# A Study of Degenerate Two-Body and Three-Body Coupled-Channel Systems -Renormalized Effective AGS Equations and Near-Threshold Resonances-

## TL;DR

This paper investigates the universal behavior of near-threshold resonances in degenerate two- and three-body coupled-channel systems, providing a renormalized effective framework and revealing unique pole dynamics relevant to exotic hadron structures.

## Contribution

It formulates an effective three-body scattering approach for degenerate channels, analyzes the universal near-threshold pole behavior, and addresses unphysical singularities through renormalization.

## Key findings

- Derived a universal relation for S-matrix poles near degenerate thresholds.
- Showed unphysical singularities can be removed by reorganization and renormalization.
- Identified a distinctive complex energy behavior different from pure two- or three-body systems.

## Abstract

Motivated by the existence of candidates for exotic hadrons whose masses are close to both of two-body and three-body hadronic thresholds lying close to each other, we study degenerate two-body and three-body coupled-channel systems. We first formulate the scattering problem of non-degenerate two-body and three-body coupled-channels as an effective three-body problem, i.e.\ effective Alt-Grassberger-Sandhas (AGS) equations. We next investigate the behavior of $S$-matrix poles near the threshold when two-body and three-body thresholds are degenerate. We solve the eigenvalue equations of the kernel of AGS equations instead of AGS equations themselves to obtain the $S$-matrix pole energy. We then face a problem of unphysical singularity: though the physical transition amplitudes have physical singularities only, the kernel of AGS equations have unphysical singularities. We show, however, that these unphysical singularities can be removed by appropriate reorganization of the scattering equations and mass renormalization. The behavior of $S$-matrix poles near the degenerate threshold is found to be universal in the sense that the complex pole energy, $E$, is determined by a real parameter, $c$, as $c - E \log{\left( - E \right)} = 0$, or equivalently, ${\rm Im} E = - \pi {\rm Re} E / \log{\mid {\rm Re} E \mid}$. This behavior is different from that of either two-body or three-body system and is characteristic in the degenerate two-body and three-body coupled-channel system. We expect that this new class of universal behavior might play a key role in understanding exotic hadrons.