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 Uls(p, kf ). This quantity is derived from the spin-dependent part of the interaction energy Σspin = i 2 ~σ ·(~q×~p )Uls(p, kf ) of a nucleon scattering off weakly inhomogeneous isospin symmetric nuclear matter. We find that iterated 1π-exchange generates at saturation density, kf0 = 272.7MeV, a spinorbit strength at p = 0 of Uls(0, kf0) ' 35 MeVfm2 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 Vls underlying the empirical spin-orbit strength Ũls = Vls r2 ls becomes a rather weak one, Vls ' 17 MeV, after the identification rls = m−1 π as suggested by the present calculation. We observe however a strong p-dependence of Uls(p, kf0) leading even to a sign change above p = 200MeV. 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 singleparticle potential UI(p, kf )+iWI(p, kf ) 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 = kf . PACS: 12.38.Bx, 21.65.+f, 24.10.Cn