Theory ofS-Wave Pion Scattering and Photoproduction at Low Energies

A fixed-source analysis of the $s$-wave pion-nucleon interaction is constructed along the lines of the Chew-Low-Wick formalism. A bilinear $s$-wave interaction of the form ${\lambda }_0\varphi ·\varphi +\lambda \tau ·(\varphi \times \pi )$ is added to the usual $p$-wave coupling ${\left(4\mathrm{}\pi \right)}^{\frac{1}{2}}\left(\frac{f}{\mu }\right)\sigma ·\nabla \tau ·\varphi$. Scattering equations are developed and solved in the one-meson approximation. Values for the renormalized coupling parameters ${\lambda }_0$ and $\lambda$ are determined which give reasonable agreement with the $s$-wave phase shifts up to ∼100-Mev pion kinetic energy. This $s$-wave interaction with the parameters fixed by the scattering analysis is then applied to the discussion of the ${\pi }^{+}$ and ${\pi }^0$ photo-production cross sections. A Kroll-Ruderman theorem is proved for the above nonlocal interaction and it is shown that the contributions to $s$-wave neutral and charged photoproduction are consistent with experiment. Other experimental implications, in particular as to the possible role of $\pi -\pi$ forces, are discussed.