Nonlocal {\it in-medium} effective interaction for nucleon scattering off isospin-asymmetric targets

We have investigated the role of isospin asymmetry of the $NN$ effective interaction in the context of $NA$ elastic scattering. To this purpose we represent the {\it in-medium} $g$ matrix as an admixture of isospin-symmetric nuclear matter and pure neutron matter solutions of Brueckner-Hartree-Fock equations for infinite nuclear matter, denoted as $g[\rho,\beta]$. We use the Argonne $v_{18}$ bare potential to represent the $NN$ interaction in free space, due to its ability to describe the $NN$ scattering amplitudes up to 350 MeV. The density-dependent isospin-asymmetric $g$ matrices are then used to calculate optical model potentials for elastic nucleon scattering off closed-shell nuclei. For this aim, we make use of the Arellano-Bauge $\delta g$-folding approach suited for an explicit treatment of nonlocal density matrices. This approach allows to account for the local isospin-asymmetry and density dependence of the {\it in-medium} $NN$ interaction. The resulting optical potentials are nonlocal since the entire nonlocal structure of the $g$ matrix is retained. We observe that including the isospin asymmetry in the $g$ matrix allows for a reasonable description of differential cross-sections at nucleon beam energies between 40 and 200 MeV. In the case of proton scattering at energies below $\sim$65 MeV, at momentum transfers $q$ below $\sim$1 fm$^{-1}$, the inclusion of neutronic-matter $g$ matrices yields better agreement with the data as compared to the case when symmetric nuclear matter $g$ matrices are used. These results provide evidence that the isospin-asymmetry in the $NN$ effective interaction yield non-negligible effects in nucleon scattering off isospin-asymmetric targets at beam energies below 65 MeV.