We propose a method to study lattice QCD at finite temperature and chemical
potential. We compare it with direct results and with the Glasgow method by
using n_f=4 QCD at Im(\mu)\neq 0. We locate the critical endpoint (E) of QCD on
the Re(\mu)-T plane. In this study we use n_f=2+1 dynamical staggered quarks
with semi-realistic masses on L_t=4 lattices.Comment: 5 pages, 3 Figures. Talk given at EPS HEP2001, July 12-18,2001
  Budapes

Fodor, Z.

Katz, S. D.

English

arXiv

We propose a method to study lattice QCD at finite temperature and chemical potential. We compare it with direct results and with the Glasgow method by using n_f=4 QCD at Im(\mu)\neq 0. We locate the critical endpoint (E) of QCD on the Re(\mu)-T plane. In this study we use n_f=2+1 dynamical staggered quarks with semi-realistic masses on L_t=4 lattices

Fodor, Z

Katz, SD

ELTE Digital Institutional Repository (EDIT)

arXiv:hep-lat/0111064v1  30 Nov 2001 PrHEP hep2001International Europhysics Conference on HEPPROCEEDINGSThe critical point of lattice QCD on the µ-T planeZ. Fodor∗Institute for Theoretical Physics, Eotvos University, Pazmany P. 1, H-1117 Budapest,HungaryE-mail: fodor@poe.elte.huS.D. KatzDESY, Deutsches Elektronen Synchrotron, Theory Group, Notkestrasse 85,D-22603 Hamburg, GermanyE-mail: katz@bodri.elte.huAbstract: We propose a method to study lattice QCD at finite temperature and chem-ical potential. We compare it with direct results and with the Glasgow method by usingnf=4 QCD at Im(µ)6=0. We locate the critical endpoint (E) of QCD on the Re(µ)-T plane. In this study we use nf=2+1 dynamical staggered quarks with semi-realisticmasses on Lt=4 lattices.QCD at finite T and µ is of fundamental importance, since it describes relevant featuresof particle physics in the early universe, in neutron stars and in heavy ion collisions. Exten-sive experimental work has been done with heavy ion collisions at CERN and Brookhavento explore the µ-T phase diagram. Note, that past, present and future heavy ion exper-iments with always higher and higher energies produce states closer and closer to the Taxis of the µ-T diagram. It is a long-standing non-perturbative question, whether a criticalpoint exists on the µ-T plane, and particularly how to tell its location theoretically [1].Let us discuss first the µ=0 case. Universal arguments [2] and lattice simulations[3] indicate that in a hypothetical QCD with a strange (s) quark mass (ms) as small asthe up (u) and down (d) quark masses (mu,d) there would be a first order finite T phasetransition. The other extreme case (nf=2) with light u/d quarks but with an infinitelylarge ms there would be no phase transition only an analytical crossover. Note, thatobservables change rapidly during a crossover, but no singularities appear. Between thetwo extremes there is a critical strange mass (mcs) at which one has a second order finite Tphase transition. Staggered lattice results on Lt=4 lattices with two light quarks and msaround the transition T (nf=2+1) indicated [4] that mcs is about half of the physical ms.Thus, in the real world we probably have a crossover.∗Speaker.PrHEP hep2001International Europhysics Conference on HEP Z. FodorReturning to a non-vanishing µ, one realizes that arguments based on a variety ofmodels (see e.g. [5, 6, 1]) predict a first order finite T phase transition at large µ. Combiningthe µ = 0 and large µ informations an interesting picture emerges on the µ-T plane. Forthe physical ms the first order phase transitions at large µ should be connected with thecrossover on the µ = 0 axis. This suggests that the phase diagram features a criticalendpoint E (with chemical potential µE and temperature TE), at which the line of firstorder phase transitions (µ > µE and T < TE) ends [1]. At this point the phase transitionis of second order and long wavelength fluctuations appear, which results in (see e.g. [7])consequences, similar to critical opalescence. Passing close enough to (µE,TE) one expectssimultaneous appearance of signatures which exhibit nonmonotonic dependence on thecontrol parameters [8], since one can miss the critical point on either of two sides.The location of E is an unambiguous, non-perturbative prediction of QCD. No abinitio, lattice analysis based on QCD was done to locate the endpoint. Crude models withms = ∞ were used (e.g. [1]) suggesting that µE ≈ 700 MeV, which should be smaller forfinite ms. The goal of our exploratory work is to show how to locate the endpoint by alattice QCD calculation. We use full QCD with dynamical nf=2+1 staggered quarks.QCD at finite µ can be formulated on the lattice [9]; however, standard Monte-Carlotechniques can not be used. The reason is that for Re(µ)6=0 the determinant of the Eu-clidean Dirac operator is complex. This fact spoils any importance sampling method.An attractive approach to alleviate the problem is the “Glasgow method” (see e.g.Ref. [10]) in which the partition function (Z) is expanded in powers of exp(µ/T ) by usingan ensemble of configurations weighted by the µ=0 action. After collecting more than 20million configurations only unphysical results were obtained. The reason is that the µ=0ensemble does not overlap enough with the finite density states of interest [11].We propose a method to reduce theFigure 1: ψ̄ψ as a function of Im(µ), for di-rect results (squares), our technique (crosses) andGlasgow-type reweighting (dots).overlap problem and determine the phasediagram in the µ-T plane (for details see[12]). The idea is to produce an ensembleof QCD configurations at µ=0 and at Tc.Then we determine the Boltzmann weights[14] of these configurations at µ 6= 0 and atT lowered to the transition temperaturesat this non-vanishing µ. Since transitionconfigurations are reweighted to transitionconfigurations a much better overlap canbe observed than by reweighting pure had-ronic configurations to transition ones [10].After illustrating the applicability of themethod we locate the critical point of QCD. (Multi-dimensional reweighting was successfulfor determining the endpoint of the hot electroweak plasma [13] e.g. on 4D lattices.)Let us study a generic system of fermions ψ and bosons φ, where the fermion Lagrange– 2 –PrHEP hep2001International Europhysics Conference on HEP Z. Fodordensity 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 staggered QCD αconsists of β, mq and µ. For some choice of the parameters α=α0 importance samplingcan be done (e.g. for Re(µ)=0). Rewriting the above equation we getZ(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 restas the measure. Changing only one parameter of the ensemble generated at α0 provides anaccurate value for some observables only for high statistics. This is ensured by rare fluctu-ations as the mismatched measure occasionally sampled the regions where the integrand islarge. This is the overlap problem. Having several parameters the set α0 can be adjustedto get a better overlap than obtained by varying only one parameter.The basic idea of the method as applied to dynamical QCD can be summarized asfollows. We study the system at Re(µ)=0 around its transition point. Using a Glasgow-type technique we calculate the determinants for each configuration for a set of µ, which,similarly to the Ferrenberg-Swendsen method [14], can be used for reweighting. The averageplaquette values can be used to perform an additional reweighting in β. Since transitionconfigurations were reweighted to transition ones a much better overlap can be observedthan by reweighting pure hadronic configurations to transition ones as done by the Glasgow-type techniques (moving along the transition line was also suggested by Ref. [17]).We have directly tested these ideas in nf=4 QCD with mq=0.05 dynamical staggeredquarks. We first collected 1200 independent V=4·63 configurations at Re(µ)=Im(µ)=0 andsome β values and used the Glasgow-reweighting and also our technique to study Re(µ)=0,Im(µ)6=0. At Re(µ)=0, Im(µ)6=0 direct simulations are possible. After performing thesedirect simulations as well, a clear comparison can be done. Figure 1 shows the predictionsof the three methods 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 larger Im(µ) thepredictions of the Glasgow method are by several standard deviations off. Based on theseexperiences we expect that our method can be applied at Re(µ)6=0.In QCD with nf staggered quarks one should change the determinants to their nf/4power in our two equations. Importance sampling works also in this case at some β and atRe(µ)=0. Since detM is complex an additional problem arises, one should choose amongthe possible Riemann-sheets of the fractional power in eq. (2). This can be done by usingthe fact that at µ = µw the ratio of the determinants is 1 and the ratio should be acontinuous function of µ. However, the continuity can only be ensured if the analyticaldependence of the determinant on µ is known [12] (the idea goes back to a method of [16]).– 3 –PrHEP hep2001International Europhysics Conference on HEP Z. FodorIn the following we keep µ real and lookFigure 2: Im(β∞0) as a function of the chemicalpotential.for the zeros of Z for complex β. At afirst order phase transition the free energy∝ logZ(β) is non-analytic. A phase transi-tion appears only in the V→ ∞ limit, butnot in a finite V . Nevertheless, Z has zerosat finite V, generating the non-analyticityof the free energy, the Lee-Yang zeros [15].These are at complex values of the parame-ters, in our case at complex β. For a systemwith a first order transition these zeros ap-proach the real axis in the V→ ∞ limit (de-tailed analysis suggests 1/V scaling). ThisV→ ∞ limit generates the non-analyticity of the free energy. For a system with crossoverZ is analytic, and the zeros do not approach the real axis in the V→ ∞ limit.At T 6=0 we used Lt=4, Ls=4,6,8 lat-Figure 3: The T-µ diagram. Direct resultsare given with errorbars. Dotted line shows thecrossover, solid line the first order transition. Thebox gives the uncertainties of the endpoint.tices. T=0 runs were done on 103· 16 lat-tices. mu,d=0.025 and ms=0.2 were ourbare quark masses. At T 6= 0 we deter-mined the complex valued Lee-Yang zeros,β0, for different V-s as a function of µ.Their V→ ∞ limit was given by a β0(V ) =β∞0+ ζ/V extrapolation. We used 14000,3600 and 840 configurations on Ls=4,6 and8 lattices, respectively. Im(β∞0) is shownon Figure 2 as a function of µ. For smallµ-s the extrapolated Im(β∞0) is inconsis-tent with a vanishing value, and predicts acrossover. Increasing µ the value of Im(β∞0)decreases, thus the transition becomes con-sistent with a first order phase transition. (Note, that systematic overshooting is a finiteV effect.) The statistical error was determined by jackknife samples of the total Ls = 4, 6and 8 ensembles. 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 3 shows the phase diagram in physical units, thus T as a function of µB, thebaryonic chemical potential (which is three times larger then the quark chemical potential).The endpoint is at TE = 160 ± 3.5 MeV, µE = 725 ± 35 MeV. At µB=0 we obtainedTc = 172± 3 MeV.We proposed a method –an overlap improving multi-parameter reweighting technique–to numerically study non-zero µ and determine the phase diagram in the T -µ plane. Ourmethod is applicable to any number of Wilson or staggered quarks. As a direct test– 4 –PrHEP hep2001International Europhysics Conference on HEP Z. Fodorwe showed that for Im(µ)6=0 the predictions of our method are in complete agreementwith the direct simulations, whereas the Glasgow method suffers from the well-knownoverlap problem. We studied the µ-T phase diagram of QCD with dynamical nf=2+1quarks. Using our method we obtained TE=160±3.5 MeV and µE=725±35 MeV for theendpoint. 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. At µ=0 weobtained Tc=172±3 MeV. Clearly, more work is needed to get the final values. One has toextrapolate in the R-algorithm and to the thermodynamic, chiral and continuum limits.This work was partially supported by Hung. Sci. grants No. OTKA-T34980/T29803/-T22929/M28413/OM-MU-708/IKTA111/NIIF. This work was in part based on the MILCcollaboration’s public lattice gauge theory code: http://physics.indiana.edu/∼sg/milc.html.References[1] 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. C60 (1999) 055205.[2] R. Pisarski and F. Wilczek, Phys. Rev. D29 (1984) 338; F. Wilczek, Int. J. Mod. Phys. A7(1992) 3911; K. Rajagopal and F. Wilczek, Nucl. Phys. B399 (1993) 395.[3] For recent reviews see: A. Ukawa, Nucl. Phys. Proc. Suppl. 53 (1997) 106; E. Laerman, ibid.63 (1998) 114; F. Karsch, ibid. 83 (2000) 14; S. Ejiri, ibid. 94 (2001) 19.[4] F.K. Brown, Phys. Rev. Lett. 65 (1990) 2491; S. Aoki et al., Nucl. Phys. Proc. Suppl. 73(1999) 459.[5] A. Barducci et al., Phys. Lett. B231 (1989) 463; Phys. Rev. D41 (1990) 1610; ibid. D49(1994) 426; S.P. Klevansky, Rev. Mod. Phys. 64 (1992) 649; M. Stephanov, Phys. Rev. Lett.76, (1996) 4472.[6] M. Alford, K. Rajagopal and F. Wilczek, Phys. Lett. B422 (1998) 247; Nucl. Phys. B537(1999) 443; R. Rapp, T. Schäfer, E.V. Shuryak and M. Velkovsky, Phys. Rev. Lett. 81 (1998)53; for a recent review with references see K. Rajagopal and F. Wilczek, hep-ph/0011333.[7] S. Borsányi et al., Phys. Rev. D64 (2001) 125011.[8] M. Stephanov, K. Rajagopal and E. Shuryak, Phys. Rev. D60 (1999) 114028.[9] P. Hasenfratz and F. Karsch, Phys. Lett. B125 (1983) 308; J. Kogut et al., Nucl. Phys. B225(1983) 93.[10] I.M. Barbour et al., Nucl. Phys. B (Proc. Suppl.) 60A (1998) 220.[11] For a recent review on QCD at finite T and µ see e.g. S. Hands, hep-lat/0109034.[12] Z. Fodor and S.D. Katz, hep-lat/0104001; hep-lat/0106002.[13] Y. Aoki et al., Phys. Rev. D60 (1999) 013001; F. Csikor, Z. Fodor and J. Heitger, Phys. Rev.Lett. 82 (1999) 21.[14] A.M. Ferrenberg, R.H. Swendsen, Phys. Rev. Lett. 63 (1989) 1195; 61 (1988) 2635.[15] C.N. Yang and T.D. Lee, Phys. Rev. 87 (1952) 404.[16] D. Toussaint, Nucl. Phys. B (Proc. Suppl.) 17 (1990) 248.[17] M. Alford, A. Kapustin and F. Wilczek, Phys. Rev. D59 (1999) 054502.– 5 –

The critical point of lattice QCD on the mu - T plane

Zoltan Fodor

Sandor D. Katz

Crossref

The critical point of QCD on the $\mu-T$ plane

Katz, S D

CERN Document Server

PrHEP hep2001International Europhysics Conference on HEPPROCEEDINGSThe critical point of lattice QCD on the µ-T planeZ. Fodor∗Institute for Theoretical Physics, Eotvos University, Pazmany P. 1, H-1117 Budapest,HungaryE-mail: fodor@poe.elte.huS.D. KatzDESY, Deutsches Elektronen Synchrotron, Theory Group, Notkestrasse 85,D-22603 Hamburg, GermanyE-mail: katz@bodri.elte.huAbstract:We propose a method to study lattice QCD at nite temperature and chem-ical potential. We compare it with direct results and with the Glasgow method by usingnf=4 QCD at Im(µ)6=0. We locate the critical endpoint (E) of QCD on the Re(µ)-T plane. In this study we use nf=2+1 dynamical staggered quarks with semi-realisticmasses on Lt=4 lattices.QCD at nite T and µ is of fundamental importance, since it describes relevant featuresof particle physics in the early universe, in neutron stars and in heavy ion collisions. Exten-sive experimental work has been done with heavy ion collisions at CERN and Brookhavento explore the µ-T phase diagram. Note, that past, present and future heavy ion exper-iments with always higher and higher energies produce states closer and closer to the Taxis of the µ-T diagram. It is a long-standing non-perturbative question, whether a criticalpoint exists on the µ-T plane, and particularly how to tell its location theoretically [1].Let us discuss rst the µ=0 case. Universal arguments [2] and lattice simulations[3] indicate that in a hypothetical QCD with a strange (s) quark mass (ms) as small asthe up (u) and down (d) quark masses (mu,d) there would be a rst order nite T phasetransition. The other extreme case (nf=2) with light u/d quarks but with an innitelylarge ms there would be no phase transition only an analytical crossover. Note, thatobservables change rapidly during a crossover, but no singularities appear. Between thetwo extremes there is a critical strange mass (mcs) at which one has a second order nite Tphase transition. Staggered lattice results on Lt=4 lattices with two light quarks and msaround the transition T (nf=2+1) indicated [4] that mcs is about half of the physical ms.Thus, in the real world we probably have a crossover.∗Speaker.PrHEP hep2001International Europhysics Conference on HEP Z. FodorReturning to a non-vanishing µ, one realizes that arguments based on a variety ofmodels (see e.g. [5, 6, 1]) predict a rst order nite T phase transition at large µ. Combiningthe µ = 0 and large µ informations an interesting picture emerges on the µ-T plane. Forthe physical ms the rst order phase transitions at large µ should be connected with thecrossover on the µ = 0 axis. This suggests that the phase diagram features a criticalendpoint E (with chemical potential µE and temperature TE), at which the line of rstorder phase transitions (µ > µE and T < TE) ends [1]. At this point the phase transitionis of second order and long wavelength fluctuations appear, which results in (see e.g. [7])consequences, similar to critical opalescence. Passing close enough to (µE,TE) one expectssimultaneous appearance of signatures which exhibit nonmonotonic dependence on thecontrol parameters [8], since one can miss the critical point on either of two sides.The location of E is an unambiguous, non-perturbative prediction of QCD. No abinitio, lattice analysis based on QCD was done to locate the endpoint. Crude models withms = 1 were used (e.g. [1]) suggesting that µE  700 MeV, which should be smaller fornite ms. The goal of our exploratory work is to show how to locate the endpoint by alattice QCD calculation. We use full QCD with dynamical nf=2+1 staggered quarks.QCD at nite µ can be formulated on the lattice [9]; however, standard Monte-Carlotechniques can not be used. The reason is that for Re(µ)6=0 the determinant of the Eu-clidean Dirac operator is complex. This fact spoils any importance sampling method.An attractive approach to alleviate the problem is the \Glasgow method" (see e.g.Ref. [10]) in which the partition function (Z) is expanded in powers of exp(µ/T ) by usingan ensemble of congurations weighted by the µ=0 action. After collecting more than 20million congurations only unphysical results were obtained. The reason is that the µ=0ensemble does not overlap enough with the nite density states of interest [11].We propose a method to reduce theFigure 1: ψψ as a function of Im(µ), for di-rect results (squares), our technique (crosses) andGlasgow-type reweighting (dots).overlap problem and determine the phasediagram in the µ-T plane (for details see[12]). The idea is to produce an ensembleof QCD congurations at µ=0 and at Tc.Then we determine the Boltzmann weights[14] of these congurations at µ 6= 0 and atT lowered to the transition temperaturesat this non-vanishing µ. Since transitioncongurations are reweighted to transitioncongurations a much better overlap canbe observed than by reweighting pure had-ronic congurations to transition ones [10].After illustrating the applicability of themethod we locate the critical point of QCD. (Multi-dimensional reweighting was successfulfor determining the endpoint of the hot electroweak plasma [13] e.g. on 4D lattices.)Let us study a generic system of fermions ψ and bosons φ, where the fermion Lagrange{ 2 {PrHEP hep2001International Europhysics Conference on HEP Z. Fodordensity is ψM(φ)ψ. Integrating over the Grassmann elds 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 staggered QCD αconsists of β, mq and µ. For some choice of the parameters α=α0 importance samplingcan be done (e.g. for Re(µ)=0). Rewriting the above equation we getZ(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 conguration) and the restas the measure. Changing only one parameter of the ensemble generated at α0 provides anaccurate value for some observables only for high statistics. This is ensured by rare fluctu-ations as the mismatched measure occasionally sampled the regions where the integrand islarge. This is the overlap problem. Having several parameters the set α0 can be adjustedto get a better overlap than obtained by varying only one parameter.The basic idea of the method as applied to dynamical QCD can be summarized asfollows. We study the system at Re(µ)=0 around its transition point. Using a Glasgow-type technique we calculate the determinants for each conguration for a set of µ, which,similarly to the Ferrenberg-Swendsen method [14], can be used for reweighting. The averageplaquette values can be used to perform an additional reweighting in β. Since transitioncongurations were reweighted to transition ones a much better overlap can be observedthan by reweighting pure hadronic congurations to transition ones as done by the Glasgow-type techniques (moving along the transition line was also suggested by Ref. [17]).We have directly tested these ideas in nf=4 QCD with mq=0.05 dynamical staggeredquarks. We rst collected 1200 independent V=463 congurations at Re(µ)=Im(µ)=0 andsome β values and used the Glasgow-reweighting and also our technique to study Re(µ)=0,Im(µ)6=0. At Re(µ)=0, Im(µ)6=0 direct simulations are possible. After performing thesedirect simulations as well, a clear comparison can be done. Figure 1 shows the predictionsof the three methods 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 larger Im(µ) thepredictions of the Glasgow method are by several standard deviations o. Based on theseexperiences we expect that our method can be applied at Re(µ)6=0.In QCD with nf staggered quarks one should change the determinants to their nf/4power in our two equations. Importance sampling works also in this case at some β and atRe(µ)=0. Since detM is complex an additional problem arises, one should choose amongthe possible Riemann-sheets of the fractional power in eq. (2). This can be done by usingthe fact that at µ = µw the ratio of the determinants is 1 and the ratio should be acontinuous function of µ. However, the continuity can only be ensured if the analyticaldependence of the determinant on µ is known [12] (the idea goes back to a method of [16]).{ 3 {PrHEP hep2001International Europhysics Conference on HEP Z. FodorIn the following we keep µ real and lookFigure 2: Im(β∞0 ) as a function of the chemicalpotential.for the zeros of Z for complex β. At arst order phase transition the free energy/ logZ(β) is non-analytic. A phase transi-tion appears only in the V! 1 limit, butnot in a nite V . Nevertheless, Z has zerosat nite V, generating the non-analyticityof the free energy, the Lee-Yang zeros [15].These are at complex values of the parame-ters, in our case at complex β. For a systemwith a rst order transition these zeros ap-proach the real axis in the V!1 limit (de-tailed analysis suggests 1/V scaling). ThisV!1 limit generates the non-analyticity of the free energy. For a system with crossoverZ is analytic, and the zeros do not approach the real axis in the V!1 limit.At T 6=0 we used Lt=4, Ls=4,6,8 lat-Figure 3: The T-µ diagram. Direct resultsare given with errorbars. Dotted line shows thecrossover, solid line the first order transition. Thebox gives the uncertainties of the endpoint.tices. T=0 runs were done on 103 16 lat-tices. mu,d=0.025 and ms=0.2 were ourbare quark masses. At T 6= 0 we deter-mined the complex valued Lee-Yang zeros,β0, for dierent V-s as a function of µ.Their V!1 limit was given by a β0(V ) =β∞0 + ζ/V extrapolation. We used 14000,3600 and 840 congurations on Ls=4,6 and8 lattices, respectively. Im(β∞0 ) is shownon Figure 2 as a function of µ. For smallµ-s the extrapolated Im(β∞0 ) is inconsis-tent with a vanishing value, and predicts acrossover. Increasing µ the value of Im(β∞0 )decreases, thus the transition becomes con-sistent with a rst order phase transition. (Note, that systematic overshooting is a niteV eect.) The statistical error was determined by jackknife samples of the total Ls = 4, 6and 8 ensembles. Our primary result is µend = 0.375(20).To set the physical scale we used a weighted average of R0, mρ andpσ. Note, that(including systematics due to nite V) we have (R0 mpi) = 0.73(6), which is at least twice,mu,d is at least four times as large as the physical values.Figure 3 shows the phase diagram in physical units, thus T as a function of µB, thebaryonic chemical potential (which is three times larger then the quark chemical potential).The endpoint is at TE = 160  3.5 MeV, µE = 725  35 MeV. At µB=0 we obtainedTc = 172 3 MeV.We proposed a method {an overlap improving multi-parameter reweighting technique{to numerically study non-zero µ and determine the phase diagram in the T -µ plane. Ourmethod is applicable to any number of Wilson or staggered quarks. As a direct test{ 4 {PrHEP hep2001International Europhysics Conference on HEP Z. Fodorwe showed that for Im(µ)6=0 the predictions of our method are in complete agreementwith the direct simulations, whereas the Glasgow method suers from the well-knownoverlap problem. We studied the µ-T phase diagram of QCD with dynamical nf=2+1quarks. Using our method we obtained TE=1603.5 MeV and µE=72535 MeV for theendpoint. 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. At µ=0 weobtained Tc=1723 MeV. Clearly, more work is needed to get the nal values. One has toextrapolate in the R-algorithm and to the thermodynamic, chiral and continuum limits.This work was partially supported by Hung. Sci. grants No. OTKA-T34980/T29803/-T22929/M28413/OM-MU-708/IKTA111/NIIF. This work was in part based on the MILCcollaboration’s public lattice gauge theory code: http://physics.indiana.edu/~sg/milc.html.References[1] 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. C60 (1999) 055205.[2] R. Pisarski and F. Wilczek, Phys. Rev. D29 (1984) 338; F. Wilczek, Int. J. Mod. Phys. A7(1992) 3911; K. Rajagopal and F. Wilczek, Nucl. Phys. B399 (1993) 395.[3] For recent reviews see: A. Ukawa, Nucl. Phys. Proc. Suppl. 53 (1997) 106; E. Laerman, ibid.63 (1998) 114; F. Karsch, ibid. 83 (2000) 14; S. Ejiri, ibid. 94 (2001) 19.[4] F.K. Brown, Phys. Rev. Lett. 65 (1990) 2491; S. Aoki et al., Nucl. Phys. Proc. Suppl. 73(1999) 459.[5] A. Barducci et al., Phys. Lett. B231 (1989) 463; Phys. Rev. D41 (1990) 1610; ibid. D49(1994) 426; S.P. Klevansky, Rev. Mod. Phys. 64 (1992) 649; M. Stephanov, Phys. Rev. Lett.76, (1996) 4472.[6] M. Alford, K. Rajagopal and F. Wilczek, Phys. Lett. B422 (1998) 247; Nucl. Phys. B537(1999) 443; R. Rapp, T. Scha¨fer, E.V. Shuryak and M. Velkovsky, Phys. Rev. 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The critical point of lattice QCD on the \mu-T plane

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