Another SMEFT story: Z′ facing new results on ε′/ε, ∆MK and K → $$ \pi \nu \overline{\nu} $$

Recently the RBC-UKQCD lattice QCD collaboration presented new results for the hadronic matrix elements relevant for the ratio ε′/ε in the Standard Model (SM) albeit with significant uncertainties. With the present knowledge of the Wilson coefficients and isospin breaking effects there is still a sizable room left for new physics (NP) contributions to ε′/ε which could both enhance or suppress this ratio to agree with the data. The new SM value for the K0 − $$ {\overline{K}}^0 $$ K ¯ 0 mass difference ∆MK from RBC-UKQCD is on the other hand by 2σ above the data hinting for NP required to suppress ∆MK. Simultaneously the most recent results for K+ → $$ {\pi}^{+}\nu \overline{\nu} $$ π + ν ν ¯ from NA62 and for KL → $$ {\pi}^0\nu \overline{\nu} $$ π 0 ν ν ¯ from KOTO still allow for significant NP contributions. We point out that the suppression of ∆MK by NP requires the presence of new CP-violating phases with interesting implications for K → $$ \pi \nu \overline{\nu} $$ πν ν ¯ , KS → μ+μ− and KL → π0ℓ+ℓ− decays. Considering a Z′-scenario within the SMEFT we analyze the dependence of all these observables on the size of NP still allowed by the data on ε′/ε. The hinted ∆MK anomaly together with the εK constraint implies in the presence of only left-handed (LH) or right-handed (RH) flavour-violating Z′ couplings strict correlation between K+ → $$ {\pi}^{+}\nu \overline{\nu} $$ π + ν ν ¯ and KL → $$ {\pi}^0\nu \overline{\nu} $$ π 0 ν ν ¯ branching ratios so that they are either simultaneously enhanced or suppressed relative to SM predictions. An anticorrelation can only be obtained in the presence of both LH and RH couplings. Interestingly, the NP QCD penguin scenario for ε′/ε is excluded by SMEFT renormalization group effects in εK so that NP effects in ε′/ε are governed by electroweak penguins. We also investigate for the first time whether the presence of a heavy Z′ 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.