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

Background: Nuclear matter fourth-order symmetry energy $E_{\text{sym,4}}\left(\rho \right)$ 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 $E_{\text{sym,4}}\left(\rho \right)$ 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 $E_{\text{sym,4}}\left(\rho \right)$ at nuclear normal density ${\rho }_0$ and its density dependence, and explore the correlation between $E_{\text{sym,4}}\left({\rho }_0\right)$ 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 $E_{\text{sym,4}}\left(\rho \right)$ and analyze the correlation of $E_{\text{sym,4}}\left({\rho }_0\right)$ with other macroscopic quantities.

Results: The value of $E_{\text{sym,4}}\left({\rho }_0\right)$ is estimated to be $1.02\pm 0.49$ MeV for the SHF model, $1.02\pm 0.50$ MeV for the eSHF model, $0.70\pm 0.60$ MeV for the GHF model, and $0.74\pm 0.63$ MeV for the MDI model. Moreover, our results indicate that the density dependence of $E_{\text{sym,4}}\left(\rho \right)$ is model dependent, especially at higher densities. Furthermore, we find that the $E_{\text{sym},4}\left({\rho }_0\right)$ has strong positive (negative) correlation with isoscalar (isovector) nucleon effective mass $m_{s,0}^{*}$ ($m_{v,0}^{*}$) at ${\rho }_0$. In particular, for the SHF and eSHF models, the $E_{\text{sym,4}}\left(\rho \right)$ is completely determined by the isoscalar and isovector nucleon effective masses $m_s^{*}\left(\rho \right)$ and $m_v^{*}\left(\rho \right)$, and the analytical expression is given.

Conclusions: In the mean-field models, the magnitude of $E_{\text{sym,4}}\left({\rho }_0\right)$ is generally less than $2\mathrm{MeV}$, and its density dependence depends on models, especially at higher densities. $E_{\text{sym,4}}\left({\rho }_0\right)$ is strongly correlated with $m_{s,0}^{*}$ and $m_{v,0}^{*}$.

Figure 1

Histogram of the number of parameter sets as a function of the value of $E_{\text{sym,4}}\left({\rho }_0\right)$, from a sample of 0.1 million parameter sets for each model. The average values of $E_{\text{sym,4}}\left({\rho }_0\right)$ are also shown.

Figure 2

Density dependence of the fourth-order symmetry enrgy $E_{\text{sym,4}}\left(\rho \right)$. The black stars with error bars are the results in this work. For comparison, the predictions of some typical interaction parameter sets in the literature, i.e., SLy4 [68], SKa [69], MSL1 [50], BSk29 [38], eMSL07 [39], eMSL09 [39], MDI with $x=1,0,-1$ [46], D1 [40], D1S [53], and D1N [52], are also included for comparison.