Determination of $\varepsilon_K$ using lattice QCD inputs

We present results for the indirect CP violation parameter $\varepsilon_K$ determined directly from the standard model using lattice QCD to fix the inputs $\hat{B}_K$, $ξ_0$, $|V_{us}|$, and $|V_{cb}|$. We use the FLAG and SWME results for $\hat{B}_K$. We use the RBC-UKQCD result for $ξ_0$ determined using the experimental value of $\varepsilon'/\varepsilon$ and the lattice result of $\mathrm{Im}\,A_2$. To set the Wolfenstein parameter $λ$, we use $|V_{us}|$, which is determined from $K_{\ell3}$ and $K_{\mu2}$ decays combined with lattice evaluations of the $K \to π\ell ν$ 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 $\bar{B} \to D^* \ell \barν$ 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$σ$. However, we observe no tension in $\varepsilon_K$ determined from inclusive $|V_{cb}|$.

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