The QCD phase diagram as a function of temperature, T, and chemical potential for baryon

number, mB, is still unknown today, due to the sign problem, which prohibits direct Monte Carlo

simulations for non-vanishing baryon density. Investigations in models sharing chiral symmetry

with QCD predict a phase diagram, in which the transition corresponds to a smooth crossover at

zero density, but which is strengthened by chemical potential to turn into a first order transition

beyond some second order critical point. This contribution reviews the lattice evidence in favour

and against the existence of a critical point

Philipsen, Owe

English

Hochschulschriftenserver - Universität Frankfurt am Main

PoS(ICHEP 2010)370The QCD phase diagram at low baryon density fromlattice simulationsOwe PhilipsenInstitut für Theoretische Physik, Goethe-Universität, Max-von-Laue-Str. 1,60438 Frankfurt am Main, GermanyE-mail: philipsen@th.physik.uni-frankfurt.deThe QCD phase diagram as a function of temperature, T, and chemical potential for baryonnumber, mB, is still unknown today, due to the sign problem, which prohibits direct Monte Carlosimulations for non-vanishing baryon density. Investigations in models sharing chiral symmetrywith QCD predict a phase diagram, in which the transition corresponds to a smooth crossover atzero density, but which is strengthened by chemical potential to turn into a ﬁrst order transitionbeyond some second order critical point. This contribution reviews the lattice evidence in favourand against the existence of a critical point.35th International Conference of High Energy Physics - ICHEP2010,July 22-28, 2010Paris FranceSpeaker.c  Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/PoS(ICHEP 2010)370QCD phase diagram Owe Philipsen* QCD critical pointcrossover 1rst0∞Real worldXHeavy quarksmu,dmsµ  QCD critical point DISAPPEAREDcrossover 1rst0∞Real worldXHeavy quarksmu,dmsµFigure 1: Left: Schematic phase transition behaviour of Nf = 2+1 QCD for different choices of quarkmasses at m = 0. On ﬁner lattices (Nt = 6;a  0:2 fm), the chiral critical line moves towards smaller quarkmasses. Middle and Right: Critical surface swept by the chiral critical line as m is turned on. Depending onits curvature, a QCD chiral critical point is present or absent.1. IntroductionThe QCD phase diagram determines the forms of nuclear matter under different conditionsgiven by temperature, T, and chemical potential for baryon number, mB =3m. Based on asymptoticfreedom, one expects at least three different regimes: hadronic (low mB;T), quark gluon plasma(high T) and colour-superconducting (high mB, low T). Whether and where these regions are sepa-rated by true phase transitions has to be determined by ﬁrst principle calculations and experiments.Since QCD is strongly coupled on nuclear scales, Monte Carlo simulations of lattice QCD are thenatural approach. Unfortunately, the so-called sign problem prohibits straightforward simulationsat ﬁnite baryon density. There are ways to circumvent this problem in an approximate way, allof them valid for m=T < 1 only [1]: reweighting, Taylor expansions in m=T about zero and sim-ulations at imaginary chemical potential, where there is no sign problem, followed by analyticcontinuation. Within the range m=T < 1, all give agreeing results for, e.g., the phase boundaryTc(m) [2]. Because of the costly ﬁnite size scaling analyses, determining the order of the transition,and hence the existence of a chiral critical point, is a much harder task.The order of the QCD phase transition at zero density depends on the quark masses, Fig. 1(left). In the limits of zero and inﬁnite masses, order parameters corresponding to the breaking ofa global symmetry can be deﬁned, and one ﬁnds ﬁrst order phase transitions at some ﬁnite temper-atures Tc(mq). On the other hand, one observes analytic crossovers for intermediate masses, sepa-rated from the ﬁrst order regions by second order lines with Z(2) (3d Ising) universality [3, 4, 5].These are called chiral and deconﬁnement critical lines, respectively. The former has been mappedout on coarse Nt = 4 (a  0:3 fm) lattices [6] and puts the physical quark mass conﬁguration inthe crossover region. The chiral critical line recedes strongly with decreasing lattice spacing [7, 8],i.e. in the continuum the physical point is deeper in the crossover region, Fig. 1 (left).When a chemical potential is switched on, the chiral critical line sweeps out a surface, asshown in Fig. 1 (right). According to standard expectations [9], for small mu;d, the critical lineshould continuously shift with m to larger quark masses until it passes through the physical point atmE, corresponding to the endpoint of the QCD phase diagram. This is depicted in Fig. 1 (middle),where the critical point is part of the chiral critical surface. However, it is also possible for thechiral critical surface to bend towards smaller quark masses, cf. Fig. 1 (right), in which case therewould be no chiral critical point or phase transition at moderate densities.2PoS(ICHEP 2010)370QCD phase diagram Owe Philipsen 0 1 2 3 4 5 6 7 8 0.85  0.9  0.95  1  1.05  1.1T/Tc(0)ρ ρ2(p/T4)ρ4(p/T4)ρ2(χB)ρ4(χB)Figure 2: Left: Imaginary part of the Lee-Yang zero closest to the real axis [10]. Right: Estimates for theradius of convergence from different observables and at different order for Nf = 2+1;Nt = 4. Dashed linescorrespond to hadron resonance gas values [13].2. Critical point for ﬁxed masses: reweighting and Taylor seriesReweighting methods at physical quark masses on Nt = 4 lattices get a signal for a criticalpoint at mEB  360 MeV [10]. A Lee-Yang zero analysis was employed, Fig. 2. For a crossover thepartition function has zeroes only off the real axis, whereas for a phase transition the zeroes accu-mulate and pinch the real axis when extrapolated to inﬁnite volume. A caveat of this calculation isthe observation that the critical point is found in the immediate neighbourhood of the onset of pioncondensation in the phase quenched theory, which is where the sign problem becomes maximal[11]. Therefore, a conﬁrmation with an independent method is desired.In principle the determination of a critical point is also possible via the Taylor expansion. Inthis case true phase transitions will be signalled by an emerging non-analyticity, or a ﬁnite radiusof convergence for the pressure series about m=T = 0, as the volume is increased. The radiusof convergence gives the distance between the expansion point and the nearest singularity of thefull function, and may be extracted from the high order behaviour of the series. Theorems ensurethat if the limit exists and asymptotically all coefﬁcients of the series are positive, then there isa singularity on the real axis. The current best attempt is based on four consecutive coefﬁcients,and a critical endpoint for the Nf = 2 theory was reported in [12]. There are also difﬁculties inthis approach. Firstly, different deﬁnitions for the radius of convergence are only unique in theasymptotic limit, but differ at ﬁnite order. Estimates at a given order are neither upper nor lowerbounds on an actual radius of convergence. Finally, one also obtains ﬁnite convergence radiusestimates from the Taylor coefﬁcients of the hadron resonance gas model, even though that modeldoes not feature a non-analytic phase transition. This is illustrated in Fig. 2 (middle) [13].3. The chiral critical surfaceFor deﬁniteness, let us consider three degenerate quarks, represented by the diagonal in thequark mass plane, Fig. 1 (left). The critical quark mass corresponding to the boundary point hasan expansion mc(m)=mc(0) = 1+c1(m=pT)2+::: Tuning to mc(0), one may evaluate the leadingcoefﬁcients of this expansion. In particular, the sign of c1 will tell us which of the scenarios isrealised. In order to guard against systematic errors, this derivative has been evaluated in two ways.One is to ﬁt the corresponding Taylor series to data generated at imaginary chemical potential, theother to compute the derivative directly [6, 14], with compatible results. One ﬁnds on Nt = 4 thatc1 =  3:3(3);c2 =  47(20) [14]. The same behaviour is found for non-degenerate quark masses.3PoS(ICHEP 2010)370QCD phase diagram Owe PhilipsenFor physical strange quark mass we calculated mu;dc (m) with c1 =  39(8) [15]. Hence, on coarselattices the region of chiral phase transitions shrinks with real chemical potential and there is nochiral critical point for mB< 500 MeV, Fig. 1 (right). Note that one also observes a weakening ofthe phase transition with m in the heavy quark case [5] as well as a weakening of the transition withisospin chemical potential [16].4. ConclusionsIn summary, signals from reweighting and radius of convergence estimates are consistent witha critical point, but their systematics does not yet allow a deﬁnite conclusion. On the other hand,following the chiral critical surface with controlled systematics tells us the chiral phase transitionweakens with moderate m, but leads us away from the physical point. Possible scenarios are: ifwe mistrust the systematics of the former methods, we would conclude for either no critical pointat all, or a critical point at larger chemical potential mB> 600 MeV, where current methods breakdown. If these calculations hold instead, their critical point at moderate densities would not belongto the chiral phase transition, but to physics unrelated to chiral symmetry breaking.This work is supported by the German BMBF, grant 06MS9150.References[1] O. Philipsen, arXiv:1009.4089 [hep-lat].[2] P. de Forcrand and S. Kratochvila, PoS LAT2005 (2006) 167 [hep-lat/0509143].[3] F. Karsch, E. 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The QCD phase diagram at low baryon density from lattice simulations

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