Nuclear matter fourth-order symmetry energy in nonrelativistic mean-field models

Abstract

Background: Nuclear matter fourth-order symmetry energy ${E}_{\text{sym,4}}(\ensuremath{\rho})$ may significantly influence the properties of neutron stars such as the core-crust transition density and pressure as well as the proton fraction at high densities. The magnitude of ${E}_{\text{sym,4}}(\ensuremath{\rho})$ is, however, largely uncertain.Purpose: Based on systematic analyses of several popular nonrelativistic energy density functionals with mean-field approximation, we estimate the value of the ${E}_{\text{sym,4}}(\ensuremath{\rho})$ at nuclear normal density ${\ensuremath{\rho}}_{0}$ and its density dependence, and explore the correlation between ${E}_{\text{sym,4}}({\ensuremath{\rho}}_{0})$ and other macroscopic quantities of nuclear matter properties.Method: We use the empirical values of some nuclear macroscopic quantities to construct model parameter sets by the Monte Carlo method for four different energy density functionals with mean-field approximation, namely, the conventional Skyrme-Hartree-Fock (SHF) model, the extended Skyrme-Hartree-Fock (eSHF) model, the Gogny-Hartree-Fock (GHF) model, and the momentum-dependent interaction (MDI) model. With the constructed samples of parameter sets, we can estimate the density dependence of ${E}_{\text{sym,4}}(\ensuremath{\rho})$ and analyze the correlation of ${E}_{\text{sym,4}}({\ensuremath{\rho}}_{0})$ with other macroscopic quantities.Results: The value of ${E}_{\text{sym,4}}({\ensuremath{\rho}}_{0})$ is estimated to be $1.02\ifmmode\pm\else\textpm\fi{}0.49$ MeV for the SHF model, $1.02\ifmmode\pm\else\textpm\fi{}0.50$ MeV for the eSHF model, $0.70\ifmmode\pm\else\textpm\fi{}0.60$ MeV for the GHF model, and $0.74\ifmmode\pm\else\textpm\fi{}0.63$ MeV for the MDI model. Moreover, our results indicate that the density dependence of ${E}_{\text{sym,4}}(\ensuremath{\rho})$ is model dependent, especially at higher densities. Furthermore, we find that the ${E}_{\text{sym},4}({\ensuremath{\rho}}_{0})$ has strong positive (negative) correlation with isoscalar (isovector) nucleon effective mass ${m}_{s,0}^{*}$ (${m}_{v,0}^{*}$) at ${\ensuremath{\rho}}_{0}$. In particular, for the SHF and eSHF models, the ${E}_{\text{sym,4}}(\ensuremath{\rho})$ is completely determined by the isoscalar and isovector nucleon effective masses ${m}_{s}^{*}(\ensuremath{\rho})$ and ${m}_{v}^{*}(\ensuremath{\rho})$, and the analytical expression is given.Conclusions: In the mean-field models, the magnitude of ${E}_{\text{sym,4}}({\ensuremath{\rho}}_{0})$ is generally less than $2\phantom{\rule{0.16em}{0ex}}\mathrm{MeV}$, and its density dependence depends on models, especially at higher densities. ${E}_{\text{sym,4}}({\ensuremath{\rho}}_{0})$ is strongly correlated with ${m}_{s,0}^{*}$ and ${m}_{v,0}^{*}$.