# Non-extensive Effects on the QCD Equation of State and Fluctuations of
Conserved Charges within Polyakov Quark Meson Model

Abdel Magied DIAB  0000-0002-5603-1952<EMAIL_ADDRESS>Faculty of
Engineering, Modern University for Technology and Information (MTI), 11571
Cairo, Egypt.

###### Abstract

The influence of non-extensive Tsallis statistics on the hadron phase
structure has been investigated using the Polyakov-quark-meson (PQM) model.
The analysis examines the non-extensive effects on the temperature dependence
of PQM order parameters, thermodynamic quantities related to the QCD equation
of state, and fluctuations of conserved charges at varying chemical
potentials. The results show that non-extensive effects have the most
significant deviations near the crossover region. The pseudo-critical
temperature $T_{\chi}(\mu_{B})$ is not a universal constant and decreases with
increasing non-extensive $q$ parameter. The chiral phase diagram of the PQM
model indicates a decrease in the behavior of the ($T_{\chi}-\mu_{B}$) plane
with increasing non-extensive $q$ parameter. The PQM model exhibits good
qualitative agreement with lattice QCD calculations. Moreover, these findings
suggest the existence of a Tsallis limit, which serves as an alternative to
the Stefan-Boltzmann (SB) limit for the massless ideal gas. The critical
endpoint (CEP) exhibits lower temperature but higher chemical potential with
increasing non-extensive $q$ parameter. Overall, this study highlights the
importance of non-extensive Tsallis statistics in characterizing the quark-
hadron phase structure of the PQM model and contributes to a deeper
understanding of non-extensive effects in the quark-hadron phase transition.

Chiral symmetries, Chiral transition, Chiral Lagrangian, Numerical
approximation and analysis, Statistical models.

###### pacs:

11.30.Rd, 11.10.Wx, 12.39.Fe, 02.60.-x, 12.40.Ee.

##  I Introduction

Power-law distributions have been extensively explored across different
scientific disciplines, including the well-known Tsallis distribution
tsallis1988possible . The Tsallis distribution, which is an extension of the
Boltzmann-Gibbs (BG) distribution with an additional parameter Tsallis:2009zex
, has gained significant attention in the past few decades. It has been widely
employed to analyze various phenomena Wilk:1999dr ; Rybczynski:2014cha ;
Lavagno:2010xu ; Lavagno:2012yn ; Trevisan:2013qoa ; Gervino:2013uoa ;
Lavagno:2013qca ; Frigori:2014fla ; Rozynek:2015iea ; Ishihara:2014txa ;
Ishihara:2017txj ; Ishihara:2018yxp ; Ishihara:2019ran ; Rozynek:2019qzq ;
Rahaman:2021xqv ; Zhao:2020xob ; Islam:2023zpl ; Rozynek:2015zca ;
Rozynek:2009zh ; Wilk:2006dv , among many others such as astrophysics
Tsallis:2012js ; Lavagno:2011zm  and optical lattice Douglas:2006zz . For
instance, the Tsallis distribution has been used to describe the momentum
distribution observed in high-energy collision experiments. Consequently, this
framework has been widely accepted to analyze transverse momentum
distributions observed in high-energy collisions. Notably, the PHENIX and STAR
collaborations at the Relativistic Heavy Ion Collider (RHIC) in BNL
STAR:2006nmo ; PHENIX:2011rvu , as well as the ALICE, ATLAS, and CMS
collaborations at the Large Hadron Collider (LHC) in CERN ALICE:2011gmo ;
ALICE:2014juv ; CMS:2010tjh ; ATLAS:2010jvh , have been at the forefront of
utilizing the Tsallis distribution in experimental particle physics. As a
result of the small size and rapid evolution of the Quark-Gluon Plasma (QGP),
it exhibits a non-uniform distribution and lacks the global equilibrium.
Consequently, certain quantities within the system display non-extensive
behavior, characterized by power-law distributions instead of exponential
distributions.

It is worth noting that BG statistics huang2009introduction ;
dalfovo1999theory  are commonly employed to describe the quantum
chromodynamics (QCD) phase transition. However, it is essential to recognize
that BG statistics can only be applied to systems in equilibrium and within
the thermodynamic limit. In relativistic heavy-ion collisions, the QGP
produced undergoes strong intrinsic fluctuations and exhibits long-range
correlations Sharma_2019 ; Bhattacharyya_2018 . The size of the QGP is
sufficiently small, and its evolution is rapid. Consequently, this system is
far from being equilibrium. It is not appropriate to apply the BG statistics
to evaluate the energies and numbers of particles in such a system exhibits
long-range correlations Stephanov:1998dy . It is evident that the introduction
of the non-extensive $q$ parameter in the PQM model plays a role in capturing
the deviations from standard Boltzmann-Gibbs (BG) statistics in equilibrium to
non-equilibrium states. The studying of the phase structure of QCD matter,
non-extensive statistics were first proposed by C. Tsallis in
tsallis1988possible , where non-extensivity is understood in the thermodynamic
sense. The so-called Tsallis statistics propose the entropy as
$S_{q}=\mbox{Tr}(\hat{\rho}-\hat{\rho}^{q})/(q-1)$ and the density matrix as
$\hat{\rho}=\exp_{q}\big{(}-E/T\big{)}/Z_{q}$, where $Z_{q}$ is the
corresponding partition function. Here, $q$ is a real number that
characterizes the non-extensive factors of such a system. The Tsallis
distribution can be characterized by two essential parameters: the non-
extensive $q$ parameter, which controls the transition between power-law and
exponential behavior, and the Tsallis temperature $T$. Additional parameters
such as mass and chemical potentials can be also included to further describe
the distribution. When the $q$-parameter exceeds one, the Tsallis distribution
exhibits a power-law behavior, whereas it converges to the conventional BG
distribution as the q-parameter approaches one.

To address the limitations of lattice QCD calculations in dealing with non-
extensive Tsallis statistics, researchers utilize various effective models to
explore the phase diagram of QCD theoretically. These models include chiral
perturbation theory Espriu:2020dge , Dyson-Schwinger Equations (DSEs)
Zhao:2019ruc ; Fischer:2018sdj ; Ayala:2011vs , based on density operator
Ishihara:2018yxp , hadron resonance gas and statistical models Kyan:2022eqp ;
Parvan:2023jnx , Nambu-Jona-Lasinio model and the Polyakov Nambu-Jona-Lasinio
(PNJL) model Rozynek:2015zca ; Zhao:2020wks ; Zhao:2020xob ; Zhao:2022ccj ;
Zhao:2023xpj , linear sigma model and Polyakov linear sigma model Shen:2017etj
; Ishihara:2016moh ; Ishihara:2018cgj .

The PQM model goes beyond merely predicting the order of the chiral phase
transition by combining two fundamental non-perturbative aspects of low-energy
QCD: confinement and spontaneous chiral symmetry breaking in a simple
approach. It provides a good interpretation of lattice QCD simulations at zero
baryon density and finite temperature. Most effective models, including the
PQM model, describe the hadron-quark phase transition based on quark degrees
of freedom. The degrees of freedom in $U(N_{f})_{r}\times U(N_{f})_{\ell}$ PQM
model encompass chiral meson states, where the number of quark flavors is
$N_{f}$. In the vacuum, the latter serve as (pseudo-) Goldstone bosons
resulting from chiral symmetry breaking Roder:2003uz . Consequently, thePQM
model can also be viewed as an effective theory for low-energy degrees of
freedom of the QCD. At high temperatures, chiral symmetry undergoes
restoration, causing the chiral partners among meson states to become
degenerate in mass. It is particularly suited to describe the chiral phase
structure of meson states Tawfik:2014gga ; Tawfik:2019rdd . In PQM model, we
can not involve the degrees of freedom of hadrons directly, but this effective
low-energy QCD model captures the confinement phase by suppressing the
excitations of the quark states Fu:2007xc . In this context, the term
”confinement” refers to the statistical suppression of the propagation of
color-triplet quarks. Similarly, the chiral symmetry is explicitly breaking
due to the nonzero current masses of quarks. However, the chiral condensate
remains a valuable indicator of the chiral crossover. The validity of the PQM
model has been confirmed through a series of works that involved comparisons
with Lattice QCD simulations, for instance, Tawfik:2014uka ; Tawfik:2016edq ;
AbdelAalDiab:2018hrx ; Tawfik:2019kaz , including the influence of the isospin
asymmetry Tawfik:2019tkp  and chiral magnetic effect Ezzelarab:2015tya ;
Tawfik:2016lih ; Tawfik:2016gye ; Tawfik:2016ihn ; Tawfik:2017cdx ;
Tawfik:2021eeb . Moreover, the Polyakov loop extended quark meson model
containing vector and axial vector mesons Kovacs:2016juc  and large
$N_{c}$-limit Kovacs:2022zcl .

The motivation behind this paper is to examine the effects of non-extensive
Tsallis statistics on the chiral phase structure, thermodynamic properties,
and fluctuations of conserved charges in QCD matter at finite temperature $T$
and chemical potentials $\mu$. To accomplish this, we extend the PQM model by
incorporating non-extensive Tsallis statistics, allowing us to compare the
results with those obtained using BG statistics.

The paper is organized as follows: Sec. II presents the formalism of the PQM
model incorporating non-extensive Tsallis statistics. In Sec. III, we analyze
the influence of the non-extensive $q$ parameter on PQM Order Parameters
(Subsec. III.1), explore the chiral phase diagram for different selections of
the non-extensive $q$ parameter, especially the location of CEP due to the
non-extensive effects (Subsec. III.2), investigate the impact on various
thermodynamic quantities with different non-extensive $q$ parameters (Subsec.
III.3), and examine the effects on the fluctuations and correlations of
conserved charges (Subsec. III.4). Finally, Sec. IV provides a brief summary
and conclusion.

##  II Non-extensive Tsallis Statistics in the Polyakov Quark Meson Model

The Lagrangian of the PQM model can be incorporated from chiral Lagrangian of
$N_{f}=3$ quark flavors and the Polyakov-loop potential as

$\displaystyle\mathcal{L}_{PQM}$ $\displaystyle=$
$\displaystyle\mathcal{L}_{\mbox{chiral}}-\mathbf{\mathcal{U}}(\phi,\bar{\phi},T)=\mathcal{L}_{\bar{\psi}\psi}+\mathcal{L}_{\mathcal{M}}-\mathbf{\mathcal{U}}(\phi,\bar{\phi},T).$
(1)

The first term of rhs of Eq. (1), gives the contributions of quarks (fermions)
with three-color degrees-of-freedom $N_{c}=3$,

$\displaystyle\mathcal{L}_{\bar{\psi}\psi}$ $\displaystyle=$
$\displaystyle\sum_{f}\overline{\psi}_{f}(i\gamma^{\mu}D_{\mu}-g\,T_{a}(\sigma_{a}+i\gamma_{5}\pi_{a}))\psi_{f},$
(2)

where $D_{\mu},\;\mu,\;g,\gamma^{\mu}$ and $\psi$ are defined as the covariant
derivative, Lorentz index, Yukawa coupling constant, chiral spinors and Dirac
spinor fields, respectively. The index $f$ refers to quark flavors as
$f=[(l=u,d),s]$. The second term in rhs of Eq. (1) shows the contributions of
mesons (bosons),

$\displaystyle\mathcal{L}_{\mathcal{M}}$ $\displaystyle=$
$\displaystyle\mathrm{Tr}\Big{[}\partial_{\nu}\mathcal{M}^{{\dagger}}\partial^{\nu}\mathcal{M}-m^{2}\mathcal{M}^{{\dagger}}\mathcal{M}\Big{]}-\lambda_{1}\,\Big{(}\mathrm{Tr}\,\Big{[}\mathcal{M}^{{\dagger}}\mathcal{M}\Big{]}\Big{)}^{2}-\lambda_{2}\,\mathrm{Tr}\Big{[}\mathcal{M}^{{\dagger}}\mathcal{M}\Big{]}^{2}+$
(3) $\displaystyle+$ $\displaystyle
c\Big{(}\mathrm{Det}[\mathcal{M}]+\mathrm{Det}[\mathcal{M}^{{\dagger}}]\Big{)}+\mathrm{Tr}\Big{[}H(\mathcal{M}+\mathcal{M}^{{\dagger}})\Big{]},\hskip
17.07164pt$

where $\mathcal{M}$ is a complex $(3\times 3)$ matrix which can be given for
nonet meson states as,
$\bar{\mathcal{M}}=\sum_{a=0}^{N_{f}^{2}-1}T_{a}(\bar{\sigma_{a}}+i\bar{\pi_{a}})$.
In U($3$) algebra, the generator operator $T_{a}=\hat{\lambda}_{a}/2$ can be
estimated from Gell-Mann matrices $\hat{\lambda}_{a}$ Weinberg:1972kfs  with
the indices $a=0,\cdots,\,8$. The parameters, $m^{2}$, $h_{l}$, $h_{s}$,
$\lambda_{1}$, $\lambda_{2}$, and $c$ are given in dependence on the mass of
the sigma meson $m_{\sigma}$ Schaefer:2008hk . The present work used the
values of these parameters at $m_{\sigma}=800~{}$MeV Schaefer:2008hk . The
chiral symmetry is explicitly broken by ($3\times 3$) matrix $H=T_{a}h_{a}$,
the explicitly symmetry breaking terms $h_{a}$ can be determined from the
partially conserved axial current (PCAC) relations Lenaghan:2000ey ;
Tawfik:2019tkp .

The third term in rhs of Eq. (1), $\mathcal{U}(\phi,\bar{\phi},T)$ represents
the potential of the Polyakov-loop variables, this potential incorporates the
color gluonic degrees-of-freedom and dynamics of the quark-gluon interactions
into the chiral Lagrangian. The potential is obtained by calculating the
thermal expectation value of a color-traced Wilson-loop in the temporal space
as $\phi=(\mathrm{Tr}_{c}\,\mathcal{P})/N_{c}$, and its hermitian conjugate
$\bar{\phi}=(\mathrm{Tr}_{c}\,\mathcal{P}^{{\dagger}})/N_{c}$, where
$\mathcal{P}$ is the matrix the Polyakov loop operator in SU$(N_{c})$ gauge
group as,

$\displaystyle\mathcal{P}(\vec{x})=P\exp\Big{(}{i\int_{0}^{1/T}d\tau
A_{0}(\vec{x},\tau)}\Big{)},$ (4)

where $A_{0},P$ are the temporal vector field and path ordering, respectively.
The expectation value of the Polyakov loops is finite and approach to the
unity (deconfined phase) at high temperatures and it vanishes or small values
at low tempertures (confined phase) Schaefer:2009ui . Therefore, this order
parameter is a good indicator of the (de)confinement transition phase
transition. Various expressions have been suggested for the potentials of the
Polyakov loop variables, which are generally considered to represent the QCD
symmetries in pure-gauge theory. For further information, Ratti:2005jh ;
Roessner:2006xn ; Schaefer:2007pw ; Fukushima:2008wg ; Lo:2013hla . In the
present work, we utilize the effective polynomial expression of Polyakov loop
variables, Eq.(5), proposed in Ref.Ratti:2005jh ; Pisarski:2000eq ;
Scavenius:2002ru  and compared with Fukushima potential, Eq.(6), as in Ref.
Fukushima:2008wg . Both of these expressions incorporate higher-order
Polyakov-loop variables and are derived from strong coupling simulations:

  * •

The polynomial form of the Polyakov-loop potential (Pot. I) Ratti:2005jh ;
Pisarski:2000eq ; Scavenius:2002ru  is given by,

$\displaystyle\mathbf{\mathcal{U}}_{\mathrm{Poly}}(\phi,\bar{\phi},T)$
$\displaystyle=$ $\displaystyle
T^{4}\Big{(}-\frac{b_{2}(T)}{2}\,\bar{\phi}\phi-\frac{b_{3}}{6}\,(\bar{\phi}^{3}+\phi^{3})+\frac{b_{4}}{4}\,(\bar{\phi}\phi)^{2}\Big{)},$
(5)

with parameters are a temperature dependent Ratti:2005jh ; Pisarski:2000eq ;
Scavenius:2002ru  as
$b_{2}(T)=a_{0}+a_{1}(T_{0}/T)+a_{2}(T_{0}/T)^{2}+a_{3}(T_{0}/T)^{3}$.

  * •

While the Fukushima version from Polyakov-loop potential (Pot. II)
Fukushima:2008wg  reads as,

$\displaystyle\mathbf{\mathcal{U}}_{\mathrm{Fuku}}(\phi,\bar{\phi},T)$
$\displaystyle=$
$\displaystyle-b\;T\left[54\,\phi\,\bar{\phi}\;\exp(-a/T)+\ln(1-6\phi\bar{\phi}-3(\phi\bar{\phi})^{2}+4(\phi^{3}+\bar{\phi}^{3}))\right],$
(6)

where $a$ and $b$ are fixed parameters Fukushima:2008wg . The partition
function can be formulated for a spatially homogeneous system in thermal
equilibrium at a finite temperature $T$ and quark chemical potentials
$\mu_{f}$. The exchange in particle and antiparticle can be governed by the
grand canonical partition function. By performing a path integral over the
(anti)quark, and meson fields, the thermodynamic grand-canonical potential can
be deduced as,

$\displaystyle\Omega(T,\mu_{f})$ $\displaystyle=$
$\displaystyle\frac{-T\;\cdot\ln{\left[\mathcal{Z}\right]}}{V}=\frac{-T}{V}\ln\mathrm{Tr}\Big{[}\exp{\Big{(}\frac{-1}{T}\int
d^{3}x(\mathcal{H})-\mu_{f}(\psi^{\dagger}\psi\Big{)}}\Big{]},$ (7)

where $\mu_{f}$ refers to the chemical potentials of the quark flavors
$f=[l=(u,d),s]$, which are associated to the conserved charge numbers of - for
instance - baryon ($B$), electric charge ($Q$) and strangeness ($S$) of each
quark flavor,

$\displaystyle\mu_{u}=\frac{\mu_{B}}{3}+\frac{2\mu_{Q}}{3},\qquad\qquad\mu_{d}=\frac{\mu_{B}}{3}-\frac{\mu_{Q}}{3},\qquad\qquad\mu_{s}=\frac{\mu_{B}}{3}-\frac{\mu_{Q}}{3}-\mu_{S}.$
(8)

Using the mean field approximation (MFA), the PQM thermodynamic potential
$\Omega$ can be obtained as,

$\displaystyle\Omega(T,\mu_{f})$ $\displaystyle=$ $\displaystyle
U(\sigma_{l},\,\sigma_{s})+\Omega_{\bar{\psi}\psi}(T,\,\mu_{f})+\mathbf{\mathcal{U}}(\phi,\bar{\phi},T).$
(9)

More details about the structure of the PQM model can be found in Refs.
Tawfik:2014gga ; Tawfik:2021eeb . The first term in rhs of Eq. (9) gives the
potential of the contribution of mesons based on non-strange $\sigma_{l}$ and
strange $\sigma_{s}$ condensate as,

$\displaystyle U(\sigma_{l},\,\sigma_{s})$ $\displaystyle=$ $\displaystyle-
h_{l}\,\sigma_{l}-h_{s}\,\sigma_{s}+\frac{m^{2}}{2}\Big{(}\sigma_{l}^{2}+\sigma_{s}^{2}\Big{)}-\frac{c}{2\sqrt{2}}\sigma_{l}^{2}\,\sigma_{s}+\frac{\lambda_{1}}{4}\Big{(}\sigma_{l}^{2}+\sigma_{s}^{2}\Big{)}^{2}+\frac{\lambda_{2}}{4}\Big{(}\frac{\sigma_{l}}{4}+\sigma_{s}^{4}\Big{)},$
(10)

where the parameters and coupling constants are listed in Eq. (10) are fixed
for $m_{\sigma}=800~{}$MeV Schaefer:2008hk . The second term in rhs of Eq. (9)
shows the potential of quark and anti-quark contributions in finite
temperature $T$ and quark chemical potential $\mu_{f}$ as Fukushima:2003fw ,

$\displaystyle\Omega_{\bar{\psi}\psi}(T,\mu_{f})$ $\displaystyle=$
$\displaystyle-2N_{c}\,N_{f}\int\frac{d^{3}k}{(2\pi)^{3}}\Big{\\{}\;E_{f}+\frac{T}{N_{c}}\;\Big{(}\ln{[\mathcal{F_{D}^{(+)}}(T,\;\mu_{f})]}+\ln{[\mathcal{F_{D}^{(-)}}(T,\;\mu_{f})]}\Big{)}\Big{\\}},$
(11)

where $\mathcal{F_{D}^{(+)}}(T,\;\mu_{f})$ and
$\mathcal{F_{D}^{(-)}}(T,\;\mu_{f})$ are defined as the Fermi-Dirac
distribution functions in the present of the dynamics of the quark-gluon
interactions by Polyakov loop variables - in a widely used notations - as,

$\displaystyle\mathcal{F_{D}^{(+)}}(T,\;\mu_{f})$ $\displaystyle=$
$\displaystyle
1+3\left(\phi\,+\bar{\phi}\;e^{-\frac{E_{f}^{(+)}}{T}}\right)e^{-\frac{E_{f}^{(+)}}{T}}+e^{-3\frac{E_{f}^{(+)}}{T}},$
(12)

where $E_{f}^{(\pm)}=E_{f}\mp\mu_{f}$ are the energy-momentum dispersion
relations, in which the upper sign is applied for quarks and the lower sign
for antiquarks. The quark-flavor energy-momentum dispersion relations are
given as $E_{f}=(k^{2}+m_{f}^{2})^{1/2}$, where $m_{f}$ is the constituent
quark masses for non-strange quark sectors $m_{l}=g\sigma_{l}/2$ and strange
quark sector $m_{s}=g\sigma_{s}/\sqrt{2}$. One can estimate that the term
$\mathcal{F_{D}^{(-)}}(T,\;\mu_{f})$, by replacing $E_{f}^{(+)}$ with
$E_{f}^{(-)}$ and the the Polyakov-loop variable $\phi$ with its conjugate
$\bar{\phi}$ and vice versa. After estimating the PQM thermodynamic potential
in MFA, one can determine the thermodynamic quantities that describe the
chiral phase structure of quark-hadron matter at finite temperatures and
finite chemical potentials. These quantities can be obtained by examining the
behaviors of the PQM order parameters.

In mean field approximation, one can estimate the quark condensates
$\bar{\sigma}_{l},\;\bar{\sigma}_{s}$ and the expectation values of the
Polyakov-loop variables $\bar{\phi},\;\bar{\bar{\phi}}$ by searching the
global minimization of the real part of the he PQM thermodynamic potential
$\mathcal{R}\;[\Omega(T,\,\mu)]$, Eq. (9), with respect to the associated
order parameter, at a saddle point,

$\displaystyle\frac{\partial\Omega}{\partial\sigma_{l}}=\frac{\partial\Omega}{\partial\sigma_{s}}=\frac{\partial\Omega}{\partial\phi}=\frac{\partial\Omega}{\partial\bar{\phi}}\Bigg{|}_{\sigma_{l}=\bar{\sigma}_{l},\sigma_{s}=\bar{\sigma}_{s},\phi=\bar{\phi},\bar{\phi}=\bar{\bar{\phi}}}=0.$
(13)

However, the Polyakov-loop is a complex matrix. Thus, the PQM thermodynamic
potential, Eq. (9), is a functional of complex variables. The physical
quantities of the imaginary part of this potential vanish Sasaki:2006ww . The
PQM order parameters are given by minimizing the real part of the
thermodynamic potential globally at a saddle point. They are determined by
solving the four equations as a complete set of equations. The effective
action turns complex, and only the saddle point can be defined, which is
generally not a global minimum. However, depending on the directions of
approach to the saddle point. Where the gap equations are well-defined, the
behaviors of PQM order parameters are analyzed.

The non-extensive Tsallis statistics is characterized by the substitution of
the conventional exponential factors with their $q$-exponential counterparts
as,

$\displaystyle f(x)$ $\displaystyle=$ $\displaystyle\exp{(x)}\longrightarrow
f_{q}(x)=\exp_{q}{(x)}=\Big{(}1+(1-q)x\Big{)}^{\frac{1}{1-q}},$ (14)
$\displaystyle g(x)$ $\displaystyle=$ $\displaystyle\ln{(x)}\longrightarrow
g_{q}(x)=\ln_{q}{(x)}=\frac{x^{1-q}-1}{1-q}.$ (15)

The non-extensive $q$ parameter indicates all possible factors does not
contain in BG Statistics. As $q$ approaches unity, the non-extensive Tsallis
statistics converges to the Boltzmann-Gibbs statistics. For this case study,
we will make two assumptions. Firstly, we will not directly incorporate non-
extensive effects on the pure Yang-Mills sector. Consequently, the Polyakov-
loop potential remains unchanged, but the non-extensive impacts on Polyakov-
loop variables are indirectly included through the saddle point, Eq. (13).
Secondly, the fixed parameters selected for the present approach remain
unchanged. At zero temperature and zero chemical potential, these parameters
are fitted to match the observed calculations. We can rely on the assumption
that these parameters can be utilized at finite temperature and quark chemical
potential. Thus,the potential of quark and anti-quark contributions, Eq. (11),
can be written within non-extensive $q$ parameter as,

$\displaystyle\Omega_{q}(T,\mu_{f})$ $\displaystyle=$
$\displaystyle-2N_{c}\,N_{f}\int\frac{d^{3}k}{(2\pi)^{3}}\Big{\\{}\;E_{f}+\frac{T}{N_{c}}\;\Big{(}\ln_{q}{[\mathcal{F_{D}^{(+)}}_{q}(T,\;\mu_{f})]}+\ln_{q}{[\mathcal{F_{D}^{(-)}}_{q}(T,\;\mu_{f})]}\Big{)}\Big{\\}},$
(16)

where

$\displaystyle\mathcal{F_{D}^{(+)}}_{q}(T,\;\mu_{f})$ $\displaystyle=$
$\displaystyle
1+3\left(\phi\,+\bar{\phi}\;e_{q}^{-\frac{E_{f}^{(+)}}{T}}\right)e_{q}^{-\frac{E_{f}^{(+)}}{T}}+e_{q}^{-3\frac{E_{f}^{(+)}}{T}}.$
(17)

The effective pattern has a complex structure and scaling properties arising
from self-energy interactions, characterized as a so-called thermofractal
system Deppman:2016fxs ; Deppman:2017fkq . This system can be described by the
Yang-Mills theory Callan:1970yg ; Symanzik:1970rt . The thermofractal
structure in its thermodynamic distributions can be be follow from the non-
extensive thermodynamics proposed by Tsallis tsallis1988possible . The
thermofractal system with quantum fields was proposed in Ref. Deppman:2019yno
. However, the thermofractal system leads to a recursive equation that
provides the running-coupling constant as given in Deppman:2019yno .

$\displaystyle
g(\epsilon_{0})=\prod_{i=1}^{2}\Big{(}1-(q-1)\frac{\epsilon_{i}}{\lambda}\Big{)}^{\frac{1}{q-1}},$
(18)

where $\lambda$ and $\epsilon_{i}$ are defined as a scale parameter and energy
the $i$th patron in the interaction. Furthermore, the beta-function can be
calculated as Deppman:2019yno ,
$\beta_{g}=-\big{(}g^{3}/16\pi^{2}\big{)}(q-1)^{-1}$, scaling the properties
of the QCD, one can obtain the beta-function of the QCD is
$\beta_{QCD}=-\big{(}g^{3}/16\pi^{2}\big{)}\Big{[}\big{(}11N_{c}/3\big{)}-\big{(}2N_{f}/3\big{)}\Big{]}$
Politzer:1974fr ; Gross:1973id . Thus, the non-extensive $q$ parameter remains
a phenomenological parameter, with $q>1$, and is considered as a measure of
non-additivity. It can be calculated in terms of $N_{c}$ and $N_{f}$ as
Deppman:2019yno ,

$\displaystyle q=1+\frac{3}{11N_{c}-2N_{f}}.$ (19)

The non-extensive $q$ parameter describes the gauge field interaction.
Moreover, it can measure the sensitivity of the effective parton to its number
degrees of freedom. In SU($3$) algebra, the non-extensive $q$ parameter is
approximated to be $q\approx 1.143$, which is in good agreement with
experimental results giving $q=1.14\pm 0.01$ Sena:2012ds ; Marques:2015mwa .
Similarly, for SU($2$) algebra, this approximation gives $q\approx 1.103$. In
the present paper, we consider $q>1$, this because the non-extensive $q$
parameter has the typical value for high energy collisions in range $1\leq
q\leq 1.2$ Marques:2015mwa ; Cleymans:2013rfq ; Li:2013kca ; Azmi:2014dwa . In
the large $N_{c}$-limit of QCD, where mesonic degrees of freedom become more
prominent, the theory undergoes simplifications. In this context, the non-
extensive $q$ parameter tends to approach one. This tendency arises from the
substantial simplification of QCD dynamics in this limit, resulting in more
additive behavior reminiscent of standard BG statistics, where $q$ equals one.
To ensure that the $q$-exponential is always a positive real function, we
assumed this condition$\big{(}1+(1-q)x\big{)}\geq 0$,

$\displaystyle\exp_{q}{(x)}=\Big{(}1+(1-q)x\Big{)}^{\frac{1}{1-q}}=0,\quad\quad\mathrm{for}\quad\Big{(}1+(1-q)x\Big{)}<0.$
(20)

This condition is known as the Tsallis cut-ff prescription Rozynek:2015zca .
To avoid this condition, one can assume,

$\displaystyle\exp_{q}{(x)}=\left\\{\begin{array}[]{cc}\Big{(}1+(1-q)x\Big{)}^{\frac{1}{1-q}},\quad\quad\mathrm{for}\quad
x\leq 0,\\\ \Big{(}1+(q-1)x\Big{)}^{\frac{1}{q-1}},\quad\quad\mathrm{for}\quad
x>0.\end{array}\right.$ (23)

Further details can be found in Ref. Rozynek:2015zca . In this study, we
investigate the chiral structure of the quark-hadron phase transition for
different selections of the non-extensive $q$ parameter at various chemical
potentials.

##  III Results and Discussion

This section explores the properties of quark-hadron phase structure using a
PQM model with non-extensive Tsallis statistics. The results focus on various
thermodynamic observables such as chiral condensates, deconfinement order
parameters, subtracted condensates, and pseudo-critical temperatures,
thermodynamic quantities, as well as fluctuations and correlations of
conserved charges at finite temperature.

###  III.1 Chiral Condensates and Deconfinement Order Parameters

Fig. 1: Temperature dependence of the normalized (non)strange chiral
condensates ($\sigma_{\ell}/\sigma_{\ell o}$) $\sigma_{s}/\sigma_{so}$ (left
panel (a, d, g)) and the order parameter of Polyakov loop variables for the
polynomial potential (middle panel (b, e, h)) and the Fukushima potential
(right panel (c, f, i)) at different selection of chemical potentials
$\mu_{f}=0.0~{}$ (upper panel) $100.0$ (middle panel) and $150.0~{}$MeV (lower
panel) are shown for different values of the non-extensive parameter $q=1.0$
(solid curves), $1.05$ (dashed curves), $1.103$ (dotted curves) and $1.143$
(dotted-dash curves), respectively.

It is instructive to provide several calculations to demonstrate the influence
of non-extensive effects on the chiral phase transition and the deconfinement
order parameters of the PQM model. In order to study the phase transition of a
system, we can minimize the thermodynamic potential with respect to the
corresponding mean field, as given by Eq. (13). Fig. 1 illustrates the effects
of the non-extensive $q$ parameter on: (left panels) the temperature
dependence of the quark constituents for normalized nonstrange and strange
condensates ($\sigma_{\ell}/\sigma_{\ell o}$, $\sigma_{s}/\sigma_{so}$);
(middle panels) the order parameter of the Polyakov-loop variables ($\phi$,
$\bar{\phi}$) for the polynomial form of the Polyakov-loop potential; and the
left panels are the same as middle panels, but for the Fukushima version, at
different selection from the quark chemical potential $\mu_{f}=0.0$ (upper
panels), $100.0$ (middle panels) and $150.0~{}$MeV (right panels). For $q=1$
gives the usual BG statistics, it is quite similar to the case of $q\neq 1.0$.
At low temperature density the PQM order parameter keep the same values for
different $q$. The left panel (a) shows that the thermal behaviors of
$\sigma_{\ell}/\sigma_{\ell o}$ and $\sigma_{s}/\sigma_{so}$ begins from the
unity as their vacuum condensates $\sigma_{\ell o}=92.5~{}$MeV and
$\sigma_{so}=94.2~{}$MeV. In this region, the impact of the non-extensive $q$
parameter appears to be negligible. However, as the non-extensive $q$
parameter increases, it appears to enhance the phase transition. The behavior
of the chiral condensates is not a sharp phase transition but a smooth
crossover. As $q$ increases, the transition occurs at a lower pseudo-critical
temperature ($T_{\chi}$). The middle (b) and right (c) panels show the
temperature dependence of the order parameters of Polyakov loop variables for
the same values of non-extensive $q$ parameter for Polynomial and Fukushima
potential, respectively. At $\mu_{f}=0.0~{}$MeV, we find that the mean values
of $\phi$ and $\bar{\phi}$ are identical, i.e. $\phi=\bar{\phi}$ for varying
temperature and non-extensive $q$ parameter values. At low temperature
regions, the confined hadron phase is dominant, and the effects of non-
extensive $q$ parameter are negligible. As the temperature increases, the
confined hadron phase is transformed into the deconfined phase. The thermal
evolution of the deconfinement phase transition appears to be smooth. We
obtain the same conclusion regarding the non-extensive effects on chiral
condensate, where the transition occurs at a smaller deconfined temperature
$T_{\phi}$. The thermal behavior of the order parameters of Polyakov loop
variables obtained in the polynomial potential is larger than that obtained in
the Fukushima potential. At higher temperatures, the expectation values of the
Polyakov loop variables for the polynomial potential (middle panels) can
exceed the unity, which is considered unphysical and is related to the free
energy of a static color charge. However, these results are in agreement with
previous studies, as found in Fu:2007xc ; Mao:2009aq ; Schaefer:2009ui ;
Gupta:2009fg . Furthermore, the Polyakov loop variables are sensitive to the
geometry and topology of spacetime, which may exhibit behavior beyond the
confined phase limits. In ansatz, the Fukushima potential replaces the higher-
order terms of the $\phi$ and $\bar{\phi}$ polynomial potential with the
logarithm of the Jacobi determinant, effectively avoiding divergences and
expectation values larger than one. Furthermore, the PQM order parameters at
$\mu_{f}=100.0$ (d), and $150.0~{}$MeV (g), it appears that the effect of
chemical potential is negligible at low-temperature region. However, as the
chemical potential increases, it enhances the phase transition. We observe
that increasing the chemical potential helps reduce the impact of the non-
extensive $q$ parameter on the phase transition, making it less severe than at
low chemical potential. Also, we notice that the behavior of the
$\sigma_{s}/\sigma_{so}$ remains longer than $\sigma_{\ell}/\sigma_{\ell o}$
which in turn is sensitive to $\mu_{f}$ and $q$ parameters. As the quark
chemical potential $\mu_{f}$ is switched on, the thermal evolution of the
order parameters for the deconfined phase transition becomes distinguishable,
see (middle panel (e, h)) and (right panel (f, i)). The mean field of Polyakov
loop variables $\phi$ (thick curves) differs from $\bar{\phi}$ (thin curves),
i.e., $\phi\neq\bar{\phi}$. It is obvious that the thermal behaviors of $\phi$
(thick curves) are evidently larger than their conjugate $\bar{\phi}$ as the
quark chemical potential increases. The values of $\phi$ and their conjugate
$\bar{\phi}$ increase as the non-extensive $q$ parameter is increased. The
increase in $\mu_{f}$ and $q$ leads to a smooth increase in the Polyakov loop
variables and enhances the deconfined phase transition. It is noteworthy that
the impact of both $\mu_{f}$ and $q$ on the transition leads to the same
conclusion: the deconfined transition occurs at a lower temperature
$T_{\phi}$.

###  III.2 Subtracted Condensates and Pseudo-Critical Temperatures

As previously mentioned, the PQM model incorporates both strange and non-
strange chiral condensates which are indicative of the chiral phase
transitions. It would be valuable to demonstrate the non-extensive effects on
the normalized net-difference or normalized subtracted condensate between the
(non)strange chiral condensates as Haas:2013qwp

$\displaystyle\Delta_{ls}(T,\;\mu_{f})=\frac{\sigma_{l}(T,\;\mu_{f})-(h_{l}/h_{s})\;\sigma_{s}(T,\;\mu_{f})}{\sigma_{\ell
o}-(h_{l}/h_{s})\;\sigma_{so}},$ (24)

where $h_{l}$ ($h_{s}$) are the explicit symmetry breaking parameters for non-
strange (strange) quark sectors which can be obtained from the Dashen-Gell-
Mann-Oakes-Renner (DGMOR) relations GellMann:1968rz ; Dashen:1969eg .

Fig. 2: The same as in Fig. 1, but for the normalized net-difference the
(non)strange chiral condensates $\Delta_{ls}$ (left panel) and the
corresponding chiral susceptibility $\mid\partial\Delta_{ls}/\partial T\mid$
(right panel). The solid band gives the Continuum-limit of lattice QCD results
Borsanyi:2010bp .  Fig. 3: The chiral phase diagram for different selections
of non-extensive $q$ parameter. The PQM results are compared with lattice QCD
simulations Bellwied:2015rza  (solid wide band) and Bazavov:2018mes  (solid
narrow band). Furthermore, these are confronted to the experimental results of
ALICE Andronic:2017pug  (open symbol) STAR experiments Das:2014qca  (closed
symbols). The trajectory of CEP positions (inside-box) is shown as star
symbols.

Figure 2 illustrates the thermal behaviors of the normalized subtracted
condensate between the (non)strange chiral condensates $\Delta_{ls}$ and the
absolute value of the corresponding chiral susceptibility
$\mid\partial\Delta_{ls}/\partial T\mid$ for different values of the non-
extensive parameter $q=1.0$ (solid curves), $1.05$ (dashed curves), $1.103$
(dotted curves) and $1.143$ (dotted-dash curves) and different values of
chemical potential $\mu_{f}=0.0$ (upper panel), $100.0$ (middle panel) and
$150.0~{}$MeV (lower panel). In the upper left-hand panel (a) of Fig. 2, the
PQM results for $q=1$ and $\mu_{f}=0.0~{}$MeV are compared with the related
lattice QCD simulation Borsanyi:2010bp . There is good agreement, especially
within the region of the phase transition. The thermal behavior of
$\Delta_{ls}$ starts from unity and rapidly decreases as temperature increases
until it becomes almost temperature independent at high temperatures. As
$q-$parameter increases, the phase transition occurs at a lower pseudo-
critical temperature ($T_{\chi}$). It is obvious that the non-extensive
effects is negligible on the normalized subtracted condensate at low and high
temperatures.

The right-hand panel of Fig. 2 shows the first derivative of the subtracted
condensate with respect to temperature in thermal and dense QCD medium for
different values of the non-extensive $q$ parameter and varying chemical
potentials $\mu_{f}$. The peak point indicates the pseudo-critical temperature
at which the phase transition occurs. The peak position is shifted to lower
temperatures as the non-extensive $q$ parameter increases. Furthermore, as the
dense medium is released, the chemical potential repairs the phase transition
towards lower temperatures. The obtained results can be used to map out the
chiral phase diagram of PQM model. As discussed before, the pseudo-critical
temperatures in PQM model, $T_{\chi}$, are estimated using different
procedures. We can now map out $T_{\chi}$ at different baryon chemical
potentials, $\mu_{B}=3\,\mu_{f}$, where the pseudo-critical temperature
$T_{\chi}$ can be estimated as,

$\displaystyle T_{\chi}=\mbox{arg\;
max}\;\Big{[}\frac{\partial\Delta_{ls}}{\partial T}\Big{]}.$ (25)

Fig. 3 illustrates the non-extensive effects on the chiral phase diagram of
the PQM model ($T_{\chi}-\mu_{B}$) plane. The PQM results at different values
of $q=1.0$ (solid curves), $1.05$ (dashed curves), $1.103$ (dotted curves) and
$1.143$ (dotted-dash curves) are compared with the lattice QCD simulations
Bellwied:2015rza  (solid wide band), Bazavov:2018mes  (solid narrow band) and
also with the experimental observations in ALICE experiment at LHC (open
symbols) Andronic:2017pug  and STAR experiment at RHIC (closed symbols)
Das:2014qca . The coordinates of ($T_{\chi},\,\mu_{B}$) were estimated by
identifying the temperature at the peak point of the chiral susceptibility for
various non-extensive $q$ parameter. The PQM results for $q=1.0$ (solid curve)
agree well with both lattice QCD calculations and experimental estimations.
The calculations of the PQM model reveal that the chiral phase diagram,
decrease with increasing the non-extensive $q$ parameter. As the non-extensive
$q$ parameter increases, the pseudo-critical temperature decreases gradually
by approximately $\sim 10\%$ compared to the previous value. The impact of
non-extensive effects on the location of the CEPs (star symbols) is
illustrated in Fig. 3 (inside-box). One can find how the positions of the CEPs
vary with the non-extensive $q$ parameter. As $q$ increases, the CEPs shift
rapidly towards higher chemical potential and lower temperature. This is
because systems with fewer particles will experience a higher value of $q$,
leading to phase transitions occurring at higher chemical potentials. The
overlap of the curves forms an approximate straight line in the direction of
lower temperature and higher chemical potential. In other words, the
characteristic CEP should not only be viewed as a universal point but also as
a critical region. Therefore, when analyzing experimental data, it is possible
to encounter a critical region rather than a well-defined CEP. Moreover, the
current search of the CEP position shows good agreement with other previous
studies Rozynek:2009zh ; Shen:2017etj ; Zhao:2023xpj .

###  III.3 Bulk Thermodynamics

Thermodynamic quantities plays role an important role in deriving the features
of hadron quark matter in thermal and dense QCD medium. Furthermore, we can
explore various thermodynamic observables from the thermodynamic pressure,
$p(T,\,\mu_{f})=-\Omega(T,\,\mu_{f})$ and the normalized interaction measure
reads as,

$\displaystyle\frac{I(T,\,\mu_{f})}{T^{4}}=\frac{\epsilon-3p}{T^{4}}=T\;\frac{\partial}{\partial
T}\Big{(}\frac{p}{T^{4}}\Big{)},$ (26)

where $\epsilon(T,\,\mu_{f})=-p(T,\,\mu_{f})+T\,s(T,\,\mu_{f})$ is the energy
density, while $s(T,\,\mu_{f})=\partial p/\partial T$ is the entropy density.
In Stefan-Boltzmann (SB) limit. The thermodynamic pressure can be determined
from the grand canonical partition function for $(N_{c}^{2}-1)$ massless
gluons and $N_{f}$ massless quark flavors as an ideal gas Schaefer:2007pw ,

$\displaystyle\frac{p_{\tiny
SB}}{T^{4}}=(N_{c}^{2}-1)\frac{\pi^{2}}{45}+N_{c}\,N_{f}\Big{(}\frac{7\pi^{2}}{180}+\frac{1}{6}\Big{[}\frac{\mu_{f}}{T}\Big{]}^{2}+\frac{1}{12\pi^{2}}\Big{[}\frac{\mu_{f}}{T}\Big{]}^{4}\Big{)}.$
(27)

The first term in rhs. of Eq. (27) refers to the gluonic contribution, the
second term is the fermonic contribution and the other terms indicates
contributions of ideal gas at finite chemical potential. Furthermore, we list
the thermodynamic quantities in massless ideal gas at finite chemical
potential Bhattacharyya:2016lrk  for non-extensive Tsallis (TS) statistics as,

$\displaystyle\frac{p_{TS}}{T^{4}}$ $\displaystyle=$
$\displaystyle\frac{g}{6\pi^{2}}\;\frac{\big{(}1-\delta
q\;\frac{\mu_{f}}{T}\big{)}^{3-\frac{1}{\delta q}}}{\big{(}1-\delta
q\big{)}\big{(}\frac{1}{2}-\delta q\big{)}\big{(}\frac{1}{3}-\delta
q\big{)}},$ $\displaystyle\frac{\epsilon_{TS}}{T^{4}}$ $\displaystyle=$
$\displaystyle\frac{g}{2\pi^{2}}\;\frac{\big{(}1-\delta
q\;\frac{\mu_{f}}{T}\big{)}^{3-\frac{1}{\delta q}}}{\big{(}1-\delta
q\big{)}\big{(}\frac{1}{2}-\delta q\big{)}\big{(}\frac{1}{3}-\delta
q\big{)}},$ $\displaystyle\frac{s_{TS}}{T^{3}}$ $\displaystyle=$
$\displaystyle\frac{g}{6\pi^{2}}\;\frac{\big{(}4-\;\frac{\mu_{f}}{T}-\;\delta
q\;\frac{\mu_{f}}{T}\big{)}\big{(}1-\delta
q\;\frac{\mu_{f}}{T}\big{)}^{2-\frac{1}{\delta q}}}{\big{(}1-\delta
q\big{)}\big{(}\frac{1}{2}-\delta q\big{)}\big{(}\frac{1}{3}-\delta
q\big{)}}.$ (28)

The degeneracy factor for massless ideal gas is defined as
$g=g_{g}+(7/8)g_{f}$. For $N_{f}=3$ quark flavors, the QGP degree-of-freedom
is $g=47.5$. Eqs. (28) are only valid for $0\leq(\delta q\equiv q-1)<1/3$ and
$\delta q<T/\mu_{f}$ as required from the consistency of the framework.

Fig. 4: (Color online) A comparison between the PQM thermodynamic quantities
at $\mu_{f}=0.0~{}$MeV, normalized pressure, entropy and interaction measure
(left-panel (a)), the speed of sound squared (middle-panel (b)) and specific
heat (right-panel (c)), with the lattice QCD simulations (symbols). The
lattice QCD data are given by (open symbols) Borsanyi:2016ksw , (closed
symbols) Borsanyi:2022qlh  and (closed squared-symbols) HotQCD:2014kol .

Figure 4 illustrates the temperature dependence of the various thermodynamic
quantities such as normalized pressure $p/T^{4}$, entropy $s/T^{3}$ and the
interaction measure $I/T^{4}$ (left panel (a)), the speed of sound squared
$C_{s}^{2}$ (middle-panel (b)) and normalized specific heat $C_{v}/T^{3}$
(right panel (c)) obtained from the PQM model at $\mu_{f}=0.0$ MeV and $q=1.0$
in the usual BG statistics. The results are presented for Fukushima potential
of Polyakov loop variables (solid curves). The PQM results exhibit good
agreement with the lattice QCD calculations (symbols) Borsanyi:2016ksw ;
Borsanyi:2022qlh ; HotQCD:2014kol , particularly in the region of the phase
transition. However, at low temperatures, the PQM results show a slight
deviation from the lattice QCD calculations Borsanyi:2016ksw ;
Borsanyi:2022qlh ; HotQCD:2014kol . On the other hand, at high temperatures,
the PQM results align well with the lattice QCD calculations Borsanyi:2016ksw
; Borsanyi:2022qlh ; HotQCD:2014kol . The inclusion of the vector meson sector
in the PQM model enhance the spatial resolution of these comparisons. It is
worth emphasizing that the speed of sound squared and the lattice QCD
simulations appear to saturate below the SB-limit (middle panel (b)), near the
pseudo-critical temperature, it has a peak and then approaches the ideal gas
value of SB-limit at high temperatures. It is worth noting that a peak appears
in the vicinity of the pseudo-critical temperature for the normalized
quantities $I/T^{4}$, $C_{v}/T^{3}$ and $C_{s}^{2}$ Subsequently, these values
tend toward the massless ideal gas limit at higher temperatures.

Fig. 5: Temperature dependence of various dimensionless thermodynamic
quantities in the PQM model include the thermodynamic pressure $p/T^{4}$ (left
panel (a)), entropy density $s/T^{3}$ (middle-panel (b)), and energy density
$\epsilon/T^{4}$ (right panel (c)). Theses thermodynamic observables are given
at $\mu_{f}=0.0~{}$MeV and presented for various values of the non-extensive
$q$ parameter: $q=1.0$ (solid curves), $q=1.05$ (dashed curves), $q=1.103$
(dotted curves), and $q=1.143$ (dotted-dash curves). Fig. 6: The same as in
Fig. 5, but for other dimensionless thermodynamic observables including
interaction measure $I/T^{4}$ (left panel (a)), specific heat $C_{v}/T^{3}$
(middle-panel (b)), and speed of sound squared $C_{s}^{2}$ (right panel (c)).

Figures 5 and 6 demonstrate the impact of the non-extensive $q$ parameter on
the temperature dependence of various dimensionless thermodynamic quantities
in the PQM model at $\mu_{f}=0.0~{}$MeV. In Fig. 5, the thermal behaviors of
the thermodynamic pressure $p/T^{4}$ (left panel (a)), entropy density
$s/T^{3}$ (middle-panel (b)), and energy density $\epsilon/T^{4}$ (right panel
(c)) are illustrated. Additionally, Fig. 6 showcases other thermodynamic
quantities: the interaction measure $I/T^{4}$ (left panel (a)), specific heat
$C_{v}/T^{3}$ (middle-panel (b)), and speed of sound $C_{s}^{2}$ (right panel
(c)), as functions of temperature $T$.

We observe a consistent increase in all thermodynamic quantities with an
increasing non-extensive $q$ parameter. Notably, the influence of the non-
extensive effect appears to be minimal in the low-temperature region, but
becomes significant as it drives the phase transition from a smooth crossover
to a prompt first-order phase transition. The thermal dependence of these
quantities indicates their sensitivity to the deviation of the non-extensive
effects, which reach a saturation value below the massless ideal gas limit
(indicated by arrows). The thermodynamic observables are very sensitive to the
deviation of the non-extensive $q$ parameter, they become larger in both
phases of hadronic and QGP phases as $q$ increases. In light of this, the
pseudo-critical temperature $T_{\chi}(\mu_{B})$ seems not a universal
constant, and decrease with increasing the non-extensive $q$ parameter. The
incorporation of non-extensive statistics into the PQM thermodynamic potential
offers a replacement for the limit of a massless ideal gas, shifting towards
the Tsallis limit instead of the SB limit. This suggests that the system,
under Tsallis statistics at high temperatures, approaches the limit of a
massless ideal gas at values exceeding the SB-limit. This limit is referred to
as the $q$-dependent limit. It appears that the effects of non-extensive
statistics on thermodynamic quantities persist even at high temperatures.

For relativistic heavy-ion collision, it is worth highlighting the correlation
between the thermodynamic observables as the speed of sound squared as,

$\displaystyle C_{s}^{2}=\frac{\partial
p}{\partial\epsilon}\mid_{s}=\frac{\partial p/\partial
T\mid_{V}}{\partial\epsilon/\partial T\mid_{V}}=\frac{s}{C_{v}},$ (29)

where $C_{v}=\partial\epsilon/\partial T\mid_{V}$ is the specific heat at
constant volume. The speed of sound squared is also related to the interaction
measure $I(T,\mu_{f})$ as,

$\displaystyle\frac{I(T,\mu_{f})}{\epsilon(T,\mu_{f})}=\frac{\epsilon(T,\mu_{f})-3p(T,\mu_{f})}{\epsilon(T,\mu_{f})}\approx
1-3\;C_{s}^{2},$ (30)

where the interaction measure can be related to the equation of state (EoS)
$\epsilon(p)$. As stated in Eq. (30), the minimum value of $C_{s}^{2}$ results
in the maximum value of the ratio $I(T,\mu_{f})/\epsilon(T,\mu_{f})$ at low
temperatures. Conversely, at high temperatures, as $C_{s}^{2}$ approaches the
SB limit, the minimum value of the ratio $I(T,\mu_{f})/\epsilon(T,\mu_{f})$ is
achieved. The right panel (c) of Fig. 6 shows the influence of the non-
extensive $q$ parameter on the speed of sound squared as a function of $T$ at
vanishing quark chemical potential. The thermal behavior of $C_{s}^{2}$ is
obtained from $(s/T^{3})/(C_{v}/T^{3})$. This panel demonstrates a relative
sensitivity to the deviation of the non-extensive $q$ parameter in both the
hadronic phase and the transition region, where it drives the phase transition
as a smooth crossover. The system transitions more rapidly towards the QGP
phase as the non-extensive $q$ parameter gradually increases. At high
temperatures, the behavior of $C_{s}^{2}$ approaches the limit of a massless
ideal gas, specifically $1/3$. The thermal behavior of $C_{s}^{2}$ exhibits an
increasing trend, particularly in the hadronic phase, as the non-extensive $q$
parameter increases. However, this trend gradually disappears as temperatures
rise. It is worth noting that the behaviors of $C_{s}^{2}$ for different $q$
parameters remain below the SB limit of $1/3$. This indicates that the
behavior of $C_{s}^{2}$ remains below the SB limit and is consistent with the
Tsallis limit at high temperatures. The speed of sound serves as a measure of
the deviation from the equation of state of a massless ideal gas, which is
represented by $\epsilon=3p$.

###  III.4 Fluctuations and Correlations of Conserved Charges

Various fluctuations (diagonal) and correlations (off-diagonal)
susceptibilities of different conserved quantum charges, $x=[B,\,Q,\,S]$, can
be determined from the normalized thermodynamic pressure of PQM model as,

$\displaystyle\chi^{BQS}_{\alpha\,\beta\,\gamma}$ $\displaystyle=$
$\displaystyle\frac{\partial^{\alpha+\beta+\gamma}\;(p(T,\hat{\mu}_{x})/T^{4})}{(\partial\hat{\mu}_{B})^{\alpha}\,(\partial\hat{\mu}_{Q})^{\beta}\,(\partial\hat{\mu}_{S})^{\gamma}\,},$
(31)

the superscripts $\alpha$, $\beta$, and $\gamma$ represent the orders of the
derivatives. Eq. (31) illustrates that the fluctuations and correlations of
conserved quantum charges can be obtained by taking derivatives of the
dimensionless thermodynamic pressure with respect to the corresponding
dimensionless chemical potential, as given in Eq. (8), where
$\hat{\mu}_{x}=\mu_{x}/T$. Diagonal (off-diagonal) susceptibilities or moment
products can be constructed, which can be linked to the measured
multiplicities of produced particles, encompassing mean, variance, skewness,
kurtosis, and other order cumulants Luo:2017faz . This section focus on the
kurtosis of net-quantum fluctuations. In particular, ratio of the second and
fourth order cumulants as, $\kappa\sigma^{2}=\chi^{x}_{4}/\chi^{x}_{2}$, which
can be related to the measured multiplicities of produced particles
Luo:2017faz .

Fig. 7: A comparison of temperature dependence of the fluctuation (left panel)
and correlations (right panel) of PQM model (solid and dashed curves) and
lattice QCD simulations (solid bands) at $\mu_{f}=0.0~{}$MeV.

In this section, we investigate the thermal behaviors of diagonal (off-
diagonal) susceptibilities of conserved charges up to the $8-$th order for
non-extensive Tsallis statistics. Figure 7 presents a comparison of the
second-order fluctuations (left panel) and correlations (right panel) with
corresponding results obtained from lattice QCD calculations Borsanyi:2011sw ;
Bazavov:2012jq ; Bazavov:2017dus  at $\mu_{f}=0.0~{}$MeV. The PQM model
results are presented in two different forms: polynomial (solid curves) and
Fukushima (dashed curves) potentials of the Polyakov loop variables, as
functions of temperature $T$ at vanishing chemical potential. Remarkably, the
PQM results demonstrate good agreement with the lattice QCD calculations
Borsanyi:2011sw ; Bazavov:2012jq ; Bazavov:2017dus , particularly in the
region of the phase transition. Moreover, the PQM results employing the
Fukushima potential (dashed curves) exhibit better alignment with the lattice
QCD calculations Borsanyi:2011sw ; Bazavov:2012jq ; Bazavov:2017dus  compared
to the polynomial potential (solid curves), we expect this result because the
Fukushima potential incorporate higher-order Polyakov-loop variables and
derived from strong coupling simulations. Furthermore, the inclusion of the
vector meson sector in the PQM model enhances the spatial resolution of these
comparisons, as discussed in our previous study on bulk thermodynamics.

In the right panel (d-f) of Fig. 7, the thermal behavior of the correlations
of conserved charges at $\mu_{f}=0.0~{}$MeV is depicted. The top panel (d)
displays the correlation between the baryon number and the electric charge,
denoted as $\chi^{BQ}_{11}$, as a function of $T$. At low temperatures, the
correlation exhibits a gradual exponential increase as $T$ rises, eventually
vanishing at high temperatures. This behavior is expected since the quarks
become effectively massless in this temperature regime Borsanyi:2011bm ;
Cheng:2008zh . The middle (e) and bottom (f) panels illustrate the
correlations between strangeness and the electric charge, denoted as
$\chi^{QS}_{11}$, and between strangeness and the baryon number, denoted as
$\chi^{BS}_{11}$. These correlations are sensitive to variations in the
strangeness degrees of freedom Koch:2005vg ; Jeon:2000wg ; Majumder:2006nq .
In the case of isospin symmetry considered in this study, these correlations
are related to each other through the quadratic strangeness fluctuations
$\chi^{S}_{2}=2\chi^{QS}_{11}-\chi^{BS}_{11}$ Jeon:2000wg ; Majumder:2006nq .
At high temperatures, the correlations $\chi^{QS}_{11}$ and $\chi^{BS}_{11}$
become approximately equal, indicating a convergence of strangeness
correlations.

The susceptibilities exhibit small values at low temperatures, indicating the
dominance of the hadronic phase in this region. As the temperature enters the
crossover region, the fluctuations rapidly increase. The temperature
dependence of fluctuations and quark number multiplicities is significantly
influenced by the chiral structure of the hadronic states. High temperatures
give rise to a new state of matter characterized by massless quarks and
gluons. As the system undergoes the transition to the deconfined phase, the
fluctuations progressively increase and approaching a saturation value,
slightly below the SB limit.

Fig. 8: The same as in Fig. 7, but for different values of non-extensive $q$
parameter as $q=1.0$ (solid curves), $1.05$ (dashed curves), $1.103$ (dotted
curves) and $1.143$ (dotted-dash curves).

Figure 8 illustrates the influence of non-extensive Tsallis statistics on the
second-order fluctuations (left panel (a-c)) and correlations (right panel
(d-f)) in the PQM model as a function of temperature ($T$) at
$\mu_{f}=0.0~{}$MeV. The results show that the second-order susceptibilities
are sensitive to variations in the $q$ parameter, with all susceptibilities
increasing as $q$ increases. At low temperatures, the non-extensive effects
are negligible as the PQM order parameter remains unaffected by the non-
extensive $q$ parameter. However, as the temperature increases, the impact of
non-extensivity becomes more pronounced, enhancing the crossover until
reaching a saturation value below the massless ideal gas limit (indicated by
arrows) at high enough temperatures. Consequently, the pseudo-critical
temperature decreases with increasing values of the non-extensive $q$
parameter. By incorporating non-extensive Tsallis statistics into the PQM
model, we introduce an alternative limit to the massless ideal gas, distinct
from the SB limit. This alternative limit, known as the Tsallis limit,
indicates that the system at high temperatures tends towards the massless
ideal gas limit beyond the SB limit. This $q$-dependent limit has been
discussed in our previous study on bulk thermodynamics.

In the left panel of Figure 8, shows the correlation between the baryon number
and the electric charge, denoted as $\chi^{BQ}{11}$ (top panel (d)), as a
function of $T$ for different values of the non-extensive $q$ parameter. We
observe a consistent increase in the thermal behavior as the non-extensive $q$
parameter increases. Additionally, the pseudo-critical temperature decreases
with increasing values of the non-extensive $q$ parameter. At low
temperatures, this thermal behavior exhibits an exponential increase as the
temperature rises. Eventually, at high temperatures, the correlation
approaches to vanish. Moreover, in the case of the correlations
$\chi^{QS}_{11}$ (middle panel (e)) and $-\chi^{BS}_{11}$ (bottom panel (f)),
they become approximately equal at high temperatures, indicating a convergence
of strangeness correlations. These correlations reach a saturation value
slightly below the limit of a massless ideal gas.

Fig. 9: The thermal dependence of the moment products $\kappa\sigma^{2}$
calculated from the PQM model for different values of non-extensive $q$
parameter as $q=1.0$ (solid curves), $1.05$ (dashed curves), $1.103$ (dotted
curves) and $1.143$ (dotted-dash curves) for (a) net-baryon, (b) net-charge
and (c) net-strangeness multiplicities compared with lattice QCD simulations
(solid band) Bazavov:2017dus  and (closed symbols) Cheng:2008zh . The solid
arrows gives the associated SB limits.  Fig. 10: The same as in Fig. 8, but
for the thermal behaviors of the higher order fluctuations,
$\chi^{x}_{4},\;\chi^{x}_{6}$ and $\chi^{x}_{8}$, where $x$ refers to the net-
baryon (left panel (a, d, c)), net-charge (middle panel (b, e, h)) and net-
strangeness (right panel (c, f, i)) multiplicities.

The PQM Lagrangian is constructed from the chiral Lagrangian coupled to the
Polyakov loop variables. This implies that at low temperatures, the effective
degrees of freedom of quark and anti-quark contributions are limited as long
as the expectation values of the Polyakov loop variables are both zero.
Consequently, excitations of one- and two-quark states are suppressed in the
partition sum. Therefore, at low temperatures, the thermodynamic pressure of
the quark and anti-quark contributions can be approximated, constituting what
is known as the Boltzmann approximation Karsch:2003zq ; Karsch:2003vd  as,

$\displaystyle\frac{p^{th}_{\psi\bar{\psi}}}{T^{4}}\sim\frac{2N_{f}}{27\pi^{2}}\;\Big{(}\frac{3m_{f}}{T}\Big{)}^{2}\;K_{2}\Big{(}\frac{3m_{f}}{T}\Big{)}\;Cosh\Big{(}\frac{3\mu_{f}}{T}\Big{)}.$
(32)

At high temperatures, it is expected that the susceptibilities will approach
those of an ideal gas of quarks. Eq. 32 refers to the Boltzmann approximation
is only valid if $\big{(}3m_{f}/T\big{)}>1$, which is satisfied for low
temperatures $T<T_{\chi}$. A similar factorization occurs in HRG models as,

$\displaystyle\frac{p^{th}_{\psi\bar{\psi}}}{T^{4}}\sim
F\Big{(}\frac{\mathcal{M}}{T}\Big{)}\;Cosh\Big{(}\frac{\mu_{B}}{T}\Big{)},$
(33)

where $\mathcal{M}=3m_{f}$ and $\mu_{B}=3\mu_{f}$, the sum is taken over all
known baryons and their resonances Ejiri:2005wq ; Redlich:2004gp ;
Karsch:2003zq ; Karsch:2003vd . The kurtosis strongly depends on the effective
degrees of freedom of quarks associated with the baryon number, (where $B=1$).
For instance, at low temperatures, the kurtosis can be expressed as
$\kappa\sigma^{2}=(3B)^{2}=9$, a relationship also supported by observations
in the HRG modelAllton:2005gk ; Ejiri:2005wq . In contrast, in the pure chiral
Quark-Meson model, where the relevant degrees of freedom consist of single
quarks at all temperatures, the kurtosis at low temperatures converges to
$\kappa\sigma^{2}=1$. It is evident that in an effective chiral model with
quark degrees of freedom, the inclusion of the Polyakov loop is imperative for
accurately describing the behavior of the kurtosis at low temperatures.

Figure 9 displays the moment products
$\kappa\sigma^{2}=\chi^{x}_{4}/\chi^{x}_{2}$ computed using the PQM model at
various values of the non-extensive parameter $q$ for (a) net-baryon, (b) net-
charge, and (c) net-strangeness multiplicities. The results have sufficient
agreement with lattice QCD simulations conducted by HOTQCD Bazavov:2017dus
(solid band) and other studies Cheng:2008zh  (closed symbols). The values of
the kurtosis at high and low temperatures differ significantly, indicating
variations near the phase transition. The substantial change in the kurtosis
between the low and high-temperature limits primarily arises from the
transformation of the relevant degrees of freedom carrying baryon number.
These degrees of freedom transition from three-quark states to single quarks.

In the left-hand panel (a) of Fig. 9, the moment product
$\kappa\sigma^{2}=\chi^{B}_{4}/\chi^{B}_{2}$ starts at unity due to the
inclusion of the Boltzmann approximation, Eq.(32), in the PQM Lagrangian
within the confinement region. As the temperature $T$ increases, there is a
rapid decrease in this behavior, attributed to the swift evolution of the
quadratic susceptibility. At high temperatures, the kurtosis reaches a
saturation level below the SB limit,
$\chi^{B}_{4}/\chi^{B}_{2}|_{SB}=2/3\pi^{2}$, and remains temperature-
independent in the deconfined phase. The temperature dependent of kurtosis is
very sensitive to variations in the $q$ parameter. At low temperatures, non-
extensive effects are negligible. However, as the temperature rises, the
influence of non-extensivity becomes increasingly pronounced, leading to an
enhancement of the crossover until it reaches to saturated plateau below the
SB limit.

The middle and right-hand panels of Fig. 9 depict the moment product
$\kappa\sigma^{2}$, following the same pattern as in the left-hand panel (a),
but for (b) net-electric charge and (c) net-strangeness multiplicity,
respectively. These PQM results are compared with lattice QCD calculations
Cheng:2008zh  (closed symbols). Once again, at low temperatures, the influence
of the Boltzmann approximation dominates during the confinement phase, and the
effects of non-extensivity are negligible. As we approach the chiral
transition, the kurtosis shows a strong dependence on the non-extensivity
parameter, extending to higher temperatures where the susceptibilities
approach those of an ideal gas, following the SB limit. There is a qualitative
agreement, especially in phase transition region to high temperatures. The
results approach the SB limits of $2/\pi^{2}$ and $6/\pi^{2}$ for net-electric
charge and net-strangeness multiplicities, respectively.

Fig. 10 illustrates the impact of non-extensive $q$ effects on the higher
order susceptibilities $\chi^{x}_{n}$ ($n=4,6,8$) for net-baryon
$\chi^{B}_{n}$ (left panel (a, d, c)), net-charge $\chi^{Q}_{n}$ (middle panel
(b, e, h)), and net-strangeness $\chi^{S}_{n}$ (right panel (c, f, i))
multiplicities as functions of temperature $T$. As the non-extensive $q$
parameter increases, the $n$-th order susceptibilities decrease. Notably, the
most significant deviations from the non-extensive effects are observed in the
higher order susceptibilities near the crossover region. Increasing the non-
extensive $q$ parameter enhances all the susceptibilities, and the pseudo-
critical temperature decreases accordingly. For $n=6$ and $8$, the higher
order susceptibilities exhibit peaks around the temperature of the phase
transition. As the non-extensive $q$ parameter increases, these
susceptibilities display oscillations in higher harmonics.It is worth
mentioning that net-strangeness exhibits higher fluctuations compared to net-
baryon and net-charge. Alternatively, it is crucial to consider the impact of
non-extensive effects when comparing theoretically calculated higher order
fluctuations with experimental measurements. Confidence in the measured system
is necessary to account for these non-extensive influences. Moreover, the PQM
results demonstrate notable characteristics, showing a smooth crossover
between the hadronic and partonic phases.

##  IV Conclusions

In conclusion, we have combined the PQM model and the non-extensive Tsallis
statistics to study the hadron-quark transition to deviations of the BG
statistics. The results highlight the influence of the non-extensive $q$
parameter on the phase transition and various thermodynamic quantities. At
vanishing chemical potential, one can find that the pseudo-critical
temperature of the chiral and deconfinement transition shifts towards a lower
temperature as the non-extensive parameter $q$ increases. Moreover, at finite
chemical potential and with an increasing $q$ parameter, the position of the
pseudo-critical temperature moves towards lower temperatures and larger
chemical potentials. This suggests that the pseudo-critical temperature is not
a universal constant; its position in the ($T_{\chi}$-$\mu_{B}$) plane may
depend on the non-extensive $q$ parameter. Furthermore, the search for the
position CEP in relativistic heavy-ion collisions may be dependent on the
parameters of non-extensive Tsallis statistics. It is worthwhile to mention
that the CEP carries lower temperature and higher chemical potential with
increasing $q$ parameter. One can consider that the location of the CEP is not
a well-defined point but rather a critical region dependent on non-extensive
effects. Notably, the impact of the non-extensive effect on PQM results seems
small in the low-temperature region but becomes significant as it propels the
phase transition from a smooth crossover to a prompt first-order phase
transition. On other hand, with an increasing $q$ parameter, the phase
transition occurs at a pseudo-critical temperature that shifts towards lower
temperatures. This implies that variations in the $q$ parameter can show an
additional phase transition. As the $q$ parameter increases, the system
transitions to the deconfined phase more rapidly.

Furthermore, the agreement between the PQM model and lattice QCD calculations
regarding various thermodynamic quantities, especially in the phase transition
region, underscores the validity of the model. The non-extensive $q$ parameter
affects on these quantities, leading to higher values as $q$ increases. The
pronounced impact of non-extensive effects occurs around the phase transition,
driving the system towards a prompt first-order phase transition from a smooth
crossover. Additionally, the incorporation of non-extensive statistics
deviates the system from the massless ideal gas limit, aligning it more
closely with the Tsallis limit. On other words, the thermodynamic quantities
at higher temperatures limit are no longer to SB limit but $q-$related Tsallis
limit. The sensitivity of thermodynamic quantities to the $q$ parameter is
evident, and the $C_{s}^{2}$ remains below the Stefan-Boltzmann limit,
consistent with the Tsallis limit at high temperatures. At higher
temperatures, the growth rate of the dimensionless entropy and specific heat
tends to be the same. Therefore, the $C_{s}^{2}$ behavior at high temperature
limit is not affected by the non-extensive $q$ parameter.

Additionally, the investigation of diagonal and off-diagonal susceptibilities
of conserved charges in the PQM model with non-extensive Tsallis statistics
yields results that align well with lattice QCD calculations, particularly
near the phase transition. The sensitivity of susceptibilities to variations
in the non-extensive $q$ parameter is demonstrated, providing valuable
insights into the behavior of the system close to the phase transition. These
susceptibilities enhance the crossover trend as the $q$ parameter increases.
Overall, this study significantly contributes to our understanding of non-
extensive effects in the hadron-quark phase structure and opens up avenues for
further research in this intriguing field. Furthermore, we studied the moments
of the net-baryon, net-electric charge and net-strangeness multiplicity,
$\kappa\sigma^{2}=\chi^{X}_{4}/\chi^{X}_{2}$, as a function $T$ at different
$q$ parameter. It is worth noting that the system transitions more rapidly
towards the deconfined phase, and it undergoes the massless ideal gas limit
more quickly as the non-extensive $q$ parameter increases.

## Acknowledgements

The author expresses his gratitude to the anonymous referees for their
invaluable feedback and constructive suggestions, which greatly enhanced the
quality of this work.

## References

  * (1) C. Tsallis,  Journal of statistical physics 52, 479 (1988).
  * (2) C. Tsallis,  Introduction to Nonextensive Statistical Mechanics: Approaching a Complex World,  Springer, New York, 2009.
  * (3) G. Wilk and Z. Wlodarczyk,  Phys. Rev. Lett. 84, 2770 (2000).
  * (4) M. Rybczynski and Z. Wlodarczyk,  Eur. Phys. J. C 74, 2785 (2014).
  * (5) A. Lavagno, D. Pigato, and P. Quarati,  J. Phys. G 37, 115102 (2010).
  * (6) A. Lavagno and D. Pigato,  (2012).
  * (7) L. A. Trevisan and C. Mirez,  Int. J. Mod. Phys. E 22, 1350044 (2013).
  * (8) G. Gervino, A. Lavagno, and D. Pigato,  J. Phys. Conf. Ser. 442, 012065 (2013).
  * (9) A. Lavagno and D. Pigato,  Physica A 392, 5164 (2013).
  * (10) R. B. Frigori,  Comput. Phys. Commun. 185, 2232 (2014).
  * (11) J. Rozynek,  EPJ Web Conf. 95, 05014 (2015).
  * (12) M. Ishihara,  Int. J. Mod. Phys. E 24, 1550085 (2015).
  * (13) M. Ishihara,  Int. J. Mod. Phys. A 33, 1850067 (2018).
  * (14) M. Ishihara,  Eur. Phys. J. A 54, 164 (2018).
  * (15) M. Ishihara,  Eur. Phys. J. A 56, 145 (2020).
  * (16) J. Rożynek and G. Wilk,  Eur. Phys. J. ST 229, 751 (2020).
  * (17) M. Rahaman, T. Bhattacharyya, and J.-e. Alam,  Int. J. Mod. Phys. A 36, 2150154 (2021).
  * (18) Y.-P. Zhao,  Phys. Rev. D 101, 096006 (2020).
  * (19) C. A. Islam,  (2023).
  * (20) J. Rozynek and G. Wilk,  Eur. Phys. J. A 52, 13 (2016),  [Erratum: Eur.Phys.J.A 52, 204 (2016)].
  * (21) J. Rozynek and G. Wilk,  J. Phys. G 36, 125108 (2009).
  * (22) G. Wilk,  Braz. J. Phys. 37, 714 (2007).
  * (23) C. Tsallis and L. J. L. Cirto,  Eur. Phys. J. C 73, 2487 (2013).
  * (24) A. Lavagno and D. Pigato,  Eur. Phys. J. A 47, 52 (2011).
  * (25) P. Douglas, S. Bergamini, and F. Renzoni,  Phys. Rev. Lett. 96, 110601 (2006).
  * (26) B. I. Abelev et al.,  Phys. Rev. C 75, 064901 (2007).
  * (27) A. Adare et al.,  Phys. Rev. C 83, 064903 (2011).
  * (28) K. Aamodt et al.,  Eur. Phys. J. C 71, 1655 (2011).
  * (29) B. B. Abelev et al.,  Phys. Lett. B 736, 196 (2014).
  * (30) V. Khachatryan et al.,  Phys. Rev. Lett. 105, 022002 (2010).
  * (31) G. Aad et al.,  New J. Phys. 13, 053033 (2011).
  * (32) K. Huang,  Introduction to statistical physics,  CRC press, 2009.
  * (33) F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari,  Reviews of modern physics 71, 463 (1999).
  * (34) S. Sharma and M. Kaur,  Physical Review D 99 (2019).
  * (35) T. Bhattacharyya et al.,  The European Physical Journal A 54 (2018).
  * (36) M. A. Stephanov, K. Rajagopal, and E. V. Shuryak,  Phys. Rev. Lett. 81, 4816 (1998).
  * (37) D. Espriu, A. Gómez Nicola, and A. Vioque-Rodríguez,  JHEP 06, 062 (2020).
  * (38) Y.-P. Zhao, R.-R. Zhang, H. Zhang, and H.-S. Zong,  Chin. Phys. C 43, 063101 (2019).
  * (39) C. S. Fischer,  Prog. Part. Nucl. Phys. 105, 1 (2019).
  * (40) A. Ayala et al.,  Phys. Rev. D 84, 056004 (2011).
  * (41) K. Kyan and A. Monnai,  Phys. Rev. D 106, 054004 (2022).
  * (42) A. S. Parvan,  (2023).
  * (43) Y.-P. Zhao, S.-Y. Zuo, and C.-M. Li,  Chin. Phys. C 45, 073105 (2021).
  * (44) Y.-P. Zhao, S.-Y. Zuo, and C.-M. Li,  Chin. Phys. C 46, 103107 (2022).
  * (45) Y.-P. Zhao, C.-Y. Wang, S.-Y. Zuo, and C.-M. Li,  Chin. Phys. C 47, 053103 (2023).
  * (46) K.-M. Shen, H. Zhang, D.-F. Hou, B.-W. Zhang, and E.-K. Wang,  Adv. High Energy Phys. 2017, 4135329 (2017).
  * (47) M. Ishihara,  Int. J. Mod. Phys. E 25, 1650066 (2016).
  * (48) M. Ishihara,  Int. J. Mod. Phys. E 28, 1950020 (2019).
  * (49) D. Roder, J. Ruppert, and D. H. Rischke,  Phys. Rev. D 68, 016003 (2003).
  * (50) A. N. Tawfik and A. M. Diab,  Phys. Rev. C 91, 015204 (2015).
  * (51) A. N. Tawfik, A. M. Diab, and M. T. Hussein,  Chin. Phys. C 43, 034103 (2019).
  * (52) W.-j. Fu, Z. Zhang, and Y.-x. Liu,  Phys. Rev. D 77, 014006 (2008).
  * (53) A. Tawfik, N. Magdy, and A. Diab,  Phys. Rev. C 89, 055210 (2014).
  * (54) A. N. Tawfik, A. M. Diab, and M. T. Hussein,  Int. J. Mod. Phys. A 31, 1650175 (2016).
  * (55) A. M. Abdel Aal Diab and A. N. Tawfik,  EPJ Web Conf. 177, 09005 (2018).
  * (56) A. N. Tawfik, C. Greiner, A. M. Diab, M. T. Ghoneim, and H. Anwer,  Phys. Rev. C 101, 035210 (2020).
  * (57) A. N. Tawfik, A. M. Diab, M. T. Ghoneim, and H. Anwer,  Int. J. Mod. Phys. A 34, 1950199 (2019).
  * (58) N. Ezzelarab, A. M. Diab, and A. N. Tawfik,  J. Phys. Conf. Ser. 668, 012102 (2016).
  * (59) A. N. Tawfik, A. M. Diab, N. Ezzelarab, and A. G. Shalaby,  Adv. High Energy Phys. 2016, 1381479 (2016).
  * (60) A. N. Tawfik, A. M. Diab, and M. T. Hussein,  J. Phys. G 45, 055008 (2018).
  * (61) A. N. Tawfik, A. M. Diab, and T. M. Hussein,  Int. J. Adv. Res. Phys. Sci. 3, 4 (2016).
  * (62) A. N. Tawfik, A. M. Diab, and M. T. Hussein,  J. Exp. Theor. Phys. 126, 620 (2018).
  * (63) A. N. Tawfik and A. M. Diab,  Eur. Phys. J. A 57, 200 (2021).
  * (64) P. Kovács, Z. Szép, and G. Wolf,  Phys. Rev. D 93, 114014 (2016).
  * (65) P. Kovács, G. Kovács, and F. Giacosa,  Phys. Rev. D 106, 116016 (2022).
  * (66) S. Weinberg,  Gravitation and Cosmology,  John Wiley and Sons, New York, 1972.
  * (67) B.-J. Schaefer and M. Wagner,  Phys. Rev. D79, 014018 (2009).
  * (68) J. T. Lenaghan, D. H. Rischke, and J. Schaffner-Bielich,  Phys. Rev. D 62, 085008 (2000).
  * (69) B.-J. Schaefer, M. Wagner, and J. Wambach,  Phys. Rev. D 81, 074013 (2010).
  * (70) C. Ratti, M. A. Thaler, and W. Weise,  Phys. Rev. D73, 014019 (2006).
  * (71) S. Roessner, C. Ratti, and W. Weise,  Phys. Rev. D75, 034007 (2007).
  * (72) B.-J. Schaefer, J. M. Pawlowski, and J. Wambach,  Phys. Rev. D76, 074023 (2007).
  * (73) K. Fukushima,  Phys. Rev. D77, 114028 (2008),  [Erratum: Phys. Rev.D78,039902(2008)].
  * (74) P. M. Lo, B. Friman, O. Kaczmarek, K. Redlich, and C. Sasaki,  Phys. Rev. D88, 074502 (2013).
  * (75) R. D. Pisarski,  Phys. Rev. D 62, 111501 (2000).
  * (76) O. Scavenius, A. Dumitru, and J. T. Lenaghan,  Phys. Rev. C 66, 034903 (2002).
  * (77) K. Fukushima,  Phys. Lett. B 591, 277 (2004).
  * (78) C. Sasaki, B. Friman, and K. Redlich,  Phys. Rev. D 75, 074013 (2007).
  * (79) A. Deppman,  Phys. Rev. D 93, 054001 (2016).
  * (80) A. Deppman, E. Megias, D. P. Menezes, and T. Frederico,  Entropy 20, 633 (2018).
  * (81) C. G. Callan, Jr.,  Phys. Rev. D 2, 1541 (1970).
  * (82) K. Symanzik,  Commun. Math. Phys. 18, 227 (1970).
  * (83) A. Deppman, E. Megias, and D. P. Menezes,  Phys. Rev. D 101, 034019 (2020).
  * (84) H. D. Politzer,  Phys. Rept. 14, 129 (1974).
  * (85) D. J. Gross and F. Wilczek,  Phys. Rev. Lett. 30, 1343 (1973).
  * (86) I. Sena and A. Deppman,  Eur. Phys. J. A 49, 17 (2013).
  * (87) L. Marques, J. Cleymans, and A. Deppman,  Phys. Rev. D 91, 054025 (2015).
  * (88) J. Cleymans et al.,  Phys. Lett. B 723, 351 (2013).
  * (89) B.-C. Li, Y.-Z. Wang, and F.-H. Liu,  Phys. Lett. B 725, 352 (2013).
  * (90) M. D. Azmi and J. Cleymans,  J. Phys. G 41, 065001 (2014).
  * (91) H. Mao, J. Jin, and M. Huang,  J. Phys. G 37, 035001 (2010).
  * (92) U. S. Gupta and V. K. Tiwari,  Phys. Rev. D 81, 054019 (2010).
  * (93) L. M. Haas, R. Stiele, J. Braun, J. M. Pawlowski, and J. Schaffner-Bielich,  Phys. Rev. D87, 076004 (2013).
  * (94) M. Gell-Mann, R. J. Oakes, and B. Renner,  Phys. Rev. 175, 2195 (1968).
  * (95) R. F. Dashen,  Phys. Rev. 183, 1245 (1969).
  * (96) S. Borsanyi et al.,  JHEP 09, 073 (2010).
  * (97) R. Bellwied et al.,  Phys. Lett. B751, 559 (2015).
  * (98) A. Bazavov et al.,  Phys. Lett. B795, 15 (2019).
  * (99) A. Andronic, P. Braun-Munzinger, K. Redlich, and J. Stachel,  Nature 561, 321 (2018).
  * (100) S. Das,  (2014),  [EPJ Web Conf.90,08007(2015)].
  * (101) T. Bhattacharyya, J. Cleymans, and S. Mogliacci,  Phys. Rev. D 94, 094026 (2016).
  * (102) S. Borsanyi et al.,  Nature 539, 69 (2016).
  * (103) S. Borsanyi et al.,  Phys. Rev. D 105, 114504 (2022).
  * (104) A. Bazavov et al.,  Phys. Rev. D 90, 094503 (2014).
  * (105) X. Luo and N. Xu,  Nucl. Sci. Tech. 28, 112 (2017).
  * (106) S. Borsanyi et al.,  JHEP 01, 138 (2012).
  * (107) A. Bazavov et al.,  Phys. Rev. D86, 034509 (2012).
  * (108) A. Bazavov et al.,  Phys. Rev. D95, 054504 (2017).
  * (109) S. Borsanyi et al.,  J. Phys. G38, 124060 (2011).
  * (110) M. Cheng et al.,  Phys. Rev. D79, 074505 (2009).
  * (111) V. Koch, A. Majumder, and J. Randrup,  Phys. Rev. Lett. 95, 182301 (2005).
  * (112) S. Jeon and V. Koch,  Phys. Rev. Lett. 85, 2076 (2000).
  * (113) A. Majumder and B. Muller,  Phys. Rev. C 74, 054901 (2006).
  * (114) F. Karsch, K. Redlich, and A. Tawfik,  Phys. Lett. B 571, 67 (2003).
  * (115) F. Karsch, K. Redlich, and A. Tawfik,  Eur. Phys. J. C 29, 549 (2003).
  * (116) S. Ejiri, F. Karsch, and K. Redlich,  Phys. Lett. B 633, 275 (2006).
  * (117) K. Redlich, F. Karsch, and A. Tawfik,  J. Phys. G 30, S1271 (2004).
  * (118) C. R. Allton et al.,  Phys. Rev. D 71, 054508 (2005).