Standard-Model Prediction of ${\epsilon }_K$ with Manifest Quark-Mixing Unitarity

The parameter ${\epsilon }_K$ describes $CP$ violation in the neutral kaon system and is one of the most sensitive probes of new physics. The large uncertainties related to the charm-quark contribution to ${\epsilon }_K$ have so far prevented a reliable standard-model prediction. We show that Cabibbo-Kobayashi-Maskawa unitarity enforces a unique form of the $|\mathrm{\Delta }S=2|$ weak effective Lagrangian in which the short-distance theory uncertainty of the imaginary part is dramatically reduced. The uncertainty related to the charm-quark contribution is now at the percent level. We present the updated standard-model prediction ${\epsilon }_K=2.16(6)(8)(15)\times {10}^{-3}$, where the errors in parentheses correspond to QCD short-distance, long-distance, and parametric uncertainties, respectively.

Figure 1

Comparison of Wilson coefficients in $u\text{−}t$ (first and third plot) and $c\text{−}t$ unitarity (second and fourth plot). Shown is the residual renormalization-scale dependence of the RI Wilson coefficients as a proxy for their theory uncertainty. In the two plots on the left the five-flavor threshold, ${\mu }_t$, is varied, while in the two on the right the three-flavor threshold, ${\mu }_c$, is varied (see text for further details).