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Recently the RBC-UKQCD lattice collaboration presented new results for thehadronic matrix elements relevant for the ratio $\varepsilon'/\varepsilon$ inthe Standard Model (SM). With the present knowledge of the Wilson coefficientsand isospin breaking effects there is still much room for new physics (NP)contributions to $\varepsilon'/\varepsilon$ which could both enhance orsuppress this ratio to agree with the data. The new SM value for the $K^0-\barK^0$ mass difference $\Delta M_K$ from RBC-UKQCD is on the other hand by$2\sigma$ above the data hinting for NP required to suppress $\Delta M_K$.Simultaneously the most recent results for $K^+\rightarrow\pi^+\nu\bar\nu$ fromNA62 and for $K_{L}\rightarrow\pi^0\nu\bar\nu$ from KOTO still allow forsignificant NP contributions. We point out that the suppression of $\Delta M_K$by NP requires the presence of new CP-violating phases with interestingimplications for $K\to\pi\nu\bar\nu$, $K_S\to\mu^+\mu^-$ and$K_L\to\pi^0\ell^+\ell^-$ decays. Considering a $Z^\prime$-scenario within theSMEFT we analyze the dependence of all these observables on the size of NPstill allowed by the data on $\varepsilon'/\varepsilon$. The NP QCD penguinscenario for $\varepsilon'/\varepsilon$ is excluded by SMEFT renormalizationgroup effects in $\varepsilon_K$ so that NP effects in$\varepsilon'/\varepsilon$ are governed by electroweak penguins. We alsoinvestigate for the first time whether the presence of a heavy $Z^\prime$ withflavour violating couplings could generate through top Yukawa renormalizationgroup effects FCNCs mediated by the SM $Z$-boson. The outcome turns out to bevery interesting.

Another SMEFT Story: $Z^\prime$ Facing New Results on $\varepsilon'/\varepsilon$, $\Delta M_K$ and $K\to\pi\nu\bar\nu$