Multiphoton amplitudes and generalized Landau-Khalatnikov-Fradkin transformation in scalar QED

We apply the worldline formalism to amplitudes in scalar quantum electrodynamics involving open scalar lines, with an emphasis on their nonperturbative gauge dependence. At the tree level, we study the scalar propagator interacting with any number of photons in configuration space as well as in momentum space. At one loop we rederive, in an efficient way, the off-shell vertex in an arbitrary dimension and any covariant gauge. Generalizing the Landau-Khalatnikov-Fradkin transformation for the nonperturbative propagator, we find simple nonperturbative transformation rules for arbitrary x-space amplitudes under a change of the covariant gauge parameter in terms of conformal cross ratios.