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For a system of fermions with a three-body contact interaction thesecond-order contributions to the energy per particle $\bar E(k_f)$ arecalculated exactly. The three-particle scattering amplitude in the medium isderived in closed analytical form from the corresponding two-loop rescatteringdiagram. We compare the (genuine) second-order three-body contribution to $\barE(k_f)\sim k_f^{10}$ with the second-order term due to the density-dependenteffective two-body interaction, and find that the latter term dominates. Theresults of the present study are of interest for nuclear many-body calculationswhere chiral three-nucleon forces are treated beyond leading order via adensity-dependent effective two-body interaction.

Exact calculation of three-body contact interaction to second order