Coulomb Corrections for Coherent Neutrino Nucleus Scattering

In this work we consider sub-leading $O(Z^2{\alpha }^3)$ corrections to coherent elastic neutrino nucleus scattering (CEvNS). These corrections are not negligible by power counting since nuclei with large coherent cross sections have sizeable nuclear charges e.g.\ $Z\alpha \sim 0.4$. We find that the corrections are much smaller than naive power counting in $Z\alpha$ would suggest and that Coulomb corrections do not substantially alter predictions for CEvNS in the Standard Model. We comment on similarities to older literature on mesonic and muonic atoms.