Theory of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>S</mml:mi></mml:math>-Wave Pion Scattering and Photoproduction at Low Energies

Abstract

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 ${\ensuremath{\lambda}}_{0}\ensuremath{\phi}\ifmmode\cdot\else\textperiodcentered\fi{}\ensuremath{\phi}+\ensuremath{\lambda}\ensuremath{\tau}\ifmmode\cdot\else\textperiodcentered\fi{}(\ensuremath{\phi}\ifmmode\times\else\texttimes\fi{}\ensuremath{\pi})$ is added to the usual $p$-wave coupling ${(4\ensuremath{\pi})}^{\frac{1}{2}}(\frac{f}{\ensuremath{\mu}})\ensuremath{\sigma}\ifmmode\cdot\else\textperiodcentered\fi{}\ensuremath{\nabla}\ensuremath{\tau}\ifmmode\cdot\else\textperiodcentered\fi{}\ensuremath{\phi}$. Scattering equations are developed and solved in the one-meson approximation. Values for the renormalized coupling parameters ${\ensuremath{\lambda}}_{0}$ and $\ensuremath{\lambda}$ are determined which give reasonable agreement with the $s$-wave phase shifts up to \ensuremath{\sim}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 ${\ensuremath{\pi}}^{+}$ and ${\ensuremath{\pi}}^{0}$ 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 $\ensuremath{\pi}\ensuremath{-}\ensuremath{\pi}$ forces, are discussed.