Method for constructing relativistic three-particle models of the pion-nucleon system

The Bakamjian-Thomas procedure is used to develop a method for constructing relativistic, instant form models of the pion-nucleon system. A limited model space is used to illustrate the method. The model space consists of a single-nucleon subspace, a pion-nucleon subspace, a two-pion-nucleon subspace, and a pion-sigma meson-nucleon subspace. A Poincaré invariant mass operator is constructed that includes vertex interactions that couple the various subspaces, as well as renormalization terms. It is shown that the pion-nucleon elastic scattering and production amplitudes can be obtained from the solution of a single, three-dimensional, integral equation of the Lippmann-Schwinger type. The effective pion-nucleon potential that appears in this equation contains contributions from direct and crossed nucleon exchanges along with sigma exchange. The production amplitudes are of the form that arises in isobar models. The elastic scattering and production amplitudes satisfy unitarity. The method developed makes it possible to extend existing coupled-channel models of the pion-nucleon system to include three-particle channels in such a way that the requirements of special relativity and unitarity are satisfied exactly.