Chiral four-body interactions in nuclear matter

An exploratory study of chiral four-nucleon interactions in nuclear and neutron matter is performed. The leading-order terms arising from pion-exchange in combination with the chiral 4π-vertex and the chiral NN3π-vertex are found to be very small. Their attractive contribution to the energy per particle stays below 0.6MeV in magnitude for densities up to ρ = 0.4 fm^-3. We consider also the fournucleon interaction induced by pion-exchange and twofold Δ-isobar excitation of nucleons. For most of the closed four-loop diagrams the occurring integrals over four Fermi spheres can either be solved analytically or reduced to easily manageable one- or two-parameter integrals. After summing the individually large contributions from 3-ring, 2-ring and 1-ring diagrams of alternating signs, one obtains at nuclear matter saturation density ρ₀ = 0.16 fm^-3 a moderate contribution of 2.35MeV to the energy per particle. The curve Ē(ρ) rises rapidly with density, approximately with the third power of ρ. In pure neutron matter the analogous chiral four-body interactions lead, at the same density ρ_n, to a repulsive contribution that is about half as strong. The present calculation indicates that long-range multi-nucleon forces, in particular those provided by the strongly coupled πNΔ-system with its small mass-gap of 293MeV, can still play an appreciable role for the equation of state of nuclear and neutron matter.