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.
Categories: hep-ph hep-ex nucl-th