Off-shell behavior of the pion-nucleon elastic amplitude

Simple models of the off-shell behavior of pion-nucleon amplitudes are reviewed. The separable potential model which includes the coupling of inelastic channels to elastic channels is derived. It is shown that the dominant, low-momentum behavior of the separable potential is determined by kinematic considerations. Recent results in the S, D, and F-wave channels of Ernst and Johnson are presented. It is argued that in P-wave channels, the presence of the nucleon pole in the amplitude requires a model that is similar to the Chew-Low model. The off-shell behavior of this amplitude is shown to differ from the separable potential model. Results of the model of Ernst and Johnson for P-wave amplitudes are presented. Off-shell effects in elastic pion-nucleon elastic scattering are examined. First, it is shown that different off-shell extrapolations are required for different choices of the basic propagator. Secondly, the dependence of elastic pion-nucleus scattering on the off-shell extrapolation and the choice of propagators is examined