Lattice QCD Calculation of Electroweak Box Contributions to Superallowed Nuclear and Neutron Beta Decays

We present the first lattice QCD calculation of the universal axial $\gamma W$-box contribution ${\mathrm{\square }}_{\gamma W}^{VA}$ to both superallowed nuclear and neutron beta decays. This contribution emerges as a significant component within the theoretical uncertainties surrounding the extraction of $\mathrm{\mid }{V}_{ud}\mathrm{\mid }$ from superallowed decays. Our calculation is conducted using two domain wall fermion ensembles at the physical pion mass. To construct the nucleon 4-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 ${\mathrm{\square }}_{\gamma W}^{VA}=3.65(8{)}_{\mathrm{l}\mathrm{a}\mathrm{t}}(1{)}_{\mathrm{P}\mathrm{T}}\times {10}^{-3}$. Consequently, this yields a slightly higher value of $\mathrm{\mid }{V}_{ud}\mathrm{\mid }=0.97386(11{)}_{\mathrm{e}\mathrm{x}\mathrm{p}\mathrm{.}}(9{)}_{\mathrm{R}\mathrm{C}}(27{)}_{\mathrm{N}\mathrm{S}}$, 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 ${\mathrm{\square }}_{\gamma W}^{VV}$, and explore its potential implications.