Fixed target experiments are the ``tool of choice'' for intensity frontier experiments, and supply the largest statistical samples for many processes.
When considering scattering with electrons, it is intuitively obvious that the atomic electrons can be treated as free and at rest. This leading
approximation, while very good, is not sufficient for precision applications, or in scenarios involving narrow resonances.
In this talk I will discuss how to properly include atomic binding corrections making use of the high-energy kinematics inherent to particle physics experiments.
I will show that for neutrino-electron and muon-electron scattering, the dominant corrections can be reduced to simple one-body matrix elements with atomic states
using the virial theorem and many-body sum rules. I will then sketch how the formalism must be modified when dealing with narrow resonances, as is relevant for
positron annihilation to (dark) long-lived particles at experiments such as SHiP, PADME, or the Fermilab Short-Basline Neutrino program.