aa r X i v : . [ h e p - ph ] O c t Quark Matter with a Chiral Chemical Potential
Marco RuggieriDepartment of Physics and Astronomy, University of Catania, Via S. Sofia 64, I-95125 Catania
In this talk, we report the results discussed in [1], related to the phase structure of hotquark matter in presence of a background of chiral charge density, n . The latter isintroduced in the simplest way possible, namely by virtue of a chemical potential, µ ,conjugated to n . In more detail, after a brief introduction and list of motivations ofthis kind of study, we discuss the interplay between chiral symmetry restoration anddeconfinement at finite µ , as well as the critical endpoint in the phase diagram andits possible relationship with the critical endpoint of the phase diagram od QuantumChromodynamics (QCD). In the talk, due to time limitation, we can emphasize fewresults related to the latter topic. Therefore we need to leave apart several applicationsof the ideas, as well as of the formalism, developed here to the physics of heavy ioncollisions, with particular reference to the Chiral Magnetic Effect [2, 3, 4]. The lattermight be relevant for the phenomenology of heavy ion collisions, because a copiousproduction of gluon configurations, the QCD sphalerons, with a finite winding numberis expected in the quark-gluon-plasma phase of QCD, see [5] and references therein.Because of the chiral Ward identity, the interaction of the sphalerons with the quarkscauses a chirality change of the latters. As a consequence, a copious production oflocal domains in which chirality is imbalanced, is expected in the quark-gluon-plasma.The critical endpoint, CP, of QCD [6] is the cornerstone of the phase diagram ofstrongly interacting matter. At CP, a crossover line and a first order line are supposedto intercept. It is thus not surprising that an intense experimental activity is nowadaysdedicated to the detection of such a point, which involves the large facilities at RHICand LHC; moreover, further experiments are expected after the development of FAIRat GSI. Several theoretical signatures of CP have been suggested [7, 8]. Despite theimportance of CP, a firm theoretical evidence of its existence is still missing. Infact, the sign problem makes the Lattice Monte Carlo simulations difficult, if notimpossible, in the large baryon-chemical potential ( µ ) region for N c = 3 [9], see [10]for a recent review. Therefore, it has not yet been possible to prove unambiguouslythe existence and the location of CP starting from first principles simulations ofgrand-canonical ensembles. Moreover, the predictions of effective models are spreadin the T − µ plane, see for example [11, 12].1nteresting overcomings of the sign problem for the quest of CP are: analyticcontinuation of data obtained at imaginary chemical potential, µ I [13, 14, 15, 16];simulations at finite isospin chemical potential, see for example [17, 18, 19]; sim-ulations in canonical, rather then grand-canonical, ensembles [20]; strong couplingexpansion of Lattice QCD [21, 22]. On the purely theoretical side, it has been sug-gested [23] that the use of orbifold equivalence in the large N c approximation of QCDcan lead to relations between the coordinates of CP at finite chemical potential, withthose at finite isospin chemical potential.In this talk, we present the idea suggested in [1] about a new theoretical way todetect the CP, by means of Lattice simulations with N c = 3. In order to accomplishthis important program, we suggest to simulate QCD with a chiral chemical potential, µ , conjugated to the chiral charge density, n = n R − n L , see [3, 24, 25, 26, 27] forprevious studies. Our idea, supported by concrete calculations within acmicroscopiceffective model, is that CP can be continued to a critical endpoint at µ = 0 and µ = 0,that we denote by CP , the latter being accessible to N c = 3 Lattice QCD simulationsof grand-canonical ensembles [3, 28]. Therefore, the detection of the former endpointvia Lattice simulations, can be considered as a signal of the existence of the latter.The model used in the calculation, namely the Nambu-Jona-Lasinio model withthe Polyakov loop [29] (PNJL model in the following) with tree level coupling amongchiral condensate and Polyakov loop [30], gives numerical relations among the coor-dinates of CP and those of CP. In particular, the critical temperature turns out tobe almost unaffected by the process of continuation; the critical value of the chemicalpotential, µ c , on the other hand turns out to be almost half of the critical chiralchemical potential, µ c .Before discussing our results, it is important to spend some word more about thechiral chemical potential. In particular, we are aware that world at finite µ shouldbe considered as a fictional one. As a matter of fact, µ cannot be considered asa true chemical potential because, in the confinement phase, the chiral condensate h qq i mixes left- and right-handed components of the quark field, leading to non-conservation of n . Moreover, the quantum chiral anomaly leads to fluctuations ofthe topological charge, which in turn causes the changes of the chiral density becauseof the Ward identity. Therefore, the point of view that we adopt is to consider µ as a mere mathematical artifice. However, the world at finite µ with N c = 3 canbe simulated on the Lattice. Therefore, it is worth to study it by grand-canonicalensemble simulations: it might furnish an evidence of the existence of the criticalendpoint in the real world. 2 The Model with the Polyakov loop
Because of its non-perturbative nature, we cannot make first principles calculationswithin QCD in the regimes to which we are interested in, namely moderate T , µ and µ . Hence we need to rely on some effective model, which is built in order to respect(at least some of) the symmetries of the QCD action. To this end, we make use ofthe Nambu-Jona-Lasinio model [31] (see [32] for reviews) improved with the Polyakovloop [29], dubbed PNJL model, which has been used many times in recent years todescribe successfully the thermodynamics of QCD with two and two-plus-one flavors,see [30, 33, 34, 35, 36, 37, 38, 39, 40] and references therein. The model is interestingbecause it allows for a self-consistent description of spontaneous chiral symmetrybreaking; even more, it allows for a simultaneous computation of quantities sensibleto confinement and chiral symmetry breaking. We restrict here to a brief summaryof the main equations; we refer to [1] for a more detailed discussion.In the PNJL model, quark propagation in the medium is described by the followinglagrangian density: L = q (cid:0) iγ µ D µ − m + µ γ γ + µγ (cid:1) q + G (cid:2) ( qq ) + ( iqγ τ q ) (cid:3) ; (1)In the above equation, q corresponds to a quark field in the fundamental representa-tion of color group SU (3) and flavor group SU (2). We have a introduced chemicalpotential for the quark number density, µ , and a pseudo-chemical potential conju-gated to chirality imbalance, µ . The chiral charge density, n = n R − n L , representsthe difference in densities of the right- and left-handed quarks. The imbalance ofchiral density can be created by instanton/sphaleron transition in QCD, see [3] andreferences therein. At finite µ , a chirality imbalance is created, namely n = 0. Forexample, in the massless limit and at zero baryon chemical potential one has [3] n = µ π + µ T . (2)If quark mass (bare or constituent) is taken into account, the relation n ( µ ) cannotbe found analytically in the general case, and a numerical investigation is needed, seefor example [24].In our computation we follow the idea implemented in [30], which brings to aPolyakov-loop-dependent coupling constant: G = g (cid:2) − α LL † − α ( L + ( L † ) ) (cid:3) , (3)The ansatz in the above equation was inspired by [42, 43] in which it was shownexplicitly that the NJL vertex can be derived in the infrared limit of QCD, it hasa non-local structure, and it acquires a non-trivial dependence on the phase of thePolyakov loop. We refer to [30] for a more detailed discussion. This idea has been3nalyzed recently in [44], where the effect of the confinement order parameter onthe four-fermion interactions and their renormalization-group fixed-point structurehas been investigated. The numerical values of α and α have been fixed in [30]by a best fit of the available Lattice data at zero and imaginary chemical potentialof Ref. [45, 46]. In particular, the fitted data are the critical temperature at zerochemical potential, and the dependence of the Roberge-Weiss endpoint on the barequark mass. The best fit procedure leads to α = α ≡ α = 0 . ± . 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 − ) (4)where ω s = q ( | p | s − µ ) + m q , (5)corresponds to the pole of the quark propagator, and F − = 1 + 3 Le − β ( ω s − µ ) + 3 L † e − β ( ω s − µ ) + e − β ( ω s − µ ) , (6) F + = 1 + 3 L † e − β ( ω s + µ ) + 3 Le − β ( ω s + µ ) + e − β ( ω s + µ ) , (7)denote the statistical confining thermal contributions to the effective potential; ω s isgiven by Equation (5), with m q = m − Gσ . Once again the vacuum fluctuation termis regularized by means of a ultraviolet cutoff, that we denote by M . The relationbetween the chiral condensate and σ in the PNJL model is σ = h qq i .We notice that the PNJL model considered here, which is dubbed Extended-PNJL in [30], has been tuned in order to reproduce quantitatively the Lattice QCDthermodynamics at zero and imaginary quark chemical potential. Hence, it representsa faithful description of QCD, in terms of collective degrees of freedom related to chiralsymmetry breaking and deconfinement.The potential term U in Eq. (4) is built by hand in order to reproduce the puregluonic lattice data with N c = 3 [33]. 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 ) − LL ) (cid:3)(cid:27) . (8)We refer to [33, 1] for the numerical values of the parameters used in this study. In Figure 1 we plot the phase diagram of the model in the µ − T plane, for thecase µ = 0. At any value of T and µ , the chiral condensate and the Polyakov loop4 P Confinement PhaseQGP Phase nd order 1 st order PNJL Μ (cid:144) T c T c (cid:144) T c Figure 1: (
Color online ). Phase diagram of the PNJL model. The scale T c = 173 . µ = 0.expectation value are computed by a minimization procedure of the full potential (4).The structure of our phase diagram is in agreement with previous model studies,see [24, 25]. Since chiral symmetry is broken explicitly by the quark mass and thephase transitions are actually crossovers, we identify the critical temperature withthat at which dL/dT is maximum. We have checked that the latter deviates from thatat which | dσ/dT | is maximum only of a few MeV, in the whole range of parametersstudied. With an abuse of nomenclature, we dub the pseudo-critical lines as secondorder and first order. It is clear from the context that the term second order transitionhas to be taken as a synonym of smooth crossover; similarly, the term first ordertransition is a synonym of discontinuous jump of the order parameters.In the Figure 1, the grey dashed line corresponds to a smooth crossover. The solidline, on the other hand, denotes the first order transition. The dot corresponds toCP . In the PNJL model we have access to the chiral condensate and to the Polyakovloop expectation value. As a consequence, we can label the phases of the model interms both of confining properties, and of chiral symmetry. In the model at hand,because of the entanglement in Equation (3), the deconfinement and chiral symmetryrestoration crossovers take place simultaneously. The region below the pseudo-criticalline is characterized by confinement and spontaneous breaking of chiral symmetry;we label this phase as the confinement phase. On the other hand, the phase abovethe critical line is identified with the Quark-Gluon-Plasma phase. In this case, CP isboth chiral and deconfinement critical endpoint. For what concerns the coordinatesof CP we find, for the PNJL model, (cid:18) µ c T c , T c T c (cid:19) = (1 . , . , CP (PNJL) , (9)5
50 100 150 Μ H MeV L Μ H MeV L H MeV L Μ H MeV L H MeV L CP CP Μ H MeV L Μ c H M e V L Figure 2: (
Color online ). Evolution of the critical endpoint in the µ − µ − T space,for the PNJL model.where T c = 173 . µ = µ = 0.Next we turn to discuss the more general case with both µ and µ different fromzero. Our scope is to show that, at least within the models, CP naturally evolvesinto CP . In particular the PNJL model, which is in quantitative agreement with theLattice at zero chemical potential, gives a numerical relation among the coordinatesof CP and CP , which might be taken as a guide to estimate the coordinates of CPin QCD, once CP is detected.In Figure 2 we collect our data on the critical point of the phase diagram in the µ − µ − T space, in the case of the PNJL model. The orange solid line is the unionof the critical points computed self-consistently at several values of µ : at any valueof µ , a point on the line corresponds to the critical point of the phase diagram in the µ − T plane. Thus the line pictorially describes the evolution of the critical point ofthe chiral model at hand, from CP to CP . In the same Figure we plot a projectionof the critical endpoint evolution curve onto the µ − µ plane, for the PNJL model.The indigo solid line corresponds to the µ -coordinate of the critical endpoint. Thecritical temperature is not so much affected when we continue CP to CP (we measurea change approximately equal to the 3%), therefore the projection in the µ − T planeis redundant. Our numerical results suggest the following relationships between the6ritical coordinates at µ = 0 and µ = 0: µ c µ c ≈ . , T c T c ≈ . , (PNJL) . (10)The model predictions (10) relate the coordinates of CP to those of CP . Inparticular, it is interesting that the critical temperature is almost unchanged in thecontinuation of CP to CP . Of course, since these results are deduced by a model, itis extremely interesting and important to study how Equation (10) is affected by thevalue of the bare quark mass, as well as by further interactions in the vector and axial-vector channels. These topics will be the subject of a forthcoming publication [47].It is worth to anticipate some of the results of [47], namely that a larger value of thequark mass, as well as the vector interaction, move CP to larger values of µ . Thecombination of these two factor, together with the finite size of the lattice cell, mightexplain the absence of CP in the Lattice simulations [28]. In this talk, we have reported on our results about the phase structure of hot quarkmatter in presence of a background of chiral charge density. Such a background isintroduced by virtue of a chemical potential, µ , conjugated to the chirality imbalance, n = n R − n L . Because of the fluctuations of the topological charge, which is connectedwith chirality imbalance in QCD via the quantum anomaly, µ should be treated asa pure mathematical artifice, and cannot be considered as a true chemical potential.This study is partly motivated by the potential applications to the Chiral MagneticEffect [2, 3, 4]. The latter might be relevant for the phenomenology of heavy ioncollisions, because a copious production of gluon configurations, the QCD sphalerons,with a finite winding number is expected in the quark-gluon-plasma phase of QCD,see [5] and references therein. Because of the chiral Ward identity, the interactionof the sphalerons with the quarks causes a chirality change of the latters. As aconsequence, a copious production of local domains in which chirality is imbalanced,is expected in the quark-gluon-plasma.After an overview on the phase diagram of hot quark matter at finite µ , obtainedwithin an effective model, we have suggested the possibility of continuation of thecritical endpoint of the phase diagram of N c = 3 QCD, CP, to a critical endpointdubbed CP at finite µ and µ = 0. The worldsheet W ≡ { µ = 0 , µ = 0 } hasthe merit that it can be simulated on the Lattice [3, 28] for N c = 3. Even if Latticeresults have been already published [28], more care should be taken in the derivationof the relation between n and µ , since n is a nonconserved quantity, hence it suffersrenormalization effects which should be taken into account.The phase structure that we have discussed here is based on the PNJL modelwith entanglement vertex, introduced in [30], which offers a description of the QCD7hermodynamics in terms of collective degrees of freedom, which is in quantitativeagreement with Lattice data at zero and imaginary chemical potential.One of our ideas is that simulations in the worldsheet W might reveal the ex-istence of a critical endpoint, CP , in the phase diagram. Then, this critical pointmight be interpreted as the continuation of the critical point which is expected tobelong to the phase diagram of real QCD, because of the continuity summarized inFig. 2. Hence it would be an indirect evidence of the existence of the critical point inreal QCD.In our calculations there are some factors that we have not included for simplicity,and that affect the location of CP . For example, the bare quark mass and the vectorinteraction move CP to higher values of µ . These observations might be helpful tounderstand why in the Lattice simulations of [28], no critical endpoint is detected.We plan to report on the aforementioned topics in the next future [47]. Acknowledgements . It is a pleasure to thank the organizers of the
EleventhWorkshop on Non-Perturbative Quantum Chromodynamics for their kind invitation.Part of this work was inspired by stimulating discussions with H. Warringa, who isacknowledged. Moreover, we acknowledge M. Chernodub, M. D’Elia, P. de Forcrand,G. Endrodi, M. Frasca, K. Fukushima, R. Gatto, T. Z. Nakano, A. Ohnishi, O.Philipsen, Y. Sakai, T. Sasaki, A. Yamamoto and N. Yamamoto for correspondenceand many discussions on the topics presented here. This work was supported by theJapan Society for the Promotion of Science under contract number P09028.
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