  Radiative corrections to elastic electron-proton scattering are analyzed in
effective field theory. A new factorization formula identifies all sources of
large logarithms in the limit of large momentum transfer, $Q^2\gg m_e^2$.
Explicit matching calculations are performed through two-loop order. A
renormalization analysis in soft-collinear effective theory is performed to
systematically compute and resum large logarithms. Implications for the
extraction of charge radii and other observables from scattering data are
discussed. The formalism may be applied to other lepton-nucleon scattering and
$e^+e^-$ annihilation processes.
