Third-order particle-hole ring contributions from contact-interactions and chiral NN-potentials

A missing piece in perturbative calculations of nuclear matter with chiral low-momentum interactions are the particle-hole ring contributions at third-order. We evaluate the 3rd-order particle-hole ring diagrams for a momentum-dependent NN-contact interaction of the Skyrme-type. By working with the antisymmetrized contact-interaction just one 4-loop diagram needs to be computed. The corresponding result for the 3-ring energy per particle anti E(k_f) is a cubic expression in the Skyrme-parameters t_i, x_i, W₀ with numerical coefficients N_j that are given by double-integrals over cubic expressions in euclidean (scalar, vector, tensor) polarizations functions Q_j(s,κ). Dimensional regularization of these integrals is realized by subtracting power-divergences and the validity of the method is checked against analytical results at 2nd order. The 3rd-order ring-energy from 1π-exchange is calculated semi-analytically and found to be very strongly attractive, with anti E(k_f0) ≅ -92 MeV. For the N³LO chiral NN-potential the 3-ring energy in symmetric nuclear matter and neutron matter is weakly attractive and it decreases for softer potentials (with lower cutoffs). The extensive computations based on partial-wave matrix elements are checked against semi-analytical treatments for model interactions.