  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 $K_S \to \mu^+ \mu^-$.
