Gravitational corrections to the scattering in a massless quantum electrodynamics

We discuss the role of gravitational corrections to the running of the electric charge through the evaluation of scattering amplitudes of charged particles in massless scalar electrodynamics. Computing the complete divergent part of the $S$-matrix amplitude for two distinct scattering processes, we show that quantum gravitational corrections do not alter the running behavior of the electric charge. Our result does not exclude the possibility that the presence of a second dimensional constant in the model (a cosmological constant or the presence of massive particles) could alter this behavior, as was proposed in earlier works.