Remarks on the complex branch points in pi-N scattering amplitude and the multiple poles structure of resonances

TL;DR

This paper explores the complex branch points in pi-N scattering amplitudes, proposing a hypothesis that their singularity structure varies smoothly with pion mass, and demonstrates how multiple poles arise from the inclusion of additional Riemann surfaces, exemplified by the Roper resonance.

Contribution

It introduces a heuristic argument linking branch points to pion mass variation and explains the origin of multiple poles in resonance structures.

Findings

Multiple poles in resonances are a mathematical consequence of additional Riemann surfaces.

The two-pole structure of the Roper resonance is explained as an example.

Branch points in the unphysical sheet are understood through the proposed hypothesis.

Abstract

A simple heuristic argument to understand the existence of branch points in the unphysical sheet for pi-N scattering amplitude is presented. It is based on a hypothesis that the singularity structure of the pi-N scattering amplitude is a smooth varying function of the pion mass. We find that, in general, multiple poles structure of a resonance is a direct mathematical consequence when additional Riemann surface is included in the study and the two-pole structure found to correspond to the Roper resonance is a good example.