Third-order particle-hole ring diagrams with contact-interactions and one-pion exchange

The third-order particle-hole ring diagrams are evaluated for a NN-contact interaction of the Skyrme type. The pertinent four-loop coefficients in the energy per particle E¯(kf)∼kf5+2n are reduced to double integrals over cubic expressions in Euclidean polarization functions. Dimensional regularization of divergent integrals is performed by subtracting power divergences and the validity of this method is checked against the known analytical results at second order. The complete O(p²) NN-contact interaction is obtained by adding two tensor terms and their third-order ring contributions are also calculated in detail. The third-order ring energy arising from long-range 1 π -exchange is computed and it is found that direct and exchange contributions are all attractive. The very large size of the three-ring energy due to point-like 1 π-exchange, E¯ (k_{f 0}) ≃ - 92 MeV at saturation density, is however in no way representative for that of realistic chiral NN-potentials. Moreover, the third-order (particle-particle and hole-hole) ladder diagrams are evaluated with the full O(p²) contact interaction, and the simplest three-ring contributions to the isospin-asymmetry energy A(kf)∼kf5 are studied.