# On-shell versus curvature mass parameter fixing schemes in the quark-meson
model and its phase diagrams

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

We compute and compare the effective potential and phase structure for the
quark-meson model in an extended mean-field approximation (e-MFA) when vacuum
one loop quark fluctuations are included and the model parameters are fixed
using different renormalization prescriptions.When the quark one loop vacuum
divergence is regularized under the minimal subtraction scheme,the model
setting of the parameter fixing using the curvature masses of the scalar and
pseudo-scalar mesons, has been termed as the quark-meson model with the vacuum
term (QMVT).However,this prescription becomes inconsistent when we notice that
the curvature mass is akin to defining the meson mass by the self-energy
evaluation at vanishing momentum.In this work,we apply the recently reported
exact prescription of the on-shell parameter fixing, to that version of quark-
meson (QM) model where the two quark flavors are coupled to the eight mesons
of the $SU(2)_{L}\times SU(2)_{R}$ linear sigma model with iso-singlet
$\sigma$ ($\eta$), iso-triplet $\vec{a_{0}}$ ($\vec{\pi}$) scalar(pseudo-
scalar) mesons.The model then becomes, the renormalized quark-meson (RQM)
model where physical (pole) masses of mesons and pion decay constant, are put
into the relation of the running mass parameter and couplings by using the on-
shell and the minimal subtraction renormalization schemes.The vacuum effective
potential plots,the phase diagrams and the order parameter temperature
variations for both the RQM model and QMVT model,are exactly identical for the
$m_{\sigma}=$616 MeV.The vacuum effective potential when the $m_{\sigma}<$ 616
MeV,is deepest for the QMVT model and an interesting trend reversal is
observed when the effective potential of the RQM model becomes deepest for the
$m_{\sigma}>$ 616 MeV.We find similar $m_{\sigma}$ dependent differences in
the nature of the RQM and QMVT model phase diagrams and the order parameter
temperature variations.Furthermore, $SU(2)_{A}$ chiral and $U(1)_{A}$ axial
symmetry breaking/restoration and their interplay can also be investigated in
this framework.

Dense QCD, chiral transition,

##  I Introduction

Strong interaction theory motivated, first quantum chromodynamics (QCD)
schematic phase diagram, appeared in 1970s Cabibbo75  which projected a low
temperature (low baryonic density) confined phase of hadrons and a high
temperature (zero baryonic density) or high baryonic density (zero
temperature) phase of quarks and gluons SveLer ; Mull ; Ortms ; Riske .Mapping
out the QCD phase diagram in all its details is still a very active area of
current research as it is not very well understood.The first-principle lattice
QCD simulationsAliKhan:2001ek ; Digal:01 ; Karsch:02 ; Fodor:03 ; Allton:05 ;
Karsch:05 ; Aoki:06 ; Cheng:06 ; Cheng:08 provide us valuable information for
the QCD phase transition but the real progress in the lattice QCD
calculations,gets seriously bogged down by the QCD action becoming complex due
to the fermion sign problemKarsch:02 when baryon density/chemical potential
becomes nonzero.The phenomenological models developed with the effective
degrees of freedomAlf ; Fukhat ,are of great help in mapping out the phase
diagram in the regions inaccessible to the lattice simulations.

The QCD Lagrangian for the two flavor of massless quarks has the
$SU(2)_{L+R}\times SU(2)_{L-R}$ symmetry.The axial(A=L-R) part of the symmetry
called the chiral symmetry,gets spontaneously broken in the low energy
hadronic vacuum of the QCD as the chiral condensate forms and one gets three
massless pions as Goldstone bosons.The chiral symmetry gets explicitly broken
as well,due to the small mass of the u and d quarks and we find light pions in
the nature.The $SU(2)_{L}\times SU(2)_{R}$ linear sigma model provide us a
good framework Roder ; fuku11 ; grahl  in which the chiral $SU(2)_{A}$
symmetry and the axial $U(1)_{A}$ symmetry breaking and restoration both can
be investigated in great detail as it enables the construction of chiral
invariant combinations using the chiral partners from the iso-singlet $\sigma$
and the iso-triplet $\vec{a_{0}}$ scalar mesons to the iso-singlet $\eta$ and
iso-triplet $\vec{\pi}$ pseudo-scalar mesons.Coupling these eight scalar and
pseudo-scalar meson degrees of freedom to the two flavor of quarks, we get the
QCD-like framework of the quark-meson (QM) model in which we can compute and
explore the QCD phase diagram.

Furthermore,at low temperatures and densities,the confinement of quarks inside
the hadrons also,gets implemented by the introduction of the Polyakov loop
where the QCD confinement is mimicked in a statistical sense by coupling the
chiral models to a constant background $SU(N_{c})$ gauge field $A_{\mu}^{a}$
Polyakov:78plb ; fuku ; benji ; Pisarski:00prd ; Vkt:06 .Using
phenomenological Polyakov loop potential ratti ; fuku2 ,the free energy
density from the gluons is added to the QM model and it becomes the PQM
model.Several investigations of the QCD phase structure/phase diagram, have
already been done in the chiral models scav ; Rischke:00 ; jakobi ; Herpay:05
; Herpay:06 ; Herpay:07 ; Kovacs:2006ym ; kahara ; Bowman:2008kc ; Fejos ;
Jakovac:2010uy ; koch ; marko , two and three flavor QM model mocsy ; bj ;
Schaefer:2006ds ; Schaefer:09  and PQM model SchaPQM2F ; SchaPQM3F ; Mao ;
TiPQM3F .

Under the “no-Dirac sea” or standard mean field approximation
(s-MFA),fermionic vacuum fluctuations and renormalization issues are neglected
altogetherscav ; mocsy ; bj ; Schaefer:2006ds ; SchaPQM2F ; kahara ;
Schaefer:09 ; SchaPQM3F ; Mao ; TiPQM3F  assuming that the redefined meson
potential parameters would reabsorb their effects.This familiar setting of the
QM model gives inconsistent result as in the chiral limit,one gets a first-
order chiral phase transition at zero baryon densities which is at odds with
the general theoretical argumentsrob ; hjss .The above inconsistency is
remedied with the proper treatment of the Dirac sea as first proposed in the
Ref. vac  and later several research papers lars ; guptiw ; schafwag12 ;
chatmoh1 ; TranAnd ; vkkr12 ; chatmoh2 ; vkkt13 ; Herbst ; Weyrich ; kovacs ;
zacchi1 ; zacchi2 ; Rai  worked out the detailed impact of including the quark
one loop vacuum correction in the two and three flavor QM/PQM model.In these
publications,it has been the standard procedure to identify the pion decay
constant with the vacuum expectation value of the sigma mean field and to use
the $\sigma$ and $\pi$ meson curvature mass,defined by the second derivative
of the thermodynamic potential at its minimum,for fixing the model parameters
where the minimal subtraction scheme has been used to properly regularize the
quark one loop vacuum divergence.Since the effective potential generates the
n-point functions of the theory at vanishing external momenta,the curvature
mass is akin to defining the meson mass by the evaluation of self-energy at
zero momentum laine ; Adhiand1 ; BubaCar ; Naylor ; fix1 .This consideration
renders the above parameter fixing procedure inconsistent.In order to make
comparisons and quantify the effect of the parameter fixing with the curvature
meson masses,we have named this model setting as the quark-meson model with
vacuum term (QMVT).

The six parameters, $\lambda_{1}$, $\lambda_{2}$, t’Hooft coupling $c$, mass
parameter $m^{2}$, explicit symmetry breaking strength $h$ and Yukawa coupling
$g$ of the QM model Lagrangian are determined by the physical values of
$m_{\sigma}$,$m_{\pi}$,eta meson mass $m_{\eta}$,isotriplet scalar meson mass
$m_{\vec{a_{0}}}$,constituent quark mass $m_{q}$ and pion decay constant
$f_{\pi}$.It is to be noted that,in most renormalization procedures,radiative
corrections to the physical quantities change their tree level relations to
the parameters of the Lagrangian.Thus the use of tree level values of the
parameters, in the effective potential calculation, becomes inconsistent.The
$\overline{\text{MS}}$ scheme running parameters depend on the renormalization
scale $\Lambda$,whereas the on-shell parameters have their tree-level values.
The correct renormalization prescription allows us to calculate the
counterterms both in the $\overline{\text{MS}}$ scheme and in the on-shell
scheme and then the renormalized parameters of the two schemes get
connected.The effective potential is then calculated using the modified
minimal subtraction procedure where the relations between the running
parameters and the on-shell parameters (physical quantities) are used as input
Adhiand1 .Adhikari and collaborators in a series of papers Adhiand1 ; Adhiand2
; Adhiand3 ; asmuAnd  used this renormalization prescription to correctly
account for the effect of Dirac sea in the context of QM model where $O(4)$
sigma model has been used for the mesonic degree of freedom (iso-singlet
scalar meson $\sigma$ and iso-triplet $\pi$).In the present work, we are
applying this prescription of the on-shell parameter fixing to that version of
the quark-meson (QM) model in which the two flavor of quarks are coupled to
the eight mesons of the $SU(2)_{L}\times SU(2)_{R}$ linear sigma model with
iso-singlet $\sigma$, iso-triplet $\vec{a_{0}}$ scalar mesons and iso-singlet
$\eta$, iso-triplet $\vec{\pi}$ pseudo-scalar mesons.We have termed this model
setting as the renormalized quark-meson (RQM) model which has the advantage of
providing us the framework in which apart from the $SU(2)_{A}$ chiral, we can
investigate the $U(1)_{A}$ axial symmetry breaking and restoration also
together with the interplay of axial $U(1)_{A}$ and $SU(2)_{A}$ chiral
symmetry.

The paper is arranged as follows.The brief formulation of the $SU(2)_{L}\times
SU(2)_{R}$ QM model is presented in the section II.The section III presents
the calculation of the effective potential of the quark-meson model with
vacuum term (QMVT) together with its parameter fixing procedure which uses the
curvature masses of the scalar and pseudo-scalar mesons. The on-shell scheme
counterterms and self-energy calculations are presented in the section
IV-A.The relations between the physical quantities and the running parameters,
are derived in the section IV-B,the derivation of the effective potential in
the RQM model is also presented in the section IV-C.The result and discussion
is presented in the section V.Finally summary and conclusion is presented in
the section VI.

##  II Model Formulation

The $SU(2)_{L}\times SU(2)_{R}$ quark-meson model formulation will be
presented in this section. In the two flavor quark meson chiral linear sigma
model,two light quarks and $SU_{V}(2)\times SU_{A}(2)$ symmetric meson fields
are coupled together.The Lagrangian of the model Roder ; fuku11 ; grahl  is
written as

$\displaystyle{\cal L_{QM}}$ $\displaystyle=$
$\displaystyle\bar{\psi}[i\gamma^{\mu}\partial_{\mu}-gt_{0}(\sigma+i\gamma_{5}\eta)$
(1)
$\displaystyle-g\vec{t}\cdot(\vec{a}+i\gamma_{5}\vec{\pi})]\psi+\cal{L(M)},$

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

$\displaystyle\psi$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}u\\\ d\end{array}\right)\;.$ (4)

The Lagrangian for meson fields is Roder

$\displaystyle\cal{L(M)}$ $\displaystyle=$
$\displaystyle\text{Tr}(\partial_{\mu}{\cal{M}}^{\dagger}\partial^{\mu}{\cal{M}}-m^{2}({\cal{M}}^{\dagger}{\cal{M}}))$
(5)
$\displaystyle-\lambda_{1}\left[\text{Tr}({\cal{M}}^{\dagger}{\cal{M}})\right]^{2}-\lambda_{2}\text{Tr}({\cal{M}}^{\dagger}{\cal{M}})^{2}$
$\displaystyle+c[\text{det}{\cal{M}}+\text{det}{\cal{M}}^{\dagger}]+\text{Tr}\left[H({\cal{M}}+{\cal{M}}^{\dagger})\right]$

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

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

$a=$0,1,2 and 3. $t_{a}$ represents the 4 generators of the $U(2)$ algebra.

$\displaystyle{\cal{M}}$ $\displaystyle=$ $\displaystyle
t_{0}(\sigma_{0}+i\pi_{0})+\vec{t}\cdot(\vec{\sigma}+i\vec{\pi})\;$ (6)
$\displaystyle=$ $\displaystyle
t_{0}(\sigma+i\eta)+\vec{t}\cdot(\vec{a}+i\vec{\pi})$

with $t_{0}=\dfrac{1}{2}\left(\begin{array}[]{c c}1&0\\\
0&1\end{array}\right),$ $t_{1}=\dfrac{1}{2}\left(\begin{array}[]{c c}0&1\\\
1&0\end{array}\right)$
, $t_{2}=\dfrac{1}{2}\left(\begin{array}[]{c c}0&-i\\\ i&0\end{array}\right),$
$t_{3}=\dfrac{1}{2}\left(\begin{array}[]{c c}1&0\\\ 0&-1\end{array}\right).$
One can rewrite the Lagrangian (5) in the form fuku11

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

further

$\displaystyle U$ $\displaystyle=$
$\displaystyle\dfrac{m^{2}}{2}(\sigma^{2}+{\vec{\pi}}^{2}+\eta^{2}+{\vec{a}}^{2})-\dfrac{c}{2}(\sigma^{2}-\eta^{2}+{\vec{\pi}}^{2}-{\vec{a}}^{2})$
(8)
$\displaystyle+\dfrac{1}{4}\left(\lambda_{1}+\dfrac{1}{2}\lambda_{2}\right)(\sigma^{2}+{\vec{\pi}}^{2}+\eta^{2}+{\vec{a}}^{2})^{2}$
$\displaystyle+\dfrac{\lambda_{2}}{2}\left((\sigma^{2}+{\vec{\pi}}^{2})(\eta^{2}+{\vec{a}}^{2})-(\sigma\eta-{\vec{\pi}}\cdot{\vec{a}})^{2}\right)$
$\displaystyle-h\sigma$

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

$\displaystyle H=t_{a}h_{a}$ (9)

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

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

$\displaystyle m^{2}_{\sigma}$ $\displaystyle=$ $\displaystyle
m^{2}-c+3(\lambda_{1}+\frac{\lambda_{2}}{2})\overline{\sigma}^{2}\;,$ (10)
$\displaystyle m^{2}_{a_{0}}$ $\displaystyle=$ $\displaystyle
m^{2}+c+(\lambda_{1}+\frac{3\lambda_{2}}{2})\overline{\sigma}^{2}\;,$ (11)
$\displaystyle m^{2}_{\eta}$ $\displaystyle=$ $\displaystyle
m^{2}+c+(\lambda_{1}+\frac{\lambda_{2}}{2})\overline{\sigma}^{2}\;,$ (12)
$\displaystyle m^{2}_{\pi}$ $\displaystyle=$ $\displaystyle
m^{2}-c+(\lambda_{1}+\frac{\lambda_{2}}{2})\overline{\sigma}^{2}\;,$ (13)
$\displaystyle m_{q}$ $\displaystyle=$ $\displaystyle\frac{g\
\overline{\sigma}}{2}$ (14)

Using (10)–(14),the parameters of the Lagrangian (5) are obtained as

$\displaystyle\lambda_{1}$ $\displaystyle=$
$\displaystyle\dfrac{m^{2}_{\sigma}+m^{2}_{\eta}-m^{2}_{a_{0}}-m^{2}_{\pi}}{2\overline{\sigma}^{2}}\;,$
(15) $\displaystyle\lambda_{2}$ $\displaystyle=$
$\displaystyle\dfrac{m^{2}_{a_{0}}-m^{2}_{\eta}}{\overline{\sigma}^{2}}\;,$
(16) $\displaystyle m^{2}$ $\displaystyle=$ $\displaystyle
m^{2}_{\pi}+\dfrac{m^{2}_{\eta}-m^{2}_{\sigma}}{2}\;,$ (17) $\displaystyle c$
$\displaystyle=$ $\displaystyle\dfrac{m^{2}_{\eta}-m^{2}_{\pi}}{2}\;,$ (18)
$\displaystyle\frac{g^{2}}{4}$ $\displaystyle=$
$\displaystyle\frac{m_{q}^{2}}{\overline{\sigma}^{2}}$ (19)

and the tree level effective potential is written as,

$\displaystyle
U(\overline{\sigma})=\frac{1}{2}m^{2}\overline{\sigma}^{2}-\frac{1}{2}c\overline{\sigma}^{2}+\frac{1}{4}\left(\lambda_{1}+\frac{1}{2}\lambda_{2}\right)\overline{\sigma}^{4}-h\overline{\sigma}\;.\
\ $ (20)

Stationarity condition for the effective potential (20), gives

$\displaystyle h$ $\displaystyle=$ $\displaystyle
m^{2}_{\pi}\overline{\sigma}$ (21)

The minimum of the effective potential at the tree level is given by
$\overline{\sigma}=f_{\pi}$.Here it is pertinent to mention that when one
reads the coefficient of $\overline{\sigma}^{2}\over 2$ as the one single
coefficient $(m^{2}-c)$ in which $m_{\eta}$ dependence cancles out and the
coefficient of $\overline{\sigma}^{4}\over 4$ as the another single
coefficient $(\lambda_{1}+\frac{\lambda_{2}}{2})$ where the $m_{\eta}$ and
$m_{a_{0}}$ dependence cancles out, we see that the tree level effective
potential of the $SU(2)_{L}\times SU(2)_{R}$ sigma model becomes equivalent to
the tree level potential of the $O(4)$ sigma model,where the degrees of
freedom are the $\sigma$ and $\pi$ only.

We are considering a spatially uniform thermodynamic system in the equilibrium
at temperature $T$ and chemical potentail $\mu_{q}\ (q=u\ \text{and}\ d)$.The
partition function is written as the path integral over the quark/antiquark
and meson fields scav ; Schaefer:09 ; TiPQM3F  ,

$\displaystyle\mathcal{Z}$ $\displaystyle=$ $\displaystyle{\rm
Tr}\exp\left[-\beta\left(\hat{\mathcal{H}}-\mu\hat{\mathcal{N}}\right)\right]\;$
(22) $\displaystyle=$
$\displaystyle\int\mathcal{D}\sigma\mathcal{D}\vec{a_{0}}\mathcal{D}\eta\mathcal{D}\vec{\pi}\int\mathcal{D}\psi\mathcal{D}\bar{\psi}\exp\left[-\int^{\beta}_{0}d\tau\int_{V}d^{3}x\right.$
$\displaystyle\left.\left(\mathcal{L}^{\mathcal{E}}_{\mathcal{QM}}+\sum_{q=u,d}\mu\bar{q}\gamma^{0}q\right)\right]$

Where $V$ is the volume of the system, $\beta=\frac{1}{T}$, and the
superscript $\mathcal{E}$ denotes the Euclidean Lagrangian.In this paper,we
assume that the masses of $u$ and $d$ quarks are equal in magnitude.Thus the
quark chemical potential of the $u$ and $d$ quarks become equal,i.e.
$\mu=\mu_{u}=\mu_{d}$.We evaluate the partition function in the mean field
approximation.We replace the meson fields by their expectation value
$<\mathcal{M}>=t_{0}\overline{\sigma}$, and neglect both the thermal and
quantum fluctuations of the meson fields,while the quarks and antiquarks are
retained as quantum fields.

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

$\displaystyle\Omega_{\rm MF}(T,\mu;\overline{\sigma})$ $\displaystyle=$
$\displaystyle-T\frac{\ln\mathcal{Z}}{V}\;$ (23) $\displaystyle=$
$\displaystyle
U(\overline{\sigma})+\Omega_{q\bar{q}}(T,\mu;\overline{\sigma}).$

The quark/antiquark contribution is given by

$\displaystyle\Omega_{q\bar{q}}(T,\mu;\overline{\sigma})$ $\displaystyle=$
$\displaystyle\Omega_{q\bar{q}}^{vac}+\Omega_{q\bar{q}}^{T,\mu}\;$ (24)
$\displaystyle\Omega_{q\bar{q}}^{vac}$ $\displaystyle=$
$\displaystyle-2N_{c}\sum_{q}\int\frac{d^{3}p}{(2\pi)^{3}}E_{q}\theta(\Lambda_{c}^{2}-\vec{p}^{2})\;$
(25) $\displaystyle\Omega_{q\bar{q}}^{T,\mu}$ $\displaystyle=$
$\displaystyle-2N_{c}\sum_{q}\int\frac{d^{3}p}{(2\pi)^{3}}T\left[\ln\left(1+e^{-E_{q}^{+}/T}\right)\right.$
(26) $\displaystyle\left.+\ln\left(1+e^{-E_{q}^{-}/T}\right)\right]\;$

The first term of the Eq. (24) denotes the fermion vacuum contribution,where
$\Lambda_{c}$ is the ultraviolet cutoff. E${}_{q}^{\pm}=E_{q}\mp\mu$ and
$E_{q}=\sqrt{p^{2}+m{{}_{q}}{{}^{2}}}$ is the flavor dependent single particle
energy of quark/antiquark, $m_{q}=\dfrac{g\overline{\sigma}}{2}$ is the mass
of the given quark flavor.

Neglecting the quark one loop vacuum term of Eq. (24) in the the standard
mean-field approximation (s-MFA),the QM model grand potential is written as,

$\displaystyle\Omega_{QM}((\overline{\sigma},T,\mu))$ $\displaystyle=$
$\displaystyle U(\overline{\sigma})+\Omega_{q\bar{q}}^{T,\mu}$
$\frac{\partial\Omega_{QM}(\overline{\sigma},T,\mu)}{\partial\overline{\sigma}}=0.$
(27)

The global minima of the grand potential in the Eq.(27),gives the chiral
condensate as a function of the temperature T and chemical potential $\mu$.

##  III QM model with Vacuum Term

We will describe the calculation of the effective potential when the quark
one-loop vacuum divergence of Eq. (24) is properly regularized using the
minimal subtraction scheme and the $\sigma$ and $\pi$ meson curvature mass
(screening mass),defined by the second derivative of the thermodynamic
potential at its minimum, has been used for fixing the model parameters.The
zero temperature quark one-loop vacuum contribution can be written as

$\displaystyle\Omega^{vac}_{q\bar{q}}$ $\displaystyle=$
$\displaystyle-2N_{c}\sum_{q}\int\frac{d^{3}p}{(2\pi)^{3}}E_{q}\;$ (28)
$\displaystyle=-2N_{c}\sum_{q}\int\frac{d^{4}p}{(2\pi)^{4}}\ln(p^{2}_{0}+E^{2}_{q})+K$

The infinite constant $K$ is dropped.When the Eq.(28) is dimensionally
regularized near three dimensions, $d=3-2\epsilon$, one gets the $\epsilon$
zeroth order potential as :

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

Redefining $\Lambda^{2}\longrightarrow\Lambda^{2}\frac{e^{\gamma_{E}}}{4\pi}$
in Eq.(III), one gets

$\displaystyle\Omega^{vac}_{q\bar{q}}$ $\displaystyle=$
$\displaystyle\frac{N_{c}}{(4\pi)^{2}}\sum_{q}m^{4}_{q}\left[\frac{1}{\epsilon}+\frac{3}{2}+\ln\left(\Lambda^{2}\over
m^{2}_{q}\right)\right]\;$ (30)

Where $\Lambda$ is renormalization scale.

The thermodynamic potential is renormalized by adding the following
counterterm to the Lagrangian of the QM model.

$\displaystyle\delta\mathcal{L}$ $\displaystyle=$
$\displaystyle\frac{N_{c}}{(4\pi)^{2}}\sum_{q}\frac{m^{4}_{q}}{\epsilon}\;.$
(31)

Now the first term of Eq.(24) is replaced by the appropriately renormalized
quark one-loop vacuum contribution of Eq.(III)

$\displaystyle\Omega^{vac}_{q\bar{q}}$ $\displaystyle=$
$\displaystyle\frac{N_{c}}{(4\pi)^{2}}\sum_{q}m^{4}_{q}\left[\frac{3}{2}+\ln\left(\Lambda^{2}\over
m^{2}_{q}\right)\right]\;$ (32)

Since the vacuum ($\mu=0$ and $T=0$) grand potential gets contributions from
$U(\overline{\sigma})$ and $\Omega^{vac}_{q\bar{q}}$, it becomes
renormalization scale dependent.

$\displaystyle\Omega^{\Lambda}(\overline{\sigma})$ $\displaystyle=$
$\displaystyle U(\overline{\sigma})+\Omega^{vac}_{q\bar{q}}\;.$ (33)

The unknown model parameters $m^{2}$,$c$,$\lambda_{1}$,$\lambda_{2}$,$h$ are
obtained using the meson curvature masses,which are found by taking the second
derivative of Eq.(33) evaluated at the minimum with respect to the different
meson fields.The details of the model parameter determination, are presented
in the Appendix (A).

When the calculated model parameters get substituted in the expression of
$U(\overline{\sigma})$,one can rewrite the Eq.(33) as:

$\displaystyle\Omega^{\Lambda}(\overline{\sigma})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(m^{2}_{s}-\dfrac{N_{c}g^{4}f^{2}_{\pi}}{2(4\pi)^{2}}\right)\overline{\sigma}^{2}-\frac{1}{2}c\overline{\sigma}^{2}+\frac{1}{4}\left(\lambda_{1}+\frac{\lambda_{2s}}{2}\right.$
(34)
$\displaystyle\left.-\frac{N_{c}g^{4}}{2(4\pi)^{2}}\ln\left(\frac{4\Lambda^{2}}{g^{2}f^{2}_{\pi}}\right)\right)\overline{\sigma}^{4}-h\overline{\sigma}$
$\displaystyle+\frac{N_{c}g^{4}\overline{\sigma}^{4}}{8(4\pi)^{2}}\left[\frac{3}{2}+\ln\left(\Lambda^{2}\over
m^{2}_{q}\right)\right]\;,$

After the rearrangement of terms,we find the complete cancellation of
renormalization scale $\Lambda$ in the vacuum grand potential.It is recast as

$\displaystyle\Omega(\overline{\sigma})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(m^{2}_{s}-\dfrac{N_{c}g^{4}f^{2}_{\pi}}{2(4\pi)^{2}}\right)\overline{\sigma}^{2}-\frac{1}{2}c\overline{\sigma}^{2}+\frac{1}{4}\left(\lambda_{1}+\frac{\lambda_{2s}}{2}\right.$
(35)
$\displaystyle\left.+\dfrac{3N_{c}g^{4}}{4(4\pi)^{2}}\right)\overline{\sigma}^{4}-h\overline{\sigma}+\frac{N_{c}g^{4}\overline{\sigma}^{4}}{8(4\pi)^{2}}\ln\left(\dfrac{f^{2}_{\pi}}{\overline{\sigma}^{2}}\right)$

Now,in the presence of appropriately renormalized quark one-loop vacuum
contribution,the thermodynamic grand potential of the quark meson model with
vacuum term (QMVT) will be written as :

$\displaystyle\Omega_{\rm QMVT}((\overline{\sigma},T,\mu))$ $\displaystyle=$
$\displaystyle\Omega(\overline{\sigma})+\Omega^{T,\mu}_{q\bar{q}}$ (36)
$\frac{\partial\Omega_{QMVT}(\overline{\sigma},T,\mu)}{\partial\overline{\sigma}}=0.$
(37)

The global minima of the grand potential in the Eq.(37),gives the PQMVT model
chiral condensate $\overline{\sigma}$ as a function of the temperature $T$ and
chemical potential $\mu$.

##  IV Renormalized Quark Meson Model

Several of the relatively recent investigations in the above detailed QMVT
model framework have used the standard procedure of equating the vacuum
expectation value of the sigma mean field to the pion decay constant and then
putting the $\sigma,\ \vec{a_{0}},\ \vec{\pi}$ and $\eta$ meson masses equal
to the their curvature (or screening) masseslars ; guptiw ; schafwag12 ;
chatmoh1 ; TranAnd ; vkkr12 ; chatmoh2 ; vkkt13 ; Herbst ; Weyrich ; kovacs ;
zacchi1 ; zacchi2 ; Rai .However,in principle the physical masses of the
mesons are given by the pole of their propagators and the residue of the pion
propagator at its pole is related to the pion decay constant BubaCar ; Naylor
; fix1 .Furthermore, since the effective potential is the generator of the
n-point functions of the theory at zero external momenta,the curvature masses
are equivalent to defining the meson masses using the evaluation of self-
energy at zero momentum laine ; Adhiand1 ; Adhiand2 ; Adhiand3 .It has been
emphasized that the pole definition is the physical and gauge invariant one
Kobes ; Rebhan .The curvature mass prescription is equivalent to the pole mass
prescription for the parameter fixing of the model in absence of Dirac sea
contributions but when the one quark loop vacuum correction is incorporated,
the screening masses of mesons start to differ from the pole masses BubaCar ;
fix1 .In view of the above considerations,it becomes necessary to use the
following detailed description of the exact on-shell parameter fixing
procedure for the renormalized quark-meson (RQM) model where physical (pole)
masses of the mesons and pion decay constant,are put into the relation of the
running mass parameter and couplings by using the on-shell and the minimal
subtraction renormalization schemesAdhiand2 ; asmuAnd .

###  IV.1 Self-Energies and Counterterms

(a) One loop self-energy diagrams for sigma particle.

(b) One loop self-energy diagrams for the $a_{0}$

Figure 1:

(a) One-loop self-energy diagrams for the eta.

(b) One-loop self-energy diagrams for the pion.

Figure 2:

Once quark one-loop corrections are taken into account, the tree level
parameters of Eqs. (15)–(19) become inconsistent unless the on-shell
renormalization prescription is used.Though the dimensional regularization is
used to regularize the divergent loop integrals in the on-shell scheme, the
counterterm choices are different from the minimal subtraction scheme.The loop
corrections to the self-energies are cancelled exactly by the suitable choice
of counterterms in the on-shell scheme.The renormalized parameters become
renormalization scale independent as couplings are evaluated on shell.The wave
functions/fields and parameters of Eq.(1) are bare quantities.The counterterms
$\delta m^{2}$, $\delta g^{2}$, $\delta\lambda_{1}$, $\delta\lambda_{2}$,
$\delta c$ and $\delta h$ for the parameters and the counterterms $\delta
Z_{\sigma}$,$\delta Z_{a_{0}}$,$\delta Z_{\eta}$,$\delta Z_{\pi}$, $\delta
Z_{\psi}$ and $\delta Z_{\overline{\sigma}}$ for the wave functions/fields are
introduced in the Lagrangian (1) where the renormalized fields and couplings
are defined as

$\displaystyle\sigma_{b}=\sqrt{Z_{\sigma}}\sigma,\
\eta_{b}=\sqrt{Z_{\eta}}\eta,\ a^{i}_{0b}=\sqrt{Z_{a_{0}}}a_{0}$ (38)
$\displaystyle\pi^{i}_{b}=\sqrt{Z_{\pi}}\pi,\ \psi_{b}=\sqrt{Z_{\psi}}\psi,\
m^{2}_{b}=Z_{m}m^{2}$ (39)
$\displaystyle\lambda_{1b}=Z_{\lambda_{1}}\lambda_{1},\
\lambda_{2b}=Z_{\lambda_{2}}\lambda_{2},\ g_{b}=\sqrt{Z_{g}}g$ (40)
$\displaystyle h_{b}=Z_{h}h,\ \ \ c_{b}=Z_{c}c,\
\overline{\sigma}_{b}=\sqrt{Z_{\overline{\sigma}}}\overline{\sigma}$ (41)

Where the $Z_{(\sigma,a_{0},\eta,\pi,\psi,\overline{\sigma})}=1+\delta
Z_{(\sigma,a_{0},\eta,\pi,\psi,\overline{\sigma})}$, denote the field strength
renormalization constant while $Z_{(m,\lambda_{1},\lambda_{2},g,h,c)}=1+\delta
Z_{(m,\lambda_{1},\lambda_{2},g,h,c)}$ denote the mass and coupling
renormalization constant.Following the Ref.Adhiand1 ; Adhiand2 ; Adhiand3 ;
asmuAnd ,counterterms $\delta m^{2}$,$\delta\lambda_{1}$, $\delta\lambda_{2}$,
$\delta c$ and $\delta g^{2}$, $\delta h$ can be expressed in terms of the
counterterms $\delta m^{2}_{\sigma}$,$\delta m^{2}_{a_{0}}$,$\delta
m^{2}_{\eta}$, $\delta m^{2}_{\pi}$ and $\delta m_{q}$,
$\delta\overline{\sigma}^{2}$.Using Eqs.(10)–(14) together with Eqs.
(38)-(41), we can write

$\displaystyle\delta\lambda_{1}$ $\displaystyle=$ $\displaystyle\frac{\delta
m^{2}_{\sigma}+\delta m^{2}_{\eta}-\delta m^{2}_{a_{0}}-\delta m^{2}_{\pi}}{2\
\overline{\sigma}^{2}}-\lambda_{1}\frac{\delta\overline{\sigma}^{2}}{\overline{\sigma}^{2}}$
(42) $\displaystyle\delta\lambda_{2}$ $\displaystyle=$
$\displaystyle\frac{\delta m^{2}_{a_{0}}-\delta
m^{2}_{\eta}}{\overline{\sigma}^{2}}-\lambda_{2}\frac{\delta\overline{\sigma}^{2}}{\overline{\sigma}^{2}}$
(43) $\displaystyle\delta c$ $\displaystyle=$ $\displaystyle\frac{\delta
m^{2}_{\eta}-\delta m^{2}_{\pi}}{2}$ (44) $\displaystyle\delta m^{2}$
$\displaystyle=$ $\displaystyle\delta m^{2}_{\pi}+\frac{\delta
m^{2}_{\eta}-\delta m^{2}_{\sigma}}{2}$ (45) $\displaystyle\frac{\delta
g^{2}}{4}$ $\displaystyle=$ $\displaystyle\frac{\delta
m^{2}_{q}}{\overline{\sigma}^{2}}-g^{2}\frac{\delta\overline{\sigma}^{2}}{4\
\overline{\sigma}^{2}}$ (46)

The one-loop correction at the pion-quark vertex is of order $N_{c}^{0}$.Hence
$Z_{\psi}=1$ and the quark self energy correction $\delta m_{q}=0$ at this
order.In consequence,we get $\sqrt{Z_{\psi}}\ \sqrt{Z_{g^{2}}\ g^{2}}=1$, and
$\frac{\delta g^{2}}{g^{2}}\ +\delta Z_{\pi}=0$.Furthermore the $\delta
m_{q}=0$ implies that $\delta g\ \overline{\sigma}/2+g\
\delta\overline{\sigma}/2=0$.Eq.(46) gives

$\displaystyle\frac{\delta\overline{\sigma}^{2}}{\overline{\sigma}^{2}}$
$\displaystyle=$ $\displaystyle-\frac{\delta g^{2}}{g^{2}}=\delta Z_{\pi}$
(47)

Now one can rewrite Eq.(42),(43) as

$\displaystyle\delta\lambda_{1}$ $\displaystyle=$ $\displaystyle\frac{\delta
m^{2}_{\sigma}+\delta m^{2}_{\eta}-\delta m^{2}_{a_{0}}-\delta m^{2}_{\pi}}{2\
\overline{\sigma}^{2}}-\lambda_{1}\delta Z_{\pi}$ (48)
$\displaystyle\delta\lambda_{2}$ $\displaystyle=$ $\displaystyle\frac{\delta
m^{2}_{a_{0}}-\delta m^{2}_{\eta}}{\overline{\sigma}^{2}}-\lambda_{2}\delta
Z_{\pi}$ (49)

The Feynman diagrams for the meson self-energies are drawn in the Figs.The
scalar $\sigma$ and $a_{0}$ mesons are represented by a solid line and a thick
solid line respectively in the Fig.1(a) and Fig.1(b) where an arrow on the
solid line denotes a quark.The corresponding self energy expressions for the
$\sigma$ and $a^{i}_{0}$ are written as

$\displaystyle\Sigma_{\sigma}(p^{2})$ $\displaystyle=$
$\displaystyle-\frac{2N_{c}g^{2}}{(4\pi)^{2}}\left[\mathcal{A}(m_{q}^{2})-\mbox{$1\over
2$}(p^{2}-4m^{2}_{q})\mathcal{B}(p^{2})\right]$ (50)
$\displaystyle+{24(\lambda_{1}+\mbox{$\lambda_{2}\over
2$})g\overline{\sigma}N_{c}m_{q}\over
m_{\sigma}^{2}}\mathcal{A}(m_{q}^{2})\;,$ $\displaystyle\Sigma_{a_{0}}(p^{2})$
$\displaystyle=$
$\displaystyle-\frac{2N_{c}g^{2}}{(4\pi)^{2}}\left[\mathcal{A}(m_{q}^{2})-\mbox{$1\over
2$}(p^{2}-4m^{2}_{q})\mathcal{B}(p^{2})\right]$ (51)
$\displaystyle+{8(\lambda_{1}+\mbox{$3\lambda_{2}\over
2$})g\overline{\sigma}N_{c}m_{q}\over
m_{\sigma}^{2}}\mathcal{A}(m_{q}^{2})\;,$

The pseudo-scalar $\eta$ and $\pi^{i}_{0}$ mesons in the Fig.2(a) and the
Fig.2(b) are drawn by a dashed line and a thick dashed line in respective
order.The corresponding self-energy expressions for the $\eta$ and
$\pi^{i}_{0}$ are given by

$\displaystyle\Sigma_{\eta}(p^{2})$ $\displaystyle=$
$\displaystyle-\frac{2N_{c}g^{2}}{(4\pi)^{2}}\left[\mathcal{A}(m_{q}^{2})-\mbox{$1\over
2$}p^{2}\mathcal{B}(p^{2})\right]$ (52)
$\displaystyle+{8(\lambda_{1}+\mbox{$\lambda_{2}\over
2$})g\overline{\sigma}N_{c}m_{q}\over
m_{\sigma}^{2}}\mathcal{A}(m_{q}^{2})\;.$ $\displaystyle\Sigma_{\pi}(p^{2})$
$\displaystyle=$
$\displaystyle-\frac{2N_{c}g^{2}}{(4\pi)^{2}}\left[\mathcal{A}(m_{q}^{2})-\mbox{$1\over
2$}p^{2}\mathcal{B}(p^{2})\right]$ (53)
$\displaystyle+{8(\lambda_{1}+\mbox{$\lambda_{2}\over
2$})g\overline{\sigma}N_{c}m_{q}\over
m_{\sigma}^{2}}\mathcal{A}(m_{q}^{2})\;,$

Figure 3: Counterterm for the two-point functions of the scalar $\bf{\sigma}$
and $\bf{a_{0}}$ meson

.

Figure 4: Counterterm for the two-point functions of the pseudo-scalar $\eta$
and $\pi$ meson.

Where the $\mathcal{A}(m^{2}_{q})$ and $\mathcal{B}(p^{2})$ are defined in the
Appendix (B) and the last term of the Eqs.(50), (51), (52) and (53) are the
contributions of the tadpole diagrams to the self energies.The counterterm
diagrams for the two-point functions of the scalar mesons $\bf{\sigma}$,
$\bf{a_{0}}$ and the pseudo-scalar mesons $\bf{\eta}$, $\bf{\pi}$ are shown
respectively in the Fig.3 and the Fig.4.

The diagram for the quark one-loop correction to the one-point function and
its counterterm diagram, is shown in Fig. 5. It can be written as

$\displaystyle\delta\Gamma^{(1)}$ $\displaystyle=$
$\displaystyle-4N_{c}gm_{q}\mathcal{A}(m_{q}^{2})+i\delta t\;,$ (54)

Figure 5: One point diagram for the sigma particle and its counterterm.

###  IV.2 Parameters with Renormalization

The vanishing of the one-point function $\Gamma^{(1)}=it=i(h-m_{\pi}^{2}\
\overline{\sigma})$ gives the tree level equation of motion $t=0$ and fixes
the classical minimum of the effective potential.The first renormalization
condition $<\sigma>=0$ requires that the one loop correction
$\delta\Gamma^{(1)}$ to the one point function is put to zero such that the
minimum of the effective potential does not change.Thus the first
renormalization condition $\delta\Gamma^{(1)}=0$ gives

$\displaystyle\delta t$ $\displaystyle=$
$\displaystyle-4iN_{c}gm_{q}\mathcal{A}(m_{q}^{2})\;.$ (55)

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

$\displaystyle\delta h$ $\displaystyle=$ $\displaystyle m^{2}_{\pi}\
\delta\overline{\sigma}+\delta m^{2}_{\pi}\ \overline{\sigma}+\delta t$ (56)

Using Eq.(47) we can write

$\displaystyle\delta h$ $\displaystyle=$ $\displaystyle\frac{1}{2}m^{2}_{\pi}\
\overline{\sigma}\delta Z_{\pi}+\delta m^{2}_{\pi}\ \overline{\sigma}+\delta
t$ (57)

We can write the inverse propagator for the scalar $\sigma,a_{0}$ and pseudo-
scalar $\pi,\eta$ mesons as

$\displaystyle
p^{2}-m_{\sigma,a_{0},\pi,\eta}^{2}-i\Sigma_{\sigma,a_{0},\pi,\eta}(p^{2}){\rm+counterterms}\;.$
(58)

The renormalized mass in the Lagrangian is put equal to the physical mass,
i.e. $m=m_{\rm pole}$111The contribution of the imaginary parts of the self-
energies into defining the mass has not been considered. when the on-shell
scheme gets implemented and we can write

$\displaystyle\Sigma(p^{2}=m_{\sigma,a_{0},\pi,\eta}^{2}){\rm+counterterms}$
$\displaystyle=$ $\displaystyle 0\;.$ (59)

Since the propagator residue is put to unity in the on shell scheme, one gets

$\displaystyle{\partial\over\partial
p^{2}}\Sigma_{\sigma,a_{0},\pi,\eta}(p^{2})\Big{|}_{p^{2}=m_{\sigma,a_{0},\pi,\eta}^{2}}{\rm+counterterms}$
$\displaystyle=$ $\displaystyle 0\;.$ (60)

The Fig. 3 and Fig. 4 diagrams for the counterterms of the two point functions
of the scalar $\sigma,a_{0}$ and pseudo-scalar $\pi,\eta$ mesons, are written
as

$\displaystyle\Sigma_{\sigma}^{\rm ct1}(p^{2})$ $\displaystyle=$
$\displaystyle i\left[\delta Z_{\sigma}(p^{2}-m_{\sigma}^{2})-\delta
m_{\sigma}^{2}\right]\;,$ (61) $\displaystyle\Sigma_{a_{0}}^{\rm ct1}(p^{2})$
$\displaystyle=$ $\displaystyle i\left[\delta
Z_{a_{0}}(p^{2}-m_{a_{0}}^{2})-\delta m_{a_{0}}^{2}\right]\;,$ (62)
$\displaystyle\Sigma_{\pi}^{\rm ct1}(p^{2})$ $\displaystyle=$ $\displaystyle
i\left[\delta Z_{\pi}(p^{2}-m_{\pi}^{2})-\delta m_{\pi}^{2}\right]\;,$ (63)
$\displaystyle\Sigma_{\eta}^{\rm ct1}(p^{2})$ $\displaystyle=$ $\displaystyle
i\left[\delta Z_{\eta}(p^{2}-m_{\eta}^{2})-\delta m_{\eta}^{2}\right]\;,$ (64)

$\displaystyle\Sigma^{\rm ct2}_{\sigma}$ $\displaystyle=$ $\displaystyle
3\Sigma^{\rm ct2}_{\pi}=3\Sigma^{\rm ct2}_{\eta}\;,$ (65) $\displaystyle=$
$\displaystyle-{24(\lambda_{1}+\mbox{$\lambda_{2}\over
2$})g\overline{\sigma}N_{c}m_{q}\over
m_{\sigma}^{2}}\mathcal{A}(m_{q}^{2})\;,$ (66) $\displaystyle\Sigma^{\rm
ct2}_{a_{0}}$ $\displaystyle=$
$\displaystyle-{8(\lambda_{1}+\mbox{$3\lambda_{2}\over
2$})g\overline{\sigma}N_{c}m_{q}\over
m_{\sigma}^{2}}\mathcal{A}(m_{q}^{2})\;,$ (67)

The counterterms of Eq.(65) completely cancel the respective tadpole
contributions to the self-energies of $\sigma,a_{0}$ and $\pi,\eta$.The
evaluation of the self-energies and their derivatives in on-shell conditions
give all the renormalization constants.When Eqs. (59),(60) and (61)–(64) are
combined,we obtain the following set of equations.

$\displaystyle\delta m_{\sigma}^{2}$ $\displaystyle=$
$\displaystyle-i\Sigma_{\sigma}(m_{\sigma}^{2})\;;\delta
Z_{\sigma}=i{\partial\over\partial
p^{2}}\Sigma_{\sigma}(p^{2})\Big{|}_{p^{2}=m_{\sigma}^{2}}\;,$ (68)
$\displaystyle\delta m_{a_{0}}^{2}$ $\displaystyle=$
$\displaystyle-i\Sigma_{a_{0}}(m_{a_{0}}^{2})\;;\delta
Z_{a_{0}}=i{\partial\over\partial
p^{2}}\Sigma_{a_{0}}(p^{2})\Big{|}_{p^{2}=m_{a_{0}}^{2}}\;,$ (69)
$\displaystyle\delta m_{\pi}^{2}$ $\displaystyle=$
$\displaystyle-i\Sigma_{\pi}(m_{\pi}^{2})\;;\delta
Z_{\pi}=i{\partial\over\partial
p^{2}}\Sigma_{\pi}(p^{2})\Big{|}_{p^{2}=m_{\pi}^{2}}\;,$ (70)
$\displaystyle\delta m_{\eta}^{2}$ $\displaystyle=$
$\displaystyle-i\Sigma_{\eta}(m_{\eta}^{2})\;;\delta
Z_{\eta}=i{\partial\over\partial
p^{2}}\Sigma_{\eta}(p^{2})\Big{|}_{p^{2}=m_{\eta}^{2}}\;.$ (71)

When the self energy (neglecting the tadpole contributions) expressions from
the Eqs. (50)–(53) are used, we get the following equations.

$\displaystyle\delta m_{\sigma}^{2}$ $\displaystyle=$ $\displaystyle
2ig^{2}N_{c}\left[\mathcal{A}(m_{q}^{2})-\mbox{$1\over
2$}(m_{\sigma}^{2}-4m_{q}^{2})\mathcal{B}(m_{\sigma}^{2})\right]\;,$ (72)
$\displaystyle\delta m_{a_{0}}^{2}$ $\displaystyle=$ $\displaystyle
2ig^{2}N_{c}\left[\mathcal{A}(m_{q}^{2})-\mbox{$1\over
2$}(m_{a_{0}}^{2}-4m_{q}^{2})\mathcal{B}(m_{a_{0}}^{2})\right]\;,$ (73)
$\displaystyle\delta m_{\pi}^{2}$ $\displaystyle=$ $\displaystyle
2ig^{2}N_{c}\left[\mathcal{A}(m_{q}^{2})-\mbox{$1\over
2$}m_{\pi}^{2}\mathcal{B}(m_{\pi}^{2})\right]\;,$ (74) $\displaystyle\delta
m_{\eta}^{2}$ $\displaystyle=$ $\displaystyle
2ig^{2}N_{c}\left[\mathcal{A}(m_{q}^{2})-\mbox{$1\over
2$}m_{\eta}^{2}\mathcal{B}(m_{\eta}^{2})\right]\;,$ (75) $\displaystyle\delta
Z_{\sigma}$ $\displaystyle=$ $\displaystyle
ig^{2}N_{c}\left[\mathcal{B}(m_{\sigma}^{2})+(m_{\sigma}^{2}-4m_{q}^{2})\mathcal{B}^{\prime}(m_{\sigma}^{2})\right]\;,$
(76) $\displaystyle\delta Z_{a_{0}}$ $\displaystyle=$ $\displaystyle
ig^{2}N_{c}\left[\mathcal{B}(m_{a_{0}}^{2})+(m_{a_{0}}^{2}-4m_{q}^{2})\mathcal{B}^{\prime}(m_{a_{0}}^{2})\right]\;,$
(77) $\displaystyle\delta Z_{\pi}$ $\displaystyle=$ $\displaystyle
ig^{2}N_{c}\left[\mathcal{B}(m_{\pi}^{2})+m_{\pi}^{2}\mathcal{B}^{\prime}(m_{\pi}^{2})\right]\;,$
(78) $\displaystyle\delta Z_{\eta}$ $\displaystyle=$ $\displaystyle
ig^{2}N_{c}\left[\mathcal{B}(m_{\eta}^{2})+m_{\eta}^{2}\mathcal{B}^{\prime}(m_{\eta}^{2})\right]\;.$
(79)

Exploiting the Eqs. (72)–(79) together with the Eqs. (42) –(47) and (57), we
find the following expressions for the counterterms in the on-shell scheme.

$\displaystyle\delta\lambda_{1\text{\tiny OS}}$ $\displaystyle=$
$\displaystyle\frac{iN_{c}g^{2}}{\overline{\sigma}^{2}}\left[-\frac{1}{2}(m^{2}_{\sigma}-4m^{2}_{q})\mathcal{B}(m^{2}_{\sigma})-\frac{1}{2}m^{2}_{\eta}\mathcal{B}(m^{2}_{\eta})+\frac{1}{2}(m^{2}_{a_{0}}-4m^{2}_{q})\mathcal{B}(m^{2}_{a_{0}})+\frac{1}{2}m^{2}_{\pi}\mathcal{B}(m^{2}_{\pi})\right]-\lambda_{1}ig^{2}N_{c}\left[\mathcal{B}(m_{\pi}^{2})\right.$
(80) $\displaystyle\left.+m_{\pi}^{2}\mathcal{B}^{\prime}(m_{\pi}^{2})\right]$
$\displaystyle=$
$\displaystyle\delta\lambda_{1\text{div}}+2\lambda_{1}\dfrac{N_{c}g^{2}}{(4\pi)^{2}}\ln\left(\frac{\Lambda^{2}}{m_{q}^{2}}\right)+\dfrac{N_{c}g^{2}}{(4\pi)^{2}}\left[\frac{(m^{2}_{\sigma}-4m^{2}_{q})\mathcal{C}(m^{2}_{\sigma})+m^{2}_{\eta}\mathcal{C}(m^{2}_{\eta})-(m^{2}_{a_{0}}-4m^{2}_{q})\mathcal{C}(m^{2}_{a_{0}})-m^{2}_{\pi}\mathcal{C}(m^{2}_{\pi})}{2\overline{\sigma}^{2}}\right.$
$\displaystyle\left.+\lambda_{1}(\mathcal{C}(m^{2}_{\pi})+m^{2}_{\pi}\mathcal{C}^{\prime}(m^{2}_{\pi}))\right]\;,$
$\displaystyle\delta\lambda_{2\text{\tiny OS}}$ $\displaystyle=$
$\displaystyle\frac{iN_{c}g^{2}}{\overline{\sigma}^{2}}\left[-(m^{2}_{a_{0}}-4m^{2}_{q})\mathcal{B}(m^{2}_{a_{0}})+m^{2}_{\eta}\mathcal{B}(m^{2}_{\eta})\right]-\lambda_{2}ig^{2}N_{c}\left[\mathcal{B}(m_{\pi}^{2})+m_{\pi}^{2}\mathcal{B}^{\prime}(m_{\pi}^{2})\right]\;,$
(81) $\displaystyle=$
$\displaystyle\delta\lambda_{2\text{div}}+\dfrac{N_{c}g^{2}}{(4\pi)^{2}}\left(2\lambda_{2}-g^{2}\right)\ln\left(\frac{\Lambda^{2}}{m_{q}^{2}}\right)$
$\displaystyle+\dfrac{N_{c}g^{2}}{(4\pi)^{2}}\left[\frac{(m^{2}_{a_{0}}-4m^{2}_{q})\mathcal{C}(m^{2}_{a_{0}})-m^{2}_{\eta}\mathcal{C}(m^{2}_{\eta})}{{\overline{\sigma}^{2}}}+\lambda_{2}(m^{2}_{\pi}\mathcal{C}^{\prime}(m^{2}_{\pi})+\mathcal{C}(m^{2}_{\pi}))\right]\;,$
$\displaystyle\delta m^{2}_{\text{\tiny OS}}$ $\displaystyle=$ $\displaystyle
2iN_{c}g^{2}\left[\mathcal{A}(m^{2}_{q})-\frac{1}{2}m^{2}_{\pi}\mathcal{B}(m^{2}_{\pi})\right]+iN_{c}g^{2}\left[-\frac{1}{2}m^{2}_{\eta}\mathcal{B}(m^{2}_{\eta})+\frac{1}{2}(m^{2}_{\sigma}-4m^{2}_{q})\mathcal{B}(m^{2}_{\sigma})\right]\;$
(82) $\displaystyle=$ $\displaystyle\delta
m^{2}_{\text{div}}+\dfrac{N_{c}g^{2}}{(4\pi)^{2}}m^{2}\ln\left(\frac{\Lambda^{2}}{m_{q}^{2}}\right)+\dfrac{N_{c}g^{2}}{(4\pi)^{2}}\left[m^{2}_{\pi}\mathcal{C}(m^{2}_{\pi})+\dfrac{m^{2}_{\eta}\mathcal{C}(m^{2}_{\eta})-(m^{2}_{\sigma}-4m^{2}_{q})\mathcal{C}(m^{2}_{\sigma})}{2}-2m^{2}_{q}\right]\;,$
$\displaystyle\delta c_{\text{\tiny OS}}$ $\displaystyle=$
$\displaystyle\frac{iN_{c}g^{2}}{2}\left[-m^{2}_{\eta}\mathcal{B}(m^{2}_{\eta})+m^{2}_{\pi}\mathcal{B}(m^{2}_{\pi})\right]\;$
(83) $\displaystyle=$ $\displaystyle\delta
c_{\text{div}}+\dfrac{N_{c}g^{2}}{(4\pi)^{2}}c\ln\left(\frac{\Lambda^{2}}{m_{q}^{2}}\right)+\dfrac{N_{c}g^{2}}{2(4\pi)^{2}}\left[m^{2}_{\eta}\mathcal{C}(m^{2}_{\eta})-m^{2}_{\pi}\mathcal{C}(m^{2}_{\pi})\right]\;,$
$\displaystyle\delta g^{2}_{\text{\tiny OS}}$ $\displaystyle=$ $\displaystyle-
iN_{c}g^{4}\left[m^{2}_{\pi}\mathcal{B}^{\prime}(m^{2}_{\pi})+\mathcal{B}(m^{2}_{\pi})\right]\;$
(84) $\displaystyle=$ $\displaystyle\delta
g^{2}_{\text{div}}+\dfrac{N_{c}g^{4}}{(4\pi)^{2}}\left[\ln\left(\frac{\Lambda^{2}}{m_{q}^{2}}\right)+\mathcal{C}(m^{2}_{\pi})+m^{2}_{\pi}\mathcal{C}^{\prime}(m^{2}_{\pi})\right]\;,$
$\displaystyle\delta h_{\text{\tiny OS}}$ $\displaystyle=$
$\displaystyle\dfrac{iN_{c}g^{2}}{2(4\pi)^{2}}h\left[m^{2}_{\pi}\mathcal{B}^{\prime}(m^{2}_{\pi})-\mathcal{B}(m^{2}_{\pi})\right]\;$
(85) $\displaystyle=$ $\displaystyle\delta
h_{\text{div}}+\dfrac{N_{c}g^{2}}{2(4\pi)^{2}}h\left[\ln\left(\frac{\Lambda^{2}}{m_{q}^{2}}\right)+\mathcal{C}(m^{2}_{\pi})-m^{2}_{\pi}\mathcal{C}^{\prime}(m^{2}_{\pi})\right]\;$
$\displaystyle\delta\overline{\sigma}^{2}_{\text{\tiny OS}}$ $\displaystyle=$
$\displaystyle
iN_{c}g^{2}\overline{\sigma}^{2}\left[m^{2}_{\pi}\mathcal{B}^{\prime}(m^{2}_{\pi})+\mathcal{B}(m^{2}_{\pi})\right]\;$
(86) $\displaystyle=$
$\displaystyle\delta\overline{\sigma}^{2}_{\text{div}}-\dfrac{N_{c}g^{2}\overline{\sigma}^{2}}{(4\pi)^{2}}\left[\ln\left(\frac{\Lambda^{2}}{m_{q}^{2}}\right)+\mathcal{C}(m^{2}_{\pi})+m^{2}_{\pi}\mathcal{C}^{\prime}(m^{2}_{\pi})\right]\;$
$\displaystyle\delta Z^{\text{\tiny OS}}_{\sigma}$ $\displaystyle=$
$\displaystyle\delta Z_{\sigma,\rm
div}-\frac{N_{c}g^{2}}{(4\pi)^{2}}\left[\ln\left(\frac{\Lambda^{2}}{m_{q}^{2}}\right)+\mathcal{C}(m_{\sigma}^{2})+(m_{\sigma}^{2}-4m_{q}^{2})\mathcal{C}^{\prime}(m_{\sigma}^{2})\right]\;$
(87) $\displaystyle\delta Z^{\text{\tiny OS}}_{a_{0}}$ $\displaystyle=$
$\displaystyle\delta Z_{a_{0},\rm
div}-\frac{N_{c}g^{2}}{(4\pi)^{2}}\left[\ln\left(\frac{\Lambda^{2}}{m_{q}^{2}}\right)+\mathcal{C}(m_{a_{0}}^{2})+(m_{a_{0}}^{2}-4m^{2}_{q})\mathcal{C}^{\prime}(m_{a_{0}}^{2})\right]\;$
(88) $\displaystyle\delta Z^{\text{\tiny OS}}_{\pi}$ $\displaystyle=$
$\displaystyle\delta Z_{\pi,\rm
div}-\frac{N_{c}g^{2}}{(4\pi)^{2}}\left[\ln\left(\frac{\Lambda^{2}}{m_{q}^{2}}\right)+\mathcal{C}(m_{\pi}^{2})+m_{\pi}^{2}\mathcal{C}^{\prime}(m_{\pi}^{2})\right]\;$
(89) $\displaystyle\delta Z^{\text{\tiny OS}}_{\eta}$ $\displaystyle=$
$\displaystyle\delta Z_{\eta,\rm
div}-\frac{N_{c}g^{2}}{(4\pi)^{2}}\left[\ln\left(\frac{\Lambda^{2}}{m_{q}^{2}}\right)+\mathcal{C}(m_{\eta}^{2})+m_{\eta}^{2}\mathcal{C}^{\prime}(m_{\eta}^{2})\right]\;$
(90)

here, $\lambda_{1}$, $\lambda_{2}$, $m^{2}$, $c$, $h$ and $g^{2}$ in
Eqs.(80)–(86) are the same as in the Eqs.(15)–(19) and Eq.(21) for the QM
model with the ’no Dirac sea’ approximation.

The $\mathcal{B}(m^{2}),\mathcal{B}^{\prime}(m^{2})$ and
$\mathcal{C}(m^{2}),\mathcal{C}^{\prime}(m^{2})$ are defined in the
Appendix(B).The divergent part of the counterterms are
$\delta\lambda_{1\text{div}}=\frac{N_{c}g^{2}2\lambda_{1}}{(4\pi)^{2}\epsilon}\;$,
$\delta\lambda_{2\text{div}}=\frac{N_{c}g^{2}}{(4\pi)^{2}\epsilon}(2\lambda_{2}-g^{2})\;$,
$\delta m^{2}_{\text{div}}=\frac{N_{c}g^{2}m^{2}}{(4\pi)^{2}\epsilon}\;$,
$\delta c_{\text{div}}=\frac{N_{c}g^{2}c}{(4\pi)^{2}\epsilon}\;$, $\delta
g^{2}_{\text{div}}=\frac{N_{c}g^{4}}{(4\pi)^{2}\epsilon}\;$,
$\delta\overline{\sigma}^{2}_{\text{div}}=-\frac{N_{c}g^{2}\overline{\sigma}^{2}}{(4\pi)^{2}\epsilon}\;$,
$\delta h_{\text{div}}=\frac{N_{c}g^{2}h}{2(4\pi)^{2}\epsilon}\;$, $\delta
Z_{\sigma,\rm div}=\delta Z_{a_{0},\rm div}=\delta Z_{\pi,\rm div}=\delta
Z_{\eta,\rm div}=-\frac{N_{c}g^{2}}{(4\pi)^{2}\epsilon}$ .For both,the on-
shell and the $\overline{\text{MS}}$ schemes, the divergent part of the
counterterms are the same, i.e.
$\delta\lambda_{1\text{div}}=\delta\lambda_{1\overline{\text{\tiny MS}}}$,
$\delta\lambda_{2\text{div}}=\delta\lambda_{2\overline{\text{\tiny MS}}}$ etc.
Since the bare parameters are independent of the renormalization scheme, we
can immediately write down the relations between the renormalized parameters
in the on-shell and MS schemes as the following

$\displaystyle\lambda_{1\overline{\text{\tiny MS}}}$ $\displaystyle=$
$\displaystyle\lambda_{1}+\delta\lambda_{1\text{\tiny
OS}}-\delta\lambda_{1\overline{\text{\tiny MS}}}\;$ (91)
$\displaystyle\lambda_{2\overline{\text{\tiny MS}}}$ $\displaystyle=$
$\displaystyle\lambda_{2}+\delta\lambda_{2\text{\tiny
OS}}-\delta\lambda_{2\overline{\text{\tiny MS}}}\;$ (92) $\displaystyle
m^{2}_{\overline{\text{\tiny MS}}}$ $\displaystyle=$ $\displaystyle
m^{2}+\delta m^{2}_{\text{\tiny OS}}-\delta m^{2}_{\overline{\text{\tiny
MS}}}\;$ (93) $\displaystyle c_{\overline{\text{\tiny MS}}}$ $\displaystyle=$
$\displaystyle c+\delta c_{\text{\tiny OS}}-\delta c_{\overline{\text{\tiny
MS}}}\;$ (94) $\displaystyle h_{\overline{\text{\tiny MS}}}$ $\displaystyle=$
$\displaystyle h+\delta h_{\text{\tiny OS}}-\delta h_{\overline{\text{\tiny
MS}}}\;$ (95)

$\displaystyle g^{2}_{\overline{\text{\tiny MS}}}$ $\displaystyle=$
$\displaystyle g^{2}+\delta g^{2}_{\text{\tiny OS}}-\delta
g^{2}_{\overline{\text{\tiny MS}}}\;$ (96)
$\displaystyle\overline{\sigma}^{2}_{\overline{\text{\tiny MS}}}$
$\displaystyle=$
$\displaystyle\overline{\sigma}^{2}+\delta\overline{\sigma}^{2}_{\text{\tiny
OS}}-\delta\overline{\sigma}^{2}_{\overline{\text{\tiny MS}}}\;.$ (97)

The minimum of the effective potential is at $\overline{\sigma}=f_{\pi}$ and
the masses have the measured value in vacuum.Using the above set of equations
together with the Eqs.(80)–(86), we can write the scale $\Lambda$ dependent
running parameters in the $\overline{\text{MS}}$ scheme as the following

$\displaystyle\lambda_{1\overline{\text{\tiny MS}}}(\Lambda)$ $\displaystyle=$
$\displaystyle\lambda_{1}+2\lambda_{1}\dfrac{N_{c}g^{2}}{(4\pi)^{2}}\ln\left(\frac{\Lambda^{2}}{m_{q}^{2}}\right)+\dfrac{N_{c}g^{2}}{(4\pi)^{2}}\left[\frac{(m^{2}_{\sigma}-4m^{2}_{q})\mathcal{C}(m^{2}_{\sigma})+m^{2}_{\eta}\mathcal{C}(m^{2}_{\eta})-(m^{2}_{a_{0}}-4m^{2}_{q})\mathcal{C}(m^{2}_{a_{0}})-m^{2}_{\pi}\mathcal{C}(m^{2}_{\pi})}{2f_{\pi}^{2}}\right.$
(98)
$\displaystyle\left.+\lambda_{1}(\mathcal{C}(m^{2}_{\pi})+m^{2}_{\pi}\mathcal{C}^{\prime}(m^{2}_{\pi}))\right]\;,$
$\displaystyle\lambda_{2\overline{\text{\tiny MS}}}(\Lambda)$ $\displaystyle=$
$\displaystyle\lambda_{2}+\dfrac{N_{c}g^{2}}{(4\pi)^{2}}\left(2\lambda_{2}-g^{2}\right)\ln\left(\frac{\Lambda^{2}}{m_{q}^{2}}\right)$
(99)
$\displaystyle+\dfrac{N_{c}g^{2}}{(4\pi)^{2}}\left[\frac{(m^{2}_{a_{0}}-4m^{2}_{q})\mathcal{C}(m^{2}_{a_{0}})-m^{2}_{\eta}\mathcal{C}(m^{2}_{\eta})}{{f_{\pi}^{2}}}+\lambda_{2}(m^{2}_{\pi}\mathcal{C}^{\prime}(m^{2}_{\pi})+\mathcal{C}(m^{2}_{\pi}))\right]\;,$
$\displaystyle m^{2}_{\overline{\text{\tiny MS}}}(\Lambda)$ $\displaystyle=$
$\displaystyle
m^{2}+\dfrac{N_{c}g^{2}}{(4\pi)^{2}}m^{2}\ln\left(\frac{\Lambda^{2}}{m_{q}^{2}}\right)+\dfrac{N_{c}g^{2}}{(4\pi)^{2}}\left[m^{2}_{\pi}\mathcal{C}(m^{2}_{\pi})+\dfrac{m^{2}_{\eta}\mathcal{C}(m^{2}_{\eta})-(m^{2}_{\sigma}-4m^{2}_{q})\mathcal{C}(m^{2}_{\sigma})}{2}-2m^{2}_{q}\right]\;,$
(100) $\displaystyle c_{\overline{\text{\tiny MS}}}(\Lambda)$ $\displaystyle=$
$\displaystyle
c+\dfrac{N_{c}g^{2}}{(4\pi)^{2}}c\ln\left(\frac{\Lambda^{2}}{m_{q}^{2}}\right)+\dfrac{N_{c}g^{2}}{2(4\pi)^{2}}\left[m^{2}_{\eta}\mathcal{C}(m^{2}_{\eta})-m^{2}_{\pi}\mathcal{C}(m^{2}_{\pi})\right]\;,$
(101) $\displaystyle h_{\overline{\text{\tiny MS}}}(\Lambda)$ $\displaystyle=$
$\displaystyle
h+\dfrac{N_{c}g^{2}}{2(4\pi)^{2}}h\left[\ln\left(\frac{\Lambda^{2}}{m_{q}^{2}}\right)+\mathcal{C}(m^{2}_{\pi})-m^{2}_{\pi}\mathcal{C}^{\prime}(m^{2}_{\pi})\right]\;,$
(102) $\displaystyle g^{2}_{\overline{\text{\tiny MS}}}(\Lambda)$
$\displaystyle=$ $\displaystyle
g^{2}+\dfrac{N_{c}g^{4}}{(4\pi)^{2}}\left[\ln\left(\frac{\Lambda^{2}}{m_{q}^{2}}\right)+\mathcal{C}(m^{2}_{\pi})+m^{2}_{\pi}\mathcal{C}^{\prime}(m^{2}_{\pi})\right]\;,$
(103) $\displaystyle\overline{\sigma}^{2}_{\overline{\text{\tiny
MS}}}(\Lambda)$ $\displaystyle=$ $\displaystyle
f_{\pi}^{2}-\dfrac{4N_{c}m^{2}_{q}}{(4\pi)^{2}}\left[\ln\left(\frac{\Lambda^{2}}{m_{q}^{2}}\right)+\mathcal{C}(m^{2}_{\pi})+m^{2}_{\pi}\mathcal{C}^{\prime}(m^{2}_{\pi})\right]\;.$
(104)

Table 1: Parameters of the QM model for $m_{\sigma}=400,500\ \text{and}\ 600\ \text{MeV}$. $m_{\sigma}(\text{MeV})$ |  $c(\text{MeV}^{2})$ |  $m^{2}(\text{MeV}^{2})$ |  $\lambda_{1}$ |  $\lambda_{2}$ |  $h(\text{MeV}^{3})$
---|---|---|---|---|---
$400$ |  $(374.28)^{2}$ |  $(297.74)^{2}$ |  $-30.61$ |  $77.51$ |  $(120.99)^{3}$
$500$ |  $(374.28)^{2}$ |  $(208.92)^{2}$ |  $-23.32$ |  $77.51$ |  $(120.99)^{3}$
$600$ |  $(374.28)^{2}$ |  $-(106.54)^{2}$ |  $-19.05$ |  $77.51$ |  $(120.99)^{3}$

$\lambda_{1}$, $\lambda_{2}$, $m^{2}$, $c$, $h$ and $g^{2}$ in Eqs.(98)–(104)
have the same tree level QM model values that we obtain after putting
$\overline{\sigma}=f_{\pi}$ in the Eqs.(15)–(19) and Eq.(21).

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

$\displaystyle\dfrac{d\lambda_{1\overline{\text{\tiny
MS}}}(\Lambda)}{d\ln(\Lambda)}$ $\displaystyle=$
$\displaystyle\dfrac{4N_{c}}{(4\pi)^{2}}g^{2}_{\overline{\text{\tiny
MS}}}\lambda_{1\overline{\text{\tiny MS}}}\;,$ (105)
$\displaystyle\dfrac{d\lambda_{2\overline{\text{\tiny
MS}}}(\Lambda)}{d\ln(\Lambda)}$ $\displaystyle=$
$\displaystyle\dfrac{2N_{c}}{(4\pi)^{2}}\left[2\lambda_{2\overline{\text{\tiny
MS}}}g^{2}_{\overline{\text{\tiny MS}}}-g^{4}_{\overline{\text{\tiny
MS}}}\right]\;,$ (106) $\displaystyle\dfrac{dm^{2}_{\overline{\text{\tiny
MS}}}(\Lambda)}{d\ln(\Lambda)}$ $\displaystyle=$
$\displaystyle\dfrac{2N_{c}}{(4\pi)^{2}}g^{2}_{\overline{\text{\tiny
MS}}}m^{2}_{\overline{\text{\tiny MS}}}\;,$ (107)
$\displaystyle\dfrac{dc_{\overline{\text{\tiny MS}}}(\Lambda)}{d\ln(\Lambda)}$
$\displaystyle=$
$\displaystyle\dfrac{2N_{c}}{(4\pi)^{2}}g^{2}_{\overline{\text{\tiny
MS}}}c_{\overline{\text{\tiny MS}}}\;,$ (108)
$\displaystyle\dfrac{dh_{\overline{\text{\tiny MS}}}(\Lambda)}{d\ln(\Lambda)}$
$\displaystyle=$
$\displaystyle\dfrac{N_{c}}{(4\pi)^{2}}g^{2}_{\overline{\text{\tiny
MS}}}h_{\overline{\text{\tiny MS}}}\;,$ (109)
$\displaystyle\dfrac{dg^{2}_{\overline{\text{\tiny MS}}}}{d\ln(\Lambda)}$
$\displaystyle=$
$\displaystyle\dfrac{2N_{c}}{(4\pi)^{2}}g^{4}_{\overline{\text{\tiny MS}}}\;,$
(110) $\displaystyle\dfrac{d\overline{\sigma}^{2}_{\overline{\text{\tiny
MS}}}}{d\ln(\Lambda)}$ $\displaystyle=$
$\displaystyle-\dfrac{2N_{c}}{(4\pi)^{2}}g^{2}_{\overline{\text{\tiny
MS}}}\overline{\sigma}^{2}_{\overline{\text{\tiny MS}}}$ (111)

Solving the differential Eqs.(105)–(111), we get the following solutions

$\displaystyle\lambda_{1\overline{\text{\tiny MS}}}(\Lambda)$ $\displaystyle=$
$\displaystyle\frac{\lambda_{10}}{\left(1-\dfrac{N_{c}g^{2}_{0}}{(4\pi)^{2}}\ln\left(\dfrac{\Lambda^{2}}{\Lambda_{0}^{2}}\right)\right)^{2}}\;,$
(112) $\displaystyle g^{2}_{\overline{\text{\tiny MS}}}(\Lambda)$
$\displaystyle=$
$\displaystyle\frac{g^{2}_{0}}{1-\dfrac{N_{c}g^{2}_{0}}{(4\pi)^{2}}\ln\left(\dfrac{\Lambda^{2}}{\Lambda^{2}_{0}}\right)}\;,$
(113) $\displaystyle\lambda_{2\overline{\text{\tiny MS}}}(\Lambda)$
$\displaystyle=$
$\displaystyle\frac{\lambda_{20}-\dfrac{N_{c}g^{4}_{0}}{(4\pi)^{2}}\ln\left(\dfrac{\Lambda^{2}}{\Lambda^{2}_{0}}\right)}{\left(1-\dfrac{N_{c}g^{2}_{0}}{(4\pi)^{2}}\ln\left(\dfrac{\Lambda^{2}}{\Lambda^{2}_{0}}\right)\right)^{2}}\;,$
(114) $\displaystyle m^{2}_{\overline{\text{\tiny MS}}}(\Lambda)$
$\displaystyle=$
$\displaystyle\frac{m^{2}_{0}}{1-\dfrac{N_{c}g^{2}_{0}}{(4\pi)^{2}}\ln\left(\dfrac{\Lambda^{2}}{\Lambda^{2}_{0}}\right)}\;,$
(115) $\displaystyle c_{\overline{\text{\tiny MS}}}(\Lambda)$ $\displaystyle=$
$\displaystyle\frac{c_{0}}{1-\dfrac{N_{c}g^{2}_{0}}{(4\pi)^{2}}\ln\left(\dfrac{\Lambda^{2}}{\Lambda^{2}_{0}}\right)}\;,$
(116) $\displaystyle h_{\overline{\text{\tiny MS}}}(\Lambda)$ $\displaystyle=$
$\displaystyle\frac{h_{0}}{\sqrt{1-\dfrac{N_{c}g^{2}_{0}}{(4\pi)^{2}}\ln\left(\dfrac{\Lambda^{2}}{\Lambda^{2}_{0}}\right)}}\;,$
(117) $\displaystyle\overline{\sigma}^{2}$ $\displaystyle=$ $\displaystyle
f^{2}_{\pi}\left[1-\frac{N_{c}g^{2}_{0}}{(4\pi)^{2}}\ln\left(\frac{\Lambda^{2}}{\Lambda^{2}_{0}}\right)\right]\;$
(118)

Where the parameters $\lambda_{10}$, $\lambda_{20}$, $g^{2}_{0}$, $m^{2}_{0}$,
$c_{0}$ and $h_{0}$, are the running parameter values at the scale
$\Lambda_{0}$.We can choose the $\Lambda_{0}$ to satisfy the following
relation

$\displaystyle\ln\left(\frac{\Lambda^{2}_{0}}{m_{q}^{2}}\right)+\mathcal{C}(m^{2}_{\pi})+m^{2}_{\pi}\mathcal{C}^{\prime}(m^{2}_{\pi})$
$\displaystyle=$ $\displaystyle 0\;.$ (119)

Now, we can calculate the parameters of Eqs.(98)–(104) at the scale
$\Lambda=\Lambda_{0}$ and find $\lambda_{10}$, $\lambda_{20}$, $g^{2}_{0}$,
$m^{2}_{0}$, $c_{0}$ and $h_{0}$.

###  IV.3 Effective Potential

Using the values of the parameters from the Eqs.(112)–(118), the vacuum
effective potential in the $\overline{\text{MS}}$ scheme can be written as

$\displaystyle\Omega_{vac}$ $\displaystyle=$ $\displaystyle
U(\overline{\sigma}_{\overline{\text{\tiny
MS}}})+\Omega^{q,vac}_{\overline{\text{\tiny MS}}}+\delta
U(\overline{\sigma}_{\overline{\text{\tiny MS}}})$ (120)

Where

$\displaystyle U(\overline{\sigma}_{\overline{\text{\tiny MS}}})$
$\displaystyle=$ $\displaystyle\frac{m_{\overline{\text{\tiny
MS}}}^{2}(\Lambda)}{2}\overline{\sigma}^{2}_{\overline{\text{\tiny
MS}}}-\frac{c_{\overline{\text{\tiny
MS}}}(\Lambda)}{2}\overline{\sigma}^{2}_{\overline{\text{\tiny MS}}}\;$ (121)
$\displaystyle+\frac{1}{4}\left(\lambda_{1\overline{\text{\tiny
MS}}}(\Lambda)+\frac{\lambda_{2\overline{\text{\tiny
MS}}}(\Lambda)}{2}\right)\overline{\sigma}^{4}_{\overline{\text{\tiny MS}}}$
$\displaystyle-h_{\overline{\text{\tiny
MS}}}(\Lambda)\overline{\sigma}_{\overline{\text{\tiny MS}}}\;,$

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

The order $\mathcal{O}(N^{2}_{c})$ terms are dropped as these are two loop
terms and we get

$\displaystyle\delta U(\overline{\sigma}_{\overline{\text{\tiny MS}}})$
$\displaystyle=$ $\displaystyle-\frac{N_{c}g^{4}_{\overline{\text{\tiny
MS}}}\overline{\sigma}^{4}_{\overline{\text{\tiny
MS}}}}{8(4\pi)^{2}}\frac{1}{\epsilon}=-\frac{2N_{c}\Delta^{4}}{(4\pi)^{2}}\frac{1}{\epsilon}$
(123)

$\displaystyle\Omega^{q,vac}_{\overline{\text{\tiny MS}}}$ $\displaystyle=$
$\displaystyle\frac{N_{c}g^{4}_{\overline{\text{\tiny
MS}}}\overline{\sigma}^{4}_{\overline{\text{\tiny
MS}}}}{8(4\pi)^{2}}\left[\frac{1}{\epsilon}+\frac{3}{2}+\ln\left(\frac{4\Lambda^{2}}{g^{2}_{\overline{\text{\tiny
MS}}}\overline{\sigma}^{2}_{\overline{\text{\tiny
MS}}}}\right)\right]=\frac{2N_{c}\Delta^{4}}{(4\pi)^{2}}\left[\frac{1}{\epsilon}+\frac{3}{2}+\ln\left(\frac{\Lambda^{2}}{\Delta^{2}}\right)\right]\;,$
(124)

One can define the sacle $\Lambda$ independent parameter
$\Delta=\frac{g_{\overline{\text{\tiny
MS}}}\overline{\sigma}_{\overline{\text{\tiny MS}}}}{2}$ using the Eq.(103)
and Eq. (104).It is instructive to write the Eq.(121) in terms of the sacle
independent $\Delta$ as

$\displaystyle U(\Delta)$ $\displaystyle=$ $\displaystyle
2\frac{m_{\overline{\text{\tiny
MS}}}^{2}(\Lambda)}{g^{2}_{\overline{\text{\tiny
MS}}}(\Lambda)}\Delta^{2}-2\frac{c_{\overline{\text{\tiny
MS}}}(\Lambda)}{g^{2}_{\overline{\text{\tiny
MS}}}(\Lambda)}\Delta^{2}+4\left(\frac{\lambda_{1\overline{\text{\tiny
MS}}}(\Lambda)}{g^{4}_{\overline{\text{\tiny
MS}}}(\Lambda)}+\frac{\lambda_{2\overline{\text{\tiny
MS}}}(\Lambda)}{2g^{4}_{\overline{\text{\tiny
MS}}}(\Lambda)}\right)\Delta^{4}-2\frac{h_{\overline{\text{\tiny
MS}}}(\Lambda)}{g_{\overline{\text{\tiny MS}}}(\Lambda)}\Delta\;$ (125)
$\displaystyle=$ $\displaystyle
2\left(\frac{m^{2}_{0}}{g^{2}_{0}}-\frac{c_{0}}{g^{2}_{0}}\right)\Delta^{2}+4\left(\frac{\lambda_{10}}{g^{4}_{0}}+\frac{\lambda_{20}}{2g^{4}_{0}}\right)\Delta^{4}-2\frac{h_{0}}{g_{0}}\Delta$

$\displaystyle\Omega_{vac}(\Delta)$ $\displaystyle=$ $\displaystyle
2\left(\frac{m^{2}_{0}}{g^{2}_{0}}-\frac{c_{0}}{g^{2}_{0}}\right)\Delta^{2}+4\left(\frac{\lambda_{10}}{g^{4}_{0}}+\frac{\lambda_{20}}{2g^{4}_{0}}\right)\Delta^{4}-2\frac{h_{0}}{g_{0}}+\frac{2N_{c}\Delta^{4}}{(4\pi)^{2}}\left[\frac{3}{2}+\ln\left(\frac{\Lambda^{2}}{\Delta^{2}}\right)\right]\;$
(126)

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

$\displaystyle\Omega_{vac}(\Delta)$ $\displaystyle=$
$\displaystyle\frac{(3m^{2}_{\pi}-m^{2}_{\sigma})f^{2}_{\pi}}{4}\left\\{1-\frac{N_{c}g^{2}}{(4\pi)^{2}}\left(\mathcal{C}(m^{2}_{\pi})+m^{2}_{\pi}\mathcal{C}^{\prime}(m^{2}_{\pi})\right)\right\\}\frac{\Delta^{2}}{m^{2}_{q}}+\frac{N_{c}g^{2}f^{2}_{\pi}}{2(4\pi)^{2}}\left\\{\frac{3m^{2}_{\pi}\mathcal{C}(m^{2}_{\pi})-(m^{2}_{\sigma}-4m^{2}_{q})\mathcal{C}(m^{2}_{\sigma})}{2}-2m^{2}_{q}\right\\}\frac{\Delta^{2}}{m^{2}_{q}}\;$
(127)
$\displaystyle+\frac{(m^{2}_{\sigma}-m^{2}_{\pi})f^{2}_{\pi}}{8}\left\\{1-\frac{N_{c}g^{2}}{(4\pi)^{2}}\left(\mathcal{C}(m^{2}_{\pi})+m^{2}_{\pi}\mathcal{C}^{\prime}(m^{2}_{\pi})\right)\right\\}\frac{\Delta^{4}}{m^{4}_{q}}+\frac{N_{c}g^{2}f^{2}_{\pi}}{(4\pi)^{2}}\left[(m^{2}_{\sigma}-4m^{2}_{q})\mathcal{C}(m^{2}_{\sigma})-m^{2}_{\pi}\mathcal{C}(m^{2}_{\pi})\over
8\right]\frac{\Delta^{4}}{m^{4}_{q}}\;$
$\displaystyle+\frac{2N_{c}\Delta^{4}}{(4\pi)^{2}}\left\\{\frac{3}{2}-\ln\left(\frac{\Delta^{2}}{m^{2}_{q}}\right)\right\\}-m^{2}_{\pi}f^{2}_{\pi}\left\\{1-\frac{N_{c}g^{2}}{(4\pi)^{2}}m^{2}_{\pi}\mathcal{C}^{\prime}(m^{2}_{\pi})\right\\}\frac{\Delta}{m_{q}}\;$

Here we point out that when we get the final expression of the RQM model
vacuum effective potential to one-loop order after renormalization and
consistent parameter fixing, the $m_{\eta}$ and $m_{a_{0}}$ dependent
correction factors cancel out. We have checked that the expression (127) turns
out to be equivalent to the expression of the vacuum effective potential
calculated in the Ref. Adhiand2  (Eq.(7) with $q=0$) and also given in the
Eq.(38) of the Ref. asmuAnd .

$\displaystyle\Omega_{RQM}(\Delta,T,\mu)$ $\displaystyle=$
$\displaystyle\frac{(3m^{2}_{\pi}-m^{2}_{\sigma})f^{2}_{\pi}}{4}\left\\{1-\frac{N_{c}g^{2}}{(4\pi)^{2}}\left(\mathcal{C}(m^{2}_{\pi})+m^{2}_{\pi}\mathcal{C}^{\prime}(m^{2}_{\pi})\right)\right\\}\frac{\Delta^{2}}{m^{2}_{q}}\;$
$\displaystyle+\frac{N_{c}g^{2}f^{2}_{\pi}}{2(4\pi)^{2}}\left\\{\frac{3m^{2}_{\pi}\mathcal{C}(m^{2}_{\pi})-(m^{2}_{\sigma}-4m^{2}_{q})\mathcal{C}(m^{2}_{\sigma})}{2}-2m^{2}_{q}\right\\}\frac{\Delta^{2}}{m^{2}_{q}}\;$
$\displaystyle+\frac{(m^{2}_{\sigma}-m^{2}_{\pi})f^{2}_{\pi}}{8}\left\\{1-\frac{N_{c}g^{2}}{(4\pi)^{2}}\left(\mathcal{C}(m^{2}_{\pi})+m^{2}_{\pi}\mathcal{C}^{\prime}(m^{2}_{\pi})\right)\right\\}\frac{\Delta^{4}}{m^{4}_{q}}\;$
$\displaystyle+\frac{N_{c}g^{2}f^{2}_{\pi}}{(4\pi)^{2}}\left[(m^{2}_{\sigma}-4m^{2}_{q})\mathcal{C}(m^{2}_{\sigma})-m^{2}_{\pi}\mathcal{C}(m^{2}_{\pi})\over
8\right]\frac{\Delta^{4}}{m^{4}_{q}}+\frac{2N_{c}\Delta^{4}}{(4\pi)^{2}}\left\\{\frac{3}{2}-\ln\left(\frac{\Delta^{2}}{m^{2}_{q}}\right)\right\\}\;$
$\displaystyle-m^{2}_{\pi}f^{2}_{\pi}\left\\{1-\frac{N_{c}g^{2}}{(4\pi)^{2}}m^{2}_{\pi}\mathcal{C}^{\prime}(m^{2}_{\pi})\right\\}\frac{\Delta}{m_{q}}-4N_{c}T\int\frac{d^{3}p}{(2\pi)^{3}}\left\\{\ln\left[1+e^{-E_{q}^{+}/T)}\right]+\ln\left[1+e^{-E_{q}^{-}/T)}\right]\right\\}\;$

One gets the chiral condensate or the parameter $\Delta$ in the RQM model by
searching the global minima of the grand potential in the Eq.(IV.3) for a
given value of temperature T and chemical potential $\mu$

$\frac{\partial\Omega_{RQM}(\Delta,T,\mu)}{\partial\Delta}=0.$ (129)

In our calculations we have used the $m_{\pi}=138.0$ MeV,$m_{a_{0}}=984.7$ MeV
and $m_{\eta}=547.0$ MeV,The Yukawa coupling $g=6.5$ and pion decay constant
$f_{\pi}=93.0$ MeV.The constituent quark mass in the vacuum
$m_{q}=\frac{gf_{\pi}}{2}=302.25$ MeV.

Table 2: Critical temperature for $m_{\sigma}=$400, 500, 600, 616 and 700 MeV. $m_{\sigma}(\text{MeV})$ |  $T_{c}\text{(QM)}$ |  $T_{c}\text{(QMVT)}$ |  $T_{c}\text{(RQM)}$
---|---|---|---
$400$ |  $113.3$ |  $143.6$ |  $131.8$
$500$ |  $130.2$ |  $157.3$ |  $145.6$
$600$ |  $147.8$ |  $173.1$ |  $169.3$
$616$ |  $150.5$ |  $175.6$ |  $175.6$
$700$ |  $166.1$ |  $189.8$ |  $203.6$

##  V Results and Discussion

(a) Effective potential for $m_{\sigma}=500\ \text{MeV}$.

(b) Effective potential for $m_{\sigma}=616\ \text{MeV}$.

(c) Effective potential for $m_{\sigma}=700\ \text{MeV}$.

Figure 6: Blue dotted line,solid red line and green dashed line respectively
depict the QM,RQM and QMVT model result.

(a) Chiral order parameter $m_{\sigma}=500\ \text{MeV}$.

(b) Chiral order parameter $m_{\sigma}=616\ \text{MeV}$.

(c) Chiral order parameter $m_{\sigma}=700\ \text{MeV}$.

Figure 7: Blue dotted line,solid red line and green dashed line present the
respective temperature variation in the QM,RQM and QMVT model. Figure 8: Phase
diagram for the $m_{\sigma}=500$ MeV in the QM, RQM and QMVT model. Figure 9:
Phase diagram for the $m_{\sigma}=616$ MeV in the QM, RQM and QMVT model.
Figure 10: Phase diagram for the $m_{\sigma}=700$ MeV in the QM, RQM and QMVT
model. Figure 11: Phase diagram for the sigma masses of 400, 500, 600, 616 and
700 MeV in the RQM model.

We have potted the normalized effective potential in the vacuum at $\mu=0$ and
$T=0$ with respect to the constituent quark mass scale independent parameter
$\Delta$ in Fig.(6) for the different model scenarios of the parameter fixing
and for the different sigma meson masses.The RQM model result is depicted by
the solid line in red,dotted line in blue shows the QM model plot and dashed
green line is plotting the QMVT model result.The effective potential plots
corresponding to the $m_{\sigma}=$ 500 MeV, 616 MeV and 700 MeV are presented
respectively in the Fig.(6(a)), (6(b)) and (6(c)).The minimum of the effective
potential in all the Figs. for every model scenario, occurs at
$\Delta=302.25=m_{q}$ MeV and its value is highest (i.e. it is most shallow)
in the case of no-sea approximation of the QM model for all the $m_{\sigma}$
values.It is evident from the Fig.(6(a)) that for the $m_{\sigma}=$ 500
MeV,the effective potential is deepest for the QMVT model parameter fixing
while the on-sell parameterization in the RQM model gives a shallower
effective potential in comparison.Interestingly as we increase the
$m_{\sigma}$ value,we notice that the RQM model effective potential becomes
deeper while the QMVT model effective potential shows an upward trend and
finally for the $m_{\sigma}=616\ \text{MeV}$,the effective potential plots for
both the model scenarios coincide with each other as shown in the
Fig.(6(b)).Increasing the $m_{\sigma}$ value beyond the 616 MeV,the effective
potential plot becomes deepest for the RQM model.The plots of Fig.(6(c)) for
the $m_{\sigma}=700\text{MeV}$,show that the effective potential becomes
deepest for the RQM model and in reversal of the trend seen in the
Fig.(6(a)),here the effective potential of the QMVT model is shallower than
that of the RQM model.

We have potted the temperature variation of the quark condensate
$\overline{\sigma}$ (which is obtianed from the $\Delta$ as the Yukawa
coupling $g$ remains the same after renormalization) at $\mu=0$ in Fig.(7) for
three values of $m_{\sigma}=$ 500,616 and 700 MeV.Here also the solid red
line,the blue dotted line and the dashed green line respectively plot the RQM
model,the QM model and the QMVT model results.In general,the chiral transition
becomes smoother due to the quark one-loop vacuum correction.For the
$m_{\sigma}=$ 500 MeV case in the Fig.(7(a)),the sharpest QM model quark
condensate temperature variation becomes more smooth for the on-shell
parameterization of the RQM model while the most smooth variation of the
condensate is seen in the QMVT model plot.Sharpest chiral crossover transition
occurs early at a pseudo-critical temperature of $T_{c}=130.2$ MeV in the QM
model and a smoother chiral crossover is witnessed for the RQM model at
$T_{c}=145.6$ MeV while a most delayed and smooth chiral crossover occurs at
$T_{c}=157.3$ MeV in the QMVT model.For the $m_{\sigma}=$ 616 MeV case in the
Fig.(7(b)),the RQM model result exactly coincides with the QMVT model result
for the temperature variation of the quark condensate and the chiral crossover
transition occurs at the same temperature of $T_{c}=175.6$ MeV.We notice that
this behaviour follows from the complete coincidence of the vacuum effective
potential plot of the RQM model with that of the QMVT model in Fig.(6(b)) for
the $m_{\sigma}=$ 616 MeV case.For the $m_{\sigma}=$ 700 MeV in the
Fig.(7(c)),the most smooth temperature variation of the quark condensate
occurs in the RQM model with a very delayed crossover transition at
$T_{c}=203.6$ MeV while we notice that a less smooth chiral crossover
transition occurs earlier at $T_{c}=189.8$ MeV in the QMVT model.We point out
that when we compare the quark condensate temperature variation of the RQM
model with that of the QMVT model,we find that for the $m_{\sigma}=$ 700 MeV
the trend becomes opposite of what we observe for the $m_{\sigma}=$ 500 MeV
case in the Fig.(7(a)) where the RQM model condensate variation is less smooth
and sharper than the QMVT model condensate variation.Here it is relevant to
remind that the vacuum effective potential depth for the RQM model when
compared with that of the QMVT model,shows the similar role reversal when the
$m_{\sigma}=$ 500 MeV plots are contrasted with the $m_{\sigma}=$ 700 MeV
plots.

We have drawn the $\mu-T$ plane phase diagram for the $m_{\sigma}=500$ MeV in
Fig.(8) with labelled line types.The QM model critical end point (CEP)
location at $\mu=$165.2 MeV, $T=$97.7 MeV shifts to a far right position in
the $\mu-T$ plane at $\mu=$299.6 MeV, $T=$29.48 MeV due to the quark one loop
vacuum correction in the QMVT model setting.Earlier studies guptiw ;
schafwag12 ; chatmoh1 ; vkkr12 ; chatmoh2  reporting similar results have
concluded that incorporating the fermionic vacuum fluctuation in the QM
model,leads to a robust and significant change in the location of CEP.Here we
point out that,the exact on-shell renormalization of the quark one-loop vacuum
fluctuation for the parameter fixing in the RQM model,gives a phase diagram in
which the CEP location $\mu=$277.3 MeV, $T=$36.2 MeV is at lower chemical
potential and higher temperature i.e. CEP shifts higher up when compared to
the position of the CEP in the QMVT model.Furthermore,the RQM model phase
diagram for the $m_{\sigma}=500$ MeV,stands in the immediate proximity of the
QM model phase diagram.

Here,we emphasize the interesting observation that the RQM model phase
diagram,depicting a crossover transition line in the whole of the $\mu-T$
plane,coincides exactly with the QMVT model phase diagram in the Fig.(9).This
overlap follows from the exact coincidence of the vacuum effective potential
plots in the Fig.(6(b)) for both the models RQM and QMVT when the
$m_{\sigma}$=616 MeV.The first order line of the QM model plot in the Fig.(9)
is ending in the critical end point (CEP) at $\mu=$223.3 MeV, $T=$94.45 MeV.In
comparison to the RQM model phase diagram,the QMVT model phase diagram,in
Fig.(10) for the $m_{\sigma}=700$ MeV,stands closer to the QM model phase
diagram.It shows the usual trend reversal when compared to the
$m_{\sigma}=500$ MeV case plots.Again this behaviour follows from the trend
reversal that we observe in the behaviour of the vacuum effective potential in
the Fig.(6(c)).We get a crossover line in the whole $\mu-T$ plane for both the
models the RQM as well as QMVT in the Fig.(10) while the first order line of
the QM model phase diagram is terminating at the critical end point (CEP) at
$\mu=$223.3 MeV, $T=$94.45 MeV.

In order to see the effect of the sigma meson mass,Fig.(11) plots the five
phase diagrams for the $m_{\sigma}$= 400 MeV, 500 MeV, 600 MeV, 616 MeV and
700 MeV in the RQM model.The CEP, $\mu$=253.5 MeV, $T$=38.2 MeV for the
$m_{\sigma}$= 400 MeV,moves rightwards to $\mu$=277.3 MeV, $T$=36.2 MeV for
the $m_{\sigma}$= 500 MeV.When the $m_{\sigma}$= 600 MeV,the CEP shifts to the
extreme right bottom of the phase diagram at $\mu$=322.5 MeV, $T$=7.2 MeV.The
phase transition line becomes a crossover in the whole of the $\mu-T$ plane
for the $m_{\sigma}$ = 616 MeV.We have also shown the crossover phase
transition line for the $m_{\sigma}$ = 700 MeV.

##  VI Summary and Conclusion

We have applied the on-shell parameter fixing prescription,to the quark-meson
(QM) model in which the two flavor of quarks are coupled to the eight mesons
of the $SU(2)_{L}\times SU(2)_{R}$ linear sigma model,and then calculated the
one-loop effective potential for the renormalized quark-meson (RQM) model
whose six running parameters $\lambda_{1}$, $\lambda_{2}$, $c$, $m^{2}$, $h$,
$g$ are determined by relating the $\overline{\text{MS}}$, on-shell schemes
and the experimental values of the quark, meson masses and pion decay
constant.After including the one quark-loop vacuum correction in the QM
model,the effective potential has been calculated also when the curvature
meson masses are used for fixing the model parameters and this model setting
has been termed as the quark-meson model with the vacuum term (QMVT).We have
computed and compared the effective potentials,the order parameter temperature
variations and the phase diagrams for the QM, RQM and QMVT model settings.

The differences and similarities for the vacuum effective potential plots in
the RQM model and the QMVT model,depend on the sigma meson mass.When we plot
the normalized effective potential with respect to the constituent quark mass
parameter $\Delta$,its depth is highest for the QMVT model if the
$m_{\sigma}=$500 MeV and the effective potential is less deep and least in
depth respectively for the RQM model and QM model.For the $m_{\sigma}=$616
MeV,the QMVT model and the RQM model effective potentials become exactly
identical to each other.For the higher $m_{\sigma}=$700 MeV,the effective
potential of the RQM model becomes most deep and the interesting trend
reversal is noticed when one contrasts it with the variation of the effective
potential for the $m_{\sigma}=$500 MeV case.Comparing the $\mu=0$ temperature
variations of the order parameter,for the QM, RQM and QMVT model settings,we
find exactly similar differences and similarities on the sigma meson mass as
observed in the nature of the corresponding normalized effective potential.

It is well reported in the earlier research literature guptiw ; schafwag12 ;
chatmoh1 ; vkkr12 ; chatmoh2 , that incorporating the quark one-loop vacuum
correction in the QMVT model setting,gives rise to a phase diagram in which
the CEP shifts towards the right side of the $\mu-T$ plane,to quite a higher
value of the chemical potential and a lower value of the temperature when one
compares it with the location of the CEP in the QM model phase diagram.We have
found that when the $m_{\sigma}=$500 MeV,the shift in the position of the CEP
observed in the RQM model,is smaller than what is observed in the QMVT model
and the RQM model phase boundary stands closer to the QM model phase
diagram.Furthermore,driven directly by the nature of the effective potential
variation,the phase boundaries depicting the crossover transition lines,for
both the models RQM and QMVT,completely overlap with each other when the
$m_{\sigma}=$616 MeV.For the higher $m_{\sigma}=$700 MeV,the crossover line of
the QMVT model phase diagram,comes closer to the QM model phase boundary.This
trend is opposite of what we see for the $m_{\sigma}=$500 MeV case when the
RQM model phase boundary stands closer to the QM model phase diagram.Again the
above behaviour is caused by the corresponding reversal in the variation of
the normalized vacuum effective potential.

## Acknowledgments

Computational support of the computing facility which has been developed by
the Nuclear Particle Physics group of the Department of Physics, University of
Allahabad (UOA) under the Center of Advanced Studies (CAS) funding of UGC,
India, is acknowledged.Department of Science and Technology,Government of
India, DST-PURSE programme Phase 2/43(C), financial support to the science
faculty of the UOA is also acknowledged.

##  Appendix A THE QMVT PARAMETER FIXING

In the QMVT model, the meson masses at $T=0$ and $\mu=0$, get contribution
from the second derivative of the pure mesonic potential $U$ and the second
derivative of $\Omega^{vac}_{q\bar{q}}$. It is written as

$\displaystyle m^{2}_{\alpha,ab}$
$\displaystyle=\left.\frac{\partial^{2}\Omega^{\Lambda}(\overline{\sigma})}{\partial\xi_{\alpha,a}\partial\xi_{\alpha,b}}\right|_{min}=$
$\displaystyle(m^{m}_{\alpha,ab})^{2}+(\delta m^{v}_{\alpha,ab})^{2}$ (130)

where $\alpha=s,p$; “$s$” stands for the scalar and “$p$” stands for the
pseudoscalar mesons and $a,b=0,1,2,3$. $m^{2}_{s,00}\equiv m^{2}_{\sigma}$;
$m^{2}_{s,11}=m^{2}_{s,22}=m^{2}_{s,33}\equiv m^{2}_{a_{0}}$ and
$m^{2}_{p,00}\equiv m^{2}_{\eta}$;
$m^{2}_{p,11}=m^{2}_{p,22}=m^{2}_{p,33}\equiv m^{2}_{\pi}$.The
$(m^{m}_{\alpha,ab})^{2}$ and $(\delta m^{v}_{\alpha,ab})^{2}$ are defined in
the similar fashion, superscript “$m$” stands for the pure mesonic
contribution and “$v$” stands for quark/qntiquark vaccum
contribution.Here$\left.\right|_{min}$ represents the global minimum of the
grand potential.

Table 3: Expressions of the curvature masses $(m^{m}_{\alpha,ab})^{2}$ are calculated from the second derivative of the pure mesonic potential as has been evaluated in Ref.Roder . $(m^{m}_{\alpha,ab})^{2}$ |  |  Meson mass found from the pure mesonic potential
---|---|---
$(m^{m}_{s,00})^{2}$ |  $(m^{m}_{\sigma})^{2}$ |  $m^{2}-c+3\left(\lambda_{1}+\frac{\lambda_{2}}{2}\right)\overline{\sigma}^{2}$
$(m^{m}_{s,11})^{2}$ |  $(m^{m}_{a_{0}})^{2}$ |  $m^{2}+c+\left(\lambda_{1}+\frac{3\lambda_{2}}{2}\right)\overline{\sigma}^{2}$
$(m^{m}_{p,00})^{2}$ |  $(m^{m}_{\eta})^{2}$ |  $m^{2}+c+\left(\lambda_{1}+\frac{\lambda_{2}}{2}\right)\overline{\sigma}^{2}$
$(m^{m}_{p,11})^{2}$ |  $(m^{m}_{\pi})^{2}$ |  $m^{2}-c+\left(\lambda_{1}+\frac{\lambda_{2}}{2}\right)\overline{\sigma}^{2}$

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

where $m^{2}_{q,\alpha a}=\frac{\partial m^{2}_{q}}{\partial\xi_{\alpha,a}}$
and $m^{2}_{q,\alpha ab}=\frac{\partial m^{2}_{q,\alpha
a}}{\partial\xi_{\alpha,b}}$.

Table 4: The last two columns present the first and second derivative of the squared quark mass summed over two quark flavor. $s/p$ |  $a$ |  $b$ |  $m^{2}_{q,\alpha a}m^{2}_{q,\alpha b}/g^{4}$ |  $m^{2}_{q,\alpha ab}/g^{2}$
---|---|---|---|---
$s$ |  $0$ |  $0$ |  $\frac{1}{2}{\overline{\sigma}}^{2}$ |  1
$s$ |  $1$ |  $1$ |  $\frac{1}{2}{\overline{\sigma}}^{2}$ |  1
$p$ |  $0$ |  $0$ |  0 |  1
$p$ |  $1$ |  $1$ |  0 |  1

Using the Table-4 in the Eq.(131) we get vacuum contributions of meson masses
as,

$\displaystyle(\delta m^{v}_{\sigma})^{2}$ $\displaystyle\equiv$
$\displaystyle(\delta
m^{v}_{s,00})^{2}=\dfrac{N_{c}g^{4}\overline{\sigma}^{2}}{2(4\pi)^{2}}\left[1+3\ln\left(\dfrac{\Lambda^{2}}{m^{2}_{q}}\right)\right]\;,\
$ (132) $\displaystyle(\delta m^{v}_{a_{0}})^{2}$ $\displaystyle\equiv$
$\displaystyle(\delta
m^{v}_{s,11})^{2}=\dfrac{N_{c}g^{4}\overline{\sigma}^{2}}{2(4\pi)^{2}}\left[1+3\ln\left(\dfrac{\Lambda^{2}}{m^{2}_{q}}\right)\right]\;,\
$ (133) $\displaystyle(\delta m^{v}_{\eta})^{2}$ $\displaystyle\equiv$
$\displaystyle(\delta
m^{v}_{p,00})^{2}=\dfrac{N_{c}g^{4}\overline{\sigma}^{2}}{2(4\pi)^{2}}\left[1+\ln\left(\dfrac{\Lambda^{2}}{m^{2}_{q}}\right)\right]\;,\
$ (134) $\displaystyle(\delta m^{v}_{\pi})^{2}$ $\displaystyle\equiv$
$\displaystyle(\delta
m^{v}_{p,11})^{2}=\dfrac{N_{c}g^{4}\overline{\sigma}^{2}}{2(4\pi)^{2}}\left[1+\ln\left(\dfrac{\Lambda^{2}}{m^{2}_{q}}\right)\right]\;.\
$ (135)

We get $(m^{m}_{\sigma})^{2}$,$(m^{m}_{\eta})^{2}$,$(m^{m}_{a_{0}})^{2}$ and
$(m^{m}_{\eta})^{2}$ after substitution of the Eqs.(132)–(135) into the
Eq.(130) as

$\displaystyle(m^{m}_{\sigma})^{2}$ $\displaystyle=$ $\displaystyle
m^{2}_{\sigma}-\dfrac{N_{c}g^{4}\overline{\sigma}^{2}}{2(4\pi)^{2}}\left[1+3\ln\left(\dfrac{\Lambda^{2}}{m^{2}_{q}}\right)\right]\;,$
(136) $\displaystyle(m^{m}_{a_{0}})^{2}$ $\displaystyle=$ $\displaystyle
m^{2}_{a_{0}}-\dfrac{N_{c}g^{4}\overline{\sigma}^{2}}{2(4\pi)^{2}}\left[1+3\ln\left(\dfrac{\Lambda^{2}}{m^{2}_{q}}\right)\right]\;,$
(137) $\displaystyle(m^{m}_{\eta})^{2}$ $\displaystyle=$ $\displaystyle
m^{2}_{\eta}-\dfrac{N_{c}g^{4}\overline{\eta}^{2}}{2(4\pi)^{2}}\left[1+\ln\left(\dfrac{\Lambda^{2}}{m^{2}_{q}}\right)\right]\;,$
(138) $\displaystyle(m^{m}_{\pi})^{2}$ $\displaystyle=$ $\displaystyle
m^{2}_{\pi}-\dfrac{N_{c}g^{4}\overline{\pi}^{2}}{2(4\pi)^{2}}\left[1+\ln\left(\dfrac{\Lambda^{2}}{m^{2}_{q}}\right)\right]\;.$
(139)

The parameters in vacuum are obtained as

$\displaystyle\lambda_{1}$ $\displaystyle=$
$\displaystyle(m^{m}_{\sigma})^{2}+(m^{m}_{\eta})^{2}-(m^{m}_{a_{0}})^{2}-(m^{m}_{\pi})^{2}\over
2f^{2}_{\pi}$ (140) $\displaystyle\lambda_{2}$ $\displaystyle=$
$\displaystyle(m^{m}_{a_{0}})^{2}-(m^{m}_{\eta})^{2}\over f^{2}_{\pi}$ (141)
$\displaystyle m^{2}$ $\displaystyle=$
$\displaystyle(m^{m}_{\pi})^{2}+\frac{(m^{m}_{\sigma})^{2}-(m^{m}_{\eta})^{2}}{2}$
(142) $\displaystyle c$ $\displaystyle=$
$\displaystyle\frac{(m^{m}_{\eta})^{2}-(m^{m}_{\pi})^{2}}{2}$ (143)

we get the parameters of the QMVT on substitution of the Eqs.(136)–(139) into
the Eqs.(140)–(143) and found that $\lambda_{1}$,$c$ of the QMVT are same with
respect to $\lambda_{1}$,$c$ of the QM.We observe change in $\lambda_{2}$ and
$m^{2}$ as,

$\displaystyle\lambda_{2}$ $\displaystyle=$
$\displaystyle\lambda_{2s}-\dfrac{N_{c}g^{4}}{(4\pi)^{2}}\ln\left(\dfrac{4\Lambda^{2}}{g^{2}f^{2}_{\pi}}\right)\;$
(144) $\displaystyle m^{2}$ $\displaystyle=$ $\displaystyle
m^{2}_{s}-\dfrac{N_{c}g^{4}f^{2}_{\pi}}{2(4\pi)^{2}}\;.$ (145)

where $\lambda_{2s}$ and $m^{2}_{s}$ are same old $\lambda_{2}$ and $m^{2}$
parameters of the QM model.

##  Appendix B INTEGRALS AND SUM INTEGRALS

The divergent loop integrals are regularized by encorporating dimensional
regularization.

$\displaystyle\int_{p}=\left(\frac{e^{\gamma_{E}}\Lambda^{2}}{4\pi}\right)^{\epsilon}\int\frac{d^{d}p}{(2\pi)^{d}}\;,$
(146)

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

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

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

$\displaystyle\mathcal{A}(m^{2}_{q})$ $\displaystyle=$
$\displaystyle\frac{im^{2}_{q}}{(4\pi)^{2}}\left[\frac{1}{\epsilon}+1+\ln\left(\frac{\Lambda^{2}}{m^{2}_{q}}\right)\right]$
(147)

$\displaystyle\mathcal{B}(p^{2})$ $\displaystyle=$
$\displaystyle\int_{k}\frac{1}{(k^{2}-m^{2}_{q})[(k+p)^{2}-m^{2}_{q})]}$ (148)
$\displaystyle=$
$\displaystyle\frac{i}{(4\pi)^{2}}\left[\frac{1}{\epsilon}+\ln\left(\frac{\Lambda^{2}}{m^{2}_{q}}\right)+C(p^{2})\right]$

$\displaystyle\mathcal{B}^{\prime}(p^{2})$ $\displaystyle=$
$\displaystyle\frac{i}{(4\pi)^{2}}C^{\prime}(p^{2})$ (149)

$\displaystyle C(p^{2})$ $\displaystyle=$
$\displaystyle\begin{dcases}2-2\sqrt{\dfrac{4m^{2}_{q}}{p^{2}}-1}\arctan\left(\dfrac{1}{\sqrt{\dfrac{4m^{2}_{q}}{p^{2}}-1}}\right)\;,&(p^{2}<4m^{2}_{q})\\\
2+\sqrt{1-\dfrac{4m^{2}_{q}}{p^{2}}}\ln\left(\dfrac{1-\sqrt{1-\dfrac{4m^{2}_{q}}{p^{2}}}}{1+\sqrt{1-\dfrac{4m^{2}_{q}}{p^{2}}}}\right),&(p^{2}>4m^{2}_{q})\end{dcases}$
(150)

$\displaystyle C^{\prime}(p^{2})$ $\displaystyle=$
$\displaystyle\begin{dcases}\frac{4m^{2}_{q}}{p^{4}\sqrt{\dfrac{4m^{2}_{q}}{p^{2}}-1}}\arctan\left(\dfrac{1}{\sqrt{\dfrac{4m^{2}_{q}}{p^{2}}-1}}\right)-\frac{1}{p^{2}}\;,&(p^{2}<4m^{2}_{q})\\\
\frac{2m^{2}_{q}}{p^{4}\sqrt{\dfrac{4m^{2}_{q}}{p^{2}}-1}}\ln\left(\dfrac{1-\sqrt{1-\dfrac{4m^{2}_{q}}{p^{2}}}}{1+\sqrt{1-\dfrac{4m^{2}_{q}}{p^{2}}}}\right)-\frac{1}{p^{2}},&(p^{2}>4m^{2}_{q})\end{dcases}$
(151)

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