Chiral Invariant Mass Constraints from HESS J1731–347 in an Extended Parity Doublet Model with Isovector Scalar Meson

Abstract

The recent discovery of an extremely light and small central compact object (CCO) within the supernova remnant HESS J1731-347, with mass $0.{77}_{-0.17}^{+0.20}{M}_{\odot }$ and radius $10.4_{-0.78}^{+0.86}$ km, is challenging our understanding of neutron stars. In this article, we identify it as an ultra-light neutron star (NS) and constrain the chiral invariant mass of nucleon $m_0$ from the observational data of NS using an extended parity doublet model with the isovector scalar meson $a_0\left(980\right)$. We show that the $1\sigma$ data from the HESS J1731-347 impose a very narrow constraint on the allowed values of $m_0$ and $L_0$ in the crossover model: $830\mathrm{MeV}\lesssim m_0\lesssim 900\mathrm{MeV}$ for $L_0=40$ MeV, and $850\mathrm{MeV}\lesssim m_0\lesssim 890\mathrm{MeV}$ for $L_0=45$ MeV. We also study the higher-order asymmertic matter properties such as the symmetry incompressibility $K_{sym}$ and the symmetry skewness $Q_{sym}$ in the presence of the $a_0$ meson. We find that $K_{sym}$ and $Q_{sym}$ are very sensitive to $m_0$ in the presence of the $a_0$ meson.

1. Introduction

The neutron star (NS) is one of the most compact objects in the universe. It is an excellent cosmic laboratory under extreme conditions for studying dense QCD matter. NSs allow us to study the equation of state (EoS) of the QCD matter in high density, which is difficult to access in the experiments. Recently, more and more NS observations are available and provide us valuable information about the EoS. For example, the NS merger event GW170817 provided insights into the mass and radius of NSs, with an estimation of approximately $1.4M_{\odot }$ and a radius of $R=11.9_{-1.4}^{+1.4}$ km [1,2]. The NS observations from NICER have also played a crucial role in advancing our understanding of NSs. The analyses [3,4] have focused on NSs with masses around $1.4M_{\odot }$ and $2.1M_{\odot }$. The results show that the radii of these NSs are rather similar for different masses, with a radius of approximately 12.45 ± 0.65 km for a $1.4M_{\odot }$ NS and 12.35 ± 0.75 km for a $2.08M_{\odot }$ NS.

Recently, a report on a central compact object (CCO) HESS J1731-347 [5] with a very small mass $M=0.{77}_{-0.17}^{+0.20}$$M_{\odot }$ and radius $R=10.4_{-0.78}^{+0.86}$ km has challenged our understanding to the NS. The observation of HESS J1731-347 implies that the NS EoS is very soft in the low-density region [6]. There are also studies that consider HESS J1731-347 as a quark star [7,8,9,10], an exotic object made from deconfined quarks rather than the usual hadronic matter. Understanding this CCO is therefore important to the study of NSs and the EoS.

Chiral symmetry and its spontaneous breaking play a fundamental role in quantum chromodynamics (QCD) and low-energy hadron physics. This symmetry breaking is responsible for the generation of the hadron masses and the mass differences between chiral partners. In dense environments, such as the interior of NSs, chiral symmetry is expected to be partially restored. Investigating how hadron masses change under such conditions can provide valuable insights into the origin of hadron masses and the properties of strongly interacting matter.

The parity doublet model (PDM) [11] is an extended linear sigma model that incorporates a parity doubling structure of nucleons. In this model, the negative-parity excited nucleon is considered as the chiral partners of the ground state nucleons, with spontaneous symmetry breaking generating the mass difference between them. When chiral symmetry is restored, these nucleons degenerate into the same mass, which is called chiral invariant mass of nucleon $m_0$. Studies such as lattice simulations [12,13,14] and QCD sum rule [15] suggest that part of the nucleon mass is independent of the chiral symmetry breaking. Both quantitative and qualitative studies of the chiral invariant mass are therefore crucial for advancing our understanding of the origin of hadron masses.

Previous analyses have attempted to constrain $m_0$ by analyzing nucleon properties in vacuum. Ref. [16] suggests that $m_0$ is smaller than 500 MeV based on an analysis of the decay width of $N\left(1535\right)$, while Ref. [17] includes higher-derivative interactions in the model, resulting in larger values of $m_0$ that are consistent with the decay width from experiments.

The PDM has also been applied to study dense mediums in several studies, such as in Refs. [18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45]. Recently, several studies [35,36,38,41,46,47] have constructed the NS EoS using an extended PDM [26,48]. In these studies, the hadronic EoS is smoothly interpolated to a NJL-type quark matter EoS, under the assumption of a crossover hadron–quark phase transition, following the approach of Ref. [49]. Reference [35] had constrained the chiral invariant mass of nucleon to 600 MeV $\lesssim m_0\lesssim 900$ MeV using the observational data of NSs given in Refs. [1,2,50,51,52,53]. Reference [54] constrained $m_0$ to 580 MeV $\lesssim m_0\lesssim 860$ MeV with the presence of the isovector scalar meson. Reference [41] showed that $m_0\simeq 850$ MeV, with the consideration of a central compact object (CCO) within the supernova remnant HESS J1731-347 [5]. The discovery of this ultra-light compact object HESS J1731-347, with a mass of approximately $0.{77}_{-0.17}^{+0.20}{M}_{\odot }$ and a radius of about $10.4_{-0.78}^{+0.86}$ km, has opened a new window for studying compact objects and provides additional constraints on the dense matter equation of state. Such light and small CCOs pose challenges to the existing theories and require careful theoretical modeling.

Recently, the effect of the isovector-scalar $a_0$(980) meson (also called the $\delta$ meson) on asymmetric matter such as NSs is gaining attention. The $a_0$(980) meson accounts for the attractive force in the isovector channel [55,56,57,58,59]. The effect of $a_0$(980) to the symmetry energy and asymmetric matter EoS was studied in Refs. [56,57,60,61,62,63,64,65] using Walecka-type relativistic mean-field (RMF) models, and in Refs. [66,67] using density-dependent RMF models. The existence of $a_0$(980) is shown to increase the symmetry energy [56,57,60,62,64,65,68], and stiffen the NS EoS [56,57,61,62,63] and asymmetric matter EoS [67]. Recently, the stiffening effect of $a_0\left(980\right)$ on the NS EoS was also confirmed in an extended PDM, and the constraint to the chiral invariant mass is obtained as 580 MeV $\lesssim m_0\lesssim 860$ MeV in Ref. [54]. The stiffening of the NS EoS due to the $a_0$ meson therefore may make the interpretation of HESS 1731-347 as an ultra-light NS difficult.

In this work, we extend previous studies by incorporating the isovector scalar meson $a_0\left(980\right)$ into the PDM and explore its effects on the properties of neutron stars, with updated NS observations including the ultra-light compact object HESS J1731-347. Through this analysis, we aim to provide tighter constraints on the chiral invariant mass $m_0$ with the consideration of HESS J1731-347. Unless otherwise stated, natural units with $\hslash =c=1$ are used in this work.

This paper is organized as follows: in Section 2.1, we review an extension of the PDM by including the isovector scalar meson $a_0\left(980\right)$ based on the chiral U(2)_L × U(2)_R symmetry with the U(1)_A anomaly included, as constructed in Ref. [54]. In Section 2.2, we construct the matter with PDM under the mean-field approximation. Then, in Section 3, we study effect of the $a_0$ meson and chiral invariant mass of nucleon $m_0$ to the asymmetric matter properties such as the symmetry incompressibility $K_{sym}$ and the symmetry skewness coefficient $Q_{sym}$. By comparing to the results to recent $K_{sym}$ constraints from experiments and theoretical works, we constrain the value of $m_0$. In Section 4, we study neutron star matter using a unified equation of state with hadron–quark crossover, and analyze the mass–radius relationship to constrain the model parameters $m_0$ and $L_0$ using recent NS observations, including the ultra-light compact object HESS J1731-347. We then compare the results from hadronic EoSs based on PDM and the constraint from nuclear matter properties to the constraints from NS observations. Finally, a summary is given in Section 5.

2. Dense Nuclear Matter with Parity Doublet Model

2.1. A Parity Doublet Model with U(2)_L × U(2)_R Symmetry

In this work, we use an parity doublet model (PDM) based on the U(2)_L × U(2)_R chiral symmetry, as constructed in Ref. [54]. The Lagrangian is given by

$$\mathcal{L}={\mathcal{L}}_N+{\mathcal{L}}_M+{\mathcal{L}}_V,$$

(1)

where ${\mathcal{L}}_N$ is the nucleons, ${\mathcal{L}}_M$ is the scalar and pseudoscalar mesons, and ${\mathcal{L}}_V$ is the vector mesons Lagrangian.

In this model, the scalar meson field M is introduced as the ${(2,2)}_{-2}$ representation under the SU(2)_L × SU(2)_R × U(1)_A symmetry, which transforms as

$$M\to e^{-2i{\theta }_A}{g}_{L}M{g}_R^{†},$$

(2)

where $g_{R,L}\in \mathrm{S}\mathrm{U}{\left(2\right)}_{R,L}$ and $e^{-2i{\theta }_A}\in \mathrm{U}{\left(1\right)}_A$. M is parameterized as

$$M=[\sigma +i\overrightarrow{\pi }·\overrightarrow{\tau }]-[\overrightarrow{a_0}·\overrightarrow{\tau }+i\eta ],$$

(3)

where $\sigma ,\overrightarrow{\pi },\overrightarrow{a_0},\eta$ are the sigma meson, pions, the lightest isovector scalar meson $a_0\left(980\right)$, and $\eta$ meson field, respectively. $\overrightarrow{\tau }$ are the Pauli matrices. The vacuum expectation value (VEV) of M is given by

$$\begin{array}{c}\hfill ⟨0\left|M\right|0⟩=\left(\begin{array}{cc}{\sigma }_0& 0\\ 0& {\sigma }_0\end{array}\right),\end{array}$$

(4)

where ${\sigma }_0=⟨0\left|\sigma \right|0⟩$ is the VEV of the $\sigma$ field equal to the pion decay constant $f_{\pi }=93$ MeV. Then, the Lagrangian ${\mathcal{L}}_M$ is given by

$${\mathcal{L}}_M={\displaystyle \frac{1}{4}}\mathrm{t}\mathrm{r}\left[{\partial }_{\mu }M{\partial }^{\mu }{M}^{†}\right]-V_M,$$

(5)

where $V_M$ is the potential for M. In the current model, $V_M$ is given by [54].

$$\begin{array}{cc}\hfill V_M=& -{\displaystyle \frac{{\overline{\mu }}^2}{4}}\mathrm{t}\mathrm{r}\left[M^{†}M\right]+{\displaystyle \frac{{\lambda }_{41}}{8}}\mathrm{t}\mathrm{r}\left[{\left(M^{†}M\right)}^2\right]\hfill \\ \hfill & -{\displaystyle \frac{{\lambda }_{42}}{16}}{\left\{\mathrm{t}\mathrm{r}\left[M^{†}M\right]\right\}}^2-{\displaystyle \frac{{\lambda }_{61}}{12}}\mathrm{t}\mathrm{r}\left[{\left(M^{†}M\right)}^3\right]\hfill \\ \hfill & -{\displaystyle \frac{{\lambda }_{62}}{24}}\mathrm{t}\mathrm{r}\left[{\left(M^{†}M\right)}^2\right]\mathrm{t}\mathrm{r}\left[M^{†}M\right]-{\displaystyle \frac{{\lambda }_{63}}{48}}{\left\{tr\left[M^{†}M\right]\right\}}^3\hfill \\ \hfill & -{\displaystyle \frac{m_{\pi }^2{f}_{\pi }}{4}}\mathrm{t}\mathrm{r}[M+M^{†}]-{\displaystyle \frac{K}{8}}\{detM+det{M}^{†}\},\hfill \end{array}$$

(6)

where terms up to the sixth order that are invariant under SU(2)_L × SU(2)_R × U(1)_A symmetry are included. In addition, a determinant-type Kobayashi–Maskawa–’t Hooft interaction is included in the current model to implement the U(1)_A anomaly.

The iso-triplet $\rho$ meson and iso-singlet $\omega$ meson are considered based on the hidden local symmetry (HLS) [69,70,71]. The HLS is introduced by performing polar decomposition of the field M as

$$\begin{array}{c}\hfill M={\xi }_L^{†}S{\xi }_R,\end{array}$$

(7)

where $S=\sigma +{\displaystyle \sum _{b=1}^3}{a}_0^b{\tau }_b/2$ is the $2\times 2$ matrix field for scalar mesons. ${\xi }_{L,R}$ are transforming as

$$\begin{array}{c}\hfill {\xi }_{L,R}\to h_{\omega }{h}_{\rho }{\xi }_{L,R}{g}_{L,R}^{†}{e}^{\pm i{\theta }_A},\end{array}$$

(8)

where $h_{\omega }\in {U\left(1\right)}_{\mathrm{HLS}}$ and $h_{\rho }\in {SU\left(2\right)}_{\mathrm{HLS}}$. We note that $e^{+i{\theta }_A}$ for ${\xi }_L$ and $e^{-i{\theta }_A}$ for ${\xi }_R$. In the unitary gauge of the HLS, ${\xi }_{L,R}$ are parameterized as

$${\xi }_R={\xi }_L^{†}=exp\left(iP/f_{\pi }\right),$$

(9)

where $P=\eta +{\displaystyle \sum _{a=1}^3}{\pi }^a{\tau }_a/2$ is the $2\times 2$ matrix field for pseudoscalar mesons. The vector mesons are the gauge bosons in HLS and transform as

$$\begin{array}{c}\hfill {\omega }_{\mu }\to h_{\omega }{\omega }_{\mu }{h}_{\omega }^{†}+{\displaystyle \frac{i}{g_{\omega }}}{\partial }_{\mu }{h}_{\omega }{h}_{\omega }^{†},\end{array}$$

(10)

$$\begin{array}{c}\hfill {\rho }_{\mu }\to h_{\rho }{\rho }_{\mu }{h}_{\rho }^{†}+{\displaystyle \frac{i}{g_{\rho }}}{\partial }_{\mu }{h}_{\rho }{h}_{\rho }^{†},\end{array}$$

(11)

where ${\omega }_{\mu }$ and ${\rho }_{\mu }={\displaystyle \sum _{a=1}^3}{\rho }_{\mu }^a{\tau }_a/2$ are the gauge bosons of SU(2)_HLS and U(1)_HLS, respectively. $g_{\omega }$ and $g_{\rho }$ are the corresponding HLS gauge coupling constants.

The HLS-invariant Lagrangian is given by

$$\begin{array}{cc}\hfill {\mathcal{L}}_V=& a_{VNN}\left[{\overline{N}}_{1l}{\gamma }^{\mu }{\xi }_L^{†}{\widehat{\alpha }}_{\Vert \mu }{\xi }_L{N}_{1l}+{\overline{N}}_{1r}{\gamma }^{\mu }{\xi }_R^{†}{\widehat{\alpha }}_{\Vert \mu }{\xi }_R{N}_{1r}\right]\hfill \\ \hfill & +a_{VNN}\left[{\overline{N}}_{2l}{\gamma }^{\mu }{\xi }_R^{†}{\widehat{\alpha }}_{\Vert \mu }{\xi }_R{N}_{2l}+{\overline{N}}_{2r}{\gamma }^{\mu }{\xi }_L^{†}{\widehat{\alpha }}_{\Vert \mu }{\xi }_L{N}_{2r}\right]\hfill \\ \hfill & +a_{0NN}\sum _{i=1,2}\left[{\overline{N}}_{il}{\gamma }^{\mu }\mathrm{t}\mathrm{r}\left[{\widehat{\alpha }}_{\Vert \mu }\right]N_{il}+{\overline{N}}_{ir}{\gamma }^{\mu }\mathrm{t}\mathrm{r}\left[{\widehat{\alpha }}_{\Vert \mu }\right]N_{ir}\right]\hfill \\ \hfill & +{\displaystyle \frac{{m_{\rho }}^2}{{g_{\rho }}^2}}\mathrm{t}\mathrm{r}\left[{\widehat{\alpha }}_{\Vert }^{\mu }{\widehat{\alpha }}_{\Vert \mu }\right]+\left({\displaystyle \frac{{m_{\omega }}^2}{8{g_{\omega }}^2}}-{\displaystyle \frac{{m_{\rho }}^2}{2{g_{\rho }}^2}}\right)\mathrm{t}\mathrm{r}\left[{\widehat{\alpha }}_{\Vert }^{\mu }\right]\mathrm{t}\mathrm{r}\left[{\widehat{\alpha }}_{\Vert \mu }\right]-{\displaystyle \frac{1}{8{g_{\omega }}^2}}\mathrm{t}\mathrm{r}\left[{\omega }^{\mu \nu }{\omega }_{\mu \nu }\right]-{\displaystyle \frac{1}{2{g_{\rho }}^2}}\mathrm{t}\mathrm{r}\left[{\rho }^{\mu \nu }{\rho }_{\mu \nu }\right]\hfill \\ \hfill & +{\lambda }_{\omega \rho }{\left(a_{VNN}+a_{0NN}\right)}^2{a}_{VNN}^2\left[{\displaystyle \frac{1}{2}}\mathrm{t}\mathrm{r}\left[{\widehat{\alpha }}_{\Vert }^{\mu }{\widehat{\alpha }}_{\Vert \mu }\right]\mathrm{t}\mathrm{r}\left[{\widehat{\alpha }}_{\Vert }^{\nu }\right]\mathrm{t}\mathrm{r}\left[{\widehat{\alpha }}_{\Vert \nu }\right]-{\displaystyle \frac{1}{4}}{\left\{\mathrm{t}\mathrm{r}\left[{\widehat{\alpha }}_{\Vert }^{\mu }\right]\mathrm{t}\mathrm{r}\left[{\widehat{\alpha }}_{\Vert \mu }\right]\right\}}^2\right],\hfill \end{array}$$

(12)

where ${\rho }^{\mu \nu }$ and ${\omega }^{\mu \nu }$ are the field strengths of the $\rho$ meson and $\omega$ meson that are given by

$$\begin{array}{cc}\hfill {\rho }_{\mu \nu }=& {\partial }_{\mu }{\rho }_{\nu }-{\partial }_{\nu }{\rho }_{\mu }-i{g}_{\rho }\left[{\rho }_{\mu },{\rho }_{\nu }\right],\hfill \\ \hfill {\omega }_{\mu \nu }=& {\partial }_{\mu }{\omega }_{\nu }-{\partial }_{\nu }{\omega }_{\mu }.\hfill \end{array}$$

(13)

${\widehat{\alpha }}_{\perp }^{\mu }$ and ${\widehat{\alpha }}_{\Vert }^{\mu }$ are the covariantized Maurer−Cartan 1-forms, defined as

$${\widehat{\alpha }}_{\perp }^{\mu }\equiv {\displaystyle \frac{1}{2i}}[D^{\mu }{\xi }_R{\xi }_R^{†}-D^{\mu }{\xi }_L{\xi }_L^{†}],$$

(14)

$${\widehat{\alpha }}_{\Vert }^{\mu }\equiv {\displaystyle \frac{1}{2i}}[D^{\mu }{\xi }_R{\xi }_R^{†}+D^{\mu }{\xi }_L{\xi }_L^{†}],$$

(15)

and the covariant derivatives of ${\xi }_{L,R}$ are given by

$$\begin{array}{c}\hfill D^{\mu }{\xi }_L={\partial }^{\mu }{\xi }_L-i{g}_{\rho }{\rho }^{\mu }{\xi }_L-i{g}_{\omega }{\omega }^{\mu }{\xi }_L+i{\xi }_L{\mathcal{L}}^{\mu }-i{\xi }_L{\mathcal{A}}^{\mu },\end{array}$$

(16)

$$\begin{array}{c}\hfill D^{\mu }{\xi }_R={\partial }^{\mu }{\xi }_R-i{g}_{\rho }{\rho }^{\mu }{\xi }_R-i{g}_{\omega }{\omega }^{\mu }{\xi }_R+i{\xi }_R{\mathcal{R}}^{\mu }+i{\xi }_R{\mathcal{A}}^{\mu },\end{array}$$

(17)

with ${\mathcal{L}}^{\mu }$, ${\mathcal{R}}^{\mu }$ and ${\mathcal{A}}^{\mu }$ being the external gauge fields corresponding to SU(2)_L × SU(2)_R × U(1)_A global symmetry.

The last term in the Lagrangian (12) is a mixing interaction of $\rho$ and $\omega$ mesons, as introduced in Ref. [54] to reduce the slope parameter [41,72].

The baryonic Lagrangian ${\mathcal{L}}_N$ based on the parity doubling [11,16] is given by

$$\begin{array}{cc}\hfill {\mathcal{L}}_{\mathcal{N}}& =\overline{N_1}i{\gamma }^{\mu }{\mathcal{D}}_{\mu }{N}_1+\overline{N_2}i{\gamma }^{\mu }{\mathcal{D}}_{\mu }{N}_2\hfill \\ \hfill & -m_0[{\overline{N}}_1{\gamma }_5{N}_2-{\overline{N}}_2{\gamma }_5{N}_1]\hfill \\ \hfill & -g_1[{\overline{N}}_{1l}M{N}_{1r}+{\overline{N}}_{1r}{M}^{†}{N}_{1l}]\hfill \\ \hfill & -g_2[{\overline{N}}_{2r}M{N}_{2l}+{\overline{N}}_{2l}{M}^{†}{N}_{2r}],\hfill \end{array}$$

(18)

where $N_{ir}={\displaystyle \frac{1+{\gamma }_5}{2}}{N}_i$ ($N_{il}={\displaystyle \frac{1-{\gamma }_5}{2}}{N}_i$) ($i=1,2$) is the left-handed (right-handed) component of the nucleon fields $N_i$, and the covariant derivatives are defined as

$$\begin{array}{cc}\hfill {\mathcal{D}}^{\mu }{N}_{1l,2r}& =\left({\partial }^{\mu }-i{\mathcal{L}}^{\mu }-i{\mathcal{V}}^{\mu }+i{\mathcal{A}}^{\mu }\right)N_{1l,2r},\hfill \\ \hfill {\mathcal{D}}^{\mu }{N}_{1r,2l}& =\left({\partial }^{\mu }-i{\mathcal{R}}^{\mu }-i{\mathcal{V}}^{\mu }-i{\mathcal{A}}^{\mu }\right)N_{1r,2l},\hfill \end{array}$$

(19)

where ${\mathcal{V}}^{\mu }$ is the external gauge field corresponding to the U(1) baryon number. $g_1$ and $g_2$ are the Yukawa couplings of the nucleon $N_i$ and $m_0$ is called the chiral invariant mass of nucleon. Two baryon fields $N_{+}$ and $N_{-}$, corresponding to the positive-parity and negative-parity nucleon fields, can be obtained by diagonalizing ${\mathcal{L}}_N$, and their vacuum masses are given by [11,16]:

$$\begin{array}{c}\hfill m_{\pm }^{\left(\mathrm{vac}\right)}={\displaystyle \frac{1}{2}}\left[\sqrt{{(g_1+g_2)}^2{\sigma }_0^2+4m_0^2}\pm (g_1-g_2){\sigma }_0\right].\end{array}$$

(20)

In the present work, we identify the fields $N_{+}$ and $N_{-}$ as the ground state N(939) and its excited state $N\left(1535\right)$.

2.2. PDM with Mean Field Approximation

In this work, the mean-field approximation is adopted as in [54]:

$$\sigma \left(x\right)\to \sigma ,\pi \left(x\right)\to 0,a_0^i\left(x\right)\to a{\delta }_{i3},\eta \left(x\right)\to 0,$$

(21)

and the matrix M is given by

$$⟨M⟩=\left(\begin{array}{cc}\sigma -a& 0\\ 0& \sigma +a\end{array}\right).$$

(22)

Then, the mean potential $V_M$ is written as

$$\begin{array}{cc}\hfill V_M=& -{\displaystyle \frac{{\overline{\mu }}_{\sigma }^2}{2}}{\sigma }^2-{\displaystyle \frac{{\overline{\mu }}_a^2}{2}}{a}^2+{\displaystyle \frac{{\lambda }_4}{4}}({\sigma }^4+a^4)+{\displaystyle \frac{{\gamma }_4}{2}}{\sigma }^2{a}^2\hfill \\ \hfill & -{\displaystyle \frac{{\lambda }_6}{6}}({\sigma }^6+15{\sigma }^2{a}^4+15{\sigma }^4{a}^2+a^6)+{\lambda }_6^{\prime }({\sigma }^2{a}^4+{\sigma }^4{a}^2)\hfill \\ \hfill & -m_{\pi }^2{f}_{\pi }\sigma ,\hfill \end{array}$$

(23)

where the parameters are redefined as

$$\begin{array}{cc}\hfill & {\overline{\mu }}_{\sigma }^2\equiv {\overline{\mu }}^2+{\displaystyle \frac{1}{2}}K,\hfill \\ \hfill & {\overline{\mu }}_a^2\equiv {\overline{\mu }}^2-{\displaystyle \frac{1}{2}}K={\overline{\mu }}_{\sigma }^2-K,\hfill \\ \hfill & {\lambda }_4\equiv {\lambda }_{41}-{\lambda }_{42},\hfill \\ \hfill & {\gamma }_4\equiv 3{\lambda }_{41}-{\lambda }_{42},\hfill \\ \hfill & {\lambda }_6\equiv {\lambda }_{61}+{\lambda }_{62}+{\lambda }_{63},\hfill \\ \hfill & {\lambda }_6^{\prime }\equiv {\displaystyle \frac{4}{3}}{\lambda }_{62}+2{\lambda }_{63}.\hfill \end{array}$$

(24)

The vector meson mean fields are given by

$${\omega }_{\mu }\left(x\right)\to \omega {\delta }_{\mu 0},{\rho }_{\mu }^i\left(x\right)\to \rho {\delta }_{\mu 0}{\delta }_{i3},$$

(25)

and the mean field Lagrangian of the vector mesons is given by

$$\begin{array}{cc}\hfill {\mathcal{L}}_V=& -g_{\omega NN}\sum _{\alpha j}{\overline{N}}_{\alpha j}{\gamma }^0\omega N_{\alpha j}-g_{\rho NN}\sum _{\alpha j}{\overline{N}}_{\alpha j}{\gamma }^0{\displaystyle \frac{{\tau }_3}{2}}\rho N_{\alpha j}\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}{m}_{\omega }^2{\omega }^2+{\displaystyle \frac{1}{2}}{m}_{\rho }^2{\rho }^2+{\lambda }_{\omega \rho }{g}_{\omega NN}^2{g}_{\rho NN}^2{\omega }^2{\rho }^2,\hfill \end{array}$$

(26)

with

$$\begin{array}{cc}\hfill g_{\omega NN}& =\left(a_{VNN}+a_{0NN}\right)g_{\omega },\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill g_{\rho NN}& =a_{VNN}{g}_{\rho }.\hfill \end{array}$$

(28)

The thermodynamic potential of the nucleons is given by

$$\begin{array}{cc}\hfill & {\mathrm{\Omega }}_N=-2\sum _{\alpha =\pm ,j=\pm }{\int }^{k_f}{\displaystyle \frac{d^{3}p}{{\left(2\pi \right)}^3}}\left[{\mu }_j^{*}-{\omega }_{\alpha j}\right].\hfill \end{array}$$

(29)

$\alpha =\pm$ denotes the parity and $j=\pm$ denotes the iso-spin of nucleons ($j=+$ for proton and $j=-$ for neutron). The effective chemical potential ${\mu }_j^{*}$ is given by

$$\begin{array}{c}\hfill {\mu }_j^{*}\equiv ({\mu }_B-g_{\omega NN}\omega )+{\displaystyle \frac{j}{2}}({\mu }_I-g_{\rho NN}\rho ).\end{array}$$

(30)

${\omega }_{\alpha j}$ is the nucleon energy as defined by ${\omega }_{\alpha j}=\sqrt{{\left(\overrightarrow{p}\right)}^2+{\left(m_{\alpha j}^{*}\right)}^2}$, where $\overrightarrow{p}$ and $m_{\alpha j}^{*}$ are the momentum and the effective mass of the nucleon. In the present model, the effective mass $m_{\alpha j}^{*}$ is given by

$$\begin{array}{c}\hfill m_{\alpha j}^{*}={\displaystyle \frac{1}{2}}\left[\sqrt{{(g_1+g_2)}^2{(\sigma -ja)}^2+4m_0^2}+\alpha (g_1-g_2)(\sigma -ja)\right].\end{array}$$

(31)

Altogether, the hadronic thermodynamic potential is

$$\begin{array}{cc}\hfill & {\mathrm{\Omega }}_H={\mathrm{\Omega }}_N\hfill \\ \hfill & -{\displaystyle \frac{{\overline{\mu }}_{\sigma }^2}{2}}{\sigma }^2-{\displaystyle \frac{{\overline{\mu }}_a^2}{2}}{a}^2+{\displaystyle \frac{{\lambda }_4}{4}}({\sigma }^4+a^4)+{\displaystyle \frac{{\gamma }_4}{2}}{\sigma }^2{a}^2\hfill \\ \hfill & -{\displaystyle \frac{{\lambda }_6}{6}}({\sigma }^6+15{\sigma }^2{a}^4+15{\sigma }^4{a}^2+a^6)+{\lambda }_6^{\prime }({\sigma }^2{a}^4+{\sigma }^4{a}^2)\hfill \\ \hfill & -m_{\pi }^2{f}_{\pi }\sigma -{\displaystyle \frac{1}{2}}{m}_{\omega }^2{\omega }^2-{\displaystyle \frac{1}{2}}{m}_{\rho }^2{\rho }^2-{\lambda }_{\omega \rho }{g}_{\omega NN}^2{g}_{\rho NN}^2{\omega }^2{\rho }^2\hfill \\ \hfill & -{\mathrm{\Omega }}_0,\hfill \end{array}$$

(32)

where the vacuum potential,

$$\begin{array}{c}\hfill {\mathrm{\Omega }}_0\equiv -{\displaystyle \frac{{\overline{\mu }}_{\sigma }^2}{2}}{f}_{\pi }^2+{\displaystyle \frac{{\lambda }_4}{4}}{f}_{\pi }^4-{\displaystyle \frac{{\lambda }_6}{6}}{f}_{\pi }^6-m_{\pi }^2{f}_{\pi }^2,\end{array}$$

(33)

is subtracted from ${\mathrm{\Omega }}_H$.

2.3. Determination of Model Parameters

In the present model, the model parameters are determined to reproduce the nuclear saturation properties and the vacuum properties of the hadrons. There are 11 parameters to be determined for a given value of chiral invariant mass $m_0$:

$$g_1,g_2,{\overline{\mu }}_{\sigma }^2,{\overline{\mu }}_a^2,{\lambda }_4,{\gamma }_4,{\lambda }_6,{\lambda }_6^{\prime },g_{\omega NN},g_{\rho NN},{\lambda }_{\omega \rho }.$$

(34)

The vacuum expectation value of $\sigma$ is taken to be ${\sigma }_0=f_{\pi }$, with the pion decay constant $f_{\pi }=$ 92.4 MeV. The Yukawa coupling constants $g_1$ and $g_2$ are determined by fitting to the nucleon masses in vacuum given in Equation (20). In this study, we identify the nucleon as $N\left(939\right)$ and its parity partner as the excited state $N^{*}\left(1535\right)$, with $m_{+}=m_N=939$ MeV and $m_{-}=m_{N^{*}}=1535$ MeV. The values of ${\overline{\mu }}_{\sigma }^2$, ${\lambda }_4$, ${\lambda }_6$, $g_{\omega NN}$ are determined from the saturation properties: saturation density $n_0$, the binding energy $B_0$, and the incompressibility $K_0$, together with the stationary condition of the potential in vacuum,

$${\overline{\mu }}_{\sigma }^2{f}_{\pi }-{\lambda }_4{f}_{\pi }^3+{\lambda }_6{f}_{\pi }^5+m_{\pi }^2{f}_{\pi }=0.$$

(35)

The value of the nuclear saturation properties are summarized in Table 1. As investigated in Refs. [54,73], terms with coefficient ${\lambda }_6^{\prime }$ are of sub-leading order in the large $N_c$ expansion and have a small effect on the matter properties. Therefore, we set ${\lambda }_6^{\prime }=0$ in this work for simplicity. The parameters ${\overline{\mu }}_a^2={\overline{\mu }}_{\sigma }^2-K$ and ${\gamma }_4$ are fitted to the meson masses and the other parameters,

$$\begin{array}{cc}\hfill K=& m_{\eta }^2-m_{\pi }^2,\hfill \\ \hfill {\gamma }_4=& {\displaystyle \frac{m_{a_0}^2+(5{\lambda }_6-2{\lambda }_6^{\prime })f_{\pi }^4+{\overline{\mu }}_a^2}{f_{\pi }^2}},\hfill \end{array}$$

(36)

where $m_{\eta }$ and $m_{a_0}$ are the masses of $\eta$ and $a_0\left(980\right)$. The values of meson masses used in this work are listed in Table 2. The values of the parameters for various $m_0$ are presented in Table 3. In the present model, the vector mixing interaction with coefficient ${\lambda }_{\omega \rho }$ are included to control behavior of the asymmetric matter at density $n_B>n_0$ beyond the saturation. The parameters $g_{\rho NN}$ and ${\lambda }_{\omega \rho }$ are related and fitted to the symmetry energy $S_0$ as well as the slope parameter $L_0$. As summarized in Ref. [74], the recent accepted value of $L_0=57.7\pm 19$ MeV. Therefore, we carry out the calculations over the range $L_0=40$–80 MeV in this work. The values of $g_{\rho NN}$ and ${\lambda }_{\omega \rho }$ are shown in Table 4 and Table 5.

Table 1. Saturation properties that are used to determine the model parameters: saturation density $n_0$, binding energy $B_0$, incompressibility $K_0$, and symmetry energy $S_0$.

${\mathit{n}}_0$ [${\mathrm{fm}}^{-3}$]	${\mathit{B}}_0$ [MeV]	${\mathit{K}}_0$ [MeV]	${\mathit{S}}_0$ [MeV]
0.16	16	240	31

Table 2. Values of meson masses and pion decay constant in the vacuum in MeV.

${\mathit{m}}_{\mathit{\pi }}$	${\mathit{m}}_{\mathit{\eta }}$	${\mathit{m}}_{{\mathit{a}}_0}$	${\mathit{m}}_{\mathit{\omega }}$	${\mathit{m}}_{\mathit{\rho }}$	${\mathit{f}}_{\mathit{\pi }}$
138	550	980	783	776	92.4

Table 3. Values of $g_1,g_2,{\overline{\mu }}_{\sigma }^2,{\overline{\mu }}_a^2,{\lambda }_4,{\gamma }_4,{\lambda }_6,g_{\omega NN}$ for $m_0$ = 600–900 MeV.

${\mathit{m}}_0$ [MeV]	600	700	800	900
$g_1$	8.48	7.81	6.99	5.96
$g_2$	14.93	14.26	13.44	12.41
${\overline{\mu }}_{\sigma }^2/f_{\pi }^2$	22.43	19.38	12.06	1.64
${\lambda }_4$	40.40	35.51	23.21	4.56
${\lambda }_6{f}_{\pi }^2$	15.75	13.90	8.93	0.69
$g_{\omega NN}$	9.14	7.31	5.66	3.52
${\overline{\mu }}_a^2/f_{\pi }^2$	−10.77	−13.82	−21.15	−31.56
${\gamma }_4$	180.45	168.18	135.97	84.38

Table 4. Values of $g_{\rho NN}$ for various $m_0$, $L_0$.

${\mathit{m}}_0$ [MeV]	600	700	800	900
$L_0=40$ MeV	15.69	14.00	12.71	11.42
$L_0=50$ MeV	15.20	13.46	12.07	10.71
$L_0=60$ MeV	14.75	12.98	11.51	10.11
$L_0=70$ MeV	14.34	12.54	11.03	9.61
$L_0=80$ MeV	13.96	12.15	10.60	9.17

Table 5. Values of ${\lambda }_{\omega \rho }$ for various $m_0$, $L_0$.

${\mathit{m}}_0$ [MeV]	600	700	800	900
$L_0=40$ MeV	0.025	0.076	0.290	2.457
$L_0=50$ MeV	0.022	0.065	0.241	1.944
$L_0=60$ MeV	0.019	0.054	0.192	1.430
$L_0=70$ MeV	0.016	0.043	0.143	0.917
$L_0=80$ MeV	0.014	0.032	0.093	0.403

3. Asymmetric Nuclear Matter Properties

The neutron star is a highly asymmetric matter, mainly composed of neutrons. Therefore, the properties of asymmetric matter such as the symmetry energy at the saturation $S_0$, the slope parameter $L_0$, the symmetry incompressibility $K_{sym}$, and the symmetry skewness coefficient $Q_{sym}$, which determine the EoS of the asymmetric matter, have a strong impact on the neutron star properties, such as their mass and radius. In this section, we compute $K_{sym}$ and $Q_{sym}$. By comparing with the recent constraint of $K_{sym}$, we constrain the chiral invariant mass of nucleon and see whether the constraints for asymmetric nuclear matter properties agree with the NS observations in the present model.

The symmetry energy at arbitrary baryon density $n_B$ is defined as

$$S\left(n_B\right)\equiv {\displaystyle \frac{1}{2}}{\displaystyle \frac{{\partial }^{2}w(x,\delta )}{\partial {\delta }^2}}{|}_{\delta =0},$$

(37)

where $w(x,\delta )\equiv {\displaystyle \frac{ϵ(n_B,n_I)}{n_B}}-m_N$ is the energy per nucleon defined from the energy density $ϵ(n_B,n_I)$. Here, $x\equiv {\displaystyle \frac{n_B-n_0}{3n_0}}$, $\delta \equiv {\displaystyle \frac{n_n-n_p}{n_B}}=-{\displaystyle \frac{2n_I}{n_B}}$. The isospin density is given by $n_I\equiv {\displaystyle \frac{n_p-n_n}{2}}$, with $n_p$ and $n_n$ being the proton and neutron densities, respectively. $K_{sym}$ and $Q_{sym}$ are defined as the coefficients of the Taylor expansion of the symmetry energy $S\left(n_B\right)$ about the saturation density $n_0$:

$$\begin{array}{cc}\hfill S\left(n_B\right)=& S_0+\left({\displaystyle \frac{n_B-n_0}{n_0}}\right){\displaystyle \frac{L_0}{3}}+{\left({\displaystyle \frac{n_B-n_0}{n_0}}\right)}^2{\displaystyle \frac{K_{sym}}{18}}+{\left({\displaystyle \frac{n_B-n_0}{n_0}}\right)}^3{\displaystyle \frac{Q_{sym}}{162}}+O\left(n_B^4\right),\hfill \end{array}$$

(38)

where

$$\begin{array}{cc}\hfill K_{sym}& =9n_0^2{\displaystyle \frac{{\partial }^{2}S}{\partial n_B^2}}{|}_{n_0},Q_{sym}=27n_0^3{\displaystyle \frac{{\partial }^{3}S}{\partial n_B^3}}{|}_{n_0}.\hfill \end{array}$$

(39)

They are the higher-order coefficients that control the high density behavior of the asymmetric nuclear matter EoS. $K_{sym}$ characterizes the curvature of the symmetry energy with respect to density, analogous to the role of the incompressibility coefficient $K_0$ in symmetric nuclear matter. $Q_{sym}$ encodes how rapidly the curvature of the symmetry energy changes as density increases. For the definition of the $S\left(n_B\right)$ and the symmetry energy parameters, such as $K_{sym}$ and $Q_{sym}$, readers are referred to the review by Lattimer [75], which also presents recent constraints on the symmetry energy parameters. The full density dependence of $S\left(n_B\right)$ in the PDM and the effect of $a_0$ meson on it is studied in our previous works [54,73].

Figure 1 shows the $K_{sym}$ as a function of $m_0$ in the models with and without the $a_0$ meson, for various values of $L_0$. We observe that the inclusion of the $a_0$ meson has a significant impact on $K_{sym}$, especially when $m_0$ is small. In particular, $K_{sym}$ becomes positive and increases rapidly as $m_0$ decreases in the $a_0$ model: $K_{sym}\sim {10}^3$ MeV when $m_0\lesssim 600$ MeV for $L_0=60$ MeV. We also note that while larger $L_0$ values lead to larger values of $K_{sym}$, its influence is weaker than that of $m_0$ when $m_0\lesssim 600$–700 MeV in the $a_0$ model. However, for larger $m_0$ values, where the sensitivity to $m_0$ diminishes, the effect of $L_0$ becomes dominant. In contrast, $K_{sym}$ shows much less variation with $m_0$ in the absence of the $a_0$ meson. The recently accepted value of $K_{sym}=-107\pm 88$ MeV, as reported in Ref. [74], is shown as the green band for comparison. We note that the value of $K_{sym}$ is still difficult to determine from the experiments and we are not going to constrain $m_0$ tightly here. However, several systematic studies of theoretical models and experiments such as [74,76,77] concluded that $K_{sym}$ is likely to be negative. In the present model, $K_{sym}<0$ when $m_0\gtrsim 650$–700 MeV, which is consistent with the constraints derived from NS observations, as discussed in Section 4.

Figure 1. $m_0$ dependence of $K_{sym}$ with $L_0=40,57.7,80$ MeV. Solid curves represent the results from the model with $a_0$ meson and dashed curves are from the model without $a_0$ meson. The green region shows the recent accepted value of $K_{sym}$ as summarized in Ref. [74].

Figure 2 shows the dependence of $Q_{sym}$ on $m_0$ in the present models with and without $a_0$ meson for different $L_0$ values. The pink band represents the range of $Q_{sym}$ estimated from Skyrme models, as summarized in Ref. [78]. Similar to the case of $K_{sym}$, the $a_0$ meson has a significant impact on $Q_{sym}$. $Q_{sym}$ has a very different $m_0$-dependence in the models with and without the $a_0$ meson. While $Q_{sym}$ is negative for a small $m_0$ in the model without the $a_0$ meson, $Q_{sym}$ is positive and considerably large when $m_0$ is small in the $a_0$ model. In particular, $Q_{sym}$ exceeds several thousand MeV in the $a_0$ model when $m_0\lesssim 600$–700 MeV, depending on the value of $L_0$, which is much larger than the typical predictions from Skyrme models.

Figure 2. $m_0$ dependence of $Q_{sym}$ with different $L_0$. Solid curve represents the model with $a_0$ meson and dashed curve represents the model without $a_0$. The pink region shows the predictions of $Q_{sym}$ from Skyrme parameterization as summarized in Ref. [78] for comparison.

Our results suggest that higher-order asymmetry properties, such as $K_{sym}$ and $Q_{sym}$, are sensitive to the existence of the $a_0$ meson. Due to the difficulties in measuring $K_{sym}$ and $Q_{sym}$ experimentally, we are not able to properly constrain the value of $m_0$ from them in this work. We believe that future constraints on them could provide valuable insight into the properties of asymmetric nuclear matter, such as the equation of state (EoS) of neutron star matter, and help further our understanding of the chiral invariant mass of the nucleon.

4. Neutron Star Matter

Neutron stars (NSs) provide unique cosmic laboratories for studying matter under extreme conditions. Recent precise measurements of NS masses and radii have significantly constrained the equation of state (EoS) of strongly interacting matter at densities beyond nuclear saturation [1,2,52,53,79]. Of particular significance is the discovery of the central compact object in the supernova remnant HESS J1731-347, characterized by an unusually low mass of approximately $0.{77}_{-0.17}^{+0.20}{M}_{\odot }$ and a radius of about $10.4_{-0.78}^{+0.86}$ km [5]. This object presents a new challenge for theoretical models that must now accommodate both massive neutron stars ($\sim 2M_{\odot }$) and this remarkably light compact object. After including the $a_0$ meson effect, we will study the implication of recent NS observations to the nucleon chiral invariant mass.

4.1. Unified EoS with Crossover Phase Transition

At densities several times of nuclear saturation density ($n_0\approx 0.16$ fm⁻³), the interior of NSs likely undergo a transition from hadronic to quark degrees of freedom. Traditional approaches often model this as a sharp first-order phase transition, producing discontinuities in the EoS [80,81,82,83,84]. However, these treatments typically rely on extrapolating hadronic and quark models far beyond their regions of established validity, leading to significant uncertainties in the predicted phase transition behavior and neutron star properties.

In our approach, we adopt a more physically motivated picture of hadron–quark continuity, where the transition occurs smoothly over a finite density range. To implement this, following Refs. [35,38,41], we expand the pressure as a function of baryon chemical potential in polynomial form $P\left({\mu }_B\right)={\sum }_{i=0}^5{C}_i{\mu }_B^i$. By imposing six boundary conditions together with the thermodynamic stability and causality conditions, we interpolate the EoS of the PDM and that of an NJL-type quark model, as constructed in Ref. [35] in the intermediate density region $2n_0\le n_B\le 5n_0$, to obtain a smoothly connected unified EoS. To account for the neutron star crust, we adopt the BPS EoS [85] at low densities ($n_B\le 0.1{\mathrm{fm}}^{-3}$), matched to our hadronic/crossover EoS at higher densities. In what follows, all neutron star EoSs are constructed with this crust–core matching procedure.

4.2. NS Mass-Radius Relation

By solving the Tolman–Oppenheimer–Volkoff (TOV) equation for spherically symmetric and static stars, we obtain the NS mass–radius (M-R) relation.

In the PDM, the chiral invariant mass $m_0$ and the slope parameter $L_0$ play crucial roles in determining the stiffness of the EoS. Larger values of $m_0$ lead to a softer EoS in the hadronic region, while larger values of the slope parameter $L_0$ result in a stiffer EoS. To illustrate these effects, we examine the NS M–R relations under different parameter combinations. In Figure 3, we fix $L_0=40$ MeV and vary the chiral invariant mass. The left panel shows the M–R relations obtained from the hadronic EoS constructed from the PDM, while the right panel presents the M–R relations from the crossover model. As $m_0$ increases, the hadronic EoS becomes progressively softer, leading to M–R curves with systematically smaller radii and smaller masses. In both the hadronic model and the crossover model, the hadronic part largely controls the radius of intermediate to low-mass NS. The crossover region, however, is strongly influenced by both the hadronic and quark sectors: a stiffer hadronic EoS enlarges the radii of NSs, while a stiffer quark EoS increases the mass of the NSs that have central densities in the crossover regime. Although a larger $m_0$ further reduces the radius of low-mass NSs and is consistent with the HESS J1731-347 observational constraint, it simultaneously makes it harder for the M–R curve to support a $2M_{\odot }$ NS. Conversely, in Figure 4, we fix $m_0=690$ MeV and vary $L_0$. The left panel displays the M–R relations obtained from the hadronic EoS, whereas the right panel shows those for the crossover model. Here, decreasing values of $L_0$ correspond to a softer EoS and smaller radii in both of the models. On the other hand, the change in $L_0$ has a limited effect on the maximum mass of the curves in the PDM. A large $L_0$ makes the EoS difficult to satisfy the observational constraint on the radius of HESS J1731-347. Furthermore, we note that the masses of the heavy NSs are primarily determined by the quark matter EoS at high densities in the crossover model. As shown in the right panels of Figure 3 and Figure 4, the masses of heavy NSs are nearly identical for EoSs with the same NJL parameters $H/G$ and $g_V/G$, regardless of the chosen values of $m_0$ and $L_0$. In contrast, the crossover EoSs can support a large maximum mass, allowing the model to satisfy constraints from massive NSs even with large $m_0$ and small $L_0$. By treating these two parameters as variables, we can investigate how recent neutron star observations constrain their allowed values in the present model.

Figure 3. (Left) Mass–radius relation for $m_0=660,850$ MeV with fixed value of $L_0=40$ MeV computed from PDM. The solid and dashed curves show the M-R relations obtained from the models with and without $a_0$ meson, respectively. (Right) Mass–radius relation for $m_0=660,850$ MeV with fixed value of $L_0=40$ MeV computed from the PDM-NJL crossover model. The PDM EoSs are connected to the quark EoS using the NJL model with parameters $(H/G,g_V/G)$ = (1.3, 0.2) (lines with point markers), (1.4, 0.5) (lines without markers), and (1.5, 0.7) (lines with square markers). See Ref. [35] for details of the NJL model.

Figure 4. (Left) Mass–radius relation for $L_0=40,57.7$ MeV with fixed value of $m_0=690$ MeV calculated from PDM. The solid and dashed curves show the M-R relations obtained from the models with and without $a_0$ meson, respectively. (Right) Mass–radius relation for $L_0=40,57.7$ MeV with fixed value of $m_0=690$ MeV calculated from the PDM-NJL crossover model. The PDM EoSs are connected to the quark EoS using the NJL model with parameters $(H/G,g_V/G)$ = (1.25, 0.1) (lines with markers) and (1.6, 1.3) (lines without markers). See Ref. [35] for details of the NJL model.

In the present models, the $a_0\left(980\right)$, also known as the $\delta$ meson, an isovector scalar meson, is incorporated into the PDM with proper treatment of the chiral symmetry. It provides an attractive contribution in the isovector channel and affects isospin asymmetric matter, such as NS matter. By comparing neutron star properties, such as mass and radius, in models with and without the $a_0$ meson, we find that the $a_0$ meson increases the radii of intermediate- to low-mass NSs and makes it more difficult for the EoS to satisfy the observational constraint on the radius of HESS J1731-347. The effect of $a_0$ is stronger when $m_0$ is smaller, due to the larger coupling. We also observe that the effect of the $a_0$ meson vanishes for heavy NSs. As shown in Figure 3 and Figure 4, the M–R curves obtained from both the PDM and the crossover model, with and without the $a_0$ meson, are almost the same for massive NSs, regardless of the value of $m_0$ and $L_0$. Therefore, the existence of light NSs with small radii, such as HESS J1731-347, place strong constraints on the chiral invariant mass in the models with the $a_0$ meson.

Our results are then compared with the observational data of HESS J1731-347 [5], GW170817 [2], PSR J0740+6620 [3], and PSR J0030+0451 [52] to constrain the value of $m_0$ and $L_0$. The plot on the left of Figure 5 shows the constraint to $m_0$ as a function of $L_0$ from the $1\sigma$ and $2\sigma$ observations of the above NSs for the PDM-NJL crossover model with the $a_0$ meson. We observe that the 1$\sigma$ neutron star (NS) constraint for the PDM-NJL crossover model is very tight, allowing only a narrow region in the $(m_0,L_0)$ plane to satisfy the condition, as indicated by the dark-blue band in the right panel of Figure 5. The $1\sigma$ constraint of $m_0$ is given by

$$830\mathrm{MeV}\lesssim m_0\lesssim 900\mathrm{MeV}$$

(40)

for $L_0=40$ MeV and

$$850\mathrm{MeV}\lesssim m_0\lesssim 890\mathrm{MeV}$$

(41)

for $L_0=45$ MeV. This implies that only very soft EoSs reproduce the $1\sigma$ NS observations, due to the small mass and radius of HESS J1731-347. At the $2\sigma$ level, however, the allowed region broadens substantially with

$$m_0\gtrsim 710\mathrm{MeV},$$

(42)

for $L_0=57.7$ MeV. Since the presence of the $a_0$ meson increases the radii of light NSs, soft EoSs with large $m_0$ are required to reproduce the observational constraint on the radius of HESS J1731-347.

Figure 5. (Left) Allowed region for $m_0$ and $L_0$ from the hadronic PDM with $a_0$ meson. (Right) Allowed region for $m_0$ and $L_0$ from the PDM-NJL crossover model with $a_0$ meson. The blue regions show the value of $m_0$ and $L_0$ and the MR relations satisfy the $1\sigma$ and $2\sigma$ constraints from the NSs observational data.

For comparison, the constraints on $m_0$ as a function of $L_0$ derived from the $1\sigma$ and $2\sigma$ observations for the hadronic PDM are also presented in the left panel of Figure 5. The constraints are significantly tighter and favor smaller values of $m_0$ compared to the crossover model. In particular, the $2\sigma$ constraint is given by the region between $650\mathrm{MeV}\lesssim m_0\lesssim 690\mathrm{MeV}$ for $L_0=40$ MeV, $670\mathrm{MeV}\lesssim m_0\lesssim 690\mathrm{MeV}$ for $L_0=45$ MeV, and $m_0\approx 690$ MeV for $L_0=50$ MeV. On the other hand, the hadronic EoS cannot satisfy the $1\sigma$ constraints, reflecting that the hadronic EoS in the present model cannot simultaneously reproduce the small radius of HESS J1731-347 while supporting a neutron star with a mass around $2M_{\odot }$.

Our results imply that a relatively large $m_0$ is required to reproduce the observational constraints from the NSs, suggesting that the contribution of the chiral symmetry breaking to the mass of nucleon is small. This finding is consistent with other studies, such as lattice simulations [12,13], which indicate that a large portion of the nucleon masses persist even near the chiral phase transition temperature. However, the nature and the origin of the chiral invariant mass remain unclear, and further studies are needed to understand the mass generation of the hadron.

5. Summary

In this work, we first studied the effect of the $a_0$ meson to the higher-order asymmetric matter properties such as the symmetry incompressibility $K_{sym}$ and the symmetry skewness $Q_{sym}$. We find that $K_{sym}$ and $Q_{sym}$ are sensitive to the chiral invariant mass of nucleon $m_0$ in the presence of the $a_0$ meson. In particular, $m_0\gtrsim 650$–700 MeV gives $K_{sym}<0$.

Then, we studied the neutron star M-R relation and gave constraints to the slope parameter $L_0$ and $m_0$. We find that the chiral invariant mass $m_0$ predominantly controls the radii of light NSs, while the slope parameter $L_0$ governs their compactness without strongly affecting the maximum mass of the M–R relations. The masses of heavy NSs are however largely affected by the quark matter EoS in the crossover model and thus insensitive to the hadronic part. We also examined the role of the $a_0\left(980\right)$ meson and found that it increases the radii of light NSs, while its effect on heavy NSs is vanishing and has a very limited effect on their masses. Consequently, The ultra-light compact object HESS J1731-347 provides particularly stringent constraints on our hadronic model parameters. With its unusually small radius and low mass, this object requires a very soft EoS in the hadronic region, which requires large $m_0$ and small $L_0$ values in our model. The 1$\sigma$ data from HESS J1731–347 impose a very narrow constraint on the allowed values of $m_0$ and $L_0$ in the crossover model. Our calculations demonstrate that for $L_0=40$ MeV, $830\mathrm{MeV}\lesssim m_0\lesssim 900\mathrm{MeV}$ and for $L_0=45$ MeV, $850\mathrm{MeV}\lesssim m_0\lesssim 890\mathrm{MeV}$, our unified EoS can simultaneously satisfy all observational constraints within the $1\sigma$ credible region, including that of HESS J1731-347 observation. In contrast, the $2\sigma$ constraint is looser, allowing $m_0\gtrsim 710\mathrm{MeV}$ for $L_0=57.7$ MeV. For comparison, we also analyzed the hadronic model and found that it fails to reproduce the $1\sigma$ observational data of the NSs. The $2\sigma$ constraint is satisfied only within a narrow range between $650\mathrm{MeV}\lesssim m_0\lesssim 690\mathrm{MeV}$ for $L_0=40$ MeV, $670\mathrm{MeV}\lesssim m_0\lesssim 690\mathrm{MeV}$ for $L_0=45$ MeV, and $m_0\approx 690$ MeV for $L_0=50$ MeV. This occurs because a soft hadronic EoS that reproduces the small radius of HESS J1731-347 is difficult to support NSs with a heavy mass ∼ $2M_{\odot }$.

In the present work, higher-order symmetry energy parameters such as $K_{sym}$ and $Q_{sym}$ are not used to impose constraints, as they are not yet well-established experimentally. Nevertheless, we emphasize that the NS constraints obtained from our crossover model are consistent with the general trends of asymmetry nuclear matter properties: our results predict $K_{sym}<0$ for $m_0\gtrsim 650$–700 MeV, in agreement with constraints inferred from neutron star observations. Future experiments on the asymmetric matter EoS will help us to further constrain the chiral invariant mass of the nucleon as well as the behavior of asymmetric matter at a high density.

Our finding suggests that, if confirmed as a neutron star, HESS J1731-347 would significantly narrow the allowed parameter space of the model, offering valuable insights into the nature of the chiral invariant mass of the nucleon and the behavior of dense asymmetric matter, which are otherwise difficult to probe in terrestrial experiments.

The current study suggests that the chiral invariant mass of the nucleon is large, in agreement with other theoretical investigations of nucleon mass. However, the nature and origin of the chiral invariant mass remain unclear, and further studies such as constructing the chiral invariant mass at the quark model level may help us to deepen our understanding of the generation of hadron mass.