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 finds substantial compensations 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}$