Determination of $\varepsilon_K$ using lattice QCD inputs

TL;DR

This paper calculates the CP violation parameter _K using lattice QCD inputs within the standard model, finding a significant discrepancy with experimental data when using exclusive _{cb} values.

Contribution

It provides a lattice QCD-based determination of _K and compares results using exclusive and inclusive _{cb}, highlighting a 3.4 discrepancy with experiment for the exclusive case.

Findings

Standard model _K prediction lower than experiment by 3.4 with exclusive _{cb}

No tension observed with inclusive _{cb}

Lattice inputs used for key CKM parameters and _K calculation

Abstract

We present results for the indirect CP violation parameter $\varepsilon_K$ determined directly from the standard model using lattice QCD to fix the inputs $\hat{B}_K$, $\xi_0$, $|V_{us}|$, and $|V_{cb}|$. We use the FLAG and SWME results for $\hat{B}_K$. We use the RBC-UKQCD result for $\xi_0$ determined using the experimental value of $\varepsilon'/\varepsilon$ and the lattice result of $\mathrm{Im}\,A_2$. To set the Wolfenstein parameter $\lambda$, we use $|V_{us}|$, which is determined from $K_{\ell3}$ and $K_{\mu2}$ decays combined with lattice evaluations of the $K \to \pi \ell \nu$ vector form factor and $f_K$. To set the Wolfenstein parameter $A$, we use the FNAL/MILC results for $|V_{cb}|$, which are determined from the exclusive decay $\bar{B} \to D^* \ell \bar{\nu}$ and the axial form factor at zero recoil. We also use the inclusive $|V_{cb}|$ obtained using the heavy quark expansion based on QCD sum rules and the OPE. We compare the results with those for exclusive $|V_{cb}|$. We find that the standard model prediction of $\varepsilon_K$ with exclusive $|V_{cb}|$ (lattice QCD results) is lower than the experimental value by 3.4$\sigma$. However, we observe no tension in $\varepsilon_K$ determined from inclusive $|V_{cb}|$.