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Relationship between the symmetry energy and the single-nucleon potential in isospin-asymmetric nucleonic matter

In this contribution, we review the most important physics presented originally in our recent publications. Some new analyses, insights and perspectives are also provided. We showed recently that the symmetry energy $E_{sym}(ρ)$ and its density slope $L(ρ)$ at an arbitrary density $ρ$ can be expressed analytically in terms of the magnitude and momentum dependence of the single-nucleon potentials using the Hugenholtz-Van Hove (HVH) theorem. These relationships provide new insights about the fundamental physics governing the density dependence of nuclear symmetry energy. Using the isospin and momentum (k) dependent MDI interaction as an example, the contribution of different terms in the single-nucleon potential to the $E_{sym}(ρ)$ and $L(ρ)$ are analyzed in detail at different densities. It is shown that the behavior of $E_{sym}(ρ)$ is mainly determined by the first-order symmetry potential $U_{sym,1}(ρ,k)$ of the single-nucleon potential. The density slope $L(ρ)$ depends not only on the first-order symmetry potential $U_{sym,1}(ρ,k)$ but also the second-order one $U_{sym,2}(ρ,k)$. Both the $U_{sym,1}(ρ,k)$ and $U_{sym,2}(ρ,k)$ at normal density $ρ_0$ are constrained by the isospin and momentum dependent nucleon optical potential extracted from the available nucleon-nucleus scattering data. The $U_{sym,2}(ρ,k)$ especially at high density and momentum affects significantly the $L(ρ)$, but it is theoretically poorly understood and currently there is almost no experimental constraints known.