-
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-
∗ [email protected]† [email protected]‡
[email protected]
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:[email protected]:[email protected]:[email protected]
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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.
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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
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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.
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