Radiative corrections to superallowed beta decays at $\mathcal O(α^2 Z)$

TL;DR

This work presents the first complete calculation of radiative corrections to superallowed $0^+ o 0^+$ beta decays at $igO( ext{$ extalpha$}^2 Z)$ using a heavy-particle EFT that treats ultrasoft photons interacting with nuclei. By reducing two-loop virtual and one-loop real-virtual amplitudes to a set of master integrals and solving them analytically, the authors obtain the outer-correction functions $oldsymbol{ exthat g}^{(2)}$ and $oldsymbol{ exthat a}^{(2)}$, which modify the phase-space factor $ar{oldsymbol{P}}$ by up to about $4 imes 10^{-3}$ for heavy isotopes and significantly reduce residual scale dependence. The results, together with the running matching coefficient $C^{(g_V)}_{ ext{eff}}$ and nuclear-structure inputs, provide a more robust, RG-resummed framework for extracting $V_{ud}$ at permille precision. The techniques and master-integral methods developed here are applicable to other observables and higher-order corrections, paving the way for more precise tests of CKM unitarity.

Abstract

We compute $\mathcal O(α^2 Z)$ radiative corrections to superallowed $β$ decays with a heavy-particle effective field theory that systematically describes the interactions of low-energy ultrasoft photons with nuclei. We calculate two-loop virtual and one-loop real-virtual amplitudes by reducing the Feynman integrals to a set of master integrals, which we solve analytically using a variety of techniques. These techniques can be applied to other phenomenologically interesting observables. The ultrasoft corrections can then be combined with contributions arising from the exchange of potential photons to obtain the complete $\mathcal O(α^2 Z)$ correction to the decay rate, with resummation of large logarithms of the electron energy times the nuclear radius. We find that $\mathcal O(α^2 Z)$ ultrasoft loops induce a relative correction to the decay rate that ranges from $0.7 \cdot 10^{-3}$ in the decay of $^{10}$C to $3.6 \cdot 10^{-3}$ in the decay of $^{54}$Co, and will thus impact the extraction of $V_{ud}$ at the permille level. We show that the inclusion of these corrections reduces the residual renormalization scale dependence of the decay rate to a negligible level, making missing ultrasoft perturbative corrections a subdominant source of theoretical error.

Figures (3)

• Figure 1: $\delta \overline{P}$ (top panel) and relative uncertainties for different beta decays ($Z$). $\delta \overline{P}$ is expected to show a linear behavior since it encodes the $\mathcal{O}(\alpha^2 Z)$ corrections computed in this work. For the same reason, in the bottom panel, the ratio of relative uncertainties, shown in blue and with scale shown on the left axis, exhibits a linear behavior. The relative uncertainty on $\overline{P}(\alpha,\alpha^2 Z)$ is shown in red, with scale on the right axis. The quadratic dependence of the relative uncertainty of the $\mathcal{O}(\alpha^2Z)$ contribution is due to missing $\mathcal{O}(\alpha^3Z^2)$ terms.
• Figure 2: Virtual-virtual diagrams contributing to $\beta$ decay at $\mathcal{O}(\alpha^2Z)$. Double, plain, and wavy lines denote nuclei, leptons and photons, respectively.
• Figure 3: Virtual-real diagrams. The notation is as in Fig. \ref{['fig:diagsvirtualvirtual']}.