Determination of ${\epsilon }_K$ using lattice QCD inputs

Abstract

We present results for the indirect CP violation parameter ${\epsilon }_K$ 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 ${\epsilon }^{\prime }/\epsilon$ and the lattice result of $\mathrm{Im}{A}_2$. To set the Wolfenstein parameter $\lambda$, we use $|V_{us}|$, which is determined from $K_{\ell 3}$ and $K_{\mu 2}$ 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 $\bar{B}\to D^{\ast }\ell \bar{\nu }$ 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 ${\epsilon }_K$ with exclusive $|V_{cb}|$ (lattice QCD results) is lower than the experimental value by 3.4$\sigma$. However, we observe no tension in ${\epsilon }_K$ determined from inclusive $|V_{cb}|$.

Comments: 7 pages and 3 figures; PoS.cls updated