Remarks on pole trajectories for resonances

TL;DR

This paper analyzes how resonance pole trajectories behave as a coupling parameter varies, highlighting differences between s-wave and higher partial waves, and discusses implications for understanding resonance structures in strong interactions.

Contribution

It reveals that pole trajectories in higher partial waves are similar regardless of dynamics, while s-wave trajectories provide insights into resonance internal structure and threshold behavior.

Findings

Higher partial wave pole trajectories are qualitatively similar.

S-wave pole trajectories reveal resonance internal structure.

Only molecular states naturally appear near thresholds without fine tuning.

Abstract

We discuss in general terms pole trajectories of resonances coupling to a continuum channel as some strength parameter is varied. It is demonstrated that, regardless the underlying dynamics, the trajectories of poles that couple to the continuum in a partial wave higher than s-wave are qualitatively the same, while in case of s-waves the pole trajectory can reveal important information on the internal structure of the resonance. In addition we show that only molecular (or extraordinary) states appear near thresholds naturally, while more compact structures need a significant fine tuning in the parameters. This study is of current relevance especially in strong interaction physics, since lattice QCD may be employed to deduce the pole trajectories for hadronic resonances as a function of the quark mass thus providing additional, new access to the structure of s-wave resonances.