#### Abstract

The gauge model with minimal scalar sector, two Higgs triplets, is presented in detail. One of the vacuum expectation values is a source of lepton-number violations and a reason for mixing among charged gauge bosons—the standard model and the bilepton gauge bosons , as well as among the neutral non-Hermitian bilepton and neutral gauge bosons—the and the new . An exact diagonalization of the neutral gauge boson sector is derived, and bilepton mass splitting is also given. Because of these mixings, the lepton-number violating interactions exist in both charged and neutral gauge boson sectors. Constraints on vacuum expectation values of the model are estimated and GeV, GeV, and TeV. In this model, there are three physical scalars, two neutral and one charged, and eight Goldstone bosons—the needed number for massive gauge bosons. The minimal scalar sector can provide all fermions including quarks and neutrinos consistent masses in which some of them require one-loop radiative corrections.

#### 1. Introduction

In spite of all the successes of the standard model, it is unlikely to be the final theory. It leaves many striking features of the physics of our world unexplained. In the following, we list some of them which leads to the model's extensions. In particular, the models with (3-3-1) gauge group are presented.

##### 1.1. Generation Problem and 3-3-1 Models

In the standard model, the fundamental fermions come in generations. In writing down the theory, one may start by first introducing just one generation, then one may repeat the same procedure by introducing copies of the first generation. Why do quarks and leptons come in repetitive structures (generations)? How many generations are there? How to understand the interrelation between generations? These are the central issues of the weak interaction physics known as the generation problem or the flavor question. Nowhere in physics this question is replied [1]. One of the most important experimental results in the past few years has been the determination of the number of these generations within the framework of the standard model. In the minimal electroweak model, the number of generations is given by the number of the neutrino species which are all massless, by definition. The number of generations is then computed from the invisible width of the : where denotes the total width, the subscript refers to hadrons, and is the width of the decay into an pair. If is the theoretical width for just one massless neutrino, the number of generations is , and recent results give a value very close to three [2, 3], but we do not understand why the number of standard model generations is three.

The answer to the generation problem may require a radical change in our approaches. It could be that the underlying objects are strings and all the low-energy phenomena will be determined by physics at the Planck scale. Grand unified theories (GUTs) have had a major impact on both cosmology and astrophysics; for cosmology they led to the inflationary scenario, while for astrophysics supernova, neutrinos were first observed in proton-decay detectors. It remains for GUTs to have impact directly on particle physics itself [4]. GUTs cannot explain the presence of fermion generations. On the other side, supersymmetry (SUSY) for the time being is an answer in search of question to be replied. It does not explain the existence of any known particle or symmetry. Some traditional approaches to the problem such as GUTs, monopoles, and higher dimensions introduce quite speculative pieces of new physics at high and experimentally inaccessible energies. Some years ago, there were hopes that the number of generations could be computed from first principles such as geometry of compactified manifolds, but these hopes did not materialize.

A very interesting alternative to explain the origin of generations comes from the cancelation of chiral anomalies of a gauge theory in all orders of perturbative expansion, which derives from the renormalizability condition. This constrains the fermion representation content. Three perturbative anomalies have been identified [5–10] for chiral gauge theories in four-dimensional space-time: (i) the triangle chiral gauge anomaly [11, 12] must be canceled to avoid violations of gauge invariance and the renormalizability of the theory; (ii) the global nonperturbative SU(2) chiral gauge anomaly, [13] which must be absent in order for the fermion integral to be defined in a gauge invariant way; and (iii) the mixed perturbative chiral gauge gravitational anomaly [14–16] which must be canceled in order to ensure general covariance. The general anomaly-free condition is where is the representation of the gauge algebra on the set of all left-handed fermion and antifermion fields put in a single column , and “Tr” denotes a sum over these fermion and antifermion species; are the coupling matrices of fermions to the current , respectively. The index runs over the dimension of a simple SU() group, , with a rank , and for the Abelian factor.

First, let us consider the relationship between anomaly cancelation and flavor problem in the standard model. The individual generations have the following structure under the (3-2-1) gauge group: The values in the parentheses denote quantum numbers based on the symmetry, where the subscripts and , respectively, indicate to the color, left-handed, and hypercharge. The electric charge operator is defined as , where with are Pauli matrices. The weak isospin group is a safe group due to the fact that However, in the case where at least one of the generators is hypercharge we have The anomaly contribution in the last condition is proportional to the sum of all fermionic discrete hypercharge values on the color, flavor, and weak hypercharge degrees of freedom: The Tr vanishes for the fermion content in the th generation because where 3 factors take into account the number of quark colors. In the last case, all the generators are hypercharge: where we used the fact that the electromagnetic vector neutral current vertices do not have anomalies. For the th generation, we have It yields that the anomaly in standard model cancels within each individual generation, but not by generations. Flavor question and anomaly-free conditions do not seem to have any connection in the standard model. This leads us to questions when going beyond this model. Are the anomalies always canceled automatically within each generation of quarks or leptons? Do the anomaly cancelation conditions have any connection with flavor puzzle?

We wish to show that some very fundamental aspects of the standard model, in particular the flavor problem, might be understood by embedding the three-generation version in a Yang-Mills theory with the semisimple gauge group with a corresponding enlargement of the lepton and quark representations [17–25]. In particular, the number of generations will be related by anomaly cancelation to the number of quark colors, and one generation of quarks will be treated differently from the two others. In the 3-2-1 low-energy limit, all three generations appear similarly and cancel anomalies separately. Let us consider the following 3-3-1 fermion representation content: The quantum numbers in the parentheses are based on the symmetry. The right-handed neutrinos and the exotic quarks and are composed along with that of the standard model. We call 3-3-1 model with right-handed neutrinos. The electric charge operator in this case takes a form with and standing for and charges, respectively. Electric charges of the exotic quarks are the same as of the usual quarks, that is, and .

The group is not safe in the sense of the standard model with the vanishing . The generators proportional to the Gell-Mann matrices close among them the Lie algebra structure: where the structure constant is totally antisymmetric, and is totally symmetric under exchange of the indices. We can normalize the -matrices such that . Therefore, and are calculated by The anomaly is proportional to in general, and of course such coefficients vanish in the case of the generators.

In the 3-3-1 model, there are six triangle anomalies which are potentially troublesome. In a self-explanatory notation, these are , and . The quantum chromodynamics anomaly is absent because the theory mentioned is vectorlike (i.e., with some unitary matrix ), and hence the condition is automatically satisfied. For any fermion representation, it satisfies the condition where is the anomaly of the conjugate representation of [26]. The pure anomaly , therefore, vanishes because there is an equal number of triplets and antitriplets in the given fermion content. The remaining anomaly-free conditions are explicitly written as follows.

(1): (2): (3): (4): where , and indicate to the charges of the left-handed lepton, quark triplets or antitriplets, the right-handed lepton, and quark singlets, respectively. It is worth noting that some 3 factors in the conditions (2), (3), and (4) take into account the number of quark colors. With the fermion content as given, it is easily checked that all the above anomaly-free conditions are satisfied. For example, let us take condition (2). We first calculate the anomaly for the first generation: . The anomaly of the second or the third generation is . It is especially interesting that this anomaly cancelation takes place between generations, unlike those of the standard model. Each individual generation possesses nonvanishing , and anomalies. Only with a matching of the number of generations with the number of quark colors does the overall anomaly vanish.

Next, let us introduce an alternative fermion content, where the three known left-handed lepton components for each generation are associated to three triplets such that (called minimal 3-3-1 model). Canceling the pure anomaly requires that there are the same number of triplets and antitriplets, thus , . The respective right-handed fields are singlets: and for the ordinary quarks; and for the exotic quarks. Similarly, to the previous 3-3-1 model, the anomalies vanish only if three generations of quarks and leptons take into account.

In a general case, we can verify that the number of
generations must be multiple of the quark-color number in order to cancel the
anomalies. On the other hand, if we suppose that the exotic quarks also
contribute to the running of the coupling constants, the asymptotic-freedom
principle requires that the number of quark generations is no more than five.
It follows that the number of generations is *just* three. This provides a first step
toward answering the flavor question. The asymmetric treatment of one
generation of quarks breaks generation universality. This might provide an
explanation of why the top quark is uncharacteristically heavy [27, 28]. An interesting alternative
feature is that the electric charge quantization in nature might also be
explained in this framework [23, 29–32]. Just enlarging to , we have thus presented the simplest gauge extension of the standard
model for the flavor question. The new models get five additional gauge bosons
contained in a gauge adjoint octet: under . The is a neutral and the two doublets are readily identifiable
from the leptonic contents as non-Hermitian bilepton gauge bosons and .
From the renormalization group analysis of the coupling
constants [17, 33], the breaking scale is estimated to be lower than some TeV in the minimal
3-3-1 model. This is due to the fact that the squared sine of the Weinberg
angle gets an upper bound, .
There is no “grand desert” in this model in comparison to GUTs. In contrast,
the energy scale in the 3-3-1 model with right-handed neutrinos is very high,
even larger than the Planck scale because of .
This version might allow the existence of a “desert." Anyway, the new physics
in these models expected arise at not too high energies. The new particles such
as the bilepton gauge bosons and exotic quarks would be determinable in the
next generation of collides.

##### 1.2. Proposal of Minimal Higgs Sector

As mentioned above, there are two main versions of
3-3-1 models—the minimal model and the model with right-handed neutrinos,
which have been subjects studied extensively over the last decade. In the
minimal 3-3-1 model [17–19], the scalar sector is quite complicated and contains
three scalar triplets and one scalar sextet. In the 3-3-1 model with
right-handed neutrinos [20–22, 34, 35], the scalar sector requires three Higgs triplets. It
is interesting to note that two Higgs triplets of this model have the same charges with two neutral components at their
top and bottom. Allowing these neutral components vacuum expectation values
(VEVs), we can reduce number of Higgs triplets to be two. Note that the
mentioned model contains very important advantage, namely, there is no new
parameter, but it contains very simple Higgs sector, therefore, the significant
number of free parameters is reduced. To mark the minimal content of the Higgs
sector, this version that includes right-handed neutrinos is going to be called
the *economical 3-3-1 model* [36–42]. The interested reader can find the
supersymmetric version in
[43–46].

This kind of model was proposed in [36] but has not got enough attention. In [37], phenomenology of this model was presented without mixing between charged gauge bosons as well as neutral ones. The mass spectrum of the mentioned scalar sector has also been presented in [36], and some couplings of the two neutral scalar fields with the charged and the neutral gauge bosons in the standard model were presented. From explicit expression for the vertex, the authors concluded that two VEVs responsible for the second step of spontaneous symmetry breaking have to be in the same range , or the theory needs an additional scalar triplet. As we will show in the following, this conclusion is incorrect.

It is well known that the electroweak symmetry breaking in the standard model is achieved via the Higgs mechanism. In the Weinberg-Salam model, there is a single complex scalar doublet, where the Higgs boson is the physical neutral Higgs scalar which is the only remaining part of this doublet after spontaneous symmetry breaking. In the extended models, there are additional charged and neutral scalar Higgs particles. The prospects for Higgs coupling measurements at the CERN Large Hadron Collider (LHC) have recently been analyzed in detail in [47]. The experimental detection of the will be great triumph of the standard model of electroweak interactions and will mark new stage in high-energy physics.

In extended Higgs models, which would be deduced in the low-energy effective theory of new physics models, additional Higgs bosons like charged and CP-odd scalar bosons are predicted. Phenomenology of these extra scalar bosons strongly depends on the characteristics of each new physics model. By measuring their properties like masses, widths, production rates, and decay branching ratios, the outline of physics beyond the electroweak scale can be experimentally determined.

The interesting feature compared with other 3-3-1 models is the Higgs physics. In the 3-3-1 models, the general Higgs sector is very complicated [48–51] and this prevents the models' predicability. The scalar sector of the considering model is one of subjects in the present work. As shown, by couplings of the scalar fields with the ordinary gauge bosons such as the photon, the , and the neutral gauge bosons, we are able to identify full content of the Higgs sector in the standard model including the neutral and the Goldstone bosons eaten by their associated massive gauge ones. All interactions among Higgs-gauge bosons in the standard model are recovered.

Production of the Higgs boson in the 3-3-1 model with right-handed neutrinos at LHC has been considered in [52]. In scalar sector of the considered model, there exists the singly-charged boson , which is a subject of intensive current studies [53, 54]. The trilinear coupling which differs at the tree level, from zero only in the models with Higgs triplets plays a special role on study phenomenology of these exotic representations. We will pay particular interest on this boson.

At the tree level, the mass matrix for the upquarks has one massless state, and in the downquark sector there are two massless ones. This calls for radiative corrections. To solve this problem, the authors in [37] have introduced the third Higgs triplet. In this sense, the economical 3-3-1 model is not realistic. In the present work, we will show that this is a mistake! Without the third one, at the one loop level, the fermions in this model, with the given set of parameters, gain a consistent mass spectrum. A numerical evaluation leads us to conclusion that in the model under consideration, there are two scales for masses of the exotic quarks.

At the tree level, the neutrino spectrum is Dirac particles with one massless and two degenerate in mass . This spectrum is not realistic under the data because there is only one squared-mass splitting. Since the observed neutrino masses are so small, the Dirac mass is unnatural. One must understand what physics gives —the mass of charged leptons. In contrast to the seesaw cases [55–62] in which the problem can be solved, in this model the neutrinos including the right-handed ones get only small masses through radiative corrections [42, 49, 63–78]. We will obtain these radiative corrections and will provide a possible explanation of natural smallness of the neutrino masses. This is not the result of a seesaw, but it is due to a finite mass renormalization arising from a very different radiative mechanism. We will show that the neutrinos can get mass not only from the standard symmetry breakdown, but also from the electroweak breaking associated with spontaneous lepton-number breaking (SLB), and even through the explicit lepton-number violating processes due to a new physics. The total neutrino mass spectrum at the one-loop level is neat and can fit the data.

This report is organized as follows. In Section 2, we give a review of the model with stressing on the gauge bosons, currents, and constraints on the new physics. The Higgs-gauge interactions and scalar content are considered in Section 3. Section 4 is devoted to fermion masses. We summarize our results and make conclusions in the last section—Section 5.

#### 2. The Economical 3-3-1 Model

We first recall the idea of constructing the model. An exact diagonalization of charged and neutral gauge boson sectors and their masses and mixings are presented. Because of the mixings, currents in this model have unusual features which are obtained then. Constraints on the parameters and some phenomena are sketched.

##### 2.1. Particle Content

The fermion content which is anomaly free is given by (1.10) like that of the 3-3-1 model with right-handed neutrinos. However, contrasting with the ordinary model in which the third generation of quarks should be discriminating [28], in the model under consideration the first generation has to be different from the two others. This results from the mass patterns for the quarks which will be derived in Section 4.

The 3-3-1 gauge group is broken spontaneously via two stages. In the first stage, it is embedded in that of the standard model via a Higgs scalar triplet: with the VEV given by In the last stage, to embed the standard model gauge symmetry in , another Higgs scalar triplet is needed: with the VEV as follows: The Yukawa interactions which induce masses for the fermions can be written in the most general form as follows: where LNC and LNV, respectively, indicate to the lepton number conserving and violating ones as shown below. Here, each part is defined by where , , and stand for indices.

The VEV gives mass for the exotic quarks and : gives mass for ,
while gives mass for and all ordinary leptons. In Section 4, we
will provide more details on analysis of fermion masses. As mentioned, is responsible for the first stage of symmetry
breaking, while the second stage is due to and ;
therefore, the VEVs in this model satisfies the constraint: The Yukawa couplings in (2.6) possess an extra global
symmetry [49, 50] which is not broken by ,
but by .
From these couplings, one can find the following lepton symmetry as in Table 1 (only the fields with nonzero are listed; all other fields have vanishing ). Here, is broken by which is behind ,
that is, *is a
kind of the SLB scale* [79–83]. It is
interesting that the exotic quarks also carry the lepton number (so-called
leptoquarks); therefore, this obviously does not commute with the gauge
symmetry. One can then construct a new conserved charge through by making a linear combination .
Applying on a lepton triplet, the coefficients will be
determined: Another useful conserved charge which is exactly not broken by , ,
and is usual baryon number: .
Both the charges and for the fermion and Higgs multiplets are
listed in Table 2.

Let us note that the Yukawa couplings of (2.7) conserve , however, violate with units which implies that these interactions are much smaller than the first ones [41]: In previous studies [19, 37, 84–86], the LNV terms of this kind have often been excluded, commonly by the adoption of an appropriate discrete symmetry. There is no reason within the 3-3-1 models why such terms should not be present.

In this model, the most general Higgs potential has very simple form: It is noteworthy that does not contain trilinear scalar couplings and conserves both the mentioned global symmetries; this makes the Higgs potential much simpler and discriminative from the previous ones of the 3-3-1 models [48–51]. This potential is closer to that of the standard model. In the next section, we will show that after spontaneous symmetry breaking, there are eight Goldstone bosons—the needed number for massive gauge ones and three physical scalar fields (one charged and two neutral). One of two physical neutral scalars is the standard model Higgs boson.

To break the gauge symmetry spontaneously, the Higgs vacuums are not singlets. Hence, nonzero values of and at the minimum value of can be easily obtained by (for details, see Section 3): It is important noting that any other choice of for the vacuum value of satisfying (2.12) gives the same physics because it is related to (2.2) by an transformation. It is worth noting that the assumed is, therefore, given in a general case. This model, however, does not lead to the formation of Majoron [79–83, 87].

##### 2.2. Gauge Bosons

The covariant derivative of a triplet is given by where the gauge fields and transform as the adjoint representations of and ,
respectively, and the corresponding gauge coupling constants .
Moreover, is fixed so that the relation is satisfied. The matrix appeared in the above covariant derivative is rewritten in a
convenient form: where .
Let us denote the following combinations: having defined charges under the
generators of the group. For the sake of convenience in further
reading, we note that and are pure real and imaginary parts of and ,
respectively: The masses of the gauge bosons in this model are
followed from The combinations and are mixing via Diagonalizing this mass matrix,
we get *physical* charged gauge
bosons: where the mixing angle is
defined by The mass eigenvalues are Because of the constraints in
(2.8), the following remarks are in order:

(1) should be very small, and then ;(2) GeV due to identification of as the boson in the standard model. Next, from (2.18), the gains mass as follows: Finally, there is a mixing among components. In the basis of these elements,
the mass matrix is given by Note that the mass Lagrangian in
this case has the form In the limit , does not mix with .
In the general case ,
the mass matrix in (2.25) contains two *exact eigenvalues* such as Thus, the and components have the same mass, and this
conclusion *contradicts the previous
analysis in* [36].
With this result, we should identify the combination of and : as *physical* neutral *non-Hermitian* gauge boson. The subscript denotes neutrality of gauge boson .
However, in the following, this subscript may be dropped. This boson caries the
lepton number with two units. Hence, it is the bilepton like those in the usual
3-3-1 model with right-handed neutrinos. From (2.22), (2.23), and (2.27), it follows
an interesting relation between the bilepton masses similar to the law of
Pythagoras: Thus, the charged bilepton is slightly heavier than the neutral one .
Remind that the similar relation in the 3-3-1 model with right-handed neutrinos
is [88]: .

Now, we turn to the eigenstate question. The eigenstates corresponding to the two values in (2.27) are determined as follows: To embed this model in the effective theory at the low energy, we follow an appropriate method in [89, 90], where the photon field couples with the lepton by strength: Therefore, the coefficient of the electromagnetic coupling constant can be identified as Using continuation of the gauge coupling constant of at the spontaneous symmetry breaking point, from which it follows The eigenstates are now rewritten as follows: where we have denoted and so forth.

The diagonalization of the mass matrix is done via three steps. In the first step, it is in the base of , where the two remaining gauge vectors are given by In this basis, the mass matrix becomes Also, in the limit does not mix with . The eigenstate is now defined by We turn to the second step. To see explicitly that the following basis is orthogonal and normalized, let us put which leads to Note that the mixing angle in this step is the same order as the mixing angle in the charged gauge boson sector. Taking into account [3] , from (2.39) we get . It is now easy to choose two remaining gauge vectors orthogonal to : Therefore, in the base of , the mass matrix has a quasi-diagonal form: with In the last step, it is trivial to diagonalize the mass matrix in (2.42). The two remaining mass eigenstates are given by where the mixing angle between and is defined by The physical mass eigenvalues are defined by Because of the condition (2.8), the angle has to be very small: In this approximation, the above physical states have masses: Consequently, can be identified as the boson in the standard model, and being the new neutral (Hermitian) gauge boson. It is important to note that in the limit the mixing angle between and is always nonvanishing. This differs from the mixing angle between the boson of the standard model and the singly-charged bilepton . Phenomenology of the mentioned mixing is quite similar to the mixing in the left-right symmetric model based on the group (the interested reader can find in [90]).

##### 2.3. Currents

The interaction among fermions with gauge bosons arises in part from

###### 2.3.1. Charged Currents

Despite neutrality, the gauge bosons belong to this section by their nature. Because of the mixing among the standard model boson and the charged bilepton as well as among () with , the new interaction terms exist as follows: where Comparing with the charged currents in the usual 3-3-1 model with right-handed neutrinos [34, 35], we get the following discrepancies:

(1)the second term in (2.52),(2)the second term in (2.53),(3)the second and the third terms in (2.54). All above-mentioned interactions are lepton-number violating and weak (proportional to or its square ). However, these couplings lead to lepton-number violations only in the neutrino sector.

###### 2.3.2. Neutral Currents

As before, in this model, a real part of the
non-Hermitian neutral mixes with the real neutral ones such as and .
This gives the *unusual* term as
follows: Despite the mixing among ,
the electromagnetic interactions *remain* the same as in the standard model
and the usual 3-3-1 model with right-handed neutrinos, that is, where runs among all the fermions of the model.

Interactions of the neutral currents with fermions have a common form: where Here, and are, respectively, the third component of the weak isospin, the charge, and the electric charge of the fermion . Note that the isospin for the fermion singlet (in the bottom of triplets) vanishes: . The values of and are listed in Tables 3 and 4.

Because of the above-mentioned mixing, the
lepton-number violating interactions mediated by neutral gauge bosons and exist in the *neutrino and the exotic quark sectors*: Again, these interactions are
very weak and proportional to .
From (2.52)–(2.54) and (2.59), we conclude that all lepton-number violating
interactions are expressed in the terms dependent only in the mixing angle
between the charged gauge bosons.

##### 2.4. Phenomenology

First of all, we should find some constraints on the
parameters of the model. There are many ways to get constraints on the mixing
angle and the charged bilepton mass .
Below we present a simple one. In our model, the boson has the following *normal main* decay modes: which are the same as in the
standard model and in the 3-3-1 model with right-handed neutrinos. Beside the
above MODES, there are additional ones which are
lepton-number violating —the model's specific feature: It is easy to compute the
tree-level decay widths as follows [91, 92]: Quantum chromodynamics radiative
corrections modify (2.62) by a multiplicative factor [3, 91, 92]: which is estimated from .
All the state masses can be ignored, the predicted total width for decay into fermions is Taking ,
and [3], in Figure 1, we have plotted as function of . From the
figure we get an upper limit: It is important to note that
this limit value on the LNV parameter is much larger than those in [50, 93, 94].

Since one of the VEVs is closely to the those in the standard model: GeV, therefore only two free VEVs exist in the considering model, namely, and . The bilepton mass limit can be obtained from the “wrong" muon decay: mediated at the tree level, by both the standard model and the singly-charged bilepton (see Figure 2). Remind that in the 3-3-1 model with right-handed neutrinos, at the lowest order, this decay is mediated only by the singly-charged bilepton . In our case, the second diagram in Figure 2 gives main contribution. Taking into account of the famous experimental data [3] we get the constraint: . Therefore, it follows that GeV.

**(a)**

**(b)**

However, the stronger bilepton mass bound of 440 GeV has been derived from consideration of experimental limit on lepton-number violating charged lepton decays [85].

In the case of , analyzing the decay width [37, 95, 96], the mixing angle is constrained by . From atomic parity violation in cesium, bounds for mass of the new exotic and the mixing angles, again in the limit , are given [37, 95, 96]: These values coincide with the bounds in the usual 3-3-1 model with right-handed neutrinos [97]. The interested reader can find in [40] for the general case of the constraints.

For our purpose, we consider the parameter—one of the most important quantities of the standard model, having a leading contribution in terms of the parameter, is very useful to get the new-physics effects. It is well-known relation between and parameter: In the usual 3-3-1 model with right-handed neutrinos, gets contribution from the oblique correction and the mixing [88]: where