Another SMEFT Story: $Z^{\mathrm{\prime }}$ Facing New Results on ${\epsilon }^{\mathrm{\prime }}\mathrm{/}\epsilon$, $\mathrm{\Delta }{M}_K$ and $K\to \pi \nu \bar{\nu }$

Recently the RBC-UKQCD lattice collaboration presented new results for the hadronic matrix elements relevant for the ratio ${\epsilon }^{\mathrm{\prime }}\mathrm{/}\epsilon$ in the Standard Model (SM). With the present knowledge of the Wilson coefficients and isospin breaking effects there is still much room for new physics (NP) contributions to ${\epsilon }^{\mathrm{\prime }}\mathrm{/}\epsilon$ which could both enhance or suppress this ratio to agree with the data. The new SM value for the $K^0-\bar K^0$massdifference$\Delta M_K$fromRBC-UKQCDisontheotherhandby$2\sigma$abovethedatahintingforNPrequiredtosuppress$\Delta M_K$. Simultaneously the most recent results for $K^{+}\to {\pi }^{+}\nu \bar{\nu }$ from NA62 and for $K_L\to {\pi }^0\nu \bar{\nu }$ from KOTO still allow for significant NP contributions. We point out that the suppression of $\mathrm{\Delta }{M}_K$ by NP requires the presence of new CP-violating phases with interesting implications for $K\to \pi \nu \bar{\nu }$, $K_S\to {\mu }^{+}{\mu }^{-}$ and $K_L\to {\pi }^0{\mathrm{\ell }}^{+}{\mathrm{\ell }}^{-}$ decays. Considering a $Z^{\mathrm{\prime }}$-scenario within the SMEFT we analyze the dependence of all these observables on the size of NP still allowed by the data on ${\epsilon }^{\mathrm{\prime }}\mathrm{/}\epsilon$. The NP QCD penguin scenario for ${\epsilon }^{\mathrm{\prime }}\mathrm{/}\epsilon$ is excluded by SMEFT renormalization group effects in ${\epsilon }_K$ so that NP effects in ${\epsilon }^{\mathrm{\prime }}\mathrm{/}\epsilon$ are governed by electroweak penguins. We also investigate for the first time whether the presence of a heavy $Z^{\mathrm{\prime }}$ with flavour violating couplings could generate through top Yukawa renormalization group effects FCNCs mediated by the SM $Z$-boson. The outcome turns out to be very interesting.