Note on the definitions of branching ratios of overlapping resonances

TL;DR

This paper examines how different definitions of branching ratios for overlapping and wide resonances affect their values, emphasizing the importance of spectral function integration and clarifying the relation between pole positions and total widths.

Contribution

It demonstrates the dependence of branching ratios on their definitions for broad and overlapping resonances and highlights the necessity of spectral function integration for accurate results.

Findings

Different definitions yield significantly different branching ratios.

Spectral function integration is essential for wide and sequential decays.

Pole positions in non-adjacent sheets do not directly give total widths.

Abstract

Branching ratios for the decay of hadrons with large width or near thresholds depend on their definition. We test different definitions and show that rather different branching ratios can be obtained. For wide resonances and for sequential decays with wide intermediate resonances, integration over the spectral functions is mandatory. The tests are performed exploiting the latest solution of the Bonn-Gatchina multi-channel analysis and published values for residues of light scalar mesons. For a resonance overlapping with a threshold, in case its pole lies in a non-adjacent sheet, we show how the total width, needed for the branching ratios, does not correspond to the imaginary part of the pole position. We use the Madrid-Krakow dispersive parameterizations to illustrate this situation with the $f_0(980)$.