Three-particle equations for the pion-nucleon system

Abstract

An analysis is carried out of a π-N-Δ field theory using projection operator techniques developed recently by the author. Relationships are derived between matrix elements of resolvents formed from bare ππN and ππΔ states and various three-particle amplitudes. These relations are exactly analogous to those obtained in potential theory between the three-particle Green’s function and the transition operators of Alt, Grassberger, and Sandhas. When the three-particle amplitudes are integrated over with appropriate weighting functions, the elastic scattering and production amplitudes are obtained. The weighting functions are the analogs of bound state wave functions for the πN and πΔ subsystems. A separation of the production amplitude into its one-fermion irre- ducible and reducible parts is derived. It is shown that the three-particle amplitudes are solutions of AlT, Grassberger, and Sandhas type of equations as extended by Kowalski to allow for a three-body interaction. When the AlT, Grassberger, and Sandhas quasiparticle method is applied to these equations, the two-particle equations developed previously by the author are recovered. If the one-fermion irreducible part of the three-body interaction is neglected a closed set of coupled nonlinear integral equations for all of the quantities of interest is obtained.