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

One of the sensitive probes of physics beyond the standard model is the test of the unitarity of the Cabbibo-Kobyashi-Maskawa (CKM) matrix. Current analysis of the first row is based on \(|V_{ud}|\) from fourteen superallowed \(0^+ \to 0^+\) 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 neutron decays. To make neutron decay competitive requires improving the measurement of neutron lifetime and the axial charge, as well as the calculation of the radiative corrections (RC) to the decay. The largest uncertainty in these RCs, which comes from the non-perturbative part of the \(\gamma W\)-box diagram and its evaluation using lattice QCD, is still not under control. Here, we show that the analogous calculations for the pion and kaon decays are robust and give \(\square_{\gamma W}^{VA}|_{\pi} = 2.810 (26) \times 10^{-3}\) and \(\square_{\gamma W}^{VA}|_{K^{0, S U(3)}} = 2.389 (17) \times 10^{-3}\) in agreement with the previous analysis carried out by Feng et al. using a different discretization of the fermion action.