Multiphoton amplitudes and generalized LKF transformation in scalar QED using the worldline formalism

We apply the worldline formalism to scalar quantum electrodynamics (QED) to find a Bern-Kosower type master formula for generalized Compton scattering, on-shell and off-shell. Moreover, we use it to study the non-perturbative gauge parameter dependence of amplitudes in scalar QED and, as our main result, find a simple non-perturbative transformation rule under changes of this parameter in x-space in terms of conformal cross ratios. This generalizes the well-known Landau-Khalatnikov-Fradkin transformation (LKFT). We also exemplify how the LKFT works in perturbation theory.