Does the Multiple-scattering Series in the Pion-deuteron Scattering Actually Converge?

TL;DR

This paper investigates the convergence of the multiple-scattering series in pion-deuteron scattering, revealing its limitations near certain scattering lengths and its relation to the formation of bound states in the three-body system.

Contribution

It demonstrates the conditions under which the multiple-scattering series converges or diverges, highlighting the impact of the two-body scattering length on the three-body system.

Findings

MSS converges for a wide range of complex scattering lengths

MSS fails to converge near the positive real axis of the scattering length

Infinite bound states form in the three-body system for real positive scattering lengths

Abstract

It is demonstrated that the well-known answer for the multiple-scattering series (MSS) for a light particle interacting to a pair of static nucleons, calculated in the Fixed Centers Approximation (FCA), works well for a wide region of the two-body complex scattering length $a$. However, this approach is not applicable in a narrow region surrounding the real positive $a$ half-axis, where the MSS does not converge. Simultaneously, for real positive $a$'s the 3-body system forms an infinite set of bound states.