We compute radiative corrections to the superallowed $\beta$ decay of ${}^{10}\mathrm{C}$ in an effective field theory approach using nuclear matrix elements obtained from quantum Monte Carlo calculations. These corrections are an important ingredient in the extraction of the Cabibbo-Kobayashi-Masakawa quark mixing matrix element $V_{ud}$, and the role of this work is to illuminate the uncertainties arising from nuclear structure. Our results provide good agreement with both the traditional extraction of $V_{ud}$, as well as with a more recent evaluation performed using the no-core shell model and a dispersion formalism. The dominant uncertainty in this approach is the presence of two unknown low-energy constants that enter into the relevant nuclear matrix elements. Future determinations of these low-energy constants—either from QCD or modeling them with two nucleon amplitudes—would improve the precision of the extraction in this formalism.

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• Beta decay
• Effective field theory
• Electroweak interactions in nuclear physics
• Nuclear structure & decays
• 6 ≤ A ≤ 19