Three particle model of the pion nucleon system
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A relativistic, three particle model of the pion-nucleon system is constructed in the instant form of relativistic quantum mechanics using the Bakamjian-Thomas procedure. The model space includes subspaces for the nucleon (N), the resonances Δ= P33 (1232), P11 (1440), D13 (1520), S11 (1535), and S31 (1620), as well as πN , πΔ , ηN , and ππN subspaces. The model specifies a Poincaré invariant mass operator that includes vertex interactions that couple the various subspaces, as well as renormalization terms. Projection operator techniques are used to reduce the equations arising from this mass operator to a set of three-dimensional Lippmann-Schwinger integral equations that couple only the πN , πΔ , and ηN channels. After a partial wave analysis these three-dimensional equations simplify to three coupled, one-dimensional integral equations for each partial wave. The mass operator interactions are derived from effective, hadronic Lagrangians that introduce a set of coupling constants. Cutoff functions are introduced to take into account the non-elementary nature of the particles in the model. These cutoff functions introduce a set of cutoff masses. The coupling constants and cutoff masses are determined by fitting a partial wave analysis of pion-nucleon elastic scattering up to a c.m. energy of W = 1550 MeV.