We present an $\mathcal{O}(\alpha )$ Standard Model calculation of the inner radiative corrections to Gamow-Teller $\beta$ decays. We find that a priori contributions arise from the photonic vertex correction and $\gamma W$ box diagram. Upon evaluation most elastic contributions vanish due to crossing symmetry or cancellation between isoscalar and isovector photonic contributions, leaving only the polarized parity-odd contribution, i.e., the Gamow-Teller equivalent of the well-known axial $\gamma W$ box contribution for Fermi decays. We show that weak magnetism contributes significantly to the Born amplitude, and consider additional hadronic contributions at low energy using a holomorphic continuation of the polarized Bjorken sum rule constrained by experimental data. We perform the same procedure for the Fermi inner radiative correction through a combination of the running of Bjorken and Gross-Llewellyn Smith sum rules. We discuss heavy flavor, higher-twist, and target mass corrections and find a significant increase at low momentum from the latter. We find ${\mathrm{\Delta }}_R^A=0.02532(22)$ and ${\mathrm{\Delta }}_R^V=0.02473(27)$ for axial and vector inner radiative corrections, respectively, resulting in ${\mathrm{\Delta }}_R^A-{\mathrm{\Delta }}_R^V=0.60(5)\times {10}^{-3}$, which allows us to extract $g_A^0$ for the first time to our knowledge. We discuss consequences for comparing experimental data to lattice calculations in beyond Standard Model fits. Further, we show how some traditional $\beta$ decay calculations contain part of this effect but fail to account for cancellations in the full $\mathcal{O}(\alpha )$ result. Finally, we correct for a double-counting instance in the isospin $T=1/2$ mirror decay extraction of $|V_{ud}|$, the up-down matrix element of the Cabibbo-Kobayashi-Maskawa matrix element, resolving a long-standing tension and leading to increased precision.

• Beta decay
• Electroweak radiative corrections
• Leptonic, semileptonic & radiative decays