Dynamical theory for theP-wave pion-nucleon interaction

We present a dynamical theory of low energy $P$-wave $\pi N$ interaction. The theory is an extension of the Chew-Low theory, with three additional features, namely, (1) inclusion of the full nucleon recoil effect, (2) inclusion of the $Z$ graphs, and (3) inclusion of $P11\mathrm{}$ channel inelasticity. The second feature is new in a $P$-wave theory based on the Low expansion. We are able to fit the $P33\mathrm{}$ and the $P11\mathrm{}$ phase shifts in the energy region $0<\mathrm{}{T}_{\pi }<250\mathrm{}$ MeV quite well. In particular, we show that it is possible to explain the change of sign of the $P11\mathrm{}$ phase shift, $T_{\pi }\sim 150$ MeV, in terms of the strong inelasticity present in the channel. For the $P13\mathrm{}$ and $P31\mathrm{}$ channels, where the phase shifts are the smallest, we do reasonably well at threshold, but not so well as the energy increases. We have also examined the structure of the off-mass-shell amplitudes in the $P11\mathrm{}$ and $P33\mathrm{}$ channels. Only the $P33\mathrm{}$ channel amplitude is found to be factorable. A major result of our work is that the $P33\mathrm{}$ form factor is very hard. If one used a monopole form, the form factor mass should be $10m_{\pi }$ or larger.