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 ∆-isobar excitation. From the corresponding three-loop Hartree and Fock diagrams we obtain an isoscalar spin-orbit strength Fso(kf ) which amounts at nuclear matter saturation density to about half of the empirical value of 90MeVfm5. The associated isovector spin-orbit strength Gso(kf ) 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 ππNN -contact interaction. We find similar (numerical) results for the isoscalar and isovector spin-orbit strengths Fso(kf ) and Gso(kf ) with a strong dominance of the p-wave part of the ππNN -contact interaction and the Hartree contribution. PACS: 12.38.Bx, 21.30.Fe, 24.10.Cn, 31.15.Ew The microscopic understanding the dynamical origin of the strong nuclear spin-orbit force is still one of the key problems in nuclear physics. The analogy with the spin-orbit interaction in atomic physics gave the hint that it could be a relativistic effect. This idea has lead to the construction of the (scalar-vector) mean-field models for nuclear structure calculations [1, 2]. In these models the nucleus is described as a collection of independent Dirac quasi-particles moving in self-consistently generated scalar and vector mean-fields. The footprints of relativity become visible through the large nuclear spin-orbit coupling which emerges in that framework naturally from the interplay of the two strong and counteracting (scalar and vector) mean-fields. The corresponding many-body calculations are usually carried out in the Hartree approximation, ignoring the negative-energy Dirac-sea. The NN-interaction underlying these models is to be considered as an effective one that is tailored to properties of finite nuclei but not constrained (completely) by the observables of free NN-scattering. On the other hand it has long been known that calculations based on Hamiltonians which contain only realistic two-nucleon potentials (thus fitting accurately all NN-phase shifts and mixing angles below the NNπ-threshold) often cannot predict the observed spin-orbit splittings of nuclear levels. In fact one of the original motivations for the Fujita-Miyazawa three-nucleon potential [3] was just the study of such spin-orbit splittings. In ref.[4] it has then been shown that one out of the Urbana family of three-nucleon forces makes a substantial contribution to the spin-orbit splitting in the nucleus N. Moreover, three-nucleon forces are actually needed in addition to realistic two-nucleon potentials in order to reproduce the correct saturation point of (isospin-symmetric) nuclear matter [5]. The long-range part of the three-nucleon interaction is generated in a natural way by two-pion exchange [6] and it can in fact be predicted by using chiral symmetry [7]. The purpose of this paper is present analytical results for the nuclear spin-orbit coupling generated by the (chiral) two-pion exchange three-nucleon interaction. In order to arrive at such results we will make use of the density-matrix expansion of Negele and Vautherin [8]. This