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

TL;DR

This paper investigates 331 models to resolve tensions in $ riangle F=2$ processes, analyzing their impact on $oldsymbol{ ext{ extmu}}^{+} ext{ extmu}^{-}$, $oldsymbol{ ext{ extmu}}^{+} ext{ extmu}^{-}$, and $oldsymbol{ ext{ extmu}}^{+} ext{ extmu}^{-}$ observables, considering recent lattice QCD results and different $|V_{cb}|$ values.

Contribution

The study demonstrates that 331 models can resolve existing tensions in flavor physics and explores their effects on key observables, highlighting the dependence on $|V_{cb}|$ and $Z'$ mass.

Findings

Tensions in $ riangle M_{s,d}$ and $oldsymbol{ ext{ extmu}}^{+} ext{ extmu}^{-}$ processes can be alleviated in 331 models.

Positive shifts in $oldsymbol{ ext{ extmu}}^{+} ext{ extmu}^{-}/oldsymbol{ ext{ extmu}}^{+} ext{ extmu}^{-}$ are possible up to $6 imes 10^{-4}$ for $M_{Z'}=3$ TeV.

NP effects are larger for $|V_{cb}|=0.040$, with potential 20% reduction in $B_s o oldsymbol{ ext{ extmu}}^{+} ext{ extmu}^{-}$ rate.

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 $\varepsilon_K$ 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 $\varepsilon^\prime/\varepsilon$ 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 $\varepsilon^\prime/\varepsilon$ 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. 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 $M_{Z^\prime}=3~TeV$ 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 $X\not=Y$, in particular NP contributions to the ratio $\varepsilon^\prime/\varepsilon$ are very strongly suppressed.