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

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 $Δ(1232)$ would be larger than its total width. By using a simple model that predicts residue phases of prominent baryons $N^*$, $Δ$, $Λ$, $Σ$, and low mass mesons, we resolve this anomaly and show that the residue cannot be a fundamental resonant property.