Joint lattice QCD–dispersion theory analysis confirms the quark-mixing top-row unitarity deficit
Chien-Yeah Seng, Xu Feng, Mikhail Gorchtein, Luchang Jin
Abstract
Recently, the first ever lattice computation of the $\ensuremath{\gamma}W$-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 $\ensuremath{\sim}3$. A recent dispersion evaluation of the $\ensuremath{\gamma}W$-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 $\ensuremath{\beta}$ decay, resulting in a deficit of the Cabibbo-Kobayashi-Maskawa (CKM) unitarity in the top row. A direct lattice computation of the $\ensuremath{\gamma}W$-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 $\ensuremath{\beta}$-decay reads ${\mathrm{\ensuremath{\Delta}}}_{R}^{V}=0.02477(24)$, in excellent agreement with the dispersion theory result ${\mathrm{\ensuremath{\Delta}}}_{R}^{V}=0.02467(22)$. Combining with other relevant information, the top-row CKM unitarity deficit persists.