Final state interactions for high energy scattering off atomic electrons

TL;DR

This paper analyzes how final state Coulomb interactions affect high energy lepton scattering off atomic electrons, showing these effects cancel at low orders and are negligible until at least third order in the fine-structure constant, impacting precision experiments.

Contribution

It introduces a resummation method for Coulomb exchanges in high energy scattering and demonstrates their cancellation at low orders, refining theoretical predictions for atomic electron scattering.

Findings

Coulomb exchanges cancel at least up to O(α^3) in the cross section.

Both initial and final state Coulomb effects are negligible until at least third order.

Transverse photon effects are suppressed by small electron velocities and rotational invariance.

Abstract

We consider the scattering of high energy leptons off bound atomic electrons focusing primarily on final state interactions i.e., the exchange of virtual photons between the outgoing energetic electron, and the heavy residual charged "debris" in the final state. These effects are inherently absent from calculations for a free electron at rest. Coulomb exchanges are enhanced by the large number of electrons in the atomic debris, and are unsuppressed by non-relativistic velocities in the debris. We find that these exchanges can be resummed using operator methods, and cancel at the level of the cross section until at least $O(\alpha^3)$. Furthermore, we argue that both final {\it and} initial state Coulomb exchanges (enhanced by the number of electrons in the atom) do not affect the cross section until at least $O(\alpha^3)$. Transverse photon couplings to non relativistic electrons are proportional to their small velocities, and rotational invariance suppresses their contribution to $O(\alpha^3)$. Our results are relevant for precision experiments involving neutrinos, electrons, positrons, and muons scattering off of atomic electrons in a fixed target.