Chiral restoration of nucleons in neutron star matter: studies based on a parity doublet model

We review the chiral variant and invariant components of nucleon masses and its consequence on the chiral restoration in extreme conditions, neutron star matter in particular. We consider a model of linear realization of chiral symmetry with the nucleon parity doublet structure that permits the chiral invariant mass, $m_0$, for positive and negative parity nucleons. Nuclear matter is constructed with the parity doublet nucleon model coupled to scalar fields $\sigma$, vector fields $(\omega, \rho)$, and to mesons with strangeness through the U(1)$_A$ anomaly. In models with large $m_0$, the nucleon mass is insensitive to the medium, and the nuclear saturation properties can be reproduced without demanding strong couplings of nucleons to scalar fields $\sigma$ and vector fields $\omega$. We confront the resulting nuclear equations of state with nuclear constraints and neutron star observations, and delineate the chiral invariant mass and effective interactions. To further examine nuclear equations of state beyond the saturation density, we supplement quark models to set the boundary conditions from the high density side. The quark models are constrained by the two-solar mass conditions, and such constraints are transferred to nuclear models through the causality and thermodynamic stability conditions. We also calculate various condensates and matter composition from nuclear to quark matter in a unified matter, by constructing a generating functional that interpolates nuclear and quark matter with external fields. Two types of chiral restoration are discussed; the one due to the positive scalar charges of nucleons, and the other triggered by the evolution of the Dirac sea. We found the U(1)$_A$ anomaly softens equations of state from low to high density.