Gauge Boson Mixing in the 3-3-1 Models with Discrete Symmetries

The mixing among gauge bosons in the 3-3-1 models with the discrete symmetries is investigated. To get tribimaximal neutrino mixing, we have to introduce sextets containing neutral scalar components with lepton number L 1, 2. Assignation of VEVs to these fields leads to the mixing of the new gauge bosons and those in the standard model. The mixing in the charged gauge bosons leads to the lepton number violating interactions of the W boson. The same situation happens in the neutral gauge boson sector.


Introduction
The experimental evidences of nonzero neutrino masses and mixing 1 have shown that the standard model of fundamental particles and interactions must be extended. Among many extensions of the standard model known today, the models based on gauge symmetry SU 3 C ⊗ SU 3 L ⊗ U 1 X called 3-3-1 models 2-9 have interesting features. First, SU 3 L 3 anomaly cancelation requires that the number of SU 3 L fermion triplets must be equal to that of antitriplets. If these multiplets are respectively enlarged from those of the standard model, the fermion family number is deduced to be a multiple of the fundamental color number, which is three, coinciding with the observation see Frampton in 2 . In addition, one family of quarks has to transform under SU 3 L differently from the other two. This can lead to an explanation why the top quark is characteristically heavy see, e.g., 10 . To complete the fundamental representations for leptons, the right-handed neutrinos or neutral fermions can be imposed which imply natural seesaw mechanisms for the neutrino small masses 11 . The 3-3-1 models can also provide a solution of electric charge quantization observed in the nature 12-16 . 2 Advances in High Energy Physics There are two typical versions of the 3-3-1 models concerning respective lepton contents. In the minimal 3-3-1 model 2-4 the lepton triplets include ordinary leptons of the standard model such as ν L , l L , l c R . The 3-3-1 model with right-handed neutrinos 5-9 introduces right-handed neutrinos into the lepton sector, that is, ν L , l L , ν c R and l R . In the framework of 3-3-1 models, to explain the smallness of neutrino masses and the tribimaximal mixing 17-20 we should propose another variant of the lepton sector such as ν L , l L , N c R and l R where N R are neutral chiral fermions carrying no lepton number called 3-3-1 model with neutral fermions , and including discrete symmetries either A 4 or S 4 21, 22 . The 3-3-1 model with neutral fermions based on S 3 flavor symmetry instead of A 4 , S 4 has been studied in 23 .
One of the most important ingredients is the sextets in which neutral scalar fields carrying lepton number L 1 or 2. Assignation of VEVs to these fields leads to the mixing among new gauge bosons and that of the SM similarly in the economical 3-3-1 model 24-26 , and such mixing leads to the lepton violating interactions. In this work we will pay attention to gauge bosons in the mentioned 3-3-1 models and give some phenomenological consequences.
The rest of this work is follows. In Section 2 we give a review of the 3-3-1 model with neutral fermions-based S 3 flavor symmetry. The other models with A 4 and S 4 can be done similarly, thus should be skip. Section 3 identifies gauge bosons and obtained the mixings among the standard model gauge bosons and the new ones. Section 4 is devoted to charged currents and give a constraint on the charged gauge boson mixing-angle. Finally we make conclusions in Section 5.

Brief Review of the Model
Before looking into the model, we provide a sketch of S 3 group theory 27, 28 . The S 3 that is a permutation group of three objects has six elements divided into three conjugacy classes. It possesses three nonequivalent irreducible representations 1, 1 of one dimension, and 2 of two dimensions. Denoting n and h as the order of class and the order of elements within each class, respectively, the character table is given by Table 1. 3 , and η 0 3 , η 0 3 , and χ 0 1 vanish. The exotic quarks get masses m U f 1 w and m D 1,2 fw. In addition, w has to be much larger than those of φ and η. Notice that the numbered subscripts are the indices of SU 3 L .
Because of the L-symmetry, the couplings ψ c L ψ L φ and ψ c L ψ L φ are suppressed. We therefore propose a new SU 3 L antisextet instead coupling to ψ c L ψ L responsible for neutrino masses. The antisextet transforms as where the numbered subscripts are the SU 3 L indices. Henceforth the indices of S 3 on scalar fields will be kept and should be understood. The VEVs of s is set as s 1 , s 2 under S 3 , where Due to the S 3 symmetry, all these VEVs are equal to each others, that is, λ 1 λ 2 , v 1 v 2 and Λ 1 Λ 2 , which can be found from the potential minimization.
With the scalar multiplets as defined, the Yukawa lagrangian is given by

2.8
Advances in High Energy Physics 5 It is easily shown that the charged leptons and ordinary quarks get consistent masses 23 . However, this case does not lead to neutrino masses and mixing consistent with the experimental data. The analysis in 21, 22 shows that i a "perturbation" is required: A possibility to derive this is to impose another antisextet s but with the VEVs being very smaller than those of s, respectively. Thus, in the followings the s should be skipped since it does not contribute at the first order. Otherwise, the s contributions start from the second order in similarity to those of s which are easily included. ii A scalar triplet ρ similar to φ must be imposed. The ρ is also skip for the same reason as s , that is, its contribution is similar to that of φ . Let us emphasis that our conclusions remain unchanged if s and ρ present. The hierarchies in the VEVs were given in 23 : In the following, the two limits are often taken into account: i the lepton-number violating parameters tend to zero, that is, λ 1,2 , u 1,2 → 0, and ii the large scales of SU 3 L symmetry break down to that of the standard model approx infinity, that is, ω, Λ 1,2 → ∞. Let us note also that v, v , u, and u are in the electroweak scale as well as the large scales all conserving the lepton number.

Gauge Bosons
The covariant derivative of a general triplet Φ is given by where the gauge fields W a and B transform as the adjoint representations of SU 3 L and U 1 X , respectively, and the corresponding gauge coupling constants g and g X . The T 9 diag 1, 1, 1 / √ 6 is chosen so that Tr T a T b δ ab /2 with a, b 1, 2, . . . , 9. The neutral gauge bosons of the theory get masses from the triplet as follows: where the subscript H denotes diagonal part of the covariant derivative: Advances in High Energy Physics The covariant derivative for an antisextet with the VEV part is 30 Let us denote the antisextet in term of the SU 3 L indices by Γ ij . Then, the mass Lagrangian due to the antisextet's contribution is given by Let us denote the following combinations: having defined charges under the generators of the SU 3 L group. For the sake of convenience in further reading, we note that W 4 and W 5 are pure real and imaginary parts of X 0 μ and X 0 * μ , respectively:

3.7
Then P μ is rewritten in a convenient form: with t ≡ g X /g.

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The covariant derivative acting on the antisextet VEV is given by 3.9 The masses of gauge bosons in this model are followed from Tr D μ s 2 D μ s 2 .

3.10
In the following, we notice that s 1 s 2 ; namely, u 1 u 2 , λ 1 λ 2 , and Λ 1 Λ 2 are taken into account. From 3.10 , the imaginary part W 5 is decoupled with mass given by In the limit λ 1 , u 1 → 0,

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The charged gauge bosons W and Y mix via

3.14
Diagonalizing this mass matrix, we get physical charged gauge bosons

3.15
The mixing angle is given by

3.17
Note that, in the limit λ 1 , u 1 → 0, the mixing angle tends to zero and the mass eigenvalues are

3.18
Advances in High Energy Physics 9 There is a mixing among the neutral gauge bosons W 3 , W 8 , B, and W 4 . The mass Lagrangian in this case has the form

3.19
In the basis of these elements, the mass matrix is given by

3.21
This mass matrix contains one exact eigenvalue:

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The associated eigenvector is

3.23
Using continuation of the gauge coupling constant g of the SU 3 L at the spontaneous symmetry breaking point, we have 2-9

3.24
In order to diagonalize the mass matrix, we choose the base of A μ , Z μ , Z μ , W 4μ , with

3.25
The new base is changed from the old by unitary matrix:

3.26
In this basis, the mass matrix M 2 becomes

3.29
It is noteworthy that in the limit u 1 0, the elements M 24 and M 34 or equivalently M 14 , M 24 , M 34 in the old base vanish. In this case, the mixing between W 4 and Z, Z disappears.
Three bosons gain masses via seesaw mechanism:

3.33
The ρ parameter in the our model is given by where δ loop gets contribution from the oblique correction depending on the masses of top quark and standard model Higgs boson 1 . The tree level correction δ tree describes the new physics as given by

3.35
It is noted that Δ M 2 22 / 0 even if ω and Λ 1 go to infinity. This is because the 33 components of antisextets and the third components of scalar triplets can be integrated out. There leave the standard model scalar doublets and triplets the submultiplets of the 3-3-1 model triplets and antisextets . Such standard model scalar triplets imply δ tree / 0 to be given by The δ tree parameter has already been given in 1 as ρ 0 − 1 from the global fit:  The interaction that provides these modes is as follows: where ν c L ≡ N R c is related to ν L via the seesaw mechanism given by ν c where the first term is due to the quark productions with α s 0.1184 chosen for the QCD radiative corrections , the second term comes from the normal modes with leptons, and the last one is for the unnormal modes. Let us choose α M Z 1/128, M W 80.399 GeV, and Γ ext W 2.085 ± 0.042 GeV 1 . The total decay width is plotted in Figure 1. From the figure, we get an upper limit on the sin θ in the model: sin θ ≤ 0. 15, 4.9 which is bigger than that given in 25, 26 . There are lepton number violating interactions in the neutral Gauge boson sector, we refer interested reader to 25, 26 .

Conclusions
In this paper, we have investigated Gauge boson sector: their mixing and masses. The vacuum expectation values u i and λ i are a source of lepton-number violations and a reason for the mixing between the charged Gauge bosons-the standard model W and the singly-charged bilepton Gauge bosons, as well as between neutral non-Hermitian X 0 and neutral Gauge bosons: the photon, the Z, and the new exotic Z . The interesting new physics compared with 3-3-1 models is the neutrino physics. Due to lepton-number violating couplings, we have many interesting consequences. We have shown that the neutrino tribimaximal mixing leads to the CPT violation. This feature will be considered in the future publication.