Different manifestations of S-matrix poles

TL;DR

This paper derives model-independent formulas for scattering observables using the S-matrix's analytical properties, clarifying the nature of resonances and their decay behaviors through Mittag-Leffler expansions and comparisons with experimental data.

Contribution

It provides new analytical expressions for cross sections and phase shifts based on S-matrix poles, extending the Breit-Wigner approximation with exact formulas and conditions for residues.

Findings

Exact expressions reduce to Breit-Wigner with corrections

Residues can be estimated analytically under certain conditions

Survival probabilities explain non-observable non-exponential decay

Abstract

Making use of the analytical properties of the $S$-matrix and a theorem of Mittag-Leffler, model independent non-relativistic expressions for cross sections in single channel elastic scattering, scattering phase shifts and survival probabilities of resonances are derived. Provided certain conditions are satisfied by the poles, the residues can also be estimated analytically. Considerations of the low energy behaviour of the $S$-matrix and cross sections reveal additional conditions on the residues of the poles appearing in the Mittag-Leffler expansions. The exact expressions for the resonant cross section and phase shift are shown to reduce to the commonly used Breit-Wigner formula plus corrections. The latter is shown to approach the exact result with the example of a meson and a baryon resonance. Finally, a comparison of the exact expressions with some realistic examples is presented. The calculations of survival probabilities in particular reveal the reason behind the non-observability of non-exponential decay.