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.

5 pages, 3 tables