We calculate the pressure (p), the energy density (epsilon) and the baryon density (n(B)) of QCD at finite temperatures (T) and chemical potentials (mu). The recently proposed overlap improving multi-parameter reweighting technique is used to determine observables at nonvanishing chemical potentials. Our results are obtained by studying n(f) =2+1 dynamical staggered quarks with semi-realistic masses on N-t = 4 lattices

Csikor, F

Egri, GI

Fodor, Z

Katz, SD

Szabo, KK

Toth, AI

English

ELTE Digital Institutional Repository (EDIT)

arXiv:hep-lat/0209114v1  12 Sep 20021ITP-Budapest 587, DESY 02-140The QCD equation of state at finite T and µF. Csikora, G.I. Egria, Z. Fodora, S.D. Katzb∗, K.K. Szabó and A.I. TóthaaInstitute for Theoretical Physics, Eötvös University, Pázmány P. 1/A, H-1117 Budapest, HungarybDeutsches Elektronen-Synchrotron DESY, Notkestr. 85, D-22607, Hamburg, GermanyWe calculate the pressure (p), the energy density (ǫ) and the baryon density (nB) of QCD at finite temperatures(T) and chemical potentials (µ). The recently proposed overlap improving multi-parameter reweighting techniqueis used to determine observables at nonvanishing chemical potentials. Our results are obtained by studyingnf=2+1 dynamical staggered quarks with semi-realistic masses on Nt = 4 lattices.weight linesbest quark−gluonplasmahadronic phase transition lineβµFigure 1. The best weight lines on the µ–β plane.In the middle we indicate the transition line. Itsfirst dotted part is the crossover region. The blobrepresents the critical endpoint, after which thetransition is of first order. The integration pathsused to calculate p are shown by the arrows alongthe β axis and the best weight lines.QCD at nonzero density is easily formulated onthe lattice by multiplying the forward/backwardlinks by exp(±µ). However, standard importancesampling based Monte-Carlo techniques can notbe used at µ 6=0. Up to now, no technique wassuggested capable of giving the equation of state(EOS) at µ 6=0, which is essential to describe thequark gluon plasma (QGP) formation at heavyion collider experiments. Results are only avail-able for µ=0 (e.g. [1,2,3]) at T 6=0.The overlap improving multi-parameterreweighting [4] opened the possibility to study∗on leave from Institute for Theoretical Physics, EötvösUniversity, Pázmány P. 1/A, H-1117 Budapest, HungaryFigure 2. p normalised by T 4 as a function ofT/Tc at µ = 0 The Nt=4 SB limit is also shown.lattice QCD at nonzero T and µ. First one pro-duces an ensemble of QCD configurations at µ=0and T 6=0. Then reweighting factors are deter-mined at µ 6= 0 and at a lowered T. The idea isexpressed in terms of the partition functionZ(β, µ,m) =∫DU exp[−Sg(β)] detM(µ,m)=∫DU exp[−Sg(β0)] detM(µ = 0,m) (1){exp[−Sg(β) + Sg(β0)]detM(µ,m)detM(µ = 0,m)},where m is the quark mass, Sg is the action ofthe gluonic field (U), at β = 6/g2 coupling. Atnonzero µ one gets a complex fermion determi-nant detM which spoils importance sampling.Thus, the first line of eq. (1), µ 6=0, is rewrittenin a way that the second line of eq. (1) is used as2Figure 3. ∆p = p(µ 6= 0, T ) − p(µ = 0, T ) nor-malised by T 4 as a function of T/Tc for µB=100,210, 330, 410 MeV and µB=530 MeV (from bot-tom to top).an integration measure (at µ=0 with importancesampling) and the curly bracket is measured oneach independent configuration and is interpretedas a weight factor {w(β, µ,m,U)}. In order tomaximise the accuracy of Z the reweighting isperformed along the best weight lines on the µ–β plane. These best weight lines are defined byminimising the spread of logw.Using the above technique, transition (orhadronic/QGP) configurations are reweighted totransition (or hadronic/QGP) configurations asillustrated by Fig. 1. The technique works fortemperatures at, below and above the transitiontemperature (Tc). By using the reweighting tech-nique, the phase diagram [4] and the location ofthe critical endpoint [5] was given. Using a Taylorexpansion around µ=0, T 6=0 for small µ can beused to determine thermal properties [6]. A dif-ferent method, analytic continuation from imagi-nary µ, confirmed results of [4,5] on the µ–T dia-gram [7].We use 4 ·N3s lattices at T 6=0 with Ns=8,10,12for reweighting and we extrapolate to V→∞ us-ing the available volumes (V ). At T=0 latticesof 24 · 143 are taken for vacuum subtraction andto connect lattice parameters to physical quanti-ties. 14 different β values are used, which corre-spond to T/Tc = 0.8, . . . , 3. Our T=0 simulationsprovided R0 and σ. The lattice spacing at Tc is≈0.25–0.30 fm. We use 2+1 flavours of dynami-Figure 4. ∆p of the QCD plasma normalised by∆p of the free gas (SB) as a function of T/Tc forthe same µB values as in Fig. 3.cal staggered quarks. While varying β (thus thetemperature) we keep the physical quark massesconstant at mud ≈ 65 MeV and ms ≈ 135 MeV(the pion to rho mass ratio is mπ/mρ ≈0.66).The determination of the equation of stateat µ 6=0 needs several observables O at non-vanishing µ values. This can be calculated byusing the weights of eq. (1)O(β, µ,m) =∑{w(β, µ,m,U)}O(β, µ,m,U)∑{w(β, µ,m,U)}.(2)p can be obtained from the partition func-tion as p=T · ∂ logZ/∂V which can be writtenas p=(T/V ) · logZ for large homogeneous sys-tems. On the lattice we can only determinethe derivatives of logZ with respect to the pa-rameters of the action (β,m, µ). Using the fol-lowing notation 〈O(β, µ,m)〉= O(β, µ,m)T 6=0 −O(β, µ = 0,m)T=0. p can be written as an inte-gral [8]:pT 4=1T 3V∫d(β,m, µ) (3)(〈∂(logZ)∂β〉,〈∂(logZ)∂m〉,〈∂(logZ)∂µ〉).The integral is by definition independent of theintegration path. The chosen integration pathsare shown on Fig 1.The energy density can be written as ǫ =(T 2/V ) ·∂(logZ)/∂T +(µT/V ) ·∂(logZ)/∂µ. Bychanging the lattice spacing T and V are simul-3Figure 5. (ǫ − 3p)/T 4 at µB=0, 210, 410 MeVand 530 MeV versus T/Tc (from bottom to top).taneously varied. The special combination ǫ− 3pcontains only derivatives with respect to a and µ:ǫ − 3pT 4= −aT 3V∂ log(Z)∂a∣∣∣∣µ+µT 3V∂(logZ)∂µ∣∣∣∣a.(4)The quark number density is n = (T/V ) ·∂ log(Z)/∂µ which can be measured directly orobtained from p (baryon density is nB=n/3 andbaryonic chemical potential is µB=3µ).We present direct lattice results on p(µ = 0, T ),∆p(µ, T ) = p(µ 6= 0, T ) − p(µ = 0, T ), ǫ(µ, T )-3p(µ, T ) and nB(µ, T ). Note, that in [9] addi-tional overall factors were used to help the phe-nomenological interpretation. Our statistical er-rorbars are rather small, sometimes smaller thanthe thickness of the lines.Fig. 2 shows p at µ=0. On Fig. 3 we present∆p/T 4 for five different µ values. Fig. 4 gives∆p(µ, T/Tc) normalised by ∆pSB ≡ ∆p(µ, T →∞). Notice the interesting scaling behaviour.∆p/∆pSB depends only on T and it is practicallyindependent of µ in the analysed region. Fig. 5shows ǫ-3p normalised by T 4, which tends to zerofor large T . Fig. 6 gives the baryonic density asa function of T/Tc for different µ-s. As it can beseen the densities exceed the nuclear density byup to an order of magnitude.An important finding concerns the applicabilityof our reweighting method: the maximal µ scaleswith the volume as µmax · a ∼ (Nt · N3s )−0.25. Ifthis behaviour persists, one could –in principle–approach the true continuum limit (a ∼ 1/Nt ∼Figure 6. nB/T3 versus T/Tc for the same µBvalues as in Fig. 3 (from bottom to top).(Nt ·N3s )−0.25, thus µmax≈const.).Future analyses should be done at smaller lat-tice spacings and quark masses. A detailed ver-sion of this work can be found elsewhere [9].Acknowledgements: This work was partiallysupported by Hungarian Scientific grants OTKA-T37615/T34980/T29803/M37071/OMFB1548/-OMMU-708. For the simulations a modifiedversion of the MILC public code was used (seehttp://physics.indiana.edu/∼sg/milc.html). Thesimulations were carried out on the Eötvös Univ.,Inst. Theor. Phys. 163 node parallel PC cluster.REFERENCES1. S. Gottlieb et al., Phys. Rev. D 55 (1997)6852.2. F. Karsch, E. Laermann and A. Peikert, Phys.Lett. B 478 (2000) 447.3. A. Ali Khan et al. [CP-PACS collaboration],Phys. Rev. D 64 (2001) 074510.4. Z. Fodor and S. D. Katz, Phys. Lett. B 534(2002) 87.5. Z. Fodor and S. D. Katz, JHEP 0203 (2002)014.6. C. R. Allton et al., hep-lat/0204010; S. Ejiriet al., hep-lat/0209012; C. Schmidt et al, hep-lat/0209009.7. P. de Forcrand and O. Philipsen, hep-lat/0205016, hep-lat/0209084.8. J. Engels et al., Phys. Lett. B 252 (1990) 625.9. Z. Fodor, S.D. Katz and K.K. Szabó, hep-lat/0208078; F. Csikor et al., in preparation.

The QCD equation of state at finite T and mu

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The QCD equation of state at finite T and mu

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