The Standard Model (SM) prediction for the ratio ε′/ε appears to be significantly below the experimental data. Also εK in the SM tends to be below the data. Any new physics (NP) removing these anomalies will first of all have impact on flavour observables in the K meson system, in particular on rare decays K+→π+νν¯\documentclass[12pt]{minimal}
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\begin{document}$$ {K}^{+}\to {\pi}^{+}\nu \overline{\nu} $$\end{document}, KL→π0νν¯\documentclass[12pt]{minimal}
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\begin{document}$$ {K}_L\to {\pi}^0\nu \overline{\nu} $$\end{document}, KL → μ+μ− and KL → π0ℓ+ℓ− and ΔMK. Restricting the operators contributing to ε′/ε to the SM ones and to the corresponding primed operators, NP contributions to ε′/ε are quite generally dominated either by QCD penguin (QCDP) operators Q6(Q6′) or electroweak penguin (EWP) operators Q8(Q8′) with rather different implications for other flavour observables. Our presentation includes general models with tree-level Z and Z′ flavour violating exchanges for which we summarize known results and add several new ones. We also briefly discuss few specific models. The correlations of ε′/ε with other flavour observables listed above allow to differentiate between models in which ε′/ε can be enhanced. Various DNA-tables are helpful in this respect. We find that simultaneous enhancements of ε′/ε, εK, ℬKL→π0νν¯\documentclass[12pt]{minimal}
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\begin{document}$$ \mathrm{\mathcal{B}}\left({K}_L\to {\pi}^0\nu \overline{\nu}\right) $$\end{document} and ℬK+→π+νν¯\documentclass[12pt]{minimal}
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\begin{document}$$ \mathrm{\mathcal{B}}\left({K}^{+}\to {\pi}^{+}\nu \overline{\nu}\right) $$\end{document} in Z scenarios are only possible in the presence of both left-handed and right-handed flavour-violating couplings. In Z′ scenarios this is not required but the size of NP effects and the correlation between ℬKL→π0νν¯\documentclass[12pt]{minimal}
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\begin{document}$$ \mathrm{\mathcal{B}}\left({K}_L\to {\pi}^0\nu \overline{\nu}\right) $$\end{document} and ℬK+→π+νν¯\documentclass[12pt]{minimal}
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\begin{document}$$ \mathrm{\mathcal{B}}\left({K}^{+}\to {\pi}^{+}\nu \overline{\nu}\right) $$\end{document} depends strongly on whether QCDP or EWP dominate NP contributions to ε′/ε. 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 εK.We point out that QCDP and EWP scenarios of NP in ε′/ε can also be uniquely distinguished by the size and the sign of NP contribution to ΔMK, elevating the importance of the precise calculation of ΔMK in the SM. We emphasize the importance of the theoretical improvements not only on ε′/ε, εK and ΔMK but also on KL → μ+μ−, KL → π0ℓ+ℓ−, and the K → ππ isospin amplitudes ReA0 and ReA2 which would in the future enrich our analysis.
