Exact calculation of three-body contact interaction to second order

TL;DR

This paper provides an exact second-order calculation of the energy contribution from three-body contact interactions in fermionic systems, highlighting the dominance of effective two-body interactions in such calculations.

Contribution

It introduces an exact analytical method for calculating second-order three-body contributions and compares them with effective two-body interactions in nuclear matter.

Findings

Second-order three-body energy scales as k_f^{10}

Effective two-body interactions dominate over three-body contributions

Analytical expressions derived for three-particle scattering amplitude

Abstract

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