Plane-Wave Packets and Their Limitation in Nonlinear Compton Scattering

We show that scattering boundary conditions are incompatible with a monochromatic radiation field for the case of nonlinear Compton scattering. We demonstrate this by showing that (a) in the monochromatic limit, gauge invariance in a given order of the expansion of the $S$ matrix is destroyed, and (b) the physical (scattering) boundary condition, that of a pulse of radiation incident on a target electron, cannot be reconstructed from the monochromatic limit of the $S$ matrix. We then proceed to show by an example that the frequency profile of the scattered radiation is a function of both the intensity and line shape of the incident field. Another interesting feature of this calculation is that the profile of the photon scattered at a fixed angle is significantly broadened in comparison with the incident line shape. The worked-out example is a simple model, that of a neutral, scalar "electron" interacting with a bilinear scalar, massless external field, which contains all the important features of nonlinear Compton scattering. While from the point of view of gauge invariance it is sufficient to treat the external radiation field as a one-dimensional wave packet, for a complete description of the problem it is necessary to describe the incident radiation (quanta) in terms of normalizable states. An estimate of the breakdown of the plane-wave approximation is included.