# Thermodynamics and phase diagrams of the Polyakov quark-meson model with on-
shell versus curvature mass parameter fixing

Suraj Kumar Rai<EMAIL_ADDRESS>Department of Physics, University of
Allahabad, Prayagraj, India-211002  Department of Physics, Acharya Narendra
Deo Kisan P.G. College, Babhnan Gonda, India-271313  Vivek Kumar Tiwari
<EMAIL_ADDRESS>Department of Physics, University of Allahabad, Prayagraj,
India-211002

###### Abstract

The Quantum Chromodynamics (QCD) phase structure has been studied using the
Polyakov-loop augmented quark-meson model (PQM) in the extended mean field
approximation (e-MFA) where the quark one-loop vacuum term is included. When
the divergent vacuum term is regularized in the minimal subtraction scheme and
the curvature meson masses are used to fix the parameters, the Polyakov quark-
meson model with the vacuum term (PQMVT) becomes inconsistent as the curvature
masses are determined by calculating the self energies at zero momentum. The
above inconsistency is remedied by the on-shell parameter fixing when the pion
decay constant and the pole masses of the mesons are put into the relation of
the couplings and running mass parameter by using the on-shell and the minimal
subtraction renormalization scheme. Combining the modified chiral effective
potential of the on-shell renormalized quark-meson model (RQM) with the
Polyakov-loop potential that mimics the physics of the confinement-
deconfinement transition, we get the renormalized Polyakov quark-meson (RPQM)
model. The phase diagrams and the thermodynamics details for the PQM, PQMVT
and RPQM model, have been computed and compared for different forms of the
Polyakov-loop potentials with and without the quark back-reaction. The results
have also been compared with the available lattice QCD data. The so called
quarkyonic phase region in the phase diagram, where the chiral symmetry is
restored but the quarks and anti-quarks are still confined, gets reduced by
the quark back-reaction in the unquenched Polyakov-loop potential. It
altogether disappears for the chemical potential dependent parameter
$T_{0}\equiv T_{0}(\mu)$ in the Log or the PolyLog-glue form of the Polyakov-
loop potential in the RPQM model.

Dense QCD, chiral transition,

##  I Introduction

The strong interaction theory, quantum chromodynamics (QCD) indicates that
under the extreme conditions of high temperatures and/or densities, there
should be a phase transition from the normal hadronic matter to a collective
form of matter known as the Quark Gluon Plasma (QGP) Cabibbo75 ; SveLer ; Mull
; Ortms ; Riske . The general properties of such a hot and dense matter are
summarized in the QCD phase diagram Cabibbo75  which can be probed by the
ultrarelativistic heavy ion collision experiments like the RHIC (BNL), LHC
(CERN) and the upcoming CBM experiments at the FAIR facility (GSI-Darmstadt).
The study of the QCD thermodynamics and its phase structure is a very active
area of current research as several issues are not yet settled. One gets
important information and insights regarding the QCD phase transition from the
first-principle lattice QCD simulations AliKhan:2001ek ; Digal:01 ; Karsch:02
; Fodor:03 ; Allton:05 ; Karsch:05 ; Aoki:06 ; Cheng:06 ; Cheng:08 ; JLange
but these calculations get seriously compromised as the QCD action becomes
complex due to the fermion sign problem Karsch:02  when the baryon
density/chemical potential is nonzero. Hence one turns to the effective models
Alf ; Fukhat  for the study of the QCD thermodynamics and its phase diagram.

The QCD Lagrangian has the global $SU_{\tiny{L+R}}(2)\times SU_{L-R}(2)$
symmetry for the two flavor of massless quarks. In the low energy hadronic
vacuum of the QCD, the chiral (axial $A=L-R$) symmetry is spontaneously broken
and one gets the chiral condensate as order parameter and the three massless
pions as Goldstone modes. The small explicit breaking of the chiral symmetry
due to the light quark masses, gives a little mass to the physical pions which
are the lightest among hadrons. The $SU_{L}(2)\times SU_{R}(2)$ linear sigma
model Roder ; fuku11 ; grahl  is a good framework to study the chiral symmetry
breaking and restoring phase transition. Coupling the iso-singlet $\sigma$,
iso-triplet $\vec{a_{0}}$ scalar mesons together with the iso-singlet $\eta$,
iso-triplet $\vec{\pi}$ pseudo-scalar mesons of the sigma model with the two
flavor of quarks, one gets the QCD-like framework of the quark-meson (QM)
model to study the QCD thermodynamics and its phase diagram in great detail.

The QCD phase structure/phase diagram, has already been studied in the chiral
models Rischke:00 ; jakobi ; Herpay:05 ; Herpay:06 ; Herpay:07 ; Kovacs:2006ym
; kahara ; Bowman:2008kc ; Fejos ; Jakovac:2010uy ; koch ; marko , two and
three flavor QM model scav ; mocsy ; bj ; Schaefer:2006ds ; Schaefer:09 . But
the QM model in the standard mean field approximation (s-MFA) gives
inconsistent result as the chiral phase transition at zero baryon densities
becomes first-order in the chiral limit which is at odds with the general
theoretical considerations rob ; hjss . This inconsistency is remedied by the
proper treatment of the Dirac sea vac  after including the quark one-loop
vacuum fluctuation. In the modified framework of the quark-meson model with
the vacuum term (QMVT), several QCD phase structure studies lars ; guptiw ;
schafwag12 ; chatmoh1 ; TranAnd ; vkkr12 ; chatmoh2 ; vkkt13 ; Herbst ;
Weyrich ; kovacs ; zacchi1 ; zacchi2 ; Rai , regularized the divergent one-
loop vacuum term in the minimal substraction scheme and after identifing the
pion decay constant with the vacuum expectation value of the sigma mean field,
fixed the model parameters using the curvature masses of the mesons. The above
parameter fixing procedure turns out to be inconsistent as one notes that the
effective potential is the generator of the n-point functions of the theory at
vanishing external momenta and the curvature mass is defined by the evaluation
of the self-energy at zero momentum laine ; Adhiand1 ; BubaCar ; Naylor ; fix1
.

The radiative corrections to the physical quantities change their tree level
relations to the parameters of the Lagrangian. Hence the use of tree level
values of the parameters for the calculation of effective potential becomes
inconsistent. One has to account for the renormalization scale $\Lambda$
dependence of the running parameters in the (modified) minimal subtraction
$\overline{\text{MS}}$ scheme while the on-shell parameters have their tree-
level values. Following the correct renormalization procedure one needs to
calculate the counterterms both in the $\overline{\text{MS}}$ scheme and in
the on-shell scheme and then connect the renormalized parameters of the two
schemes. The effective potential is then calculated using the
$\overline{\text{MS}}$ procedure where the relations between the running
parameters and the on-shell parameters (physical quantities) are used as input
Adhiand1 . Using the abovementioned renormalization prescription Adhikari and
collaborators Adhiand1 ; Adhiand2 ; Adhiand3 ; asmuAnd  correctly accounted
for the effect of Dirac sea in the QM model whose $O(4)$ sigma model has the
$\sigma$ and $\vec{\pi}$ as meson degrees of freedom. In a very recent work
RaiTiw , we have applied the exact renormalization method for the on-shell
parameter fixing to that version of the QM model in which the two flavor of
quarks are coupled to the eight mesons (iso-singlet and iso-triplet
combination of the $\sigma$ and $\vec{a_{0}}$ as scalars and the $\eta$ and
$\vec{\pi}$ as pseudo-scalars ) of the $SU_{L}(2)\times SU_{R}(2)$ sigma model
and termed this setting as the renormalized quark-meson (RQM) model.

The physics of quark confinement in the hadrons at low temperatures and
densities is implemented by the introduction of the Polyakov-loop where the
QCD confinement is mimicked in a statistical sense by coupling the chiral
models to a constant background $SU(N_{c})$ gauge field $A_{\mu}^{a}$ SveLer ;
Polyakov:78plb ; benji ; BankUka ; Pisarski:00prd ; fuku ; Vkt:06 . In such
studies, the free energy density from the gluons is added to the QM model
using the phenomenological Polyakov-loop potential ratti ; Roesnr ; fuku2 ,
and it becomes the PQM model SchaPQM2F ; SchaPQM3F ; Mao ; TiPQM3F . In the
present work, the modified chiral parts of the effective potentials for both
the on-shell renormalized quark-meson model (RQM) and the curvature mass
parameterized QMVT model, have been augmented with the physics of
confinement/deconfinement transition by including the Polyakov-loop potential
and the respective settings have been termed as the renormalized Polyakov
quark-meson model (RPQM) and the PQMVT model. In this study, we have
considered the important improvement of the Polyakov-loop potential from a
pure gauge potential to a unquenched glue potential in which backreaction
effects of the quarks are included Haas ; Redlo ; TkHerbst ; BielichP . It is
worthwhile to explore the consequences of coupling the improved chiral
effective potential with the unquenched Polyakov-loop potential because it
leads to the linkage of the chiral and deconfinement phase transitions also at
small temperatures and large chemical potentials. The abovementioned attribute
is also seen in the functional renormalization (FRG) improvement of the PQM
model when Yang-Mills Polyakov-loop potential is used Herbst ; THerbst2 . We
have computed the relative shift of the critical end point (CEP) and made the
qualitative and quantitative comparisons of the phase diagrams and
thermodynamics in the RPQM and PQMVT models when different forms of the
Polyakov-loop potentials are considered with and without quark back-reaction.

The paper is arranged as follows. Section II presents a brief formulation of
the $SU_{L}(2)\times SU_{R}(2)$ PQM model. Section II.1 describes the
different forms of the Polyakov-loop potentials while Section II.2 presents
the thermodynamic grand potential in the PQM model. Section III gives a brief
account of the Polyakov quark-meson model with the vacuum term (PQMVT). The
renormalized Polyakov quark-meson model (RPQM) effective potential has been
presented in Section IV. Section V gives results and discussion. Section V.1
discusses the order parameters and their temperature derivatives, Section V.2
describes the sigma mass and the model dependence of the phase diagrams,
Section V.3 reports the results for the quarkyonic phase in the RPQM model,
Section V.4 explains the computed results for the thermodynamic observables
while the specific heat and the speed of sound are discussed in Section V.5.
Section VI presents the summary and conclusion. Appendix A presents the
parameter fixing for the QMVT model. The essential steps of the exact on-shell
parameter fixing and the calculation of the quark one-loop effective potential
in the large $N_{c}$ limit for the renormalized quark-meson (RQM) model, have
been presented in Appendix B. The integrals are presented in Appendix C.

##  II Model Formulation

We will be combining the Polyakov-loop potential with the $SU_{L}(2)\times
SU_{R}(2)$ quark-meson model. In this model a spatially constant temporal
gauge field, two light quarks and $SU_{V}(2)\times SU_{A}(2)$ symmetric meson
fields, are coupled together. The Polyakov-loop field $\Phi$ is defined as the
thermal expectation value of color trace of the Wilson loop in temporal
direction.

$\Phi=\frac{1}{N_{c}}\langle{\mathrm{Tr}}_{c}L(\vec{x})\rangle,\qquad\qquad\bar{\Phi}=\frac{1}{N_{c}}\langle{\mathrm{Tr}}_{c}L^{\dagger}(\vec{x})\rangle\;,$
(1)

where $L$ is a matrix in the fundamental representation of the $SU_{c}(3)$
color gauge group,

$L(\vec{x})=\mathcal{P}\mathrm{exp}\left[i\int_{0}^{\beta}d\tau
A_{0}(\vec{x},\tau)\right].$ (2)

Here $\mathcal{P}$ is path ordering, $A_{0}$ is the temporal component of
vector field and $\beta=T^{-1}$ Polyakov:78plb . In accordance of Ref. ratti ;
Roesnr , we have considered a homogeneous Polyakov-loop field
$\Phi(\vec{x})=\Phi$=constant and $\bar{\Phi}(\vec{x})=\bar{\Phi}$=constant.

The model Lagrangian has quarks, mesons, couplings and Polyakov-loop potential
${\cal U}\left(\Phi,\bar{\Phi},T\right)$ as :

${\cal L}_{PQM}={\cal L}_{QM}-{\cal U}\big{(}\Phi,\bar{\Phi},T\big{)}.$ (3)

The Lagrangian of the model Roder ; fuku11 ; grahl  is written as

$\displaystyle{\cal L}_{QM}$ $\displaystyle=$
$\displaystyle\bar{\psi}[i\gamma^{\mu}\partial_{\mu}-gt_{0}(\sigma+i\gamma_{5}\eta)$
(4)
$\displaystyle-g\vec{t}\cdot(\vec{a}+i\gamma_{5}\vec{\pi})]\psi+\cal{L(M)},$

where $\psi$ is a color $N_{c}$-plet, a four-component Dirac spinor and a
flavor doublet,

$\displaystyle\psi$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}u\\\ d\end{array}\right)\;.$ (7)

The Lagrangian for meson fields is Roder

$\displaystyle\cal{L(M)}$ $\displaystyle=$
$\displaystyle\text{Tr}(\partial_{\mu}{\cal{M}}^{\dagger}\partial^{\mu}{\cal{M}}-m^{2}({\cal{M}}^{\dagger}{\cal{M}}))$
(8)
$\displaystyle-\lambda_{1}\left[\text{Tr}({\cal{M}}^{\dagger}{\cal{M}})\right]^{2}-\lambda_{2}\text{Tr}({\cal{M}}^{\dagger}{\cal{M}})^{2}$
$\displaystyle+c[\text{det}{\cal{M}}+\text{det}{\cal{M}}^{\dagger}]+\text{Tr}\left[H({\cal{M}}+{\cal{M}}^{\dagger})\right]\;,$

here, the field ${\cal{M}}$ is a complex $2\times 2$ matrix

$\displaystyle{\cal{M}}=t_{0}(\sigma+i\eta)+\vec{t}\cdot(\vec{a}+i\vec{\pi})$
(9)

with $t_{0}=\dfrac{1}{2}\left(\begin{array}[]{c c}1&0\\\
0&1\end{array}\right),$ $t_{1}=\dfrac{1}{2}\left(\begin{array}[]{c c}0&1\\\
1&0\end{array}\right),$
$t_{2}=\dfrac{1}{2}\left(\begin{array}[]{c c}0&-i\\\ i&0\end{array}\right),$
$t_{3}=\dfrac{1}{2}\left(\begin{array}[]{c c}1&0\\\ 0&-1\end{array}\right).$

One can rewrite the Lagrangian (8) in the form fuku11

$\displaystyle{\cal{L(M)}}$ $\displaystyle=$ $\displaystyle\mbox{$1\over
2$}(\partial_{\mu}\sigma\partial_{\mu}\sigma+\partial_{\mu}\vec{\pi}\cdot\partial_{\mu}\vec{\pi}+\partial_{\mu}\eta\partial_{\mu}\eta$
(10)
$\displaystyle+\partial_{\mu}\vec{a_{0}}\cdot\partial_{\mu}\vec{a_{0}})-U\;,$

further

$\displaystyle U$ $\displaystyle=$
$\displaystyle\dfrac{m^{2}}{2}(\sigma^{2}+{\vec{\pi}}^{2}+\eta^{2}+{\vec{a}_{0}}^{2})-\dfrac{c}{2}(\sigma^{2}-\eta^{2}+{\vec{\pi}}^{2}-{\vec{a}_{0}}^{2})$
(11)
$\displaystyle+\dfrac{1}{4}\left(\lambda_{1}+\dfrac{1}{2}\lambda_{2}\right)(\sigma^{2}+{\vec{\pi}}^{2}+\eta^{2}+{\vec{a}_{0}}^{2})^{2}$
$\displaystyle+\dfrac{\lambda_{2}}{2}\left((\sigma^{2}+{\vec{\pi}}^{2})(\eta^{2}+{\vec{a}_{0}}^{2})-(\sigma\eta-{\vec{\pi}}\cdot{\vec{a}_{0}})^{2}\right)$
$\displaystyle-h\sigma.$

The $2\times 2$ matrix $H$ explicitly breaks the chiral symmetry and is chosen
as

$\displaystyle H=t_{a}h_{a}\;,$ (12)

where $h_{a}$ are external fields. The field $\sigma$ acquires nonzero vacuum
expectation value (VEV), $\overline{\sigma}$, due to the spontaneous breaking
of the chiral symmetry, while the other scalar and pseudo-scalar fields
($\vec{a}_{0},\vec{\pi},\eta$) assume zero VEV. Here the two parameters
$h_{0}$ and $h_{3}$ may give rise to the explicit breaking of chiral symmetry.
We are neglecting the isospin symmetry breaking, hence we choose $h_{0}\neq 0$
and $h_{3}=0$.

The field $\sigma$ has to be shifted to $\sigma\xrightarrow{\
}\overline{\sigma}+\sigma$ as it acquires nonzero VEV. At the tree level, the
expression of the meson masses are Roder

$\displaystyle m^{2}_{\sigma}$ $\displaystyle=$ $\displaystyle
m^{2}-c+3(\lambda_{1}+\frac{\lambda_{2}}{2})\overline{\sigma}^{2}\;,$ (13)
$\displaystyle m^{2}_{a_{0}}$ $\displaystyle=$ $\displaystyle
m^{2}+c+(\lambda_{1}+\frac{3\lambda_{2}}{2})\overline{\sigma}^{2}\;,$ (14)
$\displaystyle m^{2}_{\eta}$ $\displaystyle=$ $\displaystyle
m^{2}+c+(\lambda_{1}+\frac{\lambda_{2}}{2})\overline{\sigma}^{2}\;,$ (15)
$\displaystyle m^{2}_{\pi}$ $\displaystyle=$ $\displaystyle
m^{2}-c+(\lambda_{1}+\frac{\lambda_{2}}{2})\overline{\sigma}^{2}\;,$ (16)
$\displaystyle m_{q}$ $\displaystyle=$ $\displaystyle\frac{g\
\overline{\sigma}}{2}.$ (17)

Using (13)–(17), the parameters of the Lagrangian (8) are obtained as

$\displaystyle\lambda_{1}$ $\displaystyle=$
$\displaystyle\dfrac{m^{2}_{\sigma}+m^{2}_{\eta}-m^{2}_{a_{0}}-m^{2}_{\pi}}{2\overline{\sigma}^{2}}\;,$
(18) $\displaystyle\lambda_{2}$ $\displaystyle=$
$\displaystyle\dfrac{m^{2}_{a_{0}}-m^{2}_{\eta}}{\overline{\sigma}^{2}}\;,$
(19) $\displaystyle m^{2}$ $\displaystyle=$ $\displaystyle
m^{2}_{\pi}+\dfrac{m^{2}_{\eta}-m^{2}_{\sigma}}{2}\;,$ (20) $\displaystyle c$
$\displaystyle=$ $\displaystyle\dfrac{m^{2}_{\eta}-m^{2}_{\pi}}{2}\;,$ (21)
$\displaystyle\frac{g^{2}}{4}$ $\displaystyle=$
$\displaystyle\frac{m_{q}^{2}}{\overline{\sigma}^{2}}\;.$ (22)

and the tree level effective potential is written as,

$\displaystyle
U(\overline{\sigma})=\frac{1}{2}(m^{2}-c)\overline{\sigma}^{2}+\frac{1}{4}\left(\lambda_{1}+\frac{1}{2}\lambda_{2}\right)\overline{\sigma}^{4}-h\overline{\sigma}\;,\
\ $ (23)

$\overline{\sigma}=f_{\pi}$ gives the minimum of the effective potential at
the tree level and the stationarity condition for the potential (23), gives

$\displaystyle h$ $\displaystyle=$ $\displaystyle f_{\pi}m^{2}_{\pi}.$ (24)

###  II.1 Polyakov-loop potentials

There are different possibilities for the functional form of the effective
Polyakov-loop potential $\mathcal{U}(\Phi,\bar{\Phi},T)$. Its simplest form is
constructed by finding a potential which respects all given symmetries and
includes the spontaneously broken $Z(3)$ symmetry for the system in the
deconfined phase SveLer ; benji ; BankUka . Thus the minimal content of a
Polyakov-loop potential ratti  is given by the following polynomial form

$\displaystyle\frac{\mathcal{U_{\rm Poly}}}{T^{4}}$ $\displaystyle=$
$\displaystyle-\frac{b_{2}(T)}{2}\Phi\bar{\Phi}-\frac{b_{3}}{6}(\Phi^{3}+\bar{\Phi}^{3})+\frac{b_{4}}{4}(\Phi\bar{\Phi})^{2}\;,$
(25)

the coefficients of the Eq. (25) are given by

$\displaystyle
b_{2}(T)=a_{0}+a_{1}\left(\frac{T_{0}}{T}\right)+a_{2}\left(\frac{T_{0}}{T}\right)^{2}+a_{3}\left(\frac{T_{0}}{T}\right)^{3}\;,$
(26)

where $a_{0}=6.75$, $a_{1}=-1.95$, $a_{2}=2.625$, $a_{3}=-7.44$, $b_{3}=0.75$
and $b_{4}=7.5$ .

The above Polyakov-loop potential ansatz has been enhanced by adding the
contribution that results from the integration of the $SU(3)$ group volume in
the generating functional for the Euclidean action. This integration is done
by using the Haar measure and takes the form of a Jacobian determinant. Its
logarithm is added as an effective potential to the action in the generating
functional. The positive coefficient of the logarithm term bounds the
potential from below for large $\Phi$ and $\bar{\Phi}$ and the logarithmic
form of the Polyakov-loop potential is written as fuku ; Roesnr :

$\displaystyle\frac{\mathcal{U_{\rm Log}}}{T^{4}}$ $\displaystyle=$
$\displaystyle
b(T)\ln[1-6\Phi\bar{\Phi}+4(\Phi^{3}+\bar{\Phi}^{3})-3(\Phi\bar{\Phi})^{2}]\;$
(27) $\displaystyle-\frac{1}{2}a(T)\Phi\bar{\Phi}\;.$

The parameters of the polynomial and log form of the Polyakov-loop potential
were determined ratti ; Roesnr  by fitting the lattice data for pressure,
entropy density as well as energy density and the evolution of Polyakov-loop
$<\Phi>$ on the lattice in pure gauge theory. The coefficients of the Eq. (27)
are the following Roesnr ,

$\displaystyle
a(T)=a_{0}+a_{1}\left(\frac{T_{0}}{T}\right)+a_{2}\left(\frac{T_{0}}{T}\right)^{2}\;,$
(28) $\displaystyle b(T)=b_{3}\left(\frac{T_{0}}{T}\right)^{3}\;,$ (29)

where $a_{0}=3.51$, $a_{1}=-2.47$, $a_{2}=15.2$, $b_{3}=-1.75$. Note that the
log potential has qualitative consistency with the leading order result of the
strong-coupling expansion JLange . Also, since the potential diverges for
$\Phi$, $\bar{\Phi}\longrightarrow$ 1, the Polyakov-loop always remains
smaller than 1 and approaches this value asymptotically as
$T\longrightarrow\infty$.

Ref. Redlo  took into account the Polyakov-loop fluctuations and constructed
the new Polyakov-loop effective potential in which the parameters are so
adjusted that apart from the other existing lattice data, the lattice data for
the longitudinal as well as the transverse susceptibilities are also
reproduced. They enhanced the polynomial form of the Polyakov-loop potential
with the addition of the logarithmic term to arrive at the following
expression of the PolyLog Polyakov-loop potential,

$\displaystyle\frac{\mathcal{U_{\rm PolyLog}}}{T^{4}}$ $\displaystyle=$
$\displaystyle
b(T)\ln[1-6\Phi\bar{\Phi}+4(\Phi^{3}+\bar{\Phi}^{3})-3(\Phi\bar{\Phi})^{2}]\;$
$\displaystyle+a_{2}(T)\Phi\bar{\Phi}+a_{3}(T)(\Phi^{3}+\bar{\Phi}^{3})+a_{4}(T)(\Phi\bar{\Phi})^{2}.$

The coefficients of the Eq. (II.1) PolyLog parametrization are defined as

$\displaystyle
a_{i}(T)=\frac{a^{(i)}_{0}+a^{(i)}_{1}\left(\frac{T_{0}}{T}\right)+a^{(i)}_{2}\left(\frac{T_{0}}{T}\right)^{2}}{1+a^{(i)}_{3}\left(\frac{T_{0}}{T}\right)+a_{4}^{(i)}\left(\frac{T_{0}}{T}\right)^{2}}\;$
(31) $\displaystyle
b(T)=b_{0}\left(\frac{T_{0}}{T}\right)^{b_{1}}\left[1-e^{b_{2}\left(\frac{T_{0}}{T}\right)^{b_{3}}}\right].$
(32)

The parameters are summarized in the Table 1.

Table 1: Parameters of the PolyLog Polyakov-loop potential have been taken
from the Ref. Redlo .

PolyLog |  $a^{(2)}_{0}$ |  $a^{(2)}_{1}$ |  $a^{(2)}_{2}$ |  $a^{(2)}_{3}$ |  $a^{(2)}_{4}$ |
---|---|---|---|---|---|---
|  22.07 |  -75.7 |  45.03385 |  2.77173 |  3.56403 |
|  $a^{(3)}_{0}$ |  $a^{(3)}_{1}$ |  $a^{(3)}_{2}$ |  $a^{(3)}_{3}$ |  $a^{(3)}_{4}$ |
|  -25.39805 |  57.019 |  -44.7298 |  3.08718 |  6.72812 |
|  $a^{(4)}_{0}$ |  $a^{(4)}_{1}$ |  $a^{(4)}_{2}$ |  $a^{(4)}_{3}$ |  $a^{(4)}_{4}$ |
|  27.0885 |  -56.0859 |  71.2225 |  2.9715 |  6.61433 |
|  $b_{0}$ |  $b_{1}$ |  $b_{2}$ |  $b_{3}$ |  |
|  -0.32665 |  5.8559 |  -82.9823 |  3.0 |  |

The deconfinement phase transition is first order for the pure gauge Yang-
Mills theory and $T_{c}^{\rm YM}=T_{0}=270$ MeV. The first order transition
turns to a crossover in the presence of dynamical quarks. The parameter
$T_{0}$ depends on the number of quark flavors and chemical potential in the
full dynamical QCD SchaPQM2F ; Haas ; kovacs ; BielichP ; THerbst2  as it is
linked to the mass-scale $\Lambda_{\rm QCD}$ which gets modified by the effect
of the fermionic matter fields. $T_{0}\longrightarrow T_{0}(N_{f},\mu)$ is
written as,

$\displaystyle T_{0}(N_{f},\mu)=\hat{T}\ e^{-1/(\alpha_{0}b(N_{f},\mu))}\;,$
(33)

with

$\displaystyle
b(N_{f},\mu)=\frac{1}{6\pi}(11N_{c}-2N_{f})-b_{\mu}\frac{\mu^{2}}{(\hat{\gamma}\hat{T})^{2}}.$
(34)

where the parameter $\hat{T}$ is fixed at the scale $\tau$,
$\hat{T}=T_{\tau}=1.77$ GeV and $\alpha_{0}=\alpha(\Lambda)$ at a UV scale
$\Lambda$. The $T_{0}(N_{f}=0)$ = 270 MeV gives $\alpha_{0}$ = 0.304 and
$b_{\mu}\simeq\frac{16}{\pi}N_{f}$. The parameter $\hat{\gamma}$ governs the
curvature of $T_{0}(\mu)$ with the systematic error estimation range
$0.7\lesssim\hat{\gamma}\lesssim 1$ SchaPQM2F ; THerbst2 . In our calculation,
we have taken $\hat{\gamma}=1$. The $N_{f}$, $\mu$ dependence of the $T_{0}$
accounts only partially for the unquenching of the pure gauge Polyakov-loop
potential to an effective glue potential in QCD BielichP .

For the full QCD with dynamical quarks, the Polyakov-loop potential should be
replaced by the QCD glue potential that accounts for the back-reaction of
quarks into the Polyakov-loop effective potential. Applying the FRG equations
to the QCD, Ref. Haas  compared the pure gauge potential $\mathcal{U}_{\rm
YM}$ to the “glue” potential $\mathcal{U}_{\rm glue}$ where quark polarization
was included in the gluon propagator and they found significant differences
between the two potentials. However, it was observed in their study that the
two potentials have the same shape and they can be mapped into each other by
relating the temperatures of the two systems, $T_{\rm YM}$ and $T_{\rm glue}$.
Denoting the previous equations of the Polyakov-loop potential by
$\mathcal{U}_{\rm YM}$, the improved Polyakov-loop potential $\mathcal{U}_{\rm
glue}$ can be constructed as Haas

$\displaystyle\frac{\mathcal{U}_{\rm glue}}{T^{4}_{\rm
glue}}(\Phi,\bar{\Phi},T_{\rm glue})$ $\displaystyle=$
$\displaystyle\frac{\mathcal{U}_{\rm YM}}{T^{4}_{\rm
YM}}(\Phi,\bar{\Phi},T_{\rm YM})$ (35)

here the temperature $T_{\rm glue}$ is related to $T_{\rm YM}$ as

$\displaystyle\frac{T_{\rm YM}-T^{\rm YM}_{\rm c}}{T^{\rm YM}_{\rm c}}=0.57\
\frac{T_{\rm glue}-T^{\rm glue}_{\rm c}}{T^{\rm glue}_{\rm c}}$ (36)

The $T^{\rm glue}_{\rm c}$ is the transition temperature for the unquenched
case. The coefficient 0.57 comes from the comparison of the two effective
potentials. $T^{\rm glue}_{\rm c}$ lies within a range $T^{\rm glue}_{\rm
c}\in[180,270]$. In practice, we use in the right-hand side of the Polyakov-
loop potentials where $T_{0}$ means $T^{\rm YM}_{\rm c}$, the replacement
$T\longrightarrow T^{\rm YM}_{\rm c}(1+0.57(\frac{T}{T^{\rm glue}_{\rm
c}}-1))$ where ($T\sim T_{\rm YM}$) on the left side of the arrow and ($T\sim
T_{\rm glue}$) on the right side. In our calculations, we will keep $T^{\rm
glue}_{\rm c}$ and $T_{0}$ both fixed at 208 MeV and also considered the
chemical potential dependence of $T_{0}$ and $T^{\rm glue}_{\rm c}$. The $\mu$
dependence of the $T^{\rm glue}_{\rm c}$ can be found from the Eq. (33) after
making the replacement $T_{0}(N_{f},\mu)\longrightarrow{T^{\rm glue}_{\rm
c}}(N_{f},\mu)$.

###  II.2 Thermodynamic grand potential in PQM model

In the mean-field approximation, the thermodynamic grand potential for the PQM
model is given as SchaPQM2F

$\displaystyle\Omega_{\text{\tiny
MF}}(T,\mu;\overline{\sigma},\Phi,\bar{\Phi})$ $\displaystyle=$ $\displaystyle
U(\overline{\sigma})+\Omega_{q\bar{q}}(T,\mu;\overline{\sigma},\Phi,\bar{\Phi})\;$
(37) $\displaystyle+\mathcal{U}(T,\Phi,\bar{\Phi})\;.$

The quark/antiquark contribution is given by

$\displaystyle\Omega_{q\bar{q}}(T,\mu;\overline{\sigma},\Phi,\bar{\Phi})=\Omega_{q\bar{q}}^{vac}+\Omega_{q\bar{q}}^{T,\mu}(\overline{\sigma},\Phi,\bar{\Phi})\;,$
(38)
$\displaystyle\Omega_{q\bar{q}}^{vac}=-2N_{c}\sum_{q}\int\frac{d^{3}p}{(2\pi)^{3}}E_{q}\theta(\Lambda_{c}^{2}-\vec{p}^{2})\;,$
(39)
$\displaystyle\Omega_{q\bar{q}}^{T,\mu}(\overline{\sigma},\Phi,\bar{\Phi})=-2N_{c}\sum_{q}\int\frac{d^{3}p}{(2\pi)^{3}}T\left[\ln
g_{q}^{+}+\ln g_{q}^{-}\right].\;$ (40)

The first term of the Eq. (38) denotes the fermion vacuum contribution,
regularized by the ultraviolet cutoff $\Lambda_{c}$. In presence of the
Polyakov-loop potential, the $g^{+}_{q}$ and $g^{-}_{q}$ are specified by the
trace in the color space.

$\displaystyle g^{+}_{q}$ $\displaystyle=$ $\displaystyle\left[1+3\Phi
e^{-E_{q}^{+}/T}+3\bar{\Phi}e^{-2E_{q}^{+}/T}+e^{-3E_{q}^{+}/T}\right]\;,\qquad$
(41) $\displaystyle g_{q}^{-}$ $\displaystyle=$
$\displaystyle\left[1+3\bar{\Phi}e^{-E_{q}^{-}/T}+3\Phi
e^{-2E_{q}^{-}/T}+e^{-3E_{q}^{-}/T}\right]\;.$ (42)

E${}_{q}^{\pm}=E_{q}\mp\mu_{q}$ and $E_{q}=\sqrt{p^{2}+m{{}_{q}}{{}^{2}}}$ is
the flavor dependent single particle energy of quark/antiquark and
$m_{q}=\dfrac{g\overline{\sigma}}{2}$ is the mass of the given quark flavor.
$\mu_{q}$ is quark chemical potential.

Neglecting the quark one-loop vacuum term of the Eq. (38) in the the standard
mean-field approximation (s-MFA), the PQM model grand potential is written as,

$\displaystyle\Omega_{\text{\tiny
PQM}}(T,\mu;\overline{\sigma},\Phi,\bar{\Phi})$ $\displaystyle=$
$\displaystyle
U(\overline{\sigma})+\Omega_{q\bar{q}}^{T,\mu}(\overline{\sigma},\Phi,\bar{\Phi})+\mathcal{U}(T,\Phi,\bar{\Phi})\;.$

$\frac{\partial\Omega_{\text{\tiny
PQM}}}{\partial\overline{\sigma}}=\frac{\partial\Omega_{\text{\tiny
PQM}}}{\partial{\Phi}}=\frac{\partial\Omega_{\text{\tiny
PQM}}}{\partial\bar{\Phi}}\bigg{|}_{\overline{\sigma},\Phi,\bar{\Phi}}=0.$
(44)

The global minima of the grand potential in Eq. (44) gives the
$\overline{\sigma}$, $\Phi$ and $\bar{\Phi}$ as a function of temperature and
chemical potential.

##  III PQM model with Vacuum Term

Here, we give a brief description of the effective potential calculation when
the quark one-loop vacuum divergence of the Eq. (38) is regularized in the
minimal subtraction scheme and the $\sigma$ and $\pi$ meson curvature masses
are used for fixing the model parameters. The quark one-loop vacuum
contribution of Eq. (39) is written as RaiTiw

$\displaystyle\Omega^{vac}_{q\bar{q}}$ $\displaystyle=$
$\displaystyle\frac{N_{c}}{(4\pi)^{2}}\sum_{q}m^{4}_{q}\left[\frac{1}{\epsilon}+\frac{3}{2}+\ln\left(\Lambda^{2}\over
m^{2}_{q}\right)\right]\;,$ (45)

where $\Lambda$ is renormalization scale.

Adding the counterterm
$\delta\mathcal{L}=\frac{N_{c}}{(4\pi)^{2}}\sum_{q}\frac{m^{4}_{q}}{\epsilon}$
to the Lagrangian (4), the thermodynamic potential gets renormalized. After
replacing the first term of Eq. (38) by the
$\Omega^{vac}_{q\bar{q}}=\frac{N_{c}}{(4\pi)^{2}}\sum_{q}m^{4}_{q}\left[\frac{3}{2}+\ln\left(\Lambda^{2}\over
m^{2}_{q}\right)\right]$, one gets the renormalization scale dependent chiral
part of the vacuum ($\mu=0$ and $T=0$) effective potential as

$\displaystyle\Omega^{\Lambda}(\overline{\sigma})$ $\displaystyle=$
$\displaystyle U(\overline{\sigma})+\Omega^{vac}_{q\bar{q}}\;.$ (46)

The fixing of the model parameters $m^{2}$, $c$, $\lambda_{1}$, $\lambda_{2}$
and $h$ is presented in the Appendix (A). When the calculated new parameters
are substituted in the Eq.(46) and terms are rearranged, one finds the
expression of renormalization scale $\Lambda$ independent vacuum effective
potential as:

$\displaystyle\Omega(\overline{\sigma})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(m^{2}_{s}-\dfrac{N_{c}g^{4}f^{2}_{\pi}}{2(4\pi)^{2}}\right)\overline{\sigma}^{2}-\frac{1}{2}c\overline{\sigma}^{2}+\frac{1}{4}\left(\lambda_{1}+\frac{\lambda_{2s}}{2}\right.$
(47)
$\displaystyle\left.+\dfrac{3N_{c}g^{4}}{4(4\pi)^{2}}\right)\overline{\sigma}^{4}-h\overline{\sigma}+\frac{N_{c}g^{4}\overline{\sigma}^{4}}{8(4\pi)^{2}}\ln\left(\dfrac{f^{2}_{\pi}}{\overline{\sigma}^{2}}\right)\;.$

After quark one-loop vacuum correction, the thermodynamic grand potential for
the Polyakov-loop enhanced quark-meson model with vacuum term (PQMVT) can be
written as :

$\displaystyle\Omega_{\rm\text{\tiny
PQMVT}}(T,\mu;\overline{\sigma},\Phi,\bar{\Phi})$ $\displaystyle=$
$\displaystyle\Omega(\overline{\sigma})+\Omega^{T,\mu}_{q\bar{q}}(\overline{\sigma},\Phi,\bar{\Phi})+\mathcal{U}(\Phi,\bar{\Phi})\;,$
$\frac{\partial\Omega_{\text{\tiny
PQMVT}}}{\partial\overline{\sigma}}=\frac{\partial\Omega_{\text{\tiny
PQMVT}}}{\partial{\Phi}}=\frac{\partial\Omega_{\text{\tiny
PQMVT}}}{\partial\bar{\Phi}}\bigg{|}_{\overline{\sigma},\Phi,\bar{\Phi}}=0.$
(49)

One gets the quark condensate $\overline{\sigma}$ and the Polyakov-loop
expectation values $\Phi$, $\bar{\Phi}$ by searching the global minima of the
grand potential at a given $T$ and $\mu$. It is pointed out that the pion
decay constant $f_{\pi}$ does not get renormalized because the dressing of the
meson propagator is not considered in fixing of the model parameters using the
curvature mass of the pion. The minimum of the effective potential in vacuum
($T=0$, $\mu=0$) remains fixed at $\overline{\sigma}=f_{\pi}$.

##  IV Renormalized PQM Model

The PQMVT/QMVT model investigations lars ; guptiw ; schafwag12 ; chatmoh1 ;
TranAnd ; vkkr12 ; chatmoh2 ; vkkt13 ; Herbst ; Weyrich ; kovacs ; zacchi1 ;
zacchi2 ; Rai  use the curvature (or screening) masses of the mesons to fix
the parameters while the pion decay constant $f_{\pi}$ remains unrenormalized.
We know that the poles of the mesons propagators, give their physical masses
and the $f_{\pi}$ gets correction as it is related to the residue of the pion
propagator at its pole BubaCar ; Naylor ; fix1 . The definition of the meson
curvature masses, involves the evaluation of their self-energies at zero
momentum as the effective potential is the generator of the n-point functions
of the theory at zero external momenta laine ; Adhiand1 ; Adhiand2 ; Adhiand3
. Furthermore the pole definition is the physical and gauge invariant one
Kobes ; Rebhan . If the Dirac sea contributions are neglected, the pole masses
and the curvature masses for mesons become equivalent but when the quark one-
loop vacuum correction is considered, the curvature masses of the mesons
become different from their pole masses BubaCar ; fix1 . The above
inconsistency is removed in the exact on-shell parameter fixing method of the
renormalized quark-meson (RQM) model where the physical (pole) masses of the
mesons and the pion decay constant, are put into the relation of the running
mass parameter and couplings by using the on-shell and the minimal subtraction
renormalization schemes. The derivation of the quark one-loop effective
potential in the large $N_{c}$ limit and the mathematical details of the on-
shell parameter fixing for the renormalized quark-meson (RQM) model, have been
presented in the Appendix B. Combining the RQM model chiral effective
potential in $\overline{\text{MS}}$ scheme with the Polyakov-loop potential,
we get grand thermodynamic potential for the RPQM model as,

$\displaystyle\Omega_{\text{\tiny RPQM}}(T,\mu;\Delta,\Phi,\bar{\Phi})$
$\displaystyle=$
$\displaystyle\Omega_{vac}(\Delta)+\Omega^{T,\mu}_{q\bar{q}}(\Delta,\Phi,\bar{\Phi})+\mathcal{U}(\Phi,\bar{\Phi}),$

using the expression $\Omega_{vac}(\Delta)$ of Eq. 152 given in the Appendix
B, we write the full thermodynamic potential for the RPQM model as,

$\displaystyle\Omega_{\text{RPQM}}(T,\mu;\Delta,\Phi,\bar{\Phi})$
$\displaystyle=$
$\displaystyle\frac{(3m^{2}_{\pi}-m^{2}_{\sigma})f^{2}_{\pi}}{4}\left\\{1-\frac{N_{c}g^{2}}{(4\pi)^{2}}\left(\mathcal{C}(m^{2}_{\pi})+m^{2}_{\pi}\mathcal{C}^{\prime}(m^{2}_{\pi})\right)\right\\}\frac{\Delta^{2}}{m^{2}_{q}}\;$
$\displaystyle+\frac{N_{c}g^{2}f^{2}_{\pi}}{2(4\pi)^{2}}\left\\{\frac{3m^{2}_{\pi}\mathcal{C}(m^{2}_{\pi})-(m^{2}_{\sigma}-4m^{2}_{q})\mathcal{C}(m^{2}_{\sigma})}{2}-2m^{2}_{q}\right\\}\frac{\Delta^{2}}{m^{2}_{q}}\;$
$\displaystyle+\frac{(m^{2}_{\sigma}-m^{2}_{\pi})f^{2}_{\pi}}{8}\left\\{1-\frac{N_{c}g^{2}}{(4\pi)^{2}}\left(\mathcal{C}(m^{2}_{\pi})+m^{2}_{\pi}\mathcal{C}^{\prime}(m^{2}_{\pi})\right)\right\\}\frac{\Delta^{4}}{m^{4}_{q}}\;$
$\displaystyle+\frac{N_{c}g^{2}f^{2}_{\pi}}{(4\pi)^{2}}\left[(m^{2}_{\sigma}-4m^{2}_{q})\mathcal{C}(m^{2}_{\sigma})-m^{2}_{\pi}\mathcal{C}(m^{2}_{\pi})\over
8\right]\frac{\Delta^{4}}{m^{4}_{q}}+\frac{2N_{c}\Delta^{4}}{(4\pi)^{2}}\left\\{\frac{3}{2}-\ln\left(\frac{\Delta^{2}}{m^{2}_{q}}\right)\right\\}\;$
$\displaystyle-m^{2}_{\pi}f^{2}_{\pi}\left\\{1-\frac{N_{c}g^{2}}{(4\pi)^{2}}m^{2}_{\pi}\mathcal{C}^{\prime}(m^{2}_{\pi})\right\\}\frac{\Delta}{m_{q}}+\Omega_{q\bar{q}}^{T,\mu}(\Delta,\Phi,\bar{\Phi})+\mathcal{U}(\Phi,\bar{\Phi});$
$\frac{\partial\Omega_{\text{\tiny
RPQM}}}{\partial\Delta}=\frac{\partial\Omega_{\text{\tiny
RPQM}}}{\partial{\Phi}}=\frac{\partial\Omega_{\text{\tiny
RPQM}}}{\partial\bar{\Phi}}\bigg{|}_{\Delta,\Phi,\bar{\Phi}}=0.$ (52)

Searching the global minima of the grand potential for a given $\mu$ and $T$,
one gets the quark condensate $\overline{\sigma}$
($\Delta=\frac{g_{\overline{\text{\tiny MS}}}\
\overline{\sigma}_{\overline{\text{\tiny MS}}}}{2}$), the Polyakov-loop fields
$\Phi$ and $\bar{\Phi}$. It has been explained in the Appendix B that the
minimum of the vacuum ($T=0$, $\mu=0$) effective potential remains at
$\overline{\sigma}_{\overline{\text{\tiny MS}}}=f_{\pi}$. In our calculations,
we have used the $m_{\pi}=138.0$ MeV, $m_{a_{0}}=984.7$ MeV and
$m_{\eta}=547.0$ MeV. The Yukawa coupling $g=6.5$ and pion decay constant
$f_{\pi}=93.0$ MeV. The constituent quark mass in the vacuum
$m_{q}=\frac{gf_{\pi}}{2}=302.25$ MeV.

##  V Results and Discussion

The temperature axis chiral crossover and confinement-deconfinement crossover
transition at $\mu=0$, has been thoroughly investigated and compared with, in
different model scenarios of combining, the chiral effective potential
computed in different parameter fixing schemes to the different forms of the
Polyakov-loop potential with and without quark back-reaction. The subsection A
compares the results for the temperature variations of the chiral condensate,
the Polyakov-loop condensate and their derivatives. The chiral and
deconfinement crossover transition temperatures for different model scenarios
have been computed and presented for comparison in Table 2 and Table 3. In the
subsection B, the PQM, PQMVT and RPQM model phase diagrams have been plotted
and compared with each other for different values of $m_{\sigma}$ and
different forms of the Polyakov-loop potentials with and without quark back-
reaction. The appearance and disappearance of the quarkyonic phase in
different model scenarios, have been discussed in the subsection C. In order
to understand the $\mu=0$ chiral and deconfinement transition occurring at the
temperature axis, we have computed and compared the reduced scale temperature
variations of the thermodynamic quantities namely the pressure, entropy
density, energy density and interaction measure in the subsection D while the
results for the specific heat, $C^{2}_{s}$ and $P/\epsilon$ have been
presented in the subsection E. We have also compared the results obtained for
thermodynamic observables with the available lattice QCD data.

Table 2: Pseudo-critical temperatures for the $m_{\sigma}=500\text{MeV}$ at $\mu=$0. Polyakov-loop |  Models |  $T^{\chi}_{c}(\text{MeV})$ |  $T^{\Phi}_{c}(\text{MeV})$
---|---|---|---
|  PQM |  $172.1$ |  $168.9$
Log |  PQMVT |  $187.1$ |  $168.9$
|  RPQM |  $168.6$ |  $167.9$
|  PQM |  $165.6$ |  $165.4$
PolyLog-glue |  PQMVT |  $186.3$ |  $156.1$
|  RPQM |  $175.8$ |  $154.8$
Table 3: Pseudo-critical temperatures for the $m_{\sigma}=500\text{MeV}$ in the RPQM model at $\mu=$0. Models |  $T^{\chi}_{c}(\text{MeV})$ |  $T^{\Phi}_{c}(\text{MeV})$
---|---|---
Log |  $168.6$ |  $167.9$
Log-glue |  $174.6$ |  $145.2$
Poly |  $181.6$ |  $176.6$
Poly-glue |  $174.8$ |  $159.8$
PolyLog |  $180.1$ |  $175.1$
PolyLog-glue |  $175.8$ |  $154.8$

###  V.1 Order parameters and their derivatives

(a) Normalized chiral condensate.

(b) Polyakov-loop order parameter.

Figure 1: The normalized chiral and Polyakov-loop order parameters for the
$m_{\sigma}$ =500 MeV and $\mu=0$ at $T^{\rm glue}_{c}=T_{0}=208\ \text{MeV}\
$.

(a)

(b)

(c)

Figure 2: The derivatives of the normalized chiral and Polyakov-loop order
parameters for the $\mu=$0 and $T^{\rm glue}_{c}=T_{0}=208\ \text{MeV}\ $.

Fig. 1(a) and Fig. 1(b) present the respective temperature variations of the
normalized chiral condensate ($\sigma/f_{\pi}$) and the Polyakov-loop
condensate ($\Phi$) at $\mu=0$ for the sigma mass $m_{\sigma}$ = 500 MeV in
the PQM, PQMVT and RPQM model for the Log and the PolyLog-glue form of the
Polyakov-loop potential. The term glue denotes the unquenching of the
Polyakov-loop potential when the quark back-reaction has been taken into
account. Confirming the expected pattern when the models are augmented with
the Log form of the Polyakov-loop potential, the sharper Log-PQM model chiral
transition becomes quite smooth and delayed on account of the quark one-loop
vacuum correction in the on-shell renormalized Log-RPQM model while the
curvature mass parametrization for the Log-PQMVT model gives rise to an
excessively smooth and very delayed variation of the chiral order parameter on
the temperature axis in the Fig. 1(a). When the physics of the confinement-
deconfinement transition is coupled with the physics of the chiral transition,
the unquenching of the Polyakov-loop potential in the presence of the quark
back-reaction, leads to the significant smoothing effect on the chiral
condensate and shifts its temperature variations early on the temperature
scale for the $T<190$ MeV in the PolyLog-glue : PQM, RPQM and PQMVT model.
Furthermore, the Polyakov-loop condensate temperature variations are lifted up
and get shifted early on the temperature scale for the $T<220$ MeV, due to the
effect of the quark back-reaction when compared to the corresponding PQM, RPQM
and PQMVT model temperature variations with Log form of the Polyakov-loop
potential in the Fig. 1(b).

Presenting the plots of the $\frac{\partial(\sigma/f_{\pi})}{\partial{T}}$ and
$\frac{\partial\Phi}{\partial{T}}$ versus $T$ when the $\mu=0$ and the
$m_{\sigma}=500$ MeV in the Fig. 2(a), Fig. 2(b) and Fig. 2(c) respectively
for the PQM, PQMVT and RPQM model with the Log and PolyLog-glue form of the
Polyakov-loop potential, we have compared, how the PQM model results are
changed by the effect of the quark one-loop vacuum correction in the curvature
mass parameterized PQMVT model versus the on-shell parameterized RPQM model.
We have also compared how the presence of the quark back-reaction changes the
results in one particular model. For the Log-PQM model in the Fig. 2(a), the
very sharp variation of the temperature derivative of the $\sigma/f_{\pi}$,
drives a double peak structure in the temperature variation of the
$\frac{\partial\Phi}{\partial{T}}$ whose first peak at lower temperature gives
pseudocritical temperature $T^{\Phi}_{c}=168.9$ MeV for the confinement-
deconfinement transition and its second peak coincides with the very sharp and
high peak in the $\frac{\partial(\sigma/f_{\pi})}{\partial{T}}$ variation
which gets located at the higher pseudocritical temperature for the chiral
transition $T^{\chi}_{c}=172.1$ MeV and the two transitions are separated by
the difference $T^{\chi}_{c}-T^{\Phi}_{c}=3.2$ MeV as given in the Table 2.
One notices that the double peak structure for the
$\frac{\partial\Phi}{\partial{T}}$ is smoothed out and the sharper chiral
transition also becomes quite smooth as the peak heights get significantly
reduced due to the effect of the quark back-reaction in the PolyLog-glue PQM
model and the two transitions stand very close to each other having a
difference of only 0.2 MeV as the $T^{\chi}_{c}=165.6$ MeV and the
$T^{\Phi}_{c}=165.4$ MeV. As the fermionic vacuum correction with the
curvature mass parametrization leads to excess smoothing of the chiral
transition, the $\frac{\partial(\sigma/f_{\pi})}{\partial{T}}$ variation for
the Log PQMVT model shows a very smooth double peak structure (similar to Ref.
guptiw ) in the Fig. 2(b). Here, in contrast to the Fig. 2(a), the influence
of the Log form of the Polyakov-loop potential becomes dominant and the
sharper temperature variation of the $\frac{\partial\Phi}{\partial{T}}$
generates a double peak for the $\frac{\partial(\sigma/f_{\pi})}{\partial{T}}$
variation and the separation between the chiral crossover and the confinement-
deconfinement transition is equal to the 18.2 MeV as the $T^{\chi}_{c}=187.1$
MeV and the $T^{\Phi}_{c}=168.9$ MeV in Table 2. It is to be noted that due to
the smoothing influence of the quark back-reaction for the unquenched PloyLog-
glue PQMVT model, the double peak of the
$\frac{\partial(\sigma/f_{\pi})}{\partial{T}}$ variation, gets completely
washed out and one gets largest separation of 30.2 MeV between the
confinement-deconfinement and the chiral crossover transition temperatures as
the $T^{\chi}_{c}=186.3$ MeV while the $T^{\Phi}_{c}=156.1$ MeV. The smoothing
influence of the quark one-loop vacuum correction, becomes moderate due to the
consistent on-shell parameter fixing for the RPQM model and one notices that
the temperature variations of the $\frac{\partial\Phi}{\partial{T}}$ and the
$\frac{\partial(\sigma/f_{\pi})}{\partial{T}}$ rise to almost the same height
in the Fig. 2(c) for the Log form of the Polyakov-loop potential in the RPQM
model. The $\frac{\partial(\sigma/f_{\pi})}{\partial{T}}$ variation falls
short of developing a second peak and one finds that the chiral crossover
transition at the $T^{\chi}_{c}=168.6$ MeV is very close to the confinement-
deconfinement transition occurring at the $T^{\Phi}_{c}=167.9$ MeV. Here also
in the unquenched PolyLog-glue RPQM model, the robust effect of the quark
back-reaction leads to sufficient smoothing and reduction in the heights of
the peaks of the $\frac{\partial\Phi}{\partial{T}}$ and
$\frac{\partial(\sigma/f_{\pi})}{\partial{T}}$ temperature variations and a
moderate separation of 21.0 MeV is noted between the confinement-deconfinement
and the chiral crossover transition temperatures as the $T^{\chi}_{c}=175.8$
MeV while the $T^{\Phi}_{c}=154.8$ MeV.

(a) Normalized chiral condensate.

(b) Polyakov-loop order parameter.

Figure 3: The RPQM model normalized chiral and Polyakov-loop order parameters
when $\mu=0$ and $T^{\rm glue}_{c}=T_{0}=208\ \text{MeV}\ $.

(a)

(b)

(c)

Figure 4: The derivatives of the normalized chiral and Polyakov-loop order
parameters at $\mu=$0 and $T^{\rm glue}_{c}=T_{0}=208\ \text{MeV}\ $.

The Fig. 3(a) and the Fig. 3(b) show how the respective temperature variations
of the $\sigma/f_{\pi}$ and $\Phi$, are influenced by the different forms of
the Polyakov-loop potential put in combination with the consistent formulation
of the chiral sector physics in the RPQM model. The falling patterns of the
$\sigma/f_{\pi}$ in the Fig. 3(a), shift early on the temperature scale
because of the quark back-reaction in the unquenched forms of the Polyakov-
loop potentials namely the Log-glue, Poly-glue and PolyLog-glue. The lattice
QCD data for the temperature variation of the $\Phi$ has also been plotted in
the Fig. 3(b). One can see that the RPQM model Polyakov-loop condensate
temperature variations for the Log and the PolyLog potential, are closer to
the Zantow ; Wupertal2010  lattice results when the $T<170$ MeV. For higher
temperatures, the $\Phi$ variation of the Log, the Log-glue and the PolyLog-
glue RPQM model, stand closer to the two flavor LQCD data of the Ref Zantow .

(a)

(b)

Figure 5: The PQM, PQMVT and RPQM model phase diagrams for different forms of
the Polyakov-loop potentials. The error bars of the $\pm 4$ MeV on the Log
PQMVT model crossover transition line, have been calculated as in the Ref.
guptiw .

The Polyakov-loop potential in the Log form has stronger influence as one
notes that the quite sharp peaks of the temperature variations of the
$\frac{\partial(\sigma/f_{\pi})}{\partial{T}}$ and the
$\frac{\partial\Phi}{\partial{T}}$ are of almost the same height in the Log-
RPQM model in the Fig. 4(a) while the corresponding peaks are round and
smoother respectively for the polynomial and polynomial combined with the Log
form of the Polyakov-loop potential in the Fig. 4(b) and Fig. 4(c). Since the
height of the $\frac{\partial(\sigma/f_{\pi})}{\partial{T}}$ peaks are larger
than that of the $\frac{\partial\Phi}{\partial{T}}$ peaks in the Fig. 4(b) and
Fig. 4(c), the chiral order parameter temperature variation has stronger
influence in the Poly-RPQM and PolyLog-RPQM models. Even though the height of
the $\frac{\partial\Phi}{\partial{T}}$ peak is reduced and its sharpness gets
moderated due to the quark back-reaction in the Log-glue RPQM model in the
Fig. 4(a), the Log contribution makes the influence of the Ployakov loop
potential stronger which gives rise to a double peak structure in the
temperature variation of the $\frac{\partial(\sigma/f_{\pi})}{\partial{T}}$.
The quark back-reaction has significant smoothing effect on both the order
parameters as the $\frac{\partial(\sigma/f_{\pi})}{\partial{T}}$ and the
$\frac{\partial\Phi}{\partial{T}}$ temperature variations become more flat and
rounded with reduced heights in the the Fig. 4(b) and Fig. 4(c) respectively
for the Poly-gule and the PolyLog-glue RPQM model. It is worth reminding
ourselves that the separation ($T^{\chi}_{c}-T^{\Phi}_{c}$) between the
pseudo-critical temperatures for the chiral transition and the confinement-
deconfinement transition is very small of only 0.7 MeV for the Log Polyakov-
loop potential and it increases to 5.0 MeV when one has either the Poly or the
Poly-Log form of the Ployakov-loop potential. One can see from the Table 3
that the quark back-reaction causes the largest separation of 29.4 MeV in the
Log-glue form while the separation becomes 15 MeV for the Poly-glue form and
21 MeV for the PolyLog-glue form of the unquenched Polyakov-loop potential in
the RPQM model.

###  V.2 Sigma mass and model dependence of the phase diagrams and CEPs

We have plotted and compared the phase boundaries in the chemical potential
and temperature $\mu-T$ plane for the chiral symmetry breaking-restoring phase
transitions in the Polyakov-loop enhanced chiral models namely the PQM, PQMVT
and RPQM having different forms of parametrization for the Polyakov-loop
potential. It is well known that the first order transition line shrinks and
the crossover line gets extended with the increase of sigma meson mass and
therefore the CEP shifts rightwards in the phase diagram. Below,we are
presenting phase diagrams for different sigma meson masses. Line types in all
the phase diagrams are labeled and explained in the Figs.

(a)

(b)

Figure 6: Phase diagrams in the (a) are for the PQM, PQMVT and RPQM model and
in the (b) are for the different $m_{\sigma}$ in the Log RPQM model.

The Fig 5(a) presents the phase diagram for the $m_{\sigma}=$ 400 MeV. The
complete phase boundary for the Log PQM model is a first order phase
transition line. Since the quark one-loop vacuum correction with the curvature
mass parametrization in the Log PQMVT model generates excessively smooth
chiral transition as also reported in earlier works schafwag12 ; guptiw ;
vkkr12 , one gets a longer length of the line depicting the chiral crossover
transition which terminates at the critical end point (CEP) at
$\mu_{\text{\tiny CEP}}$ = 267.2 MeV, $T_{\text{\tiny CEP}}$ = 94.77 MeV and
the phase boundary becomes the first order transition line afterwards. The on-
shell parameter fixing for the Log-RPQM model with the consistent
renormalization of the quark one-loop vacuum correction, gives rise to a
relatively moderate smoothing effect on the chiral transition and the CEP gets
located higher up in the $\mu-T$ plane at the $T_{\text{\tiny CEP}}$ = 100.1
MeV and the $\mu_{\text{\tiny CEP}}$ = 245.3 MeV. When we consider the
chemical potential dependence of the parameter $T_{0}\equiv
T_{0}(\mu)=T^{glue}_{c}(\mu)$ together with the unquenched PolyLog-glue form
of the Ployakov-loop potential in the RPQM model, the quark back-reaction
generates an additional robust smoothing influence on the chiral transition
due to which the CEP shifts downwards and gets located at $\mu_{\text{\tiny
CEP}}$ = 242.3 MeV and $T_{\text{\tiny CEP}}$ = 66.6 MeV. Note that the
influence of the quark back-reaction is quite strong in the temperature
direction as the PolyLog-glue RPQM model $T_{\text{\tiny CEP}}$ shifts down by
33.5 MeV when compared with the $T_{\text{\tiny CEP}}$ of the Log RPQM model
while the corresponding shift in the chemical potential $\mu_{\text{\tiny
CEP}}$ is only 3 MeV. This effect gives rise to the increased curvature of the
phase transition line as first reported and discussed in the Ref. BielichP .
We point out that in our recent work RaiTiw  in the on-shell renormalized
quark-meson (RQM) model where the effect of the Polyakov-loop potential is
absent, the CEP gets located in the bottom right of the $\mu-T$ plane at
$T_{\text{\tiny CEP}}$ = 38.2 MeV and $\mu_{\text{\tiny CEP}}$ = 253.5 MeV
when the $m_{\sigma}=400$ MeV. We see that the presence of Polyakov-loop
potential either in the Log or in the PolyLog-glue form in the RPQM model
leads to significant upward shift of the CEP in the $\mu-T$ plane.

The phase diagram for the $m_{\sigma}$ = 500 MeV case has been plotted in the
Fig. 5(b). We get a small crossover line ending in the critical end point at
$T_{\text{\tiny CEP}}$ = 165.1 MeV and $\mu_{\text{\tiny CEP}}$ = 98.8 MeV and
a quite long first order line for the Log-PQM model. In the Log-PQMVT model,
similar to the $m_{\sigma}=400$ MeV case, the crossover line becomes very
large at the expense of significantly shrunk first order region and the CEP
gets located at $T_{\text{\tiny CEP}}$ = 78.0 MeV, $\mu_{\text{\tiny CEP}}$ =
295.9 MeV. When we consider the unquenched PolyLog-glue form for the Ployakov-
loop potential with the chemical potential dependent parameter $T_{0}\equiv
T_{0}(\mu)=T^{glue}_{c}(\mu)$, the quark back-reaction in the PolyLog-glue
PQMVT model causes 36.2 MeV reduction in the temperature axis location of the
CEP when compared with the $T_{\text{\tiny CEP}}$ of the Log-PQMVT model as it
gets located at the $T_{\text{\tiny CEP}}$ = 41.8 MeV while the chemical
potential location remains almost the same at the $\mu_{\text{\tiny CEP}}$ =
296.0 MeV. Note that the uncertainty bar of the Log PQMVT model for finding
the $T_{c}^{\chi}$ in the chemical potential range 100-165 MeV, also
disappears in the PolyLog-glue PQMVT model and we find a well defined
crossover transition line with increased curvature. When compared to the CEP
of the Log PQMVT model, the CEP in the Log RPQM model, gets located higher up
on the temperature axis at $T_{\text{\tiny CEP}}$ = 94.1 MeV with smaller
chemical potential at $\mu_{\text{\tiny CEP}}$ = 270.6 MeV. The above result
is expected because the smoothing influence of the on-shell renormalized quark
one-loop vacuum fluctuation on the chiral transition remains moderate also
when the $m_{\sigma}=500$ MeV RaiTiw . The CEP of the Log RPQM model shifts
down in temperature by 38.5 MeV due to the effect of quark back-reaction in
the unquenched PolyLog-glue RPQM model with $T_{0}\equiv
T_{0}(\mu)=T^{glue}_{c}(\mu)$ and gets located at the $T_{\text{\tiny CEP}}$ =
55.6 MeV while the corresponding chemical potential gets located at the
$\mu_{\text{\tiny CEP}}$ = 270.9 MeV with a negligible shift. When the $\mu$
dependence of the $T_{0}$ is switched off in the PolyLog-glue RPQM model as
shown later in the Fig. 7(a), the CEP gets located at the $T_{\text{\tiny
CEP}}$ = 70.1 MeV and the $\mu_{\text{\tiny CEP}}$ = 268.1 MeV with a moderate
temperature axis shift of 24.0 MeV in reference to the CEP of the Log RPQM
model. The curvature of the phase transition line increases significantly for
all the cases where the unquenching of the Polyakov-loop potential has been
considered. In our work we get confirmation of the observation of Ref.
BielichP  that the quark back-reaction due to the unquenching of the Polyakov-
loop potential, links the chiral and deconfinement phase transitions also at
small temperatures and large chemical potentials. Here it is also relevant to
recall that for the $m_{\sigma}=500$ MeV case, the CEP for the two flavor RQM
model, lies at the $T_{\text{\tiny CEP}}$ = 36.2 MeV and the $\mu_{\text{\tiny
CEP}}$ = 277.3 MeV RaiTiw  in the absence of the Polyakov-loop effect. Thus
whatever be the form of Polyakov-loop potential when combined with the chiral
physics, it generates a noteworthy upward shift of the CEP in the $\mu-T$
plane.

Fig. 6(a) depicts the phase diagram for the $m_{\sigma}=600$ MeV case. In
comparison to the lower $\sigma$ meson masses, the CEP in the Log PQM model
shifts rightwards at $T_{\text{\tiny CEP}}$ = 159.3 MeV and $\mu_{\text{\tiny
CEP}}$ = 174.5 MeV. The entire phase boundary for the Log PQMVT model becomes
a crossover transition line. The position of the CEP for the Log RPQM model is
found at $T_{\text{\tiny CEP}}$ = 39.1 MeV and $\mu_{\text{\tiny CEP}}$ =
321.0 MeV. In the PolyLog-glue RPQM model with the $T_{0}\equiv
T_{0}(\mu)=T^{glue}_{c}(\mu)$, due to the effect of the quark back-reaction,
the CEP gets located at the $T_{\text{\tiny CEP}}$ = 24.8 MeV and the
$\mu_{\text{\tiny CEP}}$ = 318.9 MeV. In comparison to the $m_{\sigma}=400\
\text{and}\ 500$ MeV cases, here for the $m_{\sigma}=600$ MeV, we note that
the $T_{\text{\tiny CEP}}$ registers a smaller downwards shift of 14.3 MeV
with respect to the Log RPQM model $T_{\text{\tiny CEP}}$.

(a)

(b)

Figure 7: Deconfinement and chiral transition phase boundaries for the Log,
Poly and PolyLog-glue form of the Polyakov-loop potentials in the RPQM model,
(a) for the constant $T_{0}=208$ MeV and (b) for the $\mu$ dependent
$T_{0}\equiv T_{0}(\mu)=T^{glue}_{c}(\mu)$.

Fig. 6(b) presents the comparison of phase diagrams in the Log RPQM model for
the different $\sigma$ meson masses of 400, 500, 600, 616, and 700 MeV. The
location of the CEP at $T_{\text{\tiny CEP}}$= 100.1 MeV and $\mu_{\text{\tiny
CEP}}$= 245.3 MeV for the $m_{\sigma}$ = 400 MeV shifts slightly right to
$T_{\text{\tiny CEP}}$= 94.1 MeV and $\mu_{\text{\tiny CEP}}$= 270.6 MeV for
the $m_{\sigma}$ = 500 MeV. The CEP registers a significant rightward shift
when the $m_{\sigma}$ = 600 MeV and gets located at $T_{\text{\tiny CEP}}$=
39.1 MeV and $\mu_{\text{\tiny CEP}}$= 321.0 MeV. For the $m_{\sigma}\geq 616$
MeV, the entire phase boundary depicts a crossover transition. Two crossover
transition lines for the $m_{\sigma}=616\ \text{and}\ 700$ MeV have also been
shown in the Fig.

###  V.3 Quarkyonic phase and its disappearance

One can define two pseudo-critical temperatures, the $T_{c}^{\Phi}$ and the
$T_{c}^{\bar{\Phi}}$ for the deconfinement crossover transition of the fields
$\Phi$ and $\bar{\Phi}$ by identifying the respective peaks in the temperature
variations of the $\frac{\partial\Phi}{\partial T}$ and
$\frac{\partial\bar{\Phi}}{\partial T}$. Locus of the different $T_{c}^{\Phi}$
and $T_{c}^{\bar{\Phi}}$ at different chemical potentials gives the respective
phase boundaries for the crossover transition for the fields $\Phi$ and
$\bar{\Phi}$. Apart from the phase boundaries for the chiral crossover
transition, the Fig. 7(a) presents the plots of the deconfinement crossover
transition phase boundaries also for the fields $\Phi$ and $\bar{\Phi}$. Phase
diagrams for the RPQM model with Poly, Log and PolyLog-glue forms of the
Polyakov-loop potential with the constant $T_{0}=208$ MeV, have been plotted
in the Fig. 7(a) while the corresponding plots when the parameter $T_{0}$ has
the chemical potential dependence i.e. $T_{0}\equiv T_{0}(\mu)$, have been
plotted in the Fig. 7(b).

In the Fig. 7(a), when the $\mu=0$ to 130 MeV in the Poly RPQM model for the
constant $T_{0}$, the deconfinement crossover transition line for the
$\bar{\Phi}$ lies below the corresponding line for the $\Phi$ and both of
these lines get placed below the chiral crossover transition line i.e.
$T_{c}^{\bar{\Phi}}$ and $T_{c}^{\Phi}\ <T_{c}^{\chi}$. The $\Phi$ crossover
line after remaining almost coincident with the chiral crossover line in the
$\mu=130-170$ MeV range, branches out of it and the deconfinement transition
temperature $T_{c}^{\Phi}$ jumps to 173.8 MeV while the $T_{c}^{\chi}=$ 154.6
MeV when the $\mu$ becomes 175 MeV. Thus we are getting a region in the phase
diagram where the deconfinement crossover transition line for the field $\Phi$
lies significantly above the chiral crossover line and the $T_{c}^{\Phi}\
>T_{c}^{\chi}$. The abovementioned portion of the phase diagram, where the
chiral symmetry is restored but the quarks are still confined, has been
identified as the quarkyonic phase in the literature kahara ; McLerran:2007npa
; McLerran:2009npa ; Dutra:2013 . Here in the Poly RPQM model, a significantly
large region of the quarkyonic phase is obtained for the field $\Phi$ as it
starts early when the $\mu\geq\ $ 175 MeV. The $\bar{\Phi}$ crossover line
remains below the chiral transition line uptill the $\mu=205$ MeV and crosses
it between $\mu=205-210$ MeV as the $T_{c}^{\bar{\Phi}}$ jumps to 157.6 MeV
while the $T_{c}^{\chi}$ stays at 139.1 MeV and the $T_{c}^{\bar{\Phi}}\
>T_{c}^{\chi}$. Here the extent of the so called quarkyonic phase for the
field $\bar{\Phi}$ is moderate as it starts from the $\mu\geq\ $ 210 MeV. When
we consider the $\mu$ dependence of the $T_{0}\equiv T_{0}(\mu)$ for the Poly
RPQM model in the Fig. 7(b), the extent of quarkyonic phase for the $\Phi$ and
the $\bar{\Phi}$, is considerably reduced. The deconfinement transition phase
boundary for the $\Phi$ branches out of the chiral transition line for the
$\mu>$ 225 MeV while the corresponding line for the $\bar{\Phi}$ forks out
from the chiral crossover line for the $\mu>$ 255 MeV. Thus the quarkyonic
phase for the $\Phi$ begins from the $\mu>$ 225 MeV while it begins from the
$\mu>$ 255 MeV for the $\bar{\Phi}$.

(a)

(b)

Figure 8: The temperature variation of
$\frac{\partial(\sigma/f_{\pi})}{\partial T}$, $\frac{\partial\Phi}{\partial
T}$ and $\frac{\partial\bar{\Phi}}{\partial T}$ for the Log, Poly and PolyLog-
glue form of the Polyakov-loop potentials in the RPQM model at $\mu=$ 265 MeV,
(a) for the constant $T_{0}=208$ MeV and (b) for the $\mu$ dependent
$T_{0}\equiv T_{0}(\mu)=T^{glue}_{c}(\mu)$.

The two deconfinement crossover transition lines for the $\Phi$ and
$\bar{\Phi}$ in the Log RPQM model are coincident in the Fig. 7(a). Lying
below the chiral crossover transition line, these lines keeping a small
temperature difference from it in the chemical potenial range 0 – 220 MeV, get
merged with it at the $\mu=220$ MeV and remain so uptill $\mu=235$ MeV. Both
of the deconfinement crossover lines fork out of the chiral crossover
transition line together at $\mu=$235 MeV by registering a significant jump in
the deconfinement crossover temperatures for the $\Phi$ and $\bar{\Phi}$ as
the $T_{c}^{\Phi}$ becomes 142.1 MeV and the $T_{c}^{\bar{\Phi}}$ becomes
144.1 MeV at $\mu=240.0$ MeV while the chiral crossover temperature remains at
the $T_{c}^{\chi}=125.6$ MeV. Here in the Log RPQM model with the constant
$T_{0}$ case, one gets the simultaneous onset of the quarkyonic phase for both
the fields $\Phi$ and $\bar{\Phi}$ and its region is considerably smaller when
compared to the Poly RPQM model as it starts from the chemical potential
$\mu>\ 240$ MeV. When we see the plots for the Log RPQM model with chemical
potential dependent $T_{0}(\mu)$ in the Fig. 7(b), we find that the two
coincident $\Phi$ and $\bar{\Phi}$ deconfinement crossover transition lines do
not fork out of the chiral crossover transition line at higher chemical
potentials and remain merged with it. Thus the quarkyonic phase disappears in
the Log RPQM model for the chemical potential dependent $T_{0}\equiv
T_{0}(\mu)$.

For the PolyLog-glue RPQM model in the Fig. 7(a) at the constant $T_{0}$, the
deconfinement crossover transition phase boundaries for the $\Phi$ and
$\bar{\Phi}$, are completely degenerate and get placed at lower temperatures
than the chiral crossover transition line in the $\mu=$0 to 250 MeV range,
i.e. $T_{c}^{\Phi}\ =\ T_{c}^{\bar{\Phi}}\ <T_{c}^{\chi}$. The coincident
deconfinement crossover lines are crossing the chiral transition phase
boundary near the $\mu=250$ MeV and take a robust jump such that the
$T_{c}^{\Phi}=T_{c}^{\bar{\Phi}}=$125.3 MeV at the $\mu=255$ MeV while the
$T_{c}^{\chi}$ stays at 86.6 MeV. Thus one gets the quarkyonic phase and the
region of its existence is found to be the smallest in the PolyLog-gule RPQM
model as it sets in for the $\mu>255$ MeV. When we examine the PolyLog-gule
RPQM model plot for the chemical potential dependent $T_{0}\equiv
T_{0}(\mu)=T^{glue}_{c}(\mu)$ case, we do not find any region of quarkyonic
phase as the degenerate deconfinement crossover lines, completely merge with
the chiral crossover transition line for the $\mu\ \geq\ 265$ MeV. Here we
recall that Schaefer et. al.SchaPQM2F ; THerbst2  have argued that the
incidence of quarkyonic phase presents an unphysical scenario because the
deconfinement temperature should be smaller or equal to the chiral transition
temperature. Similar to our Log RPQM model finding, they have also reported
the coincidence of the chiral and deconfinement transition lines for the
entire phase diagram when the parameter $T_{0}$ is taken as the $\mu$
dependent $T_{0}(\mu)$.

In order to see how the deconfinement transition temperatures $T_{c}^{\Phi}$
and $T_{c}^{\bar{\Phi}}$ in one model setting, differ from the corresponding
chiral transition temperature $T_{c}^{\chi}$ in the region of quarkyonic
phase, the temperature variations of the
$\frac{\partial(\frac{\sigma}{f_{\pi}})}{\partial T}$,
$\frac{\partial\Phi}{\partial T}$ and $\frac{\partial\bar{\Phi}}{\partial T}$,
have been plotted for the $\mu=265$ MeV respectively in the Fig. 8(a) for the
constant $T_{0}$ case and in the Fig. 8(b) for the $\mu$ dependent case
$T_{0}\equiv T_{0}(\mu)$. When the double peak structure emerges in the
temperature variations of the $\frac{\partial\Phi}{\partial T}$ and the
$\frac{\partial\bar{\Phi}}{\partial T}$ at higher chemical potentials, one
identifies the quarkyonic phase. Since the chiral transition dynamics is the
driver of the first peak formation at a lower temperature, its location
coincides with the chiral crossover temperature $T_{c}^{\chi}$. The second
peak gets located at a higher temperature respectively for the
$\frac{\partial\Phi}{\partial T}$ and the $\frac{\partial\bar{\Phi}}{\partial
T}$ temperature variations and one gets $T_{c}^{\Phi}\ >\ T_{c}^{\chi}$ and
$T_{c}^{\bar{\Phi}}\ >\ T_{c}^{\chi}$. The $\frac{\partial\Phi}{\partial T}$
and $\frac{\partial\bar{\Phi}}{\partial T}$ temperature variations in the Fig.
8(a), show the clear structure of the double peaks in the respective red,
green and blue line plots for the PolyLog-glue, Log and Poly RPQM model where
the quarkyonic phase exists for the constant $T_{0}$ case. The first peak of
the preceding plots coincide with the peak of the
$\frac{\partial(\frac{\sigma}{f_{\pi}})}{\partial T}$ temperature variations
for the corresponding model. The temperature variations of the
$\frac{\partial\Phi}{\partial T}$ and $\frac{\partial\bar{\Phi}}{\partial T}$
in the Fig. 8(b), do not show any double peak structure in the respective red
and green line plots of the PolyLog-glue and the Log RPQM model because the
quarkyonic phase disappears for the chemical potential dependent $T_{0}\equiv
T_{0}(\mu)$. The corresponding blue line plots for the Poly RPQM model, show
the distinct double peaks due the presence of the quarkyonic phase.

(a)

(b)

Figure 9: Reduced temperature scale variation of the pressure normalized to
its Stefan-Boltzmann limit for the (a) RPQM, PQMVT and PQM model with the
PolyLog-glue form of the Polyakov-loop potential, (b) RPQM model with the Log,
Log-glue and PolyLog-glue form of the Polyakov-loop potential. The two flavor
LQCD data of the pressure have been taken from the Ref. AliKhan:2001ek .

###  V.4 Pressure, entropy density, energy density and interaction measure

The thermodynamic observables pressure, entropy, energy density and
interaction measure are sensitive to the QCD phase transition. Considering the
PolyLog-glue ansatz for the Polyakov-loop potential, the abovementioned
observables have been computed and compared to see, how the PQM model results
are changed by the quark one-loop vacuum correction in the on-shell
parametrization of the RPQM model versus the curvature mass parametrization of
the PQMVT model. It is also worthwhile to compare, how the thermodynamic
observables are influenced when one takes different form of the Polyakov-loop
potential namely the Log, Log-glue or PolyLog-glue for combining it with the
consistent chiral physics of the RPQM model.

The negative grand potential gives us the pressure,

$\displaystyle P(T,\mu)=-\Omega_{MF}(T,\mu),\;$ (53)

the pressure in vacuum has been normalized to zero, i.e. $P(0,0)=0$. When the
$\mu=0$, the ideal gas Stefan-Boltzmann limit (SB) pressure is defined as,

$\displaystyle\frac{P_{SB}}{T^{4}}=(N^{2}_{c}-1)\frac{\pi^{2}}{45}+N_{c}N_{f}\frac{7\pi^{2}}{180}.\;$
(54)

(a)

(b)

Figure 10: Reduced temperature scale variation of the entropy density
normalized to its Stefan-Boltzmann limit for the (a) RPQM, PQMVT and PQM model
with the PolyLog-glue form of the Polyakov-loop potential, (b) RPQM model with
the Log, Log-glue and PolyLog-glue form of the Polyakov-loop potential.

Fig. 9(a) presents the comparison of the PQM, RPQM and PQMVT model plots of
the normalized pressure ($\frac{p}{p_{SB}}$) on the reduced temperature scale
when the $m_{\sigma}=500$ MeV and the PolyLog-glue form of the unquenched
Polyakov-loop potential has been taken. The two flavor lattice QCD data of
pressure normalized with the corresponding SB limit (on the discretized space-
time) AliKhan:2001ek  has also been presented for the comparison. The
normalized pressure for the RPQM model, shows a decent agreement with the
lattice data in the range 0.7 to 1.25 of the $T_{c}^{\chi}$. The quark one-
loop vacuum correction leads to the increased pressure and the largest
increase of pressure is noticed in the PQMVT model. In the Fig. 9(b), one can
see that the rise of the pressure in the Log RPQM model near the chiral
transition temperature, gets significantly modified in the Log-glue and
PolyLog-glue RPQM model due to the quark back-reaction and unquenching of the
Polyakov-loop potential. The rise in the pressure is caused by the melting of
the constituent quark masses and it saturates near 80 percent of the SB limit.

(a)

(b)

Figure 11: Reduced temperature scale variation of the energy density
normalized to its Stefan-Boltzmann limit for the (a) RPQM, PQMVT and PQM model
with the PolyLog-glue form of the Polyakov-loop potential, (b) RPQM model with
the Log, Log-glue and PolyLog-glue form of the Polyakov-loop potential. The
two flavor lattice QCD data of the energy density have been taken from the
Ref. AliKhan:2001ek .

(a)

(b)

Figure 12: Reduced temperature scale variation of the interaction measure
normalized to the Stefan-Boltzmann limit of energy density for the (a) RPQM,
PQMVT and PQM model with the PolyLog-glue form of the Polyakov-loop potential,
(b) RPQM model with the Log, Log-glue and PolyLog-glue form of the Polyakov-
loop potential. The two flavor lattice QCD data of the interaction measure
have been taken from the Ref. AliKhan:2001ek .

The entropy density $s$, energy density $\epsilon$ and interaction measure
$\Delta$ are defined as

$\displaystyle s$ $\displaystyle=$
$\displaystyle-\frac{\partial\Omega}{\partial T}\;,$ (55)
$\displaystyle\epsilon$ $\displaystyle=$ $\displaystyle-P+Ts\;,$ (56)
$\displaystyle\Delta$ $\displaystyle=$ $\displaystyle\epsilon-3P\;.$ (57)

(a)

(b)

Figure 13: Reduced temperature scale variation of the specific heat at
constant volume normalized with $T^{3}$ for the (a) RPQM, PQMVT and PQM model
with the PolyLog-glue form of the Polyakov-loop potential, (b) RPQM model with
the Log, Log-glue and PolyLog-glue form of the Polyakov-loop potential.

Fig 10(a) presents the reduced temperature scale plot of the entropy density
normalized to its SB limit. Similar to the pressure, the quark one-loop vacuum
correction leads to the increased entropy density $s/s_{SB}$. When the entropy
density plot for the PolyLog-glue RPQM and PQMVT model are compared with that
of the PQM Model, one gets largest entropy density increase in the PQMVT model
while the corresponding increase in the RPQM model is moderate. Even though
the smoothing effect of the quark back-reaction is present in the PolyLog-glue
PQM model, the entropy density shows a small kink at the chiral transition
temperature ($T/T_{c}^{\chi}=1$). This kink gets smoothed out in the RPQM and
PQMVT plots. The quark back-reaction in the Log-glue and PolyLog-glue RPQM
model, causes a significant smoothing change in the rapidly increasing entropy
density plot of the Log RPQM model in the Fig 10(b). The plots of the
normalized energy density on the reduced temperature scale in the Fig 11(a)
and the Fig 11(b) show the same trend that we get for the normalized entropy
density respectively in the Fig 10(a) and Fig 10(b). The energy density plot
for the PolyLog-glue RPQM model, shows better agreement with the lattice QCD
data AliKhan:2001ek  in the temperature range $(0.7-1.1)\ T^{\chi}_{c}$ and
$(1.6-2.1)\ T^{\chi}_{c}$ .

(a)

(b)

Figure 14: Reduced temperature scale variation of the square of speed of sound
and the ratio of pressure with energy density for the (a) RPQM, PQMVT and PQM
model with the PolyLog-glue form of the Polyakov-loop potential, (b) RPQM
model with the Log, Log-glue and PolyLog-glue form of the Polyakov-loop
potential. The two flavor lattice QCD data of the $C^{2}_{s}$ have been taken
from the Ref. AliKhan:2001ek .

Fig 12(a), shows the plots of the normalized interaction measure
$\Delta/\epsilon_{\overline{\text{\tiny SB}}}$ in the RPQM, PQMVT and PQM
model having the PolyLog-glue form for the Polyakov-loop potential. The peak
of the RPQM model plot, stands a little right to that of the PQMVT and PQM
model. The RPQM model plot agrees well with the lattice QCD data points upto
$T\sim T^{\chi}_{c}$ and a point near $1.4\ T^{\chi}_{c}$ . Fig 12(b) presents
the normalized interaction measure $\Delta/\epsilon_{\overline{\text{\tiny
SB}}}$ plot for the RPQM model using Log, Log-glue and PolyLog-glue forms for
the Polyakov-loop potential. Several lattice data points at the peak of the
interaction measure lie on the line depicting the Log RPQM model result.

###  V.5 Specific heat and speed of sound

The specific heat capacity at constant volume is defined by

$\displaystyle C_{V}=\frac{\partial\epsilon}{\partial
T}\bigg{\rvert}_{V}=-T\frac{\partial^{2}\Omega}{\partial
T^{2}}\bigg{\rvert}_{V}$ (58)

Fig 13(a) shows the variation of the normalized specific heat capacity at
$\mu$ = 0 as a function of the reduced temperature when the Polylog-glue form
has been taken for the Polyakov-loop potential in the RPQM, PQMVT and PQM
model. The highest and sharpest PQM model specific heat peak at the
$T_{c}^{\chi}$, gets significantly reduced and smoothed out in the temperature
range $(0.7-1.2)\ T^{\chi}_{c}$ due to the quark one-loop vacuum fluctuations
in the PQMVT and RPQM model. The very sharp and high specific heat peak at the
$T_{c}^{\chi}$ for the Log RPQM model in the Fig 13(b), becomes quite smooth,
reduced and round due to the robust effect of the quark back-reaction when the
Log-glue or the PolyLog-glue form has been taken for the unquenching of the
Polyakov-loop potential in the RPQM model. We point out that the two peak
structure in the Log-glue RPQM model plot for the specific heat is the
consequence of the fact that the temperature variation of the chiral order
parameter has two peaks as one can see in the Fig 4(a). The specific heat
plots for all the cases approach the corresponding SB-limit at high
temperatures. Since the gluon degrees of freedom contribute differently
SchaPQM3F , the specific heat plot for the Log RPQM model in the Fig 13(b)
does not merge completely with the Log-glue and PolyLog-glue RPQM model plots
even when $T=3\ T_{c}^{\chi}$.

The speed of sound is fundamental property of the strongly interacting medium.
The square of the speed of sound at constant entropy density is defined as

$\displaystyle C^{2}_{s}=\frac{\partial
P}{\partial\epsilon}\bigg{\rvert}_{s}=\frac{\partial P}{\partial
T}\bigg{\rvert}_{V}\bigg{/}\frac{\partial\epsilon}{\partial
T}\bigg{\rvert}_{V}=\frac{s}{C_{V}}$ (59)

The speed of sound and the ratio $P/\epsilon$, have been plotted in the Fig.
14 on the reduced temperature scale. The $C_{s}^{2}$ is very close to the
$P/\epsilon$ for the $T<0.7\ T_{c}^{\chi}$ . While approaching its ideal gas
limit of $1/3$ for the $T>2.5\ T_{c}^{\chi}$, the $C_{s}^{2}$ comes close to
the $P/\epsilon$ again. Similar to the results of Ref. schafwag12 ; guptiw ,
the $C^{2}_{s}$ plot lies sufficiently above the $P/\epsilon$ plot for the
temperature interval $0.7\ T_{c}^{\chi}<T<2.5\ T_{c}^{\chi}$ except near the
$T_{c}^{\chi}$. The very sharp and a relatively less sharp dip, noticed at the
$T_{c}^{\chi}$ respectively in the $C_{s}^{2}$ and the $P/\epsilon$ plot for
the PolyLog-glue PQM model in the Fig. 14(a), becomes very smooth and
moderately smooth for the respective cases of the PQMVT and RPQM model. In the
PolyLog-glue RPQM model, the speed of sound $C_{s}^{2}$ plot, shows a better
agreement with the lattice QCD data around the $T_{c}^{\chi}$ and the minimum
value 0.078 for the ratio $P/\epsilon$ is closest to the corresponding minimum
value of 0.075 obtained from the lattice QCD in Ref. Cheng:08 . The Fig. 14(b)
shows that the sharp $C_{s}^{2}$ and $P/\epsilon$ temperature variations for
the Log RPQM model, becomes quite smooth due to the quark back-reaction in the
Log-glue and the PolyLog-glue RPQM model where the Ployakov-loop potential is
unquenched. The Log RPQM model plot for the speed of sound $C^{2}_{s}$, lies
closest to the lattice QCD data points at higher temperatures ($T>1.4\
T_{c}^{\chi}$). Similar to the findings of the Ref. schafwag12 ; guptiw , the
$C_{s}^{2}$ value is found to be less than 0.1 below the $0.5\ T^{\chi}_{c}$
(not shown in the figure) in the all of our model results. This stands in
contrast to the results of the Ref. bedanga  where a confinement model has
been applied and the value of $C^{2}_{s}\sim$ 0.2 in the vicinity of the $0.5\
T^{\chi}_{c}$ and $C^{2}_{s}\sim$ 0.15 near the $T_{c}^{\chi}$.

##  VI Summary and Conclusion

Computing the phase diagrams and the thermodynamic quantities for the PQM,
PQMVT and RPQM model, we have compared how, the quark one-loop vacuum
correction in combination of the Polyakov-loop potential, influences the
dynamics of both the chiral as well as the confinement-deconfinement
transition. Taking different parametric form for the Polyakov-loop potential
with/without the quark back-reaction and the $\mu$ dependence of the parameter
$T_{0}$, we have also made a detailed comparison of how the chiral and
Polyakov-loop condensates, their derivatives, phase diagrams and the
thermodynamic quantities, get affected in the RPQM model which has the exact
chiral effective potential. The results obtained for the different
thermodynamic quantities have also been compared with the available lattice
QCD data.

The sharp temperature variation of the chiral condensate and its temperature
derivative in the PQM model for the $\mu=$0 and $m_{\sigma}=500$ MeV, become
excessively smooth in the PQMVT model while the corresponding smoothness turns
out to be moderate when the effect of the quark one-loop vacuum correction is
computed using the exact on-shell renormalized RPQM model. The deconfinement
corssover transition occurs earlier than the chiral transition, i.e.
$T^{\Phi}_{c}<T^{\chi}_{c}$, except for the cases of the PolyLog-glue PQM and
the Log RPQM model where the $T^{\Phi}_{c}\sim T^{\chi}_{c}$. The presence of
the quark back-reaction (represented by the glue form of the Ployakov-loop
potential), in all the three chiral models, generates a significant smoothing
effect on the temperature variations of the chiral condensate, the Polyakov-
loop condensate and their derivatives. The combined effect of the quark back-
reaction and the one-loop vacuum correction for the PolyLog-glue PQMVT model,
generates the highest separation of the $(T^{\chi}_{c}-T^{\Phi}_{c})=$ 30.2
MeV between the deconfinement and the chiral crossover transition while the
corresponding separation generated for the PolyLog-glue RPQM model is
$(T^{\chi}_{c}-T^{\Phi}_{c})=$ 21.0 MeV.

The phase boundary for the RPQM model stands closer to the PQM model phase
boundary when compared to that of the PQMVT model. Since the Log PQMVT model
generates excessively smooth chiral transition similar to the earlier findings
guptiw ; schafwag12 ; chatmoh1 ; vkkr12 ; chatmoh2 ; RaiTiw , the critical end
point (CEP) respectively for the $m_{\sigma}=400$ and 500 MeV cases, gets
located at the ($T_{\text{\tiny CEP}}$ = 94.77 MeV, $\mu_{\text{\tiny CEP}}$ =
267.2 MeV) and the ( $T_{\text{\tiny CEP}}$ = 78.0 MeV, $\mu_{\text{\tiny
CEP}}$ = 295.9 MeV) positions in the right lower corner of the phase diagram.
The on-shell renormalized Log-RPQM model, gives rise to a relatively moderate
smoothing effect on the chiral transition, hence the CEP gets located higher
up in the phase diagram respectively at the ($T_{\text{\tiny CEP}}$ = 100.1
MeV, $\mu_{\text{\tiny CEP}}$ = 245.3 MeV) and the ($T_{\text{\tiny CEP}}$ =
94.1 MeV, $\mu_{\text{\tiny CEP}}$ = 270.6 MeV) for the $m_{\sigma}=400$ and
500 MeV.

When compared to the CEP in the Log RPQM model for the $m_{\sigma}=500$ MeV,
the CEP for the PolyLog-glue RPQM model with the $T_{0}\equiv T_{0}(\mu)$,
shifts down in temperature by 38.5 MeV due to the significant smoothing effect
of the quark back-reaction in the temperature direction and gets located at
the $T_{\text{\tiny CEP}}$ = 55.6 MeV while the chemical potential registers a
negligible shift as the $\mu_{\text{\tiny CEP}}$ = 270.9 MeV. Due to the above
effect, the curvature of the phase transition line increases significantly for
all the cases where the unquenching of the Polyakov-loop potential has been
considered. In our work, the observation of the Ref. BielichP  gets confirmed
that the unquenching of the Polyakov-loop potential, links the chiral and
deconfinement phase transitions at all temperatures and chemical potentials.
Comparing the results of the Polyakov-loop enhanced calculations with the two
flavor RQM model results RaiTiw  where the effect of Polyakov-loop is absent,
we note that the presence of Polyakov-loop potential either in the Log or in
the PolyLog-glue form in the RPQM model, leads to significant upward shift of
the CEP in the $\mu-T$ plane.

The occurrence of the so called quarkyonic phase, where the chiral symmetry is
restored but the quarks and anti-quarks are confined, depends on the form
taken for the Polyakov loop potential. It occurs in a large region of the
phase diagram for the Poly RPQM model while its region gets reduced for the
Log RPQM model. Its region gets significantly reduced by the quark back-
reaction in the unquenched Polyakov-loop potential. It altogether disappears
when the parameter $T_{0}$ becomes chemical potential dependent i.e.
$T_{0}\equiv T_{0}(\mu)$ in the Log or the PolyLog-glue form of the Polyakov-
loop potential in the RPQM model while it occurs with a significantly reduced
extent for the corresponding case of the Polynomial form of the Polyakov-loop
potential in the RPQM model.

The thermodynamic quantities namely the pressure, entropy density, energy
density, interaction measure, specific heat, the speed of sound and the
$P/\epsilon$ ratio have been computed for the $\mu=$0 and the $m_{\sigma}=$500
MeV. As expected their variations in the PQMVT model, are smoother than that
of the RPQM Model. The sharp reduced temperature scale variation in the Log
RPQM model, becomes quite smooth due to the quark back-reaction in the Log-
glue and the PolyLog-glue RPQM model. Comparing the results with the available
lattice QCD data, we find that near the chiral transition $(0.7-1.2)\
T^{\chi}_{c}$, the Polylog-glue RPQM model results for all the thermodynamic
quantities, are in best agreement with the lattice QCD results. The Log RPQM
model results are close to the lattice QCD results at higher temperatures. In
the PolyLog-glue RPQM model, the minimum value 0.078 for the ratio
$P/\epsilon$ is closest to the corresponding minimum value of 0.075 obtained
from the lattice QCD in Ref. Cheng:08 .

## Acknowledgments

The authors would like to thank Shinji Ejiri for providing the lattice data of
CP-PACS collaboration AliKhan:2001ek .

##  Appendix A THE QMVT PARAMETER FIXING

The tree level expression of the curvature masses of mesons for the QM model
are given by the mass matrix evaluated in Ref. Roder . In this work, the above
mass matrix is renamed as $(m^{m}_{\alpha,ab})^{2}$ where superscript $m$
stands for the contribution of the pure mesonic potential.In the QMVT
model,the meson curvature masses get modified by the quark one-loop vacuum
contribution.The total expression of the meson curvature masses in the QMVT
model is written as

$\displaystyle m^{2}_{\alpha,ab}$ $\displaystyle=$
$\displaystyle(m^{m}_{\alpha,ab})^{2}+(\delta m^{v}_{\alpha,ab})^{2}$ (60)

where $\alpha=s,p$; “$s$” stands for the scalar and “$p$” stands for the
pseudoscalar mesons and $a,b=0,1,2,3$. $m^{2}_{s,00}\equiv m^{2}_{\sigma}$;
$m^{2}_{s,11}=m^{2}_{s,22}=m^{2}_{s,33}\equiv m^{2}_{a_{0}}$ and
$m^{2}_{p,00}\equiv m^{2}_{\eta}$;
$m^{2}_{p,11}=m^{2}_{p,22}=m^{2}_{p,33}\equiv m^{2}_{\pi}$.The
$(m^{m}_{\alpha,ab})^{2}$ and $(\delta m^{v}_{\alpha,ab})^{2}$ are defined in
the similar fashion.Superscript “$v$” stands for the quark/antiquark vacuum
contribution to the curvature masses. It is written as :

$\displaystyle(\delta m^{v}_{\alpha,ab})^{2}$ $\displaystyle=$
$\displaystyle\left.\frac{\partial^{2}\Omega^{vac}_{q\bar{q}}}{\partial\xi_{\alpha,a}\partial\xi_{\alpha,b}}\right|_{min}\;,$
(61) $\displaystyle=$
$\displaystyle\sum_{q=u,d}\frac{2N_{c}}{(4\pi)^{2}}\left[\left\\{m^{2}_{q,\alpha
a}m^{2}_{q,\alpha b}+m^{2}_{q}m^{2}_{q,\alpha ab}\right\\}\right.$
$\displaystyle\left.\left\\{1+\ln\left(\frac{\Lambda^{2}}{m^{2}_{q}}\right)\right\\}-m^{2}_{q,\alpha
a}m^{2}_{q,\alpha b}\right]\;.$

where $m^{2}_{q,\alpha a}=\frac{\partial m^{2}_{q}}{\partial\xi_{\alpha,a}}$
and $m^{2}_{q,\alpha ab}=\frac{\partial m^{2}_{q,\alpha
a}}{\partial\xi_{\alpha,b}}$. When one computes the second derivative of the
Eq.(45) for the quark contribution, the full dependence of all the scalar and
pseudo-scalar meson fields, cf. Eq. (9), in the quark masses has to be
considered. The resulting quark mass matrix is diagonalized similar to the
three flavor case given in Ref. Schaefer:09 . In all the quark mass
derivatives with respect to the meson fields, the meson fields are replaced by
the non vanishing vacuum expectation value $\overline{\sigma}$ and the final
values are collected in the Table 5.

Table 4: Expressions of the curvature masses $(m^{m}_{\alpha,ab})^{2}$ are calculated from the second derivative of the pure mesonic potential as has been evaluated in Ref.Roder . $(m^{m}_{\alpha,ab})^{2}$ |  |  Meson mass found from the pure mesonic potential
---|---|---
$(m^{m}_{s,00})^{2}$ |  $(m^{m}_{\sigma})^{2}$ |  $m^{2}-c+3\left(\lambda_{1}+\frac{\lambda_{2}}{2}\right)\overline{\sigma}^{2}$ ,
$(m^{m}_{s,11})^{2}$ |  $(m^{m}_{a_{0}})^{2}$ |  $m^{2}+c+\left(\lambda_{1}+\frac{3\lambda_{2}}{2}\right)\overline{\sigma}^{2}$ ,
$(m^{m}_{p,00})^{2}$ |  $(m^{m}_{\eta})^{2}$ |  $m^{2}+c+\left(\lambda_{1}+\frac{\lambda_{2}}{2}\right)\overline{\sigma}^{2}$ ,
$(m^{m}_{p,11})^{2}$ |  $(m^{m}_{\pi})^{2}$ |  $m^{2}-c+\left(\lambda_{1}+\frac{\lambda_{2}}{2}\right)\overline{\sigma}^{2}$ ,
Table 5: Squared quark mass derivatives with respect to the meson fields evaluated at the minimum.The last two columns present the first and second derivative of the squared quark mass summed over two quark flavor. $s/p$ |  $a$ |  $b$ |  $m^{2}_{q,\alpha a}m^{2}_{q,\alpha b}/g^{4}$ |  $m^{2}_{q,\alpha ab}/g^{2}$
---|---|---|---|---
$s$ |  $0$ |  $0$ |  $\frac{1}{2}{\overline{\sigma}}^{2}$ |  1
$s$ |  $1$ |  $1$ |  $\frac{1}{2}{\overline{\sigma}}^{2}$ |  1
$p$ |  $0$ |  $0$ |  0 |  1
$p$ |  $1$ |  $1$ |  0 |  1

Using the Table-5 in the Eq.(61) we get vacuum contributions of meson masses
as,

$\displaystyle(\delta m^{v}_{\sigma})^{2}$ $\displaystyle\equiv$
$\displaystyle(\delta
m^{v}_{s,00})^{2}=\dfrac{N_{c}g^{4}\overline{\sigma}^{2}}{2(4\pi)^{2}}\left[1+3\ln\left(\dfrac{\Lambda^{2}}{m^{2}_{q}}\right)\right]\;,\
$ (62) $\displaystyle(\delta m^{v}_{a_{0}})^{2}$ $\displaystyle\equiv$
$\displaystyle(\delta
m^{v}_{s,11})^{2}=\dfrac{N_{c}g^{4}\overline{\sigma}^{2}}{2(4\pi)^{2}}\left[1+3\ln\left(\dfrac{\Lambda^{2}}{m^{2}_{q}}\right)\right]\;,\
$ (63) $\displaystyle(\delta m^{v}_{\eta})^{2}$ $\displaystyle\equiv$
$\displaystyle(\delta
m^{v}_{p,00})^{2}=\dfrac{N_{c}g^{4}\overline{\sigma}^{2}}{2(4\pi)^{2}}\left[1+\ln\left(\dfrac{\Lambda^{2}}{m^{2}_{q}}\right)\right]\;,\
$ (64) $\displaystyle(\delta m^{v}_{\pi})^{2}$ $\displaystyle\equiv$
$\displaystyle(\delta
m^{v}_{p,11})^{2}=\dfrac{N_{c}g^{4}\overline{\sigma}^{2}}{2(4\pi)^{2}}\left[1+\ln\left(\dfrac{\Lambda^{2}}{m^{2}_{q}}\right)\right]\;.\
$ (65)

We get $(m^{m}_{\sigma})^{2}$,$(m^{m}_{\eta})^{2}$,$(m^{m}_{a_{0}})^{2}$ and
$(m^{m}_{\eta})^{2}$ after substitution of the Eqs.(62)–(65) into the Eq.(60)
as,

$\displaystyle(m^{m}_{\sigma})^{2}$ $\displaystyle=$ $\displaystyle
m^{2}_{\sigma}-\dfrac{N_{c}g^{4}\overline{\sigma}^{2}}{2(4\pi)^{2}}\left[1+3\ln\left(\dfrac{\Lambda^{2}}{m^{2}_{q}}\right)\right]\;,$
(66) $\displaystyle(m^{m}_{a_{0}})^{2}$ $\displaystyle=$ $\displaystyle
m^{2}_{a_{0}}-\dfrac{N_{c}g^{4}\overline{\sigma}^{2}}{2(4\pi)^{2}}\left[1+3\ln\left(\dfrac{\Lambda^{2}}{m^{2}_{q}}\right)\right]\;,$
(67) $\displaystyle(m^{m}_{\eta})^{2}$ $\displaystyle=$ $\displaystyle
m^{2}_{\eta}-\dfrac{N_{c}g^{4}\overline{\sigma}^{2}}{2(4\pi)^{2}}\left[1+\ln\left(\dfrac{\Lambda^{2}}{m^{2}_{q}}\right)\right]\;,$
(68) $\displaystyle(m^{m}_{\pi})^{2}$ $\displaystyle=$ $\displaystyle
m^{2}_{\pi}-\dfrac{N_{c}g^{4}\overline{\sigma}^{2}}{2(4\pi)^{2}}\left[1+\ln\left(\dfrac{\Lambda^{2}}{m^{2}_{q}}\right)\right]\;.$
(69)

The parameters in vacuum are obtained as,

$\displaystyle\lambda_{1}$ $\displaystyle=$
$\displaystyle(m^{m}_{\sigma})^{2}+(m^{m}_{\eta})^{2}-(m^{m}_{a_{0}})^{2}-(m^{m}_{\pi})^{2}\over
2f^{2}_{\pi}\;,$ (70) $\displaystyle\lambda_{2}$ $\displaystyle=$
$\displaystyle(m^{m}_{a_{0}})^{2}-(m^{m}_{\eta})^{2}\over f^{2}_{\pi}\;,$ (71)
$\displaystyle m^{2}$ $\displaystyle=$
$\displaystyle(m^{m}_{\pi})^{2}+\frac{(m^{m}_{\eta})^{2}-(m^{m}_{\sigma})^{2}}{2}\;,$
(72) $\displaystyle c$ $\displaystyle=$
$\displaystyle\frac{(m^{m}_{\eta})^{2}-(m^{m}_{\pi})^{2}}{2}\;.$ (73)

we get the parameters of the QMVT on substitution of the Eqs.(66)–(69) into
the Eqs.(70)–(73) and found that $\lambda_{1}$,$c$ of the QMVT are same with
respect to $\lambda_{1}$,$c$ of the QM. We observe change in $\lambda_{2}$ and
$m^{2}$ as,

$\displaystyle\lambda_{2}$ $\displaystyle=$
$\displaystyle\lambda_{2s}-\dfrac{N_{c}g^{4}}{(4\pi)^{2}}\ln\left(\dfrac{4\Lambda^{2}}{g^{2}f^{2}_{\pi}}\right)\;,$
(74) $\displaystyle m^{2}$ $\displaystyle=$ $\displaystyle
m^{2}_{s}-\dfrac{N_{c}g^{4}f^{2}_{\pi}}{2(4\pi)^{2}}\;.$ (75)

where $\lambda_{2s}$ and $m^{2}_{s}$ are same old $\lambda_{2}$ and $m^{2}$
parameters of the QM model.Putting the value of the new parameters
$\lambda_{2}$ and $m^{2}$ in Eq. (69) one can write the expression of pion
mass independent of renormalization scale as,

$\displaystyle m^{2}_{\pi}$ $\displaystyle=$ $\displaystyle
m^{2}_{s}-\frac{N_{c}g^{4}}{2(4\pi)^{2}}(f^{2}_{\pi}-\overline{\sigma}^{2})-c+\left(\lambda_{1}+\frac{\lambda_{2s}}{2}\right)\overline{\sigma}^{2}\;$
(76)
$\displaystyle+\frac{N_{c}g^{4}}{2(4\pi)^{2}}\log\left(\frac{f^{2}_{\pi}}{\overline{\sigma}^{2}}\right)\overline{\sigma}^{2}$

##  Appendix B Fixing of the parameters in the RQM model

In the RQM model, the divergence of the first term of the Eq. (38) (as
rewritten in the Eq. (45)) , is removed by the renormalization of its
parameters. This Appendix presents the relation between the physical
quantities and the parameters of the Lagrangian (4) using the
$\overline{\text{MS}}$ and on-shell renormalization schemes RaiTiw . One
introduces the counterterms $\delta m^{2}$, $\delta g^{2}$,
$\delta\lambda_{1}$, $\delta\lambda_{2}$, $\delta c$ and $\delta h$ for the
parameters and the wave function/field counterterms $\delta
Z_{\sigma}$,$\delta Z_{a_{0}}$,$\delta Z_{\eta}$,$\delta Z_{\pi}$ and $\delta
Z_{\psi}$ in the Lagrangian (4) to define the renormalized fields and
couplings.

$\displaystyle\sigma_{b}=\sqrt{Z_{\sigma}}\sigma,\
\eta_{b}=\sqrt{Z_{\eta}}\eta,\ a^{i}_{0b}=\sqrt{Z_{a_{0}}}a_{0}$ (77)
$\displaystyle\pi^{i}_{b}=\sqrt{Z_{\pi}}\pi,\ \psi_{b}=\sqrt{Z_{\psi}}\psi,\
m^{2}_{b}=Z_{m}m^{2}$ (78)
$\displaystyle\lambda_{1b}=Z_{\lambda_{1}}\lambda_{1},\
\lambda_{2b}=Z_{\lambda_{2}}\lambda_{2},\ g_{b}=\sqrt{Z_{g}}g$ (79)
$\displaystyle h_{b}=Z_{h}h,\ \ \ c_{b}=Z_{c}c$ (80)

Where $Z_{(\sigma,a_{0},\eta,\pi,\psi)}=1+\delta
Z_{(\sigma,a_{0},\eta,\pi,\psi)}$, denote the field strength renormalization
constant while $Z_{(m,\lambda_{1},\lambda_{2},g,h,c)}=1+\delta
Z_{(m,\lambda_{1},\lambda_{2},g,h,c)}$ denote the mass and coupling
renormalization constant.

The scalar ($\sigma,a_{0}$) and pseudo-scalar ($\pi,\eta$) meson inverse
propagators with self-energy correction are written as

$\displaystyle
p^{2}-m_{\sigma,a_{0},\pi,\eta}^{2}-i\Sigma_{\sigma,a_{0},\pi,\eta}(p^{2}){\rm+counterterms}\;.$
(81)

Implementing the on-shell scheme, one puts the physical mass equal to the
renormalized mass in the Lagrangian i.e. $m=m_{\rm pole}$111The imaginary
parts of the self-energies are ignored for defining the mass. and writes

$\displaystyle\Sigma(p^{2}=m_{\sigma,a_{0},\pi,\eta}^{2}){\rm+counterterms}$
$\displaystyle=$ $\displaystyle 0\;.$ (82)

The on-shell scheme demands that the propagator residue becomes unity and one
gets

$\displaystyle{\partial\over\partial
p^{2}}\Sigma_{\sigma,a_{0},\pi,\eta}(p^{2})\Big{|}_{p^{2}=m_{\sigma,a_{0},\pi,\eta}^{2}}{\rm+counterterms}$
$\displaystyle=$ $\displaystyle 0\;.$ (83)

The quark one-loop correction to the one-point function and the tadpole
counterterm can be written as

$\displaystyle\delta\Gamma^{(1)}$ $\displaystyle=$
$\displaystyle-4N_{c}gm_{q}\mathcal{A}(m_{q}^{2})+i\delta t\;,$ (84)

when the one-point function $\Gamma^{(1)}=it=i(h-m_{\pi}^{2}\
\overline{\sigma})$ vanishes, one gets the equation of motion $t=0$ at tree
level. This should hold also at one-loop level, which results into the
renormalization condition $\delta\Gamma^{(1)}=0$.

The counterterms of the two-point functions are

$\displaystyle\Sigma_{\sigma}^{\rm ct1}(p^{2})$ $\displaystyle=$
$\displaystyle i\left[\delta Z_{\sigma}(p^{2}-m_{\sigma}^{2})-\delta
m_{\sigma}^{2}\right]\;,$ (85) $\displaystyle\Sigma_{a_{0}}^{\rm ct1}(p^{2})$
$\displaystyle=$ $\displaystyle i\left[\delta
Z_{a_{0}}(p^{2}-m_{a_{0}}^{2})-\delta m_{a_{0}}^{2}\right]\;,$ (86)
$\displaystyle\Sigma_{\pi}^{\rm ct1}(p^{2})$ $\displaystyle=$ $\displaystyle
i\left[\delta Z_{\pi}(p^{2}-m_{\pi}^{2})-\delta m_{\pi}^{2}\right]\;,$ (87)
$\displaystyle\Sigma_{\eta}^{\rm ct1}(p^{2})$ $\displaystyle=$ $\displaystyle
i\left[\delta Z_{\eta}(p^{2}-m_{\eta}^{2})-\delta m_{\eta}^{2}\right]\;,$ (88)
$\displaystyle\Sigma^{\rm ct2}_{\sigma}$ $\displaystyle=$ $\displaystyle
3\Sigma^{\rm ct2}_{a_{0}}=3\Sigma^{\rm ct2}_{\pi}=3\Sigma^{\rm ct2}_{\eta}\;,$
(89) $\displaystyle=$ $\displaystyle-{24(\lambda_{1}+\mbox{$\lambda_{2}\over
2$})g\overline{\sigma}N_{c}m_{q}\over
m_{\sigma}^{2}}\mathcal{A}(m_{q}^{2})\;,$ $\displaystyle=$
$\displaystyle-6i(\lambda_{1}+\mbox{$\lambda_{2}\over
2$})\overline{\sigma}\delta t\over m_{\sigma}^{2}\;,$ $\displaystyle\delta t$
$\displaystyle=$ $\displaystyle-4iN_{c}gm_{q}\mathcal{A}(m_{q}^{2})\;.$ (90)

The respective tadpole contributions to the $\sigma,a_{0}$ and $\pi,\eta$
self-energies are cancelled by the counterterms in Eq. (89). The on-shell
evaluation of the self-energies and their derivatives give all the
renormalization constants. Combining the Eqs. (82), (83) and (85)–(88), one
obtains the following set of equations.

$\displaystyle\delta m_{\sigma}^{2}$ $\displaystyle=$
$\displaystyle-i\Sigma_{\sigma}(m_{\sigma}^{2})\;;\delta
Z_{\sigma}=i{\partial\over\partial
p^{2}}\Sigma_{\sigma}(p^{2})\Big{|}_{p^{2}=m_{\sigma}^{2}}\;,$ (91)
$\displaystyle\delta m_{a_{0}}^{2}$ $\displaystyle=$
$\displaystyle-i\Sigma_{a_{0}}(m_{a_{0}}^{2})\;;\delta
Z_{a_{0}}=i{\partial\over\partial
p^{2}}\Sigma_{a_{0}}(p^{2})\Big{|}_{p^{2}=m_{a_{0}}^{2}}\;,$ (92)
$\displaystyle\delta m_{\pi}^{2}$ $\displaystyle=$
$\displaystyle-i\Sigma_{\pi}(m_{\pi}^{2})\;;\delta
Z_{\pi}=i{\partial\over\partial
p^{2}}\Sigma_{\pi}(p^{2})\Big{|}_{p^{2}=m_{\pi}^{2}}\;,$ (93)
$\displaystyle\delta m_{\eta}^{2}$ $\displaystyle=$
$\displaystyle-i\Sigma_{\eta}(m_{\eta}^{2})\;;\delta
Z_{\eta}=i{\partial\over\partial
p^{2}}\Sigma_{\eta}(p^{2})\Big{|}_{p^{2}=m_{\eta}^{2}}\;.$ (94)

Using the above equations and the expressions of meson self-energies given in
Ref. RaiTiw , we can write

$\displaystyle\delta m_{\sigma}^{2}$ $\displaystyle=$ $\displaystyle
2ig^{2}N_{c}\left[\mathcal{A}(m_{q}^{2})-\mbox{$1\over
2$}(m_{\sigma}^{2}-4m_{q}^{2})\mathcal{B}(m_{\sigma}^{2})\right]\;,$ (95)
$\displaystyle\delta m_{a_{0}}^{2}$ $\displaystyle=$ $\displaystyle
2ig^{2}N_{c}\left[\mathcal{A}(m_{q}^{2})-\mbox{$1\over
2$}(m_{a_{0}}^{2}-4m_{q}^{2})\mathcal{B}(m_{a_{0}}^{2})\right]\;,$ (96)
$\displaystyle\delta m_{\pi}^{2}$ $\displaystyle=$ $\displaystyle
2ig^{2}N_{c}\left[\mathcal{A}(m_{q}^{2})-\mbox{$1\over
2$}m_{\pi}^{2}\mathcal{B}(m_{\pi}^{2})\right]\;,$ (97) $\displaystyle\delta
m_{\eta}^{2}$ $\displaystyle=$ $\displaystyle
2ig^{2}N_{c}\left[\mathcal{A}(m_{q}^{2})-\mbox{$1\over
2$}m_{\eta}^{2}\mathcal{B}(m_{\eta}^{2})\right]\;,$ (98) $\displaystyle\delta
Z_{\sigma}$ $\displaystyle=$ $\displaystyle
ig^{2}N_{c}\left[\mathcal{B}(m_{\sigma}^{2})+(m_{\sigma}^{2}-4m_{q}^{2})\mathcal{B}^{\prime}(m_{\sigma}^{2})\right]\;,$
(99) $\displaystyle\delta Z_{a_{0}}$ $\displaystyle=$ $\displaystyle
ig^{2}N_{c}\left[\mathcal{B}(m_{a_{0}}^{2})+(m_{a_{0}}^{2}-4m_{q}^{2})\mathcal{B}^{\prime}(m_{a_{0}}^{2})\right]\;,$
(100) $\displaystyle\delta Z_{\pi}$ $\displaystyle=$ $\displaystyle
ig^{2}N_{c}\left[\mathcal{B}(m_{\pi}^{2})+m_{\pi}^{2}\mathcal{B}^{\prime}(m_{\pi}^{2})\right]\;,$
(101) $\displaystyle\delta Z_{\eta}$ $\displaystyle=$ $\displaystyle
ig^{2}N_{c}\left[\mathcal{B}(m_{\eta}^{2})+m_{\eta}^{2}\mathcal{B}^{\prime}(m_{\eta}^{2})\right]\;,$
(102)

where $\mathcal{A}(m^{2}_{q})$ and $\mathcal{B}(p^{2})$ are defined in the
Appendix C.

The counterterms $\delta m^{2}$,$\delta\lambda_{1}$, $\delta\lambda_{2}$,
$\delta c$ and $\delta g^{2}$, $\delta h$ can be expressed in terms of the
counterterms $\delta m^{2}_{\sigma}$, $\delta m^{2}_{a_{0}}$, $\delta
m^{2}_{\eta}$, $\delta m^{2}_{\pi}$ and $\delta m_{q}$,
$\delta\overline{\sigma}^{2}$. Using Eqs. (13)–(17) together with Eqs.
(77)–(80), we can write

$\displaystyle\delta\lambda_{1}$ $\displaystyle=$ $\displaystyle\frac{\delta
m^{2}_{\sigma}+\delta m^{2}_{\eta}-\delta m^{2}_{a_{0}}-\delta m^{2}_{\pi}}{2\
\overline{\sigma}^{2}}-\lambda_{1}\frac{\delta\overline{\sigma}^{2}}{\overline{\sigma}^{2}}\;,$
(103) $\displaystyle\delta\lambda_{2}$ $\displaystyle=$
$\displaystyle\frac{\delta m^{2}_{a_{0}}-\delta
m^{2}_{\eta}}{\overline{\sigma}^{2}}-\lambda_{2}\frac{\delta\overline{\sigma}^{2}}{\overline{\sigma}^{2}}\;,$
(104) $\displaystyle\delta c$ $\displaystyle=$ $\displaystyle\frac{\delta
m^{2}_{\eta}-\delta m^{2}_{\pi}}{2}\;,$ (105) $\displaystyle\delta m^{2}$
$\displaystyle=$ $\displaystyle\delta m^{2}_{\pi}+\frac{\delta
m^{2}_{\eta}-\delta m^{2}_{\sigma}}{2}\;,$ (106) $\displaystyle\frac{\delta
g^{2}}{4}$ $\displaystyle=$ $\displaystyle\frac{\delta
m^{2}_{q}}{\overline{\sigma}^{2}}-g^{2}\frac{\delta\overline{\sigma}^{2}}{4\
\overline{\sigma}^{2}}$ (107)

The one loop correction at the pion-quark vertex is of order $N_{c}^{0}$.
Hence $Z_{\psi}=1$ and the quark self energy correction $\delta m_{q}=0$ at
this order. In consequence, we get $Z_{\psi}\sqrt{Z_{g^{2}}\
g^{2}}\sqrt{Z_{\pi}}\approx g(1+\frac{1}{2}\frac{\delta g^{2}}{g^{2}}\
+\frac{1}{2}\delta Z_{\pi})=g$, thus $\frac{\delta g^{2}}{g^{2}}\ +\delta
Z_{\pi}=0$. Furthermore the $\delta m_{q}=0$ implies that $\delta g\
\overline{\sigma}/2+g\ \delta\overline{\sigma}/2=0$. Eq. (107) gives

$\displaystyle\frac{\delta\overline{\sigma}^{2}}{\overline{\sigma}^{2}}$
$\displaystyle=$ $\displaystyle-\frac{\delta g^{2}}{g^{2}}=\delta Z_{\pi},$
(108)

Now we can rewrite Eq. (103), (104) as

$\displaystyle\delta\lambda_{1}$ $\displaystyle=$ $\displaystyle\frac{\delta
m^{2}_{\sigma}+\delta m^{2}_{\eta}-\delta m^{2}_{a_{0}}-\delta m^{2}_{\pi}}{2\
\overline{\sigma}^{2}}-\lambda_{1}\delta Z_{\pi},$ (109)
$\displaystyle\delta\lambda_{2}$ $\displaystyle=$ $\displaystyle\frac{\delta
m^{2}_{a_{0}}-\delta m^{2}_{\eta}}{\overline{\sigma}^{2}}-\lambda_{2}\delta
Z_{\pi}.$ (110)

The equation $h=t+m_{\pi}^{2}\ \overline{\sigma}$ enables the writing of the
counterterm $\delta h$ in terms of the tadpole counterterm $\delta t$

$\displaystyle\delta h$ $\displaystyle=$ $\displaystyle m^{2}_{\pi}\
\delta\overline{\sigma}+\delta m^{2}_{\pi}\ \overline{\sigma}+\delta t.$ (111)

Using Eq. (108) we can write

$\displaystyle\delta t$ $\displaystyle=$
$\displaystyle-\frac{1}{2}m^{2}_{\pi}\ \overline{\sigma}\delta Z_{\pi}-\delta
m^{2}_{\pi}\ \overline{\sigma}+\delta h..$ (112)

Exploiting the Eqs. (95)–(102) together with the Eqs. (103) –(108) and (112),
we find the following expessions for the counterterms in the on-shell scheme.

$\displaystyle\delta\lambda_{1\text{\tiny OS}}$ $\displaystyle=$
$\displaystyle\frac{iN_{c}g^{2}}{\overline{\sigma}}\left[-\frac{1}{2}(m^{2}_{\sigma}-4m^{2}_{q})\mathcal{B}(m^{2}_{\sigma})-\frac{1}{2}m^{2}_{\eta}\mathcal{B}(m^{2}_{\eta})+\frac{1}{2}(m^{2}_{a_{0}}-4m^{2}_{q})\mathcal{B}(m^{2}_{a_{0}})+\frac{1}{2}m^{2}_{\pi}\mathcal{B}(m^{2}_{\pi})\right]-\lambda_{1}ig^{2}N_{c}\left[\mathcal{B}(m_{\pi}^{2})\right.$
(113)
$\displaystyle\left.+m_{\pi}^{2}\mathcal{B}^{\prime}(m_{\pi}^{2})\right]=\delta\lambda_{1\text{div}}+\dfrac{N_{c}g^{2}}{(4\pi)^{2}}\left[2\lambda_{1}\log\left(\frac{\Lambda^{2}}{m_{q}^{2}}\right)+\lambda_{1}(\mathcal{C}(m^{2}_{\pi})+m^{2}_{\pi}\mathcal{C}^{\prime}(m^{2}_{\pi}))\right.$
$\displaystyle\left.+\frac{(m^{2}_{\sigma}-4m^{2}_{q})\mathcal{C}(m^{2}_{\sigma})+m^{2}_{\eta}\mathcal{C}(m^{2}_{\eta})-(m^{2}_{a_{0}}-4m^{2}_{q})\mathcal{C}(m^{2}_{a_{0}})-m^{2}_{\pi}\mathcal{C}(m^{2}_{\pi})}{2\overline{\sigma}^{2}}\right]\;,$
$\displaystyle\delta\lambda_{2\text{\tiny OS}}$ $\displaystyle=$
$\displaystyle\frac{iN_{c}g^{2}}{\overline{\sigma}^{2}}\left[-(m^{2}_{a_{0}}-4m^{2}_{q})\mathcal{B}(m^{2}_{a_{0}})+m^{2}_{\eta}\mathcal{B}(m^{2}_{\eta})\right]-\lambda_{2}ig^{2}N_{c}\left[\mathcal{B}(m_{\pi}^{2})+m_{\pi}^{2}\mathcal{B}^{\prime}(m_{\pi}^{2})\right]\;,$
(114) $\displaystyle=$
$\displaystyle\delta\lambda_{2\text{div}}+\dfrac{N_{c}g^{2}}{(4\pi)^{2}}\left[\left(2\lambda_{2}-g^{2}\right)\log\left(\frac{\Lambda^{2}}{m_{q}^{2}}\right)+\frac{(m^{2}_{a_{0}}-4m^{2}_{q})\mathcal{C}(m^{2}_{a_{0}})-m^{2}_{\eta}\mathcal{C}(m^{2}_{\eta})}{{\overline{\sigma}^{2}}}+\lambda_{2}(m^{2}_{\pi}\mathcal{C}^{\prime}(m^{2}_{\pi})+\mathcal{C}(m^{2}_{\pi}))\right]\;,$
$\displaystyle\delta m^{2}_{\text{\tiny OS}}$ $\displaystyle=$ $\displaystyle
2iN_{c}g^{2}\left[\mathcal{A}(m^{2}_{q})-\frac{1}{2}m^{2}_{\pi}\mathcal{B}(m^{2}_{\pi})\right]+iN_{c}g^{2}\left[-\frac{1}{2}m^{2}_{\eta}\mathcal{B}(m^{2}_{\eta})+\frac{1}{2}(m^{2}_{\sigma}-4m^{2}_{q})\mathcal{B}(m^{2}_{\sigma})\right]\;$
(115) $\displaystyle=\delta
m^{2}_{\text{div}}+\dfrac{N_{c}g^{2}}{(4\pi)^{2}}\left[m^{2}\log\left(\frac{\Lambda^{2}}{m_{q}^{2}}\right)+m^{2}_{\pi}\mathcal{C}(m^{2}_{\pi})+\dfrac{m^{2}_{\eta}\mathcal{C}(m^{2}_{\eta})-(m^{2}_{\sigma}-4m^{2}_{q})\mathcal{C}(m^{2}_{\sigma})}{2}-2m^{2}_{q}\right]\;,$
$\displaystyle\delta c_{\text{\tiny OS}}$ $\displaystyle=$
$\displaystyle\frac{iN_{c}g^{2}}{2}\left[-m^{2}_{\eta}\mathcal{B}(m^{2}_{\eta})+m^{2}_{\pi}\mathcal{B}(m^{2}_{\pi})\right]=\delta
c_{\text{div}}+\dfrac{N_{c}g^{2}}{(4\pi)^{2}}\left[c\log\left(\frac{\Lambda^{2}}{m_{q}^{2}}\right)+m^{2}_{\eta}\mathcal{C}(m^{2}_{\eta})-m^{2}_{\pi}\mathcal{C}(m^{2}_{\pi})\right]\;,$
(116) $\displaystyle\delta g^{2}_{\text{\tiny OS}}$ $\displaystyle=$
$\displaystyle-
iN_{c}g^{4}\left[m^{2}_{\pi}\mathcal{B}^{\prime}(m^{2}_{\pi})+\mathcal{B}(m^{2}_{\pi})\right]=\delta
g^{2}_{\text{div}}+\dfrac{N_{c}g^{4}}{(4\pi)^{2}}\left[\log\left(\frac{\Lambda^{2}}{m_{q}^{2}}\right)+\mathcal{C}(m^{2}_{\pi})+m^{2}_{\pi}\mathcal{C}^{\prime}(m^{2}_{\pi})\right]\;,$
(117) $\displaystyle\delta\overline{\sigma}^{2}_{\text{\tiny OS}}$
$\displaystyle=$ $\displaystyle
iN_{c}g^{2}\overline{\sigma}^{2}\left[m^{2}_{\pi}\mathcal{B}^{\prime}(m^{2}_{\pi})+\mathcal{B}(m^{2}_{\pi})\right]=\delta\overline{\sigma}^{2}_{\text{div}}-\dfrac{N_{c}g^{4}}{(4\pi)^{2}}\left[\log\left(\frac{\Lambda^{2}}{m_{q}^{2}}\right)+\mathcal{C}(m^{2}_{\pi})+m^{2}_{\pi}\mathcal{C}^{\prime}(m^{2}_{\pi})\right]\;$
(118) $\displaystyle\delta h_{\text{\tiny OS}}$ $\displaystyle=$
$\displaystyle\dfrac{iN_{c}g^{2}}{2(4\pi)^{2}}h\left[m^{2}_{\pi}\mathcal{B}^{\prime}(m^{2}_{\pi})-\mathcal{B}(m^{2}_{\pi})\right]=\delta
h_{\text{div}}+\dfrac{N_{c}g^{2}}{2(4\pi)^{2}}h\left[\log\left(\frac{\Lambda^{2}}{m_{q}^{2}}\right)+\mathcal{C}(m^{2}_{\pi})-m^{2}_{\pi}\mathcal{C}^{\prime}(m^{2}_{\pi})\right]\;$
(119) $\displaystyle\delta Z^{\text{\tiny OS}}_{\pi}$ $\displaystyle=$
$\displaystyle\delta Z_{\pi,\rm
div}-\frac{N_{c}g^{2}}{(4\pi)^{2}}\left[\log\left(\frac{\Lambda^{2}}{m_{q}^{2}}\right)+\mathcal{C}(m_{\pi}^{2})+m_{\pi}^{2}\mathcal{C}^{\prime}(m_{\pi}^{2})\right]\;$
(120)

The $\mathcal{B}(m^{2}),\mathcal{B}^{\prime}(m^{2})$ and
$\mathcal{C}(m^{2}),\mathcal{C}^{\prime}(m^{2})$ are defined in the
appendix.The divergent part of the counterterms are
$\delta\lambda_{1\text{div}}=\frac{N_{c}g^{2}2\lambda_{1}}{(4\pi)^{2}\epsilon}\;$,
$\delta\lambda_{2\text{div}}=\frac{N_{c}g^{2}}{(4\pi)^{2}\epsilon}(2\lambda_{2}-g^{2})\;$,
$\delta m^{2}_{\text{div}}=\frac{N_{c}g^{2}m^{2}}{(4\pi)^{2}\epsilon}\;$,
$\delta c_{\text{div}}=\frac{N_{c}g^{2}c}{(4\pi)^{2}\epsilon}\;$, $\delta
g^{2}_{\text{div}}=\frac{N_{c}g^{4}}{(4\pi)^{2}\epsilon}\;$,
$\delta\overline{\sigma}^{2}_{\text{div}}=-\frac{N_{c}g^{2}\overline{\sigma}^{2}}{(4\pi)^{2}\epsilon}\;$,
$\delta h_{\text{div}}=\frac{N_{c}g^{2}h}{2(4\pi)^{2}\epsilon}\;$, $\delta
Z_{\pi,\rm div}=-\frac{N_{c}g^{2}}{(4\pi)^{2}\epsilon}$.

For both,the on-shell and the $\overline{\text{MS}}$ schemes, the divergent
part of the counterterms are the same, i.e.
$\delta\lambda_{1\text{div}}=\delta\lambda_{1\overline{\text{\tiny MS}}}$,
$\delta\lambda_{2\text{div}}=\delta\lambda_{2\overline{\text{\tiny MS}}}$ etc.
Since the bare parameters are independent of the renormalization scheme, we
can immediately write down the relations between the renormalized parameters
in the on-shell and MS schemes. We find

$\displaystyle\lambda_{1\overline{\text{\tiny MS}}}$ $\displaystyle=$
$\displaystyle\frac{Z^{\text{\tiny
OS}}_{\lambda_{1}}}{Z^{\overline{\text{\tiny
MS}}}_{\lambda_{1}}}\lambda_{1}\approx\lambda_{1}+\delta\lambda_{1\text{\tiny
OS}}-\delta\lambda_{1\overline{\text{\tiny MS}}}\;,$ (121)
$\displaystyle\lambda_{2\overline{\text{\tiny MS}}}$ $\displaystyle=$
$\displaystyle\frac{Z^{\text{\tiny
OS}}_{\lambda_{2}}}{Z^{\overline{\text{\tiny
MS}}}_{\lambda_{2}}}\lambda_{2}\approx\lambda_{2}+\delta\lambda_{2\text{\tiny
OS}}-\delta\lambda_{2\overline{\text{\tiny MS}}}\;,$ (122) $\displaystyle
m^{2}_{\overline{\text{\tiny MS}}}$ $\displaystyle=$
$\displaystyle\frac{Z^{\text{\tiny OS}}_{m^{2}}}{Z^{\overline{\text{\tiny
MS}}}_{m^{2}}}m^{2}\approx m^{2}+\delta m^{2}_{\text{\tiny OS}}-\delta
m^{2}_{\overline{\text{\tiny MS}}}\;,$ (123) $\displaystyle
c_{\overline{\text{\tiny MS}}}$ $\displaystyle=$
$\displaystyle\frac{Z^{\text{\tiny OS}}_{c}}{Z^{\overline{\text{\tiny
MS}}}_{c}}c\approx c+\delta c_{\text{\tiny OS}}-\delta
c_{\overline{\text{\tiny MS}}}\;,$ (124) $\displaystyle
h_{\overline{\text{\tiny MS}}}$ $\displaystyle=$
$\displaystyle\frac{Z^{\text{\tiny OS}}_{h}}{Z^{\overline{\text{\tiny
MS}}}_{h}}h\approx h+\delta h_{\text{\tiny OS}}-\delta
h_{\overline{\text{\tiny MS}}}\;,$ (125) $\displaystyle
g^{2}_{\overline{\text{\tiny MS}}}$ $\displaystyle=$
$\displaystyle\frac{Z^{\text{\tiny OS}}_{g^{2}}}{Z^{\overline{\text{\tiny
MS}}}_{g^{2}}}g^{2}\approx g^{2}+\delta g^{2}_{\text{\tiny OS}}-\delta
g^{2}_{\overline{\text{\tiny MS}}}\;,$ (126)
$\displaystyle{\overline{\sigma}}^{2}_{\overline{\text{\tiny MS}}}$
$\displaystyle=$ $\displaystyle\frac{Z^{\text{\tiny
OS}}_{{\overline{\sigma}}^{2}}}{Z^{\overline{\text{\tiny
MS}}}_{{\overline{\sigma}}^{2}}}{\overline{\sigma}}^{2}\approx{\overline{\sigma}}^{2}+\delta{\overline{\sigma}}^{2}_{\text{\tiny
OS}}-\delta{\overline{\sigma}}^{2}_{\overline{\text{\tiny MS}}}\;.$ (127)

The minimum of the effective potential is at $\overline{\sigma}=f_{\pi}$ and
masses have been measured in vacuum. Applying in Eqs. (113)–(119), we find the
$\Lambda$ scale dependent parameters in $\overline{\text{MS}}$ scheme

$\displaystyle\lambda_{1\overline{\text{\tiny MS}}}(\Lambda)$ $\displaystyle=$
$\displaystyle\lambda_{1}+\dfrac{N_{c}g^{2}}{(4\pi)^{2}}\left[\lambda_{1}\log\left(\frac{\Lambda^{2}}{m_{q}^{2}}\right)+\frac{(m^{2}_{\sigma}-4m^{2}_{q})C(m^{2}_{\sigma})+m^{2}_{\eta}C(m^{2}_{\eta})-(m^{2}_{a_{0}}-4m^{2}_{q})C(m^{2}_{a_{0}})-m^{2}_{\pi}C(m^{2}_{\pi})}{2f_{\pi}^{2}}\right.$
(128)
$\displaystyle\left.+\lambda_{1}(\mathcal{C}(m^{2}_{\pi})+m^{2}_{\pi}\mathcal{C}^{\prime}(m^{2}_{\pi}))\right],\qquad\quad\;$
$\displaystyle\lambda_{2\overline{\text{\tiny MS}}}(\Lambda)$ $\displaystyle=$
$\displaystyle\lambda_{2}+\dfrac{N_{c}g^{2}}{(4\pi)^{2}}\left[\left(2\lambda_{2}-g^{2}\right)\log\left(\frac{\Lambda^{2}}{m_{q}^{2}}\right)+\frac{(m^{2}_{a_{0}}-4m^{2}_{q})\mathcal{C}(m^{2}_{a_{0}})-m^{2}_{\eta}\mathcal{C}(m^{2}_{\eta})}{f_{\pi}^{2}}+\lambda_{2}(\mathcal{C}(m^{2}_{\pi})+m^{2}_{\pi}\mathcal{C}^{\prime}(m^{2}_{\pi}))\right]\;,$
(129) $\displaystyle m^{2}_{\overline{\text{\tiny MS}}}(\Lambda)$
$\displaystyle=$ $\displaystyle
m^{2}+\dfrac{N_{c}g^{2}}{(4\pi)^{2}}\left[m^{2}\log\left(\frac{\Lambda^{2}}{m_{q}^{2}}\right)+m^{2}_{\pi}\mathcal{C}(m^{2}_{\pi})+\dfrac{m^{2}_{\eta}C(m^{2}_{\eta})-(m^{2}_{\sigma}-4m^{2}_{q})C(m^{2}_{\sigma})}{2}-2m_{q}^{2}\right]\;,$
(130) $\displaystyle c_{\overline{\text{\tiny MS}}}(\Lambda)$ $\displaystyle=$
$\displaystyle
c+\dfrac{N_{c}g^{2}}{(4\pi)^{2}}\left[c\log\left(\frac{\Lambda^{2}}{m_{q}^{2}}\right)+\frac{m^{2}_{\eta}\mathcal{C}(m^{2}_{\eta})-m^{2}_{\pi}\mathcal{C}(m^{2}_{\pi})}{2}\right]\;,$
(131) $\displaystyle h_{\overline{\text{\tiny MS}}}(\Lambda)$ $\displaystyle=$
$\displaystyle
h+\dfrac{N_{c}g^{2}}{2(4\pi)^{2}}h\left[\log\left(\frac{\Lambda^{2}}{m_{q}^{2}}\right)+\mathcal{C}(m^{2}_{\pi})-m^{2}_{\pi}\mathcal{C}^{\prime}(m^{2}_{\pi})\right]\;,$
(132) $\displaystyle g^{2}_{\overline{\text{\tiny MS}}}(\Lambda)$
$\displaystyle=$ $\displaystyle
g^{2}+\dfrac{N_{c}g^{4}}{(4\pi)^{2}}\left[\log\left(\frac{\Lambda^{2}}{m_{q}^{2}}\right)+\mathcal{C}(m^{2}_{\pi})+m^{2}_{\pi}\mathcal{C}^{\prime}(m^{2}_{\pi})\right]\;,$
(133) $\displaystyle\overline{\sigma}^{2}_{\overline{\text{\tiny
MS}}}(\Lambda)$ $\displaystyle=$
$\displaystyle\overline{\sigma}^{2}-\dfrac{4N_{c}m^{2}_{q}}{(4\pi)^{2}}\left[\log\left(\frac{\Lambda^{2}}{m_{q}^{2}}\right)+\mathcal{C}(m^{2}_{\pi})+m^{2}_{\pi}\mathcal{C}^{\prime}(m^{2}_{\pi})\right]\;.$
(134)

Where the physical on-shell parameters are related with meson and quark mass
at vacuum given by Eqs.(18)–(24).

In the large-$N_{c}$ limit the parameters $\lambda_{1\overline{\text{\tiny
MS}}}$,$\lambda_{2\overline{\text{\tiny MS}}}$,$m^{2}_{\overline{\text{\tiny
MS}}}$,$c_{\overline{\text{\tiny MS}}}$,$h_{\overline{\text{\tiny MS}}}$ and
$g^{2}_{\overline{\text{\tiny MS}}}$ are running with the scale $\Lambda$ and
a set of simultaneous renormalization group equations are satisfied , which
are

$\displaystyle\dfrac{d\lambda_{1\overline{\text{\tiny
MS}}}(\Lambda)}{d\log(\Lambda)}$ $\displaystyle=$
$\displaystyle\dfrac{2N_{c}}{(4\pi)^{2}}g^{2}_{\overline{\text{\tiny
MS}}}\lambda_{1\overline{\text{\tiny MS}}}\;,$ (135)
$\displaystyle\dfrac{d\lambda_{2\overline{\text{\tiny
MS}}}(\Lambda)}{d\log(\Lambda)}$ $\displaystyle=$
$\displaystyle\dfrac{2N_{c}}{(4\pi)^{2}}\left[2\lambda_{2\overline{\text{\tiny
MS}}}g^{2}_{\overline{\text{\tiny MS}}}-g^{4}_{\overline{\text{\tiny
MS}}}\right]\;,$ (136) $\displaystyle\dfrac{dm^{2}_{\overline{\text{\tiny
MS}}}(\Lambda)}{d\log(\Lambda)}$ $\displaystyle=$
$\displaystyle\dfrac{2N_{c}}{(4\pi)^{2}}g^{2}_{\overline{\text{\tiny
MS}}}m^{2}_{\overline{\text{\tiny MS}}}\;,$ (137)
$\displaystyle\dfrac{dc_{\overline{\text{\tiny
MS}}}(\Lambda)}{d\log(\Lambda)}$ $\displaystyle=$
$\displaystyle\dfrac{2N_{c}}{(4\pi)^{2}}g^{2}_{\overline{\text{\tiny
MS}}}c_{\overline{\text{\tiny MS}}}\;,$ (138)
$\displaystyle\dfrac{dh_{\overline{\text{\tiny
MS}}}(\Lambda)}{d\log(\Lambda)}$ $\displaystyle=$
$\displaystyle\dfrac{N_{c}}{(4\pi)^{2}}g^{2}_{\overline{\text{\tiny
MS}}}h_{\overline{\text{\tiny MS}}}\;,$ (139)
$\displaystyle\dfrac{dg^{2}_{\overline{\text{\tiny MS}}}}{d\log(\Lambda)}$
$\displaystyle=$
$\displaystyle\dfrac{2N_{c}}{(4\pi)^{2}}g^{4}_{\overline{\text{\tiny MS}}}\;,$
(140) $\displaystyle\dfrac{d\overline{\sigma}^{2}_{\overline{\text{\tiny
MS}}}}{d\log(\Lambda)}$ $\displaystyle=$
$\displaystyle-\dfrac{2N_{c}}{(4\pi)^{2}}g^{2}_{\overline{\text{\tiny
MS}}}\overline{\sigma}^{2}_{\overline{\text{\tiny MS}}}$ (141)

solutions of Eq. (135)–(141) are

$\displaystyle\lambda_{1\overline{\text{\tiny MS}}}(\Lambda)$ $\displaystyle=$
$\displaystyle\frac{\lambda_{10}}{1-\frac{N_{c}g^{2}_{0}}{(4\pi)^{2}}\log\left(\dfrac{\Lambda^{2}}{\Lambda_{0}^{2}}\right)}\;,$
(142) $\displaystyle g^{2}_{\overline{\text{\tiny MS}}}(\Lambda)$
$\displaystyle=$
$\displaystyle\frac{g^{2}_{0}}{1-\dfrac{N_{c}g^{2}_{0}}{(4\pi)^{2}}\log\left(\dfrac{\Lambda^{2}}{\Lambda^{2}_{0}}\right)}\;,$
(143) $\displaystyle\lambda_{2\overline{\text{\tiny MS}}}(\Lambda)$
$\displaystyle=$
$\displaystyle\frac{\lambda_{20}-\dfrac{N_{c}g^{4}_{0}}{(4\pi)^{2}}\log\left(\dfrac{\Lambda^{2}}{\Lambda^{2}_{0}}\right)}{\left(1-\dfrac{N_{c}g^{2}_{0}}{(4\pi)^{2}}\log\left(\dfrac{\Lambda^{2}}{\Lambda^{2}_{0}}\right)\right)^{2}}\;,$
(144) $\displaystyle m^{2}_{\overline{\text{\tiny MS}}}(\Lambda)$
$\displaystyle=$
$\displaystyle\frac{m^{2}_{0}}{1-\dfrac{N_{c}g^{2}_{0}}{(4\pi)^{2}}\log\left(\dfrac{\Lambda^{2}}{\Lambda^{2}_{0}}\right)}\;,$
(145) $\displaystyle c_{\overline{\text{\tiny MS}}}(\Lambda)$ $\displaystyle=$
$\displaystyle\frac{c_{0}}{1-\dfrac{N_{c}g^{2}_{0}}{(4\pi)^{2}}\log\left(\dfrac{\Lambda^{2}}{\Lambda^{2}_{0}}\right)}\;,$
(146) $\displaystyle h_{\overline{\text{\tiny MS}}}(\Lambda)$ $\displaystyle=$
$\displaystyle\frac{h_{0}}{\sqrt{1-\dfrac{N_{c}g^{2}_{0}}{(4\pi)^{2}}\log\left(\dfrac{\Lambda^{2}}{\Lambda^{2}_{0}}\right)}}\;,$
(147) $\displaystyle\overline{\sigma}^{2}$ $\displaystyle=$ $\displaystyle
f^{2}_{\pi}\left[1-\frac{N_{c}g^{2}_{0}}{(4\pi)^{2}}\log\left(\frac{\Lambda^{2}}{\Lambda^{2}_{0}}\right)\right]\;.$
(148)

Where the parameters $\lambda_{10}$, $\lambda_{20}$, $g^{2}_{0}$, $m^{2}_{0}$,
$c_{0}$ and $h_{0}$, are the $\Lambda$ dependent parameters at the value
$\Lambda_{0}$, we consider $\Lambda_{0}$ to satisfy the relation

$\displaystyle\log\left(\frac{\Lambda^{2}_{0}}{m_{q}^{2}}\right)+\mathcal{C}(m^{2}_{\pi})+m^{2}_{\pi}\mathcal{C}^{\prime}(m^{2}_{\pi})$
$\displaystyle=$ $\displaystyle 0\;.$ (149)

###  B.1 Effective Potential

The vacuum effective potential in the $\overline{\text{MS}}$ scheme is given
by

$\displaystyle\Omega_{vac}$ $\displaystyle=$ $\displaystyle
U(\overline{\sigma}_{\overline{\text{\tiny
MS}}})+\Omega^{q,vac}_{\overline{\text{\tiny MS}}}+\delta
U(\overline{\sigma}_{\overline{\text{\tiny MS}}})\;,$ (150)

where

$\displaystyle U(\overline{\sigma}_{\overline{\text{\tiny MS}}})$
$\displaystyle=$ $\displaystyle\frac{m_{\overline{\text{\tiny
MS}}}^{2}(\Lambda)}{2}\overline{\sigma}^{2}_{\overline{\text{\tiny
MS}}}-\frac{c_{\overline{\text{\tiny
MS}}}(\Lambda)}{2}\overline{\sigma}^{2}_{\overline{\text{\tiny
MS}}}+\frac{1}{4}\left(\lambda_{1\overline{\text{\tiny
MS}}}(\Lambda)+\frac{\lambda_{2\overline{\text{\tiny
MS}}}(\Lambda)}{2}\right)\overline{\sigma}^{4}_{\overline{\text{\tiny
MS}}}-h_{\overline{\text{\tiny
MS}}}(\Lambda)\overline{\sigma}_{\overline{\text{\tiny MS}}}\;,$

$\delta U(\overline{\sigma}_{\overline{\text{\tiny
MS}}})=-\frac{N_{c}g^{4}_{\overline{\text{\tiny
MS}}}\overline{\sigma}^{4}_{\overline{\text{\tiny
MS}}}}{8(4\pi)^{2}}\frac{1}{\epsilon}$ and
$\Omega^{q,vac}_{\overline{\text{\tiny
MS}}}=\frac{N_{c}g^{4}_{\overline{\text{\tiny
MS}}}\overline{\sigma}^{4}_{\overline{\text{\tiny
MS}}}}{8(4\pi)^{2}}\left[\frac{1}{\epsilon}+\frac{3}{2}+\ln\left(\frac{4\Lambda^{2}}{g^{2}_{\overline{\text{\tiny
MS}}}\overline{\sigma}^{2}_{\overline{\text{\tiny MS}}}}\right)\right]\;$ as
in Ref. RaiTiw . One can define the scale $\Lambda$ independent parameter
$\Delta=\frac{g_{\overline{\text{\tiny
MS}}}\overline{\sigma}_{\overline{\text{\tiny MS}}}}{2}$ using the Eq. (133)
and Eq. (134) and rewrite the vacuum effective potential in terms of it as

$\displaystyle\Omega_{vac}(\Delta)$ $\displaystyle=$ $\displaystyle
2\left(\frac{m^{2}_{0}}{g^{2}_{0}}-\frac{c_{0}}{g^{2}_{0}}\right)\Delta^{2}+4\left(\frac{\lambda_{10}}{g^{4}_{0}}+\frac{\lambda_{20}}{2g^{4}_{0}}\right)\Delta^{4}-2\frac{h_{0}}{g_{0}}\Delta+\frac{2N_{c}\Delta^{4}}{(4\pi)^{2}}\left[\frac{3}{2}+\ln\left(\frac{\Lambda^{2}}{\Delta^{2}}\right)\right]\;.$
(151)

Expressing the couplings and mass parameter in terms of the Yukawa coupling,
pion decay constant and physical meson masses, one can write

$\displaystyle\Omega_{vac}(\Delta)$ $\displaystyle=$
$\displaystyle\frac{(3m^{2}_{\pi}-m^{2}_{\sigma})f^{2}_{\pi}}{4}\left\\{1-\frac{N_{c}g^{2}}{(4\pi)^{2}}\left(\mathcal{C}(m^{2}_{\pi})+m^{2}_{\pi}\mathcal{C}^{\prime}(m^{2}_{\pi})\right)\right\\}\frac{\Delta^{2}}{m^{2}_{q}}+\frac{N_{c}g^{2}f^{2}_{\pi}}{2(4\pi)^{2}}\left\\{\frac{3m^{2}_{\pi}\mathcal{C}(m^{2}_{\pi})-(m^{2}_{\sigma}-4m^{2}_{q})\mathcal{C}(m^{2}_{\sigma})}{2}-2m^{2}_{q}\right\\}\frac{\Delta^{2}}{m^{2}_{q}}\;$
(152)
$\displaystyle+\frac{(m^{2}_{\sigma}-m^{2}_{\pi})f^{2}_{\pi}}{8}\left\\{1-\frac{N_{c}g^{2}}{(4\pi)^{2}}\left(\mathcal{C}(m^{2}_{\pi})+m^{2}_{\pi}\mathcal{C}^{\prime}(m^{2}_{\pi})\right)\right\\}\frac{\Delta^{4}}{m^{4}_{q}}+\frac{N_{c}g^{2}f^{2}_{\pi}}{(4\pi)^{2}}\left[(m^{2}_{\sigma}-4m^{2}_{q})\mathcal{C}(m^{2}_{\sigma})-m^{2}_{\pi}\mathcal{C}(m^{2}_{\pi})\over
8\right]\frac{\Delta^{4}}{m^{4}_{q}}\;$
$\displaystyle+\frac{2N_{c}\Delta^{4}}{(4\pi)^{2}}\left\\{\frac{3}{2}-\ln\left(\frac{\Delta^{2}}{m^{2}_{q}}\right)\right\\}-m^{2}_{\pi}f^{2}_{\pi}\left\\{1-\frac{N_{c}g^{2}}{(4\pi)^{2}}m^{2}_{\pi}\mathcal{C}^{\prime}(m^{2}_{\pi})\right\\}\frac{\Delta}{m_{q}}\;$

Here it is relevant to remind ourselves that the Yukawa coupling and the pion
decay constant both get renormalized in the vacuum due to the dressing of the
meson propagator in the RQM model. However, the Eq. (133) gives
$g_{\overline{\text{\tiny MS}}}=g_{ren}=g$ and the Eq. (134) gives
$\overline{\sigma}_{\overline{\text{\tiny MS}}}=f_{\pi,ren}=f_{\pi}$ at the
scale $\Lambda_{0}$. Applying the stationarity condition
$\frac{\partial\Omega_{vac}(\Delta)}{\partial\Delta}$ to the Eq.(151), one
gets $h_{0}=m_{\pi,c}^{2}\ \overline{\sigma}_{\overline{\text{\tiny
MS}}}=m^{2}_{\pi}\left\\{1-\frac{N_{c}g^{2}}{(4\pi)^{2}}m^{2}_{\pi}\mathcal{C}^{\prime}(m^{2}_{\pi})\right\\}f_{\pi}$.
Note that the pion curvature mass $m_{\pi,c}$ differs from its pole mass
$m_{\pi}$ on account of the consistent parameter fixing and we have
$m_{\pi,c}^{2}=m^{2}_{\pi}\left\\{1-\frac{N_{c}g^{2}}{(4\pi)^{2}}m^{2}_{\pi}\mathcal{C}^{\prime}(m^{2}_{\pi})\right\\}$
as in Ref. fix1 . The minimum of the vacuum effective potential lies at
$\overline{\sigma}_{\overline{\text{\tiny MS}}}=f_{\pi}$.

##  Appendix C INTEGRALS

The divergent loop integrals are regularized by encorporating dimensional
regularization.

$\displaystyle\int_{p}=\left(\frac{e^{\gamma_{E}}\Lambda^{2}}{4\pi}\right)^{\epsilon}\int\frac{d^{d}p}{(2\pi)^{d}}\;,$
(153)

where $d=4-2\epsilon$ , $\gamma_{E}$ is the Euler-Mascheroni constant, and
$\Lambda$ is renormalization scale associated with the $\overline{\text{MS}}$.

$\displaystyle\mathcal{A}(m^{2}_{q})$ $\displaystyle=$
$\displaystyle\int_{p}\frac{1}{p^{2}-m^{2}_{q}}=\frac{im^{2}_{q}}{(4\pi)^{2}}\left[\frac{1}{\epsilon}+1\right.$
$\displaystyle\left.+\log(4\pi
e^{-\gamma_{E}})+\log\left(\frac{\Lambda^{2}}{m^{2}_{q}}\right)\right]\;$

we rewrite this after redefining
$\Lambda^{2}\longrightarrow\Lambda^{2}\frac{e^{\gamma_{E}}}{4\pi}$.

$\displaystyle\mathcal{A}(m^{2}_{q})$ $\displaystyle=$
$\displaystyle\frac{im^{2}_{q}}{(4\pi)^{2}}\left[\frac{1}{\epsilon}+1+\log\left(\frac{\Lambda^{2}}{m^{2}_{q}}\right)\right]$
(154) $\displaystyle\mathcal{B}(p^{2})$ $\displaystyle=$
$\displaystyle\int_{k}\frac{1}{(k^{2}-m^{2}_{q})[(k+p)^{2}-m^{2}_{q})]}$ (155)
$\displaystyle=$
$\displaystyle\frac{i}{(4\pi)^{2}}\left[\frac{1}{\epsilon}+\log\left(\frac{\Lambda^{2}}{m^{2}_{q}}\right)+C(p^{2})\right]$
$\displaystyle\mathcal{B}^{\prime}(p^{2})$ $\displaystyle=$
$\displaystyle\frac{i}{(4\pi)^{2}}C^{\prime}(p^{2})$ (156)
$\mathcal{C}(p^{2})=2-2\sqrt{\dfrac{4m^{2}_{f}}{p^{2}}-1}\arctan\left(\dfrac{1}{\sqrt{\dfrac{4m^{2}_{f}}{p^{2}}-1}}\right);\quad\mathcal{C}^{\prime}(p^{2})=\frac{4m^{2}_{f}}{p^{4}\sqrt{\dfrac{4m^{2}_{f}}{p^{2}}-1}}\arctan\left(\dfrac{1}{\sqrt{\dfrac{4m^{2}_{f}}{p^{2}}-1}}\right)-\frac{1}{p^{2}}\;,$
(157)
$\mathcal{C}(p^{2})=2+\sqrt{1-\dfrac{4m^{2}_{f}}{p^{2}}}\ln\left(\dfrac{1-\sqrt{1-\dfrac{4m^{2}_{f}}{p^{2}}}}{1+\sqrt{1-\dfrac{4m^{2}_{f}}{p^{2}}}}\right);\quad\mathcal{C}^{\prime}(p^{2})=\frac{2m^{2}_{f}}{p^{4}\sqrt{\dfrac{4m^{2}_{f}}{p^{2}}-1}}\ln\left(\dfrac{1-\sqrt{1-\dfrac{4m^{2}_{f}}{p^{2}}}}{1+\sqrt{1-\dfrac{4m^{2}_{f}}{p^{2}}}}\right)-\frac{1}{p^{2}}\;,$
(158)

The Eqs. (157) and (158) are valid with the constraints ($p^{2}<4m^{2}_{f}$)
and ($p^{2}>4m^{2}_{f}$) respectively.

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