Pion-nucleon interaction in theP11partial wave and the pion-nucleon vertex function

We have studied the pion-nucleon interaction in the $P_{11}$ channel considering its important role in the absorption and production of the pion by nuclei. Specifically, we emphasize that the total $P_{11}$ $t$ matrix (or amplitude) can be naturally decomposed into two parts: the direct (and dressed) pole part and the nonpole part. It is then easy to observe that individually these two parts can be large but tend nearly to cancel each other and produce small and negative phase shift values from threshold to $T_{\mathrm{lab}}^{\pi }\sim 170$ MeV and that at higher energies the nonpole term dominates giving positive phase shift values. We then try to find the parametrization of this partial wave $t$ matrix within the framework of the Blankenbecler-Sugar reduction. As a by-product we have obtained the $\pi \mathrm{NN}$ vertex function (with one nucleon off mass shell) and the dressed nucleon propagator.