Physics Letters B 747 (2015) 529–535
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
The dual quark condensate in local and nonlocal NJL models: An order
parameter for deconfinement?
Federico Marquez
a,∗
, Aftab Ahmad
b,c
, Michael Buballa
d
, Alfredo Raya
a,b
a
Instituto de Física, Pontificia Universidad Católica de Chile, Casilla 306, Santiago 22, Chile
b
Instituto de Física y Matemáticas, Universidad Michoacana de San Nicolás de Hidalgo, Edificio C-3, Ciudad Universitaria, Morelia 58040, Michoacán, Mexico
c
Department of Physics, Gomal University, D.I. Khan 29220, K.P.K., Pakistan
d
Theoriezentrum, Institut für Kernphysik, Technische Universität Darmstadt, Schlossgartenstr. 2, 64289 Darmstadt, Germany
a r t i c l e i n f o a b s t r a c t
Article history:
Received
17 May 2015
Accepted
11 June 2015
Available
online 16 June 2015
Editor: J.-P.
Blaizot
Keywords:
Chiral
symmetry breaking
Confinement–deconfinement
Order
parameters
We study the behavior of the dual quark condensate
1
in the Nambu–Jona-Lasinio (NJL) model and its
nonlocal variant. In quantum chromodynamics
1
can be related to the breaking of the center symmetry
and is therefore an (approximate) order parameter of confinement. The deconfinement transition is then
signaled by a strong rise of
1
as a function of temperature. However, a similar behavior is also seen
in the NJL model, which is known to have no confinement. Indeed, it was shown that in this model
the rise of
1
is triggered by the chiral phase transition. In order to shed more light on this issue, we
calculate
1
for several variants of the NJL model, some of which have been suggested to be confining.
Switching between “confining” and “non-confining” models and parametrizations we find no qualitative
difference in the behavior of
1
, namely, it always rises in the region of the chiral phase transition. We
conclude that without having established a relation to the center symmetry in a given model,
1
should
not blindly be regarded as an order parameter of confinement.
© 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP
3
.
1. Introduction
Spontaneous chiral symmetry breaking and confinement are
among the most important features of nonperturbative low-energy
quantum chromodynamics (QCD). Both properties have been stud-
ied
from first principles within lattice QCD, which nowadays pro-
vides
an excellent description of the meson and baryon spectrum
in vacuum [1]. Turning to nonzero temperature T , lattice QCD pre-
dicts
a rapid but smooth crossover from the confined phase with
broken chiral symmetry at low T to a deconfined phase with (ap-
proximately)
restored chiral symmetry at high T [2,3]. Moreover,
it is found that both transitions take place in the same temper-
ature
range. The corresponding order parameters are the chiral
quark condensate
¯
ψψ and the Polyakov-loop (PL) expectation
value [4,5], respectively, which strictly speaking belong to oppo-
site
limits of the theory. While
¯
ψψ is a strict order parameter
for chiral symmetry breaking in the limit of massless quarks, the
PL expectation value is a strict order parameter for confinement
in the limit of infinitely heavy quarks, where it can be related to
*
Corresponding author.
E-mail
address: cfmarque@uc.cl (F. Marquez).
Z(3) center-symmetry breaking and thereby to the free energy of
a static quark–antiquark pair [6,7].
The
identification of an appropriate order parameter for the
confinement–deconfinement phase transition in QCD with finite
(i.e., non-infinite) quark masses stands on less solid grounds. In
this context the authors of Ref. [8] have proposed the dual quark
condensate
1
as an alternative. Starting point is the generalized
quark condensate
¯
ψψ
φ
, which is the analogue of the usual quark
condensate, but evaluated for quark fields with twisted boundary
conditions
ψ(
x,β)=e
−iφ
ψ(
x, 0) (1)
in the imaginary time direction. Here β = 1/T is the inverse
temperature and φ ∈[0, 2π) is an arbitrary angle. Hence, physi-
cal
fermions, which obey antiperiodic boundary conditions, corre-
spond
to φ =π .
The
dual quark condensate
n
is then defined as the Fourier
transform with respect to φ,
n
=−
2π
0
dφ
2π
e
−inφ
¯
ψψ
φ
, (2)
http://dx.doi.org/10.1016/j.physletb.2015.06.031
0370-2693/
© 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by
SCOAP
3
.