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

Understand

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.