Sources of FCNC in 𝑆𝑈(3)_𝐶⊗𝑆𝑈(3)_𝐿⊗𝑈(1)_𝑋 Models

Abstract

There are different models which are based on the gauge symmetry 𝑆𝑈(3)_𝐶⊗𝑆𝑈(3)_𝐿⊗𝑈(1)_𝑋 (331), and some of them include exotic particles, and others are constructed without any exotic charges assigned to the fermionic spectrum. Each model build-up on 331 symmetry has its own interesting properties according to the representations of the gauge group used for the fermionic spectrum, that is, the main reason to explore and identify the possible sources of flavor changing neutral currents and lepton flavor violation at tree level.

1. Introduction

The standard model (SM) [1–3] has been successful to describe leptons, quarks, and their interactions. But in any case, the SM leaves open questions concerning to the electroweak symmetry breaking sector of the model, as well as the particle content of the model: why there are three generations of quarks and leptons? These questions, among others, are the motivation to consider the SM as one important attempt to understand the elementary particles of nature and their interactions but not to consider the SM as the ultimate theory of nature. A common alternative to look for new physics beyond the SM is enlarging the gauge symmetry group, one of these alternatives is the gauge symmetry 𝑆𝑈(3)_𝐶×𝑆𝑈(3)_𝐿×𝑈(1)_𝑋 (331) [4–11]. There are many motivations for this new gauge symmetry group, one of them is that there are some of models are based on 331 symmetry that explain why the family number must be three. This result is obtained from the anomaly-free condition which is satisfied when equal number of triplets and antitriplets (taking into account the 𝑆𝑈(3)_𝐶) are present and requiring the sum of all fermion charges to vanish, but even that each generation is anomalous and the anomaly cancellation is given for three generations or multiply of three. Other motivation is concerned with the feature that sin²𝜃_𝑊 in this model should be less than 1/4, it is related to the ratio of the coupling constants 𝑔^′ and 𝑔 of 𝑈(1)_𝑋 and 𝑆𝑈(3)_𝐿,𝑔^′𝑔²=sin²𝜃_𝑊1−4sin²𝜃_𝑊, (1.1) in this model, there is an energy scale at which the perturbative character is lost, and the energy scale is found using the condition sin²𝜃_𝑊=1/4 and it is order of ∼4 TeV [12].

On the other hand, in the breaking symmetry of the 331 gauge symmetry to the gauge group of the SM and then to the 𝑈(1)_𝑄, some new bosons appear such as a new neutral 𝑍^′ boson which is heavier than the SM gauge bosons and in all the 331 models it can mediate flavor changing process at tree level. In contrast, in the framework of the SM it is well known that flavor changing neutral currents (FCNC) are strongly suppressed, because they appear only at one loop level. Therefore, these FCNC processes can help to put stringent bounds on the parameter space of these kind of models [13–20]. Our aim in this work is to review the possible models that can be built on the basis of extended gauge symmetry 331 and identify the different sources of FCNC in the quark sector as well as the lepton sector.

2. 331 Models

The gauge group to be consider is 𝑆𝑈(3)_𝐶⊗𝑆𝑈(3)_𝐿⊗𝑈(1)_𝑋. Left-handed particles are into 𝑆𝑈(3)_𝐿 triplets, there are the usual quarks and leptons plus new exotic particles, and the anomaly-free condition constrains the allowed fermion representations (3 or 3^∗) and the quantum numbers. To describe the particle content of the model and to identify specific types of 331 models is important and so is how defined the electric charge operator, which can be written as a linear combination of the diagonal generators of the group𝑄=𝑇₃+𝛽𝑇₈+𝑋, (2.1) where 𝛽 is a parameter that characterizes the specific particle structure. The parameter 𝛽 can be chosen 𝛽=±√3 or 𝛽=±1/√3, obtaining 331 models with exotic electric charges or 331 models without exotic electric charges, and by exotic charges we mean charges different from those that appear in the SM framework.

Since each lepton family has three states, taking massless neutrinos, they can be arranged into 𝑆𝑈(3)_𝐿 antitriplets 𝜓^𝑇_𝑖=(𝑙⁻_𝑖,−𝜈_𝑖,𝑙⁺_𝑖), where 𝑖 is a family index. The first two components corresponds to the ordinary electroweak doublet. This model corresponds to 𝛽=√3 [4–11] for the charge operator in (2.1). Therefore each lepton family will be in the (1,3^∗)₀ representation of 𝑆𝑈(3)_𝐶⊗𝑆𝑈(3)_𝐿⊗𝑈(1)_𝑋. With these assumptions, there are no new leptons in the 331 model, and all three lepton families are treated identically. In contrast, one of the three quark families transform differently from the other two which is required to anomaly cancellation. Anomaly cancellation requires that two families of quarks transform as triplets (3,3)_−1/3, and the third one transforms as an antitriplet (3,3^∗)_2/3. The right-handed spectrum is put in singlets in the usual way (3^∗,1)_{−2/3,1/3,4/3} for the first two families and (3^∗,1)_{−5/3,−2/3,1/3} for the third one. It is worth to notice that in general the assumption that one quark family is transforming differently to the other two families is a general condition in the framework of 331 models, and it is generally assumed that the unique generation corresponds to the third generation, and then, it could explain the heavy top quark mass.

In the gauge sector, five new gauge bosons beyond the SM are found. The new gauge bosons form a complex 𝑆𝑈(2)_𝐿 doublet of dileptons (𝑌⁺⁺,𝑌⁺) with hypercharge 3 and a singlet 𝑊⁸_𝜇. The gauge boson 𝑊⁸_𝜇 mixes with the gauge boson 𝑋 from the 𝑈(1)_𝑋 to form the hypercharge 𝐵_𝜇 boson and a new neutral 𝑍^′_𝜇 boson.

In order to break the symmetry spontaneously, four Higgs multiplets are necessary. Three triplets in representations (1,3)₁, (1,3)₀, and (1,3)₋₁ for the breaking of 𝑆𝑈(2)×𝑈(1) in order to give masses to all quarks, and a sextet (1,6)₀ is required for the lepton masses [4–11].

In this first model [4–11], there are new sources of FCNC processes at tree level coming from the new 𝑍^′ boson in the quark sector, because the families are treated differently. Also, at one loop level appears new contributions coming from the charged bileptons and the charged scalar sector [21–26]. In this model, there are FCNC in the lepton sector, and they are mediated by the charged bileptons [21–26].

A possible variation of this original model is to consider a new lepton assignment using a heavy lepton 𝐸⁺ instead of the 𝑒^𝑐 and adding 𝑒^𝑐 and 𝐸⁻ as singlets [27]. With this model, it is easy to generate small neutrino masses and lepton number violation can occur and one property of this model version is that bileptons only couple standard to exotic leptons [27].

On the other hand, it is possible to obtain models based on the gauge 331 symmetry but without new exotic charges for the fermions. One version of that is the model proposed by Özer [28], where it is introduced a right-handed neutrino. A systematic study of these kind of models was done in [29, 30]. According to the 𝛽 value in (2.1), it is possible to get six different set of fermions and the fermion structure in order to avoid the quiral anomalies producing different 331 models. The fermion sets are four lepton sets and two quark sets.

The first set of leptons is𝐿₁=𝜈_𝑖𝑒⁻_𝑖𝐸⁻_𝑖𝑒⁺_𝑖𝐸⁺_𝑖(1,3)_−2/3(1,1)₁(1,1)₁ (2.2) using 𝑖=1,2,3 as the family index and 𝑒_𝑖,𝑑_𝑖, and 𝑢_𝑖 are the SM fermions and 𝐸_𝑖,𝐷_𝑖, and 𝑈_𝑖 are the exotic ones.

The second set is𝐿₂=𝑒⁻_𝑖𝜈_𝑖𝑁⁰_𝑖𝑒⁺_𝑖1,3^∗_−1/3(1,1)₁, (2.3) where there is a neutral exotic particle. For the third leptonic set,𝐿₃=𝑒⁻_𝑖𝜈_𝑖𝑁⁰₁𝐸⁻_𝑖𝑁⁰₂𝑁⁰₃𝑁⁰₄𝐸⁺_𝑖𝑒⁺_𝑖1,3^∗_−1/31,3^∗_−1/31,3^∗_2/3, (2.4) where there is a charged exotic particle and four new exotic neutral ones. Finally, for the forth set𝐿₄=𝜈_𝑖𝑒⁻_𝑖𝐸⁻_1𝑖𝐸⁻_2𝑖𝑁⁰₁𝑁⁰₂𝑁⁰₃𝐸⁻_2𝑖𝐸⁻_3𝑖𝑒⁺_𝑖𝐸⁺_1𝑖𝐸⁺_3𝑖(1,3)_−2/3(1,3)_1/3(1,3)_−2/3(1,1)₁(1,1)₁(1,1)₁, (2.5) with three exotic charged particles and three neutral.

Now, the quark sets are𝑄₁=𝑑_𝑖𝑢_𝑖𝑈_𝑖𝑑_𝑖𝑢_𝑖𝑈_𝑖3,3^∗_1/3(3,1)_1/3(3,1)_−2/3(3,1)_−2/3,𝑄₂=𝑢_𝑖𝑑_𝑖𝐷_𝑖𝑢_𝑖𝑑_𝑖𝐷_𝑖(3,3)₀(3,1)_−2/3(3,1)_1/3(3,1)_1/3. (2.6)(2.7) The anomaly contribution for each set is presented in Table 1.

Table 1 

Anomalies for the six fermion sets.

On the basis of Table 1, it is possible to build up many models asking for the anomaly free condition. There are two one family models and eight three family models, referring to how cancel out the anomalies if it is needed one family or the three families. There are two one family models composed by the sets 𝑄₂+𝐿₃ and 𝑄₁+𝐿₄. These models were studied in [29–31] and their relation with the grand unified theories established. For the three family models, there are the combinations 3𝐿₂+𝑄₁+2𝑄₂, 3𝐿₁+2𝑄₁+𝑄₂, 2(𝑄2+𝐿3)+(𝑄1+𝐿4) and 2(𝑄1+𝐿4)+(𝑄2+𝐿3), and there are other two models particularly interesting, because they treat the three family leptons completely different, they are the combinations 𝐿₁+𝐿₂+𝐿₃+𝑄₁+2𝑄₂ and 𝐿₁+𝐿₂+𝐿₄+2𝑄₁+𝑄₂ [31, 32].

In the gauge sector, there are 17 gauge bosons, one gauge boson 𝐵^𝜇 associated to 𝑈(1)_𝑋, eight gluons associated to 𝑆(3)_𝑐 and eight gauge fields from the 𝑆𝑈(3)_𝐿. The gauge bosons associated with 𝑆𝑈(3)_𝐿 transform according to the adjoint representation of the group, and they can be written as𝐖_𝜇=𝑊^𝑎_𝜇𝜆^𝑎2=12𝑊³_𝜇+1√3𝑊⁸_𝜇√2𝑊⁺_𝜇√2𝐾_1𝜇√2𝑊⁻_𝜇−𝑊³_𝜇+1√3𝑊⁸_𝜇√2𝐾_2𝜇√2𝐾_1𝜇√2𝐾_2𝜇−2√3𝑊⁸_𝜇, (2.8) where 𝜆^𝑎 are the Gell-Mann matrices, and the electric charges of 𝐾₁ and 𝐾₂ are given by 𝑄₁=1/2+√3𝛽/2 and 𝑄₂=1/2−√3𝛽/2, respectively.

In general, it is convenient to rotate the neutral gauge bosons 𝑊³_𝜇, 𝑊⁸_𝜇 and 𝐵_𝜇 into new states 𝐴_𝜇, 𝑍_𝜇, and 𝑍^′_𝜇, given by𝐴_𝜇𝑍_𝜇𝑍^′_𝜇=𝑆_𝑊𝛽𝑆_𝑊𝐶_𝑊1−𝛽²𝑇²_𝑊𝐶_𝑊−𝛽𝑆_𝑊𝑇_𝑊−𝑆_𝑊1−𝛽²𝑇²_𝑊0−1−𝛽²𝑇²_𝑊𝛽𝑇_𝑊𝑊³_𝜇𝑊⁸_𝜇𝐵_𝜇, (2.9) where the angle 𝜃_𝑊 is defined by 𝑇_𝑊=tan𝜃_𝑊=𝑔^′/𝑔²+𝛽²𝑔^′2, 𝑔, 𝑔^′ being the coupling constants associated to the groups 𝑆𝑈(3)_𝐿 and 𝑈(1)_𝑋, respectively, (𝑆_𝑊=sin𝜃_𝑊, etc.). In the new basis, 𝐴_𝜇 (the photon) is the gauge boson corresponding to the generator 𝑄, while 𝑍_𝜇 can be identified with the SM 𝑍 boson. As in the SM, the extended electroweak symmetry is spontaneously broken in 331 models by the presence of elementary scalars having nonzero vacuum expectation values [33–37]. The symmetry breakdown follows a hierarchy𝑆𝑈(3)_𝐿⊗𝑈(1)⟶_𝑋^𝑉𝑆𝑈(2)_𝐿⊗𝑈(1)⟶_𝑌^𝑣𝑈(1)_𝑄, (2.10) in which two VEV scales 𝑉 and 𝑣, with 𝑉≫𝑣, are introduced. The photon is kept as the only massless gauge boson, while the remaining neutral gauge bosons get mixed. In this way, 𝑍 and 𝑍^′ turn out to be only approximate mass eigenstates.

3. FCNC in 331 Models

First of all, the extension of the gauge group which embedded the SM group implies a new neutral gauge 𝑍^′ boson, which in general in all the 331 models presented generates FCNC at tree-level. This fact is that because in 331 models it is not possible to accommodate all the SM spectrum in multiplets with the same quantum numbers; therefore, the 𝑍^′ couplings are not universal for all the fermions, and that is the origin of a new source of FCNC. Particularly, to treat in a different manner the third generation, as is usually assumed, to the other two generations produces FCNC contributions. This property is common to all the 331 models in the quark sector. It is worth to mention that even in the left-handed couplings of the standard fermions to the 𝑍 neutral boson appear FCNC at tree level through the mixing of 𝑍−𝑍^′ and also coming from the mixing between the standard quarks and the exotic ones included in each case. Moreover, the mixing between neutral gauge bosons should take into account the gauge bosons which transform according to the adjoint representation of the 𝑆𝑈(3)_𝐿, some noted 𝐾^± and 𝐾⁰ gauge bosons for the charged sector and neutral sector (they are related to 𝐾_1,2). In order to notice these effects clearly, the Lagrangian for the new 𝑍^′ boson with a 𝛽 arbitrary is the following:ℒ^𝑍′𝑔=−^′2𝑇_𝑊𝑍^𝜇′2_𝑚=1𝐷⁰_𝑚𝛾_𝜇𝑃_𝐿√3+𝑇²_𝑊𝛽3𝑃_𝐿−2𝑃_𝑅𝐷⁰_𝑚+𝐷⁰₃𝛾_𝜇−𝑃_𝐿√3+𝑇²_𝑊𝛽3𝑃_𝐿−2𝑃_𝑅𝐷⁰₃+²_𝑚=1𝑈⁰_𝑚𝛾_𝜇𝑃_𝐿√3+𝑇²_𝑊𝛽3𝑃_𝐿+4𝑃_𝑅×𝑈⁰_𝑚+𝑈⁰₃𝛾_𝜇−𝑃_𝐿√3+𝑇²_𝑊𝛽3𝑃_𝐿+4𝑃_𝑅𝑈⁰₃+𝐿⁰𝛾_𝜇−𝑃_𝐿√3−𝑇²_𝑊𝛽𝑃_𝐿+2𝑃_𝑅×𝐿⁰+𝜈⁰𝛾_𝜇−1√3−𝑇²_𝑊𝛽𝑃_𝐿𝜈⁰+²_𝑚=1𝐽⁰_𝑚𝛾_𝜇−2𝑃_𝐿√3+𝑇²_𝑊13+3𝛽√3𝐽⁰_𝑚+𝐽⁰₃𝛾_𝜇−2𝑃_𝐿√3+𝑇²_𝑊13−3𝛽√3𝑃_𝐿−𝑃_𝑅𝐽⁰₃+𝐸⁰𝛾_𝜇2𝑃_𝐿√3+𝑇²_𝑊−13−3𝛽√3×−𝑃_𝐿+𝑃_𝑅𝐸⁰, (3.1) where 𝐷⁰=(𝑑⁰₁𝑑⁰₂𝑑⁰₃)^𝑇, 𝑈⁰=(𝑢⁰₁𝑢⁰₂𝑢⁰₃)^𝑇, 𝐿⁰=(𝑒⁰₁𝑒⁰₂𝑒⁰₃)^𝑇, 𝐸⁰=(𝐸⁰₁𝐸⁰₂𝐸⁰₃)^𝑇, and the exotic quarks 𝑗⁰_𝑖 with electric charges are given by 𝑞_𝐽1=𝑄_𝐽2=1/6+√3𝛽/2 and 𝑞_𝐽3=1/6−√3𝛽/2. There is explicitly shown the no universal couplings between the quarks 𝐷_𝑖, 𝑈_𝑖 and the 𝑍′ boson, and it is because one family is in the 3 representation, while the other two are in the 3^∗ (or vice versa). As a consequence, the FCNCs arise once the fields 𝑈_𝑖 and 𝐷_𝑖 are rotated to the mass eigenstates. The number of extra fermions up-quark type or down-quark type depends on the parameter 𝛽, for 𝛽=−1/√3 will have 𝑁_𝑈=1 and 𝑁_𝐷=2 and for 𝛽=1/√3 will have 𝑁_𝑈=2 and 𝑁_𝐷=1. Therefore, there is not only FCNC at tree level through the 𝑍′ boson, but also the usual 𝑍 boson due to the mix of these new exotic quarks with the ordinary ones. To notice this, for the case of 𝛽=+1/√3, the following definitions are useful 𝑈^𝑇₀=(𝑢⁰₁,𝑢⁰₂,𝑢⁰₃,𝑇⁰₁,𝑇⁰₂), 𝐷^𝑇₀=(𝑑⁰₁,𝑑⁰₂,𝑑⁰₃,𝐵⁰₁), 𝐸^𝑇₀=(𝑒⁰,𝜇⁰,𝜏⁰,𝐸⁰₁,𝐸⁰₂,𝐸⁰₃), and 𝑁^𝑇₀=(𝜈⁰_𝑒,𝜈⁰_𝜇,𝜈⁰_𝜏). Meanwhile, for the case 𝛽=−1/√3, the definitions are 𝑈^𝑇₀=(𝑢⁰₁,𝑢⁰₂,𝑢⁰₃,𝑇⁰₁), 𝐷^𝑇₀=(𝑑⁰₁,𝑑⁰₂,𝑑⁰₃,𝐵⁰₁,𝐵⁰₂), 𝐸^𝑇₀=(𝑒⁰,𝜇⁰,𝜏⁰), and 𝑁^𝑇₀=(𝜈⁰_𝑒,𝜈⁰_𝜇,𝜈⁰_𝜏,,𝑁⁰₁,𝑁⁰₂,𝑁⁰₃). With this vector notation, the Lagrangian for neutral currents isℒ_NC=_Ψ−𝑔𝑍^𝜇2𝐶_𝑊Ψ⁰𝛾_𝜇𝜖_Ψ⁽¹⁾_(𝐿)𝑃_𝐿Ψ⁰+Ψ⁰𝛾_𝜇𝜖_Ψ⁽¹⁾_(𝑅)𝑃_𝑅Ψ⁰−𝑔^′𝑍^′𝜇2√3𝑆_𝑊𝐶_𝑊Ψ⁰𝛾_𝜇𝜖_Ψ⁽²⁾_(𝐿)𝑃_𝐿Ψ⁰+Ψ⁰𝛾_𝜇𝜖_Ψ⁽²⁾_(𝑅)𝑃_𝑅Ψ⁰−𝑔√2Ψ⁰𝛾_𝜇𝜖_Ψ⁽³⁾_(𝐿)𝑃_𝐿Ψ⁰Re𝐾^𝜇+𝑖Ψ⁰𝛾_𝜇𝜖_Ψ⁽⁴⁾_(𝐿)𝑃_𝐿Ψ⁰Im𝐾^𝜇, (3.2) where the sum is over 𝑈₀, 𝐷₀, 𝐸₀, and 𝑁₀. The couplings 𝜖_Ψ^(1,2)_(𝐿,𝑅) depends on the parameter 𝛽. With 𝛽=±1/√3, the 𝑍⁰ interaction is𝜖_𝒰⁽¹⁾_(𝐿)=𝐶²_𝑊−𝑆²_𝑊3𝟏_(3+𝑁^±_{𝑈)×(3+𝑁}^±_𝑈)−0_(3×3)𝟏_(𝑁^±_𝑈×𝑁^±_𝑈),𝜖_𝒰⁽¹⁾_(𝑅)=−4𝑆²_𝑊3𝟏_(3+𝑁^±_{𝑈)×(3+𝑁}^±_𝑈),𝜖_𝒟⁽¹⁾_(𝐿)=−𝐶²_𝑊−𝑆²_𝑊3𝟏_(3+𝑁^±_{𝐷)×(3+𝑁}^±_𝐷)+0_(3×3)𝟏_(𝑁^±_𝐷×𝑁^±_𝐷),𝜖_𝒟⁽¹⁾_(𝑅)=+2𝑆²_𝑊3𝟏_(3+𝑁^±_{𝐷)×(3+𝑁}^±_𝐷), (3.3) where the no universality of the left handed quarks is clear, while the right-handed couplings drive for 𝜖⁽¹⁾_{𝒰,𝒟(𝑅)} are universals.

In a similar way for the 𝑍^′0 boson, the couplings are𝜖_𝒰⁽²⁾_(𝐿)=𝐶²_𝑊±𝑆²_𝑊3𝟏_(3+𝑁^±_{𝑈)×(3+𝑁}^±_𝑈)−2𝐶²_𝑊0_(2×2)𝟏_(𝑁^±_{𝑇+1)×(𝑁}^±_𝑇+1)+𝐶²_𝑊∓2𝐶²_𝑊±𝑆²_𝑊0_(3×3)𝟏_(𝑁^±_𝑈×𝑁^±_𝑈),𝜖_𝒰⁽²⁾_(𝑅)=±4𝑆²_𝑊3𝟏_(3+𝑁^±_{𝑈)×(3+𝑁}^±_𝑈),𝜖_𝒟⁽²⁾_(𝐿)=𝐶²_𝑊±𝑆²_𝑊3𝟏_(3+𝑁^±_{𝐷)×(3+𝑁}^±_𝐷)−2𝐶²_𝑊0_(2×2)𝟏_(𝑁^±_{𝐷+1)×(𝑁}^±_𝐷+1)+𝐶²_𝑊±2𝐶²_𝑊∓𝑆²_𝑊0_(3×3)𝟏_(𝑁^±_𝐷×𝑁^±_𝐷),𝜖_𝒟⁽²⁾_(𝑅)=∓2𝑆²_𝑊3𝟏_(3+𝑁^±_{𝐷)×(3+𝑁}^±_𝐷). (3.4)

At this point, it is important to mention that the couplings in (3.2)–(3.4) are in the interaction basis, thus to obtain the mass eigenstates, it is necessary to get the rotation matrices which diagonalize the mass matrices in the Yukawa sector. Therefore, the mass eigenstates 𝑈 and 𝐷 are defined by𝑈⁰_𝐿=𝑉^𝑢_𝐿𝑈_𝐿,𝐷⁰_𝐿=𝑉^𝑑_𝐿𝐷_𝐿, (3.5) with matrices 𝑉_𝐿 of dimensions (3+𝑁^±_𝑈)×(3+𝑁^±_𝑈) and (3+𝑁^±_𝐷)×(3+𝑁^±_𝐷), respectively. It is useful to write the matrices 𝑉_𝐿^𝑢,𝑑 as𝑉^𝑢_𝐿=𝑉^𝑢_0(3×3)𝑉^𝑢_{𝑋(3×𝑁±𝑈)}𝑉^𝑢_{𝑌(𝑁±𝑈×3)}𝑉_{𝑈(𝑁±𝑈×𝑁±𝑈)},𝑉^𝑑_𝐿=𝑉^𝑑_0(3×3)𝑉^𝑑_{𝑋(3×𝑁±𝐷)}𝑉^𝑑_{𝑌(𝑁±𝐷×3)}𝑉_{𝐷(𝑁±𝐷×𝑁±𝐷)}, (3.6) using submatrices in such a way that 𝑉_CKM=𝑉₀^𝑢†𝑉^𝑑₀ and in general the CKM matrix is not unitary.

In addition, the models include new gauge bosons 𝐾_𝜇 which coupled to the left handed fermions, the couplings in (3.2) for the 𝐾^𝜇₂ boson when 𝛽=1/√3 are𝜖_𝑈⁽³⁾_(𝐿)=0_2×2𝟏_2×20𝟏_2×20_2×2,𝜖_𝐷⁽³⁾_(𝐿)=0_2×2,𝜖0110_{𝐸(3)(𝐿)}=𝟏_3×3𝟏_3×3,𝜖_𝑈⁽⁴⁾_(𝐿)=0_2×2𝟏_2×20−𝟏_2×20_2×2,𝜖_𝐷⁽⁴⁾_(𝐿)=0_2×20−110,𝜖_𝐸⁽⁴⁾_(𝐿)=𝟏_3×3−𝟏_3×3. (3.7) And when 𝛽=−1/√3 for the 𝐾^𝜇₁ boson, they are𝜖_𝑈⁽³⁾_(𝐿)=0_2×20110,𝜖_𝐷⁽³⁾_(𝐿)=0_2×2−𝟏_2×20−𝟏_2×20_2×2,𝜖_𝑁⁽³⁾_(𝐿)=𝟏_3×3𝟏_3×3,𝜖_𝐷⁽⁴⁾_(𝐿)=0_2×2,𝜖01−10_{𝑈(4)(𝐿)}=0_2×2𝟏_2×20−𝟏_2×20_2×2,𝜖_𝑁⁽⁴⁾_(𝐿)=−𝟏_3×3𝟏_3×3. (3.8)

It is often assumed that the vacuum expectation values of the scalar fields are real, and this assumption implies that there is not any spontaneous CP symmetry breaking. In that case, the state Im𝐾 decouples and therefore turn into an exact mass eigenstate. However, the bosons 𝑍^𝜇, 𝑍^′𝜇 and √2Re𝐾 mix, and it is possible to get the mass basis (𝑍₁,𝑍₂, and 𝑍₃) through an orthogonal matrix which depends on the vacuum expectation values of the Higgs bosons,𝑍𝑍^′√𝑍2Re𝐾=𝑅𝑍¹₂𝑍₃, (3.9) and therefore, the Lagrangian can be rewritten asℒ_NC=−_Ψ=𝒰,𝒟𝑄_ΨΨ𝛾^𝜇Ψ𝐴_𝜇+³_𝑗,𝑘=1𝑔_𝑗Ψ𝛾^𝜇𝐸_Ψ^(𝑗)_𝐿𝑃_𝐿+𝐸_Ψ^(𝑗)_𝑅𝑃_𝑅Ψ𝑅_𝑗𝑘𝑍_𝑘𝜇𝑔+𝑖2Ψ𝛾^𝜇𝐸_Ψ⁽⁴⁾_𝐿𝑃_𝐿+𝐸_Ψ⁽⁴⁾_𝑅𝑃_𝑅Ψ√2Im𝐾_𝜇, (3.10) where 𝑄_Ψ is the electric charge and the coupling constants 𝑔_𝑗 are𝑔₁=𝑔2𝐶_𝑊,𝑔₂=𝑔′2√3𝑆_𝑊𝐶_𝑊=𝑔2√3𝐶_𝑊𝐶²_𝑊−𝛽²𝑆²_𝑊,𝑔₃=𝑔2, (3.11) and the matrices 𝐸_Ψ^(𝑖)_𝐿,𝑅 are given by𝐸_Ψ^(𝑖)_𝐿=𝑉_𝐿^Ψ†𝜖_Ψ^(𝑖)_𝐿𝑉^Ψ_𝐿,𝐸_Ψ^(𝑖)_𝑅=𝑉_𝑅^Ψ†𝜖_Ψ^(𝑖)_𝑅𝑉^Ψ_𝑅=𝜖_Ψ^(𝑖)_𝑅. (3.12)

Finally, about the sources of FCNC in the framework of the 331 models, they are two models which are very interesting, because they have some special properties from the phenomenological point of view. They are the models build up with the fermionic sets 𝐿₁+𝐿₂+𝐿₃+𝑄₁+2𝑄₂ and 𝐿₁+𝐿₂+𝐿₄+2𝑄₁+𝑄₂. They not only differentiate the quark generations, doing one family specially different, but they also do in the leptonic sector. These models will have the usual FCNC at tree level in 331 models in the quark sector through the 𝑍′ boson but also they present FCNC in the leptonic sector through the scalar fields and through the 𝑍′ boson [32]. To notice the new sources of FCNC arising in these models, the neutral current Lagrangian is going to be obtained. First of all, the spectrum should be specifiedℓ_1𝐿=𝜈₁𝑒⁻₁𝐸⁻_1𝐿,ℓ_𝑚𝐿=𝑒⁻_𝑚𝜈_𝑚𝑁⁰_𝑘𝐿,ℓ_5𝐿=𝐸⁻₂𝑁⁰₃𝑁⁰_4𝐿,ℓ_4𝐿=𝑁⁰₅𝐸⁺₂𝑒⁺_3𝐿, (3.13) where 𝑚=2,3, 𝑘=1,2 and note that one of the leptonic triplets is in the adjoint representation respect to the other two then FCNC at tree level will arise through the 𝑍′ boson. Using vector notation, the neutral current Lagrangian in this case isℒ_NC=_Ψ𝐴_𝜇Ψ⁰𝛾_𝜇𝜖^𝐴_Ψ(𝐿)𝑃_𝐿Ψ⁰+Ψ⁰𝛾_𝜇𝜖^𝐴_Ψ(𝑅)𝑃_𝑅Ψ⁰+𝑔𝑍^𝜇2𝐶_𝑊Ψ⁰𝛾_𝜇𝜖^𝑍_Ψ(𝐿)𝑃_𝐿Ψ⁰+Ψ⁰𝛾_𝜇𝜖^𝑍_Ψ(𝑅)𝑃_𝑅Ψ⁰+𝑔^′𝑍^′𝜇2√3𝑆_𝑊𝐶_𝑊Ψ⁰𝛾_𝜇𝜖_Ψ^𝑍′_(𝐿)𝑃_𝐿Ψ⁰+Ψ⁰𝛾_𝜇𝜖_Ψ^𝑍′_(𝑅)𝑃_𝑅Ψ⁰. (3.14) Defining the vector 𝐸^𝑇=(𝑒⁻₁,𝑒⁻₂,𝑒⁻₃,𝐸⁻₁,𝐸⁻₂), the couplings are𝜖^𝐴_𝐸(𝐿)=𝑔𝑆_𝑊𝐼_5×5,𝜖^𝐴_ℰ(𝑅)=𝑔𝑆_𝑊𝐼_5×5,𝜖^𝑍_𝐸(𝐿)=𝑔2𝐶_𝑊Diag𝐶_2𝑊,𝐶_2𝑊,𝐶_2𝑊,−2𝑆²_𝑊,𝐶_2𝑊,𝜖^𝑍_𝐸(𝑅)=𝑔2𝐶_𝑊Diag−2𝑆²_𝑊,−2𝑆²_𝑊,−2𝑆²_𝑊,−2𝑆²_𝑊,𝐶_2𝑊,𝜖_𝐸^𝑍′_(𝐿)=𝑔^′2√3𝑆_𝑊𝐶_𝑊Diag1,−𝐶_2𝑊,−𝐶_2𝑊,−𝐶_2𝑊,−𝐶_2𝑊,𝜖_𝐸^𝑍′_(𝑅)=𝑔^′2√3𝑆_𝑊𝐶_𝑊Diag2𝑆²_𝑊,2𝑆²_𝑊,−𝐶_2𝑊,2𝑆²_𝑊,1, (3.15) where 𝐶_2𝑊=cos(2𝜃_𝑊), and it is worthwhile to point out that the right handed couplings are not universal, and it is a new feature of this model. Usually, in the framework of the 331 models, only the left-handed couplings are not universal, but the right-handed are universal as it was shown in (3.1).

For the neutral sector, 𝑁^𝑇=(𝜈⁰₁,𝜈⁰₂,𝜈⁰₃,𝑁⁰₁,𝑁⁰₂,𝑁⁰₃,𝑁⁰₄) is defined, and the left-handed couplings are𝜖^𝐴_𝑁(𝐿)=0,𝜖^𝑍_𝑁(𝐿)=𝑔2𝐶_𝑊Diag(),𝜖1,1,1,0,0,1,0,−1_{𝑍𝑁(𝐿)}=𝑔^′2√3𝑆_𝑊𝐶_𝑊Diag1,−𝐶_2𝑊,−𝐶_2𝑊,2𝐶²_𝑊,2𝐶²_𝑊,−𝐶_2𝑊,2𝐶²_𝑊,−1. (3.16)

4. Summary

One of the most intriguing options to consider physics beyond the SM consists of extending the gauge symmetry group to 𝑆𝑈(3)_𝐶×𝑆𝑈(3)_𝐿×𝑈(1)_𝑋. There are many models which are based on the 331 symmetry, and one intriguing feature of these models is the presence of FCNC at tree level, but the source of that new interactions is not unique and depends on how the model is built up. In the Pleitez-Frampton model, the first one proposed, it was established the presence of FCNC at tree level coming from the new 𝑍′ boson and due to the different assignment of the quark representation for one of the quark families; doing the left-handed couplings between quarks and the 𝑍′ boson not universal. On the other hand, it is possible to build up models on the basis of 331 symmetry contrary to the Pleitez-Frampton model without any exotic charges for the new particles in the spectrum. These kind of models correspond to a 𝛽=±1/√3 in the electric charge operator (2.1). These models include new exotic up-quark type and down-quark type which are going to mix with the standard quarks. In one version appears five up quark type and four down quark type, and another version include four-up quark type and five-down quark type; also, these models include extra charged leptons in one case and neutral leptons in the other one. The mixing obtained is a source of FCNC at tree level when the quark fields are written in the mass basis. There are also a new source of FCNC which is coming from the mixing in the gauge sector between the bosons (𝐴,𝑍,𝑍′,𝐾). The mixing in this sector is usually reduced to the mixing between 𝑍 and 𝑍′. If we consider the mixing between the quarks and the mixing (𝑍,𝑍′), then the FCNC interactions appear through the 𝑍 and the 𝑍′ mediation. In the leptonic sector, something similar is going to happen. Finally, there are models which not only treat different the quark families but the leptonic families too. One of these models is presented and the neutral current Lagrangian obtained, and one interesting new and additional feature is the nonuniversal couplings in the right-handed sector through the 𝑍 and 𝑍′ bosons. This new contributions to the FCNC processes could help to relax the bounds obtained on the 𝑍′ Boson mass.

Acknowledgment

The authors thank Professor V. Pleitez for useful discussions. J. Duarte acknowledges IFT-Universidade Estadual Paulista for the financial support and J.-A. Rodriguez for its hospitality while this work was finished.