Eur. Phys. J. C (2015) 75:495
DOI 10.1140/epjc/s10052-015-3720-2
Regular Article - Theoretical Physics
Critical behaviors near the (tri-)critical end point of QCD
within the NJL model
Ya Lu
1
,Yi-LunDu
1
, Zhu-Fang Cui
1,3
, Hong-Shi Zong
1,2,3,a
1
Department of Physics, Nanjing University, Nanjing 210093, China
2
Joint Center for Particle, Nuclear Physics and Cosmology, Nanjing 210093, China
3
State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, CAS, Beijing 100190, China
Received: 4 August 2015 / Accepted: 2 October 2015 / Published online: 19 October 2015
© The Author(s) 2015. This article is published with open access at Springerlink.com
Abstract We investigate the dynamical chiral symmetry
breaking and its restoration at finite density and tempera-
ture within the two-flavor Nambu–Jona–Lasinio model, and
mainly focus on the critical behaviors near the critical end
point (CEP) and tricritical point (TCP) of quantum chro-
modynamics. The multi-solution region of the Nambu and
Wigner ones is determined in the phase diagram for the mas-
sive and massless current quark, respectively. We use the var-
ious susceptibilities to locate the CEP/TCP and then extract
the critical exponents near them. Our calculations reveal that
the various susceptibilities share the same critical behaviors
for the physical current quark mass, while theyshow different
features in the chiral limit.
1 Introduction
Quantum chromodynamics (QCD) is often viewed as the
basic theory of strong interactions, whose degrees of free-
dom include quarks and gluons. As a non-Abelian gauge
field theory, QCD exhibits two important features: asymp-
totic freedom and color confinement. It works well in the
large-momentum-transfer processes through the perturbative
techniques due to the feature of asymptotic freedom. How-
ever, in the small-momentum-transfer processes, the cou-
pling constant becomes so strong that problems have to be
treated by many non-perturbative methods, such as lattice
QCD [1], Nambu–Jona–Lasinio (NJL) model [2,3], Dyson–
Schwinger equations [4], and so on.
It is believed that the strongly interacting matter under-
goes some phase transition and the chiral symmetry is par-
tially restored at high temperature (T ) and/or quark chem-
ical potential (μ), from the hadronic matter to the quark–
gluon plasma (QGP). This phase transition is expected to be
a
e-mail: zonghs@nju.edu.cn
produced in on-going heavy-ion collision experiments, such
as the BNL Relativistic Heavy-Ion Collider (RHIC) and the
CERN Large Hadron Collider (LHC) [5–8]. In addition, high
density QCD matter at low temperature is anticipated in the
compact stars physics [9–11].
Lattice QCD works well for low μ and finite T .How-
ever, the lattice simulations have trouble in studying the QCD
phase transitions with finite μ due to the severe fermion sign
problem. Therefore simpler and mathematically tractable
models which respect the essential symmetries of the QCD
are necessary. NJL model captures fundamental features of
QCD itself, such as the dynamical chiral symmetry breaking
in the vacuum, and can therefore yield profound insight into
the critical behaviors associated with chiral symmetry.
Many effective theories [12–14] of QCD predict the exis-
tence of the tricritical point (TCP) and the critical end point
(CEP) in the QCD T − μ phase diagram in massless and
massive quark cases, respectively. TCP is a point where three
coexisting phases become identical simultaneously. The chi-
ral phase transition is of first order at high μ and low T , then
turns to be second order after the TCP. The second-order tran-
sition line is expected to be in the universality class of three
dimensional O(4) symmetric spin model, also called O(4)
line. With the current quark masses increasing from zero to
the physical masses, TCP emerges at the CEP at which the
first-order phase transition line ends from higher μ toward
lower μ. Meanwhile, the second-order phase transition turns
into a smooth crossover at lower μ and higher T in the phase
diagram. Different from the O(4) line, the CEP is expected to
be in the Z
2
universality class [15], this is just one of several
possible scenarios for QCD.
Both at the TCP and CEP, second-order phase transitions
are expected to occur. Therefore, the values of critical expo-
nents will be an interesting and important question. The crit-
ical exponents in mean field approximation have been pre-
dicted, by expanding the Ginzburg–Landau thermodynamic
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