Final state interactions for high energy scattering off atomic electrons

We consider the scattering of high energy leptons off bound atomic electrons focusing primarily on final state interactions 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 nonrelativistic 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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>O</a:mi><a:mo stretchy="false">(</a:mo><a:msup><a:mi>α</a:mi><a:mn>3</a:mn></a:msup><a:mo stretchy="false">)</a:mo></a:math>. Furthermore, we argue that both final initial state Coulomb exchanges (enhanced by the number of electrons in the atom) do not affect the cross section until at least <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>O</e:mi><e:mo stretchy="false">(</e:mo><e:msup><e:mi>α</e:mi><e:mn>3</e:mn></e:msup><e:mo stretchy="false">)</e:mo></e:math>. Transverse photon couplings to nonrelativistic electrons are proportional to their small velocities, and rotational invariance suppresses their contribution to <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi>O</i:mi><i:mo stretchy="false">(</i:mo><i:msup><i:mi>α</i:mi><i:mn>3</i:mn></i:msup><i:mo stretchy="false">)</i:mo></i:math>. Our results are relevant for precision experiments involving neutrinos, electrons, positrons, and muons scattering off of atomic electrons in a fixed target. Published by the American Physical Society 2024