# Nonstandard Neutrinos Interactions in a 331 Model with Minimum Higgs Sector

Authors: M. Medina; P. C. de Holanda
Journal: Advances in High Energy Physics
 (2012)

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## Abstract

We present a detailed analysis of a class of extensions to the SM Gauge chiral symmetrySU(3)C×SU(3)L×U(1)x (331 model), where the neutrino electroweak interaction with matter via charged
and neutral current is modified through new gauge bosons of the model. We found the connections
between the nonstandard contributions on 331 model with nonstandard interactions. Through
limits of such interactions in cross-section experiments, we constrained the parameters of the model,
obtaining that the new energy scale of this theory should obey V>1.3 TeV and the new bosons of
the model must have masses greater than 610 GeV.


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## Body

## 1. Introduction

Although the standard model (SM) is a good phenomenological theory, describing very well all experimental results, it leaves several unanswered questions that suggest that the SM might be an effective model at low energies, originating from a more fundamental theory. Some of the unexplained aspects in the SM are the existence of three families and lepton flavour violation observed in solar [1–5], atmospheric [6–11], and reactor [12–17] neutrino experiments. These results demonstrate that new physics is required, being interpreted as a sign of physics beyond the SM.In principle neutrinos new interactions not described by Standard Model can arise in extensions of the SM. We assume that the new physics which induces the nonstandard neutrino interactions (NSIs) [18–29] arises in some models enlarging the symmetry group where the SM is embedded. Models with larger symmetries that may allow us to understand the origin of the families have been proposed [30–34]. In some models, it is also possible to understand the number of families from the cancellation of chiral anomalies, necessary to preserve the renormalizability of the theory [35–37]. This is the case of the SU(3)C⊗SU(3)L⊗U(1)X or 331 models, which are an immediate extension of the SM [38–46]. There are a great variety of such models, which have generated new expectations and possibilities of solving several problems of the SM.Our goal is to investigate how NSI with matter can be induced by new physics generated by 331 models. Through the constraints from neutrino elastic scattering experiments on this NSI parameters, we can constrain some values expected for 331 model parameters. We find that the constraints on vacuum expectation values of the model, as well as for the mass of the new bosons, are in full agreement with the limits found in the literature, which makes this class of models a viable theory for a higher energy level.The paper is organized as follows. In Section2 we briefly review NSI and present how new interactions can contribute to new matter effects, in addition to the SM electroweak ones. In Section 3 we introduce a specific 331 model and we give the fermion gauge-boson couplings. In Section 4 we calculate the interactions involving neutrinos and how these interactions can be interpreted as new terms beyond SM. Finally, in Section 5 we summarize our main results.


## 2. Nonstandard Neutrino Interactions

One convenient way to describe neutrino new interactions with matter in the electro-weak (EW) broken phase are the so-called nonstandard neutrino interactions (NSIs), which is a very widespread and convenient way of parameterizing the effects of new physics in neutrino oscillations [18–29]. NSIs with first generation of leptons and quarks for four-fermion operators are contained in the following Lagrangian density [18–22, 24, 25, 28]:
(2.1)LeffNSI=-22GF∑f,PɛαβfP[f¯γμPf][ν̅αγμLνβ],
where GF is the Fermi constant, f=u,d,e, and P=L,R with 2L=(1-γ5),2R=(1+γ5), and the coefficients ɛαβfP encode the deviation from standard interactions between neutrinos of flavor α with component P-handed of fermions f, resulting in a neutrinos of flavor β. Then, the neutrino oscillations in the presence of nonstandard matter effects can be described by an effective Hamiltonian, parameterized as
(2.2)H̃=12E[U(0000Δm212000Δm312)U†+a(1+ɛeeɛeμɛeτɛeμ*ɛμμɛμτɛeτ*ɛμτ*ɛττ)],
where a=2GFnf, E is the neutrino energy and ɛαβ=∑f,PɛαβfPnf/ne with ne and nf the electrons and fermions f density in the medium, respectively. These parameters ɛαβ can be found in solar [22, 47], atmospheric [20, 48], accelerator [18, 19, 22, 49], and cross-section [18, 19, 21, 50, 51] neutrino data experiment.We focus on cross-section neutrino experiment, where at low energies the standard differential cross-section forναe→ναe scattering processes has the well-know form:
(2.3)dσαdT=2GFmeπ[(g1α)2+(g2α)2(1-TEν)2-g1αg2αmeTEν2],
where me is the electron mass, Eν is the incident neutrino energy, and Te is the electron recoil energy. The quantities g1α and g2α are related to the SM neutral current couplings of the electron gLe=-1/2+sin2θW and gRe=sin2θW, with sin2θW=0,23119. For νμ,τ neutrinos, which take part only in neutral current interactions, we have g1μ,τ=gLe and g2μ,τ=gRe while for electron neutrinos, which take part in both charge current (CC) and neutral current (NC) interactions, g1e=1+gLe,g2e=gRe. In the presence of nonuniversal standard interaction, the cross-section can be written in the same form of (2.3) but with g1,2α replaced by the effective nonstandard couplings g̃1α=g1α+ɛααeL and g̃2α=g2α+ɛααeR, leading to the following differential scattering cross-section [19, 21, 50, 51]
(2.4)dσαdT=2GFmeπ{(g1α+ɛααeL)2+(g2α+ɛααeR)2(1-TeEν)2-(g1α+ɛααeL)(g2α+ɛααeR)meTeEν}.


## 3. 331 Model

The success of the standard model (SM) implies that any new theory should contain the symmetrySU(3)C⊗SU(2)L⊗U(1)Y(G321) in a low energy limit. Then, it is natural that one possible modification of SM involves extensions of the representation content in matter and Higgs sector, leading to extension of symmetry group G321 to groups SU(NC)C⊗SU(m)L⊗U(1)X with SU(NC)C⊗SU(m)L⊗U(1)X⊃G321.In early 90’s, Pisano and Pleitez [38, 39] and Frampton [40] suggested an extension of the symmetry group SU(2)L⊗U(1)Y of electroweak sector to a group SU(3)L⊗U(1)X, that is, with NC=m=3. The 331 models present some interesting features; for instance, they associate the number of families to internal consistence of the theory, preserving asymptotic freedom.In these models, the SM doublets are part of triplets. In quark sector three new quarks are included to build the triplets, while in lepton sector we can use the right-handed neutrino to such role [38, 40]. Another option is to invoke three new heavy leptons, charged or not, depending on the choice of charge operator [41, 42]. In SM the electric charge operator is constructed as a combination of diagonal generators of SU(2)⊗U(1)Y. Then, it is natural to assume that this operator in SU(3)L⊗U(1)X is defined in the same way. The most general charge operator in SU(3)L⊗U(1)X is a linear superposition of diagonal generators of symmetry groups, given by
(3.1)Q≡aT3L+23bT8L+XI3,
where the group generator is defined as TiL≡λiL/2 with λiL, i=1,…,8, being the Gell-Mann matrices for SU(3)L, where the normalization chosen is Tr(λiLλjL)=2δij and I3=diag(1,1,1) is the identity matrix, and a and b are two parameters to be determined. Then the charge operator in (3.1) acts on the representations 3 and 3* of SU(3)L having the following form:
(3.2)Q[3]=diag⁡[a2+b3+X,-a2+b3+X,-2b3+X],(3.3)Q[3*]=diag⁡[-a2-b3+X,+a2-b3+X,+2b3+X],
where we have two free parameters to obtain the charge of fermions, a and b (X can be determined by anomalies cancellation). However, a=1 is necessary to obtain doublets of isospins SU(2)⊗U(1)Y correctly incorporated in the model SU(3)L⊗U(1)X [41, 42, 45]. Then we can vary b to create different models in 331 context, being a signature that differentiates such models. For b=-3/2, we have the original 331 model [38, 39].To have local gauge invariance, we have the following covariant derivative:Dμ=∂μ-i(g/2)λαWμα-igxXBμ and a total of 17 mediator bosons: one field Bμ associated with U(1)X, eight fields associated with SU(3)C, and another eight fields associated with SU(3)L, written in the following form:
(3.4)Wμ≡Wμαλα=(Wμ3+13Wμ82Wμ+2KμQ12Wμ--Wμ3+13Wμ82KμQ22Kμ-Q12Kμ-Q2-23Wμ8),
where
(3.5)Wμ±=12(W1μ∓iW2μ),Kμ±Q1=12(W4μ∓iW5μ),Kμ±Q2=12(W6μ∓iW7μ).
Therefore, charge operator in (3.2) applied over (3.4) leads to 𝒬1=1/2+b and 𝒬2=(-1/2)+b. Then the mediator bosons will have integer electric charge only if b=±1/2,±3/2,±5/2,…,±(2n+1)/2,n=0,1,2,3,…. A detailed analysis shows that if a and b are associated with the fundamental representation 3, then -a and -b will be associated with antisymmetric representation 3*.


### 3.1. The Representation Content

There are many representations for the matter content [46], for instance, b=3/2 [38]. But we note that if we accommodate the doublets of SU(2)L in the superior components of triplets and antitriplets of SU(3)L, and if we forbid exotic charges for the new fermions, we obtain from (3.2) the constrain b=±1/2 (assuming a=1). Since a negative value of b can be associated to the antitriplet, we obtain that b=1/2 is a necessary and sufficient condition to exclude exotic electric charges in fermion and boson sector [41].The fields left- and right-handed components transform underSU(3)L as triplets and singlets, respectively. Therefore the theory is chiral and can present anomalies of Alder-Bell-Jackiw [52, 53]. In a non-abelian theory, in the fermionic representation ℛ, the divergent anomaly is given by
(3.6)Aabc∝∑RTr⁡[{TLa(R),TLb(R)}TLc(R)-{TRa(R),TRb(R)}TRc(R)],
where Ta(ℛ) are the matrix representations for each group generator acting on the basis ℛ with helicity left or right. Therefore, to eliminate the pure anomaly [SU(3)L]3, we should have that 𝒜abc∝∑ℛ′Tr[{TLa(ℛ′),TLb(ℛ′)}TLc(ℛ′)]=0. We use the fact that SU(3)L has two fundamental representations, 3 and 3*, then its generators should be associated to Ta and Ta*, respectively, that is,
(3.7)∑R′Tr⁡[{TLa(R′),TLb(R′)}TLc(R′)]=∑RTr⁡[{TLa(R),TLb(R)}TLc(R)]-∑R*Tr⁡[{TLa*(R*),TLb*(R*)}TLc*(R*)],
but we know that the matrix representations for each group generator satisfies that TLa*(ℛ*)=-TLa(ℛ) [54]. So, we can see that for the anomalies to be canceled, the number of fields that transform as triplets (first term in equation above) and antitriplets under SU(3)L has to be the same; that is, two triplets quark families × 3 (color) = one antitriplet quark family × 3 (color) + 3 antitriplet lepton families. This implies that two families of quarks should transform differently than the third family, as will be discussed in next paragraph.Usually the third quark family is chosen to transform in a different way than the first two families. But we will assume that the first family transform differently, to address the fact thatmu<md,mνℓ<mℓ while mc≫ms and mt≫mb. To state this in a clearer way, we recall that in SM the SU(2)L doublets are (νℓ,ℓ)T,(u,d)T,(c,s)T,(t,b)T, with ℓ=e,μ,τ. We can see that the first component of leptons doublets and first quark family is lighter than the second component. But for the second and third quark families, the opposite occurs. Then we use this idea to justify that first quark family transform as leptons.


### 3.2. Minimal 331 Model on Scalar Sector

Among the different possibilities of 331 models, we will present a detailed study on a minimal model on scalar sector without exotic electric charges for quarks and with three new leptons without charged [41] (b=1/2), where the fermions present the following transformation structure under SU(3)C⊗SU(3)L⊗U(1)X:
(3.8)ψlL=(l-,νl,Nl0)LT~(1,3*,-13),νlR~(1,1,0),lR-~(1,1,-1),NlR0~(1,1,0),Q1L=(d,u,U1)LT~(3,3*,13),uiR~(3,1,23),diR~(3,1,-13),U1R~(3,1,23),QaL=(ua,da,Da)LT~(3,3,0),DaR~(3,1,-13),
where i=1,2,3,ℓ=e,μ,τ,a=2,3. We note that the leptons multiplets ψℓL consist of three fields ℓ={e,μ,τ}, the corresponding neutrinos νℓ={νe,νμ,ντ}, and new neutral leptons Nℓ0={Ne0,Nμ0,Nτ0}. We can also see that the multiplet associated with the first quark family Q1L consists of down and up quarks and a new quark with the same electric charge of quark up (named U1), while the multiplet associated with second (third) family QaL consists of SM quarks of second (third) family and a new quark with the same electric charge of down quark (named D2 (D3)). The numbers on parenthesis refer to the transformation properties under SU(3)C,SU(3)L, and U(1)X, respectively. With this choice, the anomalies are cancelled in a nontrivial way [55], and asymptotic freedom is guaranteed [56–59].


#### 3.2.1. Scalar Sector and the Yukawa Couplings

The scalar fields have to be coupled to fermions by the Yukawa terms, invariants underSU(3)c⊗SU(3)L⊗U(1)X. In lepton sector, these couplings can be written as
(3.9)ψ¯lLlR~(1,3,13)⊗(1,1,-1)=(1,3,-23)︸ρ*,ψ¯lLνlR~(1,3,13)⊗(1,1,0)=(1,3,13)︸η,ψ¯lLNlR0~(1,3,13)⊗(1,1,0)=(1,3,13)︸χ,
and writing only three terms in quarks sector, for example,
(3.10)Q¯1LuiR=(3*,3,-13)⊗(3,1,23)=(1,3,13)︸χ⊕(8,3,13)︸ColorHiggs,Q¯1LdiR=(3*,3,-13)⊗(3,1,-13)=(1,3,-23)︸ρ*⊕…,Q¯aLdiR=(3*,3*,0)⊗(3,1,-13)=(1,3*,-13)︸η*⊕…,….
As usual in these class of models, we impose colorless Higgs (i.e., selecting only the multiplets that transform as singlets under SU(3)C). We note that we need only three Higgs multiplets, ρ,χ, and η, to couple the different fermionic fields and generate mass through spontaneous symmetry breaking. In (3.9) and (3.10) we note that quantum numbers of triplets χ and η are the same, which leads us to consider models with two or three Higgs triplets. We will adopt the first option, two Higgs triplets, due to the simpler scalar sector in comparison with the scenario with three triplets [41–44].


### 3.3. Model with Two Higgs Triplets

For the models with two Higgs triplets, we obtain (note that in this model we assumedΦ1=χ,η e Φ2=ρ)
(3.11)Φ1=(ϕ1-,ϕ1′0,ϕ10)T~(1,3*,-13),Φ2=(ϕ20,ϕ2+,ϕ2′+)T~(1,3*,23).
Assuming the following choice to the Higgs triplets vacuum expectation value (VEV) [41] 〈Φ1〉0=(0,ϑ1,V)T and 〈Φ2〉0=(ϑ2,0,0)T, we associate V with the mass of the new fermions, which lead us to assume V≫ϑ1,ϑ2. We expand the scalar VEVs in the following way:
(3.12)ϕ10=V+Hϕ10+iAϕ102,ϕ1′0=ϑ1+Hϕ1′0+iAϕ1′02,ϕ20=ϑ2+Hϕ20+iAϕ202.
The real (imaginary) part Hϕi(Aϕi) is usually called CP-even (CP-odd) scalar field. The most general potential can be written as
(3.13)V(Φ1,Φ2)=μ12Φ1†Φ1+μ22Φ2†Φ2+λ1(Φ1†Φ1)2+λ2(Φ2†Φ2)2+λ3(Φ1†Φ1)(Φ2†Φ2)+λ4(Φ1†Φ2)(Φ2†Φ1).
Demanding that in the displaced potential V(Φ1,Φ2) the linear terms on the field should be absent, we have, in tree-level approximation, the following constraints:
(3.14)μ12+2λ1(ϑ12+V2)+λ3ϑ22=0,μ22+λ3(ϑ12+V2)+2λ2ϑ22=0.
The analysis of such equations shows that they are related to a minimum in scalar potential with the value Vmin=-ϑ24λ2-(ϑ12+V2)[(ϑ12+V2)λ1+ϑ22λ3]. Then, replacing (3.12) and (3.14) in (3.13), we can calculate the mass matrix in (Hϕ10,Hϕ20,Hϕ1′0) basis through the relation Mij2=2(∂2V(Φ1,Φ2)/∂HΦi0∂HΦj0), obtaining
(3.15)MH2=2(2λ1V2λ3ϑ2V2λ1ϑ1Vλ3ϑ2V2λ2ϑ22λ3ϑ1ϑ22λ1ϑ1Vλ3ϑ1ϑ22λ1ϑ12).
Since (3.15) has vanishing determinant, we have one Goldstone boson G1 and two massive neutral scalar fields H1 and H2 with masses (note that if λ32=4λ1λ2, we obtain two Goldstone bosons, G1 and H2, and a massive scalar field H1 with mass MH12=4[λ1(ϑ12+V2)+λ2ϑ22], where λ1λ2>0; then imposing MH12>0 leads to λ1>0 and λ2>0)
(3.16)MH1,H22=2λ1(ϑ12+V2)+2λ2ϑ22±2[λ1(ϑ12+V2)+λ2ϑ22]2+ϑ22(ϑ12+V2)(λ32-4λ1λ2),
where real values for λ’s produce positive mass to neutral scalar fields only if λ1>0 and 4λ1λ2>λ32, which implies that λ2>0. A detailed analysis shows that when V(Φ1,Φ2) in (3.13) is expanded around the most general vacuum, given by (3.12) and using constrains in (3.14), we do not obtain pseudoscalar fields AΦi0. This allows us do identify three more Goldstone bosons, G2=AΦ10,G3=AΦ20, and G4=AΦ1′0. For the mass spectrum in charged scalar sector on (ϕ1-,ϕ2+,ϕ2′+) basis, the mass matrix will be given by
(3.17)M+2=2λ4(ϑ22ϑ1ϑ2ϑ2Vϑ1ϑ2ϑ12ϑ1Vϑ2Vϑ1VV2),
with two eigenvalues equal to zero, equivalent to four Goldstone bosons G5±,G6± and two physical charged scalar fields with large masses given by λ4(ϑ12+ϑ22+V2), which leads to the constrain λ4>0.This analysis shows that, after symmetry breaking, the original twelve degrees of freedom in scalar sector leads to eight Goldstone bosons (four electrically neutral and four electrically charged), four physical scalar fields, two neutral (one of which being the SM Higgs scalar), and two charged. Eight Goldstone bosons should be absorbed by eight gauge fields as we will see in next section.


#### 3.3.1. Gauge Sector with Two Higgs Triplets

The gauge bosons interaction with matter in electroweak sector appears with the covariant derivative for a matter fieldφ as
(3.18)Dμφ=∂μ-i2gWμaλaL-igXXφBμ=∂μ-i2gMμφ,
where λaL,a=1,…,8 are Gell-Mann matrices of SU(3)L algebra and Xφ is the charge of abelian factor U(1)X of the multiplet φ in which Dμ acts. The matrix ℳμφ contains the gauge bosons with electric charges q, defined by the generic charge operator in (3.1). For b=1/2 the matrix ℳμφ will have the following form:
(3.19)Mμφ=(W3μ+W8μ3+2tXφBμ2Wμ+2Kμ+2Wμ--W3μ+W8μ3+2tXφBμ2Kμ02Kμ-2K¯μ0-2W8μ3+2tXφBμ),
where t=gx/g and nonphysical gauge bosons on nondiagonal entries, Wμ±andKμ±, are defined in (3.5) with 𝒬1=1, and
(3.20)Kμ0=12(W6μ-iW7μ),K¯μ0=12(W6μ+iW7μ).
Then for the 331 model we are considering (b=1/2), we have two neutral gauge bosons, Kμ0 and K¯μ0, and four charged gauge bosons, Wμ± and Kμ±. The three physical neutral eigenstates will be a linear combination of W3μ,W8μ, and Bμ. After breaking the symmetry with 〈Φi〉, i=1,2, and using covariant derivative Dμ=∂μ-(i/2)gℳμφ for the triplets Φi, we obtain the following masses for the charged physical fields:
(3.21)MW′2=12g2ϑ22,MK′2=12g2(ϑ12+ϑ22+V2),
and the following physical eigenstates:
(3.22)Wμ′±=1ϑ12+V2(-ϑ1Kμ±+VWμ±),Kμ′±=1ϑ12+V2(VKμ±+ϑ1Wμ±).
The neutral sector in approximation (ϑi/V)n≈0 for n>2 leads to the following masses for the neutral physical fields:
(3.23)Mphoton2=0,MKR02=12g2(V2+ϑ12),MZ2≈12g2ϑ22(3g2+4gx23g2+gx2),MZ′2≈29(V2+ϑ12)(3g2+gx2)+ϑ22(3g2+4gx2)218(3g2+gx2),MKI02=12g2(V2+ϑ12).
We can see from (3.21) and (3.23) that we have one nonmassive boson, which we associate with the photon, and four massive neutral fields, where the mass of one of them is proportional to ϑ2 while the other three have masses proportional to V (new energy scale). Therefore we can associate the field Z with SM Zμ and the fields Z′, KI0, and KR′0 with three new neutral bosons. We note that (3.23) contains two same of the eigenvalues; thus, the KI0 and KR′0 components have the same mass, and this conclusion contradicts the previous analysis in [41], but this is in agreement with [43, 44]. We also have four massive charged fields, where two of them have masses proportional to ϑ2. Thus we can associate the fields Wμ′± to the SM fields Wμ±, while the fields Kμ′± are new bosons. The eigenstates Bμ,W3μ,W8μ, and KRμo can be related to the physical eigenstates Aμ,KRμ′0,Zμ0, and Zμ′0 by
(3.24)(BμW3μW8μKRμo)=M-1(AμKRμ′0Zμ0Zμ′0).
Assuming (ϑi/V)n~0 for n>2, we obtain
(3.25)M-1=(-1tSW01tTW2CW+β1-13TW+β2SW-ϑ1VCW+β3β413SW3ϑ1V-13TWSW+β5-1tTW+β601-β7ϑ1VCW-13ϑ1tVTW),
where, again, t=gx/g and
(3.26)SW=3gx3g2+4gx2,CW=1-SW2,TW=SWCW,β1=-ϑ224tV2TW2CW-3,β2=-3ϑ224t2V2TW3CW-2,β3=-ϑ122V2CW-1,β4=-3(2CW2ϑ12+ϑ22)4tV2TWCW-2,β5=6CW4ϑ12-(3-4SW2)ϑ2243V2CW5,β6=(6CW4ϑ12+SW2ϑ22)4tV2CW4TW,β7=-2ϑ22V2.
We note that all βi are of order 𝒪((ϑi/V)2). So, assuming ϑi~𝒪(10-1)TeV, for a new energy scale of order V~10TeV, all the βi’s are negligible.


#### 3.3.2. Charged and Neutral Currents

The interaction between gauge bosons and fermions in flavor basis is given by the following Lagrangian density:(3.27)Lf=R¯iγμ(∂μ+igxBμXR)R+L¯iγμ(∂μ+igxBμXL+ig2λaWμa)L,
where R represents any right-handed singlet and L any left-handed triplet. We can write ℒf=ℒlep+ℒQ1+ℒQa, and in lepton sector, we obtain
(3.28)Llep=Llepkin+LlepCC+LlepNC,
where
(3.29)Llepkin=R¯iγμ∂μR+L¯iγμ∂μL,(3.30)LlepCC=-g2lL¯γμνlLWμ+-g2lL¯γμNlL0Kμ++h.c.,(3.31)LlepNC=gx3[lL¯γμl+νlL¯γμνlL+NlL0¯γμNlL0]Bμ+gxlR¯γμlRBμ-g23[lL¯γμlL+νlL¯γμνlL-2tNlL0¯γμNlL0]W8μ-g2νlL¯γμNlL0K0μ-g2[lL¯γμlL-νlL¯γμνlL]W3μ-g2NlL0¯γμνlLK¯μ0.
In quark sector we have that for the first family triplet X=1/3, and for the singlets d,u, and U1, we have X=-1/3,2/3 and 2/3, respectively. Then we have
(3.32)LQ1kin=Q¯1Riγμ∂μQ1R+Q¯1Liγμ∂μQ1L,LQ1CC=-g2dL¯γμuLWμ+-g2dL¯γμU1LKμ++h.c.,(3.33)LQ1NC=gx3(dR¯γμdR-2uR¯γμuR-2U1R¯γμU1R)Bμ+g2uL¯γμuLW3μ-gx3(dL¯γμdL+uL¯γμuL+U1L¯γμU1L)Bμ-g2dL¯γμdLW3μ-g2U1L¯γμuLK¯μ0-g23(dL¯γμdL+uL¯γμuL-2U1L¯γμU1L)W8μ-g2uL¯γμU1LKμ0.For second and third families we know thatX=0 for the triplets and X=2/3,-1/3 and -1/3, for the singlets u2,3,d2,3,D2,3, respectively, where u2=c,u3=t,d2=s,d3=b. Then we obtain for a=2,3(3.34)LQakin=Q¯aRiγμ∂μQaR+Q¯aLiγμ∂μQaL,LQaCC=-g2uaL¯γμdaLWμ+-g2uaL¯γμDaLKμ++h.c.,LQaNC=gx3[-2uaR¯γμuaR+daR¯γμdaR+DaR¯γμDaR]Bμ-g23[uaL¯γμuaL+daL¯γμdaL-4DaL¯γμDaL]W8μ-g2daL¯γμDaLKμ0-g2[uaL¯γμuaL-daL¯γμdaL]W3μ-g2DaL¯γμdaLK¯μ0.


## 3.1. The Representation Content

There are many representations for the matter content [46], for instance, b=3/2 [38]. But we note that if we accommodate the doublets of SU(2)L in the superior components of triplets and antitriplets of SU(3)L, and if we forbid exotic charges for the new fermions, we obtain from (3.2) the constrain b=±1/2 (assuming a=1). Since a negative value of b can be associated to the antitriplet, we obtain that b=1/2 is a necessary and sufficient condition to exclude exotic electric charges in fermion and boson sector [41].The fields left- and right-handed components transform underSU(3)L as triplets and singlets, respectively. Therefore the theory is chiral and can present anomalies of Alder-Bell-Jackiw [52, 53]. In a non-abelian theory, in the fermionic representation ℛ, the divergent anomaly is given by
(3.6)Aabc∝∑RTr⁡[{TLa(R),TLb(R)}TLc(R)-{TRa(R),TRb(R)}TRc(R)],
where Ta(ℛ) are the matrix representations for each group generator acting on the basis ℛ with helicity left or right. Therefore, to eliminate the pure anomaly [SU(3)L]3, we should have that 𝒜abc∝∑ℛ′Tr[{TLa(ℛ′),TLb(ℛ′)}TLc(ℛ′)]=0. We use the fact that SU(3)L has two fundamental representations, 3 and 3*, then its generators should be associated to Ta and Ta*, respectively, that is,
(3.7)∑R′Tr⁡[{TLa(R′),TLb(R′)}TLc(R′)]=∑RTr⁡[{TLa(R),TLb(R)}TLc(R)]-∑R*Tr⁡[{TLa*(R*),TLb*(R*)}TLc*(R*)],
but we know that the matrix representations for each group generator satisfies that TLa*(ℛ*)=-TLa(ℛ) [54]. So, we can see that for the anomalies to be canceled, the number of fields that transform as triplets (first term in equation above) and antitriplets under SU(3)L has to be the same; that is, two triplets quark families × 3 (color) = one antitriplet quark family × 3 (color) + 3 antitriplet lepton families. This implies that two families of quarks should transform differently than the third family, as will be discussed in next paragraph.Usually the third quark family is chosen to transform in a different way than the first two families. But we will assume that the first family transform differently, to address the fact thatmu<md,mνℓ<mℓ while mc≫ms and mt≫mb. To state this in a clearer way, we recall that in SM the SU(2)L doublets are (νℓ,ℓ)T,(u,d)T,(c,s)T,(t,b)T, with ℓ=e,μ,τ. We can see that the first component of leptons doublets and first quark family is lighter than the second component. But for the second and third quark families, the opposite occurs. Then we use this idea to justify that first quark family transform as leptons.


## 3.2. Minimal 331 Model on Scalar Sector

Among the different possibilities of 331 models, we will present a detailed study on a minimal model on scalar sector without exotic electric charges for quarks and with three new leptons without charged [41] (b=1/2), where the fermions present the following transformation structure under SU(3)C⊗SU(3)L⊗U(1)X:
(3.8)ψlL=(l-,νl,Nl0)LT~(1,3*,-13),νlR~(1,1,0),lR-~(1,1,-1),NlR0~(1,1,0),Q1L=(d,u,U1)LT~(3,3*,13),uiR~(3,1,23),diR~(3,1,-13),U1R~(3,1,23),QaL=(ua,da,Da)LT~(3,3,0),DaR~(3,1,-13),
where i=1,2,3,ℓ=e,μ,τ,a=2,3. We note that the leptons multiplets ψℓL consist of three fields ℓ={e,μ,τ}, the corresponding neutrinos νℓ={νe,νμ,ντ}, and new neutral leptons Nℓ0={Ne0,Nμ0,Nτ0}. We can also see that the multiplet associated with the first quark family Q1L consists of down and up quarks and a new quark with the same electric charge of quark up (named U1), while the multiplet associated with second (third) family QaL consists of SM quarks of second (third) family and a new quark with the same electric charge of down quark (named D2 (D3)). The numbers on parenthesis refer to the transformation properties under SU(3)C,SU(3)L, and U(1)X, respectively. With this choice, the anomalies are cancelled in a nontrivial way [55], and asymptotic freedom is guaranteed [56–59].


### 3.2.1. Scalar Sector and the Yukawa Couplings

The scalar fields have to be coupled to fermions by the Yukawa terms, invariants underSU(3)c⊗SU(3)L⊗U(1)X. In lepton sector, these couplings can be written as
(3.9)ψ¯lLlR~(1,3,13)⊗(1,1,-1)=(1,3,-23)︸ρ*,ψ¯lLνlR~(1,3,13)⊗(1,1,0)=(1,3,13)︸η,ψ¯lLNlR0~(1,3,13)⊗(1,1,0)=(1,3,13)︸χ,
and writing only three terms in quarks sector, for example,
(3.10)Q¯1LuiR=(3*,3,-13)⊗(3,1,23)=(1,3,13)︸χ⊕(8,3,13)︸ColorHiggs,Q¯1LdiR=(3*,3,-13)⊗(3,1,-13)=(1,3,-23)︸ρ*⊕…,Q¯aLdiR=(3*,3*,0)⊗(3,1,-13)=(1,3*,-13)︸η*⊕…,….
As usual in these class of models, we impose colorless Higgs (i.e., selecting only the multiplets that transform as singlets under SU(3)C). We note that we need only three Higgs multiplets, ρ,χ, and η, to couple the different fermionic fields and generate mass through spontaneous symmetry breaking. In (3.9) and (3.10) we note that quantum numbers of triplets χ and η are the same, which leads us to consider models with two or three Higgs triplets. We will adopt the first option, two Higgs triplets, due to the simpler scalar sector in comparison with the scenario with three triplets [41–44].


## 3.2.1. Scalar Sector and the Yukawa Couplings

The scalar fields have to be coupled to fermions by the Yukawa terms, invariants underSU(3)c⊗SU(3)L⊗U(1)X. In lepton sector, these couplings can be written as
(3.9)ψ¯lLlR~(1,3,13)⊗(1,1,-1)=(1,3,-23)︸ρ*,ψ¯lLνlR~(1,3,13)⊗(1,1,0)=(1,3,13)︸η,ψ¯lLNlR0~(1,3,13)⊗(1,1,0)=(1,3,13)︸χ,
and writing only three terms in quarks sector, for example,
(3.10)Q¯1LuiR=(3*,3,-13)⊗(3,1,23)=(1,3,13)︸χ⊕(8,3,13)︸ColorHiggs,Q¯1LdiR=(3*,3,-13)⊗(3,1,-13)=(1,3,-23)︸ρ*⊕…,Q¯aLdiR=(3*,3*,0)⊗(3,1,-13)=(1,3*,-13)︸η*⊕…,….
As usual in these class of models, we impose colorless Higgs (i.e., selecting only the multiplets that transform as singlets under SU(3)C). We note that we need only three Higgs multiplets, ρ,χ, and η, to couple the different fermionic fields and generate mass through spontaneous symmetry breaking. In (3.9) and (3.10) we note that quantum numbers of triplets χ and η are the same, which leads us to consider models with two or three Higgs triplets. We will adopt the first option, two Higgs triplets, due to the simpler scalar sector in comparison with the scenario with three triplets [41–44].


## 3.3. Model with Two Higgs Triplets

For the models with two Higgs triplets, we obtain (note that in this model we assumedΦ1=χ,η e Φ2=ρ)
(3.11)Φ1=(ϕ1-,ϕ1′0,ϕ10)T~(1,3*,-13),Φ2=(ϕ20,ϕ2+,ϕ2′+)T~(1,3*,23).
Assuming the following choice to the Higgs triplets vacuum expectation value (VEV) [41] 〈Φ1〉0=(0,ϑ1,V)T and 〈Φ2〉0=(ϑ2,0,0)T, we associate V with the mass of the new fermions, which lead us to assume V≫ϑ1,ϑ2. We expand the scalar VEVs in the following way:
(3.12)ϕ10=V+Hϕ10+iAϕ102,ϕ1′0=ϑ1+Hϕ1′0+iAϕ1′02,ϕ20=ϑ2+Hϕ20+iAϕ202.
The real (imaginary) part Hϕi(Aϕi) is usually called CP-even (CP-odd) scalar field. The most general potential can be written as
(3.13)V(Φ1,Φ2)=μ12Φ1†Φ1+μ22Φ2†Φ2+λ1(Φ1†Φ1)2+λ2(Φ2†Φ2)2+λ3(Φ1†Φ1)(Φ2†Φ2)+λ4(Φ1†Φ2)(Φ2†Φ1).
Demanding that in the displaced potential V(Φ1,Φ2) the linear terms on the field should be absent, we have, in tree-level approximation, the following constraints:
(3.14)μ12+2λ1(ϑ12+V2)+λ3ϑ22=0,μ22+λ3(ϑ12+V2)+2λ2ϑ22=0.
The analysis of such equations shows that they are related to a minimum in scalar potential with the value Vmin=-ϑ24λ2-(ϑ12+V2)[(ϑ12+V2)λ1+ϑ22λ3]. Then, replacing (3.12) and (3.14) in (3.13), we can calculate the mass matrix in (Hϕ10,Hϕ20,Hϕ1′0) basis through the relation Mij2=2(∂2V(Φ1,Φ2)/∂HΦi0∂HΦj0), obtaining
(3.15)MH2=2(2λ1V2λ3ϑ2V2λ1ϑ1Vλ3ϑ2V2λ2ϑ22λ3ϑ1ϑ22λ1ϑ1Vλ3ϑ1ϑ22λ1ϑ12).
Since (3.15) has vanishing determinant, we have one Goldstone boson G1 and two massive neutral scalar fields H1 and H2 with masses (note that if λ32=4λ1λ2, we obtain two Goldstone bosons, G1 and H2, and a massive scalar field H1 with mass MH12=4[λ1(ϑ12+V2)+λ2ϑ22], where λ1λ2>0; then imposing MH12>0 leads to λ1>0 and λ2>0)
(3.16)MH1,H22=2λ1(ϑ12+V2)+2λ2ϑ22±2[λ1(ϑ12+V2)+λ2ϑ22]2+ϑ22(ϑ12+V2)(λ32-4λ1λ2),
where real values for λ’s produce positive mass to neutral scalar fields only if λ1>0 and 4λ1λ2>λ32, which implies that λ2>0. A detailed analysis shows that when V(Φ1,Φ2) in (3.13) is expanded around the most general vacuum, given by (3.12) and using constrains in (3.14), we do not obtain pseudoscalar fields AΦi0. This allows us do identify three more Goldstone bosons, G2=AΦ10,G3=AΦ20, and G4=AΦ1′0. For the mass spectrum in charged scalar sector on (ϕ1-,ϕ2+,ϕ2′+) basis, the mass matrix will be given by
(3.17)M+2=2λ4(ϑ22ϑ1ϑ2ϑ2Vϑ1ϑ2ϑ12ϑ1Vϑ2Vϑ1VV2),
with two eigenvalues equal to zero, equivalent to four Goldstone bosons G5±,G6± and two physical charged scalar fields with large masses given by λ4(ϑ12+ϑ22+V2), which leads to the constrain λ4>0.This analysis shows that, after symmetry breaking, the original twelve degrees of freedom in scalar sector leads to eight Goldstone bosons (four electrically neutral and four electrically charged), four physical scalar fields, two neutral (one of which being the SM Higgs scalar), and two charged. Eight Goldstone bosons should be absorbed by eight gauge fields as we will see in next section.


### 3.3.1. Gauge Sector with Two Higgs Triplets

The gauge bosons interaction with matter in electroweak sector appears with the covariant derivative for a matter fieldφ as
(3.18)Dμφ=∂μ-i2gWμaλaL-igXXφBμ=∂μ-i2gMμφ,
where λaL,a=1,…,8 are Gell-Mann matrices of SU(3)L algebra and Xφ is the charge of abelian factor U(1)X of the multiplet φ in which Dμ acts. The matrix ℳμφ contains the gauge bosons with electric charges q, defined by the generic charge operator in (3.1). For b=1/2 the matrix ℳμφ will have the following form:
(3.19)Mμφ=(W3μ+W8μ3+2tXφBμ2Wμ+2Kμ+2Wμ--W3μ+W8μ3+2tXφBμ2Kμ02Kμ-2K¯μ0-2W8μ3+2tXφBμ),
where t=gx/g and nonphysical gauge bosons on nondiagonal entries, Wμ±andKμ±, are defined in (3.5) with 𝒬1=1, and
(3.20)Kμ0=12(W6μ-iW7μ),K¯μ0=12(W6μ+iW7μ).
Then for the 331 model we are considering (b=1/2), we have two neutral gauge bosons, Kμ0 and K¯μ0, and four charged gauge bosons, Wμ± and Kμ±. The three physical neutral eigenstates will be a linear combination of W3μ,W8μ, and Bμ. After breaking the symmetry with 〈Φi〉, i=1,2, and using covariant derivative Dμ=∂μ-(i/2)gℳμφ for the triplets Φi, we obtain the following masses for the charged physical fields:
(3.21)MW′2=12g2ϑ22,MK′2=12g2(ϑ12+ϑ22+V2),
and the following physical eigenstates:
(3.22)Wμ′±=1ϑ12+V2(-ϑ1Kμ±+VWμ±),Kμ′±=1ϑ12+V2(VKμ±+ϑ1Wμ±).
The neutral sector in approximation (ϑi/V)n≈0 for n>2 leads to the following masses for the neutral physical fields:
(3.23)Mphoton2=0,MKR02=12g2(V2+ϑ12),MZ2≈12g2ϑ22(3g2+4gx23g2+gx2),MZ′2≈29(V2+ϑ12)(3g2+gx2)+ϑ22(3g2+4gx2)218(3g2+gx2),MKI02=12g2(V2+ϑ12).
We can see from (3.21) and (3.23) that we have one nonmassive boson, which we associate with the photon, and four massive neutral fields, where the mass of one of them is proportional to ϑ2 while the other three have masses proportional to V (new energy scale). Therefore we can associate the field Z with SM Zμ and the fields Z′, KI0, and KR′0 with three new neutral bosons. We note that (3.23) contains two same of the eigenvalues; thus, the KI0 and KR′0 components have the same mass, and this conclusion contradicts the previous analysis in [41], but this is in agreement with [43, 44]. We also have four massive charged fields, where two of them have masses proportional to ϑ2. Thus we can associate the fields Wμ′± to the SM fields Wμ±, while the fields Kμ′± are new bosons. The eigenstates Bμ,W3μ,W8μ, and KRμo can be related to the physical eigenstates Aμ,KRμ′0,Zμ0, and Zμ′0 by
(3.24)(BμW3μW8μKRμo)=M-1(AμKRμ′0Zμ0Zμ′0).
Assuming (ϑi/V)n~0 for n>2, we obtain
(3.25)M-1=(-1tSW01tTW2CW+β1-13TW+β2SW-ϑ1VCW+β3β413SW3ϑ1V-13TWSW+β5-1tTW+β601-β7ϑ1VCW-13ϑ1tVTW),
where, again, t=gx/g and
(3.26)SW=3gx3g2+4gx2,CW=1-SW2,TW=SWCW,β1=-ϑ224tV2TW2CW-3,β2=-3ϑ224t2V2TW3CW-2,β3=-ϑ122V2CW-1,β4=-3(2CW2ϑ12+ϑ22)4tV2TWCW-2,β5=6CW4ϑ12-(3-4SW2)ϑ2243V2CW5,β6=(6CW4ϑ12+SW2ϑ22)4tV2CW4TW,β7=-2ϑ22V2.
We note that all βi are of order 𝒪((ϑi/V)2). So, assuming ϑi~𝒪(10-1)TeV, for a new energy scale of order V~10TeV, all the βi’s are negligible.


### 3.3.2. Charged and Neutral Currents

The interaction between gauge bosons and fermions in flavor basis is given by the following Lagrangian density:(3.27)Lf=R¯iγμ(∂μ+igxBμXR)R+L¯iγμ(∂μ+igxBμXL+ig2λaWμa)L,
where R represents any right-handed singlet and L any left-handed triplet. We can write ℒf=ℒlep+ℒQ1+ℒQa, and in lepton sector, we obtain
(3.28)Llep=Llepkin+LlepCC+LlepNC,
where
(3.29)Llepkin=R¯iγμ∂μR+L¯iγμ∂μL,(3.30)LlepCC=-g2lL¯γμνlLWμ+-g2lL¯γμNlL0Kμ++h.c.,(3.31)LlepNC=gx3[lL¯γμl+νlL¯γμνlL+NlL0¯γμNlL0]Bμ+gxlR¯γμlRBμ-g23[lL¯γμlL+νlL¯γμνlL-2tNlL0¯γμNlL0]W8μ-g2νlL¯γμNlL0K0μ-g2[lL¯γμlL-νlL¯γμνlL]W3μ-g2NlL0¯γμνlLK¯μ0.
In quark sector we have that for the first family triplet X=1/3, and for the singlets d,u, and U1, we have X=-1/3,2/3 and 2/3, respectively. Then we have
(3.32)LQ1kin=Q¯1Riγμ∂μQ1R+Q¯1Liγμ∂μQ1L,LQ1CC=-g2dL¯γμuLWμ+-g2dL¯γμU1LKμ++h.c.,(3.33)LQ1NC=gx3(dR¯γμdR-2uR¯γμuR-2U1R¯γμU1R)Bμ+g2uL¯γμuLW3μ-gx3(dL¯γμdL+uL¯γμuL+U1L¯γμU1L)Bμ-g2dL¯γμdLW3μ-g2U1L¯γμuLK¯μ0-g23(dL¯γμdL+uL¯γμuL-2U1L¯γμU1L)W8μ-g2uL¯γμU1LKμ0.For second and third families we know thatX=0 for the triplets and X=2/3,-1/3 and -1/3, for the singlets u2,3,d2,3,D2,3, respectively, where u2=c,u3=t,d2=s,d3=b. Then we obtain for a=2,3(3.34)LQakin=Q¯aRiγμ∂μQaR+Q¯aLiγμ∂μQaL,LQaCC=-g2uaL¯γμdaLWμ+-g2uaL¯γμDaLKμ++h.c.,LQaNC=gx3[-2uaR¯γμuaR+daR¯γμdaR+DaR¯γμDaR]Bμ-g23[uaL¯γμuaL+daL¯γμdaL-4DaL¯γμDaL]W8μ-g2daL¯γμDaLKμ0-g2[uaL¯γμuaL-daL¯γμdaL]W3μ-g2DaL¯γμdaLK¯μ0.


## 3.3.1. Gauge Sector with Two Higgs Triplets

The gauge bosons interaction with matter in electroweak sector appears with the covariant derivative for a matter fieldφ as
(3.18)Dμφ=∂μ-i2gWμaλaL-igXXφBμ=∂μ-i2gMμφ,
where λaL,a=1,…,8 are Gell-Mann matrices of SU(3)L algebra and Xφ is the charge of abelian factor U(1)X of the multiplet φ in which Dμ acts. The matrix ℳμφ contains the gauge bosons with electric charges q, defined by the generic charge operator in (3.1). For b=1/2 the matrix ℳμφ will have the following form:
(3.19)Mμφ=(W3μ+W8μ3+2tXφBμ2Wμ+2Kμ+2Wμ--W3μ+W8μ3+2tXφBμ2Kμ02Kμ-2K¯μ0-2W8μ3+2tXφBμ),
where t=gx/g and nonphysical gauge bosons on nondiagonal entries, Wμ±andKμ±, are defined in (3.5) with 𝒬1=1, and
(3.20)Kμ0=12(W6μ-iW7μ),K¯μ0=12(W6μ+iW7μ).
Then for the 331 model we are considering (b=1/2), we have two neutral gauge bosons, Kμ0 and K¯μ0, and four charged gauge bosons, Wμ± and Kμ±. The three physical neutral eigenstates will be a linear combination of W3μ,W8μ, and Bμ. After breaking the symmetry with 〈Φi〉, i=1,2, and using covariant derivative Dμ=∂μ-(i/2)gℳμφ for the triplets Φi, we obtain the following masses for the charged physical fields:
(3.21)MW′2=12g2ϑ22,MK′2=12g2(ϑ12+ϑ22+V2),
and the following physical eigenstates:
(3.22)Wμ′±=1ϑ12+V2(-ϑ1Kμ±+VWμ±),Kμ′±=1ϑ12+V2(VKμ±+ϑ1Wμ±).
The neutral sector in approximation (ϑi/V)n≈0 for n>2 leads to the following masses for the neutral physical fields:
(3.23)Mphoton2=0,MKR02=12g2(V2+ϑ12),MZ2≈12g2ϑ22(3g2+4gx23g2+gx2),MZ′2≈29(V2+ϑ12)(3g2+gx2)+ϑ22(3g2+4gx2)218(3g2+gx2),MKI02=12g2(V2+ϑ12).
We can see from (3.21) and (3.23) that we have one nonmassive boson, which we associate with the photon, and four massive neutral fields, where the mass of one of them is proportional to ϑ2 while the other three have masses proportional to V (new energy scale). Therefore we can associate the field Z with SM Zμ and the fields Z′, KI0, and KR′0 with three new neutral bosons. We note that (3.23) contains two same of the eigenvalues; thus, the KI0 and KR′0 components have the same mass, and this conclusion contradicts the previous analysis in [41], but this is in agreement with [43, 44]. We also have four massive charged fields, where two of them have masses proportional to ϑ2. Thus we can associate the fields Wμ′± to the SM fields Wμ±, while the fields Kμ′± are new bosons. The eigenstates Bμ,W3μ,W8μ, and KRμo can be related to the physical eigenstates Aμ,KRμ′0,Zμ0, and Zμ′0 by
(3.24)(BμW3μW8μKRμo)=M-1(AμKRμ′0Zμ0Zμ′0).
Assuming (ϑi/V)n~0 for n>2, we obtain
(3.25)M-1=(-1tSW01tTW2CW+β1-13TW+β2SW-ϑ1VCW+β3β413SW3ϑ1V-13TWSW+β5-1tTW+β601-β7ϑ1VCW-13ϑ1tVTW),
where, again, t=gx/g and
(3.26)SW=3gx3g2+4gx2,CW=1-SW2,TW=SWCW,β1=-ϑ224tV2TW2CW-3,β2=-3ϑ224t2V2TW3CW-2,β3=-ϑ122V2CW-1,β4=-3(2CW2ϑ12+ϑ22)4tV2TWCW-2,β5=6CW4ϑ12-(3-4SW2)ϑ2243V2CW5,β6=(6CW4ϑ12+SW2ϑ22)4tV2CW4TW,β7=-2ϑ22V2.
We note that all βi are of order 𝒪((ϑi/V)2). So, assuming ϑi~𝒪(10-1)TeV, for a new energy scale of order V~10TeV, all the βi’s are negligible.


## 3.3.2. Charged and Neutral Currents

The interaction between gauge bosons and fermions in flavor basis is given by the following Lagrangian density:(3.27)Lf=R¯iγμ(∂μ+igxBμXR)R+L¯iγμ(∂μ+igxBμXL+ig2λaWμa)L,
where R represents any right-handed singlet and L any left-handed triplet. We can write ℒf=ℒlep+ℒQ1+ℒQa, and in lepton sector, we obtain
(3.28)Llep=Llepkin+LlepCC+LlepNC,
where
(3.29)Llepkin=R¯iγμ∂μR+L¯iγμ∂μL,(3.30)LlepCC=-g2lL¯γμνlLWμ+-g2lL¯γμNlL0Kμ++h.c.,(3.31)LlepNC=gx3[lL¯γμl+νlL¯γμνlL+NlL0¯γμNlL0]Bμ+gxlR¯γμlRBμ-g23[lL¯γμlL+νlL¯γμνlL-2tNlL0¯γμNlL0]W8μ-g2νlL¯γμNlL0K0μ-g2[lL¯γμlL-νlL¯γμνlL]W3μ-g2NlL0¯γμνlLK¯μ0.
In quark sector we have that for the first family triplet X=1/3, and for the singlets d,u, and U1, we have X=-1/3,2/3 and 2/3, respectively. Then we have
(3.32)LQ1kin=Q¯1Riγμ∂μQ1R+Q¯1Liγμ∂μQ1L,LQ1CC=-g2dL¯γμuLWμ+-g2dL¯γμU1LKμ++h.c.,(3.33)LQ1NC=gx3(dR¯γμdR-2uR¯γμuR-2U1R¯γμU1R)Bμ+g2uL¯γμuLW3μ-gx3(dL¯γμdL+uL¯γμuL+U1L¯γμU1L)Bμ-g2dL¯γμdLW3μ-g2U1L¯γμuLK¯μ0-g23(dL¯γμdL+uL¯γμuL-2U1L¯γμU1L)W8μ-g2uL¯γμU1LKμ0.For second and third families we know thatX=0 for the triplets and X=2/3,-1/3 and -1/3, for the singlets u2,3,d2,3,D2,3, respectively, where u2=c,u3=t,d2=s,d3=b. Then we obtain for a=2,3(3.34)LQakin=Q¯aRiγμ∂μQaR+Q¯aLiγμ∂μQaL,LQaCC=-g2uaL¯γμdaLWμ+-g2uaL¯γμDaLKμ++h.c.,LQaNC=gx3[-2uaR¯γμuaR+daR¯γμdaR+DaR¯γμDaR]Bμ-g23[uaL¯γμuaL+daL¯γμdaL-4DaL¯γμDaL]W8μ-g2daL¯γμDaLKμ0-g2[uaL¯γμuaL-daL¯γμdaL]W3μ-g2DaL¯γμdaLK¯μ0.


## 4. Neutrinos Interactions with Matter in 331 Model

It is well known that neutrino oscillation phenomenon in a material medium, as the sun, earth, or in a supernova, can be quite different from the oscillation that occurs in vacuum, since the interactions in the medium modify the dispersion relations of the particles traveling through it [60]. From the macroscopic point of view, the modifications of neutrino dispersion relations can be represented in terms of a refractive index or an effective potential. And according to [60, 61], the effective potential can be calculated from the amplitudes of coherent elastic scattering in relativistic limit.In the present 331 model, the coherent scattering will be induced by neutral currents, NC, mediated by bosonsZμ′0,Zμ0, and KRμ′0 and by charged currents, CC, mediated by bosons Wμ′± and Kμ′±. Following [61], we calculate in next sections the neutrino effective potentials in coherent scattering.


### 4.1. Charged Currents

The first term of (3.30) shows that the interaction of charged leptons with neutrinos occurs only through the gauge bosons Wμ±; then, by (3.22) we obtain that the interaction through charged bosons is given by
(4.1)-g2lL¯γμνlLWμ+=-Vg2ϑ12+V2lL¯γμνlLWμ′±-gϑ12ϑ12+V2lL¯γμνlLKμ′±.
The amplitude for the neutrino elastic scattering with charged leptons in tree level through CC is given by (note from (4.1) that only left-handed leptons interact with neutrinos, as in SM)
(4.2)Lint⁡cc=-(-Vg2ϑ12+V2)2lL¯(p1)γμνlL(p2)-igμλ(p2-p1)2-MW2ν̅lL(p3)γλlL(p4)-(-gϑ12ϑ12+V2)2lL¯(p1)γμνlL(p2)-igμλ(p2-p1)2-MK2ν̅lL(p3)γλlL(p4).
For low energies MW′2,MK′2≫(p2-p1)2, the effective Lagrangian is given by
(4.3)Leffcc≈-g22(ϑ12+V2)(V2MW′2+ϑ12MK′2)[l¯L(p1)γμlL(p4)][ν¯lL(p3)γμνlL(p2)],
where we used the Fierz transformation [62] to go from (4.2) to (4.3). Replacing (3.21) in (4.3), we obtain
(4.4)-Leffcc≈[1ϑ22-ϑ12V2ϑ22+(ϑ12V4)K′+O(1V4)]〈l¯γμ(1-γ5)2l〉{ν¯lL(p)γμνlL(p)},
where we used ()K′ to denote the term that appears from the new charged boson. We can see that for a new energy scale V≫ϑ1 the term that comes from the new boson does not contribute to the process, as expected, since the new charged boson Kμ′± has a mass of the order of the new energy scale of the theory (see (3.21)).Now, since usual matter has only leptons from first family, we will restrain our calculations to the neutrino interactions with first family standard model particles. The term〈〉 in (4.4) can be calculated following [61], where we have the correspondence 〈e¯γμγ5e〉~ spin, 〈e¯γie〉~ velocity, and 〈e¯γ0e〉~ne, where ne is the electronic density. Assuming nonpolarized medium and vanishing average velocity, we obtain that (4.4) can be written as
(4.5)Leffcc≈-[12ϑ22-ϑ122V2ϑ22+(ϑ122V4)K′+O(V-4)]neν¯eLγ0νeL.
The modifications on electronic neutrino dispersion relations can be represented by the following effective potential:
(4.6)VCCe≈12ϑ22ne-ϑ122V2ϑ22ne+(ϑ122V4)K′ne+O(V-4).
Disregarding the term ()K′ since we are assuming V≫ϑi, and remembering that in Section 3.3.1 we associated boson W′ with SM boson W, we can easily associate
(4.7)2GF≈12ϑ22-ϑ122V2ϑ22.
We note that (4.7) gives limits for the VEV of one of the Higgs triplets. Under assumption ϑ1,ϑ2≪V, we can write GF≈(1/22ϑ22)(1-ϑ12/V2), from which we can see that the maximum value of ϑ22 is achieved when we consider (ϑ12/V2)→0, in which replacing GF=1.16637(1)×10-5Gev-2 leads to
(4.8)ϑ2≲174.105GeV.


### 4.2. Neutral Current

The Lagrangian for neutrino elastic scattering with fermionsf=e,u,d through NC is given by
(4.9)-Lint⁡NC=f¯(p1)γμ(gz′Lf+gz′Rf)f(p2)-igμλ(p2-p1)2-Mz′2ν¯lL(p3)γλgνz′νlL(p4)+f¯(p1)γμ(gzLf+gzRf)f(p2)-igμλ(p2-p1)2-Mz2ν¯lL(p3)γλgνzνlL(p4)+f¯(p1)γμ(gk′Lf+gk′Rf)f(p2)-igμλ(p2-p1)2-Mk′2ν¯lL(p3)γλgνk′νlL(p2).
For low energies, we have that Mk′2,Mz2,Mz′2≫(p2-p1)2 with p3=p4=p and (4.9), and following the same procedure of Section 4.1, we obtain
(4.10)LeffNC≈-∑P=L,R(gz′PfGνz′Mz′2+gzPfGνzMz2+gk′PfGνk′Mk′2)12nfν¯lLγ0νlL.


#### 4.2.1. Leptons Sector

From (3.31) and (3.24), we obtain that for the known neutral leptons
(4.11)gx3ν¯lLγμνlLBμ=ν¯lLγμνlL[-g3SWAμ+(g3TW2CW+gx3β1)Zμ0-gx3(13TW-β2)Zμ′0],(4.12)g2ν¯lLγμνlLW3μ=ν¯lLγμνlL[g2SWAμ-gϑ12VKRμ′0+g(CW+β3)2Zμ0+gβ42Zμ′0],(4.13)-g23ν¯lLγμνlLW8μ=ν¯lLγμνlL[-g6SWAμ-gϑ12VKRμ′0+(g6SW2CW-gβ523)Zμ0+g23(1tTW-β6)Zμ′0].
By (4.11), (4.12), and (4.13), we obtain that vertex interactions with neutrinos can be written as
(4.14)ν¯lLγμνlLAμ∝0,(4.15)ν¯lLγμνlLKRμ′0∝-gϑ1V≡GνK′,(4.16)ν¯lLγμνlLZμ0∝12gCW-1+η1≡GνZ,(4.17)ν¯lLγμνlLZμ′0∝(3g-2gxt63t)TW+η2≡GνZ′,
where
(4.18)η1=-4gtCW2ϑ12+gx(1-2SW2)ϑ228tV2CW5,η2=gt(1-4CW2)ϑ1223V2CWSW-(-gt3+2gt3CW2+8gt3CW4+6gxSW4)ϑ22243t2V2CW5SW.
We note from (4.14) that neutrinos do not interact electrically, as expected. For charged leptons, from (3.31) and (3.24), we obtain
(4.19)gx3lL¯γμlLBμ=lL¯γμlL[gx3(13TW-β2)Zμ′0-g3SWAμ+(g3TW2CW+gx3β1)Zμ0-gx3(13TW-β2)Zμ′0],-g2l¯LγμlLW3μ=l¯LγμlL[-g2SWAμ+gϑ12VKRμ′0-g(CW+β3)2Zμ0-gβ42Zμ′0],-g23l¯LγμlLW8μ=l¯LγμlL[-g6SWAμ-gϑ12VKRμ′0+(g6SW2CW-gβ523)Zμ0+g23(1tTW-β6)Zμ′0],gxlR¯γμlRBμ=lR¯γμlR[gx(13TW-β2)Zμ′0-gSWAμ+(gTW2CW+gxβ1)Zμ0-gx(13TW-β2)Zμ′0],
and therefore
(4.20)l¯γμlAμ∝-gSW,(4.21)lL¯γμlLKRμ′0∝0≡gk′Ll=gk′Rl,(4.22)lL¯γμlLZμ0∝12g(-1+TW2)CW+η3≡gzLl,lR¯γμlRZμ0∝gTW2CW+η5≡gzRl,lL¯γμlLZμ′0∝163t(3g-2tgx)TW+η4≡gz′Ll,(4.23)lR¯γμlRZμ′0∝-gx3TW+η6≡gz′Rl,
where
(4.24)η3=(-1+2CW2)gxϑ228tV2CW5,η4=(gt3(1+2CW2)2-12gt3SW2CW2-6gxSW4)243t2V2CW5SW,η5=-gxϑ224tV2CW3TW2,η6=-3gxϑ224t2V2CW2TW3,
and, again, t=gx/g. We note that by (4.20) we can make the association gSW=|e|. Then for f=e, (4.15)–(4.17) and (4.21)–(4.23) lead to
(4.25)Leff-eNC≈-∑P=L,R12(gz′PeGνz′Mz′2+gzPeGνzMz2+gk′PeGνk′Mk′2)neν¯lLγ0νlL≈-{[TW4144t2gx2V2(3g-2tgx)2+TW28V2(1-TW2)+12(12ϑ22-ϑ122V2ϑ22)(1-2CW2)]L+[TW4(2tgx-3g)24tgxV2-TW44V2+(12ϑ22-ϑ122V2ϑ22)SW2]R}neν¯lLγ0νlL.
Since intermediate neutral bosons in (4.9) do not distinguish between different lepton flavors, the interaction through NC with electron is described by the following effective potential:
(4.26)VNCe=VNCμ=VNCτ=VNCl,=VNClL+VNClR,
where
(4.27)VNClL=[TW4144t2gx2V2(3g-2tgx)2+TW28V2(1-TW2)+12(12ϑ22-ϑ12V2ϑ22)(1-2CW2)]ne,VNClR=[TW4(2tgx-3g)24tgxV2-TW44V2+(12ϑ22-ϑ122V2ϑ22)SW2]ne,
and index ℓ refers to neutrino flavor. We note that the potential through CC comes from interactions of electron neutrinos with left-handed electrons, while the effective potential through NC comes from left- and right-handed electrons.Considering both NC and CC, we can write the effective potential felt by neutrinos asVℓ=VℓL+VℓR, where
(4.28)VlL=(12ϑ22-ϑ122V2ϑ22)δelne+VNClL,VlR=VNClR.
Comparing with SM expression for such potential:
(4.29)VNCl=-2GF(12-2SW2)ne,VCCe=2GFne,
we can find that
(4.30)VlL=VlL+[TW4144t2gx2V2(3g-2tgx)2+TW28V2(1-TW2)]ne,VlR=VNClR+[TW4(2tgx-3g)24tgxV2-TW44V2]ne,
where we adopt in what follow, the convention that V denotes SM-like part of the model; thus, the new terms beyond SM [] can be associated with the parameters ɛ′s in NSI [63]. So, in the approximation (ϑi/V)n≈0, for n>2, we obtain
(4.31)ɛlleL≈(1-2SW2)ϑ228V2CW4,(4.32)ɛlleR≈-SW2(1+2SW2)ϑ224V2CW4.
We note that on limit V→∞, we recover SM. The NSIs are a subleading interaction, as expected. By (4.31) and (4.32), we obtain ɛℓℓeR≈-2SW2ɛℓℓeL-(ϑ22/V2)TW4.


#### 4.2.2. Quarks Sector

For the quarks of the first family, the Lagrangian density in (3.33) describes the interactions with gauge bosons W3μ,W8μ, and Bμ; then, by (3.24) and (3.25) we obtain the following interactions for up quarks:
(4.33)-gx3uL¯γμuLBμ=uL¯γμuL[gx3(13TW-β2)Zμ′0g3SWAμ-gx3(1tTW2CW+β1)Zμ0+gx3(13TW-β2)Zμ′0],g2uL¯γμuLW3μ=uL¯γμuL[g2SWAμ-gϑ12VKRμ′0+g(CW+β3)2Zμ0+gβ42Zμ′0],-g23uL¯γμuLW8μ=uL¯γμuL[-g6SWAμ-gϑ12VKRμ′0+g23(13TWSW-β5)Zμ0+g23(1tTW-β6)Zμ′0],-2gx3uR¯γμuRBμ=uR¯γμuR[2g3SWAμ-2gx3(1tTW2CW+β1)Zμ0+2gx3(1tTW-β6)Zμ′0].
The couplings quark-quark-boson for the first family are given by
(4.34)uL¯γμuLAμ∝23gSW,(4.35)uR¯γμuRAμ∝23gSW,(4.36)uL¯γμuLKRμ′0∝-gϑ1V≡gk′Lu,uR¯γμuRKRμ′0∝0≡gk′Ru,uL¯γμuLZμ0∝16g(3-TW2)CW+ζ1≡gzLu,uR¯γμuRZμ0∝-23gTW2CW+ζ3≡gzRu,uL¯γμuLZμ′0∝163t(3g+2tgx)TW≡gz′Lu,uR¯γμuRZμ′0∝233gxTW+ζ4≡gz′Ru,
where
(4.37)ζ1=gx(-12CW4ϑ12+(1+2CW2)ϑ22)24tV2CW5,ζ2=12gt3CW4(1-4CW2)ϑ12+(gt3(1-2CW2-8CW4)+6gxSW4)ϑ22243t2V2CW5SW,ζ3=g6SW2ϑ22CW5V2,ζ4=gxSW3ϑ2223t2V2CW5.
We note that (4.34) and (4.35) reflect the fact that quarks interact electrically through photons with coupling constant QfsinθW, as in SM. The effective Lagrangian at low energies for neutrino interaction with quarks up through neutral currents are given by (4.10) with f=u:
(4.38)Lquark,uNC≈-12∑P=L,R(gz′PuGνz′Mz′2+gzPuGνzMz2+gk′PuGνk′Mk′2)nuν¯lLγ0νlL≈-{[124V2(3+TW4)+TW4144t4V2(9-4t4)+(12ϑ22-ϑ122V2ϑ22)(12-23SW2)-ϑ124V2ϑ22]L+[TW46V2+TW4(3g-2tgx)36tgxV2-23(12ϑ22-ϑ122V2ϑ22)SW2]R}nuν¯lLγ0νlL,
where nu is the up quarks average density.SM predictions, using result of (4.7), can be written as
(4.39)VNCu=VNCuL+VNCuR=(12ϑ22-ϑ122V2ϑ22)(12-43SW2)nu,
where
(4.40)VNCuL=(12ϑ22-ϑ122V2ϑ22)(12-23SW2)nu,VNCuR=-23(12ϑ22-ϑ122V2ϑ22)SW2nu.
By comparison, we obtain
(4.41)VNCuL≈VNCuL+[124V2(3+TW4)+TW4144t4V2(9-4t4)-ϑ124V2ϑ22]nu,VNCuR≈VNCuR+[TW46V2+TW4(3g-2tgx)36tgxV2]nu.
Then we can say that ɛℓℓu=ɛℓℓuL+ɛℓℓuR, where
(4.42)ɛlluL≈-ϑ122V2+ϑ2224V2CW4(9-8SW2),ɛlluR≈ϑ226V2SW2CW4.
Again, we obtain universal NSI, as for the electrons. We note that ɛℓℓuL=-(ϑ12/2V2)+(3ϑ22/8V2CW4)-2ɛℓℓuR and in the limit V→∞ we recover SM.For down quarks by (3.33) and (3.24), we obtain that
(4.43)-gx3dL¯γμdLBμ=dL¯γμdL[gx3(13TW-β2)Zμ′0g3SWAμ-gx3(1tTW2CW+β1)Zμ0+gx3(13TW-β2)Zμ′0],-g2dL¯γμdLW3μ=dL¯γμdL[-gSW2Aμ+gϑ12VKRμ′0-g(CW+β3)2Zμ0-gβ42Zμ′0],-g23dL¯γμdLW8μ=dL¯γμdL[-gSW6Aμ+g23(13TWSW-β5)Zμ0+-gϑ12VKRμ′0+g23(1tTW-β6)Zμ′0],gx3dR¯γμdRBμ=dL¯γμdL[gx3(-13TW+β2)Zμ′0-gSW3Aμ+gx3(1tTW2CW+β1)Zμ0+gx3(-13TW+β2)Zμ′0].(4.44)dL¯γμdLAμ∝-13gSW,dR¯γμdRAμ∝-13gSW,dL¯γμdLKRμ′0∝0≡gk′Ld,dR¯γμdRKRμ′0∝0≡gk′Rd,dL¯γμdLZμ0∝-16g(3+TW2)CW+ζ5≡gzLd,dR¯γμdRZμ0∝g3TW2CW+ζ7≡gzRd,dL¯γμdLZμ′0∝163t(3g+2tgx)TW+ζ6≡gz′Ld,uR¯γμuRZμ′0∝-133gxTW+ζ8≡gz′Rd,
where
(4.45)ζ5=gϑ2224V2CW5(3-2SW2),ζ6=(-1+3CW2+6CW4-8CW6)243V2CW5SW3,ζ7=-gSW2ϑ2212V2CW5,ζ8=-gxSW3ϑ2243t2V2CW5.
Then by (4.10) for f=d, we obtain the following effective Lagrangian for NC:
(4.46)Lquark,dNC≈-(gz′VdGνz′Mz′2+gzVdGνzMz2+gk′VdGνk′Mk′2)ndν¯lLγ0νlL≈-{[(3SW2-2SW4)24V2CW4+(9-4t4)144t4V2TW4+(12ϑ22-ϑ122V2ϑ22)(-12+13SW2)]L+[-SW224V2CW4+13(12ϑ22-ϑ122V2ϑ22)SW2]R}ndν¯lLγ0νlL,
and the effective potential felt by neutrinos when crossing a medium composed by a density nd of down quarks is VNCd=VNCdL+VNCdR, where
(4.47)VNCdL≈[(3SW2-2SW4)24V2CW4+(9-4t4)144t4V2TW4+(12ϑ22-ϑ122V2ϑ22)(-12+13SW2)]nd,(4.48)VNCdR≈[-SW224V2CW4+13(12ϑ22-ϑ122V2ϑ22)SW2]nd.
Then we can easily see that in SM the NC effective potential for neutrinos in a d-quark medium, using result of (4.7), will be given by
(4.49)VNCd=VNCdL+VNCdR≈-(12ϑ22-ϑ122V2ϑ22)(12-23SW2)nd,VNCdL=(12ϑ22-ϑ122V2ϑ22)(-12+13SW2)nd,(4.50)VNCdR=13(12ϑ22-ϑ122V2ϑ22)SW2nd.
Then from (4.47)–(4.50), we obtain
(4.51)VNCdL≈VNCdL+[(3SW2-2SW4)24V2CW4+(9-4t4)144t4V2TW4]nd,VNCdR≈VNCdR-SW224V2CW4nd,
and neglecting terms of order (ϑi/V)n, for n>2, we obtain that ɛℓℓd=ɛℓℓdL+ɛℓℓdR, where
(4.52)ɛlldL≈ϑ2224V2CW4(3-2SW2),(4.53)ɛlldR≈-SW2ϑ2212V2CW4.
Then we obtain ɛℓℓdL≈(ϑ22/8V2CW4)+ɛℓℓdR. Note that again in limit V→∞ we recover the SM.


## 4.1. Charged Currents

The first term of (3.30) shows that the interaction of charged leptons with neutrinos occurs only through the gauge bosons Wμ±; then, by (3.22) we obtain that the interaction through charged bosons is given by
(4.1)-g2lL¯γμνlLWμ+=-Vg2ϑ12+V2lL¯γμνlLWμ′±-gϑ12ϑ12+V2lL¯γμνlLKμ′±.
The amplitude for the neutrino elastic scattering with charged leptons in tree level through CC is given by (note from (4.1) that only left-handed leptons interact with neutrinos, as in SM)
(4.2)Lint⁡cc=-(-Vg2ϑ12+V2)2lL¯(p1)γμνlL(p2)-igμλ(p2-p1)2-MW2ν̅lL(p3)γλlL(p4)-(-gϑ12ϑ12+V2)2lL¯(p1)γμνlL(p2)-igμλ(p2-p1)2-MK2ν̅lL(p3)γλlL(p4).
For low energies MW′2,MK′2≫(p2-p1)2, the effective Lagrangian is given by
(4.3)Leffcc≈-g22(ϑ12+V2)(V2MW′2+ϑ12MK′2)[l¯L(p1)γμlL(p4)][ν¯lL(p3)γμνlL(p2)],
where we used the Fierz transformation [62] to go from (4.2) to (4.3). Replacing (3.21) in (4.3), we obtain
(4.4)-Leffcc≈[1ϑ22-ϑ12V2ϑ22+(ϑ12V4)K′+O(1V4)]〈l¯γμ(1-γ5)2l〉{ν¯lL(p)γμνlL(p)},
where we used ()K′ to denote the term that appears from the new charged boson. We can see that for a new energy scale V≫ϑ1 the term that comes from the new boson does not contribute to the process, as expected, since the new charged boson Kμ′± has a mass of the order of the new energy scale of the theory (see (3.21)).Now, since usual matter has only leptons from first family, we will restrain our calculations to the neutrino interactions with first family standard model particles. The term〈〉 in (4.4) can be calculated following [61], where we have the correspondence 〈e¯γμγ5e〉~ spin, 〈e¯γie〉~ velocity, and 〈e¯γ0e〉~ne, where ne is the electronic density. Assuming nonpolarized medium and vanishing average velocity, we obtain that (4.4) can be written as
(4.5)Leffcc≈-[12ϑ22-ϑ122V2ϑ22+(ϑ122V4)K′+O(V-4)]neν¯eLγ0νeL.
The modifications on electronic neutrino dispersion relations can be represented by the following effective potential:
(4.6)VCCe≈12ϑ22ne-ϑ122V2ϑ22ne+(ϑ122V4)K′ne+O(V-4).
Disregarding the term ()K′ since we are assuming V≫ϑi, and remembering that in Section 3.3.1 we associated boson W′ with SM boson W, we can easily associate
(4.7)2GF≈12ϑ22-ϑ122V2ϑ22.
We note that (4.7) gives limits for the VEV of one of the Higgs triplets. Under assumption ϑ1,ϑ2≪V, we can write GF≈(1/22ϑ22)(1-ϑ12/V2), from which we can see that the maximum value of ϑ22 is achieved when we consider (ϑ12/V2)→0, in which replacing GF=1.16637(1)×10-5Gev-2 leads to
(4.8)ϑ2≲174.105GeV.


## 4.2. Neutral Current

The Lagrangian for neutrino elastic scattering with fermionsf=e,u,d through NC is given by
(4.9)-Lint⁡NC=f¯(p1)γμ(gz′Lf+gz′Rf)f(p2)-igμλ(p2-p1)2-Mz′2ν¯lL(p3)γλgνz′νlL(p4)+f¯(p1)γμ(gzLf+gzRf)f(p2)-igμλ(p2-p1)2-Mz2ν¯lL(p3)γλgνzνlL(p4)+f¯(p1)γμ(gk′Lf+gk′Rf)f(p2)-igμλ(p2-p1)2-Mk′2ν¯lL(p3)γλgνk′νlL(p2).
For low energies, we have that Mk′2,Mz2,Mz′2≫(p2-p1)2 with p3=p4=p and (4.9), and following the same procedure of Section 4.1, we obtain
(4.10)LeffNC≈-∑P=L,R(gz′PfGνz′Mz′2+gzPfGνzMz2+gk′PfGνk′Mk′2)12nfν¯lLγ0νlL.


### 4.2.1. Leptons Sector

From (3.31) and (3.24), we obtain that for the known neutral leptons
(4.11)gx3ν¯lLγμνlLBμ=ν¯lLγμνlL[-g3SWAμ+(g3TW2CW+gx3β1)Zμ0-gx3(13TW-β2)Zμ′0],(4.12)g2ν¯lLγμνlLW3μ=ν¯lLγμνlL[g2SWAμ-gϑ12VKRμ′0+g(CW+β3)2Zμ0+gβ42Zμ′0],(4.13)-g23ν¯lLγμνlLW8μ=ν¯lLγμνlL[-g6SWAμ-gϑ12VKRμ′0+(g6SW2CW-gβ523)Zμ0+g23(1tTW-β6)Zμ′0].
By (4.11), (4.12), and (4.13), we obtain that vertex interactions with neutrinos can be written as
(4.14)ν¯lLγμνlLAμ∝0,(4.15)ν¯lLγμνlLKRμ′0∝-gϑ1V≡GνK′,(4.16)ν¯lLγμνlLZμ0∝12gCW-1+η1≡GνZ,(4.17)ν¯lLγμνlLZμ′0∝(3g-2gxt63t)TW+η2≡GνZ′,
where
(4.18)η1=-4gtCW2ϑ12+gx(1-2SW2)ϑ228tV2CW5,η2=gt(1-4CW2)ϑ1223V2CWSW-(-gt3+2gt3CW2+8gt3CW4+6gxSW4)ϑ22243t2V2CW5SW.
We note from (4.14) that neutrinos do not interact electrically, as expected. For charged leptons, from (3.31) and (3.24), we obtain
(4.19)gx3lL¯γμlLBμ=lL¯γμlL[gx3(13TW-β2)Zμ′0-g3SWAμ+(g3TW2CW+gx3β1)Zμ0-gx3(13TW-β2)Zμ′0],-g2l¯LγμlLW3μ=l¯LγμlL[-g2SWAμ+gϑ12VKRμ′0-g(CW+β3)2Zμ0-gβ42Zμ′0],-g23l¯LγμlLW8μ=l¯LγμlL[-g6SWAμ-gϑ12VKRμ′0+(g6SW2CW-gβ523)Zμ0+g23(1tTW-β6)Zμ′0],gxlR¯γμlRBμ=lR¯γμlR[gx(13TW-β2)Zμ′0-gSWAμ+(gTW2CW+gxβ1)Zμ0-gx(13TW-β2)Zμ′0],
and therefore
(4.20)l¯γμlAμ∝-gSW,(4.21)lL¯γμlLKRμ′0∝0≡gk′Ll=gk′Rl,(4.22)lL¯γμlLZμ0∝12g(-1+TW2)CW+η3≡gzLl,lR¯γμlRZμ0∝gTW2CW+η5≡gzRl,lL¯γμlLZμ′0∝163t(3g-2tgx)TW+η4≡gz′Ll,(4.23)lR¯γμlRZμ′0∝-gx3TW+η6≡gz′Rl,
where
(4.24)η3=(-1+2CW2)gxϑ228tV2CW5,η4=(gt3(1+2CW2)2-12gt3SW2CW2-6gxSW4)243t2V2CW5SW,η5=-gxϑ224tV2CW3TW2,η6=-3gxϑ224t2V2CW2TW3,
and, again, t=gx/g. We note that by (4.20) we can make the association gSW=|e|. Then for f=e, (4.15)–(4.17) and (4.21)–(4.23) lead to
(4.25)Leff-eNC≈-∑P=L,R12(gz′PeGνz′Mz′2+gzPeGνzMz2+gk′PeGνk′Mk′2)neν¯lLγ0νlL≈-{[TW4144t2gx2V2(3g-2tgx)2+TW28V2(1-TW2)+12(12ϑ22-ϑ122V2ϑ22)(1-2CW2)]L+[TW4(2tgx-3g)24tgxV2-TW44V2+(12ϑ22-ϑ122V2ϑ22)SW2]R}neν¯lLγ0νlL.
Since intermediate neutral bosons in (4.9) do not distinguish between different lepton flavors, the interaction through NC with electron is described by the following effective potential:
(4.26)VNCe=VNCμ=VNCτ=VNCl,=VNClL+VNClR,
where
(4.27)VNClL=[TW4144t2gx2V2(3g-2tgx)2+TW28V2(1-TW2)+12(12ϑ22-ϑ12V2ϑ22)(1-2CW2)]ne,VNClR=[TW4(2tgx-3g)24tgxV2-TW44V2+(12ϑ22-ϑ122V2ϑ22)SW2]ne,
and index ℓ refers to neutrino flavor. We note that the potential through CC comes from interactions of electron neutrinos with left-handed electrons, while the effective potential through NC comes from left- and right-handed electrons.Considering both NC and CC, we can write the effective potential felt by neutrinos asVℓ=VℓL+VℓR, where
(4.28)VlL=(12ϑ22-ϑ122V2ϑ22)δelne+VNClL,VlR=VNClR.
Comparing with SM expression for such potential:
(4.29)VNCl=-2GF(12-2SW2)ne,VCCe=2GFne,
we can find that
(4.30)VlL=VlL+[TW4144t2gx2V2(3g-2tgx)2+TW28V2(1-TW2)]ne,VlR=VNClR+[TW4(2tgx-3g)24tgxV2-TW44V2]ne,
where we adopt in what follow, the convention that V denotes SM-like part of the model; thus, the new terms beyond SM [] can be associated with the parameters ɛ′s in NSI [63]. So, in the approximation (ϑi/V)n≈0, for n>2, we obtain
(4.31)ɛlleL≈(1-2SW2)ϑ228V2CW4,(4.32)ɛlleR≈-SW2(1+2SW2)ϑ224V2CW4.
We note that on limit V→∞, we recover SM. The NSIs are a subleading interaction, as expected. By (4.31) and (4.32), we obtain ɛℓℓeR≈-2SW2ɛℓℓeL-(ϑ22/V2)TW4.


### 4.2.2. Quarks Sector

For the quarks of the first family, the Lagrangian density in (3.33) describes the interactions with gauge bosons W3μ,W8μ, and Bμ; then, by (3.24) and (3.25) we obtain the following interactions for up quarks:
(4.33)-gx3uL¯γμuLBμ=uL¯γμuL[gx3(13TW-β2)Zμ′0g3SWAμ-gx3(1tTW2CW+β1)Zμ0+gx3(13TW-β2)Zμ′0],g2uL¯γμuLW3μ=uL¯γμuL[g2SWAμ-gϑ12VKRμ′0+g(CW+β3)2Zμ0+gβ42Zμ′0],-g23uL¯γμuLW8μ=uL¯γμuL[-g6SWAμ-gϑ12VKRμ′0+g23(13TWSW-β5)Zμ0+g23(1tTW-β6)Zμ′0],-2gx3uR¯γμuRBμ=uR¯γμuR[2g3SWAμ-2gx3(1tTW2CW+β1)Zμ0+2gx3(1tTW-β6)Zμ′0].
The couplings quark-quark-boson for the first family are given by
(4.34)uL¯γμuLAμ∝23gSW,(4.35)uR¯γμuRAμ∝23gSW,(4.36)uL¯γμuLKRμ′0∝-gϑ1V≡gk′Lu,uR¯γμuRKRμ′0∝0≡gk′Ru,uL¯γμuLZμ0∝16g(3-TW2)CW+ζ1≡gzLu,uR¯γμuRZμ0∝-23gTW2CW+ζ3≡gzRu,uL¯γμuLZμ′0∝163t(3g+2tgx)TW≡gz′Lu,uR¯γμuRZμ′0∝233gxTW+ζ4≡gz′Ru,
where
(4.37)ζ1=gx(-12CW4ϑ12+(1+2CW2)ϑ22)24tV2CW5,ζ2=12gt3CW4(1-4CW2)ϑ12+(gt3(1-2CW2-8CW4)+6gxSW4)ϑ22243t2V2CW5SW,ζ3=g6SW2ϑ22CW5V2,ζ4=gxSW3ϑ2223t2V2CW5.
We note that (4.34) and (4.35) reflect the fact that quarks interact electrically through photons with coupling constant QfsinθW, as in SM. The effective Lagrangian at low energies for neutrino interaction with quarks up through neutral currents are given by (4.10) with f=u:
(4.38)Lquark,uNC≈-12∑P=L,R(gz′PuGνz′Mz′2+gzPuGνzMz2+gk′PuGνk′Mk′2)nuν¯lLγ0νlL≈-{[124V2(3+TW4)+TW4144t4V2(9-4t4)+(12ϑ22-ϑ122V2ϑ22)(12-23SW2)-ϑ124V2ϑ22]L+[TW46V2+TW4(3g-2tgx)36tgxV2-23(12ϑ22-ϑ122V2ϑ22)SW2]R}nuν¯lLγ0νlL,
where nu is the up quarks average density.SM predictions, using result of (4.7), can be written as
(4.39)VNCu=VNCuL+VNCuR=(12ϑ22-ϑ122V2ϑ22)(12-43SW2)nu,
where
(4.40)VNCuL=(12ϑ22-ϑ122V2ϑ22)(12-23SW2)nu,VNCuR=-23(12ϑ22-ϑ122V2ϑ22)SW2nu.
By comparison, we obtain
(4.41)VNCuL≈VNCuL+[124V2(3+TW4)+TW4144t4V2(9-4t4)-ϑ124V2ϑ22]nu,VNCuR≈VNCuR+[TW46V2+TW4(3g-2tgx)36tgxV2]nu.
Then we can say that ɛℓℓu=ɛℓℓuL+ɛℓℓuR, where
(4.42)ɛlluL≈-ϑ122V2+ϑ2224V2CW4(9-8SW2),ɛlluR≈ϑ226V2SW2CW4.
Again, we obtain universal NSI, as for the electrons. We note that ɛℓℓuL=-(ϑ12/2V2)+(3ϑ22/8V2CW4)-2ɛℓℓuR and in the limit V→∞ we recover SM.For down quarks by (3.33) and (3.24), we obtain that
(4.43)-gx3dL¯γμdLBμ=dL¯γμdL[gx3(13TW-β2)Zμ′0g3SWAμ-gx3(1tTW2CW+β1)Zμ0+gx3(13TW-β2)Zμ′0],-g2dL¯γμdLW3μ=dL¯γμdL[-gSW2Aμ+gϑ12VKRμ′0-g(CW+β3)2Zμ0-gβ42Zμ′0],-g23dL¯γμdLW8μ=dL¯γμdL[-gSW6Aμ+g23(13TWSW-β5)Zμ0+-gϑ12VKRμ′0+g23(1tTW-β6)Zμ′0],gx3dR¯γμdRBμ=dL¯γμdL[gx3(-13TW+β2)Zμ′0-gSW3Aμ+gx3(1tTW2CW+β1)Zμ0+gx3(-13TW+β2)Zμ′0].(4.44)dL¯γμdLAμ∝-13gSW,dR¯γμdRAμ∝-13gSW,dL¯γμdLKRμ′0∝0≡gk′Ld,dR¯γμdRKRμ′0∝0≡gk′Rd,dL¯γμdLZμ0∝-16g(3+TW2)CW+ζ5≡gzLd,dR¯γμdRZμ0∝g3TW2CW+ζ7≡gzRd,dL¯γμdLZμ′0∝163t(3g+2tgx)TW+ζ6≡gz′Ld,uR¯γμuRZμ′0∝-133gxTW+ζ8≡gz′Rd,
where
(4.45)ζ5=gϑ2224V2CW5(3-2SW2),ζ6=(-1+3CW2+6CW4-8CW6)243V2CW5SW3,ζ7=-gSW2ϑ2212V2CW5,ζ8=-gxSW3ϑ2243t2V2CW5.
Then by (4.10) for f=d, we obtain the following effective Lagrangian for NC:
(4.46)Lquark,dNC≈-(gz′VdGνz′Mz′2+gzVdGνzMz2+gk′VdGνk′Mk′2)ndν¯lLγ0νlL≈-{[(3SW2-2SW4)24V2CW4+(9-4t4)144t4V2TW4+(12ϑ22-ϑ122V2ϑ22)(-12+13SW2)]L+[-SW224V2CW4+13(12ϑ22-ϑ122V2ϑ22)SW2]R}ndν¯lLγ0νlL,
and the effective potential felt by neutrinos when crossing a medium composed by a density nd of down quarks is VNCd=VNCdL+VNCdR, where
(4.47)VNCdL≈[(3SW2-2SW4)24V2CW4+(9-4t4)144t4V2TW4+(12ϑ22-ϑ122V2ϑ22)(-12+13SW2)]nd,(4.48)VNCdR≈[-SW224V2CW4+13(12ϑ22-ϑ122V2ϑ22)SW2]nd.
Then we can easily see that in SM the NC effective potential for neutrinos in a d-quark medium, using result of (4.7), will be given by
(4.49)VNCd=VNCdL+VNCdR≈-(12ϑ22-ϑ122V2ϑ22)(12-23SW2)nd,VNCdL=(12ϑ22-ϑ122V2ϑ22)(-12+13SW2)nd,(4.50)VNCdR=13(12ϑ22-ϑ122V2ϑ22)SW2nd.
Then from (4.47)–(4.50), we obtain
(4.51)VNCdL≈VNCdL+[(3SW2-2SW4)24V2CW4+(9-4t4)144t4V2TW4]nd,VNCdR≈VNCdR-SW224V2CW4nd,
and neglecting terms of order (ϑi/V)n, for n>2, we obtain that ɛℓℓd=ɛℓℓdL+ɛℓℓdR, where
(4.52)ɛlldL≈ϑ2224V2CW4(3-2SW2),(4.53)ɛlldR≈-SW2ϑ2212V2CW4.
Then we obtain ɛℓℓdL≈(ϑ22/8V2CW4)+ɛℓℓdR. Note that again in limit V→∞ we recover the SM.


## 4.2.1. Leptons Sector

From (3.31) and (3.24), we obtain that for the known neutral leptons
(4.11)gx3ν¯lLγμνlLBμ=ν¯lLγμνlL[-g3SWAμ+(g3TW2CW+gx3β1)Zμ0-gx3(13TW-β2)Zμ′0],(4.12)g2ν¯lLγμνlLW3μ=ν¯lLγμνlL[g2SWAμ-gϑ12VKRμ′0+g(CW+β3)2Zμ0+gβ42Zμ′0],(4.13)-g23ν¯lLγμνlLW8μ=ν¯lLγμνlL[-g6SWAμ-gϑ12VKRμ′0+(g6SW2CW-gβ523)Zμ0+g23(1tTW-β6)Zμ′0].
By (4.11), (4.12), and (4.13), we obtain that vertex interactions with neutrinos can be written as
(4.14)ν¯lLγμνlLAμ∝0,(4.15)ν¯lLγμνlLKRμ′0∝-gϑ1V≡GνK′,(4.16)ν¯lLγμνlLZμ0∝12gCW-1+η1≡GνZ,(4.17)ν¯lLγμνlLZμ′0∝(3g-2gxt63t)TW+η2≡GνZ′,
where
(4.18)η1=-4gtCW2ϑ12+gx(1-2SW2)ϑ228tV2CW5,η2=gt(1-4CW2)ϑ1223V2CWSW-(-gt3+2gt3CW2+8gt3CW4+6gxSW4)ϑ22243t2V2CW5SW.
We note from (4.14) that neutrinos do not interact electrically, as expected. For charged leptons, from (3.31) and (3.24), we obtain
(4.19)gx3lL¯γμlLBμ=lL¯γμlL[gx3(13TW-β2)Zμ′0-g3SWAμ+(g3TW2CW+gx3β1)Zμ0-gx3(13TW-β2)Zμ′0],-g2l¯LγμlLW3μ=l¯LγμlL[-g2SWAμ+gϑ12VKRμ′0-g(CW+β3)2Zμ0-gβ42Zμ′0],-g23l¯LγμlLW8μ=l¯LγμlL[-g6SWAμ-gϑ12VKRμ′0+(g6SW2CW-gβ523)Zμ0+g23(1tTW-β6)Zμ′0],gxlR¯γμlRBμ=lR¯γμlR[gx(13TW-β2)Zμ′0-gSWAμ+(gTW2CW+gxβ1)Zμ0-gx(13TW-β2)Zμ′0],
and therefore
(4.20)l¯γμlAμ∝-gSW,(4.21)lL¯γμlLKRμ′0∝0≡gk′Ll=gk′Rl,(4.22)lL¯γμlLZμ0∝12g(-1+TW2)CW+η3≡gzLl,lR¯γμlRZμ0∝gTW2CW+η5≡gzRl,lL¯γμlLZμ′0∝163t(3g-2tgx)TW+η4≡gz′Ll,(4.23)lR¯γμlRZμ′0∝-gx3TW+η6≡gz′Rl,
where
(4.24)η3=(-1+2CW2)gxϑ228tV2CW5,η4=(gt3(1+2CW2)2-12gt3SW2CW2-6gxSW4)243t2V2CW5SW,η5=-gxϑ224tV2CW3TW2,η6=-3gxϑ224t2V2CW2TW3,
and, again, t=gx/g. We note that by (4.20) we can make the association gSW=|e|. Then for f=e, (4.15)–(4.17) and (4.21)–(4.23) lead to
(4.25)Leff-eNC≈-∑P=L,R12(gz′PeGνz′Mz′2+gzPeGνzMz2+gk′PeGνk′Mk′2)neν¯lLγ0νlL≈-{[TW4144t2gx2V2(3g-2tgx)2+TW28V2(1-TW2)+12(12ϑ22-ϑ122V2ϑ22)(1-2CW2)]L+[TW4(2tgx-3g)24tgxV2-TW44V2+(12ϑ22-ϑ122V2ϑ22)SW2]R}neν¯lLγ0νlL.
Since intermediate neutral bosons in (4.9) do not distinguish between different lepton flavors, the interaction through NC with electron is described by the following effective potential:
(4.26)VNCe=VNCμ=VNCτ=VNCl,=VNClL+VNClR,
where
(4.27)VNClL=[TW4144t2gx2V2(3g-2tgx)2+TW28V2(1-TW2)+12(12ϑ22-ϑ12V2ϑ22)(1-2CW2)]ne,VNClR=[TW4(2tgx-3g)24tgxV2-TW44V2+(12ϑ22-ϑ122V2ϑ22)SW2]ne,
and index ℓ refers to neutrino flavor. We note that the potential through CC comes from interactions of electron neutrinos with left-handed electrons, while the effective potential through NC comes from left- and right-handed electrons.Considering both NC and CC, we can write the effective potential felt by neutrinos asVℓ=VℓL+VℓR, where
(4.28)VlL=(12ϑ22-ϑ122V2ϑ22)δelne+VNClL,VlR=VNClR.
Comparing with SM expression for such potential:
(4.29)VNCl=-2GF(12-2SW2)ne,VCCe=2GFne,
we can find that
(4.30)VlL=VlL+[TW4144t2gx2V2(3g-2tgx)2+TW28V2(1-TW2)]ne,VlR=VNClR+[TW4(2tgx-3g)24tgxV2-TW44V2]ne,
where we adopt in what follow, the convention that V denotes SM-like part of the model; thus, the new terms beyond SM [] can be associated with the parameters ɛ′s in NSI [63]. So, in the approximation (ϑi/V)n≈0, for n>2, we obtain
(4.31)ɛlleL≈(1-2SW2)ϑ228V2CW4,(4.32)ɛlleR≈-SW2(1+2SW2)ϑ224V2CW4.
We note that on limit V→∞, we recover SM. The NSIs are a subleading interaction, as expected. By (4.31) and (4.32), we obtain ɛℓℓeR≈-2SW2ɛℓℓeL-(ϑ22/V2)TW4.


## 4.2.2. Quarks Sector

For the quarks of the first family, the Lagrangian density in (3.33) describes the interactions with gauge bosons W3μ,W8μ, and Bμ; then, by (3.24) and (3.25) we obtain the following interactions for up quarks:
(4.33)-gx3uL¯γμuLBμ=uL¯γμuL[gx3(13TW-β2)Zμ′0g3SWAμ-gx3(1tTW2CW+β1)Zμ0+gx3(13TW-β2)Zμ′0],g2uL¯γμuLW3μ=uL¯γμuL[g2SWAμ-gϑ12VKRμ′0+g(CW+β3)2Zμ0+gβ42Zμ′0],-g23uL¯γμuLW8μ=uL¯γμuL[-g6SWAμ-gϑ12VKRμ′0+g23(13TWSW-β5)Zμ0+g23(1tTW-β6)Zμ′0],-2gx3uR¯γμuRBμ=uR¯γμuR[2g3SWAμ-2gx3(1tTW2CW+β1)Zμ0+2gx3(1tTW-β6)Zμ′0].
The couplings quark-quark-boson for the first family are given by
(4.34)uL¯γμuLAμ∝23gSW,(4.35)uR¯γμuRAμ∝23gSW,(4.36)uL¯γμuLKRμ′0∝-gϑ1V≡gk′Lu,uR¯γμuRKRμ′0∝0≡gk′Ru,uL¯γμuLZμ0∝16g(3-TW2)CW+ζ1≡gzLu,uR¯γμuRZμ0∝-23gTW2CW+ζ3≡gzRu,uL¯γμuLZμ′0∝163t(3g+2tgx)TW≡gz′Lu,uR¯γμuRZμ′0∝233gxTW+ζ4≡gz′Ru,
where
(4.37)ζ1=gx(-12CW4ϑ12+(1+2CW2)ϑ22)24tV2CW5,ζ2=12gt3CW4(1-4CW2)ϑ12+(gt3(1-2CW2-8CW4)+6gxSW4)ϑ22243t2V2CW5SW,ζ3=g6SW2ϑ22CW5V2,ζ4=gxSW3ϑ2223t2V2CW5.
We note that (4.34) and (4.35) reflect the fact that quarks interact electrically through photons with coupling constant QfsinθW, as in SM. The effective Lagrangian at low energies for neutrino interaction with quarks up through neutral currents are given by (4.10) with f=u:
(4.38)Lquark,uNC≈-12∑P=L,R(gz′PuGνz′Mz′2+gzPuGνzMz2+gk′PuGνk′Mk′2)nuν¯lLγ0νlL≈-{[124V2(3+TW4)+TW4144t4V2(9-4t4)+(12ϑ22-ϑ122V2ϑ22)(12-23SW2)-ϑ124V2ϑ22]L+[TW46V2+TW4(3g-2tgx)36tgxV2-23(12ϑ22-ϑ122V2ϑ22)SW2]R}nuν¯lLγ0νlL,
where nu is the up quarks average density.SM predictions, using result of (4.7), can be written as
(4.39)VNCu=VNCuL+VNCuR=(12ϑ22-ϑ122V2ϑ22)(12-43SW2)nu,
where
(4.40)VNCuL=(12ϑ22-ϑ122V2ϑ22)(12-23SW2)nu,VNCuR=-23(12ϑ22-ϑ122V2ϑ22)SW2nu.
By comparison, we obtain
(4.41)VNCuL≈VNCuL+[124V2(3+TW4)+TW4144t4V2(9-4t4)-ϑ124V2ϑ22]nu,VNCuR≈VNCuR+[TW46V2+TW4(3g-2tgx)36tgxV2]nu.
Then we can say that ɛℓℓu=ɛℓℓuL+ɛℓℓuR, where
(4.42)ɛlluL≈-ϑ122V2+ϑ2224V2CW4(9-8SW2),ɛlluR≈ϑ226V2SW2CW4.
Again, we obtain universal NSI, as for the electrons. We note that ɛℓℓuL=-(ϑ12/2V2)+(3ϑ22/8V2CW4)-2ɛℓℓuR and in the limit V→∞ we recover SM.For down quarks by (3.33) and (3.24), we obtain that
(4.43)-gx3dL¯γμdLBμ=dL¯γμdL[gx3(13TW-β2)Zμ′0g3SWAμ-gx3(1tTW2CW+β1)Zμ0+gx3(13TW-β2)Zμ′0],-g2dL¯γμdLW3μ=dL¯γμdL[-gSW2Aμ+gϑ12VKRμ′0-g(CW+β3)2Zμ0-gβ42Zμ′0],-g23dL¯γμdLW8μ=dL¯γμdL[-gSW6Aμ+g23(13TWSW-β5)Zμ0+-gϑ12VKRμ′0+g23(1tTW-β6)Zμ′0],gx3dR¯γμdRBμ=dL¯γμdL[gx3(-13TW+β2)Zμ′0-gSW3Aμ+gx3(1tTW2CW+β1)Zμ0+gx3(-13TW+β2)Zμ′0].(4.44)dL¯γμdLAμ∝-13gSW,dR¯γμdRAμ∝-13gSW,dL¯γμdLKRμ′0∝0≡gk′Ld,dR¯γμdRKRμ′0∝0≡gk′Rd,dL¯γμdLZμ0∝-16g(3+TW2)CW+ζ5≡gzLd,dR¯γμdRZμ0∝g3TW2CW+ζ7≡gzRd,dL¯γμdLZμ′0∝163t(3g+2tgx)TW+ζ6≡gz′Ld,uR¯γμuRZμ′0∝-133gxTW+ζ8≡gz′Rd,
where
(4.45)ζ5=gϑ2224V2CW5(3-2SW2),ζ6=(-1+3CW2+6CW4-8CW6)243V2CW5SW3,ζ7=-gSW2ϑ2212V2CW5,ζ8=-gxSW3ϑ2243t2V2CW5.
Then by (4.10) for f=d, we obtain the following effective Lagrangian for NC:
(4.46)Lquark,dNC≈-(gz′VdGνz′Mz′2+gzVdGνzMz2+gk′VdGνk′Mk′2)ndν¯lLγ0νlL≈-{[(3SW2-2SW4)24V2CW4+(9-4t4)144t4V2TW4+(12ϑ22-ϑ122V2ϑ22)(-12+13SW2)]L+[-SW224V2CW4+13(12ϑ22-ϑ122V2ϑ22)SW2]R}ndν¯lLγ0νlL,
and the effective potential felt by neutrinos when crossing a medium composed by a density nd of down quarks is VNCd=VNCdL+VNCdR, where
(4.47)VNCdL≈[(3SW2-2SW4)24V2CW4+(9-4t4)144t4V2TW4+(12ϑ22-ϑ122V2ϑ22)(-12+13SW2)]nd,(4.48)VNCdR≈[-SW224V2CW4+13(12ϑ22-ϑ122V2ϑ22)SW2]nd.
Then we can easily see that in SM the NC effective potential for neutrinos in a d-quark medium, using result of (4.7), will be given by
(4.49)VNCd=VNCdL+VNCdR≈-(12ϑ22-ϑ122V2ϑ22)(12-23SW2)nd,VNCdL=(12ϑ22-ϑ122V2ϑ22)(-12+13SW2)nd,(4.50)VNCdR=13(12ϑ22-ϑ122V2ϑ22)SW2nd.
Then from (4.47)–(4.50), we obtain
(4.51)VNCdL≈VNCdL+[(3SW2-2SW4)24V2CW4+(9-4t4)144t4V2TW4]nd,VNCdR≈VNCdR-SW224V2CW4nd,
and neglecting terms of order (ϑi/V)n, for n>2, we obtain that ɛℓℓd=ɛℓℓdL+ɛℓℓdR, where
(4.52)ɛlldL≈ϑ2224V2CW4(3-2SW2),(4.53)ɛlldR≈-SW2ϑ2212V2CW4.
Then we obtain ɛℓℓdL≈(ϑ22/8V2CW4)+ɛℓℓdR. Note that again in limit V→∞ we recover the SM.


## 5. Results

In last sections we saw that in 331 model we chose, all NSI parameters are universal and diagonal and will not affect oscillation experiments. However, measurements of cross-section will be sensitive to such parameters, through modifications ongiα [51]. We will now compare our results with those obtained in cross-section measurements. We will assume sin2θW=0.23149(13).In Table1 we can see that constrains in ɛℓℓeP lead to V2>4.7ϑ22., while the constrains in ɛℓℓuR lead to V2>21.7ϑ22, and the constrains in ɛℓℓdP (|ɛμμdL|<0.003) lead to V2>60ϑ22. If ϑ2 has its maximum value of 174.105 GeV, then V≳1.3 TeV. We note also that by |ɛμμuL|<0.003 we obtain |ϑ22-ϑ12|<0.006V2; then, for V~1.3 TeV and ϑ2=174 GeV, we obtain 142 GeV<ϑ1<201 GeV. We therefore cannot predict any hierarchy to the VEV’s ϑ1 and ϑ2. Based on those results, we obtain the following inferior limits for the new gauge bosons masses:
(5.1)MKI=MZ′>610GeV,MK′>613GeV,MKR>740GeV.Table 1
Values for NSI in 331 model and experimental limits taken of the strongest constraints on these parameters are given in [18, 19, 21, 22].
331 ModelExp. 90% C.L.-0.14<ɛeeeL<0.09ɛℓℓeL≈(1-2SW2)ϑ228V2CW40.114(ϑ22V2)-0.033<ɛμμeL<0.055-0.6<ɛττeL<0.4-0.03<ɛeeeR<0.18ɛℓℓeR≈-2SW2ɛℓℓeL-ϑ22V2TW4-0.143(ϑ22V2)-0.040<ɛμμeR<0.053-0.4<ɛττeR<0.6-1<ɛeeuL<0.3ɛℓℓuL≈-ϑ122V2+ϑ2224V2CW4(9-8SW2)0.50(ϑ22-ϑ12V2)|ɛμμuL|<0.003|ɛττuL|<1.4-0.4<ɛeeuR<0.7ɛℓℓuR≈ϑ226V2SW2CW40.065(ϑ22V2)-0.008<ɛμμuR<0.003|ɛττuR|<3-0.3<ɛeedL<0.3ɛℓℓdL≈ϑ2224V2CW4(3-2SW2)0.179(ϑ22V2)|ɛμμdL|<0.003|ɛττdL|<1.1-0.6<ɛeedR<0.5ɛℓℓdR≈-SW2ϑ2212V2CW4-0.033(ϑ22V2)-0.008<ɛμμdR<0.015|ɛττdR|<6


## 6. Conclusion

We presented in this work a procedure to show that models with extended gauge symmetriesSU(3)C×SU(3)L×U(1)X can lead to neutrino nonstandard interactions, respecting the Standard Model Gauge symmetry SU(3)C×SU(2)L×U(1)Y, without spoiling the available experimental data and reproducing the known phenomenology at low energies. We also have shown that with an assumption about a mass hierarchy for the Higgs triplets VEV’s we could qualitatively address the mass hierarchy problem in standard model. Finally we obtained limits for the triplets VEV’s based on limits for NSI in cross-section experiments.We believe that the class of model presented here is an interesting theoretical possibility to look for new physics beyond SM. We restrained our work to a simple scenario, but flavor-changing interactions can be naturally introduced in the model, leading to new constraints on NSI.



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*Source: 763829-2012-02-22.xml*