Hot Quark Matter with an Axial Chemical Potential
aa r X i v : . [ h e p - ph ] F e b Hot Quark Matter with an Axial Chemical Potential
Raoul Gatto and Marco Ruggieri Departement de Physique Theorique, Universite de Geneve, CH-1211 Geneve 4, Switzerland. Department of Physics and Astronomy, University of Catania, Via S. Sofia 64, I-95125 Catania
We analyze the phase diagram of hot quark matter in presence of an axial chemical potential, µ . The latter is introduced to mimic the chirality transitions induced, in hot Quantum Chromody-namics, by the strong sphaleron configurations. In particular, we study the curvature of the criticalline at small µ , the effects of a finite quark mass and of a vector interaction. Moreover, we buildthe mixed phase at the first order phase transition line, and draw the phase diagram in the chiraldensity and temperature plane. We finally compute the full topological susceptibility in presence ofa background of topological charge. PACS numbers: 12.38.Aw, 12.38.Mh, 12.38.LgKeywords: Effective Models of QCD, Axial Chemical Potential, Phase Structure of QCD.
I. INTRODUCTION
Understanding the properties of strongly interact-ing matter, in extreme conditions of high temperatureand/or large baryon density, is very important to get adeeper knowledge of our universe, at the macroscopic aswell as at the microscopic levels. For example, very hotmatter, with estimated temperature of the order of 10 Kelvin, is produced in heavy ion collisions (HICs) experi-ments immediately after the collision, see for example [1]for an indirect measurement of such a temperature by thePHENIX collaboration at Brookhaven National Labora-tory (BNL). As a consequence, it is crucial to make theo-retical investigations on the phase structure of QuantumChromodynamics (QCD), the theory of strong interac-tions, in conditions of high temperature, in order to giveproper interpretation of experimental data, at the sametime suggesting new phenomena to look for.The theoretical knowledge on the thermodynamics ofQCD at zero baryon density is fixed by Lattice simu-lations, which allow first principle numerical computa-tions of the relevant thermodynamical quantities. In fact,Lattice simulations performed by independent groupsshow that in the range of temperature (140 , /g evolution in arange of β = 6 N c /g ≤ ≡ β s is consistent with LatticeMontecarlo simulations, for which the continuum limit isrealized for β ≈
6, hence not too far from β s .The lack of feasible first principle calculations of theQCD thermodynamic properties at finite baryon chemi-cal potential invokes the use of other theoretical strate- gies to investigate the structure of QCD in this regime.The main strategy is the use of some model. Among theseveral models, the Nambu-Jona-Lasinio (NJL) model [7]is very popular, see [8] for reviews. In the NJL model, theQCD gluon-mediated interactions are replaced by effec-tive interactions among quarks, which are built in orderto respect the global symmetries of QCD. Under someapproximations, it is possible to derive the NJL modeleffective interaction kernel from first principles QCD,see [9, 10]. The common feature of the effective modelsis that they share (most of) the symmetries of the QCDLagrangian. At the same time, they are able to describespontaneous chiral symmetry breaking, as well as otherkinds of breaking patterns expected at large chemical po-tential, in a self-consistent way. Moreover, it has beenshown how it is possible to extend the chiral models in or-der to compute quantities which are sensible to confinemtproperties of a given phase [11], see [12–24, 60, 61] for re-cent studies. One of the advantage of these models is thatthey allow for analytic or semi-analytic computations atzero as well as at finite chemical potential, at least atthe one-loop level; thus they do not rely (in general) onsimulations, and are not affected by the complex value ofthe quark determinant, allowing speculations about thestructure of QCD in regimes where first principles cal-culations are not feasible. After standard bosonizationprocedure, the chiral and deconfinement transitions aredescribed in terms of collective fields which take someexpectation value, and whose quantum fluctuations de-scribe physical particles ( σ and π mesons in the simplestversion of the model).It has been suggested that very strong magnetic fieldsare produced during the very first moments of a noncen-tral heavy ion collision [25–27]; this has motivated sev-eral studies about the effect of a strong magnetic back-ground on the QCD phase structure, see [28–39] and ref-erences therein. Moreover, since the temperature of thefireball produced by the collision is very high, a copi-ous production of topological gluon configurations (i.e.,the QCD sphalerons) with finite winding number is ex-pected [40], which induce locally chirality imbalance inthe hot plasma as a natural consequence of the QCDWard identity. The combined effect of the latter andof the magnetic field induces an electric current alongthe direction of the magnetic field. This effect, calledthe Chiral Magnetic Effect (CME) [25, 41], leads to theevent-by-event separation of electric charges with respectto the reaction plane, which is a parity ( P ) as well as a CP -odd effect. Experimental data obtained by the STARcollaboration at BNL seem to point towards the directionof charge separation in collisions [42], even if the inter-pretation of such data in terms of the CME is still underdebate [27, 43, 44].Because of the expected production of chirality imbal-ance in the quark-gluon-plasma phase of QCD, and be-cause of its potential relevance for the physics of heavyion collisions, it is of interest to study how chirality modi-fies the structure of QCD itself. This is the main scope ofthis article. In particular, we continue the study of [45–48] in which chirality was induced by an axial chemi-cal potential, µ , conjugated to chirality. Besides theapplications to the CME, which are very interesting ontheir own because of the potential relevance for the phe-nomenology of heavy ion collisions, the theory at finite µ is interesting because it does not suffer from the signproblem; as a consequence, grandcanonical ensembles atfinite µ can be simulated on the Lattice [49, 50]. Thesestudies might be helpful to understand the structure ofthe QCD phase diagram at finite baryon chemical po-tential. For example, in [47] it has been suggested thatthe critical endpoint of the QCD phase diagram might bedetected in Lattice simulations at finite µ . The detailsabout the theoretical framework will be given in SectionII of the article. Here, we wish to stress the noveltiesembraced by our study.Firstly, we investigate the effect of the vector inter-action, as well as the finite current quark mass, on thelocation of the critical endpoint. Such aspects should betaken into account if a comparison with the Lattice datais desirable, but they have not been studied in [47]. More-over, it is of interest to analyze analytically the effect ofthe axial chemical potential on the chiral condensate, andon the curvature of the critical line for restoration of chi-ral symmetry. The latter aspect is quite interesting, sinceit shows how a competition between the vacuum termand the thermal excitations compete and eventually leadto a lowering of the critical temperature. This reduc-tion of the critical temperature was found numerically inthe previous model studies [45–47] but it was not inves-tigated in detail. Thirdly, we are interested to the phasediagram in the n − T plane. This might be interestingbecause n is connected to the topological charge density,because of the integrated Ward identity. With an abuseof nomenclature, we call the latter as canonical ensembleformulation, in analogy with the case of QCD at finitebaryon density. Finally, we compute the full topologicalsusceptibility at zero as well as at finite µ , as a functionof temperature. The latter part of our investigation hassome overlap with the model study of Ref. [19], wheretopological susceptibility with a background of topolog- ical charge is computed. In the case of Ref. [19], thebackground topological charge is introduced by adding a θ − term to the QCD action.The plan of the paper is as follows. In Section II weintroduce the axial chemical potential. In Section IIIwe summarize the model we use in our calculations. InSection IV we perform a perturbative analysis at small µ at zero as well as at finite temperature, computingthe dependence of the chiral condensate and of the criti-cal temperature on the axial chemical potential. In Sec-tion V we relate the phase diagrams in the canonicaland grand-canonical ensembles, and build explicitly themixed phase. In Section VI we study the effect of thebare quark mass, and of a vector interaction, on the lo-cation of the critical endpoint of the phase diagram inthe µ − T plane. Finally, we present our conclusions inSection VII. II. THE AXIAL CHEMICAL POTENTIAL
In this Section we define the axial chemical poten-tial, µ . It has been already discussed in several refer-ences, see [41, 45–47, 51] and references therein. There-fore we limit ourselves to the basic definitions and tofix our notation. First of all we introduce the chirality, N = N R − N L , as the imbalance between right- andleft-handed quarks. In QCD, change of chirality at zeroas well as at finite temperature can be related directlyto the topology of nonperturbative gluon configurationswith a finite winding number, Q W , via the integratedWard identity, N = n V = 2 N f Q W , (1)where N f is the number of flavors considered, and wehave assumed that before the interaction with the gluonconfiguration, chirality was zero. In Equation (1) we haveintroduced the chiral density n . The above equation is aconsequence of the strong chiral anomaly; Q W is relatedto the topological charge of the given gluon configuration, Q W = g π Z d xF µνa ˜ F aµν . (2)At high temperature, a copious production of gluonconfigurations with nonvanishing winding number is ex-pected, see for example [40] and references therein. Theseare the strong (that is, QCD) sphalerons. Therefore, be-cause of the existence of QCD sphalerons, chirality canbe produced in the high temperature phase of QCD.The simplest way to treat quark matter with net chiral-ity in effective models, is to introduce an axial chemicalpotential, µ , conjugated to chiral density [41, 45–47].At the lagrangian level, this amounts to add the chiraldensity operator, µ ¯ ψγ γ ψ , (3)to the lagrangian density. This procedure is similar tothe one usually adopted to study systems at finite baryondensity: in the latter case, we introduce a baryon chem-ical potential, µ , that induces a net baryon density, n .For example, in the very high temperature phase of twoflavor QCD, the following relation holds [41]: n = µ π + µ T , (4)which shows how a chiral chemical potential induces anet chirality in the system. In the general case, one needsto compute the relation between n and µ numerically,see [45, 46] and Section V of this article.Before going ahead, it is necessary to make few re-marks. Firstly, we are aware that the axial chemicalpotential cannot be considered as a true chemical po-tential. As a matter of fact, µ is conjugated to n , orto the topological charge because of the Ward identity.The latter is a not conserved quantity in QCD becauseof the quantum anomaly. In the common picture of theQCD vacuum with µ = 0, the vanishing average value ofthe topological charge is understood as the formation ofseveral local domains, each one characterized by a finitetopological charge; the probability to create a domainwith charge Q W is the same as the probability to cre-ate a domain with charge − Q W . As a consequence, ina volume much larger than the typical domain size, theaverage value of the topological charge sums up to zero.However, flucuations of the topological charge can changethe value of the charge of any local domain. The formalrole of the axial chemical potential is to reproduce thelocal domains of the QCD vacuum with a net topologicalcharge as equilibrium states. Because of the fluctuationsof the topological charge, this description is meaningfulas long as the time scale is shorter than the inverse of thetopological changing transition rate [49]. This interestingtheoretical question has been analyzed also in [52], whereit is claimed that a proper combination of the chiral den-sity and of a Chern-Simon term is a conserved quantity,and it is precisely this quantity that should be coupledto the axial chemical potential. We will devote a moredetailed study of this problem to a next article.Another remark is that the theory with µ = 0 is asign-free theory. It is well known that QCD with threecolors suffers the sign problem: namely, the fermion de-terminant of QCD with three colors is complex at finitequark chemical potential, making the usual Montecarlosampling of configurations in the Lattice simulations notpossible when the quark chemical potential is larger thanthe temperature (see [4] for a review). On the other hand,the theory at finite µ does not suffer the sign problem.As a matter of fact, γ D ( µ ) γ = D † ( µ ), where D cor-responds to the Dirac operator with µ = 0. As a conse-quence, the fermion determinant is real and positive at µ = 0, and grand canonical ensembles with finite µ canbe simulated on the Lattice [41]. Indeed, some Latticesimulation at µ = 0 has been already performed to studythe chiral magnetic effect on the Lattice, see [49, 50]. Inthese references, a preliminary study of the phase struc-ture in the µ − T plane is also addressed. III. THE MODEL
In this Section, we describe the model that we usein calculations, namely the Nambu-Jona-Lasinio modelimproved with the Polyakov loop (PNJL in the follow-ing) [11]. In the PNJL model, quark propagation in themedium is described by the following lagrangian density: L = ¯ q ( iγ µ D µ − m ) q + G h (¯ qq ) + ( i ¯ qγ τ q ) i ; (5)here q is the quark Dirac spinor in the fundamental repre-sentation of the flavor SU (2) and the color group; τ cor-respond to the Pauli matrices in flavor space. A sum overcolor and flavor is understood. The covariant derivativeembeds the QCD coupling with the background gluonfield which is related to the Polyakov loop, see below.The PNJL model has access to the expectation valueof the Polyakov loop, L , which is sensible to confinementor deconfinement properties of a given phase. In orderto compute h L i we introduce a static, homogeneous andEuclidean background temporal gluon field, A = iA = iλ a A a , coupled minimally to the quarks via the QCDcovariant derivative. Then W = Tr c exp ( iβλ a A a ) and L = W/
3, where β = 1 /T . In the Polyakov gauge, whichis convenient for this study, A = iλ φ + iλ φ ; moreover,for simplicity we take L = L † from the beginning asin [12], which implies A = 0.In our computation we follow the idea implementedin [21], which brings to a Polyakov-loop-dependent cou-pling constant: G = g (cid:2) − α LL † − α ( L + ( L † ) ) (cid:3) , (6)The ansatz in the above equation was inspired by [9] inwhich it was shown explicitly that the NJL vertex can bederived in the infrared limit of QCD, it has a non-localstructure, and it acquires a non-trivial dependence on thephase of the Polyakov loop. This idea has been analyzedrecently in [53], where the effect of the confinement or-der parameter on the four-fermion interactions and theirrenormalization-group fixed-point structure has been in-vestigated. The numerical values of α and α have beenfixed in [21] by a best fit of the available Lattice dataat zero and imaginary chemical potential of Ref. [54, 55].In particular, the fitted data are the critical temperatureat zero chemical potential, and the dependence of theRoberge-Weiss endpoint on the bare quark mass. Thebest fit procedure leads to α = α ≡ α = 0 .
2, which arethe values we adopt in this article.In the one-loop approximation, the effective potentialof this model is given by [47] V = U ( L, L † , T ) + σ G − N c N f X s = ± Z d p (2 π ) ω s − N f β X s = ± Z d p (2 π ) log ( F + F − ) , (7)where σ = ¯ qq is a scalar collective field representing themean field and the quantum fluctuations of the operator¯ qq , and G is defined in Eq. (6); moreover, ω s = q ( | p | s − µ ) + m q , (8)corresponds to the pole of the quark propagator, and m q = m − σG is the constituent quark mass; the index s denotes the helicity projection. In Eq. (7) we haveintroduced the functions F − = 1 + 3 Le − β ( ω s − µ ) + 3 L † e − β ( ω s − µ ) + e − β ( ω s − µ ) , (9) F + = 1 + 3 L † e − β ( ω s + µ ) + 3 Le − β ( ω s + µ ) + e − β ( ω s + µ ) , (10)which are responsible for the statistical confinementproperty of the model at low temperature [11].In right hand side of the first line of Equation (7) themomentum integral corresponds to the vacuum quarkfluctuations contribution to the thermodynamic poten-tial. We treat the divergence in this term phenomeno-logically, introducing a momentum cutoff, Λ, in the vac-uum term; the numerical value of Λ will be then fixedby requiring that the quark condensate, the pion massand the pion decay constant computed in the model arein agreement with the phenomenological values. Beforegoing ahead, it is useful to remind that within the quark-meson model a renormalization procedure is feasible [17],the model itself being renormalizable. The renormaliz-ability of the model might be useful to remove the cutoffeffects that, instead, appear in the NJL model (see thediscussion in the next Sections). On the other hand, therenormalization program of the energy density at finite µ is not trivial, since a nonvanishing µ induces furtherdivergences in the theory, as it can be argued by an in-spection of the µ − dependence of the vacuum energy,see for example Eq. (14). Hence, proper renormalizationconditions should be adopted.The potential term U in Eq. (7) is built by hand inorder to reproduce the pure gluonic lattice data with N c = 3 [12]. We adopt the following logarithmic form, U [ L, ¯ L, T ] = T (cid:26) − a ( T )2 ¯ LL + b ( T ) ln (cid:2) − LL + 4( ¯ L + L ) −
3( ¯ LL ) (cid:3)(cid:27) , (11)with three model parameters (one of four is constrainedby the Stefan-Boltzmann limit), a ( T ) = a + a (cid:18) T T (cid:19) + a (cid:18) T T (cid:19) , (12) b ( T ) = b (cid:18) T T (cid:19) . (13)The standard choice of the parameters reads a = 3 . a = − . a = 15 . b = − .
75. The parameter T in Eq. (11) sets the deconfinement scale in the puregauge theory. In absence of dynamical fermions one has T = 270 MeV. However, dynamical fermions induce adependence of this parameter on the number of activeflavors [16]. For the case of two light flavors to whichwe are interested here, we take T = 190 MeV as in [21].Also for the remaining parameters we follow [21] and takeΛ = 631 . m = 5 . g = 5 . × − MeV − .We notice that the PNJL model considered here, whichis dubbed Extended-PNJL in [21], has been tuned in or-der to reproduce quantitatively the Lattice QCD thermo-dynamics at zero and imaginary quark chemical poten-tial. Hence, it represents a faithful description of QCD,in terms of collective degrees of freedom related to chiralsymmetry breaking and deconfinement. IV. PERTURBATIVE ANALYSIS
In the case of small µ , we can make some analyti-cal and semianalytical estimate of the effect of the ax-ial chemical potential on quark condensation, and on thecritical line. In this Section we restrict our analysis to thepure NJL model case, corresponding to take L = L † = 1in Eq. (7), and α = α = 0 in Eq. (6). This simpli-fies the numerical analysis. On the other hand, the re-sults obtained here will not be modified qualitatively bythe Polyakov loop, since the effect of the latter is justa suppression of colored states below T c . Moreover, wework in the chiral limit; this simplification allows to de-fine rigorously the chiral phase transition, and computeunambiguously the critical temperature. A. Zero temperature: chiral condensate
To begin with, we consider the zero temperature case,and we compute the shift of the chiral condensate in-duced by µ , showing that the chiral chemical potentialacts as a catalyzer of chiral symmetry breaking. In fact,the µ − dependent zero temperature correction to the ef-fective potential is given by V = − N c N f π µ m q F (cid:16) m q Λ (cid:17) , (14)where we have assumed µ ≪ m q ; in Eq. (14) we haveintroduced the function F ( x ) = log 1 + √ x x − √ x . (15)It is easy to prove that F ( x ) is always positive, thus mak-ing V <
0. Hence, the energy density of a broken phaseand µ = 0 is smaller than that of a phase with unbrokensymmetry and the same value of the chiral condensate.We can use Eq. (14) to analyze the perturbative solu-tion of the gap equation at zero temperature. As a matterof fact, for small values of µ we can look for a solutionof the gap equation in the form m q = ¯ m q + δm q , where¯ m q satisfies the gap equation at µ = 0, and δm q corre-sponds to the µ -dependent contribution. An elementarycomputation shows that δm q = − M σ (cid:18) ∂V ∂m q (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) m q = ¯ m q , (16)where we have defined M σ ≡ (cid:0) ∂ V /∂m q (cid:1) m q = ¯ m q and V ≡ V ( µ = 0). By definition, M σ >
0, since ¯ m q corresponds to a minimum of the effective potential at µ = 0. Moreover, for m q ≪ Λ we find (cid:0) ∂V /∂m q (cid:1) ∝ m q log ( m q / Λ) which is negative (we have verified thatthe sign does not change as long as m q ≤ Λ). We con-clude that δm q >
0, showing that µ favors the sponta-neous breaking of chiral symmetry at zero temperature. B. Finite temperature: chiral condensate
We can extend the analysis of the previous subsection,to the case of finite temperature. For the perturbativesolution of the gap equation, the derivative of the effec-tive potential with respect to the quark mass is needed,see Eq. (16). At finite temperature, the µ -dependentcontribution of the effective potential is V = V + V T with V given by Eq. (14). Thus, Eq. (16) is replaced by δm q = − M σ (cid:18) ∂V ∂m q + ∂V T ∂m q (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) m q = ¯ m q . (17)The expression of V T is not informative, thus it is notnecessary to report it here. Its derivative with respect tothe quark mass is more interesting, ∂V T ∂m q = N c N f µ G ( m q , T ) . (18)In the above equation, G ( m q , T ) corresponds to a func-tion defined in terms of a convergent numerical integral.In Fig. 1, we plot G as a function of temperature, for twodifferent values of the quark mass. We find that G > µ -dependentcontribution of the thermal fluctuations tends to reducethe value of the constituent quark mass (i.e., of the chiralcondensate). C. Critical line
We compute in this subsection the critical tempera-ture, T c , as a function of µ , and show at the same timethat the chiral phase transition is of the second order.To this end we perform an expansion of the effective po-tential near T c . In fact, the order parameter around T c is small enough that a series expansion in powers of σ/T is justified. Since we restrict ourselves to the case of H MeV L G H M e V L FIG. 1. Function G of Eq. (18) versus temperature, for twovalues of the constituent quark mass, m q . Orange solid linecorresponds to m q = 50 MeV; Indigo dashed line correspondsto the case m q = 350 MeV. the pure NJL model, there is no dependence from thePolyakov loop in the NJL coupling constant, G .One can write V = V + α G σ + α G σ + . . . , (19)where the dots denote higher order terms; V is the po-tential at σ = 0: it is independent on the condensate,thus it is just a number which does not affect the physicsof the problem. We notice that because of our defini-tion of the σ field, which has the dimension of a cubicmass, we have extracted the proper powers of the NJLcoupling constant from the definitions of the coefficients,in order to give the common mass dimension to the coef-ficients themselves. At the second order transition point T = T c one has α = 0 and α >
0. Therefore, to de-termine the critical temperature as a function of µ it isenough to determine the zeros of α in the µ − T plane.Next, we compute α to check that the transition pointscorrespond to a second order phase transition.Computing the second derivative of the effective po-tential in Equation (7) we find, after some algebra, α = α , + α , µ , (20)where α , = 2 G − N c N f π Λ + 2 N c N f T , (21) α , = 4 N c N f π (cid:18) log T Λ + c (cid:19) , (22)and the numerical factor c = 1 .
36 arises from a conver-gent numerical integral.At small µ the equation α = 0, which determines thecritical temperature, can be solved perturbatively: fromthe condition α , = 0 we find T c = (cid:18) π − N c N f G (cid:19) / ; (23)using the parameters of the model we find T c = 174 MeV.Then, writing T c = T c + δT and solving α , + α , µ = 0for δT we find T c = T c (cid:18) − µ N c N f ( T c ) α , ( T c ) (cid:19) . (24)Using the parameters given above we find α , ( T c ) =0 .
17, which implies that the critical temperature is a de-creasing function of µ . Hence, a finite µ favors thedisordered phase at finite temperature.The last step is to check the sign of α , to be confidentthat the transition is of the second order. Since we makea perturbative analysis at µ ≪ T , we may assume µ infinitesimal of the same order of σ , hence terms of theorder of O ( µ σ ) can be regarded as O ( σ ) and can beneglected. At the leading order, it is therefore enough tocompute the coefficient at µ = 0, α = 48 N c N f π (cid:18) log Λ T − b (cid:19) , (25)where b = 0 .
90 arises from a convergent integral com-puted numerically. Using the values of Λ and T c we findthat α = 11 .
35 at T = T c ; being it positive, the phasetransition is of the second order.Before going ahead, it is useful to summarize the re-sults of this Section. Eq. (17) represents the correction tothe solution of the gap equation due to µ = 0. The firstand second addenda in the right hand side of the equationcorrespond to the zero temperature and finite tempera-ture contributions, respectively. At zero temperature,only the first contribution survives, and it is negative,leading to a positive shift of the constituent quark mass.On the other hand, the contribution of thermal fluctua-tions is positive. Thus at a given temperature, there is acompetition between the effect of µ on the vacumm andthermal contributions to the gap equation. In passing,we notice that if vacuum fluctuations were neglected inthe thermodynamic potential, then the thermal contribu-tion in Eq. (17) would lower the constituent quark massat finite temperature as µ is switched on. Finally, theresults in Equations (22), (24) and (25) show that at zeroand small µ , the chiral phase transition is of the secondorder, and the critical temperature is a decreasing func-tion of the chiral chemical potential. This scenario forthe critical line is in agreement with the results of [45–47], where a numerical procedure of minimization of theone-loop potential has been adopted. V. GRANDCANONICAL AND CANONICALPHASE DIAGRAMS
In this Section, we describe the expected phase struc-ture of QCD at finite temperature and axial chemicalpotential, as predicted by the PNJL model. The grand-canonical phase diagram, already discussed in [47], cor-responds to the map of the different phases in the µ − T plane. On the other hand, for the case of the canoni-cal ensemble phase diagram, µ is replaced by the chiraldensity n , the latter being defined as n = − ∂ Ω ∂µ , (26)where the derivative has to be computed at the globalminimum of the grand potential.The computation of n in the model needs some care.In particular, at m q = 0 the divergence of the vacuumenergy at finite µ is transmitted to n . As a matter offact, n can be formally split into a vacuum and a thermalparts. The thermal part is convergent, and its derivativeis finite as well, therefore it does not lead to any difficulty.On the other hand, the T = 0 contribution is divergent,and its derivative with respect to µ is divergent as wellin the case m q = 0. This can be realized easily if we takefor a moment the limit Λ ≫ µ , m q , in which we find n = N c N f π (cid:20) µ + 2 m q µ (cid:18) log 2Λ m q − (cid:19)(cid:21) . (27)In the case m q = 0 we obtain the result of [41], which isnot affected by the ultarviolet divergence. Because of thelatter, we cannot take the limit Λ → ∞ in the calculationof the phase space integrals which are involved in the ex-pressions of n . Therefore in the numerical computationof n , we cutoff the vacuum contribution at p = Λ unlessotherwise stated, for internal consistency.Before the discussion of the results, it is useful to com-ment briefly on the expected effect of the finite cutoffon n . If µ ≫ m , which is appropriate in the quark-gluon plasma phase, we can neglect the mass term in thevacuum contribution of the fermion determinant to n ,which we call C : C = N c N f π (cid:2) µ θ (Λ − µ ) + Λ θ ( µ − Λ) (cid:3) . (28)We notice that the term proportional to Λ in the aboveequation does not appear in Eq. (27) since in the latterwe have taken the limit Λ ≫ µ . As soon as µ > Λ,the vacuum contribution to the chiral density saturatesbecause of the existence of a cutoff in the theory, as shownby Eq. (28). In the numerical calculations, we expectthat the saturation effect will be softened by the thermalfluctuation contributions. As a consequence we expectto measure some mild saturation effect as µ ≈ Λ. Thissaturation is observed on the Lattice as well, see [49, 50].In last analysis, this is not a serious trouble, since weexpect the model to be consistent only when masses aresmaller than the cutoff, in this case for regime µ < Λ.In Fig. 2 we plot the normalized chiral density, ρ W ≡ n / N f , as a function of µ for several values of the tem-perature. In the Figure, T c = 173 . µ = 0. At large values of µ wemeasure the expected saturation of ρ W . This saturationis not physical but a mere artifact of our regularization
100 200 300 400 500 600 700 Μ H MeV L Ρ W H f m - L FIG. 2. Normalized chiral density, ρ W ≡ n / N f , as a func-tion of the chiral chemical potential, at several values of thetemperature. From right to left, temperature is equal to0 . T c , 0 . T c , 0 . T c , 0 . T c , T c , 1 . T c respectively. scheme. Hence we will consider trustable only the resultsin the range µ < Λ; when the results outside the abovespecified range are shown, it is done only for complete-ness.Next we turn to the phase structure. The grandcanon-ical phase diagram has been computed in [47]. It is use-ful, however, to briefly summarize here the results of [47],in order to facilitate the comparison with the canonicalphase diagram. In the upper panel of Fig 3 we plot thephase diagram in the µ − T plane. The indigo dashed linecorresponds to the chiral and deconfinement crossover;the solid line denotes the first order phase transition. Theindigo dot is the critical endpoint. The scale T c = 173 . µ = 0.The critical line in the phase diagram in Fig. 3 is iden-tified with the peak of the derivative dL/dT . Within fewMeV, we have found that the latter coincides with thelocation of the peak of | dσ/dT | , for the values of µ an-alyzed in this work. Thus, within the model ad hand,the deconfinement and the chiral symmetry restorationtake place simultaneously. In last analysis, this is relatedto the explicit dependence of the NJL coupling on thePolyakov loop, and to the mechanism that leads to de-confinement in the model. As a matter of fact, at finitetemperature the one- and two-quark contributions to thegrand potential act as efficient Z − breaking terms, whichin turn favor the L = 0 state. As a consequence, the ef-fective value of the NJL coupling which is responsiblefor the spontaneous chiral symmetry breaking is reducedwith respect to the zero temperature case, as soon as L = 0, see Eq. (6). Thus, it is expected that deconfine-ment in the extended PNJL model implies chiral sym-metry restoration, as it is confirmed by the numericalresults. The inverse assertion, namely that chiral sym-metry restoration implies deconfinement, is not true andindeed it is not necessarily realized in the model, unlessan explicit backreaction on the Polyakov loop effectivepotential is introduced, see for example [14, 16, 24]. Μ (cid:144) T c T (cid:144) T c Confinement Phase QGP Phase
Crossover 1 st order Ρ W H fm - L T (cid:144) T c FIG. 3. (color online).
Upper panel: phase diagram in thegrandcanonical ensemble. The indigo dashed line correspondsto the chiral and deconfinement crossover; the solid line de-notes the first order phase transition. The indigo dot is thecritical endpoint. The scale T c = 173 . µ = 0. Lower panel: phase di-agram in the temperature-chiral density plane. Solid linescorrespond to the boundaries of the mixed phase. On the leftline the volume fraction, c , of hadron phase is c = 1; on theright line we find c = 0. Dashed line denotes the chiral anddeconfinement crossovers. Dot-dashed line corresponds to thevalues of T ( ρ W ) at which c = 1 /
2. In the portion of the phasediagram below the dotted line we find µ > Λ. Finally, theindigo dot denotes the critical endpoint.
The grandcanonical phase diagram can be translatedto a canonical one, by replacing µ with ρ W . This pro-gram is easily accomplished once we use Eq. (26) to com-pute ρ W , once the physical values of h ¯ qq i and L areknown. As already explained in the Introduction, theuse of the term canonical has to be taken as an abuseof nomenclature. Indeed n is not a conserved quantity,because of the fluctuations of the topological charge inQCD and the Ward identity. Thus, discussing about n is meaningful only if the time of observation of the sys-tem is smaller than the typical time needed to observea fluctuation of the topological charge, the latter beingrelated to the inverse of the sphaleron transition rate.In the lower panel of Fig. 3 we plot the phase diagramin the ρ W − T plane. In the figure, the indigo dashedline denotes the chiral and deconfinement crossovers (thetwo coincide within numerical uncertainties within thismodel calculation). The indigo dot denotes the criticalendpoint. Its coordinates in the phase diagram are ρ c = 0 .
518 fm − , T c = 167 MeV . (29)The orange solid lines correspond to the boundaries ofthe mixed phase, which develops in correspondence ofthe first order phase transition line of the grancanoni-cal phase diagram. The composition of the mixed phaseat the temperature T can be evaluated easily as follows.Given the total charge density ρ W , then the volume frac-tions of the ordered, c , and disordered, d , phases have tosatisfy cρ + dρ = ρ W with c + d = 1; here ρ , denotethe values of the topological charge density on the twoboundaries of the mixed phase region. It follows then c = ρ − ρ W ρ − ρ ; (30)on the left line the volume fraction c = 1, correspondingto a homogeneous pure ordered phase; on the right line c = 0, corresponding to a pure disordered phase. Forreference, in Fig. 3 we plot a pink dot-dashed line whichcorresponds to the values of T ( ρ W ) at which c = 1 / µ > Λ, in which cutoff artifactsare relevant. At small temperatures the green line isalmost vertical, because the topological charge densityturns out to be very insensitive of temperature, see Fig. 2.
VI. INFLUENCE OF THE QUARK MASS ANDOF THE VECTOR INTERACTION
In this Section, we wish to investigate the quantitativeeffect of a nonphysical bare quark mass, as well as of thepresence of the vector interaction, on the critical line inthe µ − T plane. In particular, we wish to compute theevolution of the critical endpoint coordinates as a func-tion of the bare quark mass, and of the coupling strengthin the vector channel. A. The effect of the bare quark mass
In this Section, we study the effects of the bare quarkmass on the location of the critical endpoint CP . Esti-mation of the effect of the bare quark mass is very im-portant, because Lattice simulations at µ = 0 nowadaysare performed with a numerical value of the quark masswhich is larger than the physical value. In Lattice simula-tions, the non physical value of the quark mass is reflectedinto a non physical value of the pion mass. In [49, 50],the value of the pion mass in the vacuum is m π ≈ (cid:144) m Μ c , T c FIG. 4. (
Color online ). Evolution of the critical endpointcoordinates as a function of the bare quark mass. Solid or-ange line corresponds to the critical value of the axial chem-ical potential; dashed indigo line corresponds to the criticalendpoint temperature. Axial chemical potential and temper-ature are measured in units of the endpoint coordinates at m = m , Phys . orange line corresponds to the critical value of the axialchemical potential; the dashed indigo line corresponds tothe critical endpoint temperature. Axial chemical poten-tial and temperature are measured in units of the end-point coordinates at m = m , Phys with m , Phys = 5 . m π ≈
139 MeV. For compar-ison, at the largest value of quark mass considered herewe find m π ≈
400 MeV.Our results show that the critical endpoint tempera-ture, T c , is not so much affected by the quark mass. Aswe will show in the next Section, this inertia of T c is alsopresent when we switch on a vector interaction. On theother hand, the critical endpoint axial chemical potential, µ c , is strongly affected by the quark mass. In more detail,the larger value of m , the larger µ c . This is quite easyto understand naively, since the bare quark mass turnsthe chiral transition to a crossover. At µ = 0, the chiralcrossover is smoothed as m is increased. Therefore, it isnatural to expect that a larger value of µ is necessary toexperience a first order phase transition, if m > m , Phys .This might partly explain why the critical endpoint CP is not yet detected in Lattice simulations [49, 50]. B. The role of a vector interaction
In this Section we briefly comment on the role of avector interaction on the phase structure of the model.To this end, we add to the lagrangian density the term L V = − G V h(cid:0) ¯ ψγ µ ψ (cid:1) + (cid:0) ¯ ψγ µ γ ψ (cid:1) i ; (31)we do not include the interaction in the triplet channel,since at the one-loop level it gives rise to terms whichcouple the isospin density, δn = n d − n u , to the isospindensity operator; these are not relevant as long as we donot introduce the isospin chemical potential.At the mean field level one has L V = G V n − G V n ¯ ψγ ψ + G V n − G V n ¯ ψγ γ ψ ; (32)as usual, we have defined n = h ψ † ψ i and n = h ψ † γ ψ i .Thus at the mean field level, the chemical potentials forquark number density and for chiral density are shiftedbecause of the vector interaction, µ → µ − G V n , (33) µ → µ − G V n . (34)The previous equations are useful to grasp the effectof G V = 0 on the phase structure. As a matter of fact,if G V > µ is smoothed, and thecritical endpoint moves to higher values of the chemi-cal potential. On the same footing, because of Eq. (34),the effective chiral chemical potential is smaller than µ .Therefore we expect that the critical endpoint coordi-nate moves to higher values of µ compared to the case G V = 0.This reasonings are confirmed by our concrete numer-ical computations. In the case of G V = 0, the valueof n has to be computed self-consistently by means ofthe number equation (26), which has to be solved, at anygiven value of µ and T , together with the gap equations, ∂ Ω ∂σ = ∂ Ω ∂L = 0 . (35)In Fig. 5 we plot the expectation value of the Polyakovloop in the pseudocritical range as a function of temper-ature, for several values of the strength of the couplingin the vector channel, at µ = 300 MeV. At G V = 0,this value of µ is slightly above µ c . In the figure, solidline corresponds to G V = 0; dashed line corresponds to G V = 0 . G ; dot-dashed line corresponds to G V = 0 . G ;dotted line corresponds to G V = 0 . G ; finally, shortdashed line corresponds to G V = G . As expected, therole of the vector interaction is to smooth the phase tran-sition in comparison with the latter at G V = 0. In-deed, the first order phase transition at µ = 300 MeVat G V = 0 is turned by the vector interaction into acrossover at the same value of µ .In Fig. 6 we plot the critical endpoint coordinates, µ c and T c , as a function of the ratio G V /G . Blue dashedline corresponds to the critical temperature; red solidline corresponds to the critical chiral chemical potential.We find that the temperature of the critical endpointis poorly affected by the value of G V in the range thatwe have examined; on the other hand, the critical valueof the chiral chemical potential is quite sensitive to thevector coupling. H MeV L L FIG. 5. (
Color online ). Expectation value of the Polyakovloop in the pseudocritical range as a function of temperature,for several values of the strength of the coupling in the vectorchannel, at µ = 300 MeV. Black solid line corresponds to G V = 0. Green dashed line corresponds to G V = 0 . G . Bluedot-dashed line corresponds to G V = 0 . G . Orange dottedline corresponds to G V = 0 . G . Finally, indigo short dashedline corresponds to G V = G . V (cid:144) G0.250.50.7511.251.51.752 Μ c , T c FIG. 6. (
Color online ). Critical endpoint coordinates as afunction of the ratio G V /G . Indigo dashed line correspondsto the critical temperature; orange solid line corresponds tothe critical chiral chemical potential. Axial chemical poten-tial and temperature are measured in units of the endpointcoordinates at m = m , Phys . It is useful to comment about the similarities betweenthe phase structure at finite µ , which turns out from ourcomputation, and the one well established (within effec-tive models) at finite baryon chemical potential, µ . Inthis article, we have considered only one type of conden-sate, namely the chiral condensate, which characterizesthe symmetry breaking pattern at finite T and µ in thefermion sector of our model. Besides a smaller curva-ture of the critical line in the case of the chiral chemicalpotential, compared with that at finite µ , we do not findqualitative difference in the phase structure. Also, the ef-fect of the vector interaction, as well as of a finite current0quark mass, is very similar, qualitatively, in the cases offinite µ and finite µ . Some difference between the twophase diagrams might arise from different types of con-densates. For example, an important point that we havenot considered in our article, for the sake of simplicity,is the introduction of a diquark condensate, which mightappear at finite mu5. For the case of QCD at finite µ ,such a condensate is expected to be developed at verylarge µ [62–64]. However, this possibility deserves furtherstudy and needs to be checked numerically by dynamicalcomputation of the condensates. We leave this point toa future project. VII. TOPOLOGICAL SUSCEPTIBILITY
In this Section we compute the topological suscepti-bility at finite temperature, in presence of a backgroundchiral density. Topological susceptibility in QCD is de-fined as the correlator of the topological charge at zeromomentum; it can be computed from the QCD partitionfunction via the relation χ = ∂ Ω ∂θ (cid:12)(cid:12)(cid:12)(cid:12) θ =0 , (36)where the θ -angle is introduced by adding the followingterm to the lagrangian density: θ g π ε µνρσ F aµν F aρσ . (37)It is well known that one can get rid of the term inEq. (37) in the QCD action by means of a chiral ro-tation. However, after the chiral transformation, the θ dependence of the QCD partition function appears ex-plicitely in the quark part of the QCD action. A modelstudy having some overlap with our study can be foundin [19], where the full topological susceptibility in pres-ence of a background of topological charge is computed.In the case of [19], the topological charge is introducedby adding the θ − term to the lagrangian density, which inturns acts as a source for the topological charge. The rel-evance of a finite θ in [19] lies on the possibility that fortemperatures higher than Λ QCD , sphaleron transitionsinduce a spacetime dependent θ -angle [51].A low energy relation connects the vacuum chiral con-densate to the topological susceptibilty; in the theory of N f light flavors the relation reads [56, 57] χ = |h ¯ qq i| X f m f − + O (cid:18) m f Λ QCD (cid:19) , (38)where h ¯ qq i is the common value of the quark condensatesfor the light quarks. Equation (38) shows that the topo-logical susceptibility is proportional to the product of thelight flavor masses; thus, in the equation it is manifestthe fact that in a theory in which at least one massless (cid:144) T c Χ (cid:144) H M e V L FIG. 7. Topological susceptibility as a function of temper-ature, for several values of µ . Black data correspond to µ = 0; orange data correspond to µ = 300 MeV. The greendotted lines correspond to the right hand side of Eq. (38). flavor exists, one has χ = 0. This was proved also byWard identities in [58].As stated before, a chiral rotation transmits the θ -dependence of the QCD action to the quark sector. Afterthe rotation, the quarks acquire a complex mass term.Besides, the condensates that are not invariant underthe axial rotation are mixed among themselves. Thisscenario can be implemented within the effective mod-els to compute θ − dependent quantities, see for exam-ple [59]. Beside this, the θ -dependent action can be usedto compute χ by virtue of Eq. (36). A detailed discussionabout the implementation of the θ − dependent lagrangianwithin the chiral model with the Polyakov loop can befound in [19, 20]. In this article it is enough to men-tion that, in order to introduce the θ angle in the modeland compute the topological susceptibility, it is enoughto change M = m − σG into the grand potential with s(cid:20) m cos (cid:18) θ (cid:19) − σG (cid:21) + m sin (cid:18) θ (cid:19) . (39)It is worth to mention here that generally speaking,at finite θ other condensates might develop (and indeedthey do develop, see [19, 20]). The presence of thesecondensates makes the substitution (39) not sufficient forthe complete treatment of the θ angle within the chiralmodels. However, such new condensates vanish in thetheory at θ = 0, to which we are interested; neglectingthem from the beginning does not change the derivativeof Ω at θ = 0. Therefore, the replacement in Eq. (39)is sufficient for the purpose of computing the topologicalsusceptibility within this model.In Fig. 7 we plot the topological susceptibility as afunction of temperature, for several values of µ . At zerotemperature and chiral chemical potential we find χ = (79 .
97 MeV) , (40)1which is in agreement with the large N c prediction inEq. (38), which gives χ = (80 . with two lightflavors at T = µ = 0.The relation (38) is actually satisfied within the chiralmodel also at T = 0 and µ = 0. Indeed, a straightfor-ward computation shows that χ = − m N c N f π σ I , (41)where I = G Z Λ0 " p dp p ( p + µ ) + ( m − σG ) + µ → − µ ;(42)moreover, from the conditions ∂ Ω /∂σ = 0 and σ = 2 h ¯ uu i we have h ¯ uu i = − N c N f π σ I + O (cid:18) m m q (cid:19) . (43)Here m q corresponds to the constituent quark mass. Acomparison between Eqs. (41) and (43) leads to χ = − m h ¯ uu i + O (cid:18) m m q (cid:19) , (44)which is in agreement with Eq. (38) when the latter iscomputed for two degenerate flavors. On the other hand,at finite temperature we measure some discrepancy be-tween the theoretical prediction given by Eq. (38), cor-responding to the green dotted line in Fig. 7, and thenumerical results obtained within the PNJL model (solidlines in the same figure). VIII. CONCLUSIONS
In this article, we have reported our new results aboutthe structure of hot quark matter in a background of achiral density, the latter induced by a finite axial chemicalpotential µ . Firstly, we have analyzed analytically theeffect of the axial chemical potential on the chiral con-densate, and on the critical temperature for restorationof chiral symmetry. We restricted ourselves to the caseof the NJL model, and to the chiral limit. However, evenwithin these simplifications, we are able to understandthe shape of the critical line at finite µ . Secondly, we have discussed the phase diagram in thecanonical ensemble formulation, in which µ is replacedby the chiral density, n . This might be interesting be-cause n in QCD is connected locally to the topologicalcharge density, because of the integrated Ward identity.As a consequence, it might be of interest to predict thenumerical value of the topological charge density alongthe critical line, as well as at the critical endpoint.Thirdly, we have computed the effect of the vector in-teraction, as well as of the finite current quark mass, onthe location of the critical endpoint. Such aspects shouldbe taken into account if a comparison with the Latticedata is desirable, and extend the study started in [47].As a final investigation, we have computed the fulltopological susceptibility (i.e., which takes into accountboth the pure gauge and the dynamical fermion contri-bution) at zero as well as at finite µ , as a function oftemperature. We find that the Di Vecchia-Leutwyler-Smilga-Veneziano (DLSV) relation [56, 57] is satisfied atfinite µ in the confinement phase of the model. At largetemperature, above the critical temperature, we measurea deviation from the DLSV relation, both at zero andat finite µ ; this can be understood within the model,since terms of the order of m /m q which are negligible inthe confinement phase, become important in the quark-gluon-plasma phase.It is interesting to ask wether the work presented herecan be improved. In our opinion, several directions arepossible for future research. As a first step, it would beimportant to investigate from a theoretical point of view,how to couple correctly µ to a conserved quantity inQCD. Along this line, the work in Ref. [52] seems quiteenlightening. Secondly, it is of interest to include thepossibility of other condensates, among them a diquarkcondensate, in analogy to the situation of QCD at verylarge density. We plan to report on these, as well asrelated, topics in the next future. ACKNOWLEDGMENTS
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