Beyond Leading Logarithms in $g_V$: The Semileptonic Weak Hamiltonian at $\mathcal{O}(\alpha\,\alpha_s^2)$

We present the first next-to-leading-logarithmic QCD analysis of the electromagnetic corrections to the semileptonic weak Hamiltonian, including the mixed $\mathcal{O}(\alpha\,\alpha_s^2)$ 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 $\Delta^V_R = 2.436(16)\%$ can be systematically refined using input from lattice QCD and perturbation theory and improves the consistency of first-row CKM unitarity tests.