Superallowed ($0^{+}\to 0^{+}$) 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 pointlike 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, $E_m/R^{-1}$. 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, $\log (2E_m/m)$. 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 40 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.

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• Beta decay
• Cabibbo–Kobayashi–Maskawa matrix
• Effective field theory
• Quantum electrodynamics