Recent developments on direct $CP$ violation in the kaon system and connection to $K \to \pi \nu \bar{\nu}$ measurements

The first lattice result from the RBC and UKQCD Collaborations and improved perturbative calculations of $\varepsilon^{\prime}_K / \varepsilon_K$ have implied that the Standard-Model (SM) expectation deviates from measured values at the $2.8\,\sigma$ level. Since $\varepsilon^{\prime}_K / \varepsilon_K$ comes from $CP$-violating FCNC and is significantly suppressed in the SM, the discrepancy can be explained easily in several new physics (NP) models. In this contribution, it is shown that correlations with the other rare decays, especially $K\to \pi \nu \overline{\nu}$ and $K_S \to \mu^+ \mu^-$, are crucial for discrimination of the NP models. These channels can be probed precisely in the future by the NA62 and KOTO experiments for $K\to \pi \nu \overline{\nu}$ and LHCb experiment for $K_S \to \mu^+ \mu^-$.