We determined the curvature of the phase transition line in the mu-T plane
using a Taylor expansion in mu. The Polyakov loop and the strange quark number
susceptibility were measured to locate the pseudocritical line. The analysis
was carried out on Nt=4,6,8,10 lattices generated with a Symanzik improved
gauge and stout-link improved (2+1) flavour staggered fermion action using
physical quark masses.Comment: 7 pages, 5 figures; format adjuste

Fodor, Zoltán

Guse, Christa

Katz, Sándor D.

Szabó, Kálmán K.

English

arXiv

Fodor, Zoltan

Katz, Sandor D

Szabo, Kalman K

ELTE Digital Institutional Repository (EDIT)

arXiv:0712.2702v2  [hep-lat]  19 Dec 2007Curvature of the phase transition line in the µ-TplaneZoltán Fodor, abc Christa Guse ∗,a Sándor D. Katz ab and Kálmán K. Szabó aaDepartment of Physics, University of Wuppertal,Gauss Strasse 20, D-42119, Wuppertal, GermanybInstitute for Theoretical Physics, Eötvös University,Pazmany 1, H-1117 Budapest, HungarycJohn von Neumann Institute for Computing (NIC),DESY, D-15738, Zeuthen / FZJ, D-52425, Juelich, GermanyE-mail: fodor@bodri.elte.hu,guse@physik.uni-wuppertal.de,katz@bodri.elte.hu, szaboka@general.elte.huWe determined the curvature of the phase transition line in theµ-T plane using a Taylor expansionin µ . The Polyakov loop and the strange quark number susceptibility were measured to locatethe pseudocritical line. The analysis was carried out onNt = 4,6,8,10 lattices generated with aSymanzik improved gauge and stout-link improved (2+1) flavour staggered fermion action usingphysical quark masses.The XXV International Symposium on Lattice Field TheoryJuly 30 - August 4 2007Regensburg, Germany∗Speaker.c© Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licen. http://pos.sissa.it/Curvature of the phase transition line in theµ-T plane Christa Guse1. IntroductionThe thermodynamics of strongly interacting matter at finitetemperature has been studied ex-tensively. It has been shown that the QCD transition is a cross-over [1]. This result is of importancefor the physics of the early universe. It is also interestingo examine the QCD phase diagram atfinite chemical potentialµB 6= 0 which would be relevant for the description of heavy ion colli-sions. Lattice simulations in this regime have been hindered by the sign problem ever since thefirst attempts [2]. The sign problem originates from the factthat the fermion matrixM loses itsγ5 hermiticity at finite chemical potentialM 6= γ5M†γ5. Thus the fermion determinant det(M) be-comes complex and spoils any method based on importance sampling. There are several differentmethods to circumvent this problem for smallµB based on either performing simulations at imag-inary µB and then analytically continuing the results to the physically interesting case of realµB[3] or reweighting samples produced with a real and positivedet(M) [4, 5]. For our calculationswe used a reweighting technique, where all observables measured on theµB = 0 configurations areexpanded inµB. The analysis was performed using the same configurations asi [6].2. Reweighting and Taylor expansionThe reweighting technique uses configurations generated atµB = 0 (where the fermion deter-minant is real and positive), to access observablesO at finite chemical potential.〈O〉 =∫DU O[det(M(µB))nf /4]e−Sg(β)∫DU[det(M(µB))nf /4]e−Sg(β)=∫DU O ω ′[det(M(0))nf /4]e−Sg(β0)∫DU ω ′[det(M(0))nf /4]e−Sg(β0)=〈O ω ′〉0〈ω ′〉0ω ′ =det(M(µB))nf /4det(M(0))nf /4e−(Sg(β)−Sg(β0)) (2.1)The expectation value〈O〉 at µB andβ is rewritten as the ratio of two expectation values atµB = 0andβ0. The chemical potential range where the reweighting is effectiv is limited by the decreasingoverlap between the sample generated atµB = 0 and the target ensemble atµB 6= 0. As the distri-bution of the sample becomes narrower with increasing volume the overlap problem gets also moresevere, thus requiring higher statistics when approachingt e continuum limit. It is quite expensiveto calculate the determinant det(M(µB)) exactly, especially for large lattices. For that reason wedecided to use a Taylor expansion inµ . This technique has been used several times, especially bythe Bielefeld-Swansea Collaboration (for a recent paper with sixth order expansion see [7]). Forgluonic observables which do not depend on the chemical potential xplicitly this expansion is veryeasy to perform.〈O〉= 〈O〉0+2 〈OO1〉0 µq+[〈O(2O21+O2)〉0−〈O〉0〈2O21+O2〉0]µ2q (2.2)withO1 =14∂∂ µq[lndet(M(µq))]∣∣∣∣µq=0=14tr(M−1M′)(2.3)2Curvature of the phase transition line in theµ-T plane Christa GuseandO2 =14∂ 2∂ µ2q[lndet(M(µq))]∣∣∣∣∣µq=0=14tr(M−1M′′−M−1M′M−1M′)(2.4)One can also note, that atµq = 0 the odd order derivatives of lndet(M(µq)) are purely imaginaryand the even order derivatives are real. This follows from the fermion matrix identity:M†(µq) = γ5M (−µq)γ5 (2.5)Therefore it is clear that for a real observable the linear term in the expansion is zero.3. ObservablesAll observables that we compute in this work namely the Polyakov loop and the strange quarknumber susceptibility do not depend explicitly onµB and can thus be expanded like gluonic ob-servables.3.1 Polyakov loopThe Polyakov loop is given by the trace over a product of gaugelinks along a line in timedirection:P=1N3s∑xtr [U4 (x,0)U4 (x,1) . . .U4 (x,Nt −1)] (3.1)In pure gauge theory a change in the expectation value of the Polyakov loop to a non-vanishingvalue signals the spontaneous breakdown of the Z(3) symmetry and thus the onset of deconfine-ment. The Polyakov loop is also related to the quark-antiquark free energy at infinite separation:|〈P〉|2 = exp(−∆Fqq̄(r → ∞)) (3.2)Here∆Fqq̄ denotes the difference of the free energies of the quark-gluon plasma with and withoutthe quark-antiquark pair. It is then possible to renormalize the Polyakov loop by renormalizing thefree energy of the quark-antiquark pair. For that purpose one uses the renormalization conditionVR(r0) = 0 to renormalize the potential atT = 0 for each lattice spacing [6]. As the ultravioletdivergencies are the same atT = 0 as at finiteT the same renormalization shift can then be used torenormalize the free energy. The renormalized Polyakov loop is given by|〈PR〉|= |〈P〉| exp(V(r0)/(2T)) , (3.3)whereV(r0) is the unrenormalizedT = 0 potential obtained from Wilson-loops. The transitiontemperature can be defined as the peak in the temperature derivativ of the Polyakov loop, that isthe inflection point of the Polyakov loop curve.We also note that the linear term in the Taylor expansion of the Polyakov loop is zero. Thiscan be shown by using the CPT symmetry [8]. It is enough to showt at〈P〉µ = 〈P〉∗µ . This followsfrom the CPT transformation of the expectation value of the Polyakov loop:〈P〉µ = 〈P〉CPTµ =∫DUCPTe−Sg(UCPT)det(Mµ(UCPT))PCPT=∫DU e−Sg(U)[det(Mµ(U))]∗P∗ =(〈P〉µ)∗(3.4)3Curvature of the phase transition line in theµ-T plane Christa Guse3.2 Strange quark number susceptibilityThe strange quark number susceptibilityχs is defined byχsT2=1TV∂ 2 logZ∂ µ2s∣∣∣∣µs=0(3.5)It can be related to the event-by-event fluctuations in heavy-ion experiments. Asχs has a welldefined continuum limit, renormalization is not necessary.The transition temperature can be de-fined as the peak in the temperature derivative of the strangequark number susceptibility, that isthe inflection point of the susceptibility curve.As the strange quark number susceptibility is a real observable the linear term in theµB ex-pansion vanishes as was shown above.4. Simulation SetupWe used a Symanzik improved gauge and stout-link improved staggered fermionic lattice ac-tion in order to reduce taste violation [9]. The configurations were generated with an exact RHMCalgorithm. We determined a line of constant physics (LCP) using physical masses for the lightquarksmu,d as well as for the strange quarkms. The LCP was fixed by settingmK/ fK andmK/mπto their physical values [6]. We used four different latticespacingsNt = 4,6,8,10 with an aspectratio of Ns/Nt = 4. The scale was fixed byfK and its unambiguity checked by calculatingmK∗,fπ andr0. The operatorsO1 andO2 were computed using a random noise estimator for the traces.The number of random vectors was chosen to give an error of thesame size as the statistical error.It was between 50 and 160 in all cases.5. Results00.511.522.533.5135 145 155 165 175〈P〉T [MeV]µa= 0.0µa= 0.100.20.40.60.811.2135 145 155 165 175χsT [MeV]µa= 0.0µa= 0.1Figure 1: On the left hand side the temperature dependence of the expectation value of the Polyakov loopis shown forµ a= 0 andµ a= 0.1. The graph on the right hand side depicts the temperature depen nce ofthe strange quark number susceptibility forµ a= 0 andµ a= 0.1. The results were obtained on theNt = 6lattices.4Curvature of the phase transition line in theµ-T plane Christa GuseFigure 1 shows the Polyakov loop and the strange quark numbersusceptibility forµ a= 0 andµ a = 0.1 at Nt = 6. We want to compute the curvatureκ of the phase transition line atµB = 0,which we define byTc(µB) = Tc(1−κµ2BT2c)(5.1)that isκ =−TcdTc(µB)dµ2B∣∣∣∣µB=0. (5.2)Here it is assumed that the phase transition line can be well described by a purely quadratic term inµB, whereas the fourth order term is negligible. Now we usedTc(µB)dµ2B∣∣∣∣µB=0=∂O∂ µ2B∣∣∣∣µB=0/∂O∂T∣∣∣∣T=Tc(0)(5.3)to directly extract the curvature from∂O/∂ µ2B and∂O/∂T. HereO stands for the Polyakov loop orthe strange quark number susceptibility respectively. Figure 2 shows−TcdTc(µB)/dµ2B computedby dTc(µB)/dµ2B = ∂ χs/∂ µ2B (∂ χs/∂T)−1 for theNt = 4 lattices. We use an average over the pointsnearTc(µB = 0) to extract the curvature atTc and can thus increase the statistics.00.010.020.030.040.05140 145 150 155 160−TcdTc(µB)dµ2BT [MeV]Figure 2: This plot shows the−TcdTc(µB)/dµ2B computed bydTc(µB)/dµ2B = ∂ χs/∂ µ2B(∂ χs/∂T)−1 fortheNt = 4 lattices.In our case∂O/∂ µ2B is directly given by the quadratic term in theµ expansion. Whereas∂O/∂T can be extracted from a fit to the Polyakov loop and the quark number susceptibility nearTc. For one point we also calculated∂O/∂ µ2B by exact determinant evaluation [10] and found aperfect agreement.We determine the error of the curvature by carrying out the whole analysis for differentjackknife-samples and by varying the fit range.To extract the continuum limit ofκ one needs at least three points in the scaling region. It wasestablished in [6] that ourNt = 4 lattices aren’t in this region. Thus we use theNt = 6,8,10 results5Curvature of the phase transition line in theµ-T plane Christa Guse1411421431441451461471480 20 40 60 80 100 120T [MeV]µB [MeV]usingχsusing〈P〉00.010.020.030.040.050 0.04 0.08 0.12κa2 [fm2]usingχsusing〈P〉Figure 3: The left plot shows the phasediagram in theµB-T plane for the lattice spacingNt = 4. The redcurves give the crossover line computed by finding the inflection point of the Polyakov loop, the black onecorresponds to the computations using the strange quark number susceptibility. The big errorbars at the startof the curve denote the total error ofTc(µ = 0), the smaller errorbars include just the error of the shift∆T.As the phasediagram was extracted from a Taylor expansion its only correct to leading order. On the righthand side the lattice spacing dependence of the curvatureκ is displayed. No safe continuum extrapolation ispossible with our present statistics. Here the red and blacklines correspond to the results obtained from thePolyakov loop and the strange quark number susceptibility respectively.for the extrapolation, see Figure 3. Unfortunately the present tatistics for theNt = 10 lattices isvery limited and thus the errors are large. It is an ongoing project to increase the statistics to give areliable continuum value for the curvature.6. SummaryWe have computed the curvatureκ of theµ-T pseudocritical line atµB = 0. The simulationswere performed at physical quark masses withnf = 2+1 flavours for 4 different lattice spacings.We used a Taylor expansion inµB of the Polyakov loop and the strange quark mass susceptibilityand determined their inflection points in order to locate thecrossover line.AcknowledgmentsThe computations were carried out on the 370 processor PC cluster of Eötvös University, onthe 1024 processor PC cluster of Wuppertal University, on the 107 node PC cluster equipped withGraphical Processing Units at Wuppertal University [11] and o the Blue-Gene/L at FZ Jülich. Weused a modified version of the publicly available MILC code [12] with next-neighbour communi-cation architecture for PC-clusters [13]. Partial supportof grants of DFG F0 502/1, EU I3HP andOTKA AT049652 is acknowledged.6Curvature of the phase transition line in theµ-T plane Christa GuseReferences[1] Y. Aoki, G. Endrodi, Z. Fodor, S. D. Katz, K. K. 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Curvature of the phase transition line in the mu-T plane

Curvature of the phase transition line in the mu-T plane

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