New Physics Patterns in $\varepsilon^\prime/\varepsilon$ and $\varepsilon_K$ with Implications for Rare Kaon Decays and $\Delta M_K$

The SM prediction for the ratio $\varepsilon^\prime/\varepsilon$ appears to be significantly below the experimental data. Also $\varepsilon_K$ in the SM tends to be below the data. Any NP removing these anomalies will first of all have impact on flavour observables in the $K$ meson system, in particular on rare $K$ decays and $\Delta M_K$. Restricting the operators contributing to $\varepsilon^\prime/\varepsilon$ to the SM ones and to the corresponding primed operators, NP contributions are quite generally dominated either by QCD penguin (QCDP) operators $Q_6(Q_6^\prime)$ or electroweak penguin (EWP) operators $Q_8(Q_8^\prime)$ with rather different implications for other flavour observables. We discuss general models with tree-level $Z$ and $Z^\prime$ flavour violating exchanges and few specific models. We find that simultaneous enhancements of $\varepsilon^\prime/\varepsilon$, $\varepsilon_K$, $\mathcal{B}(K_L\to\pi^0\nu\bar\nu)$ and $\mathcal{B}(K^+\to\pi^+\nu\bar\nu)$ in $Z$ scenarios are only possible in the presence of both LH and RH flavour-violating couplings. In $Z^\prime$ scenarios this is not required but the size of NP effects and the correlation between $\mathcal{B}(K_L\to\pi^0\nu\bar\nu)$ and $\mathcal{B}(K^+\to\pi^+\nu\bar\nu)$ depends strongly on whether QCDP or EWP dominate NP contributions to $\varepsilon^\prime/\varepsilon$. In the QCDP case possible enhancements of both branching ratios are much larger than for EWP scenario and take place only on the branch parallel to the Grossman-Nir bound, which is in the case of EWP dominance only possible in the absence of NP in $\varepsilon_K$. QCDP and EWP scenarios of NP in $\varepsilon^\prime/\varepsilon$ can also be uniquely distinguished by the size and the sign of NP contribution to $\Delta M_K$, elevating the importance of the precise calculation of $\Delta M_K$ in the SM.