Phenomenology of the 3-3-1-1 model

# Phenomenology of the 3-3-1-1 model

P. V. Dong Institute of Physics, Vietnam Academy of Science and Technology, 10 Dao Tan, Ba Dinh, Hanoi, Vietnam    D. T. Huong Institute of Physics, Vietnam Academy of Science and Technology, 10 Dao Tan, Ba Dinh, Hanoi, Vietnam    Farinaldo S. Queiroz Department of Physics and Santa Cruz Institute for Particle Physics, University of California, Santa Cruz, CA 95064, USA    N. T. Thuy Department of Physics and IPAP, Yonsei University, Seoul 120-479, Korea
July 12, 2019
###### Abstract

In this work we discuss a new (3-3-1-1) gauge model that overhauls the theoretical and phenomenological aspects of the known 3-3-1 models. Additionally, we sift the outcome of the 3-3-1-1 model from precise electroweak bounds to dark matter observables. We firstly advocate that if the number is conserved as the electric charge, the extension of the standard model gauge symmetry to the 3-3-1-1 one provides a minimal, self-contained framework that unifies all the weak, electromagnetic and interactions, apart from the strong interaction. The -parity (similar to the -parity) arises as a remnant subgroup of the broken 3-3-1-1 symmetry. The mass spectra of the scalar and gauge sectors are diagonalized when the scale of the 3-3-1-1 breaking is compatible to that of the ordinary 3-3-1 breaking. All the interactions of the gauge bosons with the fermions and scalars are obtained. The standard model Higgs () and gauge () bosons are realized at the weak scales with consistent masses despite of their mixings with the heavier particles, respectively. The 3-3-1-1 model provides two dark matters which are stabilized by the -parity conservation: one fermion which may be either a Majorana or Dirac fermion and one complex scalar. We conclude that in the fermion dark matter setup the gauge boson resonance sets the dark matter observables, whereas in the scalar one the Higgs portal dictates them. The standard model GIM mechanism works in the model because of the -parity conservation. Hence, the dangerous flavor changing neutral currents due to the ordinary and exotic quark mixing are suppressed, while those coming from the non-universal couplings of the and gauge bosons are easily evaded. Indeed, the and mixings limit  TeV and  TeV, respectively, while the LEPII searches provide a quite close bound . The violation of the CKM unitarity due to the loop effects of the and gauge bosons is negligible.

###### pacs:
12.10.-g, 12.60.Cn, 12.60.Fr

## I Introduction

The standard model pdg () has been extremely successful. However, it describes only about 5% mass-energy density of our universe. There remain around 25% dark matter and 70% dark energy that are referred as the physics beyond the standard model. In addition, the standard model cannot explain the nonzero small masses and mixing of the neutrinos, the matter-antimatter asymmetry of the universe, and the inflationary expansion of the early universe. On the theoretical side, the standard model cannot show how the Higgs mass is stabilized against radiative corrections, what makes the electric charges exist in discrete amounts, and why there are only the three generations of fermions observed in the nature.

Among the standard model’s extensions for the issues, the recently-proposed (3-3-1-1) gauge model has interesting features 3311 (). (i) The theory arises as a necessary consequence of the 3-3-1 models 331m (); 331r (); dongfla () that respects the conservation of lepton and baryon numbers. (ii) The number is naturally gauged because it is a combination of the and charges. And, the resulting theory yields an unification of the electroweak and interactions, apart from the strong interaction. (iii) The right-handed neutrinos are emerged as fundamental fermion constituents, and consequently the small masses of the active neutrinos are generated by the type I seesaw mechanism. (iv) The -parity which has the form similarly to the -parity in supersymmetry is naturally resulted as a conserved remnant subgroup of the broken 3-3-1-1 gauge symmetry. (v) The dark matter automatically exists in the model that is stabilized due to the -parity. It is the lightest particle among the new particles that characteristically have wrong lepton numbers transforming as odd fields under the -parity (so-called -particles). The dark matter candidate may be a neutral fermion () or a neutral complex scalar ().

The 3-3-1-1 model includes all the good features of the 3-3-1 models. Namely, the number of fermion families is just three as a consequence of anomaly cancelation and QCD asymptotic freedom condition anoma (). The third quark generation transforms under differently from the first two. This explains why the top quark is uncharacteristically-heavy tquark (). The strong problem is solved by just its particle content with an appropriate Peccei-Quinn symmetry palp (). The electric charge quantization is due to a special structure of the gauge symmetry and fermion content ecq (). Additionally, it also provides the mentioned dark matter candidates similarly to farinaldoDM1 (); farinaldoDM2 (). The 3-3-1-1 model can solve the potential issues of the 3-3-1 models because the unwanted interactions and vacuums that lead to the dangerous tree-level flavor changing neutral currents (FCNCs) ponce () as well as the violation cpt () are all suppressed due to the -parity conservation 3311 ().

In the previous work 3311 (), the proposal of the 3-3-1-1 model with its direct consequence—the dark matter has been given. In the current work, we will deliver a detailed study of this new model. Particularly, we consider the new physics consequences besides the dark matter that are implied by the new extended sectors beyond those of the 3-3-1 model. These sectors include the new neutral gauge boson () as associated with and the new scalar () as required for the totally breaking with necessary mass generations. The totally breaking that consequently breaks the symmetry, where the is a residual charge related to the charge and a generator, can happen closely to the 3-3-1 breaking scale of TeV order. This leads to a finite mixing and interesting interplay between the new neutral gauge bosons such as the of the 3-3-1 model and the of . Notice that our previous work considers only a special case when the breaking scale is very high like the GUT one gutscale () as an example so that the new physics over the ordinary 3-3-1 symmetry is decoupled, which has neglected its imprint at the low energy 3311 (). Indeed, the stability of the proton is already ensured by the 3-3-1-1 gauge symmetry, there is no reason why that scale is not presented at the 3-3-1 scale. Similarly to the new neutral gauge bosons, there is an interesting mixing among the new neutral scalars that are used to break the above symmetry kinds, the 3-3-1 and the .

It is interesting to note that the new scalars and new gauge bosons as well as the new fermions can give significant contributions to the production and decay of the standard model Higgs boson. They might also modify the well-measured standard model couplings such as those of the photon, and bosons with the fermions. There exist the hadronic FCNCs due to the contribution of the new neutral gauge bosons. These gauge bosons can also take part in the electron-positron collisions such as the LEPII and ILC as well as in the dark matter observables. The presence of the new neutral gauge bosons also induces the apparent violation of the CKM unitarity. In some case, the new scalar responsible for the breaking may act as an inflaton. The decays of some new particles can solve the matter-antimatter asymmetry via leptogenesis mechanisms.

The scope of this work is given as follows. The 3-3-1-1 model will be calculated in detail. Namely, the scalar potential and the gauge boson sector are in a general case diagonalized. All the interactions of the gauge bosons with the fermions as well as with the scalars are derived. The new physics processes through the FCNCs, the LEPII collider, the violation of the CKM unitarity as well as the dark matter observables are analyzed. Particularly, we will perform a phenomenological study of the dark matter taking into account the current data as well as the new contributions of the physics at that have been kept in 3311 (). The constraints on the new gauge boson and dark matter masses are also obtained.

The rest of this work is organized as follows. In Sec. II, we give a review of the model. Secs. III and IV are respectively devoted to the scalar and gauge sectors. In Sec. V we obtain all the gauge interactions of the fermions and scalars. Sec. VI is aimed at studying the new physics processes and constraints. Finally, we summarize our results and make concluding remarks in Sec. VII.

## Ii A review of the 3-3-1-1 model

The 3-3-1-1 model 3311 () is based on the gauge symmetry,

 SU(3)C⊗SU(3)L⊗U(1)X⊗U(1)N, (1)

where the first three groups are the ordinary gauge symmetry of the 3-3-1 models 331r (); 331m (); dongfla (), while the last one is a necessary gauge extension of the 3-3-1 models that respects the conservation of lepton () and baryon () numbers. Indeed, the 3-3-1 symmetry and symmetry do not commute and also nonclose algebraically. To be concrete, for a lepton triplet (see below), we have , which is not commuted with the generators as for . It is easily checked that

 [B−L,T4±iT5]=∓(T4±iT5)≠0, [B−L,T6±iT7]=∓(T6±iT7)≠0.

The non-closed algebras can be deduced from the fact that in order for to be some generator of , we have a linear combination () and thus , which is invalid for the lepton triplet, , even for other particle multiplets. In other words, and by themselves do not make a symmetry under which our theory based on is manifest. Therefore, to have a closed algebra, we must introduce at least a new Abelian charge so that is a residual symmetry of closed group , i.e. , where the embedding coefficients are given below (the existence of can also be understood by a current algebra approach for and similarly to the case of hyper-charge when we combine with to perform the electroweak symmetry). Note that cannot be identified as (that defines the electric charge operator) because they generally differ for the particle multiplets (see below); thus they are independent charges. As a fact, the normal Lagrangian of the 3-3-1 models (including the gauge interactions, minimal Yukawa Lagrangian and minimal scalar potential) always preserves a Abelian symmetry that along with realizes as a conserved (non-commuting) residual charge, which has actually been investigated in the literature and given in terms of and where is 3-3-1 model-class dependent and 3311 (); lepto331 (). Note also that a violation in due to some unwanted interaction, by contrast, would lead to the corresponding violation in and vice versa. Because are gauged charges, and must be gauged charges (by contrast, are global which is incorrect). The gauging of is a consequence of the non-commuting between and (which is unlike the standard model case). And, the theory is only consistent if it includes as a gauge symmetry which also necessarily makes the resulting theory free from all the nontrivial leptonic and baryonic anomalies 3311 (). Otherwise, the 3-3-1 models must contain (abnormal) interactions that explicitly violate (or ). Equivalently, the 3-3-1 models are only survival if is not a symmetry of such theories, actually recognized as an approximate symmetry, which has explicitly shown in dongdongdm (). To conclude, assuming that the charge is conserved (that is respected by the experiments, the standard model, even the typical 3-3-1 models pdg (); 331m (); 331r (); dongfla ()), the Abelian factor must be included so that the algebras are closed that is needed for a self-consistent theory. Apart from the strong interaction with group, the framework thus presents an unification of the electroweak and interactions, in the same manner of the standard model electroweak theory for the weak and electromagnetic ones.

The two Abelian factors of the 3-3-1-1 symmetry associated with the group correspondingly determine the electric charge and operators as residual symmetries, given by

 Q=T3−1√3T8+X,B−L=−2√3T8+N, (2)

where , and are the charges of , and , respectively (the charges will be denoted by ). Note that the above and definitions embed the 3-3-1 model with neutral fermions dongfla () in the considering theory. However, the coefficients of might be different depending on which class of the 3-3-1 models is embedded in lepto331 ().

The is conserved responsible for the electromagnetic interaction, whereas the must be broken so that the gauge boson gets a large enough mass to escape from the detectors. Indeed, the is broken down to a parity (i.e., a symmetry),

 P=(−1)3(B−L)+2s=(−1)−2√3T8+3N+2s, (3)

which consequently makes “wrong particles” become stabilized, providing dark matter candidates 3311 (). We see that this -parity has an origin as a residual symmetry of the broken gauge symmetry, which is unlike the -symmetry in supersymmetry susy (). That being said, the parity is automatically existed, and due to its nature it will play an important role in the model besides stabilizing the dark matter candidates as shown throughout the text.

The fermion content of the 3-3-1-1 model that is anomaly free is given as 3311 ()

 ψaL = ⎛⎜⎝νaLeaL(NaR)c⎞⎟⎠∼(1,3,−1/3,−2/3), (4) νaR ∼ (1,1,0,−1),eaR∼(1,1,−1,−1), (5) QαL = (6) uaR ∼ (3,1,2/3,1/3),daR∼(3,1,−1/3,1/3), (7) UR ∼ (3,1,2/3,4/3),DαR∼(3,1,−1/3,−2/3), (8)

where the quantum numbers located in the parentheses are defined upon the gauge symmetries , respectively. The family indices are and .

The exotic fermions , and have been included to complete the fundamental representations of the group, respectively. By the embedding, their electric charges take usual values, , and . However, their charges get values, , and , which are abnormal in comparison to those of the standard model particles. These exotic fermions including the following bosons of this kind have ordinary baryon numbers, however, possessing anomalous lepton numbers as well as being odd under the parity (see Table 1 in more detail) 3311 (). Such particles are generally called as the wrong-lepton particles (or -particles for short) and the parity is thus named as the -parity. Whereas, all other particles of the model including the standard model ones (which have both the ordinary baryon and ordinary lepton numbers or only differing from the ordinary lepton number by an even lepton number as just the scalar given below) are even under the -parity, and they can be considered as ordinary particles.

Let us remind that the neutral fermions might have left-handed counterparts, , transforming as singlets under any gauge symmetry group including the . By this view, the are truly sterile which is unlike the as usually considered in the literature. Interestingly, the sterile fermions are -particles like the . If the are not included, the are Majorana fermions. Otherwise, the presence of the yields as generic fermions (which may be Dirac ones). Further, we will exploit this matter by deriving the dark matter observables for the cases of the Dirac or Majorana fermions.

To break the gauge symmetry and generate the masses for the particles in a correct way, the 3-3-1-1 model needs the following scalar multiplets 3311 ():

 η = ⎛⎜ ⎜⎝η01η−2η03⎞⎟ ⎟⎠∼(1,3,−1/3,1/3),ρ=⎛⎜ ⎜⎝ρ+1ρ02ρ+3⎞⎟ ⎟⎠∼(1,3,2/3,1/3), χ = ⎛⎜ ⎜⎝χ01χ−2χ03⎞⎟ ⎟⎠∼(1,3,−1/3,−2/3),ϕ∼(1,1,0,2), (9)

with the VEVs that conserve and being respectively given by

 (10)

The VEVs of break only to , which leaves the invariant. The breaks as well as the that defines the -parity, , with the form as given 3311 (). It provides also the mass for the gauge boson as well as the Majorana masses for . Note that the , and are the -particles, while the others including are not (i.e., as the ordinary particles). The electrically-neutral fields and cannot develop a VEV due to the -parity conservation. To keep a consistency with the standard model, we suppose .

Up to the gauge fixing and ghost terms, the Lagrangian of the 3-3-1-1 model is given by

 L = ∑fermion multiplets¯ΨiγμDμΨ+∑scalar multiplets(DμΦ)†(DμΦ) (11) −14GiμνGμνi−14AiμνAμνi−14BμνBμν−14CμνCμν −V(ρ,η,χ,ϕ)+LYukawa,

with the covariant derivative

 Dμ=∂μ+igstiGiμ+igTiAiμ+igXXBμ+igNNCμ, (12)

and the field strength tensors

 Giμν = ∂μGiν−∂νGiμ−gsfijkGjμGkν, Aiμν = ∂μAiν−∂νAiμ−gfijkAjμAkν, Bμν = ∂μBν−∂νBμ,Cμν=∂μCν−∂νCμ. (13)

The denotes fermion multiplets such as , , and so on, whereas the stands for scalar multiplets, , , and . The coupling constants () and the gauge bosons () are defined as coupled to the generators (), respectively. It is noted that in a mass basis the bosons are associated with , the photon is with , and the , are with generators that are orthogonal to . All these fields including the and gluons are even under the -parity. However, the new non-Hermitian gauge bosons, as coupled to and as coupled to , are the -particles.

The scalar potential and Yukawa Lagrangian as mentioned above are obtained as follows 3311 ()

 LYukawa = heab¯ψaLρebR+hνab¯ψaLηνbR+h′νab¯νcaRνbRϕ+hU¯Q3LχUR+hDαβ¯QαLχ∗DβR (14) +hua¯Q3LηuaR+hda¯Q3LρdaR+hdαa¯QαLη∗daR+huαa¯QαLρ∗uaR+H.c., V(ρ,η,χ,ϕ) = μ21ρ†ρ+μ22χ†χ+μ23η†η+λ1(ρ†ρ)2+λ2(χ†χ)2+λ3(η†η)2 (15) +λ4(ρ†ρ)(χ†χ)+λ5(ρ†ρ)(η†η)+λ6(χ†χ)(η†η) +λ7(ρ†χ)(χ†ρ)+λ8(ρ†η)(η†ρ)+λ9(χ†η)(η†χ)+(fϵmnpηmρnχp+H.c.) +μ2ϕ†ϕ+λ(ϕ†ϕ)2+λ10(ϕ†ϕ)(ρ†ρ)+λ11(ϕ†ϕ)(χ†χ)+λ12(ϕ†ϕ)(η†η).

Because of the 3-3-1-1 gauge symmetry, the Yukawa Lagrangian and scalar potential as given take the standard forms that contain no lepton-number violating interactions.

If such violating interactions as well as nonzero VEVs of and were presented as in the 3-3-1 model, they would be the sources for the hadronic FCNCs at tree level ponce (). The FCNC problem is partially solved by the 3-3-1-1 symmetry and -parity conservation. Also, the presence of the and VEVs would imply a mass hierarchy between the real and imaginary components of the gauge boson due to their different mixings with the neutral gauge bosons. This leads to the violation that is experimentally unacceptable cpt (). The violation encountered with the 3-3-1 model is thus solved by the 3-3-1-1 symmetry and -parity conservation too.

Table 1 lists all the model particles with their parity values explicitly provided. The lepton numbers have also been included for a convenience in reading. However, the baryon numbers were not listed since they can be obtained as usual (all the quarks , , and have , whereas the other particles have ).

As shown in 3311 (), the gauge boson cannot be a dark matter. However, the neutral fermion (a combination of fields) or the neutral complex scalar (a combination of and fields) can be dark matter whatever one of them is the lightest wrong-lepton particle (LWP) in agreement with farinaldoDM2 ().

The fermion masses that are obtained from the Yukawa Lagrangian after the gauge symmetry breaking have been presented in 3311 () in detail. Below, we will calculate the masses and physical states of the scalar and gauge boson sectors when the scale of the breaking is comparable to the scale of the 3-3-1 breaking, which has been neglected in 3311 (). Also, all the gauge interactions of fermions and scalars as well as the constraints on the new physics are derived. We stress again that in the regime the and 3-3-1 symmetries decouple; whereas, when those scales become comparable, the new physics associated with the and that of the 3-3-1 model are correlated, possibly happening at the TeV scale, to be all proved by the LHC or the ILC project.

## Iii Scalar sector

Since the -parity is conserved, only the neutral scalar fields that are even under this parity symmetry can develop the VEVs as given in (10). We expand the fields around these VEVs as

 η=⟨η⟩+η′=⎛⎜ ⎜⎝u√200⎞⎟ ⎟⎠+⎛⎜ ⎜ ⎜ ⎜ ⎜⎝S1+iA1√2η−2S′3+iA′3√2⎞⎟ ⎟ ⎟ ⎟ ⎟⎠,ρ=⟨ρ⟩+ρ′=⎛⎜ ⎜⎝0v√20⎞⎟ ⎟⎠+⎛⎜ ⎜ ⎜⎝ρ+1S2+iA2√2ρ+3⎞⎟ ⎟ ⎟⎠, (16)
 χ=⟨χ⟩+χ′=⎛⎜ ⎜⎝00ω√2⎞⎟ ⎟⎠+⎛⎜ ⎜ ⎜ ⎜ ⎜⎝S′1+iA′1√2χ−2S3+iA3√2⎞⎟ ⎟ ⎟ ⎟ ⎟⎠,ϕ=⟨ϕ⟩+ϕ′=Λ√2+S4+iA4√2, (17)

where in each expansion the first term and last term are denoted as the VEVs and physical fields, respectively. Note that and are -even while those with primed signs, and , are -odd. There is no mixing between the -even and -odd fields due to the -parity conservation. On the other hand, the parameter in the scalar potential can be complex (the remaining parameters such as ’s and ’s are all real). However, its phase can be removed by redefining the fields appropriately. Consequently, the scalar potential conserves the symmetry. Assuming that the symmetry is also conserved by the vacuum, the VEVs and can simultaneously be considered as the real parameters by this work. There is no mixing between the scalars (-even) and pseudoscalars (-odd) due to the conservation.

To find the mass spectra of the scalar fields, let us expand all the terms of the potential up to the second order contributions of the fields:

 μ21(ρ†ρ) = μ21(⟨ρ⟩†⟨ρ⟩+⟨ρ⟩†ρ′+ρ′†⟨ρ⟩+ρ′†ρ′) = μ21(v22+vS2+ρ+1ρ−1+ρ+3ρ−3+S22+A222), μ22(χ†χ) = μ22(ω22+ωS3+χ−2χ+2+S′21+A′21+S23+A232), μ23(η†η) = μ23(u22+uS1+η−2η+2+S21+A21+S′23+A′232), μ2(ϕ†ϕ) = μ2(Λ22+ΛS4+S24+A242),
 λ(ϕ†ϕ)2 = λ[Λ44+Λ2S24+Λ3S4+Λ22(S24+A24)+interaction], λ1(ρ†ρ)2 = λ1[v44+v2S22+v3S2+v2(ρ+1ρ−1+ρ+3ρ−3+S22+A222)+interaction], λ2(χ†χ)2 = λ2[ω44+ω2S23+ω3S3+ω2(χ−2χ+2+S′21+A′21+S23+A232)+interaction], λ3(η†η)2 = λ3[u44+u2S21+u3S1+u2(η−2η+2+S21+A21+S′23+A′232)+interaction],
 λ4(ρ†ρ)(χ†χ) = λ4[v2ω24+ωv22S3+vω22S2+vωS2S3+v22(χ−2χ+2+S′21+A′21+S23+A232) +ω22(ρ+1ρ−1+ρ+3ρ−3+S22+A222)+interaction], λ5(ρ†ρ)(η†η) = λ5[v2u24+uv22S1+vu22S2+vuS1S2+v22(η−2η+2+S21+A21+S′23+A′232) +u22(ρ+1ρ−1+ρ+3ρ−3+S22+A222)+interaction], λ6(χ†χ)(η†η) = λ6[ω2u24+uω22S1+ωu22S3+uωS1S3+ω22(η−2η+2+S21+A21+S′23+A′232) +u22(χ+2χ−2+S′21+A′21+S23+A232)+interaction],
 λ7(ρ†χ)(χ†ρ) = λ72(vχ−2+ωρ−3)(ωρ+3+vχ+2)+interaction, λ8(ρ†η)(η†ρ) = λ82(vη−2+uρ−1)(uρ+1+vη+2)+interaction, λ9(χ†η)(η†χ) = λ9[ω2(S′3+iA′3)+u2(S′1−iA′1)][u2(S′1+iA′1)+ω2(S′3−iA′3)]+interaction,
 λ10(ϕ†ϕ)(ρ†ρ) = λ10[Λ2v24+vΛ22S2+Λv22S4+vΛS2S4+v22(S24+A242) +Λ22(ρ+1ρ−1+ρ+3ρ−3+S22+A222)+interaction], λ11(ϕ†ϕ)(χ†χ) = λ11[Λ2ω24+ωΛ22S3+Λω22S4+ωΛS3S4+ω22(S24+A242) +Λ22(χ+2χ−2+S′21+A′21+S23+A232)+interaction], λ12(ϕ†ϕ)(η†η) = λ12[Λ2u24+uΛ22S1+Λu22S4+uΛS1S4+u22(S24+A242) +Λ22(η+2η−2+S21+A21+S′23+A′232)+interaction],
 fϵmnpηmρnχp+H.c. = f[uvω√2+uv√2S3+uω√2S2+vω√2S1+u√2(S2S3−A2A3 −ρ+3χ−2−ρ−3χ+2)+v√2(S1S3−A1A3−S′1S′3+A′1A′3) +ω√2(S1S2−A1A2−η−2ρ+1−η+2ρ−1)]+interaction.

The scalar potential that is summed of all the terms above can be rearranged as

 V(ρ,η,χ,ϕ)=Vmin+Vlinear+Vmass+Vinteraction, (18)

where the interactions as stored in need not to be explicitly obtained. The contains the terms that are independent of the scalar fields,

 Vmin = μ21v22+μ22ω22+μ23u22+μ2Λ22+λ21v44+λ22ω44+λ23u44+λ2Λ44 +λ24v2ω24+λ25v2u24+λ26u2ω24+λ210v2Λ24+λ211Λ2ω24+λ212u2Λ24+fuvω√2,

which contributes to the vacuum energy only. It does not affect to the physical processes.

The includes all the terms that linearly depend on the scalar fields,

 Vlinear = S1[uμ23+λ3u3+12λ5uv2+12λ6uω2+√22fvω+12λ12uΛ2] (19) +S2[vμ21+λ1v3+12λ4vω2+1