Nuclear matter fourth-order symmetry energy in the relativistic mean field models

Within the nonlinear relativistic mean field model, we derive the analytical expression of the nuclear matter fourth-order symmetry energy $E_{sym,4}(\rho)$. Based on two accurately calibrated interactions FSUGold and IU-FSU, our results show that the value of $E_{sym,4}(\rho)$ at normal nuclear matter density $\rho_{0}$ is generally less than 1 MeV, confirming the empirical parabolic approximation to the equation of state for asymmetric nuclear matter at $\rho_{0}$. On the other hand, we find that the $E_{sym,4}(\rho)$ may become nonnegligible at high densities. Furthermore, the analytical form of the $E_{sym,4}(\rho)$ provides the possibility to study the higher-order effects on the isobaric incompressibility of asymmetric nuclear matter, i.e., $K_{sat}(\delta)=K_{0}+K_{sat,2}\delta ^{2}+K_{sat,4}\delta ^{4}+\mathcal{O}(\delta ^{6})$ where $\delta =(\rho_{n}-\rho_{p})/\rho $ is the isospin asymmetry, and we find that the value of $K_{sat,4}$ is generally small compared with that of the $K_{sat,2}$. In addition, we study the effects of the $E_{sym,4}(\rho)$ on the proton fraction $x_{p}$ and the core-crust transition density $\rho_{t}$ and pressure $P_{t}$ in neutron stars. Interestingly, we find that, compared with the results from the empirical parabolic approximation, including the $E_{sym,4}(\rho)$ contribution can significantly enhance the $x_{p}$ at high densities and strongly reduce the $\rho_{t}$ and $P_{t}$ in neutron stars, demonstrating that the widely used empirical parabolic approximation may cause large errors in determining the $x_{p}$ at high densities as well as the $\rho_{t}$ and $P_{t}$ in neutron stars within the nonlinear relativistic mean field model, consistent with previous nonrelativistic calculations.