Probabilistic interpretation of compositeness relation for resonances

TL;DR

This paper develops a probabilistic framework to interpret the compositeness of resonances using pole residues, enabling meaningful compositeness relations with positive coefficients for certain resonances.

Contribution

It introduces a new formalism to extract probabilistic compositeness relations for resonances from pole residues, extending previous narrow resonance results.

Findings

A sum rule for compositeness derived from pole residues.

Positive coefficients for compositeness in certain resonance cases.

Application to various resonances demonstrating the formalism.

Abstract

Bound, antibound and resonance states are associated to poles in the on-shell partial wave amplitudes. We show here that from the residues of the pole a rank 1 projection operator associated with any of these states can be extracted, in terms of which a sum rule related to the composition of the state can be derived. Although typically it involves complex coefficients for the compositeness and elementariness, except for the bound state case, we demonstrate that one can formulate a meaningful compositeness relation with only positive coefficients for resonances whose associated Laurent series in the variable s converges in a region of the physical axis around Re(s_P), with s_P the pole position of the resonance. It is also shown that this result can be considered as an analytical extrapolation in s_P of the clear narrow resonance case. We exemplify this formalism to study the two-body components of several resonances of interest.