331 Models Facing the Tensions in $\Delta F=2$ Processes with the Impact on ${\epsilon }^{\prime }/\epsilon$, $B_s\to {\mu }^{+}{\mu }^{-}$ and $B\to K^{\ast }{\mu }^{+}{\mu }^{-}$

Abstract

Motivated by the recently improved lattice QCD results on the hadronic matrix elements entering $\Delta M_{s,d}$ in $B_{s,d}^0-{\bar{B}}_{s,d}^0$ mixings and the resulting increased tensions between $\Delta M_{s,d}$ and ${\epsilon }_K$ in the Standard Model and CMFV models, we demonstrate that these tensions can be removed in 331 models based on the gauge group $SU(3{)}_C\times SU(3{)}_L\times U(1{)}_X$ both for $M_{Z^{\prime }}$ in the LHC reach and well beyond it. But the implied new physics (NP) patterns in $\Delta F=1$ observables depend sensitively on the value of $|V_{cb}|$. Concentrating the analysis on three 331 models that have been selected by us previously on the basis of their performance in electroweak precision tests and ${\epsilon }^{\prime }/\epsilon$ we illustrate this for $|V_{cb}|=0.042$ and $|V_{cb}|=0.040$. We find that these new lattice data still allow for positive shifts in ${\epsilon }^{\prime }/\epsilon$ up to $6\times 10^{-4}$ for $M_{Z^{\prime }}=3TeV$ for both values of $$|V_{cb}|$ but for $M_{Z^{\prime }}=10TeV$ only for $|V_{cb}|=0.040$ such shifts can be obtained. NP effects in $B_s\to {\mu }^{+}{\mu }^{-}$ and in the Wilson coefficient $C_9$ are significantly larger in all three models for the case of $|V_{cb}|=0.040$. In particular in two models the rate for $B_s\to {\mu }^{+}{\mu }^{-}$ can be reduced by NP by $20\%$ for $M_{Z^{\prime }}=3TeV$ resulting in values in the ballpark of central values from CMS and LHCb. In the third model a shift in $C_9$ up to $C_9^{\text{NP}}=-0.5$ is possible. We also consider the simplest 331 model, analyzed recently in the literature, in which $X=Y$, the usual hypercharge. We find that in this model NP effects in flavour observables are much smaller than in the three models with $X\ne Y$, in particular NP contributions to the ratio ${\epsilon }^{\prime }/\epsilon$ are very strongly suppressed.

Comments: LaTeX, 24 pages, 16 figures. Discussion on $V_{ub}$ and $V_{cb}$ improved, two more figures added, conclusions unchanged