Z-Z' Mixing and Z-Mediated FCNCs in SU(3)_C x SU(3)_L x U(1)_X Models

Most of the existing analyses of FCNC processes in the 331 models, based on the gauge group SU(3)_C x SU(3)_L x U(1)_X, take only into account tree-level exchanges of a new heavy neutral gauge boson Z'. However due to the Z-Z' mixing also corresponding contributions from Z boson are present that are usually neglected. We calculate the impact of these contributions on Delta F=2 processes and rare K, B_s and B_d decays for different values of a parameter beta, which distinguishes between various 331 models and for different fermion representations under the SU(3)_L group. We find a general expression for the Z-Z' mixing in terms beta, M_Z, M_Z' and tan(bar{beta}), familiar from 2 Higgs Doublet models, that differs from the one quoted in the literature. We study in particular the models with beta=+-n/sqrt{3} with n=1,2 which have recently been investigated by us in the context of new data on B_{s,d}->mu^+ mu^- and B_d->K^*(K)mu^+ mu^-. We find that these new contributions can indeed be neglected in the case of Delta F=2 transitions and decays, like B_d->K^*mu^+mu^-, where they are suppressed by the small vectorial Z coupling to charged leptons. However the contributions of tree-level Z exchanges to decays sensitive to axial-vector couplings, like B_{s,d}->mu^+ mu^- and B_d->K mu^+ mu^-, and those with neutrinos in the final state, like b->s nu bar{nu} transitions, K^+->pi^+ nu bar{nu} and K_L->pi^0 nu bar{nu} cannot be generally neglected with size of Z contributions depending on beta, tan(bar{beta}) and M_Z'. We analyze for the first time the ratio epsilon'/epsilon in these models including both Z' and Z contributions. Our analysis of electroweak precision observables within 331 models demonstrates transparently that the interplay of NP effects in electroweak precision observables and those in flavour observables could allow in the future to identify the favourite 331 model.


Introduction
An interesting class of dynamical models are the 331 models based on the gauge group SU (3) C × SU (3) L × U (1) X [1,2]. Detailed analyses of FCNC processes in these models have been presented by us in [3,4]. Selection of earlier analyses of various aspects of these models related to flavour physics can be found in [5][6][7][8][9][10][11][12][13][14][15]. For other variants of 331 models see [16][17][18]. We briefly recall here only a few aspects of these models relevant for the analysis in this paper. As will be discussed in section 2.2, fermion representations under SU (3) L transformations can be chosen in several ways. However, requirement of anomaly cancelation and asymptotic freedom of QCD imposes that, for example, if two quark generations transform as triplets, the remaining one should be an antitriplet. An interesting relation connects the electric charge Q to the generators T 3 , T 8 of SU (3) and the generator X of U (1) X : Q = T 3 + βT 8 + X, introducing the parameter β that plays a key role in this class of models.
Having an enlarged gauge group with respect to the SM, a number of new gauge bosons is present, whose charge depend on the value of β. However, independently of it, a new neutral gauge boson Z is always present and can mediate FCNC at tree level in the quark sector. The Higgs sector is also enlarged. In particular, three Higgs triplets are present. Among these, two give masses to up and down type quarks, and the relative size of their vacuum expectation values will be important for our subsequent discussion. We shall introduce them later in Section 2. Finally, also new heavy fermions are predicted to exist, but they do not play any role in our study and we shall not consider them any longer.
The nice feature of these models is a small number of free parameters which is lower than present in general Z scenarios with left-handed flavour violating couplings to quarks considered in [19,20]. This allows to find certain correlations between different meson systems which is not possible in the general case. Indeed the strength of the relevant Z couplings to down-quarks is governed in these models by two mixing parameters, two CP-violating phases and the parameter β which defines a given 331 model up to the choice of fermion representations [8,14] and determines the charges of new heavy fermions and gauge bosons as we have already mentioned above.
Thus for a given M Z and β there are only four free parameters to our disposal. In particular for a given β, the diagonal couplings of Z to quarks and leptons are fixed. Knowing these couplings simplifies the analysis significantly, increasing simultaneously the predictive power of the theory.
• The model with β = − √ 3, advocated in particular in [26] in the context of B d → K * µ + µ − anomalies, has several problems originating in the presence of a Landau singularity for sin 2 θ W = 0.25. The same problem is found in the case β = √ 3. Therefore we will not consider them here.
We would like to emphasize here that these results have been obtained by assigning the fermions to specific fermion representations under SU (3) L and that even for a given β the results listed above can change if the choice of representations is different. While it is known that the phenomenology of 331 models depends on the choice of fermion representations [8,14] we recall some aspects of it below as this freedom has interesting consequences in the context of our analysis. Moreover, it clarifies certain differences between the analyses in [8,14] and [3,4] 1 .
However, our previous analyses and to our knowledge all analyses of FCNC processes in 331 models neglected contributions from tree-level Z boson exchanges. Such contributions can be generated in 331 models by the Z − Z mixing but were expected to be very small as according to general analyses [27,28] this mixing should be at most of O(10 −3 ). In the absence of this mixing the Z couplings remain for a given electric quark charge flavour universal and in contrast to Z gauge boson, there are no FCNCs mediated by Z boson at tree-level.
The goal of the present paper is to investigate, whether the neglect of Z boson FCNCs generated by Z − Z mixing in the 331 models in question is really justified. It will turn out that this is not always the case and we will investigate what is the impact of these new contributions on our results in [4]. Fortunately it will turn out that the determination of the allowed ranges for the parameters of the 331 models through ∆F = 2 processes is unaffected by these new contributions. The same applies to our analysis of the anomalies in B d → K * µ + µ − . On the other hand in other decays considered by us Z contributions can be as large as Z contributions so that for certain parameters and models the two contributions can cancel each other.
At this point we would like to emphasize that even if 331 models would not survive future flavour precision tests, they offer a very powerful laboratory to study not only correlations between various flavour observables but also between flavour observables and electroweak precision observables. One of the goals of our paper is to exhibit these correlations transparently.
Our paper is organized as follows. In Section 2 we summarize some aspects of 331 models and present the general expression for the Z −Z mixing that differs from the one quoted in the literature [8,14]. We also analyze for which processes and for which values of β the resulting FCNCs mediated by Z boson are relevant. We will frequently refer to our previous papers [3,4], where all the details on the models considered can be found. In particular in the Appendix A in [4] a compendium of all Z couplings and Z couplings including their numerical values can be found. We will not repeat this compendium here but we will use it extensively in order to find out already in this section where the neglect of flavour violating Z exchanges is justified and where they have to be taken into account. In Section 3 we show how our results in [3,4] are modified through the inclusion of Z contributions. In Section 4 we present for the first time the analysis of ε /ε in 331 models and its correlation with rare K decays. In Section 5 we reconsider the effects of Z − Z mixing in electroweak precision observables and discuss correlations between flavour and electroweak precision observables in 331 models in question. We conclude in Section 6.
2 Z − Z Mixing in 331 Models

Basic Formulae for Z − Z Mixing
Among the new heavy particles in 331 models the most important role in flavour physics is played by a new Z boson originating in the additional U (1) X factor in the extended gauge group. The electroweak symmetry breaking is discussed in several papers quoted above and we will not repeat it here. It suffices to state that after the mass eigenstates for the SM fields, the photon and the Z boson have been constructed through appropriate rotation, there remains still small mixing between Z and Z so that the heavy mass eigenstates are really Z 1 µ = cos ξZ µ + sin ξZ µ , Z 2 µ = − sin ξZ µ + cos ξZ µ .
As sin ξ is estimated to be at most O(10 −3 ) this mixing is usually neglected in FCNC processes so that the two mass eigenstates are simply Z 1 µ = Z µ and Z 2 µ = Z µ . Consequently only Z µ has flavour violating couplings in the mass eigenstate basis for quarks as a result of different transformation properties of the third generation under the extended gauge group. The flavour violating couplings of Z are parametrized by complex couplings ∆ ij L (Z ) with i, j = d, s, b in the present paper. When the small but non-vanishing mixing represented by sin ξ is taken into account, not only the flavour violating couplings of the mass eigenstate Z 1 µ to quarks are generated but also its flavour diagonal couplings to SM fermions differ from the ones of the SM Z boson. Explicitly we have for i = j where in order not to modify the notation in flavour violating observables relative to our previous papers we will still use Z for Z 1 and Z for Z 2 with masses M Z and M Z , respectively. The small shifts in the masses of these gauge bosons relative to the case sin ξ = 0 are irrelevant in flavour violating processes. For flavour diagonal couplings to fermions (generically denoted with f ) we have with In the calculations of flavour violating effects we can neglect the mixing effects in these couplings so that we can simply set as in our previous papers, but in the discussion of electroweak precision tests in Section 5 we have to keep mixing effects in (4). Following [29] in this case we will use for the modified diagonal Z couplings to fermions The second term in this equation allows then as we will see in the course of our analysis to select by means of electroweak precision observables the favourite 331 models. Now the flavour violating Z couplings to quarks, for the three meson systems K, B d and B s , depend on the elements v ij of a mixing matrix V L . Being proportional to v * 32 v 31 , v * 33 v 31 and v * 33 v 32 , respectively, they depend only on four new parameters (explicit formulae are given in [3]):s 13 ,s 23 , δ 1 , δ 2 .
Heres 13 ands 23 are positive definite and δ i in the range [0, 2π]. Therefore for fixed M Z and β, the Z contributions to all processes analyzed by us depend only on these parameters implying very strong correlations between NP contributions to various observables. Indeed, the B d system involves only the parameterss 13 and δ 1 while the B s system depends ons 23 and δ 2 . Moreover, stringent correlations between observables in B d,s sectors and in the kaon sector are found since kaon physics depends ons 13 ,s 23 and δ 2 − δ 1 . A very constraining feature of this models is that the diagonal couplings of Z to quarks and leptons are fixed for a given β, except for a weak dependence on M Z due to running of sin 2 θ W .
As the mass M Z and flavour diagonal Z-couplings to all SM fermions are known, the model is also predictive after the inclusion of Z − Z mixing, although one additional parameter, tanβ, enters the game. This mixing has been calculated in [8] in terms of the SU (3) L and U (1) X gauge couplings and the relevant vacuum expectations values but for our purposes it is useful to express it in terms of measurable quantities and β. Repeating this calculation we find an important expression where with v 2 ± given in terms of the vacuum expectation values of two Higgs triplets ρ and η as follows v 2 As the Higgs system responsible for the breakdown of the SM group has the structure of a two Higgs doublet model and the triplets ρ and η are responsible for the masses of up-quarks and down-quarks respectively one can express the parameter a in terms of the usual tanβ where we introduced a bar to distinguish the usual angle β from the parameter β in 331 models. We have then Thus for tanβ = 1 the parameter a = 0 which simplifies the formula for sin ξ relating uniquely its sign to the sign of β. On the other hand in the large tanβ limit we find a = −1 and in the low tanβ limit one has a = 1.
We have emphasized in [4] that the couplings ∆ ij L (Z ) should be evaluated at µ = M Z and this implies that the s 2 W entering these couplings should be evaluated at µ = M Z . In evaluating ∆ ij L (Z) by means of (3) such Z -couplings should be used. However, as the Z − Z mixing is generated in the process of the SM electroweak symmetry breaking, in evaluating sin ξ by means of (10) and subsequently ∆ ij L (Z) by means of (3) the value of s 2 W at µ = M Z should be used.
Our result for sin ξ differs from the one that one would obtain from the formula given in [8,14] by expressing it in terms of s W , c W , M Z and M Z . We find opposite overall sign and the factor √ 3 in front of the parameter a that is missing in [8,14] 2 . The difference in sign is important for the interference between NP contributions from Z and Z exchanges and consequently for the pattern of NP effects. It is also crucial for the interplay of flavour physics with electroweak precision tests and should also have an impact on the analyses of Z − Z mixing effects in [8,9,14]. But these correlations depend also on the value of tanβ and we will see this explicitly below.
The expression in (10) tells us indeed that sin ξ is very small but one should remember that the propagator suppression of FCNC transitions in the case of Z is by a factor of M 2 Z /M 2 Z stronger than in the case of Z at the amplitude level. Therefore we should make a closer look at the values of sin ξ and Z couplings to leptons as functions of β and tanβ and compare them with the known Z couplings to fermions in order to decide whether Z boson contributions to FCNC processes can be neglected or not. However first we have to elaborate on the choice of fermion representations.

Choice of Fermion Representations
As already emphasized in [8,14] the choice of β does not uniquely specify the phenomenology of the 331 model considered which further depends on the choice of fermion representations under SU (3) L . Here we discuss some aspects of this dependence that are relevant for our study.
Our choice of representations in [3,4] under SU (3) L can be summarized as follows. The first two generations of quarks are put into triplets (3) while the third one into the antitriplet (3 * ): The corresponding right handed quarks are singlets. The anomaly cancellation then requires that leptons are put into antitriplets: We refer to this choice as F 1 .
On the other hand in [8,14] the triplets and antitriplets are interchanged relative to our choice. That is the first two quark generations are in antitriplets while the third one in a triplet. Therefore leptons are also in triplets. We call this fermion assignment F 2 3 . The important two features to be remembered for our discussion below is that for a given β: • The expression for sin ξ in (10) is independent of whether F 1 or F 2 is used.
• On the other hand as evident from the comparison of our compendium for Z couplings to fermions in [4] with Table 4 of [14] the signs in front of β in these couplings are changed when going from F 1 to F 2 . This property can be derived from the action of the relevant operatorQ W on triplet and antitriplet. See formulae in Section 2 of [3].
These observations have the following important phenomenological implications given here first without FCNCs due to Z boson: • In F 1 scenario the models with β = −2/ √ 3 and β = −1/ √ 3 are useful for the explanation of the anomalies in B d → K * µ + µ − because with F 1 representations the coupling ∆ µμ V (Z ) is large. On the other hand the models with β = 2/ √ 3 and β = 1/ √ 3 having significant ∆ µμ A (Z ) coupling provide interesting NP effects in B s,d → µ + µ − .
• While these two scenarios cannot be distinguished by flavour observables when only Z contributions are considered they can be distinguished when Z boson contributions are taken into account. This originates in the fact that the sin ξ entering the ∆ ij L (Z) couplings in (3) does depend on the sign of β but does not depend on whether F 1 scenario or F 2 scenario is considered. In other words the invariance in flavour observables under the transformations present in the absence of Z − Z mixing is broken by this mixing. We will see this explicitly in our numerical analysis below.
• As a particular sign of β could be favoured by flavour conserving observables, in particular electroweak precision tests, this feature allows in principle to determine whether the representation F 1 or F 2 is favoured by nature. We will see this explicitly in Section 5.

∆F = 2 Processes
In the models considered only SM ∆F = 2 operator (i.e. that change flavour quantum number by two units, as for example in neutral meson mixing) in each meson system is present and the effects of NP in all ∆F = 2 transitions can be compactly summarized by generally flavour dependent shifts ∆S in the SM one loop function S that is flavour independent. However due to the relation (3) the flavour dependence of the shifts ∆S due to Z and Z contributions is the same. Consequently for all meson systems the ratio of the shifts in S due to Z and Z is given universally as follows: As B(β, a) ≤ 1.1 in all four models considered by us, it follows that Z contributions to all ∆F = 2 transitions can be neglected. This is good news: the determination of the allowed values of the new parameters (9) by means of ∆F = 2 processes remains unmodified relative to our analyses in [3,4].

∆F = 1 Processes
It should be noted that in ∆F = 2 processes the flavour violating coupling of Z enters twice which resulted in sin 2 ξ dependence in the ∆F = 2 amplitudes. However, in ∆F = 1 amplitudes (implying a change by one unit of flavour quantum number, as in weak decays) it appears only once, whereas the dependence on the mass of the exchanged gauge boson remains unchanged. Again the flavour dependence in the vertex involving quarks is the same for Z and Z and as the operators in each systems are also the same, the ratio of the amplitudes A (Z) and A (Z ) takes a very simple form:   Table 1: sin ξ, R µµ V,A,L , W µµ V,A,L , ∆ µµ,νν V,A,L and R ε from (54) for different β and tanβ = 1(5) in scenario F 1 for fermion representations and for M Z = 3 TeV. R ε is defined in (53).
where k = L, R, A, V and stands either for charged leptons or neutrinos in the final state. The remarkable property of this formula is its independence on M Z . Consequently the ratios R k with known Z couplings to leptons are only functions of β and of the parameter a or equivalently tanβ. In addition they depend on whether the representation F 1 or F 2 is considered.
The ratios R k give us the information on the importance of Z contributions relatively to Z contributions but in order to get the full picture, in particular in view of the dependence of NP effects on the choice of fermion representations, it is useful to consider the quantities which will directly enter the phenomenological expressions.
In Table 1 we show the values of sin ξ, R k and W k relevant for the couplings ∆ µµ V , ∆ µµ A and ∆ νν L in scenario F 1 for fermion representations for the four values of β and two values of tanβ = 1 (5). The corresponding results for scenario F 2 are given in Table 2 and for tanβ = 0.2 and F 1 (F 2 ) in Table 3. In these tables we fix M Z = 3 TeV, as we do in our numerical analysis.
We observe the following features: • In the case of decays involving the coupling ∆ µµ V (Z) the contributions from Z boson can be neglected due its small vectorial coupling to charged leptons. The large values of R V µµ for β = 1/ √ 3 in the case of F 1 and β = −1/ √ 3 in the case of F 2 do not imply large contribution of Z boson as in this case Z contribution is  Table 3: sin ξ, R µµ V,A,L , W µµ V,A,L and R ε from Eq. (54) for different β and tanβ = 0.2 in scenario F 1 (F 2 ) for fermion representations and for M Z = 3 TeV. R ε is defined in (53).
negligible. This is good news. The explanation of B d → K * µ + µ − anomalies with Z contributions presented in [4] remains basically unmodified.
• But in the case of B s,d → µ + µ − , B d → Kµ + µ − and decays with neutrinos in the final state, like b → sνν transitions, K + → π + νν and K L → π 0 νν Z contributions cannot be generally neglected but the size of the additional contributions depends on β and tanβ.
• Comparing Tables 1 and 2 we observe that indeed the symmetry (17) is broken by Z − Z mixing. This is also seen in Table 3.
• Finally, we emphasize that the pattern of NP effects is governed by the sign of sin ξ in (10).
In the next section we will investigate these new contributions numerically.

Preliminaries
The inclusion of flavour violating effects from Z in the observables analyzed in [4] amounts to replacing the shifts due to NP in SM one-loop functions, given in the formulae (22)- (27) in that paper, by the following ones. Defining one has for decays and for Similarly for b → qνν transitions governed by the function X one finds and for K + → π + νν and K L → π 0 νν The corrections from NP to the Wilson coefficients C 9 and C 10 that weight the semileptonic operators in the effective hamiltonian relevant for b → sµ + µ − transitions and used in the recent literature are given as follows As seen in these equations C NP 9 involves leptonic vector coupling of Z while C NP 10 the axial-vector one.

Numerical Results for B Decays
Before presenting our results we recall the present SM value for where we have also shown the latest average of the results from LHCb and CMS [23][24][25]. The agreement of the SM prediction with the data for B s → µ + µ − in (28) is remarkable, although the rather large experimental error still allows for sizable NP contributions with the ones suppressing the branching ratio relative to its SM value being favoured.
As far as the anomalies in B d → K * µ + µ − [21,22] are concerned a number of analyses, of which we only quote three [32][33][34], indicate that Re(C NP 9 ) = −1.0 ± 0.5. Thus in the plots presented below the results with are favoured by the present data.
The results for various observables in 331 models with fermion representations F 1 have been presented in Figs. 5-17 in [4]. The present analysis shows how the latter results are modified when Z boson contributions are included and when the fermion representations F 2 are considered instead of F 1 .
First of all taking into account that the Z contributions can be neglected in all ∆F = 2 transitions and in the coefficient C 9 the results in Figs. 7 and 8 in [4] remain basically unchanged as they involve only ∆F = 2 observables and C 9 . On the other hand in the processes in which NP is governed by the shifts in (22)- (25) and C 10 we find that modifications can be sizable, in particular when two observables taking part in the correlation are affected by Z contributions in a rather different manner.
In order to have appropriate comparison with the results in [4] we use the same treatment of CKM parameters and hadronic uncertainties as in the latter paper so that the difference between various correlations are only due to differences in NP contributions. For this reason we do not list the input parameters that can be found in Table 3 of that paper.
• The results of [4] and those new ones in fermion representations F 2 that include only Z contributions are presented in black.
• The results that include both Z and Z contributions are given in colours that distinguish between the values of C Bs = ∆M s /(∆M s ) SM .
• As for a given β the contributions from Z boson depend on tanβ, we show the results for tanβ = 1 in light colours, for tanβ = 5 in darker colours and for tanβ = 0.2 in gray colours.
• Finally we show the results for the four different values β in question and the fermion representations F 1 and F 2 .
The results of this extensive numerical analysis are shown in Figs. 2-6. While with the comments just made these figures are self-explanatory, we would like to emphasize the most interesting features in them:  • From the present perspective, ignoring at first the constraints from electroweak precision observables, the most interesting model is the one with β = −2/ √ 3 and fermion representations F 1 considered also by us in [4]. It allows to bring the theory closer to the data on B d → K * µ + µ − and B s → µ + µ − than it is possible in the remaining models. In particular the inclusion of Z boson contributions allows to suppress B(B s → µ + µ − ) by (15 − 20)% below its SM value, which is not possible if only Z contributions are present. But this suppression is only significant for tanβ > 1.0 and is clearly visible for tanβ = 5.0. On the other hand for tanβ = 0.2 there is a destructive interference between Z and Z so that in this case NP effects in B(B s → µ + µ − ) turn out to be small.
• On the other hand if B d → K * µ + µ − anomaly disappeared but future more precise data would definitely show that B(B s → µ + µ − ) is significantly below its SM value, other models, in particular the one with β = −2/ √ 3 but fermion representations F 2 , would be favoured. Further tests would come from future measurements of decays with neutrinos in the final state.
• As seen in Figs. 4 and 5 in the case of neutrinos in the final state the dependence on tanβ is opposite to the case of B s,d → µ + µ − and Z and Z contributions interfere constructively for small tanβ but for large tanβ they cancel each other to a large extent.
• There is no specific correlations between the branching ratios for B s → µ + µ − and B d → µ + µ − decays and this implies significant departures from CMFV relation Figure 4: Colours as in Fig. 2 between their branching ratios. We show as an example in Fig. 6 the results for values of tanβ = 1, 5, 0.2 and fermion representations F 1 . The case without Z −Z mixing, that is pure Z contributions, represented by the black regions allows to see that the presence of Z boson contributions in both decays represented by departure from these areas can be significant as could be deduced from previous results.
• As seen in Fig. 7 Z boson contributions have significant effect on the size of CP violation in B s → µ + µ − that originates from a non-vanishing imaginary part of C N P 10 , in constrast to C SM 10 that is real. As the tests of CP-violating effects in this decay are in the distant future we only show the results for F 1 in this case.
Finally, we look at the B d → Kµ + µ − decay and its correlation with Re(C NP 9 ). The interest in the analysis of this decay lies in the fact that in contrast to B s → µ + µ − and B d → K * µ + µ − that in 331 models are sensitive only to C NP 10 and C NP 9 , respectively, the branching ratio for B d → Kµ + µ − depends on both coefficients. Moreover, lattice calculations of the relevant form factors are making significant progress here [35,36] and the importance of this decay will increase in the future.
Neglecting the interference between NP contributions the formula for the differential branching ratio confined to large q 2 region (15 GeV 2 ≤ q 2 ≤ 22 GeV 2 ) 4 reduces in 331 models in units of 1/ GeV 2 to The relevant Wilson coefficients are given in (26) and (27). This formula describes triple correlation between B d → Kµ + µ − , Re(C NP 9 ) and B(B s → µ + µ − ) which constitutes an important test for the models in question. In the absence of Z − Z mixing this triple correlation involving the rate for B d → Kµ + µ − at large q 2 can be found in Fig. 13 in [4] for the case of F 1 representations.
In Fig. 8 we generalize this result to include Z − Z mixing and in Fig. 9 we present corresponding results for F 2 . As in the case of Figs. 2 and 3 we obtain straight lines with with lighter colours showing when B(B s → µ + µ − ) is enhanced and with darker ones when it is suppressed with respect to the SM prediction. There are two striking differences between these results an those in Figs. 2-5. The effects of Z − Z mixing in Figs. 8 and 9 are significantly smaller and consequently the symmetry in (17) is less broken.
At this point we would like to emphasize that all the results described until now do not take into account the constraints from electroweak precision tests. In Section 5 we will analyze which lines in Figs. 2-5, 8 and 9 survive the latter tests and which not. In any case with M Z = 3 TeV, as demonstrated in [4], the bounds from LEP-II and LHC are satisied.

Numerical Results for Rare K Decays
In our recent paper [38] we have reemphasized the strong dependence of rare K decay branching ratios on the values of the elements of the CKM matrix |V ub | and |V cb |. This dependence is particularly strong in the case of K L → π 0 νν as seen in Table 3 of that paper. While in [38] we have studied six scenarios for |V ub | and |V cb | in 331 models most of these scenarios are ruled out by ε K . In fact as already pointed out in [3] NP effects in ε K are rather small when constraints from B 0 d,s −B d,s mixing are taken into account. Therefore the 331 models can only be made consistent with data on ε K for values of |V ub | and |V cb | for which the SM prediction for ε K is rather close to this data. Then only scenarios d) and f ) in [38] d) survive the ε K constraint in 331 models as then for central values of remaining parameters |ε K | = 2.35 × 10 −3 and |ε K | = 2.25 × 10 −3 , respectively. This is close to the experimental value |ε K | = 2.23 × 10 −3 so that there is no problem in fitting the data. Now in [4] and in the analysis of B decays in the present paper we have used the values which implies |ε K | = 2.17 × 10 −3 that is rather close to the choice f ) above and as it also allows to satisfy the ε K constraint, we will use these values for rare K decays and ε /ε. Figure 8: Correlation of the differential branching ratio for B d → Kµ + µ − ) [15,22] versus Re(C NP 9 ) for β = ±1/ √ 3 and β = ±2/ √ 3 and F 1 setting M Z = 3 TeV and different values of tanβ with colour coding given in (32). B(B s → µ + µ − ) ≤ B(B s → µ + µ − ) SM (darker colours) and B(B s → µ + µ − ) ≥ B(B s → µ + µ − ) SM (lighter colours) . The gray regions show the experimental range in (31).
In Figs. 10 and 11 we show various correlations between rare decay branching ratios for the four 331 models considered for the F 1 case, three values of tanβ = 1, 5, 0.2 and the case without Z − Z mixing which represents sole Z contributions. While these plots are self-explanatory, in particular when considered simultaneously with Table 1, we would like to emphasize a number of most important features in them. These are: • A rather striking feature is the cancellation of Z and Z contributions to the branching ratios for K + → π + νν and K L → π 0 νν for tanβ = 5.0 so that in this case one obtains basically the SM prediction independently of the value of β. On the contrary in the case of K L → µ + µ − for tanβ = 5.0 and in particular for negative β NP effects are enhanced through Z − Z mixing. This difference between decays with neutrino and muons in the final state has been also seen in the case of B decays.
• A different behaviour is observed for tanβ = 1.0. In the case of K + → π + νν and K L → π 0 νν for models with β = −2/ √ 3 and β = −1/ √ 3 there is a destructive interference between Z and Z contributions decreasing somewhat NP effects due to Z exchange alone. For β = 2/ √ 3 and β = 1/ √ 3, on the other hand, the corresponding interference is constructive and NP effects are increased. On the contrary the effects of Z − Z mixing for tanβ = 1.0 in K L → µ + µ − are rather small.
• On the whole NP effects except for the case of K L → π 0 νν and tanβ ≤ 1 are rather small and it will be difficult to distinguish them from SM expectations unless parametric uncertainties decrease by much and experimental data will be very precise.
Considering the case of F 2 representations one can deduce from Table 2 that the symmetry in (17) is significantly broken in rare K decays but the size of NP effects is similar to the F 1 case. As an example we show in Fig. 12 the correlation between the branching ratios for K + → π + νν and K L → π 0 νν for F 2 . In particular for β = 2/ √ 3 and tanβ = 0.2 NP effects are significant.
But the main message from our analysis of rare B and K decays is that neglecting Z contributions in decays governed by axial vector couplings to muons or left-handed couplings to neutrinos is not justified and observing significant NP effects in B s,d → µ + µ − would imply tanβ > 1 and only small effects in K L → π 0 νν and K + → π + νν. On the other hand confirming SM predictions for B s,d → µ + µ − to high degree would in 331 models for M Z still allow for modest by significant departures from SM expectations for Figure 10: K + → π + νν versus K L → π 0 νν for different values of β for tanβ = 1.0 (cyan), tanβ = 5.0 (darker cyan), tanβ = 0.2 (purple) and without Z − Z mixing (blue) for F 1 representations.
Again the results just described do not take into account the constraints from electroweak precision tests and it will be interesting to see in Section 5 the impact of the latter tests on them.

Preliminaries
Recently we have presented a new analysis of ε /ε within the SM and models with treelevel Z and Z boson exchanges [38]. Several of the formulae presented in that paper can be directly used in the context of the 331 models and consequently our presentation will be rather brief. In 331 models we have where the formula for the SM contribution, an update of the original one in [39], is given in (53) of [38].

Z Contribution
This case is simple as the only thing to be done is to introduce shifts in the functions X, Y and Z that enter the SM model contribution to ε /ε. Using the results of Section 7 in [38] together with the relation (3) we find where g = 0.652 and λ t = V td V * ts . Replacing then the functions X 0 (x t ), Y 0 (x t ) and Z 0 (x t ) by in the formula (53) for ε /ε in [38] allows to take automatically the first two contributions in (36) in 331 models into account.

Z Contribution
Using the general formulae for the flavour diagonal Z couplings to quarks in [3,4] we find that in LO as far as penguin operators are concerned the only non-vanishing Wilson Figure 12: K + → π + νν versus K L → π 0 νν as in Fig. 11 but for F2.
coefficients at µ = M Z are the ones of the known QCD penguin operator Q 3 , the known electroweak penguin operator Q 7 as well as of the operator which is present due to different couplings of Z to the third generation of quarks. Their coefficients are given in 331 models as follows with f (β) defined in (11). We recall that these results are valid for fermion representation F 1 . For F 2 one just has to reverse the sign in front of β according to the rules outlined in Section 2.2. The coefficients of these three operators at µ = M Z are of the same order. Yet, when QCD renormalization group effects are included and the size of hadronic matrix elements relevant for ε /ε are taken into account we find that the Q 3 andQ 3 contributions can be neglected leaving the left-right electroweak penguin operators as the only relevant operators in Z contribution to ε /ε in 331 models. However, even if at LO the Wilson coefficient of Q 8 vanishes at µ = M Z , at µ = m c used in our numerical analysis of ε /ε its Wilson coefficient C 8 (m c ) is of the same order as C 7 (m c ). Indeed the relevant one-loop anomalous dimension matrix in the (Q 7 , Q 8 ) basis is given bŷ Performing the renormalization group evolution from M Z to m c = 1.3 GeV we find using explicit formulae in [38] Due to the large element (1, 2) in the matrix (44) and the large anomalous dimension of the Q 8 operator represented by the (2, 2) element in (44), the two coefficients are comparable in size. But the matrix element Q 7 2 is colour suppressed which is not the case of Q 8 2 and within a good approximation we can neglect the contributions of Q 7 . In summary, it is sufficient to keep only Q 8 contributions in ε /ε in 331 models.
The relevant isospin amplitude A 2 in K → ππ decays necessary to calculate Z contribution to ε /ε is given as follows where [38] The final expression for Z contributions is given by where [42] ω + = (4.1 ± 0.1) × 10 −2 .
As NP contributions to ε /ε, both due to Z and Z , are dominated by the operator Q 8 , the ratio of these contributions depends with high accuracy simply on the ratio of C 8 (m c ) in these two contributions. This allows then to derive a simple relation where (s 2 W = 0.23116) This expression is valid for fermion representation F 1 . For F 2 for a given β one should just remove the overall minus sign in this expression as the β in front of the parenthesis enters the Z coupling. The expression in the parenthesis comes from Z − Z mixing and is independent of the fermion representation. The factor 0.53 summarizes the difference in renormalization group effects for Z and Z contributions. QCD renormalization of Q 8 gives alone 0.56 and additional small suppression comes from the running of electroweak parameters. We list the values of R ε in the last row of Tables 1-3. Evidently Z dominates NP contributions to ε /ε implying that Z − Z mixing effects are small in this ratio. The two exceptions are the case of β = −1/ √ 3 and tanβ = 5 and the case of β = 1/ √ 3 and tanβ = 0.2 for which Z contribution reaches 50% of the Z one.

Numerical Analysis of ε /ε
In Fig. 13 we show ε /ε versus ε K . We make the following observations: • 331 models for all values of β are consistent with the data for ε /ε provided B (1/2) 6 ≈ 1.0 represented by red colour.
• Z − Z mixing effects are for most parameters small as already expected on the basis of the formula (54) and the values of R ε in Tables 1-3.
In Fig. 14 we show ε /ε versus B(K L → π 0 νν) for fermion representations F 1 . We observe that the correlation between these two observables is very strict as ε /ε is linear in the imaginary parts of the flavour violating Z and Z couplings and NP contribution to B(K L → π 0 νν) is dominated by the interference of the SM and NP amplitude. Consequently is also linear in these couplings. Two interesting features of these plots should be emphasized • while for negative β the ratio ε /ε decreases with increasing B(K L → π 0 νν), for positive β it increases with increasing B(K L → π 0 νν). The latter property is rather rarly found in other extensions of the SM.
• The effects of Z − Z mixing are clearly visible, in particular for β > 0. They originate dominantly in the ones present in K L → π 0 νν.
• As already known from Fig. 13 the agreement of the theory and the data is best for B

Preliminaries
The modifications of the Z boson couplings due to Z − Z mixing in 331 models can be tested through electroweak precision measurements at LEP-I and SLD and such an analysis can be found in [9] and recently in [43]. As our formula for sin ξ differs from the one used in these papers we want to analyze the impact of Z − Z mixing on most important electroweak precision observables (EWPO) and study their dependence on β, tanβ and the choice of fermion representations. Thus our next goal is to construct three tables for the shifts of a number of EWPO relative to SM predictions. This will eventually allow us to investigate the correlations between NP effects in EWPO and flavour observables. In these tables we will set M Z = 3 TeV. As the shifts in question are inversely proportional to M 2 Z it is straight forward to find out what happens for other values of M Z .
Transparent analyses of the effects of Z − Z mixing in EWPO can be found in [29,44,45]. We will follow here the general analysis in [29] and to this end it is useful to note that our couplings of a given fermion f to Z and Z differ from the couplings v f S,N and a f S,N used in that paper by an overall factor: For Z couplings we then have where T f 3L is the third component of the weak isospin of the fermion f and Q f its electric charge. Our effective Z couplings in (7) are then related to the analogously defined ones in [29] simply as follows Due to the Z − Z mixing the parameter, defined by receives tree-level contribution ∆ M which for M Z M Z is given by [29,44,45] This shift is strictly positive and O(M 2 Z /M 2 Z ). Consequently it is of the same order as the Z − Z mixing effects in the effective Z couplings in (7) so that it has to be taken into account. On the other hand we conclude on the basis of [46] that in 331 models studied by us the oblique contributions involving new heavy charged gauge bosons, scalars and fermions can be neglected when their masses are in the few TeV regime. Consequently the shift in ∆ due to NP in these models is dominated by ∆ M given above.
Keeping fixed as input parameters α(M Z ), G F and M Z , the effective Weinberg angle in (59) is modified due to the shift in as follows [29,44] It should be noted that this shift is strictly negative and this property is valid for any Z model. Yet, as we will see below, this does not mean that the shift in the so-called effective sin 2 θ l eff is negative in any Z model as the contributions to Z couplings that are proportional to sin ξ are clearly model dependent. Some aspects of this dependence has been already analyzed in the context of 331 models in [43]. Our improved formula for sin ξ in (10) and the gained insight on the correlations between Z − Z mixing and FCNCs processes allows us to have another look at this issue thereby also generalizing the analysis in [43].
Next denoting a given observable by O the shift due to Z −Z mixing can be linearized in ∆ M and sin ξ as follows [29,45] Here the coefficients A O are universal and depend only on the SM parameters and couplings. On the other hand B O depend on the diagonal Z -couplings in (7). Direct Z contributions to EWPO are negligible. We will list the general formulae for the coefficients A O and B O below but rather then giving numerical values for them as done in [29] we will calculate the shifts in EWPO for different values of β, tanβ and the fermion representations F 1 and F 2 for M Z = 3 TeV. To this end we will proceed as follows • We will calculate a number of EWPO as functions of the effective diagonal Zcouplings in (7) in 331 models using tree-level formulae. In doing this one should remember that in addition to the direct contribution of Z to the effective Z couplings leading to the second term in (62) also the shift in sin 2 θ W entering the vector Z coupling has to be taken into account. That is in the Z coupling to a fermion f one should make additional replacement: with ∆(sin 2 θ W ) given in (61). This contribution to the shift δO is represented by the first term (62).
• Denoting the result for a given observable calculated in this manner by O(ξ) we can simply calculate the shift δO due to NP by subtracting the pure SM contributions at tree level: As the mixing effects are very small the higher order terms generated by this numerical procedure are tiny and consequently one would obtain quite generally very similar results by using the linearized form in (62) as done in [29,45]. Moreover as stressed in [29] in certain models such a linearized expression could not be sufficiently precise and using (64) from the start takes such effects into account.
• For the SM contributions we will use the most recent values for EWPO that include electroweak radiative corrections. Even if the interference terms between SM and NP contributions will not include these corrections, this procedure is justified in view of the smallness of NP effects in 331 models considered by us.
• Comparing SM results with the data we will be able to identify the pattern of deviations from SM predictions and see for which values of β, tanβ and fermion representations the 331 models can improve the agreement of the theory with the data and what does this imply for our analysis of flavour violating effects.

Basic Formulae for EWPO
The tree level formula for the partial widths is given by with N c = 3 for quarks and N c = 1 for leptons. The additional overall factor relative to formula (2.8) in [29] takes into account the difference in the definitions of the vector and axial-vector couplings summarized above. Defining next Γ f = Γ(Z →f f ) we have While we do not distinguish between for b quark contributions and contributions from lighter quarks as such effects are taken into account in the full SM contribution, we separate NP b quark contribution to Γ h from the one of d and s as transformation properties of the third generation of quarks under SU (3) L are different compared to the the first two generations. See the Lagrangian (63) in [3]. This difference has to be taken into account in all observables involving the b quark. Of interest are then the ratios R f and the peak cross sections σ f p defined as follows Note that the definition of R differs from the one of R c and R b . For the asymmetries A f we have and for the forward-backward asymmetry We would like to warn the reader that similar to [29] our vector and axial-vector couplings for Z and Z are defined in terms of left-handed and right-handed ones as follows: Consequently the axial-vector couplings differ by sign from the ones used in PDG. This implies that also asymmetries A f differ by sign from PDG and in Table 7 when quoting PDG values we adjusted their definition to our. From these formulae it is straightforward to derive general analytical formulae for the coefficients A O and B O in (62), valid in any Z model. We first find in the case of the partial width Γ f where the second term in A Γ f is the universal correction from the shift in in the overall factor in (65).
In the case of the asymmetry A f the corresponding expressions read Our results in (71) and (72 agree with the ones one would derive from formulae (6.5)-(6.6) and (7.4)-(7.7) in [44] by dividing them by Γ f and A f , respectively. For R b and R c we find respectively and In these formulae Γ f and Γ h are just tree-level SM contributions. We observe that indeed the coefficients A O depend only on SM couplings and the first term in (62) feels the presence of Z only through sin ξ, while the second one has additional dependence on the diagonal Z couplings. While these formulae allow an easy comparison with the analyses in [29,44,45], in numerical calculations it is easier to use directly (64).

Numerical Results
In Tables 4-6 we list the shifts in a number of observables as functions of β, tanβ for the fermion representations F 1 and F 2 . In Table 7 we summarize SM predictions for these observables, the corresponding data and the pulls as presented after Higgs discovery in [47].
In what follows it will be useful to denote by MI, with a given I=1,..24, a particular 331 model in which β, tanβ and fermion representations are fixed. The index I numbers the column in Tables 4-6 corresponding to a given model in order of its appearance. Thus we deal with 24 models that we specify in Table 8 to make their identification easier. For instance M5 denotes the model with β = 1/ √ 3, tanβ = 1 and F 1 .   Table 4: Values of the shifts in EWPO for different β and tanβ = 1(5) in scenario F 1 for fermion representations. We also give the values of Ω 331 defined in (76) and δ sin 2 θ eff from (83).
In order to judge the quality of a given model and compare it with the performance of the SM we define for each observable O i the pulls P SM i and P 331 i as follows The pulls P SM i are the usual ones and their values are given in the last column in Table 7. In principle in order to calculate such pulls for every 331 model MI considered by us we would have to repeat the fit of [47] for all models including also other observables that are sensitive to new charged gauge bosons. Such an analysis is clearly beyond the scope of our paper. As O i are fixed in a given MI, that is do not introduce new parameters, we expect that such a simplified procedure should give us a correct, even if rough, picture of what is going on.
In order to identify favourite 331 models, as far as electroweak observables are concerned, we define the measures (13.51), The first SM value is based on Table 7 using the average of LEP and SLD values for A while the one in the parentheses is obtained by using as curiosity only the LEP value.   Table 5: Values of the shifts in EWPO for different β and tanβ = 1(5) in scenario F 2 for fermion representations. We also give the values of Ω 331 defined in (76) and δ sin 2 θ eff from (83).
The values of Ω 331 for 331 models are given in Tables 4-6. The models with smallest Ω 331 are favoured, while the ones with with largest Ω 331 disfavoured. Inspecting these results we make first the following observations when the average of LEP and SLD values for A is used.
• All models give small contributions to A c and R c and consequently agree with the data.
• All models give rather small contributions to A b and consequently cannot remove the small pull present in the SM. Moreover, M21 and M23 soften the agreement of the theory with data.
• The sizable discrepancy of the SM with the data on R b cannot be removed in any model. It can be softened by (0.3 − 0.4)σ in M2 and M10 and visibly increased in models M21-M24.
We concentrate then our discussion on the remaining observables, where NP effects turn out to be larger. We find • In most models NP effects in Γ Z are small in agreement with data. Deviations larger than 1σ are only found in M2, M10 and M23.
• In the case of σ h the agreement with data is significantly improved in the case of   • Interestingly in the case of R all the models with tanβ = 0.2 improve the agreement of the theory with data and this also applies to M7 and M15, while the remaining models slightly worsen the agreement.
• On the contrary in the case of A FB and A the models with tanβ = 1.0 and tanβ = 5.0 are performing much better than the ones with tanβ = 0.2. Moreover they perform better than the SM. However, this depends, as discussed below, whether one takes into account the SLD value for A or not.
• Concerning δA c FB and δA b FB the favourite models listed below introduce only a very small shift to the SM values not improving the status of the theory, while several models, in particular M2, M4, M7, M10, M12, M21 and M23 worsen the agreement with the data. This is also the case of two favoured models below, M14 and M16 but they compensate it through better results for σ h than obtained within the SM.    with the first five performing better than the SM while the last two having basically the same Ω 331 . The models with odd index I correspond to tanβ = 1.0 and the ones with even one to tanβ = 5.0. We list the pulls P 331 i in these seven models in Table 9. As seen in this table all favoured models improve the agreement of the theory with  Table 9: Pulls for the seven selected 331 models.
data on σ h . This is in particular the case of M9 and this fact is primarily responsible why M9 wins the competition as it does reasonably well for other observables. We also observe that all favoured models, except M8, improve the agreement of theory with data on A FB and A . However the models M14 and M16 which perform best in this respect are not the top models as they perform worse than the first ones on other observables which basically do not provide any improvements on these asymmetries. The reason is that M14 and M16 experience difficulties with δA c FB and δA b FB , as stated above, making the agreement of the theory with data to be worse than in the SM.
Yet, as discussed below, our analysis confirms the general findings of Richard [43] that in 331 models the departure of the data on A FB and A from their SM values is correlated within 331 models with the B d → K * µ + µ − anomaly, even if the model M2 which he studied is basically excluded by other electroweak data when correct expression for sin ξ is used. This is in particular the case of Γ Z and A b FB . Otherwise this model is interesting as it removes the discrepancy of the SM with the data on σ h .
An important result of our analysis is the weak performance of models with tanβ = 0.2, although the models M17-M20 cannot be excluded. The models which definitely have difficulties with electroweak precision data are M2, M4, M7, M10, M21, M22, M23, M24, (disfavoured) Looking at these results we conclude that from the point of view of electroweak precision tests the following combinations of the values of β, tanβ and fermion representations are favoured.
It should be emphasized that all these seven models pass the present electroweak tests for M Z = 3 TeV as well as the SM or in five cases even better than it. Even if the models with β = ±2/ √ 3 perform slightly better than the ones with β = ±1/ √ 3 it is not possible on the basis of Ω 331 alone to identify the winner among them. On the other hand one day this should be possible on the basis of the Table 9 once various questions related to measurements at LEP and SLD will be clarified. As demonstrated below flavour physics can offer definite help in this context.
In the latter context it is interesting to observe that if the LEP result for A would be the true one five models on the list of favourites in (77) with the first three performing better than the SM. M14 and M16 are not present anymore on this list because they favoured SLD result. Instead M13 and in particular M17 with tanβ = 0.2 are among favourites. As we will discuss below M17 has a unique property among the favourites as far as flavour physics is concerned.

The issue of sin 2 θ eff
In testing the SM one can define sin 2 θ eff by using SM tree-level expression for A . This parameter is most precisely extracted from the data on A and A b FB but also from A FB . Unfortunately the determinations of sin 2 θ eff from these observables are really not in agreement with each other. On one hand in the case of A we have [43,48] and from forward-backward asymmetries A FB and A b FB one finds respectively sin 2 θ eff = 0.23099(53), sin 2 θ eff = 0.23221 (29).
Consequently there is some preference for the positive shift in sin 2 θ eff relative to the best SM value. Until now we did not look at sin 2 θ eff in 331 models and calculated A FB , A , A b FB and other observables directly to judge the quality of a given model on the basis of them. But it is instructive to calculate the shift of sin 2 θ eff due to NP contributions for 24 models considered by us using the operative definition [48] δ sin 2 θ eff = 1 where in the effective vector couplings the shift (63) has to be included. An extensive discussion of sin 2 θ eff can be found in [48] where further references can be found. See also [43]. In writing (83) we have adjusted the sign in the formula (8.3) in [48] to our definition of the axial-vector coupling. The shift in δ sin 2 θ eff in 331 models comes first from the shift of sin 2 θ W which as seen in (61) is negative in 331 models and in any Z model. But in addition to this shift, which comes from ∆ M and affects only vector couplings, both vector and axialvector couplings receive modifications from the mixing with Z , that is the shifts in the couplings proportional to sin ξ and involving Z couplings.
The result of this exercise is given in the last rows in Tables 4-6. The striking feature is that out of 24 models 19 give a negative shift of sin 2 θ eff , while only 7 a positive one. These are M8, M13, M15, M17 and M19.
It is not surprising that M8, M13 and M17 perform here so well as they are on the list of LEP favourites in (79). What is remarkable that M8 is the only among our favourite models in (77) that gives a positive shift of sin 2 θ eff . But this shift being 1×10 −5 is totally negligible. Yet, this model similar to all models with positive shift is fully compatible with the experimental value in (82).
We also note that models M14 and M16 on our list of favourites give where we have added the corresponding shifts to the SM value in (82). These values are within 1.5σ from the central value of sin 2 θ eff from the SLD. We also confirm qualitatively the finding in [43] that the model M2 provides a shift that results in sin 2 θ eff close to the SLD result. We find 0.23122 and 0.23048 for tanβ = 1.0 and tanβ = 5.0, respectively. For tanβ ≈ 3.0 the SLD result can be well reproduced. However, this model has problems with other observables as we have seen above.
While we do not think that just looking at sin 2 θ eff offers a fully transparent test of a given extension of the SM, the rather different values of this parameter extracted from different observables calls for future improved measurements of EWPO which hopefully one day will be possible at a future ILC. This, as stressed in [43], could offer powerful tests of 331 models.

Implications for Flavour Physics
Our analysis of EWPO identified a group of seven models among 24 considered by us and the question arises whether flavour physics could distinguish between them. As we will now discuss our analysis in previous sections demonstrates this rather clearly and having the plots presented there we can summarize how the seven models in question could be distinguished by flavour observables in the coming flavour precision era.
Let us first summarize the main message from our analysis of EWPO which is related to the fact that the values tanβ = 0.2 are disfavoured: • Significant NP effects in B and K decays to neutrinos seem rather unlikely, even if the branching ratio for K L → π 0 νν could still be modified by 15%. In turn due to the correlation with ε /ε also NP effects in this ratio are predicted to be small implying that B (1/2) 6 ≈ 1.0 in order to agree with data. On the other hand as discussed in our recent paper [38] the required precise value of this parameter depends on the values of |V cb | and |V ub |.
• The most interesting NP effects are found in B → K * µ + µ − and B s,d → µ + µ − and we will confine the discussion of the favourite models in (77) to these decays.
The selection of favourite models and the comments just made imply that it is sufficient to look at Figs. 2 and 3 for F 1 and F 2 , respectively. The results for the seven favourite models can be found there with only the results in the upper left panel in Fig. 2 being disfavoured. In the remaining seven panels in these two figures, the results for our favourite models are simply identified by selecting the tanβ = 1.0 line (lighter colours) in the case of β < 0 and the tanβ = 5.0 line (darker colours) in the case of β > 0. The main implications for rare B decays are then as follows: • A significant suppression of B(B s → µ + µ − ) and significant negative shift in ReC NP 9 cannot take place simultaneously. This would be possible in M2 but this model belongs to disfavoured ones by our EWPO analysis.
• For softening the B d → K * µ + µ − anomaly the most interesting is the model M16, that is the model with F 2 , β = 2/ √ 3 and tanβ = 5.0. See upper right panel in Fig. 3. The usual statements present in the literature [4,26,43] that the 331 models with negative β are most powerful in this case apply to F 1 representations. But as our analysis shows the model M2 with β = −2/ √ 3 considered by us in [4] is disfavoured by the EWPO analysis.
• If the anomaly in question remains but decreases with time also model M3 (left lower panel in Fig. 2) and M14 (right lower panel in Fig. 3) would be of interest.
• The remaining four models, in fact the four top models on our list of favourites in (77), do not provide any explanation of B d → K * µ + µ − anomaly but are interesting for B s,d → µ + µ − decays. These are M6, M8, M9 and M11, the first two with F 1 and the last two with F 2 fermion representation. The differences between these models as far as B s → µ + µ − is concerned are most transparently seen in Figs. 4 and 5 from which we conclude that the strongest suppression of the rate for B s → µ + µ − can be achieved in M8 and M9. M8 result is shown in the right upper panels in Figs. 2 and 4 and M9 result in the left upper panels of in Figs. 3 and 5. In fact these two models are the two leaders on the list of favourites in (77). The suppression of the B s → µ + µ − rate is smaller in M6 and M11 as seen in lower right and lower left panels in Figs. 2 and 3, respectively.
We observe that flavour physics can clearly distinguish between the favourite models selected by EWPO analysis. We summarize it in Fig. 15, where only the results in the seven favourite models are shown. If the B d → K * µ + µ − anomaly will persist the winner will be M16 which is represented in this figure by the dark red line in the upper right panel. If it disappears but suppression of the rate for B s → µ + µ − will be required the winners will be M8 and M9, the blue and red lines in the upper right and upper left panel in Fig. 15, respectively. It is interesting that the combination of future flavour data and EWPO tests can rather clearly indentify one or two favourite 331 models among 24 cases considered by us.
An interesting situation would also arise if the LEP result for A would turn out to be the correct one. In this case M14 and M16 are no longer among favourites as seen in (79) and are replaced by M13 and M17. M13 is a good replacement for M14 as it also allows moderate softening of the B d → K * µ + µ − anomaly as seen in the right lower panel in Fig. 3. But more interesting is M17. Indeed as seen in the upper left panel in Fig. 2 for tanβ = 0.2 (gray line) the B d → K * µ + µ − anomaly can be softened as much as Figure 15: B(B s → µ + µ − ) versus Re(C NP 9 ) for the favourite models from Tab. 9 for F1 (blue) and F2 (red) and tanβ = 1(5) with lighter (darker) colours.
it was possible in the case of M16. But as seen in the upper left panel in Fig. 10 (purple line) in this model with F 1 representations also significant NP effects in K L → π 0 νν are possible. Smaller NP effects, as seen in Fig. 12, are found in the case of F 2 but M 18 is not among the favourite models anyway. All this again shows an important interplay between flavour observables and electroweak precision tests.

Summary and Conclusions
In this paper we have addressed the question whether in 331 models the FCNCs due to Z tree-level exchanges generated through Z − Z mixing could play any significant role in flavour physics. Actually it is known from the flavour analyses of Randall-Sundrum models with custodial protection (RSc) [49,50], that while ∆F = 2 processes are governed by heavy Kaluza-Klein gauge bosons with and without colour, NP contributions in ∆F = 1 processes are governed by induced right-handed flavour-violating Z couplings.
Here we analyzed several 331 models which have a much smaller number of parameters than RSc and this allows to see more transparently various NP effects than in the latter scenario. As our basic formula for the Z − Z in 331 model differs from the one found in the literature [8,14,43] we have reconsidered some aspects of constraints from EWPO. Moreover we have identified for the first time various correlations between flavour and electroweak physics that depend on the parameters of 331 model, in particular, β, tanβ, M Z and the fermion representations.
As far as flavour physics is concerned our main findings are as follows: • NP contributions to ∆F = 2 transitions and decays like B → K * + − are governed by Z tree-level exchanges. Therefore for these processes our analysis in [4] remains unchanged. But as we summarize below our analysis of EWPO casts some shadow on some of these models.
• On the other hand for B s,d → µ + µ − decays Z contributions can be important. We find that for tanβ = 5.0 these contributions interfere constructively with Z contributions enhancing NP effects, while for low tanβ = 0.2 Z contributions practically cancel the ones from Z . Similar dependence on tanβ is found for • Similarly Z boson tree-level contributions to B s,d and K decays with neutrinos in the final state can be relevant but in this case the tanβ dependence is opposite to the one found for B s,d → µ + µ − . We find that for tanβ = 5.0 these contributions practically cancel the ones from Z but for low tanβ = 0.2 Z contributions interfere constructively with Z contributions enhancing NP effects. In particular as seen in Figs. 10 and 11 in the case of K L → π 0 νν NP effects can amount to 30% at the level of the branching ratio when the constraints from EWPO are not taken into account.
• As a result of this opposite dependence on tanβ the correlations between decays with muons and neutrinos in the final state exhibit significant dependence on tanβ and can serve to determine this parameter in the future. See in particular Figs. 4 and 5.
• Our analysis of ε /ε is to our knowledge the first one in 331 models. Including both Z and Z contributions we find that the former dominate. But NP effects are not large and in order to fit the data B (1/2) 6 ≈ 1.0 is favoured.
• We also find a strict correlation between ε /ε and B(K L → π 0 νν). The interesting feature here, as seen in Fig. 14, is the decrease of ε /ε with increasing B(K L → π 0 νν) for negative β and its increase with increasing B(K L → π 0 νν) for positive β.
• Performing the analysis for different fermion representations we find that for certain observables this dependence is significant. As an example the comparison of the plots in Figs. 2 and 3 demonstrates the breakdown of the invariance under (17) by Z − Z mixing.
As far as electroweak physics is concerned our main findings are as follows: • Seven among 24 combinations of β, tanβ and fermion representation F 1 or F 2 provide better or equally good description of the electroweak precision data compared with the SM. However, none of these models allows for the explanation of the 2.5σ departures of A b FB and R b from the SM but several of them improve significantly the agreement of the theory with the average of SLD and LEP data for A l .
• Among these models none of them allows to simultaneously suppress the rate for B s → µ + µ − and soften the B d → K * µ + µ − anomaly.
• On the other there are few models which either suppress the rate for B s → µ + µ − or soften the B d → K * µ + µ − anomaly.
• None of these models allows significant NP effects in B and K decays with neutrinos in the final state although departures by 15% relative to the SM prediction for the rate of K L → π 0 νν are still possible.
• Assuming that the LEP result for A is the correct one, we have found that in this case NP effects in K L → π 0 νν are larger than when both LEP and SLD results are taken into account.
Our analysis shows that the interplay of flavour physics and EWPO tests can significantly constrain NP models, in particular those with not too many free parameters. We are looking forward to coming years in order to see whether the 331 models will survive improved flavour data and in particular whether Z will be discovered at the LHC. The correlations presented by us should allow to monitor transparently future developments in the data.