Low-Energy Theorems for Photon Processes

We review and discuss various methods of obtaining low-energy theorems for photon processes: (1) Low's method, (2) the tensor method, and (3) the $S$-matrix approach. The purely kinematical nature of these theorems is emphasized, for they are found to follow primarily from the identification of the correct amplitudes free of kinematical singularities and zeros. Gauge invariance serves to inform us of the presence of additional kinematical zeros in certain physical amplitudes, so that the unknown continuum contribution is suppressed relative to the known singular Born terms arising from single-particle exchange at the physical threshold. Besides the well-known low-energy theorems specifying Compton amplitudes to first order in the photon frequency, one can show that some pieces of the amplitude satisfy higher-order theorems; in fact, all $2J+1$ multipoles of a spin-$J$ target have an associated low-energy theorem. We explicitly establish a low-energy theorem for the quadrupole moment of a $J=1\mathrm{}$ target to supplement the known theorems for the total charge and magnetic moment. An additional theorem for photopion production is obtained along with the well-known Kroll-Ruderman theorem, and serves to specify the $E_{2^{-}}$ multipole at threshold.