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 $\bar E(k_f) \sim k_f^{5+2n}$ 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 ${\cal O}(p^2)$ 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 $1\pi$-exchange is computed and it is found that direct and exchange contributions are all attractive. The very large size of the pion-ring energy, $\bar E(k_{f0})\simeq -92\,$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 ${\cal O}(p^2)$ contact interaction and the simplest three-ring contributions to the isospin-asymmetry energy $A(k_f)\sim k_f^5$ are studied.