Beyond Leading Logarithms in $g_V$: The Semileptonic Weak Hamiltonian at $\mathcal{O}(\hat{I}\pm \hat{I}{\pm }_s^2)$

Abstract

We present the first next-to-leading-logarithmic QCD analysis of the electromagnetic corrections to the semileptonic weak Hamiltonian, including the mixed $\mathcal{O}(\hat{I}\pm \hat{I}{\pm }_s^2)$ corrections to the vector coupling $g_V$. The analysis combines the evaluation of three-loop anomalous dimensions and two-loop matching corrections with a consistent factorization of short-distance QCD effects. The latter is implemented through a scheme change based on a $d$-dimensional operator product expansion performed inside the loop integrals. The resulting renormalization-group--improved expression for the radiative correction $\hat{I}{”}_R^V=2.436(16)\%$ can be systematically refined using input from lattice QCD and perturbation theory and improves the consistency of first-row CKM unitarity tests.

Minor typos corrected and update in the numerical analysis, where the running of alpha electromagnetic between the charm and tau mass is now included. Overall impact is negligible compared to residual higher-order uncertainties. Conclusions unchanged