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

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.