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 $F_{\rm so}(k_f)$ which amounts at nuclear matter saturation density to about half of the empirical value of $90 $MeVfm$^5$. The associated isovector spin-orbit strength $G_{\rm so}(k_f)$ comes out 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 $\pi\pi NN$-contact interaction. We find similar (numerical) results for the isoscalar and isovector spin-orbit strengths $F_{\rm so}(k_f)$ and $G_{\rm so}(k_f)$ with a strong dominance of the p-wave part of the $\pi\pi NN$-contact interaction and the Hartree contribution