General mathematical analysis on multiple solutions of interfering resonances combinations

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.

Figure 1

Diagrams showing the relationship between $P$ and $P^{\prime }$ [diagram (a)] and their reciprocals [diagram (b)].

Figure 2

An example of interference between two Breit-Wigner distributions. The solid curve is the total cross section, and the dashed curves are the individual contributions from the resonances.

Figure 3

Cross sections with interference between four resonances. Diagram (a) corresponds to input parameters, and diagrams (b)–(h) denote solutions II–VIII estimated with the present methods.

Figure 4

Interference between a resonance and an exponential background. The dashed and solid curves show the individual contributions and the cross sections obtained with the input parameters [diagram (a)] and the new amplitudes obtained with the present method [diagram (b)].

Figure 5

The same as Fig. 4, but for polynomial background.