Updated Standard Model Prediction for $K \to \pi \nu \bar{\nu}$ and $\epsilon_K$

The rare $K \to \pi \nu \bar{\nu}$ decay modes and the parameter $\epsilon_K$ that measures CP violation in Kaon mixing are sensitive probes of physics beyond the standard model. In this article we provide the updated standard-model prediction for the rare decay modes in detail, and summarise the status of standard-model prediction of $\epsilon_K$. We find $\text{BR}(K^+ \to \pi^+ \nu \bar \nu) = 7.73(61) \times 10^{-11}$ and $\text{BR}(K_L \to \pi^0 \nu \bar \nu) = 2.59(29) \times 10^{-11}$. The uncertainties are dominated by parametric input.