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 nuclear spin-orbit strength U_ls(p, k_f). This quantity is derived from the spin-dependent part of the interaction energy Σ_spin = 2/i_σ^→· (q ^→ x p ^→) U_ls (p, k_f) of a nucleon scattering off weakly inhomogeneous isospin symmetric nuclear matter. We find that iterated 1π-exchange generates at saturation density, k_f0 = 272.7 MeV, a spin-orbit strength at p = 0 of U_ls (0, k_f0)≃ 35 MeV fm², in perfect agreement with the empirical value used in the shell model. This novel spin-orbit strength is neither of relativistic nor of short range origin. The potential V_ls underlying the empirical spin-orbit strength ^U_ls = V_lsr_ls² becomes a rather weak one, V_ls ≃ 17 MeV, after the identification r_ls = m_π^-1 as suggested by the present calculation. We observe, however, a strong p-dependence of U_ls(p, k_f0) leading even to a sign change above p = 200 MeV. This and other features of the emerging spin-orbit Hamiltonian which go beyond the usual shell model parametrization leave questions about the ultimate relevance of the spin-orbit interaction generated by 2π-exchange for a finite nucleus. We also calculate the complex-valued isovector single-particle potential U_I(p, k_f) + i W_I (p, k_f) in isospin asymmetric nuclear matter proportional to τ₃ (N - Z)/(N + Z). For the real part we find reasonable agreement with empirical values and the imaginary part vanishes at the Fermi-surface p = k_f.