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Motivated by the recently improved lattice QCD results on the hadronic matrixelements entering $\Delta M_{s,d}$ in $B_{s,d}^0-\bar B_{s,d}^0$ mixings andthe resulting increased tensions between $\Delta M_{s,d}$ and $\varepsilon_K$in the Standard Model and CMFV models, we demonstrate that these tensions canbe removed in 331 models based on the gauge group $SU(3)_C\times SU(3)_L\timesU(1)_X$ both for $M_{Z^\prime}$ in the LHC reach and well beyond it. But theimplied new physics (NP) patterns in $\Delta F=1$ observables dependsensitively on the value of $|V_{cb}|$. Concentrating the analysis on three 331models that have been selected by us previously on the basis of theirperformance in electroweak precision tests and $\varepsilon^\prime/\varepsilon$we illustrate this for $|V_{cb}|=0.042$ and $|V_{cb}|=0.040$. We find thatthese new lattice data still allow for positive shifts in$\varepsilon^\prime/\varepsilon$ up to $6\times 10^{-4}$ for  $M_{Z^\prime}=3~TeV$ for both values of $$|V_{cb}|$ but for$M_{Z^\prime}=10~TeV$ only for $|V_{cb}|=0.040$ such shifts can be obtained. NPeffects in $B_s\to\mu^+\mu^-$ and in the Wilson coefficient $C_9$ aresignificantly larger in all three models for the case of $|V_{cb}|=0.040$. Inparticular in two models the rate for $B_s\to\mu^+\mu^-$ can be reduced by NPby $20\%$ for $M_{Z^\prime}=3~TeV$ resulting in values in the ballpark ofcentral 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. Wefind that in this model NP effects in flavour observables are much smaller thanin the three models with $X\not=Y$, in particular NP contributions to the ratio  $\varepsilon^\prime/\varepsilon$ are very strongly suppressed.

the journal of high energy physics/the journal of high energy physics

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