Reducible chiral four-body interactions in nuclear matter

The method of unitary transformations generates five classes of leading-order reducible chiral four-nucleon interactions which involve pion-exchanges and a spin-spin contact-term. Their first-order contributions to the energy per particle of isospin-symmetric nuclear matter and pure neutron matter are evaluated in detail. For most of the closed four-loop diagrams the occurring integrals over four Fermi-spheres can be reduced to easily manageable one- or two-parameter integrals. One observes substantial cancelations among the different contributions arising from 2-ring and 1-ring diagrams. Altogether, the net attraction generated by the chiral four-nucleon interaction does not exceed values of $-1.3$\,MeV for densities $\rho<2\rho_0$.