Rethinking Partial Widths: Unitary Mixing and the $\Delta (1232)$ Pole Residue

Abstract

The extracted $\pi N$ partial decay width of the $\Delta (1232)$ systematically exceeds its total width ($2|r|>\Gamma$). We demonstrate this anomaly is a natural consequence of S-matrix unitary mixing. Because exact multi-channel shadow poles are distant and model-dependent, we utilize a heuristic elastic model -- treating the overlapping $\Delta (1600)$ as fully elastic -- to isolate the core mechanism. We show that evaluating a perturbing S-matrix at a state's complex pole systematically inflates the residue magnitude. This proof of principle confirms complex residues reflect global amplitude topology rather than isolated intrinsic properties, challenging naive interpretations of branching fractions.

arXiv comment5 pages, 3 tables