Coherent enhancement of QED cross sections in electromagnetic backgrounds

We introduce form factors that relate the amplitude of a quantum electrodynamics (QED) process in vacuum to its corresponding background-field process. The latter is characterized by a reduced S-matrix element where one or more photon field operators are replaced by classical background fields. In the associated Feynman diagram, external photon lines are supplanted with lines representing the c-number field. This modifies the cross section by factors proportional to powers of the Fourier amplitude of the classical field (and its complex conjugate). We demonstrate this explicitly by comparing different reaction channels of low-energy photon-photon scattering in a classical background. We find that background field cross sections typically undergo coherent enhancement and for some reaction channels display a more favorable scaling with center-of-mass energy compared to the vacuum process. Similar coherent enhancement may be found for leading-order pair annihilation to one photon, but this competes with kinematic suppression. This suppression can be minimized by using an x-ray free electron laser as the classical background.