Resolving the $\Delta(1232)$ partial width anomaly: Complex pole residue is not a fundamental resonance property

TL;DR

This paper challenges the common assumption that the complex pole residue directly represents a resonance's partial width, demonstrating that this identification leads to inconsistencies such as the $
2322$ partial width exceeding its total width.

Contribution

The authors show that the complex pole residue is not a fundamental property of resonances, resolving the $
2322$ partial width anomaly with a simple model.

Findings

Residue phases vary and are not directly proportional to partial widths.

The partial width cannot be accurately derived from the residue magnitude.

The $
2322$ partial width anomaly is explained by residue phase effects.

Abstract

The resonant properties of excited hadrons are commonly identified with the complex pole positions and residues of the scattering amplitude. The mass and total decay width are given by position, whereas the partial width is given by the magnitude of the residue. If this identification was correct, the partial width of famous $\Delta(1232)$ would be larger than its total width. By using a simple model that predicts residue phases of prominent baryons $N^*$, $\Delta$, $\Lambda$, $\Sigma$, and low mass mesons, we resolve this anomaly and show that the residue cannot be a fundamental resonant property.