Effective Three-Boson Interactions using a Separable Potential

TL;DR

This paper derives and solves the three-body scattering equations for systems with finite-range separable potentials, revealing new scaling laws and analytic forms for inelastic and elastic scattering processes.

Contribution

It introduces a new integral equation framework for three-body scattering with separable potentials, avoiding renormalization issues present in contact interactions.

Findings

Retrieved analytic form of inelastic scattering amplitude.

Formulated a new scaling law for elastic three-body scattering.

Demonstrated the effectiveness of separable potentials in three-body problems.

Abstract

Effective field theories (EFTs) are widely used to study many-body systems by describing two-body interactions using zero-ranged contact potentials. However, when extended to three-body processes, these contact interactions lead to divergences due to the absence of an intrinsic length scale. In EFT, this is typically resolved by introducing a zero-ranged three-body interaction, which can be renormalized to make the low-energy physics independent of the short-distance physics. However, when the two-body potential has a finite range, such as in separable potentials, there is no need for such renormalization. In this work, we derive the integral equation for the three-body scattering amplitude for separable potentials, and solve it in the strongly-interacting regime. With our model, we retrieve the known analytic form of the scattering amplitude for inelastic scattering processes and formulate a new scaling law for elastic three-body scattering processes.