Exploring axial $U(1)$ restoration in a modified 2+1 flavor Polyakov quark meson model

TL;DR

This paper investigates how temperature-dependent axial U(1) symmetry restoration occurs in a modified 2+1 flavor Polyakov quark meson model, showing significant effects on meson masses and mixing angles near the chiral transition.

Contribution

It introduces a temperature-dependent Kobayashi-Maskawa-'t Hooft determinant term in the model, revealing novel insights into U(1) symmetry restoration and meson behavior at high temperatures.

Findings

Significant decrease in $ ext{m}_{ ext{η}'}$ mass near T=176 MeV.

U(1) symmetry restoration occurs at approximately 1.75 times the chiral transition temperature.

Meson mass degeneracy and mixing angle changes indicate effective symmetry restoration.

Abstract

We report on the $U_{A}(1)$ symmetry restoration resulting due to temperature dependence of the coefficient c(T) for the Kobayashi-Maskawa-'t Hooft determinant (KMT) term in a modified 2+1 flavor Polyakov loop quark meson model having fermionic vacuum correction term (PQMVT).Temperature dependence of KMT coupling c(T) drives the non-strange condensate melting to significantly smaller temperatures in comparison to the constant c case.Further due to c(T), $m_{\eta^{\prime}}$ decreases from its vacuum value by 220 MeV near T=176 MeV after the chiral transition ($T_{c}^{\chi}=154.9$ MeV).This is similar to the $\eta^{\prime}$ in-medium mass drop of at least 200 MeV as reported by Csorgo and Vertesi in Ref (Phys Rev Lett 105:182301,2010; Phys Rev C 83:054903,2011), as an experimental signature of the effective restoration of $U_A(1)$ symmetry. The pseudoscalar mixing angle $\theta_p$ achieves anti-ideal mixing in the influence of c(T). The $\eta $ meson becomes light quark system ($ \eta_{NS} $) at T=176 MeV and changes its identity with $ \eta' $ meson which becomes strange quark system ($ \eta_{S} $). The degenerated temperature variations of $\sigma$, $\pi$ meson masses merges with the temperature variations of the masses of degenerated $a_{0}$, $\eta$ mesons near 275 MeV. It means that for c(T) when $m_{\sigma}=400$ MeV, the $U_{A}(1)$ restoration takes place at 1.75 $T_{c}^{\chi}$=275 MeV.

Tables

Table 1. Table 1: Expressions for ( m α , a ​ b m ) 2 superscript subscript superscript 𝑚 m 𝛼 𝑎 𝑏 2 ({m^{\text{m}}_{\alpha,ab}})^{2} in the left half of the table are the vacuum meson masses calculated from the second derivatives of the pure mesonic potential U( σ x , σ y subscript 𝜎 𝑥 subscript 𝜎 𝑦 \sigma_{x},\sigma_{y} ). Evaluated expressions of mass modifications ( δ ​ m α , a ​ b v ) 2 superscript 𝛿 subscript superscript 𝑚 v 𝛼 𝑎 𝑏 2 ({\delta m^{\text{v}}_{\alpha,ab}})^{2} due to the fermionic vacuum correction are given in the right half. Symbols used in the expressions are defined as x = σ x 𝑥 subscript 𝜎 𝑥 x=\sigma_{x} , y = σ y 𝑦 subscript 𝜎 𝑦 y=\sigma_{y} , X = ( 1 + 4 ​ log ⁡ ( g ​ σ x 2 ​ M ) ) 𝑋 1 4 𝑔 subscript 𝜎 𝑥 2 M X=\left(1+4\log\left(\frac{g\sigma_{x}}{2\text{M}}\right)\right) and Y = ( 1 + 4 ​ log ⁡ ( g ​ σ y 2 ​ M ) ) 𝑌 1 4 𝑔 subscript 𝜎 𝑦 2 M Y=\left(1+4\log\left(\frac{g\sigma_{y}}{\sqrt{2}\text{M}}\right)\right) .Here, when α 𝛼 \alpha =s, the (11) element gives squared mass of the scalar a 0 subscript 𝑎 0 a_{0} meson which is degenerate with the (22) and (33) elements. Similarly, the (44) element which is degenerate with (55), (66) and (77) elements, gives the squared κ 𝜅 \kappa meson mass. The squared σ 𝜎 \sigma and f 0 subscript 𝑓 0 f_{0} meson masses are obtained by diagonalizing the scalar mass matrix in the (00)-(88) sector and we get a scalar mixing angle θ S subscript 𝜃 𝑆 \theta_{S} . We have a completely analogous situation for the pseudo scalar sector ( α 𝛼 \alpha = p) with the following identification; the (11) element gives the squared pion mass and the squared kaon mass is given by the (44) element.

	Meson masses calculated from pure mesonic potential		Fermionic vacuum correction in meson masses
${\left(m_{a_0}^{\text{m}}\right)}^2$	$m^2+{\lambda }_1\left(x^2+y^2\right)+\frac{3{\lambda }_2}{2}{x}^2+\frac{\sqrt{2}c}{2}y$	${\left(\delta m_{s,11}^{\text{v}}\right)}^2$	$-\frac{N_c{g}^4}{64{\pi }^2}{x}^2\left(4+3X\right)$
${\left(m_{\kappa }^{\text{m}}\right)}^2$	$m^2+{\lambda }_1\left(x^2+y^2\right)+\frac{{\lambda }_2}{2}\left(x^2+\sqrt{2}xy+2y^2\right)+\frac{c}{2}x$	${\left(\delta m_{s,44}^{\text{v}}\right)}^2$	$-\frac{N_c{g}^4}{64{\pi }^2}\left(\frac{x+\sqrt{2}y}{x^2-2y^2}\right)\left(x^{3}X-2\sqrt{2}{y}^{3}Y\right)$
${\left(m_{s,00}^{\text{m}}\right)}^2$	$m^2+\frac{{\lambda }_1}{3}\left(7x^2+4\sqrt{2}xy+5y^2\right)+{\lambda }_2\left(x^2+y^2\right)-\frac{\sqrt{2}c}{3}\left(\sqrt{2}x+y\right)$	${\left(\delta m_{s,00}^{\text{v}}\right)}^2$	$-\frac{N_c{g}^4}{96{\pi }^2}\left(3\left(x^{2}X+y^{2}Y\right)+4\left(x^2+y^2\right)\right)$
${\left(m_{s,88}^{\text{m}}\right)}^2$	$m^2+\frac{{\lambda }_1}{3}\left(5x^2-4\sqrt{2}xy+7y^2\right)+{\lambda }_2\left(\frac{x^2}{2}+2y^2\right)+\frac{\sqrt{2}c}{3}\left(\sqrt{2}x-\frac{y}{2}\right)$	${\left(\delta m_{s,88}^{\text{v}}\right)}^2$	$-\frac{N_c{g}^4}{96{\pi }^2}\left(\frac{3}{2}\left(x^{2}X+4y^{2}Y\right)+2\left(x^2+4y^2\right)\right)$
${\left(m_{s,08}^{\text{m}}\right)}^2$	$\frac{2{\lambda }_1}{3}\left(\sqrt{2}{x}^2-xy-\sqrt{2}{y}^2\right)+\sqrt{2}{\lambda }_2\left(\frac{x^2}{2}-y^2\right)+\frac{c}{3\sqrt{2}}\left(x-\sqrt{2}y\right)$	${\left(\delta m_{s,08}^{\text{v}}\right)}^2$	$-\frac{N_c{g}^4}{8\sqrt{2}{\pi }^2}\left(\frac{1}{4}\left(x^{2}X-2y^{2}Y\right)+\frac{1}{3}\left(x^2-2y^2\right)\right)$
${\left(m_{\pi }^{\text{m}}\right)}^2$	$m^2+{\lambda }_1\left(x^2+y^2\right)+\frac{{\lambda }_2}{2}{x}^2-\frac{\sqrt{2}c}{2}y$	${\left(\delta m_{p,11}^{\text{v}}\right)}^2$	$-\frac{N_c{g}^4}{64{\pi }^2}{x}^{2}X$
${\left(m_K^{\text{m}}\right)}^2$	$m^2+{\lambda }_1\left(x^2+y^2\right)+\frac{{\lambda }_2}{2}\left(x^2-\sqrt{2}xy+2y^2\right)-\frac{c}{2}x$	${\left(\delta m_{p,44}^{\text{v}}\right)}^2$	$-\frac{N_c{g}^4}{64{\pi }^2}\left(\frac{x-\sqrt{2}y}{x^2-2y^2}\right)\left(x^{3}X+2\sqrt{2}{y}^{3}Y\right)$
${\left(m_{p,00}^{\text{m}}\right)}^2$	$m^2+{\lambda }_1\left(x^2+y^2\right)+\frac{{\lambda }_2}{3}\left(x^2+y^2\right)+\frac{c}{3}\left(2x+\sqrt{2}y\right)$	${\left(\delta m_{p,00}^{\text{v}}\right)}^2$	$-\frac{N_c{g}^4}{96{\pi }^2}\left(x^{2}X+y^{2}Y\right)$
${\left(m_{p,88}^{\text{m}}\right)}^2$	$m^2+{\lambda }_1\left(x^2+y^2\right)+\frac{{\lambda }_2}{6}\left(x^2+4y^2\right)-\frac{c}{6}\left(4x-\sqrt{2}y\right)$	${\left(\delta m_{p,88}^{\text{v}}\right)}^2$	$-\frac{N_c{g}^4}{192{\pi }^2}\left(x^{2}X+4y^{2}Y\right)$
${\left(m_{p,08}^{\text{m}}\right)}^2$	$\frac{\sqrt{2}{\lambda }_2}{6}\left(x^2-2y^2\right)-\frac{c}{6}\left(\sqrt{2}x-2y\right)$	${\left(\delta m_{p,08}^{\text{v}}\right)}^2$	$-\frac{N_c{g}^4}{96\sqrt{2}{\pi }^2}\left(x^{2}X-2y^{2}Y\right)$

Table 2. Table 2: The squared masses of scalar and pseudo scalar mesons which are obtained after the diagonalization of the 00-88 sector of mass matrix. The meson masses in the non strange σ N ​ S ​ ( η N ​ S ) subscript 𝜎 𝑁 𝑆 subscript 𝜂 𝑁 𝑆 \sigma_{NS}(\eta_{NS}) and strange σ S ​ ( η S ) subscript 𝜎 𝑆 subscript 𝜂 𝑆 \sigma_{S}(\eta_{S}) basis are given in the last two rows.

	Scalar Meson Masses		Pseudo scalar Meson Masses
$m_{\sigma }^2$	$m_{s,00}^2{\cos }^2{\theta }_s+m_{s,88}^2{\sin }^2{\theta }_s+2m_{s,08}^2\sin {\theta }_s\cos {\theta }_s$	$m_{{\eta }^{\prime }}^2$	$m_{p,00}^2{\cos }^2{\theta }_p+m_{p,88}^2{\sin }^2{\theta }_p+2m_{p,08}^2\sin {\theta }_p\cos {\theta }_p$
$m_{f_0}^2$	$m_{s,00}^2{\sin }^2{\theta }_s+m_{s,88}^2{\cos }^2{\theta }_s-2m_{s,08}^2\sin {\theta }_s\cos {\theta }_s$	$m_{\eta }^2$	$m_{p,00}^2{\sin }^2{\theta }_p+m_{p,88}^2{\cos }^2{\theta }_p-2m_{p,08}^2\sin {\theta }_p\cos {\theta }_p$
$m_{{\sigma }_{NS}}^2$	$\frac{1}{3}\left(2m_{s,00}^2+m_{s,88}^2+2\sqrt{2}{m}_{s,08}^2\right)$	$m_{{\eta }_{NS}}^2$	$\frac{1}{3}\left(2m_{p,00}^2+m_{p,88}^2+2\sqrt{2}{m}_{p,08}^2\right)$
$m_{{\sigma }_S}^2$	$\frac{1}{3}\left(m_{s,00}^2+2m_{s,88}^2-2\sqrt{2}{m}_{s,08}^2\right)$	$m_{{\eta }_S}^2$	$\frac{1}{3}\left(m_{p,00}^2+2m_{p,88}^2-2\sqrt{2}{m}_{p,08}^2\right)$

Table 3. Table 3: Parameters for different values of sigma meson mass m σ subscript 𝑚 𝜎 m_{\sigma} . We have chosen a combination of sigma meson mass and Polyakov loop potential parameter ( m σ , T g ​ l ​ u ​ e c subscript 𝑚 𝜎 superscript subscript 𝑇 𝑔 𝑙 𝑢 𝑒 𝑐 m_{\sigma},T_{glue}^{c} ) and found for this combination, the values of the parameters T 1 subscript 𝑇 1 T_{1} and b 1 subscript 𝑏 1 b_{1} which give the temperature dependence of KMT coupling c ​ ( T ) 𝑐 𝑇 c(T) by obtaining the most suitable coincidence of the temperature variations of the subtracted chiral condensate Δ l , s ​ ( T ) subscript Δ 𝑙 𝑠 𝑇 \Delta_{l,s}(T) calculated in our PQMVT-II model with the Wuppertal group lattice QCD data for the same.

$m_{\sigma }\left[MeV\right]$	$T_{glue}^c\left[MeV\right]$	$c\left[MeV\right]$	$m^2\left[Me{V}^2\right]$	${\lambda }_1$	${\lambda }_{2s}$	$h_x\left[Me{V}^3\right]$	$h_y\left[Me{V}^3\right]$	$T_1\left[MeV\right]$	$b_1\left[MeV\right]$
$400$	$240$	$4807.84$	${\left(282.67\right)}^2$	$-8.18$	$46.48$	${\left(120.73\right)}^3$	${\left(336.41\right)}^3$	$108$	$90$
$500$	$240$	$4807.84$	${\left(171.73\right)}^2$	$-5.29$	$46.48$	${\left(120.73\right)}^3$	${\left(336.41\right)}^3$	$105$	$78$
$600$	$240$	$4807.84$	$-{\left(183.73\right)}^2$	$-1.689$	$46.48$	${\left(120.73\right)}^3$	${\left(336.41\right)}^3$	$105$	$62$

Table 4. Table 4: The table of characteristic temperature (pseudo critical temperature) for the chiral transition in the non-strange sector T c χ superscript subscript 𝑇 𝑐 𝜒 T_{c}^{\chi} , strange sector T s χ superscript subscript 𝑇 𝑠 𝜒 T_{s}^{\chi} and the confinement-deconfinement transition T c Φ superscript subscript 𝑇 𝑐 Φ T_{c}^{\Phi} , in the PQMVT-I and PQMVT-II models. ± plus-or-minus \pm gives the temperature range near T s χ superscript subscript 𝑇 𝑠 𝜒 T_{s}^{\chi} over which the rather flat and broad second peak of the strange condensate derivative ∂ σ y ∂ T subscript 𝜎 𝑦 𝑇 \frac{\partial\sigma_{y}}{\partial T} , shows a distinct change of about 0.1 percent of the numerical value of the second peak height.

	PQMVT-I	PQMVT-II	PQMVT-I	PQMVT-II	PQMVT-I	PQMVT-II
	$m_{\sigma }$=400 MeV	$m_{\sigma }$=400 MeV	$m_{\sigma }$=500 MeV	$m_{\sigma }$=500 MeV	$m_{\sigma }$=600 MeV	$m_{\sigma }$=600 MeV
	$T_{glue}^c$=240 MeV	$T_{glue}^c$=240 MeV	$T_{glue}^c$=240 MeV	$T_{glue}^c$=240 MeV	$T_{glue}^c$=240 MeV	$T_{glue}^c$=240 MeV
$T_c^{\chi }$(MeV)	$175.6$	$154.9$	$187.9$	$158.3$	$201.3$	$159.0$
$T_s^{\chi }$(MeV)	$261.7\pm 2$	$265.2\pm 2$	$267.9\pm 3.4$	$268.9\pm 3.3$	$274.7\pm 3.5$	$275.7\pm 3.5$
$T_c^{\Phi }$(MeV)	$155.3$	$145.8$	$154.3$	$146.1$	$155.3$	$146.1$

Equations

$$\Phi (\vec{x})=\frac{1}{N_c}⟨{\mathrm{Tr}}_{c}L(\vec{x})⟩,\bar{\Phi }(\vec{x})=\frac{1}{N_c}⟨{\mathrm{Tr}}_c{L}^{†}(\vec{x})⟩$$

$$L(\vec{x})=\mathcal{P}\exp \left[i{\int }_0^{\beta }d\tau A_0(\vec{x},\tau )\right]$$

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

$$\displaystyle{\cal L}_{QM}=\bar{q_{f}}\big{(}i\gamma^{\mu}D_{\mu}-g\;T_{a}\big{(}\sigma_{a}+i\gamma_{5}\pi_{a}\big{)}\big{)}q_{f}+{\cal L}_{m}$$

$${\mathcal{L}}_m$$

$$M=T_a{\xi }_a=T_a({\sigma }_a+i{\pi }_a)$$

$$H=T_a{h}_a$$

$$\begin{gathered} S(T)=exp[-{\pi }^2{\rho }^2{T}^2(\frac{2}{3}{N}_c+\frac{1}{3}{N}_f)] \\ =exp[-\frac{T^2}{b^2}] \end{gathered}$$

$$c(T)=\{\begin{array}{ll}c(0),& (T<T_1)\\ c(0)exp[-\frac{(T-T_1{)}^2}{b_1^2}],& (T>T_1)\end{array}$$

$$\frac{{\mathcal{U}}_{\text{log}}\left(\Phi ,\bar{\Phi },T\right)}{T^4}$$

$$a(T)=a_0+a_1\left(\frac{T_0}{T}\right)+a_2{\left(\frac{T_0}{T}\right)}^2,b(T)=b_3{\left(\frac{T_0}{T}\right)}^3.$$

$$\displaystyle a_{0}=3.51,\ a_{1}=-2.47,\ a_{2}=-2.47,\ \text{and}\ b_{3}=-1.75\$$

$$\frac{{\mathcal{U}}_{\text{glue}}\left(\Phi ,\bar{\Phi },T_{\text{glue}}\right)}{T_{\text{glue}}^4}$$

$$\frac{T_{\text{YM}}-T_{\text{YM}}^c}{T_{\text{YM}}^c}=0.57\frac{T_{\text{glue}}-T_{\text{glue}}^c}{T_{\text{glue}}^c}$$

$$\mathcal{Z}$$

$${\Omega }_{\mathrm{MF}}(T,{\mu }_f)=-\frac{T\ln Z}{V}$$

$$U({\sigma }_x,{\sigma }_y)=\frac{m^2}{2}\left({\sigma }_x^2+{\sigma }_y^2\right)-h_x{\sigma }_x-h_y{\sigma }_y-\frac{c}{2\sqrt{2}}{\sigma }_x^2{\sigma }_y$$

$$\displaystyle+\frac{\lambda_{1}}{2}\sigma_{x}^{2}\sigma_{y}^{2}+\frac{1}{8}\left(2\lambda_{1}+\lambda_{2}\right)\sigma_{x}^{4}+\frac{1}{8}\left(2\lambda_{1}+2\lambda_{2}\right)\sigma_{y}^{4}\$$

$${\sigma }_x$$

$${\sigma }_y$$

$$m_x=g\frac{{\sigma }_x}{2},m_y=g\frac{{\sigma }_y}{\sqrt{2}}$$

$${\Omega }_{\bar{q}q}(T,{\mu }_f)={\Omega }_{q\bar{q}}^{\mathrm{vac}}+{\Omega }_{q\bar{q}}^{\mathrm{T}}=-2\sum _{f=u,d,s}$$

$$\displaystyle\int\frac{d^{3}p}{(2\pi)^{3}}\Big{[}{N_{c}E_{f}}\theta(\Lambda^{2}-\vec{p}^{\,2})+T\{\ln g_{f}^{+}+\ln g_{f}^{-}\}\Big{]}$$

$$\displaystyle g_{f}^{+}=\Big{[}1+3\Phi e^{-E_{f}^{+}/T}+3\bar{\Phi}e^{-2E_{f}^{+}/T}+e^{-3E_{f}^{+}/T}\Big{]}\,$$

$$\displaystyle g_{f}^{-}=\Big{[}1+3\bar{\Phi}e^{-E_{f}^{-}/T}+3\Phi e^{-2E_{f}^{-}/T}+e^{-3E_{f}^{-}/T}\Big{]}\,$$

$$E_f=\sqrt{p^2+m{}_f{}^2}$$

$${\Omega }_{q\bar{q}}^{\mathrm{vac}}=-\sum _{f=u,d,s}\frac{N_c}{8{\pi }^2}{m}_f^4\ln \left(\frac{m_f}{\text{M}}\right)$$

$${\Omega }^{\mathrm{M}}({\sigma }_x,{\sigma }_y)=U({\sigma }_x,{\sigma }_y)+{\Omega }_{q\bar{q}}^{\mathrm{vac}}$$

$$\Omega ({\sigma }_x,{\sigma }_y)=\frac{m^2}{2}\left({\sigma }_x^2+{\sigma }_y^2\right)-h_x{\sigma }_x-h_y{\sigma }_y-\frac{c}{2\sqrt{2}}{\sigma }_x^2{\sigma }_y$$

$$+\frac{{\lambda }_1}{2}{\sigma }_x^2{\sigma }_y^2+\frac{{\lambda }_1}{4}\left({\sigma }_x^4+{\sigma }_y^4\right)+\frac{\left({\lambda }_{2\text{v}}+n\right)}{8}\left({\sigma }_x^4+2{\sigma }_y^4\right)$$

$$-\frac{n{\sigma }_x^4}{2}\log \left(\frac{{\sigma }_x}{\left(2f_K-f_{\pi }\right)}\right)-n{\sigma }_y^4\log \left(\frac{\sqrt{2}{\sigma }_y}{\left(2f_K-f_{\pi }\right)}\right)$$

$${\Omega }_{\mathrm{MF}}(T,{\mu }_f;{\sigma }_x,{\sigma }_y,\Phi ,\bar{\Phi })$$

$${\frac{\partial {\Omega }_{\mathrm{MF}}}{\partial {\sigma }_x}=\frac{\partial {\Omega }_{\mathrm{MF}}}{\partial {\sigma }_y}=\frac{\partial {\Omega }_{\mathrm{MF}}}{\partial \Phi }=\frac{\partial {\Omega }_{\mathrm{MF}}}{\partial \bar{\Phi }}|}_{{\sigma }_x={\bar{\sigma }}_x,{\sigma }_y={\bar{\sigma }}_y,\Phi =\bar{\Phi }}=0.$$

Full text

††journal: Eur. Phys. J. Plus

1 Introduction

The normal hadronic matter under the extreme conditions of high temperature and/or density, turns into a collective form of matter known as the Quark Gluon Plasma (QGP) when the individual hadrons dissolve into their quark and gluon constituents Shuryak ; Rafelski ; Svetitsky ; Ortmanns:96ea ; Muller ; Rischke:03 . Relativistic heavy ion collision experiments at RHIC (BNL), LHC (CERN) and the future CBM experiments at the FAIR facility (GSI-Darmstadt) aim to create and study such a collective state of matter. Study of the different aspects of this phase transition, is a tough and challenging task because Quantum Chromodynamics (QCD), the theory of strong interaction, becomes nonperturbative in the low energy limit.

It is well known that the quantum chromodynamics (QCD) Lagrangian has the global $SU_{V}(3)\times SU_{A}(3)\times U_{V}(1)\times U_{A}(1)$ symmetry. The $U_{V}(1)$ symmetry gives baryon number conservation. The QCD vacuum reveals itself through the process of spontaneous breakdown of chiral symmetry $SU_{A}(3)$ for the massless quarks in the low energy limit of QCD . Chiral condensate works as an order parameter and one gets eight massless pseudoscalar mesons as Goldstone bosons for the low energy hadronic vacuum of the QCD. For small quark masses in the QCD, the mesons become pseudo-Goldstone bosons with small non zero masses in comparison to other hadrons . The chiral condensate becomes zero at higher temperature $T_{c}$ in the chiral limit of zero quark mass and this leads to chiral symmetry restoring phase transition. For 2+1 flavor of light quark masses, the phase transition turns into a smooth crossover. Lattice QCD simulations show that the pseudo-critical temperature for such chiral crossover is $T_{c}=154\pm 9$ LQCDWB2 ; HotLQCDL .

QCD also has $U_{A}(1)$ axial symmetry at the classical level which is broken explicitly by the quantum anomaly as shown by ’t Hooft tHooft:76prl . In QCD vacuum, Instanton effects explicitly break the $U_{A}(1)$ to $Z_{A}(3)$. $U_{A}(1)$ symmetry breaking can be understood in terms of Kobayashi-Maskawa-’t Hooft (KMT) determinant tHooft:76prl ; Kobaya describing the flavor mixing six quark interaction that gets contributions from fluctuations in the topological charge. Flavor mixing removes the degeneracy among several mesons.The pseudoscalar singlet $\eta^{\prime}$ meson acquires a mass of about 1 GeV. In the absence of $U_{A}(1)$ anomaly, $\eta^{\prime}$ would have been degenerate with $\pi$ in $U(3)$ as a ninth Goldstone boson in the spontaneously broken chirally symmetric phase. Since the fluctuations in the topological charge are suppressed at temperatures far above the chiral crossover temperature $T_{c}$ Gross ; Yaffe ; TSchaf ,it is expected that the axial anomaly vanishes and $U_{A}(1)$ symmetry gets effectively restored for temperature $T_{U(1)}>>T_{c}$.Shuryak Shuryak1 has discussed two scenarios : the complete chiral symmetry $SU_{A}(3)\times U_{A}(1)$ is restored in scenario one for $T_{U(1)}>>T_{c}$ well inside the quark gluon plasma region and in scenario two $T_{U(1)}\sim T_{c}$. Further, it has been argued that mass splitting between pion and pseudoscalar singlet meson could become small immediately above the chiral crossover Jkapusta . In Ref Csorgo ; Vertesi , it is reported that the in-medium mass of pseudoscalar singlet meson drops by at least 200 MeV which supposedly is an experimental signature of effective restoration of $U_{A}(1)$ symmetry .

Some lattice QCD calculations have addressed the issue of effective restoration of $U_{A}(1)$ symmetry. Recently $U_{A}(1)$ restoration was explored by computing the degeneration of pion and $a_{0}$-meson screening masses using improved $p^{4}$ staggered fermions ChengLQCD . In a two-flavor simulation with overlap and domain wall fermionsCossuL ; CossuLQCD , the correlators of pion and pseudo-scalar singlet meson have been observed to be degenerate in the chiral symmetric phase . For the 2+1 domain wall fermions, Ref TBHATTA ; BazLQCD find effective restoration of $U_{A}(1)$ at a larger temperature of 196 MeV while Refs Sharma ; Dick do not see restoration even at 1.5 times the crossover temperature using highly improved staggered fermions. The interplay of $U_{A}(1)$ axial symmetry restoration and chiral crossover transition has been explored in several effective model settings like Dyson-Schwinger approach using quark-gluon interaction Smekal ; Alfok ; Benic ; SBenic , in the Nambu-Jona-Lasinio (NJL) model Kunihiro ; Costa:04 ; Costa:05 ; Costa:09 ; Chen ; Zhang ; Powell ; Ruivo ; Bratovic and linear sigma modelsBielich:00prl ; Rischke:00 ; Fejos ; Eser which become quark-meson (QM) models Schaefer:09 ; Schaefer:08ax ; Mitter ; Herbst when coupled with quarks , in the Polyakov loop augmented Nambu-Jona-Lasinio (PNJL) models and Polyakov loop enhanced quark-meson (PQM) models. PNJL and PQM models combine the features of spontaneous breakdown of both chiral symmetry as well as the center $Z(3)$ symmetry of QCD in one single model (see for example Costa:09 ; Ruivo ; Schaefer:07 ; gupta ; Schaefer:09ax ; Schaefer:09wspax ; H.mao09 ; Herbst:2010rf ; Pawl:Schaef ; Marko:2010cd ; kahara ; Digal:01 ; Pisarski:00prd ; Dumitru ; Ratti:06 ; Ratti:07 ; Haas ; Aichelin ; Kovacs ; Ghosh ; Hansen:07 ; Ratti:07npa ; Tamal:06 ; Sasaki:07 ; Hell:08 ; Abuki:08 ; Ciminale:07 ; Fu:07 ; Kfuku:04plb ; Fukushima:08d77 ; Fukushima:08d78 ; Fukushima:09 ; Contrera ; nonlocal ; Odilon ; Contrera:NLM ; Hiller ). In these models chiral condensate and Polyakov loop are simultaneously coupled to the quark degrees of freedom. In the high temperature/density regime, QCD vacuum shows color confinement-deconfinement phase transition for the infinitely heavy quarks where the center symmetry $Z(3)$ of color gauge group $SU_{c}(3)$ gets spontaneously broken and the expectation value of the Wilson line (Polyakov loop) serves as the order parameter Polyakov:78plb as it is related to the free energy of a static color charge . Since the center symmetry is always explicitly broken in the presence of dynamical quarks in the system, one regards the Polyakov loop as an approximate order parameter for the confinement-deconfinement transition Pisarski:00prd .

The effect of confinement physics i.e that of the Polyakov loop potential on the relation of $SU(3)$ chiral symmetry and $U_{A}(1)$ axial symmetry above chiral crossover temperature has been explored in the 2+1 flavor PQM model in Ref.gupta .Earlier,in the 2+1 flavor QM model, Schaefer et. al.Schaefer:09 ; Schaefer:08ax had studied the interplay of these symmetries at higher temperatures. When renormalized fermionic vacuum fluctuation is included in their thermodynamic potential, the two flavor QM model and PQM model Skokov:2010sf ; Vivek:12 ; VKTR , become effective QCD-like because now models can reproduce the second order chiral phase transition at $\mu=0$ as expected from the universality arguments Wilczek for the two massless flavors of QCD. In earlier works of implementing fermionic vacuum correction in the 2+1 flavor PQM model, the renormalization scale independence of the model parameters and meson masses used to be implemented numerically Schaef:12 ; Sandeep . Recently, explicit renormalization scale independent analytical expressions for the model parameters, meson masses and mixing angles have been found for the 2+1 flavour PQM model in the Ref. Trvk where the effect of fermionic vacuum correction, on the interplay of $SU_{A}(3)$ chiral symmetry and $U_{A}(1)$ symmetry restoration has been explored. We will be considering a modification of this model in the present work.

It is very relevant to ask if the $U_{A}(1)$ symmetry,which is broken in the vacuum by instantons at the quantum level, is still broken in the chiral symmetric phase Meggiolaro ; fukushima:01 .If the $U_{A}(1)$ symmetry breaking strength decreases with temperature, it is pertinent to ask what fraction of it remains above the critical temperature and whether and when this symmetry is restored Ruivo . It has been pointed out that there are phenomenological consequences for the nature of phase transition depending on the degree of anomaly present at the critical temperature BazLQCD . Temperature dependence in the form of decreasing exponential for the axial anomaly term was originally proposed by Kunihiro in Ref. Kunihiro and later its phenomenological effects for $U_{A}(1)$ restoration was explored by Costa et. al. in Ref. Costa:04 ; Costa:05 for the NJL model. Another form of exponentially decreasing temperature dependence of anomaly coefficient and its consequences for the $U_{A}(1)$ symmetry restoration has been explored in Ref. Ruivo in the context of SU(2) EPNJL model.

Here, we point out that the evolution of $U_{A}(1)$ anomaly and its relation to the chiral symmetry in the extended linear sigma model, QM model has also been investigated in the powerful framework of nonperturbative functional renormalization group (FRG) method Fejos ; Eser ; Mitter ; Yin ; Kamikado ; Jiang ; Fejos:16 ; Heller ; Rennecke ; Fejos:18 where the bosonic and fermionic quantum and thermal fluctuations are included in the FRG effective action through running the RG scale from the ultraviolet scale to the infrared limit, which, as an advantage, can automatically guarantee the Nambu-Goldstone theorem in the symmetry breaking phase. The thermal evolution of $U_{A}(1)$ factor when explored by the chiral invariant expansion technique Fejos , shows that mesonic fluctuations not only produce condensate dependent anomaly coefficient, but depending on region of the parameter space, they can either strengthen or weaken the KMT determinant term as the temperature increases. In Ref. Fejos:16 ; Fejos:18 , it was found that mesonic fluctuations strengthen the anomaly towards $T_{c}$. While recent lattice simulations, show that the anomaly does not restore until around $1.5\ T_{c}$ Sharma ; Dick . In this light, it becomes an important question whether temperature dependence of KMT coupling that arises from the instanton effects, can offset the effect of the mesonic fluctuations.

In the present work, we consider a temperature dependence of the coefficient c(T) for the Kobayashi-Maskawa-’t Hooft (KMT) determinant term in the 2+1 flavor PQM model with fermionic vacuum correction (PQMVT). Phenomenological form of strong temperature suppression for the KMT coupling strength was inferred from the lattice data on ($\pi,a_{0}$) screening massesChengLQCD . Its effect on the $U_{A}(1)$ symmetry restoration has already been studied in the 2+1 flavor entanglement-PNJL (EPNJL) model Ishii ; Ishiky where they find degeneration in the masses of $a_{0}$ and $\pi$ mesons at T=180 MeV=$1.1\ T_{c}$ with $T_{c}=163$ MeV which means $U_{A}(1)$ restoration takes places immediately after chiral crossover. We do not have such investigations in the PQM model for meson masses calculated from the curvature of effective potential. It is worthwhile to have such a detailed investigation in the PQMVT model also for obtaining qualitative and quantitative trends of $U_{A}(1)$ symmetry restoration that may facilitate comparison with other studies. Further it is to be noted that the mesons in the PQM model are the degrees of freedom included in the Lagrangian form the outset while those are generated by some prescription in the NJL model Costa:05 and the $\eta^{\prime}$ is not a well defined quantity fukushima:01 which becomes unbound for higher temperatures.

This paper has the following arrangement . Sec.2, explains the formulation of the model. In subsection 2.1, we have explained the introduction of temperature dependence for KMT coupling. In subsection 2.2, the used form of Polyakov loop potential has been explained. The Sec.3 describes the mean field grand potential with fermionic vacuum correction. The Sec.4 gives the model formulae of meson masses and mixing angles in the vacuum and the finite temperature/density medium. The Sec. 5 gives the details of parameter fixing. In Sec.6, we will be discussing the numerical results and plots for temperature variations of meson masses and mixing angles. Summary and conclusion is presented in the last Sec.7.

2 Model Formulation

In the model gupta ; Schaefer:09wspax ; H.mao09 ; Schaefer:09ax , 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

[TABLE]

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

[TABLE]

Here $\mathcal{P}$ is path ordering, $A_{0}$ is the temporal component of vector field and $\beta=T^{-1}$ Polyakov:78plb . In accordance to Ref. Ratti:06 ; Ratti:07 we consider an homogeneous Polyakov loop field $\Phi(\vec{x})=\Phi$=constant and $\bar{\Phi}(\vec{x})=\bar{\Phi}$=constant.

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

[TABLE]

where the Lagrangian in Quark Meson Chiral Sigma model

[TABLE]

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. $\lambda_{a}$ are Gell-Mann matrices in the color space, $a$ runs from $1\cdots 8$. $q_{f}=(u,d,s)^{T}$ denotes the quarks of three flavors and three colors. $T_{a}$ represent 9 generators of $U(3)$ flavor symmetry with $T_{a}=\frac{\lambda_{a}}{2}$ and $a=0,1\dots~{}8$, here $\lambda_{a}$ are standard Gell-Mann matrices in flavor space with $\lambda_{0}=\sqrt{\frac{2}{3}}\ \bf 1$. $g$ is the flavor blind Yukawa coupling that couples the three flavor of quarks with nine mesons in the scalar ($\sigma_{a},J^{P}=0^{+}$) and pseudo scalar ($\pi_{a},J^{P}=0^{-}$) sectors. The quarks acquire mass though spontaneous breakdown of chiral symmetry .

The mesonic part of the Lagrangian has the following form

[TABLE]

The chiral field $M$ is a $3\times 3$ complex matrix comprising of the nine scalars $\sigma_{a}$ and the nine pseudo scalar $\pi_{a}$ mesons.

[TABLE]

The generators follow $U(3)$ 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 standard antisymmetric and symmetric structure constants respectively with $f_{ab0}=0$ and $d_{ab0}=\sqrt{\frac{2}{3}}\ {\bf 1}\ \delta_{ab}$ and matrices are normalized as ${\mathrm{Tr}}(T_{a}T_{b})=\frac{\delta_{ab}}{2}$. c is the KMT coupling, a parameter considered in general as constant.

The $SU_{L}(3)\times SU_{R}(3)$ chiral symmetry is explicitly broken by the explicit symmetry breaking term

[TABLE]

Here $H$ is a $3\times 3$ matrix with nine external parameters. The $\xi$ field which denotes both the scalar as well as pseudo scalar mesons, picks up the nonzero vacuum expectation value, $\bar{\xi}$ for the scalar mesons due to the spontaneous breakdown of the chiral symmetry while the pseudo scalar mesons have zero vacuum expectation value. Since $\bar{\xi}$ must have the quantum numbers of the vacuum, explicit breakdown of the chiral symmetry is only possible with three nonzero parameters $h_{0}$, $h_{3}$ and $h_{8}$. We are neglecting isospin symmetry breaking hence we choose $h_{0}$, $h_{8}\neq 0$. This leads to the $2+1$ flavor symmetry breaking scenario with nonzero condensates $\bar{\sigma_{0}}$ and $\bar{\sigma_{8}}$.

2.1 Temperature dependence of KMT Coupling

The density of the instantons gets screened by the medium at finite temperature $T$ . Since the coupling $c$ of the KMT interaction is proportional to the instanton density , it gets reduced as $T$ increases. Such suppression $S(T)\equiv c(T)/c(0)$ was discussed by Pisarski and Yaffe for high temperatures Yaffe , say $T\geq 2T_{c}$ for the chiral crossover transition pseudocritical temperature $T_{c}$, by calculating the Debye-type screening

[TABLE]

Here $N_{f}(N_{c})$ is the number of flavors (colors) and the typical instanton radius $\rho$ is about $\frac{1}{3}$ fm, hence the suppression parameter b is about $0.70T_{c}$ for $N_{f}=N_{c}=3$. The temperature dependence of the instanton density was theoretically estimated by the instanton-liquid model Shuryak1 after Pisarski-Yaffe discussion on $S(T)$ but this estimation is applicable only for $T\geq 2T_{c}$ . Due to this reason, Ruivo et. al. in Ref. Ruivo used a phenomenological Wood-Saxon form $1/(1+exp((T-T_{1}^{\prime})/b_{1}^{\prime}))$ with two parameters ${T_{1}^{\prime}}$ and ${b_{1}^{\prime}}$ for the instanton suppression $S(T)$ factor. Further, lattice QCD study suggests that at $T\sim 1.5\ T_{c}$, the origin of $U_{A}(1)$ breaking results due to the dilute gas of weakly interacting instantons and anti-instantons Sharma ; Dick . It would be interesting to explore, how the suppression of instanton density and the diminishing of the KMT coupling in the vicinity of the chiral crossover transition temperature $T_{c}$ (say $T\sim 0.7\ T_{c}$), influences the restoration of chiral symmetry and its interplay with the $U_{A}(1)$ symmetry restoration in the temperature range $T\sim(0.7\ T_{c}\to 2.0\ T_{c})$ .

In our model, we consider the following temperature dependence for the coefficient c(T) :

[TABLE]

The above parameterization of KMT coupling strength was introduced very recently for exploring the $U_{A}(1)$ symmetry restoration in the context of the entanglement-PNJL (EPNJL) model Ishii ; Ishiky where the parameters $T_{1}$ and $b_{1}$ are determined from LQCD data on pion and $a_{0}$-meson screening masses . The $T$ dependence of Eq. (9) is consistent with the Pisarski and Yaffe suppression of the instanton density for high $T\geq 2T_{c}$ and is similar to the Woods-Saxon form Yaffe ; Gross . In our model setting, we shall determine the parameters $T_{1}$ and $b_{1}$ by fitting the subtracted chiral order parameter obtained in our calculation to the Wuppertal group LQCD data LQCDWB2 for the same . We have considered temperature dependence of subtracted chiral order parameter for fitting it with the LQCD data because other quantities depend on it and it is simple to understand and numerically implement in our model calculation. Further the physics of temperature dependent reduction in axial anomaly is interlinked with the chiral symmetry restoration and confinement-deconfinement transition.

2.2 Polyakov Loop Potential

The effective Polyakov loop potential ${\cal U}\left(\Phi,\bar{\Phi},T\right)$ is constructed such that it reproduces thermodynamics of pure glue theory on the lattice for temperatures upto about twice the deconfinement phase transition temperature. The functional form of Polyakov loop potential has various possibilities. The simplest Polyakov loop effective potential as introduced in Pisarski:00prd ; Dumitru ; Ratti:06 has a polynomial form. The polynomial form of Polyakov loop potential in Ref. Ratti:06 produces some unwanted properties such as negative Susceptibilities Sasaki:07 . We are using logarithmic parameterization of Polyakov loop potential which comes from the SU(3) Haar measure of the group integration Ratti:07 and the negative Susceptibilities problem is also absent in this case . The logarithmic Polyakov loop potential is the following

[TABLE]

where the parameters are the following

[TABLE]

where constants have the values

[TABLE]

For pure gauge Yang Mills theory, the deconfinement phase transition is first order and the critical temperature is $T_{0}=270$ MeV. In the presence of dynamical quarks, the first order transition turns to a crossover and $T_{0}$ gets reduced depending on number of quark flavours and chemical potential Schaefer:07 ; Herbst:2010rf ; Pawl:Schaef ; Haas . As given in Ref. Schaefer:07 , at zero chemical potential $T_{0}=187$ for the 2+1 quark flavors. We should note that the change of $T_{0}$ does not modify the functional form of the Polyakov loop effective potential. Due to this, the expectation value of the Polyakov loop potential fails to coincide with the minimum of the effective potential because $\Phi(T)$ is found by the minimization of the grand-canonical potential (which includes the quark-antiquark contribution also) and not of only $\cal U_{\text{YM}}$ One has to look for a possible effect of quarks on the Polyakov loop effective potential out of its minimum Aichelin .

In order to find the gluonic contribution to the pressure $P_{\text{gluon}}$ = $-\cal{U_{\text{YM}}}$ $(T;\Phi(T)\bar{\Phi}(T))$, one should introduce quark back reaction onto $\cal{U}$, not limited to their effect at the minimum of the potential. In the context of the functional renormalization group (FRG) equations applied to QCD, Ref. Haas has compared the pure gauge effective potential $\cal U_{\text{YM}}$ to the “glue” potential $\cal U_{\text{glue}}$ where quark polarization has been included in the gluon propagator. They have found significant differences between the two effective potentials. However, the study observed that the two potentials have the same shape and that they can be mapped into each other by relating the temperatures of the two systems , $T_{\text{YM}}$ and $T_{\text{glue}}$ respectively. Denoting by $\cal U_{\text{YM}}$ the potential in Eq. (10), the improved Polyakov loop potential was constructed as Haas

[TABLE]

where the relation between temperature $T_{\text{glue}}$ and $T_{\text{YM}}$ is given by

[TABLE]

Here, the critical temperature $T_{\text{YM}}^{c}=270$ MeV and $T_{\text{glue}}^{c}$ is the transition temperature for the unquenched case. The coefficient 0.57 comes from the comparison of the two effective potentials. $T_{\text{glue}}^{c}$ lies within a range $T_{\text{glue}}^{c}\in[180,270]$. In practice , we use in the right-hand side of Eq. (10) where $T_{0}$ means $T_{\text{YM}}^{c}$, the replacement $T\rightarrow T_{\text{YM}}^{c}(1+0.57(\frac{T}{T_{\text{glue}}^{c}}-1))$ where $(T\equiv T_{\text{YM}})$ on the left side of arrow and $(T\equiv T_{\text{glue}})$ on the right side . In our calculation, we shall use several values of $T_{\text{glue}}^{c}$ in the range to fit the lattice data for different parameters depending on different values of $\sigma$ meson mass.

3 Grand Potential in the Mean Field Approach

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

[TABLE]

where $V$ is the three dimensional volume of the system, $\beta=\frac{1}{T}$ and the superscript $\mathcal{E}$ denotes the euclidean Lagrangian. For three quark flavors, in general, the three quark chemical potentials are different. In this work, we assume that $SU_{V}(2)$ symmetry is preserved and neglect the small difference in masses of $u$ and $d$ quarks. Thus the quark chemical potential for the $u$ and $d$ quarks become equal $\mu_{x}=\mu_{u}=\mu_{d}$. The strange quark chemical potential is $\mu_{y}=\mu_{s}$.

Here, the partition function is evaluated in the mean-field approximation Schaefer:08ax ; gupta ; Schaefer:09 . We replace meson fields by their expectation values $\langle M\rangle=T_{0}\bar{\sigma_{0}}+T_{8}\bar{\sigma_{8}}$ and neglect both thermal as well as quantum fluctuations of meson fields while quarks and anti quarks are retained as quantum fields. Now following the standard procedure as given in Refs. Kapusta_Gale ; Schaefer:07 ; Ratti:06 ; Kfuku:04plb , 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,

[TABLE]

The mesonic potential $U(\sigma_{x},\sigma_{y})$ is obtained from the $U(\sigma_{0},\sigma_{8})$ after transforming the original singlet-octet (0, 8) basis of condensates to the non strange-strange basis $(x,y)$ as in Refs. Rischke:00 ; Schaefer:09 ; gupta ; Schaef:12 . We write the mesonic potential as

[TABLE]

where

[TABLE]

The chiral symmetry breaking external fields ($h_{x}$, $h_{y}$) are written in terms of ($h_{0}$, $h_{8}$) analogously. In our model when $c$ becomes $c(T)$, $U(\sigma_{x},\sigma_{y})$ becomes $U(\sigma_{x},\sigma_{y},T)$.

Further the non strange and strange quark/antiquark decouple and the quark masses are

[TABLE]

Quarks become massive in symmetry broken phase because of non zero vacuum expectation values of the condensates. The quark/antiquark contribution Skokov:2010sf ; Trvk , is written as

[TABLE]

The first term of the Eq. (20) represents the fermion vacuum one loop contribution, regularized by the ultraviolet cutoff $\Lambda$. The expressions $g_{f}^{+}$ and $g_{f}^{-}$ in the presence of Polyakov loop potential are defined in the second term after taking trace over the color space

[TABLE]

[TABLE]

E${}_{f}^{\pm}=E_{f}\mp\mu_{f}$ and $E_{f}$ is the flavor dependent single particle energy of quark/antiquark and $m{{}_{f}}$ is the mass of the given quark flavor.

[TABLE]

The first term of Eq. (20) gives renormalized fermion vacuum loop correction at one-loop order  Quiros:1999jp ; Skokov:2010sf ; Trvk as :

[TABLE]

The Polyakov loop potential and the temperature dependent part of the quark-antiquark contribution to the grand potential in Eq.(15) vanishes at $T=0$ and $\mu=0$. Now the grand potential in vacuum becomes the renormalization scale M dependent as Trvk :

[TABLE]

The logarithmic M dependence of the term $\Omega_{q\bar{q}}^{\rm vac}$ generates a renormalization scale M dependent part $\lambda_{2\text{M}}$ in the expression of the parameter $\lambda_{2}=\lambda_{2s}+n+\lambda_{2+}+\lambda_{\text{2M}}$. $\lambda_{2s}$ is the same old $\lambda_{2}$ parameter of the QM/PQM model in Ref. Rischke:00 ; Schaefer:09 ; gupta . Here, $n=\frac{N_{c}g^{4}}{32\pi^{2}}$, $\lambda_{2+}=\frac{n{f_{\pi}}^{2}}{f_{K}\left(f_{K}-f_{\pi}\right)}\log\{\frac{2f_{K}-f_{\pi}}{f_{\pi}}\}$ and $\lambda_{\text{2M}}=4n\log\{\frac{g\left(2f_{K}-f_{\pi}\right)}{2\text{M}}\}$. After substituting this value of $\lambda_{2}$ in the expression of U($\sigma_{x},\sigma_{y}$) and rearranging the terms, we find that the logarithmic M dependence of $\lambda_{2}$ contained in $\lambda_{\text{2M}}$, completely cancels the scale dependence of all the terms in $\Omega_{q\bar{q}}^{\rm vac}$. The vacuum effective potential now becomes free of any renormalization scale dependence. It is expressed as Trvk

[TABLE]

here, $\lambda_{2\text{v}}=\lambda_{2s}+\lambda_{2+}$. The calculation of vacuum meson masses from the effective potential also shows that the scale M dependence completely cancels out from their expressions. The explicit derivations of scale independent meson masses are given in the appendix B of Ref. Trvk . Again when $c$ becomes $c(T)$, $\Omega(\sigma_{x},\sigma_{y})$ becomes $\Omega(\sigma_{x},\sigma_{y},T)$.

Now the thermodynamic grand potential in the Polyakov Quark Meson Model with fermion vacuum correction term (PQMVT) will be written as

[TABLE]

One can get the quark condensates $\sigma_{x}$, $\sigma_{y}$ and Polyakov loop expectation values $\Phi$, $\bar{\Phi}$ by searching the minimum of the grand potential for a given value of temperature $T$ and chemical potential $\mu_{f}$.

[TABLE]

4 Meson Masses and Mixing Angles

The curvature of the grand potential in Eq.(15) at the minimum gives the finite temperature scalar and pseudo scalar meson mass matrices.

[TABLE]

The subscript $\alpha=$ s, p ; s stands for scalar and p stands for pseudo scalar mesons and $a,b=0\cdots 8$.

[TABLE]

Where $({m^{\text{m}}_{\alpha,ab}})^{2}$ is the tree level mass matrix, $({\delta m^{\text{v}}_{\alpha,ab}})^{2}$ and $({\delta m^{\text{T}}_{\alpha,ab}})^{2}$ are the contributions from fermionic vacuum and thermal fluctuations, respectively.

The mass modification in vacuum ($T=0$, $\mu=0$), due to the fermionic vacuum correction will be given by

[TABLE]

Here $m^{2}_{f,a}\equiv\partial m^{2}_{f}/\partial\xi_{\alpha,a}$ denotes the first partial derivative and $m^{2}_{f,ab}\equiv\partial m^{2}_{f,a}/\partial\xi_{\alpha,b}$ signifies the second partial derivative of the squared quark mass with respect to the meson fields $\xi_{\alpha,b}$. These derivatives are evaluated in the non strange-strange basis and given in the Table III of Ref.Schaefer:09 . The mass modifications given in Eq.(31) have been evaluated Trvk ; Sandeep and different expressions of $({\delta m^{\text{v}}_{\alpha,ab}})^{2}$ are collected in the Table 1 for all the mesons of scalar and pseudo-scalar nonet. The vacuum mass expressions $({m^{\text{m}}_{\alpha,ab}})^{2}$ for these mesons as originally evaluated from the second derivative of pure mesonic potential in Ref. Rischke:00 ; Schaefer:09 , are also given in this Table. Further the diagonalization of the (00)-(88) sector of the pseudo scalar mass matrix, gives the squared masses of $\eta$ and $\eta^{\prime}$ and analogously, we get a pseudo-scalar mixing angle $\theta_{P}$. The mass expressions $({m^{\text{m}}_{\alpha,ab}})^{2}$ have a renormalization scale M dependence in the PQMVT model due to the parameter $\lambda_{2}$. This dependence gets completely canceled by the already existing scale M dependence in the mass modifications $({\delta m^{\text{v}}_{\alpha,ab}})^{2}$ and the final expressions of vacuum meson masses $m_{\alpha,ab}^{2}$ which is equal to $({m^{\text{m}}_{\alpha,ab}})^{2}+({\delta m^{\text{v}}_{\alpha,ab}})^{2}$ , are free of any renormalization scale dependence as shown explicitly in the appendix B of Ref.Trvk .

The correction to the meson mass matrix due to the fermionic thermal fluctuations in the presence of Polyakov loop as originally evaluated in Ref.gupta in the PQM model is written as:

[TABLE]

The notations $A_{f}^{\pm}$ and $B_{f}^{\pm}$ have the following definitions

[TABLE]

[TABLE]

and $B_{f}^{\pm}={3(A_{f}^{\pm}})^{2}-C_{f}^{\pm}$, where we again define

[TABLE]

[TABLE]

In the present work we are considering symmetric quark matter and net baryon number to be zero.

The diagonalization of (0,8) component of mass matrix gives the masses of $\sigma$ and $f_{0}$ mesons in scalar sector and the masses of $\eta^{\prime}$ and $\eta$ in pseudo scalar sector. The scalar mixing angle $\theta_{s}$ and pseudo scalar mixing angle $\theta_{p}$ are given by

[TABLE]

The appendix C of Ref.Schaefer:09 contains all transformation details of the mixing for the (0,8) basis that generates the physical basis of the scalar ($\sigma$,$f_{0}$) and pseudo-scalar ($\eta^{\prime}$, $\eta$) mesons. This appendix also explains the ideal mixing, and gives the details of formulae by which the physical mesons transform into the pure strange or non-strange quark system of mesons.

5 Model Parameter Fixing

The six unknown parameters $\text{m}^{2}$, $\lambda_{1},\lambda_{2}$, $h_{x},h_{y}$ and $c$ in the mesonic potential U($\sigma_{x},\sigma_{y}$), are fixed in the vacuum by six experimentally known quantities. The known values of $m_{\pi}$, $m_{K}$, the average squared mass of $\eta$ and $\eta^{\prime}$ mesons, $m_{\eta}^{2}+m_{\eta^{\prime}}^{2}$, and the pion and kaon decay constants $f_{\pi}$ and $f_{K}$ and finally the $m_{\sigma}$ are used to obtain the value six parameters in vacuum Rischke:00 ; Schaefer:09 .

In the PQMVT model calculations, the vacuum mass expressions that determine $\lambda_{2}$ and c are {widetext}

$m^{2}_{\pi}=({m^{\text{m}}_{\pi}})^{2}+({\delta m^{\text{v}}_{p,11}})^{2}$, $m^{2}_{K}=({m^{\text{m}}_{K}})^{2}+({\delta m^{\text{v}}_{p,44}})^{2}$ and $m^{2}_{\eta}+m^{2}_{\eta^{\prime}}=m^{2}_{p,00}+m^{2}_{p,88}$ where $m^{2}_{p,00}=({m^{\text{m}}_{p,00}})^{2}+({\delta m^{\text{v}}_{p,00}})^{2}$ and $m^{2}_{p,88}=({m^{\text{m}}_{p,88}})^{2}+({\delta m^{\text{v}}_{p,88}})^{2}$. We can write $m^{2}_{\eta}+m^{2}_{\eta^{\prime}}=({m^{\text{m}}_{\eta}})^{2}+({m^{\text{m}}_{\eta^{\prime}}})^{2}+({\delta m^{\text{v}}_{p,00}})^{2}+({\delta m^{\text{v}}_{p,88}})^{2}$ where $({m^{\text{m}}_{\eta}})^{2}+({m^{\text{m}}_{\eta^{\prime}}})^{2}=({m^{\text{m}}_{p,00}})^{2}+({m^{\text{m}}_{p,88}})^{2}$. Using mass modification expressions $({\delta m^{\text{v}}_{\alpha,ab}})^{2}$ given in the Table 1, we write

[TABLE]

The $f_{\pi}$ and $f_{K}$ give vacuum condensates according to the partially conserved axial vector current (PCAC) relation. The x =$\sigma_{x}=f_{\pi}$ and y =$\sigma_{y}=\left(\frac{2f_{K}-f_{\pi}}{\sqrt{2}}\right)$ at $T=0$. The parameters $\lambda_{2}$ and c in vacuum are obtained as:

[TABLE]

[TABLE]

When expressions of $({m^{\text{m}}_{\pi}})^{2}$, $({m^{\text{m}}_{K}})^{2}$ and $\left(({m^{\text{m}}_{\eta}})^{2}+({m^{\text{m}}_{\eta^{\prime}}})^{2}\right)$ from Eq.(38) are substituted in Eq.(39) and Eq.(40) and the vacuum value of the condensates are used, the final rearrangement of terms yields:

[TABLE]

[TABLE]

We note that the $\lambda_{2s}$ is equal to the earlier $\lambda_{2}$ parameter determined in the QM/PQM model calculations in Ref. Rischke:00 ; Schaefer:09 ; gupta . We have used the numerical values $f_{\pi}=92.4$  MeV, $f_{K}=113.0$  MeV, $m_{\pi}=138.0$  MeV, $m_{K}=496.0$  MeV,$m_{\eta}=539.0$  MeV and $m_{\eta^{\prime}}=963.0$  MeV . In the present calculation, the proper renormalization of fermionic vacuum, leads to the augmentation of $\lambda_{2s}$ by the addition of a term (n+$\lambda_{2+}$) and further, we get a renormalization scale M dependent contribution $\lambda_{\text{2M}}$ in the expression of the $\lambda_{2}$ in Eq.(5).

We get the complete cancellation of M dependence in the evaluation of $c$ also and finally its value turns out to be the same as in the QM model. The scale M independent expression of $m^{2}_{\pi}$ obtained in the appendix B of Ref.Trvk can be used with x=$f_{\pi}$ and y=$(\frac{2f_{K}-f_{\pi}}{\sqrt{2}})$, to express $m^{2}$ in terms of $\lambda_{1}$.

[TABLE]

When we use the formula of $m^{2}_{\sigma}$ in the Table 2 of the appendix B of RefTrvk (with the vacuum values of the masses $m^{2}_{s,00}$, $m^{2}_{s,88}$, $m^{2}_{s,08}$ and the mixing angle $\theta_{S}$) and substitute the above expression of $m^{2}$ in it, we will get the numerical value of $\lambda_{1}$ for different values of $m_{\sigma}$ when we put $m_{\pi}$=138 MeV . The explicit symmetry breaking parameters $h_{x}$ and $h_{y}$ in the nonstrange-strange basis are related to the pion and kaon masses by the ward identities Rischke:00 ; Schaefer:09 .

[TABLE]

The last parameter , the Yukawa coupling g, is fixed from the nonstrange constituent quark mass

[TABLE]

The Yukawa coupling value $g=6.5$ has been fixed from the non strange constituent quark mass $m_{q}=300$ MeV in vacuum ($T=0,\mu=0)$. This predicts the vacuum strange quark mass $m_{s}\simeq 433$ MeV. We have chosen a combination of sigma meson mass and Polyakov loop potential parameter ($m_{\sigma},T_{glue}^{c}$) and found for this combination, the values of the parameters $T_{1}$ and $b_{1}$ which give the temperature dependence of KMT coupling $c(T)$ by obtaining the most suitable coincidence of the temperature variations of the subtracted chiral condensate $\Delta_{l,s}(T)$ calculated in our PQMVT-II model with the Wuppertal group lattice QCD data for the same. Fixing $T_{glue}^{c}$ at 240 MeV, we obtain a good coincidence of $\Delta_{l,s}(T)$ temperature variation with the LQCD data when temperature dependence of $c(T)$ in PQMVT-II model becomes steep and steeper with small and smaller values of the parameters $T_{1}$ and $b_{1}$ for large and larger values of the sigma meson mass $m_{\sigma}$.

6 Temperature Dependence of c(T) and Chiral Restoration

We are exploring, how the temperature dependence of $U_{A}(1)$ breaking coupling strength c(T) in the $2+1$ flavor Polyakov quark meson model with the fermionic vacuum correction term (PQMVT), influences the chiral symmetry restoration and the axial $U_{A}(1)$ symmetry restoration. For comparing the results in scenario one, when $U_{A}(1)$ breaking coupling strength c is constant, we call the PQMVT model as PQMVT-I while in scenario two, when $U_{A}(1)$ breaking coupling strength c(T) is temperature dependent, we call PQMVT model as PQMVT-II. The temperature variations of nonstrange condensate, strange condensate and Polyakov loop expectation value are quite sensitive to the sigma meson mass in both the model scenarios. Table 3, shows values of all the parameters where $\lambda_{1}$ , $m^{2}$ change significantly depending on the $m_{\sigma}$ value. We have fitted the Wuppertal group LQCD data for the subtracted chiral order parameter LQCDWB2 with the corresponding $\Delta_{ls}$ temperature variation in the PQMVT-II model for fixed value of $T_{glue}^{c}$= 240 MeV and three different values of $m_{\sigma}$. When $m_{\sigma}$=400 MeV, the parameter set $T_{1}$=108 MeV and $b_{1}$=90 MeV for the temperature dependence of $c(T)$ in the PQMVT-II model calculation, gives a good coincidence of $\Delta_{ls}$ temperature variation with the LQCD data. For $m_{\sigma}$=500 MeV, we obtain a good fit to the LQCD data when the temperature dependence of $c(T)$ becomes steeper with smaller $T_{1}$=105 MeV and $b_{1}$=78 MeV. For larger $m_{\sigma}$ = 600 MeV, in order to find a good fit of the $\Delta_{ls}$ temperature variation with the LQCD data, the temperature dependence of $c(T)$ becomes steepmost with still smaller $T_{1}$=105 MeV and $b_{1}$=62 MeV in the PQMVT-II model.

6.1 Effect of c(T) on Condensates

On solving the gap equations Eq.(28) at zero chemical potential, we get the temperature dependence of the Polyakov loop expectation value $\Phi$, non strange and strange condensates . The inflection point of these order parameters respectively give the characteristic temperature (pseudo-critical temperature) for the confinement - deconfinement transition $T_{c}^{\Phi}$, the chiral transition in the non-strange $T_{c}^{\chi}$ and strange sector $T_{s}^{\chi}$. Table 4 shows the various pseudo-critical temperatures in two model scenarios for different values of $m_{\sigma}$. We can not have direct comparison of non-strange condensate $<\sigma_{x}>$ and strange condensate $<\sigma_{y}>$ with the lattice data . In the limit of zero light quark mass (chiral limit), QCD has a chiral symmetry and at finite temperature, a true phase transition occurs. To eliminate quadratic divergences in the linear quark mass dependent correction to the chiral condensate, one calculates a suitable combination of light and strange quark condensates at finite temperature. This quantity is further normalized by the corresponding combination of condensates calculated at $T=0$ MeV cheng2008 . Subtracted chiral condensate $\Delta_{l,s}(T)$ is such an observable which works as an order parameter for the chiral symmetry breaking. It is written as

[TABLE]

For $T=0$, the condensate $\sigma_{x0}$ = 92.4 MeV while the $\sigma_{y0}$ = 94.5 MeV, we have plotted the temperature variations of ($\frac{\sigma_{x}}{\sigma_{x0}}$) , ($\frac{\sigma_{y}}{\sigma_{y0}}$) and $\Phi$ in Fig.1(a) for $m_{\sigma}=400$ MeV and the Polyakov loop potential parameter $T_{glue}^{c}=240$ MeV. We have shown the temperature variation of the $\Delta_{l,s}(T)$ in the Fig.1(b). We obtained the values of parameters $T_{1}=\ 108\ \text{MeV}\ =\ .70\ T_{c}$ and $b_{1}=\ 90\ \text{MeV}\ =\ .58\ T_{c}$ where $T_{c}=\ 154$ MeV for the temperature variation of c(T) by fitting the Wuppertal group lattice data for the $\Delta_{l,s}(T)$ LQCDWB2 to our calculations in model PQMVT-II when $m_{\sigma}=400$ MeV and $T_{glue}^{c}=240$ MeV. $T_{c}$ denotes pseudocritical temperature for chiral crossover obtained from LQCD data. The solid deep red line in Fig.1(b) for the PQMVT-II model, shows a very good agreement of $\Delta_{l,s}(T)$ temperature variation with lattice QCD data LQCDWB2 . Table 3 shows parameters for the temperature dependence of c(T) for different values of $\sigma$ meson mass. The dash dotted line in green, shows temperature variation of the $\Delta_{l,s}(T)$ for $T_{glue}^{c}=260$ MeV in the PQMVT-II model while dash line in deep blue shows $\Delta_{l,s}(T)$ for $T_{glue}^{c}=220$ MeV. The right most line in magenta dots denotes $\Delta_{l,s}(T)$ temperature variation in the PQMVT-I model for constant c.

The dash line in green in Fig.1(a), shows the scaled non-strange condensate temperature variation in the PQMVT-I model for constant c and its inflection point gives the pseudo-critical temperature as $T_{c}^{\chi}=\ 175.6$ MeV while the dotted line in green shows the corresponding strange condensate temperature variation. The second peak of the strange condensate derivative $\frac{\partial\sigma_{y}}{\partial T}$ is rather flat and broad and it gives $T_{s}^{\chi}=\ 261.7\pm 2$ MeV where $\pm$ denotes a distinct change of about 0.1 percent in the numerical value of the second peak height. In PQMVT-II model, for the temperature dependent c(T), the deep red solid line shows a large quantitative and a little sharper qualitative change in the temperature variation of non-strange condensate when compared to the PQMVT-I scenario because the temperature variation of $\frac{\partial\sigma_{x}}{\partial T}$ shows a a little sharper and higher peak (Fig not shown in the text) which occurs quite early at $T_{c}^{\chi}=\ 154.9$ MeV in the PQMVT-II scenario. The deep red dash dotted line, shows the corresponding strange condensate temperature variation in the PQMVT-II model, it shows large change in comparison to the PQMVT-I model for the temperature range 125-210 MeV. The second peak of the strange condensate gives the pseudo-critical temperature $T_{s}^{\chi}=265.2\pm 2$ MeV for the PQMVT-II model.

Since we have taken $\mu=0$ in our computations, we get $\Phi$  =  $\bar{\Phi}$ for the Polyakov loop expectation value. We recall that the logarithmic contribution in the improved ansatz of the polyakov loop potential Haas in Eq. (12) avoids the $\Phi$ expectation value higher than one. It represents more effective description of the gluon dynamics. The inflection point of the temperature variation of the Polyakov loop expectation value, shown by the green line with filled triangles in Fig.1(a) , gives confinement-deconfinement transition pseudocritical temperature $T_{c}^{\Phi}=155.3$ MeV, in the PQMVT-I model for constant c. The red line with hollow circles,show the temperature variation of the Polyakov loop field $\Phi$ in the PQMVT-II model scenario for temperature dependent c(T) whose inflection point gives $T_{c}^{\Phi}=145.8$ MeV when $m_{\sigma}$=400 MeV. $T_{glue}^{c}=240$ MeV for all the cases. When $m_{\sigma}$=500 MeV, $T_{c}^{\Phi}=154.3$ MeV in PQMVT-I model while $T_{c}^{\Phi}=146.1$ MeV in PQMVT-II scenario. $T_{c}^{\Phi}=155.3$ MeV in PQMVT-I model and $T_{c}^{\Phi}=146.1$ MeV in PQMVT-II scenario for $m_{\sigma}$=600 MeV as shown in Table 4.

Higher values of $m_{\sigma}$ makes the non strange condensate melting smoother and shifts it to quite high temperatures in the PQMVT-I model. The $\Delta_{l,s}(T)$ temperature variation for the constant c also shifts to quite high temperatures as shown by dash double dot line in green for $m_{\sigma}$=500 MeV and long thin dash line in blue for $m_{\sigma}$=600 MeV in Fig.2(a) where for comparison, we have repeated the dotted line plot in magenta color for $m_{\sigma}$=400 MeV (already given in Fig.1(b)). The pseudocritical temperature $T_{c}^{\chi}$, which is 175.6 MeV for $m_{\sigma}$=400 MeV and constant c in PQMVT-I scenario, shifts to $T_{c}^{\chi}$=187.9 MeV for $m_{\sigma}$=500 MeV and $T_{c}^{\chi}$=201.3 MeV for $m_{\sigma}$=600 MeV. Here $T_{glue}^{c}=240$ MeV for all the cases. This effect of higher $m_{\sigma}$ on $\Delta_{l,s}(T)$ temperature variation, is completely offset by having a sharper and steeper decrease of c(T) at lower $T_{1}$ and $b_{1}$ values in the PQMVT-II scenario and we could find a good fit to the Wuppertal group LQCD data for $\Delta_{l,s}(T)$ also for $m_{\sigma}$=500 and 600 MeV. The LQCD data points within the error bars for the $\Delta_{ls}(T)$ coincide quite well with the corresponding calculations having the temperature dependence of coupling c(T) as shown in Fig.2(a) by solid red line for $m_{\sigma}$=400 MeV,$T_{1}$=108 MeV=.70$T_{c}$ and $b_{1}$=90 MeV=.58$T_{c}$, dash dot line in green for $m_{\sigma}$=500 MeV, $T_{1}$=105 MeV=.68$T_{c}$ and $b_{1}$=78 MeV=.51$T_{c}$ and thick small dash line in blue for $m_{\sigma}$=600 MeV, $T_{1}$=105 MeV=.68$T_{c}$ and $b_{1}$=62 MeV=.40$T_{c}$. In the PQMVT-II model scenario, the pseudocritical temperature $T_{c}^{\chi}$=154.9 MeV for $m_{\sigma}$=400 MeV is quite close to the $T_{c}^{\chi}$=158.3 MeV for $m_{\sigma}$=500 MeV and $T_{c}^{\chi}$=159.0 MeV for $m_{\sigma}$=600 MeV. Since the temperature dependence of KMT coupling causes quite early and a little sharper melting of the non-strange condensate in the PQMVT-II scenario, we have plotted the temperature variation of c(T) in Fig.2(b). When $m_{\sigma}$ = 400 MeV, solid red line shows temperature variation of c(T) for parameters $T_{1}$=108 MeV and $b_{1}$=90 MeV. The temperature variation of c(T) becomes steeper with the parameters $T_{1}$=105 MeV and $b_{1}$=78 MeV for $m_{\sigma}$=500 MeV as shown by dash dot line in green. In order to find a good fit to the $\Delta_{l,s}(T)$ variation with the LQCD data and offset the effect of higher value of $m_{\sigma}$=600 MeV, the temperature variation of c(T) becomes steepmost in the PQMVT-II scenario for the parameters $T_{1}$=105 MeV and $b_{1}$=62 MeV as plotted by thick blue dash line in Fig.2(b).

In all the situations, confinement-deconfinement transition occurs earlier than the chiral transition as $T_{c}^{\Phi}<T_{c}^{\chi}$ but this difference is least of about 9 MeV for the PQMVT-II model with temperature dependent c(T) when $m_{\sigma}=400$ MeV, $T_{glue}^{c}=240$ MeV while with this parameter set for the constant c scenario in the model PQMVT-I, this difference is large 20.3 MeV. The difference $T_{c}^{\chi}-T_{c}^{\Phi}$ is largest of about 46 MeV for the PQMVT-I model with constant c when $m_{\sigma}=600$ MeV, $T_{glue}^{c}=240$ MeV while with this parameter set for the temperature dependent c(T) in the PQMVT-II scenario, this difference is only of 12.9 MeV. The larger value of sigma meson mass badly spoils the closeness of chiral crossover with the confinement-deconfinement transition while temperature dependence of anomaly coefficient c(T) brings the confinement-deconfinement crossover transition closer to the chiral crossover transition. Due to temperature dependence of c(T), the chiral crossover transition temperature $T_{c}^{\chi}$ gets quite reduced in the PQMVT-II model when compared to the constant c scenario, respectively by 20.7 MeV (for $m_{\sigma}=400$ MeV), 33.6 MeV (for $m_{\sigma}=500$ MeV) and 42.3 MeV (for $m_{\sigma}=600$ MeV) while $T_{glue}^{c}$ is fixed at 240 MeV. The Physics of axial anomaly is microscopically interconnected with the confinement-deconfinement transition and chiral crossover transition in the coupled equations of our model. Temperature dependence of c(T) triggers early confinement-deconfinement transition which gives rise to early chiral crossover transition .

6.2 Meson Mass Variations and the $U_{A}(1)$ Restoration

Temperature variations of the meson masses ($\sigma,\pi$) and ($\eta^{\prime},a_{0}$) in the PQMVT-I model with constant c has been presented in the left panel (Fig.3(a)) while the corresponding temperature variations for the temperature dependent KMT coupling strength c(T) has been shown in the right panel (Fig.3(b)) for the PQMVT-II model setting when $m_{\sigma}=400$ MeV, $T_{glue}^{c}=240$ MeV. Dash dot line in green shows the temperature variation of $\pi$ meson, dash line in magenta colour shows the $\sigma$ mass temperature variation while solid line with hollow circles in black, shows temperature variation of $\eta^{\prime}$ meson mass and solid red line depicts temperature variation of $a_{0}$ meson mass. Showing the trend of chiral symmetry restoring transition in Fig.3(a) $\sigma$ degenerates with its chiral partner $\pi$ near 204 MeV and $a_{0}$ meson degenerates with the chiral partner $\eta^{\prime}$ near 231 MeV for the PQMVT-I model. Afterwards, the degenerated $\sigma$, $\pi$ meson mass line, shows a poor converging trend towards the degenerated $a_{0}$, $\eta^{\prime}$ meson mass line, these two lines merge with each other for very large temperature of about 990 MeV$\sim 5.6T_{c}^{\chi}$ (not shown in the Fig.3(a)). Persistence of $U_{A}(1)$ symmetry in PQMVT-I upto such large temperature is not tenable because instanton density is supposed to vanish near $2\ T_{c}^{\chi}$.

We find that due to the effect of c(T) in the PQMVT-II scenario as shown in Fig.3(b), the $\eta^{\prime}$ meson mass decreases from its vacuum value by 220 MeV near T=176 MeV after the chiral transition (at $T_{c}^{\chi}=154.9$ MeV) temperature. This is similar to the $\eta^{\prime}$ in-medium mass drop of at least 200 MeV as reported by Csorgo and Vertesi in Ref Csorgo ; Vertesi , as an experimental signature of the effective restoration of $U_{A}(1)$ symmetry. In the PQMVT-I model for constant c case also as shown in Fig.3(a), the $\eta^{\prime}$ meson mass decreseas by 170 MeV near T=200 MeV after the chiral transition temperature (at $T_{c}^{\chi}=175.6$ MeV). We have shown the temperature variation of $\eta$ meson also in Fig.3(b) by the blue line with filled triangles for the temperature dependent coupling c(T), in the PQMVT-II model. We point out that the temperature variation in the mass of $\eta^{\prime}$ shows a discontinuous drop of about 118 MeV at 176 MeV. This happens because, the temperature variation of the pseudoscalar mixing angle presented later in Fig.5(b) shows a discontinuous jump from $-45^{\circ}$ to $+45^{\circ}$ at the same temperature. Corresponding discontinuous jump of 118 MeV in the temperature variation of the mass of $\eta$ meson is seen exactly at the same temperature in Fig.3(b). The $\eta$ meson becomes light quark system ($\eta_{NS}$) at T=176 MeV and changes its identity with $\eta^{\prime}$ meson which becomes strange quark system ($\eta_{S}$) as shown in Fig.6(b).

The mass of $\sigma$ meson, degenerates with that of $\pi$ near 189 MeV and mass of $a_{0}$ meson, degenerates with that of $\eta$ (which has changed identity with $\eta^{\prime}$) near 193 MeV. We note that the $m_{\sigma}$ temperature variation in Fig.3(b), shows an increase after T=108 MeV (equal to $T_{1}$ where the temperature dependence of c(T) begins) and reaches maximum at T=136.7 MeV registering an increase of about 42.5 MeV which is an uncommon behavior, afterwards it shows the usual behavior and becomes degenerate with the $m_{\pi}$ temperature variation. The origin of this pattern is consequence of decrease in c(T) which affects the squared masses $m_{s,00}^{2}$, $m_{s,88}^{2}$ and $m_{s,08}^{2}$ Trvk in such a way that the expression of $m_{\sigma}^{2}$ on its numerical computation, first shows an uncommon increase and then decreases with temperature. Further, We emphasize here that for the temperature dependence of c(T) in the PQMVT-II scenario, the chiral partner of $a_{0}$ meson is $\eta$ meson rather than the $\eta^{\prime}$ meson, similar to the very recent findings of the FRG investigation under the $\text{LPA}^{\prime}$+Y (full effective potential with the running of Yukawa coupling and non vanishing anomalous dimensions) approximation in Ref. Rennecke and the vector meson extended linear sigma model investigation in Ref. Kovacs . This finding differs also from the result of earlier studies in the linear sigma model Rischke:00 , QM model Schaefer:09 , PQM model gupta , PQMVT model Trvk as well as an FRG investigation with the QM model under LPA (Local potential approximation) Mitter . Such a change of identity from $\eta^{\prime}$ to $\eta$ has also been found for density dependent KMT coupling $g_{D}(\rho)$ in the NJL model investigation in Ref. Costa:05 .

The temperature variations of the masses of the degenerated $\sigma$, $\pi$ meson line completely, merges with the temperature variations of the masses of degenerated $a_{0}$, $\eta$ meson line near 275 MeV. It means that for the parameters of the PQMVT-II model when $m_{\sigma}=400$ MeV, $T_{glue}^{c}=240$ MeV, the $U_{A}(1)$ restoration takes place around 275 MeV which is equal to 1.75 $T_{c}^{\chi}$. We point out that mesonic thermal and quantum fluctuations investigated under FRG framework strengthen the $U_{A}(1)$ anomaly near $T_{c}^{\chi}$ Fejos ; Fejos:16 ; Rennecke . In Ref. Jiang , when fluctuations are included in the calculation of topological susceptibility, fluctuation induced part of the topological susceptibility remains remarkably large (upto T=250 MeV) in the chiral symmetry restored phase. It shows noticeable survival of $U_{A}(1)$ anomaly upto 250 MeV. In the EPNJL model studies by Ishii et. al. Ishii , the temperature variations of the screening masses of $a_{0}$ meson and $\pi$ mesons matches with the lattice QCD data from Ref. ChengLQCD and masses become degenerate near T=180 MeV where they find $T_{c}^{\chi}$=163 MeV , it means $U_{A}(1)$ symmetry is restored near $T=1.1\ T_{c}^{\chi}$. However, recent lattice simulations Sharma ; Dick , do not observe $U_{A}(1)$ symmetry restoration even at $T=1.5\ T_{c}^{\chi}$ using highly improved staggered fermions.

Fig.4(a) in the left panel, shows the temperature variations of masses of ($\eta$, $f_{0}$) and ($K$,  $\kappa$) for the constant c case in the PQMVT model while the right panel Fig.4(b) presents the corresponding temperature variations of masses in the PQMVT-II model for the temperature dependent coupling c(T). Dash dot line in magenta shows the temperature variation of $K$ meson, solid blue color line with filled triangles, shows the $\eta$ mass temperature variation while dash line in red , presents temperature variation of $\kappa$ meson mass and solid deep blue line shows temperature variation of $f_{0}$ meson mass. Mass variation lines of $K$, $\kappa$ and $\eta$ merge with each other near 339 MeV while $f_{0}$ meson mass variation almost merges with these degenerated three lines near 375 MeV in the PQMVT-I model. In contrast, in Fig.4(b) in the right panel, for the temperature dependent c(T) in the PQMVT-II model, mass variation lines of $K$, $\kappa$ and $\eta^{\prime}$ ($\eta$ becomes $\eta^{\prime}$ as strange quark system $\eta_{S}$ at T=176 MeV after showing jump) merge early with each other near 270 MeV while $f_{0}$ meson mass variation merges with these degenerated three lines near 367 MeV. Here $\eta^{\prime}$ rather than $\eta$ becomes chiral partner of $f_{0}$ meson similar to the FRG study result in Ref. Rennecke under the $\text{LPA}^{\prime}$+Y approximation. We remind that most of the changes in the strange condensate,takes place in the temperature range 125 MeV to 210 MeV due to temperature dependent parameterizations of KMT coupling c(T) in the model PQMVT-II and the melting trend of the strange condensate variation after 210 MeV in Fig.1(a) matches almost exactly with the corresponding strange condensate variation in the PQMVT-I model with constant c. Due to this, $f_{0}$ meson mass variation in the PQMVT-I model in Fig.4(a) looks similar to the temperature variation of $f_{0}$ mass in the Fig.4(b) in PQMVT-II model.

6.3 Meson Mixing Angle Variations

We will now be exploring the nature of the scalar $\theta_{s}$ and pseudo scalar $\theta_{p}$ mixing angles temperature variations. In Fig.5(a) in the left panel, the dash dotted line in green shows the pseudoscalar mixing angle $\theta_{p}$ temperature variations while the solid line in deep red shows scalar mixing angle $\theta_{s}$ temperature variations for the PQMVT-I model with constant coupling c. In Fig.5(b) in the right panel, the same line types respectively, show the $\theta_{p}$ and $\theta_{s}$ temperature variations for the PQMVT-II model with temperature dependent KMT coupling c(T). As in Refs. Schaefer:09 ; gupta ; Trvk , in the low temperature chiral symmetry broken phase , the nonstrange and strange quark mixing is strong and one gets almost constant pseudoscalar mixing angle $\theta_{p}=-5^{\circ}$ in both the Fig.5(a) and Fig.5(b) . As temperature increases after 150 MeV, the $\theta_{p}$ variation in the PQMVT-I model develops a dip around pseudocritical temperature $T_{c}^{\chi}=175.6$ MeV and then smoothly starts approaching the ideal mixing angle $\theta_{p}\to\arctan{\frac{1}{\sqrt{2}}}\sim 35^{\circ}$ which gets fully achieved for temperatures around 350 MeV in the high temperature chiral symmetry restored phase.

The ($\eta$,$\eta^{\prime}$) basis of mesons is related to the flavour nonstrange-strange ($\eta_{NS}$,$\eta_{S}$) basis through the mixing angle $\phi_{p}$ where $\phi_{p}=\theta_{p}+\text{arctan}\sqrt{2}\sim\theta_{p}+54.74^{\circ}$ as explained in the appendix C of Ref. Schaefer:09 . Thus when $\theta_{p}\sim 35^{\circ}$, the $\phi_{p}$ becomes $90^{\circ}$. Due to this the $\eta$ and $\eta^{\prime}$ mesons, respectively become a pure strange $\eta_{S}$ and non strange light $\eta_{NS}$ quark system in PQMVT-I. Mass formula for $m_{\eta_{NS}}$ and $m_{\eta_{S}}$ are given in the Table 2. The $m_{\eta^{\prime}}$ variation shown by the black line with hollow circles merges with $m_{\eta_{NS}}$ variation depicted by the deep red line near 236 MeV in PQMVT-I model in Fig.6(a) while the $m_{\eta}$ variation shown by the solid line in blue color with filled triangles merges with the $m_{\eta_{S}}$ variation shown by the dash line in green near the same temperature and further these two degenerated lines merge with each other near very high temperature of about 810 MeV (not shown in the Fig.6(a)).

In the PQMVT-II model, in complete contrast, in Fig.5(b), the $\theta_{p}$ variation starts a sharp dip after 120 MeV due to the effect of the temperature dependent coupling $c(T)$ and at 176 MeV, the pseudoscalar mixing angle $\theta_{p}$ jumps from $-45^{\circ}$ to $+45^{\circ}$. $\theta_{p}$ is calculated from tan$2\theta_{p}$ which remains positive in first and third quadrant. When $\theta_{p}$ jumps from $-45^{\circ}$ to $+45^{\circ}$, the observed temperature variation of $\theta_{p}$ afterwards from $+45^{\circ}$ to ideal mixing $35.26^{\circ}$ in first quadrant actually results due to the variation of $\theta_{p}$ from $-45^{\circ}$ to anti-ideal mixing $-54.74^{\circ}$ in the fourth quadrant. In consequence $\phi_{p}$ changes from $49.74^{\circ}$ near T=0 MeV and $9.74^{\circ}$ at T=176 MeV to $0^{\circ}$ near T=250 MeV. Due to this $\eta^{\prime}$ temperature variation of the mass, shows a drop of 118 MeV at T= 176 MeV in Fig.6(b) and changes its identity with $\eta$ meson mass variation which shows a jump of 118 MeV at the same temperature. $m_{\eta^{\prime}}$ temperature variation degenerates with the strange quark system $m_{\eta_{S}}$ temperature variation after change of its identity at T= 176 MeV while $m_{\eta}$ temperature variation degenerates with $m_{\eta_{NS}}$ light quark system temperature variation. The anti-ideal mixing when $\phi_{p}\to 0$ is achieved near 250 MeV in PQMVT-II scenario.

We remind ourselves again that for the temperature dependence of c(T) in the PQMVT-II scenario, the behavior of the pseudoscalar mixing angle $\theta_{p}$ or $\phi_{p}$ temperature variation is similar to their corresponding behavior in recent findings of the FRG investigation under the $\text{LPA}^{\prime}$+Y approximation in Ref. Rennecke and the vector meson extended linear sigma model investigation in Ref. Kovacs . This finding differs also from the result of earlier studies in the linear sigma model Rischke:00 , QM model Schaefer:09 , PQM model gupta , PQMVT model Trvk as well as an FRG investigation with the QM model under LPA (Local potential approximation) Mitter . Such a jump for pseudoscalar mixing angle $\theta_{p}$ from $-45^{\circ}$ to $+45^{\circ}$, has also been reported for density dependent KMT coupling $g_{D}(\rho)$ in the NJL model investigation in Ref. Costa:05 . In contrast to Fig.6(a) of PQMVT-I model, the $m_{\eta^{\prime}}$ and $m_{\eta_{S}}$ degenerated lines further become degenerate with the $m_{\eta}$ and $m_{\eta_{NS}}$ degenerated lines i.e. all the four lines merge with each other near 375 MeV in Fig.6(b) for the PQMVT-II model.

The scalar mixing angle $\theta_{s}$ in vacuum ($T=0$) is about $8^{\circ}$ in both the cases of PQMVT-I model and PQMVT-II model in Fig.5(a) and Fig.5(b) respectively. The $\theta_{s}$ starts growing towards its ideal value for temperature about 25 MeV higher than the $T_{c}^{\chi}=175.6$ MeV, in the PQMVT-I model in Fig.5(a) and approaches the ideal mixing angle very smoothly in the higher temperature chirally symmetric phase. The effect of temperature dependent coupling c(T) modifies the temperature variation of scalar mixing angle $\theta_{s}$ in the PQMVT-II model. Here, the $\theta_{s}$ starts approaching its ideal value for temperature about 20 MeV higher than the $T_{c}^{\chi}=154.9$ MeV in Fig.5(b) and for higher temperatures in the chirally symmetric phase, the scalar mixing angle first achieves its ideal value $35^{\circ}$ and then drops down to $\theta_{s}\sim-53^{\circ})$ for about 450 MeV temperature. This pattern is already reported and discussed in Ref.Schaefer:09 ; gupta for the Quark Meson (QM) model and Polyakov Quark Meson (PQM) model .

In Fig.7(a) for the PQMVT-I model, we notice that the $f_{0}$ meson mass temperature variation shown by the solid deep blue line degenerates with the pure strange quark system $\sigma_{S}$ mass temperature variation shown by the solid red line with hollow circle near 196 MeV while the $\sigma$ meson mass temperature variation shown by the dash line in magenta becomes identical with the pure non-strange quark system $\sigma_{NS}$ mass variation shown by the green dots line near the temperature 209 MeV . Further these two degenerated lines show a mild mass convergence trend and merge with each other near very high temperature of 720 MeV (not shown in the Fig.7(a)) . For temperature dependent coupling c(T) in the PQMVT-II model, in Fig.7(b), we find that the $f_{0}$ meson mass temperature variation degenerates with the pure strange quark system $\sigma_{S}$ mass temperature variation near 167 MeV and the $\sigma$ meson mass temperature variation merges with the pure non-strange quark system $\sigma_{NS}$ mass variation near the temperature 175 MeV. In quite contrast to PQMVT-I model, these two lines show a strong mass convergence trend and merge with each other near T=385 MeV.

7 Summary and Conclusion

In the present work, we have explored, how the temperature dependence of KMT coupling strength c(T) in the PQMVT-II model which we call as PQMVT-I model for constant coupling c , modifies the interplay of chiral symmetry and the $U_{A}(1)$ symmetry restoration. We computed the subtracted chiral order parameter $\Delta_{l,s}(T)$ temperature variation in the PQMVT-II scenario and fitted it to the Wuppertal group LQCD data from which they obtained the pseudo-critical temperature for chiral transition as $T_{c}=157(3)$ MeV LQCDWB2 . Temperature dependence of KMT coupling causes quite early and a little sharper melting of the non-strange condensate in the PQMVT-II scenario while larger values of $m_{\sigma}$ makes this melting smoother and shifts it to quite high temperatures in the PQMVT-I model. In comparison to PQMVT-I model, the nonstrange condensate and strange condensate temperature variations for the PQMVT-II scenario, are found to have the largest difference in the temperature range 125-210 MeV. In all the situations, confinement-deconfinement transition occurs earlier than the chiral transition as $T_{c}^{\Phi}<T_{c}^{\chi}$ but this difference is least of about 9 MeV for the PQMVT-II model with temperature dependent c(T) when $m_{\sigma}=400$ MeV, $T_{glue}^{c}=240$ MeV while with this parameter set for the constant c scenario in the model PQMVT-I, this difference is large 20.3 MeV. The larger value of sigma meson mass badly spoils the closeness of chiral crossover with the confinement-deconfinement transition while temperature dependence of anomaly coefficient c(T) brings the confinement-deconfinement crossover transition closer to the chiral crossover transition.

In the constant c PQMVT-I scenario, the degenerated $\sigma$, $\pi$ meson mass line, shows a poor converging trend towards the degenerated $a_{0}$, $\eta^{\prime}$ meson masses line, these two lines merge with each other for very large temperature of about 990 MeV$\sim 5.6T_{c}^{\chi}$. Persistence of $U_{A}(1)$ symmetry to such large temperature is not tenable because instanton density is most likely to approach zero near $2\ T_{c}^{\chi}$. We notice that due to the effect of c(T) in the PQMVT-II scenario, the $\eta^{\prime}$ meson mass decreases from it vacuum value by 220 MeV near T=176 MeV after the chiral transition (at $T_{c}^{\chi}=154.9$ MeV) temperature. This is similar to the $\eta^{\prime}$ in-medium mass drop of at least 200 MeV as reported by Csorgo and Vertesi in Ref Csorgo ; Vertesi , as an experimental signature of the effective restoration of $U_{A}(1)$ symmetry. For temperature dependence of the KMT coupling, mass variations of the degenerated $\sigma$, $\pi$ mesons merges completely with the temperature variations of the masses of degenerated $a_{0}$, $\eta$ mesons ( note here that chiral partner of $a_{0}$ is $\eta$ rather than the $\eta^{\prime}$) near 275 MeV. It means that for the parameters of the PQMVT-II model when $m_{\sigma}=400$ MeV, the $U_{A}(1)$ restoration takes place around 275 MeV which is equal to 1.75 $T_{c}^{\chi}$. We note that mesonic thermal and quantum fluctuations investigated under FRG framework strengthen the $U_{A}(1)$ anomaly near $T_{c}^{\chi}$ Fejos ; Fejos:16 ; Rennecke . In the EPNJL model studies by Ishii et. al. Ishii , the temperature variations of the screening masses of $a_{0}$ meson and $\pi$ mesons matches with the lattice QCD data from Ref. ChengLQCD and masses become degenerate near T=180 MeV where they find $T_{c}^{\chi}$=163 MeV , it means $U_{A}(1)$ symmetry is restored near $T=1.1\ T_{c}^{\chi}$. However, recent lattice simulations Sharma ; Dick , do not observe $U_{A}(1)$ symmetry restoration even at $T=1.5\ T_{c}^{\chi}$ using highly improved staggered fermions. In the PQMVT-II model, mass variation lines of $K$, $\kappa$ and $\eta^{\prime}$ ($\eta$ becomes $\eta^{\prime}$ as strange quark system $\eta_{S}$ at T=176 MeV) merge early with each other near 270 MeV while $f_{0}$ meson mass variation merges with these degenerated three lines near 367 MeV. Here $\eta^{\prime}$ rather than $\eta$ becomes chiral partner of $f_{0}$ meson.

In the PQMVT-II model, due to the temperature dependent coupling $c(T)$, the pseudoscalar mixing angle $\theta_{p}$ jumps from $-45^{\circ}$ to $+45^{\circ}$ at 176 MeV, afterwards the $\theta_{p}$ temperature variation from $+45^{\circ}$ to ideal mixing $35.26^{\circ}$ in first quadrant actually results due to the variation of $\theta_{p}$ from $-45^{\circ}$ to anti-ideal mixing $-54.74^{\circ}$ in the fourth quadrant, since $\theta_{p}$ is calculated from tan$\ 2\theta_{p}$ which is positive in first and third quadrant. Due to this $\eta^{\prime}$ temperature variation of the mass, drops by 118 MeV at T= 176 MeV and changes its identity with $\eta$ meson mass variation which shows a jump of 118 MeV at the same temperature. The $\eta$ meson becomes light quark system ($\eta_{NS}$) at T=176 MeV and changes its identity with $\eta^{\prime}$ meson which becomes strange quark system ($\eta_{S}$). These results are similar to the recent findings of the FRG investigation under the $\text{LPA}^{\prime}$+Y approximation in Ref. Rennecke and the vector meson extended linear sigma model investigation in Ref. Kovacs and differ from the results of earlier studies in the linear sigma model Rischke:00 , QM model Schaefer:09 , PQM model gupta , PQMVT model Trvk as well as an FRG investigation with the QM model under LPA (Local potential approximation) Mitter . Such a change of identity from $\eta^{\prime}$ to $\eta$ has also been reported for density dependent KMT coupling $g_{D}(\rho)$ in the NJL model investigation in Ref. Costa:05 .

In PQMVT-I, $\eta$ becomes $\eta_{S}$ and $\eta^{\prime}$ becomes $\eta_{NS}$ due to ideal mixing, these two degenerated lines merge with each other near very high temperature of about 810 MeV. In PQMVT-II model due to anti-ideal mixing, the $m_{\eta^{\prime}}$ and $m_{\eta_{S}}$ degenerated lines further merge with the $m_{\eta}$ and $m_{\eta_{NS}}$ degenerated lines near 375 MeV. In PQMVT-I model,the $f_{0}$ becomes $\sigma_{S}$ and $\sigma$ becomes $\sigma_{NS}$. These two lines merge with each other near very high temperature of 720 MeV .In the PQMVT-II model also, the $f_{0}$ becomes $\sigma_{S}$ and $\sigma$ becomes $\sigma_{NS}$ but these two lines merge near T= 385 MeV.

Acknowledgements.

Computational support of the computing facility which has been developed by the Nuclear Particle Physics group of the Physics Department, Allahabad University under the Center of Advanced Studies (CAS) funding of UGC, India, is acknowledged.