Isospin Symmetry Breaking in Non-Perturbative QCD ^†

Abstract

At finite isospin chemical potential ${\mu }_I$, the tension between measured decays and partial branching ratios of neutral and charged bosons as functions of dimuon mass squared and the Standard Model (SM) isospin asymmetry can be analyzed in nonperturbative QCD-effective models, for instance, the Polyakov linear sigma-model. With almost first-principle derivation of the explicit isospin symmetry breaking, namely, ${\overline{\sigma }}_3=f_{K^{\pm }}-f_{K^0}$ the isospin sigma field, and $h_3=m_{a_0}^2\left(f_{K^{\pm }}-f_{K^0}\right)$ the third generator of the matrix of the explicit symmetry breaking $H=T_a{h}_a$. $f_{K^{\pm }}$ and $f_{K^0}$ are decay constants of $K^{\pm }$ and $K^0$, respectively. $m_{a_0}$ is the mass of $a_0$ meson. Accordingly, the QCD phase structure could be extended to finite ${\mu }_I$. With the thermal and density dependence of $a_0$, $f_{K^{\pm }}$, and $f_{K^0}$, ${\overline{\sigma }}_3$ and $h_3$ are accordingly expressed in dependence on the temperatures and the chemical potentials. We find that the resulting critical chiral temperatures $T_{\chi }$ decrease with the increase in ${\mu }_B$ and/or ${\mu }_I$. We conclude that the ($T_{\chi }-{\mu }_I$) boundary has almost the same structure as that of the ($T_{\chi }-{\mu }_B$) plane.

1. Introduction

At finite isospin chemical potential ${\mu }_I$, the lattice Quantum ChromoDynamics (QCD) simulations have real and positive action so that the Monte Carlo (MC) techniques are well applicable [1]. On the other hand, at finite baryon chemical potential ${\mu }_B$, the action becomes complex, i.e., sign problem [2]. Both types of nonperturbative simulations share common features, for example, deconfinement, hadronization, hadron-parton phase transition, Silverblaze phenomenon, and Bose–Einstein condensations. Hence, even qualitative if not quantitative conclusions on nonperturbative QCD, at finite ${\mu }_I$, are of great importance [3]. This might go beyond the current limitations and offer a theoretical framework for the Bose–Einstein condensations and the yet-still-hypothetical superconducting phases [4]. Thereby, the QCD phase diagram is enriched [5]. Characterizing the possible imbalance between the charged pions degrees-of-freedom [6] in dense quark matter, such as the neutron stars [7,8], is an astrophysical example of the chiral isospin asymmetry.

The QCD-like effective models, such as the Ployakov linear-sigma model (PLSM), offer a complementary approach to the nonperturbative QCD [9,10,11,12]. The present script introduces a novel study suggesting first-principle derivations of a new set of PLSM parameters, namely, $h_3$ and $bar{\sigma }_3$ the isospin sigma field. As a result of the spontaneous symmetry breaking in the QCD-like effective nonperturbative QCD approach, the PLSM, the mean value of the field $\Phi$, $\langle \Phi \rangle$, and that of its conjugate $\langle {\Phi }^{†}\rangle$ could be related to the quantum numbers of the vacuum [13]. Therefore, while the mean values of ${\overline{\pi }}_a$ vanish, that of the quark condensates $\overline{{\sigma }_a}$ remain finite. Concretely, $\overline{{\sigma }_0}\ne \overline{{\sigma }_3}\ne \overline{{\sigma }_8}\ne 0$ corresponding to the diagonal generators $U\left(3\right)$. On the other hand, the isospin symmetry is broken in SU(2) though finite quark condensates $\overline{{\sigma }_3}$[13] and finite $\mathrm{Tr}[H(\Phi +{\Phi }^{†})$ in the PLSM Langrangian. Thus, the symmetry generators $h_a$ are conjectured to break the isospin symmetry, as $H=T_a{h}_a$. Therefore, the finite diagonal components of $h_0,h_3,h_8$ lead to finite condensates $\overline{{\sigma }_0},\overline{{\sigma }_3}\mathrm{and}\overline{{\sigma }_8}$ corresponding to the three quark flavors. Accordingly, the masses of the three quark flavors are no longer degenerate, i.e., $m_u\ne m_d\ne m_s$. It is obvious that the nature likely prefers such a configuration. By converting the three quark condensates through the orthogonal basis transformation from the original basis, ${\sigma }_0,{\sigma }_3$, and ${\sigma }_8$ to pure up (${\sigma }_u$), down (${\sigma }_d$), and strange (${\sigma }_s$) quark flavor basis, we obtain,

$$\begin{array}{c}\hfill \left[\begin{array}{c}\overline{{\sigma }_u}\\ \overline{{\sigma }_d}\\ \overline{{\sigma }_s}\end{array}\right]=\frac{1}{\sqrt{3}}\left[\begin{array}{ccc}\sqrt{2}& 1& 1\\ \sqrt{2}& -1& 1\\ 1& 0& -\sqrt{2}\end{array}\right]\left[\begin{array}{c}\overline{{\sigma }_0}\\ \overline{{\sigma }_3}\\ \overline{{\sigma }_8}\end{array}\right].\end{array}$$

(1)

In light of this, the masses of u-, d-, and s-quarks can be expressed as,

$$\begin{array}{c}\hfill m_u=\frac{g}{2}{\sigma }_u,m_d=\frac{g}{2}{\sigma }_d,m_s=\frac{g}{\sqrt{2}}{\sigma }_s.\end{array}$$

(2)

In the present calculations, we introduce PLSM calculations for the thermodynamic properties and thereby the chiral QCD phase structure, at finite isospin asymmetry. Firstly, the effects of finite isospin asymmetry on differentiation between the nonstrange (light) condensates of u- and d-quark shall be analyzed. Secondly, as a result of the isospin symmetry breaking, ${\overline{\sigma }}_3$ should have a nonzero value because ${\sigma }_u={\sigma }_l+{\sigma }_3$ and ${\sigma }_d={\sigma }_l-{\sigma }_3$. To this end, we estimate the pure mesonic potential for $N_f$ quark flavors, Equation (14), as functions of temperatures and chemical potentials.

To the author’s best knowledge, there is so-far no reliable estimation for the generator $h_3$. Following the assumption of refs. [14,15], we first assume that the violation of the isospin symmetry is negligibly small, i.e., $h_3\to 0$, see

Table 1. Various LSM parameters fixed at $m_{\sigma }=800$MeV and $h_3=0$ [16].

${\mathit{m}}_{\mathit{\sigma }}$ [MeV]	c [MeV]	${\mathit{h}}_{\mathit{ud}}$ [MeV${}^3$]	${\mathit{h}}_3$ [MeV${}^3$]	${\mathit{h}}_{\mathit{s}}$ [MeV${}^3$]	${\mathit{m}}^2$ [MeV${}^2$]	${\mathit{\lambda }}_1$	${\mathit{\lambda }}_2$
800	$4807.84$	${\left(120.73\right)}^3$	0	${\left(336.41\right)}^3$	$-{\left(306.26\right)}^2$	$13.49$	$46.48$

. In this limit, one assumes that the u- and d-quark condensates can be given as ${\sigma }_u={\sigma }_l+{\sigma }_3$ and ${\sigma }_d={\sigma }_l-{\sigma }_3$, where ${\sigma }_l$ is the nonstrange light condensate ${\sigma }_l=({\sigma }_u+{\sigma }_d)/2$ for non-degenerated masses of the light quarks, Equation (1). The main contribution of the present script is revising such an assumption. Finite $h_3$, the third generator of the matrix of the explicit symmetry breaking $H=T_a{h}_a$, and ${\overline{\sigma }}_3$, the isospin sigma field, considerably contribute to the isospin symmetry breaking. We are not just assigning finite values to $h_3$ and ${\overline{\sigma }}_3$. We rather introduce a theoretical framework for the thermal and dense dependence of both $h_3$ and ${\overline{\sigma }}_3$.

The present script is organized as follows. In Section 2, the formalism is outlined. The effective nonperturbative QCD approach, the SU(3) Polyakov linear-sigma model (PLSM), at vanishing $h_3$, is introduced in Section 2.1. The isospin symmetry breaking based on suggesting finite $h_3$ is discussed in Section 2.2. In Section 3, we investigate the impacts of the isospin asymmetry on the QCD phase transition(s). Last but not least, Section 4 is devoted to the conclusions.

2. Formalism

The limitation of the MC techniques to vanishing baryon chemical potential in the non-perturbative lattice QCD promotes the utilization of QCD-like effective approaches. In the present study, we focus on the Polyakov linear-sigma model (PLSM). Various QCD quantities, including the thermodynamics of conserved charges and chiral quark-hadron phase transitions, can be estimated at least qualitatively [15,17,18,19,20,21,22,23,24,25].

Here, we aim at analyzing how the finite isospin asymmetry should be integrated in the chiral models, PLSM, especially that the nonperturbative lattice QCD simulations—in contrast to the finite baryon chemical potential—are reliable, at finite isospin chemical potential. Having done this allows the characterization of the thermodynamic properties of the nonperturbative QCD and thereby the mapping out of the QCD phase diagram. With finite isospin asymmetry, the generic chemical potentials of both light quarks are no longer degenerate. The PLSM thermodynamics in thermal and dense medium shall be confronted to the recent lattice QCD simulations. To this end, we aim at presenting a general expression of the chiral limit, at finite isospin asymmetry.

2.1. PLSM at Finite Chemical Potential

In flat Minkowski space, the Lagrangian of the linear-sigma model (LSM) with $N_f$ quark flavors and Polyakov-loop potential would be summarized as

$$\begin{array}{c}\hfill {\mathcal{L}}_{PLSM}={\mathcal{L}}_{\psi }+{\mathcal{L}}_m-\mathcal{U}(\varphi ,\overline{\varphi },T),\end{array}$$

(3)

where ${\mathcal{L}}_{\psi }$ represent the quarks (fermions), while ${\mathcal{L}}_m$ represents the mesons (bosons), and $\mathcal{U}(\varphi ,\overline{\varphi },T)$ represents the Polyakov-loop potential contributions. For $N_c$, the color degrees of freedom, the contributions of the quarks (fermions) read

$$\begin{array}{ccc}\hfill {\mathcal{L}}_{\psi }& =& \sum _f{\overline{\psi }}_f(i{\gamma }^{\mu }{D}_{\mu }-g{T}_a({\sigma }_a+i{\gamma }_5{\pi }_a)){\psi }_f,\hfill \end{array}$$

(4)

where $\psi$ are the Dirac spinor fields; $f=[u,d,s]$ are the quark flavors; and $D_{\mu },\mu ,{\gamma }^{\mu }$ and g, respectively, represent the covariant derivative, Lorentz index, chiral spinors, and Yukawa coupling constant. The contributions of the mesons (bosons) are given as,

$$\begin{array}{ccc}\hfill {\mathcal{L}}_m& =& \mathrm{Tr}({\partial }_{\nu }{\Phi }^{†}{\partial }^{\nu }\Phi -m^2{\Phi }^{†}\Phi )-{\lambda }_1{\left[\mathrm{Tr}({\Phi }^{†}\Phi )\right]}^2\hfill \\ & -& {\lambda }_2\mathrm{Tr}{({\Phi }^{†}\Phi )}^2+c[\mathrm{Det}(\Phi )+\mathrm{Det}\left({\Phi }^{†}\right)]+\mathrm{Tr}\left[H(\Phi +{\Phi }^{†})\right],\hfill \end{array}$$

(5)

where $\Phi$ is the nonet meson $(3\times 3)$-matrix,

$$\begin{array}{ccc}\hfill \overline{\Phi }& =& \sum _{a=0}^{N_f^2-1}{T}_a(\overline{{\sigma }_a}+i\overline{{\pi }_a}).\hfill \end{array}$$

(6)

and $T_a={\widehat{\lambda }}_a/2$ is a generator operator in U(3) algebra. $T_a$ can be determined from the Gell–Mann matrices ${\widehat{\lambda }}_a$ with $a=0,\cdots ,8$ [26].

So far, we have various LSM parameters to be fixed, namely, $m^2$, $h_l$, $h_s$, ${\lambda }_1$, ${\lambda }_2$, and c. They are strongly dependent on the mass of the sigma meson $m_{\sigma }$[16]. In the present calculations, we assumed that $m_{\sigma }=800$MeV. Table 1 summarizes all these parameters [16]. The present script aims at determining another set of parameters, namely, $h_3$ and ${\overline{\sigma }}_3$, the third generator of the matrix of the explicit symmetry breaking H and isospin sigma field, respectively. In this regard, we recall that the same symbol $h_3$ was used in refs. Refer to [27,28] for the dimensionless coupling constant distinguishing between u and d quark flavors.

The third type of contributions to the PLSM Lagrangian, Equation (3), represents the Polyakov-loop potential responsible for the gluonic degrees of freedom and the dynamics of the quark-gluon interactions, $\mathcal{U}(\varphi ,\overline{\varphi },T)$. The Polyakov-loop potentials are suggested to characterize the QCD symmetries in pure-gauge theory [18,19,23,29,30]. For example, based on strong coupling simulations and by including higher-order Polyakov-loop variables, we have

$$\begin{array}{ccc}\hfill {\mathcal{U}}_{\mathrm{Fuku}}(\varphi ,\overline{\varphi },T)& =& -bT\left[54\varphi \overline{\varphi }exp(-a/T)+ln(1-6\varphi \overline{\varphi }-3{\left(\varphi \overline{\varphi }\right)}^2+4({\varphi }^3+{\overline{\varphi }}^3))\right].\hfill \end{array}$$

(7)

In temporal space, the thermal expectation value of the color traced Wilson loop, also known as the Polyakov-loop variable, reads

$$\begin{array}{ccc}\hfill \varphi & =& \left({\mathrm{Tr}}_c\mathcal{P}\right)/N_c,\hfill \end{array}$$

(8)

$$\begin{array}{ccc}\hfill \overline{\varphi }& =& \left({\mathrm{Tr}}_c{\mathcal{P}}^{†}\right)/N_c,\hfill \end{array}$$

(9)

where $\mathcal{P}$ are the Polyakov loops.

Then, in the mean-field approximation, the PLSM grand-canonical potential can be expressed as

$$\begin{array}{ccc}\hfill \Omega (T,{\mu }_f)=\frac{-T·ln\left[\mathcal{Z}\right]}{V}& =& {\Omega }_{\overline{\psi }\psi }(T,{\mu }_f)+U({\sigma }_u,{\sigma }_d,{\sigma }_s)+{\mathcal{U}}_{\mathrm{Fuku}}(\varphi ,\overline{\varphi },T),\hfill \end{array}$$

(10)

where ${\mu }_f$ are the chemical potentials of the three quark flavors. For the conserved charge baryon B, strangeness S, electric charge Q, and isospin I, the quark chemical potentials are composed as

$$\begin{array}{ccc}\hfill {\mu }_u& =& \frac{{\mu }_B}{3}+\frac{2{\mu }_Q}{3}+\frac{{\mu }_I}{2},\hfill \end{array}$$

(11)

$$\begin{array}{ccc}\hfill {\mu }_d& =& \frac{{\mu }_B}{3}-\frac{{\mu }_Q}{3}-\frac{{\mu }_I}{2},\hfill \end{array}$$

(12)

$$\begin{array}{ccc}\hfill {\mu }_s& =& \frac{{\mu }_B}{3}-\frac{{\mu }_Q}{3}-{\mu }_S,\hfill \end{array}$$

(13)

accordingly, the mesonic contributions to the LSM potential $U({\sigma }_u,{\sigma }_d,{\sigma }_s)$ can be determined by substituting Equation (6), the mesonic field, into Equation (5)

$$\begin{array}{ccc}\hfill U({\sigma }_u,{\sigma }_d,{\sigma }_s)& =& \frac{m^2}{4}\left[{\sigma }_u^2+{\sigma }_d^2+2{\sigma }_s^2\right]-\frac{c}{2\sqrt{2}}{\sigma }_u{\sigma }_d{\sigma }_s+\frac{{\lambda }_1}{16}{\left({\sigma }_u^2+{\sigma }_d^2+2{\sigma }_s^2\right)}^2\hfill \\ & +& \frac{{\lambda }_2}{16}\left({\sigma }_u^4+{\sigma }_d^4+4{\sigma }_s^4\right)-h_{ud}\frac{{\sigma }_u+{\sigma }_d}{2}-h_3\frac{{\sigma }_u-{\sigma }_d}{2}-h_s{\sigma }_s,\hfill \end{array}$$

(14)

where $h_3$ shall be derived in Section 2.2.

The contributions of quarks and antiquarks to the PLSM potential are given as [18,31,32,33]

$$\begin{array}{ccc}\hfill {\Omega }_{\overline{\psi }\psi }(T,{\mu }_f)& =& -2T\sum _{f=u,d,s}{\int }_0^{\infty }\frac{d^3\overrightarrow{P}}{{\left(2\pi \right)}^3}ln\left[1+n_{q,f}(T,{\mu }_f)\right]+ln\left[1+n_{\overline{q},f}(T,{\mu }_f)\right].\hfill \end{array}$$

(15)

It is evident that $n_{\overline{q},f}(T,{\mu }_f)$ is identical to $n_{q,f}(T,{\mu }_f)$ with a double replacement, namely, $-{\mu }_f$ by $+{\mu }_f$ and the order parameter of the Ployakov-loop field $\varphi$ by its conjugate $\overline{\varphi }$ or vice versa.

$$\begin{array}{ccc}\hfill n_{q,f}(T,{\mu }_f)& =& 3\left(\varphi +\overline{\varphi }{e}^{-\frac{E_f-{\mu }_f}{T}}\right)\times e^{-\frac{E_f-{\mu }_f}{T}}+e^{-3\frac{E_f-{\mu }_f}{T}}.\hfill \end{array}$$

(16)

$E_f={({\overrightarrow{P}}^2+m_f^2)}^{1/2}$ gives the energy-momentum dispersion relation corresponding to the quark and antiquark, where $m_f$ is the mass of fth quark flavor. Equation (10) expresses the grand canonical potential, which in mean-field approximation derives the various physical quantities characterizing the QCD thermodynamics and thereby the QCD phase structure in thermal and dense medium.

2.2. Isospin Asymmetry and Meson Potential

As discussed in the previous section, the isospin asymmetry SU(2) is broken, at finite ${\overline{\sigma }}_3$[13], not only $\overline{{\sigma }_3}$, which breaks the isospin symmetry, but also the potential of pure mesonic contributions in SU($N_f$), which can be rewritten as [14],

$$\begin{array}{ccc}\hfill U\left(\overline{\sigma }\right)& =& \left(\frac{m^2}{2}-h_a\right){\overline{\sigma }}_a-3{\mathcal{G}}_{abc}{\overline{\sigma }}_b{\overline{\sigma }}_c-\frac{4}{3}{\mathcal{F}}_{abcd}{\overline{\sigma }}_b{\overline{\sigma }}_c{\overline{\sigma }}_d,\hfill \end{array}$$

(17)

where the coefficients ${\mathcal{G}}_{abc}$ and ${\mathcal{F}}_{abcd}$ are given as [14]

$$\begin{array}{ccc}\hfill {\mathcal{G}}_{abc}& =& \frac{c}{6}\left[d_{abc}-\frac{3}{2}\left(d_{0bc}{\delta }_{a0}+d_{a0c}{\delta }_{b0}+d_{ab0}{\delta }_{c0}\right)+\frac{9}{2}{d}_{000}{\delta }_{a0}{\delta }_{b0}{\delta }_{c0}\right],\hfill \end{array}$$

(18)

$$\begin{array}{ccc}\hfill {\mathcal{F}}_{abcd}& =& \frac{{\lambda }_1}{4}\left[{\delta }_{ab}{\delta }_{cd}+{\delta }_{ad}{\delta }_{cd}+{\delta }_{ac}{\delta }_{bd}\right]+\frac{{\lambda }_2}{8}\left[d_{abn}{d}_{ncd}+d_{adn}{d}_{nbc}+d_{acn}{d}_{nbd}\right].\hfill \end{array}$$

(19)

We notice that the explicitly symmetry breaking terms, $h_0,h_3$, and $h_8$, can be determined by minimizing the potential, Equation (17), on tree level, $\partial U\left(\overline{\sigma }\right)/\partial {\overline{\sigma }}_a=0$. Concretely, both $h_0$ and $h_8$ can be determined from the partially conserved axial current (PCAC) relations.

The generator operator ${\widehat{T}}_a={\widehat{\lambda }}_a/2$ in U(3) is obtained from Gell–Mann matrices ${\widehat{\lambda }}_a$ [26] with the indices running as $a=0,\cdots ,8$. From U(3) algebra, we have

$$\begin{array}{ccc}\hfill \left[{\widehat{T}}_a,{\widehat{T}}_b\right]& =& i{f}_{abc}{\widehat{T}}_c,\hfill \end{array}$$

(20)

$$\begin{array}{ccc}\hfill \left\{{\widehat{T}}_a,{\widehat{T}}_b\right\}& =& i{d}_{abc}{\widehat{T}}_c,\hfill \end{array}$$

(21)

where $f_{abc}$ and $d_{abc}$, respectively, are the standard antisymmetric and symmetric structure constants of SU(3). In this regard, the symmetric structure constant $d_{abc}$ can be defined as

$$\begin{array}{ccc}\hfill d_{abc}& =& \frac{1}{4}\mathit{Tr}\left[\left\{{\widehat{\lambda }}_a,{\widehat{\lambda }}_b\right\}{\widehat{\lambda }}_c\right],\hfill \end{array}$$

(22)

$$\begin{array}{ccc}\hfill d_{ab0}& =& \sqrt{\frac{2}{3}}{\delta }_{ab}.\hfill \end{array}$$

(23)

In PCAC relation, the decay constant $f_a$ is related to the symmetric structure constant as

$$\begin{array}{c}\hfill f_a=d_{aab}{\overline{\sigma }}_a.\end{array}$$

(24)

Accordingly, the decay constants of the charged and neutral pion mesons ($f_{{\pi }^{\pm }}=f_1,f_{{\pi }^0}=f_3$) and the kaon meson ($f_{K^{\pm }}=f_4,f_{K^0}=f_6$) are given as

$$\begin{array}{ccc}\hfill f_{{\pi }^0}& =& f_{{\pi }^{\pm }}=\sqrt{\frac{2}{3}}{\overline{\sigma }}_0+\frac{1}{\sqrt{3}}{\overline{\sigma }}_8,\hfill \end{array}$$

(25)

$$\begin{array}{ccc}\hfill f_{K^{\pm }}& =& \sqrt{\frac{2}{3}}{\overline{\sigma }}_0+\frac{1}{2}{\overline{\sigma }}_3-\frac{1}{2\sqrt{3}}{\overline{\sigma }}_8,\hfill \end{array}$$

(26)

$$\begin{array}{ccc}\hfill f_{K^0}& =& \sqrt{\frac{2}{3}}{\overline{\sigma }}_0-\frac{1}{2}{\overline{\sigma }}_3-\frac{1}{2\sqrt{3}}{\overline{\sigma }}_8,\hfill \end{array}$$

(27)

where the isospin sigma field, ${\overline{\sigma }}_3$, is the difference between the decay constants of neutral and charged kaon mesons as

$$\begin{array}{ccc}\hfill {\overline{\sigma }}_3& =& f_{K^{\pm }}-f_{K^0}.\hfill \end{array}$$

(28)

From the experimental and recent lattice review on physical constants [34,35,36], $f_{{\pi }^{\pm }}=f_{{\pi }^0}=92.4\mathrm{MeV}$ and $f_{K^{\pm }}=113\mathrm{MeV},f_{K^0}=113.453\mathrm{MeV}$. Then, we suggest expressions for both $h_0$ and $h_8$,

$$\begin{array}{ccc}\hfill h_0& =& \frac{1}{\sqrt{6}}\left(m_{\pi }^2{f}_{\pi }+2m_K^2{f}_K\right),\hfill \end{array}$$

(29)

$$\begin{array}{ccc}\hfill h_8& =& \frac{2}{\sqrt{3}}\left(m_{\pi }^2{f}_{\pi }-m_K^2{f}_K\right).\hfill \end{array}$$

(30)

The explicit symmetry breaking term, $h_3$, the third generator of the matrix of the explicit symmetry breaking $H=T_a{h}_a$, can be deduced from $\partial U\left(\overline{\sigma }\right)/\partial {\overline{\sigma }}_3=0$,

$$\begin{array}{ccc}\hfill h_3& =& \left[m^2+\frac{c}{\sqrt{6}}\overline{{\sigma }_0}-\frac{c}{\sqrt{3}}\overline{{\sigma }_8}+{\lambda }_1\left({\overline{{\sigma }_0}}^2+{\overline{{\sigma }_3}}^2+{\overline{{\sigma }_8}}^2\right)+\right.\hfill \\ & & \left.{\lambda }_2\left({\overline{{\sigma }_0}}^2+\frac{{\overline{{\sigma }_3}}^2}{2}+\frac{{\overline{{\sigma }_8}}^2}{2}+\sqrt{2}\overline{{\sigma }_0}\overline{{\sigma }_8}\right)\right]\overline{{\sigma }_3},\hfill \end{array}$$

(31)

where the square brackets $[\cdots ]$ is the squared mass of the $a_0$ meson, Equation (28). Then, $h_3$ can be expressed as

$$\begin{array}{ccc}\hfill h_3& =& m_{a_0}^2\left(f_{K^{\pm }}-f_{K^0}\right),\hfill \end{array}$$

(32)

As discussed in the Introduction, the finite isospin asymmetry leads to degenerate masses of the quark flavors, namely, $m_u\ne m_d\ne m_s$. At nonvanishing $h_3$, the PLSM parameters listed in Table 1 are revised,

Table 2. The revised values of the LSM parameters, Equation (5), at $m_{\sigma }=800$MeV [16].

${\mathit{m}}_{\mathit{\sigma }}$ [MeV]	c [MeV]	${\mathit{h}}_{\mathit{ud}}$ [MeV${}^3$]	${\mathit{h}}_3$ [MeV${}^3$]	${\mathit{h}}_{\mathit{s}}$ [MeV${}^3$]	${\mathit{m}}^2$ [MeV${}^2$]	${\mathit{\lambda }}_1$	${\mathit{\lambda }}_2$
800	$4807.84$	${\left(120.73\right)}^3$	$-{\left(78.31\right)}^3$	${\left(336.41\right)}^3$	$-{\left(306.26\right)}^2$	$13.49$	$46.48$

.

3. Results and Discussion

For a reliable differentiation between u- and d-quark condensates, the influences of the finite isospin asymmetry on the corresponding PLSM chiral condensates and on the deconfinement order parameters should be estimated [37]. It was pointed out that the impacts of finite isospin asymmetry enhances the PLSM thermodynamics, especially with the increase in temperatures. The dependence of the PLSM critical chiral temperatures on the isospin asymmetry obviously maps out the QCD phase structure. The critical chiral temperature decreases as the isospin asymmetry increases. We also conclude that the PLSM results on the critical temperatures reproduce the recent lattice QCD simulations well [38,39], Figure 1 For a reliable comparison, both temperature and isospin chemical potential are normalized to the critical temperature and the pion mass, respectively, as follows. For lattice QCD: $m_{\pi }=400.0$MeV and $T_{\chi }^{{\mu }_I=0}=164$MeV. For PLSM: $m_{\pi }=138$MeV and $T_{\chi }^{{\mu }_I=0}=210$MeV.

At finite ${\mu }_I$, the potential impacts of $bar{\sigma }_3$ and $h_3$ are taken into consideration. The procedure to determine the critical chiral temperature goes as follows.

• Both u- and d-quark chiral condensates become distinguishable. As the temperature approaches a critical value, the normalized nonstrange condensates are split into two different curves. At this point, the critical chiral temperatures, $T_{\chi }$, can be at least qualitatively estimated. We notice that the value of the resulting $T_{\chi }$ decreases with increasing ${\mu }_I$.
• Both Ployakov-loop variables $\varphi$ and $\overline{\varphi }$ also become distinguishable. Increasing ${\mu }_I$ decreases $\overline{\varphi }$ but increases $\varphi$. Moreover, both $\varphi$ and $\overline{\varphi }$ become more distinguishable with a further increase in ${\mu }_I$.
• Both nonstrange quark susceptibilities become distinguishable as well. The critical chiral temperature $T_{\chi }$ is positioned in the middle of the deconfinement phase transition. The resulting $T_{\chi }$ decreases with the increase in ${\mu }_I$.

The Figure 1 illustrates that similar to the lattice QCD simulations [38,39], the critical temperature decreases with increasing ${\mu }_I$. Because the PLSM results reproduce the available lattice QCD calculations well, it is possible to predict the tendency at larger ${\mu }_I$. Furthermore, when comparing our results for Figure 1, with the QCD phase structure in ($T_{\chi }$-${\mu }_B$) plane reported in refs. [10,40,41,42], good similarity can be concluded.

4. Conclusions

We have studied the Ployakov linear-sigma model with three quark flavors and U${\left(1\right)}_A$ anomaly, at finite isospin asymmetry. The finite isospin asymmetry emerges nonvanishing diagonal generators ${\sigma }_0\ne {\sigma }_3\ne {\sigma }_8\ne 0$ of the mean sigma-fields $\overline{{\sigma }_a}$. This means that the SU$\left(2\right)$ isospin asymmetry is broken through ${\sigma }_3$, i.e., ${\sigma }_u={\sigma }_l+{\sigma }_3$ and ${\sigma }_d={\sigma }_l-{\sigma }_3$ [14,43,44].

From the thermal and dense dependence of the quark condensates ${\sigma }_u$, ${\sigma }_d$, and ${\sigma }_s$, and the Polyakov-loop variables $\varphi$ and $\overline{\varphi }$, fruitful information, especially about the QCD chiral phase transition, becomes available. All of these quantities can be estimated by minimizing the real part of thermodynamic potential, $\mathcal{R}e[\Omega (T,{\mu }_f)]$, Equation (10). We conclude the chiral QCD phase transition, the dependence of the critical chiral temperatures on the isospin chemical potential, looks very similar to the QCD phase transition, the dependence of the critical temperatures on the baryon chemical potential, i.e., the ($T_{\chi }$-${\mu }_I$) plane looks very similar to the ($T_{\chi }$-${\mu }_B$) plane. We conclude that the critical chiral temperatures are not universally constant. This apparently depends on the quark flavors and the isospin chemical potentials.