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

Abstract

We report recent progress on ${\epsilon }_K$ evaluated directly from the standard model (SM) with lattice QCD inputs such as ${\hat{B}}_K$, $|V_{cb}|$, $|V_{us}|$, $|V_{ud}|$, ${\xi }_0$, ${\xi }_2$, ${\xi }_{\text{LD}}$, $f_K$, and $m_c$. We find that the standard model with exclusive $|V_{cb}|$ and lattice QCD inputs describes only 66\% of the experimental value of $|{\epsilon }_K|$ and does not explain its remaining 34\%, which corresponds to a strong tension in $|{\epsilon }_K|$ 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 $|V_{cb}|$ obtained using the heavy quark expansion based on the QCD sum rule approach.

Comments: 11 pages, 8 tables, 4 figures, Lattice 2023 proceedings. arXiv admin note: substantial text overlap with arXiv:2301.12375, arXiv:2202.11473. references are added