The equilibrium thermodynamic properties of the SU(N) plasma at finite
temperature are studied non-perturbatively in the large-N limit, via lattice
simulations. We present high-precision numerical results for the pressure,
trace of the energy-momentum tensor, energy density and entropy density of
SU(N) Yang-Mills theories with N=3, 4, 5, 6 and 8 colors, in a temperature
range from 0.8T_c to 3.4T_c (where T_c denotes the critical deconfinement
temperature). The results, normalized according to the number of gluons, show a
very mild dependence on N, supporting the idea that the dynamics of the
strongly-interacting QCD plasma could admit a description based on large-N
models. We compare our numerical data with general expectations about the
thermal behavior of the deconfined gluon plasma and with various theoretical
descriptions, including, in particular, the improved holographic QCD model
recently proposed by Kiritsis and collaborators. We also comment on the
relevance of an AdS/CFT description for the QCD plasma in a phenomenologically
interesting temperature range where the system, while still strongly-coupled,
approaches a `quasi-conformal' regime characterized by approximate scale
invariance. Finally, we perform an extrapolation of our results to the N to
$\infty$ limit.Comment: 1+38 pages, 13 eps figures; v2: added reference

C. E. DeTar

J. M. Maldacena

M. Laine

M. Teper

Marco Panero

English

arXiv

Crossref

Thermodynamics of the QCD Plasma and the Large-
                    
                      N

CiteSeerX

Thermodynamics of the QCD plasma and the large-N limit

Panero, Marco

Institutional Research Information System University of Turin

Thermodynamics of the QCD Plasma and the Large-N LimitMarco Panero*Institute for Theoretical Physics, ETH Zu¨rich, 8093 Zu¨rich, Switzerlandand Institute for Theoretical Physics, University of Regensburg, 93040 Regensburg, Germany(Received 20 August 2009; published 3 December 2009)The equilibrium thermodynamic properties of the SUðNÞ plasma are studied nonperturbatively in thelarge-N limit, via high-precision lattice simulations at temperatures from 0:8Tc to 3:4Tc (Tc being thecritical deconfinement temperature). Our results for SUðNÞ Yang-Mills theories with N ¼ 3, 4, 5, 6, and 8colors show a very mild dependence on N, supporting the idea that the QCD plasma could be described bymodels based on the large-N limit. We compare our data with various theoretical descriptions, including,in particular, the improved holographic QCD model proposed by Kiritsis and collaborators. We alsocomment on the relevance of an AdS/CFT description in a phenomenologically interesting temperaturerange where the system, while still strongly coupled, becomes approximately scale invariant. Finally, weextrapolate our results to the N ! 1 limit.DOI: 10.1103/PhysRevLett.103.232001 PACS numbers: 12.38.Gc, 11.15.Pg, 11.25.Tq, 12.38.MhIntroduction.—The high-energy heavy-ion collisions atSPS and RHIC showed evidence for ‘‘a new state ofmatter’’ [1], which reaches rapid thermalization, and ischaracterized by very low viscosity values, making it anearly ideal fluid [2]. On the theoretical side, however, theunderstanding of such strongly interacting quark-gluonplasma is still an open issue [3], and investigation fromthe first principles of QCD largely relies on the numericalapproach on the lattice [4]. Although several unquenchedstudies of finite-temperature QCD have appeared in recentyears, the thermodynamics of the pure-gauge sector is stillrelevant from a fundamental perspective, as it captures theessential qualitative features of the deconfinement phe-nomenon, is characterized by a well-defined theoreticalsetup, is computationally less demanding and not hinderedby the technical difficulties of simulations with dynamicalfermions. Finally, it is relevant for the large-N limit, inwhich the theory undergoes dramatic analytical simplifi-cations [5] and entails connections with string theory,playing a crucial roˆle in the conjectured AdS/CFT corre-spondence [6] and in models of strongly interacting gaugetheories based on gravity duals [7]. All such models aimingat a description of N ¼ 3 QCD [or Yang-Mills (YM)theory] in terms of predictions based on the large-N limitimplicitly rely on the assumption that the features of theN ¼ 3 theory are ‘‘close enough’’ to those of its N ¼ 1counterpart. While a priori this assumption is not guaran-teed to be true, there is strong numerical evidence that thisis indeed the case [8]. The aim of the work reported herewas to study nonperturbatively the equilibrium thermody-namic properties at finite temperature in SUðNÞ YM theo-ries with N ¼ 3, 4, 5, 6, and 8 colors, via latticesimulations; similar studies include Refs. [9]. In particular,here we compare the SUðNÞ lattice results with the im-proved holographic QCD (IHQCD) model recently pro-posed by Kiritsis and collaborators [10]; we also discussthe deficit of the entropy density s with respect to itsStefan-Boltzmann (SB) limit s0 in a temperature regimewhere the Yang-Mills plasma, while still strongly interact-ing, approaches a ‘‘quasiconformal’’ regime characterizedby approximate scale invariance (and a related AdS/CFTprediction for the large-N limit of the N ¼ 4 super-symmetric YM theory [11]). Next, we also investigatethe possibility that the trace anomaly of the deconfinedSUðNÞ plasma  receives contributions proportional to T2,possibly related to a dimension-two condensate [12].Finally, we present an extrapolation of our results for thepressure p, trace anomaly , energy density  and entropydensity s to the N ! 1 limit.Lattice formulation.—We simulated SUðNÞYM theorieswith N ¼ 3, 4, 5, 6, and 8 colors, regularized on a four-dimensional Euclidean hypercubic, isotropic lattice ofspacing a, spacelike volume V ¼ ðaNsÞ3 at temperatureT ¼ 1=ðaNtÞ. The system dynamics was given by theWilson gauge action [13] and the Markov chains weregenerated combining heat-bath steps for SU(2) subgroups[14] with full-SUðNÞ overrelaxation updates [15]. Thephysical scale for the SU(3) gauge group was set usingthe r0 values taken from Ref. [16]. For the other SUðN > 3Þgroups, we set the scale interpolating high-precision mea-surements of the string tension  [17] or (at the largest values only) using the method discussed in Ref. [18].The trace anomaly  ¼  3p was measured from dif-ferences of plaquette expectation values at T ¼ 0 (from alattice of size N4s ) and at finite T: ¼ T5 @@TpT4¼ 6a4@@ lnaðhUhi0  hUhiTÞ: (1)The pressure (relative to its T ¼ 0 vacuum value) wasdetermined using the ‘‘integral method’’ [19]:p ¼ 6a4Z 0d0ðhUhiT  hUhi0Þ; (2)PRL 103, 232001 (2009) P HY S I CA L R EV I EW LE T T E R Sweek ending4 DECEMBER 20090031-9007=09=103(23)=232001(4) 232001-1  2009 The American Physical Societywhile the energy and entropy densities were obtained from ¼ þ 3p and s ¼ ðþ 4pÞ=T, respectively.Results.—Our results for =T4, p=T4, =T4, and s=T3,normalized to the respective SB limits [20] are displayed inFig. 1, and reveal a very weak dependence on the grouprank. It is thus natural to compare these data with theanalogous curves obtained in the IHQCD model [10],which is a holographic AdS/QCD model based on anEinstein-dilaton gravity theory in five dimensions. Thismodel involves a dilaton potential which reproducesasymptotic freedom with a logarithmically running cou-pling in the ultraviolet (UV) and linear confinement with adiscrete mass gap in the infrared (IR) limit of the dualSUðNÞ gauge theory, capturing most of its nonperturbativefeatures (both at zero and finite temperature) at a quanti-tative level. Strictly speaking, the IHQCD model is ex-pected to hold in the large-N limit only, while at finite None expects corrections; in particular, the calculations inthe gravity model neglect string interactions, that are ex-pected to become important above a cutoff scale—which,for the parameters used in Refs. [10], for SU(3) would beapproximately equal to 2.5 GeV. Nevertheless, Fig. 1shows that the agreement between our SUðNÞ results andthe IHQCDmodel is very good for all the groups. Note thatin Refs. [10], by looking at the comparison with the N ¼ 3lattice results for =T4 from Ref. [21], it was pointed outthat the slight discrepancy between the improved holo-graphic QCD model and the lattice results in the regionof the peak [which for the SU(3) lattice data is located atT ’ 1:1Tc, and is slightly lower than the IHQCD curve]was likely to be a finite-volume lattice artifact. The firstpanel in Fig. 1 indeed confirms this: our results for theSU(3) gauge group are consistent with Ref. [21], while the=T4 maximum for the N > 3 gauge groups is larger andlocated closer to Tc. This is related to the fact that thedeconfining phase transition, which is a weakly first-orderone for SU(3), becomes stronger when N is increased[22]—and correspondingly the value of the correlationlength at the critical point gets shorter. While the IHQCDmodel accounts for the running of the coupling, comparingSUðNÞ results with predictions directly derived from con-formal models, such as N ¼ 4 SYM theory, is lessstraightforward: the left panel of Fig. 2 shows that theregime where the QCD plasma is strongly coupled is farfrom conformality, and the SUðNÞ plasma approaches aregime characterized by approximate scale invariance onlyat temperatures about 3Tc. Note that, around that tempera-ture, the plasma is still far from the Stefan-Boltzmann limit(top right corner of the diagram), still strongly interacting,and, interestingly, the entropy density deficit with respectto the free limit is very close to the AdS/CFT prediction forthe large-N limit of N ¼ 4 supersymmetric YM in thestrongly interacting regime [11]:ss0¼ 34þ 4532ð3Þð2Þ3=2 þ . . . (3)if one identifies the value of the renormalized ’t Hooft cou-pling in theMS scheme with the  parameter of the super-symmetric model (right panel of Fig. 2) [24]. Another issuewe investigated is the possibility that, in the temperaturerange between Tc and 3Tc, the trace anomaly may receivenonperturbative contributions proportional to T2 [12]:T4’ AT2þ B: (4)From our results, this indeed seems to be a general featureof all the gauge groups studied in this work, as shown in theleft-hand side panel of Fig. 3 [25]. Finally, the right-handside panel of Fig. 3 shows an extrapolation of our results tothe N ! 1 limit. This is based on the following parame-trization for the trace anomaly [26]:zT4¼1 1f1þ exp½ðT=TcÞf1f2 g2f3T2cT2þ f4T4cT4; (5)with z ¼ ðN2  1Þ2=45. The results for  from the vari-ous gauge groups are fitted to Eq. (5), and the resultingparameters are extrapolated to the large-N limit, as func-tions of N2 [27]. The extrapolated parameters are dis-played in Table I, where the first error is statistical and thesecond is an estimate of the overall systematic uncertain-ties involved in our calculation (which include discretiza-tion effects, finite-volume effects, systematic uncertaintiesin setting the scale, and in the extrapolation to the large-Nlimit). Note that f2 ! 0 for N ! 1, which is consistentwith the strongly first-order nature of the deconfining0.5 1 1.5 2 2.5 3 3.5T / Tc0.250.50.7511.251.51.75SU(3)SU(4)SU(5)SU(6)SU(8)IHQCD∆ / T4, normalized to the SB limit of p / T 40.5 1 1.5 2 2.5 3 3.5T / Tc0.20.40.60.81SU(3)SU(4)SU(5)SU(6)SU(8)IHQCDp / T 4, normalized to the SB limit0.5 1 1.5 2 2.5 3 3.5T / Tc0.20.40.60.81SU(3)SU(4)SU(5)SU(6)SU(8)IHQCDε / T 4, normalized to the SB limit0.5 1 1.5 2 2.5 3 3.5T / Tc0.20.40.60.81SU(3)SU(4)SU(5)SU(6)SU(8)IHQCDs / T3, normalized to the SB limitFIG. 1 (color online). Results for the trace anomaly, pressure, energy density, and entropy density evaluated in the various SUðNÞgauge groups. All quantities are normalized to their Stefan-Boltzmann (SB) limits (except for =T4, which is normalized to the SBlimit of p=T4). The results are also compared with the curves obtained using the IHQCD model [10] (yellow solid lines).PRL 103, 232001 (2009) P HY S I CA L R EV I EW LE T T E R Sweek ending4 DECEMBER 2009232001-2transition in the large-N limit and with the expectation thatin this limit the equilibrium thermodynamic quantitiesconsidered here vanish for T < Tc (note the N2 normal-ization factor). From the parameters thus extrapolated, weobtain the curves for the large-N limit of =ðN2T4Þ, ofp=ðN2T4Þ (by numerical integration of the latter over lnT),and of =ðN2T4Þ and s=ðN2T3Þ (through linear combina-tions of the former two quantities): they are shown in theright panel of Fig. 3, where we also show the large-N limitof the latent heat Lh calculated in Ref. [28], namely:L1=4h N1=2T1c ¼ 0:766ð40Þ. Our estimate for the samequantity is fully consistent: L1=4h N1=2T1c ¼ 0:759ð19Þ.Conclusions.—The high-precision lattice study of finite-temperature SUðNÞ gauge theories with N ¼ 3, 4, 5, 6, and8 colors presented here, reveals that the main equilibriumthermodynamic observables (per gluon) have a very weakN dependence—except for a tendency towards a morestrongly first-order deconfining transition when N is in-creased, as it is expected on general grounds [22]—and thatthe SU(3) results are close to the N ! 1 limit. This isimportant for AdS/CFT or holographic QCD models,which aim at providing an analytical description of thestrongly interacting QCD plasma produced in relativisticcollisions of heavy ions, using techniques based on themathematical simplifications occurring in the large-Nlimit. We compared our results with the IHQCD model[10], finding very good agreement, and with predictions forthe entropy density deficit from N ¼ 4 supersymmetricYM using the AdS/CFT correspondence, in particular, in atemperature regime where the SUðNÞ plasma gets nearlyscale invariant, while still strongly interacting. Our dataalso show that the T2 behavior observed in the SU(3) traceanomaly [12] appears to be a generic feature of all theories0 0.2 0.4 0.6 0.8 1(Tc / T)200.511.52SU(3)SU(4)SU(5)SU(6)SU(8)∆ / T4, normalized to the SB limit of p / T41 1.5 2 2.5 3T / Tc00.20.40.60.81 p / ( N2T4 )∆ / ( N2T4 )ε /  ( N2T4 )s / ( N2T3 )Extrapolation to the large-N limitFIG. 3 (color online). Left panel: the =T4 ratio, plotted against ðTc=TÞ2, is compatible with the behavior described by Eq. (4) in therange ðTc=TÞ2  0:9. Right panel: extrapolation of p=ðN2T4Þ (black solid curve), =ðN2T4Þ (red dashed curve), =ðN2T4Þ (blue dash-dotted curve) and s=ðN2T3Þ (green dotted curve) to the N ! 1 limit; the error bars (including statistical and systematic uncertainties)at some reference temperatures are also shown. The horizontal bars on the right-hand side of this plot show the SB limits for thepressure, energy density, and entropy density, from bottom to top; the maroon arrow denotes the large-N limit of the latent heat Lh, ascalculated in Ref. [28].0 0.5 1 1.5 2 2.5 3ε  / T 4, normalized to 1 / 3 of its SB limit00.20.40.60.81p / T4 , norm. to the SB limit SU(3)SU(4)SU(5)SU(6)SU(8)conformal limitweak-couplingp(ε) equation of state5 5.5 6 6.5 7 7.5 8 8.5 9λ00.20.40.60.81s / s0SU(3)SU(4)SU(5)SU(6)SU(8)AdS/CFTEntropy density vs. ’t Hooft couplingFIG. 2 (color online). Left panel: the equation of state of the SUðNÞ plasma, displayed as pðÞ, reveals strong deviations from thedotted straight line corresponding to conformally invariant models, which is approached only at temperatures around 3Tc; the dashedline is the weak-coupling expansion for SU(3) [23]. Right panel: the entropy density (normalized to its value in the free limit) as afunction of the running ’t Hooft coupling in the MS scheme; the dashed line is the corresponding prediction (at the first perturbativeorder around the strong-coupling limit) in N ¼ 4 SYM [11], obtained identifying the  parameter with the renormalized YMcoupling.PRL 103, 232001 (2009) P HY S I CA L R EV I EW LE T T E R Sweek ending4 DECEMBER 2009232001-3investigated here (and of the large-N limit thereof). Theextrapolation of our results for N ! 1 strongly suggeststhe possibility that the hot QCD plasma admits a verysimple (yet to be discovered) theoretical description inthis limit.The calculations presented in this Letter were performedon machines in the University of Regensburg. The authoracknowledges support from the Alexander von HumboldtFoundation and from INFN, and thanks B. Bringoltz,M. Caselle, Ph. de Forcrand, J. Erdmenger, M. Fromm,F. Gliozzi, U. Gu¨rsoy, E. Kiritsis, I. Kirsch, A. Kurkela,M. Laine, B. Lucini, F. Nitti, G. D. Torrieri, and P. Weiszfor correspondence, discussions, and encouragement. Theauthor also thanks the authors of Refs. [10] for providingthe numerical values obtained in the improved holographicQCD model for various thermodynamic quantities, whichare reproduced in Fig. 1. The University of Regensburghosts the Collaborative Research Center SFB/TR 55‘‘Hadron Physics from Lattice QCD.’’*panero@phys.ethz.ch[1] U.W. Heinz and M. Jacob, arXiv:nucl-th/0002042; K.Adcox et al. (PHENIX Collaboration), Nucl. Phys.A757, 184 (2005); I. Arsene et al. (BRAHMSCollaboration), Nucl. Phys. A757, 1 (2005); B. B. Backet al., Nucl. Phys. A757, 28 (2005); J. Adams et al. (STARCollaboration), Nucl. Phys. A757, 102 (2005).[2] P. F. Kolb and U.W. Heinz, arXiv:nucl-th/0305084.[3] E. Shuryak, Prog. Part. Nucl. Phys. 62, 48 (2009).[4] C. E. DeTar, Proc. Sci., LATTICE2008 (2008) 001; C. E.DeTar and U.M. Heller, Eur. Phys. J. A 41, 405 (2009);M. Laine, Proc. Sci. LATTICE2009 (2009) 006.[5] G. ’t Hooft, Nucl. Phys. B72, 461 (1974); E. Witten, Nucl.Phys. B160, 57 (1979); M. U¨nsal and L.G. Yaffe, Phys.Rev. D 78, 065035 (2008); D. Auckly and S. Koshkin,Geom. Topol. Monogr. 8, 195 (2006).[6] J.M. Maldacena, Adv. Theor. Math. Phys. 2, 231 (1998);Int. J. Theor. Phys. 38, 1113 (1999).[7] J. Erdmenger, N. Evans, I. Kirsch, and E. Threlfall, Eur.Phys. J. A 35, 81 (2008); D. Mateos, Classical QuantumGravity 24, S713 (2007); S. S. Gubser and A. Karch,arXiv:0901.0935.[8] M. Teper, Proc. Sci., LATTICE2008 (2008) 022.[9] B. Bringoltz and M. Teper, Phys. Lett. B 628, 113 (2005);S. Datta and S. Gupta, Nucl. Phys. A830, 749c (2009).[10] U. Gu¨rsoy and E. Kiritsis, J. High Energy Phys. 02(2008) 032; U. Gu¨rsoy, E. Kiritsis, and F. Nitti, J. HighEnergy Phys. 02 (2008) 019; U. Gu¨rsoy, E. Kiritsis,L. Mazzanti, and F. Nitti, Phys. Rev. Lett. 101, 181601(2008); Nucl. Phys. B820, 148 (2009); U. Gu¨rsoy,E. Kiritsis, G. Michalogiorgakis, and F. Nitti,arXiv:0906.1890[11] S. S. Gubser, I. R. Klebanov, and A.A. Tseytlin, Nucl.Phys. B534, 202 (1998);[12] R. D. Pisarski, Phys. Rev. D 74, 121703(R) (2006); Prog.Theor. Phys. Suppl. 168, 276 (2007); E. Megı´as, E. RuizArriola, and L. L. Salcedo, J. High Energy Phys. 01 (2006)073.[13] K. G. Wilson, Phys. Rev. D 10, 2445 (1974).[14] A. D. Kennedy and B. J. Pendleton, Phys. Lett. B 156, 393(1985); N. Cabibbo and E. Marinari, Phys. Lett. B 119,387 (1982).[15] J. Kiskis, R. Narayanan, and H. Neuberger, Phys. Lett. B574, 65 (2003); Ph. de Forcrand and O. Jahn, arXiv:hep-lat/0503041; S. Du¨rr, Comput. Phys. Commun. 172, 163(2005).[16] S. Necco and R. Sommer, Nucl. Phys. B622, 328(2002).[17] B. Lucini, M. Teper, and U. Wenger, J. High Energy Phys.06 (2004) 012; B. Lucini and M. Teper, Phys. Lett. B 501,128 (2001).[18] C. Allton, M. Teper, and A. Trivini, J. High Energy Phys.07 (2008) 021.[19] J. Engels, J. Fingberg, F. Karsch, D. Miller, and M. Weber,Phys. Lett. B 252, 625 (1990).[20] Since the SB limit of  is zero, we conventionally nor-malize =T4 to the SB limit of p=T4.[21] G. Boyd, J. Engels, F. Karsch, E. Laermann, C. Legeland,M. Lu¨tgemeier, and B. Petersson, Nucl. Phys. B469, 419(1996).[22] B. Lucini, M. Teper, and U. Wenger, Phys. Lett. B 545,197 (2002); J. High Energy Phys. 01 (2004) 061; J. HighEnergy Phys. 02 (2005) 033; K. Holland, M. Pepe, andU.-J. Wiese, Nucl. Phys. B694, 35 (2004); Nucl. Phys. B,Proc. Suppl. 129–130, 712 (2004); M. Pepe, Nucl. Phys.B, Proc. Suppl. 141, 238 (2005).[23] A. Hietanen, K. Kajantie, M. Laine, K. Rummukainen,and Y. Schro¨der, Phys. Rev. D 79, 045018 (2009).[24] Note, however, that such identification is not justifieda priori, and is prone to scheme dependence; further-more, it should be noted that, at relatively small valuesof the ’t Hooft coupling , subleading corrections notincluded in Eq. (3) could be non-negligible.[25] Yet, the precision of our data is not sufficient to rule outthe possibility that the trace anomaly may actually bedescribed by a more complicated functional form (pos-sibly involving logarithms of perturbative origin).[26] A. Bazavov et al., Phys. Rev. D 80, 014504 (2009).[27] We used constant-plus-linear extrapolations in N2, ex-cept for f2, for which we also included a quadratic term.[28] B. Lucini, M. Teper, and U. Wenger, J. High Energy Phys.02 (2005) 033.TABLE I. Results of the large-N extrapolation of the parame-ters appearing in Eq. (5). For each parameter, the first (second)error is the statistical (systematic) one.Parameter Extrapolated valuef1 0.9918(20)(87)f2 0.0090(5)(58)f3 1.768(36)(71)f4 0:244ð46Þð29ÞPRL 103, 232001 (2009) P HY S I CA L R EV I EW LE T T E R Sweek ending4 DECEMBER 2009232001-4

The equilibrium thermodynamic properties of the SU(N) plasma are studied nonperturbatively in the large-N limit, via high-precision lattice simulations at temperatures from 0.8T(c) to 3.4T(c) (T-c being the critical deconfinement temperature). Our results for SU(N) Yang-Mills theories with N=3, 4, 5, 6, and 8 colors show a very mild dependence on N, supporting the idea that the QCD plasma could be described by models based on the large-N limit. We compare our data with various theoretical descriptions, including, in particular, the improved holographic QCD model proposed by Kiritsis and collaborators. We also comment on the relevance of an AdS/CFT description in a phenomenologically interesting temperature range where the system, while still strongly coupled, becomes approximately scale invariant. Finally, we extrapolate our results to the N ->infinity limit

University of Regensburg Publication Server

Thermodynamics of the QCD Plasma and the Large-NLimit

http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.312.6800

Thermodynamics of the QCD plasma and the large-N limit

Abstract

Similar works

Full text

Available Versions

Crossref

CiteSeerX

Institutional Research Information System University of Turin

University of Regensburg Publication Server