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.