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Constraining the neutron-proton effective mass splitting using empirical constraints on the density dependence of nuclear symmetry energy around normal density

According to the Hugenholtz-Van Hove theorem, nuclear symmetry energy \esym and its slope \lr at an arbitrary density $ρ$ are determined by the nucleon isovector (symmetry) potential \usym and its momentum dependence $\frac{\partial U_{sym}}{\partial k}$. The latter determines uniquely the neutron-proton effective k-mass splitting $m^*_{n-p}(ρ,δ)\equiv (m_{\rm n}^*-m_{\rm p}^*)/m$ in neutron-rich nucleonic matter of isospin asymmetry $δ$. Using currently available constraints on the \es0 and \l0 at normal density $ρ_0$ of nuclear matter from 28 recent analyses of various terrestrial nuclear laboratory experiments and astrophysical observations, we try to infer the corresponding neutron-proton effective k-mass splitting $m^*_{n-p}(ρ_0,δ)$. While the mean values of the $m^*_{n-p}(ρ_0,δ)$ obtained from most of the studies are remarkably consistent with each other and scatter very closely around an empirical value of \emass$=0.27\cdotδ$, it is currently not possible to scientifically state surely that the \emass is positive within the present knowledge of the uncertainties. Quantifying, better understanding and then further reducing the uncertainties using modern statistical and computational techniques in extracting the \es0 and \l0 from analyzing the experimental data are much needed.