Updated evaluation of ${\epsilon }_K$ in the Standard Model with lattice QCD inputs

Abstract

We report a strong tension in ${\epsilon }_K$ at the $4\sigma$ level between the experimental value and the theoretical value calculated directly from the standard model using lattice QCD inputs such as ${\hat{B}}_K$, $|V_{cb}|$, $|V_{us}|$, ${\xi }_0$, ${\xi }_2$, ${\xi }_{\text{LD}}$, $F_K$, and $m_c$. The standard model with lattice QCD inputs describes only 70% of the experimental value of ${\epsilon }_K$, and does not explain its remaining 30%. We also find that this tension disappears when we use the inclusive value of $|V_{cb}|$ (results of the heavy quark expansion based on QCD sum rules) to determine ${\epsilon }_K$. This tension is highly correlated with the present discrepancy between the exclusive and inclusive values of $|V_{cb}|$. In order to resolve, in part, the issue with $|V_{cb}|$, it would be highly desirable to have a comprehensive re-analysis over the entire set of experimental data on the $\bar{B}\to D^{\ast }\ell \bar{\nu }$ decays using an alternative parametrization of the form factors, such as the BGL parametrization, and a comparison with results of the CLN method.

Comments: 17 pages, 12 figures, references updated, title changed