Modeling chiral criticality
MModeling chiral criticality ∗ Bengt Friman
GSI Helmholtzzentrum f¨ur Schwerionenforschung, D-64291 Darmstadt, Germany andG´abor Andr´as Alm´asi
GSI Helmholtzzentrum f¨ur Schwerionenforschung, D-64291 Darmstadt, Germany andKrzysztof Redlich
ExtreMe Matter Institute EMMI, GSI, D-64291 Darmstadt, GermanyUniversity of Wroc(cid:32)law, Institute of Theoretical Physics, PL 50-204 Wroc(cid:32)law,PolandWe discuss the critical properties of net-baryon-number fluctuations atthe chiral restoration transition in matter at non-zero temperature and net-baryon density. The chiral dynamics of quantum chromodynamics (QCD)is modeled by the Polykov-loop extended Quark-Meson Lagrangian, thatincludes the coupling of quarks to temporal gauge fields. The FunctionalRenormalization Group is employed to account for the criticality at thephase boundary. We focus on the ratios of the net-baryon-number cumu-lants, χ nB , for 1 ≤ n ≤
4. The results are confronted with recent exper-imental data on fluctuations of the net-proton number in nucleus-nucleuscollisions.PACS numbers: PACS numbers come here
1. Introduction
Lattice QCD (LQCD) results imply that at vanishing and small valuesof baryon chemical potential, µ B , strongly interacting matter undergoes asmooth crossover transition from the hadronic phase to the quark-gluon ∗ Presented at CPOD 2016 in Wroclaw (1) a r X i v : . [ h e p - ph ] M a r Friman-CPOD printed on October 9, 2018 plasma, where the spontaneously broken chiral symmetry is restored [1, 2].The order of the transition in the limit of massless u and d quarks is a subtleissue, and still under debate [3]. Here we assume that the transition in thislimit is second order, belonging to the O (4) universality class. Owing to thesign problem, the nature of the transition at higher net-baryon densities isnot settled by first principle LQCD studies. However, in effective models ofQCD, it is found that, at sufficiently large µ B , the systems can exhibit a firstorder chiral phase transition. The endpoint of this conjectured transitionline in the ( T, µ B )- plane, is the chiral critical point (CP) [4, 5]. At the CP,the system exhibits a 2 nd order phase transition, which belongs to the Z (2)universality class [6].Due to the restriction of present LQCD calculations to small net-baryondensities, effective models that belong to the same universality class as QCD,e.g., the Polyakov-loop extended, Nambu-Jona-Lasinio (PNJL) [7–9] andQuark-Meson (PQM) models [10–15], have been employed to study thechiral phase transition for a broad range of thermal parameters.One of the strategic goals of current experimental and theoretical stud-ies of chiral symmetry restoration in QCD is to unravel the phase diagramof strongly interacting matter and to clarify whether a chiral CP exists. Adedicated research program at RHIC, the beam energy scan, has been estab-lished to explore these issues in collisions of heavy ions at relativistic energies[16]. By varying the beam energy at RHIC, the properties of strongly inter-acting matter in a broad range of net-baryon densities, correponding to awide range in baryon chemical potential, 20 MeV < µ B <
500 MeV [17, 18],can be studied. To study the phase structure, the fluctuations of conservedcharges have been proposed as probes [19–23]. These are experimentallyaccessible and reflect the criticality of the chiral transition.First data on net-proton-number fluctuations, which are used as a proxyfor fluctuations of the net-baryon number, have been obtained in heavy-ion collisions by the STAR Collaboration at RHIC energies [24–26]. TheSTAR data on the variance, skewness and kurtosis of net proton number,are intriguing and have stimulated a lively discussion on their physics originand interpretation [27].In the following, we focus on the properties and systematics of the cu-mulants of net-baryon-number fluctuations near the chiral phase bound-ary. Our studies are performed in the Polykov-loop extended Quark-Mesonmodel which includes coupling of quarks to temporal gauge fields. To ac-count for the O (4) and Z (2) critical fluctuations near the phase boundary,we employ the Functional Renormalization Group [28–30]. We present re-sults for ratios of cumulants obtained on the phase boundary and on afreezeout line determined by fitting the skewness ratio, following [31, 32].These are confronted with the corresponding experimental values of the riman-CPOD printed on October 9, 2018 STAR collaboration.
2. The Polyakov–quark-meson model
The PQM model is a low energy effective approximation to QCD for-mulated in terms of the light quark q = ( u, d ) as well as scalar and thepseudoscalar meson φ = ( σ, (cid:126)π ) fields. The quarks are coupled to the back-ground Euclidean gluon field A µ , with vanishing spatial components, whichis linked to the Polyakov loopΦ = 1 N c (cid:28) Tr c P exp (cid:18) i (cid:90) β dτ A (cid:19)(cid:29) . (1)The resulting Lagrangian of the model reads L = ¯ q ( iγ µ D µ − g ( σ + iγ (cid:126)τ(cid:126)π )) q + 12 ( ∂ µ σ ) + 12 ( ∂ µ (cid:126)π ) − U m ( σ, (cid:126)π ) − U (Φ , ¯Φ; T ) − m ω ω , (2)where D µ = ∂ µ − iA µ . The parameters of the mesonic potential U m ( σ, (cid:126)π ) = λ σ + (cid:126)π ) + m σ + (cid:126)π ) − Hσ, (3)are tuned to vacuum properties of the σ and (cid:126)π mesons. We use the Polyakov-loop potential U (Φ , ¯Φ; T ) determined in [33].We compute the thermodynamics of this model, including fluctuations ofthe scalar and pseudoscalar meson fields within the framework of the FRGmethod. The Polyakov loop is treated on the mean-field level. Its valueits tuned such that a stationary point of the thermodynamic potential isreached at the end of the RG calculation.In the FRG framework, the effective average action Γ k , which inter-polates between the classical and the full quantum action, is obtained bysolving the renomalization group flow equation [28] ∂ k Γ k [ φ ] = 12 Tr (cid:20)(cid:16) Γ (2) k [ φ ] + R k (cid:17) − ∂ k R k (cid:21) , (4)where φ denotes the quantum fields considered, Tr is a trace over the fields,over momentum and over all internal indices. The regulator function R k suppresses fluctuations at momenta below k . Thus, effects of fluctuationsof quantum fields are included shell by shell in momentum space, startingfrom a UV cutoff scale Λ. We employ the optimized regulator introducedby Litim [34]. Details of the calculation can be found in [32]. Friman-CPOD printed on October 9, 2018
3. Net-baryon-number cumulants and the phase boundary
The chiral Lagrangian introduced above shares the chiral critical proper-ties with QCD. In particular, at moderate values of the chemical potential,the PQM model exhibits a chiral transition belonging to the O (4) universal-ity class. For larger values of µ , it reveals a Z (2) critical endpoint, followedby a first order phase transition [6]. Consequently, the PQM model embod-ies the generic phase structure expected for QCD, with the universal O (4)and Z (2) criticality encoded in the scaling functions. Furthermore, due tothe coupling of the quarks to the background gluon fields, the PQM modelincorporates ”statistical confinement”, i.e., the suppression of quark and di-quark degrees of freedom in the low temperature, chirally broken phase [7].Consequently, by studying fluctuations of conserved charges in the PQMmodel, one can explore the influence of chiral symmetry restoration and of”statistical confinement” on the cumulants in different sections of the chiralphase boundary. The baryon- and quark-number cumulants of order n , χ nB and χ nq , and the baryon-number cumulant ratios, χ n,mB , are defined as χ nB = χ nq n = 13 n T − n ∂ n Ω( T, µ q ) ∂µ nq , χ n,mB = χ nB ( T, µ B ) χ mB ( T, µ B ) . (5)In the following we focus on ratios of net-baryon-number susceptibilitiesthat can be related to experimentally measurable quantities: χ , B ( T, µ B ) = M / σ , χ , B ( T, µ B ) = S B σ / M , χ , B ( T, µ B ) = κσ (6)where M is the mean, σ the variance, S B the skewness and κ the kurtosisof the net-baryon-number distribution.At vanishing chemical potential, all odd susceptibilities of the net baryonnumber vanish, owing to the baryon-antibaryon symmetry. In addition, inthe O (4) universality class, the second and fourth order cumulants remainfinite at the phase transition temperature at µ q = 0 in the chiral limit,implying that only sixth and higher order susceptibilities diverge. Thus,for physical quark masses, only higher order cumulants, χ nB with n > O (4) criticality at µ q = 0 [23]. A further consequence of thebaryon-antibaryon symmetry is the equality of the ratios χ m − , n − B = χ m, nB (7)for any integer m and n ≥ µ q = 0. For χ , B and χ , B , the equality atsmall µ q can also be seen by comparing the right panel of Fig. 1 to the leftpanel of Fig. 2.At finite net-baryon density the singularity at the O (4) critical line isstronger than at µ q = 0. Thus, in this case the third-order cumulant and all riman-CPOD printed on October 9, 2018 Phase Boundary Μ q (cid:144) T c T (cid:144) T c Χ B1 (cid:144) Χ B2 - Phase Boundary Μ q (cid:144) T c T (cid:144) T c Χ B3 (cid:144) Χ B1 Fig. 1. Contour plots of the ratios χ B /χ B and χ B /χ B in the ( T, µ q )-plane, com-puted in the PQM model. The broken lines indicate the location of the chiralcrossover phase boundary. -
10 0.150.150.3 0.50.50.9 1.2 1.21.5 2 25
Phase Boundary Μ q (cid:144) T c T (cid:144) T c Χ B4 (cid:144) Χ B2 Χ B4 (cid:144) Χ B2 g s2 f Π Χ Ch T (cid:144) T c Fig. 2. Left: contour plot of the kurtosis ratio, χ B /χ B , in the ( T, µ q ) plane.Right: temperature dependence of ratios of net-baryon-number cumulants, χ B /χ B = χ B /χ B in the PQM model at µ q = 0. higher-order ones diverge at the O (4) line. The second order cumulant χ B remains finite, and diverges at the tricritical point and for non-zero quarkmasses, at the CP.In Fig. 1 we show contour plots of the ratios χ , B and χ , B in the ( T, µ q )plane. As noted above, all odd cumulants vanish at µ q = 0. Consequently, χ , B | µ q =0 = 0 for any T , while the ratio χ , B | µ q =0 is non-vanishing. Asindicated in Fig. 1, the ratio χ , B decreases with temperature, and dependsweakly on the chemical potential. Thus, the ratio χ , B can be used as ameasure of the temperature.In Fig. 2 we show contour plots of the ratios χ , B and results on thetemperature dependence of the ratio χ , B = χ , B of net-baryon-number sus-ceptibilities at µ q = 0, together with the variance of the chiral condensate, Friman-CPOD printed on October 9, 2018 χ ch . The location of the maximum of the chiral susceptibility, χ ch , definesthe pseudo-critical temperature, T c .At small µ q /T , the properties of the first four susceptibilities, χ nB with n = 1 , ..,
4, and consequently their ratios, near the chiral crossover aredominantly affected by the coupling of the quarks to the Polyakov loop,and the resulting statistical confinement. The critical chiral dynamics, i.e.the O (4) and Z (2) criticality at the chiral crossover transition and at theCP, respectively, unfolds at larger µ/T . Near the CP, there is a strongvariation of the cumulants with T and µ q , which increases with the orderof cumulants.
4. Net-baryon cumulant ratios and freeze-out in heavy ioncollisions
In heavy-ion collisions, the thermal fireball formed in the quark-gluonplasma phase undergoes expansion and passes through the QCD phaseboundary at some point ( µ q , T ), which depends on the collision energy, √ s . Analysis of ratios of particle multiplicities indicate that at high beamenergies (small values of µ q /T ), the freeze out occurs at or just below thephase boundary. Thus, the beam energy dependence of net-baryon-numbersusceptibilities can provide insight into the structure of the QCD phase dia-gram and information on the existence and location of the CP. Consequently,it is of phenomenological interest to compute the properties of fluctuationsof conserved charges along the chiral phase boundary. Since there, the crit-ical structure and the relations between different susceptibilities are by andlarge governed by the universal scaling functions, the generic behavior ofratios of net-baryon-number susceptibilities can be explored also in modelcalculations.A comparison of results obtained in the PQM model with data, requiresa correspondence between the collision energy √ s and the thermal param-eters ( µ q , T ). Here we employ the phenomenological relation, obtained byanalysing the freeze-out conditions in terms of the hadron-resonance-gasmodel (HRG) [17, 18]. We then use the resulting dependence of µ B and T on √ s to assign a value for the ratio µ q /T to each of the STAR beamenergies. We note that, for µ q /T <
1, the phenomenological freeze-outline coincides within errors with the crossover phase boundary obtained inlattice QCD [2, 36]. This motivates a comparison of model results on net-baryon-number fluctuations near the phase boundary with data. A similaranalysis was first done using LQCD results in Ref. [31].In Fig. 3, we show the STAR data on net-proton-number susceptibilityratios and the corresponding PQM model results on net-baryon-numberfluctuations computed along the phase boundary. The model results for the riman-CPOD printed on October 9, 2018 Χ B1 (cid:144) Χ B2 Χ B3 (cid:144) Χ B2 Χ B1 (cid:144) Χ B2 Χ B3 (cid:144) Χ B2 Tanh @ Μ B (cid:144) T D Μ q (cid:144) T s @ GeV D Χ B4 (cid:144) Χ B2 Χ B3 (cid:144) Χ B1 Χ B4 (cid:144) Χ B2 Χ B3 (cid:144) Χ B1 - Μ q (cid:144) T s @ GeV D Fig. 3. Ratios of cumulants of net-baryon-number fluctuations in the PQM model,computed along the chiral phase boundary, for four sets of model parameters [32].Also shown are the preliminary STAR data [25, 26], assuming the relation betweenthe ratio ( µ q /T ) and the collision energy obtained by analysing the chemical free-zout conditions [18, 35, 36]. The green dashed line in the left figure shows thebaseline result for χ , B , tanh(3 µ q /T ). Χ B1 (cid:144) Χ B2 Χ B3 (cid:144) Χ B2 Χ B1 (cid:144) Χ B2 Χ B3 (cid:144) Χ B2 Μ q (cid:144) T s @ GeV D Χ B4 (cid:144) Χ B2 Χ B3 (cid:144) Χ B1 Χ B4 (cid:144) Χ B2 Χ B3 (cid:144) Χ B1 - Μ q (cid:144) T s @ GeV D Fig. 4. Ratios of cumulants of net-baryon-number fluctuations in the PQM modelalong the freezeout line, obtained by fitting χ B /χ B to the STAR data. The foursets of model parameters used and the preliminary STAR data shown, are the sameas in Fig. 3. ratios χ , B , χ , B and χ , B are in qualitative agreement with the data in thewhole energy range. For the kurtosis ratio, χ , B , this is the case also up tothe SPS energy, i.e., for √ s ≥
20 GeV. However, for µ q /T > .
5, the dataon the kurtosis ratio exhibits a qualitatively different dependence on µ q /T than expected for the critical behavior of χ , B , as the CEP is approachedalong the phase boundary.In the comparison of model predictions with data in Fig. 3, we as- Friman-CPOD printed on October 9, 2018 sume, that the freezout of the net-baryon-number fluctuations tracks thechiral phase boundary. Clearly, this assumption provides a qualitative un-derstanding of the data. In order to obtain a more quantitative description,we follow Refs. [31, 37, 38], and determine the freezout conditions by fittingthe data on the χ , B ratio, using the √ s -dependence of µ q /T obtained fromthe fit of the HRG model to particle multiplicities [18, 35, 36].In Fig. 4 we show the fluctuation ratios along the freezeout line that isfixed through the skewness data. The model results are obtained for foursets of initial conditions introduced in [32]. Fig. 4 clearly shows that, alongthe freezeout line, the spread of all fluctuations ratios considered for thevarious parameter sets is much weaker than that observed in Fig. 3 alongthe phase boundary. This indicates that moderate changes of the sigmamass and modifications of the form of the Polyakov loop potential may leadto a shift in the temperature scale but essentially no change of the relativestructure of the cumulant ratios.The results presented in Fig. 4 clearly show that the model providesa very good description of the data on χ , B and χ , B . Also the kurtosisdata, obtained at higher collision energies, are consistent with model results.However, at √ s <
20 GeV the latter again exhibits a different trend, withthe data increasing rapidly at lower energies, while the model result keepsdecreasing.The comparison of model results on ratios of net-baryon-number sus-ceptibilities with the STAR data in Figs. 3 and 4 shows that the data, withthe exception of kurtosis at low energies, follow general trends expected dueto critical chiral dynamics and general considerations. We note that theratios of net-baryon-number susceptibilities near the phase boundary in-volving net-baryon-number cumulants χ nB with n ≥ O (4) and Z (2) universality classes, respec-tively. This observation indicates, that by measuring fluctuations of con-served charges in heavy-ion collisions, we are indeed probing the QCD phaseboundary, and thus accumulating evidence for chiral symmetry restoration.However, as discussed above, there are several uncertainties and as-sumptions which must be thoroughly understood before the QCD phaseboundary can be pinned down with confidence. Possible contributions tofluctuation observables from effects not related to critical phenomena, likee.g. baryon-number conservation [39] and volume fluctuations [40, 41] arebeing explored. We mention in particular the rather strong sensitivity ofhigher order net-proton-number cumulants on the transverse momentumrange imposed in the analysis of the STAR data. Nevertheless, it is in-triguing that the dynamics of this model provides a good description of theSTAR data (except for χ B at the lowest energies), without all these effectsof non-critical origin. It remains an important task to assess the effect of EFERENCES theses additional sources of fluctuations in the whole energy range probedby the experiments. Acknowledgments
We acknowledge stimulating discussions with Peter Braun-Munzinger,Frithjof Karsch and Nu Xu. The work of B.F. and K.R. was partly sup-ported by the Extreme Matter Institute EMMI. K. R. also acknowledgespartial supports of the Polish National Science Center (NCN) under Mae-stro grant DEC-2013/10/A/ST2/00106. G. A. acknowledges the support ofthe Hessian LOEWE initiative through the Helmholtz International Centerfor FAIR (HIC for FAIR).
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