Relativistic mean field

In this paper, we employ the framework of RMF models to extract information on the symmetry energy in symmetric nuclear matter (SNM) systems. RMF models provide a comprehensive description of both nuclear matter and finite nuclei and are widely used to study neutron star properties. To better categorize the parametrizations associated with RMF models, Ref. [53] defined three distinct types: (i) nonlinear, (ii) density-dependent, and (iii) point-coupling models.

The Lagrangians for the different RMF models are

1. Nonlinear (NL) model $\begin{array}{c}{\mathcal{L}}_{\text{NL}}=\overline{\text{Ψ}}\left[i{\gamma }_{\mu }{\partial }^{\mu }-m_{\text{N}}\right]\text{Ψ}+\frac{1}{2}\left({\partial }_{\mu }\sigma {\partial }^{\mu }\sigma -m_{\sigma }^2{\sigma }^2\right)-U(\sigma )\\ -\frac{1}{4}{\omega }_{\mu \nu }{\omega }^{\mu \nu }+\frac{1}{2}{m}_{\omega }^2{\omega }_{\mu }{\omega }^{\mu }+\frac{1}{4}{\zeta }^4{\left({\omega }_{\mu }{\omega }^{\mu }\right)}^2\\ -\frac{1}{4}{\rho }_{\mu \nu }{\rho }^{\mu \nu }+\frac{1}{2}{m}_{\rho }^2{\rho }_{\mu }{\rho }^{\mu }+\frac{1}{2}\left({\partial }_{\mu }\delta {\partial }^{\mu }\delta -m_{\delta }^2{\delta }^2\right)\\ +g_{\sigma \text{NN}}\overline{\text{Ψ}}\text{Ψ}\sigma -g_{\omega \text{NN}}\overline{\text{Ψ}}{\gamma }_{\mu }\text{Ψ}{\omega }^{\mu }\\ -g_{\rho \text{NN}}\overline{\text{Ψ}}{\gamma }_{\mu }\tau \cdot \text{Ψ}{\rho }^{\mu }+g_{\delta \text{NN}}\overline{\text{Ψ}}\tau \cdot \text{Ψ}\delta \\ +g_{\sigma }{g}_{\omega }^2\sigma {\omega }_{\mu }{\omega }^{\mu }\left({\alpha }_1+\frac{1}{2}{\alpha }_1^{\text{'}}{g}_{\sigma }\right)\\ +g_{\sigma }{g}_{\rho }^2\sigma {\rho }_{\mu }{\rho }^{\mu }\left({\alpha }_2+\frac{1}{2}{\alpha }_2^{\text{'}}{g}_{\sigma }\right)\\ +\frac{1}{2}{\alpha }_3^{\text{'}}{g}_{\omega }^2{g}_{\rho }^2{\omega }_{\mu }{\omega }^{\mu }{\rho }_{\mu }{\rho }^{\mu }\mathrm{,}\end{array}$ (1) 2. Density-dependence (DD) model $\begin{array}{c}{\mathcal{L}}_{\text{DD}}=\overline{\text{Ψ}}\left[i{\gamma }_{\mu }{\partial }^{\mu }-m_{\text{N}}\right]\text{Ψ}+\frac{1}{2}\left({\partial }_{\mu }\sigma {\partial }^{\mu }\sigma -m_{\sigma }^2{\sigma }^2\right)\\ -\frac{1}{4}{\omega }_{\mu \nu }{\omega }^{\mu \nu }+\frac{1}{2}{m}_{\omega }^2{\omega }_{\mu }{\omega }^{\mu }-\frac{1}{4}{\rho }_{\mu \nu }{\rho }^{\mu \nu }+\frac{1}{2}{m}_{\rho }^2{\rho }_{\mu }{\rho }^{\mu }\\ +\frac{1}{2}\left({\partial }_{\mu }\delta {\partial }^{\mu }\delta -m_{\delta }^2{\delta }^2\right)+{\text{Γ}}_{\sigma }\left(\rho \right)\overline{\text{Ψ}}\text{Ψ}\sigma -{\text{Γ}}_{\omega }\left(\rho \right)\overline{\text{Ψ}}{\gamma }_{\mu }\text{Ψ}{\omega }^{\mu }\\ -{\text{Γ}}_{\rho }\left(\rho \right)\overline{\text{Ψ}}{\gamma }_{\mu }\tau \cdot \text{Ψ}{\rho }^{\mu }+{\text{Γ}}_{\delta }\left(\rho \right)\overline{\text{Ψ}}\tau \cdot \text{Ψ}\delta \mathrm{,}\end{array}$ (2) 3. Point-coupling (PC) model $\begin{array}{c}{\mathcal{L}}_{\text{PC}}=\overline{\text{Ψ}}\left[i{\gamma }_{\mu }{\partial }^{\mu }-m_{\text{N}}\right]\text{Ψ}-\frac{{\alpha }_{\text{S}}}{2}{\left(\overline{\text{Ψ}}\text{Ψ}\right)}^2\\ -\frac{{\alpha }_{\text{V}}}{2}\left(\overline{\text{Ψ}}{\gamma }_{\mu }\text{Ψ}\right)\left(\overline{\text{Ψ}}{\gamma }^{\mu }\text{Ψ}\right)-\frac{{\alpha }_{\text{TV}}}{2}\left(\overline{\text{Ψ}}{\gamma }_{\mu }\tau \text{Ψ}\right)\cdot \left(\overline{\text{Ψ}}{\gamma }^{\mu }\tau \text{Ψ}\right)\\ -\frac{{\alpha }_{\text{TS}}}{2}\left(\overline{\text{Ψ}}\tau \text{Ψ}\right)\cdot \left(\overline{\text{Ψ}}\tau \text{Ψ}\right)\\ -\frac{{\beta }_{\text{S}}}{3}{\left(\overline{\text{Ψ}}\text{Ψ}\right)}^3-\frac{{\gamma }_{\text{S}}}{4}{\left(\overline{\text{Ψ}}\text{Ψ}\right)}^4-\frac{{\gamma }_{\text{V}}}{4}{\left(\overline{\text{Ψ}}{\gamma }_{\mu }\text{Ψ}\overline{\text{Ψ}}{\gamma }^{\mu }\text{Ψ}\right)}^2\\ -\frac{{\alpha }_{\text{TV}}}{4}{\left(\overline{\text{Ψ}}{\gamma }_{\mu }\tau \text{Ψ}\cdot \overline{\text{Ψ}}{\gamma }^{\mu }\tau \text{Ψ}\right)}^2\\ +\left[{\eta }_1+{\eta }_2\left(\overline{\text{Ψ}}\text{Ψ}\right)\right]\left(\overline{\text{Ψ}}\text{Ψ}\right)\left(\overline{\text{Ψ}}{\gamma }_{\mu }\text{Ψ}\right)\left(\overline{\text{Ψ}}{\gamma }^{\mu }\text{Ψ}\right)\\ -{\eta }_3\left(\overline{\text{Ψ}}\text{Ψ}\right)\left(\overline{\text{Ψ}}{\gamma }_{\mu }\tau \text{Ψ}\right)\cdot \left(\overline{\text{Ψ}}{\gamma }^{\mu }\tau \text{Ψ}\right)\mathrm{,}\end{array}$ (3) where ωμν and ${\rho }_{\mu \nu }$ are defined by ${\partial }_{\mu }{\omega }_{\nu }-{\partial }_{\nu }{\omega }_{\mu }$ and ${\partial }_{\mu }{\rho }_{\nu }-{\partial }_{\nu }{\rho }_{\mu }$, respectively. In Eq. 1, $U(\sigma )=\frac{1}{3}{g}_2{\sigma }^3+\frac{1}{4}{g}_3{\sigma }^4$ is the nonlinear potential of the σ field.

As an example, we use the nonlinear RMF model to present the expressions of relevant quantities [51-53]. In the RMF model, meson fields can be replaced with their expectation values: $\begin{array}{l}\sigma \to <\sigma >=\overline{\sigma }\mathrm{,}{\omega }^{\mu }\to <{\omega }^{\mu }>={\overline{\omega }}^0\\ {\rho }^{\mu }\to <{\rho }^{\mu }>={\overline{\rho }}_3^0\mathrm{,}{\delta }^{\mu }\to <{\delta }^{\mu }>={\overline{\delta }}_3^0\end{array}$ (4) The equation of motions (EOMs) for nucleons and mesons were derived from the Lagrangian density: $\left[{\gamma }^{\mu }i{\partial }_{\mu }-g_{\omega }{\overline{\omega }}^0-g_{\rho }{\overline{\rho }}_3^0{\tau }_3-\left(m_{\text{N}}-g_{\sigma }\overline{\sigma }-g_{\delta }{\overline{\delta }}_3{\tau }_3\right)\right]\text{Ψ}=0$ (5) $\begin{array}{c}{m}_{\sigma }^2\overline{\sigma }=g_{\sigma }{\rho }_{\text{s}}-g_2{\overline{\sigma }}^2-g_3{\overline{\sigma }}^3+g_{\sigma }{g}_{\omega }^2{\left({\overline{\omega }}^0\right)}^2\left({\alpha }_1+{\alpha }_1^{\text{'}}{g}_{\sigma }\overline{\sigma }\right)\\ +g_{\sigma }{g}_{\rho }^2{\left({\overline{\rho }}_3^0\right)}^2\left({\alpha }_2+{\alpha }_2^{\text{'}}{g}_{\sigma }\overline{\sigma }\right)\end{array}$ (6) $\begin{array}{c}{m}_{\omega }^2{\overline{\omega }}^0=g_{\omega }\rho -\zeta g_{\omega }^4{\left({\overline{\omega }}^0\right)}^3-g_{\sigma }{g}_{\omega }^2\overline{\sigma }{\overline{\omega }}^0\left(2{\alpha }_1+{\alpha }_1^{\text{'}}{g}_{\sigma }\overline{\sigma }\right)\\ -{\alpha }_3^{\text{'}}{g}_{\omega }^2{g}_{\rho }^2{\left({\overline{\rho }}_3^0\right)}^2{\overline{\omega }}^0\end{array}$ (7) $\begin{array}{c}{m}_{\rho }^2{\overline{\rho }}_3^0=g_{\rho }{\rho }_3-g_{\sigma }{g}_{\rho }^2\overline{\sigma }{\overline{\rho }}_3^0\left(2{\alpha }_2+{\alpha }_2^{\text{'}}{g}_{\sigma }\overline{\sigma }\right)\\ -{\alpha }_3^{\text{'}}{g}_{\omega }^2{g}_{\rho }^2{\overline{\rho }}_3^0{\left({\overline{\omega }}^0\right)}^2\end{array}$ (8) $m_{\delta }^2{\overline{\delta }}_3=g_{\delta }{\rho }_{\text{s3}}\mathrm{,}$ (9) where the different densities are defined as ${\rho }_{\text{s}}=\langle \overline{\text{Ψ}}\text{Ψ}\rangle ={\rho }_{\text{sn}}+{\rho }_{\text{sp}}\mathrm{,}$ (10) $\rho =\langle \overline{\text{Ψ}}{\gamma }^0\text{Ψ}\rangle ={\rho }_{\text{n}}+{\rho }_{\text{p}}\mathrm{,}$ (11) ${\rho }_{\text{s3}}=\langle \overline{\text{Ψ}}{\tau }_3\text{Ψ}\rangle ={\rho }_{\text{sp}}-{\rho }_{\text{sn}}\mathrm{,}$ (12) ${\rho }_3=\langle \overline{\text{Ψ}}{\gamma }^0{\tau }_3\text{Ψ}\rangle ={\rho }_{\text{p}}-{\rho }_{\text{n}}\mathrm{,}$ (13) Filling up to the Fermi momenta $k_{\text{F}\mathrm{,}i}$ for i = n or p in the nuclear matter, the neutron (n) and proton (p) scalar and baryon densities are given by $\begin{array}{c}{\rho }_{\text{s}i}=\frac{C(i)}{{\left(2\pi \right)}^3}{\displaystyle {\int }_{k<k_{\text{F}i}}}{d}^{3}k\frac{m_i^{*}}{\sqrt{k^2+m_i^{*2}}}\\ =\frac{m_i^{*}}{2{\pi }^2}\left[k_{\text{F}i}{E}_{\text{F}i}^{*}-m_i^{*2}\text{ln}\frac{k_{\text{F}i}+E_{\text{F}i}^{*}}{m_{\text{i}}^{*2}}\right]\mathrm{,}\end{array}$ (14) ${\rho }_i=\frac{C(i)}{{\left(2\pi \right)}^3}{\displaystyle {\int }_{k<k_{\text{F}i}}}{d}^{3}k=\frac{{\left(k_{\text{F}i}\right)}^3}{3{\pi }^2}\mathrm{,}$ (15) where the degeneracy factor C(i=n,p)=2, and $E_{\text{F}i}^{*}=\sqrt{k_{\text{F}i}^2+m_i^{*2}}$ is the Fermi energy of neutrons and protons. Here, $m_i^{*}$ is the Dirac effective mass of nucleons, given by the relation $m_{\text{n}}^{*}=m_{\text{N}}-g_{\sigma }\overline{\sigma }+g_{\delta }{\overline{\delta }}_3\mathrm{,}$ (16) $m_{\text{p}}^{*}=m_{\text{N}}-g_{\sigma }\overline{\sigma }-g_{\delta }{\overline{\delta }}_3\mathrm{.}$ (17) The eigenvalues of the neutrons and protons from the Dirac equation are $e_{\text{n}}=g_{\omega }{\overline{\omega }}^0-g_{\rho }{\overline{\rho }}_3^0+\sqrt{k_{\text{n}}^{2*}+m_{\text{n}}^{*2}}\mathrm{,}$ (18) $e_{\text{p}}=g_{\omega }{\overline{\omega }}^0+g_{\rho }{\overline{\rho }}_3^0+\sqrt{k_{\text{p}}^{2*}+m_{\text{p}}^{*2}}\mathrm{.}$ (19) The expressions for the energy density and pressure are obtained from the given Lagrangian using the energy-momentum tensor relation given by $T^{\mu \nu }={\displaystyle \sum _i\frac{\partial \mathcal{L}}{\partial \left({\partial }_{\mu }{\varphi }_i\right)}}{\partial }_{\nu }{\varphi }_i-g^{\mu \nu }\mathcal{L}\mathrm{,}$ (20) where ϕi runs over all the possible fields. The energy density ϵ and pressure P can be obtained from the energy-momentum tensor: $\begin{array}{c}{ϵ}_{\text{NL}}=\langle T^{00}\rangle \\ =\frac{1}{2}{m}_{\sigma }^2{\overline{\sigma }}^2+\frac{1}{3}{g}_2{\overline{\sigma }}^3+\frac{1}{4}{g}_3{\overline{\sigma }}^4-\frac{1}{2}{m}_{\omega }^2{\left({\overline{\omega }}^0\right)}^2\\ -\frac{\zeta }{4}{g}_{\omega }^4{\left({\overline{\omega }}^0\right)}^4+g_{\omega }{\overline{\omega }}^0\rho -\frac{1}{2}{m}_{\rho }^2{\left({\overline{\rho }}_3^0\right)}^2+g_{\rho }{\overline{\rho }}_3^0{\rho }_3\\ +\frac{1}{2}{m}_{\delta }^2{\overline{\delta }}_3^2-g_{\sigma }{g}_{\omega }^2\overline{\sigma }{\left({\overline{\omega }}^0\right)}^2\left({\alpha }_1+\frac{1}{2}{\alpha }_1^{\text{'}}{g}_{\sigma }\overline{\sigma }\right)\\ -g_{\sigma }{g}_{\rho }^2\overline{\sigma }{\left({\overline{\rho }}_3^0\right)}^2\left({\alpha }_2+\frac{1}{2}{\alpha }_2^{\text{'}}{g}_{\sigma }\overline{\sigma }\right)-\frac{1}{2}{\alpha }_3^{\text{'}}{g}_{\omega }^2{g}_{\rho }^2{\left({\overline{\rho }}_3^0\right)}^2{\left({\overline{\omega }}^0\right)}^2\\ +\frac{1}{4}\left[3E_{\text{Fn}}^{*}{\rho }_{\text{n}}+m_{\text{n}}^{*}{\rho }_{\text{s n}}\right]+\frac{1}{4}\left[3E_{\text{F p}}^{*}{\rho }_p+m_{\text{p}}^{*}{\rho }_{\text{s p}}\right]\mathrm{,}\end{array}$ (21) and $\begin{array}{c}{P}_{\text{NL}}=\frac{1}{3}{\displaystyle \sum _{i=1}^3}\langle T^{ii}\rangle \\ =-\frac{1}{2}{m}_{\sigma }^2{\overline{\sigma }}^2-\frac{1}{3}{g}_2{\overline{\sigma }}^3-\frac{1}{4}{g}_3{\overline{\sigma }}^4\\ +\frac{1}{2}{m}_{\omega }^2{\left({\overline{\omega }}^0\right)}^2+\frac{\zeta }{4}{g}_{\omega }^4{\left({\overline{\omega }}^0\right)}^4+\frac{1}{2}{m}_{\rho }^2{\left({\overline{\rho }}_3^0\right)}^2\\ -\frac{1}{2}{m}_{\delta }^2{\overline{\delta }}_3^2+g_{\sigma }{g}_{\omega }^2\overline{\sigma }{\left({\overline{\omega }}^0\right)}^2\left({\alpha }_1+\frac{1}{2}{\alpha }_1^{\text{'}}{g}_{\sigma }\overline{\sigma }\right)\\ +g_{\sigma }{g}_{\rho }^2\overline{\sigma }{\left({\overline{\rho }}_3^0\right)}^2\left({\alpha }_2+\frac{1}{2}{\alpha }_2^{\text{'}}{g}_{\sigma }\overline{\sigma }\right)+\frac{1}{2}{\alpha }_3^{\text{'}}{g}_{\omega }^2{g}_{\rho }^2{\left({\overline{\rho }}_3^0\right)}^2{\left({\overline{\omega }}^0\right)}^2\\ +\frac{1}{4}\left[E_{\text{Fn}}^{*}{\rho }_{\text{n}}-m_{\text{n}}^{*}{\rho }_{\text{sn}}\right]+\frac{1}{4}\left[E_{\text{Fp}}^{*}{\rho }_{\text{p}}-m_{\text{p}}^{*}{\rho }_{\text{sp}}\right]\mathrm{.}\end{array}$ (22) The same calculations for the density-dependence and point-coupling models are given in Refs. [55-59].

The binding energy per particle in asymmetric nuclear matter can be expressed as follows: $E(\rho \mathrm{,}\alpha )=\frac{ϵ}{\rho }-m_{\text{N}}=E_0\left(\rho \right)+S(\rho ){\alpha }^2+O\left({\alpha }^4\right)\mathrm{,}$ (23) Here, the isoscalar term $E_0\left(\rho \right)=E(\rho \mathrm{,}\alpha =0)$ is the binding energy per nucleon in SNM, and the isovector term $S(\rho )$ is the symmetry energy. $\rho ={\rho }_{\text{n}}+{\rho }_{\text{p}}$ is the nuclear matter density, and $\alpha =\left({\rho }_{\text{n}}-{\rho }_{\text{p}}\right)/\left({\rho }_{\text{n}}+{\rho }_{\text{p}}\right)$ is the isospin asymmetry. The nuclear symmetry energy $S(\rho )$ is defined as $S(\rho )=\frac{1}{2}\frac{{\partial }^{2}E(\rho \mathrm{,}\alpha )}{\partial {\alpha }^2}{|}_{\alpha =0}\mathrm{.}$ (24) The symmetry energy is expanded in terms of $\left(\rho -{\rho }_0\right)/3{\rho }_0$ as $S(\rho )=J+\frac{L}{3{\rho }_0}\left(\rho -{\rho }_0\right)+\cdots ，$ (25) where $J=S\left({\rho }_0\right)$ is the symmetry energy, and $L=3{\rho }_0\frac{\partial S}{\partial \rho }{|}_{\rho ={\rho }_0}$ is the slope of the symmetry energy at the saturation density.

For SNM, $m_{\text{n}}^{*}=m_{\text{p}}^{*}=m_{\text{N}}^{*}$ because ${\overline{\delta }}_3$ vanishes. The symmetry energies of the RMF models are expressed as follows: $S{\left(\rho \right)}_{\text{NL}}=\frac{k_{\text{F}}^2}{6E_{\text{F}}^{*}}+\frac{1}{2}\rho \frac{g_{\rho }^2}{m_{\rho }^{*2}}-\frac{1}{2}\rho \left(\frac{\frac{g_{\delta }^2}{m_{\delta }^2}{m}_{\text{N}}^{*2}}{E_{\text{F}}^{*2}\left[1+\frac{g_{\delta }^2}{m_{\delta }^2}A\left(\rho \mathrm{,}{m}_{\text{N}}^{*}\right)\right]}\right)\mathrm{,}$ (26) $S{\left(\rho \right)}_{\text{DD}}=\frac{k_{\text{F}}^2}{6E_{\text{F}}^{*}}+\frac{1}{2}\rho \frac{{\text{Γ}}_{\rho }^2}{m_{\rho }^2}-\frac{1}{2}\rho \left(\frac{\frac{{\text{Γ}}_{\delta }^2}{m_{\delta }^2}{m}_{\text{N}}^{*2}}{E_{\text{F}}^{*2}\left[1+\frac{{\text{Γ}}_{\delta }^2}{m_{\delta }^2}A\left(\rho \mathrm{,}{m}_{\text{N}}^{*}\right)\right]}\right)\mathrm{,}$ (27) $\begin{array}{l}S{\left(\rho \right)}_{\text{PC}}=\frac{k_{\text{F}}^2}{6E_{\text{F}}^{*}}+\frac{1}{2}{\alpha }_V\rho +{\eta }_3{\rho }_{\text{s}}\rho \\ +\frac{1}{2}{\alpha }_{\text{TS}}\rho \left(\frac{m_{\text{N}}^{*2}}{E_{\text{F}}^{*2}\left[1-{\alpha }_{TS}A\left(\rho \mathrm{,}{m}_{\text{N}}^{*}\right)\right]}\right)\mathrm{,}\end{array}$ (28) where $m_{\rho }^{*2}=m_{\rho }^2+g_{\sigma }{g}_{\rho }^2\overline{\sigma }\left(2{\alpha }_2+{\alpha }_2^{\text{'}}{g}_{\sigma }\overline{\sigma }\right)+{\alpha }_3^{\text{'}}{g}_{\omega }^2{g}_{\rho }^2{\left({\overline{\omega }}^0\right)}^2$, and $A\left(\rho \mathrm{,}{m}_{\text{N}}^{*}\right)=3\left(\frac{{\rho }_s}{m_{\text{N}}^{*}}-\frac{\rho }{E_{\text{F}}^{*}}\right)\mathrm{.}$ (29) The expressions of the slope of symmetry energy (L) of the various RMF models are $\begin{array}{c}{L}_{\text{NL}}=\frac{k_{\text{F}}^2}{3E_{\text{F}}^{*}}\left(1-\frac{k_{\text{F}}^2}{2E_{\text{F}}^{*2}}-\frac{k_{\text{F}}^3{m}_{\text{N}}^{*}}{E_{\text{F}}^{*2}{\pi }^2}\frac{\partial m_{\text{N}}^{*}}{\partial \rho }\right)\\ +\frac{3g_{\rho }^2}{2m_{\rho }^{*2}}\rho \left(1-\frac{1}{m_{\rho }^{*2}}\frac{\partial m_{\rho }^{*2}}{\partial \rho }\rho \right)\\ -\frac{1}{2}\rho \left(\frac{\frac{g_{\delta }^2}{m_{\delta }^2}{m}_{\text{N}}^{*2}}{E_{\text{F}}^{*2}\left[1+\frac{g_{\delta }^2}{m_{\delta }^2}A\left(\rho \mathrm{,}{m}_{\text{N}}^{*}\right)\right]}\right)\\ \times \{3-\frac{2k_{\text{F}}^2}{E_{\text{F}}^{*2}}+6\left(1-\frac{m_{\text{N}}^{*2}}{E_{\text{F}}^{*2}}\right)\frac{\rho }{m_{\text{N}}^{*}}\frac{\partial m_{\text{N}}^{*}}{\partial \rho }\\ -3\frac{g_{\delta }^2}{m_{\delta }^2}\frac{1}{1+\frac{g_{\delta }^2}{m_{\delta }^2}A}[2A\left(\frac{\rho }{m_N^{*}}\frac{\partial m_N^{*}}{\partial \rho }\right)\\ +\rho \frac{k_{\text{F}}^2}{E_{\text{F}}^{*3}}\left(1-3\frac{\rho }{m_{\text{N}}^{*}}\frac{\partial m_{\text{N}}^{*}}{\partial \rho }\right)]\}\mathrm{,}\end{array}$ (30) $\begin{array}{c}{L}_{\text{DD}}=\frac{k_{\text{F}}^2}{3E_{\text{F}}^{*}}\left(1-\frac{k_{\text{F}}^2}{2E_{\text{F}}^{*2}}-\frac{k_{\text{F}}^3{m}_{\text{N}}^{*}}{E_{\text{F}}^{*2}{\pi }^2}\frac{\partial m_{\text{N}}^{*}}{\partial \rho }\right)\\ +\frac{3{\text{Γ}}_{\rho }^2}{2m_{\rho }^2}\rho \left(1+6\frac{\rho }{{\text{Γ}}_{\rho }}\frac{\partial {\text{Γ}}_{\rho }}{\partial \rho }\right)\\ -\frac{1}{2}\rho \left(\frac{\frac{{\text{Γ}}_{\delta }^2}{m_{\delta }^2}{m}_{\text{N}}^{*2}}{E_{\text{F}}^{*2}\left[1+\frac{{\text{Γ}}_{\delta }^2}{m_{\delta }^2}A(\rho \mathrm{,}{m}_{\text{N}}^{*})\right]}\right)\\ \times \{3+6\frac{\rho }{{\text{Γ}}_{\delta }}\frac{\partial {\text{Γ}}_{\delta }}{\partial \rho }-\frac{2k_{\text{F}}^2}{E_{\text{F}}^{*2}}+6\left(1-\frac{m_{\text{N}}^{*2}}{E_{\text{F}}^{*2}}\right)\frac{\rho }{m_{\text{N}}^{*}}\frac{\partial m_{\text{N}}^{*}}{\partial \rho }\\ -3\frac{{\text{Γ}}_{\delta }^2}{m_{\delta }^2}\frac{1}{1+\frac{{\text{Γ}}_{\delta }^2}{m_{\delta }^2}A}[2A\left(\frac{\rho }{{\text{Γ}}_{\delta }}\frac{\partial {\text{Γ}}_{\delta }}{\partial \rho }+\frac{\rho }{m_{\text{N}}^{*}}\frac{\partial m_{\text{N}}^{*}}{\partial \rho }\right)\\ +\rho \frac{k_{\text{F}}^2}{E_{\text{F}}^{*3}}\left(1-3\frac{\rho }{m_{\text{N}}^{*}}\frac{\partial m_{\text{N}}^{*}}{\partial \rho }\right)]\}\mathrm{,}\end{array}$ (31) $\begin{array}{c}{L}_{\text{PC}}=\frac{k_{\text{F}}^2}{3E_{\text{F}}^{*}}\left(1-\frac{k_{\text{F}}^2}{2E_{\text{F}}^{*2}}-\frac{k_{\text{F}}^3{m}_{\text{N}}^{*}}{E_{\text{F}}^{*2}{\pi }^2}\frac{\partial m_{\text{N}}^{*}}{\partial \rho }\right)\\ +\frac{3}{2}{\alpha }_{\text{V}}\rho +3{\eta }_3{\rho }_{\text{s}}\rho +3{\eta }_3{\rho }^2\frac{\partial {\rho }_{\text{s}}}{\partial \rho }\\ +\frac{1}{2}{\alpha }_{\text{TS}}\rho \left(\frac{m_{\text{N}}^{*2}}{E_{\text{F}}^{*2}\left[1-{\alpha }_{\text{TS}}A\left(\rho \mathrm{,}{m}_{\text{N}}^{*}\right)\right]}\right)\\ \times \{3-\frac{2k_{\text{F}}^2}{E_{\text{F}}^{*2}}+6\left(1-\frac{m_{\text{N}}^{*2}}{E_{\text{F}}^{*2}}\right)\frac{\rho }{m_{\text{N}}^{*}}\frac{\partial m_{\text{N}}^{*}}{\partial \rho }\\ +3{\alpha }_{\text{TS}}\frac{1}{1-{\alpha }_{\text{TS}}A}[2A\left(\frac{\rho }{m_{\text{N}}^{*}}\frac{\partial m_{\text{N}}^{*}}{\partial \rho }\right)\\ +\rho \frac{k_{\text{F}}^2}{E_{\text{F}}^{*3}}\left(1-3\frac{\rho }{m_{\text{N}}^{*}}\frac{\partial m_{\text{N}}^{*}}{\partial \rho }\right)]\}\mathrm{.}\end{array}$ (32) In this paper, we utilize 180 interaction parameter sets selected from those used in RMF models [53]. These parameter sets satisfy the incompressibility at saturation densities K₀ within the range of 200 to 300 A MeV by computing the distribution of isoscalar monopole strength in ²⁰⁸Pb with relativistic models [60]. The interaction parameter sets are listed below, and detailed information is available in Ref. [53]:

(i)165 nonlinear RMF models (E [61], ER [61], NL1 [51], NL3 [62], NL3-II [62], NL3^* [63], NL4 [64], NLC [54], NLB1 [51], NLB2 [51], NLRA1 [65], NLS [66], P-067 [67], P-070 [67], P-075 [67], P-080 [67], GL1 [68], GL2 [68], GL3 [68], GL4 [68], GL5 [68], GL6 [68], GL7 [68], GL8 [68], GL82 [69], GL9 [68], GM1 [70], GM2 [70], GM3 [70], GPSa [71], GPSb [71], NLρA [72], NLρB [72], RMF301 [73], RMF302 [73], RMF303 [73], RMF304 [73], RMF305 [73], RMF306 [73], RMF307 [73], RMF308 [73], RMF309 [73], RMF310 [73], RMF311 [73], RMF312 [73], RMF313 [73], RMF314 [73], RMF315 [73], RMF316 [73], RMF317 [73], RMF401 [73], RMF402 [73], RMF403 [73], RMF404 [73], RMF405 [73], RMF406 [73], RMF407 [73], RMF408 [73], RMF409 [73], RMF410 [73], RMF411 [73], RMF412 [73], RMF413 [73], RMF414 [73], RMF415 [73], RMF416 [73], RMF417 [73], RMF418 [73], RMF419 [73], RMF420 [73], RMF421 [73], RMF422 [73], RMF423 [73], RMF424 [73], RMF425 [73], RMF426 [73], RMF427 [73], RMF428 [73], RMF429 [73], RMF430 [73], RMF431 [73], RMF432 [73], RMF433 [73], RMF434 [73], Q1 [74], G1 [74], G2 [74], SMFT2 [75], DJM [75], S271 [76], Z271 [76], SRK3M5 [77], HD [78],HC [78], MS1 [79], MS3 [80], XS [80], NLSV1 [81], NLSV2 [81], TM1 [82], PK1 [83], hybrid [84], Z271^* [85], G2^* [85], BKA20 [86], BKA22 [86], BKA24 [86], FSUGOLD [9], FSUGOLD4 [87], FSUGOLD5 [87], FSUGZ00 [88], FSUGZ03 [88], FSUGZ06 [88], IU-FSU [89], NL3V1 [90], NL3V2 [90], NL3V3 [90], NL3V4 [90], NL3V5 [90], NL3V6 [90], S271V1 [90], S271V2 [90], S271V3 [90], S271V4 [90], S271V5 [90], S271V6 [90], Z271S1 [90], Z271S2 [90], Z271S3 [90], Z271S4 [90], Z271S5 [90], Z271S6 [90], Z271V1 [90], Z271V2 [90], Z271V3 [90], Z271V4 [90], Z271V5 [90], Z271V6 [90], TM1* [91], BSR1 [92], BSR2 [92], BSR3 [92], BSR4 [92], BSR5 [92], BSR6 [92], BSR7 [92], BSR8 [92], BSR9 [92], BSR10 [92], BSR11 [92], BSR12 [92], BSR13 [92], BSR14 [92], BSR15 [92], BSR16 [92], BSR17 [92], BSR18 [92], BSR19 [92], BSR20 [92], BSR21 [92], SVI-1 [93], SVI-2 [93], SIG-OM [94], NLρδ A [72], NLρδ B [72]);

(ii) Nine density-dependent RMF models (TW99 [95], DD-ME1 [96], PKDD [83], DD-ME2 [58], DD [97], DD-F [98], DD2 [99], DDMEδ [59], DDRHρδ [59]);

(iii)6 point-coupling RMF models (FA3 [57], FA4 [57], FZ3 [57], VZ3 [57], PC-F1 [55], PC-F3 [55]).

Equation of state of a neutron star

For different density regions of a neutron star, different forms of the EoS are employed in this paper as follows.

(i) For the outer crust of a neutron star, the EoS provided by Baym, Pethick, and Sutherland (BPS) [100] is adopted for densities in the range ${\rho }_{\text{min}}\le {\rho }_{\text{outer}}$. Here, ρ_min is the minimum density (ρ_min =4.73×10^-15 fm^-3), with the corresponding energy density ${ϵ}_{\text{min}}\text{=}4.38\times {10}^{\text{-}12}$ MeV, fm^-3 and pressure P_min=6.3×10^-25 MeV fm^-3. The outer crust density is ρ_outer=2.57×10^-4 fm^-3 with an energy density ${ϵ}_{\text{outer}}=0.24$ MeV fm^-3 and pressure P_outer=4.86×10^-4 MeV, fm^-3. In this region, because of the heavy nuclei, primarily around the iron mass number, a Coulomb lattice coexists in β-equilibrium (i.e., equilibrium with respect to weak interaction processes) with an electron gas in the neutron star outer crust [100].

(ii) For the inner crust, ${\rho }_{\text{outer}}<\rho \le {\rho }_{\text{T}}$, the EoS takes the form $P=A+B{ϵ}^{4/3}$ from Refs. [101, 102], where two constants A and B are adjusted to match the outer crust EoS to that of a liquid core at the crust-core transition density ρ_T (ρ_T=0.5ρ₀ [103]).

(iii) In the liquid core region, also referred to as the outer core (OC) region, the density range is ${\rho }_{\text{T}}<\rho \le {\rho }_{\text{OC}}$ (${\rho }_{OC}=3{\rho }_0$), where the EOS is obtained using the RMF. This region requires the system to be in β-equilibrium and is composed of protons, neutrons, electrons, and muons. For β equilibrium to be attained in nuclear matter, the following electroweak processes must occur: $\text{n}\to \text{p}+{\text{e}}^{-}+{\overline{\nu }}_{\text{e}}$, $\text{p}+{\text{e}}^{-}\to \text{n}+{\nu }_{\text{e}}$, ${\text{e}}^{-}\to {\mu }^{-}+{\nu }_{\text{e}}+{\overline{\nu }}_{\mu }$, $\text{p}+{\mu }^{-}\to \text{n}+{\nu }_{\mu }$, and $\text{n}\to \text{p}+{\mu }^{-}+{\nu }_{\mu }$. The β-equilibrated matter must satisfy the following conditions for the chemical potentials: ${\mu }_{\text{n}}-{\mu }_{\text{p}}={\mu }_{\text{e}}\mathrm{,}{\mu }_{\text{e}}={\mu }_{\mu }\mathrm{,}$ (33) where ${\mu }_{{\nu }_{\text{i}}}=0$, $i=\text{e}\mathrm{,}\mu$. At zero temperature, the neutron and proton chemical potentials are ${\mu }_{\text{n}}=e_{\text{n}}$ and ${\mu }_{\text{p}}=e_{\text{p}}$, respectively. For relativistic degenerate electrons, ${\mu }_{\text{e}}=\sqrt{m_{\text{e}}^2+k_{\text{Fe}}^2}=\sqrt{m_{\text{e}}^2+{\left(3{\pi }^2{x}_{\text{e}}\rho \right)}^{2/3}}，$ (34) ${\mu }_{\mu }=\sqrt{m_{\mu }^2+k_{\text{F}\mu }^2}=\sqrt{m_{\mu }^2+{\left(3{\pi }^2{x}_{\mu }\rho \right)}^{2/3}}\mathrm{,}$ (35) where m_e=0.511 MeV, and $m_{\mu }=0.105$ GeV. With the particle fraction $x_i={\rho }_i/\rho$ ($i=\text{n}\mathrm{,}\text{p}\mathrm{,}\text{e}\mathrm{,}\mu$), $x_{\text{p}}=x_{\text{e}}+x_{\mu }$ satisfies charge neutrality.

(iv) For the neutron star inner core (IC) region at a high density (ρ > ρ_OC), studies have incorporated exotic particles such as hyperons, Δs, quarks, and dark matter into the core of massive neutron stars at densities exceeding 3ρ₀ [104-115]. However, owing to the significant uncertainties in the interactions between nucleons and exotic particles (e.g., hyperons, Δs, and quarks), and the unclear composition of the neutron star core, a polytropic EoS is employed instead of explicitly modeling all possible exotic particles. In this paper, a piecewise polytropic EoS of the form $P_i\left(\rho \right)={\kappa }_i{\rho }^{{\gamma }_i}$ [116-118] is used to smoothly extend the EoS to the density region of the IC of the neutron star (ρ > ρ_OC). The polytropic EoS is constructed as follows: $P_i={\kappa }_i{\rho }^{{\gamma }_i}\mathrm{,}\text{with}{\kappa }_i\approx {\kappa }_{i-1}$ (36) ${\gamma }_i=\frac{\text{ln}\left(P_i/P_{i-1}\right)}{\text{ln}\left({\rho }_i/{\rho }_{i-1}\right)}\mathrm{,}{P}_i=P_{i-1}{\left(\frac{{\rho }_i}{{\rho }_{i-1}}\right)}^{{\gamma }_i}\mathrm{,}$ (37) ${ϵ}_i=\left({ϵ}_{i-1}-\frac{P_{i-1}}{{\gamma }_i-1}\right){\left(\frac{P_i}{P_{i-1}}\right)}^{1/{\gamma }_i}+\frac{P_i}{{\gamma }_i-1}\mathrm{,}$ (38) where the set of dividing densities ${\rho }_{\text{OC}}={\rho }_1<{\rho }_2<\cdots$, and ${\rho }_{i+1}-{\rho }_i=\text{Δ}\rho =0.05{\rho }_0$.

In summary, the EoS of neutron star matter is $P(ϵ)=\{\begin{array}{ll}{P}_{\text{BPS}}\left(ϵ\right)\hfill & \text{for }{\rho }_{\text{min}}\le \rho \le {\rho }_{\text{outer}}\hfill \\ A+B{ϵ}^{4/3}\hfill & \text{for }{\rho }_{\text{outer}}<\rho \le {\rho }_{\text{T}}\hfill \\ P_{\text{RMF}}\left(ϵ\right)\hfill & \text{for }{\rho }_{\text{T}}<\rho \le {\rho }_{\text{OC}}\hfill \\ P_{\text{IC}}\left(ϵ\right)\hfill & \text{for }\rho >{\rho }_{\text{OC}}\mathrm{,}\hfill \end{array}$ where $P=P_{\text{N}}+P_{\text{e}}+P_{\mu }$, $ϵ={ϵ}_{\text{N}}+{ϵ}_{\text{e}}+{ϵ}_{\mu }$, and the total pressure and energy density should include the leptons. The thermodynamically stable EoS must satisfy $\frac{\partial P}{\partial ϵ}\ge 0$. The adiabatic speed of sound can be expressed as $c_{\text{s}}^2={\left(\frac{v_{\text{s}}}{c}\right)}^2=\frac{\partial P}{\partial ϵ}<\mathrm{1,}$ (39) where ϵ is the energy density of the β-stable nuclear matter. For the causality condition, i.e., the speed of sound is always less than that of light v_s<c.

Tolman–Oppenheimer–Volkov equation

The structure of a neutron star is obtained by solving the TOV equation derived from General Relativity [119-121]. The TOV equations are $\frac{\text{d}P}{\text{d}r}=-\frac{GMϵ}{r^2}\frac{(1+P/ϵ)(1+4\pi r^{3}P/M)}{1-2GM/r}\mathrm{,}$ (40) $\frac{\text{d}M}{\text{d}r}=4\pi r^2ϵ\mathrm{,}$ (41) where G=6.707×10^-45 MeV^-2 is the gravitational constant, r is the distance from the core of the star, $P=P(r)$ is the pressure, and $M=M(r)$ is the mass with radius r.

The in-spiral phase of the two merging neutron stars creates strong tidal gravitational fields, resulting in the deformation of the multipolar structure of the star. The deforming effects are quantified through the tidal deformability parameter Λ, which relates the induced mass quadruple moment ${\mathcal{Q}}_{ij}$ to the time-independent external tidal field ${\mathcal{E}}_{ij}$ through the relation [122-124]: ${\mathcal{Q}}_{ij}=-k_2\frac{2R^5}{3G}{\mathcal{E}}_{ij}\mathrm{.}$ (42) Here, k₂ is the Love number, which can be obtained from the solution of the first-order differential equation [124] $\frac{\text{d}y}{\text{d}r}=-\frac{y^2}{r}-\frac{y}{r}\frac{\left(r-4\pi G{r}^3\left(ϵ-p\right)\right)}{r-2GM}-rQ\mathrm{,}$ (43) $\begin{array}{c}Q=\frac{4\pi r\left[G\left(5ϵ+9p+(ϵ+p)/c_s^2\right)-\frac{3}{2}\frac{1}{\pi r^2}\right]}{1-2GM/r}\\ -{\left[\frac{2G\left(M+4\pi p{r}^3\right)}{r(r-2GM)}\right]}^2\mathrm{,}\end{array}$ (44) $\begin{array}{c}{k}_2=\frac{8}{5}{\beta }^5{\left(1-2\beta \right)}^2\left[2-y_R+2\beta \left(y_R-1\right)\right]\\ \times \{2\beta \left[6-3y_R+3\beta \left(5y_R-8\right)\right]\\ +4{\beta }^3\left[13-11y_R+\beta \left(3y_R-2\right)+2{\beta }^2\left(1+y_R\right)\right]\\ {+3{\left(1-2\beta \right)}^2\left[2-y_R+2\beta \left(y_R-1\right)\right]ln(1-2\beta )\}}^{-1}\mathrm{,}\end{array}$ (45) where $y_R=y(R)$, and $\beta =GM/R$. Eqs. 40, 41, 43 are solved using the boundary conditions at the center of the star, $M(0)=0$, $P(0)=P_{\text{c}}$, and $y(0)=2$, where P_c is the central pressure. Varying P_c yields all possible stars for a given EoS. Thus, $P(R)=0$ (vacuum pressure being set to zero) defines the radius of star R and the total gravitational mass of the star is $M(R)$, which is simply denoted by M in the following.

The dimensionless deformability Λ is defined as follows: $\text{Λ}=\frac{2k_2}{3}{\left(\frac{R}{GM}\right)}^5\mathrm{.}$ (46)