Ladder resummation of spin 1/2 fermion many-body systems with arbitrary partial-wave content

TL;DR

This paper develops a ladder resummation method for calculating the energy density of spin 1/2 fermion many-body systems using arbitrary two-body scattering amplitudes, including partial-wave decomposition and contact interactions.

Contribution

It introduces a partial-wave expansion framework for in-medium scattering amplitudes and provides a renormalized solution for contact interactions in various regularization schemes.

Findings

Derived expressions for energy density using partial-wave amplitudes.

Provided renormalized results in cutoff regularization.

Analyzed S- and P-wave interactions near the unitary limit.

Abstract

We resum the ladder diagrams for the calculation of the energy density $\cal{E}$ of a spin 1/2 fermion many-body system in terms of arbitrary vacuum two-body scattering amplitudes. The partial-wave decomposition of the in-medium two-body scattering amplitudes is developed, and the expression for calculating $\cal{E}$ in a partial-wave amplitude expansion is also given. The case of contact interactions is completely solved and is shown to provide renormalized results, expressed directly in terms of scattering data parameters, within cutoff regularization in a wide class of schemes. $S$- and $P$-wave interactions are considered up to including the first three-terms in the effective-range expansion, paying special attention to the parametric region around the unitary limit.