Neutron star matter based on a parity doublet model including the $a_0\left(980\right)$ meson

We study the effect of the isovector-scalar meson $a_0$(980) on the properties of nuclear matter and the neutron star (NS) matter by constructing a parity doublet model with including the $a_0$ meson based on the chiral SU(2)${}_L\times {\text{SU}\left(2\right)}_R$ symmetry. We also include the $\omega -\rho$ mixing contribution to adjust the slope parameter at the saturation. We find that, when the chiral invariant mass of nucleon $m_0$ is smaller than about 800 MeV, the existence of $a_0$(980) enlarges the symmetry energy by strengthening the repulsive $\rho$ meson coupling. On the other hand, for large $m_0$ where the Yukawa coupling of $a_0$(980) to nucleon is small, the symmetry energy is reduced by the effect of $\omega -\rho$ mixing. We then construct the equation of state (EoS) of a neutron star matter to obtain the mass-radius relation of NS. We find that, in most choices of $m_0$, the existence of $a_0$(980) stiffens the EoS and makes the radius of NS larger. We then constrain the chiral invariant mass of nucleon from the observational data of NS, and find that $580\text{MeV}\lesssim m_0\lesssim 860\text{MeV}$ for $L_0=57.7$ MeV.

Figure 1

Strength of the Yukawa couplings of $a_0$ meson, $|g_{a_{0}NN}|=|g_{\sigma NN}|$, as a function of the chiral invariant mass $m_0$.

Figure 2

Symmetry energy $S\left(n_B\right)$ for $m_0=500$–900 MeV and $L_0=40$–80 MeV. Solid curves represent the $S\left(n_B\right)$ of the model including $a_0\left(980\right)$ with ${\lambda }_6^{\prime }=0$, while dashed curves show the results of the model without $a_0\left(980\right)$.

Figure 3

Density dependence of $S_k\left(n_B\right)$ for $m_0=500–900$ MeV and $L_0=57.7$ MeV.

Figure 4

Density dependence of $S_{a_0}\left(n_B\right)$ for $m_0=500–900$ MeV and $L_0=57.7$ MeV.

Figure 5

Density dependence of $\frac{\partial m_{+n}^{*}}{\partial n_I}{|}_{n_I=0}$ for $m_0=500$–900 MeV and $L_0=57.7$ MeV.

Figure 6

Density dependence of $S_{\rho }\left(n_B\right)$ for $m_0=500–900$ MeV and $L_0=57.7$ MeV.

Figure 7

Relation between $g_{\rho }$ and $|g_{a_{0}NN}|$ for $L_0=57.7$ MeV.

Figure 8

Density dependence of the symmetry energy $S\left(n_B\right)$ of the model including $a_0\left(980\right)$ meson for several choices of $m_0$ with $L_0=40$–80 MeV. Solid, dash-dotted, and dotted curves show the $S\left(n_B\right)$ with ${\lambda }_6^{\prime }=0,{\lambda }_6$, and $-{\lambda }_6$, respectively.

Figure 9

Comparison of $S\left(2n_0\right)$ with other models: Refs. [54] (brown crosses), [67] (pink), [68] (grey), and [69] (cyan).

Figure 10

Comparison of pressure of the NSM, $P_{\mathrm{NSM}}\left(n_B\right)$ with $a_0$(980) (solid curves) and without $a_0\left(980\right)$ (dashed curves) for several values of $m_0$ with $L_0=57.7$ MeV. Dash-dotted and dotted curves show the EoS with ${\lambda }_6^{\prime }={\lambda }_6$ and $-{\lambda }_6$, respectively.

Figure 11

Allowed range of NJL parameters ($H/G,g_v/G$) for several choices of $m_0$ with $L_0=57.7$ MeV, ${\lambda }_6^{\prime }=0$. $\blacksquare$ shows the allowed parameters for the model with $a_0\left(980\right)$, while $\blacksquare$ shows the allowed parameters for the model without $a_0\left(980\right)$. represents the parameters that are allowed for both of the models. The color indicates the maximum mass of the corresponding neutron star in the unit of solar mass.

Figure 12

Examples of interpolated pressure of neutron star matter with $a_0$(980) for $m_0=700$ MeV, $L_0=57.7$ MeV, and ${\lambda }_6^{\prime }=0$.

Figure 13

Speed of sound $c_s$ corresponding to the interpolated pressure in Fig. 12.

Figure 14

$M-R$ relations computed from models with and without the existence of $a_0\left(980\right)$ for different $m_0$ with $L_0=57.7$ MeV. Solid curves represent the $M-R$ relations from the model with the $a_0\left(980\right)$ meson and dashed curves the ones of the model without $a_0\left(980\right)$. Parameters ($H/G,g_v/G$) of the NJL-type quark model are chosen as to be the same for a specific $m_0$, (1.55,1.3) for $m_0=600$ MeV, (1.6,1.3) for $m_0=700$ MeV, and (1.55,1) for $m_0=800$ MeV, respectively.

Figure 15

$M-R$ relations computed from models including $a_0$(980) with different choices of ${\lambda }_6^{\prime }$. Solid curves represent the results for ${\lambda }_6^{\prime }=0$, dashed-dotted curves for ${\lambda }_6^{\prime }=+{\lambda }_6$, and dotted curves for ${\lambda }_6^{\prime }=-{\lambda }_6$. NJL parameters ($H/G,g_v/G$) are chosen as (1.55,1.3) for $m_0=600$ MeV, (1.6,1.3) for $m_0=700$ MeV, and (1.55,1) for $m_0=800$ MeV. The slope parameter is fixed as $L_0=57.7$ MeV.