Pauli nonlocality and the nucleon effective mass

A study of the nucleon mean-field potential in nuclear matter (NM) is done within an extended Hartree-Fock (HF) formalism, using the CDM3Y6 density dependent version of the M3Y interaction which is associated with the nuclear incompressibility $K\simeq 252$ MeV. The momentum dependence of nucleon optical potential (OP) in NM at the saturation density ${\rho }_0$ is shown to be due mainly to its exchange term up to $k\approx 2{\mathrm{fm}}^{-1}$, so that the Pauli nonlocality is expected to be the main origin of the nucleon effective mass at low momenta. Because nucleons in neutron-rich NM at $\rho \approx {\rho }_0$ are either weakly bound or unbound by the in-medium nucleon-nucleon interaction, the determination of the effective mass of nucleon scattered on targets with neutron excess at low energies should be of interest for the mean-field studies of neutron star matter. For this purpose, the folding model is used to calculate the nonlocal nucleon OP for the optical model analysis of elastic nucleon scattering on ${}^{40,48}\mathrm{Ca}, {}^{90}\mathrm{Zr}$, and ${}^{208}\mathrm{Pb}$ targets at energies $E<50$ MeV, to probe the model reliability and validate the Wentzel-Kramers-Brillouin (WKB) local approximation to obtain the local folded nucleon OP. The nucleon effective mass $m^{*}$ is then carefully deduced from the momentum dependence of the local folded nucleon OP which results from the Pauli nonlocality of the exchange term. The obtained $m^{*}$ values agree well with the nucleon effective mass given by the extended HF calculation of the single-particle potential in asymmetric NM. The neutron-proton effective mass splitting determined at $\rho \approx {\rho }_0$ from the central strength of the real folded nucleon OP for ${}^{48}\mathrm{Ca}, {}^{90}\mathrm{Zr}$, and ${}^{208}\mathrm{Pb}$ targets has been found to depend linearly on the neutron-proton asymmetry parameter as $m_{n-p}^{*}\approx (0.167\pm 0.018)\delta$, in a good agreement with the recent empirical constraints.