First-principles completion of the electroweak radiative correction to semileptonic kaon decays: the full physical-point $O(\alpha)$ lattice QCD computation and the unknown $O(e^2p^4)$ chiral contributions to $\delta_{\mathrm{EM}}^{K\ell}$

The extraction of the CKM matrix element $|V_{us}|$ from semileptonic kaon decays $K\to\pi\ell\nu_\ell(\gamma)$ is limited by the long-distance electromagnetic radiative correction $\delta_{\mathrm{EM}}^{K\ell}$. A lattice QCD computation of the axial $\gamma W$-box diagram in the flavor SU(3) limit pinned the leading chiral low-energy constants but left open the completion of the correction from first principles: the unknown $O(e^2p^4)$ chiral perturbation theory terms, the extension to physical quark masses and to the complete set of $O(\alpha)$ diagrams (vector and axial $\gamma W$ boxes beyond the forward limit, inelastic intermediate states, and the three-current contribution), and the reduction of residual lattice systematics. The open problem is to determine the complete inclusive $O(\alpha)$ correction for $K_{\ell 3}$ at the physical mass point from QCD+QED directly, at the few$\times10^{-4}$ precision required by first-row CKM unitarity tests, removing the residual reliance on chiral power-counting estimates.

Background

The Cabibbo–Kobayashi–Maskawa (CKM) matrix describes quark flavor mixing in the Standard Model, and the unitarity of its first row, $|V_{ud}|^2+|V_{us}|^2+|V_{ub}|^2=1$, is among the sharpest low-energy tests of the theory; the current determinations show a deficit at about the $2.8\sigma$ level (the "Cabibbo angle anomaly"), which could signal new physics such as right-handed quark couplings. Semileptonic kaon decays $K\to\pi\ell\nu_\ell$ (denoted $K_{\ell 3}$) provide the cleanest determination of $|V_{us}|$, but the extraction requires, besides experimental rates and the lattice vector form factor $f_+^{K\pi}(q^2)$, the electroweak radiative corrections (RC) that connect the measured inclusive rate (virtual photons plus bremsstrahlung) to the underlying weak decay rate; the long-distance electromagnetic piece of this correction is conventionally denoted $\delta_{\mathrm{EM}}^{K\ell}$ (often written $\delta_K^\ell$ in the literature). In Sirlin's representation of the $O(\alpha)$ amplitude, the process-independent short-distance piece is perturbative, while the long-distance pieces are: (i) the correction $\delta F_\mu$ to the hadronic weak matrix element, involving the "generalized Compton tensor" $T_{\mu\nu}$, a time-ordered product of the electromagnetic and weak currents whose nonperturbative content requires lattice QCD (a Monte Carlo discretization of QCD on a space-time grid); and (ii) the $\gamma W$-box diagram, a photon–$W$-boson loop with the same nonperturbative kernel. In parallel, chiral perturbation theory (ChPT) — the low-energy effective field theory of QCD organized in derivatives $p$ and photon couplings $e$ — expresses the same corrections order by order in powers of $e^2p^n$, with unknown low-energy constants (LECs) that parametrize short-distance hadron physics. Ma, Feng, Gorchtein, Jin and Seng [Phys. Rev. D 103 (2021) 114503] computed the axial $\gamma W$-box correction for $K^0\to\pi^+e\bar\nu_e$ on the lattice in the flavor SU(3) limit (equal light-quark masses, $m_K=m_\pi$), where the relevant four-point functions simplify, obtaining $\square_{\gamma W}^{VA}|_{K^{0},SU(3)}=2.437(44)\times10^{-3}$ and thereby determining the leading $O(e^2p^2)$ LEC combination $-\frac{8}{3}X_1+\tilde X_6^{\mathrm{phys}}=19.6(1.0)\times10^{-3}$ from first principles, in agreement with the previously used minimal-resonance-model estimate $20.3(14.3)\times10^{-3}$ but with far smaller uncertainty. A later re-analysis by Seng, Galviz, Gorchtein and Meißner [JHEP 07 (2022) 071], combining this lattice input with dispersive form-factor data within a hybrid Sirlin+ChPT framework, achieved $10^{-4}$ precision on $\delta_{\mathrm{EM}}^{K\ell}$ and performed the awaited global $|V_{us}|$ update. However, that framework still estimates several hadronic pieces by chiral power counting rather than computing them: the box is evaluated only in the forward limit ($p_i=p_f$) at unphysical masses, non-forward corrections, inelastic contributions, the three-current "three-point function" piece $\delta F_3$, and the unknown $O(e^2p^4)$ ChPT terms and LECs all remain explicit entries in the current uncertainty budget. An independent cross-discretization lattice computation by Yoo et al. [Phys. Rev. D 108 (2023) 034508] agrees with the SU(3)-limit kaon result but likewise evaluates the kaon box only at the SU(3)-symmetric point. What remains is therefore the completion of the radiative-correction program from first principles.

Problem Statement

Determine, from first principles (lattice QCD+QED combined with perturbative matching at large loop momentum), the complete inclusive $O(\alpha)$ electroweak radiative correction to the semileptonic kaon decay rates $K\to\pi\ell\nu_\ell(\gamma)$ at the physical quark masses — comprising the virtual-photon loops and bremsstrahlung together with the full $\gamma W$-box contribution (axial and vector parts, beyond the forward kinematic limit, including inelastic intermediate states and the three-current contribution $\delta F_3$) — together with the $O(e^2p^4)$ chiral perturbation theory contributions to the long-distance electromagnetic correction $\delta_{\mathrm{EM}}^{K\ell}$, such that $\delta_{\mathrm{EM}}^{K\ell}$ is determined from first principles with an error budget demonstrably controlled at the few$\times10^{-4}$ level required for the $|V_{us}|$ extraction and first-row CKM unitarity test, without residual reliance on model estimates of chiral low-energy constants or uncomputed higher-order chiral terms. The several requested results form one overarching objective: replacing every estimated entry in the current hybrid uncertainty budget of $\delta_{\mathrm{EM}}^{K\ell}$ — the non-forward box, inelastic intermediate states, the three-current contribution $\delta F_3$, the $O(e^2p^4)$/LEC chiral terms, and the residual lattice systematics (discretization, finite volume, splicing to perturbative QCD at large loop momentum $Q^2$) — by a first-principles determination. Each of these entries is a mandatory part of that objective.

References

1. Peng-Xiang Ma, Xu Feng, Mikhail Gorchtein, Lu-Chang Jin, Chien-Yeah Seng, "Lattice QCD calculation of the electroweak box diagrams for the kaon semileptonic decays", Phys. Rev. D 103 (2021) 114503, DOI 10.1103/PhysRevD.103.114503, arXiv:2102.12048, https://arxiv.org/abs/2102.12048
2. Xu Feng, Mikhail Gorchtein, Lu-Chang Jin, Peng-Xiang Ma, Chien-Yeah Seng, "First-principles calculation of electroweak box diagrams from lattice QCD", Phys. Rev. Lett. 124 (2020) 192002, DOI 10.1103/PhysRevLett.124.192002, arXiv:2003.09798, https://arxiv.org/abs/2003.09798
3. Jun-Sik Yoo, Tanmoy Bhattacharya, Rajan Gupta, Santanu Mondal, Boram Yoon, "Electroweak box diagram contribution for pion and kaon decay from lattice QCD", Phys. Rev. D 108 (2023) 034508, DOI 10.1103/PhysRevD.108.034508, arXiv:2305.03198, https://arxiv.org/abs/2305.03198
4. Chien-Yeah Seng, Daniel Galviz, Mikhail Gorchtein, Ulf-G. Meißner, "Improved Ke3 radiative corrections sharpen the Kμ2–Kl3 discrepancy", JHEP 11 (2021) 172, DOI 10.1007/JHEP11(2021)172, arXiv:2103.04843, https://arxiv.org/abs/2103.04843
5. Chien-Yeah Seng, Daniel Galviz, Mikhail Gorchtein, Ulf-G. Meißner, "Complete theory of radiative corrections to Kℓ3 decays and the Vus update", JHEP 07 (2022) 071, DOI 10.1007/JHEP07(2022)071, arXiv:2203.05217, https://arxiv.org/abs/2203.05217
6. M. Gorbahn, S. Jäger, F. Moretti, E. van der Merwe, "Semileptonic weak Hamiltonian to O(αα_s) in momentum-space subtraction schemes", JHEP 01 (2023) 159, DOI 10.1007/JHEP01(2023)159, arXiv:2209.05289, https://arxiv.org/abs/2209.05289
7. Chien-Yeah Seng, "Semileptonic kaon decays and the precise determination of Vus", PoS(CD2024) 024 (2025), proceedings of the 11th International Workshop on Chiral Dynamics, DOI 10.22323/1.479.0024, arXiv:2502.04721, https://arxiv.org/abs/2502.04721
8. Peng-Xiang Ma, Xu Feng, Mikhail Gorchtein, Lu-Chang Jin, Keh-Fei Liu, Chien-Yeah Seng, Bi-Geng Wang, Zhao-Long Zhang, "Lattice QCD Calculation of Electroweak Box Contributions to Superallowed Nuclear and Neutron Beta Decays", Phys. Rev. Lett. 132 (2024) 191901, DOI 10.1103/PhysRevLett.132.191901, arXiv:2308.16755, https://arxiv.org/abs/2308.16755