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One of the sensitive probes of physics beyond the standard model is the testof the unitarity of the Cabbibo-Kobyashi-Maskawa (CKM) matrix. Current analysisof the first row is based on $|V_{ud}|$ from fourteen superallowed $0^+ \to0^+$ nuclear $\beta$ decays and $|V_{ud}|$ from the kaon semileptonic decay, $K\to \pi \ell \nu_\ell$. Modeling the nuclear effects in the $0^+ \to 0^+$decays is a major source of uncertainty, which would be absent in neutrondecays. To make neutron decay competitive requires improving the measurement ofneutron lifetime and the axial charge, as well as the calculation of theradiative corrections (RC) to the decay. The largest uncertainty in these RCs,which comes from the non-perturbative part of the $\gamma W$-box diagram andits evaluation using lattice QCD, is still not under control. Here, we showthat the analogous calculations for the pion and kaon decays are robust andgive $\square_{\gamma W}^{VA}|_{\pi} = 2.810 (26) \times 10^{-3}$ and$\square_{\gamma W}^{VA}|_{K^{0, S U(3)}} = 2.389 (17) \times 10^{-3}$ inagreement with the previous analysis carried out by Feng et al. using adifferent discretization of the fermion action.

Electroweak box diagram contribution for pion and kaon decay from lattice QCD