Electroweak box diagrams on the lattice for pion and neutron decay

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 ${\mathrm{\square }}_{\gamma W}^{VA}{\mathrm{\mid }}_{\pi }=2.820(28)\times {10}^{-3}$ and ${\mathrm{\square }}_{\gamma W}^{VA}{\mathrm{\mid }}_K=2.384(17)\times {10}^{-3}$.