$\delta$ meson effects on neutron stars in the modified quark-meson coupling model

The properties of neutron stars are investigated by including $\delta$ meson field in the Lagrangian density of modified quark-meson coupling model. The $\Sigma^-$ population with $\delta$ meson is larger than that without $\delta$ meson at the beginning, but it becomes smaller than that without $\delta$ meson as the appearance of $\Xi^-$. The $\delta$ meson has opposite effects on hadronic matter with or without hyperons: it softens the EOSes of hadronic matter with hyperons, while it stiffens the EOSes of pure nucleonic matter. Furthermore, the leptons and the hyperons have the similar influence on $\delta$ meson effects. The $\delta$ meson increases the maximum masses of neutron stars. The influence of $(\sigma^*,\phi)$ on the $\delta$ meson effects are also investigated.