Three-body scattering problem in the fixed center approximation: the case of attraction

TL;DR

This paper investigates three-body scattering involving a light particle and a bound heavy pair using the fixed center approximation, revealing ambiguities in the amplitude that depend on a parameter and showing equivalence of different solution methods.

Contribution

It introduces a comprehensive analysis of three-body scattering with attraction, demonstrating the equivalence of coordinate and momentum space approaches and clarifying the role of a three-body parameter.

Findings

Scattering amplitude depends on a single real parameter.

Coordinate and momentum space solutions are equivalent.

The approach aligns with three-body contact interaction methods.

Abstract

We study the scattering of a light particle on a bound pair of heavy particles (e.g., the deuteron) within the fixed center approximation in the case of light-heavy attraction, solving the integral equation for the three-body Green's function both in the coordinate and in the momentum space. The results for the three-body scattering amplitude appear to be ambiguous -- they depend on a single real parameter. This parameter may be fixed by a three-body input, e.g., the three-body scattering length. We also solve the integral equation for the three-body Green function in the momentum space, introducing a finite cut-off. We show that all three approaches are equivalent. We also discuss how our approach to the problem matches with the introduction of three-body contact interaction as done by other authors.