Isovector nuclear spin-orbit interaction from chiral pion-nucleon dynamics

Using the two-loop approximation of chiral perturbation theory, we calculate the momentum and density-dependent isovector nuclear spin-orbit strength V_ls(p,k_f). This quantity is derived from the spin-dependent part of the interaction energy Σ_spin=i/2σ→· (q→×p→)[U_ls(p,k_f)- V_ls(p,k_f)τ₃δ] of a nucleon scattering off weakly inhomogeneous isospin-asymmetric nuclear matter. We find that iterated 1π-exchange generates at saturation density, k_f0=272.7 MeV, an isovector nuclear spin-orbit strength at p=0 of V_ls(0,k_f0)≃50 MeV fm². This value is about 1.4 times the analogous isoscalar nuclear spin-orbit strength U_ls(0,k_f0)≃35 MeV fm² generated by the same two-pion exchange diagrams. We also calculate several relativistic 1/M-corrections to the isoscalar nuclear spin-orbit strength. In particular, we evaluate the contributions from irreducible two-pion exchange to U_ls(p,k_f). The effects of the three-body diagrams constructed from the Weinberg-Tomozawa ππNN-contact vertex on the isoscalar nuclear spin-orbit strength are computed. We find that such relativistic 1/M-corrections are less than 20% of the isoscalar nuclear spin-orbit strength generated by iterated one-pion-exchange, in accordance with the expectation from chiral power counting.