Final-state interactions in a system of three pions

Abstract

Subenergy unitarity methods are used to analyze the vacuum-to-three-pion matrix element of the axial-vector current. The isobar expansion together with dispersion relations in the isobar subenergy variables provides a representation which describes the interactions among the three final pions in the ${1}^{+}$ and ${0}^{\ensuremath{-}}$ states. Linear integral equations in a single variable are obtained for the isobar amplitudes. The coupled system of $\ensuremath{\epsilon}\ensuremath{\pi}$ and $\ensuremath{\rho}\ensuremath{\pi}$ channels is considered. A phenomenological viewpoint is adopted from which a modular picture of the isobar amplitudes emerges. The amplitudes are products of process-dependent factors and so-called resolvents common to any process with the same final state. Integral equations for the resolvents are obtained and solved by means of a basis-function method. The solutions in the ${1}^{+}$ state exhibit a remarkable fluctuation as a function of the three-pion mass when this variable is near the region of the ${A}_{1}$ resonance. The possible impact of this effect on the experimental ${A}_{1}$ situation is briefly discussed.