Model for the Two-Pion and Three-Pion Resonances

Abstract

A relativistic model of $\ensuremath{\pi}\ensuremath{-}\ensuremath{\pi}$ interaction is proposed from analogy with field theory. The main assumption in the model is that the interaction kernel is separable in momentum space. A Frazer-Fulco type resonance formula for the isovector $2\ensuremath{\pi}$ resonance is exactly derived on this model and its parameters are determined from the observed $2\ensuremath{\pi}$ resonance at 750 MeV.The $3\ensuremath{\pi}$ problem is then solved with this model of $\ensuremath{\pi}\ensuremath{-}\ensuremath{\pi}$ interaction and the exact isoscalar $3\ensuremath{\pi}$ wave function expressed in terms of a single-parameter function which satisfies a one-dimensional integral equation for the cases of "scalar" and "axial vector" forms of the wave function. It is found that it is possible to understand the observed energy (780 MeV) of the $3\ensuremath{\pi}$ resonance, on the above model, only for the case of the axial vector wave function and not for the scalar case. The model thus predicts a vector isoscalar meson at the observed energy.