2023 Update of ${\epsilon }_K$ with lattice QCD inputs

We report recent progress on ${\epsilon }_K$ evaluated directly from the standard model (SM) with lattice QCD inputs such as ${\widehat{B}}_K$, $\mathrm{\mid }{V}_{cb}\mathrm{\mid }$, $\mathrm{\mid }{V}_{us}\mathrm{\mid }$, $\mathrm{\mid }{V}_{ud}\mathrm{\mid }$, ${\xi }_0$, ${\xi }_2$, ${\xi }_{\text{LD}}$, $f_K$, and $m_c$. We find that the standard model with exclusive $\mathrm{\mid }{V}_{cb}\mathrm{\mid }$ and lattice QCD inputs describes only 66% of the experimental value of $\mathrm{\mid }{\epsilon }_K\mathrm{\mid }$ and does not explain its remaining 34%, which corresponds to a strong tension in $\mathrm{\mid }{\epsilon }_K\mathrm{\mid }$ at the $4.9\sigma \sim 3.9\sigma$ level between the SM theory and experiment. We also find that this tension disappears when we use the inclusive value of $\mathrm{\mid }{V}_{cb}\mathrm{\mid }$ obtained using the heavy quark expansion based on the QCD sum rule approach.