2024 Update on ${\epsilon }_K$ with lattice QCD inputs

We report recent progress on ${\epsilon }_K$ evaluated directly from the standard model (SM) with lattice QCD inputs such as ${\widehat{B}}_K$, exclusive $\mathrm{\mid }{V}_{cb}\mathrm{\mid }$, $\mathrm{\mid }{V}_{us}\mathrm{\mid }$, $\mathrm{\mid }{V}_{ud}\mathrm{\mid }$, ${\xi }_0$, ${\xi }_2$, ${\xi }_{\text{LD}}$, $f_K$, and $m_c$. We find that the standard model with exclusive $\mathrm{\mid }{V}_{cb}\mathrm{\mid }$ and lattice QCD inputs describes only $2\mathrm{/}3\cong 65\mathrm{\%}$ of the experimental value of $\mathrm{\mid }{\epsilon }_K\mathrm{\mid }$ and does not explain its remaining 35%, which represents a strong tension in $\mathrm{\mid }{\epsilon }_K\mathrm{\mid }$ at the $5.1\sigma \sim 4.1\sigma$ 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. We also report results for $\mathrm{\mid }{\epsilon }_K\mathrm{\mid }$ obtained using the Brod-Gorbahn-Stamou (BGS) method for ${\eta }_i$ of $u-t$ unitarity, which leads to even a stronger tension of $5.7\sigma \sim 4.2\sigma$ with lattice QCD inputs.