The chiral critical surface is a surface of second order phase transitions bounding the region of

first order chiral phase transitions for small quark masses in the fmu;d;ms;mg parameter space.

The potential critical endpoint of the QCD (T;m)-phase diagram is widely expected to be part of

this surface. Since for m = 0 with physical quark masses QCD is known to exhibit an analytic

crossover, this expectation requires the region of chiral transitions to expand with m for a chiral

critical endpoint to exist. Instead, on coarse Nt = 4 lattices, we find the area of chiral transitions

to shrink with m, which excludes a chiral critical point for QCD at moderate chemical potentials

mB < 500 MeV. First results on finer Nt = 6 lattices indicate a curvature of the critical surface

consistent with zero and unchanged conclusions. We also comment on the interplay of phase

diagrams between the Nf = 2 and Nf = 2+1 theories and its consequences for physical QCD

Philipsen, Owe

English

Hochschulschriftenserver - Universität Frankfurt am Main

PoS(CPOD 2009)026Towards a determination of the chiral criticalsurface of QCDOwe Philipsen†Westfälische Wilhelms-Universität Münster, 48149 Münster, GermanyE-mail: ophil@uni-muenster.deThe chiral critical surface is a surface of second order phase transitions bounding the region ofﬁrst order chiral phase transitions for small quark masses in the fmu;d;ms;mg parameter space.The potential critical endpoint of the QCD (T;m)-phase diagram is widely expected to be part ofthis surface. Since for m = 0 with physical quark masses QCD is known to exhibit an analyticcrossover, this expectation requires the region of chiral transitions to expand with m for a chiralcritical endpoint to exist. Instead, on coarse Nt = 4 lattices, we ﬁnd the area of chiral transitionsto shrink with m, which excludes a chiral critical point for QCD at moderate chemical potentialsmB < 500 MeV. First results on ﬁner Nt = 6 lattices indicate a curvature of the critical surfaceconsistent with zero and unchanged conclusions. We also comment on the interplay of phasediagrams between the Nf = 2 and Nf = 2+1 theories and its consequences for physical QCD.5th International Workshop on Critical Point and Onset of Deconﬁnement - CPOD 2009,June 08 - 12 2009Brookhaven National Laboratory, Long Island, New York, USASpeaker.†In collaboration with Ph. de Forcrandc  Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/PoS(CPOD 2009)026Chiral critical surface of QCD Owe Philipsenphys.point00N  = 2N  = 3N  = 1fffm  ss mGauge  m   , m u1st2nd orderO(4) ?2nd orderZ(2)2nd orderZ(2)crossover1st d tric∞∞Pure 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0  0.01  0.02  0.03  0.04amsamu,dNf=2+1physical pointmstric - C mud2/5Figure 1: Left: Schematic phase transition behaviour of Nf = 2+1 QCD for different choices of quarkmasses (mu;d;ms) at m = 0. Right: Measured chiral critical line on an Nt = 4 lattice with staggered fermions[13].1. IntroductionThe QCD phase diagram has been the subject of intense research over the last ten years.Based on asymptotic freedom, one expects at least three different forms of nuclear matter: conﬁnedhadronic matter (low mB;T), quark gluon plasma (high T) and colour-superconducting matter (highmB, low T). Whether and where these regions are separated by true phase transitions has to bedetermined by ﬁrst principle calculations and experiments. Since QCD is strongly coupled onscales of nuclear matter, Monte Carlo simulations of lattice QCD are presently the only viableapproach.Unfortunately, the so-called sign problem prohibits straightforward simulations at ﬁnite bary-on density, thus our expectations for the QCD phase diagram are largely founded on model cal-culations. Since 2001, several ways have been designed to circumvent the sign problem in anapproximate way, all of them valid for m=T < 1 only [1, 2]. Within this range, those methods givequantitatively agreeing results for, e.g., the calculation of Tc(m) [3]. Many phenomenologicallyinteresting quantities like screening masses, the thermodynamical pressure, quark number suscep-tibilities etc. are thus theoretically controlled at moderate quark densities. On the other hand,because of the intricate and costly ﬁnite size scaling analyses involved, determining the order ofthe QCD phase transition, and hence the existence of a chiral critical point, is a much harder task.Here we discuss the problem of determining the order of the ﬁnite temperature phase transition bylattice simulations in the extended parameter space fmu;d;ms;T;mg.2. The chiral critical line at m = 0The order of the QCD ﬁnite temperature phase transition as a function of quark masses isdepicted in Fig. 1, for m = 0 (left). In the limits of zero and inﬁnite quark masses (lower left andupper right corners), order parameters corresponding to the breaking of the global chiral and centresymmetries, respectively, can be deﬁned, and one numerically ﬁnds ﬁrst order phase transitions2PoS(CPOD 2009)026Chiral critical surface of QCD Owe Philipsen 0 0.5 1 1.5 2 2.5 3 3.5 4-1 -0.5  0  0.5  1First orderCrossoverV1V2 > V1V3 > V2 > V1V=οο" 1.3 1.4 1.5 1.6 1.7 1.8 1.9 0.018  0.021  0.024  0.027  0.03  0.033  0.036B4amL=8L=12L=16IsingFigure 2: Left: Schematic behaviour of the Binder cumulant as a function of the Nf = 3 quark mass atﬁnite and inﬁnite volume. Right: Data obtained on Nt = 4 lattices with staggered fermions [13].at small and large quark masses at some ﬁnite temperatures Tc(mu;d;ms). On the other hand, oneobserves an analytic crossover at intermediate quark masses, with second order boundary linesseparating these regions. Both lines have been shown to belong to the Z(2) universality class ofthe 3d Ising model [4, 5, 6]. Since the line on the lower left marks the boundary of the quark massregion featuring a chiral phase transition, it is referred to as chiral critical line.However, the nature of the Nf = 2 chiral transition is far from being settled. Wilson fermionsappear to see O(4) scaling [7], while staggered actions are inconsistent with O(4) and O(2) (for thediscretised theory) [8]. A recent ﬁnite size scaling analysis using staggered fermions with unprece-dented lattice sizes was performed in [9]. Again, these data appear inconsistent with O(4)/O(2),and the authors conclude a ﬁrst order transition to be a possibility. A different conclusion wasreached in [10], in which cQCD was investigated numerically. This is a staggered action modiﬁedby an irrelevant term such as to allow simulations in the chiral limit. The authors ﬁnd their datacompatible with those of an O(2) spin model on moderate to small volumes, which would indicatelarge ﬁnite volume effects in the other simulations. Finally, from universality of chiral models it isknown that the order of the chiral transition is related to the strength of the UA(1) anomaly [11].In a model constructed to have the right symmetry with a tunable anomaly strength, it has recentlybeen demonstrated non-perturbatively that both scenarios are possible, with a strong anomaly re-quired for the chiral phase transition to be second order [12]. Should the chiral transition turn outto be ﬁrst order, the likely modiﬁcation of Fig. 1 (left) would be the disappearance of the tricriticalpoint, with the chiral critical line intersecting the Nf = 2 axis at some ﬁnite mu;d and being Z(2) allthe way.A convenient observable to locate and identify the second order boundary lines is the BindercumulantB4(X)  h(X  hXi)4i=h(X  hXi)2i2; X = ¯ yy : (2.1)It has to be evaluated at the (pseudo-)critical coupling bc(m;m), i.e. on the phase boundary deﬁnedby a vanishing third moment of the ﬂuctuation, h(X  hXi)3ijbc = 0. In the inﬁnite volume limit,B4 !1;3 for a ﬁrst order transition or crossover, respectively. At the second order transition, B4 !1:604 dictated by the 3d Ising universality class to which the chiral critical line belongs. On ﬁnite3PoS(CPOD 2009)026Chiral critical surface of QCD Owe Philipsen* QCD critical pointcrossover 1rst0∞Real worldXHeavy quarksmu,dmsµ  QCD critical point DISAPPEAREDcrossover 1rst0∞Real worldXHeavy quarksmu,dmsµFigure 3: Critical surface swept by the chiral critical line as m is turned on. Depending on the curvature, aQCD chiral critical point is present or absent. For heavy quarks the curvature has been determined [6] andthe ﬁrst order region shrinks with m.lattices this step function gets smeared out to an analytic function, as shown in Fig. 2 (left). Theintersection points of different volumes serve as estimators of the critical value, B4(bc(mc);mc). Ascan of B4(bc(m);m) in the Nf = 3 theory is shown in Fig. 2 (right). Also shown is a simultaneousﬁtofallthreevolumestoaTaylorexpansionofthecumulantaroundthecriticalpointandexploitingthe theoretically known behaviour under ﬁnite size scaling,B4(m;L) = 1:604+bL1=n(m mc)+::: (2.2)Such a ﬁt allows to check whether the chosen volumes are large enough to be consistent with ﬁnitesize scaling, as well as extracting the universality class from the values of B4 at the intersectionpoint and the scaling exponent, n = 0:63 in the case of 3d Ising.Following this recipe by ﬁxing ms and then scanning in mu;d, the chiral critical line has recentlybeen mapped out on Nt = 4 lattices [13], Fig. 1 (right). In agreement with expectations, the criticalline steepens when approaching the chiral limit. Assuming the Nf = 2 chiral transition to be in theO(4) universality class implies a tricritical point on the ms-axis, Fig. 1 (left). The data are consistentwith tricritical scaling [14] of the critical line with mu;d and we estimate mtrics  2:8Tc. However,this value is extremely cut-off sensitive and likely smaller in the continuum, cf. Sec.4.3. The chiral critical surfaceWhenachemicalpotentialisswitchedon, thechiralcriticallinesweepsoutasurface, asshownin Fig. 3. According to standard expectations in the literature [14], for small mu;d, the critical lineshould continuously shift with m to larger quark masses until it passes through the physical pointat mE, corresponding to the endpoint in the QCD (T;m) phase diagram. This is depicted in Fig. 3(left), where the critical point is part of the chiral critical surface. However, it is also possible forthe chiral critical surface to bend towards smaller quark masses, Fig. 3 (right), in which case therewould be no chiral critical point or phase transition at moderate densities. For deﬁniteness, let usconsider three degenerate quarks, represented by the diagonal in the quark mass plane. The critical4PoS(CPOD 2009)026Chiral critical surface of QCD Owe Philipsen-4-3.5-3-2.5-2-1.5-1-0.5 0 0  0.002  0.004  0.006  0.008  0.01µi2∆B4/∆µi2B4(pbp)finite µi, fit (µ2+µ4)fit µ2fit (µ2+µ4)-120-100-80-60-40-20 0 20 0  0.0002  0.0004  0.0006  0.0008  0.001  0.0012(aµi)2∆B4/∆(aµi)2B4(pbp)fit µ2fit (µ2+µ4)Figure 4: m2-dependence of the Binder cumulant on the chiral critical line for Nf = 3 (left) [15] and Nf =2+1 with physical strange quark mass (right), both for Nt = 4.quark mass corresponding to the boundary point has an expansionmc(m)mc(0)= 1+åk=1ck mpT2k: (3.1)A strategy to learn about the chiral critical surface is now to tune the quark mass to mc(0) andevaluate the leading coefﬁcients of this expansion. In particular, the sign of c1 will tell us which ofthe scenarios in Fig. 1 is realised.The curvature of the critical surface in lattice units is directly related to the behaviour of theBinder cumulant via the chain rule,damcd(am)2 =  ¶B4¶(am)2¶B4¶am 1: (3.2)While the second factor is sizeable and easy to evaluate, the m-dependence of the cumulant isexcessively weak and requires enormous statistics to extract. In order to guard against systematicerrors, this derivative has been evaluated in two independent ways. One is to ﬁt the correspondingTaylor series of B4 in powers of m=T to data generated at imaginary chemical potential [13, 15], theother to compute the derivative directly and without ﬁtting via the ﬁnite difference quotient [15],¶B4¶(am)2 = lim(am)2!0B4(am) B4(0)(am)2 : (3.3)Because the required shift in the couplings is very small, it is adequate and safe to use the originalMonte Carlo ensemble for amc(0);m = 0 and reweight the results by the standard Ferrenberg-Swendsen method. Moreover, by reweighting to imaginary m the reweighting factors remain realpositive and close to 1.The results of these two procedures based on 20 and 5 million trajectories on 834, respec-tively, areshowninFig.4(left). Theerrorbandrepresentstheﬁrstcoefﬁcientfromﬁtstoimaginarym data, while the data points represent the ﬁnite difference quotient extrapolated to zero. Both re-sults are consistent, and the slope permits and extraction of the subleading m4 coefﬁcient, whilethe combination of all data also constrains the sign of the m6 term. After continuum conversion5PoS(CPOD 2009)026Chiral critical surface of QCD Owe Philipsen-150-100-50 0 50 100 0  0.0002  0.0004  0.0006  0.0008  0.001  0.0012(aµi)2∆B4/∆(aµi)2B4(pbp)fit µ2fit (µ2+µ4)Figure 5: Left: The chiral critical line moves to smaller quark masses with decreasing lattice spacing.Right: m2-dependence of the Binder cumulant on the chiral critical line for Nf = 3;Nt = 6the result for Nf = 3 is c1 =  3:3(3);c2 =  47(20);c3 < 0 [15]. The same behaviour is found fornon-degenerate quark masses. Tuning the strange quark mass to its physical value, we calculatedmu;dc (m) with c1 =  39(8) and c2 < 0, Fig. 4 (right). Hence, on coarse Nt = 4 lattices, the regionof chiral phase transitions shrinks as a real chemical potential is turned on, and there is no chiralcritical point for mB< 500 MeV. Note that one also observes a weakening of the phase transitionwith m in the heavy quark case [6], in recent model studies of the light quark regime [18, 19], aswell as a weakening of the transition with isospin chemical potential [20].4. First steps towards the continuum, Nt = 6The largest uncertainty in these calculations by far is due to the coarse lattice spacing a  0:3fm on Nt =4 lattices. First steps towards the continuum are currently being taken on Nt =6;a0:2fm. At m = 0, the chiral critical line is found to recede strongly with decreasing lattice spacing[16, 17]: for Nf = 3, on the critical point mp(Nt = 4)=mp(Nt = 6)  1:8. Thus, in the continuumthe gap between the physical point and the chiral critical line is much wider than on coarse lattices,as indicated in Fig. 5 (left). Preliminary results for the curvature of the critical surface, Fig. 5(right), result in c1 = 7(14); 17(18) for a LO,NLO extrapolation in m2, respectively. Thus thesign of the curvature is not yet constrained. But even if positive, its absolute size appears too smallto make up for the shift of the chiral critical line towards smaller quark masses, and one wouldagain conclude for absence of a chiral critical point below mB< 500 MeV in this approximation.Higher order terms with large coefﬁcients would be needed to change this picture.Note that on current lattices cut-off effects appear to be larger than ﬁnite density effects, hencedeﬁnite conclusions for continuum physics cannot yet be drawn. A general ﬁnding is the steepnessof the critical surface, making the location of a possible critical endpoint extremely quark masssensitive, and hence difﬁcult to determine accurately.5. The interplay between Nf = 2;3 and Nf = 2+1Let us ﬁnally comment on the importance of understanding both the Nf = 2;3 as well as theirconnection before concluding anything for physical QCD. Fig. 6 (left) shows the scenario that6PoS(CPOD 2009)026Chiral critical surface of QCD Owe Philipsen1rst ordercrossovermu,dmsµX 1rst0∞1rst orderReal worldHeavy quarks1rst ordercrossovermu,dmsµX 1rst0∞1rst orderReal worldHeavy quarksFigure 6: Two possible scenarios for connected and disconnected triple lines in the mu;d = 0-plane.is used in the literature when the argument for the expected QCD phase diagram with a criticalendpoint is made [14]. The tricritical point at m = 0, which we have discussed here, is expected tobe analytically connected by a tricritical line, upon varying ms, to the tricritical point at some ﬁnitem in the two-ﬂavour case. This picture arises plausibly if the chiral critical surface behaves as inFig. 3 (left). However if, as on our coarse lattices, Fig. 3 (right) is realised, the situation might wellbe as in Fig. 6 (right). Even if the chiral Nf = 2 theory does feature a tricritical point at ﬁnite m,it need not be connected to the chiral critical surface, and nothing follows for the physical pointwithout additional information.Furthermore, the shrinking of the critical quark masses with diminishing lattice spacing makesit likely that a potential tricritical point, Fig. 1, also moves from a large value for the strange quarkmass, mu;d = 0;mtrics  2:8T on Nt = 4 [13], towards smaller values in the continuum limit. Inparticular, it is possible to have a situation for which mu;d = 0;mtrics < mphyss . In this case the chiralcritical surface we have been discussing here would not be responsible for a possible critical point,regardless of its curvature, but another surface emanating from the O(4)-chiral limit. Again, bothof these scenarios depend on conclusively understanding the situation in the Nf = 2 chiral limit.6. ConclusionsThe determination of the order of the QCD ﬁnite temperature phase transition as a func-tion of quark chemical potential is a maximally difﬁcult problem. Besides the sign-problem, thestrong quark mass and ﬂavour dependence are responsible for potentially rich structures in thefmu;d;s;T;mg parameter space which are compute-expensive to disentangle due to the required in-tricate ﬁnite size scaling analyses in the light quark mass regime. The existing results show that weneed to control various limiting cases of QCD as well as their connection to Nf = 2+1 in orderto understand if there is a critical point in the QCD phase diagram, and to which critical surface itbelongs.Acknowledgements:This work is partially supported by Deutsche Forschungsgemeinschaft, project PH 158/3-1.7PoS(CPOD 2009)026Chiral critical surface of QCD Owe PhilipsenReferences[1] O. Philipsen, Eur. Phys. J. ST 152 (2007) 29 [arXiv:0708.1293 [hep-lat]]. PoS LAT2005 (2006) 016[arXiv:hep-lat/0510077].[2] C. 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Towards a determination of the chiral critical surface of QCD

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