly raman scattered is not valid for current estimates of the ch@xmath4vertical distribution , which implies only a 4% raman contribution . comparisons with iue , hst , and groundbased observations confirm that high altitude haze absorption is reducing neptune s geometric albedo by @xmath36% in the 0.22 - 0.26 @xmath2 m range and by @xmath313% in the 0.35 - 0.45 @xmath2 m range . a sample haze model with 0.2 optical depths of 0.2-@xmath2 m radius particles between 0.1 and 0.8 bars fits reasonably well , but is not a unique solution . we used accurate calculations to evaluate several approximations of raman scattering . the karkoschka ( 1994 , _ icarus * 111 * _ , 174 - 192 ) method of applying raman corrections to calculated spectra and removing raman effects from observed spectra is shown to have limited applicability and to undercorrect the depths of weak ch@xmath4 absorption bands . the relatively large q - branch contribution observed by karkoschka is shown to be consistent with current estimates of raman cross sections . the wallace ( 1972 , _ astrophys . j. * 176 * _ , 249 - 257 ) approximation , produces geometric albedo values @xmath35% low as originally proposed , but can be made much more accurate by including a scattering contribution from the vibrational transition . the original pollack ( 1986 , _ icarus * 65 * _ , 442 - 466 ) approximation is inaccurate and unstable , but can be greatly improved by several simple modifications . a new approximation based on spectral tuning of the effective molecular single scattering albedo provides low errors for zenith angles below 70 in a clear atmosphere , although intermediate clouds present problems at longer wavelengths .

1504.02726

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 .

1701.05709

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 .

0708.1123

henri poincar introduced the idea of a cross section to a flow to study the 3-body problem . a global cross section to a flow @xmath0 on a manifold @xmath1 is a codimension one submanifold @xmath2 of @xmath1 such that @xmath2 intersects every orbit of @xmath0 transversely . it is natural to ask whether any given non - singular flow admits one . if @xmath2 is a global cross section for @xmath0 , it is not hard to check that every orbit which starts on @xmath2 returns to @xmath2 after some positive time , defining the poincar first - return map @xmath3 . the analysis of @xmath0 can then be reduced to the study of the map @xmath4 , which in principle can be an easier task . the flow can be reconstructed from the poincar map by suspending it ( cf . , @xcite ) . the object of this paper is to investigate the existence of global cross sections to volume - preserving anosov flows . recall that a non - singular flow @xmath5 on a closed ( compact and without boundary ) riemannian manifold @xmath1 is called if there exists an invariant splitting @xmath6 of the tangent bundle of @xmath1 and uniform constants @xmath7 , @xmath8 and @xmath9 such that the @xmath10 is spanned by the infinitesimal generator @xmath11 of the flow and for all @xmath12 , @xmath13 , and @xmath14 , we have @xmath15 and @xmath16 where @xmath17 denotes the derivative ( or tangent map ) of @xmath18 . we call @xmath19 and @xmath20 the and ; @xmath21 and @xmath22 are called the and . it is well - known @xcite that all of them are hlder continuous and uniquely integrable @xcite . the corresponding foliations will be denoted by @xmath23 , and @xmath24 . they are also hlder continuous in the sense that each one admits hlder foliation charts . this means that if @xmath25 ( @xmath26 ) is @xmath27 , then every point in @xmath1 lies in a @xmath27 chart @xmath28 such that in @xmath29 the local @xmath25-leaves are given by @xmath30 , where @xmath31 is a @xmath27 homeomorphism and @xmath32 is the dimension of @xmath25 . the leaves of all invariant foliations are as smooth as the flow . see also @xcite for a discussion of regularity of hlder foliations . [ [ sec : textbfprevious - work ] ] * related work . * + + + + + + + + + + + + + + + the first results on the existence of global cross sections to anosov flows were proved by plante in @xcite . he showed that if @xmath33 is a uniquely integrable distribution or equivalently , if the foliations @xmath34 and @xmath35 are jointly integrable - holonomy between local @xmath36-leaves takes local @xmath34-leaves to @xmath34-leaves . ] , then the anosov flow admits a global cross section . sharp @xcite showed that a transitive anosov flow admits a global cross section if it is not homologically full ; this means that for every homology class @xmath37 there is a closed @xmath0-orbit @xmath38 whose homology class equals @xmath39 . ( this is equivalent to the condition that there is _ no _ fully supported @xmath0-invariant ergodic probability measure whose asymptotic cycle in the sense of schwartzman @xcite is trivial . ) along different lines bonatti and guelman @xcite showed that if the time - one map of an anosov flow can be @xmath40 approximated by axiom a diffeomorphisms , then the flow is topologically equivalent to the suspension of an anosov diffeomorphism . let @xmath41 and @xmath42 . if @xmath43 or @xmath44 , the anosov flow is said to be of codimension one . in the discussion that follows we always assume @xmath43 . in @xcite ghys proved the existence of global cross sections for codimension one anosov flows in the following cases : ( 1 ) if @xmath45 is @xmath40 and @xmath46 ( in this case the global cross section has constant return time ) ; if ( 2 ) the flow is volume - preserving , @xmath47 and @xmath36 is of class @xmath48 . this was generalized by the author in @xcite and @xcite where we showed that a codimension one anosov flow admits a global cross section if any of the following assumptions is satisfied : ( 1 ) @xmath49 is lipschitz ( in the sense that it is locally spanned by lipschitz vector fields ) and @xmath47 ; ( 2 ) the flow is volume - preserving , @xmath50 , and @xmath49 is @xmath27-hlder for _ all _ @xmath51 ( 3 ) the flow is volume - preserving , @xmath47 , and @xmath52 is of class @xmath53 for _ all _ @xmath54 . note that all the regularity assumptions above require that the invariant bundles be smoother than they usually are : @xmath49 is generically only hlder continuous and in the codimension one case , @xmath52 is generically only @xmath53 for some small @xmath55 . see @xcite and @xcite . the goal of this paper is to establish the following result . let @xmath5 be a volume - preserving anosov flow on a closed riemannian manifold @xmath1 and let @xmath56 be the smaller of the hlder exponents of @xmath34 and @xmath36 . if @xmath57 then @xmath0 admits a global cross section . * the condition has a chance of being satisfied only if @xmath58 is much smaller than @xmath59 . if @xmath60 , then by reversing time it is easy to show that @xmath61 also implies the existence of a global cross section , where @xmath39 is the minimum of the hlder exponents of @xmath20 and @xmath62 . * if the flow is of codimension one with @xmath43 , then reduces to @xmath63 it is well - known ( cf . , @xcite and @xcite ) that the center stable bundle @xmath52 and strong unstable bundle @xmath20 of a volume - preserving anosov flow in dimensions @xmath47 are both @xmath64 . thus if @xmath49 is lipschitz as in @xcite or @xmath27 , for all @xmath54 , as in @xcite , then is clearly satisfied . if @xmath52 is @xmath65 for all @xmath66 as in @xcite , then it is not hard to show that @xmath19 is necessarily of class @xmath27 for all @xmath54 , which again implies . therefore , in the case of volume - preserving codimension one anosov flows , our result implies all the previously known criteria for the existence of global cross sections . * in the early 1970 s , prompted by a dearth of examples , a. verjovsky conjectured @xcite that every codimension one anosov flow in dimensions @xmath47 admits a global cross section . the importance of verjovsky s conjecture stems from the fact that codimension one anosov diffeomorphisms were classified by franks @xcite and newhouse @xcite who showed that every such diffeomorphism is topologically conjugate to a linear hyperbolic automorphism of a torus . therefore , the affirmation of verjovsky s conjecture would yield a complete classification of codimension one anosov flows in dimensions @xmath67 . + progress towards verjovsky s conjecture was made in the early 1980 s by plante @xcite and armendariz @xcite who showed that the conjecture holds if the fundamental group of the manifold is solvable . by the work of asaoka @xcite it follows that it suffices to prove the conjecture for volume - preserving flows , since any topologically transitive codimension one anosov flow is topologically equivalent to a volume - preserving one . as of this writing , the conjecture remains open . throughout this paper smooth will mean of class @xmath68 . + [ [ outline - of - the - proof . ] ] * outline of the proof . * + + + + + + + + + + + + + + + + + + + + + + + the main idea of the proof of the theorem is to find a smooth closed 1-form @xmath69 such that @xmath70 , where @xmath11 is the infinitesimal generator of the anosov flow . it is not hard to see that this immediately implies the existence of a global cross section ( cf . , [ sec : proof ] ) . to construct @xmath69 , we use the fact that for any @xmath32-form @xmath71 , the @xmath72-distance from @xmath71 to the space of closed @xmath32-forms is bounded above by the @xmath72 norm @xmath73 ; see proposition [ prop : forms ] . it therefore suffices to construct a smooth 1-form @xmath71 such that @xmath74 , for all @xmath75 , since proposition [ prop : forms ] then yields a smooth closed 1-form @xmath69 such that @xmath70 . we will actually construct a smooth 1-form @xmath71 such that @xmath76 and @xmath77 . the construction of @xmath71 is divided into two steps . in the first step , we find an initial candidate for @xmath71 such that the norm of its exterior derivative restricted to @xmath52 is small , while its total norm blows up in a way controlled by the hlder exponent @xmath78 . more precisely , for each @xmath79 we construct a smooth 1-form @xmath80 on @xmath1 such that @xmath81 , @xmath82 , and @xmath83 , where @xmath84 and @xmath85 are positive constants independent of @xmath86 . this is achieved by carefully building smooth local cross sections and the corresponding flow boxes ; cf . , [ sec : local - sections ] . in the second step , we pull back @xmath80 by @xmath87 for suitable @xmath88 to make the norm of its exterior derivative small in the remaining directions . for this , we use an estimate ( see lemma [ lem : backward ] ) on the growth of @xmath89 , for @xmath88 , @xmath12 and @xmath13 . assuming , we then show that there exist @xmath79 and @xmath90 such that @xmath91 has the desired properties . here we recall a standard technique for approximating locally integrable functions by smooth ones called or ( see @xcite ) . define the standard mollifier @xmath92 by @xmath93 where @xmath94 is chosen so that @xmath95 . for every @xmath79 , set @xmath96 . note that the support of @xmath97 is contained in the ball @xmath98 of radius @xmath86 centered at the origin and that @xmath99 . for a locally integrable function @xmath100 define @xmath101 [ prop : reg ] assume that @xmath102 is locally integrable . then : 1 . @xmath103 . 2 . if @xmath104 , then @xmath105 . 3 . if @xmath106 is continuous , then @xmath107 , uniformly as @xmath108 . if @xmath109 @xmath110 , then @xmath111 , where the @xmath27-norm is the sum of its sup - norm and its best hlder constant . if @xmath109 , then @xmath112 where @xmath113 and @xmath114 denotes the maximum of the sup - norms of the partial derivatives of @xmath115 . proofs of ( a ) , ( b ) and the first part of ( c ) can be found in @xcite . for the second part of ( c ) , we have @xmath116 \ , dy \right\rvert \\ & \leq { \left\lvertu\right\rvert } { \varepsilon}^\theta \int_{b(0,{\varepsilon } ) } \eta_{\varepsilon}(y ) \ , dy \\ & = { \left\lvertu\right\rvert}_{c^\theta } { \varepsilon}^\theta . \end{aligned}\ ] ] if @xmath117 , then the same estimates hold with @xmath78 replaced by @xmath118 . observe that since @xmath97 has compact support , @xmath119 for @xmath120 . note also that @xmath121 assuming @xmath109 , we obtain ( d ) : @xmath122 \frac{{\partial}\eta_{\varepsilon}}{{\partial}x_i}(y ) \ , dy \right\rvert , \\ & \leq { \left\lvertu\right\rvert}_{c^\theta } \ , { \varepsilon}^\theta \int_{b(0,{\varepsilon } ) } \left\lvert \frac{{\partial}\eta_{\varepsilon}}{{\partial}x_i}(y ) \right\rvert \ , dy \\ & = { \left\lvertu\right\rvert}_{c^\theta } \ , { \varepsilon}^\theta \int_{b(0,{\varepsilon } ) } \frac{1}{{\varepsilon}^{n+1 } } \left\lvert \frac{{\partial}\eta}{{\partial}x_i}\left(\frac{y}{{\varepsilon}}\right ) \right\rvert \ , dy \\ & \overset{z = \frac{y}{{\varepsilon}}}{= } { \left\lvertu\right\rvert}_{c^\theta } \ , { \varepsilon}^\theta \cdot \frac{1}{{\varepsilon } } \int_{b(0,1 ) } \left\lvert \frac{{\partial}\eta}{{\partial}x_i}(z ) \right\rvert \ , dz , \\ & \leq { \left\lvertd\eta\right\rvert}_{l^1 } { \left\lvertu\right\rvert}_{c^\theta } \ , { \varepsilon}^{\theta-1}. \qedhere \end{aligned}\ ] ] [ cor : mollify ] let @xmath1 be a compact manifold without boundary . fix a finite atlas @xmath123 of @xmath1 . if @xmath124 is @xmath27 and @xmath125 for some @xmath126 , then there exist @xmath127 and a family of smooth approximations @xmath115 @xmath128 of @xmath106 such that : 1 . @xmath115 is defined on @xmath129 where @xmath130 . 2 . @xmath131 . 3 . @xmath132 . here @xmath133 depends only on @xmath134 and @xmath135 . the family @xmath136 \circ \varphi_j$ ] has the desired properties . we can take @xmath137 to be any positive number such that for @xmath138 , the sets @xmath139 defined above are non - empty . in this section we construct a finite covering of the manifold by smooth flow boxes defined by carefully chosen smooth local cross sections . let @xmath75 be arbitrary and choose a @xmath34-foliation chart @xmath28 containing @xmath140 . this means that @xmath141 is a @xmath27-homeomorphism such that the local @xmath34-leaves in @xmath29 are given by @xmath142 where @xmath143 . we can also arrange that the local @xmath36-leaves in @xmath29 are defined by @xmath144 so that in each local @xmath36-leaf in @xmath29 @xmath34 is given by @xmath145 . furthermore , we can take @xmath29 so that its closure is contained in a _ smooth _ chart for @xmath1 . this condition will allow us to mollify continuous functions defined on @xmath29 without having to shrink the domain . even though @xmath146 is only hlder , the flow invariance of @xmath34 implies that each @xmath147 is differentiable with respect to @xmath11 . since @xmath11 is tangent to the @xmath36 leaves , it follows that @xmath148 , for @xmath149 . furthermore , the restriction of @xmath34 to @xmath36-leaves is as smooth as the flow , so @xmath150 is smooth on the local @xmath36-leaves in @xmath29 . since @xmath11 is uniformly transverse to @xmath34 it is clear that @xmath151 and by continuity there exists @xmath152 such that @xmath153 on @xmath29 . define @xmath154 where @xmath155 denotes the local @xmath25-leaf in @xmath29 ( for @xmath156 ) . then @xmath157 is a hlder continuous local cross section for the flow . ( this makes sense , since the intersection of @xmath157 with each local @xmath36-leaf is a local @xmath34-leaf , hence smooth and transverse to the flow . ) let @xmath2 be a slightly smaller compact subset of @xmath157 containing @xmath140 . we claim that there exists @xmath158 , depending on @xmath2 and @xmath29 , such that @xmath159 is a hlder continuous flow box contained in @xmath29 . assume the contrary ; it follows that the map @xmath160 fails to be 11 on @xmath161 , for any @xmath158 . thus there exist sequences @xmath162 and @xmath163 such that @xmath164 , @xmath165 , @xmath166 and @xmath167 , for all @xmath32 . since @xmath168 and @xmath169 lie in the same local @xmath36-leaf , it follows that @xmath170 . on the other hand , by compactness of @xmath2 , @xmath162 has a subsequence @xmath171 which converges to some @xmath172 . this implies @xmath173 , which contradicts @xmath174 . therefore , @xmath175 exists ; we will call it the _ length _ of the continuous flow box @xmath176 ( although a more appropriate name would be half - length ) . define @xmath177 by @xmath178 for @xmath179 and @xmath180 . it is clear that @xmath181 is @xmath27 , @xmath182 and @xmath181 is constant on the local @xmath34-leaves in @xmath176 . next we approximate @xmath183 and @xmath176 by smooth objects with similar properties . [ lem : tau ] there exists @xmath127 such that for every @xmath184 there exist an open set @xmath185 and a smooth function @xmath186 with the following properties : 1 . 2 . @xmath188 , for all @xmath189 and @xmath12 , where @xmath190 is a constant independent of @xmath86 , @xmath191 , and @xmath192 . @xmath193 , where @xmath194 is a constant independent of @xmath86 . the hausdorff distance between @xmath176 and @xmath195 tends to zero , as @xmath108 . for the sake of notational simplicity , we will write @xmath196 ( @xmath197 ) to mean that there exists a constant @xmath198 independent of @xmath197 such that @xmath199 , for all @xmath197 . since @xmath29 is contained in a smooth chart for @xmath1 , by corollary [ cor : mollify ] there exists @xmath137 such that for each @xmath200 there is a family @xmath201 ( @xmath184 ) of smooth approximations of @xmath147 satisfying @xmath202 note that @xmath115 is defined on all of @xmath29 . denote by @xmath203 the foliation of @xmath139 defined by @xmath204 it is easy to see that @xmath203 is smooth and the ( largest principal ) angle between @xmath205 and @xmath203 is @xmath206 . let @xmath207 then @xmath208 is a smooth local cross section and there exists @xmath209 such that the set @xmath210 is a smooth flow box for @xmath11 contained in @xmath139 . it is clear that the length @xmath211 of @xmath195 is close to the length @xmath175 of @xmath176 ; we can also take @xmath212 . define @xmath213 by @xmath214 for all @xmath215 and @xmath216 . clearly , @xmath217 is smooth and @xmath187 , proving ( a ) . with local cross section @xmath208 . ] we will first show that ( b ) holds along @xmath208 . let @xmath215 be arbitrary . since the angle between @xmath218 and @xmath219 is @xmath206 , it follows that the angle between @xmath220 and @xmath219 is also @xmath206 . see figure [ fig : flowbox ] . then the fact that @xmath221 on @xmath220 implies @xmath222 , as desired . to extend this to all points in @xmath195 we will use the invariance of @xmath223 with respect to the flow : @xmath224 , whenever both sides are defined . for @xmath215 , @xmath225 and @xmath226 , we have @xmath227 where we used @xmath212 . this proves ( b ) for all @xmath189 . to prove ( c ) , we work in a smooth coordinate system in which @xmath228 . since @xmath2 is a @xmath27 hypersurface , it is locally the graph of a @xmath27 function @xmath4 such that @xmath229 , for all @xmath230 is some open set in @xmath231 . similarly , @xmath208 is locally the graph of a smooth function @xmath232 such that @xmath233 , for all @xmath230 in some open set in @xmath231 . since @xmath234 , as @xmath108 , forces @xmath235 . differentiating @xmath233 with respect to @xmath236 for @xmath237 and using @xmath238 , we obtain @xmath239 thus ( c ) holds on @xmath208 ; we can extend it to @xmath195 using the flow invariance of @xmath223 as in the proof of ( b ) . part ( d ) holds by construction . by the above analysis and compactness , we can cover @xmath1 by finitely many hlder flow boxes @xmath240 , each of which is equipped with a local hlder cross section @xmath241 . we can approximate each @xmath241 by a smooth cross section @xmath242 as above and obtain smooth flow boxes @xmath243 and smooth functions @xmath244 satisfying the properties from lemma [ lem : tau ] ; namely , @xmath245 where the constants @xmath246 are independent of @xmath86 . let @xmath247 ( not to be confused with the constants @xmath248 in lemma [ lem : tau ] ) . by lemma [ lem : tau ] ( d ) , there exists @xmath127 such that for all @xmath184 the sets @xmath249 cover @xmath1 . we will consider @xmath250 differential @xmath32-forms @xmath71 , with @xmath251 . we denote the @xmath72 norm of @xmath71 on @xmath1 by @xmath252 : @xmath253 where @xmath254 is is the operator norm of @xmath255 as a @xmath32-linear map @xmath256 . consider first a differential form @xmath257 on @xmath258 $ ] , where @xmath259 is the coordinate in @xmath260 $ ] . denote by @xmath261 \to m$ ] and @xmath262 \to [ 0,1]$ ] the obvious projections . since @xmath263 ) = t_p m \oplus t_t [ 0,1],\ ] ] any differential @xmath32-form on @xmath258 $ ] can be uniquely written as @xmath264 where @xmath265 if some @xmath266 is in the kernel of @xmath267 and @xmath69 is a @xmath268-form with the analogous property ( i.e. , @xmath269 , for every `` vertical '' vector @xmath270)$ ] , where @xmath271 denotes contraction by @xmath192 ) . define a @xmath268-form @xmath272 on @xmath1 by @xmath273 where @xmath274 $ ] is defined by @xmath275 . it is well - known ( cf . , @xcite ) that @xmath276 considering @xmath277 as a linear operator from @xmath278)$ ] to @xmath279 , both equipped with the @xmath72 norm , it is not hard to see that @xmath280 we claim : [ prop : forms ] let @xmath71 be a @xmath250 differential @xmath32-form ( @xmath281 ) on a closed manifold @xmath1 . then @xmath282 in other words , @xmath73 is an upper bound on the distance from @xmath71 to the space of closed forms . the inequality also holds for continuous forms which admit a continuous exterior derivative . first , let us show that the result holds on any manifold @xmath1 which is smoothly contractible to a point @xmath283 via @xmath284 \to m$ ] , where @xmath285 and @xmath286 , for all @xmath287 . since @xmath288 is the identity map of @xmath1 and @xmath289 is the constant map @xmath283 , it follows that @xmath290 applying to @xmath291 , we obtain @xmath292 using , we obtain @xmath293 therefore , the statement of the theorem holds for contractible @xmath1 . let @xmath1 now be any closed manifold and @xmath71 a @xmath250 @xmath32-form on @xmath1 , @xmath281 . cover @xmath1 by contractible open sets @xmath294 . denote the operator @xmath277 restricted to forms on @xmath295 by @xmath296 and let @xmath297 be the restriction of @xmath71 to @xmath295 . define a @xmath32-form @xmath69 on @xmath1 by requiring that the restriction of @xmath69 to @xmath295 be equal to @xmath298 . we claim that @xmath69 is well - defined and closed . indeed , @xmath299 on @xmath300 , and @xmath301 for every @xmath32-form @xmath257 defined on @xmath302 $ ] . thus on @xmath300 , we have @xmath303 , so @xmath69 is well - defined . since @xmath69 is locally exact , it follows that it is closed . by , we obtain @xmath304 this completes the proof of the proposition . denote by @xmath305 the smooth volume form preserved by the flow and let @xmath306 be the riemannian metric which induces @xmath305 . our goal is to show that relative to some riemannian metric the area of the parallelogram @xmath307 grows as @xmath308 , where @xmath12 , @xmath13 , and @xmath309 . to do this , it will be convenient to switch from the original riemannian metric @xmath306 to a new metric @xmath310 with respect to which @xmath11 is a unit vector and @xmath311 is an orthogonal splitting . this metric can in general be only continuous and the corresponding volume form @xmath312 may not be invariant with respect to the flow . we will show that this does not present a problem . let @xmath310 be as above and let @xmath312 be the riemannian volume form induced by @xmath310 . since @xmath305 and @xmath312 are both volume forms , there exists a positive continuous function @xmath313 such that @xmath314 . let @xmath315 denote the norms of tangent vectors ( and their wedge products ) with respect to @xmath306 and @xmath310 by @xmath316 and @xmath317 , respectively . by compactness of @xmath1 there exist @xmath318 such that @xmath319 for all @xmath320 . observe that for @xmath12 , @xmath321 , and @xmath14 , we have @xmath322 where @xmath323 . it is easy to check that @xmath324 for all @xmath325 . thus for any @xmath326-dimensional parallelepiped @xmath327 in a tangent space to @xmath1 and @xmath325 , we have @xmath328 [ lem : backward ] if @xmath5 is a volume preserving anosov flow with constants defined in and , then @xmath329 for all @xmath12 , @xmath13 and @xmath14 . let @xmath12 , @xmath13 and @xmath14 be arbitrary . set @xmath330 and @xmath331 . choose vectors @xmath332 and @xmath333 such that , relative to @xmath310 , @xmath334 is an orthogonal basis of the corresponding tangent space and @xmath335 are all of unit length , for @xmath336 . then : @xmath337 in the remainder of the paper we will always be working with @xmath310 as the underlying riemannian metric on @xmath1 ; the norms of tangent vectors and differential forms are taken relative to @xmath310 and will be denoted by the symbol @xmath316 ( thus slightly abusing notation for the sake of keeping it less cumbersome ) . we will construct a smooth closed 1-form @xmath69 such that @xmath338 . assuming for a moment that such a form has been found , the proof can be completed as follows . define @xmath339 then @xmath340 is an anosov vector field @xcite and @xmath341 . thus the lie derivative of @xmath69 with respect to @xmath340 satisfies @xmath342 which implies that @xmath69 is invariant with respect to the flow . it follows that its kernel @xmath343 is an invariant codimension one distribution transverse to the flow , so @xmath343 is forced to be the sum @xmath344 of the strong stable and strong unstable bundles of @xmath340 . since @xmath69 is closed , @xmath345 is uniquely integrable , so by plante @xcite , @xmath340 admits a global cross section @xmath2 , which is also a global cross section for @xmath11 . so it remains to construct a smooth 1-form @xmath69 with @xmath70 , which will be done in three steps . in the first two steps we construct a smooth 1-form @xmath71 such that @xmath346 the third step consists of approximating @xmath71 by a smooth closed 1-form using proposition [ prop : forms ] . + let @xmath79 be arbitrary . as in [ sec : local - sections ] , for @xmath347 , we can cover @xmath1 be smooth flow boxes @xmath348 , with @xmath349 independent of @xmath86 , such that with respect to the hausdorff distance each @xmath350 is close to a fixed open set @xmath351 . in addition , we have smooth functions @xmath244 such that @xmath352 where @xmath248 are constants independent of @xmath86 . let @xmath353 be a smooth partition of unity subordinate to the cover @xmath354 . since @xmath349 and the sizes of the sets @xmath350 are independent of @xmath86 , there is a constant @xmath355 also independent of @xmath86 such that @xmath356 for all @xmath184 . define @xmath357 * @xmath81 ; * @xmath358 , for all @xmath184 ; * @xmath359 , for all unit vectors @xmath360 and @xmath184 , where @xmath84 is a constant independent of @xmath86 ; * @xmath83 , for every @xmath184 , where @xmath85 is a constant independent of @xmath86 . part ( a ) is clear . part ( b ) follows easily from the second inequality in . to prove ( c ) , first note that if @xmath361 is a bilinear form on an inner product space @xmath362 which splits into two orthogonal subspaces @xmath363 and @xmath364 and @xmath365 , then @xmath366 we fix @xmath75 and take @xmath367 , @xmath368 and @xmath369 . since @xmath370 if @xmath371 are unit vectors , then @xmath372 and @xmath373 where we used @xmath374 and @xmath375 . thus by we can take @xmath376 in ( c ) . finally , and @xmath377 imply @xmath378 so ( d ) holds with @xmath379 . observe that if @xmath361 is a bilinear form as in the proof of the previous lemma and @xmath384 , for all @xmath385 , then @xmath386 fix @xmath75 and take @xmath387 , @xmath388 , @xmath389 and @xmath390 . since @xmath391 and @xmath392 , for @xmath88 , lemmas [ lem : xi0 ] and [ lem : backward ] imply @xmath393 note that the second inequality holds because if @xmath394 and @xmath13 , then @xmath395 is largest if @xmath12 . observe also that @xmath396 is smaller than @xmath397 ( @xmath398 ) since the flow contracts strong unstable vectors in negative time . by the previous lemma , it suffices to show that the following system of inequalities @xmath402 admits a solution @xmath403 . solving the first inequality for @xmath259 we obtain @xmath404 the second inequality is equivalent to @xmath405 there exists @xmath259 with these properties if and only if we can find @xmath184 such that @xmath406 since @xmath407 is fixed and @xmath86 is small , this is possible if @xmath408 which is equivalent to . choose @xmath184 and @xmath88 such that @xmath401 and set @xmath409 by proposition [ prop : forms ] there exists a smooth closed 1-form @xmath69 such that @xmath410 since @xmath76 , it follows that @xmath70 . this completes the construction of @xmath69 and concludes the proof of the theorem .

we provide a new criterion for the existence of a global cross section to a volume - preserving anosov flow . the criterion is expressed in terms of expansion and contraction rates of the flow and is more general than the previous results of similar kind .

1403.2672

among biometrics , fingerprints are probably the best - known and widespread because of the fingerprint properties : universality , durability and individuality . unfortunately it has been shown that fingerprint scanners are vulnerable to presentation attacks with an artificial replica of a fingerprint . therefore , it is important to develop countermeasures to those attacks . numerous methods have been proposed to solve the susceptibility of fingerprint devices to attacks by spoof fingers . one primary countermeasure to spoofing attacks is called `` liveness detection '' or presentation attack detection . liveness detection is based on the principle that additional information can be garnered above and beyond the data procured and/or processed by a standard verification system , and this additional data can be used to verify if an image is authentic . liveness detection uses either a hardware - based or software - based system coupled with the authentication program to provide additional security . hardware - based systems use additional sensors to gain measurements outside of the fingerprint image itself to detect liveness . software - based systems use image processing algorithms to gather information directly from the collected fingerprint to detect liveness . these systems classify images as either live or fake . since 2009 , in order to assess the main achievements of the state of the art in fingerprint liveness detection , university of cagliari and clarkson university organized the first fingerprint liveness detection competition . the first international fingerprint liveness detection competition ( livdet ) 2009 @xcite , provided an initial assessment of software systems based on the fingerprint image only . the second , third and fourth liveness detection competitions ( livdet 2011 @xcite , 2013 @xcite and 2015 @xcite ) were created in order to ascertain the progressing state of the art in liveness detection , and also included integrated system testing . this paper reviews the previous livdet competitions and how they have evolved over the years . section 2 of this paper describes the background of spoofing and liveness detection . section 3 details the methods used in testing for the livdet competitions as well as descriptions of the datasets that have generated from the competition so far . section 4 discusses the trends across the competitions reflecting advances in the state of the art . section 5 concludes the paper and discusses the future of the livdet competitions . the concept of spoofing has existed for some time now . research into spoofing can be seen beginning in 1998 from research conducted by d. willis and m. lee where six different biometric fingerprint devices were tested against fake fingers and it was found that four of the six were susceptible to spoofing attacks @xcite . this research was approached again in 2000 - 2002 by multiple institutions including ; putte and kuening as well as matsumoto et al . putte et al . examined different types of scanning devices as well as different ways of counterfeiting fingerprints @xcite . the research presented by these researchers looked at the vulnerability of spoofing . in 2001 , kallo ( et al . ) looked at a hardware solution to liveness detection ; while in 2002 , schuckers delved into using software approaches for liveness detection @xcite . liveness detection , with either hardware - based or software - based systems , is used to check if a presented fingerprint originates from a live person or an artificial finger . usually the result of this analysis is a score used to classify images as either live or fake . many solutions have been proposed to solve the vulnerability of spoofing @xcite . bozhao tan et al . has proposed a solution based on ridge signal and valley noise analysis @xcite . this solution examines the perspiration patterns along the ridge and the patterns of noise in the valleys of images @xcite . it was proposed that since live fingers sweat , but spoof fingers do not , the live fingerprint will look `` patchy '' compared to a spoof @xcite . also it was proposed that due to the properties of a spoof material , spoof fingers will have granules in the valleys that live fingers will not have @xcite . pietro coli et al . examined static and dynamic features of collected images on a large data set of images @xcite . there are two general forms of creating artificial fingers , the cooperative method and non - cooperative method . in the cooperative method the subject pushes their finger into a malleable material such as dental impression material , plastic , or wax creating a negative impression of the fingerprint as a mold , see figure [ fig : mold ] . the mold is then filled with a material , such as gelatin , playdoh or silicone . this cast can be used to represent a finger from a live subject , see figure [ fig : fakefinger ] . the non - cooperative method involves enhancing a latent fingerprint left on a surface , digitizing it through the use of a photograph , and finally printing the negative image on a transparency sheet . this printed image can then be made into a mold , for example , by etching the image onto a printed circuit board ( pcb ) which can be used to create the spoof cast as seen on figure [ fig : latentpcb ] . most competitions focus on matching , such as the fingerprint verification competition held in 2000 , 2002 , 2004 and 2006 @xcite and the icb competition on iris recognition ( icir2013 ) @xcite . however , these competitions did not consider spoofing . the liveness detection competition series was started in 2009 and created a benchmark for measuring liveness detection algorithms , similar to matching performance . at that time , there had been no other public competitions held that has examined the concept of liveness detection as part of a biometric modality in deterring spoof attacks . in order to understand the motivation of organizing such a competition , we observed that the first trials to face with this topic were often carried out with home - made data sets that were not publicly available , experimental protocols were not unique , and the same reported results were obtained on very small data sets . we pointed out these issues in @xcite . therefore , the basic goal of livdet has been since its birth to allow researchers testing their own algorithms and systems on publicly available data sets , obtained and collected with the most updated techiques to replicate fingerprints enabled by the experience of clarkson and cagliari laboratories , both active on this problem since 2000 and 2003 , respectively . at the same time , using a competition " instead of simply releasing data sets , could be assurance of a free - of - charge , third - party testing using a sequestered test set . ( clarkson and cagliari has never took part in livdet as competitors , due to conflict of interest . ) livdet 2009 provided results which demonstrated the state of the art at that time @xcite for fingerprint systems . livdet continued in 2011 , 2013 and 2015 @xcite and contained two parts : evaluation of software - based systems in part 1 : algorithms , and evaluation of integrated systems in part 2 : systems . fingerprint will be the focus of this paper . however , livdet 2013 also included a part 1 : algorithms for the iris biometric @xcite and is continuing in 2015 . since 2009 , evaluation of spoof detection for facial systems was performed in the competition on counter measures to 2-d facial spoofing attacks , first held in 2011 and then held a second time in 2013 . the purpose of this competition is to address different methods of detection for 2-d facial spoofing @xcite . the competition dataset consisted of 400 video sequences , 200 of them real attempts and 200 attack attempts @xcite . a subset was released for training and then another subset of the dataset was used for testing purposes . [ cols="<,>,>,>,>,>,>,>",options="header " , ] table [ tab : easynessdegree ] shows the subjective evaluation on the easiness of obtaining a good spoof from the combination of the same materials of table [ tab : qualitydegree ] . this evaluation depends on the solidification time of the adopted material , the level of difficulty in separating mold and cast without destroying one of them or both , the natural dryness or wetness level of the related spoof . tables [ tab : qualitydegree ] , [ tab : easynessdegree ] show that , from a practical viewpoint , many materials are difficult to manage when fabricating a fake finger . in many cases , the materials with this property also exhibit a low subjective quality level . therefore , thanks to this lesson , the livdet competition challenge participants with images coming from the spoofs obtained with the best and most `` potentially dangerous '' materials . the materials choice is made on the basis of the best trade off between the criteria pointed out in tables [ tab : qualitydegree ] , [ tab : easynessdegree ] and the objective quality values output by quality assessment algorithms such as nfiq . what reported has been confirmed along the four livdet editions . in particular the ferrfake and ferrlive rates for each differing quality levels support the idea that the images quality level is correlated with the error rate decrease . the error rates for each range of quality levels for dermalog in livdet 2011 fingerprint part 1 : algorithms is shown in figure [ fig : ferrderm ] , as an example . the graphs showcase from images of only quality level 1 up to all quality levels being shown . as lower quality spoof images were added , ferrfake generally decreased . for all images which included the worst quality images , the error rates were less consistent likely due to the variability in low quality spoofs . the percentage of images at each quality level for two representative datasets for livdet 2011 , 2013 , and 2015 , respectively , are given in figures [ fig : imageperc ] , [ fig : spoofperc ] , and [ fig : quality2015 ] . the crossmatch dataset had high percentages of the data being in the top two quality levels in both livdet 2011 and 2013 . the swipe dataset had many images that were read as being of lower quality which could be seen in the data itself because of the difficulty in collecting spoof data on the swipe device . since its first edition in 2009 , the fingerprint liveness detection competition was aimed to allow research centres and companies a fair and independent assessment of their anti - spoofing algorithms and systems . we have seen over time an increasing interest for this event , and the general recognition for the enormous amount of data made publicly available . the number of citations that livdet competitions have collected is one of the tangible signs of such interest ( about 100 citations according to google scholar ) and further demonstrates the benefits that the scientific community has received from livdet events . the competition results show that liveness detection algorithms and systems strongly improved their performance : from about 70% classification accuracy achieved in livdet 2011 , to 90% classification accuracy in livdet 2015 . this result , obtained under very difficult conditions like the ones of the consensual methodology of fingerprints replication , is comparable with that obtained in livdet 2013 ( first two data sets ) , where the algorithms performance was tested under the easier task of fingerprints replication from latent marks . moreover , the two challenges characterizing the last edition , namely , the presence of 1000 dpi capture device and the evaluation against unknown " spoofing materials , further contributed to show the great improvement that researchers achieved on these issues : submitted algorithms performed very well on both 500 and 1000 dpi capture devices , and some of them also exhibited a good robustness degree against never - seen - before attacks . results reported on fusion also shows that the liveness detection could further benefit from the combination of multiple features and approaches . a specific section on algorithms and systems fusion might be explicitly added to a future livdet edition . there is a dark side of the moon , of course . it is evident that , despite the remarkable results reported in this paper , there is a clear need of further improvements . current performance for most submissions are not yet good enough for embedding a liveness detection algorithm into fingerprint verification system where the error rate is still too high for many real applications . in the authors opinion , discovering and explaining benefits and limitations of the currently used features is still an issue whose solution should be encouraged , because only the full understanding of the physical process which leads to the finger s replica and what features extraction process exactly does will shed light on the characteristics most useful for classification . we are aware that this is a challenging task , and many years could pass before seeing concrete results . however , we believe this could be the next challenge for a future edition of livdet , the fingerprint liveness detection competition . the first and second author had equal contributions to the research . this work has been supported by the center for identification technology research and the national science foundation under grant no . 1068055 , and by the project computational quantum structures at the service of pattern recognition : modeling uncertainty " [ crp-59872 ] funded by regione autonoma della sardegna , l.r . 7/2007 , bando 2012 . marcialis , et al . , first international fingerprint liveness detection competition livdet 2009 . d. yambay , et al . , livdet 2011 - fingerprint liveness detection competition 2011 , 5th iapr / ieee int . conf . on biometrics ( icb 2012 ) , new delhi ( india ) , march , 29th , april , 1st , 2012 . marcialis , et al . , livdet 2013 fingerprint liveness detection competition 2013 , 6th iapr / ieee int . conf . on biometrics ( icb2013 ) , madrid ( spain ) , june , 4th , june , 7 , 2013 . marcialis , et al . , livdet 2015 fingerprint liveness detection competition 2015 , 7th ieee int . conf . on biometrics : theory , applications and systems ( btas 2015 ) , in press . d. yambay , et al . , livdet - iris 2013-iris liveness detection competition 2013 . ieee international joint conference on biometrics ( ijcb 2014 ) , 2014 . c. sousedik , and c. busch , presentation attack detection methods for fingerprint recognition systems : a survey , iet biometrics , 2014 . p. coli , g.l . marcialis , and f. roli , vitality detection from fingerprint images : a critical survey , ieee / iapr 2nd international conference on biometrics icb 2007 , august , 27 - 29 , 2007 , seoul ( korea ) , s .- w . lee and s. li eds . , springer lncs 4642 , pp.722 - 731 . d. willis , m. lee , six biometric devices point the finger at security . biometrics under our thumb , network computing , june 1998 . van der putte , t. and keuning , j. : biometrical fingerprint recognition : do nt get your fingers burned , smart card reserch and advanced applications , ifip tc8/wg8.8 fourth working conference on smart card research and advanced applications , pp . 289 - 303 ( 2001 ) . t. matsumoto , h. matsumoto , k. yamada , and s. hoshino , impact of artificial gummy fingers on fingerprint systems , in proceedings of spie , 4677 , optical security and counterfeit deterence techniques iv , yokohama , japan . peter kallo , imre kiss , andras podmaniczky , all of budapest , janos talosi , negykanizsa , all of ( hu ) . `` detector for recognizing the living character of a finger in a fingerprint recognizing apparatus '' patent us 6,175641 , jan . 16 , 2001 . schuckers sac . spoofing and anti - spoofing measures . information security technical report , vol 7 . 4 , pages 56 - 62 , 2002 . bozhao tan , stephanie schuckers , spoofing protection for fingerprint scanner by fusing ridge signal and valley noise , pattern recognition , volume 43 , issue 8 , august 2010 , pages 2845 - 2857 , issn 0031 - 3203 , doi : 10.1016/j.patcog.2010.01.023 . p. coli , g.l . marcialis , and f. roli , fingerprint silicon replicas : static and dynamic features for vitality detection using an optical capture device , international journal of image and graphics , world scientific , 8 ( 4 ) 495 - 512 , 2008 . cappelli , raffaele , et al . fingerprint verification competition 2006 . " biometric technology today 15.7 ( 2007 ) : 7 - 9 . alonso - fernandez , fernando , and josef bigun . halmstad university submission to the first icb competition on iris recognition ( icir2013 ) . " ( 2013 ) . chakka , murali mohan , et al . competition on counter measures to 2-d facial spoofing attacks . " biometrics ( ijcb ) , 2011 international joint conference on . ieee , 2011 . c. watson , m. garris , e. tabassi , c. wilson , r. mccabe , s. janet , k. ko . user s guide to nist biometric image software . national institute of standards and technology . j. galbally , et al . , a high performance fingerprint liveness detection method based on quality related features , future gener . 28 , 1 ( january 2012 ) , 311 - 321 . doi=10.1016/j.future.2010.11.024 b. biggio , et al . , security evaluation of biometric authentication systems under real spoofing attacks , in biometrics , iet , vol.1 , no.1 , pp.11 - 24 , march 2012 e. marasco , and c. sansone , combining perspiration- and morphology - based static features for fingerprint liveness detection , pattern recognition letters , volume 33 , issue 9 , 1 july 2012 , pages 1148 - 1156 , issn 0167 - 8655 j. galbally , et al . , image quality assessment for fake biometric detection : application to iris , fingerprint , and face recognition , in image processing , ieee transactions on , vol.23 , no.2 , pp.710 - 724 , feb . 2014 , doi : 10.1109/tip.2013.2292332 e. marasco , and c. sansone , an anti - spoofing technique using multiple textural features in fingerprint scanners , in biometric measurements and systems for security and medical applications ( bioms ) , 2010 ieee workshop on , vol . , no . , pp.8 - 14 , 9 - 9 sept . 2010 , doi : 10.1109/bioms.2010.5610440 l. ghiani , et al . , experimental results on fingerprint liveness detection , in proceedings of the 7th international conference on articulated motion and deformable objects ( amdo 2012 ) , springer - verlag , berlin , heidelberg , 210 - 218 d. gragnaniello , et al . , wavelet - markov local descriptor for detecting fake fingerprints , electronics letters , 2014 , 50 , ( 6 ) , p. 439 - 441 , doi : 10.1049/el.2013.4044 , iet digital library p. b. patil , and h. shabahat , an anti - spoofing technique using multiple textural features in fingerprint scanner , international journal of electrical , electronics and computer engineering r. nogueira , et al . , evaluating software - based fingerprint liveness detection using convolutional networks and local binary patterns , in biometric measurements and systems for security and medical applications ( bioms ) proceedings , 2014 ieee workshop on , vol . , pp.22 - 29 , 17 - 17 oct . 2014 doi : 10.1109/bioms.2014.6951531 y. jiang , and l. xin , spoof fingerprint detection based on co - occurrence matrix , international journal of signal processing , image processing and pattern recognition ( 2015 ) . x. jia , et al . , multi - scale local binary pattern with filters for spoof fingerprint detection , information sciences , volume 268 , 1 june 2014 , pages 91 - 102 , issn 0020 - 0255 , http://dx.doi.org/10.1016/j.ins.2013.06.041 . d. gragnaniello , et al . , local contrast phase descriptor for fingerprint liveness detection , pattern recognition , volume 48 , issue 4 , april 2015 , pages 1050 - 1058 , issn 0031 - 3203 , http://dx.doi.org/10.1016/j.patcog.2014.05.021 . n. poh , et al . , anti - forensic resistant likelihood ratio computation : a case study using fingerprint biometrics , in signal processing conference ( eusipco ) , 2014 proceedings of the 22nd european , vol . , no . , pp.1377 - 1381 , 1 - 5 sept . 2014 a. f. sequeira , and j. s. cardoso , fingerprint liveness detection in the presence of capable intruders , sensors . 2015 , 15(6):14615 - 14638 . g. fumera , et al . , multimodal antispoofing in biometric recognition systems , in handbook of biometric antispoofing , s. marcel , m. nixon , and s. li ( eds . ) , springer , pp . 145 - 164 , doi : 10.1007/978 - 1 - 4471 - 6524 - 89 , 2014 n. poh , et al . , toward an attack - sensitive tamper - resistant biometric recognition with a symmetric matcher : a fingerprint case study , in computational intelligence in biometrics and identity management ( cibim ) , 2014 ieee symposium on , vol . , no . , pp.175 - 180 , 9 - 12 dec . 2014 , doi : 10.1109/cibim.2014.7015460 l. ghiani , et al . , fingerprint liveness detection using binarized statistical image features , in biometrics : theory , applications and systems ( btas ) , 2013 ieee sixth international conference on , vol . , no . , pp.1 - 6 , sept . 29 2013-oct . 2 2013 , doi : 10.1109/btas.2013.6712708 x. jia , et al . , multi - scale block local ternary patterns for fingerprints vitality detection , in biometrics ( icb ) , 2013 international conference on , vol . , no . , pp.1 - 6 , 4 - 7 june 2013 , doi : 10.1109/icb.2013.6612964 g.l . marcialis , et al . , large scale experiments on fingerprint liveness detection , joint iapr int . work . on structural and statistical pattern recognition ( spr & sspr 2012 ) , hiroshima ( japan ) , november , 7 - 9 , 2012 , springer lncs 7625 , pp . 501 - 509 , 2012 y. zhang , et al . , fake fingerprint detection based on wavelet analysis and local binary pattern , biometric recognition ( ccbr 2014 ) , 8833 : 191 - 198 2014 a. rattani , et al . , open set fingerprint spoof detection across novel fabrication materials , in information forensics and security , ieee transactions on , vol.10 , no.11 , pp.2447 - 2460 , nov . 2015 , doi : 10.1109/tifs.2015.2464772 p. johnson , and s. schuckers , fingerprint pore characteristics for liveness detection , proceedings of the international conference of the biometrics special interest group ( biosig ) , darmstadt , germany , 1012 september 2014 ; pp . x. jia , et al . , one - class svm with negative examples for fingerprint liveness detection , biometric recognition . springer international publishing , 2014 . 216 - 224 . c. gottschlich , et al . , fingerprint liveness detection based on histograms of invariant gradients , biometrics ( ijcb ) , 2014 ieee international joint conference on . ieee , 2014 .

a spoof attack , a subset of presentation attacks , is the use of an artificial replica of a biometric in an attempt to circumvent a biometric sensor . liveness detection , or presentation attack detection , distinguishes between live and fake biometric traits and is based on the principle that additional information can be garnered above and beyond the data procured by a standard authentication system to determine if a biometric measure is authentic . the goals for the liveness detection ( livdet ) competitions are to compare software - based fingerprint liveness detection and artifact detection algorithms ( part 1 ) , as well as fingerprint systems which incorporate liveness detection or artifact detection capabilities ( part 2 ) , using a standardized testing protocol and large quantities of spoof and live tests . the competitions are open to all academic and industrial institutions which have a solution for either software - based or system - based fingerprint liveness detection . the livdet competitions have been hosted in 2009 , 2011 , 2013 and 2015 and have shown themselves to provide a crucial look at the current state of the art in liveness detection schemes . there has been a noticeable increase in the number of participants in livdet competitions as well as a noticeable decrease in error rates across competitions . participants have grown from four to the most recent thirteen submissions for fingerprint part 1 . fingerprints part 2 has held steady at two submissions each competition in 2011 and 2013 and only one for the 2015 edition . the continuous increase of competitors demonstrates a growing interest in the topic . fingerprint , liveness detection , biometric

1609.01648

here we provide a brief review of differential geometry concepts that we will use further . for more detailed treatment of the subject see , for example , @xcite . a topological hausdorff space @xmath6 with a countable base is called an @xmath2-dimensional * manifold * ( without boundary ) , @xmath19 , if , for any point @xmath20 , there exists an open neighborhood @xmath21 of the point @xmath22 , open subset @xmath23 , and homeomorphism @xmath24 . the pair @xmath25 is called a * coordinate chart * or * local coordinates*. we always assume that all manifolds we deal with are smooth . that is , given an @xmath2-dimensional manifold @xmath6 and two coordinate patches @xmath25 and @xmath26 , such that @xmath27 , the corresponding * transition map * @xmath28 is smooth ( infinitely differentiable ) . let @xmath6 be a manifold and @xmath29 be a function on it . the function @xmath30 is * continuously differentiable * on @xmath6 if , for any point @xmath20 and any coordinate patch @xmath25 , containing the point @xmath22 , the function @xmath31 is continuously differentiable . if @xmath30 is continuously differentiable on @xmath6 , we will write @xmath32 . in the same manner one can give definitions of @xmath33-times differentiable functions @xmath34 and infinitely differentiable functions @xmath35 , as well as differentiable mappings @xmath36 . we will denote the space of smooth functions with compact support by @xmath37 . let @xmath6 be an @xmath2-dimensional manifold . a * vector * @xmath38 at a point @xmath20 is a linear differential operator of the first order @xmath39 . in local coordinates it can be written as @xmath40(p ) \,,\end{gathered}\ ] ] where @xmath41 are components of @xmath38 . the vector @xmath38 is also called the * tangent * or * contravariant vector*. the set of all tangent vectors to the manifold @xmath6 at the point @xmath22 is called the * tangent space * and is denoted by @xmath42 . the union of all tangent spaces to the manifold @xmath6 at all points is called the * tangent bundle * and is denoted by @xmath43 . a * vector field * ( or a * section of the tangent bundle * ) @xmath44 on the manifold @xmath6 is a mapping @xmath45 , such that , for any point @xmath20 , there holds @xmath46 . below we follow the standard einstein convention of summation over repeated indices from one to the dimension of the manifold . where it is necessary , we will specify limits of the summation explicitly . in local coordinates a vector field @xmath44 may be written as @xmath47 with @xmath48 being an @xmath49-th vector of the coordinate basis in the tangent space @xmath42 , defined by @xmath50 where @xmath51 are local coordinates . we will say that the vector field @xmath44 is smooth and will write @xmath52 , if all its components @xmath53 are smooth functions in all coordinate patches . one can show that , although we use local coordinates to introduce various geometrical objects , they have invariant geometrical meaning which does not depend on the choice of a coordinate patch . all our results will be formulated in the invariant form . let @xmath6 be a manifold . the * cotangent space * @xmath54 at a point @xmath20 is the dual space to @xmath42 , i.e. the space of all linear functionals on @xmath42 . its elements are called * cotangent vectors * , * covariant vectors * or * covectors*. the union of all cotangent spaces at all points is called the * cotangent bundle * and is denoted by @xmath55 . a * covector field * ( also called a * @xmath56-form * or a * section of the cotangent bundle * ) @xmath44 on the manifold @xmath6 is a mapping @xmath57 , such that , for any point @xmath20 , there holds @xmath58 . the basis in a cotangent space , dual to the coordinate basis @xmath48 , is denoted by @xmath59 and is defined by the condition @xmath60 , where @xmath61 is the kronecker symbol . as well as for vector fields , we define the set of all smooth covector fields @xmath62 . let @xmath1 be a smooth mapping on a manifold @xmath6 . let @xmath20 and @xmath63 . let @xmath51 be local coordinates in the point @xmath22 , @xmath64 be local coordinates in the point @xmath65 , and @xmath66 be the coordinate functions of the mapping @xmath15 . the * differential * of the mapping @xmath15 at the point @xmath22 is a linear mapping @xmath67 with components @xmath68 the * tensor product * of two functionals @xmath69 and @xmath70 is a functional @xmath71 defined by @xmath72 for any @xmath73 and @xmath74 . a * tensor @xmath75 of the type @xmath76 * at a point @xmath22 of a manifold @xmath6 is a linear functional @xmath77 in the coordinate basis it is given by @xmath78 the set of all tensors of the type @xmath76 at all points of the manifold @xmath6 is called the * @xmath76-tensor bundle * and is denoted by @xmath79 . a smooth * @xmath76-tensor field * is a smooth mapping assigning to each point @xmath20 a tensor of the type @xmath76 . the set of all smooth @xmath76-tensor fields is denoted by @xmath80 . when @xmath81 or @xmath82 in the above definition equals zero , we will not write it , that is , @xmath83 and @xmath84 . for any @xmath85 let @xmath86 be the set of all permutations of the numbers @xmath87 . a transposition is a permutation , which exchanges only two numbers , leaving all others fixed . every permutation can be represented as a composition of transpositions . such a representation is not unique , but the number of transpositions in it is an invariant @xmath88 . the signature of a permutation @xmath89 , denoted by @xmath90 , is @xmath56 , if it is represented by a composition of even number of transpositions , and @xmath91 otherwise . for any permutation @xmath89 and any tensor @xmath92 we define the tensor @xmath93 by @xmath94 we will say that @xmath92 is a * symmetric tensor * , if @xmath95 for any @xmath89 . let @xmath96 be the space of symmetric @xmath97-tensors . we define the * operator of symmetrization * @xmath98 by @xmath99 we will also use parentheses to indicate symmetrization : @xmath100 if it is necessary to exclude some indices from symmetrization , we will delimit them by @xmath101 , that is , @xmath102 we will say that @xmath92 is an * antisymmetric tensor * , if @xmath103 for any @xmath89 . let @xmath104 be the space of antisymmetric @xmath97-tensors ( also called * @xmath105-forms * ) . it follows from the definition , that , for any @xmath2-dimensional manifold @xmath6 , only spaces @xmath106 with @xmath107 are non - trivial , that is , they contain non - zero tensors . we define the * operator of antisymmetrization * ( alternation ) @xmath108 by @xmath109 we will also use square brackets to indicate alternation : @xmath110}.\end{gathered}\ ] ] the * exterior * or * wedge product * @xmath111 is defined by @xmath112 when it is not confusing , we will write @xmath113 , @xmath114 , and @xmath115 instead of @xmath116 , @xmath117 , and @xmath118 respectively . we say that a manifold @xmath6 is equipped with a * riemannian metric * @xmath13 if @xmath13 is a smooth symmetric tensor field of the type @xmath119 and it defines a positive definite non - degenerate inner product in each tangent space @xmath42 . we will call the pair @xmath3 a * riemannian manifold*. on a riemannian manifold we can use the metric for raising and lowering tensor indices : @xmath120 where @xmath121 are components of the inverse matrix to @xmath122 , i.e. @xmath123 . the determinant of the matrix @xmath122 we will denote by @xmath124 or by @xmath125 . for integration over a manifold @xmath6 we need to define a volume form . an @xmath2-dimensional manifold @xmath6 is * orientable * , if there exists a constant sign @xmath2-form @xmath126 . in case of an orientable riemannian @xmath2-dimensional manifold @xmath3 there is a special volume form generated by the metric . it is called the * riemannian volume form * and is given by @xmath127 where @xmath128 is equal to @xmath56 for positively oriented basis @xmath48 and @xmath91 otherwise . the volume form changes sign whenever a basis changes the orientation ; such a geometrical object is called a * pseudoform*. we will always work with orientable manifolds and will use only positively oriented bases without specifying it explicitly . in this case @xmath129 and we will omit it henceforth . when computing integrals over a manifold , it is usually necessary to work in several coordinate charts . for using formulas , written in local coordinates , a partition of unity over a manifold is used for splitting an integral into parts , such that each part can be covered by a single chart . let @xmath3 be a riemannian manifold . the * christoffel symbols * @xmath130 are defined by @xmath131 the * covariant derivative * ( also called the * connection , compatible with the metric * , or * levi - civita connection * ) is an operator @xmath132 , defined by @xmath133 for a given point @xmath20 and a vector @xmath134 , the covariant derivative in the direction of the vector @xmath38 is @xmath135 we will denote covariant derivatives by indices after a semicolon , @xmath136 the components of the * riemann curvature * tensor are defined by @xmath137 the riemann curvature tensor has the following properties : @xmath138 the last two equations are called * bianci identities*. for the commutator of covariant derivatives there holds @xmath139}= \sum_{l=1}^s r^{i_l}{}_{mpk } t^{i_1 \ldots i_{l-1 } m i_{l+1 } \ldots i_s}_{j_1 \ldots j_q } -\sum_{l=1}^q r^{m}{}_{j_{l } pk } t^{i_1 \ldots i_s}_{j_1 \ldots j_{l-1 } m j_{l+1 } \ldots j_q } \,.\end{gathered}\ ] ] contraction of two indices of the riemann curvature tensor gives the * ricci curvature * tensor @xmath140 and contraction of all indices gives the * scalar curvature * @xmath141 a * geodesic * on a riemannian manifold @xmath3 is a critical curve @xmath142\to m$ ] of the * length functional * @xmath143 one can show that geodesics satisfy the equation @xmath144 which in local coordinates takes the form @xmath145 since we consider smooth manifolds , the theory of ordinary differential equations guarantees that , for any point @xmath20 and any vector @xmath146 , there exists an @xmath147 , such that equation has a unique solution @xmath148 , @xmath149 , satisfying initial conditions @xmath150 and @xmath151 . on a smooth compact manifold without boundary this solution exists for any @xmath152 . let @xmath3 be a smooth compact riemannian manifold without boundary . for any point @xmath20 the * exponential map * @xmath153 is defined by @xmath154 where @xmath148 is the solution of the equation with the initial conditions @xmath150 and @xmath151 . for non - compact manifolds the exponential map is well defined only in some neighborhood of @xmath155 . the local geometry of a smooth riemannian manifold @xmath3 can be described by the world function ( see , for example , @xcite ) . the world function is a real - valued non - negative function @xmath156 defined in a neighborhood of the diagonal of @xmath157 as follows . let @xmath158 be a fixed point . we fix a sufficiently small neighborhood of @xmath159 in @xmath6 so that every point @xmath160 in this neighborhood can be connected by a single geodesic to the point @xmath159 , which means that the exponential map is injective within this neighborhood . let @xmath161 be the geodesic radius of such a neighborhood . the * injectivity radius * of the manifold @xmath6 , denoted by @xmath162 , is defined as the infimum of @xmath161 over the whole manifold : @xmath163 we assume that the injectivity radius is strictly positive , i.e. @xmath164 . for smooth compact manifolds this is always the case . for non - compact manifolds , this will be one of our assumptions ( among other assumptions specified below ) . the * world function * @xmath165 is defined as half the square of the length of the geodesic connecting the points @xmath160 and @xmath159 . let @xmath166 be a parametrization of the geodesic connecting the points @xmath160 and @xmath159 with @xmath81 being the natural parameter ( arc length ) along the geodesic . let @xmath167 and @xmath168 . then @xmath169 is the unit tangent vector at the point @xmath170 of the geodesic . the world function satisfies the equations @xcite @xmath171 and @xmath172 it is easy to see that @xmath173 if two points @xmath174 can not be connected by a single geodesic , it is still possible to define the * distance * between @xmath160 and @xmath159 as the infimum of the length functional over all smooth curves connecting @xmath160 and @xmath159 , if they belong to the same connected component . the distance between two submanifolds @xmath175 we define as @xmath176 now we will describe the construction of the laplace operator on a riemannian manifold @xmath3 . for details see @xcite . we define the hilbert space of square integrable functions @xmath177 to be the completion of the space @xmath37 of smooth functions with compact support in the norm induced by the @xmath178-inner product @xmath179 the metric @xmath13 defines the inner product in every tangent space @xmath42 and induces in a natural way the inner product in the dual space @xmath54 which we will also denote by @xmath13 . it enables one to define the @xmath180-inner product in the space @xmath181 of smooth sections of the cotangent bundle @xmath182 with compact support by @xmath183 and consider a completion of @xmath181 in the norm induced by this inner product . we will denote this completion by @xmath184 . the covariant derivative on the space of smooth functions is the mapping @xmath185 . since @xmath186 and @xmath37 is dense in @xmath177 , the operator @xmath187 is densely defined . thus , we can define the adjoint operator @xmath188 by @xmath189 where @xmath190 and @xmath191 . now we define the * laplace operator * on @xmath37 by @xmath192 it is a symmetric operator , i.e. @xmath193 for any @xmath194 . moreover , it is essentially self - adjoint , which means that there is a unique self - adjoint extension of @xmath195 . to simplify notation , we will denote this extension by the same symbol . in local coordinates the expression for the laplacian is @xmath196 notice , that the leading symbol of the laplacian is negative definite : @xmath197 for any @xmath198 and any @xmath199 . therefore , the leading symbol of the operator @xmath200 is positive definite . we will call a differential operator of order @xmath105 over an @xmath2-dimensional manifold @xmath6 without boundary * elliptic * , if there exist constants @xmath201 , such that for any @xmath198 and any @xmath202 its leading symbol @xmath203 satisfies the inequality @xmath204 where @xmath205 . let @xmath206 be a hilbert space and @xmath207 be a linear operator on it . let @xmath208 . if the inverse of the operator @xmath209 exists , we will call it the * resolvent * and denote it by @xmath210 . there are the following possibilities @xcite : 1 . @xmath211 exists , is bounded , and is densely defined in @xmath206 . then @xmath212 is called a * regular value * of the operator @xmath213 . the set of all regular values is called the * resolvent set * and is denoted by @xmath214 . 2 . @xmath211 does not exist . that is , the kernel of the operator @xmath209 is non - trivial , @xmath215 . then @xmath212 is called an * eigenvalue * of the operator @xmath213 . the vector space @xmath216 is called the * eigenspace * corresponding to the eigenvalue @xmath212 , and any non - zero vector @xmath217 is called an * eigenvector * , corresponding to the eigenvalue @xmath212 . if the space @xmath216 is finite - dimensional , then @xmath218 is called the * multiplicity * of @xmath212 . the set of all eigenvalues is called the * point * ( or * discrete * ) * spectrum * and is denoted by @xmath219 . @xmath211 exists and is densely defined in @xmath206 , but is unbounded . the set of all such @xmath212 is called the * continuous spectrum * and is denoted by @xmath220 . 4 . @xmath211 exists , but is not densely defined in @xmath206 . the set of all such @xmath212 is called the * residual spectrum * and is denoted by @xmath221 . the * spectrum * @xmath222 of the operator @xmath213 is the union of discrete , continuous , and residual spectra , @xmath223 clearly , the spectrum is the complement of the resolvent set , @xmath224 . if @xmath206 is a finite dimensional space , then all points of the spectrum of the operator @xmath213 are eigenvalues , that is , @xmath225 , but it may not be the case if @xmath206 is infinite - dimensional . both operators @xmath195 and @xmath226 are elliptic . there is the following well - known theorem about the spectrum of an elliptic self - adjoint differential operator with a positive definite leading symbol @xcite . let @xmath3 be a compact riemannian manifold without boundary and @xmath227 be an elliptic self - adjoint differential operator with a positive definite leading symbol . then : 1 . the spectrum of the operator @xmath226 is real , discrete , and bounded from below . 2 . all eigenspaces of the operator @xmath226 are finite - dimensional . eigenfunctions of the operator @xmath226 are smooth and form an orthonormal basis in @xmath177 . more generally , a second - order elliptic differential operator acting on smooth sections of a vector bundle is called a * laplace type operator * , if it has a scalar leading symbol . in this section we will describe the heat kernel and other spectral functions on a riemannian manifold @xmath3 . for details see @xcite . the * heat kernel * @xmath228 of a laplace type operator @xmath226 on an @xmath2-dimensional riemannian manifold without boundary @xmath6 is the integral kernel of the heat semigroup operator @xmath229 and can be defined also as the fundamental solution of the heat equation @xmath230 for @xmath231 , with the initial condition @xmath232 where @xmath233 is the dirac distribution . near the diagonal set of @xmath157 the heat kernel has the form @xcite @xmath234 where @xmath235 is the van vleck - morette determinant , @xmath236 and the function @xmath237 has the asymptotic expansion as @xmath4 @xmath238 ( asymptotic equivalence will be defined in section [ se : laplacemethod ] . ) for a compact manifold @xmath6 the heat semigroup operator @xmath239 is a bounded trace - class operator on the hilbert space @xmath177 , with the trace @xmath240 given by @xmath241 where each eigenvalue @xmath242 , @xmath85 , is repeated as many times as its multiplicity . this trace is also called the * trace of the heat kernel * and is one of the spectral functions of the operator @xmath226 . some other spectral functions may be expressed in terms of the trace of the heat kernel by means of integral transforms . the * distribution * ( or * counting * ) * function * @xmath243 is defined as the number of eigenvalues below @xmath244 @xmath245 and is given by @xmath246 for @xmath247 , @xmath85 , where @xmath248 is a positive constant and @xmath249 is a heaviside distribution . for @xmath250 we have @xmath251 the * density function * @xmath252 is defined as the derivative of the distribution function @xmath253 and is given by @xmath254 let @xmath208 be such that @xmath255 . then the operator @xmath256 is positive . let @xmath257 be such that @xmath258 ( this condition is necessary for convergence of the following integral and sum ) . the * generalized @xmath259-function * @xmath260 is defined as the trace of a complex power of the operator @xmath256 @xmath261 and is given by @xmath262 the generalized @xmath259-function is an analytical function of @xmath81 for @xmath263 and can be continued analytically to a meromorphic function in @xmath264 . in particular , it is analytic at @xmath265 and , thus , it is possible to define the * functional determinant * of the operator @xmath256 by @xmath266 these spectral functions are very useful in studying the spectrum of the operator @xmath226 and , if known exactly , they determine the spectrum . this is not valid for their asymptotic expansions and there are examples of operators with the same asymptotic series of spectral functions , but different spectra @xcite . in principle , all these functions are equivalent to each other , but the heat kernel is more convenient for practical purposes , since it is a smooth function , while the distribution and density functions are singular . in this work we will develop a generalized laplace method for computing asymptotics of integrals over manifolds . here we present necessary information for both one- and multi - dimensional laplace methods . for proofs of cited theorems and lemmas see , for example , @xcite . functions @xmath267 and @xmath268 are * asymptotically equivalent * as @xmath4 , denoted by @xmath269 , if @xmath270 for any @xmath271 . let @xmath272)$ ] . a point @xmath273 is called a * non - degenerate critical point * of the function @xmath274 , if @xmath275 and @xmath276 . [ th : laplacem1d ] let @xmath277)$ ] . let the function @xmath274 attain its global minimum on @xmath278 $ ] at an interior point @xmath279 . let the point @xmath280 be the only point where the global minimum is attained , that is , @xmath281 for any @xmath282 , and assume that @xmath280 is a non - degenerate critical point . then there exists an asymptotic expansion as @xmath4 @xmath283 the coefficients @xmath284 depend polynomially on derivatives of the function @xmath285 at the point @xmath280 , on derivatives of the function @xmath274 of order higher than two at the point @xmath280 , and on @xmath286^{-1/2}$ ] . for computing the coefficients @xmath284 in theorem [ th : laplacem1d ] , the following result for standard gaussian integrals is useful . [ le : gaussianintegral1d ] let @xmath287 . then for any @xmath288 , there holds @xmath289 a theorem similar to theorem [ th : laplacem1d ] holds in the multi - dimensional case . let @xmath290 be a bounded open set . let @xmath291 . a point @xmath292 is called a * non - degenerate critical point * of the function @xmath274 , if @xmath293 and the hessian matrix of the function @xmath274 at the point @xmath280 , that is , the matrix with entries @xmath294 , is either strictly positive or strictly negative definite . [ th : laplacemmd ] let @xmath290 be a bounded open set . let @xmath295 . let the function @xmath274 attain its global minimum on @xmath296 at a point @xmath280 . let the point @xmath280 be the only point where the global minimum is attained and assume that @xmath280 is a non - degenerate critical point . then there exists an asymptotic expansion as @xmath4 @xmath297 the coefficients @xmath284 depend polynomially on derivatives of the function @xmath285 at the point @xmath280 , on derivatives of the function @xmath274 of order higher than two at the point @xmath280 , and on the inverse of the hessian matrix @xmath298^{-1}$ ] . [ le : gaussianintegralmd ] let @xmath213 be a strictly positive symmetric matrix and @xmath299 . then for any @xmath300 , @xmath301 , there holds @xmath302 in this section we will introduce deformed spectral functions , following @xcite , and formulate the problem considered in this work . let @xmath3 be a smooth compact riemannian manifold without boundary , @xmath1 be an isometric mapping on it , and @xmath0 be the fixed point set of the mapping @xmath15 , that is , @xmath303 the mapping @xmath15 naturally defines the operator @xmath304 by @xmath305 the composition of the operator @xmath15 and the heat semigroup operator defines an operator @xmath306 whose kernel is @xmath307 which we will call the deformed heat kernel . the * deformed heat trace * is @xmath308 it can be used to define other deformed spectral functions of the laplace operator on the manifold @xmath6 . namely , the * deformed distribution function * @xmath309 the * deformed density function * @xmath310 the * deformed @xmath259-function * @xmath311 and the * deformed functional determinant * of the operator @xmath256 @xmath312 where we have used the same notation as in section [ se : spectralfunctions ] . these deformed spectral functions were considered in @xcite . the following theorem is known @xcite for the asymptotic expansion of the deformed heat trace . [ th : ae - isometry ] let @xmath6 be a smooth compact riemannian manifold without boundary . let @xmath1 be an isometry on @xmath6 . let @xmath0 be the fixed point set of @xmath15 . then : 1 . @xmath0 is a disjoined union of connected submanifolds @xmath7 , @xmath8 ; 2 . as @xmath4 @xmath313 where @xmath314 and @xmath315 are scalar invariants on @xmath7 . donnelly @xcite computed the coefficients @xmath16 and @xmath17 of this asymptotic expansion . asymptotic expansions of this kind are used for proofs of the atiyah - singer - lefschetz formulas for compact group actions and index theorems . our goal is to consider a more general problem . instead of the isometry we will consider a smooth mapping @xmath15 ( not necessarily an isometry ) with the fixed point set @xmath0 , satisfying the following assumptions : 1 . the fixed point set @xmath0 consists of finitely many disjoint connected components @xmath7 , @xmath8 . each connected component @xmath7 is a smooth compact submanifold of @xmath6 without boundary . each connected component @xmath7 is a `` non - degenerate '' fixed point set . we will formulate this assumption precisely for all considered cases in section [ se : planecurve ] ( p. ) , section [ se : planepoint ] ( p. ) , and section [ se : curvedpoint ] ( p. ) . a theorem analogous to theorem [ th : ae - isometry ] holds for such mappings as well @xcite . all these assumptions are automatically satisfied for isometries . we will develop a technique for computation of the asymptotic expansion coefficients and compute explicitly the coefficients @xmath16 , @xmath17 , and @xmath18 . we will say that two subsets @xmath213 and @xmath316 of the manifold @xmath6 are separated if @xmath317 . it follows from the above assumptions , that all components @xmath7 are separated , that is , @xmath318 since the asymptotic expansion of the deformed heat trace @xmath319 is determined by a neighborhood of @xmath0 and all connected components @xmath7 are isolated , it is possible to consider each component separately . the whole asymptotic expansion is equal to the sum of asymptotic expansions generated by each component . so , without loss of generality and in order to simplify notation , we will assume that @xmath0 consists of only one connected component . we will also consider non - compact manifolds @xmath6 . in this case our additional assumption will be that the injectivity radius of @xmath6 is non - zero . in this chapter we will derive reduction formulas for contraction of symmetric tensors , that allow us to express multi - dimensional gaussian integrals in a symmetrization - free form . our main result in this chapter is lemma [ le : contraction ] , which gives an explicit expression of this kind for a general case . let @xmath320 be the * symmetrized tensor product * of symmetric covariant tensors , defined by @xmath321 let @xmath322 be a symmetric @xmath323-tensor . let @xmath324 be the operator of contraction of symmetric covariant tensors with @xmath322 , defined by @xmath325 by definition , @xmath326 for any @xmath213 from @xmath327 or @xmath328 . a * multi - index * @xmath329 of the order @xmath33 is a @xmath33-tuple @xmath330 of non - negative integers . we will use the following notation : @xmath331 let @xmath332 and @xmath329 be a multi - index of the order @xmath33 , such that @xmath333 . let @xmath334 , @xmath335 , be symmetric tensors . we will be interested in explicit formulas for the full contraction of the symmetrized product of these tensors with the tensor @xmath322 , that is , in scalar expressions of the form @xmath336 by `` explicit formulas '' we mean expressions without symmetrization of indices . we will develop a technique for obtaining such formulas and then apply it to some particular cases that will be needed in the following chapters . by definition , we have @xmath337 where @xmath338 we need to combine all similar terms in the sum , that arise because of the symmetry of the tensors @xmath339 and @xmath322 , as well as the possibility to permute factors in the tensor product of @xmath322 s . with each term @xmath340 we will associate a graph @xmath341 . every graph @xmath342 has @xmath33 distinct vertices of degrees @xmath343 , corresponding to the tensors @xmath339 , and @xmath344 edges , corresponding to the factors of the tensor @xmath322 . an edge between vertices @xmath49 and @xmath345 means that an index of the tensor @xmath339 is contracted with an index of the tensor @xmath346 by one of the factors @xmath322 . let @xmath347 be the set of all such graphs . for every permutation @xmath348 the term @xmath340 has a unique graph @xmath341 associated with it . thus , the mapping @xmath349 is well defined . from the definition of the mapping @xmath206 it follows : 1 . given a graph @xmath350 it is possible to construct many terms @xmath340 or , in other words , to find many permutations @xmath348 , such that @xmath351 . 2 . for any graph @xmath350 all permutations @xmath352 generate similar terms @xmath340 in the sum . 3 . for any two different graphs @xmath353 and any two permutations @xmath354 , @xmath355 terms @xmath356 and @xmath357 are not similar . so , the mapping @xmath206 is surjective , but not injective . therefore , we have a partition of @xmath358 into equivalence classes @xmath359 all permutations @xmath285 from the same class @xmath360 generate similar terms @xmath340 . our goal is now to list all elements of @xmath347 and to count how many similar terms @xmath340 in the sum correspond to each graph . that is , for any graph @xmath361 we want to find the number of elements in the equivalence class @xmath360 . we will denote it by @xmath362 we will identify a graph @xmath361 with its adjacency matrix @xmath363 , which is a square matrix of order @xmath33 . by definition , its entry @xmath364 is equal to the number of edges between the vertices @xmath49 and @xmath345 , that is , to the number of indices of the tensor @xmath339 contracted with indices of the tensor @xmath346 by factors of @xmath322 . it is easy to see , that every matrix @xmath361 has the following properties : 1 . for any @xmath49 and @xmath345 the entry @xmath364 is a non - negative integer from @xmath365 to @xmath366 . the matrix @xmath342 is symmetric , that is , for any @xmath49 and @xmath345 there holds @xmath367 . 3 . for any @xmath49 the diagonal coefficient @xmath368 is even . 4 . for any @xmath49 the sum of all entries of the @xmath49-th row , as well as the sum of all entries of the @xmath49-th column , is equal to the rank of the tensor @xmath339 : @xmath369 conversely , every matrix of the order @xmath33 having all these properties is an element of the set @xmath347 . let @xmath361 . for computing @xmath370 we select an arbitrary element @xmath371 and the corresponding term @xmath340 . we can generate new terms associated with @xmath342 by using symmetry of the tensors @xmath322 and @xmath339 , and permuting factors in the tensor product of @xmath322 s . it gives us @xmath372 terms . however , some terms will differ just by names of dummy indices . below we compute how many of these generated terms are the same ( up to names of dummy indices ) . 1 . if @xmath373 , then simultaneous permutation of two indices of the contracted pair in the tensors @xmath322 and @xmath339 does not change the term . for example : @xmath374 it reduces the number of terms by a factor of @xmath375 . note , that @xmath376 is the number of self - loops at the vertex @xmath49 , and @xmath377 is the number of self - loops in the whole graph @xmath342 . 2 . if @xmath378 , then simultaneous permutation of @xmath376 contracting factors of @xmath322 and @xmath376 contracted pairs of indices of the tensor @xmath339 does not change the term . for example : @xmath379 it reduces the number of terms by a factor of @xmath380 . 3 . if @xmath381 for @xmath382 , then simultaneous permutation of @xmath364 indices of the tensors @xmath339 and @xmath346 contracted with the tensor @xmath322 , and factors of @xmath322 , does not change the term . for example : @xmath383 it reduces the number of terms by a factor of @xmath384 . taking all this into account , the number of terms associated with @xmath342 is @xmath385 note , that these coefficients have the following property : @xmath386 so , it is natural to define weights or probabilities of graphs @xmath342 by @xmath387 and they have the property @xmath388 thus , we have proved the following lemma . [ le : contraction ] let @xmath389 . let @xmath329 be a multi - index of the order @xmath33 , such that @xmath333 . let @xmath334 , @xmath335 , be symmetric @xmath390-tensors . then @xmath391 where @xmath392 is an arbitrary permutation from the set @xmath360 and the coefficients @xmath393 are given by @xmath394 now we will apply lemma [ le : contraction ] to a few particular cases . matrices in the following corollaries are listed in the `` decreasing row - wise lexicographical order '' . [ co : contraction33 ] let @xmath395 . then @xmath396 where @xmath397 are terms associated with the adjacency matrices @xmath398 listed below : @xmath399 [ co : contraction35 ] let @xmath400 , @xmath401 . then @xmath402 where @xmath397 are terms associated with the adjacency matrices @xmath398 listed below : @xmath403 [ co : contraction44 ] let @xmath404 , @xmath405 . then @xmath406 where @xmath397 are terms associated with the adjacency matrices @xmath398 listed below : @xmath407 [ co : contraction334 ] let @xmath395 , @xmath408 . then @xmath409 where @xmath397 are terms associated with the adjacency matrices @xmath398 listed below : @xmath410 \gamma_{(4)}&= \begin{pmatrix } 2 & 0 & 1\\ 0 & 0 & 3\\ 1 & 3 & 0 \end{pmatrix } , & \gamma_{(5)}&= \begin{pmatrix } 0 & 3 & 0\\ 3 & 0 & 0\\ 0 & 0 & 4 \end{pmatrix } , & \gamma_{(6)}&= \begin{pmatrix } 0 & 2 & 1\\ 2 & 0 & 1\\ 1 & 1 & 2 \end{pmatrix } , \\[2ex ] \gamma_{(7)}&= \begin{pmatrix } 0 & 1 & 2\\ 1 & 2 & 0\\ 2 & 0 & 2 \end{pmatrix } , & \gamma_{(8)}&= \begin{pmatrix } 0 & 1 & 2\\ 1 & 0 & 2\\ 2 & 2 & 0 \end{pmatrix } , & \gamma_{(9)}&= \begin{pmatrix } 0 & 0 & 3\\ 0 & 2 & 1\\ 3 & 1 & 0 \end{pmatrix}.\end{aligned}\ ] ] [ co : contraction3333 ] let @xmath411 . then @xmath412 where @xmath397 are terms associated with the adjacency matrices @xmath398 listed below : @xmath413\\[2ex ] \gamma_{(4)}&= \begin{pmatrix } 2 & 1 & 0 & 0\\ 1 & 0 & 1 & 1\\ 0 & 1 & 2 & 0\\ 0 & 1 & 0 & 2 \end{pmatrix } , & \gamma_{(5)}&= \begin{pmatrix } 2 & 1 & 0 & 0\\ 1 & 0 & 1 & 1\\ 0 & 1 & 0 & 2\\ 0 & 1 & 2 & 0 \end{pmatrix } , & \gamma_{(6)}&= \begin{pmatrix } 2 & 1 & 0 & 0\\ 1 & 0 & 0 & 2\\ 0 & 0 & 2 & 1\\ 0 & 2 & 1 & 0 \end{pmatrix } , \displaybreak[0]\\[2ex ] \gamma_{(7)}&= \begin{pmatrix } 2 & 0 & 1 & 0\\ 0 & 2 & 1 & 0\\ 1 & 1 & 0 & 1\\ 0 & 0 & 1 & 2 \end{pmatrix } , & \gamma_{(8)}&= \begin{pmatrix } 2 & 0 & 1 & 0\\ 0 & 2 & 0 & 1\\ 1 & 0 & 2 & 0\\ 0 & 1 & 0 & 2 \end{pmatrix } , & \gamma_{(9)}&= \begin{pmatrix } 2 & 0 & 1 & 0\\ 0 & 2 & 0 & 1\\ 1 & 0 & 0 & 2\\ 0 & 1 & 2 & 0 \end{pmatrix } , \displaybreak[0]\\[2ex ] \gamma_{(10)}&= \begin{pmatrix } 2 & 0 & 1 & 0\\ 0 & 0 & 2 & 1\\ 1 & 2 & 0 & 0\\ 0 & 1 & 0 & 2 \end{pmatrix } , & \gamma_{(11)}&= \begin{pmatrix } 2 & 0 & 1 & 0\\ 0 & 0 & 1 & 2\\ 1 & 1 & 0 & 1\\ 0 & 2 & 1 & 0 \end{pmatrix } , & \gamma_{(12)}&= \begin{pmatrix } 2 & 0 & 1 & 0\\ 0 & 0 & 0 & 3\\ 1 & 0 & 2 & 0\\ 0 & 3 & 0 & 0 \end{pmatrix } , \displaybreak[0]\\[2ex ] \gamma_{(13)}&= \begin{pmatrix } 2 & 0 & 0 & 1\\ 0 & 2 & 1 & 0\\ 0 & 1 & 2 & 0\\ 1 & 0 & 0 & 2 \end{pmatrix } , & \gamma_{(14)}&= \begin{pmatrix } 2 & 0 & 0 & 1\\ 0 & 2 & 1 & 0\\ 0 & 1 & 0 & 2\\ 1 & 0 & 2 & 0 \end{pmatrix } , & \gamma_{(15)}&= \begin{pmatrix } 2 & 0 & 0 & 1\\ 0 & 2 & 0 & 1\\ 0 & 0 & 2 & 1\\ 1 & 1 & 1 & 0 \end{pmatrix } , \displaybreak[0]\\[2ex ] \gamma_{(16)}&= \begin{pmatrix } 2 & 0 & 0 & 1\\ 0 & 0 & 3 & 0\\ 0 & 3 & 0 & 0\\ 1 & 0 & 0 & 2 \end{pmatrix } , & \gamma_{(17)}&= \begin{pmatrix } 2 & 0 & 0 & 1\\ 0 & 0 & 2 & 1\\ 0 & 2 & 0 & 1\\ 1 & 1 & 1 & 0 \end{pmatrix } , & \gamma_{(18)}&= \begin{pmatrix } 2 & 0 & 0 & 1\\ 0 & 0 & 1 & 2\\ 0 & 1 & 2 & 0\\ 1 & 2 & 0 & 0 \end{pmatrix } , \displaybreak[0]\\[2ex ] \gamma_{(19)}&= \begin{pmatrix } 0 & 3 & 0 & 0\\ 3 & 0 & 0 & 0\\ 0 & 0 & 2 & 1\\ 0 & 0 & 1 & 2 \end{pmatrix } , & \gamma_{(20)}&= \begin{pmatrix } 0 & 3 & 0 & 0\\ 3 & 0 & 0 & 0\\ 0 & 0 & 0 & 3\\ 0 & 0 & 3 & 0 \end{pmatrix } , & \gamma_{(21)}&= \begin{pmatrix } 0 & 2 & 1 & 0\\ 2 & 0 & 1 & 0\\ 1 & 1 & 0 & 1\\ 0 & 0 & 1 & 2 \end{pmatrix } , \displaybreak[0]\\[2ex ] \gamma_{(22)}&= \begin{pmatrix } 0 & 2 & 1 & 0\\ 2 & 0 & 0 & 1\\ 1 & 0 & 2 & 0\\ 0 & 1 & 0 & 2 \end{pmatrix } , & \gamma_{(23)}&= \begin{pmatrix } 0 & 2 & 1 & 0\\ 2 & 0 & 0 & 1\\ 1 & 0 & 0 & 2\\ 0 & 1 & 2 & 0 \end{pmatrix } , & \gamma_{(24)}&= \begin{pmatrix } 0 & 2 & 0 & 1\\ 2 & 0 & 1 & 0\\ 0 & 1 & 2 & 0\\ 1 & 0 & 0 & 2 \end{pmatrix } , \displaybreak[0]\\[2ex ] \gamma_{(25)}&= \begin{pmatrix } 0 & 2 & 0 & 1\\ 2 & 0 & 1 & 0\\ 0 & 1 & 0 & 2\\ 1 & 0 & 2 & 0 \end{pmatrix } , & \gamma_{(26)}&= \begin{pmatrix } 0 & 2 & 0 & 1\\ 2 & 0 & 0 & 1\\ 0 & 0 & 2 & 1\\ 1 & 1 & 1 & 0 \end{pmatrix } , & \gamma_{(27)}&= \begin{pmatrix } 0 & 1 & 2 & 0\\ 1 & 2 & 0 & 0\\ 2 & 0 & 0 & 1\\ 0 & 0 & 1 & 2 \end{pmatrix } , \displaybreak[0]\\[2ex ] \gamma_{(28)}&= \begin{pmatrix } 0 & 1 & 2 & 0\\ 1 & 0 & 1 & 1\\ 2 & 1 & 0 & 0\\ 0 & 1 & 0 & 2 \end{pmatrix } , & \gamma_{(29)}&= \begin{pmatrix } 0 & 1 & 2 & 0\\ 1 & 0 & 0 & 2\\ 2 & 0 & 0 & 1\\ 0 & 2 & 1 & 0 \end{pmatrix } , & \gamma_{(30)}&= \begin{pmatrix } 0 & 1 & 1 & 1\\ 1 & 2 & 0 & 0\\ 1 & 0 & 2 & 0\\ 1 & 0 & 0 & 2 \end{pmatrix } , \displaybreak[0]\\[2ex ] \gamma_{(31)}&= \begin{pmatrix } 0 & 1 & 1 & 1\\ 1 & 2 & 0 & 0\\ 1 & 0 & 0 & 2\\ 1 & 0 & 2 & 0 \end{pmatrix } , & \gamma_{(32)}&= \begin{pmatrix } 0 & 1 & 1 & 1\\ 1 & 0 & 2 & 0\\ 1 & 2 & 0 & 0\\ 1 & 0 & 0 & 2 \end{pmatrix } , & \gamma_{(33)}&= \begin{pmatrix } 0 & 1 & 1 & 1\\ 1 & 0 & 1 & 1\\ 1 & 1 & 0 & 1\\ 1 & 1 & 1 & 0 \end{pmatrix } , \displaybreak[0]\\[2ex ] \gamma_{(34)}&= \begin{pmatrix } 0 & 1 & 1 & 1\\ 1 & 0 & 0 & 2\\ 1 & 0 & 2 & 0\\ 1 & 2 & 0 & 0 \end{pmatrix } , & \gamma_{(35)}&= \begin{pmatrix } 0 & 1 & 0 & 2\\ 1 & 2 & 0 & 0\\ 0 & 0 & 2 & 1\\ 2 & 0 & 1 & 0 \end{pmatrix } , & \gamma_{(36)}&= \begin{pmatrix } 0 & 1 & 0 & 2\\ 1 & 0 & 2 & 0\\ 0 & 2 & 0 & 1\\ 2 & 0 & 1 & 0 \end{pmatrix } , \displaybreak[0]\\[2ex ] \gamma_{(37)}&= \begin{pmatrix } 0 & 1 & 0 & 2\\ 1 & 0 & 1 & 1\\ 0 & 1 & 2 & 0\\ 2 & 1 & 0 & 0 \end{pmatrix } , & \gamma_{(38)}&= \begin{pmatrix } 0 & 0 & 3 & 0\\ 0 & 2 & 0 & 1\\ 3 & 0 & 0 & 0\\ 0 & 1 & 0 & 2 \end{pmatrix } , & \gamma_{(39)}&= \begin{pmatrix } 0 & 0 & 3 & 0\\ 0 & 0 & 0 & 3\\ 3 & 0 & 0 & 0\\ 0 & 3 & 0 & 0 \end{pmatrix } , \displaybreak[0]\\[2ex ] \gamma_{(40)}&= \begin{pmatrix } 0 & 0 & 2 & 1\\ 0 & 2 & 1 & 0\\ 2 & 1 & 0 & 0\\ 1 & 0 & 0 & 2 \end{pmatrix } , & \gamma_{(41)}&= \begin{pmatrix } 0 & 0 & 2 & 1\\ 0 & 2 & 0 & 1\\ 2 & 0 & 0 & 1\\ 1 & 1 & 1 & 0 \end{pmatrix } , & \gamma_{(42)}&= \begin{pmatrix } 0 & 0 & 2 & 1\\ 0 & 0 & 1 & 2\\ 2 & 1 & 0 & 0\\ 1 & 2 & 0 & 0 \end{pmatrix } , \displaybreak[0]\\[2ex ] \gamma_{(43)}&= \begin{pmatrix } 0 & 0 & 1 & 2\\ 0 & 2 & 1 & 0\\ 1 & 1 & 0 & 1\\ 2 & 0 & 1 & 0 \end{pmatrix } , & \gamma_{(44)}&= \begin{pmatrix } 0 & 0 & 1 & 2\\ 0 & 2 & 0 & 1\\ 1 & 0 & 2 & 0\\ 2 & 1 & 0 & 0 \end{pmatrix } , & \gamma_{(45)}&= \begin{pmatrix } 0 & 0 & 1 & 2\\ 0 & 0 & 2 & 1\\ 1 & 2 & 0 & 0\\ 2 & 1 & 0 & 0 \end{pmatrix } , \displaybreak[0]\\[2ex ] \gamma_{(46)}&= \begin{pmatrix } 0 & 0 & 0 & 3\\ 0 & 2 & 1 & 0\\ 0 & 1 & 2 & 0\\ 3 & 0 & 0 & 0 \end{pmatrix } , & \gamma_{(47)}&= \begin{pmatrix } 0 & 0 & 0 & 3\\ 0 & 0 & 3 & 0\\ 0 & 3 & 0 & 0\\ 3 & 0 & 0 & 0 \end{pmatrix}.\end{aligned}\ ] ] we will also need formulas for contractions of equal tensors . [ co : contraction3x4 ] let @xmath400 . then @xmath414 where @xmath397 are defined as in corollary [ co : contraction3333 ] . when some or all tensors in the contraction are equal , some adjacency matrices generate similar terms . namely , if tensors @xmath339 and @xmath346 in are equal , and matrices @xmath415 and @xmath416 can be obtained one from another by exchanging @xmath49-th and @xmath345-th rows , and @xmath49-th and @xmath345-th columns , then @xmath415 and @xmath416 generate similar terms . by direct comparison , the matrices @xmath398 from corollary [ co : contraction3333 ] split into eight groups , such that all matrices in each group generate similar terms . numbers @xmath49 for groups of equivalent matrices @xmath398 are listed below : [ cols= " < , < " , ] by adding the coefficients of the terms @xmath397 in the formula separately for each group listed above , we obtain the coefficients in . in the same way one can obtain the following corollary . [ co : contraction3x24 ] let @xmath400 , @xmath405 . then @xmath417 where @xmath397 are defined as in corollary [ co : contraction334 ] . in this chapter we consider the plane , @xmath418 . we provide new proofs for asymptotic expansion theorems [ th : expansion - plane - curve ] and [ th : expansion - plane - point ] . these proofs will give us a way to compute the asymptotic expansion coefficients . the coefficients @xmath17 , @xmath18 in lemma [ le : coefficients - plane - curve ] , and the coefficients @xmath419 , @xmath420 in lemma [ le : coefficients - plane - point ] are computed explicitly for the first time . the injectivity radius of the plane is infinite , @xmath421 , and the world function is defined globally , @xmath422 . we will use the euclidean metric in @xmath423 and local coordinates @xmath424 . in cartesian coordinates the world function is given by @xmath425 and the laplacian by @xmath426 the heat kernel @xmath228 has the well - known form @xmath427 let @xmath428 be a smooth mapping and let @xmath0 , @xmath429 , and @xmath430 be defined as in section [ se : framework ] by , , . let @xmath431 be a real - valued function defined by @xmath432 since the world function @xmath156 is defined globally , the function @xmath274 is well - defined . obviously , @xmath433 for any @xmath434 and @xmath435 on @xmath0 . since @xmath436 , there are only two possible cases : @xmath437 , that is , @xmath0 is a curve ; and @xmath438 , that is , @xmath0 is a point . we will consider these two cases separately . let @xmath437 . let @xmath439 be a smooth unit normal vector field along the curve @xmath0 and @xmath440 be the corresponding normal derivative . we define functions @xmath441 by @xmath442 since @xmath443 and @xmath444 , it follows that @xmath445 and @xmath446 vanish . [ th : expansion - plane - curve ] let @xmath15 be a smooth mapping @xmath428 . let @xmath0 be its fixed point set : @xmath447 . let @xmath274 be a real - valued function @xmath431 defined by @xmath448 . let @xmath15 satisfy the assumptions : 1 . @xmath0 is a smooth compact connected one - dimensional submanifold of @xmath449 without boundary . there exist constants @xmath450 , such that for any @xmath451 , if @xmath452 , then @xmath453^p .\end{gathered}\ ] ] 3 . @xmath454 at all points of @xmath0 , where @xmath455 are defined as in . then there exists the asymptotic expansion as @xmath4 @xmath456 where the coefficients @xmath457 are locally computable in the form @xmath458 and @xmath459 are scalars on @xmath0 depending polynomially on @xmath460 and @xmath461 . assumptions ( @xmath15.1)(@xmath15.3 ) mean that the fixed point set @xmath0 is a smooth closed curve without self - intersections , the function @xmath462 grows sufficiently fast at infinity and it is non - degenerate near @xmath0 . let @xmath424 be cartesian coordinates in @xmath423 . we have @xmath463 where @xmath464 is the standard lebesgue measure . the proof of the theorem will be based on a generalized laplace method and involves the following steps : 1 . reducing the integration in the deformed heat trace @xmath319 to an integral over a suitable neighborhood @xmath465 of @xmath0 . 2 . choosing a special tangent normal coordinate system in @xmath465 . 3 . scaling the normal variable by @xmath466 . 4 . expanding the integrand in the power series in @xmath466 . computing integrals over the normal variable . let the fixed point set @xmath0 be described by the equations @xmath467 where @xmath468 is a parameter on @xmath0 . the vector field @xmath469 is tangent to @xmath0 . in a sufficiently small neighborhood of @xmath0 we can introduce local coordinates for @xmath423 in the following way . we construct a family of geodesics @xmath470 in @xmath423 , such that at each point of @xmath0 the tangent vector to the geodesic is equal to @xmath439 , the smooth unit vector field , normal to @xmath0 . we choose the parameter @xmath471 in a natural way , so that it is the signed distance to @xmath0 along the geodesic . that is , we have the differential equation @xmath472 with the initial conditions @xmath473 \label{eq : geodesicsic2 } { \dfrac{{\partial}^ { } { x^\alpha}}{{\partial}{z}^{}}}\big|_{z=0}&=h^\alpha(y ) \,.\end{aligned}\ ] ] we restrict ourselves to a tubular neighborhood @xmath465 of @xmath0 with @xmath474 for sufficiently small @xmath248 , so that the change of coordinates @xmath475 is not degenerate . we will specify the exact condition later . the solution of the equations is given by @xmath476 at every point in @xmath465 we have : @xmath477 due to the choice of @xmath471 , we also have : @xmath478 i.e. the vectors @xmath479 and @xmath480 form an orthogonal system . finally , the volume form in the coordinates @xmath481 is @xmath482^{1/2 } dy\wedge dz={{\varepsilon}}_{\alpha\beta}{\dfrac{{\partial}^ { } { x^\alpha}}{{\partial}{y}^{}}}{\dfrac{{\partial}^ { } { x^\beta}}{{\partial}{z}^ { } } } dy\wedge dz\\ & = \left[\gamma(y)\right]^{1/2 } \ , [ 1+z\ , { \varkappa}(y ) ] \ , dy\wedge dz,\end{aligned}\ ] ] where @xmath483 is the standard levi - civita symbol with @xmath484 , @xmath485 , @xmath486 is the induced metric on @xmath0 and @xmath487^{-1/2 } \ , { { \varepsilon}}_{\alpha\beta}{\dfrac{{\partial}^ { } { h^\alpha}}{{\partial}{y}^ { } } } h^\beta\,.\end{gathered}\ ] ] now we see , that it is enough to take @xmath248 less than @xmath488 . since @xmath0 is compact , this infimum is positive . since @xmath462 is a non - negative function , the asymptotic expansion of the deformed heat trace @xmath319 will be determined by a neighborhood of points where @xmath435 , i.e. by @xmath465 : [ le : reductionoftheareaofintegration ] there holds as @xmath4 @xmath489 let @xmath490 be such constants that the estimation holds . let @xmath491 be the ball with radius @xmath492 centered at the origin : @xmath493 . we split @xmath319 into the sum of three integrals : @xmath494 where @xmath495 we will show that @xmath496 and @xmath497 are exponentially small as @xmath4 , i.e. they are asymptotically equivalent to zero and the asymptotics of @xmath319 is determined by @xmath498 only . since @xmath499 is compact and @xmath500 on this set , it is separated from zero : @xmath501 for all @xmath502 . hence , as @xmath4 , we have @xmath503 for @xmath497 we use to estimate the integrand at infinity : @xmath504^p\right)}\\ = & ( 2t)^{-1 } \int_{r}^{\infty}r dr { \exp\left(-\dfrac{c r^{2p}}{4t}\right)}\\ = & ( 2t)^{-1 } \dfrac{1}{2p } \left(\dfrac{4t}{c}\right)^{1/p } \gamma\left(\frac{1}{p},\frac{cr^{2p}}{4t}\right),\end{aligned}\ ] ] where @xmath505 is the incomplete @xmath342-function : @xmath506 it is well known ( see , e.g. @xcite ) that as @xmath507 @xmath508 hence , we have as @xmath4 @xmath509 therefore , @xmath510 . so , after the change of coordinates @xmath511 in @xmath465 we get : @xmath512 by scaling the variable @xmath513 we obtain @xmath514 now we expand the integrand in the last formula in the power series in @xmath515 . below we compute the series expansion for @xmath516 . we define @xmath517 then @xmath518 since @xmath519 is the fixed point set of the mapping @xmath15 , we have @xmath520 therefore , @xmath521 where @xmath522 & = \dfrac{{\partial}^k \psi^\alpha}{{\partial}x^{\nu_1}\dots{\partial}x^{\nu_k } } \big|_{z=0 } h^{\nu_1}\dots h^{\nu_k}.\end{aligned}\ ] ] now we obtain @xmath523 where @xmath524 we consider a non - degenerate case ( see the assumption ( @xmath15.3 ) ) , when @xmath525 for any point of @xmath0 . in this case @xmath526 everywhere on @xmath0 . it is worth mentioning , that if @xmath527 somewhere on @xmath0 , that is , the first non - vanishing term of the taylor series of @xmath528 is of degree higher than 2 , then the asymptotics of @xmath319 as @xmath4 will be substantially different . note also , that since @xmath0 is compact , this restriction means that @xmath529 for some constant @xmath530 . the next several coefficients are : @xmath531 now we rewrite the integral for @xmath319 in the form @xmath532 where @xmath533 our next step is to expand @xmath316 in the power series in @xmath466 @xmath534 and compute integrals over @xmath471 . it is easy to see that in this taylor series : 1 . all coefficients @xmath535 and @xmath536 are polynomials in @xmath471 , @xmath537 , and @xmath455 , @xmath538 . integer powers of @xmath539 have coefficients @xmath535 with even powers of @xmath471 only . half - integer powers of @xmath539 have coefficients @xmath536 with odd powers of @xmath471 only . all @xmath536 will disappear after integration over @xmath471 , since they are odd functions , so we do not need to keep track of them . for computing integrals involving @xmath535 we will use the following result . [ le : giasymptotics1d ] let @xmath540 . then there holds as @xmath4 @xmath541 we have as @xmath4 @xmath542 here we used the asymptotic expansion . therefore , by substituting the expansion into and using lemma [ le : giasymptotics1d ] , we obtain @xmath543 where @xmath544 and @xmath545 due to lemma [ le : gaussianintegral1d ] , integration over @xmath471 will bring inverse powers of @xmath546 . this completes the proof of theorem [ th : expansion - plane - curve ] . now we will compute the first several coefficients @xmath459 . by expanding the function @xmath547 , given by , in the power series in @xmath466 , we obtain @xmath548 therefore , by using formula , lemma [ le : gaussianintegral1d ] , and performing straightforward computations , we get the following lemma . [ le : coefficients - plane - curve ] let conditions of theorem [ th : expansion - plane - curve ] be satisfied . then @xmath549 , \\ a_2&=(4\pi s_2)^{-1/2 } \left [ \left ( -\frac{1}{2 } s_5 { \varkappa}- \frac{1}{12 } s_6 \right ) s_2^{-3 } \right . + \left ( \frac{35}{12 } s_3 s_4 { \varkappa}+ \frac{7}{12 } s_3 s_5 + \frac{35}{96 } s_4 ^ 2 \right ) s_2^{-4}\\ & + \left.\left ( -\frac{35}{12 } s_3 ^ 3 { \varkappa}-\frac{35}{16 } s_3 ^ 2 s_4 \right ) s_2^{-5 } + \frac{385}{288 } s_3 ^ 4 s_2^{-6 } \right],\end{aligned}\ ] ] where coefficients @xmath455 are given by . in this section we consider the case when the fixed point set @xmath0 is just a point , that is , @xmath438 . [ th : expansion - plane - point ] let @xmath15 be a smooth mapping @xmath428 . let @xmath0 be its fixed point set : @xmath447 . let @xmath274 be a real - valued function @xmath431 defined by @xmath448 . let @xmath15 satisfy the assumptions : 1 . @xmath0 is a one - point set , @xmath550 . 2 . there exist constants @xmath450 such that for any @xmath451 , if @xmath452 , then @xmath551^p .\end{gathered}\ ] ] 3 . @xmath552 is a non - degenerate critical point of the function @xmath274 . then there exists the asymptotic expansion as @xmath4 @xmath553 where scalars @xmath554 depend polynomially on derivatives of the function @xmath274 and the inverse of its hessian matrix , evaluated at @xmath552 . before the proof of the theorem , let us note that , given the fact that coefficients @xmath554 are polynomials in derivatives of the function @xmath274 , it is possible to write down their general form by using dimensional considerations @xcite . the deformed heat trace @xmath319 is dimensionless , so the expansion on the right hand side of must also be dimensionless . from the heat equation , @xmath539 has the dimension @xmath178 , so the dimension of @xmath554 is @xmath555 . the function @xmath274 has the dimension of @xmath178 ( @xmath556 dimension of length ) , since it is equal to the world function . its @xmath105-th derivative with respect to the space variables has the dimension @xmath557 . in particular , its second derivative is dimensionless , and all derivatives of order higher than two have dimensions of negative powers of @xmath556 . in order to construct scalars @xmath554 , it is necessary to contract all indices of derivatives with indices of the inverse of the hessian matrix of the function @xmath274 , which we will denote by @xmath322 . the order of this contraction can be different , therefore , the coefficients @xmath554 are linear combinations of invariants of the form @xmath558 where @xmath559 is a multi - index , @xmath560 , and @xmath561 is a permutation of numbers @xmath562 . note , that @xmath563 for any @xmath49 , since @xmath554 depends polynomially only on derivatives of the function @xmath274 of order higher than two ( besides , the function @xmath274 itself and its first derivative vanish on @xmath0 ) . let us introduce the length , the order , and the dimension of the term @xmath75 ( we omit arguments ) : @xmath564 in other words , the length is the number of derivatives of the function @xmath274 in the term @xmath75 , the order is the sum of orders of all derivatives , and the dimension is the ( negative ) space - dimension of @xmath75 . clearly , @xmath565 now we are ready to write down the general formula for the coefficients @xmath554 : @xmath566 where @xmath567 are some numerical coefficients . note , that this sum is finite for any @xmath33 . the idea of this proof is very similar to the one of theorem [ th : expansion - plane - curve ] , but in this case we need to use the multi - dimensional version of the laplace method . in the same way , due to lemma [ le : reductionoftheareaofintegration ] , we can reduce the area of integration to a small neighborhood @xmath465 of @xmath0 and introduce there a system of cartesian coordinates @xmath568 with the origin at @xmath552 . then we expand the function @xmath274 in the taylor series in @xmath471 , scale the variables @xmath569 and obtain @xmath570 where @xmath571 again , @xmath316 has the expansion @xmath572 with coefficients @xmath535 and @xmath536 being polynomials in @xmath568 and derivatives of @xmath274 of order higher than two . and there is no need to keep track of @xmath573 since they do not contribute to the asymptotic expansion of the deformed heat trace @xmath319 . since @xmath552 is a non - degenerate critical point of the function @xmath274 , its hessian matrix @xmath574 is invertible . we will denote the inverse matrix by @xmath322 . it satisfies the equation @xmath575 by substituting into the integral and applying lemma [ le : gaussianintegralmd ] , we conclude that @xmath576 where @xmath577 due to lemma [ le : gaussianintegralmd ] , integration over @xmath471 will bring factors of @xmath322 . this completes the proof of theorem [ th : expansion - plane - point ] . now we will compute the first several coefficients @xmath554 . in order to simplify formulas , we will use the following compact notation for the ( symmetrized ) derivatives of the function @xmath274 : @xmath578 by expanding the function @xmath579 , given by , in the power series in @xmath466 , we obtain @xmath580 in terms of the mapping @xmath15 and its derivatives , derivatives of the functions @xmath274 are @xmath581\\ s_{(\alpha\beta\gamma ) } & = -3\ , g_{\mu(\alpha } { \phi^{\mu}}_{\beta\gamma ) } + 3\ , g_{\mu\nu } { \phi^{\mu}}_{(\alpha}{\phi^{\nu}}_{\beta\gamma ) } \ , , \displaybreak[0]\\ s_{(\alpha\beta\gamma\delta ) } & = -4\ , g_{\mu(\alpha } { \phi^{\mu}}_{\beta\gamma\delta ) } + 4\ , g_{\mu\nu } { \phi^{\mu}}_{(\alpha}{\phi^{\nu}}_{\beta\gamma\delta ) } + 3\ , g_{\mu\nu } { \phi^{\mu}}_{(\alpha\beta}{\phi^{\nu}}_{\gamma\delta ) } \ , , \displaybreak[0]\\ s_{(\alpha\beta\gamma\delta{{\varepsilon } } ) } & = -5\ , g_{\mu(\alpha } { \phi^{\mu}}_{\beta\gamma\delta{{\varepsilon } } ) } + 5\ , g_{\mu\nu } { \phi^{\mu}}_{(\alpha}{\phi^{\nu}}_{\beta\gamma\delta{{\varepsilon } } ) } + 10\ , g_{\mu\nu } { \phi^{\mu}}_{(\alpha\beta}{\phi^{\nu}}_{\gamma\delta{{\varepsilon } } ) } \,,\\ s_{(\alpha\beta\gamma\delta{{\varepsilon}}\zeta ) } & = -6\ , g_{\mu(\alpha } { \phi^{\mu}}_{\beta\gamma\delta{{\varepsilon}}\zeta ) } + 6\ , g_{\mu\nu } { \phi^{\mu}}_{(\alpha}{\phi^{\nu}}_{\beta\gamma\delta{{\varepsilon}}\zeta ) } + 15\ , g_{\mu\nu } { \phi^{\mu}}_{(\alpha\beta}{\phi^{\nu}}_{\gamma\delta{{\varepsilon}}\zeta ) } \nonumber \displaybreak[0]\\ & + 10\ , g_{\mu\nu } { \phi^{\mu}}_{(\alpha\beta\gamma}{\phi^{\nu}}_{\delta{{\varepsilon}}\zeta ) } \ , . \label{eq : s6-plane - point}\end{aligned}\ ] ] these formulas can be obtained as a particular case of , where all derivatives of the world function vanish except for the second order . let @xmath582 be the operator of contraction of a symmetric tensor with the matrix @xmath322 : @xmath583 by using formula , lemma [ le : gaussianintegralmd ] , and performing straightforward computations , we get @xmath584 a_1&=\sqrt{\det q } \left [ -\frac{1}{4}\ , \tau^2 \left ( s_{(4 ) } \right ) + \frac{5}{12}\ , \tau^3 \left ( s_{(3)}\vee s_{(3 ) } \right ) \right],\\[1ex ] a_2&=\sqrt{\det q } \left [ -\frac{1}{12}\ , \right . \tau^3 \left ( s_{(6 ) } \right ) + \frac{7}{12}\ , \tau^4 \left ( s_{(3)}\vee s_{(5 ) } \right ) + \frac{35}{96}\ , \tau^4 \left ( s_{(4)}\vee s_{(4 ) } \right ) \\[1ex ] & -\frac{35}{16}\ , \tau^5 \left ( s_{(3)}\vee s_{(3)}\vee s_{(4 ) } \right ) + \left . \frac{385}{288}\ , \tau^6 \left ( s_{(3)}\vee s_{(3)}\vee s_{(3)}\vee s_{(3 ) } \right ) \right].\end{aligned}\ ] ] since expressions involving symmetrization are inconvenient for practical computations , we will expand them in a symmetrization - free form . we will use the following notation @xmath585\nu_1\ldots\nu_m}=\left(\tau^k\left(s_{(2k+m)}\right)\right)_{\nu_1\ldots\nu_m } = q^{\mu_1\mu_2}\ldots q^{\mu_{2k-1}\mu_{2k}}s_{\mu_1\mu_2\ldots\mu_{2k-1}\mu_{2k}\nu_1\ldots\nu_m } \ \label{eq : contracteds}\end{gathered}\ ] ] that is , the number in brackets is the number of indices contracted with @xmath322 . by using corollaries [ co : contraction33 ] , [ co : contraction35 ] , [ co : contraction44 ] , [ co : contraction3x24 ] , and [ co : contraction3x4 ] , we obtain the following lemma . [ le : coefficients - plane - point ] let conditions of theorem [ th : expansion - plane - point ] be satisfied . then @xmath586\\[1ex ] a_1&=\sqrt{\det q } \left [ -\frac{1}{4 } s_{[4 ] } + \frac{1}{4 } s_{[2]\nu_1 } s_{[2]\nu_2 } q^{\nu_1\nu_2 } + \frac{1}{6 } s_{\nu_1\nu_2\nu_3 } s_{\nu_4\nu_5\nu_6 } q^{\nu_1\nu_4 } q^{\nu_2\nu_5 } q^{\nu_3\nu_6 } \right ] , \displaybreak[0]\\[1ex ] a_2&=\sqrt{\det q } \left [ -\frac{1}{12 } \right . s_{[6 ] } + \frac{1}{4 } s_{[2]\nu_1 } s_{[4]\nu_2 } q^{\nu_1\nu_2 } + \frac{1}{3 } s_{\nu_1\nu_2\nu_3 } s_{[2]\nu_4\nu_5\nu_6 } q^{\nu_1\nu_4 } q^{\nu_2\nu_5 } q^{\nu_3\nu_6 } \displaybreak[0]\\[1ex ] & + \frac{1}{32 } s_{[4 ] } s_{[4 ] } + \frac{1}{4 } s_{[2]\nu_1\nu_2 } s_{[2]\nu_3\nu_4 } q^{\nu_1\nu_3 } q^{\nu_2\nu_4 } \displaybreak[0]\\[1ex ] & + \frac{1}{12 } s_{\nu_1\nu_2\nu_3\nu_4 } s_{\nu_5\nu_6\nu_7\nu_8 } q^{\nu_1\nu_5 } q^{\nu_2\nu_6 } q^{\nu_3\nu_7 } q^{\nu_4\nu_8 } -\frac{1}{16 } s_{[2]\nu_1 } s_{[2]\nu_2 } s_{[4 ] } q^{\nu_1\nu_2 } \displaybreak[0]\\[1ex ] & -\frac{1}{2 } s_{[2]\nu_1 } s_{\nu_2\nu_3\nu_4 } s_{[2]\nu_5\nu_6 } q^{\nu_1\nu_2 } q^{\nu_3\nu_5 } q^{\nu_4\nu_6 } -\frac{1}{4 } s_{[2]\nu_1 } s_{[2]\nu_2 } s_{[2]\nu_3\nu_4 } q^{\nu_1\nu_3 } q^{\nu_2\nu_4 } \displaybreak[0]\\[1ex ] & -\frac{1}{3 } s_{[2]\nu_1 } s_{\nu_2\nu_3\nu_4 } s_{\nu_5\nu_6\nu_7\nu_8 } q^{\nu_1\nu_5 } q^{\nu_2\nu_6 } q^{\nu_3\nu_7 } q^{\nu_4\nu_8 } \displaybreak[0]\\[1ex ] & -\frac{1}{24 } s_{\nu_1\nu_2\nu_3 } s_{\nu_4\nu_5\nu_6 } s_{[4 ] } q^{\nu_1\nu_4 } q^{\nu_2\nu_5 } q^{\nu_3\nu_6 } \displaybreak[0]\\[1ex ] & -\frac{1}{2 } s_{\nu_1\nu_2\nu_3 } s_{\nu_4\nu_5\nu_6 } s_{[2]\nu_7\nu_8 } q^{\nu_1\nu_4 } q^{\nu_2\nu_5 } q^{\nu_3\nu_7 } q^{\nu_6\nu_8 } \displaybreak[0]\\[1ex ] & -\frac{1}{2 } s_{\nu_1\nu_2\nu_3 } s_{\nu_4\nu_5\nu_6 } s_{\nu_7\nu_8\nu_9\nu_{10 } } q^{\nu_1\nu_4 } q^{\nu_2\nu_7 } q^{\nu_3\nu_8 } q^{\nu_5\nu_9 } q^{\nu_6\nu_{10 } } \displaybreak[0]\\[1ex ] & + \frac{1}{32 } s_{[2]\nu_1 } s_{[2]\nu_2 } s_{[2]\nu_3 } s_{[2]\nu_4 } q^{\nu_1\nu_2 } q^{\nu_3\nu_4 } \displaybreak[0]\\[1ex ] & + \frac{1}{24 } s_{[2]\nu_1 } s_{[2]\nu_2 } s_{\nu_3\nu_4\nu_5 } s_{\nu_6\nu_7\nu_8 } q^{\nu_1\nu_2 } q^{\nu_3\nu_6 } q^{\nu_4\nu_7 } q^{\nu_5\nu_8 } \displaybreak[0]\\[1ex ] & + \frac{1}{4 } s_{[2]\nu_1 } s_{\nu_2\nu_3\nu_4 } s_{\nu_5\nu_6\nu_7 } s_{[2]\nu_8 } q^{\nu_1\nu_2 } q^{\nu_3\nu_5 } q^{\nu_4\nu_6 } q^{\nu_7\nu_8 } \displaybreak[0]\\[1ex ] & + \frac{1}{12 } s_{[2]\nu_1 } s_{\nu_2\nu_3\nu_4 } s_{[2]\nu_5 } s_{[2]\nu_6 } q^{\nu_1\nu_2 } q^{\nu_3\nu_5 } q^{\nu_4\nu_6 } \displaybreak[0]\\[1ex ] & + \frac{1}{2 } s_{[2]\nu_1 } s_{\nu_2\nu_3\nu_4 } s_{\nu_5\nu_6\nu_7 } s_{\nu_8\nu_9\nu_{10 } } q^{\nu_1\nu_2 } q^{\nu_3\nu_5 } q^{\nu_4\nu_8 } q^{\nu_6\nu_9 } q^{\nu_7\nu_{10 } } \displaybreak[0]\\[1ex ] & + \frac{1}{72 } s_{\nu_1\nu_2\nu_3 } s_{\nu_4\nu_5\nu_6 } s_{\nu_7\nu_8\nu_9 } s_{\nu_{10}\nu_{11}\nu_{12 } } q^{\nu_1\nu_4 } q^{\nu_2\nu_5 } q^{\nu_3\nu_6 } q^{\nu_7\nu_{10 } } q^{\nu_8\nu_{11 } } q^{\nu_9\nu_{12 } } \displaybreak[0]\\[1ex ] & + \frac{1}{4 } s_{\nu_1\nu_2\nu_3 } s_{\nu_4\nu_5\nu_6 } s_{\nu_7\nu_8\nu_9 } s_{\nu_{10}\nu_{11}\nu_{12 } } q^{\nu_1\nu_4 } q^{\nu_2\nu_5 } q^{\nu_3\nu_7 } q^{\nu_6\nu_{10 } } q^{\nu_8\nu_{11 } } q^{\nu_9\nu_{12 } } \displaybreak[0]\\[1ex ] & \left . + \frac{1}{6 } s_{\nu_1\nu_2\nu_3 } s_{\nu_4\nu_5\nu_6 } s_{\nu_7\nu_8\nu_9 } s_{\nu_{10}\nu_{11}\nu_{12 } } q^{\nu_1\nu_4 } q^{\nu_2\nu_7 } q^{\nu_3\nu_{10 } } q^{\nu_5\nu_8 } q^{\nu_6\nu_{11 } } q^{\nu_9\nu_{12 } } \right],\end{aligned}\ ] ] where tensors @xmath587\nu_1\ldots\nu_m}$ ] are given by , , and @xmath322 is the inverse matrix to @xmath574 . in this chapter we will consider a curved manifold @xmath6 , and a smooth mapping @xmath1 on it , such that its fixed point set @xmath0 is zero - dimensional , that is , it is just a single point . we provide a new proof for asymptotic expansion theorem [ th : expansion - curved - point ] . this proof will give us a way to compute the asymptotic expansion coefficients . the coefficients @xmath419 , @xmath420 in lemma [ le : coefficients - curved - point ] , and the coefficient @xmath420 in lemma [ le : coefficients - curved - point - isometry ] are computed explicitly for the first time . [ th : expansion - curved - point ] let @xmath3 be an @xmath2-dimensional compact smooth riemannian manifold without boundary . let @xmath15 be a smooth mapping @xmath1 . let @xmath0 be its fixed point set : @xmath588 . let @xmath15 satisfy the assumptions : 1 . @xmath0 is a one - point set , @xmath589 . 2 . @xmath590 . then there exists the asymptotic expansion as @xmath4 @xmath591 where the coefficients @xmath554 are scalar invariants depending polynomially on covariant derivatives of the curvature of the metric @xmath13 , symmetrized covariant derivatives of the differential @xmath14 of the mapping @xmath15 , and the matrix @xmath592 evaluated at @xmath593 . the idea of this proof is the same as for a flat manifold . we will choose a convenient system of coordinates , reduce the integral over the whole manifold @xmath6 to the integral over a neighborhood of the fixed point set @xmath0 , expand integrand in taylor series at @xmath593 , and compute the asymptotics of the integral by reducing it to the standard gaussian integrals , which are known exactly . in this proof we will make several references to some computations in the upcoming sections . however , it does not affect rigor , since those computations do not use this theorem . let @xmath274 be a real - valued function , defined in some neighborhood of @xmath0 by @xmath594 . obviously , @xmath593 is a point of minimum of the function @xmath274 , so @xmath595 and @xmath596 . we will show below ( section [ se : symderofs ] , ) that @xmath597 or , in the matrix notation , @xmath598 therefore , the matrix @xmath322 defined by is nothing but @xmath599 . from now on for simplicity we will omit a semicolon in covariant derivatives of scalar functions , for example , @xmath600 , @xmath601 , etc . let @xmath465 be an open geodesic ball with center at the point @xmath593 and small enough radius @xmath248 , so that the functions @xmath156 and @xmath274 are well - defined in @xmath465 . since @xmath602 is compact , there exists a positive constant @xmath530 , such that for any point @xmath603 we have @xmath604 from the last inequality and the standard off - diagonal estimates for the heat kernel in @xcite it follows that as @xmath4 @xmath605 therefore , @xmath606 and according to we have @xmath607 now we will choose a convenient coordinate system in @xmath465 and expand the integrand of in the covariant taylor series at @xmath593 . let @xmath160 be a point in @xmath465 . due to , the components of the tangent vector @xmath608 to the geodesic connecting the points @xmath593 and @xmath160 at the point @xmath593 in the direction of the point @xmath160 and with the norm equal to the length of this geodesic are given by @xmath609 according to , we have @xmath610 one can use this vector to define new coordinates in @xmath465 by assigning to a point @xmath160 with coordinates @xmath424 new coordinates @xmath568 equal to the components of the vector @xmath611 in the coordinates @xmath424 . let @xmath612 be such a coordinate mapping . since @xmath465 is a geodesic ball with radius @xmath248 , the image of @xmath465 is @xmath613 the differentials are related by @xmath614 thus , for the measures we obtain @xmath615 so , the riemannian volume element has in new coordinates the form @xmath616 after the change of variables @xmath617 integral takes the form @xmath618 let @xmath30 be a smooth function on the manifold @xmath6 . then in the neighborhood @xmath465 of the fixed point @xmath593 this function can be expanded in covariant taylor series @xcite @xmath619 where @xmath620 are the coordinates introduced in the previous section . since this expansion involves only covariant derivatives of the function @xmath30 and the coordinates @xmath620 are covariant derivatives of the world function , all parts of this expression are written in an invariant form . note also , that all covariant derivatives of the function @xmath30 are symmetrized , which makes very convenient for practical computations . we will use the notation @xmath621 we will expand each factor of the integrand in in the taylor series at @xmath593 separately and then multiply the expansions . note , that we will need to compute symmetrized covariant derivatives of functions of the form @xmath622 at the point @xmath593 , which has the property @xmath623 . the function @xmath296 has asymptotic expansion in powers of @xmath539 . let @xmath624 that is , the first index stands for the coefficient of the power series in @xmath539 , and the second one for the coefficient of the covariant taylor series in @xmath471 . it is convenient to represent the van fleck - morette determinant in the form @xcite @xmath625 the function @xmath259 has a known expansion in a taylor series with the first two coefficients being equal to zero @xcite , @xmath626 let @xmath627 and @xmath628 we will show in section [ se : vfmdet ] that @xmath629 . the taylor series of the function @xmath274 also starts from the second term , @xmath630 now we rescale the coordinates @xmath631 then all terms in covariant taylor series are transformed as @xmath632 and , hence , they become asymptotic expansions in @xmath466 . by substituting expansions from the previous section and into , we obtain @xmath633 where @xmath634 and @xmath635 note , that terms @xmath636 and @xmath637 depend on the mapping @xmath15 , while @xmath638 does not . let @xmath535 be the coefficients of a power series in @xmath466 for the function @xmath316 , @xmath639 it is easy to see , that the coefficients @xmath535 are polynomials in @xmath471 . moreover , they are polynomials in @xmath640 and @xmath641 . from @xcite and our computations in section [ se : curvepointcomputation ] it follows , that @xmath640 and @xmath641 can be expressed in terms of covariant derivatives of the curvature of the metric @xmath13 and symmetrized covariant derivatives of the differential of the mapping @xmath15 . note , also , that half - integer ( integer ) powers of @xmath539 correspond to polynomials of odd ( even ) degree in @xmath471 . by substituting into , we obtain @xmath642 in the same way , as in lemma [ le : giasymptotics1d ] , we expand the integration over the set @xmath643 to the integration over the whole @xmath644 , i.e. one can show that @xmath645 for both odd and even @xmath33 . after this operation all terms with half - integer powers of @xmath539 disappear , since they contain integrals of odd polynomials over a symmetric set . therefore , we have @xmath646 by comparing this expression with , we finally obtain @xmath647 in this expression : 1 . @xmath648 , where @xmath574 is the positive - definite quadratic form , defined by . @xmath649 are polynomials in @xmath471 , defined by . all integrals in are standard gaussian integrals which can be computed by using lemma [ le : gaussianintegralmd ] . 4 . @xmath649 are polynomials in covariant derivatives of the curvature of the metric @xmath13 and symmetrized covariant derivatives of the differential @xmath14 of the mapping @xmath15 . due to lemma [ le : gaussianintegralmd ] , gaussian integrals in depend on the inverse of the matrix @xmath574 , that is , the matrix @xmath322 . this completes the proof of the theorem . in the remaining part of the chapter we will develop an algorithm for computation of the coefficients @xmath554 in terms of covariant derivatives of the differential @xmath14 and the curvature of the metric @xmath13 . we will compute coefficients @xmath651 , and @xmath420 explicitly . in the next section we describe tools for construction of taylor series of two - point functions . let @xmath652 be a two - point function defined on a manifold @xmath6 . we will denote by @xmath653 $ ] its coincidence limit , that is , @xmath654(x)=\lim_{x'\to x } f(x , x ' ) \,.\end{gathered}\ ] ] obviously , covariant derivatives of the function @xmath30 at different points commute with each other . we will use primes for indices of covariant derivatives at the point @xmath159 . for coincidence limits of mixed derivatives of the function @xmath30 there holds @xcite @xmath655=[f_{\ldots}]_{;\alpha}-[f_{\ldots;\alpha } ] \,,\end{gathered}\ ] ] where dots stand for a fixed set of some indices and may include other covariant derivatives . by using this property repeatedly , we obtain @xmath656 & = [ f_{\ldots}]_{;\alpha\beta } -[f_{\ldots;\alpha}]_{;\beta } -[f_{\ldots;\beta}]_{;\alpha } + [ f_{\ldots;\beta\alpha } ] \,,\\ \label{eq : coincder3 } [ f_{\ldots;\alpha'\beta'\gamma ' } ] & = [ f_{\ldots}]_{;\alpha\beta\gamma } -[f_{\ldots;\gamma}]_{;\alpha\beta } -[f_{\ldots;\beta}]_{;\alpha\gamma } -[f_{\ldots;\alpha}]_{;\beta\gamma } \\ \nonumber & + [ f_{\ldots;\gamma\beta}]_{;\alpha } + [ f_{\ldots;\gamma\alpha}]_{;\beta } + [ f_{\ldots;\beta\alpha}]_{;\gamma } -[f_{\ldots;\gamma\beta\alpha } ] \,.\end{aligned}\ ] ] one can easily obtain the following general formula of this kind for @xmath657 $ ] . let @xmath658 and @xmath659 be its power set . for any @xmath660 , such that @xmath661 and @xmath662 , let @xmath663 be the increasing sequence of indices @xmath664 and @xmath665 be the decreasing sequence of indices @xmath666 . then @xmath667 = \sum_{i\in { \ensuremath{\mathbb{p}}}({\ensuremath{\mathbb{n}}}_m)}\ ( -1)^{{\left|i\right|}}\ , [ f_{\ldots;\underleftarrow{\mu(i)}}]_{;\underrightarrow{\mu(j ) } } \,,\end{gathered}\ ] ] where @xmath668 . let @xmath30 be a two - point function @xmath669 . in this section we will develop an algorithm for computing symmetrized covariant derivatives of the function @xmath670 . we will agree , that indices with primes denote covariant differentiation of the function @xmath30 with respect to the second argument and indices without primes with respect to the first argument . obviously , derivatives with respect to different arguments commute with each other . let @xmath671 be the set of two - point tensors of the type @xmath672 at a point @xmath160 and of the type @xmath673 at a point @xmath159 , that is , the set of linear mappings @xmath674 if some of the numbers @xmath675 are equal to zero , we will omit them . the external tensor product of two - point tensors @xmath676 and @xmath677 is the two - point tensor @xmath678 , defined by @xmath679 we define a mapping @xmath680 by @xmath681 it is easy to see , that the @xmath2-th symmetrized derivative of the function @xmath13 is a linear combination of terms of the form @xmath682 where numbers @xmath683 are non - negative integers , such that @xmath684 a simple computation shows that there exist @xmath685 such terms . let numbers @xmath686 be the coefficients of this linear combination , that is , @xmath687 we will derive a recursive formula for these coefficients . note , that the term @xmath688 of the @xmath2-th derivative can be obtained from the following terms of the @xmath689-th derivative : 1 . @xmath690 , by differentiating @xmath30 with respect to the first argument , if @xmath691 . 2 . @xmath692 , by differentiating @xmath30 with respect to the second argument , if @xmath693 and @xmath694 . 3 . @xmath695 , by differentiating @xmath14 , if @xmath696 . let @xmath697 , if one or more arguments are negative . the above rules give us @xmath698 if @xmath699 , and @xmath700 if @xmath694 . the obvious initial condition for this recursion is @xmath701 one can also get a non - recursive formula for the coefficients @xmath686 . let @xmath19 and @xmath683 be non - negative integers , satisfying . then @xmath702 it is not hard to check that the non - symmetrized covariant derivative @xmath703 is the sum of all terms of the form @xmath704 for all possible values of @xmath683 and all permutations @xmath285 of numbers @xmath705 satisfying certain conditions , that we will determine below . symmetrization of indices @xmath706 makes all terms with fixed @xmath683 the same . hence , the coefficient @xmath686 is the number of all possible permutations @xmath285 . let @xmath707 be defined for @xmath335 by @xmath708 note that indices @xmath709 correspond to the derivative of @xmath30 , index @xmath710 corresponds to the @xmath49-th factor @xmath14 , and indices @xmath711 correspond to the derivative of this factor . this observation leads us to the following conditions on the permutation @xmath285 : 1 . @xmath712 ; 2 . @xmath713 ; 3 . @xmath714 . in words , indices on every factor and the first indices of factors @xmath14 must be in the `` increasing alphabetical order . so , there are @xmath715 ways to choose indices corresponding to the derivative of @xmath30 . after this step @xmath716 is determined uniquely and there are @xmath717 ways to choose indices corresponding to the derivative of the first factor @xmath14 . by continuing this reasoning , we get @xmath718 \times \binom{n - m-2-j_1}{j_2 } \ldots \binom{n - m-(k-1)-j_1-\ldots - j_{k-2}}{j_{k-1 } } \,.\end{gathered}\ ] ] after straightforward transformations , we get the desired formula . now we will apply the technique developed in section [ se : derdefdiag ] to the computation of symmetrized covariant derivatives of the function @xmath719 up to the sixth order . since we are interested in values of the derivatives on the fixed point set @xmath0 , all appearing derivatives of the world function will be actually equal to their coincidence limits . we will take into account the following properties of the world function @xcite : 1 . the world function is symmetric in its arguments . 2 . the first two derivatives with respect to the same argument commute . 3 . @xmath720=0 $ ] . 4 . @xmath721=0 $ ] . @xmath722=-[\sigma_{\alpha\beta'}]=g_{\alpha\beta}$ ] . 6 . @xmath723=[\sigma_{\alpha\beta\gamma'}]=0 $ ] . @xmath724=0 $ ] , @xmath538 . 8 . @xmath725=[\sigma_{\alpha'(\mu_1\ldots\mu_k)}]=0 $ ] , @xmath726 . the recursive algorithm , described above , for computing the coefficients @xmath686 was realized as the python script listed in appendix [ ap : script ] . this script also generates preliminary expressions for derivatives of the function @xmath274 in the latex format . for the derivatives of the differential @xmath14 of the mapping @xmath15 , evaluated at the point @xmath593 , the notation @xmath727 will be used . the first six symmetrized derivatives of @xmath274 are @xmath728 \psi^{\mu}{}_{\gamma } \psi^{\nu}{}_{\delta ) } \,,\\ s_{(\alpha\beta\gamma\delta{{\varepsilon}})}&= -5\ , \psi_{(\alpha\beta\gamma\delta{{\varepsilon } } ) } + 5\ , \psi_{\mu(\alpha } \psi^{\mu}{}_{\beta\gamma\delta{{\varepsilon } } ) } + 10\ , \psi_{\mu(\alpha\beta } \psi^{\mu}{}_{\gamma\delta{{\varepsilon } } ) } \displaybreak[0]\\ & + 30\ , [ \sigma_{\mu'\nu'(\alpha\beta } ] \psi^{\mu}{}_{\gamma } \psi^{\nu}{}_{\delta{{\varepsilon } } ) } \displaybreak[0]\\ & + [ \sigma_{\mu'\nu'\xi'(\alpha } ] \big\ { 5\ , \psi^{\mu}{}_{\beta } \psi^{\nu}{}_{\gamma } \psi^{\xi}{}_{\delta{{\varepsilon } } ) } + 25\ , \psi^{\mu}{}_{\beta } \psi^{\nu}{}_{\gamma\delta } \psi^{\xi}{}_{{{\varepsilon } } ) } \big\ } \displaybreak[0]\\ & + 10\ , [ \sigma_{\mu'\nu'(\alpha\beta\gamma } ] \psi^{\mu}{}_{\delta } \psi^{\nu}{}_{{{\varepsilon } } ) } + 10\ , [ \sigma_{\mu'\nu'\xi'(\alpha\beta } ] \psi^{\mu}{}_{\gamma } \psi^{\nu}{}_{\delta } \psi^{\xi}{}_{{{\varepsilon } } ) } \,,\\ s_{(\alpha\beta\gamma\delta{{\varepsilon}}\zeta)}&= -6\ , \psi_{(\alpha\beta\gamma\delta{{\varepsilon}}\zeta ) } + 6\ , \psi_{\mu(\alpha } \psi^{\mu}{}_{\beta\gamma\delta{{\varepsilon}}\zeta ) } + 15\ , \psi_{\mu(\alpha\beta } \psi^{\mu}{}_{\gamma\delta{{\varepsilon}}\zeta ) } + 10\ , \psi_{\mu(\alpha\beta\gamma } \psi^{\mu}{}_{\delta{{\varepsilon}}\zeta ) } \displaybreak[0]\\ & + [ \sigma_{\mu'\nu'(\alpha\beta } ] \big\ { 60\ , \psi^{\mu}{}_{\gamma } \psi^{\nu}{}_{\delta{{\varepsilon}}\zeta ) } + 45\ , \psi^{\mu}{}_{\gamma\delta } \psi^{\nu}{}_{{{\varepsilon}}\zeta ) } \big\ } \displaybreak[0]\\ & + [ \sigma_{\mu'\nu'\xi'(\alpha } ] \big\ { 6\ , \psi^{\mu}{}_{\beta } \psi^{\nu}{}_{\gamma } \psi^{\xi}{}_{\delta{{\varepsilon}}\zeta ) } + 42\ , \psi^{\mu}{}_{\beta } \psi^{\nu}{}_{\gamma\delta } \psi^{\xi}{}_{{{\varepsilon}}\zeta ) } \displaybreak[0]\\ & + 54\ , \psi^{\mu}{}_{\beta } \psi^{\nu}{}_{\gamma\delta{{\varepsilon } } } \psi^{\xi}{}_{\zeta ) } + 48\ , \psi^{\mu}{}_{\beta\gamma } \psi^{\nu}{}_{\delta{{\varepsilon } } } \psi^{\xi}{}_{\zeta ) } \big\ } \displaybreak[0]\\ & + [ \sigma_{\mu'\nu'\xi'\pi ' } ] \big\ { 3\ , \psi^{\mu}{}_{(\alpha } \psi^{\nu}{}_{\beta } \psi^{\xi}{}_{\gamma\delta } \psi^{\pi}{}_{{{\varepsilon}}\zeta ) } + 9\ , \psi^{\mu}{}_{(\alpha } \psi^{\nu}{}_{\beta\gamma } \psi^{\xi}{}_{\delta } \psi^{\pi}{}_{{{\varepsilon}}\zeta ) } \displaybreak[0]\\ & + 18\ , \psi^{\mu}{}_{(\alpha } \psi^{\nu}{}_{\beta\gamma } \psi^{\xi}{}_{\delta{{\varepsilon } } } \psi^{\pi}{}_{\zeta ) } + 15\ , \psi^{\mu}{}_{(\alpha\beta } \psi^{\nu}{}_{\gamma\delta } \psi^{\xi}{}_{{{\varepsilon } } } \psi^{\pi}{}_{\zeta ) } \big\ } \displaybreak[0]\\ & + 60\ , [ \sigma_{\mu'\nu'(\alpha\beta\gamma } ] \psi^{\mu}{}_{\delta } \psi^{\nu}{}_{{{\varepsilon}}\zeta ) } \displaybreak[0]\\ & + [ \sigma_{\mu'\nu'\xi'(\alpha\beta } ] \big\ { 15\ , \psi^{\mu}{}_{\gamma } \psi^{\nu}{}_{\delta } \psi^{\xi}{}_{{{\varepsilon}}\zeta ) } + 75\ , \psi^{\mu}{}_{\gamma } \psi^{\nu}{}_{\delta{{\varepsilon } } } \psi^{\xi}{}_{\zeta ) } \big\ } \displaybreak[0]\\ & + [ \sigma_{\mu'\nu'\xi'\pi'(\alpha } ] \big\ { 6\ , \psi^{\mu}{}_{\beta } \psi^{\nu}{}_{\gamma } \psi^{\xi}{}_{\delta } \psi^{\pi}{}_{{{\varepsilon}}\zeta ) } + 12\ , \psi^{\mu}{}_{\beta } \psi^{\nu}{}_{\gamma } \psi^{\xi}{}_{\delta{{\varepsilon } } } \psi^{\pi}{}_{\zeta ) } \displaybreak[0]\\ & + 42\ , \psi^{\mu}{}_{\beta } \psi^{\nu}{}_{\gamma\delta } \psi^{\xi}{}_{{{\varepsilon } } } \psi^{\pi}{}_{\zeta ) } \big\ } \displaybreak[0]\\ & + 15\ , [ \sigma_{\mu'\nu'(\alpha\beta\gamma\delta } ] \psi^{\mu}{}_{{{\varepsilon } } } \psi^{\nu}{}_{\zeta ) } + 20\ , [ \sigma_{\mu'\nu'\xi'(\alpha\beta\gamma } ] \psi^{\mu}{}_{\delta } \psi^{\nu}{}_{{{\varepsilon } } } \psi^{\xi}{}_{\zeta ) } \displaybreak[0]\\ & + 15\ , [ \sigma_{\mu'\nu'\xi'\pi'(\alpha\beta } ] \psi^{\mu}{}_{\gamma } \psi^{\nu}{}_{\delta } \psi^{\xi}{}_{{{\varepsilon } } } \psi^{\pi}{}_{\zeta ) } \ , . \ ] ] the second derivative can also be written in the form @xmath729 we need the following coincidence limits for derivatives of the world function : @xmath730 \,,\ [ \sigma_{\mu'\nu'\xi'\alpha } ] \,,\ [ \sigma_{\mu'\nu'\xi'\pi ' } ] \,,\\ [ \sigma_{\mu'\nu'\xi'\alpha\beta } ] \,,\ [ \sigma_{\mu'\nu'\xi'\pi'\alpha } ] \,,\\ [ \sigma_{\mu'\nu'(\alpha\beta\gamma\delta ) } ] \,,\ [ \sigma_{(\mu'\nu'\xi')(\alpha\beta\gamma ) } ] \,.\end{gathered}\ ] ] we will reduce them to the limits which are known @xcite @xmath731&= -\frac{1}{3}\ , r_{\alpha(\mu_1|\beta|\mu_2 ) } \,,\displaybreak[0]\\[1ex ] [ \sigma_{\alpha'\beta ( \mu_1\mu_2\mu_3)}]&= -\frac{1}{2}\ , r_{\alpha(\mu_1|\beta|\mu_2;\mu_3 ) } \,,\displaybreak[0]\\[1ex ] [ \sigma_{\alpha'\beta ( \mu_1\mu_2\mu_3\mu_4)}]&= -\frac{3}{5}\ , r_{\alpha(\mu_1|\beta|\mu_2;\mu_3\mu_4 ) } -\frac{7}{15}\ , r_{\alpha(\mu_1|\gamma|\mu_2 } r^\gamma{}_{\mu_3|\beta|\mu_4 ) } \,.\end{aligned}\ ] ] for this purpose we will use properties of coincidence limits , , expression for the commutator of covariant derivatives , symmetry properties of the curvature tensor , and bianci identities , . after straightforward but somewhat cumbersome computation we have : @xmath732 & = -\frac{2}{3}\ , r_{\alpha(\mu|\beta|\nu ) } \,,\displaybreak[0]\\ [ \sigma_{\mu'\nu'\xi'\alpha } ] & = \frac{2}{3}\ , r_{\alpha(\mu|\xi|\nu ) } \,,\displaybreak[0]\\ [ \sigma_{\mu'\nu'\xi'\pi ' } ] & = -\frac{2}{3}\ , r_{\pi(\mu|\xi|\nu ) } \,,\displaybreak[0]\\ [ \sigma_{\mu'\nu'\xi'\alpha\beta } ] & = \frac{1}{4}\ , r_{(\mu|\xi|\nu)(\alpha;\beta ) } -\frac{5}{12}\ , r_{(\mu|\alpha|\nu)\beta;\xi } -\frac{1}{12}\ , r_{(\alpha|\xi|\beta)(\mu;\nu ) } \,,\displaybreak[0]\\ [ \sigma_{\mu'\nu'\xi'\pi'\alpha } ] & = \frac{5}{6}\ , r_{(\mu|\alpha|\nu)(\xi;\pi ) } -\frac{1}{6}\ , r_{(\xi|\alpha|\pi)(\mu;\nu ) } \,,\displaybreak[0]\\ [ \sigma_{\mu'\nu'(\alpha\beta\gamma\delta ) } ] & = -\frac{2}{5}\ , r_{\mu(\alpha|\nu|\beta;\gamma\delta ) } -\frac{8}{15}\ , r _ { a ( \alpha|\mu|\beta } r^a{}_{\gamma|\nu|\delta ) } \,,\displaybreak[0]\\ [ \sigma_{(\mu'\nu'\xi')(\alpha\beta\gamma ) } ] & = \left ( -\frac{3}{10}\ , r^{(\pi}{}_{(\alpha}{}^{\varrho}{}_{\beta;}{}^{\tau)}{}_{\gamma ) } -\frac{4}{3}\ , r_a{}^{(\pi}{}_{(\alpha}{}^{\varrho}r^{|a|}{}_\beta{}^{\tau)}{}_{\gamma ) } \right ) g_{\pi\mu } g_{{\varrho}\nu } g_{\tau\xi } \,.\end{aligned}\ ] ] detailed computation is given in appendix [ ap : clofwf ] . by combining obtained expressions , we finally have @xmath733\\ & -20\ , r_{(\alpha|\mu|\beta|\nu } \psi^{\mu}{}_{\gamma } \psi^{\nu}{}_{\delta{{\varepsilon } } ) } + \frac{10}{3}\ , r_{(\mu|\xi|\nu)(\alpha } \big\ { \psi^{\mu}{}_{\beta } \psi^{\nu}{}_{\gamma } \psi^{\xi}{}_{\delta{{\varepsilon } } ) } + 5\ , \psi^{\mu}{}_{\beta } \psi^{\nu}{}_{\gamma\delta } \psi^{\xi}{}_{{{\varepsilon } } ) } \big\ } \nonumber \displaybreak[0]\\ & + 10\ , \psi_{\mu(\alpha\beta } \psi^{\mu}{}_{\gamma\delta{{\varepsilon } } ) } -5\ , \psi_{(\alpha\beta\gamma\delta{{\varepsilon } } ) } + 5\ , \psi_{\mu(\alpha } \psi^{\mu}{}_{\beta\gamma\delta{{\varepsilon } } ) } \,,\\ s_{(\alpha\beta\gamma\delta{{\varepsilon}}\zeta)}&= -\big\ { 6\ , r_{\mu(\alpha|\nu|\beta;\gamma\delta } + 8\ , r _ { { \varrho}(\alpha|\mu|\beta } r^{\varrho}{}_{\gamma|\nu|\delta } \big\ } \psi^{\mu}{}_{{{\varepsilon } } } \psi^{\nu}{}_{\zeta ) } \nonumber \displaybreak[0]\\ & -\frac{2}{3}\big\ { 9\ , r_{\mu(\alpha|\nu|\beta;|\xi|\gamma } + 40\ , r _ { { \varrho}\mu(\alpha|\nu| } r^{\varrho}{}_{\beta|\xi|\gamma } \big\ } \psi^{\mu}{}_{\delta } \psi^{\nu}{}_{{{\varepsilon } } } \psi^{\xi}{}_{\zeta ) } \nonumber \displaybreak[0]\\ & -\big\ { 6\ , r_{\mu(\alpha|\nu|\beta;|\xi\pi| } + 8\ , r _ { { \varrho}\mu\nu(\alpha } r^{\varrho}{}_{|\xi\pi|\beta } \big\ } \psi^{\mu}{}_{\gamma } \psi^{\nu}{}_{\delta } \psi^{\xi}{}_{{{\varepsilon } } } \psi^{\pi}{}_{\zeta ) } \nonumber \displaybreak[0]\\ & -30\ , r_{\mu(\alpha|\nu|\beta;\gamma } \psi^{\mu}{}_{\delta } \psi^{\nu}{}_{{{\varepsilon}}\zeta ) } \nonumber \displaybreak[0]\\ & + \frac{5}{8}\ , \big\ { 6\ , r_{(\mu|\xi|\nu)(\alpha;\beta } -10\ , r_{\mu(\alpha|\nu|\beta;|\xi| } -r_{\xi(\alpha|\mu|\beta;|\nu| } -r_{\xi(\alpha|\nu|\beta;|\mu| } \big\ } \nonumber \displaybreak[0]\\ & \times \big\ { \psi^{\mu}{}_{\gamma } \psi^{\nu}{}_{\delta } \psi^{\xi}{}_{{{\varepsilon}}\zeta ) } + 5\ , \psi^{\mu}{}_{\gamma } \psi^{\nu}{}_{\delta{{\varepsilon } } } \psi^{\xi}{}_{\zeta ) } \big\ } \nonumber \displaybreak[0]\\ & + \big\ { 5\ , r_{(\mu}{}^{\varrho}{}_{\nu)(\xi;\pi ) } -r_{(\xi}{}^{\varrho}{}_{\pi)(\mu;\nu ) } \big\ } g_{{\varrho}(\alpha } \big\ { \psi^{\mu}{}_{\beta } \psi^{\nu}{}_{\gamma } \psi^{\xi}{}_{\delta } \psi^{\pi}{}_{{{\varepsilon}}\zeta ) } \nonumber \displaybreak[0]\\ & + 2\ , \psi^{\mu}{}_{\beta } \psi^{\nu}{}_{\gamma } \psi^{\xi}{}_{\delta{{\varepsilon } } } \psi^{\pi}{}_{\zeta ) } + 7\ , \psi^{\mu}{}_{\beta } \psi^{\nu}{}_{\gamma\delta } \psi^{\xi}{}_{{{\varepsilon } } } \psi^{\pi}{}_{\zeta ) } \big\ } \nonumber \displaybreak[0]\\ & -10\ , r_{\mu(\alpha|\nu|\beta } \big\ { 4\ , \psi^{\mu}{}_{\gamma } \psi^{\nu}{}_{\delta{{\varepsilon}}\zeta ) } + 3\ , \psi^{\mu}{}_{\gamma\delta } \psi^{\nu}{}_{{{\varepsilon}}\zeta ) } \big\ } \nonumber \displaybreak[0]\\ & + 4\ , r_{(\mu|\xi|\nu)(\alpha } \big\ { \psi^{\mu}{}_{\beta } \psi^{\nu}{}_{\gamma } \psi^{\xi}{}_{\delta{{\varepsilon}}\zeta ) } + 7\ , \psi^{\mu}{}_{\beta } \psi^{\nu}{}_{\gamma\delta } \psi^{\xi}{}_{{{\varepsilon}}\zeta ) } \nonumber \displaybreak[0]\\ & + 9\ , \psi^{\mu}{}_{\beta } \psi^{\nu}{}_{\gamma\delta{{\varepsilon } } } \psi^{\xi}{}_{\zeta ) } + 8\ , \psi^{\mu}{}_{\beta\gamma } \psi^{\nu}{}_{\delta{{\varepsilon } } } \psi^{\xi}{}_{\zeta ) } \big\ } \nonumber \displaybreak[0]\\ & -2\ , r_{\xi(\mu|\pi|\nu ) } \big\ { \psi^{\mu}{}_{(\alpha } \psi^{\nu}{}_{\beta } \psi^{\xi}{}_{\gamma\delta } \psi^{\pi}{}_{{{\varepsilon}}\zeta ) } + 3\ , \psi^{\mu}{}_{(\alpha } \psi^{\nu}{}_{\beta\gamma } \psi^{\xi}{}_{\delta } \psi^{\pi}{}_{{{\varepsilon}}\zeta ) } \nonumber \displaybreak[0]\\ & + 6\ , \psi^{\mu}{}_{(\alpha } \psi^{\nu}{}_{\beta\gamma } \psi^{\xi}{}_{\delta{{\varepsilon } } } \psi^{\pi}{}_{\zeta ) } + 5\ , \psi^{\mu}{}_{(\alpha\beta } \psi^{\nu}{}_{\gamma\delta } \psi^{\xi}{}_{{{\varepsilon } } } \psi^{\pi}{}_{\zeta ) } \big\ } \nonumber \displaybreak[0]\\ & -6\ , \psi_{(\alpha\beta\gamma\delta{{\varepsilon}}\zeta ) } + 6\ , \psi_{\mu(\alpha } \psi^{\mu}{}_{\beta\gamma\delta{{\varepsilon}}\zeta ) } + 15\ , \psi_{\mu(\alpha\beta } \psi^{\mu}{}_{\gamma\delta{{\varepsilon}}\zeta ) } + 10\ , \psi_{\mu(\alpha\beta\gamma } \psi^{\mu}{}_{\delta{{\varepsilon}}\zeta ) } \label{eq : s6curvature } \ , . \ ] ] by using the general algorithm for computing coincidence limits of the heat kernel coefficients and their symmetrized covariant derivatives , developed in @xcite , we obtain @xmath734 & = 1 \,,\displaybreak[0]\\ [ \omega_1 ] & = -\frac{1}{6}\ , r \,,\displaybreak[0]\\ [ \omega_{1;\alpha } ] & = -\frac{1}{12}\ , r_{;\alpha } \,,\displaybreak[0]\\ [ \omega_{1;\alpha\beta } ] & = -e_{\alpha\beta } \,,\displaybreak[0]\\ [ \omega_2 ] & = \frac{1}{36}\ , r^2 + \frac{1}{15}\ , r_{;\mu}{}^{\mu } -\frac{1}{90}\ , r_{\mu\nu } r^{\mu\nu } + \frac{1}{90}\ , r_{\mu\nu\xi\pi } r^{\mu\nu\xi\pi } \,,\end{aligned}\ ] ] where @xmath735 so , for the coefficients @xmath736 at the point @xmath593 we have @xmath737\\ a_1(x_0 ) & = -\frac{1}{6}\ , r \,,\displaybreak[0]\\ a_{1;\alpha}(x_0 ) & = -\frac{1}{12}\ , r_{;\alpha } -\frac{1}{12}\ , r_{;\mu } \psi^\mu{}_\alpha \,,\displaybreak[0]\\ a_{1;\alpha\beta}(x_0 ) & = -e_{\alpha\beta } -\frac{1}{6}\ , r_{;\mu(\alpha } \psi^\mu{}_{\beta ) } + 2\,e_{\mu(\alpha } \psi^\mu{}_{\beta ) } -e_{\mu\nu } \psi^\mu{}_{(\alpha } \psi^\nu{}_{\beta ) } -\frac{1}{12}\ , r_{;\mu } \psi^\mu{}_{(\alpha\beta ) } \,,\displaybreak[0]\\ a_2(x_0 ) & = \frac{1}{36}\ , r^2 + \frac{1}{15}\ , r_{;\mu}{}^{\mu } -\frac{1}{90}\ , r_{\mu\nu } r^{\mu\nu } + \frac{1}{90}\ , r_{\mu\nu\xi\pi } r^{\mu\nu\xi\pi } \,.\end{aligned}\ ] ] for the function @xmath259 in representation we have @xcite @xmath738 & = 0 \,,\\ [ \zeta_\alpha ] & = 0 \,,\\ [ \zeta_{\alpha\beta } ] & = \frac{1}{6}\ , r_{\alpha\beta } \,,\\ [ \zeta_{(\alpha\beta\gamma ) } ] & = \frac{1}{4}\ , r_{(\alpha\beta;\gamma ) } \,,\\ [ \zeta_{(\alpha\beta\gamma\delta ) } ] & = \frac{3}{10}\ , r_{(\alpha\beta;\gamma\delta ) } + \frac{1}{15}\ , r _ { \mu ( \alpha| \nu |\beta } r^\mu{}_\gamma{}^\nu{}_{\delta ) } \,.\end{aligned}\ ] ] in the same way , as for the world function , we obtain non - symmetrized derivatives of the function @xmath259 from these expressions for symmetrized derivatives . after performing straightforward transformations , we have @xmath739 & = \frac{1}{4}\ , r_{(\alpha\beta;\gamma ) } \,,\\ [ \zeta_{\alpha\beta\gamma\delta } ] & = f_{\alpha\beta\gamma\delta } \,,\end{aligned}\ ] ] where @xmath740 by using the method developed in section [ se : derdefdiag ] , for derivatives of the function @xmath741 at the point @xmath593 we obtain @xmath742\\ & -\frac{1}{2}\ , r_{\mu(\alpha;\beta } \psi^{\mu}{}_{\gamma ) } + \frac{1}{4}\ , r_{\mu\nu;(\alpha } \psi^{\mu}{}_{\beta } \psi^{\nu}{}_{\gamma ) } -\frac{1}{2}\ , r_{\mu(\alpha;|\nu| } \psi^{\mu}{}_{\beta } \psi^{\nu}{}_{\gamma ) } \displaybreak[0]\\ & + \frac{1}{4}\ , r_{\mu\nu;\xi } \psi^{\mu}{}_{(\alpha } \psi^{\nu}{}_{\beta } \psi^{\xi}{}_{\gamma ) } \,,\\ \eta_{(\alpha\beta\gamma\delta ) } ( x_0 ) & = -\frac{2}{3}\ , r_{\mu(\alpha } \psi^{\mu}{}_{\beta\gamma\delta ) } + \frac{1}{6}\ , r_{\mu\nu } \big\ { 4\ , \psi^{\mu}{}_{(\alpha } \psi^{\nu}{}_{\beta\gamma\delta ) } + 3\ , \psi^{\mu}{}_{(\alpha\beta } \psi^{\nu}{}_{\gamma\delta ) } \big\ } \displaybreak[0]\\ & + r_{(\alpha\beta;|\mu| } \psi^{\mu}{}_{\gamma\delta ) } + 2\ , r_{\mu\nu;(\alpha } \psi^{\mu}{}_{\beta } \psi^{\nu}{}_{\gamma\delta ) } + r_{\mu\nu;(\alpha\beta } \psi^{\mu}{}_{\gamma } \psi^{\nu}{}_{\delta ) } \displaybreak[0]\\ & -\frac{1}{2}\ , r_{(\alpha\beta;|\mu } \psi^{\mu}{}_{\gamma\delta ) } -r_{\mu(\alpha;\beta } \psi^{\mu}{}_{\gamma\delta ) } -r_{\mu\nu;(\alpha } \psi^{\mu}{}_{\beta } \psi^{\nu}{}_{\gamma\delta ) } \displaybreak[0]\\ & -2\ , r_{\mu(\alpha;|\nu| } \psi^{(\mu}{}_{\beta } \psi^{\nu)}{}_{\gamma\delta ) } + \frac{3}{2}\ , r_{(\mu\nu;\xi ) } \psi^{\mu}{}_{(\alpha } \psi^{\nu}{}_{\beta } \psi^{\xi}{}_{\gamma\delta ) } + r_{(\alpha\beta;\gamma|\mu| } \psi^{\mu}{}_{\delta ) } \displaybreak[0]\\ & -r_{\mu\nu;(\alpha\beta } \psi^{\mu}{}_{\gamma } \psi^{\nu}{}_{\delta ) } -2\ , r_{\mu(\alpha;|\nu|\beta } \psi^{\mu}{}_{\gamma } \psi^{\nu}{}_{\delta ) } + r_{\mu\nu;\xi(\alpha } \psi^{\mu}{}_{\beta } \psi^{\nu}{}_{\gamma } \psi^{\xi}{}_{\delta ) } \displaybreak[0]\\ & + \frac{3}{10}\ , r_{(\alpha\beta;\gamma\delta ) } + \frac{1}{15}\ , r _ { \mu ( \alpha| \nu |\beta } r^\mu{}_\gamma{}^\nu{}_{\delta ) } -4\ , f_{(\alpha\beta\gamma|\mu| } \psi^{\mu}{}_{\delta ) } \displaybreak[0]\\ & + 6\ , f_{\mu\nu(\alpha\beta } \psi^{\mu}{}_{\gamma } \psi^{\nu}{}_{\delta ) } -4\ , f_{\mu\nu\xi(\alpha } \psi^{\mu}{}_{\beta } \psi^{\nu}{}_{\gamma } \psi^{\xi}{}_{\delta ) } \displaybreak[0]\\ & + \bigg\ { \frac{3}{10}\ , r_{\mu\nu;\xi\pi } + \frac{1}{15}\ , r _ { { \varrho}\mu \theta \nu } r^{\varrho}{}_\xi{}^\theta{}_{\pi } \bigg\ } \psi^{\mu}{}_{(\alpha } \psi^{\nu}{}_{\beta } \psi^{\xi}{}_{\gamma } \psi^{\pi}{}_{\delta ) } \,.\end{aligned}\ ] ] by direct expansion we get the coefficients @xmath743 , @xmath744 , and @xmath745 , defined by ( all objects are computed at the point @xmath593 and we will omit arguments ) @xmath746\\ b_2 & = -{\widetilde}{a}_{1(0 ) } -h_2 + \frac{1}{2}\ , h_1 ^ 2 \,,\displaybreak[0]\\ b_4 & = \frac{1}{2}\ , { \widetilde}{a}_{2(0 ) } -{\widetilde}{a}_{1(2 ) } + h_1 { \widetilde}{a}_{1(1 ) } + ( h_2-\frac{1}{2}\ , h_1 ^ 2)\ , { \widetilde}{a}_{1(0 ) } \\ & -h_4 + h_1 h_3 + \frac{1}{2}\ , h_2 ^ 2 -\frac{1}{2}\ , h_1 ^ 2 h_2 + \frac{1}{24}\ , h_1 ^ 4 \,.\end{aligned}\ ] ] due to and taking into account that @xmath747 , we have @xmath748\\ h_2 & = \dfrac{1}{2}\,{\widetilde}{s}_{(4 ) } + 2\,{\widetilde}{\zeta}_{(2 ) } - { \widetilde}{\eta}_{(2 ) } \,,\displaybreak[0]\\ h_3 & = \dfrac{1}{2}\,{\widetilde}{s}_{(5 ) } + 2\,{\widetilde}{\zeta}_{(3 ) } - { \widetilde}{\eta}_{(3 ) } \,,\displaybreak[0]\\ h_4 & = \dfrac{1}{2}\,{\widetilde}{s}_{(6 ) } + 2\,{\widetilde}{\zeta}_{(4 ) } - { \widetilde}{\eta}_{(4 ) } \,.\end{aligned}\ ] ] therefore , @xmath746\\ b_2 & = -a_1 - 2\,{\widetilde}{\zeta}_{(2 ) } + { \widetilde}{\eta}_{(2 ) } -\dfrac{1}{2}\,{\widetilde}{s}_{(4 ) } + \frac{1}{8}\ , \left({\widetilde}{s}_{(3)}\right)^2 \,,\displaybreak[0]\\ b_4 & = \frac{1}{2}\ , a_2 -{\widetilde}{a}_{1(2 ) } + 2\ , a_1 { \widetilde}{\zeta}_{(2 ) } - a_1 { \widetilde}{\eta}_{(2 ) } + \dfrac{1}{2}\ , a_1 { \widetilde}{s}_{(4 ) } - 2\,{\widetilde}{\zeta}_{(4 ) } + { \widetilde}{\eta}_{(4 ) } + \dfrac{1}{2}\ , { \widetilde}{s}_{(3 ) } { \widetilde}{a}_{1(1 ) } \displaybreak[0]\\ & + 2\ , \left({\widetilde}{\zeta}_{(2)}\right)^2 + \frac{1}{2}\ , \left({\widetilde}{\eta}_{(2)}\right)^2 -{\widetilde}{\zeta}_{(2 ) } { \widetilde}{\eta}_{(2 ) } -\dfrac{1}{2}\,{\widetilde}{s}_{(6 ) } + \dfrac{1}{2}\ , { \widetilde}{s}_{(4 ) } { \widetilde}{\zeta}_{(2 ) } -\dfrac{1}{4}\ , { \widetilde}{s}_{(4 ) } { \widetilde}{\eta}_{(2 ) } \displaybreak[0]\\ & -\frac{1}{8}\ , a_1 \left({\widetilde}{s}_{(3)}\right)^2 + { \widetilde}{s}_{(3)}{\widetilde}{\zeta}_{(3 ) } -\frac{1}{2}\,{\widetilde}{s}_{(3)}{\widetilde}{\eta}_{(3 ) } + \dfrac{1}{4}\,{\widetilde}{s}_{(5)}{\widetilde}{s}_{(3 ) } + \dfrac{1}{8}\ , \left({\widetilde}{s}_{(4)}\right)^2 \displaybreak[0]\\ & -\frac{1}{4}\ , \left({\widetilde}{s}_{(3)}\right)^2 { \widetilde}{\zeta}_{(2 ) } + \frac{1}{8}\ , \left({\widetilde}{s}_{(3)}\right)^2 { \widetilde}{\eta}_{(2 ) } -\dfrac{1}{16}\ , { \widetilde}{s}_{(4 ) } \left({\widetilde}{s}_{(3)}\right)^2 + \frac{1}{384}\ , \left({\widetilde}{s}_{(3)}\right)^4 \,.\end{aligned}\ ] ] since integrals in are gaussian , it is convenient to introduce a gaussian average for a function @xmath30 , defined on @xmath644 , by @xmath749 where @xmath750 the gaussian average is normalized so that @xmath751 . recall that @xmath752 , so takes the form @xmath753 due to lemma [ le : gaussianintegralmd ] , for polynomials in @xmath471 we have @xmath754 where @xmath322 is the matrix inverse to @xmath574 and it is given by . let @xmath582 be the operator of contraction of a symmetric tensor with the matrix @xmath322 : @xmath583 by using formula and performing straightforward computations , we get @xmath755\\[2ex ] a_1 & = { \varrho}\bigg [ -a_1 -2\ , \tau \left ( \zeta_{(2 ) } \right ) + \tau \left ( \eta_{(2 ) } \right ) -\frac{1}{4}\ , \tau^2 \left ( s_{(4 ) } \right ) + \frac{5}{12}\ , \tau^3 \left ( s_{(3)}\vee s_{(3 ) } \right ) \bigg ] \,,\displaybreak[0]\\[2ex ] a_2 & = { \varrho}\bigg [ \frac{1}{2}\ , a_2 -\tau \left ( a_{1(2 ) } \right ) + 2\ , a_1 \tau \left ( \zeta_{(2 ) } \right ) -a_1 \tau \left ( \eta_{(2 ) } \right ) \displaybreak[0]\\ & + \dfrac{1}{4}\ , a_1 \tau^2 \left ( s_{(4 ) } \right ) -\tau^2 \left ( \zeta_{(4 ) } \right ) + \frac{1}{2}\ , \tau^2 \left ( \eta_{(4 ) } \right ) + \tau^2 \left ( s_{(3 ) } \vee a_{1(1 ) } \right ) \displaybreak[0]\\ & + 6\ , \tau^2 \left ( \zeta_{(2 ) } \vee \zeta_{(2 ) } \right ) + \frac{3}{2}\ , \tau^2 \left ( \eta_{(2 ) } \vee \eta_{(2 ) } \right ) -3\ , \tau^2 \left ( \zeta_{(2 ) } \vee \eta_{(2 ) } \right ) \displaybreak[0]\\ & -\dfrac{1}{12}\ , \tau^3 \left ( s_{(6 ) } \right ) + \dfrac{5}{4}\ , \tau^3 \left ( s_{(4 ) } \vee \zeta_{(2 ) } \right ) -\dfrac{5}{8}\ , \tau^3 \left ( s_{(4 ) } \vee \eta_{(2 ) } \right ) \displaybreak[0]\\ & -\frac{5}{12}\ , a_1 \tau^3 \left ( s_{(3 ) } \vee s_{(3 ) } \right ) + \frac{10}{3}\ , \tau^3 \left ( s_{(3 ) } \vee \zeta_{(3 ) } \right ) -\frac{5}{3}\ , \tau^3 \left ( s_{(3 ) } \vee \eta_{(3 ) } \right ) \displaybreak[0]\\ & + \dfrac{7}{12}\ , \tau^4 \left ( s_{(5 ) } \vee s_{(3 ) } \right ) + \dfrac{35}{96}\ , \tau^4 \left ( s_{(4 ) } \vee s_{(4 ) } \right ) \displaybreak[0]\\ & -\frac{35}{6}\ , \tau^4 \left ( s_{(3 ) } \vee s_{(3 ) } \vee \zeta_{(2 ) } \right ) + \frac{35}{12}\ , \tau^4 \left ( s_{(3 ) } \vee s_{(3 ) } \vee \eta_{(2 ) } \right ) \displaybreak[0]\\ & -\dfrac{35}{16}\ , \tau^5 \left ( s_{(4 ) } \vee s_{(3 ) } \vee s_{(3 ) } \right ) \displaybreak[0]\\ & + \frac{385}{288}\ , \tau^6 \left ( s_{(3 ) } \vee s_{(3 ) } \vee s_{(3 ) } \vee s_{(3 ) } \right ) \bigg ] \,.\end{aligned}\ ] ] all terms in these expressions were evaluated above . their substitution gives expressions for the computed coefficients in terms of the curvature , the differential @xmath14 of the mapping @xmath15 , and their covariant derivatives . [ le : coefficients - curved - point ] let conditions of theorem [ th : expansion - curved - point ] be satisfied . then @xmath755\\[2ex ] a_1 & = { \varrho}\bigg\ { h^{(1 ) } + h^{(1)}_{(\alpha\beta ) } q^{\alpha\beta } + h^{(1)}_{(\alpha\beta\gamma\delta ) } q^{\alpha\beta } q^{\gamma\delta } + h^{(1)}_{(\alpha\beta\gamma\delta{{\varepsilon}}\zeta ) } q^{\alpha\beta } q^{\gamma\delta } q^{{{\varepsilon}}\zeta } \bigg\ } \,,\displaybreak[0]\\[2ex ] a_2 & = { \varrho}\bigg\ { h^{(2 ) } + h^{(2)}_{(\alpha\beta ) } q^{\alpha\beta } + h^{(2)}_{(\alpha\beta\gamma\delta ) } q^{\alpha\beta } q^{\gamma\delta } + h^{(2)}_{(\alpha\beta\gamma\delta{{\varepsilon}}\zeta ) } q^{\alpha\beta } q^{\gamma\delta } q^{{{\varepsilon}}\zeta } \displaybreak[0]\\ & + h^{(2)}_{(\alpha\beta\gamma\delta{{\varepsilon}}\zeta\eta\theta ) } q^{\alpha\beta } q^{\gamma\delta } q^{{{\varepsilon}}\zeta } q^{\eta\theta } + h^{(2)}_{(\alpha\beta\gamma\delta{{\varepsilon}}\zeta\eta\theta{\varkappa}\lambda ) } q^{\alpha\beta } q^{\gamma\delta } q^{{{\varepsilon}}\zeta } q^{\eta\theta } q^{{\varkappa}\lambda } \displaybreak[0]\\ & + h^{(2)}_{(\alpha\beta\gamma\delta{{\varepsilon}}\zeta\eta\theta{\varkappa}\lambda\psi\omega ) } q^{\alpha\beta } q^{\gamma\delta } q^{{{\varepsilon}}\zeta } q^{\eta\theta } q^{{\varkappa}\lambda } q^{\psi\omega } \bigg\ } , \end{aligned}\ ] ] where @xmath756\\[2ex ] h^{(1)}_{\alpha\beta } & = -\frac{1}{6}\ , r_{\alpha\beta } -\frac{1}{3}\ , r_{\mu\alpha } \psi^{\mu}{}_{\beta } + \frac{1}{6}\ , r_{\mu\nu } \psi^{\mu}{}_{\alpha } \psi^{\nu}{}_{\beta } \,,\displaybreak[0]\\[2ex ] h^{(1)}_{\alpha\beta\gamma\delta } & = r_{\alpha\mu\beta\nu } \psi^{\mu}{}_{\gamma } \psi^{\nu}{}_{\delta } + \psi_{\alpha\beta\gamma\delta } -\frac{3}{4}\ , \psi_{\mu\alpha\beta } \psi^{\mu}{}_{\gamma\delta } -\psi_{\mu\alpha } \psi^{\mu}{}_{\beta\gamma\delta } \,,\displaybreak[0]\\[2ex ] h^{(1)}_{\alpha\beta\gamma\delta{{\varepsilon}}\zeta } & = \frac{15}{4}\ , \psi_{\alpha\beta\gamma } \psi_{\delta{{\varepsilon}}\zeta } -\frac{15}{2}\ , \psi_{\mu\alpha } \psi^{\mu}{}_{\beta\gamma } \psi_{\delta{{\varepsilon}}\zeta } + \frac{15}{4}\ , \psi_{\mu\alpha } \psi^{\mu}{}_{\beta\gamma}\psi_{\nu\delta } \psi^{\nu}{}_{{{\varepsilon}}\zeta } \,,\displaybreak[0]\\[2ex ] h^{(2 ) } & = \frac{1}{72}\ , r^2 + \frac{1}{30}\ , r_{;\mu}{}^{\mu } -\frac{1}{180}\ , r_{\mu\nu } r^{\mu\nu } + \frac{1}{180}\ , r_{\mu\nu\xi\pi } r^{\mu\nu\xi\pi } \,,\displaybreak[0]\\[2ex ] h^{(2)}_{\alpha\beta } & = -\frac{1}{36}\ , r r_{\alpha\beta } + e_{\alpha\beta } + \bigg ( \frac{1}{6}\ , r_{;\mu\alpha } -\frac{1}{18}\ , r r_{\mu\alpha } -2\,e_{\mu\alpha } \bigg ) \psi^\mu{}_{\beta } \displaybreak[0]\\ & + \bigg ( \frac{1}{36}\ , r r_{\mu\nu } + e_{\mu\nu } \bigg ) \psi^\mu{}_{\alpha } \psi^\nu{}_{\beta } + \frac{1}{12}\ , r_{;\mu } \psi^\mu{}_{\alpha\beta } \,,\displaybreak[0]\\[2ex ] h^{(2)}_{\alpha\beta\gamma\delta } & = \frac{1}{8}\ , r_{\alpha\beta } r_{\gamma\delta } -\frac{1}{30}\ , r _ { \mu \alpha \nu \beta } r^\mu{}_\gamma{}^\nu{}_{\delta } -\frac{3}{20}\ , r_{\alpha\beta;\gamma\delta } \displaybreak[0]\\ & + \bigg ( \frac{1}{2}\ , r_{\alpha\beta;\gamma\mu } -2\ , f_{\alpha\beta\gamma\mu } \bigg ) \psi^{\mu}{}_{\delta } \displaybreak[0]\\ & + \bigg ( \frac{1}{6}\ , r_{\mu\alpha } r_{\nu\beta } + \frac{1}{6}\ , r r_{\mu\alpha\nu\beta } -r_{\mu\alpha;\nu\beta } + 3\ , f_{\mu\nu\alpha\beta } \bigg ) \psi^{\mu}{}_{\gamma } \psi^{\nu}{}_{\delta } \displaybreak[0]\\ & + \bigg ( \frac{1}{2}\ , r_{\mu\nu;\xi\alpha } -\frac{1}{6}\ , r_{\mu\alpha } r_{\nu\xi } -2\ , f_{\mu\nu\xi\alpha } \bigg ) \psi^{\mu}{}_{\beta } \psi^{\nu}{}_{\gamma } \psi^{\xi}{}_{\delta } \displaybreak[0]\\ & + \bigg ( \frac{1}{24}\ , r_{\mu\nu } r_{\xi\pi } + \frac{1}{30}\ , r _ { { \varrho}\mu \tau \nu } r^{\varrho}{}_\xi{}^\tau{}_{\pi } + \frac{3}{20}\ , r_{\mu\nu;\xi\pi } \bigg ) \psi^{\mu}{}_{\alpha } \psi^{\nu}{}_{\beta } \psi^{\xi}{}_{\gamma } \psi^{\pi}{}_{\delta } \displaybreak[0]\\ & + \frac{1}{4}\ , r_{;\alpha } \psi_{\beta\gamma\delta } + \bigg ( \frac{1}{4}\ , r_{\alpha\beta;\mu } -\frac{1}{2}\ , r_{\mu\alpha;\beta } \bigg ) \psi^{\mu}{}_{\gamma\delta } -\frac{1}{4}\ , r_{;\alpha } \psi_{\mu\beta } \psi^{\mu}{}_{\gamma\delta } \displaybreak[0]\\ & + \frac{1}{4}\ , r_{;\mu } \psi^\mu{}_\alpha \psi_{\beta\gamma\delta } + \bigg ( \frac{1}{2}\ , r_{\mu\nu;\alpha } -\frac{1}{2}\ , r_{\mu\alpha;\nu } -\frac{1}{2}\ , r_{\nu\alpha;\mu } \bigg ) \psi^{\mu}{}_{\beta } \psi^{\nu}{}_{\gamma\delta } \displaybreak[0]\\ & -\frac{1}{4}\ , r_{;\mu } \psi^\mu{}_\alpha \psi_{\nu\beta } \psi^{\nu}{}_{\gamma\delta } + \bigg ( \frac{1}{4}\ , r_{\mu\nu;\xi } + \frac{1}{2}\ , r_{\xi\mu;\nu } \bigg ) \psi^{\mu}{}_{\alpha } \psi^{\nu}{}_{\beta } \psi^{\xi}{}_{\gamma\delta } \displaybreak[0]\\ & + \frac{1}{6}\ , r \psi_{\alpha\beta\gamma\delta } -\frac{1}{3}\ , r_{\mu\alpha } \psi^{\mu}{}_{\beta\gamma\delta } -r \bigg ( \frac{1}{6}\ , \psi_{\mu\alpha } \psi^{\mu}{}_{\beta\gamma\delta } + \frac{1}{8}\ , \psi_{\mu\alpha\beta } \psi^{\mu}{}_{\gamma\delta } \bigg ) \displaybreak[0]\\ & + r_{\mu\nu } \bigg ( \frac{1}{3}\ , \psi^{\mu}{}_{\alpha } \psi^{\nu}{}_{\beta\gamma\delta } + \frac{1}{4}\ , \psi^{\mu}{}_{\alpha\beta } \psi^{\nu}{}_{\gamma\delta } \bigg ) \,,\displaybreak[0]\\[2ex ] h^{(2)}_{\alpha\beta\gamma\delta{{\varepsilon}}\zeta } & = \bigg ( \dfrac{2}{3}\ , r_{\xi\alpha\mu\beta } r^\xi{}_{\gamma\nu\delta } -\dfrac{5}{12}\ , r_{\mu\alpha\nu\beta } r_{\gamma\delta } + \dfrac{1}{2}\ , r_{\mu\alpha\nu\beta;\gamma\delta } \bigg ) \psi^{\mu}{}_{{{\varepsilon } } } \psi^{\nu}{}_{\zeta } \displaybreak[0]\\ & + \bigg ( \dfrac{20}{9}\ , r_{\pi\mu\alpha\nu } r^\pi{}_{\beta\xi\gamma } -\dfrac{5}{6}\ , r_{\mu\alpha\nu\beta } r_{\xi\gamma } + \dfrac{1}{2}\ , r_{\mu\alpha\nu\beta;\xi\gamma } \bigg ) \psi^{\mu}{}_{\delta } \psi^{\nu}{}_{{{\varepsilon } } } \psi^{\xi}{}_{\zeta } \displaybreak[0]\\ & + \bigg ( \dfrac{5}{12}\ , r_{\mu\alpha\nu\beta } r_{\xi\pi } + \dfrac{2}{3}\ , r _ { { \varrho}\mu\alpha\nu } r^{\varrho}{}_{\xi\beta\pi } + \dfrac{1}{2}\ , r_{\mu\alpha\nu\beta;\xi\pi } \bigg ) \psi^{\mu}{}_{\gamma } \psi^{\nu}{}_{\delta } \psi^{\xi}{}_{{{\varepsilon } } } \psi^{\pi}{}_{\zeta } \displaybreak[0]\\ & + \frac{7}{4}\ , r_{\alpha\beta;\gamma } \bigg ( \psi_{\mu\delta } \psi^{\mu}{}_{{{\varepsilon}}\zeta } -\psi_{\delta{{\varepsilon}}\zeta } \bigg ) + \bigg ( \frac{3}{4}\ , r_{\alpha\beta;\mu } -\frac{3}{2}\ , r_{\mu\alpha;\beta } \bigg ) \psi^{\mu}{}_{\gamma } \psi_{\delta{{\varepsilon}}\zeta } \displaybreak[0]\\ & + \dfrac{5}{2}\ , r_{\mu\alpha\nu\beta;\gamma } \psi^{\mu}{}_{\delta } \psi^{\nu}{}_{{{\varepsilon}}\zeta } + \bigg ( \frac{3}{2}\ , r_{\mu\alpha;\beta } -\frac{3}{4}\ , r_{\alpha\beta;\mu } \bigg ) \psi^{\mu}{}_{\gamma } \psi_{\nu\delta } \psi^{\nu}{}_{{{\varepsilon}}\zeta } \displaybreak[0]\\ & + \bigg ( \frac{3}{4}\ , r_{\mu\nu;\alpha } -\frac{3}{2}\ , r_{\alpha\mu;\nu } \bigg ) \psi^{\mu}{}_{\beta } \psi^{\nu}{}_{\gamma } \psi_{\delta{{\varepsilon}}\zeta } \displaybreak[0]\\ & + \bigg ( \dfrac{5}{4}\ , r_{\xi\mu\alpha\nu;\beta } + \frac{15}{4}\ , r_{\xi\alpha\mu\beta;\nu } \bigg ) \psi^{\mu}{}_{\gamma } \psi^{\nu}{}_{\delta } \psi^{\xi}{}_{{{\varepsilon}}\zeta } \displaybreak[0]\\ & + \bigg ( \frac{3}{2}\ , r_{\alpha\mu;\nu } -\frac{3}{4}\ , r_{\mu\nu;\alpha } \bigg ) \psi^{\mu}{}_{\beta } \psi^{\nu}{}_{\gamma } \psi_{\xi\delta } \psi^{\xi}{}_{{{\varepsilon}}\zeta } \displaybreak[0]\\ & + \frac{3}{4}\ , r_{\mu\nu;\xi } \psi^{\mu}{}_{\alpha } \psi^{\nu}{}_{\beta } \psi^{\xi}{}_{\gamma } \psi_{\delta{{\varepsilon}}\zeta } + r_{\mu\alpha\nu\pi;\xi } \psi^{\mu}{}_{\beta } \psi^{\nu}{}_{\gamma } \psi^{\xi}{}_{\delta } \psi^{\pi}{}_{{{\varepsilon}}\zeta } \displaybreak[0]\\ & -\frac{3}{4}\ , r_{\mu\nu;\xi } \psi^{\mu}{}_{\alpha } \psi^{\nu}{}_{\beta } \psi^{\xi}{}_{\gamma } \psi_{\pi\delta } \psi^{\pi}{}_{{{\varepsilon}}\zeta } -\dfrac{5}{12}\ , r_{{{\varepsilon}}\zeta } \psi_{\alpha\beta\gamma\delta } + \frac{5}{8}\ , r \psi_{\alpha\beta\gamma } \psi_{\delta{{\varepsilon}}\zeta } \displaybreak[0]\\ & + r_{\alpha\beta } \bigg ( \dfrac{5}{16}\ , \psi_{\mu\gamma\delta } \psi^{\mu}{}_{{{\varepsilon}}\zeta } + \dfrac{5}{12}\ , \psi_{\mu\gamma } \psi^{\mu}{}_{\delta{{\varepsilon}}\zeta } \bigg ) \displaybreak[0]\\ & -r_{\mu\alpha } \bigg ( \dfrac{5}{6}\ , \psi^{\mu}{}_{\beta } \psi_{\gamma\delta{{\varepsilon}}\zeta } + \frac{3}{2}\ , \psi^{\mu}{}_{\beta\gamma } \psi_{\delta{{\varepsilon}}\zeta } \bigg ) \displaybreak[0]\\ & + r_{\mu\alpha\nu\beta } \bigg ( \dfrac{10}{3}\ , \psi^{\mu}{}_{\gamma } \psi^{\nu}{}_{\delta{{\varepsilon}}\zeta } + \dfrac{5}{2}\ , \psi^{\mu}{}_{\gamma\delta } \psi^{\nu}{}_{{{\varepsilon}}\zeta } \bigg ) -\frac{5}{4}\ , r \psi_{\mu\alpha } \psi^{\mu}{}_{\beta\gamma } \psi_{\delta{{\varepsilon}}\zeta } \displaybreak[0]\\ & + r_{\mu\alpha } \bigg ( \dfrac{5}{8}\ , \psi^{\mu}{}_{\beta } \psi_{\nu\gamma\delta } \psi^{\nu}{}_{{{\varepsilon}}\zeta } + \dfrac{5}{6}\ , \psi^{\mu}{}_{\beta } \psi_{\nu\gamma } \psi^{\nu}{}_{\delta{{\varepsilon}}\zeta } + \dfrac{3}{2}\ , \psi^{\mu}{}_{\beta\gamma } \psi_{\pi\delta } \psi^{\pi}{}_{{{\varepsilon}}\zeta } \bigg ) \displaybreak[0]\\ & + r_{\mu\nu } \bigg ( \dfrac{5}{12}\ , \psi^{\mu}{}_{\alpha } \psi^{\nu}{}_{\beta } \psi_{\gamma\delta{{\varepsilon}}\zeta } + \frac{3}{2}\ , \psi^{\mu}{}_{\alpha } \psi^{\nu}{}_{\beta\gamma } \psi_{\delta{{\varepsilon}}\zeta } \bigg ) \displaybreak[0]\\ & + r_{\mu\xi\nu\alpha } \bigg ( \dfrac{7}{6}\ , \psi^{\mu}{}_{\beta } \psi^{\nu}{}_{\gamma } \psi^{\xi}{}_{\delta{{\varepsilon}}\zeta } + \dfrac{3}{2}\ , \psi^{\mu}{}_{\beta } \psi^{\nu}{}_{\gamma\delta } \psi^{\xi}{}_{{{\varepsilon}}\zeta } \bigg ) \displaybreak[0]\\ & + \frac{5}{8}\ , r \psi_{\mu\alpha } \psi^{\mu}{}_{\beta\gamma } \psi_{\nu\delta } \psi^{\nu}{}_{{{\varepsilon}}\zeta } + \dfrac{1}{4}\ , r_{\mu\xi\nu\pi } \psi^{\mu}{}_{\alpha } \psi^{\nu}{}_{\beta } \psi^{\xi}{}_{\gamma\delta } \psi^{\pi}{}_{{{\varepsilon}}\zeta } \displaybreak[0]\\ & -r_{\mu\nu } \bigg ( \dfrac{5}{16}\ , \psi^{\mu}{}_{\alpha } \psi^{\nu}{}_{\beta } \psi_{\xi\gamma\delta } \psi^{\xi}{}_{{{\varepsilon}}\theta } + \dfrac{5}{12}\ , \psi^{\mu}{}_{\alpha } \psi^{\nu}{}_{\beta } \psi_{\xi\gamma } \psi^{\xi}{}_{\delta{{\varepsilon}}\theta } \displaybreak[0]\\ & + \frac{3}{2}\ , \psi^{\mu}{}_{\alpha } \psi^{\nu}{}_{\beta\gamma } \psi_{\xi\delta } \psi^{\xi}{}_{{{\varepsilon}}\zeta } \bigg ) + \dfrac{1}{2}\ , \psi_{\alpha\beta\gamma\delta{{\varepsilon}}\zeta } -\dfrac{1}{2}\ , \psi_{\mu\alpha } \psi^{\mu}{}_{\beta\gamma\delta{{\varepsilon}}\zeta } \displaybreak[0]\\ & -\dfrac{5}{4}\ , \psi_{\mu\alpha\beta } \psi^{\mu}{}_{\gamma\delta{{\varepsilon}}\zeta } -\dfrac{5}{6}\ , \psi_{\mu\alpha\beta\gamma } \psi^{\mu}{}_{\delta{{\varepsilon}}\zeta } \,,\displaybreak[0]\\[2ex ] h^{(2)}_{\alpha\beta\gamma\delta{{\varepsilon}}\zeta\eta\theta } & = \dfrac{35}{6}\ , r_{\alpha\mu\beta\nu } r_{{{\varepsilon}}\xi\zeta\pi } \psi^{\mu}{}_{\gamma } \psi^{\nu}{}_{\delta } \psi^{\xi}{}_{\eta } \psi^{\pi}{}_{\theta } + \dfrac{35}{4}\ , r_{\mu\alpha\nu\beta;\gamma } \psi^{\mu}{}_{\delta } \psi^{\nu}{}_{{{\varepsilon } } } \psi_{\zeta\eta\theta } \displaybreak[0]\\ & + \dfrac{35}{4}\ , r_{\mu\alpha\nu\beta;\xi } \psi^{\mu}{}_{\gamma } \psi^{\nu}{}_{\delta } \psi^{\xi}{}_{{{\varepsilon } } } \psi_{\zeta\eta\theta } -\dfrac{35}{4}\ , r_{\mu\alpha\nu\beta;\gamma } \psi^{\mu}{}_{\delta } \psi^{\nu}{}_{{{\varepsilon } } } \psi_{\pi\zeta } \psi^{\pi}{}_{\eta\theta } \displaybreak[0]\\ & -\dfrac{35}{4}\ , r_{\mu\alpha\nu\beta;\xi } \psi^{\mu}{}_{\gamma } \psi^{\nu}{}_{\delta } \psi^{\xi}{}_{{{\varepsilon } } } \psi_{\pi\zeta } \psi^{\pi}{}_{\eta\theta } -\frac{35}{4}\ , r_{\mu\alpha } \psi^{\mu}{}_{\beta } \psi_{\gamma\delta{{\varepsilon } } } \psi_{\zeta\eta\theta } \displaybreak[0]\\ & + r_{\alpha\beta } \bigg ( \frac{35}{4}\ , \psi_{\mu\gamma } \psi^{\mu}{}_{\delta{{\varepsilon } } } \psi_{\zeta\eta\theta } -\frac{35}{8}\ , \psi_{\gamma\delta{{\varepsilon } } } \psi_{\zeta\eta\theta } \bigg ) \displaybreak[0]\\ & + r_{\mu\alpha\nu\beta } \bigg ( \dfrac{35}{3}\ , r_{\mu\alpha\nu\beta } \psi^{\mu}{}_{\gamma } \psi^{\nu}{}_{\delta } \psi_{{{\varepsilon}}\zeta\eta\theta } + 35\ , \psi^{\mu}{}_{\gamma } \psi^{\nu}{}_{\delta{{\varepsilon } } } \psi_{\zeta\eta\theta } \bigg ) \displaybreak[0]\\ & -\frac{35}{8}\ , r_{\alpha\beta } \psi_{\mu\gamma } \psi^{\mu}{}_{\delta{{\varepsilon } } } \psi_{\nu\zeta } \psi^{\nu}{}_{\eta\theta } + \frac{35}{2}\ , r_{\mu\alpha } \psi^{\mu}{}_{\beta } \psi_{\nu\gamma } \psi^{\nu}{}_{\delta{{\varepsilon } } } \psi_{\zeta\eta\theta } \displaybreak[0]\\ & + \frac{35}{8}\ , r_{\mu\nu } \psi^{\mu}{}_{\alpha } \psi^{\nu}{}_{\beta } \psi_{\gamma\delta{{\varepsilon } } } \psi_{\zeta\eta\theta } -r_{\mu\alpha\nu\beta } \bigg ( 35\ , \psi^{\mu}{}_{\gamma } \psi^{\nu}{}_{\delta{{\varepsilon } } } \psi_{\pi\zeta } \psi^{\pi}{}_{\eta\theta } \displaybreak[0]\\ & + \dfrac{35}{4}\ , \psi^{\mu}{}_{\gamma } \psi^{\nu}{}_{\delta } \psi_{\xi{{\varepsilon}}\zeta } \psi^{\xi}{}_{\eta\theta } + \dfrac{35}{3}\ , \psi^{\mu}{}_{\gamma } \psi^{\nu}{}_{\delta } \psi_{\xi{{\varepsilon } } } \psi^{\xi}{}_{\zeta\eta\theta } \bigg ) \displaybreak[0]\\ & + \frac{35}{4}\ , r_{\mu\xi\nu\alpha } \psi^{\mu}{}_{\beta } \psi^{\nu}{}_{\gamma } \psi^{\xi}{}_{\delta{{\varepsilon } } } \psi_{\zeta\eta\theta } -\frac{35}{4}\ , r_{\mu\alpha } \psi^{\mu}{}_{\beta } \psi_{\nu\gamma } \psi^{\nu}{}_{\delta{{\varepsilon } } } \psi_{\xi\zeta } \psi^{\xi}{}_{\eta\theta } \displaybreak[0]\\ & -\frac{35}{4}\ , r_{\mu\nu } \psi^{\mu}{}_{\alpha } \psi^{\nu}{}_{\beta } \psi_{\xi\gamma } \psi^{\xi}{}_{\delta{{\varepsilon } } } \psi_{\zeta\eta\theta } -\frac{35}{4}\ , r_{\mu\xi\nu\alpha } \psi^{\mu}{}_{\beta } \psi^{\nu}{}_{\gamma } \psi^{\xi}{}_{\delta{{\varepsilon } } } \psi_{\pi\zeta } \psi^{\pi}{}_{\eta\theta } \displaybreak[0]\\ & + \frac{35}{8}\ , r_{\mu\nu } \psi^{\mu}{}_{\alpha } \psi^{\nu}{}_{\beta } \psi_{\xi\gamma } \psi^{\xi}{}_{\delta{{\varepsilon } } } \psi_{\pi\zeta } \psi^{\pi}{}_{\eta\theta } + \dfrac{35}{6}\ , \psi_{\alpha\beta\gamma\delta } \psi_{{{\varepsilon}}\zeta\eta\theta } \displaybreak[0]\\ & + \dfrac{35}{4}\ , \psi_{\alpha\beta\gamma\delta{{\varepsilon } } } \psi_{\zeta\eta\theta } -\dfrac{35}{2}\ , \psi_{\mu\alpha\beta } \psi^{\mu}{}_{\gamma\delta{{\varepsilon } } } \psi_{\zeta\eta\theta } -\dfrac{35}{4}\ , \psi_{\mu\alpha } \psi^{\mu}{}_{\beta\gamma\delta{{\varepsilon } } } \psi_{\zeta\eta\theta } \displaybreak[0]\\ & -\dfrac{35}{4}\ , \psi_{\alpha\beta\gamma\delta{{\varepsilon } } } \psi_{\pi\zeta } \psi^{\pi}{}_{\eta\theta } -\dfrac{35}{4}\ , \psi_{\mu\alpha\beta } \psi^{\mu}{}_{\gamma\delta } \psi_{{{\varepsilon}}\zeta\eta\theta } -\dfrac{35}{3}\ , \psi_{\mu\alpha } \psi^{\mu}{}_{\beta\gamma\delta } \psi_{{{\varepsilon}}\zeta\eta\theta } \displaybreak[0]\\ & + \dfrac{35}{2}\ , \psi_{\mu\alpha\beta } \psi^{\mu}{}_{\gamma\delta{{\varepsilon } } } \psi_{\pi\zeta } \psi^{\pi}{}_{\eta\theta } + \dfrac{35}{4}\ , \psi_{\mu\alpha } \psi^{\mu}{}_{\beta\gamma\delta{{\varepsilon } } } \psi_{\pi\zeta } \psi^{\pi}{}_{\eta\theta } \displaybreak[0]\\ & + \frac{105}{32}\ , \psi_{\mu\alpha\beta } \psi^{\mu}{}_{\gamma\delta } \psi_{\xi{{\varepsilon}}\zeta } \psi^{\xi}{}_{\eta\theta } + \dfrac{35}{6}\ , \psi_{\mu\alpha } \psi^{\mu}{}_{\beta\gamma\delta } \psi_{\xi{{\varepsilon } } } \psi^{\xi}{}_{\zeta\eta\theta } \displaybreak[0]\\ & + \dfrac{35}{4}\ , \psi_{\mu\alpha\beta } \psi^{\mu}{}_{\gamma\delta } \psi_{\xi{{\varepsilon } } } \psi^{\xi}{}_{\zeta\eta\theta } \,,\displaybreak[0]\\[2ex ] h^{(2)}_{\alpha\beta\gamma\delta{{\varepsilon}}\zeta\eta\theta{\varkappa}\lambda } & = \dfrac{315}{4}\ , \bigg [ r_{\mu\alpha\nu\beta } \bigg ( \psi^{\mu}{}_{\gamma } \psi^{\nu}{}_{\delta } \psi_{{{\varepsilon}}\zeta\eta } \psi_{\theta{\varkappa}\lambda } -2\ , \psi^{\mu}{}_{\gamma } \psi^{\nu}{}_{\delta } \psi_{\xi{{\varepsilon } } } \psi^{\xi}{}_{\zeta\eta } \psi_{\theta{\varkappa}\lambda } \displaybreak[0]\\ & + \psi^{\mu}{}_{\gamma } \psi^{\nu}{}_{\delta } \psi_{\xi{{\varepsilon } } } \psi^{\xi}{}_{\zeta\eta } \psi_{\pi\theta } \psi^{\pi}{}_{{\varkappa}\lambda } \bigg ) + \psi_{\alpha\beta\gamma } \psi_{\delta{{\varepsilon}}\zeta } \psi_{\eta\theta{\varkappa}\lambda } \displaybreak[0]\\ & -\frac{3}{4}\ , \psi_{\alpha\beta\gamma } \psi_{\delta{{\varepsilon}}\zeta } \psi_{\mu\eta\theta } \psi^{\mu}{}_{{\varkappa}\lambda } -\psi_{\alpha\beta\gamma } \psi_{\delta{{\varepsilon}}\zeta } \psi_{\mu\eta } \psi^{\mu}{}_{\theta{\varkappa}\lambda } \displaybreak[0]\\ & -2\ , \psi_{\mu\alpha } \psi^{\mu}{}_{\beta\gamma } \psi_{\delta{{\varepsilon}}\zeta } \psi_{\eta\theta{\varkappa}\lambda } + \frac{3}{2}\ , \psi_{\mu\alpha } \psi^{\mu}{}_{\beta\gamma } \psi_{\delta{{\varepsilon}}\zeta } \psi_{\nu\eta\theta } \psi^{\nu}{}_{{\varkappa}\lambda } \displaybreak[0]\\ & + 2\ , \psi_{\mu\alpha } \psi^{\mu}{}_{\beta\gamma } \psi_{\delta{{\varepsilon}}\zeta } \psi_{\nu\eta } \psi^{\nu}{}_{\theta{\varkappa}\lambda } + \psi_{\mu\alpha } \psi^{\mu}{}_{\beta\gamma } \psi_{\nu\delta } \psi^{\nu}{}_{{{\varepsilon}}\zeta } \psi_{\eta\theta{\varkappa}\lambda } \displaybreak[0]\\ & -\frac{3}{4}\ , \psi_{\mu\alpha } \psi^{\mu}{}_{\beta\gamma } \psi_{\nu\delta } \psi^{\nu}{}_{{{\varepsilon}}\zeta } \psi_{\xi\eta\theta } \psi^{\xi}{}_{{\varkappa}\lambda } -\psi_{\mu\alpha } \psi^{\mu}{}_{\beta\gamma } \psi_{\nu\delta } \psi^{\nu}{}_{{{\varepsilon}}\zeta } \psi_{\xi\eta } \psi^{\xi}{}_{\theta{\varkappa}\lambda } \bigg ] \,,\displaybreak[0]\\[2ex ] h^{(2)}_{\alpha\beta\gamma\delta{{\varepsilon}}\zeta\eta\theta{\varkappa}\lambda\psi\omega } & = \frac{3465}{32}\ , \bigg [ \psi_{\alpha\beta\gamma } \psi_{\delta{{\varepsilon}}\zeta } \psi_{\eta\theta{\varkappa } } \psi_{\lambda\psi\omega } -4\ , \psi_{\mu\alpha } \psi^{\mu}{}_{\beta\gamma } \psi_{\delta{{\varepsilon}}\zeta } \psi_{\eta\theta{\varkappa } } \psi_{\lambda\psi\omega } \displaybreak[0]\\ & + 6\ , \psi_{\mu\alpha } \psi^{\mu}{}_{\beta\gamma } \psi_{\nu\delta } \psi^{\nu}{}_{{{\varepsilon}}\zeta } \psi_{\eta\theta{\varkappa } } \psi_{\lambda\psi\omega } \displaybreak[0]\\ & -4\ , \psi_{\mu\alpha } \psi^{\mu}{}_{\beta\gamma } \psi_{\nu\delta } \psi^{\nu}{}_{{{\varepsilon}}\zeta } \psi_{\xi\eta } \psi^{\xi}{}_{\theta{\varkappa } } \psi_{\lambda\psi\omega } \displaybreak[0]\\ & + \psi_{\mu\alpha } \psi^{\mu}{}_{\beta\gamma } \psi_{\nu\delta } \psi^{\nu}{}_{{{\varepsilon}}\zeta } \psi_{\xi\eta } \psi^{\xi}{}_{\theta{\varkappa } } \psi_{\pi\lambda } \psi^{\pi}{}_{\psi\omega } \bigg ] \,,\\ \psi^{\mu}{}_{\nu_1\nu_2\ldots\nu_k}&=(d\phi)^{\mu'}{}_{\nu_1;\nu_2\ldots\nu_k}(x_0 ) \,,\\ { \varrho}&={\left|\det \big(i - d\phi(x_0)\big)\right|}^{-1 } \,,\end{aligned}\ ] ] and @xmath757 , @xmath758 are given by , respectively . if the mapping @xmath15 is an isometry , all derivatives of @xmath14 vanish ( see , for example , @xcite ) and from lemma [ le : coefficients - curved - point ] we get the following lemma . [ le : coefficients - curved - point - isometry ] let conditions of theorem [ th : expansion - curved - point ] be satisfied . then @xmath755\\[2ex ] a_1 & = { \varrho}\bigg\ { \frac{1}{6}\ , r -\bigg [ \frac{1}{6}\ , r_{\alpha\beta } + \frac{1}{3}\ , r_{\mu\alpha } \psi^{\mu}{}_{\beta } -\frac{1}{6}\ , r_{\mu\nu } \psi^{\mu}{}_{\alpha } \psi^{\nu}{}_{\beta } \bigg ] q^{\alpha\beta } \displaybreak[0]\\ & + r_{\alpha\mu\beta\nu } \psi^{\mu}{}_{\gamma } \psi^{\nu}{}_{\delta } q^{(\alpha\beta } q^{\gamma\delta ) } \bigg\ } \,,\displaybreak[0]\\[2ex ] a_2 & = { \varrho}\bigg\ { \frac{1}{72}\ , r^2 + \frac{1}{30}\ , r_{;\mu}{}^{\mu } -\frac{1}{180}\ , r_{\mu\nu } r^{\mu\nu } + \frac{1}{180}\ , r_{\mu\nu\xi\pi } r^{\mu\nu\xi\pi } \displaybreak[0]\\ & + \bigg [ e_{\alpha\beta } -2\,e_{\mu\alpha } \psi^\mu{}_{\beta } + e_{\mu\nu } \psi^\mu{}_{\alpha } \psi^\nu{}_{\beta } + \frac{1}{6}\ , r_{;\mu\alpha } \psi^\mu{}_{\beta } \displaybreak[0]\\ & -\frac{1}{36}\ , r r_{\alpha\beta } -\frac{1}{18}\ , r r_{\mu\alpha } \psi^{\mu}{}_{\beta } + \frac{1}{36}\ , r r_{\mu\nu } \psi^{\mu}{}_{\alpha } \psi^{\nu}{}_{\beta } \bigg ] q^{\alpha\beta } \displaybreak[0]\\ & + \bigg [ -\frac{3}{20}\ , r_{\alpha\beta;\gamma\delta } + \frac{1}{2}\ , r_{\alpha\beta;\gamma\mu } \psi^{\mu}{}_{\delta } -r_{\mu\alpha;\nu\beta } \psi^{\mu}{}_{\gamma } \psi^{\nu}{}_{\delta } \displaybreak[0]\\ & + \frac{1}{2}\ , r_{\mu\nu;\xi\alpha } \psi^{\mu}{}_{\beta } \psi^{\nu}{}_{\gamma } \psi^{\xi}{}_{\delta } + \frac{3}{20}\ , r_{\mu\nu;\xi\pi } \psi^{\mu}{}_{\alpha } \psi^{\nu}{}_{\beta } \psi^{\xi}{}_{\gamma } \psi^{\pi}{}_{\delta } \displaybreak[0]\\ & + \frac{1}{8}\ , r_{\alpha\beta } r_{\gamma\delta } + \frac{1}{6}\ , r_{\mu\alpha } r_{\xi\gamma } \psi^{\mu}{}_{\beta } \psi^{\xi}{}_{\delta } -\frac{1}{6}\ , r_{\mu\alpha } r_{\xi\pi } \psi^{\mu}{}_{\beta } \psi^{\xi}{}_{\gamma } \psi^{\pi}{}_{\delta } \displaybreak[0]\\ & + \frac{1}{24}\ , r_{\mu\nu } r_{\xi\pi } \psi^{\mu}{}_{\alpha } \psi^{\nu}{}_{\beta } \psi^{\xi}{}_{\gamma } \psi^{\pi}{}_{\delta } + \frac{1}{6}\ , r r_{\alpha\mu\beta\nu } \psi^{\mu}{}_{\gamma } \psi^{\nu}{}_{\delta } \displaybreak[0]\\ & -\frac{1}{30}\ , r _ { \mu \alpha \nu \beta } r^\mu{}_\gamma{}^\nu{}_{\delta } + \frac{1}{30}\ , r _ { { \varrho}\mu \tau \nu } r^{\varrho}{}_\xi{}^\tau{}_{\pi } \psi^{\mu}{}_{\alpha } \psi^{\nu}{}_{\beta } \psi^{\xi}{}_{\gamma } \psi^{\pi}{}_{\delta } \displaybreak[0]\\ & -2\ , f_{\alpha\beta\gamma\mu } \psi^{\mu}{}_{\delta } + 3\ , f_{\mu\nu\alpha\beta } \psi^{\mu}{}_{\gamma } \psi^{\nu}{}_{\delta } -2\ , f_{\mu\nu\xi\alpha } \psi^{\mu}{}_{\beta } \psi^{\nu}{}_{\gamma } \psi^{\xi}{}_{\delta } \bigg ] q^{(\alpha\beta } q^{\gamma\delta ) } \displaybreak[0]\\ & + \bigg [ -\dfrac{5}{12}\ , r_{\alpha\mu\beta\nu } r_{{{\varepsilon}}\zeta } \psi^{\mu}{}_{\gamma } \psi^{\nu}{}_{\delta } -\dfrac{5}{6}\ , r_{\alpha\mu\beta\nu } r_{\xi{{\varepsilon } } } \psi^{\xi}{}_{\zeta } \psi^{\mu}{}_{\gamma } \psi^{\nu}{}_{\delta } \displaybreak[0]\\ & + \dfrac{5}{12}\ , r_{\alpha\mu\beta\nu } r_{\xi\pi } \psi^{\xi}{}_{{{\varepsilon } } } \psi^{\pi}{}_{\zeta } \psi^{\mu}{}_{\gamma } \psi^{\nu}{}_{\delta } + \dfrac{2}{3}\ , r_{\xi\alpha\mu\beta } r^\xi{}_{\gamma\nu\delta } \psi^{\mu}{}_{{{\varepsilon } } } \psi^{\nu}{}_{\zeta } \displaybreak[0]\\ & + \dfrac{20}{9}\ , r_{\pi\mu\alpha\nu } r^\pi{}_{\beta\xi\gamma } \psi^{\mu}{}_{\delta } \psi^{\nu}{}_{{{\varepsilon } } } \psi^{\xi}{}_{\zeta } + \dfrac{2}{3}\ , r _ { { \varrho}\mu\nu\alpha } r^{\varrho}{}_{\xi\pi\beta } \psi^{\mu}{}_{\gamma } \psi^{\nu}{}_{\delta } \psi^{\xi}{}_{{{\varepsilon } } } \psi^{\pi}{}_{\zeta } \displaybreak[0]\\ & + \dfrac{1}{2}\ , r_{\mu\alpha\nu\beta;\gamma\delta } \psi^{\mu}{}_{{{\varepsilon } } } \psi^{\nu}{}_{\zeta } + \dfrac{1}{2}\ , r_{\mu\alpha\nu\beta;\xi\gamma } \psi^{\mu}{}_{\delta } \psi^{\nu}{}_{{{\varepsilon } } } \psi^{\xi}{}_{\zeta } \displaybreak[0]\\ & + \dfrac{1}{2}\ , r_{\mu\alpha\nu\beta;\xi\pi } \psi^{\mu}{}_{\gamma } \psi^{\nu}{}_{\delta } \psi^{\xi}{}_{{{\varepsilon } } } \psi^{\pi}{}_{\zeta } \bigg ] q^{(\alpha\beta } q^{\gamma\delta } q^{{{\varepsilon}}\zeta ) } \displaybreak[0]\\ & + \dfrac{35}{6}\ , r_{\alpha\mu\beta\nu } r_{{{\varepsilon}}\xi\zeta\pi } \psi^{\mu}{}_{\gamma } \psi^{\nu}{}_{\delta } \psi^{\xi}{}_{\eta } \psi^{\pi}{}_{\theta } q^{(\alpha\beta } q^{\gamma\delta } q^{{{\varepsilon}}\zeta } q^{\eta\theta ) } \bigg\ } , \\ \psi^{\mu}{}_{\nu}&=(d\phi)^{\mu'}{}_{\nu}(x_0 ) \,,\\ { \varrho}&={\left|\det \big(i - d\phi(x_0)\big)\right|}^{-1 } \,,\end{aligned}\ ] ] and @xmath757 , @xmath758 are given by , respectively . in this section we compare our expression for the coefficient @xmath419 in a particular case when the mapping @xmath15 is an isometry @xmath759 q^{\alpha\beta } \nonumber \displaybreak[0]\\ & + r_{\alpha\mu\beta\nu } \psi^{\mu}{}_{\gamma } \psi^{\nu}{}_{\delta } q^{(\alpha\beta } q^{\gamma\delta ) } \bigg\ } \label{eq : a_1-isometry}\end{aligned}\ ] ] with the result obtained by donnelly in @xcite : @xmath760 \,,\end{gathered}\ ] ] where @xmath761 and all tensors are evaluated at the point @xmath593 . from it follows that @xmath762 since @xmath763 and @xmath764 , we have @xmath765 and takes the form @xmath766\\ & + \frac{1}{6}\ , r_{\mu\nu } ( b^\mu{}_\xi b^{\nu \xi}-2\,b^{\mu\nu}+\delta^{\mu\nu } ) \displaybreak[0]\\ & + r_{\alpha\mu\beta\nu } b^{(\alpha}{}_\xi b^{\beta |\xi| } \big(b^\mu{}_\pi-\delta^\mu{}_\pi \big ) \big(b^{\nu)\pi}-\delta^{\nu)\pi}\big ) \bigg\ } .\end{aligned}\ ] ] after performing straightforward computations , we get the same expression as . let us summarize the results obtained in this thesis . our goal was to study the geometry of the fixed point set @xmath0 of a smooth mapping @xmath15 on a riemannian manifold @xmath3 by computing the asymptotic expansion as @xmath4 of the trace of the deformed heat kernel @xmath5 of the laplace operator on @xmath6 . the fixed point set @xmath0 was assumed to be a smooth compact submanifold of @xmath6 . multi - dimensional gaussian integrals are expressed in terms of full contractions of the symmetrized products of symmetric tensors with a symmetric ( 2,0)-tensor . in chapter [ ch : symmetrictensors ] we developed a technique for obtaining symmetrization - free formulas for such contracted products . the described algorithm can be easily implemented for automatic computations . in chapter [ ch : flatmanifolds ] we considered the case of a flat manifold , @xmath418 . we showed the existence of the asymptotic expansion in the form @xmath767 where @xmath768 and @xmath769 are scalar invariants depending polynomially on derivatives of the function @xmath719 , and the inverse matrix of its hessian matrix . we developed a generalized laplace method for computing the coefficients @xmath459 and computed explicitly the coefficients @xmath16 , @xmath17 , and @xmath18 for both zero- and one - dimensional fixed point sets @xmath0 . we used the results from chapter [ ch : symmetrictensors ] to obtain a symmetrization - free form in the zero - dimensional case . finally , in chapter [ ch : curvedmanifolds ] we obtained the asymptotic expansion as @xmath4 for an arbitrary curved manifold @xmath6 and a zero - dimensional fixed point set @xmath589 in the form @xmath770 where @xmath554 are scalar invariants depending polynomially on covariant derivatives of the curvature of the metric @xmath13 , symmetrized covariant derivatives of the differential @xmath14 of the mapping @xmath15 , and the matrix @xmath771 evaluated at the point @xmath593 . the coefficients @xmath554 are expressed in terms of symmetrized covariant derivatives of functions of the form @xmath772 . we developed an algorithm for computation of such symmetrized covariant derivatives and realized it as the python script listed in appendix [ ap : script ] . we computed explicitly the coefficients @xmath773 , @xmath419 , and @xmath420 . in a particular case , when the mapping @xmath15 is an isometry , our result for the coefficient @xmath419 coincides with the former result of donnelly @xcite . .... # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # computes symmetrized covariant derivatives of the function # s(x)=\sigma(x,\phi(x ) ) # on the fixed point set of the mapping \phi(x ) . # # input : # sseparator - a separator which will be inserted between all terms # in the output # nderivativeorder - the order of the derivative to be computed # # uses indices from lsalphabet for derivatives and from # lsprimealphabet for dummy indices in contractions # # creates in the current directory file named " output.tex " which # contains symmetrized derivatives of the function s(x ) up to # nderivativeorder written in latex notation . # # properties of the world functions , taken into account : # [ \sigma]=0 # [ \sigma_\alpha]=0 # [ \sigma_{\alpha\beta\gamma}]=0 # [ \sigma_{(\alpha_1\dots\alpha_n)}]=0 , n>=3 # [ \sigma_{\mu(\alpha_1\dots\alpha_n)}]=0 , n>=2 # # an existing file with the same name will be overwritten . # # copyright ( c ) andrey novoseltsev , 2005 # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # def nextdistribution(n , m , k , ai ) : if ai[0]==n - m - k+1 : return false ai0old = ai[0 ] ai[0]=1 for i in range(1,k ) : if ai[i]!=1 : ai[i]=ai[i]-1 ai[i-1]=ai0old+1 return true for n in range(1,nderivativeorder+1 ) : for m in range(n ) : for k in range(1,n - m+1 ) : ai = array('h ' ) for i in range(1,k ) : ai.append(1 ) ai.append(n-m-k+1 ) while 1==1 : # compute the current coefficient coef=0 if m!=0 : coef = coef+dcoefficients[m-1,k , tuple(ai ) ] if ai[k-1]==1 : coef = coef+dcoefficients[m , k-1,tuple(ai)[0:k-1 ] ] for i in range(k ) : if ai[i]>1 : ai[i]=ai[i]-1 coef = coef+dcoefficients[m , k , tuple(ai ) ] ai[i]=ai[i]+1 dcoefficients[m , k , tuple(ai)]=coef # proceed to the next coefficient if not nextdistribution(n , m , k , ai ) : break dcoefficients[n,0,()]=1 foutput.write ( " s=0 " + " \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n " + " s_{"+lsalphabet[0]+"}=0 " + " \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n " + " s_{("+lsalphabet[0]+lsalphabet[1]+")}= " + " g_{"+lsalphabet[0]+lsalphabet[1]+ " } " + " -2\ , " + spsi+"_{("+lsalphabet[0]+lsalphabet[1]+ " ) } " + " + " + spsi+"_{"+lsprimealphabet[0]+"("+lsalphabet[0]+ " } " + spsi+"^{"+lsprimealphabet[0]+"}{}_{"+lsalphabet[1]+ " ) } " ) for n in range(3,nderivativeorder+1 ) : foutput.write("\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n " ) foutput.write("s_ { ( " ) for i in range(n ) : foutput.write(lsalphabet[i ] ) foutput.write(")}=\n " ) nnumberofterms=0 for t in range(2,n+1 ) : for m in range(t-1,-1,-1 ) : k = t - m ai = array('h ' ) for i in range(1,k ) : ai.append(1 ) ai.append(n-m-k+1 ) while 1==1 : # check if the current term should be written if t==3 or ( k==1 and m>=2 ) or ( k>=n-1 and k>=3 ) : if not nextdistribution(n , m , k , ai ) : break else : continue # write the current term if nnumberofterms!=0 : foutput.write(sseparator ) if k==1 and m==1 : foutput.write("- " ) elif nnumberofterms!=0 : foutput.write("+ " ) nnumberofterms = nnumberofterms+1 if dcoefficients[m , k , tuple(ai)]!=1 : foutput.write(str(dcoefficients[m,k,tuple(ai ) ] ) + " \ , ") if t==2 : foutput.write("g_ { " ) for i in range(k ) : foutput.write(lsprimealphabet[i ] ) if m!=0 : foutput.write ( " ( " ) for i in range(m ) : foutput.write(lsalphabet[i ] ) foutput.write ( " } ") else : foutput.write("["+ssigma+"_ { " ) for i in range(k ) : foutput.write(lsprimealphabet[i]+ " ' " ) if m!=0 : foutput.write ( " ( " ) for i in range(m ) : foutput.write(lsalphabet[i ] ) foutput.write ( " } ] ") foutput.write(spsi+"^{"+lsprimealphabet[0]+"}{}_ { " ) if m==0 : foutput.write ( " ( " ) for i in range(m , m+ai[0 ] ) : foutput.write(lsalphabet[i ] ) nextindex = m+ai[0 ] for i in range(1,k ) : foutput.write ( " } " + spsi+"^{"+lsprimealphabet[i ] + " } { } _ { " ) for j in range(nextindex , nextindex+ai[i ] ) : foutput.write(lsalphabet[j ] ) nextindex = nextindex+ai[i ] foutput.write ( " ) } " ) # proceed to the next coefficient if not nextdistribution(n , m , k , ai ) : break foutput.write("\n% number of terms : " + str(nnumberofterms ) ) @xmath774\\ & + g_{\mu\nu } \psi^{\mu}{}_{(\alpha } \psi^{\nu}{}_{\beta\gamma ) } \displaybreak[0]\\ & + 2\ , g_{\mu\nu } \psi^{\mu}{}_{(\alpha\beta } \psi^{\nu}{}_{\gamma ) } \,,\\ s_{(\alpha\beta\gamma\delta)}&= -4\ , g_{\mu(\alpha } \psi^{\mu}{}_{\beta\gamma\delta ) } \displaybreak[0]\\ & + g_{\mu\nu } \psi^{\mu}{}_{(\alpha } \psi^{\nu}{}_{\beta\gamma\delta ) } \displaybreak[0]\\ & + 3\ , g_{\mu\nu } \psi^{\mu}{}_{(\alpha\beta } \psi^{\nu}{}_{\gamma\delta ) } \displaybreak[0]\\ & + 3\ , g_{\mu\nu } \psi^{\mu}{}_{(\alpha\beta\gamma } \psi^{\nu}{}_{\delta ) } \displaybreak[0]\\ & + 6\ , [ \sigma_{\mu'\nu'(\alpha\beta } ] \psi^{\mu}{}_{\gamma } \psi^{\nu}{}_{\delta ) } \,,\\ s_{(\alpha\beta\gamma\delta{{\varepsilon}})}&= -5\ , g_{\mu(\alpha } \psi^{\mu}{}_{\beta\gamma\delta{{\varepsilon } } ) } \displaybreak[0]\\ & + g_{\mu\nu } \psi^{\mu}{}_{(\alpha } \psi^{\nu}{}_{\beta\gamma\delta{{\varepsilon } } ) } \displaybreak[0]\\ & + 4\ , g_{\mu\nu } \psi^{\mu}{}_{(\alpha\beta } \psi^{\nu}{}_{\gamma\delta{{\varepsilon } } ) } \displaybreak[0]\\ & + 6\ , g_{\mu\nu } \psi^{\mu}{}_{(\alpha\beta\gamma } \psi^{\nu}{}_{\delta{{\varepsilon } } ) } \displaybreak[0]\\ & + 4\ , g_{\mu\nu } \psi^{\mu}{}_{(\alpha\beta\gamma\delta } \psi^{\nu}{}_{{{\varepsilon } } ) } \displaybreak[0]\\ & + 10\ , [ \sigma_{\mu'\nu'(\alpha\beta } ] \psi^{\mu}{}_{\gamma } \psi^{\nu}{}_{\delta{{\varepsilon } } ) } \displaybreak[0]\\ & + 20\ , [ \sigma_{\mu'\nu'(\alpha\beta } ] \psi^{\mu}{}_{\gamma\delta } \psi^{\nu}{}_{{{\varepsilon } } ) } \displaybreak[0]\\ & + 5\ , [ \sigma_{\mu'\nu'\xi'(\alpha } ] \psi^{\mu}{}_{\beta } \psi^{\nu}{}_{\gamma } \psi^{\xi}{}_{\delta{{\varepsilon } } ) } \displaybreak[0]\\ & + 10\ , [ \sigma_{\mu'\nu'\xi'(\alpha } ] \psi^{\mu}{}_{\beta } \psi^{\nu}{}_{\gamma\delta } \psi^{\xi}{}_{{{\varepsilon } } ) } \displaybreak[0]\\ & + 15\ , [ \sigma_{\mu'\nu'\xi'(\alpha } ] \psi^{\mu}{}_{\beta\gamma } \psi^{\nu}{}_{\delta } \psi^{\xi}{}_{{{\varepsilon } } ) } \displaybreak[0]\\ & + 10\ , [ \sigma_{\mu'\nu'(\alpha\beta\gamma } ] \psi^{\mu}{}_{\delta } \psi^{\nu}{}_{{{\varepsilon } } ) } \displaybreak[0]\\ & + 10\ , [ \sigma_{\mu'\nu'\xi'(\alpha\beta } ] \psi^{\mu}{}_{\gamma } \psi^{\nu}{}_{\delta } \psi^{\xi}{}_{{{\varepsilon } } ) } \,,\\ s_{(\alpha\beta\gamma\delta{{\varepsilon}}\zeta)}&= -6\ , g_{\mu(\alpha } \psi^{\mu}{}_{\beta\gamma\delta{{\varepsilon}}\zeta ) } \displaybreak[0]\\ & + g_{\mu\nu } \psi^{\mu}{}_{(\alpha } \psi^{\nu}{}_{\beta\gamma\delta{{\varepsilon}}\zeta ) } \displaybreak[0]\\ & + 5\ , g_{\mu\nu } \psi^{\mu}{}_{(\alpha\beta } \psi^{\nu}{}_{\gamma\delta{{\varepsilon}}\zeta ) } \displaybreak[0]\\ & + 10\ , g_{\mu\nu } \psi^{\mu}{}_{(\alpha\beta\gamma } \psi^{\nu}{}_{\delta{{\varepsilon}}\zeta ) } \displaybreak[0]\\ & + 10\ , g_{\mu\nu } \psi^{\mu}{}_{(\alpha\beta\gamma\delta } \psi^{\nu}{}_{{{\varepsilon}}\zeta ) } \displaybreak[0]\\ & + 5\ , g_{\mu\nu } \psi^{\mu}{}_{(\alpha\beta\gamma\delta{{\varepsilon } } } \psi^{\nu}{}_{\zeta ) } \displaybreak[0]\\ & + 15\ , [ \sigma_{\mu'\nu'(\alpha\beta } ] \psi^{\mu}{}_{\gamma } \psi^{\nu}{}_{\delta{{\varepsilon}}\zeta ) } \displaybreak[0]\\ & + 45\ , [ \sigma_{\mu'\nu'(\alpha\beta } ] \psi^{\mu}{}_{\gamma\delta } \psi^{\nu}{}_{{{\varepsilon}}\zeta ) } \displaybreak[0]\\ & + 45\ , [ \sigma_{\mu'\nu'(\alpha\beta } ] \psi^{\mu}{}_{\gamma\delta{{\varepsilon } } } \psi^{\nu}{}_{\zeta ) } \displaybreak[0]\\ & + 6\ , [ \sigma_{\mu'\nu'\xi'(\alpha } ] \psi^{\mu}{}_{\beta } \psi^{\nu}{}_{\gamma } \psi^{\xi}{}_{\delta{{\varepsilon}}\zeta ) } \displaybreak[0]\\ & + 18\ , [ \sigma_{\mu'\nu'\xi'(\alpha } ] \psi^{\mu}{}_{\beta } \psi^{\nu}{}_{\gamma\delta } \psi^{\xi}{}_{{{\varepsilon}}\zeta ) } \displaybreak[0]\\ & + 24\ , [ \sigma_{\mu'\nu'\xi'(\alpha } ] \psi^{\mu}{}_{\beta\gamma } \psi^{\nu}{}_{\delta } \psi^{\xi}{}_{{{\varepsilon}}\zeta ) } \displaybreak[0]\\ & + 18\ , [ \sigma_{\mu'\nu'\xi'(\alpha } ] \psi^{\mu}{}_{\beta } \psi^{\nu}{}_{\gamma\delta{{\varepsilon } } } \psi^{\xi}{}_{\zeta ) } \displaybreak[0]\\ & + 48\ , [ \sigma_{\mu'\nu'\xi'(\alpha } ] \psi^{\mu}{}_{\beta\gamma } \psi^{\nu}{}_{\delta{{\varepsilon } } } \psi^{\xi}{}_{\zeta ) } \displaybreak[0]\\ & + 36\ , [ \sigma_{\mu'\nu'\xi'(\alpha } ] \psi^{\mu}{}_{\beta\gamma\delta } \psi^{\nu}{}_{{{\varepsilon } } } \psi^{\xi}{}_{\zeta ) } \displaybreak[0]\\ & + [ \sigma_{\mu'\nu'\xi'\pi ' } ] \psi^{\mu}{}_{(\alpha } \psi^{\nu}{}_{\beta } \psi^{\xi}{}_{\gamma } \psi^{\pi}{}_{\delta{{\varepsilon}}\zeta ) } \displaybreak[0]\\ & + 3\ , [ \sigma_{\mu'\nu'\xi'\pi ' } ] \psi^{\mu}{}_{(\alpha } \psi^{\nu}{}_{\beta } \psi^{\xi}{}_{\gamma\delta } \psi^{\pi}{}_{{{\varepsilon}}\zeta ) } \displaybreak[0]\\ & + 4\ , [ \sigma_{\mu'\nu'\xi'\pi ' } ] \psi^{\mu}{}_{(\alpha } \psi^{\nu}{}_{\beta\gamma } \psi^{\xi}{}_{\delta } \psi^{\pi}{}_{{{\varepsilon}}\zeta ) } \displaybreak[0]\\ & + 5\ , [ \sigma_{\mu'\nu'\xi'\pi ' } ] \psi^{\mu}{}_{(\alpha\beta } \psi^{\nu}{}_{\gamma } \psi^{\xi}{}_{\delta } \psi^{\pi}{}_{{{\varepsilon}}\zeta ) } \displaybreak[0]\\ & + 3\ , [ \sigma_{\mu'\nu'\xi'\pi ' } ] \psi^{\mu}{}_{(\alpha } \psi^{\nu}{}_{\beta } \psi^{\xi}{}_{\gamma\delta{{\varepsilon } } } \psi^{\pi}{}_{\zeta ) } \displaybreak[0]\\ & + 8\ , [ \sigma_{\mu'\nu'\xi'\pi ' } ] \psi^{\mu}{}_{(\alpha } \psi^{\nu}{}_{\beta\gamma } \psi^{\xi}{}_{\delta{{\varepsilon } } } \psi^{\pi}{}_{\zeta ) } \displaybreak[0]\\ & + 10\ , [ \sigma_{\mu'\nu'\xi'\pi ' } ] \psi^{\mu}{}_{(\alpha\beta } \psi^{\nu}{}_{\gamma } \psi^{\xi}{}_{\delta{{\varepsilon } } } \psi^{\pi}{}_{\zeta ) } \displaybreak[0]\\ & + 6\ , [ \sigma_{\mu'\nu'\xi'\pi ' } ] \psi^{\mu}{}_{(\alpha } \psi^{\nu}{}_{\beta\gamma\delta } \psi^{\xi}{}_{{{\varepsilon } } } \psi^{\pi}{}_{\zeta ) } \displaybreak[0]\\ & + 15\ , [ \sigma_{\mu'\nu'\xi'\pi ' } ] \psi^{\mu}{}_{(\alpha\beta } \psi^{\nu}{}_{\gamma\delta } \psi^{\xi}{}_{{{\varepsilon } } } \psi^{\pi}{}_{\zeta ) } \displaybreak[0]\\ & + 10\ , [ \sigma_{\mu'\nu'\xi'\pi ' } ] \psi^{\mu}{}_{(\alpha\beta\gamma } \psi^{\nu}{}_{\delta } \psi^{\xi}{}_{{{\varepsilon } } } \psi^{\pi}{}_{\zeta ) } \displaybreak[0]\\ & + 20\ , [ \sigma_{\mu'\nu'(\alpha\beta\gamma } ] \psi^{\mu}{}_{\delta } \psi^{\nu}{}_{{{\varepsilon}}\zeta ) } \displaybreak[0]\\ & + 40\ , [ \sigma_{\mu'\nu'(\alpha\beta\gamma } ] \psi^{\mu}{}_{\delta{{\varepsilon } } } \psi^{\nu}{}_{\zeta ) } \displaybreak[0]\\ & + 15\ , [ \sigma_{\mu'\nu'\xi'(\alpha\beta } ] \psi^{\mu}{}_{\gamma } \psi^{\nu}{}_{\delta } \psi^{\xi}{}_{{{\varepsilon}}\zeta ) } \displaybreak[0]\\ & + 30\ , [ \sigma_{\mu'\nu'\xi'(\alpha\beta } ] \psi^{\mu}{}_{\gamma } \psi^{\nu}{}_{\delta{{\varepsilon } } } \psi^{\xi}{}_{\zeta ) } \displaybreak[0]\\ & + 45\ , [ \sigma_{\mu'\nu'\xi'(\alpha\beta } ] \psi^{\mu}{}_{\gamma\delta } \psi^{\nu}{}_{{{\varepsilon } } } \psi^{\xi}{}_{\zeta ) } \displaybreak[0]\\ & + 6\ , [ \sigma_{\mu'\nu'\xi'\pi'(\alpha } ] \psi^{\mu}{}_{\beta } \psi^{\nu}{}_{\gamma } \psi^{\xi}{}_{\delta } \psi^{\pi}{}_{{{\varepsilon}}\zeta ) } \displaybreak[0]\\ & + 12\ , [ \sigma_{\mu'\nu'\xi'\pi'(\alpha } ] \psi^{\mu}{}_{\beta } \psi^{\nu}{}_{\gamma } \psi^{\xi}{}_{\delta{{\varepsilon } } } \psi^{\pi}{}_{\zeta ) } \displaybreak[0]\\ & + 18\ , [ \sigma_{\mu'\nu'\xi'\pi'(\alpha } ] \psi^{\mu}{}_{\beta } \psi^{\nu}{}_{\gamma\delta } \psi^{\xi}{}_{{{\varepsilon } } } \psi^{\pi}{}_{\zeta ) } \displaybreak[0]\\ & + 24\ , [ \sigma_{\mu'\nu'\xi'\pi'(\alpha } ] \psi^{\mu}{}_{\beta\gamma } \psi^{\nu}{}_{\delta } \psi^{\xi}{}_{{{\varepsilon } } } \psi^{\pi}{}_{\zeta ) } \displaybreak[0]\\ & + 15\ , [ \sigma_{\mu'\nu'(\alpha\beta\gamma\delta } ] \psi^{\mu}{}_{{{\varepsilon } } } \psi^{\nu}{}_{\zeta ) } \displaybreak[0]\\ & + 20\ , [ \sigma_{\mu'\nu'\xi'(\alpha\beta\gamma } ] \psi^{\mu}{}_{\delta } \psi^{\nu}{}_{{{\varepsilon } } } \psi^{\xi}{}_{\zeta ) } \displaybreak[0]\\ & + 15\ , [ \sigma_{\mu'\nu'\xi'\pi'(\alpha\beta } ] \psi^{\mu}{}_{\gamma } \psi^{\nu}{}_{\delta } \psi^{\xi}{}_{{{\varepsilon } } } \psi^{\pi}{}_{\zeta ) } \ , . \ ] ] @xmath775 & = [ \sigma_{\alpha\mu'\nu'\xi ' } ] \,,\\ [ \sigma_{\alpha\mu'(\nu'\xi ' ) } ] & = \frac{1}{2 } ( [ \sigma_{\alpha\mu'\nu'\xi'}]+[\sigma_{\alpha\mu'\xi'\nu ' } ] ) = [ \sigma_{\alpha\mu'\nu'\xi'}]-\frac{1}{2}[\sigma_{\alpha a ' } r^{a'}{}_{\mu'\nu'\xi ' } ] \,,\\ [ \sigma_{\alpha\mu'\nu'\xi ' } ] & = [ \sigma_{\alpha\mu'(\nu'\xi ' ) } ] - \frac{1}{2}\ , r_{\alpha\mu\nu\xi } = -\frac{1}{3}\ , r_{\alpha(\nu|\mu|\xi ) } - \frac{1}{2}\ , r_{\alpha\mu\nu\xi } \\ & = -\frac{1}{6}\ , ( r_{\alpha\nu\mu\xi}+r_{\alpha\xi\mu\nu } ) - \frac{1}{2}\ , r_{\alpha\mu\nu\xi } \\ & = \frac{1}{6}\ , ( r_{\alpha\nu\xi\mu}+r_{\alpha\mu\nu\xi}+r_{\alpha\nu\xi\mu } ) + \frac{1}{2}\ , r_{\alpha\mu\xi\nu } \\ & = \frac{1}{3}\ , ( r_{\alpha\nu\xi\mu}+r_{\alpha\mu\xi\nu } ) = \frac{2}{3}\ , r_{\alpha(\mu|\xi|\nu ) } \,,\\ [ \sigma_{\mu'\nu'\alpha\beta } ] & = -[\sigma_{\mu'\nu'\beta'\alpha } ] = -\frac{2}{3}\ , r_{\alpha(\mu|\beta|\nu ) } \,,\\ [ \sigma_{\mu'\nu'\xi'\pi ' } ] & = -[\sigma_{\mu'\nu'\xi'\pi } ] = -\frac{2}{3}\ , r_{\pi(\mu|\xi|\nu ) } \,.\end{aligned}\ ] ] @xmath776 & = [ \sigma_{\alpha\mu'\nu'\xi'\pi ' } ] \,,\displaybreak[0]\\ [ \sigma_{\alpha\mu'(\nu'\xi'\pi ' ) } ] & = \frac{1}{6}\ , ( [ \sigma_{\alpha\mu'\nu'\xi'\pi ' } ] + [ \sigma_{\alpha\mu'\nu'\pi'\xi ' } ] + [ \sigma_{\alpha\mu'\xi'\nu'\pi ' } ] \displaybreak[0]\\ & + [ \sigma_{\alpha\mu'\xi'\pi'\nu ' } ] + [ \sigma_{\alpha\mu'\pi'\nu'\xi ' } ] + [ \sigma_{\alpha\mu'\pi'\xi'\nu ' } ] ) \displaybreak[0]\\ & = \frac{1}{3}\ , ( [ \sigma_{\alpha\mu'\nu'\xi'\pi ' } ] + [ \sigma_{\alpha\mu'\xi'\nu'\pi ' } ] + [ \sigma_{\alpha\mu'\pi'\nu'\xi ' } ] ) \displaybreak[0]\\ & = \frac{2}{3}\ , [ \sigma_{\alpha\mu'\nu'\xi'\pi ' } ] -\frac{1}{3}\ , [ ( \sigma_{\alpha a ' } r^{a'}{}_{\mu'\nu'\xi'})_{;\pi ' } ] \displaybreak[0]\\ & + \frac{1}{3}\ , [ \sigma_{\alpha\mu'\nu'\pi'\xi ' } ] -\frac{1}{3}\ , [ ( \sigma_{\alpha a ' } r^{a'}{}_{\mu'\nu'\pi'})_{;\xi ' } ] \displaybreak[0]\\ & = [ \sigma_{\alpha\mu'\nu'\xi'\pi ' } ] + \frac{1}{3}\ , r_{\alpha\mu\nu\xi;\pi } + \frac{1}{3}\ , r_{\alpha\mu\nu\pi;\xi } \,,\displaybreak[0]\\ [ \sigma_{\alpha\mu'\nu'\xi'\pi ' } ] & = [ \sigma_{\alpha\mu'(\nu'\xi'\pi ' ) } ] -\frac{2}{3 } r_{\alpha\mu\nu(\xi;\pi ) } = -\frac{1}{2}\ , r_{\alpha(\nu|\mu|\xi;\pi ) } -\frac{2}{3 } r_{\alpha\mu\nu(\xi;\pi ) } \displaybreak[0]\\ & = -\frac{1}{6}\ , r_{\alpha\nu\mu(\xi;\pi ) } -\frac{1}{6}\ , r_{\alpha(\xi|\mu\nu|;\pi ) } -\frac{1}{6}\ , r_{\alpha(\xi|\mu|\pi);\nu } -\frac{2}{3 } r_{\alpha\mu\nu(\xi;\pi ) } \displaybreak[0]\\ & = \frac{5}{6}\ , r_{(\mu|\alpha|\nu)(\xi;\pi ) } -\frac{1}{6}\ , r_{(\xi|\alpha|\pi)(\mu;\nu ) } \,,\displaybreak[0]\\ [ \sigma_{\mu'\nu'\xi'\alpha\beta } ] & = [ \sigma_{\mu'\nu'\xi'\alpha}]_{;\beta } -[\sigma_{\mu'\nu'\xi'\beta'\alpha } ] \\ & = \frac{2}{3}\ , r_{\alpha(\mu|\xi|\nu);\beta } + \frac{5}{6}\ , r_{\alpha(\mu\nu)(\xi;\beta ) } -\frac{1}{6}\ , r_{\alpha(\xi\beta)(\mu;\nu ) } \\ & = \frac{2}{3}\ , r_{\alpha(\mu|\xi|\nu);\beta } + \frac{5}{12}\ , r_{\alpha(\mu\nu)\xi;\beta } + \frac{5}{12}\ , r_{\alpha(\mu\nu)\beta;\xi } \\ & -\frac{1}{12}\ , r_{\alpha\xi\beta(\mu;\nu ) } -\frac{1}{12}\ , r_{\alpha\beta\xi(\mu;\nu ) } \\ & = \frac{2}{3}\ , r_{(\mu|\xi|\nu)(\alpha;\beta ) } + \frac{5}{12}\ , r_{\xi(\mu\nu)(\alpha;\beta ) } + \frac{5}{12}\ , r_{\alpha(\mu\nu)\beta;\xi } -\frac{1}{12}\ , r_{(\alpha|\xi|\beta)(\mu;\nu ) } \\ & = \frac{1}{4}\ , r_{(\mu|\xi|\nu)(\alpha;\beta ) } -\frac{5}{12}\ , r_{(\mu|\alpha|\nu)\beta;\xi } -\frac{1}{12}\ , r_{(\alpha|\xi|\beta)(\mu;\nu ) } \,.\end{aligned}\ ] ] @xmath777 & = -[\sigma_{\mu'(\alpha\beta\gamma\delta)\nu } ] \,,\displaybreak[0]\\ [ \sigma_{\mu'\alpha(\beta\gamma\delta\nu ) } ] & = \frac{1}{4}\ , ( [ \sigma_{\mu'\alpha(\beta\gamma\delta)\nu } ] + [ \sigma_{\mu'\alpha(\beta\gamma|\nu|\delta ) } ] + [ \sigma_{\mu'\alpha(\beta|\nu|\gamma\delta ) } ] + [ \sigma_{\mu'\alpha\nu(\beta\gamma\delta ) } ] ) \displaybreak[0]\\ & = \frac{1}{4}\ , [ \sigma_{\mu'\alpha(\beta\gamma\delta)\nu } ] + \frac{1}{4}\ , [ \sigma_{\mu'\alpha(\beta\gamma|\nu|\delta ) } ] + \frac{1}{2}\ , [ \sigma_{\mu'\alpha(\beta|\nu|\gamma\delta ) } ] \displaybreak[0]\\ & -\frac{1}{4}\ , [ ( \sigma_{\mu ' a } r^a{}_{\alpha(\beta|\nu|})_{;\gamma\delta ) } ] \displaybreak[0]\\ & = \frac{1}{4}\ , [ \sigma_{\mu'\alpha(\beta\gamma\delta)\nu } ] + \frac{3}{4}\ , [ \sigma_{\mu'\alpha(\beta\gamma|\nu|\delta ) } ] -\frac{1}{2}\ , [ ( \sigma_{\mu'a(\beta } r^a{}_{|\alpha|\gamma|\nu|})_{;\delta ) } ] \displaybreak[0]\\ & -\frac{1}{2}\ , [ ( \sigma_{\mu'\alpha a } r^a{}_{(\beta\gamma|\nu|})_{;\delta ) } ] -\frac{1}{4}\ , [ \sigma_{\mu ' a ( \beta\gamma } ] r^a{}_{|\alpha|\delta)\nu } + \frac{1}{4}\ , r_{\mu\alpha(\beta|\nu|;\gamma\delta ) } \displaybreak[0]\\ & = [ \sigma_{\mu'\alpha(\beta\gamma\delta)\nu } ] -\frac{3}{4}\ , [ \sigma_{\mu ' a ( \beta\gamma } ] r^a{}_{|\alpha|\delta)\nu } -\frac{3}{4}\ , [ \sigma_{\mu'\alpha a ( \beta } ] r^a{}_{\gamma\delta)\nu } \displaybreak[0]\\ & -\frac{3}{4}\ , [ \sigma_{\mu'\alpha(\beta|a| } ] r^a{}_{\gamma\delta)\nu } -\frac{1}{2}\ , [ \sigma_{\mu'a(\beta\gamma } ] r^a{}_{|\alpha|\delta)\nu } \displaybreak[0]\\ & -\frac{1}{2}\ , [ \sigma_{\mu'\alpha a ( \beta } ] r^a{}_{\gamma\delta)\nu } -\frac{1}{4}\ , [ \sigma_{\mu ' a ( \beta\gamma } ] r^a{}_{|\alpha|\delta)\nu } + \frac{1}{4}\ , r_{\mu\alpha(\beta|\nu|;\gamma\delta ) } \displaybreak[0]\\ & = [ \sigma_{\mu'\alpha(\beta\gamma\delta)\nu } ] -\frac{3}{2}\ , [ \sigma_{\mu ' a ( \beta\gamma } ] r^a{}_{|\alpha|\delta)\nu } -\frac{5}{4}\ , [ \sigma_{\mu'\alpha a ( \beta } ] r^a{}_{\gamma\delta)\nu } \displaybreak[0]\\ & -\frac{3}{4}\ , [ \sigma_{\mu'\alpha(\beta|a| } ] r^a{}_{\gamma\delta)\nu } + \frac{1}{4}\ , r_{\mu\alpha(\beta|\nu|;\gamma\delta ) } \,,\displaybreak[0]\\ [ \sigma_{\mu'\alpha(\beta\gamma\delta)\nu } ] & = [ \sigma_{\mu'\alpha(\beta\gamma\delta\nu ) } ] + \frac{3}{2}\,[\sigma_{\mu ' a ( \beta\gamma } ] r^a{}_{|\alpha|\delta)\nu } + \frac{5}{4}\,[\sigma_{\mu'\alpha a ( \beta } ] r^a{}_{\gamma\delta)\nu } \displaybreak[0]\\ & + \frac{3}{4}\,[\sigma_{\mu'\alpha(\beta|a| } ] r^a{}_{\gamma\delta)\nu } -\frac{1}{4}\,r_{\mu\alpha(\beta|\nu|;\gamma\delta ) } \displaybreak[0]\\ & = -\frac{3}{5}\ , r_{\mu(\beta|\alpha|\gamma;\delta\nu ) } -\frac{7}{15}\ , r_{\mu(\beta| a |\gamma } r^a{}_{\delta|\alpha|\nu ) } -\frac{1}{2}\ , r_{\mu(\beta| a |\gamma } r^a{}_{|\alpha|\delta)\nu } \displaybreak[0]\\ & -\frac{5}{24}\ , r_{\mu a \alpha(\beta } r^a{}_{\gamma\delta)\nu } -\frac{5}{24}\ , r_{\mu(\beta|\alpha a| } r^a{}_{\gamma\delta)\nu } -\frac{5}{8}\ , r_{\mu\alpha a ( \beta } r^a{}_{\gamma\delta)\nu } \displaybreak[0]\\ & -\frac{1}{8}\ , r_{\mu ( \beta|\alpha a | } r^a{}_{\gamma\delta)\nu } -\frac{1}{8}\ , r_{\mu a \alpha(\beta } r^a{}_{\gamma\delta)\nu } -\frac{3}{8}\ , r_{\mu\alpha(\beta| a | } r^a{}_{\gamma\delta)\nu } \displaybreak[0]\\ & -\frac{1}{4}\,r_{\mu\alpha(\beta|\nu|;\gamma\delta ) } \displaybreak[0]\\ & = -\frac{3}{5}\ , r_{\mu(\beta|\alpha|\gamma;\delta\nu ) } -\frac{1}{4}\,r_{\mu\alpha(\beta|\nu|;\gamma\delta ) } -\frac{7}{15}\ , r_{\mu(\beta| a |\gamma } r^a{}_{\delta|\alpha|\nu ) } \displaybreak[0]\\ & -\frac{1}{2}\ , r_{\mu(\beta| a |\gamma } r^a{}_{|\alpha|\delta)\nu } -\frac{1}{3}\ , r_{\mu a \alpha(\beta } r^a{}_{\gamma\delta)\nu } -\frac{1}{3}\ , r_{\mu(\beta|\alpha a| } r^a{}_{\gamma\delta)\nu } \displaybreak[0]\\ & -\frac{5}{8}\ , r_{\mu\alpha a ( \beta } r^a{}_{\gamma\delta)\nu } -\frac{3}{8}\ , r_{\mu\alpha(\beta| a | } r^a{}_{\gamma\delta)\nu } \,,\displaybreak[0]\\ [ \sigma_{\mu'(\alpha\beta\gamma\delta)\nu } ] & = -\frac{3}{20}\ , r_{\mu(\alpha\beta|\nu|;\gamma\delta ) } -\frac{1}{4}\ , r_{\mu(\alpha\beta|\nu|;\gamma\delta ) } -\frac{7}{60}\ , r_{\mu(\alpha| a |\beta } r^a{}_{\gamma\delta)\nu } \displaybreak[0]\\ & -\frac{1}{2}\ , r_{\mu(\alpha| a |\beta } r^a{}_{\gamma\delta)\nu } -\frac{1}{3}\ , r_{\mu(\alpha\beta| a | } r^a{}_{\gamma\delta)\nu } -\frac{5}{8}\ , r_{\mu(\alpha| a |\beta } r^a{}_{\gamma\delta)\nu } \displaybreak[0]\\ & -\frac{3}{8}\ , r_{\mu(\alpha\beta| a | } r^a{}_{\gamma\delta)\nu } \displaybreak[0]\\ & = -\frac{2}{5}\ , r_{\mu(\alpha\beta|\nu|;\gamma\delta ) } -\frac{149}{120}\ , r_{\mu(\alpha| a |\beta } r^a{}_{\gamma\delta)\nu } -\frac{17}{24}\ , r_{\mu(\alpha\beta| a | } r^a{}_{\gamma\delta)\nu } \displaybreak[0]\\ & = \frac{2}{5}\ , r_{\mu(\alpha|\nu|\beta;\gamma\delta ) } + \frac{8}{15}\ , r _ { a ( \alpha|\mu|\beta } r^a{}_{\gamma|\nu|\delta ) } \,,\displaybreak[0]\\ [ \sigma_{\mu'\nu'\xi'(\alpha\beta\gamma ) } ] & = [ \sigma_{\mu'\nu'(\alpha\beta\gamma)}]_{;\xi } -[\sigma_{\mu'\nu'(\alpha\beta\gamma)\xi } ] \displaybreak[0]\\ & = [ \sigma_{\mu'\nu'(\alpha\beta\gamma)}]_{;\xi } -[\sigma_{\mu'(\alpha\beta\gamma)\xi}]_{;\nu } + [ \sigma_{\mu'(\alpha\beta\gamma)\xi\nu } ] \,,\displaybreak[0]\\ [ \sigma_{\mu'(\alpha\beta\gamma\xi)\nu } ] & = \frac{1}{4}\ , ( [ \sigma_{\mu'(\alpha\beta\gamma)\xi\nu } ] + [ \sigma_{\mu'(\alpha\beta|\xi|\gamma)\nu } ] + [ \sigma_{\mu'(\alpha|\xi|\beta\gamma)\nu } ] + [ \sigma_{\mu'\xi(\alpha\beta\gamma)\nu } ] ) \displaybreak[0]\\ & = \frac{1}{4}\ , [ \sigma_{\mu'(\alpha\beta\gamma)\xi\nu } ] + \frac{1}{4}\ , [ \sigma_{\mu'(\alpha\beta|\xi|\gamma)\nu } ] + \frac{1}{2}\ , [ \sigma_{\mu'(\alpha|\xi|\beta\gamma)\nu } ] \displaybreak[0]\\ & = \frac{1}{4}\ , [ \sigma_{\mu'(\alpha\beta\gamma)\xi\nu } ] + \frac{3}{4}\ , [ \sigma_{\mu'(\alpha\beta|\xi|\gamma)\nu } ] -\frac{1}{2}\ , [ ( \sigma_{\mu ' a } r^a{}_{(\alpha\beta|\xi|})_{;\gamma)\nu } ] \displaybreak[0]\\ & = [ \sigma_{\mu'(\alpha\beta\gamma)\xi\nu } ] -\frac{3}{2}\ , [ ( \sigma_{\mu ' a ( \alpha } r^a{}_{\beta\gamma)\xi})_{;\nu } ] \displaybreak[0]\\ & -\frac{1}{2}\ , [ \sigma_{\mu ' a ( \alpha|\nu| } ] r^a{}_{\beta\gamma)|\xi } + \frac{1}{2}\ , r_{\mu(\alpha\beta|\xi|;\gamma)\nu } \displaybreak[0]\\ & = [ \sigma_{\mu'(\alpha\beta\gamma)\xi\nu } ] -2\ , [ \sigma_{\mu ' a ( \alpha|\nu| } ] r^a{}_{\beta\gamma)\xi } + \frac{1}{2}\ , r_{\mu(\alpha\beta|\xi|;\gamma)\nu } \,,\displaybreak[0]\\ [ \sigma_{\mu'(\alpha\beta\gamma)\xi\nu } ] & = [ \sigma_{\mu'(\alpha\beta\gamma\xi)\nu } ] + 2\ , [ \sigma_{\mu ' a ( \alpha|\nu| } ] r^a{}_{\beta\gamma)\xi } -\frac{1}{2}\ , r_{\mu(\alpha\beta|\xi|;\gamma)\nu } \displaybreak[0]\\ & = \frac{2}{5}\ , r_{\mu(\alpha|\nu|\beta;\gamma\xi ) } + \frac{8}{15}\ , r _ { a ( \alpha|\mu|\beta } r^a{}_{\gamma|\nu|\xi ) } \displaybreak[0]\\ & + \frac{4}{3}\ , r_{(\mu| a |\nu)(\alpha } r^a{}_{\beta\gamma)\xi } -\frac{1}{2}\ , r_{\mu(\alpha\beta|\xi|;\gamma)\nu } \,,\displaybreak[0]\\ [ \sigma_{\mu'\nu'\xi'(\alpha\beta\gamma ) } ] & = -\frac{5}{12}\ , r_{(\alpha|\mu|\beta|\nu|;\gamma)\xi } -\frac{1}{12}\ , r_{\mu(\alpha|\nu|\beta;\gamma)\xi } + \frac{1}{12}\ , r_{(\alpha| \mu \xi|\beta;\gamma)\nu } \displaybreak[0]\\ & -\frac{5}{12}\ , r_{(\alpha| \mu |\beta|\xi|;\gamma)\nu } + \frac{2}{5}\ , r_{\mu(\alpha|\nu|\beta;\gamma\xi ) } + \frac{8}{15}\ , r _ { a ( \alpha|\mu|\beta } r^a{}_{\gamma|\nu|\xi ) } \displaybreak[0]\\ & + \frac{4}{3}\ , r_{(\mu| a |\nu)(\alpha } r^a{}_{\beta\gamma)\xi } -\frac{1}{2}\ , r_{\mu(\alpha\beta|\xi|;\gamma)\nu } \displaybreak[0]\\ & = -\frac{1}{2}\ , r_{\mu(\alpha|\nu|\beta;\gamma)\xi } + \frac{2}{5}\ , r_{\mu(\alpha|\nu|\beta;\gamma\xi ) } \displaybreak[0]\\ & + \frac{8}{15}\ , r _ { a ( \alpha|\mu|\beta } r^a{}_{\gamma|\nu|\xi ) } + \frac{4}{3}\ , r_{(\mu| a |\nu)(\alpha } r^a{}_{\beta\gamma)\xi } \displaybreak[0]\\ & = -\frac{1}{2}\ , r_{\mu(\alpha|\nu|\beta;\gamma)\xi } + \frac{1}{10}\ , r_{\mu(\alpha|\nu|\beta;\gamma)\xi } + \frac{1}{10}\ , r_{\mu(\alpha|\nu|\beta;|\xi|\gamma ) } \displaybreak[0]\\ & + \frac{1}{10}\ , r_{\mu(\alpha|\nu\xi|;\beta\gamma ) } + \frac{1}{10}\ , r_{\mu\xi\nu(\alpha;\beta\gamma ) } + \frac{2}{15}\ , r _ { a ( \alpha|\mu|\beta } r^a{}_{\gamma)\nu\xi } \displaybreak[0]\\ & + \frac{2}{15}\ , { a ( \alpha|\mu|\beta } r^a{}_{|\xi\nu|\gamma ) } + \frac{2}{15}\ , r _ { a ( \alpha|\mu\xi } r^a{}_{\beta|\nu|\gamma ) } + \frac{2}{15}\ , r _ { a \xi\mu(\alpha } r^a{}_{\beta|\nu|\gamma ) } \displaybreak[0]\\ & + \frac{4}{3}\ , r_{(\mu| a |\nu)(\alpha } r^a{}_{\beta\gamma)\xi } \,.\end{aligned}\ ] ] symmetrization in @xmath778 is assumed in all terms of the following expression , @xmath779 & = -\frac{2}{5}\ , r_{\mu(\alpha|\nu|\beta;\gamma)\xi } + \frac{1}{10}\ , r_{\mu(\alpha|\nu|\beta;|\xi|\gamma ) } -\frac{4}{3}\ , r_{a \mu(\alpha|\nu| } r^a{}_{\beta|\xi|\gamma ) } \displaybreak[0]\\ & = -\frac{2}{5}\ , r_{\mu(\alpha|\nu|\beta;|\xi|\gamma ) } + \frac{2}{5}\ , r _ { a ( \alpha|\nu|\beta } r^a{}_{|\mu\xi|\gamma ) } + \frac{2}{5}\ , r_{\mu a \nu ( \alpha } r^a{}_{\beta|\xi|\gamma ) } \displaybreak[0]\\ & + \frac{1}{10}\ , r_{\mu(\alpha|\nu|\beta;|\xi|\gamma ) } -\frac{4}{3}\ , r_{a \mu(\alpha|\nu| } r^a{}_{\beta|\xi|\gamma ) } \displaybreak[0]\\ & = -\frac{3}{10}\ , r_{\mu(\alpha|\nu|\beta;|\xi|\gamma ) } -\frac{4}{3}\ , r_{a \mu(\alpha|\nu| } r^a{}_{\beta|\xi|\gamma ) } \,.\end{aligned}\ ] ] estrada , r. and fulling , s. a. , distributional asymptotic expansions of spectral functions and of the associated green kernels , _ electron . j. differential equations _ , * 7 * ( 1999 ) , arxiv : funct - an/9710003 .

in this thesis we study the geometry of the fixed point set @xmath0 of a smooth mapping @xmath1 on an @xmath2-dimensional riemannian manifold @xmath3 by computing the asymptotic expansion as @xmath4 of the trace of the deformed heat kernel @xmath5 of the laplace operator on @xmath6 . we assume that the fixed point set is a union of connected components @xmath7 , @xmath8 , each of which is a smooth compact submanifold of @xmath6 with dimension @xmath9 , @xmath10 . there exists the asymptotic expansion as @xmath4 @xmath11 where @xmath12 are scalar invariants on @xmath7 depending on covariant derivatives of the curvature of the metric @xmath13 and symmetrized covariant derivatives of the differential @xmath14 of the mapping @xmath15 , evaluated on @xmath7 . due to the localization principle , it is possible to compute the expansion for each component separately . we develop a generalized laplace method for computing the coefficients @xmath12 in this expansion and compute the coefficients @xmath16 , @xmath17 , and @xmath18 explicitly in the following cases : 1 . @xmath6 is a flat two - dimensional manifold , @xmath0 is a zero- or one - dimensional component of the fixed point set . 2 . @xmath6 is an @xmath2-dimensional curved manifold , @xmath0 is a zero - dimensional component of the fixed point set . we also develop algorithms for computing further coefficients and expressing them in a symmetrization - free form . i would like to express my gratitude to my scientific advisor prof . ivan g. avramidi for arousing my interest in differential geometry and his area of work . for endless hours he spent with me during my years at new mexico tech exploring possible research directions and helping to improve my background where it lacked . he was always ready to answer my questions and it was just impossible to wish for a better mentor ! thanks to my scientific advisor prof . igor b. simonenko at rostov state university , russia . his inexhaustible enthusiasm inspired me a lot during all my six years there , and our prolonged discussions helped me to develop a mathematical style of thinking . thanks to don clewett and especially rachael defibaugh - chvez for reading drafts of this thesis and making a lot of useful comments and corrections .

math0507453

the established evidence that neutrinos oscillate and possess small masses @xcite necessitates physics beyond the standard model ( sm ) , which could manifest itself at the cern large hadron collider ( lhc ) and/or in low energy experiments which search for the lepton flavour violation @xcite . consequently , models of neutrino mass generation which can be probed at present and forthcoming experiments are of great phenomenological interest . neutrinos may obtain mass via the vacuum expectation value ( vev ) of a neutral higgs boson in an isospin triplet representation @xcite . a particularly simple implementation of this mechanism of neutrino mass generation is the `` higgs triplet model '' ( htm ) in which the sm lagrangian is augmented solely by @xmath10 which is a @xmath1 triplet of scalar particles with hypercharge @xmath11 @xcite . in the htm , the majorana neutrino mass matrix @xmath12 ( @xmath13 ) is given by the product of a triplet yukawa coupling matrix @xmath14 and a triplet vev ( @xmath15 ) . consequently , the direct connection between @xmath14 and @xmath12 gives rise to phenomenological predictions for processes which depend on @xmath14 because @xmath16 has been restricted well by neutrino oscillation measurements @xcite . a distinctive signal of the htm would be the observation of doubly charged higgs bosons ( @xmath0 ) whose mass ( @xmath17 ) may be of the order of the electroweak scale . such particles can be produced with sizeable rates at hadron colliders in the processes @xmath4 @xcite and @xmath18 @xcite . the first searches for @xmath0 at a hadron collider were carried out at the fermilab tevatron , assuming the production channel @xmath4 and decay @xmath19 . the mass limits @xmath20 @xcite were derived , with the strongest limits being for @xmath21 @xcite . the branching ratios ( brs ) for @xmath19 depend on @xmath14 and are predicted in the htm in terms of the parameters of the neutrino mass matrix @xcite . detailed quantitative studies of br(@xmath19 ) in the htm have been performed in @xcite with particular emphasis given to their sensitivity to the majorana phases and the absolute neutrino mass i.e. parameters which can not be probed in neutrino oscillation experiments . a study on the relation between br(@xmath19 ) and the neutrinoless double beta decay can be seen in @xcite . simulations of the detection prospects of @xmath0 at the lhc with @xmath22 previously focussed on @xmath4 only @xcite , but recent studies now include the mechanism @xmath18 @xcite . the first search for @xmath0 at the lhc with @xmath23 @xcite has recently been performed for both production mechanisms @xmath4 and @xmath18 , for the decay channels @xmath19 and @xmath24 . in phenomenological studies of the htm , for simplicity it is sometimes assumed that @xmath0 and @xmath25 are degenerate , with a mass @xmath26 which arises from a bilinear term @xmath27 in the scalar potential . in this scenario the only possible decay channels for @xmath0 are @xmath19 and @xmath28 , and the branching ratios are determined by the magnitude of @xmath15 . however , quartic terms in the scalar potential break the degeneracy of @xmath0 and @xmath25 , and induce a mass splitting @xmath29 , which can be of either sign . if @xmath30 then a new decay channel becomes available for @xmath0 , namely @xmath31 . some attention has been given to the decay @xmath31 , and it has been shown that it can be the dominant channel over a wide range of values of @xmath32 and @xmath15 @xcite , even for @xmath33 . another scenario is the case of @xmath34 , which would give rise to a new decay channel for the singly charged scalar , namely @xmath2 . this possibility has been mentioned in the context of the htm in @xcite only . we will perform the first study of the magnitude of its branching ratio , as well as quantify its contribution to the production of @xmath0 at the lhc . has also been briefly mentioned in @xcite in the context of a model with an isospin 3/2 multiplet with hypercharge @xmath35 , which also includes triply charged higgs bosons . ] the decay rate for @xmath2 is easily obtained from the corresponding expression for the decay rate for @xmath31 , and thus one expects that @xmath2 will be sizeable over a wide range of values of @xmath32 and @xmath15 . we point out for the first time that the decay @xmath2 would give rise to an alternative way to produce @xmath0 in pairs ( @xmath36 ) , namely by the production mechanism @xmath18 followed by @xmath37 . production of @xmath36 can give rise to a distinctive signature of four leptons ( @xmath38 ) , and simulations and searches of this channel currently only assume production via the process @xmath4 . our work is organised as follows . in section ii we describe the theoretical structure of the htm . in section iii the decay @xmath2 is introduced . section iv contains our numerical analysis of the magnitude of the cross section for @xmath36 which originates from production via @xmath18 followed by the decay @xmath2 . conclusions are given in section v. in the htm @xcite a @xmath11 complex @xmath1 isospin triplet of scalar fields is added to the sm lagrangian . such a model can provide majorana masses for the observed neutrinos without the introduction of @xmath1 singlet neutrinos via the gauge invariant yukawa interaction : @xmath39 here @xmath40 is a complex and symmetric coupling , @xmath41 is the dirac charge conjugation operator , @xmath42 is the pauli matrix , @xmath43 is a left - handed lepton doublet , and @xmath10 is a @xmath44 representation of the @xmath11 complex triplet fields : @xmath45 a non - zero triplet vacuum expectation value @xmath46 gives rise to the following mass matrix for neutrinos : @xmath47 the necessary non - zero @xmath48 arises from the minimisation of the most general @xmath49 invariant higgs potential @xcite , which is written ) by using @xmath50 ^ 2 - \text{tr}[(\delta^\dagger \delta)^2]$ ] and @xmath51 . ] as follows @xcite ( with @xmath52 ) : @xmath53 ^ 2 + \lambda_3{\rm det } ( \delta^\dagger\delta ) \nonumber \\ & & + \lambda_4(\phi^\dagger\phi){\rm tr}(\delta^\dagger\delta ) + \lambda_5(\phi^\dagger\tau_i\phi){\rm tr}(\delta^\dagger\tau_i \delta)+\left ( { 1\over \sqrt 2}\mu(\phi^ti\tau_2\delta^\dagger\phi ) + \text{h.c . } \right ) . \label{higgs_potential}\end{aligned}\ ] ] here @xmath54 in order to ensure @xmath55 which spontaneously breaks @xmath56 to @xmath57 , and @xmath58 is the mass term for the triplet scalars . in the model of gelmini - roncadelli @xcite the term @xmath59 is absent , which leads to spontaneous violation of lepton number for @xmath60 . the resulting higgs spectrum contains a massless triplet scalar ( majoron , @xmath61 ) and another light scalar ( @xmath7 ) . pair production via @xmath62 would give a large contribution to the invisible width of the @xmath63 and this model was excluded at the cern large electron positron collider ( lep ) . the inclusion of the term @xmath64 ) @xcite explicitly breaks lepton number @xmath65 when @xmath10 is assigned @xmath66 , and eliminates the majoron . thus the scalar potential in eq . ( [ higgs_potential ] ) together with the triplet yukawa interaction of eq . ( [ trip_yuk ] ) lead to a phenomenologically viable model of neutrino mass generation . for small @xmath67 , the expression for @xmath15 resulting from the minimisation of @xmath68 is : @xmath69 for large @xmath26 compared to @xmath70 one has @xmath71 , which is sometimes referred to as the `` type ii seesaw mechanism '' and would naturally lead to a small @xmath15 . recently there has been much interest in the scenario of light triplet scalars ( @xmath72 ) within the discovery reach of the lhc , for which eq . ( [ tripletvev ] ) leads to @xmath73 . in extensions of the htm the term @xmath64 ) may arise in various ways : i ) it can be generated at tree level via the vev of a higgs singlet field @xcite ; ii ) it can arise at higher orders in perturbation theory @xcite ; iii ) it can originate in the context of extra dimensions @xcite . an upper limit on @xmath15 can be obtained from considering its effect on the parameter @xmath74 . in the sm @xmath75 at tree - level , while in the htm one has ( where @xmath76 ) : @xmath77 the measurement @xmath78 leads to the bound @xmath79 , or @xmath80 . production mechanisms which depend on @xmath15 ( i.e. @xmath81 and fusion via @xmath82 @xcite ) are not competitive with the processes @xmath4 and @xmath18 at the energies of the fermilab tevatron , but such mechanisms can be the dominant source of @xmath0 at the lhc if @xmath83 and @xmath84 . at the 1-loop level , @xmath15 must be renormalised and explicit analyses lead to bounds on its magnitude similar to the above bound from the tree - level analysis , e.g. see @xcite . the scalar eigenstates in the htm are as follows : i ) the charged scalars @xmath0 and @xmath9 ; ii ) the cp - even neutral scalars @xmath85 and @xmath7 ; iii ) a cp - odd neutral scalar @xmath8 . the doubly charged @xmath0 is entirely composed of the triplet scalar field @xmath86 , while the remaining eigenstates are in general mixtures of the doublet and triplet fields . however , such mixing is proportional to the triplet vev , and hence small _ even if _ @xmath15 assumes its largest value of a few gev . @xcite . ] therefore @xmath87 are predominantly composed of the triplet fields , while @xmath85 is predominantly composed of the doublet field and plays the role of the sm higgs boson . the scale of squared masses of @xmath88 are determined by @xmath89 with mass splittings of order @xmath90 @xcite : @xmath91 the degeneracy @xmath92 can be understood by the fact that the higgs potential is invariant under a global @xmath93 for @xmath10 ( @xmath65 conservation ) when one neglects the trilinear term proportional to @xmath94 . the mass hierarchy @xmath95 is obtained for @xmath96 , and the opposite hierarchy @xmath97 is obtained for @xmath98 . in general , one would not expect degenerate masses for @xmath88 , but instead one of the above two mass hierarchies . the sign of @xmath99 is not fixed by theoretical requirements of vacuum stability of the scalar potential @xcite , although @xmath100 is necessary to ensure that @xmath101 and @xmath102 in eq . ( [ eq : mh ] ) are positive . therefore the decays channels @xmath103 and @xmath104 are possible in the htm . the potential importance of the decay channel @xmath105 ( for @xmath106 ) has not been quantified in the htm . for this decay to be kinematically open @xcite one needs the mass hierarchy where @xmath0 is the lightest of the triplet scalars ( @xmath107 ) , which is obtained for @xmath96 . for the opposite mass hierarchy with @xmath98 ( @xmath97 ) the related decay @xmath108 was shown to be important in the htm in @xcite . the expression for the decay width of @xmath2 is easily obtained from the expression for @xmath108 by merely interchanging @xmath17 and @xmath109 . after summing over all fermion states for @xmath110 , excluding the @xmath111 quark , the decay rate is given by @xmath112 where @xmath113 and the analytical expression for @xmath114 can be found in @xcite ( see also @xcite ) . note that this decay mode does not depend on @xmath15 . in eq . ( [ hhwdecay ] ) we take @xmath115 and @xmath116 to be massless , which is a good approximation as long as the mass splitting between @xmath17 and @xmath109 is above the mass of the charmed hadrons ( @xmath117 ) . in our numerical analysis we will be mostly concerned with sizeable mass splittings , @xmath118 . the other possible decays for @xmath9 are @xmath24 , @xmath119 , @xmath120 ( where @xmath85 is the sm - like scalar field ) and @xmath121 . explicit expressions for the decay widths of these channels can be found in the literature ( e.g. @xcite ) and they are presented below . the decay width for @xmath24 is given by @xmath122 note that @xmath123 has no dependence on the neutrino mixing angles because @xmath124 , where @xmath125 ( @xmath126 ) are neutrino masses . the decay widths for the channels which are proportional to @xmath127 are expressed as follows : @xmath128 ^ 3 , \label{eq : gam_hp_wz}\end{aligned}\ ] ] @xmath129 ^ 3 , \label{eq : gam_hp_wh}\end{aligned}\ ] ] @xmath130 @xmath131 the decay @xmath120 is caused by two small mixings of scalar fields . one is the mixing angle @xmath132 between @xmath133 and @xmath134 , and the other is the mixing angle @xmath135 between @xmath136 and @xmath137 . if @xmath138 , then one has @xmath139 in eq . ( [ eq : gam_hp_wh ] ) . since we are interested in the case where the exotic scalars have masses of the electroweak scale , we do not take a very large @xmath26 . however , we assume @xmath139 for simplicity , which can be achieved by @xmath140 . the decay @xmath141 is mediated by the small @xmath133 component of @xmath9 through @xmath142 . for @xmath143 , @xmath144 is comparable to @xmath145 and @xmath146 . these three decay widths in eq . ( [ eq : gam_hp_wz])-([eq : gam_hp_tb ] ) are greater than @xmath147 for @xmath148 while @xmath147 dominates for @xmath149 . it has already been shown that the decay @xmath31 can be the dominant decay channel for the doubly charged scalar over a wide range of values of @xmath150 and @xmath15 @xcite , even for @xmath33 . hence we expect a similar result for the decay @xmath2 for the singly charged scalar . the branching ratio @xmath151 will be maximised with respect to @xmath15 if @xmath152 which is achieved for @xmath153 . a numerical study of the magnitude of @xmath151 is presented in the next section . we now emphasise an important phenomenological difference between the distinct scenarios of a sizeable branching ratio for the decay channels @xmath31 ( for @xmath154 ) and @xmath2 ( for @xmath96 ) . the decay @xmath31 is expected to weaken the discovery potential of @xmath0 at the lhc , because it would reduce the branching ratio of a channel like @xmath155 ( which is otherwise the dominant channel for @xmath156 , and enjoys low sm backgrounds ) . we note that there has been no simulation of the detection prospects of @xmath31 , and its signature would be different to that of the standard decay channels @xmath19 and @xmath28 . in contrast , we point out that the decay @xmath157 could actually _ improve _ the discovery potential of @xmath0 at the lhc . from the production mechanism @xmath18 the decay mode @xmath157 would give rise to pair production ( @xmath36 ) of doubly charged higgs bosons . we believe that this additional way to produce @xmath0 has not been discussed before . in this scenario @xmath0 is the lightest of the triplet scalars , and its only possible decay channels are @xmath19 and @xmath28 , with branching ratios determined by the magnitude of @xmath15 . these two branching ratios can be of the same order of magnitude for @xmath153 , as can be seen in fig . [ fig : br_hdoub ] where we fix @xmath158 and @xmath159 ( similar figures can be found in @xcite ) . in the range of @xmath160 , one has @xmath161 for @xmath162 , while for @xmath163 one has br(@xmath164 . in simulations of pair production of @xmath0 it is assumed that the production channel @xmath4 is the only mechanism . if @xmath156 then the decay channel @xmath155 is dominant , and four - lepton signatures ( @xmath165 ) would be possible . studies have shown that the standard model background for the @xmath165 signature @xcite is considerably smaller than that for the signature of @xmath166 @xcite , and at present it is assumed that the @xmath165 signature can only arise from @xmath4 . the importance of the production mechanism @xmath18 has been appreciated for the @xmath166 signature , in which the decay @xmath167 is assumed @xcite . for the case of a sizeable branching ratio for @xmath157 we point out that the production mechanism @xmath18 can also contribute to the @xmath165 signature , which is the signature with lowest background . searches for four leptons originating from @xmath36 have already been performed by the tevatron @xcite and lhc @xcite . if br(@xmath157 ) were sizeable we would expect a strengthening of the derived limit on @xmath17 . in this section we quantify the magnitude of the number of pair - produced @xmath36 arising from the process @xmath18 with decay @xmath157 , and make a comparison with the conventional mechanism @xmath4 . the important parameters for our analyses are @xmath15 , @xmath109 , and @xmath17 . we take @xmath168 or @xmath169 and show results as functions of @xmath15 and @xmath109 . the decay branching ratios of @xmath9 also depend on two undetermined parameters , @xmath170 and @xmath171 ( one of the neutrino masses ) . these are fixed as @xmath172 and @xmath173 in our numerical analysis . note that @xmath170 only enters through the decay width for @xmath174 . neutrino oscillation experiments @xcite provide a measurement of two neutrino mass differences , @xmath175 , and we use the following values : @xmath176 , @xmath177 . although @xmath178 ( referred to as `` normal mass ordering '' ) is also assumed in our analysis , our results do not change significantly for @xmath179 because the neutrino masses are almost degenerate for @xmath180 . the experimental bound @xmath181 gives a stringent constraint on @xmath182 and @xmath17 . @xcite or @xmath183 @xcite . see also @xcite . ] assuming naively @xmath184 for @xmath173 , the bound on @xmath185 can be translated into the constraint @xmath186 . therefore , we use @xmath187 in order to satisfy this constraint for @xmath168 . + in fig . [ fig : br_hsing ] we show the brs of @xmath9 decays into @xmath188 ( red solid ) , @xmath189 ( blue dashed ) , @xmath190 ( green dotted ) , @xmath191 ( magenta dot - dashed ) , and @xmath192 ( cyan dot - dot - dashed ) as a function of @xmath193 for various values of @xmath15 , fixing @xmath168 and @xmath194 . the range of @xmath109 in the figures corresponds to @xmath195 , which easily satisfies the perturbative constraint @xmath196 . very large mass splittings ( e.g. @xmath197 ) are constrained by measurements of electroweak precision observables , but the mass splittings in fig . [ fig : br_hsing ] are compatible with the analyses in @xcite ( which are for models with a @xmath198 triplet ) . in fig . [ fig : br_hsing](a ) we fix @xmath199 , for which @xmath200 . one can see that @xmath105 competes with @xmath201 , with all other decay channels being negligible . for @xmath202 , @xmath105 becomes the dominant decay channel . in fig . [ fig : br_hsing](b ) we fix @xmath158 , and @xmath105 becomes the dominant decay channel for much smaller mass splittings , @xmath203 . in fig . [ fig : br_hsing](c ) we fix @xmath204 , for which the competing decays are @xmath205 , @xmath206 and @xmath207 . in this scenario the decay @xmath105 becomes the dominant channel for @xmath208 . in fig . [ fig : br_contour ] we show contours of br(@xmath2 ) in the plane @xmath209 $ ] . the red solid , green dashed , and blue dotted lines correspond to contours of @xmath210 , 0.9 , and 0.99 , respectively . the br is maximised at around @xmath211 , as expected . it is clear from fig . [ fig : br_hsing ] and fig . [ fig : br_contour ] that the decay of @xmath9 into @xmath0 can be dominant in a wide region of the parameter space of the htm even if the two - body decay into @xmath212 ( for @xmath213 ) is forbidden kinematically . moreover , for @xmath214 ( i.e. when the four - lepton signal arising from the decay of @xmath36 is dominant ) the magnitude of @xmath215 becomes very large if @xmath216 . we now study the magnitude of the number of pair - produced @xmath36 which originate from @xmath217 followed by the decay @xmath2 . we define the variable @xmath218 as follows : @xmath219 in fig . [ fig : brsig ] we show the behaviour of @xmath220 with respect to @xmath109 for several values of @xmath15 . in fig . [ fig : brsig]a we take @xmath168 and @xmath221 , and in fig . [ fig : brsig]b we take @xmath222 and @xmath223 . we use cteq6l1 parton distribution functions @xcite . the range of @xmath109 in fig . [ fig : brsig]b corresponds to @xmath224 . the horizontal dot - dashed line corresponds to the case of @xmath225 , i.e. the magnitude of @xmath226 alone . the red solid , green dashed , and blue dotted lines are the results with @xmath214 , @xmath227 , and @xmath228 , respectively . the red solid line ( for which @xmath229 ) shows that the extra contribution from @xmath2 can enhance the number of four - lepton events by a factor of 2 ( at @xmath230 in fig . [ fig : brsig]a ) and 2.4 ( at @xmath231 in fig . [ fig : brsig]b ) . for @xmath211 , around which @xmath232 can still be sizeable ( see fig . [ fig : br_hdoub ] ) , the enhancement factor for pair - produced @xmath36 can be as large as 2.6 in fig . [ fig : brsig]a and 2.8 in fig . [ fig : brsig]b . for @xmath204 the enhancement of pair - produced @xmath36 is interpreted as an increase in the number of @xmath233 events , because @xmath234 . the shape of the curves is caused by the different dependence of the cross section and br on the mass splitting @xmath32 . as @xmath109 increases , the cross section of @xmath235 is unaffected but the cross section of @xmath217 decreases . however , a larger mass splitting is favourable from the point of view of the br . finally , we note that a pair of @xmath0 can also be produced from other production mechanisms , namely @xmath236 , @xmath237 , @xmath238 , and @xmath239 . although the contribution from @xmath240 in eq . ( [ eq : hpphm ] ) is the most important one because of the mass hierarchy @xmath241 and its linear dependence on br@xmath242 , the above mechanisms can give a significant contribution to the number of pair - produced @xmath0 , as will be described qualitatively below . naively , one would expect the next most important mechanism to be @xmath236 because its contribution to the production of @xmath36 scales as @xmath243 as follows : @xmath244 ^ 2 .\end{aligned}\ ] ] however , the couplings for @xmath245 and @xmath246 are about a half of those for @xmath247 and @xmath248 , respectively . the interference between @xmath249 and @xmath250 is destructive for @xmath236 production while it is constructive for @xmath36 production . tables [ tab : lhc7 ] and [ tab : lhc14 ] show that @xmath251 is smaller than @xmath226 by a factor of @xmath252 , even for @xmath253 ( see e.g. @xcite ) . moreover , @xmath254 is suppressed relative to @xmath218 by an extra factor of br when @xmath255 . therefore the contribution from @xmath251 to the production of @xmath36 is considerably less than the qcd @xmath256 factor for @xmath235 ( which is known to be around 1.25 at the lhc @xcite ) . it turns out that the production of @xmath237 and @xmath238 are numerically more important than @xmath257 , despite their contributions scaling as @xmath258 . the narrow width approximation for contributions from @xmath7 and @xmath8 with @xmath92 is rather complicated because of their interference . we define the variables @xmath259 and @xmath260 as follows : @xmath261 ^ 2 , \label{eq : hph0 } \\ x_3^\prime & \equiv & \left\ { \sigma({{pp\to w^ * \to h^+h^0 } } ) + \sigma({{pp\to w^ * \to h^-h^0 } } ) \right\ } \nonumber\\ & & \hspace*{50 mm } \times { \text{br}}_-\ , [ { \text{br}}(h^\pm \to h^{\pm\pm } w^*)]^2 , \label{eq : hph0 - 2 } \\ { \text{br}}_\pm & \equiv & { \text{br}}(h^0\to h^\pm w^ * ) + { \text{br}}(a^0\to h^\pm w^ * ) \nonumber\\ & & \hspace*{20 mm } { } \pm \frac { 4 { \text{br}}(h^0\to h^\pm w^ * ) { \text{br}}(a^0\to h^\pm w^ * ) } { { \text{br}}(h^0\to h^\pm w^ * ) + { \text{br}}(a^0\to h^\pm w^ * ) } , \end{aligned}\ ] ] where we used @xmath262 because @xmath263 . the interesting point is that @xmath260 is for the process which gives _ same - sign _ @xmath264 ( with @xmath265 ) and @xmath266 ( with @xmath267 ) while @xmath259 is for @xmath36 production . since @xmath260 arises as the breaking effect of the lepton number ( @xmath10 has @xmath268 ) , it vanishes for @xmath269 , for which the total decay widths satisfy @xmath270 , namely @xmath271 . this means that the _ same - sign _ @xmath272 would not give the _ same - sign _ @xmath165 signal because @xmath273 is small for a large @xmath15 where @xmath260 could be sizeable . a pair of @xmath0 ( _ same - sign _ or different sign ) is provided by @xmath274 , which is proportional to @xmath275 $ ] ; the factor of 2 compensates the fact that the sum of the cross sections in eq . ( [ eq : hph0 ] ) is a half of the sum in eq . ( [ eq : hpphm ] ) for @xmath276 as shown in tables [ tab : lhc7 ] and [ tab : lhc14 ] . although @xmath277 ( likewise for @xmath7 ) and the maximum value of each is @xmath278 , this is compensated by @xmath279 in @xmath274 . since the partial decay widths of @xmath7 and @xmath8 depend on the scalar masses and @xmath15 in a way which is very similar to the partial decay widths of @xmath9 ( see e.g. @xcite ) , the analogies of fig . [ fig : br_contour ] for @xmath280 and @xmath281 would show a similar quantitative behaviour as fig . [ fig : br_contour ] . and @xmath5 were also mentioned as a source of @xmath9 in @xcite . ] thus the main difference between @xmath218 and @xmath274 would be the phase space factor because we take @xmath282 . the contribution of @xmath274 to the production of a pair of @xmath0 would be sizeable for @xmath153 , where the relevant brs in eq . ( [ eq : hph0 ] ) could be very large for a small mass splitting . moreover , the contribution of @xmath274 would not be so small even for large mass splittings e.g. @xmath283 and @xmath284 ( which give @xmath285 ) , for which the brs in eq . ( [ eq : hph0 ] ) could be maximal . the last mechanisms ( which scale as @xmath286 ) are @xmath287 ^ 2 , \\ x_4^\prime & \equiv & \sigma({{pp\to z^ * \to h^0 a^0}})\ , { \text{br}}_+\ , { \text{br}}_-\ , [ { \text{br}}(h^\pm\to h^{\pm\pm } w^*)]^2 , \\ x_4^{\prime\prime } & \equiv & \sigma({{pp\to z^ * \to h^0 a^0}})\ , { \text{br}}_-^2\ , [ { \text{br}}(h^\pm\to h^{\pm\pm } w^*)]^2 .\end{aligned}\ ] ] note that @xmath288 gives a pair of _ same - sign _ @xmath0 ( being proportional to @xmath289 , like @xmath260 ) and its magnitude is negligible for small @xmath15 . although both of @xmath290 and @xmath291 give @xmath36 , @xmath291 also vanishes for @xmath269 because it is sensitive to @xmath292 i.e. it is quadratic in lepton number violation . the phase space suppression ( @xmath241 ) ensures that @xmath293 is much smaller than @xmath226 for the case of a large mass splitting with @xmath221 . therefore , for @xmath290 to be important a large mass splitting with @xmath223 or a small mass splitting for @xmath153 are preferred . we note that the detection efficiencies for the above mechanisms ( @xmath218 , @xmath254 , @xmath259 and @xmath290 ) would in general be different from that of the well - studied mechanism @xmath235 because of the extra @xmath294 . we defer a detailed study to a future work . . production cross sections of a pair of exotic higgs bosons ( @xmath295 ) from off - shell gauge bosons ( @xmath296 ) in the htm at the lhc with @xmath221 . we take @xmath284 and we use a relation @xmath297 ; @xmath298 , @xmath299 , @xmath300 for @xmath301 , @xmath302 , @xmath303 , respectively . [ cols="^,^,^,^,^,^,^,^ " , ] doubly charged higgs bosons ( @xmath0 ) , which arise in the higgs triplet model ( htm ) of neutrino mass generation , are being searched for at the tevatron and at the lhc . we showed that @xmath0 can be produced from the decay of a singly charged higgs boson ( @xmath25 ) via @xmath304 , which can have a large branching ratio in a wide region of the parameter space of the htm . from the production mechanism @xmath305 , the above decay would give rise to pair production @xmath36 , with a number of events which can be comparable to that from the conventional mechanism @xmath4 . current simulations and searches for @xmath36 at the tevatron / lhc assume production solely from @xmath4 . the contribution from @xmath18 with decay @xmath304 would be an additional source of pair - produced @xmath0 , which should enhance the detection prospects in this channel ( e.g. four - lepton signatures if the decay mode @xmath19 is dominant ) . we also pointed out that production mechanisms involving the neutral triplet scalars ( @xmath7,@xmath8 ) of the htm can contribute to pair production @xmath36 through the decay chain @xmath306 followed by @xmath304 . we advocate dedicated simulations of @xmath18 with the decay @xmath304 ( and the analogous mechanisms with neutral scalars ) , and a comparison with @xmath4 . we thank mayumi aoki and koji tsumura for useful discussions . a.g.a was supported by a marie curie incoming international fellowship , fp7-people-2009-iif , contract no . 252263 . the work of h.s . was supported in part by the sasakawa scientific research grant from the japan science society and grant - in - aid for young scientists ( b ) no . 23740210 . y. fukuda _ et al . _ [ super - kamiokande collaboration ] , phys . lett . * 81 * , 1562 ( 1998 ) . y. kuno and y. okada , rev . phys . * 73 * , 151 ( 2001 ) ; m. raidal _ et al . _ , eur . j. c * 57 * , 13 ( 2008 ) . w. konetschny and w. kummer , phys . b * 70 * , 433 ( 1977 ) . r. n. mohapatra and g. senjanovic , phys . * 44 * , 912 ( 1980 ) . m. magg and c. wetterich , phys . b * 94 * , 61 ( 1980 ) ; g. lazarides , q. shafi and c. wetterich , nucl . b * 181 * , 287 ( 1981 ) . j. schechter and j. w. f. valle , phys . d * 22 * , 2227 ( 1980 ) . t. p. cheng and l. f. li , phys . d * 22 * , 2860 ( 1980 ) . b. t. cleveland _ et al . _ , astrophys . j. * 496 * , 505 ( 1998 ) ; w. hampel _ et al . _ [ gallex collaboration ] , phys . b * 447 * , 127 ( 1999 ) ; j. n. abdurashitov _ et al . _ [ sage collaboration ] , j. exp . phys . * 95 * , 181 ( 2002 ) [ zh . * 122 * , 211 ( 2002 ) ] ; j. hosaka _ et al . _ [ super - kamkiokande collaboration ] , phys . d * 73 * , 112001 ( 2006 ) ; b. aharmim _ et al . _ [ sno collaboration ] , phys . * 101 * , 111301 ( 2008 ) ; c. arpesella _ et al . _ [ the borexino collaboration ] , phys . * 101 * , 091302 ( 2008 ) . y. ashie _ et al . _ [ super - kamiokande collaboration ] , phys . rev . d * 71 * , 112005 ( 2005 ) ; j. l. raaf [ super - kamiokande collaboration ] , j. phys . * 136 * , 022013 ( 2008 ) . m. h. ahn _ [ k2k collaboration ] , phys . d * 74 * , 072003 ( 2006 ) ; p. adamson _ et al . _ [ minos collaboration ] , phys . rev . * 101 * , 131802 ( 2008 ) . m. apollonio _ et al . _ [ chooz collaboration ] , eur . j. c * 27 * , 331 ( 2003 ) . s. abe _ et al . _ [ kamland collaboration ] , phys . * 100 * , 221803 ( 2008 ) . v. d. barger , h. baer , w. y. keung and r. j. n. phillips , phys . d * 26 * , 218 ( 1982 ) . j. f. gunion , j. grifols , a. mendez , b. kayser and f. i. olness , phys . d * 40 * , 1546 ( 1989 ) ; j. f. gunion , c. loomis and k. t. pitts , econf * c960625 * , lth096 ( 1996 ) [ arxiv : hep - ph/9610237 ] . m. muhlleitner and m. spira , phys . d * 68 * , 117701 ( 2003 ) . t. han , b. mukhopadhyaya , z. si and k. wang , phys . d * 76 * , 075013 ( 2007 ) . k. huitu , j. maalampi , a. pietila and m. raidal , nucl . b * 487 * , 27 ( 1997 ) . b. dion , t. gregoire , d. london , l. marleau and h. nadeau , phys . d * 59 * , 075006 ( 1999 ) . a. g. akeroyd and m. aoki , phys . d * 72 * , 035011 ( 2005 ) . d. e. acosta _ et al . _ [ cdf collaboration ] , phys . rev . lett . * 93 * , 221802 ( 2004 ) ; v. m. abazov _ et al . _ [ d0 collaboration ] , phys . rev . lett . * 93 * , 141801 ( 2004 ) ; v. m. abazov _ et al . _ [ d0 collaboration ] , phys . * 101 * , 071803 ( 2008 ) . t. aaltonen _ et al . _ [ the cdf collaboration ] , phys . rev . * 101 * , 121801 ( 2008 ) . e. ma , m. raidal and u. sarkar , phys . * 85 * , 3769 ( 2000 ) ; e. ma , m. raidal and u. sarkar , nucl . phys . b * 615 * , 313 ( 2001 ) . e. j. chun , k. y. lee and s. c. park , phys . b * 566 * , 142 ( 2003 ) . j. garayoa and t. schwetz , jhep * 0803 * , 009 ( 2008 ) . a. g. akeroyd , m. aoki and h. sugiyama , phys . rev . d * 77 * , 075010 ( 2008 ) . m. kadastik , m. raidal and l. rebane , phys . d * 77 * , 115023 ( 2008 ) . p. fileviez perez , t. han , g. y. huang , t. li and k. wang , phys . d * 78 * , 015018 ( 2008 ) . s. t. petcov , h. sugiyama and y. takanishi , phys . rev . d * 80 * , 015005 ( 2009 ) . g. azuelos , k. benslama and j. ferland , j. phys . g * 32 * , 73 ( 2006 ) ; t. rommerskirchen and t. hebbeker , j. phys . g * 34 * , n47 ( 2007 ) . g. b. gelmini and m. roncadelli , phys . b * 99 * , 411 ( 1981 ) . j. schechter and j. w. f. valle , phys . d * 25 * , 774 ( 1982 ) ; m. a. diaz , m. a. garcia - jareno , d. a. restrepo and j. w. f. valle , nucl . b * 527 * , 44 ( 1998 ) ; a. g. akeroyd , m. a. diaz , m. a. rivera and d. romero , phys . d * 83 * , 095003 ( 2011 ) . t. blank and w. hollik , nucl . b * 514 * , 113 ( 1998 ) ; m. c. chen , s. dawson and t. krupovnickas , phys . d * 74 * , 035001 ( 2006 ) ; p. h. chankowski , s. pokorski and j. wagner , eur . j. c * 50 * , 919 ( 2007 ) ; m. c. chen , s. dawson and c. b. jackson , phys . d * 78 * , 093001 ( 2008 ) . p. dey , a. kundu and b. mukhopadhyaya , j. phys . g * 36 * , 025002 ( 2009 ) . a. g. akeroyd and c. w. chiang , phys . d * 81 * , 115007 ( 2010 ) . r. godbole , b. mukhopadhyaya and m. nowakowski , phys . b * 352 * , 388 ( 1995 ) ; k. m. cheung , r. j. n. phillips and a. pilaftsis , phys . d * 51 * , 4731 ( 1995 ) ; k. huitu , j. laitinen , j. maalampi and n. romanenko , nucl . b * 598 * , 13 ( 2001 ) . a. g. akeroyd , m. aoki and h. sugiyama , phys . d * 79 * , 113010 ( 2009 ) . t. fukuyama , h. sugiyama and k. tsumura , jhep * 1003 * , 044 ( 2010 ) . j. pumplin , d. r. stump , j. huston , h. l. lai , p. m. nadolsky and w. k. tung , jhep * 0207 * , 012 ( 2002 ) ; d. stump , j. huston , j. pumplin , w. k. tung , h. l. lai , s. kuhlmann and j. f. owens , jhep * 0310 * , 046 ( 2003 ) .

the existence of doubly charged higgs bosons ( @xmath0 ) is a distinctive feature of the higgs triplet model ( htm ) , in which neutrinos obtain tree - level masses from the vacuum expectation value of a neutral scalar in a triplet representation of @xmath1 . we point out that a large branching ratio for the decay of a singly charged higgs boson to a doubly charged higgs boson via @xmath2 is possible in a sizeable parameter space of the htm . from the production mechanism @xmath3 the above decay mode would give rise to pair production of @xmath0 , with a cross section which can be comparable to that of the standard pair - production mechanism @xmath4 . we suggest that the presence of a sizeable branching ratio for @xmath2 could significantly enhance the detection prospects of @xmath0 in the four - lepton channel . moreover , the decays @xmath5 and @xmath6 from production of the neutral triplet scalars @xmath7 and @xmath8 would also provide an additional source of @xmath9 , which can subsequently decay to @xmath0 .

1105.2209

for all thermodynamical systems , the macroscopic quantities have a fluctuation because of the statistical nature . according to the law of large numbers , the fluctuation is negligible for large system , which means the probability distribution concentrates near the expectation @xcite . but for small systems , the macroscopic quantity spreads in a wide range , which urges us to explore more on the distribution of the quantity . the probability distribution of the work done to the system under a certain process is usually referred as work function . work function , together with work fluctuation of small system have attracted much attention recently @xcite . work function also relates non - equilibrium qualities with the equilibrium ones @xcite . for example , jarzynski equality relates the non - equilibrium work @xmath0 with helmholtz free energy @xmath1 through @xmath2 . in such discussions , the work fluctuation becomes a vital issue because it gives us information about the error in the estimation of @xmath3 in practice . therefore understanding the work function @xcite , as well as suppressing the corresponding work fluctuation are very important for small systems . researchers are making significant progress on work function . some recent researches @xcite compare the work function of adiabatic and non - adiabatic process under quantum scheme . results show that adiabatic process owns smaller work fluctuation . this result is not surprising , because adiabatic process will keep the population on each state invariant , or in other words , eliminate transitions between the eigenstates of the system . however , conventional adiabatic process requires the parameter changing slowly , and due to this reason , it will take a comparatively long time period in practice . thus one of our motivations is to speed up adiabatic process . to be more precise , in quantum case , we hope to eliminate the transition between states even if the parameter changes rapidly . and in classical case , we will keep the action variable , a classical analog of quantum number invariant as time evolves . we notice that in both cases , we are trying to accomplish a transitionless feature . based on the previous works of transitionless driving @xcite , we develop a method to achieved this goal in both quantum and classical cases by adding a control field to the system . with this approach , the system effectively undergoes an adiabatic process in a short time period , which is definitely a powerful tool for practical purpose . based on recent works on work function and jarzynski equality , we digest deeper on this topic , and use an extra driving field to achieve the so - called fast - forward adiabatic process . in the mean time , the fast - forward adiabatic process could retain all the features of the work function and work fluctuation of conventional adiabatic process with a carefully chosen control field . one amazing result is the estimation of @xmath3 converges much faster in practice with such control field . fast - forward adiabatic process also has potential applications in technology aspect . recent research on quantum otto engine @xcite is faced with choices between efficiency and output power . in the conventional scheme , non - adiabatic cycles have smaller efficiency but larger output power , compared with adiabatic cycles . qualitatively , non - adiabatic cycles have larger work fluctuation thus might not be very efficient ; but they can be performed within arbitrarily short duration time , thus the output power could be very large . however , if we remember the previously mentioned remarkable features of our fast - forward adiabatic process , we realize that it minimizes the duration time and work fluctuation at the same time . follow the same logic , in later chapters we could see how our fast - forward adiabatic process helps the quantum engine to achieve the maximum efficiency and output power at the same time . in the rest of this report , we will first review both quantum and classical adiabatic theorem in the second chapter , followed by the formal definitions and discussions on work function and work fluctuation in the third chapter . after that , we will introduce our original work on classical fast - forward adiabatic process , including the formal solution of control field and application in 1-d harmonic oscillator . work functions of adiabatic and non - adiabatic processes will be compared in analytical and numerical manner . next , for the quantum fast - forward adiabatic process , we will follow berrys approach of transitionless driving . furthermore , we will consider its work function and compare it with quantum non - adiabatic process in a similar way . last but not least , we will show some dramatic application of our fast - forward adiabatic process , including increasing the converging speed of @xmath3 and improving the performance of quantum engine . adiabatic process plays an important role in modern quantum mechanics . because of the population - invariant nature of adiabatic process , it is widely used in quantum optics and atomic physics in both theoretical @xcite@xcite and experimental aspect @xcite . besides that , there are some very fundamental signatures of a quantum system , for example , berrys phase , can only be described and measured when the system undergoes a cyclic adiabatic process . adiabatic theorem points out one way of realizing the adiabatic process . it tells us that a system usually undergoes an adiabatic process when the parameters of the system are changing slowly . thus slowly changing the parameters becomes the most common approach to adiabatic process . such approach will be referred as conventional adiabatic process in the rest of this article . in this chapter , we will review both quantum and classical adiabatic theorem to explain why the changing rate of parameter matters . particularly for classical adiabatic theorem , before constructing fast - adiabatic process , we hope to introduce an unfamiliar tool called action - angle variables and make analog with the more familiar quantum version . we will illustrate the quantum adiabatic theorem for the system with only one time - dependent parameter . systems with more parameters are quite similar . for such system , hamiltonian is represented by @xmath4 , where @xmath5 is the time - dependent parameter . notice here we write it as @xmath6 rather than @xmath7 . this is because in later chapters we will modify @xmath6 and hope the notations to be consistent with each other . the state at time @xmath8 satisfies @xmath9 and instantaneous eigenstates of @xmath4 are given by @xmath10 the general solution of @xmath11 can be expanded using eigenstates of @xmath4 at time @xmath8 , i.e. , @xmath12 where @xmath13 is the dynamical phase of state @xmath14 at @xmath8 . plug ( [ c1generalsolution ] ) into ( [ c1shordinger ] ) , @xmath15 here we omit @xmath8 in time - dependent terms @xmath16 and @xmath17 for convenience . differentiating ( [ c1staticstate ] ) on both sides gives us @xmath18 and multiplying @xmath19 @xmath20 on the left then gives @xmath21 where @xmath22 are short hand notation for @xmath23 , and we further assume the system is non - degenerated @xmath24 . and ( [ c1coefficient ] ) becomes @xmath25 when @xmath26(compared with the level spacing @xmath27 ) , @xmath28 thus @xmath29 notice @xmath30 is purely imaginary , as @xmath31 , we immediately know @xmath32 , i.e. _ adiabatic approximation _ holds . this approximation means there is no transition between eigenstates of @xmath33 . and slowly changing parameter is the common approach . in the previous derivations , we have shown that if @xmath34 changes slowly , adiabatic approximation holds , i.e. the probability of a state falling in a certain instantaneous eigenstate of @xmath4 is constant . this implies that there is no transition between states since probability is conserved . however , if the initial state to be a mixed state @xmath35 , @xmath36 where @xmath37 and @xmath38 . if adiabatic approximation holds , it is easy to conclude that the state at time @xmath8 will be @xmath39 and the probability of falling in @xmath40 is a still constant . in other words , for an quantum ensemble undergoing quantum adiabatic process , the population on each energy level will not change as time evolves . here we would like to distinguish the above quantum adiabatic process from the semistatic adiabatic process in thermodynamics . consider a gibbs canonical ensemble with partition function @xmath41 , @xmath42 thus a quantum adiabatic process turn the state into @xmath43 meanwhile , for a gibbs ensemble undergoing a seimistatic adiabatic process , the system is always in equilibrium , hence @xmath44 where @xmath45 is the partition function when parameter @xmath34 is @xmath5 . in general , since the energy level spacing between states is not fixed , @xmath46 , and the quantum adiabatic process discussed in the adiabatic theorem is not the equilibrium thermodynamical adiabatic process . the canonical ensemble in classical adiabatic theory encounters the same problem . in the rest of this article , when we use adiabatic process , we are actually referring to the adiabatic process described by either quantum or classical adiabatic theorem , rather than the equilibrium thermodynamical one . classical adiabatic theorem is based on a special set of canonical coordinates called action - angle variables . in this section we will start with canonical transformation , followed by a brief introduction about the mathematics of the action - angle variables . we will also make analogues between action variable and quantum number to clarify the relationship between classical and quantum adiabatic theorems . given a time - dependent hamiltonian @xmath47 , let @xmath48 representing the system energy . the the action - angle is defined by @xmath49 the integral is integrated over a periodic @xmath50 . for a system has more degrees of freedom , the existence of such integral requires the system to be separable @xcite , hence action - angle variable might not exist for some systems . if action - angle variables exist , ( [ c1action ] ) implies that @xmath51 is independent of @xmath52 , @xmath53 notice that although @xmath54 and @xmath55 has the same value , they have different dependent variables , thus we use tilde to distinguish one from the other . the advantage of action - angle variable is , if the hamiltonian is time - independent ( @xmath56 ) , @xmath57 becomes our new hamiltonian under @xmath58 . remember @xmath57 is independent of @xmath52 , we have @xmath59 i.e. @xmath60 is constant as time evolves . now let us come back to the physical interpretation of @xmath60 . equation ( [ c1action ] ) reminds us of _ bohr - sommerfeld quantization _ in old quantum theory , which obeys @xmath61 where @xmath62 are quantum numbers in old quantum theory . a famous result is the quantization of angular momentum @xmath63 in bohr model , @xmath64 one important property of quantum adiabatic theorem is that an energy eigenstate remains on the corresponding instantaneous energy eigenstate of the system , or in other words , the quantum number @xmath65 is invariant . so it is quite natural to require @xmath66 in classical adiabatic theorem , when @xmath47 is changing slowly . in order to derive classical adiabatic theorem , we need to know the new hamiltonian @xmath67 when the coordinate is changed from @xmath68 to @xmath69 . according to ( [ c1independent ] ) , when @xmath54 is time independent , @xmath70 , which could greatly simplify our calculation . however , to carry out the transformation under time - dependent @xmath47 , we need to find the so - called type - ii generating function @xmath71 @xcite . here we will skip this step , and show the subsequent steps . explicit example in 1-d harmonic oscillator will be shown in the next chapter . once we get the generating function @xmath71 , the relations between coordinates @xmath68 and@xmath69 are given by @xmath72 in principle , with the above equations , we can solve @xmath68 as functions of @xmath58 , i.e. @xmath73 and @xmath74 . hamiltonian @xmath47 in @xmath68 coordinate and @xmath67 in @xmath75 coordinate are related by @xmath76\biggr|_{p = p(i,\theta,\lambda),\ q = q(i,\theta,\lambda ) } { \nonumber \\}&= & h_0(p(i,\theta,\lambda),q(i,\theta,\lambda)),\lambda ) + \left[\left ( { \partial f_2(i , q,\lambda)\over \partial t}\right)\biggr|_{i , q}\dot \lambda\right]\biggr|_{q = q(i,\theta,\lambda ) } { \nonumber \\}&= & \tilde h_0(i,\theta,\lambda ) + \left[\left ( { \partial f_2(i , q,\lambda)\over \partial t}\right)\biggr|_{i,\ q}\dot \lambda\right]\biggr|_{q = q(i,\theta,\lambda)}\end{aligned}\ ] ] from ( [ c1cyclic ] ) we know @xmath55 is independent of @xmath52 , therefore we obtain @xmath77\biggr|_{q = q(i,\theta,\lambda ) } \label{c1k}\ ] ] the dynamics of @xmath60 is given by @xmath78\biggr|_{q = q(i,\theta,\lambda))}\right)\biggr|_{i\ const } { \nonumber \\}&= & -\dot \lambda { \partial \over \partial \theta } \left(\left[{\partial f_2(i , q,\lambda)\over \partial \lambda}\dot \lambda\right]\biggr|_{q = q(i,\theta,\lambda))}\right)\biggr|_{i\ const } \label{c1idot}\end{aligned}\ ] ] here , if we cheat a little bit , we could claim that @xmath79 when @xmath80 approached 0 . in addition , we notice that if @xmath54 is time - independent , i.e. @xmath81 , equation ( [ c1idot ] ) is reduced to @xmath82 , which is consistent with ( [ c1independent ] ) . in fact , the classical adiabatic theorem requires one more step . this step is used to guarantee when @xmath83 , ( [ c1idot ] ) does not result in significant change of @xmath60 as time accumulates . to prove this , we actually require @xmath84 , where @xmath85 is the inherent angular frequency of the system . an complete deviation will be given in goldsteins textbook @xcite . work function is an important concept for small systems . it is defined as the probability distribution of the work done to the system during a certain process . one may be wondering why there is distribution of work . as we know , in general , for a thermodynamical system , the macroscopic quantities have fluctuations . however , due to the large number of particles ( @xmath86 ) in large systems , the fluctuation is negligible , and the probability distribution of the macroscopic quantities is quite close to a @xmath87-function . but the case is quite different for small systems . for example , for a canonical ensemble , the ratio between the fluctuation and average of the system energy is given @xcite by @xmath88 where @xmath89 is the particles in the system . for a system whose @xmath89 is large enough , the fluctuation is negligible . in contrary , for small systems , the macroscopic quantity spreads in a wide range . in order to describe the work precisely , we need to take work distribution and work fluctuation into consideration . this chapter mainly discusses the work function and work fluctuation . we will introduce work function under both classical and quantum scheme . in each section , we will first give the definition of work and the general form of work function . at last we will briefly introduce jarzynski equality . before defining the work function , we should make it clear what is the work done to the system in classical mechanics . in this project we will follow jarzynski approach of inclusive work @xcite@xcite . notice we always assume the system is not in contact with heat reservoir during the whole process , so the work is simply the energy difference of final and initial state . assume time @xmath8 varies from @xmath90 to @xmath91 , and hamiltonian @xmath92 is given , then in principle we could solve for the trajectory of the system , @xmath93 here the system is not restricted to 2 degrees of freedom , and @xmath94 can have arbitrary many components . given any initial condition @xmath95 of the system , during the action time @xmath91 of the process , work @xmath0 can be worked out as a function @xmath96 of @xmath95 , @xmath97 this equation is totally deterministic . for the system to have fluctuation on work @xmath0 , the initial condition must be given in the form of classical ensemble @xmath98 . @xmath98 is the probability distribution of initial condition in phase space . then the probability density @xmath99 of work satisfies @xmath100 or equivalently , @xmath101 @xmath102 is the dirac - delta function , and @xmath103 under the integral means that the integration is performed over the entire phase space @xmath103 . for quantum ensemble , the case is slightly different . let @xmath104 be the eigenenergies of the system at @xmath105 , similarly for @xmath106 . first consider a system starts with a pure state @xmath107 , there are several final states the system might fall in . the transition probability of initial state @xmath107 to final state @xmath108 is @xmath109 where @xmath110 is the time - evolution from 0 to @xmath91 . the work done is simply @xmath111 . thus for this single state @xmath107 , the work function is discretized , @xmath112\bigr).\ ] ] it is easy to verify that @xmath113 is normalized . similarly , lets now consider a quantum ensemble , i.e. a mixed state @xmath114 in order to measure the work done to the system , we need to measure the system energy at both @xmath105 and @xmath115 to get the exact system energy difference . in mathematics , this is equivalent to projecting the mixed state onto instantaneous energy eigenstates of @xmath116 . the corresponding @xmath99 is @xmath117\bigr ) . \label{c1quantumwf}\ ] ] jarzynski equality is an important equation relating non - equilibrium and equilibrium process under a fixed temperature . a brief idea of the classical theory is stated below . remember we define work in jarzynskis approach ( [ c1solution ] ) , i.e. , @xmath118 for a certain trajectory with @xmath95 as initial condition . thus for a system start with a gibbs canonical ensemble @xmath119 the expectation @xmath120 notice that @xmath121 is a canonical transformation , thus the jacobian is equal to @xmath122 , and @xmath123 where @xmath124 is the partition function when @xmath125 . furthermore , we know the relation between helmholtz free energy and partition function , @xmath126 plug into ( [ c3interjar ] ) , and jarzynski equality emerges , @xmath127 this equation is very powerful in the sense that it relates non - equilibrium quality @xmath0 with equilibrium quality @xmath1 , with regardless of work function @xmath99 . in practical aspect , if we want to measure the free energy difference between two equilibrium state , we only need to prepare the gibbs canonical ensemble , randomly pick a sample from it , and change the parameter @xmath34 from @xmath128 to @xmath129 ( might be very fast ) . we do not even need to wait for the final state turning to equilibrium . as if we repeat such non - equilibrium process for enough times and measure the work done during the process , we could estimate @xmath130 through jarzynski equality . for example , if we hope to measure the free energy increase when the length of a protein is changed @xmath131 , a conventional way is stretching the protein very slowly , so that the process can be regarded as semi - static . such process costs much time . but with jarzynski equality , we need only to stretch the protein by length @xmath131 and measure the work @xmath0 in this process . average of @xmath132 will give us estimation of @xmath130 . however , we notice that ( [ c3jar ] ) only contains information about the average , and it does not tell us information on work fluctuation . for some system , the work fluctuation might be very large , and the expectation value @xmath3 converges slowly . later we will show how fast - forward adiabatic process helps @xmath3 to converge faster , by shrinking the work fluctuation . in the introduction chapters we have briefly discussed the pros and cons of the conventional adiabatic process . conventional adiabatic process suppresses the work fluctuation significantly according to lutz @xcite . on the other hand , it takes comparatively longer time since parameter must change slowly enough . so far there is no literal discussion focused on overcoming such difficulty in classical cases . however , our pioneering work on fast - forward adiabatic process fills in this gap . it does not only boost up the speed , but also suppresses the work fluctuation compare with non - adiabatic process with the same speed . this chapter will explain the details of our original work in classical theory . this chapter is aimed to show how to guarantee action @xmath60 invariant even if the parameter changes fast . basically we will ensure this by adding a control field @xmath133 onto the original hamiltonian @xmath54 . an explicit example of 1-d classical harmonic oscillator will be shown for different processes . we will then compare their work functions , hence show how the control field @xmath133 suppresses the work fluctuation of a gibbs canonical ensemble . simulation results will also be shown at the end of the chapter . assume we have obtained the type - ii generating function @xmath134 , we can immediately get the relation between @xmath68 and @xmath58 . according to ( [ c1k ] ) and ( [ c1idot ] ) @xmath135\biggr|_{q = q(i,\theta,\lambda ) } \label{c2k}\ ] ] and @xmath136\biggr|_{q = q(i,\theta,\lambda))}\right)\biggr|_i\approx 0 , \label{c2idot}\ ] ] when @xmath137 adiabatic approximation holds . however , if @xmath138 is not negligible , ( [ c2idot ] ) is the only term term which might change the value of @xmath60 . this is resulted from the @xmath52 dependence of the second term in ( [ c2k ] ) . our method of speeding up the adiabatic process ( keep @xmath60 constant ) is very straight forward . we will add a control field @xmath139 to @xmath140 , such that the new hamiltonian @xmath141 is @xmath52 independent , which leads to @xmath142 one obvious solution is @xmath143\biggr|_{q = q(i,\theta,\lambda)},\ ] ] so that @xmath144 which is @xmath52 independent . in explicit examples , if necessary , we could transform it back to @xmath68 coordinate to get a more familiar physical picture . since we are much concerned about the work function , it is necessary for us to specify which ensemble we are working with . we will use gibbs canonical ensemble as our start point , because it is the most natural ensemble to deal with and also simple to prepare in experiment . then for this classical ensemble , the partition function at @xmath105 is @xmath145 where @xmath146 is the conventional inverse temperature . distribution of initial momentum @xmath147 and position @xmath148 is @xmath149 by ( [ c3classwork ] ) , the work function is @xmath150 in this section we will show the results for 1-d classical harmonic oscillator . the original hamiltonian @xmath54 is given by @xmath151 where @xmath152 plays the role of @xmath5 . we will first derive @xmath133 and work function under a certain @xmath152 . we will then give the work function under a process without @xmath133 . adiabatic and sudden limit will also be given for comparison . first calculate action @xmath60 . follow ( [ c1action ] ) and ( [ c1cyclic ] ) , let @xmath153 , @xmath154 or simply @xmath155 as illustrated previously , we also need to find the type - ii generating function @xmath134 . as ( [ c1generate ] ) indicates , @xmath156 thus @xmath157 and @xmath52 is given by @xmath158 or @xmath159 by ( [ c1k ] ) @xmath160\biggr|_{q = q(i,\theta,\omega ) } { \nonumber \\}&=&\omega i + \dot \omega\left [ \int { \partial \over \partial \omega}\sqrt{2m\omega i - m^2\omega^2 q^2}dq\right ] \biggr|_{q = q(i,\theta,\omega ) } { \nonumber \\}&=&\omega i + \dot \omega\left[\int { ( m i -m^2 \omega q^2)dq\over \sqrt{2m\omega i - m^2\omega^2 q^2}}\right ] \biggr|_{q = q(i,\theta,\omega ) } { \nonumber \\}&=&\omega i + \dot \omega\left[\int { mi\over \sqrt{2m\omega i}}{1-{m\omega\over i } q^2 dq\over \sqrt{1 - { m\omega\over 2i } q^2}}\right ] \biggr|_{q = q(i,\theta,\omega ) } { \nonumber \\}&=&\omega i + \dot \omega\left[\int { i\over\omega}{1-{m\omega\over i } q^2 d\sqrt{m\omega\over 2i}q\over \sqrt{1 - { m\omega\over 2i } q^2}}\right ] \biggr|_{q = q(i,\theta,\omega ) } \nonumber\end{aligned}\ ] ] substitute @xmath161 with @xmath162 , @xmath163 \biggr|_{q = q(i,\theta,\omega ) } { \nonumber \\}&=&\omega i+{\dot \omega i\over \omega}\left[\int{(1 - 2\sin^2s ) d s}\right ] \biggr|_{q = q(i,\theta,\omega ) } { \nonumber \\}&=&\omega i+{\dot \omega i\over \omega}\left[\int{\cos(2s ) d s}\right ] \biggr|_{q = q(i,\theta,\omega ) } { \nonumber \\}&=&\omega i+{\dot \omega i\over 2\omega}\left[\sin ( 2\arcsin(\sqrt{m\omega\over 2i}q ) ) \right ] \biggr|_{q = q(i,\theta,\omega ) } { \nonumber \\}({\rm plug\ in \ } q = \sqrt{2i\over m\omega}\sin\theta)\ \ \ & = & \omega i+{\dot \omega i\over 2\omega}\sin(2\theta ) . \label{c2kfinal}\end{aligned}\ ] ] thus an obvious control field @xmath139 to make @xmath164 @xmath52 independent is @xmath165 and as we know , @xmath166 so control field in @xmath68 coordinate is @xmath167 before working out the work function @xmath99 explicitly , in order to compare the fast - forward adiabatic process with other processes , we hope to choose our @xmath152 such that @xmath168 reduces to @xmath54 at the start ( @xmath169 ) and the end ( @xmath170 ) of the process , i.e. @xmath171 . we will always assume this requirement is satisfied . with such @xmath85 , the canonical ensemble @xmath35 ( or , our initial probability distribution of @xmath95 ) is given by @xmath172 which implies @xmath173 here @xmath174 is the jacobian matrix , and its determinate equals to @xmath122 since the transformation is canonical . and partition function at @xmath169 is @xmath175 since @xmath176 at @xmath177 , the work done to the system is @xmath178 although @xmath133 vanished at both end , it still does work to the system . so here the work is the total work of both @xmath54 and @xmath133 . next , plug ( [ c2workfunction1 ] ) , ( [ c2workfunction2 ] ) and ( [ c2workfunction3 ] ) into ( [ c2classicalwf ] ) , @xmath179 which is an exponential distribution . both the expectation and standard deviation are @xmath180 . as we know , conventional adiabatic process has infinitely long duration @xmath91 as the parameter is changing infinitely slow . we now consider finite - time process with arbitrary finite duration @xmath91 . by adiabatic theorem , such finite - time process can be reduced to conventional adiabatic process as if we choose a sufficiently long duration @xmath91 . first of all , although @xmath133 is not used in this section , to compare work with fast forward adiabatic process , we hope @xmath152 could lead to @xmath181 . such that @xmath182 i.e. the definition work is consistent . one way of ensuring this is @xmath183 where @xmath184 is a real number and @xmath65 is an integer . it is easy to verify that @xmath185 , which implies @xmath171 . in the context , if not specify we will choose @xmath186 for convenience . the advantage of choosing @xmath186 is that @xmath187 actually @xmath184 stands for factor of @xmath85 being increased . under such realization of @xmath152 , the dynamics is @xmath188q,\label{c2pdot } \\ \dot q & = & { \partial h_0 \over \partial p } = { p\over m } . \label{c2qdot}\end{aligned}\ ] ] differentiate ( [ c2qdot ] ) with respect to @xmath8 and plug it into ( [ c2pdot ] ) @xmath189q(t ) = 0 . \label{c2diffeqn1}\ ] ] let @xmath190 , or @xmath191 we get @xmath192q(x ) = 0,\ ] ] or @xmath193q(x ) = 0.\ ] ] then define @xmath194 we get @xmath195q(x ) = 0 , \label{c2diffeqn2}\ ] ] which is exactly _ mathieus differential equation_. the independent solutions are called _ mathieu sin _ and _ mathieu cos _ , which satisfies @xmath196 here @xmath197 denotes the derivative with respect to @xmath198 . the solution with initial condition @xmath95 is @xmath199 now lets consider the work done to the system . we would like to emphasize that the following arguments hold in general , here we simply use mathieu function as an example . let @xmath16 and @xmath200 be solutions of 1-d harmonic oscillator with time - dependent @xmath152 . @xmath201 and @xmath202 satisfies @xmath203 since in our example we have chosen our @xmath152 , so the corresponding solutions are @xmath204 and solutions with initial condition @xmath95 are @xmath205 like ( [ c2pq ] ) indicates . consider ( [ c1solution ] ) , since we are not considering @xmath133 , @xmath206 is our full hamiltonian , @xmath207 ^ 2+{m \omega_{f}^2\over 2}\left[q_0 c(\tau ) + { p_0\over m } s(\tau)\right]^2 { \nonumber \\}&\ & -{1\over 2m}p_0 ^ 2 - { m \omega_0 ^ 2\over 2}q_0 ^ 2 { \nonumber \\}&=&k { \beta\over 2m}p_0 ^ 2 + l { \beta m\omega_0 ^ 2\over 2}q_0 ^ 2 + m\beta\omega_0 p_0 q_0 , \label{c2workdone}\end{aligned}\ ] ] where @xmath208 is the conventional inverse temperature , and @xmath209 } , { \nonumber \\}l & \equiv&{1\over \beta } { \left[{\dot c^2(\tau)\over \omega_0 ^ 2 } + { \omega_{final}^2 \over \omega_0 ^ 2}c^2(\tau)-1 \right ] } , { \nonumber \\}m & \equiv&{1\over \beta \omega_0 } { \left[\dot c(\tau ) \dot s(\tau)+ \omega_{f}^2 c(\tau)s(\tau ) \right]}.\end{aligned}\ ] ] or in canonical quadratic form , @xmath210 where @xmath211 , @xmath63 , @xmath212 consist of @xmath213 , @xmath214 , @xmath215 , @xmath216 and system constants , therefore @xmath211 , @xmath63 , @xmath212 are independent of @xmath217 . in our case they contain mathieu functions , in a more general case we only need to replace the mathieu function with other special functions , and could always express @xmath96 as a canonical quadratic form of @xmath95 . because the matrix consists of @xmath211 , @xmath63 and @xmath212 is symmetric , there exists an orthonormal matrix @xmath218 , s.t . @xmath219 if we define @xmath220 @xmath96 can be expressed as @xmath221 doing such complicated transformation helps working out our work function . notice that @xmath218 is orthonormal , thus @xmath222 and work function is given by @xmath223 here we use the result @xmath224 , and additional @xmath225 factor is the jacobian from @xmath95 to @xmath226 . to finish the following calculation , we will further assume that matrix consists of @xmath211 , @xmath63 and @xmath212 is positive - definite when @xmath227 , i.e. its two eigenvalues @xmath228 . because of this , we can alway make the following transformation @xmath229 with @xmath230 and @xmath231 . easy to calculate the jacobian from @xmath232 to @xmath233 is @xmath234 , thus @xmath235}d\phi { \nonumber \\}&=&\exp{\left[-{\mu_++\mu_-\over 2\mu_+\mu_-}w\right]}\int_0^{4\pi}{1\over4\pi\sqrt{\mu_+\mu_-}}\exp{\left[{\mu_+-\mu_-\over 2\mu_+\mu_-}w\cos(2\phi)\right]}d2\phi { \nonumber \\}&=&4\exp{\left[-{\mu_++\mu_-\over 2\mu_+\mu_-}w\right]}\int_0^{\pi}{1\over4\pi\sqrt{\mu_+\mu_-}}\exp{\left[{\mu_+-\mu_-\over 2\mu_+\mu_-}w\cos(\phi\rq{})\right]}d\phi\rq { } { \nonumber \\}&=&{1\over\sqrt{\mu_+\mu_-}}\exp{\left[-{\mu_++\mu_-\over 2\mu_+\mu_-}w\right]}i_0\left[{\mu_+-\mu_-\over 2\mu_+\mu_-}w\right ] \ \ \ \ \ ( w\ge 0 ) , \label{c4nonadia}\end{aligned}\ ] ] where @xmath236 is the modified bessel function of the first kind , with parameter @xmath237 . and we use the formula @xmath238 this is the case for @xmath239 , and @xmath240 . if @xmath241 , @xmath99 has a similar expression with @xmath242 , except @xmath243 will be less than @xmath90 . although we have got the compact form of work function under finite - time process , it is not easy to calculate @xmath244 and higher order moment . and the algebraic relation between @xmath243 and mathieu function is quite complicated in our case . because of this , in later sections we will use numerical and simulation results to compare finite - time process with fast - forward adiabatic process . for conventional adiabatic process , @xmath245 . thus partition function at @xmath169 is @xmath246 and @xmath247 since @xmath176 at @xmath177 , the work done to the system is @xmath248 and work function is @xmath249 we notice the work function of adiabatic process is identical with ( [ c21drho ] ) , the work function of fast - forward adiabatic process . in fact , for any system , if the control field vanishes at the start and the end of the process , the work function should be identical with the conventional adiabatic process . this is due to the @xmath52-independence of @xmath55 , i.e. , the initial distribution ( gibbs canonical ensemble @xmath35 ) and work alone a specific path @xmath96 are both independent of @xmath52 . in addition , the dynamics of @xmath60 are the same in both process : @xmath250 is constant . thus the work functions are identical . now lets work on system undergoing a sudden change in @xmath85 at @xmath105 this sudden change condition is an extreme case of non - adiabatic process . since the duration of this sudden change is infinitely small , the position @xmath50 and momentum @xmath251 are not changed , i.e. @xmath252 suppose @xmath85 is increased from @xmath253 to @xmath254 , the work is @xmath255 the initial partition function @xmath256 is still @xmath257 , but this time we will express the distribution under @xmath68 coordinates , @xmath258.\ ] ] the work function is @xmath259\delta[w-{m\over 2}(\omega_{f}^2-\omega_0 ^ 2)q_0 ^ 2]dp_0dq_0 , \end{aligned}\ ] ] noting there are @xmath260 distinct root of @xmath261 , @xmath2622\times \biggr|{1\over 2\sqrt{w{m\over 2}(\omega_{f}^2-\omega_0 ^ 2)}}\biggr|dp_0 { \nonumber \\}&=&\int\limits_{\gamma}{1\over\pi}\sqrt{\beta \omega_0 ^ 2 \over w(\omega_{f}^2-\omega_0 ^ 2 ) } \sqrt{\beta \over 2m}\exp\left[{-\beta ( { p^2\over 2 m } + { \omega_0 ^ 2\over \omega_{f}^2-\omega_0 ^ 2}w)}\right ] dp_0 { \nonumber \\}&=&\sqrt{1\over\pi w } \sqrt{\beta \omega_0 ^ 2 \over ( \omega_{f}^2-\omega_0 ^ 2 ) } \exp\left[-{\beta\omega_0 ^ 2\over \omega_{f}^2-\omega_0 ^ 2}w\right ] \label{c4suddenwork}\end{aligned}\ ] ] for @xmath240 . the expectation of @xmath0 under sudden change is @xmath263 dw { \nonumber \\}&= & { 1\over\sqrt\pi } { \omega_{f}^2-\omega_0 ^ 2 \over\beta \omega_0 ^ 2 } \gamma({3\over 2 } ) { \nonumber \\}&=&{1\over 2 } { \omega_{f}^2-\omega_0 ^ 2 \over\beta \omega_0 ^ 2 } . \label{c4suddenwork}\end{aligned}\ ] ] here @xmath103 stand for gamma function , not the phase space . standard deviation of @xmath0 is calculated through @xmath264 dw { \nonumber \\}&= & { 1\over\sqrt\pi } \left({\omega_{f}^2-\omega_0 ^ 2 \over\beta \omega_0 ^ 2 } \right)^2\gamma({5\over 2 } ) { \nonumber \\}&= & { 3\over 4}\left({\omega_{f}^2-\omega_0 ^ 2 \over\beta \omega_0 ^ 2 } \right)^2,\end{aligned}\ ] ] and standard deviation @xmath265 is @xmath266 and higher order moments can be easily obtained through gamma function . although we have got the analytic solution for work function , it might be difficult to measure the work fluctuation for distributions like ( [ c4nonadia ] ) . therefore we will basically use numerical results to compare different processes . for this section , we will fix the parameter @xmath267 in our @xmath268 . remember @xmath269 we set @xmath270 , @xmath271 and @xmath272 . here @xmath91 is an important parameter . it is the total duration of our process , therefore it together with @xmath85 describes adiabaticity of the process . small @xmath273 means the process tends to be non - adiabatic , while large @xmath273 means adiabatic process . simulation follows the following steps . we first fix @xmath274 , and then randomly take samples according to @xmath275 . next we solve the corresponding differential equation numerically with initial condition @xmath276 , and then calculate the work @xmath0 . this procedure is repeated 1 million times , so that the histogram of @xmath0 will be a good estimation of work function @xmath99 . we simulate the following @xmath277 processes . process 1 . fast - forward adiabatic process with control field @xmath133 . we choose @xmath278 , such that @xmath279 . this example is aimed to show we can keep the process adiabatic even if the parameter changes rapidly . the work function should be the same with the one of conventional adiabatic process process 2 . non - adiabatic process without @xmath133 . we choose @xmath278 , and @xmath280 in finite - time process . this example will show the work function of extremely fast and non - adiabatic process . process 3 . conventional adiabatic process without @xmath133 . we choose @xmath281 , and @xmath282 in finite - time process . in this case , classical adiabatic theorem approximately holds , thus it could be regarded as conventional adiabatic process . we first compare the work function from process 1 and process 3 . this is to convince the reader that fast - forward and conventional adiabatic process share the same work function . , @xmath283 . blue line is the theoretical work function given by([c21drho ] ) . ] . blue line is the theoretical work function given by ( [ c4convenadw ] ) . ] figure [ p(w)fastad ] and [ p(w)ad ] illustrate the work function of two adiabatic processes . the blue and gray bars are histogram of @xmath0 which estimates the work function . blue lines indicate two identical theoretical work function ( [ c21drho ] ) and ( [ c4convenadw ] ) . we observe that these two distributions coincide with each other . actually , we must split them into two graph in order to distinguish them from each other . therefore we are quite confident that fast - forward adiabatic process has the identical work function with conventional adiabatic process . here comes the crucial part . we are going to compare the work function of fast - forward adiabatic process ( process 1 ) and non - adiabatic process ( process 2 ) in this section . at the end of this section , we will reach our conclusion saying fast - forward adiabatic process has smaller work fluctuation , or equivalently , our control field @xmath133 significantly suppresses the work fluctuation . . blue one is for fast - forward adiabatic process ( process 1 ) , @xmath284 . red one is for non - adiabatic process ( process 2 ) , @xmath284 . theoretical predictions are given by ( [ c21drho ] ) and ( [ c4nonadia ] ) respectively . corresponding expectation and standard deviation are listed in table [ table ] . ] .list of expectation @xmath244 and standard deviation @xmath285 from simulation of fast - forward adiabatic process and non - adiabatic process . theoretical predictions of @xmath244 and @xmath285 are also listed for comparison purpose . [ cols="^,^,^,^,^",options="header " , ] [ table2 ] from previous chapters we know two outstanding features of our fast - forward adiabatic process and control field . first one is the control field greatly suppresses the work fluctuation . second one is that it allows adiabatic process to be performed arbitrarily fast , which saves much time in practice . in this chapter we are going to discuss how these two features are applied in jarzynski equality and quantum engine . , @xmath274 . @xmath286 . red curve is for non - adiabatic process , blue one for fast - forward adiabatic process . the middle horizontal line is the theoretical value of @xmath3 , @xmath287 . ] from chapter three we know that jarzynski relates the non - equilibrium quantity work @xmath0 with the equilibrium quantity free energy @xmath1 through @xmath288 in chapter three we also discussed that jarzynski equality contains only information about the average , and does not contain information about the work fluctuation . so if the work fluctuation is very large , converging rate of @xmath3 might be very slow . in such cases , if we add the control field @xmath133 , the non - equilibrium process can be completed with smaller work fluctuation and within the same time . figure [ fig : convergingrate ] shows the comparison between simulation results of non - adiabatic process without control field or fast - forward adiabatic process . from this figure , we find that the estimated @xmath3 of fast - forward adiabatic process approached theoretical value much faster . it comes stable around the theoretical value @xmath289 after @xmath290 trajectories . however , the non - adiabatic one comes to @xmath289 at the end of simulation @xmath291 trajectories . hence @xmath133 could speed up converging of jarzyski average @xmath3 by reducing the work fluctuation . the quantum engine we are going to discuss is based on otto cycle considered by lutz @xcite . we suppose the system is a previously discussed quantum harmonic oscillator . the cycle consists of four consecutive steps as shown in [ fig : otto ] . suppose the start point @xmath292 is a gibbs canonical ensemble . isentropic compression @xmath293 . the angular frequency is increased during time @xmath294 , and the system is isolated from any heat reservoir . the time - evolution is unitary , thus the von neumann entropy is constant . notice @xmath295 is no longer an equilibrium state , thus we use @xmath296 instead of exact temperature . hot isochore @xmath297 . the angular frequency of the system is fixed , and meanwhile the system is weakly coupled with a heat reservoir at @xmath298 . thus state @xmath201 is a canonical ensemble . the relaxation time is @xmath299 . isentropic expansion @xmath300 . @xmath85 is decreased to @xmath301 during time @xmath302 while system is isolated from heat reservoir . cold isochore @xmath303 . similar to 2 . time duration is @xmath304 . the authors consider two cases under classical limit : angular frequency in 1 and 3 is changed slowly ( conventional adiabatic limit ) or fast ( sudden change limit ) , and then consider the efficiency at maximum average output for a cycle . here , for convenience we assume the oscillator is actually a classical one . there are two reasons . first , the results of both classical oscillator and quantum oscillator under classical limit turn out to be the same . it is not surprising that classical and quantum process share the same work function under classical limit @xmath305 . second , many bio - motors are actually classical engines as they are under room temperature , which will kill most quantum effects . thus it is reasonable to apply the results of classical fast - forward adiabatic process to quantum engines under classical limit . we first calculate the adiabatic limit case . average energy of state @xmath306 is @xmath307 , which is obvious . and by ( [ c21drho ] ) , the average work done in 1 is @xmath308 . thus average energy of state @xmath295 is @xmath309 . since @xmath310 , the heat received from hight - temperature reservoir is @xmath311 . we could get @xmath312 in the similar way . thus the average work done in one cycle is @xmath313 noting the total time is @xmath314 , and output @xmath315 . the tricky part is , @xmath316 are dominant , because 1 and 3 are adiabatic process . therefore minimizing total work in one cycle is actually maximizing the output . after simple mathematics , we find the maximum power occurs at @xmath317 . the efficiency at maximum output is given by @xmath318 the power adiabatic limit is very low since @xmath316 tends to infinity . the sudden limit case is quite similar . using ( [ c4suddenwork ] ) @xmath319 , and @xmath320 . then @xmath321 , and @xmath322 . this time , under sudden change limit , total time is @xmath323 as @xmath324 . we further assume @xmath323 is approximately constant for any combination of @xmath325 . this is reasonable as they are system relaxation time . so minimizing total work @xmath326 gives us @xmath327{\beta_1/\beta_2}$ ] . and efficiency at maximum output is @xmath328 the power of sudden change is much bigger than adiabatic limit , but the efficiency is less than half of @xmath329 . in previous section we show that designers of quantum engine have to choose between efficiency and work output : sudden change provides higher output while conventional adiabatic process provides higher efficiency . now lets apply our fast - forward adiabatic process to 1 and 3 . obviously the efficiency at maximum output should be the same for both conventional and fast - forward adiabatic process , since they share the same work function , expectation and fluctuation , etc . but the output can be improved up to @xmath330 because the fast - forward process can be performed arbitrarily fast . next compare it with the sudden change limit . we know the efficiency is improved more than twice , how about the output power ? for fast - forward adiabatic process , maximum power occurs at @xmath317 , the absolute value of total work is latexmath:[\[\begin{aligned } for sudden change , maximum power is at @xmath327{\beta_1/\beta_2}$ ] , latexmath:[\[\begin{aligned } which means fast - forward adiabatic process doubles both the efficiency and output if we assume @xmath323 is constant . besides the output and efficiency issue , there are other advantages of fast - forward adiabatic process . quantum engine , as the name suggests , is engine work on small scale system . due to the size of the system , the fluctuation is not negligible . large work fluctuation , especially negative work in quantum non - adiabatic work function , might lead to a fluctuated output . however , quite uniform output of a heat engine is always one important industrial requirement , and fast - forward adiabatic process suppresses the fluctuation , i.e. provides a much more uniform output . in this report , we discuss the fast - forward adiabatic process , particularly its effect in suppressing the work fluctuation in details . we review both the classical and quantum adiabatic theorems , which describe the slowly changing feature of the conventional adiabatic process . we emphasize and rigorously define the work function of small system , as well as the work fluctuation . in classical aspect , we construct our original control field under the most general condition , which could turn a non - adiabatic process into a fast - forward process . we calculate an explicit example in a time - dependent harmonic oscillator to illustrate how to construct the control field analytically . numerical and simulated results are performed in order to compare the work functions of different processes . in quantum aspect , we follow the works of berry and lutz on transitionless driving and non - adiabatic process . we make our contribution by comparing them . we propose to use control field to make a fast non - adiabatic process adiabatic , which effectively suppresses the work fluctuation . we verify our arguments again using a time - dependent harmonic oscillator example . the toy model also reveals physical intrinsics , for example , how the quantum nature affects the work function . based on our formalism and examples in fast - forward adiabatic process , we conjecture that the work fluctuation argument holds in general , which is resulted from the nature of adiabatic assumption . there are many applications of the fast - forward adiabatic processes . in this report we only briefly touch two of them . the first one is the jarzynski equality which links the thermo - average of the work function and the change of free energy . using fast - forward adiabatic processes , the equality converges much faster . the second application is a quantum engine based on an otto cycle . the application of fast - forward adiabatic processes not only maximizes the power output by speeding up the otto cycle , but also increases the efficiency of the engine . the results in this report can be easily realized in models other than the one - dimensional harmonic oscillator , for example , the two - level system . the fast - forward adiabatic processes can be also applied to quantum engines based on other cycles . k. huang , _ statistical mechanics _ ( wiley , 1987 ) , 2nd ed . c. jarzynski and d. k. wojcik , phys lett . 92,230602 ( 2004 ) . m. campisi , p. hanggi , and p. talkner , rev . 83,771 ( 2011 ) . c. jarzynsk , phys . 78.2690(1997 ) . m. campisi , p. talkner , and p. hanggi , phys . 102 , 210401 ( 2009 ) . s. deffner and e. lutz , phys . e 77 , 021128 ( 2008 ) . m. v. berry , j. phys . a : math . theor . 42 365303 ( 2009 ) . o.abah , j. rossnagel , g. jacob , s. deffner , f. schmidt - kaler , k. singer , and e , lutz , phys . 109 , 203006 ( 2012 ) . xi chen , i. lizuain , a. ruschhaupt , d. gury - odelin , and j. g. muga , phys . 105 , 123003 ( 2010 ) . xi chen , e. torrontegui , d. stefanatos , jr - shin li , and j. g. muga , phys . rev . a 84 , 043415 ( 2011 ) . m. a. bouchiat and c. bouchiat , phys . a 83 , 052126 ( 2011 ) . jiangbin gong , lecture notes of _ advanced dynamics_. h. goldstein , c. poole , and j. safko , _ classical mechanics _ ( pearson , 2002 ) , 3rd ed . k. husimi , prog . theo . phys . 9 , 381 ( 1953 ) .

adiabatic processes are important for studying the dynamics of a time - dependent system . conventionally , the adiabatic processes can only be achieved by varying the system slowly . we speed up both classical and quantum adiabatic processes by adding control protocols . in classical systems , we work out the control protocols by analyzing the classical adiabatic approximation . in quantum systems , we follow the idea of transitionless driving by berry [ _ j . phys . a : math . theor . _ * 42 * 365303 ( 2009 ) ] . such fast - forward adiabatic processes can be performed at arbitrary fast speed , and in the meanwhile reduce the work fluctuation . in both systems , we use a time - dependent harmonic oscillator model to work out explicitly the work function and the work fluctuation in three types of processes : fast - forward adiabatic processes , adiabatic processes , and non - adiabatic processes . we show the significant reduction on work fluctuation in fast - forward adiabatic process . we further illustrate how the fast - forward process improved the converging rate of the jarzynski equality between the work function and the free energy . as an application , we show that the fast - forward process not only maximizes the output power but also improve the efficiency of a quantum engine .

1305.4207

[ 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 . .

1206.6664

one of the main ingredients necessary to study few - body nuclear systems is a realistic description of the nuclear interaction . a number of nucleon - nucleon ( @xmath1 ) potentials has been determined in the recent years . they all reproduce the deuteron binding energy and fit a large set of @xmath1 scattering data below the pion - production threshold with a @xmath2/datum of about 1 . among these potentials , we will consider in the present study only the `` phenomenological '' model of ref . @xcite ( av18 ) , and a model based on chiral symmetry derived in ref . @xcite ( n3lo - idaho ) . among the many features of these two models , we note only that the av18 is a local @xmath1 potential model , with a strong short - range repulsion and tensor component , while the n3lo - idaho is a non - local @xmath1 potential model , with a softer short - range repulsion and tensor component than the av18 . as a consequence of these differences , it is interesting to test these potential models studying light nuclear systems . in these systems , a further contribution to the realistic nuclear hamiltonian model comes from the three - nucleon interaction ( tni ) . several models of tni s have been proposed . they are mainly based on the exchange of pions among the three nucleons , as the urbana ix ( uix ) tni @xcite , which will be considered in the present study . the more recent tni models studied within the chiral approach @xcite and the extension of the uix model known as the illinois tni @xcite will be considered in a near future . a second crucial ingredient in the study of light nuclear systems is the technique used to solve the @xmath0-body schrdinger equation . several methods have been developed in the past years ( see refs . @xcite for a review ) . among them , we consider in the present study the technique known as the hyperspherical harmonics ( hh ) method , which will be briefly described in the following section . in sec . [ sec : res ] , the results for the @xmath3 and @xmath4 scattering lengths will be presented and compared with the available experimental data . the nuclear wave function for an @xmath0-body system can be written as @xmath5 where @xmath6 is a suitable complete set of states , and @xmath7 is an index denoting the set of quantum numbers necessary to completely determine the basis elements . in the present work , the functions @xmath6 have been written in terms of hh functions both in configuration - space or in momentum - space @xcite . the unknown coefficients @xmath8 of eq . ( [ eq : psi ] ) are obtained applying the rayleigh - ritz ( kohn ) variational principle for the bound ( scattering ) state problem . then , the matrix elements of the different operators of the hamiltonian are calculated , working in coordinate- or in momentum - space depending on what is more convenient . thus , the problem is reduced to an eigenvalue - eigenvector problem ( system of algebraic linear equations ) , which can be solved with standard numerical techniques @xcite . the @xmath3 and @xmath4 doublet and quartet scattering lengths obtained with the non - local n3lo - idaho @xcite @xmath1 interaction , with or without the inclusion of the uix tni @xcite , are given in table 1 , and compared with the available experimental data @xcite . also shown are the results obtained with the local av18 @xcite @xmath1 interaction and the av18/uix potential model for a comparison @xcite . note that in the case of the n3lo - idaho / uix model , the parameter in front of the spin - isospin independent part of the uix tni has been rescaled by a factor of 0.384 to fit the triton binding energy . in this way , the triton , @xmath9he , and @xmath10he binding energies are 8.481 mev , 7.730 mev , and 28.534 mev , respectively . furthermore , the n3lo - idaho and n3lo - idaho / uix results shown in the table are accurate at the 10@xmath11 fm level . in fact , the convergence of the hh expansion has been tested with a procedure similar to the one used in ref . @xcite for the @xmath0=3 and 4 bound states observables . from inspection of the table we can conclude that : ( i ) both the @xmath3 and @xmath4 quartet scattering lengths are very little model - dependent . also , they are not affected by the inclusion of the tni . the trend shown by the av18 and av18/uix results has been found also in the case of the non - local n3lo - idaho and n3lo - idaho / uix potential models . ( ii ) the @xmath3 doublet scattering length is very sensitive to the choice of the @xmath1 potential model , when no tni is included . however , once the tni is included , and therefore the triton binding energy is well reproduced , @xmath12 becomes model - independent . this is a well - known feature , related to the fact that @xmath12 and the triton binding energy are linearly correlated ( the so - called phillips line @xcite ) . ( iii ) the @xmath4 doublet scattering length is positive and quite model - dependent , if only the two - nucleon interaction is included . once the tni is added , @xmath13 becomes very little and negative . some model - dependence remains , but the problem of extrapolating to zero energy the experimental results makes impossible any meaningful comparison between theory and experiment . in conclusion , the application of the hh method to treat the low - energy scattering problem using non - local @xmath1 interactions has been found successful . both @xmath3 and @xmath4 systems have been considered , with the full inclusion of the coulomb interaction , in the second case . a similar investigation for the @xmath0=4 scattering lengths has been reported in ref . . further work at higher energies is currently underway .

the structure of @xmath0=3 low - energy scattering states is described using the hyperspherical harmonics method with realistic hamiltonian models , consisting of two- and three - nucleon interactions . both coordinate and momentum space two - nucleon potential models are considered .

0712.1113

all the well - established particles can be categorized using the constituent quark model which describes light mesons as bound states of @xmath2 pairs , and baryons as bound 3-quarks states . on the other hand , high energy experiments have shown a more complicated internal structure of mesons and baryons made of a swarms of quarks , anti - quarks and gluons . it is then natural to ask wether particles with more complex configurations exists , like for example 5-quarks ( @xmath3 ) states , where the @xmath4 has different flavor than the others quarks . these states , with quark content other than @xmath2 or @xmath5 are termed as _ exotics_. + the idea of exotics has in fact been proposed since the early 70 s but the experimental signals for exotic baryons were so controversial that never rised to a level of certainty sufficient for the particle data group s tables @xcite . till , in its 1988 review the particle data group officially put the subject to sleep @xcite . + although the lack of clear evidence of exotic particles , theoretical work on this subject was continued by several authors on the basis of quark and bag models @xcite and on the skyrme model @xcite . using the latter one , praszalowicz @xcite provided the first estimate of the mass of the lightest exotic state , @xmath6 mev , and in 1997 diakonov , petrov and polyakov @xcite , in the framework of the chiral quark soliton model , predicted an antidecuplet of 5-quarks baryons , with spin and parity @xmath7 illustrated in fig . [ fig : decupletto ] . the lowest mass member is an isosinglet state , dubbed @xmath1 , with quark configuration ( @xmath8 ) giving s=+1 , with mass @xmath9 gev and width of around 15 mev . + invariant mass measured by the leps collaboration @xcite in @xmath10 events.,title="fig:",width=275 ] + invariant mass measured by the leps collaboration @xcite in @xmath10 events.,title="fig:",width=275 ] experimental evidence for a s=+1 baryon resonance with mass 1.54 gev and width less than 25 mev has been reported for the first time by the leps collaboration at spring-8 @xcite in the photoproduction on neutron bound in a carbon target . immedialely after , several other experimental groups analyzing previously obtained data , have found this exotic baryon in both his decaying channels @xmath11 and @xmath12 @xcite . the properties of the observed candidate pentaquark signals obtained studying different reactions with different experimental methods , are summarized in table [ table:1 ] . .summary table of the experimental results of the different @xmath1 experiments ( first column ) . the @xmath1 decay channels studied are reported in the second column ; mass , width and statistical significance of the measured signals in columns 3 to 5 . [ cols= " < , < , < , < , < " , ] + the g11 experiment run soon after the _ g10 _ one and finished to take data at the end of july 2004 . data were taken using a 40 cm length liquid hydrogen target and tagged photons in the enrgy range ( 0.8 - 3.8 ) gev . the new longer target , necessary to achieve the goal of this experiment , needed a new start counter detector around the target itself to improve event triggering and particle identification . under this conditions an integrated luminosity of 80 @xmath13 was achieved . the detector calibration is underway and the data quality check of the clas setup is shown in fig . [ fig : g11 ] where the @xmath14 invariant mass spectrum , based on a small fraction of the statistics , in the @xmath15 reaction clearly shows the @xmath16 peak . the reaction channels under study are : @xmath17 , @xmath18 , @xmath19 , @xmath20 , @xmath21 , @xmath22 , and @xmath23 . + while the goal of the _ g11 _ experiment is primarly to check the existence of the @xmath1 and possible excited states on a proton target , the _ super - g _ experiment will be a comprehensive study of exotic baryons from a proton target with a maximum photon energy of about 5.5 gev . due to the broad kinematic coverage for a variety of channels , it will measure spin , decay angular distributions and reaction mechanism of the produced particles . another goal of the _ super g _ experiment is to try to verify the existence of exotic cascades reported by na49 @xcite . the experiment is scheduled to run in the @xmath24 half of 2005 . + invariant mass spectrum in the @xmath15 reaction , showing the @xmath16 peak . ( preliminary clas data.),title="fig:",width=279 ] + invariant mass spectrum in the @xmath15 reaction , showing the @xmath16 peak . ( preliminary clas data.),title="fig:",width=260 ] as mantioned above , observation of other 5-quarks states belonging to the antidecuplet of fig . [ fig : decupletto ] , came from na49 @xcite which found the @xmath25 and the @xmath26 at a mass of 1.86 gev . nevertheless , up to date , no other experiments have been able to confirm these observations . + the goal of the _ eg3 _ experiment is to measure the production of pentaquark cascade states using a 5.7 gev electron beam incident on a thin deuterium target ( 0.5 cm length ) but without detecting the scattered electron . this untagged virtual photon beam is necessary to achieve sufficient sensitivity to the expected small cross sections . in this case , missing mass technique can not be used and the method requires the direct reconstruction of the cascades using their decay products . the sequence of weakly decaying daughter particles provides a powerful tool to pick out the reactions of interest . the main goal of the experiment will be to search for @xmath27 , @xmath28 and @xmath29 . other decay mode are detectable with lower sensitivity . using the available theoretical estimate for the production cross section of 10 nb , the detection of 460 @xmath30 particles is expected during a 20 day run . together with the estimation for the background levels , this represents a statistical significant result of @xmath31 . the experiment is schedule to start to take data in november 2004 . a key question in non - perturbative qcd is the structure of hadrons . the existence of baryon states beyond the minimal @xmath32 configuration is one of the open questions of strong interaction physics . while such states are not prohibited by qcd , no experimental evidence had been found until recently . the first evidence of a narrow resonance with a quark content ( @xmath33 ) and so with strangeness s=+1 , named @xmath1 , was reported by the leps collaboration . this observation has been confirmed by another nine experimental groups , with various projectiles and targets . there are however other experiments where @xmath1 has not been seen . in addition , all the signals have rather low statistical precision and there are inconsistencies in the measured masses and widths . thus , at this time the existence of a narrow pentaquark state is not fully confirmed . + the question of whether pentaquarks exist can only be solved by a second generation high statistics experiments . the clas collaboration at jafferson lab is currently pursuing this goal . high statistics searches for exotic baryons on hydrogen and deuterium target and in various final states have been started in march 2004 and will going on till next year . data for two of these experiments are already in hand and results are expected by the end of the year . 9 m. roos et al . , `` review of particle properties . particle data group '' , phys . b 111 , ( 1982 ) 1 . b.k . jennings and k. maltman , phys . rev . d 69 , ( 2004 ) 094020 ; hep - ph/0308286 . yost et al.,``review of particle properties . particle data group '' , phys . b 204 , ( 1988 ) 1 . r.l . jaffe , proc . topical conference on baryon resonances , oxford , july 1976 , slac - pub-1774 . a. manohar , nucl . b 248 , ( 1984 ) 19 . m. chemtob , nucl . b 256 , ( 1985 ) 600 . m. praszalowicz , `` workshop on skyrmions and anomalies '' , m. jezabek and m. praszalowicz editors , world scientific , 1987 , page 112 . d. diakonv , v. petrov , m.v . polyakov , z. phys . a359 , ( 1997 ) 305 . t. nakano et al . , phys . 91 , ( 2003 ) 012002 ; hep - ex/0301020 . v. barmin et al . 66 , ( 2003 ) 1715 ; hep - ex/0304040 . v. koubarovsky and s. stepanyan , proceedings of @xmath34 conference on the intersections of particle and nuclear physics ( cipanp2003 ) , new york , ny , usa , may 19 - 24 , 2003 ; hep - ex/0307088 . s. stepanyan et al . 91 , ( 2003 ) 252001 ; hep - ex/0307018 . v. koubarovsky et al . 92 , ( 2004 ) 032001 ; ibidem 92 ( 2004 ) 049902(e ) ; hep - ex/0311046 . j. barth et al . , phys . b572 , ( 2003 ) 127 ; hep - ex/0307083 . a. airapetian et al . , phys . b585 , ( 2004 ) 213 ; hep - ex/0312044 . asratyan , a.g . dolgolenko , m.a . kubantsev , phys . atom . 67 , ( 2004 ) 682 ; yad . fiz . 67 , ( 2004 ) 704 ; hep - ex/0309042 . a. aleev et al . , hep - ex/0401024 . m. abdel - bary et al . , phys . b595 , ( 2004 ) 127 ; hep - ex/0403011 . s. chekanov et al . , phys . b591 , ( 2004 ) 7 ; hep - ex/0403051 , hep - ex/0405013 . bai et al . , hep - ex/0402012 ; k.t . knoepfle et al . , hep - ex/0403020 ; m.i . adamovich et al . , hep - ex/0405042 ; c. pinkeburg hep - ex/0404001 ; babar , hep - ex/0408064 ; yu.m . antipov et al . , hep - ex/0407026 ; s. schaet et al . , phys . b 599 , ( 2004 ) 1 ; cdf , e690 , lep , focus , hypercp ( qnp2004 conference presentation , still unpublished ) . sober et al . , nucl . inst . & meth . a440 , ( 2000 ) 263 . b. mecking et al . a503/3 , ( 2003 ) 513 . k. hicks et al . , jlab proposal e03 - 113 . m. battaglieri et al . , jlab proposal e04 - 021 e.s . smith et al . , jlab proposal e04 - 010 d. weygand et al . , jlab proposal e04 - 017 c. alt et al . , phys . 92 , ( 2004 ) 042003 .

the existence of an anti - decuplet of pentaquark particles has been predicted some year ago within the chiral soliton model . in the last year , several experimental groups have reported evidence for a s=+1 baryon resonance , with mass ranging from 1.52 and 1.55 gev and width less than 25 mev , by looking at the invariant mass of the @xmath0 system . this resonance , has been identified with the lowest mass of the anti - decuplet , the @xmath1 . at the same time , there are a number of experiment , mostly at high energies , that report null results . + an overview of the experimental results so far obtained will be given here together with a review of the second generation experiments currently ongoing and planned at jefferson lab hall b.

hep-ex0409057

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

quant-ph0410136

a non - equilibrated state of matter triggered by an ultrashort light pulse is acquiring great interest from both fundamental and application points of view @xcite . the impact by the pulse can induce a variety of phenomena including coherent oscillations , ultrafast phase transitions , and femtosecond laser ablations that are applicable to micromachining @xcite , thin film growth @xcite , and clinical surgery @xcite . the pulse may drive a solid state into warm - dense matter @xcite , the extreme temperature and pressure of which mimic the conditions in the core of planets and stars @xcite . the out - of - equilibrium phenomena can be studied by femtosecond pump - and - probe methods @xcite , in which a probe pulse snapshots the sample impacted by a pump pulse . time - resolved photoemission spectroscopy ( trpes ) has become a powerful tool to investigate the non - equilibrium properties of solid state matter from an electronic perspective @xcite . in trpes , a probing pulse has the photon energy that exceeds the work function , so that it can generate photoelectrons through a one - photon process : the distribution of the photoelectrons in energy and angle carries the information of the electronic structures . nowadays , the investigations are mostly done at the pump fluence @xmath1 of @xmath310 @xmath4j / cm@xmath2 based on ti : sapphire pulsed lasers @xcite operating at 10@xmath6 - 10@xmath7 khz repetition . these include : ultrafast perturbation / melt of charge - density waves accompanying coherent oscillations @xcite ; ultrafast modification @xcite and phase transition @xcite of correlated insulators ; investigations into graphene @xcite , graphitic materials @xcite , semiconductors @xcite , and cuprate superconductors @xcite ; disclosure of novel states @xcite , dynamics @xcite , and functions @xcite in topological insulators ; and others @xcite . one of the natural pathways to understand the far - from - equilibrium phenomena is to approach them from the near - equilibrium , or mildly non - equilibrated states induced by a weak pump pulse . besides , dynamics induced by a low pump fluence can be interesting on its own @xcite . in order to discern the small variations induced by the low - fluence pump in trpes , it becomes practical to achieve higher repetition rate , or increase the number of pump - probe events per unit time , and improve the signal - to - noise ratio ( s / n ) of the dataset . note , there is a limit in improving s / n by increasing the probe intensity . when too intense , the probe pulse generates a bunch of photoelectrons that repel each other through space - charge effects , thereby resulting in an undesired broadening in their distribution . fiber lasers have emerged as a powerful high - repetition femtosecond light source @xcite . optical fibers doped with rare - earth ions can be a lasing medium @xcite , and oscillators made thereof can be mode locked @xcite to generate pulses as short as sub-30 fs @xcite at the typical repetition rate higher than @xmath020 mhz @xcite . amplification for generating high - photon - energy pulses can also be done by doped fibers @xcite as well as by others such as external cavities @xcite . the amplified pulses can be strong enough to be non - linearly converted up to 108 ev at 78 mhz @xcite . the fiber lasers thus foresee the high - repetition - rate ultrafast spectroscopy in the deep - to - extreme ultraviolet region @xcite ; already , they have been nourishing the ultrafast spectroscopy in the low - photon - energy regions such as thz @xcite and multi - photon - photoemission methods @xcite . besides , fiber lasers are compact because of the flexibility , stable due to the all - solid nature , and cost effective owing to the economics scale of the telecommunications industry . the paper describes what we believe is the first trpes apparatus based on a fiber laser system . one of the design concepts was to achieve a high repetition by utilizing yb - doped fibers . we adopted a mode - locked yb : fiber oscillator and a yb : fiber amplifier for generating 1.2-ev pump and 5.9-ev probe pulses at the 95-mhz repetition . non - linear crystals were used for generating the deep - ultraviolet probe , so that the laser system is all solid in nature . the repetition rate thus achieved is higher than the @xmath880 mhz of the ti : sapphire - based trpes apparatus @xcite . in addition , the energy resolution of 11.3 mev is comparable to 10.5 mev achieved in a 250-khz trpes apparatus @xcite which took considerations into the uncertainty principle ; that is , to our knowledge , the yb : fiber - based apparatus is the second to demonstrate the sub-20-mev resolution in trpes . we show that the combination of the high repetition and high energy resolution is advantageous in detecting the subtle spectral changes induced by a weak pump fluence . besides , the yb : fiber laser system is compact and transportable , so that it can be easily installed into an existing trpes apparatus , as we shall demonstrate herein . the paper is organized as follows . after the present introduction ( section [ intro ] ) , we describe the setup and performance of the apparatus in section [ setup ] : the yb : fiber laser system is sketched in [ layout ] ; the energy ( 11.3 mev ) and time resolutions ( @xmath9310 fs ) are described in [ spacemirror ] and [ timer ] , respectively . the mitigation of the space - charge effects is also described in [ spacemirror ] . in section [ trpes ] , we present trpes datasets of bismuth ( bi ) : trpes at @xmath1 as small as 30 nj / mm@xmath10 is demonstrated in [ polybi ] for a polycrystalline bi film ; in [ bifilm ] , we show the results of a highly - oriented bi film grown on graphite , the pump - and - probe response of which turned out to be useful for cross - correlating the pump and probe beams in real time even at the low pump fluence . finally , discussions are presented in section [ conclusion ] . in appendix [ appendix_spacemirror ] , we present discussions of the mirror - charge effects that manifest when the fluence of the probe is low ; in appendix [ appendix_bi ] , we describe the method to fabricate the highly - oriented bi thin films . the layout of the trpes setup is sketched in figure [ fig_setup ] . the laser system consists of an oscillator , an amplifier , a compressor , a frequency converter , and a section where the delay between the pump and probe pulses is controlled . schematic of the 95-mhz yb : fiber laser based trpes apparatus . ld : laser diode ; wdm : wavelength division multiplexer ; pbs : polarized beam splitter ; dm : dichroic mirror ; smf : single - mode fiber ; dcf : double - clad fiber ; trg : transmission gratings ; @xmath11/2 and @xmath11/4 : half and quarter wave plates , respectively.,width=294 ] the 95-mhz oscillator is a ring cavity composed of a yb - doped single - mode fiber pumped by a 976-nm diode laser @xcite . mode locking occurs passively due to the non - linear polarization evolution . a pair of gratings optimizes the group - delay dispersion in the cavity . the oscillator delivers pulses with a central wavelength of 1062 nm , a bandwidth of 22 nm , and an average power of @xmath020 mw . the seeding pulses are amplified in a yb - doped double - clad fiber pumped by a 976-nm diode laser . note , the fiber amplification is done without any reduction in the repetition rate , which is contrasted to the regenerative amplifications often adopted in ti : sapphire laser systems . we amplify the seed up to @xmath03 w , or up to @xmath030 nj / pulse , in order to generate sufficiently high flux of the fifth - harmonics probe ; see later . the amplified output is then compressed by using a pair of transmission gratings . the amplified - and - compressed pulse was characterized by using the second harmonic generation ( shg ) frequency - resolved optical gating method ( frog ) . the left and right panels in figure [ fig_frog](a ) display the experimental and retrieved shg - frog spectrograms , respectively ; the latter was generated by using a standard frog algorithm @xcite . the retrieved intensity and phase of the pulse in time and frequency domains are shown in figures [ fig_frog](b ) and [ fig_frog](c ) , respectively . the full width at half the maximum ( fwhm ) of the pulse is 140 fs , as estimated through fitting the profile by a gaussian function [ figure [ fig_frog](b ) ] , and the central wavelength is 1040 nm [ figure [ fig_frog](c ) ] . laser pulse characterized by the shg - frog method . ( a ) experimental ( left ) and typical retrieved ( right ) shg - frog spectrograms . the frog error @xcite of the retrieved image was 5.34@xmath1210@xmath13 . the retrieved intensity and phase of the pulse in time ( b ) and frequency ( c ) domains . , width=294 ] the frequency up - conversion of the fundamental @xmath14 into the fifth harmonics 5@xmath14 is done by using two @xmath15-bab@xmath16o@xmath17 ( bbo ) and a lib@xmath18o@xmath19 ( lbo ) non - linear crystals . @xmath20 is done by the lbo type i , and @xmath21 is done by the bbo type i. finally , 4@xmath14 and @xmath14 ( remainder after the first bbo ) are mixed at the second bbo type i. the output of 5@xmath14 is as high as 0.3 mw . the thicknesses of the lbo , first bbo , and second bbo crystals are 2.0 , 0.1 , and 0.1 mm , respectively . the focal lengths of the lens before the lbo and first bbo are both 30 mm , while those focusing the 4@xmath14 and @xmath14 onto the second bbo are 75 and 100 mm , respectively . the beam after the second bbo is split into 5@xmath14 ( probe ) and @xmath14 ( pump ) by using a dichroic mirror , and the two beams are sent into the pump - and - probe delay section . the yb : fiber laser system is docked to an ultra - high vacuum ( @xmath32 @xmath12 10@xmath22 torr ) chamber equipped with a hemispherical electron analyzer that has an acceptance angle of @xmath2318@xmath24 along the entrance - slit direction ( scienta - omicron , r4000 ) . the analyzer chamber is shared by a trpes apparatus based on a ti : sapphire laser system delivering 1.5-ev pump and 5.8-ev probe pulses at 250-khz repetition @xcite . because the probe beam of the yb : fiber and ti : sapphire systems happens to fall into the same photon energy region , the probe beam line is also shared by both systems . the arrival of the pump and probe pulses at the sample position is controlled by a delay stage inserted in the pump beam line . the pump and probe beams are finally joined at a dichroic mirror , and directed into the analyzer chamber through a caf@xmath16 window . the spot sizes of the pump and probe beams at the sample position can be measured and tuned by using a pin hole located inside the analyzer chamber @xcite . the oscillator is enclosed in a box having a footprint of 20 @xmath12 20 cm@xmath10 . the amplifier , compressor , frequency converter , and part of the delay stage section are compacted on a board of 45 @xmath12 60 cm@xmath10 . the fiber laser system was developed and characterized on a vibration - free table and then transferred to a table next to the analyzer chamber ; that is , the desk - top - sized laser system is transportable . the space - charge broadening is an unwanted effect in photoemission spectroscopy , and has to be taken care of particularly when the probe light is pulsed . the flux of the probe beam has to be lowered until the spectral broadening becomes sufficiently small . here , taking advantage that the apparatus is equipped with both 95-mhz yb : fiber and 250-khz ti : sapphire laser systems , we demonstrate that , when using the same amount of the probing photons per second , the 95-mhz system is less prone to the space - charge effects compared to the 250-khz system . we also describe the energy resolution of the apparatus . space charge and mirror charge effects . ( a , b ) fermi edge of gold recorded by the 250-khz ti : sapphire ( a ) and by the 95-mhz yb : fiber ( b ) laser systems at various photoemission count rates . in ( b ) , some of the spectra are vertically shifted for clarity . the diameter @xmath26 of the probe at the sample position was set to 130 @xmath4 m . ( c ) enlarged view of ( b ) . ( d ) shifts of the fermi edge as functions of the photoemission count rate . the cases for @xmath26 = 90 and 130 @xmath4 m are plotted for the 95-mhz setup . ( e ) fermi edge of gold recorded by the 95-mhz yb : fiber laser system . the fit ( see text ) shows the energy resolution of 11.3 mev . , width=325 ] when comparing the probe power dependency between the 250-khz and 95-mhz systems , it becomes important to keep the conditions other than the repetition rate as constant as possible . first of all , the probing photon energy of the fiber - based system is higher only by 0.08 ev than that of the ti : sapphire system . therefore , the variations between the two setups in the matrix element , propagation velocity of the photoelectron bunch , and spectral window between the fermi cutoff and work - function cutoff ( appendix [ appendix_spacemirror ] ) can be regarded as marginal . in both setups , the spot diameter @xmath26 at the sample position was tuned to 130 @xmath4 m by utilizing the pin hole attached next to the sample @xcite . we also quickly switched the system from ti : sapphire to yb : fiber during the measurement in order to minimize the change of the condition of the sample surface as well as that of the analyzer . the whole dataset to be presented in figure [ fig_sc ] was acquired within one day . under these settings , we regarded the photoemission count rate from the gold sample as the measure of the probe - beam power , or the number of the probing photons per second , although in arbitrary units ; see below . the data were acquired in a normal - emission geometry . the probe was @xmath27 polarized and the incidence angle was 45@xmath24 . figures [ fig_sc](a ) and [ fig_sc](b ) respectively show the fermi edge of polycrystalline gold recorded at various probe power values by the 250-khz and 95-mhz systems . we took the photoemission count rate at 50 @xmath23 15 mev below the fermi level ( @xmath28 ) as a measure of the probe power that was typically in the sub - mw range . the edge recorded by the 250-khz system [ [ fig_sc](a ) ] broadened and shifted into higher kinetic energy upon increasing the power because of the space - charge effects apparently , such a shift - and - broadening is small in the dataset recorded by the 95-mhz system [ [ fig_sc](b ) and [ fig_sc](c ) ] . the results demonstrate that the space - charge effects are less pronounced in the high - repetition 95-mhz system . the probe - power - dependent shifts of the fermi edge recorded by the 250-khz and 95-mhz systems are summarized in figure [ fig_sc](d ) : the shifts were estimated by fitting the edge by fermi - dirac functions convoluted with gaussian functions . we here note that the low probing photon energy of @xmath06 ev is advantageous for reducing the space - charge effects when the number of the photoelectrons consisting the bunch is concerned . at @xmath06 ev , only the electrons bound near the valence - band maximum can be emitted as photoelectrons , because the spectral cutoff due to the work function is located in the valence - band region ; see appendix [ appendix_spacemirror ] . therefore , the number of the valence electrons that can be photo - emitted is intrinsically small at @xmath06 ev compared to the cases for higher photon energies . qualitatively , the suppression of the space charge effects at low probing photon energy is acknowledged in time - resolved multi - photon photoelectron spectroscopy @xcite , although quantitative comparison to the present results is difficult because the photon energy of the pulses as well as other conditions is different in general . looking closer into the dataset recorded by the 95-mhz system [ [ fig_sc](c ) ] , we observe that , upon increasing the probe power , the edge initially shifts into low kinetic energies before shifting into high kinetic energies . when the spot diameter @xmath26 was reduced from 130 to 90 @xmath4 m , the magnitude of the initial shift into lower kinetic energies was increased , as shown in figure [ fig_sc](d ) . the shift into low kinetic energy upon increasing the probe power is most likely attributed to mirror - charge effects @xcite that manifested when the space - charge effects were reduced . for further discussion , see appendix [ appendix_spacemirror ] . we estimate the energy resolution of the apparatus by recording the fermi edge of gold . figure [ fig_sc](e ) shows the spectrum recorded at @xmath29 = 5 k. the width of the entrance slit was set to 0.2 mm . here , the probe power was reduced so that the space- and mirror - charge effects were small . by fitting the spectrum to a fermi - dirac function convoluted with a gaussian function , the energy resolution [ fwhm of the gaussian ] was estimated to be 11.3 @xmath23 0.3 mev . the energy resolution is comparable to that of the 250-khz ti : sapphire - based system ( 10.5 mev ) @xcite , whose design concept was to achieve a high energy resolution by considering the uncertainty principle : a compromise has to be done in the time resolution when pursuing the energy resolution . we here describe the time resolution of the apparatus . the resolution can be estimated by performing trpes on a highly oriented pyrolytic graphite ( hopg ) . we utilize the rise of the spectral intensity upon the arrival of the pump pulse , which occurs quasi - instantaneously when seen at the @xmath0100-fs resolution @xcite . the hopg sample was cleaved by the scotch - tape method ( see appendix [ appendix_bi ] ) in a vacuum chamber for sample preparation . then , the sample was directly transferred to the analyzer chamber without exposing the sample to air . the trpes dataset was recorded at room temperature . the delay stage of the pump was repetitively scanned until a sufficient s / n was achieved in the whole dataset @xcite . figure [ fig_hopg](a ) shows a logarithmic spectral intensity mapped in energy and delay plane . upon the arrival of the pump pulse , the spectral intensity is spread into the unoccupied side , and then shows recovery . in figure [ fig_hopg](b ) , we plot the variation of the intensity in the energy window [ 0.02 , 0.20 ev ] as a function of the pump - probe delay @xmath30 . by fitting the temporal profile to a two - exponential function convoluted with a gaussian as done in ref . @xcite , the time resolution ( fwhm of the gaussian ) is estimated to be 310 fs . time resolution estimated by trpes of graphite . ( a ) logarithmic spectral intensity mapped in energy and delay plane . ( b ) spectral - weight variation at [ 0.02 , 0.20 ev ] as a function of delay . the fit ( see text ) shows the time resolution of 310 fs . , width=325 ] bi has played a pivotal role in the history of solid - state physics , and also continues to attract both scientific and industrial attentions @xcite . being an exemplary semi - metal , bi is the first matter whose fermi surface was revealed experimentally @xcite . studies of the ultrafast phenomena in bi induced by femtosecond pulses shed light into coherent oscillations @xcite and ultrafast bond - breaking dynamics @xcite . in addition , bi single crystals exhibit novel metallic surface states , as revealed by angle - resolved photoemission spectroscopy ( arpes ) studies @xcite . since bi is a heavy element , valence electrons are subjected to large spin - orbit interaction that results in intriguing phenomena : ( 1 ) surface states of bi exhibit rashba - type spin splittings @xcite . ( 2 ) bi adatoms can turn the surface of an ordinary semiconductor into exotic metal that shows giant rashba splittings @xcite . ( 3 ) a bi - layer bi is a candidate of a two - dimensional topological insulator , owing to the the band inversion induced by the spin - orbit interaction @xcite . ( 4 ) a three - dimensional topological insulator was realized in bi alloyed by antimony @xcite , a materials system also well known for its high thermoelectric performance . here , we show trpes datasets of bi films recorded by the apparatus . first , we show the trpes dataset of a polycrystalline bi film at the low pump fluence of 30 nj / mm@xmath2 , and demonstrate the high s / n achieved by the high repetition rate . then , we show the time - resolved arpes ( tarpes ) dataset recorded on the 111 face of bi @xcite : the surface electrons exhibited a giant pump - and - probe response , which turned out to be useful for tuning the spatio - temporal overlap of the pump and probe beams in real time even when the pump fluence is as low as 30 nj / mm@xmath2 . a polycrystalline bi film was prepared by evaporating bi on a copper plate in vacuum . the trpes dataset was recorded at room temperature . the diameter of the pump and probe beams were 250 and @xmath9150 @xmath4 m , respectively , which were calibrated by using the pin - hole method @xcite . the pump fluence @xmath1 was set to @xmath030 nj / mm@xmath2 . the dataset were acquired in a normal - emission geometry . the probe was @xmath27 polarized and the incidence angle was 45@xmath24 . trpes of polycrystalline bi at the pump fluence as small as 30 nj / mm@xmath2 . ( a , b ) spectra recorded at @xmath31 0.56 ( a ) and @xmath32 0.56 ps ( b ) . ( c ) pump - induced variation of the spectral intensity at various energies . the energy window was set to @xmath2325 mev . fitting functions ( see text ) are overlaid . ( d ) decay time as a function of energy . the vertical bars indicate @xmath231 standard deviation , and the horizontal bars indicate the energy window of @xmath2325 mev . , width=325 ] figures [ fig_polybi](a ) and [ fig_polybi](b ) respectively show the trpes datasets of @xmath31 0.56 and @xmath32 0.56 ps . upon the arrival of the pump pulse , the intensity in the unoccupied side increases [ [ fig_polybi](a ) ] and then recovers gradually [ [ fig_polybi](b ) ] . one can discern the pump - induced variation in the spectra as small as 10@xmath33 of the intensity at @xmath28 , thanks to the high repetition rate of the yb : fiber laser enabling the high s / n ratio . figure [ fig_polybi](c ) shows pump - induced variation of the spectral intensity at various energies in the unoccupied side ( @xmath34 ) . clearly , the recovery slows on approach to @xmath28 . the recovery time @xmath35 of the carriers is estimated by fitting the temporal profile at each energy by an exponential function @xmath36 convoluted by a gaussian representing the time resolution . @xmath35 as a function of @xmath37 is plotted in figure [ fig_polybi](d ) . @xmath35 increases when @xmath38 approaches to @xmath28 . we thus succeed in detecting the energy - dependent carrier dynamics induced by the pump fluence as small as 30 nj / mm@xmath2 . here we present the tarpes dataset of bi thin film grown on a hopg substrate . vacuum - evaporated bi on the surface of hopg forms bi micro - crystals with the 111 face oriented normal to the surface @xcite . this enabled us to investigate the pump - induced response of the rashba - split surface states on bi(111 ) @xcite . for the method to fabricate the thin film , see appendix [ appendix_bi ] . tarpes of highly - oriented bi(111 ) grown on graphite . ( a ) tarpes images . the upper panels show band dispersions recorded at various delay - time values , and the lower panels show the images of the pump - induced difference . ( b , c ) spectra averaged over the emission angle of @xmath2310@xmath24 . the upper and lower panels respectively show the averaged spectra recorded at @xmath30 @xmath8 0.74 and @xmath32 0.74 ps . linear and logarithmic scales are adopted for the intensity axes in ( b ) and ( c ) , respectively . , width=325 ] tarpes of bi(111)/hopg was done at @xmath29 = 10 k at a pump fluence of @xmath39 30 nj / mm@xmath2 . the upper and lower panels of figure [ fig_bitrar](a ) respectively display tarpes images and their difference to the averaged image before pumped . in the images , dispersions of the rashba - split surface states are observed around the normal emission ( emission angle 0@xmath24 ) , which is consistent to those presented in ref . @xcite . upon the arrival of the pump pulse , the spectral intensity is spread into the unoccupied side , and then shows recovery . the spectra averaged over the emission - angle region of @xmath2310@xmath24 are shown in figures [ fig_bitrar](b ) and [ fig_bitrar](c ) ; the former ( latter ) adopts linear ( logarithmic ) scale in the intensity axis . the pump - induced variation into the unoccupied side ( figure [ fig_bitrar ] ) is surprisingly larger than that observed on the polycrystalline bi film evaporated on a copper plate ( figure [ fig_polybi ] ) . the signal appearing in the unoccupied side is easily observed in real time on the screen monitoring the multi - channel plate where the photoelectrons are detected . the large pump - and - probe response of the surface states of bi(111 ) is found to be very useful in maximizing the spatial overlap of the pump and probe beams at the sample position even when the pump fluence is as low as @xmath030 nj / mm@xmath10 ( @xmath40 @xmath03.0 @xmath4j / cm@xmath10 ) , which corresponds to the pump power of 140 mw for the 250-@xmath4 m beam diameter and 95-mhz repetition . if @xmath1 could be raised to @xmath050 @xmath4j / cm@xmath10 , which is a level easily achieved by the 250-khz ti : sapphire - based system @xcite , we could have used the response seen in the spectrum of hopg as the real - time monitor for aligning the beams @xcite . however , such a high pump fluence is not reachable by the present @xmath03-w yb : fiber laser system , because it corresponds to the pump power of @xmath412 w. moreover , such a high load would heat up the sample . while hopg is established as the reference sample for characterizing the pump and probe beams in the 250-khz system @xcite , we find bi(111)/hopg as a counterpart in the 95-mhz system . besides , b(111)/hopg is easy to fabricate and has high surface stability , as described in appendix [ appendix_bi ] : these characteristics facilitated us to adopt b(111)/hopg as the reference sample for trpes at low @xmath1 . we described the trpes apparatus based on an all - solid yb : fiber laser system delivering 1.2-ev pump and 5.9-ev probe at the 95-mhz repetition . the energy and time resolutions were 11.3 mev and 310 fs , respectively . the high repetition as well as the high energy resolution facilitated us to discern the subtle spectral changes induced by a weak pump pulse , as demonstrated by the trpes of polycrystalline bi film : the spectral intensity variation of 10@xmath33 to the intensity at @xmath28 was detected . concerning the investigations into the non - equilibrated state of solid state matter , the high repetition in the pump - and - probe method works most nicely when the pump fluence is low . if a high fluence pump was done at a high repetition , the sample would suffer from heatings and ablations . the general perspective is that the high ( low ) repetition is suited for investigating the weakly ( intensively ) pumped states . thus , the high - repetition trpes achieved by the implementation of yb : fibers complements the ti : sapphire - based trpes operating at lower repetition rates . the use of the high - repetition - rate yb : fiber laser system resulted in the improved s / n in the trpes datasets and expands the pathways to perform trpes in the very low excitation limit , the conditions of which are of major importance for the studies close to equilibrium that might reveal dynamics otherwise hidden or masked at stronger excitation conditions . the setup may be suitable for investigating novel dynamics mildly induced by a weak pump , such as the oscillations in superconductors in its superconducting phase @xcite and those in cuprates in its pseudogap phase @xcite . we also demonstrated the compatibility of the yb : fiber laser system to the ti : sapphire - based trpes . first of all , the all - solid yb : fiber laser system is compact and transportable . this enabled us to set the system next to the ti : sapphire laser system . besides , a variety of optics components can be shared by the two systems , because the photon energies of the probes are similar . moreover , the methods as well as concepts developed for operating ti : sapphire - based trpes measurements @xcite were readily applicable . these include : the pin - hole method for directing the pump and probe beams to the sample position at a controlled beam size ; the method to visualize the infrared pump and ultraviolet probe beams simultaneously ; the know - hows to stabilize the laser , temperature , and sample position ; the repetitive delay scanning during the data acquisition ; and to use not - too - short pulses to keep a fair resolution in energy . a method developed uniquely for yb : fiber trpes was to use bi(111)/hopg as a reference sample . the giant response of the surface states on bi(111 ) enabled us to cross - correlate the pump and probe pulses at the sample position in real time even when the pump fluence was as low as @xmath42 30 nj / mm@xmath2 . we hope that the descriptions would facilitate yb : fiber trpes to be an accessible option to those based on ti : sapphire lasers , and would expand the pathways for investigating the ultrafast phenomena in matter . this work was supported by photon and quantum basic research coordinated development program from mext , nedo , and jsps kakenhi no . 26800165 and no . the shift and broadening of the spectrum as a function of the probing photon flux occur due to the combination of the space- and mirror - charge effects @xcite . the effects depend on a variety of parameters such as @xcite : ( 1 ) the number of the photoelectrons @xmath43 forming a bunch ; ( 2 ) the pulse length ; ( 3 ) the size and shape of the probe beam ; ( 4 ) the energy distribution of the photoelectrons that depends on the valence band structure of the sample and photon energy of the probe ; and ( 5 ) the conductivity of the sample . concerning the shift , the space - charge and mirror - charge effects , respectively , cause an upward and downward shift of the fermi cutoff in kinetic energy ; the latter occurs because the photoelectron bunch is effectively attracted by its own mirror charge induced in the metal sample . the dominance of the downward shift of the cutoff in the low - flux region , as seen in figures [ fig_sc](c ) and [ fig_sc](d ) , can be understood by considering @xmath43 @xmath44 1 : in the limiting case , there is no space - charge effects but only the mirror - charge effects . to date , however , only upward shifts have been reported when the duration of the probing pulses are in the femtoseconds @xcite ; downward shift is reported only when the pulse duration is @xmath060 ps @xcite . thus , the observation of the downward shift resolves the mystery why the signatures of mirror - charge effects were not observed for the trpes probe in the femtoseconds . the successful observation presumably owes to the followings : ( 1 ) the photon energy of the probe is as low as 5.9 ev , so that only the electrons bound near @xmath28 can contribute to the photoelectrons ; that is , @xmath43 is intrinsically small compared to the cases when the probes have higher photon energies ; see fig . [ fig_photoneng ] . ( 2 ) the high energy resolution of @xmath920 mev and high s / n facilitated us to discern the downward shift as small as 1 mev . the observation of the downward shift in femtosecond trpes nicely connects into the regime of mutli - photon photoelectron spectroscopy where the mirror - charge effects dominate over the space - charge effects @xcite because it usually deals with very low - kinetic - energy photoelectrons generated by the low - fluence pulses . density of states , photon energy of the probe @xmath45 , and spectral cutoff due to the work function . the work - function cutoff occurs in the valence - band region when @xmath45 is low and fulfills the condition @xmath46 ( upper two panels ) . here , @xmath47 ( @xmath05 ev ) is the work function , and @xmath48 ( @xmath010 ev ) is the width of the valence band in energy . as @xmath45 is increased ( from top to bottom ) , the gray shaded energy window @xmath49 expands , and the fermi cutoff shifts away from the work - function cutoff in the valence - band spectrum . @xmath50 is the number of the valence electrons that can contribute to form the photoelectron bunch when the probing photon energy is @xmath51 . @xmath50 increases until @xmath52 15 ev is reached ( lower two panels ; note , @xmath53 ) . , width=264 ] we describe the method to obtain the highly - oriented bi(111 ) micro - crystalline thin film . vacuum - evaporated bi on cleaved surface of hopg forms micro - crystals with the 111 face oriented normal to the surface @xcite . because hopg consists of micron - sized graphite sheets randomly oriented in plane as shown in figure [ fig_bigrowth](a ) , the bi micro - crystals grown on hopg also have random in - plane orientation , as we describe below . first , we obtain a clean surface of hopg by using the scotch - tape method @xcite . the tape attached on hopg is peeled off in the vacuum chamber for sample preparation . the vacuum can be at the level of @xmath05 @xmath12 10@xmath54 torr . in figure [ fig_bigrowth](b ) , we show the snapshots taken during the removal of the tape . note , the demonstration is done in air . to our experience , the tape is compatible to ultrahigh vacuum unless baked . the scotch - tape method nicely works as long as the cleaving is done around the room temperature . at cryogenic temperatures , the tape hardens , so that it becomes less successful to have a good cleave . next , we evaporate bi on the exposed surface of hopg . the evaporator consists of an alumina crucible heated by a tungsten filament , and is equipped in the preparation chamber . the typical thickness of the film , after @xmath030-min evaporation , is more than 1000 , as judged by the color of the copper sample holder deposited by bi . the vacuum level can be of the order of @xmath05 @xmath12 10@xmath54 torr during the deposition ; that is , an ultra - high vacuum of @xmath5510 @xmath56 torr is not necessary . subsequently , the sample is transferred into the analyzer chamber without exposure to air . here , we anneal the sample following a procedure described in figure [ fig_bigrowth](c ) . the sample is heated by using a ceramic heater , whose main usage is to control the temperature of the sample during arpes measurements . in figure [ fig_bigrowth](d ) , we show arpes images monitored during the heating - and - cooling sequence . the image recorded on the as - deposited sample [ the left - most column in figure [ fig_bigrowth](d ) ] already shows bands typical to bi(111 ) surface , although some other features also exist ; see later . upon raising the temperature across @xmath0340 k , the bands in the image become sharp , indicating that the film is annealed . similar annealing occurs around 340 k in a bi film deposited on si(111)-7@xmath127 surface @xcite . we heat the sample up to 365 k to ensure a homogeneous annealing . then the sample is cooled down to 305 k. one can see that some features in the image are removed after the annealing , such as the band occurring near @xmath28 around the normal emission ( emission angle around 0@xmath57 ) . as the sample is further cooled down to 5 k , the bands in the image become even more sharp [ right - most column of figure [ fig_bigrowth](d ) ] . in figure [ fig_bigrowth](e ) , we show constant - energy maps of the angular distributions of the photoelectrons emitted from the bi film grown on hopg . a debye - ring - like distribution is observed in the maps . this shows that the film consists of bi micro - crystals whose 111 faces are oriented normal to the surface , while their orientations being random in plane . the highly oriented structure is similar to that of hopg , as shown in figure [ fig_bigrowth](a ) . the hexagonally - warped band dispersions of bi(111 ) are thus averaged out in the azimuth to result in the ring - shaped photoelectron distributions in the maps . we find that the surface of the bi film thus obtained is extraordinary stable . the bands in the arpes image were observable even after the sample had been kept in the preparation chamber for two months . even when the surface had acquired residual gas in the vacuum chamber at cryogenic temperatures , the sharp bands in the arpes image were recovered after raising the sample temperature to 300 k. however , once the sample was exposed to air , the bands were not observable any more . the high stability of the surface , besides the ease in the sample preparation , make the highly - oriented bi(111)/hopg a convenient reference sample for trpes . we use the tarpes signal of the film when searching for and fine - tuning the spatio - temporal overlap of the pump and probe pulses at the focal point of the electron analyzer , as described in sec . [ bifilm ] . 114ifxundefined [ 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 * * , ( ) http://stacks.iop.org/0034-4885/76/i=3/a=036502 [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) \doibase http://dx.doi.org/10.1063/1.1329904 [ * * , ( ) ] @noop * * , ( ) \doibase http://dx.doi.org/10.1063/1.3125049 [ * * , ( ) ] link:\doibase 10.1364/oe.19.019169 [ * * , ( ) ] \doibase http://dx.doi.org/10.1063/1.3600901 [ * * , ( ) ] \doibase http://dx.doi.org/10.1063/1.3700190 [ * * , ( ) ] \doibase http://dx.doi.org/10.1063/1.4772070 [ * * , ( ) ] http://stacks.iop.org/1347-4065/51/i=7r/a=072401 [ * * , ( ) ] \doibase [ * * , ( ) ] \doibase http://dx.doi.org/10.1016/j.elspec.2014.04.013 [ * * , ( ) ] \doibase http://dx.doi.org/10.1063/1.4903347 [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) \doibase http://dx.doi.org/10.1016/j.elspec.2015.12.002 [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1103/physrevlett.112.087402 [ * * , ( ) ] @noop * * , ( ) link:\doibase 10.1103/physrevlett.113.216401 [ * * , ( ) ] @noop * * , ( ) \doibase http://dx.doi.org/10.1063/1.4871381 [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1103/physrevlett.112.257401 [ * * , ( ) ] link:\doibase 10.1103/physrevb.92.184303 [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1103/physrevlett.108.117403 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.109.127401 [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1002/lpor.200710041 [ * * , ( ) ] link:\doibase 10.1364/josab.27.000b63 [ * * , ( ) ] @noop * * , ( ) link:\doibase 10.1103/physrevlett.7.444 [ * * , ( ) ] @noop * * ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1364/oe.16.007055 [ * * , ( ) ] @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 * * , ( ) link:\doibase 10.1103/physrevb.80.113309 [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1103/physrevlett.109.016801 [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1103/physrevb.80.245407 [ * * , ( ) ]

the paper describes a time - resolved photoemission ( trpes ) apparatus equipped with a yb - doped fiber laser system delivering 1.2-ev pump and 5.9-ev probe pulses at the repetition rate of 95 mhz . time and energy resolutions are 11.3 mev and @xmath0310 fs , respectively ; the latter is estimated by performing trpes on a highly oriented pyrolytic graphite ( hopg ) . the high repetition rate is suited for achieving high signal - to - noise ratio in trpes spectra , thereby facilitating investigations of ultrafast electronic dynamics in the low pump fluence ( @xmath1 ) region . trpes of polycrystalline bismuth ( bi ) at @xmath1 as low as 30 nj / mm@xmath2 is demonstrated . the laser source is compact and is docked to an existing trpes apparatus based on a 250-khz ti : sapphire laser system . the 95-mhz system is less prone to space - charge broadening effects compared to the 250-khz system , which we explicitly show in a systematic probe - power dependency of the fermi cutoff of polycrystalline gold . we also describe that the trpes response of an oriented bi(111)/hopg sample is useful for fine - tuning the spatial overlap of the pump and probe beams even when @xmath1 is as low as 30 nj / mm@xmath2 .

1612.03026

cepheid variables are arguably the most important stellar standard candles to calibrate the range of a few kiloparsecs out to some 30 megaparsecs on the extragalactic distance ladder . the hst key project on the extragalactic distance scale ( freedman et al . 2001 ) has used cepheid photometry in optical v and i bands in some 25 nearby resolved spiral and irregular galaxies to improve on the value of the hubble constant . there are reasons to believe that cepheids are even better standard candles when they are used in the near - infrared regime . yet , only few galaxies have to date cepheid observations in the near - infrared , and usually few variables have been observed in these galaxies which have not been sufficient to significantly improve on the distance determinations which have been made in optical bands . a recent example is the work of the hst key project team which used the nicmos camera onboard hst to obtain follow - up observations of a number of cepheids in several of the key project galaxies which had previously been surveyed for cepheids in vi passbands ( macri et al . 2001 ) . in previous papers , we have reported on the discovery of more than a hundred cepheid variables in the sculptor group spiral ngc 300 ( pietrzyski et al . 2002 ) from an optical wide - field imaging survey , and we have derived the distance to this galaxy from the vi light curves of a long - period subsample of these cepheids ( gieren et al . this latter work resulted in a more accurate determination of the distance of ngc 300 than the previous studies of freedman et al . ( 1992 , 2001 ) which only used the limited number of cepheids with sparsely sampled light curves known at the time . ngc 300 , at a distance of about 2 mpc , is a key galaxy in our ongoing _ araucaria project _ in which we seek to improve the calibration of the environmental dependences of several stellar distance indicators ( e.g. gieren et al . 2001 ) to the point that the distances to nearby , resolved galaxies can be measured to 5 percent or better . apart from cepheid variables , our group has also been studying the usefulness of blue supergiants as a spectroscopic distance indicator in this galaxy , with very encouraging preliminary results ( bresolin et al . 2002 ; kudritzki et al . recent deep hst / acs imaging in several fields in ngc 300 obtained by us in bvi filters will further allow us to obtain an improved estimate of the distance of ngc 300 from the tip of the red giant branch magnitude . it is by comparing the distances of a galaxy coming from such independent methods , and doing this for a sample of galaxies with widely different environmental properties that we hope to filter out and calibrate one or several techniques of distance measurement able to yield the desired high accuracy . while the cepheid period - luminosity ( pl ) relation in optical bandpasses is already a powerful tool for distance determination , especially when the problem of the appropriate reddening corrections is minimized by the application of wesenheit magnitudes ( e.g. fouqu et al . 2003 ; storm et al . 2004 ) ) , one can expect that the use of cepheids in the near - infrared ( nir ) will lead to the most accurate distance work with these variables . this is for three well - known reasons : first , absorption corrections in the nir are small in comparison to their sizes in optical passbands , which is important because cepheids as young stars are usually found in dusty regions in their host galaxies ; second , the intrinsic dispersion of the pl relation due to the finite width of the cepheid instability strip decreases with increasing wavelength , and is in k only about half of the intrinsic dispersion of the pl relation in the v band ; and third and very importantly , even random - phase cepheid magnitudes in a nir band can already produce a distance determination which can compete in accuracy with the one coming from optical cepheid photometry using full , well - sampled light curves . this latter fact is a consequence of the decreasing amplitudes of cepheid light curves towards the infrared . since cepheid light curves also become increasingly more symmetric and stable in their shapes from the optical towards the k band , it is possible to determine accurate corrections to single - phase nir observations to derive the mean magnitude of a cepheid in a nir band with very good accuracy . this has been demonstrated , for instance , by nikolaev et al . ( 2004 ) , and more recently by our group ( soszyski et al . 2005 ) using a different approach . since in the araucaria project our interest is to boost the accuracy of the stellar standard candles we are investigating to their highest possible levels , our prime interest with cepheid variables is to use them in the near - infrared , and to calibrate the effect environmental parameters , most importantly metallicity , might have on the cepheid nir pl relations . our strategy is therefore to find a substantial number of cepheids in each target galaxy from wide - field optical surveys ( where they are most easily discovered ) , and obtain follow - up nir photometry at one or two epochs for a selected subsample of cepheids which covers a broad range in periods to allow the construction of accurate nir pl relations . these can then be compared to the corresponding nir pl relations in the large magellanic cloud ( lmc ) which are now very well established in jhk bands by the recent work of persson et al . ngc 300 is the first galaxy of our project for which we report such nir cepheid photometry ; this work will be followed by similar studies for the other local group and sculptor group target galaxies in our project . the observations , and the reduction and calibration of the data will be described in section 2 . in section 3 we will derive the pl relations in the j and k bands from our data and use them , together with the information from the optical v and i bands , to estimate an improved reddening for ngc 300 , and to determine an improved distance to the galaxy . in section 4 we discuss our results , and in section 5 we will summarize the results of this study and draw some conclusions . we have obtained nir images in j and ks filters for 3 fields in ngc 300 with the vlt and the isaac nir camera / spectrometer of the european southern observatory on cerro paranal . the coordinates of the field centers are given in table 1 , and the location of the selected fields in the galaxy is shown in fig . the data were obtained in service mode on six nights in the period july 28-september 05 , 2003 . each 2.5 x 2.5 arcmin field was observed in jks on 2 _ different _ nights to cover two different pulsation phases for the cepheids in the fields , with the aim to improve the determination of their mean magnitudes . the pixel scale in our chosen setup was 0.148 arcsec / pix . all images were obtained under sub - arcsec seeing conditions , and for most images the seeing ranged from 0.3 - 0.5 arcsec , a crucial advantage for the photometry in the crowded fields observed . weather conditions were photometric for at least one of the two observations of a given field . jhk standard stars on the ukirt system from the list of hawarden et al . ( 2001 ) were observed on the photometric nights together with the target fields in ngc 300 to allow an accurate photometric calibration of the data . we observed our target fields in a dithering mode , limiting each individual integration in j and ks to 30 s and 15 s , respectively . the total exposure time for a given target field on a given night was 11.5 min in j and 54 min in k , long enough to reach even the fainter cepheids in our fields previously discovered in our optical survey with a s / n high enough to allow photometry with reasonably small random errors ( @xmath3 0.08 mag ) . in order to achieve an accurate sky subtraction for our frames , we regularly observed random comparison fields located slightly outside the galaxy where the stellar density was very low , compared to our target fields , using the autojitteroffset template . our fields were chosen as to contain 16 cepheid variables from the catalog of pietrzyski et al . ( 2002 ) , 14 of them with periods longer than 10 days ; the longest - period cepheid in our sample has a period of 83 days . the use of such long - period cepheids makes our distance result independent of problems due to the possible contamination of the sample with overtone pulsators ( see section 4 ) . sky subtraction and flatfielding of the frames was performed with the eclipse package . the psf model was obtained in an iterative way as described by pietrzyski , gieren and udalski ( 2002 ) . the final aperture correction for a given frame was adopted as a median from aperture corrections obtained for 10 - 20 carefully selected stars . the typical rms scatter for the aperture corrections derived in this manner was about 0.02 mag . in order to calibrate the photometry onto the standard system , aperture photometry for the standard stars was performed by choosing an aperture of 14 pixels . having relatively few standard star observations on a given photometric night , typically 5 , we adopted mean extinction coefficients for paranal as published on the eso webpage , and the color coefficients established by pietrzyski , gieren and udalski ( 2002 ) . this way we derived zero points in the j and k bands from each standard . the final zero point in a band was adopted as the mean of the zero points obtained from all standards . > from the residuals of the standard star observations , the zero point uncertainty of the photometry is estimated to be @xmath40.04 mag . in table 2 , we present the journal of calibrated j and k magnitudes of the 16 cepheids we observed . for all cepheids except cep017 , cep032 and cep116 two observations at different phases of their pulsation cycles were obtained . the v - band phases of the observations were calculated with the ephemeris given in gieren et al . ( 2004 ) , and are given in table 2 . in the last two columns of table 2 , we show the intensity mean magnitudes of the cepheids which we derived by applying a correction to the observed single - phase magnitudes . the corrections to mean magnitudes were calculated using the actual v - band phases , and the v and i light curve amplitudes of the cepheids from our previous optical work together with template light curves in j and k. a detailed description of the procedure we use to correct the single - phase j- and k - band observations of cepheid variables to obtain the corresponding mean magnitudes has been presented in soszyski et al . ( 2005 ) , and the reader is referred to that paper for details . in short , our correction procedure makes use of the known amplitudes of a cepheid in v and i to predict the correction to its mean near - infrared magnitude from an observation obtained at a known v - band phase . since for all of our present cepheids accurate periods and optical light curves have been established from our previous work , we were able to apply this model to each of the cepheids in our sample . soszyski et al . ( 2005 ) have demonstrated that our adopted procedure reproduces the mean j and k magnitudes from single - phase j and k observations of a cepheid with a typical accuracy of 0.02 - 0.03 mag if the light curves of this cepheid are accurately determined in v and i bands , and its period is well established . this accuracy is superior to that obtained from the approach of nikolaev et al . it is seen from the data in table 2 that the mean magnitudes in j and k derived from the two observations at different phases do indeed agree very well for most of the cepheids , with differences between the two independent determinations of the mean j and k magnitudes for each star which are just a few hundredths of a mag , which is in the order of the uncertainties of the single j and k measurements . for a very few cepheids in the sample , the agreement between the two independent determinations of the mean magnitude is slightly worse ; this is likely due to a larger uncertainty on their periods , and a minor quality of their v and i light curves , or to the fact that for some cepheids we have light curves only in v , but not in i ( see pietrzynski et al . 2002 ; gieren et al . 2004 ) . in table 3 , we show the final , adopted intensity mean j and k magnitudes for our ngc 300 cepheid sample , obtained from a weighted mean of the two individual determinations for each star . also given are the respective mean errors of these values which are about 0.03 mag for the brightest cepheids in the sample and increase slightly toward the fainter , shorter - period variables . the periods of the cepheids were taken from our previous optical studies . in fig . 2 , we show the j- and k - band pl relations based on the data in table 3 . in this figure , we have retained the two individual determinations of the mean j and k measurements for each cepheid ( except the 3 stars having observations at only one epoch ) , in order to demonstrate the very good agreement between the individual determinations of the mean magnitudes , which again shows the high efficiency of our correction procedure mentioned before . in fig . 3 , we plot the adopted , final mean magnitudes of the cepheids against log p. these are the final j- and k - band cepheid pl relations determined from our current data for ngc 300 . in our distance determination from these data , we omit the two shortest - period cepheids in the sample which have periods less than 10 days . the faintest cepheid , cep116 , has by far the largest uncertainty on its j and k measurements and is excluded for this reason . it could also be an overtone pulsator , given its short period of 5.98 days . the star cep104 ( p=7.73 days ) is strongly overluminous in the k band , but not so in the j band . a possible explanation for this behavior is the presence of a bright , unresolved red companion star . given the very strong ( about 1 mag ) deviation of this star from the mean k - band pl relation , far beyond the observed scatter of this relation , we prefer to exclude it from our analysis . we retain for the distance determination all other 14 stars which span a period range from 13.5 to 83.0 days , yielding an excellent period baseline for the pl fits . weighted least - squares fits to the mean j and k data of these 14 cepheids yield slopes of the pl relations of @xmath5 in j and @xmath6 in k , with 1 @xmath7 uncertainties of 0.047 and 0.044 , respectively . these values have to be compared with the slopes of the pl relations in j and k determined by persson et al . ( 2004 ) for lmc cepheids , which are @xmath8 and @xmath9 , respectively . while in the k band the agreement of the slope of the pl relation defined by the cepheids in ngc 300 and in the lmc is excellent , the agreement in the j band is less good , but there is still marginal consistency . in order to derive the distance of ngc 300 relative to the lmc , we therefore performed weighted least - squares fits to the data points forcing the slopes to the corresponding lmc values from persson et al . this yields the following results : @xmath10 @xmath11 according to hawarden et al . ( 2001 ) , the j and k magnitudes on the nicmos system , on which the persson et al . ( 2004 ) data were obtained , are fainter than the ukirt magnitudes by a color - independent and constant 0.034 @xmath0 0.004 mag in j , and by 0.015 @xmath0 0.007 mag in k. on the nicmos system , our pl zero points are therefore 24.288 in j , and 23.932 in k. using the zero points of the lmc pl relations as determined by persson et al . ( 16.336 in j , and 16.036 in k ) , we then obtain a difference of the distance moduli between ngc 300 and the lmc of 7.952 in the j band , and 7.896 in the k band . assuming an lmc distance of 18.50 mag , we find the distance modulus of ngc 300 to be 26.452 in j , and 26.396 in k. to correct these values for interstellar absorption , we adopted the reddening law of schlegel et al . ( 1998 ) which yields @xmath12=0.902 e(b - v ) , and @xmath13=0.367 e(b - v ) . while the foreground galactic reddening toward ngc 300 is only 0.025 mag ( burstein and heiles 1984 ) , there is clearly some additional dust absorption produced inside ngc 300 , visible on hst images of the galaxy . in our previous determination of the distance to ngc 300 from optical vi data , we had found evidence for an additional reddening of 0.05 mag ( gieren et al . 2004 ) by comparing the distance moduli derived for the galaxy in v , i and the reddening - independent wesenheit ( w ) band . with the distance moduli determined in the near - infrared j and k bands in this study , we can improve on this value , by fitting the relationship @xmath14 with the known values of the ratios @xmath15 of total to selective absorption in vijk bands ( 3.24 , 1.96 , 0.902 and 0.367 from schlegel et al . 1998 ) , and with the reddened distance moduli @xmath16 we have determined in vijk ( 26.670 and 26.577 in v , i ( gieren et al . 2004 ) , and 26.452 and 26.396 in j and k , this paper ) . the best - fitting relation yields @xmath17 @xmath18 from the slope and intersect of the least - squares fit , which is shown in fig . 4 . extending the wavelength coverage of our photometry to the near - infrared , we therefore find a total reddening to ngc 300 which is slightly larger than the value of 0.075 we had found from the optical study alone . our best value for the reddening - corrected distance to ngc 300 from all bands is 26.367 mag , corresponding to 1.88 mpc . its uncertainty will be discussed in the next section . the values in table 4 show that the reddening - corrected distance moduli of ngc 300 in the vijk passbands , calculated with e(b - v)=0.096 as derived above , do all agree extremely well with our adopted true distance modulus of 26.367 mag . it is worth to note that the very small dispersion of the points in fig . 4 seem to indicate that the reddening laws in ngc 300 and in the galaxy must be very similar . in this section , we discuss how different sources of random and systematic error may affect our distance result for ngc 300 . we will not discuss the probably largest single systematic uncertainty , the distance to the lmc , which is discussed in a number of recent papers , including feast ( 2003 ) , walker ( 2003 ) , and benedict et al . ( 2002 ) . in agreement with our previous work , we have adopted @xmath1 ( lmc)=18.50 . should future work prove that a different value is more appropriate , our distance result can be easily adapted to this new value . an obvious source of random uncertainty is the photometric noise on our magnitude measurements . the relatively low scatter of the data points about the mean pl relation in both j and k seen in fig . 3 , which is comparable to the scatter of the individual magnitudes of lmc cepheids around the mean relations ( see persson et al . 2004 ; their figure 3 ) , demonstrates that the random errors on our magnitude measurements are small enough to not significantly increase the observed dispersions of the pl relations in ngc 300 . from the data and their random errors given in tables 1 and 2 , including the calculation of the mean magnitudes of the cepheids from data obtained at only 2 epochs , we estimate that the effect of the random photometric errors on the derived distance modulus does not exceed @xmath0 0.03 mag . the systematic uncertainty of our photometric zero points in both passbands is estimated to be @xmath0 0.04 mag , as discussed in section 3 of this paper . one issue of concern is the distribution of the cepheids in our selected sample over the pulsational instability strip . in principle , and particularly with a small number of cepheids it could happen that all stars tend to lie close to the blue or red edge of the instability strip , rather than having positions randomly distributed across the strip . this could introduce an effect on the zero point of the derived pl relation , even if the true slope remains unaffected , which would translate directly into a corresponding systematic error on the distance modulus derived from these data . from this point of view , a larger number of cepheids in our sample would have been desirable to reduce the effect of this possibly non - homogeneous filling of the instability strip . in fig . 5 , we have plotted the distribution of the cepheids in ngc 300 over the instability strip in the v , b - v color - magnitude diagram . the full width of the instability strip on this diagram is about 0.5 mag , and it is clearly appreciated that the cepheids selected for the near - infrared followup in this paper are distributed more or less randomly over the strip , with no systematic concentration towards the blue , or the red edge . we therefore conclude that a non - homogeneous filling of the instability strip is not an important source of error in our present study , and estimate that the contribution to the total random error of our derived distance modulus from non - homogeneous filling of the instability strip for our selected cepheid sample does not exceed @xmath00.03 mag . regarding the selection of our final cepheid sample for the distance solution , we already stated why the two shortest - period cepheids were omitted from the analysis . keeping only stars with periods longer than 10 days secures a sample with reasonably small random photometric errors , and at the same time assures that there are no overtone pulsators in the sample which would tend to make our zero points systematically too bright ( and hence the distance of the galaxy too small ) . in order to investigate the sensitivity of our derived distance modulus on the chosen cutoff period , we repeated the fits retaining only the 10 cepheids with periods longer than 20 days . the effect of this change of the cutoff period on the zero points of the pl fits is less than 0.02 mag in both bands , confirming that the choice of the cutoff period is not a cause of concern in our analysis . in other words , it means that our distance result is not affected in any significant way by a lutz - kelker bias . as we have shown , our data define slopes of the pl relation in j and k which are consistent with the very accurate slopes derived by persson et al . ( 2004 ) for a sample of nearly 100 lmc cepheids with periods ranging from 3 to 100 days . the period range of the ngc 300 cepheids selected for our present study is bracketted by the lmc cepheids which provide the fiducial pl relations , so there is no reason of a concern referring to a systematic difference of the mean periods of the program cepheid sample , and of the cepheid sample from which the fiducial relations in j and k were obtained . as a matter of fact , the only truly accurate cepheid pl relations in j and k bands available at the present time , in any galaxy , are those measured in the lmc by persson et al . we note , however , that the galactic near - infrared pl relations established from the near - infrared surface brightness technique as recently revised by gieren et al . ( 2005 ) do agree very well with the lmc relations of persson et al . , and have similarly small dispersions . the 2mass data , while existing for a larger sample of lmc cepheids , are clearly less accurate because they are based on shallow single - phase observations while the persson et al . data are based on full j and k light curves constructed from data of smaller photometric noise than those of the 2mass survey . the use of the persson et al . pl relations to derive the relative distance of ngc 300 with respect to the lmc is therefore the most reasonable approach , at the present time . the effect of metal abundances on the slopes and zero points of cepheid pl relations in optical and near - infrared photometric passbands is currently a debated issue . in particular , very little _ empirical _ work has so far been carried out in the near - infrared . in optical bands , evidence has been mounting that the _ slope _ of the pl relation is independent of metallicity , particularly with the recent result of gieren et al . ( 2005 ) that the galactic cepheid pl relation derived from the infrared surface brightness technique seems to have a slope consistent with the lmc and smc pl relations determined by the ogle - ii project ( udalski et al . 1999 ; udalski 2000 ) . the effect of metallicity on the zero points of optical pl relations seems to be less well constrained , at the present time , although recent progress has been made by the work of storm et al . ( 2004 ) , and sakai et al . ( 2004 ) . in near - infrared bands , the only accurate pl relations existing to date are those measured by persson et al . ( 2004 ) for the lmc . fortunately , the work on the metal abundances of blue supergiant stars in ngc 300 by bresolin et al . ( 2002 ) , and more recently by urbaneja et al . ( 2005 ) has shown that the average metallicity of young stars in ngc 300 is @xmath2 -0.3 dex , which is very close to the average metallicity of cepheid variables in the lmc , -0.34 @xmath0 0.15 dex ( luck et al . 1998 ) . it seems therefore reasonable to assume that by using the lmc pl relations as the fiducial relations , our distance modulus result for ngc 300 should be practically independent of metallicity effects , even if future work should prove that there _ is _ a measurable effect of metallicity on the slope and/or zero point of the cepheid pl relations in j and k. we remark that the determination of the size of such metallicity effects on infrared cepheid pl relations is one of the goals of the araucaria project which we will address once we will have measured the j- and k - band cepheid pl relations for more target galaxies of different metallicities . we believe that with our approach to use the reddened distance moduli obtained in v , i , j and k together with the reddening law of schlegel et al . ( 1998 ) to constrain the total reddening towards ngc300 , including the intrinsic contribution to the reddening , we have obtained a very accurate result for the total reddening which has improved our previous estimate ( gieren et al . 2004 ) which was based on optical photometry alone . this claim is supported by the excellent agreement of the values for the absorption - corrected distance moduli which we obtain for each of the bands ( see table 4 ) . slight possible changes in the adopted reddening law would not change our results , and their agreement among the different bands , in any significant way . if we adopted as the true distance modulus of ngc 300 only the result coming from the two infrared bands , the change would be less than 0.01 mag , and therefore not significant . we believe that with our accurate measurement of cepheid mean magnitudes in both optical and near - infrared bands we are controlling the effect of reddening on our distance result to a point that any remaining effect is insignificant . in our previous paper on the distance of ngc 300 from optical photometry ( gieren et al . 2004 ) , the effect on the cepheid magnitudes due to unresolved companion stars was extensively discussed , and it was concluded that the distance determination was not significantly affected by this problem . a confirmation of this conclusion has very recently come from vi photometry of some 25 cepheids in ngc 300 obtained with the _ hubble space telescope _ and acs camera in six fields in ngc 300 , which agrees very well with the ground - based photometry of these stars ( bresolin et al . 2005 , in preparation ) . since our near - infrared images have an even better spatial resolution and are deeper than the optical images used in our previous study , and our selected fields for the near - infrared follow - up are not in the very dense central region of ngc 300 , we expect that crowding is even less a problem in the determination of the j and k magnitudes of our program cepheids than it was in the determination of their v- and i - band magnitudes . from the previous discussion , we conclude that the value of our derived true distance modulus of ngc 300 has a total random error of 0.04 mag , and a total systematic uncertainty of about 0.03 mag ( which is by @xmath19 smaller than the @xmath20.04 mag uncertainty of each of the individual photometric zero points in the four bands we have used for the calculation of the true distance modulus ) . this estimate for the systematic error does , however , not include the contribution coming from the uncertainty on the adopted lmc distance of 18.5 mag , which is clearly the dominant source of systematic error on our derived distance to ngc 300 . as our final result from vijk photometry of cepheid variables in ngc 300 , we then find a true distance modulus of ngc 300 of @xmath1=26.37 @xmath0 0.04(random ) @xmath0 0.03(systematic ) mag ( excluding the systematic uncertainty on the lmc distance , which is difficult to estimate , and scaling our distance result to an adopted lmc true distance modulus of 18.50 ) . our derived distance value is more accurate than our previous determination of @xmath1=26.43 mag which we had derived from photometry in vi bands alone . butler et al . ( 2004 ) have recently found a distance modulus for ngc 300 of 26.56 @xmath0 0.07(random)@xmath0 0.13 ( systematic ) from hst - based photometry of the i - band magnitude of the tip of the red giant branch . however , in their analysis they only used a galactic foreground reddening of 0.013 mag to calculate the absorption correction . using our improved value of e(b - v)=0.096 mag , which takes into account the intrinsic reddening in ngc 300 , the butler et al . value changes to 26.397 , in excellent agreement with the value derived from the cepheids in this paper . in the study of bresolin ( 2005 , in preparation ) , we will derive a trgb distance to ngc 300 of significantly reduced uncertainty , as compared to the butler et al . result , given that our hst i - band images are going about 2 mag deeper than those of butler , and were obtained for six acs fields . this study will provide a definitive measurement of the trgb magnitude in ngc 300 which will then be compared to our cepheid - based result . our new and definitive distance result for ngc 300 from cepheid variables is also in excellent agreement with the earlier result of madore et al . ( 1987 ) who had found a true distance modulus of 26.35 @xmath0 0.25 mag from near - infrared ( h - band ) photometry of two long - period cepheids in ngc 300 . while the revised ngc 300 cepheid distance obtained by the hst key project team ( freedman et al . 2001 ) from optical photometry of 26.53 @xmath0 0.07 mag ( also tied to a lmc distance modulus of 18.50 , as our present result in this paper ) seems at first glance in marginal disagreement with our result , this is not the case . freedman et al . used , as butler et al . ( 2004 ) in their trgb study , only the galactic foreground reddening to correct their ngc 300 distance modulus result for interstellar absorption . using our improved reddening value derived in this paper , their result changes to 26.31 @xmath0 0.07 , again in very good agreement with our result from a combination of optical and near - infrared photometry of the cepheids in ngc 300 . as a conclusion , all recent determinations of the distance of ngc 300 from cepheids and red giants agree with our determination within the combined 1 @xmath7 uncertainties if consistent reddening corrections are applied . it therefore seems that the _ relative _ distance of ngc 300 with respect to the lmc is now determined with an accuracy of about 3 percent . we have measured accurate near - infrared magnitudes in the j and k bands for 16 cepheid variables in ngc 300 with well - known periods and optical lightcurves . mean magnitudes were derived from our two - phase observations by using the correction procedure of soszyski et al . fits to the observed period - luminosity relations were made adopting the slopes derived from the lmc cepheids by persson et al . ( 2004 ) . by combining the values of the distance moduli derived in the optical vi and near - infrared jk bands , we have determined e(b - v)=0.096 @xmath0 0.006 as an accurate value for the total reddening appropriate for the ngc 300 cepheids . applying this reddening value to correct the observed distance moduli for absorption , we obtain extremely consistent values for the true distance modulus of ngc 300 from each photometric band . our result is @xmath1(ngc 300)=26.37 @xmath0 0.04 ( random ) @xmath00.03 ( systematic ) mag , or ( 1.88 @xmath0 0.05 ) mpc . the random error of this result is dominated by the photometric noise of our observations and the relatively small sample size , whereas the systematic uncertainty is dominated by the uncertainty on the photometric zero points . our systematic uncertainty estimate does not include the contribution from the distance of the lmc , which we adopted as 18.50 mag . effects due to metallicity , reddening and blending are small and do not significantly affect the accuracy of our distance measurement . our distance result is also unaffected by a lutz - kelker bias related to the chosen period cutoff in the pl relation . the insensitivity of our distance determination to ngc 300 to metallicity is a consequence of near - identical metallicities of the young stellar populations in ngc 300 and in the lmc . applying the fiducial lmc pl relations in j and k to the cepheids in more metal - poor or metal - rich systems could cause a systematic effect on the derived distance if the zero points of the pl relations in j and/or k are affected by metallicity . similar work on galaxies with a range of metallicities should produce improved constraints on this important question . such studies are underway in our araucaria project . our distance determination for ngc 300 from combined optical / near - infrared photometry of cepheid variables is in very good agreement with the value determined from h - band photometry of 2 cepheids by madore et al . ( 1987 ) , and with the more recent value of freedman et al . ( 2001 ) from optical photometry when our improved reddening is applied to correct for interstellar absorption inside ngc 300 . our result is also in agreement with the distance determined by butler et al . ( 2004 ) from i - band photometry of the trgb in ngc 300 , if the additional reddening inside ngc 300 we have determined in this paper is taken into account . it therefore appears that the distance of ngc 300 relative to the lmc is now determined with an accuracy of about @xmath0 3% , mainly due to the improved accuracy we have been able to achieve in our present work . given the generally very good agreement between the mean j and k magnitudes of the cepheids derived from the individual single - phase observations , it will be a better strategy in the future to observe each field just once , and double the number of observed fields , and therefore approximately double the number of cepheids for the determination of the pl relation of a galaxy . this will help to reduce the random error due to a non - homogeneous filling of the instability strip without significantly increasing the photometric noise on the mean magnitudes . we are grateful to the staff on paranal who conducted the observations reported in this paper in service mode , with their usual great expertise . wg , gp , and dm gratefully acknowledge financial support for this work from the chilean center for astrophysics fondap 15010003 . support from the polish kbn grant no 2p03d02123 and bst grant for warsaw university observatory is also acknowledged . ccccccccccc cep003 & 2452861.88669 & 0.544 & 17.939 & 0.022 & 2452861.79693 & 0.543 & 17.502 & 0.020 & 17.941 & 17.519 cep003 & 2452873.80867 & 0.688 & 18.069 & 0.021 & 2452873.89290 & 0.689 & 17.591 & 0.018 & 18.036 & 17.572 cep004 & 2452865.79330 & 0.766 & 18.382 & 0.034 & 2452865.88891 & 0.767 & 17.940 & 0.030 & 18.307 & 17.875 cep004 & 2452887.77461 & 0.061 & 18.280 & 0.010 & 2452887.68768 & 0.060 & 17.807 & 0.022 & 18.322 & 17.819 cep007 & 2452848.92857 & 0.938 & 19.306 & 0.022 & 2452848.85179 & 0.936 & 18.868 & 0.034 & 19.205 & 18.718 cep007 & 2452860.85828 & 0.214 & 19.102 & 0.038 & 2452860.76916 & 0.212 & 18.634 & 0.033 & 19.246 & 18.747 cep008 & 2452848.92857 & 0.362 & 19.098 & 0.016 & 2452848.85179 & 0.360 & 18.615 & 0.032 & 19.126 & 18.648 cep008 & 2452860.85828 & 0.658 & 19.094 & 0.040 & 2452860.76916 & 0.656 & 18.624 & 0.033 & 19.055 & 18.600 cep011 & 2452865.79330 & 0.776 & 19.874 & 0.071 & 2452865.88891 & 0.779 & 19.157 & 0.060 & 19.608 & 18.920 cep011 & 2452887.77461 & 0.394 & 19.416 & 0.016 & 2452887.68768 & 0.392 & 18.808 & 0.038 & 19.525 & 18.978 cep015 & 2452865.79330 & 0.918 & 19.679 & 0.067 & 2452865.88891 & 0.921 & 19.309 & 0.061 & 19.495 & 19.081 cep015 & 2452887.77461 & 0.599 & 19.654 & 0.019 & 2452887.68768 & 0.596 & 19.064 & 0.042 & 19.603 & 19.097 cep017 & 2452873.80867 & 0.433 & 19.896 & 0.060 & 2452873.89290 & 0.436 & 19.373 & 0.055 & 19.944 & 19.474 cep025 & 2452848.92857 & 0.822 & & & 2452848.85179 & 0.819 & 19.438 & 0.067 & 9.999 & 19.180 cep025 & 2452860.85828 & 0.321 & 19.984 & 0.067 & 2452860.76916 & 0.317 & 19.304 & 0.057 & 20.141 & 19.488 cep028 & 2452865.79330 & 0.586 & 19.819 & 0.069 & 2452865.88891 & 0.590 & 19.243 & 0.055 & 19.788 & 19.264 cep028 & 2452887.77461 & 0.536 & 19.775 & 0.030 & 2452887.68768 & 0.533 & 19.130 & 0.058 & 19.790 & 19.200 cep030 & 2452861.88669 & 0.544 & 19.947 & 0.060 & 2452861.79693 & 0.540 & 19.396 & 0.054 & 19.947 & 19.445 cep030 & 2452873.80867 & 0.081 & 19.912 & 0.074 & 2452873.89290 & 0.085 & 19.399 & 0.063 & 19.998 & 19.424 cep032 & 2452865.79330 & 0.034 & 19.925 & 0.097 & 2452865.88891 & 0.038 & 19.912 & 0.087 & 19.827 & 19.776 cep059 & 2452861.88669 & 0.918 & 20.796 & 0.102 & 2452861.79693 & 0.912 & 20.454 & 0.087 & 20.639 & 20.287 cep059 & 2452873.80867 & 0.737 & 20.970 & 0.126 & 2452873.89290 & 0.743 & 20.544 & 0.118 & 20.823 & 20.450 cep069 & 2452861.88669 & 0.330 & 20.654 & 0.082 & 2452861.79693 & 0.323 & 20.143 & 0.076 & 20.836 & 20.327 cep069 & 2452873.80867 & 0.206 & 20.691 & 0.101 & 2452873.89290 & 0.212 & 20.194 & 0.096 & 20.920 & 20.371 cep072 & 2452861.88669 & 0.325 & 20.460 & 0.069 & 2452861.79693 & 0.318 & 19.744 & 0.062 & 20.563 & 19.847 cep072 & 2452873.80867 & 0.207 & 20.434 & 0.090 & 2452873.89290 & 0.214 & 19.781 & 0.084 & 20.562 & 19.880 cep104 & 2452848.92857 & 0.291 & 21.239 & 0.064 & 2452848.85179 & 0.281 & 20.165 & 0.083 & 21.321 & 20.243 cep104 & 2452860.85828 & 0.835 & 21.176 & 0.086 & 2452860.76916 & 0.824 & 20.028 & 0.073 & 21.050 & 19.918 cep116 & 2452861.88669 & 0.843 & 21.418 & 0.186 & 2452861.79693 & 0.828 & 21.179 & 0.172 & 21.185 & 20.972 ccccccccccc cep003 & 1.919 & 17.989 & 0.026 & 17.545 & 0.025 cep004 & 1.872 & 18.314 & 0.028 & 17.847 & 0.028 cep007 & 1.635 & 19.225 & 0.031 & 18.733 & 0.032 cep008 & 1.605 & 19.090 & 0.030 & 18.624 & 0.031 cep011 & 1.551 & 19.566 & 0.042 & 18.949 & 0.041 cep015 & 1.509 & 19.549 & 0.041 & 19.089 & 0.043 cep017 & 1.455 & 19.944 & 0.067 & 19.474 & 0.063 cep025 & 1.379 & 20.141 & 0.073 & 19.334 & 0.049 cep028 & 1.364 & 19.789 & 0.043 & 19.232 & 0.045 cep030 & 1.347 & 19.973 & 0.052 & 19.434 & 0.047 cep032 & 1.323 & 19.827 & 0.102 & 19.776 & 0.092 cep059 & 1.163 & 20.731 & 0.084 & 20.368 & 0.076 cep069 & 1.134 & 20.878 & 0.068 & 20.349 & 0.065 cep072 & 1.130 & 20.562 & 0.061 & 19.864 & 0.056 cep104 & 0.888 & 21.186 & 0.058 & 20.081 & 0.059 cep116 & 0.777 & 21.185 & 0.188 & 20.972 & 0.175

we have obtained deep near - infrared images in j and k filters of three fields in the sculptor galaxy ngc 300 with the eso vlt and isaac camera . for 16 cepheid variables in these fields , we have determined j and k magnitudes at two different epochs , and have derived their mean magnitudes in these bands . the slopes of the resulting period - luminosity relations are in very good agreement with the slopes of these relations measured in the lmc by persson et al . fitting the lmc slopes to our data , we have derived distance moduli in j and k. using these values together with the values derived in the optical v and i bands in our previous work , we have determined an improved total reddening for ngc 300 of e(b - v)=0.096 @xmath0 0.006 mag , which yields extremely consistent values for the absorption - corrected distance modulus of the galaxy from vijk bands . our distance result for ngc 300 from this combined optical / near infrared cepheid study is @xmath1=26.37 @xmath0 0.04 ( random ) @xmath0 0.03 ( systematic ) mag and is tied to an adopted true lmc distance modulus of 18.50 mag . both random and systematic uncertainties are dominated by photometric errors , while errors due to reddening , metallicity effects and crowding are less important . our distance determination is consistent with the earlier result from near - infrared ( h - band ) photometry of two cepheids in ngc 300 by madore et al . , but far more accurate . our distance value also agrees with the hst key project result of freedman et al . , and with the recent distance estimate for ngc 300 from butler et al . from the trgb i - band magnitude when our improved reddening is used to calculate the absorption corrections . our distance results from the different optical and near - infrared bands indicate that the reddening law in ngc 300 must be very similar to the galactic one . with the result from this work , the distance of ngc 300 relative to the lmc seems now determined with an accuracy of @xmath2 @xmath0 3 percent . the distance to this nearby sculptor galaxy is therefore now known with higher accuracy than that of most ( nearer ) local group galaxies .

astro-ph0503626

the multi - channel kondo effect@xcite has been the subject of intensive theoretical @xcite and experimental@xcite studies , which is characterized by unusual non - fermi liquid behaviors . its applications are now extended not only to standard dilute magnetic alloys , but also to quantum dots , etc . thus far , theoretical and experimental studies on the multi - channel kondo effect have been focused on a _ static _ kondo impurity , which has been related to the measurements of the specific heat , the spin susceptibility , the resistivity , etc . this naturally motivates us to address a question whether such a nontrivial phenomenon can be observed in dynamically generated situations . the photoemission and the inverse photoemission may be one of the key experiments to study non - fermi liquid behaviors , which reveal the dynamics of a single hole or electron suddenly created in the system . we here propose _ the dynamically induced multi - channel kondo effect _ , when an electron is emitted from ( or added to ) the kondo impurity by the photoemission ( inverse photoemission ) . a remarkable point is that the ground state of the system is assumed to be a completely screened kondo singlet , and non - fermi liquid properties are generated by an electron or hole suddenly created . we study low - energy critical properties of the spectrum by using the exact solution of the multi - channel kondo model @xcite combined with boundary conformal field theory ( cft)@xcite . we analyze the one - particle green function for the impurity to show its typical non - fermi liquid behavior . it is further demonstrated that this effect can be observed even in a homogeneous system without impurities . to show this explicitly , we apply the analysis to the photoemission spectrum in a quantum spin chain with spin @xmath0 . this paper is organized as follows . in 2 we briefly illustrate the idea of the dynamically induced multi - channel kondo effect , and derive low - energy scaling forms of the one - particle green function . we discuss non - fermi liquid properties in the spectrum by exactly evaluating the critical exponents . in 3 the analysis is then applied to the photoemission spectrum for a quantum spin chain . brief summary is given in 4 . we note that preliminary results on this issue have been reported in ref . let us consider the spin-@xmath1 kondo impurity which is _ completely screened _ by conduction electrons with @xmath2 channels . the impurity spin is assumed to be composed of @xmath3 electrons by the strong hund coupling . to study the core - electron photoemission spectrum , we start with spectral properties of the impurity green function , @xmath4 > \cr & & { \qquad \ } { \displaystyle { \quad } \atop { \displaystyle = g^{>}(t ) + g^{<}(t ) , } } \label{1}\end{aligned}\ ] ] where @xmath5 is the annihilation operator for one of core electrons which compose the impurity spin and t is the conventional time - ordered product . here , @xmath6 ( @xmath7 ) is the green function , which is restricted in @xmath8 . for the photoemission , we consider @xmath7 . to be specific , we discuss the case that a core electron is emitted as depicted in fig . 1 ( a ) , for which the binding energy @xmath9 ( measured from the fermi energy ) is assumed to be larger than the band width @xmath10 . then in the excited state the overscreening system is generated , which is referred to as _ the dynamically induced overscreening kondo effect_.@xcite at the low - energy regime around @xmath11 , we may express the operator as @xmath12 where @xmath13 is the corresponding boundary operator@xcite in boundary cft , which characterizes the boundary critical phenomena . it is known that the fermi - edge singularity@xcite is reformulated by the boundary operator,@xcite in which nontrivial effects for the overscreening kondo effect are incorporated in @xmath14 . we write down the one - particle green function @xmath7 as , @xmath15 on the other hand , for the inverse photoemission , an added electron is combined with the local spin @xmath1 to form higher spin @xmath16 by the strong hund - coupling , as shown in fig . 1 ( b ) . then we may write @xmath17 , where @xmath18 is the energy cost to make @xmath16 spin , and @xmath19 is another boundary operator which controls the undersreening kondo effect induced by the inverse photoemission . we have @xmath20 in order to evaluate the critical exponents , we now employ the idea of finite - size scaling@xcite in cft . the scaling form of the correlators @xmath21 and @xmath22 are given by @xmath23 in the long - time asymptotic region . according to the finite - size scaling , the boundary dimensions @xmath24 and @xmath25 are read from the lowest excitation energy @xmath26 , @xmath27 with @xmath28 , where @xmath29 corresponds to the system size of one dimension in the radial direction . we thus end up with the relevant scaling forms as @xmath30 where @xmath31 that @xmath32 represents @xmath33 and @xmath34 . in both cases , the spectral functions have power - law edge singularity due to the _ dynamically induced multi - channel kondo effect _ , which will be shown to exhibit non - fermi liquid properties . we now discuss low - energy critical properties by exactly evaluating @xmath24 and @xmath25 . to this end , we consider the multi - channel kondo model , @xmath35 where @xmath36 is the creation operator for conduction electrons with spin @xmath37 and orbital indices , @xmath38 . the exact solution of this model@xcite is expressed in terms of the bethe equations for spin rapidities @xmath39 and charge rapidities @xmath40 , @xmath41 where @xmath42 is the number of electrons and @xmath43 is the one - dimensional system size . it is assumed that the impurity with spin @xmath44 is completely screened in the ground state . then , the core - level photoemission suddenly reduces the impurity spin , thus inducing the overscreening kondo effect with @xmath45 . as for the underscreening effect induced by the inverse photoemission , the condition is replaced by @xmath46 . at zero temperature the ground - state properties are described by the @xmath47th order string solutions,@xcite @xmath48 where @xmath49 and @xmath50 which is restricted by @xmath51 . here , @xmath3 represents the number of orbitals . it is well known that a naive application of finite - size techniques based on the string hypothesis turns out to fail for the overscreening case at zero magnetic field.@xcite this difficulty comes from an improper treatment of the @xmath52 symmetry sector in terms of the string solutions . however , as long as the finite magnetic field is concerned , we can use the bethe equations to describe its critical properties . we will separately discuss the case of zero - magnetic field , by incorporating @xmath52 sector correctly.@xcite by applying standard procedures to eq.([7 ] ) , it is straightforward to exactly evaluate the lowest excitation energy in magnetic fields , for which one of the impurity electrons is assumed to be removed from the system , @xmath53 although the above finite - size correction is apparently similar to that for 1d solvable systems with a static impurity or boundaries,@xcite _ the final - state interaction _ induced by photoemission is included in the present case.@xcite thus all the features which are governed by the dynamical kondo effect can be read from this quantity . by applying the finite - size scaling in eq . ( [ 4 ] ) , the critical exponent @xmath54 in eq . ( [ 5 ] ) is now obtained as , @xmath55 where @xmath56 is the charge scattering phase shift , which is caused by a created hole as in the ordinary fermi edge singularity.@xcite this term depends on the detail of potential scattering . it is mentioned that the second term with the phase shift @xmath57 is caused by the kondo effect , which is explicitly evaluated as , @xmath58 where @xmath59 here @xmath60 and @xmath61 are determined by @xmath62 the key quantity , @xmath57 , is obtained by using wiener - hopf method , @xcite @xmath63 in fig . 2 we display the critical exponent @xmath64 for the overscreening case as a function of the magnetic field . note that the magnetic - field dependence of @xmath65 without @xmath66 is determined by the dynamical kondo effect , because the charge scattering phase shift @xmath56 does not depend on magnetic fields . particularly in weak magnetic field , @xmath67 , the obtained exponent behaves as @xmath68 it is seen that the phase shift , @xmath57 , gives rise to the anomalous magnetic - field dependence of the exponent . this non - fermi liquid behavior is characteristic of the overscreening effect.@xcite another interesting feature in the overscreening effect appears at @xmath69 . we recall here that the symmetry is enhanced from u(1 ) to su(2 ) at @xmath69 , for which the boundary dimension @xmath70 for the spin sector is analytically obtained by employing fusion rules hypothesis proposed by affleck and ludwig , @xcite @xmath71 which is a typical conformal dimension for level-@xmath3 su(2 ) kac - moody algebra . note that the critical exponent shows a discontinuity at @xmath69 , @xmath72 which is caused by the fact that @xmath52 symmetric sector is massless only at @xmath69,@xcite as already mentioned . we now move to the underscreening case induced by the inverse photoemission . the calculated critical exponent @xmath73 is shown as a function of magnetic fields in fig . 3 . for weak magnetic field , the obtained exponent @xmath74 behaves as @xmath75 which is characteristic of the underscreening system . in contrast to the overscreening case , there is no discontinuity in the exponent in this case , @xmath76 . this completes a general description of the dynamically induced kondo effect . an important point to be emphasized is that this kind of phenomenon may be observed not only for impurity systems but also for other related quantum systems , which will be explicitly discussed in the next section . we now wish to demonstrate that the dynamically induced kondo effect proposed here may be observed for gapless quantum spin systems _ which do not possess impurities_. as an example , we consider an integrable antiferromagnetic spin chain with spin @xmath0 , for which the exact solution is available even for the case with doped holes.@xcite the photoemission suddenly removes one electron from the spin system , and thus bears an _ impurity site _ with spin @xmath77 . in the final state , this impurity spin is screened by host spins , and as a result the overscreening kondo effect may be dynamically induced . it is remarkable that the induced impurity in this case can move through the lattice via the exchange interaction , and the edge singularity is thus governed by a _ mobile _ multichannel kondo impurity.@xcite let us consider the gapless @xmath78 spin chain with a mobile @xmath79 impurity as an example . although in more general situations including non - integrable models , the higher spin should be a half - odd integer , the present treatment can be straightforwardly extended to such cases . we here consider an integrable spin chain derived by the quantum inverse scattering method , @xcite @xmath80 \right ) \nonumber\end{aligned}\ ] ] where the spin @xmath81 with @xmath82 or 1/2 , and @xmath83 permutes the spin on sites @xmath84 and @xmath85 . in order to deal with the excited states when an electron is emitted from the spin chain , we write down the bethe equations for the spin-@xmath1 chain with _ one hole _ being doped , @xmath86 where @xmath87 is the number of down spins and @xmath88 are spin rapidities . note that the hole rapidity @xmath89 appears in the above equation , which characterizes a massive charge excitation suddenly created . thus , @xmath89 specifies the hole momentum @xmath90 . it is seen that the above spin - charge scattering term , which describes the final - state interaction , corresponds to the impurity term in eq . ( [ 7 ] ) . the manipulation illustrated in the previous section enables us to exactly calculate the scaling dimension for the one - particle green function via the finite - size corrections,@xcite @xmath91 the quantity @xmath92 often referred to as the dressed charge @xcite is given by @xmath93 where @xmath94 and the cut - off parameter @xmath95 is related to the magnetization @xmath96 , @xmath97 the density function @xmath98 is determined by the following integral equation , @xmath99 we stress that two key quantities @xmath100 and @xmath101 , which contain the effect of a mobile impurity , are introduced in eq . ( [ 20 ] ) , @xmath102 where @xmath103 here we have introduced the phase function , @xmath104 these quantities are alternatively represented in terms of the phase shifts @xmath105 and @xmath106 at the left and right fermi points in massless spin excitations : @xmath107 . we mention that the asymmetric phase shift @xmath101 is inherent in a _ mobile _ kondo impurity , different from a localized impurity in 2 . note that the scaling dimension @xmath108 depends on the hole momentum @xmath90 through the asymmetric phase shift . let us now discuss low - energy critical properties in the photoemission spectra . we write down the one - particle green function which depends on the momentum @xmath90 , @xmath109 with @xmath110 , where @xmath108 is the scaling dimension in eq . ( [ 20 ] ) and @xmath111 is the dispersion of the charge excitation generated by the photoemission . in fig . 4 we show the obtained critical exponent as a function of the momentum @xmath90 for the @xmath78 case . this anomalous power - law behavior and the momentum dependence of @xmath112 are caused by a suddenly induced _ mobile _ kondo impurity . as we discussed in 2 , the discontinuity of the exponent @xmath113 at @xmath114 is caused by the @xmath52 symmetric sector.@xcite . in this way , the dynamically induced kondo effect proposed here may be expected to be observed in photoemission experiments for quantum spin chains . in order for this phenomenon to be observed , there may be several problems to be resolved ; preparation of a proper 1d sample , experiments with high resolution , etc . anyway , it is desired to experimentally find or synthesize rather ideal spin chain systems without the magnetic order even at low temperatures . in summary we have proposed the multi - channel kondo effect dynamically induced by the photoemission and the inverse photoemission , for which the ground state is a completely screened kondo singlet . by studying low - energy critical properties in the photoemission spectra , it has been found that the anomalous behavior generated by this effect is indeed characteristic of the multi - channel kondo system . in particular , we have demonstrated that the idea proposed here can be directly applied to homogeneous quantum spin systems without any impurity . it has been shown in this case that a _ mobile _ kondo impurity suddenly created by the photoemission gives rise to the momentum - dependent anomalous exponent . although we have mainly focused on the photoemission spectra for magnetic impurity systems in this paper , it should be noted that the idea can be directly applied to the photoemission spectrum in quantum dot systems , for which a multilevel quantum dot with the hund coupling plays a role of the kondo impurity . in this connection , it is also interesting to apply a similar idea to the optical absorption spectra in a quantum dot , as recently demonstrated . @xcite in such cases , not only the linear but also the non - linear optics play an important role,@xcite which may provide interesting phenomena related to the dynamically induced kondo effect . this work was partly supported by a grant - in - aid from the ministry of education , science , sports and culture , japan . 99 p. nozi@xmath115res and blandin , j. phys . ( paris ) * 41 * , 193 ( 1980 ) . n. andrei and c. destri , phys . lett . * 52 * , 364 ( 1984 ) , a. m. tsvelik and p. b. wiegmann , z. phys . b * 54 * , 201 ( 1984 ) . d. l. cox , phys . rev . lett . * 59 * , 1240 ( 1991 ) ; d. l. cox and a. zawadowski , _ exotic kondo effects in metals _ , ( taylor and francis , 1999 ) . p. schlottmann and p. d. sacramento , _ adv . _ * 42 * , 641 ( 1993 ) . i. affleck and a. w. w. ludwig , nucl . b * 360 * , 641 ( 1991 ) . a. w. w. ludwig and i. affleck , phys . . lett . * 67 * , 3160 ( 1991 ) ; i. affleck and a. w. w. ludwig , phys . b * 48 * , 7397 ( 1993 ) . c. l. seaman et.al . , * 67 * , 2882 ( 1991 ) . a. a. belavin , a. m. polyakov and a. b. zamolodchikov , nucl . b * 241 * , 333 ( 1984 ) . j. l. cardy , nucl . b * 270 * , 186 ( 1986 ) . t. fujii and n. kawakami , physica b * 281 * , 406 ( 2000 ) . j. l. cardy , nucl . b * 324 * , 581 ( 1989 ) . i. affleck and a. w. w. ludwig , phys . lett . * 67 * , 161 ( 1991 ) . k. d. schotte and u. schotte , phys . rev . * 182 * , 479 ( 1969 ) . i. affleck and a. w. w. ludwig , j. phys . a : math . gen . * 27 * , 5375 ( 1994 ) . reshetikhin , lett . * 7 * , 205 ( 1983 ) . a. klumper and p. a. pearce , physica a * 183 * , 304 ( 1992 ) . a. kuniba , t. nakanishi and j. suzuki , int . j. mod . phys . a * 9 * , 5215 ( 1994 ) ; * 9 * , 5267 ( 1994 ) . s. fujimoto and n. kawakami , phys . b * 52 * , r10132 ( 1995 ) . h. schulz , j. phys . c * 18 * , 581 ( 1985 ) . h. asakawa and m. suzuki , j. phys . a * 29 * , 225 ( 1996 ) ; a * 30 * , 3741 ( 1997 ) . a. a. zvyagin and p. schlottmann , phys . rev . b * 54 * , 15191 ( 1996 ) . s. fujimoto and n. kawakami , phys . b * 54 * , 5784 ( 1996 ) . g. bedrftig , f. essler and h. frahm , phys . 77 * , 5098 ( 1996 ) ; nucl . b * 489 * , 697 ( 1997 ) . h. shiroishi and m. wadati , j. phys . . jpn . * 66 * , 1 ( 1997 ) . t. deguchi and r. ye , j. phys . a * 30 * 8129 ( 1997 ) . t. fujii , y. tsukamoto and n. kawakami , j. phys . jpn . * 66 * , 2552 ( 1997 ) ; * 68 * , 151 ( 1999 ) . a. b , zamolodchikov and v. a. fateev , sov . jetp * 62 * , 215 ( 1985 ) . a. m. tsvelik , nucl . b * 305 * [ fs23 ] , 675 ( 1988 ) . i. affleck and a. w. w. ludwig , nucl . b * 352 * , 849 ( 1991 ) . h. frahm , m. p. pfannm@xmath116ller and a. m. tsvelik , phys . lett . * 81 * , 2116 ( 1998 ) ; h. frahm , nucl . phys . b * 559 * , 613 ( 1999 ) . y. tsukamoto , t. fujii and n. kawakami , phys . b * 58 * , 3633 ( 1998 ) ; eur . j. b * 5 * , 479 ( 1998 ) . f. c. alcaratz and m. j. martins , j. phys . a : math . gen . * 22 * , 1829 ( 1989 ) . s. sorella and a. parola , phys . lett . * 76 * , 4604 ( 1996 ) ; phys . rev . b * 57 * , 6444 ( 1998 ) n. m. bogoliubov , a. g. izergin and v. e. korepin , nucl . b * 275 * , 687 ( 1986 ) . h. frahm and v. korepin : phys . b * 42 * ( 1990 ) 10553 . n. kawakami and s .- k . yang : phys . lett . a * 148 * ( 1990 ) 359 ; n. kawakami and s .- k . yang : phys . rev * 65 * ( 1990 ) 2309 . k. kikoin and y. avishi , cond - mat/0002444 t. fujii , a. furusaki , n. kawakami and m. sigrist , in preparation

we study how the multi - channel kondo effect is dynamically induced to affect the photoemission and the inverse photoemission spectrum when an electron is emitted from ( or added to ) the completely screened kondo impurity with spin @xmath0 . the spectrum thereby shows a power - law edge singularity characteristic of the multi - channel kondo model . we discuss this anomalous behavior by using the exact solution of the multi - channel kondo model and boundary conformal field theory . the idea is further applied to the photoemission for quantum spin systems , in which the edge singularity is controlled by the dynamically induced overscreening effect with a _ mobile _ kondo impurity .

cond-mat0005495

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 .

1003.1883

x - ray spectra of bhxrbs show evidence for a two - phase structure to the accretion flow , an optically thick , geometrically thin accretion disc @xcite giving rise to a blackbody component in the x - ray spectrum , and a hot optically - thin component , modelled as a power law . the relative strengths of these two components define the appearance of different ` states ' ( e.g. @xcite ) . in the hard state , which we focus on in this paper , the power - law emission dominates the total luminosity . it has been suggested that the power - law is produced by an inner , optically - thin advection dominated accretion flow ( adaf ) ( e.g. @xcite ) , which replaces the inner optically thick disc at low accretion rates , and extends down to the innermost stable circular orbit ( isco ) . this implies that the optically thick disc is truncated at some transition radius . alternatively , the optically thick and optically thin components may co - exist over some range of radii , e.g. if the thin disc is sandwiched by a hot flow or corona @xcite . the corona may in turn evaporate the innermost regions of the disc at low accretion rates ( e.g. @xcite ) , so that the optically thin flow is radially separated from the optically thick disc as for the adaf model . it has also been suggested @xcite that a cold thin accretion disc extends close to the black hole in the hard - state . in this model , most of the accretion power is transported away from the disc to power a strong outflowing corona and jet . since the adaf and corona perform similar roles in that they upscatter soft photons to produce the observed power - law , we shall henceforth refer to both components interchangeably as the corona , without necessarily favouring either picture . in the hard state , the variability of the power - law continuum offers further clues to the structure of the accretion flow . studies of the timing properties of hard state bhxrbs show that the frequencies of broad lorentzian features in their power - spectral density functions ( psds ) correlate with the strength of the reflection features as well as the steepness of the power law continuum @xcite . these correlations can naturally be explained if the lorentzian frequencies correspond to a characteristic timescale at the disc truncation radius , e.g. the viscous time - scale , so that as the truncation radius increases the lorentzian frequency decreases , with disc reflection and compton cooling of the optically thin hot flow by disc photons decreasing accordingly . in this picture , the truncation radius of the thin disc acts to generate the signals of the lowest - frequency lorentzian in the psd , while the highest - frequency lorentzians may be generated at the innermost radius of the hot inner coronal flow , i.e. , at the isco @xcite . regardless of whether the corona is radially or vertically separated from the thin disc , photons upscattered by the corona should interact with the disc . this interaction gives rise to reflection features in the x - ray spectrum , including fluorescent iron line emission and a reflection continuum due to compton scattering off the disc material . an often - neglected consideration is that a significant fraction of the photons interacting with the disc are absorbed and the disc heated in a process known as thermal reprocessing . provided that the disc subtends at least a moderate solid angle as seen by the corona , this effect should be particularly significant in the hard state , where the coronal power law continuum dominates the total luminosity . when the power - law luminosity impinging on the disc is high compared to the disc luminosity due to internal heating , then a significant fraction of the disc blackbody emission should be reprocessed and will therefore track variations of the power law continuum . the anticipated correlated variations of the blackbody and power - law emission can be studied using variability spectra , e.g. the rms spectrum , which show only the variable components of the spectrum @xcite . if the geometry is such that the observed power - law produces reprocessed blackbody emission by x - ray heating the disc , then both power - law and blackbody components should appear together in the variability spectra . furthermore , by selecting different time ranges covered by these variability spectra ( i.e. , analogous to the method of fourier - resolved spectrosopy , @xcite ) , it is possible to determine whether the low - frequency part of the psd has a different origin to the high - frequency part in terms of the contributions of blackbody and power - law components . if the optically thick disc does drive the low - frequency lorentzian , we predict that the blackbody component should be stronger in the corresponding variability spectrum . in this work , we examine the variability spectra of two hard state black hole x - ray binaries , swift j1753.5 - 0127 and gx 339 - 4 , which have good _ xmm - newton _ data for studying simultaneous variations of disc and power - law . previous analyses of time - averaged spectra for these _ xmm - newton _ observations have shown the existence of the expected blackbody components , together with relativistically broadened reflection features , which have been used to argue that the disc is truncated at significantly smaller radii than previously thought , perhaps only a few gravitational radii @xcite . however , see @xcite for arguments in favour of disc truncation at larger radii . in the following section , we describe the observations and data reduction . in section [ anres ] we show the soft and hard - band psds obtained from the data , and present a technique to produce a type of rms spectrum , the ` covariance spectrum ' which we use to identify the variable spectral components for each hard state source . in particular we show that , although both power - law and disc blackbody emission are correlated , as expected from thermal reprocessing , the disc is relatively more variable than the power - law on longer time - scales ( corresponding to the low - frequency lorentzian ) , contrary to what we would expect from simple reprocessing models . in section [ discuss ] we discuss the interpretation of our results and present further analysis to suggest that the low - frequency lorentzian corresponds to fluctuations intrinsic to the disc . a summary of our conclusions is given in section [ conc ] . swift j1753.5 - 0127 was observed during revolution 1152 for 42 ks by _ xmm - newton _ epic - pn on 2006 march 24 in _ pn - timing _ mode using the medium optical filter . the events list was screened using the perl script xmmcleankaa / xselect / xmmclean ] to select only events with flag=0 and pattern @xmath0 4 . examination of light curves showed no evidence for background flaring . using sas version 7.0 , evselect was used to extract the mean spectrum , following the procedure of @xcite , using event positions between 20 and 56 in rawx and using the full rawy range . the sas tasks arfgen and rmfgen were used to generate the ancillary response file ( arf ) and redistribution matrix file ( rmf ) . gx 339 - 4 was observed by _ xmm - newton _ on 2004 march 16 during revolutions 782 and 783 , again in _ pn - timing _ mode using the medium optical filter . the sas command evselect was used to filter the events on time to avoid background flaring , producing a total combined exposure of 127 ks . data reduction was very similar to that for swift j1753.5 - 0127 , using sas version 7.0 , but care was taken to avoid pile - up . successive columns from the centre of the image were excised in rawx and spectra extracted until no discernable difference in spectral shape could be identified between successive selections on rawx , implying that pile - up is no longer significant . the epic - pn data was deemed free of pile - up using extraction regions in rawx from columns 30 to 36 and 40 to 46 ( i.e. , columns 37 to 39 inclusive were excised ) . background spectra were selected in rawx from columns 10 to 18 over the full rawy range . to generate an appropriate arf for the data made in this way , it was necessary to use arfgen to generate an arf for the full region in rawx from columns 30 to 46 and generate a second arf from the spectrum of the excluded region ( rawx columns 37 to 39 ) and then subtract the latter from the former using the command addarf . for both sources , the ftool grppha was used to scale the background spectra and to ensure that a minimum of 20 counts were in each bin for @xmath1 fitting . the epic - pn covers the energy range from 0.2 - 10 kev , but the epic calibration status document recommends restricting the fit to energies greater than 0.5 kev . to be conservative , our epic - pn spectral fits were restricted to the range 0.7 to 10.0 kev . we extracted data from _ rxte _ observations which were contemporaneous with the _ xmm - newton _ observations . we used the high time and spectral - resolution pca event mode data to extract mean spectra and make rms and covariance spectra using the same time binning as the epic - pn data ( see section [ method ] ) . throughout this work we use _ rxte _ pca data in the 3 - 25 kev range , and fit these together with the corresponding epic - pn spectra , tying all fit parameters together but allowing _ rxte _ spectral fits to be offset by a constant factor with respect to the simultaneous epic - pn fits , to allow for the difference in flux calibration between the pca and epic - pn instruments , and also for the fact that slight flux differences may result from the fact that the _ rxte _ observations covered shorter intervals than the _ xmm - newton _ data . a 1 per cent systematic error was assumed in all spectral fits to account for uncertainties in instrumental response and cross - calibration . the signal to noise of _ rxte _ hexte data was not sufficient to perform the covariance analysis we describe in section 3.2 . before describing the spectral analysis method and results , we first examine the timing properties of each source in soft and hard spectral bands using the power - spectral density function ( psd ) . for each source , we used the epic - pn data to generate light curves with 1 ms time binning in two energy bands : 0.5 - 1 kev ( soft ) and 2 - 10 kev ( hard ) . only complete segments of 131 s duration were used to construct the psds , segments with gaps ( which can be common at high count rates in timing mode ) were skipped over . the resulting poisson - noise - subtracted psds are shown in figure [ psds ] . it is clear from the figure that the psds are significantly different in shape between the two bands . in gx 339 - 4 , the hard - band psd shows two broad components , with the higher - frequency component appearing to be shifted to even higher frequencies in the hard band . in swift j1753.5 - 0127 , soft and hard psds appear , within the noise , to overlap more closely at higher frequencies . however , in both sources the soft band psd shows relatively larger low - frequency power compared to the psd components at frequencies above @xmath2 hz . this @xmath3 extra low - frequency power at lower x - ray energies may be associated with the soft excess emission , e.g. if the disc is varying more than the power - law on longer time - scales . alternatively , there could be power - law spectral variability that applies only on longer time - scales and causes the soft band variability amplitude to be enhanced , i.e. , due to steepening of the power - law spectral slope on longer time - scales . to determine the origin of the extra low - frequency power in the soft band , we must carry out a spectral analysis of the variations on different time - scales , the methodology for which we describe in the next section . when fitting models to x - ray spectra , it is typical to use mean x - ray spectra that only describe the time - averaged spectral shape of a source . such fits say nothing about the way the different spectral components such as disc blackbody and power - law vary with respect to each other in time . by looking at the absolute amplitude of variations in count rate as a function of energy we can construct ` variability spectra ' which pick out only the time - varying components . one technique is to construct fourier - frequency resolved spectra , obtaining a psd for each individual energy channel and integrating the psd over a given fourier - frequency range in order to measure the variance in that channel , which is used to obtain the rms and so construct the spectrum due to components which vary over that frequency range ( e.g. @xcite ) . this approach is attractive in that it allows the user to pick out complex patterns of spectral variability where components have different time - scales of variation , as may be implied by the soft and hard band psds shown in figure [ psds ] . here we will consider only two time - scales of variability , corresponding roughly to the high and low - frequency parts of the psd which show significant relative differences between the soft and hard bands . to approximate the more complex fourier - resolved approach we will measure over two time - scale ranges a variant of the ` rms spectrum ' , which measures the absolute root - mean - squared variability ( rms ) as a function of energy ( e.g. see @xcite ) . producing the rms spectrum involves allocating each photon event to a time and energy bin , dividing the light curve into segments consisting of @xmath4 time bins per segment and then working out the variance in each segment for each energy bin according to the following standard formula : @xmath5 where @xmath6 is the count rate in the @xmath7 bin and @xmath8 is the mean count rate in the segment . the expectation of poisson noise variance , given by the average squared - error @xmath9 , is subtracted , leaving the ` excess variance ' , @xmath10 , in each segment . the excess variances can then be averaged over all of the segments of the light curve . the square root of the average excess variance plotted against energy forms the rms spectrum . by selecting the time bin size and the segment size , we can isolate different time - scales of variability , effectively replicating the fourier resolved approach for the two time - scale ranges that we are interested in . for this purpose , we choose two combinations of bin size and segment size . to look at variations on shorter time - scales we choose 0.1 s time bins measured in segments of 4 s ( i.e. , 40 bins long ) , i.e. , covering the frequency range 0.25 - 5 hz where @xmath11 is the time bin size . ] . for longer time - scale regions we use 2.7 s bins in segments of 270 s , i.e. , covering the range 0.0037 - 0.185 hz note that these two frequency ranges do not overlap and also cover the two parts of both source psds which show distinct behaviour in soft and hard bands . there is , however , a problem with the rms spectrum when signal to noise is low , e.g. at higher energies . it is possible for the expectation value of the poisson variance term to be larger than the measured average variance term , producing negative average excess variances . if this is the case , it is not possible to calculate the rms at these energies and this introduces a bias towards the statistically higher - than - average realisations of rms values , which can still be recorded . in order to overcome these problems we have developed a technique called the ` covariance spectrum ' . the covariance is calculated according to the formula : @xmath12 where @xmath13 now refers to the light curve for a ` reference band ' running over some energy range where the variability signal - to - noise is large . in this work , we use reference bands of 14 kev for epic - pn data and 35 kev for pca data . in other words , the covariance spectrum is to the rms spectrum what the cross - correlation function of a time series is to its auto - correlation function . the covariance spectrum therefore does not suffer from the same problems as the rms spectrum , as no poisson error term has to be subtracted , since uncorrelated noise tends to cancel out and any negative residuals do not affect the calculation . to remove the reference band component of the covariance , and produce a spectrum in count - rate units , we obtain the normalised covariance for each channel using : @xmath14 where @xmath15 is the excess variance of the reference band . therefore , the only requirement for there being a valid , unbiased value of covariance at a given energy is that the reference excess variance is not negative . this is usually the case , since the reference band is chosen to include those energies with the largest absolute variability . when the covariance is being calculated for an energy channel inside the reference band , the channel of interest is removed from the reference band . the reasoning behind this is that if the channel of interest is duplicated in the reference band , the poisson error contribution for that channel will not cancel and will contaminate the covariance . one can think of the covariance technique as applying a matched filter to the data , where the variations in the good signal - to - noise reference band pick out much weaker correlated variations in the energy channel of interest that are buried in noise . in this way the covariance spectrum picks out the components of the energy channel of interest that are correlated with those in the reference band . it is important to note that the covariance spectrum only picks out the correlated variability component and is therefore a more appropriate measure than the rms spectrum in constraining the reprocessing of hard photons to soft photons , which will result in correlated variations . when the raw counts rms and covariance spectra are overlaid , as in figure [ fig : covrms ] , they match closely indicating that the reference band is well correlated with all other energies ( the spectral ` coherence ' is high , e.g. see @xcite ) . another advantage of the matched filter aspect of the covariance spectrum is that it leads to smaller statistical errors than the rms spectrum . specifically , the errors are given by : @xmath16=\sqrt{\frac{\sigma_{\rm xs , x}^2 \overline{\sigma_{\rm err , y}^2}+\sigma_{\rm xs , y}^2 \overline{\sigma_{\rm err , x}^2}+\overline{\sigma_{\rm err , x}^2}~\overline{\sigma_{\rm err , y}^2}}{nm\sigma_{\rm xs , y}^2}}\ ] ] where @xmath17 denotes the number of segments and subscripts x and y identify excess variances and poisson variance terms for the channel of interest and reference band respectively . this error equation can be derived simply from the bartlett formula for the error on the zero - lag cross - correlation function , assuming that the source light curves have unity intrinsic coherence @xcite . by comparison with equation b2 of @xcite , using the relation : @xmath18 ( which is true because the reference band has good signal to noise ) it can be shown that the errors on the covariance are smaller than corresponding errors on the rms values . finally , we note that , since the covariance is analogous to the zero - lag cross - correlation function , it could be affected by intrinsic time lags in the data . using measurements of the cross - spectral phase - lags between various energy bands , we have confirmed that the lags between hard and soft band variations are smaller than the time bin sizes used to make the long and short time - scale covariance spectra . thus , intrinsic time lags will have no effect on our results . we first consider the covariance spectra of swift j1753.5 - 0127 , and use xspec v12 @xcite to fit the long and short - time - scale data together with the mean spectrum in order to identify the origin of the additional long - time - scale variability which can be seen in the soft band psd . the covariance spectra do not show sufficient signal - to - noise to detect the rather weak iron line present in the mean spectrum of this source @xcite , so for simplicity we fit only a simple power - law and multicolour disc blackbody diskbb , together with neutral absorption . we fit the short and long - time - scale covariance and mean spectra simultaneously , tying the absorbing column density and renormalising constant for pca data to be the same for the mean and covariance spectra . an f - test showed that the disk blackbody temperature does not change significantly between spectra , so that was also tied to be the same for all spectra . the remaining parameters were allowed to be free between the covariance and mean spectra . for those parameters that were free to vary , epic - pn and pca values were tied together . the @xmath1 of the final fit was 1802 for 1948 degrees of freedom ( d.o.f . ) and full fit parameters are listed in table [ tab : swift ] . the best - fitting unfolded spectra and the corresponding data / model ratios are shown in figure [ fig : swifteeufspec ] . to interpret the fits to the covariance spectra , consider the case where the observed psds have identical shapes in both hard and soft bands ( they may have different normalisations ) . in this case , the ratio of soft to hard - band variability amplitudes measured over the same time - scale range will be identical for any given time - scale range . therefore , since ( for high - coherence variations ) the covariance spectrum quantifies variability amplitude as a function of energy , the shape of the covariance spectrum will be independent of time - scale . on the other hand , if the soft band contains more long - time - scale variability relative to short - time - scale variability than the hard band , the long - time - scale covariance spectrum will appear _ softer _ than the short - time - scale covariance spectrum : lower energies show correspondingly greater variability on long time - scales and therefore larger fluxes in the covariance spectra . just such an effect is seen in the model fits to the covariance spectra : the long - time - scale covariance spectrum is softer than the short - time - scale covariance spectrum , because the disc blackbody normalisation is higher on long time - scales . the power - law slope is remarkably similar on both long and short time - scales however . therefore the additional long - term variability in the soft - band psd seems to result from _ additional _ variability of the disc blackbody , not any extra power - law variability ( e.g. due to spectral pivoting ) . .fit parameters for mean and covariance spectra of swift j1753.5 - 0127 for the model constant*phabs*(diskbb+powerlaw ) [ cols="<,^,^,^",options="header " , ] for the definitions of the parameters listed in the first six rows see table [ tab : swift ] . the additional parameters shown are for the relativistically smeared reflection ( from top to bottom ) : disc innermost radius ( units of @xmath19 ) ; disc inclination ( fixed to be the same in both kdblur and hrefl ) ; gaussian line energy ; gaussian normalisation ( photons @xmath20 s@xmath21 ) ; covering fraction of the reflection ( where 1.0 corresponds to @xmath22 steradians ) . [ tab : gx ] the fits to the covariance spectra for gx 339 - 4 show a similar pattern to that seen for swift j1753.5 - 0127 , in that the long time - scale covariance spectrum is softer than the short time - scale covariance spectrum because of a significantly stronger disc blackbody component , while their power - law indices are very similar . we can confirm this interpretation using a plot of the covariance ratio , which is shown in figure [ fig : gxcovratios ] , and shows a similar rise at low energies to that seen for swift j1753.5 - 0127 , which underlines the interpretation that the disc blackbody component is the main contributor to the additional variability seen at low frequencies in the soft band in gx 339 - 4 . the reader is referred to sections 4.1 and 4.4 for a consistent interpretation of the short time - scale covariance spectrum in gx 339 - 4 and swift j1753.5 - 0127 . note that the apparent emission feature around 2 kev is probably an instrumental effect , possibly related to mild pile - up in timing mode , since it disappears when larger regions of rawx are excised from the data . due to the 1 per cent systematic included in the spectral fitting , this feature has no effect on the fit results . based on the ratio of component normalisations to those in the mean spectrum , the fractional rms values for the disc component are 39 per cent and 30 per cent over the long and short time - scale ranges respectively . the corresponding power - law fractional rms values at 1 kev are 41 per cent and 40 per cent . using the same approach , we can also define a fractional rms for the iron line emission for long and short time - scales , at @xmath23 per cent in each case . interestingly , the @xmath24 per cent increase in iron line rms over that of the power - law emission ( which drives the line variability ) is comparable to the increase in reflection covering fraction from the mean to covariance spectra . these results may imply the presence of an additional constant power - law component which dilutes the power - law fractional variability but does not contribute to reflection . it is also interesting to note that the power - law component in the covariance spectra for gx 339 - 4 is softer than in the mean spectrum , i.e. , the difference is in the opposite sense to that seen in swift j1753.5 - 0127 . we have shown that disc blackbody emission contributes significantly to the x - ray variability spectra in the hard state of the black hole candidates swift j1753.5 - 0127 and gx 339 - 4 , and moreover , that the disc emission is the origin of the additional soft band variability seen on longer time - scales in both sources , which manifests itself as an enhanced low - frequency component in the psd . in this section , we discuss the evidence for a connection between disc and power - law variability through x - ray reprocessing , and then consider two possible explanations for the enhanced disc variability on long time - scales , in terms of geometry changes or fluctuations intrinsic to the accretion disc . finally we will compare our results and interpretation to the wider picture of different accretion states . the power - law emission clearly dominates the x - ray luminosity in both sources , as can be seen in figures [ fig : swifteeufspec ] and [ fig : gx339eeufspec ] . in this situation , if the disc sees a reasonable fraction of the power - law emission , we should expect that x - ray heating of the disc , i.e. , thermal reprocessing of power - law emission , will produce a significant fraction of the observed disc luminosity . in fact , the thermal reprocessed emission is directly related to the disc reflection component in the spectrum : if a fraction of incident power - law luminosity @xmath25 is reflected by the disc ( both through compton reflection and emission line fluorescence ) , then a fraction @xmath26 must be absorbed and will be reprocessed into thermal blackbody radiation . the reflected fraction @xmath25 depends on disc ionisation state but simple exploration of the disc reflection models in xspec shows that it is typically 30 - 40 per cent of the incident luminosity , so that around 60 - 70 per cent of the incident luminosity is reprocessed into disc blackbody emission . for typical iron k@xmath27 line equivalent widths ( around 1 kev with respect to the reflection continuum ) , we then expect the line flux to be of order 1 per cent of the thermally reprocessed flux . we might reasonably assume that the blackbody component in the short time - scale covariance spectrum is produced by thermal reprocessing of the varying power - law , which also drives the line emission in the same spectrum . the unabsorbed disc flux is @xmath28 erg @xmath20 s@xmath21 , and the iron line flux is @xmath29 erg @xmath20 s@xmath21 which is 1.3 per cent of the reprocessed flux , i.e. , consistent with a reprocessing origin for the thermal emission , at least on short time - scales . in swift j1753.5 - 0127 , the disc blackbody emission is considerably weaker than in gx 339 - 4 , and correspondingly we would expect a relatively weak iron line , with a few tens of ev equivalent width , which is only just consistent with the lower - limits on line strengths reported by @xcite and @xcite . it is possible that the disc in swift j1753.5 - 0127 is substantially ionised ( e.g. see @xcite ) which would enhance line emission relative to the absorbed ( and hence thermally reprocessed ) emission . since it is likely that there is substantial x - ray heating of the disc , one must interpret the observed blackbody normalisations in the mean spectra with caution . the disc emissivity may be more centrally concentrated than the theoretically expected @xmath30 law , and so the normalisations indicate better the emitting surface area and can not be simply translated to an inner radius . nonetheless , the inferred emitting areas are still relatively small , implying distance scales of tens to hundreds of km ( assuming distances @xmath31 kpc and 6 - 15 kpc for swift j1753.5 - 0127 and gx 339 - 4 respectively ; @xcite ) . we also note here that although the need for disc blackbody emission to explain the spectrum of swift j1753.5 - 0127 has been questioned by @xcite , the model - independent covariance ratio plots in figure [ fig : swiftcovratios ] show that a distinct soft component must be present in order to explain the difference in the shapes of the covariance spectra . it is interesting to note that the power - law indices of the covariance spectra are different to those of the mean spectra , but in an opposite sense for each of the two sources considered here : compared to the mean spectrum swift j1753.5 - 0127 shows a harder power - law in the covariance spectra , while gx 339 - 4 shows a softer power - law . the difference may be caused by flux - dependent spectral pivoting or steepening , so that as flux increases the spectrum gets softer in gx 339 - 4 , increasing the covariance at soft energies relative to the mean , while the opposite effect occurs in swift j1753.5 - 0127 ( it hardens as it gets brighter ) . the difference in behaviour may be related to the source luminosity : if they lie at similar distances swift j1753.5 - 0127 is at least a factor 10 less luminous than gx 339 - 4 , implying a significantly lower accretion rate . correlations between flux and spectral - hardness have been seen in bhxrb hard states , and interestingly the sign of the correlation appears to switch over from negative to positive at low luminosities , both on short time - scales @xcite and in the long - term global correlation @xcite . the same switch in flux - hardness correlation could be related to the different power - law behaviour of swift j1753.5 - 0127 and gx 339 which we see here . the same pattern appears on both long and short time - scales , which show almost identical power - law indices in their covariance spectra , so that the effect of the power - law spectral variability will be to change the normalisation of the psd , but not the shape . since most of the power - law luminosity will be found at tens of kev ( assuming a thermal cutoff at around 100 kev ) , spectral steepening with flux will cause the observed gx 339 - 4 0.5 - 10 kev variability amplitude to be enhanced compared to the true luminosity variations . conversely , observed variations in swift j1753.5 - 0127 will be smaller than the total luminosity variations . therefore , the fractional luminosity variations for swift j1753.5 - 0127 and gx 339 - 4 may be similar , but gx 339 - 4 shows a significantly greater normalisation in the psds shown in figure [ psds ] . we have seen that it is likely that reprocessing of the power - law drives at least some of the disc variability seen in hard state sources , and possibly all of it on time - scales @xmath32 s. however , model - independent covariance ratio plots and spectral fitting show that the covariance spectra of both sources demonstrate increased disc blackbody variability with respect to the power law on longer timescales . there are several possible explanations for this pattern , but the key thing that any successful model needs to achieve is an increase in disk variability on longer timescales without a concomitant rise in power law variability . also , it is important to note that the additional disc variability must still be correlated with power - law variations , because the enhanced disc variations appear in the covariance spectrum , which is identical to the rms spectrum over the high signal - to - noise energy range covered by the disc ( i.e. , coherence is unity ) . if the disc variations were independent of the power - law they would cancel to some extent , since they would be uncorrelated with the power - law component , and covariance would be smaller than the rms . thus the blackbody variations map on to power - law variations but with larger amplitude . one possibility is to change the geometry of the system on longer time - scales . for example , a variable coronal scale height on longer timescales could lead to changes in the solid angle of disc heated by the power - law , thus increasing the disc blackbody variability amplitude relative to the power - law . weaker correlated power - law variation could then be produced if scale - height correlates with power - law luminosity . alternatively , correlated power - law variations could be due to variable seed photon numbers from the disc due to the enhanced variable heating , but in either case the additional power - law variability must be of smaller amplitude than the observed blackbody variability . regardless of these model - dependent arguments , one can make a simple observational test of the variable - geometry model , by comparing the variability of reflection on long and short time - scales . gx 339 - 4 shows significant reflection features in its covariance spectra , so any variation in coronal geometry should manifest itself as increased variability in the reflection components as the disk sees the varying power law . however , the spectral fit parameters given in table [ tab : gx ] indicate that there is little change in reflection amplitude between long and short time - scales , both in terms of reflection covering fraction and iron line equivalent width ( e.g. ratio of line flux to power - law normalisation , which is meaningful because the power - law indices are so similar ) . to highlight this similarity in the covariance spectra , we show in figure [ fig : gxrefl ] the data / model ratios for the pca spectra with the reflection components taken out of the fit . the spectra demonstrate no significant change in the reflection continuum and associated iron line . this implies that if there is any change in geometry , it is small , and the increase in the thermal component of the variability spectra on longer timescales has a different origin . to place this result on a more rigorous statistical footing , we show in figure [ fig : contour ] a contour plot of the best - fitting short and long - time - scale reflection covering fractions , which are obtained only from fits to the pca data , which are most sensitive to the reflection continuum . if variable geometry is the cause of the enhanced long - term blackbody variability , we would expect the covering fraction to show a similar enhancement in variability on long time - scales . the dashed line in the figure shows the largest ratio of short - to - long - time - scale covering fraction which crosses the 99 per cent confidence contour , with a value of 0.81 , placing a 99 per cent confidence upper limit of 23 per cent on any increase in the covering fraction on long time - scales compared to short time - scales ( the 90 per cent confidence upper limit on any increase is 12 per cent ) . in contrast , the long - time - scale blackbody variability amplitude increases by 30 per cent compared to that on shorter time - scales , which can not be explained by the permitted increase in variable reflection at greater than the 99 per cent confidence level . thus , in gx 339 - 4 at least , we can rule out long - time - scale changes in coronal geometry as a viable explanation for the enhanced blackbody variability amplitude . note also that the same arguments apply to any other non - geometric arguments which seek to produce the extra long - term disc variability by varying the power - law contribution as seen by the disc , e.g through variable beaming of the power - law towards the disc . due to the weakness of the iron line in swift j1753.5 - 0127 it is not possible to place similar constraints on coronal geometry changes in this source , although by analogy we expect the same interpretation to apply . having established that variable coronal geometry can not explain the extra blackbody variability on longer timescales , we are left to consider the possibility that the variations are intrinsic to the disc itself . perhaps the simplest possibility is that the fluctuations are due to accretion rate fluctuations in the disc which undergo viscous damping before they reach the corona . such damping is expected in thin discs ( e.g. see @xcite ) , but will not be as significant in geometrically thick flows , which might correspond to the corona . therefore one can envisage that the long time - scale variability , e.g. corresponding to the low - frequency lorentzian in the psd , is generated in the thin disc , producing relatively large amplitude blackbody variations but being damped before reaching an inner coronal emitting region ( perhaps inside the innermost radius of the thin disc ) . the correlated power - law variability could then arise either from the residual undamped variations in accretion rate which reach the corona , or be driven by seed photon variations from the disc . if the inner radius of the disc were to fluctuate on long time - scales , then the varying disc area would also introduce extra blackbody variability . this model is in some sense analogous to that of changes in coronal scale height , since any change in disc area will vary the solid angle of disc seen by the corona which will cause an increase in reflection as well as correlated power - law variability driven by seed photon variations . one could mitigate these effects if the coronal properties were linked to those of the disc inner - radius , e.g. as inner radius decreases , so does coronal scale - height . this situation might be expected if the corona is formed by evaporation of the disc @xcite , so that condensation of the disc will drain and cool the coronal plasma , leading to a reduction in scale - height . we have established that the hard - state variability on time - scales greater than seconds , corresponding to the low - frequency lorentzian psd component , is very probably produced by variations intrinsic to the accretion disc , perhaps in the form of propagating accretion rate fluctuations , as envisaged by @xcite to explain the broad shapes of observed psds . these variations then manifest as weaker power - law variations , either through propagation of accretion fluctuations to the corona @xcite , or through variations of seed photons from the disc , which are compton upscattered in the corona . on shorter time - scales , the blackbody variations are probably mostly produced by x - ray heating of the disc by the power - law , which is a required outcome of the x - ray reflection directly observed in these systems . the exact split between direct emission from intrinsic disc variations and x - ray heating is difficult to judge , since some residual intrinsic variations may remain on short time - scales , and x - ray heating will also contribute on long time - scales . but in the case of intrinsic disc accretion fluctuations , the variable x - ray heating is itself a product of those fluctuations , and so any intrinsic disc variations on long time - scales must be large , at least comparable to the fractional rms of blackbody emission , i.e. , 40 per cent in both sources . it is interesting to contrast the large intrinsic disc variability in these hard state sources with that in the soft states , where the disc emission dominates the bolometric luminosity . the soft states are well - known for showing very weak , if any , variability @xcite , and the strongest variability which is seen , e.g. in cyg x-1 , is associated with the power - law , with the disc being remarkably constant @xcite . therefore it seems likely that hard state discs are inherently unstable compared to soft state discs . this difference may represent just another observable distinction between hard and soft states , but it is interesting to speculate that it may play a more primary role in creating the other observed differences , such as a strong corona and jet formation in the hard state . certainly , it seems likely that intrinsic disc variability plays an important role in determining the psd shape in the hard state . for instance , the low energy psd of grs 1915 + 105 in the @xmath33-class hard intermediate state @xcite shows a low frequency component which disappears at higher energies , possibly indicating that this source is demonstrating intrinsic disc variability . it is worth noting here that x - ray / optical studies of agn also show evidence for reprocessing on short time - scales and intrinsic disc variations on longer time - scales @xcite . however these agn are relatively luminous and radio - quiet and so are likely analogues of bhxrb soft states , perhaps indicating that disc - stability shows a mass - dependence , e.g. related to the transition between gas- and radiation - pressure dominated discs @xcite . our results strongly suggest that disc variations are responsible for the low - frequency component in the hard state psd . a number of authors have suggested that the low - frequency lorentzian corresponds to the viscous time - scale of the inner , truncation radius of the thin disc ( e.g. @xcite ) , a view which is consistent with our results . the higher - frequency psd components may then be produced in the corona , which is likely to be geometrically thick so will show naturally shorter variability time - scales . the viscous time - scale scales with scale - height ( @xmath34 ) over radius ( @xmath35 ) as @xmath36 . assuming the ratio of disc scale - height to radius @xmath37 ( where @xmath27 is the viscosity parameter ) , the predicted inner disc radius corresponding to the observed low - frequency psd peak around @xmath38 hz is 15 r@xmath39 ( see @xcite ) . however , this radius may be even smaller for smaller @xmath40 , which would make the disc truncation radius consistent with the results from fits to the iron line ( this work , and @xcite ) . however , in the latter case the corona would need to be very compact , or seed photon variations from intrinsic disc variability would modulate the power - law with a similar amplitude to the disc . in either case , assuming the disc is thin and that it varies on the viscous time - scale , its inner radius must be relatively small . such a picture is very different from earlier models for the hard state , where the disc is highly truncated and power - law emission is produced by a very extended corona or adaf ( e.g. @xcite ) . spectral fits to observations of bhxrb sources in the hard state show increasing evidence for both power law and black body components . in this work we have explored the hard - state variability of two sources with known soft excesses , swift j1753.5 - 0127 and gx 339 - 4 . our findings are summarised below . 1 . we have introduced a new spectral analysis technique , the covariance spectrum , which measures the correlated variability in different energy bands . this technique overcomes the problems of low signal - to - noise and bias associated with the rms spectrum and has smaller statistical errors . 2 . psds of the two sources demonstrate larger low - frequency power in the soft band . 3 . the longer time - scale ( 2.7 - 270 s ) covariance spectra of both sources are softer than the short time - scale ( 0.1 - 4 s ) covariance spectra , due to additional disc variability , i.e. , extra disc variability occurs on longer time - scales without a concomitant rise in power law variability on such timescales . however , the coherence of the rms and covariance spectra show clearly that disc variations are not independent of the power law variations the strength of reflection features that are detected in the short time - scale covariance spectra of gx 339 - 4 are consistent with the observed blackbody variations on those time - scales being driven by thermal reprocessing of the power - law emission absorbed by the disc . however , the reflection covering fraction and iron line equivalent width show little change between short and long time - scales , implying that additional reprocessing , due to coronal geometry change , is not responsible for the additional blackbody variability seen on longer time - scales . the extra blackbody variability seen on longer time - scales appears to be intrinsic to the accretion disc itself , giving rise to the extra low - frequency power in the psd . this represents the first clear evidence that the low - frequency lorentzian component in hard state psds is produced by disc variability . models invoking damped mass accretion rate variations or oscillations in the disc truncation radius can satisfactorily explain the observed pattern of variability . 6 . the implication of such variations occurring in a thin disc on viscous timescales , is that the disc truncation radius is @xmath41 r@xmath39 . this work highlights the importance of measuring spectral variability on a range of time - scales . mean spectra , which describe the average properties of a source , provide no information on how different spectral components are related to one another as a function of time . by using the covariance spectra we have been able to disentangle the correlated spectral components in these two sources , identify thermal reprocessing as the mechanism by which variability is correlated in different bands , produce model - independent evidence for additional blackbody variability on longer time - scales and therefore associate intrinsic disc variability with the low frequency lorentzian feature seen in hard - state psds . we would like to thank beike hiemstra and the anonymous referee for useful comments . we are grateful to maria daz trigo for providing the gx 339 - 4 epic - pn events files and helpful advice . tw is supported by an stfc postgraduate studentship grant , and pu is supported by an stfc advanced fellowship . this research has made use of data obtained from the high energy astrophysics science archive research center ( heasarc ) , provided by nasa s goddard space flight center , and also made use of nasa s astrophysics data system . 1 arvalo , p. , uttley , p. , kaspi , s. , breedt , e. , lira , p. , m@xmath42hardy , i. m. , 2008 , mnras , 389 , 1479 arnaud , k. a. 1996 , astronomical data analysis software and systems v , 101 , 17 axelsson , m. , hjalmarsdotter , l. , borgonovo , l. , larsson , s. , 2008 , a&a , 490 , 253 bartlett , m.s . , 1955 , an introduction to stochastic processes , cup , cambridge box , g.e.p . , jenkins , g.m . , 1976 , time series analysis : forecasting and control , 2nd edn . holden - day , san francisco breedt , e. , et al . , 2009 , mnras in press ( arxiv0812.0810 ) churazov , e. , gilfanov , m. , & revnivtsev , m. , 2001 , mnras , 321 , 759 done , c. , gierliski , m. , & kubota , a. , 2007 , a&arv , 15 , 1 esin a. a. , mcclintock j. e. , narayan r. , 1997 , apj , 489 , 865 gierliski , m. , done , c. , page , k. , 2008 , mnras , 388 , 753 gilfanov , m. , churazov , e. , & revnivtsev , m. , 1999 , a&a , 352 , 182 hiemstra , b. , soleri , p. , mendez , m. , belloni , t. , mostafa , r. , wijnands , r. , 2009 , mnras in press ( arxiv0901.2255 ) homan , j. , wijnands , r. , van der klis , m. , belloni , t. , van paradijs , j. , klein - wolt , m. , fender , r. , & mndez , m. , 2001 , apjs , 132 , 377 hynes , r. i. , steeghs , d. , casares , j. , charles , p. a. , obrien , k. , 2004 , apj , 609 , 317 lyubarskii , y. e. , 1997 , mnras , 292 , 679 maccarone , t. j. , gallo , e. , fender , r. , 2003 , mnras , 345 , l19 malzac , j. , 2007 , apss , 311 , 149 m@xmath42hardy , i. m. , krding , e. , knigge , c. , uttley , p. , fender , r. p. , 2006 , nature , 444 , 730 mayer , m. , & pringle , j. e. , 2007 , mnras , 376 , 435 meyer - hofmeister , e. , meyer , f. 2003 , a&a , 402 , 1013 miller j. m. , homan j. , steeghs d. , rupen m. , hunstead r. w. , wijnands r. , charles p. a. , fabian a. c. , 2006 , apj , 653 , 525 miller j. m. , homan j. , miniutti g. , 2006 , apj , 652 , l113 narayan , r. , & yi , i. , 1994 , apj , 428 , l13 reis , r. c. , fabian , a. c. , ross , r. r. , miniutti , g. , miller , j. m. , reynolds , c. , 2008 , mnras , 387 , 1489 reis , r. c. , fabian , a. c. , ross , r. r. , miller , j. m. , 2009 , mnras in press ( arxiv0902.1745 ) remillard r. a. , mcclintock j. e. , 2006 , ara&a , 44 , 49 revnivtsev , m. , gilfanov , m. , & churazov , e. , 1999 , a&a , 347 , l23 revnivtsev , m. , gilfanov , m. , & churazov , e. , 2001 , a&a , 380 , 520 rodriguez , j. , corbel , s. , hannikainen , d. c. , belloni , t. , paizis , a. , & vilhu , o. , 2004 , apj , 615 , 416 shakura n. i. , sunyaev r. a. , 1973 , a&a , 24 , 337 uttley , p. , edelson , r. , m@xmath42hardy , i. m. , peterson , b. m. , markowitz , a. , 2003 , apj , 584 , l53 vaughan , b. a. , nowak , m. a. , 1997 , apj , 474 , l43 vaughan , s. , edelson , r. , warwick , r. s. , uttley , p. , 2003 , mnras , 345 , 1271 witt , h. j. , czerny , b. , zycki , p. t. , 1997 , mnras , 286 , 848 wu , q. , gu , m. , 2008 , apj , 682 , 212 zurita , c. , durant , m. , torres , m. a. p. , shahbaz , t. , casares , j. , steeghs , d. , 2008 , apj , 681 , 1458

_ xmm - newton _ x - ray spectra of the hard state black hole x - ray binaries ( bhxrbs ) swift j1753.5 - 0127 and gx 339 - 4 show evidence for accretion disc blackbody emission , in addition to hard power - laws . the soft and hard band power - spectral densities ( psds ) of these sources demonstrate variability over a wide range of time - scales . however , on time - scales of tens of seconds , corresponding to the putative low - frequency lorentzian in the psd , there is additional power in the soft band . to interpret this behaviour , we introduce a new spectral analysis technique , the ` covariance spectrum ' , to disentangle the contribution of the x - ray spectral components to variations on different time - scales . we use this technique to show that the disc blackbody component varies on all time - scales , but varies more , relative to the power - law , on longer time - scales . this behaviour explains the additional long - term variability seen in the soft band . comparison of the blackbody and iron line normalisations seen in the covariance spectra in gx 339 - 4 implies that the short - term blackbody variations are driven by thermal reprocessing of the power - law continuum absorbed by the disc . however , since the amplitude of variable reflection is the same on long and short time - scales , we rule out reprocessing as the cause of the enhanced disc variability on long time - scales . therefore we conclude that the long - time - scale blackbody variations are caused by instabilities in the disc itself , in contrast to the stable discs seen in bhxrb soft states . our results provide the first observational evidence that the low - frequency lorentzian feature present in the psd is produced by the accretion disc . [ firstpage ]

0905.0587

solar and atmospheric neutrino experiments have observed for a long time anomalies that are commonly interpreted as evidences in favor of neutrino oscillations with mass squared differences @xmath0 respectively ( see refs.@xcite ) . more recently , the accelerator lsnd experiment has reported the observation of @xmath1 and @xmath2 appearance @xcite with a mass - squared difference @xmath3 the lsnd evidence in favor of neutrino oscillations has not been confirmed by other experiments , but it has not been excluded either . awaiting an independent check of the lsnd result , that will probably come soon from the miniboone experiment @xcite , it is interesting to consider the possibility that the results of solar , atmospheric and lsnd experiments are due to neutrino oscillations . in this case , the existence of the three mass - squared differences ( [ dm2sun])([dm2lsnd ] ) with different scales implies that there are at least four massive neutrinos ( three massive neutrinos are not enough because the three @xmath4 s have different scales and do not add up to zero ) . since the mass - squared differences ( [ dm2sun])([dm2lsnd ] ) have been obtained by analyzing separately the data of each type of experiment ( solar , atmospheric and lsnd ) in terms of two - neutrino mixing , it is legitimate to ask if three different mass squared are really necessary to fit the data . the answer is `` yes '' , as explained in section [ three ] . although the precise measurement of the invisible width of the @xmath5 boson has determined that there are only three active flavor neutrinos , @xmath6 , @xmath7 , @xmath8 , the possible existence of at least four massive neutrinos is not a problem , because in general flavor neutrinos are not mass eigenstates , _ i.e. _ there is _ neutrino mixing _ ( see , _ e.g. _ , ref.@xcite ) . in general , the left - handed component @xmath9 of a flavor neutrino field is a linear combination of the left - handed components @xmath10 of neutrino fields with masses @xmath11 : @xmath12 , where @xmath13 is the unitary neutrino mixing matrix . the number of massive neutrinos is only constrained to be @xmath14 . following the old principle known as _ occam razor _ , we consider the simplest case of four massive neutrinos that allows to explain all data with neutrino oscillations @xcite . in this case , in the flavor basis the usual three active neutrinos @xmath6 , @xmath7 , @xmath8 , are associated with a sterile neutrino , @xmath15 , that is a singlet of the electroweak group . taking into account the measured hierarchy @xmath16 there are only six types of possible four - neutrino schemes , which are shown in fig.[4schemes ] . these six schemes are divided in two classes : 3 + 1 and 2 + 2 . in both classes there are two groups of neutrino masses separated by the lsnd gap , of the order of 1 ev , such that the largest mass - squared difference generates the oscillations observed in the lsnd experiment : @xmath17 ( where @xmath18 ) . in 3 + 1 schemes there is a group of three neutrino masses separated from an isolated mass by the lsnd gap . in 2 + 2 schemes there are two pairs of close masses separated by the lsnd gap . the numbering of the mass eigenvalues in fig . [ 4schemes ] is conveniently chosen in order to have always solar neutrino oscillations generated by @xmath19 . in 3 + 1 schemes atmospheric neutrino oscillations are generated by @xmath20 , whereas in 2 + 2 schemes they are generated by @xmath21 . in 1999 the 3 + 1 schemes were rather strongly disfavored by the experimental data , with respect to the 2 + 2 schemes @xcite . in june 2000 the lsnd collaboration presented the results of a new improved analysis of their data , leading to an allowed region in the @xmath22@xmath4 plane ( @xmath23 is the two - generation mixing angle ) that is larger and shifted towards lower values of @xmath22 , with respect to the 1999 allowed region . this implies that the 3 + 1 schemes are now marginally compatible with the data . therefore , in section [ 3 + 1 ] i discuss the 3 + 1 schemes , that have been recently revived @xcite . in section [ 2 + 2 ] i discuss the 2 + 2 schemes , that are still favored by the data . let us consider the general expression of the probability of @xmath24 transitions in vacuum valid for any number of massive neutrinos : @xmath25 where @xmath26 is the source - detector distance , @xmath27 is the neutrino energy , and @xmath28 is anyone of the mass eigenstate indices ( a phase common to all terms in the sum in eq.([prob ] ) is irrelevant ) . if all the phases @xmath29 s are very small , oscillations are not observable because the probability reduces to @xmath30 . since the lsnd experiment has the smallest average @xmath31 , of the order of @xmath32 , at least one @xmath33 , denoted by @xmath34 , must be larger than about @xmath35 in order to generate the observed @xmath1 and @xmath2 lsnd transitions , whose measured probability is of the order of @xmath36 . solar neutrino experiments observe large transitions of @xmath6 s into other states , with an average probability of about 1/2 . these transitions can not be generated by a @xmath37 because they should have been observed by the long - baseline chooz experiment @xcite . hence , at least another @xmath33 smaller than about @xmath38 , denoted by @xmath39 , is needed for the oscillations of solar neutrinos . the necessary existence of at least a third @xmath33 for atmospheric neutrino oscillations is less obvious , but can be understood by noticing that a dependence of the transition probability from the energy @xmath27 and/or from the distance @xmath26 is observable only if at least one phase @xmath29 is of order one . indeed , all the exponentials with phase @xmath40 can be approximated to one , whereas all the exponentials with phase @xmath41 are washed out by the averages over the energy resolution of the detector and the uncertainty in the source - detector distance . since the super - kamiokande atmospheric neutrino experiment measures a variation of the oscillation probability for @xmath42 ( see ref.@xcite ) , there must be at least one @xmath33 in the range @xmath43 , which is out of the ranges allowed for @xmath39 and @xmath34 . therefore , at least a third @xmath33 , denoted by @xmath44 , is needed for atmospheric neutrino oscillations . this argument is supported by a detailed calculation presented in ref.@xcite . 3 + 1 schemes . dotted and dashed lines : @xmath45 from bugey @xcite and chooz @xcite . solid lines enclose the allowed regions . , scaledwidth=45.0% ] in the following sections we discuss some phenomenological aspects of the four - neutrino schemes in fig.[4schemes ] , in which there are three mass squared differences with the hierarchy ( [ hierarchy ] ) indicated by the data . 3 + 1 schemes . dotted and dashed lines : @xmath46 from cdhs @xcite and super - kamiokande @xcite . solid lines : allowed regions . , scaledwidth=45.0% ] in 3 + 1 schemes the amplitude of @xmath24 and @xmath47 transitions in short - baseline neutrino oscillation experiments ( equivalent to the usual @xmath22 in the two - generation case ) is given by ( see , for example , ref.@xcite ) @xmath48 and the oscillation amplitude ( again equivalent to the usual two - generation @xmath22 ) in short - baseline @xmath49 disappearance experiments is given by @xmath50 3 + 1 schemes . very thick solid line : allowed regions . thick solid line : disappearance bound ( [ 31bound ] ) . dotted line : lsnd 2000 allowed regions at 90% cl @xcite . solid line : lsnd 2000 allowed regions at 99% cl @xcite . broken dash - dotted line : bugey exclusion curve at 90% cl @xcite . vertical dash - dotted line : chooz exclusion curve at 90% cl @xcite . long - dashed line : karmen 2000 exclusion curve at 90% cl @xcite . short - dashed line : bnl - e776 exclusion curve at 90% cl @xcite . , scaledwidth=45.0% ] 3 + 1 schemes with @xmath51 . solid lines enclose the allowed regions . long dashed line : chorus exclusion curve at 90% cl @xcite . short dashed line : nomad exclusion curve at 90% cl @xcite . dotted line : cdhs exclusion curve at 90% cl @xcite . , scaledwidth=45.0% ] short - baseline @xmath52 and @xmath7 disappearance experiments put rather stringent limits @xmath53 and @xmath54 for @xmath55 in the lsnd - allowed region . taking into account also the results of solar and atmospheric neutrino experiments , eq.([16 ] ) implies that @xmath56 and @xmath57 are small ( see ref.@xcite and references therein ) : @xmath58 as shown by the dashed and dotted lines in figs.[uel4 ] and [ umu4 ] . these limits imply that the amplitude @xmath59 , equivalent to the usual @xmath22 in short - baseline @xmath2 and @xmath1 experiments , is very small : @xmath60 so small to be at the border of compatibility with the oscillations observed in the lsnd experiment . figure [ 31-amuel - allowed ] shows the comparison of the bound ( [ 31bound ] ) with the lsnd allowed region , taking into account also the exclusion curves exclusion curves of the karmen @xcite and bnl - e776 @xcite experiments . one can see that there are four regions that are marginally allowed , denoted by r1 , r2 , r3 , r4 . let us denote by @xmath61 the lower limit for @xmath62 in the four allowed regions in fig.[31-amuel - allowed ] . then , from @xmath63 and the upper bounds ( [ 24 ] ) , one can derive lower limits for @xmath56 and @xmath57 : @xmath64 the upper and lower limits ( [ 24 ] ) and ( [ 124 ] ) for @xmath56 and @xmath57 determine the allowed regions enclosed by solid lines in figs.[uel4 ] and [ umu4 ] . summarizing the general properties of 3 + 1 schemes obtained so far , from fig.[uel4 ] we know that @xmath56 is very small , of the order of @xmath65 , and from fig.[umu4 ] we know that in the regions r2 , r3 , r4 @xmath57 is also very small , of the order of @xmath65 , whereas in the region r1 @xmath57 is relatively large , @xmath66 . on the other hand , the mixings of @xmath8 and @xmath15 with @xmath67 are unknown . the authors of ref.@xcite considered the interesting possibility that @xmath68 _ i.e. _ that the isolated neutrino @xmath67 practically coincides with @xmath15 . notice , however , that @xmath69 can not be exactly equal to one , because lsnd oscillations require that @xmath56 and @xmath57 do not vanish , as shown in figs.[uel4 ] and [ umu4 ] , and unitarity implies that @xmath70 . the possibility ( [ fate ] ) is attractive because it represents a perturbation of the standard three - neutrino mixing in which a mass eigenstate is added , that mixes mainly with the new sterile neutrino @xmath15 and very weakly with the standard active neutrinos @xmath6 , @xmath7 , @xmath8 . in this case , the usual phenomenology of three - neutrino mixing in solar and atmospheric neutrino oscillation experiments is practically unchanged : the atmospheric neutrino anomaly would be explained by dominant @xmath71 transitions , with possible sub - dominant @xmath72 transitions constrained by the chooz bound , and the solar neutrino problem would be explained by an approximately equal mixture of @xmath73 and @xmath74 transitions ( see , for example , ref.@xcite ) . an appealing characteristic of this scenario is the practical absence of transitions of solar and atmospheric neutrinos into sterile neutrinos , that seems to be favored by the latest data ( see @xcite ) . 2 + 2 schemes . see caption of fig.[31-amuel - allowed ] . , scaledwidth=45.0% ] 2 + 2 schemes . dotted and dashed lines : upper limit from bugey @xcite and chooz @xcite . the regions marked by `` a '' enclosed by solid lines are allowed . , scaledwidth=45.0% ] 2 + 2 schemes . solid , dotted and dashed lines : limits from cdhs @xcite , super - kamiokande @xcite and lsnd @xcite . the allowed regions are marked by `` a '' . , scaledwidth=45.0% ] another interesting possibility has been considered in ref.@xcite : @xmath75 this could be obtained , for example , in the hierarchical scheme i ( see fig . [ 4schemes ] ) with an appropriate symmetry keeping the sterile neutrino very light , _ i.e. _ mostly mixed with the lightest mass eigenstates . notice that nothing forbids @xmath69 to be even zero exactly . the possibility ( [ large ] ) is interesting because if it is realized there are relatively large @xmath71 and @xmath74 transitions in short - baseline neutrino oscillation experiments , that could be observed in the near future . this is due to the fact that the unitarity of the mixing matrix implies that @xmath76 is large ( @xmath77 in the regions r2 , r3 , r4 and @xmath78 in the region r1 ) . therefore , the amplitudes @xmath79 and @xmath80 of short - baseline @xmath71 and @xmath74 oscillations are suppressed only by the smallness of @xmath57 and @xmath56 and lie just below the upper limits imposed by the negative results of short - baseline @xmath7 and @xmath52 disappearance experiments . figure [ amuta ] shows the allowed regions in the @xmath81@xmath55 plane . one can see that the region r4 is excluded by the negative results of the chorus @xcite and nomad @xcite experiments . the other three regions are possible and predict relatively large oscillation amplitudes that could be observed in the near future , especially the two regions r2 and r3 in which @xmath82 . an unattractive feature of this scenario is its predictions of large @xmath83 transitions of atmospheric neutrinos , that appear to be disfavored by the latest data ( see @xcite ) . the two 2 + 2 schemes in fig . [ 4schemes ] are favored by the data because they do not suffer the constraint imposed by the thick solid line in fig.[31-amuel - allowed ] , that is valid only in 3 + 1 schemes . therefore , all the part of the lsnd region in the @xmath59@xmath84 plane that is not excluded by other experiments is allowed , as shown in fig.[22-amuel - allowed ] . for this reason , the phenomenology of 2 + 2 schemes has been studied in many articles @xcite . figures [ sel ] and [ smu ] show the limits on the mixing of @xmath6 and @xmath7 obtained from the results of short - baseline , solar and atmospheric experiments @xcite . from fig . [ sel ] one can see that the mixing of @xmath6 with @xmath85 and @xmath67 , whose mass - squared difference @xmath86 generates atmospheric neutrino transitions , is very small , leading to a suppression of oscillations of @xmath6 s in atmospheric and long - baseline experiments @xcite . the mixing of @xmath8 and @xmath15 is almost unknown , with weak limits obtained in recent fits of solar @xcite and atmospheric data @xcite . for example , it is possible that both solar @xmath6 s and atmospheric @xmath7 s oscillate into approximately equal mixtures of @xmath8 and @xmath15 s . in the future it may be possible to exclude the scheme a if it will be established with confidence that the effective number of neutrinos in big - bang nucleosynthesis is less that four . in this case @xmath87 @xcite and solar and atmospheric neutrino oscillations occur , respectively , through the decoupled channels @xmath88 and @xmath71 . it has been shown that in this scenario the small mass splitting in scheme a between @xmath85 and @xmath67 is incompatible with radiative corrections @xcite and the effective majorana mass in neutrinoless double - beta decay in scheme a is at the border of compatibility with the experimental limit @xcite . four - neutrino mixing is a realistic possibility ( if the solar , atmospheric and lsnd anomalies are due to neutrino oscillations ) . it is rather complicated , but very interesting , both for theory and experiments , because : it has a rich phenomenology ; the existence of a sterile neutrino is far beyond the standard model , hinting for exciting new physics ; there are several observable oscillation channels in short - baseline and long - baseline experiments ; cp violation may be observable in long - baseline experiments @xcite .

the main features of four - neutrino 3 + 1 and 2 + 2 mixing schemes are reviewed , after a discussion on the necessity of at least four massive neutrinos if the solar , atmospheric and lsnd anomalies are due to neutrino oscillations .

hep-ph0012236

deuterium is understood to be only produced in significant amount during primordial big bang nucleosynthesis ( bbn ) and thoroughly destroyed in stellar interiors . deuterium is thus a key element in cosmology and in galactic chemical evolution ( see e.g. audouze & tinsley 1976 ) . indeed , its primordial abundance is the best tracer of the baryonic density parameter of the universe @xmath7 , and the decrease of its abundance during the galactic evolution should trace the amount of star formation ( among other astrophysical interests ) . in the galactic ism , d / h measurements made toward hot stars have suggested variations : imaps observations toward @xmath8 ori led to a low value ( jenkins _ et al . _ 1999 ) , confirming the previous analysis by laurent _ et al . _ ( 1979 ) from _ copernicus _ observations , while toward @xmath9 vel they led to a high value ( sonneborn _ et al . _ 2000 this seems to indicate that in the ism , within few hundred parsecs , d / h may vary by more than a factor @xmath10 . in the nearby ism , the case of g191b2b was studied in detail ( see the most recent analysis by lemoine _ et al . _ 2002 ) and the evaluation toward capella ( linsky _ et al . _ 1995 ) taken as a reference . their comparison provided , for a while , a possible case for d / h variations within the local ism . concerning g191b2b , lemoine _ et al . _ ( 2002 ) have shown that the total @xmath11(h@xmath0i ) column density evaluation was greatly perturbed by the possible addition of two broad and weak h@xmath0i components . such components , able to mimic the shape of the lyman @xmath12 damping wings , can induce an important decrease of the evaluated @xmath11(h@xmath0i ) . to illustrate this point , the error bar estimation on @xmath11(h@xmath0i ) from all previously published studies considered as the extremes of a 2@xmath4 limit was of the order of dex 0.07 , while including the lemoine _ et al . _ ( 2002 ) analysis enlarged the error bar to about dex 0.37 . this huge change has , of course , a considerable impact on any d / h evaluation . this raises two crucial questions . first , is that situation typical of g191b2b alone and possibly due to an unexpected shape of the core of the stellar lyman @xmath12 profile improperly described by the theoretical models ? second , if weak h@xmath0i features are present in the ism , to what extent are evaluations toward other targets affected ? from the combination of _ stis _ echelle observations ( spectrograph on board the hubble space telescope , hst ) and _ fuse _ ones ( the far ultraviolet spectroscopic explorer , moos _ et al . _ , 2000 ) , lemoine _ et al . _ ( 2002 ) have found through iterative fitting process ( with the owens.f fitting program developed by martin lemoine and the french fuse team ) that three interstellar absorption components are present along the line of sight and that two additional broad and weak h@xmath0i components could be added , detected only over the lyman @xmath12 line ( negligible over the lyman @xmath13 line ) but important enough to strongly perturb the total h@xmath0i column density evaluation . within the local ism , it has been shown that such additional hi absorptions are often present ; they have been interpreted either as cloud interfaces with the hot gas within the local ism ( bertin _ et al _ 1995 ) or as `` hydrogen walls '' , signature of the shock interaction between the solar wind ( or stellar wind ) and the surrounding ism ( linsky , 1998 ) . this latter heliospheric absorption has been modeled by wood _ et al . _ ( 2000 ) and a prediction derived in the direction of g191b2b ( see figure 9 of lemoine _ et al . _ 2002 ) . most of the predicted absorption is expected in the saturated core of the observed interstellar line but some weak absorption ( @xmath14 of the continuum ) might extend over several tenths of angstroms on the red side of the line , due to the neutral hydrogen atoms seen behind the shock in the downwind direction where g191b2b is located . it was found that the combination of two broad and weak hi components can easily reproduce the model prediction . if real , besides the three interstellar absorptions , a fourth component representing the bulk of the predicted absorption and a fifth one for the broad and shallow extended red wing are needed . this is exactly what lemoine _ et al . _ ( 2002 ) have found . in the course of determining the minimum number of components ( each defined by its hi column density @xmath1 , its velocity @xmath15 , its temperature @xmath16 and turbulence broadening @xmath17 ) needed to fit the data , lemoine _ et al . _ ( 2002 ) completed the @xmath18test which uses the fisher - snedecor law describing the probability distribution of @xmath3 ratio . what is tested is the probability that the decrease of the @xmath3 with additional components is not simply due to the increase of free parameters . the result gives a probability @xmath19 and @xmath20 that a fourth and a fifth hi component are respectively not required by the data . these low probabilities of non occurence strongly suggest that lemoine _ ( 2002 ) have indeed detected the heliospheric absorption downwind in the direction of g191b2b . note however that this heliospheric complex absorption profile is simulated by two components whose physical meaning in terms of hydrogen content and/or temperature is not clear . furthermore , the photospheric lyman @xmath12 stellar core is difficult to evaluate ( see discussion in e.g. lemoine _ et al . _ 2002 ) and is slightly red - shifted relative to the ism absorptions ; this result may very well be simply related to the use of a white dwarf as background target star . the detailed analysis of the capella line of sight could directly test the heliospheric hypothesis . if the two additional components present along the g191b2b line of sight are as a matter of fact due to an heliospheric phenomenon , it is an extremely local signature ( within few hundreds of astronomical units to be compared to the few tens of parsecs lines of sight lengths ) which should be also present along the capella sight - line , both stars being separated by only @xmath21 on the sky , and similar in shape to the structure predicted and observed in the direction of g191b2b . if that description is correct , we are expecting an extra absorption reasonably represented by two additional components , a main one mostly lost within the ism absorption core and a weak one extending over several tenths of angstroms on the red side of the line , again due to the neutral hydrogen atoms seen behind the shock in the downwind direction where both g191b2b and capella are located . recently , young _ et al . _ ( 2002 ) analysed new obervations obtained at lyman @xmath13 , lyman @xmath22 and the whole lyman series with _ fuse_. the precise lyman @xmath12 stellar profile compatible with all lyman lines and with the data sets obtained at different phases of the capella binary system ( see also linsky _ et al . _ 1995 ) was reevaluated ( wood , 2001 ) and is used here as a reference profile , s@xmath23 . we thus revisited the fits completed over the lyman @xmath12 line as observed toward capella with the best available data set , i.e. the one obtained with the _ ghrs _ ( the goddard high resolution spectrograph on board _ hst _ ) . the study by linsky _ ( 1995 ) , essentially confirmed by vidal ( 1998 ) , shows that only one interstellar component ( the local interstellar cloud , lic , also seen toward g191b2b ) is needed on that line of sight . this very simple structure strengthens the capella case as the simplest one where d / h can be very well evaluated . however , vidal madjar _ et al . _ ( 1998 ) have already noted that an additional weak and broad hi component was required to better reproduce the profile ; this was a first indication of the presence of an heliospheric absorption toward capella . in fact , we were able to show that , as in the case of g191b2b , the addition of one or two weak and broad hi components ( together with the very weak geocoronal component present at a known velocity but not shown on figure [ cap - contvar ] for clarity ) improves the @xmath3 . more precisely we fitted the ghrs data assuming that the stellar continuum was s@xmath23 . adding successively to the fit one then two free hi components ( the added hi components have only three free parameters , velocity @xmath15 , column density @xmath1 and width * t * , since the thermal @xmath16 or turbulent broadening @xmath17 act in an undifferentiated manner when only one species is observed ) we obtained the following @xmath3/degree of freedom(d.o.f . ) values : for only the lic component and the geocorona , 844.89/716 ; for one additional component , 831.17/713 ; for two additional components , 822.68/710 . the @xmath18test probabilities that these two additional components are not required by the data are respectively @xmath24 and @xmath25 . the first one is clearly needed here ( its correlated parameter ranges according to different possible solutions similar in terms of @xmath3 are : @xmath26 @xmath15 ( km s@xmath27 ) @xmath28 ; @xmath29 @xmath1 ( @xmath30 ) @xmath31 ; @xmath32 * t * ( k ) @xmath33 ) but unlike in the case of g191b2b , the second one corresponding to the weaker and broader one ( parameter ranges are : @xmath34 @xmath15 ( km s@xmath27 ) @xmath35 ; @xmath36 @xmath1 ( @xmath30 ) @xmath37 ; @xmath38 * t * ( k ) @xmath39 ) is less strongly needed . these ranges are certainly compatible with the corresponding estimated values in the direction of g191b2b ( see figure 10 of lemoine _ et al . _ 2002 ) . to search for the possible impact of the choice of the continuum on the evaluation of @xmath1(hi ) , we fixed this value and looked for the best fitted solutions while the stellar continuum we used s@xmath23 was allowed for some variations by multiplying it by a low order polynomial ( 8@xmath40 order ) which coefficients were free to vary along with all components parameters . results are shown in figure [ cap - contvar ] . slight changes of the continuum shape by no more than @xmath41 lead to nearly identical @xmath3 values , with @xmath1(hi ) varying from @xmath42 to @xmath43 which corresponds to a change in d / h from @xmath44 to @xmath45 . this is clearly a larger range ( @xmath46 ) than the one previously claimed ( @xmath47 , linsky _ et al . . the situation could be even worse since we do not know how far the capella continuum could be away from s@xmath23 . the question is thus to evaluate if the capella lyman @xmath12 stellar continuum shape is estimated to better than @xmath41 or not ? it is true that having a binary system can help constraining the continuum shape as linsky _ ( 1995 ) did , but their whole approach requires that the lyman @xmath12 stellar profiles of both g1 iii and g8 iii stars are invariant with phase and time . in fact , from the study of 120 _ iue _ echelle spectra , ayres _ ( 1993 ) have shown on one hand that the line fluxes were surprisingly stable , but on the other hand that whichever way they process the data , obvious variations were seen . these seem to be related to variations of the blue peak of the g1 iii dominant stellar lyman @xmath12 line . they found that in the 19811986 interval , the line shape at phase 0.25 of the system was quite stable and similar to the one recorded with the _ ghrs _ in 1991 ( at a @xmath41 level ) . earlier spectra taken in 1980 or later ones observed after 1986 look quite different . this very careful study shows that with the _ iue _ sensitivity level of @xmath48 , variations are clearly detected . since linsky _ ( 1995 ) used _ ghrs _ observations at two different phases of the system ( 0.26 and 0.80 ) taken respectively in april 1991 and in september 1993 , i.e. two and a half year apart , it is difficult to ascertain that the lyman @xmath12 profile evaluated for each stellar component is well controlled . because of the very careful analysis made by linsky _ ( 1995 ) , it may be possible that the stellar lyman @xmath12 profiles are relatively well evaluated but certainly not at a level better than @xmath41 as previously mentioned . thus , an heliospheric absorption is also detected on the capella sight line ; furthermore , even in such a simple ism configuration ( a unique component ) , it appears impossible to tightly constrain the total hi column density in that direction . we have shown that , for two lines of sight , @xmath1(hi ) can not be evaluated with a high accuracy . column densities on both sight lines are very similar , of the order of few times @xmath49 . for lower column densities , the situation should be worse since then the possible absorption signature of the weak components is becoming relatively more and more important and the lyman @xmath12 line is getting closer to the flat part of the curve of growth where column densities are indeed difficult to evaluate . note however that the hst / euve comparison completed by linsky _ et al . _ ( 2000 ) shows that often @xmath1(hi ) values derived from the ghrs and euve data ( not sensitive to weak hi ) are in good agreement , implying that heliospheric absorption ( or other hot components ) do nt necessarily ruin lyman - alpha analyses in a dramatic way . but clearly counter examples leave that question open due to possible systematics related to the evaluation of total hi below the lyman limit in the euve domain . on the contrary , one could guess that for larger column densities the situation should improve since the lyman @xmath12 damping wings are becoming broader and the signature of the weak features may disapear in the line core . just above @xmath5 , the reliability of the d / h values is greatly enhanced if the studied gas is demonstrably warm ( 6000 k ; for a thorough discussion see york , 2001 ) ; as one goes above @xmath6 , credibility increases unless either cold gas components are hidden in the warm di but still affect the hi damping wings or weak hi features at high velocity are present . the unknown referee further stressed this point through an impressive report . he mentionned that at @xmath1(hi)=@xmath50 , the half - intensity point of pure damping lyman @xmath12 and lyman @xmath13 profiles , located respectively at velocity shifts of 274 km s@xmath27 and 55 km s@xmath27 , should be the place where a putative high - velocity feature could have a strong perturbing influence on the damping profile . however only very few high velocity ism components were detected above 120 km s@xmath27 . this could lead to the inverse impression that for larger column densities , the estimation through the lyman @xmath13 line should be more questionable than the one made at lyman @xmath12 . as a matter of fact , in both the @xmath22 cas and @xmath51 pup lines of sight , the hi column density was discrepant when derived through the lyman @xmath12 or the lyman @xmath13 line ( see below ) . however in both cases the lyman @xmath13 estimations of @xmath1(hi ) are smaller , in contradiction with the formerly suggested cause since the most perturbed evaluation by additional absorptions should lead instead to larger column densities . high velocity ism components essentially observed below 120 km s@xmath27were only searched for through other lines and species than hi at lyman @xmath12 the strongest transition of the most abundant element . for instance , cowie _ et al . _ ( 1979 ) reported lyman @xmath8 and lyman @xmath52 hi ism absorptions up to about 105 km / s for @xmath53 ori . from their study , the referee evaluated that a shock at 274 km s@xmath27 should produce either a very broad ( @xmath54 km s@xmath27 ) and undetectable ( maximum depth of @xmath55 ) post - shock absorption signature or should originate in a region far downstream from the front where the gas has cooled and compressed enough to allow recombination of the h atoms , i.e. a shock from a supernova explosion entering the radiative phase . in this second case however , he estimated from cowie and york ( 1978 ) and spitzer ( 1978 ) that such signatures should occur only very near a known supernova event , i.e. within about 30 to 60 pc for standard ism and sn values . thus , that looks unlikely too . on the other hand , high - velocity gas could be generated by the target stars . while gry , lamers and vidal madjar ( 1984 ) seem to detect most of the activity at velocities below 100 km / s , they nevertheless identified a transient component at -150 km / s toward @xmath9 vel through lyman @xmath8 and another one at -220 km / s toward @xmath53 ori , through lyman @xmath52 . note also that this survey was completed over a limited spectral domain scanned with the copernicus instrument , and not at lyman @xmath12 , _ i.e. _ with a relatively limited sensitivity . one might argue that there are some risks that stellar ejecta could influence the lyman @xmath12 measurements ; but the observers have a good defense : multiple observations at very different epochs . this strategy was invoked by jenkins _ ( 1999 ) and sonneborn _ et al . _ ( 2000 ) in their studies of d / h toward @xmath8 ori , @xmath51 pup and @xmath9 vel . their findings are thus pretty convincing in this regard . all the above stated arguments should mitigate our concern that small amounts of hi at high velocities are a likely source of confusion for the flanks of the damping profiles for @xmath1(hi ) of the order of or greater than @xmath56 @xmath30 . one however should recall the two lines of sight for which the hi column density was discrepant when derived through the lyman @xmath12 or the lyman @xmath13 line : * @xmath22 cas + in bohlin , savage and drake ( 1978 ) , lyman @xmath12 only ( _ copernicus _ ) @xmath1(hi ) = @xmath57 + in ferlet _ et al . _ ( 1980 ) , core of lyman @xmath13 only ( _ copernicus _ ) @xmath1(hi ) = @xmath58 + in diplas and savage ( 1994 ) , from lyman @xmath12 only ( _ iue _ ) @xmath1(hi ) = @xmath59 * @xmath51 pup + in bohlin ( 1975 ) from lyman @xmath12 only ( _ copernicus _ ) @xmath1(hi ) = @xmath60 + in vidal madjar _ et al . _ ( 1977 ) , from lyman @xmath13 only ( _ copernicus _ ) @xmath1(hi ) = @xmath61 + in diplas and savage ( 1994 ) , from lyman @xmath12 only ( _ iue _ ) @xmath1(hi ) = @xmath62 + in sonneborn _ et al . _ ( 2000 ) , from lyman @xmath12 only ( _ iue _ ) @xmath1(hi ) = @xmath63 since lyman @xmath13 is less sensitive to weak hi features , it is interesting to note that in both cases the lyman @xmath13 evaluation is slightly below the lyman @xmath12 one , a possible indication of a similar effect for column densities of the order of @xmath6 . note however that these two cases are marginally convincing since the different evaluations are still compatible within the error bars . therefore , if this effect is indeed real for relatively large column densities , it means that some of the d / h values may be underestimated and thus the higher d / h ratios may be favoured . this further shows the importance of the @xmath64 vel estimation ( sonneborn _ et al . according to our conclusion that there is lower precision for @xmath1(hi ) measurements within the local ism , i.e. at low column densities , which induces large error bars on the corresponding d / h estimations : + * the @xmath1(hi ) evaluation in the direction of capella is relatively less accurate than previously claimed and is of the order of log @xmath1(hi ) @xmath65 leading to d / h @xmath66 ; * an average d / h ratio may exist in the local ism but should be larger than previously evaluated since locally @xmath1(hi ) could be overestimated by as much as about 20% ; this does affect arguments about local variability ; * d / o should be a better tracer of d variations as originally suggested by timmes _ ( 1997 ) , and directly verified in the lism ( moos _ et al . _ 2002 ; hbrard _ et al . _ 2002a , 2002b ) and further confirmed by the stability of o / h over longer path length ( meyer _ et al . _ 1998 ; andr _ et al . _ 2002 ) , provided all sources of errors related to the oxygen measurements are well understood . finally , we note also that a similar systematic effect has been pointed out by pettini and bowen ( 2001 ) for the evaluation of d / h in quasars absorption line systems . again as in our study , the systems presenting the highest values of @xmath1(hi ) are derived from the damping wings of the lyman @xmath12 line , which also includes all hi in close proximity of the hydrogen at the line center of deuterium ( burles , 2001 ) , while those for which @xmath1(hi ) is evaluated from the discontinuity at the lyman limit present smaller column densities and larger d / h evaluations . we would like to thank brian wood who kindly provided to us the capella lyman @xmath12 stellar profile he has evaluated for his own study of that line of sight as well as jeff linsky and jeff kruk for constructive comments . we are also pleased to warmly thank don york for about twenty seven years of exciting collaboration and martin lemoine for the last decade of common enlightening work . we deeply thank our unknown referee whose report was nearly as long as this letter and briefly summarized here .

the deuterium abundance evaluation in the direction of capella has for a long time been used as a reference for the local interstellar medium ( ism ) within our galaxy . we show here that broad and weak h@xmath0i components could be present on the capella line of sight , leading to a large new additional systematic uncertainty on the @xmath1(h@xmath0i ) evaluation . the d / h ratio toward capella is found to be equal to @xmath2 with almost identical @xmath3 for all the fits ( this range includes only the systematic error ; the 2 @xmath4 statistical one is almost negligible in comparison ) . it is concluded that d / h evaluations over h@xmath0i column densities below @xmath5 ( even perhaps below @xmath6 if demonstrated by additional observations ) may present larger uncertainties than previously anticipated . it is mentionned that the d / o ratio might be a better tracer for d@xmath0i variations in the ism as recently measured by the far ultraviolet spectroscopic explorer ( _ fuse _ ) .

astro-ph0204386

in this paper , all graphs considered are finite , simple and undirected . we use @xmath5 , @xmath6 , @xmath7 and @xmath2 to denote the vertex set , the edge set , the minimum degree and the maximum degree of a graph @xmath1 , respectively . denote @xmath8 and @xmath9 . let @xmath10 ( or @xmath11 for simple ) denote the degree of vertex @xmath12 . a @xmath13- , @xmath14- and @xmath15-@xmath16 is a vertex of degree @xmath13 , at least @xmath13 and at most @xmath13 , respectively . any undefined notation follows that of bondy and murty @xcite . a graph @xmath1 is @xmath0-immersed into a surface if it can be drawn on the surface so that each edge is crossed by at most one other edge . in particular , a graph is @xmath0-planar if it is @xmath0-immersed into the plane ( i.e. has a plane @xmath0-immersion ) . the notion of @xmath0-planar - graph was introduced by ringel @xcite in the connection with problem of the simultaneous coloring of adjacent / incidence of vertices and faces of plane graphs . ringel conjectured that each @xmath0-planar graph is @xmath17-vertex colorable , which was confirmed by borodin @xcite . recently , albertson and mohar @xcite investigated the list vertex coloring of graphs which can be @xmath0-immersed into a surface with positive genus . borodin , et al . @xcite considered the acyclic vertex coloring of @xmath0-planar graphs and proved that each @xmath0-planar graph is acyclically @xmath18-vertex colorable . the structure of @xmath0-planar graphs was studied in @xcite by fabrici and madaras . they showed that the number of edges in a @xmath0-planar graph @xmath1 is bounded by @xmath19 . this implies every @xmath0-planar graph contains a vertex of degree at most @xmath20 . furthermore , the bound @xmath20 is the best possible because of the existence of a @xmath20-regular @xmath0-planar graph ( see fig.1 in @xcite ) . in the same paper , they also derived the analogy of kotzig theorem on light edges ; it was proved that each @xmath21-connected @xmath0-planar graph @xmath1 contains an edge such that its endvertices are of degree at most @xmath18 in @xmath1 ; the bound @xmath18 is the best possible . the aim of this paper is to exhibit a detailed structure of @xmath0-planar graphs which generalizes the result that every @xmath0-planar graph contains a vertex of degree at most @xmath20 in section 2 . by using this structure , we answer two questions on light graphs posed by fabrici and madaras @xcite in section 3 and give a linear upper bound of acyclic edge chromatic number of @xmath0-planar graphs in section 4 . to begin with , we introduce some basic definitions . let @xmath1 be a @xmath0-planar graph . in the following , we always assume @xmath1 has been drawn on a plane so that every edge is crossed by at most one another edge and the number of crossings is as small as possible ( such a dawning is called to be @xmath22 ) . so for each pair of edges @xmath23 that cross each other at a crossing point @xmath24 , their end vertices are pairwise distinct . let @xmath25 be the set of all crossing points and let @xmath26 be the non - crossed edges in @xmath1 . then the @xmath27 @xmath28 @xmath29 @xmath30 of @xmath1 is the plane graph such that @xmath31 and @xmath32 . thus the crossing points in @xmath1 become the real vertices in @xmath30 all having degree four . for convenience , we still call the new vertices in @xmath30 crossing vertices and use the notion @xmath33 to denote the set of crossing vertices in @xmath30 . a simple graph @xmath1 is @xmath34 if every cycle of length @xmath35 has an edge joining two nonadjacent vertices of the cycle . we say @xmath36 is a @xmath37 @xmath38 of a @xmath0-planar graph @xmath1 if @xmath36 is obtained from @xmath1 by the following operations . * step 1*. for each pair of edges @xmath39 that cross each other at a point @xmath40 , add edges @xmath41 and @xmath42 `` close to @xmath40 '' , i.e. so that they form triangles @xmath43 and @xmath44 with empty interiors . * step 2*. delete all multiple edges . * step 3*. if there are two edges that cross each other then delete one of them . * step 4*. triangulate the planar graph obtained after the operation in step 3 in any way . * step 5*. add back the edges deleted in step 3 . note that the associated planar graph @xmath45 of @xmath36 is a special triangulation of @xmath30 such that each crossing vertex remains to be of degree four . also , each vertex @xmath46 in @xmath45 is incident with just @xmath47 @xmath21-faces . denote @xmath48 to be the neighbors of @xmath46 in @xmath45 ( in a cyclic order ) and use the notations @xmath49 , @xmath50 , where @xmath51 and @xmath52 is taken modulo @xmath53 . in the following , we use @xmath54 to denote the number of crossing vertices which are adjacent to @xmath46 in @xmath45 . then we have the following observations . since their proofs of them are trivial , we omit them here . in particular , the second observation uses the facts that @xmath36 admits no multiple edge and the drawing of @xmath36 minimizes the number of crossing . [ obs ] for a canonical triangulation @xmath36 of a @xmath0-planar simple graph @xmath1 , we have \(1 ) any two crossing vertices are not adjacent in @xmath45 . \(2 ) if @xmath55 , then @xmath56 . \(3 ) if @xmath57 , then @xmath58 . \(4 ) if @xmath59 , then @xmath60 . let @xmath61 and @xmath62 be a crossing vertex in @xmath45 such that @xmath63 . then by the definitions of @xmath64 and @xmath65 , we have @xmath66 . furthermore , the path @xmath67 in @xmath45 corresponds to the original edge @xmath68 with a crossing point @xmath62 in @xmath36 . let @xmath69 be the neighbor of @xmath46 in @xmath36 so that @xmath70 crosses @xmath68 at @xmath62 in @xmath36 . by the definition of @xmath45 , we have @xmath71 . we call @xmath69 the @xmath72-@xmath73 of @xmath46 in @xmath36 and @xmath74 the @xmath75-@xmath76 of @xmath46 in @xmath36 . other neighbors of @xmath46 in @xmath36 are called @xmath77-@xmath76 . sometimes when we say mirror vertex , image vertex and normal vertex , we refer to mirror neighbor , image neighbor and normal neighbor of @xmath46 . the triangle @xmath78 in @xmath36 is called the @xmath72-@xmath79 incident with @xmath46 . since the neighbors of @xmath46 in @xmath45 can be listed in a cyclic order , via replacing the crossing vertex by the mirror vertex incident with it , the neighbors of @xmath46 in @xmath36 can be also listed in a cyclic order . note that different crossing vertices are adjacent to different mirror vertices since multiple edges are forbidden in @xmath1 . let @xmath80 be the neighbors of @xmath46 in @xmath36 in a cyclic order . we define @xmath81 , @xmath82 and @xmath83 , where @xmath52 is taken modulo @xmath84 . note that @xmath36 is a canonical triangulation of @xmath1 . then @xmath85 is a cycle which is called the @xmath27 @xmath86 of @xmath46 , denoted by @xmath87 . we call the path @xmath88 a @xmath89 of @xmath87 if ( a ) the elements of @xmath90 are image neighbors of @xmath46 ; ( b ) the elements of @xmath91 are mirror neighbors of @xmath46 ; ( c ) the triangles in @xmath36 of the form @xmath92 where @xmath93 are mirror triangles incident with @xmath46 and ( d ) @xmath94 and @xmath95 are not mirror neighbors of @xmath46 . then @xmath96 of a segment @xmath97 is defined to be the number of mirror triangles incident with @xmath46 using vertices in @xmath98 , denoted by @xmath99 . then we easily have @xmath100 . of a 1-planar graph @xmath1,title="fig:",width=604,height=207 ] + now we show the main result in this section . [ structure ] let @xmath1 be a @xmath0-planar simple graph . then there exists a vertex @xmath46 in @xmath1 with exactly @xmath13 neighbors @xmath101 satisfying @xmath102 such that one of the following is true . ( @xmath1031 ) @xmath104 , ( @xmath1032 ) @xmath105 with @xmath106 , ( @xmath1033 ) @xmath107 with @xmath108 and @xmath109 , ( @xmath1034 ) @xmath110 with @xmath111 , @xmath112 and @xmath113 , ( @xmath1035 ) @xmath114 with @xmath115 , @xmath116 , @xmath117 and @xmath118 , ( @xmath1036 ) @xmath119 with @xmath120 , @xmath121 , @xmath122 , @xmath123 and @xmath124 . the theorem is proved by contradiction . let @xmath1 be a simple @xmath0-planar graph with a fixed embedding in the plane , and suppose @xmath1 is a counterexample to the theorem . note that if we add a new edge @xmath125 between two nonadjacent vertices in @xmath1 so that @xmath126 is still 1-planar graph , then @xmath126 shall also be a counterexample to the theorem . hence in the following , without loss of generality , we always assume @xmath1 is 2-connected and @xmath127 , where @xmath36 is a canonical triangulation of @xmath1 that has been draw on a plane properly . in other words , @xmath1 is just a canonical triangulation of itself . so in the next , there is no necessity to distinguish the two notions @xmath1 and @xmath36 , and when we say @xmath30 , we also refer to @xmath45 . on the other hand , by the definition of associated plane graph , one can observe that @xmath128 when @xmath46 is not a crossing vertex . so in the detail proof below , we either need not to distinguish @xmath129 and @xmath10 when @xmath46 is a vertex of @xmath1 , in which case we only use the notation @xmath11 to represent both @xmath129 and @xmath10 . for a fixed vertex @xmath46 of @xmath1 , we define @xmath130 ( @xmath131 in short ) to be the number of neighbors of @xmath46 in @xmath1 which are of degree @xmath52 . for a vertex set @xmath132 , let @xmath133 . denote @xmath134 and @xmath135 . for a subgraph @xmath136 of @xmath1 , @xmath137 represents the number of @xmath52-vertices contained in @xmath136 . let @xmath87 be the associated cycle of @xmath46 . suppose @xmath87 has @xmath138 segments , denoted by @xmath139 ( in this cyclic order ) . denote @xmath140 . let @xmath141 ( @xmath142 ) . we call the vertex set @xmath143 the @xmath144 of the associated cycle @xmath87 . for each segment @xmath97 , we define a graph @xmath145 so that @xmath146 and @xmath147 . let @xmath148 . by the definition of @xmath97 , we have @xmath149 . ( see fig . [ cal ] ) . a triangle @xmath150 in @xmath1 is @xmath151 if @xmath152 . otherwise we say that @xmath150 is @xmath153 . note that for the vertex @xmath46 described above , there are @xmath154 mirror triangles incident with it ( recall the definition of the parameter @xmath154 ) . now , suppose @xmath155 of them are heavy and @xmath156 of them are light . we divide all the light mirror triangles incident with @xmath46 into three classes . class slowromancap1@. triangles in the form @xmath150 such that @xmath157 is mirror vertex and @xmath158 are image vertices with @xmath159 and @xmath160 . class slowromancap2@. triangles in the form @xmath150 such that @xmath157 is mirror vertex and @xmath158 are image vertices with @xmath161 . class slowromancap3@. triangles in the form @xmath150 such that @xmath162 . denote the number of triangles belonging to class slowromancap1@ , slowromancap2@ and slowromancap3@ by @xmath163 and @xmath164 , where @xmath165 and @xmath166 . * claim 1*. @xmath167 . since each heavy mirror triangle incident with @xmath46 covers at least one @xmath168-vertex , there are at least @xmath169 @xmath168-vertices contained in heavy mirror triangles . and this lower bound is reachable only if each heavy mirror triangle covers exactly one @xmath168-vertex such that each pair of incident mirror triangles share one common @xmath168-vertex . for each class slowromancap2@ light mirror triangle @xmath150 such that @xmath157 is mirror vertex and @xmath158 are image vertices with @xmath170 , since ( @xmath1031)-(@xmath1034 ) are forbidden in @xmath1 , we have @xmath171 and another three neighbors of @xmath172 are all @xmath173-vertices . let @xmath87 be the associated cycle of @xmath46 . then @xmath174 . if @xmath175 , let @xmath176 . then @xmath177 is a normal vertex . so @xmath177 could be incident with at most two image vertices of degree no more than five . if @xmath178 , let @xmath179 . then @xmath180 is a heavy mirror triangle with two @xmath168-vertices . in either case , we would account at least @xmath181 new @xmath168-vertices which are not counted in the above step . hence , we have @xmath182 . * claim 2*. there is an integer @xmath183 such that @xmath184 . since each class slowromancap1@ light mirror triangle contains two vertices either of degree @xmath17 or of degree @xmath20 and each class slowromancap3@ light mirror triangle contains three vertices either of degree @xmath17 or of degree @xmath20 , we deduce that @xmath185 . * claim 3*. @xmath186 . since ( @xmath1031 ) is forbidden , we have @xmath187 and @xmath188 . note that @xmath189 . we deduce from claim 1 and claim 2 that @xmath186 . * claim 4*. @xmath190 . recall the definitions of @xmath191 and @xmath192 where @xmath142 . each @xmath145 contains @xmath193 mirror triangles incident with @xmath46 . suppose @xmath194 of them are light and @xmath195 of them are heavy . then @xmath196 . since ( @xmath1033 ) is forbidden , no light mirror triangle contains @xmath197-vertex . so the neighbors of @xmath46 in @xmath1 with degree @xmath197 are all contained either in the heavy mirror triangles or in the interval of @xmath87 . note that all the image vertices contained in @xmath198 are of degree at least five ( see figure [ cal ] ) , so if there is a 4-vertex in @xmath198 , then it must be a mirror vertex . in view of this , one can easily claim that @xmath198 contains at most @xmath195 @xmath197-vertices . furthermore , if @xmath199 , @xmath200 and @xmath201 , then @xmath202 are all @xmath173-vertices since ( @xmath1033 ) is forbidden in @xmath1 . so the triangles @xmath203 and @xmath204 are both heavy . then one can similarly claim that @xmath205 contains at most @xmath206 @xmath197-vertices , where @xmath207 . by ( 2 ) of observation [ obs ] and the definition of @xmath145 , if there are @xmath21-vertices on @xmath87 , they must be on the interval . suppose there are @xmath208 @xmath21-vertices in @xmath192 where @xmath209 . if @xmath210 , then @xmath192 contains at least @xmath211 non-@xmath197-vertices since ( @xmath1032 ) and ( @xmath1033 ) are forbidden . here , note that neither @xmath212 nor @xmath213 can be @xmath21-vertex by ( 2 ) of observation [ obs ] since each image vertex is adjacent to a crossing vertex in @xmath30 . so @xmath214 if @xmath210 . if @xmath199 , then @xmath215 . so the above inequation on @xmath216 holds unless @xmath200 and @xmath201 . in this special case , this inequation becomes @xmath217 indeed , but on the other hand , we have @xmath218 and @xmath219 by the former arguments . note that @xmath220 and @xmath221 . so we can deduce that @xmath222 . hence , we have @xmath190 . now we apply the discharging method to the associated planar graph @xmath30 of @xmath1 . since @xmath30 is a planar graph and @xmath223 . by euler s formula , we have @xmath224 now we define @xmath225 to be the initial charge of @xmath226 . let @xmath227 for each vertex @xmath12 and let @xmath228 and for each face @xmath229 . it follows that @xmath230 . we now redistribute the initial charge @xmath225 and form a new charge @xmath231 for each @xmath226 by discharging method . since our rules only move charge around , and do not affect the sum , we have @xmath232 . a @xmath21-face in @xmath30 is @xmath233 if it is incident with one crossing vertex . our discharging rules are defined as follows : ( @xmath2341 ) each non - special @xmath21-face in @xmath30 receives @xmath235 from each vertex incident with it ; ( @xmath2342 ) each special @xmath21-face in @xmath30 receive @xmath236 from each non - crossing vertex incident with it ; ( @xmath2343 ) each vertex @xmath46 in @xmath1 with @xmath237 sends @xmath238 to each adjacent @xmath20-vertex in @xmath1 ; ( @xmath2344 ) each vertex @xmath46 in @xmath1 with @xmath239 sends @xmath240 to each adjacent @xmath20-vertex and @xmath241 to each adjacent @xmath17-vertex in @xmath1 ; ( @xmath2345 ) each vertex @xmath46 in @xmath1 with @xmath242 sends @xmath243 to each adjacent @xmath20-vertex , @xmath244 to each adjacent @xmath17-vertex and @xmath245 to each adjacent @xmath246-vertex in @xmath1 ; ( @xmath2346 ) each vertex @xmath46 in @xmath1 with @xmath247 sends @xmath248 to each adjacent @xmath20-vertex , @xmath249 to each adjacent @xmath17-vertex , @xmath235 to each adjacent @xmath246-vertex and @xmath250 to each adjacent @xmath197-vertex in @xmath1 ; ( @xmath2347 ) each vertex @xmath46 in @xmath1 with @xmath251 sends @xmath252 to each adjacent @xmath20-vertex , @xmath235 to each adjacent @xmath17-vertex , @xmath253 to each adjacent @xmath246-vertex , @xmath254 to each adjacent @xmath197-vertex and @xmath255 to each adjacent @xmath21-vertex in @xmath1 . let @xmath256 be a face of @xmath30 . then @xmath257 . if @xmath256 is non - special , then by ( @xmath2341 ) , @xmath258 . if @xmath256 is special , then by observation [ obs ] , @xmath256 is incident with two non - crossing vertices . by ( @xmath2342 ) , we have @xmath259 . let @xmath46 be a vertex of @xmath1 . since ( @xmath1031 ) is forbidden , we have @xmath260 . suppose @xmath46 is a @xmath53-vertex and has @xmath53 neighbors @xmath261 in @xmath1 where @xmath262 . in the following , we show @xmath263 for each such a vertex . suppose @xmath264 . since ( @xmath1032 ) is forbidden , @xmath46 is adjacent three @xmath173-vertices . note that @xmath46 is not incident with any special @xmath21-face by ( 2 ) of observation [ obs ] . by ( @xmath2341 ) and ( @xmath2347 ) , we have @xmath265 . suppose @xmath266 . since @xmath58 by ( 3 ) of observation [ obs ] , @xmath46 is incident with at most two special @xmath21-faces . if @xmath267 , then by ( @xmath2341 ) , ( @xmath2342 ) and ( @xmath2346 ) , we have @xmath268 . if @xmath108 , then @xmath269 since ( @xmath1033 ) is forbidden so by ( @xmath2347 ) we have @xmath270 . suppose @xmath271 . if @xmath272 , then by ( @xmath2341 ) , ( @xmath2342 ) , ( @xmath2345 ) and ( 4 ) of observation [ obs ] , we have @xmath273 . so we may assume @xmath111 . if @xmath274 , then by ( @xmath2341 ) , ( @xmath2342 ) and ( @xmath2346 ) , we have @xmath275 . so we may assume @xmath112 . then @xmath276 for otherwise ( @xmath1034 ) occurs . in this case , by ( @xmath2341 ) , ( @xmath2342 ) and ( @xmath2347 ) , we also have @xmath277 . suppose @xmath278 . if @xmath279 , then by ( @xmath2341 ) , ( @xmath2342 ) and ( @xmath2344 ) , we have @xmath280 . so we may assume @xmath115 . if @xmath281 , then by ( @xmath2341 ) , ( @xmath2342 ) and ( @xmath2345 ) , we have @xmath282 . so we may assume @xmath116 . if @xmath283 , then by ( @xmath2341 ) , ( @xmath2342 ) and ( @xmath2346 ) , we have @xmath284 . so we may assume @xmath285 . then @xmath286 for otherwise ( @xmath1035 ) occurs . in this case , by ( @xmath2341 ) , ( @xmath2342 ) and ( @xmath2347 ) , we also have @xmath287 . suppose @xmath288 . if @xmath289 , then by ( @xmath2341 ) , ( @xmath2342 ) , ( @xmath2343 ) and ( 4 ) of observation [ obs ] , we have @xmath290 . so we may assume @xmath120 . if @xmath291 , then by ( @xmath2341 ) , ( @xmath2342 ) and ( @xmath2344 ) , we have @xmath292 . so we may assume @xmath121 . if @xmath293 , then by ( @xmath2341 ) , ( @xmath2342 ) and ( @xmath2345 ) , we have @xmath294 . so we may assume @xmath295 . if @xmath296 , then by ( @xmath2341 ) , ( @xmath2342 ) and ( @xmath2346 ) , we have @xmath297 . so we may assume @xmath123 . then @xmath298 for otherwise ( @xmath1036 ) occurs . in this case , by ( @xmath2341 ) , ( @xmath2342 ) and ( @xmath2347 ) , we also have @xmath299 . suppose @xmath300 . then by ( @xmath2341)-(@xmath2348 ) , we have @xmath301 . suppose @xmath302 . recall that @xmath154 is the number of mirror triangles incident with @xmath46 . so @xmath303 . note that @xmath304 and @xmath305 . by ( @xmath2341 ) , ( @xmath2342 ) and ( @xmath2343 ) , we have @xmath306 . suppose @xmath307 . note that @xmath304 and @xmath308 . by ( @xmath2341),(@xmath2342 ) and ( @xmath2344 ) , we have @xmath309 . suppose @xmath310 . note that @xmath304 . by ( @xmath2341),(@xmath2342 ) , ( @xmath2345 ) and claims 2 , 3 , we have @xmath311 . suppose @xmath312 . note that @xmath304 . by ( @xmath2341),(@xmath2342 ) , ( @xmath2346 ) and claims 2 , 3 , 4 , we have @xmath313 . suppose @xmath314 . note that @xmath304 . by ( @xmath2341),(@xmath2342 ) , ( @xmath2347 ) and claims 2 , 3 , 4 , we have @xmath315 . by the above arguments , we have @xmath316 , a contradiction . hence we have proved the theorem . let @xmath317 be a family of graphs and let @xmath136 be a connected graph . let @xmath318 be the smallest integer with the property that each graph @xmath319 contains a subgraph @xmath320 such that @xmath321 . if such an integer does not exist , we write @xmath322 . we say that the graph @xmath136 is @xmath151 in the family @xmath317 if @xmath323 . by @xmath324 , we denote the set of light graphs in the family @xmath317 . in the next , @xmath325 denotes a path with @xmath13 vertices and @xmath326 denotes a star with maximum degree @xmath13 . we use the notation @xmath327 for the family of all @xmath0-planar graphs of minimum degree at least @xmath328 . in @xcite , fabrici and madaras showed that @xmath329 and @xmath330 and posed a few of open problems . two of them are stated as follows . is @xmath331 true ? is @xmath332 true ? in this section , we partially answer these two questions by applying the results in section 2 . let @xmath1 be a simple @xmath0-planar graph with minimum degree @xmath333 . then @xmath1 contains a @xmath21-path with all vertices of degree at most @xmath334 . by theorem [ structure ] , @xmath1 contains one of the configuration in \{(@xmath1033),(@xmath1034),(@xmath1035),(@xmath1036 ) } described in section @xmath335 . in each case , we will find a path @xmath336 in @xmath1 such that @xmath337 . similarly we can prove an analogous theorem . let @xmath1 be a simple @xmath0-planar graph with minimum degree @xmath338 . then @xmath1 contains a @xmath21-star with all vertices of degree at most @xmath334 . hence we have the following many corollaries . . @xmath339 . @xmath340 . a mapping @xmath341 from @xmath6 to the sets of colors @xmath342 is called a @xmath22 @xmath13-@xmath343 @xmath344 of @xmath1 provided any two adjacent edges receive different colors . the @xmath343 @xmath345 @xmath346 @xmath347 is the minimum number of colors needed to color the edges of g properly . a @xmath22 @xmath13-@xmath343 @xmath344 @xmath341 of @xmath1 is called an @xmath348 @xmath13-@xmath343-@xmath344 of @xmath1 if there are no bichromatic cycles in @xmath1 under the coloring @xmath341 . the smallest number of colors such that @xmath1 has an acyclic edge coloring is called the @xmath348 @xmath345 @xmath346 of @xmath1 , denoted by @xmath349 . acyclic edge coloring was introduced by alon et al . @xcite , and they presented a linear upper bound on @xmath349 . it was proved that @xmath350 holds for every graph , which was later improved to @xmath351 by molloy and reed @xcite . for planar graph @xmath1 , a. fiedorowicz et al . @xcite proved that @xmath352 . recently , hou et al . @xcite gave a better upon bound . they showed that @xmath353 holds for each planar graph . let @xmath354 be an edge coloring of @xmath1 . for any vertex @xmath12 , we define @xmath355 . in this section , we consider the acyclic edge coloring of @xmath0-planar graphs . let @xmath360 and @xmath361 . suppose @xmath362 . by the minimality of @xmath1 , the graph @xmath363 has an acyclic @xmath3-edge coloring @xmath354 with color set @xmath87 . let @xmath364 and @xmath365 . for the edge @xmath366 , we remain @xmath367 . note that @xmath368 . then @xmath369 is an acyclic @xmath3-edge coloring of @xmath1 , a contradiction . so @xmath370 . let @xmath371 . then @xmath372 has an acyclic @xmath3-edge coloring @xmath354 with color set @xmath87 . now we let @xmath373 . since @xmath374 and @xmath375 , @xmath376 . now we color @xmath377 by @xmath378 . it is easy to see that @xmath379 . for the edge @xmath380 , we also remain @xmath367 . then @xmath369 is again an acyclic @xmath3-edge coloring of @xmath1 , a contradiction . in this case , @xmath1 has one of the five configurations \{(@xmath1032),(@xmath1033),(@xmath1034),(@xmath1035),(@xmath1036 ) } which are described in theorem [ structure ] . let @xmath381 , @xmath382 , @xmath383 , @xmath384 and @xmath385 . suppose @xmath1 contains the @xmath386-th configuration , where @xmath387 . if @xmath388 , let @xmath389 . otherwise , let @xmath371 . then @xmath372 has an acyclic @xmath3-edge coloring @xmath354 with color set @xmath87 . if @xmath388 , let @xmath390 otherwise , let @xmath391 . now we color @xmath392 by a color @xmath393 . note that @xmath374 and @xmath394 , we have @xmath395 . let @xmath396 and @xmath397 where @xmath398 . then we color @xmath399 in turn as follows . let @xmath400 . if @xmath401 , let @xmath402 . otherwise we let @xmath403 . at last , for each @xmath404 , we let @xmath405 . for the edge @xmath380 , we still remain @xmath367 . note that @xmath406 , @xmath407 and @xmath408 . so this coloring @xmath369 does exist . it is easy to check that @xmath369 is proper and acyclic . so we have constructed a new coloring @xmath369 which is an acyclic @xmath3-edge coloring of @xmath1 , a contradiction . this completes the proof of theorem [ acyclic ] .

a graph is called @xmath0-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge . in this paper , we establish a local property of @xmath0-planar graphs which describes the structure in the neighborhood of small vertices ( i.e. vertices of degree no more than seven ) . meanwhile , some new classes of light graphs in @xmath0-planar graphs with the bounded degree are found . therefore , two open problems presented by fabrici and madaras [ the structure of 1-planar graphs , discrete mathematics , 307 , ( 2007 ) , 854865 ] are solved . furthermore , we prove that each @xmath0-planar graph @xmath1 with maximum degree @xmath2 is acyclically edge @xmath3-choosable where @xmath4 . _ keywords _ : @xmath0-planar graph ; light graph ; acyclic edge coloring . please cite this published article as : _ x. zhang , g. liu , j .- l . wu . structural properties of 1-planar graphs and an application to acyclic edge coloring . scientia sinica mathematica , 2010 , 40 , 10251032_.

1008.5000

99 taylor j h 1994 _ rev . mod . phys . _ * 66 * 711 url ` http://www.ligo.caltech.edu/ ` ; url ` http://www.virgo.infn.it/ ` ; + url ` http://www.geo600.uni-hannover.de/ ` ; url ` http://tamago.mtk.nao.ac.jp/ ` ; url ` http://www.gravity.uwa.edu.au/aigo/aigo.html ` ; url ` http://sam.phys.lsu.edu/ ` ; url ` http://www.auriga.lnl.infn.it/ ` ; url ` http://www.gravity.uwa.edu.au/bar/bar.html ` braginski v b and menskii m b 1971 _ zh . pisma _ * 13 * 585 [ 1971 _ jetp lett . _ * 13 * 417 ] ; lupanov g a 1967 _ zh . eksp . fiz . _ * 52 * 118 [ 1967 _ sov . phys .. -jetp _ * 25 * 76 ] ; braginski v b _ et al _ 1973 _ zh . eksp . fiz . _ * 65 * 1729 [ 1974 _ sov . phys .- jetp _ * 38 * 865 ] ; grishchuk l p and sazhin m v 1975 _ zh . fiz . _ * 68 * 1569 [ 1976 _ sov . phys .- jetp _ * 41 * 787 ] ; balakin a b and ignatev yu g 1983 _ phys . lett . a _ * 96 * 10 ; kolosnitsyn n i 1994 _ zh . eksp . teor . pisma _ * 60 * 69 [ 1994 _ jetp lett . _ * 60 * 73 ] ; cruise a m 2000 _ class . quantum grav . _ * 17 * 2525 ; url ` http://www.sr.bham.ac.uk/research/gravity `

we present a proposal for a gravitational wave detector , based on the excitation of an electromagnetic mode in a resonance cavity . the mode is excited due to the interaction between a large amplitude electromagnetic mode and a quasi - monochromatic gravitational wave . the minimum metric perturbation needed for detection is estimated to the order @xmath0 using current data on superconducting niobium cavities . using this value together with different standard models predicting the occurrence of merging neutron star or black hole binaries , the corresponding detection rate is estimated to 120 events per year , with a ` table top ' cavity of a few meters length . [ [ section ] ] during the last decades the quest for detecting gravitational waves has intensified . the efforts have been inspired by the indirect evidence for gravitational radiation @xcite , advances in technology and the prospects of obtaining new useful astrophysical information through the development of gravitational wave astronomy @xcite . a number of ambitious detector projects are already in operation or being built all over the world , for example ligo and allegro in usa , virgo , auriga and geo 600 in europe , tama 300 in japan , aigo and niobe in australia @xcite . furthermore , there are well developed plans to use space based gravitational wave detectors , i.e. , the lisa project @xcite . the detection mechanisms are basically of a mechanical nature in the cases above , but there have also been several proposals for electromagnetic detection mechanisms @xcite . in the present paper we will investigate a detection mechanism based on the interaction of electromagnetic modes and gravitational radiation in a cavity with highly conducting walls . the main feature of our proposed gravitational wave detector is that it supports two electromagnetic eigenmodes with nearby eigenfrequencies , a possibility that has previously been discussed in refs . @xcite . if one eigenmode is excited initially ( called the pump - mode ) , and a quasi - monochromatic gravitational wave with a frequency equal to the eigenmode frequency difference reaches our system , a new electromagnetic eigenmode can be excited due to the gravitational - electromagnetic wave interaction . the coupling mechanism is similar in principle to the wave interaction processes described in , e.g. , ref . @xcite . the low - frequency nature of the gravitational modes , as compared to the electromagnetic resonance frequencies , at first seem to greatly limit the efficiency of a cavity with nearby frequencies . to get a large gravitationally induced mode - coupling for a simple cavity geometry , the estimated cavity dimensions become prohibitively large , i.e. , comparable to the wavelength of the gravitational wave . solutions to this problem has been found by refs . @xcite , who has considered a gravitational wave detector consisting of two coupled cavities . cavities based on these principles have been built , and experimental results are presented in refs . @xcite . in this letter we consider a single cavity with a variable crossection . the main purpose of varying the crossection is the following . for a cavity with dimensions much smaller than the wavelength of the gravitational wave , the suggested geometry greatly magnifies the gravitational wave induced mode - coupling . in our present work we have simulated the effect of a varying crossection , by considering a cavity filled with three different dielectrics , in order to be able to perform most of the calculations analytically . it is straightforward to make a semi - quantitative translation of our results to the case of a vacuum cavity with a varying crossection . using current data on the latter type of cavity @xcite , we estimate the minimum detection level of the metric perturbation to the value @xmath1 , where we have considered an inspiraling neutron star or black hole pair as a gravitational wave source . if such a level of sensitivity can be reached , neutron star or black hole binaries close to collapse could be detected at a distance @xmath2 . adopting data from ref . @xcite for the occurrence of compact binary mergers , we obtain the estimate 120 detection events per year . in vacuum , a linearized gravitational wave can be represented by @xmath3 where @xmath4 , and @xmath5 in standard notation . neglecting terms proportional to derivatives of @xmath6 and @xmath7 , the wave equation for the magnetic field is @xcite @xmath8{\boldsymbol{b } } = \left [ h_+\left ( \frac{\partial^2}{\partial y^2 } - \frac{\partial^2}{\partial z^2 } \right ) + h_{\times}\frac{\partial^2}{\partial y\partial z } \right]{\boldsymbol{b } } , \label{waveb}\ ] ] and similarly for the electric field . here @xmath9 is the index of refraction . for the moment , we will neglect mechanical effects , i.e. , effects which are associated with the varying coordinates of the walls due to the restoring forces of the cavity . the coupling of two electromagnetic modes and a gravitational wave in a cavity will depend strongly on the geometry of the electromagnetic eigenfunctions . we can greatly magnify the coupling , as compared to a rectangular prism geometry , by varying the cross - section of the cavity , or by filling the cavity partially with a dielectric medium . the former case is of more interest from a practical point of view , since a vacuum cavity implies better detector performance , but we will consider the latter case since it can be handled analytically . _ however _ , we will show how to make a semi - quantitative translation of our results to the case of a varying cavity cross - section . specifically , we choose a rectangular cross - section ( side lengths @xmath10 and @xmath11 ) , and we divide the length of the cavity into three regions . region 1 has length @xmath12 ( occupying the region @xmath13 ) and a refractive index @xmath14 . region 2 has length @xmath15 ( occupying the region @xmath16 ) , with a refractive index @xmath17 , while region 3 consists of vacuum and has length @xmath18 ( occupying the region @xmath19 ) . the cavity is supposed to have positive coordinates , with one of the corners coinciding with the origin . furthermore , we require that @xmath20 , and that the wave number in region 2 is less than in region 1 . the reason for this arrangement is twofold . firstly , we want to obtain a large coupling between the wave modes , and secondly we want an efficient filtering of the eigenmode with the lower frequency in region three . the first step is to analyze the linear eigenmodes in this system . the simplest modes are of the type [ region1 ] @xmath21e^{-i\omega t } , \\ b_{z } = \widetilde{b}_{zj}\cos \left ( \frac{m\pi x}{x_0}\right ) \sin [ k_{j}z + \varphi_{j}]e^{-i\omega t } , \\ b_{x } = -\frac{k_{j}x_0}{m\pi } \widetilde{b}_{zj}\sin \left ( \frac{m\pi x}{x_0}% \right ) \cos [ k_{j}z + \varphi_{j}]e^{-i\omega t } , \end{aligned}\ ] ] in regions @xmath22 , @xmath23 and @xmath24 , respectively , where the wave in region 3 is a standing wave , and @xmath25 is the mode number . in region 3 we may also have a decaying wave [ region2b ] @xmath26e^{-i\omega t } , \\ b_{z } = \widetilde{b}_{z3}\cos \left ( \frac{m\pi x}{x_0}\right ) \sinh [ k_{3}z+\varphi _ { 3}]e^{-i\omega t } , \\ b_{x } = -\frac{k_{3}x_0}{m\pi } \widetilde{b}_{z3}\sin \left ( \frac{m\pi x}{x_0}% \right ) \cosh [ k_{3}z+\varphi _ { 3}]e^{-i\omega t } . \end{aligned}\ ] ] using standard boundary conditions , the wave numbers are calculated for an eigenmode , and the relation between the amplitudes in the three regions is found , and thereby the mode profile . we are interested in the shift from decaying to oscillatory behavior in region 3 . denote the highest frequency mode which is decaying in region 3 with index @xmath27 , and the wave number and decay coefficient with @xmath28 , @xmath29 and @xmath30 respectively . similarly , the next frequency , which is oscillatory in both regions , is denoted by index @xmath31 . if we have @xmath32 ( and @xmath25 the same ) these two frequencies will be very close , and a gravitational wave which has a frequency equal to the difference between the electromagnetic modes causes a small coupling between these modes . an example of two such eigenmodes is shown in fig . 1 . , @xmath33 , @xmath34 , @xmath35 and @xmath36 . @xmath37 is the solid line and @xmath38 is the dotted line , and @xmath39 . ] we define the eigenmodes to have the form @xmath40 , where @xmath41 is a time - dependent amplitude and the normalized eigenmodes @xmath42 satisfy @xmath43 . we let all electromagnetic field components be of the form @xmath44 , where c.c . stands for complex conjugate , and the indices stand for the eigenmodes discussed above . the gravitational perturbation can be approximated by @xmath45 , where we neglect the spatial dependency , since the gravitational wavelength is assumed to be much longer than all of the cavity dimensions . during a certain interval in time , the frequency matching condition @xmath46 will be approximately fulfilled . given the wave equation ( [ waveb ] ) , and the above ansatz we find after integrating over the length of the cavity @xmath47 where @xmath48 and we have added a phenomenological linear damping term represented by @xmath49 . thus we note that for the given geometry , only the @xmath50-polarization gives a mode - coupling . calculations of the eigenmode parameters show that @xmath51 may be different from zero when @xmath52 , and generally @xmath51 of the order of unity can be obtained , see fig . 1 for an example . from eq . ( [ excitation - eq ] ) , we find that the saturated value of the gravitationally excited mode is @xmath53 in fig . 1 it is shown that we can get an appreciable mode - coupling constant @xmath51 for a cavity filled with materials with different dielectric constants , and it is of interest whether or not this can be achieved in a vacuum cavity . as seen by eq . ( [ coupling - int ] ) , the coupling is essentially determined by the wave numbers of the modes , given by @xmath54 . thus by adjusting the width @xmath10 in a vacuum cavity , we may get the same variations in the wave numbers as when varying the index of refraction @xmath9 . the translation of our results to a vacuum cavity with a varying width is not completely accurate , however . when varying @xmath10 , the mode - dependence on @xmath55 and @xmath56 does not exactly factorize , in particular close to the change in width . moreover , the contribution to the coupling @xmath57 in each section becomes proportional to the corresponding volume , and thereby also to the cross - section . however , since most of the contribution to the integral in eq . ( [ coupling - int ] ) comes from region 1 , our results can still be approximately translated to the case of a vacuum cavity , by varying @xmath10 instead of @xmath9 such as to get the same wavenumber as in our above example . we denote the minimum detection level of the excited mode with @xmath58 and the maximum allowed field in the cavity with @xmath59 . the magnetic field @xmath58 is related to the minimum number of photons @xmath60 needed for detection by @xmath61 . furthermore , we have @xmath62 @xmath63 , where @xmath64 is the quality factor of the cavity , and thus we obtain , using eq . ( [ saturation - eq ] ) , @xmath65 before giving estimates of the parameter values , we will investigate certain other effects that may limit the detection efficiency . massive binaries produce monochromatic radiation to a good approximation , but close to merging , the frequency will be increasing rather rapidly which means that the phase matching will be lost . the cavity is designed to detect the frequency @xmath66 , and we have assumed that the signal varies as @xmath67 , but in reality we have @xmath68 , where for simplicity we assume @xmath69 , @xmath70 at @xmath71 . the coherence time @xmath72 is roughly defined by @xmath73 . thus , @xmath74^{1/2}$ ] provided @xmath75 . using newtonian calculations of two masses @xmath76 in circular orbits around the center of mass , complemented by the quadrupole formula for gravitational radiation , we find @xmath77 close to binary merging , the time of coherent interaction will be shorter than the photon life time , and in that regime the growth of the excited mode is limited by decoherence rather than the damping due to a finite quality factor of the cavity . however , formula ( [ hplus ] ) can still be applied if we simply replace @xmath78 by @xmath79 . to be able to estimate the number of photons needed for detection , we must study various sources of noise . the simplest effect is direct thermal excitation of photons in mode @xmath31 . since each mode has a thermal energy level of order @xmath80 , the number of such photons is @xmath81 . however , we will also have a contribution associated with thermal variations in the length of the cavity . a standard model for the variations in length @xmath82 is @xcite @xmath83 where @xmath84 is the eigenfrequency of longitudinal oscillations ( of the order of the acoustic velocity in the cavity divided by the length ) , @xmath85 is the mechanical quality factor associated with these oscillations , and @xmath86 is the stochastic acceleration due to the thermal motion , giving rise to a random walk in the oscillation amplitude . first we note that the amplitude of the length variations with the gravitational frequency is @xmath87 . we will consider the case when @xmath88 , which implies that _ the amplitude _ of the gravitational length perturbation is essentially unaffected by the restoring force of the cavity , as is the number of gravitationally generated photons . the thermal fluctuation contribution to the right hand side of eq.([excitation - eq ] ) , via the coupling to the pump wave , becomes proportional to @xmath89 $ ] . since @xmath90 , the longitudinal oscillations give rise to slightly off - resonant ( i.e. , driven ) fields with a frequency difference @xmath91 compared to mode @xmath31 . however , due to the stochastic changes in amplitude and/or phase of the longitudinal oscillation ( where the time - scale for significant changes is given by @xmath92 ) , a contribution to mode @xmath31 of the order @xmath93 is also made during a single oscillation period @xmath94 . here @xmath25 is the mass taking part in the longitudinal oscillation . during a time of the order @xmath95 , this contribution adds up to approximately @xmath96 by a random walk process . using @xmath97 , the condition for the gravitational contribution @xmath98 to be larger than that of the thermal fluctuations can be written @xmath99 assume that we want to reach a sensitivity @xmath100 . we let @xmath101 , @xmath102 , @xmath103 , and @xmath104 which gives @xmath105 . furthermore , we take @xmath106 and let @xmath107 . we assume @xmath108 together with @xmath109 . from ( [ h_therm_cond ] ) , we then find that we need a mechanical quality factor @xmath110 to reach the desired sensitivity . moreover , assuming the necessary number of photons for detection to be @xmath111 ( well above the direct electromagnetic noise level of a few photons ) , @xmath112 and @xmath113 @xcite , we need an electromagnetic quality factor @xmath114 to reach the desired sensitivity @xmath115 . note that the coherence time is slightly longer than the photon life - time , as needed . following the example given above for calculating the coherence time , i.e. , a binary consisting of two compact objects , each of one solar mass @xmath116 , separated by a distance @xmath117 , the amplitude of the metric perturbation at a distance @xmath118 is given by @xmath119 . for definiteness we choose @xmath120 corresponding to @xmath106 , and thus for @xmath121 of the order of @xmath122 we obtain the maximum observational distance @xmath123 . using data from ref . @xcite , we deduce that the number of galaxies within the observational distance @xmath2 is of the order @xmath124 . combining these figures with the expected number of compact binary mergers per galaxy and million years @xcite , we obtain a detection rate in the interval @xmath125 events per year ( the uncertainty is due to different models used for the birth of compact binaries ) . our proposal for a gravitational wave detector is to a large extent based on currently available technology @xcite , and our requirements are moderate given the performance of some existing microwave cavities . for example , values of @xmath126 has been reported in ref . @xcite and quantum non - demolition measurements of single microwave photons have been made in ref.@xcite . furthermore , the key performance parameters of the superconducting niobium cavities , i.e. , the quality factor and the maximum allowed field strength before field emission , have been improving over the years , suggesting that the detection sensitivity can be increased even further . a sensitivity @xmath127 seems extremely good , but , on the other hand , idealizations have been made when making the estimate . in addition to any effects induced by the gravitational wave , the walls generally vibrate slightly due to the electromagnetic forces exerted by the large pump field . while these later oscillations clearly will be larger than the variations in length that are directly due to the gravitational wave , the associated nonlinearities will be harmonics of the pump frequency , and thus such effects do not couple to the other eigenmodes of the cavity . furthermore , we have assumed that the detection of the excited mode is not much affected by the presence of the pump signals . even though mode @xmath27 is partially filtered out in region 3 , the small frequency shift between the two electromagnetic modes may pose a certain difficulty in this respect : in particular , a very narrow bandwidth ( pump ) antenna signal must be used , in order to exclude the slightest initial perturbation at the gravitationally excited frequency @xmath128 . however , although there are technical difficulties in constructing an electromagnetic detector , the real advantage is the possibility to reduce the size of the devise . in the example presented above , the length of the cavity has been taken to be @xmath129 , and the cross section roughly @xmath130 . this alone could prove to be useful when trying to set up new gravitational wave observatories . we thank ulf jordan for helpful discussions .

astro-ph0207458

understanding how galaxies form and evolve into the objects we observe today remains one of the most fundamental quest of astrophysical research . even if the field is still at its beginnings , the use of numerical approaches to study how galaxies are assembled in a cosmological hierarchical scenario from primordial fluctuations , seems promising . the main advantage of these approaches ( namely hydrodynamical simulations ) is that physics is introduced at a most general level , and the dynamical processes relevant to galaxy assembly ( i.e. , collapse , gas infall , interactions , mergers , instabilities ... ) emerge naturally , rather than by assumption , and can be followed in detail . only the star forming processes need to be modelled . these considerations emphasize the interest of hydrodynamical simulations as an outstanding working tool to learn about galaxy formation and evolution from primordial fluctuations . for other approaches to this problem , see manrique et al . ( in preparation ) , firmani & avila - reese ( 2000 ) and the references therein . as discs are naturally produced in the collapse of a dissipative system embedded in a gravitational field , learning how to get well - behaved discs ( i.e. , similar to those observed in spirals ) in such simulations is one important step towards understanding galaxy assembly in general . according to the standard model of disc formation ( fall & efstathiou 1980 ) , disc - like structures with observable counterparts can form provided that the diffuse gas in the dark matter halo cools and collapses conserving its specific angular momentum , @xmath1 . in hierarchical scenarios , halo gas cooling occurs with @xmath1 conservation in the _ quiescent _ phases of galaxy assembly , producing extended , thin discs with an exponential mass density profile in the equatorial plane ( dalcanton , spergel & summers 1997 ) . but _ violent _ phases play also a fundamental role in these scenarios . during these phases , the system undergoes interactions and merger events , that can destroy discs that had possibly formed in previous quiescent phases . thus far , no hydrodynamical simulation of galaxy formation in fully consistent hierarchical cosmological scenarios had provided extended discs , with structural and dynamical properties similar to those observed in spiral galaxies ( see , however , recent work by sommer - larsen & dolgov 1999 ) . the objects formed were either a ) gaseous and too concentrated , when no stellar formation processes are considered at all ( navarro & benz 1991 ; navarro , frenk & white 1995 ; evrard , summers & davis 1994 ; vedel , hellsten & sommer - larsen 1994 ; navarro & steinmetz 1997 ; sommer - larsen , gelato & vedel 1999 ; weil , eke & efstathiou 1998 and references therein ) , or b ) rather stellar and without discs , when these processes have been included ( navarro & steinmetz 2000 ; thacker & couchman 2000 ) . in the former cases , objects are too concentrated due to an excessive @xmath1 loss in violent events ; this is the so - called disc angular momentum catastrophe [ damc ] . in the latter cases , as discussed by tissera , siz & domnguez - tenreiro ( in preparation , hereafter tsdt ) the particular star formation implementations used lead to a too early gas transformation into stars , leaving no gas to cool and form discs at lower @xmath2 . the effects of star formation have also been considered by katz ( 1992 ) and steinmetz & mller ( 1995 ) who obtained , in both cases , a three component system that resembles a spiral galaxy . however , their simulations treat only the case of a semi - cosmological modellization of the collapse of a constant density perturbation in solid - body rotation . sommer - larsen & dolgov ( 1999 ) have succeeded in avoiding excessive @xmath1 loss in warm dark matter scenarios . nevertheless , their discs still do not match observations very well . this inability to reproduce observed discs in fully consistent cosmological simulations , represents clearly a major pitfall of present - day numerical simulations as a working tool to learn about galaxy formation and evolution . in a series of papers ( domnguez - tenreiro , tissera & siz 1998 ; siz , tissera & domnguez - tenreiro 1999 ; tsdt ) it has been shown that the object properties at the end of violent phases of evolution critically depend on the structural stability of the disc - like objects involved in interaction and merger events , on the one hand , and on the availability of gas to regenerate a disc structure after the last violent episode , on the other hand . concerning stability , cold thin discs are known to be very unstable against non - axisymmetric modes ( athanassoula & sellwood 1986 ; binney & tremaine 1987 ; barnes & hernquist 1991 , 1992 ; martinet 1995 and references therein ; mihos & hernquist 1994 , 1996 ; mo , mao & white 1998 ) . massive haloes can stabilize discs , but not every halo is able to stabilize the pure exponential disc that would form from @xmath1-conserving diffuse gas collapse in its last quiescent phase . in these cases , a central compact bulge is needed to ensure stability ( christodoulou , shlosman & tohline 1995 ; van den bosch 1998 , 2000 ) . otherwise , non - axisymmetric instabilities could be easily triggered by interactions and mergers , causing important @xmath1 losses ( that is , damcs ) , followed by strong gas inflow . in that series of papers we also show that compact central bulges form when star - forming processes are considered in hierarchical hydrodynamical simulations , provided that the particular star formation implementation used does not lead to a too early gas exhaustion ( i.e. , _ inefficient _ star formation , see silk 1999 ) . the physical processes involved in disc formation and stability when an inefficient schmidt law - like stellar formation algorithm is implemented in these simulations are discussed in detail in tsdt . the aim of this paper is to show that discs identified in galaxy - like objects produced in these hierarchical hydrodynamical simulations are , at a structural and dynamical level , similar to those observed in spirals . to this end , we analyze the structural and dynamical properties of a sample of 29 disc - like objects formed in three different such simulations . we focus on those properties that can be constrained with available data for observed spiral galaxies . these data are taken principally from the compilations of galaxy structural parameters of broeils ( 1992 , hereafter b92 ) , de jong ( 1996 ) and courteau ( 1996 , 1997b , hereafter c97b ) . the paper is organized as follows : @xmath3[simu ] describes the simulations . the general characteristics of disc - like objects at @xmath4 are outlined in @xmath3[objects ] , and their bulge - disc decomposition and rotation curves are discussed in @xmath3[budi ] and @xmath3[rocu ] , respectively . finally , a summary and conclusions are presented in @xmath3[sumcon ] . we consider three simulations , s1 , s1b , and s1c , from different realizations of a given model for the primordial spectrum of the density fluctuation field . in each case , we follow the evolution of @xmath5 gas plus dark matter particles in a periodic box of comoving side 10 mpc ( @xmath6 km s@xmath7 mpc@xmath7 with @xmath8 ) using a sph code coupled to a high resolution ap3 m code ( thomas & couchman 1992 ; see tissera , lambas & abadi 1997 for details on sph algorithm implementation ) . the initial distributions of positions and velocities of the @xmath5 particles are consistent with a standard cdm cosmology , with @xmath9 , @xmath10 and @xmath11 ; one out of ten of them is randomly chosen to be a gas particle . these initial distributions are set by means of the action algorithm , kindly provided by e. bertschinger . all , dark , gas and stellar particles have the same mass , @xmath12 m@xmath13 . ap3m - sph integrations were carried out from @xmath14 to the present using fixed time steps @xmath15 years . the gravitational softening length is 3 kpc and the minimum allowed smoothing length is 1.5 kpc . these simulations include a star formation prescription described by tissera et al . ( 1997 ; see tissera 2000 for a discussion on the resulting star formation history ) . the gas cools due to radiative cooling . we use the approximation for the cooling function by dalgarno & mccray ( 1972 ) for a primordial mixture of hydrogen and helium . gas particles are transformed into stars if they are cold ( @xmath16 k ) , dense ( @xmath17 g @xmath18 ) , in a convergent flow and satisfy jean s instability criterion . when a gas particle satisfies these conditions , it is transformed into a star particle after a time interval ( @xmath19 ) over which its gas mass is being converted into stars according to @xmath20 where @xmath21 is the star formation efficiency , @xmath22 is a characteristic time - scale assumed to be proportional to the dynamical time of the particle ( @xmath23 ) and @xmath24 is the time interval over which 99 per cent of the gas mass in a particle is expected to be transformed into stars ( navarro & white 1993 ) . we adopt a low star formation [ sf ] efficiency with @xmath25 ( for a discussion of the inefficient sf implementation , see tsdt ) . no supernovae explosion effects have been considered . however , this low @xmath21 value avoids the quick depletion of gas at high @xmath2 , leaving enough remanent gas to form disc - like structures at low @xmath2 . to some extent , this could mimic the effects of energy injection from supernovae explosions . the treatment of supernova feedback in numerical simulations is still an open and complex problem that remains to be properly solved . galaxy - like objects are identified at their virial radius , @xmath27 , at @xmath0 . dark matter haloes formed in the s1 simulation have been extensively studied in tissera & domnguez - tenreiro ( 1998 , hereafter tdt98 ) . these haloes , as well as those formed in s1b and s1c simulations , are resolved with a relatively high number of particles ( see the number of dark mass particles , @xmath28 , per halo , in table [ general ] ) . the main baryonic objects embedded in these dark haloes can be identified by applying a friends - of - friends ( fof ) algorithm to the baryonic particle distribution , with a linking length of 10 per cent the mean total interparticle separation in the box . baryonic objects identified in the simulation at @xmath0 are galaxy - like objects ( glos ) that span a range of morphological characteristics : disc - like objects ( hereafter , dlos ) , spheroids and irregular objects . dlos have external extended , populated discs , consisting mostly of gas , central stellar bulge - like concentration and , in some cases , stars in a thick disc . spheroids are mostly composed of stars forming a relaxed , spheroidal configuration , with a small amount of gas forming a small central disc . irregular objects do not have a defined morphology , and in most cases they are the end product of a recent merger . in this paper only dlos will be analyzed . only those glos with a total baryon number @xmath29 larger than @xmath30 have been considered is defined unambiguously as @xmath31 , where @xmath32 is the baryon number as given by the friends - of - friends algorithm for a given glo . however , some glos have close satellites that are included in the friends - of - friends algorithm , even if these satellites are dynamically and structurally different entities from the glo itself . in this case , the actual glo baryon number is @xmath33 , with @xmath34 the number of baryonic particles in the @xmath35-th satellite entering in @xmath32 . ] . this criterion has been chosen as a compromise : taking @xmath36 much higher than 150 would result into too few dlos in our sample ; taking @xmath36 much lower than 150 would result into dlos composed of too few particles to be properly analyzed . out of our 3 simulations , we are able to identify different numbers of discs ( 9 , 13 and 7 , for a total sample of 29 objects ) and spheroids or irregular objects ( 5 , 7 and 11 , not studied in this paper ) following the previous mass - limit criterion . dlos are labelled by the three coordinates ( in 10@xmath37mesh units ) that identify the cell where the centre of mass of their host halo is found in the total box . when several haloes ( hosting either dlos , spheroids or irregulars ) are found in a given cell , a greek letter is added to distinguish among them . some haloes host more than one baryonic object ; asterisks in table [ general ] identify haloes that include more than one baryonic object ( either dlo , spheroid or irregular ) . in this case , the different components are distinguished by means of a capital letter in order of decreasing mass . for instance , dlo # 165c ( in s1b ) belongs to a massive halo , where the more massive objects # 165a ( spheroid ) and # 165b ( irregular ) are also found ; dlos # 233a and # 233b ( in s1 ) , on the one side , and # 643@xmath38a and # 643@xmath38b ( in s1b ) , on the other side , share the same halo . the virial radius , @xmath27 , maximum limiting radius , @xmath39 ( see @xmath3[budi ] ) , dark , star and gas particle number , @xmath28 , @xmath40 and @xmath41 , respectively , within spheres of radii @xmath27 and @xmath39 for each of the 29 dlos of the sample are listed in table [ general ] . in fig . [ dlos](a ) we plot , for the @xmath35-th baryonic particle inside the halo hosting dlo # 242 , the cosine of the angle formed by its position and velocity vectors , @xmath42 and @xmath43 , versus @xmath44 , its distance to the dlo mass centre . in fig . [ dlos](b ) we plot @xmath45 versus @xmath44 , where @xmath46 is the component of the angular momentum per unit mass of the @xmath35-th particle parallel to @xmath47 ( disc specific total angular momentum ) . the solid line is @xmath48 , where @xmath49 is the circular velocity at projected distance @xmath50 ( see @xmath3[rocu ] ) . in these figures , stars stand for stellar particles , circles for gas particles and open symbols for counter - rotating particles ( i.e. , with @xmath51 ) of any kind . these figures show that most gas particles placed at @xmath52 kpc move coherently along circular trajectories ( so that they have @xmath53 , see fig . [ dlos](a ) ) on the equatorial plane ( and so they verify @xmath54 ) , that is , they have @xmath55 , with a small dispersion around this value ( see fig . [ dlos](b ) ; @xmath56 if the @xmath35-th particle is on the equatorial plane ) : the particles form a _ cold thin disc_. in contrast with disc gas particles , gas particles placed at @xmath57 kpc ( hereafter , halo gas particles ) do not show any order : roughly half of them are in counter - rotation ( open circles ) , their @xmath58 $ ] and their @xmath45 takes any value under the solid line . concerning stars , we see that most of them are found at @xmath59 kpc , where they form a central compact relaxed structure , i.e. , a bulge , with @xmath60 without any preferred direction and very low @xmath45 ; in some dlos , as figs . [ dlos](a ) and [ dlos](b ) illustrate , some stars are placed at @xmath61 kpc in some orderly fashion , forming a sort of thick disc . baryons forming other dlos in our sample follow similar behaviour patterns . in fig . [ jmvsm ] we plot the specific total angular momentum at @xmath0 versus mass for dark haloes , @xmath62 ( open symbols ) , for the inner 83 per cent of the disc gas mass ( i.e. , the mass fraction enclosed by @xmath63 in a purely exponential disc , where @xmath64 is the disc scale length , see @xmath3[budi ] ) , @xmath65 ( filled symbols ) , and for the stellar component of the dlos in our simulations , @xmath66 ( starred symbols ) . we see that @xmath65 is of the same order as @xmath62 , so that these gas particles have collapsed conserving , on average , their angular momentum . moreover , dlos formed in our simulations are inside the box defined by observed spiral discs in this plot ( fall 1983 ) . in contrast , @xmath66 is much smaller than either @xmath62 or @xmath65 , meaning that the stellar component in the central parts has formed out of gas that had lost an important amount of its angular momentum . a detailed description of the physical processes leading to dlo configuration at @xmath0 we have just described is given in domnguez - tenreiro et al . ( 1998 ) and tsdt . here these processes are only briefly summarized : the net effect of shocks and cooling in the quiescent phases of evolution on disordered halo gas particles , is that they force them to a coherent rotation with global specific angular momentum conservation . consequently , in the quiescent phases of the evolution , gas particles tend to settle at the inner halo regions , and if the gravitational potential has some centre of axial ( or central ) symmetry at these regions , gas particles will move on a plane on circular orbits around this centre , forming a cold thin disc . most stars in the central bulges have been formed out of gas particles that had been involved in an inflow event with high @xmath1 loss ; these compact central bulges create a gravitational potential that is spherically symmetric on scales of a few kpc , ensuring @xmath1 conservation . this stabilizes the system against gas inflows in future violent phases of the evolution . in fact , in the next major merger event , those gas particles involved in the merger and with high @xmath1 will be placed on an intermediate disc ; in their turn , the stellar bulge of the smaller dlo involved in the event is eventually destroyed by strong tidal forces ; then , incomplete orbital angular momentum loss puts most of its stars at some distance of the centre , where , after some relaxation , they form a kind of thick disc . finally , most of the disc external gas particles are supplied by infall of particles , either belonging to baryonic clumps ( satellites ) or diffuse component , completing the dlo assembly . because of the sf implementation used , gas in discs are not transformed into stars . then , simulated discs are mostly composed of gas . observed discs in spirals are mostly stellar , but the stellar discs inherit the structural and dynamical characteristics of the gaseous discs out of which disc stars have formed , so that , for the sake of comparison , the value of the parameters describing those properties can be safely determined from the simulated gaseous discs . a proper implementation of star formation with supernovae feedback should lead to a self - regulated star formation that would allow the formation of both compact stellar bulges and thin stellar discs . observationally , the structural parameters of spirals are obtained from a fit to the shape of the surface _ brightness _ profile . the galaxy luminosity profile is often decomposed into a bulge ( b ) and disc ( d ) component , such as @xmath67 each individual profile can be modelled using the parametrization from srsic ( 1968 ) , @xmath68 , \ { \rm{for } \hspace{0.5 cm } c = { \rm b , d } } , \label{surprof}\ ] ] and where @xmath69 is a shape parameter taken to be @xmath70 for the disc component ( i.e. , a pure exponential law ) and that is left as a free parameter for the bulge component . the de vaucouleurs profile for spheroids corresponds to a choice of @xmath71 . @xmath72 and @xmath73 are the central surface brightness and scale length for the bulge and disc components . the same profile form has been fit to the projected _ mass _ density of baryons , @xmath74 , for the dlos in our sample . @xmath74 is calculated by averaging concentric rings centred at the centre of mass of each dlo . projections on the plane normal to the total angular momentum of each dlo ( disc plane ) have been used . these projected densities are binning dependent and somewhat noisy . to circumvent these problems , the _ integrated projected _ mass density in concentric cylinders of radius @xmath50 and mass @xmath75 , @xmath76 has been used as fitting function , instead of the projected mass density itself . cumulative histograms @xmath77 are constructed by adding equal height steps at those positions @xmath78 where baryon particles are placed . [ mcyl ] shows these @xmath79 histograms for two dlos . simulated discs , as well as discs observed in spirals , do not have a sharp end . we have taken as maximum limiting radius for the simulated discs @xmath80 . ] which corresponds to a limiting or truncation radius in surface brightness of @xmath81 mag arcsec@xmath82 assuming that discs have @xmath83 mag arcsec@xmath82 . as minimum limiting radius we have taken @xmath84 kpc . these changes affect mainly the @xmath85 parameter values , that in the less favoured case , @xmath86 kpc , change on average only by less than a 10 per cent ; @xmath64 variations , on the other hand , are on average lower than the corresponding monte carlo errors due to finite sampling ( see fig . [ rbrd ] ) . ] . because the total baryon number inside @xmath39 for each object is known ( see table [ general ] ; this is equivalent to knowing the _ total _ dlo mass ) , we are left with four free fitting parameters among @xmath69 , @xmath87 , @xmath88 , @xmath89 and @xmath64 . an updated version of the minuit software from cern library has been used to make the fits . to compare the size of bulges with different shape ( i.e. , with different bulge shape parameter @xmath69 ) the @xmath88 parameter is meaningless because its value , as determined from the fitting procedure , depends on @xmath69 . in fact , when bulges with the same physical scale and different shape are considered , @xmath88 varies by orders of magnitude when @xmath69 varies by a factor of some few units ( see eqs . ( [ reffrb ] ) and ( [ reffrb2 ] ) below ) . to circumvent this shortcoming , an _ effective _ bulge scale length or half light radius , @xmath85 , is often used by observers . in our case , it will be defined through the projected bulge mass profile by the condition @xmath90 , where @xmath91 is the bulge mass ( i.e. , it is a half mass radius ) . @xmath85 is related to @xmath88 through : @xmath92 where @xmath93 and @xmath94 are the incomplete and complete gamma functions . for @xmath70 , eq . ( [ reffrb ] ) gives @xmath95 ; for @xmath71 one finds @xmath96 . for arbitrary @xmath69 , eq . ( [ reffrb ] ) can be approximated by @xmath97 the results for the effective bulge scale length , @xmath85 , and disc scale length , @xmath64 , corresponding to the optimal bulge shape parameter , @xmath98 , and , also , to @xmath99 ( exponential bulge profile ) are given in table [ buldis ] , together with their corresponding @xmath100 per degree of freedom parameter . for comparison , table [ buldis ] also gives the disc scale length and the @xmath100 per degree of freedom for a de vaucouleurs profile ( @xmath101 ) . note that the disc scale parameter , @xmath64 , is , in most cases , nearly independent of the bulge shape parameter ( either @xmath98 , @xmath70 or @xmath71 ) . knowing the length and mass scale parameters for simulated bulges and discs allows us to determine their _ mass _ bulge - to - disc ratios , @xmath102 these ratios are given in table [ buldis ] . this table shows another interesting parameter characterizing dlo structure : the _ transition _ radius , @xmath103 , where @xmath104 . in fig . [ mcyl ] we have drawn the best fits to the integrated projected baryon mass density , @xmath105 , for dlos # 242 and # 643@xmath38a . a solid line arrow marks @xmath103 ; a dashed line arrow marks the lower limit of the fitting interval @xmath84 kpc . this figure illustrates the good quality achieved in the fittings , even for the regions where @xmath106 , and hence the stability of the bulge / disc parameter determination against changes in @xmath107 , as mentioned before . we see in table [ buldis ] that bulges for most dlos are more adequately modelled with an exponential profile ( @xmath99 ) than with a de vaucouleurs profile ( @xmath101 ) . this agrees with observational results for early and late - type spiral galaxies ( e.g. andredakis , peletier & balcells 1995 ; courteau , de jong & broeils 1996 [ cdjb96 ] ; moriondo , giovanelli & haynes 1999 [ mgh99 ] ) . furthermore , andredakis et al . ( 1995 ) and cdjb96 have found that the bulge shape parameter is correlated with spiral type . in particular , cdjb96 find that most sb sc and all high surface brightness spiral galaxies of later type are best fitted with a @xmath70 profile , most sa sab galaxies are best modelled with a @xmath108 bulge , and only a small fraction of late type spirals have bulges that follow the de vaucouleurs law . this distribution of shape parameters , @xmath98 , is most similar to that of dlos in table [ buldis ] . based on the grid of shape parameters in andredakis et al . ( 1995 ) and cdjb96 , most of our simulated dlos seem to be of type @xmath109 or greater ( see @xmath98 in table [ buldis ] ) . the dependence of the observed @xmath69 values on the morphological type suggests a difference in the formation and evolution of the bulge component . the values of the @xmath88 and @xmath64 parameters , corresponding to a double exponential fit , are plotted in fig . [ rbrd ] for all the dlos in our sample ; squares , triangles and circles correspond to dlos formed in s1 , s1b and s1c simulations , respectively . the error bars in the top left corner of the figure are the typical dispersions in the @xmath88 ( vertical ) and @xmath64 ( horizontal ) parameters values , arising from finite sampling and calculated through monte carlo realizations of the dlo profiles , as given by eqs . ( [ surdes ] ) and ( [ surprof ] ) , with @xmath110 particles and @xmath70 . for comparison , in fig . [ rbrd ] we also show the @xmath88 and @xmath64 values given by cdjb96 and courteau ( 1997a ) from one - dimensional luminosity decompositions of sb sc spirals ( points ) . we get average bulge / disc scale length ratios of @xmath111 for dlos in s1 , @xmath112 for dlos in s1b and @xmath113 for dlos in s1c . the ratio for the whole sample of 29 dlos is @xmath114 , which compares well with the values found by cdjb96 in the @xmath115-band , 0.13 @xmath116 0.07 , and de jong ( 1996 ) at @xmath117 alone , 0.09 @xmath116 0.04 . mgh99 also find @xmath118 . in all cases , consistency between available data and our dlo sample is good . note that we miss dlos with small values of @xmath88 and/or @xmath64 as compared with the range of values quoted by these authors . it simply reflects our lack of dynamical resolution to resolve smaller structures . in fig . [ rbrd ] we compare scale lengths obtained from one - dimensional surface _ brightness _ decompositions with scales obtained from projected _ mass _ density decompositions with bulge shape parameter @xmath70 . as compared with the disc , the bulge is typically less prominent in surface brightness than in projected mass density , since its mass - to - light ratio ( @xmath119 ) is likely to be higher than the disc mass - to - light ratio ( @xmath120 ) . it is not clear , a priori , if this fact makes the previous comparisons meaningless . to find out if this is indeed the case , one would have to compare the values of the @xmath88 and @xmath64 parameters as obtained from a fit to the surface brightness profile of a given dlo with those obtained from a fit to its projected mass density profile . since this is not possible , in order to assess the feasibility of the previous comparisons , we adopt the following method : assuming that both @xmath119 and @xmath120 do not depend on @xmath50 , a given dlo with double exponential @xmath121 profile characterized by values of @xmath88 , @xmath64 , @xmath122 and @xmath123 , would have a surface brightness profile , @xmath124 , given by eqs . ( [ surdes ] ) and ( [ surprof ] ) with the same @xmath88 and @xmath64 parameters , but with @xmath125 and @xmath126 . the previous comparisons will make sense if fittings to monte carlo realizations of a given @xmath121 profile and to monte carlo realizations of its surface brightness counterpart , @xmath124 , with the same number of particles , lead to the same values of the @xmath88 and @xmath64 parameters . this test has been performed for @xmath121 corresponding to dlos # 242 and # 545 ( from s1 ) in the sample . results are shown in fig . [ rbrdtest - exp ] for @xmath127 = 1 , 3 and 5 , from where we see that consistent values of @xmath88 and @xmath64 are obtained ( within errors resulting from finite dlo sampling ) . therefore , the mass and length structural parameters are decoupled , making the comparisons valid . this compatibility between the simulated mass scale lengths and observed luminosity scale lengths in the @xmath115 and @xmath117 bands , suggests that the projected light density ( i.e. , stars ) follows the projected mass density . this could be expected if the star formation rate in a disc were proportional to the baryon surface mass density ( see kennicutt 1998 and silk 1999 for a discussion ) . the _ mass _ @xmath128 ratios given in table [ buldis ] are plotted in fig . [ noptbd ] versus the corresponding @xmath98 . note that no correlation is observed between @xmath128 and @xmath98 . these _ mass _ @xmath128 ratios are considerably higher than the corresponding _ luminosity _ @xmath128 ratios of late - type systems . but it is difficult to decide at the present whether the central baryon concentrations found in dlos are excessive or not , as both , disc and bulge , mass - to - light ratios , would be required to compare with observed _ luminosity _ @xmath128 ratios . in fact , the only disc galaxy where the masses of the disc and bulge / central component can be estimated directly is the milky way . sommer - larsen & dolgov ( 1999 ) obtain @xmath129 0.2 0.4 , for a disc scale length of 2.5 3.0 kpc . for other galaxies , some estimations give @xmath130 12 ( in solar units , for @xmath131 ; courteau & rix 1999 , hereafter cr99 ) . values in these ranges would make the _ mass _ @xmath128 ratios obtained in the simulations roughly compatible with _ @xmath128 ratios from observations , but more accurate measures of @xmath119 and @xmath120 are required before a conclusion can be reached . it is worthwhile to note that decompositions of extended rotation curves ( rcs ) of some galaxies from broeils compilation ( b92 ) , using sub - maximal discs ( bottema 1993 , 1997 ; cr99 ) , yield _ mass _ @xmath128 ratios of about 1 ( see rhee 1996 ) . also , mgh99 find that 5 out of 23 sa scd galaxies for which they obtain a reliable fit to the rotation curve ( rc ) for both the bulge and the disc component , have _ mass _ @xmath128 ratios of about 1 or larger . note , however , that the majority of these decompositions are ill - constrained , as different combinations of bulge , disc , and halo profiles yield equally good agreement with the data ( van albada et al . 1985 , cr99 ) . current observational uncertainties on @xmath119 and @xmath120 preclude any consistency check for our derived baryon mass distributions . for more stringent constraints , one must turn instead to dynamical observables such as rcs . the shapes of spiral rcs are determined by their three dimensional total ( i.e. , bulge , disc and dark matter halo ) mass distribution . the rcs of the simulated objects have been constructed by adding up in quadrature the contributions to circular rotation from the bulge , the disc ( both of them formed by baryons , either gas or stars , see @xmath3[objects ] ) , so that @xmath132 , and the dark matter halo , @xmath133 , @xmath134 we have adopted a softened plummer potential ( evrard et al . 1994 ) @xmath135 where @xmath136 and @xmath137 are the total baryonic and dark matter masses , respectively , inside a sphere or radius @xmath115 . the adaptive ap3 m scheme produces an effective dynamical gravitational softening at @xmath0 of @xmath138 kpc at the central region of the dlos . the rcs for several dlos in our sample are given in fig . [ rotcur ] . points are the tangential velocity component of the baryonic particles ( projected on the disc plane ) , solid lines are @xmath139 , dashed lines are @xmath140 and point lines @xmath141 . note that the halo component , @xmath140 , gives rise to an almost _ flat _ contribution to the rcs beyond the very inner regions . these rcs fit very well the tangential velocity component of the baryonic particles up to @xmath142 , suggesting that these particles are in rotationally supported equilibrium within the potential well produced by both the dark haloes and the baryons themselves ( tdt98 ) . on scales encompassing the baryonic object ( i.e. , @xmath143 ) , dlos have rcs that are declining in their outer parts , and some of them have central spikes produced by the bulge , similar to those found in some observed galaxies with _ mass _ @xmath144 or larger ( for example , ngc 6674 and ngc 7331 in rhee 1996 , or ugc 89 , ugc 1013 and ugc 1238 in mgh99 ) . it is customary to try to explain the shapes of rcs as a combination of various ` fundamental plane ' parameters such as global size ( i.e. , a spatial and a mass scale ; rubin et al . 1985 ) , central mass concentration ( rhee & van albada 1996 ) , and luminosity ( persic , salucci & stel 1996 ) . here , the rc shapes for simulated dlos are parametrized as a function of disc scale length , @xmath64 , as spatial scale , rotation velocity at @xmath145 is the radius at which a _ purely _ exponential disc would reach its maximum circular velocity . ] , @xmath146 , as mass size , and central mass concentration , via the maximum or peak rotation velocity , @xmath147 , and the radius where this is reached , @xmath148 ( c97b ) . some combinations of these parameters have also been explored , such as the logarithmic slope , @xmath149 ( casertano & van gorkom 1991 [ cvg91 ] ; b92 ; de blok , mcgaugh & van der hulst 1996 ) . the @xmath149 is observationally defined as the slope of a straight line that fits the rc in log log scale , from @xmath150 to @xmath151 ( this last point is usually taken to be the last measured rc point , but see c97b for variations on this operational procedure ) . for simulated objects in our sample , the rcs are nearly linear in log log scale for @xmath152 , so that @xmath149 does not appreciably change for @xmath153 . the values of these parameters measured in simulated dlos have been compared with those measured in rcs from : a ) sb sc field spirals drawn from the courteau sample , with long - slit h@xmath154 spectra and @xmath115-band photometry ( see c97b ) , and b ) sb sc spirals from broeils compilation ( b92 ) of extended rcs with surface photometry . we keep only galaxies with @xmath155 km s@xmath7 ( only one of broeils galaxies has @xmath156 in the range @xmath157 $ ] km s@xmath7 ) . apart from these parameters characterizing the shapes of rcs at scales of the baryonic objects , other interesting parameters at halo scales ( i.e. , @xmath158 ) , are the velocity at the virial radius @xmath159 and the spin parameter , @xmath160 , with @xmath161 , @xmath162 and @xmath163 the total angular momentum , energy and mass for all particles inside @xmath27 . note that both @xmath164 and @xmath165 are not observable . in table [ dynamical ] we give @xmath164 , @xmath165 and the rc parameters for our dlo sample . note that the @xmath165 values are within their expected range for a standard cdm scenario ( warren et al . 1992 ; dalcanton et al . 1997 ; lemson & kauffmann 1998 ) . in fig . [ lambrd ] we plot @xmath64 versus @xmath166 . a correlation appears in the sense that haloes with higher spin host more extended discs . in the framework of semianalytical models of quiescent _ pure _ ( i.e. , bulgeless ) disc formation with @xmath1 conservation one approximately has @xmath167 ( see dalcanton et al . 1997 ; mo et al . 1998 ) ; we find an important dispersion due to the loss of angular momentum by _ bulge _ particles involved in damcs in the violent phases of the evolution . the dlo values of the parameters @xmath64 and @xmath156 are consistent with observations ( see tables [ buldis ] and [ dynamical ] and figs . [ rbrd ] and [ rdv22 ] ) . by virtue of selecting dlos with @xmath168 , we find high rc amplitudes mostly with @xmath155 km s@xmath7 . our simulated dlos are either intermediate spirals ( 100 km s@xmath169 km s@xmath7 , 9 objects ) , large bright spirals ( @xmath170 km s@xmath7 , @xmath171 kpc , 18 objects ) , or compact bright spirals ( @xmath170 km s@xmath7 , @xmath172 kpc , 2 objects ) . our sample does not include any dwarf galaxy . as expected , @xmath156 is correlated with @xmath164 , albeit with a dispersion . this dispersion is provided by the particularities of the evolutionary history of each dlo in the sample , namely , the number and characteristics of the interactions and merger events involved in its assembly ( see tdt98 for a discussion ) . the mass distribution of real spirals and their decomposition into bulge , disc and halo components can not be inferred uniquely from observed rcs , as different decompositions yield equally good agreement with the data ( van albada et al . 1985 ; broeils & courteau 1997 ; cr99 ) . one has to postulate different scalings ( mass - to - light ratios ) between the bulge and disc components in order to infer the mass distribution of the halo . common constraints for ( m / l)@xmath173 include a ) the so - called ` maximum disc ( or maximal light ) hypothesis ' ( van albada & sancisi 1986 ) , which postulates that the halo dark mass component needed to fit the rcs should be minimum , or , equivalently , that the contribution of the disc and bulge to the inner parts of rcs should be maximum , and b ) sub - maximal discs with @xmath174 . this constraint is obtained either by matching the vertical velocity dispersion , scale height and scale length of a thin exponential disc which yields the maximum disc rotation ( bottema 1993 ) , or , independently , showing that the tully fisher relation of bright spirals is independent of surface brightness ( cr99 ) . a maximum disc is defined as @xmath175 , at the 95 per cent confidence level ( sackett 1997 ) , corresponding to @xmath176 . to the sub - maximal discs described above corresponds a @xmath177 of @xmath178 . one notes that this choice of sub - maximal disc corresponds to half the amount of luminous matter at @xmath150 compared to the maximum disc case . let us now consider our simulations . @xmath179 can be measured directly once @xmath64 is determined for each dlo ( assuming that @xmath180 ) and the different hypotheses above can be tested . results are given in table [ dynamical ] for both the optimal and double exponential fits . @xmath181 is correlated with @xmath156 ( and , also , with @xmath164 , albeit with more dispersion ) . in fig . [ vratio ] we plot the @xmath182 ratios versus @xmath164 for dlos in our sample ( double exponential fits ; no substantial changes occur taking instead the values corresponding to optimal fits ) . the shaded area shows the 95 per cent interval for maximum discs according to sackett ( 1997 ) . only dlos # 126 and # 545 ( from s1 ) are inside this area , as they have @xmath183 0.765 and 0.763 , respectively . these dlos are somewhat peculiar : dlo # 126 has a particularly populated disc ; dlo # 545 has two satellites close to it . we see that most dlos lie inside the allowed interval for sub - maximal discs . the mean value for the simulated dlos is @xmath184 , in excellent agreement with bottema ( 1993 , 1997 ) and cr99 . in fig . [ vratio ] only a slight correlation appears between @xmath182 and @xmath164 with an important dispersion , in the sense that less massive haloes have slightly lower values of @xmath182 . this presumably reflects the fact that less massive haloes are somewhat more centrally concentrated than more massive ones ( navarro , frenk & white 1995b ; see also tdt98 ) . the dispersion , on the other hand , reflects the particularities of the assembly of each dlo in the sample . to quantify the importance of dark matter ( or baryons ) in the inner regions of dlos we have used the @xmath185 parameter ( tdt98 ) . we define @xmath185 as the radius where @xmath186 , such that rcs are dynamically dominated by baryons for @xmath187 , and dark matter dominated @xmath188 ( see table [ dynamical ] ) . note that most dlos have @xmath189 and only a few have @xmath190 ; note also that there is no correlation between @xmath185 and @xmath64 . in fig . [ rcross ] we plot @xmath191 versus @xmath156 ; some correlation appears , in the sense that more massive objects are less dark matter dominated at their centres . this correlation disappears when @xmath156 is normalized to the halo global velocity , @xmath164 , indicating that most of the correlation seen in fig . [ rcross ] is provided by the halo total mass . in fact , as stated above , less massive haloes are known to be more centrally concentrated than more massive ones . in figs . [ lsrd](a ) and [ lsrd](b ) , we plot @xmath149 versus the disc scale lengths @xmath64 and versus @xmath156 , respectively , for the dlos and , as a comparison , the objects of cvg91 compilation in the same dynamical range . we see that dlos with @xmath170 km s@xmath7 do have observational counterparts , i.e. , their @xmath149 parameters take values that are found in observed spirals with the same range in @xmath156 , even if we miss objects with @xmath149 between , say , @xmath192 and @xmath193 . those with @xmath194 km s@xmath7 have excessively low @xmath149s , but recall that only one galaxy in cvg91 has @xmath195 $ ] km s@xmath7 . moreover , according to cvg91 , @xmath149s are correlated with the morphological type . our dlos would be @xmath196 , in consistence with the range in @xmath197 type previously found from the bulge shape parameters . it is worthwhile to note that the value of the @xmath149 is almost independent on how baryonic mass is distributed inside @xmath150 , particularly on how it is shared between the bulge and the disc ( it changes on average by only 10 per cent when the bulge mass is halved conserving the baryon number inside @xmath150 ) . the fact that @xmath149s are negative may reflect in part the fact that dark matter haloes in standard cdm scenarios are too concentrated , leading to declining rotation curves ( navarro , frenk & white 1998 ) . in fact , the declining shape of dlo rcs results from the flat halo contribution to rcs ( see tdt98 for a detailed discussion ) and the dynamic baryon dominance in the inner dlo regions . declining rcs have been observed in spiral galaxies . early examples can be found in bosma ( 1978 , 1981 ) , van moorsel ( 1982 ) . then , cvg91 , b92 and bosma ( 1999 ) , among others , also found other examples . cvg91 suggest that this is a common feature of compact bright spiral galaxies , and also frequent in large bright spirals ( although see b92 who shows , contrary to cvg91 , that declining rcs do not correlate with the disc scale length ) . dubinski , mihos & hernquist ( 1999 ) have found that the declining character of rcs is a necessary condition to form long tidal tails in galaxy mergers as those observed in some interacting galaxy systems . in fig . [ vrmax ] we plot @xmath147 versus the @xmath198 ratio for the 29 dlos in our sample . for comparison , we also give the peak velocities in of sb sc spirals from table 4.1 in rc96 ( most of them from b92 compilation ) , with @xmath198 ratios taken from c97b . we also plot the @xmath147 and @xmath198 ratios measured in the courteau sample of optical rcs ( c97b ) . we confirm the previous finding that dlos in table [ general ] have observational counterparts , even if the smaller ones are somewhat too concentrated as compared with courteau data . note , however , that @xmath148 is difficult to measure in dlos and the measurement hard to assess in spiral galaxies . we present the results of a detailed and careful comparison between the parameters characterizing the structural and dynamical properties of a sample of 29 simulated dlos and those measured in observed spiral galaxies . these dlos have been identified in three fully consistent hierarchical hydrodynamical simulations , where an _ inefficient _ schmidt law - like algorithm to model the stellar formation process has been implemented . in this paper we have been only concerned with disc _ structural _ and _ dynamical _ properties , because they keep the imprints of the dynamical and hydrodynamical processes that , together with their interplay with star formation , are key for galaxy formation and evolution . the comparison of dlos in our sample with spiral discs has been a two step work . first , dlo bulge - disc decomposition has been performed . this yields the bulge effective scale length and shape parameters , @xmath85 and @xmath98 , the disc scale length , @xmath64 , and the mass bulge - to - disc ratio , @xmath128 , among other dlo parameters . after having tested the robustness of the fitting procedure , the following results have been obtained : 1 . the scale lengths , @xmath85 and @xmath64 , and their ratio @xmath199 , are consistent with available data ( cdjb96 ; de jong 1996 ; mgh99 ) . the distribution of bulge shape parameters is similar to that found by cdjb96 for their sample of sa sc galaxies . the mass @xmath128 ratios are somewhat high as compared to luminosity @xmath128 ratios , but current observational uncertainties on the bulge and disc mass - to - light ratios make it difficult to draw any conclusion from this comparison . in a second step , dlo rcs have been analyzed . dlo gas particles placed at distances between 2 kpc and 30 kpc to the centre move roughly along circular orbits on the equatorial plane . they are in centrifugal supported equilibrium within the potential well produced by the total mass distribution ( tdt98 ) . rotation curve shapes have been parametrized through @xmath64 ( or @xmath145 ) as spatial scale , @xmath200 as mass size , and the maximum or peak velocity , @xmath147 , and the radius where this is reached , @xmath148 . these parameters have been compared with those measured at : a ) sb sc field spirals drawn from the courteau sample , with long - slit h@xmath154 spectrum and @xmath115-band photometry ( c97b ) , and b ) sb sc spirals from broeils compilation ( b92 ) of extended rcs with surface photometry . the main results follow : 1 . in contrast to findings in other fully - consistent hydrodynamical simulations ( e.g. , navarro & steinmetz 2000 ; thacker & couchman 2000 ) , dlo @xmath156 velocities have been found to be consistent with observational data . this is a consequence of disc formation with @xmath1 conservation . the @xmath64 and @xmath156 values obtained indicate that dlos in our sample are either large bright spirals ( 18 ) , intermediate spirals ( 9 ) , or compact bright spirals ( 2 ) . the average relative contribution of baryons to @xmath156 in our sample is @xmath201 , in very good agreement with bottema ( 1993 , 1997 ) and cr99 , if we take @xmath202 . most dlos have been found to have sub - maximal discs ; only two of them have been found to lie inside the 95 per cent confidence interval for maximum discs according to sackett ( 1997 ) . the previous agreement also shows that the amount of baryon mass that has ended up inside @xmath203 is not excessive , again as a consequence of @xmath1 conservation . 4 . the @xmath191 parameter ratio is a measure of the relative amount of dark matter at the inner dlo zones . some correlation between these amounts and the circular velocity at @xmath150 has been found , in the sense that less massive objects tend to be more dark matter dominated in their central regions . 5 . concerning the parameters that give a measure of the central concentrations , the comparison of the @xmath147 versus @xmath204 plots to b92 data confirms again that dlos have observational counterparts , even if the smaller dlos are somewhat too concentrated as compared to courteau data ( c97b ) . the logarithmic slope , @xmath149 , measures the rc slopes for @xmath152 , where , as stated , in most cases dark matter is already dynamically dominant . dlos with @xmath170 km s@xmath7 occupy the same zone in the @xmath149 versus @xmath156 and versus @xmath150 plots as galaxies studied by cvg91 in the same dynamical range ; those with @xmath194 km s@xmath7 do not show the tendency of less massive galaxies in cvg91 to have higher @xmath149 . this may be a consequence of the strongly concentrated mass distribution of dark haloes in standard cdm scenarios . a concern is in order regarding numerical resolution . dlos are resolved with a relatively low number of particles . in contrast , dark matter haloes are described with a much better resolution . an inappropriate low gas resolution would result in an unphysical gas heating that could halt the gas collapse ( navarro & steinmetz 1997 ) . however , some works suggest that it is an inadequate resolution in the dark matter halo component that may produce the larger undesired numerical artifacts ( steinmetz & white 1997 ) . in fact , it appears that a well - resolved dark matter halo , even if the number of gas particles is lower ( but more than , say , one hundred ) , gives rise to a well - represented ( or well - resolved ) gas density profile , being this point the most important for both the hydrodynamics and the tracking of the star formation history . moreover , to make sure that the populated and extended discs in the simulations do not result from unphysical gas heating or smoothing , we have run a higher resolution simulation ( @xmath5 particles in a periodic box of 5 mpc , with cosmological and star formation parameters similar to those in s1 , s1b and s1c ; hereafter hrs ) . only one disc with mass comparable to those in these lower resolution simulations forms . it has 1713 + 1380 gas+star particles and its halo 36112 dark matter particles . its analysis has shown that it is populated and extended , that its structural and dynamical characteristics are also compatible with observations ( see table 1 and fig . 1 in domnguez - tenreiro et al . 1998 , where we show that the @xmath88 and @xmath64 parameters and their ratio take a typical value , and the specific angular momentum has been conserved ) and that the physical processes leading to its formation are essentially the same as those that are at work in s1 , s1b or s1c . in conclusion , the comparison between dlos produced in our simulations and observational data allows us to affirm that they have counterparts in the real world . this agreement suggests that the process operating in fall & efstathiou ( 1980 ) standard model for disc formation ( i.e. , gas cooling and collapse with specific angular momentum , @xmath1 , conservation ) is also at work in the _ quiescent _ phases of dlo formation in these simulations , resulting in discs with exponential density profiles , as predicted by dalcanton et al . however , in domnguez - tenreiro et al . ( 1998 ) and siz et al . ( 1999 ) it is shown that _ violent _ episodes ( i.e. , interaction and merger events ) also occur and play an important role in dlo assembly . in particular , it is shown that dark matter haloes formed in these simulations are not always able to stabilize the _ pure _ exponential disc that would form at its central region according to fall & efstathiou and dalcanton et al . scenarios ( see also van den bosch 2000 ) . to provide the right conditions for disc regeneration to occur after the last violent episode of dlo assembly , a compact central bulge is needed ; this will ensure the axisymmetric character of the gravitational potential well at scales of some kpcs at all times , avoiding excessive @xmath1 losses in violent events . in tsdt these arguments are developed , and , moreover , we prove that a second condition is necessary for disc regeneration : the availability of gas at low @xmath2 , that is , it is necessary that the sf algorithm implementation does not result in a too early gas exhaustion . the good behaviour of our dlo sample as compared with observations suggests that the _ inefficient _ sf algorithm used in the simulations has met both requirements . the global agreement with observations we have found , at a structural and dynamical level , also represents an important step towards making numerical approaches more widely used to learn about galaxy formation and evolution in a cosmological framework , i.e. , from primordial fluctuations . these approaches will be particularly useful if , as expected , future improvements in numerical methods and computers speed allow to numerically simulate the different aspects of galaxy assembly with an increasingly high degree of realism . it is a pleasure to thank e. bertschinger for providing us with some software subroutines , j. f. navarro for interesting discussions and j. silk for his valuable comments on our work . we wish to acknowledge as well the referee , j. sommer - larsen , for his suggestions and interesting comments . this work was supported in part by dges ( spain ) through grants number pb93 - 0252 and pb96 - 0029 . as was also supported by dges through fellowships . pbt thanks dges , and conicet and anpcyt ( argentina ) for their financial support . we are indebted to the centro de computacin cientfica ( universidad autnoma de madrid ) and to the oxford university for providing the computational support to perform this work . lllrrrrrlrrr & & & & & & & + simulation & dlo # & @xmath27 & @xmath39 & & @xmath28&@xmath41&@xmath40 & & @xmath28&@xmath41&@xmath40 + s1 & 126 & 239 . & 25.9 & & 2709 & 248 & 157 & & 516 & 185 & 157 + & 143 & 194 . & 37.5 & & 1482 & 153 & 47 & & 473 & 110 & 47 + & 233a & 310 . * & 28.5 & ( & 6027 & 583 & 164 & ) & 506 & 201 & 95 + & 233b & 310 . * & 25.4 & ( & 6027 & 583 & 164 & ) & 400 & 148 & 69 + & 242 & 316 . & 33.4 & & 6325 & 621 & 281 & & 959 & 339 & 278 + & 333 & 296 . & 49.7 & & 5213 & 509 & 217 & & 1337 & 338 & 212 + & 531 & 235 . & 30.9 & & 2573 & 275 & 79 & & 589 & 224 & 79 + & 544 & 312 . & 43.4 & & 6170 & 555 & 230 & & 1304 & 349 & 215 + & 545 & 328 . & 26.2 & & 7131 & 671 & 252 & & 794 & 381 & 238 + + s1b & 054 & 252 . & 35.9 & & 3076 & 299 & 138 & & 611 & 171 & 138 + & 110 & 254 . & 30.3 & & 3145 & 268 & 174 & & 648 & 163 & 174 + & 116@xmath154 & 303 . & 73.0 & & 5357 & 455 & 270 & & 1806 & 291 & 269 + & 116@xmath38 & 178 . & 38.1 & & 1029 & 171 & 30 & & 460 & 138 & 30 + & 164 & 222 . & 29.2 & & 2107 & 233 & 62 & & 496 & 160 & 62 + & 165c & 467 . * & 39.1 & ( & 19522 & 1188 & 1578 & ) & 556 & 113 & 56 + & 211 & 186 . & 43.5 & & 1236 & 157 & 12 & & 468 & 130 & 12 + & 212 & 261 . & 49.4 & & 3419 & 226 & 237 & & 1171 & 192 & 234 + & 633a & 312 . * & 40.9 & ( & 5890 & 371 & 401 & ) & 1068 & 188 & 325 + & 643@xmath154 & 317 . & 47.4 & & 6176 & 447 & 373 & & 1217 & 180 & 358 + & 643@xmath38a & 307 . * & 38.1 & ( & 5583 & 518 & 270 & ) & 695 & 217 & 213 + & 643@xmath38b & 307 . * & 32.4 & ( & 5583 & 518 & 270 & ) & 351 & 146 & 55 + & 643@xmath205 & 232 . & 40.8 & & 2409 & 234 & 97 & & 683 & 181 & 97 + + s1c & 106 & 217 . & 29.5 & & 1966 & 196 & 76 & & 419 & 122 & 76 + & 324 & 191 . & 36.6 & & 1322 & 173 & 22 & & 395 & 127 & 22 + & 342 & 280 . & 34.3 & & 4214 & 470 & 114 & & 770 & 311 & 107 + & 444 & 267 . & 33.8 & & 3727 & 352 & 99 & & 737 & 243 & 99 + & 454 & 215 . & 20.4 & & 1907 & 184 & 70 & & 383 & 149 & 70 + & 556 & 265 . & 31.0 & & 3605 & 257 & 229 & & 707 & 178 & 229 + & 631 & 215 . & 18.9 & & 1917 & 193 & 59 & & 322 & 145 & 59 + + distances are given in kpc . an asterisk indicates that the corresponding dlo is embedded in a double or multiple halo , i.e. , that one or more other baryonic objects ( either disc - like or not ) exist within @xmath27 . this is to be taken into account on evaluating the total number of halo particles ( numbers in parentheses ) . 126 & 2.6 & 0.90 & 6.3 & 1.83 & 5.72 & 4.46 & & 1.09 & 5.8 & 1.48 & 10.40 & 3.50 & & 6.5 & 6.35 + 143 & 3.2 & 0.58 & 10.9 & 1.45 & 1.61 & 4.54 & & 0.91 & 10.1 & 1.36 & 2.30 & 3.52 & & 11.1 & 1.53 + 233a & 1.3 & 1.00 & 6.5 & 1.25 & 5.52 & 3.57 & & 1.08 & 6.4 & 1.22 & 5.55 & 3.43 & & 7.1 & 5.81 + 233b & 0.6 & 1.05 & 5.6 & 1.22 & 9.09 & 2.77 & & 0.89 & 5.7 & 1.23 & 9.55 & 2.89 & & 5.8 & 12.54 + 242 & 1.2 & 1.25 & 7.5 & 1.23 & 15.96 & 4.19 & & 1.29 & 7.5 & 1.21 & 16.19 & 4.06 & & 7.9 & 42.04 + 333 & 1.9 & 1.36 & 12.0 & 1.72 & 5.30 & 6.59 & & 1.50 & 11.1 & 1.58 & 11.40 & 5.36 & & 12.8 & 11.98 + 531 & 0.9 & 1.70 & 6.9 & 2.02 & 3.85 & 5.24 & & 1.69 & 6.9 & 2.07 & 4.00 & 5.39 & & 9.3 & 19.57 + 544 & 2.3 & 1.05 & 10.6 & 1.79 & 2.44 & 5.82 & & 1.26 & 9.7 & 1.61 & 9.05 & 4.54 & & 11.3 & 4.06 + 545 & 1.1 & 1.26 & 5.9 & 1.23 & 3.01 & 3.78 & & 1.27 & 5.9 & 1.22 & 3.11 & 3.74 & & 6.4 & 20.24 + + 054 & 1.6 & 0.89 & 8.4 & 2.99 & 2.94 & 4.80 & & 1.04 & 8.0 & 2.82 & 3.65 & 4.13 & & 9.7 & 4.32 + 110 & 1.8 & 0.60 & 7.0 & 2.54 & 2.64 & 3.46 & & 0.78 & 6.8 & 2.47 & 3.28 & 3.13 & & 7.2 & 3.46 + 116@xmath154 & 2.3 & 0.48 & 16.4 & 1.40 & 13.97 & 4.12 & & 0.76 & 16.3 & 1.39 & 14.55 & 3.50 & & 16.5 & 14.79 + 116@xmath38 & 1.3 & 1.26 & 8.7 & 1.70 & 1.75 & 4.97 & & 1.34 & 8.5 & 1.64 & 1.93 & 4.60 & & 10.1 & 3.10 + 164 & 1.3 & 1.19 & 7.2 & 2.47 & 1.70 & 4.83 & & 1.25 & 6.9 & 2.36 & 1.99 & 4.44 & & 9.9 & 4.64 + 165c & 1.6 & 0.93 & 9.0 & 2.37 & 1.08 & 4.67 & & 1.06 & 8.7 & 2.28 & 1.28 & 4.15 & & 9.8 & 1.71 + 211 & 1.8 & 1.35 & 14.0 & 1.42 & 0.75 & 6.47 & & 1.49 & 13.0 & 1.36 & 1.19 & 5.40 & & 17.0 & 1.30 + 212 & 2.3 & 0.53 & 11.2 & 1.95 & 4.93 & 4.04 & & 0.80 & 11.1 & 1.91 & 6.30 & 3.46 & & 11.3 & 5.49 + 633a & 1.1 & 0.72 & 9.0 & 3.01 & 4.69 & 3.33 & & 0.74 & 9.2 & 2.98 & 4.33 & 3.32 & & 9.6 & 6.62 + 643@xmath154 & 1.4 & 1.19 & 10.9 & 3.40 & 3.14 & 5.91 & & 1.28 & 10.6 & 3.27 & 5.07 & 5.30 & & 13.2 & 19.61 + 643@xmath38a & 2.8 & 0.86 & 9.4 & 2.36 & 3.18 & 5.75 & & 1.13 & 8.5 & 2.02 & 10.91 & 4.22 & & 10.0 & 3.70 + 643@xmath38b & 1.4 & 1.28 & 7.6 & 1.98 & 1.46 & 5.02 & & 1.35 & 7.3 & 1.87 & 1.96 & 4.54 & & 9.5 & 3.45 + 643@xmath205 & 1.7 & 1.18 & 13.2 & 2.43 & 2.90 & 6.56 & & 1.32 & 11.9 & 2.32 & 4.33 & 5.30 & & 17.8 & 5.37 + + 106 & 2.0 & 1.06 & 7.5 & 2.69 & 1.18 & 5.51 & & 1.20 & 6.6 & 2.25 & 2.47 & 4.22 & & 8.2 & 1.70 + 324 & 1.2 & 1.42 & 8.3 & 1.96 & 0.80 & 5.22 & & 1.46 & 8.2 & 1.91 & 0.84 & 4.99 & & 10.1 & 2.56 + 342 & 1.3 & 1.43 & 8.2 & 1.45 & 2.52 & 5.09 & & 1.46 & 7.7 & 1.38 & 2.87 & 4.60 & & 9.2 & 9.95 + 444 & 1.3 & 1.41 & 7.9 & 2.02 & 3.49 & 5.34 & & 1.46 & 7.6 & 1.92 & 4.22 & 4.89 & & 10.9 & 11.14 + 454 & 1.0 & 1.23 & 4.6 & 3.14 & 2.33 & 4.26 & & 1.24 & 4.6 & 3.10 & 2.34 & 4.20 & & 10.2 & 6.45 + 556 & 2.1 & 0.86 & 7.6 & 3.70 & 1.64 & 5.37 & & 1.06 & 6.9 & 3.17 & 6.17 & 4.13 & & 8.2 & 3.70 + 631 & 0.9 & 1.13 & 4.1 & 2.36 & 0.97 & 3.50 & & 1.11 & 4.2 & 2.43 & 0.97 & 3.62 & & 18.0 & 2.78 + llllccccccccc & & & & & & & & & & + dlo # & @xmath164 & & @xmath185 & @xmath149 & @xmath148 & @xmath147 & & @xmath156 & @xmath181&&@xmath156 & @xmath181 + 126 & 114 . & 0.039 & 16.6 & -0.176 & 10.6 & 216 . & & 210 . & 157 . & & 206 . & 153 . + 143 & 98.5 & 0.045 & 8.94 & -0.172 & 7.98 & 156 . & & 148 . & 83.0 & & 147 . & 80.6 + 233a & 157 . * & 0.063 * & 14.0 & -0.139 & 9.45 & 203 . & & 197 . & 139 . & & 197 . & 139 . + 233b & 157 . * & 0.063 * & 12.9 & -0.085 & 6.93 & 179 . & & 176 . & 125 . & & 176 . & 125 . + 242 & 160 . & 0.069 & 19.3 & -0.168 & 6.33 & 279 . & & 257 . & 186 . & & 257 . & 186 . + 333 & 150 . & 0.079 & 16.2 & -0.108 & 7.25 & 250 . & & 221 . & 143 . & & 218 . & 139 . + 531 & 118 . & 0.011 & 12.7 & -0.192 & 7.70 & 217 . & & 206 . & 140 . & & 206 . & 140 . + 544 & 158 . & 0.023 & 17.8 & -0.104 & 5.69 & 254 . & & 228 . & 153 . & & 225 . & 148 . + 545 & 166 . & 0.049 & 18.6 & -0.159 & 8.66 & 278 . & & 270 . & 204 . & & 269 . & 204 . + + 054 & 126 . & 0.122 & 16.4 & -0.152 & 5.64 & 213 . & & 189 . & 132 . & & 187 . & 129 . + 110 & 127 . & 0.044 & 12.2 & -0.172 & 6.74 & 237 . & & 215 . & 148 . & & 214 . & 147 . + 116@xmath154 & 151 . & 0.067 & 16.8 & -0.097 & 6.46 & 249 . & & 206 . & 121 . & & 206 . & 121 . + 116@xmath38 & 88.8 & 0.055 & 8.80 & -0.214 & 8.11 & 167 . & & 155 . & 92.6 & & 153 . & 92.0 + 164 & 111 . & 0.038 & 9.63 & -0.193 & 6.88 & 199 . & & 186 . & 120 . & & 184 . & 118 . + 165c & 228 . * & 0.043 * & 7.70 & -0.136 & 7.43 & 176 . & & 160 . & 92.7 & & 160 . & 92.6 + 211 & 92.7 & 0.095 & 7.29 & -0.189 & 7.15 & 153 . & & 136 . & 70.8 & & 135 . & 69.0 + 212 & 130 . & 0.062 & 13.2 & -0.155 & 5.64 & 243 . & & 213 . & 132 . & & 213 . & 131 . + 633a & 156 . * & 0.067 * & 19.9 & -0.161 & 5.78 & 265 . & & 227 . & 160 . & & 227 . & 161 . + 643@xmath154 & 158 . & 0.086 & 20.6 & -0.118 & 5.78 & 275 . & & 222 . & 153 . & & 222 . + 643@xmath38a & 154 . * & 0.067 * & 20.3 & -0.184 & 7.43 & 234 . & & 209 . & 149 . & & 204 . & 144 . + 643@xmath38b & 154 . * & 0.067 * & 15.1 & -0.166 & 8.11 & 170 . & & 157 . & 110 . & & 156 . & 108 . + 643@xmath205 & 116 . & 0.054 & 12.7 & -0.174 & 8.53 & 205 . & & 176 . & 105 . & & 171 . & 100 . + + 106 & 108 . & 0.075 & 12.4 & -0.167 & 5.64 & 177 . & & 169 . & 116 . & & 167 . & 110 . + 324 & 95.2 & 0.045 & 8.53 & -0.201 & 8.66 & 160 . & & 147 . & 89.9 & & 147 . & 88.8 + 342 & 140 . & 0.018 & 15.1 & -0.152 & 9.76 & 235 . & & 220 . & 153 . & & 217 . & 149 . + 444 & 133 . & 0.055 & 14.0 & -0.135 & 6.74 & 225 . & & 207 . & 141 . & & 207 . & 139 . + 454 & 107 . & 0.034 & 10.6 & -0.154 & 6.88 & 205 . & & 197 . & 141 . & & 197 . & 141 . + 556 & 130 . & 0.063 & 15.4 & -0.167 & 6.46 & 248 . & & 226 . & 160 . & & 222 . & 155 . + 631 & 107 . & 0.025 & 9.63 & -0.178 & 6.33 & 199 . & & 196 . & 140 . & & 196 . & 141 . + + distances are given in kpc ; velocities in km s@xmath7 . asterisks stand for double or multiple haloes , so that their global parameters ( with an asterisk in this table ) might not reflect a direct property of the corresponding dlo .

we present results from a careful and detailed analysis of the structural and dynamical properties of a sample of 29 disc - like objects identified at @xmath0 in three ap3m - sph fully consistent cosmological simulations . these simulations are realizations of a cdm hierarchical model , where an _ inefficient _ schmidt law - like algorithm to model the stellar formation process has been implemented . we focus on properties that can be constrained with available data from observations of spiral galaxies , namely , the bulge and disc structural parameters and the rotation curves . comparison with data from broeils ( 1992 ) , de jong ( 1996 ) and courteau ( 1996 , 1997 ) gives satisfactory agreement , in contrast with previous findings using other codes . this suggests that the stellar formation implementation we have used has succeeded in forming compact bulges that stabilize disc - like structures in the violent phases of their assembly , while in the quiescent phases the gas has cooled and collapsed according with the fall & efstathiou standard model of disc formation . galaxies : evolution galaxies : formation galaxies : structure galaxies : spiral hydrodynamics methods : numerical cosmology : theory dark matter

astro-ph0102011

atoms scattered out of bose - einstein condensates can be an object of benchmark tests of various quantum - mechanical models . a prominent example is a collision of two counter - propagating condensates @xcite . during the collision , which takes place at super - sonic velocity , atoms are scattered into initially empty modes , and description of such process requires fully quantum treatment . this can be done semi - analytically in the bogoliubov approximation @xcite or numerically in more general cases @xcite . the analysis reveals strong correlations between the scattered atoms @xcite and sub - poissonian fluctuations of the opposite - momentum atom counts @xcite . therefore , the many - body atomic states created in the collisions could have potential application for ultra - precise sub shot - noise atomic interferometry @xcite . a different relevant example of atom scattering out of a coherent cloud takes place in a spin-1 condensate @xcite . in this case , a single stationary matter - wave is prepared in a zeeman substate with @xmath0 . a two - body interaction can change the spin projection of the colliding pair into @xmath1 . recently , it has been demonstrated @xcite that produced atomic pairs are usefully entangled from atom - interferometry point of view . here we concentrate on another pair production process , namely the raman scattering @xcite . in this case , an ultra - cold atomic cloud is illuminated with a strong laser beam . as a result , an inter - atomic transition leads to creation of a correlated stokes photon and atomic excitation . the scattered pairs are correlated analogously to those produced in the condensate- or spin - changing collisions . raman scattering is similar to the elastic rayleigh process @xcite , though the stokes photons have different energy then the incident light . this process has been widely studied theoretically @xcite and observed experimentally in ultra - cold samples @xcite and bose - einstein condensates @xcite . in this work we consider a different source of raman - scattered particles , namely the quasi - condensate , which forms in strongly elongated traps @xcite . due to non - zero temperature of the gas , phase fluctuations occur and they limit the spatial coherence of the system . this , in turn , has influence on the scattering process . we demonstrate that one can determine the temperature of the parent cloud from both the density and the second order correlation function of the scattered atoms . , absorbs strongly detuned pump photon with wave - vector @xmath2 . the absorption is accompanied by spontaneous emission of a stokes photon with wave - vector @xmath3 . as a result , the atom undergoes a transition @xmath4 . ] the paper is organized as follows . in section ii we formulate the 3-dimensional problem and introduce the hamiltonian for the process of raman scattering . we derive the heisenberg equations for atoms and photons and introduce the relevant correlation functions . in section iii , basing on perturbative solution of the atomic dynamics , we calculate the one - body density matrix both in the position and momentum representation . in section iv we discuss the method for incorporating the phase fluctuations due to non - zero temperature of the quasi - condensate . using this approach , we calculate the density of scattered atoms both after long expansion time , i.e. in the far - field regime , and when the expansion time is short . then , we turn to the second order correlation function . we show , how the temperature influences its peak height as well as the width . some details of calculations are presented in the appendices . the process of raman scattering takes places when an atom in a three - level lambda configuration is illuminated with an intense pump beam . as a result of interaction with light , the atom absorbs a photon from the pump and undergoes an effective transition @xmath5 accompanied by spontaneous emission of a `` stokes '' photon shown in fig . [ fig : levels ] . to model the phenomenon , we assume that the pump can be described classically as @xmath6 where @xmath7 is its amplitude , @xmath2 is the central wave - vector and @xmath8 . when this frequency is strongly detuned from the transition @xmath9 , the upper level can be adiabatically eliminated . as a result , the process can be regarded as creation of a quantum of atomic excitation @xmath4 together with an emission of a stokes photon . we describe the quantum state of the atoms and stokes photons using two field operators @xmath10 where @xmath11 is the @xmath4 transition frequency . the operator @xmath12 annihilates an atomic excitation with momentum @xmath13 , while the index @xmath14 runs over all the @xmath15 atoms in the cloud . if the majority of atoms occupy @xmath16 , one can apply the holstein - primakoff approximation @xcite , and accordingly @xmath17 satisfies bosonic commutation relations . moreover , @xmath18 is the field operator of the stokes photons . when a large number of atoms @xmath15 occupy a single - particle state , one can replace summation over separate particles in eq.([def_b ] ) with an integral over the quasi - condensate wave - function @xmath19 . the effective hamiltonian for the process of raman scattering is @xmath20 , where @xmath21 is the free part , with @xmath22 . also , @xmath23 is centered around the stokes frequency @xmath24 . the interaction hamiltonian @xmath25 governs the desired process , where an atomic excitation is created together with the stokes photon . the coupling function @xmath26 is expressed in terms of a fourier transform of the product of the quasi - condensate and pump beam fields , @xmath27 . after many scattering events , the photons will form a sphere of radius @xmath28 denoted here by the dashed circle . due to momentum conservation , atoms scatter onto a sphere of radius @xmath28 as well , shifted by @xmath29 due to absorption of the pump photon . the width of the gray ring occupied by the atoms represents the uncertainty resulting from the momentum spread of the parent quasi - condensate . ] where the coupling constant @xcite is equal to @xmath30 here , @xmath31 is the atomic dipole moment associated with the @xmath32 transition and @xmath33 is the dielectric constant . note that in eq . ( [ hint ] ) we have neglected the interaction of the scattered atoms with the mean - field of the quasi - condensate . below we make further , physically well justified simplifications . first , we choose the pump envelope @xmath7 to be time - independent , which corresponds to a common situation of square pulses . moreover , the spatial extent of the pump usually vastly exceeds the size of the quasi - condensate . since the duration of the pump pulse is much shorter than the characteristic time scale of @xmath19 dynamics , the quasi - condensate wave - function can be taken constant and the coupling function in a frame of reference moving with velocity @xmath34 reads @xmath35 here , @xmath36 is a fourier transform of wave - function @xmath37 . we can now derive the set of coupled heisenberg equations of motion for the stokes and atomic field resulting from the hamiltonian ( [ ham ] ) , [ heis ] @xmath38 these equations are a starting point for the analysis of the second order correlation function of scattered atoms , defined as @xmath39 since equations ( [ heis ] ) are linear and the initial state of scattered atoms and photons is a vacuum , then @xmath40 is a function of the ( one - body ) density matrix , which reads @xmath41 its diagonal part @xmath42 represents the momentum distribution of scattered atoms . in the following section we derive analytical expressions for the density matrix in momentum and position representations treating the atom - photon interactions in the perturbative manner . when the number of scattered atom - photon pairs is small , one can solve eq . ( [ heis_b ] ) perturbatively in the coupling constant @xmath43 defined in eq . ( [ h0 ] ) , @xmath44 where @xmath45 . as we argue in appendix [ app_time_dep_b ] , since @xmath46 , the first order solution can be written as @xmath47 this expression is used to calculate the first order correlation function ( [ g1 ] ) of scattered atoms . the measurement of positions of scattered atoms is performed as follows . first , the initial wave - packet of the quasi - condensate interacts with the pump beam for a time @xmath48 and atoms scatter out of the mother cloud . then , the system freely expands for a time @xmath49 , and finally the positions of atoms are recorded . if @xmath49 is sufficiently long and the system reaches the far - field regime , positions of atoms @xmath50 are related to their wave - vectors @xmath51 by @xmath52 . in the present work we compare the two possible experimental situations , when the system either is or is not in the far field . in the former case , as we argued above , it is sufficient to calculate the density matrix ( [ g1 ] ) in the momentum space just after the interaction ends . in the latter , we provide an expression for @xmath53 as a function of expansion time in position space . ) for @xmath54 . the gray ring represents the halo of scattered atoms , in analogy to fig . [ fig : schematic_halo ] . when @xmath55 , as in the main part of the picture , the integration samples the majority of the atomic cloud , leading to value of the peak height . ( a ) when @xmath56 , the integration samples the tails of the quasi - condensate and thus the density of scattered atoms does not vanish . ( b ) however , when @xmath57 , the tails do not contribute to the integral anymore and the density drops rapidly with growing @xmath58 . ] in order to calculate the momentum - dependent density matrix , note that for typical interaction times , the `` sinc '' function appearing in eq . ( [ b_time_dependence ] ) , which is peaked around @xmath59 , has much smaller width than the fourier transform of the condensate function , which , via eq . ( [ h_def_fourier ] ) , enters @xmath60 . therefore , one can fix the length of the photon wave - vector to be equal to @xmath28 . using the definition from eq . ( [ g1 ] ) and the solution from eq . ( [ b_time_dependence ] ) we obtain the density matrix in the momentum representation , @xmath61 where we omitted an irrelevant phase factor and @xmath62 . the integration is performed over all the directions of the unit vector @xmath63 . since the above perturbative expression , apart from a trivial scaling of @xmath14 with @xmath48 , is time - independent , we have skipped the time argument of @xmath53 . all the intermediate steps leading to the above solution are presented in appendix [ app_g1 ] . by setting @xmath64 we obtain the density of the scattered atoms @xmath65 which is directly related to the momentum distribution of the quasi - condensate . integration over all directions of stokes photon momentum @xmath66 is characteristic for a spontaneous regime , where photons scatter isotropically . note also that since the quasi - condensate wave - function from eq . ( [ h_def_fourier ] ) is expressed in the reference frame moving with the velocity @xmath34 , in the laboratory frame scattered atoms form a halo centered around @xmath2 vector , see fig . [ fig : schematic_halo ] . for different temperatures and normalized by the value of the peak height . each curve is an average over 400 realizations of the phase noise . the black solid line is calculated for the quasi - condensate at @xmath67nk , the dotted blue line for @xmath68200nk and the dashed red line for @xmath68960nk.,scaledwidth=45.0% ] to deal with situations when @xmath49 is not sufficiently long for the system to enter the far - field regime we provide an expression for the density matrix in position space as a function of the expansion time @xmath49 , which up to an irrelevant phase factor reads @xmath69 \cdot s\left[\tilde\psi\left(\frac{{{\mathbf r}}_2 m}{\hbar t_f}+ k_s { \mathbf{n}}\right)\right]^\star.\nonumber \label{g1_position}\end{aligned}\ ] ] here , the functional @xmath70 is given by @xmath71 = \int\!\!d(\delta{{\mathbf k}})\ e^{-i\frac{\hbar ( \delta k)^2}{2m}t_f } \tilde\psi({{\mathbf k}}+ \delta{{\mathbf k } } ) . \label{def_s}\ ] ] the derivation of eq . ( [ g1_position ] ) is presented in detail in the appendix [ app_position_rep ] . note that ( [ g1_position ] ) resembles ( [ g1_momentum ] ) except that @xmath36 is replaced with @xmath72 $ ] . the atomic density is obtained by setting @xmath73 in eq . ( [ g1_position ] ) and reads @xmath74\right|^2.\ ] ] we also underline that for sufficiently long @xmath49 , @xmath75\sim\tilde\psi({{\mathbf k}})$ ] , so the far field is reached . in this section we calculate the correlation functions ( [ g1_momentum ] ) and ( [ g1_position ] ) using realistic experimental parameters . first , we briefly describe a numerical method for simulating phase fluctuations present in a strongly elongated ultra - cold bosonic gas . for different temperatures and normalized by the value of the peak height . the time of flight is @xmath76ms . each curve is an average over 400 realizations of the phase noise . the black solid line is calculated for the quasi - condensate at @xmath67nk , the dotted blue line for dashed @xmath68200nk and the dashed red line for @xmath68960nk . , scaledwidth=45.0% ] we apply the above model to the process of raman scattering of atoms from @xmath77 metastable @xmath78he bosons with the atomic mass @xmath79 kg and the scattering length @xmath80 m . the atoms are confined in a harmonic potential with the radial and the axial frequencies equal to @xmath81 , @xmath82 . such an elongated gas is called a `` quasi - condensate '' due to presence of the phase fluctuations along the @xmath83 axis @xcite . to account for the quasi - condensate fluctuations , we use the method introduced in @xcite . first , we evaluate numerically the density profile of the pure condensate by finding a ground state of the stationary gross - pitaevskii equation @xmath84 where @xmath85 and @xmath86 is the chemical potential . next , we construct the quasi - condensate wave function @xmath87 by imprinting a phase @xmath88 onto the pure condensate function , @xmath89 , where @xmath90 here , @xmath91 is the energy of the low - lying axial excitations and @xmath92 is the jacobi polynomial . also , @xmath93 is the axial density width given by thomas - fermi approximation . as a function of temperature . different curves correspond to averaging over various number of realizations . clearly , @xmath94 and the inequality is saturated only for @xmath95 . ] the phase fluctuations result from randomness of @xmath96 , which is drawn from a gaussian distribution with zero mean and the variance given by occupation of the @xmath97-th mode @xmath98 consequently , the temperature of the gas enters the dynamics of the raman scattering . due to presence of the @xmath83-dependent phase factor ( [ phase ] ) in the quasi - condensate wave - function , the momentum distribution along the @xmath83 axis broadens as the temperature grows . the quasi - condensate is illuminated with an intense laser beam with the wave - vector equal to @xmath99 . since the stokes photon and the pump wave - vectors are similar , we set @xmath100 . our final simplification regards the form of the condensate density profile . a simple numerical check shows that the ground state of eq . ( [ gp ] ) can be approximated by a gaussian function in the radial direction , so that @xmath101 the axial function @xmath102 is found numerically by setting @xmath103 in eq . ( [ gp ] ) and the gaussian fit gives @xmath104 . all the numerical results presented below are obtained by calculating the relevant physical quantity for a single realization of the phase noise @xmath88 and then averaging over many such realizations . we can now estimate the free expansion time @xmath49 , at which the system enters the far - field regime in the @xmath83 direction . the velocity spread of the quasi - condensate @xmath105 is approximately 2 mm / s , while the initial size is @xmath106 mm . therefore , the far - field condition would be @xmath107 s. let us now comment on the consistency of the above approximations . equation ( [ phase ] ) was derived in @xcite under assumption , that the quasi - condensate has a thomas - fermi density profile in all three dimensions , valid when the non - linear term dominates the gross - pitaevskii equation . to simplify our calculations , we model radial wave - functions with gaussians , which is true in the quasi-1d limit , when the radial trapping potential dominates over the non - linear term . however , when we plot the numerically evaluated ground state of eq . ( [ gp ] ) , it turns out to be in an intermediate regime , and could be equally well modeled with either a gaussian or a thomas - fermi shape . therefore , the above method can be regarded as an appropriate approach in such a transitional case . since our main goal is to demonstrate the general behavior of the density and the correlation functions of scattered atoms as a function of @xmath108 , we believe that this approximate method is sufficiently precise for the purpose . first , we present the numerical results for the momentum distribution of scattered atoms , as given by eq . ( [ dens ] ) . we use the reference frame co - moving with a velocity @xmath34 , hence according to fig . [ fig : schematic_halo ] the density is centered around @xmath109 with the radius equal to @xmath28 . we investigate the momentum density as a function of @xmath110 in a vicinity of @xmath55 . this quantity , via eq . ( [ dens ] ) , samples the @xmath83-dependence of the momentum distribution of the quasi - condensate and therefore may provide some information on its temperature . in fig . [ den_int ] we schematically show which @xmath111 vectors contribute to the integral eq . ( [ dens ] ) with @xmath54 . when @xmath112 , as in the main part of the figure , the integration runs approximately through the center of the cloud . when @xmath113 , as in the inset ( a ) , the tails of the quasi - condensate still contribute to the density so the integral does not vanish rapidly as we move away from @xmath112 . on the other hand , when @xmath114 , shown in ( b ) , the intergal does not sample the tails anymore and the density of scattered atoms quickly drops with growing @xmath110 . this simple graphical interpretation is readily confirmed in fig . [ kz_den ] , where we present the result of numerical integration of eq . ( [ dens ] ) with the quasi - condensate function obtained for various temperatures and averaged over 400 realizations of the phase noise . as expected , for @xmath115nk the density is peaked around @xmath116 , and is largely extended for @xmath113 , while it drops immediately as @xmath114 . as the temperature grows , the density widens substantially due to increased width of the quasi - condensate in the momentum space . therefore , the density of scattered atoms , when measured along the @xmath110 axis , could be used to determine the temperature of the mother cloud . next , we investigate the dependence of the atomic density measured in position space after a typical expansion time of @xmath117ms @xcite . at this time , as argued in the previous section , the system has not yet entered the far - field regime . we set @xmath118 in eq . ( [ dens_pos ] ) to make a direct comparison with previous results and evaluate the integral numerically . we observe that when the expansion time is finite , contrary to the far - field regime considered above , the density is less sensitive to the temperature . also , we notice that for high @xmath108 results in both cases are more similar . on the length of the wave - vector @xmath119 in a vicinity of @xmath120 . the black solid line is calculated for @xmath115nk , the dotted blue for @xmath121nk while the dashed red for @xmath122nk . the number of realizations was 400 for each curve.,scaledwidth=45.0% ] this is because at large temperatures , the momentum distribution of the quasi - condensate broadens and so the far - field condition is satisfied at earlier times . we now investigate the impact of the phase fluctuations on the correlations of scattered atoms . we begin with the far - field expression ( [ g1_momentum ] ) and using eq . ( [ g2 ] ) we calculate the normalized second order correlation function @xmath123 where @xmath124 denotes averaging over many realizations of the phase @xmath88 . in order to be consistent with the results of the previous section , we concentrate on a region of wave - vectors close to @xmath120 . namely , we set @xmath125 and @xmath126 and analyze @xmath127 . first , note that for @xmath128 , according to eq . ( [ g2 ] ) we have @xmath129 , where @xmath130 . since the variance @xmath131 is non - negative , then @xmath94 . the inequality is saturated only in the absence of noise fluctuations . to picture the impact of the temperature on the height of the peak of the second order correlation function , in fig . [ fig_peak ] we plot @xmath132 for various temperatures . we clearly notice the change of the height of the peak as soon as @xmath133 . in the position space in a vicinity of @xmath134 as a function of @xmath135 . the black solid line is calculated for @xmath115nk , the dotted blue for @xmath121nk while the dashed red for @xmath122nk . the number of realizations was 400 for each curve.,scaledwidth=45.0% ] next , in fig . [ kz_corr ] we plot @xmath127 as a function of @xmath119 for various temperatures averaged for 400 realizations . we observe that apart from the change of the peak height @xmath132 , the wings of the correlation function broaden , due to increased momentum width of the quasi - condensate at higher temperatures . when @xmath95 , the correlation function oscillates in the momentum space . this behavior is determined by a fourier transform of the thomas - fermi profile in the @xmath110 direction . at larger temperatures , the phase fluctuations smear out the fringes . we now switch to the finite expansion time regime and evaluate the normalized second order correlation function in position space . namely , we use the definition ( [ g2 ] ) , where the @xmath53 function is calculated using eq . ( [ g1_position ] ) , giving @xmath136 in analogy to the far - field case , we set @xmath137 and @xmath138 . in fig . [ z_corr ] we plot @xmath139 as a function of @xmath135 for three different values of temperature @xmath108 and @xmath140 ms . equivalently to the results obtained in the momentum space , we have @xmath141 and agian we observe that this equality is saturated only for @xmath95 . however , differently from the previous case , the oscillations of the correlation function at low temperatures are not present , since the thomas - fermi profile is a smooth function of @xmath83 . also , the broadeding of the correlation function is much less pronounced . we have analyzed in detail the properties of the field of atoms scattered out of a quasi - condensate in the raman process . we have demonstrated that the density of scattered atoms , when measured in the far - field regime , strongly depends on the temperature of the quasi - condensate . however , this dependence is much weaker , when the expansion time is finite . furthermore , we have calculated the second order correlation function in both expansion time regimes . in each case , @xmath142 broadens with growing @xmath108 , although in the latter the effect is less pronounced . the presence of the temperature - induced phase fluctuations can be also deduced from the peak height of @xmath142 . while for the pure condensate , @xmath143 , this value can substantially rise at higher @xmath108 . in summary , the measurements of the position of scattered atoms could provide some information on the temperature of the mother quasi - condensate . nevertheless , physical quantities such as the density or the correlation functions do not change drastically in a wide range of temperatures @xmath144\,\mu$]k . if the experiment is aimed at determining the temperature of the quasi - condensate , it requires very high spatial resolution low detection noise and long expansion time . note that in our calculations we have neglected the possible impact of the atomic transition rules on the field of scattered atoms . in the case of particular atomic transitions , due to polarization of the stokes field , some scattering directions are forbidden . we underline , that such effect could be easily taken into account by modifying the coupling function @xmath145 . we aregrateful to wojciech wasilewski , marie bonneau , denis boiron and chris westbrook for fruitful discussions . acknowledges the foundation for polish science international ph.d . projects program co - financed by the eu european regional development fund . j. ch . acknowledges foundation for polish science international team program co - financed by the eu european regional development fund . p.z . acknowledges the support of polish ministry of science and higher education program `` mobility plus '' . m.t . acknowledges financial support of the national science centre . in this appendix we present details of the derivation of eq.([b_time_dependence ] ) . first , we introduce a solution of eq . ( [ heis_a ] ) in absence of coupling , i.e. @xmath146 this expression , when inserted into eq . ( [ heis_b ] ) , gives a first order equation of motion for the atomic field , which reads @xmath147 where @xmath148 . integration over time gives @xmath149 the typical values of the kinetic energy of scattered atoms are @xmath150 , while the photon energies are of the order of @xmath151 . since @xmath152 is of the order of the inverse of `` compton wavelength '' of an atom one can drop the dependence on the atom energy in the `` sinc '' function . we now express the above equation in terms of operator @xmath17 and arrive at eq . ( [ b_time_dependence ] ) . in this appendix we derive eq.([g1_momentum ] ) . using the perturbative solution from eq . ( [ b_time_dependence ] ) and the definition of @xmath53 from eq . ( [ g1 ] ) , we obtain up to an irrelevant phase factor @xmath153 typically the duration of the pump pulse is of the order of @xmath154s . for such value , @xmath155 is much smaller than the width of the coupling function @xmath156 , which , via eq . ( [ h_def_fourier ] ) , is related to the fourier transform of the quasi - condensate function . since @xmath157 is centered around @xmath158 , we can set @xmath159 in the coupling function and perform the integral over @xmath160 . this way , we obtain @xmath161 where @xmath162 , @xmath163 and @xmath164 denotes integration over a solid angle pointed by @xmath63 . using the definition of the coupling function from eq . ( [ h_def_fourier ] ) we arrive at eq . ( [ g1_momentum ] ) . in this appendix , we calculate the first order correlation function in the position space after @xmath49 time of the free expansion . we employ a reasonable approximation , that the pump duration time @xmath48 is much shorter than @xmath49 . in this case , the position - dependent correlation function @xmath165 is simply given by the following fourier transform of eq . ( [ app_g1 ] ) @xmath166 where @xmath167 and @xmath168 changing the variables to @xmath169 gives eq . ( [ g1_position ] ) , up to an irrelevant phase factor . m. kozuma , l. deng , e. w. hagley , j. wen , r. lutwak , k. helmerson , s. l. rolston , and w. d. phillips , phys . lett . * 82 * , 871 ( 1999 ) j. m. vogels , k. xu , and w. ketterle , phys . lett . * 89 * , 020401 ( 2002 ) a. perrin , h. chang , v. krachmalnicoff , m. schellekens , d. boiron , a. aspect , and c. i. westbrook , phys . rev . lett . * 99 * , 150405 ( 2007 ) vv . krachmalnicoff , j .- c . jaskula , m. bonneau , v. leung , g. b. partridge , d. boiron , c. i. westbrook , p. deuar , p. zi , m. trippenbach , and k. v. kheruntsyan , phys . lett . * 104 * , 150402 ( 2010 ) j .- c . jaskula , m. bonneau , g. b. partridge , v. krachmalnicoff , p. deuar , k. v. kheruntsyan , a. aspect , d. boiron , and c. i. westbrook , phys . lett . * 105 * , 190402 ( 2010 ) wu rugway , s. s. hodgman , r. g. dall , m. t. johnsson , and a. g. truscott , phys . * 107 * , 075301 ( 2011 ) r. a. williams , l. j. leblanc , k. jimnez - garca , m. c. beeler , a. r. perry , w. d. phillips , and i. b. spielman , science * 335 * , 314 ( 2012 ) r. bach , m. trippenbach , and k. rzaewski , phys . a * 65 * , 063605 ( 2002 ) p. zi , j. chwedeczuk , a. veitia , k. rzaewski , and m. trippenbach , phys . lett . * 94 * , 200401 ( 2005 ) p. zi , j. chwedeczuk , and m. trippenbach , phys . a * 73 * , 033602 ( 2006 ) j. chwedeczuk , p. zi , k. rzaewski , and m. trippenbach , phys . * 97 * , 170404 ( 2006 ) j. chwedeczuk , p. zi , m. trippenbach , a. perrin , v. leung , d. boiron , and c. i. westbrook , phys . , 053605 ( 2008 ) a. a. norrie , r. j. ballagh , and c. w. gardiner , phys . rev . lett . * 94 * , 040401 ( 2005 ) p. deuar and p. d. drummond , phys . 98 * , 120402 ( 2007 ) p. deuar , phys . * 103 * , 130402 ( 2009 ) p. deuar , j. chwedeczuk , m. trippenbach , and p. zi , phys . rev . a * 83 * , 063625 ( 2011 ) p. deuar , p. zi , j. chwedeczuk and m. trippenbach , eur . j. d * 65 * , 19 ( 2011 ) a. d. cronin , j. schmiedmayer , and david e. pritchard , rev . mod . phys . * 81 * , 1051 ( 2009 ) and references therein c. klempt , o. topic , g. gebreyesus , m. scherer , t. henninger , p. hyllus , w. ertmer , l. santos , and j. j. arlt , phys . rev * 103 * , 195302 ( 2009 ) c. klempt , o. topic , g. gebreyesus , m. scherer , t. henninger , p. hyllus , w. ertmer , l. santos , and j. j. arlt , phys . * 104 * , 195303 ( 2010 ) b. lcke , m. scherer , j. kruse , l. pezz , f. deuretzbacher , p. hyllus , o. topic , j. peise , w. ertmer , j. arlt , l. santos , a. smerzi and c. klempt , science * 11 * , 773 - 776 ( 2011 ) klemens hammerer , anders s. srensen and eugene s. polzik , rev . * 82 * , 1041 ( 2010 ) c. simon _ . j. d * 58 * , 1 ( 2010 ) s. inouye , a. p. chikkatur , d. m. stamper - kurn , j. stenger , d. e. pritchard , w. ketterle , science * 285 * , 571 ( 1999 ) j. stenger , s. inouye , d.m . stamper - kurn , a.p . chikkatur , d.e . pritchard , w. ketterle , appl . b * 69 * , 347 ( 1999 ) m. g. moore and p. meystre , phys . rev . lett . * 85 * , 5026 ( 2000 ) m. g. payne and l. deng , phys . rev . a * 64 * , 031802(r ) ( 2001 ) xu xu , xiaoji zhou , and xuzong chen , phys . a * 79 * , 033605 ( 2009 ) tun wang and s. f. yelin , phys . rev . a * 72 * , 043804 ( 2005 ) h. uys and p. meystre , phys . a * 77 * , 063614 ( 2008 ) yutaka yoshikawa , yoshio torii , and takahiro kuga , phys . lett . * 94 * , 083602 ( 2005 ) dominik schneble , gretchen k. campbell , erik w. streed , micah boyd , david e. pritchard , and wolfgang ketterle , phys . a * 69 * , 041601(r ) ( 2004 ) yutaka yoshikawa , toshiaki sugiura , yoshio torii , and takahiro kuga , phys . a * 69 * , 041603(r ) ( 2004 ) yutaka yoshikawa , kazuyuki nakayama , yoshio torii , and takahiro kuga , phys . lett . * 99 * , 220407 ( 2007 ) l. e. sadler , j. m. higbie , s. r. leslie , m. vengalattore , and d. m. stamper - kurn , phys . * 98 * , 110401 ( 2007 ) s. dettmer _ et al_. , phys . lett . * 87 * , 160406 ( 2001 ) d.s . petrov , g.v . shlyapnikov and j.t.m . walraven , phys . rev . lett . * 87 * , 050404 ( 2001 ) d.s . petrov , d.m . gangardt and g.v . shlyapnikov , j. phys . iv france * 116 * , 3 ( 2004 ) t. holstein and h. primakoff , phys . rev . * 58 * , 1098 ( 1940 ) m. raymer , i. walmsley , j. mostowski and b. sobolewska , phys . a * 32 * , 332 ( 1985 )

it is demonstrated that measurements of positions of atoms scattered from a quasi - condensate in a raman process provide information on the temperature of the parent cloud . in particular , the widths of the density and second order correlation functions are sensitive to the phase fluctuations induced by non - zero temperature of the quasi - condensate . it is also shown how these widths evolve during expansion of the cloud of scattered atoms . these results are useful for planning future raman scattering experiments and indicate the degree of spatial resolution of atom - position measurements necessary to detect the temperature dependence of the quasi - condensate .

1203.0894

analytic models and numerical simulations of clusters of galaxies have been used to predict the existence of scaling relations between various observable quantities , such as the well - known luminosity ( @xmath3 ) - temperature ( @xmath4 ) and mass ( @xmath5 ) - temperature relations , where and , respectively . however , it is now fairly well established that x - ray properties of clusters do not scale in such a fashion . most notable of these is the relationship , which is observed to be much steeper than predicted , ( e.g. , markevitch 1998 ; allen & fabian 1998 ; arnaud & evrard 1999 ) . considerable effort has recently been directed towards explaining why the observed relations deviate from their predicted scalings ( e.g. , tozzi & norman 2001 ; dav et al . 2001 ; babul et al . 2002 , hereafter bblp02 ) . in particular , it is the @xmath6 relation that has grabbed most of the spotlight because there is a wealth of published observational studies on the luminosities and temperatures of clusters with which to compare models and simulations . however , another important scaling relation is the cluster gas mass relation . neumann & arnaud ( 2001 ) have suggested that a deviation from the self - similar scaling of @xmath7 might `` explain '' the observed deviation in the @xmath8 relation . indeed , a number of observational studies have indicated that the relation is much steeper , with @xmath9 ( vikhlinin et al . 1999 ; mohr et al . 1999 , hereafter mme99 ; neumann & arnaud 2001 ) . if the gas density profile is roughly self - similar , this does lead to consistency with the observed relation . however , we still need a _ physical explanation _ for why the relationship between a cluster s gas mass and its temperature deviates from its self - similar scaling . expressing the total gas mass within the cluster as , a steepening of the relation can be interpreted as a dependence of @xmath10 on cluster mass . that is , if , as suggested by the self - similar model , then the observed relation implies that . a varying gas mass fraction is expected if the efficiency of galaxy formation varies systematically across clusters of different mass . observational support for this has been claimed recently by bryan ( 2000 ) . however , this is still controversial , and there is no compelling evidence for a variation of @xmath10 with cluster temperature ( but see arnaud & evrard 1999 ; mme99 ) . this is especially true for the systems that we are specifically interested in : hot clusters with @xmath0 kev . this is apparent , for example , in figure 1 ( top ) of balogh et al . ( 2001 ) , who carry out an accounting of stars and gas to estimate the fraction of cooling baryons in clusters . moreover , roussel , sadat , & blanchard ( 2000 ) have carried out a careful analysis of group and cluster x - ray data to estimate @xmath10 directly and have found no trends . more recently , grego et al . ( 2001 ) have analysed sunyaev - zeldovich effect observations of 18 hot clusters and have also found no correlations between a hot cluster s gas mass fraction and its temperature . finally , observational studies of the _ total _ cluster mass ( @xmath5 ) - temperature relation have indicated that @xmath11 ( horner et al . 1999 ; ettori & fabian 1999 ; nevalainen et al . 2000 ; finoguenov et al . 2001 ) , which , given the observed @xmath12 relation , is consistent with @xmath10 being constant . theoretically , it is only now becoming possible to reliably investigate the dependence of @xmath10 on temperature with the inclusion of radiative cooling , star formation , feedback , and other relevant processes in numerical simulations ( e.g. , lewis et al . 2000 ; pearce et al . 2000 ; muanwong et al . 2001 ; dav et al . 2001 ) . as of yet , however , there is little agreement in the approaches adopted to model these processes and prevent the so - called cooling crisis ( compare , for example , the findings of lewis et al . 2000 with those of pearce et al . 2000 ) . this is not surprising . as discussed in detail by balogh et al . ( 2001 ) , attempting to model the effects of cooling across the wide range of halo masses found in clusters is inherently very difficult . the addition of `` sub - grid '' processes , such as star formation and feedback , further complicates matters . thus , the effects that these additional physical processes have on the gas mass fraction of clusters will not be fully realized until such issues are resolved . in this paper , however , we show that the observed variation of the @xmath13 relation(s ) arises quite naturally within the class of models that invoke preheating of the intracluster medium during the early stages of cluster formation . in these models , @xmath10 is constant on cluster scales ( @xmath0 kev ) , and the self - similarity is instead broken by an entropy floor generated by early non - gravitational heating events . preheating has previously been shown to bring consistency between a number of other observed and predicted scaling relations for groups and clusters ( e.g. , bblp02 ) , and therefore one might expect that the @xmath13 relation should also be modified . the preheating model was originally put forward by kaiser ( 1991 ) and has subsequently been investigated by a number of authors ( e.g. , evrard & henry 1991 , bower 1997 , cavaliere et al . 1997 ; 1998 ; 1999 ; balogh et al . 1999 , wu , fabian , & nulsen 2000 ; loewenstein 2000 , tozzi & norman 2001 ; borgani et al . 2001 ; thomas et al . 2002 ; bblp02 ) . if the icm is injected with enough thermal energy , the hot x - ray emitting gas will become decoupled from the dark halo potential and break the self - similar scaling relations . the best estimates suggest that a substantial amount of energy ( @xmath14 1 kev per particle ) is required to reproduce the observed relations ( mainly the @xmath6 relation ) . it is not yet known what source(s ) could inject such a large amount of energy into the icm . both galactic winds ( driven by supernovae ) and ejecta from active galactic nuclei have been proposed , but because of the complexity of the physics , the exact details have yet to be worked out . for an in - depth discussion of potential sources of preheating and of alternative possibilities for reproducing the observed relations we refer the reader to bblp02 . in this paper , we adopt the physically motivated , analytic model developed in bblp02 to explore the impact of cluster preheating on the @xmath12 relation . in comparison with the @xmath6 and @xmath15 relations , it has drawn very little attention by theoretical studies . the only studies to have examined the effects of entropy injection on the @xmath16 relation to date are loewenstein ( 2000 ) and bialek et al . ( 2001 ) . to be specific , loewenstein ( 2000 ) considered models where the entropy injection occurs at the centers of groups and clusters , after the latter have formed whereas bialek et al ( 2001 ) , like bblp02 , investigated preheated models . for reasons that will be described below ( in 5 ) , we believe our work greatly improves upon both of these studies . the prevailing apathy by theorists is perhaps due in part to a near absence of published observational studies on the gas masses of clusters . however , in light of the recent important observations discussed above ( e.g. , vikhlinin et al . 1999 ; mme99 ; neumann & arnaud 2001 ) and the new influx of high resolution data from the _ chandra _ and _ xmm - newton _ x - ray satellites , which will likely provide even tighter constraints , we believe that a thorough examination of the @xmath13 relation is timely . the models we consider below were developed in a flat @xmath17-cdm cosmology with @xmath18 , @xmath19 , and a nucleosynthesis value @xmath20 ( burles & tytler 1998 ) . they are computed for a number of different preheating levels , corresponding to entropy constants of @xmath21 = 100 , 200 , 300 , & 427 kev @xmath2 . these span the range required to match the observed @xmath6 relations of groups and hot clusters ( e.g. , ponman et al . 1999 ; lloyd - davies et al . 2000 ; tozzi & norman 2001 ; bblp02 ) . for the purposes of comparison , we also implement an `` isothermal '' model ( see section 2.3 in bblp02 ) , which mimics the self - similar result deduced from numerical simulations ( e.g. , evrard et al . since an in - depth discussion of the preheated cluster models can be found in bblp02 , we present only a brief description of the models here . the preheated models can be summarized as follows : the dominant dark matter component , which is unaffected by the energy injection , collapses and virializes to form bound halos . the distribution of the dark matter in such halos is assumed to be the same as found in recent ultra - high resolution numerical simulations ( moore et al . 1998 ; klypin et al . 1999 ; lewis et al . 2000 ) and is described by @xmath22 where n = 1.4 , @xmath23 is the profile normalization , and @xmath24 is the scale radius . while the dark component is unaffected by energy injection , the collapse of the baryonic component is hindered by the pressure forces induced by preheating . if the maximum infall velocity due purely to gravity of the dark halo is subsonic , the flow will be strongly affected by the pressure and it will not undergo accretion shocks . it is assumed that the baryons will accumulate onto the halos _ isentropically _ at the adiabatic bondi accretion rate ( as described in balogh et al . this treatment , however , is only appropriate for low mass halos . if the gravity of the dark halos is strong enough ( as it is expected to be in the hot clusters being considered here ) so that the maximum infall velocity is transonic or supersonic , the gas will experience an additional ( generally dominant ) entropy increase due to accretion shocks . in order to trace the shock history of the gas , a detailed knowledge of the merger history of the cluster / group is required but is not considered by bblp02 . instead , it is assumed that at some earlier time the most massive cluster progenitor will have had a mass low enough such that shocks were negligible in its formation , similar to the low mass halos discussed above . this progenitor forms an isentropic core of radius @xmath25 at the cluster center . the entropy of gas outside of the core , however , will be affected by shocks . recent high resolution numerical simulations suggest that the entropy profile for gas outside this core can be adequately represented by a simple analytic expression given by @xmath26 ( lewis et al . 2000 ) , where @xmath27 for the massive , hot clusters ( @xmath0 kev ) of interest here ( tozzi & norman 2001 ; bblp02 ) . it should be noted that in the case of these massive systems , the accretion of gas is limited by gravitational infall , and hence they accrete their full compliment of baryons [ i.e. , @xmath28 . it is assumed that the mass of baryons locked up in stars is negligible ( as suggested by , for example , roussel et al . 2000 ; balogh et al . 2001 ) . following this prescription and specifying the parameters @xmath29 , @xmath30 , and @xmath31 ( as discussed in bblp02 ) completely determines the models . under all conditions , the gas is assumed to be in hydrostatic equilibrium within the dark halo potential . the effects of radiative cooling are neglected by these models . preheating will affect the @xmath12 relation in two ways : ( 1 ) by altering the temperature profile and increasing the emission - weighted gas temperature of a cluster and ( 2 ) by altering the gas density profile and reducing the gas mass in the cluster core . we are interested in the strength of these effects and whether or not they can be distinguished by current or future observational data . first , we consider the effect of preheating on the temperature of a cluster . figure 1 is a plot of @xmath4 as a function of entropy floor level ( i.e. , @xmath21 ) for three clusters of different _ total _ masses . the thin line represents a cluster with @xmath32 , the next thickest line represents a cluster with @xmath33 , and the thickest line represents a cluster with @xmath34 . as expected , the gas temperature of a cluster increases as the level of preheating is increased . on average , an increase in @xmath4 of about 1 kev ( 10 - 25% ) occurs when a cluster is preheated to the level of @xmath35 kev @xmath2 ( over the range 3 kev @xmath36 kev ) . this effect will primarily manifest itself as a normalization shift in the @xmath12 relation . figure 2 presents the dimensionless gas density profile of a cluster with @xmath37 kev ( left panel ) and a cluster with @xmath38 kev ( right panel ) as a function of the level of preheating . the dot - dashed line is the self - similar result ( i.e. , isothermal model of bblp02 ) . the long - dashed , short - dashed , dotted , and solid lines represent the preheated models of bblp02 with @xmath21 = 100 , 200 , 300 , and 427 kev @xmath2 , respectively . preheating reduces the gas density , and therefore the gas mass , in the central regions of a cluster . in 4 , we investigate the @xmath12 relation within three different radii : @xmath39 mpc and @xmath40 mpc and @xmath41 ( the radius within which the mean dark matter mass density is 500 times the mean critical density @xmath42 at @xmath43 = 0 ) . these radii are indicated in figure 2 by the open squares , pentagons , and triangles , respectively . clearly , the effect on the @xmath12 relation will be strongest when @xmath44 is evaluated within @xmath45 mpc . 0.1 in furthermore , because @xmath39 mpc is a fixed radius that samples different fractions of the virial radius ( @xmath46 ) for clusters of different temperature , the ( fractional ) reduction in gas mass within that radius 0.1 in will be largest for the lowest temperature systems . this will lead to both a normalization shift and a steepening of the @xmath12 relation . to illustrate how strong the effect is , we examine the reduction of gas mass within @xmath39 mpc , @xmath40 mpc and @xmath41 for a low mass cluster with a _ total _ mass of @xmath47 and for a high mass cluster with @xmath48 . we find that when the low mass cluster has undergone preheating at the level of @xmath49 kev @xmath2 it has @xmath50 more mass in gas within @xmath51 mpc than when the same cluster has undergone preheating at the level of @xmath52 kev @xmath2 . using the same test on the @xmath48 cluster , however , yields a difference of only 22% . when we probe the larger radius @xmath53 mpc , we find the effect is less pronounced ( as expected ) . the difference in @xmath44 between the @xmath49 kev @xmath2 model and @xmath52 kev @xmath2 model is 8% for the low mass cluster and 7% for the high mass cluster . finally , when the gas mass is evaluated within @xmath41 , the difference is 4% for the low mass cluster as opposed to 2% for the high mass cluster . in summary , preheating will significantly affect the @xmath12 relation by increasing the emission - weighted gas temperature of clusters . whether or not the relation is also affected by the reduction of gas mass in the cores of clusters depends on within which radius @xmath44 is evaluated and what temperature regime is being probed . the effect will be strongest for low temperature systems and when @xmath44 is probed within small radii ( e.g. , @xmath39 mpc ) . an evaluation of the @xmath13 relation within large radii , such as @xmath41 , however , probes the integrated properties of a cluster and will be sensitive only to the temperature shift . in the next section , we compare the results of the bblp02 preheated models to genuine observational data . as we show below , only models with @xmath35 kev @xmath2 are consistent with the data . in figure 3 we present the @xmath12 relation as predicted by the bblp02 preheated models within @xmath41 . the radius @xmath41 is typically comparable in size to the _ observed _ radius of a cluster and represents the boundary between the inner , virialized region and recently accreted , still settling outer region of a cluster ( evrard et al . thus , as already mentioned , the @xmath54 relation can be regarded as a probe of the integrated properties of a cluster and can be directly compared with the self - similar result of @xmath55 . in figures 4 and 5 we present the @xmath12 relation as predicted by the bblp02 preheated models within the fixed radii @xmath39 mpc and @xmath53 mpc , respectively . as mentioned above , the determination of the @xmath12 relation within some fixed radius , such as @xmath39 mpc or @xmath56 mpc , can be used as an indirect probe of the gas density profiles of clusters because it samples different fractions of the virial radius for clusters of different temperature . for the purposes of clarity , we discuss the @xmath12 relation at these three radii separately . the solid squares in figure 3 represent the gas mass determinations of mme99 within @xmath41 using surface brightness profile fitting ( with isothermal @xmath58 models ) of _ rosat _ position sensitive proportional counter data and mean emission - weighted temperatures from the literature , for clusters with @xmath59 kev and whose error bars are 1 kev or smaller . we compare this with the self - similar result represented by the `` isothermal '' model of bblp02 ( dot - short - dashed line ) . finally , the long - dashed , short - dashed , dotted , and solid lines represent the preheated models of bblp02 with @xmath21 = 100 , 200 , 300 , and 427 kev @xmath2 , respectively . the thick dot - long - dashed line represents the predictions of the best - fit heated model of loewenstein ( 2000 ) . this model is discussed further in 5.1 . it is readily apparent that only the preheated models of bblp02 with @xmath60 kev @xmath2 have a reasonable chance of being consistent with the data of mme99 . the normalization clearly indicates that the observed gas temperature of clusters with a given gas mass is hotter than predicted by models with entropy floors of @xmath61 kev @xmath2 . we note that this discrepancy may be remedied by assuming a smaller value of @xmath62 . however , a similar offset , in the same sense , is seen in the correlation with _ total _ dark matter mass and gas temperature ( horner et al . 1999 ; nevalainen et al . 2000 ; finoguenov et al . 2001 ) . this will not be reconciled by lowering @xmath62 . the reason why the preheated models with @xmath60 kev @xmath2 are better able to match the normalization of the observational data than models with @xmath63 kev @xmath2 is , as mentioned above , because an increase in the amount of preheating directly leads to an increase in the emission - weighted gas temperature . we have attempted to quantify how well ( or poorly ) the preheated and self - similar models match the observational data . we have fit both the theoretical results and the observational data with simple linear models of the form @xmath64 over the range 3 kev @xmath65 kev . for the theoretical results , we calculate the best - fit slope and intercept using the ordinary least squares ( ols ) test . we stress that the results of these fits , which are presented in table 1 , are only valid for clusters with @xmath0 kev . at lower temperatures , the role of preheating becomes much more important [ as @xmath44 becomes less than @xmath66 and as a result , the relations steepen dramatically . for example , the preheated model with @xmath52 kev @xmath2 is well approximated by a power - law with @xmath67 over the range 3 kev @xmath65 kev but is significantly steeper over the range 1 kev @xmath68 kev with @xmath69 . _ thus , it is absolutely essential that comparisons between theoretical models and observations are done over the same range in temperatures . _ to fit the observational data of mme99 , we have used a linear regression technique that takes into account measurement errors in both coordinates as well as intrinsic scatter ( the bces test of akritas & bershady 1996 ) . as a consistency check , we have also employed 10,000 monte carlo bootstrap simulations . no significant deviations between the two tests were found . the results of the linear regression fits to the observational data are also presented in table 1 . for all 38 clusters taken from mme99 , we derive a best fit that is inconsistent with the results of _ all _ of the theoretical models considered at greater than the 90% confidence level@xmath70 . however , as is apparent from figure 3 , the slope and intercept of the best - fit line are sure to be heavily dependent upon the two low temperature clusters with the lowest measured gas masses ( and gas mass fractions ) : the hya i cluster ( abell 1060 ) and the cen cluster ( abell 3526 ) . a number of other studies ( both optical and x - ray ) have also identified very unusual properties in both clusters . for example , fitchett & merritt ( 1988 ) were unable to fit a spherical equilibrium model to the kinematics of galaxies in the core of hya i. they suggest that substructure is present and is likely why hya i does not lie along the @xmath71 relation for galaxy clusters . more recently , furusho et al . ( 2001 ) have found that the metal abundance distribution implies that the gas in hya i is well - mixed ( i.e. , it does not contain an obvious metallicity gradient ) , suggesting that a major merger event may have occurred sometime after the enrichment of the icm . measurements of the bulk motions of the intracluster gas in the cen cluster ( through doppler shifting of x - ray spectral lines ) reveal strange gas velocity gradients indicative of a large merger event in the not too distant past ( dupke & bregman 2001 ) . this picture has also been supported by furusho et al . ( 2001 ) , who found large temperature variations across the cluster s surface . thus , neither hya i nor cen can be regarded as typical `` relaxed '' clusters and are probably not representative of the majority of low temperature systems . one way to ameliorate the impact of the two clusters would be to increase the number of systems of this temperature . however , there are very few published gas mass estimates of cool clusters and groups within @xmath41 . x - ray emission from groups is usually only detected out to a small fraction of this radius . the one study that does present group gas masses for a radius at fixed overdensity , roussel et al . ( 2000 ) , does so for @xmath72 and is not directly comparable to the results presented in figure 3 . also , in that study , gas masses were determined by extrapolating the surface brightnesses far outside the limiting radius for which x - ray emission was actually detected . this can lead to biases in determining group / cluster properties ( see mulchaey 2000 ; balogh et al . 2001 ) . in recognition of the above , we have tried removing hya i and cen from the sample and fitting the remaining 36 clusters using the same procedure . we find that the preheated model with @xmath52 kev @xmath2 is then consistent with the data at the 90% level . the @xmath21 = 200 and 300 kev @xmath2 models are marginally inconsistent with the mme99 data . the isothermal model is ruled out at @xmath73 confidence irrespective of whether these clusters are dropped or not . the other two observational studies that have investigated the @xmath12 relation , neumann & arnaud ( 2001 ) and vikhlinin et al . ( 1999 ) , unfortunately did not present gas mass determinations within @xmath41 for individual clusters in a table or graphically . they did , however , present their best - fit values for the slope of the relation . these were deduced from samples of clusters that have temperatures spanning roughly the same range as that considered in figure 3 . the best - fit slopes of the preheated models are shallower than the best - fit claimed by neumann & arnaud ( 2001 ) of @xmath69 for a sample of 15 hot clusters . however , an estimate of the uncertainty on this result was not reported ; thus we are unable to say whether this result is inconsistent with the predictions of the preheated models . the predicted slopes of all four preheated models studied here are in excellent agreement with the findings of vikhlinin et al . ( 1999 ) , who report @xmath74 for their sample of 39 clusters . we also note that the results of neumann & arnaud ( 2001 ) and vikhlinin et al . ( 1999 ) differ significantly from the predictions of the self - similar model . in summary , we find that the class of models that invoke preheating are much better able to match the observed @xmath57 of hot clusters than that of the isothermal self - similar model , which is ruled out with a high level of confidence . a careful analysis of the mme99 data also suggests that only those models that invoke a `` high '' level of energy injection ( i.e. , @xmath75 kev @xmath2 ) are able to match observations . the solid triangles and pentagons in figure 4 represent the gas mass determinations within @xmath39 mpc of peres et al . ( 1998 ) and white et al . ( 1997 ) , respectively . these data were obtained using surface brightness profile fitting of _ rosat _ data ( peres et al . 1998 ) and _ einstein _ data ( white et al . 1997 ) and emission - weighted temperatures from the literature , for clusters with @xmath59 kev and whose error bars are 1 kev or smaller . the predictions of the isothermal self - similar model are represented by the dot - dashed line . once again , the long - dashed , short - dashed , dotted , and solid lines represent the preheated models of bblp02 with @xmath21 = 100 , 200 , 300 , and 427 kev @xmath2 , respectively . in spite of the scatter , it is apparent that only those preheated models with entropy floors of @xmath35 kev @xmath2 are consistent with the 57 clusters plotted in figure 4 . as with the @xmath57 relation , the normalization of the self - similar model and the preheated model with @xmath49 kev @xmath2 suggests that icm is observed to be much hotter than predicted by either of these models . fitting both the theoretical predictions and observational data in a manner identical to that presented in the previous subsection , we find that only the preheated models with @xmath76 kev @xmath2 have both slopes and intercepts that are consistent with the observational data ( see table 2 ) . on the basis of normalization ( intercept ) , the self - similar model is ruled out with greater than 99% confidence . in 3 we briefly discussed the potential of the gas density profile to affect the @xmath44(@xmath39 mpc ) - @xmath4 relation . this effect is obvious in figure 4 , with mild breaks at @xmath77 kev for the @xmath52 kev @xmath2 model and at @xmath78 kev for the @xmath79 kev @xmath2 model . with the large scatter obscuring any potential breaks in the @xmath12 relation , all we can conclude is that the data are consistent with predicted profiles of the bblp02 preheated models with @xmath35 kev @xmath2 . the exact nature of the scatter in figure 4 is unclear . while some of the scatter is likely attributable to the large uncertainties in the temperature measurements made using _ einstein _ , _ ginga _ , and _ exosat _ data , some of it may also be due to unresolved substructure ( e.g. , cooling flows ) and point sources . such issues become particularly important when investigating the central regions of clusters as opposed to its integrated properties . indeed , new high - resolution data obtained by _ chandra _ support this idea ( see , e.g. , stanford et al . we anticipate that future data obtained by both _ chandra _ and _ xmm - newton _ will place much tighter constraints on the @xmath44(@xmath39 mpc ) - @xmath4 relation and possibly even allow one to probe the mild break in the relationship predicted by the preheated models . the solid pentagons in figure 5 represent the gas mass determinations of white et al . ( 1997 ) within @xmath53 mpc using surface profile fitting of _ einstein _ data and emission - weighted gas temperatures from the literature , for clusters with @xmath59 kev and whose error bars are 1 kev or smaller . again , the predictions of the isothermal self - similar model are represented by the dot - dashed line while the long - dashed , short - dashed , dotted , and solid lines represent the preheated models with @xmath21 = 100 , 200 , 300 , and 427 kev @xmath2 , respectively . a linear regression fit to the 20 clusters taken from white et al . ( 1997 ) yields a best - fit slope and intercept that is consistent with only the @xmath21 = 300 and 427 kev @xmath2 preheated models ( 90% confidence ; see table 3 ) . once again , the isothermal model is ruled out with greater than 99% confidence . this follows the same general trend discovered in the previous tests . as expected , the influence of the gas density profile on the @xmath44(@xmath56 mpc ) - @xmath4 is minimal . only a very modest break is detectable at @xmath80 kev for the @xmath52 kev @xmath2 preheated model . like the @xmath57 relation , this relation is mostly sensitive to the temperature shift ( at least for clusters with @xmath0 kev ) . only two other theoretical studies have examined the effects on the @xmath16 relation of entropy injection into the icm : loewenstein ( 2000 ) and bialek et al . both studies investigated the @xmath57 relation and demonstrated that entropy injection does , indeed , steepen the relation , in agreement with the present work ( however , neither implemented the @xmath12 at fixed radii test ) . these studies suggest that models that produce an entropy floor with a level that is consistent with measurements of groups ( @xmath81 kev @xmath2 ; ponman et al . 1999 ; lloyd - davies et al . 2000 ) are capable of matching the observations of even hot clusters ( up to 10 kev ) . this is in apparent conflict with the results presented in 3 that suggest that a high entropy floor of @xmath35 kev @xmath2 is required to match the observations of hot clusters . a low value of the entropy floor is also in apparent conflict with a number of other studies that have focused mainly on the @xmath82 relation of hot clusters . for example , da silva et al . ( 2001 ) , tozzi & norman ( 2001 ) and bblp02 have all concluded that such low levels of entropy injection do not bring consistency between observations and theoretical models of _ hot clusters_. as such , a closer analysis of loewenstein ( 2000 ) and bialek et al . ( 2001 ) studies is warranted . to model the observed deviations of the cluster x - ray scaling relations , loewenstein ( 2000 ) has constructed a suite of hydrostatic polytropic models ( which are normalized to observations of high - temperature clusters and numerical simulations ) , and then modified them by adding various amounts of heat per particle at the cluster center . strictly speaking , the loewenstein ( 2000 ) models can not be characterized as _ preheated _ models , since the injection of entropy into the icm occurs after the cluster has formed . thus , a straightforward comparison between the loewenstein ( 2000 ) and bblp02 models is not trivial . however , success in matching the @xmath83 relation ( the data of mme99 ) is claimed by loewenstein ( 2000 ) for a model that `` produces an entropy - temperature relation with the observed entropy floor at @xmath84 kev @xmath2 . '' regardless of how the entropy floor actually arose , this contradicts the results presented in 3 , which suggest that an entropy floor of @xmath1 kev @xmath2 is required to match the observations . can the analysis of loewenstein ( 2000 ) and that of the present work be reconciled ? a closer investigation of figure 4 of loewenstein ( 2000 ) reveals that first of all , his heated models were not compared to the actual data but rather to points that represent mme99 s best - fit power - law match to their data . second , this power - law relationship was assumed to hold true and hence , extrapolated to span a wider range in temperatures than considered by mme99 . of the 45 clusters studied by mme999 , only one had a temperature below 3 kev ( it was 2.41 kev ) , yet loewenstein ( 2000 ) compared his heated models to the best - fit relation of mme99 over the range 1 kev @xmath65 kev . as previously mentioned , however , entropy injection preferentially affects low - temperature systems and , therefore , extrapolating scaling relations derived from high - temperature systems down to the low - temperature regime is not safe . in figure 3 , we compare the best - fit heated model of loewenstein ( 2000 ) ( his @xmath85 model , as the thick dot - long - dashed line ) with the predictions of bblp02 models and the data of mme99 . the plot clearly demonstrates that his best - fit model does not match the data of mme99 nearly as well as the bblp02 preheated models with @xmath86 kev @xmath2 , especially at the high temperature end . the difference in temperature ranges examined by loewenstein ( 2000 ) and the present study ( whose range of temperatures were purposely chosen to match the observational data ) has likely led to an underestimation of the entropy floor in these clusters by loewenstein ( 2000 ) . we once again re - iterate that it is extremely important that comparisons between theoretical models and observations are done over the same range in temperatures . in similarity to the present work , bialek et al . ( 2001 ) investigated the impact of preheating on the @xmath12 relation for a number of different levels of entropy injection , spanning the range 0 kev @xmath2 @xmath87 kev @xmath2 . fitting their @xmath57 simulation data over the range 2 kev @xmath88 kev , which is similar ( but not identical ) to the mme99 sample , they claim success in matching the observations of mme99 for models with entropy injection at the level of 55 kev @xmath2 @xmath89 kev @xmath2 , at least on the basis of slope . their models with higher levels of entropy injection , apparently , predict relations much too steep to be consistent with the data of mme99 . these predictions are inconsistent with the results of the bblp02 analytic models with similar levels of entropy injection ( e.g. , for @xmath90 kev @xmath2 , bblp02 predict @xmath91 while bialek et al . find @xmath92 ) . however , we believe the difference in the predictions ( and conclusions ) of bialek et al . ( 2001 ) and the present work can be reconciled . as noted by neumann & arnaud ( 2001 ) , bialek et al . ( 2001 ) have simulated very few hot clusters and , although they fit their @xmath57 simulation data over a range similar to mme99 , the results are too heavily weighted by the cool clusters ( @xmath93 kev ) to be properly compared with the data of mme99 . as an example , we consider their `` s6 '' sample of 12 clusters that have @xmath94 kev @xmath2 . according to the present study , this model should give a reasonably good fit to the mme99 observational data , much better than that of a model with @xmath95 kev @xmath2 . although the normalization of the s6 model is in excellent agreement with the mme99 data ( as is apparent in table 3 of bialek et al . and the general trends in their figure 1 ) they rule this model out based on the fact that the predicted slope is 2.67 , much steeper than the 1.98 found by mme99 . however , a closer analysis reveals that the fraction of cool clusters in the simulation data set is much higher than fraction of cool clusters in the mme99 sample . for example , in the mme99 sample of 45 clusters , only one cluster has a temperature below 3 kev . in the bialek et al . ( 2001 ) s6 set , however , 5 of the 12 clusters have temperatures below 3 kev . in addition , the mean temperature of clusters in the mme99 sample is @xmath96 5.5 kev , while it is only about 3.8 kev in the bialek et al . ( 2001 ) s6 data set . as previously mentioned , preheating preferentially affects low temperature systems and , therefore comparisons between theory and observations should be done over the same range in temperatures . to illustrate the problems of comparing theoretical models and observations that span different temperature ranges , we tried to reproduce the fit of bialek et al . ( 2001 ) to their s6 data set . we used data presented in their table 2 for clusters with @xmath97 kev ( we use their preferred `` processed '' temperatures ) and fit it with a linear model and found @xmath98 . this is slightly different from the value listed in their table 3 , presumably because table 2 is based on data within @xmath72 while table 3 is based on data within @xmath41 ( they note that a change of up to 6% in the predicted slope can occur when switching between the two ) . to match the conditions of the present work , we then discarded all simulated cluster data below 3 kev ( the mean temperature for the remaining 7 clusters was then @xmath99 kev , similar to the mme99 data ) and found a best fit of @xmath100 . this is excellent agreement with the results of mme99 and only marginally inconsistent with the bblp02 models of similar entropy injection . what about their favored models ? we have tried the same type of test on their s3 data set ( @xmath101 kev @xmath2 ) . fitting all simulated clusters with @xmath102 kev ( mean temperature of 3.8 kev ) we find @xmath103 , which is in good agreement with the results of mme99 . when we remove all clusters below 3 kev ( mean temperature of 4.9 kev ) , however , the best fit is @xmath104 . in this case , the best - fit relation is not very constraining . it is even indistinguishable from the self - similar result . it is apparent from their figure 1 , however , that the predicted normalization for this model ( and all other low entropy models ) does not match the observations of mme99 . this is noted by the authors themselves . they claim the difference in the zero point may be resolved by reducing the baryon fraction by @xmath105 . as we noted earlier , however , a similar normalization offset is also seen in the total cluster mass - temperature ( @xmath15 ) relation and this can not be resolved by reducing the baryon fraction . this suggests that the problem lies with the temperature , rather than the gas mass . alternatively , bialek et al . also suggest that rescaling their simulations for @xmath106 km s@xmath107 mpc@xmath107 ( instead of 80 km s@xmath107 mpc@xmath107 ) would bring consistency between the normalization of this model and the observations . this would be true only if the baryon fraction was held fixed at 0.1 and not rescaled for the new cosmology . given that they assume @xmath18 , this would imply @xmath108 which is roughly 30% lower than observed in quasar absorption spectra ( burles & tytler 1998 ) . thus , while the normalization offset between their theoretical model and the observations of mme99 is directly reduced by decreasing the value of @xmath109 , it is indirectly increased by roughly the same proportion through the increased value of @xmath62 . in summary , as with the loewenstein ( 2000 ) models , we find that the difference in the results and conclusions of bialek et al . ( 2001 ) and the present work can be explained on the basis that different temperature ranges were examined . in particular , we have shown that the fraction of cool clusters in bialek et al.s simulated data set is much larger than that found in the mme99 sample and this has likely led to an underestimation of the entropy floor in these clusters . in order to safely and accurately compare the preheated models of bblp02 with observations we have paid special attention to only those hot clusters with @xmath0 kev . as such , we believe our comparison is more appropriate . motivated by a number of observational studies that have suggested that the @xmath16 relation of clusters of galaxies is inconsistent with the self - similar result of numerical simulations and by the launch of the _ chandra _ and _ xmm - newton _ satellites , which will greatly improve the quality of the observed @xmath110 relation , we have implemented the analytic model of bblp02 to study the impact of preheating on @xmath12 relation . the predictions of the model have previously been shown to be in very good agreement with observations ( e.g. , @xmath6 relation and @xmath71 relation ) . in agreement with the previous theoretical studies of loewenstein ( 2000 ) and bialek et al . ( 2001 ) , our analysis indicates that injecting the intracluster medium with entropy leads to a steeper relationship than predicted by the self - similar result of numerical simulations of clusters that evolve through the effects of gravity alone . loewenstein ( 2000 ) and bialek et al . ( 2001 ) have found that models that produce an entropy floor of @xmath111 kev @xmath2 , which is consistent with measurements of galaxy groups , are capable of reproducing the @xmath12 relation of hot clusters . this is inconsistent with our analysis , which indicates that a `` high '' level of entropy injection ( @xmath35 kev @xmath2 ) is required to match the observational data of hot clusters of white et al . ( 1997 ) , peres et al . ( 1998 ) , and mme99 . it is also inconsistent with bblp02 s best - fit value of @xmath112 kev @xmath2 found via an investigation of the @xmath82 relation of both groups and hot clusters . they note that the strongest constraints for a high entropy floor comes from hot clusters . moreover , a high value of @xmath21 , one that is inconsistent with the predictions of the best - fit models of loewenstein ( 2000 ) and bialek et al . ( 2001 ) , has also been reported by tozzi & norman ( 2001 ) . finally , da silva et al . ( 2001 ) used numerical simulations with a `` low '' value of @xmath113 kev @xmath2 ( which is similar to the predictions of the best - fit models of loewenstein 2000 and bialek et al . 2001 ) and found that they _ could not _ reproduce the observed x - ray scaling relations . our result , on the other hand , is consistent with the results of bblp02 , tozzi & norman ( 2001 ) , and da silva et al . as discussed in 5 , we believe the difference between the studies of loewenstein ( 2000 ) and bialek et al . ( 2001 ) and present work can be explained by considering the difference in temperature ranges studied . in particular , we have focused only on hot clusters in an attempt to match the majority of the observational data as closely as possible . the results and conclusions of the other two studies , however , are strongly influenced by their low temperature model data . we have proposed that the @xmath12 relation can be used as a probe of the gas density profiles of clusters if it is evaluated at different fixed radii . this is a new test . the preheated models of bblp02 predict a mild break in the scaling relations when small fixed radii ( such as @xmath114 mpc ) are used . the scatter in the current observational data is consistent with the predictions of the bblp02 models with @xmath35 kev @xmath2 ; however , the exact shape of the gas density profiles is not tightly constrained . we anticipate that large samples of clusters observed by _ chandra _ and _ xmm - newton _ will place much stronger constraints on the gas density profiles of clusters and allow for further testing of the preheating scenario . finally , the high level of energy injection inferred from our analysis has important implications for the possible sources of this excess entropy . valageas & silk ( 1999 ) , balogh et al . ( 1999 ) , and wu et al . ( 2000 ) have all shown that galactic winds driven by supernovae can only heat the intracluster / intergalactic medium at the level of @xmath115 kev per particle . this is lower than the @xmath116 kev per particle result found here . thus , if the bblp02 preheated models provide an accurate description of the icm , supernovae winds alone can not be responsible for the excess entropy . it has also been speculated that quasar outflows may be responsible ( e.g. , valageas & silk 1999 ; nath & roychowdhury 2002 ) . this remains an open possibility . the role of radiative cooling also remains an open issue . recently , it has been suggested that _ both radiative cooling and preheating together _ could be actively involved in shaping the x - ray scaling relations ( e.g. , voit & bryan 2001 ; voit et al . 2002 ) . radiative cooling ( and subsequent star formation ) would serve to remove the lowest entropy gas , which in turn would help to compress the highest entropy gas , thus increasing the emission - weighted gas temperature and steepening the @xmath16 relation ( cf . the discussion of entropy in the cool+sf simulation of lewis et al . 2000 ) . in this way , the combination of cooling and preheating may reduce the best - fit entropy level , perhaps even to a level that can be provided by supernovae winds ( voit et al . further study is required to determine the relative roles that both preheating and cooling have on cluster evolution . we would like to thank mike loewenstein and the anonymous referee for many useful comments and suggestions . i. g. m. is supported by a postgraduate fellowship from the natural sciences and engineering research council of canada ( nserc ) and by the petrie fellowship at the university of victoria . he also acknowledges additional assistance in the form of a john criswick travel bursary . a. b. is supported by an nserc operating grant and m. l. b. is supported by a pparc rolling grant for extragalactic astronomy and cosmology at the university of durham . akritas , m. g. , & bershady , m. a. 1996 , apj , 470 , 706 allen , s. w. , & fabian , a. c. 1998 , mnras , 297 , l57 arnaud , m. , & evrard , a. e. 1999 , mnras , 305 , 631 babul , a. , balogh , m. l. , lewis , g. f. , & poole , g. b. 2002 , mnras , 330 , 329 balogh , m. l. , babul , a. , & patton , d. r. 1999 , mnras 307 , 463 balogh , m. l. , pearce , f. r. , bower , r. g. , & kay , s. t. 2001 , mnras , 326 , 1228 bialek , j. j. , evrard , a. e. , & mohr , j. j. 2001 , apj , 555 , 597 borgani , s. , et al . 2001 , apj , 559 , l71 bower , r. g. 1997 , mnras , 288 , 355 bryan , g. l. 2000 , apj , 544 , l1 burles , s. , & tytler , d. 1998 , apj , 499 , 699 cavaliere , a. , menci , n. , & tozzi , p. 1997 , apj , 484 , l21 cavaliere , a. , menci , n. , & tozzi , p. 1998 , apj , 501 , 493 cavaliere , a. , menci , n. , & tozzi , p. 1999 , mnras , 308 , 599 da silva , a.c . , 2001 , apjl , 561 , 15 dav , r. , katz , n. , hernquist , l. , & weinberg d. 2001 , in sesto 2001 - tracing cosmic evolution with galaxy clusters ( astro - ph/0109394 ) dupke , r. a. , & bregman , j. n. 2001 , 562 , 266 ettori , s. , & fabian , a. c. 1999 , 305 , 834 evrard , a. e. , & henry , j. p. 1991 , apj , 383 , 95 evrard , a. e. , metzler , c. a. , & navarro , j. f. 1996 , apj , 469 , 494 finoguenov , a. , reiprich , t. h. , & bhringer , h. 2001 , a&a , 368 , 749 fitchett , m. , & merritt , d. 1988 , apj , 335 , 18 furusho , t. , et al . 2001 , pasj , 53 , 421 grego , l. , et al . 2001 , apj , 552 , 2 horner , d. j. , mushotzky , r. f. , & scharf , c. a. 1999 , apj , 520 , 78 kaiser , n. 1991 , apj , 383 , 104 klypin , a. a. , gttlober , s. , kravstov , a. v. , & khokholov , a. m. 1999 , apj , 516 , 530 lewis , g. f. , babul , a. , katz , n. , quinn , t. , hernquist , l. , & weinberg , d. 2000 , apj , 536 , 623 lloyd - davies , e. j. , ponman , t. j. , & cannon , d. b. 2000 , mnras , 315 , 689 loewenstein , m. 2000 , apj , 532 , 17 markevitch , m. 1998 , apj , 504 , 27 mohr , j.j . , mathiesen , b. , & evrard , a.e . 1999 , apj , 517,627 moore , b. , et al . 1998 , apj , 499 , l5 muanwong , o. , et al . 2001 , apj , 552 , l27 mulchaey , j. s. 2000 , ara&a , 38 , 289 nath , b. b. , & roychowdhury s. 2002 , mnras , in press neumann , d. m. , & arnaud , m. 2001 , a&a , 373 , 33 nevalainen , j. , markevitch , m. , & forman , w. 2000 , apj , 532 , 694 pearce , f. r. , thomas , p. a. , couchman , h. m. p. , & edge a. c. 2000 , mnras , 317 , 1029 peres , c.b . , et al . 1998 , mnras , 298 , 416 ponman , t. j. , cannon , d. b. , & navarro , j. f. 1999 , nature , 397 , 135 roussel , h. , sadat , r. , & blanchard , a. 2000 , a&a , 361 , 429 stanford , s. a. , et al . 2001 , apj , 552 , 504 thomas , p. a. , muanwong , o. , kay , s. t. , & liddle , a. r. 2002 , mnras , 330 , l48 tozzi , p. , & norman , c. 2001 , apj , 546 , 63 valageas , p. , & silk , j. 1999 , a&a , 350 , 725 vikhlinin , a. , forman , w. , & jones , c. 1999 , apj , 525 , 47 voit , m. g. , & bryan , g. l. 2001 , nature , 414 , 425 voit , m. g. , bryan , g. l. , balogh , m. l. , & bower , r. g. 2002 , apj , submitted white , d. a. , jones , c. , & forman , w. 1997 , mnras , 292 , 419 wu , k. k. s. , fabian , a. c. , & nulsen , p. e. j. 2000 , mnras , 318 , 889 lccc isothermal model & 0 & 1.50 & 12.72 + preheated models & 100 & 1.65 & 12.54 + & 200 & 1.66 & 12.48 + & 300 & 1.67 & 12.45 + & 427 & 1.68 & 12.42 + mme99 data & ? & @xmath117 & @xmath118 + mme99 data minus 2 clusters & ? & @xmath119 & @xmath120 + lccc isothermal model & 0 & 0.84 & @xmath121 + preheated models & 100 & @xmath122 & @xmath123 + & 200 & @xmath124 & @xmath125 + & 300 & @xmath126 & @xmath127 + & 427 & @xmath128 & @xmath129 + peres et al . and white et al . data & ? & @xmath130 & @xmath131 +

recent x - ray observations have been used to demonstrate that the cluster gas mass - temperature relation is steeper than theoretical self - similar predictions drawn from numerical simulations that consider the evolution of the cluster gas through the effects of gravity and shock heating alone . one possible explanation for this is that the gas mass fraction is not constant across clusters of different temperature , as is usually assumed . observationally , however , there is no compelling evidence for gas mass fraction variation , especially in the case of hot clusters . seeking an alternative physical explanation for the observed trends , we investigate the role in the cluster gas mass - temperature relation of the preheating of the intracluster medium by some arbitrary source for clusters with emission - weighted mean temperatures of @xmath0 kev . making use of the physically - motivated , analytic models developed in 2002 by babul and coworkers , we find that preheating does , indeed , lead to a steeper relation . this is in agreement with previous theoretical studies on the relation . however , in apparent conflict with these studies , we argue that a `` high '' level of entropy injection is required to match observations . in particular , an entropy floor of @xmath1 kev @xmath2 is required . we also present a new test , namely , the study of the relation within different fixed radii . this allows one to indirectly probe the density profiles of clusters , since it samples different fractions of the virial radius for clusters of different temperature . this test also confirms that a high level of preheating is required to match observations .

astro-ph0203189

we study elementary geometric operations on triangles defined as follows . let @xmath0 be a triangle , and @xmath1 be a real number . let @xmath2 , and @xmath3 be division points of the edges @xmath4 , and @xmath5 by @xmath6 respectively , namely , @xmath7 let @xmath8 ( @xmath9 or @xmath10 ) be the intersection of the lines @xmath11 and @xmath12 ( @xmath12 and @xmath13 or @xmath13 and @xmath11 respectively ) . define _ equisection operators _ @xmath14 and @xmath15 , where @xmath15 can be defined when @xmath16 , by @xmath17 the operators @xmath14 have been studied in articles such as @xcite , _ et . al . _ ] 0.4 cm ] in this note we study the equivalence relation ( denoted by @xmath18 ) of the set of triangles ( denoted by @xmath19 ) generated by similarity and @xmath20 , which we shall call _ equisectional equivalence_. the equivalence relation generated by similarity and @xmath21 shall be called _ rational equisectional equivalence _ and denoted by @xmath22 . we say two triangles @xmath23 and @xmath24 are _ equisectionally equivalent _ ( or _ rational equisectionally equivalent _ ) if @xmath25 ( or @xmath26 respectively ) . we remark that we use the term `` similarity '' as the equivalence under orientatipon preserving homothetic transformation in this article . we say two triangles are reversely similar if they are equivalent under orientation reversing homothetic transformation . nakamura and oguiso introduced the moduli space of similarity classes of triangles in @xcite , which is a strong tool for the study of @xmath14 and @xmath15 . using their results ( explained in section [ section_no ] ) , we give ( projective ) geometric characterization of equisectionally equivalent triangles . namely , two triangles with a common base , say @xmath27 , with the third vertices , say @xmath28 and @xmath29 , in the same side of the base are equisectionally equivalent if and only if @xmath28 and @xmath29 are on the same circle of apollonius with foci being two vertices ( denoted by @xmath30 and @xmath31 ) of regular triangles with the common base @xmath27 . therefore , each equisectional equivalence class with a given base @xmath27 corresponds to a circle of apollonius with foci @xmath30 and @xmath31 . it is an element of a hyperbolic pencil of circles defined by @xmath30 and @xmath31 from a projective geometric viewpoint . we then study properties of triangles of the following three special types , right triangles , isosceles triangles , and trianges with sides in arithmetic progression ( which shall be denoted by _ sap _ triangles ) , that appear in the same equisectional equivalence class . there are ( at most ) two similarity classes of such triangles for each type , which are reversely similar in the case of right or sap triangles , or the base angles of which satisfy @xmath32 in the case of isosceles triangles . for each type we explicitly give the ratio @xmath1 such that @xmath14 maps one to the other in the same equisectional equivalence class , which implies that a pair of triangles @xmath23 and @xmath24 of one of the above special types with rational edges satisfies @xmath33 if and only if @xmath34 . we finally study compass and straightedge constructibility of @xmath1 for a given pair of triangles . [ def_alpha ] let @xmath35 be a triangle . let @xmath36 be a half plane containing @xmath28 with boundary the line @xmath37 , and @xmath30 and @xmath38 be two points ( @xmath39 ) such that @xmath40 and @xmath41 are regular triangles . define @xmath42 ( @xmath43 ) and @xmath44 by @xmath45 where @xmath46 means the area of @xmath0 . we remark that both @xmath42 and @xmath44 are independent of the choice of the base of the triangle . a locus of points @xmath47 such that @xmath48 is a given positive constant is a circle , called a _ circle of apollonius with foci @xmath30 and @xmath38_. put @xmath49 note that @xmath50 when @xmath23 is a regular triangle . the quantity @xmath51 takes the value @xmath52 if and only if @xmath23 is a regular triangle , and approaches @xmath53 as @xmath23 becomes thinner and thinner . in that sense , it can be considered as measuring how far a triangle is from a regular triangle . [ main_theorem ] given two triangles @xmath35 and @xmath54 . let @xmath29 be a point in @xmath36 such that @xmath55 is similar to @xmath56 . then the following conditions are equivalent : 1 . @xmath23 is equisectionally equivalent to @xmath57 . @xmath58 , in other words , @xmath29 is on the circle of apollonius with foci @xmath30 and @xmath59 that passes through @xmath28 . 3 . @xmath60 . 4 . let @xmath61 and @xmath62 be points in @xmath36 such that @xmath63 and @xmath64 are similar to @xmath0 in such a way that each vertex of @xmath63 or @xmath64 corresponds to a vertex of @xmath0 in the same sequential order through the similarity ( figure [ fig_ap_circ_three_pts ] ) . then @xmath29 is on the circle that passes through @xmath65 , and @xmath62 . when @xmath23 is a regular triangle we agree that the circle through @xmath65 , and @xmath62 consists of a single point . and @xmath59 ] 0.8 cm and @xmath59 ] the set of circles of apollonius with foci @xmath30 and @xmath59 is called a _ hyperbolic pencil _ of circles defined by @xmath30 and @xmath38 ( or a _ poncelet pencil _ with _ limit points _ ( or _ poncelet points _ ) @xmath30 and @xmath38 ) . it consists of circles that are orthogonal to any circle passing through @xmath30 and @xmath38 ( figure [ pencil ] left ) . a set of circles through @xmath30 and @xmath38 is called an _ elliptic pencil _ ( or a _ pencil of circles with base points _ ) . let @xmath67 be the set of similarity classes of triangles and @xmath68 $ ] denote the similarity class of a triangle @xmath23 . nakamura and oguiso s result implies that the sets of similarity classes of equisectionally equivalent triangles form a codimension @xmath53 foliation of @xmath67 with a unique singularity @xmath52 that corresponds to regular triangles . we study the intersection of each leaf and another codimension @xmath53 subspace of @xmath67 which is the set of similarity classes of one of the following three special triangles , isosceles triangles , right triangles , and sap triangles ( i.e. , triangles with sides in arithmetic progression ) ( the reader is referred to @xcite for the properties of sap triangles ) . [ cor_right ] two right triangles are equisectionally equivalent if and only if they are either similar or reversely similar . any triangle @xmath23 is equisectionally equivalent to a right triangle if and only if @xmath69 . [ cor_sap ] two sap triangles are equisectionally equivalent if and only if they are either similar or reversely similar . any trianle is equisectionally equivalent to such a triangle . [ prop_isosceles ] two isosceles triangles @xmath23 and @xmath24 are equisectionally equivalent if and only if either they are similar or the base angles satisfy @xmath70 . when @xmath68\ne[\delta']$ ] , @xmath71=[\delta']$ ] if and only if @xmath72 or @xmath73 . any triangle is equisectionally equivalent to an isosceles triangle . corollaries [ cor_right ] and [ cor_sap ] imply the importance of an equisectional operator that maps a similarity class of a given triangle to that of its mirror image . [ lemma ] let @xmath74 denote a triangle with side lengths @xmath75 in the anti - clockwise order . then @xmath76=[\delta_{ba}]$ ] if and only if @xmath77 [ thm_rationality ] suppose @xmath23 and @xmath24 are isosceles ( or right or sap ) triangles such that all the sides are rational numbers . if @xmath78 then @xmath1 is also rational , namely , @xmath33 if and only if @xmath34 . nakamura and oguiso gave a bijection between @xmath67 and the open unit disc @xmath79 in @xmath80 , and showed that the set of equisection operators @xmath81 acts on @xmath79 as rotations . let us first introduce the result of nakamura - oguiso @xcite . we work in the complex plane @xmath80 . let @xmath82 be the upper half plane @xmath83 , @xmath84 and @xmath79 be the open unit discs in variables @xmath85 and @xmath86 respectively . put @xmath87 . define @xmath88 , and @xmath89 by @xmath90 then we have the following commutative diagram : @xmath91 let us fix the base of a triangle to be @xmath92 $ ] so that a triangle can be identified by the vertex @xmath93 . suppose @xmath35 is similar to the triangle @xmath94 . since the three choices of the base , @xmath95 , or @xmath5 corresponds to @xmath96 , or @xmath97 , there is a bijection @xmath98 ( @xcite ) given by @xmath99)=g\circ f(z)=\left(\frac{z-\rho}{z-\rho^{-1}}\right)^3\,.\ ] ] let us call @xmath79 the _ nakamura - oguiso moduli space _ of the similarity classes of triangles . from the construction of the moduli space and the property of linear fractional transformations , it follows that the set of isosceles triangles is expressed by a real axis in @xmath79 ( explained in section [ section_proofs ] ) , and reversely similar triangles by complex conjugate numbers , as was pointed out in @xcite . the equisection operators and the equisectional equivalence relation on @xmath67 , denoted by the same symbols , can be naturally induced from those on @xmath19 . we shall express the operators on @xmath79 given by @xmath100 and @xmath101 simply by @xmath14 and @xmath15 respectively . since our @xmath14 and @xmath15 are equal to @xmath102 and @xmath103 in @xcite respectively , theorem 1 of @xcite implies [ thm_no](@xcite ) the operator @xmath14 acts on @xmath79 as a rotation by angle @xmath104 and @xmath15 by angle @xmath105 . theorem [ thm_no ] implies that @xmath18 is in fact an equivalence relation , that @xmath33 if and only if there is a real numbers @xmath1 such that @xmath24 is similar to @xmath106 , and that the equivalence relation generated by similarity and @xmath107 is identical with the equisectional equivalence . the following corollaries can be obtained by simple computation . [ cor_t_q_id](@xcite ) 1 . @xmath108 if and only if @xmath109 , or @xmath53 . 2 . @xmath110 and @xmath111 if and only if @xmath72 or @xmath73 . [ cor_t_q_equal ] @xmath112 if and only if @xmath113 [ cor_t_q_inverse](@xcite ) @xmath114 if and only if @xmath115 the six functions of @xmath1 which appear in the right hand sides in corollaries [ cor_t_q_equal ] and [ cor_t_q_inverse ] form a non - abelian group with the operation of composition , which is isomorphic to the full permutation group of three elements . [ cor_t_q_composition ] given real numbers @xmath1 and @xmath116 . @xmath117 if and only if @xmath118 the formula implies that @xmath119 is the rotation by angle @xmath120.\ ] ] since @xmath121 \hspace{0.7cm}(\mbox{modulo } \>2\pi),\ ] ] substitution @xmath122 and @xmath123 implies that if we put @xmath124 then @xmath117 . the other two values for @xmath125 can be obtained by corollary [ cor_t_q_equal ] . corollaries [ cor_t_q_id ] ( 1 ) , [ cor_t_q_inverse ] , and [ cor_t_q_composition ] show that the rational equisectional equivalence @xmath22 is in fact an equivalence relation . in what follows , we restrict ourselves to the case of non - regular triangles . * proof of theorem * [ main_theorem ] . * * theorem [ thm_no ] shows that a set of equisectionally equivalent triangles corresponds to a circle with center @xmath52 in @xmath79 . therefore , the formula implies @xmath126\sim[\triangle z'01]\longleftrightarrow \left|\left(\frac{z-\rho}{z-\rho^{-1}}\right)^3\right|=\left|\left(\frac{z'-\rho}{z'-\rho^{-1}}\right)^3\right| \longleftrightarrow \frac{|z-\rho|}{|z-\rho^{-1}|}=\frac{|z'-\rho|}{|z'-\rho^{-1}|},\ ] ] which proves the equivalence between ( 1 ) and ( 2 ) . since @xmath0 , @xmath127 , and @xmath128 are similar , the above argument implies that a circle of apollonius with foci @xmath129 and @xmath130 that passes through @xmath28 also passes through @xmath61 and @xmath62 , which proves the equivalence of ( 1 ) and ( 4 ) . the equivalence between ( 2 ) and ( 3 ) follows from computation . since @xmath131 @xmath132 which , translated to a scale - invariant statement , is equivalent to ( 3 ) . we remark that the formula implies @xmath133 . let us give projective geometric explanation of the equivalence between ( 1 ) and ( 2 ) . the set of lines through @xmath52 , which are considered as circles through @xmath52 and @xmath134 , is an elliptic pencil of circles defined by @xmath52 and @xmath134 , and the set of concentric circles with center @xmath52 is a hyperbolic pencil of circles defined by @xmath52 and @xmath134 . they are mutually orthogonal . a linear fractional transformation is a conformal map ( i.e. , it preserves the angles ) that maps circles ( which include lines that can be considered as circles through @xmath134 ) to circles , and hence it maps an elliptic pencil ( or a hyperbolic pencil ) of circles defined by a pair of points to an elliptic pencil ( or a hyperbolic pencil ) defined by a pair of corresponding points . since @xmath135 is a linear fractional transformation which maps @xmath52 and @xmath134 to @xmath129 and @xmath130 , it maps the set of lines through @xmath52 to an elliptic pencil consisting of the circles through @xmath129 and @xmath130 , and the set of concentric circles with center @xmath52 to a hyperbolic pencil defined by @xmath129 and @xmath130 , which consists of circles of apollonius with foci @xmath129 and @xmath130 ( figure [ pencil ] ) . now it follows from the construction of nakamura - oguiso moduli space that the vertices @xmath136 of equisectionally equivalent triangles @xmath94 form a circle of apollonius with foci @xmath129 and @xmath130 , which proves the equivalence between ( 1 ) and ( 2 ) . and @xmath130 , and a pair of a hyperbolic pencil ( right , black ) and an elliptic pencil ( right , blue ) of circles defined by @xmath52 and @xmath134 . the former is mapped to the latter by a linear fractional transformation that maps @xmath129 and @xmath130 to @xmath52 and @xmath134 respectively . in each case , a circle in a hyperbolic pencil is orthogonal to a circle in an elliptic pencil . ] in what follows , it sometimes makes things easier to work in a a fundamental domain of @xmath137 , @xmath138 which corresponds to studying triangles with the longest edge ( one of the longest edges ) being fixed to @xmath139 $ ] . let us explain why the isosceles triangles corresponds to @xmath140 in the real axis in the nakamura - oguiso moduli space @xmath79 . [ def_gamma ] let @xmath141 be a circle through three points @xmath142 , and @xmath143 . when one of the three points is @xmath134 , @xmath141 is a line . we assume that @xmath141 is oriented by the cyclic order of @xmath142 , and @xmath143 . in @xmath144 , a vertex of an isosceles triangle lies either on the line @xmath145 or on the circle @xmath146 . since @xmath147 is a linear fractional transformation with @xmath148 it maps a circle @xmath146 ( @xmath149 ) to the real axis ( @xmath150 ) , another circle @xmath151 ( @xmath152 ) to a line joining @xmath52 and @xmath153 ( @xmath154 ) , a line @xmath145 ( @xmath155 ) to a line segment joining @xmath52 and @xmath129 ( @xmath156 ) , the real axis ( @xmath150 ) to the unit circle ( @xmath157 ) , and @xmath144 to one third of the open unit disc @xmath158 . therefore , the images of @xmath159 of @xmath160 and @xmath161 are @xmath162 and @xmath163 $ ] respectively . * proof of proposition * [ prop_isosceles ] . * * we work in the nakamura - oguis moduli space @xmath79 . the isosceles triangles correspond to the real axis in @xmath79 . each circle with center @xmath52 , which corresponds to a set of equisectionally equivalent non - regular triangles , intersects the real axis in two points . it proves the third statement . it also proves that if @xmath14 satisfies @xmath78 with two isosceles triangles @xmath23 and @xmath24 then either @xmath68=[\delta']$ ] or @xmath14 is a rotation by @xmath164 on @xmath79 . theorem [ thm_no ] shows that @xmath14 is a rotation by angle @xmath165 for some @xmath166 if and only if @xmath72 or @xmath73 , which proves the second statement . the equation @xmath70 follows directly from the fact that @xmath167=t_{1/3}([\delta])$ ] as is illustrated in figure [ isosceles3 ] . ] * proof of corollary * [ cor_right ] . * * in the fundamental domain @xmath144 , a vertex @xmath136 of a right triangle @xmath94 lies on the upper half hemi - circle @xmath168 , which intersects any circle of apollonius in at most two points , which are symmetric in the line @xmath169 . the extremal value of @xmath51 is given by a right isosceles triangle . * proof of corollary * [ cor_sap ] . * * in the fundamental domain @xmath144 , a non - regular triangle such that the ratio of the edge lengths is @xmath170 corresponds to @xmath171 let @xmath172 be a curve @xmath173 . we show that each of @xmath172 , say @xmath174 , intersects any circle of apollonius with foci @xmath129 and @xmath130 , @xmath175 , in exactly one point . firstly , since @xmath174 is an open curve joining @xmath129 and a point @xmath176 on the real axis , it must intersect @xmath175 . secondly , if we put @xmath177 , ( or @xmath178 ) of @xmath159 is the intersection of @xmath79 and the upper half plane ( or the lower half plane respectively ) . ] then @xmath179 , and hence @xmath180 . let @xmath181 , then , as @xmath182 is an increasing function and @xmath183 a decreasing function of @xmath184 , we have @xmath185 on @xmath174 , whereas @xmath186 can be expressed as a graph of an increasing function . therefore @xmath174 intersects @xmath175 in at most one point . we remark that the statement can also be proved by computation showing that @xmath187 is a monotonely increasing function of @xmath184 with @xmath188 and @xmath189 . * proof of lemma * [ lemma ] . * * suppose @xmath190 is expressed by a complex number @xmath191 @xmath192 in the fundamental domain @xmath144 . since @xmath193 and @xmath194 , @xmath195 and @xmath196 satisfy @xmath197 y^2&=&\displaystyle \frac14\left(b^2-(a-1)^2\right)\left((a+1)^2-b^2\right ) . \end{array } \label{xy_ab}\ ] ] let @xmath198 and @xmath199 . then @xmath200 substitution of gives @xmath201 since @xmath74 is a mirror image of @xmath190 , we have @xmath202 , and since @xmath203 , we have @xmath204 on the other hand , @xmath14 acts on @xmath79 as a rotation by @xmath205 , where @xmath206 . suppose @xmath207 it means @xmath208 , which implies @xmath209 modulo @xmath164 and hence @xmath210 modulo @xmath211 , which implies @xmath212 , i.e. , @xmath213)=[\delta_{ba}]$ ] . the equation gives @xmath214 the other two values follow from the above by corollary [ cor_t_q_equal ] . * proof of theorem * [ thm_rationality ] . * * when @xmath68=[\delta']$ ] the conclusion follows from corollary [ cor_t_q_id ] . when @xmath68\ne[\delta']$ ] the conclusion follows from proposition [ prop_isosceles ] for isosceles triangles and from corollaries [ cor_right ] , [ cor_sap ] and lemma [ lemma ] for right and sap triangles . [ constructibility ] given two triangles @xmath35 and @xmath215 . whether @xmath23 is equisectionally equivalent to @xmath24 or not can be determined using a straightedge and compass , and if the answer is affirmative , a real number @xmath1 such that @xmath216)=[\delta']$ ] is compass - and - straightedge constructible . 1 . two points @xmath30 and @xmath59 ( @xmath39 ) such that both @xmath40 and @xmath217 are regular triangles . 2 . a vertex @xmath29 in @xmath36 such that @xmath218=[\triangle a'b'c']$ ] . 3 . an oriented circle @xmath219 and another oriented circle @xmath220 ( see definition [ def_gamma ] ) . 4 . a circle of apollonius @xmath175 with foci @xmath30 and @xmath59 that passes through @xmath28 , since the center is the intersection of a bisector of the edge @xmath27 and a tangent line of @xmath221 at point @xmath28 . a decision whether @xmath33 or not , since the answer is affirmative if and only if @xmath222 . 1 . the signed angle @xmath223 ( @xmath224 ) at point @xmath30 from the oriented circle @xmath219 to @xmath220 at least one @xmath166 such that @xmath225 3 . the value @xmath1 which is given by @xmath226,\ ] ] where @xmath227 is given by the proceeding step . we explain the process ( 6),(7),(8 ) . by working in the fundamental domain @xmath144 , we may assume that @xmath228 , and @xmath229 . by formulae and , we want a real number @xmath1 such that @xmath230 since the right hand side divided by @xmath231 is equal to the singed angle @xmath232 at @xmath52 from the oriented line @xmath233 to the oriented line @xmath234 , and a linear fractional transformation @xmath135 maps @xmath235 , and @xmath134 to @xmath236 , and @xmath130 respectively , the signed angle @xmath232 is equal to the signed angle @xmath223 at point @xmath129 from the oriented circle @xmath237 to the oriented circle @xmath238 . if @xmath1 is given by , then @xmath239 and therefore , @xmath240 ( modulo @xmath211 ) , which means .

we study equivalence relation of the set of triangles generated by similarity and operation on a triangle to get a new one by joining division points of three edges with the same ratio . using the moduli space of similarity classes of triangles introduced by nakamura and oguiso , we give characterization of equivalent triangles in terms of circles of apollonius ( or hyperbolic pencil of circles ) and properties of special equivalent triangles . we also study rationality problem and constructibility problem . 2010 _ mathematics subject classification _ : 51m04 .

1608.08087

over the last decade , optical and x - ray observations made with the _ hubble space telescope _ ( e.g. , * ? ? ? * ; * ? ? ? * ) and the _ chandra _ x - ray observatory ( e.g. , * ? ? ? * ) have produced exquisite images of extragalactic kiloparsec - scale jets , completely changing our understanding of their properties . currently , more than 70 ( 30 ) extragalactic jets and hotspots are known in the x - ray ( optical ) ; ] all but the few brightest jets were discovered by _ ( _ hst _ ) . the origin of the broad - band spectral energy distributions ( seds ) of large - scale quasar jets , constructed using _ hst _ and _ chandra _ data , are the subject of active debate ( for reviews , see * ? ? ? * ; * ? ? ? * ) . in luminous quasars with x - ray jets extending from the quasar nucleus out to hundreds of kiloparsec ( e.g. , * ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , the x - ray intensity relative to the radio synchrotron flux is generally too high to be explained by a synchrotron - self - compton model unless there is a huge deviation from equipartition @xcite . the radio , optical , and x - ray fluxes of a jet knot generally trace a peaked , inflected broad - band spectrum , which rules out the interpretation of x - rays as due to synchrotron radiation from a single population of electrons . an alternative scenario , inverse - compton ( ic ) scattering of cmb photons by high - energy electrons ( @xmath2 ) in a highly relativistic jet with bulk lorentz factor @xmath3 ( the beamed ic model : * ? ? ? * ; * ? ? ? * ) initially seemed a more natural way to explain the observed x - ray emission , but this process is also not free of problems ( e.g. , * ? ? ? * ) . finally , it is possible that the x - rays arise from synchrotron radiation from extremely energetic protons @xcite . determining which of these emission mechanisms produces the observed x - ray jets in powerful quasars is a strong motivation for more observations of radio - loud quasars , and has resulted in a rapid increase in the number of known x - ray and optical jets . a new window to explore extragalactic large - scale jets has been opened by the _ spitzer space telescope _ , which is capable of detecting jet infrared emission thanks to the excellent sensitivity of the infrared array camera ( irac ; * ? ? ? * ) and multiband imaging photometer ( mips ; * ? ? ? the first example was the detection of infrared synchrotron radiation from jet knots in the quasar pks 0637@xmath0752 with the _ spitzer _ irac at wavelengths of 3.6 and @xmath1 @xcite . in terms of the beamed ic model , the infrared bandpass is particularly interesting since the bulk - comptonization bump produced by cold electrons is expected to appear in the infrared @xcite . the absence of such features in the pks 0637@xmath0752 jet rules out the jet model dynamically dominated by @xmath4 pairs in the guise of the beamed ic model @xcite . the mips observations of the jet in centaurus a @xcite , and most recently , the results from irac and mips imaging photometry of the jet in m 87 @xcite have been reported , which also demonstrate the power of _ spitzer _ to study jet emissions in lower power jets ( see also * ? ? ? * for the m87 jet ) . now the three great observatories collectively offer the possibility to identify the radiation mechanisms operating in powerful quasar jets . in fact , when combined with the data from the _ vla _ , _ hubble _ , and _ @xcite , the _ spitzer _ observation of the bright quasar 3c 273 shed new light on the riddle of x - ray jets @xcite . the _ spitzer _ irac photometry of the jet knots in 3c 273 indicated a two - component spectral energy distribution : a radio - to - infrared synchrotron component and a separate optical - to - x - ray component . the latter also seems likely to be of synchrotron origin , given the similar polarization of optical and radio light . the optical polarization , however , has not yet been measured with high precision , so this conclusion is not yet firm . perhaps such a double synchrotron scenario is applicable to the radiation output from many quasar jets . in this paper , we present _ spitzer _ irac imaging of the powerful jet in the luminous quasar pks 1136@xmath0135 . together with data from the _ vla _ , _ hst _ , and _ @xcite , our infrared photometry makes the sed of the pks 1136@xmath0135 jet the most detailed and best constrained among _ lobe - dominated _ quasars . the jet in the quasar pks 1136@xmath0135 is reminiscent of the 3c 273 jet @xcite , demonstrating anti - correlation between radio and x - ray brightness , such that the radio intensity increases toward a hotspot while x - ray flux decreases . applying the beamed ic model to the x - ray emission , this has recently been interpreted to imply _ deceleration _ of the jet @xcite . here we analyze the multiwavelength jet emission in the light of the double synchrotron scenario recently outlined for the 3c 273 jet @xcite . the redshift of pks 1136@xmath0135 is @xmath5 , so we adopt a luminosity distance of @xmath6 , for a concordance cosmology with @xmath7 , @xmath8 , and @xmath9 . the angular scale of @xmath10 corresponds to 6.4 kpc . we observed pks 1136@xmath0135 with _ spitzer _ irac @xcite on 2005 june 10 as part of our cycle-1 general observer program ( _ spitzer _ program i d 3586 ) . we used the pair of 3.6 and @xmath1 arrays , observing the same sky simultaneously . the pixel size in both arrays is @xmath11 . the point - spread functions ( psfs ) are @xmath12 and @xmath13 ( fwhm ) for the 3.6 and @xmath1 bands , respectively . the photometry with irac is calibrated to an accuracy of @xmath14 @xcite . we obtained a total of 50 frames per irac band , each with a 30-s frame time . the pipeline process ( version s14.0.0 ) at the _ spitzer _ science center yielded 50 calibrated images ( basic calibrated data ) . these well - dithered frames were combined into a mosaic image with a pixel size of @xmath15 using mopex @xcite , which removes spurious sources such as cosmic rays and moving objects based on inter - frame comparisons . finally , in order to align the irac image with respect to the vla image , we set the center of the quasar in the irac image at the core position in the vla image . in this way irac astrometry is estimated to be accurate to @xmath16 . the infrared fluxes from the quasar core of pks 1136@xmath0135 were measured as 2.2 and 4.0 mjy in the 3.6 and @xmath1 bands , respectively ( table [ tbl-1 ] ) . these values do not reach the saturation limits of the irac arrays . assuming that the infrared spectrum has a power law with @xmath17 , as consistent with the ratio of the 3.6 and @xmath1 fluxes , the luminosity in the 310 @xmath18 band amounts to @xmath19 . in the model of @xcite , the infrared emission is assumed to arise from the inner jet , while the optical / uv emission originates in the accretion disk . our infrared measurement is consistent with that model . if this is the case , the infrared fluxes determine the apparent luminosity of the synchrotron component . inspection of the @xmath20 irac images clearly reveals the presence of an infrared jet that traces the extended radio jet . the irac @xmath1 image also shows the jet emission , although its quality is worse than the @xmath21 image . the extended psf of the bright quasar core is significant at the location of the jet , amounting to @xmath22 of the jet flux . in order to separate the jet infrared emission from the contaminating psf wings of the core , we subtracted the psf wings by making use of the psf templates in a way similar to @xcite . figure [ fig : image ] shows the irac image of pks 1136@xmath0135 in the @xmath21 band after subtraction of the psf wings of the quasar core . infrared counterparts of the jet knots are clearly visible in the irac image , located @xmath23 from the core ( knot b through hotspot hs ) . a possible counterpart of knot a is marginally found in the irac image , but it may be due to our incomplete psf subtraction , so we regard the knot a flux only as an upper limit . the hotspot at the counter - jet side also suffers from the uncertainties of psf subtraction ; we do not find an infrared counterpart and place an upper limit of @xmath24 at @xmath20 . also , we do not detect extended infrared emissions from the radio lobes . to construct the broad - band seds of jet knots using multifrequency data , we derived infrared flux densities of some knot features at 3.6 and @xmath1 . given the small separation of adjacent knots , typically @xmath25 , we utilized a psf - fitting method @xcite . specifically , the jet image was fitted with a series of irac psfs , fixing the psf centers at the knot positions in the vla 22 ghz radio image . however , this procedure did not fully determine knot by knot fluxes due mainly to a multiplicity of knots from c to e. we then fixed the flux density of knot e at @xmath26 and @xmath27 . we derived the systematic errors due to this treatment by changing the knot e flux in a range of @xmath28 and @xmath29 . the systematic errors are then added in quadrature to statistical ones . the photometric results determined in this way are listed in table [ tbl-1 ] , where a combined flux is reported for knots c , d , and e ( referred as knots cde ) . crrrrrrrrr core & 463 & & & @xmath30 & @xmath31 & & @xmath32 & 170 & + a & & @xmath33 & & @xmath34 & @xmath35 & @xmath36 & @xmath37 & @xmath38 & @xmath39 + b & @xmath40 & @xmath41 & @xmath42 & @xmath43 & @xmath44 & @xmath45 & @xmath46 & @xmath47 & @xmath48 + cde & @xmath49 & @xmath50 & @xmath51 & @xmath52 & @xmath53 & @xmath54 & @xmath55 & @xmath56 & @xmath57 + hs & @xmath58 & @xmath59 & @xmath60 & @xmath61 & @xmath62 & @xmath63 & @xmath64 & @xmath65 & + in figure [ fig : great ] we present a three - color multifrequency image based on _ spitzer _ infrared ( _ red _ ) , _ hst _ optical ( _ green _ ) , and _ chandra _ x - rays ( _ blue _ ) . the vla 8.5 ghz radio contours in a logarithmic scale are overlaid . the infrared photometry at @xmath21 is illustrated as a series of best - fitted psfs of every knot after artificially shrinking the widths of the psfs ( @xmath10 fwhm ) to restore a resolution similar to the x - rays . at the positions of optically bright knots a and b we plot a _ green _ dot of @xmath66 to illustrate clear detections with the _ hst _ stis and acs @xcite . chandra _ x - ray image in 0.38 kev band is smoothed with a gaussian kernel of @xmath67 . the jet morphology , or brightness pattern along the jet , in the infrared appears to be different from emission at shorter wavelengths . it is interesting to note that optically - bright features ( knots a and b ) are not particularly prominent in the infrared , while an infrared - bright knot d does not emit correspondingly strong optical light . the difference in the brightness pattern between the infrared and optical suggests a dramatic change of the spectral shape in the infrared - to - optical band along the jet . this emphasizes the importance of measuring infrared fluxes , together with the optical fluxes , in quasar jets . putting hotspot hs aside , the jet knots can be divided up into two parts : the _ inner knots _ ( a and b ) and the _ outer knots _ ( c , d , and e ) . the inner knots are bright in both the optical and x - rays , and as such they are high - energy dominated . " indeed the inner knots are most luminous in the x - rays , while the outer knots seem to have a luminosity peak in the far - infrared . to clarify this , we present the seds of the jet knots in the following section . ccccc parameter & knot a & knot b & knots cde & hotspot hs + @xmath68 & 0.65 & 0.65 & 0.85 & 1.0 + @xmath69 ( hz ) & @xmath70 & @xmath71 & @xmath72 & @xmath73 + @xmath74 ( @xmath75 ) & @xmath76 & @xmath77 & @xmath78 & @xmath79 + @xmath80 & 0.75 & 0.70 & 0.50 & + @xmath81 ( @xmath75 ) & @xmath82 & @xmath83 & @xmath84 & + we construct the broad - band seds of the jet knots using the new _ spitzer _ irac data added to the _ vla _ , _ hst _ , and _ chandra _ data already presented in @xcite . once again , the _ spitzer _ irac provides crucial photometric points in characterizing the emissions of quasar jets @xcite . table [ tbl-1 ] summarizes the flux densities obtained with _ spitzer _ , _ hst _ , and _ chandra_. the radio fluxes measured with the vla at frequencies of 4.9 , 8.46 , and 22.5 ghz are taken from @xcite . infrared photometry with the _ spitzer _ irac is described in [ irac ] . in the optical , we list the flux densities from the _ hst _ acs in two filters @xcite : f475w ( @xmath85 ) and f814w ( @xmath86 ) . we employed a somewhat different method for x - ray photometry compared with @xcite . specifically , we here define two energy bands for photometry : 12 kev and 36 kev . integrating over the entire jet ( from knot a to hotspot hs ) we found 203 and 57 counts in the 12 and 36 bands , respectively ( also , 147 counts in the 0.61 kev band ) . the nominal energy @xmath87 for each band is the mean energy weighted by the effective area : @xmath88 1.56 kev and 4.35 kev . we reanalyzed the _ chandra _ data sets that are fully described in @xcite , and constructed flux images ( in units of @xmath89 ) in the two bands , to which aperture photometry was applied . the object apertures for knots a / b and hotspot are a circle with a radius of @xmath90 centered on each feature , while that for knots cde is a box enclosing knots c , d , and e. to correct for the interstellar absorption of @xmath91 , we multiplied the measured fluxes by a factor of 1.04 for the 12 kev band . the fudge factor slightly depends on the source spectrum such that it ranges from 1.040 to 1.048 for a power law with photon index @xmath92 . the photometry - based spectra are found to be consistent with the usual spectral analysis presented by @xcite . the photometric method is convenient when combining lower frequency data . care should be taken in interpreting the x - ray fluxes . first , the x - ray emission downstream of knot b does not show well discernible knots , and therefore the x - ray flux of knots cde may not be related directly to the lower frequency fluxes . second , in knot a there is an offset of @xmath93 between the soft and hard x - ray peaks @xcite . the x - ray spectrum derived for knot a may be contaminated by some unrelated emission , which makes the spectral shape inconsistent with a power law . in fig . [ fig : sed ] , we present the seds from radio to x - rays for knots a , b , cde , and hotspot hs . we note that the infrared fluxes we have measured with the _ spitzer _ irac fill central points in the radio - to - x - ray seds of the jet knots , setting an important constraint on models of the broad - band emission . the _ inner knots _ ( a and b ) radiate most strongly in the x - ray . also , significant ( and probably flatter ) optical emission is observed exclusively from the _ inner knots_. the possible difference in slopes of the optical continuum between the _ inner _ and _ outer knots _ @xcite suggest different origins ; the optical emission in the _ inner knots _ may belong to the same spectral component responsible for the x - ray emission ( as suggested by * ? ? ? * for knot a ) , while in the outer knots the optical emission is related to the radio synchrotron component . these spectral characteristics are analogous to those of the jet in 3c 273 @xcite , for which we identified two spectral components : ( 1 ) the low - energy synchrotron spectrum extending from radio to infrared , and ( 2 ) the high - energy component arising in the optical and smoothly connecting to the x - ray flux . we argued that the second component is likely to be of synchrotron origin as well , thus forming double - synchrotron spectra , because of the similarity of polarization between optical and radio emission ( though the degrees of optical polarization are not established yet ) . given the similarity to 3c 273 , we model the radio - to - x - ray seds phenomenologically by the following function , namely , a double power - law with an exponential cutoff : @xmath94 .\ ] ] the first term ( @xmath95 ) of the right - hand - side accounts for the low - energy spectrum and the second term ( @xmath96 ) describes the high - energy part . as there are no firm indication of spectral steepening at x - rays , we set an arbitrary cutoff at @xmath97 hz for the second component . for the inverse - compton model described later ( [ sec : iccmb ] ; see fig . [ fig : iccmb ] ) , however , the high - energy cutoff is located far beyond the x - ray domain , reaching very high - energy gamma - rays . table [ tbl-2 ] lists the spectral parameters that well describe the knot seds shown in fig . [ fig : sed ] . instead of the normalization , @xmath98 , we present the total apparent luminosity ( without taking account of relativistic beaming ) defined as @xmath99 for each component , where the low - energy boundary of the integral is set at @xmath100 hz and @xmath101 hz for the first and second component , respectively . knot b , the most x - ray luminous one , has a luminosity of @xmath102 for the high - energy component , which indeed exceeds the luminosity of the low - energy component . the spectral index of the low - energy component is in the range of @xmath103 , which is determined by the radio spectra , as listed in table [ tbl-2 ] . the cutoff frequency of the low - energy component , as constrained by the irac fluxes , is @xmath104 hz in the cases of knots b , cde , and hotspot hs ; @xmath105 hz for knot a. the second spectral index is found to be @xmath106 . while the low - energy emission from quasar jets is undoubtedly of synchrotron origin , the production mechanism(s ) of strong x - rays remains a matter of debate ( e.g. , * ? ? ? three radiation models have been proposed to explain the high - energy component : inverse - compton scattering by radio - emitting electrons , synchrotron radiation by a second electron population , and synchrotron radiation by energetic protons . in this section we consider each of these in turn . there are reasons to think that jets are relativistic on large scales @xcite and thus beaming may be important . the doppler beaming factor is defined as @xmath107^{-1}$ ] where @xmath108 is the velocity of the jet , @xmath109 is the bulk lorentz factor of the jet , and @xmath110 is the observing angle with respect to the jet direction . a one - sided jet with two - sided hotspots in this source is a sign of substantial relativistic beaming . the doppler factor is , however , not well constrained for this jet . throughout this section , all physical quantities are referred to a jet co - moving frame unless otherwise specified . the exceptions to this are the direction and velocity of the jet itself . it is assumed that a relativistically moving blob ( as noted by knot or hotspot ) has spherical geometry with a radius of @xmath111 kpc ( in the jet frame of reference ) occupied homogeneously with relativistic particles and magnetic fields ( one - zone " model with a filling factor of unity ) . before we discuss the high - energy emission component of the _ jet knots _ ( like knot b ) , we derive a lower limit on the magnetic field strength in hotspot hs , based on the _ absence _ of the high - energy emission . we do not introduce beaming effects for hotspot hs ( @xmath112 ) . synchrotron photons are compton upscattered by synchrotron - emitting electrons themselves , namely synchrotron self - compton " ( ssc ) , which is known to be a primary component for the high - energy emission in bright hotspots such as those in cygnus a @xcite . in fact , since the average energy density of synchrotron photons ( see * ? ? ? * ) in hotspot hs , @xmath113 , largely exceeds the cmb energy density of @xmath114 , the ssc flux well exceeds the ic / cmb flux in the hotspot of pks 1136@xmath0135 . we found a lower limit on magnetic field as @xmath115 in order not to violate the x - ray upper limit by the ssc emission . with an equipartition magnetic filed of @xmath116 where the energy density of relativistic electrons is equal to that of magnetic fields , the one - zone ssc flux is an order of magnitude lower than the x - ray upper limit . we first discuss the second , x - ray - dominated component of the sed seen in the jet knots ( knot b , in particular ) in the framework of the beamed inverse - compton model @xcite , which has been advocated in the previous papers for this jet @xcite . in this model , it is assumed that the jet has a highly relativistic bulk velocity , with a lorentz factor @xmath3 out to distances of hundreds of kiloparsecs , so that the cmb field _ seen _ by electrons is enhanced by a factor of @xmath117 in the jet comoving frame . the amplified cmb can be compton up - scattered by high - energy electrons of @xmath118 into the observed x - rays . this model automatically requires that the jet direction is close to the line - of - sight , @xmath119 . ( such a condition , and therefore the beamed ic model itself , would not be favorable for a lobe - dominated quasar like pks 1136@xmath0135 . later in this section , we shall briefly discuss this issue . ) it is interesting to test the beamed ic / cmb hypothesis , particularly for the pks 1136@xmath0135 jet . the jet exhibits the evolution of multiwavelength emission along the jet , namely the increase of radio brightness accompanied by the decrease of x - ray brightness towards downstream . as detailed in @xcite , such a behavior can be interpreted , within the framework of the beamed ic radiation , as being due to deceleration of the jet @xcite . specifically , @xcite concluded that the bulk lorentz factor of the jet decreases from @xmath120 ( knot b ) to @xmath121 ( knot e ) . if the beamed ic scenario is confirmed , we gain a new probe of the flow structure of the large - scale jets . let us apply a synchrotron plus beamed ic / cmb model of @xcite to the broadband sed of knot b. we also include ssc calculation for completeness . the both models adopt one - zone " emission , assuming an emitting blob with homogeneously filled with relativistic particles and magnetic fields . it is interesting to note that in the case of no - beaming ( @xmath112 ) , again , the average energy density of synchrotron photons in the knot , @xmath122 , exceeds the cmb energy density . the energy distribution of radiating electrons is assumed to be of the form @xmath123 , having a low - energy cutoff at @xmath124 and a high - energy exponential cutoff at @xmath125 , where @xmath126 denotes the electron s lorentz factor . the number index of electrons is set to be @xmath127 based on a typical radio index in this jet ( @xmath128 ) , and the high - energy cutoff is determined to be @xmath129 to produce the infrared flux . the magnetic field strength is adopted as @xmath130 , where @xmath131 g ; this relation ensures equipartition between relativistic electrons and magnetic fields , and keeps the peak frequency of the synchrotron radiation , @xmath132 . the bulk lorentz factor itself does not explicitly affect the discussion made here and it is adopted to be @xmath133 . figure [ fig : icssc ] shows the ic / cmb and ssc x - ray fluxes as a function of @xmath134 , given the radio flux observed for knot b. since the normalization of the electron distribution scales as @xmath135 and the beaming pattern of the ic / cmb emission follows @xmath136 @xcite , the ic / cmb flux scales as @xmath137 . the ssc flux scales as @xmath138 , where @xmath139 . as such , if the jet is heavily beamed , @xmath140 , the ic / cmb flux is dominant . it reaches the observed x - ray flux at @xmath141 ; the corresponding sed is shown in the top panel of fig . [ fig : iccmb ] . the ssc becomes a main component for @xmath142 . even if we invoke a particle - dominated case of @xmath143 , the ssc model requires a significant de - beaming of @xmath144 to explain the observed x - rays , which is quite unlikely as the on - axis version of such jet emission becomes unacceptably luminous . the radio - emitting electrons emit gev @xmath126-rays through the ic / cmb process ( fig . [ fig : iccmb ] ) . the total radiative output is dominated by the multi - gev @xmath126-rays and their predicted flux just reaches the sensitivity offered by the upcoming glast satellite ( see * ? ? ? * for the case of 3c 273 ) . the quasar core is expected to show a similar level of @xmath126-ray emission @xcite ; steadiness and hard spectrum are signatures of the large - scale emission . the observation of pks 1136@xmath0135 ( or quasar jets in general ) with glast may be able to test the ic / cmb hypothesis . note that the sub - tev domains are subject to intergalactic absorption , which makes it difficult to investigate the jet emission with future ground - based cherenkov telescopes . the doppler factor of @xmath145 required by the ic / cmb model implies an uncomfortably small angle between the jet and the line - of - sight , @xmath146 . @xcite also derived the maximum angle permitted as @xmath147 for this jet . the small angle would make the jet quite long , @xmath148 , and the total source extent would be as large as the largest quasar . the core - to - lobe flux ratio at 5 ghz , @xmath149 , is an indicator of jet orientation . in the case of 1136@xmath0135 , the core - to - lobe ratio of @xmath150 is obtained @xcite , thus designated as a _ lobe - dominated _ quasar . according to @xcite , a sample of radio - loud quasars with a median value of @xmath151 have an average angle to the line - of - sight of @xmath152 with simple beaming models . also , the mean angle for steep - spectrum radio quasars is estimated as @xmath152 @xcite . in this respect , a very small viewing angle @xmath146 for a lobe - dominated quasar like pks 1136@xmath0135 is not favored in the light of simple unification schemes of radio - loud quasars and radio galaxies @xcite ; a similar statement was made by @xcite . however , it should be kept in mind that _ chandra_-detected jets can be biased toward smaller jet angles , which alleviates a problem of the jet angle . we need a more direct means to distinguish currently proposed models ( see [ sec : discri ] ) . the large - scale jet in pks 1136@xmath0135 is in many respects similar to the well - known jet of quasar 3c 273 . as @xcite have pointed out , in both jets , the radio emission brightens monotonically towards the terminal hotspot , while the upstream knots are brighter in x - rays . the two - component seds are also similar ( cf , * ? ? ? . in the case of 3c 273 , since both the radio and optical emission are linearly polarized to a similar degree and in the same direction , it seems reasonable that the high - energy component , which contributes more than half the optical flux , should also arise from the synchrotron process @xcite . therefore we discuss the optical - to - x - ray component of 1136@xmath0135 in terms of synchrotron radiation produced by a second population of high - energy electrons . the suggestion of the presence of a second synchrotron component has been made in previous studies of other large - scale jets ( e.g. , * ? ? ? ? * ; * ? ? ? let the energy distribution of _ accelerated _ electrons be @xmath153 for an energy interval of interest . the power - law slope of electrons is derived from the radiation slope as @xmath154 [ see eq . ( [ eq : fnu ] ) ] in a synchrotron - cooling regime as appropriate for optical and x - ray emission ; the _ cooled _ electrons responsible for the observed radiation have a steeper number index of @xmath155 ( see e.g. , * ? ? ? then the spectral index of the high - energy component , @xmath106 corresponds to @xmath156 . however , within the framework of diffusive shock acceleration , the hardest possible electron distribution of acceleration would be @xmath157 , in other words , @xmath158 ( see * ? ? ? the inferred index @xmath156 would violate such a theoretical limit . this brings into question the idea that the second electron distribution is formed through the diffusive shock acceleration or that there is a second electron distribution at all . note that the jet in 3c 273 has spectral index of @xmath159 corresponding to @xmath160 , which was considered to be compatible with the shock acceleration theory @xcite . the second synchrotron component in the pks 1136@xmath0135 jet may instead be due to turbulent acceleration operating in the shear layers @xcite . the coexistence of two distinct types of acceleration , shock and turbulent acceleration , in the knots would naturally give rise to a double - synchrotron spectrum . more importantly , unlike the shock acceleration , a hard spectrum , @xmath161 , can be expected to form in the case of turbulent acceleration ( i.e. , second - order fermi acceleration ) . in fully turbulent shear layers , the time scale of turbulent acceleration can be estimated as @xmath162 , where @xmath163 is an electron gyroradius and @xmath164 denotes the alfvn velocity . the turbulent acceleration has to compete against synchrotron losses , which is presumed to dominate over ic losses , with a timescale of @xmath165 where @xmath166 is the thomson cross section , and @xmath167 is the energy density of the magnetic field . by equating @xmath168namely , balancing the acceleration and synchrotron loss rates one obtains the maximum attainable energy limited by synchrotron losses : @xmath169 where @xmath170 . to account for the observed x - rays with synchrotron radiation , one needs @xmath171 ( see * ? ? ? * ) , thus requiring @xmath172 . this condition is reasonable with typical jet parameters . finally , we argue the possibility that the optical - to - x - ray emission may be due to synchrotron radiation by very high energy _ protons _ @xcite . this model requires that protons in the jet are somehow accelerated to very high energies , @xmath173 or more . also , the magnetic field strength must be of the order @xmath174 , so that energy equipartition between the relativistic _ protons _ and magnetic fields can be ( roughly ) realized . the knots and hotspots in the relativistic jets of powerful quasars and radio galaxies are indeed one of a few potential sites of cosmic - ray acceleration up to @xmath175 @xcite . for example , the turbulent acceleration in the shear layer may be able to accelerate such ultra high energy protons @xcite . the characteristic frequency and cooling time of proton synchrotron radiation can be written as @xmath176 \left ( \frac{b}{\rm mg}\right ) \left ( \frac{e_{p}}{10^{18}\ \rm ev } \right)^2\ \rm hz,\ ] ] and @xmath177 respectively @xcite . ( note that the synchrotron cooling time is much shorter than the photo - meson cooling time at energies relevant here . ) a total energy content of the magnetic field @xmath178 integrated over a spherical knot with a radius of @xmath179 amounts to @xmath180 . if we assume equipartition between radiating protons and magnetic fields , @xmath181 , the x - ray luminosity of synchrotron radiation by ultra - high - energy protons of @xmath173 can be estimated roughly as @xmath182 where @xmath183 denotes the fraction of kinetic energy of protons responsible for the x - ray emission . relativistic beaming is not taken into account in this estimate . roughly speaking , the luminosity scales with @xmath184 and @xmath134 such that @xmath185 . the expected luminosity agrees with typical ( apparent ) luminosity of the jet knots of quasars , @xmath186 , if @xmath187 . thus , very high energy protons rather than a second population of electrons , may explain the second synchrotron component extending from the optical to x - rays provided that both ultra - high - energy protons of @xmath173 and a magnetic field of @xmath174 are present in the jet knots . under the proton - synchrotron hypothesis , the low - energy synchrotron emission may be accounted for by accompanying electrons . to be specific , we modeled the high - energy spectral component of knot b with proton synchrotron radiation . we adopt mild beaming of @xmath188 ( @xmath189 ) . the low - energy spectral component was simultaneously modeled by electron synchrotron radiation with taking account of the effects of significant synchrotron cooling . it is assumed that during @xmath190 years , protons and electrons are injected continuously into the emission volume ( @xmath191 ) with the distributions characterized by number index @xmath192 and @xmath193 and by the maximum energy @xmath194 and @xmath195 : @xmath196 . in the bottom panel of fig . [ fig : iccmb ] , we present the broadband sed reproduced by the proton synchrotron model with the equipartition magnetic field of @xmath197 , the index of the power - law distribution of _ accelerated _ particles @xmath198 , and the maximum energy of @xmath199 and @xmath200 . the injection rate of electrons , in terms of energy content @xmath201 , is about 5% of that of protons . the injected electrons suffer from severe synchrotron cooling . even radio - emitting electrons have a cooling time of only @xmath202 years , which requires _ in - situ _ acceleration of electrons . the infrared - emitting electrons have a lifetime of @xmath203 years , implying that each knot is a currently ( within @xmath204 years ) active site of particle acceleration . the energy density of protons @xmath205 , which is equal to @xmath206 , can be calculated by @xmath207 with @xmath208 , since synchrotron cooling of protons is not effective . the kinetic power of the jet , estimated as @xmath209 , amounts to @xmath210 . this estimate is not sensitive to the choice of @xmath211 as long as the equipartition condition is fulfilled . a large power has to be carried by the protons and magnetic fields in the jet to produce the x - ray emission via a proton - synchrotron process . the black hole mass that powers the jet is estimated to be @xmath212 @xcite using the empirical relation to the width of the h@xmath213 line and the optical continuum luminosity . the black hole mass corresponds to the eddington luminosity of @xmath214 . therefore , the proton synchrotron model requires the jet to be _ super - eddington _ , @xmath215 . this issue would pose a problem for the proton synchrotron model , though it may be alleviated by assuming that the knots present the locations of _ power peaks _ due to modulated activity of the central engine . we emphasize here that optical polarimetry can be an effective way of discriminating the radiation models responsible for the optical - to - x - ray emission of the jet in pks 1136@xmath0135 , and of quasar jets in general . in the beamed ic interpretation , if the optical fluxes belong to the ic component , the optical and x - ray emission are due to compton up - scattering off the amplified cmb by high - energy electrons of @xmath216 ( optical ) and @xmath217 ( x - ray ) . unlike in the case of synchrotron models , the x - rays are expected to be _ unpolarized _ and the optical light is nearly unpolarized at most a few percent of polarization ( uchiyama & coppi , in preparation ) ( in the jet frame ) exists , namely bulk comptonization " , can yield in principle a high degree of polarization for a jet angle @xmath218 @xcite . for @xmath219 of a few as relevant here , however , the polarization is largely suppressed . ] . precise polarization measurements in the optical can in principle verify ( or discard ) the beamed ic model . unfortunately , there have been no useful polarization observations of quasar jets with _ hst _ so far . only for 3c 273 , early _ hst _ polarimetry of the jet was done but with the pre - costar foc and low significances @xcite . as was recently performed for nearby radio galaxies by @xcite , new , deep polarimetry of quasar jets on large - scales is highly encouraged , although the optical emission from the jet knots is generally faint , say @xmath220 @xcite . we have detected the infrared emission from the kiloparsec - scale jet in the quasar pks 1136@xmath0135 with the _ spitzer space telescope_. using the new _ spitzer _ data together with deep _ chandra _ , _ hubble _ , and multi - frequency vla observations from @xcite , we construct the broadband spectral energy distributions along the large - scale jet . the seds of the jet knots are comprised of two components : radio - to - infrared synchrotron emission and a separate high - energy component responsible for x - rays , which may extend down to the optical . in total , each component has a similar apparent luminosity of the order of @xmath221 . the origin of the high - energy component has significant implications to the properties of particle acceleration in a relativistic jet as well as jet dynamics . we consider three radiation models that have been proposed to explain the high - energy component : inverse - compton scattering in a highly relativistic jet , synchrotron radiation by a second electron population , and synchrotron radiation by highly energetic protons . in terms of the double synchrotron scenario , a flat spectrum of the second synchrotron component , @xmath222 , would turn down diffusive shock acceleration as the mechanism of electron acceleration to very high energies , @xmath223 tev . instead turbulent acceleration in the shear layers may explain the second population of electrons giving rise to the flat spectrum . in the beamed ic interpretation , a jet orientation with respect to the observer of @xmath146 is required , which is not quite favorable given the small core - to - lobe ratio of this object . future optical polarimetry or @xmath126-ray observations with glast may be able to test the beamed ic model . the proton synchrotron model requires a relatively large magnetic field of @xmath224 mg , and consequently large kinetic power of @xmath225 , with efficient acceleration of protons to @xmath226 . if this is the case , one can probe particle acceleration at very high energies , higher than any other known objects in the universe . we thank the referee , eric perlman , for his careful reading of the manuscript and valuable suggestions . this work is based on observations made with the spitzer space telescope , which is operated by the jet propulsion laboratory , california institute of technology under nasa contract 1407 . support for this work was provided by nasa through contract number rsa 1265389 issued by jpl / caltech . the national radio astronomy observatory is operated by associated universities , inc . under a cooperative agreement with the national science foundation . aharonian , f. a. 2002 , , 332 , 215 atoyan , a. m. , & dermer , c. d. 2004 , , 613 , 151 bahcall , j. n. , kirhakos , s. , schneider , d. p. , davis , r. j. , muxlow , t. w. b. , garrington , s. t. , conway , r. g. , & unwin , s. c. 1995 , , 452 , l91 band , d. l. , & grindlay , j. e. 1985 , , 298 , 128 begelman , m. c. , & sikora , m. 1987 , , 322 , 650 brookes , m. h. , lawrence , c. r. , keene , j. , stern , d. , gorijan , v. , werner , m. , & charmandaris , v. 2006 , , 646 , l41 celotti , a. , ghisellini , g. , & chiaberge , m. 2001 , , 321 , l1 chartas , g. , et al . 2000 , , 542 , 655 cheung , c. c. , stawarz , . , & siemiginowska , a. 2006 , , 650 , 679 crane , p. , et al . 1993 , , 402 , l37 dermer , c. d. 1995 , , 446 , l63 dermer , c. d. , & atoyan , a. m. 2002 , , 568 , l81 fazio , g. , et al.2004 , , 154 , 10 gambill , j. k. , sambruna , r. m. , chartas , g. , cheung , c. c. , maraschi , l. , tavecchio , f. , urry , c. m. , & pesce , j. e. 2003 , , 401 , 505 garrington , s. t. , leahy , j. p. , conway , r. g. , & laing , r. a. 1988 , , 331 , 147 georganopoulos , m. , & kazanas , d. 2004 , , 604 , l81 georganopoulos , m. , kazanas , d. , perlman , e. , & stecker , f. w. 2005 , , 625 , 656 georganopoulos , m. , kirk , j. g. , & mastichiadis , a. 2001 , , 561 , 111 georganopoulos , m. , perlman , e. , kazanas , d. , & mcenery j.2006 , , 653 , 5 hardcastle , m. j. , birkinshaw , m. , & worrall , d. m. 2001 , , 326 , 1499 hardcastle , m. j. , kraft , r. p. , & worrall , d. m. 2006 , , 368 , l15 harris , d. e. , carilli , c. l. , perley , r. a. 1994 , , 367 , 713 harris , d. e. , & krawczynski , h. 2006 , , 44 , 463 harris , d. e. , mossman , a. e. , & walker , r. c. 2004 , , 615 , 161 hillas , a. m.1984 , , 22 , 425 jester , s. , harris , d. e. , marshall , h. l. , & meisenheimer , k. 2006 , , 648 , 900 jester , s. , rser , h .- j . , meisenheimer , k. , & perley , r. a. 2002 , , 385 , l27 jester , s. , rser , h .- j . , meisenheimer , k. , & perley , r. a. 2005 , , 431 , 477 jorstad , s. g. , & marscher , a. p. 2004 , , 614 , 615 kataoka , j. , leahy , j. p. , edwards , p. g. , kino , m. , takahara , f. , serino , y. , kawai , n. , & martel , a. r. 2003 , , 410 , 833 kataoka , j. , & stawarz , . 2005 , , 622 , 797 laurent - muehleisen , s. a. , kollgaard , r. i. , ryan , p. j. , feigelson , e. d. , brinkmann , w. , siebert , j.1997 , , 122 , 235 makovoz , d. , & marleau , f. r. 2005 , , 117 , 1113 malkov , m. a. , & drury , l. oc.2001 , rep . prog . phys . , 64 , 429 marshall , h. l. , et al.2005 , , 156 , 13 oshlack , a. y. k. n. , webster , r. l. , & whiting , m. t. 2002 , , 576 , 81 ostrowski , m. 2000 , , 312 , 579 perlman , e. s. , et al . 2006 , , 651 , 735 perlman , e. s. , et al.2007 , submitted to rieke , g. h. , et al.2004 , , 154 , 25 reach , w. t. , et al.2005 , , 117 , 978 sambruna , r. m. , gambill , j. k. , maraschi , l. , et al.2004 , , 608 , 698 sambruna , r. m. , gliozzi , m. , donato , d. , maraschi , l. , tavecchio , f. , cheung , c. c. , urry , c. m. wardle , j. f. c. 2006a , , 641 , 717 sambruna , r. m. , gliozzi , m. , tavecchio , f. , maraschi , l. , & foschini , l. 2006b , , 652 , 146 sambruna , r. m. , maraschi , l. , tavecchio , f. , et al.2002 , , 571 , 206 schwartz , d. a. , et al.2000 , , 540 , l69 siemiginowska , a. , stawarz , . , cheung , c. c. , harris , d. e. , sikora , m. , aldcroft , t. l. , & bechtold , j. 2006 , , in press ( astro - ph/0611406 ) shi , y. , rieke , g. h. , hines , d. c. , gordon , k. d. , egami , e. 2006 , , in press ( astro - ph/0610494 ) stawarz , . 2003 , chinese journal of astronomy and astrophysics supplement , 3 , 383 stawarz , . , & ostrowski , m. 2002 , , 578 , 763 tavecchio , f. , maraschi , l. , sambruna , r. m. , gliozzi , m. , cheung , c. c. , wardle , j. f. c. , urry , c. m. 2006 , , 641 , 732 tavecchio , f. , maraschi , l. , sambruna , r. m. , urry , c. m. 2000 , , 544 , l23 thomson , r. c. , mackay , c. d. , & wright , a. e. 1993 , , 365 , 133 uchiyama , y. , urry , c. m. , van duyne , j. , cheung , c. c. , sambruna , r. m. , takahashi , t. , tavecchio , f. , & maraschi , l. 2005 , , 631 , l113 uchiyama , y. , urry , c. m. , cheung , c. c. , jester , s. , van duyne , j. , coppi , p. , sambruna , r. m. , takahashi , t. , tavecchio , f. , & maraschi , l. 2006 , , 648 , 910 urry , c. m. , & padovani , p. 1995 , , 107 , 803

we present _ spitzer _ irac imaging of the large - scale jet in the quasar pks 1136@xmath0135 at wavelengths of 3.6 and @xmath1 , combined with previous _ vla _ , _ hst _ , and _ chandra _ observations . we clearly detect infrared emission from the jet , resulting in the most detailed multifrequency data among the jets in lobe - dominated quasars . the spectral energy distributions of the jet knots have significant variations along the jet , like the archetypal jet in 3c 273 . the infrared measurements with irac are consistent with the previous idea that the jet has two spectral components , namely ( 1 ) the low - energy synchrotron spectrum extending from radio to infrared , and ( 2 ) the high - energy component responsible for the x - ray flux . the optical fluxes may be a mixture of the two components . we consider three radiation models for the high - energy component : inverse compton scattering of cosmic microwave background ( cmb ) photons by radio - emitting electrons in a highly relativistic jet , synchrotron radiation by a second distinct electron population , and synchrotron radiation by ultra high energy protons . each hypothesis leads to important insights into and constraints on particle acceleration in the jet , as well as the basic physical properties of the jet such as bulk velocity , transporting power , and particle contents .

astro-ph0703295

although the m dwarfs are the most numerous stars in our galaxy , the mass , metalicity and age dependencies of their stellar luminosities and radii are poorly calibrated . the reason is the selection effect that plays against the detection of fainter and smaller stars . less than 20 binaries with low - mass dm components have empirically - determined masses , radii , luminosities and temperatures ( see section [ sec : global ] , table [ tab : stars ] ) . as a result the mass - luminosity relation is determined by only a few low - mass stars . this deficiency hindered the development of the models for the cool dense atmospheres of the m dwarfs . it is established that all available models underestimate the radii ( by around 1015 per cent ) and overestimate the temperatures ( by 200300 k ) of short - period binaries with dm components @xcite . the northern sky variability survey ( nsvs ) contains a great number of photometric data @xcite that allows searching of variable stars and determination of their periods and types of variability . a multiparametric method for search for variable objects in large datasets was tested on the nsvs @xcite and as a result many eclipsing stars were discovered . one of them was gsc 2314 - 0530 @xmath17 nsvs 6550671 ( @xmath14=02@xmath18 , @xmath19=+@xmath20 ) . on the base of the nsvs photometry obtained in 19992000 we derived the ephemeris : @xmath21 and built its light curve ( fig . [ fig : nsvs ] ) . we found that this star has been assigned also as swasp j022050.85 + 332047.6 according to the superwasp photometric survey @xcite . @xcite reported its coincidence with the _ rosat _ x - ray source 1rxs j022050.7 + 332049 . initially gsc 2314 - 0530 attracted our interest by its short orbital period because there were only several systems with non - degenerate components and periods below the short - period limit of 0.22 days @xcite : gsc 1387 - 0475 with @xmath22 d @xcite , asas j071829 - 0336.7 with @xmath23 d @xcite , the star v34 in the globular cluster 47 tuc with @xmath24 d @xcite and bw3 v38 with orbital period @xmath25 d @xcite . when we established that the components of gsc 2314 - 0530 were dm stars our interest increased and we undertook intensive photometric and spectral observations in order to determine its global parameters and to add a new information for the dm stars as well as for the short - period binaries . the ccd photometry of gsc 2314 - 0530 in @xmath0 bands was carried out at rozhen national astronomical observatory with the 2-m rcc telescope equipped with versarray ccd camera ( 1300 @xmath26 1340 pixels , 20 @xmath27 m pixel , field of 5.25 @xmath26 5.35 arcmin ) as well as with the 60-cm cassegrain telescope using the fli pl09000 ccd camera ( 3056 @xmath26 3056 pixels , 12 @xmath27 m pixel , field of 17.1 @xmath26 17.1 arcmin ) . the average photometric precision per data point was 0.005 0.008 mag for the 60-cm telescope and 0.002 0.003 mag for the 2-m telescope . table [ tab : log1 ] presents the journal of our photometric observations . it should be noted that the observations on 2009 december 30 are synchronous in the @xmath0 colors . [ cols="^,^,^,^,>,>,^",options="header " , ] the small orbital angular momentum is characteristic feature of all short - period systems ranging from cvs to cb that seem to be old , being at later stages of the angular momentum loss evolution as a result of the period decrease . we calculated the orbital angular momentum of the target by the expression @xcite @xmath28 where @xmath29 is in days and @xmath30 are in solar units . the obtained value @xmath31 of gsc 2314 - 0530 is considerably smaller than those of the rs cvn binaries and detached systems which have @xmath32 . the orbital angular momentum of gsc 2314 - 0530 is smaller even than those of the contact systems which have @xmath33 . it is bigger only than those of the short - period cvs of su uma type . the small orbital angular momentum of gsc 2314 - 0530 implies existence of past episode of angular momentum loss during the binary evolution . it means also that gsc 2314 - 0530 is not pre - ms object . this conclusion is supported by the values of @xmath34 of its components . the x - ray emission of the stellar coronae are directly related to the presence of magnetic fields and consequently gives information about the efficiency of the stellar dynamo . @xcite established that the x - ray luminosity decreased for later m stars while the ratio @xmath35 did not change significantly from m0 to m6 . as a result he proposed the ratio @xmath35 as most relevant measure of activity of m dwarfs . @xcite found that the upper boundary of @xmath35 for late m stars is @xmath36 . besides all indicators of stellar activity in the optical ( surface inhomogeneities , emission lines , flares ) the star gsc 2314 - 0530 shows also x - ray emission ( it is identified as _ rosat _ x - ray source 1rxs j022050.7 + 332049 ) and x - ray flares . on the basis of the measured x - ray flux @xmath37 ergs @xmath38 @xmath39 of gsc 2314 - 0530 at quiescence @xcite and derived distance 59 pc we calculated its x - ray luminosity @xmath40 ergs s@xmath41 . this value is at the upper boundary @xmath42 29 for dm stars @xcite . the value @xmath43 of gsc 2314 - 0530 is almost at the upper boundary of this ratio and considerably bigger than those of the m dwarfs studied by @xcite and @xcite . it is known that the activity and angular momentum loss tend to be saturated at high - rotation rates @xcite . due to its short period and high activity gsc 2314 - 0530 is perhaps an example of such saturation . our observed field ( fig . [ fig : chart ] ) contains the weak star usno - b1 1233 - 0046425 . we called it twin due to the same tangential shift as our target star gsc 2314 - 0530 . table [ tab : colors ] presents the proper motion and the colors of twin according to the catalogue nomad . usno - b1 1233 - 0046425 has @xmath44 corresponding to temperature less than 3200 k. we suspect that our `` twins '' may form visual binary . the angular distance between them of 61 arcsec corresponds to linear separation around 3500 au for distance of 59 pc . such a supposition is reasonable because it is known that the short - period close binaries often are triple systems @xcite . particularly , the object tres her0 - 07621 from our table [ tab : stars ] has a red stellar neighbor at a distance 8 arcsec with close proper motion @xcite . the check of the supposition if twin is physical companion of gsc 2314 - 0530 needs astrometric observations of the `` twins '' . the analysis of our photometric and spectral observations of the newly discovered eclipsing binary gsc 2314 - 0530 allows us to derive the following conclusions : \(1 ) this star is the shortest - period binary with dm components which period is below the short - period limit . \(2 ) by simultaneous radial velocity solution and light curve solution we determined the global parameters of gsc 2314 - 0530 : inclination @xmath11 ; orbital separation @xmath12 r@xmath5 ; masses @xmath4 m@xmath5 and @xmath6 m@xmath5 ; radii @xmath7 r@xmath5 and @xmath8 r@xmath5 ; temperatures @xmath2 k and @xmath3 k ; luminosities @xmath9 l@xmath5 and @xmath10 l@xmath5 ; distance @xmath13 pc . \(3 ) we derived empirical relations mass@xmath16 , mass radius and mass temperature on the basis of the parameters of known binaries with low - mass dm components . \(4 ) the distorted light curve of gsc 2314 - 0530 were reproduced by two cool spots on the primary component . the next sign of the activity of gsc 2314 - 0530 is the strong h@xmath14 emission of its components . moreover we registered 6 flares of gsc 2314 - 0530 . half of them occurred at the phases of maximum visibility of the larger stable cool spot on the primary . the analysis of all appearances of magnetic activity revealed existence of long - lived active area on the primary of gsc 2314 - 0530 . the high activity of the target is natural consequence of the fast rotation and low temperatures of its components . our study of the newly discovered short - period eclipsing binary gsc 2314 - 0530 presents a next small step toward understanding dme stars and adds a new information to the poor statistic of the low - mass dm stars . recently they became especially interesting as appropriate targets for planet searches due to the relative larger transit depths . the research was supported partly by funds of projects do 02 - 362 of the bulgarian scientific foundation . this research make use of the simbad and vizier databases , operated at cds , strasbourg , france , and nasa s astrophysics data system abstract service . the authors are very grateful to the anonymous referee for the valuable notes and advices . becker a. et al . , 2008 , mnras , 386 , 416 blake c. et al . , 2008 , aj , 684 , 635 bopp b.w . , 1974 , apj , 193 , 389 caillault j. et al . , 1986 , apj , 304 , 318 cakirly o. , ibanoglu c. , 2010 , mnras , 401 , 1141 coughlin j. , shaw j. , 2007 , j. of southeastern assoc . for res . in astr . , 1 , 7 creevey o. , benedict g. , brown t. et al . , 2005 , apjl , 625 , 127 delfosse x. et al . , 1999 , a&a , 341 , l63 dimitrov d.p . , 2009 , bulgaj , 12 , 49 everett m. , howell s. , 2001 , pasp , 113 , 1428 fuhrmeister b. , schmitt j. , 2003 , a&a , 403 , 247 hebb l. et al . , 2006 , aj , 131 , 555 irwin j. et al . , 2009 , apj , 701 , 1436 landolt a. , 1992 , aj , 104 , 340 leggett s. , 2000 , apjs , 82 , 351 lenz p. , breger m. , 2005 , coast , 146 , 53 leung k. , schneider d. , 1978 , aj , 83,618 lopez - morales m. , ribas i. , 2005 , apj , 131 , 555 lopez - morales m. et al . , 2006 , arxiv : astro - ph/0610225v1 maceroni c. , montalban j. , 2004 , a&a , 426 , 577 maceroni c. , rucinski s.m . , 1997 , pasp , 109 , 782 maceroni c. et al . , 1994 , a&a , 288 , 529 metcalfe t. et al . , 1996 , apj , 456 , 356 mullan d. , macdonald j. , 2001 , apj , 559 , 353 norton a.j . , et al . , 2007 , a&a , 467 , 785 pollacco d. , et al . 2006 , pasp , 118 , 1407 popper d. , ulrich r. , 1977 , apj , 212 , l131 pribulla t. , rucinski s. , 2006 , aj 131 , 2986 pribulla t. , vanko m. , hambalek l. , 2009 , ibvs , no.5886 pra a. , zwitter t. , 2005 , apj , 628 , 426 ribas i. , 2003 , a&a , 398 , 239 rosner r. et al . , 1981 , apj , 249 , l5 rucinski s.m . , 1992 , aj , 103 , 960 rucinski s.m . , 1984 , a&a , 132 , l9 rucinski s.m . , 2007 , mnras , 382 , 393 rucinski s.m . , pribulla t. , 2008 , mnras , 388 , 1831 schlegel d. , finkbeiner d. , davis m. , 1998 , apj , 500 , 525 schmitt j. , fleming t. , giampapa m. , 1995 , apj , 450 , 392 stauffer j.r . , hartmann l.w . , 1986 , apjs , 61 , 531 stepien k. , 2006 , acta astr . , 56 , 347 stetson p. , 2000 , pasp , 112 , 925 torres g. , ribas i. , 2002 , apj , 567 , 1140 vandenberg d. , clem j , 2003 , aj , 126 , 778 van hamme w. , 1993 , aj , 106 , 2096 vida k. , olah k. , kovari zs . , bartus j. , 2009 , aips , 1094 , 812 vilhu o. , walter f. , 1987 , apj 321 , 958 voges w. , aschenbach b. , boller t. , et al . , 1999 , a&a 349 , 389 weldrake d.t.f . , sackett p.d . , bridges t.j . , freeman k.c . , 2004 , aj , 128 , 736 worden s.p . , schneeberg t.j . , giampapa m.s . , 1981 , apjs , 46 , 159 wozniak p.r . , vestrand c.w . , akerlof r. , et al . , 2004 , aj , 127 , 2436 young t. et al . , 2006 , mnras , 370 , 1529 zacharias n. , monet d. , levine s. , et al . , 2005 , aas , 205 , 4815

ccd photometric observations in @xmath0 colors and spectroscopic observations of the newly discovered eclipsing binary gsc 2314 - 0530 ( nsvs 6550671 ) with dme components and very short period of @xmath1 days are presented . the simultaneous light - curve solution and radial velocity solution allows to determine the global parameters of gsc 2314 - 0530 : @xmath2 k ; @xmath3 k ; @xmath4 m@xmath5 ; @xmath6 m@xmath5 ; @xmath7 r@xmath5 ; @xmath8 r@xmath5 ; @xmath9 l@xmath5 ; @xmath10 l@xmath5 ; @xmath11 ; @xmath12 r@xmath5 ; @xmath13 pc . the chromospheric activity of its components is revealed by strong emission in the h@xmath14 line ( with mean @xmath15 ) and observed several flares . empirical relations mass@xmath16 , mass radius and mass temperature are derived on the basis of the parameters of known binaries with low - mass dm components . [ firstpage ] binaries : eclipsing binaries : spectroscopic stars : activity stars : fundamental parameters stars : late - type stars : low - mass

1005.0260

in this paper , we present a unified description of a measurement process of quantum observables together with the amplification process associated with it . for this purpose , we recall the essence of micro - macro duality @xcite as a mathematical expression of the general idea of quantum - classical correspondence which plays crucial roles . in this context , we note that the ` boundary ' between the quantum and classical levels can be found in the notion of a sector , in terms of which we can understand , in a clear - cut manner , the mutual relations between the microscopic quantum world and the macroscopic classical levels . to define a sector , we classify representations and states of a c*-algebra @xmath0 of quantum observables according to the _ quasi - equivalence _ @xmath1 @xcite defined by the unitary equivalence of representations @xmath2 _ up to multiplicity _ , which is equivalent to the isomorphism of von neumann algebras @xmath3 of representatoins @xmath4 and @xmath5 . sector _ or a _ pure phase _ in the physical context is then defined by a quasi - equivalence class of _ factor _ representations and states corresponding to a von neumann algebra with a trivial centre , which is a minimal unit among quasi - equivalence classes . representations belonging to different sectors @xmath6 and @xmath7 are mutually _ disjoint _ with no non - zero intertwiners : namely , if @xmath8 is an intertwiner from @xmath6 to @xmath9 defined as a bounded operator @xmath8 from the representation space @xmath10 of @xmath6 to that @xmath11 of @xmath9 satisfying the relation @xmath12 ( @xmath13 ) , then it vanishes , @xmath14 . if @xmath15 is not a factor representation belonging to one sector , it is called a _ mixed phase . _ in the standard situations where separable hilbert spaces are used , a mixed phase can uniquely be decomposed into a direct sum ( or integral ) of sectors , through the spectral decomposition of its non - trivial centre @xmath16 of @xmath17 which is a commutative von neumann algebra admitting a ` simultaneous diagonalization ' . each sector contained in @xmath15 is faithfully parametrized by the gelfand spectrum @xmath18 of the centre @xmath19 . thus , commutative classical observables belonging to the centre physically play the role of _ macroscopic order parameters _ and the central spectrum @xmath18 can be regarded as the _ classifying space of sectors _ to register faithfully all the sectors contained in @xmath15 . in this way , we find in a _ mixed phase _ @xmath15 the coexistence of quantum ( = _ _ intra - sectorial _ _ ) and classical systems , the latter of which describes an _ inter - sectorial _ structure in terms of order parameters constituting the centre @xmath19 . in this way , the ` boundary ' and the gap between the quantum world described by non - commutative algebras of quantum variables and the classical levels with commutative algebras of order parameters can be identified with a ( _ superselection _ ) _ sector structure _ consisting of a family of sectors or pure phases @xcite . since a single sector or a pure phase corresponds to a ( quasi - equivalence class of ) factor representation @xmath15 of a c*-algebra @xmath0 of quantum observables , its _ intra - sectorial _ structure , the structure inside of a sector , is described by the observables belonging to the factor von neumann algebra @xmath20 corresponding to @xmath15 . in this and the next sections , we recapitulate the essence of the general scheme to analyze the intra - sectorial structure @xcite . because of the non - commutativity of @xmath21 , what can be experimentally observed through a measurement is up to a certain maximal abelian subalgebra ( masa , for short ) @xmath22 ( with @xmath23 the commutant of @xmath24 ) of @xmath21 : elements of a masa @xmath24 can be regarded as macroscopic observables to visualize some aspects of the microscopic structure of a sector in the macroscopic form of @xmath25 . in fact , a tensor product @xmath26 ( acting on the tensor product hilbert space @xmath27 ) has a centre given by @xmath28 , and hence , the spectrum @xmath25 of a masa @xmath24 to be measured can be understood as parametrizing a _ conditional sector structure _ of the composite system @xmath26 of the observed system @xmath21 and @xmath24 , the latter of which can be identified with the measuring apparatus @xmath24 in the simplified version @xcite of ozawa s measurement scheme @xcite . this picture of conditional sector structure is consistent with the physical essence of a measurement process as ` classicalization ' of some restricted aspects @xmath24(@xmath29 ) of a quantum system , conditional on the coupling @xmath26 of @xmath21 with the apparatus identified with @xmath24 . to implement a physical process to measure the observables in @xmath24 , we need to specify a dynamical coupling between the observed and measuring systems , which is accomplished by choosing such a unitary group @xmath30 in @xmath24 as generating @xmath24 , i.e. , @xmath31 . in the standard situation where the relevant hilbert space is separable , the abelian von neumann algebra @xmath24 on it is generated by a single element , and hence , we can assume without loss of generality that @xmath30 is a locally compact abelian lie group . because of the commutativity of @xmath30 , the group characters @xmath32 of @xmath30 , @xmath33 (: 1-dimensional torus ) s.t . @xmath34 , @xmath35 , constitute the dual goup @xmath36 satisfying the fourier - pontryagin duality @xmath37 . since the restriction @xmath38 to @xmath39 of an _ algebraic character _ @xmath40 of @xmath24 is naturally a _ group character _ of @xmath30 , a canonical embedding @xmath41 can be defined by @xmath42 . as the masa @xmath22 is the fixed - point subalgebra @xmath43 of @xmath21 under the adjoint action of @xmath30 , our discussion can also be related with the galois - theoretical context of the duality between w*-dynamical systems @xmath44 and @xmath45 and between the associated crossed products @xmath46 and @xmath47 , where the co - action @xmath48 of @xmath30 dual to @xmath49 can be identified with an action of @xmath36 : @xmath50 . this co - action @xmath51 plays important roles in the reconstruction of quantum ( microscopic ) systems from the classical macroscopic data . we show that the above measurement coupling can be specified by means of a kac - takesaki operator @xcite ( k - t operator , for short ) , one of the central notions in harmonic analysis ( where it is called a fundamental operator in @xcite and a multiplicative unitary in @xcite ) . in what follows this operator is seen to play essential roles in our whole scheme to unify both measurement and amplification processes . in the regular representation of the group @xmath30 , a k - t operator @xmath52 is defined by @xmath53 for @xmath54 with @xmath55 the haar measure of @xmath30 , characterized by the pentagonal and intertwining relations : @xmath56 where the suffices @xmath57 indicate the places in the tensor product @xmath58 on which the operators act . the simplest form of the action @xmath49 , @xmath59 , of @xmath30 on @xmath21 is given by the adjoint action @xmath60 , as commonly found in many discussions on the measurement processes . this corresponds physically to such an approximation to the coupled dynamics of the composite system @xmath26 that the hamiltonian @xmath61 intrinsic to the observed system is neglected but the bilinear coupling @xmath62 is kept between the system observables @xmath63 ( @xmath64 ) and the external forces @xmath65 ( @xmath66 ) . to retain the effects of the dynamics intrinsic to the observed system , we take here a more general form of the action @xmath44 of the measuring system than the adjoint one under the assumption that @xmath49 is unitarily implemented , @xmath67 ( @xmath68 , @xmath69 ) , by a unitary representation @xmath70 of @xmath30 on the standard representation hilbert space @xmath71 of @xmath21 . then the representation @xmath72 of @xmath52 corresponding to @xmath73 is defined by@xmath74 satisfying the pentagonal and intertwining relations:@xmath75 the meaning of @xmath72 can be seen in the following heuristic expression in dirac s bra - ket notation : @xmath76 this unitary operator @xmath72 provides the coupling between the observed and measuring systems precisely required for measuring the observables in @xmath24 . for this purpose , we examine the action of its fourier transform on the state vectors of the composite system belonging to @xmath77 . first , in terms of the fourier transform @xmath78 for @xmath79 , the fourier transform @xmath80 of the k - t operator @xmath52 on @xmath81 is defined , which turns out just to be the k - t operator of the dual group @xmath36 ( equipped with the plancherel measure @xmath82 ) satisfying and characterized by the relations : @xmath83 similarly , the fourier transform of @xmath84 is defined by @xmath85 . owing to the snag theorem due to the abelianness of @xmath30 , its unitary representation @xmath86 admits the spectral decomposition @xmath87 , corresponding to which @xmath88 has the spectral decompostion given by@xmath89 in the dirac notation , the action of @xmath88 on @xmath90 is given for @xmath91 , @xmath92 , by@xmath93 to understand the physical meaning of the above quantities , we introduce some such vocabularies @xcite as ` probe system ' and ` neutral position ' in measurement processes : the former means the _ microscopic end _ of the measuring apparatus _ at its microscopic contact point _ with the observed system , and the latter the _ initial _ ( _ microscopic _ ) _ state of the probe system corresponding to the macroscopically stable position of the measuring pointer realized when the apparatus is isolated_. to see clearly the essence of the formulation , we assume that @xmath94 is discrete ( or , equivalently , @xmath30 is compact ) ; then we can plug into @xmath95 and @xmath92 in eq.([eqn : ft ] ) , respectively , the group identity @xmath96 and such an eigenstate @xmath97 as @xmath98 ( @xmath99 ) of @xmath40 , which gives@xmath100 namely , corresponding to the eigenstate @xmath101 of @xmath24 found in the observed system , the coupling unitary @xmath88 causes such a state change as @xmath102 in the probe system . for such a generic state as @xmath103 of the observed system , therefore , we obtain@xmath104 that is , the unitary operator @xmath88 creates from a decoupled state @xmath105 of @xmath26 a ` perfect correlation ' @xcite between states of the observed system and of the probe system , which is just required for transmitting the information from the observed system to the probe system . when the group @xmath30 is not compact with @xmath36 not being discrete , the identity element @xmath96 is not represented by a normalized vector , @xmath106 , but we can choose an invariant mean @xmath107 over @xmath30 owing to the amenability of the abelian group @xmath30 which plays the physically equivalent roles of the neutral position @xmath108 . as all what can be realized in this case is known @xcite to be the _ approximate measurements _ , the formula corresponding to eq.([eqn : ft2 ] ) can be given by eq . ( [ eqn : ft ] ) and by the use of @xmath107 as seen below in eq . ( [ instru ] ) . in this way the k - t operators are seen to fullfil the necessary tasks for materializing the physical essence of measurements in the mathematical formulation : the k - t operator @xmath72 determines the coupling between the observed and the measuring systems and its fourier transform @xmath88 given by eq.([eqn : ft ] ) establishes the ` perfect correlation ' @xcite . integrating all the ingredients relevant to our measurement scheme , we define an instrument @xmath109 as a completely positive operation - valued measure as follows : @xmath110 where @xmath111 s.t . @xmath112 is an initial state of the observed system , @xmath113 an arbitrary probability measure with respect to which the spectral measure @xmath114 of @xmath70 is absolutely continuous : @xmath115 , and @xmath116 the indicator function of a borel set @xmath117 to which the measured values of @xmath24 belongs . the spectral measure @xmath114 is just the _ effect _ of the measurement , from which our k - t operator @xmath88 can be reconstructed by @xmath118 . in this sense , the _ three notions _ , the k - t operator @xmath88 , the effect @xmath114 and the instrument @xmath119 , _ are all mutually equivalent_. the most important essence of the statistical interpretation in the measurement processes is summarized in this notion of instrument as follows : the probability distribution for measured values of observables in @xmath24 to be found in a borel set @xmath117 is given by @xmath120 and , associated with this , the initial state @xmath121 of the observed system is changed by the read - out of measured values in @xmath122 into a final state given in such a form as @xmath123 @xcite , according to which a process of the so - called ` reduction of wave packets ' is described . incidentally , the reason for the relevance of the _ fourier transform _ from @xmath84 to @xmath124 can naturally be understood in relation with the duality between the ( algebra of ) observables and the states : when the group @xmath30 acts on the algebra @xmath21 of the observed system , the corresponding states can be parametrized by @xmath36 as eigenstates w.r.t . the action @xmath70 of @xmath30 , which should also be read out as the measured values . by means of the instrument @xmath109 , a measurement process is described as the process of state changes due to the measurement coupling @xmath73 which transforms an _ initial _ state @xmath121 of the observed system _ decoupled _ from the probe system into _ final _ ones of the same nature , in parallel with the _ scattering processes _ described in terms of the _ incoming _ and _ outgoing _ asymptotic states of free particles . the algebra describing the composite system is the tensor algebra @xmath125 realized in the _ initial _ and _ final _ stages , respectively , before and after the measuring processes according to the _ switching - on _ and _ -off of the coupling _ @xmath73 . as incoming and outgoing asymptotic fields , @xmath126 and @xmath127 , in quantum field theory are interpolated by _ interacting heisenberg fields _ @xmath128 , we can consider a similar description of the composite system of @xmath21 and @xmath24 with the coupled dynamics @xmath49 incorporated at the level of the algebra which interpolates the initial and final decoupled system @xmath26 . this is given by the notion of the crossed products @xmath46 of the algebra due to the action @xmath49 of @xmath30 on @xmath21 , in terms of which the effect of the measuring coupling in the measurement process can be seen in such a form as @xmath129 , in parallel with the scattering processes , @xmath130 . in terms of the k - t operators , the crossed product @xmath131 as an important notion in the fourier - galois duality is defined on @xmath132 in the following two equivalent ways : either as a von neumann algebra @xmath133 generated by the fourier transform @xmath134 of @xmath21-valued @xmath135-functions @xmath136 with the convolution product , @xmath137 , mapped by @xmath138 into @xmath139 , or , as a von neumann algebra @xmath140 generated by @xmath141 and by @xmath142 these two versions are related by the mapping @xmath143 , @xmath144 which can be understood as the schrdinger and heisenberg pictures : the former @xmath145 is in the schrdinger picture with _ unchanged _ microscopic observables @xmath146 and with the _ coupling _ @xmath147 _ to change macroscopic states _ , while , in the latter , all the coupling effects are concentrated in the observables @xmath148 in contrast to the _ kinematical changes _ of macroscopic _ states _ caused by @xmath149 . in the case of the instrument , the effects of the measurement coupling @xmath88 are encoded in the form of _ macroscopic state _ _ changes _ recorded in the spectrum of the non - trivial centre @xmath150 of @xmath26 , playing the same roles as the order parameters to specify sectors in the inter - sectorial context . for these reasons , the most natural physical essence of the formalism in terms of an instrument @xmath109 can be found in the _ interaction picture _ , whose _ coupling _ term @xmath151 is responsible for deforming the decoupled algebra @xmath152 into the above crossed product @xmath153 . to clarify the natural meaning of the above scheme , we note a useful analogy of the duality coupling to the familiar _ complementarity of dna _ between a(denine ) and t(hymine ) and between g(uanine ) and c(ytosine ) , repectively : the role of the coupling between @xmath154 and @xmath155 in the k - t operator @xmath118 is just similar to that of the complementarity of a - t and g - c , as the former implements the _ transcription _ of the data @xmath156 in the object system to the probe system in the form of @xmath157 similarly to the latter case . at this point , we note that the above standard description of measurement processes in terms of an instrument implicitly presupposes that the _ quantum_-theoretical processes , @xmath158 and @xmath159 , taking place at the _ microscopic _ contact point of the observed and the probe systems can be directly interpreted as the measured data @xmath156 identifiable with a position of the measuring pointer visible at the _ macroscopic level_. there exist certain mathematical and/or physical gaps between these two levels which need be filled up : to adjust theoretical descriptions to the realistic experimental situations , we need to discuss how these changes of probe systems dynamically propagate into macroscopic motions of the measuring pointer . this is just the problem of the _ amplification _ processes to amplify the invisible quantum state changes in the probe system into the macroscopic data registered in some visible form of suitable order parameters . ( continuuing the above analogy to the dna , the aspect of amplification can naturally be compared with the process of _ pcr_[= polymer chain reaction ] to amplify the sequential data of dna . ) in the next section , we formulate its general and abstract essence in mathematical terms , by which the notion of the instrument need be supplemented . in view of the inevitable noises in the actual experiment situations , it is also necessary to show how the relevant information survives to reach the macroscopically visible level , which requires the estimates of the disturbance terms in the form of adiabaticity condition as will be done in [ example ] . we note here such a remarkable property inherent in the regular representation of @xmath36 as the mutual quasi - equivalence , @xmath160 ( @xmath161 ) , among its arbitrary tensor powers @xmath162 , as seen by the repeated use of the intertwining relation @xmath163 of the k - t operator @xmath164 : @xmath165 on this basis , we can formulate a dynamical process of amplification @xcite in terms of a unitary action @xmath166 of @xmath167 on the tensor algebra @xmath168 with @xmath169defined by @xmath170 which is similar to the formulation of quantum markov chain due to accardi @xcite . when @xmath36 is discrete , this process can be seen in a more clear - cut way in the schrdinger picture:@xmath171 , \end{aligned}\ ] ] where @xmath103 is a generic state @xmath92 of the observed system . according to the general basic idea of ` quantum - classical correspondence ' , a classical macroscopic object is to be identified with a _ condensed state of infinite number of quanta _ , as well exemplified by the macroscopic magnetization of ising or heisenberg ferromagnets described by the aligned states @xmath172 of ` infinite number ' @xmath173 of microscopic spins . likewise , the states @xmath174 and latexmath:[$|\gamma\rangle^{\otimes n}:=\underset { n}{\underbrace{|\gamma\rangle\otimes|\gamma\rangle\otimes\cdots\otimes interpreted as representing macroscopic positions of the measuring pointer corresponding , respectively , to the initial and final probe states parametrized by @xmath108 and @xmath32 . thus the above repeated action @xmath176 of the k - t operator @xmath164 describes a cascade process or a domino effect of ` decoherence ' , which , triggered by the initial data @xmath177 of the observed system , amplifies a probe state change @xmath178 of the measuring pointer . in view of the above aspects , we define a unified version of the instrument combined with the amplification process : @xmath179 in terms of which we can give an affirmative answer to the question posed at the end of the previous section , 2 , concerning the realistic meaning of the quantity @xmath122 as the actual data to be read out from the measuring pointer . to this end , we show the equality @xmath180 between the usual and the above instruments as follows : assuming the discreteness of @xmath36 for simplicity , we calculate for @xmath181 , @xmath182 which reduces for @xmath183 to such a familiar result as @xmath184 since @xmath185 gives precisely the probability of finding a _ macroscopic _ state @xmath186 , we have observed just the agreement of the probability distributions between the one arising from the microscopic system - probe coupling and the final result realized through the amplification process . this fact ensures the pertinence of instruments for the description of measurements , giving a clear - cut version of quantum - classical correspondence . the unitarity of the above amplification process is guaranteed by the quasi - equivalence relations among arbitrary tensor powers @xmath187 of the regular representation @xmath188 of @xmath36 . it can also explain the possibility of the recurrent quantum interference even after the contact of a quantum system with the measuring apparatus when the number @xmath189 of repetition need not be regarded as a real infinity . this point is evident from eq.([ldp ] ) which is valid independently of @xmath190 . in general , the problem as to whether the situation is made ` completely ' classical or not depends highly on the relative configurations among many large or small numbers , which can consistently be described in the framework of the non - standard analysis ( see , for instance , @xcite ) . in close relation to this , it is also interesting to note that the above amplification process is related to a lvy process through its ` infinite divisibility ' as follows : similarly to the affine property @xmath191 ( @xmath192 ) of a map @xmath193 defined on a convex set following from the additivity @xmath194 , we can extrapolate the relation @xmath195 ( @xmath196 ) into @xmath197 , which means the infinite divisibility @xmath198 ( @xmath199 ) of the process induced by the above transformation . in this way , we see that simple individual measurements with definite measured values are connected without gaps with discrete and/or continuous repetitions of measurements @xcite . if this formulation exhausts the essence of the problem , the remaining tasks reduce to its physical and/or technical implementation through suitable choices of the media connecting the microscopic contact point between the system and the apparatus to the measuring pointer . in such contexts , we need to examine some aspects concerning the stability of the information transmitted from microscopic to macroscopic levels , as will be seen in the next section . in this section we apply the scheme developed so far to the experimental situation of stern - gerlach type to check the validity of its general essence and to attain a deeper understanding of it through the concrete example . we will find also the necessity of some generalization or modification for adapting the scheme to actual situations . the essence of stern - gerlach experiments @xcite can be found in the coupling between the ( spin and/or orbital ) angular momentum of the quantum particles ( such as atoms or electrons ) and the inhomogeneous external magnetic field , according to which the _ microscopic _ differences in the _ _ quantized directions of angular momentum are amplified into the _ macroscopic _ distance of the arriving points of the particle . for simplicity , we consider here the spin @xmath200 @xmath201 of an electron ( with spin @xmath202 ) , whose associated magnetic moment @xmath203@xmath200 couples to the magnetic field via the interaction term @xmath203@xmath204@xmath205 : through the @xmath206-dependence of @xmath207 due to its inhomogeneity , this coupling causes the orbital change of the electron according to its spin direction ( _ up _ or _ down _ ) with respect to the defined axis ( see fig.[setting ] in [ s - g ] ) . thus the magnetic field @xmath207 is seen to play a double role ; the coupling @xmath208@xmath200@xmath209 causes , on the one hand , the _ spectral decomposition _ of the quantum spin @xmath200 , and it causes , on the other hand , the _ amplification process _ through its dependence on @xmath206 . through the process , we can ` see ' the quantum spin variable of the electron as the separation of its spatial orbits ( or , more directly , the arriving points on the screen ) . thus the two states @xmath210 and @xmath211 , respectively , of spin up and down , can be distinguished through the amplification process caused by the stern - gerlach measurement apparatus . in [ amplification ] the amplification process was formulated in its idealized abstract form in terms of the homogeneous repetition by a k - t operator . in the present case of stern - gerlach experiment , however , the coupling between the electron and the inhomogeneous magnetic field depends on the position of the moving electron , owing to which the unitary coupling term @xmath212@xmath200@xmath213 $ ] depends on the position @xmath206 of the electron along its trajectory . at the same time , any amplification processes can not get rid of _ noise effects _ to disturb the ideal separations between upward and downward electron beams corresponding to macroscopically distinguishable states @xmath214 and @xmath215 , respectively . for these reasons , it is necessary to examine whether the possible spin - flips during the travel of electron through the magnetic field can sufficiently be suppressed . otherwise , frequent spin - flips may destroy the meaningful connection between the spin variables of the electrons and the points on the screen to detect them . therefore , to ensure the distinguishability and the stability in the separations of final results , some physical conditions need be supplemented to ensure that these ` error probability ' is small enough . this can be understood as a kind of ` adiabaticity condition ' related with the validity of adiabatic approximation to treat the varying and fluctuating background field . the standard setting of the stern - gerlach experiment is shown below ( see fig.[setting ] to illustrate the apparatus ) ; we prepare a given type of metal which emits the electron beam through the thermal oscillation . the thermal electronic beam enters in the inhomogeneous magnetic field @xmath207 generated between magnetic poles which covers a spatial region with a length scale of the order of a meter . the orbital motion of each electron is bent upward or downward according to the directions of its spin coupled to the magnetic field ; the microscopic state determined by the direction of electron spin as an invisible _ internal _ degree of freedom is thus converted into the visible macroscopic form of spatial separations of the spots on the screen caused by the electrons . from here , we focus on the situation for detecting the spin direction consisting of an electron with spin @xmath202 , mass @xmath216 , charge @xmath217 , magnetic moment @xmath203(@xmath218 with magnetic permeability @xmath219 of vacuum ) and of the external magnetic field whose direction is supposed to be fixed in the @xmath220-axis . for applying our general scheme , we should proceed in the following steps : \0 ) to find the algebra which describes the physical system . \1 ) to extract the basic ingredients relevant to micro - macro duality ( masa , unitary group and their duals ) from the algebra found in 0 ) . \2 ) to identify the k - t operator in terms of these ingredients . [ [ to - find - the - algebra - which - describes - the - physical - system . ] ] 0 ) to find the algebra which describes the physical system . + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + the physical variables of the electron constitute the algebra @xmath221 consisting of the spin variables @xmath222 and the algebra @xmath223 of the canonical commutation relations ( a ccr algebra , for short , or , a heisenberg algebra ) generated , respectively , by pauli matrices @xmath224 and by the spatial coordinates @xmath225 and the momenta @xmath226 . according to the general framework in [ intro ] , we can take @xmath227 as the algebra describing the system to be observed ( as a von neumann algebra of type i ) . [ [ to - extract - basic - ingredients - relevant - to - micro - macro - duality . ] ] 1 ) to extract basic ingredients relevant to micro - macro duality . + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + we can find the masa as@xmath228 up to unitary conjugacy , where @xmath229 denotes the set of @xmath230 diagonal matrices @xmath231{cc}\alpha & 0\\ 0 & \beta \end{array } \right ) $ ] ( @xmath232 ) . this algebra is generated by the group @xmath233 of its unitary elements:@xmath234 the dual objects are also determined as follows : spectrum:@xmath235 dual group : @xmath236where @xmath237 consists of compactly supported @xmath238-valued step functions on @xmath239 , namely , each element @xmath240 takes a constant integer value @xmath241 on each @xmath242 of a finite number of non - intersecting borel sets @xmath243 in @xmath239 and vanishes outside of @xmath244 : @xmath245{c}c_{i}\text { for } x\in\delta_{i},\\ 0\text { otherwise . } \end{array } \right . $ ] we note that it is possible to extract the information on the spin degrees of freedom of the observed system from the spin algebra only , ignoring the orbital part described by the ccr . in this context , the relevant masa @xmath246 is just the _ cartan subalgebra _ of the lie algebra @xmath247 ( as is familiar in the theory of semi - simple lie algebras ) , where the spectrum @xmath248 can be identified with its root system . physically they correspond to the spin up / down states with respect to the @xmath220-axis . in contrast to @xmath25 having no identity element in itself , we can identify the unit element @xmath249 of the dual group @xmath250 as the _ neutral position _ of the measuring system , which can also be identified with the haar measure @xmath251 of @xmath252 or the constant function @xmath253 on @xmath252 . while this neutral position does not exist as a position of measuring pointer , operationally it represents a situation of _ no click _ on either of upper or lower detector . generic states of electron spin to be measured are represented by arbitrary superpositions @xmath254 @xmath255 of two eigenstates @xmath256 of @xmath257 . according to the result in [ measurement ] , the coefficient @xmath258 gives the transition amplitude from the above ` state ' of neutral position ( of the measuring pointer ) to either of the ` amplified ' macroscopic states @xmath259 and @xmath260 . [ [ to - identify - the - k - t - operator . ] ] 2 ) to identify the k - t operator . + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + our aim here is to understand the role of the coupling hamiltonian @xmath208@xmath200@xmath261 in relation with a k - t operator and its associated instrument . for this purpose , we consider a ( trivial ) vector bundle @xmath262 over a base space @xmath239 spanned by the electron coordinates @xmath206 with a fibre @xmath263 describing spin states of the electron at @xmath206 ; @xmath264 has group actions on its base space and its standard fibre , respectively , by the 3-dimensional motion group @xmath265 and by the spin rotations @xmath266 , where @xmath267 means the semi - direct product w.r.t . the adjoint action of @xmath268 on @xmath269 it is important here to note that @xmath264 is a homogeneous bundle over the homogeneous space @xmath270 , according to which a representation of @xmath271 can be induced from that of its subgroup @xmath272 . therefore , the geometry involved in the stern - gerlach experiment ( as an intra - sectorial version ) can be related to the measurement scheme @xcite for a _ sector bundle _ @xmath273 over @xmath274 consisting of the _ degenerate vacua _ associated to a spontaneous symmetry breaking of @xmath271 into an unbroken subgroup @xmath272 with the standard fibre @xmath275 describing the sector structure associated with @xmath272:@xmath276{ccc}\mathcal{m}^{h}\rtimes\widehat{g}\simeq\mathcal{m}\rtimes\widehat{(h\backslash g ) } & \longrightarrow & \begin{array } [ c]{c}\text{read - out data in } spec(\text{centre})\\ \text{(i)}=g / h\text { : degenerate vacua}\end{array } \\ \uparrow\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } & & \\\begin{array } [ c]{c}\hat{g}\curvearrowright\lbrack\mathcal{m}\rtimes h\simeq\mathcal{m}^{h}]\\ \text { : coupling ( i ) \ \ \ \ \ \ \ \ } \end{array } & \longrightarrow & \begin{array } [ c]{c}\text{read - out data in } spec(\text{centre})\\ \text{(ii)}=\hat{h}\text { : sectors on a vacuum}\end{array } \\ \text { \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \uparrow & & \\ \text{coupling ( ii ) : } h\curvearrowright\mathcal{m } & & \end{array } \right ] .\ ] ] the interpretation of each step of ( i ) and ( ii ) in this diagram is just in parallel with our measurement scheme : the unbroken subgroup @xmath272 acts on the algebra @xmath21 of observables of the system through the coupling ( ii ) , according to which the associated sector structure over a fixed vacuum can be read off ( ii ) in terms of @xmath275 realized as the spectrum of the centre of @xmath277 , and , similarly , the coupling ( i ) to implement the co - action of @xmath271 on the crossed product @xmath277 makes it possible to observe the sector structure ( i ) of the degenerate vacua parametrized by @xmath274 . from this viewpoint , the interaction hamiltonian @xmath200@xmath278@xmath257@xmath279{cc}\mu b_{z}(\mathbf{x } ) & 0\\ 0 & -\mu b_{z}(\mathbf{x } ) \end{array } \right ) $ ] can be interpreted as follows : the coupling term @xmath280@xmath257@xmath281 $ ] exhibits , via spectral decomposition , the ` sector ' structure @xmath282 parametrized by the roots @xmath283 of @xmath284 similarly to the above ( ii ) within a fibre . when we recall the @xmath206-dependence of @xmath285 , the aspects ( i ) of the degenerate vacua as condensed states shows up in relation with the base space @xmath286 . to see this , we consider such an approximation of the inhomogeneus magnetic field @xmath287 as @xmath288 this allows us to interpret the above coupling term @xmath289@xmath257@xmath281 $ ] ( for a time interval @xmath290 ) as another k - t operator relevant to ( i ) : @xmath291 & = \left ( \begin{array } [ c]{cc}\exp[\frac{i}{\hbar}\mu b_{z}(\mathbf{x})\delta t ] & 0\\ 0 & \exp[-\frac{i}{\hbar}\mu b_{z}(\mathbf{x})\delta t ] \end{array } \right ) \\ & \simeq e^{\frac{i}{\hbar}\sigma_{z}\mu b_{0}\delta t}\left ( \begin{array } [ c]{cc}e^{\frac{i}{\hbar}\mu b_{1}z\delta t } & 0\\ 0 & e^{-\frac{i}{\hbar}\mu b_{1}z\delta t}\end{array } \right ) , \end{aligned}\ ] ] which describes the ( co-)action of the @xmath220-axis @xmath292 on @xmath21 to generate @xmath293 ( an _ augmented algebra _ introduced in @xcite ) . to understand this , it is sufficient to note that the exponent @xmath294 of matrix elements @xmath295 in the above coupling unitary @xmath296 can be seen as the spectral value of the k - t operator @xmath297 corresponding to the ( generalized ) eigenvalue @xmath298 of the momentum operator @xmath299 : @xmath300 in the context of group representations , two representations of @xmath301 are induced from the two representations of @xmath284 corresponding to the eigenvalues @xmath282 , which are restricted to another subgroup @xmath239 and then to the @xmath220-axis @xmath302 , corresponding to ( approximately ) plane waves with @xmath303 , which reach the upper / lower detectors , respectively : @xmath304{ccc } & \text{\ \ \ \ } g=\mathbb{r}^{3}\underset{ad}{\rtimes}su(2 ) & \\ & & \\ & \text{\ \ \ \ \ } \text{(induction : ) } \nearrow \text{\ \ \ \ \ \ \ \ \ \ \ \ } \searrow\text { (: restriction ) } & \\ & & \\ & \sigma_{z}\longrightarrow\text{\ \ \ } g/\mathbb{r}^{3}=su(2)\underset { \text{helgason duality}}{\longleftrightarrow}g / h=\mathbb{r}^{3}\text{\ \ } \longrightarrow\hat{p}_{z}\text{\ . } & \end{array}\ ] ] in this way , the spin @xmath200 and the orbital motion described by @xmath305 are coupled by the inhomogeneity of the external magnetic field @xmath306 , according to which the microscopic directions @xmath282 of the former is _ amplified _ into the macroscopic directions @xmath303 in the orbital motion . these latter directions can be understood as the ` amplified ' states , @xmath307 and @xmath260 with the upper / lower points on the target screen . it is remarkable that the coupling unitary @xmath308 $ ] characteristic of the stern - gerlach experiment contains the two kinds of k - t operators , the one , @xmath309 , to couple the quantum observable @xmath200 with the angle variable @xmath310 and the other one , @xmath311 , corresponding to the translations @xmath312 of @xmath220 due to the @xmath220-dependence of @xmath306 , the latter of which is responsible for the direct amplification of the former coupling . this explains a dynamical mechanism to transcribe the information on the spin direction into the momentum change in the orbital motion of the electron , which allows us to achieve the quantitative estimation as shown above . aside from the stern - gerlach case , a unitary coupling of the similar nature has been found in @xcite . our focus here is , however , to clarify the universal essence of such couplings _ via external fields _ , which seems impossible without the use of k - t operators . in the above discussion for deriving the momentum change of the electron , we neglected such secondary effects as the terms come from @xmath313 or @xmath314 . since these effects are outside the scope of the above _ ideal _ situation of amplification , we need to estimate them as correction terms in the next step . without the necessity to develop the general method for treating these secondary terms , we already know some of typical methodology for these estimation ; in some cases ( including the stern - gerlach case ) it would be called ` adiabaticity conditions ' . for stern - gerlach experiment , this condition can be interpreted as the one under which the effect of _ spin - flips _ caused by the factor @xmath315 remains small enough compared with that of @xmath316 . in this section , we confirm that the adiabaticity condition surely gives the consistency in the present context by an elementary discussion . ` adiabatic perturbation ' originally means a coupling of a quantum system with an external force which changes the system slowly enough in comparison to the typical time scales of intrinsic transitions among quantum states but whose changes _ along the direction of condensed order parameters _ can eventually accumulate into a visible size . the general essence of the adiabaticity can be formulated in such a condition as @xmath317 , in terms of the rate of change of the matrix elements of hamiltonian @xmath272 defined by @xmath318 between the initial and final states with the energies @xmath319 and @xmath320 , respectively . the physical meaning of the quantity @xmath321 can be understood by the following reformulation of it : @xmath322 with @xmath323 which sets up the standard time scale for the comparison . the requirement @xmath317 can now be understood as the self - consistency condition for a process to change the values of the order parameters describing a given inter - sectorial structure of the quantum - classical composite system , without destroying the whole sector structure : if the change rate @xmath324 of the hamiltonian is very small , it should be almost perpendicular to the main ` tangential direction ' of the changes caused by the external force in favour of the change in the order parameters . therefore , @xmath290 can be interpreted as the ` almost intrinsic ' time scale of the microscopic motions of the intra - sectorial quantum system put in a background with slowly changing order parameters , in which @xmath325 can represent , for instance , the frequency of the light emitted in the transition . then the numerator in @xmath326 is the change @xmath327 of the matrix element of @xmath272 from the initial @xmath328 to final states @xmath193 _ caused by the adiabatic perturbation _ during the time interval @xmath290 , which is to be compared with the denominator @xmath329 given by the energy _ _ _ _ difference _ almost intrinsic to the quantum system_. going back to the stern - gerlach case , the interaction hamiltonian is given by @xmath330 the decomposition of the external magnetic field into its @xmath220-component @xmath316 and the remaining @xmath313 can be understood as the one into the directions to preserve and to disturb the sector structure according to the eigenvalues of @xmath257 . therefore , the dominant term in this hamiltonian to disturb the spin direction due to the spin - flips is identified with @xmath331 the size of the effect due to this term should be estimated to preserve the visibility aspect due to @xmath316 . as each trajectory of electron can be considered as a smooth curve in @xmath239 parameterized by the _ time _ parameter @xmath332 , the time derivative of @xmath333 is calculated as @xmath334 here we introduce an approximation @xmath335 . in terms of a basis of eigenstates of @xmath257 , we can estimate and obtain a representation of off - diagonal matrix elements@xmath336 under the assumption that the velocity @xmath337 of the electron can be replaced by the typical velocity @xmath338 of thermal electrons . owing to the first condition for @xmath339 to be adiabatic , the derivative of the external magnetic field can be approximated in the context of the estimate by @xmath340 , where @xmath341 represents the range in which the magnetic field exists . in terms of the larmor frequency of the thermal electron @xmath342 , the changing rate in which we are interested is essentially given by @xmath343 in the use of the rotation - free condition @xmath344 of the magnetic field @xmath345 . thus the adiabaticity condition @xmath317 can be written down as @xmath346 this inequality is nothing but the condition imposed on the arrangement of external magnetic field in order to guarantee the ideal amplification of spin variables . in this paper we have formulated a unified scheme of measurement and amplification processes based on the notion of micro - macro duality . in this context , the duality relation ( or , in more general contexts , adjunction ) between @xmath21 as the microscopic system and @xmath25 as the macroscopic observational data controlled by the k - t operator has played the essential role , on the basis of which we have obtained a clear understanding of how microscopic states are amplified into macroscopic level as discussed in [ amplification ] . we hope that this essence of amplification processes will shed some new lights on various problems involving different scales or levels ( especially , ` micro ' and ` macro ' ) such as the coexistence of different phases and their boundaries , the problem of emergence of macroscopic structures from microscopic worlds , and so on . one of the authors ( i. o. ) would like to express his sincere thanks to prof . m. ohya , prof . l. accardi and prof . t. hida for their encouragements . both of the authors are very grateful to mr . h. ando , mr . t. hasebe and mr . h. saigo for their valuable discussions in the early stage of the work . ojima , i. , a unified scheme for generalized sectors based on selection criteria order parameters of symmetries and of thermality and physical meanings of adjunctions , open systems and information dynamics , * 10 * ( 2003 ) , 235 - 279 . ojima , i. and takeori , m , how to observe quantum fields and recover them from observational data ? takesaki duality as a micro - macro duality , open systems and information dynamics , * 14 * , 307 - 318 ( 2007 ) ( math - ph/0604054 ) . tatsuuma , n. , a duality theory for locally compact groups , j. math . kyoto univ . * 6 * ( 1967 ) , 187 - 217 ; takesaki , m. , a characterization of group algebras as a converse of tannaka - stinespring - tatsuuma duality theorem , amer . j. math . * 91 * ( 1969 ) , 529 - 564 . the earlier version of the mathematical formulation of amplification processes proposed by one of the authors ( i.o . ) can be found in the following articles : ojima , i. , lvy process and innovation theory in the context of micro - macro duality , a brief summary of talks at the 5th lvy seminar ( 2006 ) , edited by t. hida ; ojima , i. , micro - macro duality and emergence of macroscopic levels , quantum probability and white noise analysis , * 21 * , 217 - 228 ( 2008 ) ( math - ph/07052945 ) . ojima , i. and tanaka , s. , state preparation , wave packet reduction and repeated measurements ( in japanese ) , part iii , chapater 2 , pp . 235 - 243 in _ quantum information and evolution dynamics _ , ed . by ohya , m. and ojima , i. , makino - shoten ( 1996 ) .

a unified scheme for quantum measurement processes is formulated on the basis of micro - macro duality as a mathematical expression of the general idea of _ quantum - classical correspondence_. in this formulation , we can naturally accommodate the amplification processes necessary for magnifying quantum state changes at the microscopic end of the probe system into the macroscopically visible motion of the measuring pointer . its essence is exemplified and examined in the concrete model of the stern - gerlach experiment for spin measurement , where the helgason duality controlling the radon transform is seen to play essential roles .

0810.3400

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 .

1108.1350

in general , observed astrophysical objects are characterized by a non - spherically symmetric distribution of mass and by rotation . in many cases , like ordinary planets and satellites , it is possible to neglect the deviations from spherical symmetry and the frame dragging effect , so that the gravitational field can be described by the exterior schwarzschild solution . in fact , the three classical tests of general relativity make use of the schwarzschild spacetime in order to describe gravity within the solar system@xcite . in the case of strong gravitational fields , however , the deviation from spherical symmetry and the rotation become important and must be taken into account , at least to some extent . the first metric describing the exterior field of a slowly rotating slightly deformed object was found by hartle and thorne@xcite in 1968 . alternative methods were proposed independently by fock and abdildin@xcite and sedrakyan and chubaryan@xcite . only recently , it was shown that in fact all these approaches are equivalent from a mathematical point of view@xcite . at the level of the interpretation of the parameters entering the metric used in each approach , certain differences can appear which could make a particular approach more suitable for the investigation of certain problems . for the purpose of the present work , it is convenient to use the hartle - thorne formalism which leads to an approximate metric describing , up to the first order in the quadrupole and the second order in the angular momentum , the exterior gravitational field of a rotating deformed object . we will use in this work the hartle - thorne metric in the form presented by bini et al.@xcite which in geometrical units is given by @xmath0dt^2 \nonumber\\ & & + \left(1-\frac{2{m}}{r}\right)^{-1}\left[1 - 2k_2p_2(\cos\theta)-2\left(1-\frac{2{m}}{r}\right)^{-1}\frac{j^{2}}{r^4}\right]dr^2 \nonumber\\ & & + r^2[1 - 2k_3p_2(\cos\theta)](d\theta^2+\sin^2\theta d\phi^2)-4\frac{j}{r}\sin^2\theta dt d\phi\ , \end{aligned}\ ] ] where @xmath1 here @xmath2 is legendre polynomial of the first kind , @xmath3 are the associated legendre functions of the second kind determined as @xmath4 , \nonumber\\ q_{2}^{2}(x)&=&(x^{2}-1)\left[\frac{3}{2}\ln\frac{x+1}{x-1}-\frac{3x^{3}-5x}{(x^{2}-1)^2}\right],\end{aligned}\ ] ] and the constants @xmath5 , @xmath6 and @xmath7 are the total mass , angular momentum and mass quadrupole moment of the rotating object , respectively ( for more details see refs . and ) . is related to the mass quadrupole moment defined by hartle and thorne@xcite through @xmath8 . ] the approximate kerr metric@xcite in boyer - lindquist coordinates can be obtained from the above hartle - thorne metric after setting @xmath9 and making a coordinate transformation implicitly given by @xmath10\ , \nonumber\\ \theta&=&\theta+\frac{a^2}{2r^2}\left(1+\frac{2m}{r}\right)\sin\theta\cos\theta.\end{aligned}\ ] ] the kerr metric is important to investigate the physical processes taking place around rotating black holes , i.e. , the source with probably the strongest possible gravitational field . the role of rotation is essential in the physics of accretion disks and energy extraction from a black hole . moreover , depending on the direction of the rotation , the radius of the accretion disk can be larger or smaller with respect to the schwarzschild case . the situation changes when one involves compact objects such as white dwarfs , neutron stars and quark stars as they have additional parameters to be taken into account . the combination of the strong field with the quadrupolar deformation of the source plays a pivotal role when one considers the motion of test particles . there exist many exact solutions that include a quadrupole parameter . the importance of the quadrupole moment in the astrophysical context has been emphasized in several works@xcite . however , most analysis must be performed numerically due to the complexity of the exact metrics . the advantage of considering the hartle - thorne approximate solution is that several physical quantities can be calculated analytically which facilitates their study . we will prove below that this is possible for a particular set of geodesics . in this work , we are interested in studying the motion of test particles in the hartle - thorne spacetime . therefore , we will perform both analytical and numerical analysis of the timelike and lightlike geodesic equations . in particular , we are interested in comparing the effects of the quadrupole and angular momentum parameters within the approximation allowed by the hartle - thorne metric . in this work , we will make use the timelike normalization condition @xmath11 which for equatorial circular geodesics is equivalent to @xmath12 sometimes the following convenient notations are used for the four - velocity of equatorial circular geodesics @xmath13 where @xmath14 is the normalization factor and @xmath15 is the orbital angular velocity . using the fact that the hartle - thorne solution possesses two killing vector fields @xmath16 and @xmath17 , which determine two constants of motion , from the geodesic equations for equatorial circular orbits ( @xmath18 and @xmath19 , we obtain @xmath20 where a comma indicates partial differentiation . then , a straightforward computation yields @xmath21 ^ 2-\left[\frac{d\phi(s)}{ds}\right]^2\right\}-\frac{2m^2(r-2m)}{r^3}\frac{dt(s)}{ds}\frac{d\phi(s)}{ds}j \nonumber\\ & & \qquad\qquad\qquad+\left\{a_{1}(r)\left[\frac{dt(s)}{ds}\right]^2+a_{2}(r)\left[\frac{d\phi(s)}{ds}\right]^2\right\}j^2 \nonumber\\ & & \qquad\qquad\qquad\qquad\qquad\qquad + \left\{a_{3}(r)\left[\frac{dt(s)}{ds}\right]^2+a_{4}(r)\left[\frac{d\phi(s)}{ds}\right]^2\right\}q = 0 , \end{aligned}\ ] ] where we introduced the dimensionless quantities @xmath22 and @xmath23 and new functions defined as @xmath24 this expansion in terms of the quadrupole and angular momentum parameters can be used to derive analytical expressions for the parameters that characterize the orbits of the test particles . starting from the @xmath26 component of the geodesic equation and using the fact that @xmath26 and @xmath27 are both constant on circular equatorial orbits , one easily derives the following expression for the angular velocity : @xmath28 performing an analysis similar to the one carried out in the previous section , we finally obtain @xmath29,\ ] ] where @xmath30^{-1}[48m^7 - 80m^6r+4m^5r^2 + 42m^4r^3\nonumber\\ & & -40m^3r^4 - 10m^2r^5 - 15mr^6 + 15r^7]-f(r ) , \nonumber\\ f_{3}(r)&=&-\frac{5(6m^4 - 8m^3r-2m^2r^2 - 3mr^3 + 3r^4)}{16m^2r(r-2m)}+f(r ) , \nonumber\\ f(r)&=&\frac{15(r^3 - 2m^3)}{32m^3}\ln\frac{r}{r-2m}.\end{aligned}\ ] ] this expression for the orbital angular velocity of a test particle at the equatorial plane can be used to determine the mass shedding limit of the source in general relativity@xcite . moreover , in x - ray astronomy , the orbital angular velocity is associated with the upper frequency of the quasiperiodic oscillations@xcite . the analytic expression obtained here leads to results that are in agreement with those obtained by using pure numerical methods . in the case of lightlike geodesics , we can use the expression for the norm of the corresponding 4-velocity to calculate its components . evaluating @xmath32 and @xmath33 directly from the hartle - thorne line element , we then obtain for the orbital angular velocity : @xmath34,\ ] ] where @xmath35 the specific energy per unit mass @xmath37 is usually used to estimate the radius of marginally ( innermost ) bound orbits of test particles . thus , this radius determines the stable region which is essential for the formation of accretion disks . evaluating @xmath38 from the line element , one obtains @xmath39 @xmath40,\ ] ] where @xmath41^{-1}[144m^8 - 144m^7r-28m^6r^2 + 122m^5r^3\nonumber\\ & & + 184m^4r^4 - 685m^3r^5 + 610m^2r^6 - 225mr^7 + 30r^8]+h(r ) , \nonumber \\ h_{3}(r)&=&\frac{5(r - m)(6m^3 + 20m^2r-21mr^2 + 6r^3)}{16mr(r-2m)(r-3m)}-h(r ) , \nonumber\\ h(r)&=&\frac{15r(8m^2 - 7mr+2r^2)}{32m^2(r-3m)}\ln\frac{r}{r-2m}.\end{aligned}\ ] ] as we will see below , the specific angular momentum for test particles per unit energy @xmath43 is crucial for the determination of the marginally ( innermost ) stable orbits of test particles forming accretion disks . calculating @xmath44 and @xmath38 , we obtain the following analytic expression for @xmath43 @xmath45 @xmath46,\ ] ] where @xmath47^{-1}[96m^8 - 112m^7r-8m^6r^2 + 72m^5r^3\nonumber \\ & & -18m^4r^4 - 220m^3r^5 + 260m^2r^6 - 105mr^7 + 15r^8]-g(r ) , \nonumber \\ g_{3}(r)&=&\frac{5(6m^4 - 22m^2r^2 + 15mr^3 - 3r^4)}{16m^2 r(r-2m)}+g(r ) , \nonumber \\ g(r)&=&\frac{15(2m^3 + 4m^2r-4mr^2+r^3)}{32m^3}\ln\frac{r}{r-2m}.\end{aligned}\ ] ] the normalization condition @xmath48 gives the photon orbit , @xmath49 , where @xmath50 is the photon four - momentum and for circular orbits @xmath51 . indeed the normalization condition @xmath52 gives the orbital angular velocity for the photon @xmath53 , but @xmath54 remains arbitrary . to determine the photon orbit , @xmath49 , first one should use the above expression for @xmath53 and then evaluate the four - acceleration @xmath55 . for a circular geodesic @xmath56 , and only from this condition one can determine @xmath49 . note , alternatively it is also convenient to use the condition @xmath57 to find @xmath49 . moreover , in order to determine the radius of the innermost ( marginally ) bound circular orbits @xmath58 one should use the condition @xmath59 . in addition , the condition @xmath60 allows one to find the radius of the innermost stable circular orbits , @xmath61 . here we used the methods of perturbation theory and the results of these calculations are : @xmath62,\\ r_{mb}&=&4m\left[1\mp\frac{1}{2}j+\left(\frac{8033}{256}-45\ln2\right)j^2+\left(-\frac{1005}{32}+45\ln2\right)q\right],\\ r_{ms}&=&6m\left[1\pm\frac{2}{3}\sqrt{\frac{2}{3}}j+\left(-\frac{251903}{2592}+240\ln\frac{3}{2}\right)j^2+\left(\frac{9325}{96}-240\ln\frac{3}{2}\right)q\right ] . \qquad\end{aligned}\ ] ] finally , we mention that using the hartle - thorne line element it is possible to derive the radial and vertical fundamental frequencies.@xcite the application of these frequencies to the observed quasiperiodic oscillations from the low - mass x - ray binaries has been considered on the basis of the relativistic precession model.@xcite all the epicyclic frequencies have been derived in previous works.@xcite with the method proposed in this work we obtained equivalent results after applying the redefinition @xmath63 or @xmath64 . it is convenient to investigate the motion of test particles numerically in the hartle - thorne spacetime as the full set of equations is cumbersome even for the equatorial plane . we select different values for the parameters of the source and initial conditions for test particles to consider all types of trajectories . the results of the numerical integration of timelike geodesics are shown in figs . [ plotq]-[ploth2 ] . [ cols= " < , > " , ] in fig . [ plotq ] ( left panel ) , we analyze the motion of a test particle in the field of a static and deformed source . as one can see , the quadrupole parameter generates different deviations from the schwarzschild spacetime , depending on its value and sign . the frame dragging effect is illustrated in fig . [ plotq ] ( right panel ) for a spherically symmetric source . the drag strengthens as the test particles approach the source . the influence of the frame dragging effect on the circular motion has considered in ref . for one revolution of the test particle . by analyzing the behavior of the test particles with the certain initial conditions , it is possible to select the values of @xmath65 and @xmath66 in order to recover circular geodesics , i.e. , the effects caused by the deformation of the source can be balanced by its rotation and vice - versa ( see ref . ) . unbound orbits are shown in fig [ ploth1 ] ( left panel ) , where we change only the initial angular component of the velocity @xmath67 . the remaining quantities are fixed for the sake of comparison . for different values of @xmath66 , we obtain the plot of fig . [ ploth1 ] ( right panel ) , whereas in fig . [ ploth2 ] ( left panel ) we present the results for different values of @xmath65 . due to this interplay between the initial conditions of the test particles and the parameters of the source , one can construct all kind of geodesics . an example of a double loop trajectory , which is missing in classical physics , is shown in fig . [ ploth2 ] ( right panel ) . from here we conclude that the parameters of the geodesic motion can be used to determine the main parameters of the source such as @xmath68 , @xmath65 and @xmath66 . in this work , we have explored geodesics in the hartle - thorne spacetime both analytically and numerically . we considered the geodesics on the equatorial plane and investigated the role of the quadrupole parameter , as well as the frame dragging effect on the motion of test particles . we investigated bounded and unbounded orbits varying the initial conditions of the test particles and the main parameters of the source . we conclude that using different combinations of both initial conditions and main parameters , one can generate many different geodesic curves . in all our computations we used the methods of perturbation theory . our results have the same order of approximation as the harte - thorne solution . namely , we derived the expressions for the orbital angular velocity , energy , orbital angular momentum for test particles and orbital angular velocity for photons . in turn , with the help of these expressions we obtained the radii of the innermost bound , innermost stable and photon orbits . all the analytic expressions obtained here and in refs . and are in agreement with the results of refs . and , if one redefines @xmath64 . we briefly discussed some applications of our theoretical results in the astrophysical context . in fact , the results presented in this paper can be applied to study the physics of accretion disks , the motion of test particles near the source and the epicyclic frequencies ; all these aspects are of high relevance and importance in relativistic astrophysics and x - ray astronomy . for instance , using epicyclic frequencies and quasiperiodic oscillation data , one can test the strong field regime of general relativity , determine the parameters of the gravitational source and test the equations of state of compact objects . in a future work , we expect to apply the analytic expressions we obtained in the present work in the context of observational astrophysics . furthermore , the investigation of the stability of the geodesics and the structure of the accretion disks are crucial to understand the physical properties of the hartle - thorne spacetime . we expect to perform such an analysis in a future work by applying the procedure shown , for instance , in ref . . this work was supported by the ministry of education and science of the republic of kazakhstan grants no . 3101/gf4 ipc-11/2015 and no . 1597/gf3 ipc-30 .

we investigate equatorial geodesics in the gravitational field of a rotating and deformed source described by the approximate hartle - thorne metric . in the case of massive particles , we derive within the same approximation analytic expressions for the orbital angular velocity , the specific angular momentum and energy , and the radii of marginally stable and marginally bound circular orbits . moreover , we calculate the orbital angular velocity and the radius of lightlike circular geodesics . we study numerically the frame dragging effect and the influence of the quadrupolar deformation of the source on the motion of test particles . we show that the effects originating from the rotation can be balanced by the effects due to the oblateness of the source .

1510.02016

lens - source relative proper motions @xmath5 are frequently measured in planetary microlensing events , but to date there are no published measurements of the source proper motion itself in these events ( @xmath6 ) . this may seem surprising at first sight because the source is almost always visible whereas the lens is typically invisible . in fact , however , @xmath7 is generally both more useful and easier to measure than @xmath6 . source - lens proper motions can be measured essentially whenever there are significant finite - source effects in the event @xcite because the source - lens crossing time @xmath8 is directly measurable from the light curve , while the angular size of the source can be extracted from its dereddened color and magnitude @xcite , which in turn can be extracted by placing the source on an instrumental color - magnitude diagram @xcite . the most important application of @xmath7 is not the proper - motion itself , but rather that it immediately yields the einstein radius , @xmath9 where @xmath10 is the einstein timescale ( measurable from the event ) , @xmath11 is the lens mass , and @xmath12 is the lens - source relative parallax . therefore , @xmath13 usefully constrains a combination of the lens mass and distance . however , @xmath7 does often play a role at the next level . because @xmath11 and @xmath14 are not determined independently , one normally must make a bayesian estimate of these quantities , using inputs from a galactic model @xcite , which can include priors on @xmath7 . in principle , the bayesian analysis could also include priors on @xmath6 if this quantity were measured . there are two reasons why this has not been done yet . first , in many cases , the posterior probabilities would not be strongly impacted by this additional prior . second , and probably more important , it is remarkably difficult to measure @xmath6 in most cases . here we present a new method to measure @xmath6 , which is tailored to meet the challenges of the faint , moderately blended sources typical of microlensing events seen toward the galactic bulge . we are motivated to develop this method by , and specifically apply it to , the planetary microlensing event moa-2011-blg-262/ogle-2011-blg-0703 . this event has a short timescale @xmath0days , a very high ( or high ) relative proper motion @xmath15 ( or @xmath16 for the competing , nearly equally likely , microlensing model ) , and a companion / host mass ratio @xmath1 @xcite . these parameters are , in themselves , consistent with either a lens that contains a stellar host with a jovian - class planet in the galactic bulge , or a brown dwarf ( or possibly a jovian planet ) with an earth - class `` moon '' . in the former case ( stellar host in the bulge ) , the very high @xmath7 that is measured in the microlensing event would almost certainly require combinations of abnormally high lens and source proper motions . that is , if the source were moving slowly , it would be quite unusual for the lens proper motion to be large enough to account for the high relative proper motion by itself . by contrast , if the lens were in the foreground disk ( and so of much lower mass ) , its proper motion relative to the frame of the galactic bulge could easily be high enough to explain the observed @xmath7 . thus , for this event , it would be important to actually measure @xmath6 . the optical gravitational lensing experiment ( ogle ) is a long - term photometric sky survey focused on finding and characterizing microlensing events in the galaxy . the first phase of the ogle project began in april 1992 and the project continues to this date with its fourth phase currently being executed @xcite . ogle monitors brightness of hundreds of millions of stars toward the galactic bulge with high cadence , using dedicated 1.3 m warsaw telescope at las campanas observatory , chile . every night between 100 and 200 1.4 deg@xmath17 exposures are being taken . in addition to the performed real - time reduction and analysis , all the science frames are archived in their full integrity . these constitute an unprecedented data set for various astronomical studies . the decades - long time span of the ogle monitoring provides us with a unique opportunity to precisely measure proper motions of many stars in the galactic bulge , including the source star of this very interesting microlensing event . proper motion studies of the galactic bulge have been previously carried out using the ogle data . for example , @xcite measured proper motions for over @xmath18 stars over 11 deg@xmath17 using the ogle - ii data from 1997 to 2000 . however , this survey was restricted to resolved stars , @xmath19 . in the present case , the source magnitude is @xmath20 ( as determined from the microlens models ) , which would be close to the photometric detection limit even if the source was relatively isolated . in fact , the source is blended with a brighter star , and was not recognized as an independent star in the reference image , prior to the event . hence , a new technique is required to measure the proper motion , which is described in the next section . consider a difference of two images that has been generated by standard difference image analysis ( dia,@xcite ) . that is , the images have been geometrically aligned to a common frame of reference stars , photometrically aligned to the same mean flux level , and one has been convolved with a kernel function to mimic the point spread function ( psf ) of the other . the usual purpose of this procedure is to detect stars whose flux has changed between the two epochs . these will appear as relatively isolated psfs on an otherwise flat background ( beside the noise , cosmic rays , masked regions , satellites , etc . ) . however , let us now consider the case that there have been no flux changes but only position changes . for simplicity , we begin by assuming that the psf is an isotropic gaussian @xmath21 where @xmath22 is the gaussian width . let us now assume that one star has been displaced by a distance @xmath23 in the direction @xmath24 , while all other stars have remained fixed . if we further assume that @xmath25 , then it is straightforward to show that the difference profile will have the form @xmath26 where @xmath27 and @xmath28 is the flux of the star . thus , the difference image will have an anti - symmetric dipole profile , whose form is always the same ( @xmath29 ) . the amplitude of this dipole is given by the product @xmath30 , and the direction is simply the direction of motion . note that the maximum of the dipole profile lies at @xmath31 , i.e. , @xmath32 from the star center . moreover its height relative to the peak of the original star is @xmath33 thus , if several stars have moved , then the difference image will contain several dipoles , all with the same form , but with different amplitudes and pointing in different directions . see figure [ fig : dia ] . in principle , then , it is possible to determine the proper motion of a star by measuring the height of the dipole relative to the original image and applying equation ( [ eqn : max ] ) . in practice , the utility of this approach is limited to fairly restricted conditions . first , if the star is truly isolated , it is usually more convenient to simply measure the star at many epochs in the usual fashion . on the other hand , if the star is heavily blended , it will still produce a dipole as given by equation ( [ eqn : dipole ] ) but there will be two problems . first , the star s flux @xmath28 can not usually be disentangled from that of other stars within the psf . hence , when the dipole amplitude ( @xmath30 ) is measured , one can not extract @xmath23 from it . second , other stars within the psf may also have moved , each in its own direction @xmath34 and each with own amplitude @xmath35 . when several stars are in the same psf , all that can be observed is the vector sum of these dipoles : @xmath36 . now , for microlensed sources , the first of these problems is actually easily solved because the microlensing fit returns the source flux @xmath37 as one of its parameters . however , the second problem remains . this means that the technique is actually applicable only to moderately blended sources in which the blending can be reasonably well understood . before undertaking such an application , we note that the above formalism is easily extended to asymmetric gaussian profiles,@xmath38 where @xmath39 is the inverse covariance matrix , @xmath40 is now the 2-dimensional coordinate , and where we use einstein summation convention . then the dipole is given by @xmath41 of course , one might also consider non - gaussian profiles , but since almost all the signal comes from within a few @xmath42 , while the deviations from a gaussian are not understood well enough to characterize @xmath43 at sub - pixel scales , this level of complexity is generally not warranted . the difference image in dia can be constructed either by taking the second epoch image as an `` image '' and the first epoch image as a `` reference image '' or vice versa . in the latter case , the measured dipole direction should be flipped to obtain the real direction of the proper motion . since the `` reference image '' will be convolved with a kernel to match its psf to the psf of the `` image '' , one should choose the lower - seeing image out of the two for the reference image . before the dia procedure can be performed , the images should be aligned to a common pixel grid . this is done by picking a set of bright common stars in both images , finding polynomial or spline interpolation of the coordinates and resampling one image to the other image s pixel grid @xcite . this step ensures that both images cover the same region of the sky modulo the quality of the transformation . the differences should be small compared to the fitting regions of the dia kernel ( domains ) . other advantage this procedure brings is compensation of the majority of the differences of field distortion between the two epochs . thus , the dia kernel can have less free parameters to achieve similar quality of subtraction . it is extremely important to understand what degrees of freedom the dia kernel has before any conclusions about the proper motion can be made . one example would be a kernel transforming a gaussian psf into a gaussian psf with the same centroid . if the images were not carefully aligned to remove all of the field of view deformations the subtraction could turn out poor . by introduction of the additional parameters to the mathematical model of the psf , describing smooth shifts of its center across the field , subtraction would be greatly improved . @xcite dia uses a series of gaussians multiplied by polynomials for the mathematical model of the psf . the difference of two gaussian profiles displaced by a small distance @xmath44 ( @xmath45 ) is equal to the gaussian multiplied by the first order polynomial ( _ cf . _ ( [ eqn : dipole ] ) ) . hence , if we allow a psf model to be constructed from a gaussian multiplied by the first order polynomial , with parameters being a function of position on the image , we effectively allow for the smooth profile centers displacements ( by some non negligible fraction of the profile width ) . this approach allows for compensation of the field distortions , but at the same time , it potentially smoothes the gradients in the stars motions across the field . let us consider a globular cluster covering some fraction of the field in front of the galactic background . stars in the cluster would have different expected median velocities than the background stars . if , in a region of the frame , the percentage of cluster members is significant , their motion between two epochs could be seen as a field deformation near this region . hence , careful initial alignment of the images should only use the field stars . we also note , that the dia is specifically designed to reduce signal on the subtracted image within the freedom of the psf model . thus , without care in setting up the dia procedure , it is very likely , that the kernel parameters will try to compensate for the effective bulk motion of the part of the field containing cluster members effectively reducing observed dipole signal for the cluster members and introducing false signal in the field stars . in the field of view studied in this paper we do not expect any unusual velocity gradients . however , we expect that the bright stars sample would consist of a mixture of bulge stars and disk stars . disk stars are expected to have on average proper motions of couple of @xmath46 in the direction of galactic rotation , with respect to the bulge stars . for a reference point of our proper motion measurement we choose the median proper motion of the red clump ( rc ) giants , which are identified on the color - magnitude diagram ( cmd ) . we use the rc giant stars for two reasons . first , they belong to the bulge system so their median proper motion is not affected by the galactic rotation and can serve as an approximation of the median motion of the whole bulge system . second , the red clump giants are bright , abundant and easy to identify on the cmd . we initially align two epoch images using the predefined sample of rc giants . after performing the dia , we measure the shift of centroids of those stars that is likely introduced by the procedure . we take the `` reference '' epoch and measure the positions of the reference stars . then , we take `` convolved reference image '' ( it is produced by the dia in order to match the reference image to the target image before the subtraction ) and measure the positions of the same sample of stars . the median shift between the measured positions indicates the total displacement the dia procedure introduced , and hence , should be subtracted from any dipole measurements to form the final proper motion measurement . we use a series of images obtained with the warsaw telescope during the third and fourth phase of the ogle project . we select eight best seeing images near the beginning of the ogle - iii ( mean epoch @xmath47 @xmath48 @xmath49 ) and 11 of the best seeing ( unmagnified ) images from ogle - iv ( mean epoch @xmath47 @xmath50 ) . note that ogle - iii and ogle - iv have identical pixel scales : 259 mas px@xmath51 . we stack each set of images . these stacked images , which have similar , but not identical seeing fwhm @xmath52px , are offset by 7.65 years in mean epoch . they are presented in figure [ fig : images ] . as shown in figure [ fig : images ] , the source of this event is moderately blended with a brighter neighboring star even in the excellent - seeing images used to construct the two stacks . in fact , there is one image with extremely good seeing on which the source appears isolated , but the single - to - noise ratio ( s / n ) of a single image is too low to perform accurate astrometry or photometry . instead , we find the microlensing source position from difference images at the time of the magnification event ( as is standard practice ) , when it can be easily observed ( _ cf . _ fig [ fig : images ] ) . then , we determine the source flux ( @xmath37 ) from the microlens fit and subtract the resulting source profile from the second - epoch stacked image in order to investigate the structure of the blending star(s ) alone @xcite namely , measure positions and brightness . there is no detectable remaining flux at the source position . we thereby place an upper limit on the blend flux at the source position of @xmath53 , implying @xmath54 . for stars in the bulge , this implies absolute magnitude to be @xmath55 ( where we take the reddening @xmath56 ) . in particular , this allows only for bulge lenses with @xmath57 . using equation ( [ eqn : thetae ] ) , the measured @xmath58 ( @xmath59 ) for the very - high @xmath7 ( high @xmath7 ) model implies that the allowed mass range corresponds to lens - source separations @xmath60 ( @xmath61 ) . in the case of the former ( very - high @xmath7 ) model the substantial fraction of the available phase space is ruled out , thus putting significant constraints upon ( but certainly not ruling out ) a bulge lens . in addition , this flux limit restricts the presence of distant hosts in `` free - floating - planet '' scenarios . for example , if the primary lens is assumed to be a 10-jupiter - mass object , then it would have a relative parallax @xmath62mas , and therefore be at a distance @xmath63kpc . the flux limit would then imply @xmath64 , i.e. , @xmath65 for the putative host . however , our central concern here is to measure or place limits on the proper motion of the source , @xmath6 . we notice that there is no evident dipole at the position of the source in the difference image in figure [ fig : dia ] . to place limits , we must fit for a dipole at this location . because the source is partially blended with the neighboring star , we must fit simultaneously for two dipoles , one at each location . in fact , to be conservative , for each ( 2-d ) trial value of the source dipole , we consider all possible values for the neighbor dipole , and choose the one that gives the best overall fit . we assign the resulting @xmath66 to this trial value . each dipole amplitude @xmath67 is divided by the flux ( known from the microlens fit ) to obtain the displacement @xmath68 , and hence the proper motion @xmath69 , where @xmath70yr is the time elapsed between the two mean epochs . to model the psf we use equations ( [ eqn : psf_asym ] ) and ( [ eqn : dipole_asym ] ) with @xmath71 , as measured with the dophot photometry package @xcite , with the @xmath44-axis aligned to that of the ogle - iii camera , i.e. , equatorial north - south . the resulting contours are shown in figure [ fig : pm ] . origin in figure [ fig : pm ] was adjusted for the shift that was introduced by the dia ( see section [ sec : frame ] ) so it is now consistent with the median proper motion of the bugle stars . in our @xmath72 work subfield we have identified 511 rc giants . the median shift of this set of bulge stars was measured between `` reference images '' and `` convolved reference image '' to be ( 0.0095 , 0.0174 ) pixels in ( north , east ) direction . this is the equivalent to ( -0.336 , 0.583 ) @xmath46 proper motion in the galactic north and east direction . to test this shift - correction procedure we artificially resample target image by introducing simple shifts with values between @xmath73 and @xmath74 pixels in both axes . the resulting subtracted image is virtually the same and yields same measurements of the proper motion of the source star , showing that the dia kernel easily absorbed the artificial shift . the value of the displacement measured by the shift - correction procedure , recovers both , the original values quoted above , plus any artificial shift we have introduced . we note that the contours shown in figure [ fig : pm ] are elongated along an axis that is approximately aligned with galactic north - south . this is because the neighboring star lies along the galactic east - west axis . we can understand this analytically by approximating the psf as axisymmetric . without loss of generality , we can then assume that the two stars are separated by a distance @xmath75 along the @xmath44-axis . their dipoles can then be represented ( in the global coordinate system ) by @xmath76 \exp(-[-(x\pm a/2)^2 + y^2]/2\sigma^2)\over 2\pi\sigma^4 } \label{eqn : dip_pm}\ ] ] where @xmath77 are the orientations of the two dipoles relative to the direction of their separation . one then finds @xmath78 and @xmath79\cos\phi_+\cos\phi_- + \sin\phi_+\sin\phi_- \over \exp(a^2/4\sigma^2 ) } \label{eqn : diracrat}\ ] ] it is straightforward to show that for uniform noise per pixel ( background - limited case ) the ratio of the correlated errors to the error estimates that would hold for an isolated source are @xmath80 then , for each possible orientation of the source dipole @xmath81 , we must choose the orientation of the neighbor - dipole that maximizes equation ( [ eqn : diracrat ] ) [ and so eq . ( [ eqn : corerr ] ) ] . differentiation yields , @xmath82\cot\phi_+$ ] , from which one obtains , @xmath83e^{-(1+k)}\biggr\}^{-1/2 } , \qquad k\equiv { a^2\over 2\sigma^2 } - 1 . \label{eqn : errcont}\ ] ] it is clear from equation ( [ eqn : errcont ] ) that the error contour will be aligned either parallel or perpendicular to the direction of the neighbor , depending on whether @xmath84 is larger or smaller than unity . that is , @xmath85^{-2 } = 1 - k^2\exp[-(1+k)]$ ] and @xmath86^{-2 } = 1 -\exp[-(1+k)]$ ] . note that the contour will not be perfectly elliptical but will generally approximate an ellipse . in the present case @xmath87 , so @xmath88 and the contour is aligned perpendicular to the direction of the neighbor . however , since @xmath89\sim 0.36 $ ] , the overall deviation from circular symmetry is only about 14% in this case . we measured the proper motion of the source star in the microlensing event moa-2011-blg-262 using new dipole - fitting method performed on the difference image of two epochs separated by @xmath90 yrs . the result is @xmath91 in a ( north , east ) galactic coordinate frame . the new method yields a measurement of proper motion of the star that is close to the photometric detection limit . in fact , the subject star was not even discovered in the initial photometry of the photometric reference image for the field , due to being faint and close to another star . the star s exact position and brightness was measured during the microlensing event . additionally , this new approach allows to marginalize over an unknown motion of the partially blended neighboring star . the importance of this measurement for our particular microlensing event lies mainly in the fact that the obtained confidence regions exclude the source being a high velocity star and show that it follows the typical bulge kinematics . it is compatible with both microlensing models and only increases _ a priori _ lensing probability of the very - high @xmath92 microlensing model ( @xmath93 ) by a factor of 1.5 when compared to the high @xmath92 microlensing model ( @xmath94 ) . however , more importantly , it heavily disfavors lens - in - the - bulge scenario for the former and moderately disfavors it for the later . the stellar lens with the jupiter - mass planetary companion located in the galactic bulge is , _ a priori _ , much more likely explanation for the moa-2011-blg-262 microlensing event , than the close - by jupiter - mass planet with the earth - mass moon in the galactic disk . by reducing the likelihood of the lens being in the bulge , our measurement brings those two explanations on par . acknowledge support of the space exploration research fund of the ohio state university . the ogle project has received funding from the european research council under the european community s seventh framework programme ( fp7/20072013)/erc grant agreement no . 246678 to a.u .

we develop a new method to measure source proper motions in microlensing events , which can partially overcome problems due to blending . it takes advantage of the fact that the source position is known precisely from the microlensing event itself . we apply this method to the event moa-2011-blg-262 , which has a short timescale @xmath0day , a companion mass ratio @xmath1 and a very high or high lens - source relative proper motion @xmath2 or @xmath3 ( for two possible models ) . these three characteristics imply that the lens could be a brown dwarf or a massive planet with a roughly earth - mass `` moon '' . the probability of such an interpretation would be greatly increased if it could be shown that the high lens - source relative proper motion was primarily due to the lens rather than the source . based on the long - term monitoring data of the galactic bulge from the optical gravitational lensing experiment ( ogle ) , we measure the source proper motion that is small , @xmath4 in a ( north , east ) galactic coordinate frame . these values are then important input into a bayesian analysis of the event presented in a companion paper by bennett et al .

1312.7297

there has been an increasing interest over the last decade in performing large - scale simulations of colloidal systems , proteins , micelles and other biological assemblies . simulating such systems , and the phenomena that take place in them , typically requires a description of dynamical events that occur over a wide range of time scales . nearly all simulations of such systems to date are based on following the microscopic time evolution of the system by integration of the classical equations of motion . usually , due to the complexity of intermolecular interactions , this integration is carried out in a step - by - step numerical fashion producing a time ordered set of phase - space points ( a _ trajectory _ ) . this information can then be used to calculate thermodynamic properties , structural functions or transport coefficients . an alternative approach , which has been employed in many contexts , is to use step potentials to approximate intermolecular interactions while affording the analytical solution of the dynamics @xcite . the simplification in the interaction potential can lead to an increase in simulation efficiency since the demanding task of calculating forces is reduced to computing momentum exchanges between bodies at the instant of interaction . this approach is called event - driven or _ discontinuous molecular dynamics _ ( dmd ) . in the dmd approach , various components of the system interact via discontinuous forces , leading to impulsive forces that act at specific moments of time . as a result , the motion of particles is free of inter - molecular forces between impulsive _ events _ that alter the trajectory of bodies via discontinuous jumps in the momenta of the system at discrete interaction times . to determine the dynamics , the basic interaction rules of how the ( linear and angular ) momenta of the body are modified by collisions must be specified . for molecular systems with internal degrees of freedom it is straightforward to design fully - flexible models with discontinuous potentials , but dmd simulations of such systems are often inefficient due to the relatively high frequency of internal motions@xcite . this inefficiency is reflected by the fact that most collision events executed in a dmd simulation correspond to intra rather than inter - molecular interactions . on the other hand , much of the physics relevant in large - scale simulations is insensitive to details of intra - molecular motion at long times . for this reason , methods of incorporating constraints into the dynamics of systems with continuous potentials have been developed that eliminate high frequency internal motion , and thus extend the time scales accessible to simulation . surprisingly , relatively little work has appeared in the literature on incorporating such constraints into dmd simulations . the goal of this paper is to extend the applicability of dmd methods to include constrained systems and to outline efficient methods that are generally applicable in the simulations of semi - flexible and rigid bodies interacting via discontinuous potentials . in contrast to systems containing only simple spherical particles @xcite , the application of dmd methods to rigid - body systems is complicated by two main challenges . the first challenge is to analytically solve the dynamics of the system so that the position , velocity , or angular velocity of any part of the system can be obtained exactly . this is in principle possible for a rigid body moving in the absence of forces and torques , even if it does not possess an axis of symmetry which facilitates its motion . however , an explicit solution suitable for numerical implementation seems to be missing in the literature ( although partial answers are abundant @xcite ) . for this reason , we will present the explicit solution here . armed with a solution of the dynamics of all bodies in the system , one can calculate the collision times in an efficient manner , and in some instances , analytically . the second challenge is to determine how the impulsive forces lead to discontinuous jumps in the momenta of the interacting bodies . for complicated rigid or semi - flexible bodies , the rules for computing the momentum jumps are not immediately obvious . it is clear however that these jumps in momenta must be consistent with basic conservation laws connected to symmetries of the underlying lagrangian characterizing the dynamics . often the basic lagrangian is invariant to time and space translations , and rotations , and , hence , the rules governing collisions must explicitly obey energy , momentum , and angular momentum constraints . such conservation laws can be utilized as a guide to derive the proper collision rules . a first attempt to introduce constraints into an event - driven system was carried out by ciccotti and kalibaeva@xcite , who studied a system of rigid , diatomic molecules ( mimicking liquid nitrogen ) . furthermore , non - spherical bodies of a special kind were treated by donev _ et al._@xcite by assuming that all rotational motion in between interaction events was that of a spherically symmetric body . more recently , a spherically symmetric hard - sphere model with four tetrahedral short ranged ( sticky ) interactions ( mimicking water ) has been studied by de michele _ et al._@xcite with an event - driven molecular dynamics simulation method similar to the most basic scheme presented in this paper . this work primarily focuses on the phase diagram of this `` sticky '' water model as a prototype of network forming molecular systems . our purpose , in contrast , is to discuss a general framework that allows one to carry out event - driven dmd simulations in the presence of constraints and , in particular , for fully general rigid bodies . the methodology is applicable to modeling the correct dynamics of water molecules in aqueous solutions@xcite as well as other many body systems . the paper is organized as follows . section [ calculation ] discusses the equations of motions in the presence of constraints and sec . [ includingcollisions ] discusses the calculation and scheduling of collision times . the collision rules are derived in sec . [ rules ] . in sec . [ ensembles ] it is shown how to sample the canonical and microcanonical ensembles and how to handle subtle issues concerning missing events that are particular to event - driven simulations . finally , conclusions are presented in sec . [ conclusions ] . the motion of rigid bodies can be considered to be a special case of the dynamics of systems under a minimal set of @xmath0 time - independent holonomic constraints ( i.e.dependent only on positions ) that fix all intra - body distances : @xmath1 where the index @xmath2 runs over all constraints present in the system and @xmath3 is a generalized vector whose components are the set of all cartesian coordinates of the @xmath4 total particles in the system . for fully rigid bodies , the number of constraints @xmath0 can easily be calculated by noting that the number of spatial degrees of freedom of an @xmath5-particle body is @xmath6 in 3 dimensions , while only 6 degrees of freedom are necessary to completely specify the position of all components of a rigid body : 3 degrees of freedom for the center of mass of the object and 3 degrees of freedom to specify its orientation with respect to some arbitrary fixed reference frame . there are therefore @xmath7 constraint equations for a single rigid body with @xmath5 point masses . below , these point masses will be referred to as _ atoms _ while the body as a whole will be called a _ molecule_. a typical constraint equation fixes the distance between atoms @xmath8 and @xmath9 in the molecule to be some value , @xmath10 , i.e. , @xmath11 the equations of motion for the system follow from hamilton s principle of stationary action , which states that the motion of the system over a fixed time interval is such that the variation of the line integral defining the action @xmath12 is zero : @xmath13 where the lagrangian @xmath14 in the presence of the constraints is written in cartesian coordinates as @xmath15 where @xmath16 is the interaction potential . for clarity throughout this paper , the einstein summation convention will be used for sums over repeated _ greek _ indices i.e. @xmath17 , whereas the sum over atom indices will be written explicitly . in eq . , the parameters @xmath18 are lagrange multipliers to enforce the distance constraints @xmath19 . the resulting equations of motion are : @xmath20 for an elementary discussion of constrained dynamics in the lagrangian formulation of mechanics , we refer to ref . . when there are no interactions , such as for a single molecule , the potential @xmath21 and eq becomes @xmath22 these equations of motion must be supplemented by equations for the @xmath0 lagrange multipliers @xmath18 , which are functions of time . although the @xmath18 are not functions of @xmath3 and @xmath23 in a mathematical sense , it will be shown below that once the equations are solved they can be expressed in terms of @xmath24 and @xmath23 . note that the equations of motion show that even in the absence of an external potential , the motion of the point masses ( atoms ) making up a rigid body ( molecule ) are non - trivial due to the emergence of a _ constraint force _ @xmath25 . in fortuitous cases , the time dependence of the lagrange multipliers is relatively simple and can be solved for by taylor expansion of the lagrange multipliers in time @xmath26 . to evaluate the time derivatives of the multipliers , one can use time derivatives of the initial constraint conditions , which must vanish to all orders . the result is a hierarchy of equations , which , at order @xmath27 , is linear in the unknown @xmath27th time derivatives @xmath28 but depends on the lower order time derivatives @xmath29 , @xmath30 , @xmath31 . in exceptional circumstances , this hierarchy naturally truncates . for example , for a rigid diatomic molecule with a single bond length constraint , one finds that the hierarchy truncates at order @xmath32 , and the lagrange multiplier is a constant @xcite . however this is not the typical case . alternatively , since the constraints @xmath33 are to be satisfied at all times @xmath26 , and not just at time zero , their time derivatives are zero at all times . from the first time derivative @xmath34 one sees that the initial velocities @xmath35 must obey @xmath36 for each constraint condition @xmath2 . the lagrange multipliers can be determined by the condition that the second derivatives of all the constraints vanish so that @xmath37 yielding a linear equation for the lagrange multipliers that can be solved in matrix form as @xmath38 where @xmath39 it may be shown that with @xmath18 given by eq . , all higher time derivatives of @xmath19 are automatically zero . as eq . shows , in general the lagrange multipliers are dependent on both the positions @xmath3 and the velocities @xmath23 of the particles . to see that this makes the dynamics non - hamiltonian , the equations of motion can be cast into hamiltonian - like form using @xmath40 , i.e. , @xmath41 where it is apparent that the forces in the system depend on the momentum through @xmath42 in eq . . there exists no hamiltonian that generates these equations of motion.@xcite since the underlying dynamics of the system is non - hamiltonian , the statistical mechanics of the constrained system is potentially more complex . in general , phase - space averages have to be defined with respect to a metric that is invariant to the standard measure of hamiltonian systems , but @xmath43 is not conserved under the dynamics and the standard form of the liouville equation does not hold @xcite . in general , there is a phase - space compressibility factor @xmath44 associated with the lack of conservation of the measure that is given by minus the divergence of the flow in phase space . it may be shown that@xcite @xmath45 where @xmath46 is the determinant of the matrix @xmath47 defined in eq . ( [ zdef ] ) . the compressibility factor is related to the invariant phase - space metric @xmath48 with@xcite @xmath49 statistical averages are therefore defined for the non - hamiltonian system as@xcite @xmath50 where @xmath51 is the probability density for the unconstrained system and @xmath52 is the partition function for the constrained system , given by @xmath53 although the invariant metric is non - uniform for many constrained systems , for entirely rigid systems the @xmath54 matrix is a function only of the point masses and fixed distances . hence the term @xmath55 acts as a multiplicative factor which cancels in the averaging process . although the solution of the dynamics of constrained systems via time - independent holonomic constraints is intellectually appealing and useful in developing a formal statistical mechanics for these systems , it is often difficult to analytically solve for the values of the lagrange multipliers at arbitrary times . one therefore often resorts to numerical solutions of the multipliers in iterative form , using algorithms such as shake@xcite . such an approach is not really consistent with the principles of dmd , in which a computationally efficient means of calculating event times is one of the great advantages of the method . for fully - constrained , rigid bodies , it is more sensible to apply other , equivalent , approaches , such as the principal axis or quaternion methods , to calculate analytically the evolution of the system in the absence of external forces . the basic simplification in the dynamics of rigid bodies results from the fact that the general motion of a rigid body can be decomposed into a translation of the center of mass of the body plus a rotation about the center of mass . the orientation of the body relative to its center of mass is described by the relation between the so - called _ body frame _ , in which a set of axes are held fixed with the body as it moves , and the fixed external _ laboratory frame_. the two frames of reference can be connected by an orthogonal transformation , such that the position of an atom @xmath8 in a rigid body can be written at an arbitrary time @xmath26 as : @xmath56 where @xmath57 is the position of atom @xmath8 in the body frame ( which is independent of time ) , @xmath58 is the center of mass , and the matrix @xmath59 is the orthogonal matrix that converts coordinates in the body frame to the laboratory frame . note that matrix - vector and matrix - matrix multiplication will be implied throughout the paper . the matrix @xmath59 is the transpose of @xmath60 , which converts coordinates from the laboratory frame to the body frame at time @xmath26 . the elements composing the columns of the matrix @xmath59 are simply the coordinates of the axes in the body frame written in the laboratory frame . note that eq . ( [ cartesianpositions ] ) implies that the relative vector @xmath61 satisfies @xmath62 here as well as below , we have dropped the explicit time dependence for most time dependent quantities with the exception of quantities at time zero or at a time that is integrated over . one sees that in order to determine the location of different parts of the body in the laboratory frame , the rotation matrix @xmath63 must be specified . this matrix satisfies a differential equation that will now be derived and subsequently solved . before doing so , it will be useful to restate some properties of rotation matrices and establish some notation to be used below . formally , a rotation matrix @xmath64 is an orthogonal matrix with determinant one and whose its inverse is equal to its transpose @xmath65 . any rotation can be specified by a rotation axis @xmath66 and an angle @xmath67 over which to rotate . here @xmath68 is a unit vector , so that one may also say that any non - unit vector @xmath69 can be used to specify a rotation , where its norm is equal to the angle @xmath67 and its direction is equal to the axis @xmath68 . according to rodrigues formula , the matrix corresponding to this rotation is@xcite @xmath70 the derivation of the differential equation for @xmath63 starts by taking the time derivative of eq . , yielding @xmath71 from elementary classical mechanics@xcite , it is known that this relative velocity can also be written as @xmath72 where @xmath73 is the angular velocity vector in the lab frame . since both eq . and eq . are true for any vector @xmath74 , it follows that @xmath75 is the matrix representation of a cross product with the angular velocity @xmath73 , i.e. , @xmath76 multiplying eq . on the right with @xmath77 and taking the transpose on both sides ( note that @xmath78 is antisymmetric ) yields @xmath79 this equation involves the angular velocity in the laboratory frame , but the rotational equations of motion are more easily solved in the body frame . the angular velocity vector transforms to the body frame according to @xmath80 for any rotation @xmath63 and vector @xmath81 one has @xmath82 , hence one can write @xmath83 substituting eq . into eq . yields the differential equation for @xmath63 : @xmath84 although the choice of body frame is arbitrary , perhaps the most convenient choice of axes for the body are the so - called principal axes in which the moment of inertia tensor @xmath85 is diagonal , i.e. , @xmath86 . choosing this reference frame as the body frame , the representation of the components of the angular momentum @xmath87 is @xmath88 where @xmath89 and @xmath90 are the principal moments of inertia and principal components of the angular velocity . the time dependence of the principal components of the angular velocity may be obtained from the standard expression for the torque in the laboratory frame : @xmath91 . \label{transformangularmomentum}\end{aligned}\ ] ] where eq . was used in the last equality . transforming eq . to the principal axis frame gives euler s equations of motion for a rigid body @xmath92 where @xmath93 are the components of the torque @xmath94 in the body frame . note that even in the absence of any torque , the principal components of the angular velocity are in general time dependent . once the angular velocity @xmath95 is known , it can be substituted into eq . for the matrix @xmath96 . the general solution of eq . is of the form @xmath97 where @xmath98 is a rotation matrix itself which ` propagates ' the orientation @xmath99 to the orientation at time @xmath26 . @xmath98 satisfies the same equation as @xmath63 , but with initial condition @xmath100 . by integrating this equation , one can obtain an expression for @xmath98 . at first glance , it may seem that @xmath98 can only be written as a formal expression containing a time - ordered exponential . however , for the torque - free case @xmath101 , the conservation of angular momentum and energy and the orthogonality of the matrix @xmath98 can be used to derive the following explicit expression@xcite ( implicitly also found in ref . ): @xmath102 here @xmath103 and @xmath104 are two rotation matrices . the matrix @xmath103 rotates @xmath105 to @xmath87 and can be written as @xmath106 where @xmath107 and @xmath108 while @xmath109 . the matrix @xmath104 can be expressed , using the notation in eq . , as @xmath110 where the angle @xmath111 is given by @xmath112 with @xmath113 the angle @xmath111 can be interpreted as an angle over which the body rotates . if the body rotates one way , the laboratory frame as seen from the body frame rotates in the opposite way , which explains the minus sign in eq . . for the derivation of eqs . - we refer to ref . . similar equations , but in a special reference frame , can be found in ref . . in the following , the solution of eq . with @xmath101 for bodies of differing degrees of symmetry will be analyzed and then used to obtain explicit expressions for the matrix @xmath98 as a function of time and of the initial angular velocity in the body frame @xmath114 . for the case of a spherical rotor in which all three moments of inertia are equal , @xmath115 , the form of the euler equations is particularly simple : @xmath116 . it is therefore clear that all components of the angular velocity in the body frame are conserved , as are those of the angular momentum . as a result , @xmath103 in eq . is equal to the identity matrix . a second consequence is that @xmath117 in eq . is constant , so that @xmath118 where @xmath117 may be rewritten , using @xmath115 , as @xmath119 . therefore eqs . and give @xmath120 corresponding to a rotation by an angle of @xmath121 around the axis @xmath122 . for the case of a symmetric top for which @xmath123 , one can solve the euler equations in terms of simple sines and cosines , since eq . becomes @xmath124 where @xmath125 is the precession frequency . the full solution of the euler equations is given by @xmath126 using eq . and the fact that @xmath127 and @xmath128 are conserved in this case , one can easily show that @xmath103 is given by @xmath129 and one can determine @xmath117 from eq . : @xmath130 } { i_1 ^ 2{\tilde{\omega}_{1}}^2(0 ) + i_1 ^ 2{\tilde{\omega}_{2}}^2(0 ) } = \frac{l}{i_1}.\ ] ] this is a constant so that @xmath131 . thus @xmath132 and one gets from eq . : @xmath133 if all the principal moments of inertia are distinct , the time dependence of the angular velocity @xmath95 involves elliptic functions@xcite . while this may seem complicated , efficient standard numerical routines exist to evaluate these functions@xcite . more challenging is the evaluation of the matrix @xmath98 . while its exact solution has been known for more than 170 years@xcite , it is formulated even in more recent texts@xcite in terms of undetermined constants and using complex algebra , which hinders its straightforward implementation in a numerical simulation . it is surprisingly difficult to find an explicit formula in the literature for the matrix @xmath98 as a function of the initial conditions , which is the form needed in dmd simulations . for this reason , the explicit general solution for @xmath98 will briefly be presented here in terms of general initial conditions . the details of the derivation can be found elsewhere@xcite . following jacobi@xcite , it is useful to adopt the convention that @xmath134 is the moment of inertia intermediate in magnitude ( i.e. , either @xmath135 or @xmath136 ) and one chooses the overall ordering of magnitudes , such that : @xmath137 where @xmath138 is the rotational kinetic energy @xmath139 and @xmath140 is the norm of the angular momentum @xmath141 . without this convention some quantities defined below would be complex valued , which is numerically inconvenient and inefficient . note that in a simulation molecules will often be assigned a specific set of physical inertial moments with fixed order , i.e. not depending on the particular values of @xmath138 and @xmath140 . a simple way to nevertheless adopt the convention in eq . is to introduce internal variables @xmath142 , @xmath143 and @xmath144 which differ when necessary from the physical ones by a rotation given by the rotation matrix @xmath145 this matrix interchanges the @xmath146 and @xmath147 directions and reversed the @xmath148 direction , and is equal to its inverse . the euler equations can be solved because there are two conserved quantities @xmath149 and @xmath150 which allow @xmath151 and @xmath152 to be expressed in terms of @xmath153 , at least up to a sign which the quadratic conserved quantities can not prescribe . in this way the three coupled equations are reduced to a single ordinary differential equation for @xmath154 , from which @xmath26 can be solved as an integral over @xmath153 : this is an incomplete elliptic integral of the first kind@xcite . to get @xmath153 as a function of @xmath26 , one needs its inverse , which is the elliptic function @xmath155@xcite . without giving further details , the solution of the euler equations is given by@xcite @xmath156 here @xmath157 and @xmath158 are also elliptic functions@xcite , while the @xmath159 are the extreme ( maximum or minimum ) values of the @xmath160 and are given by @xmath161 where @xmath162 is the sign of @xmath146 . furthermore , in eq . the _ precession frequency _ @xmath163 is given by @xmath164 the elliptic functions are periodic functions of their first argument , and look very similar to the sine , cosine and constant function . they furthermore depend on the _ elliptic parameter _ @xmath165 ( or elliptic modulus @xmath166 ) , which determines how closely the elliptic functions resemble their trigonometric counterparts , and which is given by @xmath167 ( solid line ) , @xmath155 ( bold dashed line ) , @xmath158 ( dotted line ) for @xmath168 ( @xmath169 , @xmath170 , @xmath171 ) . also plotted are the cosine ( short dashed line ) and sine ( thin short dashed line ) with the same period , for comparison . ] by matching the values of @xmath153 at time zero , one can determine the integration constant @xmath172 : @xmath173 where @xmath174 is the incomplete elliptic integral of the first kind@xcite @xmath175 in fact , @xmath176 is simply the inverse of this function . as a result of the ordering convention in eq . , the parameter @xmath165 in eq . is guaranteed to be less than one , which is required in order that @xmath177 in eq . not be complex - valued . three more numbers can be derived from the elliptic parameter @xmath165 which play an important role in the elliptic functions . these are the _ quarter - period _ @xmath178 , the _ complementary quarter - period _ @xmath179 and the _ nome _ @xmath180 , which is the parameter in various series expansions . the period of the elliptic functions @xmath157 and @xmath155 is equal to @xmath181 , while that of @xmath158 is @xmath182 . these elliptic functions have the following fourier series@xcite : @xmath183 note that the right - hand side of eqs . - depends on @xmath165 through @xmath184 and @xmath185 $ ] . for @xmath186 , one gets @xmath187 and @xmath157 , @xmath155 and @xmath158 reduce to @xmath188 , @xmath189 and @xmath190 , respectively . the constancy of @xmath191 is reminiscent of the conservation of @xmath192 in the case of the symmetric top , and , indeed , for @xmath123 , @xmath186 according to eq . . typical values for @xmath193 are quite small , hence often the elliptic function @xmath155 , @xmath157 and @xmath158 resemble the @xmath188 , @xmath189 and a constant function with value one ( as e.g. in fig . [ cnsndn ] ) . for small values of @xmath193 , the series expressions for the elliptic functions converge quickly ( although this is not the best way to compute the elliptic functions@xcite ) . having given the solutions of the euler equations , we now turn to the solution of eq . as given by eqs . - . the expression on the right - hand side of eq . isnot a constant in this case but involves elliptic functions . despite this difficulty , the integral can still be performed using some properties of elliptic functions , with the result@xcite @xmath194 the constants @xmath195 , @xmath196 and the periodic function @xmath197 can be expressed using the theta function @xmath198@xcite as @xmath199 where we have used the definition @xmath200 the equations - involve complex values which are not convenient for numerical evaluation . using the known series expansions of the theta function @xmath201 and its logarithmic derivative@xcite in terms of the nome @xmath193 , these equations may be rewritten in a purely real form . in fact , one readily obtains the sine and cosine of @xmath111 , which are all that is needed in eqs . and , @xmath202 with @xmath203 while the constant @xmath195 is @xmath204 + n\pi,\ ] ] where @xmath205 if @xmath206 , @xmath207 if @xmath208 and @xmath209 , and @xmath210 if @xmath208 and @xmath211 . finally , the constant @xmath196 is given by@xcite @xmath212 , \label{c2}\ ] ] where @xmath213 . the series expansion in @xmath193 in eq . convergences for @xmath214 . because @xmath215 ( cf . eq . ) , one has @xmath216 , and the series always converges . since @xmath193 is typically small , the convergence is rarely very slow ( e.g. for convergence up to relative order @xmath217 one needs @xmath218 terms ) . note that since the constants @xmath195 and @xmath196 depend only on the initial angular velocities , they only need to be calculated once at the beginning of the motion of a free rigid body . on the other hand , the series expansions in eqs . and , which have to be evaluated any time the positions are desired , have extremely fast convergence due to the @xmath219 appearing in these expressions ( for example , unless @xmath220 , the series converges up to @xmath221 occurs taking only three terms ) . there are efficient routines to calculate the functions @xmath157 , @xmath155 @xmath158 and @xmath174 , see e.g. refs . , and the series in eqs . , and converge , the former two quite rapidly in fact . therefore , despite an apparent preference in the literature for conventional numerical integration of the equations of motion via many successive small time steps even for torque - free cases , the analytical solution can be used to calculate the same quantities in a computationally more efficient manner requiring only the evaluation of special functions . the gain in efficiency should be especially pronounced in applications in which many evaluations at various times could be needed , such as in the root searches in discontinuous molecular dynamics ( see below ) . if the interaction potential between atoms @xmath8 and @xmath9 is assumed to be discontinuous , say of the form @xmath222 then rigid molecules interacting via this potential evolve freely until there is a change in the potential energy of the system and an _ interaction event _ or _ collision event _ occurs . the time at which an event occurs is governed by a collision indicator function @xmath223 defined such that at time @xmath224 , @xmath225 . here , the time dependence of @xmath226 and @xmath227 can be obtained using the results of sec . [ calculation ] . the simplest example of this kind of system consists of two hard spheres of diameter @xmath10 located at positions @xmath226 and @xmath228 . if two spheres are approaching , when they get to a distance @xmath10 from one another , the potential energy would change from @xmath229 to @xmath230 if they kept approaching one another . as this eventually would lead to a violation of energy conservation , the spheres bounce off one another in a _ hard - core collision _ at time @xmath224 , where @xmath224 is determined by the criterion @xmath231 , i.e. by the zeros of the collision indicator function . another kind of interaction event , with @xmath232 and @xmath229 finite , will be called a _ square - well collision _ because the potential then has a square well shape . to find the times at which collisions take place , the zeros of the collision indicator functions must be determined , which generally has to be done numerically . the calculation of the collision times of non - penetrating rigid objects is an important aspect of manipulating robotic bodies , and is also an important element of creating realistic animation sequences . as a result , many algorithms have been proposed in these contexts to facilitate the event time search@xcite . the search for the earliest collision event time can be facilitated using screening strategies to decide when rigid bodies may overlap@xcite . usually , these involve placing the bodies in bounding boxes and using an efficient method to determine when bounding boxes intersect . the simplest way to do this in a simulation of rigid molecules is to place each molecule in the smallest sphere around its center of mass containing all components of the molecule@xcite . the position of the sphere is determined by the motion of the center of mass , while any change in orientation of the rigid molecule occurs within the sphere . collisions between rigid molecules can therefore only occur when their encompassing spheres overlap , and the time at which this occurs can be calculated analytically for any pair of molecules . this time serves as a useful point to begin a more detailed search for collision events ( see below ) . similarly , one can also calculate the time at which the spheres no longer overlap , and use these event times to bracket a possible root of the collision indicator function . it is crucial to make the time bracketing as tight as possible in any implementation of dmd with numerical root searches because the length of the time bracketing interval determines the required number of evaluations of the positions and velocities of the atoms , and therefore plays a significant role in the efficiency of the overall procedure . the simplest reliable and reasonably efficient means of detecting a root is to perform a _ grid search _ that looks for changes in sign of @xmath233 , i.e. , one looks at @xmath234 and @xmath235 for successive @xmath5 . the time points @xmath236 will be called the _ grid points_. when a time interval in between two grid points is found in which a sign change of @xmath233 occurs , the newton - raphson algorithm@xcite can be called to numerically determine the root with arbitrary accuracy . since the newton - raphson method requires the calculation of first time derivatives , one must also calculate , for any time @xmath26 , the derivative @xmath237 , where the notation @xmath238 and @xmath239 has been used . such time derivatives are readily evaluated using eqs . ( [ cartesianpositions ] ) and ( [ step1vel ] ) . unfortunately , while the newton - raphson method is a very efficient algorithm for finding roots , it can be somewhat unstable when one is searching for the roots of an oscillatory function . for translating and rotating rigid molecules , the collision indicator function is indeed oscillatory due to the periodic motion of the relative orientation of two colliding bodies . it is particularly easy to miss so - called _ grazing collisions _ when the grid search interval @xmath240 is too large , in which case the indicator function is positive in two consecutive points of the grid search , yet nonetheless `` dips '' below zero in the grid interval . it is important that no roots are missed , for a missed root can lead to a different , even infinite energy ( but see sec . [ ensembles ] below ) . to reduce the frequency of missing grazing collisions to zero , a vanishingly small grid interval @xmath240 would be required . of course such a scheme is not practical , and one must balance the likelihood of missing events with practical considerations since several collision indicator functions need to be evaluated at each point of the grid . clearly the efficiency of the root search algorithm significantly depends on the magnitude of grid interval . to save computation time , a coarser grid can be utilized if a means of handling grazing collisions is implemented . since the collision indicator function has a local extremum ( maximum or minimum , depending on whether @xmath241 is initially smaller or larger than @xmath242 ) at some time near the time of a grazing collision , a reasonable strategy to find these kind of collision events is to determine the extremum of the indicator function in cases in which the indicator function @xmath233 itself does not change sign on the interval but its derivative @xmath243 does . furthermore , since the indicator function at the grid points near a grazing collision is typically small , it is fruitful to search only for extrema when the indicator function at one of the grid points lies below some threshold value@xcite . to find the local extrema of the indicator function , any simple routine of locating the extrema of a non - linear function can be utilized . for example , brent s minimization method@xcite , which is based on a parabolic interpolation of the function , is a good choice for sufficiently smooth one - dimensional functions . once the extremum is found , it is a simple matter to decide whether or not a real collision exists by checking the sign of @xmath233 . once the root has been bracketed ( either through a sign change of @xmath233 during the grid search or after searching for an extremum ) , one can simply use the newton - raphson algorithm to find the root to desired accuracy , typically within only a few iterations . the time value returned by the newton - raphson routine needs to be in the bracketed interval and @xmath244 if @xmath233 was initially positive and @xmath245 if it was initially negative . if those criteria are not satisfied , the newton - raphson algorithm has clearly failed and a less efficient but more reliable method is needed to track down the root . for example , the van wijngaarden - dekker - brent method@xcite , which combines bisection and quadratic interpolation , is guaranteed to converge if the function is known to have a root in the interval under analysis . in the previous section is was shown how to determine the time @xmath246 at which two atoms @xmath8 and @xmath9 collide under the assumption that there is no other earlier collision . this we will call a _ possible collision event_. in a dmd simulation , once the possible collision events at times @xmath246 have been computed for all possible collision pairs @xmath8 and @xmath9 , the earliest event @xmath247 should be selected . after the collision event between atoms @xmath248 and @xmath249 has been executed ( according to the rules derived in the next section ) , the next earliest collision should be performed . however , because the velocities of atoms of the molecules involved in the collision have changed , the previously computed collision times involving these molecules are no longer valid . the next event in the sequence can be determined and performed only after these collision times have been recomputed . this process describes the basic strategy of dmd , which without further improvements would be needlessly inefficient . for if @xmath250 is the number of possible collision events , finding the earliest time would require @xmath251 checks , and @xmath252 , while the number of invalidated collisions that have to be recomputed after each collision would be @xmath253 . since the number of collisions in the system per unit of physical time also grows with @xmath4 , the cost of a simulation for a given physical time would be @xmath254 for the computation of collision times and @xmath255 for finding the first collision event@xcite . fortunately , there are ways to significantly reduce this computational cost@xcite . the first technique , also used in molecular dynamics simulations of systems interacting with continuous potentials , reduces the number of possible collision times that have to be computed by employing a _ cell division _ of the system@xcite . note that while the times of certain interaction events ( e.g. involving only the molecule s center of mass ) can be expressed in analytical form and thus computed very efficiently , the atom - atom interactions have , in general , an orientational dependence and the possible collision time has to be found by means of a numerical root search as explained in the previous section . as a consequence , the most time consuming task in a dmd simulation with rigid bodies is the numerical root search for the collision times . one can however minimize the required number of collision time computations by dividing the system into a cell structure and sorting all molecules into these cells according to the positions of their centers of mass . each cell has a diameter of at least the largest `` interaction diameter '' of a molecule as measured from its center of mass . as a result , molecules can only collide if they are in the same cell or in an adjacent cell , so the number of collision events to determine and to recompute after a collision is much smaller . in this technique , the sorting of molecules into cells is only done initially , while the sorting is dynamically updated by introducing a _ cell - crossing event _ for each molecule that is also stored@xcite . since the center of mass of a molecule performs linear motion between collision events , one can express its cell - crossing time analytically and therefore the numerical computation of that time is very fast . the second technique reduces the cost of finding the earliest event time . it consists of storing possible collision and cell - crossing events in a time - ordered structure called a _ binary tree_. for details we refer to refs . and ( alternative event scheduling algorithms exist@xcite but it is not clear which technique is generally the most efficient@xcite . ) finally , a third standard technique is to update the molecules positions and velocities only at collisions ( and possibly upon their crossing the periodic boundaries ) , while storing the time of their last collision as a property of the molecule called its _ _ local clock__@xcite . whenever needed , the positions and velocities at later times can be determined from the exact solution of force - free and torque - free motion of the previous sec . [ calculation ] . the use of cell divisions , a binary event tree to manage the events , and local clocks is a standard practice in dmd simulations and largely improves the simulation s efficiency@xcite . to see this , note that in each step of the simulation one picks the earliest event from the tree , which scales as @xmath256 for randomly balanced trees@xcite . if it is a cell - collision event , it is then performed and subsequently @xmath257 collisions and cell crossings are recomputed and added to the event tree ( @xmath258 ) . if it is a crossing event , the corresponding molecule is put in its new cell , new possible collision and crossing events are computed ( @xmath259 ) and added to the tree ( @xmath256 ) . then the program progresses to the next event . since still @xmath253 real events take place per unit of physical time , one sees that using these techniques , the computational cost per unit of physical time due to the computation of possible collisions and cell crossing times scales as @xmath253 instead of @xmath254 , while the cost due to the event scheduling is @xmath260 per unit of physical time instead of @xmath255 a huge reduction . contrary to what their scaling may suggest , one often finds that the cost of the computation of collision times greatly dominates the scheduling cost for finite @xmath4 . this is due to fact that the computations of many of the collision times requires numerical root searches , although some can , and should , be done analytically . thus , to gain further computational improvements , one has to improve upon the efficiency of the numerical search for collision event times . a non - standard time - saving technique that we have developed for this purpose is to use _ virtual collision events_. in this case , the grid search ( see sec . [ includingcollisions ] ) for a possible collision time of atoms @xmath8 and @xmath9 is carried out only over a fixed small number of grid points , thus limiting the scope of the root search to a small search interval . if no collisions are detected in this search interval , a virtual collision event is scheduled in the binary event tree , much as if it were a possible future collision at the time of the last grid point that was investigated . if the point at which the grid search is curtailed is rather far in the future , it is likely this virtual event will not be executed because the atoms @xmath8 and @xmath9 probably will have collided with other atoms beforehand . thus , computational work has been saved by stopping the grid search after a few grid points . every now and then however , atoms @xmath8 and @xmath9 will not have collided with other atoms at the time at which the grid search was stopped . in this case , the virtual collision event in the tree is executed , which entails continuing the root search from the point at which the search was previously truncated . the continued search again may not find a root in a finite number of grid points and schedule another virtual collision , or it may now find a collision . in either case the new event is scheduled in the tree . this virtual collision technique avoids the unnecessary computation of a collision time that is so far in the future that it will not be executed in all likelihood anyway , while at the same time ensuring that if , despite the odds , that collision is to happen nonetheless , it is indeed found and correctly executed . the trade - off of this technique is that the event tree is substantially larger , slowing down the event management . due to the high cost of numerical root searches however , the simulations presented in the accompanying paper showed that using virtual collision events yields an increase in efficiency between 25% and 110% , depending mainly on the system size . at each moment of collision , the impulsive forces and torques lead to discontinuous jumps in the momenta and angular momenta of the colliding bodies . in the presence of constraints , there are two equivalent ways of deriving the rules governing these changes . in the first approach , the dynamics are treated by applying constraint conditions to cartesian positions and momenta . this approach is entirely general and is suited for both constrained rigid and non - rigid motion . in its generality , it is unnecessarily complicated for purely rigid systems and is not suitable for continuum bodies . the second approach , suitable for rigid bodies only , uses the fact that only six degrees of freedom , describing the center of mass motion and orientational dynamics are required to fully describe the dynamics of an arbitrary rigid body . the derivation therefore consists of prescribing a collision process in terms of impulsive changes to the velocity of the center of mass and impulsive changes to the angular velocity . the general collision process in systems with discontinuous potentials can be seen as a limit of the collision process for continuous systems in which the interaction potential becomes infinitely steep . a useful starting point for deriving the collision rules is therefore to consider the effect of a force applied to the overall change in the momentum of any atom @xmath27 : @xmath261 where @xmath262 is the total force acting on atom @xmath27 and @xmath263 . furthermore , here and below the pre and post - collision values of a quantity @xmath264 are denoted by @xmath264 and @xmath265 , respectively . for discontinuous systems , the intermolecular forces are impulsive and occur only at an instantaneous collision time @xmath224 . when atoms @xmath8 and @xmath9 collide , the interaction potential @xmath16 depends only on the scalar distance @xmath266 between those atoms , so that the force on an arbitrary atom @xmath27 is given by ( without summation over @xmath8 and @xmath9 ) @xmath267 note that this is non - zero only for the atoms involved in the collision , as expected . given that the force is impulsive , it may be written as @xmath268 where the scalar @xmath269 is the magnitude of the impulse ( to be determined ) on atom @xmath270 in the collision . in general , the constraint forces on the right - hand side of eq . ( [ eleq ] ) must also have an impulsive component whenever intermolecular forces are instantaneous in order to maintain the rigid body constraints at all times . we account for this by writing the lagrange multipliers as @xmath271 because @xmath18 enters into the equations of motion for all atoms @xmath27 involved in the constraint @xmath19 , there is an effect of this impulsive constraint force on all those atoms . thus , one can write for the force on a atom @xmath27 when atoms @xmath8 and @xmath9 collide : @xmath272 . \label{forces}\end{gathered}\ ] ] substituting eq . ( [ forces ] ) into ( [ deltap1 ] ) , one finds that the term proportional to @xmath273 vanishes in the limit that the time interval @xmath240 approaches zero , so that the post - collision momenta @xmath274 are related to the pre - collision momenta @xmath275 by @xmath276 note that at the instant of collision @xmath277 , the positions of all atoms @xmath3 remain the same ( only their momenta change ) so that there is no ambiguity in the right - hand side of eq . as to whether to take the @xmath3 before or after the collision . it is straightforward to show that due to the symmetry of the interaction potential , the total linear momentum and angular momentum of the system are conserved by the collision rule eq.([deltap2 ] ) for arbitrary values of the unknown scalar functions @xmath269 and @xmath278 . in addition to these constants of the motion , the collision rule must also conserve total energy and preserve the constraint conditions , @xmath279 and @xmath280 , before and after the collision . the first constraint condition is trivially satisfied at the collision time , since the positions are not altered at the moment of contact . the second constraint condition allows the scalar @xmath278 to be related to the value of @xmath269 using eq . before and after the collision , since we must have @xmath281 inserting eq . ( [ deltap2 ] ) into eq . , one gets @xmath282 solving this linear equation for @xmath278 gives @xmath283 where the @xmath284 matrix was defined in eq . ( [ zdef ] ) . note that if atoms @xmath8 and @xmath9 are on different bodies , a given constraint @xmath285 involves either one or the other atom ( or neither ) , so at least one of the two terms on the right - hand side of eq . is then zero . equation can now be written as @xmath286 where @xmath287 is a function of the phase - space coordinate as determined by eq . and is independent of @xmath269 . finally , the scalar @xmath269 can be determined by employing energy conservation , @xmath288 where @xmath289 denotes the discontinuous change in the potential energy at the collision time . inserting the expression in ( [ changep ] ) into and using eq . , one gets a quadratic equation for the scalar @xmath269 , @xmath290 for finite values of @xmath291 , the value of @xmath269 is therefore @xmath292 where the physical solution corresponds to the positive ( negative ) root if @xmath293 ( @xmath294 ) , provided @xmath295 . if this latter condition is not met , there is not enough kinetic energy to overcome the discontinuous barrier , and the system experiences a hard - core scattering , with @xmath296 , so that eq . gives @xmath297 . once the value of @xmath269 has been computed , the discrete changes in momenta or velocities are easily computed using eq . ( [ changep ] ) . the solution method outlined above can be applied to semi - flexible as well as rigid molecular systems , but is not very suitable for rigid , continuous bodies composed of an infinite number of point particles . for perfectly rigid molecules , a more convenient approach is therefore to analyze the effect of impulsive collisions on the center of mass and angular coordinates of the system , which are the minimum number of degrees of freedom required to specify the dynamics of rigid systems . the momentum of the center of mass @xmath298 and the angular momentum @xmath299 of rigid molecule @xmath270 are affected by the impulsive collision via @xmath300 where @xmath301 and @xmath302 are the moment of inertia tensor and the angular velocity of body @xmath270 in the laboratory frame , respectively . note that they are related to their respective quantities in the principal axis frame ( body frame ) via the matrix @xmath303 ( now associated with the body @xmath270 ) : @xmath304 to derive specific forms for the impulsive changes @xmath305 and @xmath306 , one may either calculate the impulsive force and torque acting on the center of mass and angular momentum , leading to @xmath307 and @xmath308 , where @xmath309 and @xmath310 are the points at which the forces are applied on body @xmath270 and @xmath311 , respectively , while @xmath312 , and @xmath269 should be obtained from energy conservation . to understand this better and make a connection with the previous section , one may equivalently view the continuum rigid body as a limit of a non - continuum rigid body composed of @xmath5 constrained point particles , and use the expressions derived in the previous section for the changes in momenta of the constituents . in the latter approach , it is convenient to switch the notation for the positions and momenta of the atoms from @xmath227 and @xmath313 to @xmath314 and @xmath315 , which indicate the position and momentum of particle @xmath8 on body @xmath270 , respectively . using this notation and considering a collision between particle @xmath8 on body @xmath270 and particle @xmath9 on body @xmath311 , eq . ( [ changep ] ) can be written as @xmath316 , \label{changepa}\end{aligned}\ ] ] where @xmath317 is the unit vector along the direction of the vector @xmath318 connecting atom @xmath8 on body @xmath270 with its colliding partner @xmath9 on body @xmath311 . thus , noting that @xmath319 , where @xmath320 is the total mass of body @xmath270 , and using eq . ( [ changepa ] ) , one finds that @xmath321 since @xmath322 similarly , one finds that @xmath323 where @xmath324 . comparing with eq . , it is evident that @xmath325 note that @xmath326 is a matrix inverse . once again the impulsive changes are directly proportional to @xmath269 , and the change of the angular velocity of body @xmath311 in the laboratory frame due to the collision can be calculated analogously . to determine the scalar @xmath269 , one again uses the conservation of total energy ( @xmath327 ) to see that @xmath328 inserting eqs . ( [ changepcom ] ) and ( [ changeomega ] ) into the energy equation above yields , after some manipulation , a quadratic equation for @xmath269 of the form of eq . ( [ quadratic ] ) , with @xmath329 where @xmath330 with @xmath331 for a spherically symmetric system , @xmath332 , and @xmath333 . any event - driven molecular dynamics simulation relies on the assumption that no collision is ever missed . however , collisions will be missed whenever the time difference between two nearby events is on the order of ( or smaller than ) the time error of the scheduled events , which indicates that there is still a finite chance that a collision is missed even when event times are calculated in a simulation starting from an analytic expression , due to limits on machine precision . although this subtle issue is not very important in a hard sphere system , in the present context it is of interest . indeed , the extensive use of numerical root searches for the event time calculations combined with the need for computational efficiency demands a lower precision in the time values of collision events ( typically a precision of @xmath334 instead of @xmath335 for analytical roots ) . in this section , it will be shown how to handle missed collisions in the context of the hybrid monte carlo scheme ( hmc ) . in general , the hmc method @xcite combines the monte carlo method with molecular dynamics to construct a sequence of independent configurations @xmath336 , distributed according to the canonical probability density @xmath337 , \label{canonicalprob}\ ] ] where @xmath338 is the configurational integral , @xmath339 is boltzmann s constant , and @xmath340 is the temperature . in the present context , this method can be implemented as follows : initially , a new set of momenta @xmath341 is selected by choosing a random center of mass momentum @xmath342 and angular velocity @xmath73 for each molecule from the gaussian distribution @xmath343.\ ] ] the system is then evolved deterministically through phase - space for a fixed time @xmath344 according to the equations of motion . this evolution defines a mapping of phase - space given by @xmath345 . the resulting phase space point @xmath346 and trajectory segment are then accepted with probability @xmath347 \right\ } , \label{probacceptance}\ ] ] where @xmath348 and @xmath349 this algorithm generates a markov chain of configurations with an asymptotic distribution given by the stationary canonical distribution defined in eq . ( [ canonicalprob ] ) provided that the phase space trajectory is _ time reversible _ and _ _ area preserving__@xcite . since free translational motion is time reversible , and the reversibility of the rotational equations of motion is evident from eq . ( [ pmat ] ) , the first requirement is satisfied . furthermore , since the invariant phase space metric is uniform for fully rigid bodies ( see eq . in sec . [ constraineddynamics ] ) , the _ area preserving _ condition is also satisfied . ideally , a dmd simulation satisfies @xmath350 so that according to eq . ( [ probacceptance ] ) every trajectory segment is accepted . in the less ideal , more realistic case in which collisions are occasionally missed , the hmc scheme provides a rigorous way of accepting or rejecting the segment . if a hard - core collision has been missed and the configuration at the end of a trajectory segment has molecules in unphysical regions of phase space where the potential energy is infinite , then @xmath351 and the new configuration and trajectory segment are always rejected . on the other hand , if only a square - well interaction has been missed , @xmath352 at the end of the trajectory segment is finite and there is a non - zero probability of accepting the trajectory . an analogous strategy can be devised to carry out microcanonical averages . in this case , the assignment of new initial velocities in the first step is still done randomly but in such a way that the total kinetic energy of the system remains constant . such a procedure can be carried out by exchanging center of mass velocities between randomly chosen pairs of molecules . the system is evolved dynamically through phase space for a fixed time @xmath344 and the new phase space point @xmath346 is accepted according to @xmath353 where @xmath352 is given by eq . ( [ deltah ] ) . clearly , the case @xmath354 only occurs when a collision has been missed , and in such a case the trajectory segment is never accepted . it should be emphasized that in the hmc scheme , a new starting configuration for a segment of time evolution is chosen only after every dmd time interval @xmath344 . an algorithm in which a new configuration is selected only after a collision is missed is likely to violate detailed balance , and is therefore not a valid monte - carlo scheme . on the other hand , the length of the trajectory segments @xmath344 in the hmc method outlined above can be chosen to be slightly larger than the relevant relaxation time of the system . such a choice allows one to use the deterministic phase space trajectory to compute time - dependent correlation functions from the _ exact _ dynamics of the system without rejecting a significant fraction of the trajectory segments . in this paper we have shown how to carry out discontinuous molecular dynamics simulations for arbitrary semi - flexible and rigid molecules . for semi - flexible bodies , the dynamics and collision rules have been derived from the principles of constrained lagrangian mechanics . the implementation of an efficient dmd method for semi - flexible systems is hindered by the fact that in almost all cases the equations of motion must be propagated numerically in an event searching algorithm so that the constraints are enforced at all times . nonetheless , such a scheme can be realized using the shake@xcite or rattle@xcite algorithms in combination with the root searching methods outlined here . the dynamics of a system of completely rigid molecules interacting through discontinuous potentials is more straightforward . for such a system , the euler equations for rigid body dynamics can be used to calculate the free evolution of a general rigid object . this analytical solution enables the design of efficient numerical algorithms for the search for collision events . in addition , the collision rules for calculating the discontinuous changes in the components of the center of mass velocity and angular momenta have been obtained for arbitrary bodies interacting through a point based on conservation principles . furthermore , the sampling of the canonical and microcanonical ensembles , as well as the handling of missed collisions , has also been discussed in the context of a hybrid monte carlo scheme . from an operational standpoint , the difference between the method of dmd and molecular dynamics using continuous potentials in rigid systems lies in the fact that the dmd approach does not require the calculation of forces and sequential updating of phase space coordinates at discrete ( and short ) time intervals since the response of the system to an impulse can be computed analytically . instead , the computational effort focuses on finding the precise time at which such impulses exert their influence . the basic building block outlined here for the numerical computation of collision times is a grid search , for which the positions of colliding atoms on a given pair of molecules need to be computed at equally spaced points in time . as outlined in sec . [ includingcollisions ] , this can be done efficiently starting with a completely explicit analytical form of the motion of a torque - free rigid body , without which the equations of motions would have to be integrated numerically . an efficient implementation of the dmd technique to find the time collision events should make use of a ) a large grid step combined with a threshold scenario to catch pathological cases , b ) sophisticated but standard techniques such as binary event trees , cell divisions , and local clocks , and c ) a new technique of finding collision times numerically that involves truncating the grid search and scheduling virtual collision events . on a fundamental level , it is natural to wonder whether the ` stepped ' form of a discontinuous potential could possibly model any realistic interaction . such concerns are essentially academic , since it is always possible to approximate a given interaction potential with as many ( small ) steps as one would like in order to approximate a given potential to any desired level of accuracy@xcite . of course , the drawback to mimicking a smooth potential with a discontinuous one with many steps is that the number of ` collision ' events that occur in the system per unit time scales with the number of steps in the potential . hence , one would expect that the efficiency of the simulation scales roughly inversely with the number of steps in the interaction potential . nonetheless , the issue is a practical one : how small can the number of steps in the interaction potential be such that one still gets a good description of the physics under investigation ? in the accompanying paper@xcite , we will see for benzene and methane that it takes surprisingly few steps ( e.g. a hard core plus a square - well interaction ) to get results which are very close to those of continuous molecular dynamics . additionally , we compare the efficiency of such simulations to simulations based on standard molecular dynamics methods . the authors would like to acknowledge support by grants from the natural sciences and engineering research council of canada ( nserc ) . to be more precise , no hamiltonian exists with @xmath228 and @xmath313 as conjugate variables . otherwise , @xmath355 should be equal to @xmath356 and @xmath357 should be equal to @xmath358 , whence @xmath359 , but eq . violates this relation . the threshold value is found by trial and error ; a trial simulation with a very small threshold value is run which is sure to miss a collision at some point due to a grazing collision ; the collision indicator function around the time of this grazing collision is inspected and the threshold value is adjusted such that this collision will not be missed . new trial simulation are run , and the threshold adjusted , until the frequency of missed grazing collisions is acceptable . this assumes no optimization whatsoever . it is easy to reduce the cost of finding the first possible collision times to @xmath254 per unit physical time , however , by storing only the first collision for every given particle@xcite .

a general framework for performing event - driven simulations of systems with semi - flexible or rigid bodies interacting under impulsive torques and forces is outlined . two different approaches are presented . in the first , the dynamics and interaction rules are derived from lagrangian mechanics in the presence of constraints . this approach is most suitable when the body is composed of relatively few point masses or is semi - flexible . in the second method , the equations of rigid bodies are used to derive explicit analytical expressions for the free evolution of arbitrary rigid molecules and to construct a simple scheme for computing interaction rules . efficient algorithms for the search for the times of interaction events are designed in this context , and the handling of missed interaction events is discussed .

cond-mat0607527

here we present the method we used to calculate the fano factor for the tunnel exclusion process in the sierpiski lattice and in the random - resistor network . we follow the master equation approach of refs . the two - dimensionality of our networks requires a more elaborate bookkeeping , which we manage by means of the hamiltonian formalism of ref . @xcite . we consider a network of @xmath53 sites , each of which is either empty or singly occupied . two sites are called adjacent if they are directly connected by at least one bond . a subset @xmath54 of the @xmath53 sites is connected to the source and a subset @xmath55 is connected to the drain . each of the @xmath56 possible states of the network is reached with a certain probability at time @xmath10 . we store these probabilities in the @xmath56-dimensional vector @xmath57 . its time evolution in the tunnel exclusion process is given by the master equation @xmath58 where the matrix @xmath59 contains the tunnel rates . the normalization condition can be written as @xmath60 , in terms of a vector @xmath61 that has all @xmath62 components equal to 1 . this vector is a left eigenstate of @xmath59 with zero eigenvalue @xmath63 because every column of @xmath59 must sum to zero in order to conserve probability . the right eigenstate with zero eigenvalue is the stationary distribution @xmath64 . all other eigenvalues of @xmath59 have a real part @xmath65 . we store in the vector @xmath66 the conditional probabilities that a state is reached at time @xmath10 after precisely @xmath67 charges have entered the network from the source . because the source remains occupied , a charge which has entered the network can not return back to the source but must eventually leave through the drain . one can therefore use @xmath67 to represent the number of transfered charges . the time evolution of @xmath66 reads @xmath68 where @xmath69 has been decomposed into a matrix @xmath70 containing all transitions by which @xmath67 does not change and a matrix @xmath71 containing all transitions that increase @xmath67 by 1 . the probability @xmath72 that @xmath67 charges have been transferred through the network at time @xmath10 represents the counting statistics . it describes the entire statistics of current fluctuations . the cumulants @xmath73 are obtained from the cumulant generating function @xmath74.\ ] ] the average current and fano factor are given by @xmath75 the cumulant generating function can be expressed in terms of a laplace transformed probability vector @xmath76 as @xmath77 transformation of eq . gives @xmath78 where we have introduced the counting matrix @xmath79 the cumulant generating function follows from @xmath80 the long - time limit of interest for the fano factor can be implemented as follows @xcite . let @xmath81 be the eigenvalue of @xmath82 with the largest real part , and let @xmath83 be the corresponding ( normalized ) right eigenstate , @xmath84 since the largest eigenvalue of @xmath85 is zero , we have @xmath86 ( note that @xmath87 is the stationary distribution @xmath88 introduced earlier . ) in the limit @xmath89 only the largest eigenvalue contributes to the cumulant generating function , @xmath90 = \mu(\chi).\ ] ] the construction of the counting matrix @xmath82 is simplified by expressing it in terms of raising and lowering operators , so that it resembles a hamiltonian of quantum mechanical spins @xcite . first , consider a single site with the basis states @xmath91 ( vacant ) and @xmath92 ( occupied ) . we define , respectively , raising and lowering operators @xmath93 we also define the electron number operator @xmath94 and the hole number operator @xmath95 ( with @xmath96 the @xmath97 unit matrix ) . each site @xmath98 has such operators , denoted by @xmath99 , @xmath100 , @xmath101 , and @xmath102 . the matrix @xmath82 can be written in terms of these operators as @xmath103 where all tunnel rates have been set equal to unity . the first sum runs over all ordered pairs @xmath104 of adjacent sites . these are hermitian contributions to the counting matrix . the second sum runs over sites in @xmath105 connected to the source , and the third sum runs over sites in @xmath106 connected to the drain . these are non - hermitian contributions . it is easy to convince oneself that @xmath85 is indeed @xmath59 of eq . , since every possible tunneling event corresponds to two terms in eq . : one positive non - diagonal term responsible for probability gain for the new state and one negative diagonal term responsible for probability loss for the old state . in accordance with eq . , the full @xmath82 differs from @xmath59 by a factor @xmath107 at the terms associated with charges entering the network . in view of eq . , the entire counting statistics in the long - time limit is determined by the largest eigenvalue @xmath81 of the operator . however , direct calculation of that eigenvalue is feasible only for very small networks . our approach , following ref . @xcite , is to derive the first two cumulants by solving a hierarchy of linear equations . we will now express @xmath81 in terms of @xmath112 . we start from the definition . if we act with @xmath61 on the left - hand - side of eq . we obtain @xmath113 in the second equality we have used eq . [ which holds since @xmath114 . acting with @xmath61 on the the right - hand - side of eq . we obtain just @xmath81 , in view of eq .. hence we arrive at @xmath115 to obtain @xmath116 we set up a system of linear equations starting from @xmath120 commuting @xmath101 to the right , using the commutation relations @xmath121 = s_i^+$ ] and @xmath122 = - s_i^-$ ] , we find @xmath123 the notation @xmath124 means that the sum runs over all sites @xmath125 adjacent to @xmath98 . the number @xmath126 is the total number of bonds connected to site @xmath98 ; @xmath127 of these bonds connect site @xmath98 to the source . in order to compute @xmath109 we set @xmath128 in eq . , use eq . to set the left - hand - side to zero , and solve the resulting symmetric sparse linear system of equations , @xmath129 this is the first level of the hierarchy . substitution of the solution into eq . gives the average current @xmath5 . to calculate the fano factor via eq . we also need @xmath130 . we take eq . , substitute eq . for @xmath81 , differentiate and set @xmath128 to arrive at @xmath131 to find @xmath110 we note that @xmath132 and commute @xmath101 to the right . setting @xmath128 provides the second level of the hierarchy of linear equations , @xmath133 the number @xmath134 is the number of bonds connecting sites @xmath98 and @xmath125 if they are adjacent , while @xmath135 if they are not adjacent . 99 i. webman , phys . . lett . * 47 * , 1496 ( 1981 ) . s. alexander and r. orbach , j. phys . ( paris ) * 43 * , l625 ( 1982 ) . d. ben - avraham and s. havlin , j. phys . a * 15 * , l691 ( 1982 ) . y. gefen , a. aharony , and s. alexander , phys . lett . * 50 * , 77 ( 1983 ) . r. rammal and g. toulouse , j. phys . ( paris ) * 44 * , l13 ( 1983 ) . s. havlin and d. ben - avraham , adv . phys . * 36 * , 695 ( 1987 ) . m. b. isichenko , rev . * 64 * , 961 ( 1992 ) . d. stauffer and a. aharony , _ introduction to percolation theory _ ( taylor and francis , 1994 ) . s. redner , arxiv:0710.1105 . b. i. shklovskii and a. l. efros , _ electronic properties of doped semiconductors _ ( springer , 1984 ) . a. bunde and s. havlin , editors , _ fractals and disordered systems _ ( springer , 1996 ) . s. a. trugman , phys . b 27 , 7539 ( 1983 ) . v. v. cheianov , v. i. falko , b. l. altshuler , and i. l. aleiner , arxiv:0706.2968 . j. martin , n. akerman , g. ulbricht , t. lohmann , j. h. smet , k. von klitzing , and a. yacoby , nature phys . * 4 * , 144 ( 2008 ) . e. h. hwang , s. adam , and s. dassarma , phys . 98 * , 186806 ( 2007 ) . graphene is a single layer of carbon atoms , forming a two - dimensional honeycomb lattice . electrical conduction is provided by overlapping @xmath136-orbitals , with on average one electron per @xmath136-orbital in undoped graphene . electron puddles have a little more than one electron per @xmath136-orbital ( _ n_-type doping ) , while hole puddles have a little less than one electron per @xmath136-orbital ( _ p_-type doping ) . an electron gas is called `` degenerate '' if the average occupation number of a quantum state is either close to unity or close to zero . it is called `` nondegenerate '' if the average occupation number is much smaller than unity for all states . m. blanter and m. bttiker , phys . rep . * 336 * , 1 ( 2000 ) . c. w. j. beenakker and c. schnenberger , physics today , may 2003 , p. 37 . l. dicarlo , j. r. williams , y. zhang , d. t. mcclure , and c. m. marcus , arxiv:0711.3206 . r. danneau , f. wu , m. f. craciun , s. russo , m. y. tomi , j. salmilehto , a. f. morpurgo , and p. j. hakonen , arxiv:0711.4306 . j. tworzydo , b. trauzettel , m. titov , a. rycerz , and c. w. j. beenakker , phys . * 96 * , 246802 ( 2006 ) . p. san - jose , e. prada , and d. s. golubev , phys . b * 76 * , 195445 ( 2007 ) . c. h. lewenkopf , e. r. mucciolo , and a. h. castro neto , arxiv:0711.3202 . c. w. j. beenakker and m. bttiker , phys . rev . b * 46 * , 1889 ( 1992 ) . k. e. nagaev , phys . lett . a * 169 * , 103 ( 1992 ) . r. rammal , j. phys . ( paris ) * 45 * , l1007 ( 1984 ) . a. hansen and m. nelkin , phys . b * 33 * , 649 ( 1986 ) . b. fourcade and a .- m . s. tremblay , phys . b * 34 * , 7802 ( 1986 ) . v. v. kuznetsov , e. e. mendez , x. zuo , g. l. snider , and e. t. croke , phys . rev . lett . * 85 * , 397 ( 2000 ) . f. e. camino , v. v. kuznetsov , e. e. mendez , m. e. gershenson , d. reuter , p. schafmeister , and a. d. wieck , phys . b * 68 * , 073313 ( 2003 ) . y. a. kinkhabwala , v. a. sverdlov , a. n. korotkov , and k. k. likharev , j. phys . matter * 18 * , 1999 ( 2006 ) . v. nazarov , phys . lett . * 73 * , 134 ( 1994 ) . e. v. sukhorukov and d. loss , phys . lett . * 80 * , 4959 ( 1998 ) . w. sierpiski , c. r. acad . paris * 160 * , 302 ( 1915 ) . r. c. liu , p. eastman , and y. yamamoto , solid state comm . * 102 * , 785 ( 1997 ) . roche , b. derrida , and b. douot , eur . j. b * 43 * , 1434 ( 2005 ) . d. a. bagrets and yu . v. nazarov , phys . b * 67 * , 085316 ( 2003 ) . g. m. schtz , in _ phase transitions and critical phenomena _ , edited by c. domb and j. l. lebowitz ( academic , 2001 ) .

by solving a master equation in the sierpiski lattice and in a planar random - resistor network , we determine the scaling with size @xmath0 of the shot noise power @xmath1 due to elastic scattering in a fractal conductor . we find a power - law scaling @xmath2 , with an exponent depending on the fractal dimension @xmath3 and the anomalous diffusion exponent @xmath4 . this is the same scaling as the time - averaged current @xmath5 , which implies that the fano factor @xmath6 is scale independent . we obtain a value @xmath7 for anomalous diffusion that is the same as for normal diffusion , even if there is no smallest length scale below which the normal diffusion equation holds . the fact that @xmath8 remains fixed at @xmath9 as one crosses the percolation threshold in a random - resistor network may explain recent measurements of a doping - independent fano factor in a graphene flake . diffusion in a medium with a fractal dimension is characterized by an anomalous scaling with time @xmath10 of the root - mean - squared displacement @xmath11 . the usual scaling for integer dimensionality @xmath12 is @xmath13 , independent of @xmath12 . if the dimensionality @xmath3 is noninteger , however , an anomalous scaling @xmath14 with @xmath15 may appear . this anomaly was discovered in the early 1980 s @xcite and has since been studied extensively ( see refs . @xcite for reviews ) . intuitively , the slowing down of the diffusion can be understood as arising from the presence of obstacles at all length scales characteristic of a selfsimilar fractal geometry . a celebrated application of the theory of fractal diffusion is to the scaling of electrical conduction in random - resistor networks ( reviewed in refs.@xcite ) . according to ohm s law , the conductance @xmath16 should scale with the linear size @xmath0 of a @xmath12-dimensional network as @xmath17 . in a fractal dimension the scaling is modified to @xmath18 , depending both on the fractal dimensionality @xmath3 and on the anomalous diffusion exponent @xmath4 . at the percolation threshold , the known @xcite values for @xmath19 are @xmath20 and @xmath21 , leading to a scaling @xmath22 . this almost inverse - linear scaling of the conductance of a planar random - resistor network contrasts with the @xmath0-independent conductance @xmath23 predicted by ohm s law in two dimensions . all of this body of knowledge applies to classical resistors , with applications to disordered semiconductors and granular metals @xcite . the quantum hall effect provides one quantum mechanical realization of a random - resistor network @xcite , in a rather special way because time - reversal symmetry is broken by the magnetic field . very recently @xcite , cheianov , falko , altshuler , and aleiner announced an altogether different quantum realization in zero magnetic field . following experimental @xcite and theoretical @xcite evidence for electron and hole puddles in undoped graphene @xcite , cheianov et al . modeled this system by a degenerate electron gas in a random - resistor network . they analyzed both the high - temperature classical resistance , as well as the low - temperature quantum corrections , using the anomalous scaling laws in a fractal geometry . these very recent experimental and theoretical developments open up new possibilities to study quantum mechanical aspects of fractal diffusion , both with respect to the pauli exclusion principle and with respect to quantum interference ( which are operative in distinct temperature regimes ) . to access the effect of the pauli principle one needs to go beyond the time - averaged current @xmath5 ( studied by cheianov et al . @xcite ) , and consider the time - dependent fluctuations @xmath24 of the current in response to a time - independent applied voltage @xmath25 . these fluctuations exist because of the granularity of the electron charge , hence their name `` shot noise '' ( for reviews , see refs . @xcite ) . shot noise is quantified by the noise power @xmath26 and by the fano factor @xmath6 . the pauli principle enforces @xmath27 , meaning that the noise power is smaller than the poisson value @xmath28which is the expected value for independent particles ( poisson statistics ) . the investigation of shot noise in a fractal conductor is particularly timely in view of two different experimental results @xcite that have been reported recently . both experiments measure the shot noise power in a graphene flake and find @xmath27 . a calculation @xcite of the effect of the pauli principle on the shot noise of undoped graphene predicted @xmath7 in the absence of disorder , with a rapid suppression upon either _ p_-type or _ n_-type doping . this prediction is consistent with the experiment of danneau et al . @xcite , but the experiment of dicarlo et al.@xcite gives instead an approximately _ doping - independent _ @xmath8 near @xmath9 . computer simulations @xcite suggest that disorder in the samples of dicarlo et al . might cause the difference . motivated by this specific example , we study here the fundamental problem of shot noise due to anomalous diffusion in a fractal conductor . while _ equilibrium _ thermal noise in a fractal has been studied previously @xcite , it remains unknown how anomalous diffusion might affect the _ nonequilibrium _ shot noise . existing studies @xcite of shot noise in a percolating network were in the regime where _ inelastic _ scattering dominates , leading to hopping conduction , while for diffusive conduction we need predominantly _ elastic _ scattering . we demonstrate that anomalous diffusion affects @xmath1 and @xmath5 in such a way that the fano factor ( their ratio ) becomes scale independent as well as independent of @xmath3 and @xmath4 . anomalous diffusion , therefore , produces the same fano factor @xmath7 as is known @xcite for normal diffusion . this is a remarkable property of diffusive conduction , given that hopping conduction in a percolating network does not produce a scale - independent fano factor @xcite . our general findings are consistent with the doping independence of the fano factor in disordered graphene observed by dicarlo et al . @xcite . to arrive at these conclusions we work in the experimentally relevant regime where the temperature @xmath29 is sufficiently high that the phase coherence length is @xmath30 , and sufficiently low that the inelastic length is @xmath31 . quantum interference effects can then be neglected , as well as inelastic scattering events . the pauli principle remains operative if the thermal energy @xmath32 remains well below the fermi energy , so that the electron gas remains degenerate . we first briefly consider the case that the anomalous diffusion on long length scales is preceded by normal diffusion on short length scales . this would apply , for example , to a percolating cluster of electron and hole puddles with a mean free path @xmath33 which is short compared to the typical size @xmath34 of a puddle . we can then rely on the fact that @xmath7 for a conductor of any shape , provided that the normal diffusion equation holds locally @xcite , to conclude that the transition to anomalous diffusion on long length scales must preserve the one - third fano factor . this simple argument can not be applied to the more typical class of fractal conductors in which the normal diffusion equation does not hold on short length scales . as representative for this class , we consider fractal lattices of sites connected by tunnel barriers . the local tunneling dynamics then crosses over into global anomalous diffusion , without an intermediate regime of normal diffusion . lower panel : electrical conduction through a sierpiski lattice . this is a deterministic fractal , constructed by recursively removing a central triangular region from an equilateral triangle . the recursion level @xmath35 quantifies the size @xmath36 of the fractal in units of the elementary bond length @xmath34 ( the inset shows the fourth recursion ) . the conductance @xmath37 ( open dots , normalized by the tunneling conductance @xmath38 of a single bond ) and shot noise power @xmath1 ( filled dots , normalized by @xmath39 ) are calculated for a voltage difference @xmath25 between the lower - left and lower - right corners of the lattice . both quantities scale as @xmath40 ( solid lines on the double - logarithmic plot ) . the fano factor @xmath41 rapidly approaches @xmath9 , as shown in the upper panel . ] a classic example is the sierpiski lattice @xcite shown in fig.[fig_sier ] ( inset ) . each site is connected to four neighbors by bonds that represent the tunnel barriers , with equal tunnel rate @xmath42 through each barrier . the fractal dimension is @xmath43 and the anomalous diffusion exponent is @xcite @xmath44 . the pauli exclusion principle can be incorporated as in ref . @xcite , by demanding that each site is either empty or occupied by a single electron . tunneling is therefore only allowed between an occupied site and an adjacent empty site . a current is passed through the lattice by connecting the lower left corner to a source ( injecting electrons so that the site remains occupied ) and the lower right corner to a drain ( extracting electrons so that the site remains empty ) . the resulting stochastic sequence of current pulses is the `` tunnel exclusion process '' of ref . @xcite . the statistics of the current pulses can be obtained exactly ( albeit not in closed form ) by solving a master equation @xcite . we have calculated the first two cumulants by extending to a two - dimensional lattice the one - dimensional calculation of ref . @xcite . to manage the added complexity of an extra dimension we found it convenient to use the hamiltonian formulation of ref . @xcite . the hierarchy of linear equations that we need to solve in order to obtain @xmath5 and @xmath1 is derived in the appendix . the deviation of the fano factor from @xmath9 scales to zero as a power law for the sierpiski lattice ( triangles ) and for the random - resistor network ( circles ) . ] the results in fig . [ fig_sier ] demonstrate , firstly , that the shot noise power @xmath1 scales as a function of the size @xmath0 of the lattice with the same exponent @xmath45 as the conductance ; and , secondly , that the fano factor @xmath8 approaches @xmath9 for large @xmath0 . more precisely , see fig.[fig_fano ] , we find that @xmath46 scales to zero as a power law , with @xmath47 for our largest @xmath0 . same as fig . [ fig_sier ] , but now for the random - resistor network of disordered graphene introduced by cheianov et al . @xcite . the inset shows one realization of the network for @xmath48 ( the data points are averaged over @xmath49 such realizations ) . the alternating solid and dashed lattice sites represent , respectively , the electron ( _ n _ ) and hole ( _ p _ ) puddles . horizontal bonds ( not drawn ) are _ p - n _ junctions , with a negligibly small conductance @xmath50 . diagonal bonds ( solid and dashed lines ) each have the same tunnel conductance @xmath38 . current flows from the left edge of the square network to the right edge , while the upper and lower edges are connected by periodic boundary conditions . this plot is for undoped graphene , corresponding to an equal fraction of solid ( _ n - n _ ) and dashed ( _ p - p _ ) bonds . ] turning now to the application to graphene mentioned in the introduction , we have repeated the calculation of shot noise and fano factor for the random - resistor network of electron and hole puddles introduced by cheianov et al . @xcite . the results , shown in fig . [ fig_puddles ] , demonstrate that the shot noise power @xmath1 scales with the same exponent @xmath51 as the conductance @xmath16 ( solid lines in the lower panel ) , and that the fano factor @xmath8 approaches @xmath9 for large networks ( upper panel ) . this is a random , rather than a deterministic fractal , so there remains some statistical scatter in the data , but the deviation of @xmath8 from @xmath9 for the largest lattices is still @xmath52 ( see the circular data points in fig . [ fig_fano ] ) . in conclusion , we have found that the universality of the one - third fano factor , previously established for normal diffusion @xcite , extends to anomalous diffusion as well . this universality might have been expected with respect to the fractal dimension @xmath3 ( since the fano factor is dimension independent ) , but we had not expected universality with respect to the anomalous diffusion exponent @xmath4 . the experimental implication of the universality is that the fano factor remains fixed at @xmath9 as one crosses the percolation threshold in a random - resistor network thereby crossing over from anomalous diffusion to normal diffusion . this is consistent with the doping - independent fano factor measured in a graphene flake by dicarlo et al . @xcite . a discussion with l. s. levitov motivated this research , which was supported by the netherlands science foundation nwo / fom . we also acknowledge support by the european community s marie curie research training network under contract mrtn - ct-2003 - 504574 , fundamentals of nanoelectronics .

0803.2709

classification of networked data is a quite attractive field with applications in computer vision , bioinformatics , spam detection and text categorization . in recent years networked data have become widespread due to the increasing importance of social networks and other web - related applications . this growing interest is pushing researchers to find scalable algorithms for important practical applications of these problems . + in this paper we focus our attention on a task called _ node classification _ , often studied in the semi - supervised setting @xcite . recently , different teams studied the problem from a theoretic point of view with interesting results . for example @xcite developed on - line fast predictors for weighted and unweighted graphs and herbster et al . developed different versions of the perceptron algorithm to classify the nodes of a graph ( @xcite ) . @xcite introduced a game - theoretic framework for node classification . we adopt the same approach and , in particular , we obtain a scalable algorithm by finding a nash equilibrium on a special instance of their game . the main difference between our algorithm and theirs is the high scalability achieved by our approach . this is really important in practice , since it makes possible to use our algorithm on large scale problems . given a weighted graph @xmath0 , a labeling of @xmath1 is an assignment @xmath2 where @xmath3 . + we expect our graph to respect a notion of regularity where adjacent nodes often have the same label : this notion of regularity is called _ homophily_. most machine learning algorithms for node classification ( @xcite ) adopt this bias and exploit it to improve their performances . + the learner is given the graph @xmath1 , but just a subset of @xmath4 , that we call training set . the learner s goal is to predict the remaining labels minimizing the number of mistakes . @xcite introduce also an irregularity measure of the graph @xmath1 , for the labeling @xmath4 , defined as the ratio between the sum of the weights of the edges between nodes with different labels and the sum of all the weights . intuitively , we can view the weight of an edge as a similarity measure between two nodes , we expect highly similar nodes to have the same label and edges between nodes with different labels being `` light '' . based on this intuition , we may assign labels to non - training nodes so to minimize some function of the induced weighted cut . in the binary classification case , algorithms based on min - cut have been proposed in the past ( for example @xcite ) . generalizing this approach to the multiclass case , naturally takes us to the _ multi - way cut _ ( or multi - terminal cut see @xcite ) problem . given a graph and a list of terminal nodes , find a set of edges such that , once removed , each terminal belongs to a different component . the goal is to minimize the sum of the weights of the removed edges . + unfortunately , the multi - way cut problem is max snp - hard when the number of terminals is bigger than two ( @xcite ) . furthermore , efficient algorithms to find the multi - way cut on special instances of the problem are known , but , for example , it is not clear if it is possible to reduce a node classification problem on a tree to a multi - way cut on a tree . in this section we describe the game introduced by @xcite that , in a certain sense , aims at distributing over the nodes the cost of approximating the multi - way cut . this is done by expressing the labels assignment as a nash equilibrium . we have to keep in mind that , since this game is non - cooperative , each player maximizes its own payoff disregarding what it can do to maximize the sum of utilities of all the players ( the so - called social welfare ) . the value of the multi - way cut is strongly related to the value of the social welfare of the game , but in the general case a nash equilibrium does not give any guarantee about the collective result . + in the graph transduction game ( later called gtg ) , the graph topology is known in advance and we consider each node as a player . each possible label of the nodes is a pure strategy of the players . since we are working in a batch setting , we will have a train / test split that induces two different kind of players : * * determined players*(@xmath5 ) those are nodes with a known label ( train set ) , so in our game they will be players with a fixed strategy ( they do not change their strategy since we can not change the labels given as training set ) * * undetermined players*(@xmath6 ) those that do not have a fixed strategy and can choose whatever strategy they prefer ( we have to predict their labels ) the game is defined as @xmath7 , where @xmath8 is the set of players , @xmath9 is the joint strategy space ( the cartesian product of all strategy sets @xmath10 ) , and @xmath11 is the combined payoff function which assigns a real valued payoff @xmath12 to each pure strategy profile @xmath13 and player @xmath14 . a mixed strategy of player @xmath14 is a probability distribution @xmath15 over the set of the pure strategies of @xmath16 . each pure strategy @xmath17 corresponds to a mixed strategy where all the strategies but the @xmath17-th one have probability equals to zero . we define the utility function of the player @xmath16 as @xmath18 where @xmath19 is the probability of @xmath20 . we assume the payoff associated to each player is additively separable ( this will be clear in the following lines ) . this makes gtg a member of a subclass of the multi - player games called polymatrix games . for a pure strategy profile @xmath21 , the payoff function of every player @xmath14 is : @xmath22 where @xmath23 means that @xmath16 and @xmath24 are neighbors , this can be written in matrix form as @xmath25 where @xmath26 is the partial payoff matrix between @xmath16 and @xmath24 , defined as @xmath27 , where @xmath28 is the identity matrix of size @xmath29 and @xmath30 represent the element of @xmath31 at row @xmath15 and column @xmath4 . the utility function of each player @xmath32 can be re - written as follows : [ cols= " > , < " , ] the results of our experiments , shown in table [ t : multi ] , are not conclusive , but we can observe some interesting trends : * it is not really clear which one between gtg - ess and labprop is the most accurate algorithm , but anyway @xmath33 is always competitive with them . * @xmath33 is always much better than wmv . as expected wmv works better on `` not too sparse '' graphs such ghgraph , but even in this case it is outperformed by @xmath33 . * gtg - ess and labprop s time complexity did not permit us to run them in a reasonable amount of time with our computational resources . we introduced a novel scalable algorithm for multiclass node classification in arbitrary weighted graphs . our algorithm is motivated within a game theoretic framework , where test labels are expressed as the nash equilibrium of a certain game . in practice , mucca works well even on binary problems against competitors like label propagation and shazoo that have been specifically designed for the binary setting . several questions remain open . for example , committees of mucca predictors work well but we do not know whether there are better ways to aggregate their predictions . also , given their common game - theoretic background , it would be interesting to explore possible connections between committees of mucca predictors and gtg - ess .

we introduce a scalable algorithm , mucca for multiclass node classification in weighted graphs . unlike previously proposed methods for the same task , mucca works in time linear in the number of nodes . our approach is based on a game - theoretic formulation of the problem in which the test labels are expressed as a nash equilibrium of a certain game . however , in order to achieve scalability , we find the equilibrium on a spanning tree of the original graph . experiments on real - world data reveal that mucca is much faster than its competitors while achieving a similar predictive performance .

1112.4344