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}|$butfor$M_{Z^\prime}=10~TeV$onlyfor$|V_{cb}|=0.040$suchshiftscanbeobtained.NPeffectsin$B_s\toμ^+μ^-$andintheWilsoncoefficient$C_9$aresignificantlylargerinallthreemodelsforthecaseof$|V_{cb}|=0.040$.Inparticularintwomodelstheratefor$B_s\toμ^+μ^-$canbereducedbyNPby$20\%$for$M_{Z^\prime}=3~TeV$resultinginvaluesintheballparkofcentralvaluesfromCMSandLHCb.Inthethirdmodelashiftin$C_9$upto$C_9^\text{NP}=-0.5$ispossible.Wealsoconsiderthesimplest331model,analyzedrecentlyintheliterature,inwhich$X=Y$,theusualhypercharge.WefindthatinthismodelNPeffectsinflavourobservablesaremuchsmallerthaninthethreemodelswith$X\not=Y$,inparticularNPcontributionstotheratio$\varepsilon^\prime/\varepsilon$ are very strongly suppressed.

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

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• Updated Standard Model Prediction for ${\epsilon }^{\prime }/\epsilon$
• $c\to u\nu \bar{\nu }$ transitions of $B_c$ mesons: 331 model facing Standard Model null tests
• Impact of Lattice QCD on CKM Phenomenology
• Direct CP violation in $K\to \pi \pi$ decays and supersymmetry