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
fifteen superallowed $0^+\!\! \to 0^+ $ nuclear $\beta$ decays and
$|V_{us}|$ 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 comes from the
non-perturbative part of the $\gamma W$-box diagram, and lattice QCD provides
a first principle method for its evaluation.
Our calculations, using lattice configurations generated with highly
improved staggered quarks by the MILC Collaboration, show that
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} \Big|_{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.