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

TL;DR

This work delivers the first next-to-leading-logarithmic QCD analysis of electromagnetic corrections to the semileptonic weak Hamiltonian, incorporating mixed $O(\alpha\,\alpha_s^2)$ corrections to the vector coupling $g_V$ within a carefully factorized EFT framework. By combining three-loop anomalous dimensions with two-loop matching and implementing a scheme-based OPE subtraction inside loop integrals, the authors achieve a renormalization-group–improved prediction for the radiative correction $\Delta^V_R = 2.432(16)\%$. The approach introduces regularization-independent quantities that ensure scheme- and scale-independence, enabling more reliable extractions of $|V_{ud}|$ and improved tests of first-row CKM unitarity. The results, including lattice inputs for the hadronic box, reduce theoretical uncertainties and bring CKM unitarity into better agreement, guiding future precision determinations of fundamental weak interaction parameters.

Abstract

We present the first next-to-leading-logarithmic QCD analysis of the electromagnetic corrections to the semileptonic weak Hamiltonian, including the mixed $\mathcal{O}(α\,α_s^2)$ 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 $Δ^V_R = 2.432(16)\%$ can be systematically refined using input from lattice QCD and perturbation theory and improves the consistency of first-row CKM unitarity tests.

Figures (3)

• Figure 1: Feynman diagrams for the computation of the Green's function $\langle O \rangle_{\mu_0}$, i.e. the $\mu_0$-subtracted scheme. All diagrams are evaluated in dimensional regularization. The diagram on the right-hand side represents the OPE subtraction $\langle O \rangle_{S}$, while the diagram on the left-hand side also contributes to the $\overline{\text{MS}}$ renormalization. The latter diagrams cancel in the scheme-matching equation.
• Figure 2: Both panels show the residual scale dependence on $\mu$ (left panel) and $\mu_W$ (right panel) for $g_V(m_n)$ at LL and NLL in QCD. The impact of the Lattice uncertainties in $\Box^{V<}_{\gamma W}(Q_0^2)$ are shown for the NLO curve as the grey region. Leading Log QCD corrections are absent in the no LL$_s$ curve.
• Figure 3: Determination of $V_{us}$ from CKM unitarity using the $\text{LL}_s$ and $\text{NLL}_s$ determinations of $V_{ud}$ for superallowed nuclear beta decays ($0^+ \to 0^+$), compared with the free neutron decay at $\text{NLL}_s$ and with $K_{\ell 2}$ and $K_{\ell 3}$ decays, including their average. The NLL result reduces the theoretical uncertainty of $\Delta^V_R$ and restores consistency with first-row CKM unitarity. Leading Log QCD corrections are absent in the no $\text{LL}_s$ interval.