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.