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

Abstract

Recently the RBC-UKQCD lattice collaboration presented new results for the hadronic matrix elements relevant for the ratio ${\epsilon }^{\prime }/\epsilon$ 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 ${\epsilon }^{\prime }/\epsilon$ which could both enhance or suppress this ratio to agree with the data. The new SM value for the $K^0-{\bar{K}}^0$ 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 $K^{+}\to {\pi }^{+}\nu \bar{\nu }$ 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 $\Delta M_K$ 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 $K_L\to {\pi }^0{\ell }^{+}{\ell }^{-}$ decays. Considering a $Z^{\prime }$-scenario within the SMEFT we analyze the dependence of all these observables on the size of NP still allowed by the data on ${\epsilon }^{\prime }/\epsilon$. The NP QCD penguin scenario for ${\epsilon }^{\prime }/\epsilon$ is excluded by SMEFT renormalization group effects in ${\epsilon }_K$ so that NP effects in ${\epsilon }^{\prime }/\epsilon$ are governed by electroweak penguins. We also investigate for the first time whether the presence of a heavy $Z^{\prime }$ 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.

Comments: 35 pages, 12 figures; v2: added references, results unchanged