Bound state corrections and high-energy scattering

Many fundamental-physics experiments scatter high energy beams off of fixed targets composed of ordinary matter i.e., atoms. When considering the scattering off of atomic electrons we often make the approximation that the electron is free and at rest, however one can ask how good this approximation really is? This becomes especially important in the face of demanding precision goals of certain experiments. For example the planned MuonE experiment will attempt to measure the shape of  $\mu e \rightarrow \mu e$ scattering as a function of angle with a precision of 10 ppm. In this talk I will explain how to systematically include bound-state corrections arising from the difference between a free-and-at-rest electron and those bound in atomic orbitals. When the final state of the atom is not measured, a surprisingly simple and elegant formula can be obtained that reduces the leading order corrections to a single atomic matrix element. New developments related to Coulomb corrections for inelastic systems will also be discussed.  Based on (arXiv:2403.12184, 2407.21752).