Singularities in Partial-Wave Amplitudes for Two Ingoing and Two Outgoing Particles

The singularities which appear in the energy plane of a partial-wave amplitude are investigated for the general process with two ingoing and two outgoing particles. These lie either on the real axis or on curves symmetrical about the real axis. The equations of these curves and the conditions under which they occur are obtained; also the ranges in the energy spectra of the crossed channels to which they correspond. These general results are applied to the particular cases of pion-nucleon scattering and pion photoproduction from nucleons.