arX

iv:1

003.

0047

v2 [

hep-

ph]

15

Jun

2010

YITP-10-3

Chiral magnetic effect in the PNJL model

Kenji Fukushima∗ and Marco Ruggieri†

Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan

Raoul Gatto‡

Departement de Physique Theorique, Universite de Geneve, CH-1211 Geneve 4, Switzerland

We study the two-flavor Nambu–Jona-Lasinio model with the Polyakov loop (PNJL model) inthe presence of a strong magnetic field and a chiral chemical potential µ5 which mimics the effectof imbalanced chirality due to QCD instanton and/or sphaleron transitions. Firstly we focus onthe properties of chiral symmetry breaking and deconfinement crossover under the strong magneticfield. Then we discuss the role of µ5 on the phase structure. Finally the chirality charge, electriccurrent, and their susceptibility, which are relevant to the Chiral Magnetic Effect, are computed inthe model.

PACS numbers: 12.38.Aw,12.38.Mh

I. INTRODUCTION

Quantum Chromodynamics (QCD) is widely believedto be the theory of the strong interactions. Investiga-tions on its rich vacuum structure and how the QCD vac-uum can be modified in extreme environment are amongthe major theoretical challenges in modern physics. Itis in particular an interesting topic to study how non-perturbative features of QCD are affected by thermal ex-citations at high temperature T and/or by baryon-richconstituents at large baryon (quark) chemical potentialµq. Such research on hot and dense QCD is importantnot only from the theoretical point of view but also fornumerous applications to the physics problems of theQuark-Gluon Plasma (copiously produced in relativisticheavy-ion collisions), ultra-dense and cold nuclear/quarkmatter as could exist in the interior of compact stellarobjects, and so on.The most intriguing non-perturbative aspects of the

QCD vacuum at low energy are color confinement andspontaneous breakdown of chiral symmetry. In recentyears our knowledge on (some parts of) the QCD phasediagram has increased noticeably because of significantdevelopments of the lattice QCD simulations (see [1–4] for several examples and see also references therein).At zero µq, except for some reports [2], the numeri-cal simulations have almost established that two QCDphase transitions (crossovers) take place simultaneouslyat nearly the same temperature; one for quark deconfine-ment and another for restoration of chiral symmetry (thelatter being always broken because of finite bare quarkmasses, strictly speaking). It is still however under de-bate whether two crossovers should occur at exactly thesame temperature, however.Once a finite µq is turned on, the Monte-Carlo simu-

lation in three-color QCD on the lattice cannot be per-

∗ fuku@yukawa.kyoto-u.ac.jp† ruggieri@yukawa.kyoto-u.ac.jp‡ raoul.gatto@unige.ch

formed straightforwardly because of the (in)famous signproblem [5]. To overcome this problem several tech-niques have been developed such as the multi-parameterreweighting method [6], Taylor expansion [7], density ofstate method [8], analytical continuation from the imag-inary chemical potential [9], the complex Langevin dy-namics [10], etc.In addition to hot and dense QCD with T and µq,

the effect of a strong magnetic field B on the QCD vac-uum structure is also a very interesting subject. It wouldbe of academic interest to speculate modification of thevacuum structure of a non-Abelian quantum field theoryunder strong external fields. Besides, more importantly,this kind of investigation has realistic relevance to phe-nomenology in relativistic heavy-ion collisions in whicha strong magnetic field is produced in non-central col-lisions [11, 12]. In particular, the results obtained bythe UrQMD model [12] show that eB created in non-central Au-Au collisions can be as large as eB ≈ 2m2π(i.e. B ∼ 1018Gauss) for the top collision energy at RHIC,namely

√sNN

= 200 GeV. Moreover, an estimate withthe energy reachable at LHC,

√sNN

≈ 4.5 TeV, giveseB ≈ 15m2π for the Pb-Pb collision according to Ref. [12].Hence, heavy-ion collisions provide us with a most in-triguing laboratory available on the Earth in order tostudy the effect of extremely strong magnetic fields onthe QCD vacuum.Concerning the (electromagnetic) magnetic field effect

on the QCD vacuum structure, there have been manyinvestigations and it has been recognized that B plays arole as a catalyzer of dynamical chiral symmetry break-ing [13–16]. The QCD vacuum properties have been alsostudied by means of so-called holographic QCD mod-els [17]. The relation between the dynamics of QCD in astrong magnetic field and non-commutative field theoriesis investigated in Ref. [18].A phenomenologically interesting consequence from

the strong B in heavy-ion collisions is what is termed theChiral Magnetic Effect (CME) [11, 19]. The underlyingphysics of the CME is the axial anomaly and topologicalobjects in QCD. Analytical and numerical investigations

http://arxiv.org/abs/1003.0047v2mailto:fuku@yukawa.kyoto-u.ac.jpmailto:ruggieri@yukawa.kyoto-u.ac.jpmailto:raoul.gatto@unige.ch

2

have demonstrated that the sphaleron transition occursat a copious rate at high temperature unlike instantonsthat are thermally suppressed [20, 21]. Sphalerons arefinite-energy solutions of the Minkowskian equations ofmotion in the pure gauge sector and they appear not onlyin the electroweak theory but also in QCD [22]. Theycarry a finite winding number QW which is defined as

QW =g2

32π2

∫

d4x Tr[FF̃ ] , (1)

where F and F̃ denote respectively the field strength ten-sor and its dual. Sphalerons connect two distinct classicalvacua of the theory with different Chern-Simons numbersin Minkowskian space-time. It is possible through thecoupling with fermions in the theory to relate the changeof chirality, NR − NL, to the winding number by virtueof the Adler-Bell-Jackiw anomaly relation,

(NR −NL)t=+∞ − (NR −NL)t=−∞ = −2QW . (2)

The r.h.s. of Eq. (2) is the integral over space-time of∂µj

µ5 , where j

µ5 represents the anomalous flavor-singlet

axial current. The physical picture that arises fromEq. (2) is that in the presence of topological excitationssuch as instantons and sphalerons with a given QW , andstarting with a system of quarks with NR = NL, an un-balance between left-handed and right-handed quarks isproduced. Such an unbalance can lead to observable ef-fects to probe topological P- and CP-odd excitations. Anexperimental observable sensitive to local P- and CP-odd effects has been proposed in Ref. [23]. Recently theSTAR collaboration presented the conclusive observationof charge azimuthal correlations [24] possibly resultingfrom the CME with local P- and CP-violation.The intuitive picture of the CME is as follows. In a

strong magnetic field B, quarks are polarized along thedirection of B. Let us suppose that B is along the pos-itive z axis (that is conventionally taken as the y axisin the context of heavy-ion collisions). Neglecting quarkmasses, which is a good approximation for u and d quarksin the high-T chiral restored phase, the chirality is aneigenvalue to label the quarks. Then, right-handed uquarks should have both their spin and momentum par-allel to B and left-handed u quarks should have theirspin parallel to B and momentum anti-parallel to B.Obviously the same reasoning applies to d quarks. IfNR = NL, then the current that would originate fromthe motion of left-handed quarks is exactly cancelled bythat of right-handed quarks. If NR 6= NL which is ex-pected from the anomaly relation (2), on the other hand,a finite net current is produced. Therefore, if quarks ex-perience a strong magnetic field in a domain where thetopological transition occurs, a net current is producedlocally.The CME has been investigated in the chiral effective

model [25] as well as in the holographic QCD model [17].The chiral magnetic conductivity is calculated withoutgluon interactions in Ref. [26]. In [27] the electric-current

susceptibility under a homogeneous magnetic field, whichcan be related to the fluctuation of the electric-chargeasymmetry measured by the STAR collaboration, hasbeen computed in the same way. The first lattice-QCDstudy of the CME has been performed by the ITEP lat-tice group [28] in the color-SU(2) quench approxima-tion. Moreover, the Connecticut group [29] performeda lattice-QCD study of the CME with 2 + 1 dynamicalquark flavors.

This article is devoted to the study of the two-flavorNambu–Jona-Lasinio model with the Polyakov loop cou-pling (PNJL model) in a strong magnetic field. ThePNJL model has been introduced in Refs. [30, 31] to in-corporate deconfinement physics into the NJL model [32].The main addition to the NJL model is a backgroundgluon field in the Euclidean temporal direction. Thebackground field is related to the expectation value ofthe traced Polyakov loop, Φ, which is known to be an or-der parameter for the deconfinement transition in a puregauge theory [33]. There are many theoretical studies re-lated to different aspects of the PNJL model; see for ex-ample Ref. [34]. See Refs. [35] for a related study withinthe Polyakov-Quark-Meson model, and [36] for an inves-tigation within QCD with imaginary chemical potential.

We work in the chiral limit throughout the paper, inwhich the definition of the chiral critical temperature hasno ambiguity. Firstly we focus on the effect of B on chi-ral symmetry restoration at finite temperature. As it willbe clear soon, our results support the role of the externalB as a catalyzer of dynamical symmetry breaking; thecritical temperature increases with increasing B. Natu-rally the (pseudo)critical temperature for deconfinementcrossover is less sensitive to the presence of B becausethere is no direct coupling between photons and gluons.Hence, the PNJL model predicts that at large enoughB, a substantial range of temperature will open at whichquark matter is deconfined but chiral symmetry is stilldynamically broken. See [37] for a related study.

Also we shall discuss the effects of a finite chiral chem-ical potential µ5 on the phase structure within the PNJLmodel. This µ5 mimics the topologically induced changesin chirality charges N5 = NR−NL that are naturally ex-pected by the QCD anomaly relation. The relevant quan-tity in a microscopic picture is rather the total chiralitycharge N5 but for technical reasons it is easier to work inthe grand-canonical ensemble by treating µ5 (see Ref. [38]for an alternative description based on the flux-tube pic-ture), which is to be interpreted as the time derivative of

the θ angle of the strong interactions; µ5 = θ̇/(2Nf). Be-sides the phase diagram from the PNJL model, we com-pute quantities that are relevant to the CME, namely, theinduced electric current density, its susceptibility, and thechiral charge density n5 together with its susceptibility.

This paper is organized as follows; in Sec. II we givea detailed description of the model we are using. InSecs. III and IV we present and discuss our numericalresults from the model. Finally, in Sec. V we draw ourconclusions.

3

II. MODEL WITH MAGNETIC FIELD AND

CHIRAL CHEMICAL POTENTIAL

In this section we analyze the interplay between thechiral phase transition and the deconfinement crossoverat large B using the PNJL model. Here we considertwo-flavor quark matter in the chiral limit since the chi-ral phase transition is a true phase transition only in thechiral limit, and then and only then Tc can be identifiedunambiguously by vanishing chiral condensate. The chi-ral limit in the two-flavor sector is not far from the phys-ical world in which the current quark masses are a fewMeV, almost negligible as compared to the temperature.Moreover, we are interested in studying the situation inthe presence of chirality charge density. In the grand-canonical ensemble we can introduce the chirality chargeby virtue of the associated chemical potential µ5 in thefollowing way.The Lagrangian density of the model we consider is

given by

L = ψ̄(

iγµDµ + µ5γ

0γ5)

ψ

+G[

(

ψ̄ψ)2

+(

ψ̄iγ5τψ)2]

− U [Φ, Φ̄, T ] , (3)

where the covariant derivative embeds the quark cou-pling to the external magnetic field and to the back-ground gluon field as well, as we will see explicitly below.We note that µ5 couples to the chiral density operator

N5 = ψ̄γ0γ5ψ = ψ†RψR − ψ†LψL, hence n5 = 〈N5〉 6= 0can develop when µ5 6= 0. The mean-field Lagrangian isthen given by

L = ψ̄(

iγµDµ −M + µ5γ0γ5

)

ψ − U [Φ, Φ̄, T ] , (4)

where M = −2σ with σ = G〈ψ̄ψ〉 = G(〈ūu〉+ 〈d̄d〉).In Eq. (4) Φ, Φ̄ correspond to the normalized traced

Polyakov loop and its Hermitean conjugate respectively,Φ = (1/Nc)TrL and Φ̄ = (1/Nc)TrL

†, with the Polyakovloop matrix,

L = P exp(

i

∫ β

0

A4 dτ

)

, (5)

where β = 1/T .The potential term U [Φ, Φ̄, T ] in Eq. (4) is built by

hand in order to reproduce the pure gluonic latticedata [34]. Among several different potential choices [39]we adopt the following logarithmic form [31, 34],

U [Φ, Φ̄, T ] = T 4{

−a(T )2

Φ̄Φ

+ b(T ) ln[

1− 6Φ̄Φ + 4(Φ̄3 +Φ3)− 3(Φ̄Φ)2]

}

,

(6)

with three model parameters (one of four is constrained

by the Stefan-Boltzmann limit),

a(T ) = a0 + a1

(

T0T

)

+ a2

(

T0T

)2

,

b(T ) = b3

(

T0T

)3

.

(7)

The standard choice of the parameters reads [34];

a0 = 3.51 , a1 = −2.47 , a2 = 15.2 , b3 = −1.75 .(8)

The parameter T0 in Eq. (6) sets the deconfinement scalein the pure gauge theory, i.e. Tc = 270 MeV.We assume a homogeneous magnetic field, B, along

the positive z axis. The eigenvalues of the Dirac operatorcan be derived by the Ritus method [40], which are [19];

ω2s =M2 +

[

|p|+ s µ5sgn(pz)]2, (9)

apart from (the phases of) the Polyakov loop, where s =±1, p2 = p2z + 2|qfB|k with k a non-negative integerlabelling the Landau level.The thermodynamic al potential Ω in the mean-field

approximation in the presence of an Abelian chromo-magnetic field has been considered in many literatures,Ref. [41] for example. The expression for an electromag-netic B can be obtained in the same way. Here we simplywrite the final result;

Ω = U + σ2

G−Nc

∑

f=u,d

|qfB|2π

∑

s,k

αsk

∫ ∞

−∞

dpz2π

f2Λ ωs(p)

− 2T∑

f=u,d

|qfB|2π

∑

s,k

αsk

∫ ∞

−∞

dpz2π

× ln(

1 + 3Φe−βωs + 3 Φ̄e−2βωs + e−3βωs)

. (10)

Here the above definition of Ω is different from the stan-dard grand potential in thermodynamics by a volumefactor V . The quasi-particle dispersion ωs is given byEq. (9). The spin degeneracy factor is

αsk =

δs,+1 for k = 0, qB > 0 ,δs,−1 for k = 0, qB < 0 ,1 for k 6= 0 .

(11)

Before going ahead further, one may wonder why we in-troduce only one order parameter for the chiral symmetrybreaking even though the magnetic field breaks isospinsymmetry. Since qu 6= qd, one could suspect that the ef-fects of B on 〈ūu〉 and on 〈d̄d〉 are different. This is notthe case, however, in the present model in the mean-fieldapproximation. As a matter of fact, even in the presenceof B 6= 0, the thermodynamic potential (10) dependsonly on σ ∝ 〈ūu〉+ 〈d̄d〉. This is so only when the four-fermion interaction is Eq. (3) with equal mixing of theU(1)A-symmetric and U(1)A-breaking terms. Hence, therelevant quantity for the chiral symmetry breaking is justone condensate, namely σ, even for B 6= 0, and there is

4

no need to introduce two independent condensates in thisspecial case. Even when we consider more general four-fermion interaction, the isospin breaking effect is onlynegligibly small.The vacuum part of the thermodynamic potential,

Ω(T = 0), is ultraviolet divergent. This divergence istransmitted to the gap equations. Thus we must spec-ify a scheme to regularize this divergence. The choiceof the regularization scheme is a part of the model def-inition and, nevertheless, the physically meaningful re-sults should not depend on the regulator eventually. Inthe case with a strong magnetic field the sharp momen-tum cutoff suffers from cutoff artifact since the contin-uum momentum is replaced by the discrete Landau quan-tized one. To avoid cutoff artifact, in this work, weuse a smooth regularization procedure by introducing aform factor fΛ(p) in the diverging zero-point energy. Ourchoice of fΛ(p) is as follows;

fΛ(p) =

√

Λ2N

Λ2N + |p|2N , (12)

where we specifically chooseN = 10. In theN → ∞ limitthe above fΛ(p) is reduced to the sharp cutoff functionθ(Λ − |p|). Since the thermal part of Ω is not divergent,we do not need to introduce a regularization function.

III. PHASE STRUCTURE WITH CHIRAL

CHEMICAL POTENTIAL

In this section we firstly focus on the system at µ5 = 0and discuss the role of the magnetic field as a catalyzer ofthe dynamical chiral symmetry breaking. We also ana-lyze the interplay between chiral symmetry restorationand deconfinement crossover as the strength of B in-creases.

A. Results at µ5 = 0 – chiral symmetry breakingand deconfinement

We analyze, within the PNJL model, the response ofquark matter to B at µ5 = 0. In particular we are inter-ested in the interplay between chiral symmetry restora-tion and deconfinement crossover in the presence of amagnetic field which leads to the so-called chiral mag-netic catalysis [14]. Our model parameter set is

Λ = 620 MeV , GΛ2 = 2.2 . (13)

These parameters correspond to fπ = 92.4 MeV and thevacuum chiral condensate 〈ūu〉1/3 = −245.7 MeV, andthe constituent quark mass M = 339 MeV. The criticaltemperature for chiral restoration in the NJL part atB =0 is Tc ≈ 190 MeV. We set the deconfinement scaleT0 in the Polyakov loop potential (see Eq. (6)) as T0 =270 MeV, which is the value of the known deconfinementtemperature in the pure SU(3) gauge theory.

0.6 0.8 1 1.2

T/Tc

50

100

150

200

250

300

Ch

ira

l C

on

de

nsa

te

[MeV]

eB=0

eB=4mp2

eB=10mp2

eB=20mp2

0.6 0.8 1 1.2

T/Tc

0.2

0.4

0.6

0.8

Poly

ako

v L

oo

p

eB=0

eB=4mp2

eB=10mp2

eB=20mp2

FIG. 1. Absolute value of the chiral condensate |〈ūu〉1/3| (up-per panel) and expectation value of the Polyakov loop (lowerpanel) as a function of T computed at several values of eB(in unit of m2π). In this model Tc = 228 MeV at µ5 = B = 0.

In Fig. 1 we plot the absolute value of the chiral con-densate 〈ūu〉1/3 (upper panel) and expectation value ofthe Polyakov loop (lower panel) as a function of T com-puted at several values of eB (expressed in unit of m2π).The chiral condensate 〈ūu〉 and the Polyakov loop Φ arethe solution of the gap equations ∂Ω/∂σ = ∂Ω/∂Φ = 0in the model at hand.Figure 1 is interesting for several reasons. First of all,

the role of B as a catalyzer of chiral symmetry breakingis evident. Indeed, the chiral condensate and thus con-stituent quark mass increase in the whole T region as eBis raised (for graphical reasons, we have plotted our re-sults starting from T = 100 MeV. There is neverthelessno significant numerical difference between the T = 0and T = 100 MeV results). This behavior is in the cor-rect direction consistent with the well-known magneticcatalysis revealed in Ref. [14] and also discussed recentlyin Ref. [41] in a slightly different context of the PNJL-model study on the response of quark matter to externalchromomagnetic fields.Secondly, we observe that the deconfinement crossover

5

is only marginally affected by the magnetic field. Wecan identify the deconfinement Tc by the inflection pointof Φ as a function of T . This simple procedure givesresults nearly in agreement with those obtained by thepeak position in the Polyakov loop susceptibility, whichis a common prescription to locate the so-called pseudo-critical temperature. Also we can identify the deconfine-ment Tc with the temperature at which Φ = 0.5. We notethat Tc in Fig. 1 is the chiral Tc = 228 MeV where thechiral condensate vanishes, but not the deconfinement Tcin both figures.From Fig. 1 we notice that, increasing eB from 4m2π

to 20m2π, the shift of the chiral transition temperature∆Tχ ≈ 20 MeV, while the shift of the deconfinementcrossover temperature is as small as ∆TΦ ≈ 5 MeV.Hence, the chiral phase transition is more easily in-fluenced by the magnetic field than the deconfinementas anticipated. Consequently, under a strong magneticfield, there opens a substantially wide T -window in whichquarks are deconfined but chiral symmetry is still spon-taneously broken. This result, valid for µ5 = 0, does notnecessarily hold, in general, for µ5 6= 0, as we will see inthe next subsection.

B. Results at µ5 6= 0 – suppression on the chiralcondensate

We next turn to the study of the effect of a finite µ5on the QCD phase transitions using the PNJL model.We recall that µ5 cannot be a true chemical potentialsince its conjugate variable n5 is only approximately con-served due to axial anomaly. Nevertheless, because thetime derivative of the strong θ angle translates into µ5, asexplained in the introduction section, µ5 itself is a phys-ically meaningful quantity. We specifically look into thebehavior of the critical line for chiral symmetry restora-tion, which is well defined in the chiral limit, at differ-ing B while keeping µ5 fixed. This study will be useful,among other things, in order to understand the relationbetween the chirality density n5 and µ5 that we computenumerically in later discussions.One effect of µ5 6= 0 is lowering of the critical tem-

perature of the chiral phase transition. This is evidentfrom the upper panels of Fig. 2. Firstly we discuss thecase of eB = 5m2π corresponding to the left upper andleft lower panels of Fig. 2. We see that increasing µ5at low T results in slight enhancement of the chiral con-densate. As T approaches Tc, however, the chiral phasetransition at larger µ5 takes earlier place below Tc. As aresult of the coupling between the chiral condensate andthe Polyakov loop, the deconfinement crossover as shownin the lower panels of Fig. 2 is also shifted earlier as µ5becomes greater.In view of the right upper and right lower panels of

Fig. 2 for large eB = 10m2π the µ5-effect on the chiralcondensate at low T is less visible. This is understoodfrom the fact that the chiral magnetic catalysis effect

is predominant over the minor enhancement due to µ5.In contrast, as T is increased toward Tc, the qualitativebehavior of the shift in the critical temperature is justthe same as what we have seen previously for eB = 5m2π.An interesting prediction from the PNJL model is that,

at a given value of eB, there exists a critical µ5, abovewhich the chiral phase transition becomes first order.In the case eB = 5m2π as shown in Fig. 2 the criti-cal µ5 is found between 300 ∼ 400 MeV. As a mat-ter of fact, at µ5 = 400 MeV, the chiral condensateand the Polyakov loop both exhibit discontinuity at thecritical temperature. We see that, as compared to theµ5 = 300 MeV case, the slopes of the chiral condensateand the Polyakov loop sharply change as a function of Tfor the µ5 = 400 MeV case. Hence, our data plotted inFig. 2 suggest the existence of a critical µ5 in the range300 MeV < µc5 < 400 MeV at which the weakly first-order transition becomes a true second-order one. Thephase diagram in the µ5-T plane has a tricritical point(TCP) accordingly. We will discuss more on the TCPin the next subsection. We notice that this picture isqualitatively robust regardless of the chosen value of eB,as is already implied from Fig. 2. The first-order phasetransition in the high-mu5 and low T region leads to adiscontinuity in the chirality density as a function of µ5.This point will be also addressed in some details in thenext section.As a final remark in this subsection we note that, in

Figs. 1 and 2, the slope of the curve quickly changes atsome point (at T/Tc = 1 for eB = 0 for example). Thisis because of the second-order phase transition which isthe case for chiral restoration in the chiral limit.

C. Phase diagram

The results we have revealed so far can be summarizedinto the phase diagram in the µ5-T plane. In the upperpanel of Fig. 3 we show the phase diagram at eB = 5m2π.In the lower panel for comparison we plot the phase di-agram at eB = 15m2π. The line represents the chiralphase transition. It is of second order for small valuesof µ5 (shown by a thin line) and becomes of first orderat large µ5 (shown by a thick line). The location of theTCP on the phase diagram depends only slightly on eB,while the topology of the phase diagram is not sensitiveto the magnetic field.The general effect of µ5 is to lower the chiral transition

temperature. One may argue that the critical line canhit T = 0 eventually at very large µ5, though the PNJLmodel is of no use at such large µ5 because the ultravioletcutoff causes unphysical artifacts. The locations of theTCP are estimated from the PNJL model as

(µ5, T ) ≈ (400 MeV, 200 MeV) , for eB = 5m2π , (14)(µ5, T ) ≈ (370 MeV, 200 MeV) , for eB = 15m2π . (15)

6

0.6 0.8 1 1.2

T/Tc

50

100

150

200

250

300C

hir

al C

on

de

nsa

te

m5�0m5�200m5�300m5����

eB 5mp2

[MeV]

0.6 0.8 1 1.2

T/Tc

50

100

150

200

250

300

Ch

ira

l C

on

de

nsa

te

m5�0m5�200m5�300m5��

eB 10mp2

[MeV]

0.6 0.8 1 1.2

T/Tc

0.2

0.4

0.6

0.8

Poly

ako

v L

oo

p

m5�0m5�200m5���m5�400

eB 5mp2

0.6 0.8 1 1.2

T/Tc

0.2

0.4

0.6

0.8

Poly

ako

v L

oo

p

m5�0m5�200m5����m5�400

eB 10mp2

FIG. 2. Absolute value of the chiral condensate |〈ūu〉1/3| (upper panel) and expectation value of the Polyakov loop (lowerpanel) as a function of T computed at several values of eB (in unit of m2π) and µ5 (in unit of MeV).

IV. CHIRALITY CHARGE, ELECTRIC

CURRENT, AND SUSCEPTIBILITIES

In this section we show our results for quantities rele-vant to the Chiral Magnetic Effect (CME). We numeri-cally compute the chiral density n5 and its susceptibilityχ5 as a function of µ5 and eB. Also we calculate the cur-rent density j3 along the direction of B and its suscep-tibility χJ . Finally, we use the result n5(µ5) to evaluatej3 as a function of n5.

A. Chirality density and its susceptibility

The axial anomaly relates the topological charge QWto the chirality charge N5 with N5 = n5V where V =LxLyLz is the volume of topological domains. We canread n5 from

n5 = −∂Ω

∂µ5. (16)

It is useful information to relate n5 and µ5 for varioustemperatures and magnetic field strength. In the nextsection we will use the results of n5(µ5) to express thecurrent density as a function of the chirality density.

The relation between n5 and µ5 can be found analyti-cally only in simple limiting cases [19]. In general one hasto determine it numerically using an effective model. Weshow n5(µ5) for eB = 5m

2π at three temperatures around

Tc in Fig. 4. The qualitative picture is hardly modifiedeven if we change the magnetic field.

From Fig. 3 we can read the critical temperature atµ5 = 0 that is Tc = 228 MeV. At temperatures well be-low Tc, as seen in the T = 160 MeV case in Fig. 4, thediscontinuity associated with the first-order phase transi-tion with respect to 〈ūu〉1/3 and Φ is conveyed to the rela-tion n5(µ5), which is a typical manifestation of the mixedphase at critical µ5. Naturally, as T gets larger, the chi-rality density as a function of µ5 becomes smoother, sincethe chiral phase transition is of second order at higher Tas is clear from Fig. 3.

It is interesting to compute the chirality charge sus-

7

100 200 300 400 500

m5 [MeV]

50

100

150

200

250T

[M

eV

]

TCP

eB=5mp2

100 200 300 400 500

m5 [MeV]

50

100

150

200

250

T [M

eV

] TCP

eB=15mp2

FIG. 3. (Upper panel) Phase diagram in the µ5-T plane ob-tained at eB = 5m2π . The thin line represents a second-orderchiral phase transition and the thick one a first-order transi-tion. Below the line, chiral symmetry is spontaneously bro-ken, while chiral symmetry is restored above the line. The la-bel “TCP” denotes the tricritical point. (Lower panel) Phasediagram for eB = 15m2π .

0 100 200 300 400 500 600

m5 [MeV]

0

2

4

6

8

10

n5

eB=5mp2

T=160MeV

T=220MeV

T=240MeV

[fm ]-3

FIG. 4. Chirality density n5 (in unit of fm−3) as a function

of µ5 (in unit of MeV) at eB = 5m2

π for several values of T .

ceptibility χ5, as well as n5, defined as

χ5 = 〈n25〉 − 〈n5〉2 = −1

βV

∂2Ω

∂µ25, (17)

where β = 1/T and V the volume. We note that thisdefinition of the susceptibility is different from that inRef. [27] by V . It should be mentioned that we takea numerical derivative to compute χ5 including implicitdependence in Φ and σ. In Fig. 5 we plot χ5 as a functionof eB for several µ5 values. The upper panel correspondsto µ5 = 0, middle one to µ5 = 200 MeV, and lowerone to µ5 = 400 MeV. For completeness, in the rightpanels of the same figure 5 we plot the chiral condensate|〈ūu〉1/3| for the same T and same µ5. The oscillations inχ5 are artificial results because of the momentum cutoffΛ. As shown in Ref. [41], choosing a regulator whichis smoother than used in this work, the oscillations ofthe various quantities could be erased. The qualitativepicture is, nevertheless, unchanged even with a differentregulator. For this reason we do not perform a systematicstudy here on the cutoff scheme dependence.A notable aspect is the suppression of the chirality-

charge fluctuations at large T and large eB. This isevident, for example, in the result with µ5 = 0 andT = 1.1Tc in Fig. 5. As long as eB is small, χ5 isa monotonously increasing function of eB as expectednaively. When eB reaches a critical value around 20m2π,however, χ5 has a pronounced peak and then decreaseswith increasing eB, which is a result of mixture with di-verging chiral susceptibility at the chiral phase transition.It should be mentioned that χ5 at µ5 = 0 (as shown inthe upper left panel of Fig. 5) does not diverge at thecritical eB since the mixing with the chiral susceptibilityis vanishing due to µ5 = 0. This behavior below andabove the chiral critical point can be easily understoodin terms of the chiral symmetry breaking by virtue ofthe magnetic field. As a matter of fact, at T > Tc thechiral condensate stays zero identically as long as eB issmall enough, leading to zero quasiparticle masses. OnceeB exceeds a critical value, the chiral symmetry is bro-ken spontaneously even at high T (see the upper rightpanel of Fig. 5) and the quasiparticle masses can thenjump to a substantially large number then. Such dy-namical quark masses result in appreciable suppressionof the chirality-charge fluctuations. As it will be shownin the next section, this interesting and intuitively un-derstandable effect appears in the current susceptibilityas well.

B. Current density and its susceptibility

The current density j3 (and its susceptibility as well) isthe most important quantity to compute for the ChiralMagnetic Effect [27]. It corresponds to the charge perunit volume that moves in the direction of the appliedmagnetic field in a domain where an instanton/sphalerontransition takes place, which causes chirality change of

8

0 10 20 30 40

eB/mp2

0.025

0.05

0.075

0.1

0.125

0.15bVc

5

m5=0

T=0T=0.95 TcT=1.10 Tc

[GeV ]2

0 10 20 30 40

eB/mp2

0

50

100

150

200

250

300

Ch

ira

l C

on

de

nsa

te

m5=0

[MeV]

m5=200MeV

0 10 20 30 400.00

0.05

0.10

0.15

0.20

eB/mp2

bVc

5

[GeV ]2

m5=200MeV

0 10 20 30 400

50

100

150

200

250

300

eB/mp2

Chiral C

ondensate

[MeV]

m5=400MeV

[GeV ]2

0 10 20 30 400.00

0.05

0.10

0.15

0.20

0.25

0.30

eB/mp2

bVc

5

m5=400MeV

0 10 20 30 400

50

100

150

200

250

300

eB/.mp2

Ch

ira

l C

on

de

nsa

te

[MeV]

FIG. 5. (Left panels) Chirality charge susceptibility χ5 as a function of eB (in unit of m2

π) for several temperatures. The chiralchemical potential is chosen as µ5 = 0, 200 MeV, and 400 MeV, respectively, from the upper to the lower panels. The solidline corresponds to T = 0, the dashed line T = 0.95Tc, and the dot-dashed line T = 1.1Tc. Here Tc = 228 MeV is the criticaltemperature in this model at µ5 = B = 0. (Right panels) Absolute value of the chiral condensate 〈ūu〉

1/3 as a function of eB.The line styles are the same defined in the left upper panel.

quarks. The current has been computed analytically inRef. [19] in four different ways.

To compute the current density along the magneticfield, i.e. j3 = q〈ψ̄γ3ψ〉, we follow the common proce-dure to add an external homogeneous vector potentialalong the magnetic field, A3, coupled to the fermion field.

Then,

j3 = −∂Ω

∂A3

∣

∣

∣

∣

A3=0

. (18)

The derivative of the thermodynamic potential in thepresence of a background field is computed in the follow-ing way. The coupling of quarks to A3 is achieved by

9

shifting pz in Eq. (10) as pz → pz + qfA3. After puttingregularization in the momentum integral with an ultra-violet cutoff Λ (we know that the current is ultravioletfinite, hence the choice of the regularization method doesnot affect the final result) we change the order of the mo-mentum integral and the derivative with respect to A3.Then we make use of the following replacement,

∂

∂A3→ qf

d

dpz, (19)

to obtain,

j3 = Nc∑

f=u,d

qf|qfB|2π

∑

s,k

αks

∫ Λ

−Λ

dpz2π

d

dpz[ωs(p) + · · · ] .

(20)The ellipsis represents irrelevant matter terms. Aftersumming over the spin s, the contribution of the inte-grand from the Landau levels with n > 0 turns out tobe an odd function of pz. Therefore, only the lowestLandau level gives a non-vanishing contribution to thecurrent and we get from the surface contribution [19],

j3 = Nc∑

f=u,d

q2fµ5B

2π2=

5µ5e2B

6π2, (21)

which is certainly ultraviolet finite as it should be. Gen-erally speaking we should utilize a gauge-invariant regu-larization. Nevertheless the above (21) indicates that anaive momentum cutoff can reproduce a correct expres-sion for the anomalous chiral magnetic current.The current density as given by Eq. (21) does not de-

pend on quark mass explicitly, and on temperature ei-ther. The reason is that the current is generated by theaxial anomaly and it receives contributions only from theultraviolet momentum regions (as the above derivationshows), and so it is insensitive to any infrared energyscales. Also, the Polyakov loop does not appear explic-itly in Eq. (21). This is easy to understand; the Polyakovloop is a thermal coupling between quark excitations andthe gluonic medium, and thus the Polyakov loop only en-ters the thermal part of Ω. Since the current originatesfrom the anomaly, however, the thermal part of Ω justdrops off for the current generation. The effect of thePolyakov loop will appear implicitly through the relationbetween µ5 to n5.To confirm that our numerical prescription works well,

we have computed j3 by means of Eq. (18) with Ω given inEq. (10). In Fig. 6 we show the results from our numericalcomputation as a function of µ5. In the figure we haveplotted the normalized current,

j̃3 =

(

5µ0 e2B

6π2

)−1

j3 , (22)

with a choice of µ0 = 1 GeV, which we defined so to makethe comparison transparent at a glance. In Fig. 6 thedashed line represents j̃3 at T = 160 MeV; on the other

200 400 600 800 1000

m5 [MeV]

0.2

0.4

0.6

0.8

1

j 3

eB=5mp2

T=160MeV

T=240MeV

~

2 4 6 8 10

n5 [fm ]

0.0

0.25

0.5

j 3

eB=5mp2

T=160MeV

T=220MeV

T=240MeV

-3

[e fm ]-3

FIG. 6. (Upper panel) Normalized current density, j̃3 =(5µ0 e

2B/6π2)−1j3 with µ0 = 1 GeV, as a function of µ5at two different temperatures (below and above Tc). (Lowerpanel) Current density as a function of n5 for eB = 5m

2

π

computed at three different temperatures.

hand, the dot-dashed line the case at T = 240 MeV. Wenotice that our numerical results are perfectly in agree-ment with Eq. (21). We conclude that our numericalprocedure correctly reproduces the expected dependenceof j3 on µ5 with the correct coefficient insensitive to in-frared scales regardless of whether T is below or aboveTc.

The result shown in the upper panel of Fig. 6 givesus confidence in our numerical procedure but the figureitself is not yet more informative than Eq. (21). We ex-press now j3 as a function of n5 using Eq. (21) and theresults discussed in the previous section. The result ofthis computation is shown in the lower panel of Fig. 6, inwhich we plot the (not normalized) current density (mea-sured in fm−3) as a function of n5 (measured in fm

−3),at eB = 5m2π and at three different temperatures.

From Fig. 6 we notice that, at a fixed value of n5, thelarger the temperature is, the smaller j3 becomes. Thisseemingly counter intuitive result is easy to understand.As a matter of fact, as the temperature gets larger, the

10

Μ5=0

T=0T=0.95 TcT=1.10 Tc

0 10 20 30 40-0.002

0.000

0.002

0.004

0.006

0.008

0.010

eBmΠ2

ΒVΧ

JHG

eV2L

Μ5=200 MeV

T=0T=0.95 TcT=1.10 Tc

0 10 20 30 40-0.002

0.000

0.002

0.004

0.006

0.008

0.010

eBmΠ2

ΒVΧ

JHG

eV2L

Μ5=400 MeV

T=0T=0.95 TcT=1.10 Tc

0 10 20 30 40-0.002

0.000

0.002

0.004

0.006

0.008

0.010

eBmΠ2

ΒVΧ

JHG

eV2L

FIG. 7. Subtracted current susceptibility, βV χ̄J , as a functionof eB (in unit of m2π) for several different values of T (in unitof Tc = 228 MeV) and µ5 (measured in MeV).

corresponding µ5 for a given n5 should decrease becauseof more abundant thermal particles at higher tempera-ture. Since j3 depends solely on µ5, a higher temperaturerequires a larger n5 to give the same j3.Besides j3, another interesting quantity is the current

susceptibility defined by

χJ = 〈j23〉 − 〈j3〉2 = −1

βV

∂2Ω

∂A23

∣

∣

∣

∣

A3=0

. (23)

If we naively use the above definition (23) for the cut-off model like the PNJL model, χJ is non-zero propor-tional to Λ2 even at T = B = 0 as discussed in Ref. [27].

Μ5=200 MeV

T=0T=0.95 TcT=1.10 Tc

0 10 20 30 40-0.002

0.000

0.002

0.004

0.006

0.008

0.010

eBmΠ2

ΒVΧ

JHG

eV2L

FIG. 8. Subtracted current susceptibility with a smootherregulator with N = 5 in Eq. (12).

This is in contradiction with the gauge invariance, whichrequires the above susceptibility to vanish because thecurrent susceptibility is nothing but the 33-componentof the photon polarization tensor at zero momentum.Therefore, in order to deal with the physically meaning-ful quantity, we subtract the vacuum part from the aboveequation and compute,

χ̄J = χJ(µ5, B, T )− χJ (µ5, 0, 0) . (24)

to fulfill the requirement that photons are unscreened atT = B = 0 regardless of any value of µ5.We plot our results for βV χ̄J as a function of eB in

Fig. 7 at µ5 = 0 (upper panel), µ5 = 200 MeV (middlepanel), and µ5 = 400 MeV (lower panel). The oscilla-tions in the susceptibility behavior are an artifact of themomentum cutoff. In these plots Tc = 228 MeV denotesthe critical temperature for chiral symmetry restorationat µ5 = B = 0.Let us first focus on the case at µ5 = 0. At T = 0

and T = 0.95Tc the system is in the broken phase with〈ūu〉 6= 0 over the whole range of eB. On the otherhand, at the temperature T = 1.1Tc, the system is inthe chiral symmetric phase for eB smaller than a criticalvalue. There is a phase transition from the symmetric tothe broken phase with increasing eB. This transition isdriven by the presence of the magnetic field as the catal-ysis of chiral symmetry breaking, as mentioned before.The effect of the phase transition leads to a cusp in thesusceptibility χJ as a function of eB. We also notice thatthere seems to exist a range in eB in which χ̄J < 0. Thisis a mere artifact of the momentum cutoff, which causesunphysical fluctuations in χ̄J . The qualitative picture issimilar also at µ5 6= 0.To distinguish physically meaningful information from

cutoff artifacts, we have computed χJ using a smootherregulator with N = 5 in Eq. (12). We have readjustedthe NJL parameters to keep the physical quantities (fπand 〈ūu〉) unchanged. Figure 8 is the result in whichoscillations are suppressed and χJ > 0 for any T and B.

11

In view of Figs. 7 and 8 we can conclude that it iscertainly a physical effect that the chiral phase transi-tion critically affects the susceptibility χJ as well as χ5.As shown in Ref. [27] the susceptibility difference be-tween the longitudinal and transverse directions has anorigin in the axial anomaly and is insensitive to the in-frared information. Nevertheless, χJ (and transverse χ

TJ

too) should be largely enhanced near the chiral phasetransition through mixing with the divergent chiral sus-ceptibility, which is not constrained by anomaly. Suchenhancement in χJ would ease the confirmation of theCME signals at experiment.

V. CONCLUSIONS

In this article we have considered several aspects re-lated to the response of quark matter to an externalmagnetic field. Quark matter has been modelled by thePolyakov extended version of the Nambu–Jona-Lasinio(PNJL) model, in which the QCD interaction amongquarks is replaced by effective four-fermion interactions.In the PNJL model, besides the quark-antiquark con-densate which is responsible for the dynamical chiralsymmetry breaking in the QCD vacuum, it is possibleto compute the expectation value of the Polyakov loop,which is a relevant indicator for the quark deconfinementcrossover.In our study, we have firstly focused on the effect of a

strong magnetic field on chiral symmetry restoration atfinite temperature. Our results show the effect of the ex-ternal field as a catalyzer of dynamical symmetry break-ing. Moreover, the critical temperature increases as thestrength of B is increased. This behavior is in agree-

ment with the previous studies on magnetic catalysis inNJL-like models.

We have also discussed the effects of a chiral chemicalpotential, µ5, on the phase structure of the model. Thechiral chemical potential mimics the chirality induced bytopological excitations according to the QCD anomalyrelation. Instead of working at fixed chirality N5, wehave worked in the grand-canonical ensemble introducingµ5, i.e. the chemical potential conjugate to N5. Besidesthe phase diagram of the model, summarized in Fig. 3,we have computed several quantities that are relevant forthe Chiral Magnetic Effect (CME). That is, we have com-puted the current density j3 and its susceptibility χJ aswell as the chiral charge density n5 and its susceptibilityχ5.

As a future project it is indispensable to extend ouranalysis to the 2 + 1 flavors and tune the PNJL modelparameters to reproduce the correct Tc and thermo-dynamic properties, which would enable us to makea serious comparison with the dynamical lattice-QCDdata [29], and furthermore, it would be possible to givemore pertinent prediction on the physical observables.

ACKNOWLEDGMENTS

M. R. acknowledges discussions with H. Abuki andS. Nicotri and K. F. thanks E. Fraga for discussions. Thework of M. R. is supported by JSPS under the contractnumber P09028. K. F. is supported by Japanese MEXTgrant No. 20740134 and also supported in part by YukawaInternational Program for Quark Hadron Sciences. Thenumerical calculations were carried out on Altix3700 BX2at YITP in Kyoto University.

[1] P. de Forcrand and O. Philipsen, JHEP 0701, 077(2007) [arXiv:hep-lat/0607017]; JHEP 0811, 012 (2008)[arXiv:0808.1096 [hep-lat]]; PoS LATTICE2008, 208(2008) [arXiv:0811.3858 [hep-lat]].

[2] Y. Aoki, Z. Fodor, S. D. Katz and K. K. Szabo, Phys.Lett. B 643, 46 (2006) [arXiv:hep-lat/0609068]; Y. Aoki,S. Borsanyi, S. Durr, Z. Fodor, S. D. Katz, S. Krieg andK. K. Szabo, JHEP 0906, 088 (2009) [arXiv:0903.4155[hep-lat]].

[3] A. Bazavov et al., Phys. Rev. D 80, 014504 (2009)[arXiv:0903.4379 [hep-lat]].

[4] M. Cheng et al., arXiv:0911.3450 [hep-lat].[5] S. Muroya, A. Nakamura, C. Nonaka and

T. Takaishi, Prog. Theor. Phys. 110, 615 (2003)[arXiv:hep-lat/0306031]; K. Splittorff, PoS LAT2006,023 (2006) [arXiv:hep-lat/0610072]; K. Splittorff andJ. J. M. Verbaarschot, Phys. Rev. Lett. 98, 031601 (2007)[arXiv:hep-lat/0609076]; K. Fukushima and Y. Hidaka,Phys. Rev. D 75, 036002 (2007) [arXiv:hep-ph/0610323];J. C. R. Bloch and T. Wettig, JHEP 0903, 100 (2009)[arXiv:0812.0324 [hep-lat]].

[6] Z. Fodor and S. D. Katz, JHEP 0203, 014 (2002)[arXiv:hep-lat/0106002].

[7] C. R. Allton et al., Phys. Rev. D 71, 054508 (2005)[arXiv:hep-lat/0501030]; R. V. Gavai and S. Gupta,Phys. Rev. D 78, 114503 (2008) [arXiv:0806.2233 [hep-lat]].

[8] J. Ambjorn, K. N. Anagnostopoulos, J. Nishimura andJ. J. M. Verbaarschot, JHEP 0210, 062 (2002)[arXiv:hep-lat/0208025]; Z. Fodor, S. D. Katzand C. Schmidt, JHEP 0703, 121 (2007)[arXiv:hep-lat/0701022].

[9] M. D’Elia and M. P. Lombardo, Phys. Rev. D 67,014505 (2003) [arXiv:hep-lat/0209146]; Phys. Rev. D70, 074509 (2004) [arXiv:hep-lat/0406012]; M. D’Elia,F. Di Renzo and M. P. Lombardo, Phys. Rev. D 76,114509 (2007) [arXiv:0705.3814 [hep-lat]]; P. de For-crand and O. Philipsen, Nucl. Phys. B 673, 170 (2003)[arXiv:hep-lat/0307020].

[10] G. Aarts, Phys. Rev. Lett. 102, 131601 (2009)[arXiv:0810.2089 [hep-lat]].

[11] D. E. Kharzeev, L. D. McLerran and H. J. Warringa,Nucl. Phys. A 803, 227 (2008) [arXiv:0711.0950 [hep-

http://arxiv.org/abs/hep-lat/0607017http://arxiv.org/abs/0808.1096http://arxiv.org/abs/0811.3858http://arxiv.org/abs/hep-lat/0609068http://arxiv.org/abs/0903.4155http://arxiv.org/abs/0903.4379http://arxiv.org/abs/0911.3450http://arxiv.org/abs/hep-lat/0306031http://arxiv.org/abs/hep-lat/0610072http://arxiv.org/abs/hep-lat/0609076http://arxiv.org/abs/hep-ph/0610323http://arxiv.org/abs/0812.0324http://arxiv.org/abs/hep-lat/0106002http://arxiv.org/abs/hep-lat/0501030http://arxiv.org/abs/0806.2233http://arxiv.org/abs/hep-lat/0208025http://arxiv.org/abs/hep-lat/0701022http://arxiv.org/abs/hep-lat/0209146http://arxiv.org/abs/hep-lat/0406012http://arxiv.org/abs/0705.3814http://arxiv.org/abs/hep-lat/0307020http://arxiv.org/abs/0810.2089http://arxiv.org/abs/0711.0950

12

ph]].[12] V. Skokov, A. Y. Illarionov and V. Toneev, Int. J. Mod.

Phys. A 24, 5925 (2009) [arXiv:0907.1396 [nucl-th]].[13] S. P. Klevansky and R. H. Lemmer, Phys. Rev. D 39,

3478 (1989); H. Suganuma and T. Tatsumi, Annals Phys.208, 470 (1991); I. A. Shushpanov and A. V. Smilga,Phys. Lett. B 402, 351 (1997) [arXiv:hep-ph/9703201];D. N. Kabat, K. M. Lee and E. J. Weinberg, Phys. Rev.D 66, 014004 (2002) [arXiv:hep-ph/0204120]; T. Ina-gaki, D. Kimura and T. Murata, Prog. Theor. Phys.111, 371 (2004) [arXiv:hep-ph/0312005]; T. D. Cohen,D. A. McGady and E. S. Werbos, Phys. Rev. C 76,055201 (2007) [arXiv:0706.3208 [hep-ph]]; K. Fukushimaand H. J. Warringa, Phys. Rev. Lett. 100, 032007(2008) [arXiv:0707.3785 [hep-ph]]; J. L. Noronha andI. A. Shovkovy, Phys. Rev. D 76, 105030 (2007)[arXiv:0708.0307 [hep-ph]].

[14] V. P. Gusynin, V. A. Miransky and I. A. Shovkovy,Nucl. Phys. B 462, 249 (1996) [arXiv:hep-ph/9509320];Nucl. Phys. B 563, 361 (1999) [arXiv:hep-ph/9908320];G. W. Semenoff, I. A. Shovkovy and L. C. R. Wi-jewardhana, Phys. Rev. D 60, 105024 (1999)[arXiv:hep-th/9905116]; V. A. Miransky andI. A. Shovkovy, Phys. Rev. D 66, 045006 (2002)[arXiv:hep-ph/0205348].

[15] E. S. Fraga and A. J. Mizher, Phys. Rev. D 78, 025016(2008) [arXiv:0804.1452 [hep-ph]]; Nucl. Phys. A 831, 91(2009) [arXiv:0810.5162 [hep-ph]]; J. K. Boomsma andD. Boer, arXiv:0911.2164 [hep-ph].

[16] K. G. Klimenko, Theor. Math. Phys. 89, 1161 (1992)[Teor. Mat. Fiz. 89, 211 (1991)]; K. G. Klimenko, Z.Phys. C 54, 323 (1992); K. G. Klimenko, Theor. Math.Phys. 90, 1 (1992) [Teor. Mat. Fiz. 90, 3 (1992)].

[17] A. V. Zayakin, JHEP 0807, 116 (2008) [arXiv:0807.2917[hep-th]]; G. Lifschytz and M. Lippert, Phys. Rev.D 80, 066005 (2009) [arXiv:0904.4772 [hep-th]]; Phys.Rev. D 80, 066007 (2009) [arXiv:0906.3892 [hep-th]];H. U. Yee, JHEP 0911, 085 (2009) [arXiv:0908.4189[hep-th]]; B. Sahoo and H. U. Yee, arXiv:0910.5915[hep-th]; A. Rebhan, A. Schmitt and S. A. Stricker,arXiv:0909.4782 [hep-th]; S. l. Cui, Y. h. Gao, Y. Seo,S. j. Sin and W. s. Xu, arXiv:0910.2661 [hep-th];E. D’Hoker and P. Kraus, arXiv:0911.4518 [hep-th].

[18] E. V. Gorbar, S. Homayouni and V. A. Miransky, Phys.Rev. D 72, 065014 (2005) [arXiv:hep-th/0503028].

[19] K. Fukushima, D. E. Kharzeev and H. J. Warringa, Phys.Rev. D 78, 074033 (2008) [arXiv:0808.3382 [hep-ph]].

[20] P. Arnold and L. D. McLerran, Phys. Rev. D 37, 1020(1988).

[21] G. D. Moore, Phys. Lett. B 412, 359 (1997)[arXiv:hep-ph/9705248]; arXiv:hep-ph/0009161;D. Bodeker, G. D. Moore and K. Rummukainen,Phys. Rev. D 61, 056003 (2000) [arXiv:hep-ph/9907545].

[22] L. D. McLerran, E. Mottola and M. E. Shaposhnikov,Phys. Rev. D 43, 2027 (1991).

[23] S. A. Voloshin, Phys. Rev. C 70, 057901 (2004)[arXiv:hep-ph/0406311].

[24] B. I. Abelev et al. [STAR Collaboration],arXiv:0909.1717 [nucl-ex]; Phys. Rev. Lett. 103,251601 (2009) [arXiv:0909.1739 [nucl-ex]].

[25] S. i. Nam, Phys. Rev. D 80, 114025 (2009)[arXiv:0911.0509 [hep-ph]].

[26] D. E. Kharzeev and H. J. Warringa, Phys. Rev. D 80,034028 (2009) [arXiv:0907.5007 [hep-ph]].

[27] K. Fukushima, D. E. Kharzeev and H. J. Warringa,arXiv:0912.2961 [hep-ph].

[28] P. V. Buividovich, M. N. Chernodub, E. V. Luschevskayaand M. I. Polikarpov, Phys. Rev. D 80, 054503 (2009)[arXiv:0907.0494 [hep-lat]].

[29] M. Abramczyk, T. Blum, G. Petropoulos and R. Zhou,arXiv:0911.1348 [hep-lat].

[30] P. N. Meisinger and M. C. Ogilvie, Phys. Lett. B 379,163 (1996) [arXiv:hep-lat/9512011].

[31] K. Fukushima, Phys. Lett. B 591, 277 (2004)[arXiv:hep-ph/0310121].

[32] U. Vogl and W. Weise, Prog. Part. Nucl. Phys. 27, 195(1991); S. P. Klevansky, Rev. Mod. Phys. 64, 649 (1992);T. Hatsuda and T. Kunihiro, Phys. Rept. 247, 221 (1994)[arXiv:hep-ph/9401310]; M. Buballa, Phys. Rept. 407,205 (2005) [arXiv:hep-ph/0402234].

[33] A. M. Polyakov, Phys. Lett. B 72, 477 (1978);L. Susskind, Phys. Rev. D 20, 2610 (1979); B. Svetitskyand L. G. Yaffe, Nucl. Phys. B 210, 423 (1982); B. Svetit-sky, Phys. Rept. 132, 1 (1986).

[34] C. Ratti, M. A. Thaler and W. Weise, Phys. Rev. D73, 014019 (2006) [arXiv:hep-ph/0506234]; E. Megias,E. Ruiz Arriola and L. L. Salcedo, Phys. Rev. D 74,114014 (2006) [arXiv:hep-ph/0607338]; Eur. Phys. J.A 31, 553 (2007) [arXiv:hep-ph/0610163]; S. Roess-ner, C. Ratti and W. Weise, Phys. Rev. D 75, 034007(2007) [arXiv:hep-ph/0609281]; C. Sasaki, B. Frimanand K. Redlich, Phys. Rev. D 75, 074013 (2007)[arXiv:hep-ph/0611147]; S. K. Ghosh, T. K. Mukher-jee, M. G. Mustafa and R. Ray, Phys. Rev. D 77,094024 (2008) [arXiv:0710.2790 [hep-ph]]; W. j. Fu,Z. Zhang and Y. x. Liu, Phys. Rev. D 77, 014006 (2008)[arXiv:0711.0154 [hep-ph]]; M. Ciminale, R. Gatto,N. D. Ippolito, G. Nardulli and M. Ruggieri, Phys. Rev.D 77, 054023 (2008) [arXiv:0711.3397 [hep-ph]]; Y. Sakai,K. Kashiwa, H. Kouno and M. Yahiro, Phys. Rev.D 77, 051901 (2008) [arXiv:0801.0034 [hep-ph]]; Phys.Rev. D 78, 036001 (2008) [arXiv:0803.1902 [hep-ph]];K. Kashiwa, H. Kouno and M. Yahiro, arXiv:0908.1213[hep-ph]; K. Fukushima, Phys. Rev. D 77, 114028 (2008)[Erratum-ibid. D 78, 039902 (2008)] [arXiv:0803.3318[hep-ph]]; H. Abuki, R. Anglani, R. Gatto, G. Nar-dulli and M. Ruggieri, Phys. Rev. D 78, 034034 (2008)[arXiv:0805.1509 [hep-ph]]; H. Abuki, M. Ciminale,R. Gatto and M. Ruggieri, Phys. Rev. D 79, 034021(2009) [arXiv:0811.1512 [hep-ph]]; T. Hell, S. Roessner,M. Cristoforetti and W. Weise, Phys. Rev. D 79, 014022(2009) [arXiv:0810.1099 [hep-ph]]; arXiv:0911.3510 [hep-ph].

[35] B. J. Schaefer, J. M. Pawlowski and J. Wambach, Phys.Rev. D 76, 074023 (2007) [arXiv:0704.3234 [hep-ph]].

[36] J. Braun, L. M. Haas, F. Marhauser and J. M. Pawlowski,arXiv:0908.0008 [hep-ph].

[37] N. O. Agasian and S. M. Fedorov, Phys. Lett. B 663, 445(2008) [arXiv:0803.3156 [hep-ph]].

[38] K. Fukushima, D. E. Kharzeev and H. J. Warringa,arXiv:1002.2495 [hep-ph].

[39] J. Wambach, B. J. Schaefer and M. Wagner,arXiv:0911.0296 [hep-ph].

[40] V. I. Ritus, Annals Phys. 69, 555 (1972); Sov. Phys.JETP 48, 788 (1978) [Zh. Eksp. Teor. Fiz. 75, 1560(1978)].

[41] L. Campanelli and M. Ruggieri, Phys. Rev. D 80, 034014(2009) [arXiv:0905.0853 [hep-ph]].

http://arxiv.org/abs/0907.1396http://arxiv.org/abs/hep-ph/9703201http://arxiv.org/abs/hep-ph/0204120http://arxiv.org/abs/hep-ph/0312005http://arxiv.org/abs/0706.3208http://arxiv.org/abs/0707.3785http://arxiv.org/abs/0708.0307http://arxiv.org/abs/hep-ph/9509320http://arxiv.org/abs/hep-ph/9908320http://arxiv.org/abs/hep-th/9905116http://arxiv.org/abs/hep-ph/0205348http://arxiv.org/abs/0804.1452http://arxiv.org/abs/0810.5162http://arxiv.org/abs/0911.2164http://arxiv.org/abs/0807.2917http://arxiv.org/abs/0904.4772http://arxiv.org/abs/0906.3892http://arxiv.org/abs/0908.4189http://arxiv.org/abs/0910.5915http://arxiv.org/abs/0909.4782http://arxiv.org/abs/0910.2661http://arxiv.org/abs/0911.4518http://arxiv.org/abs/hep-th/0503028http://arxiv.org/abs/0808.3382http://arxiv.org/abs/hep-ph/9705248http://arxiv.org/abs/hep-ph/0009161http://arxiv.org/abs/hep-ph/9907545http://arxiv.org/abs/hep-ph/0406311http://arxiv.org/abs/0909.1717http://arxiv.org/abs/0909.1739http://arxiv.org/abs/0911.0509http://arxiv.org/abs/0907.5007http://arxiv.org/abs/0912.2961http://arxiv.org/abs/0907.0494http://arxiv.org/abs/0911.1348http://arxiv.org/abs/hep-lat/9512011http://arxiv.org/abs/hep-ph/0310121http://arxiv.org/abs/hep-ph/9401310http://arxiv.org/abs/hep-ph/0402234http://arxiv.org/abs/hep-ph/0506234http://arxiv.org/abs/hep-ph/0607338http://arxiv.org/abs/hep-ph/0610163http://arxiv.org/abs/hep-ph/0609281http://arxiv.org/abs/hep-ph/0611147http://arxiv.org/abs/0710.2790http://arxiv.org/abs/0711.0154http://arxiv.org/abs/0711.3397http://arxiv.org/abs/0801.0034http://arxiv.org/abs/0803.1902http://arxiv.org/abs/0908.1213http://arxiv.org/abs/0803.3318http://arxiv.org/abs/0805.1509http://arxiv.org/abs/0811.1512http://arxiv.org/abs/0810.1099http://arxiv.org/abs/0911.3510http://arxiv.org/abs/0704.3234http://arxiv.org/abs/0908.0008http://arxiv.org/abs/0803.3156http://arxiv.org/abs/1002.2495http://arxiv.org/abs/0911.0296http://arxiv.org/abs/0905.0853