2024 Update on $\varepsilon_K$ with lattice QCD inputs

TL;DR

This paper updates the calculation of the CP violation parameter  in the standard model using lattice QCD inputs, revealing a significant tension with experimental values when using exclusive |V_{cb}|, which diminishes with inclusive |V_{cb}|.

Contribution

It provides the latest lattice QCD-based evaluation of , highlighting a persistent tension with experimental data and comparing different methods of determining |V_{cb}|.

Findings

Standard model with exclusive |V_{cb}| explains ~65% of experimental |.

Significant tension (~5.1 to 4.1 sigma) between theory and experiment.

Using inclusive |V_{cb}| reduces the tension, but alternative methods increase it.

Abstract

We report recent progress on $\varepsilon_K$ evaluated directly from the standard model (SM) with lattice QCD inputs such as $\hat{B}_K$, exclusive $|V_{cb}|$, $|V_{us}|$, $|V_{ud}|$, $\xi_0$, $\xi_2$, $\xi_\text{LD}$, $f_K$, and $m_c$. We find that the standard model with exclusive $|V_{cb}|$ and lattice QCD inputs describes only $2/3 \cong 65\%$ of the experimental value of $|\varepsilon_K|$ and does not explain its remaining 35\%, which represents a strong tension in $|\varepsilon_K|$ 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 $|V_{cb}|$ obtained using the heavy quark expansion based on the QCD sum rule approach. We also report results for $|\varepsilon_K|$ 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.