We present the first lattice QCD calculation of the universal axial $\gamma W$-box contribution ${\square }_{\gamma W}^{VA}$ to both superallowed nuclear and neutron beta decays. This contribution emerges as a significant component within the theoretical uncertainties surrounding the extraction of $|V_{ud}|$ from superallowed decays. Our calculation is conducted using two domain wall fermion ensembles at the physical pion mass. To construct the nucleon four-point correlation functions, we employ the random sparsening field technique. Furthermore, we incorporate long-distance contributions to the hadronic function using the infinite-volume reconstruction method. Upon performing the continuum extrapolation, we arrive at ${\square }_{\gamma W}^{VA}=3.65(7{)}_{\mathrm{lat}}(1{)}_{\mathrm{PT}}\times {10}^{-3}$. Consequently, this yields a slightly higher value of $|V_{ud}|=0.97386(11{)}_{\exp }(9{)}_{\mathrm{RC}}(27{)}_{\mathrm{NS}}$, reducing the previous $2.1\sigma$ tension with the CKM unitarity to $1.8\sigma$. Additionally, we calculate the vector $\gamma W$-box contribution to the axial charge $g_A$, denoted as ${\square }_{\gamma W}^{VV}$, and explore its potential implications.

• Received 1 September 2023
• Revised 23 January 2024
• Accepted 16 April 2024

DOI:https://doi.org/10.1103/PhysRevLett.132.191901

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Published by the American Physical Society