[Paper Review] A mathematical solve on the three-interfering-resonances' parameters

Han Xu, Chuyang Shen|arXiv (Cornell University)|Feb 5, 2018

Scientific Measurement and Uncertainty Evaluation17 citations

TL;DR

This paper presents a mathematical framework to systematically identify and derive all four equivalent solutions in fits involving three interfering resonances described by coherent Breit-Wigner amplitudes. Using constraint equations between solutions, it enables numerical reconstruction of all solutions from a single fitted solution, verified via toy Monte Carlo simulations with excellent agreement between derived and directly fitted results.

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.

Motivation & Objective

• To address the multiple-solution problem in fitting three interfering resonances, where multiple parameter sets yield identical goodness-of-fit.
• To determine whether all four equivalent solutions can be derived mathematically from a single known solution when using coherent Breit-Wigner amplitudes.
• To provide a systematic mathematical method—based on constraint equations—for numerically reconstructing all solutions from one fitted solution.
• To verify the correctness of the derived constraint equations and methodology using numerical simulations with realistic resonance forms.

Proposed method

• Formulates a general mathematical model for three coherent amplitudes as |g(x) + z₁f(x) + z₂h(x)|²/d, where z₁, z₂ are complex coupling parameters.
• Derives algebraic constraint equations (Eq. II) that relate the four equivalent solutions by linking the ratios of amplitudes F(x) = f(x)/g(x) and H(x) = h(x)/g(x).
• Applies the constraint equations to both nonrelativistic and relativistic Breit-Wigner amplitude forms, enabling numerical reconstruction of other solutions from a known one.
• Uses toy Monte Carlo simulations to generate 100,000-event data samples based on one input solution, then fits the same data to recover all four solutions.
• Compares the mathematically derived solutions with those obtained via direct maximum likelihood fitting, confirming consistency in resonance masses, widths, and coupling strengths.
• Extends the method to more complex amplitude forms by preserving the functional relations in Eqs. (7), (11), and (II), allowing numerical solution reconstruction even when analytical formulae are unavailable.

Experimental results

Research questions

• RQ1 Can all four equivalent solutions in a three-interfering-resonance fit be mathematically derived from a single known solution?
• RQ2 What algebraic constraints govern the relationships between the four solutions when using coherent Breit-Wigner amplitudes?
• RQ3 How can the method be generalized to relativistic and complex amplitude forms beyond simple Breit-Wigner functions?
• RQ4 To what extent do numerically reconstructed solutions match those obtained via direct fitting in realistic simulation scenarios?
• RQ5 Can the constraint equations be used to ensure completeness in experimental resonance parameter extraction when multiple solutions exist?

Key findings

• Four equivalent solutions exist for a fit involving three interfering resonances described by coherent Breit-Wigner amplitudes, all sharing identical resonance masses and widths but differing in relative phases and coupling strengths.
• The constraint equations derived (e.g., Eq. II) successfully relate all four solutions, enabling numerical reconstruction of the remaining three from a single known solution.
• In the relativistic Breit-Wigner case, the method reconstructed solutions with relative phase differences of approximately 1.06 rad (e.g., π/2 ≈ 1.57 vs. 2.63 rad), and coupling strengths (BΓ) varied significantly (e.g., BΓ for g-state: 1.00 vs. 2.85 in different solutions).
• The numerical reconstruction method achieved excellent agreement with direct fitting: for example, the phase φ_f was reconstructed as 2.63 (fit) vs. 2.63 (derived), and φ_h as 6.14 (fit) vs. 6.14 (derived), with χ²/dof ≈ 1.0 for all solutions.
• The method was validated using two toy Monte Carlo data samples, confirming that solutions derived via constraint equations matched those from maximum likelihood fitting within statistical uncertainties.
• For complex amplitude forms, the functional relations in Eqs. (7), (11), and (II) remain valid, allowing numerical reconstruction of all solutions even when analytical formulae are intractable.