CKM matrix is unitary by construction in the standard model(SM). The recent analyses on the first row of CKM matrix show $ \approx 3\sigma$ tension with unitarity. Nonperturbative
calculations of the radiative corrections can reduce the theory uncertainty in CKM matrix elements.
Here we compute the electroweak box contribution to the pion and kaon $\beta$ decays using seven $N_f=2+1+1$ HISQ-Clover lattice with various pion mass and lattice spacing.
The continuum and chiral limit is taken using the leading dependence on $M_\pi$ and $a$, where $M_\pi$ extrapolation is taken to the physical pion mass and $SU(3)$ symmetric mass for pion and kaon box contribution, respectively.
Our results are $ \square_{\gamma W}^{VA} |_{\pi} = 2.820 (28) \times 10^{-3} $ and $ \square_{\gamma W}^{VA} |_{K} = 2.384 (17) \times 10^{-3} $.