Coulomb corrections in the extraction of the proton radius

Multi-photon exchange contributions are important in extracting the proton charge radius from elastic electron--proton scattering. So far, only diagrams associated with the exchange of a second photon have been evaluated. At the very low $Q^2$ values relevant to the radius extraction, higher order contributions may become important. We evaluate these corrections in the Effective Momentum Approximation, which includes the Coulomb interaction to all orders, and find small corrections with a strong $Q^2$ dependence at low $Q^2$ and large scattering angles. This suggests that the higher order terms may be important in the evaluation of the proton magnetic radius.