The $Z\alpha^2$ correction to superallowed beta decays in effective field theory and implications for $|V_{ud}|$

Superallowed ($0^+\rightarrow0^+$) beta decays currently provide the most precise extraction of quark mixing in the Standard Model. Their interpretation as a measurement of $|V_{ud}|$ relies on a reliable first-principles computation of QED radiative corrections expressed as a series in $Z\alpha$ and $\alpha$. In this work, we provide the first model-independent result for two-loop, $O(Z\alpha^2)$, long-distance radiative corrections where the nuclei are treated as heavy point-like particles. We use renormalization group analysis to obtain new results at $O(Z\alpha^3)$ for the coefficient of double-logarithms in the ratio of the maximal beta energy to the inverse nuclear size, $\Em/R^{-1}$. We use the Kinoshita-Lee-Nauenberg theorem to obtain new results at $O(Z^2\alpha^3)$ for the coefficient of logarithms in the ratio of maximal beta energy to the electron mass, $\log(2\Em/\me)$. We identify a structure-dependent, and therefore short-distance, contribution to the traditional $Z\alpha^2$ correction that should be revisited.. We provide the first comprehensive update to the long-distance corrections in almost forty years and comment on the impact of our findings for extractions of $|V_{ud}|$. We find that shifts in the long-distance corrections are $2.5\times$ larger than past estimates of their uncertainty, $1.5\times$ larger than the statistical uncertainty from the combined fit of superallowed decays, and about $1/2$ the size of estimated systematic error, which stems dominantly from nuclear structure effects.