Meson-exchange $\pi N$ models in three-dimensional Bethe-Salpeter formulation

The pion-nucleon scattering is investigated by using several three-dimensional reduction schemes of the Bethe-Salpeter equation for a model Lagrangian involving $\pi ,$ $N,$ $\Delta ,$ $\rho ,$ and $\sigma$ fields. It is found that all of the resulting meson-exchange models can give similar good descriptions of the $\pi N$ scattering data up to 400 MeV. However they have significant differences in describing the $\pi \mathrm{NN}$ and $\pi N\Delta$ form factors and the $\pi N$ off-shell t-matrix elements. We point out that these differences can be best distinguished by investigating the near threshold pion production from nucleon-nucleon collisions and pion photoproduction on the nucleon. The consequences of using these models to investigate various pion-nucleus reactions are also discussed.