${\epsilon }^{\mathrm{\prime }}\mathrm{/}\epsilon$ in 331 Models

Motivated by the recent findings that the ratio ${\epsilon }^{\mathrm{\prime }}\mathrm{/}\epsilon$ in the Standard Model (SM) appears to be significantly below the data we investigate whether the necessary enhancement of this ratio can be obtained in 331 models, in which new physics (NP) contributions to ${\epsilon }^{\mathrm{\prime }}\mathrm{/}\epsilon$ and other flavour observables are dominated by tree-level exchanges of a $Z^{\mathrm{\prime }}$. NP contributions to ${\epsilon }^{\mathrm{\prime }}\mathrm{/}\epsilon$ in these models are governed by the electroweak operator $Q_8$ for which the hadronic matrix element is rather well known so that our analysis of NP contributions is subject to much smaller uncertainties than within the SM. We consider seven 331 models selected in our earlier analysis on the basis of electroweak precision data. Imposing the constraints from $\mathrm{\Delta }F=2$ transitions we find that only three of these models can provide a significant positive shift in ${\epsilon }^{\mathrm{\prime }}\mathrm{/}\epsilon$ up to $6\times {10}^{-4}$ for $M_{Z^{\mathrm{\prime }}}=3$ TeV. Two of them allow simultaneously a supression of $\mathcal{B}(B_s\to {\mu }^{+}{\mu }^{-})$ by $20\mathrm{\%}$, bringing the theory closer to the data without any significant impact on the Wilson coefficient $C_9$. The third one provides also the shift $\mathrm{\Delta }{C}_9=-0.6$, softening the anomalies in $B\to K^{\ast }{\mu }^{+}{\mu }^{-}$, without any significant impact on $B_{s}\to \mu^+\mu^-$\mathrm{.}NPeffectsinrare$K$decaysandin$B\to K(K^*)\nu\bar\nu$ turn out to be small. The flavour structure of 331 models implies that even for $M_{Z^{\mathrm{\prime }}}=30$ Tev a shift of ${\epsilon }^{\mathrm{\prime }}\mathrm{/}\epsilon$ up to $8\times 10^{-4}$andasignificantshiftin$\varepsilon_K$ can be obtained, while the effects in other flavour observables are small.