Analysis of rescattering effects in $3\pi$ final states

TL;DR

This paper investigates the significance of crossed-channel rescattering effects in three-pion decay systems, providing amplitude decompositions and quantifying the minimum data needed to detect these effects in Dalitz-plot analyses.

Contribution

It introduces a method to incorporate crossed-channel rescattering effects using Khuri-Treiman equations and quantifies their detectability in experimental data.

Findings

Rescattering effects are significant for certain quantum numbers and masses.

A minimum number of events is identified to detect rescattering effects.

Kinematic effects can enhance or dilute rescattering signals.

Abstract

Decays into three particles are often described in terms of two-body resonances and a non-interacting spectator particle. To go beyond this simplest isobar model, crossed-channel rescattering effects need to be accounted for. We quantify the importance of these rescattering effects in three-pion systems for different decay masses and angular-momentum quantum numbers. We provide the amplitude decompositions for four decay processes with total $J^{PC} = 0^{--}$, $1^{--}$, $1^{-+}$, and $2^{++}$, all of which decay predominantly as $\rho\pi$ states. Two-pion rescattering is described in terms of an Omn\`es function, which incorporates the $\rho$ resonance. Inclusion of crossed-channel effects is achieved by solving the Khuri-Treiman integral equations. The unbinned log-likelihood estimator is used to determine the significance of the rescattering effects beyond two-body resonances; we compute the minimum number of events necessary to unambiguously find these in future Dalitz-plot analyses. Kinematic effects that enhance or dilute the rescattering are identified for the selected set of quantum numbers and various masses.