A mathematical review on the multiple-solution problem

TL;DR

This paper mathematically analyzes the multiple-solution problem in high energy physics data fitting, proving the existence of exactly two solutions for certain models and providing a numerical method to find both, emphasizing the importance of considering all solutions for accurate physics interpretation.

Contribution

It offers a rigorous proof of the dual solutions in fitting with two Breit-Wigner functions and introduces a numerical method to derive the second solution from the first.

Findings

Exactly two non-trivial solutions exist for the specified fit model.

A numerical method effectively finds the second solution from the first.

The solutions are related by analytical formulae, ensuring consistency.

Abstract

The recent multiple-solution problem in extracting physics information from a fit to the experimental data in high energy physics is reviewed in a mathematical viewpoint. All these multiple solutions were found via a fit process previously, while in this letter we prove that if the sum of two coherent Breit-Wigner functions is used to fit the measured distribution, there should be two and only two non-trivial solutions, and they are related to each other by analytical formulae. For real experimental measurements in more complicated situations, we also provide a numerical method to derive the other solution from the already obtained one. The excellent consistency between the exact solution obtained this way and the fit process justifies the method. From our results it is clear that the physics interpretation should be very different depending on which solution is selected. So we suggest that all the experimental measurements with potential multiple solutions be re-analyzed to find the other solution because the result is not complete if only one solution is reported.