Regge Poles and the Photoproduction of Pions

The photoproduction of pions, $\gamma +N\to \pi +N$, is examined from the standpoint of the Regge pole hypothesis and the Mandelstam representation. The asymptotic behavior of the forward scattering amplitudes is determined in terms of the Regge trajectories of the $\gamma +\pi \to N+\overline{N}$ channel. The $\rho$, $\phi$, and $\omega$ trajectories are included in the description of the photoproduction of neutral pions, whereas only the $\rho$ and $\pi$ trajectories contribute to the photoproduction of charged pions. In the case of backward scattering, asymptotic representations of the scattering amplitudes are controlled by the trajectories of the $\gamma +N\to \pi +N$ channel by crossing. Finally, generalized Pomeranchuk relations are established for the differential cross sections in the forward and backward directions for the various charge configurations of the photoproduction channel. In particular, we have the following interesting results: (1) The differential cross sections for $\gamma +p\to n+{\pi }^{+}$ and $\gamma +n\to p+{\pi }^{-}$ are asymptotically equal in the forward and backward directions; (2) the differential cross sections for $\gamma +p\to p+{\pi }^0$ and $\gamma +n\to n+{\pi }^0$ are asymptotically equal in the backward direction.