PHYSICAL REVIEW C 90, 025203 (2014)

Importance of asymptotic freedom for the pseudocritical temperature in magnetized quark matter

R. L. S. Farias,1,2,* K. P. Gomes,1,† G. Krein,3,‡ and M. B. Pinto4,§

1Departamento de Ciências Naturais, Universidade Federal de São João Del Rei, 36301-000, São João Del Rei, MG, Brazil
2Departamento de Fı́sica, Universidade Federal de Santa Maria, 97105-900, Santa Maria, RS, Brazil

3Instituto de Fı́sica Teórica, Universidade Estadual Paulista, Rua Dr. Bento Teobaldo Ferraz, 271 - Bloco II, 01140-070 São Paulo, SP, Brazil
4Departamento de Fı́sica, Universidade Federal de Santa Catarina, 88040-900 Florianópolis, Santa Catarina, Brazil

(Received 23 April 2014; revised manuscript received 17 June 2014; published 11 August 2014)

Although asymptotic freedom is an essential feature of QCD, it is absent in effective chiral quark models like
the Nambu–Jona-Lasinio and linear sigma models. In this work we advocate that asymptotic freedom plays a
key role in the recently observed discrepancies between results of lattice QCD simulations and quark models
regarding the behavior of the pseudocritical temperature Tpc for chiral-symmetry restoration in the presence of
a magnetic field B. We show that the lattice predictions that Tpc decreases with B can be reproduced within
the Nambu–Jona-Lasinio model if the coupling constant G of the model decreases with B and the temperature.
Without aiming at numerical precision, we support our claim by considering a simple ansatz for G that mimics
the asymptotic-freedom behavior of the QCD coupling constant 1/αs ∼ ln(eB/�2

QCD) for large values of B.

DOI: 10.1103/PhysRevC.90.025203 PACS number(s): 21.65.Qr, 25.75.Nq, 11.30.Rd, 11.10.Wx

I. INTRODUCTION

The investigation of the effects produced by a magnetic
field in the phase diagram of strongly interacting matter
became a subject of great interest in recent years. The recent
motivation stems mainly from the fact that strong magnetic
fields may be produced in noncentral heavy-ion collisions [1–
3]—for an updated discussion, see Ref. [4]. Strong magnetic
fields are also present in magnetars [5,6] and might have played
an important role in the physics of the early universe [7].
At vanishing baryon density and magnetic field, lattice QCD
simulations [8,9] predict that there is a crossover transition at a
pseudocritical temperature Tpc. More recent lattice simulations
[10–13] indicate that such a crossover persists in the presence
of a magnetic field and, in addition, confirm predictions made
a few years ago [14–16] of the phenomenon of magnetic
catalysis (MC), in that the quark condensate is enhanced in the
presence of the magnetic field at vanishing temperatures. These
lattice results also lend support to calculations using effective
models, e.g., those of Refs. [17–21]. There are, however,
marked disagreements between lattice results and model
calculations regarding the dependence of Tpc on the strength
B of the magnetic field. Specifically, the lattice results of
Refs. [12,13], performed with 2 + 1 quark flavors and physical
pion-mass values, predict an inverse catalysis, in that Tpc

decreases with B, while effective models predict an increase
of Tpc with B—an exception is a calculation [22] based on the
MIT bag model which, on the other hand, misses the crossover
nature of the transition. Earlier model evaluations have been
mainly performed considering the two flavor Nambu–Jona-
Lasinio model (NJL) [23] and the linear sigma model [17]
within the mean-field approximation (MFA). Subsequently,

*ricardofarias@ufsj.edu.br
†karinaponcianogomes@gmail.com
‡gkrein@ift.unesp.br
§marcus.benghi@ufsc.br

these evaluations have been considered at more sophisticated
levels by coupling degrees of freedom related to the Polyakov
loop (PNJL models) [18] and including strangeness [19].
Calculations have also performed beyond the MFA by using
the functional renormalization group [24,25]. Nevertheless,
despite those refinements in model calculations, no qualitative
changes in the behavior of Tpc with B have been observed.

Possible explanations for the disagreement concerning the
dependence of Tpc on B have been recently given in Refs. [26–
29]. A particularly insightful discussion is the one of Ref. [28]:
the authors of this reference argue that the inverse catalysis
is the result of the back reaction of the gluons due to the
coupling of the magnetic field to the sea quarks. Such a back
reaction is naturally implemented in the so-called entangled
PNJL model [30,31], in that the four-quark coupling of the
NJL model is made dependent on the Polyakov loop. Although
such a model is also unable [32] to produce inverse catalysis,
a very recent study [33] has shown that, if the lattice data
[13] are fit by making the pure-gauge critical temperature T0,
a parameter of PNJL models, depend on B, the model gives
inverse catalysis. A similar recent study [34] in the context of a
quark-meson (QM) model has concluded that the only way to
get inverse catalysis would be to let the Yukawa quark-meson
coupling increase with B, but no qualitative agreement with
lattice simulations for the phase diagram Tpc(B) is obtained.
Moreover, the situation does not improve when this model
is extended with the Polyakov loop (PQM) and the model
parameter T0 is made B dependent like in Ref. [33].

From this discussion, it is apparent that the failure of
effective models in providing inverse catalysis can be traced
to a more fundamental level by recalling that the couplings in
those models do not run with the magnetic field, in contrast
with QCD whose coupling, as Miransky and Shovkovy have
shown [35], decreases for large B in a manifestation of the
asymptotic-freedom phenomenon. Therefore, in the present
work we make use of an ansatz that makes G a running
coupling with B and T , very much like the strong-coupling
runs in QCD. As a result, we will show that, at T = 0, the

0556-2813/2014/90(2)/025203(6) 025203-1 ©2014 American Physical Society

http://dx.doi.org/10.1103/PhysRevC.90.025203


FARIAS, GOMES, KREIN, AND PINTO PHYSICAL REVIEW C 90, 025203 (2014)

model realizes magnetic catalysis, and at T �= 0, it realizes
inverse catalysis, in qualitative agreement with the lattice
simulations of Refs. [12,13]. For zero magnetic field, the idea
of using an ansatz to mimic high-temperature lattice results for
the effective QCD coupling within the NJL model is not new;
it was implemented in Ref. [36] to study the decoupling of
the pion at high temperatures. A similar ansatz was used very
recently in Ref. [37] in a study of the quark spectral density
in a strongly coupled quark-gluon plasma in the context of the
QCD Dyson–Schwinger equations at finite temperature and
density—for a review, see Ref. [38].

Our motivation is easily understood by recalling that, in the
NJL model, the quark mass gap and the quark condensate are
related by M ∼ −G〈ψ̄f ψf 〉, where 〈ψ̄f ψf 〉 grows with B, as
a manifestation of MC. When G does not run with B and T ,
the quark mass gap and Tpc also grow with B. One way to
circumvent this problem has been recently suggested by Kojo
and Su [27], who consider an effective interaction with infrared
enhancement and ultraviolet suppression. Our proposal is that
basically the same effect can be obtained by assuming that
the NJL quark-quark coupling decreases with B and T , as
does the QCD coupling. As a result, the NJL model results
turn out to be in qualitative agreement with the recent lattice
results; the condensate grows in accordance with MC and the
pseudocritical temperature for the chiral transition decreases
with B. Although this agreement comes out in a particular
model implementation, it lends support to to the suggestion
that the inverse catalysis is the result of the back reaction of
the sea quarks to the magnetic field [28].

On a wider perspective, the motivation for tuning highly
simplified models of the NJL type to reproduce the decreas-
ing of Tpc with B obtained from much more fundamental
nonperturbative lattice QCD calculations comes within the
same spirit as in previous studies using such models. The
situation here is similar to the calculation of the hadron
masses using lattice QCD simulations: while presently correct
numbers for the masses are generated by the simulations,
there is no information on the physical mechanisms that
generate those numbers from gluon and (essentially massless)
quark fields. In this respect, the NJL model, in particular,
provides physical insight into the mechanism behind mass
generation via dynamical chiral-symmetry breaking. In the
present context, one is able to understand the decreasing of
Tpc with B by using a magnetic-field-induced running of G,

which is interpreted as being due to the indirect coupling of
B to the gluons whose physics is similar to that of asymptotic
freedom.

The paper is organized as follows: In the next section we
discuss the quark condensate considering the two-flavor NJL
model for hot and magnetized quark matter in the MFA. The
running of the coupling G is introduced in the same section.
Then, in Sec. III, we discuss our numerical results while our
conclusions and perspectives are presented in Sec. IV.

II. RUNNING COUPLING IN MAGNETIZED
NAMBU–JONA-LASINIO MODEL

The standard two flavor NJL model is defined by a fermionic
Lagrangian density given by [39]

LNJL = ψ̄(i/∂ − m)ψ + G[(ψ̄ψ)2 − (ψ̄γ5 �τψ)2], (2.1)

where ψ represents a flavor iso-doublet (u and d flavors) and
Nc-plet of quark fields, while �τ are isospin Pauli matrices—a
sum over flavor and color degrees of freedom is implicit. The
Lagrangian density in Eq. (2.1) is invariant under (global)
U(2)f × SU(Nc) and, when m = 0, the theory is also invariant
under chiral SU(2)L × SU(2)R . Within the NJL model a sharp
cutoff � is generally used as an ultraviolet regulator and, since
the model is not renormalizable, one has to fix � to physical
quantities. The phenomenological values of quantities such as
the pion mass mπ , the pion decay constant fπ , and the quark
condensate 〈ψ̄f ψf 〉 are used to fix G, �, and m. Here, we
choose the set � = 650 MeV and G = 5.022 GeV−2 with
m = 5.5 MeV in order to reproduce fπ = 93 MeV, mπ =
140 MeV, and 〈ψ̄f ψf 〉1/3 = −250 MeV in the vacuum.

The expressions for the thermodynamic potential of the
model in the MFA are well documented in the literature; see,
e.g., Refs. [23,40]. For results beyond the MFA, see Ref. [41].
Thus, we refrain from repeating them here and start with the
expression for the gap equation [21]:

Mf = mf − 2G
∑
f

〈ψ̄f ψf 〉, (2.2)

where 〈ψ̄f ψf 〉 represents the quark condensate of flavor f

〈ψ̄f ψf 〉 = −NcMf

2π2

⎧⎨
⎩�

√
�2 + M2

f − M2
f

2
ln

⎡
⎣

(
� +

√
�2 + M2

f

)2

M2
f

⎤
⎦

⎫⎬
⎭

− NcMf

2π2
|qf |B

{
ln[	(xf )] − 1

2
ln(2π ) + xf − 1

2
(2xf − 1) ln(xf )

}

+ NcMf

2π2

∞∑
k=0

αk|qf |B
∫ ∞

−∞

dpz

Ep,k(B)

{
1

e[Ep,k(B)]/T + 1

}
, (2.3)

where Ep, k(B) = (p2
z + 2k|qf |B + M2

f )1/2, xf = M2
f /(2|qf |

B), and αk = 2 − δ0k . In addition, |qf | is the absolute value of
the quark electric charge; |qu| = 2e/3, |qd | = e/3, with e =
1/

√
137 representing the electron charge—we use Gaussian

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IMPORTANCE OF ASYMPTOTIC FREEDOM FOR THE . . . PHYSICAL REVIEW C 90, 025203 (2014)

natural units where 1 GeV2 = 1.44 × 1019 G. Note also that
here we have taken the chemical equilibrium condition by
setting μu = μd = μ. Details of the manipulations leading to
this equation can be found in the appendix of Ref. [23]. Note
that the condensates for the flavors u and d are different due to
their different electric charges. Remark also that, in principle,
one should have two coupled gap equations for the two distinct
flavors: Mu = mu − 2G(〈ψ̄uψu〉 + 〈ψ̄dψd〉) and Md = md −
2G(〈ψ̄uψu〉 + 〈ψ̄dψd〉). However, in the two-flavor case, the
different condensates contribute to Mu and Md in a symmetric
way and, since mu = md = m, one can write Mu = Md = M .

Reference [13] presents results for (�u + �d )/2 and �u −
�d , where �f = �f (B,T ) is defined as

�f (B,T ) = 2mf

m2
πf 2

π

[〈ψ̄f ψf 〉 − 〈ψ̄f ψf 〉0] + 1, (2.4)

with 〈ψ̄f ψf 〉0 being the quark condensate at T = 0 and B = 0.
Let us recall the important result by Miransky and

Shovkovy [35] that, for sufficiently strong magnetic fields,
eB 	 �2

QCD, the leading-order running of the QCD coupling
constant αs is given by

1

αs

∼ b ln
eB

�2
QCD

, (2.5)

where b = (11Nc − 2Nf )/(12π ), and the energy scale
√

eB
is fixed up to a factor of order 1. Motivated by this result,
we propose for the NJL coupling, at T = 0, the interpolating
formula

G(B) = G0

1 + α ln
(
1 + β eB

�2
QCD

) , (2.6)

with G0 = 5.022 GeV−2, which is the value of the coupling
at B = 0. The free parameters α and β are fixed to obtain a
reasonable description of the lattice average (�u + �d )/2 for
T = 0. In principle, for the values of B presently considered
in lattice simulations, there is no reason for G(B) to have
a logarithm-running like the large-B running of αs(B). The
proposed parametrization for the running of G with B is mo-
tivated, primarily, by the assumption that similar physics that
drives asymptotic freedom is also responsible for the decrease
of the effective coupling with the magnetic field; namely, the
back reaction of gluons via the coupling of the magnetic
field on the sea quarks. In addition, although the precise B
dependence of the effective coupling for values of eB not much
larger that �2

QCD is not known, the proposed parametrization
involves a minimum number of fitting parameters, extrapolates
smoothly to the large-B running of the quark-gluon running in
QCD, and can be tested when larger values of B will be used
in lattice simulations. As we will show in the following, only
a decrease (not necessarily in a logarithmic way) of G with B
can account for the recent lattice results at T = 0.

At high temperatures, αs also runs as the inverse of
ln(T/�QCD). However, the values of T used in the lattice
simulations of Refs. [12,13], T � �QCD, are not high enough
to justify the use of such a running for G. Since the explicit
form in which αs runs with B and T is presently not known,

eB 1.0 GeV 2

eB 0.8 GeV 2

eB 0.6 GeV 2

eB 0.4 GeV 2

eB 0.2 GeV 2

eB 0.0 GeV 2

0 50 100 150 200
0.0

0.5

1.0

1.5

2.0

2.5

T MeV

u
d

2

FIG. 1. (Color online) The condensate average as a function of
the temperature. Data points are from the lattice simulations of
Ref. [13].

we consider a T dependence for G as

G (B,T ) = G (B)

(
1 − γ

|eB|
�2

QCD

T

�QCD

)
, (2.7)

where γ is fixed to obtain a reasonable description of the
temperature dependence of the lattice average (�u + �d )/2 at
the highest temperatures. Notice that terms proportional to T ,
T 2, (BT )2, . . . could be considered; however, it turned out
that they are not needed to obtain a reasonable fit to the lattice
data for (�u + �d )/2. It should be clear that, for the values of
B used in the lattice simulations, any function G(B) that gives
an effective coupling that decreases with B can fit the lattice
data for (�u + �d )/2 for the magnetic-field values used in the
simulations if a sufficient number of free parameters are used.

III. NUMERICAL RESULTS

Before presenting our results, we note that Refs. [12,13]
used mu = md = 5.5 MeV, mπ = 135 MeV, and fπ =
86 MeV in the multiplicative factor mf /(m2

πf 2
π ) in Eq. (2.4) to

make �f (B,T ) dimensionless. The fact that these values differ
from the ones we use is of no significance for comparison
purposes since they only set a general scale. We also consider
�QCD = 200 MeV.

Figure 1 displays the condensate average (�u + �d )/2.
We obtain a good fit of the data at T = 0 with α = 2 and
β = 0.000 327 in Eq. (2.6). Figure 1 reveals that the magnetic
catalysis is naturally reproduced at T = 0. In addition, the
T dependence is also reasonably well reproduced—here we
used γ = 0.0175 in Eq. (2.7). Having fixed our parameters,
we proceed in the analyses of other quantities.

The difference �u − �d is displayed in Fig. 2. Although
there is a small deviation from the lattice results for the highest
values of B, the overall agreement is quite impressive, given
the simplicity of the model.

In this work we consider the physical point with nonzero
current quark masses so that, at high temperatures, the
model displays a crossover where chiral symmetry is partially
restored. In this case, one can only establish a pseudocritical
temperature which depends on the observable used to define

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FARIAS, GOMES, KREIN, AND PINTO PHYSICAL REVIEW C 90, 025203 (2014)

eB 1.0 GeV 2

eB 0.8 GeV 2

eB 0.6 GeV 2

eB 0.4 GeV 2

eB 0.2 GeV 2

0 50 100 150 200
0.0

0.2

0.4

0.6

0.8

1.0

1.2

T MeV

u
d

FIG. 2. (Color online) The condensate difference as a function of
temperature. Data points are from the lattice simulations of Ref. [13].

it. Here, we use the location of the peaks for the vacuum nor-
malized quark condensates, where the thermal susceptibilities
are given by

χT = −mπ

∂σ

∂T
, (3.1)

with σ being defined by

σ = 〈ψ̄uψu〉(B,T ) + 〈ψ̄dψd〉(B,T )

〈ψ̄uψu〉(B,0) + 〈ψ̄dψd〉(B,0)
. (3.2)

In Fig. 3 we plot the thermal susceptibility defined by
Eq. (3.1) as a function of the temperature for different values
of the magnetic field. The figure clearly indicates the decrease
of Tpc for increasing values of the magnetic field. Again, the
effects of asymptotic freedom seem to be a rather important
feature to conciliate results obtained with the NJL model and
lattice simulations.

Finally, in Fig. 4 we present the results for the pseudocritical
Tpc temperature as a function of the magnetic field. It is
clearly seen in this figure that the pseudocritical temperature

eB 1.0 GeV 2

eB 0.8 GeV 2

eB 0.6 GeV 2

eB 0.4 GeV 2

eB 0.2 GeV 2

eB 0.0

60 80 100 120 140 160 180 200
0.0

0.5

1.0

1.5

2.0

2.5

3.0

T MeV

Χ T

FIG. 3. (Color online) The normalized thermal susceptibility as
a function of the temperature for different values of the magnetic
field B.

0.0 0.2 0.4 0.6 0.8 1.0
130

140

150

160

170

180

eB GeV2

T p
c
M
eV

FIG. 4. The pseudocritical temperature for the chiral transition of
magnetized quark matter as a function of magnetic field.

decreases as B increases, in (qualitative) agreement with the
lattice results of Refs. [12,13].

The phase diagram shown in Fig. 4 reproduces qualitatively
the lattice results. The figure shows a Tpc dependence with B
that starts linearly at small B, with almost no curvature, while
the lattice results have an almost-vanishing linear term and
a noticeable curvature. One may view this as a consequence
of the rather simple form used for the running of G(B,T ).
However, it is important to note that our ansatz avoids the
undesired “turnover” effect seen in NJL models with a B-
independent coupling, in which at intermediate values of B,
after an initial decrease, Tpc starts to increase with B—see,
e.g., Refs. [33,34]. A better quantitative agreement with the
lattice phase diagram can certainly be obtained by using a
G(B,T ) with more fitting parameters.

IV. CONCLUSIONS AND PERSPECTIVES

In this work we have considered the two-flavor NJL model
for hot and magnetized quark matter within the MFA. With the
aim of understanding discrepancies between effective model
predictions and recent lattice results regarding the behavior of
the chiral-transition pseudocritical temperature as a function
of B, we examined the effect of introducing a running
coupling G motivated by asymptotic freedom. Effective quark
theories such as the NJL model can be motivated by QCD
by integrating-out gluonic degrees of freedom. Although
some features of confinement can be enforced by means of
extending the model with the Polyakov loop, the running with
energy scales of the effective coupling, such as, e.g., due
to asymptotic freedom, is lost. In general, a decrease of G
with the temperature does not modify the qualitative features
regarding the chiral phase transition [36]. However, as we have
discussed, the same does not seem to be the case in the presence
of a strong magnetic field when the formation of a quark
condensate 〈ψ̄f ψf 〉 is enhanced by the magnetic catalysis
effect. Generically, the quark mass gap at zero temperature
sets the pseudocritical temperature at which chiral symmetry
is (partially) restored. Since the effective quark mass M in
the NJL model, given by the gap equation M ∼ −G〈ψ̄f ψf 〉,

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IMPORTANCE OF ASYMPTOTIC FREEDOM FOR THE . . . PHYSICAL REVIEW C 90, 025203 (2014)

increases with B, one would intuitively expect that Tpc should
also increase; but this is not confirmed by lattice simulations.
Here, we have shown that one way to conciliate the results is
to suppose that G decreases with B and T so as to moderate
the increase of M and, in turn, to the decrease of Tpc with B in
qualitative agreement with the Tpc(B) obtained by the lattice
simulations of Refs. [12,13].

As future work, one can improve the ansatz so as to
get a better quantitative agreement with the lattice results.
Next, the model could be used in a plethora of situations
in order to analyze the behavior of physical quantities such
as decay constants, meson-quark couplings, among others
which can be tested in lattice simulations. One could also
extend the calculations to the finite-chemical-potential do-
main, which currently is inaccessible to lattice calculations,
and investigate eventual additional running of G with μ. This
is particularly interesting, because a recent study has shown
[42] that the coexistence chemical potential decreases with
B. Such a study is important in the context of the physics of
magnetars.

Also, it would be interesting to further investigate why
the results for Tpc(B) obtained with G(B,T ) and T0(B) in
the NJL and entangled PNJL models, respectively, are in
qualitative agreement with the QCD lattice results while those
obtained with the QM and PQM models are not. Another
interesting new perspective is the investigation of the effect
of a G(B,T ) in the generation of spin-one condensates [43].
Such spin-one condensates give rise to a dynamical anomalous
magnetic moment for the fermions and lead to an increased

critical temperature for chiral symmetry restoration for B and
T independent NJL coupling.

In conclusion, our assumption of the decrease of the effec-
tive four-quark coupling with B and T mimicking asymptotic
freedom in QCD [35] represents a concrete implementation of
the back reaction of the sea quarks and confirms its potential
importance on explaining the inverse catalysis as stressed in
the recent literature [27,28]. In this respect, a first-principles
nonperturbative framework to study the effects of a magnetic
field in the continuum is provided by the Dyson–Schwinger
equations at finite temperature and density [38]. The coupling
of the magnetic field to the sea quarks is naturally taken into
account via the coupled system of integral equations for the
quark and gluon propagators. Very recently, this approach was
used to study fermion mass generation in the presence of an
external magnetic field in QED [44,45] and QCD [27,46,47].
The extension of such studies to finite temperatures would
shadow light on the problem of inverse magnetic catalysis
beyond the framework of effective chiral models.

ACKNOWLEDGMENTS

We thank Gergely Endrodi for discussions and also for
providing the lattice data of the up- and down-quark conden-
sates. We also thank Eduardo Fraga for bringing Ref. [34]
to our attention. This work was partially supported by CNPq,
FAPEMIG (R.L.S.F), FAPESP (G.K.), FAPESC (M.B.P.), and
CAPES (K.P.G). R.L.S.F. would like to thank J. Noronha and
R.O. Ramos for discussions on related matters.

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