Dispersion Relations for Mesonic 3-Body Decays

Abstract

We present dispersive analyses of mesonic three-body decay amplitudes based on the fundamental principles of analyticity, unitarity, and crossing. In this framework the leading final-state interactions are fully taken into account, with the resulting amplitudes being solely dependent on the respective two-body scattering phase shifts of the final-state mesons. The first part focuses on $\eta^{\prime}\to\eta\pi\pi$ decays. This decay offers several features of interest: due to final-state interactions it can be used to constrain $\pi\eta$ scattering; in the soft-pion limit two Adler zeros are predicted; and the neutral decay channel $\eta^{\prime}\to\eta\pi^{0}\pi^{0}$ shows a cusp effect within the physical decay region. In the second part we study the quark-mass dependence of $\omega\to3\pi$ decays. We rely on the quark-mass-dependent scattering phase shift for the $\pi\pi$ $P$-wave extracted from unitarized chiral perturbation theory. The described formalism may be used as an extrapolation tool for lattice QCD calculations of three-pion decays, for which $\omega\to3\pi$ can serve as a paradigm case.