Contribution of the rho meson and quark sub-structure to the nuclear spin-orbit potential

The microscopic origin of the spin-orbit (SO) potential in terms of sub-baryonic degrees of freedom is explored and discussed for application to nuclei and hyper-nuclei. We thus develop a chiral relativistic approach where the coupling to the scalar and vector meson fields are controlled by the quark substructure. This approach suggests that the isoscalar and isovector density dependence of the SO potential can be used to test the microscopic ingredients which are implemented in the relativistic framework: the quark sub-structure of the nucleon in its ground-state and its coupling to the rich meson sector where the $\rho$ meson plays a crucial role. This is also in line with the Vector Dominance Model (VDM) phenomenology and the known magnetic properties of the nucleons. We explore predictions based on Hartree and Hartree-Fock mean field, as well as various scenarios for the $\rho$-nucleon coupling, ranked as weak, medium and strong, which impacts the isoscalar and isovector density dependence of the SO potential. We show that a medium to strong $\rho$ coupling is essential to reproduce Skyrme phenomenology in $N=Z$ nuclei as well as its isovector dependence.