We analyze the role played by the Polyakov loop in the description of the
chiral phase transition within the framework of nonlocal SU(3) chiral models
with flavor mixing. We show that its presence provides a substantial
enhancement of the predicted critical temperature, bringing it to a better
agreement with the most recent results of lattice calculations.Comment: 9 pages, 1 figur

Allton

Aoki

Bernard

Blaschke

Bowler

Broniowski

Cheng

Daniel Gómez Dumm

Duhau

Dumitru

Fodor

Fukushima

General

Gomez Dumm

Gustavo A. Contrera

Hatsuda

Karsch

Klevansky

Megias

Meisinger

Norberto N. Scoccola

Parappilly

Plant

Ratti

Rezaeian

Ripka

Roberts

Roessner

Scarpettini

Schafer

Vogl

Physics Letters B

English

arXiv

Crossref

Nonlocal SU(3) chiral quark models at finite temperature: The role of the Polyakov loop

AbstractWe analyze the role played by the Polyakov loop in the description of the chiral phase transition within the framework of nonlocal SU(3) chiral models with flavor mixing. We show that its presence provides a substantial enhancement of the predicted critical temperature, bringing it to a better agreement with the most recent results of lattice calculations

Contrera, Gustavo A.

Dumm, Daniel Gómez

Scoccola, Norberto N.

Elsevier - Publisher Connector 

Physics Letters B 661 (2008) 113–117
www.elsevier.com/locate/physletb
Nonlocal SU(3) chiral quark models at finite temperature:
The role of the Polyakov loop
Gustavo A. Contrera a,b, Daniel Gómez Dumm c,d, Norberto N. Scoccola a,d,e,∗
a Physics Department, Comisión Nacional de Energía Atómica, Av. Libertador 8250, (1429) Buenos Aires, Argentina
b Universidad Nacional de San Martín, Argentina
c IFLP, CONICET—Departamento de Física, Universidad Nacional de La Plata, C.C. 67, (1900) La Plata, Argentina
d CONICET, Rivadavia 1917, (1033) Buenos Aires, Argentina
e Universidad Favaloro, Solís 453, (1078) Buenos Aires, Argentina
Received 30 October 2007; received in revised form 7 January 2008; accepted 28 January 2008
Available online 7 February 2008
Editor: J.-P. Blaizot
Abstract
We analyze the role played by the Polyakov loop in the description of the chiral phase transition within the framework of nonlocal SU(3) chiral
models with flavor mixing. We show that its presence provides a substantial enhancement of the predicted critical temperature, bringing it to a
better agreement with the most recent results of lattice calculations.
© 2008 Elsevier B.V.
PACS: 12.39.Ki; 11.30.Rd; 12.38.Aw
Open access under CC BY license.The detailed knowledge of the phase diagram for strongly
interacting matter has become an issue of great interest in re-
cent years, both from the theoretical and experimental points of
view. On the theoretical side, even if a significant progress has
been made on the development of ab initio calculations such as
lattice QCD [1–3], these are not yet able to provide a full un-
derstanding of the QCD phase diagram due to the well-known
difficulties of dealing with finite chemical potentials. In this sit-
uation, it is important to develop effective models that show
consistency with lattice results and can be extrapolated into re-
gions not accessible by lattice calculation techniques. In previ-
ous works [4–7] the study of the phase diagram of SU(2) chiral
quark models that include nonlocal interactions [8] has been un-
dertaken. These theories can be viewed as nonlocal extensions
of the widely studied Nambu–Jona-Lasinio model [9]. In fact,
nonlocality arises naturally in the context of several success-
ful approaches to low-energy quark dynamics as, for example,
* Corresponding author at: Physics Department, Comisión Nacional de En-
ergía Atómica, Av. Libertador 8250, (1429) Buenos Aires, Argentina.
E-mail address: scoccola@tandar.cnea.gov.ar (N.N. Scoccola).0370-2693 © 2008 Elsevier B.V.
doi:10.1016/j.physletb.2008.01.069
Open access under CC BY license.the instanton liquid model [10] and the Schwinger–Dyson re-
summation techniques [11]. Lattice QCD calculations [12] also
indicate that quark interactions should act over a certain range
in momentum space. Moreover, several studies [13,14] have
shown that nonlocal chiral quark models provide a satisfactory
description of hadron properties at zero temperature and den-
sity. On the other hand, when looking at the description of the
chiral phase transition, it has been noticed that for zero chemical
potential these SU(2) models lead to a rather low critical tem-
perature T 0cr in comparison with lattice results [4,5]. The aim of
the present work is to go one step beyond these previous analy-
ses, studying the finite temperature behavior of nonlocal chiral
models that include mixing with active strangeness degrees of
freedom, and taking care of the effect of gauge interactions by
coupling the quarks with the Polyakov loop. The inclusion of
the Polyakov loop has been considered recently in the context
of NJL-like models (so-called PNJL models) [15–19], serving
as an order parameter for the deconfinement transition. In par-
ticular, this has been done in Ref. [20] in the framework of a
nonlocal two-flavor model, focusing on the analysis of mesonic
correlations. Here we show that within nonlocal SU(3) effec-
tive models with flavor mixing the coupling to the Polyakov
114 G.A. Contrera et al. / Physics Letters B 661 (2008) 113–117loop leads to a significant increase of the chiral phase transition
temperature T 0cr, which is otherwise as low as in SU(2) sym-
metric models. As stated, this is a desired effect in order to be
in better agreement with lattice expectations [21].
We deal here with the nonlocal covariant SU(3) quark model
described in Ref. [22], including now the coupling to the
Polyakov loop. The corresponding Euclidean effective action
is given by
SE =
∫
d4x
{
ψ¯(x)[−iγμDμ + mˆ]ψ(x)
− G
2
[
jSa (x) j
S
a (x) + jPa (x) jPa (x)
]
− H
4
Tabc
[
jSa (x)j
S
b (x)j
S
c (x) − 3 jSa (x)jPb (x)jPc (x)
]
(1)+ U(Φ[A(x)])},
where ψ is a chiral U(3) vector that includes the light quark
fields, ψ ≡ (uds)T , and mˆ = diag(mu,md,ms) stands for the
current quark mass matrix. For simplicity we consider the
isospin symmetry limit, in which mu = md = m¯. The currents
j
S,P
a (x) are given by
(2)jSa (x) =
∫
d4y d4z r(y − x)r(x − z)ψ¯(y)λaψ(z),
(3)jPa (x) =
∫
d4y d4z r(y − x)r(x − z)ψ¯(y)iγ5λaψ(z),
where the form factor r(x − y) is local in momentum space,
namely
(4)r(x − y) =
∫
d4p
(2π)4
e−i(x−y)pr(p),
and the matrices λa , with a = 0, . . . ,8, are the usual eight
Gell-Mann 3 × 3 matrices—generators of SU(3)—plus λ0 =√
2/313×3. The constants Tabc in the ’t Hooft term accounting
for flavor-mixing are defined by
(5)Tabc = 13!ijkmnl(λa)im(λb)jn(λc)kl .
The coupling to the Polyakov loop has been implemented
in Eq. (1) through the covariant derivative Dμ ≡ ∂μ − iAμ.
In what follows, we assume that the quarks move in a back-
ground color gauge field φ = iA0 = igδμ0Gμa λa/2, where Gμa
are the SU(3) color gauge fields. Then the traced Polyakov
loop, which is taken as order parameter of confinement, is given
by Φ = 13 Tr exp(iφ). We will work in the so-called Polyakov
gauge, in which the matrix φ is given a diagonal representation
φ = φ3λ3 +φ8λ8, which leaves only two independent variables,
φ3 and φ8. Finally, the action (1) also includes an effective po-
tential U , which accounts for the gauge field self-interactions.
To proceed we consider the grand canonical thermodynami-
cal potential of the model within the mean field approximation.
Using the standard Matsubara formalism we getΩMFA(T ) = −2T
∑
f,c
∫
d3p
(2π)3
∞∑
n=−∞
Tr ln
[
ω2nc + Σ2f c
(
ω2nc
)]
− 1
2
[∑
f
(
σ¯f S¯f + G2 S¯
2
f
)
+ H
2
S¯uS¯d S¯s
]
(6)+ U(Φ¯, T ),
where f = (u, d, s), c = (r, g, b), and we have used the def-
inition ω2nc = (ωn − φc)2 + p2, ωn = (2n + 1)πT being the
usual Matsubara frequencies. The quantities φc are defined by
the relation φ = diag(φr ,φg,φb). The constituent masses Σfc
are here momentum-dependent quantities, given by
(7)Σfc
(
ω2nc
)= mf + σ¯f r2(ω2nc).
Within the stationary phase approximation, the mean field val-
ues of the auxiliary fields S¯f turn out to be related with the
mean field values of the scalar fields σ¯f by [22]
σ¯u + GS¯u + H2 S¯uS¯s = 0,
(8)σ¯s + GS¯s + H2 S¯
2
u = 0.
The mean field effective potential U(Φ¯, T ), which accounts
for the Polyakov loop dynamics, can be fitted by taking into
account group theory constraints together with lattice results,
from which one can estimate the temperature dependence. Fol-
lowing Ref. [19] we take
U(Φ,T ) =
[
−1
2
a(T )Φ2
(9)+ b(T ) ln(1 − 6Φ2 + 8Φ3 − 3Φ4)]T 4,
with the corresponding definitions of a(T ) and b(T ). Owing to
the charge conjugation properties of the QCD Lagrangian [23],
the mean field value of the Polyakov loop field Φ is expected
to be a real quantity. Assuming that φ3 and φ8 are real-valued
fields [19], this implies φ¯8 = 0, Φ¯ = [2 cos(φ¯3/T ) + 1]/3.
For finite current quark masses the quark contribution to
ΩMFA(T ) turns out to be divergent. To regularize it we fol-
low the same prescription as in previous works [6]. Namely, we
subtract from ΩMFA(T ) the quark contribution in the absence
of fermion interactions, and then we add it in a regularized
form, i.e., after the subtraction of an infinite, T -independent
contribution. From the minimization of this regularized thermo-
dynamical potential, it is possible now to obtain a set of three
coupled “gap” equations that determine the mean field values
σ¯u, σ¯s and φ¯3 at a given temperature.
We are also interested in the estimation of chiral conden-
sates, which are given by the vacuum expectation values 〈u¯u〉 =
〈d¯d〉 and 〈s¯s〉. As usual, they can be obtained by varying ΩMFA
with respect to the corresponding current quark masses. The ex-
plicit regularized expression for a quark condensate 〈f¯ f 〉 reads
G.A. Contrera et al. / Physics Letters B 661 (2008) 113–117 115〈f¯ f 〉 = 4
∑
c
∫
d3p
(2π)3
{
−T
∞∑
n=−∞
[
Σfc
(
ω2nc
)
ω2nc + Σ2f c(ω2nc)
(10)− mf
ω2nc + m2f
]
+ mf
Ef
(n+f c + n−f c)
}
,
where n±f c = {1+exp[(Ef ± iφ¯c)/T ]}−1, with Ef =
√
p2 + m2f .
In what follows we analyze the chiral phase transition for a
definite form factor, taking into account the temperature depen-
dence of effective masses, chiral condensates and susceptibili-
ties. For simplicity, we have considered a Gaussian form factor
(11)r(p2)= exp(−p2/2Λ2),
where Λ is a free parameter of the model, playing the role of
an ultraviolet cut-off momentum scale. This parameter, as well
as quark current masses and couplings in Eq. (1), can be cho-
sen so as to reproduce the empirical values of meson properties
at T = 0. We take into account the analysis in Ref. [22], where
the parameters are fixed to obtain the empirical values of meson
masses mπ , mK and mη′ , together with the pion decay con-
stant fπ . As shown in that work, in this way one can get a good
description of the light pseudoscalar meson phenomenology at
zero temperature. For definiteness we will work here with the
parameter set GI in Ref. [22]. Namely, we use
m¯ = 8.5 MeV, ms = 223 MeV, Λ = 709 MeV,
(12)GΛ2 = 10.99, HΛ5 = −295.3.
The corresponding numerical results are given in Fig. 1,
where we show the behavior of various physical magnitudes
as functions of the temperature within the described nonlo-
cal SU(3) model. Left (right) panels correspond to the model
without (with) the inclusion of the Polyakov loop. In the up-
per panels we show the behavior of the mean field values σ¯u
and σ¯s (solid and dotted lines, respectively). For the sake of
comparison, the curves have been normalized to the respective
values at T = 0, which are found to be σ¯ 0u = 304 MeV and
σ¯ 0s = 427 MeV (notice that at T = 0 there is no effect of the
Polyakov loop). It is clearly seen that in both cases the SU(2)
chiral restoration proceeds as a smooth crossover, whereas there
is an enhancement of the order of 80 MeV in the correspond-
ing critical temperature when the effect of the Polyakov loop
is taken into account. In order to properly define the values of
these critical temperatures, we consider the chiral susceptibility
χq , defined as
(13)χq =
(
∂〈q¯q〉
∂mq
)
T =cte
.
The phase transition temperature can be defined as the point
where the susceptibility shows a peak, the sharpness of this
peak serving as a measure of the steepness of the crossover [6].
The curves showing the behavior of the susceptibilities in the
nonlocal SU(3) models are plotted in the central panels of
Fig. 1. As already mentioned, it is seen that the inclusion
of the Polyakov loop leads to an enhancement in the criti-
cal SU(2) chiral restoration temperature, which brings it to a
better agreement with the most recent lattice values T 0(Lat)cr =160–200 MeV [21]. Our results are (see solid lines in the cen-
tral panels of Fig. 1)
Nonlocal SU(3) model without Polyakov loop:
T 0cr = 114 MeV.
Nonlocal SU(3) model including Polyakov loop:
T 0cr = 198 MeV.
In addition, it is seen that after the inclusion of the Polyakov
loop there is an enhancement in the sharpness of the peak. The
restoration of the SU(2) chiral symmetry can be also observed
when looking to the behavior of the chiral condensates with the
temperature, which is shown in the lower panels of Fig. 1. At
T = 0 we find 〈u¯u〉1/3 = −211 MeV and 〈s¯s〉1/3 = −186 MeV.
As usual, the restoration of the SU(3) chiral symmetry is much
less pronounced, due to the larger current mass of the strange
quark. In fact, both in absence or presence of the Polyakov loop
a broad peak is observed in the corresponding chiral susceptibil-
ity χs . This is shown by the dotted lines in the central panels of
Fig. 1, where the values of χs have been normalized by a fac-
tor 20 to be comparable with the corresponding values of χu.
The peaks in χs are found to be placed at about 190 (230) MeV
for the model without (with) the Polyakov loop. It is interesting
to notice that due to flavor mixing effects the light quark con-
densates 〈u¯u〉 exhibit a second drop at those temperatures. In
addition, owing to flavor mixing some signal of the SU(2) chiral
restoration can be noticed in the behavior of χs : one can observe
a shoulder at about 115 MeV for the model without Polyakov
loop (barely noticeable in the figure), and a small peak at about
200 MeV for the model with Polyakov loop. Finally, it is also
important to analyze the behavior of the mean field value of
the traced Polyakov loop Φ¯ . As stated, the latter can be inter-
preted as an order parameter of the deconfinement transition,
Φ¯ = 0 and Φ¯ = 1 being related to confined and deconfined
quarks, respectively. Correspondingly, a susceptibility associ-
ated with the Polyakov loop can be defined as χΦ = dΦ¯/dT .
In the nonlocal SU(3) theory under consideration, the behav-
ior of Φ¯ and χΦ with the temperature are shown by the dashed
curves in the upper and central right panels of Fig. 1, respec-
tively. By comparison with the behavior of the mean field value
σ¯u, it can be seen that even if the deconfinement transition goes
smoother than the SU(2) chiral restoration, both transitions are
found to occur at the same temperature region, located at about
T = 200 MeV.
The analysis presented above corresponds to the specific
case of a covariant nonlocal SU(3) model, in which we have
considered a separable four-fermion interaction. Such an inter-
action is motivated by the instanton liquid model [10], where it
arises from the nonlocal structure of the QCD vacuum. For sim-
plicity we have used here a Gaussian-like form factor, choosing
a definite parameter set to get a good agreement with meson
phenomenology at zero temperature. In order to determine the
range of validity of the conclusions extracted from our analy-
sis, we have also considered different parameter sets (within
the range allowed by meson phenomenology), and different
shapes for the form factors such as, e.g., Lorentzian func-
tions. We have also analyzed an alternative way of including
116 G.A. Contrera et al. / Physics Letters B 661 (2008) 113–117Fig. 1. Behavior of various physical magnitudes as functions of the temperature. Left (right) panels correspond to the model in the absence (presence) of the coupling
to the Polyakov loop. The upper panels display the behavior of the mean field values σ¯q (normalized to their value at T = 0) and the traced Polyakov loop Φ¯ , while
central and bottom panels show the behavior of the susceptibilities χq , χΦ and the quark–antiquark condensates, respectively. For large values of T , the behavior
of condensates corresponds to that of a system of free fermions with nonvanishing mass.the nonlocality based on an effective one-gluon exchange pic-
ture of strong interactions (see Ref. [24] for a detailed com-
parison of model features in the SU(2) case at T = 0). It
comes out that the results are qualitatively similar in all these
cases.
To summarize, we have analyzed the role played by the
Polyakov loop in the description of the chiral phase transi-
tion within the framework of nonlocal SU(3) chiral models
with flavor mixing. We have found that its presence provides
a substantial enhancement of the predicted critical temperature,
bringing it to a better agreement with the most recent results of
lattice calculations. Another interesting effect of the coupling to
the Polyakov loop is that the phase transition becomes steeper,
showing a sharper peak in the chiral susceptibility. We noticethat, as it was done for the two-flavor case in Ref. [20], it would
be of great interest to extend the present SU(3) model analysis
beyond the mean field approximation, considering the effect of
mesonic correlations. Other tasks to be addressed are the analy-
sis of the behavior of meson properties with the temperature,
and the extension of all these studies in the presence of finite
chemical potentials. We hope to report on these issues in forth-
coming publications.
Acknowledgements
This work has been supported in part by CONICET and
ANPCyT (Argentina), under grants PIP 6009, PIP 6084 and
PICT04-03-25374.
G.A. Contrera et al. / Physics Letters B 661 (2008) 113–117 117References
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Nonlocal SU(3) chiral quark models at finite temperature: The role of the Polyakov loop 

We analyze the role played by the Polyakov loop in the description of the chiral phase transition within the framework of nonlocal SU(3) chiral models with flavor mixing. We show that its presence provides a substantial enhancement of the predicted critical temperature, bringing it to a better agreement with the most recent results of lattice calculations.Facultad de Ciencias ExactasInstituto de Física La Plat

Contrera, Gustavo Aníbal

Gómez Dumm, Daniel

El Servicio de Difusión de la Creación Intelectual

Physics Letters B 661 (2008) 113–117
www.elsevier.com/locate/physletb
Nonlocal SU(3) chiral quark models at finite temperature:
The role of the Polyakov loop
Gustavo A. Contrera a,b, Daniel Gómez Dumm c,d, Norberto N. Scoccola a,d,e,∗
a Physics Department, Comisión Nacional de Energía Atómica, Av. Libertador 8250, (1429) Buenos Aires, Argentina
b Universidad Nacional de San Martín, Argentina
c IFLP, CONICET—Departamento de Física, Universidad Nacional de La Plata, C.C. 67, (1900) La Plata, Argentina
d CONICET, Rivadavia 1917, (1033) Buenos Aires, Argentina
e Universidad Favaloro, Solís 453, (1078) Buenos Aires, Argentina
Received 30 October 2007; received in revised form 7 January 2008; accepted 28 January 2008
Available online 7 February 2008
Editor: J.-P. Blaizot
Abstract
We analyze the role played by the Polyakov loop in the description of the chiral phase transition within the framework of nonlocal SU(3) chiral
models with flavor mixing. We show that its presence provides a substantial enhancement of the predicted critical temperature, bringing it to a
better agreement with the most recent results of lattice calculations.
© 2008 Elsevier B.V.
PACS: 12.39.Ki; 11.30.Rd; 12.38.Aw
Open access under CC BY license.The detailed knowledge of the phase diagram for strongly
interacting matter has become an issue of great interest in re-
cent years, both from the theoretical and experimental points of
view. On the theoretical side, even if a significant progress has
been made on the development of ab initio calculations such as
lattice QCD [1–3], these are not yet able to provide a full un-
derstanding of the QCD phase diagram due to the well-known
difficulties of dealing with finite chemical potentials. In this sit-
uation, it is important to develop effective models that show
consistency with lattice results and can be extrapolated into re-
gions not accessible by lattice calculation techniques. In previ-
ous works [4–7] the study of the phase diagram of SU(2) chiral
quark models that include nonlocal interactions [8] has been un-
dertaken. These theories can be viewed as nonlocal extensions
of the widely studied Nambu–Jona-Lasinio model [9]. In fact,
nonlocality arises naturally in the context of several success-
ful approaches to low-energy quark dynamics as, for example,
* Corresponding author at: Physics Department, Comisión Nacional de En-
ergía Atómica, Av. Libertador 8250, (1429) Buenos Aires, Argentina.
E-mail address: scoccola@tandar.cnea.gov.ar (N.N. Scoccola).0370-2693 © 2008 Elsevier B.V.
doi:10.1016/j.physletb.2008.01.069
Open access under CC BY license.the instanton liquid model [10] and the Schwinger–Dyson re-
summation techniques [11]. Lattice QCD calculations [12] also
indicate that quark interactions should act over a certain range
in momentum space. Moreover, several studies [13,14] have
shown that nonlocal chiral quark models provide a satisfactory
description of hadron properties at zero temperature and den-
sity. On the other hand, when looking at the description of the
chiral phase transition, it has been noticed that for zero chemical
potential these SU(2) models lead to a rather low critical tem-
perature T 0cr in comparison with lattice results [4,5]. The aim of
the present work is to go one step beyond these previous analy-
ses, studying the finite temperature behavior of nonlocal chiral
models that include mixing with active strangeness degrees of
freedom, and taking care of the effect of gauge interactions by
coupling the quarks with the Polyakov loop. The inclusion of
the Polyakov loop has been considered recently in the context
of NJL-like models (so-called PNJL models) [15–19], serving
as an order parameter for the deconfinement transition. In par-
ticular, this has been done in Ref. [20] in the framework of a
nonlocal two-flavor model, focusing on the analysis of mesonic
correlations. Here we show that within nonlocal SU(3) effec-
tive models with flavor mixing the coupling to the Polyakov
114 G.A. Contrera et al. / Physics Letters B 661 (2008) 113–117loop leads to a significant increase of the chiral phase transition
temperature T 0cr, which is otherwise as low as in SU(2) sym-
metric models. As stated, this is a desired effect in order to be
in better agreement with lattice expectations [21].
We deal here with the nonlocal covariant SU(3) quark model
described in Ref. [22], including now the coupling to the
Polyakov loop. The corresponding Euclidean effective action
is given by
SE =
Z
d4x
½
ψ¯(x)[−iγμDμ + mˆ]ψ(x)
− G
2
£
jSa (x) j
S
a (x) + jPa (x) jPa (x)
¤
− H
4
Tabc
£
jSa (x)j
S
b (x)j
S
c (x) − 3 jSa (x)jPb (x)jPc (x)
¤
(1)+ U¡Φ£A(x)¤¢¾,
where ψ is a chiral U(3) vector that includes the light quark
fields, ψ ≡ (uds)T , and mˆ = diag(mu,md,ms) stands for the
current quark mass matrix. For simplicity we consider the
isospin symmetry limit, in which mu = md = m¯. The currents
j
S,P
a (x) are given by
(2)jSa (x) =
Z
d4y d4z r(y − x)r(x − z)ψ¯(y)λaψ(z),
(3)jPa (x) =
Z
d4y d4z r(y − x)r(x − z)ψ¯(y)iγ5λaψ(z),
where the form factor r(x − y) is local in momentum space,
namely
(4)r(x − y) =
Z
d4p
(2π)4
e−i(x−y)pr(p),
and the matrices λa , with a = 0, . . . ,8, are the usual eight
Gell-Mann 3 × 3 matrices—generators of SU(3)—plus λ0 =√
2/313×3. The constants Tabc in the ’t Hooft term accounting
for flavor-mixing are defined by
(5)Tabc = 13!²ijk²mnl(λa)im(λb)jn(λc)kl .
The coupling to the Polyakov loop has been implemented
in Eq. (1) through the covariant derivative Dμ ≡ ∂μ − iAμ.
In what follows, we assume that the quarks move in a back-
ground color gauge field φ = iA0 = igδμ0Gμa λa/2, where Gμa
are the SU(3) color gauge fields. Then the traced Polyakov
loop, which is taken as order parameter of confinement, is given
by Φ = 13 Tr exp(iφ). We will work in the so-called Polyakov
gauge, in which the matrix φ is given a diagonal representation
φ = φ3λ3 +φ8λ8, which leaves only two independent variables,
φ3 and φ8. Finally, the action (1) also includes an effective po-
tential U , which accounts for the gauge field self-interactions.
To proceed we consider the grand canonical thermodynami-
cal potential of the model within the mean field approximation.
Using the standard Matsubara formalism we getΩMFA(T ) = −2T
X
f,c
Z
d3p
(2π)3
∞X
n=−∞
Tr ln
£
ω2nc + Σ2f c
¡
ω2nc
¢¤
− 1
2
"X
f
µ
σ¯f S¯f + G2 S¯
2
f
¶
+ H
2
S¯uS¯d S¯s
#
(6)+ U(Φ¯, T ),
where f = (u, d, s), c = (r, g, b), and we have used the def-
inition ω2nc = (ωn − φc)2 + Ep2, ωn = (2n + 1)πT being the
usual Matsubara frequencies. The quantities φc are defined by
the relation φ = diag(φr ,φg,φb). The constituent masses Σfc
are here momentum-dependent quantities, given by
(7)Σfc
¡
ω2nc
¢= mf + σ¯f r2¡ω2nc¢.
Within the stationary phase approximation, the mean field val-
ues of the auxiliary fields S¯f turn out to be related with the
mean field values of the scalar fields σ¯f by [22]
σ¯u + GS¯u + H2 S¯uS¯s = 0,
(8)σ¯s + GS¯s + H2 S¯
2
u = 0.
The mean field effective potential U(Φ¯, T ), which accounts
for the Polyakov loop dynamics, can be fitted by taking into
account group theory constraints together with lattice results,
from which one can estimate the temperature dependence. Fol-
lowing Ref. [19] we take
U(Φ,T ) =
·
−1
2
a(T )Φ2
(9)+ b(T ) ln¡1 − 6Φ2 + 8Φ3 − 3Φ4¢¸T 4,
with the corresponding definitions of a(T ) and b(T ). Owing to
the charge conjugation properties of the QCD Lagrangian [23],
the mean field value of the Polyakov loop field Φ is expected
to be a real quantity. Assuming that φ3 and φ8 are real-valued
fields [19], this implies φ¯8 = 0, Φ¯ = [2 cos(φ¯3/T ) + 1]/3.
For finite current quark masses the quark contribution to
ΩMFA(T ) turns out to be divergent. To regularize it we fol-
low the same prescription as in previous works [6]. Namely, we
subtract from ΩMFA(T ) the quark contribution in the absence
of fermion interactions, and then we add it in a regularized
form, i.e., after the subtraction of an infinite, T -independent
contribution. From the minimization of this regularized thermo-
dynamical potential, it is possible now to obtain a set of three
coupled “gap” equations that determine the mean field values
σ¯u, σ¯s and φ¯3 at a given temperature.
We are also interested in the estimation of chiral conden-
sates, which are given by the vacuum expectation values hu¯ui =
hd¯di and hs¯si. As usual, they can be obtained by varying ΩMFA
with respect to the corresponding current quark masses. The ex-
plicit regularized expression for a quark condensate hf¯ f i reads
G.A. Contrera et al. / Physics Letters B 661 (2008) 113–117 115hf¯ f i = 4
X
c
Z
d3p
(2π)3
(
−T
∞X
n=−∞
·
Σfc
¡
ω2nc
¢
ω2nc + Σ2f c(ω2nc)
(10)− mf
ω2nc + m2f
¸
+ mf
Ef
(n+f c + n−f c)
)
,
where n±f c = {1+exp[(Ef ± iφ¯c)/T ]}−1, with Ef =
q
Ep2 + m2f .
In what follows we analyze the chiral phase transition for a
definite form factor, taking into account the temperature depen-
dence of effective masses, chiral condensates and susceptibili-
ties. For simplicity, we have considered a Gaussian form factor
(11)r¡p2¢= exp¡−p2/2Λ2¢,
where Λ is a free parameter of the model, playing the role of
an ultraviolet cut-off momentum scale. This parameter, as well
as quark current masses and couplings in Eq. (1), can be cho-
sen so as to reproduce the empirical values of meson properties
at T = 0. We take into account the analysis in Ref. [22], where
the parameters are fixed to obtain the empirical values of meson
masses mπ , mK and mη0 , together with the pion decay con-
stant fπ . As shown in that work, in this way one can get a good
description of the light pseudoscalar meson phenomenology at
zero temperature. For definiteness we will work here with the
parameter set GI in Ref. [22]. Namely, we use
m¯ = 8.5 MeV, ms = 223 MeV, Λ = 709 MeV,
(12)GΛ2 = 10.99, HΛ5 = −295.3.
The corresponding numerical results are given in Fig. 1,
where we show the behavior of various physical magnitudes
as functions of the temperature within the described nonlo-
cal SU(3) model. Left (right) panels correspond to the model
without (with) the inclusion of the Polyakov loop. In the up-
per panels we show the behavior of the mean field values σ¯u
and σ¯s (solid and dotted lines, respectively). For the sake of
comparison, the curves have been normalized to the respective
values at T = 0, which are found to be σ¯ 0u = 304 MeV and
σ¯ 0s = 427 MeV (notice that at T = 0 there is no effect of the
Polyakov loop). It is clearly seen that in both cases the SU(2)
chiral restoration proceeds as a smooth crossover, whereas there
is an enhancement of the order of 80 MeV in the correspond-
ing critical temperature when the effect of the Polyakov loop
is taken into account. In order to properly define the values of
these critical temperatures, we consider the chiral susceptibility
χq , defined as
(13)χq =
µ
∂hq¯qi
∂mq
¶
T =cte
.
The phase transition temperature can be defined as the point
where the susceptibility shows a peak, the sharpness of this
peak serving as a measure of the steepness of the crossover [6].
The curves showing the behavior of the susceptibilities in the
nonlocal SU(3) models are plotted in the central panels of
Fig. 1. As already mentioned, it is seen that the inclusion
of the Polyakov loop leads to an enhancement in the criti-
cal SU(2) chiral restoration temperature, which brings it to a
better agreement with the most recent lattice values T 0(Lat)cr =160–200 MeV [21]. Our results are (see solid lines in the cen-
tral panels of Fig. 1)
Nonlocal SU(3) model without Polyakov loop:
T 0cr = 114 MeV.
Nonlocal SU(3) model including Polyakov loop:
T 0cr = 198 MeV.
In addition, it is seen that after the inclusion of the Polyakov
loop there is an enhancement in the sharpness of the peak. The
restoration of the SU(2) chiral symmetry can be also observed
when looking to the behavior of the chiral condensates with the
temperature, which is shown in the lower panels of Fig. 1. At
T = 0 we find hu¯ui1/3 = −211 MeV and hs¯si1/3 = −186 MeV.
As usual, the restoration of the SU(3) chiral symmetry is much
less pronounced, due to the larger current mass of the strange
quark. In fact, both in absence or presence of the Polyakov loop
a broad peak is observed in the corresponding chiral susceptibil-
ity χs . This is shown by the dotted lines in the central panels of
Fig. 1, where the values of χs have been normalized by a fac-
tor 20 to be comparable with the corresponding values of χu.
The peaks in χs are found to be placed at about 190 (230) MeV
for the model without (with) the Polyakov loop. It is interesting
to notice that due to flavor mixing effects the light quark con-
densates hu¯ui exhibit a second drop at those temperatures. In
addition, owing to flavor mixing some signal of the SU(2) chiral
restoration can be noticed in the behavior of χs : one can observe
a shoulder at about 115 MeV for the model without Polyakov
loop (barely noticeable in the figure), and a small peak at about
200 MeV for the model with Polyakov loop. Finally, it is also
important to analyze the behavior of the mean field value of
the traced Polyakov loop Φ¯ . As stated, the latter can be inter-
preted as an order parameter of the deconfinement transition,
Φ¯ = 0 and Φ¯ = 1 being related to confined and deconfined
quarks, respectively. Correspondingly, a susceptibility associ-
ated with the Polyakov loop can be defined as χΦ = dΦ¯/dT .
In the nonlocal SU(3) theory under consideration, the behav-
ior of Φ¯ and χΦ with the temperature are shown by the dashed
curves in the upper and central right panels of Fig. 1, respec-
tively. By comparison with the behavior of the mean field value
σ¯u, it can be seen that even if the deconfinement transition goes
smoother than the SU(2) chiral restoration, both transitions are
found to occur at the same temperature region, located at about
T = 200 MeV.
The analysis presented above corresponds to the specific
case of a covariant nonlocal SU(3) model, in which we have
considered a separable four-fermion interaction. Such an inter-
action is motivated by the instanton liquid model [10], where it
arises from the nonlocal structure of the QCD vacuum. For sim-
plicity we have used here a Gaussian-like form factor, choosing
a definite parameter set to get a good agreement with meson
phenomenology at zero temperature. In order to determine the
range of validity of the conclusions extracted from our analy-
sis, we have also considered different parameter sets (within
the range allowed by meson phenomenology), and different
shapes for the form factors such as, e.g., Lorentzian func-
tions. We have also analyzed an alternative way of including
116 G.A. Contrera et al. / Physics Letters B 661 (2008) 113–117Fig. 1. Behavior of various physical magnitudes as functions of the temperature. Left (right) panels correspond to the model in the absence (presence) of the coupling
to the Polyakov loop. The upper panels display the behavior of the mean field values σ¯q (normalized to their value at T = 0) and the traced Polyakov loop Φ¯ , while
central and bottom panels show the behavior of the susceptibilities χq , χΦ and the quark–antiquark condensates, respectively. For large values of T , the behavior
of condensates corresponds to that of a system of free fermions with nonvanishing mass.the nonlocality based on an effective one-gluon exchange pic-
ture of strong interactions (see Ref. [24] for a detailed com-
parison of model features in the SU(2) case at T = 0). It
comes out that the results are qualitatively similar in all these
cases.
To summarize, we have analyzed the role played by the
Polyakov loop in the description of the chiral phase transi-
tion within the framework of nonlocal SU(3) chiral models
with flavor mixing. We have found that its presence provides
a substantial enhancement of the predicted critical temperature,
bringing it to a better agreement with the most recent results of
lattice calculations. Another interesting effect of the coupling to
the Polyakov loop is that the phase transition becomes steeper,
showing a sharper peak in the chiral susceptibility. We noticethat, as it was done for the two-flavor case in Ref. [20], it would
be of great interest to extend the present SU(3) model analysis
beyond the mean field approximation, considering the effect of
mesonic correlations. Other tasks to be addressed are the analy-
sis of the behavior of meson properties with the temperature,
and the extension of all these studies in the presence of finite
chemical potentials. We hope to report on these issues in forth-
coming publications.
Acknowledgements
This work has been supported in part by CONICET and
ANPCyT (Argentina), under grants PIP 6009, PIP 6084 and
PICT04-03-25374.
G.A. Contrera et al. / Physics Letters B 661 (2008) 113–117 117References
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Nonlocal SU(3) chiral quark models at finite temperature: the role of
  the Polyakov loop

Nonlocal SU(3) chiral quark models at finite temperature: the role of the Polyakov loop

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