331 Models Facing the Tensions in Δ F = 2 Processes with the Impact on ε ′ /ε , B s → μ + μ − and B → K ∗ μ + μ −

Motivated by the recently improved results from the fermilab lattice and milc collaborations on the hadronic matrix elements entering δm s,d in $$ {b}_{s,d}^0-{overline{b}}_{s,d}^0 $$ mixings and the resulting increased tensions between δm s,d and ε k in the standard model (sm) and cmfv models, we demonstrate that these tensions can be removed in 331 models based on the gauge group su(3) c × su(3) l × u(1) x both for m z ′ in the lhc reach and well beyond it. but the implied new physics (np) patterns in δf = 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 ε′ /ε 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 ε ′ /ε up to 6 × 10 −4 for m z ′ = 3 tev and |v ub | = 0.0036 for both values of |v cb | but for m z ′ = 10 tev only for |v cb | = 0.040 such shifts can be obtained. for |v ub | = 0.0042 maximal shifts in ε′ /ε increase to ≃ 7×10 −4. np effects in b s → μ + μ − 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 → μ + μ − can be reduced by np by 20% for m z ′ = 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 np  − 0.5 is possible. for |v cb | = 0.042, np effects in b s → μ + μ − and in c 9 are by at least a factor of two smaller. for m z ′ = 10 tev np effects in b s → μ + μ − and c 9, independently of |v cb |, are at most at the level of a few percent. 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 ≠ y , in particular np contributions to the ratio ε ′ /ε are very strongly suppressed. our analysis exhibits the important role of lattice qcd and of precise values of ckm parameters, in particular |v cb |, for quark flavour phenomenology beyond the sm. it also demonstrates exceptional role of δf = 2 observables and of ε ′ /ε in testing high energy scales beyond the lhc.