2024 Update on ${\epsilon }_K$ 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 ${\hat{B}}_K$, exclusive $|V_{cb}|$, $|V_{us}|$, $|V_{ud}|$, $\hat{I}{¾}_0$, $\hat{I}{¾}_2$, $\hat{I}{¾}_{\text{LD}}$, $f_K$, and $m_c$. We find that the standard model with exclusive $|V_{cb}|$ and lattice QCD inputs describes only $2/3\cong 65\%$ of the experimental value of $|{\epsilon }_K|$ and does not explain its remaining 35\%, which represents a strong tension in $|{\epsilon }_K|$ at the $5.1\ddot{I}ƒ\sim 4.1\ddot{I}ƒ$ 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. We also report results for $|{\epsilon }_K|$ obtained using the Brod-Gorbahn-Stamou (BGS) method for $\hat{I}{\cdot }_i$ of $u-t$ unitarity, which leads to even a stronger tension of $5.7\ddot{I}ƒ\sim 4.2\ddot{I}ƒ$ with lattice QCD inputs.

11 pages, 4 figures, Lattice 2024 proceeding. arXiv admin note: substantial text overlap with arXiv:2312.02986, arXiv:2301.12375