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Recent calculations have pointed to a 2.8 $\sigma$ tension between data on$\epsilon^{\prime}_K / \epsilon_K$ and the standard-model (SM) prediction.Several new physics (NP) models can explain this discrepancy, and such NPmodels are likely to predict deviations of $\mathcal{B}(K\to \pi \nu\overline{\nu})$ from the SM predictions, which can be probed precisely in thenear future by NA62 and KOTO experiments. We present correlations between$\epsilon^{\prime}_K / \epsilon_K$ and $\mathcal{B}(K\to \pi \nu\overline{\nu})$ in two types of NP scenarios: a box dominated scenario and a$Z$-penguin dominated one. It is shown that different correlations arepredicted and the future precision measurements of $K \to \pi \nu\overline{\nu}$ can distinguish both scenarios.

Correlations of $\epsilon^{\prime}_K/\epsilon_K$ with $K \to \pi \nu \overline{\nu}$ in Models of New Physics