Hard-photon theorems for radiative pion-proton scattering

Low's prescription for imposing gauge invariance is used to construct a reasonably general form of the amplitude for ${\pi }^{\pm }p\to {\pi }^{\pm }p\gamma$, excluding magnetic moment radiation from the internal strongly interacting structure. The explicit $k$ dependence of the amplitude is exhibited and the breakdown of Low's result (that the first two terms in the expansion in $k$ are model-independent) is examined in the hard-photon regime. We then consider possible hard-photon theorems and a program to determine unknown form factors in the off-shell scattering amplitude, in the proton magnetic moment, and in the radiation of the resonant $\pi p$ system, e.g., magnetic moment radiation from the $\Delta \mathrm{}\left(1236\right)\mathrm{}$.