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We report the Standard Model evaluation of the indirect CP violationparameter $\varepsilon_K$ using inputs determined from lattice QCD: the kaonbag parameter $\hat{B}_K$, $\xi_0$, $|V_{us}|$ from the $K_{\ell 3}$ and$K_{\mu 2}$ decays, and $|V_{cb}|$ from the axial current form factor for theexclusive decay $\bar{B} \to D^* \ell \bar{\nu}$ at zero-recoil. Thetheoretical expression for $\varepsilon_K$ is thoroughly reviewed to give anestimate of the size of the neglected corrections, including long distanceeffects. The Wolfenstein parametrization $(|V_{cb}|, \lambda, \bar{\rho},\bar{\eta})$ is adopted for CKM matrix elements which enter through the shortdistance contribution of the box diagrams. For the central value, we take theUnitarity Triangle apex $(\bar{\rho}, \bar{\eta})$ from the angle-only fit ofthe UTfit collaboration and use $V_{us}$ as an independent input to fix$\lambda$. We find that the Standard Model prediction of $\varepsilon_K$ withexclusive $V_{cb}$ (lattice QCD results) is lower than the experimental valueby $3.4\sigma$. However, with inclusive $V_{cb}$ (results of the heavy quarkexpansion), there is no gap between the Standard Model prediction of$\varepsilon_K$ and its experimental value. For the calculation of$\varepsilon_K$, we perform the renormalization group running to obtain$\eta_{cc}$ at next-to-next-to-leading-order; we find$\eta_{cc}^\mathrm{NNLO}=1.72(27)$.

Standard Model evaluation of $\varepsilon_K$ using lattice QCD inputs for $\hat{B}_K$ and $V_{cb}$