# The EconomicalSU(3)C⊗SU(3)L⊗U(1)X Model

Authors: P. V. Dong; H. N. Long
Journal: Advances in High Energy Physics
 (2008)

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## Abstract

TheSU(3)C⊗SU(3)L⊗U(1)X gauge model with minimal scalar sector,
two Higgs triplets, is presented in detail. One of the vacuum expectation values u is a source of lepton-number violations and a reason for mixing among charged
gauge bosons—the standard model W± and the bilepton gauge bosons Y±, as well
as among the neutral non-Hermitian bilepton X0 and neutral gauge bosons—the Z and the new Z′. 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 u≃𝒪(1) GeV, v≃vweak=246 GeV, and ω≃𝒪(1) 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.


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## Body

## 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 withSU(3)C⊗SU(3)L⊗U(1)X (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 Z0: (1.1)Γinv≡ΓZ0−(Γh+∑lΓl), where ΓZ0 denotes the total width, the subscript h refers to hadrons, and Γl(l=e,μ,τ) is the width of the Z0 decay into an ll¯ pair. If Γν is the theoretical width for just one massless
neutrino, the number of generations is Ngen=Nν=Γinv/Γν,
and recent results give a value very close to three Ngen=2.99±0.03 [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 (1.2)Aijk≡Tr[{Ti,Tj}Tk]=∑representationsTr[{TLi,TLj}TLk−{TRi,TRj}TRk]=0,where Ti is the representation of the gauge algebra on
the set of all left-handed fermion and antifermion fields put in a single
column ψ,
and “Tr” denotes a sum over these fermion and antifermion species; TL̲,Ri are the coupling matrices of fermions ψL,R to the current Jμi=ψ¯LγμTLiψL+ψ¯RγμTRiψR,
respectively. The i index runs over the dimension of a simple SU(n) group, i=1,2,…,n2−1,
with a rank n−1,
and i=0 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 theSU(3)C⊗SU(2)L⊗U(1)Y (3-2-1) gauge group: (1.3)(νaL,laL)~(1,2,−1),laR~(1,1,−2),(uaL,daL)~(3,2,13),uaR~(3,1,43),daR~(3,1,−23). The values in the parentheses
denote quantum numbers based on the (SU(3)C,SU(2)L,U(1)Y) symmetry, where the subscripts C,L, and Y,
respectively, indicate to the color, left-handed, and hypercharge. The electric
charge operator is defined as Q=T3+(1/2)Y,
where Ti=(1/2)σi(i=1,2,3) with σi are Pauli matrices. The weak isospin group
SU(2)L is a safe group due to the fact that (1.4)Tr[{σi,σj}σk]=2δijTr[σk]=0. However, in the case where at
least one of the generators is hypercharge we have (1.5)Tr[σiYY]∝Tr[σi]=0,Tr[{σi,σj}Y]=2δijTr[Y]. 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: (1.6)Tr[Y]=∑lepton(YL+YR)+∑quark(YL+YR). The Tr [Y] vanishes for the fermion content in the ath generation because (1.7)∑lepton(YL+YR)=Y(νaL)+Y(laL)+Y(laR)=−4,∑quark(YL+YR)=3[Y(uaL)+Y(daL)+Y(uaR)+Y(daR)]=+4, where 3
factors take into
account the number of quark colors. In the last case, all the generators are
hypercharge: (1.8)Tr[Y3]∝Tr[Q2T3−QT32], where we used the fact that the
electromagnetic vector neutral current vertices do not have anomalies. For the ath generation, we have (1.9)∑lepton(Q2T3−QT32)=[(0)2(12)−(0)(12)2]+[(−1)2(−12)−(−1)(−12)2]=−14,∑quark(Q2T3−QT32)=3[(23)2(12)−(23)(12)2]+3[(−13)2(−12)−(−13)(−12)2]=+14. 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 theSU(3)C⊗SU(3)L⊗U(1)X 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: (1.10)ψaL=(νaLlaLνaRc)~(1,3,−13),laR~(1,1,−1),a=1,2,3,Q1L=(u1Ld1LUL)~(3,3,13),QαL=(dαL−uαLDαL)L~(3,3*,0),α=2,3,uaR~(3,1,23),daR~(3,1,−13),UR~(3,1,23),DαR~(3,1,−13). The quantum numbers in the
parentheses are based on the (SU(3)C,SU(3)L,U(1)X) symmetry. The right-handed neutrinos νR and the exotic quarks U and Dα 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 Q=T3−(1/3)T8+X with Ti=1/2λi(i=1,2,…,8)  and X standing for SU(3)L and U(1)X charges, respectively. Electric charges of the
exotic quarks are the same as of the usual quarks, that is, qU=2/3 and qDα=−1/3.TheSU(3)L group is not safe in the sense of the standard model SU(2)L with the vanishing Tr[{σi,σj},σk]=0.
The SU(3)L generators proportional to the Gell-Mann matrices close among them the
Lie algebra structure: (1.11)[λi,λj]=2ifijkλk,{λi,λj}=43δij+2dijkλk, where the structure constant fijk is totally antisymmetric, and dijk is totally symmetric under exchange of the
indices. We can normalize the λ-matrices such that Tr[λiλj]=2δij.
Therefore, fijk and dijk are calculated by (1.12)fijk=14iTr[[λi,λj]λk],dijk=14Tr[{λi,λj}λk]. The anomaly is proportional to dijk in general, and of course such coefficients
vanish in the case of the SU(2)L generators.In the 3-3-1 model, there are six triangle anomalies
which are potentially troublesome. In a self-explanatory notation, these are(3C)3,(3C)2X,(3L)3,(3L)2X,X3, and (graviton)2X.
The quantum chromodynamics anomaly (3C)3 is absent because the theory mentioned is
vectorlike (i.e., TLi=U−1TRiU with some unitary matrix U), and hence the
condition Aijk=0 is automatically
satisfied. For any D fermion representation, it satisfies the
condition A(D)=−A(D*), where A(D*) is the anomaly of the conjugate representation
of D [26]. The pure SU(3)L anomaly (3L)3,
therefore, vanishes because there is an equal number of triplets 3L and antitriplets 3L* in the given fermion content. The remaining
anomaly-free conditions are explicitly written as follows.(1)
Tr[SU(3)C]2[U(1)X]=0: (1.13)3∑generationXqL−∑generation∑singletXqR=0;(2)
Tr[SU(3)L]2[U(1)X]=0: (1.14)∑generationXlL+3∑generationXqL=0;(3)
Tr[U(1)X]3=0: (1.15)3∑generation(XlL)3+9∑generation(XqL)3−3∑generation∑singlet(XqR)3−∑generation∑singlet(XlR)3=0;(4)
Tr[graviton]2[U(1)X]=0: (1.16)3∑generationXlL+9∑generationXqL−3∑generation∑singletXqR−∑generation∑singletXlR=0, where XlL,XqL,XlR,
and XqR indicate to the U(1)X 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 3L2X anomaly for the first generation: −1/3+3×(1/3)=2/3.
The anomaly of the second or the third generation is −1/3+3×0=−1/3.
It is especially interesting that this anomaly cancelation takes place between
generations, unlike those of the standard model. Each individual generation
possesses nonvanishing (3L)3,(3L)2X,X3,
and (gravion)2X 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 threeSU(3)L triplets such that (νaL,laL,laRc)T~(1,3,0) (called minimal 3-3-1 model). Canceling the
pure SU(3)L anomaly requires that there are the same number of triplets and antitriplets,
thus Q1L=(u1L,d1L,JL)T~(3,3,2/3), QαL=(dαL,−uαL,JαL)T~(3,3*,−1/3).
The respective right-handed fields are singlets: uaR~(3,1,2/3) and daR~(3,1,−1/3) for the ordinary quarks; JR~(3,1,5/3) and JαR~(3,1,−4/3) for the exotic quarks. Similarly, to the
previous 3-3-1 model, the (3L)3,(3L)2X,X3 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 isjust 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 SU(2)L to SU(3)L, 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: 8=3+(2+2)+1 under SU(2)L. The 1 is a neutral Z′ and the two doublets are readily identifiable
from the leptonic contents as non-Hermitian bilepton gauge bosons (X,Y)T and (X*,Y*).
From the renormalization group analysis of the coupling
constants [17, 33], the SU(3)L 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 θW gets an upper bound, sin2θW<1/4.
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 sin2θW<3/4.
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 Z′ 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 U(1)X 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 W and the neutral Z gauge bosons in the standard model were
presented. From explicit expression for the ZZH vertex, the authors concluded that two VEVs
responsible for the second step of spontaneous symmetry breaking have to be in
the same range u~v,
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
bosonH 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 H 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 W,
and the neutral Z gauge bosons, we are able to identify full
content of the Higgs sector in the standard model including the neutral H 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 H2±,
which is a subject of intensive current studies [53, 54]. The trilinear coupling ZW±H∓ 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~hνv.
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 hνv≪hlv—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 SU(3)L⊗U(1)X 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 Section2, 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.


## 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 Z0: (1.1)Γinv≡ΓZ0−(Γh+∑lΓl), where ΓZ0 denotes the total width, the subscript h refers to hadrons, and Γl(l=e,μ,τ) is the width of the Z0 decay into an ll¯ pair. If Γν is the theoretical width for just one massless
neutrino, the number of generations is Ngen=Nν=Γinv/Γν,
and recent results give a value very close to three Ngen=2.99±0.03 [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 (1.2)Aijk≡Tr[{Ti,Tj}Tk]=∑representationsTr[{TLi,TLj}TLk−{TRi,TRj}TRk]=0,where Ti is the representation of the gauge algebra on
the set of all left-handed fermion and antifermion fields put in a single
column ψ,
and “Tr” denotes a sum over these fermion and antifermion species; TL̲,Ri are the coupling matrices of fermions ψL,R to the current Jμi=ψ¯LγμTLiψL+ψ¯RγμTRiψR,
respectively. The i index runs over the dimension of a simple SU(n) group, i=1,2,…,n2−1,
with a rank n−1,
and i=0 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 theSU(3)C⊗SU(2)L⊗U(1)Y (3-2-1) gauge group: (1.3)(νaL,laL)~(1,2,−1),laR~(1,1,−2),(uaL,daL)~(3,2,13),uaR~(3,1,43),daR~(3,1,−23). The values in the parentheses
denote quantum numbers based on the (SU(3)C,SU(2)L,U(1)Y) symmetry, where the subscripts C,L, and Y,
respectively, indicate to the color, left-handed, and hypercharge. The electric
charge operator is defined as Q=T3+(1/2)Y,
where Ti=(1/2)σi(i=1,2,3) with σi are Pauli matrices. The weak isospin group
SU(2)L is a safe group due to the fact that (1.4)Tr[{σi,σj}σk]=2δijTr[σk]=0. However, in the case where at
least one of the generators is hypercharge we have (1.5)Tr[σiYY]∝Tr[σi]=0,Tr[{σi,σj}Y]=2δijTr[Y]. 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: (1.6)Tr[Y]=∑lepton(YL+YR)+∑quark(YL+YR). The Tr [Y] vanishes for the fermion content in the ath generation because (1.7)∑lepton(YL+YR)=Y(νaL)+Y(laL)+Y(laR)=−4,∑quark(YL+YR)=3[Y(uaL)+Y(daL)+Y(uaR)+Y(daR)]=+4, where 3
factors take into
account the number of quark colors. In the last case, all the generators are
hypercharge: (1.8)Tr[Y3]∝Tr[Q2T3−QT32], where we used the fact that the
electromagnetic vector neutral current vertices do not have anomalies. For the ath generation, we have (1.9)∑lepton(Q2T3−QT32)=[(0)2(12)−(0)(12)2]+[(−1)2(−12)−(−1)(−12)2]=−14,∑quark(Q2T3−QT32)=3[(23)2(12)−(23)(12)2]+3[(−13)2(−12)−(−13)(−12)2]=+14. 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 theSU(3)C⊗SU(3)L⊗U(1)X 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: (1.10)ψaL=(νaLlaLνaRc)~(1,3,−13),laR~(1,1,−1),a=1,2,3,Q1L=(u1Ld1LUL)~(3,3,13),QαL=(dαL−uαLDαL)L~(3,3*,0),α=2,3,uaR~(3,1,23),daR~(3,1,−13),UR~(3,1,23),DαR~(3,1,−13). The quantum numbers in the
parentheses are based on the (SU(3)C,SU(3)L,U(1)X) symmetry. The right-handed neutrinos νR and the exotic quarks U and Dα 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 Q=T3−(1/3)T8+X with Ti=1/2λi(i=1,2,…,8)  and X standing for SU(3)L and U(1)X charges, respectively. Electric charges of the
exotic quarks are the same as of the usual quarks, that is, qU=2/3 and qDα=−1/3.TheSU(3)L group is not safe in the sense of the standard model SU(2)L with the vanishing Tr[{σi,σj},σk]=0.
The SU(3)L generators proportional to the Gell-Mann matrices close among them the
Lie algebra structure: (1.11)[λi,λj]=2ifijkλk,{λi,λj}=43δij+2dijkλk, where the structure constant fijk is totally antisymmetric, and dijk is totally symmetric under exchange of the
indices. We can normalize the λ-matrices such that Tr[λiλj]=2δij.
Therefore, fijk and dijk are calculated by (1.12)fijk=14iTr[[λi,λj]λk],dijk=14Tr[{λi,λj}λk]. The anomaly is proportional to dijk in general, and of course such coefficients
vanish in the case of the SU(2)L generators.In the 3-3-1 model, there are six triangle anomalies
which are potentially troublesome. In a self-explanatory notation, these are(3C)3,(3C)2X,(3L)3,(3L)2X,X3, and (graviton)2X.
The quantum chromodynamics anomaly (3C)3 is absent because the theory mentioned is
vectorlike (i.e., TLi=U−1TRiU with some unitary matrix U), and hence the
condition Aijk=0 is automatically
satisfied. For any D fermion representation, it satisfies the
condition A(D)=−A(D*), where A(D*) is the anomaly of the conjugate representation
of D [26]. The pure SU(3)L anomaly (3L)3,
therefore, vanishes because there is an equal number of triplets 3L and antitriplets 3L* in the given fermion content. The remaining
anomaly-free conditions are explicitly written as follows.(1)
Tr[SU(3)C]2[U(1)X]=0: (1.13)3∑generationXqL−∑generation∑singletXqR=0;(2)
Tr[SU(3)L]2[U(1)X]=0: (1.14)∑generationXlL+3∑generationXqL=0;(3)
Tr[U(1)X]3=0: (1.15)3∑generation(XlL)3+9∑generation(XqL)3−3∑generation∑singlet(XqR)3−∑generation∑singlet(XlR)3=0;(4)
Tr[graviton]2[U(1)X]=0: (1.16)3∑generationXlL+9∑generationXqL−3∑generation∑singletXqR−∑generation∑singletXlR=0, where XlL,XqL,XlR,
and XqR indicate to the U(1)X 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 3L2X anomaly for the first generation: −1/3+3×(1/3)=2/3.
The anomaly of the second or the third generation is −1/3+3×0=−1/3.
It is especially interesting that this anomaly cancelation takes place between
generations, unlike those of the standard model. Each individual generation
possesses nonvanishing (3L)3,(3L)2X,X3,
and (gravion)2X 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 threeSU(3)L triplets such that (νaL,laL,laRc)T~(1,3,0) (called minimal 3-3-1 model). Canceling the
pure SU(3)L anomaly requires that there are the same number of triplets and antitriplets,
thus Q1L=(u1L,d1L,JL)T~(3,3,2/3), QαL=(dαL,−uαL,JαL)T~(3,3*,−1/3).
The respective right-handed fields are singlets: uaR~(3,1,2/3) and daR~(3,1,−1/3) for the ordinary quarks; JR~(3,1,5/3) and JαR~(3,1,−4/3) for the exotic quarks. Similarly, to the
previous 3-3-1 model, the (3L)3,(3L)2X,X3 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 isjust 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 SU(2)L to SU(3)L, 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: 8=3+(2+2)+1 under SU(2)L. The 1 is a neutral Z′ and the two doublets are readily identifiable
from the leptonic contents as non-Hermitian bilepton gauge bosons (X,Y)T and (X*,Y*).
From the renormalization group analysis of the coupling
constants [17, 33], the SU(3)L 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 θW gets an upper bound, sin2θW<1/4.
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 sin2θW<3/4.
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 Z′ 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 U(1)X 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 W and the neutral Z gauge bosons in the standard model were
presented. From explicit expression for the ZZH vertex, the authors concluded that two VEVs
responsible for the second step of spontaneous symmetry breaking have to be in
the same range u~v,
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
bosonH 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 H 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 W,
and the neutral Z gauge bosons, we are able to identify full
content of the Higgs sector in the standard model including the neutral H 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 H2±,
which is a subject of intensive current studies [53, 54]. The trilinear coupling ZW±H∓ 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~hνv.
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 hνv≪hlv—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 SU(3)L⊗U(1)X 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 Section2, 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:(2.1)χ=(χ10χ2−χ30)~(1,3,−13) with the VEV given by (2.2)〈χ〉=12(u0ω). In the last stage, to embed the
standard model gauge symmetry in SU(3)C⊗U(1)Q,
another Higgs scalar triplet is needed: (2.3)ϕ=(ϕ1+ϕ20ϕ3+)~(1,3,23) with the VEV as follows: (2.4)〈ϕ〉=12(0v0). The Yukawa interactions which induce masses for the
fermions can be written in the most general form as follows: (2.5)ℒY=ℒLNC+ℒLNV, where LNC and LNV, respectively,
indicate to the lepton number conserving and violating ones as shown below.
Here, each part is defined by (2.6)ℒLNC=hUQ¯1LχUR+hαβDQ¯αLχ*DβR+hadQ¯1LϕdaR+hαauQ¯αLϕ*uaR+hablψ¯aLϕlbR+habνϵpmn(ψ¯aLc)p(ψbL)m(ϕ)n+H.c.,(2.7)ℒLNV=sauQ¯1LχuaR+sαadQ¯αLχ*daR+sαDQ¯1LϕDαR+sαUQ¯αLϕ*UR+H.c., where p, m,
and n stand for SU(3)L indices.The VEVω gives mass for the exotic quarks U and Dα: u gives mass for u1,dα,
while v gives mass for uα,d1, and all ordinary leptons. In Section 4, we
will provide more details on analysis of fermion masses. As mentioned, ω is responsible for the first stage of symmetry
breaking, while the second stage is due to u and v;
therefore, the VEVs in this model satisfies the constraint: (2.8)u2,v2≪ω2. The Yukawa couplings in (2.6) possess an extra global
symmetry [49, 50] which is not broken by v,ω,
but by u.
From these couplings, one can find the following lepton symmetry L as in Table 1 (only the fields with nonzero L are listed; all other fields have vanishing L). Here, L is broken by u which is behind L(χ10)=2,
that is, uis 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 L obviously does not commute with the gauge
symmetry. One can then construct a new conserved charge ℒ through L by making a linear combination L=xT3+yT8+ℒI.
Applying L on a lepton triplet, the coefficients will be
determined: (2.9)L=43T8+ℒI. Another useful conserved charge ℬ which is exactly not broken by u, v,
and ω is usual baryon number: B=ℬI.
Both the charges ℒ and ℬ for the fermion and Higgs multiplets are
listed in Table 2.Table 1
Nonzero lepton numberL of the model particles.
FieldνaLlaL,RνaRcχ10χ2−ϕ3+UL,RDαL,RL11−122−2−22Table 2
ℬ and ℒ charges of the model multiplets.
MultipletχϕQ1LQαLuaRdaRURDαRψaLlaRℬ-charge0013131313131300ℒ-charge43−23−232300−22131Let us note that the Yukawa couplings of (2.7) conserve ℬ,
however, violate ℒ with ±2 units which implies that these interactions
are much smaller than the first ones [41]: (2.10)sau,sαad,sαD,sαU≪hU,hαβD,had,hαau. 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:(2.11)V(χ,ϕ)=μ12χ†χ+μ22ϕ†ϕ+λ1(χ†χ)2+λ2(ϕ†ϕ)2+λ3(χ†χ)(ϕ†ϕ)+λ4(χ†ϕ)(ϕ†χ). It is noteworthy that V(χ,ϕ) 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 notSU(3)L⊗U(1)X singlets. Hence, nonzero values of χ and ϕ at the minimum value of V(χ,ϕ) can be easily obtained by (for details, see
Section 3): (2.12)χ†χ≡u2+ω22=λ3μ22−2λ2μ124λ1λ2−λ32,(2.13)ϕ†ϕ≡v22=λ3μ12−2λ1μ224λ1λ2−λ32. It is important noting that any
other choice of u,ω for the vacuum value of χ satisfying (2.12) gives the same physics because
it is related to (2.2) by an SU(3)L⊗U(1)X transformation. It is worth noting that the
assumed u≠0 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(2.14)Dμ=∂μ−igTiWiμ−igXT9XBμ≡∂μ−i𝒫μ, where the gauge fields Wi and B transform as the adjoint representations of SU(3)L and U(1)X,
respectively, and the corresponding gauge coupling constants g,gX.
Moreover, T9=(1/6)diag(1,1,1) is fixed so that the relation Tr(TiTj)=(1/2)δij(i,j=1,2,…,9)  is satisfied. The 𝒫μ matrix appeared in the above covariant derivative is rewritten in a
convenient form: (2.15)𝒫μ=g2(W3μ+W8μ3+t23XBμ2Wμ′+2Xμ′02Wμ′−−W3μ+W8μ3+t23XBμ2Yμ′−2Xμ′0*2Yμ′+−2W8μ3+t23XBμ), where t≡gX/g.
Let us denote the following combinations: (2.16)Wμ′±≡W1μ∓iW2μ2,Yμ′∓≡W6μ∓iW7μ2,Xμ′0≡W4μ−iW5μ2 having defined charges under the
generators of the SU(3)L group. For the sake of convenience in further
reading, we note that W4 and W5 are pure real and imaginary parts of Xμ′0 and Xμ′0*,
respectively: (2.17)W4μ=12(Xμ′0+Xμ′0*),W5μ=i2(Xμ′0−Xμ′0*). The masses of the gauge bosons in this model are
followed from (2.18)ℒmassGB=(Dμ〈ϕ〉)†(Dμ〈ϕ〉)+(Dμ〈χ〉)†(Dμ〈χ〉)=g24(u2+v2)Wμ′−W′+μ+g24(ω2+v2)Y′μ−Y′+μ+g2uω4(Wμ′−Y′+μ+Yμ′−W′+μ)+g2v28(−W3μ+13W8μ+t2323Bμ)2+g2u28(W3μ+13W8μ−t1323Bμ)2+g2ω28(−23W8μ−t1323Bμ)2+g2uω42(W3μ+13W8μ−t1323Bμ)(X′0μ+X′0*μ)+g2uω42(−23W8μ−t1323Bμ)(X′0μ+X′0*μ)+g216(u2+ω2){(Xμ′0+Xμ′0*)2+[i(Xμ′0−Xμ′0*)]2}. The combinations W′ and Y′ are mixing via (2.19)ℒmassCG=g24(Wμ′−,Yμ′−)(u2+v2uωuωω2+v2)(W′+μY′+μ). Diagonalizing this mass matrix,
we get physical charged gauge
bosons: (2.20)Wμ=cosθWμ′−sinθYμ′,Yμ=sinθWμ′+cosθYμ′, where the mixing angle is
defined by (2.21)tanθ=uω. The mass eigenvalues are (2.22)MW2=g2v24,(2.23)MY2=g24(u2+v2+ω2). Because of the constraints in
(2.8), the following remarks are in order:(1)
θ should be very small, and then Wμ≃Wμ′,Yμ≃Yμ′;(2)
v≃vweak=246 GeV due to identification of W as the W boson in the standard model. Next, from (2.18), the W5 gains mass as follows: (2.24)MW52=g24(ω2+u2). Finally, there is a mixing among W3,W8,B,W4 components. In the basis of these elements,
the mass matrix is given by (2.25)M2=g24(u2+v2u2−v23−2t36(u2+2v2)2uωu2−v2313(4ω2+u2+v2)2t9(2ω2−u2+2v2)−23uω−2t36(u2+2v2)2t9(2ω2−u2+2v2)2t227(ω2+u2+4v2)−8t36uω2uω−23uω−8t36uωu2+ω2). Note that the mass Lagrangian in
this case has the form (2.26)ℒmassNG=12VTM2V,VT≡(W3,W8,B,W4). In the limit u→0, W4 does not mix with W3,W8,B.
In the general case u≠0,
the mass matrix in (2.25) contains two exact eigenvalues such as (2.27)Mγ2=0,MW4′2=g24(ω2+u2). Thus, the W4′ and W5 components have the same mass, and this
conclusion contradicts the previous
analysis in [36].
With this result, we should identify the combination of W4′ and W5: (2.28)2Xμ0=W4μ′−iW5μ as physical neutral non-Hermitian gauge boson. The subscript 0 denotes neutrality of gauge boson X.
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: (2.29)MY2=MX2+MW2. Thus, the charged bilepton Y is slightly heavier than the neutral one X.
Remind that the similar relation in the 3-3-1 model with right-handed neutrinos
is [88]: |MY2−MX2|≤mW2.Now, we turn to the eigenstate question. The
eigenstates corresponding to the two values in (2.27) are determined as
follows: (2.30)Aμ=118+4t2(3t−t320),W4μ′=11+4tan22θ(tan2θ3tan2θ01). 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: (2.31)ℒintEM=−3gX18+4t2l¯γμlAμ. Therefore, the coefficient of
the electromagnetic coupling constant can be identified as (2.32)3gX18+4t2=e. Using continuation of the gauge
coupling constant g of SU(3)L at the spontaneous symmetry breaking
point, (2.33)g=g[SU(2)L]=esW from which it follows (2.34)t=32sW3−4sW2. The eigenstates are now
rewritten as follows: (2.35)Aμ=sWW3μ+cW(−tW3W8μ+1−tW23Bμ),W4μ′=t2θ1+4t2θ2W3μ+3t2θ1+4t2θ2W8μ+11+4t2θ2W4μ, where we have denoted sW≡sinθW,t2θ≡tan2θ and so forth.The diagonalization of the mass matrix is done via
three steps. In the first step, it is in the base of(Aμ,Zμ,Zμ′,W4μ),
where the two remaining gauge vectors are given by (2.36)Zμ=cWW3μ−sW(−tW3W8μ+1−tW23Bμ),Zμ′=1−tW23W8μ+tW3Bμ. In this basis, the mass matrix M2 becomes (2.37)M′2=g24(00000u2+v2cW2c2Wu2−v2cW23−4sW22uωcW0c2Wu2−v2cW23−4sW2v2+4cW4ω2+c2W2u2cW2(3−4sW2)−2uωcW3−4sW202uωcW−2uωcW3−4sW2u2+ω2). Also, in the limit u→0,W4μ does not mix with Zμ,Zμ′.
The eigenstate W4μ′ is now defined by (2.38)W4μ′=t2θcW1+4t2θ2Zμ+4cW2−1t2θcW1+4t2θ2Zμ′+11+4t2θ2W4μ. We turn to the second step. To see explicitly that the
following basis is orthogonal and normalized, let us put (2.39)sθ′≡t2θcW1+4t2θ2, which leads to (2.40)W4μ′=sθ′Zμ+cθ′[tθ′4cW2−1Zμ′+1−tθ′2(4cW2−1)W4μ]. Note that the mixing angle in
this step θ′ is the same order as the mixing angle in the
charged gauge boson sector. Taking into account [3] sW2≃0.231,
from (2.39) we get sθ′≃2.28sθ.
It is now easy to choose two remaining gauge vectors orthogonal to W4μ′: (2.41)𝒵μ=cθ′Zμ−sθ′[tθ′4cW2−1Zμ′+1−tθ′2(4cW2−1)W4μ],𝒵μ′=1−tθ′2(4cW2−1)Zμ′−tθ′4cW2−1W4μ. Therefore, in the base of (Aμ,𝒵μ,𝒵μ′,W4μ′), the mass matrix M′2 has a quasi-diagonal form: (2.42)M′′2=(00000m𝒵2m𝒵𝒵′200m𝒵𝒵'2m𝒵'20000g24(u2+ω2)) with (2.43)m𝒵2=(1+3t2θ2)u2+(1+4t2θ2)v2−t2θ2ω24g−2[cW2+(3−4sW2)t2θ2],m𝒵𝒵′2=1+4t2θ2{[c2W+(3−4sW2)t2θ2]u2−v2−(3−4sW2)t2θ2ω2}4g−23−4sW2[cW2+(3−4sW2)t2θ2],m𝒵'2=[c2W2+(3−4s2W2)t2θ2]u2+v2+[4cW4+(1+4cW2)(3−4sW2)t2θ2]ω24g−2(3−4sW2)[cW2+(3−4sW2)t2θ2]. In the last step, it is trivial to diagonalize the
mass matrix in (2.42). The two remaining mass eigenstates are given by (2.44)Zμ1=cφ𝒵μ−sφ𝒵μ′,Zμ2=sφ𝒵μ+cφ𝒵μ′, where the mixing angle φ between 𝒵 and 𝒵′ is defined by (2.45)t2φ={[(3−4sW2)(1+4t2θ2)]1/2{[c2W+(3−4sW2)t2θ2]u2−v2−(3−4sW2)t2θ2ω2}}×{[2sW4−1+(8sW4−2sW2−3)t2θ2]u2−[c2W+2(3−4sW2)t2θ2]v2+[2cW4+(8sW4+9c2W)t2θ2]ω2}−1. The physical mass eigenvalues
are defined by (2.46)MZ12=[2g−2(3−4sW2)]−1{cW2(u2+ω2)+v2−[cW2(u2+ω2)+v2]2+(3−4sW2)(3u2ω2−u2v2−v2ω2)},MZ22=[2g−2(3−4sW2)]−1{cW2(u2+ω2)+v2+[cW2(u2+ω2)+v2]2+(3−4sW2)(3u2ω2−u2v2−v2ω2)}. Because of the condition (2.8), the angle φ has to be very small: (2.47)t2φ≃−3−4sW2[v2+(11−14sW2)u2]2cW4ω2. In this approximation, the above
physical states have masses: (2.48)MZ12≃g24cW2(v2−3u2),(2.49)MZ22≃g2cW2ω23−4sW2. Consequently, Z1 can be identified as the Z boson in the standard model, and Z2 being the new neutral (Hermitian) gauge boson.
It is important to note that in the limit u→0 the mixing angle φ between 𝒵 and 𝒵′ is always nonvanishing. This differs from the
mixing angle θ between the W boson of the standard model and the
singly-charged bilepton Y.
Phenomenology of the mentioned mixing is quite similar to the WL−WR mixing in the left-right symmetric model based
on the SU(2)R⊗SU(2)L⊗U(1)B−L group (the interested reader can find in
[90]).


### 2.3. Currents

The interaction among fermions with gauge bosons
arises in part from(2.50)iψ¯γμDμψ=kinematicterms+HCC+HNC.


#### 2.3.1. Charged Currents

Despite
neutrality, the gauge bosonsX0,X0* belong to this section by their nature.
Because of the mixing among the standard model W boson and the charged bilepton Y as well as among (X0+X0*) with (W3,W8,B),
the new interaction terms exist as follows: (2.51)HCC=g2(JWμ−Wμ++JYμ−Yμ++JXμ0*Xμ0+H.c.), where (2.52)JWμ−=cθ(ν¯aLγμlaL+u¯aLγμdaL)−sθ(ν¯aLcγμlaL+U¯Lγμd1L+u¯αLγμDαL),(2.53)JYμ−=cθ(ν¯aLcγμlaL+U¯Lγμd1L+u¯αLγμDαL)+sθ(ν¯aLγμlaL+u¯aLγμdaL),(2.54)JXμ0*≃(1−t2θ2)(ν¯aLγμνaLc+u¯1LγμUL−D¯αLγμdαL)−t2θ2(ν¯aLcγμνaL+U¯Lγμu1L−d¯αLγμDαL)+t2θ1+4t2θ2×(ν¯aγμνa+u¯1Lγμu1L−U¯LγμUL−d¯αLγμdαL+D¯αLγμDαL). 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 sinθ or its square sin2θ). 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 neutralX′0 mixes with the real neutral ones such as Z and Z′.
This gives the unusual term as
follows: (2.55)HNC=eAμJμEM+ℒNC+ℒunnormalNC. Despite the mixing among W3,W8,B,W4,
the electromagnetic interactions remain the same as in the standard model
and the usual 3-3-1 model with right-handed neutrinos, that is, (2.56)JμEM=∑fqff¯γμf, where f runs among all the fermions of the model.Interactions of the neutral currents with fermions
have a common form:(2.57)ℒNC=g2cWf¯γμ[gkV(f)−gkA(f)γ5]fZμk,k=1,2, where (2.58)g1V(f)=cφ{T3(fL)−3t2θ2X(fL)+[(3−8sW2)t2θ2−2sW2]Q(f)}(1+4t2θ2)[1+(3−tW2)t2θ2]−sφ[(4cW2−1)T3(fL)+3cW2X(fL)−(3−5sW2)Q(f)](4cW2−1)[1+(3−tW2)t2θ2],g1A(f)=cφ[T3(fL)−3t2θ2(X−Q)(fL)](1+4t2θ2)[1+(3−tW2)t2θ2]−sφ[(4cW2−1)T3(fL)+3cW2(X−Q)(fL)](4cW2−1)[1+(3−tW2)t2θ2],g2V(f)=g1V(f)(cφ→sφ,sφ→−cφ),g2A(f)=g1A(f)(cφ→sφ,sφ→−cφ). Here, T3(fL),X(fL), and Q(f) are, respectively, the third component of the
weak isospin, the U(1)X charge, and the electric charge of the fermion fL.
Note that the isospin for the SU(2)L fermion singlet (in the bottom of triplets)
vanishes: T3(fL)=0.
The values of g1V(f),g1A(f) and g2V(f) are listed in Tables 3 and 4.Table 3
TheZμ1→f¯f couplings.
fg1V(f)g1A(f)νacφ−sφ(4cW2−1)(1+4t2θ2)2(1+4t2θ2)[1+(3−tW2)t2θ2]cφ(4cW2−1)(1+4t2θ2)+sφ2(4cW2−1)[1+(3−tW2)t2θ2]la(3−4cW2)[cφ(4cW2−1)(1+4t2θ2)+sφ]2(4cW2−1)[1+(3−tW2)t2θ2]−cφ(4cW2−1)(1+4t2θ2)+sφ2(4cW2−1)[1+(3−tW2)t2θ2]u1cφ4cW2−1[3(1+2t2θ2)−8sW2(1+4t2θ2)]−sφ(3+2sW2)1+4t2θ26(4cW2−1)(1+4t2θ2)[1+(3−tW2)t2θ2]cφ4cW2−1(1+2t2θ2)−sφc2W1+4t2θ22(4cW2−1)(1+4t2θ2)[1+(3−tW2)t2θ2]d1(1−4cW2)[cφ(4cW2−1)(1+4t2θ2)+sφ]6(4cW2−1)[1+(3−tW2)t2θ2]−cφ(4cW2−1)(1+4t2θ2)+sφ2(4cW2−1)[1+(3−tW2)t2θ2]uα(3−8sW2)[cφ(4cW2−1)(1+4t2θ2)+sφ]6(4cW2−1)[1+(3−tW2)t2θ2]cφ(4cW2−1)(1+4t2θ2)+sφ2(4cW2−1)[1+(3−tW2)t2θ2]dαcφ4cW2−1[(1−4cW2)(1+4t2θ2)+6t2θ2]+sφ(1+2cW2)1+4t2θ26(4cW2−1)(1+4t2θ2)[1+(3−tW2)t2θ2]−cφ4cW2−1(1+2t2θ2)−sφc2W1+4t2θ22(4cW2−1)(1+4t2θ2)[1+(3−tW2)t2θ2]Ucφ4cW2−1[3t2θ2−4sW2(1+4t2θ2)]+sφ(3−7sW2)1+4t2θ23(4cW2−1)(1+4t2θ2)[1+(3−tW2)t2θ2]cφ4cW2−1t2θ2+sφcW21+4t2θ2(4cW2−1)(1+4t2θ2)[1+(3−tW2)t2θ2]Dαcφ4cW2−1[2sW2(1+4t2θ2)−3t2θ2]−sφ(3−5sW2)1+4t2θ23(4cW2−1)(1+4t2θ2)[1+(3−tW2)t2θ2]−cφ4cW2−1t2θ2+sφcW21+4t2θ2(4cW2−1)(1+4t2θ2)[1+(3−tW2)t2θ2]Table 4
TheZμ2→f¯f couplings.
fg2V(f)g2A(f)νasφ+cφ(4cW2−1)(1+4t2θ2)2(1+4t2θ2)[1+(3−tW2)t2θ2]sφ(4cW2−1)(1+4t2θ2)−cφ2(4cW2−1)[1+(3−tW2)t2θ2]la(3−4cW2)[sφ(4cW2−1)(1+4t2θ2)−cφ]2(4cW2−1)[1+(3−tW2)t2θ2]−sφ(4cW2−1)(1+4t2θ2)−cφ2(4cW2−1)[1+(3−tW2)t2θ2]u1sφ4cW2−1[3(1+2t2θ2)−8sW2(1+4t2θ2)]+cφ(3+2sW2)1+4t2θ26(4cW2−1)(1+4t2θ2)[1+(3−tW2)t2θ2]sφ4cW2−1(1+2t2θ2)+cφc2W1+4t2θ22(4cW2−1)(1+4t2θ2)[1+(3−tW2)t2θ2]d1(1−4cW2)[sφ(4cW2−1)(1+4t2θ2)−cφ]6(4cW2−1)[1+(3−tW2)t2θ2]−sφ(4cW2−1)(1+4t2θ2)−cφ2(4cW2−1)[1+(3−tW2)t2θ2]uα(3−8sW2)[sφ(4cW2−1)(1+4t2θ2)−cφ]6(4cW2−1)[1+(3−tW2)t2θ2]sφ(4cW2−1)(1+4t2θ2)−cφ2(4cW2−1)[1+(3−tW2)t2θ2]dαsφ4cW2−1[(1−4cW2)(1+4t2θ2)+6t2θ2]−cφ(1+2cW2)1+4t2θ26(4cW2−1)(1+4t2θ2)[1+(3−tW2)t2θ2]−sφ4cW2−1(1+2t2θ2)+cφc2W1+4t2θ22(4cW2−1)(1+4t2θ2)[1+(3−tW2)t2θ2]Usφ4cW2−1[3t2θ2−4sW2(1+4t2θ2)]−cφ(3−7sW2)1+4t2θ23(4cW2−1)(1+4t2θ2)[1+(3−tW2)t2θ2]sφ4cW2−1t2θ2−cφcW21+4t2θ2(4cW2−1)(1+4t2θ2)[1+(3−tW2)t2θ2]Dαsφ4cW2−1[2sW2(1+4t2θ2)−3t2θ2]+cφ(3−5sW2)1+4t2θ23(4cW2−1)(1+4t2θ2)[1+(3−tW2)t2θ2]−sφ4cW2−1t2θ2−cφcW21+4t2θ2(4cW2−1)(1+4t2θ2)[1+(3−tW2)t2θ2]Because of the above-mentioned mixing, the
lepton-number violating interactions mediated by neutral gauge bosonsZ1 and Z2 exist in the neutrino and the exotic quark sectors: (2.59)ℒunnormalNC=−gt2θgkV(ν)2(ν¯aLγμνaLc+u¯1LγμUL−D¯αLγμdαL)Zμk+H.c. Again, these interactions are
very weak and proportional to sinθ.
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
angleθ and the charged bilepton mass MY.
Below we present a simple one. In our model, the W boson has the following normal main decay modes: (2.60)W−→lνl(l=e,μ,τ),↘ucd,ucs,ucb,(u→c), 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 (ΔL=2)—the model's specific feature: (2.61)W−→lνl(l=e,μ,τ). It is easy to compute the
tree-level decay widths as follows [91, 92]: (2.62)ΓBorn(W→lνl)=g2cθ28MW6π(1−x)(1−x2−x22)≃cθ2αMW12sW2,ΓBorn(W→lνl)=g2sθ28MW6π(1−x)(1−x2−x22)≃sθ2αMW12sW2,x≡ml2MW2,∑colorΓBorn(W→uicdj)=3g2cθ28MW6π|Vij|2[1−2(x+x¯)+(x−x¯)2]1/2×[1−x+x¯2−(x−x¯)22]≃cθ2αMW4sW2|Vij|2,x≡mdj2MW2,x¯≡muic2MW2. Quantum chromodynamics radiative
corrections modify (2.62) by a multiplicative factor [3, 91, 92]: (2.63)δQCQ=1+αs(MZ)π+1.409αs2π2−12.77αs3π3≃1.04, which is estimated from αs(MZ)≃0.12138.
All the state masses can be ignored, the predicted total width for W decay into fermions is (2.64)ΓWtot=1.04αMW2sW2(1−sθ2)+αMW4sW2. Taking α(MZ)≃1/128,MW=80.425GeV,sW2=0.2312,
and ΓWtot=2.124±0.041GeV [3], in Figure 1, we have plotted ΓWtot as function of sθ. From the
figure we get an upper limit: (2.65)sinθ≤0.08. It is important to note that
this limit value on the LNV parameter u/ω is much larger than those in [50, 93, 94].Figure 1
W width as function of sinθ,
and the horizontal lines are an upper and a lower limit.Since one of the VEVs is closely to the those in the
standard model:v≃vweak=246 GeV, therefore only two free VEVs exist in the
considering model, namely, u and ω.
The bilepton mass limit can be obtained from the “wrong" muon decay: (2.66)μ−→e−νeνμ, mediated at the tree level, by
both the standard model W and the singly-charged bilepton Y (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 Y.
In our case, the second diagram in Figure 2 gives main contribution. Taking into
account of the famous experimental data [3] (2.67)Rmuon≡Γ(μ−→e−νeνμ)Γ(μ−→e−νeνμ)<1.2%90%CL, we get the constraint: Rmuon≃MW4/MY4.
Therefore, it follows that MY≥230 GeV.Feynman
diagram for the wrong muon decayμ−→e−νeνμ.
(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 ofu→0, analyzing the Z decay width [37, 95, 96], the Z−Z′ mixing angle is constrained by −0.0015≤φ≤0.001.
From atomic parity violation in cesium, bounds for mass of the new exotic Z′ and the Z−Z′ mixing angles, again in the limit u→0,
are given [37, 95, 96]: (2.68)−0.00156≤φ≤0.00105,MZ2≥2.1TeV. 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 u≠0 of the constraints.For our purpose, we consider theρ parameter—one of the most important
quantities of the standard model, having a leading contribution in terms of the T parameter, is very useful to get the
new-physics effects. It is well-known relation between ρ and T parameter: (2.69)ρ=1+αT. In the usual 3-3-1 model with
right-handed neutrinos, T gets contribution from the oblique correction
and the Z−Z′ mixing [88]: (2.70)TRHN=TZZ′+Toblique, where TZZ′≃(tan2φ/α)(MZ22/MZ12−1) is negligible for MZ′ less than 1 TeV; Toblique depends on masses of the top quark and the
standard model Higgs boson. Again, at the tree level and the limit (2.8), from
(2.22) and (2.48) we get an expression for the ρ parameter in the considering model: (2.71)ρ=MW2cW2MZ12=v2v2−3u2≃1+3u2v2. Note that (2.71) has only one free
parameter u,
since v is very close to the VEV in the standard
model. Neglecting the contribution from the usual 3-3-1 model with right-handed
neutrinos and taking into account the experimental data [3] ρ=0.9987±0.0016,
we get the constraint on u parameter by u/v≤0.01 which leads to u≤2.46 GeV. This means that u is much smaller than v,
as expected.It seems that theρ parameter, at the tree level, in this model,
is favorable to be bigger than one and this is similar to the case of the
models contained heavy Z′ [98].The interesting new physics compared with other 3-3-1
models is the neutrino physics. Due to lepton-number violating couplings, we
have the following interesting consequences.(1) Processes withΔL=±2
From the charged currents, we have the following
lepton-number violatingΔL=±2 decays such as (2.72)μ−→e−νeνμ,μ−→e−νeνμ,(μcanbereplacedbyτ) in which both the standard model W boson and charged bilepton Yμ− are in intermediate states (see Figure 3). Here, the main
contribution arises from the first diagram. Note that the wrong muon decay
violates only family lepton-number, that is, ΔL=0,
but not lepton number at all as in (2.72). The decay rates are given by (2.73)Rrare≡Γ(μ−→e−νeνμ)Γ(μ−→e−νeνμ)=Γ(μ−→e−νeνμ)Γ(μ−→e−νeνμ)≃sθ2. Taking sθ=0.08,
we get Rrare≃6×10−3.
This rate is the same as the wrong muon decay one. Interesting to note that,
the family lepton-number violating processes (2.74)νiνi→νjνj,(i≠j) are mediated not only by the
non-Hermitian bilepton X but also by the Hermitian neutral Z1,Z2 (see Figure 4).Feynman
diagram forμ−→e−νeνμ.
(a)(b)Feynman
diagram forνiνi→νjνj(i≠j=e,μ,τ).
(a)(b)The first diagram in Figure4 exists also in the 3-3-1
model with right-handed neutrinos, but the second one does not appear there.(2)Lepton-Number Violating Kaon Decays
Next, let us consider the lepton-number violating decay
[3]: (2.75)K+→π0+e+νe<3×10−3at90%CL. This decay can be explained in
the considering model as the subprocess given below: (2.76)s→u+e+νe. This process is mediated by the
standard model W boson and the charged bilepton Y.
Amplitude of the considered process is proportional to sinθ: (2.77)M(s→u+e+νe)≃sin2θ2MW2(1−MW2MY2). Next, let us consider the
“normal decay" [3]: (2.78)K+→π0+e+νe(4.87±0.06)% with amplitude (2.79)M(s→u+e+νe)≃1MW2. From (2.77) and (2.79), we get (2.80)Rkaon≡Γ(s→u+e+νe)Γ(s→u+e+νe)≃sin2θ.In the framework of this model, we derive the
following decay modes with rates:(2.81)Rkaon=Γ(K+→π0+e+νe)Γ(K+→π0+e+νe)≃Γ(K+→π0+μ+νμ)Γ(K+→π0+μ+νμ)≃sin2θ≤6×10−3. Note that the similar
lepton-number violating processes exist in the  SU(2)R⊗SU(2)L⊗U(1)B−L model (for details, see [90]).


### 2.5. Summary

In this section, we have presented the 3-3-1 model
with the minimal scalar sector (only two Higgs triplets). This version belongs
to the 3-3-1 model without exotic charges (charges of the exotic quarks are2/3 and −1/3). The spontaneous symmetry breakdown is
achieved with only two Higgs triplets. One of the VEVs u is a source of lepton-number violations and a
reason for the mixing between the charged gauge bosons—the standard model W and the singly-charged bilepton gauge bosons
as well as between neutral non-Hermitian X0 and neutral gauge bosons: the Z and the new exotic Z′.
At the tree level, masses of the charged gauge bosons satisfy the law of
Pythagoras MY2=MX2+MW2 and in the limit ω≫u,v,
the ρ parameter gets additional contribution
dependent only on u/v.
Thus, this leads to u≪v,
and there are three quite different scales for the VEVs of the model: one is
very small u≃O(1) GeV—a lepton-number violating parameter; the
second v is close to the standard model one: v≃vweak=246 GeV; and the last is in the range of new physics
scale about O(1) TeV.In difference with the usual 3-3-1 model with
right-handed neutrinos, in this model the first family of quarks should be
distinctive of the two others.The exact diagonalization of the neutral gauge boson sector
is derived. Because of the parameteru,
the lepton-number violation happens only in neutrino but not in charged lepton
sector. It is interesting to note that despite the above-mentioned mixing, the
electromagnetic current remains unchanged. In this model, the lepton-number
changing (ΔL=±2) processes exist but only in the neutrino
sector.It is worth mentioning on the advantage of the
considered model: the new mixing angle between the charged gauge bosonsθ is connected with one of the VEVs u—the parameter of lepton-number violations.
There is no new parameter, but it contains very simple Higgs sector, hence the
significant number of free parameters is reduced.The model contains of new
kinds of interactions in the neutrino sector. Hence, neutrino physics in this
model is very rich. We will turn to further studies on neutrino masses and
mixing in Section4.


## 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:(2.1)χ=(χ10χ2−χ30)~(1,3,−13) with the VEV given by (2.2)〈χ〉=12(u0ω). In the last stage, to embed the
standard model gauge symmetry in SU(3)C⊗U(1)Q,
another Higgs scalar triplet is needed: (2.3)ϕ=(ϕ1+ϕ20ϕ3+)~(1,3,23) with the VEV as follows: (2.4)〈ϕ〉=12(0v0). The Yukawa interactions which induce masses for the
fermions can be written in the most general form as follows: (2.5)ℒY=ℒLNC+ℒLNV, where LNC and LNV, respectively,
indicate to the lepton number conserving and violating ones as shown below.
Here, each part is defined by (2.6)ℒLNC=hUQ¯1LχUR+hαβDQ¯αLχ*DβR+hadQ¯1LϕdaR+hαauQ¯αLϕ*uaR+hablψ¯aLϕlbR+habνϵpmn(ψ¯aLc)p(ψbL)m(ϕ)n+H.c.,(2.7)ℒLNV=sauQ¯1LχuaR+sαadQ¯αLχ*daR+sαDQ¯1LϕDαR+sαUQ¯αLϕ*UR+H.c., where p, m,
and n stand for SU(3)L indices.The VEVω gives mass for the exotic quarks U and Dα: u gives mass for u1,dα,
while v gives mass for uα,d1, and all ordinary leptons. In Section 4, we
will provide more details on analysis of fermion masses. As mentioned, ω is responsible for the first stage of symmetry
breaking, while the second stage is due to u and v;
therefore, the VEVs in this model satisfies the constraint: (2.8)u2,v2≪ω2. The Yukawa couplings in (2.6) possess an extra global
symmetry [49, 50] which is not broken by v,ω,
but by u.
From these couplings, one can find the following lepton symmetry L as in Table 1 (only the fields with nonzero L are listed; all other fields have vanishing L). Here, L is broken by u which is behind L(χ10)=2,
that is, uis 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 L obviously does not commute with the gauge
symmetry. One can then construct a new conserved charge ℒ through L by making a linear combination L=xT3+yT8+ℒI.
Applying L on a lepton triplet, the coefficients will be
determined: (2.9)L=43T8+ℒI. Another useful conserved charge ℬ which is exactly not broken by u, v,
and ω is usual baryon number: B=ℬI.
Both the charges ℒ and ℬ for the fermion and Higgs multiplets are
listed in Table 2.Table 1
Nonzero lepton numberL of the model particles.
FieldνaLlaL,RνaRcχ10χ2−ϕ3+UL,RDαL,RL11−122−2−22Table 2
ℬ and ℒ charges of the model multiplets.
MultipletχϕQ1LQαLuaRdaRURDαRψaLlaRℬ-charge0013131313131300ℒ-charge43−23−232300−22131Let us note that the Yukawa couplings of (2.7) conserve ℬ,
however, violate ℒ with ±2 units which implies that these interactions
are much smaller than the first ones [41]: (2.10)sau,sαad,sαD,sαU≪hU,hαβD,had,hαau. 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:(2.11)V(χ,ϕ)=μ12χ†χ+μ22ϕ†ϕ+λ1(χ†χ)2+λ2(ϕ†ϕ)2+λ3(χ†χ)(ϕ†ϕ)+λ4(χ†ϕ)(ϕ†χ). It is noteworthy that V(χ,ϕ) 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 notSU(3)L⊗U(1)X singlets. Hence, nonzero values of χ and ϕ at the minimum value of V(χ,ϕ) can be easily obtained by (for details, see
Section 3): (2.12)χ†χ≡u2+ω22=λ3μ22−2λ2μ124λ1λ2−λ32,(2.13)ϕ†ϕ≡v22=λ3μ12−2λ1μ224λ1λ2−λ32. It is important noting that any
other choice of u,ω for the vacuum value of χ satisfying (2.12) gives the same physics because
it is related to (2.2) by an SU(3)L⊗U(1)X transformation. It is worth noting that the
assumed u≠0 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(2.14)Dμ=∂μ−igTiWiμ−igXT9XBμ≡∂μ−i𝒫μ, where the gauge fields Wi and B transform as the adjoint representations of SU(3)L and U(1)X,
respectively, and the corresponding gauge coupling constants g,gX.
Moreover, T9=(1/6)diag(1,1,1) is fixed so that the relation Tr(TiTj)=(1/2)δij(i,j=1,2,…,9)  is satisfied. The 𝒫μ matrix appeared in the above covariant derivative is rewritten in a
convenient form: (2.15)𝒫μ=g2(W3μ+W8μ3+t23XBμ2Wμ′+2Xμ′02Wμ′−−W3μ+W8μ3+t23XBμ2Yμ′−2Xμ′0*2Yμ′+−2W8μ3+t23XBμ), where t≡gX/g.
Let us denote the following combinations: (2.16)Wμ′±≡W1μ∓iW2μ2,Yμ′∓≡W6μ∓iW7μ2,Xμ′0≡W4μ−iW5μ2 having defined charges under the
generators of the SU(3)L group. For the sake of convenience in further
reading, we note that W4 and W5 are pure real and imaginary parts of Xμ′0 and Xμ′0*,
respectively: (2.17)W4μ=12(Xμ′0+Xμ′0*),W5μ=i2(Xμ′0−Xμ′0*). The masses of the gauge bosons in this model are
followed from (2.18)ℒmassGB=(Dμ〈ϕ〉)†(Dμ〈ϕ〉)+(Dμ〈χ〉)†(Dμ〈χ〉)=g24(u2+v2)Wμ′−W′+μ+g24(ω2+v2)Y′μ−Y′+μ+g2uω4(Wμ′−Y′+μ+Yμ′−W′+μ)+g2v28(−W3μ+13W8μ+t2323Bμ)2+g2u28(W3μ+13W8μ−t1323Bμ)2+g2ω28(−23W8μ−t1323Bμ)2+g2uω42(W3μ+13W8μ−t1323Bμ)(X′0μ+X′0*μ)+g2uω42(−23W8μ−t1323Bμ)(X′0μ+X′0*μ)+g216(u2+ω2){(Xμ′0+Xμ′0*)2+[i(Xμ′0−Xμ′0*)]2}. The combinations W′ and Y′ are mixing via (2.19)ℒmassCG=g24(Wμ′−,Yμ′−)(u2+v2uωuωω2+v2)(W′+μY′+μ). Diagonalizing this mass matrix,
we get physical charged gauge
bosons: (2.20)Wμ=cosθWμ′−sinθYμ′,Yμ=sinθWμ′+cosθYμ′, where the mixing angle is
defined by (2.21)tanθ=uω. The mass eigenvalues are (2.22)MW2=g2v24,(2.23)MY2=g24(u2+v2+ω2). Because of the constraints in
(2.8), the following remarks are in order:(1)
θ should be very small, and then Wμ≃Wμ′,Yμ≃Yμ′;(2)
v≃vweak=246 GeV due to identification of W as the W boson in the standard model. Next, from (2.18), the W5 gains mass as follows: (2.24)MW52=g24(ω2+u2). Finally, there is a mixing among W3,W8,B,W4 components. In the basis of these elements,
the mass matrix is given by (2.25)M2=g24(u2+v2u2−v23−2t36(u2+2v2)2uωu2−v2313(4ω2+u2+v2)2t9(2ω2−u2+2v2)−23uω−2t36(u2+2v2)2t9(2ω2−u2+2v2)2t227(ω2+u2+4v2)−8t36uω2uω−23uω−8t36uωu2+ω2). Note that the mass Lagrangian in
this case has the form (2.26)ℒmassNG=12VTM2V,VT≡(W3,W8,B,W4). In the limit u→0, W4 does not mix with W3,W8,B.
In the general case u≠0,
the mass matrix in (2.25) contains two exact eigenvalues such as (2.27)Mγ2=0,MW4′2=g24(ω2+u2). Thus, the W4′ and W5 components have the same mass, and this
conclusion contradicts the previous
analysis in [36].
With this result, we should identify the combination of W4′ and W5: (2.28)2Xμ0=W4μ′−iW5μ as physical neutral non-Hermitian gauge boson. The subscript 0 denotes neutrality of gauge boson X.
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: (2.29)MY2=MX2+MW2. Thus, the charged bilepton Y is slightly heavier than the neutral one X.
Remind that the similar relation in the 3-3-1 model with right-handed neutrinos
is [88]: |MY2−MX2|≤mW2.Now, we turn to the eigenstate question. The
eigenstates corresponding to the two values in (2.27) are determined as
follows: (2.30)Aμ=118+4t2(3t−t320),W4μ′=11+4tan22θ(tan2θ3tan2θ01). 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: (2.31)ℒintEM=−3gX18+4t2l¯γμlAμ. Therefore, the coefficient of
the electromagnetic coupling constant can be identified as (2.32)3gX18+4t2=e. Using continuation of the gauge
coupling constant g of SU(3)L at the spontaneous symmetry breaking
point, (2.33)g=g[SU(2)L]=esW from which it follows (2.34)t=32sW3−4sW2. The eigenstates are now
rewritten as follows: (2.35)Aμ=sWW3μ+cW(−tW3W8μ+1−tW23Bμ),W4μ′=t2θ1+4t2θ2W3μ+3t2θ1+4t2θ2W8μ+11+4t2θ2W4μ, where we have denoted sW≡sinθW,t2θ≡tan2θ and so forth.The diagonalization of the mass matrix is done via
three steps. In the first step, it is in the base of(Aμ,Zμ,Zμ′,W4μ),
where the two remaining gauge vectors are given by (2.36)Zμ=cWW3μ−sW(−tW3W8μ+1−tW23Bμ),Zμ′=1−tW23W8μ+tW3Bμ. In this basis, the mass matrix M2 becomes (2.37)M′2=g24(00000u2+v2cW2c2Wu2−v2cW23−4sW22uωcW0c2Wu2−v2cW23−4sW2v2+4cW4ω2+c2W2u2cW2(3−4sW2)−2uωcW3−4sW202uωcW−2uωcW3−4sW2u2+ω2). Also, in the limit u→0,W4μ does not mix with Zμ,Zμ′.
The eigenstate W4μ′ is now defined by (2.38)W4μ′=t2θcW1+4t2θ2Zμ+4cW2−1t2θcW1+4t2θ2Zμ′+11+4t2θ2W4μ. We turn to the second step. To see explicitly that the
following basis is orthogonal and normalized, let us put (2.39)sθ′≡t2θcW1+4t2θ2, which leads to (2.40)W4μ′=sθ′Zμ+cθ′[tθ′4cW2−1Zμ′+1−tθ′2(4cW2−1)W4μ]. Note that the mixing angle in
this step θ′ is the same order as the mixing angle in the
charged gauge boson sector. Taking into account [3] sW2≃0.231,
from (2.39) we get sθ′≃2.28sθ.
It is now easy to choose two remaining gauge vectors orthogonal to W4μ′: (2.41)𝒵μ=cθ′Zμ−sθ′[tθ′4cW2−1Zμ′+1−tθ′2(4cW2−1)W4μ],𝒵μ′=1−tθ′2(4cW2−1)Zμ′−tθ′4cW2−1W4μ. Therefore, in the base of (Aμ,𝒵μ,𝒵μ′,W4μ′), the mass matrix M′2 has a quasi-diagonal form: (2.42)M′′2=(00000m𝒵2m𝒵𝒵′200m𝒵𝒵'2m𝒵'20000g24(u2+ω2)) with (2.43)m𝒵2=(1+3t2θ2)u2+(1+4t2θ2)v2−t2θ2ω24g−2[cW2+(3−4sW2)t2θ2],m𝒵𝒵′2=1+4t2θ2{[c2W+(3−4sW2)t2θ2]u2−v2−(3−4sW2)t2θ2ω2}4g−23−4sW2[cW2+(3−4sW2)t2θ2],m𝒵'2=[c2W2+(3−4s2W2)t2θ2]u2+v2+[4cW4+(1+4cW2)(3−4sW2)t2θ2]ω24g−2(3−4sW2)[cW2+(3−4sW2)t2θ2]. In the last step, it is trivial to diagonalize the
mass matrix in (2.42). The two remaining mass eigenstates are given by (2.44)Zμ1=cφ𝒵μ−sφ𝒵μ′,Zμ2=sφ𝒵μ+cφ𝒵μ′, where the mixing angle φ between 𝒵 and 𝒵′ is defined by (2.45)t2φ={[(3−4sW2)(1+4t2θ2)]1/2{[c2W+(3−4sW2)t2θ2]u2−v2−(3−4sW2)t2θ2ω2}}×{[2sW4−1+(8sW4−2sW2−3)t2θ2]u2−[c2W+2(3−4sW2)t2θ2]v2+[2cW4+(8sW4+9c2W)t2θ2]ω2}−1. The physical mass eigenvalues
are defined by (2.46)MZ12=[2g−2(3−4sW2)]−1{cW2(u2+ω2)+v2−[cW2(u2+ω2)+v2]2+(3−4sW2)(3u2ω2−u2v2−v2ω2)},MZ22=[2g−2(3−4sW2)]−1{cW2(u2+ω2)+v2+[cW2(u2+ω2)+v2]2+(3−4sW2)(3u2ω2−u2v2−v2ω2)}. Because of the condition (2.8), the angle φ has to be very small: (2.47)t2φ≃−3−4sW2[v2+(11−14sW2)u2]2cW4ω2. In this approximation, the above
physical states have masses: (2.48)MZ12≃g24cW2(v2−3u2),(2.49)MZ22≃g2cW2ω23−4sW2. Consequently, Z1 can be identified as the Z boson in the standard model, and Z2 being the new neutral (Hermitian) gauge boson.
It is important to note that in the limit u→0 the mixing angle φ between 𝒵 and 𝒵′ is always nonvanishing. This differs from the
mixing angle θ between the W boson of the standard model and the
singly-charged bilepton Y.
Phenomenology of the mentioned mixing is quite similar to the WL−WR mixing in the left-right symmetric model based
on the SU(2)R⊗SU(2)L⊗U(1)B−L group (the interested reader can find in
[90]).


## 2.3. Currents

The interaction among fermions with gauge bosons
arises in part from(2.50)iψ¯γμDμψ=kinematicterms+HCC+HNC.


### 2.3.1. Charged Currents

Despite
neutrality, the gauge bosonsX0,X0* belong to this section by their nature.
Because of the mixing among the standard model W boson and the charged bilepton Y as well as among (X0+X0*) with (W3,W8,B),
the new interaction terms exist as follows: (2.51)HCC=g2(JWμ−Wμ++JYμ−Yμ++JXμ0*Xμ0+H.c.), where (2.52)JWμ−=cθ(ν¯aLγμlaL+u¯aLγμdaL)−sθ(ν¯aLcγμlaL+U¯Lγμd1L+u¯αLγμDαL),(2.53)JYμ−=cθ(ν¯aLcγμlaL+U¯Lγμd1L+u¯αLγμDαL)+sθ(ν¯aLγμlaL+u¯aLγμdaL),(2.54)JXμ0*≃(1−t2θ2)(ν¯aLγμνaLc+u¯1LγμUL−D¯αLγμdαL)−t2θ2(ν¯aLcγμνaL+U¯Lγμu1L−d¯αLγμDαL)+t2θ1+4t2θ2×(ν¯aγμνa+u¯1Lγμu1L−U¯LγμUL−d¯αLγμdαL+D¯αLγμDαL). 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 sinθ or its square sin2θ). 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 neutralX′0 mixes with the real neutral ones such as Z and Z′.
This gives the unusual term as
follows: (2.55)HNC=eAμJμEM+ℒNC+ℒunnormalNC. Despite the mixing among W3,W8,B,W4,
the electromagnetic interactions remain the same as in the standard model
and the usual 3-3-1 model with right-handed neutrinos, that is, (2.56)JμEM=∑fqff¯γμf, where f runs among all the fermions of the model.Interactions of the neutral currents with fermions
have a common form:(2.57)ℒNC=g2cWf¯γμ[gkV(f)−gkA(f)γ5]fZμk,k=1,2, where (2.58)g1V(f)=cφ{T3(fL)−3t2θ2X(fL)+[(3−8sW2)t2θ2−2sW2]Q(f)}(1+4t2θ2)[1+(3−tW2)t2θ2]−sφ[(4cW2−1)T3(fL)+3cW2X(fL)−(3−5sW2)Q(f)](4cW2−1)[1+(3−tW2)t2θ2],g1A(f)=cφ[T3(fL)−3t2θ2(X−Q)(fL)](1+4t2θ2)[1+(3−tW2)t2θ2]−sφ[(4cW2−1)T3(fL)+3cW2(X−Q)(fL)](4cW2−1)[1+(3−tW2)t2θ2],g2V(f)=g1V(f)(cφ→sφ,sφ→−cφ),g2A(f)=g1A(f)(cφ→sφ,sφ→−cφ). Here, T3(fL),X(fL), and Q(f) are, respectively, the third component of the
weak isospin, the U(1)X charge, and the electric charge of the fermion fL.
Note that the isospin for the SU(2)L fermion singlet (in the bottom of triplets)
vanishes: T3(fL)=0.
The values of g1V(f),g1A(f) and g2V(f) are listed in Tables 3 and 4.Table 3
TheZμ1→f¯f couplings.
fg1V(f)g1A(f)νacφ−sφ(4cW2−1)(1+4t2θ2)2(1+4t2θ2)[1+(3−tW2)t2θ2]cφ(4cW2−1)(1+4t2θ2)+sφ2(4cW2−1)[1+(3−tW2)t2θ2]la(3−4cW2)[cφ(4cW2−1)(1+4t2θ2)+sφ]2(4cW2−1)[1+(3−tW2)t2θ2]−cφ(4cW2−1)(1+4t2θ2)+sφ2(4cW2−1)[1+(3−tW2)t2θ2]u1cφ4cW2−1[3(1+2t2θ2)−8sW2(1+4t2θ2)]−sφ(3+2sW2)1+4t2θ26(4cW2−1)(1+4t2θ2)[1+(3−tW2)t2θ2]cφ4cW2−1(1+2t2θ2)−sφc2W1+4t2θ22(4cW2−1)(1+4t2θ2)[1+(3−tW2)t2θ2]d1(1−4cW2)[cφ(4cW2−1)(1+4t2θ2)+sφ]6(4cW2−1)[1+(3−tW2)t2θ2]−cφ(4cW2−1)(1+4t2θ2)+sφ2(4cW2−1)[1+(3−tW2)t2θ2]uα(3−8sW2)[cφ(4cW2−1)(1+4t2θ2)+sφ]6(4cW2−1)[1+(3−tW2)t2θ2]cφ(4cW2−1)(1+4t2θ2)+sφ2(4cW2−1)[1+(3−tW2)t2θ2]dαcφ4cW2−1[(1−4cW2)(1+4t2θ2)+6t2θ2]+sφ(1+2cW2)1+4t2θ26(4cW2−1)(1+4t2θ2)[1+(3−tW2)t2θ2]−cφ4cW2−1(1+2t2θ2)−sφc2W1+4t2θ22(4cW2−1)(1+4t2θ2)[1+(3−tW2)t2θ2]Ucφ4cW2−1[3t2θ2−4sW2(1+4t2θ2)]+sφ(3−7sW2)1+4t2θ23(4cW2−1)(1+4t2θ2)[1+(3−tW2)t2θ2]cφ4cW2−1t2θ2+sφcW21+4t2θ2(4cW2−1)(1+4t2θ2)[1+(3−tW2)t2θ2]Dαcφ4cW2−1[2sW2(1+4t2θ2)−3t2θ2]−sφ(3−5sW2)1+4t2θ23(4cW2−1)(1+4t2θ2)[1+(3−tW2)t2θ2]−cφ4cW2−1t2θ2+sφcW21+4t2θ2(4cW2−1)(1+4t2θ2)[1+(3−tW2)t2θ2]Table 4
TheZμ2→f¯f couplings.
fg2V(f)g2A(f)νasφ+cφ(4cW2−1)(1+4t2θ2)2(1+4t2θ2)[1+(3−tW2)t2θ2]sφ(4cW2−1)(1+4t2θ2)−cφ2(4cW2−1)[1+(3−tW2)t2θ2]la(3−4cW2)[sφ(4cW2−1)(1+4t2θ2)−cφ]2(4cW2−1)[1+(3−tW2)t2θ2]−sφ(4cW2−1)(1+4t2θ2)−cφ2(4cW2−1)[1+(3−tW2)t2θ2]u1sφ4cW2−1[3(1+2t2θ2)−8sW2(1+4t2θ2)]+cφ(3+2sW2)1+4t2θ26(4cW2−1)(1+4t2θ2)[1+(3−tW2)t2θ2]sφ4cW2−1(1+2t2θ2)+cφc2W1+4t2θ22(4cW2−1)(1+4t2θ2)[1+(3−tW2)t2θ2]d1(1−4cW2)[sφ(4cW2−1)(1+4t2θ2)−cφ]6(4cW2−1)[1+(3−tW2)t2θ2]−sφ(4cW2−1)(1+4t2θ2)−cφ2(4cW2−1)[1+(3−tW2)t2θ2]uα(3−8sW2)[sφ(4cW2−1)(1+4t2θ2)−cφ]6(4cW2−1)[1+(3−tW2)t2θ2]sφ(4cW2−1)(1+4t2θ2)−cφ2(4cW2−1)[1+(3−tW2)t2θ2]dαsφ4cW2−1[(1−4cW2)(1+4t2θ2)+6t2θ2]−cφ(1+2cW2)1+4t2θ26(4cW2−1)(1+4t2θ2)[1+(3−tW2)t2θ2]−sφ4cW2−1(1+2t2θ2)+cφc2W1+4t2θ22(4cW2−1)(1+4t2θ2)[1+(3−tW2)t2θ2]Usφ4cW2−1[3t2θ2−4sW2(1+4t2θ2)]−cφ(3−7sW2)1+4t2θ23(4cW2−1)(1+4t2θ2)[1+(3−tW2)t2θ2]sφ4cW2−1t2θ2−cφcW21+4t2θ2(4cW2−1)(1+4t2θ2)[1+(3−tW2)t2θ2]Dαsφ4cW2−1[2sW2(1+4t2θ2)−3t2θ2]+cφ(3−5sW2)1+4t2θ23(4cW2−1)(1+4t2θ2)[1+(3−tW2)t2θ2]−sφ4cW2−1t2θ2−cφcW21+4t2θ2(4cW2−1)(1+4t2θ2)[1+(3−tW2)t2θ2]Because of the above-mentioned mixing, the
lepton-number violating interactions mediated by neutral gauge bosonsZ1 and Z2 exist in the neutrino and the exotic quark sectors: (2.59)ℒunnormalNC=−gt2θgkV(ν)2(ν¯aLγμνaLc+u¯1LγμUL−D¯αLγμdαL)Zμk+H.c. Again, these interactions are
very weak and proportional to sinθ.
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.


## 2.3.1. Charged Currents

Despite
neutrality, the gauge bosonsX0,X0* belong to this section by their nature.
Because of the mixing among the standard model W boson and the charged bilepton Y as well as among (X0+X0*) with (W3,W8,B),
the new interaction terms exist as follows: (2.51)HCC=g2(JWμ−Wμ++JYμ−Yμ++JXμ0*Xμ0+H.c.), where (2.52)JWμ−=cθ(ν¯aLγμlaL+u¯aLγμdaL)−sθ(ν¯aLcγμlaL+U¯Lγμd1L+u¯αLγμDαL),(2.53)JYμ−=cθ(ν¯aLcγμlaL+U¯Lγμd1L+u¯αLγμDαL)+sθ(ν¯aLγμlaL+u¯aLγμdaL),(2.54)JXμ0*≃(1−t2θ2)(ν¯aLγμνaLc+u¯1LγμUL−D¯αLγμdαL)−t2θ2(ν¯aLcγμνaL+U¯Lγμu1L−d¯αLγμDαL)+t2θ1+4t2θ2×(ν¯aγμνa+u¯1Lγμu1L−U¯LγμUL−d¯αLγμdαL+D¯αLγμDαL). 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 sinθ or its square sin2θ). 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 neutralX′0 mixes with the real neutral ones such as Z and Z′.
This gives the unusual term as
follows: (2.55)HNC=eAμJμEM+ℒNC+ℒunnormalNC. Despite the mixing among W3,W8,B,W4,
the electromagnetic interactions remain the same as in the standard model
and the usual 3-3-1 model with right-handed neutrinos, that is, (2.56)JμEM=∑fqff¯γμf, where f runs among all the fermions of the model.Interactions of the neutral currents with fermions
have a common form:(2.57)ℒNC=g2cWf¯γμ[gkV(f)−gkA(f)γ5]fZμk,k=1,2, where (2.58)g1V(f)=cφ{T3(fL)−3t2θ2X(fL)+[(3−8sW2)t2θ2−2sW2]Q(f)}(1+4t2θ2)[1+(3−tW2)t2θ2]−sφ[(4cW2−1)T3(fL)+3cW2X(fL)−(3−5sW2)Q(f)](4cW2−1)[1+(3−tW2)t2θ2],g1A(f)=cφ[T3(fL)−3t2θ2(X−Q)(fL)](1+4t2θ2)[1+(3−tW2)t2θ2]−sφ[(4cW2−1)T3(fL)+3cW2(X−Q)(fL)](4cW2−1)[1+(3−tW2)t2θ2],g2V(f)=g1V(f)(cφ→sφ,sφ→−cφ),g2A(f)=g1A(f)(cφ→sφ,sφ→−cφ). Here, T3(fL),X(fL), and Q(f) are, respectively, the third component of the
weak isospin, the U(1)X charge, and the electric charge of the fermion fL.
Note that the isospin for the SU(2)L fermion singlet (in the bottom of triplets)
vanishes: T3(fL)=0.
The values of g1V(f),g1A(f) and g2V(f) are listed in Tables 3 and 4.Table 3
TheZμ1→f¯f couplings.
fg1V(f)g1A(f)νacφ−sφ(4cW2−1)(1+4t2θ2)2(1+4t2θ2)[1+(3−tW2)t2θ2]cφ(4cW2−1)(1+4t2θ2)+sφ2(4cW2−1)[1+(3−tW2)t2θ2]la(3−4cW2)[cφ(4cW2−1)(1+4t2θ2)+sφ]2(4cW2−1)[1+(3−tW2)t2θ2]−cφ(4cW2−1)(1+4t2θ2)+sφ2(4cW2−1)[1+(3−tW2)t2θ2]u1cφ4cW2−1[3(1+2t2θ2)−8sW2(1+4t2θ2)]−sφ(3+2sW2)1+4t2θ26(4cW2−1)(1+4t2θ2)[1+(3−tW2)t2θ2]cφ4cW2−1(1+2t2θ2)−sφc2W1+4t2θ22(4cW2−1)(1+4t2θ2)[1+(3−tW2)t2θ2]d1(1−4cW2)[cφ(4cW2−1)(1+4t2θ2)+sφ]6(4cW2−1)[1+(3−tW2)t2θ2]−cφ(4cW2−1)(1+4t2θ2)+sφ2(4cW2−1)[1+(3−tW2)t2θ2]uα(3−8sW2)[cφ(4cW2−1)(1+4t2θ2)+sφ]6(4cW2−1)[1+(3−tW2)t2θ2]cφ(4cW2−1)(1+4t2θ2)+sφ2(4cW2−1)[1+(3−tW2)t2θ2]dαcφ4cW2−1[(1−4cW2)(1+4t2θ2)+6t2θ2]+sφ(1+2cW2)1+4t2θ26(4cW2−1)(1+4t2θ2)[1+(3−tW2)t2θ2]−cφ4cW2−1(1+2t2θ2)−sφc2W1+4t2θ22(4cW2−1)(1+4t2θ2)[1+(3−tW2)t2θ2]Ucφ4cW2−1[3t2θ2−4sW2(1+4t2θ2)]+sφ(3−7sW2)1+4t2θ23(4cW2−1)(1+4t2θ2)[1+(3−tW2)t2θ2]cφ4cW2−1t2θ2+sφcW21+4t2θ2(4cW2−1)(1+4t2θ2)[1+(3−tW2)t2θ2]Dαcφ4cW2−1[2sW2(1+4t2θ2)−3t2θ2]−sφ(3−5sW2)1+4t2θ23(4cW2−1)(1+4t2θ2)[1+(3−tW2)t2θ2]−cφ4cW2−1t2θ2+sφcW21+4t2θ2(4cW2−1)(1+4t2θ2)[1+(3−tW2)t2θ2]Table 4
TheZμ2→f¯f couplings.
fg2V(f)g2A(f)νasφ+cφ(4cW2−1)(1+4t2θ2)2(1+4t2θ2)[1+(3−tW2)t2θ2]sφ(4cW2−1)(1+4t2θ2)−cφ2(4cW2−1)[1+(3−tW2)t2θ2]la(3−4cW2)[sφ(4cW2−1)(1+4t2θ2)−cφ]2(4cW2−1)[1+(3−tW2)t2θ2]−sφ(4cW2−1)(1+4t2θ2)−cφ2(4cW2−1)[1+(3−tW2)t2θ2]u1sφ4cW2−1[3(1+2t2θ2)−8sW2(1+4t2θ2)]+cφ(3+2sW2)1+4t2θ26(4cW2−1)(1+4t2θ2)[1+(3−tW2)t2θ2]sφ4cW2−1(1+2t2θ2)+cφc2W1+4t2θ22(4cW2−1)(1+4t2θ2)[1+(3−tW2)t2θ2]d1(1−4cW2)[sφ(4cW2−1)(1+4t2θ2)−cφ]6(4cW2−1)[1+(3−tW2)t2θ2]−sφ(4cW2−1)(1+4t2θ2)−cφ2(4cW2−1)[1+(3−tW2)t2θ2]uα(3−8sW2)[sφ(4cW2−1)(1+4t2θ2)−cφ]6(4cW2−1)[1+(3−tW2)t2θ2]sφ(4cW2−1)(1+4t2θ2)−cφ2(4cW2−1)[1+(3−tW2)t2θ2]dαsφ4cW2−1[(1−4cW2)(1+4t2θ2)+6t2θ2]−cφ(1+2cW2)1+4t2θ26(4cW2−1)(1+4t2θ2)[1+(3−tW2)t2θ2]−sφ4cW2−1(1+2t2θ2)+cφc2W1+4t2θ22(4cW2−1)(1+4t2θ2)[1+(3−tW2)t2θ2]Usφ4cW2−1[3t2θ2−4sW2(1+4t2θ2)]−cφ(3−7sW2)1+4t2θ23(4cW2−1)(1+4t2θ2)[1+(3−tW2)t2θ2]sφ4cW2−1t2θ2−cφcW21+4t2θ2(4cW2−1)(1+4t2θ2)[1+(3−tW2)t2θ2]Dαsφ4cW2−1[2sW2(1+4t2θ2)−3t2θ2]+cφ(3−5sW2)1+4t2θ23(4cW2−1)(1+4t2θ2)[1+(3−tW2)t2θ2]−sφ4cW2−1t2θ2−cφcW21+4t2θ2(4cW2−1)(1+4t2θ2)[1+(3−tW2)t2θ2]Because of the above-mentioned mixing, the
lepton-number violating interactions mediated by neutral gauge bosonsZ1 and Z2 exist in the neutrino and the exotic quark sectors: (2.59)ℒunnormalNC=−gt2θgkV(ν)2(ν¯aLγμνaLc+u¯1LγμUL−D¯αLγμdαL)Zμk+H.c. Again, these interactions are
very weak and proportional to sinθ.
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
angleθ and the charged bilepton mass MY.
Below we present a simple one. In our model, the W boson has the following normal main decay modes: (2.60)W−→lνl(l=e,μ,τ),↘ucd,ucs,ucb,(u→c), 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 (ΔL=2)—the model's specific feature: (2.61)W−→lνl(l=e,μ,τ). It is easy to compute the
tree-level decay widths as follows [91, 92]: (2.62)ΓBorn(W→lνl)=g2cθ28MW6π(1−x)(1−x2−x22)≃cθ2αMW12sW2,ΓBorn(W→lνl)=g2sθ28MW6π(1−x)(1−x2−x22)≃sθ2αMW12sW2,x≡ml2MW2,∑colorΓBorn(W→uicdj)=3g2cθ28MW6π|Vij|2[1−2(x+x¯)+(x−x¯)2]1/2×[1−x+x¯2−(x−x¯)22]≃cθ2αMW4sW2|Vij|2,x≡mdj2MW2,x¯≡muic2MW2. Quantum chromodynamics radiative
corrections modify (2.62) by a multiplicative factor [3, 91, 92]: (2.63)δQCQ=1+αs(MZ)π+1.409αs2π2−12.77αs3π3≃1.04, which is estimated from αs(MZ)≃0.12138.
All the state masses can be ignored, the predicted total width for W decay into fermions is (2.64)ΓWtot=1.04αMW2sW2(1−sθ2)+αMW4sW2. Taking α(MZ)≃1/128,MW=80.425GeV,sW2=0.2312,
and ΓWtot=2.124±0.041GeV [3], in Figure 1, we have plotted ΓWtot as function of sθ. From the
figure we get an upper limit: (2.65)sinθ≤0.08. It is important to note that
this limit value on the LNV parameter u/ω is much larger than those in [50, 93, 94].Figure 1
W width as function of sinθ,
and the horizontal lines are an upper and a lower limit.Since one of the VEVs is closely to the those in the
standard model:v≃vweak=246 GeV, therefore only two free VEVs exist in the
considering model, namely, u and ω.
The bilepton mass limit can be obtained from the “wrong" muon decay: (2.66)μ−→e−νeνμ, mediated at the tree level, by
both the standard model W and the singly-charged bilepton Y (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 Y.
In our case, the second diagram in Figure 2 gives main contribution. Taking into
account of the famous experimental data [3] (2.67)Rmuon≡Γ(μ−→e−νeνμ)Γ(μ−→e−νeνμ)<1.2%90%CL, we get the constraint: Rmuon≃MW4/MY4.
Therefore, it follows that MY≥230 GeV.Feynman
diagram for the wrong muon decayμ−→e−νeνμ.
(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 ofu→0, analyzing the Z decay width [37, 95, 96], the Z−Z′ mixing angle is constrained by −0.0015≤φ≤0.001.
From atomic parity violation in cesium, bounds for mass of the new exotic Z′ and the Z−Z′ mixing angles, again in the limit u→0,
are given [37, 95, 96]: (2.68)−0.00156≤φ≤0.00105,MZ2≥2.1TeV. 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 u≠0 of the constraints.For our purpose, we consider theρ parameter—one of the most important
quantities of the standard model, having a leading contribution in terms of the T parameter, is very useful to get the
new-physics effects. It is well-known relation between ρ and T parameter: (2.69)ρ=1+αT. In the usual 3-3-1 model with
right-handed neutrinos, T gets contribution from the oblique correction
and the Z−Z′ mixing [88]: (2.70)TRHN=TZZ′+Toblique, where TZZ′≃(tan2φ/α)(MZ22/MZ12−1) is negligible for MZ′ less than 1 TeV; Toblique depends on masses of the top quark and the
standard model Higgs boson. Again, at the tree level and the limit (2.8), from
(2.22) and (2.48) we get an expression for the ρ parameter in the considering model: (2.71)ρ=MW2cW2MZ12=v2v2−3u2≃1+3u2v2. Note that (2.71) has only one free
parameter u,
since v is very close to the VEV in the standard
model. Neglecting the contribution from the usual 3-3-1 model with right-handed
neutrinos and taking into account the experimental data [3] ρ=0.9987±0.0016,
we get the constraint on u parameter by u/v≤0.01 which leads to u≤2.46 GeV. This means that u is much smaller than v,
as expected.It seems that theρ parameter, at the tree level, in this model,
is favorable to be bigger than one and this is similar to the case of the
models contained heavy Z′ [98].The interesting new physics compared with other 3-3-1
models is the neutrino physics. Due to lepton-number violating couplings, we
have the following interesting consequences.(1) Processes withΔL=±2
From the charged currents, we have the following
lepton-number violatingΔL=±2 decays such as (2.72)μ−→e−νeνμ,μ−→e−νeνμ,(μcanbereplacedbyτ) in which both the standard model W boson and charged bilepton Yμ− are in intermediate states (see Figure 3). Here, the main
contribution arises from the first diagram. Note that the wrong muon decay
violates only family lepton-number, that is, ΔL=0,
but not lepton number at all as in (2.72). The decay rates are given by (2.73)Rrare≡Γ(μ−→e−νeνμ)Γ(μ−→e−νeνμ)=Γ(μ−→e−νeνμ)Γ(μ−→e−νeνμ)≃sθ2. Taking sθ=0.08,
we get Rrare≃6×10−3.
This rate is the same as the wrong muon decay one. Interesting to note that,
the family lepton-number violating processes (2.74)νiνi→νjνj,(i≠j) are mediated not only by the
non-Hermitian bilepton X but also by the Hermitian neutral Z1,Z2 (see Figure 4).Feynman
diagram forμ−→e−νeνμ.
(a)(b)Feynman
diagram forνiνi→νjνj(i≠j=e,μ,τ).
(a)(b)The first diagram in Figure4 exists also in the 3-3-1
model with right-handed neutrinos, but the second one does not appear there.(2)Lepton-Number Violating Kaon Decays
Next, let us consider the lepton-number violating decay
[3]: (2.75)K+→π0+e+νe<3×10−3at90%CL. This decay can be explained in
the considering model as the subprocess given below: (2.76)s→u+e+νe. This process is mediated by the
standard model W boson and the charged bilepton Y.
Amplitude of the considered process is proportional to sinθ: (2.77)M(s→u+e+νe)≃sin2θ2MW2(1−MW2MY2). Next, let us consider the
“normal decay" [3]: (2.78)K+→π0+e+νe(4.87±0.06)% with amplitude (2.79)M(s→u+e+νe)≃1MW2. From (2.77) and (2.79), we get (2.80)Rkaon≡Γ(s→u+e+νe)Γ(s→u+e+νe)≃sin2θ.In the framework of this model, we derive the
following decay modes with rates:(2.81)Rkaon=Γ(K+→π0+e+νe)Γ(K+→π0+e+νe)≃Γ(K+→π0+μ+νμ)Γ(K+→π0+μ+νμ)≃sin2θ≤6×10−3. Note that the similar
lepton-number violating processes exist in the  SU(2)R⊗SU(2)L⊗U(1)B−L model (for details, see [90]).


## 2.5. Summary

In this section, we have presented the 3-3-1 model
with the minimal scalar sector (only two Higgs triplets). This version belongs
to the 3-3-1 model without exotic charges (charges of the exotic quarks are2/3 and −1/3). The spontaneous symmetry breakdown is
achieved with only two Higgs triplets. One of the VEVs u is a source of lepton-number violations and a
reason for the mixing between the charged gauge bosons—the standard model W and the singly-charged bilepton gauge bosons
as well as between neutral non-Hermitian X0 and neutral gauge bosons: the Z and the new exotic Z′.
At the tree level, masses of the charged gauge bosons satisfy the law of
Pythagoras MY2=MX2+MW2 and in the limit ω≫u,v,
the ρ parameter gets additional contribution
dependent only on u/v.
Thus, this leads to u≪v,
and there are three quite different scales for the VEVs of the model: one is
very small u≃O(1) GeV—a lepton-number violating parameter; the
second v is close to the standard model one: v≃vweak=246 GeV; and the last is in the range of new physics
scale about O(1) TeV.In difference with the usual 3-3-1 model with
right-handed neutrinos, in this model the first family of quarks should be
distinctive of the two others.The exact diagonalization of the neutral gauge boson sector
is derived. Because of the parameteru,
the lepton-number violation happens only in neutrino but not in charged lepton
sector. It is interesting to note that despite the above-mentioned mixing, the
electromagnetic current remains unchanged. In this model, the lepton-number
changing (ΔL=±2) processes exist but only in the neutrino
sector.It is worth mentioning on the advantage of the
considered model: the new mixing angle between the charged gauge bosonsθ is connected with one of the VEVs u—the parameter of lepton-number violations.
There is no new parameter, but it contains very simple Higgs sector, hence the
significant number of free parameters is reduced.The model contains of new
kinds of interactions in the neutrino sector. Hence, neutrino physics in this
model is very rich. We will turn to further studies on neutrino masses and
mixing in Section4.


## 3. Higgs-Gauge Boson Interactions

We first obtain the scalar fields and mass spectra.
The couplings of the scalar fields with the ordinary gauge bosons are presented
then. Cross section for the production of the charged Higgs boson at LHC
is calculated.


### 3.1. Higgs Potential

The Higgs potential in the model under consideration
is given by (2.11). Let us first shift the Higgs fields into physical ones: (3.1)χ=(χ1P0+u2χ2−χ3P0+ω2),ϕ=(ϕ1+ϕ2P0+v2ϕ3+). The subscript P denotes physical fields as in the usual treatment.
However, in the following, this subscript will be dropped. By substitution of
(3.1) into (2.11), the potential becomes (3.2)V(χ,ϕ)=μ12[(χ10*+u2)(χ10+u2)+χ2+χ2−+(χ30*+ω2)(χ30+ω2)]+μ22[ϕ1−ϕ1++(ϕ20*+v2)(ϕ20+v2)+ϕ3−ϕ3+]+λ1[(χ10*+u2)(χ10+u2)+χ2+χ2−+(χ30*+ω2)(χ30+ω2)]2+λ2[ϕ1−ϕ1++(ϕ20*+v2)(ϕ20+v2)+ϕ3−ϕ3+]2+λ3[(χ10*+u2)(χ10+u2)+χ2+χ2−+(χ30*+ω2)(χ30+ω2)]×[ϕ1−ϕ1++(ϕ20*+v2)(ϕ20+v2)+ϕ3−ϕ3+]+λ4[(χ10*+u2)ϕ1++χ2+(ϕ20+v2)+(χ30*+ω2)ϕ3+]×[ϕ1−(χ10+u2)+(ϕ20*+v2)χ2−+ϕ3−(χ30+ω2)]. From the above expression, we
get constraint equations at the tree level: (3.3)μ12+λ1(u2+ω2)+λ3v22=0,μ22+λ2v2+λ3(u2+ω2)2=0. The nonzero values of χ and ϕ at the potential minimum as mentioned can be
easily derived from these equations to yield the given (2.12) and (2.13).Sinceu is a parameter of lepton-number violation,
therefore the terms linear in u violate the latter. Applying the constraint
(3.3), we get the minimum value, mass terms, lepton-number conserving,
and violating interactions as follows: (3.4)V(χ,ϕ)=Vmin+VmassN+VmassC+VLNC+VLNV, where (3.5)Vmin=−λ24v4−14(u2+ω2)[λ1(u2+ω2)+λ3v2],VmassN=λ1(uS1+ωS3)2+λ2v2S22+λ3v(uS1+ωS3)S2,(3.6)VmassC=λ42(uϕ1++vχ2++ωϕ3+)(uϕ1−+vχ2−+ωϕ3−),(3.7)VLNC=λ1(χ†χ)2+λ2(ϕ†ϕ)2+λ3(χ†χ)(ϕ†ϕ)+λ4(χ†ϕ)(ϕ†χ)+2λ1ωS3(χ†χ)+2λ2vS2(ϕ†ϕ)+λ3vS2(χ†χ)+λ3ωS3(ϕ†ϕ)+λ42(vχ2−+ωϕ3−)(χ†ϕ)+λ42(vχ2++ωϕ3+)(ϕ†χ),(3.8)VLNV=2λ1uS1(χ†χ)+λ3uS1(ϕ†ϕ)+λ42u[ϕ1−(χ†ϕ)+ϕ1+(ϕ†χ)]. In the above equations, we have
dropped the subscript P and used χ=(χ10,χ2−,χ30)T,ϕ=(ϕ1+,ϕ20,ϕ3+)T.
Moreover, we have expanded the neutral Higgs fields as (3.9)χ10=S1+iA12,χ30=S3+iA32,ϕ20=S2+iA22. In the literature, the real
parts (Si,i=1,2,3) are also called CP-even scalar and the
imaginary part (Ai,i=1,2,3)—CP-odd scalar. In this paper, for short, we
call them scalar and pseudoscalar field, respectively. As expected, the
lepton-number violating part VLNC is linear in u and trilinear in scalar fields. These
couplings will be also a source for lepton-number violations such as the mass
spectra of quarks including exotic ones as well as neutrino Majorana masses,
but given at higher-order corrections.In the pseudoscalar sector, all the fields are
Goldstone bosons:G1=A1,G2=A2, and G3=A3 (cf. (3.5)). The
scalar fields S1,S2, and S3 gain masses via (3.5), thus we get one
Goldstone boson G4 and two neutral physical fields—the standard
model H0 and the new H10 with masses: (3.10)mH02=λ2v2+λ1(u2+ω2)−[λ2v2−λ1(u2+ω2)]2+λ32v2(u2+ω2)≃4λ1λ2−λ322λ1v2,(3.11)MH102=λ2v2+λ1(u2+ω2)+[λ2v2−λ1(u2+ω2)]2+λ32v2(u2+ω2)≃2λ1ω2. In term of original fields, the
Goldstone and Higgs fields are given by (3.12)G4=11+tθ2(S1−tθS3),H0=cζS2−sζ1+tθ2(tθS1+S3),H10=sζS2+cζ1+tθ2(tθS1+S3), where (3.13)t2ζ≡λ3MWMXλ1MX2−λ2MW2. From (3.11), it follows that mass
of the new Higgs boson MH10 is related to mass of the bilepton gauge X0 (or Y± via the law of Pythagoras) through (3.14)MH102=8λ1g2MX2[1+𝒪(MW2MX2)]=2λ1sW2παMX2[1+𝒪(MW2MX2)]≈18.8λ1MX2. Here, we have used α=1/128 and sW2=0.231.In the charged Higgs sector, the mass terms for(ϕ1,χ2,ϕ3) are given by (3.6), thus there are two
Goldstone bosons and one physical scalar field: (3.15)H2+≡1u2+v2+ω2(uϕ1++vχ2++ωϕ3+) with mass (3.16)MH2+2=λ42(u2+v2+ω2)=2λ4MY2g2=sW2λ42παMY2≃4.7λ4MY2. The two remaining Goldstone
bosons are (3.17)G5+=11+tθ2(ϕ1+−tθϕ3+),G6+=1(1+tθ2)(u2+v2+ω2)[v(tθϕ1++ϕ3+)−ω(1+tθ2)χ2+]. Thus, all the pseudoscalars are eigenstates and
massless (Goldstone). Other fields are related to the scalars in the weak basis
by the linear transformations: (3.18)(H0H10G4)=(−sζsθcζ−sζcθcζsθsζcζcθcθ0−sθ)(S1S2S3),(H2+G5+G6+)=1ω2+cθ2v2(ωsθvcθωcθcθω2+cθ2v20−sθω2+cθ2v2vs2θ2−ωvcθ2)(ϕ1+χ2+ϕ3+). With the two Higgs triplets of the model, there are
twelve real scalar components. Eight of the gauge symmetries of SU(3)L⊗U(1)X are spontaneously broken, which eliminates
just eight Goldstone bosons associated with these fields. It leaves over just
four massive scalar particles as obtained (one charged and two natural). There
is no Majoron field in this model which contrasts to the 3-3-1 model with
right-handed neutrinos [99, 100]. Let us remind the reader that among the Goldstone
bosons there are four fields carrying the lepton number, but they can be gauged
away by an unitary transformation [87].From (3.10) and (3.11), we come to the previous result
in [36]: (3.19)λ1>0,λ2>0,4λ1λ2>λ32. Equation (3.16) shows that the
mass of the charged Higgs boson H2± is proportional to those of the charged
bilepton Y through a coefficient of Higgs self-interaction λ4>0.
Analogously, this happens for the standard-model-like Higgs boson H0(MH0~MW) and the new H10(MH10~MX).
Combining (3.19) with the constraint (3.3), we get a consequence: λ3 is negative (λ3<0). Let us remind the reader that the couplings λ4,1,2 are fixed by the Higgs boson masses and λ3,
where λ3 defines the splitting ΔmH2≃−[λ32/(2λ1)]v2 from the standard
model prediction.To finish this section, let us comment on our physical
Higgs bosons. In the effective approximationw≫v,u,
from (3.18), it follows that (3.20)H0~S2,H10~S3,G4~S1,H2+~ϕ3+,G5+~ϕ1+,G6+~χ2+. This means that, in the
effective approximation, the charged boson H2− is a scalar bilepton (with lepton number L=2), while the neutral scalar bosons H0 and H10 do not carry lepton number (with L=0).


### 3.2. Higgs-Standard Model Gauge Couplings

There are a total of 9 gauge bosons in theSU(3)L⊗U(1)X group and 8 of them are massive. As shown in
the previous section, we have got just 8 massless Goldstone bosons—the
justified number for the model. One of the neutral scalars is identified with
the standard model Higgs boson; therefore, its couplings to ordinary gauge
bosons such as the photon, the Z,
and the W± bosons have to have, in the effective limit,
usual known forms. To search Higgs bosons at future high-energy colliders, one
needs their couplings with ordinary particles, specially with the gauge bosons
in the standard model.The interactions among the gauge bosons and the Higgs
bosons arise in part from(3.21)∑Y=χ,ϕ(DμY)†(DμY). In the following, the summation
over Y is default and only the terms giving
interested couplings are explicitly displayed. The covariant derivative is
given by (2.14): (3.22)Dμ=∂μ−i𝒫μ≡∂μ−i𝒫μNC−i𝒫μCC, where the matrices 𝒫μNC and 𝒫μCC are written as (3.23)𝒫μNC=g2(W3μ+W8μ3+t23XBμ0yμ0−W3μ+W8μ3+t23XBμ0yμ0−2W8μ3+t23XBμ),(3.24)𝒫μCC=g2(0cθWμ++sθYμ+Xμ0cθWμ−+sθYμ−0cθYμ−−sθWμ−Xμ0*cθYμ+−sθWμ+0). Let us recall that t=gX/g=32sW/3−4sW2,tanθ=u/ω, and
Wμ±,Yμ±,
and Xμ0 are the physical fields. The existence of yμ is a consequence of mixing among the real part (Xμ0*+Xμ0) with W3μ,W8μ,
and Bμ;
and its expression is determined from the mixing matrix U given in Appendix A.1: (3.25)yμ≡U42Zμ+U43Zμ′+(U44−1)(Xμ0*+Xμ0)2, where (3.26)U42=−tθ′(cφ1−4sθ′2cW2−sφ4cW2−1),U43=−tθ′(sφ1−4sθ′2cW2+cφ4cW2−1),U44=1−4sθ′2cW2. First, we consider the relevant couplings of the
standard model W boson with the Higgs and Goldstone bosons. The
trilinear couplings of the pair W+W− with the neutral scalars are given by (3.27)(𝒫μCC〈χ〉)†(𝒫CCμχ)+(𝒫μCC〈ϕ〉)†(𝒫CCμϕ)+H.c.=g2v2Wμ+W−μS2. Because of S2 is a combination of only H and H10,
therefore, there are two couplings which are given in Table 5.Table 5
Trilinear
coupling constants ofW+W− with neutral Higgs bosons.
VertexCouplingW+W−Hg22vcζW+W−H10g22vsζCouplings of the singleW with two Higgs bosons exist in (3.28)i(Y†𝒫μCC∂μY−∂μY†𝒫μCCY)=ig2Wμ−[Y2*(cθ∂μY1−sθ∂μY3)−∂μY2*(cθY1−sθY3)]+H.c.=ig2Wμ−[χ2+(cθ∂μχ10−sθ∂μχ30)−∂μχ2+(cθχ10−sθχ30)+ϕ20*(cθ∂μϕ1+−sθ∂μϕ3+)−∂μϕ20*(cθϕ1+−sθϕ3+)]+H.c. The resulting couplings of the
single W boson with two scalar fields are listed in
Table 6, where we have
used a notation A∂μ↔B=A(∂μB)−(∂μA)B.
Vanishing couplings are (3.29)𝒱(W−H2+H0)=𝒱(W−H2+H10)=𝒱(W−H0G6+)=𝒱(W−H10G6+)=𝒱(W−H2+G2)=𝒱(W−G6+G2)=0. Quartic couplings of W+W− with two scalar fields
arise in part from (3.30)(𝒫μCCY)+(𝒫CCμY)=g22Wμ+W−μ[χ2+χ2−+cθ2χ10*χ10+sθ2χ30*χ30−cθsθ(χ10*χ30+χ10χ30*)+ϕ20*ϕ20+cθ2ϕ1−ϕ1++sθ2ϕ3−ϕ3+−cθsθ(ϕ1+ϕ3−+ϕ1−ϕ3+)]. With the help of (A.3) and
(A.4), we get the interested couplings of W+W− with two scalars which are listed in Table 7. Our
calculation give following vanishing couplings: (3.31)𝒱(W+W−H2+G5−)=𝒱(W+W−G5+G6−)=𝒱(W+W−H0G40)=𝒱(W+W−H10G40)=0. Now, we turn to the couplings of neutral gauge bosons
with Higgs bosons. In this case, the interested couplings exist in (3.32)i(Y†𝒫μNC∂μY−∂μY†𝒫μNCY)=−ig2{W3μ(∂μχ10*χ10−∂μχ2+χ2−+∂μϕ1−ϕ1+−∂μϕ20*ϕ20)+W8μ3(∂μχ10*χ10+∂μχ2+χ2−+∂μϕ1−ϕ1++∂μϕ20*ϕ20−2∂μχ30*χ30−2∂μϕ3−ϕ3+)+t23Bμ[−13(∂μχ10*χ10+∂μχ2+χ2−+∂μχ30*χ30)+23(∂μϕ1−ϕ1++∂μϕ20*ϕ20+∂μϕ3−ϕ3+)]+yμ(∂μχ10*χ30+∂μχ30*χ10+∂μϕ1−ϕ3++∂μϕ3−ϕ1+)}+H.c. It can be checked that, as
expected, the photon Aμ does not interact with neutral Higgs bosons.
Other vanishing couplings are (3.33)𝒱(AH2+G5−)=𝒱(AH2+G6−)=𝒱(AG6+G5−)=0,𝒱(AAH0)=𝒱(AAH10)=𝒱(AAG4)=0,𝒱(AZH0)=𝒱(AZH10)=𝒱(AZG4)=0,𝒱(AZ′H0)=𝒱(AZ′H10)=𝒱(AZ′G4)=0. The nonzero electromagnetic
couplings are listed in Table 8. It should be
noticed that the electromagnetic interaction is diagonal, that is, the nonzero
couplings in this model always have a form: (3.34)ieqHAμH*∂μ↔H. For the Z bosons, the following observation is
useful: (3.35)W3μ=U12Zμ+⋯,W8μ=U22Zμ+⋯,Bμ=U32Zμ+⋯,yμ=U42Zμ+⋯. Here, (3.36)U12=cφcθ′cW,U22=cφ(sW2−3cW2sθ′2)−sφ(1−4sθ′2cW2)(4cW2−1)3cWcθ′,U32=−tW(cφ4cW2−1+sφ1−4sθ′2cW2)3cθ′ are elements in the mixing
matrix of the neutral gauge bosons given in Appendix A.1. From (3.32) and (3.35),
it follows that the trilinear couplings of the single Z with charged Higgs bosons exist in part from
the Lagrangian terms: (3.37)−ig2Zμ[(U12−U223+t323U32)∂μχ2−χ2++(U12+U223+2t323U32)∂μϕ1−ϕ1++(−23U22+2t323U32)∂μϕ3−ϕ3++U42(∂μϕ1−ϕ3++∂μϕ3−ϕ1+)]+H.c. From (3.37), we get trilinear
couplings of the Z with the charged Higgs bosons which are listed
in Table 9. The limit
sign (→) in the Tables is the effective one.Table 6
Trilinear coupling constants ofW− with two Higgs bosons.
VertexCouplingVertexCouplingWμ−H2+∂μ↔G4igvcθ2ω2+cθ2v2Wμ−G6+∂μ↔G1gcθω2ω2+cθ2v2Wμ−G5+∂μ↔H−igcζ2Wμ−G5+∂μ↔G2−g2Wμ−G6+∂μ↔G4igω2ω2+cθ2v2Wμ−G5+∂μ↔H10−ig2sζWμ−H2+∂μ↔G1−gvcθ22ω2+cθ2v2Wμ−G6+∂μ↔G30−gsθω2ω2+cθ2v2Wμ−H2+∂μ↔G3gvs2θ4ω2+cθ2v2Table 7
Nonzero quartic coupling constants ofW+W− with Higgs bosons.
VertexCouplingVertexCouplingW+W−H2+H2−g2cθ2v22(ω2+v2cθ2)W+W−G10G10g2cθ22W+W−G5+G5−g22W+W−G30G30g2sθ22W+W−G6+G6−g2ω22(ω2+cθ2v2)W+W−G40G40g22W+W−H2+G6−−g2cθvω2(ω2+cθ2v2)W+W−HH10g2s2ζ4W+W−HHg2cζ22W+W−G10G30−g2s2θ4W+W−H10H10g2sζ22W+W−G20G20g22Table 8
Trilinear electromagnetic coupling constants
ofAμ with two Higgs bosons.
VertexAμH2−∂μ↔H2+AμG5−∂μ↔G5+AμG6−∂μ↔G6+CouplingieieieTable 9
Trilinear coupling constants ofZμ with two charged Higgs bosons.
VertexCouplingZμH2−∂μ↔H2+ig2(ω2+v2cθ2){(v2cθ2+ω2sθ2)U12+[ω2(1−3cθ2)−v2cθ2]U223+(v2cθ2+2ω2)t323U32+ω2s2θU42}→−igsWtWZμG5−∂μ↔G5+ig2[cθ2U12+(1−3sθ2)U223+2t323U32−s2θU42]→ig2cW(1−2sW2)ZμG6−∂μ↔G6+ig2(ω2+cθ2v2){(ω2+v2sθ2cθ2)U12+[v2cθ2(1−3cθ2)−ω2]U223+t323(ω2+2v2cθ2)U32+2v2sθcθ3U42}→ig2cW(1−2sW2)ZμH2−∂μ↔G5+igω4ω2+cθ2v2(s2θU12+3s2θU22+2c2θU42)→0ZμH2−∂μ↔G6+igωvcθ2(ω2+cθ2v2)[−cθ2U12+(2−3cθ2)U223+t323U32+s2θU42]→0ZμG5−∂μ↔G6+igvcθ4ω2+cθ2v2(s2θU12+3s2θU22+2c2θU42)→0In the effective limit, theZG5G5 vertex gets an exact expression as in the
standard model. Hence, G5 can be identified with the charged Goldstone
boson in the standard model (GW+).Now, we search couplings of the singleZμ boson with neutral scalar fields. With the
help of the following equations, (3.38)χ10∂μ↔χ10*=iG1∂μ↔S1,χ30∂μ↔χ30*=iG3∂μ↔S3,ϕ20∂μ↔ϕ20*=iG2∂μ↔S2,∂μχ10*χ30+∂μχ30*χ10=12[∂μS1S3+∂μS3S1+∂μG1G3+∂μG3G1+iG3∂μ↔S1+iG1∂μ↔S3], the necessary parts of
Lagrangian are (3.39)g2[(U12+U223−t323U32)G1∂μ↔S1+U42G1∂μ↔S3+(−23U22−t323U32)×G3∂μ↔S3+U42G3∂μ↔S1+(−U12+U223+2t323U32)G2∂μ↔S2]. The resulting couplings are
listed in Table 10. we conclude that G2 should be identified to GZ in the standard model. For the Z′ boson, the following remark is again
helpful: (3.40)W3μ=U13Z′μ+⋯,W8μ=U23Z′μ+⋯,Bμ=U33Z′μ+⋯,yμ=U43Z′μ+⋯, where (3.41)U13=sφcθ′cW,U23=sφ(sW2−3cW2sθ′2)+cφ(1−4sθ′2cW2)(4cW2−1)3cWcθ′,U33=−tW(sφ4cW2−1−cφ1−4sθ′2cW2)3cθ′. Thus, with the replacement Z→Z′ one just replaces column 2 by 3,
for example, trilinear coupling constants of the Zμ′ with two neutral Higgs bosons are given in
Table 11.Table 10
Trilinear coupling constants ofZμ with two neutral Higgs bosons.
VertexCouplingZμG1∂μ↔H−gsζ2[(U12+U223−t323U32)sθ+U42cθ]→0ZμG2∂μ↔Hg2(−U12+U223+2t323U32)cζ→−g2cWZμG3∂μ↔Hgsζ2[(23U22+t323U32)cθ−U42sθ]→0ZμG1∂μ↔H10gcζ2[(U12+U223−t323U32)sθ+U42cθ]→0ZμG2∂μ↔H10g2(−U12+U223+2t323U32)sζ→0ZμG3∂μ↔H10−gcζ2[(23U22+t323U32)cθ−U42sθ]→0ZμG1∂μ↔G4g2[(U12+U223−t323U32)cθ−U42sθ]→g2cWZμG2∂μ↔G40ZμG3∂μ↔G4g2[(23U22+t323U32)sθ+U42cθ]→0Table 11
Trilinear coupling constants ofZμ′ with two neutral Higgs bosons.
VertexCouplingZ′μG1∂μ↔H−gsζ2[(U13+U233−t323U33)sθ+U43cθ]→0Z′μG2∂μ↔Hg2(−U13+U233+2t323U33)cζ→g2cW4cW2−1Z′μG3∂μ↔Hgsζ2[(23U23+t323U33)cθ−U43sθ]→0Z′μG1∂μ↔H10gcζ2[(U13+U233−t323U33)sθ+U43cθ]→0Z′μG2∂μ↔H10g2(−U13+U233+2t323U33)sζ→0Z′μG3∂μ↔H10−gcζ2[(23U23+t323U33)cθ−U43sθ]→−gcW4cW2−1Z′μG1∂μ↔G4g2[(U13+U233−t323U33)cθ−U43sθ]→gc2W2cW4cW2−1Z′μG2∂μ↔G40Z′μG3∂μ↔G4g2[(23U23+t323U33)sθ+U43cθ]→0Next, we search couplings of two neutral gauge bosons
with scalar fields which arise in part from(3.42)Y+𝒫μNC𝒫NCμY=g24{[Y1*(A11μA11μ+yμyμ)+Y3*(A11μyμ+A33μyμ)]Y1+A22μA22μ×Y2*Y2+[Y1*(A11μyμ+A33μyμ)+Y3*(A33μA33μ+yμyμ)]Y3}=g24{[χ10*(A11μχA11μχ+yμyμ)+χ30*(A11μχyμ+A33μχyμ)]χ10+[χ10*(A11μχyμ+A33μχyμ)+χ30*(A33μχA33μχ+yμyμ)]χ30+[ϕ1−(A11μϕA11μϕ+yμyμ)+ϕ3−(A11μϕyμ+A33μϕyμ)]ϕ1++[ϕ1−(A11μϕyμ+A33μϕyμ)+ϕ3−(A33μϕA33μϕ+yμyμ)]ϕ3++(A22μχA22μχ)χ2+χ2−+(A22μϕA22μϕ)ϕ20*ϕ20}. Here, Aiiμ(i=1,2,3) is a diagonal element in the matrix (2/g)𝒫μNC which is dependent on the U(1)X charge: (3.43)A11μχ=W3μ+W8μ3−t323Bμ,A11μϕ=W3μ+W8μ3+2t323Bμ,A22μχ=−W3μ+W8μ3−t323Bμ,A22μϕ=−W3μ+W8μ3+2t323Bμ,A33μχ=−2W8μ3−t323Bμ,A33μϕ=−2W8μ3+2t323Bμ. Quartic couplings of two Z with neutral scalar fields are given by (3.44)g24{[χ10*(A11μχA11μχ+yμyμ)+χ30*(A11μχyμ+A33μχyμ)]χ10+[χ10*(A11μχyμ+A33μχyμ)+χ30*(A33μχA33μχ+yμyμ)]χ30+(A22μϕA22μϕ)ϕ20*ϕ20}=g24{(A11μχA11μχ+yμyμ)χ10*χ10+(A33μχA33μχ+yμyμ)χ30*χ30+(A11μχyμ+A33μχyμ)(χ10*χ30+χ30*χ10)+(A22μϕA22μϕ)ϕ20*ϕ20}. In this case, the couplings are
listed in Table 12.Table 12
Quartic coupling constants ofZZ with two scalar bosons.
VertexCouplingZZG1G1g22[(U12+U223−t323U32)2+U422]→g22cW2ZZG2G2g22(−U12+U223+2t323U32)2→g22cW2ZZG3G3g22[(23U22+t323U32)2+U422]→0ZZG1G3g22(U12−U223−2t323U32)U42→0ZZHHg22{sζ2[sθ2(U12+U223−t323U32)2+cθ2(23U22+t323U32)2+U422+s2θU42(U12−U223−2t323U32)]+cζ2(U12−U223−2t323U32)2}→g22cW2ZZH10H10g22{cζ2[sθ2(U12+U223−t323U32)2+cθ2(23U22+t323U32)2+U422+s2θU42(U12−U223−2t323U32)]+sζ2(U12−U223−2t323U32)2}→0ZZG4G4g22[cθ2(U12+U223−t323U32)2+sθ2(23U22+t323U32)2−s2θ(U12−U223−2t323U32)U42+U422]→g22cW2ZZHH1−g2s2ζ4[sθ2(U12+U223−t323U32)2+cθ2(23U22+t323U32)2+U422−(U12−U223−2t323U32)2+s2θ(U12−U223−2t323U32)U42]→0ZZHG4−g2sζ4(U12−U223−2t323U32)[2c2θU42+s2θ(U12+3U22)]→0ZZH1G4g2cζ4(U12−U223−2t323U32)[2c2θU42+s2θ(U12+3U22)]→0Trilinear couplings of the pairZZ with one scalar field are obtained via the
following terms: (3.45)g24[vS2A22μϕA22μϕ+uS1A11μχA11μχ+ωS3A33μχA33μχ+(uS1+ωS3)yμyμ−(ωS1+uS3)yμA22μϕ]. The obtained couplings are given
in Table 13.Table 13
Trilinear coupling constants ofZZ with one scalar bosons.
VertexCouplingZZHg22[vcζ(U12−U223−2t323U32)2−usζsθ(U12+U223−t323U32)2−ωsζcθU422−ωsζcθ(23U22+t323U32)2−2ωsζsθ(U12−U223−2t323U32)U42]→g2v2cW2ZZH10g22[vsζ(U12−U223−2t323U32)2+ucζsθ(U12+U223−t323U32)2+ωcζcθU422+ωcζcθ(23U22+t323U32)2+2ωcζsθ(U12−U223−2t323U32)U42]→0ZZG4g2ω2[sθ(U12+3U22)+c2θcθU42][U12−U223−2t323U32]→0Because of (3.40), for the ZZ′ couplings with scalar fields, the above
manipulation is good enough. For example, Table 12 is replaced by Table 14.Table 14
Trilinear coupling constants ofZZ′ with one scalar bosons.
VertexCouplingZZ′Hg22[vcζ(U12−U223−2t323U32)(U13−U233−2t323U33)−usζsθ×(U12+U223−t323U32)(U13+U233−t323U33)−ωsζcθ(23U22+t323U32)×(23U23+t323U33)−ωsζcθU42U43−ωsζsθ(U12−U223−2t323U32)U43−ωsζsθ(U13−U233−2t323U33)U42]→g2vc2W2cW4cW2−1ZZ′H10g22[vsζ(U12−U223−2t323U32)(U13−U233−2t323U33)+ucζsθ×(U12+U223−t323U32)(U13+U233−t323U33)+ωcζcθ(23U22+t323U32)×(23U23+t323U33)+ωcζcθU42U43+ωcζsθ(U12−U223−2t323U32)U43+ωcζsθ(U13−U233−2t323U33)U42]→0ZZ′G4g2ωsθ2[(U12+U223−t323U32)(U13+U233−t323U33)−(23U22+t323U32)(23U23+t323U33)+cot2θU42×(U13−U233−2t323U33)+cot2θU43(U12−U223−2t323U32)]→0Now, we turn to the interested couplingZW±H2∓ arisen in part from (3.46)Y+𝒫μNC𝒫CCμY+H.c.=g222{Wμ−A22μY2*(cθY1−sθY3)+Wμ+[(cθA11μ−sθyμ)Y1*+(cθyμ−sθA33μ)Y3*]Y2}+H.c. For our Higgs triplets, one
gets (3.47)g222{Wμ−[A22χμχ2+(cθχ10−sθχ30)+A22ϕμϕ20*(cθϕ1+−sθϕ3+)]+Wμ+χ2−[(cθA11χμ−sθyμ)χ10*+(cθyμ−sθA33χμ)χ30*]+Wμ+ϕ20[(cθA11ϕμ−sθyμ)ϕ1−+(cθyμ−sθA33ϕμ)ϕ3−]}+H.c. From (3.47), the trilinear couplings of the W boson
with one scalar and one neutral gauge bosons exist in a part:(3.48)g24Wμ+{vϕ1−[cθ(23W8μ+4t323Bμ)−sθyμ]+vϕ3−[cθyμ−sθ(−W3μ−W8μ3+4t323Bμ)]+ωχ2−[sθ(W3μ+3W8μ)+c2θcθyμ]}+H.c. From the above equation, we get
necessary nonzero couplings, which are listed in Table 15. Vanishing
couplings are (3.49)𝒱(AW+H2−)=𝒱(AW+G6−)=0. Equation (3.49) is consistent
with an evaluation in [53], where authors neglected the diagrams with the γW±H∓ vertex.Table 15
Trilinear coupling
constants of neutral gauge bosons withW+ and the charged scalar boson.
VertexCouplingAW+G5−g22vsWZW+H2−g2vω2ω2+cθ2v2[sθcθ(U12+3U22)+c2θU42]Z′W+H2−g2vω2ω2+cθ2v2[sθcθ(U13+3U23)+c2θU43]→0ZW+G5−g2v4[−sθ2U12+(2−3sθ2)U223+4t323U32−s2θU42]→−g22vsWtWZW+G6−g2(v2cθ2−ω2)8cθω2+cθ2v2[s2θ(U12+3U22)+2c2θU42]→0From (3.24), it follows that, to get couplings of the
bilepton gauge boson Y+ with ZH2−,
one just makes in (3.48) the replacement cθ→−sθ,sθ→cθ.Finally, we can identify the scalar fields in the
considered model with that in the standard model as follows:(3.50)H↔h,G5+↔GW+,G2↔GZ. In the effective limit ω≫v,u our Higgs can be represented as (3.51)χ=(12u+GX0GY−12(ω+H10+iGZ′)),ϕ=(GW+12(v+h+iGZ)H2+), where G3~GZ′,G6−~GY− and (3.52)G4+iG1~2GX0 are the Goldstone boson of the
massive gauge bosons Z′,Y−, and X0,
respectively. Note that identification in (3.52) is possible due to the fact
that both scalar and pseudoscalar parts of χ10 are massless. In addition, the pseudoscalar
part is decoupled from others, while its scalar part mixes in the same as in
the gauge boson sector.We emphasize again that,
in the effective approximation, all Higgs gauge boson couplings in the standard
model arerecovered (see Table 16). In
contradiction with the previous analysis in [36], the condition u~v or introduction of the third triplet are not
necessary.Table 16
The standard model coupling constants in the effective limit.
VertexCouplingVertexCouplingWWhhg22GWGWAieWWhg22vWWGZGZg22WGWh−ig2WWGWGWg22WGWGZg2ZZhg22cW2vZZhhg22cW2ZZGZGZg22cW2AWGWg22vsWZWGW−g22vsWtWZGZh−g2cWZGWGWig2cW(1−2sW2)


### 3.3. Production ofH2± Via WZ Fusion at LHC

The possibility to detect the neutral Higgs boson in
the minimal version ate+e− colliders was considered in [101] and production of the
standard model-like neutral Higgs boson at LHC was considered in [52]. This section is devoted to
production of the charged H2± at the CERN LHC.Let us firstly discuss on the mass of this Higgs
boson. Equation (3.16) gives us a connection between its mass and those of the
singly-charged bilepton Y through the coefficient of Higgs self-coupling λ4.
Note that in the considered model the neutrino Majorana masses exist only in
the loop levels. To keep these masses in the experimental range, the mass of MH2± can be taken in the electroweak scale with λ4~0.01 (see the next section). From (3.16), taking the
lower limit for MY to be 1 TeV, the mass of H2± is in range of 200 GeV.Taking into account that, in the effective
approximation,H2− is the bilepton, we get the dominant decay
channels as follows: (3.53)H2−→lνl,Uda,Dαua,↘ZW−,Z′W−,XW−,ZY−. Assuming that masses of the
exotic quarks (U,Dα) are larger than MH2±,
we come to the fact that the hadron modes are absent in decay of the charged
Higgs boson. Due to that the Yukawa couplings of H2±l∓ν are very small, the main decay modes of the are in the second line
of (3.53). Note that the charged Higgs bosons in doublet models such as
two-Higgs doublet model or minimal supersymmetric standard model
have both hadronic and leptonic modes [54]. This is a specific feature
of the model under consideration.Because of the exoticX,Y,Z′ gauge bosons are heavy, the coupling of a
singly-charged Higgs boson (H2±) with the weak gauge bosons, H2±W∓Z,
may dominate. Here, it is of particular importance for the electroweak symmetry
breaking. Its magnitude is directly related to the structure of the extended
Higgs sector under global symmetries [102–106]. This coupling can appear at the tree level in models
with scalar triplets, while it is induced at the loop level in multiscalar
doublet models. The coupling, in our model, differs from zero at the tree level
due to the fact that the H2± belongs to a triplet.Thus, for the charged Higgs bosonH2±,
it is important to study the couplings given by the interaction
Lagrangian: (3.54)ℒint=fZWHH2±Wμ∓Zμ, where fZWH,
at tree level, is given in Table 15. The same as in [53], the dominant rate is due
to the diagram connected with the W and Z bosons. Putting necessary matrix elements in
Table 15 , we get (3.55)fZWH=−g2vωs2θ4ω2+cθ2v2cφ−sφ(4cW2−1)(1+4t2θ2)(1+4t2θ2)[cW2+(4cW2−1)t2θ2]. Thus, the form factor, at the
tree-level, is obtained by (3.56)F≡fZWHgMW=−ωs2θ[cφ−sφ(4cW2−1)(1+4t2θ2)]2(ω2+cθ2v2)(1+4t2θ2)[cW2+(4cW2−1)t2θ2]. The decay width of H2±→Wi±Zi,
where i=L,T representing,
respectively, the longitudinal and transverse polarizations is given by
[53]: (3.57)Γ(H2±→Wi±Zi)=MH2±λ1/2(1,w,z)16π|Mii|2, where λ(1,w,z)=(1−w−z)2−4wz,w=MW2/MH2±2 and z=MZ2/MH2±2.
The longitudinal and transverse contributions are given in terms of F by (3.58)|MLL|2=g24z(1−w−z)2|F|2,|MTT|2=2g2w|F|2. For the case of MH2±≫MZ,
we have |MTT|2/|MLL|2~8MW2MZ2/MH2±4 which implies that the decay into a
longitudinally polarized weak boson pair dominates that into a transversely
polarized one. The form factor F and mixing angle tφ are presented in Table 17, where we have used sW2=0.2312,v=246GeV,ω=3TeV(orMY=1TeV) as the typical values to get five cases
corresponding with the sθ values under the constraint (2.65).Table 17
Values ofF,tφ, and MH2±max for given sθ.
sθ0.080.050.020.0090.005tφ−0.0329698−0.0156778−0.00598729−0.00449063−0.00422721F−0.087481−0.0561693−0.022803−0.0102847−0.00571598MH2±max[GeV]17001300700420320Next, let us study the impact of theH2±W∓Z vertex on the production cross section of pp→W±*Z*X→H2±X which is a pure electroweak process with high pT jets going into the forward and backward
directions from the decay of the produced scalar boson without color flow in
the central region. The hadronic cross section for pp→H2±X via W±Z fusion is expressed in the effective vector
boson approximation [107–109] by (3.59)σeff(s,MH2±2)≃16π2λ(1,w,z)MH2±3∑λ=T,LΓ(H2±→Wλ±Zλ)τdℒdτ|pp/Wλ±Zλ, where τ=MH2±2/s,
and (3.60)dℒdτ|pp/Wλ±Zλ=∑ij∫τ1dτ′τ′∫τ′1dxxfi(x)fj(τ′x)dℒdξ|qiqj/Wλ±Zλ with τ′=s^/s and ξ=τ/τ′.
Here, fi(x) is the parton structure function for the ith quark, and (3.61)dℒdξ|qiqj/WT±ZT=c64π41ξln(s^MW2)ln(s^MZ2)[(2+ξ)2ln(1ξ)−2(1−ξ)(3+ξ)],dℒdξ|qiqj/WL±ZL=c16π41ξ[(1+ξ)ln(1ξ)+2(ξ−1)], where c=(g4cθ2/16cW2)[g1V2(qj)+g1A2(qj)] with g1V(qj),g1A(qj) for quark qj is given in [38, Table I]. Using CTEQ6L
[110], in Figure 5, we
have plotted σeff(s,MH2±2) at s=14TeV,
as a function of the Higgs boson mass corresponding five cases in Table 17.Figure 5
Hadronic cross section ofW±Z fusion process as a function of the charged
Higgs boson mass for five cases of sinθ.
Horizontal line is discovery limit (25 events).Assuming discovery limit of 25 events corresponding to
the horizontal line, and taking the integrated luminosity of300fb−1 [111], from the figure, we come to conclusion that, for sθ=0.08 (the line on top), the charged Higgs boson H2± with mass larger than 1700 GeV, cannot be seen
at the LHC. These limiting masses are denoted by MH2±max and listed in Table 17. If the mass of the
above-mentioned Higgs boson is in range of 200 GeV and sθ=0.08,
the cross section can exeed 260fb,
that is, 78000 of H2± can be produced at the integrated LHC
luminosity of 300fb−1.
This production rate is about ten times larger than those in [53]. The cross sections
decrease rapidly as mass of the Higgs boson increases from 200 GeV to 400 GeV.


### 3.4. Summary

In this section, we have considered the scalar sector
in the economical 3-3-1 model. The model contains eight Goldstone bosons—the
justified number of the massless ones eaten by the massive gauge bosons.
Couplings of the standard model-like gauge bosons such as of the photon, theZ,
and the new Z′ gauge bosons with physical Higgs ones are also
given. From these couplings, the standard model-like Higgs boson as well as
Goldstone ones are identified. In the effective approximation, full content of
scalar sector can be recognized. The CP-odd part of Goldstone associated with
the neutral non-Hermitian bilepton gauge bosons GX0 is decoupled, while its CP-even counterpart
has the mixing by the same way in the gauge boson sector. Despite the mixing
among the photon with the non-Hermitian neutral bilepton X0 as well as with the Z and the Z′ gauge bosons, the electromagnetic couplings
remain unchanged.It is worth mentioning that masses of all physical
Higgs bosons are related to that of gauge bosons through the coefficients of
Higgs self-interactions. All gauge scalar boson couplings in the standard model
are recovered. The coupling of the photon with the Higgs bosons are diagonal.It should be mentioned that in [36], to get nonzero coupling ZZh at the tree level, the authors suggested the
following solution: (i) u~v or (ii) by introducing the third Higgs scalar
with VEV (~v). This problem does not happen in our
consideration.After all we focused attention to the singly-charged
Higgs bosonH2± with mass proportional to the bilepton mass MY through the coefficient λ4.
Mass of the H2± is estimated in a range of 200 GeV. This
boson, in difference with those arisen in the Higgs doublet models, does not
have the hadronic and leptonic decay modes. The trilinear coupling ZW±H2∓ which differs, at the tree level, while the
similar coupling of the photon γW±H2∓ as expected, vanishes. In the model under
consideration, the charged Higgs boson H2± with mass larger than 1700 GeV cannot be seen
at the LHC. If the mass of the above-mentioned Higgs boson is in range of 200 GeV, however, the cross section can exceed 260 fb, that is, 78000 of H2± can be produced at the LHC for the luminosity
of 300fb−1.
By measuring this process, we can obtain useful information to determine the
structure of the Higgs sector.


## 3.1. Higgs Potential

The Higgs potential in the model under consideration
is given by (2.11). Let us first shift the Higgs fields into physical ones: (3.1)χ=(χ1P0+u2χ2−χ3P0+ω2),ϕ=(ϕ1+ϕ2P0+v2ϕ3+). The subscript P denotes physical fields as in the usual treatment.
However, in the following, this subscript will be dropped. By substitution of
(3.1) into (2.11), the potential becomes (3.2)V(χ,ϕ)=μ12[(χ10*+u2)(χ10+u2)+χ2+χ2−+(χ30*+ω2)(χ30+ω2)]+μ22[ϕ1−ϕ1++(ϕ20*+v2)(ϕ20+v2)+ϕ3−ϕ3+]+λ1[(χ10*+u2)(χ10+u2)+χ2+χ2−+(χ30*+ω2)(χ30+ω2)]2+λ2[ϕ1−ϕ1++(ϕ20*+v2)(ϕ20+v2)+ϕ3−ϕ3+]2+λ3[(χ10*+u2)(χ10+u2)+χ2+χ2−+(χ30*+ω2)(χ30+ω2)]×[ϕ1−ϕ1++(ϕ20*+v2)(ϕ20+v2)+ϕ3−ϕ3+]+λ4[(χ10*+u2)ϕ1++χ2+(ϕ20+v2)+(χ30*+ω2)ϕ3+]×[ϕ1−(χ10+u2)+(ϕ20*+v2)χ2−+ϕ3−(χ30+ω2)]. From the above expression, we
get constraint equations at the tree level: (3.3)μ12+λ1(u2+ω2)+λ3v22=0,μ22+λ2v2+λ3(u2+ω2)2=0. The nonzero values of χ and ϕ at the potential minimum as mentioned can be
easily derived from these equations to yield the given (2.12) and (2.13).Sinceu is a parameter of lepton-number violation,
therefore the terms linear in u violate the latter. Applying the constraint
(3.3), we get the minimum value, mass terms, lepton-number conserving,
and violating interactions as follows: (3.4)V(χ,ϕ)=Vmin+VmassN+VmassC+VLNC+VLNV, where (3.5)Vmin=−λ24v4−14(u2+ω2)[λ1(u2+ω2)+λ3v2],VmassN=λ1(uS1+ωS3)2+λ2v2S22+λ3v(uS1+ωS3)S2,(3.6)VmassC=λ42(uϕ1++vχ2++ωϕ3+)(uϕ1−+vχ2−+ωϕ3−),(3.7)VLNC=λ1(χ†χ)2+λ2(ϕ†ϕ)2+λ3(χ†χ)(ϕ†ϕ)+λ4(χ†ϕ)(ϕ†χ)+2λ1ωS3(χ†χ)+2λ2vS2(ϕ†ϕ)+λ3vS2(χ†χ)+λ3ωS3(ϕ†ϕ)+λ42(vχ2−+ωϕ3−)(χ†ϕ)+λ42(vχ2++ωϕ3+)(ϕ†χ),(3.8)VLNV=2λ1uS1(χ†χ)+λ3uS1(ϕ†ϕ)+λ42u[ϕ1−(χ†ϕ)+ϕ1+(ϕ†χ)]. In the above equations, we have
dropped the subscript P and used χ=(χ10,χ2−,χ30)T,ϕ=(ϕ1+,ϕ20,ϕ3+)T.
Moreover, we have expanded the neutral Higgs fields as (3.9)χ10=S1+iA12,χ30=S3+iA32,ϕ20=S2+iA22. In the literature, the real
parts (Si,i=1,2,3) are also called CP-even scalar and the
imaginary part (Ai,i=1,2,3)—CP-odd scalar. In this paper, for short, we
call them scalar and pseudoscalar field, respectively. As expected, the
lepton-number violating part VLNC is linear in u and trilinear in scalar fields. These
couplings will be also a source for lepton-number violations such as the mass
spectra of quarks including exotic ones as well as neutrino Majorana masses,
but given at higher-order corrections.In the pseudoscalar sector, all the fields are
Goldstone bosons:G1=A1,G2=A2, and G3=A3 (cf. (3.5)). The
scalar fields S1,S2, and S3 gain masses via (3.5), thus we get one
Goldstone boson G4 and two neutral physical fields—the standard
model H0 and the new H10 with masses: (3.10)mH02=λ2v2+λ1(u2+ω2)−[λ2v2−λ1(u2+ω2)]2+λ32v2(u2+ω2)≃4λ1λ2−λ322λ1v2,(3.11)MH102=λ2v2+λ1(u2+ω2)+[λ2v2−λ1(u2+ω2)]2+λ32v2(u2+ω2)≃2λ1ω2. In term of original fields, the
Goldstone and Higgs fields are given by (3.12)G4=11+tθ2(S1−tθS3),H0=cζS2−sζ1+tθ2(tθS1+S3),H10=sζS2+cζ1+tθ2(tθS1+S3), where (3.13)t2ζ≡λ3MWMXλ1MX2−λ2MW2. From (3.11), it follows that mass
of the new Higgs boson MH10 is related to mass of the bilepton gauge X0 (or Y± via the law of Pythagoras) through (3.14)MH102=8λ1g2MX2[1+𝒪(MW2MX2)]=2λ1sW2παMX2[1+𝒪(MW2MX2)]≈18.8λ1MX2. Here, we have used α=1/128 and sW2=0.231.In the charged Higgs sector, the mass terms for(ϕ1,χ2,ϕ3) are given by (3.6), thus there are two
Goldstone bosons and one physical scalar field: (3.15)H2+≡1u2+v2+ω2(uϕ1++vχ2++ωϕ3+) with mass (3.16)MH2+2=λ42(u2+v2+ω2)=2λ4MY2g2=sW2λ42παMY2≃4.7λ4MY2. The two remaining Goldstone
bosons are (3.17)G5+=11+tθ2(ϕ1+−tθϕ3+),G6+=1(1+tθ2)(u2+v2+ω2)[v(tθϕ1++ϕ3+)−ω(1+tθ2)χ2+]. Thus, all the pseudoscalars are eigenstates and
massless (Goldstone). Other fields are related to the scalars in the weak basis
by the linear transformations: (3.18)(H0H10G4)=(−sζsθcζ−sζcθcζsθsζcζcθcθ0−sθ)(S1S2S3),(H2+G5+G6+)=1ω2+cθ2v2(ωsθvcθωcθcθω2+cθ2v20−sθω2+cθ2v2vs2θ2−ωvcθ2)(ϕ1+χ2+ϕ3+). With the two Higgs triplets of the model, there are
twelve real scalar components. Eight of the gauge symmetries of SU(3)L⊗U(1)X are spontaneously broken, which eliminates
just eight Goldstone bosons associated with these fields. It leaves over just
four massive scalar particles as obtained (one charged and two natural). There
is no Majoron field in this model which contrasts to the 3-3-1 model with
right-handed neutrinos [99, 100]. Let us remind the reader that among the Goldstone
bosons there are four fields carrying the lepton number, but they can be gauged
away by an unitary transformation [87].From (3.10) and (3.11), we come to the previous result
in [36]: (3.19)λ1>0,λ2>0,4λ1λ2>λ32. Equation (3.16) shows that the
mass of the charged Higgs boson H2± is proportional to those of the charged
bilepton Y through a coefficient of Higgs self-interaction λ4>0.
Analogously, this happens for the standard-model-like Higgs boson H0(MH0~MW) and the new H10(MH10~MX).
Combining (3.19) with the constraint (3.3), we get a consequence: λ3 is negative (λ3<0). Let us remind the reader that the couplings λ4,1,2 are fixed by the Higgs boson masses and λ3,
where λ3 defines the splitting ΔmH2≃−[λ32/(2λ1)]v2 from the standard
model prediction.To finish this section, let us comment on our physical
Higgs bosons. In the effective approximationw≫v,u,
from (3.18), it follows that (3.20)H0~S2,H10~S3,G4~S1,H2+~ϕ3+,G5+~ϕ1+,G6+~χ2+. This means that, in the
effective approximation, the charged boson H2− is a scalar bilepton (with lepton number L=2), while the neutral scalar bosons H0 and H10 do not carry lepton number (with L=0).


## 3.2. Higgs-Standard Model Gauge Couplings

There are a total of 9 gauge bosons in theSU(3)L⊗U(1)X group and 8 of them are massive. As shown in
the previous section, we have got just 8 massless Goldstone bosons—the
justified number for the model. One of the neutral scalars is identified with
the standard model Higgs boson; therefore, its couplings to ordinary gauge
bosons such as the photon, the Z,
and the W± bosons have to have, in the effective limit,
usual known forms. To search Higgs bosons at future high-energy colliders, one
needs their couplings with ordinary particles, specially with the gauge bosons
in the standard model.The interactions among the gauge bosons and the Higgs
bosons arise in part from(3.21)∑Y=χ,ϕ(DμY)†(DμY). In the following, the summation
over Y is default and only the terms giving
interested couplings are explicitly displayed. The covariant derivative is
given by (2.14): (3.22)Dμ=∂μ−i𝒫μ≡∂μ−i𝒫μNC−i𝒫μCC, where the matrices 𝒫μNC and 𝒫μCC are written as (3.23)𝒫μNC=g2(W3μ+W8μ3+t23XBμ0yμ0−W3μ+W8μ3+t23XBμ0yμ0−2W8μ3+t23XBμ),(3.24)𝒫μCC=g2(0cθWμ++sθYμ+Xμ0cθWμ−+sθYμ−0cθYμ−−sθWμ−Xμ0*cθYμ+−sθWμ+0). Let us recall that t=gX/g=32sW/3−4sW2,tanθ=u/ω, and
Wμ±,Yμ±,
and Xμ0 are the physical fields. The existence of yμ is a consequence of mixing among the real part (Xμ0*+Xμ0) with W3μ,W8μ,
and Bμ;
and its expression is determined from the mixing matrix U given in Appendix A.1: (3.25)yμ≡U42Zμ+U43Zμ′+(U44−1)(Xμ0*+Xμ0)2, where (3.26)U42=−tθ′(cφ1−4sθ′2cW2−sφ4cW2−1),U43=−tθ′(sφ1−4sθ′2cW2+cφ4cW2−1),U44=1−4sθ′2cW2. First, we consider the relevant couplings of the
standard model W boson with the Higgs and Goldstone bosons. The
trilinear couplings of the pair W+W− with the neutral scalars are given by (3.27)(𝒫μCC〈χ〉)†(𝒫CCμχ)+(𝒫μCC〈ϕ〉)†(𝒫CCμϕ)+H.c.=g2v2Wμ+W−μS2. Because of S2 is a combination of only H and H10,
therefore, there are two couplings which are given in Table 5.Table 5
Trilinear
coupling constants ofW+W− with neutral Higgs bosons.
VertexCouplingW+W−Hg22vcζW+W−H10g22vsζCouplings of the singleW with two Higgs bosons exist in (3.28)i(Y†𝒫μCC∂μY−∂μY†𝒫μCCY)=ig2Wμ−[Y2*(cθ∂μY1−sθ∂μY3)−∂μY2*(cθY1−sθY3)]+H.c.=ig2Wμ−[χ2+(cθ∂μχ10−sθ∂μχ30)−∂μχ2+(cθχ10−sθχ30)+ϕ20*(cθ∂μϕ1+−sθ∂μϕ3+)−∂μϕ20*(cθϕ1+−sθϕ3+)]+H.c. The resulting couplings of the
single W boson with two scalar fields are listed in
Table 6, where we have
used a notation A∂μ↔B=A(∂μB)−(∂μA)B.
Vanishing couplings are (3.29)𝒱(W−H2+H0)=𝒱(W−H2+H10)=𝒱(W−H0G6+)=𝒱(W−H10G6+)=𝒱(W−H2+G2)=𝒱(W−G6+G2)=0. Quartic couplings of W+W− with two scalar fields
arise in part from (3.30)(𝒫μCCY)+(𝒫CCμY)=g22Wμ+W−μ[χ2+χ2−+cθ2χ10*χ10+sθ2χ30*χ30−cθsθ(χ10*χ30+χ10χ30*)+ϕ20*ϕ20+cθ2ϕ1−ϕ1++sθ2ϕ3−ϕ3+−cθsθ(ϕ1+ϕ3−+ϕ1−ϕ3+)]. With the help of (A.3) and
(A.4), we get the interested couplings of W+W− with two scalars which are listed in Table 7. Our
calculation give following vanishing couplings: (3.31)𝒱(W+W−H2+G5−)=𝒱(W+W−G5+G6−)=𝒱(W+W−H0G40)=𝒱(W+W−H10G40)=0. Now, we turn to the couplings of neutral gauge bosons
with Higgs bosons. In this case, the interested couplings exist in (3.32)i(Y†𝒫μNC∂μY−∂μY†𝒫μNCY)=−ig2{W3μ(∂μχ10*χ10−∂μχ2+χ2−+∂μϕ1−ϕ1+−∂μϕ20*ϕ20)+W8μ3(∂μχ10*χ10+∂μχ2+χ2−+∂μϕ1−ϕ1++∂μϕ20*ϕ20−2∂μχ30*χ30−2∂μϕ3−ϕ3+)+t23Bμ[−13(∂μχ10*χ10+∂μχ2+χ2−+∂μχ30*χ30)+23(∂μϕ1−ϕ1++∂μϕ20*ϕ20+∂μϕ3−ϕ3+)]+yμ(∂μχ10*χ30+∂μχ30*χ10+∂μϕ1−ϕ3++∂μϕ3−ϕ1+)}+H.c. It can be checked that, as
expected, the photon Aμ does not interact with neutral Higgs bosons.
Other vanishing couplings are (3.33)𝒱(AH2+G5−)=𝒱(AH2+G6−)=𝒱(AG6+G5−)=0,𝒱(AAH0)=𝒱(AAH10)=𝒱(AAG4)=0,𝒱(AZH0)=𝒱(AZH10)=𝒱(AZG4)=0,𝒱(AZ′H0)=𝒱(AZ′H10)=𝒱(AZ′G4)=0. The nonzero electromagnetic
couplings are listed in Table 8. It should be
noticed that the electromagnetic interaction is diagonal, that is, the nonzero
couplings in this model always have a form: (3.34)ieqHAμH*∂μ↔H. For the Z bosons, the following observation is
useful: (3.35)W3μ=U12Zμ+⋯,W8μ=U22Zμ+⋯,Bμ=U32Zμ+⋯,yμ=U42Zμ+⋯. Here, (3.36)U12=cφcθ′cW,U22=cφ(sW2−3cW2sθ′2)−sφ(1−4sθ′2cW2)(4cW2−1)3cWcθ′,U32=−tW(cφ4cW2−1+sφ1−4sθ′2cW2)3cθ′ are elements in the mixing
matrix of the neutral gauge bosons given in Appendix A.1. From (3.32) and (3.35),
it follows that the trilinear couplings of the single Z with charged Higgs bosons exist in part from
the Lagrangian terms: (3.37)−ig2Zμ[(U12−U223+t323U32)∂μχ2−χ2++(U12+U223+2t323U32)∂μϕ1−ϕ1++(−23U22+2t323U32)∂μϕ3−ϕ3++U42(∂μϕ1−ϕ3++∂μϕ3−ϕ1+)]+H.c. From (3.37), we get trilinear
couplings of the Z with the charged Higgs bosons which are listed
in Table 9. The limit
sign (→) in the Tables is the effective one.Table 6
Trilinear coupling constants ofW− with two Higgs bosons.
VertexCouplingVertexCouplingWμ−H2+∂μ↔G4igvcθ2ω2+cθ2v2Wμ−G6+∂μ↔G1gcθω2ω2+cθ2v2Wμ−G5+∂μ↔H−igcζ2Wμ−G5+∂μ↔G2−g2Wμ−G6+∂μ↔G4igω2ω2+cθ2v2Wμ−G5+∂μ↔H10−ig2sζWμ−H2+∂μ↔G1−gvcθ22ω2+cθ2v2Wμ−G6+∂μ↔G30−gsθω2ω2+cθ2v2Wμ−H2+∂μ↔G3gvs2θ4ω2+cθ2v2Table 7
Nonzero quartic coupling constants ofW+W− with Higgs bosons.
VertexCouplingVertexCouplingW+W−H2+H2−g2cθ2v22(ω2+v2cθ2)W+W−G10G10g2cθ22W+W−G5+G5−g22W+W−G30G30g2sθ22W+W−G6+G6−g2ω22(ω2+cθ2v2)W+W−G40G40g22W+W−H2+G6−−g2cθvω2(ω2+cθ2v2)W+W−HH10g2s2ζ4W+W−HHg2cζ22W+W−G10G30−g2s2θ4W+W−H10H10g2sζ22W+W−G20G20g22Table 8
Trilinear electromagnetic coupling constants
ofAμ with two Higgs bosons.
VertexAμH2−∂μ↔H2+AμG5−∂μ↔G5+AμG6−∂μ↔G6+CouplingieieieTable 9
Trilinear coupling constants ofZμ with two charged Higgs bosons.
VertexCouplingZμH2−∂μ↔H2+ig2(ω2+v2cθ2){(v2cθ2+ω2sθ2)U12+[ω2(1−3cθ2)−v2cθ2]U223+(v2cθ2+2ω2)t323U32+ω2s2θU42}→−igsWtWZμG5−∂μ↔G5+ig2[cθ2U12+(1−3sθ2)U223+2t323U32−s2θU42]→ig2cW(1−2sW2)ZμG6−∂μ↔G6+ig2(ω2+cθ2v2){(ω2+v2sθ2cθ2)U12+[v2cθ2(1−3cθ2)−ω2]U223+t323(ω2+2v2cθ2)U32+2v2sθcθ3U42}→ig2cW(1−2sW2)ZμH2−∂μ↔G5+igω4ω2+cθ2v2(s2θU12+3s2θU22+2c2θU42)→0ZμH2−∂μ↔G6+igωvcθ2(ω2+cθ2v2)[−cθ2U12+(2−3cθ2)U223+t323U32+s2θU42]→0ZμG5−∂μ↔G6+igvcθ4ω2+cθ2v2(s2θU12+3s2θU22+2c2θU42)→0In the effective limit, theZG5G5 vertex gets an exact expression as in the
standard model. Hence, G5 can be identified with the charged Goldstone
boson in the standard model (GW+).Now, we search couplings of the singleZμ boson with neutral scalar fields. With the
help of the following equations, (3.38)χ10∂μ↔χ10*=iG1∂μ↔S1,χ30∂μ↔χ30*=iG3∂μ↔S3,ϕ20∂μ↔ϕ20*=iG2∂μ↔S2,∂μχ10*χ30+∂μχ30*χ10=12[∂μS1S3+∂μS3S1+∂μG1G3+∂μG3G1+iG3∂μ↔S1+iG1∂μ↔S3], the necessary parts of
Lagrangian are (3.39)g2[(U12+U223−t323U32)G1∂μ↔S1+U42G1∂μ↔S3+(−23U22−t323U32)×G3∂μ↔S3+U42G3∂μ↔S1+(−U12+U223+2t323U32)G2∂μ↔S2]. The resulting couplings are
listed in Table 10. we conclude that G2 should be identified to GZ in the standard model. For the Z′ boson, the following remark is again
helpful: (3.40)W3μ=U13Z′μ+⋯,W8μ=U23Z′μ+⋯,Bμ=U33Z′μ+⋯,yμ=U43Z′μ+⋯, where (3.41)U13=sφcθ′cW,U23=sφ(sW2−3cW2sθ′2)+cφ(1−4sθ′2cW2)(4cW2−1)3cWcθ′,U33=−tW(sφ4cW2−1−cφ1−4sθ′2cW2)3cθ′. Thus, with the replacement Z→Z′ one just replaces column 2 by 3,
for example, trilinear coupling constants of the Zμ′ with two neutral Higgs bosons are given in
Table 11.Table 10
Trilinear coupling constants ofZμ with two neutral Higgs bosons.
VertexCouplingZμG1∂μ↔H−gsζ2[(U12+U223−t323U32)sθ+U42cθ]→0ZμG2∂μ↔Hg2(−U12+U223+2t323U32)cζ→−g2cWZμG3∂μ↔Hgsζ2[(23U22+t323U32)cθ−U42sθ]→0ZμG1∂μ↔H10gcζ2[(U12+U223−t323U32)sθ+U42cθ]→0ZμG2∂μ↔H10g2(−U12+U223+2t323U32)sζ→0ZμG3∂μ↔H10−gcζ2[(23U22+t323U32)cθ−U42sθ]→0ZμG1∂μ↔G4g2[(U12+U223−t323U32)cθ−U42sθ]→g2cWZμG2∂μ↔G40ZμG3∂μ↔G4g2[(23U22+t323U32)sθ+U42cθ]→0Table 11
Trilinear coupling constants ofZμ′ with two neutral Higgs bosons.
VertexCouplingZ′μG1∂μ↔H−gsζ2[(U13+U233−t323U33)sθ+U43cθ]→0Z′μG2∂μ↔Hg2(−U13+U233+2t323U33)cζ→g2cW4cW2−1Z′μG3∂μ↔Hgsζ2[(23U23+t323U33)cθ−U43sθ]→0Z′μG1∂μ↔H10gcζ2[(U13+U233−t323U33)sθ+U43cθ]→0Z′μG2∂μ↔H10g2(−U13+U233+2t323U33)sζ→0Z′μG3∂μ↔H10−gcζ2[(23U23+t323U33)cθ−U43sθ]→−gcW4cW2−1Z′μG1∂μ↔G4g2[(U13+U233−t323U33)cθ−U43sθ]→gc2W2cW4cW2−1Z′μG2∂μ↔G40Z′μG3∂μ↔G4g2[(23U23+t323U33)sθ+U43cθ]→0Next, we search couplings of two neutral gauge bosons
with scalar fields which arise in part from(3.42)Y+𝒫μNC𝒫NCμY=g24{[Y1*(A11μA11μ+yμyμ)+Y3*(A11μyμ+A33μyμ)]Y1+A22μA22μ×Y2*Y2+[Y1*(A11μyμ+A33μyμ)+Y3*(A33μA33μ+yμyμ)]Y3}=g24{[χ10*(A11μχA11μχ+yμyμ)+χ30*(A11μχyμ+A33μχyμ)]χ10+[χ10*(A11μχyμ+A33μχyμ)+χ30*(A33μχA33μχ+yμyμ)]χ30+[ϕ1−(A11μϕA11μϕ+yμyμ)+ϕ3−(A11μϕyμ+A33μϕyμ)]ϕ1++[ϕ1−(A11μϕyμ+A33μϕyμ)+ϕ3−(A33μϕA33μϕ+yμyμ)]ϕ3++(A22μχA22μχ)χ2+χ2−+(A22μϕA22μϕ)ϕ20*ϕ20}. Here, Aiiμ(i=1,2,3) is a diagonal element in the matrix (2/g)𝒫μNC which is dependent on the U(1)X charge: (3.43)A11μχ=W3μ+W8μ3−t323Bμ,A11μϕ=W3μ+W8μ3+2t323Bμ,A22μχ=−W3μ+W8μ3−t323Bμ,A22μϕ=−W3μ+W8μ3+2t323Bμ,A33μχ=−2W8μ3−t323Bμ,A33μϕ=−2W8μ3+2t323Bμ. Quartic couplings of two Z with neutral scalar fields are given by (3.44)g24{[χ10*(A11μχA11μχ+yμyμ)+χ30*(A11μχyμ+A33μχyμ)]χ10+[χ10*(A11μχyμ+A33μχyμ)+χ30*(A33μχA33μχ+yμyμ)]χ30+(A22μϕA22μϕ)ϕ20*ϕ20}=g24{(A11μχA11μχ+yμyμ)χ10*χ10+(A33μχA33μχ+yμyμ)χ30*χ30+(A11μχyμ+A33μχyμ)(χ10*χ30+χ30*χ10)+(A22μϕA22μϕ)ϕ20*ϕ20}. In this case, the couplings are
listed in Table 12.Table 12
Quartic coupling constants ofZZ with two scalar bosons.
VertexCouplingZZG1G1g22[(U12+U223−t323U32)2+U422]→g22cW2ZZG2G2g22(−U12+U223+2t323U32)2→g22cW2ZZG3G3g22[(23U22+t323U32)2+U422]→0ZZG1G3g22(U12−U223−2t323U32)U42→0ZZHHg22{sζ2[sθ2(U12+U223−t323U32)2+cθ2(23U22+t323U32)2+U422+s2θU42(U12−U223−2t323U32)]+cζ2(U12−U223−2t323U32)2}→g22cW2ZZH10H10g22{cζ2[sθ2(U12+U223−t323U32)2+cθ2(23U22+t323U32)2+U422+s2θU42(U12−U223−2t323U32)]+sζ2(U12−U223−2t323U32)2}→0ZZG4G4g22[cθ2(U12+U223−t323U32)2+sθ2(23U22+t323U32)2−s2θ(U12−U223−2t323U32)U42+U422]→g22cW2ZZHH1−g2s2ζ4[sθ2(U12+U223−t323U32)2+cθ2(23U22+t323U32)2+U422−(U12−U223−2t323U32)2+s2θ(U12−U223−2t323U32)U42]→0ZZHG4−g2sζ4(U12−U223−2t323U32)[2c2θU42+s2θ(U12+3U22)]→0ZZH1G4g2cζ4(U12−U223−2t323U32)[2c2θU42+s2θ(U12+3U22)]→0Trilinear couplings of the pairZZ with one scalar field are obtained via the
following terms: (3.45)g24[vS2A22μϕA22μϕ+uS1A11μχA11μχ+ωS3A33μχA33μχ+(uS1+ωS3)yμyμ−(ωS1+uS3)yμA22μϕ]. The obtained couplings are given
in Table 13.Table 13
Trilinear coupling constants ofZZ with one scalar bosons.
VertexCouplingZZHg22[vcζ(U12−U223−2t323U32)2−usζsθ(U12+U223−t323U32)2−ωsζcθU422−ωsζcθ(23U22+t323U32)2−2ωsζsθ(U12−U223−2t323U32)U42]→g2v2cW2ZZH10g22[vsζ(U12−U223−2t323U32)2+ucζsθ(U12+U223−t323U32)2+ωcζcθU422+ωcζcθ(23U22+t323U32)2+2ωcζsθ(U12−U223−2t323U32)U42]→0ZZG4g2ω2[sθ(U12+3U22)+c2θcθU42][U12−U223−2t323U32]→0Because of (3.40), for the ZZ′ couplings with scalar fields, the above
manipulation is good enough. For example, Table 12 is replaced by Table 14.Table 14
Trilinear coupling constants ofZZ′ with one scalar bosons.
VertexCouplingZZ′Hg22[vcζ(U12−U223−2t323U32)(U13−U233−2t323U33)−usζsθ×(U12+U223−t323U32)(U13+U233−t323U33)−ωsζcθ(23U22+t323U32)×(23U23+t323U33)−ωsζcθU42U43−ωsζsθ(U12−U223−2t323U32)U43−ωsζsθ(U13−U233−2t323U33)U42]→g2vc2W2cW4cW2−1ZZ′H10g22[vsζ(U12−U223−2t323U32)(U13−U233−2t323U33)+ucζsθ×(U12+U223−t323U32)(U13+U233−t323U33)+ωcζcθ(23U22+t323U32)×(23U23+t323U33)+ωcζcθU42U43+ωcζsθ(U12−U223−2t323U32)U43+ωcζsθ(U13−U233−2t323U33)U42]→0ZZ′G4g2ωsθ2[(U12+U223−t323U32)(U13+U233−t323U33)−(23U22+t323U32)(23U23+t323U33)+cot2θU42×(U13−U233−2t323U33)+cot2θU43(U12−U223−2t323U32)]→0Now, we turn to the interested couplingZW±H2∓ arisen in part from (3.46)Y+𝒫μNC𝒫CCμY+H.c.=g222{Wμ−A22μY2*(cθY1−sθY3)+Wμ+[(cθA11μ−sθyμ)Y1*+(cθyμ−sθA33μ)Y3*]Y2}+H.c. For our Higgs triplets, one
gets (3.47)g222{Wμ−[A22χμχ2+(cθχ10−sθχ30)+A22ϕμϕ20*(cθϕ1+−sθϕ3+)]+Wμ+χ2−[(cθA11χμ−sθyμ)χ10*+(cθyμ−sθA33χμ)χ30*]+Wμ+ϕ20[(cθA11ϕμ−sθyμ)ϕ1−+(cθyμ−sθA33ϕμ)ϕ3−]}+H.c. From (3.47), the trilinear couplings of the W boson
with one scalar and one neutral gauge bosons exist in a part:(3.48)g24Wμ+{vϕ1−[cθ(23W8μ+4t323Bμ)−sθyμ]+vϕ3−[cθyμ−sθ(−W3μ−W8μ3+4t323Bμ)]+ωχ2−[sθ(W3μ+3W8μ)+c2θcθyμ]}+H.c. From the above equation, we get
necessary nonzero couplings, which are listed in Table 15. Vanishing
couplings are (3.49)𝒱(AW+H2−)=𝒱(AW+G6−)=0. Equation (3.49) is consistent
with an evaluation in [53], where authors neglected the diagrams with the γW±H∓ vertex.Table 15
Trilinear coupling
constants of neutral gauge bosons withW+ and the charged scalar boson.
VertexCouplingAW+G5−g22vsWZW+H2−g2vω2ω2+cθ2v2[sθcθ(U12+3U22)+c2θU42]Z′W+H2−g2vω2ω2+cθ2v2[sθcθ(U13+3U23)+c2θU43]→0ZW+G5−g2v4[−sθ2U12+(2−3sθ2)U223+4t323U32−s2θU42]→−g22vsWtWZW+G6−g2(v2cθ2−ω2)8cθω2+cθ2v2[s2θ(U12+3U22)+2c2θU42]→0From (3.24), it follows that, to get couplings of the
bilepton gauge boson Y+ with ZH2−,
one just makes in (3.48) the replacement cθ→−sθ,sθ→cθ.Finally, we can identify the scalar fields in the
considered model with that in the standard model as follows:(3.50)H↔h,G5+↔GW+,G2↔GZ. In the effective limit ω≫v,u our Higgs can be represented as (3.51)χ=(12u+GX0GY−12(ω+H10+iGZ′)),ϕ=(GW+12(v+h+iGZ)H2+), where G3~GZ′,G6−~GY− and (3.52)G4+iG1~2GX0 are the Goldstone boson of the
massive gauge bosons Z′,Y−, and X0,
respectively. Note that identification in (3.52) is possible due to the fact
that both scalar and pseudoscalar parts of χ10 are massless. In addition, the pseudoscalar
part is decoupled from others, while its scalar part mixes in the same as in
the gauge boson sector.We emphasize again that,
in the effective approximation, all Higgs gauge boson couplings in the standard
model arerecovered (see Table 16). In
contradiction with the previous analysis in [36], the condition u~v or introduction of the third triplet are not
necessary.Table 16
The standard model coupling constants in the effective limit.
VertexCouplingVertexCouplingWWhhg22GWGWAieWWhg22vWWGZGZg22WGWh−ig2WWGWGWg22WGWGZg2ZZhg22cW2vZZhhg22cW2ZZGZGZg22cW2AWGWg22vsWZWGW−g22vsWtWZGZh−g2cWZGWGWig2cW(1−2sW2)


## 3.3. Production ofH2± Via WZ Fusion at LHC

The possibility to detect the neutral Higgs boson in
the minimal version ate+e− colliders was considered in [101] and production of the
standard model-like neutral Higgs boson at LHC was considered in [52]. This section is devoted to
production of the charged H2± at the CERN LHC.Let us firstly discuss on the mass of this Higgs
boson. Equation (3.16) gives us a connection between its mass and those of the
singly-charged bilepton Y through the coefficient of Higgs self-coupling λ4.
Note that in the considered model the neutrino Majorana masses exist only in
the loop levels. To keep these masses in the experimental range, the mass of MH2± can be taken in the electroweak scale with λ4~0.01 (see the next section). From (3.16), taking the
lower limit for MY to be 1 TeV, the mass of H2± is in range of 200 GeV.Taking into account that, in the effective
approximation,H2− is the bilepton, we get the dominant decay
channels as follows: (3.53)H2−→lνl,Uda,Dαua,↘ZW−,Z′W−,XW−,ZY−. Assuming that masses of the
exotic quarks (U,Dα) are larger than MH2±,
we come to the fact that the hadron modes are absent in decay of the charged
Higgs boson. Due to that the Yukawa couplings of H2±l∓ν are very small, the main decay modes of the are in the second line
of (3.53). Note that the charged Higgs bosons in doublet models such as
two-Higgs doublet model or minimal supersymmetric standard model
have both hadronic and leptonic modes [54]. This is a specific feature
of the model under consideration.Because of the exoticX,Y,Z′ gauge bosons are heavy, the coupling of a
singly-charged Higgs boson (H2±) with the weak gauge bosons, H2±W∓Z,
may dominate. Here, it is of particular importance for the electroweak symmetry
breaking. Its magnitude is directly related to the structure of the extended
Higgs sector under global symmetries [102–106]. This coupling can appear at the tree level in models
with scalar triplets, while it is induced at the loop level in multiscalar
doublet models. The coupling, in our model, differs from zero at the tree level
due to the fact that the H2± belongs to a triplet.Thus, for the charged Higgs bosonH2±,
it is important to study the couplings given by the interaction
Lagrangian: (3.54)ℒint=fZWHH2±Wμ∓Zμ, where fZWH,
at tree level, is given in Table 15. The same as in [53], the dominant rate is due
to the diagram connected with the W and Z bosons. Putting necessary matrix elements in
Table 15 , we get (3.55)fZWH=−g2vωs2θ4ω2+cθ2v2cφ−sφ(4cW2−1)(1+4t2θ2)(1+4t2θ2)[cW2+(4cW2−1)t2θ2]. Thus, the form factor, at the
tree-level, is obtained by (3.56)F≡fZWHgMW=−ωs2θ[cφ−sφ(4cW2−1)(1+4t2θ2)]2(ω2+cθ2v2)(1+4t2θ2)[cW2+(4cW2−1)t2θ2]. The decay width of H2±→Wi±Zi,
where i=L,T representing,
respectively, the longitudinal and transverse polarizations is given by
[53]: (3.57)Γ(H2±→Wi±Zi)=MH2±λ1/2(1,w,z)16π|Mii|2, where λ(1,w,z)=(1−w−z)2−4wz,w=MW2/MH2±2 and z=MZ2/MH2±2.
The longitudinal and transverse contributions are given in terms of F by (3.58)|MLL|2=g24z(1−w−z)2|F|2,|MTT|2=2g2w|F|2. For the case of MH2±≫MZ,
we have |MTT|2/|MLL|2~8MW2MZ2/MH2±4 which implies that the decay into a
longitudinally polarized weak boson pair dominates that into a transversely
polarized one. The form factor F and mixing angle tφ are presented in Table 17, where we have used sW2=0.2312,v=246GeV,ω=3TeV(orMY=1TeV) as the typical values to get five cases
corresponding with the sθ values under the constraint (2.65).Table 17
Values ofF,tφ, and MH2±max for given sθ.
sθ0.080.050.020.0090.005tφ−0.0329698−0.0156778−0.00598729−0.00449063−0.00422721F−0.087481−0.0561693−0.022803−0.0102847−0.00571598MH2±max[GeV]17001300700420320Next, let us study the impact of theH2±W∓Z vertex on the production cross section of pp→W±*Z*X→H2±X which is a pure electroweak process with high pT jets going into the forward and backward
directions from the decay of the produced scalar boson without color flow in
the central region. The hadronic cross section for pp→H2±X via W±Z fusion is expressed in the effective vector
boson approximation [107–109] by (3.59)σeff(s,MH2±2)≃16π2λ(1,w,z)MH2±3∑λ=T,LΓ(H2±→Wλ±Zλ)τdℒdτ|pp/Wλ±Zλ, where τ=MH2±2/s,
and (3.60)dℒdτ|pp/Wλ±Zλ=∑ij∫τ1dτ′τ′∫τ′1dxxfi(x)fj(τ′x)dℒdξ|qiqj/Wλ±Zλ with τ′=s^/s and ξ=τ/τ′.
Here, fi(x) is the parton structure function for the ith quark, and (3.61)dℒdξ|qiqj/WT±ZT=c64π41ξln(s^MW2)ln(s^MZ2)[(2+ξ)2ln(1ξ)−2(1−ξ)(3+ξ)],dℒdξ|qiqj/WL±ZL=c16π41ξ[(1+ξ)ln(1ξ)+2(ξ−1)], where c=(g4cθ2/16cW2)[g1V2(qj)+g1A2(qj)] with g1V(qj),g1A(qj) for quark qj is given in [38, Table I]. Using CTEQ6L
[110], in Figure 5, we
have plotted σeff(s,MH2±2) at s=14TeV,
as a function of the Higgs boson mass corresponding five cases in Table 17.Figure 5
Hadronic cross section ofW±Z fusion process as a function of the charged
Higgs boson mass for five cases of sinθ.
Horizontal line is discovery limit (25 events).Assuming discovery limit of 25 events corresponding to
the horizontal line, and taking the integrated luminosity of300fb−1 [111], from the figure, we come to conclusion that, for sθ=0.08 (the line on top), the charged Higgs boson H2± with mass larger than 1700 GeV, cannot be seen
at the LHC. These limiting masses are denoted by MH2±max and listed in Table 17. If the mass of the
above-mentioned Higgs boson is in range of 200 GeV and sθ=0.08,
the cross section can exeed 260fb,
that is, 78000 of H2± can be produced at the integrated LHC
luminosity of 300fb−1.
This production rate is about ten times larger than those in [53]. The cross sections
decrease rapidly as mass of the Higgs boson increases from 200 GeV to 400 GeV.


## 3.4. Summary

In this section, we have considered the scalar sector
in the economical 3-3-1 model. The model contains eight Goldstone bosons—the
justified number of the massless ones eaten by the massive gauge bosons.
Couplings of the standard model-like gauge bosons such as of the photon, theZ,
and the new Z′ gauge bosons with physical Higgs ones are also
given. From these couplings, the standard model-like Higgs boson as well as
Goldstone ones are identified. In the effective approximation, full content of
scalar sector can be recognized. The CP-odd part of Goldstone associated with
the neutral non-Hermitian bilepton gauge bosons GX0 is decoupled, while its CP-even counterpart
has the mixing by the same way in the gauge boson sector. Despite the mixing
among the photon with the non-Hermitian neutral bilepton X0 as well as with the Z and the Z′ gauge bosons, the electromagnetic couplings
remain unchanged.It is worth mentioning that masses of all physical
Higgs bosons are related to that of gauge bosons through the coefficients of
Higgs self-interactions. All gauge scalar boson couplings in the standard model
are recovered. The coupling of the photon with the Higgs bosons are diagonal.It should be mentioned that in [36], to get nonzero coupling ZZh at the tree level, the authors suggested the
following solution: (i) u~v or (ii) by introducing the third Higgs scalar
with VEV (~v). This problem does not happen in our
consideration.After all we focused attention to the singly-charged
Higgs bosonH2± with mass proportional to the bilepton mass MY through the coefficient λ4.
Mass of the H2± is estimated in a range of 200 GeV. This
boson, in difference with those arisen in the Higgs doublet models, does not
have the hadronic and leptonic decay modes. The trilinear coupling ZW±H2∓ which differs, at the tree level, while the
similar coupling of the photon γW±H2∓ as expected, vanishes. In the model under
consideration, the charged Higgs boson H2± with mass larger than 1700 GeV cannot be seen
at the LHC. If the mass of the above-mentioned Higgs boson is in range of 200 GeV, however, the cross section can exceed 260 fb, that is, 78000 of H2± can be produced at the LHC for the luminosity
of 300fb−1.
By measuring this process, we can obtain useful information to determine the
structure of the Higgs sector.


## 4. Fermion Masses

We first give some comments on the charged lepton masses and set conventions. The neutrino and quark masses are correspondingly considered.


### 4.1. Charged-Lepton Masses

The charged
leptons(l=e,μ,τ) gain masses via the following couplings:(4.1)ℒYl=hablψ¯aLϕlbR+H.c.The mass matrix is, therefore,
followed by(4.2)Ml=−v2(h11lh12lh13lh21lh22lh23lh31lh32lh33l),which of course is the same as
in the standard model and thus gives consistent masses for the charged leptons
[37].For the sake of simplicity, in the following, we can suppose
that the Yukawa coupling of charged leptonshl is flavor diagonal, thus la(a=1,2,3) are mass eigenstates respective to the mass
eigenvalues ma=−(v/2)haal.For convenience in further reading, we present the Yukawa interactions of (2.6) and (2.7) in terms by Feynman diagrams in Figures 6, 7, and 8, where the Hermitian adjoint ones are not displayed. The Higgs boson self-couplings are depicted in Figure 9.Lepton Yukawa couplings.
(a)(b)Relevant lepton-number conserving quark Yukawa couplings.
(a)(b)(c)(d)Lepton-number violating quark Yukawa couplings.
(a)(b)(c)(d)Higgs boson self-couplings.
(a)(b)(c)(d)


### 4.2. Neutrino Masses

First, we
present mass mechanisms for the neutrinos. Next, detailed calculations and
analysis of the neutrino mass spectrum are given. The experimental constraints
on the couplinghν are also considered.


#### 4.2.1. Neutrino Mass Mechanisms

In the considering model, the possible different mass
mechanisms for the neutrinos can be summarized through the three dominantSU(3)C⊗SU(3)L⊗U(1)X-invariant effective operators as follows
[112, 113]:(4.3)OabLNC=ψ¯aLcψbLϕ,(4.4)OabLNV=(χ*ψ¯aLc)(χ*ψbL),(4.5)OabSLB=(χ*ψ¯aLc)(ψbLϕχ),
where the Hermitian adjoint
operators are not displayed. It is worth noting that they are also all the
performable operators with the mass dimensionality d≤6 responsible for the neutrino masses. The
difference among the mass mechanisms can be verified through the operators.
Both (4.3) and (4.5) conserve ℒ,
while (4.4) violates this charge with two units. Since d(OLNC)=4 and L〈ϕ〉=0,
(4.3) provides only Dirac masses for the neutrinos which can be obtained at the
tree level through the Yukawa couplings in (2.6). Since d(OSLB)=6 and (L〈χ〉)p≠0 for p=1 vanish for other
cases, (4.5) provides both Dirac and Majorana masses for the neutrinos through
radiative corrections mediated by the model particles. The masses induced by
(4.3) are given by the standard SU(2)L⊗U(1)Y symmetry breaking via the VEV v.
However, those by (4.5) are obtained from both the stages of SU(3)L⊗U(1)X breaking achieved by the VEVs u, ω,
and v.Note that the LNV interactions in (2.7) are due to
quarks. Hence, they do not give contribution to LNV of the leptons such as of
the neutrinos. Except the LNV couplings of (2.7), all the remaining interactions
of the model (lepton Yukawa couplings (2.6), Higgs self-couplings (2.11), etc.)
conserve ℒ.
This means that the operator (4.4) of LNV cannot be mediated by particles of
the model; in other words, it must be introduced by hands. As a fact, the
economical 3-3-1 model including the alternative versions [17–22] are only extensions beyond
the standard model in the scales of orders of TeV [40, 114, 115]. Hence, it is expected that
the operator in (4.4) has to be mediated by heavy particles of an underlined
new physics at a scale ℳ much greater than ω which have been followed in various of grand
unified theories (GUTs) [37, 112, 113, 116–125]. Thus, in this model the neutrinos can get mass from
three very different sources widely ranging over the mass scales: u~𝒪(1)GeV, v≈246GeV, ω~𝒪(1)TeV,
and ℳ~𝒪(1016)GeV.We remind that, in the former version [20–22], the authors in [126] have considered operators
of the type (4.4), however, under a discrete symmetry [22, 37]. As shown in Section 4, the
current model is realistic, and such a discrete symmetry is not needed because
as a fact that the model will fail if it is enforced. In addition, if such
discrete symmetries are not discarded, the important mass contributions for the
neutrinos mediated by model particles are then suppressed. For example, in this
case the remaining operators (4.3) and (4.5) will be removed. With the only
operator (4.4), the three active neutrinos will get effective zero masses under
a type II seesaw [55–62] (see
below). However, this operator occupies a particular importance in this
version.Alternatively, in such model, the authors in [49] have examined two-loop
corrections to (4.4) by the aid of explicit LNV Higgs self-couplings, and using
a fine tuning for the tree-level Dirac masses of (4.3) down to current values.
However, as mentioned, this is not the case in the considering model because
our Higgs potential (2.11) conserves ℒ.
We know that one of the problems of the 3-3-1 model with RH neutrinos is
associated with the Dirac mass term of neutrinos. In the following, we will
show that if such a fine tuning is done to get small values for these terms,
then the mass generation of neutrinos mediated by model particles is not able,
or the results will be trivial. This is in contradiction with [49]. In the next, the large
bare Dirac masses for the neutrinos, which are as of charged fermions of a
natural result from standard symmetry breaking, will be studied.


#### 4.2.2. Neutrino Mass Matrix

The operatorsOLNC, OSLB,
and OLNV (including their Hermitian adjoint) will
provide the masses for the neutrinos: the first responsible for tree-level
masses, the second for one-loop corrections, and the third for contributions of
heavy particles.Tree-Level Dirac Masses
From the Yukawa
couplings in (2.6), the tree-level mass Lagrangian for the neutrinos is obtained
by [127, 128]:(4.6)ℒmassLNC=habνν¯aRνbL〈ϕ20〉−habνν¯aLcνbRc〈ϕ20〉+H.c.=2〈ϕ20〉habνν¯aRνbL+H.c.=−(MD)abν¯aRνbL+H.c.=−12(ν¯aLc,ν¯aR)(0(MDT)ab(MD)ab0)(νbLνbRc)+H.c.=−12X¯LcMνXL+H.c.,where habν=−hbaν is due to Fermi statistics. The MD is the mass matrix for the Dirac
neutrinos:(4.7)(MD)ab≡−2vhabν=(−MDT)ab=(0−A−BA0−CBC0),where(4.8)A≡2heμνv,B≡2heτνv,C≡2hμτνv.This mass matrix has been
rewritten in a general basis XLT≡(νeL,νμL,ντL,νeRc,νμRc,ντRc):(4.9)Mν≡(0MDTMD0).The tree-level neutrino spectrum, therefore, consists
of only Dirac fermions. Since habν is antisymmetric in a and b,
the mass matrix MD gives one neutrino massless and two others
degenerate in mass 0, −mD, mD, where mD≡(A2+B2+C2)1/2.
This mass spectrum is not realistic under the data; however, it will be
severely changed by the quantum corrections; the most general mass matrix can
then be written as follows:(4.10)Mν=(MLMDTMDMR),where ML,R (vanished at the
tree level) and MD get possible corrections.If such a tree-level contribution dominates the
resulting mass matrix (after corrections), the model will provide an
explanation about a large splitting eitherΔmatm2≫Δmsol2 or ΔmLSND2≫Δmatm, sol2 [3] (see also [49]). Hence, we need a fine-tuning at the tree level
[49] either mD~(Δmatm2)1/2(~5×10−2eV) or mD~(ΔmLSND2)1/2(~eV) [3]. Without loss of generality, assuming that heμν~heτν~hμτν,
we get then hν~10−13(or10−12).
The coupling hν in this case is so small and, therefore, this
fine tuning is not natural [129, 130]. Indeed, as shown below, since hν enters the
dominant corrections from (4.5) for ML,R,
these terms ML,R get very small values which are not large
enough to split the degenerate neutrino masses into a realistic spectrum. (The
largest degenerate splitting in squared mass is still much smaller than Δmsol2~8×10−5eV2 [3].) In addition, in this case, the Dirac masses get
corrections trivially.The above problem can be solvedjust by the LNV operator (4.4), and then
the operator (4.5) obtaining the contributions from particles in the model is
suppressed (for details, see [126]). However, we do not consider the above solution in
this work. This implies that the tree-level Dirac mass term for the neutrinos
by its naturalness should be treated as those as of the usual charged fermions
resulted of the standard symmetry breaking, say, hν~he(~10−6) [129, 130]. It turns out that this term is regarded as a large
bare quantity and unphysical. Under the interactions, they will of course
change to physical masses. In the following, we will obtain such finite renormalizations (for more details,
see [131]) in the
masses of neutrinos.One-Loop Level Dirac and Majorana Masses
The operator
(4.5) and its Hermitian adjoint arise from the radiative corrections mediated
by the model particles and give contributions to Majorana and Dirac mass terms ML, MR,
and MD for the neutrinos. The Yukawa couplings of the
leptons in (2.6) and the relevant Higgs self-couplings in (2.11) are explicitly
rewritten as follows:(4.11)ℒYlept=2habνν¯aLclbLϕ3+−2habνν¯aRlbLϕ1++hablν¯aLlbRϕ1++hablν¯aRclbRϕ3++habll¯aLlbRϕ20+H.c.,ℒHrelv=λ3ϕ1−ϕ1+(χ10*χ10+χ30*χ30)+λ3ϕ3−ϕ3+(χ10*χ10+χ30*χ30)+λ4ϕ1−ϕ1+χ10*χ10+λ4ϕ3−ϕ3+χ30*χ30+λ4ϕ3−ϕ1+χ10*χ30+λ4ϕ1−ϕ3+χ30*χ10.The one-loop corrections to the
mass matrices ML of νL, MR of νR,
and MD of ν are, therefore, given in Figures 10, 11, and 12, respectively.The one-loop corrections for the mass
matrixML.
(a)(b)The one-loop corrections for the
mass matrixMR.
(a)(b)The one-loop corrections for the
mass matrixMD.
(a)(b)Radiative Corrections toML and MR
With the Feynman rules at hand [127, 128], ML is obtained by(4.12)−i(ML)abPL=∫d4p(2π)4(i2hacνPL)i(p+mc)p2−mc2(ihcdlv2PR)i(p+md)p2−md2×(ihbdl*PL)−1(p2−mϕ12)(p2−mϕ32)(iλ4uω2)+∫d4p(2π)4(ihacl*PL)i(−p+mc)p2−mc2(ihdclv2PR)i(−p+md)p2−md2×(i2hbdνPL)−1(p2−mϕ12)(p2−mϕ32)(iλ4uω2).Because the Yukawa couplings of
the charged leptons are flavor diagonal, (4.12) becomes(4.13)(ML)ab=i2λ4uωvhabν[mb2I(mb2,mϕ32,mϕ12)−ma2I(ma2,mϕ32,mϕ12)],(a,bnotsummed),where the integral I(a,b,c) is given in Appendix B.In the effective approximation (2.8), identifications
are given by ϕ3±~H2± and ϕ1±~GW± [39], where H2± and GW± as above mentioned are the charged bilepton
Higgs boson and the Goldstone boson associated with W± boson, respectively. For the masses, we have
also mϕ32≃mH22(≃(λ4/2)ω2) and mϕ12≃0.
Using (B.5), the integrals are given by(4.14)I(ma2,mϕ32,mϕ12)≃−i16π21ma2−mH22[1−mH22ma2−mH22lnma2mH22],a=e,μ,τ.Consequently, the mass matrix
(4.13) becomes(4.15)(ML)ab≃2λ4uωhabν16π2v[mH22(ma2−mb2)(mb2−mH22)(ma2−mH22)+ma2mH22(ma2−mH22)2lnma2mH22−mb2mH22(mb2−mH22)2lnmb2mH22]≃2λ4uωhabν16π2vmH22[ma2(1+lnma2mH22)−mb2(1+lnmb2mH22)],where the last approximation
(4.15) is kept in the orders up to 𝒪[(ma,b2/mH22)2].
Since mH22≃(λ4/2)ω2,
it is worth noting that the resulting ML is not explicitly dependent on λ4,
however, proportional to tθ=u/ω (the mixing angle between the W boson and the singly-charged bilepton gauge
boson Y [38]), 2vhabν (the tree-level Dirac mass term of neutrinos),
and mH2 in the logarithm scale. Here, the VEV v≈vweak and the charged-lepton masses  ma(a=e,μ,τ) have the well-known values. Let us note that ML is symmetric and has vanishing diagonal
elements.For the corrections toMR,
it is easily to check that the relationship (MR)ab=−(ML)ab is exact at the one-loop level. (This result
can be derived from Figure 11 in a general case without imposing any
additional condition on hl, hν,
and further.) Combining this result with (4.15), the mass matrices are
explicitly rewritten as follows:(4.16)(ML)ab=−(MR)ab≃(0frf0trt0),where the elements are obtained
by(4.17)f≡(2vheμν){(tθ8π2v2)[me2(1+lnme2mH22)−mμ2(1+lnmμ2mH22)]},r≡(2vheτν){(tθ8π2v2)[me2(1+lnme2mH22)−mτ2(1+lnmτ2mH22)]},t≡(2vhμτν){(tθ8π2v2)[mμ2(1+lnmμ2mH22)−mτ2(1+lnmτ2mH22)]}.It can be checked that f, r, t are much smaller than those of MD.
To see this, we can take me≃0.51099MeV, mμ≃105.65835MeV, mτ≃1777MeV, v≃246GeV, u≃2.46GeV, ω≃3000GeV,
and mH2≃700GeV(λ4~0.11) [38–40], which give us then(4.18)f≃(2vheμν)(3.18×10−11),r≃(2vheτν)(5.93×10−9),t≃(2vhμτν)(5.90×10−9),where the second factors rescale
negligibly with ω~1–10TeV and mH2~200–2000GeV.
This thus implies that(4.19)|ML,R||MD|~10−9,which can be checked with the
help of |M|≡(M†M)1/2.
In other words, the constraint is given as follows(4.20)|ML,R|≪|MD|. With the above results at hand, we can then get the
masses by studying diagonalization of the mass matrix (4.10), in which, the
submatrices ML,R and MD satisfying the constraint (4.20) are given by
(4.16) and (4.7), respectively. In calculation, let us note that since MD has one vanishing eigenvalue, Mν does not possess the pseudo-Dirac property in
all three generations [132], despite being very close to those because the remaining
eigenvalues do. As a fact, we will see that Mν contains a combined framework of the seesaw
[55–62] and the
pseudo-Dirac [132–142]. To
get mass, we can suppose that hν is real and, therefore, the matrix iMD is Hermitian: (iMD)†=iMD (4.7). The Hermitianity for ML,R is also followed by (4.16). Because the
dominant matrix is MD (4.20), we first diagonalize it by biunitary
transformation [131]:(4.21)ν¯aR=ν¯iR(−iU)ia†,νbL=UbjνjL,(i,j=1,2,3),(4.22)Mdiag≡diag(0,−mD,mD)=(−iU)†MDU,mD=A2+B2+C2,where the matrix U is easily obtained by(4.23)U=1mD2(A2+C2)(C2(A2+C2)iBC−AmDBC−iAmD−B2(A2+C2)i(A2+C2)(A2+C2)A2(A2+C2)iAB+CmDAB+iCmD).Resulted by the
anti-Hermitianity of MD,
it is worth noting that Mν in the case of vanishing ML,R (4.9) is indeed diagonalized by the following
unitary transformation:(4.24)V=12(UU−iUiU). A new basis (ν1,ν2,…,ν6)LT≡V†XLT,
which is different from (νjL,νiRc)T of (4.21), is therefore performed. The neutrino
mass matrix (4.10) in this basis becomes(4.25)V†MνV=(Mdiagϵϵ−Mdiag),(4.26)ϵ≡U†MLU,ϵ†=ϵ, where the elements of ϵ are obtained by(4.27)ϵ11=ϵ22=ϵ33=0,(4.28)ϵ12=iϵ13*={[ABmD+iC(A2−B2+C2)]f+[(C2−A2)mD+2iABC]r+[iA(A2−B2+C2)−BCmD]t}[mD22(A2+C2)]−1,(4.29)ϵ23={(A2+C2)[(CmD−iAB)t−(AmD+iBC)f]−[B(A2−C2)mD+iAC(A2+2B2+C2)]r}[mD2(A2+C2)]−1.
Let us remind the reader that
(4.27) is exactly given at the one-loop level ML (4.13) without imposing any approximation on
this mass matrix. Interchanging the positions of component fields in the basis (ν1,ν2,…,ν6)LT by a permutation transformation P†≡P23P34,
that is, νp→(P†)pqνq(p,q=1,2,…,6) with(4.30)P†=(100000000100010000001000000010000001),the mass matrix (4.25) can be
rewritten as follows:(4.31)P†(V†MνV)P=(0000ϵ12ϵ1300ϵ12ϵ13000ϵ21−mD00ϵ230ϵ310mDϵ320ϵ2100ϵ23mD0ϵ310ϵ3200−mD). It is worth noting that in (4.31), all the off-diagonal
components |ϵ| are much smaller than the eigenvalues |±mD| due to Condition (4.20). The degenerate
eigenvalues 0, −mD,
and +mD (each twice) are now splitting into three
pairs with six different values, two light and four heavy. The two neutrinos of
first pair resulted by the 0 splitting have very small masses as a result
of exactly what a seesaw does [55–62], that is, the off-diagonal block contributions to
these masses are suppressed by the large pseudo-Dirac masses of the lower-right
block. The suppression in this case is different from the usual ones [55–62] because it needs only the
pseudo-Dirac particles [132–142] with the masses mD of the electroweak scale instead of extremely
heavy RH Majorana fields, and that the Dirac masses in those mechanisms are now
played by loop-induced f, r, t (4.17) as a result of the SLB u/ω.
Therefore, the mass matrix (4.31) is effectively decomposed into MS for the first pair of light neutrinos (νS) and MP for the last two pairs of heavy pseudo-Dirac
neutrinos (νP):(4.32)(ν1,ν4,ν2,ν3,ν5,ν6)LT→(νS,νP)LT=Veff†(ν1,ν4,ν2,ν3,ν5,ν6)LT,Veff†(P†V†MνVP)Veff=diag(MS,MP),where Veff, MS,
and MP get the approximations:(4.33)Veff≃(1ℰ−ℰ+1),ℰ≡(00ϵ12ϵ13ϵ12ϵ1300)(−mD00ϵ230mDϵ3200ϵ23mD0ϵ3200−mD)−1,MS≃−ℰ(0ϵ210ϵ31ϵ210ϵ310),MP≃(−mD00ϵ230mDϵ3200ϵ23mD0ϵ3200−mD).The mass matrices MS and MP,
respectively, give exact eigenvalues as follows:(4.34)mS±=±2Im(ϵ13ϵ13ϵ32)mD2−ϵ232≃±2Im(ϵ13ϵ13ϵ32mD2),(4.35)mP±=−mD±|ϵ23|,mP′±=mD±|ϵ23|.
In this case, the mixing
matrices are collected into (νS±,νP±,νP'±)LT=V±†(νS,νP)LT,
where the V± is obtained by(4.36)V±=12(1−1000011000000κ−κ000000110000κ−κ001100),κ≡ϵ23|ϵ23|=exp(iargϵ23).It is to be noted that the
degeneration in the Dirac one |±mD| is now splitting severally.From (4.35), we see that the four large pseudo-Dirac
masses for the neutrinos are almost degenerate. In addition, the resulting
spectrum (4.34), (4.35) yields two largest squared mass splittings, respectively,
proportional to mD2 and 4mD|ϵ23|. From (4.29) and (4.18), we can evaluate |ϵ23|≃3.95×10−9mD≪mD (where A~B~C~mD/3 is understood). Because the splitting 4mD|ϵ23| is still much smaller than Δmsol2,
this therefore implies that the fine tuning, as mentioned, is not realistic.
(In detail, in Table 18, we give the numerical values of these fine tunings,
where the parameters are given as before (4.18).)Table 18
The values forhν and two largest splittings in squared mass.
Fine tuninghνmD2(eV2)4mD|ϵ23|(eV2)mD2~Δmatm28.30×10−142.50×10−33.95×10−11mD2~ΔmLSND21.66×10−121.001.58×10−8Similarly, for the two small masses, we can also
evaluate|mS±|≃4.29×10−28mD.
This shows that the masses mS± are very much smaller than the splitting |ϵ23|.
This also implies that the two light neutrinos in this case are hidden for any mD value of pseudo-Dirac neutrinos. Let us see
the sources of the problem why these masses are so small: (i) vanishing of all
the elements of left-upper block of (4.31); (ii) in (4.34) the resulting masses
are proportional to |ϵ|3/mD2,
but not to |ϵ|2/mD as expected from (4.31). It turns out that this
is due to the antisymmetric of habν enforcing on the tree-level Dirac-mass matrix
and the degenerate of MR=−ML of the one-loop level left-handed (LH) and RH
Majorana-mass matrices. It can be easily checked that such degeneration in
Majorana masses remains up to higher-order radiative corrections as a result of
treating the LH and RH neutrinos in the same gauge triplets with the model
Higgs content. For example, by the aid of (4.5) the degeneration retains up to
any higher-order loop.Radiative Corrections toMD
As mentioned, the mass matrixMD requires the one-loop corrections as given in
Figure 12, and the contributions are easily obtained as follows:(4.37)−i(MDrad)abPL=∫d4p(2π)4(−i2hacνPL)i(p+mc)p2−mc2(ihcdlv2PR)i(p+md)p2−md2×(ihbdl*PL)−1(p2−mϕ12)2(iλ3u2+ω22+iλ4u22)+∫d4p(2π)4(ihacl*PL)i(−p+mc)p2−mc2(ihdclv2PR)i(−p+md)p2−md2×(i2hbdνPL)−1(p2−mϕ32)2(iλ3u2+ω22+iλ4ω22).We
rewrite(4.38)(MDrad)ab=−i2habνv{[λ3(u2+ω2)+λ4u2]mb2I(mb2,mϕ12)+[λ3(u2+ω2)+λ4ω2]ma2I(ma2,mϕ32)},(a,bnotsummed),where I(a,b) is given in (B.13). With the help of (B.14),
the approximation for (4.38) is obtained by(4.39)(MDrad)ab≃−habν82π2v{[λ3(u2+ω2)+λ4u2]+[λ3(u2+ω2)+λ4ω2]ma2mH22}=−2habν(λ3ω216π2v)[1+(1+λ4λ3)(u2ω2+ma2mH22)+𝒪(u4ω4,ma,b4mH24)].Because of the constraint (2.8),
the higher-order corrections 𝒪(⋯) can be neglected; thus MDrad is rewritten as follows:(4.40)(MDrad)ab=−2habν(λ3ω216π2v)(1+δa),δa≡(1+λ4λ3)(u2ω2+ma2mH22),where δa is of course an infinitesimal coefficient,
that is, |δ|≪1.
Again, this implies also that if the fine tuning is done the resulting
Dirac-mass matrix get trivially. It is due to the fact that the contribution of
the term associated with δa in (4.40) is then very small and neglected; the
remaining term gives an antisymmetric resulting Dirac-mass matrix, that is,
therefore, unrealistic under the data.With this result, it is worth noting that the
scale(4.41)|λ3ω216π2v|of the radiative Dirac masses
(4.40) is in the orders of the scale v of the tree-level Dirac masses (4.7). Indeed,
if one puts |(λ3ω2)/(16π2v)|=v and takes |λ3|~0.1–1,
then ω~3–10TeV as expected in the constraints [40, 114, 115]. The resulting Dirac-mass
matrix which is combined of (4.7) and (4.40), therefore, gets two typical
examples of the bounds: (i) (λ3ω2)/(16π2v)+v~𝒪(v);
(ii) (λ3ω2)/(16π2v)+v~𝒪(0).
The first case (i) yields that the status on the masses of neutrinos as given
above is remained unchanged and therefore is also trivial as mentioned. In the
last case (ii), the combination of (4.7) and (4.40) gives(4.42)(MD)ab=2habν(vδa).It is interesting that in this
case the scale v for the Dirac masses (4.7) gets naturally a
large reduction, and we argue that this is not a fine tuning. Because the large
radiative mass term in (4.40) is canceled by the tree-level Dirac masses, we
mean this as a finite renormalization in the masses of neutrinos. It is also
noteworthy that, unlike the case of the tree-level mass term (4.7), the mass
matrix (4.42) is now nonantisymmetric in a and b.
Among the three eigenvalues of this matrix, we can check that one vanishes
(since detMD=0) and two others massive are now nondegenerate (splitting). Let us recall
that in the first case (i) the degeneration of the two nonzero eigenvalues are,
however, retained because the combination of (4.7) and (4.40) is proportional to habνv.In contrast to (4.19), in this case there is no large
hierarchy between ML,R and MD.
To see this explicitly, let us take the values of the parameters as given
before (4.18), thus λ3≃−1.06 and the coefficients δa are evaluated by(4.43)δe≃6.03×10−7,δμ≃6.23×10−7,δτ≃6.28×10−6.Hence, we get(4.44)|ML,R||MD|~10−2–10−3.With the values given in (4.43),
the quantities hν and mD can be evaluated through the mass term (4.42);
the neutrino data imply that hν and mD are in the orders ofhe andme—the Yukawa coupling and mass of electron,
respectively.Because of the Condition (4.44) and the vanishing of
one eigenvalue of MD,
we can repeat the procedure as given above to diagonalize the full matrix Mν with MD given by (4.42) and ML,R by (4.16). First, we can easily find a mixing
matrix V as in (4.24); Second, in the new basis we
obtain the seesaw form as in (4.31); Finally, the resulting mixing matrix and
masses for the neutrinos are derived. It is worth checking that the two largest
squared mass splittings as given before can be approximately applied on this
case of (4.44) such as (mD|δ|)2 and 4(mD|δ|)|ϵ|,
and seeing that they fit naturally the data.Mass Contributions from Heavy Particles
There remain now two questions not yet answered: (i)
the degeneration ofMR=−ML;
(ii) the hierarchy of ML,R and MD (4.44) can be continuously reduced? As
mentioned, we will prove that the new physics gives us the solution.The mass Lagrangian for the neutrinos given by the
operator (4.4) can be explicitly written as follows:(4.45)ℒmassLNV=sabνℳ−1(〈χ†〉ψ¯aLc)(〈χ†〉ψbL)+H.c.=sabνℳ−1(u2ν¯aLc+ω2ν¯aR)(u2νbL+ω2νbRc)+H.c.=−12X¯LcMνnewXL+H.c.,where the mass matrix for the neutrinos
is obtained by(4.46)Mνnew≡−(u2ℳsνuωℳsνuωℳsνω2ℳsν),in which the coupling sabν is symmetric in a and b.
For convenience in reading, let us define the submatrices of (4.46) to be MLnew, MDnew,
and MRnew similar to that of (4.10). Because of the
condition u2≪uω≪ω2,
the corresponding submatrices MLnew, MDnew,
and MRnew of (4.46) get the right hierarchies and the two
questions as mentioned are solved
simultaneously .Intriguing comparisons betweensν and hν are given in order:
(1)
hν conserves the lepton number; sν violates this charge;(2)
hν is antisymmetric and enforcing on the
Dirac-mass matrix; sν is symmetric and breaks this property;(3)
hν preserves the degeneration of MR=−ML; sν breaks the MR=−ML;(4)
a pair of(sν,hν) in the lepton sector will complete the rule
played by the quark couplings (sq,hq) (see below);(5)
hν defines the interactions in the standard model
scale v; sν gives those in the GUT scale ℳ.
Let us now take the values ℳ≃1016GeV, ω≃3000GeV, u≃2.46GeV,
and sν~𝒪(1) (perhaps smaller), the submatrices MLnew≃−6.05×10−7sνeV and MDnew≃−7.38×10−4sνeV can give contributions (to the diagonal
components of ML and MD,
resp.) but very small. It is noteworthy that the last one MRnew≃−0.9sνeV can dominateMR.To summarize, in this model the neutrino mass matrix
is combined byMν+Mνnew,
where the first term is defined by (4.10) and the last term by (4.46); the
submatrices of Mν are given in (4.16) and (4.42), respectively.
Depending on the strength of the new physics
coupling sν,
the submatrices of the last term, MLnew and MDnew,
are included or removed.


##### 4.2.3. Some Remarks from Experimental Constraints

Conventional neutrino oscillations are insensitive to
the absolute scale of neutrino masses. Although the latter will be tested
directly in high-sensitivity tritium beta decay studies and neutrinoless double
beta decay(0νββ) as well as by its effects on the cosmic
microwave background and the large-scale structure of the Universe
[143, 144]. With the present of
sterile neutrinos in this model, the experimental constraints on their masses
may be also important and give us bounds on several parameters such as the
coupling hν and δa.If the liquid scintillator neutrino detector
experiment is confirmed, the sterile-neutrino masses will get some values in
range of eV. In this case, the couplinghν is also in orders of he.
The X-ray measurements yield an upper limit of sterile neutrino mass [145] ms<6.3keV.
For all the other cosmological constraints, the sterile neutrino masses are in
the range [146, 147] 2keV<ms<8keV.
In such cases, the coupling hν will get bounds in orders of hμ,τ.It is well known that the radiative mass generation
can also induce the large lepton flavor violating processes such asμ→eγ as the similar one-loop effect. The possible
one-loop diagrams for this process are depicted in Figure 13. Suppose that mY2,mH22≫mW2=g2v2/4 [38], we get the approximation [148]:(4.47)Br(μ→eγ)≡Γ(μ→eγ)Γ(μ→eνeνμ)≃3sW48π3α(hμτν*heτν)2. Since Br(μ→eγ)<1.2×10−11, α=1/128 and sW2=0.2312 [3], the coupling hν is bounded by hν<3.47×10−3,
where hν≡heτν=hμτν set is understood. Our above result, hν~he,
satisfies this constraint. It can be shown that the value for hν also satisfies constraints from such processes
as μ→3e and μe conversion (for more details, see [149]).One-loop contributions to the lepton flavor violating decayμ→eγ.
(a)(b)(c)(d)


#### 4.3. Quark Masses

First, we present the general quark mass spectrum. Some details on the one-loop quark masses are given then.


##### 4.3.1. Quark Mass Spectra

Note that in [37], the authors have considered the fermion mass
spectrum under the Z2 discrete symmetry which discards the LNV
interactions. Here, the couplings of (2.7) in such case are forbidden. Then, it
can be checked that some quarks remain massless up to two-loop level. To solve
the mass problem of the quarks, the authors in [37] have shown that one-third scalar triplet has to be
added to the resulting model. In the following, we show that it is not
necessary. The Z2 is not introduced and thus the third one is
not required. The LNV Yukawa couplings are vital for the economical 3-3-1
model.The Yukawa couplings in (2.6) and (2.7) give the mass
Lagrangian for the upquarks (quark sector with electric charge qup=2/3):(4.48)ℒupmass=hU2(u¯1Lu+U¯Lω)UR+sau2(u¯1Lu+U¯Lω)uaR−v2u¯αL(hαauuaR+sαUUR)+H.c.Consequently, we obtain the mass
matrix for the upquarks (u1,u2,u3,U) as follows:(4.49)Mup=12(−s1uu−s2uu−s3uu−hUuh21uvh22uvh23uvs2Uvh31uvh32uvh33uvs3Uv−s1uω−s2uω−s3uω−hUω).Because the first and the last
rows of the matrix (4.49) are proportional, the tree-level upquark spectrum
contains a massless one!Similarly, for the downquarks (qdown=−1/3), we get the following mass Lagrangian:(4.50)ℒdownmass=hαβD2(d¯αLu+D¯αLω)DβR+sαad2(d¯αLu+D¯αLω)daR+v2d¯1L(haddaR+sαDDαR)+H.c.Hence, we get mass matrix for
the downquarks (d1,d2,d3,D2,D3): (4.51)Mdown=−12(h1dvh2dvh3dvs2Dvs3Dvs21dus22dus23duh22Duh23Dus31dus32dus33duh32Duh33Dus21dωs22dωs23dωh22Dωh23Dωs31dωs32dωs33dωh32Dωh33Dω).We see that the second and
fourth rows of matrix in (4.51) are proportional, while the third and the last
are the same. Hence, in this case there are two massless eigenstates.The masslessness of the tree-level quarks in both the
sectors calls radiative corrections (the so-called mass problem of quarks).
These corrections start at the one-loop level. The diagrams in Figure14 contribute the upquark spectrum, while Figure 15 gives the
downquarks. Let us note the reader that the quarks also get some one-loop
contributions in the case of the Z2 symmetry enforcing [37]. The careful study of this
radiative mechanism shows that the one-loop quark spectrum is consistent.Figure 14
One-loop contributions to the upquark mass matrix (4.49).Figure 15
One-loop contributions to the downquark mass matrix (4.51).


##### 4.3.2. Typical Examples of the One-Loop Corrections

To provide the quarks masses, in the following we can
suppose that the Yukawa couplings are flavor diagonal. Then, theu2 and u3 states are mass eigenstates corresponding to
the mass eigenvalues:(4.52)m2=h22uv2,m3=h33uv2.The u1 state mixes with the exotic U in terms of one submatrix of the mass matrix
(4.48):(4.53)MuU=−12(s1uuhUus1uωhUω).This matrix contains one
massless quark ~u1, m1=0,
and the remaining exotic quark ~U with the mass of the scale ω.Similarly, for the downquarks, thed1 state is a mass eigenstate corresponding to
the eigenvalue:(4.54)m1′=−h1dv2.The pairs (d2,D2) and (d3,D3) are decoupled,
while the quarks of each pair mix via the mass submatrices, respectively,(4.55)Md2D2=−12(s22duh22Dus22dωh22Dω),(4.56)Md3D3=−12(s33duh33Dus33dωh33Dω).
These matrices contain the
massless quarks ~d2 and d3 corresponding to m2′=0 and m3′=0,
and two exotic quarks ~D2 and D3 with the masses of the scale ω.With the help of the constraint (2.8), we identify m1, m2,
and m3,
respectively, to those of the u1=u, u2=c,
and u3=t quarks. The downquarks d1, d2,
and d3 are, therefore, corresponding to d, s,
and b quarks. Unlike the usual 3-3-1 model with
right-handed neutrinos, where the third family of quarks should be
discriminating [28],
in the model under consideration the first family has to be different from the
two others.The mass matrices (4.53), (4.55), and (4.56) remain the
tree-level properties for the quark spectra—one massless in the upquark
sector and two in the downquark sector. From these
matrices, it is easily to verify that the conditions in (2.8) and (2.10) are
satisfied. First, we consider radiative corrections to the upquark masses.Upquarks
In the previous studies [19, 37, 84–86], the LNV interactions have
often been excluded, commonly by the adoption of an appropriate discrete
symmetry. Let us remind that there is no reason within the 3-3-1 model to
ignore such interactions. The experimental limits on processes which do not
conserve total lepton numbers such as neutrinoless double
beta decay [150, 151] require them to be small.If the Yukawa Lagrangian is restricted toℒLNC [37], then the mass matrix (4.53) becomes(4.57)MuU=−12(0hUu0hUω).In this case, only the element (MuU)12 gets an one-loop correction defined by Figure
16. Other elements remain unchanged under this one-loop effect.Figure 16
One-loop contribution underZ2 to the upquark mass matrix (4.57).The Feynman
rules gives us(4.58)−i(MuU)12PR=∫d4p(2π)4(ihUPR)i(p+MU)p2−MU2(−iMUPL)i(p+MU)p2−MU2(ihUPR)×−1(p2−Mχ12)(p2−Mχ32)(i4λ1)uω2.Thus, we get(4.59)(MuU)12=−2iuωλ1MU(hU)2∫d4p(2π)4p2(p2−MU2)2(p2−Mχ32)(p2−Mχ12)≡−2iuωλ1MU(hU)2I(MU2,Mχ32,Mχ12).The integral I(a,b,c) with a,b≫c is given in the B. Following [39], we conclude that in an
effective approximation, MU2,Mχ32≫Mχ12.
Hence, we have(4.60)(MuU)12≃−λ1tθMU34π2[MU2−Mχ32+Mχ32ln(Mχ32/MU2)(MU2−Mχ32)2]~u≡−12R(MU).The resulting mass matrix is
given by(4.61)MuU=−12(0hUu+R0hUω).We see that one quark remains
massless as the case of the tree-level spectrum. This result keeps up to
two-loop level and can be applied to the downquark sector as well as in the
cases of nondiagonal Yukawa couplings. Therefore, under the Z2,
it is not able to provide consistent masses for the quarks.If the full Yukawa Lagrangian is used, the LNV
couplings must be enough small in comparison with the usual couplings [see
(2.10)]. Combining (2.8) and (2.10), we have(4.62)hUω≫hUu,s1uω≫s1uu.In this case, the element (MuU)11 of (4.53) gets the radiative correction
depicted in Figure 17. The resulting
mass matrix is obtained by(4.63)MuU=−12(s1u(u+RhU)hUus1uωhUω).In contradiction with the first
case, the mass of u quark is now nonzero and given by(4.64)mu≃s1u2hUR. Let us note that the matrix (4.63) gives an eigenvalue
in the scale of (1/2)hUω which can be identified with that of the
exotic quark U.
In effective approximation [39], the mass for the Higgs χ3 is defined by Mχ32≃2λ1ω2.
Hereafter, for the parameters, we use the following values λ1=2.0, tθ=0.08 as mentioned, and ω=10TeV.
The mass value for the u quark is as function of s1u and hU.
Some values of the pair (s1u,hU) which give consistent masses for the u quark are listed in Table 19.Table 19
Mass for theu quark as function of (s1u,hU).
hU21.510.50.1s1u0.00020.00030.00040.0010.01mu [MeV]2.2072.5652.2462.3752.025Figure 17
One-loop contribution to the upquark mass matrix (4.53).Note that the mass values in Table19 for the u quark are in good consistence with the data
given in [3]: mu∈1.5÷4MeV.Downquarks
For the downquarks, the constraint,(4.65)hααDω≫hααDu,sααdω≫sααdu,should be applied. In this case,
three elements (MdαDα)11, (MdαDα)12,
and (MdαDα)21 will get radiative corrections. The relevant
diagrams are depicted in Figure 18. It
is worth noting that Diagram 18(c) exists even in the case of the Z2 symmetry. The contributions are given by(4.66)(MdαDα)11=−sααd2hααDR(MDα),(MdαDα)21=−4iλ1sααdhααDMDα3I(MDα2,Mχ32,Mχ32)=−λ1sααdMDα34π2hααD[MDα2+Mχ32(MDα2−Mχ32)2−2MDα2Mχ32(MDα2−Mχ32)3lnMDα2Mχ32]≡−12R′(MDα),(MdαDα)12=−12R(MDα).We see that two last terms are
much larger than the first one. This is responsible for the masses of the
quarks d2 and d3.
At the one-loop level, the mass matrix for the downquarks is given by(4.67)MdαDα=−12(sααd(u+RhααD)hααDu+Rsααdω+R′hααDω). We remind the reader that a matrix (see also [131])(4.68)(acbD)with D≫b,c≫a has two eigenvalues:(4.69)x1≃[a2−2bcaD+b2c2−(b2+c2)a2D2]1/2,x2≃D.Therefore, the mass matrix in
(4.67) gives an eigenvalue in the scale of D≡(1/2)hααDω which is of the exotic quark Dα′.
Here, we have another eigenvalue for the mass of dα′:(4.70)mdα′=hααDu+R2hααDω{R′2−(sααd)2(hααD)2[(sααdω+R′)2+(hααDu+R)2]}1/2. Let us remember that Mχ32≃2λ1ω2 and the parameters λ1=2.0, tθ=0.08 and ω=10TeV as given above are used in this case; mdα is a function of sααd and hααD.
We take the value hααD=2.0 for both the sectors, α=2 and α=3.
If s22d=0.015,
we get then the mass of the so-called s quark:(4.71)ms=99.3MeV.For the downquark of the third
family, we put s33d=0.7.
Then, the mass of the b quark is obtained by(4.72)mb=4.4GeV. We emphasize again that (4.71) and (4.72) are in good
consistence with the data given in [3]: ms~95±25MeV and mb~4.70±0.07GeV.One-loop contributions to the downquark mass matrix (4.55) or (4.56).
(a)(b)(c)


### 4.4. Summary

The basic motivation of this section is to present the
answer to one of the most crucial questions: whether within the framework of
the model based onSU(3)C⊗SU(3)L⊗U(1)X gauge group contained minimal Higgs sector
with right-handed neutrinos, all fermions including quarks and neutrinos can gain
the consistent masses.In this model, the masses of neutrinos are given by
three different sources widely ranging over the mass scales including the GUT's
and the small VEVu of spontaneous lepton breaking. At the tree
level, there are three Dirac neutrinos: one massless and two degenerate with
the masses in the order of the electron mass. At the one-loop level, a possible
framework for the finite renormalization of the neutrino masses is obtained.
The Dirac masses obtain a large reduction; the Majorana mass types get
degenerate in MR=−ML;
all these masses are in the bound of the data. It is emphasized that the above
degeneration is a consequence of the fact that the left-handed and right-handed
neutrinos in this model are in the same gauge triplets. The new physics
including the 3-3-1 model is strongly signified.
The degenerations and hierarchies among the mass types are completely removed
by heavy particles.The resulting mass matrix for the neutrinos consists
of two partsMν+Mνnew:
the first is mediated by the model particles, and the last is due to the new
physics. Upon the contributions of Mνnew,
the different realistic mass textures can be produced. For example, neglecting
the last term, the pseudo-Dirac patterns can be obtained. In another scenario,
that the bare coupling hν of Dirac masses
gets higher values, for example, in orders of hμ,τ,
the VEV ω can be picked up to an enough large value (~𝒪(104–105)TeV) so that the type II seesaw spectrum is
obtained. Such features deserve further study. We have also shown that the
lepton flavor violating processes such as μ→eγ, μ→3e,
and μe conversion get the consistent values in the
bounds of the current experiments.In the first section, we have shown that, in the
considered model, there are three quite different scales of vacuum expectation
values:ω~𝒪(1)TeV,v≈246GeV,
and u~𝒪(1)GeV.
In this section, we have added a new characteristic property, namely, there are
two types of Yukawa couplings with different strengths: the LNC coupling h's and the LNV ones s's satisfying the condition s≪h.
With the help of these key properties, the mass spectrum of quarks is
consistent without introducing the third scalar triplet. With the given set of
parameters, the numerical evaluation shows that in this model, masses of the
exotic quarks also have different scales, namely, the U exotic quark (qU=2/3) gains mass mU≈700 GeV, while the Dα exotic quarks (qDα=−1/3) have masses in the TeV scale: mDα∈10÷80 TeV.Let us summarize our results.(1)
At the tree level.(a)
All charged leptons gain masses similar to that in the standard model.(b)
One neutrino is massless and the other two are degenerate in masses.(c)
Three
quarksu1,d2,d3 are massless.(d)
Allexotic quarks gain masses proportional toω—the VEV of the first step of symmetry breaking.(2)
At the one-loop level.(a)
All above-mentioned fermions gain masses.(b)
The light quarks gain masses proportional tou—the LNV parameter.(c)
The
exotic quark masses are separated:mU≈700GeV,mDα∈10÷80TeV.(d)
There exist two types of Yukawa couplings: the LNC and LNV with quite different strengths.
With the positive answer, the economical version becomes one of the very attractive models beyond the standard model.


## 4.1. Charged-Lepton Masses

The charged
leptons(l=e,μ,τ) gain masses via the following couplings:(4.1)ℒYl=hablψ¯aLϕlbR+H.c.The mass matrix is, therefore,
followed by(4.2)Ml=−v2(h11lh12lh13lh21lh22lh23lh31lh32lh33l),which of course is the same as
in the standard model and thus gives consistent masses for the charged leptons
[37].For the sake of simplicity, in the following, we can suppose
that the Yukawa coupling of charged leptonshl is flavor diagonal, thus la(a=1,2,3) are mass eigenstates respective to the mass
eigenvalues ma=−(v/2)haal.For convenience in further reading, we present the Yukawa interactions of (2.6) and (2.7) in terms by Feynman diagrams in Figures 6, 7, and 8, where the Hermitian adjoint ones are not displayed. The Higgs boson self-couplings are depicted in Figure 9.Lepton Yukawa couplings.
(a)(b)Relevant lepton-number conserving quark Yukawa couplings.
(a)(b)(c)(d)Lepton-number violating quark Yukawa couplings.
(a)(b)(c)(d)Higgs boson self-couplings.
(a)(b)(c)(d)


## 4.2. Neutrino Masses

First, we
present mass mechanisms for the neutrinos. Next, detailed calculations and
analysis of the neutrino mass spectrum are given. The experimental constraints
on the couplinghν are also considered.


### 4.2.1. Neutrino Mass Mechanisms

In the considering model, the possible different mass
mechanisms for the neutrinos can be summarized through the three dominantSU(3)C⊗SU(3)L⊗U(1)X-invariant effective operators as follows
[112, 113]:(4.3)OabLNC=ψ¯aLcψbLϕ,(4.4)OabLNV=(χ*ψ¯aLc)(χ*ψbL),(4.5)OabSLB=(χ*ψ¯aLc)(ψbLϕχ),
where the Hermitian adjoint
operators are not displayed. It is worth noting that they are also all the
performable operators with the mass dimensionality d≤6 responsible for the neutrino masses. The
difference among the mass mechanisms can be verified through the operators.
Both (4.3) and (4.5) conserve ℒ,
while (4.4) violates this charge with two units. Since d(OLNC)=4 and L〈ϕ〉=0,
(4.3) provides only Dirac masses for the neutrinos which can be obtained at the
tree level through the Yukawa couplings in (2.6). Since d(OSLB)=6 and (L〈χ〉)p≠0 for p=1 vanish for other
cases, (4.5) provides both Dirac and Majorana masses for the neutrinos through
radiative corrections mediated by the model particles. The masses induced by
(4.3) are given by the standard SU(2)L⊗U(1)Y symmetry breaking via the VEV v.
However, those by (4.5) are obtained from both the stages of SU(3)L⊗U(1)X breaking achieved by the VEVs u, ω,
and v.Note that the LNV interactions in (2.7) are due to
quarks. Hence, they do not give contribution to LNV of the leptons such as of
the neutrinos. Except the LNV couplings of (2.7), all the remaining interactions
of the model (lepton Yukawa couplings (2.6), Higgs self-couplings (2.11), etc.)
conserve ℒ.
This means that the operator (4.4) of LNV cannot be mediated by particles of
the model; in other words, it must be introduced by hands. As a fact, the
economical 3-3-1 model including the alternative versions [17–22] are only extensions beyond
the standard model in the scales of orders of TeV [40, 114, 115]. Hence, it is expected that
the operator in (4.4) has to be mediated by heavy particles of an underlined
new physics at a scale ℳ much greater than ω which have been followed in various of grand
unified theories (GUTs) [37, 112, 113, 116–125]. Thus, in this model the neutrinos can get mass from
three very different sources widely ranging over the mass scales: u~𝒪(1)GeV, v≈246GeV, ω~𝒪(1)TeV,
and ℳ~𝒪(1016)GeV.We remind that, in the former version [20–22], the authors in [126] have considered operators
of the type (4.4), however, under a discrete symmetry [22, 37]. As shown in Section 4, the
current model is realistic, and such a discrete symmetry is not needed because
as a fact that the model will fail if it is enforced. In addition, if such
discrete symmetries are not discarded, the important mass contributions for the
neutrinos mediated by model particles are then suppressed. For example, in this
case the remaining operators (4.3) and (4.5) will be removed. With the only
operator (4.4), the three active neutrinos will get effective zero masses under
a type II seesaw [55–62] (see
below). However, this operator occupies a particular importance in this
version.Alternatively, in such model, the authors in [49] have examined two-loop
corrections to (4.4) by the aid of explicit LNV Higgs self-couplings, and using
a fine tuning for the tree-level Dirac masses of (4.3) down to current values.
However, as mentioned, this is not the case in the considering model because
our Higgs potential (2.11) conserves ℒ.
We know that one of the problems of the 3-3-1 model with RH neutrinos is
associated with the Dirac mass term of neutrinos. In the following, we will
show that if such a fine tuning is done to get small values for these terms,
then the mass generation of neutrinos mediated by model particles is not able,
or the results will be trivial. This is in contradiction with [49]. In the next, the large
bare Dirac masses for the neutrinos, which are as of charged fermions of a
natural result from standard symmetry breaking, will be studied.


### 4.2.2. Neutrino Mass Matrix

The operatorsOLNC, OSLB,
and OLNV (including their Hermitian adjoint) will
provide the masses for the neutrinos: the first responsible for tree-level
masses, the second for one-loop corrections, and the third for contributions of
heavy particles.Tree-Level Dirac Masses
From the Yukawa
couplings in (2.6), the tree-level mass Lagrangian for the neutrinos is obtained
by [127, 128]:(4.6)ℒmassLNC=habνν¯aRνbL〈ϕ20〉−habνν¯aLcνbRc〈ϕ20〉+H.c.=2〈ϕ20〉habνν¯aRνbL+H.c.=−(MD)abν¯aRνbL+H.c.=−12(ν¯aLc,ν¯aR)(0(MDT)ab(MD)ab0)(νbLνbRc)+H.c.=−12X¯LcMνXL+H.c.,where habν=−hbaν is due to Fermi statistics. The MD is the mass matrix for the Dirac
neutrinos:(4.7)(MD)ab≡−2vhabν=(−MDT)ab=(0−A−BA0−CBC0),where(4.8)A≡2heμνv,B≡2heτνv,C≡2hμτνv.This mass matrix has been
rewritten in a general basis XLT≡(νeL,νμL,ντL,νeRc,νμRc,ντRc):(4.9)Mν≡(0MDTMD0).The tree-level neutrino spectrum, therefore, consists
of only Dirac fermions. Since habν is antisymmetric in a and b,
the mass matrix MD gives one neutrino massless and two others
degenerate in mass 0, −mD, mD, where mD≡(A2+B2+C2)1/2.
This mass spectrum is not realistic under the data; however, it will be
severely changed by the quantum corrections; the most general mass matrix can
then be written as follows:(4.10)Mν=(MLMDTMDMR),where ML,R (vanished at the
tree level) and MD get possible corrections.If such a tree-level contribution dominates the
resulting mass matrix (after corrections), the model will provide an
explanation about a large splitting eitherΔmatm2≫Δmsol2 or ΔmLSND2≫Δmatm, sol2 [3] (see also [49]). Hence, we need a fine-tuning at the tree level
[49] either mD~(Δmatm2)1/2(~5×10−2eV) or mD~(ΔmLSND2)1/2(~eV) [3]. Without loss of generality, assuming that heμν~heτν~hμτν,
we get then hν~10−13(or10−12).
The coupling hν in this case is so small and, therefore, this
fine tuning is not natural [129, 130]. Indeed, as shown below, since hν enters the
dominant corrections from (4.5) for ML,R,
these terms ML,R get very small values which are not large
enough to split the degenerate neutrino masses into a realistic spectrum. (The
largest degenerate splitting in squared mass is still much smaller than Δmsol2~8×10−5eV2 [3].) In addition, in this case, the Dirac masses get
corrections trivially.The above problem can be solvedjust by the LNV operator (4.4), and then
the operator (4.5) obtaining the contributions from particles in the model is
suppressed (for details, see [126]). However, we do not consider the above solution in
this work. This implies that the tree-level Dirac mass term for the neutrinos
by its naturalness should be treated as those as of the usual charged fermions
resulted of the standard symmetry breaking, say, hν~he(~10−6) [129, 130]. It turns out that this term is regarded as a large
bare quantity and unphysical. Under the interactions, they will of course
change to physical masses. In the following, we will obtain such finite renormalizations (for more details,
see [131]) in the
masses of neutrinos.One-Loop Level Dirac and Majorana Masses
The operator
(4.5) and its Hermitian adjoint arise from the radiative corrections mediated
by the model particles and give contributions to Majorana and Dirac mass terms ML, MR,
and MD for the neutrinos. The Yukawa couplings of the
leptons in (2.6) and the relevant Higgs self-couplings in (2.11) are explicitly
rewritten as follows:(4.11)ℒYlept=2habνν¯aLclbLϕ3+−2habνν¯aRlbLϕ1++hablν¯aLlbRϕ1++hablν¯aRclbRϕ3++habll¯aLlbRϕ20+H.c.,ℒHrelv=λ3ϕ1−ϕ1+(χ10*χ10+χ30*χ30)+λ3ϕ3−ϕ3+(χ10*χ10+χ30*χ30)+λ4ϕ1−ϕ1+χ10*χ10+λ4ϕ3−ϕ3+χ30*χ30+λ4ϕ3−ϕ1+χ10*χ30+λ4ϕ1−ϕ3+χ30*χ10.The one-loop corrections to the
mass matrices ML of νL, MR of νR,
and MD of ν are, therefore, given in Figures 10, 11, and 12, respectively.The one-loop corrections for the mass
matrixML.
(a)(b)The one-loop corrections for the
mass matrixMR.
(a)(b)The one-loop corrections for the
mass matrixMD.
(a)(b)Radiative Corrections toML and MR
With the Feynman rules at hand [127, 128], ML is obtained by(4.12)−i(ML)abPL=∫d4p(2π)4(i2hacνPL)i(p+mc)p2−mc2(ihcdlv2PR)i(p+md)p2−md2×(ihbdl*PL)−1(p2−mϕ12)(p2−mϕ32)(iλ4uω2)+∫d4p(2π)4(ihacl*PL)i(−p+mc)p2−mc2(ihdclv2PR)i(−p+md)p2−md2×(i2hbdνPL)−1(p2−mϕ12)(p2−mϕ32)(iλ4uω2).Because the Yukawa couplings of
the charged leptons are flavor diagonal, (4.12) becomes(4.13)(ML)ab=i2λ4uωvhabν[mb2I(mb2,mϕ32,mϕ12)−ma2I(ma2,mϕ32,mϕ12)],(a,bnotsummed),where the integral I(a,b,c) is given in Appendix B.In the effective approximation (2.8), identifications
are given by ϕ3±~H2± and ϕ1±~GW± [39], where H2± and GW± as above mentioned are the charged bilepton
Higgs boson and the Goldstone boson associated with W± boson, respectively. For the masses, we have
also mϕ32≃mH22(≃(λ4/2)ω2) and mϕ12≃0.
Using (B.5), the integrals are given by(4.14)I(ma2,mϕ32,mϕ12)≃−i16π21ma2−mH22[1−mH22ma2−mH22lnma2mH22],a=e,μ,τ.Consequently, the mass matrix
(4.13) becomes(4.15)(ML)ab≃2λ4uωhabν16π2v[mH22(ma2−mb2)(mb2−mH22)(ma2−mH22)+ma2mH22(ma2−mH22)2lnma2mH22−mb2mH22(mb2−mH22)2lnmb2mH22]≃2λ4uωhabν16π2vmH22[ma2(1+lnma2mH22)−mb2(1+lnmb2mH22)],where the last approximation
(4.15) is kept in the orders up to 𝒪[(ma,b2/mH22)2].
Since mH22≃(λ4/2)ω2,
it is worth noting that the resulting ML is not explicitly dependent on λ4,
however, proportional to tθ=u/ω (the mixing angle between the W boson and the singly-charged bilepton gauge
boson Y [38]), 2vhabν (the tree-level Dirac mass term of neutrinos),
and mH2 in the logarithm scale. Here, the VEV v≈vweak and the charged-lepton masses  ma(a=e,μ,τ) have the well-known values. Let us note that ML is symmetric and has vanishing diagonal
elements.For the corrections toMR,
it is easily to check that the relationship (MR)ab=−(ML)ab is exact at the one-loop level. (This result
can be derived from Figure 11 in a general case without imposing any
additional condition on hl, hν,
and further.) Combining this result with (4.15), the mass matrices are
explicitly rewritten as follows:(4.16)(ML)ab=−(MR)ab≃(0frf0trt0),where the elements are obtained
by(4.17)f≡(2vheμν){(tθ8π2v2)[me2(1+lnme2mH22)−mμ2(1+lnmμ2mH22)]},r≡(2vheτν){(tθ8π2v2)[me2(1+lnme2mH22)−mτ2(1+lnmτ2mH22)]},t≡(2vhμτν){(tθ8π2v2)[mμ2(1+lnmμ2mH22)−mτ2(1+lnmτ2mH22)]}.It can be checked that f, r, t are much smaller than those of MD.
To see this, we can take me≃0.51099MeV, mμ≃105.65835MeV, mτ≃1777MeV, v≃246GeV, u≃2.46GeV, ω≃3000GeV,
and mH2≃700GeV(λ4~0.11) [38–40], which give us then(4.18)f≃(2vheμν)(3.18×10−11),r≃(2vheτν)(5.93×10−9),t≃(2vhμτν)(5.90×10−9),where the second factors rescale
negligibly with ω~1–10TeV and mH2~200–2000GeV.
This thus implies that(4.19)|ML,R||MD|~10−9,which can be checked with the
help of |M|≡(M†M)1/2.
In other words, the constraint is given as follows(4.20)|ML,R|≪|MD|. With the above results at hand, we can then get the
masses by studying diagonalization of the mass matrix (4.10), in which, the
submatrices ML,R and MD satisfying the constraint (4.20) are given by
(4.16) and (4.7), respectively. In calculation, let us note that since MD has one vanishing eigenvalue, Mν does not possess the pseudo-Dirac property in
all three generations [132], despite being very close to those because the remaining
eigenvalues do. As a fact, we will see that Mν contains a combined framework of the seesaw
[55–62] and the
pseudo-Dirac [132–142]. To
get mass, we can suppose that hν is real and, therefore, the matrix iMD is Hermitian: (iMD)†=iMD (4.7). The Hermitianity for ML,R is also followed by (4.16). Because the
dominant matrix is MD (4.20), we first diagonalize it by biunitary
transformation [131]:(4.21)ν¯aR=ν¯iR(−iU)ia†,νbL=UbjνjL,(i,j=1,2,3),(4.22)Mdiag≡diag(0,−mD,mD)=(−iU)†MDU,mD=A2+B2+C2,where the matrix U is easily obtained by(4.23)U=1mD2(A2+C2)(C2(A2+C2)iBC−AmDBC−iAmD−B2(A2+C2)i(A2+C2)(A2+C2)A2(A2+C2)iAB+CmDAB+iCmD).Resulted by the
anti-Hermitianity of MD,
it is worth noting that Mν in the case of vanishing ML,R (4.9) is indeed diagonalized by the following
unitary transformation:(4.24)V=12(UU−iUiU). A new basis (ν1,ν2,…,ν6)LT≡V†XLT,
which is different from (νjL,νiRc)T of (4.21), is therefore performed. The neutrino
mass matrix (4.10) in this basis becomes(4.25)V†MνV=(Mdiagϵϵ−Mdiag),(4.26)ϵ≡U†MLU,ϵ†=ϵ, where the elements of ϵ are obtained by(4.27)ϵ11=ϵ22=ϵ33=0,(4.28)ϵ12=iϵ13*={[ABmD+iC(A2−B2+C2)]f+[(C2−A2)mD+2iABC]r+[iA(A2−B2+C2)−BCmD]t}[mD22(A2+C2)]−1,(4.29)ϵ23={(A2+C2)[(CmD−iAB)t−(AmD+iBC)f]−[B(A2−C2)mD+iAC(A2+2B2+C2)]r}[mD2(A2+C2)]−1.
Let us remind the reader that
(4.27) is exactly given at the one-loop level ML (4.13) without imposing any approximation on
this mass matrix. Interchanging the positions of component fields in the basis (ν1,ν2,…,ν6)LT by a permutation transformation P†≡P23P34,
that is, νp→(P†)pqνq(p,q=1,2,…,6) with(4.30)P†=(100000000100010000001000000010000001),the mass matrix (4.25) can be
rewritten as follows:(4.31)P†(V†MνV)P=(0000ϵ12ϵ1300ϵ12ϵ13000ϵ21−mD00ϵ230ϵ310mDϵ320ϵ2100ϵ23mD0ϵ310ϵ3200−mD). It is worth noting that in (4.31), all the off-diagonal
components |ϵ| are much smaller than the eigenvalues |±mD| due to Condition (4.20). The degenerate
eigenvalues 0, −mD,
and +mD (each twice) are now splitting into three
pairs with six different values, two light and four heavy. The two neutrinos of
first pair resulted by the 0 splitting have very small masses as a result
of exactly what a seesaw does [55–62], that is, the off-diagonal block contributions to
these masses are suppressed by the large pseudo-Dirac masses of the lower-right
block. The suppression in this case is different from the usual ones [55–62] because it needs only the
pseudo-Dirac particles [132–142] with the masses mD of the electroweak scale instead of extremely
heavy RH Majorana fields, and that the Dirac masses in those mechanisms are now
played by loop-induced f, r, t (4.17) as a result of the SLB u/ω.
Therefore, the mass matrix (4.31) is effectively decomposed into MS for the first pair of light neutrinos (νS) and MP for the last two pairs of heavy pseudo-Dirac
neutrinos (νP):(4.32)(ν1,ν4,ν2,ν3,ν5,ν6)LT→(νS,νP)LT=Veff†(ν1,ν4,ν2,ν3,ν5,ν6)LT,Veff†(P†V†MνVP)Veff=diag(MS,MP),where Veff, MS,
and MP get the approximations:(4.33)Veff≃(1ℰ−ℰ+1),ℰ≡(00ϵ12ϵ13ϵ12ϵ1300)(−mD00ϵ230mDϵ3200ϵ23mD0ϵ3200−mD)−1,MS≃−ℰ(0ϵ210ϵ31ϵ210ϵ310),MP≃(−mD00ϵ230mDϵ3200ϵ23mD0ϵ3200−mD).The mass matrices MS and MP,
respectively, give exact eigenvalues as follows:(4.34)mS±=±2Im(ϵ13ϵ13ϵ32)mD2−ϵ232≃±2Im(ϵ13ϵ13ϵ32mD2),(4.35)mP±=−mD±|ϵ23|,mP′±=mD±|ϵ23|.
In this case, the mixing
matrices are collected into (νS±,νP±,νP'±)LT=V±†(νS,νP)LT,
where the V± is obtained by(4.36)V±=12(1−1000011000000κ−κ000000110000κ−κ001100),κ≡ϵ23|ϵ23|=exp(iargϵ23).It is to be noted that the
degeneration in the Dirac one |±mD| is now splitting severally.From (4.35), we see that the four large pseudo-Dirac
masses for the neutrinos are almost degenerate. In addition, the resulting
spectrum (4.34), (4.35) yields two largest squared mass splittings, respectively,
proportional to mD2 and 4mD|ϵ23|. From (4.29) and (4.18), we can evaluate |ϵ23|≃3.95×10−9mD≪mD (where A~B~C~mD/3 is understood). Because the splitting 4mD|ϵ23| is still much smaller than Δmsol2,
this therefore implies that the fine tuning, as mentioned, is not realistic.
(In detail, in Table 18, we give the numerical values of these fine tunings,
where the parameters are given as before (4.18).)Table 18
The values forhν and two largest splittings in squared mass.
Fine tuninghνmD2(eV2)4mD|ϵ23|(eV2)mD2~Δmatm28.30×10−142.50×10−33.95×10−11mD2~ΔmLSND21.66×10−121.001.58×10−8Similarly, for the two small masses, we can also
evaluate|mS±|≃4.29×10−28mD.
This shows that the masses mS± are very much smaller than the splitting |ϵ23|.
This also implies that the two light neutrinos in this case are hidden for any mD value of pseudo-Dirac neutrinos. Let us see
the sources of the problem why these masses are so small: (i) vanishing of all
the elements of left-upper block of (4.31); (ii) in (4.34) the resulting masses
are proportional to |ϵ|3/mD2,
but not to |ϵ|2/mD as expected from (4.31). It turns out that this
is due to the antisymmetric of habν enforcing on the tree-level Dirac-mass matrix
and the degenerate of MR=−ML of the one-loop level left-handed (LH) and RH
Majorana-mass matrices. It can be easily checked that such degeneration in
Majorana masses remains up to higher-order radiative corrections as a result of
treating the LH and RH neutrinos in the same gauge triplets with the model
Higgs content. For example, by the aid of (4.5) the degeneration retains up to
any higher-order loop.Radiative Corrections toMD
As mentioned, the mass matrixMD requires the one-loop corrections as given in
Figure 12, and the contributions are easily obtained as follows:(4.37)−i(MDrad)abPL=∫d4p(2π)4(−i2hacνPL)i(p+mc)p2−mc2(ihcdlv2PR)i(p+md)p2−md2×(ihbdl*PL)−1(p2−mϕ12)2(iλ3u2+ω22+iλ4u22)+∫d4p(2π)4(ihacl*PL)i(−p+mc)p2−mc2(ihdclv2PR)i(−p+md)p2−md2×(i2hbdνPL)−1(p2−mϕ32)2(iλ3u2+ω22+iλ4ω22).We
rewrite(4.38)(MDrad)ab=−i2habνv{[λ3(u2+ω2)+λ4u2]mb2I(mb2,mϕ12)+[λ3(u2+ω2)+λ4ω2]ma2I(ma2,mϕ32)},(a,bnotsummed),where I(a,b) is given in (B.13). With the help of (B.14),
the approximation for (4.38) is obtained by(4.39)(MDrad)ab≃−habν82π2v{[λ3(u2+ω2)+λ4u2]+[λ3(u2+ω2)+λ4ω2]ma2mH22}=−2habν(λ3ω216π2v)[1+(1+λ4λ3)(u2ω2+ma2mH22)+𝒪(u4ω4,ma,b4mH24)].Because of the constraint (2.8),
the higher-order corrections 𝒪(⋯) can be neglected; thus MDrad is rewritten as follows:(4.40)(MDrad)ab=−2habν(λ3ω216π2v)(1+δa),δa≡(1+λ4λ3)(u2ω2+ma2mH22),where δa is of course an infinitesimal coefficient,
that is, |δ|≪1.
Again, this implies also that if the fine tuning is done the resulting
Dirac-mass matrix get trivially. It is due to the fact that the contribution of
the term associated with δa in (4.40) is then very small and neglected; the
remaining term gives an antisymmetric resulting Dirac-mass matrix, that is,
therefore, unrealistic under the data.With this result, it is worth noting that the
scale(4.41)|λ3ω216π2v|of the radiative Dirac masses
(4.40) is in the orders of the scale v of the tree-level Dirac masses (4.7). Indeed,
if one puts |(λ3ω2)/(16π2v)|=v and takes |λ3|~0.1–1,
then ω~3–10TeV as expected in the constraints [40, 114, 115]. The resulting Dirac-mass
matrix which is combined of (4.7) and (4.40), therefore, gets two typical
examples of the bounds: (i) (λ3ω2)/(16π2v)+v~𝒪(v);
(ii) (λ3ω2)/(16π2v)+v~𝒪(0).
The first case (i) yields that the status on the masses of neutrinos as given
above is remained unchanged and therefore is also trivial as mentioned. In the
last case (ii), the combination of (4.7) and (4.40) gives(4.42)(MD)ab=2habν(vδa).It is interesting that in this
case the scale v for the Dirac masses (4.7) gets naturally a
large reduction, and we argue that this is not a fine tuning. Because the large
radiative mass term in (4.40) is canceled by the tree-level Dirac masses, we
mean this as a finite renormalization in the masses of neutrinos. It is also
noteworthy that, unlike the case of the tree-level mass term (4.7), the mass
matrix (4.42) is now nonantisymmetric in a and b.
Among the three eigenvalues of this matrix, we can check that one vanishes
(since detMD=0) and two others massive are now nondegenerate (splitting). Let us recall
that in the first case (i) the degeneration of the two nonzero eigenvalues are,
however, retained because the combination of (4.7) and (4.40) is proportional to habνv.In contrast to (4.19), in this case there is no large
hierarchy between ML,R and MD.
To see this explicitly, let us take the values of the parameters as given
before (4.18), thus λ3≃−1.06 and the coefficients δa are evaluated by(4.43)δe≃6.03×10−7,δμ≃6.23×10−7,δτ≃6.28×10−6.Hence, we get(4.44)|ML,R||MD|~10−2–10−3.With the values given in (4.43),
the quantities hν and mD can be evaluated through the mass term (4.42);
the neutrino data imply that hν and mD are in the orders ofhe andme—the Yukawa coupling and mass of electron,
respectively.Because of the Condition (4.44) and the vanishing of
one eigenvalue of MD,
we can repeat the procedure as given above to diagonalize the full matrix Mν with MD given by (4.42) and ML,R by (4.16). First, we can easily find a mixing
matrix V as in (4.24); Second, in the new basis we
obtain the seesaw form as in (4.31); Finally, the resulting mixing matrix and
masses for the neutrinos are derived. It is worth checking that the two largest
squared mass splittings as given before can be approximately applied on this
case of (4.44) such as (mD|δ|)2 and 4(mD|δ|)|ϵ|,
and seeing that they fit naturally the data.Mass Contributions from Heavy Particles
There remain now two questions not yet answered: (i)
the degeneration ofMR=−ML;
(ii) the hierarchy of ML,R and MD (4.44) can be continuously reduced? As
mentioned, we will prove that the new physics gives us the solution.The mass Lagrangian for the neutrinos given by the
operator (4.4) can be explicitly written as follows:(4.45)ℒmassLNV=sabνℳ−1(〈χ†〉ψ¯aLc)(〈χ†〉ψbL)+H.c.=sabνℳ−1(u2ν¯aLc+ω2ν¯aR)(u2νbL+ω2νbRc)+H.c.=−12X¯LcMνnewXL+H.c.,where the mass matrix for the neutrinos
is obtained by(4.46)Mνnew≡−(u2ℳsνuωℳsνuωℳsνω2ℳsν),in which the coupling sabν is symmetric in a and b.
For convenience in reading, let us define the submatrices of (4.46) to be MLnew, MDnew,
and MRnew similar to that of (4.10). Because of the
condition u2≪uω≪ω2,
the corresponding submatrices MLnew, MDnew,
and MRnew of (4.46) get the right hierarchies and the two
questions as mentioned are solved
simultaneously .Intriguing comparisons betweensν and hν are given in order:
(1)
hν conserves the lepton number; sν violates this charge;(2)
hν is antisymmetric and enforcing on the
Dirac-mass matrix; sν is symmetric and breaks this property;(3)
hν preserves the degeneration of MR=−ML; sν breaks the MR=−ML;(4)
a pair of(sν,hν) in the lepton sector will complete the rule
played by the quark couplings (sq,hq) (see below);(5)
hν defines the interactions in the standard model
scale v; sν gives those in the GUT scale ℳ.
Let us now take the values ℳ≃1016GeV, ω≃3000GeV, u≃2.46GeV,
and sν~𝒪(1) (perhaps smaller), the submatrices MLnew≃−6.05×10−7sνeV and MDnew≃−7.38×10−4sνeV can give contributions (to the diagonal
components of ML and MD,
resp.) but very small. It is noteworthy that the last one MRnew≃−0.9sνeV can dominateMR.To summarize, in this model the neutrino mass matrix
is combined byMν+Mνnew,
where the first term is defined by (4.10) and the last term by (4.46); the
submatrices of Mν are given in (4.16) and (4.42), respectively.
Depending on the strength of the new physics
coupling sν,
the submatrices of the last term, MLnew and MDnew,
are included or removed.


#### 4.2.3. Some Remarks from Experimental Constraints

Conventional neutrino oscillations are insensitive to
the absolute scale of neutrino masses. Although the latter will be tested
directly in high-sensitivity tritium beta decay studies and neutrinoless double
beta decay(0νββ) as well as by its effects on the cosmic
microwave background and the large-scale structure of the Universe
[143, 144]. With the present of
sterile neutrinos in this model, the experimental constraints on their masses
may be also important and give us bounds on several parameters such as the
coupling hν and δa.If the liquid scintillator neutrino detector
experiment is confirmed, the sterile-neutrino masses will get some values in
range of eV. In this case, the couplinghν is also in orders of he.
The X-ray measurements yield an upper limit of sterile neutrino mass [145] ms<6.3keV.
For all the other cosmological constraints, the sterile neutrino masses are in
the range [146, 147] 2keV<ms<8keV.
In such cases, the coupling hν will get bounds in orders of hμ,τ.It is well known that the radiative mass generation
can also induce the large lepton flavor violating processes such asμ→eγ as the similar one-loop effect. The possible
one-loop diagrams for this process are depicted in Figure 13. Suppose that mY2,mH22≫mW2=g2v2/4 [38], we get the approximation [148]:(4.47)Br(μ→eγ)≡Γ(μ→eγ)Γ(μ→eνeνμ)≃3sW48π3α(hμτν*heτν)2. Since Br(μ→eγ)<1.2×10−11, α=1/128 and sW2=0.2312 [3], the coupling hν is bounded by hν<3.47×10−3,
where hν≡heτν=hμτν set is understood. Our above result, hν~he,
satisfies this constraint. It can be shown that the value for hν also satisfies constraints from such processes
as μ→3e and μe conversion (for more details, see [149]).One-loop contributions to the lepton flavor violating decayμ→eγ.
(a)(b)(c)(d)


### 4.3. Quark Masses

First, we present the general quark mass spectrum. Some details on the one-loop quark masses are given then.


#### 4.3.1. Quark Mass Spectra

Note that in [37], the authors have considered the fermion mass
spectrum under the Z2 discrete symmetry which discards the LNV
interactions. Here, the couplings of (2.7) in such case are forbidden. Then, it
can be checked that some quarks remain massless up to two-loop level. To solve
the mass problem of the quarks, the authors in [37] have shown that one-third scalar triplet has to be
added to the resulting model. In the following, we show that it is not
necessary. The Z2 is not introduced and thus the third one is
not required. The LNV Yukawa couplings are vital for the economical 3-3-1
model.The Yukawa couplings in (2.6) and (2.7) give the mass
Lagrangian for the upquarks (quark sector with electric charge qup=2/3):(4.48)ℒupmass=hU2(u¯1Lu+U¯Lω)UR+sau2(u¯1Lu+U¯Lω)uaR−v2u¯αL(hαauuaR+sαUUR)+H.c.Consequently, we obtain the mass
matrix for the upquarks (u1,u2,u3,U) as follows:(4.49)Mup=12(−s1uu−s2uu−s3uu−hUuh21uvh22uvh23uvs2Uvh31uvh32uvh33uvs3Uv−s1uω−s2uω−s3uω−hUω).Because the first and the last
rows of the matrix (4.49) are proportional, the tree-level upquark spectrum
contains a massless one!Similarly, for the downquarks (qdown=−1/3), we get the following mass Lagrangian:(4.50)ℒdownmass=hαβD2(d¯αLu+D¯αLω)DβR+sαad2(d¯αLu+D¯αLω)daR+v2d¯1L(haddaR+sαDDαR)+H.c.Hence, we get mass matrix for
the downquarks (d1,d2,d3,D2,D3): (4.51)Mdown=−12(h1dvh2dvh3dvs2Dvs3Dvs21dus22dus23duh22Duh23Dus31dus32dus33duh32Duh33Dus21dωs22dωs23dωh22Dωh23Dωs31dωs32dωs33dωh32Dωh33Dω).We see that the second and
fourth rows of matrix in (4.51) are proportional, while the third and the last
are the same. Hence, in this case there are two massless eigenstates.The masslessness of the tree-level quarks in both the
sectors calls radiative corrections (the so-called mass problem of quarks).
These corrections start at the one-loop level. The diagrams in Figure14 contribute the upquark spectrum, while Figure 15 gives the
downquarks. Let us note the reader that the quarks also get some one-loop
contributions in the case of the Z2 symmetry enforcing [37]. The careful study of this
radiative mechanism shows that the one-loop quark spectrum is consistent.Figure 14
One-loop contributions to the upquark mass matrix (4.49).Figure 15
One-loop contributions to the downquark mass matrix (4.51).


#### 4.3.2. Typical Examples of the One-Loop Corrections

To provide the quarks masses, in the following we can
suppose that the Yukawa couplings are flavor diagonal. Then, theu2 and u3 states are mass eigenstates corresponding to
the mass eigenvalues:(4.52)m2=h22uv2,m3=h33uv2.The u1 state mixes with the exotic U in terms of one submatrix of the mass matrix
(4.48):(4.53)MuU=−12(s1uuhUus1uωhUω).This matrix contains one
massless quark ~u1, m1=0,
and the remaining exotic quark ~U with the mass of the scale ω.Similarly, for the downquarks, thed1 state is a mass eigenstate corresponding to
the eigenvalue:(4.54)m1′=−h1dv2.The pairs (d2,D2) and (d3,D3) are decoupled,
while the quarks of each pair mix via the mass submatrices, respectively,(4.55)Md2D2=−12(s22duh22Dus22dωh22Dω),(4.56)Md3D3=−12(s33duh33Dus33dωh33Dω).
These matrices contain the
massless quarks ~d2 and d3 corresponding to m2′=0 and m3′=0,
and two exotic quarks ~D2 and D3 with the masses of the scale ω.With the help of the constraint (2.8), we identify m1, m2,
and m3,
respectively, to those of the u1=u, u2=c,
and u3=t quarks. The downquarks d1, d2,
and d3 are, therefore, corresponding to d, s,
and b quarks. Unlike the usual 3-3-1 model with
right-handed neutrinos, where the third family of quarks should be
discriminating [28],
in the model under consideration the first family has to be different from the
two others.The mass matrices (4.53), (4.55), and (4.56) remain the
tree-level properties for the quark spectra—one massless in the upquark
sector and two in the downquark sector. From these
matrices, it is easily to verify that the conditions in (2.8) and (2.10) are
satisfied. First, we consider radiative corrections to the upquark masses.Upquarks
In the previous studies [19, 37, 84–86], the LNV interactions have
often been excluded, commonly by the adoption of an appropriate discrete
symmetry. Let us remind that there is no reason within the 3-3-1 model to
ignore such interactions. The experimental limits on processes which do not
conserve total lepton numbers such as neutrinoless double
beta decay [150, 151] require them to be small.If the Yukawa Lagrangian is restricted toℒLNC [37], then the mass matrix (4.53) becomes(4.57)MuU=−12(0hUu0hUω).In this case, only the element (MuU)12 gets an one-loop correction defined by Figure
16. Other elements remain unchanged under this one-loop effect.Figure 16
One-loop contribution underZ2 to the upquark mass matrix (4.57).The Feynman
rules gives us(4.58)−i(MuU)12PR=∫d4p(2π)4(ihUPR)i(p+MU)p2−MU2(−iMUPL)i(p+MU)p2−MU2(ihUPR)×−1(p2−Mχ12)(p2−Mχ32)(i4λ1)uω2.Thus, we get(4.59)(MuU)12=−2iuωλ1MU(hU)2∫d4p(2π)4p2(p2−MU2)2(p2−Mχ32)(p2−Mχ12)≡−2iuωλ1MU(hU)2I(MU2,Mχ32,Mχ12).The integral I(a,b,c) with a,b≫c is given in the B. Following [39], we conclude that in an
effective approximation, MU2,Mχ32≫Mχ12.
Hence, we have(4.60)(MuU)12≃−λ1tθMU34π2[MU2−Mχ32+Mχ32ln(Mχ32/MU2)(MU2−Mχ32)2]~u≡−12R(MU).The resulting mass matrix is
given by(4.61)MuU=−12(0hUu+R0hUω).We see that one quark remains
massless as the case of the tree-level spectrum. This result keeps up to
two-loop level and can be applied to the downquark sector as well as in the
cases of nondiagonal Yukawa couplings. Therefore, under the Z2,
it is not able to provide consistent masses for the quarks.If the full Yukawa Lagrangian is used, the LNV
couplings must be enough small in comparison with the usual couplings [see
(2.10)]. Combining (2.8) and (2.10), we have(4.62)hUω≫hUu,s1uω≫s1uu.In this case, the element (MuU)11 of (4.53) gets the radiative correction
depicted in Figure 17. The resulting
mass matrix is obtained by(4.63)MuU=−12(s1u(u+RhU)hUus1uωhUω).In contradiction with the first
case, the mass of u quark is now nonzero and given by(4.64)mu≃s1u2hUR. Let us note that the matrix (4.63) gives an eigenvalue
in the scale of (1/2)hUω which can be identified with that of the
exotic quark U.
In effective approximation [39], the mass for the Higgs χ3 is defined by Mχ32≃2λ1ω2.
Hereafter, for the parameters, we use the following values λ1=2.0, tθ=0.08 as mentioned, and ω=10TeV.
The mass value for the u quark is as function of s1u and hU.
Some values of the pair (s1u,hU) which give consistent masses for the u quark are listed in Table 19.Table 19
Mass for theu quark as function of (s1u,hU).
hU21.510.50.1s1u0.00020.00030.00040.0010.01mu [MeV]2.2072.5652.2462.3752.025Figure 17
One-loop contribution to the upquark mass matrix (4.53).Note that the mass values in Table19 for the u quark are in good consistence with the data
given in [3]: mu∈1.5÷4MeV.Downquarks
For the downquarks, the constraint,(4.65)hααDω≫hααDu,sααdω≫sααdu,should be applied. In this case,
three elements (MdαDα)11, (MdαDα)12,
and (MdαDα)21 will get radiative corrections. The relevant
diagrams are depicted in Figure 18. It
is worth noting that Diagram 18(c) exists even in the case of the Z2 symmetry. The contributions are given by(4.66)(MdαDα)11=−sααd2hααDR(MDα),(MdαDα)21=−4iλ1sααdhααDMDα3I(MDα2,Mχ32,Mχ32)=−λ1sααdMDα34π2hααD[MDα2+Mχ32(MDα2−Mχ32)2−2MDα2Mχ32(MDα2−Mχ32)3lnMDα2Mχ32]≡−12R′(MDα),(MdαDα)12=−12R(MDα).We see that two last terms are
much larger than the first one. This is responsible for the masses of the
quarks d2 and d3.
At the one-loop level, the mass matrix for the downquarks is given by(4.67)MdαDα=−12(sααd(u+RhααD)hααDu+Rsααdω+R′hααDω). We remind the reader that a matrix (see also [131])(4.68)(acbD)with D≫b,c≫a has two eigenvalues:(4.69)x1≃[a2−2bcaD+b2c2−(b2+c2)a2D2]1/2,x2≃D.Therefore, the mass matrix in
(4.67) gives an eigenvalue in the scale of D≡(1/2)hααDω which is of the exotic quark Dα′.
Here, we have another eigenvalue for the mass of dα′:(4.70)mdα′=hααDu+R2hααDω{R′2−(sααd)2(hααD)2[(sααdω+R′)2+(hααDu+R)2]}1/2. Let us remember that Mχ32≃2λ1ω2 and the parameters λ1=2.0, tθ=0.08 and ω=10TeV as given above are used in this case; mdα is a function of sααd and hααD.
We take the value hααD=2.0 for both the sectors, α=2 and α=3.
If s22d=0.015,
we get then the mass of the so-called s quark:(4.71)ms=99.3MeV.For the downquark of the third
family, we put s33d=0.7.
Then, the mass of the b quark is obtained by(4.72)mb=4.4GeV. We emphasize again that (4.71) and (4.72) are in good
consistence with the data given in [3]: ms~95±25MeV and mb~4.70±0.07GeV.One-loop contributions to the downquark mass matrix (4.55) or (4.56).
(a)(b)(c)


## 4.2.1. Neutrino Mass Mechanisms

In the considering model, the possible different mass
mechanisms for the neutrinos can be summarized through the three dominantSU(3)C⊗SU(3)L⊗U(1)X-invariant effective operators as follows
[112, 113]:(4.3)OabLNC=ψ¯aLcψbLϕ,(4.4)OabLNV=(χ*ψ¯aLc)(χ*ψbL),(4.5)OabSLB=(χ*ψ¯aLc)(ψbLϕχ),
where the Hermitian adjoint
operators are not displayed. It is worth noting that they are also all the
performable operators with the mass dimensionality d≤6 responsible for the neutrino masses. The
difference among the mass mechanisms can be verified through the operators.
Both (4.3) and (4.5) conserve ℒ,
while (4.4) violates this charge with two units. Since d(OLNC)=4 and L〈ϕ〉=0,
(4.3) provides only Dirac masses for the neutrinos which can be obtained at the
tree level through the Yukawa couplings in (2.6). Since d(OSLB)=6 and (L〈χ〉)p≠0 for p=1 vanish for other
cases, (4.5) provides both Dirac and Majorana masses for the neutrinos through
radiative corrections mediated by the model particles. The masses induced by
(4.3) are given by the standard SU(2)L⊗U(1)Y symmetry breaking via the VEV v.
However, those by (4.5) are obtained from both the stages of SU(3)L⊗U(1)X breaking achieved by the VEVs u, ω,
and v.Note that the LNV interactions in (2.7) are due to
quarks. Hence, they do not give contribution to LNV of the leptons such as of
the neutrinos. Except the LNV couplings of (2.7), all the remaining interactions
of the model (lepton Yukawa couplings (2.6), Higgs self-couplings (2.11), etc.)
conserve ℒ.
This means that the operator (4.4) of LNV cannot be mediated by particles of
the model; in other words, it must be introduced by hands. As a fact, the
economical 3-3-1 model including the alternative versions [17–22] are only extensions beyond
the standard model in the scales of orders of TeV [40, 114, 115]. Hence, it is expected that
the operator in (4.4) has to be mediated by heavy particles of an underlined
new physics at a scale ℳ much greater than ω which have been followed in various of grand
unified theories (GUTs) [37, 112, 113, 116–125]. Thus, in this model the neutrinos can get mass from
three very different sources widely ranging over the mass scales: u~𝒪(1)GeV, v≈246GeV, ω~𝒪(1)TeV,
and ℳ~𝒪(1016)GeV.We remind that, in the former version [20–22], the authors in [126] have considered operators
of the type (4.4), however, under a discrete symmetry [22, 37]. As shown in Section 4, the
current model is realistic, and such a discrete symmetry is not needed because
as a fact that the model will fail if it is enforced. In addition, if such
discrete symmetries are not discarded, the important mass contributions for the
neutrinos mediated by model particles are then suppressed. For example, in this
case the remaining operators (4.3) and (4.5) will be removed. With the only
operator (4.4), the three active neutrinos will get effective zero masses under
a type II seesaw [55–62] (see
below). However, this operator occupies a particular importance in this
version.Alternatively, in such model, the authors in [49] have examined two-loop
corrections to (4.4) by the aid of explicit LNV Higgs self-couplings, and using
a fine tuning for the tree-level Dirac masses of (4.3) down to current values.
However, as mentioned, this is not the case in the considering model because
our Higgs potential (2.11) conserves ℒ.
We know that one of the problems of the 3-3-1 model with RH neutrinos is
associated with the Dirac mass term of neutrinos. In the following, we will
show that if such a fine tuning is done to get small values for these terms,
then the mass generation of neutrinos mediated by model particles is not able,
or the results will be trivial. This is in contradiction with [49]. In the next, the large
bare Dirac masses for the neutrinos, which are as of charged fermions of a
natural result from standard symmetry breaking, will be studied.


## 4.2.2. Neutrino Mass Matrix

The operatorsOLNC, OSLB,
and OLNV (including their Hermitian adjoint) will
provide the masses for the neutrinos: the first responsible for tree-level
masses, the second for one-loop corrections, and the third for contributions of
heavy particles.Tree-Level Dirac Masses
From the Yukawa
couplings in (2.6), the tree-level mass Lagrangian for the neutrinos is obtained
by [127, 128]:(4.6)ℒmassLNC=habνν¯aRνbL〈ϕ20〉−habνν¯aLcνbRc〈ϕ20〉+H.c.=2〈ϕ20〉habνν¯aRνbL+H.c.=−(MD)abν¯aRνbL+H.c.=−12(ν¯aLc,ν¯aR)(0(MDT)ab(MD)ab0)(νbLνbRc)+H.c.=−12X¯LcMνXL+H.c.,where habν=−hbaν is due to Fermi statistics. The MD is the mass matrix for the Dirac
neutrinos:(4.7)(MD)ab≡−2vhabν=(−MDT)ab=(0−A−BA0−CBC0),where(4.8)A≡2heμνv,B≡2heτνv,C≡2hμτνv.This mass matrix has been
rewritten in a general basis XLT≡(νeL,νμL,ντL,νeRc,νμRc,ντRc):(4.9)Mν≡(0MDTMD0).The tree-level neutrino spectrum, therefore, consists
of only Dirac fermions. Since habν is antisymmetric in a and b,
the mass matrix MD gives one neutrino massless and two others
degenerate in mass 0, −mD, mD, where mD≡(A2+B2+C2)1/2.
This mass spectrum is not realistic under the data; however, it will be
severely changed by the quantum corrections; the most general mass matrix can
then be written as follows:(4.10)Mν=(MLMDTMDMR),where ML,R (vanished at the
tree level) and MD get possible corrections.If such a tree-level contribution dominates the
resulting mass matrix (after corrections), the model will provide an
explanation about a large splitting eitherΔmatm2≫Δmsol2 or ΔmLSND2≫Δmatm, sol2 [3] (see also [49]). Hence, we need a fine-tuning at the tree level
[49] either mD~(Δmatm2)1/2(~5×10−2eV) or mD~(ΔmLSND2)1/2(~eV) [3]. Without loss of generality, assuming that heμν~heτν~hμτν,
we get then hν~10−13(or10−12).
The coupling hν in this case is so small and, therefore, this
fine tuning is not natural [129, 130]. Indeed, as shown below, since hν enters the
dominant corrections from (4.5) for ML,R,
these terms ML,R get very small values which are not large
enough to split the degenerate neutrino masses into a realistic spectrum. (The
largest degenerate splitting in squared mass is still much smaller than Δmsol2~8×10−5eV2 [3].) In addition, in this case, the Dirac masses get
corrections trivially.The above problem can be solvedjust by the LNV operator (4.4), and then
the operator (4.5) obtaining the contributions from particles in the model is
suppressed (for details, see [126]). However, we do not consider the above solution in
this work. This implies that the tree-level Dirac mass term for the neutrinos
by its naturalness should be treated as those as of the usual charged fermions
resulted of the standard symmetry breaking, say, hν~he(~10−6) [129, 130]. It turns out that this term is regarded as a large
bare quantity and unphysical. Under the interactions, they will of course
change to physical masses. In the following, we will obtain such finite renormalizations (for more details,
see [131]) in the
masses of neutrinos.One-Loop Level Dirac and Majorana Masses
The operator
(4.5) and its Hermitian adjoint arise from the radiative corrections mediated
by the model particles and give contributions to Majorana and Dirac mass terms ML, MR,
and MD for the neutrinos. The Yukawa couplings of the
leptons in (2.6) and the relevant Higgs self-couplings in (2.11) are explicitly
rewritten as follows:(4.11)ℒYlept=2habνν¯aLclbLϕ3+−2habνν¯aRlbLϕ1++hablν¯aLlbRϕ1++hablν¯aRclbRϕ3++habll¯aLlbRϕ20+H.c.,ℒHrelv=λ3ϕ1−ϕ1+(χ10*χ10+χ30*χ30)+λ3ϕ3−ϕ3+(χ10*χ10+χ30*χ30)+λ4ϕ1−ϕ1+χ10*χ10+λ4ϕ3−ϕ3+χ30*χ30+λ4ϕ3−ϕ1+χ10*χ30+λ4ϕ1−ϕ3+χ30*χ10.The one-loop corrections to the
mass matrices ML of νL, MR of νR,
and MD of ν are, therefore, given in Figures 10, 11, and 12, respectively.The one-loop corrections for the mass
matrixML.
(a)(b)The one-loop corrections for the
mass matrixMR.
(a)(b)The one-loop corrections for the
mass matrixMD.
(a)(b)Radiative Corrections toML and MR
With the Feynman rules at hand [127, 128], ML is obtained by(4.12)−i(ML)abPL=∫d4p(2π)4(i2hacνPL)i(p+mc)p2−mc2(ihcdlv2PR)i(p+md)p2−md2×(ihbdl*PL)−1(p2−mϕ12)(p2−mϕ32)(iλ4uω2)+∫d4p(2π)4(ihacl*PL)i(−p+mc)p2−mc2(ihdclv2PR)i(−p+md)p2−md2×(i2hbdνPL)−1(p2−mϕ12)(p2−mϕ32)(iλ4uω2).Because the Yukawa couplings of
the charged leptons are flavor diagonal, (4.12) becomes(4.13)(ML)ab=i2λ4uωvhabν[mb2I(mb2,mϕ32,mϕ12)−ma2I(ma2,mϕ32,mϕ12)],(a,bnotsummed),where the integral I(a,b,c) is given in Appendix B.In the effective approximation (2.8), identifications
are given by ϕ3±~H2± and ϕ1±~GW± [39], where H2± and GW± as above mentioned are the charged bilepton
Higgs boson and the Goldstone boson associated with W± boson, respectively. For the masses, we have
also mϕ32≃mH22(≃(λ4/2)ω2) and mϕ12≃0.
Using (B.5), the integrals are given by(4.14)I(ma2,mϕ32,mϕ12)≃−i16π21ma2−mH22[1−mH22ma2−mH22lnma2mH22],a=e,μ,τ.Consequently, the mass matrix
(4.13) becomes(4.15)(ML)ab≃2λ4uωhabν16π2v[mH22(ma2−mb2)(mb2−mH22)(ma2−mH22)+ma2mH22(ma2−mH22)2lnma2mH22−mb2mH22(mb2−mH22)2lnmb2mH22]≃2λ4uωhabν16π2vmH22[ma2(1+lnma2mH22)−mb2(1+lnmb2mH22)],where the last approximation
(4.15) is kept in the orders up to 𝒪[(ma,b2/mH22)2].
Since mH22≃(λ4/2)ω2,
it is worth noting that the resulting ML is not explicitly dependent on λ4,
however, proportional to tθ=u/ω (the mixing angle between the W boson and the singly-charged bilepton gauge
boson Y [38]), 2vhabν (the tree-level Dirac mass term of neutrinos),
and mH2 in the logarithm scale. Here, the VEV v≈vweak and the charged-lepton masses  ma(a=e,μ,τ) have the well-known values. Let us note that ML is symmetric and has vanishing diagonal
elements.For the corrections toMR,
it is easily to check that the relationship (MR)ab=−(ML)ab is exact at the one-loop level. (This result
can be derived from Figure 11 in a general case without imposing any
additional condition on hl, hν,
and further.) Combining this result with (4.15), the mass matrices are
explicitly rewritten as follows:(4.16)(ML)ab=−(MR)ab≃(0frf0trt0),where the elements are obtained
by(4.17)f≡(2vheμν){(tθ8π2v2)[me2(1+lnme2mH22)−mμ2(1+lnmμ2mH22)]},r≡(2vheτν){(tθ8π2v2)[me2(1+lnme2mH22)−mτ2(1+lnmτ2mH22)]},t≡(2vhμτν){(tθ8π2v2)[mμ2(1+lnmμ2mH22)−mτ2(1+lnmτ2mH22)]}.It can be checked that f, r, t are much smaller than those of MD.
To see this, we can take me≃0.51099MeV, mμ≃105.65835MeV, mτ≃1777MeV, v≃246GeV, u≃2.46GeV, ω≃3000GeV,
and mH2≃700GeV(λ4~0.11) [38–40], which give us then(4.18)f≃(2vheμν)(3.18×10−11),r≃(2vheτν)(5.93×10−9),t≃(2vhμτν)(5.90×10−9),where the second factors rescale
negligibly with ω~1–10TeV and mH2~200–2000GeV.
This thus implies that(4.19)|ML,R||MD|~10−9,which can be checked with the
help of |M|≡(M†M)1/2.
In other words, the constraint is given as follows(4.20)|ML,R|≪|MD|. With the above results at hand, we can then get the
masses by studying diagonalization of the mass matrix (4.10), in which, the
submatrices ML,R and MD satisfying the constraint (4.20) are given by
(4.16) and (4.7), respectively. In calculation, let us note that since MD has one vanishing eigenvalue, Mν does not possess the pseudo-Dirac property in
all three generations [132], despite being very close to those because the remaining
eigenvalues do. As a fact, we will see that Mν contains a combined framework of the seesaw
[55–62] and the
pseudo-Dirac [132–142]. To
get mass, we can suppose that hν is real and, therefore, the matrix iMD is Hermitian: (iMD)†=iMD (4.7). The Hermitianity for ML,R is also followed by (4.16). Because the
dominant matrix is MD (4.20), we first diagonalize it by biunitary
transformation [131]:(4.21)ν¯aR=ν¯iR(−iU)ia†,νbL=UbjνjL,(i,j=1,2,3),(4.22)Mdiag≡diag(0,−mD,mD)=(−iU)†MDU,mD=A2+B2+C2,where the matrix U is easily obtained by(4.23)U=1mD2(A2+C2)(C2(A2+C2)iBC−AmDBC−iAmD−B2(A2+C2)i(A2+C2)(A2+C2)A2(A2+C2)iAB+CmDAB+iCmD).Resulted by the
anti-Hermitianity of MD,
it is worth noting that Mν in the case of vanishing ML,R (4.9) is indeed diagonalized by the following
unitary transformation:(4.24)V=12(UU−iUiU). A new basis (ν1,ν2,…,ν6)LT≡V†XLT,
which is different from (νjL,νiRc)T of (4.21), is therefore performed. The neutrino
mass matrix (4.10) in this basis becomes(4.25)V†MνV=(Mdiagϵϵ−Mdiag),(4.26)ϵ≡U†MLU,ϵ†=ϵ, where the elements of ϵ are obtained by(4.27)ϵ11=ϵ22=ϵ33=0,(4.28)ϵ12=iϵ13*={[ABmD+iC(A2−B2+C2)]f+[(C2−A2)mD+2iABC]r+[iA(A2−B2+C2)−BCmD]t}[mD22(A2+C2)]−1,(4.29)ϵ23={(A2+C2)[(CmD−iAB)t−(AmD+iBC)f]−[B(A2−C2)mD+iAC(A2+2B2+C2)]r}[mD2(A2+C2)]−1.
Let us remind the reader that
(4.27) is exactly given at the one-loop level ML (4.13) without imposing any approximation on
this mass matrix. Interchanging the positions of component fields in the basis (ν1,ν2,…,ν6)LT by a permutation transformation P†≡P23P34,
that is, νp→(P†)pqνq(p,q=1,2,…,6) with(4.30)P†=(100000000100010000001000000010000001),the mass matrix (4.25) can be
rewritten as follows:(4.31)P†(V†MνV)P=(0000ϵ12ϵ1300ϵ12ϵ13000ϵ21−mD00ϵ230ϵ310mDϵ320ϵ2100ϵ23mD0ϵ310ϵ3200−mD). It is worth noting that in (4.31), all the off-diagonal
components |ϵ| are much smaller than the eigenvalues |±mD| due to Condition (4.20). The degenerate
eigenvalues 0, −mD,
and +mD (each twice) are now splitting into three
pairs with six different values, two light and four heavy. The two neutrinos of
first pair resulted by the 0 splitting have very small masses as a result
of exactly what a seesaw does [55–62], that is, the off-diagonal block contributions to
these masses are suppressed by the large pseudo-Dirac masses of the lower-right
block. The suppression in this case is different from the usual ones [55–62] because it needs only the
pseudo-Dirac particles [132–142] with the masses mD of the electroweak scale instead of extremely
heavy RH Majorana fields, and that the Dirac masses in those mechanisms are now
played by loop-induced f, r, t (4.17) as a result of the SLB u/ω.
Therefore, the mass matrix (4.31) is effectively decomposed into MS for the first pair of light neutrinos (νS) and MP for the last two pairs of heavy pseudo-Dirac
neutrinos (νP):(4.32)(ν1,ν4,ν2,ν3,ν5,ν6)LT→(νS,νP)LT=Veff†(ν1,ν4,ν2,ν3,ν5,ν6)LT,Veff†(P†V†MνVP)Veff=diag(MS,MP),where Veff, MS,
and MP get the approximations:(4.33)Veff≃(1ℰ−ℰ+1),ℰ≡(00ϵ12ϵ13ϵ12ϵ1300)(−mD00ϵ230mDϵ3200ϵ23mD0ϵ3200−mD)−1,MS≃−ℰ(0ϵ210ϵ31ϵ210ϵ310),MP≃(−mD00ϵ230mDϵ3200ϵ23mD0ϵ3200−mD).The mass matrices MS and MP,
respectively, give exact eigenvalues as follows:(4.34)mS±=±2Im(ϵ13ϵ13ϵ32)mD2−ϵ232≃±2Im(ϵ13ϵ13ϵ32mD2),(4.35)mP±=−mD±|ϵ23|,mP′±=mD±|ϵ23|.
In this case, the mixing
matrices are collected into (νS±,νP±,νP'±)LT=V±†(νS,νP)LT,
where the V± is obtained by(4.36)V±=12(1−1000011000000κ−κ000000110000κ−κ001100),κ≡ϵ23|ϵ23|=exp(iargϵ23).It is to be noted that the
degeneration in the Dirac one |±mD| is now splitting severally.From (4.35), we see that the four large pseudo-Dirac
masses for the neutrinos are almost degenerate. In addition, the resulting
spectrum (4.34), (4.35) yields two largest squared mass splittings, respectively,
proportional to mD2 and 4mD|ϵ23|. From (4.29) and (4.18), we can evaluate |ϵ23|≃3.95×10−9mD≪mD (where A~B~C~mD/3 is understood). Because the splitting 4mD|ϵ23| is still much smaller than Δmsol2,
this therefore implies that the fine tuning, as mentioned, is not realistic.
(In detail, in Table 18, we give the numerical values of these fine tunings,
where the parameters are given as before (4.18).)Table 18
The values forhν and two largest splittings in squared mass.
Fine tuninghνmD2(eV2)4mD|ϵ23|(eV2)mD2~Δmatm28.30×10−142.50×10−33.95×10−11mD2~ΔmLSND21.66×10−121.001.58×10−8Similarly, for the two small masses, we can also
evaluate|mS±|≃4.29×10−28mD.
This shows that the masses mS± are very much smaller than the splitting |ϵ23|.
This also implies that the two light neutrinos in this case are hidden for any mD value of pseudo-Dirac neutrinos. Let us see
the sources of the problem why these masses are so small: (i) vanishing of all
the elements of left-upper block of (4.31); (ii) in (4.34) the resulting masses
are proportional to |ϵ|3/mD2,
but not to |ϵ|2/mD as expected from (4.31). It turns out that this
is due to the antisymmetric of habν enforcing on the tree-level Dirac-mass matrix
and the degenerate of MR=−ML of the one-loop level left-handed (LH) and RH
Majorana-mass matrices. It can be easily checked that such degeneration in
Majorana masses remains up to higher-order radiative corrections as a result of
treating the LH and RH neutrinos in the same gauge triplets with the model
Higgs content. For example, by the aid of (4.5) the degeneration retains up to
any higher-order loop.Radiative Corrections toMD
As mentioned, the mass matrixMD requires the one-loop corrections as given in
Figure 12, and the contributions are easily obtained as follows:(4.37)−i(MDrad)abPL=∫d4p(2π)4(−i2hacνPL)i(p+mc)p2−mc2(ihcdlv2PR)i(p+md)p2−md2×(ihbdl*PL)−1(p2−mϕ12)2(iλ3u2+ω22+iλ4u22)+∫d4p(2π)4(ihacl*PL)i(−p+mc)p2−mc2(ihdclv2PR)i(−p+md)p2−md2×(i2hbdνPL)−1(p2−mϕ32)2(iλ3u2+ω22+iλ4ω22).We
rewrite(4.38)(MDrad)ab=−i2habνv{[λ3(u2+ω2)+λ4u2]mb2I(mb2,mϕ12)+[λ3(u2+ω2)+λ4ω2]ma2I(ma2,mϕ32)},(a,bnotsummed),where I(a,b) is given in (B.13). With the help of (B.14),
the approximation for (4.38) is obtained by(4.39)(MDrad)ab≃−habν82π2v{[λ3(u2+ω2)+λ4u2]+[λ3(u2+ω2)+λ4ω2]ma2mH22}=−2habν(λ3ω216π2v)[1+(1+λ4λ3)(u2ω2+ma2mH22)+𝒪(u4ω4,ma,b4mH24)].Because of the constraint (2.8),
the higher-order corrections 𝒪(⋯) can be neglected; thus MDrad is rewritten as follows:(4.40)(MDrad)ab=−2habν(λ3ω216π2v)(1+δa),δa≡(1+λ4λ3)(u2ω2+ma2mH22),where δa is of course an infinitesimal coefficient,
that is, |δ|≪1.
Again, this implies also that if the fine tuning is done the resulting
Dirac-mass matrix get trivially. It is due to the fact that the contribution of
the term associated with δa in (4.40) is then very small and neglected; the
remaining term gives an antisymmetric resulting Dirac-mass matrix, that is,
therefore, unrealistic under the data.With this result, it is worth noting that the
scale(4.41)|λ3ω216π2v|of the radiative Dirac masses
(4.40) is in the orders of the scale v of the tree-level Dirac masses (4.7). Indeed,
if one puts |(λ3ω2)/(16π2v)|=v and takes |λ3|~0.1–1,
then ω~3–10TeV as expected in the constraints [40, 114, 115]. The resulting Dirac-mass
matrix which is combined of (4.7) and (4.40), therefore, gets two typical
examples of the bounds: (i) (λ3ω2)/(16π2v)+v~𝒪(v);
(ii) (λ3ω2)/(16π2v)+v~𝒪(0).
The first case (i) yields that the status on the masses of neutrinos as given
above is remained unchanged and therefore is also trivial as mentioned. In the
last case (ii), the combination of (4.7) and (4.40) gives(4.42)(MD)ab=2habν(vδa).It is interesting that in this
case the scale v for the Dirac masses (4.7) gets naturally a
large reduction, and we argue that this is not a fine tuning. Because the large
radiative mass term in (4.40) is canceled by the tree-level Dirac masses, we
mean this as a finite renormalization in the masses of neutrinos. It is also
noteworthy that, unlike the case of the tree-level mass term (4.7), the mass
matrix (4.42) is now nonantisymmetric in a and b.
Among the three eigenvalues of this matrix, we can check that one vanishes
(since detMD=0) and two others massive are now nondegenerate (splitting). Let us recall
that in the first case (i) the degeneration of the two nonzero eigenvalues are,
however, retained because the combination of (4.7) and (4.40) is proportional to habνv.In contrast to (4.19), in this case there is no large
hierarchy between ML,R and MD.
To see this explicitly, let us take the values of the parameters as given
before (4.18), thus λ3≃−1.06 and the coefficients δa are evaluated by(4.43)δe≃6.03×10−7,δμ≃6.23×10−7,δτ≃6.28×10−6.Hence, we get(4.44)|ML,R||MD|~10−2–10−3.With the values given in (4.43),
the quantities hν and mD can be evaluated through the mass term (4.42);
the neutrino data imply that hν and mD are in the orders ofhe andme—the Yukawa coupling and mass of electron,
respectively.Because of the Condition (4.44) and the vanishing of
one eigenvalue of MD,
we can repeat the procedure as given above to diagonalize the full matrix Mν with MD given by (4.42) and ML,R by (4.16). First, we can easily find a mixing
matrix V as in (4.24); Second, in the new basis we
obtain the seesaw form as in (4.31); Finally, the resulting mixing matrix and
masses for the neutrinos are derived. It is worth checking that the two largest
squared mass splittings as given before can be approximately applied on this
case of (4.44) such as (mD|δ|)2 and 4(mD|δ|)|ϵ|,
and seeing that they fit naturally the data.Mass Contributions from Heavy Particles
There remain now two questions not yet answered: (i)
the degeneration ofMR=−ML;
(ii) the hierarchy of ML,R and MD (4.44) can be continuously reduced? As
mentioned, we will prove that the new physics gives us the solution.The mass Lagrangian for the neutrinos given by the
operator (4.4) can be explicitly written as follows:(4.45)ℒmassLNV=sabνℳ−1(〈χ†〉ψ¯aLc)(〈χ†〉ψbL)+H.c.=sabνℳ−1(u2ν¯aLc+ω2ν¯aR)(u2νbL+ω2νbRc)+H.c.=−12X¯LcMνnewXL+H.c.,where the mass matrix for the neutrinos
is obtained by(4.46)Mνnew≡−(u2ℳsνuωℳsνuωℳsνω2ℳsν),in which the coupling sabν is symmetric in a and b.
For convenience in reading, let us define the submatrices of (4.46) to be MLnew, MDnew,
and MRnew similar to that of (4.10). Because of the
condition u2≪uω≪ω2,
the corresponding submatrices MLnew, MDnew,
and MRnew of (4.46) get the right hierarchies and the two
questions as mentioned are solved
simultaneously .Intriguing comparisons betweensν and hν are given in order:
(1)
hν conserves the lepton number; sν violates this charge;(2)
hν is antisymmetric and enforcing on the
Dirac-mass matrix; sν is symmetric and breaks this property;(3)
hν preserves the degeneration of MR=−ML; sν breaks the MR=−ML;(4)
a pair of(sν,hν) in the lepton sector will complete the rule
played by the quark couplings (sq,hq) (see below);(5)
hν defines the interactions in the standard model
scale v; sν gives those in the GUT scale ℳ.
Let us now take the values ℳ≃1016GeV, ω≃3000GeV, u≃2.46GeV,
and sν~𝒪(1) (perhaps smaller), the submatrices MLnew≃−6.05×10−7sνeV and MDnew≃−7.38×10−4sνeV can give contributions (to the diagonal
components of ML and MD,
resp.) but very small. It is noteworthy that the last one MRnew≃−0.9sνeV can dominateMR.To summarize, in this model the neutrino mass matrix
is combined byMν+Mνnew,
where the first term is defined by (4.10) and the last term by (4.46); the
submatrices of Mν are given in (4.16) and (4.42), respectively.
Depending on the strength of the new physics
coupling sν,
the submatrices of the last term, MLnew and MDnew,
are included or removed.


### 4.2.3. Some Remarks from Experimental Constraints

Conventional neutrino oscillations are insensitive to
the absolute scale of neutrino masses. Although the latter will be tested
directly in high-sensitivity tritium beta decay studies and neutrinoless double
beta decay(0νββ) as well as by its effects on the cosmic
microwave background and the large-scale structure of the Universe
[143, 144]. With the present of
sterile neutrinos in this model, the experimental constraints on their masses
may be also important and give us bounds on several parameters such as the
coupling hν and δa.If the liquid scintillator neutrino detector
experiment is confirmed, the sterile-neutrino masses will get some values in
range of eV. In this case, the couplinghν is also in orders of he.
The X-ray measurements yield an upper limit of sterile neutrino mass [145] ms<6.3keV.
For all the other cosmological constraints, the sterile neutrino masses are in
the range [146, 147] 2keV<ms<8keV.
In such cases, the coupling hν will get bounds in orders of hμ,τ.It is well known that the radiative mass generation
can also induce the large lepton flavor violating processes such asμ→eγ as the similar one-loop effect. The possible
one-loop diagrams for this process are depicted in Figure 13. Suppose that mY2,mH22≫mW2=g2v2/4 [38], we get the approximation [148]:(4.47)Br(μ→eγ)≡Γ(μ→eγ)Γ(μ→eνeνμ)≃3sW48π3α(hμτν*heτν)2. Since Br(μ→eγ)<1.2×10−11, α=1/128 and sW2=0.2312 [3], the coupling hν is bounded by hν<3.47×10−3,
where hν≡heτν=hμτν set is understood. Our above result, hν~he,
satisfies this constraint. It can be shown that the value for hν also satisfies constraints from such processes
as μ→3e and μe conversion (for more details, see [149]).One-loop contributions to the lepton flavor violating decayμ→eγ.
(a)(b)(c)(d)


## 4.2.3. Some Remarks from Experimental Constraints

Conventional neutrino oscillations are insensitive to
the absolute scale of neutrino masses. Although the latter will be tested
directly in high-sensitivity tritium beta decay studies and neutrinoless double
beta decay(0νββ) as well as by its effects on the cosmic
microwave background and the large-scale structure of the Universe
[143, 144]. With the present of
sterile neutrinos in this model, the experimental constraints on their masses
may be also important and give us bounds on several parameters such as the
coupling hν and δa.If the liquid scintillator neutrino detector
experiment is confirmed, the sterile-neutrino masses will get some values in
range of eV. In this case, the couplinghν is also in orders of he.
The X-ray measurements yield an upper limit of sterile neutrino mass [145] ms<6.3keV.
For all the other cosmological constraints, the sterile neutrino masses are in
the range [146, 147] 2keV<ms<8keV.
In such cases, the coupling hν will get bounds in orders of hμ,τ.It is well known that the radiative mass generation
can also induce the large lepton flavor violating processes such asμ→eγ as the similar one-loop effect. The possible
one-loop diagrams for this process are depicted in Figure 13. Suppose that mY2,mH22≫mW2=g2v2/4 [38], we get the approximation [148]:(4.47)Br(μ→eγ)≡Γ(μ→eγ)Γ(μ→eνeνμ)≃3sW48π3α(hμτν*heτν)2. Since Br(μ→eγ)<1.2×10−11, α=1/128 and sW2=0.2312 [3], the coupling hν is bounded by hν<3.47×10−3,
where hν≡heτν=hμτν set is understood. Our above result, hν~he,
satisfies this constraint. It can be shown that the value for hν also satisfies constraints from such processes
as μ→3e and μe conversion (for more details, see [149]).One-loop contributions to the lepton flavor violating decayμ→eγ.
(a)(b)(c)(d)


## 4.3. Quark Masses

First, we present the general quark mass spectrum. Some details on the one-loop quark masses are given then.


### 4.3.1. Quark Mass Spectra

Note that in [37], the authors have considered the fermion mass
spectrum under the Z2 discrete symmetry which discards the LNV
interactions. Here, the couplings of (2.7) in such case are forbidden. Then, it
can be checked that some quarks remain massless up to two-loop level. To solve
the mass problem of the quarks, the authors in [37] have shown that one-third scalar triplet has to be
added to the resulting model. In the following, we show that it is not
necessary. The Z2 is not introduced and thus the third one is
not required. The LNV Yukawa couplings are vital for the economical 3-3-1
model.The Yukawa couplings in (2.6) and (2.7) give the mass
Lagrangian for the upquarks (quark sector with electric charge qup=2/3):(4.48)ℒupmass=hU2(u¯1Lu+U¯Lω)UR+sau2(u¯1Lu+U¯Lω)uaR−v2u¯αL(hαauuaR+sαUUR)+H.c.Consequently, we obtain the mass
matrix for the upquarks (u1,u2,u3,U) as follows:(4.49)Mup=12(−s1uu−s2uu−s3uu−hUuh21uvh22uvh23uvs2Uvh31uvh32uvh33uvs3Uv−s1uω−s2uω−s3uω−hUω).Because the first and the last
rows of the matrix (4.49) are proportional, the tree-level upquark spectrum
contains a massless one!Similarly, for the downquarks (qdown=−1/3), we get the following mass Lagrangian:(4.50)ℒdownmass=hαβD2(d¯αLu+D¯αLω)DβR+sαad2(d¯αLu+D¯αLω)daR+v2d¯1L(haddaR+sαDDαR)+H.c.Hence, we get mass matrix for
the downquarks (d1,d2,d3,D2,D3): (4.51)Mdown=−12(h1dvh2dvh3dvs2Dvs3Dvs21dus22dus23duh22Duh23Dus31dus32dus33duh32Duh33Dus21dωs22dωs23dωh22Dωh23Dωs31dωs32dωs33dωh32Dωh33Dω).We see that the second and
fourth rows of matrix in (4.51) are proportional, while the third and the last
are the same. Hence, in this case there are two massless eigenstates.The masslessness of the tree-level quarks in both the
sectors calls radiative corrections (the so-called mass problem of quarks).
These corrections start at the one-loop level. The diagrams in Figure14 contribute the upquark spectrum, while Figure 15 gives the
downquarks. Let us note the reader that the quarks also get some one-loop
contributions in the case of the Z2 symmetry enforcing [37]. The careful study of this
radiative mechanism shows that the one-loop quark spectrum is consistent.Figure 14
One-loop contributions to the upquark mass matrix (4.49).Figure 15
One-loop contributions to the downquark mass matrix (4.51).


### 4.3.2. Typical Examples of the One-Loop Corrections

To provide the quarks masses, in the following we can
suppose that the Yukawa couplings are flavor diagonal. Then, theu2 and u3 states are mass eigenstates corresponding to
the mass eigenvalues:(4.52)m2=h22uv2,m3=h33uv2.The u1 state mixes with the exotic U in terms of one submatrix of the mass matrix
(4.48):(4.53)MuU=−12(s1uuhUus1uωhUω).This matrix contains one
massless quark ~u1, m1=0,
and the remaining exotic quark ~U with the mass of the scale ω.Similarly, for the downquarks, thed1 state is a mass eigenstate corresponding to
the eigenvalue:(4.54)m1′=−h1dv2.The pairs (d2,D2) and (d3,D3) are decoupled,
while the quarks of each pair mix via the mass submatrices, respectively,(4.55)Md2D2=−12(s22duh22Dus22dωh22Dω),(4.56)Md3D3=−12(s33duh33Dus33dωh33Dω).
These matrices contain the
massless quarks ~d2 and d3 corresponding to m2′=0 and m3′=0,
and two exotic quarks ~D2 and D3 with the masses of the scale ω.With the help of the constraint (2.8), we identify m1, m2,
and m3,
respectively, to those of the u1=u, u2=c,
and u3=t quarks. The downquarks d1, d2,
and d3 are, therefore, corresponding to d, s,
and b quarks. Unlike the usual 3-3-1 model with
right-handed neutrinos, where the third family of quarks should be
discriminating [28],
in the model under consideration the first family has to be different from the
two others.The mass matrices (4.53), (4.55), and (4.56) remain the
tree-level properties for the quark spectra—one massless in the upquark
sector and two in the downquark sector. From these
matrices, it is easily to verify that the conditions in (2.8) and (2.10) are
satisfied. First, we consider radiative corrections to the upquark masses.Upquarks
In the previous studies [19, 37, 84–86], the LNV interactions have
often been excluded, commonly by the adoption of an appropriate discrete
symmetry. Let us remind that there is no reason within the 3-3-1 model to
ignore such interactions. The experimental limits on processes which do not
conserve total lepton numbers such as neutrinoless double
beta decay [150, 151] require them to be small.If the Yukawa Lagrangian is restricted toℒLNC [37], then the mass matrix (4.53) becomes(4.57)MuU=−12(0hUu0hUω).In this case, only the element (MuU)12 gets an one-loop correction defined by Figure
16. Other elements remain unchanged under this one-loop effect.Figure 16
One-loop contribution underZ2 to the upquark mass matrix (4.57).The Feynman
rules gives us(4.58)−i(MuU)12PR=∫d4p(2π)4(ihUPR)i(p+MU)p2−MU2(−iMUPL)i(p+MU)p2−MU2(ihUPR)×−1(p2−Mχ12)(p2−Mχ32)(i4λ1)uω2.Thus, we get(4.59)(MuU)12=−2iuωλ1MU(hU)2∫d4p(2π)4p2(p2−MU2)2(p2−Mχ32)(p2−Mχ12)≡−2iuωλ1MU(hU)2I(MU2,Mχ32,Mχ12).The integral I(a,b,c) with a,b≫c is given in the B. Following [39], we conclude that in an
effective approximation, MU2,Mχ32≫Mχ12.
Hence, we have(4.60)(MuU)12≃−λ1tθMU34π2[MU2−Mχ32+Mχ32ln(Mχ32/MU2)(MU2−Mχ32)2]~u≡−12R(MU).The resulting mass matrix is
given by(4.61)MuU=−12(0hUu+R0hUω).We see that one quark remains
massless as the case of the tree-level spectrum. This result keeps up to
two-loop level and can be applied to the downquark sector as well as in the
cases of nondiagonal Yukawa couplings. Therefore, under the Z2,
it is not able to provide consistent masses for the quarks.If the full Yukawa Lagrangian is used, the LNV
couplings must be enough small in comparison with the usual couplings [see
(2.10)]. Combining (2.8) and (2.10), we have(4.62)hUω≫hUu,s1uω≫s1uu.In this case, the element (MuU)11 of (4.53) gets the radiative correction
depicted in Figure 17. The resulting
mass matrix is obtained by(4.63)MuU=−12(s1u(u+RhU)hUus1uωhUω).In contradiction with the first
case, the mass of u quark is now nonzero and given by(4.64)mu≃s1u2hUR. Let us note that the matrix (4.63) gives an eigenvalue
in the scale of (1/2)hUω which can be identified with that of the
exotic quark U.
In effective approximation [39], the mass for the Higgs χ3 is defined by Mχ32≃2λ1ω2.
Hereafter, for the parameters, we use the following values λ1=2.0, tθ=0.08 as mentioned, and ω=10TeV.
The mass value for the u quark is as function of s1u and hU.
Some values of the pair (s1u,hU) which give consistent masses for the u quark are listed in Table 19.Table 19
Mass for theu quark as function of (s1u,hU).
hU21.510.50.1s1u0.00020.00030.00040.0010.01mu [MeV]2.2072.5652.2462.3752.025Figure 17
One-loop contribution to the upquark mass matrix (4.53).Note that the mass values in Table19 for the u quark are in good consistence with the data
given in [3]: mu∈1.5÷4MeV.Downquarks
For the downquarks, the constraint,(4.65)hααDω≫hααDu,sααdω≫sααdu,should be applied. In this case,
three elements (MdαDα)11, (MdαDα)12,
and (MdαDα)21 will get radiative corrections. The relevant
diagrams are depicted in Figure 18. It
is worth noting that Diagram 18(c) exists even in the case of the Z2 symmetry. The contributions are given by(4.66)(MdαDα)11=−sααd2hααDR(MDα),(MdαDα)21=−4iλ1sααdhααDMDα3I(MDα2,Mχ32,Mχ32)=−λ1sααdMDα34π2hααD[MDα2+Mχ32(MDα2−Mχ32)2−2MDα2Mχ32(MDα2−Mχ32)3lnMDα2Mχ32]≡−12R′(MDα),(MdαDα)12=−12R(MDα).We see that two last terms are
much larger than the first one. This is responsible for the masses of the
quarks d2 and d3.
At the one-loop level, the mass matrix for the downquarks is given by(4.67)MdαDα=−12(sααd(u+RhααD)hααDu+Rsααdω+R′hααDω). We remind the reader that a matrix (see also [131])(4.68)(acbD)with D≫b,c≫a has two eigenvalues:(4.69)x1≃[a2−2bcaD+b2c2−(b2+c2)a2D2]1/2,x2≃D.Therefore, the mass matrix in
(4.67) gives an eigenvalue in the scale of D≡(1/2)hααDω which is of the exotic quark Dα′.
Here, we have another eigenvalue for the mass of dα′:(4.70)mdα′=hααDu+R2hααDω{R′2−(sααd)2(hααD)2[(sααdω+R′)2+(hααDu+R)2]}1/2. Let us remember that Mχ32≃2λ1ω2 and the parameters λ1=2.0, tθ=0.08 and ω=10TeV as given above are used in this case; mdα is a function of sααd and hααD.
We take the value hααD=2.0 for both the sectors, α=2 and α=3.
If s22d=0.015,
we get then the mass of the so-called s quark:(4.71)ms=99.3MeV.For the downquark of the third
family, we put s33d=0.7.
Then, the mass of the b quark is obtained by(4.72)mb=4.4GeV. We emphasize again that (4.71) and (4.72) are in good
consistence with the data given in [3]: ms~95±25MeV and mb~4.70±0.07GeV.One-loop contributions to the downquark mass matrix (4.55) or (4.56).
(a)(b)(c)


## 4.3.1. Quark Mass Spectra

Note that in [37], the authors have considered the fermion mass
spectrum under the Z2 discrete symmetry which discards the LNV
interactions. Here, the couplings of (2.7) in such case are forbidden. Then, it
can be checked that some quarks remain massless up to two-loop level. To solve
the mass problem of the quarks, the authors in [37] have shown that one-third scalar triplet has to be
added to the resulting model. In the following, we show that it is not
necessary. The Z2 is not introduced and thus the third one is
not required. The LNV Yukawa couplings are vital for the economical 3-3-1
model.The Yukawa couplings in (2.6) and (2.7) give the mass
Lagrangian for the upquarks (quark sector with electric charge qup=2/3):(4.48)ℒupmass=hU2(u¯1Lu+U¯Lω)UR+sau2(u¯1Lu+U¯Lω)uaR−v2u¯αL(hαauuaR+sαUUR)+H.c.Consequently, we obtain the mass
matrix for the upquarks (u1,u2,u3,U) as follows:(4.49)Mup=12(−s1uu−s2uu−s3uu−hUuh21uvh22uvh23uvs2Uvh31uvh32uvh33uvs3Uv−s1uω−s2uω−s3uω−hUω).Because the first and the last
rows of the matrix (4.49) are proportional, the tree-level upquark spectrum
contains a massless one!Similarly, for the downquarks (qdown=−1/3), we get the following mass Lagrangian:(4.50)ℒdownmass=hαβD2(d¯αLu+D¯αLω)DβR+sαad2(d¯αLu+D¯αLω)daR+v2d¯1L(haddaR+sαDDαR)+H.c.Hence, we get mass matrix for
the downquarks (d1,d2,d3,D2,D3): (4.51)Mdown=−12(h1dvh2dvh3dvs2Dvs3Dvs21dus22dus23duh22Duh23Dus31dus32dus33duh32Duh33Dus21dωs22dωs23dωh22Dωh23Dωs31dωs32dωs33dωh32Dωh33Dω).We see that the second and
fourth rows of matrix in (4.51) are proportional, while the third and the last
are the same. Hence, in this case there are two massless eigenstates.The masslessness of the tree-level quarks in both the
sectors calls radiative corrections (the so-called mass problem of quarks).
These corrections start at the one-loop level. The diagrams in Figure14 contribute the upquark spectrum, while Figure 15 gives the
downquarks. Let us note the reader that the quarks also get some one-loop
contributions in the case of the Z2 symmetry enforcing [37]. The careful study of this
radiative mechanism shows that the one-loop quark spectrum is consistent.Figure 14
One-loop contributions to the upquark mass matrix (4.49).Figure 15
One-loop contributions to the downquark mass matrix (4.51).


## 4.3.2. Typical Examples of the One-Loop Corrections

To provide the quarks masses, in the following we can
suppose that the Yukawa couplings are flavor diagonal. Then, theu2 and u3 states are mass eigenstates corresponding to
the mass eigenvalues:(4.52)m2=h22uv2,m3=h33uv2.The u1 state mixes with the exotic U in terms of one submatrix of the mass matrix
(4.48):(4.53)MuU=−12(s1uuhUus1uωhUω).This matrix contains one
massless quark ~u1, m1=0,
and the remaining exotic quark ~U with the mass of the scale ω.Similarly, for the downquarks, thed1 state is a mass eigenstate corresponding to
the eigenvalue:(4.54)m1′=−h1dv2.The pairs (d2,D2) and (d3,D3) are decoupled,
while the quarks of each pair mix via the mass submatrices, respectively,(4.55)Md2D2=−12(s22duh22Dus22dωh22Dω),(4.56)Md3D3=−12(s33duh33Dus33dωh33Dω).
These matrices contain the
massless quarks ~d2 and d3 corresponding to m2′=0 and m3′=0,
and two exotic quarks ~D2 and D3 with the masses of the scale ω.With the help of the constraint (2.8), we identify m1, m2,
and m3,
respectively, to those of the u1=u, u2=c,
and u3=t quarks. The downquarks d1, d2,
and d3 are, therefore, corresponding to d, s,
and b quarks. Unlike the usual 3-3-1 model with
right-handed neutrinos, where the third family of quarks should be
discriminating [28],
in the model under consideration the first family has to be different from the
two others.The mass matrices (4.53), (4.55), and (4.56) remain the
tree-level properties for the quark spectra—one massless in the upquark
sector and two in the downquark sector. From these
matrices, it is easily to verify that the conditions in (2.8) and (2.10) are
satisfied. First, we consider radiative corrections to the upquark masses.Upquarks
In the previous studies [19, 37, 84–86], the LNV interactions have
often been excluded, commonly by the adoption of an appropriate discrete
symmetry. Let us remind that there is no reason within the 3-3-1 model to
ignore such interactions. The experimental limits on processes which do not
conserve total lepton numbers such as neutrinoless double
beta decay [150, 151] require them to be small.If the Yukawa Lagrangian is restricted toℒLNC [37], then the mass matrix (4.53) becomes(4.57)MuU=−12(0hUu0hUω).In this case, only the element (MuU)12 gets an one-loop correction defined by Figure
16. Other elements remain unchanged under this one-loop effect.Figure 16
One-loop contribution underZ2 to the upquark mass matrix (4.57).The Feynman
rules gives us(4.58)−i(MuU)12PR=∫d4p(2π)4(ihUPR)i(p+MU)p2−MU2(−iMUPL)i(p+MU)p2−MU2(ihUPR)×−1(p2−Mχ12)(p2−Mχ32)(i4λ1)uω2.Thus, we get(4.59)(MuU)12=−2iuωλ1MU(hU)2∫d4p(2π)4p2(p2−MU2)2(p2−Mχ32)(p2−Mχ12)≡−2iuωλ1MU(hU)2I(MU2,Mχ32,Mχ12).The integral I(a,b,c) with a,b≫c is given in the B. Following [39], we conclude that in an
effective approximation, MU2,Mχ32≫Mχ12.
Hence, we have(4.60)(MuU)12≃−λ1tθMU34π2[MU2−Mχ32+Mχ32ln(Mχ32/MU2)(MU2−Mχ32)2]~u≡−12R(MU).The resulting mass matrix is
given by(4.61)MuU=−12(0hUu+R0hUω).We see that one quark remains
massless as the case of the tree-level spectrum. This result keeps up to
two-loop level and can be applied to the downquark sector as well as in the
cases of nondiagonal Yukawa couplings. Therefore, under the Z2,
it is not able to provide consistent masses for the quarks.If the full Yukawa Lagrangian is used, the LNV
couplings must be enough small in comparison with the usual couplings [see
(2.10)]. Combining (2.8) and (2.10), we have(4.62)hUω≫hUu,s1uω≫s1uu.In this case, the element (MuU)11 of (4.53) gets the radiative correction
depicted in Figure 17. The resulting
mass matrix is obtained by(4.63)MuU=−12(s1u(u+RhU)hUus1uωhUω).In contradiction with the first
case, the mass of u quark is now nonzero and given by(4.64)mu≃s1u2hUR. Let us note that the matrix (4.63) gives an eigenvalue
in the scale of (1/2)hUω which can be identified with that of the
exotic quark U.
In effective approximation [39], the mass for the Higgs χ3 is defined by Mχ32≃2λ1ω2.
Hereafter, for the parameters, we use the following values λ1=2.0, tθ=0.08 as mentioned, and ω=10TeV.
The mass value for the u quark is as function of s1u and hU.
Some values of the pair (s1u,hU) which give consistent masses for the u quark are listed in Table 19.Table 19
Mass for theu quark as function of (s1u,hU).
hU21.510.50.1s1u0.00020.00030.00040.0010.01mu [MeV]2.2072.5652.2462.3752.025Figure 17
One-loop contribution to the upquark mass matrix (4.53).Note that the mass values in Table19 for the u quark are in good consistence with the data
given in [3]: mu∈1.5÷4MeV.Downquarks
For the downquarks, the constraint,(4.65)hααDω≫hααDu,sααdω≫sααdu,should be applied. In this case,
three elements (MdαDα)11, (MdαDα)12,
and (MdαDα)21 will get radiative corrections. The relevant
diagrams are depicted in Figure 18. It
is worth noting that Diagram 18(c) exists even in the case of the Z2 symmetry. The contributions are given by(4.66)(MdαDα)11=−sααd2hααDR(MDα),(MdαDα)21=−4iλ1sααdhααDMDα3I(MDα2,Mχ32,Mχ32)=−λ1sααdMDα34π2hααD[MDα2+Mχ32(MDα2−Mχ32)2−2MDα2Mχ32(MDα2−Mχ32)3lnMDα2Mχ32]≡−12R′(MDα),(MdαDα)12=−12R(MDα).We see that two last terms are
much larger than the first one. This is responsible for the masses of the
quarks d2 and d3.
At the one-loop level, the mass matrix for the downquarks is given by(4.67)MdαDα=−12(sααd(u+RhααD)hααDu+Rsααdω+R′hααDω). We remind the reader that a matrix (see also [131])(4.68)(acbD)with D≫b,c≫a has two eigenvalues:(4.69)x1≃[a2−2bcaD+b2c2−(b2+c2)a2D2]1/2,x2≃D.Therefore, the mass matrix in
(4.67) gives an eigenvalue in the scale of D≡(1/2)hααDω which is of the exotic quark Dα′.
Here, we have another eigenvalue for the mass of dα′:(4.70)mdα′=hααDu+R2hααDω{R′2−(sααd)2(hααD)2[(sααdω+R′)2+(hααDu+R)2]}1/2. Let us remember that Mχ32≃2λ1ω2 and the parameters λ1=2.0, tθ=0.08 and ω=10TeV as given above are used in this case; mdα is a function of sααd and hααD.
We take the value hααD=2.0 for both the sectors, α=2 and α=3.
If s22d=0.015,
we get then the mass of the so-called s quark:(4.71)ms=99.3MeV.For the downquark of the third
family, we put s33d=0.7.
Then, the mass of the b quark is obtained by(4.72)mb=4.4GeV. We emphasize again that (4.71) and (4.72) are in good
consistence with the data given in [3]: ms~95±25MeV and mb~4.70±0.07GeV.One-loop contributions to the downquark mass matrix (4.55) or (4.56).
(a)(b)(c)


## 4.4. Summary

The basic motivation of this section is to present the
answer to one of the most crucial questions: whether within the framework of
the model based onSU(3)C⊗SU(3)L⊗U(1)X gauge group contained minimal Higgs sector
with right-handed neutrinos, all fermions including quarks and neutrinos can gain
the consistent masses.In this model, the masses of neutrinos are given by
three different sources widely ranging over the mass scales including the GUT's
and the small VEVu of spontaneous lepton breaking. At the tree
level, there are three Dirac neutrinos: one massless and two degenerate with
the masses in the order of the electron mass. At the one-loop level, a possible
framework for the finite renormalization of the neutrino masses is obtained.
The Dirac masses obtain a large reduction; the Majorana mass types get
degenerate in MR=−ML;
all these masses are in the bound of the data. It is emphasized that the above
degeneration is a consequence of the fact that the left-handed and right-handed
neutrinos in this model are in the same gauge triplets. The new physics
including the 3-3-1 model is strongly signified.
The degenerations and hierarchies among the mass types are completely removed
by heavy particles.The resulting mass matrix for the neutrinos consists
of two partsMν+Mνnew:
the first is mediated by the model particles, and the last is due to the new
physics. Upon the contributions of Mνnew,
the different realistic mass textures can be produced. For example, neglecting
the last term, the pseudo-Dirac patterns can be obtained. In another scenario,
that the bare coupling hν of Dirac masses
gets higher values, for example, in orders of hμ,τ,
the VEV ω can be picked up to an enough large value (~𝒪(104–105)TeV) so that the type II seesaw spectrum is
obtained. Such features deserve further study. We have also shown that the
lepton flavor violating processes such as μ→eγ, μ→3e,
and μe conversion get the consistent values in the
bounds of the current experiments.In the first section, we have shown that, in the
considered model, there are three quite different scales of vacuum expectation
values:ω~𝒪(1)TeV,v≈246GeV,
and u~𝒪(1)GeV.
In this section, we have added a new characteristic property, namely, there are
two types of Yukawa couplings with different strengths: the LNC coupling h's and the LNV ones s's satisfying the condition s≪h.
With the help of these key properties, the mass spectrum of quarks is
consistent without introducing the third scalar triplet. With the given set of
parameters, the numerical evaluation shows that in this model, masses of the
exotic quarks also have different scales, namely, the U exotic quark (qU=2/3) gains mass mU≈700 GeV, while the Dα exotic quarks (qDα=−1/3) have masses in the TeV scale: mDα∈10÷80 TeV.Let us summarize our results.(1)
At the tree level.(a)
All charged leptons gain masses similar to that in the standard model.(b)
One neutrino is massless and the other two are degenerate in masses.(c)
Three
quarksu1,d2,d3 are massless.(d)
Allexotic quarks gain masses proportional toω—the VEV of the first step of symmetry breaking.(2)
At the one-loop level.(a)
All above-mentioned fermions gain masses.(b)
The light quarks gain masses proportional tou—the LNV parameter.(c)
The
exotic quark masses are separated:mU≈700GeV,mDα∈10÷80TeV.(d)
There exist two types of Yukawa couplings: the LNC and LNV with quite different strengths.
With the positive answer, the economical version becomes one of the very attractive models beyond the standard model.


## 5. Conclusion

Finally, this is the time to mention some developments
of the model as reported on this work [36–42]. The idea to give VEVs at
the top and bottom elements of χ triplet was given in [36]. Some consequences such as
the atomic parity violation, Z−Z′ mixing angle and Z′ mass were studied [37]. However, in the
above-mentioned works, the W−Y and W4−Z−Z′ mixings were excluded. To solve the
difficulties such as the standard model coupling ZZh or quark masses, the third scalar triplet was
introduced. Thus, the scalar sector was no longer minimal and the economical in
this sense was unrealistic!In the beginning of the last year, there was a new
step in development of the model. In [38], the correct identification of non-Hermitian bilepton
gauge boson X0 was established. The W−Y mixing as well as W4, Z, Z′ one were also entered into couplings among
fermions and gauge bosons. The lepton-number violating interactions exist in
both charged and neutral gauge boson sectors. However, the lepton-number
violation happens only in the neutrino and exotic quarks sectors, but not in
the charged lepton sector. The scalar sector was studied in [39], and all gauge-Higgs
couplings were presented, and all similar ones in the standard model were
recovered. The Higgs sector contains eight Goldstone bosons—the needed number
for massive gauge ones of the model. Interesting to note that the CP-odd part of Goldstone associated with the
neutral non-Hermitian gauge boson GX0 is decoupled, while its CP-even counterpart has the mixing by the same
way in the gauge boson sector.In [40], the deviation δQW of the weak charge from its standard model
prediction due to the mixing of the W boson with the charged bilepton Y as well as of the Z boson with the neutral Z′ and the real part of the non-Hermitian neutral
bilepton X0 is established.The model is consistent with the effective theory and
new experiments because it can provide all fermions including the quarks and
neutrinos with the consistent masses [41, 42]. The exotic quarks and new bosons get masses in order
of TeV. There are two different scales of exotic quark masses: mU≈700GeV,mDα∈10÷80TeV.It is worth mentioning on advantage of the model: the
new mixing angle between the charged gauge bosonsθ is connected with one of the VEVs u—the parameter of lepton-number violations.
There is no new parameter, but it contains very simple Higgs sector, hence the
significant number of free parameters is reduced. The Higgs self-couplings λ1,2,4 are constrained by the scalar masses, but the
remainder λ3 is fixed by the neutrino masses [42]. This means also that the
generation of the neutrino masses leads to a shift in mass of the Higgs boson
from the standard model prediction.The model is rich in physics because it includes the
right-handed neutrinos, exotic quarks, and new bosons and also gives an possible
explanation of the generation question, electric charge quantization, and
current neutrino mass problem. The supersymmetric
version has been considered [43–46]. The new physics is at TeV scale, therefore, the
results can be verified in the next generation of collides such as LHC and ILC.



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