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

TL;DR

This work advances the precision of long-distance QED radiative corrections for superallowed beta decays by formulating a heavy-particle EFT that yields a model-independent, RG-improved calculation through O(Zα^2) and beyond. It shows a factorization of real-virtual photons at leading power and provides the first two-loop, massless-electron corrections, including KLN-constrained logarithms, to update the outer corrections. The authors offer a comprehensive numerical update across ten nuclei, quantify the shifts in Ft and |V_ud|, and compare their results to the traditional Towner–Hardy framework via a careful scheme translation. These results sharpen the theoretical uncertainties on |V_ud| and point to a concrete path for integrating short-distance effects and structure-dependent pieces in a unified, minimal-subtraction–consistent scheme.

Abstract

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 $Zα$ and $α$. In this work, we provide the first model-independent result for two-loop, $O(Zα^2)$, long-distance radiative corrections where the nuclei are treated as heavy point-like particles. We use renormalization group analysis to obtain new results at $O(Zα^3)$ 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 $O(Z^2α^3)$ 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 $Zα^2$ 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.

Figures (2)

• Figure 1: Feynman diagrams that correct the real-photon amplitude at $O(Z\alpha)$. The circle denotes the weak vertex, and we use the Feynman rules discussed in Borah:2024ghnPlestid:2024eib that make $Z$-counting manifest.
• Figure 2: A reproduction of Fig. 3 of Towner and Hardy Hardy:2020qwl for the $\mathcal{F}t$ values of the ten nuclei considered in this work. Error bars show purely experimental errors, and do not account for theoretical uncertainties related to radiative corrections. The data points of Towner and Hardy are shown in blue (dark gray). The same data points multiplied by the "total shift" column of \ref{['tab:conversion formula']} are shown in orange (light gray).