The charm of 331

We perform a detailed analysis of flavour changing neutral current processes in the charm sector in the context of 331 models. As pointed out recently, in the case of $Z^\prime$ contributions in these models there are no new free parameters beyond those already present in the $B_{d,s}$ and $K$ meson systems analyzed in the past. As a result, definite ranges for new Physics (NP) effects in various charm observables could be obtained. While generally NP effects turn out to be small, in a number of observables they are much larger than the tiny effects predicted within the Standard Model. In particular we find that the branching ratio of the mode $D^0 \to \mu^+ \mu^-$, despite remaining tiny, can be enhanced by 6 orders of magnitude with respect to the SM. We work out correlations between this mode and rare $B_{d,s}$ and $K$ decays. We also discuss neutral charm meson oscillations and CP violation in the charm system. In particular, we point out that 331 models provide new weak phases that are a necessary condition to have non-vanishing CP asymmetries. In the case of $\Delta A_{CP}$, the difference between the CP asymmetries in $D^0 \to K^+ K^-$ and $D^0 \to \pi^+ \pi^-$, we find that agreement with experiment can be obtained provided that two conditions are verified: the phases in the ranges predicted in 331 models and large hadronic matrix elements.


Introduction
The role of charm in particle physics can hardly be overemphasized [1]. Not only historically the discovery of J/ψ, and hence of the charm quark, was so important to be ascribed the name of November revolution, but even in much more recent times, the observation of charmed states with properties not expected in the quark model requires the development of a new hadron spectroscopy, to incorporate all hints of exotic states in a predictive framework.
However, the peculiar role of charm brings in also difficulties from the theory point of view. The first reason lies in the fact that the charm quark is not as heavy as the beauty quark to allow a completely reliable use of the heavy quark techniques efficiently developed for beauty hadrons. On the other hand, it cannot either be considered a light quark, so that the theoretical predictions are very often affected by uncertainties difficult to control because of long distance dynamics.
Among the observables in the charm sector showing this kind of problems there are the neutral meson mixing, i.e. D 0 − D 0 oscillation, and CP violation, as well as rare charm decays governed by penguin diagrams [2]. Two facts make the Standard Model (SM) treatment of these processes rather challenging. The first one is that loop-induced processes in the charm case get contributions from internal down-type quarks. Since the loop functions depend on the ratios x i = m 2 i /M 2 W , the smallness of these ratios for i = d, s, b makes them all very close to zero and therefore almost equal, resulting in almost perfect GIM cancellation. Another point is that the CKM elements involving charm have tiny imaginary parts, which is a challenge for CP violation studies if the SM was the whole story. Yet, we know that the SM is not the whole story. Hence, the strong suppression of some processes within SM, namely lepton flavour violating decays and the electric dipole moments, can even be an advantage in the search for new physics (NP) because of the small SM background. In the case of rare flavour changing neutral current (FCNC) processes induced by the c → u transition the effectiveness of the GIM mechanism results in so tiny branching ratios in the SM that one can consider them as null tests: their observation with a less severe degree of suppression with respect to SM would be a signal of NP.
Deviations from the SM predictions in the flavour sector have been observed in several cases: they are usually referred to as flavour anomalies. Such tensions deal with loopinduced processes, as in the ratios R K ( * ) and in the so-called P 5 distribution in b → s + − processes, and very recently in (g − 2) µ . They also affect tree-level charged current processes, as in the ratios R(D ( * ) ) in semileptonic B decay and in the analogous ratios involving corresponding decays of other beauty hadrons [3,4]. In view of this, it is natural to look for deviations in other processes, and the purpose of this paper is to explore possible deviations from the SM expectations in the charm sector. A SM extension that could be promising from this point of view is the 331 model, whose main features we briefly describe in the next Section. The reason is the presence in this model of a new neutral gauge boson Z that can mediate tree-level FCNC. A few new parameters are introduced, affecting the interaction Lagrangian of SM fermions with Z . As observed in [5], it is remarkable that the parameters governing the flavour violating coupling cuZ , which enter in FCNC charm transitions, up to CKM factors are the same parameters describing the sdZ and bsZ couplings. This is a rather powerful feature: it allows us to exploit the experimental constraints in the kaon and in the beauty systems to place bounds on rare charm transitions and to establish correlations among observables in the three sectors.
The plan of the paper is the following. In Section 2 we present the basic features of the 331 model and of its variants. Section 3 is devoted to D 0 − D 0 mixing, CP violation and rare charm decays in the SM and in 331 models. The numerical results and the correlations with kaon and B observables are presented in Section 4. In the last Section we summarize our findings.

331 model
The extension of the SM gauge group to SU (3) c × SU (3) L × U (1) X , proposed in [6,7], defines a class of models generically referred to as 331 models. In these models two spontaneous symmetry breakings occur: first to the SM group SU (3) c × SU (2) L × U (1) Y and then to SU (3) c ×U (1) Q . The larger group leads to the introduction of 5 additional gauge bosons, as well as new fermions. Indeed, with the enlargement SU (2) L → SU (3) L the left-handed SM fermions are components of triplets or antitriplets rather than doublets. As a consequence, a third partner is usually needed, in general a new heavy fermion. A very appealing feature of 331 models is that the requirement of anomaly cancelation together with that of asymptotic freedom of QCD constrains the number of generations to be equal to the number of colours, making the 331 able to predict the existence of just three generations of ordinary quarks 1 . Moreover, quark generations should transform differently under the action of SU (3) L . In particular, two quark generations should transform as triplets, one as an antitriplet. The common choice is that the latter is the third generation. The different role of the third generation with respect to the others could possibly be at the origin of the large top quark mass.
The relation between the electric charge Q and T 3 and T 8 , two of the SU (3) generators, and X, the generator of U (1) X , introduces a parameter β. This is a key parameter that defines the specific variant of the model. Among the new gauge bosons, four of them Y Q ± Y and V Q ± V have properties that depend on the value of β: they have integer charges when β is multiple of 1 √ 3 and of √ 3. On the other hand, in all the model variants a new neutral gauge boson Z is present. A very appealing feature of the 331 models is that Z mediates tree-level flavour changing neutral currents in the quark sector, while its couplings to leptons are diagonal and universal. An extended Higgs sector also comprises three SU (3) L triplets and one sextet. Finally, a relation exists between the U (1) X gauge coupling g X and the SU (3) L coupling g: where θ W is the Weinberg angle. In analogy to the SM, quark mass eigenstates are obtained through rotation of flavour eigenstates by means of two unitary matrices U L (for up-type quarks) and V L (for downtype ones) that satisfy the relation V CKM = U † L V L , as in the SM case. However, while in the SM V CKM appears only in charged current interactions and U L and V L never appear individually, in the 331 models it is possible to get rid of only one matrix, U L or V L , by expressing it in terms of V CKM and the other one; the remaining rotation matrix enters in the Z couplings to quarks. Choosing V L as the surviving rotation matrix, one can parametrize it as [8] (3) In the 331 Lagrangian density the term describing the Z interaction with ordinary fermions reads: where s W ≡ sin θ W , c W ≡ cos θ W , q u (q d ) denotes an up (down)-type quark (i, j are generation indices), and v ij , u ij are the elements of the V L and U L matrices, respectively. Using the parametrization in Eq. (3) and the Feynman rules for Z couplings to quarks from Eq. (4) [8,9], one finds that the B d system involves only the parameterss 13 and δ 1 , while the B s system involves onlys 23 and δ 2 . Kaon physics depends ons 13 ,s 23 and δ 2 − δ 1 , so that stringent correlations between observables in B d,s sectors and in the kaon sector can be established.
Exploiting this feature, correlations among several quark flavour observables in B d , B s and K decays have been studied in [8][9][10][11][12]. Moreover, the relation produces additional correlations between B d , B s , K decays and the transitions induced by up-type quark FCNC decays mediated by Z , which occur in 331 models. In particular, as it follows from Eq. (4), correlations exist between observables in the charm sector and observables for B d , B s and K. This observation has been exploited to relate the c → uνν processes, such as The parameter a introduced in Eq. (6) is defined in terms of the vacuum expectation values of two Higgs triplets ρ and η, as follows: With the same notation of the two Higgs doublet models, a is expressed in terms of the parameter tanβ as 2 : A detailed analysis of the impact of the Z −Z mixing in several 331 models distinguished by different values of β, including the constraints from electroweak precision observables, is presented in [10]. Here, we analyze the impact on the charm sector, in particular on D 0 − D 0 mixing, on CP violation in D 0 system and on rare D 0 transitions 3 .
3 Charm observables in the SM and in 331 model Due to off-diagonal terms in M, mass eigenstates do not coincide with the flavour eigenstates D 0 , D 0 , but are given by 2 The overline distinguishesβ from the parameter β characterizing the 331 models. 3 The mass difference in the neutral D system has been studied in 331 models with β = ±1/ √ 3 considering the contribution to FCNC of the mixing between ordinary and exotic quarks [13]. The effects of new scalars have been investigated in [14], while FCNC processes in 331 models with right-handed neutrinos have been considered in [15]. CP parity transforms D 0 , D 0 into each other modulo a phase that can be chosen in such a way that Mass and width differences are usually normalized to the average width of the two mass eigenstates: with In the 2021 online report the HFLAV group quoted the summary of the charm-mixing data obtained from global fits in which CP violation is allowed [16]: New measurements have been provided by the LHCb collaboration allowing for CP violation in mixing and in the interference between mixing and decay. The best fit point is [17]: It appears then that in the D system x D ∼ y D or Γ 12 ∼ M 12 , whereas in the B system |Γ 12 /M 12 | 1. For future prospects see [18,19]. In the limit of (approximate) CP symmetry x D , y D > 0 implies that the CP even state (D 1 ) is slightly heavier and shorter lived than the CP odd one (D 2 ) (unlike for neutral kaons).
In the SM the D 0 − D 0 mixing originates in the box diagrams with W exchange and internal down-type quarks, as opposed to B 0 d,s −B 0 d,s and K 0 −K 0 mixings where the uptype quarks are exchanged, in particular the top-quark. As a result, the mass differences ∆M s,d predicted by the SM turn out to be in the ballpark of their experimental values. This also holds for ∆M K , although in this case sizable long-distance contributions are present which are still not fully under control [19].
so that this contribution can be easily evaluated. One finds that, due to strong GIM suppression, x D and y D are by at least one order of magnitude below the experimental values in (19), so that either long-distance or new physics contributions are responsible for their measured values. Many predictions for x D , y D in the SM have been worked out, which however are affected by large uncertainties, as discussed in Section 3 of [19]. As in the SM, M 12 corresponds to the dispersive part and Γ 12 to the absorptive part of the box diagrams, which are both strongly suppressed by the GIM mechanism. The short-distance SM contributions to them are both real to a very good approximation, and this is also the case of the long-distance contributions. This implies that, unless NP contributions to x D and y D in a given model are in the ballpark of their values in (19), the best chance to search for NP in D 0 − D 0 mixing is through CP violation, represented by the departure of |q/p| from unity and the non-vanishing phase φ in the measured range in (20). If NP is taken into account, other operators beyond Q 1 can contribute. The experimental values in (19) and (20) provide strong constraints to the extensions of the SM in which left-right, scalar and tensor operators are present [20][21][22][23]. Indeed, such operators have large hadronic matrix elements and strongly enhanced Wilson coefficients through QCD renormalization group effects, In the 331 model only the operator Q 1 is involved, as in the SM, but this time its Wilson coefficient receives the contribution from tree-level Z exchange. However, even such tree-level contributions are below the experimental values of x D and y D when the contraints from the other quark sectors are included. As a matter of fact in the 331 model, if all other contributions to the mixing are ignored, at the scale M Z x D is given by: where F D is the D 0 decay constant and B 1 parametrizes the deviation of the matrix element of the four quark operator Q 1 = (ū L γ µ c L )(ū L γ µ c L ) from the vacuum saturation approximation. The coupling ∆ uc L (Z ) reads: The elements u ij can be obtained using Eqs. (3) and (5): Hence, x 331 D depends on the four parameterss 13 , δ 1 ands 23 , δ 2 which govern B d and B s decays, respectively, and which altogether govern the K decays.
When evolved down to m c , the RG running gives [24,25]: The evolution can be performed including NLO QCD corrections by means of the results in [26], but here we restrict the analysis to the leading order. The reason is that, as discussed in Section 4, the numerical value of x 331 D resulting from (23), (27) is below the experimental measurement in (19) by about one order of magnitude when the constraints ons 13 , δ 1 ,s 23 and δ 2 are considered.
The small value of x 331 D leads us to conjecture that both SM long-distance and 331 contributions are responsible for the measured values of x D and y D . The 331 term is pivotal in providing the complex phases producing a deviation of |q/p| from unity and a non-vanishing mixing phase φ. This is at the basis of the analysis strategy in Section 4: We use the measured values of x D and y D in Eq. (19), interpreting them as resulting from the contributions of both LD in the SM and of 331. Then, we predict observables and correlations, as those involving |q/p| and φ.

CP asymmetries
CP violation in the charm system can be tested in various ways. Here we consider the time-dependent CP asymmetries where f is a CP eigenstate The phase ϕ f is phase-convention independent since it depends only on the relative phase between q/p and A f /A f : where In the case of a non-negligible CP phase ξ f in the decay amplitude In this case, as x D , y D 1, it is appropriate to consider the CP asymmetry in the limit of a small t [27]: whereτ D = 1/Γ. On the other hand, when the phase ξ f is negligible, as happens in the SM and, as we shall see, also in the 331 models, one has and with S f is analogous to S ψK S and S ψφ in the B d and B s system, respectively. However, in the B system y x and |q/p| 1, so that the above result simplifies considerably, leaving only the second term and allowing the measurement of the phase φ.
Finally, we consider the semileptonic asymmetry to wrong sign leptons with X = K ( * ) , . . . , which represents CP violation in mixing in ∆C = 1 transitions. The last expression comes from the condition ||q/p| − 1| 1.

Correlations
Two interesting correlations have been derived in [28], which read with our conventions [27]: In the limit |q/p| − 1 1, x D ∼ y D , the relation (39) simplifies to [28] sin Inserting this result in (37) and (38) one finds for ξ f = 0: It should be emphasized that this relation is only valid if in our convention ξ f = 0 independently of f , that is no CP phase in decay amplitudes. The experimental violation of the relation (41) would then imply the occurrence of CP violation in the decay (direct CP violation) [28]. In the case of a significant phase ξ f one finds: As ξ f is phase convention dependent, the presence of direct CP violation would be signalled by finding S f 1 − S f 2 = 0 for two different final states f 1 and f 2 . We shall analyze all these relations within 331 models in Section 4, but we want to have first a look at time-independent CP-asymmetries.

∆A
The first observation of CP violation in the decays of neutral charm mesons was presented by the LHCb collaboration. For the difference between the CP asymmetries in D 0 → K + K − and D 0 → π + π − , the measurement is [29] ∆A CP = (−15.4 ± 2.9) × 10 −4 .
With an excellent approximation this difference measures direct CP violation. Within the SM this result has been explained invoking non perturbative effects enhancing the penguin contributions [30][31][32][33][34][35][36][37][38][39][40]. Other studies envisage the possibility that some NP could be responsible for this result [41][42][43][44][45][46][47][48] (also supported by the analysis in [49]). A discussion of the LHCb data and of the future prospects can be found in [2]. Here we analyze the asymmetry in 331 models. The modes D 0 → K + K − and D 0 → π + π − are single Cabibbo suppressed. These are of the kind D 0 (D 0 ) → f with f a CP eigenstate that can be produced both in D 0 , the CP asymmetry is defined as Note that in contrast to (29) this asymmetry is time-independent. When two amplitudes (at least) contribute to each decay, one can write [31] A where 1 are the difference between strong and weak phases, respectively. If one of the amplitudes dominates, i.e. r f 1, one finds: When NP is present the previous relations can be generalized by adding a new amplitude A f,N P = |A f,N P |e i δ N P e i φ N P to (46): Consequently we have: In the SM both D 0 → K + K − and D 0 → π + π − have a tree-level contribution A tree SM , proportional to λ s = V * cs V us for f = K + K − and to λ d = V * cd V ud for f = π + π − . Penguin contributions are also present with internal down-type quark exchange and hence proportional to λ D = V * cD V uD with D = d, s, b. We denote such contributions as A P,D SM . Exploiting the CKM unitarity relation λ d + λ s + λ b = 0, the general structure of the SM amplitude reads: In 331 models new contributions from tree-level Z exchanges are present. Except for different hadronic matrix elements and the change of the coupling ∆ sd L (Z ) to ∆ uc L (Z ) given in Eq. (24), the structure of the Z contribution to the effective Hamiltonian at M Z is as in the case of the ratio ε /ε [10]. While flavour-changing Z couplings involve only left-handed quarks, flavour conserving ones involve both left and right-handed ones. Flavour conserving Z couplings to left-handed quarks are with f (β) given in (7). Numerically, the second term in Eqs. (52)-(53) is a factor of O(10 −5 ) smaller than the first one and can be neglected, so that we are left with the universal coupling Moreover the couplings to right-handed quarks read Calculating the tree-level Z exchange diagrams we find for both D → K + K − and D → π + π − decays the effective Hamiltonian at M Z : with the penguin operators Q 3 and Q 7 also present in the SM and defined in Appendix A.
The new operator arises from the different couplings of Z to the third generation of quarks; however, its contribution, only through RG effects, is negligible because b and t quarks do not appear in the low energy effective theory relevant for D decays. The coefficients C 3 , C 7 and C 3 are expressed in terms of the couplings in (54), (55) and (56): The RG evolution to the scale µ c m c generates other operators. In particular, the full set of operators with the structure of QCD (Q 3 − Q 6 ) and of the EW penguins (Q 7 , Q 8 ) is generated, as described in Appendix A. Using short-hand notation for the various hadronic matrix elements the 331 contributions to the decay amplitudes read: where µ c O(m c ). The values of the coefficients C i (µ c ) are collected in Appendix A. We use such expressions in the numerical analysis in Section 4.
Neglecting SM contributions, we find for the coefficients in 331: and Using (67) the D 0 → + − branching fraction reads: In the SM we find, in agreement with [50]: where Y (x b ) is the one-loop function that in B s → µ + µ − appears like Y (x t ) [51]. Since m b M W , we have to an excellent approximation: The SM prediction is then in correspondence of the central values for the input parameters. The present experimental upper bound is B(D 0 → µ + µ − ) < 6.2 × 10 −9 at 90% C.L. [52]. 4

Numerical results in 331
A strategy for the numerical analysis of flavour observables in 331 model has been outlined in [11]. The model parameters are bound imposing that ∆M B d , S J/ψK S and ∆M Bs , S J/ψφ lie in their experimental ranges within 2σ. In the kaon sector we require ε K ∈ [1.6, 2.5] × 10 −3 and ∆M K varying between [0.75, 1.25] × (∆M K ) SM : using V ub in Table 1 this corresponds to (∆M K ) SM = 0.0047 ps −1 . We refer to [8] for the formulae for the various observables in the SM and in 331 models. We collect in Table 1 the theory and experimental input parameters used in the present study. In the case of the CKM matrix, Table 1 displays the four entries that we set as independent parameters; all the other CKM parameters are derived from these ones. In particular, |V cb | is fixed at the central value in the Table, while |V ub | and γ are chosen within the range quoted in [52]. Among the other CKM parameters, we obtain for |V cd |, |V ud | and |V cs | relevant for our study: |V cd | = 0.2251, |V ud | = 0.9743 and |V cs | = 0.9735. The allowed ranges for the parameterss 13 , δ 1 ,s 23 , δ 2 are shown in Fig. 1 for M Z ∈ [1, 5] TeV. The regions in the parameter plane (s 13 , δ 1 ) are obtained imposing the constraints on ∆M B d and S J/ψK S whose measurements are in Table 1, the regions for (s 23 , δ 2 ) are obtained using the measured ∆M Bs and S Jψφ in the same Table. All panels in Fig. 1 show two ranges for the phases δ 1,2 which are independent of the value of M Z ; the ranges for the parameterss 13(23) depend on M Z .
The observables analyzed in the following are computed varyings 13 , δ 1 ,s 23 , δ 2 in their allowed ranges, and selecting only the values for which the constraints from ∆F = 2 observables in the kaon sector are also satisfied.    It is interesting to consider the correlation between the asymmetry S f (for η f = +1) and the semileptonic asymmetry a SL (D 0 ) in Eqs. (37), (38). It is depicted in Fig. 3   parameters fixed in the analysis of ∆A CP described below, are included in the expression (42). A semileptonic asymmetry a SL at the per-cent level is permitted, while |S f | does not exceed O(10 −4 ).
of the hadronic matrix elements Q i KK and Q i ππ . The simplest approach to evaluate such quantities, the naive factorization (NF) prescription, allows to express the matrix elements of the relevant operators in terms of Q 1 f : with N c the number of colors. In (83) the chiral factors are The 331 contribution to ∆A CP can be worked out using these relations and varying the 331 parameters in the allowed oases. The largest value is obtained for β = −2/ √ 3: |(∆A 331 CP ) max | 5.6 × 10 −5 . The prediction for the sign cannot be provided without an information on the strong phases.
However, the accuracy of naive factorization is doubtful, in particular for charm. On the other hand, no information on the hadronic matrix elements is available from QCD methods as lattice QCD. For this reason, we follow a different approach. We consider the ratios as additional quantities to be constrained using data. In the numerical analysis, together with the 331 parameterss 13 ,s 23 , δ 1 , δ 2 , we scan the space of Large values of r f correspond to large values of the hadronic matrix elements, a statement that can be made more precise if the difference ∆A 331 CP is considered, as in Fig. 6. When the ratios R f i are varied in the ranges quoted above, for the variants with β = ±2/ √ 3 it is possible to find a set of parameters (the 331 parameters plus ratios of hadronic matrix elements) giving ∆A CP in agreement with data. For larger values of the R f i also for β = ±1/ √ 3 ∆A CP can be obtained. An insight into the required size of the ratios of the hadronic matrix elements can be gained considering β = −2/ √ 3 for which, as shown in Fig. 6, the 331 result for ∆A CP has an overlap with the experimental range. The largest value Figure 4: Correlation between the ratior K + K − and phaseφ K + K − in (79) in 331 models with the same parameters of Fig. 2, varying the ratios of the operator matrix elements in (84) as described in the text.
of |∆A CP | is obtained in correspondence of the ratios R K + K − i : and of the ratios R π + π − i : The conclusion is that large individual CP asymmetries and ∆A CP can be obtained provided that two conditions are verified: the phases are in the ranges obtained in 331 models and the ratios of the hadronic matrix elements are large, close to the values in (87) and (88). This last condition can be verified by future explicit calculations, namely using lattice QCD. Figure 5: Correlation between the ratior π + π − and phaseφ π + π − in (80) in 331 models, as described in the text.
4.3 D 0 → µ + µ − , correlations with B and K observables We shall next investigate correlations of charm observables with those in the B and K systems, a striking possibility within the 331 models.
In Fig. 7 we show the correlation between the S f asymmetry in (37) and the mixinginduced CP asymmetries S J/ψK S and S J/ψφ in the B d and B s systems. The message from these plots is clear. CP violation in the charm system is by orders of magnitude Figure 6: Range for ∆A 331 CP in 331 models. ∆A 331 CP is obtained for each β using the ratiosr K + K − , r π + π − and phasesφ K + K − ,φ π + π − in Figs  smaller than in the B d and B s systems. Even in the B s system S f is roughly by two orders of magnitude smaller than S J/ψφ .
In Fig. 8 we show the correlation between x 331 D and B(D 0 → µ + µ − ) in 331 for the four variants with β considered in this paper and M Z = 1 TeV. The SM contribution is set to zero in both observables. We note that B(D 0 → µ + µ − ) can reach values of order 10 −14 , that is by roughly six orders of magnitude larger than the result (74) obtained within the SM but still much smaller than the experimental bound.
It is worth considering the impact on these results of the measurement where B(B s → µ + µ − ) is the average time integrated branching ratio [61]: with A ∆Γ = 2 Re[λ µ + µ − ] 1 + |λ µ + µ − | 2 and y s = τ Bs ∆Γ s 2 = 0.0064±0.004 [52]. λ µ + µ − is analogous to for this further suppression is evident from Fig. 9 where we show the correlation between B(B s → µ + µ − ) and B(D 0 → µ + µ − ) in 331 models with parameters as in Fig.8. Indeed, to suppress the SM branching ratio for B s → µ + µ − to achieve a better agreement with the experimental measurement (89), a smaller B(D 0 → µ + µ − ) is implied. This would not be the case if the data for B(B s → µ + µ − ) was above the SM expectation.
In Figs. 10 and 11 we show the correlations of B(D 0 → µ + µ − ) with the branching ratios for K + → π + νν and K L → π 0 νν, respectively. We observe that the largest values of B(D 0 → µ + µ − ) are found for the smallest 331 contributions to both kaon decays. The largest contributions to these decays in 331 models amount to roughly ±10% of the SM value and these modest effects could be tested one day, in particular if the future data turned out to depart significantly from the SM predictions within the SM central value represented by black dots in these figures.
The pattern turns out to be different in Figs. 12 and 13, where we show the correlations of B(D 0 → µ + µ − ) with branching ratios for K L → µ + µ − and K S → µ + µ − , respectively. Without the B(B s → µ + µ − ) constraint the largest 331 contributions to K L → µ + µ − and K S → µ + µ − are accompanied by largest contributions to B(D 0 → µ + µ − ). This is in particular seen in the case of K L → µ + µ − in Fig. 12. However, when the data on B(B s → µ + µ − ) are imposed, 331 effects in D 0 → µ + µ − are dwarfed as already seen in Fig. 9.

Conclusions
We have performed a detailed analysis of FCNC processes in the charm sector in the context four variants of 331 models. The motivation for this analysis arose from the observation [5] that in the case of Z contributions there are no new free parameters beyond those present already in the B d,s and K meson systems. As a result definite ranges for NP effects in various charm observables could be obtained.
In the SM FCNC in the charm system are very strongly GIM suppressed so that a contribution of Z that appears already at tree-level could in principle imply NP effects in the charm sector by orders of magnitude larger than it is possible within the SM. In fact if we performed our analysis first for the charm system, ignoring correlations with B d,s and K meson systems, very large effects would be possible. However, as our analysis demonstrates most of these effects would be subsequently dwarfed by imposing the contraints from B d,s and K meson systems that were analyzed in the past [8][9][10][11][12].
In particular we find very small effects in D 0 −D 0 mixing although the effects are still larger than short-distance contributions within the SM.
Our results are presented in the plots in Section 4 and in the accompanying comments. Here we list the most remarkable results.
• As seen in Fig. 3 the semileptonic asymmetry a SL (D 0 ) can be as large as ±5 × 10 −2 and therefore much larger than in the SM.
• As seen in Fig. 9, the correlation between B(B s → µ + µ − ) and B(D 0 → µ + µ − ) implies the suppression of the latter branching ratio if a better agreement with the experimental data for B(B s → µ + µ − ) is to be achieved. Yet, despite this suppression the branching ratio B(D 0 → µ + µ − ) can reach values of a few 10 −15 which is by six orders larger than its SM value.
• The correlations between B(D 0 → µ + µ − ) and the branching ratios for K + → π + νν, K L → π 0 νν and K L,S → µ + µ − show patterns that depend on the specific 331 model considered. However, if future data on theoretically clean decays K + → π + νν and K L → π 0 νν and also short distance part in K S → µ + µ − will show deviations from SM expectations by more than 15% the 331 models will not be able to explain them.
• The D 0 → π + π − , K + K − CP asymmetries and ∆A CP can be sizable for phases in the ranges provided in 331 model, however the ratios of the matrix elements in Eq. (84) should be large, with a size as provided in (87),(88). This possibility requires a nonperturbative calculation of the matrix elements.
We are looking forward to the improved experimental data on all observables discussed by us to see whether 331 models will survive these tests.