Single-nucleon potential decomposition of the nuclear symmetry energy

The nuclear symmetry energy $E_{\mathrm{sym}}\left(\rho \right)$ and its density slope $L\left(\rho \right)$ can be decomposed analytically in terms of the single-nucleon potential in isospin asymmetric nuclear matter. Using three popular nuclear effective interaction models which have been extensively used in nuclear structure and reaction studies, namely, the isospin and momentum-dependent MDI interaction model, the Skyrme-Hartree-Fock approach, and the Gogny-Hartree-Fock approach, we analyze the contribution of different terms in the single-nucleon potential to $E_{\mathrm{sym}}\left(\rho \right)$ and $L\left(\rho \right)$. Our results show that the observed different density behaviors of $E_{\mathrm{sym}}\left(\rho \right)$ for different interactions are essentially due to the variation of the symmetry potential $U_{\mathrm{sym},1}(\rho ,k)$. Furthermore, we find that the contribution of the second-order symmetry potential $U_{\mathrm{sym},2}(\rho ,k)$ to $L\left(\rho \right)$ generally cannot be neglected. Moreover, our results demonstrate that the magnitude of $U_{\mathrm{sym},2}(\rho ,k)$ is generally comparable with that of $U_{\mathrm{sym},1}(\rho ,k)$, indicating that the second-order symmetry potential $U_{\mathrm{sym},2}(\rho ,k)$ may have significant corrections to the widely used Lane approximation for the single-nucleon potential in extremely neutron rich or proton rich nuclear matter.