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

Abstract

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.

1. 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].

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] 𝑈^HPS=⎛⎜⎜⎜⎜⎜⎜⎜⎝2√61√30−1√61√31√2−1√61√3−1√2⎞⎟⎟⎟⎟⎟⎟⎟⎠,(1.1) 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 𝐴₄ or 𝑆₄ [21, 22]. The 3-3-1 model with neutral fermions based on 𝑆₃ flavor symmetry instead of 𝐴₄, 𝑆₄ 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 𝑆₃ flavor symmetry. The other models with 𝐴₄ and 𝑆₄ 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.

2. Brief Review of the Model

Before looking into the model, we provide a sketch of 𝑆₃ group theory [27, 28]. The 𝑆₃ 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.

Table 1 

Character table of 𝑆₃, where 𝜒 stands for character of representation and 𝐶 for class.

We will work in the basis that 2 is complex (see, e.g. [27]). Decomposition rules are 1⊗1=1(11),1⊗1=1(11),1⊗1=11(11),⊗2=2(11,12),1⊗2=22(11,−12),⊗2=1(12+21)⊕1(12−21)⊕2(22,11).(2.1) Here the first and second factors of the terms appearing in the parentheses indicate to the multiplet components of the first and second representations given in l.h.s, respectively. In this basis, the conjugation rules are given by 2^∗1^∗,2^∗=22^∗,1^∗,1^∗1^∗=11^∗,1^∗1^∗=11^∗.(2.2)

The lepton number in the 3-3-1 model with 𝑆₃ symmetry [23] does not commute with the gauge symmetry. It is thus better to work with a new lepton charge ℒ related to the lepton number 𝐿 by diagonal matrices 𝐿=𝑥𝑇₃+𝑦𝑇₈+ℒ. Applying 𝐿 to the lepton triplet with the notation that 𝐿(𝑁_𝑅)=0, the coefficients are defined as 𝑥=0, 𝑦=2/√3, and thus 𝐿=(2/√3)𝑇₈+ℒ [29]. The leptons and quarks under [SU(3)_𝐿,𝑈(1)_𝑋,𝑈(1)_ℒ,𝑆₃] symmetries correspondingly transform as follows: 𝜓_1𝐿=𝜈_1𝐿,𝑙_1𝐿,𝑁^𝑐_1𝑅^𝑇∼13,−3,23,1,𝑙_1𝑅∼1,−1,1,1,𝜓_𝛼𝐿=𝜈_𝛼𝐿,𝑙_𝛼𝐿,𝑁^𝑐_𝛼𝑅^𝑇∼13,−3,23,2,𝑙_𝛼𝑅∼1,−1,1,2,𝑄_1𝐿=𝑢_1𝐿,𝑑_1𝐿,𝑈_𝐿^𝑇∼13,31,−3,1,𝑢_1𝑅∼21,3,0,1,𝑑_1𝑅∼11,−3,0,1,𝑈_𝑅∼21,3,−1,1,𝑄_𝛼𝐿=𝑑_𝛼𝐿,−𝑢_𝛼𝐿,𝐷_𝛼𝐿^𝑇∼3^∗1,0,3,2,𝑢_𝛼𝑅∼21,3,0,2,𝑑_𝛼𝑅∼11,−3,0,2,𝐷_𝛼𝑅∼11,−3,1,2,(2.3) where 𝛼=2,3 is a family index of the last two lepton and quark families, which are in order defined as the components of 2 representations.

To generate masses for the charged leptons, we need two scalar multiplets: ⎛⎜⎜⎜⎜⎝𝜙𝜙=𝜙⁺¹⁰₂𝜙⁺₃⎞⎟⎟⎟⎟⎠∼23,31,−3,1,𝜙=⎛⎜⎜⎜⎜⎝𝜙₁⁺𝜙₂⁰𝜙₃⁺⎞⎟⎟⎟⎟⎠∼23,31,−3,1,(2.4) with VEVs ⟨𝜙⟩=(0,𝑣,0)^𝑇 and ⟨𝜙⟩=(0,𝑣,0)^𝑇. To generate masses for quarks, we additionally acquire the following scalar multiplets: 𝜒=𝜒⁰₁,𝜒⁻₂,𝜒⁰₃^𝑇∼13,−3,23,1,𝜂=𝜂⁰₁,𝜂⁻₂,𝜂⁰₃^𝑇∼13,−31,−3,1,𝜂=𝜂₁⁰,𝜂₂⁻,𝜂₃^0𝑇∼13,−31,−3,1.(2.5) Suppose that the VEVs of 𝜂, 𝜂, and 𝜒 are 𝑢, 𝑢 and 𝑤, where 𝑢=⟨𝜂⁰₁⟩, 𝑢=⟨𝜂₁⁰⟩, 𝑤=⟨𝜒⁰₃⟩, and ⟨𝜂⁰₃⟩, ⟨𝜂₃⁰⟩, and ⟨𝜒⁰₁⟩ vanish. The exotic quarks get masses 𝑚_𝑈=𝑓₁𝑤 and 𝑚_𝐷1,2=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_𝐿𝜓_𝐿𝜙 and 𝜓^𝑐_𝐿𝜓_𝐿𝜙 are suppressed. We therefore propose a new SU(3)_𝐿 antisextet instead coupling to 𝜓^𝑐_𝐿𝜓_𝐿 responsible for neutrino masses. The antisextet transforms as ⎛⎜⎜⎜⎜⎝𝑠𝑠=⁰¹¹𝑠⁺₁₂𝑠⁰₁₃𝑠⁺₁₂𝑠⁺⁺₂₂𝑠⁺₂₃𝑠⁰₁₃𝑠⁺₂₃𝑠⁰₃₃⎞⎟⎟⎟⎟⎠∼6^∗,234,−3,2,(2.6) where the numbered subscripts are the SU(3)_𝐿 indices. Henceforth the indices of 𝑆₃ on scalar fields will be kept and should be understood. The VEVs of 𝑠 is set as (⟨𝑠₁⟩,⟨𝑠₂⟩) under 𝑆₃, where ⟨𝑠_𝑖⎛⎜⎜⎜⎜⎝𝜆⟩=^𝑖0𝑣_𝑖𝑣000_𝑖0Λ_𝑖⎞⎟⎟⎟⎟⎠(𝑖=1,2).(2.7) Due to the 𝑆₃ symmetry, all these VEVs are equal to each others, that is, 𝜆₁=𝜆₂, 𝑣₁=𝑣₂ and Λ₁=Λ₂, which can be found from the potential minimization.

With the scalar multiplets as defined, the Yukawa lagrangian is given by ℒ_𝑌=ℎ₁𝜓_1𝐿𝜙𝑙_1𝑅+ℎ𝜓_2𝐿𝑙_2𝑅+𝜓_3𝐿𝑙_3𝑅𝜙+ℎ𝜓_3𝐿𝑙_3𝑅−𝜓_2𝐿𝑙_2𝑅𝜙+𝑓₁𝑄_1𝐿𝜒𝑈_𝑅+𝑓𝑄_𝐿𝜒^∗𝐷_𝑅+ℎ^𝑢₁𝑄_1𝐿𝜂𝑢_1𝑅+ℎ^𝑑𝑄_𝐿𝜂^∗𝑑_𝑅+ℎ^𝑑𝑄_𝐿𝜂^∗𝑑_𝑅+ℎ^𝑑₁𝑄_1𝐿𝜙𝑑_1𝑅+ℎ^𝑢𝑄_𝐿𝜙^∗𝑢_𝑅+ℎ^𝑢𝑄_𝐿𝜙^∗𝑢_𝑅+12𝑥𝜓^𝑐_2𝐿𝜓_2𝐿𝑠₁+𝜓^𝑐_3𝐿𝜓_3𝐿𝑠₂+12𝑦𝜓^𝑐_1𝐿𝜓_2𝐿𝑠₂+𝜓_3𝐿𝑠₁+H.c.(2.8) 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: 𝜆₁≃𝜆₂,𝑣₁≃𝑣₂,Λ₁≃Λ₂.(2.9) 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]: 𝜆₁,𝜆₂<𝑢₁,𝑢₂<𝑣,𝑣,𝑢,𝑢<𝜔,Λ₁,Λ₂.(2.10) 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.

3. Gauge Bosons

The covariant derivative of a general triplet Φ is given by 𝐷_𝜇=𝜕_𝜇+𝑖𝑔𝑇_𝑎𝑊_𝑎𝜇+𝑖𝑔_𝑋𝑇₉𝑋𝐵_𝜇≡𝜕_𝜇−𝑖𝒫_𝜇,(3.1) 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 𝑇₉=diag(1,1,1)/√6 is chosen so that Tr(𝑇_𝑎𝑇_𝑏)=𝛿_𝑎𝑏/2 with 𝑎,𝑏=1,2,…,9. The neutral gauge bosons of the theory get masses from the triplet as follows: ℒ^Φ_mass=𝐷^𝐻_𝜇⟨Φ⟩⁺𝐷^𝐻𝜇⟨Φ⟩,(3.2) where the subscript 𝐻 denotes diagonal part of the covariant derivative: 𝐷^𝐻_𝜇=𝜕_𝜇+𝑖𝑔𝑇₃𝑊³_𝜇+𝑖𝑔𝑇₈𝑊⁸_𝜇+𝑖𝑔_𝑋𝑇₉𝑋_𝜒𝐵_𝜇.(3.3)

The covariant derivative for an antisextet with the VEV part is [30] 𝐷_𝜇⟨𝑠_𝑖⟩=−𝑖𝑔2𝐴^𝑎_𝜇𝜆^∗_𝑎⟨𝑠_𝑖⟩+⟨𝑠_𝑖⟩𝐴^𝑎_𝜇𝜆_𝑎^∗𝑇+𝑖𝑔_𝑋𝑇₉𝑋𝐵_𝜇⟨𝑠⟩.(3.4)

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 ℒ^Γ_mass=𝐷^𝐻_𝜇⟨Γ⟩_𝑖𝑗⁺𝐷^𝐻𝜇⟨Γ⟩_𝑖𝑗,(3.5)

Let us denote the following combinations: 𝑊_𝜇^±≡𝑊_1𝜇∓𝑖𝑊_2𝜇√2,𝑌_𝜇^∓≡𝑊_6𝜇∓𝑖𝑊_7𝜇√2,𝑋_𝜇⁰≡𝑊_4𝜇−𝑖𝑊_5𝜇√2(3.6) having defined charges under the generators of the SU(3)_𝐿 group. For the sake of convenience in further reading, we note that 𝑊₄ and 𝑊₅ are pure real and imaginary parts of 𝑋_𝜇⁰ and 𝑋_𝜇^0∗, respectively: 𝑊_4𝜇=1√2𝑋_𝜇⁰+𝑋_𝜇^0∗,𝑊_5𝜇=𝑖√2𝑋_𝜇⁰−𝑋_𝜇^0∗.(3.7) Then 𝒫_𝜇 is rewritten in a convenient form: 𝑔2⎛⎜⎜⎜⎜⎜⎜⎜⎝𝑊_3𝜇+1√3𝑊_8𝜇+𝑡23𝑋𝐵_𝜇√2𝑊_𝜇⁺√2𝑋_𝜇⁰√2𝑊_𝜇⁻−𝑊_3𝜇+1√3𝑊_8𝜇+𝑡23𝑋𝐵_𝜇√2𝑌_𝜇⁻√2𝑋_𝜇^0∗√2𝑌_𝜇⁺−2√3𝑊_8𝜇+𝑡23𝑋𝐵_𝜇⎞⎟⎟⎟⎟⎟⎟⎟⎠(3.8) with 𝑡≡𝑔_𝑋/𝑔.

The covariant derivative acting on the antisextet VEV is given by 𝐷_𝜇⟨𝑠_𝑖⟩₁₁=−𝑖𝑔𝜆_𝑖𝑊_3𝜇+𝜆_𝑖1√3𝑊_8𝜇−𝑡𝜆_𝑖2313𝐵_𝜇+√2𝑢_𝑖𝑋⁰_𝜇,𝐷_𝜇⟨𝑠_𝑖⟩₁₂=−𝑖𝑔√2𝜆_𝑖𝑊⁻_𝜇+𝑢_𝑖𝑌⁻_𝜇,𝐷_𝜇⟨𝑠_𝑖⟩₁₃=−𝑖𝑔2𝑢_𝑖𝑊_3𝜇−𝑢_𝑖1√3𝑊_8𝜇−𝑡𝑢_𝑖2323𝐵_𝜇+√2Λ_𝑖𝑋⁰_𝜇+√2𝜆_𝑖𝑋_𝜇^0∗,𝐷_𝜇⟨𝑠_𝑖⟩₂₂𝐷=0,_𝜇⟨𝑠_𝑖⟩₂₃=−𝑖𝑔√2𝑢_𝑖𝑊⁻_𝜇+Λ_𝑖𝑌⁻_𝜇,𝐷_𝜇⟨𝑠_𝑖⟩₃₃=𝑖𝑔2Λ_𝑖1√3𝑊_8𝜇+𝑡Λ_𝑖2313𝐵_𝜇−√2𝑢_𝑖𝑋_𝜇^∗0,𝐷_𝜇⟨𝑠_𝑖⟩₁₂=𝐷_𝜇⟨𝑠_𝑖⟩₂₁,𝐷_𝜇⟨𝑠_𝑖⟩₁₃=𝐷_𝜇⟨𝑠_𝑖⟩₃₁,𝐷_𝜇⟨𝑠_𝑖⟩₂₃=𝐷_𝜇⟨𝑠_𝑖⟩₃₂.(3.9) The masses of gauge bosons in this model are followed from ℒ^GB_mass=𝐷_𝜇⟨𝜙⟩⁺𝐷^𝜇⟨𝜙⟩+𝐷_𝜇𝜙⁺𝐷^𝜇𝜙+𝐷_𝜇⟨𝜒⟩⁺𝐷^𝜇+𝐷⟨𝜒⟩_𝜇⟨𝜂⟩⁺𝐷^𝜇⟨𝜂⟩+𝐷_𝜇𝜂⁺𝐷^𝜇𝜂+Tr𝐷_𝜇⟨𝑠₁⟩⁺𝐷^𝜇⟨𝑠₁⟩+Tr𝐷_𝜇⟨𝑠₂⟩⁺𝐷^𝜇⟨𝑠₂⟩.(3.10) In the following, we notice that ⟨𝑠₁⟩=⟨𝑠₂⟩; namely, 𝑢₁=𝑢₂, 𝜆₁=𝜆₂, and Λ₁=Λ₂ are taken into account.

From (3.10), the imaginary part 𝑊₅ is decoupled with mass given by 𝑀²_𝑊5=𝑔²216𝑢²₁+4𝜆²₁−8Λ₁𝜆₁+4Λ²₁+𝜔²+𝑢²+𝑢².(3.11) In the limit 𝜆₁,𝑢₁→0, 𝑀²_𝑊5=𝑔²2𝑢²+𝑢²+𝜔²+4Λ²₁.(3.12)

The charged gauge bosons 𝑊 and 𝑌 mix via ℒ^CG_mass=𝑔²4𝑊⁻_𝜇,𝑌⁻_𝜇𝑀²_𝑊𝑌𝑊⁺_𝜇,𝑌⁺_𝜇^𝑇,(3.13) where 𝑀²_𝑊𝑌⎛⎜⎜⎝𝑣=2²+𝑣²+𝑢²+𝑢²+4𝑢²₁+4𝜆²₁4Λ₁𝑢₁+𝜆₁𝑢₁4Λ₁𝑢₁+𝜆₁𝑢₁𝑣²+𝑣²+𝜔²+4Λ²₁+4𝑢²₁⎞⎟⎟⎠.(3.14) Diagonalizing this mass matrix, we get physical charged gauge bosons 𝑊⁻_𝜇=cos𝜃𝑊_𝜇⁻+sin𝜃𝑌_𝜇⁻,𝑌⁻_𝜇=−sin𝜃𝑊_𝜇⁻+cos𝜃𝑌_𝜇⁻.(3.15) The mixing angle is given by 4Λtan𝜃=¹𝑢₁+𝜆₁𝑢₁𝜔²+4Λ²₁−𝑢²−𝑢′²−4𝜆²₁∼𝑢₁Λ₁,(3.16) provided that 𝜔²∼Λ²₁≫𝑢²,𝑢²,𝑢²₁,𝜆²₁. The mass eigenvalues are 𝑀²_𝑊=𝑔²4𝑣²+𝑣²+2𝑢²+2𝑢²+𝜔²+4𝜆²₁+4Λ²₁−𝑣²+𝑣²−𝜔²²+16𝜆⁴₁+16Λ⁴₁+128𝜆₁Λ₁𝑢²₁+8Λ²₁𝜔²−𝑣²−𝑣²+8𝑢²₁−8𝜆²₁4Λ²₁𝜔²−𝑣²−𝑣²−8𝑢²₁^1/2,𝑀²_𝑌=𝑔²4𝑣²+𝑣²+2𝑢²+2𝑢²+𝜔²+4𝜆²₁+4Λ²₁+𝑣²+𝑣²−𝜔²²+16𝜆⁴₁+16Λ⁴₁+128𝜆₁Λ₁𝑢²₁+8Λ²₁𝜔²−𝑣²−𝑣²+8𝑢²₁−8𝜆²₁4Λ²₁𝜔²−𝑣²−𝑣²−8𝑢²₁^1/2.(3.17) Note that, in the limit 𝜆₁,𝑢₁→0, the mixing angle tends to zero and the mass eigenvalues are 𝑀²_𝑊=𝑔²2𝑣²+𝑣²+𝑢²+𝑢²,𝑀²_𝑌=𝑔²2𝑣²+𝑣²+𝜔²+4Λ²₁.(3.18)

There is a mixing among the neutral gauge bosons 𝑊₃,𝑊₈,𝐵, and 𝑊₄. The mass Lagrangian in this case has the form ℒ^NG_mass=12𝑉^𝑇𝑀²𝑉𝑉,_𝑇≡𝑊₃,𝑊₈,𝐵,𝑊₄.(3.19) In the basis of these elements, the mass matrix is given by 𝑀²=𝑔²4⎛⎜⎜⎜⎜⎜⎜⎜⎝𝑀²₁₁𝑀²₁₂𝑀²₁₃𝑀²₁₄𝑀²₁₂𝑀²₂₂𝑀²₂₃𝑀²₂₄𝑀²₁₃𝑀²₂₃𝑀²₃₃𝑀²₃₄𝑀²₁₄𝑀²₂₄𝑀²₃₄𝑀²₄₄⎞⎟⎟⎟⎟⎟⎟⎟⎠,(3.20) where 𝑀²₁₁=24𝑢²₁+8𝜆²₁+𝑢²+𝑢′²+𝑣²+𝑣′²,𝑀²₂₂=234𝑢²₁+8𝜆²₁+𝑢²+𝑢′²+32Λ²₁+𝑣²+𝑣′²+4𝜔²,𝑀²₃₃=4𝑡²2716𝑢²₁+8𝜆²₁+𝑢²+𝑢²+8Λ²₁+4𝑣²+4𝑣′²+𝜔²,𝑀²₄₄=2𝜔²+𝑢²+𝑢′²+16𝑢²₁+4Λ²₁+4𝜆²₁+8Λ₁𝜆₁,𝑀²₁₂=2√3−4𝑢²₁+8𝜆²₁+𝑢²+𝑢′²−𝑣²−𝑣′²,𝑀²₁₃2=−323𝑡8𝑢²₁+8𝜆²₁+𝑢²+𝑢²+2𝑣²+2𝑣²,𝑀²₁₄=83𝑢₁𝜆₁+𝑢₁Λ₁,𝑀²₂₃=2√29𝑡8𝑢²₁+8𝜆²₁−𝑢²−𝑢²+16Λ²₁+2𝑣²+2𝑣²+2𝜔²,𝑀²₂₄=8√3𝑢₁𝜆₁−5𝑢₁Λ₁,𝑀²₃₄=−32323𝑡𝑢₁𝜆₁+𝑢₁Λ₁.(3.21)

This mass matrix contains one exact eigenvalue: 𝑀²_𝛾=0.(3.22) The associated eigenvector is 𝐴_𝜇=1√18+4𝑡²⎛⎜⎜⎜⎜⎜⎜⎝√3√3𝑡−𝑡20⎞⎟⎟⎟⎟⎟⎟⎠.(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√𝑡=2𝑠_𝑊3−4𝑠²_𝑊.(3.24)

In order to diagonalize the mass matrix, we choose the base of (𝐴_𝜇,𝑍_𝜇,𝑍_𝜇,𝑊_4𝜇), with 𝑍_𝜇=𝑐_𝑊𝑊_3𝜇−𝑠_𝑊⎛⎜⎜⎝−𝑡_𝑊√3𝑊_8𝜇+𝑡1−^2𝑊3𝐵_𝜇⎞⎟⎟⎠,𝑍_𝜇=𝑡1−^2𝑊3𝑊_8𝜇+𝑡_𝑊√3𝐵_𝜇.(3.25) The new base is changed from the old by unitary matrix: ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝𝑠𝑈=^𝑊𝑐_𝑊−𝑐00_𝑊𝑡_𝑊√3𝑠_𝑊𝑡_𝑊√3𝑡1−^2𝑊30𝑐_𝑊𝑡1−^2𝑊3−𝑠_𝑊𝑡1−^2𝑊3𝑡_𝑊√30⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠0001.(3.26) In this basis, the mass matrix 𝑀² becomes 𝑀²=𝑈⁺𝑀²𝑔𝑈=²4⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝00000𝑀²₂₂𝑀²₂₃𝑀²₂₄0𝑀²₂₃𝑀²₃₃𝑀²₃₄0𝑀²₂₄𝑀²₃₄𝑀²₄₄⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠.(3.27) In the approximation 𝜆²₁,𝑢²₁≪Λ²₁, we have 𝑀²₂₂=2𝑢²+𝑢′²+𝑣²+𝑣′²1𝑐²_𝑊,𝑀²₂₃=−22𝑠²_𝑊𝑢²+𝑢′²−𝑣²+𝑣′²√𝛼₀𝑐²_𝑊,𝑀²₂₄=8𝑢₁Λ₁𝑐_𝑊,𝑀²₃₃=2𝑢²+𝑢′²𝑐²_𝑊𝛼₀−2𝑣²+𝑣′²𝛼₀𝑐²_𝑊+8𝜔²𝑐²_𝑊𝛼₀+64Λ²₁𝑐²_𝑊𝛼₀,𝑀²₃₄=−8𝑥₀𝑢₁Λ₁𝑐_𝑊√𝛼₀,𝑀²₄₄=2𝜔²+𝑢²+𝑢²+4Λ₁𝜆₁+4Λ²₁,(3.28) with 𝑥₀=4𝑐²_𝑊,𝛼+1₀=14𝑐²_𝑊.−1(3.29) It is noteworthy that in the limit 𝑢₁=0, the elements 𝑀₂₄ and 𝑀₃₄ (or equivalently 𝑀₁₄,𝑀₂₄,𝑀₃₄ in the old base) vanish. In this case, the mixing between 𝑊₄ and 𝑍,𝑍 disappears.

Three bosons gain masses via seesaw mechanism: 𝑀²_𝑍=𝑀²₂₂−𝑀^off𝑇𝑀²_2𝑋2⁻¹𝑀^off,𝑀²_2𝑋2≈𝑀²_2𝑋2,(3.30) where 𝑀^off=⎛⎜⎜⎝𝑀²₂₃𝑀²₂₄⎞⎟⎟⎠,𝑀²_2𝑋2=⎛⎜⎜⎝𝑀²₃₃𝑀²₃₄𝑀²₃₄𝑀²₄₄⎞⎟⎟⎠.(3.31) We have then 𝑀²_𝑍=𝑔²4𝑀²₂₂−𝑀²₂₃²𝑀²₄₄−2𝑀²₂₃𝑀²₂₄𝑀²₃₄+𝑀²₂₄²𝑀²₃₃𝑀²₃₃𝑀²₄₄−𝑀²₃₄²=𝑔²2𝑐²_𝑊𝑢²+𝑢′²+𝑣²+𝑣′²−Δ_𝑀²₂₂,(3.32) where Δ_𝑀²₂₂=𝑔²𝛼²₀322𝑥₀𝑥₁+𝑥₃𝑢²₁Λ²₁+𝑥²₁𝑥₂𝑥₂𝑥₃−32𝑥²₀𝑢²₁Λ²₁,𝑥₁=𝑐_2𝑊𝑢²+𝑢′²−𝑣²+𝑣′²,𝑥₂=𝜔²+𝑢²+𝑢′²+4Λ₁𝜆₁+4Λ²₁,𝑥₃=𝑐_2𝑊𝑢²+𝑢′²+𝑣²+𝑣′²+4𝑐⁴_𝑊𝜔²+32𝑐⁴_𝑊Λ²₁.(3.33)

The 𝜌 parameter in the our model is given by 𝜌=1+𝛿_tree+𝛿_loop,(3.34) 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 𝛿_tree=𝑀²_𝑊𝑐²_𝑊𝑀²_𝑍𝑐−1≃^2𝑊Δ_𝑀²₂₂𝑀²_𝑊.(3.35) It is noted that Δ_𝑀²₂₂≠0 even if 𝜔 and Λ₁ 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 𝛿_tree≃8𝑔²𝑐²_𝑊4𝑐²_𝑊−1²𝑢²₁𝑀²_𝑊≠0.(3.36) The 𝛿_tree parameter has already been given in [1] as 𝜌₀−1 from the global fit: 𝛿_tree=0.0008^+0.0017_−0.0007.(3.37) Hence 0.8403GeV²≤Δ_𝑀²₂₂≤21.0065GeV²,(3.38)or 0.239GeV≤𝑢₁≤1.197GeV,(3.39) where we have used 𝑐²_𝑊=0.769 and 𝑀_𝑊=80.384GeV.

Diagonalizing the mass matrix 𝑀²_2𝑋2, we get new gauge bosons:𝑍_𝜇=cos𝜙𝑍_𝜇+sin𝜙𝑊_4𝜇,𝑊_4𝜇=−sin𝜙𝑍_𝜇+cos𝜙𝑊_4𝜇.(3.40) The mixing angle is defined by tan𝜙=2𝑀²₃₄𝑀²₄₄−𝑀²₃₃+𝑀²₄₄−𝑀²₃₃²+4𝑀²₃₄².(3.41) Substituting (3.28) into (3.41), we get tan𝜙=−2√𝛼₁𝑢₁Λ₁⎧⎪⎨⎪⎩𝛼₂𝑢²+𝑢²+𝛼₃𝑣²+𝑣²−2𝛼₀𝑤²+8𝜆₁Λ₁−8𝑥₀𝛼₀Λ²₁+𝛼₂𝑢²+𝑢²+𝛼₃𝑣²+𝑣²−2𝛼₀𝑤²+8𝜆₁Λ₁−8𝑥₀𝛼₀Λ²₁²+4𝛼₁𝑢²₁Λ²₁^1/2⎫⎪⎬⎪⎭⁻¹≃4𝑐²_𝑊−1𝑐_𝑊𝑢₁Λ₁,(3.42) provided that 𝜔²∼Λ²₁≫𝑢²,𝑢²,𝑣²,𝑣²,𝑢²₁,𝜆²₁, where 𝛼₁=64𝑥²₀𝛼₀𝑐²_𝑊,𝛼₂=−2+6𝑐²_𝑊𝑐²_𝑊𝛼₀,𝛼₃=−2𝑐²_𝑊𝛼₀.(3.43) The physical mass eigenvalues are defined by 𝑀²_{𝑍′′,𝑊′4𝜇}=𝑔²4𝑀²₄₄+𝑀²₃₃∓𝑀²₄₄−𝑀²₃₃²+4𝑀²₃₄²2=𝑔²8𝛼₄𝑢²+𝑢′²−𝛼₃𝑣²+𝑣′²+𝛼₅𝑤²+8𝜆₁Λ₁+𝛼₆Λ²₁±𝛼₂𝑢²+𝑢′²+𝛼₃𝑣²+𝑣²−2𝛼₀𝑤²+8𝜆₁Λ₁−8𝑥₀𝛼₀Λ²₁²+4𝛼₁𝑢²₁Λ²₁(3.44)with 𝛼₄=2−10𝑐²_𝑊+16𝑐⁴_𝑊𝑐²_𝑊𝛼₀,𝛼₅=16𝑐²_𝑊−2𝛼₀,𝛼₆=812𝑐²_𝑊−1𝛼₀.(3.45)

In the limit 𝜆₁,𝑢₁→0, we have 𝑀²_𝑍′′=𝑔²𝑐²_2𝑊𝑢²+𝑢²+𝑣²+𝑣²+4𝑐⁴_𝑊𝜔²+32𝑐⁴_𝑊Λ²₁2𝑐²_𝑊𝛼₀,𝑀²_𝑊′4=𝑔²2𝑢²+𝑢²+𝜔²+4Λ²₁.(3.46) Thus the 𝑊₄ and 𝑊₅ components have the same mass. With this result, we should identify the combination of 𝑊′₄ and 𝑊₅√2𝑋⁰_𝜇=𝑊_4𝜇−𝑖𝑊_5𝜇(3.47) as physical neutral non-Hermitian gauge boson. The subscript 0 denotes neutrality of gauge boson 𝑋. However, to get tribimaximal mixing, the previous limit is not valid [21, 22]. This means that neutrino tribimaximal mixing leads to the masses of 𝑋⁰ and 𝑋^0† to be different. Consequence of this fact is that there is CPT violation [1, 31] in the model under consideration. We will return to this problem in the future work.

In the limit 𝜔²∼Λ²₁≫𝑢²,𝑢²,𝑣²,𝑣²,𝑢²₁,𝜆²₁ (or 𝜔,Λ₁→∞), the mixings between the charged gauge bosons 𝑊−𝑌 and the neutral ones 𝑊₄−𝑍 are in the same order since from (3.16) and (3.42) they are proportional to 𝑢₁/Λ₁. In addition, from (3.46), 𝑀²_𝑍′′≃2𝑔²(𝜔²+16𝑐²_𝑊Λ¹₁) is bigger than 𝑀²_𝑊4≃(𝑔²/2)(𝜔²+4Λ²₁) (or 𝑀²_𝑋0). It is also verified that |𝑀²_𝑌−𝑀²_𝑋0|<𝑀²_𝑊. In that limit, the masses of 𝑋⁰ and 𝑌 degenerate.

Note that the formulas for masses and mixing of gauge bosons previously presented, are common for the 3-3-1 models with more complicated Higgs sector such as with 𝐴₄ or 𝑆₄ discrete symmetries.

4. Charged Currents

The interaction among fermions with gauge bosons arises from part 𝑖𝜓𝛾_𝜇𝐷^𝜇𝜓=kinematicterms+𝐻^CC+𝐻^NC.(4.1)

Similarly in the economical 3-3-1 model, despite neutrality, the gauge bosons 𝑋⁰ and 𝑋^0∗ belong to this section by their nature. Because of the mixing among the SM 𝑊 boson and the charged bilepton 𝑌 as well as among (𝑋⁰+𝑋^0∗) with (𝑊₃,𝑊₈,𝐵), the new terms exist the same as the economical 3-3-1 model [25, 26]: 𝐻^CC=𝑔√2𝐽_𝑊^𝜇−𝑊⁺_𝜇+𝐽_𝑌^𝜇−𝑌⁺_𝜇+𝐽_𝑋^𝜇0∗𝑋⁰_𝜇+H.c.,(4.2) where 𝐽_𝑊^𝜇−=𝑐_𝜃𝜈_𝑖𝐿𝛾^𝜇𝑒_𝑖𝐿+𝑢_𝑖𝐿𝛾^𝜇𝑑_𝑖𝐿+𝑠_𝜃𝜈^𝑐_𝑖𝐿𝛾^𝜇𝑒_𝑖𝐿+𝑈_𝐿𝛾^𝜇𝑑_1𝐿+𝑢_𝛼𝐿𝛾^𝜇𝐷_𝛼𝐿,𝐽_𝑌^𝜇−=𝑐_𝜃𝜈^𝑐_𝑖𝐿𝛾^𝜇𝑒_𝑖𝐿+𝑈_𝐿𝛾^𝜇𝑑_1𝐿+𝑢_𝛼𝐿𝛾^𝜇𝐷_𝛼𝐿−𝑠_𝜃𝜈_𝑖𝐿𝛾^𝜇𝑒_𝑖𝐿+𝑢_𝑖𝐿𝛾^𝜇𝑑_𝑖𝐿,𝐽_𝑋^𝜇0∗=1−𝑡²_2𝜃𝜈_𝑖𝐿𝛾^𝜇𝜈^𝑐_𝑖𝐿+𝑢_1𝐿𝛾^𝜇𝑈_𝐿−𝐷_𝛼𝐿𝛾^𝜇𝑑_𝛼𝐿−𝑡²_2𝜃𝜈^𝑐_𝑖𝐿𝛾^𝜇𝜈_𝑖𝐿+𝑈_𝐿𝛾^𝜇𝑢_1𝐿−𝑑_𝛼𝐿𝛾^𝜇𝐷_𝛼𝐿+𝑡_2𝜃1+4𝑡²_2𝜃𝜈_𝑖𝛾^𝜇𝜈_𝑖+𝑢_1𝐿𝛾^𝜇𝑢_1𝐿−𝑈_𝐿𝛾^𝜇𝑈_𝐿−𝑑_𝛼𝐿𝛾^𝜇𝑑_𝛼𝐿+𝐷_𝛼𝐿𝛾^𝜇𝐷_𝛼𝐿.(4.3)

All aforementioned interactions are lepton-number violating and weak (proportional to sin𝜃 or its square sin²𝜃). However, these couplings lead to lepton-number violations only in the neutrino sector.

Let us consider some constraints on the parameters of the model; one of the ways to do that is the consideration for 𝑊 decay. In our model, the 𝑊 boson has the following normal main decay modes: 𝑊⁻⟶𝑙𝜈̃_𝑙(𝑙=𝑒,𝜇,𝜏),↘𝑢^𝑐𝑑,𝑢^𝑐𝑠,𝑢^𝑐𝑏,(𝑢→𝑐),(4.4) which are the same as in the SM and in the 331 with right-handed neutrinos. Beside the aforementioned modes, there are additional ones which are lepton-number violating (Δ𝐿=2) the model’s specific feature: 𝑊⁻⟶𝑙𝜈_𝑙(𝑙=𝑒,𝜇,𝜏).(4.5) The interaction that provides these modes is as follows: _╱𝑔ℒ=√2𝑠_𝜃𝜈^𝑐_𝑖𝐿𝛾^𝜇𝑒_𝑖𝐿,(4.6) where 𝜈^𝑐_𝐿≡(𝑁_𝑅)^𝑐 is related to 𝜈_𝐿 via the seesaw mechanism given by 𝜈^𝑐_𝐿=𝑀_𝐷𝑀_𝑅⁻¹𝜈_𝐿. Here 𝑀_𝑅 and 𝑀_𝐷 are right-handed Majorana and Dirac mass matrices (due to the contribution of 𝑠), respectively, which can be derived from the Yukawa Lagrangian above to yield 𝑀_𝐷𝑀_𝑅⁻¹=(𝑢₁/Λ₁)diag(1,1,1). On the other hand, from (3.16) we have 𝑠_𝜃≃tan𝜃≃𝑢₁/Λ₁ if Λ₁ is largest among the VEVs. It is therefore that 𝜈^𝑐_𝐿≃𝑠_𝜃𝜈_𝐿 and _╱𝑔ℒ=√2𝑠²_𝜃𝜈_𝑖𝐿𝛾^𝜇𝑒_𝑖𝐿.(4.7) The total decay width of 𝑊 is given by [25, 26] Γ_𝑊=1.039𝛼𝑀_𝑊2𝑠²_𝑊1−𝑠²_𝜃+𝛼𝑀_𝑊4𝑠²_𝑊1−𝑠²_𝜃+𝛼𝑀_𝑊4𝑠²_𝑊𝑠⁴_𝜃,(4.8)

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 sin𝜃 in the model: sin𝜃≤0.15,(4.9) which is bigger than that given in [25, 26].

Figure 1 

The total decay width of 𝑊 [GeV] is depicted as a function of sin(𝜃). The two horizontal lines are the upper and lower limits, respectively.

There are lepton number violating interactions in the neutral Gauge boson sector, we refer interested reader to [25, 26].

5. 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 𝑋⁰ 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.

Acknowledgment

This work was supported in part by the National Foundation for Science and Technology Development (NAFOSTED) of Vietnam under Grant no. 103.01–2011.63.