Preserving unitarity for overlapping multichannel states: Breit-Wigner versus K matrix -- comparison, advantages and disadvantages

TL;DR

This paper compares the K-matrix and Breit-Wigner methods for describing overlapping resonances, highlighting their advantages and disadvantages, and proposes a way to construct a unitary S-matrix with Breit-Wigner functions.

Contribution

It introduces a method to make Breit-Wigner amplitudes unitary by accounting for interference, enabling direct physical interpretation and background inclusion.

Findings

The K-matrix guarantees unitarity but has parameters not directly related to resonances.

The Breit-Wigner approach has parameters with direct experimental meaning.

A new method to construct a unitary S-matrix with Breit-Wigner functions is demonstrated.

Abstract

The K-matrix method is often used to describe overlapping resonances. It guarantees the unitarity of the scattering matrix but its parameters are not resonances masses and widths. It is also unclear how to separate resonant and background contributions and to describe background in terms of phase shifts. The Breit-Wigner (BW) approach operates with parameters having direct experimental meaning but a simple sum of the BW functions is not unitary. We show how to construct the unitary S-matrix by taking into account the interference of the BW functions. The method is simple and straightforward, background can be added to resonance amplitudes in the standard quantum mechanics form. In examples we give a comparison between the K-matrix and unitary BW-approaches.