Effective Field Theory for Two-Body Systems with Shallow S-Wave Resonances

TL;DR

This paper develops an effective field theory framework for two-body scattering involving shallow S-wave resonances, providing a systematic expansion that captures low-energy poles and their properties.

Contribution

It introduces a systematic effective field theory approach for S-wave resonances with a focus on low-energy poles and renormalization constraints.

Findings

Systematic expansion for S-matrix with two low-energy poles

Leading order restrictions imply non-positive effective range

Explicit next-to-leading order calculations match toy model results

Abstract

Resonances are of particular importance to the scattering of composite particles in quantum mechanics. We build an effective field theory for two-body scattering which includes a low-energy $S$-wave resonance. Our starting point is the most general Lagrangian with short-range interactions. We demonstrate that these interactions can be organized into various orders so as to generate a systematic expansion for an $S$ matrix with two low-energy poles. The pole positions are restricted by renormalization at leading order, where the common feature is a non-positive effective range. We carry out the expansion explicitly to next-to-leading order and illustrate how it systematically accounts for the results of a toy model -- a spherical well with a delta shell at its border.