Equation of State for Hyperonic Neutron-Star Matter in SU(3) Flavor Symmetry

Abstract

Using a relativistic mean-field model calibrated to finite-nucleus observables and bulk properties of dense nuclear matter, we investigate hyperonic neutron-star matter within an SU(3) flavor symmetry scheme. To retain SU(6)-based couplings within SU(3) flavor symmetry, we introduce a quartic $\varphi$ self-interaction and $\varphi$-$\rho$ mixing. We demonstrate the roles of ${\alpha }_v$ ($F/(F+D)$ ratio), ${\theta }_v$ (mixing angle), and $z_v$ (singlet-to-octet coupling ratio) in SU(3)-invariant vector-meson couplings. It is found that $z_v$ predominantly controls the maximum mass of a neutron star, and $2M_{\odot }$ neutron stars can be supported for $z_v\le 0.15$. The ${\alpha }_v$ also helps sustain large masses, whereas ${\theta }_v$ has a smaller effect on neutron-star properties. This SU(3) framework reconciles nuclear and astrophysical constraints and offers a plausible resolution of the hyperon puzzle.

1. Introduction

Recent multimessenger observations have significantly advanced our understanding of the dense matter equation of state (EoS). In particular, precise measurements of neutron-star radii from the Neutron star Interior Composition ExploreR (NICER) mission [1,2,3,4,5], together with the constraints on the tidal deformability from the binary neutron-star merger event, GW170817 [6,7,8], have placed stringent limits on the properties of dense matter at supranuclear densities. These astrophysical constraints, combined with the discovery of massive pulsars with masses around or above $2M_{\odot }$ [9,10,11,12], strongly influence the theoretical modeling of neutron-star matter and provide important clues to its microscopic composition.

The possible appearance of hyperons in the core of a neutron star is a natural outcome of the Pauli exclusion principle at high densities, where the conversion of high-momentum nucleons into hyperons becomes energetically favorable. However, their onset generally softens the nuclear EoS, leading to a substantial reduction in the maximum mass of a neutron star, $M_{\max }$. This discrepancy between theoretical predictions and astrophysical observations is widely known as the hyperon puzzle.

A variety of approaches have been proposed to address this problem. In non-relativistic frameworks, repulsive three-body forces among baryons have been introduced to provide additional stiffness at high densities, as demonstrated in variational calculations and Brueckner–Hartree–Fock (BHF) studies [13,14,15,16]. In relativistic calculations, several extensions have been explored using relativistic mean-field (RMF) approaches with the additional strange mesons [17,18,19,20] and/or explicit three-baryon couplings [21,22], relativistic Hartree–Fock (RHF) models including Fock terms and tensor couplings [23,24,25,26,27], and Dirac–Brueckner–Hartree–Fock (DBHF) calculations [28,29,30].

The RMF model is a covariant energy-density functional framework that effectively describes baryonic interactions in terms of meson exchange [31,32,33], but it does not explicitly include quark and gluon degrees of freedom originating from microscopic dynamics based on quantum chromodynamics (QCD). Therefore, it should be regarded as a phenomenological realization of the strong interaction, complementary to QCD-motivated approaches such as the lattice QCD [34,35,36], chiral effective field theory ($\chi$EFT) [37,38], Dyson–Schwinger equation formalism [39,40], Nambu–Jona–Lasinio (NJL) model [41,42,43], quark–meson coupling (QMC) model [44,45,46,47,48], and quarkyonic matter [49,50,51].

One of the most practical and effective approaches proposed to resolve the hyperon puzzle in RMF models is the adoption of SU(3) flavor symmetry for determining the vector-meson couplings to the baryon octet [52,53,54,55,56,57,58]. This approach enables the tuning of model parameters with empirical hypernuclear data and hyperon potentials in nuclear matter, introducing an additional repulsive force via the $\varphi$ meson. Such flexibility can delay the onset of hyperons and maintain a sufficiently stiff EoS, which can support $2M_{\odot }$ neutron stars. However, only a few studies have attempted to simultaneously account for both results from nuclear experiments and astrophysical observations. In particular, the nuclear EoS at high densities often shows a substantial mismatch with the constraints obtained in heavy-ion collisions [59,60,61].

The effective nucleon mass, $M_N^{\ast }$, plays a crucial role in determining the global properties of neutron stars. As discussed in Choi et al. [62] and Li et al. [63], a lower $M_N^{\ast }$ generally leads to a stiffer EoS, thereby increasing $M_{\max }$ to values exceeding $2M_{\odot }$. However, this reduction in $M_N^{\ast }$ also tends to enhance the dimensionless tidal deformability, $\Lambda$, making it challenging to reproduce the constraints from GW170817 [7,8]. This dual sensitivity of the EoS to $M_N^{\ast }$ highlights its pivotal role in balancing the competing requirements from both $M_{\max }$ and $\Lambda$ observations. Moreover, recent model-independent analysis suggests a much smaller $\Lambda$ than most current estimates [64]. Such a stringent constraint is particularly difficult to accommodate in the standard RMF-based EoSs, posing a serious challenge to developing a unified description of dense nuclear matter consistent with astrophysical observations.

In the present study, we extend our previous work in Miyatsu et al. [53]. We first build an RMF model that satisfies neutron-star radii and $\Lambda$ from NICER and GW170817, as well as the characteristics of finite nuclei in SU(6) spin–flavor symmetry. The SU(3) flavor symmetry approach, incorporating the effects of strange mesons, is then applied to the calibrated RMF parameters to examine how $M_{\max }$ changes. We aim to clarify the role of modified vector-meson couplings in balancing the stiffness of the hyperonic neutron-star EoS under the combined constraints from both astrophysical and terrestrial data.

This paper is organized as follows. Section 2 reviews the extended RMF model based on Quantum Hadrodynamics [65]. The SU(3) extension of the vector-meson couplings is described in Section 3. Section 4 outlines the model construction and the determination of parameter sets consistent with both nuclear experiments and neutron-star observations. The properties of neutron stars in SU(3) flavor symmetry are presented in Section 5. Finally, Section 6 summarizes the main results and conclusions.

2. Lagrangian Density in SU(3) Flavor Symmetry

We extend the standard Lagrangian density in the RMF approximation to include not only the $\sigma$, ${\omega }^{\mu }$, $\mathit{\delta }$, and ${\mathit{\rho }}^{\mu }$ mesons but also the strange mesons, namely the isoscalar, Lorentz-scalar (${\sigma }^{\ast }$), and Lorentz-vector (${\varphi }^{\mu }$) mesons. Since charge neutrality and $\beta$ equilibrium conditions are imposed in neutron-star matter, leptons are also included in the system. The total Lagrangian density is thus chosen to be [53,62,66,67]

$$\begin{array}{cc}\hfill \mathcal{L}& =\sum _{B=N,\Lambda ,\Sigma ,\Xi }{\overline{\psi }}_B\left[i{\gamma }_{\mu }{\partial }^{\mu }-M_B^{\ast }(\sigma ,\mathit{\delta },{\sigma }^{\ast })-g_{\omega B}{\gamma }_{\mu }{\omega }^{\mu }-g_{\rho B}{\gamma }_{\mu }{\mathit{\rho }}^{\mu }·{\mathit{I}}_B-g_{\varphi B}{\gamma }_{\mu }{\varphi }^{\mu }\right]{\psi }_B\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}\left({\partial }_{\mu }\sigma {\partial }^{\mu }\sigma -m_{\sigma }^2{\sigma }^2\right)+{\displaystyle \frac{1}{2}}{m}_{\omega }^2{\omega }_{\mu }{\omega }^{\mu }-{\displaystyle \frac{1}{4}}{W}_{\mu \nu }{W}^{\mu \nu }\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}\left({\partial }_{\mu }\mathit{\delta }·{\partial }^{\mu }\mathit{\delta }-m_{\delta }^2\mathit{\delta }·\mathit{\delta }\right)+{\displaystyle \frac{1}{2}}{m}_{\rho }^2{\mathit{\rho }}_{\mu }·{\mathit{\rho }}^{\mu }-{\displaystyle \frac{1}{4}}{\mathit{R}}_{\mu \nu }·{\mathit{R}}^{\mu \nu }\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}\left({\partial }_{\mu }{\sigma }^{\ast }{\partial }^{\mu }{\sigma }^{\ast }-m_{{\sigma }^{\ast }}^2{\sigma }^{\ast 2}\right)+{\displaystyle \frac{1}{2}}{m}_{\varphi }^2{\varphi }_{\mu }{\varphi }^{\mu }-{\displaystyle \frac{1}{4}}{P}_{\mu \nu }{P}^{\mu \nu }\hfill \\ \hfill & -U_{\mathrm{NL}}(\sigma ,{\omega }^{\mu },{\mathit{\rho }}^{\mu },{\varphi }^{\mu })+\sum _{\ell =e,\mu }{\overline{\psi }}_{\ell }\left(i{\gamma }_{\mu }{\partial }^{\mu }-m_{\ell }\right){\psi }_{\ell },\hfill \end{array}$$

(1)

where ${\psi }_B$ and ${\psi }_{\ell }$ denote baryon (B) and lepton (ℓ) fields, respectively, and the sum runs $B\in \{N,\Lambda ,\Sigma ,\Xi \}$. The baryon fields are expressed as ${\psi }_N=\left(\begin{array}{c}p\\ n\end{array}\right)$, ${\psi }_{\Lambda }=\left(\begin{array}{c}\Lambda \end{array}\right)$, ${\psi }_{\Sigma }=\left(\begin{array}{c}-{\Sigma }^{+}\\ {\Sigma }^0\\ {\Sigma }^{-}\end{array}\right)$, and ${\psi }_{\Xi }=\left(\begin{array}{c}-{\Xi }^0\\ {\Xi }^{-}\end{array}\right)$. The effective baryon mass is given by

$$M_B^{\ast }(\sigma ,\mathit{\delta },{\sigma }^{\ast })=M_B-g_{\sigma B}\sigma -g_{\delta B}\mathit{\delta }·{\mathit{I}}_B-g_{{\sigma }^{\ast }B}{\sigma }^{\ast },$$

(2)

with $M_B$ being the free mass and ${\mathit{I}}_B$ being the isospin matrix for B. The isospin matrix for B is defined as ${\mathit{I}}_N={\mathit{I}}_{\Xi }=\left({\tau }_1,{\tau }_2,{\tau }_3\right)$, ${\mathit{I}}_{\Lambda }=\mathbf{0}$, and ${\mathit{I}}_{\Sigma }=\left(I_1,I_2,I_3\right)$ with ${\tau }_1=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$, ${\tau }_2=\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right)$, ${\tau }_3=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$, ${\displaystyle I_1=\frac{1}{\sqrt{2}}\left(\begin{array}{ccc}0& 1& 0\\ 1& 0& 1\\ 0& 1& 0\end{array}\right)}$, ${\displaystyle I_2=\frac{i}{\sqrt{2}}\left(\begin{array}{ccc}0& -i& 0\\ i& 0& -i\\ 0& i& 0\end{array}\right)}$, and $I_3=\left(\begin{array}{ccc}1& 0& 0\\ 0& 0& 0\\ 0& 0& -1\end{array}\right)$. The $\sigma$-B, $\omega$-B, $\delta$-B, $\rho$-B, ${\sigma }^{\ast }$-B, and $\varphi$-B coupling constants are, respectively, denoted by $g_{\sigma B}$, $g_{\omega B}$, $g_{\delta B}$, $g_{\rho B}$, $g_{{\sigma }^{\ast }B}$, and $g_{\varphi B}$. The covariant derivatives for the vector-meson fields are expressed as $W_{\mu \nu }={\partial }_{\mu }{\omega }_{\nu }-{\partial }_{\nu }{\omega }_{\mu }$, ${\mathit{R}}_{\mu \nu }={\partial }_{\mu }{\mathit{\rho }}_{\nu }-{\partial }_{\nu }{\mathit{\rho }}_{\mu }$, and $P_{\mu \nu }={\partial }_{\mu }{\varphi }_{\nu }-{\partial }_{\nu }{\varphi }_{\mu }$. In addition, a nonlinear potential in Equation (1) is supplemented as follows:

$$\begin{array}{cc}\hfill U_{\mathrm{NL}}(\sigma ,{\omega }^{\mu },{\mathit{\rho }}^{\mu },{\varphi }^{\mu })& ={\displaystyle \frac{1}{3}}{g}_2{\sigma }^3+{\displaystyle \frac{1}{4}}{g}_3{\sigma }^4-{\displaystyle \frac{1}{4}}{c}_3{\left({\omega }_{\mu }{\omega }^{\mu }\right)}^2-{\displaystyle \frac{1}{4}}{f}_3{\left({\varphi }_{\mu }{\varphi }^{\mu }\right)}^2\hfill \\ \hfill & -{\Lambda }_{\omega \rho }\left({\omega }_{\mu }{\omega }^{\mu }\right)\left({\mathit{\rho }}_{\nu }·{\mathit{\rho }}^{\nu }\right)-{\Lambda }_{\varphi \rho }\left({\varphi }_{\mu }{\varphi }^{\mu }\right)\left({\mathit{\rho }}_{\nu }·{\mathit{\rho }}^{\nu }\right).\hfill \end{array}$$

(3)

The first and second terms in Equation (3) are introduced to provide a quantitative description of the ground-state properties of symmetric nuclear matter [68,69]. We introduce the quartic $\omega$ self-interaction and $\omega$-$\rho$ mixing, which only affects the characteristics of $N\ne Z$ finite nuclei and isospin-asymmetric nuclear matter [70,71,72,73,74]. Furthermore, the quartic self-interaction of the $\varphi$ meson and the $\varphi$-$\rho$ mixing are taken into account to sustain the model parameters determined in SU(6) symmetry when the vector fields are extended to the SU(3) flavor symmetry scheme [75].

In the present study, the hadron and lepton masses in free space are taken as follows: $M_N=939$ MeV, $M_{\Lambda }=1116$ MeV, $M_{\Sigma }=1193$ MeV, $M_{\Xi }=1318$ MeV, $m_{\sigma }=465$ MeV, $m_{\omega }=783$ MeV, $m_{\delta }=980$ MeV, $m_{\rho }=775$ MeV, $m_{{\sigma }^{\ast }}=990$ MeV, $m_{\varphi }=1019$ MeV, $m_e=0.511$ MeV, and $m_{\mu }=106$ MeV. Note that the $\sigma$-meson mass, $m_{\sigma }$, is adjusted to reproduce the ground-state properties of several closed-shell nuclei (see Section 4).

In mean-field approximation, the meson fields are replaced by the mean-field values: $\overline{\sigma }$, $\overline{\omega }$, $\overline{\delta }$, $\overline{\rho }$, ${\overline{\sigma }}^{\ast }$, and $\overline{\varphi }$. The equations of motion for the meson fields in uniform matter are thus given by

$$\begin{array}{cc}\hfill m_{\sigma }^2\overline{\sigma }+g_2{\overline{\sigma }}^2+g_3{\overline{\sigma }}^3& =\sum _B{g}_{\sigma B}{\rho }_B^s,\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill m_{\omega }^2\overline{\omega }+c_3{\overline{\omega }}^3+2{\Lambda }_{\omega \rho }\overline{\omega }{\overline{\rho }}^2& =\sum _B{g}_{\omega B}{\rho }_B,\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill m_{\delta }^2\overline{\delta }& =\sum _B{g}_{\delta B}{\rho }_B^s{\left({\mathit{I}}_B\right)}_3,\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill m_{\rho }^2\overline{\rho }+2{\Lambda }_{\omega \rho }{\overline{\omega }}^2\overline{\rho }+2{\Lambda }_{\varphi \rho }{\overline{\varphi }}^2\overline{\rho }& =\sum _B{g}_{\rho B}{\rho }_B{\left({\mathit{I}}_B\right)}_3,\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill m_{{\sigma }^{\ast }}^2{\overline{\sigma }}^{\ast }& =\sum _B{g}_{{\sigma }^{\ast }B}{\rho }_B^s,\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill m_{\varphi }^2\overline{\varphi }+f_3{\overline{\varphi }}^3+2{\Lambda }_{\varphi \rho }\overline{\varphi }{\overline{\rho }}^2& =\sum _B{g}_{\varphi B}{\rho }_B,\hfill \end{array}$$

(9)

where the scalar density, ${\rho }_B^s$, and the baryon density, ${\rho }_B$, read

$$\begin{array}{cc}\hfill {\rho }_B^s& ={\displaystyle \frac{1}{{\pi }^2}}{\int }_0^{k_{F_B}}dk{k}^2{\displaystyle \frac{M_B^{\ast }}{\sqrt{k^2+M_B^{\ast 2}}}},\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill {\rho }_B& ={\displaystyle \frac{k_{F_B}^3}{3{\pi }^2}},\hfill \end{array}$$

(11)

with $k_{F_B}$ being the Fermi momentum for $B\in \{p,n,\Lambda ,{\Sigma }^{+},{\Sigma }^0,{\Sigma }^{-},{\Xi }^0,{\Xi }^{-}\}$. The total energy density, $\epsilon$, and pressure, P, in neutron-star matter are expressed as

$$\begin{array}{cc}\hfill \epsilon & =\sum _B{\epsilon }_B+{\epsilon }_M+\sum _{\ell }{\epsilon }_{\ell },\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill P& =\sum _B{P}_B+P_M+\sum _{\ell }{P}_{\ell },\hfill \end{array}$$

(13)

where the baryon (B), meson (M), and lepton (ℓ) contributions to $\epsilon$ and P are given by

$$\begin{array}{cc}\hfill {\epsilon }_B& ={\displaystyle \frac{1}{{\pi }^2}}{\int }_0^{k_{F_B}}dk{k}^2\sqrt{k^2+M_B^{\ast 2}},\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill {\epsilon }_M& ={\displaystyle \frac{1}{2}}\left(m_{\sigma }^2{\overline{\sigma }}^2+m_{\omega }^2{\overline{\omega }}^2+m_{\delta }^2{\overline{\delta }}^2+m_{\rho }^2{\overline{\rho }}^2+m_{{\sigma }^{\ast }}^2{\overline{\sigma }}^{\ast 2}+m_{\varphi }^2{\overline{\varphi }}^2\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{3}}{g}_2{\overline{\sigma }}^3+{\displaystyle \frac{1}{4}}{g}_3{\overline{\sigma }}^4+{\displaystyle \frac{3}{4}}{c}_3{\overline{\omega }}^4+{\displaystyle \frac{3}{4}}{f}_3{\overline{\varphi }}^4+3{\Lambda }_{\omega \rho }{\overline{\omega }}^2{\overline{\rho }}^2+3{\Lambda }_{\varphi \rho }{\overline{\varphi }}^2{\overline{\rho }}^2,\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill {\epsilon }_{\ell }& ={\displaystyle \frac{1}{{\pi }^2}}{\int }_0^{k_{F_{\ell }}}dk{k}^2\sqrt{k^2+m_{\ell }^2},\hfill \end{array}$$

(16)

and

$$\begin{array}{cc}\hfill P_B& ={\displaystyle \frac{1}{3{\pi }^2}}{\int }_0^{k_{F_B}}dk{\displaystyle \frac{k^4}{\sqrt{k^2+M_B^{\ast 2}}}},\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill P_M& =-{\displaystyle \frac{1}{2}}\left(m_{\sigma }^2{\overline{\sigma }}^2-m_{\omega }^2{\overline{\omega }}^2+m_{\delta }^2{\overline{\delta }}^2-m_{\rho }^2{\overline{\rho }}^2+m_{{\sigma }^{\ast }}^2{\overline{\sigma }}^{\ast 2}-m_{\varphi }^2{\overline{\varphi }}^2\right)\hfill \\ \hfill & -{\displaystyle \frac{1}{3}}{g}_2{\overline{\sigma }}^3-{\displaystyle \frac{1}{4}}{g}_3{\overline{\sigma }}^4+{\displaystyle \frac{1}{4}}{c}_3{\overline{\omega }}^4+{\displaystyle \frac{1}{4}}{f}_3{\overline{\varphi }}^4+{\Lambda }_{\omega \rho }{\overline{\omega }}^2{\overline{\rho }}^2+{\Lambda }_{\varphi \rho }{\overline{\varphi }}^2{\overline{\rho }}^2,\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill P_{\ell }& ={\displaystyle \frac{1}{3{\pi }^2}}{\int }_0^{k_{F_{\ell }}}dk{\displaystyle \frac{k^4}{\sqrt{k^2+m_{\ell }^2}}}.\hfill \end{array}$$

(19)

3. SU(3) Symmetry in the Vector-Meson Couplings

To study the properties of neutron stars with hyperons (Y), it is important to extend SU(6) spin–flavor symmetry based on the non-relativistic quark model to the more general SU(3) flavor symmetry [19,23,24,25,26,52,53,54,55,56,57,58]. Restricting the discussion to three quark flavors (up, down, and strange), SU(3) symmetry can be regarded as a symmetry group of strong interaction. To consider combinations of the meson–baryon couplings, it is convenient to employ the SU(3)-invariant interaction Lagrangian. Using the matrix representations for the baryon octet, B, and meson nonet (singlet state, $M_1$, and octet state, $M_8$), the interaction Lagrangian can be written as the sum of three terms: one arising from the coupling of the meson singlet to the baryon octet (S term) and the other two terms from the interaction of the meson octet and the baryons—one being the antisymmetric (F) term and the other being the symmetric (D) term [76,77]:

$${\mathcal{L}}_{\mathrm{int}}=-g_8\sqrt{2}\left[\alpha \mathrm{Tr}\left(\left[\overline{B},M_8\right]B\right)+(1-\alpha )\mathrm{Tr}\left(\left\{\overline{B},M_8\right\}B\right)\right]-g_1{\displaystyle \frac{1}{\sqrt{3}}}\mathrm{Tr}\left(\overline{B}B\right)\mathrm{Tr}\left(M_1\right),$$

(20)

where $g_1$ and $g_8$ are, respectively, the coupling constants for the meson singlet and octet states, and $\alpha$ ($0\le \alpha \le 1$) is known as the $F/(F+D)$ ratio.

Here we focus on the vector-meson couplings to the octet baryons, because, as usual, the other coupling constants can be determined so as to reproduce the observed properties of nuclear matter and hypernuclei. When SU(3) symmetry is applied to the isovector vector mesons, the Fock term is, in fact, necessary to reproduce the observed symmetry energy [24]. The physical $\omega$ and $\varphi$ mesons are described in terms of the pure singlet, $|1\rangle$, and octet, $|8\rangle$, states as

$$\begin{array}{cc}\hfill \omega & =cos{\theta }_v|1\rangle +sin{\theta }_v|8\rangle ,\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill \varphi & =-sin{\theta }_v|1\rangle +cos{\theta }_v|8\rangle ,\hfill \end{array}$$

(22)

with ${\theta }_v$ being the mixing angle for the vector mesons. The matrix representations for $M_1$, $M_8$, and B are expressed as

$$\begin{array}{cc}\hfill M_1& ={\displaystyle \frac{1}{\sqrt{3}}}\mathrm{diag}\left({\varphi }_1,{\varphi }_1,{\varphi }_1\right),\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill M_8& =\left(\begin{array}{ccc}{\rho }^0/\sqrt{2}+{\omega }_8/\sqrt{6}& {\rho }^{+}& K^{\ast +}\\ {\rho }^{-}& -{\rho }^0/\sqrt{2}+{\omega }_8/\sqrt{6}& K^{\ast 0}\\ K^{\ast -}& {\overline{K}}^{\ast 0}& -2{\omega }_8/\sqrt{6}\end{array}\right),\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill B& =\left(\begin{array}{ccc}{\Sigma }^0/\sqrt{2}+\Lambda /\sqrt{6}& {\Sigma }^{+}& p\\ {\Sigma }^{-}& -{\Sigma }^0/\sqrt{2}+\Lambda /\sqrt{6}& n\\ {\Xi }^{-}& {\Xi }^0& -2\Lambda /\sqrt{6}\end{array}\right),\hfill \end{array}$$

(25)

with ${\varphi }_1$ (${\omega }_8$) being the singlet (octet) state.

In SU(3) symmetry, all possible combinations of the couplings are then determined by four parameters: the singlet and octet coupling constants, $g_1$ and $g_8$, the $F/(F+D)$ ratio for the vector mesons, ${\alpha }_v$, and the mixing angle, ${\theta }_v$. With the coupling ratio defined by $z_v=g_8/g_1$, the relations of the coupling constants for the $\omega$ and $\varphi$ mesons in SU(3) symmetry can be expressed as

$$\begin{array}{cc}\hfill g_{\omega \Lambda }& ={\displaystyle \frac{1-\frac{2}{\sqrt{3}}{z}_v(1-{\alpha }_v)tan{\theta }_v}{1-\frac{1}{\sqrt{3}}{z}_v(1-4{\alpha }_v)tan{\theta }_v}}{g}_{\omega N},\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill g_{\omega \Sigma }& ={\displaystyle \frac{1+\frac{2}{\sqrt{3}}{z}_v(1-{\alpha }_v)tan{\theta }_v}{1-\frac{1}{\sqrt{3}}{z}_v(1-4{\alpha }_v)tan{\theta }_v}}{g}_{\omega N},\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill g_{\omega \Xi }& ={\displaystyle \frac{1-\frac{1}{\sqrt{3}}{z}_v(1+2{\alpha }_v)tan{\theta }_v}{1-\frac{1}{\sqrt{3}}{z}_v(1-4{\alpha }_v)tan{\theta }_v}}{g}_{\omega N},\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill g_{\varphi N}& =-{\displaystyle \frac{tan{\theta }_v+\frac{1}{\sqrt{3}}{z}_v(1-4{\alpha }_v)}{1-\frac{1}{\sqrt{3}}{z}_v(1-4{\alpha }_v)tan{\theta }_v}}{g}_{\omega N},\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill g_{\varphi \Lambda }& =-{\displaystyle \frac{tan{\theta }_v+\frac{2}{\sqrt{3}}{z}_v(1-{\alpha }_v)}{1-\frac{1}{\sqrt{3}}{z}_v(1-4{\alpha }_v)tan{\theta }_v}}{g}_{\omega N},\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill g_{\varphi \Sigma }& =-{\displaystyle \frac{tan{\theta }_v-\frac{2}{\sqrt{3}}{z}_v(1-{\alpha }_v)}{1-\frac{1}{\sqrt{3}}{z}_v(1-4{\alpha }_v)tan{\theta }_v}}{g}_{\omega N},\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill g_{\varphi \Xi }& =-{\displaystyle \frac{tan{\theta }_v+\frac{1}{\sqrt{3}}{z}_v(1+2{\alpha }_v)}{1-\frac{1}{\sqrt{3}}{z}_v(1-4{\alpha }_v)tan{\theta }_v}}{g}_{\omega N}.\hfill \end{array}$$

(32)

The $\rho$-Y coupling constants are given by

$$g_{\rho \Lambda }=0, g_{\rho \Sigma }=2{\alpha }_v{g}_{\rho N}, g_{\rho \Xi }=-(1-2{\alpha }_v)g_{\rho N}.$$

(33)

Conventionally, vector couplings are taken with ${\alpha }_v=1$, under which the contribution from the D term is ignored. In the present study, by relaxing this assumption and treating ${\alpha }_v$ as a free parameter, we are able to examine the impact of both the F and D terms. Although an additional interaction between $\Lambda$ and ${\Sigma }^0$ via the $\rho$ meson should be treated in SU(3) symmetry under the relation $g_{\rho \Lambda \Sigma }={\displaystyle \frac{2}{\sqrt{3}}}(1-{\alpha }_v)g_{\rho N}$ [78], we neglect its effect in the present study since our interest lies in the $NN$ and $NY$ interactions.

We summarize again the physical meanings of the parameters in SU(3) flavor symmetry. Specifically, ${\alpha }_v$ represents the $F/(F+D)$ ratio, which controls the relative strength of the antisymmetric and symmetric couplings; ${\theta }_v$ is the singlet–octet mixing angle that determines the physical composition of the $\omega$ and $\varphi$ mesons; and $z_v$ denotes the singlet-to-octet coupling ratio, which governs the overall magnitude of the vector-meson couplings.

4. Model Parameters

4.1. Model Construction and Parameters for Nucleons

Here, we construct the effective interaction that accounts not only for the characteristics of finite nuclei but also for the properties of dense nuclear matter. As a first step, model optimization is performed so as to fit the experimental data for binding energy per nucleon, $B/A$, and charge radius, $R_{\mathrm{ch}}$, of several, closed-shell nuclei in SU(6) symmetry [66,67,70]. The characteristics of several finite nuclei and the coupling constants for N are listed in

Table 1. Theoretical predictions for ground-state properties of several closed-shell nuclei in SU(6) symmetry. Experimental data for the binding energy per nucleon, $B/A$, and charge radius, $R_{\mathrm{ch}}$, refer to Wang et al. [81] and Angeli and Marinova [82], respectively. The neutron skin thickness, $R_{\mathrm{skin}}$, is defined as the difference between the root-mean-square radii of point neutrons and protons in a nucleus. As explained in Refs. [66,67], it is difficult to understand the PREX-2 and CREX results simultaneously in the present study [83,84].

Nucleus	$\mathit{B}/\mathit{A}$ (MeV)		${\mathit{R}}_{\mathbf{ch}}$ (fm)		${\mathit{R}}_{\mathbf{skin}}$ (fm)
	Theory	Exp.	Theory	Exp.
16O	7.98	7.98	2.74	2.70	$-0.03$
40Ca	8.55	8.55	3.47	3.48	$-0.05$
48Ca	8.48	8.67	3.50	3.48	0.20
68Ni	8.61	8.68	3.90	3.89	0.21
90Zr	8.62	8.71	4.29	4.27	0.09
132Sn	8.37	8.35	4.74	4.71	0.28
208Pb	7.86	7.87	5.54	5.50	0.21

and

Table 2. Coupling constants for N in SU(6) and SU(3) symmetry. Since the ${\sigma }^{\ast }$ meson is mainly composed of an $s\overline{s}$ pair, we assume that the ${\sigma }^{\ast }$ meson does not couple to N due to the OZI rule, namely $g_{{\sigma }^{\ast }N}=0$ [18]. In the limit of the ideal mixing, the $F/(F+D)$ ratio, mixing angle, and singlet-to-octet coupling ratio are, respectively, given by ${\alpha }_v^{id}=1.00$, ${\theta }_v^{id}={tan}^{-1}(1/\sqrt{2})\simeq 35.{26}^{\circ }$, and $z_v^{id}=1/\sqrt{6}\simeq 0.4082$. In SU(3) symmetry, we present the cases for ${\theta }_v={\theta }_v^{id}$ from the ideal mixing (case A) and ${\theta }_v=36.5^{\circ }$ from the Particle Data Group (PDG) (case B) [85]. The parameter $g_2$ is in fm⁻¹.

Sym.	Case	$({\mathit{\alpha }}_{\mathit{v}},{\mathit{\theta }}_{\mathit{v}},{\mathit{z}}_{\mathit{v}})$	${\mathit{g}}_{\mathit{\sigma }\mathit{N}}$	${\mathit{g}}_{\mathit{\omega }\mathit{N}}$	${\mathit{g}}_{\mathit{\delta }\mathit{N}}$	${\mathit{g}}_{\mathit{\rho }\mathit{N}}$	${\mathit{g}}_{\mathit{\varphi }\mathit{N}}$	${\mathit{g}}_2$	${\mathit{g}}_3$	${\mathit{c}}_3$	${\mathit{f}}_3$	${\mathbf{\Lambda }}_{\mathit{\omega }\mathit{\rho }}$	${\mathbf{\Lambda }}_{\mathit{\varphi }\mathit{\rho }}$
SU(6)		$({\alpha }_v^{id},{\theta }_v^{id},z_v^{id})$	8.36	10.77	6.71	7.45	—	11.89	$-18.91$	10.00	—	241.06	—
SU(3)	A	$({\alpha }_v^{id},{\theta }_v^{id},0.00)$	8.36	9.46	6.71	7.45	$-6.69$	11.89	$-18.91$	12.97	125.85	241.10	408.76
		$({\alpha }_v^{id},{\theta }_v^{id},0.25)$	8.36	10.63	6.71	7.45	$-2.23$	11.89	$-18.91$	10.27	1096.49	241.07	408.02
		$({\alpha }_v^{id},{\theta }_v^{id},0.50)$	8.36	10.74	6.71	7.45	1.06	11.89	$-18.91$	10.06	4637.76	241.06	407.25
	B	$(0.25,36.5^{\circ },0.15)$	8.36	9.36	6.71	7.45	$-6.93$	11.89	$-18.91$	13.24	117.40	241.10	408.76
		$(0.50,36.5^{\circ },0.15)$	8.36	9.74	6.71	7.45	$-5.98$	11.89	$-18.91$	12.25	156.82	241.10	408.66
		$(0.75,36.5^{\circ },0.15)$	8.36	10.05	6.71	7.45	$-5.05$	11.89	$-18.91$	11.50	220.33	241.09	408.61

. The bulk properties of nuclear matter at the saturation density, $n_0=0.15$ fm⁻³, are calculated as follows: the effective nucleon mass $M_N^{\ast }/M_N=0.70$, binding energy per nucleon $E_0(n_0)=-16.35$ MeV, nuclear incompressibility $K_0=240$ MeV, nuclear symmetry energy $E_{\mathrm{sym}}(n_0)=32.5$ MeV, slope parameter $L=50$ MeV, and curvature parameter $K_{\mathrm{sym}}=-210.67$ MeV. We emphasize that the $\delta$-N coupling constant, $g_{\delta N}$, is determined to reproduce the proton–neutron effective mass splitting, $M_p^{\ast }-M_n^{\ast }\simeq 100$ MeV, at $n_0$ [79,80]. Furthermore, the self-interaction of the $\omega$ meson is introduced to satisfy the constraints from elliptical flow data and kaon production data in heavy-ion collisions, particularly the pressure range of symmetric nuclear matter at high densities [59,60,61].

As a second step, we determine the coupling constants for N in SU(3) flavor symmetry. According to Equation (29), the $\varphi$ meson, in addition to the $\omega$ meson, influences the nuclear EoS in SU(3) symmetry. To preserve the same saturation properties—$E_0(n_0)$, $K_0$, $E_{\mathrm{sym}}(n_0)$, L, and $K_{\mathrm{sym}}$—as in the SU(6) framework, we adjust the vector-meson coupling constants, $g_{\omega N}$, $g_{\varphi N}$, $c_3$, and ${\Lambda }_{\omega \rho }$, by introducing two additional couplings, $f_3$ and ${\Lambda }_{\varphi \rho }$ (see Equation (3)). Several parameter sets in SU(3) symmetry are also listed in Table 2. When the quartic $\varphi$-meson self-interaction and $\varphi$-$\rho$ mixing terms are omitted, the extension to SU(3) flavor symmetry slightly disturbs the original SU(6) saturation conditions, particularly $K_0$ and L [53].

4.2. Coupling Constants for Hyperons

Using Equations (26)–(33), the hyperon coupling constants for the vector mesons, $g_{\omega Y}$, $g_{\rho Y}$, and $g_{\varphi Y}$, are automatically determined by ${\alpha }_v$, ${\theta }_v$, and $z_v$. On the other hand, those for the scalar mesons are fixed so as to satisfy the empirical data on potential depths. We here adopt the single-baryon potential based on the so-called Schrödinger-equivalent potential [86,87]:

$$U_B={\Sigma }_B^s-{\Sigma }_B^0+{\displaystyle \frac{1}{2M_B}}{\left({\Sigma }_B^s-{\Sigma }_B^0\right)}^2,$$

(34)

where the scalar (s) and time (0) components of baryon self-energy are written as

$$\begin{array}{cc}\hfill {\Sigma }_B^s& =-g_{\sigma B}\overline{\sigma }-g_{\delta B}\overline{\delta }{\left({\mathit{I}}_B\right)}_3-g_{{\sigma }^{\ast }B}{\overline{\sigma }}^{\ast },\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill {\Sigma }_B^0& =-g_{\omega B}\overline{\omega }-g_{\rho B}\overline{\rho }{\left({\mathit{I}}_B\right)}_3-g_{\varphi B}\overline{\varphi }.\hfill \end{array}$$

(36)

As for the $\sigma$-Y couplings, we employ the recently updated values of the potential depths in symmetric nuclear matter (SNM), $U_{\Lambda }^{(\mathrm{SNM})}(n_0)=-27.7$ MeV, $U_{\Sigma }^{(\mathrm{SNM})}(n_0)=+30$ MeV, and $U_{\Xi }^{(\mathrm{SNM})}(n_0)=-21$ MeV [88,89,90,91,92,93,94]. The ${\sigma }^{\ast }$-$\Lambda$ and ${\sigma }^{\ast }$-$\Xi$ coupling constants, $g_{{\sigma }^{\ast }\Lambda }$ and $g_{{\sigma }^{\ast }\Xi }$, are calculated using the conventional relations, $U_{\Xi }^{(\Xi )}(n_0)\simeq 2U_{\Lambda }^{(\Lambda )}(n_0/2)$, with the $\Lambda \Lambda$ potential from the Nagara event, $U_{\Lambda }^{(\Lambda )}(n_0/2)=-5$ MeV [17,95]. In addition, we use the potential depth in pure neutron matter (PNM), $U_{{\Xi }^{-}}^{(\mathrm{PNM})}=+6$ MeV, from the Lattice QCD result by the Hadrons to Atomic nuclei from Lattice QCD (HAL QCD) Collaboration to get $g_{\delta \Xi }$ because ${\Xi }^{-}$ plays an important role in supporting a massive neutron star [96,97,98]. The other couplings are given by the relations based on the quark model: $g_{\delta \Lambda }=0$, $g_{\delta \Sigma }=2g_{\delta N}$, and $g_{{\sigma }^{\ast }\Sigma }=g_{{\sigma }^{\ast }\Lambda }$. The coupling constants for Y in SU(6) and SU(3) symmetry are listed in

Table 3. Coupling constants for Y in SU(6) and SU(3) symmetry. We assume that $g_{\delta \Lambda }=0$, $g_{\delta \Sigma }=2g_{\delta N}$, and $g_{{\sigma }^{\ast }\Sigma }=g_{{\sigma }^{\ast }\Lambda }$. For detail, see the text.

Sym.	Case	$({\mathit{\alpha }}_{\mathit{v}},{\mathit{\theta }}_{\mathit{v}},{\mathit{z}}_{\mathit{v}})$	${\mathit{g}}_{\mathit{\sigma }\mathit{\Lambda }}$	${\mathit{g}}_{\mathit{\sigma }\mathit{\Sigma }}$	${\mathit{g}}_{\mathit{\sigma }\mathit{\Xi }}$	${\mathit{g}}_{\mathit{\delta }\mathit{\Xi }}$	${\mathit{g}}_{{\mathit{\sigma }}^{\ast }\mathit{\Lambda }}$	${\mathit{g}}_{{\mathit{\sigma }}^{\ast }\mathit{\Xi }}$
SU(6)		$({\alpha }_v^{id},{\theta }_v^{id},z_v^{id})$	5.12	3.41	2.77	7.07	$-3.74$	$-9.99$
SU(3)	A	$({\alpha }_v^{id},{\theta }_v^{id},0.00)$	7.27	5.55	7.06	5.88	— a	— a
		$({\alpha }_v^{id},{\theta }_v^{id},0.25)$	6.05	4.34	4.64	6.55	$-2.37$	$-7.50$
		$({\alpha }_v^{id},{\theta }_v^{id},0.50)$	4.64	2.92	1.80	7.45	$-4.02$	$-9.29$
	B	$(0.25,36.5^{\circ },0.15)$	7.07	5.75	6.87	$-1.09$	— a	— a
		$(0.50,36.5^{\circ },0.15)$	6.94	5.55	6.58	$-5.20$	— a	— a
		$(0.75,36.5^{\circ },0.15)$	6.79	5.27	6.20	5.13	— a	— a

^a Because the $\sigma$ meson contribution already gives $U_{\Lambda }^{(\Lambda )}(n_0/2)\le -5$ MeV, the additional attractive force due to the ${\sigma }^{\ast }$ meson is not required.

.

In the present RMF approach, the coupling constants are not arbitrarily chosen but are determined so as to reproduce the characteristics of finite, closed-shell nuclei and saturation properties of nuclear matter. Therefore, the uncertainties of these parameters are already strongly constrained by empirical nuclear data. Although a quantitative uncertainty analysis is beyond the scope of the present study, we have confirmed that a moderate variation in the key coupling constants does not alter qualitative conclusions regarding the equation of state and neutron star properties.

5. Neutron-Star Properties in SU(3) Flavor Symmetry

The charge neutrality and $\beta$ equilibrium conditions are generally imposed in the discussion of neutron-star properties. Since the neutron-star radii are remarkably sensitive to the low-density EoS that covers the crust region, we employ the realistic EoS for nonuniform matter, in which neutron-rich nuclei and neutron drips out of the nuclei are considered using the Thomas–Fermi calculation with the uniform nuclear EoS based on RHF approximation [99]. As for the crust–core phase transition, we consider the thermo-dynamical method, and it occurs at $n_B=0.089$ fm⁻³ in all cases [67], where the total baryon density is given by $n_B={\sum }_B{\rho }_B$.

The neutron-star EoSs and mass–radius relations of neutron stars are presented in Figure 1 and Figure 2, respectively. As is well known, the appearance of hyperons softens the EoS and thereby drastically reduces $M_{\max }$ in SU(6) symmetry. In the SU(3) flavor symmetry scheme, the smaller $z_v$ hardens the EoS and enhances $M_{\max }$, and the $2M_{\odot }$ limit can be sufficiently satisfied for ${\alpha }_v=0.00$ ($M_{\max }=2.10M_{\odot }$). Note that as we set $M_N^{\ast }/M_N=0.70$ at $n_0$, the neutron-star radius and dimensionless tidal deformability at the typical mass of a neutron star, $R_{1.4}$ and ${\Lambda }_{1.4}$, show relatively small values ($R_{1.4}=12.56$ km and ${\Lambda }_{1.4}=494$), and thus ${\Lambda }_{1.4}$ is consistent with the severe constraints from GW170817, ${\Lambda }_{1.4}={190}_{-120}^{+390}$ and ${\Lambda }_{1.4}=265.{18}_{-104.38}^{+237.88}$ [7,8,64].

The correlation between ${\theta }_v$ and $z_v$ in $M_{\max }$ is displayed in Figure 3. We here fix the ideal mixing for ${\alpha }_v$. It is clearly seen that ${\theta }_v$ shows less impact on $M_{\max }$, while $z_v$ strongly affects $M_{\max }$ and the smaller $z_v$ is favorable to support $2M_{\odot }$ neutron stars. When ${\alpha }_v$ is restricted by the realistic mixing angle, ${\alpha }_v^{id}=1.00$, the $2M_{\odot }$ neutron stars are supported for $z_v\le 0.15$, which is slightly smaller than the result from the Nijmegen extended-soft-core (ESC) model, $z_v^{\mathrm{ESC}}=0.195$ [100]. The irregular contour structures observed around the central band are attributed to the rapid onset of hyperons. This behavior arises because, near the SU(6) value, the coupling $g_{\varphi N}$ becomes small, and in order to reproduce the same saturation properties as in SU(6) symmetry, $f_3$ in Equation (3) must increase significantly (see Table 2). Consequently, the enhanced contribution of this term leads to the sudden change in hyperon production.

In Figure 4, we also show the correlation between ${\alpha }_v$ and $z_v$ in $M_{\max }$ with the fixed mixing angle from PDG, ${\theta }_v=36.5^{\circ }$ [85]. Similarly to the case of the fixed ${\alpha }_v$ shown in Figure 3, $z_v$ mainly governs the behavior of $M_{\max }$. Nevertheless, ${\alpha }_v$ still plays a qualitative role by setting the stiffness of the neutron-star EoS. The variation in ${\alpha }_v$ shifts the overall magnitude of $M_{\max }$, although it is empirically indicated as ${\alpha }_v=1$. Thus, $M_{\max }$ is primarily driven by $z_v$, while ${\alpha }_v$ provides a secondary but non-negligible adjustment to $2M_{\odot }$ neutron stars.

To investigate the composition of the neutron-star core, the chemical potential, ${\mu }_B=\sqrt{k_{F_B}^2+M_B^{\ast 2}}-{\Sigma }_B^0$, and partial fractions, $Y_i={\rho }_i/n_B$, are presented in Figure 5. As we consider the slightly deeper potentials for $\Xi$ in SNM and PNM, $U_{\Xi }^{(\mathrm{SNM})}(n_0)=-21$ MeV and $U_{{\Xi }^{-}}^{(\mathrm{PNM})}=+6$ MeV in the present study, ${\Xi }^{-}$ appears quickly compared with the previous work in Miyatsu et al. [53]. In the cases for $z_v=0.25$ and $0.50$, $\Lambda$ and ${\Xi }^{-}$ are generated at the almost same density and their components are enhanced at high densities. Ultimately, the softening of the neutron-star EoS gives rise to the well-known hyperon puzzle. In contrast, the smaller $z_v$ delays the $\Lambda$ onset, and then ${\mu }_n$ continues to increase monotonically at high densities. Consequently, it enables supporting $2M_{\odot }$ by preventing rapid hyperon production, although ${\Sigma }^{-}$ is likely to appear at lower densities through ${\mu }_{{\Sigma }^{-}}={\mu }_n+{\mu }_{e^{-}}$.

The effect of ${\alpha }_v$ on the hyperon generation is demonstrated in Figure 6, showing the cases for ${\alpha }_v=0.25$, $0.50$, and $0.75$ with ${\theta }_v=36.5^{\circ }$ and $z_v=0.15$. As already explained in Figure 5, the $2M_{\odot }$ constraint is satisfied in all cases by means of the small $z_v$. As ${\alpha }_v$ decreases, the ${\Sigma }^{-}$ onset occurs at low densities, and then ${\Sigma }^{-}$ appears below $2n_0$ for ${\alpha }_v=0.25$. It is interesting that with decreasing ${\alpha }_v$, the chemical potential splitting between ${\mu }_{{\Xi }^0}$ and ${\mu }_{{\Xi }^{-}}$ is enhanced, leading to a situation in which ${\Xi }^0$, rather than ${\Xi }^{-}$, is produced in the core of a neutron star.

The squared speed of sound in neutron-star matter is illustrated in Figure 7. We show two types of speed of sound: one is the adiabatic form, $c_s^2={\left.{\displaystyle \frac{\partial P}{\partial \epsilon }}\right|}_{Y_p}$, and the other is the equilibrium form, $c_e^2={\displaystyle \frac{dP}{d\epsilon }}$ [101,102,103]. Taking hyperons into account, the difference between $c_s^2$ and $c_e^2$ becomes more pronounced. As explained in Motta et al. [104] and Ye et al. [105], an abrupt decrease appears around $3n_0$, which corresponds to the threshold for the first hyperon (see Figure 5). Moreover, for $z_v\ge 0.25$, $c_e^2$ exhibits a sharp decline at high densities due to the rapid emergence of ${\Xi }^0$.

6. Summary and Conclusions

Under SU(3) flavor symmetry, we have studied the properties of neutron-star matter with hyperons using the RMF model. The model parameters are calibrated to reproduce finite-nucleus observables as well as bulk properties of dense nuclear matter. To simultaneously satisfy the astrophysical constraints on ${\Lambda }_{1.4}$ from GW170817 and the experimental data on the dense nuclear EoS from heavy-ion collisions, we have fixed a relatively large $M_N^{\ast }$ at $n_0$, $M_N^{\ast }/M_N=0.70$, and included a quartic self-interaction of the $\omega$ meson. The $\delta$ meson is incorporated to reproduce the proton–neutron effective mass splitting suggested by DBHF calculations. We further introduce the additional interactions, ${\left({\varphi }_{\mu }{\varphi }^{\mu }\right)}^2$ and $\left({\varphi }_{\mu }{\varphi }^{\mu }\right)\left({\mathit{\rho }}_{\nu }·{\mathit{\rho }}^{\nu }\right)$, so that SU(6)-based couplings can be retained within the SU(3) scheme. The hyperon–meson couplings are determined by applying SU(3) flavor symmetry to the vector mesons and by constraining the scalar couplings with empirical potential depths in matter.

We have analyzed how ${\alpha }_v$, ${\theta }_v$, and $z_v$ affect $M_{\max }$ in the SU(3)-invariant vector-meson couplings. It has been found that $z_v$ predominantly controls $M_{\max }$, and the $2M_{\odot }$ neutron stars are supported for $z_v\le 0.15$. The ${\alpha }_v$ also contributes significantly to sustaining large masses, whereas the variation in ${\theta }_v$ has a minor impact on neutron-star properties. Within the SU(3) framework, ${\Sigma }^{-}$ may become as important as, or even more significant than, $\Lambda$ and ${\Xi }^{-}$, as shown in Figure 6. This highlights the need for a more precise determination of $U_{\Sigma }^{(\mathrm{SNM})}$ through detailed analyses of hypernuclei. Overall, the SU(3)-based approach yields a consistent description of hyperonic neutron-star matter that reconciles nuclear data with astrophysical observations, offering a plausible resolution of the long-standing hyperon puzzle.