Some General Relations between the Photoproduction and Scattering ofπMesons

With a partial wave analysis of the photomeson cross sections are combined the principle of charge independence, the hypothesis of time reversibility, and the unitarity of the scattering matrix. This leads to a natural starting point for the study of the photo cross sections. It also leads to some close relations between the photoproduction and scattering of pions in that the complex phases of the matrix elements for photoproduction are explicitly given in terms of the scattering phase shifts. One consequence of this is that there must be an $S$-wave contribution to the ${\pi }^0$ photoproduction on whose amplitude a lower limit can be given in terms of the $S$-wave pion-nucleon scattering. A second, independent lower limit on the $S$-wave term for the ${\pi }^0$ cross sections can be expressed in terms of the $\frac{{\pi }^{-}}{{\pi }^{+}}$ ratio. Several other nontrivial conditions are imposed on the cross sections.