Three-body spin-orbit forces from chiral two-pion exchange

Using chiral perturbation theory, we calculate the density-dependent spin-orbit coupling generated by the two-pion exchange three-nucleon interaction involving virtual Δ-isobar excitation. From the corresponding three-loop Hartree and Fock diagrams we obtain an isoscalar spin-orbit strength F_so(kf) which amounts at nuclear matter saturation density to about half of the empirical value of 90 MeV fm⁵. The associated isovector spin-orbit strength G_so(k_f) comes out to be about a factor of 20 smaller. Interestingly, this three-body spin-orbit coupling is not a relativistic effect but independent of the nucleon mass M. Furthermore, we calculate the three-body spin-orbit coupling generated by two-pion exchange on the basis of the most general chiral ππNN-contact interaction. We find similar (numerical) results for the isoscalar and isovector spin-orbit strengths F_so(k_f) and G_so(k_f) with a strong dominance of the p-wave part of the ππNN-contact interaction and the Hartree contribution.