# Phase structure of the on-shell parametrized 2+1 flavor Polyakov quark-meson
model

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

Augmenting the improved chiral effective potential of the on-shell
renormalized 2+1 flavour quark-meson (RQM) model with the Polyakov-loop
potential that accounts for the deconfinement transition, we get the Quantum
Chromodynamics (QCD) like framework of the renormalized Polyakov quark-meson
(RPQM) model. When the divergent quark one-loop vacuum term is included in the
effective potential of the quark-meson (QM) model, its tree level parameters
or the parameters fixed by the use of meson curvature masses, become
inconsistent as the curvature masses involve the self energy evaluations at
zero momentum. Using the modified minimal subtraction method, the consistent
chiral effective potential for the RQM model has been calculated after
relating the counterterms in the on-shell (OS) scheme to those in the
$\overline{\text{MS}}$ scheme and finding the relations between the
renormalized parameters of both the schemes where the physical (pole) masses
of the $\pi,K,\eta$ and $\eta^{\prime}$ pseudo-scalar mesons and the scalar
$\sigma$ meson, the pion and kaon decay constants, have been put into the
relation of the running couplings and mass parameter. Using the RPQM model and
the PQM Model with different forms for the Polyakov-loop potentials in the
presence or the absence of the quark back-reaction, we have computed and
compared the effect of the consistent quark one-loop correction and the quark
back-reaction on the scaled chiral order parameter, the QCD phase diagrams and
the different thermodynamic quantities. The results have been compared with
the 2+1 flavor lattice QCD data from the Wuppertal-Budapest collaboration
{JHEP 09,73(2010); PLB 730,99(2014)} and the HotQCD collaboration {PRD
90,094503(2014)}.

Dense QCD, chiral transition,

##  I Introduction

The hadronic matter under the extreme conditions of high temperatures and/or
densities, gets dissolved into its quark and gluon constituents and the Quark
Gluon Plasma (QGP) Cabibbo75 ; SveLer ; Mull ; Ortms ; Riske  is formed as
predicted by the strong interaction theory quantum chromodynamics (QCD). The
QCD phase diagram Cabibbo75  and the general properties of QGP are suject
matter of intensive investigation for the ultra-relativistic heavy ion
collision experiments like the RHIC (BNL), LHC (CERN) and the upcoming CBM
experiments at the FAIR facility (GSI-Darmstadt). 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  give us important
information and insights for the QCD phase transition that occurs on the
temperature axis but when the baryon density is nonzero, these calculations
get severely hampered as the QCD action becomes complex due to the fermion
sign problem Karsch:02 . The QCD-like effective theory models Alf ; Fukhat
built upon the symmetries of the QCD give us the much needed framework in
which the QCD phase structure and its thermodynamics can be explored in great
details.

The QCD Lagrangian has the global $SU_{\tiny{L+R}}(3)\times SU_{L-R}(3)$
symmetry for the three massless quarks. The chiral (axial $A=L-R$) symmetry
gets spontaneously broken in the low energy vacuum of the QCD and one gets the
non-strange and the strange chiral condensates as the order parameters with
eight massless pseudoscalar bosons as Goldstone modes. Since the small masses
of the $u$ and $d$ quarks cause a small explicit breaking while a relatively
large mass of the $s$ quark generates a larger explicit breaking of the chiral
symmetry, the three pions are light while the four kaons and one eta are
heavier in nature. Due to the instanton effects, the $U_{A}(1)$ axial symmetry
also gets explicitly broken to the $Z_{A}(N_{f})$ at the quantum level
tHooft:76prl . The $\eta^{\prime}$ meson does not remain a massless Goldstone
boson even when the quarks are massless as it acquires a mass of about 1 GeV
due to the $U_{A}(1)$ axial anomaly. Coupling the nine scalar and nine pseudo-
scalar mesons of the three flavor linear sigma model Rischke:00  with the two
light quarks $u,d$ and the one heavier s quark, one gets the effective theory
framework of the 2+1 flavor quark-meson (QM) model Schaefer:09 .

Several investigations of the QCD phase structure, have already been done in
the chiral models Rischke:00 ; Roder ; fuku11 ; grahl ; 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 . When the effect of the Dirac sea gets
neglected in the standard mean field approximation (s-MFA), the QM model
studies look inconsistent because in the chiral limit, the chiral phase
transition at zero baryon density turns first-order which is ruled out by the
general theoretical arguments rob ; hjss . The inclusion of the quark one-loop
vacuum fluctuation in the QM model vac , removes the above inconsistency. In
several of the QCD phase structure studies carried out in the ambit of the
quark-meson model with the quark one-loop vacuum term lars ; guptiw ;
schafwag12 ; chatmoh1 ; vkkr12 ; TranAnd ; chatmoh2 ; vkkt13 ; Herbst ;
Weyrich ; kovacs ; zacchi1 ; zacchi2 ; Rai , the model parameters are fixed by
using the curvature masses of the mesons while the pion and kaon decay
constant are fixed by the vacuum expectation value of the non-strange and
strange chiral condensates. This parameter fixing turns out to be inconsistent
because the effective potential generates the n-point functions of the theory
at vanishing external momenta and the curvature masses are defined by the
self-energy evaluations at zero momentum laine ; Adhiand1 ; BubaCar ; Naylor ;
fix1 .

The tree level relations between the physical quantities and the parameters of
the Lagrangian, are changed by the radiative corrections to the physical
quantities. Hence the use of the tree level parameters in the effective
potential becomes inconsistent. The running parameters in the
$\overline{\text{MS}}$ scheme have the renormalization scale $\Lambda$
dependence while the on-shell parameters have their tree-level values.
Following the correct renormalization prescription, when the counterterms
calculated in the $\overline{\text{MS}}$ scheme are put into the relations of
the counterterms in the on-shell scheme, one finds the relations between the
different renormalized parameters of the two schemes. The relations between
the on-shell parameters (physical quantities) and the running
$\overline{\text{MS}}$ parameters are used as input when the effective
potential is calculated using the modified minimal subtraction procedure
Adhiand1 . Adhikari and collaborators Adhiand1 ; Adhiand2 ; Adhiand3 ; asmuAnd
in a series of papers, used the above renormalization prescription for
including the quark one-loop vacuum correction in the two flavor QM model
which uses the $O(4)$ sigma model with the $\sigma$ and $\vec{\pi}$ mesons. In
a recent QCD phase structure study, we implemented RaiTiw  the on-shell
parameter fixing method in the QM model where the two quark flavors are
coupled to the eight mesons of the $SU_{L}(2)\times SU_{R}(2)$ linear sigma
model. Here, we have calculated the consistent chiral effective potential for
the on-shell renormalized 2+1 flavor quark-meson (RQM) model.

The introduction of the Polyakov-loop implements the physics of quark
confinement inside the hadrons as the QCD confinement is mimicked in a
statistical sense when the chiral models are coupled to a constant background
$SU(N_{c})$ gauge field $A_{\mu}^{a}$ SveLer ; Polyakov:78plb ; benji ;
BankUka ; Pisarski:00prd ; fuku ; Vkt:06 . Using the phenomenological
Polyakov-loop potential ratti ; Roesnr ; fuku2 , when the free energy density
from the gluons is added to the QM model, it becomes the PQM model SchaPQM2F ;
SchaPQM3F ; Mao ; TiPQM3F . In the present work, combining the Polyakov-loop
potential with the consistent chiral effective potential calculated for the
2+1 flavor RQM Model, we get the QCD like framework of the renormalized
Polyakov quark-meson model (RPQM). We have considered different
parametrizations for the Polyakov-loop potentials along with the recent
improvement in the form of the Polyakov-loop potential from a pure gauge
potential to a unquenched glue potential in which the quark back-reaction
effects are included Haas ; Redlo ; TkHerbst ; BielichP . It is important to
explore how the QCD phase structure computed in the present RPQM model
framework, gets modified by the unquenching of the Polyakov-loop potential
because it leads to the linkage of the chiral and deconfinement phase
transitions also at small temperatures and large chemical potentials BielichP
. The above feature is also noticed in the studies having the functional
renormalization (FRG) improvement of the PQM model when the Yang-Mills
Polyakov-loop potential is used Herbst ; THerbst2 . In order to know the
effect of the consistent quark one-loop vacuum correction, we have calculated
the QCD phase diagrams and the QCD thermodynamics in the RPQM model and
compared the results with that of the PQM model. Taking different forms of the
Polyakov-loop potentials with and without quark back-reaction, we have
computed and compared how the phase structure gets influenced by the different
implementations of the confinement-deconfinement physics in the RPQM model.

The organization of the paper is as follows. Section II presents a brief
formulation of the $SU_{L}(3)\times SU_{R}(3)$ PQM model. The different forms
of the Polyakov-loop potentials are described in the sub-section II.1, while
the sub-section II.2 gives the thermodynamic grand potential in the PQM model
and the parameters of the PQM model are determined in the sub-section II.3.
Section III presents the technical details of the on-shell renormalization of
the parameters in the renormalized Polyakov quark-meson model. Sub-section
III.1 gives the description of the counterterms of the Lagrangian and the
self-energies of the mesons, sub-section III.2 presents the renormalization of
the model parameters and the effective potential of the RPQM model has been
derived in the sub-section III.3. Results and discussion are presented in the
section IV. The comparisons of the thermodynamic quantities with the lattice
QCD data, have been discussed in the sub-section IV.1 and the sub-section IV.2
illustrates and compares the phase diagrams and their CEP position in
different model scenarios. Section V provides the summary and conclusion. The
Appendix A contains the important integrals and expressions of the factors.

##  II Model Formulation

In the model TiPQM3F ; Schaefer:09wspax ; Mao ; SchaPQM3F , three flavor of
quarks are coupled to the $SU_{V}(3)\times SU_{A}(3)$ symmetric mesonic fields
together with temporal component of gauge field represented by the Polyakov
loop potential. Thermal expectation value of color trace of Wilson loop in
temporal direction defines the Polyakov loop field $\Phi$ as

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

where $L(\vec{x})$ 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 . As in the Ref. ratti ; Roesnr
, the homogeneous Polyakov loop fields $\Phi(\vec{x})=\Phi$=constant and
$\bar{\Phi}(\vec{x})=\bar{\Phi}$=constant.

The Polyakov quark-meson (PQM) model Lagrangian is written in terms of the
quarks, mesons, their couplings and the Polyakov loop potential ${\cal
U}\left(\Phi,\bar{\Phi},T\right)$ as

$\displaystyle{\cal L_{PQM}}$ $\displaystyle=$ $\displaystyle{\cal
L_{QM}}-{\cal U}\big{(}\Phi,\bar{\Phi},T\big{)},$ (3) $\displaystyle{\cal
L_{QM}}$ $\displaystyle=$
$\displaystyle\bar{\psi}[i\gamma^{\mu}D_{\mu}-g\;T_{a}\big{(}\sigma_{a}+i\gamma_{5}\pi_{a}\big{)}]\psi+\cal{L(M)},$
(4)

here $\psi$ is a four-component Dirac spinor, a flavor triplet and a color
$N_{c}$-plet quark field.

$\displaystyle\psi$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}u\\\ d\\\ s\end{array}\right)\;.$ (8)

The three flavor of quarks are coupled to the nine scalar
($\sigma_{a},J^{P}=0^{+}$) and nine pseudo-scalar ($\pi_{a},J^{P}=0^{-}$)
mesons by the flavor blind Yukawa coupling $g$. Quarks couple with the uniform
temporal background gauge field as the following
$D_{\mu}=\partial_{\mu}-iA_{\mu}$ and $A_{\mu}=\delta_{\mu 0}A_{0}$ (Polyakov
gauge), where $A_{\mu}=g_{s}A^{a}_{\mu}\lambda^{a}/2$ with vector potential
$A^{a}_{\mu}$ for color gauge field. $g_{s}$ is the $SU_{c}(3)$ gauge
coupling.

The Lagrangian for the meson fields is written as Schaefer:09 ; Roder ;
TiPQM3F

$\displaystyle\cal{L(M)}$ $\displaystyle=$
$\displaystyle\text{Tr}(\partial_{\mu}{\cal{M}}^{\dagger}\partial^{\mu}{\cal{M}}-m^{2}({\cal{M}}^{\dagger}{\cal{M}}))$
(9)
$\displaystyle-\lambda_{1}\left[\text{Tr}({\cal{M}}^{\dagger}{\cal{M}})\right]^{2}-\lambda_{2}\text{Tr}({\cal{M}}^{\dagger}{\cal{M}})^{2}$
$\displaystyle+c[\text{det}{\cal{M}}+\text{det}{\cal{M}}^{\dagger}]+\text{Tr}\left[H({\cal{M}}+{\cal{M}}^{\dagger})\right]\;,$

the field ${\cal{M}}$ is a $3\times 3$ complex matrix containing the nine
scalars $\sigma_{a}$ and the nine pseudo-scalar $\pi_{a}$ mesons.

$\displaystyle{\cal{M}}$ $\displaystyle=$ $\displaystyle
T_{a}\xi_{a}=T_{a}(\sigma_{a}+i\pi_{a})\;.$ (10)

The $T_{a}$ in the above are 9 generators of $U(3)$ with
$T_{a}=\frac{\lambda_{a}}{2}$ where $a=0,1,\dots~{},8$. The $\lambda_{a}$ are
standard Gell-Mann matrices with $\lambda_{0}=\sqrt{\frac{2}{3}}\
{\mathbb{I}}_{3\times 3}$. The generators follow the $U(3)$ algebra
$\left[T_{a},T_{b}\right]=if_{abc}T_{c}$ and
$\left\\{T_{a},T_{b}\right\\}=d_{abc}T_{c}$ where $f_{abc}$ and $d_{abc}$ are
the standard antisymmetric and symmetric structure constants respectively with
$f_{ab0}=0$ and $d_{ab0}=\sqrt{\frac{2}{3}}\ \delta_{ab}$ and matrices are
normalized as $\text{Tr}(T_{a}T_{b})=\frac{\delta_{ab}}{2}$. The
$SU_{L}(3)\times SU_{R}(3)$ chiral symmetry is explicitly broken by the
following term

$\displaystyle H=T_{a}h_{a}\;.$ (11)

$H$ is a $3\times 3$ matrix with 9 parameters. Due to the spontaneous chiral
symmetry breaking, the field $\xi$ takes the nonzero vacuum expectation value,
$\bar{\xi}$. Since $\bar{\xi}$ must have the quantum numbers of the vacuum,
only three nonzero parameters $h_{0}$, $h_{3}$ and $h_{8}$ can cause the
explicit breakdown of the chiral symmetry. Neglecting isospin symmetry
breaking, we have taken $h_{0}$, $h_{8}\neq 0$. One gets the $2+1$ flavor
symmetry breaking scenario having the two nonzero condensates
$\bar{\sigma_{0}}$ and $\bar{\sigma_{8}}$. In addition to the $h_{0}$ and
$h_{8}$, the five other parameters of the model at tree-level are the quartic
coupling constants $\lambda_{1}$ and $\lambda_{2}$, squared mass parameter
$m^{2}$, a Yukawa coupling $g$ and the coefficient of the t’Hooft determinant
term $c$ that models the $U_{A}(1)$ axial anomaly of the QCD vacuum.

###  II.1 Polyakov-loop potentials

Different forms of the Polyakov-loop effective potential
$\mathcal{U}(\Phi,\bar{\Phi},T)$, have been used in the literature to study
the deconfinement phase transition. One constructs its simplest form by
finding a potential which respects all the given symmetries and accounts for
the spontaneously broken $Z(3)$ symmetry for the system in the deconfined
phase SveLer ; benji ; BankUka . The following polynomial form gives the
minimal content of the Polyakov-loop potential

$\displaystyle\frac{\mathcal{U_{\rm Poly}}}{T^{4}}$ $\displaystyle=$
$\displaystyle-\frac{b_{2}(T)}{2}\Phi\bar{\Phi}-\frac{b_{3}}{6}(\Phi^{3}+\bar{\Phi}^{3})+\frac{b_{4}}{4}(\Phi\bar{\Phi})^{2}\;,$
(12)

the coefficients of the Eq. (12) are given by

$\displaystyle
b_{2}(T)=a_{0}+a_{1}\left(\frac{T_{0}}{T}\right)+a_{2}\left(\frac{T_{0}}{T}\right)^{2}+a_{3}\left(\frac{T_{0}}{T}\right)^{3}\;,$
(13)

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 Polyakov-loop potential in the above is improved by adding the
contribution coming from the integration of the $SU(3)$ group volume in the
generating functional for the Euclidean action. The Haar measure is used to
perform this integration which 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 :

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

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. (14)
are the following Roesnr ,

$\displaystyle
a(T)=a_{0}+a_{1}\left(\frac{T_{0}}{T}\right)+a_{2}\left(\frac{T_{0}}{T}\right)^{2}\;,$
(15) $\displaystyle b(T)=b_{3}\left(\frac{T_{0}}{T}\right)^{3}\;,$ (16)

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

Accounting for the Polyakov-loop fluctuations, the new Polyakov-loop effective
potential was constructed in the Ref. Redlo  where the parameters get so
adjusted that in addition to the other existing lattice data, the longitudinal
as well as the transverse susceptibilities are also reproduced. After the
addition of the logarithmic term to the polynomial form of the Polyakov-loop
potential, the new expression of the PolyLog Polyakov-loop potential in the
above work has been found as the following.

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

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

$\displaystyle
a_{i}(T)=\frac{a^{(i)}_{0}+a^{(i)}_{1}\left(\frac{T_{0}}{T}\right)+a^{(i)}_{2}\left(\frac{T_{0}}{T}\right)^{2}}{1+a^{(i)}_{3}\left(\frac{T_{0}}{T}\right)+a_{4}^{(i)}\left(\frac{T_{0}}{T}\right)^{2}}\;$
(18) $\displaystyle
b(T)=b_{0}\left(\frac{T_{0}}{T}\right)^{b_{1}}\left[1-e^{b_{2}\left(\frac{T_{0}}{T}\right)^{b_{3}}}\right].$
(19)

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 |  |

For the pure gauge Yang-Mills theory, the deconfinement phase transition is
first order and $T_{c}^{\rm YM}=T_{0}=270$ MeV. When the dynamical quarks are
present, the first order transition becomes a crossover. 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. The number of flavor and the chemical potential
dependence of $T_{0}\longrightarrow T_{0}(N_{f},\mu)$ is written as,

$\displaystyle T_{0}(N_{f},\mu)=\hat{T}\ e^{-1/(\alpha_{0}b(N_{f},\mu))}\;,$
(20)

with

$\displaystyle
b(N_{f},\mu)=\frac{1}{6\pi}(11N_{c}-2N_{f})-b_{\mu}\frac{\mu^{2}}{(\hat{\gamma}\hat{T})^{2}}.$
(21)

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 . Massive flavors
lead to suppression factors of the order $T_{0}^{2}/(T^{2}_{0}+m^{2})$ in the
$\beta$-function. For 2+1 flavors and a current strange quark mass $m_{s}\sim$
150 MeV, one obtains $T_{0}(2+1)$=187 MeVSchaPQM2F ; THerbst2 .

When the back-reaction of the quarks is accounted for, in the full QCD with
dynamical quarks, the Polyakov-loop potential gets replaced by the QCD glue
potential. The Ref. Haas  applied the FRG equations to the QCD and 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 that the two potentials are of 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

$\displaystyle\frac{\mathcal{U}_{\rm glue}}{T^{4}_{\rm
glue}}(\Phi,\bar{\Phi},T_{\rm glue})$ $\displaystyle=$
$\displaystyle\frac{\mathcal{U}_{\rm YM}}{T^{4}_{\rm
YM}}(\Phi,\bar{\Phi},T_{\rm YM})$ (22)

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

$\displaystyle\frac{T_{\rm YM}-T^{\rm YM}_{\rm c}}{T^{\rm YM}_{\rm c}}=0.57\
\frac{T_{\rm glue}-T^{\rm glue}_{\rm c}}{T^{\rm glue}_{\rm c}}$ (23)

The transition temperature for the unquenched case is the $T^{\rm glue}_{\rm
c}$. 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, one uses the replacement $T\longrightarrow
T^{\rm YM}_{\rm c}(1+0.57(\frac{T}{T^{\rm glue}_{\rm c}}-1))$ in the right-
hand side of the Polyakov-loop potentials where $T_{0}$ means $T^{\rm YM}_{\rm
c}$ and ($T\sim T_{\rm YM}$) on the left side of the arrow while ($T\sim
T_{\rm glue}$) on the right side. In our calculations, we have taken $T^{\rm
glue}_{\rm c}$ and $T_{0}$ both fixed at 187 MeV for the 2+1 quark flavor as
in the Ref. SchaPQM2F ; Haas .

###  II.2 Grand Potential in the Mean Field Approach

We have the spatially uniform system that is in the thermal equilibrium at
temperature $T$ and quark chemical potential $\mu_{f}(f=u,d,s)$. The partition
function is calculated by performing the path integral over the
quark/antiquark and meson fields Schaefer:09 ; TiPQM3F

$\displaystyle\mathcal{Z}$ $\displaystyle=$
$\displaystyle\mathrm{Tr\,exp}[-\beta(\hat{\mathcal{H}}-\sum_{f=u,d,s}\mu_{f}\hat{\mathcal{N}}_{f})]$
(24) $\displaystyle=$
$\displaystyle\int\prod_{a}\mathcal{D}\sigma_{a}\mathcal{D}\pi_{a}\int\mathcal{D}\psi\mathcal{D}\bar{\psi}\;\mathrm{exp}\bigg{[}-\int_{0}^{\beta}d\tau\int_{V}d^{3}x$
$\displaystyle\bigg{(}\mathcal{L_{QM}^{E}}+\sum_{f=u,d,s}\mu_{f}\bar{\psi}_{f}\gamma^{0}\psi_{f}\bigg{)}\bigg{]}\;.$

Here $\beta=\frac{1}{T}$ and the volume of the system in the three dimension
is $V$. There will be three chemical potentials for the three flavor of quarks
and the $SU_{V}(2)$ symmetry is assumed to be preserved as the small
difference in the mass of $u$ and $d$ quark gets neglected. Hence the quark
chemical potential for the $u$ and $d$ quarks is equal $\mu_{u}=\mu_{d}$ and
the strange quark chemical potential is $\mu_{s}$.

In the standard mean-field approximation scav ; Schaefer:09 ; TiPQM3F , the
partition function is calculated by replacing the meson fields with their
vacuum expectation values $\langle
M\rangle=T_{0}\bar{\sigma_{0}}+T_{8}\bar{\sigma_{8}}$ and neglecting the
thermal as well as quantum fluctuations of the meson fields while retaining
the quarks and antiquarks as quantum fields. Now following the standard
procedure as given in Refs. Kapusta_Gale ; SchaPQM2F ; ratti ; fuku , one can
obtain the expression of grand potential as the sum of pure gauge field
contribution ${\cal U}\left(\Phi,\bar{\Phi},T\right)$, meson contribution and
quark/antiquark contribution evaluated in the presence of Polyakov loop,

$\displaystyle\Omega_{\rm MF}(T,\mu)$ $\displaystyle=$ $\displaystyle
U(\overline{\sigma_{0}},\overline{\sigma_{8}})+\Omega_{q\bar{q}}(T,\mu;\overline{\sigma_{0}},\overline{\sigma_{8}},\Phi,\bar{\Phi})\;$
(25) $\displaystyle+\mathcal{U}(T,\Phi,\bar{\Phi})\;.$

The 2 + 1 flavor case can be explored after implementing the basis
transformation of the condensates and the external fields from the original
singlet octet (0, 8) basis to the nonstrange strange basis
($\overline{\sigma}_{x}$, $\overline{\sigma}_{y}$)

$\displaystyle\overline{\sigma}_{x}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{2}{3}}\bar{\sigma}_{0}+\frac{1}{\sqrt{3}}\bar{\sigma}_{8},$
(26) $\displaystyle\overline{\sigma}_{y}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{3}}\bar{\sigma}_{0}-\sqrt{\frac{2}{3}}\bar{\sigma}_{8}.$
(27)

The grand potential is written in $\overline{\sigma}_{x}$,
$\overline{\sigma}_{y}$ basis as,

$\displaystyle\Omega_{\rm MF}(T,\mu)$ $\displaystyle=$ $\displaystyle
U(\overline{\sigma}_{x},\overline{\sigma}_{y})+\mathcal{U}(T,\Phi,\bar{\Phi})\;$
(28)
$\displaystyle+\Omega_{q\bar{q}}(T,\mu;\overline{\sigma}_{x},\overline{\sigma}_{y},\Phi,\bar{\Phi})\;.$

The external fields ($h_{x}$, $h_{y}$) can be written in terms of the
($h_{0}$, $h_{8}$) by similar expressions. Since the nonstrange and strange
quark/antiquark decouple, the quark masses are written as,

$\displaystyle m_{u}=g\frac{\overline{\sigma}_{x}}{2},\qquad
m_{s}=g\frac{\overline{\sigma}_{y}}{\sqrt{2}}\;.$ (29)

One writes the tree level effective potential in the nonstrange-strange basis
as the following.

$\displaystyle U(\overline{\sigma}_{x},\overline{\sigma}_{y})$
$\displaystyle=$
$\displaystyle\frac{m^{2}}{2}\left(\overline{\sigma}_{x}^{2}+\overline{\sigma}_{y}^{2}\right)-h_{x}\overline{\sigma}_{x}-h_{y}\overline{\sigma}_{y}-\frac{c}{2\sqrt{2}}\overline{\sigma}_{x}^{2}\overline{\sigma}_{y}$
(30)
$\displaystyle+\frac{\lambda_{1}}{2}\overline{\sigma}_{x}^{2}\overline{\sigma}_{y}^{2}+\frac{1}{8}\left(2\lambda_{1}+\lambda_{2}\right)\overline{\sigma}_{x}^{4}$
$\displaystyle+\frac{1}{8}\left(2\lambda_{1}+2\lambda_{2}\right)\overline{\sigma}_{y}^{4}\
.$

Applying the stationarity conditions $\frac{\partial
U(\overline{\sigma}_{x},\overline{\sigma}_{y})}{\partial\overline{\sigma}_{x}}=0=\frac{\partial
U(\overline{\sigma}_{x},\overline{\sigma}_{y})}{\partial\overline{\sigma}_{y}}$
to the effective potential (30), one gets

$\displaystyle h_{x}$ $\displaystyle=$ $\displaystyle\overline{\sigma}_{x}\
m^{2}_{\pi}\,$ (31) $\displaystyle h_{y}$ $\displaystyle=$
$\displaystyle{\biggl{\\{}\frac{\sqrt{2}}{2}(m_{K}^{2}-m_{\pi}^{2})\overline{\sigma}_{x}+m_{K}^{2}\overline{\sigma}_{y}\biggr{\\}}}\;.\quad$
(32)

The tree level curvature masses of the pions, kaons and other mesons in the QM
model are given by the mass matrix $(m_{\alpha,ab})^{2}$ evaluated in Ref.
Rischke:00 ; Schaefer:09 . Here $\alpha=$ s, p; “s” stands for the scalar and
“p” stands for the pseudo-scalar mesons and $a,b=0,1,2,\cdots,8$. In the
scalar sector, the $a_{0}$ meson mass is given by the 11 element (degenerate
with the 22 and 33 elements) and the $\kappa$ meson mass is given by the 44
element (degenerate with the 55, 66 and 77 elements). The $\sigma$ and $f_{0}$
meson masses are found by diagonalizing the (00)-(88) sector of the scalar
mass matrix. In exactly analogous manner for the pseudoscalar sector
$m^{2}_{\text{p},11}=m^{2}_{\text{p},22}=m^{2}_{\text{p},33}\equiv
m^{2}_{\pi}$ and
$m^{2}_{\text{p},44}=m^{2}_{\text{p},55}=m^{2}_{\text{p},66}=m^{2}_{\text{p},77}\equiv
m^{2}_{K}$. Diagonalization of the pseudoscalar (00)-(88) sector of the mass
matrix gives us the masses of the physical $\eta$ and $\eta^{\prime}$ mesons.
The Table 2 contains masses of all the mesons.

Table 2: Meson masses calculated from the second derivative of the grand
potential at its minimum as given in Ref. Schaefer:09 ; Herpay:06

|  Scalar meson masses |  |  Pseudo-scalar meson masses
---|---|---|---
$(m_{a_{0}})^{2}$ |  $m^{2}+\lambda_{1}(\overline{\sigma}_{x}^{2}+\overline{\sigma}_{y}^{2})+\frac{3\lambda_{2}}{2}\overline{\sigma}_{x}^{2}+\frac{\sqrt{2}c}{2}\overline{\sigma}_{y}$ |  $(m_{\pi})^{2}$ |  $m^{2}+\lambda_{1}(\overline{\sigma}_{x}^{2}+\overline{\sigma}_{y}^{2})+\frac{\lambda_{2}}{2}\overline{\sigma}_{x}^{2}-\frac{\sqrt{2}c}{2}\overline{\sigma}_{y}$
$(m_{\kappa})^{2}$ |  $m^{2}+\lambda_{1}(\overline{\sigma}_{x}^{2}+\overline{\sigma}_{y}^{2})+\frac{\lambda_{2}}{2}(\overline{\sigma}_{x}^{2}+\sqrt{2}\overline{\sigma}_{x}\overline{\sigma}_{y}+2\overline{\sigma}_{y}^{2})+\frac{c}{2}\overline{\sigma}_{x}$ |  $(m_{K})^{2}$ |  $m^{2}+\lambda_{1}(\overline{\sigma}_{x}^{2}+\overline{\sigma}_{y}^{2})+\frac{\lambda_{2}}{2}(\overline{\sigma}_{x}^{2}-\sqrt{2}\overline{\sigma}_{x}\overline{\sigma}_{y}+2\overline{\sigma}_{y}^{2})-\frac{c}{2}\overline{\sigma}_{x}$
$(m_{s,00})^{2}$ |  $m^{2}+\frac{\lambda_{1}}{3}(7\overline{\sigma}_{x}^{2}+4\sqrt{2}\overline{\sigma}_{x}\overline{\sigma}_{y}+5\overline{\sigma}_{y}^{2})+\lambda_{2}(\overline{\sigma}_{x}^{2}+\overline{\sigma}_{y}^{2})-\frac{\sqrt{2}c}{3}(\sqrt{2}\overline{\sigma}_{x}+\overline{\sigma}_{y})$ |  $(m_{p,00})^{2}$ |  $m^{2}+\lambda_{1}(\overline{\sigma}_{x}^{2}+\overline{\sigma}_{y}^{2})+\frac{\lambda_{2}}{3}(\overline{\sigma}_{x}^{2}+\overline{\sigma}_{y}^{2})+\frac{c}{3}(2\overline{\sigma}_{x}+\sqrt{2}\overline{\sigma}_{y})$
$(m_{s,88})^{2}$ |  $m^{2}+\frac{\lambda_{1}}{3}(5\overline{\sigma}_{x}^{2}-4\sqrt{2}\overline{\sigma}_{x}\overline{\sigma}_{y}+7\overline{\sigma}_{y}^{2})+\lambda_{2}(\frac{\overline{\sigma}_{x}^{2}}{2}+2\overline{\sigma}_{y}^{2})+\frac{\sqrt{2}c}{3}(\sqrt{2}\overline{\sigma}_{x}-\frac{\overline{\sigma}_{y}}{2})$ |  $(m_{p,88})^{2}$ |  $m^{2}+\lambda_{1}(\overline{\sigma}_{x}^{2}+\overline{\sigma}_{y}^{2})+\frac{\lambda_{2}}{6}(\overline{\sigma}_{x}^{2}+4\overline{\sigma}_{y}^{2})-\frac{c}{6}(4\overline{\sigma}_{x}-\sqrt{2}\overline{\sigma}_{y})$
$(m_{s,08})^{2}$ |  $\frac{2\lambda_{1}}{3}(\sqrt{2}\overline{\sigma}_{x}^{2}-\overline{\sigma}_{x}\overline{\sigma}_{y}-\sqrt{2}\overline{\sigma}_{y}^{2})+\sqrt{2}\lambda_{2}(\frac{\overline{\sigma}_{x}^{2}}{2}-\overline{\sigma}_{y}^{2})+\frac{c}{3\sqrt{2}}(\overline{\sigma}_{x}-\sqrt{2}\overline{\sigma}_{y})$ |  $(m_{p,08})^{2}$ |  $\frac{\sqrt{2}\lambda_{2}}{6}(\overline{\sigma}_{x}^{2}-2\overline{\sigma}_{y}^{2})-\frac{c}{6}(\sqrt{2}\overline{\sigma}_{x}-2\overline{\sigma}_{y})$
$(m_{s,xx})^{2}$ |  $m^{2}+3(\lambda_{1}+\frac{\lambda_{2}}{2})\overline{\sigma}_{x}^{2}+\lambda_{1}\overline{\sigma}_{y}^{2}-\frac{c}{\sqrt{2}}\overline{\sigma}_{y}$ |  $(m_{p,xx})^{2}$ |  $m^{2}+(\lambda_{1}+\frac{\lambda_{2}}{2})\overline{\sigma}_{x}^{2}+\lambda_{1}\overline{\sigma}_{y}^{2}+\frac{c}{\sqrt{2}}\overline{\sigma}_{y}$
$(m_{s,yy})^{2}$ |  $m^{2}+\lambda_{1}\overline{\sigma}_{x}^{2}+3(\lambda_{1}+\lambda_{2})\overline{\sigma}_{y}^{2}$ |  $(m_{p,yy})^{2}$ |  $m^{2}+\lambda_{1}\overline{\sigma}_{x}^{2}+(\lambda_{1}+\lambda_{2})\overline{\sigma}_{y}^{2}$
$(m_{s,xy})^{2}$ |  $2\lambda_{1}\overline{\sigma}_{x}\overline{\sigma}_{y}-\frac{c}{\sqrt{2}}\overline{\sigma}_{x}$ |  $(m_{p,xy})^{2}$ |  $\frac{c}{\sqrt{2}}\overline{\sigma}_{x}$
$m_{\sigma}^{2}$ |  $\frac{1}{2}(m_{s,00}^{2}+m_{s,88}^{2})-\frac{1}{2}\sqrt{(m_{s,00}^{2}-m_{s,88}^{2})^{2}+4m_{s,00}^{4}}$ |  $m_{\eta}^{2}$ |  $\frac{1}{2}(m_{p,00}^{2}+m_{p,88}^{2})-\frac{1}{2}\sqrt{(m_{p,00}^{2}-m_{p,88}^{2})^{2}+4m_{p,00}^{4}}$
$m_{f_{0}}^{2}$ |  $\frac{1}{2}(m_{s,00}^{2}+m_{s,88}^{2})+\frac{1}{2}\sqrt{(m_{s,00}^{2}-m_{s,88}^{2})^{2}+4m_{s,00}^{4}}$ |  $m_{\eta^{\prime}}^{2}$ |  $\frac{1}{2}(m_{p,00}^{2}+m_{p,88}^{2})+\frac{1}{2}\sqrt{(m_{p,00}^{2}-m_{p,88}^{2})^{2}\ +4m_{p,00}^{4}}$

The quark/antiquark contribution is given by

$\displaystyle\Omega_{q\bar{q}}(T,\mu;\overline{\sigma}_{x},\overline{\sigma}_{y},\Phi,\bar{\Phi})=\Omega_{q\bar{q}}^{vac}+\Omega_{q\bar{q}}^{T,\mu}\;,$
(33)
$\displaystyle\Omega_{q\bar{q}}^{vac}=-2N_{c}\sum_{q}\int\frac{d^{3}p}{(2\pi)^{3}}E_{q}\theta(\Lambda_{c}^{2}-\vec{p}^{2})\;,$
(34)
$\displaystyle\Omega_{q\bar{q}}^{T,\mu}=-2N_{c}\sum_{q}\int\frac{d^{3}p}{(2\pi)^{3}}T\left[\ln
g_{f}^{+}+\ln g_{f}^{-}\right].\;$ (35)

The first term of the Eq. (33) is the contribution of the fermion vacuum
fluctuation, where the $\Lambda_{c}$ is the ultraviolet cutoff.In presence of
the Polyakov loop potential, the $g^{+}_{f}$ and $g^{-}_{f}$ are specified by
the trace in the color space.

$\displaystyle g^{+}$ $\displaystyle=$ $\displaystyle\left[1+3\Phi
e^{-E_{q}^{+}/T}+3\bar{\Phi}e^{-2E_{q}^{+}/T}+e^{-3E_{q}^{+}/T}\right]\;,\qquad$
(36) $\displaystyle g_{q}^{-}$ $\displaystyle=$
$\displaystyle\left[1+3\bar{\Phi}e^{-E_{q}^{-}/T}+3\Phi
e^{-2E_{q}^{-}/T}+e^{-3E_{q}^{-}/T}\right]\;.$ (37)

$E_{f}^{\pm}=E_{f}\mp\mu_{f}$ and $E_{f}=\sqrt{p^{2}+m{{}_{f}}{{}^{2}}}$ is
the flavor dependent single particle energy of the quark/antiquark,
$m_{u}=m_{d}=\frac{g\overline{\sigma}_{x}}{2}$ is the mass of the light quarks
$u$, $d$ and strange quark mass is
$m_{s}=\frac{g\overline{\sigma}_{y}}{\sqrt{2}}$. For the present work, it is
assumed that $\mu_{u}=\mu_{d}=\mu_{s}=\mu$.

The quark one-loop vacuum term of the Eq. (33) is neglected in the standard
mean-field approximation (s-MFA), and the grand potential of the PQM model is
written as,

$\displaystyle\Omega_{\rm
PQM}(T,\mu,\overline{\sigma}_{x},\overline{\sigma}_{y},\Phi,\bar{\Phi})$
$\displaystyle=$ $\displaystyle
U(\overline{\sigma}_{x},\overline{\sigma}_{y})+\mathcal{U}(\rm
T,\Phi,\bar{\Phi})\;$ (38)
$\displaystyle+\Omega_{q\bar{q}}(T,\mu;\overline{\sigma}_{x},\overline{\sigma}_{y},\Phi,\bar{\Phi})\;.$

The grand minima of the thermodynamic potential given by the Eq. (38) gives
the $\overline{\sigma}_{x}$, $\overline{\sigma}_{y}$, $\Phi$ and $\bar{\Phi}$
by,

$\frac{\partial\Omega_{\rm
PQM}}{\partial\overline{\sigma}_{x}}=\frac{\partial\Omega_{\rm
PQM}}{\partial\overline{\sigma}_{y}}=\frac{\partial\Omega_{\rm
PQM}}{\partial\Phi}=\frac{\partial\Omega_{\rm PQM}}{\partial\bar{\Phi}}=0.$
(39)

###  II.3 Parameters of the QM model

The experimental values of the pion and kaon mass, the average squared mass of
the pseudo-scalars $\eta$ and $\eta^{\prime}$ mesons
($m_{\eta}^{2}+m_{\eta^{\prime}}^{2}$), the scalar $\sigma$ meson mass
$m_{\sigma}$ and the pion and kaon decay constants $f_{\pi}$ and $f_{K}$, are
used as the input Rischke:00 ; Schaefer:09  for determining the six model
parameters $m^{2}$, $\lambda_{1}$, $\lambda_{2}$, $c$, $h_{x}$ and $h_{y}$.

The values of the condensates in the vacuum are
$\overline{\sigma}_{x}=f_{\pi}$ and
$\overline{\sigma}_{y}=(2f_{K}-f_{\pi})/\sqrt{2}$ according to the partially
conserved axial-vector current relation (PCAC). The above values of
condensates at the $T=0,\mu=0$, give the minimum of the vacuum effective
potential in the Eq. (39). The parameters $\lambda_{2}$ and $c$ in the vacuum
are given by,

$\displaystyle\lambda_{2}$ $\displaystyle=$ $\displaystyle\dfrac{2\
}{(\overline{\sigma}_{x}^{2}+4\overline{\sigma}_{y}^{2})(\sqrt{2}\overline{\sigma}_{y}-\overline{\sigma}_{x})}\left[(3\sqrt{2}\overline{\sigma}_{y})m_{K}^{2}-\right.$
(40)
$\displaystyle\left.(\sqrt{2}\overline{\sigma}_{y}+2\overline{\sigma}_{x})m_{\pi}^{2}-(\sqrt{2}\overline{\sigma}_{y}-\overline{\sigma}_{x})(m_{\eta}^{2}+m_{\eta^{{}^{\prime}}}^{2})\right]\;,$
$\displaystyle c$ $\displaystyle=$
$\displaystyle\dfrac{2(m^{2}_{K}-m^{2}_{\pi})}{(\sqrt{2}\overline{\sigma}_{y}-\overline{\sigma}_{x})}-\sqrt{2}\
\overline{\sigma}_{y}\ \lambda_{2}\;.$ (41)

The mass parameter $m^{2}$ gets cancelled in the difference of the $\sigma$
and $\pi$ mass squares $(m_{\sigma}^{2}-m_{\pi}^{2})$. The difference
$(m_{\sigma}^{2}-m_{\pi}^{2})$, depends on the parameters $\lambda_{1}$,
$\lambda_{2}$ and $c$. When the $\lambda_{2}$ and $c$ calculated from the
above equations are put into the expression of $(m_{\sigma}^{2}-m_{\pi}^{2})$
with the $\overline{\sigma}_{x}=f_{\pi}$ and
$\overline{\sigma}_{y}=(2f_{K}-f_{\pi})/\sqrt{2}$, one gets the vacuum value
of the parameter $\lambda_{1}$. The mass parameter $m^{2}$ can be obtained
from the expression of the $m_{\pi}^{2}$ as the following.

$\displaystyle m^{2}$ $\displaystyle=$ $\displaystyle
m^{2}_{\pi}-\lambda_{1}(\overline{\sigma}_{x}^{2}+\overline{\sigma}_{y}^{2})-\frac{\lambda_{2}}{2}\overline{\sigma}_{x}^{2}+\frac{c}{\sqrt{2}}\overline{\sigma}_{y}\;.$
(42)

Putting the vacuum values of $m_{\pi}^{2}$, $\lambda_{1}$, $\lambda_{2}$, $c$,
$\overline{\sigma}_{x}$ and $\overline{\sigma}_{y}$ in the Eq. (42), one gets
the value of the mass parameter $m^{2}$. Putting the $\overline{\sigma}_{x}$
and $\overline{\sigma}_{y}$ values in the Eq. (31), one finds

$\displaystyle h_{x}$ $\displaystyle=$ $\displaystyle f_{\pi}m^{2}_{\pi}\
\text{and}\ h_{y}=\sqrt{2}f_{K}\ m_{K}^{2}-\frac{1}{\sqrt{2}}f_{\pi}\
m_{\pi}^{2}\;.$ (43)

The Yukawa coupling gets fixed from the non-strange constituent quark mass as
$g=\frac{2m_{u}}{f_{\pi}}$. For the $f_{\pi}=92.4$ MeV and the $m_{u}\sim
300.3$ MeV, one gets the $g\sim 6.5$. The strange quark constituent mass
becomes $m_{s}\sim 334.34$ MeV. The experimental value of $m_{\eta}=547.5$ MeV
and $m_{\eta^{\prime}}=957.78$ MeV. In Ref. Schaefer:09 , the parameter
$\lambda_{2}$ is determined by taking the $m_{\eta}=539$ MeV and
$m_{\eta^{\prime}}=963$ MeV as input because the sum of the squared masses
$m_{\eta}^{2}+m_{\eta^{\prime}}^{2}=(539)^{2}+(963)^{2}$ is almost equal to
the $(547.5)^{2}+(957.78)^{2}$ and the calculated parameters reproduce
$m_{\eta}=539$ MeV and $m_{\eta^{\prime}}=963$ MeV in the output.

##  III Renormalized Polyakov Quark Meson Model

Several of the recent QM/PQM model investigations with quark one-loop vacuum
correction lars ; guptiw ; schafwag12 ; chatmoh1 ; vkkr12 ; TranAnd ; chatmoh2
; vkkt13 ; Herbst ; Weyrich ; kovacs ; zacchi1 ; zacchi2 ; Rai  have fixed the
model parameters by the use of the curvature (or screening) masses of the
$\pi,K,\eta,\eta^{\prime}$ and $\sigma$ mesons where the pion decay constant
has been identified as the vacuum expectation value of non-strange condensate
and the vacuum strange condensate gets related to the pion and kaon decay
constant. However, it is well known that the physical masses are given by the
poles of the meson propagators and the residue of the pion propagator at its
pole gets related to the pion decay constant BubaCar ; Naylor ; fix1 .
Furthermore, the definition of the curvature mass of the meson, involves the
evaluation of its self-energy at the zero momentum laine ; Adhiand1 ; Adhiand2
; Adhiand3  as one knowns that the effective potential is the generator of the
n-point functions of the theory at zero external momenta. Also one notes that
the pole definition is the physical and gauge invariant one Kobes ; Rebhan .
In the absence of the Dirac sea contributions, the pole mass becomes
equivalent to the curvature mass for the model parameter fixing but when the
quark one-loop vacuum correction is present, the pole masses of the mesons
become different from their screening masses BubaCar ; fix1 . In view of the
above considerations, it becomes important to use of the exact on-shell
parameter fixing method for the renormalized Polyakov enhanced quark meson
(RPQM) model where the physical (pole) masses of the mesons, the pion and kaon
decay constants are put into the relation of the running mass parameter and
couplings by using the on-shell and the minimal subtraction renormalization
prescriptions Adhiand2 ; asmuAnd ; RaiTiw . In a very recent work vkkr22 , one
of us has calculated the consistent effective potential and the on-shell
renormalized parameters for the 2+1 flavor quark-meson (QM) where the quark
one-loop vacuum fluctuation is properly renormalized. The above calculation of
the exact effective potential and the on-shell renormalization of the seven
parameters $m^{2},\ \lambda_{1},\ \lambda_{2},\ c,\ h_{x},\ h_{y}$ for the 2+1
flavor RPQM model are presented and reproduced below.

###  III.1 Description of the counterterms and self-energies

The tree level parameters of the Eqs. (40)–(43) become inconsistent after
including the quark one-loop vacuum correction unless the on-shell
renormalization scheme is used. In the on-shell scheme, one uses the
dimensional regularization to regularize the divergent loop integrals but the
counterterm are chosen differently from the minimal subtraction scheme. The
appropriate choice of counterterms in the on-shell scheme enables the exact
cancellation of the loop corrections to the self-energies. The renormalized
parameters become renormalization scale independent as the couplings are
evaluated on-shell. The wave functions/fields and parameters of the Eq. (4)
are bare quantities.

The counterterms $\delta Z_{\pi}$, $\delta Z_{K}$, $\delta Z_{\sigma}$,
$\delta Z_{\eta}$, $\delta Z_{\eta}^{\prime}$, $\delta Z_{\psi}$ $\delta
Z_{\overline{\sigma}_{x}}$ and $\delta Z_{\overline{\sigma}_{y}}$ for the wave
functions/fields and the counterterms $\delta m^{2}$, $\delta\lambda_{1}$,
$\delta\lambda_{2}$, $\delta c$, $\delta h_{x}$, $\delta h_{y}$ and $\delta
g^{2}$, for the parameters, are introduced in the Lagrangian (4) where the
couplings and renormalized fields are defined as,

$\displaystyle\pi^{i}_{b}$ $\displaystyle=$ $\displaystyle\sqrt{Z_{\pi}}\
\pi^{i},\ K_{b}=\sqrt{Z_{K}}\ K,\ \eta_{b}=\sqrt{Z_{\eta}}\ \eta,$ (44)
$\displaystyle\eta^{\prime}_{b}$ $\displaystyle=$
$\displaystyle\sqrt{Z_{\eta}^{\prime}}\ \eta^{\prime},\
\sigma_{b}=\sqrt{Z_{\sigma}}\ \sigma,\ m^{2}_{b}=Z_{m}\ m^{2}$ (45)
$\displaystyle\psi_{b}$ $\displaystyle=$ $\displaystyle\sqrt{Z_{\psi}}\ \psi,\
\lambda_{1b}=Z_{\lambda_{1}}\ \lambda_{1},\ \lambda_{2b}=Z_{\lambda_{2}}\
\lambda_{2},$ (46) $\displaystyle\ g_{b}$ $\displaystyle=$
$\displaystyle\sqrt{Z_{g}}\ g,\ h_{xb}=Z_{h_{x}}\ h_{x},\ h_{yb}=Z_{h_{y}}\
h_{y},$ (47) $\displaystyle c_{b}$ $\displaystyle=$ $\displaystyle Z_{c}\ c,\
{\overline{\sigma}_{x}}_{b}=\sqrt{Z_{\overline{\sigma}_{x}}}\
\overline{\sigma}_{x},\
{\overline{\sigma}_{y}}_{b}=\sqrt{Z_{\overline{\sigma}_{y}}}\
\overline{\sigma}_{y}\;.$ (48)

Here the
$Z_{(\pi,K,\eta,\eta^{\prime},\sigma,\psi,\overline{\sigma}_{x},\overline{\sigma}_{y})}=1+\delta
Z_{(\pi,K,\eta,\eta^{\prime},\sigma,\psi,\overline{\sigma}_{x},\overline{\sigma}_{y})}$,
identify the field strength renormalization constants while
$Z_{(m,\lambda_{1},\lambda_{2},g,h_{x},h_{y},c)}=1+\delta
Z_{(m,\lambda_{1},\lambda_{2},g,h_{x},h_{y},c)}$ signify the mass and coupling
renormalization constants. One-loop corrections to the quark fields and the
quark masses are zero because in the large $N_{c}$ limit, the $\pi$ and
$\sigma$ loops that may renormalize the quark propagators are of the order
$N_{c}^{0}$. Hence the $Z_{\psi}=1$ and the respective quark self energy
corrections for the non-strange and the strange quarks are $\delta m_{u}=0$
and $\delta m_{s}=0$. Since, the pion-quark $\pi\overline{\psi}\psi$ vertex is
of order $N_{c}^{0}$, the one-loop correction gets neglected. As a result, one
gets $Z_{\psi}\ \sqrt{Z_{g}\ g^{2}}\sqrt{Z_{\pi}}\approx
g(1+\frac{1}{2}\frac{\delta g^{2}}{g^{2}}+\frac{1}{2}\delta Z_{\pi})=g$. Thus
$\frac{\delta g^{2}}{g^{2}}\ +\delta Z_{\pi}=0$. The $\delta m_{u}=0$ and
$\delta m_{s}=0$ implies that $\delta g\ \overline{\sigma}_{x}/2+g\
\delta\overline{\sigma}_{x}/2=0$ and $\delta g\
\overline{\sigma}_{y}/\sqrt{2}+g\ \delta\overline{\sigma}_{y}/\sqrt{2}=$ 0\.
One gets
$\delta\overline{\sigma}_{x}/\overline{\sigma}_{x}=\delta\overline{\sigma}_{y}/\overline{\sigma}_{y}=-\delta
g/g$ which can be written as

$\displaystyle\frac{\delta\overline{\sigma}_{x}^{2}}{\overline{\sigma}_{x}^{2}}$
$\displaystyle=$
$\displaystyle\frac{\delta\overline{\sigma}_{y}^{2}}{\overline{\sigma}_{y}^{2}}=-\frac{\delta
g^{2}}{g^{2}}=\delta Z_{\pi}\;.$ (49)

Following the Refs. Adhiand1 ; Adhiand2 ; Adhiand3 ; asmuAnd ; RaiTiw ; vkkr22
and using the Eqs. (44)-(48) with the Eqs. (40) and (41), the counterterm
$\delta\lambda_{2}$ can be expressed in terms of the counterterms $\delta
m^{2}_{\pi}$, $\delta m^{2}_{K}$, $\delta m^{2}_{\eta}$ , $\delta
m^{2}_{\eta^{\prime}}$ and $\delta Z_{\pi}$ while the $\delta c$ is expressed
in terms of the $\delta m^{2}_{\pi}$, $\delta m^{2}_{K}$, $\delta Z_{\pi}$ and
the preceding $\delta\lambda_{2}$. One gets the following expressions for the
$\delta\lambda_{2}$ and $\delta c$.

$\displaystyle\delta\lambda_{2}$ $\displaystyle=$
$\displaystyle\frac{2}{(\overline{\sigma}_{x}^{2}+4\overline{\sigma}_{y}^{2})(\sqrt{2}\
\overline{\sigma}_{y}-\overline{\sigma}_{x})}\biggl{[}(3\sqrt{2}\overline{\sigma}_{y})\delta
m_{K}^{2}-(\sqrt{2}\overline{\sigma}_{y}+2\overline{\sigma}_{x})\;$ (50)
$\displaystyle\delta
m_{\pi}^{2}-(\sqrt{2}\overline{\sigma}_{y}-\overline{\sigma}_{x})(\delta
m_{\eta}^{2}+\delta m_{\eta^{{}^{\prime}}}^{2})\biggr{]}-\lambda_{2}\delta
Z_{\pi}\;,$

$\displaystyle\delta c$ $\displaystyle=$ $\displaystyle\dfrac{2(\delta
m^{2}_{K}-\delta m^{2}_{\pi})}{(\sqrt{2}\
\overline{\sigma}_{y}-\overline{\sigma}_{x})}-\sqrt{2}\ \overline{\sigma}_{y}\
\delta\lambda_{2}\;$ (51) $\displaystyle-(2\sqrt{2}\ \overline{\sigma}_{y}\
\lambda_{2}+c)\ \frac{\delta Z_{\pi}}{2}\;.$

Knowing the $\delta\lambda_{2}$ and $\delta c$, and using the expression of
($\delta m_{\sigma}^{2}-\delta m_{\pi}^{2}$), the counterterm
$\delta\lambda_{1}$ can be written after some algebraic manipulations as the
following.

$\displaystyle\delta\lambda_{1}=\frac{\delta\lambda_{1\text{{\tiny
NUMI}}}}{\lambda_{1\text{{\tiny DENOM}}}}-\lambda_{1}\ \delta Z_{\pi};\;$ (52)
$\displaystyle\lambda_{1\text{{\tiny
DENOM}}}=\biggl{(}\sqrt{(m^{2}_{s,00}-m^{2}_{s,88})^{2}+4m^{4}_{s,08}}\biggr{)}\
(\overline{\sigma}_{x}^{2}+\overline{\sigma}_{y}^{2})-\frac{(m^{2}_{s,00}-m^{2}_{s,88})}{3}(\overline{\sigma}_{x}^{2}+4\sqrt{2}\overline{\sigma}_{x}\overline{\sigma}_{y}-\overline{\sigma}_{y}^{2})\;$
$\displaystyle\qquad\quad\quad-\frac{4m^{2}_{s,08}}{3}\
(\sqrt{2}\overline{\sigma}_{x}^{2}-\overline{\sigma}_{x}\overline{\sigma}_{y}-\sqrt{2}\overline{\sigma}_{y}^{2})\;$
(53) $\displaystyle\delta\lambda_{1\text{{\tiny
NUMI}}}=\sqrt{(m^{2}_{s,00}-m^{2}_{s,88})^{2}+4m^{4}_{s,08}}\biggl{(}\delta
m_{\sigma}^{2}-\delta
m_{\pi}^{2}\biggr{)}-\biggl{\\{}\delta\lambda_{2}\frac{(\overline{\sigma}_{x}^{2}+6\overline{\sigma}_{y}^{2})}{4}+\delta
c\frac{\sqrt{2}\
\overline{\sigma}_{y}}{4}\biggr{\\}}\sqrt{(m^{2}_{s,00}-m^{2}_{s,88})^{2}+4m^{4}_{s,08}}\;$
$\displaystyle\qquad\quad+\biggl{\\{}\delta\lambda_{2}\frac{(\overline{\sigma}_{x}^{2}-2\overline{\sigma}_{y}^{2})}{4}-\delta
c\frac{\sqrt{2}\
(4\sqrt{2}\overline{\sigma}_{x}+\overline{\sigma}_{y})}{12}\biggr{\\}}(m^{2}_{s,00}-m^{2}_{s,88})+\biggl{\\{}\delta\lambda_{2}\sqrt{2}(\overline{\sigma}_{x}^{2}-2\overline{\sigma}_{y}^{2})+\delta
c\frac{\sqrt{2}\
(\overline{\sigma}_{x}-\sqrt{2}\overline{\sigma}_{y})}{3}\biggr{\\}}m^{2}_{s,08}\;$
$\displaystyle\qquad\quad-\lambda_{2}\ \delta
Z_{\pi}\Biggl{\\{}\frac{(\overline{\sigma}_{x}^{2}+6\overline{\sigma}_{y}^{2})}{4}\sqrt{(m^{2}_{s,00}-m^{2}_{s,88})^{2}+4m^{4}_{s,08}}-\frac{(\overline{\sigma}_{x}^{2}-2\overline{\sigma}_{y}^{2})}{4}(m^{2}_{s,00}-m^{2}_{s,88})-\sqrt{2}(\overline{\sigma}_{x}^{2}-2\overline{\sigma}_{y}^{2})m^{2}_{s,08}\Biggr{\\}}\;$
$\displaystyle\qquad\quad-c\frac{\delta
Z_{\pi}}{2}\Biggl{\\{}\frac{\sqrt{2}\overline{\sigma}_{y}}{4}\sqrt{(m^{2}_{s,00}-m^{2}_{s,88})^{2}+4m^{4}_{s,08}}+\frac{\sqrt{2}\
(4\sqrt{2}\overline{\sigma}_{x}+\overline{\sigma}_{y})}{12}(m^{2}_{s,00}-m^{2}_{s,88})+\frac{\sqrt{2}\
(\sqrt{2}\overline{\sigma}_{y}-\overline{\sigma}_{x})}{3}m^{2}_{s,08}\Biggr{\\}}\;$
(54) The counterterm $\delta m^{2}$ is written in terms of the $\delta
m_{\pi}^{2}$, $\delta\lambda_{1}$, $\delta\lambda_{2}$, $\delta c$ and $\delta
Z_{\pi}$ $\displaystyle\delta m^{2}=\delta m^{2}_{\pi}-\delta\lambda_{1}\
(\overline{\sigma}_{x}^{2}+\overline{\sigma}_{y}^{2})-\
\frac{(\delta\lambda_{2})\ \overline{\sigma}_{x}^{2}}{2}+\frac{\delta c\
\overline{\sigma}_{y}}{\sqrt{2}}-\delta Z_{\pi}\biggl{\\{}\lambda_{1}\
(\overline{\sigma}_{x}^{2}+\overline{\sigma}_{y}^{2})+\ \frac{\lambda_{2}\
\overline{\sigma}_{x}^{2}}{2}-\frac{c\
\overline{\sigma}_{y}}{2\sqrt{2}}\biggr{\\}}\;.$ (55)

The Feynman diagrams of the self energy and tadpole contributions for the
scalar particles are presented in the Fig. (1(a)) while the Fig. (1(b))
presents the corresponding counter term diagrams. The Fig. (2(a)) presents the
Feynman diagrams of the self energy and tadpol contributions for the pseudo-
scalar particles and the corresponding diagrams for the counter terms are
shown in the Fig. (2(b)). The on-shell parameter fixing requires, the self
energies of the scalar sigma $\sigma$, pseudo-scalar eta ($\eta$), eta-prime
($\eta^{\prime}$), pion ($\pi$) and kaon ($K$). The self energy correction of
the scalar $\sigma$ is calculated in terms of the self energy corrections
$\Sigma_{\text{s},00}(p^{2})$, $\Sigma_{\text{s},88}(p^{2})$ and
$\Sigma_{\text{s},08}(p^{2})$ while the pseudo-scalar $\eta$ and
$\eta^{\prime}$ self energy corrections are found in terms of the self energy
corrections $\Sigma_{\text{p},00}(p^{2})$, $\Sigma_{\text{p},88}(p^{2})$ and
$\Sigma_{\text{p},08}(p^{2})$. The scalar and pseudo-scalar self energies
expressions are given below.

$\displaystyle\Sigma_{\text{s},00}(p^{2})=-\frac{2}{3}N_{c}g^{2}\left[2\mathcal{A}(m^{2}_{u})-(p^{2}-4m^{2}_{u})\mathcal{B}(p^{2},m_{u})\right]-\frac{1}{3}N_{c}g^{2}\left[2\mathcal{A}(m^{2}_{s})-(p^{2}-4m^{2}_{s})\mathcal{B}(p^{2},m_{s})\right]+\Sigma^{tad}_{\text{s},00}\quad\;,$
(56)
$\displaystyle\Sigma_{\text{s},11}(p^{2})=-N_{c}g^{2}\left[2\mathcal{A}(m^{2}_{u})-(p^{2}-4m^{2}_{u})\mathcal{B}(p^{2},m_{u})\right]+\Sigma^{tad}_{\text{s},11}\;,$
(57)
$\displaystyle\Sigma_{\text{s},44}(p^{2})=-N_{c}g^{2}\left[\mathcal{A}(m^{2}_{u})+\mathcal{A}(m^{2}_{s})-(p^{2}-(m_{u}+m_{s})^{2})\mathcal{B}(p^{2},m_{u},m_{s})\right]+\Sigma^{tad}_{\text{
s},44}\;,$ (58)
$\displaystyle\Sigma_{\text{s},88}(p^{2})=-\frac{1}{3}N_{c}g^{2}\left[2\mathcal{A}(m^{2}_{u})-(p^{2}-4m^{2}_{u})\mathcal{B}(p^{2},m_{u})\right]-\frac{2}{3}N_{c}g^{2}\left[2\mathcal{A}(m^{2}_{s})-(p^{2}-4m^{2}_{s})\mathcal{B}(p^{2},m_{s})\right]+\Sigma^{tad}_{\text{s},88}\;,$
(59)
$\displaystyle\Sigma_{\text{s},08}(p^{2})=-\frac{\sqrt{2}}{3}N_{c}g^{2}\left[2\mathcal{A}(m^{2}_{u})-(p^{2}-4m^{2}_{u})\mathcal{B}(p^{2},m_{u})\right]+\frac{\sqrt{2}}{3}N_{c}g^{2}\left[2\mathcal{A}(m^{2}_{s})-(p^{2}-4m^{2}_{s})\mathcal{B}(p^{2},m_{s})\right]+\Sigma^{tad}_{\text{s},08}\;,$
(60)

(a) One-loop self energy and tadpole diagrams.

(b) One-loop self energy and tadpole counterterm diagrams.

Figure 1: The solid line represents scalar particles and an arrow on the solid
line denotes a quark.

(a) One-loop self energy and tadpole diagrams.

(b) One-loop self energy and tadpole counterterm diagrams.

Figure 2: The dash line represents pseudo-scalar particles and an arrow on the
solid line denotes a quark.

$\displaystyle\Sigma_{\text{p},00}(p^{2})$ $\displaystyle=$
$\displaystyle-\frac{2}{3}N_{c}g^{2}\left[2\mathcal{A}(m^{2}_{u})-p^{2}\mathcal{B}(p^{2},m_{u})\right]-\frac{1}{3}N_{c}g^{2}\left[2\mathcal{A}(m^{2}_{s})-p^{2}\mathcal{B}(p^{2},m_{s})\right]+\Sigma^{tad}_{\text{p},00}\;,$
(61) $\displaystyle\Sigma_{\text{p},11}(p^{2})$ $\displaystyle=$
$\displaystyle\Sigma_{\pi}(p^{2})=-N_{c}g^{2}\left[2\mathcal{A}(m^{2}_{u})-p^{2}\mathcal{B}(p^{2},m_{u})\right]+\Sigma^{tad}_{\text{
p},11}\;,$ (62) $\displaystyle\Sigma_{\text{ p},44}(p^{2})$ $\displaystyle=$
$\displaystyle\Sigma_{K}(p^{2})=-N_{c}g^{2}\left[\mathcal{A}(m^{2}_{u})+\mathcal{A}(m^{2}_{s})-\left\\{p^{2}-(m_{u}-m_{s})^{2}\right\\}\mathcal{B}(p^{2},m_{u},m_{s})\right]+\Sigma^{tad}_{\text{p},44}\;,$
(63) $\displaystyle\Sigma_{\text{p},88}(p^{2})$ $\displaystyle=$
$\displaystyle-\frac{1}{3}N_{c}g^{2}\left[2\mathcal{A}(m^{2}_{u})-p^{2}\mathcal{B}(p^{2},m_{u})\right]-\frac{2}{3}N_{c}g^{2}\left[2\mathcal{A}(m^{2}_{s})-p^{2}\mathcal{B}(p^{2},m_{s})\right]+\Sigma^{tad}_{\text{
p},88}\;,$ (64) $\displaystyle\Sigma_{\text{ p},08}(p^{2})$ $\displaystyle=$
$\displaystyle-\frac{\sqrt{2}}{3}N_{c}g^{2}\left[2\mathcal{A}(m^{2}_{u})-p^{2}\mathcal{B}(p^{2},m_{u})\right]+\frac{\sqrt{2}}{3}N_{c}g^{2}\left[2\mathcal{A}(m^{2}_{s})-p^{2}\mathcal{B}(p^{2},m_{s})\right]+\Sigma^{tad}_{\text{
p},08}\;.$ (65)

Figure 3: One point diagram for the nonstrange scalar and its counterterm.
Figure 4: One point diagram for the strange scalar and its counterterm.

The Fig. (3) shows the diagram of the one-point function and its counterterm
for the quark one-loop correction to the non-strange component of the scalar
$\sigma$. One can write it as,

$\displaystyle\delta\Gamma_{x}^{(1)}=-4\ N_{c}\ g\ m_{u}\
\mathcal{A}(m_{u}^{2})+i\delta t_{x}\;.$ (66)

The Fig. (4) depicts the diagram of the one-point function and its counter-
term for the quark one-loop correction to the strange component of the scalar
$\sigma$. It is written as,

$\displaystyle\delta\Gamma_{y}^{(1)}=-2\sqrt{2}\ N_{c}\ g\ m_{s}\
\mathcal{A}(m_{s}^{2})+i\delta t_{y}\;.$ (67)

###  III.2 Renormalization of the parameters

The respective one-point functions
$\Gamma^{(1)}_{x}=it_{x}=i(h_{x}-m_{\pi}^{2}\overline{\sigma}_{x})$ and
$\Gamma^{(1)}_{y}=it_{y}=i{\tiny{\biggl{\\{}h_{y}-\frac{\sqrt{2}}{2}(m_{K}^{2}-m_{\pi}^{2})\
\overline{\sigma}_{x}-m_{K}^{2}\overline{\sigma}_{y}\biggr{\\}}}}$ for the
non-strange and the strange degree of freedom become zero and one gets two
tree level equations of motion $t_{x}=0$ and $t_{y}=0$. Hence the classical
minimum of the effective potential gets fixed. The first renormalization
condition for the non-strange $<\overline{\sigma}_{x}>=0$ and the strange
degree of freedom $<\overline{\sigma}_{y}>=0$ demands that the respective one-
loop corrections $\delta\Gamma^{(1)}_{x}$ and $\delta\Gamma^{(1)}_{y}$ to the
one point functions, are put to zero such that the minimum of the effective
potential does not change. Thus the $\delta\Gamma^{(1)}_{x}=0$ and
$\delta\Gamma^{(1)}_{y}=0$ give us

$\displaystyle\delta t_{x}$ $\displaystyle=$ $\displaystyle-4i\ N_{c}\ g\
m_{u}\ \mathcal{A}(m_{u}^{2})\;,$ (68) $\displaystyle\delta t_{y}$
$\displaystyle=$ $\displaystyle-2\sqrt{2}i\ N_{c}\ g\ m_{s}\
\mathcal{A}(m_{s}^{2})\;.$ (69)

The equation $h_{x}=t_{x}+m_{\pi}^{2}\ \overline{\sigma}_{x}$ and
$h_{y}=t_{y}+{\tiny{\biggl{\\{}\frac{\sqrt{2}}{2}(m_{K}^{2}-m_{\pi}^{2})\
\overline{\sigma}_{x}+m_{K}^{2}\ \overline{\sigma}_{y}\biggr{\\}}}}$, are used
to write the counter-terms $\delta h_{x}$ and $\delta h_{y}$ in terms of the
corresponding tadpole counter-terms $\delta t_{x}$ and $\delta t_{y}$ as the
following

$\displaystyle{\hskip-56.9055pt}\delta h_{x}$ $\displaystyle=$ $\displaystyle
m^{2}_{\pi}\ \delta\overline{\sigma}_{x}+\delta m^{2}_{\pi}\
\overline{\sigma}_{x}+\delta t_{x},\qquad\qquad\qquad\qquad\;$ (70)
$\displaystyle{\hskip-56.9055pt}\delta h_{y}$ $\displaystyle=$
$\displaystyle{\biggl{\\{}\frac{\sqrt{2}}{2}(m_{K}^{2}-m_{\pi}^{2})\
\delta\overline{\sigma}_{x}+\frac{\sqrt{2}}{2}(\delta m_{K}^{2}-\delta
m_{\pi}^{2})\ \overline{\sigma}_{x}\biggr{\\}}}$ (71)
$\displaystyle+m_{K}^{2}\ \delta\overline{\sigma}_{y}+\delta m_{K}^{2}\
\overline{\sigma}_{y}+\delta t_{y}\;.$

Using the Eq. (49), one writes

$\displaystyle\delta h_{x}$ $\displaystyle=$
$\displaystyle\frac{1}{2}m^{2}_{\pi}\ \overline{\sigma}_{x}\ \delta
Z_{\pi}+\delta m^{2}_{\pi}\ \overline{\sigma}_{x}+\delta
t_{x},\qquad\qquad\qquad\qquad\;$ (72) $\displaystyle{\hskip-42.67912pt}\delta
h_{y}$ $\displaystyle=$
$\displaystyle{\biggl{\\{}\frac{\sqrt{2}}{2}(m_{K}^{2}-m_{\pi}^{2})\
\overline{\sigma}_{x}\ \frac{\delta Z_{\pi}}{2}+\frac{\sqrt{2}}{2}(\delta
m_{K}^{2}-\delta m_{\pi}^{2})\ \overline{\sigma}_{x}\biggr{\\}}}$ (73)
$\displaystyle+m_{K}^{2}\ \overline{\sigma}_{y}\ \frac{\delta
Z_{\pi}}{2}+\delta m_{K}^{2}\ \overline{\sigma}_{y}+\delta t_{y}.$

The pseudo-scalar $\pi$ and $K$ meson inverse propagators are defined as

$\displaystyle
p^{2}-m_{\pi,K}^{2}-i\Sigma_{\pi,K}(p^{2}){\rm+counterterms}\;.$ (74)

The physical states of the scalar $\sigma$ and $f_{0}$ and the pseudo-scalar
$\eta$ and $\eta^{\prime}$, are obtained by the mixing of the $00$ and $88$
components for the scalar (s) and pseudo-scalar (p) particles. The inverse
propagator is given by the $2\times 2$ matrix which results due to the mixing
of the $00$ and $88$ components. Solving the equation that one gets after
putting the determinant of the matrix equal to zero, the negative root of the
solution gives the inverse propagator of the physical $\sigma$ in the scalar
and the physical $\eta$ in the pseudo-scalar channel. The positive root of the
solution gives the inverse propagator of the physically observed particles
$f_{0}$ and $\eta^{\prime}$ in the respective scalar and the pseudo-scalar
channel.

$\textnormal{Det}\left.\begin{pmatrix}p^{2}-m^{2}_{\text{s(p)},00}-i\Sigma_{\text{s(p)},00}(p^{2})\,\,&-m^{2}_{\text{s(p)},08}-i\Sigma_{\text{s(p)},08}(p^{2})\\\
-m^{2}_{\text{s(p)},08}-i\Sigma_{\text{s(p)},08}(p^{2})&p^{2}-m^{2}_{\text{s(p)},88}-i\Sigma_{\text{s(p)},88}(p^{2})\,\,\end{pmatrix}\right.=0\;.$
(75)

One gets two solutions for the $p^{2}$

$\displaystyle
p^{2}=\frac{1}{2}\Biggl{[}{\tiny{\Biggl{(}\biggl{\\{}m^{2}_{\text{s(p)},00}+i\Sigma_{\text{s(p)},00}(p^{2})\biggr{\\}}+\biggl{\\{}m^{2}_{\text{s(p)},88}+i\Sigma_{\text{s(p)},88}(p^{2})\biggr{\\}}\Biggr{)}}}\mp$
$\displaystyle\qquad{\tiny{\sqrt{\Biggl{(}\biggl{\\{}m^{2}_{\text{s(p)},00}+i\Sigma_{\text{s(p)},00}(p^{2})\biggr{\\}}-\biggl{\\{}m^{2}_{\text{s(p)},88}+i\Sigma_{\text{s(p)},88}(p^{2})\biggr{\\}}\Biggr{)}^{2}+4\Biggl{(}m^{2}_{\text{s(p)},08}+i\Sigma_{\text{s(p)},08}(p^{2})\Biggr{)}^{2}}}}\Biggr{]}\;.$
(76) Neglecting the higher order ($N_{c}^{2}$) terms like
$\tiny{\biggl{\\{}\Sigma_{\text{s(p)},00}(p^{2}))-\Sigma_{\text{s(p)},88}(p^{2})\biggr{\\}}^{2}}$
and $\Sigma_{\text{s(p)},08}^{2}(p^{2})$ in self energy corrections, the above
expression is written as $\displaystyle
p^{2}=\frac{1}{2}\Biggl{[}\biggl{(}m^{2}_{\text{s(p)},00}+m^{2}_{\text{s(p)},88}\biggr{)}\mp{\tiny{\sqrt{\Biggl{(}m^{2}_{\text{s(p)},00}-m^{2}_{\text{s(p)},88}\Biggr{)}^{2}+4\Biggl{(}m^{2}_{\text{s(p)},08}\Biggr{)}^{2}}}}\Biggr{]}+\frac{1}{2}\Biggl{[}\Biggl{(}i\Sigma_{\text{s(p)},00}(p^{2})+i\Sigma_{\text{s(p)},88}(p^{2})\Biggr{)}\mp$
$\displaystyle{\hskip-2.84544pt}\frac{1}{\sqrt{(m^{2}_{\text{s(p)},00}-m^{2}_{\text{s(p)},88})^{2}+4m^{4}_{\text{s(p)},08}}}\biggl{\\{}\biggl{(}i\Sigma_{\text{s(p)},00}(p^{2})-i\Sigma_{\text{s(p)},88}(p^{2})\biggr{)}(m^{2}_{\text{s(p)},00}-m^{2}_{\text{s(p)},88})+4i\Sigma_{\text{s(p)},08}(p^{2})\
m^{2}_{\text{s(p)},08}\biggr{\\}}\Biggr{]}\;.$ Negative root of the Eq.
(III.2) is equal to the sum of the mass and self energy correction for the
scalar $\sigma$ (pseudoscalar $\eta$) $\displaystyle
p^{2}=m_{\sigma(\eta)}^{2}+i\Sigma_{\sigma(\eta)}(p^{2})\ \text{where}\
m_{\sigma(\eta)}^{2}=\frac{1}{2}\Biggl{[}\biggl{(}m^{2}_{\text{s(p)},00}+m^{2}_{\text{s(p)},88}\biggr{)}-{\tiny{\sqrt{\Biggl{(}m^{2}_{\text{s(p)},00}-m^{2}_{\text{s(p)},88}\Biggr{)}^{2}+4\Biggl{(}m^{2}_{\text{s(p)},08}\Biggr{)}^{2}}}}\Biggr{]}\
\text{and}$
$\displaystyle\Sigma_{\sigma(\eta)}(p^{2})=\frac{1}{2}\Biggl{[}\Sigma_{\text{s(p)},00}(p^{2})+\Sigma_{\text{s(p)},88}(p^{2})-\frac{1}{\sqrt{(m^{2}_{\text{s(p)},00}-m^{2}_{\text{s(p)},88})^{2}+4m^{4}_{\text{s(p)},08}}}$
$\displaystyle\qquad\quad\qquad\quad\biggl{\\{}\biggl{(}\Sigma_{\text{s(p)},00}(p^{2})-\Sigma_{\text{s(p)},88}(p^{2})\biggr{)}(m^{2}_{\text{s(p)},00}-m^{2}_{\text{s(p)},88})+4\Sigma_{\text{s(p)},08}(p^{2})\
m^{2}_{\text{s(p)},08}\biggr{\\}}\Biggr{]}\;.$ (78)

Positive root of the Eq. (III.2) is equal to the sum of the mass and self
energy correction for the scalar $f_{0}$ (pseudoscalar $\eta^{\prime}$)

$\displaystyle
p^{2}=m_{f_{0}(\eta^{\prime})}^{2}+i\Sigma_{f_{0}(\eta^{\prime})}(p^{2})\
\text{where}\
m_{f_{0}(\eta^{\prime})}^{2}=\frac{1}{2}\Biggl{[}\biggl{(}m^{2}_{\text{s(p)},00}+m^{2}_{\text{s(p)},88}\biggr{)}+{\tiny{\sqrt{\Biggl{(}m^{2}_{\text{s(p)},00}-m^{2}_{\text{s(p)},88}\Biggr{)}^{2}+4\Biggl{(}m^{2}_{\text{s(p)},08}\Biggr{)}^{2}}}}\Biggr{]}\
\text{and}$
$\displaystyle\Sigma_{f_{0}(\eta^{\prime})}(p^{2})=\frac{1}{2}\Biggl{[}\Sigma_{\text{s(p)},00}(p^{2})+\Sigma_{\text{s(p)},88}(p^{2})+\frac{1}{\sqrt{(m^{2}_{\text{s(p)},00}-m^{2}_{\text{s(p)},88})^{2}+4m^{4}_{\text{s(p)},08}}}$
$\displaystyle\qquad\quad\qquad\quad\biggl{\\{}\biggl{(}\Sigma_{\text{s(p)},00}(p^{2})-\Sigma_{\text{s(p)},88}(p^{2})\biggr{)}(m^{2}_{\text{s(p)},00}-m^{2}_{\text{s(p)},88})+4\Sigma_{\text{s(p)},08}(p^{2})\
m^{2}_{\text{s(p)},08}\biggr{\\}}\Biggr{]}\;.$ (79)

One can write the inverse propagator for the scalar $\sigma$ and the pseudo-
scalar $\eta,\ \eta^{\prime}$ mesons as

$\displaystyle
p^{2}-m_{\sigma,\eta,\eta^{\prime}}^{2}-i\Sigma_{\sigma,\eta,\eta^{\prime}}(p^{2}){\rm+counterterms}\;.$
(80)

In the Lagrangian, the renormalized mass is made equal to the physical mass,
i.e. $m=m_{\rm pole}$ 111The contributions of the imaginary parts of the self-
energies for defining the mass are neglected. when one implements the on-shell
scheme. One writes

$\displaystyle\Sigma(p^{2}=m_{\sigma,\eta,\eta^{\prime},\pi,K}^{2}){\rm+counterterms}$
$\displaystyle=$ $\displaystyle 0\;.$ (81)

The propagator residue is set to 1 in the on-shell scheme and one gets

$\displaystyle{\partial\over\partial
p^{2}}\Sigma_{\sigma,\eta,\eta^{\prime},\pi,K}(p^{2})\Big{|}_{p^{2}=m_{\sigma,\eta,\eta^{\prime},\pi,K}^{2}}$
$\displaystyle{\rm+counterterms}$ $\displaystyle=$ $\displaystyle 0\;.$ (82)

Using the diagrams of the Fig. 1(b) and Fig. 2(b), the counterterms of the two
point functions of the scalar and pseudo-scalar mesons can be written as

$\displaystyle\Sigma_{\sigma}^{\rm ct1}(p^{2})$ $\displaystyle=$
$\displaystyle i\left[\delta Z_{\sigma}(p^{2}-m_{\sigma}^{2})-\delta
m_{\sigma}^{2}\right]\;,$ (83) $\displaystyle\Sigma_{\pi}^{\rm ct1}(p^{2})$
$\displaystyle=$ $\displaystyle i\left[\delta
Z_{\pi}(p^{2}-m_{\pi}^{2})-\delta m_{\pi}^{2}\right]\;,$ (84)
$\displaystyle\Sigma_{K}^{\rm ct1}(p^{2})$ $\displaystyle=$ $\displaystyle
i\left[\delta Z_{K}(p^{2}-m_{K}^{2})-\delta m_{K}^{2}\right]\;,$ (85)
$\displaystyle\Sigma_{\eta}^{\rm ct1}(p^{2})$ $\displaystyle=$ $\displaystyle
i\left[\delta Z_{\eta}(p^{2}-m_{\eta}^{2})-\delta m_{\eta}^{2}\right]\;,$ (86)
$\displaystyle\Sigma_{\eta^{\prime}}^{\rm ct1}(p^{2})$ $\displaystyle=$
$\displaystyle i\left[\delta
Z_{\eta^{\prime}}(p^{2}-m_{\eta^{\prime}}^{2})-\delta
m_{\eta^{\prime}}^{2}\right]\;.$ (87)

The scalar and pseudo-scalar self energy tadpole contributions, have two
independent terms proportional to $N_{c}gm_{u}\mathcal{A}(m_{u}^{2})$ and
$N_{c}gm_{s}\mathcal{A}(m_{s}^{2})$ respectively as given in the Appendix B of
the Ref. vkkr22 . The tadpole counterterms $\Sigma^{\rm ct2}$ for the scalar
and pseudo-scalar particles are chosen (negative of the respective tadpole
contributions to the scalar and pseudo-scalar self energies) such that they
completely cancel the respective tadpole contributions to the self-energies.
When the self-energies and their derivatives are evaluated in the on-shell
conditions, one gets all the renormalization constants. Combining the Eqs.
(81), (III.2) and (83)–(87), one finds the set of equations given below :

$\displaystyle{\hskip-14.22636pt}\delta m_{\pi}^{2}$ $\displaystyle=$
$\displaystyle-i\Sigma_{\pi}(m_{\pi}^{2})\;;\delta
Z_{\pi}=i{\partial\over\partial
p^{2}}\Sigma_{\pi}(p^{2})\Big{|}_{p^{2}=m_{\pi}^{2}}\;,$ (88)
$\displaystyle{\hskip-14.22636pt}\delta m_{K}^{2}$ $\displaystyle=$
$\displaystyle-i\Sigma_{K}(m_{K}^{2})\;;\delta Z_{K}=i{\partial\over\partial
p^{2}}\Sigma_{K}(p^{2})\Big{|}_{p^{2}=m_{K}^{2}}\;,$ (89)
$\displaystyle{\hskip-14.22636pt}\delta m_{\eta}^{2}$ $\displaystyle=$
$\displaystyle-i\Sigma_{\eta}(m_{\eta}^{2})\;;\delta
Z_{\eta}=i{\partial\over\partial
p^{2}}\Sigma_{\eta}(p^{2})\Big{|}_{p^{2}=m_{\eta}^{2}}\;,$ (90)
$\displaystyle{\hskip-14.22636pt}\delta m_{\eta^{\prime}}^{2}$
$\displaystyle=$
$\displaystyle-i\Sigma_{\eta^{\prime}}(m_{\eta^{\prime}}^{2})\;;\delta
Z_{\eta^{\prime}}=i{\partial\over\partial
p^{2}}\Sigma_{\eta^{\prime}}(p^{2})\Big{|}_{p^{2}=m_{\eta^{\prime}}^{2}}\;,$
(91) $\displaystyle{\hskip-14.22636pt}\delta m_{\sigma}^{2}$ $\displaystyle=$
$\displaystyle-i\Sigma_{\sigma}(m_{\sigma}^{2})\;;\delta
Z_{\sigma}=i{\partial\over\partial
p^{2}}\Sigma_{\sigma}(p^{2})\Big{|}_{p^{2}=m_{\sigma}^{2}}\;.$ (92)

When the self energy (neglecting the tadpole contributions) expressions from
the Eqs. (62), (63), (78) and (79) are used, we get the following set of
equations.

$\displaystyle\delta
m_{\pi}^{2}=2ig^{2}N_{c}[\mathcal{A}(m_{u}^{2})-\mbox{$1\over
2$}{m_{\pi}^{2}}\mathcal{B}(m_{\pi}^{2},m_{u})]\;,$ (93) $\displaystyle\delta
m_{K}^{2}=ig^{2}N_{c}\biggl{[}\mathcal{A}(m_{u}^{2})+\mathcal{A}(m_{s}^{2})-\\{m_{K}^{2}-(m_{u}-m_{s})^{2}\\}\mathcal{B}(m_{K}^{2},m_{u},m_{s})\biggr{]}\;,$
(94)

$\displaystyle\delta
m_{\eta}^{2}=\frac{-i}{2}\Biggl{[}\Sigma_{\text{p},00}(m_{\eta}^{2})+\Sigma_{\text{p},88}(m_{\eta}^{2})-\frac{1}{\sqrt{(m^{2}_{\text{p},00}-m^{2}_{\text{p},88})^{2}+4m^{4}_{\text{p},08}}}$
$\displaystyle\qquad\quad\biggl{\\{}\biggl{(}\Sigma_{\text{p},00}(m_{\eta}^{2})-\Sigma_{\text{p},88}(m_{\eta}^{2})\biggr{)}(m^{2}_{\text{p},00}-m^{2}_{\text{p},88})+4\Sigma_{\text{p},08}(m_{\eta}^{2})\
m^{2}_{\text{p},08}\biggr{\\}}\Biggr{]}\;,$ (95) $\displaystyle\delta
m_{\eta}^{2}=ig^{2}N_{c}\Biggl{[}\biggl{\\{}\mathcal{A}(m_{u}^{2})+\mathcal{A}(m_{s}^{2})-\mbox{$1\over
2$}{m_{\eta}^{2}}\mathcal{B}(m_{\eta}^{2},m_{u})-\mbox{$1\over
2$}{m_{\eta}^{2}}\mathcal{B}(m_{\eta}^{2},m_{s})\biggr{\\}}$
$\displaystyle\qquad\quad-\frac{\tiny{\biggl{\\{}(m^{2}_{\text{p},00}-m^{2}_{\text{p},88})+4\sqrt{2}m^{2}_{\text{p},08}\biggr{\\}}}}{3\sqrt{(m^{2}_{\text{p},00}-m^{2}_{\text{p},88})^{2}+4m^{4}_{\text{p},08}}}\biggl{\\{}\mathcal{A}(m_{u}^{2})-\mathcal{A}(m_{s}^{2})-\mbox{$1\over
2$}{m_{\eta}^{2}}\mathcal{B}(m_{\eta}^{2},m_{u})+\mbox{$1\over
2$}{m_{\eta}^{2}}\mathcal{B}(m_{\eta}^{2},m_{s})\biggr{\\}}\Biggr{]}\;,$ (96)
$\displaystyle\delta
m_{\eta^{\prime}}^{2}=\frac{-i}{2}\Biggl{[}\Sigma_{\text{p},00}(m_{\eta^{\prime}}^{2})+\Sigma_{\text{p},88}(m_{\eta^{\prime}}^{2})+\frac{1}{\sqrt{(m^{2}_{\text{p},00}-m^{2}_{\text{p},88})^{2}+4m^{4}_{\text{p},08}}}$
$\displaystyle\qquad\quad\biggl{\\{}\biggl{(}\Sigma_{\text{p},00}(m_{\eta^{\prime}}^{2})-\Sigma_{\text{p},88}(m_{\eta^{\prime}}^{2})\biggr{)}(m^{2}_{\text{p},00}-m^{2}_{\text{p},88})+4\Sigma_{\text{p},08}(m_{\eta^{\prime}}^{2})\
m^{2}_{\text{p},08}\biggr{\\}}\Biggr{]}\;,$ (97) $\displaystyle\delta
m_{\eta^{\prime}}^{2}=ig^{2}N_{c}\Biggl{[}\biggl{\\{}\mathcal{A}(m_{u}^{2})+\mathcal{A}(m_{s}^{2})-\mbox{$1\over
2$}{m_{\eta^{\prime}}^{2}}\mathcal{B}(m_{\eta^{\prime}}^{2},m_{u})-\mbox{$1\over
2$}{m_{\eta^{\prime}}^{2}}\mathcal{B}(m_{\eta^{\prime}}^{2},m_{s})\biggr{\\}}$
$\displaystyle\qquad\quad+\frac{\tiny{\biggl{\\{}(m^{2}_{\text{p},00}-m^{2}_{\text{p},88})+4\sqrt{2}m^{2}_{\text{p},08}\biggr{\\}}}}{3\sqrt{(m^{2}_{\text{p},00}-m^{2}_{\text{p},88})^{2}+4m^{4}_{\text{p},08}}}\biggl{\\{}\mathcal{A}(m_{u}^{2})-\mathcal{A}(m_{s}^{2})-\mbox{$1\over
2$}{m_{\eta^{\prime}}^{2}}\mathcal{B}(m_{\eta^{\prime}}^{2},m_{u})+\mbox{$1\over
2$}{m_{\eta^{\prime}}^{2}}\mathcal{B}(m_{\eta^{\prime}}^{2},m_{s})\biggr{\\}}\Biggr{]}\;,$
(98) $\displaystyle\delta
m_{\sigma}^{2}=\frac{-i}{2}\Biggl{[}\Sigma_{\text{s},00}(m_{\sigma}^{2})+\Sigma_{\text{s},88}(m_{\sigma}^{2})-\frac{1}{\sqrt{(m^{2}_{\text{s},00}-m^{2}_{\text{s},88})^{2}+4m^{4}_{\text{s},08}}}$
$\displaystyle\qquad\quad\biggl{\\{}\biggl{(}\Sigma_{\text{s},00}(m_{\sigma}^{2})-\Sigma_{\text{s},88}(m_{\sigma}^{2})\biggr{)}(m^{2}_{\text{s},00}-m^{2}_{\text{s},88})+4\Sigma_{\text{s},08}(m_{\sigma}^{2})\
m^{2}_{\text{s},08}\biggr{\\}}\Biggr{]}\;,$ (99) $\displaystyle\delta
m_{\sigma}^{2}=ig^{2}N_{c}\Biggl{[}\biggl{\\{}\mathcal{A}(m_{u}^{2})+\mathcal{A}(m_{s}^{2})-\mbox{$1\over
2$}{(m_{\sigma}^{2}-4m_{u}^{2})}\mathcal{B}(m_{\sigma}^{2},m_{u})-\mbox{$1\over
2$}{(m_{\sigma}^{2}-4m_{s}^{2})}\mathcal{B}(m_{\sigma}^{2},m_{s})\biggr{\\}}-$
$\displaystyle\qquad\quad\frac{\tiny{\biggl{\\{}(m^{2}_{\text{s},00}-m^{2}_{\text{s},88})+4\sqrt{2}m^{2}_{\text{s},08}\biggr{\\}}}}{3\sqrt{(m^{2}_{\text{s},00}-m^{2}_{\text{s},88})^{2}+4m^{4}_{\text{s},08}}}\biggl{\\{}\mathcal{A}(m_{u}^{2})-\mathcal{A}(m_{s}^{2})-\frac{(m_{\sigma}^{2}-4m_{u}^{2})}{2}\mathcal{B}(m_{\sigma}^{2},m_{u})+\frac{(m_{\sigma}^{2}-4m_{s}^{2})}{2}\mathcal{B}(m_{\sigma}^{2},m_{s})\biggr{\\}}\Biggr{]}\;,$
(100) $\displaystyle\delta
Z_{\pi}=ig^{2}N_{c}\left[\mathcal{B}(m_{\pi}^{2},m_{u})+m_{\pi}^{2}\mathcal{B}^{\prime}(m_{\pi}^{2},m_{u})\right]\;,$
(101) $\displaystyle\delta
Z_{K}=ig^{2}N_{c}\left[\mathcal{B}(m_{K}^{2},m_{u},m_{s})+(m_{K}^{2}-(m_{u}-m_{s})^{2})\mathcal{B}^{\prime}(m_{K}^{2},m_{u},m_{s})\right]\;,$
(102) $\displaystyle\delta
Z_{\eta}=\frac{ig^{2}N_{c}}{2}\Biggl{[}\biggl{\\{}\mathcal{B}(m_{\eta}^{2},m_{u})+\mathcal{B}(m_{\eta}^{2},m_{s})+m_{\eta}^{2}\
\mathcal{B}^{\prime}(m_{\eta}^{2},m_{u})+m_{\eta}^{2}\
\mathcal{B}^{\prime}(m_{\eta}^{2},m_{s})\biggr{\\}}$
$\displaystyle\qquad\quad+\frac{\tiny{\biggl{\\{}(m^{2}_{\text{p},00}-m^{2}_{\text{p},88})+4\sqrt{2}m^{2}_{\text{p},08}\biggr{\\}}}}{3\sqrt{(m^{2}_{\text{p},00}-m^{2}_{\text{p},88})^{2}+4m^{4}_{\text{p},08}}}\biggl{\\{}-\mathcal{B}(m_{\eta}^{2},m_{u})+\mathcal{B}(m_{\eta}^{2},m_{s})-m_{\eta}^{2}\
\mathcal{B}^{\prime}(m_{\eta}^{2},m_{u})+m_{\eta}^{2}\
\mathcal{B}^{\prime}(m_{\eta}^{2},m_{s})\biggr{\\}}\Biggr{]}\;,$ (103)
$\displaystyle\delta
Z_{\eta^{\prime}}=\frac{ig^{2}N_{c}}{2}\Biggl{[}\biggl{\\{}\mathcal{B}(m_{\eta^{\prime}}^{2},m_{u})+\mathcal{B}(m_{\eta^{\prime}}^{2},m_{s})+m_{\eta^{\prime}}^{2}\
\mathcal{B}^{\prime}(m_{\eta^{\prime}}^{2},m_{u})+m_{\eta^{\prime}}^{2}\
\mathcal{B}^{\prime}(m_{\eta^{\prime}}^{2},m_{s})\biggr{\\}}$
$\displaystyle\qquad\quad-\frac{\tiny{\biggl{\\{}(m^{2}_{\text{p},00}-m^{2}_{\text{p},88})+4\sqrt{2}m^{2}_{\text{p},08}\biggr{\\}}}}{3\sqrt{(m^{2}_{\text{p},00}-m^{2}_{\text{p},88})^{2}+4m^{4}_{\text{p},08}}}\biggl{\\{}-\mathcal{B}(m_{\eta^{\prime}}^{2},m_{u})+\mathcal{B}(m_{\eta^{\prime}}^{2},m_{s})-m_{\eta^{\prime}}^{2}\
\mathcal{B}^{\prime}(m_{\eta^{\prime}}^{2},m_{u})+m_{\eta^{\prime}}^{2}\
\mathcal{B}^{\prime}(m_{\eta^{\prime}}^{2},m_{s})\biggr{\\}}\Biggr{]}\;,$
(104) $\displaystyle\delta
Z_{\sigma}=\frac{ig^{2}N_{c}}{2}\Biggl{[}\biggl{\\{}\mathcal{B}(m_{\sigma}^{2},m_{u})+\mathcal{B}(m_{\sigma}^{2},m_{s})+(m_{\sigma}^{2}-4m_{u}^{2})\mathcal{B}^{\prime}(m_{\sigma}^{2},m_{u})+(m_{\sigma}^{2}-4m_{s}^{2})\mathcal{B}^{\prime}(m_{\sigma}^{2},m_{s})\biggr{\\}}+$
$\displaystyle\qquad\quad\frac{\tiny{\biggl{\\{}(m^{2}_{\text{s},00}-m^{2}_{\text{s},88})+4\sqrt{2}m^{2}_{\text{s},08}\biggr{\\}}}}{3\sqrt{(m^{2}_{\text{s},00}-m^{2}_{\text{s},88})^{2}+4m^{4}_{\text{s},08}}}\biggl{\\{}\mathcal{B}(m_{\sigma}^{2},m_{s})-\mathcal{B}(m_{\sigma}^{2},m_{u})-(m_{\sigma}^{2}-4m_{u}^{2})\mathcal{B}^{\prime}(m_{\sigma}^{2},m_{u})+(m_{\sigma}^{2}-4m_{s}^{2})\mathcal{B}^{\prime}(m_{\sigma}^{2},m_{s})\biggr{\\}}\Biggr{]}\;.$
(105)

The expressions of the field renormalization constants for the $\pi$, $K$,
$\eta$, $\eta^{\prime}$ and $\sigma$ are given above. However, one needs to
have the simplified expression of the $\delta Z_{\pi}$ only in the
calculations below. Substituting the expressions of $\delta Z_{\pi},\delta
m_{K}^{2},\ \delta m_{\pi}^{2},\ \delta m_{\eta}^{2}$ and $\delta
m_{\eta^{\prime}}^{2}$ from the above in the Eq.(50), the $\delta\lambda_{2}$
can be written as

$\displaystyle\delta\lambda_{2\text{\tiny OS}}=\dfrac{2\ iN_{c}\ g^{2}\
}{(\overline{\sigma}_{x}^{2}+4\overline{\sigma}_{y}^{2})(\sqrt{2}\overline{\sigma}_{y}-\overline{\sigma}_{x})}\Biggl{[}(3\sqrt{2}\overline{\sigma}_{y})\biggl{\\{}A(m_{u}^{2})+A(m_{s}^{2})-\biggl{(}m_{K}^{2}-(m_{s}-m_{u})^{2}\biggr{)}\mathcal{B}(m^{2}_{K},m_{u},m_{s})\biggr{\\}}\;$
(106)
$\displaystyle\qquad\quad-(\sqrt{2}\overline{\sigma}_{y}+2\overline{\sigma}_{x})\biggl{\\{}2A(m_{u}^{2})-m_{\pi}^{2}\mathcal{B}(m^{2}_{\pi},m_{u})\biggr{\\}}-(\sqrt{2}\overline{\sigma}_{y}-\overline{\sigma}_{x})\Biggl{\\{}2A(m_{u}^{2})+2A(m_{s}^{2})-\frac{m_{\eta}^{2}}{2}\biggl{\\{}\mathcal{B}(m^{2}_{\eta}\
,m_{u})\;$ $\displaystyle\qquad\quad+\mathcal{B}(m^{2}_{\eta}\ ,\
m_{s})\biggr{\\}}-\frac{m_{\eta^{{}^{\prime}}}^{2}}{2}\biggl{\\{}\mathcal{B}(m^{2}_{\eta^{{}^{\prime}}}\
,\ m_{u})+\mathcal{B}(m^{2}_{\eta^{{}^{\prime}}}\ ,\
m_{s})\biggr{\\}}-\frac{\biggl{(}m^{2}_{p,00}-m^{2}_{p,88}+4\sqrt{2}m^{2}_{p,08}\biggr{)}}{6\
\biggl{(}\sqrt{(m^{2}_{p,00}-m^{2}_{p,88})^{2}+4m^{4}_{p,08}\biggr{)}}}$
$\displaystyle\qquad\quad\Biggl{\\{}m_{\eta}^{2}\biggl{\\{}-\mathcal{B}(m^{2}_{\eta}\
,m_{u})+\mathcal{B}(m^{2}_{\eta}\ ,\
m_{s})\biggr{\\}}-m_{\eta^{{}^{\prime}}}^{2}\biggl{\\{}-\mathcal{B}(m^{2}_{\eta^{{}^{\prime}}}\
,\ m_{u})+\mathcal{B}(m^{2}_{\eta^{{}^{\prime}}}\ ,\
m_{s})\biggr{\\}}\Biggr{\\}}\Biggr{]}\;,$ (107)
$\displaystyle\delta\lambda_{2\text{\tiny
OS}}=\delta\lambda_{2\text{div}}+\lambda_{2\text{{\tiny FIN}}}\;,$ (108)
$\displaystyle\lambda_{2\text{{\tiny
FIN}}}=\dfrac{N_{c}g^{2}}{(4\pi)^{2}}(\lambda_{2}-g^{2})\ln\biggl{(}\frac{\Lambda^{2}}{m_{u}^{2}}\biggr{)}+\dfrac{N_{c}g^{2}}{(4\pi)^{2}}\dfrac{2}{(\overline{\sigma}_{x}^{2}+4\overline{\sigma}_{y}^{2})}\Biggl{[}\dfrac{(\sqrt{2}\overline{\sigma}_{y}+2\overline{\sigma}_{x})}{(\sqrt{2}\overline{\sigma}_{y}-\overline{\sigma}_{x})}\Biggl{\\{}m^{2}_{u}-m^{2}_{s}\biggl{\\{}1-2\ln\biggl{(}\frac{m_{s}}{m_{u}}\biggr{)}\biggr{\\}}$
$\displaystyle\qquad\quad-
m_{\pi}^{2}\mathcal{C}(m^{2}_{\pi},m_{u})\Biggr{\\}}+\dfrac{3\sqrt{2}\
\overline{\sigma}_{y}}{(\sqrt{2}\overline{\sigma}_{y}-\overline{\sigma}_{x})}\biggl{\\{}m_{K}^{2}-(m_{s}-m_{u})^{2}\biggr{\\}}\mathcal{C}(m^{2}_{K},m_{u},m_{s})-\frac{m_{\eta}^{2}}{2}\biggl{\\{}\mathcal{C}(m^{2}_{\eta}\
,m_{u})+\mathcal{C}(m^{2}_{\eta}\ ,\ m_{s})\;$
$\displaystyle\qquad\quad-2\ln\biggl{(}\frac{m_{s}}{m_{u}}\biggr{)}\biggr{\\}}-\frac{m_{\eta^{{}^{\prime}}}^{2}}{2}\biggl{\\{}\mathcal{C}(m^{2}_{\eta^{{}^{\prime}}}\
,\ m_{u})+\mathcal{C}(m^{2}_{\eta^{{}^{\prime}}}\ ,\
m_{s})-2\ln\biggl{(}\frac{m_{s}}{m_{u}}\biggr{)}\biggr{\\}}+\frac{\biggl{(}m^{2}_{p,00}-m^{2}_{p,88}+4\sqrt{2}m^{2}_{p,08}\biggr{)}}{6\
\biggl{(}\sqrt{(m^{2}_{p,00}-m^{2}_{p,88})^{2}+4m^{4}_{p,08}}\biggr{)}}$
$\displaystyle\qquad\quad\Biggl{\\{}m_{\eta}^{2}\biggl{\\{}\mathcal{C}(m^{2}_{\eta}\
,m_{u})-\mathcal{C}(m^{2}_{\eta}\ ,\
m_{s})+2\ln\biggl{(}\frac{m_{s}}{m_{u}}\biggr{)}\biggr{\\}}-m_{\eta^{{}^{\prime}}}^{2}\biggl{\\{}\mathcal{C}(m^{2}_{\eta^{{}^{\prime}}}\
,\ m_{u})-\mathcal{C}(m^{2}_{\eta^{{}^{\prime}}}\ ,\
m_{s})+2\ln\biggl{(}\frac{m_{s}}{m_{u}}\biggr{)}\biggr{\\}}\Biggr{\\}}\Biggr{]}\;,$
(109) Substituting the expressions of $\delta Z_{\pi},\ \delta m_{K}^{2},\
\delta m_{\pi}^{2},\ \delta\lambda_{2}$ in the Eq. (51), the $\delta c$ is
written as $\displaystyle\delta c_{\text{\tiny OS}}=\dfrac{2\ i\ N_{c}\
g^{2}}{(\sqrt{2}\
\overline{\sigma}_{y}-\overline{\sigma}_{x})}\biggl{\\{}A(m_{u}^{2})+A(m_{s}^{2})-\biggl{(}m_{K}^{2}-(m_{s}-m_{u})^{2}\biggr{)}\mathcal{B}(m^{2}_{K},m_{u},m_{s})-2A(m_{u}^{2})+m_{\pi}^{2}\mathcal{B}(m^{2}_{\pi},m_{u})\biggr{\\}}$
$\displaystyle\qquad\quad\ \ -\sqrt{2}\ \overline{\sigma}_{y}\
\delta\lambda_{2\text{\tiny OS}}-(2\sqrt{2}\ \overline{\sigma}_{y}\
\lambda_{2}+c)\frac{\delta Z_{\pi}}{2}\ ;\ \ \ \ \ \delta c_{\text{\tiny
OS}}=\delta c_{\text{div}}+c_{\text{\tiny{FINTOT}}}\ \ \ ;\ \ \ \ \
c_{\text{\tiny{FINTOT}}}=-\sqrt{2}\ \overline{\sigma}_{y}\
\lambda_{2\text{{\tiny FIN}}}+c_{\text{{\tiny FIN}}}\;,$ (110) $\displaystyle
c_{\text{{\tiny
FIN}}}=\dfrac{N_{c}g^{2}}{(4\pi)^{2}}\Biggl{[}\biggl{\\{}c+\sqrt{2}\overline{\sigma}_{y}(\lambda_{2}-g^{2})\biggr{\\}}\ln\biggl{(}\frac{\Lambda^{2}}{m_{u}^{2}}\biggr{)}+\dfrac{2}{(\sqrt{2}\overline{\sigma}_{y}-\overline{\sigma}_{x})}\Biggl{\\{}\biggl{\\{}m_{K}^{2}-(m_{s}-m_{u})^{2}\biggr{\\}}\mathcal{C}(m^{2}_{K},m_{u},m_{s})-m_{\pi}^{2}\mathcal{C}(m^{2}_{\pi},m_{u})\Biggr{\\}}\;$
$\displaystyle\qquad\quad-\dfrac{g^{2}}{2}(\sqrt{2}\overline{\sigma}_{y}+\overline{\sigma}_{x})+\dfrac{2g^{2}\overline{\sigma}_{y}^{2}}{(\sqrt{2}\overline{\sigma}_{y}-\overline{\sigma}_{x})}\ln\biggl{(}\frac{m_{s}}{m_{u}}\biggr{)}\Biggr{]}\;.$
(111)

Using the Eq. (52) and substituting the expressions of $\delta Z_{\pi},\
\delta m_{\sigma}^{2},\ \delta m_{\pi}^{2},\ \delta\lambda_{2}$ and $\delta c$
in the Eq. (54), the $\delta\lambda_{1}$ is written as

$\displaystyle\delta\lambda_{1\text{\tiny OS}}=\frac{\lambda_{1\text{{\tiny
NUMOS}}}}{\lambda_{1\text{{\tiny DENOM}}}}-\lambda_{1}\delta Z_{\pi}\;,$ (112)
$\displaystyle\lambda_{1\text{{\tiny
NUMOS}}}=iN_{c}g^{2}\Biggl{[}\biggl{(}\sqrt{(m^{2}_{s,00}-m^{2}_{s,88})^{2}+4m^{4}_{s,08}}\biggr{)}\biggl{\\{}A(m_{u}^{2})+A(m_{s}^{2})-\biggl{(}\frac{m_{\sigma}^{2}-4m_{u}^{2}}{2}\biggr{)}\mathcal{B}(m^{2}_{\sigma},m_{u})-\biggl{(}\frac{m_{\sigma}^{2}-4m_{s}^{2}}{2}\biggr{)}$
$\displaystyle\qquad\quad\ \ \ \
\mathcal{B}(m^{2}_{\sigma},m_{s})-2A(m_{u}^{2})+m_{\pi}^{2}\mathcal{B}(m^{2}_{\pi},m_{u})\biggr{\\}}-\biggl{(}\frac{m^{2}_{s,00}-m^{2}_{s,88}+4\sqrt{2}m^{2}_{s,08}}{3}\biggr{)}\biggl{\\{}A(m_{u}^{2})-A(m_{s}^{2})-\biggl{(}\frac{m_{\sigma}^{2}-4m_{u}^{2}}{2}\biggr{)}$
$\displaystyle\qquad\quad\ \ \
\mathcal{B}(m^{2}_{\sigma},m_{u})+\biggl{(}\frac{m_{\sigma}^{2}-4m_{s}^{2}}{2}\biggr{)}\mathcal{B}(m^{2}_{\sigma},m_{s})\biggr{\\}}\Biggr{]}+\frac{(m^{2}_{s,00}-m^{2}_{s,88})}{12}\biggl{\\{}(3\overline{\sigma}_{x}^{2}-6\overline{\sigma}_{y}^{2})\delta\lambda_{2\text{\tiny
OS}}-\sqrt{2}(4\sqrt{2}\overline{\sigma}_{x}+\overline{\sigma}_{y})\delta
c_{\text{\tiny OS}}\biggr{\\}}$ $\displaystyle\qquad\quad\ \
-\frac{\sqrt{2}m^{2}_{s,08}}{3}\biggl{\\{}(2\overline{\sigma}_{y}^{2}-\overline{\sigma}_{x}^{2})\delta\lambda_{2\text{\tiny
OS}}+(\sqrt{2}\overline{\sigma}_{y}-\overline{\sigma}_{x})\delta
c_{\text{\tiny
OS}}\biggl{\\}}-\frac{1}{4}\sqrt{(m^{2}_{s,00}-m^{2}_{s,88})^{2}+4m^{4}_{s,08}}\biggl{\\{}(\overline{\sigma}_{x}^{2}+6\overline{\sigma}_{y}^{2})\delta\lambda_{2\text{\tiny
OS}}+\sqrt{2}\overline{\sigma}_{y}\delta c_{\text{\tiny OS}}\biggl{\\}}$
$\displaystyle\qquad\quad\ \ +\delta
Z_{\pi}\Biggl{[}\frac{(m^{2}_{\sigma_{00}}-m^{2}_{s,88})}{12}\biggl{\\{}(3\overline{\sigma}_{x}^{2}-6\overline{\sigma}_{y}^{2})\lambda_{2}-\sqrt{2}(4\sqrt{2}\overline{\sigma}_{x}+\overline{\sigma}_{y})\frac{c}{2}\biggr{\\}}-\frac{\sqrt{2}m^{2}_{s,08}}{3}\biggl{\\{}(2\overline{\sigma}_{y}^{2}-\overline{\sigma}_{x}^{2})\lambda_{2}+(\sqrt{2}\overline{\sigma}_{y}-\overline{\sigma}_{x})\frac{c}{2}\biggl{\\}}$
$\displaystyle\qquad\quad\ \
-\frac{1}{4}\biggl{(}\sqrt{(m^{2}_{s,00}-m^{2}_{s,88})^{2}+4m^{4}_{s,08}}\biggr{)}\biggl{\\{}(\overline{\sigma}_{x}^{2}+6\overline{\sigma}_{y}^{2})\lambda_{2}+\sqrt{2}\overline{\sigma}_{y}\frac{c}{2}\biggl{\\}}\Biggr{]}\;,$
(113) $\displaystyle\delta\lambda_{1\text{\tiny
OS}}=\delta\lambda_{1\text{div}}+\lambda_{1\text{{\tiny
FIN}}};\qquad\lambda_{1\text{{\tiny FIN}}}=\frac{\lambda_{1\text{{\tiny
NUMF}}}}{\lambda_{1\text{{\tiny DENOM}}}};\qquad\lambda_{1\text{{\tiny
NUMF}}}=\lambda_{1\text{{\tiny NUMF-I}}}+\lambda_{1\text{{\tiny NUMF-II}}}\;,$
(114) $\displaystyle\text{expression of }\ \lambda_{1\text{{\tiny DENOM}}}\
\text{is given in the Eq.~{}(\ref{lam1de})}\;,$
$\displaystyle\lambda_{1\text{{\tiny
NUMF-I}}}=\frac{(m^{2}_{s,00}-m^{2}_{s,88})}{12}\biggl{\\{}(3\overline{\sigma}_{x}^{2}+8\sqrt{2}\overline{\sigma}_{x}\overline{\sigma}_{y}-4\overline{\sigma}_{y}^{2})\lambda_{2\text{{\tiny
FIN}}}-\sqrt{2}(4\sqrt{2}\overline{\sigma}_{x}+\overline{\sigma}_{y})c_{\text{{\tiny
FIN}}}\biggr{\\}}-\frac{\sqrt{2}m^{2}_{s,08}}{3}\biggl{\\{}(\sqrt{2}\overline{\sigma}_{y}-\overline{\sigma}_{x})c_{\text{{\tiny
FIN}}}\;$ $\displaystyle\qquad\quad\ \ \
+(4\overline{\sigma}_{y}^{2}+\sqrt{2}\overline{\sigma}_{x}\overline{\sigma}_{y}-3\overline{\sigma}_{x}^{2})\lambda_{2\text{{\tiny
FIN}}}\biggr{\\}}-\biggl{(}\frac{1}{4}\sqrt{(m^{2}_{s,00}-m^{2}_{s,88})^{2}+4m^{4}_{s,08}}\biggr{)}\
\biggl{\\{}(\overline{\sigma}_{x}^{2}+4\overline{\sigma}_{y}^{2})\lambda_{2\text{{\tiny
FIN}}}+\sqrt{2}\overline{\sigma}_{y}c_{\text{{\tiny FIN}}}\biggr{\\}}\;,$
(115) $\displaystyle\lambda_{1\text{{\tiny NUMF-
II}}}=\dfrac{N_{c}g^{2}}{(4\pi)^{2}}\Biggl{[}\biggl{(}\sqrt{(m^{2}_{s,00}-m^{2}_{s,88})^{2}+4m^{4}_{s,08}}\biggr{)}\biggl{\\{}\dfrac{g^{2}}{4}(\overline{\sigma}_{x}^{2}-2\overline{\sigma}_{y}^{2})+(m^{2}_{\sigma}-m^{2}_{\pi}-m^{2}_{u}-3m^{2}_{s})\ln\biggl{(}\frac{\Lambda^{2}}{m_{q}^{2}}\biggr{)}\;$
$\displaystyle\qquad\quad\ \
+2m^{2}_{s}\ln\biggl{(}\frac{m_{s}}{m_{u}}\biggr{)}+\frac{(m_{\sigma}^{2}-4m^{2}_{u})}{2}\
\mathcal{C}(m^{2}_{\sigma}\ ,m_{u})+\frac{(m_{\sigma}^{2}-4m^{2}_{s})}{2}\
\biggl{(}\mathcal{C}(m^{2}_{\sigma}\
,m_{s})-2\ln\biggl{(}\frac{m_{s}}{m_{u}}\biggr{)}\biggr{)}-m_{\pi}^{2}\mathcal{C}(m^{2}_{\pi}\
,m_{u})\biggr{\\}}\;$ $\displaystyle\qquad\quad\ \
-\biggl{(}\dfrac{(m^{2}_{s,00}-m^{2}_{s,88})+4\sqrt{2}m^{2}_{s,08}}{3}\biggr{)}\biggl{\\{}\dfrac{g^{2}}{4}(2\overline{\sigma}_{y}^{2}-\overline{\sigma}_{x}^{2})\biggl{(}1+3\ln\biggl{(}\frac{\Lambda^{2}}{m_{q}^{2}}\biggr{)}\biggr{)}+(m^{2}_{\sigma}-6\
m^{2}_{s}\ )\ln\biggl{(}\frac{m_{s}}{m_{u}}\biggr{)}\;$
$\displaystyle\qquad\quad\ \ +\frac{(m_{\sigma}^{2}-4m^{2}_{u})}{2}\
\mathcal{C}(m^{2}_{\sigma}\ ,m_{u})-\frac{(m_{\sigma}^{2}-4m^{2}_{s})}{2}\
\mathcal{C}(m^{2}_{\sigma}\ ,m_{s})\biggr{\\}}\Biggl{]}\;,$ (116)
$\displaystyle\delta m^{2}_{\text{\tiny
OS}}=iN_{c}g^{2}\biggl{\\{}2A(m_{u}^{2})-m_{\pi}^{2}\mathcal{B}(m^{2}_{\pi},m_{u})\biggl{\\}}-\delta\lambda_{1\text{\tiny
OS}}(\overline{\sigma}_{x}^{2}+\overline{\sigma}_{y}^{2})-\delta\lambda_{2\text{\tiny
OS}}\frac{\overline{\sigma}_{x}^{2}}{2}+\frac{\delta c_{\text{\tiny
OS}}\overline{\sigma}_{y}}{\sqrt{2}}-\delta
Z_{\pi}\biggl{\\{}\lambda_{1}(\overline{\sigma}_{x}^{2}+\overline{\sigma}_{y}^{2})+\lambda_{2}\frac{\overline{\sigma}_{x}^{2}}{2}-\frac{c\overline{\sigma}_{y}}{2\sqrt{2}}\biggr{\\}}\;,$
$\displaystyle\delta m^{2}_{\text{\tiny OS}}=\delta
m^{2}_{\text{div}}+m^{2}_{\text{\tiny{FIN}}}\;,$ (117) $\displaystyle
m^{2}_{\text{\tiny{FIN}}}=\dfrac{N_{c}g^{2}}{(4\pi)^{2}}\Biggl{[}-2m_{u}^{2}+(m_{\pi}^{2}-2m_{u}^{2})\ln\biggl{(}\frac{\Lambda^{2}}{m_{u}^{2}}\biggr{)}+m_{\pi}^{2}\mathcal{C}(m^{2}_{\pi}\
,m_{u})\Biggr{]}-\Biggl{[}\lambda_{1\text{\tiny{FIN}}}(\overline{\sigma}_{x}^{2}+\overline{\sigma}_{y}^{2})+\lambda_{2\text{\tiny{FIN}}}\frac{\overline{\sigma}_{x}^{2}}{2}-c_{\text{\tiny{FINTOT}}}\frac{\overline{\sigma}_{y}}{\sqrt{2}}\Biggr{]}\;,$
(118) $\displaystyle\delta h_{x\text{\tiny
OS}}=-\frac{i}{2}N_{c}g^{2}m^{2}_{\pi}\
\overline{\sigma}_{x}\left[\mathcal{B}(m^{2}_{\pi},m_{u})-m^{2}_{\pi}\mathcal{B}^{\prime}(m^{2}_{\pi},m_{u})\right]\;,$
(119) $\displaystyle\delta h_{x\text{\tiny OS}}=\delta
h_{x\text{div}}+h_{x\text{\tiny{FIN}}}\;,$ (120) $\displaystyle
h_{x\text{\tiny{FIN}}}=\dfrac{N_{c}g^{2}}{2(4\pi)^{2}}h_{x}\left[\ln\left(\frac{\Lambda^{2}}{m_{u}^{2}}\right)+\mathcal{C}(m^{2}_{\pi},m_{u})-m^{2}_{\pi}\mathcal{C}^{\prime}(m^{2}_{\pi},m_{u})\right]\;,$
(121) $\displaystyle\delta h_{y\text{\tiny
OS}}=iN_{c}g^{2}\Biggl{[}\left(\frac{\sqrt{2}}{2}\overline{\sigma}_{x}-\overline{\sigma}_{y}\right)\left\\{\mathcal{A}(m^{2}_{s})-\mathcal{A}(m^{2}_{u})\right\\}-\left(\frac{\sqrt{2}}{2}\overline{\sigma}_{x}+\overline{\sigma}_{y}\right)\left\\{m^{2}_{K}-(m_{u}-m_{s})^{2}\right\\}\mathcal{B}(m^{2}_{K},m_{u},m_{s})$
$\displaystyle\qquad\quad\
+\left(\frac{\sqrt{2}}{2}\overline{\sigma}_{x}+\overline{\sigma}_{y}\right)\frac{m^{2}_{K}}{2}\left[\mathcal{B}(m^{2}_{\pi},m_{u})+m^{2}_{\pi}\mathcal{B}^{\prime}(m^{2}_{\pi},m_{u})\right]+\frac{\sqrt{2}}{4}\overline{\sigma}_{x}\
m^{2}_{\pi}\left[\mathcal{B}(m^{2}_{\pi},m_{u})-m^{2}_{\pi}\mathcal{B}^{\prime}(m^{2}_{\pi},m_{u})\right]\Biggr{]}\;,$
(122) $\displaystyle\delta h_{y\text{\tiny OS}}=\delta
h_{y\text{div}}+h_{y\text{\tiny{FIN}}}\;,$ (123) $\displaystyle
h_{y\text{\tiny{FIN}}}=\dfrac{N_{c}g^{2}}{(4\pi)^{2}}\Biggl{[}\dfrac{h_{y}}{2}\biggl{\\{}\ln\left(\frac{\Lambda^{2}}{m_{u}^{2}}\right)-\mathcal{C}(m^{2}_{\pi},m_{u})-m^{2}_{\pi}\mathcal{C}^{\prime}(m^{2}_{\pi},m_{u})\biggr{\\}}-\dfrac{\sqrt{2}\
h_{x}}{2}\mathcal{C}(m^{2}_{\pi},m_{u})$
$\displaystyle\qquad\quad+\left(\dfrac{\sqrt{2}\
\overline{\sigma}_{x}}{2}-\overline{\sigma}_{y}\right)\
\biggl{\\{}m_{u}^{2}-m_{s}^{2}+2m_{s}^{2}\ln\left(\frac{m_{s}}{m_{u}}\right)\biggr{\\}}+\left(\dfrac{\sqrt{2}\
\overline{\sigma}_{x}}{2}+\overline{\sigma}_{y}\right)\biggl{\\{}m_{K}^{2}-(m_{s}-m_{u})^{2}\biggr{\\}}\mathcal{C}(m^{2}_{K},m_{u},m_{s})\Biggr{]}\;,$
(124) $\displaystyle\delta Z^{\text{\tiny OS}}_{\pi}=\delta Z_{\pi,\rm
div}-\frac{N_{c}g^{2}}{(4\pi)^{2}}\left[\ln\left(\frac{\Lambda^{2}}{m_{u}^{2}}\right)+\mathcal{C}(m_{\pi}^{2},m_{u})+m_{\pi}^{2}\mathcal{C}^{\prime}(m_{\pi}^{2},m_{u})\right]\;,$
(125) $\displaystyle\delta g^{2}_{\text{\tiny
OS}}=-iN_{c}g^{4}\left[m^{2}_{\pi}\mathcal{B}^{\prime}(m^{2}_{\pi},m_{u})+\mathcal{B}(m^{2}_{\pi},m_{u})\right]=\delta
g^{2}_{\text{div}}+\dfrac{N_{c}g^{4}}{(4\pi)^{2}}\left[\ln\left(\frac{\Lambda^{2}}{m_{u}^{2}}\right)+\mathcal{C}(m^{2}_{\pi},m_{u})+m^{2}_{\pi}\mathcal{C}^{\prime}(m^{2}_{\pi},m_{u})\right]\;,$
(126) $\displaystyle\delta{\overline{\sigma}_{x}}^{2}_{\text{\tiny
OS}}=iN_{c}g^{2}\overline{\sigma}_{x}^{2}\left[m^{2}_{\pi}\mathcal{B}^{\prime}(m^{2}_{\pi},m_{u})+\mathcal{B}(m^{2}_{\pi},m_{u})\right]=\delta{\overline{\sigma}_{x}}^{2}_{\text{div}}-\dfrac{N_{c}g^{2}\overline{\sigma}_{x}^{2}}{(4\pi)^{2}}\left[\ln\left(\frac{\Lambda^{2}}{m_{u}^{2}}\right)+\mathcal{C}(m^{2}_{\pi},m_{u})+m^{2}_{\pi}\mathcal{C}^{\prime}(m^{2}_{\pi},m_{u})\right]\;,$
(128) $\displaystyle\delta{\overline{\sigma}_{y}}^{2}_{\text{\tiny
OS}}=iN_{c}g^{2}\overline{\sigma}_{y}^{2}\left[m^{2}_{\pi}\mathcal{B}^{\prime}(m^{2}_{\pi},m_{u})+\mathcal{B}(m^{2}_{\pi},m_{u})\right]=\delta{\overline{\sigma}_{y}}^{2}_{\text{div}}-\dfrac{N_{c}g^{2}{\overline{\sigma}_{y}}^{2}}{(4\pi)^{2}}\left[\ln\left(\frac{\Lambda^{2}}{m_{u}^{2}}\right)+\mathcal{C}(m^{2}_{\pi},m_{u})+m^{2}_{\pi}\mathcal{C}^{\prime}(m^{2}_{\pi},m_{u})\right]\;.$
(129) The common factor in the r.h.s. of the above four equations is defined
as
$\displaystyle\text{SCF}=\left[\ln\left(\frac{\Lambda^{2}}{m_{u}^{2}}\right)+\mathcal{C}(m^{2}_{\pi},m_{u})+m^{2}_{\pi}\mathcal{C}^{\prime}(m^{2}_{\pi},m_{u})\right]\;.$
(130)

The $\mathcal{A}(m^{2}_{f}),\ \mathcal{B}(m^{2},m_{f}),\
\mathcal{B}(m^{2},m_{u},m_{s}),\ \mathcal{B}^{\prime}(m^{2},m_{f})$,
$\mathcal{C}(m^{2},m_{f})$, $\mathcal{C}(m^{2},m_{u},m_{s})$,
$\mathcal{C}^{\prime}(m^{2},m_{f})$ and
$\mathcal{C}^{\prime}(m^{2},m_{u},m_{s})$ are defined in the Appendix (A). The
divergent part of the counterterms are
$\delta\lambda_{2\text{div}}=\frac{N_{c}g^{2}}{(4\pi)^{2}\epsilon}(2\lambda_{2}-g^{2})\;$,
$\delta c_{\text{div}}=\frac{3N_{c}g^{2}c}{2(4\pi)^{2}\epsilon}\;$,
$\delta\lambda_{1\text{div}}=\frac{N_{c}g^{2}\lambda_{1}}{(4\pi)^{2}\epsilon}\;$,
$\delta m^{2}_{\text{div}}=\frac{N_{c}g^{2}m^{2}}{(4\pi)^{2}\epsilon}\;$,
$\delta h_{x\text{div}}=\frac{N_{c}g^{2}h_{x}}{2(4\pi)^{2}\epsilon}\;$,
$\delta h_{y\text{div}}=\frac{N_{c}g^{2}h_{y}}{2(4\pi)^{2}\epsilon}\;$,
$\delta g^{2}_{\text{div}}=\frac{N_{c}g^{4}}{(4\pi)^{2}\epsilon}\;$,
$\delta{\overline{\sigma}_{x}}^{2}_{\text{div}}=-\frac{N_{c}g^{2}\overline{\sigma}_{x}^{2}}{(4\pi)^{2}\epsilon}\;$,
$\delta{\overline{\sigma}_{y}}^{2}_{\text{div}}=-\frac{N_{c}g^{2}\overline{\sigma}_{y}^{2}}{(4\pi)^{2}\epsilon}\;$,
$\delta Z_{\pi,\rm div}=-\frac{N_{c}g^{2}}{(4\pi)^{2}\epsilon}$ . 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.

Due to the renormalization scheme independence of the bare parameters, one can
write down the relations between the renormalized parameters in the on-shell
and the $\overline{\text{MS}}$ schemes as given below

$\displaystyle\lambda_{2\overline{\text{\tiny MS}}}$ $\displaystyle=$
$\displaystyle\lambda_{2}+\delta\lambda_{2\text{\tiny
OS}}-\delta\lambda_{2\overline{\text{\tiny MS}}}\;,$ (131) $\displaystyle
c_{\overline{\text{\tiny MS}}}$ $\displaystyle=$ $\displaystyle c+\delta
c_{\text{\tiny OS}}-\delta c_{\overline{\text{\tiny MS}}}\;,$ (132)
$\displaystyle\lambda_{1\overline{\text{\tiny MS}}}$ $\displaystyle=$
$\displaystyle\lambda_{1}+\delta\lambda_{1\text{\tiny
OS}}-\delta\lambda_{1\overline{\text{\tiny MS}}}\;,$ (133) $\displaystyle
m^{2}_{\overline{\text{\tiny MS}}}$ $\displaystyle=$ $\displaystyle
m^{2}+\delta m^{2}_{\text{\tiny OS}}-\delta m^{2}_{\overline{\text{\tiny
MS}}}\;,$ (134) $\displaystyle h_{x}{\overline{\text{\tiny MS}}}$
$\displaystyle=$ $\displaystyle h_{x}+\delta h_{x}{\text{\tiny OS}}-\delta
h_{x}{\overline{\text{\tiny MS}}}\;,$ (135) $\displaystyle
h_{y}{\overline{\text{\tiny MS}}}$ $\displaystyle=$ $\displaystyle
h_{y}+\delta h_{y}{\text{\tiny OS}}-\delta h_{y}{\overline{\text{\tiny
MS}}}\;,$ (136) $\displaystyle g^{2}_{\overline{\text{\tiny MS}}}$
$\displaystyle=$ $\displaystyle g^{2}+\delta g^{2}_{\text{\tiny OS}}-\delta
g^{2}_{\overline{\text{\tiny MS}}}\;,$ (137)
$\displaystyle{\overline{\sigma}_{x}}^{2}_{\overline{\text{\tiny MS}}}$
$\displaystyle=$
$\displaystyle\overline{\sigma}_{x}^{2}+\delta{\overline{\sigma}_{x}}^{2}_{\text{\tiny
OS}}-\delta{\overline{\sigma}_{x}}^{2}_{\overline{\text{\tiny MS}}}\;,$ (138)
$\displaystyle{\overline{\sigma}_{y}}^{2}_{\overline{\text{\tiny MS}}}$
$\displaystyle=$
$\displaystyle\overline{\sigma}_{y}^{2}+\delta{\overline{\sigma}_{y}}^{2}_{\text{\tiny
OS}}-\delta{\overline{\sigma}_{y}}^{2}_{\overline{\text{\tiny MS}}}\;.$ (139)

The vacuum effective potential minimum lies at $\overline{x}=f_{\pi}$ and
$\overline{y}=\frac{(2f_{K}-f_{\pi})}{\sqrt{2}}$. Using the above set of
equations together with the Eqs. (108), (110), (114), (117), (120), (123) and
(125)–(129), one writes the scale $\Lambda$ dependent running parameters in
the $\overline{\text{MS}}$ scheme as the following

$\displaystyle\lambda_{2\overline{\text{\tiny MS}}}(\Lambda)$ $\displaystyle=$
$\displaystyle\lambda_{2}+\lambda_{2\text{\tiny{FIN}}}\;,$ (140)
$\displaystyle c_{\overline{\text{\tiny MS}}}(\Lambda)$ $\displaystyle=$
$\displaystyle c+c_{\text{\tiny{FINTOT}}}\;,$ (141)
$\displaystyle\lambda_{1\overline{\text{\tiny MS}}}(\Lambda)$ $\displaystyle=$
$\displaystyle\lambda_{1}+\lambda_{2\text{\tiny{FIN}}}\;,$ (142)
$\displaystyle m^{2}_{\overline{\text{\tiny MS}}}(\Lambda)$ $\displaystyle=$
$\displaystyle m^{2}+m^{2}_{\text{\tiny{FIN}}}\;,$ (143) $\displaystyle
h_{x}{\overline{\text{\tiny MS}}}(\Lambda)$ $\displaystyle=$ $\displaystyle
h_{x}+h_{x\text{\tiny{FIN}}}\;,$ (144) $\displaystyle
h_{y}{\overline{\text{\tiny MS}}}(\Lambda)$ $\displaystyle=$ $\displaystyle
h_{y}+h_{y\text{\tiny{FIN}}}\;,$ (145) $\displaystyle
g^{2}_{\overline{\text{\tiny MS}}}(\Lambda)$ $\displaystyle=$ $\displaystyle
g^{2}+\frac{N_{c}g^{4}}{(4\pi)^{2}}\text{SCF}\;,$ (146)
$\displaystyle{\overline{\sigma}_{x}}^{2}_{\overline{\text{\tiny
MS}}}(\Lambda)$ $\displaystyle=$ $\displaystyle
f_{\pi}^{2}-\dfrac{4N_{c}m^{2}_{u}}{(4\pi)^{2}}\text{SCF}\;,$ (147)
$\displaystyle{\overline{\sigma}_{y}}^{2}_{\overline{\text{\tiny
MS}}}(\Lambda)$ $\displaystyle=$
$\displaystyle\left(\frac{2f_{K}-f_{\pi}}{\sqrt{2}}\right)^{2}-\dfrac{2N_{c}m^{2}_{s}}{(4\pi)^{2}}\text{SCF}\;.$
(148)

In the Eqs. (140)–(146), the parameters $\lambda_{2}$, $c$, $\lambda_{1}$,
$m^{2}$, $h_{x}$, $h_{y}$ and $g^{2}$, have the tree level values of the QM
model when the $\overline{\sigma}_{x}=f_{\pi}$ and the
$\overline{\sigma}_{y}=\frac{(2f_{K}-f_{\pi})}{\sqrt{2}}$ in the vacuum.

The parameters $\lambda_{2\overline{\text{\tiny MS}}}$,
$c_{\overline{\text{\tiny MS}}}$, $\lambda_{1\overline{\text{\tiny MS}}}$,
$m^{2}_{\overline{\text{\tiny MS}}}$, $h_{x\overline{\text{\tiny MS}}}$,
$h_{y\overline{\text{\tiny MS}}}$ and $g^{2}_{\overline{\text{\tiny MS}}}$ in
the large-$N_{c}$ limit, are running with the scale $\Lambda$ and satisfy the
following set of simultaneous renormalization group equations

$\displaystyle\dfrac{d\lambda_{2\overline{\text{\tiny
MS}}}(\Lambda)}{d\ln(\Lambda)}$ $\displaystyle=$
$\displaystyle\dfrac{2N_{c}}{(4\pi)^{2}}\left[2\lambda_{2\overline{\text{\tiny
MS}}}g^{2}_{\overline{\text{\tiny MS}}}-g^{4}_{\overline{\text{\tiny
MS}}}\right]\;,$ (149) $\displaystyle\dfrac{dc_{\overline{\text{\tiny
MS}}}(\Lambda)}{d\ln(\Lambda)}$ $\displaystyle=$
$\displaystyle\dfrac{2N_{c}}{(4\pi)^{2}}g^{2}_{\overline{\text{\tiny
MS}}}c_{\overline{\text{\tiny MS}}}\;,$ (150)
$\displaystyle\dfrac{d\lambda_{1\overline{\text{\tiny
MS}}}(\Lambda)}{d\ln(\Lambda)}$ $\displaystyle=$
$\displaystyle\dfrac{4N_{c}}{(4\pi)^{2}}g^{2}_{\overline{\text{\tiny
MS}}}\lambda_{1\overline{\text{\tiny MS}}}\;,$ (151)
$\displaystyle\dfrac{dm^{2}_{\overline{\text{\tiny
MS}}}(\Lambda)}{d\ln(\Lambda)}$ $\displaystyle=$
$\displaystyle\dfrac{2N_{c}}{(4\pi)^{2}}g^{2}_{\overline{\text{\tiny
MS}}}m^{2}_{\overline{\text{\tiny MS}}}\;,$ (152)
$\displaystyle\dfrac{dh_{x\overline{\text{\tiny
MS}}}(\Lambda)}{d\ln(\Lambda)}$ $\displaystyle=$
$\displaystyle\dfrac{N_{c}}{(4\pi)^{2}}g^{2}_{\overline{\text{\tiny
MS}}}h_{x\overline{\text{\tiny MS}}}\;,$ (153)
$\displaystyle\dfrac{dh_{y\overline{\text{\tiny
MS}}}(\Lambda)}{d\ln(\Lambda)}$ $\displaystyle=$
$\displaystyle\dfrac{N_{c}}{(4\pi)^{2}}g^{2}_{\overline{\text{\tiny
MS}}}h_{y\overline{\text{\tiny MS}}}\;,$ (154)
$\displaystyle\dfrac{dg^{2}_{\overline{\text{\tiny MS}}}}{d\ln(\Lambda)}$
$\displaystyle=$
$\displaystyle\dfrac{2N_{c}}{(4\pi)^{2}}g^{4}_{\overline{\text{\tiny MS}}}\;,$
(155) $\displaystyle\dfrac{d{\overline{\sigma}_{x}}^{2}_{\overline{\text{\tiny
MS}}}}{d\ln(\Lambda)}$ $\displaystyle=$
$\displaystyle-\dfrac{2N_{c}}{(4\pi)^{2}}g^{2}_{\overline{\text{\tiny
MS}}}{\overline{\sigma}_{x}}^{2}_{\overline{\text{\tiny MS}}}\;,$ (156)
$\displaystyle\dfrac{d{\overline{\sigma}_{y}}^{2}_{\overline{\text{\tiny
MS}}}}{d\ln(\Lambda)}$ $\displaystyle=$
$\displaystyle-\dfrac{2N_{c}}{(4\pi)^{2}}g^{2}_{\overline{\text{\tiny
MS}}}{\overline{\sigma}_{y}}^{2}_{\overline{\text{\tiny MS}}}\;.$ (157)

The differential the Eqs. (149)–(157) have the following solutions.

$\displaystyle{\hskip-14.22636pt}\lambda_{2\overline{\text{\tiny
MS}}}(\Lambda)$ $\displaystyle=$
$\displaystyle\frac{\lambda_{20}-\dfrac{N_{c}g^{4}_{0}}{(4\pi)^{2}}\ln\left(\dfrac{\Lambda^{2}}{\Lambda^{2}_{0}}\right)}{\left(1-\dfrac{N_{c}g^{2}_{0}}{(4\pi)^{2}}\ln\left(\dfrac{\Lambda^{2}}{\Lambda^{2}_{0}}\right)\right)^{2}}\;,$
(158) $\displaystyle{\hskip-14.22636pt}c_{\overline{\text{\tiny
MS}}}(\Lambda)$ $\displaystyle=$
$\displaystyle\frac{c_{0}}{\sqrt{\left[1-\dfrac{N_{c}g^{2}_{0}}{(4\pi)^{2}}\ln\left(\dfrac{\Lambda^{2}}{\Lambda^{2}_{0}}\right)\right]^{3}}}\;,$
(159) $\displaystyle{\hskip-14.22636pt}\lambda_{1\overline{\text{\tiny
MS}}}(\Lambda)$ $\displaystyle=$
$\displaystyle\frac{\lambda_{10}}{\left(1-\dfrac{N_{c}g^{2}_{0}}{(4\pi)^{2}}\ln\left(\dfrac{\Lambda^{2}}{\Lambda_{0}^{2}}\right)\right)^{2}}\;,$
(160) $\displaystyle m^{2}_{\overline{\text{\tiny MS}}}(\Lambda)$
$\displaystyle=$
$\displaystyle\frac{m^{2}_{0}}{1-\dfrac{N_{c}g^{2}_{0}}{(4\pi)^{2}}\ln\left(\dfrac{\Lambda^{2}}{\Lambda^{2}_{0}}\right)}\;,$
(161) $\displaystyle{\hskip-14.22636pt}h_{x\overline{\text{\tiny
MS}}}(\Lambda)$ $\displaystyle=$
$\displaystyle\frac{h_{x0}}{\sqrt{1-\dfrac{N_{c}g^{2}_{0}}{(4\pi)^{2}}\ln\left(\dfrac{\Lambda^{2}}{\Lambda^{2}_{0}}\right)}}\;,$
(162) $\displaystyle{\hskip-14.22636pt}h_{y\overline{\text{\tiny
MS}}}(\Lambda)$ $\displaystyle=$
$\displaystyle\frac{h_{y0}}{\sqrt{1-\dfrac{N_{c}g^{2}_{0}}{(4\pi)^{2}}\ln\left(\dfrac{\Lambda^{2}}{\Lambda^{2}_{0}}\right)}}\;,$
(163) $\displaystyle{\hskip-14.22636pt}g^{2}_{\overline{\text{\tiny
MS}}}(\Lambda)$ $\displaystyle=$
$\displaystyle\frac{g^{2}_{0}}{1-\dfrac{N_{c}g^{2}_{0}}{(4\pi)^{2}}\ln\left(\dfrac{\Lambda^{2}}{\Lambda^{2}_{0}}\right)}\;,$
(164)
$\displaystyle{\hskip-14.22636pt}{\overline{\sigma}_{x}^{2}}_{\overline{\text{\tiny
MS}}}$ $\displaystyle=$ $\displaystyle
f^{2}_{\pi}\left[1-\frac{N_{c}g^{2}_{0}}{(4\pi)^{2}}\ln\left(\frac{\Lambda^{2}}{\Lambda^{2}_{0}}\right)\right]\;,$
(165)
$\displaystyle{\hskip-14.22636pt}{\overline{\sigma}_{y}}^{2}_{\overline{\text{\tiny
MS}}}$ $\displaystyle=$
$\displaystyle\frac{(2f_{K}-f_{\pi})^{2}}{2}\left[1-\frac{N_{c}g^{2}_{0}}{(4\pi)^{2}}\ln\left(\frac{\Lambda^{2}}{\Lambda^{2}_{0}}\right)\right]\;.$
(166)

Here the $\lambda_{10}$, $\lambda_{20}$, $g^{2}_{0}$, $m^{2}_{0}$, $c_{0}$,
$h_{x0}$ and $h_{y0}$ are the values of the running parameters at the scale
$\Lambda_{0}$. The $\Lambda_{0}$ can be chosen to satisfy the relation,

$\displaystyle\ln\left(\frac{\Lambda^{2}_{0}}{m_{u}^{2}}\right)+\mathcal{C}(m^{2}_{\pi})+m^{2}_{\pi}\mathcal{C}^{\prime}(m^{2}_{\pi})$
$\displaystyle=$ $\displaystyle 0\;.$ (167)

The parameters of the Eqs. (140)–(148) are obtained at the scale
$\Lambda=\Lambda_{0}$.

###  III.3 Derivation of the effective potential

The values of the parameters in the Eqs. (158)–(164) can be used to find
expression of the vacuum effective potential in the $\overline{\text{MS}}$
scheme as,

$\displaystyle\Omega_{vac}$ $\displaystyle=$ $\displaystyle
U({\overline{\sigma}_{x}}_{\overline{\text{\tiny
MS}}},{\overline{\sigma}_{y}}_{\overline{\text{\tiny
MS}}})+\Omega^{q,vac}_{\overline{\text{\tiny MS}}}+\delta
U({\overline{\sigma}_{x}}_{\overline{\text{\tiny
MS}}},{\overline{\sigma}_{y}}_{\overline{\text{\tiny MS}}})\;,$ (168)

where

$\displaystyle U({\overline{\sigma}_{x}}_{\overline{\text{\tiny
MS}}},{\overline{\sigma}_{y}}_{\overline{\text{\tiny MS}}})$ $\displaystyle=$
$\displaystyle\frac{m_{\overline{\text{\tiny
MS}}}^{2}}{2}\left({\overline{\sigma}_{x}}_{\overline{\text{\tiny
MS}}}^{2}+{\overline{\sigma}_{y}}_{\overline{\text{\tiny
MS}}}^{2}\right)-h_{x\overline{\text{\tiny MS}}}\
{\overline{\sigma}_{x}}_{\overline{\text{\tiny MS}}}-h_{y\overline{\text{\tiny
MS}}}{\overline{\sigma}_{y}}_{\overline{\text{\tiny
MS}}}-\frac{c_{\overline{\text{\tiny
MS}}}}{2\sqrt{2}}{\overline{\sigma}_{x}}_{\overline{\text{\tiny
MS}}}^{2}{\overline{\sigma}_{y}}_{\overline{\text{\tiny
MS}}}+\frac{\lambda_{1\overline{\text{\tiny
MS}}}}{2}{\overline{\sigma}_{x}}_{\overline{\text{\tiny
MS}}}^{2}{\overline{\sigma}_{y}}_{\overline{\text{\tiny MS}}}^{2}$ (169)
$\displaystyle+\frac{2\lambda_{1\overline{\text{\tiny
MS}}}+\lambda_{2\overline{\text{\tiny MS}}}}{8}\
{\overline{\sigma}_{x}}_{\overline{\text{\tiny
MS}}}^{4}+\frac{2\lambda_{1\overline{\text{\tiny
MS}}}+2\lambda_{2\overline{\text{\tiny MS}}}}{8}\
{\overline{\sigma}_{y}}_{\overline{\text{\tiny MS}}}^{4}\;,$
$\displaystyle\delta U({\overline{\sigma}_{x}}_{\overline{\text{\tiny
MS}}},{\overline{\sigma}_{y}}_{\overline{\text{\tiny MS}}})$ $\displaystyle=$
$\displaystyle\frac{\delta m_{\overline{\text{\tiny
MS}}}^{2}}{2}\left(\overline{\sigma}_{x\overline{\text{\tiny
MS}}}^{2}+\overline{\sigma}_{y\overline{\text{\tiny
MS}}}^{2}\right)+\frac{m_{\overline{\text{\tiny
MS}}}^{2}}{2}\left(\delta\overline{\sigma}_{x\overline{\text{\tiny
MS}}}^{2}+\delta\overline{\sigma}_{y\overline{\text{\tiny
MS}}}^{2}\right)-\delta h_{x\overline{\text{\tiny
MS}}}\overline{\sigma}_{x\overline{\text{\tiny MS}}}-h_{x\overline{\text{\tiny
MS}}}\ \delta{\overline{\sigma}_{x}}_{\overline{\text{\tiny
MS}}}\quad\quad\quad\;$ $\displaystyle-\delta h_{y\overline{\text{\tiny MS}}}\
\overline{\sigma}_{y\overline{\text{\tiny MS}}}-h_{y\overline{\text{\tiny
MS}}}\ \delta{\overline{\sigma}_{y\overline{\text{\tiny MS}}}}-\frac{\delta
c_{\overline{\text{\tiny MS}}}}{2\sqrt{2}}\
{\overline{\sigma}_{x}}_{\overline{\text{\tiny MS}}}^{2}\
{\overline{\sigma}_{y}}_{\overline{\text{\tiny
MS}}}-\frac{c_{\overline{\text{\tiny MS}}}}{2\sqrt{2}}\
(\delta{\overline{\sigma}_{x}}_{\overline{\text{\tiny MS}}}^{2}\
{\overline{\sigma}_{y}}_{\overline{\text{\tiny
MS}}}+{\overline{\sigma}_{x}}_{\overline{\text{\tiny MS}}}^{2}\
\delta{\overline{\sigma}_{y}}_{\overline{\text{\tiny MS}}})$
$\displaystyle+\frac{\delta\lambda_{1\overline{\text{\tiny
MS}}}}{2}{\overline{\sigma}_{x}}_{\overline{\text{\tiny MS}}}^{2}\
{\overline{\sigma}_{y}}_{\overline{\text{\tiny
MS}}}^{2}+\frac{\lambda_{1\overline{\text{\tiny
MS}}}}{2}(\delta{\overline{\sigma}_{x}}_{\overline{\text{\tiny MS}}}^{2}\
{\overline{\sigma}_{y}}_{\overline{\text{\tiny
MS}}}^{2}+{\overline{\sigma}_{x}}_{\overline{\text{\tiny MS}}}^{2}\
\delta{\overline{\sigma}_{y}}_{\overline{\text{\tiny
MS}}}^{2})+(\frac{2\delta\lambda_{1\overline{\text{\tiny
MS}}}+\delta\lambda_{2\overline{\text{\tiny MS}}}}{8})\
{\overline{\sigma}_{x}}_{\overline{\text{\tiny MS}}}^{4}\;$
$\displaystyle+(\frac{2\lambda_{1\overline{\text{\tiny
MS}}}+\lambda_{2\overline{\text{\tiny MS}}}}{8})\
\delta{\overline{\sigma}_{x}}_{\overline{\text{\tiny
MS}}}^{4}+(\frac{2\delta\lambda_{1\overline{\text{\tiny
MS}}}+2\delta\lambda_{2\overline{\text{\tiny MS}}}}{8})\
{\overline{\sigma}_{y}}_{\overline{\text{\tiny
MS}}}^{4}+(\frac{2\lambda_{1\overline{\text{\tiny
MS}}}+2\lambda_{2\overline{\text{\tiny MS}}}}{8})\
\delta{\overline{\sigma}_{y}}_{\overline{\text{\tiny MS}}}^{4}\;,$

Dropping the two loop terms ($\mathcal{O}(N^{2}_{c})$), we get

$\displaystyle\delta U({\overline{\sigma}_{x}}_{\overline{\text{\tiny
MS}}},{\overline{\sigma}_{y}}_{\overline{\text{\tiny MS}}})$ $\displaystyle=$
$\displaystyle-\frac{N_{c}g^{4}_{\overline{\text{\tiny
MS}}}({\overline{\sigma}_{x}}^{4}_{\overline{\text{\tiny
MS}}}+2{\overline{\sigma}_{y}}^{4}_{\overline{\text{\tiny
MS}}})}{8(4\pi)^{2}}\frac{1}{\epsilon}=-\frac{N_{c}(2\Delta_{x}^{4}+\Delta_{y}^{4})}{(4\pi)^{2}}\frac{1}{\epsilon}\;.$
(171)

One writes the quark one-loop vacuum correction for the two non-strange quark
and the one strange quark flavor as,

$\displaystyle\Omega^{q,vac}_{\overline{\text{\tiny MS}}}$ $\displaystyle=$
$\displaystyle\frac{N_{c}g^{4}_{\overline{\text{\tiny
MS}}}{\overline{\sigma}_{x}}^{4}_{\overline{\text{\tiny
MS}}}}{8(4\pi)^{2}}\left[\frac{1}{\epsilon}+\frac{3}{2}+\ln\left(\frac{4\Lambda^{2}}{g^{2}_{\overline{\text{\tiny
MS}}}{\overline{\sigma}_{x}}_{\overline{\text{\tiny
MS}}}^{2}}\right)\right]+\frac{N_{c}g^{4}_{\overline{\text{\tiny
MS}}}2{\overline{\sigma}_{y}}^{4}_{\overline{\text{\tiny
MS}}}}{8(4\pi)^{2}}\left[\frac{1}{\epsilon}+\frac{3}{2}+\ln\left(\frac{2\Lambda^{2}}{g^{2}_{\overline{\text{\tiny
MS}}}{\overline{\sigma}_{y}}^{2}_{\overline{\text{\tiny MS}}}}\right)\right]$
(172) $\displaystyle=$
$\displaystyle\frac{2N_{c}\Delta_{x}^{4}}{(4\pi)^{2}}\left[\frac{1}{\epsilon}+\frac{3}{2}+\ln\left(\frac{\Lambda^{2}}{\Delta_{x}^{2}}\right)\right]+\frac{N_{c}\Delta_{y}^{4}}{(4\pi)^{2}}\left[\frac{1}{\epsilon}+\frac{3}{2}+\ln\left(\frac{\Lambda^{2}}{\Delta_{y}^{2}}\right)\right]\;.$

The scale $\Lambda$ independent parameters
$\Delta_{x}=\frac{g_{\overline{\text{\tiny MS}}}\
{\overline{\sigma}_{x}}_{\overline{\text{\tiny MS}}}}{2}$ and
$\Delta_{y}=\frac{g_{\overline{\text{\tiny MS}}}\
{\overline{\sigma}_{y}}_{\overline{\text{\tiny MS}}}}{\sqrt{2}}$ are defined
by the use of the Eqs. (146), (147) and (148). The Eq. (169) is written in
terms of the scale independent $\Delta_{x}$ and $\Delta_{y}$ as

$\displaystyle U(\Delta_{x},\Delta_{y})=\frac{m_{\overline{\text{\tiny
MS}}}^{2}(\Lambda)}{g^{2}_{\overline{\text{\tiny
MS}}}(\Lambda)}(2\Delta_{x}^{2}+\Delta_{y}^{2})-2\frac{h_{x\overline{\text{\tiny
MS}}}(\Lambda)}{g_{\overline{\text{\tiny
MS}}}(\Lambda)}\Delta_{x}-\sqrt{2}\frac{h_{y\overline{\text{\tiny
MS}}}(\Lambda)}{g_{\overline{\text{\tiny
MS}}}(\Lambda)}\Delta_{y}-2\frac{c_{\overline{\text{\tiny
MS}}}(\Lambda)}{g^{3}_{\overline{\text{\tiny MS}}}(\Lambda)}\Delta_{x}^{2}\
\Delta_{y}+4\frac{\lambda_{1\overline{\text{\tiny
MS}}}(\Lambda)}{g^{4}_{\overline{\text{\tiny MS}}}(\Lambda)}\Delta_{x}^{2}\
\Delta_{y}^{2}$ $\displaystyle{\hskip
56.9055pt}+2\frac{(2\lambda_{1\overline{\text{\tiny
MS}}}+\lambda_{2\overline{\text{\tiny MS}}})}{g^{4}_{\overline{\text{\tiny
MS}}}(\Lambda)}\ \Delta_{x}^{4}+\frac{(\lambda_{1\overline{\text{\tiny
MS}}}+\lambda_{2\overline{\text{\tiny MS}}})}{g^{4}_{\overline{\text{\tiny
MS}}}(\Lambda)}\ \Delta_{y}^{4}\;,$ (173) $\displaystyle
U(\Delta_{x},\Delta_{y})=\frac{m^{2}_{0}}{g^{2}_{0}}(2\Delta_{x}^{2}+\Delta_{y}^{2})-2\frac{h_{x0}}{g_{0}}\Delta_{x}-\sqrt{2}\frac{h_{y0}}{g_{0}}\Delta_{y}-2\frac{c_{0}}{g^{3}_{0}}\Delta_{x}^{2}\
\Delta_{y}+4\frac{\lambda_{10}}{g^{4}_{0}}\Delta_{x}^{2}\ \Delta_{y}^{2}\ $
$\displaystyle{\hskip
56.9055pt}+2\frac{(2\lambda_{10}+\lambda_{20})}{g^{4}_{0}}\
\Delta_{x}^{4}+\frac{(\lambda_{10}+\lambda_{20})}{g^{4}_{0}}\
\Delta_{y}^{4}\;,$ (174)
$\displaystyle\Omega_{vac}(\Delta_{x},\Delta_{y})=\frac{m^{2}_{0}}{g^{2}_{0}}(2\Delta_{x}^{2}+\Delta_{y}^{2})-2\frac{h_{x0}}{g_{0}}\Delta_{x}-\sqrt{2}\frac{h_{y0}}{g_{0}}\Delta_{y}-2\frac{c_{0}}{g^{3}_{0}}\Delta_{x}^{2}\
\Delta_{y}+4\frac{\lambda_{10}}{g^{4}_{0}}\Delta_{x}^{2}\
\Delta_{y}^{2}+2\frac{(2\lambda_{10}+\lambda_{20})}{g^{4}_{0}}\Delta_{x}^{4}\
$ $\displaystyle{\hskip
56.9055pt}+\frac{(\lambda_{10}+\lambda_{20})}{g^{4}_{0}}\Delta_{y}^{4}+\frac{2N_{c}\Delta_{x}^{4}}{(4\pi)^{2}}\left[\frac{3}{2}+\ln\left(\frac{\Lambda^{2}}{\Delta_{x}^{2}}\right)\right]+\frac{N_{c}\Delta_{y}^{4}}{(4\pi)^{2}}\left[\frac{3}{2}+\ln\left(\frac{\Lambda^{2}}{\Delta_{y}^{2}}\right)\right]\;.$
(175)

Expressing the mass parameter and the couplings in terms of the physical
masses of mesons, the pion decay constant, the kaon decay constant and Yukawa
coupling, we write

$\displaystyle\Omega_{vac}(\Delta_{x},\Delta_{y})$ $\displaystyle=$
$\displaystyle\frac{(m^{2}+m^{2}_{\text{\tiny{FIN}}})}{2}\left\\{f^{2}_{\pi}\left(\frac{\Delta_{x}^{2}}{m^{2}_{u}}\right)+\frac{(2f_{K}-f_{\pi})^{2}}{2}\left(\frac{\Delta_{y}^{2}}{m^{2}_{s}}\right)\right\\}-(h_{x}+h_{x\text{\tiny{FIN}}})f_{\pi}\left(\frac{\Delta_{x}}{m_{u}}\right)-(h_{y}+h_{y\text{\tiny{FIN}}})\frac{(2f_{K}-f_{\pi})}{\sqrt{2}}\;$
$\displaystyle\left(\frac{\Delta_{y}}{m_{s}}\right)-\frac{(c+c_{\text{\tiny{FINTOT}}})}{4}f^{2}_{\pi}(2f_{K}-f_{\pi})\left(\frac{\Delta_{x}^{2}}{m^{2}_{u}}\right)\left(\frac{\Delta_{y}}{m_{s}}\right)+\frac{(\lambda_{1}+\lambda_{1\text{\tiny{FIN}}})}{4}\
f^{2}_{\pi}(2f_{K}-f_{\pi})^{2}\left(\frac{\Delta_{x}^{2}}{m^{2}_{u}}\right)\left(\frac{\Delta_{y}^{2}}{m^{2}_{s}}\right)\;$
$\displaystyle+\frac{\\{2(\lambda_{1}+\lambda_{1\text{\tiny{FIN}}})+(\lambda_{2}+\lambda_{2\text{\tiny{FIN}}})\\}}{8}f^{4}_{\pi}\left(\frac{\Delta_{x}^{4}}{m^{4}_{u}}\right)+\frac{\\{(\lambda_{1}+\lambda_{1\text{\tiny{FIN}}})+(\lambda_{2}+\lambda_{2\text{\tiny{FIN}}})\\}}{16}(2f_{K}-f_{\pi})^{4}\left(\frac{\Delta_{y}^{4}}{m^{4}_{s}}\right)$
$\displaystyle+\frac{2N_{c}\Delta_{x}^{4}}{(4\pi)^{2}}\left[\frac{3}{2}-\ln\left(\frac{\Delta_{x}^{2}}{m^{2}_{u}}\right)-\mathcal{C}(m^{2}_{\pi})-m^{2}_{\pi}\mathcal{C}^{\prime}(m^{2}_{\pi})\right]+\frac{N_{c}\Delta_{y}^{4}}{(4\pi)^{2}}\left[\frac{3}{2}-\ln\left(\frac{\Delta_{y}^{2}}{m^{2}_{u}}\right)-\mathcal{C}(m^{2}_{\pi})-m^{2}_{\pi}\mathcal{C}^{\prime}(m^{2}_{\pi})\right]\;.$

One notes that due to the dressing of the meson propagator in the on-shell
scheme of the RQM model, the pion decay constant, the kaon decay constant and
Yukawa coupling get renormalized in the vacuum. But the Eqs. (146), (147) and
(148) at the scale $\Lambda_{0}$ give us $g_{\overline{\text{\tiny
MS}}}=g_{ren}=g$, ${\overline{\sigma}_{x}}_{\overline{\text{\tiny
MS}}}=f_{\pi,ren}=f_{\pi}$ and ${\overline{\sigma}_{y}}_{\overline{\text{\tiny
MS}}}=\frac{2f_{K,ren}-f_{\pi,ren}}{\sqrt{2}}=\frac{2f_{K}-f_{\pi}}{\sqrt{2}}$.
When the stationarity condition
$\frac{\partial\Omega_{vac}(\Delta_{x},\Delta_{y})}{\partial\Delta_{x}}=0$ is
applied to the Eq. (III.3) in the non-strange direction, one gets
$h_{x0}=m_{\pi,c}^{2}\ {\overline{\sigma}_{x}}_{\overline{\text{\tiny
MS}}}=m^{2}_{\pi}\left\\{1-\frac{N_{c}g^{2}}{(4\pi)^{2}}m^{2}_{\pi}\mathcal{C}^{\prime}(m^{2}_{\pi})\right\\}f_{\pi}$.
Hence the pion curvature mass
$m_{\pi,c}^{2}=m^{2}_{\pi}\left\\{1-\frac{N_{c}g^{2}}{(4\pi)^{2}}m^{2}_{\pi}\mathcal{C}^{\prime}(m^{2}_{\pi})\right\\}$.
The implementation of the stationarity condition
$\frac{\partial\Omega_{vac}(\Delta_{x},\Delta_{y})}{\partial\Delta_{y}}=0$ in
the strange direction, gives $\quad
h_{0y}=\left({\frac{{\overline{\sigma}_{x}}_{\overline{\text{\tiny
MS}}}}{\sqrt{2}}+{\overline{\sigma}_{y}}_{\overline{\text{\tiny
MS}}}}\right)m^{2}_{K,c}-\frac{{\overline{\sigma}_{x}}_{\overline{\text{\tiny
MS}}}}{\sqrt{2}}m^{2}_{\pi,c}$=$\sqrt{2}f_{K}m^{2}_{K,c}-\frac{f_{\pi}}{\sqrt{2}}m^{2}_{\pi,c}$.
One finds the expression of the kaon curvature mass $m_{K,c}^{2}$ as given
below in the Eq. (177) by using the expression of $h_{y\overline{\text{\tiny
MS}}}(\Lambda_{0})=h_{y0}$ in the Eq. (145). It is worth emphasizing that the
pion curvature mass $m_{\pi,c}$ (as in Ref. fix1 ) and the kaon curvature mass
are different from their pole masses $m_{\pi}$ and $m_{K}$ due to the
consistent on-shell parameter fixing. The minimum of the effective potential
remains fixed at ${\overline{\sigma}_{x}}_{\overline{\text{\tiny
MS}}}=f_{\pi}$ and ${\overline{\sigma}_{y}}_{\overline{\text{\tiny
MS}}}=\frac{(2f_{K}-f_{\pi})}{\sqrt{2}}$ .

Table 3: Parameters of the different model scenarios. The RQM model parameters are obtained by putting the $\Lambda=\Lambda_{0}$ in the Eqs. (140)–(145). Model |  $m_{\sigma}(\text{MeV})$ |  $\lambda_{2}$ |  $c(\text{MeV}^{2})$ |  $\lambda_{1}$ |  $m^{2}(\text{MeV}^{2})$ |  $h_{x}(\text{MeV}^{3})$ |  $h_{y}(\text{MeV}^{3})$
---|---|---|---|---|---|---|---
|  $400$ |  46.43 |  4801.82 |  -5.89 |  $(494.549)^{2}$ |  $(120.73)^{3}$ |  $(336.43)^{3}$
QM |  $500$ |  46.43 |  4801.82 |  -2.69 |  $(434.305)^{2}$ |  $(120.73)^{3}$ |  $(336.43)^{3}$
|  $600$ |  46.43 |  4801.82 |  1.141 |  $(342.139)^{2}$ |  $(120.73)^{3}$ |  $(336.43)^{3}$
|  $400$ |  34.88 |  7269.20 |  1.45 |  $(442.447)^{2}$ |  $(119.53)^{3}$ |  $(323.32)^{3}$
RQM |  $500$ |  34.88 |  7269.20 |  3.676 |  $(396.075)^{2}$ |  $(119.53)^{3}$ |  $(323.32)^{3}$
|  $600$ |  34.88 |  7269.20 |  8.890 |  $(256.506)^{2}$ |  $(119.53)^{3}$ |  $(323.32)^{3}$

$\leavevmode\resizebox{446.2658pt}{}{
$m^{2}_{K,c}=m^{2}_{K}\left[1-\frac{N_{c}g^{2}}{(4\pi)^{2}}\left\\{\mathcal{C}(m^{2}_{\pi},m_{u})+m^{2}_{\pi}\mathcal{C}^{\prime}(m^{2}_{\pi},m_{u})-\left(1-\frac{(m_{s}-m_{u})^{2}}{m_{K}^{2}}\right)\mathcal{C}(m^{2}_{K},m_{u},m_{s})+\left(1-\frac{f_{\pi}}{f_{K}}\right)\frac{m^{2}_{u}-m_{s}^{2}+2m_{s}^{2}\ln\left(\frac{m_{s}}{m_{u}}\right)}{m^{2}_{K}}\right\\}\right]$}\;.$
(177)

Combining the above RQM model vacuum effective potential with the thermal
contributions of quarks-antiquarks and the Polyakov loop potential, we write
the grand thermodynamic potential of the RPQM model as the following,

$\displaystyle\Omega_{RPQM}(\Delta_{x},\Delta_{y},\Phi,\bar{\Phi},T,\mu)$
$\displaystyle=$
$\displaystyle\frac{(m^{2}+m^{2}_{\text{\tiny{FIN}}})}{2}\left\\{f^{2}_{\pi}\left(\frac{\Delta_{x}^{2}}{m^{2}_{u}}\right)+\frac{(2f_{K}-f_{\pi})^{2}}{2}\left(\frac{\Delta_{y}^{2}}{m^{2}_{s}}\right)\right\\}-(h_{x}+h_{x\text{\tiny{FIN}}})f_{\pi}\left(\frac{\Delta_{x}}{m_{u}}\right)\;$
(178)
$\displaystyle{\hskip-56.9055pt}-(h_{y}+h_{y\text{\tiny{FIN}}})\frac{(2f_{K}-f_{\pi})}{\sqrt{2}}\left(\frac{\Delta_{y}}{m_{s}}\right)-\frac{(c+c_{\text{\tiny{FINTOT}}})}{4}f^{2}_{\pi}(2f_{K}-f_{\pi})\left(\frac{\Delta_{x}^{2}}{m^{2}_{u}}\right)\left(\frac{\Delta_{y}}{m_{s}}\right)+\frac{(\lambda_{1}+\lambda_{1\text{\tiny{FIN}}})}{4}\
f^{2}_{\pi}(2f_{K}-f_{\pi})^{2}\;$
$\displaystyle{\hskip-56.9055pt}\left(\frac{\Delta_{x}^{2}}{m^{2}_{u}}\right)\left(\frac{\Delta_{y}^{2}}{m^{2}_{s}}\right)+\frac{\\{2(\lambda_{1}+\lambda_{1\text{\tiny{FIN}}})+(\lambda_{2}+\lambda_{2\text{\tiny{FIN}}})\\}}{8}f^{4}_{\pi}\left(\frac{\Delta_{x}^{4}}{m^{4}_{u}}\right)+\frac{\\{(\lambda_{1}+\lambda_{1\text{\tiny{FIN}}})+(\lambda_{2}+\lambda_{2\text{\tiny{FIN}}})\\}}{16}(2f_{K}-f_{\pi})^{4}\left(\frac{\Delta_{y}^{4}}{m^{4}_{s}}\right)$
$\displaystyle{\hskip-56.9055pt}+\frac{2N_{c}\Delta_{x}^{4}}{(4\pi)^{2}}\left[\frac{3}{2}-\ln\left(\frac{\Delta_{x}^{2}}{m^{2}_{u}}\right)-\mathcal{C}(m^{2}_{\pi})-m^{2}_{\pi}\mathcal{C}^{\prime}(m^{2}_{\pi})\right]+\frac{N_{c}\Delta_{y}^{4}}{(4\pi)^{2}}\left[\frac{3}{2}-\ln\left(\frac{\Delta_{y}^{2}}{m^{2}_{u}}\right)-\mathcal{C}(m^{2}_{\pi})-m^{2}_{\pi}\mathcal{C}^{\prime}(m^{2}_{\pi})\right]\;$
$\displaystyle{\hskip-56.9055pt}+\mathcal{U}(T,\Phi,\bar{\Phi})+\Omega_{q\bar{q}}(T,\mu;\Delta_{x},\Delta_{y},\Phi,\bar{\Phi})\;.$

One gets the nonstrange condensate $\Delta_{x}$, strange condensate
$\Delta_{y}$, $\Phi$ and $\bar{\Phi}$ in the RPQM model by searching the
global minimum of the grand potential in the Eq. (178) for a given value of
temperature $T$ and chemical potential $\mu$

$\frac{\partial\Omega_{RPQM}}{\partial\Delta_{x}}=\frac{\partial\Omega_{RPQM}}{\partial\Delta_{y}}=\frac{\partial\Omega_{RPQM}}{\partial\Phi}=\frac{\partial\Omega_{RPQM}}{\partial\bar{\Phi}}=0$
(179)

In our calculations, we have used the $m_{\pi}=138.0$ MeV, $m_{K}=496$ MeV.
Here in the RQM model, fixing the
$m_{\eta}^{2}+m_{\eta^{\prime}}^{2}=(547.5)^{2}+(957.78)^{2}$ and then taking
the $\eta$ mass as 527.58 MeV, one gets the ${\eta^{\prime}}$ mass equal to
968.89 MeV. The pole mass $m_{\eta}=527.58$ MeV and $m_{\eta^{\prime}}=968.89$
MeV have been used for calculating the self energy corrections (for
$\eta,\eta^{\prime}$) and fixing of the parameters in the on-shell scheme
because it has been checked that when the masses are calculated with the new
set of renormalized parameters and respective self energy corrections are
added, the same pole masses are reproduced.

(a) Normalized non-strange chiral condensate.

(b) Normalized strange chiral condensate.

Figure 5: Temperature variation of the $\sigma_{x}/\sigma_{x0}$ and the
$\sigma_{y}/\sigma_{y0}$ for the $m_{\sigma}=500$ MeV at $\mu=0$. The
$\sigma_{x0}=f_{\pi}$ and $\sigma_{y0}=\frac{2f_{K}-f_{\pi}}{\sqrt{2}}$

##  IV Results and Discussion

The non-strange and strange direction chiral crossover transition and the
confinement-deconfinement transition occurring on the temperature axis at
$\mu=0$ are being explored here in detail. The pseudo-critical temperature for
the non-strange, the strange chiral crossover transition, $T^{\chi}_{c}$,
$T^{s}_{c}$ and the confinement-deconfinement transition, $T^{\Phi}_{c}$ are
obtained by identifying the peaks in the respective temperature direction
variation of the temperature derivative
$\frac{\partial(\overline{\sigma}_{x}/{\overline{\sigma}_{x}}_{0})}{\partial
T}$,
$\frac{\partial(\overline{\sigma}_{y}/{\overline{\sigma}_{y}}_{0})}{\partial
T}$ and $\frac{\partial\Phi}{\partial T}$.

Figure 6: Temperature variation of the
$\frac{\partial\overline{\sigma}_{x}/{\overline{\sigma}_{x}}_{0}}{\partial T}$
and $\frac{\partial\Phi}{\partial T}$ for the $m_{\sigma}=500$ MeV at $\mu=0$.

The Fig.5(a) and the Fig.5(b), show the respective temperature variations of
the normalized non-strange and strange chiral condensate at $\mu=0$ MeV and
$m_{\sigma}=500$ MeV for the PQM and the RPQM model with the Log and the
PolyLog-glue form of the Polyakov-loop potential. The term glue stands for the
unquenching of the Polyakov loop potential in the presence of the quark back-
reaction at $\mu=0$. Though, the temperature variation of the non-strange
condensate in the PolyLog-glue PQM model in the Fig. 5(a), shifts early on the
temperature scale due to the quark back-reaction, it is sharper than that of
the Log PQM model because, the early occuring weaker deconfinement transition
at the $T_{c}^{\Phi}=138.8$ MeV (from the Table 4) in the PolyLog-glue PQM
model, does not influence its chiral transition (at the $T_{c}^{\chi}=155.2$
MeV) that much while the Log PQM model relatively stronger deconfinement
transition which occurs at the $T_{c}^{\Phi}=149.9$ MeV influences the chiral
transition at the $T_{c}^{\chi}=159.3$ MeV. The above behavior can be inferred
from the temperature variation of the
$\frac{\partial(\overline{\sigma}_{x}/{\overline{\sigma}_{x}}_{0})}{\partial
T}$ and the $\frac{\partial\Phi}{\partial T}$ in the Log and the PolyLog-glue
PQM model in the Fig. 6. The temperature variation of the non-strange
condensate becomes significantly smoother due to the quark one-loop vacuum
correction in the Log RPQM model as shown the Fig 5(a) and it also gets
influenced by Log form of the Polyakov loop potential which causes a stronger
deconfinement transition. The sharper temperature variation of the
$\frac{\partial\Phi}{\partial T}$ and it’s peak at the $T_{c}^{\Phi}=148.1$
MeV has a strong influence on the chiral transition, because the
$\frac{\partial(\overline{\sigma}_{x}/{\overline{\sigma}_{x}}_{0})}{\partial
T}$ temperature variation peaks at the adjacent $T_{c}^{\chi}=148.9$ MeV and
develops a small very slowly decreasing shoulder like structure thereafter
(falling short of developing the second peak). Therefore the finding of the
$T_{c}^{\chi}$ has an in-built ambiguity of $\sim+9.0$ MeV. Due to the
combined effect of the quark one-loop vacuum correction and the quark back-
reaction, one gets most smooth temperature variation of the non-strange
condensate in the Fig. 5(a) for the PolyLog-glue RPQM model. The temperature
variation of the $\frac{\partial\Phi}{\partial T}$ is very smooth and well
seperated from the quite smooth temperature variation of the
$\frac{\partial(\overline{\sigma}_{x}/{\overline{\sigma}_{x}}_{0})}{\partial
T}$ and the deconfinement transition occurs early as
$\frac{\partial\Phi}{\partial T}$ temperature variation peaks at
$T_{c}^{\Phi}=136.6$ MeV while the temperature variation of the
$\frac{\partial(\overline{\sigma}_{x}/{\overline{\sigma}_{x}}_{0})}{\partial
T}$ peaks later giving $T_{c}^{\chi}=157.3$ MeV.

Table 4: Pseudo-critical temperatures at $\mu=$0. Polyakov-loop |  Models | $m_{\sigma}=400\ \text{MeV}$ | $m_{\sigma}=500\ \text{MeV}$ | $m_{\sigma}=600\ \text{MeV}$
---|---|---|---|---
|  |  $T^{\chi}_{c}(\text{MeV})$ |  $T^{s}_{c}(\text{MeV})$ |  $T^{\Phi}_{c}(\text{MeV})$ |  $T^{\chi}_{c}(\text{MeV})$ |  $T^{s}_{c}(\text{MeV})$ |  $T^{\Phi}_{c}(\text{MeV})$ |  $T^{\chi}_{c}(\text{MeV})$ |  $T^{s}_{c}(\text{MeV})$ |  $T^{\Phi}_{c}(\text{MeV})$
Log |  PQM |  $149.5$ |  $244.3$ |  $149.5$ |  $159.3$ |  $252.1$ |  $149.9$ |  $171.9$ |  $261.7$ |  $150.5$
|  RPQM |  $146.4$ |  $222.8$ |  $146.0$ |  $148.9$ |  $225.6$ |  $148.1$ |  $181.1$ |  $233.8$ |  $150.1$
PolyLog-glue |  PQM |  $142.5$ |  $245.1$ |  $142.4$ |  $155.2$ |  $252.3$ |  $138.8$ |  $169.5$ |  $261.2$ |  $129.1$
|  RPQM |  $145.6$ |  $225.6$ |  $133.6$ |  $157.3$ |  $227.8$ |  $136.6$ |  $179.6$ |  $235.3$ |  $138.6$

Strange condensate temperature variations for both the Log and the PolyLog-
glue PQM models have a small and sharp kink like structure near the non-
strange chiral transition temperature $T_{c}^{\chi}$. The above kink like
stucture gets smoothed out and hence disappears due to effect of quark one-
loop vacuum correction in the RPQM model. The quark back-reaction present in
the PolyLog-glue PQM model causes a noticeable decrease in the strange
condensate in the temperature range 120 to 165 MeV. After $T>165$ MeV, the
strange condensate temperature variation for the PolyLog-glue PQM model
becomes degenerate with that of the Log PQM model. Note that the strange
direction explicit symmetry breaking strength $h_{y}$ (see Table 3) becomes
weaker by relatively large amount (in comparison to $h_{x}$) as the kaon
curvature mass $m_{K,c}=467.99$ MeV becomes smaller than it’s pole mass
$m_{K}=$ 496 MeV after renormalization of the parameters in the RPQM model.
Hence, the melting of the strange condensate gets significantly enhanced due
to the effect of quark one-loop vacuum correction in the RPQM model. In the
temperature range 110 to 185 MeV around the non-strange chiral crossover
transition temperature ($T^{\chi}_{c}$), the melting of the strange condensate
in the Log RPQM model differs significantly from its largest melting noticed
in the PolyLog-glue RPQM model. The strange condensate temperature variation
for the Log and the PolyLog-glue RPQM model merge with each other for $T>180$
MeV. The earliest chiral transition in the strange direction occurs in the Log
RPQM model at $T_{c}^{s}=225.6$ MeV while the strange direction chiral
transition for the Log PQM model occurs at the $T_{c}^{s}$= 252.1 MeV. The
Table 4 summarizes the $\mu=0$ transition pseudocritical temperatures
$T^{\chi}_{c}$, $T^{s}_{c}$ and $T_{c}^{\Phi}$ for $m_{\sigma}=$ 400, 500 and
600 MeV in the PQM and the RPQM model with the Log and the PolyLog-glue form
of the Polyakov-loop potential.

(a) Subtracted chiral condensate $\Delta_{ls}$.

(b) The Polyakov-loop order parameter $\Phi$.

Figure 7: Temperature variation of the $\Delta_{ls}$ and the $\Phi$ for the
$m_{\sigma}=500$ MeV at $\mu=0$. The lattice data of the WBLQCD-I for the
$\Delta_{ls}$ and the $\Phi$, has been taken from the Ref. Wupertal2010  in
the continum limit. The Cheng et.al. LQCD data for the $\Phi$ has been taken
from the Ref. Cheng:06  for the $N_{\tau}=6$ in the p4-action.

The lattice data can not be directly compared with the temperature variations
of the non-strange and strange condensate, $<\overline{\sigma}_{x}>$ and
$<\overline{\sigma}_{y}>$. A suitable combination of the light and strange
quark condensates is calculated to eliminate the quadratic divergences in the
linear quark mass dependent correction to the chiral condensate. The above
quantity is further normalized by the corresponding combination of condensates
calculated at $T=0$ MeV Cheng:08 . Thus one finds, the chiral symmetry
breaking order parameter called the subtracted chiral condensate
$\Delta_{l,s}(T)$ as given below.

$\Delta_{l,s}(T)=\frac{<\overline{\sigma}_{x}>(T)-(\frac{h_{x}}{h_{y}})<\overline{\sigma}_{y}>(T)}{<\overline{\sigma}_{x}>(0)-(\frac{h_{x}}{h_{y}})<\overline{\sigma}_{y}>(0)}.$
(180)

The subtracted chiral condensate temperature variations (at $\mu=0$) have been
calculated for the PQM and the RPQM model with the Log and the PolyLog-glue
form of the Polyakov loop potential, and compared with the Wuppertal-Budapest
collaboration lattice QCD (WBLQCD-I) data of the $\Delta_{ls}$ Wupertal2010
in the Fig. 7(a). We find that the calculation of the $\Delta_{ls}$
temperature variation in our PolyLog-glue RPQM model gives the closest fit to
the WBLQCD-I data. The Fig.7(b) presents the comparison of the WBLQCD-I
Wupertal2010  and the lattice data in Ref. Cheng:08  for the reduced
temperature scale variation of the Polyakov loop with the temperature
variation of the Polyakov loop condensate obtained in the PQM and RPQM model
calculations having the Log and the PolyLog-glue form for the Polyakov loop
potential. In model calculations the reduced temperature $T/T_{c}$ has been
obtained by taking $T_{c}=T_{c}^{\Phi}$. The lattice data Cheng:08  for the
Polyakov loop in the temperature range 0.7 $T_{c}$ to 1.7 $T_{c}$ shows quite
a close resemblance with the Polyakov loop temperature variation that we get
in our PolyLog-glue RPQM model calculation. The WBLQCD-I Wupertal2010  data
for the Polyakov loop lie significantly below the temperature variation of the
Polyakov loop condensate that one finds in the all model calculations.

###  IV.1 Comparison of thermodynamic quantities with the lattice QCD data

We will compare the thermodynamic quantities: pressure, entropy density,
energy density, specific heat and speed of sound in the Log PQM model and the
RPQM model with the Log and the PolyLog-glue form of the Polyakov-loop
potential. The QCD phase transition has strong influence on these
thermodynamic quantities.

The pressure of a QCD system can be defined as negative of the grand
potential.

$P(T,\mu)=-\Omega(T,\mu)$ (181)

the pressure is normalized to zero in the vacuum, i.e. $P(0,0)=0$.

Figure 8: Temperature variation of the pressure for the $m_{\sigma}=500$ MeV
at $\mu$=0. The lattice data of the pressure, in the continuum limit of the
HotQCD-I and WBLQCD-II, have been taken from the Ref. HotQCD2014  and the Ref.
WB2014  respectively. The Cheng et.al. LQCD data has been taken from the Ref.
Cheng:06  for the $N_{\tau}=6$ in the p4-action. Here $T_{c}=T_{c}^{\chi}$.

The reduced temperature scale variation of the pressure for the PolyLog-glue
RPQM model (where the quark back-reaction and the quark one-loop vacuum
correction, both are present) in the Fig. 8, stands close to the lattice QCD
data of the pressure that has been reported by the WBLQCD-II WB2014  and the
HotQCD-I HotQCD2014  collaboration. The lattice data of Ref. Cheng:08  for the
pressure, are in exact agreement with the temperature variation of the
pressure for the PolyLog-glue RPQM model below $T=1.1\ T_{c}$ and above this
temperature, the lattice data become closer to the temperature variation of
the pressure calculated in the Log PQM model.

In order to find the thermodynamic quantities other than pressure, one needs
to take temperature derivative of the grand potential. The entropy density
$s$, the energy density $\epsilon$ and the interaction measure $\Delta$ at the
zero quark chemical potential, are defined by

$\displaystyle s$ $\displaystyle=$
$\displaystyle-\frac{\partial\Omega}{\partial T}\;,$ (182)
$\displaystyle\epsilon$ $\displaystyle=$ $\displaystyle-P+Ts\;,$ (183)
$\displaystyle\Delta$ $\displaystyle=$ $\displaystyle\epsilon-3P\;.$ (184)

(a) Energy density

(b) Entropy density

Figure 9: Temperature variation of the energy density and entropy density for
the $m_{\sigma}=500$ MeV at $\mu$=0. The lattice data of the energy density
and entropy density, in the continuum limit of the HotQCD-I and WBLQCD-II,
have been taken from the the Ref. HotQCD2014  and the Ref. WB2014
respectively. The Cheng et.al. LQCD data has been taken from the Ref. Cheng:06
for the $N_{\tau}=6$ in the p4-action. Here $T_{c}=T_{c}^{\chi}$.

The respective temperature variations of the energy density and the entropy
density in the Fig. 9(a) and the Fig. 9(b), either in the PQM or in the RPQM
model with the Log form of the Polyakov-loop potential, increase very rapidly
near the $T_{c}$ and attain saturation after the temperature 1.5 $T_{c}$. The
Ref. Cheng:08  lattice data of the energy density (entropy density), are close
to the energy density (entropy density) temperature variation of the Log PQM
model when the $T<1.6\ T_{c}$ and afterwards, the LQCD data become closer to
the energy density (entropy density) obtained in the PolyLog-glue RPQM model
calculation. The lattice QCD data of the WBLQCD-II WB2014  and the HotQCD-I
HotQCD2014  collaborations for the energy density (entropy density)