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

Abstract

We report the Standard Model evaluation of the indirect CP violation parameter ${\epsilon }_K$ using inputs determined from lattice QCD: the kaon bag 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 the exclusive decay $\bar{B}\to D^{\ast }\ell \bar{\nu }$ at zero-recoil. The theoretical expression for ${\epsilon }_K$ is thoroughly reviewed to give an estimate of the size of the neglected corrections, including long distance effects. The Wolfenstein parametrization $(|V_{cb}|,\lambda ,\bar{\rho },\bar{\eta })$ is adopted for CKM matrix elements which enter through the short distance contribution of the box diagrams. For the central value, we take the Unitarity Triangle apex $(\bar{\rho },\bar{\eta })$ from the angle-only fit of the UTfit collaboration and use $V_{us}$ as an independent input to fix $\lambda$. We find that the Standard Model prediction of ${\epsilon }_K$ with exclusive $V_{cb}$ (lattice QCD results) is lower than the experimental value by $3.4\sigma$. However, with inclusive $V_{cb}$ (results of the heavy quark expansion), there is no gap between the Standard Model prediction of ${\epsilon }_K$ and its experimental value. For the calculation of ${\epsilon }_K$, we perform the renormalization group running to obtain ${\eta }_{cc}$ at next-to-next-to-leading-order; we find ${\eta }_{cc}^{\mathrm{NNLO}}=1.72(27)$.

Comments: 20 pages, 3 figures, references augmented, typos fixed