331 Model Predictions for Rare $B$ and $K$ Decays, and $\Delta F=2$ Processes: an Update

Motivated by the improved results from the HPQCD lattice collaboration on the hadronic matrix elements entering $\Delta M_{s,d}$ in $B_{s,d}^0-\bar B_{s,d}^0$ mixings and the increase of the experimental branching ratio for $B_s\to\mu^+\mu^-$, we update our 2016 analysis of various flavour observables in four 331 models, M1, M3, M13 and M16 based on the gauge group $SU(3)_C\times SU(3)_L\times U(1)_X$. These four models, which are distinguished by the quantum numbers, are selected among 24 331 models through their consistency with the electroweak precision tests and simultaneously by the relation $C_9^\text{NP}=-b\, C_{10}^\text{NP}$ with $b\ge 2$, which after new result on $B_s\to\mu^+\mu^-$ from CMS is favoured over the popular relation $C_9^\text{NP}=- C_{10}^\text{NP}$ predicted by several leptoquark models. In this context we investigate in particular the dependence of various observables on $|V_{cb}|$, varying it in the broad range $[0.0386,\,0.043]$, that encompasses both its inclusive and exclusive determinations. Imposing the experimental constraints from $\varepsilon_K$, $\Delta M_s$, $\Delta M_d$ and the mixing induced CP asymmetries $S_{\psi K_S}$ and $S_{\psi K_S}$, we investigate for which values of $|V_{cb}|$ the four models can be made compatible with these data and what is the impact on $B$ and $K$ branching ratios. In particular we analyse NP contributions to the Wilson coefficients $C_9$ and $C_{10}$ and the decays $B_{s,d}\to\mu^+\mu^-$, $K^+\rightarrow\pi^+\nu\bar\nu$ and $K_L\rightarrow\pi^0\nu\bar\nu$. This allows us to illustrate how the value of $|V_{cb}|$ determined together with other parameters of these models is infected by NP contributions and compare it with the one obtained recently under the assumption of the absence of NP in $\varepsilon_K$, $\Delta M_s$, $\Delta M_d$ and $S_{\psi K_S}$.