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Research ObjectivesThe MILC Collaboration is engaged in a broad research program in Quantum Chromodynamics (QCD).This research addresses fundamental questions in high energy and nuclear physics,and is directly related to major experimental programs in thesefields.It includes studies of the mass spectrum of strongly interacting particles,the weak interactions of these particles,and the behavior of strongly interacting matter under extreme conditions.The Standard Model of High Energy Physics encompasses our current knowledge of the funda-mental interactions of subatomic physics.It consists of two quantumfield theories:the Weinberg-Salaam theory of electromagnetic and weak interactions,and QCD,the theory of the strong interac-tions.The Standard Model has been enormously successful in explaining a wealth of data produced in accelerator and cosmic ray experiments over the past thirty years;however,our knowledge of it is incomplete because it has been difficult to extract many of the most interesting predictions of QCD,those that depend on the strong coupling regime of the theory,and therefore require non-perturbative calculations.At present,the only means of carrying out non-perturbative QCD calculations fromfirst principles and with controlled errors,is through large scale numerical sim-ulations within the framework of lattice gauge theory.These simulations are needed to obtain a quantitative understanding of the physical phenomena controlled by the strong interactions,to de-termine a number of the fundamental parameters of the Standard Model,and to make precise tests of the Standard Model’s range of validity.Despite the many successes of the Standard Model,it is believed by high energy physicists that to understand physics at the shortest distances,a more general theory,which unifies all four of the fundamental forces of nature,will be required.The Standard Model is expected to be a limiting case of this more general theory,just as classical mechanics is a limiting case of the more general quantum mechanics.A central objective of the experimental program in high energy physics,and of lattice QCD simulations,is to determine the range of validity of the Standard Model,and to search for new physics beyond it.Thus,QCD simulations play an important role in efforts to obtain a deeper understanding of the fundamental laws of physics.QCD is formulated in the four-dimensional space-time continuum;however,in order to carry out numerical calculations one must reformulate it on a lattice or grid.It should be emphasized that the lattice formulation of QCD is not merely a numerical approximation to the continuum formu-lation.The lattice regularization of QCD is every bit as valid as continuum regularizations.The lattice spacing a establishes a momentum cutoffπ/a that removes ultraviolet divergences.Stan-dard renormalization methods apply,and in the perturbative regime they allow a straightforward conversion of lattice results to any of the standard continuum regularization schemes.Lattice QCD calculations proceed in two steps.In thefirst,one uses importance sampling tech-niques to generate gauge configurations,which are representative samples from the Feynman path integrals that define QCD.These configurations are saved,and in the second step they are used to calculate a wide variety of physical quantities.It is necessary to generate configurations with a range of lattice spacings,and then perform extrapolations to the zero lattice spacing limit.Fur-thermore,the computational cost of calculations rises as the masses of the quarks,the fundamental constituents of strongly interacting matter,decrease.Until recently,it has been too expensive to carry out calculations with the masses of the two lightest quarks,the up and the down,set to their physical values.Instead,one has performed calculations for a range of up and down quark masses, and extrapolated to their physical values guided by chiral perturbation theory,an effectivefield theory that determines how physical quantities depend on the masses of the lightest quarks.The extrapolations in lattice spacing(continuum extrapolation)and quark mass(chiral extrapolation) are the major sources of systematic errors in QCD calculations,and both must be under control in order to obtain trustworthy results.In our current simulations,we are,for thefirst time,working at or near the physical masses of the up and down quarks.The gauge configurations produced in these simulations greatly reduce,and will eventually eliminate,the systematic errors associatedwith the chiral extrapolation.A number of different formulations of QCD on the lattice are currently in use by lattice gauge theorists,all of which are expected to give the same results in the continuum limit.In recent years, major progress has been made in thefield through the development of improved formulations(im-proved actions)which reducefinite lattice spacing artifacts.Approximately twelve years ago,we developed one such improved action called asqtad[1],which significantly increased the accuracy of our simulations for a given amount of computing resources.We have used the asqtad action to generate an extensive library of gauge configurations with small enough lattice spacings and light enough quark masses to perform controlled calculations of a number of physical quantities. Computational resources provided by the DOE and NSF have enabled us to complete our program of generating asqtad gauge configurations.These configurations are publicly available,and have been used by us and by other groups to study a wide range of physical phenomena of importance in high energy and nuclear physics.Ours was thefirst set of full QCD ensembles that enabled control over both the continuum and chiral extrapolations.We have published a review paper describing the asqtad ensembles and the many calculations that were performed with them up to2009[2]. Over the last decade,a major component of our work has been to use our asqtad gauge config-urations to calculate quantities of importance to experimental programs in high energy physics. Particular emphasis was placed on the study of the weak decays and mixings of strongly interact-ing particles in order to determine some of the least well known parameters of the standard model and to provide precise tests of the standard model.The asqtad ensembles have enabled the calcu-lation of a number of physical quantities to a precision of1%–5%,and will enable many more quantities to be determined to this precision in the coming years.These results are already having an impact on experiments in high energy physics;however,in some important calculations,partic-ularly those related to tests of the standard model,higher precision is needed than can be provided by the existing asqtad ensembles.In order to obtain the required precision,we are now working with the Highly Improved Staggered Quark(HISQ)action developed by the HPQCD Collabora-tion[3].We have performed tests of scaling in the lattice spacing using HISQ valence quarks with gauge configurations generated with HISQ sea quarks[4].We found that lattice artifacts for the HISQ action are reduced by approximately a factor of2.5from those of the asqtad action for the same lattice spacing,and taste splittings in the pion masses are reduced by approximately a factor of three,which is sufficient to enable us to undertake simulations with the mass of the Goldstone pion at or near the physical pion mass.(“Taste”refers to the different ways one can construct the same physical particle in the staggered quark formalism.Although particles with different tastes become identical in the continuum limit,their masses can differ atfinite lattice spacing).More-over,the improvement in the quark dispersion relation enables us to include charm sea quarks in the simulations.The properties of the HISQ ensembles are described in detail in Ref.[5],and the first physics calculations using the physical quark mass ensembles in Refs.[6,7,8].The current status of the HISQ ensemble generation project is described at the link HISQ Lattice Generation and some initial calculations with them at Recent Results.The HISQ action also has major advan-tages for the study of QCD at high temperatures,so we have started to use it in our studies of this subject.Projects using the HISQ action will be a major component of our research for the next several years.Our research is currently focused on three major areas:1)the properties of light pseudoscalar mesons,2)the decays and mixings of heavy-light mesons,3)the properties of strongly interacting matter at high temperatures.We briefly discuss our research in each of these areas at the link Recent Results.References[1]The MILC Collaboration:C.Bernard et al.,Nucl.Phys.(Proc.Suppl.),60A,297(1998);Phys.Rev.D58,014503(1998);G.P.Lepage,Nucl.Phys.(Proc.Suppl.),60A,267(1998);Phys.Rev.D59,074501(1999);Kostas Orginos and Doug Toussaint(MILC),Nucl.Phys.(Proc.Suppl.),73,909(1999);Phys.Rev.D59,014501(1999);Kostas Orginos,Doug Tou-ssaint and R.L.Sugar(MILC),Phys.Rev.D60,054503(1999);The MILC Collaboration:C.Bernard et al.,Phys.Rev.D61,111502(2000).[2]The MILC Collaboration: A.Bazavov et al.,Rev.Mod.Phys.82,1349-1417(2010)[arXiv:0903.3598[hep-lat]].[3]The HPQCD/UKQCD Collaboration: E.Follana et al.,Phys.Rev.D73,054502(2007)[arXiv:hep-lat/0610092].[4]The MILC Collaboration: A.Bazavov al.,Phys.Rev.D82,074501(2010)[arXiv:1004.0342].[5]The MILC Collaboration: A.Bazavov al.,Phys.Rev.D87,054505(2013)[arXiv:1212.4768].[6]The MILC Collaboration: A.Bazavov et al.,Phys.Rev.Lett.110,172003(2013)[arXiv:1301.5855].[7]The Fermilab Lattice and MILC Collaborations:A.Bazavov,et al.,Phys.Rev.Lett.112,112001(2014)[arXiv:1312.1228].[8]The MILC Collaboration:A.Bazavov et al.,Proceedings of Science(Lattice2013)405(2013)[arXiv:1312.0149].。
a r X i v :h e p -l a t /9709021v 1 9 S e p 1997Dual variables for the SU (2)lattice gauge theory at finite temperatureSrinath CheluvarajaTheoretical Physics GroupTata Institute of Fundamental ResearchHomi Bhabha Road,Mumbai 400005,IndiaWe study the three-dimensional SU (2)lattice gauge theory at finite temperature using an observable which is dual to the Wilson line.This observable displays a behaviour which is the reverse of that seen for the Wilson line.It is non-zero in the confined phase and becomes zero in the deconfined phase.At large distances,it’s correlation function falls offexponentially in the deconfined phase and remains non-zero in the confined phase.The dual variable is non-local and has a string attached to it which creates a Z (2)interface in the system.It’s correlation function measures the string tension between oppositely oriented Z (2)domains.The construction of this variable can also be made in the four-dimensional theory where it measures the surface tension between oppositely oriented Z (2)domains.e-mail:srinath@theory.tifr.res.in1Dual variables have played an important role in statistical mechanical systems[1].These variables display a behaviour which is the opposite of that seen for the order parameters.They are non-zero in the disordered phase and remain zero in the ordered phase.Hence they are commonly referred to as disorder variables.Unlike the order parameters which are local observables and measure long range order in a statistical mechanical system,the dual variables are non-local and are sensitive to disordering effects which often arise as a consequence of topological excitations supported by a system-like vortices,magnetic monopoles etc.Disorder variables for the U(1)LGT have been studied recently[2].In this paper we study thefinite temperature properties of the three-dimensional SU(2)lattice gauge theory using an observable which is dual to the Wilson line.We explain the sense in which this is dual to the Wilson line and show that it’s behaviour is the reverse of that observed for the Wilson line.Unlike the Wilson line which creates a static quark propagating in a heat bath,the dual variable creates a Z(2)interface in the system.The definition of this variable can also be extended to the four-dimensional theory.Before we consider the three-dimensional SU(2)lattice gauge theory let us briefly recall the construction of the dual variable for the two-dimensional Ising model[3].The variable dual to the spin variableσ( n) is denoted byµ(⋆ n)and is defined on the dual lattice.This variable which is shown in Fig.1has a string attached to it which pierces the bonds connecting the spin variables.The position of the string is notfixed and it can be varied using a Z(2)(σ( n)→−σ( n))transformation.The average value of the dual variable is defined asZ(˜K)<µ(⋆ n)>=The dual variableµ(⋆ n)thus creates an interface beginning from⋆ n.It has the following behaviour at high and low temperatures[3]<µ(⋆ n)>≈1for K small<µ(⋆ n)>≈0for K large.It is in this sense that the variableµ(⋆ n)is dual to the variableσ( n)which behaves as<σ( n)>≈0for K small<σ( n)>≈1for K large.The spin and dual correlation functions satisfy the relation<µ(⋆ n)µ(⋆ n′)>K>>1=<σ( n)σ( n′)>K<<1.(3) Using theσ→−σtransformation it can be shown that the correlation function of theµ’s is independent of the shape of the string joining⋆ n and⋆ n′.The variablesσ( n)andµ(⋆ n)satisfy the algebraσ( n)µ(⋆ n)=µ(⋆ n)σ( n)exp(iω),(4) whereω=0if the variableσdoes not lie on a bond pierced by the string attached toµ(⋆ n)andω=πotherwise.The above considerations generalize easily to the three-dimensional Z(2)gauge theory.The dual variables are again defined on the sites of the dual lattice and the string attached to them will now pierce plaquettes instead of bonds.Whenever a plaquette is pierced by a string the coupling constant changes sign just as in the case of the Ising model.One can similarly define correlation functions of these variables.Since the three-dimensional Z(2)gauge theory is dual to the the three-dimensional Ising model,the correlation functions of these variables will have a behaviour which is the reverse of the spin-spin correlation function in the three-dimensional Ising model.For the case of the SU(2)lattice gauge theory which is our interest here,the definition of these variables is more involved.However,since Z(2)is a subgroup of SU(2)one can define variables which are dual to the Z(2)degrees of freedom by following the same prescription as in the three-dimensional Z(2)gauge theory.The relevance and effectiveness of these variables will depend3on the role played by the Z(2)degrees of freedom in the SU(2)lattice gauge theory.The role of the center degrees of freedom in the SU(2)lattice gauge theory was also examined in[4].Since thefinite temperature transition in SU(N)lattice gauge theories is governed by the center(Z(N) for SU(N))degrees of freedom[5],we expect these variables to be useful in studying this transition.The usual analysis offinite temperature lattice gauge theories is carried out by studying the behaviour of the Wilson line which becomes non-zero across thefinite temperature transition[5].The non-zero value of the Wilson line indicates deconfinement of static quarks.The spatial degrees of freedom undergo no dramatic change across the transition and only serve to produce short-range interactions between the Wilson lines. Thus one gets an effective theory of Wilson lines in one lower dimension[6].The deconfinement transition can be monitored by either measuring the expectation value of the Wilson line or by looking at the behaviour of the Wilson line correlation function[7].In the confining phase,the correlation function is(for| n− n′| large)<L( n)L( n′)>≈exp(−σT| n− n′|)(5) while in the deconfining phase<L( n)L( n′)>≈constant.(6) We define the variableµ(⋆ n)on the dual lattice site⋆ n asµ(⋆ n)=Z(˜β)2ptr U(p).(8)The variablesµ(⋆ n)and L( n)satisfy the algebra4L( n)µ(⋆ n)=µ(⋆ n)L( n)exp(iω)(9) whereω=0if the plaquette pierced by the string attached toµ(⋆ n)is not touching any of the links belonging to L( n)andω=πif the plaquette makes contact with any of the links of L( n).The variables µ(⋆ n)and L( n)satisfy the same algebra as theσandµvariables in the Ising model.This is the same as the algebra of the order and disorder variables in[8].Note that this algebra is only satisfied if the string is taken to be in the spatial direction.The location of the string can again be changed by local Z(2) transformations.The correlation function of the dual variables is defined to beZ(˜β)<µ(⋆ x)µ(⋆ y)>=)Nτ n n′J( n− n′)trL( n)trL( n′).(12)2The term which gives this contribution is shown in Fig.2.When we calculate the correlation function in Eq.10(where x and y are only separated in space)using this approximation,one plaquette occurring in this diagram will contribute with the opposite sign(shown shaded in Figure.2)and will cause the bond between n and n′to have a coupling with the opposite sign.In Eq.12J( n− n′)contains the sign induced5on the bond.This feature will persist for every diagram contributing to the effective two-dimensional Ising model and it’s effect will be to create a disorder line from x to y.Thus this correlation function will behave exactly like the disorder variable in the two-dimensional Ising model and at large distances will fall offexponentially in the ordered phase and will approach a constant value in the disordered phase.We expect it to behave(for large| x− y|)as<µ( x)µ( y)>≈exp(−| x− y|/ξ)β>βcr<µ( x)µ( y)>≈µ2β<βcrWriting the above correlation function as<µ( x)µ( y)>=exp(−βτ(F( x− y))(13) we can interpret F as the free energy of an interface of length| x− y|.The inverse temperature is denoted byβτto distinguish it from the gauge theory couplingβ.In the ordered phase the interface energy increases linearly with the length of the interface while in the disordered phase it is independent of the length.In thefinite temperature system high temperature results in the ordering of the Wilson lines and low temperature results in the disordering of the Wilson lines.Therefore the dual variables will display ordering at low temperatures and disordering at high temperatures.A direct measurement of the dual variable results in large errors because the dual variable is the expo-nential of a sum of plaquettes andfluctuates greatly.We have directly measured the dual variable and the correlation function and found that they fall to zero at high temperatures and remain non-zero at low temperatures.Since the measurement had large errors we prefer to use the method in[11]where a similar problem was encountered in the measurement of the disorder variable in the U(1)LGT.Instead of directly measuring the correlation function we measure∂ln<µ>ρ( x, y)=−where p′denotes the plaquettes which are dual to the string joining x and y.In our case this quantity directly measures the free energy of the Z(2)interface between x and y.Hence we expect it to increase linearly with the interface length in the deconfining phase and approach a constant value in the confining phase.Also this variable is like any other statistical variable and is easier to measure numerically.The variableρcan be used to directly measure the interface string tension between oppositely oriented Z(2) domains.The behaviour of the quantityρis shown in Fig.3and Fig.4.In the confined phaseρapproaches a constant value at large distances while it increases linearly with distance in the deconfined phase.The slope of the straight line in Fig.3gives the interface string tension.The calculation ofρwas made on a 12∗∗23lattice with200000iterations.The values ofβused were2.5in the confined phase and5.5in the deconfined phase.The deconfinement transition on the Nτ=3lattice occurs atβ=4.1[10].The errors were estimated by blocking the data.We would now like to point out a few applications of these dual variables.The mass gap in the high temperature phase is determined by studying the large distance behaviour of the Wilson line correlation function.Since the Wilson line correlation function remains non-zero in the deconfined phase the long distance part is subtracted out to get the leading exponential.The dual variable correlation function already displays an exponential fall offin the high temperature phase and provides us with another method of estimating the mass gap.Also,since dual variables reverse the roles of strong and weak coupling,they provide an alternate way of looking at the system which may be convenient to address certain questions. In this case they can be used to determine the string tension between oppositely oriented Z(2)domains in the SU(2)gauge theory.The surface tension between oppositely oriented Z(2)domains in the four-dimensional theory has been calculated semi-classically in[12].The above construction of the dual variable can also be made in four dimensions.The only difference is that in four dimensions the dual variables are defined on loops in the dual lattice.The spatial string in three-dimensions is replaced by a spatial surface which has the loops as it’s the boundary.The dual variables are functionals of the surface bounding the loops.The correlation function of the dual variables is defined to be<µ(C,C′)>=<exp(−β p′tr U(p))>(16)7where the summation is over all plaquettes which are dual to the surface joining C and C′.Since the surface is purely spatial the plaquettes contributing to the summation are all space-time plaquettes.This correlation function will fall of exponentially as the area of the surface joining C and C′in the deconfined phase and will approach a constant value in the confined phase.A similar measurement ofρcan be used to determine the surface tension between oppositely oriented Z(2)domains in the four-dimensional gauge theory.8........................X FIG.1.Dual variable in the Ising model.10n n′333333 FIG.3.ρin the deconfining phase.12333333 FIG.4.ρin the confining phase.13。
a r X i v :c o n d -m a t /9503139v 1 27 M a r 1995Singularity of the density of states in the two-dimensional Hubbard model from finitesize scaling of Yang-Lee zerosE.Abraham 1,I.M.Barbour 2,P.H.Cullen 1,E.G.Klepfish 3,E.R.Pike 3and Sarben Sarkar 31Department of Physics,Heriot-Watt University,Edinburgh EH144AS,UK 2Department of Physics,University of Glasgow,Glasgow G128QQ,UK 3Department of Physics,King’s College London,London WC2R 2LS,UK(February 6,2008)A finite size scaling is applied to the Yang-Lee zeros of the grand canonical partition function for the 2-D Hubbard model in the complex chemical potential plane.The logarithmic scaling of the imaginary part of the zeros with the system size indicates a singular dependence of the carrier density on the chemical potential.Our analysis points to a second-order phase transition with critical exponent 12±1transition controlled by the chemical potential.As in order-disorder transitions,one would expect a symmetry breaking signalled by an order parameter.In this model,the particle-hole symmetry is broken by introducing an “external field”which causes the particle density to be-come non-zero.Furthermore,the possibility of the free energy having a singularity at some finite value of the chemical potential is not excluded:in fact it can be a transition indicated by a divergence of the correlation length.A singularity of the free energy at finite “exter-nal field”was found in finite-temperature lattice QCD by using theYang-Leeanalysisforthechiral phase tran-sition [14].A possible scenario for such a transition at finite chemical potential,is one in which the particle den-sity consists of two components derived from the regular and singular parts of the free energy.Since we are dealing with a grand canonical ensemble,the particle number can be calculated for a given chem-ical potential as opposed to constraining the chemical potential by a fixed particle number.Hence the chem-ical potential can be thought of as an external field for exploring the behaviour of the free energy.From the mi-croscopic point of view,the critical values of the chemical potential are associated with singularities of the density of states.Transitions related to the singularity of the density of states are known as Lifshitz transitions [15].In metals these transitions only take place at zero tem-perature,while at finite temperatures the singularities are rounded.However,for a small ratio of temperature to the deviation from the critical values of the chemical potential,the singularity can be traced even at finite tem-perature.Lifshitz transitions may result from topological changes of the Fermi surface,and may occur inside the Brillouin zone as well as on its boundaries [16].In the case of strongly correlated electron systems the shape of the Fermi surface is indeed affected,which in turn may lead to an extension of the Lifshitz-type singularities into the finite-temperature regime.In relating the macroscopic quantity of the carrier den-sity to the density of quasiparticle states,we assumed the validity of a single particle excitation picture.Whether strong correlations completely distort this description is beyond the scope of the current study.However,the iden-tification of the criticality using the Yang-Lee analysis,remains valid even if collective excitations prevail.The paper is organised as follows.In Section 2we out-line the essentials of the computational technique used to simulate the grand canonical partition function and present its expansion as a polynomial in the fugacity vari-able.In Section 3we present the Yang-Lee zeros of the partition function calculated on 62–102lattices and high-light their qualitative differences from the 42lattice.In Section 4we analyse the finite size scaling of the Yang-Lee zeros and compare it to the real-space renormaliza-tion group prediction for a second-order phase transition.Finally,in Section 5we present a summary of our resultsand an outlook for future work.II.SIMULATION ALGORITHM AND FUGACITY EXPANSION OF THE GRAND CANONICALPARTITION FUNCTIONThe model we are studying in this work is a two-dimensional single-band Hubbard HamiltonianˆH=−t <i,j>,σc †i,σc j,σ+U i n i +−12 −µi(n i ++n i −)(1)where the i,j denote the nearest neighbour spatial lat-tice sites,σis the spin degree of freedom and n iσis theelectron number operator c †iσc iσ.The constants t and U correspond to the hopping parameter and the on-site Coulomb repulsion respectively.The chemical potential µis introduced such that µ=0corresponds to half-filling,i.e.the actual chemical potential is shifted from µto µ−U412.(5)This transformation enables one to integrate out the fermionic degrees of freedom and the resulting partition function is written as an ensemble average of a product of two determinantsZ ={s i,l =±1}˜z = {s i,l =±1}det(M +)det(M −)(6)such thatM ±=I +P ± =I +n τ l =1B ±l(7)where the matrices B ±l are defined asB ±l =e −(±dtV )e −dtK e dtµ(8)with V ij =δij s i,l and K ij =1if i,j are nearestneigh-boursand Kij=0otherwise.The matrices in (7)and (8)are of size (n x n y )×(n x n y ),corresponding to the spatial size of the lattice.The expectation value of a physical observable at chemical potential µ,<O >µ,is given by<O >µ=O ˜z (µ){s i,l =±1}˜z (µ,{s i,l })(9)where the sum over the configurations of Ising fields isdenoted by an integral.Since ˜z (µ)is not positive definite for Re(µ)=0we weight the ensemble of configurations by the absolute value of ˜z (µ)at some µ=µ0.Thus<O >µ= O ˜z (µ)˜z (µ)|˜z (µ0)|µ0|˜z (µ0)|µ0(10)The partition function Z (µ)is given byZ (µ)∝˜z (µ)N c˜z (µ0)|˜z (µ0)|×e µβ+e −µβ−e µ0β−e −µ0βn (16)When the average sign is near unity,it is safe to as-sume that the lattice configurations reflect accurately thequantum degrees of freedom.Following Blankenbecler et al.[1]the diagonal matrix elements of the equal-time Green’s operator G ±=(I +P ±)−1accurately describe the fermion density on a given configuration.In this regime the adiabatic approximation,which is the basis of the finite-temperature algorithm,is valid.The situa-tion differs strongly when the average sign becomes small.We are in this case sampling positive and negative ˜z (µ0)configurations with almost equal probability since the ac-ceptance criterion depends only on the absolute value of ˜z (µ0).In the simulations of the HSfields the situation is dif-ferent from the case of fermions interacting with dynam-ical bosonfields presented in Ref.[1].The auxilary HS fields do not have a kinetic energy term in the bosonic action which would suppress their rapidfluctuations and hence recover the adiabaticity.From the previous sim-ulations on a42lattice[3]we know that avoiding the sign problem,by updating at half-filling,results in high uncontrolledfluctuations of the expansion coefficients for the statistical weight,thus severely limiting the range of validity of the expansion.It is therefore important to obtain the partition function for the widest range ofµ0 and observe the persistence of the hierarchy of the ex-pansion coefficients of Z.An error analysis is required to establish the Gaussian distribution of the simulated observables.We present in the following section results of the bootstrap analysis[17]performed on our data for several values ofµ0.III.TEMPERATURE AND LATTICE-SIZEDEPENDENCE OF THE YANG-LEE ZEROS The simulations were performed in the intermediate on-site repulsion regime U=4t forβ=5,6,7.5on lat-tices42,62,82and forβ=5,6on a102lattice.The ex-pansion coefficients given by eqn.(14)are obtained with relatively small errors and exhibit clear Gaussian distri-bution over the ensemble.This behaviour was recorded for a wide range ofµ0which makes our simulations reli-able in spite of the sign problem.In Fig.1(a-c)we present typical distributions of thefirst coefficients correspond-ing to n=1−7in eqn.(14)(normalized with respect to the zeroth power coefficient)forβ=5−7.5for differ-entµ0.The coefficients are obtained using the bootstrap method on over10000configurations forβ=5increasing to over30000forβ=7.5.In spite of different values of the average sign in these simulations,the coefficients of the expansion(16)indicate good correspondence between coefficients obtained with different values of the update chemical potentialµ0:the normalized coefficients taken from differentµ0values and equal power of the expansion variable correspond within the statistical error estimated using the bootstrap analysis.(To compare these coeffi-cients we had to shift the expansion by2coshµ0β.)We also performed a bootstrap analysis of the zeros in theµplane which shows clear Gaussian distribution of their real and imaginary parts(see Fig.2).In addition, we observe overlapping results(i.e.same zeros)obtained with different values ofµ0.The distribution of Yang-Lee zeros in the complexµ-plane is presented in Fig.3(a-c)for the zeros nearest to the real axis.We observe a gradual decrease of the imaginary part as the lattice size increases.The quantitative analysis of this behaviour is discussed in the next section.The critical domain can be identified by the behaviour of the density of Yang-Lee zeros’in the positive half-plane of the fugacity.We expect tofind that this density is tem-perature and volume dependent as the system approaches the phase transition.If the temperature is much higher than the critical temperature,the zeros stay far from the positive real axis as it happens in the high-temperature limit of the one-dimensional Ising model(T c=0)in which,forβ=0,the points of singularity of the free energy lie at fugacity value−1.As the temperature de-creases we expect the zeros to migrate to the positive half-plane with their density,in this region,increasing with the system’s volume.Figures4(a-c)show the number N(θ)of zeros in the sector(0,θ)as a function of the angleθ.The zeros shown in thesefigures are those presented in Fig.3(a-c)in the chemical potential plane with other zeros lying further from the positive real half-axis added in.We included only the zeros having absolute value less than one which we are able to do because if y i is a zero in the fugacity plane,so is1/y i.The errors are shown where they were estimated using the bootstrap analysis(see Fig.2).Forβ=5,even for the largest simulated lattice102, all the zeros are in the negative half-plane.We notice a gradual movement of the pattern of the zeros towards the smallerθvalues with an increasing density of the zeros nearθ=πIV.FINITE SIZE SCALING AND THESINGULARITY OF THE DENSITY OF STATESAs a starting point for thefinite size analysis of theYang-Lee singularities we recall the scaling hypothesis forthe partition function singularities in the critical domain[11].Following this hypothesis,for a change of scale ofthe linear dimension LLL→−1),˜µ=(1−µT cδ(23)Following the real-space renormalization group treatmentof Ref.[11]and assuming that the change of scaleλisa continuous parameter,the exponentαθis related tothe critical exponentνof the correlation length asαθ=1ξ(θλ)=ξ(θ)αθwe obtain ξ∼|θ|−1|θ|ναµ)(26)where θλhas been scaled to ±1and ˜µλexpressed in terms of ˜µand θ.Differentiating this equation with respect to ˜µyields:<n >sing =(−θ)ν(d −αµ)∂F sing (X,Y )ν(d −αµ)singinto the ar-gument Y =˜µαµ(28)which defines the critical exponent 1αµin terms of the scaling exponent αµof the Yang-Lee zeros.Fig.5presents the scaling of the imaginary part of the µzeros for different values of the temperature.The linear regression slope of the logarithm of the imaginary part of the zeros plotted against the logarithm of the inverse lin-ear dimension of the simulation volume,increases when the temperature decreases from β=5to β=6.The re-sults of β=7.5correspond to αµ=1.3within the errors of the zeros as the simulation volume increases from 62to 82.As it is seen from Fig.3,we can trace zeros with similar real part (Re (µ1)≈0.7which is also consistentwith the critical value of the chemical potential given in Ref.[22])as the lattice size increases,which allows us to examine only the scaling of the imaginary part.Table 1presents the values of αµand 1αµδ0.5±0.0560.5±0.21.3±0.3∂µ,as a function ofthe chemical potential on an 82lattice.The location of the peaks of the susceptibility,rounded by the finite size effects,is in good agreement with the distribution of the real part of the Yang-Lee zeros in the complex µ-plane (see Fig.3)which is particularly evident in the β=7.5simulations (Fig.4(c)).The contribution of each zero to the susceptibility can be singled out by expressing the free energy as:F =2n x n yi =1(y −y i )(29)where y is the fugacity variable and y i is the correspond-ing zero of the partition function.The dotted lines on these plots correspond to the contribution of the nearby zeros while the full polynomial contribution is given by the solid lines.We see that the developing singularities are indeed governed by the zeros closest to the real axis.The sharpening of the singularity as the temperature de-creases is also in accordance with the dependence of the distribution of the zeros on the temperature.The singularities of the free energy and its derivative with respect to the chemical potential,can be related to the quasiparticle density of states.To do this we assume that single particle excitations accurately represent the spectrum of the system.The relationship between the average particle density and the density of states ρ(ω)is given by<n >=∞dω1dµ=ρsing (µ)∝1δ−1(32)and hence the rate of divergence of the density of states.As in the case of Lifshitz transitions the singularity of the particle number is rounded at finite temperature.However,for sufficiently low temperatures,the singular-ity of the density of states remains manifest in the free energy,the average particle density,and particle suscep-tibility [15].The regular part of the density of states does not contribute to the criticality,so we can concentrate on the singular part only.Consider a behaviour of the typedensity of states diverging as the−1ρsing(ω)∝(ω−µc)1δ.(33)with the valueδfor the particle number governed by thedivergence of the density of states(at low temperatures)in spite of thefinite-temperature rounding of the singu-larity itself.This rounding of the singularity is indeedreflected in the difference between the values ofαµatβ=5andβ=6.V.DISCUSSION AND OUTLOOKWe note that in ourfinite size scaling analysis we donot include logarithmic corrections.In particular,thesecorrections may prove significant when taking into ac-count the fact that we are dealing with a two-dimensionalsystem in which the pattern of the phase transition islikely to be of Kosterlitz-Thouless type[23].The loga-rithmic corrections to the scaling laws have been provenessential in a recent work of Kenna and Irving[24].In-clusion of these corrections would allow us to obtain thecritical exponents with higher accuracy.However,suchanalysis would require simulations on even larger lattices.The linearfits for the logarithmic scaling and the criti-cal exponents obtained,are to be viewed as approximatevalues reflecting the general behaviour of the Yang-Leezeros as the temperature and lattice size are varied.Al-though the bootstrap analysis provided us with accurateestimates of the statistical error on the values of the ex-pansion coefficients and the Yang-Lee zeros,the smallnumber of zeros obtained with sufficient accuracy doesnot allow us to claim higher precision for the critical ex-ponents on the basis of more elaboratefittings of the scal-ing behaviour.Thefinite-size effects may still be signifi-cant,especially as the simulation temperature decreases,thus affecting the scaling of the Yang-Lee zeros with thesystem rger lattice simulations will therefore berequired for an accurate evaluation of the critical expo-nent for the particle density and the density of states.Nevertheless,the onset of a singularity atfinite temper-ature,and its persistence as the lattice size increases,areevident.The estimate of the critical exponent for the diver-gence rate of the density of states of the quasiparticleexcitation spectrum is particularly relevant to the highT c superconductivity scenario based on the van Hove sin-gularities[25],[26],[27].It is emphasized in Ref.[25]thatthe logarithmic singularity of a two-dimensional electrongas can,due to electronic correlations,turn into a power-law divergence resulting in an extended saddle point atthe lattice momenta(π,0)and(0,π).In the case of the14.I.M.Barbour,A.J.Bell and E.G.Klepfish,Nucl.Phys.B389,285(1993).15.I.M.Lifshitz,JETP38,1569(1960).16.A.A.Abrikosov,Fundamentals of the Theory ofMetals North-Holland(1988).17.P.Hall,The Bootstrap and Edgeworth expansion,Springer(1992).18.S.R.White et al.,Phys.Rev.B40,506(1989).19.J.E.Hirsch,Phys.Rev.B28,4059(1983).20.M.Suzuki,Prog.Theor.Phys.56,1454(1976).21.A.Moreo, D.Scalapino and E.Dagotto,Phys.Rev.B43,11442(1991).22.N.Furukawa and M.Imada,J.Phys.Soc.Japan61,3331(1992).23.J.Kosterlitz and D.Thouless,J.Phys.C6,1181(1973);J.Kosterlitz,J.Phys.C7,1046(1974).24.R.Kenna and A.C.Irving,unpublished.25.K.Gofron et al.,Phys.Rev.Lett.73,3302(1994).26.D.M.Newns,P.C.Pattnaik and C.C.Tsuei,Phys.Rev.B43,3075(1991);D.M.Newns et al.,Phys.Rev.Lett.24,1264(1992);D.M.Newns et al.,Phys.Rev.Lett.73,1264(1994).27.E.Dagotto,A.Nazarenko and A.Moreo,Phys.Rev.Lett.74,310(1995).28.A.A.Abrikosov,J.C.Campuzano and K.Gofron,Physica(Amsterdam)214C,73(1993).29.D.S.Dessau et al.,Phys.Rev.Lett.71,2781(1993);D.M.King et al.,Phys.Rev.Lett.73,3298(1994);P.Aebi et al.,Phys.Rev.Lett.72,2757(1994).30.E.Dagotto, A.Nazarenko and M.Boninsegni,Phys.Rev.Lett.73,728(1994).31.N.Bulut,D.J.Scalapino and S.R.White,Phys.Rev.Lett.73,748(1994).32.S.R.White,Phys.Rev.B44,4670(1991);M.Veki´c and S.R.White,Phys.Rev.B47,1160 (1993).33.C.E.Creffield,E.G.Klepfish,E.R.Pike and SarbenSarkar,unpublished.Figure CaptionsFigure1Bootstrap distribution of normalized coefficients for ex-pansion(14)at different update chemical potentialµ0for an82lattice.The corresponding power of expansion is indicated in the topfigure.(a)β=5,(b)β=6,(c)β=7.5.Figure2Bootstrap distributions for the Yang-Lee zeros in the complexµplane closest to the real axis.(a)102lat-tice atβ=5,(b)102lattice atβ=6,(c)82lattice at β=7.5.Figure3Yang-Lee zeros in the complexµplane closest to the real axis.(a)β=5,(b)β=6,(c)β=7.5.The correspond-ing lattice size is shown in the top right-hand corner. Figure4Angular distribution of the Yang-Lee zeros in the com-plex fugacity plane Error bars are drawn where esti-mated.(a)β=5,(b)β=6,(c)β=7.5.Figure5Scaling of the imaginary part ofµ1(Re(µ1)≈=0.7)as a function of lattice size.αm u indicates the thefit of the logarithmic scaling.Figure6Electronic susceptibility as a function of chemical poten-tial for an82lattice.The solid line represents the con-tribution of all the2n x n y zeros and the dotted line the contribution of the six zeros nearest to the real-µaxis.(a)β=5,(b)β=6,(c)β=7.5.。
当前LatticeQCD的国内外研究现状Z 4430 Belle 在道发现一个共振结构如果实验上确认其存在，则最小的夸克组分应该是其质量与和的阈十分接近量子数还没有确定。
一些理论家把它解释为可能的 S-wave分子态或者四夸克态候选者。
对此，我们进行了两方面的研究：计算的散射；用四夸克场算符计算可能的质量谱，用单粒子态和两粒子态有限个点上对体积的依赖关系来判断所得到的态是散射态还是束缚态。
这两方面的都得到了一些初步结果。
散射根据Luscher公式，在有限的格点上，两粒子散射系统的能量、两粒子自由能量以及低能散射的散射长度之间有如下关系散射能量两粒子自由能量散射长度约化质量空间体积 a 0.204fm 400confs. a0.144fm 400confs. Beta 2.5的结果 Beta 2.8的结果表示 A1 A2 B1 B2 E 2.5 0.370 33 0.494 60 0.397 64 0.432 63 0.267 33 2.8 0.624 56 0.507 76 0.523 77 0.517 78 0.667 60 手征外推后散射长度（单位：fm 的结果：初步结论：在所有的情况下，的S-wave散射的散射长度为正，说明之间的相互作用是吸引性的。
至于这种吸引性的相互作用的强弱、以及能否构成束缚态，还需要进一步的理论分析。
小结：井冈山长征抗战战略防御战略相持战略进攻解放战争建国萌芽生存成长成功 2.我们的方针制胜三大法宝：群众路线：CLQCD努力工作，合作攻关。
理论联系实际：密切结合BES试验，关注国际实验新结果。
统一战线：希望大家多多支持，多多建议。
1.我们所处的阶段谢谢！基态是单粒子态还是两粒子态？可能的判据：Weight的体积效应。
单粒子态：两粒子态：当前Lattice QCD的国内外研究现状陈莹中国科学院高能物理研究所报告内容一、Lattice QCD简介二、国际研究现状三、国内研究现状四、关于X 3872 和Z 4430 五、小结一、Lattice QCD 简介 Wick 转动Euclidean 空间QCD作用量路径积分时空离散化连续时空四维超立方格点体系无限自由度有限自由度路径积分量子化生成泛函: 物理观测量: 算符的真空平均值淬火近似 quenched approximation : 物理含义: 不考虑夸克真空极化图,即忽略海夸克效应 Monte Carlo 模拟重点抽样：根据Boltzmann 分布产生由有限数量的位形构成的统计系综，计算可观测量的系综平均值，样本越大，统计误差越小。
量子色动力学相变的临界行为量子色动力学（Quantum Chromodynamics，简称QCD）是描述强相互作用的理论，它在粒子物理中起着重要的作用。
在高温和高能量密度条件下，QCD相变会导致强子系统中的自由夸克和胶子相互作用的改变，这种相变被称为量子色动力学相变（Quantum Chromodynamics Phase Transition）。
本文将讨论量子色动力学相变的临界行为。
1. 引言量子色动力学是标准模型的一部分，它描述了夸克和胶子之间的相互作用。
在冷却高温夸克胶子等离子体时，会发生从强子相到夸克-胶子等离子体的相变。
在相变过程中，系统的热力学性质发生了显著变化，这种变化被称为临界现象。
量子色动力学相变的临界行为一直是研究者关注的焦点。
2. 临界行为的表征量子色动力学相变的临界行为可以通过临界指数来表征。
临界指数是指在临界点附近各种物理量的行为方式。
其中，最常用的是比热容、磁化率和相关长度的临界指数。
3. 临界指数（1）比热容的临界指数在量子色动力学相变的临界点附近，比热容的行为可以用下式描述：C_v \sim |T - T_c|^{-\alpha}$$式中，$C_v$为比热容，$T$为温度，$T_c$为临界温度，$\alpha$为比热容的临界指数。
临界指数$\alpha$的数值决定了比热容在临界点附近的行为。
（2）磁化率的临界指数磁化率是描述系统磁现象的物理量，它在临界点附近的行为可以用下式表示：$$\chi \sim |T - T_c|^{-\gamma}$$式中，$\chi$为磁化率，$\gamma$为磁化率的临界指数。
磁化率的临界指数$\gamma$决定了磁化率在临界点附近的变化行为。
（3）相关长度的临界指数临界点附近的相关长度也能够描述临界行为。
相关长度可以用下式表示：$$\xi \sim |T - T_c|^{-\nu}式中，$\xi$为相关长度，$\nu$为相关长度的临界指数。
美国连续束电子加速器的能量升级到12Ge V 的科学与实验设备石 宗 仁(中国原子能科学研究院 北京 102413)摘 要 驱动美国托马斯杰斐逊国家加速器装置(Thomas Jeffers on Nati onal Accelerat or Facility,简称JLab )的能量升级到12GeV 的科学是研究胶子激发和色禁闭的起因,研究原子核的构件核子是如何由夸克和胶子构成的,研究原子核的结构及寻找新物理等的一门科学.实验设备是12Ge V 的加速器、各种超导磁谱仪及极化靶等.在能量为12Ge V 的加速器中,将采用深度遍举过程和极化实验.关键词 胶子激发,色禁闭,普适的部分子分布,连续束电子加速器,超导磁谱仪,深度遍举过程,极化The exper i m ent a l equi p mentfor the 12GeV upgrade of CEBAF a t J LabSH I Zong 2Ren(China Institute of A to m ic Energy,B eijing 102413,China )Abstract The upgrade at JLab of the continuous electr on beam accelerator facility to 12GeV was based on the requirements for research on gluonic excitations and the origin of quark confinement,how nucleons are built up fr om quarks and gluons,the structure of nuclei,and new physics .The upgraded experi mental equipment includes the 12GeV accelerator,superconducting magnetic s pectr ometers and polarized targets .Keywords gluonic excitations,color confinement,generalized parton distributi ons,continuous electr on beam accelerator,superconducting magnetic s pectr ometer,deep exclusive p rocesses,polarizati on2006-03-10收到初稿,2006-04-28修回 Email:zrshi@iris .ciae .ac .cn1 引言核物理学是一门研究原子核性质及其结构的基础科学,是核工程的物理基础,是研究物质起源和演化的基础,也是研究星际燃料来源的基础,是理论与实验紧密结合、不断深入和发展、需要长期研究的科学.它在国防、能源、医学、材料分析及其改性、环境科学和空间探索等方面都已发挥了重要的作用.在21世纪,核物理学将面临新的机遇和挑战.美国将核物理学作为国家级的科学,是美国技术大厦的支柱之一,也是培养人材的摇篮.在1979年,美国能源部和美国国家自然科学基金委员会的核科学咨询委员会第一次制定了核科学的长远规划LRP (a Long range p lan for nuclear science ),它评估过去、展望和计划未来.在随后的1983、1989、1996和2002年都做过LRP .在1989年的LRP [1]中,将标准模型(standard model,简称S M )作为核物理学的理论基础,其中强相互作用理论是量子色动力学(quantu m chr omo dyna m ics,QCD )它标志了核物理学新的里程碑.在文献[1]中,也明确地提出了原子核组成的四个层次:原子核类似于有表面振动和集体转动的液滴;原子核由核子组成;原子核由核子、介子和核子激发态等强子组成,也称为介子-重子模型(mes 2on-bary on model);原子核由夸克和胶子组成等.前两个层次构成了低能核物理.前三个层次统称传统核物理(conventi onal nuclear physics).在夸克和胶子的层次上研究强子和原子核的性质称为现代核物理(conte mporary nuclear physics).在2002年的LRP[2]中,明确地提出现代核物理包含5个科学目标:(1)核子的QCD结构及其相互作用;(2)原子核的结构及其稳定性;(3)高能量密度的QCD及热核物质的性质;(4)在核天体中元素的起源及物质的演化;(5)寻求新物理.美国托马斯杰斐逊国家加速器装置(Thomas Jeffers on Nati onal Accelerat or Facility,简称JLab)是基于第一、第二和第五个科学目标.总之,现代核物理以标准模型为理论基础,在夸克和胶子的层次上研究核物质的性质.当前研究非微扰QCD、色禁闭(col or confin ment)机制以及QCD 真空的结构是重要的科学任务.它的研究内容已经大大地超出了传统的范围.当然,人们最终希望用QCD统一地描述强子和原子核及其在各种极端条件下核物质的性质,它将是21世纪核物理学面临的机遇和挑战.为此,对构成核物理实验的炮弹(入射粒子束)、靶、探测器、电子学线路和数据获取及其处理程序,以及实验方法等都提出了新的要求.首先,人们需要多种类型的加速器,以提供参于电磁、弱和强相互作用的高品质的多种初级和次级粒子束(初级的如电子、质子和重离子,次级的如γ射线、反质子和中微子等),从而探知强子的电磁、弱和强相互作用方面的结构及极端条件下核物质的性质.其中高品质电子加速器具有头等重要的作用.这是由于电磁相互作用在理论上清楚,可微扰计算,大多数反应是单步过程,具有远高于弱相互作用的强度,以及通过调节电子转移四动量的平方(Q2)可改变空间和时间分辨率等优点.借改变Q2,能够观察到在核子内组分夸克模型CQM(constituent quark model)和部分子模型(part on model)之间以及在原子核内4个层次及其间的过渡性质.在1956年,Hofstadter等人[3]通过电子弹性散射,测量了质子内部的电荷和电流分布,说明了质子不是点粒子,而是有结构的,由此Hofstadter获得了诺贝尔奖;在1968年,Friedman等人[4]在斯坦福直线加速器中心(Stanf ord linear accelerat or center, S LAC)通过电子深度非弹散射(deep inelastic scat2 tering,D I S)发现了质子内部有后来称为夸克的微小颗粒存在,为此获得了诺贝尔奖;Prescott等人[5,6]利用极化电子-非极化靶,测量电子极化不对称度,证明了弱相互作用中存在中性流;在1983,年欧洲合作组Aubert等人[7]利用电子在原子核上的D I S,发现了原子核的结构不同于自由核子集体的结构,它称为E MC效应;A sh man等人[8]利用极化电子-极化靶,测量极化不对称度,发现了夸克对质子自旋的贡献约20%,而相对论CQM预言约70%,这是著名的至今仍在研究的热门课题“自旋危机”(s p in cri2 sis).大量事实已证明,电子是研究强子和原子核性质的强有力的探针.在虚光子携带能量等于零的B reit框架中,虚光子的波长λ=h/(2πQ),Q越大,λ越小,空间分辨率也越高.用高能量的电子能获得大的Q值.由于反应截面反比于Q n,一般n=4,所以Q越大,截面越小,为补偿小截面,需要强束流.实验上,由于需要采用对夸克和胶子场算符矩阵元灵敏的遍举和极化测量,所以分别要求电子束流的占空因子(duty fac2 t or)～100%(也称连续束),以及电子自旋具有取向的极化束.总之,需要高能、强流、极化和连续束的电子加速器.现在,J lab具有6Ge V、强流、极化、连续电子束的加速器装置(continuos electr on bea m accelerat or fa2 cility,简称CE BAF).随着QCD的进展,如色禁闭的流管模型被第一原理的格点(lattice)规范QCD计算所证实,新的包含丰富强子结构信息的普适的部分子分布(generalized part on distributi ons,GP D)理论的出现等,驱使人们采用12Ge V或更高能量的CE2 BAF.目前,由于技术上的成熟,将6Ge V提高到12Ge V是现实的,预计2010年即可实现.12Ge V将为人们提供新的运动学的区域,打开许多新的前所未有的研究窗口,以及在已有的运动学区域提供统计不确定度小的实验数据.在12Ge V,将采用深度遍举过程DEP(deep exclusive p r ocesse)和极化实验.本文第2节介绍驱动12Ge V的核科学,第3节是JLab的发展历史,加速器和超导磁谱仪等的现状和未来,在结束语中简单地介绍国际上在核物理方面正在筹建的大型加速器.在附录Ⅰ,简要地介绍了S M和QCD,在附录Ⅱ,介绍了实验上相关的问题:遍举和极化测量,运动学变量和反应类型,观测量和因子化定理等.本文主要参考了JLab在2004年6月的“CE BAF12Ge V改进的科学和实验装置的预概念设计报告”[9],及有关的文献.2 驱动12Ge V 的核科学胶子激发和色禁闭的起因、原子核构件是如何由夸克和胶子构成的、原子核物理和检验对称性及寻找新物理等四个方面的科学动机驱动了12GeV.下面分别叙述它们,但着重说明前两个.2.1 胶子激发和色禁闭的起因通过研究胶子的性质及其在强子中的作用,能得出色禁闭的起因,而寻找和研究含有胶子自由度的混合介子(hybrids )则是重要的手段.在JLab 新建的D 厅里,胶子激发(gluonic excitati ons )合作组(简称Glue X 组)将通过线极化的实光子与核子相互作用产生混合介子,利用密封性的谱仪测量混合介子的各种衰变产物.实验上将得到普通介子、轻夸克的混合介子、胶球(glueball )、介子分子态(mes on -mes on molecules )的质量谱,特别是奇异混合介子的质量谱[9,10].2.1.1 流管-色禁闭的起因早在1970年,Na mbu Y 在芝加哥大学没有发表的报告中,谈到在粒子内部夸克是由弦联系在一起的.在1985年,Isgur 和Pat on [11]提出了胶子的流管(gluonic flux -tube )模型.在1995年,Bali 等人[12]用格点QCD 计算表明,介子中的正反价夸克间形成了由胶子构成的流管.图1表示出量子电动力学QE D 和QCD 的场线和力同介子的正反价夸克间距的关系.图的上部分显示出QED 单位面积的电力线数和力反比于r 2,而势能将反比于r ,r 越大,势能越小.图的下部分显示出QCD 单位面积的色力线数和力与r 无关,势能将随r 线性增加,r 越大,势能越大.QCD 和QE D 的性质截然不同.色禁闭起源于流管的形成,GlueX 实验将检验它是否正确.2.1.2 普通介子的JPC根据CQM ,在夸克和反夸克构成的普通介子中,胶子自由度被冻结了,普通介子用qq _表示.qq _的总角动量J =L +S,宇称量子数P =(-1)L +1,电荷共轭量子数C =(-1)L +S.其中L 是正反夸克的相对轨道角动量;总自旋S =s 1+s 2,s 1和s 2分别是等于1/2的正反夸克的自旋.当L =S =0时,J PC=0-+,对应于九重赝标量介子(p seudoscalar mes on )π、η、η′和K;当L =0,S =1时,J PC =1--,对应于九重矢量介子(vect or mes on )ρ、ω、<和K 3.图1 在QCD 和QED 中的场线和力同正反价夸克间距的关系具有整数自旋J 、宇称P =(-1)J的称为自然宇称的粒子,P =(-1)J +1的称为非自然宇称的粒子.自然性(naturality )量子数τ=P (-1)J,自然宇称和非自然宇称粒子的τ分别为+1和-1.赝标量介子的τ=-1,矢量介子的τ=+1.2.1.3 混合介子的J PCHCQM 不能描述用qq _g 表示的混合介子,它的存在表明,在低能QCD 有价胶子自由度.流管基态的角动量L ′=0,最低的胶子集体激发态是L ′=1,它相当流管的顺时针和逆时针的两种转动态,两者的线性组合构成了J PC G =1-+和1+-的退化态.J PC与J PCG 耦合可得到qq _g 的J PCH ,见表1.从表1能够看出,J PC G 与L =S =0的J PC耦合产生的J PCH =1--和1++,在S =1的J PC中有其副本.J PCG 与S =1的JPC耦合产生的J PCH 有两类:一类是τ=-1,如0-+,2-+,1+-,…,在S =0的J PC 中有其副本;另一类τ=+1,如1-+,0+-,2+-,…,在J PC中没有副本.J PCH 与J PC相同的称为普通的qq _g,不同的称为奇异(exotic )的qq _g .由此得出,奇异的qq _g 能够通过S =1的矢量介子的胶子集体激发产生,但不能通过S =0的赝标量介子产生.在高能,由于光子可看作是由自旋平行的虚的正2反夸克构成的矢量介子,所以它能够产生奇异的qq _g .在表1也列出了L ≠0的J PCH .由于普通的qq _g 与qq _能够产生组态混合,所以实验上很难区分两者.如果在实验上找到了奇异的qq _g,如J PCH =1-+,那么qq _g 就被唯一的确定了.格点QCD 已经预言了轻夸克qq _g 的谱及其衰变方式,表1 J PC 和J PCHSL012JPC00-+1+-2-+11--(0,1,2)++(1,2,3)--J PCH01--,1++(0,1,2)++,(0,1,2)--(1,2,3)--,(1,2,3)++1(0,1,2)-+,(0,1,2)+-(1;0,1,2;1,2,3)+-,(1;0,1,2;1,2,3)-+(0,1,2;1,2,3;2,3,4)-+,(0,1,2;1,2,3;2,3,4)+-式,其中最低的1-+的奇异q q _g 质量为1.9—2.0Ge V.图2表示在2.7Ge V /c2以下的q q _、胶球、q q _g和介子分子态的质量谱.从图可以看出,在1.3—2.7Ge V /c2的区间,除奇异qq _g 具有0+-、1-+、2+-量子数外,其他的都没有,所以实验上找到具有此量子数的介子就是奇异的qq _g.图2 普通介子、胶球、混合介子和分子态的质量谱2.1.4 光致反应除光子等效矢量介子外,由于能够得到高品质、高强度、线极化的光子束,所以用光致反应产生qq _g,特别是奇异的qq _g,是有效和现实的.光致反应γ↑N →X ↑N ′,s =(p γ+p p )2,(1)t =(p γ-p X )2,(2)其中X 是具有确定J P的qq _g;↑表示γ射线是线极化的,及在实验上测量X 的极化;N 和N ′分别是初态核子和末态重子;洛伦兹标量s 和t 分别是在质心系的能量和四动量转移,称为Mandelstam 变量.在图3的t 道反应中,当没有重子数交换时,可以通过交换介子或坡密子(pomer on )等实现.在小t 时,主要是交换π+[13].实验上光子和核子靶分别是线极化和非极化的,测量反冲核子(但不测其极化)并测量X 衰变产物的角分布.通过角分布能够导出X 的极化方向和极化度.当交换坡密子时反应截面σ与随s 无关;当交换介子时,σ随s 增大而减少,σ与t 满足指数衰减关系:σ～e -α│t │.图3 光致产生混合介子的t 道反应2.1.5 线极化光子使用线极化光子的意义在于通过测量X 的极化,如果交换粒子的自然性知道了,就能知道X 的自然性,反之亦然;X 的极化方向与光子的线极化方向是相互关联的,并与X 的自然性有关.线极化光子的波函数为│x 〉=(│-1〉-│+1〉)/2,(3)│y 〉=i (│-1〉+│+1〉)/2,(4)其中│-1〉和│+1〉分别是左和右手圆极化光子的本征态.在光子和X 的三动量方向构成的产生平面里,│x 〉和│y 〉分别是在其内和与其垂直的线极化光子.由于|x 〉和|y 〉分别是左和右圆极化光子的差与和,所以它们对应的幅度分别是〈J X λX |T |J γλγ,J ex λex 〉与〈J X λX |T |J γ-λγ,J ex λex 〉的差与和.J ex 和λex 分别是交换粒子粒子的角动量和螺旋性.螺旋性定义为粒子自旋在其运动方向上的投影,其取值为-s 至s .在相互作用顶点,根据宇称和角动量守恒,〈J X λX |T |J γλγ,J exλex 〉∞τ〈J X -λX |T |J γ-λγ,J ex -λex 〉[14,15],其中τ=τX τex .从测量X 的极化,可判定τ=+1或τ=-1,于是如果X 的τX 知道了,交换粒子的τex 就知道了,反之亦然.当t 道交换π+时,如果τX =-1,X 的线极化方向与光子的线极化方向平行;如果τX =+1,X 的线极化方向与光子的线极化方向垂直[13].非极化和圆极化光子不具备这种性质.2.1.6 分波分析分析实验数据的重要工具是分波分析P WA (partial wave analysis ).螺旋性的反应幅度能够多极展开:T λ3λ4;λ1λ2=16πρJ ≥μ(2J +1)T J λ3λ4;λ1λ2(s )d Jλλ′(θs ),(5)λ=λ1-λ2,λ′=λ3-λ4,m =max (|λ|,|λ′|),其中T Jλ3λ4;λ1λ2是分波的系数;λ1、λ2、λ3和λ4分别是光子、N 、X 和N ′的螺旋性;J 是分波的阶数;d J λλ′(θs )是转动矩阵,d J 00(θs )=P J(θs );在质心坐标系中,λ和λ′分别是碰撞前后总角动量在运动方向上的投影.通过拟合角分布和各种极化观测量,确定分波系数,然后得到共振参数及J PC等.P WA 具有固有的缺点:当增加分波时,强分波串到弱分波,这种现象称为Dannachie 的连续模糊性或称泄漏.为了减少泄漏,建立密封性、质量分辨好、允许高计数率、对各种衰变方式灵敏、效率随角度变化均匀的探测器是必要的.D 厅的探测器正是根据这些要求设计的.2.2原子核构件核子的基本结构核子是原子核的基本构件,它的质量、自旋及相互作用性质直接取决于其内部的夸克和胶子的运动,它可用附录(Ⅰ)的(1)式描述,研究核子结构是QCD 的突出任务.在1994年以后,Muller [16],J i [17]和Radyush 2kin [18]等人提出了新的普适于硬的遍举过程的GP D ,这些过程如:深度虚光子的康普顿散射DVCS (deep ly virtual Co mp t on scattering ),ep →ep γ;深度虚介子产生过程DVMP (deep ly virtual mes on p r oduc 2ti on ),ep →epm;深度虚光子的双轻子产生,ep →ep ll 等.DVCS 和DVMP 的手袋(handbag )见图4,从DVCS 的手袋图看出,电子发射的虚光子γ3从质子击出一个夸克,高速夸克传播距离z 发射实光子γ后返回质子.在DVMP 中,高速夸克发射的胶子转变成正反夸克对qq _,其中q 返回质子,q _与被击中的夸克形成赝标量或矢量介子.GP D 打开了新的研究非微扰QCD 的窗口.它是图4 DVCS 和DVMP 的手袋图形状因子FF (for m fact or )、部分子分布函数P DF(part on distributi on functi ons )、分布幅度的统一.它给出了许多新的预言:夸克轨道角动量同GP D 的关系;矢量介子产生反应对胶子灵敏;冲击参数分布能够给出纵向动量比例x 和冲击参数b ⊥关联的三维图像,称为核子的断层扫描[19];通过DVCS 和BH (Bethe -Heitler )相干项可以得到反应幅度的相位参数,实现强子的全息照相[20];GP D 的二次矩是引力理论中的能量-动量张量形状因子等.由于它包含了丰富的强子结构信息,所以利用12Ge V 测量GP D 是重点之一.在12Ge V ,通过单举、半单举和遍举及极化测量,研究价夸克极化的P DF 、微扰QCD 的组分标度、强子螺旋性守恒、核子自旋结构、横动量相关的分布函数、高级t w ist 效应、普适的G DH (gerasi m ov drell hearn )求和规则、夸克-强子二重性等广泛的课题.2.2.1 GP D在硬反应(hard reacti on )中,夸克场算符[21]为O (x )=1/4π・∫d z -exp (xP +z -)q _(-z /2)Γi W [-z /2,z /2]q (z /2)|z +=0,z =0,(6)Γi =γ+,γ+γ5,σ+jγ5,W [-z /2,z /2]=P exp (-i g ・∫d sz μA μ(sz )),其中P +是强子动量的正分量;xP +是激活夸克动量的正分量,x =(x i +x f )/2,x i 和x f 分别是初、末态激活夸克携带强子正动量的比例;γ+、γ+γ5、σ+j γ5分别是矢量、轴矢量和张量的D irac 算符;q _(-z /2)和q (z /2)分别是在位置-z /2和z /2的共轭夸克和夸克的场算符;W ils on 连线W 保证了算符的规范不变性,积分从-z /2到z /2,A μ是胶子场.当取光锥规范时,Aμ=0,W ils on 连线等于1.算符O (x )是用光锥坐标表示的,光锥坐标定义为,υ±=υ0±υ3,υ=(υ1,υ2);夸克q 与其共轭夸克q _的间距,即虚光子吸收点和光子或介子的发射点的间距z 满足光锥条件:z 2=2z +z --z 2=0(1/Q 2),z +～M x B /Q 2,z -～1/M x B .夸克场算符的强子矩阵元为F i (x,ξ,t )=∫d z-/4πexp (xP +z -)〈p ′,s ′|q _(-z /2)Γi q (z /2)|p,s 〉,(7)其中p,s 和p ′,s ′分别表示初、末态强子的动量和自旋的极化.在弹性散射、D I S 和DEP 中,矩阵元分别对应:z =0,p ′≠p ;z ≠0,p ′=p 和z ≠0,p ′≠p 三种情况.z =0和z ≠0分别称为局部和非局部算符.矩阵元可用矢量、轴矢量和张量的基展开,在t w ist -2的层次上,Γi =γ+,F i (x,ξ,t,Q 2)=1/2P +[H (x,ξ,t )u_γ+u +E (x,ξ,t )u _(i σ+αΔα/2m )u ],(8)Γi =γ+γ5,F i (x,ξ,t,Q 2)=1/2P +[H _(x,ξ,t )u _γ+γ5u +E _(x,ξ,t )u _(γ5Δ+/2m )u ],(9)Γi =σ+j γ5,F i (x,ξ,t,Q 2) =-i /2P+[H T (x,ξ,t )u _σ+jγ5u + H _T (x,ξ,t )u _(ε+j αβΔαP β/m 2)u +E T (x,ξ,t )u _(ε+jαβΔαγβ/2m )u +E _T (x,ξ,t )u _(ε+jαβP αγβ/m )u ],(10)其中2P +=(p ′+p )+,2ξ=(x f -x i )=(p ′-p )+/P +,t =Δ2,Δ=p ′-p .H 、E 和H _、E _分别是自旋平均和螺旋性相关的手征偶(chiral even )的GP D,H T 、H _T 、E T 和E _T 分别是横向自旋相关的手征奇(chiralodd )的GP D,共8个GP D.它们是洛伦兹标量x 、ξ、t和Q 2的函数.在B j orken 极限,ξ=x B /(2-x B ).ξ和x 的范围在-1和1之间.-ξ≥x ≥-1、-ξ≤x ≤ξ和ξ≤x ≤1分别是反夸克区、中心区和夸克区,中心区的GP D 是强子内部存在介子的概率.对于胶子可写成类似的算符和GP D[22].DVCS 对夸克味量子数不灵敏,其反应截面∝Q-4.在DVMP 中,产生的赝标量介子选H _和E _及其对自旋灵敏,矢量介子选择H 和E 及其对自旋不灵敏,DVMP 的反应截面∝Q -6[23].(1)GP D 同P DF 和FF 的关系P DF 是GP D 在ξ=t =0的极限,即H (x,0,0)=f 1(x ),H _(x,0,0)=g 1(x ),H T (x,0,0)=h 1(x ),(11)其中f 1(x )、g 1(x )和h 1(x )分别是在非极化、纵向和横向极化的核子中存在相应极化的夸克具有x 的概率.实验上f 1(x )和g 1(x )已经测量了.由于h 1(x )是手征奇的,实验上很难测量,人们正期待德国GSI 未来极化质子和反质子的对撞实验.定义GP D 的Mellin 矩=∫d xxn -1GP D.它将非局部夸克和胶子的场算符转换成局部算符,便于用格点QC D 计算.核子电磁的FF 可表示成n =1的矩,F 1(t )=∑e q∫d xH q(x,ξt )F 2(t )=∑eq∫d xE q(x,ξ,t )(12)轴矢量和张量FF 同GP D 有类似的关系.在实光子的康普顿散射中,相应的FF 是GP D 的n =0的矩:R V (t )=∑e q ∫d x 1xH q(x,ξ,t ),R T (t )=∑e q∫d x 1xE q (x,ξ,t ),R A (t )=∑eq∫d x 1xH _q(x,ξ,t ),(13)所有的积分限从-1到+1.(2)季向东的求和规则当n =2时得到季向东[17]的求和规则:J q =12Δρq -L q =12∫1-1x d x[H q (x,ξ,0)+E q(x,ξ,0)],(14)其中J q、Δρq 和L q 分别是味q 夸克的总角动量、自旋和轨道角动量.通过测量H q 和E q能够计算出J q ,扣出从D I S 实验得到的Δρq ,可求出L q.最新的测量结果表明,胶子对核子自旋的贡献很小[24],所以L q对解答自旋危机是十分关键的.(3)冲击参数分布冲击参数分布I P D (i m pact para meter distribu 2ti on )是在ξ=0,Δ⊥≠0,Δ∥=0时的GP D 的富利叶变换[21].如自旋平均的I P Dh (x,b ⊥)=∫d 2Δexp (i b ⊥Δ⊥)H (x,ξ=0,Δ⊥)/(2π)2,(15)其中b ⊥是相对横动量中心R ⊥=ρi x i r i ⊥的距离,x k 和r k ⊥分别是第k 个部分子的纵向动量及其横向位置.8个GP D 对应的I P D 具有概率解释.在图5,从左至右,FF 是二维b ⊥的函数,P DF 是一维x 的函数,I P D 则将两者联立起来得到三维分布.测量核子的I P D 将使人们对核子结构有突破性的认识.图5 FF 、P DF 和I P D的关系图6 在非极化和横向X 方向极化质子中非极化夸克的I P D (4)I P D 的极化效应从I P D 得到了两种新的极化效应[25]:第一,在横向X 方向极化的核子中,味q 非极化的密度q X (x,b ⊥)不是轴对称的,并在Y 方向移位.它可写城如下形式:q X (x,b ⊥)=q (x,b ⊥)-5/5b Y [E q (x,b ⊥)]/2M 4π2,E q (x,b ⊥)=∫d 2Δ⊥E q (x,0,-Δ⊥)ex p (-i b ⊥Δ⊥)(16)其中q (x,b ⊥)是在非极化核子中的非极化夸克的I P D.在图6左侧的两个图分别是在非极化和X 方向极化的质子中,纵向动量x =0.5的u 夸克的I P D ,非极化时I P D 是轴对称的,极化时I P D 是非轴对称的,并在Y 方向移位.在图6右侧的两个图分别是d 夸克对应的I P D.d 夸克比u 夸克的变形大,两者的移位方向相反.平均移位和方向正比于该夸克的反常磁矩k q 及其符号,平均移位可表示成:d q Y =∫d x ∫d 2b ⊥q (x,b ⊥)b Y =1/(2M )∫d xE q (x,0,0)=k q /2M,(17)其中M 是核子的质量,d qY ～0.2f m.第二,在非极化的核子中,味q 的夸克横向极化的密度分布q i (x,b ⊥)=-s i e ij9/9b j [2H _q (x,b _)+E T (x,b ⊥)]/2M 4π2,(18)其中s 是夸克自旋的分量,i,j =X 或Y .q i (x,b ⊥)具有类似的变形.在文献[26]中,明确地给出了I P D 与横动量相关的时间反演不守恒的P DF 的关系:f ⊥1T ∴-E ′,h ⊥1∴-(E ′T +2H _″T ),h ⊥1T ∴2H _″T ,其中f ′=(5/5b 2)f,f ″=(5/5b 2)2f .I P D 极化效应说明了SS A起源于夸克密度分布变形和位移,它是强子自旋物理的焦点之一,并有待实验上的验证.(5)GP D 的测量方法实验上将采用两类遍举极化实验测量GP D.第一类利用高Q 2的弹性散射和共振跃迁、高-t 值的康普顿散射和高-t 值及低Q 2的电产生介子等反应得到相应的FF,它们同GP D 的关系见(12)和(13)式.反应概图分别表示在图7中的(a ),(b ),(c )和(d ).在图7(a )中,核子内的一个价夸克吸收入射的虚(实)光子后,立刻返回核子,并迅速地将其能动量传递给其他的价夸克,最后核子获得了能动量;在图7(b )中,高能动量的夸克返回核子,使核子处在激发态;在图7(c )和7(d )中,高能动量的夸克飞行距离z 后放射一个实光子或介子并返回核子.第一类测量将约束GP D 在高-t值和小b ⊥的行为.第二类是利用高Q 2和小-t 值的DVCS 和DV MP 等直接地测量GP D (见图4).图7 形状因子反应的概图2.2.2 大Q2物理大Q2物理的内容很多,主要是研究从非微扰向微扰的过渡.以μpG p E/G p M同Q2关系为例说明J lab的重要成果及大Q2的必要.在e↑p→ep′↑弹性极化迁移反应中,在单光子交换下,质子电磁形状因子的比G p E/G p M正比于反冲质子p′的横向和纵极化度P T和P L,G p E/G p M=-P T/P L・(E+E′)tan(θ/2)(2M)-1,(19)其中E和E′是入射和出射电子的能量,θ是入射和出射电子间的夹角.在图8中,绿点表示μpG p E/G p M,它随着Q2增大线性地减少,说明在质子内电荷和电流的分布不同[27,28],其中μp是质子的磁矩.过去通过非极化的Rosenbluth分离方法得到的μpG p E/G p M在1. 0附近,说明电荷和电流的分布相同.极化和非极化两种实验方法得出两种截然不同的结论,在实验和理论上产生了极大的反响.在实验上,JLab再次用Rosenbluth方法测量说明过去的结果是正确的.在理论上,Guichon等人[29]提出了除单光子交换外还有双光子交换,双光子交换部分对非极化的结果贡献大,对极化的贡献小.目前,实验上正在测量单-双光子交换在电子散射中的比例.现在需要知道Q2等于多少μpG p E/G p M才能达到微扰QCD的预言值?在12Ge V,Q2可达14Ge V2. 2.2.3 大x物理从图12对DVCS的运动学复盖范围能够看出,强流12Ge V为研究大x物理提供了新的机遇.根据大量非极化质子的实验数据,在x=0.5—1.0的区域,海夸克(sea quark)的贡献可以忽略,是一个纯的价夸克区.在该区,目的是研究价夸克和胶子的动力学,从而检验S U(6)对称的价夸克模型、S U(6)破缺、微扰QCD的强子螺旋性守恒等理论.另外,为高能强子-强子碰撞和寻找新粒子及新物理提供数据.过去在x>0.3的区域,由于价夸克的分布概率也很小,所以实验数据很少,即便有,数据的不确定度也很图8 GpE/GpM同Q2的关系大,不能鉴别理论模型.图9显示出中子纵向极化不对称A n1同x的关系,绿点是6Ge V的数据,红点是A 厅计划测量的数据[30].图9 A n1同x的关系2.3 原子核物理原子核物理分为研究原子核本身性质和将原子核作为研究QCD的实验室等两个方面.前者研究强子在核介质中的改性,核子-介子自由度在什么标度是可靠的,核子-介子自由度如何过渡到夸克-胶子自由度,核子-核子相互作用中的长程(>2f m)张量力和短程排斥力的QCD基础等;后者研究色透明性,夸克强子化的时空特性,高密度夸克分布的性质等.图10表示出在原子核内两个核子重迭的概率很高,在重叠区域核子密度达到～5—10倍于正常值,它为研究高密度和夸克部分去禁闭及中子星的性质提供了难得的条件.实验上通过测量x >1的P DF 能够认识高密度的性质.图10 在原子核内两个核子重迭2.4 检验对称性和寻找新的标准模型JLab 将通过Pri m akoff 效应测量轻的赝标量介子π0,η,η′→γγ的衰变宽度和跃迁形状因子,研究手征自发破缺和手征反常[31].世界上,通过极高能的反应寻找新粒子及新相互作用力,和在低能比较S M 预言的参数与测量值的偏离等两种方法检验S M 和寻求新物理.JLab 将通过低能宇称不守恒的电弱相干散射测量质子的弱荷,以及电弱混合角sin 2θW与S M 预言相比较.3 美国托马斯杰斐逊国家加速器装置(简称JLab )的状况 JLab 属于美国能源部,由美国东南大学研究协会管理.它是NS AC 在1979年的LRP 中优先推荐建造的占空因子为100%、能量为4Ge V 的电子加速器,于1987年在V iginia Ne wport Ne ws 建造,它是工作在液氦温度的超导铌射频共振腔加速电子的直线加速器,该加速器于1994年夏运行,在1996年电子能量提高到6Ge V.在2010年电子能量将达到12Ge V ,以后能量提高到24Ge V 也是可能的.目前,JLab 的CE 2BAF 是世界上最大的超导射频直线加速器,它的束流品质超过了其他连续电子束加速器.除初级电子束外,通过纵向极化电子在重金属上的韧致辐射和在薄片晶体上的相干韧致辐射,分别产生圆极化和线极化的实光子.韧致辐射后的电子在磁偶极场的作用下偏转到电子焦面探测器,电子信号与光子反应产物信号符合,从电子能量能够确定对应光子的能量,这种光子称为标记光子.在20世纪90年代初,世界上出现了连续束电子加速器,如德国Mainz 大学的0.85Ge V 的MAM I (Mainz M icr otr on ),美国M I T 大学的1.1Ge V 的Bates 、德国Bonn 大学的3.5Ge V 的E LS A (Electr on Stretcher Accelerat or )及荷兰N I KHEF (Nati onal I nsti 2tute f or Nuclear Physics and H igh -Energy Physics )的0.9Ge V 的AmPS (Am sterda m Pulse Stretcher ).除MA 2M I 是电子感应方式实现连续束外,其余都是贮存拉长环(st orage /stretcher )方式.目前,N I KHEF 已关闭,Bates 于2005年结束了核物理计划,MAM I 的能量将提高到1.5Ge V ,ELS A 还在运行.3.1 加速器6Ge V 和未来12Ge V CE BAF 的结构和实验厅的布局见图11.经预直线加速器,能量为45Me V 的电子注入到80m 长的超导直线加速段,能量达到0.56Ge V ,经弧形轨道arc 进入另一个80m 的超导直线加速段,能量达1.12Ge V.在2个直线加速段回转5次,电子能量达到5.6Ge V.在A 、B 、C 3个实验厅能同时工作,并各自可以工作在不同的电子能量,能量分别是1.1、2.2、3.3、4.4和5.6Ge V.其束流品质为:平均束流强度I =200μA,发散度ε～2×10-9m ・rad,能散度σE /E =2×10-5.用极化的激光照射拉伸或非拉伸的Ga A s 光阴极,通过光电效应产生纵向极化度P e=70%—85%的电子.光子的能量及其分辨率分别为E γ<6GeV 和ΔE γ/E γ=10-3.电子的纵向极化度利用莫特(Mott )、Moller 和康普顿散射极化仪测量.图11 JLab 加速器的结构和实验厅的布局美国核科学家在建造4Ge V 时,就认识到其能量是低的,因而在直线加速段的隧道内留有一定的空间.刚开始运行时,即1994年,就酝酿提高能量的物理和需要的设备.在2001年2月,形成了之《驱动。
a r X i v :c o n d -m a t /0609680v 1 [c o n d -m a t .s t a t -m e c h ] 27 S e p 2006Effective Temperature Characterizing the Folding of a ProteinNaoko NakagawaDepartment of Mathematical Sciences,Ibaraki University,Mito,Ibaraki 310-8512,Japan(Dated:February 6,2008)The time sequences of the molecular dynamics simulation for the folding process of a protein is analyzed with the inherent structure landscape which focuses on configurational dynamics of the system.Time dependent energy and entropy for inherent structures are introduced and from these quantities an effective temperature is defined.The effective temperature follows the time evolution of a slow relaxation process and reaches the bath temperature when the system is equilibrated.We show that the nonequilibrium system is described by two temperatures,one for fast vibration and the other for slow configurational relaxation,while the equilibrium system is by one temperature.The proposed formalism is applicable widely for the systems with many metastable states.PACS numbers:05.70.Ln 05.90.+mSlow relaxations of non-equilibrium systems are a very general process which are typically observed in glasses.The focus of the study for glasses has been put on descrip-tions of their instantaneous state,or their aging proper-ties [1],often characterized by an effective temperature deduced from the violation of the fluctuation-dissipation theorem [2,3,4,5].On another hand,for various slow re-laxations a major interest is how the relaxation proceeds.This is the point that we want to investigate in this let-ter by showing how an effective temperature that follows the time evolution of a slowly relaxing non-equilibrium system can be introduced.The effective temperature in glasses is,in most cases,defined from the ratio between equilibrium and non-equilibrium response to a perturba-tion,and therefore it is observable-dependent [6].On the contrary,the effective temperature that we introduce in this letter is a one-time property of the system,which evolves towards the equilibrium temperature as the sys-tem relaxes.To illustrate the usefulness of the concept,we have chosen to consider proteins,which fold from random coil states to their own native conformation in µs or m s,while their vibrational dynamics takes place in time scales of the order of f s or p s.The folding of a protein provides a nice example of an out-of-equilibrium complex physical system which evolves in an experimentally and numeri-cally accessible time scale.Proteins can be investigated in the spirit of glasses by characterizing their inherent structures,i.e.the local minima of their energy landscape [7].We showed that the density of states in the Inherent Structure Landscape (ISL)of a model protein depends exponentially on their energy [7],so that the model has some similarities with the simple models of glasses such as spin glasses [8],Lennard-Jones clusters [9]and the trap model [6].This letter shows that the ISL is not only useful to study the equilibrium properties of a protein [7]but can also be used in practice to characterize its slow evolution towards the folded configuration.Model:We consider an off-lattice G¯o protein model with a slight frustration [7].The protein is reduced to itsbackbone without side chains,each residue being repre-sented by its C αatom.The effective potential between the C αelements in the backbone is designed from the experimentally determined native structure.We study proteinG which has 56residues [10].At low temperature,the model protein folds to the native structure,while at high temperature it denatures.Folding-unfolding transi-tion occurs at T f ,and around T d ≈0.4T f a dynamical transition is observed below which the fluctuations of the protein structure are dramatically reduced.In numerical investigation,molecular dynamics simu-lations with Langevin equations are applied,where a de-natured initial condition is prepared by a simulation at 1.7T f at which the protein is in the random coil state.A sudden cooling to T <T f is applied at t =0.For each value of T ,200initial conditions are sampled.Definitions of quantities:Slow dynamics comes from itinerant transitions among local energy well while the fast dynamics comes from rapid vibrations inside each well.Such slow dynamics can be characterized by “in-herent structures”[11],in which the set of conformations belonging to the same basin labeled by the index αare mapped to its local energy minimum with energy e α.Let w (Γ,t,T )be the probability density for the system to be at point Γ,up to d Γ,in phase space at time t and let Γαbe the region corresponding to a basin α.The probability weight to observe the basin αisw α(t,T )=Γαw (Γ,t,T )d Γ,(1)whereαw α(t,T )= 1.The probability weight w αv (Γ,t,T )normalized in the basin αis introduced asw (Γ,t,T )=w α(t,T )·w αv (Γ,t,T ).(2)To be consistent with Eq.(1), Γαw αv(Γ,t,T )d Γ=1.For each point Γ,the value of the energy e (Γ)is de-termined from the model potential.When the point Γbelongs to basin α,the energy can be rewritten as e (Γ)=e α+∆V α(Γ),where ∆V α(Γ)is the potential at2 pointΓmeasured from the minimum eα.The mean en-ergy U at time t is given byU(t,T)≡ e(Γ)w(Γ,t,T)dΓ=U IS(t,T)+ αUαv(t,T)wα(t,T).(3)This expression splits the energy into two compo-nents,U IS(t,T)≡ αeαwα(t,T)and Uαv(t,T)≡ Γα∆Vα(Γ)wαv(Γ,t,T)dΓ.U IS(t,T)is the mean inherent structure energy at time t which reflects slow structural changes due to the transition among the basins while Uαv(t,T)is the mean vibrational energy inside basinα. Since the inherent structure energies eαtake discrete values,the entropy for the inherent structures can be defined with the definition of quantum statistical physics,S IS(t,T)≡−k B αwα(t,T)log wα(t,T).(4)The total entropy S(t,T)splits into S IS(t,T)and a vi-brational contribution Sαv(t,T)according toS(t,T)≡−k B w(Γ,t,T)log w(Γ,t,T)dΓ=S IS(t,T)+ αSαv(t,T)wα(t,T).(5)Here Sαv(t,T)is the vibrational entropy at the basinαsuch that Sαv(t,T)≡−k B Γαwαv(Γ,t,T)log wαv(Γ,t,T). We add that so-called configuration entropy is not nec-essarily equal to S IS even in canonical equilibrium,S IS,can(T)=k B logΩIS(U IS,can(T)),(6) because the value of eαcan largely deviate from the mean value U IS,can(T)in small systems such as proteins.Here ΩIS(eα)is the density of states for the inherent struc-tures,which can be determined from a numerical simu-lation in equilibrium situation as shown in[7].In analogy to the equilibrium case,we define a time-dependent effective temperature for inherent structures as1∂U IS(t,T)/∂T,(7)where we expect T e(t,T)→T as the relaxation proceeds, and T e=T in thermal equilibrium.Slow relaxation in Inherent structure landscape:The energetic aspects of the folding process are demon-strated in Fig.1,in which the mean vibrational en-ergy over the phase space is defined as U v(t,T)≡ αUαv(t,T)wα(t,T).As are shown in Fig.1,the mean inherent structure energy U IS(t,T)relaxes logarithmi-cally in time.Such logarithmic relaxations are typically10201e+06 1e+07 1e+08U ISk B T f-------timeT/T f =0.340.55U vk B T f N r-------0.510.831e61e71e8FIG.1:Time evolution of mean inherent structure energy U IS and(inset)mean vibrational energy U v.In the case0.55T f> T d,the folding process reaches thermal equilibrium around t=2×107,whereas in0.34T f<T d it does not within the computation time.U v is in the equilibrium value which is slightly larger than the harmonic approximation3/2k B N r T (indicated as0.51and0.83for T/T f=0.34and0.55).seen in glassy systems,which suggests intrinsic glassy properties for proteins even above the dynamical transi-tion temperature T d.On the other hand,U v(t,T)is almost constant,cor-responding to its equilibrium value at temperature T, although the probability weight wα(t,T)is expected to significantly depend on time as the folding proceeds.This is possible if the vibrational energy Uαv(t,T)is approxi-mately the same for all basinsαand the vibrational mo-tions reach equilibrium much faster than the time scale of the folding.The value of Sαv(t,T)is also expected to be equal to its equilibrium value,which could be approx-imately independent ofα,similarly to Uαv(t,T).S IS(t,T)cannot be accessible to numerical simulation because the number of basins in the whole phase space is too large to determine wα(t,T)for eachαand each t. Instead of determining wα(t,T)for all basins,let us in-troduce a probability P(eα,t,T)to observe the inherent structure energy eα,P(eα,t,T)= dΓw(Γ,t,T)δ(e min(Γ)−eα).(8)Here e min(Γ)is the quenched energy from the pointΓ.If Γbelongs to the basinα,e min(Γ)=eα.Since the initial conditions in numerical simulation are randomly selected from the random coil state,let us assume that the weight is approximately the same for basins with the same minimum energy eα,wα(t,T)≃P(eα,t,T)/ΩIS(eα).(9) Then,S IS(t,T)≃−k B deαP(eα,t,T)log P(eα,t,T)32040212< t x 10-4<214210< t x 10-4<21228< t x 10-4<21026< t x 10-4<2824< t x 10-4<26e α / k B Tf T / T f = 0.342040212< t x 10-4<214210< t x 10-4<21228< t x 10-4<21026< t x 10-4<2824< t x 10-4<26e α / k B T fT / T f = 0.55FIG.2:Time evolution of the probability distributionP (e α,t,T )for the inherent structure.The horizontal axis is inherent structure energy e αscaled by k B T f .T =0.34T f and 0.55T f correspond to T <T d and T >T d ,respectively.The right bottom figure shows the equilibrium distribution at T =0.55T f .At t =0the protein has been cooled to T .0 4 8 121620 240481216U IS / k B T fS IS / k BT/T f =0.340.55EquiribriumFIG.3:Time evolution in the plot on S IS vs U IS for t >106.T /T f =0.34and 0.55.The line is the relation between S IS and U IS in thermal equilibrium.Higher temperatures corre-sponds to the right-upper part of the line.which is accessible to numerical simulations.Examples of the time evolution of P (e α,t,T )in Fig.2show that the distribution is strongly dependent on time after the sudden cooling.The energy range that the dis-tribution spans becomes smaller as time proceeds,and the width of the distribution decreases significantly.As is expected P (e α,t,T )is considerably different from the equilibrium distribution (bottom figure for T/T f =0.55).For each time t ,we can obtain numerically U IS (t,T )and S IS (t,T )following the definition (10).If we plot S IS (t,T )as a function of U IS (t,T )for various t ,we get the time evolution for the relaxation process shown in Fig.3.For comparison we plot the line corresponding to thermal equilibrium cases over a wide range of T .This comparison makes it clear that the time evolution in a fixed cooling temperature T proceeds in parallel to the line for thermal equilibrium from high to low tempera-T fT fTδU IS /δT δS IS /δT(b)0.60.8107108timeT e / T f(a)(0.34,0.41)(0.41,0.48)(0.55,0.59)0.55+0.592FIG.4:(a)Time evolution of T e (t,(T 1+T 2)/2)in eq.(7)estimated by {S IS (t,T 2)−S IS (t,T 1)}/(T 2−T 1)and {U IS (t,T 2)−U IS (t,T 1)}/(T 2−T 1).Here (T 1,T 2)=(0.34T f ,0.41T f ),(0.41T f ,0.48T f )and (0.55T f ,0.59T f ).The dotted horizontal line is (T 1+T 2)/2for (0.55T f ,0.59T f ),which is expected to be a convergence line of T e (t,0.57T f ).(b)Confirmation of thermodynamic relation in Eq.(??)in thermal equilibrium condition.48 12 160 4 8 12 16U e / k BT f U IS / k B T f(0.34, 0.41)(0.41, 0.48)(0.55, 0.59)FIG.5:(a)U e (t,T )vs U IS (t,T ).Here T =(T 1+T 2)/2with (T 1,T 2)=(0.34T f ,0.41T f ),(0.41T f ,0.48T f )and (0.55T f ,0.59T f ).tures.Let us study this evolution by the effective tem-perature T e (t,T )defined in eq.(7).In equilibrium,S IS and U IS work well to determine the temperature of the system as is demonstrated in Fig.4(b),where T e for the canonical ensemble with ΩIS (e α)has the expected prop-erty,T e =T .In the relaxation process,T e decreases with time from a higher temperature.When T is suffi-ciently high,we can observe that T e converges to a value corresponding to T after the relaxation to equilibrium (T ≃0.55T f in Fig.4(a)).Thus the features observed in Fig.2or Fig.3in the relaxation process are reflected in T e (t,T ).The value of T e (t,T )is expected to give an approxima-tion of P (e α,t,T )to an equilibrium distribution at the temperature T e .To check it,we compare the approxi-mated mean IS-energy U e (t,T )with U IS (t,T )in Fig.5,4whereU e(t,T)≡ deαeαΩIS(eα)k B T e(t,T) .(11)Thefigure displays the tendency U e(t,T)≃U IS(t,T), which suggests that T e can give afirst approximation for the distribution P(eα,t,T).The deviation of the points from the line in Fig.5indicates that the states on the way of folding do not coincide strictly with any equilib-rium state.Correspondingly,the points in Fig.3deviates slightly from the equilibrium line.Discussion:We have introduced time dependent en-ergies and entropies for inherent structures to analyze the slow folding process of a model protein.After a sudden cooling,we found that the probability distribution of the inherent structures is widely spread in the early stage and becomes narrower as time proceeds.This evolution is successfully characterized by the time evolution of the effective temperature Eq.(7),as shown in Fig.4(a). Although the approximation of the distribution by T e is not complete,we note that even thisfirst approxima-tion is not obtained if we treat the total probability distri-bution without separating the inherent structure modes and the vibrational modes.As argued in Fig.1,the vibra-tional modes are reaching local equilibrium in the early stage.Thus there is a temperature difference between the fast dynamics and the slow dynamics.Thermody-namic frameworks including two temperatures T and T e could be proposed[12].It might be interesting to study various slow dynamics adopting such two temperatures. When T is lower than T d,relaxation becomes very slow as is expected,but it would relax after a long period because the present model is smallfinite system without extensibity.We expect a convergence tendency of T e to T with time.This could be in contrast to spin glasses, since the shape of P(eα,t,T)for a spin glass[5]does not change in the slow relaxation process.Then T e might be obtained as a constant larger than T,which could be compared with the effective temperature in the violation offluctuation-dissipation system.The effective temperature defined in the present pa-per is different from the known effective temperature in glasses.For instance,Sciortino and Targlia[4]introduced an effective temperature according to the inherent struc-ture energy and configuration entropy in Lennard-Jones systems.They successfully showed that their effective temperature coincides with the one defined by the viola-tion offluctuation-dissipation relation.These effective temperatures are the parameter to indicate the prop-erty of the individual system.Even though the system is very close to equilibrium,their effective temperature does not converge to the surrounding temperature.On the other hand,the effective temperature of Eq.(7)is de-fined to indicate the relaxation process of the system.It becomes equal to the surrounding temperature if the system reaches equilibrium.Finally we would like to point out that this study em-phasizes the interest of the Inherent Structure Landscape for protein.We showed earlier[7]that,contrary to the free-energy landscape which is only a useful concept,the density of states of the ISL can be obtained in practice for a protein model,and that it characterizes the equilib-rium properties of the protein to a good accuracy.Here we show that the ISL is not only useful in equilibrium, but can be used to characterize the slow relaxation with a time-dependent effective temperature.The assumption used in this letter,that Uαv and Sαv does not depend on α,appears to be good for the protein case.It can be relaxed for more complex systems[13].Note:Since the present model possesses at least two funnel which is well separated from the others,we choose the relaxation process to a funnel of which global mini-mum energy state corresponds to a misfolded state.Its minimum energy state resembles the native state but a α-helix is bent compared to the native state.In this funnel,relaxation proceeds in a time scale accessible to numerical simulations if the temperature is not too low. Acknowledgement:N.N.thanks M.Peyrard for useful discussion and critical reading of the manuscript.She also thank MEXT KAKENHI(No.16740217)for sup-port.[1]Slow relaxations and nonequilibrium dynamics in con-densed matter,J.L.Barrat et al Eds.,EDP Sciences;Springer-Verlag(2003).[2]L.F.Cugliandolo and J.Kurchan,J.Phys.A,27(1994)5749.[3]L.F.Cugliandolo,J.Kurchan and L.Peliti,Phys.Rev.E,55(1997)3898.[4]F.Sciortino and P.Tartaglia,Phys.Rev.Lett.86(2001)107.[5]A.Crissanti and F.Ritort,J.Phys:Condens.Matter14(2002)1381.[6]S.Fielding and P.Sollich,Phys.Rev.Lett.88(2002)05603.[7]N.Nakagawa and M.Peyrard,Proc.Nat.Acad.Sci.,103(2006)5279.[8]A.J.Bray and M.A.Moore,J.Phys.C:Solid StatePhys.14(1981)1313.[9]J.P.K.Doye and D.J.Wales,J.Chem.Phys.102(1995)9659.[10]H.M.Berman et al,Nucleic Acids Research,28(2000)235.[11]F.H.Stillinger and T.A.Weber,Phys.Rev.A,25(1982)978.[12]Th.M.Nieuwenhuizen,Phys.Rev.Lett.80(1998)5580.[13]N.Nakagawa and M.Peyrard,in preparation.。
a r X i v :h e p -l a t /9612011v 1 14 D e c 19961Finite -Temperature QCD on the LatticeAkira Ukawa aaInstitute of Physics,University of Tsukuba,Tsukuba,Ibaraki 305,JapanRecent developments in finite-temperature studies of lattice QCD are reviewed.Topics include (i)tests of improved actions for the pure gauge system,(ii)scaling study of the two-flavor chiral transition and restoration of U A (1)symmetry with the Kogut-Susskind quark action,(iii)present understanding of the finite-temperature phase structure for the Wilson quark action.New results for finite-density QCD are briefly discussed.1.IntroductionFinite-temperature studies of lattice QCD have been pursued over a number of years.Quite clearly the pure gauge system is the best under-stood of the entire subject.The system has a well-established first-order deconfinement transi-tion[1],and extensive and detailed results are al-ready available for a number of thermodynamic quantities[2].Nonetheless many new studies have been made for this system recently.The purpose is to examine to what extent cutoffeffects in ther-modynamic quantities are reduced for improved actions as compared to the plaquette action which had been used almost exclusively in the past.Full QCD thermodynamics with the Kogut-Susskind quark action has also been investigated extensively in the past.A basic question for this system is the order of chiral phase transition for light quarks.For the system with two flavors,finite-size analyses carried out around 1989-1990indicated an absence of phase transition down to the quark mass m q /T ≈0.05[3],and a more re-cent study[4]attempted to find direct evidence for the second-order nature of the transition,as sug-gested by the sigma model analysis in the contin-uum[5],through scaling analyses.Scaling studies have been continued this year to establish the uni-versality nature of the transition on a firm basis.Another issue discussed at the Symposium is the question of restoration of U A (1)symmetry at the chiral transition.Results have been presented for equation of state both without and with use of improved actions.Studies of thermodynamics with the Wilsonquark action is much less developed compared to that for the Kogut-Susskind quark action.Past simulations found a number of unexpected fea-tures,which made even an understanding of the phase structure a non-trivial problem[6].Re-cently,however,considerable light has been shed on this problem through an analysis based on the view that the critical line of vanishing pion mass marks the point of a second-order phase transi-tion which spontaneously breaks parity and flavor symmetry[7].Some new work with improved ac-tions,which was initiated a few years[8],has also been made this year.In this article we review recent studies of finite-temperature lattice QCD.In Sec.2we summarize results for the pure gauge deconfinement tran-sition obtained with a variety of improved ac-tions.Results for the two-flavor chiral transition for the Kogut-Susskind quark action are discussed in Sec.3with the main part devoted to scaling analyses of the order of the transition.In Sec 4we describe recent progress on the phase struc-ture analysis for full QCD with the Wilson quark action.This year’s results for finite density are briefly discussed in Sec 5.Our summary and con-clusions are presented in Sec.6.2.Recent work on pure gauge system In Table 1we list recent studies of the pure gauge system using improved actions.Among ac-tions constructed through renormalization group,RG(1,2)[18]includes 1×2loop in addition to the plaquette.The action FP is an 8-parameter ap-proximation to the fixed point action[9]contain-2Table1Recent work on pure gauge system with improved actions.Argument forβc means spatial volume,µ(L)the torelon mass for spatial size L and N t the temporal lattice size.RG-improvedS(1,2)tree[14]βc(4N t)3,4,5,6[15]βc(∞),σ,ǫ,p,σI4S(1,2)tadpole[15]βc(∞),σ,ǫ,p,σI4S(2,2)tree[16,15]βc(∞),σ,ǫ,p4 SLW tadpole[17]βc(2N t),µ(L)2,3,4[13]βc(∞),µ(L)2,3σA basic quantity for the pure gauge system is√the ratio T c/3 aT c=0.25where values for six types of ac-tions are available.Among those belonging tothe category of Symanzik improvement,we ob-√serve that T c/σ≈0.64.If we take√T c/σ(L)Another quantity often used for testing im-provement with simulations on small latticesis the torelon massµ(L)extracted from thePolyakov loop correlator on a lattice of spatial sizeL.Definingσ(L)=µ(L)/L we compile in Fig.2results for the ratio T c/4ref.size mq5up to 123,but stays constant within errors be-tween 123and 163both at m q =0.025and 0.0125−0.01(see Fig.11and 12in the second paper of ref.[27]).The saturation implies the ab-sence of a phase transition down to m q ≈0.01.Since this quark mass is quite small,correspond-ing to m π/m ρ≈0.2at the point of the transition βc ≈5.27,it was thought that the result is con-sistent with the transition being of second order at m q =0as suggested by the sigma model anal-ysis[5].3.1.1.Scaling analysis of susceptibilities One can attempt to examine if the transition is of second order employing the method of scaling analysis.Let us define the susceptibilities χm and χt,i (i =f,σ,τ)byχm =V(qq 2 (3)χt,f =V [ qD 0q − qD 0q ](4)χt,i=V [qq P i ],i =σ,τ(5)with V =L 3N t ,D 0the temporal component of the Dirac operator,and P σ,τthe spatial and tem-poral plaquette.For a given quark mass m q ,let g −2c (m q )be the peak position of χm as a func-tion of the coupling constant g −2,and let χmax m and χmax t,i(i =f,σ,τ)be the peak height.For a second-order transition,these quantities are ex-pected to scale toward m q →0asg −2c (m q )=g −2c (0)+c g m z gq(6)χmax m =c m m −z m q (7)χmax t,i=c t,i m −z t,iq,i =f,σ,τ(8)Let us note that χt,i (i =f,σ,τ)are three parts ofthe susceptibility χt =V [ qq ǫ ]with ǫthe energy density[4].The leading exponent is therefore given by z t =Max(z t,f ,z t,σ,z t,τ).Natural values to expect for the exponents z g ,z m and z t at a finite lattice spacing are those of O (2)≈U (1)corresponding to the exact sym-metry group of the Kogut-Susskind action.How-ever,sufficiently close to the continuum limit where flavor breaking effects are expected to dis-appear,they may take the values for O (4)≈SU (2)⊗SU (2)which is the group of chiral sym-metry for N f =2in the continuum.One should also remember that mean-field exponents controlthe scaling behavior not too close to the transi-tion.A possibility of mean-field exponents arbi-trarily close to the critical point has also been discussed[30].The initial scaling study was carried out by Karsch and Laermann[4]employing an 83×4lat-tice and m q =0.02,0.0375,pared to the O (4)values their results for exponents show a good agreement of z m ,a 50%larger value for z g and a value twice larger for z t .Comparison with O (2)and mean-field exponents is similar since they are not too different from the O (4)values.This work had limitations in several respects:(i)the scaling formulae are valid for a spatial size large enough compared to the correlation length.At m q =0.02the pion correlation length equals ξπ≈3.Whether the spatial size of L =8em-ployed is sufficiently large has to be examined.(ii)The size of the scaling region in terms of quark mass is a priori not known.Hence the behavior for smaller quark masses should be ex-plored to check if the results are not affected by sub-leading and analytic terms in an expansion of susceptibilities in m q .(iii)In the original work the noisy estimator with a single noise vector was employed to estimate disconnected double quark loop contributions.This introduces contamina-tion from connected diagrams and local contact terms,which has to be removed.Other fac-tors such as step size of the hybrid R algorithm and stopping condition for the solver of Kogut-Susskind matrix could also affect the value of sus-ceptibilities.For these reasons the Bielefeld group has con-tinued their study[28],and the JLQCD Collabo-ration[29]has started their own work last year.As one sees in Table 2run parameters of new simulations are chosen to examine the points (i)and (ii)above.In order to deal with (iii)Biele-feld group worked out the correction formula for the case of the single noise vector.They also employed the method of multiple noise vectors for some of the runs.JLQCD employed the method of wall source without gauge fixing[31],and removed contamination by a correction for-mula.At present both groups have accumulated (5−10)×103trajectories of unit length with a small step size of δτ=(1−1/2)m q for each6z m 0.790.792/30.79(4) 1.05(8)0.93(9)0.70(3) 1.01(11)1.02(7)L =8to L =12−16is evident,with the val-ues for larger sizes sizably deviating from either O (4),O (2)or the mean-field predictions.For z g the deviation seems less apparent though full data are not yet available.A puzzling nature of the values of exponents becomes clearer if we translate them into the more basic thermal and magnetic exponents y t and y h using the relations,z g =y ty h,z t =y ty h−1(9)withd =3the space dimension.The values in Table 3are reasonably consistent with the rela-tion z g +z m =z t +1which follow from (9).We observe that z m ≈1.0(1)obtained for larger spa-tial lattices implies y h ≈3.0(3)to be compared with the O (4)value 2.49,while y h =d =3is ex-pected for a first-order phase transition.For the thermal exponent we find y t ≈2.4(3)if we take z t ≈0.8(1)or y t ≈2.7(3)for z t ≈0.9(1),which is substantially larger than the O (4)value of 1.34.One may think of various possibilities for the reason leading to these values of exponents.(i)The most conventional would be that the influence of sub-leading and analytic terms is still sizable at the range of quark mass explored.(ii)Another possibility,suggested by the value yh ≈d for L =12and 16,is that a disconti-7 nuityfixed point with y h=d[33]controlling thefirst-ordertransition along the line m q=0in thelow-temperature phase is strongly influencing the scaling behavior.The transition is of second or-der in this case.Whether the deviation of y t from any of the expected values can be explained is not clear,however.(iii)The transition is of second or-der with the exponents close to but not equal tod.This would mean a significant departure fromthe universality concepts,stepping even beyond the suggestion of mean-field exponents arbitrarilyclose to the critical point[30].(iv)The transition is offirst order.In this case,the value of quarkmass m q=m c q at which thefirst-order transitionterminates would have to be small or even vanish since the scaling formula is derived under the as-sumption of a transition taking place at a singlepoint at m q=0.Concerning the possibility(iv),results ofpresent data examined fromfinite-size scaling point of view are as follows.As we alreadypointed out,χm for afixed value of m q staysconstant for L=12−16down to m q=0.02. Results for other susceptibilities exhibit a similarbehavior.Thus a phase transition does not existfor m q≥0.02as concluded in the previous stud-ies[26,27].At m q=0.01the susceptibilities in-crease by a factor3between L=8and16.Runsfor L=12are needed to see if the increase is con-sistent with a linear behavior in volume expected for afirst-order transition.We have to conclude that scaling analyses ofsusceptibilities carried out so far do not allow adefinite conclusion.Much further work,possibly with a quark mass smaller than has been explored so far,is needed to elucidate the nature of the chiral transition for N f=2.3.1.2.Scaling analysis of chiral order pa-rameterFor a second-order transition the singular partof the chiral order parameter is expected to scale asqq generated on a123×6 lattice with m q=0.025and0.0125.Adding an analytic term of form m q(c0+c1/g2+c2/g4), thefit was found acceptable for O(4)and also for the mean-field scaling function.Extrapolat-ing to the limit m q=0,the results differ signifi-cantly between the two cases,however(see Fig.2 of ref.[35]).We note that the results of the present MILC analysis do not contradict those of susceptibili-ties:the quark mass used for this work corre-sponds to m q≈0.02−0.04on an83×4lattice, for which case the exponents found from suscepti-bilities are similar to the O(4)values.We further remind,however,that the exponents exhibit a sig-nificant size dependence.This means that stud-ies with larger lattice sizes and smaller m q are required to explore the nature of the two-flavor transition from scaling of the chiral order param-eter.3.2.Restoration of U A(1)symmetryFor sufficiently high temperatures topologically non-trivial gauge configurations are suppressed, leading to restoration of U A(1)symmetry.To what extent U A(1)symmetry is restored close to the chiral transition is an interesting question. Three groups[35,37,28]examined the problem using the susceptibility defined byχUA(1)= d4x( π(x)· π(0) − a0(x)· a0(0) )(10)which should vanish at m q=0if U A(1)sym-metry is restored.In Fig.5we plot the m q de-pendence of this quantity obtained by the MILC Collaboration[35]and the Columbia group[37]. Both results are taken in the high temperature phase corresponding to T/T c≈1.2−1.3.While the data appear to extrapolate linearly to zero at m q=0(dotted lines)[37],it is more reason-89that the critical line should be defined by the van-ishing of the quark mass m q at zero temperature, where m q is defined through chiral Ward iden-tity[42,50,51].They reported that the crossing pointβct with this definition of the critical lineis located in the region of strong coupling on anN t=4lattice,e.g.,βct≈3.9−4.0for N f=2. For the phase diagram based on this result seeref.[6].This phase diagram,however,has an unsatis-factory feature.It has been observed[48,49]that physical observables do not exhibit any singu-lar behavior across the critical line in the high temperature phase.This means that the region K≥K c(β),usually thought unphysical,is not distinct from the high temperature phase,be-ing analytically connected to it.Hence one can cross from the low-to the high-temperature phase through the part of the critical line belowβ=βct, which is not a line offinite-temperature transi-tion.Clearly the phase diagram above does not cap-ture the full aspect of the phase structure.Re-cent investigations indicate that a more natural understanding of the phase structure is provided by a different view on the critical line proposed by Aoki some time ago[7].In the following we review the phase structure based on this view. Let us note that a slightly different phase struc-ture has been discussed in ref.[52].The phase structure for general values of N f up to N f=300 has also been examined recently[53].4.2.Spontaneous breakdown of parity-flavor symmetry and massless pionIn order to illustrate the basic idea,let us con-sider an effective sigma model for lattice QCD with the Wilson quark action with N f=2.The effective lagrangian may be written asL eff=(∇µ π)2+(∇µσ)2+a π2+bσ2+ (11)where the coefficients a and b differ reflecting ex-plicit breaking of chiral symmetry due to the Wil-son term.We know that the pion mass vanishes as a=m2π∝K c−K toward the critical line,while σstays massive,i.e.,b=m2σ>0at K≈K c.If K increases beyond K c,the coefficient a becomes negative.Hence we expect the pionfield to de-velop a vacuum expectation value π =0.The condensate spontaneously breaks parity andfla-vor symmetry.Let us note that pion is not the Nambu-Goldstone boson of spontaneously broken chiral symmetry in this view.Instead it represents the massless mode of a parity-flavor breaking second-order phase transition which takes place at K=K c.We expect it to become the Nambu-Goldstone boson of chiral symmetry in the contin-uum limit,however,as chiral symmetry breaking effects disappear in this limit.The idea above has been explicitly tested for the two-dimensional Gross-Neveu model formu-lated with the Wilson action[7].An analytic so-lution in the large N limit shows spontaneous breakdown of parity for K≥K c(β).Another important result of the solution is that the criti-cal line forms three spikes,which reach the weak-coupling limit g=0at1/2K=+2,0,−2.This structure arises from the fact that the doublers at the conventional continuum limit(g,1/2K)= (0,2)become physical massless modes at1/2K= 0and−2.A close similarity of the Gross-Neveu model and QCD regarding the asymptotic freedom and chiral symmetry aspects leads one to expect a similar phase structure for the case of QCD ex-cept that the critical line will formfive spikes reaching the continuum limit because of differ-ence in dimensions[7].Evidence supporting sucha phase structure is summarized in ref.[54].4.3.Finite-temperature phase structure For afinite temporal lattice size N t correspond-ing to afinite temperature,the above considera-tion can be naturally extend by defining the crit-ical line as the line of vanishing pion screening mass determined from the pion propagator for large spatial separations.In Fig.6the critical line for the two-dimensional Gross-Neveu model calculated in the large N limit is plotted for N t=∞,16,8,4,2 starting from the outermost curve and moving to-ward inside.The result shows that the location of the critical line as defined above depends on N t.Another important point is that the spikes formed by the critical line moves away from the101.5 1.00.50g21-1-21/2K Figure 6.Critical line in (g,1/2K )plane for the two-dimensional Gross-Neveu model for the tem-poral size N t =∞,16,8,4,2(from outside to in-side)[55].weak-coupling limit as N t decreases.Simulations to examine if lattice QCD has a similar structure of the critical line at finite temperatures have been made recently for the case of N f =2[55]and 4[56]on an 83×4lat-tice.The results are summarized as follows:(i)For both systems the conventional critical line turns back toward strong coupling forming a cusp,whose tip is located at β≈4.0for N f =2and β≈1.8for N f = 4.The cusp repre-sents one of five cusps expected for lattice QCD.(ii)Parity and flavor symmetry are spontaneously broken inside the cusp.Simulations have been made for the N f =2system with an exter-nal field term δS W =2KH nψγ5τ3ψ =0of the parity-flavor order parameter and vanishing of π±masslim H →0m π±=0expected inside the cusp[56].Concerning the relation between the thermal line and the critical line,we recall that the pion mass vanishes all along the critical line.This suggests that the region close to the critical line is in the cold phase even after the critical line turns back toward strong coupling,and hence the thermal line cannot cross the critical line.Since numerical estimates show that the thermal line comes close to the turning point of the cusp,the natural possibility is that the thermal line runs past the tip of the cusp and continues toward larger values of K .Results of measurement of thermodynamic quantities provide support ofthis0.150.200.250.30K11Indeed strongfirst-order signals have been ob-served for the case of N f=3[48]and4[56]awayfrom the critical line,as shown by solid squares in Fig.7,in contrast to a crossover behavior rep-resented by open squares seen for N f=2.How-ever,thefirst-order transition for N f=4weak-ens closer to the critical line,apparently turninginto a smooth crossover before reaching the region around the cusp of the critical line as indicated by open rectangles[56].While parallel data arenot yet available for N f=3,results of the QCD-PAX Collaboration[48]also appears to indicate aweakening of thefirst-order transition.A possible reason for this unexpected behav-ior is that breaking of chiral symmetry due tothe Wilson term,which becomes stronger asβdecreases along the thermal line,smoothens the first-order transition.Another possibility is thatthefirst-order transition for N f=3and4ob-served so far is a lattice artifact sharing its originwith the sharpening of the crossover atβ≈5.0 found by the MILC Collaboration for N f=2[49]. Some support for this interpretation is given bya recent study of the QCDPAX Collaboration for the N f=3system with an improved gauge ac-tion[57].So far they have not found clearfirst-order signals in the region where the plaquette action shows a clearfirst-order behavior.In either case,if chiral transition in the con-tinuum is indeed offirst order for N f=3and 4,it will emerge only when the cusp moves suffi-ciently toward weak-coupling with an increase of the temporal size N t.4.5.Continuum limitWe expect the cusp of thefinite-temperature critical line to grow toward weak coupling as N tincreases.In the limit N t=∞it should con-verge to the zero-temperature critical line which reachesβ=∞.Since the thermal line is locatedon the weak-coupling side of the cusp for afinite N t,it will be pinched by the tip of the cusp at(β,K)=(∞,1/8)as N t→∞.We expect chiral phase transition in the continuum to emerge in this limit.In order to extract continuum proper-ties of the chiral transition,we then need a sys-tematic study of thermodynamic quantities in the neighborhood of the thermal line when it runs close to the tip of the cusp as a function of N t. Simulations,however,indicate that the cusp moves only very slowly as N t increases.For the N f=2case,current estimates of the position of the tip of the cusp isβ≈4.0for N t=4[55], 4.0−4.2for N t=6[48],4.2−4.3for N t=8[56] and4.5−5.0even for N t=18[48].A recent work also reports an absence of parity-broken phase aboveβ=5.0on symmetric lattices up to the size104[60].For N f=4the values are even lower:β≈1.80for N t=4and2.2−2.3for N t=8[56].These estimates indicate that a very large temporal size will be needed for the cusp to move into the scaling region(e.g.,β≥5.5for N f=2)as long as one employs the Wilson quark action together with the plaquette action for the gauge part.We emphasize that this result has an impor-tant implication also for spectrum calculations at zero temperature.Since the location of the cusp is determined by the smaller of the spatial and temporal size,the critical line will be shifted or may even be absent unless lattice size is taken sufficiently large.Therefore hadron masses cal-culated on a lattice of small spatial size and ex-trapolated toward the position of the critical line might involve significant systematic uncertainties.4.6.Studies with improved actionsThe problems discussed above indicate the presence of sizable cutoffeffects when the Wil-son quark action is used in conjunction with the plaquette action.A way to alleviate this problem is to employ improved actions.This approach has been pursued by the QCDPAX Collabora-tion[8,57],replacing the plaquette action with an improved gauge action RG(1,2)[18].This year the MILC Collaboration reported simulations with the action SLW tadpole for the gauge part and the tadpole-improved clover action for the quark part[58].Results with the tree-level clover ac-tion keeping the plaquette action are also avail-able[59].Thus there are data for four types of action combinations,unimproved and improved both for the gauge and quark actions,to make a comparative study of improvement.An indication from such a comparison is that improving the gauge action substantially removes12cutoffeffects.An inflection of the critical line seen for the plaquette action atβ≈4−5becomes absent with improvement of the gauge action[8], while it still seems to remains if only the Wilson quark action is replaced by the clover action[59]. Also an intermediate sharpening of the thermal transition seen for the plaquette action atβ≈5.0[49]is not observed for improved actions[8,58]. Another point to note is that the lattice spacing at the coupling constant where the thermal line approaches the critical line has a similar value mρa≈1on an N t=4lattice for all of the four action combinations.This means that studies of physical quantities are needed to assess reduction of cutoffeffects with improved actions.Interest-ing results have already been obtained for scaling of the chiral order parameter[8,61],and work with the critical temperature is being pursued[8,58].5.Results infinite density studiesIt has long been known that the quenched ap-proximation breaks down for a non-zero quark chemical potentialµin that a transition takes place atµ≈mπ/2rather than atµ≈m N/3[62, 63].While the importance of the phase of the quark determinant has been made clear, the mechanism how the quenched approximation breaks has not been fully explained.Recently Stephanov[64],employing a random matrix model of the quark determinant[65]and a replica formulation of quenched approximation, traced back the failure of the quenched approxi-mation to the non-uniformity of the limit of the replica number n→0forµ=0andµ>0.He has also shown that the quenched approximation is valid for the theory in which a quarkχin the conjugate representation is added to each quark q.Formation of a condensateχq≈mπin such a theory explains the occurance of transition at µ≈mπ/2.Barbour and collaborators reported new results in full QCD simulations[66].With the method of fugacity expansion[67]runs were carried out for fourflavors of quarks on64and84lattices at β=5.1with the Kogut-Susskind quark action. They found an onset of non-zero baryon number at a small value ofµ,e.g.,µc≈0.1at m q=0.01. For comparison the MT c Collaboration reported m N=1.10(6)and mπ=0.290(6)at a slightly larger coupling ofβ=5.15at m q=0.01[68].It is not yet clear if these results mean that an early onset of transitionµc≈mπ/2also holds for full QCD or reflect computational problems of the method employed for the simulation.6.ConclusionsMuch work has been made infinite tempera-ture studies of lattice QCD encompassing a num-ber of subjects during the last year.Tests of improved actions made for the pure gauge system indicate a possibility that accurate results for thermodynamics in the continuum may be obtained with simulations carried out with a moderately large temporal size.In full QCD studies much progress has been made in understanding the phase structure for the Wilson quark action.On the other hand, new problems have also been encountered,mak-ing it necessary to reexamine conclusions reached in previous studies.These are the unexpected val-ues of exponents for N f=2found in scaling stud-ies of susceptibilities with the Kogut-Susskind quark action,and theflavor dependence of or-der of chiral transition with the Wilson quark ac-tion.Elucidating these problems is important for reaching an understanding of the nature of chiral phase transition,which is consistent between the Kogut-Susskind and Wilson quark actions. Some progress has been made in QCD atfinite density.A puzzling result reported from the lat-est simulation shows,however,that we are still far from understanding this difficult subject.In closing we point out that most work in full QCD during the past several years have concen-trated on the case of N f degenerate quarks,espe-cially for N f=2.While a variety of basic prob-lems we have encountered for this case has to be clarified with further work,we should also recall that nature corresponds to the case of N f=2+1 with a heavier strange quark.A delicate change of phase that might possibly result from its pres-ence,as suggested in the continuum sigma model analysis[5],makes it important to enlarge previ-。
