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

TL;DR

This paper introduces a geometric framework for understanding light hadron resonances in the complex plane, linking the residue phase to the threshold position and revealing differences between vector and scalar resonances.

Contribution

It demonstrates that the residue phase is mainly governed by a geometric phase related to the pole and threshold, highlighting the role of Adler zeros in scalar resonances.

Findings

Vector resonances align with the geometric baseline

Scalar resonances deviate by 10-15 degrees due to Adler zeros

Threshold position critically influences the complex-plane structure

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.