# General mathematical analysis on multiple solutions of interfering resonances combinations

## TL;DR

This paper provides a mathematical analysis explaining why multiple equally good solutions occur when fitting cross sections with multiple interfering resonances, revealing the underlying structure and offering a method to find all solutions.

## Contribution

It derives the source of multiple solutions in resonance fitting problems and introduces a general method to identify all solutions with equal fit quality.

## Key findings

- Number of solutions for n+1 resonances is 2^n.
- The multiplicity depends on zeros of amplitudes in the complex plane.
- A simple method to generate all solutions from a known one.

## Abstract

When fitting cross sections with several resonances or interfering background and resonances, one usually obtains multiple solutions of parameters with equal fitting quality. In the present work, we find the source of multiple solutions for a combination of several resonances or interfering background and resonances by analyzing the mathematical structure of the Breit-Wigner function. We find that there are $2^n$ fitting solutions with equal quality for $n+1$ resonances, and the multiplicity of the interfering background and resonances depends on zeros of the amplitudes in the complex plane. We provide a simple, general method to infer all other solutions with equal fitting quality from a known solution.