Recently, the first ever lattice computation of the $gamma W$-box radiative correction to the rate of the semileptonic pion decay allowed for a reduction of the theory uncertainty of that rate by a factor of $sim3$. A recent dispersion evaluation of the $gamma W$-box correction on the neutron also led to a significant reduction of the theory uncertainty, but shifted the value of $V_{ud}$ extracted from the neutron and superallowed nuclear $beta$ decay, resulting in a deficit of the CKM unitarity in the top row. A direct lattice computation of the $gamma W$-box correction for the neutron decay would provide an independent cross-check for this result but is very challenging. Before those challenges are overcome, we propose a hybrid analysis, converting the lattice calculation on the pion to that on the neutron by a combination of dispersion theory and phenomenological input. The new prediction for the universal radiative correction to free and bound neutron $beta$-decay reads $Delta_R^V=0.02477(24)$, in excellent agreement with the dispersion theory result $Delta_R^V=0.02467(22)$. Combining with other relevant information, the top-row CKM unitarity deficit persists.
Download