Elastic phase shift analysis reveals the geometric origin of the residue phase

Abstract

We show that the complex-plane structure of light hadron resonances is governed by a unified geometric framework where the threshold position plays a decisive role. By applying this framework to $\pi \pi$, $\pi K$, and $\pi N$ phase shifts, we show that the residue phase $\theta$ is primarily determined by the geometric phase ${\delta }_0$ (the angle between pole and real axis seen from the threshold). While vector resonances exhibit excellent alignment with this geometric baseline, scalar resonances show systematic deviations of $10^{\circ }$--$15^{\circ }$, which we identify as the dynamical imprint of Adler zeros.

Comments: Significant revision and extension of arXiv:2005.11564 [hep-ph] and the previous version of this paper. Includes updated analysis, additional authors, and improved methodology as validated in Phys. Lett. B 872 (2026) 140136