We propose a method to study lattice QCD at finite temperature (T) and
chemical potential (\mu). We compare the method with direct results and with
the Glasgow method by using n_f=4 QCD at Im(\mu)\neq0. We locate the critical
endpoint (E) of QCD on the Re(\mu)-T plane. We use n_f=2+1 dynamical staggered
quarks with semi-realistic masses on L_t=4 lattices. Our results are based on
{\cal{O}}(10^3-10^4) configurations.Comment: 8 pages, 4 figures. Talk given at the XXX. Hirschegg Workshop on
  Ultrarelativistic Heavy-Ion Collisions, January 13-19, 200

Fodor, Z.

Katz, S. D.

English

arXiv

Fodor, Z

Katz, SD

ELTE Digital Institutional Repository (EDIT)

arXiv:nucl-th/0201071v1  25 Jan 2002FINITE TEMPERATURE AND CHEMICALPOTENTIAL IN LATTICE QCDAND ITS CRITICAL POINTZ. FODOR∗1 AND S.D. KATZ‡2∗Institute for Theoretical Physics, Eötvös University, Pázmány P. 1/A, H-1117,Budapest, Hungary‡Deutsches Elektronen-Synchrotron DESY, Notkestr. 85, D-22607,Hamburg, GermanyAbstractWe propose a method to study lattice QCD at finite temperature (T)and chemical potential (µ). We compare the method with direct resultsand with the Glasgow method by using nf=4 QCD at Im(µ)6=0. Welocate the critical endpoint (E) of QCD on the Re(µ)-T plane. We usenf=2+1 dynamical staggered quarks with semi-realistic masses on Lt=4lattices. Our results are based on O(103 − 104) configurations.QCD at finite T and/or µ is of fundamental importance, since it describesrelevant features of particle physics in the early universe, in neutron stars andin heavy ion collisions. Extensive experimental work has been done with heavyion collisions at CERN and Brookhaven to explore the µ-T phase boundary (cf.[1]). Note, that past, present and future heavy ion experiments with alwayshigher and higher energies produce states closer and closer to the T axis of theµ-T diagram. It is a long-standing question, whether a critical point exists onthe µ-T plane, and particularly how to tell its location theoretically [2].Let us discuss first the µ=0 case. Universal arguments [3] and lattice simu-lations [4] indicate that in a hypothetical QCD with a strange (s) quark mass(ms) as small as the up (u) and down (d) quark masses (mu,d) there would be afirst order finite T phase transition. The other extreme case (nf=2) with lightu/d quarks but with an infinitely large ms there would be no phase transitiononly a crossover. Observables change rapidly during a crossover, but no singu-larities appear. Between the two extremes there is a critical strange mass (mcs)1email: fodor@poe.elte.hu2on leave from Institute for Theoretical Physics, Eötvös University, Pázmány P. 1/A,H-1117, Budapest, Hungaryemail: katz@bodri.elte.huat which one has a second order finite T phase transition. Staggered latticeresults on Lt=4 lattices with two light quarks and ms around the transition T(nf=2+1) indicated [5] that mcs is about half of the physical ms. Thus, in thereal world we probably have a crossover.Returning to a non-vanishing µ, one realizes that arguments based on avariety of models (see e.g. [6, 7, 2]) predict a first order finite T phase transitionat large µ. Combining the µ = 0 and large µ informations an interestingpicture emerges on the µ-T plane. For the physical ms the first order phasetransitions at large µ should be connected with the crossover on the µ = 0axis. This suggests that the phase diagram features a critical endpoint E(with chemical potential µE and temperature TE), at which the line of firstorder phase transitions (µ > µE and T < TE) ends [2]. At this point the phasetransition is of second order and long wavelength fluctuations appear, whichresults in (see e.g. [8]) consequences, similar to critical opalescence. Passingclose enough to (µE ,TE) one expects simultaneous appearance of signatureswhich exhibit nonmonotonic dependence on the control parameters [9], sinceone can miss the critical point on either of two sides.The location of E is an unambiguous, non-perturbative prediction of QCD.No ab initio, lattice analysis based on QCD was done to locate the endpoint.Crude models withms = ∞ were used (e.g. [2]) suggesting that µE ≈ 700 MeV,which should be smaller for finite ms. The goal of our exploratory work is toshow how to locate the endpoint by a lattice QCD calculation. We use fullQCD with dynamical nf=2+1 staggered quarks.QCD at finite µ can be given on the lattice [10]; however, standard Monte-Carlo techniques fail. At Re(µ) 6=0 the determinant of the Euclidean Diracoperator is complex, which spoils any importance sampling method.Several suggestions were studied in detail to solve the problem. We list afew of them (for a recent review see Ref. [11]).In the large gauge coupling limit a monomer-dimer algorithm was used [12].For small gauge coupling an attractive approach is the “Glasgow method” [13]in which the partition function is expanded in powers of exp(µ/T ) by usingan ensemble of configurations weighted by the µ=0 action. After collectingmore than 20 million configurations only unphysical results were obtained:a premature onset transition. The reason is that the µ=0 ensemble does notoverlap sufficiently with the states of interest. Another possibility is to separatethe absolute value and the phase of the fermionic determinant and use theformer to generate configurations and the latter in observables [14].At imaginary µ the measure remains positive and standard Monte Carlotechniques apply. The canonical partition function can be obtained by aFourier transform [15, 16]. In this technique the dominant source of errorsis the Fourier transform rather than the poor overlap. One can also use thefact that the partition function away from the transition line should be ananalytic function of µ, and the fit for imaginary µ values could be analyticallycontinued to real values of µ [17]. At T sufficiently above the transition, bothreal and imaginary µ can be studied by dimensionally reducing QCD [18].Hamiltonian formulation may also help studying the problem [19].Glasgow methodnew methodµβFig. 1: Schematic difference between the present and the Glasgow methods.An attractive approach to alleviate the problem is the “Glasgow method”(see e.g. Ref. [13]) in which the partition function (Z) is expanded in powers ofexp(µ/T ) by using an ensemble of configurations weighted by the µ=0 action.After collecting more than 20 million configurations only unphysical resultswere obtained. The reason is that the µ=0 ensemble does not overlap enoughwith the finite density states of interest [20].We propose a method to reduce the overlap problem and determine thephase diagram in the µ-T plane (for details see [21]). The idea is to producean ensemble of QCD configurations at µ=0 and at Tc. Then we determinethe Boltzmann weights [23] of these configurations at µ 6= 0 and at T low-ered to the transition temperatures at this non-vanishing µ. Since transitionconfigurations are reweighted to transition ones a much better overlap can beobserved than by reweighting pure hadronic configurations to transition ones[13]. Since the original ensemble is collected at µ=0 we do not expect to be ableto decsribe the physics of the large µ region with ie.g. exotic colour supercon-ductivity. Fortunately, the typical µ values at present heavy ion acceleratorsare smaller than the region we cover (for the freeze-out condition see e.g. [24]).After illustrating the applicability of the method we locate the criticalpoint of QCD. (Multi-dimensional reweighting was successful for determiningthe endpoint of the hot electroweak plasma [22] e.g. on 4D lattices.)Let us study a generic system of fermions ψ and bosons φ, where the fermionLagrange density is ψ̄M(φ)ψ. Integrating over the Grassmann fields we get:Z(α) =∫Dφ exp[−Sbos(α, φ)] detM(φ, α), (1)where α denotes a set of parameters of the Lagrangian. In the case of staggeredQCD α consists of β, mq and µ. For some choice of the parameters α=α0importance sampling can be done (e.g. for Re(µ)=0). Rewriting eq. (1)Z(α) =∫Dφ exp[−Sbos(α0, φ)] detM(φ, α0){exp[−Sbos(α, φ) + Sbos(α0, φ)]detM(φ, α)detM(φ, α0)}. (2)We treat the curly bracket as an observable (measured on each configuration)and the rest as the measure. Changing only one parameter of the ensemblegenerated at α0 provides an accurate value for some observables only for highstatistics. This is ensured by rare fluctuations as the mismatched measureoccasionally sampled the regions where the integrand is large. This is theoverlap problem. Having several parameters the set α0 can be adjusted to geta better overlap than obtained by varying only one parameter.Fig. 2: ψ̄ψ as a function of Im(µ), for direct results (squares), our technique (crosses) andGlasgow-type reweighting (dots).The basic idea of the method as applied to dynamical QCD can be summa-rized as follows. We study the system at Re(µ)=0 around its transition point.Using a Glasgow-type technique we calculate the determinants for each con-figuration for a set of µ, which, similarly to the Ferrenberg-Swendsen method[23], can be used for reweighting. The average plaquette values can be used toperform an additional reweighting in β. Since transition configurations werereweighted to transition ones a much better overlap can be observed than byreweighting pure hadronic configurations to transition ones as done by theGlasgow-type techniques. The differences between the two methods are shownin Figure 1. Moving along the transition line was also suggested by Ref. [16].Fig. 3: Im(β∞0) as a function of the chemical potential.We have directly tested these ideas in nf=4 QCD with mq=0.05 dynamicalstaggered quarks. We first collected 1200 independent V=4·63 configurationsat Re(µ)=Im(µ)=0 and some β values and used the Glasgow-reweighting andalso our technique to study Re(µ)=0, Im(µ) 6=0. At Re(µ)=0, Im(µ) 6=0 directsimulations are possible. After performing these direct simulations as well, aclear comparison can be done. Figure 2 shows the predictions of the threemethods for the average quark condensates at β=5.085 as a function of Im(µ).The predictions of our method agree with the direct results, whereas for largerIm(µ) the predictions of the Glasgow method are by several standard devia-tions off. We expect that our method can be applied at Re(µ) 6=0.In QCD with nf staggered quarks one should change the determinants totheir nf/4 power in our two equations. Importance sampling works also in thiscase at some β and at Re(µ)=0. Since detM is complex an additional problemarises, one should choose among the possible Riemann-sheets of the fractionalpower in eq. (2). This can be done by using [21] the fact that at µ=µw theratio of the determinants is 1 and it should be a continuous function of µ.In the following we keep µ real and look for the zeros of Z for complex β.At a first order phase transition the free energy ∝ logZ(β) is non-analytic.A phase transition appears only in the V→ ∞ limit, but not in a finite V .Nevertheless, Z has zeros at finite V, generating the non-analyticity of the freeenergy, the Lee-Yang zeros [25]. These are at complex parameters (e.g. β).For a system with first order transition these zeros approach the real axis asV→ ∞ by a 1/V scaling. This V→ ∞ limit generates the non-analyticity ofthe free energy. For a system with crossover Z is analytic, and the zeros donot approach the real axis as V→ ∞.Fig. 4: The T-µ diagram. Direct results are given with errorbars. Dotted line shows thecrossover, solid line the first order transition. The box gives the uncertainties of the endpoint.At T6=0 we used Lt=4, Ls=4,6,8 lattices. T=0 runs were done on 103·16 lattices. mu,d=0.025 and ms=0.2 were our bare quark masses. At T 6= 0we determined the complex valued Lee-Yang zeros, β0, for different V-s asa function of µ. Their V→ ∞ limit was given by a β0(V ) = β∞0 + ζ/Vextrapolation. We used 14000, 3600 and 840 configurations on Ls=4,6 and 8lattices, respectively. Im(β∞0) is shown on Figure 3 as a function of µ. Forsmall µ-s the extrapolated Im(β∞0) is inconsistent with a vanishing value, andpredicts a crossover. Increasing µ the value of Im(β∞0) decreases, thus thetransition becomes consistent with a first order phase transition (overshootingis a finite V effect). Our primary result is µend = 0.375(20).To set the physical scale we used a weighted average of R0, mρ and√σ.Note, that (including systematics due to finite V) we have (R0 ·mπ) = 0.73(6),which is at least twice, mu,d is at least four times as large as the physical values.Figure 4 shows the phase diagram in physical units, thus T as a functionof µB, the baryonic chemical potential (which is three times larger then thequark chemical potential). The endpoint is at TE = 160 ± 3.5 MeV, µE =725± 35 MeV. At µB=0 we obtained Tc = 172± 3 MeV.We proposed a method –an overlap improving multi-parameter reweightingtechnique– to numerically study non-zero µ and determine the phase diagramin the T -µ plane. Our method is applicable to any number of Wilson orstaggered quarks. As a direct test we showed that for Im(µ) 6=0 the predictionsof our method are in complete agreement with the direct simulations, whereasthe Glasgow method suffers from the well-known overlap problem. We studiedthe µ-T phase diagram of QCD with dynamical nf=2+1 quarks. Using ourmethod we obtained TE=160±3.5 MeV and µE=725±35 MeV for the endpoint.Though µE is too large to be studied at RHIC or LHC, the endpoint wouldprobably move closer to the µ=0 axis when the quark masses get reduced. Atµ=0 we obtained Tc=172±3 MeV. More work is needed to get the final valuesby extrapolating in the R-algorithm and to the thermodynamic, chiral andcontinuum limits. The details of the presented results can be found in [21].This work was partially supported by grants OTKA-T34980/T29803/-M37071/OM-MU-708/IKTA111/NIIF and in part based on the MILC col-laboration’s lattice code: http://physics.indiana.edu/∼sg/milc.html.References[1] P. Braun-Munzinger, J. Stachel, nucl-th/0112051.[2] M. Halasz et al. Phys. Rev. D58 (1998) 096007; J. Berges, K. Rajagopal,Nucl. Phys. B538 (1999) 215. M. Stephanov, K. Rajagopal, E. Shuryak,Phys. Rev. Lett. 81 (1998) 4816; T.M. Schwarz, S.P. Klevansky, G. Papp,Phys. Rev. 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Finite temperature and chemical potential in lattice QCD and its critical point

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