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 Delta -isobar excitation. From the corresponding three-loop Hartree and Fock diagrams we obtain an isoscalar spin-orbit strength F-so(k(f)) which amounts at nuclear matter saturation density to about half of the empirical value of 90 MeV fm(5). 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 pipiNN -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 pipiNN -contact interaction and the Hartree contribution.