A mathematical solution for the parameters of three interfering resonances

Abstract

The multiple-solution problem in determining the three-interfering-resonances' parameters from a fit to an experimentally measured distribution is considered in a mathematical viewpoint. In this paper it is shown that there are four numerical solutions for the fit with three coherent Breit-Wigner functions. Although the explicit analytical formulae can not be derived in this case, we provide some constraint equations between the four solutions. For the cases of nonrelativistic and relativistic Breit-Wigner forms of amplitude functions, numerical method is provided to derive the other solutions from the already obtained one based on the obtained constraint equations. In real experimental measurements with more complicated amplitude forms similar to Breit-Wigner functions, the same method can be deduced and performed to get numerical solutions. The well agreement between the solved solutions using this mathematical method and those from the fit directly verifies the correctness of the supplied constraint equations and mathematical methodology.

General mathematical analysis on multiple solutions of interfering resonances combinations

“…As indicated in Refs. [3,5] the masses and widths of the resonances involved in two sets of parameters are identical. Thus, we only need to determine how to obtain another set of parameters f R and φ R from the known f R and φ R .…”

“…In Ref. [5], they extended to the case of three resonances and obtained some constraint equations between four different solutions. In the present work, we further extend the investigation to the case of an arbitrary number of resonances.…”