# Challenges for the 3-3-1 models

V. Pleitez<EMAIL_ADDRESS>https://orcid.org/0000-0001-5279-8438.
Instituto de Física Teórica–Universidade Estadual Paulista
R. Dr. Bento Teobaldo Ferraz 271, Barra Funda
São Paulo - SP, 01140-070, Brazil

(06/09/2022 )

###### Abstract

The so-called 3-3-1 models have been widely studied in the literature.
However, in almost all cases, as with other models, just simplified versions
have been considered. On the other hand, the LHC has not neither confirmed nor
rejected any model. In this paper we discuss some aspects of the 3-3-1 models
that must be investigated, or be re-analyzed, firstly theoretically and,
eventually, experimentally. Some of these issues are independent of the
representation content of the particular model, that is, they are independent
of the value of the $\beta$ parameter. However, others strongly depend on the
particular model considered.

##  I Introduction

It is very well known that there are questions which are still open in the
context of the standard model (SM), and they are ranging from neutrinos
physics, astrophysics, and cosmology. Just to mention the oldest ones: dark
matter [1] (however see [2]) and dark energy [3], neutrino masses [4] and
matter-antimatter asymmetry in the universe [5]. All those issues have been
already observed but not explained. There are some possible anomalies i.e.,
observables that are not in total agreement with the SM predictions, for
instance, muon $(g-2)_{\mu}$ [6], $B$-decays [7], the Cabibbo angle anomaly
[8], just to mention some of them. Although it is not clear that these
anomalies will be confirmed in the near future, we have to pay attention to
them [9].

Most proposed solutions to explain the possible deviations from the
predictions of the SM have been made independently of the model, or using
effective operators and, usually each one targeting a specific problem.
Moreover, if a new sort of particle is found the respective experiment will
not be able, in a first moment, discern between different models that have
this sort of particle. For instance, in searching an extra singly charged
vector boson $W^{\prime}$, it is assumed that the main decay is $W^{\prime}\to
t\bar{b}$ [10]. However, in some models the decay is $W^{\prime}\to\bar{t}X$
or $W^{\prime}\to bY$, where $X$ and $Y$ may be quarks with exotic electric
charge, say $5/3$ or $-4/3$.

It is also important to emphasize that the phenomenological analyses which are
inspired in a particular model, usually are carried out without taking into
account the matrices (unitary or not) present in all vertices of the model.
Although this is easy to understand because the calculations are already
complicated, we cannot forget that the SM would be ruled out if it did not
have the possibility of accomodate the CKM matrix. In the same way, no model
should be discarded just by placing constraints in the mass of only one of the
particles present in the model and a few more parameters. The usual approaches
are also justified since no new particle has been discovered until now.
However, this scheme produces a patchwork of possible solutions over distinct
issues lacking unity and consistency themselves. Nevertheless, sooner or
later, the experimental data will have to be confronted with at least one
specific model, which will be renormalizable and free of anomalies. Or, maybe
a new paradigm in elementary particle physics would arise, probably one which
will not based in gauge symmetries.

Among all the extensions of the SM proposed in the literature, we do not know
yet if one of them (if any) is realized in nature. However, it is still
possible that new physics be confirmed at energies of some TeVs. Some
possibilities are the following:

  * •

Multi-Higgs extension of the SM [11, 12],

  * •

Left-right (symmetric or not) models [13, 14, 15, 16],

  * •

3-3-1 models [17, 18, 19, 20, 21]

  * •

A combination of the previous ones [22, 23],

  * •

Grand unification [24],

  * •

Supersymmetric version of one of the previous ones [25, 26, 27],

  * •

None of the above.

In fact, it is still possible, as pointed out above, that the SM is the last
model based on gauge principle, and a new paradigm would arise in the physics
of the elementary particles and their interactions. Here, of course, we will
not consider this possibility. In fact, we will focus only on models in which
the gauge symmetries are $SU(3)_{C}\otimes SU(3)_{L}\otimes U(1)_{X}$, called
3-3-1 models, with different representation contents. All of them are free of
anomalies only when the three generations are considered at once, and not
generation per generation as is the case in the SM. Due to the representation
content, all these models have a lot of parameters and we will consider here
their origin and why they, sooner or later, they will have to be consider when
studying their phenomenology.

The text is organized as follows. In Sec. II we consider general features of
these models, for example the motivation to extend the SM symmetry to 3-3-1 is
considered in Sec. II.1; in Sec. II.2 we define the electric charge operator
and the parameter $\beta$ (which allow us to distinguish different models); in
Sec. II.3 we consider two of the most studied 3-3-1 models, one with
$\beta=-\sqrt{3}$ in Sec. II.3.1 and the other with $\beta=-1/\sqrt{3}$ in
Sec. II.3.2; in Sec. II.4 we show that models with the same $\beta$ may have
different phenomenology, i.e., they are different models because they have
different representation contents. The main predictions of the models are
considered in Sec II.5. In Sec. III we show the representation content of the
model with $\beta=-\sqrt{3}$ projected onto the symmetry of the SM, includes
many extensions of the SM with several scalar multiplets, and also new quarks
and leptons which are singlets under the SM symmetries. In Sec. IV we discuss
some of the old models with the same gauge symmetries arguing why they cannot
be consider of the 3-3-1 type. Sec. V is devoted to different challenges that,
in our opinion, will have to be confronted with this type of models, but of
course, not only with them. The origin of the parameters of the m331 model is
considered in Sec. VI, for quarks and leptons in Secs. VI.1 and VI.2,
respectively. The challenges in the quark sector are discussed with more
detail in Sec. VII; the direct search for exotics quarks in Subec. VII.1 and
their indirect effect in the possible existence of exotic hadrons in Subsec.
VII.2.

Sec. VIII is devoted to the challenges in the lepton sector, neutrino masses
in Sec. VIII.1; in Sec. VIII.2 we consider rare processes (not confirmed) that
motivated the old models with the same gauge symmetry but with different
representation content; in Sec. VIII.3 we consider some, at present, academic
issues in neutrino experiments. In Sec. IX we review a non trivial solution to
the problem of how the model goes into the SM for the known particles at tree
level. The last section is devoted to our conclusions.

##  II The models

###  II.1 From 3-2-1 to 3-3-1

The 3-3-1 models are interesting because they provide a partial answer to the
problem of the origin of three generations [17, 18, 19, 20]: these models are
free of anomalies only if the number of generation is three or multiple of
three. This solution to the problem of the number of families is not
definitive since it has been changed to: who orders these representation
contents? However, the fact that 3-3-1 models allow us to predict that number
has to be appreciated. After all, most models do not have this property.

We think that these models deserve more attention since they include particles
that appear also in other extensions of the SM, for instance as we stressed
above: several scalar fields in doublets and triplets of $SU(2)\otimes
U(1)_{Y}$, new neutral and charged vector fields, and last, but not least, new
vector-like fermions (in some cases with exotic electric charge) that are
singlets of the SM symmetries.

In these models the defining sector is the leptonic one. After having chosen
this sector, quarks are adjusted so that the model is free of anomalies. In
fact, these models can be considered not just as extensions of the SM, but as
alternatives to it. For example, we can say that the degrees of freedom
already discovered are those that coincide with those of the SM. In order to
appreciate this, let us recall the lepton representation content of the SM
where the lepton sector transform under $SU(3)_{c}\otimes SU(2)_{L}\otimes
U(1)_{Y}$, as left-handed doublets and right-handed singlets:

$\displaystyle\left(\begin{array}[]{c}\nu^{\prime}_{l}\\\
l^{\prime}\end{array}\right)_{L}\sim(\mathbf{1},\textbf{2},-1),\quad
l^{\prime}_{l_{R}}=(l^{\prime c}_{l})_{L}\sim(\mathbf{1},\textbf{1},-2).$ (3)

where we define $Q=(\tau_{3}+Y)/2$, being $\tau_{3}$ the resepctive Pauli
matrix and $Y$ the hypercharge. Primed fields denote symmetry eigenstates.

It is interesting that instead of Eq. (3), we can put all the known leptons
together in a triplet (the adjoint repsesentation of $SU(2)$ is discarded
since in this case the only neutral vector is the photon [28]). For instance
in the minimal 3-3-1 model (m331) leptons transform as [18, 19, 20]:

$SU(2)_{L}\to SU(3)_{L}:\left(\begin{array}[]{c}\nu_{l^{\prime}}\\\
l^{\prime}\\\ l^{\prime c}\end{array}\right)_{L}\sim\textbf{3},$ (4)

Does not seem more natural as a model of leptons? After all, if all the
leptons are massless before the spontaneous symmetry breaking, why not put
them all together in the same multiplet? This model is called minimal because
only the already known leptons are included. Quarks, as we said above, must be
adapted in such a way that the model is free of anomalies only if three
generations are considered and $SU(3)_{L}\to SU(3)_{L}\otimes U(1)_{X}$.
Notice that for leptons the symmetry is just SU(3), since X=0 for them, and
electric charge is quantized in the sense that $\textrm{Tr}Q=0$. The
$U(1)_{X}$ factor is needed to include quarks and it seems that the
quantization of the electric charge is lost. Notwithstanding, if other
conditions are considered it has been shown that electric charge is quantized
independently of the nature of neutrino, Dirac or Majorana [29].

In fact, it is interesting to recall that some of the first models of leptons
were based on gauge theories with $SU(3)$ and considered the leptons in
triplets [30] but in the manner of Konopinski and Mahmoud: the leptons are
$(\nu_{e}\,e^{-}\,\mu^{+})$ while the anti-leptons are
$(\nu^{c}_{e}\,e^{+}\,\mu^{-})$ [31]. Even in Ref. [32], most of it is
dedicated to a lepton model with $SU(3)$ symmetry, leaving only for the end to
consider the model that was known as the Weinberg-Salam-Glashow electroweak
model, with the representation content in Eq. (3) with the gauge symmetry
$SU(2)_{L}\otimes U(1)_{Y}$ [33, 34]. This was the beginning of the now called
standard electroweak model which was builded by many people, theoretically and
experimentally [35] and which is now a textbook theme.

On the other hand, if right-handed neutrinos do exist, a possible
representation content of the SM with this type of neutrinos is

$\displaystyle\left(\begin{array}[]{c}\nu_{l^{\prime}}\\\
l^{\prime}\end{array}\right)_{L}\sim(\mathbf{1},\textbf{2},-1)\quad
l^{\prime}_{R}\sim(\mathbf{1},\textbf{1},-2),\;\;(\nu_{l^{\prime}R})^{c}=(\nu^{c}_{l^{\prime}})_{L}\sim(\mathbf{1},\textbf{1},0),$
(7)

we have the option of the $\nu_{R}$331 model [36, 37]:

$SU(2)_{L}\otimes U(1)_{Y}\to SU(3)_{L}\otimes
U(1)_{X}:\left(\begin{array}[]{c}\nu_{l^{\prime}}\\\ l^{\prime}\\\
\nu^{c}_{l^{\prime}}\end{array}\right)_{L}\sim(\textbf{3},-1/3),\quad
l^{\prime}_{R}\sim(\textbf{1},-1).$ (8)

Again, quarks are accommodated in such a way that the model is anomaly free.

There are several possibilities for the representation content in the 3-3-1
models. As we said before, the defining sector is the leptonic one. The
electric charge operator, $Q$, depends mainly on two parameters, $\beta$ and
$X$, and, secondly for a given $\beta$ we can obtain a different
representation content by choosing different values of $X$. Notice that, like
the SM, the 3-3-1 models are chiral.

###  II.2 The electric charge operator

In principle, the electric charge operator of the models are parametrized by
three parameters $a,\beta$ and $X$:

$\displaystyle Q_{\beta}(X)=aT_{3}+\beta T_{8}+X\mathbf{1},$ (9)

the $\beta$ parameter introduced in Refs. [38, 40, 39]. Hereafter we use
$a=1$, and $\beta=\pm\sqrt{3},\pm\,1/\sqrt{3}$:

$Q_{-\sqrt{3}}(X)=\left(\begin{array}[]{ccc}X&0&0\\\ 0&-1+X&0\\\
0&0&1+X\end{array}\right)\;(a),\;Q_{-\frac{1}{\sqrt{3}}}(X)\\!=\\!\left(\begin{array}[]{ccc}\frac{1}{3}+X&0&0\\\
0&-\frac{2}{3}+X&0\\\ 0&0&\frac{1}{3}+X\end{array}\right)\,(b)$ (10)

and

$Q_{\sqrt{3}}(X)=\left(\begin{array}[]{ccc}1+X&0&0\\\ 0&X&0\\\
0&0&-1+X\end{array}\right)\,(a),\;Q_{\frac{1}{\sqrt{3}}}(X)\\!=\\!\left(\begin{array}[]{ccc}\frac{2}{3}+X&0&0\\\
0&-\frac{1}{3}+X&0\\\ 0&0&-\frac{1}{3}+X\end{array}\right)\,(b)$ (11)

With the notation of Eq. (9), the two most studied models are those with
$\beta=-\sqrt{3}$ and $\beta=-1/\sqrt{3}$. Anyway, the other models [41, 42]
can be obtained of the latter ones.

###  II.3 3-3-1 models with $\beta=-\sqrt{3}$ and $\beta=-1/\sqrt{3}$

We consider the full representation content of the 3-3-1 models with
$\beta=-\sqrt{3}$ and $\beta=-1/\sqrt{3}$ recalling that a model is defined
not only by the gauge (and global) symmetries but also by its representation
content. Below, all fields are symmetry eigenstates but we omit any special
notation for them.

####  II.3.1 Minimal 331 model

Below we omit the prime but all the fields are symmetry eigenstates. Let us
consider the representation content of the m331 model with $\beta=-\sqrt{3}$
and leptons with $X=0$ [18, 19, 20]:

Leptons:

$\Psi_{lL}=\left(\begin{array}[]{c}\nu_{l}\\\ l^{-}\\\
l^{+}\end{array}\right)_{L}\sim(\mathbf{1},\mathbf{3},0).$ (12)

Here $l=e,\mu,\tau$. $\nu_{lR}\\!\\!\sim({\bf 1},{\bf 1},0)$ can be added if
necessary.

Quarks:

$Q_{mL}=\left(\begin{array}[]{c}d_{m}\\\ -u_{m}\\\
j_{m}\end{array}\right)_{L}\sim({\bf 3},{\bf 3}^{*},-1/3),\quad m=1,2;\quad
Q_{3L}=\left(\begin{array}[]{c}u_{3}\\\ d_{3}\\\
J\end{array}\right)_{L}\sim({\bf 3},{\bf 3},2/3)$ (13)

and right-handed singlets $u_{\alpha R}\sim({\bf 3},{\bf 1},2/3)$, $d_{\alpha
R}\sim({\bf 3},{\bf 1},-1/3),\,\alpha=1,2,3$, $j_{mR}\sim({\bf 3},{\bf
1},-4/3)$, and $J_{R}\sim({\bf 3},{\bf 1},5/3)$. Notice that the electric
charge of the quarks $j_{m}$ and $J$ are (in units of $|e|$) are -4/3 and 5/3,
respectively.

Scalars:

$\eta=\left(\begin{array}[]{c}\eta^{0}\\\ \eta^{-}_{1}\\\
\eta^{+}_{2}\end{array}\right)\sim({\bf 1},{\bf
3},0),\;\rho=\left(\begin{array}[]{c}\rho^{+}\\\ \rho^{0}\\\
\rho^{++}\end{array}\right)\sim({\bf 1},{\bf
3},1),\;\chi=\left(\begin{array}[]{c}\chi^{-}\\\ \chi^{--}\\\
\chi^{0}\end{array}\right)\sim({\bf 1},{\bf 3},-1),$ (14)

and the sextet $\textbf{S}\sim({\bf 1},{\bf 6},0)$.

This model, as we said before, will be dubbed m331 for short. It is also
possible the following representation contents: lepton triplets
$(\nu_{l}\,l^{-}E^{+}_{l})^{T}_{L}$, singlets $l^{-}_{R},E^{+}_{R}$
($\nu_{R}$); being $E_{l}$ new charged heavy leptons. In this case the
introduction of the sextet in not mandatory. The model is usually called 3-3-1
model with heavy leptons, or 331HL for short [21, 43]. In both cases the
quarks sector is the same.

####  II.3.2 Model with right-handed neutrinos in the lepton triplets

In this case $\beta=-1/\sqrt{3}$, and leptons have $X=-1/3$ [36, 37], the
minimal representation content is:

Leptons:

$\Psi_{lL}=\left(\begin{array}[]{c}\nu_{l}\\\ l^{-}\\\
\nu^{c}_{l}\end{array}\right)_{L}\sim(\mathbf{1},\mathbf{3},-1/3).$ (15)

Here $l=e,\mu,\tau$; and $l_{R}\sim({\bf 1},{\bf 1},-1))$.

Quarks:

$Q_{mL}=\left(\begin{array}[]{c}d_{m}\\\ -u_{m}\\\
d^{\prime}_{m}\end{array}\right)_{L}\sim({\bf 3},{\bf 3}^{*},-1/3),\quad
m=1,2;\quad Q_{3L}=\left(\begin{array}[]{c}u_{3}\\\ d_{3}\\\
u^{\prime}\end{array}\right)_{L}\sim({\bf 3},{\bf 3},2/3),$ (16)

and right-handed singlets $u_{\alpha R}\sim({\bf 3},{\bf 1},2/3)$,
$\alpha=1,2,3,4$, and $d_{\gamma R}\sim({\bf 3},{\bf
1},-1/3),\,\gamma=1,2,3,4,5$.

Scalars:

$\eta=\left(\begin{array}[]{c}\eta^{+}_{1},\\\ \eta^{0},\\\
\eta^{+}_{2}\end{array}\right)\sim({\bf 1},{\bf
3},2/3),\;\rho=\left(\begin{array}[]{c}\rho^{0}_{1},\\\ \rho^{-},\\\
\rho^{0}_{2}\end{array}\right)\sim({\bf 1},{\bf 3},-1/3),$ (17)

A third triplet $\chi=(\chi^{0}\,\chi^{-}\,\chi^{0})^{T}\sim({\bf 1},{\bf
3},-1/3)$ may be included to avoid the flavor changing neutral currents in the
Higgs sector. A different model is obtained if the lepton triplets are
$(\nu_{l}\,l\,N)_{L}$ and singlets $l_{R}$, $\nu_{R}$ and $N_{R}$ [17]. This
model is usually called 3-3-1 model with right-handed neutrinos, but $N_{L}$
may be considered a lepton not an anti-lepton.

Next, we will show that models with the same values of $\beta$ and $X$, but
different leptonic representation content, have different phenomenology in
this sector.

###  II.4 Same $\beta$ and $X$ and different phenomemnology

Models with the same value of $\beta$ and $X$, have different phenomenology in
the lepton sector because they may have different representation content in
the lepton and scalar sectors. For instance if $\beta=-\sqrt{3}$ and $X=0$
there are two possibilities:

$\displaystyle\left(\begin{array}[]{c}\nu_{l}\\\ l^{-}\\\
l^{+}\end{array}\right)_{L}\;\;(a),\quad\left(\begin{array}[]{c}\nu_{l}\\\
l\\\ E^{+}_{l}\end{array}\right)_{L}\;\;(b),$ (24)

model (a) does not need any singlets $\nu_{lR}$ but they can be included if
necessary; (b) needs the singlets $l^{-}_{R}$ and $E^{+}_{lR}$ and right-
handed neutrinos may also be added if necessary. Notice also that $E^{+}_{l}$
may be considered a lepton, then lepton number is conserved in model (b) but
not in (a). Similar to the Konopinski and Mahmoud scheme [31]: leptons are
$\nu,\,l^{-}$ and $\mu^{+}\to E^{+}_{l}$; anti-leptons are $\nu^{c},\,l^{+}$
and $\mu^{-}\to E^{-}_{l}$. This point was not considered in Ref. [44]. The
minimal scalar sector are different also: Model (a) needs besides the three
triplets, a scalar sextet, while model (b) only three triplets.

Similarly, two models with $\beta=-1/\sqrt{3}$ and $X=-1/3$ are different from
the phenomenological point of view in the lepton and in the scaralr sectors:

$\displaystyle\left(\begin{array}[]{c}\nu_{l}\\\ l\\\
\nu^{c}_{l}\end{array}\right)_{L}\;\;(a),\quad\left(\begin{array}[]{c}\nu_{l}\\\
l\\\ N_{l}\end{array}\right)_{L}\;\;(b),$ (31)

model (a) needs only the singlet $l^{-}_{R}$ (each one per generation) and a
scalar sextet; model (b) needs the singlets $l^{-}_{R},N_{lR}$ (per
generation) and $\nu_{lR}$ may be introduced if necessary. Again, $N_{l}$ may
be a lepton. For instance, effects in $(g-2)_{\mu}$ for the same value of
$\beta$ but different particle contents see [45]. Model (a) needs a sextet to
give mass to neutrinos, model (b) needs only triplets.

###  II.5 Some predictions of the models

The minimal versions of these models have interesting phenomenological
consequences. For example

  1. (a)

Explain the number of generations (multiple of three or just three if
asymptotic freedom is valid for the exotic quarks) [18, 17].

  2. (b)

Explain the quantization of the electric charge [29], for 3-4-1 models see
[46]; and Ref. [47] for $3\\!\\!-\\!\\!n\\!\\!-\\!\\!1$ models.

  3. (c)

They incorporate (almost) naturally the Peccei-Quinn symmetry [48, 49, 50,
51].

  4. (d)

Models with $\beta=-\sqrt{3}$ [18, 19, 20] predicts quarks with exotic
electric charge $F^{Q}=J^{5/3},j^{-4/3}$ and doubly charged vector boson,
$U^{++}_{\mu}$, carrying lepton number $L=-2$ [52], and models with
$\beta=-1/\sqrt{3}$ predict new $u$\- and $d$-type quarks and a neutral vector
boson $\mathcal{Z}^{0}$ carrying also two units of lepton number [17, 36, 37].

  5. (e)

If we assume that the condition

$\frac{g_{X}}{g_{3L}}=\frac{s^{2}_{W}}{1+\frac{4}{\sqrt{3}}\beta s^{2}_{W}}$
(32)

is valid for any energy, then, the models with $\beta=-\sqrt{3}$ explain why
$\sin^{2}\theta_{W}<1/4$ [20, 53] which occurs at energies of the order of the
TeVs [54] or $\sin^{2}\theta_{W}>1/4$ [55]. For models with
$\beta=-1/\sqrt{3}$ the condition is when $\sin^{2}\theta_{W}$ is lower or
greater than $3/4$ which occurs at a rather high energy.

  6. (f)

They incorporate many of the multi-Higgs extensions of the standard model,
singlets (neutral or charged), several doublets, and triplets [56].

  7. (g)

All these models accommodate new sources of $C\\!P$ violation [57, 58, 59].

  8. (h)

They have a neutral real vector boson $Z^{\prime}$ which has interesting
contribution to flavor physics [60, 61].

  9. (i)

The model has candidates for dark matter [62, 63]. For instance, if active
neutrinos are pure Majorana particles, the right-handed neutrinos are not
necessary in the m331 and they may be introduced just as dark matter
candidates [64].

  10. (j)

These models allow partial dynamical symmetry breaking [65, 66].

  11. (k)

In some versions of the models there are new source of $|\Delta L|=2$
processes [44]. In the m331 model these effects occur even without the scalar
sextet in the model. For instance there are new contribution to the
$(\beta\beta)_{0\nu}$ decay [52].

##  III The standard model projection

Although the 3-3-1 model contain the complete representation of the SM, six
quarks and six leptons, the $W^{\pm}$ and $Z^{0}$ bosons, their projection in
the symmetry of the SM also includes, as it was emphasized above, multiplets
that, although they are not in the minimal version of SM, they have been
considered in some of its more motivated extensions: several Higgs doublets;
fields singlets of $SU(2)$: neutral and charged scalars, and also new quarks
and leptons. Moreover, doublets of vector and scalar fields carrying lepton
$L$ number and exotic electric charge. In terms of the $SU(2)_{L}\otimes
U(1)_{Y}$ symmetries, the representation contents of any 3-3-1 model are as is
shown in Figure 1.

Figure 1: Projection of the representation content of 331 models into the SM
symmetries.

In this section we consider the projection of the representation content of
the m331 model ($\beta=-\sqrt{3},\;X=0$) on $SU(3)_{c}\otimes SU(2)_{L}\otimes
U(1)_{Y}$. Firstly, let us consider the SM projection of the already known
quarks and leptons. We have to consider the three generations (recall that the
models are anomalous with only one or two generations).

Leptons are as in Eq. (3) while quarks transform as follows

$\displaystyle Q_{L}=\left(\begin{array}[]{l}u_{i}\\\
d_{i}\end{array}\right)_{L}\sim(\mathbf{3},\mathbf{2},1/3),\;\;u_{iR}\sim(\mathbf{3},\mathbf{2},4/3),\;d_{iR}\sim(\mathbf{3},\mathbf{1},-2/3),$
(35)

$i=1,2,3$. The models have also vector-like quarks singlets of $SU(2)_{L}$. In
particular, the m331 model have vector-like quarks

$\displaystyle J_{L,R}\sim(\mathbf{3},\mathbf{1},10/3),\quad
j_{m(L,R)}\sim(\mathbf{3},\mathbf{1},-8/3),\;m=1,2,$ (36)

with electric charge $+5/3$ and $-4/3$, respectively.

The scalar sector has the following multiplets transforming under the SM
symmetries. Scalar doublets:

$\displaystyle\left(\begin{array}[]{c}\eta^{0}\\\
-\eta^{-}\end{array}\right),\;\left(\begin{array}[]{c}\rho^{+}\\\
\rho^{0}\end{array}\right),\;\left(\begin{array}[]{c}s^{+}_{2}\\\
s^{0}_{2}\end{array}\right),$ (43)

with $Y=-1,+1,+1$, respectively. A scalar triplet $T$ of $SU(2)$ with $Y=2$,

$\mathbf{T}=\vec{\tau}\cdot\vec{t}=\left(\begin{array}[]{cc}T^{+}&\sqrt{2}T^{++}\\\
\sqrt{2}T^{0}&-T^{+}\end{array}\right),$ (44)

scalar singlets $\eta^{+}_{2},\rho^{++},\chi^{0},S^{--}_{2}$ with
$Y=+2,+4,0,+4$, respectively. Moreover, there are two doublets with $Y=+3$ of
vector and scalar fields, $\mathbf{V}_{\mu}$, and $\mathbf{\Xi}$,
respectively:

$\mathbf{V}_{\mu}=\left(\begin{array}[]{c}U^{++}_{\mu}\\\ V^{+}_{\mu}\\\
\end{array}\right),\quad\;\mathbf{\Xi}=\left(\begin{array}[]{c}\chi^{++}\\\
\chi^{+}\end{array}\right).$ (45)

The vector doublet $\mathbf{V}_{\mu}$ is in an octet of $SU(3)$ of gauge
bosons but, after the breakdown of symmetry they become matter fields. In
general, the doubly charged scalar appear as member of a triplet $Y=2$.
Finally, there is a neutral boson $Z^{\prime}_{\mu}$ which gain mass by the
VEV of the singlet $\chi^{0}$.

The interactions involving the vector bosons in Eq. (45) with the known quarks
are

$\mathcal{L}_{Vj(J)}=\overline{Q}_{iL}K_{im}\gamma^{\mu}j_{mR}\mathbf{V}_{\mu}+\overline{Q}_{iL}\gamma^{\mu}K_{i}J_{R}\mathbf{\tilde{V}}_{\mu}+H.c.,$
(46)

where $i=1,2,3$, $m=1,2$ and we have defined
$\mathbf{\tilde{V}_{\mu}}=i\tau_{2}\mathbf{V}^{*}_{\mu}$. As these
interactions are inspired in the m331 model, we define
$K_{im}=(g/\sqrt{2})\delta_{im}$ and $k_{i}=g/\sqrt{2}$. In the mass
eigenstates basis (unless a factor of $g/\sqrt{2}$) we write the interactions
above as

$\displaystyle\mathcal{L}_{Vj(J)}$ $\displaystyle=$
$\displaystyle\bar{u}_{iL}(V^{u\dagger}_{L}O)_{ik}\gamma^{\mu}j_{kR}U^{++}_{\mu}-\bar{d}_{iL}(V^{d\dagger}_{L}O)_{ik}j_{kR}V^{+}_{\mu}$
(47) $\displaystyle+$
$\displaystyle\bar{u}_{iL}(V^{u\dagger}_{L})_{iJ}\gamma^{\mu}J_{L}V^{-}_{\mu}-\bar{d}_{iL}(V^{d\dagger}_{L})_{iJ}\gamma^{\mu}J^{c}_{L}U^{--}_{\mu}+H.c.$

$i=1,2,3;\,k=1,2$ and the index $J$ is fixed.

Notice that these vector-like quarks have transitions to single charged vector
bosons as

$\displaystyle b\to j+V^{+}_{\mu},\quad t\to J+V^{-}_{\mu},$ (48)

i.e., quarks $j$ and $J$ couples only with quarks $d(u)$ thorughout
$V^{+}_{\mu}$ not with $W^{+}_{\mu}$; and the transitions to doubly charged
vector bosons are

$\displaystyle t\to j+U^{++}_{\mu},\quad b\to J+U^{--}_{\mu}.$ (49)

The vector doublet $\mathbf{V}_{\mu}$ couples also with leptons (unless a
factor of $g/\sqrt{2}$):

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\bar{L}\gamma^{\mu}l^{c}_{L}\mathbf{V}_{\mu}+H.c.$ (50)
$\displaystyle=$
$\displaystyle\bar{\nu}_{iL}\gamma^{\mu}(V^{\nu}_{L}V^{l*}_{R})_{ij}l^{c}_{jL}V^{-}_{\mu}-\bar{l}_{iL}\gamma^{\mu}(V^{l\dagger}_{L}V^{l*}_{R})_{ij}l^{c}_{jL}U^{--}_{\mu}+H.c.,$

where in the second line we have written the interactions in the mass basis.

Notice the transitions like

$\nu_{L}\to l^{+}_{L}+V^{-}_{\mu},\quad l^{-}_{L}\to l^{+}_{L}+U^{--}_{\mu}.$
(51)

The scalar $\mathbf{\Xi}$ couples only with quarks as follows:

$-\mathcal{L}_{\Xi
j(J)}=\bar{Q}_{nL}G^{j}_{nm}j_{mR}\epsilon_{kl}\mathbf{\Xi}^{*}_{l}+\bar{Q}_{kL}G^{J}_{k}J_{R}\mathbf{\Xi}+H.c.,$
(52)

or, in terms of the mass eigenstates

$\displaystyle-\mathcal{L}_{\Xi j(J)}$ $\displaystyle=$
$\displaystyle\bar{u}_{iL}(V^{u\dagger}_{L}G^{j}O)_{ij}j_{jR}\chi^{++}-\bar{d}_{iL}(V^{d\dagger}_{L}G^{j}O)_{ij}j_{jR}\chi^{+}$
(53) $\displaystyle+$
$\displaystyle\bar{u}_{iL}(V^{u\dagger}G^{J})_{i}J_{R}\chi^{-}+\bar{d}_{iL}(V^{d\dagger}_{L}G^{J})_{i}J_{R}\chi^{--}+H.c.,$

and we have transitions as

$t_{L}\to j_{R}+\chi^{++},\quad b\\_LtoJ_{R}+\chi^{--},$ (54)

and

$t_{L}\to J_{R}+\chi^{-},\quad b_{L}\to j_{R}+\chi^{+}.$ (55)

Since all 3-3-1 models have a real neutral vector boson $Z^{\prime 0}_{\mu}$
and a complex neutral scalar $\chi^{0}$ we have

$\mathcal{L}_{Z}=(\partial_{\mu}+i\xi_{\chi}Z^{\prime
0})\chi^{0*}(\partial_{\mu}-i\xi_{\chi}Z^{\prime 0})\chi^{0},$ (56)

The neutral vector field cannot has a kinetic mixing with $A$ or $Z$ since
these term does not exist in the m331. Hence there is a parity-like
conservation. Under this operation $Z^{\prime 0},\chi^{0}$ and exotic fields
are odd while the field that also belong to the SM spectra are even.

The scalar potential includes besides the doublets with $|Y|=1$, the complex
triplet and also the singlets $\eta^{-}_{2},\rho^{++},\chi^{0}$, and the
doublet $\mathbf{\Xi}$.

Moreover, it was shown in Ref. [52] that the model is invariant under local
conservation of $B+L$. The new fields carry an extra quantum number related to
$X=B+L$:

$\displaystyle X(J)=X(j_{m})=+\frac{7}{3},\quad
X(V^{-},U^{--},\chi^{-},\chi^{--},\eta^{+}_{2})=+2,\quad
X(H^{--}_{2},\rho^{++},T)=-2,$ (57)

and the particles belonging to the SM carry their usual $B$ and $L$ charges.

Summaryzing, the SM projection of the 3-3-1 models are into

$SU(2)_{L}\otimes U(1)_{Y}\rtimes U(1)_{B+L}.$ (58)

Notice that the $U(1)_{B+L}$ factor is a semidirect product.

The above interactions are considered in of the context of the symmetries in
Eq. (58). Notice that all the extra fields, but the doublets
$(U^{++}_{\mu}\,V^{+}_{\mu})$, have been already considered as extensions of
the SM. For instance

  * •

The SM plus three scalar doublets with $Y=+1$ [67].

  * •

A singly charged singlet scalar with $Y=4$, $\eta^{+}_{2}$, inducing Majorana
masses for neutrino at 1-loop level. Under certain conditions this mechanism
can be implemented in the m331 model [68, 69].

  * •

A doubly charged singlet $S^{--}_{1}$ has been considered for generating
Majorana neutrino masses at the 2-loop level [70, 71].

  * •

A complex triplet $T$ generating the type-II see saw model [72, 73].

  * •

The model with $\beta=-\sqrt{3}$ and $X=0$ but with heavy charged leptons that
transform as singlet under $SU(2)_{L}\otimes U(1)_{Y}$, $E_{lL},E_{lR}$ [21]
could implement the see-saw mechanism for lepton masses and explain the muon
$(g−2)$ [74].

  * •

The scalar doublet $\mathbf{\Xi}$ in Eq. (45), which is part of a triplet of
$SU(3)$, has also been considered as extension of the SM [75], for instance,
in a new scotogenic model [76].

Since all these extensions of the SM are embedded in the m331, the fact that
these models have many scalar fields may be not a problem, but an interesting
feature.

In models with $\beta=-1/\sqrt{3}$ the standard model projection implies
singlet fermions: neutral $N_{L}$, or $(\nu_{L})^{c}$, and quarks
$u^{\prime}_{4}$ and $d^{\prime}_{4},d^{\prime}_{5}$ (symmetry eigenstates).
In the these models it is possible to have mixing among all quarks of the same
charge or, the mixing can be avoided if discrete symmetries are introduced. We
stress that as remarked in Sec. II.4: a model with $(\nu^{c}_{l})_{L}$ in the
triplet is a different model from that with one with neutral fermion,
$N_{lL}$, in each triplet.

At this point, we can ask ourselves, what defines a 3-3-1 model? Of course,
the 331 symmetry is necessary but not sufficient, as we already said, a model
is characterized not just by its symmetries but also by its particle content.
Firstly, they are chiral models, secondly, the triangle anomalies are canceled
out only with a number of generations multiple of 3, or equal to 3 if the
additional quarks contribute to the running of $\alpha_{QCD}$; Finally, they
have a projection into the SM which includes besides the particles of the SM,
some of its more motivated extensions, as in shown in Fig. 1. These
requirements are what define the 3-3-1 models. Moreover, the fact that the
projection into the 3-2-1 symmetry include the more studied extensions of the
SM gives to them a rationale.

##  IV old electroweak models with $SU(3)\otimes U(1)$ symmetries

It is good at this point to make an interlude to comment on some electroweak
models with $SU(3)\otimes U(1)$ symmetries proposed in the early 1970s. These
models are not of the type now called 3-3-1 models although they have the same
electroweak symmetries and the charge operator defined as in Eq. (9) they are
anomalous or are not chiral and, last but not least, have not the projection
into the SM with the three known generations. These models have
$\beta=+1/\sqrt{3}$, with the electric charge operator given in Eq. (11)(b)
but different values of $X$.

  * •

Two models, one with $SU(3)\otimes U(1)$ and the other with $SU(3)\otimes
SU(3)\otimes U(1)\otimes U(1)$ was proposed in Ref. [77]. The model has
$X=-2/3$ for leptons but only two generations of leptons (including two heavy
leptons) are considered, the left-handed fields are in triplets, and the
right-handed components of the charged fields in singlets. In the quark sector
the known left-handed quarks $u,d,s$ have $X=0$ and are in triplet; the
respective right-handed components are in singlets. Obviusly the model is
anomalous. In the chiral $SU(3)\otimes SU(3)$ model both left-handed and
right-handed are in triplets of each $SU(3)$ group. The representation content
of this chiral model is the same as in the first model. For this reason the
model is also anomalous. In fact, that the model should be renormalizable,
modulo anomalies, was recognized in Ref. [78]. Besides, these models do not
have the projection into the SM as in Fig. 1 and Sec. III.

  * •

In Ref. [79] it was proposed an electroweak model with $SU(3)\otimes U(1)$
gauge symmetry, and with the values of $\beta$ and $X$ for leptons as in the
first model of Ref. [77], but now three lepton and two quark generations are
consider; the leptons of each chirality are in triplets with $X=-2/3$ and
there are also left-handed neutral singlets. The left- and right- handed
quarks $u,d,s$ are in triplets. There are extra heavy fermions which are odd
under a $R$ symmetry. Although the model is free of anomalies since the model
is not quiral i.e, is a vector model, it does not have the same $SU(2)\otimes
U(1)$ representation content, for instance right-handed electron belongs to a
triplet together with two other heavy leptons. Similarly with the right-handed
muon. The scalar sector consists of one triplet with $X=-2/3$, and one octet
with $X=0$. A similar model but strictly vector-like triplets was proposed in
Ref. [80].

  * •

The $SU(3)\otimes U(1)$ of Ref. [81] considered leptons in anti-triplets with
$X=-1/3$ and quarks in triplets with $X=0$. In this model the left- and right-
handed electron, and also the muons, are in anti-triplets; but the left-handed
tau lepton is also in a triplet but its right-handed components are in
singlet. Although the model is anomaly free (because of its vector-like
structure) it does not has the SM projection in Sec. III.

  * •

The model of Ref. [82] has the following representation content: a quark
family consists of one triplet with $X=0$ and one singlet of each chirality. A
lepton family consists of one triplet of each chirality, and one $X=0$ left-
handed singlet. The model is free of nomalies due to the quasi-vectorlike
character but again it has not the projection into the SM representation as in
Fig. 1.

The motivation of these models was the possibility that the events with three
muon do exist (see below). We see that the old models with $SU(3)\otimes U(1)$
symmetry cannot be called 3-3-1 models in the sense defined at the end of Sec.
III: either they were anomalous as in Ref. [77], or they are anomaly free
because of the vector-like character. In fact, the first model with simetry
$SU(3)_{L}\otimes U(1)_{X}$ and $\beta=-1/\sqrt{3}$, in which anomalies were
cancel out because the number of triplets is equal to the number of
antitriplets was proposed in Ref. [83]. However, although the model has six
quarks, they are two with electric charge 2/3, and four -$1/3$. That is, the
model consider only two of the known generations: $(u,d,s,c)$ and
$(\nu_{e},e,\nu_{\mu},\mu)$ plus exotic fermions. In order to cancel the
anomalies is necessary add unkown heavy leptons and the antileptons in $SU(2)$
doublets. Hence the model has not the SM projection as in Sec. III. Moreover,
in this model if we add the known third family of quarks, it must be vector-
like under the SM group in order not to spoil the anomalies cancelation [41].
In any case the models do not have the projection into the known standard
model with three chiral generations.

We stress that, the so called 3-3-1 models are chiral models and the anomalies
are canceled out among the three generations with the number of 3 being equal
to the number of $\textbf{3}^{*}$ and apropriately chosen $X$ charges and
also, the projections into SM the known particles must be as it was shown in
Sec. III.

##  V Some Challenges

It is well known that models beyond the SM (BSM) have a rich particle spectrum
and, in general, they have new interactions which depend on many parameters.
For instance, mixing matrices with new $C\\!P$ violating phases, Yukawa
couplings and the mass of the extra degrees of freedom. The 3-3-1 models are
not exception of this, they have many new parameters and fields, but that also
makes them phenomenologically robust. Let us remember that the SM would have
already been discarded if the Cabibbo angle had not been proposed first, and
the CKM matrix later. The radiative corrections were also very important to
make the SM consistent with the experimental data, and they will probably also
be important in physics BSM when the right model (if any) be discovered.

Going back to the 3-3-1 models, here we will concentrate in some issues that
although most of them have already been considered in literature, we think
that the last word has not been given yet. Probably, one day it may be
necessary to reconsider them. They are

  * •

The search of exotic quarks or leptons

  * •

Exotic baryons

  * •

Neutrino masses

  * •

The old processes that were not confirmed

  * •

Confusion among neutrinos and antineutrinos

Before considering in more detail the challenges in the quark and lepton
sectors, we discuss the origin of the parameters of the m331 model.

##  VI Parameters of the m331 model

In order to understand why there are many parameters in this sort of models,
we summarize the quark and lepton in the context of the m331 model. The model
with $\beta=1/\sqrt{3}$ is also briefly considered.

The quark sector is considered in Subec. VI.1 and leptons in Subec. VI.2.

###  VI.1 Quarks

In all 3-3-1 models, since two quark generations transform different from the
third one, the model needs several scalar triplets. In the m331 model, the
triplets $\eta,\rho$, and $\chi$ are needed in order to give appropriate mass
to all quarks and the sextet $S\sim(\textbf{6},0)$ coupling only with leptons,
and the triplet $\eta$, allow to leptons to acquiere masses. All of them are
given in Subsec. II.3.1.

The Yukawa interactions in that sector are given by

$\displaystyle-\mathcal{L}_{Yq}$ $\displaystyle=$
$\displaystyle\overline{Q_{mL}}\,[G_{m\alpha}U_{\alpha
R}\rho^{*}+\tilde{G}_{m\alpha}D_{\alpha
R}\eta^{*}]+\overline{Q_{3L}}\,[F_{3\alpha}U_{\alpha
R}\eta+\tilde{F}_{3\alpha}D_{\alpha R}\rho]$ (59) $\displaystyle+$
$\displaystyle\overline{Q_{mL}}\,G^{j}_{mn}j_{nR}\chi^{*}+g_{J}\,\overline{Q_{3L}}\,J_{R}\chi+H.c.$

The mass matrices for the quarks $U$ and $D$ arise when $\rho\to
v_{\rho}/\sqrt{2},\eta\to v_{\eta}/\sqrt{2}$ and $\chi\to v_{\chi}/\sqrt{2}$
(we consider all VEVs real) and both have contributions of two triplets $\eta$
and $\rho$. Hence there are FCNC mediated by neutral scalars in the quark
sector [84]. In Eq. (59) all states are symmetry eigenstates. Since there are
two quarks with $Q=(-4/3)|e|$, they mix by a Cabibbo-like $2\times 2$ matrix.
Only the quark $J$ does not mix because it is the only one quark with
$Q=(5/3)|e|$.

####  VI.1.1 Quark-neutral scalar interactions

Firstly, we observe that from Eq. (59) we obtain the following mass matrices

$\displaystyle M^{U}=\left(\begin{array}[]{ccc}rG_{11}&rG_{12}&rG_{13}\\\
rG_{21}&rG_{22}&rG_{23}\\\
F_{31}&F_{32}&F_{33}\end{array}\right)|v_{\eta}|,\;\;M^{D}=\left(\begin{array}[]{ccc}r^{-1}\tilde{G}_{11}&r^{-1}\tilde{G}_{12}&r^{-1}\tilde{G}_{13}\\\
r^{-1}\tilde{G}_{21}&r^{-1}\tilde{G}_{22}&r^{-1}\tilde{G}_{23}\\\
\tilde{F}_{31}&\tilde{F}_{32}&\tilde{F}_{33}\end{array}\right)|v_{\rho}|.$
(66)

where $r=|v_{\rho}|/|v_{\eta}|$. We see that, unless an appropriate symmetry
is introduced, say to have the $u$-type quarks in the diagonal basis from the
very start, we proceed as usual: Denote $U^{\prime}_{L,R},D^{\prime}_{L,R}$:
the eigenstates of the symmetry and $U_{L,R},D_{L,R}$ the mass eigenstates.
For instance, $U^{\prime}_{L}=(u^{\prime}\,c^{\prime}\,t^{\prime})$ and so on.
These fields are related to each other by the unitary matrices
$U_{L,R}=V^{U}_{L,R}U^{\prime}_{L,R}$ and
$D_{L,R}=V^{D}_{L,R}D^{\prime}_{L,R}$ and they diagonalized the respective
mass matrices:

$\displaystyle\mathbf{V^{D\dagger}_{L}}M^{D}\mathbf{V^{D}_{R}}=\hat{M}^{D},\quad\mathbf{V^{U\dagger}_{L}}M^{U}\mathbf{V^{U}_{R}}=\hat{M}^{U},\quad\Large{\mathbf{V_{CKM}=V_{L}^{D\dagger}}}\mathbf{V_{L}^{U}},$
(67)

with

$\hat{M}^{U}=\textrm{diag}(m_{u},m_{c},m_{t}),\quad\hat{M}^{D}=\textrm{diag}(m_{d},m_{s},m_{b}).$
(68)

Let us recall how the CKM matrix arises in the context of the SM. In this
model the so called GIM mechanism is automatically implemented: there is no
FCNC neither with the Higgs, $H^{0}$, nor with the vector $Z^{0}$. For this
reason the $V^{U,D}_{L}$ matrices survive only through the combination in the
CKM matrix, $\mathbf{V_{CKM}=V^{D\dagger}_{L}V^{U}_{L}}$. Usually
$V^{U}_{L}=\mathbf{1}$ is used. Moreover, the matrices $V^{U,D}_{R}$
completely disappear from the Lagrangian. In fact, in the SM we can use the
mass matrices of the quarks being diagonal from the beginning and place the
CKM matrix by hand.

However, in the m331 model the matrices $\mathbf{V^{Q}_{L}}$ and
$\mathbf{V^{Q}_{R}}$, $\mathbf{Q}=U,D$, survive in combinations that are
different from those appearing in the definition of the CKM matrix. For
instance, in the quark-neutral scalar interactions in the quarks mass
eigenstates basis are written as

$-\mathcal{L}^{nc}_{Yq}=\overline{U}_{L}\mathbf{K}^{U}U_{R}+\overline{D}_{L}\mathbf{K}^{D}D_{R}+H.c.,$
(69)

with
$\mathbf{K}^{U}=\mathbf{V^{U}_{L}}\mathcal{Z}^{U}\mathbf{V^{U\dagger}_{R}}$
and
$\mathbf{K}^{D}=\mathbf{V^{D}_{L}}\mathcal{Z}^{D}\mathbf{V^{D\dagger}_{R}}$,
and $\mathcal{Z}^{U,D}$ are $3\times 3$ matrizes involving the Yukawa
couplings $G$ and $F$ in Eq. (59) and are given by [86],

$\displaystyle\mathcal{Z}^{U}=\left(\begin{array}[]{ccc}G_{11}\rho^{0}&G_{12}\rho^{0}&G_{13}\rho^{0}\\\
G_{21}\rho^{0}&G_{22}\rho^{0}&G_{23}\rho^{0}\\\
F_{31}\eta^{0}&F_{32}\eta^{0}&F_{33}\eta^{0}\end{array}\right),\;\;\mathcal{Z}^{D}=\left(\begin{array}[]{ccc}\tilde{G}_{11}\eta^{0}&\tilde{G}_{12}\eta^{0}&\tilde{G}_{13}\eta^{0}\\\
\tilde{G}_{21}\eta^{0}&\tilde{G}_{22}\eta^{0}&\tilde{G}_{23}\eta^{0}\\\
\tilde{F}_{31}\rho^{0}&\tilde{F}_{32}\rho^{0}&\tilde{F}_{33}\rho^{0}\end{array}\right).$
(76)

The mass terms in Eq. (66) appears from Eq. (76), when
$\sqrt{2}x^{0}=v_{x}+\textrm{Re}\,x^{0}+i\textrm{Im}\,x^{0}$, with
$x^{0}=\eta^{0},\rho^{0}$, with $\textrm{Re}\,x^{0}$ and $\textrm{Im}\,x^{0}$
are still symmetry eigenstates. For instance,
$\textrm{Re}\,\eta^{0}=\sum_{i}U_{\eta
i}h^{0}_{i},\;\textrm{Re}\rho^{0}=\sum_{i}U_{\rho i}h^{0}_{i}$ with
$h^{0}_{i}$ being mass eigenstates with mass $m_{i}$. It means that FCNC
mediated by neutral scalars is a prediction in the minimal version of the
model, at most they may be controlled not avoided.

####  VI.1.2 Quark-charged scalar interactions

The quark-charged scalar interactions in this model are

$-\mathcal{L}^{cc}_{Yq}=\overline{D}_{L}\mathcal{K}^{DU}U_{R}+\overline{U}_{L}\mathcal{K}^{UD}D_{R}+H.c.,$
(77)

where
$\mathcal{K}^{DU}=\mathbf{V^{U}_{L}}\mathcal{Z}^{DU}\mathbf{V^{U\dagger}_{R}}$
and
$\mathcal{K}^{UD}=\mathbf{V^{D}_{L}}\mathcal{Z}^{UD}\mathbf{V^{D\dagger}_{R}}$
with [86]

$\displaystyle\mathcal{Z}^{DU}=\left(\begin{array}[]{ccc}G_{11}\rho^{-}&G_{12}\rho^{-}&G_{13}\rho^{-}\\\
G_{21}\rho^{-}&G_{22}\rho^{-}&G_{23}\rho^{-}\\\
\tilde{F}_{31}\eta^{-}_{1}&\tilde{F}_{32}\eta^{-}_{1}&\tilde{F}_{33}\eta^{-}_{1}\end{array}\right),\quad\mathcal{Z}^{UD}=\left(\begin{array}[]{ccc}\tilde{G}_{11}\eta^{+}_{1}&\tilde{G}_{12}\eta^{+}_{1}&\tilde{G}_{13}\eta^{+}_{1}\\\
\tilde{G}_{21}\eta^{+}_{1}&\tilde{G}_{22}\eta^{+}_{1}&\tilde{G}_{23}\eta^{+}_{1}\\\
F_{31}\rho^{+}&F_{32}\rho^{+}&F_{33}\rho^{+}\end{array}\right).$ (84)

Notice that in interactions in Eqs. (69) and (77) the matrices $V^{U,D}_{L,R}$
survive separately.

####  VI.1.3 Quark-vector interactions

The m331 model has eight vector bosons, the neutral ones: the photon,
$A_{\mu}$, $Z_{\mu}$ and $Z^{\prime}_{\mu}$; and the charged ones:
$W^{\pm}_{\mu}$, $V^{\pm}_{\mu}$ and $U^{\pm\pm}_{\mu}$. The Lagrangian terms
for interactions among charged gauge bosons and quarks may be written as
follows:

  * •

$D$-type and $U$-type quarks interact as in the SM:

$\mathcal{L}_{Wq}=\frac{g}{\sqrt{2}}\overline{D_{Li}}\,\gamma^{\mu}\left(\mathbf{V_{CKM}}\right)_{ij}U_{Lj}W^{-}_{\mu}+H.c,\quad
i,j=1,2,3.$ (85)

  * •

$J$ and $U$-type quarks:

$\mathcal{L}_{VJ}=\frac{g}{\sqrt{2}}\overline{J_{L}}\,\gamma^{\mu}(\mathbf{V_{L}^{U}})_{3j}U_{Lj}V^{+}_{\mu}+H.c.$
(86)

  * •

$J$ and $D$-type quarks:

$\mathcal{L}_{JU}=\frac{g}{\sqrt{2}}\overline{J_{L}}\,\gamma^{\mu}(\mathbf{V_{L}^{D}})_{3j}D_{Lj}\mathcal{U}^{++}_{\mu}+H.c.,\quad
j=1,2,3$ (87)

  * •

$j$-type and $D$-type quarks:

$\mathcal{L}_{jV}=-\frac{g}{\sqrt{2}}\overline{j_{Lm}}\,\gamma^{\mu}(\mathbf{O^{\dagger}V_{L}^{D}})_{mj}D_{Lj}V^{-}_{\mu}+H.c.,\quad
m=1,2,\;j=1,2,3.$ (88)

  * •

$j$-type and $U$-type quarks:

$\mathcal{L}_{jU}=\frac{g}{\sqrt{2}}\overline{j_{Lm}}\,\gamma^{\mu}(\mathbf{O^{\dagger}V_{L}^{U}})_{mj}U_{Lj}\mathcal{U}^{--}_{\mu}+H.c.,\quad
m=1,2,\;j=1,2,3.$ (89)

The model has neutral currents coupled to $Z^{\prime}$ which are proportional
to [85]

$\displaystyle\mathcal{L}_{Z^{\prime}}$ $\displaystyle\propto$
$\displaystyle[\overline{U_{L}}\gamma^{\mu}\mathbf{V^{U\dagger}_{L}Y^{U}_{L}V^{U}_{L}}U_{L}+\overline{D_{L}}\gamma^{\mu}\mathbf{V^{D}_{L}Y^{D}_{L}V^{D}_{L}}D_{L}$
(90) $\displaystyle+$
$\displaystyle\bar{U}_{R}\gamma^{\mu}_{L}Y^{U}_{R}U_{R}+\bar{D}_{R}\gamma^{\mu}_{L}Y^{D}_{R}U_{R}]Z^{\prime}_{\mu}$

where we have defined

$Y^{U}_{L}=Y^{D}_{L}\propto\textrm{Diag}(1-2s^{2}_{W},\,1-2s^{2}_{W},\,-1),$
(91)

then, there are FCNC in the left-handed interactions but the right-handed
neutral currents with $Z^{\prime}$ are diagonal in the flavour space since in
this case we have $Y^{U}_{R}\propto\textbf{1}$ and
$Y^{D}_{R}\propto\textbf{1}$.

We see that all the matrices, $\mathbf{V_{CKM}}$, $\mathbf{V^{U}_{L}}$ and
$\mathbf{V^{D}_{L}}$, also survive separately in the left-handed interactions
in Eq. (85) - (90). Moreover, as we show above, in the Yukawa interactions in
Eq. (77) the matrizes $\mathbf{V^{U}_{R}}$ and $\mathbf{V^{D}_{R}}$, appear
separately. We see that the origin of the parameters in the quark sector is
Eq. (59) in which two Higgs triplets contribute the quark masses.

All this serves, at least, to make evident the beautiful simplicity of the
standard electroweak model. However, it is possible that this simplicity could
be its greatest weakness when dealing with processes that do not fit in its
context.

We can wonder ourselves, what are the values of each entry of the matrices
$V^{U,D}_{L,R}$? and in the lepton sector, as we will se below,
$V^{\nu}_{L},V^{l}_{L}$ and $V^{l}_{R}$. How can we calculate something with
so many parameters? A possibility is the following. Let’s go back to the Eq.
(67) and calculate $V^{U,D}_{L,R}$ solving simultaneously the following
equations

$\displaystyle
V^{D\dagger}_{L}M^{D}M^{D\dagger}V^{D}_{L}=V^{D\dagger}_{R}M^{D\dagger}M^{D}V^{D}_{R}=(\hat{M}^{D})^{2},$
$\displaystyle
V^{U\dagger}_{L}M^{U}M^{U\dagger}V^{U}_{L}=V^{U\dagger}_{R}M^{U\dagger}M^{U}V^{U}_{R}=(\hat{M}^{U})^{2},$
(92)

where $\hat{M}^{U}$ and $\hat{M}^{D}$ are defined in Eq. (68), and at the same
time using the numerical values of the quark masses, the CKM matrix, and the
Yukawa couplings, $G,\tilde{G},F\tilde{F}$ in Eq. (66), as input. Knowing the
values of all the entries of the matrices $V^{U,D}_{L,R}$, only the masses of
the extra particles remain free parameters. The problem with this approach is
that there are many (realistic) solutions for the entries of the
$V^{U,D}_{L,R}$ matrices.

Using this method in the processes with $\Delta F=2$ and $\Delta F=1$, we note
that, besides the FCNC via $Z^{\prime}$, the contributions of the neutral
(pseudo) scalars may be important since, for some region of the parameter
space, it may be negative interference. Taken into account only the
contribution of $Z^{\prime}$ to the neutral mesons $\Delta M_{M}$ we have
$M_{Z^{\prime}}>5$ TeV. However, by considering also the pseudoscalar
contributions it is possible to obtain
$M_{Z^{\prime}}\stackrel{{\scriptstyle>}}{{\sim}}2.3$ TeV [86]. See Fig. 2.

Figure 2: Lower bounds on $\Delta M_{M}$ taken into account the contribution
of $Z^{\prime}$ and a neutral scalar [86].

Another possibility, complementary to the previous one, is the following: we
can consider the matrices $V^{U,D}_{L,R}$, and the masses of the extra
particles (scalars,vectors and fermions), as free parameters. Then, as it was
done in the SM, after a lot of phenomenology using known and rare decays
(possible anomalies), $\mathbf{(g-2)}_{\mu}$, … it will be possible to
determine all the entries of the mixing matrices $V^{U,D}_{L,R}$, but without
forgetting that, unlike the SM case, in the end, these matrices have to
diagonalize the respective mass matrix. Of course, we can begin in a model-
independent way by considering most known baryon and meson decays, and other
processes, then continue with all the possible flavor anomalies. In this case
we can get rid of one the matrices, $V^{U}_{L}$ or $V^{D}_{L}$, by expressing
one of them in terms of $V_{CKM}$ and the other one [87]. However, the
matrices $V^{U}_{R}$ and $V^{D}_{R}$ still survive independently in the
Lagrangian.

###  VI.2 Leptons

Again, here we will consider only the m331 model [18, 19, 20] in which the
minimal version only includes the already known leptons. As in the case of
quarks, also in the leptonic sector the 3-3-1 models have many parameters.
Although the interactions with $Z$ and $Z^{\prime}$ are diagoanl in the
flavour space (the three generations of lepton transform in the same way under
the electroweak symmetries $SU(3)_{L}\otimes U(1)_{X}$) there are FCNC in the
scalar sector. The origin of the parameters is, as in the case of quarks, the
different contributions to the mass matrices of charged leptons. If the masses
are generated at the tree level, the Yukawa interactions include the triplet
$\eta\sim(\textbf{3},0)$, in Sec. VI.1, and the sextet $S\sim(\textbf{6},0)$
under the respective group factors. The components of the sextet are as
follows

$S=\left(\begin{array}[]{ccc}s^{0}_{1}&\frac{s^{-}_{1}}{\sqrt{2}}&\frac{s^{+}_{2}}{\sqrt{2}}\\\
\frac{s^{-}_{1}}{\sqrt{2}}&S^{--}_{1}&\frac{s^{0}_{2}}{\sqrt{2}}\\\
\frac{s^{+}_{2}}{\sqrt{2}}&\frac{s^{0}_{2}}{\sqrt{2}}&S^{++}_{2}\end{array}\right).$
(93)

The Yukawa interactions in this model are give by

$-2\mathcal{L}_{Yl}=\overline{(\psi_{aiL})^{c}}G^{\eta}_{ab}\epsilon_{ijk}\psi_{bjL}\eta_{k}+\overline{(\psi_{aiL})^{c}}G^{S}_{ab}\psi_{bjL}S_{ij}+H.c.$
(94)

where $G^{\eta}(G^{S})$ is a anti-symmetric (symmetric) matrix in the flavor
space.

The scalar sextet in Eq. (93) has two neutral components, $s^{0}_{1}$ and
$s^{0}_{2}$. If both gain a non zero VEV, $v_{s_{1}}$ and $v_{s_{2}}$
respectively, we obtain the lepton mass matrices

$M^{\nu}=v_{s_{1}}G^{S}_{ab},\quad
M^{l}_{ab}=\frac{v_{\eta}}{\sqrt{2}}G^{\eta}_{ab}+\frac{v_{s_{2}}}{2}G^{S}_{ab},$
(95)

it means that

$M^{l}=\frac{1}{\sqrt{2}}\left(v_{\eta}G^{\eta}+v_{s_{2}}\frac{M^{\nu}}{\sqrt{2}\,v_{s_{1}}}\right),$
(96)

and since $M^{\nu}/\sqrt{2}\,v_{s_{1}}\approx O(1)$, we see that
$v_{s_{2}}>v_{\eta}|G^{\eta}|$ in order that the symmetric matrix could be the
main contribution to the charged lepton masses. Notice that, since there are
two source for the charged lepton masses, there are flavor changing neutral
interactions in the scalar sector. We can not assume the charged lepton masses
as being in some diagonal basis. This is a prediction of the model.

Next, let us consider the interactions in this sector sector.

####  VI.2.1 Lepton-neutral scalar interactions

Omitting generation indices (scalar fields are still symmetry eigenstates)

$\displaystyle-\mathcal{L}^{nc}_{Yl}=\bar{l}_{R}(V^{l\dagger}_{R}G^{\eta}\eta^{0}+G^{S}s^{0}_{2})V^{l}_{L}l^{\prime}_{L}+\overline{(\nu_{L})^{c}}V^{\nu
T}G^{S}V^{\nu}\nu_{L}s^{0}_{1}+H.c.$ (97)

As in the quark sector we have to diagonalize the matrix $M^{l}$ in Eq. (96)
by a bi-unitary transformation: $V^{l\dagger}_{R}M^{l}V^{l}_{L}=\hat{M}^{l}$
where $\hat{M}^{l}=\textrm{diag}(m_{e},m_{\mu},m_{\tau})$; and $V^{\nu
T}M^{\nu}V^{\nu}=\hat{M}^{\nu}$ [88], with
$\hat{M}^{\nu}=\textrm{diag}(m_{1},m_{2},m_{3})$ in the neutrinos sector. The
matrices $M^{\nu}$ and $M^{l}$ are given in Eq. (95). Hence, we have three
irredutivel matrices: $V^{l}_{L},V^{l}_{R}$ and $V^{\nu}$; or
$V^{l}_{R},V_{L}$ and introducing the PMNS matrix
$V^{\nu}_{L}=V^{l}_{L}V_{PMNS}$. We stress that $V^{l}_{L}$ cannot be assumed
as being the unity matrix. Below we use the first option i.e., the PMNS matrix
appears in the lagrangian only when the right combination defined above
appears in the interactions with the vector $W^{\pm}$.

####  VI.2.2 Lepton-charged scalar interactions

These interactions are given by (charged scalar fields are still symmetry
eigenstates)

$\displaystyle-\mathcal{L}^{cc}_{Yl}=\overline{l_{R}}\left[(K^{\eta}_{R})\eta^{-}_{1}+(K^{S}_{R})\nu_{L}\frac{s^{-}_{2}}{\sqrt{2}}\right]\nu_{bL}+\overline{(\nu_{L})^{c}}\left[(K^{\eta}_{L})\eta^{+}_{2}+(K^{S}_{L})\frac{s^{+}_{1}}{\sqrt{2}}\right]l_{L}$
(98)

where we have omitted generation indices and defined
$K^{\eta}_{R}=V^{l\dagger}_{R}G^{\eta}V^{\nu}$, $K^{\eta}_{L}=V^{\nu
T}G^{\eta}V^{l}_{L}$; $K^{S}_{R}=V^{l\dagger}_{R}G^{S}V^{\nu}$ and
$K^{S}_{L}=V^{\nu\,T}G^{S}V^{l}_{L}$. As in the quark sector the matrices
$V^{l}_{L,R}$ and $V^{\nu}$ survive separately in the Lagrangian. We are also
assuming that there is no right-handed neutrinos.

####  VI.2.3 Lepton-vector interactions

In the lepton sector we have the interactions with singly charged vector
bosons. The $\nu-l-W$ interaction

$\mathcal{L}_{\nu
lW}=i\frac{g}{2\sqrt{2}}\left[\overline{\nu_{L}}\,\mathbf{e^{i\phi_{l}}\,V^{\dagger}_{PMNS}}\,\gamma^{\mu}l_{L}+\overline{(l^{c})_{R}}\,\mathbf{e^{{}^{-}i\phi_{l}}V^{*}_{PMNS}}\gamma^{\mu}(\nu^{c})_{R}\right]W^{+}_{\mu}+H.c.,$
(99)

and the only difference with the SM is the phase $e^{i\phi_{l}}$, here as
usual

$V_{PMNS}=V^{l\dagger}_{L}V^{\nu}_{L},$ (100)

and since neutrinos are Majorana particles, the $V_{PMNS}$ matrix have one
Dirac and two Majorana phases.

There are also $\nu-l-V$ currents

$\displaystyle\mathcal{L}_{\nu
lV}=i\frac{g}{2\sqrt{2}}\left[\overline{(l^{c})_{L}}\,\mathbf{V^{*}_{LR}}\gamma^{\mu}\nu_{L}-\overline{(\nu^{c})_{R}}\,\mathbf{V_{LR}}\gamma^{\mu}l_{R}\right]V^{+}_{\mu}+H.c.$
(101)

where we have defined

$\mathbf{V_{LR}}=V^{\nu\dagger}_{L}V^{l*}_{R}.$ (102)

Hence, in the m331 model $V^{\nu}\not=V_{PMNS}$ always, and $V^{l}_{L}$ and
$V^{l}_{R}$ appear in Eq. (101) in the combination showing in Eq. (102). We
recall that $V^{\nu}$ appears also in the Yukawa interactions in Eq. (98).
Notice that since the charged lepton mass matrix in Eq. (96) has a
contribution of an anti-symmetric matrix, it is not possible to assume it
diagonal from the very beginning, hence we cannot put $V^{l}_{R}=\mathbf{1}$
in the interactions in Eq. (101). This would be possible if the interactions
leptons with the scalar triplet $\eta$ is forbidden, say by an appropriate
discrete symmetry. However, in this case, the charged leptons mass matrix is
diagonalized by the same unitary matriz that the neutrino mass matrix, and the
PMNS is trivial.

Then, as in the quark sector, it is necessary to known what are the values of
the entries of the matrices $V^{\nu}$, $V^{l}_{L}$ and $V^{l}_{R}$ if we want
to make a realistic phenomenology of this model.

It is possible to have mixing bewteem $W^{\pm}_{\mu}$ and $V^{\pm}_{\mu}$
(this occurs if neutral boson in the sextet $v_{s_{1}}\not=0$ [162]). In
general, the global phase $e^{i\phi_{l}}$ in the charged current coupled with
the $W^{\pm}_{\mu}$, see Eq. (99), cannot be neglected anymore. The
elimination of the global phase is possible only in the context of the SM
where there is only one charged current coupled to $W^{\pm}_{\mu}$. In this
case there is a contribution to the lepton electric dipole moment at the one-
loop level. The same occurs in other models with several singly charged vector
bosons as the left-right symmetric models [15].

In this model there are also interactions between charged leptons and doubly
charged vector bosons given by the lagrangian

$\displaystyle\mathcal{L}^{U}=-\frac{g}{2\sqrt{2}}\,\overline{l^{c}_{a}}\left[\gamma^{\mu}\mathbf{A_{ab}}-\gamma_{5}\gamma^{\mu}\mathbf{S_{ab}}\right]l_{b}\,U^{++}_{\mu},$
(103)

$\mathbf{A}=\mathbf{\bar{V}}_{RL}-\mathbf{\bar{V}^{T}}_{RL}$ and
$\mathbf{S}=\mathbf{\bar{V}}_{RL}+\mathbf{\bar{V}^{T}}_{RL}$ are antisymmetric
and symmetric matrices, respectively; and

$\mathbf{\bar{V}}_{RL}=(V^{l}_{R})^{T}V^{l}_{L},$ (104)

Finally, leptons couple universally with neutral vector bosons. For instance
$Z^{\prime}$-lepton interactions $Z^{\prime}$ is (univarsally) leptophobic the
m331 model

$f^{l}_{V}=-f^{l}_{A}=-\frac{\sqrt{3}}{6}\sqrt{1-4s^{2}_{W}}.$ (105)

which are valid in some conditions discussed in Sec. IX.

##  VII Challenges in the quark sector

Here, we will place some topics in the quark sector that in the future
probably will deserve to be studied in more detail than what have been done so
far.

###  VII.1 In search of exotic quarks

Flavor physics is capable of unveiling smaller distances than those that are
possible in accelerators like the LHC. In fact, it may provide an insight into
scales as short as $10^{-21}$ m (Zeptouniverse), corresponding to energy scale
of 200 TeV or even shorter distance scales [90].

The search of vector-like singlet quark with exotic electric charge $-4/3,5/3$
has already been done in several colliders. When looking for this type of
quarks, with electric charge $-4/3$, it is usually assumed $X_{q}\to b+W^{+}$
but data are consistent with the standard model (charge $q=+2/3$) and exclude
the existence of an exotic quark with -$4/3$ electric charge and mass of the
conventional top quark at the 99% confidence level [91]. Quarks with charge
$5/3$ has also been searched in processes as $X_{5/3}\to W^{+}+t$, being the
$X_{5/3}$ mass, with right-handed (left-handed) couplings, below 1.32 (1.30)
TeV excluded at 95% confidence level [92].

However, in the m331 model the transitions involving these exotic quarks are,
see Eqs. (86)-(87):

$\displaystyle j_{-4/3}\to V^{-}+b,\quad j_{-4/3}\to U^{--}+t,$ $\displaystyle
J_{5/3}\to V^{+}+t,\quad J_{5/3}\to U^{++}+b.$ (106)

How important is having $V^{-}$ or $U^{--}$ instead of $W^{-}$?

We would like to emphasize that, from D0 [93] and CDF [94], to the LHC [10,
95], the search for $W^{\prime}$ has been done through the decay
$W^{\prime}\to t+\bar{b}$. This is also the case in phenomenological analyses
[96]. Of course, there are models in which this decay does exist. However, in
the m331 model the decays are $V^{+}\to j_{-4/3}+\bar{b}$ or $V^{+}\to
j_{5/3}+\bar{t}$. Moreover in Ref. [96] $W^{\prime}$ was considered that
interacts with leptons with left-handed current, and the decay is
$W^{\prime+}\to\tau^{+}_{R}+\nu_{L}$, but in the m331 the decay involving the
extra charged vector boson is $V^{+}\to\tau^{+}_{L}+\nu^{c}_{R}$ i.e., through
a right-handed current.

###  VII.2 Do exotic bound states exist?

The 3-3-1 models predict, under certain circumstances, the existence of new
bound states that do not exist in the context of SM. For example, in the model
with heavy leptons (331HL) the lightest of these leptons decay into the knwon
leptons only throught a charged scalar belonging to a triplet, $\eta^{-}_{1}$.
Then, the lightest lepton may be stable enough to form atoms with one of the
known charged leptons, the electron for example: $E^{+}e^{-}$.

Figure 3: Fundamental representation of the $SU(3)_{H}$.

On the other hand, it has been observed that in the m331 and the 331HL models
(both with $\beta=-\sqrt{3}$ and $X=0$) there is one quark with electric
charge $5/3$ and two with $-4/3$ (in units of $|e|$). Hence it is possible the
existence of a replica of the flavor $SU(3)_{F}$ symmetry of the light quarks
$u,d,s$ but now with the heavy quarks $J,j_{1},j_{2}$ which would imply the
existence of hadrons with exotic quarks [97]. It means that $SU(3)_{F}\to
SU(3)_{H}$ making $u,d,s\rightarrow J,j_{1},j_{2}$. See Fig. 3.

In this case, exotic hadrons are predicted with electric charge $\pm(3,4,5)$
with

$Q=\kappa I_{3H}+\frac{Y_{H}}{2},\quad\kappa=3,\quad Y_{H}=B+S_{H}$ (107)

Exotic baryons are for instance those in Tables 1 and 2. Their decays are as
follows:

$\Theta^{+5}_{3J}\to V^{+}+\Theta^{+4}_{2Jt}\to
2l^{+}+2\nu+\Theta^{+4}_{Jtt}\to...3l^{+}+3\nu+3t\to...$

Model with $\beta=-1/\sqrt{3}$ may have an $SU(3)_{H}$ with $u_{4}$ and
$d_{4},d_{5}$, then the electric charge are the same as the known mesons and
baryons.

Baryon | $Y_{H}$ | $Q/|e|$
---|---|---
$\Theta_{3J}$ | $0$ | +5
$\Theta_{2j_{1}j_{2}}$ | $-2$ | $-4$
Table 1: Examples of exotic baryons in models with $\beta=-\sqrt{3}$.  Meson | $Y_{H}$ | $Q/|e|$
---|---|---
$\Pi^{0}_{\bar{J}J}$ | $1/3$ | 0
$\Pi_{\bar{J}j_{1}}$ | $1/3$ | -3
$\Pi^{\prime}_{\bar{J}j_{2}}$ | -3 | -3
$\Pi^{\prime\prime}_{\bar{j}_{1}j_{2}}$ | -3 | 0
Table 2: Examples of exotic mesons in models with $\beta=-\sqrt{3}$.

All these exotic hadrons may be long lived, with $\tau\sim 10^{-11}-10^{-14}$
s (depending on the mass of the exotic quark and vector bosons). Is it
possible that these new degrees of freedom are waiting for us in a regime in
which the theory is no more perturbative?

##  VIII Challenges in the lepton sector

Neutrino physics in 3-3-1 models has been widely studied in the literature.
Here we only want to mention some aspects of this topic that we believe should
be better studied in the future.

###  VIII.1 Neutrino masses

The lepton mass matrices are given in Eq. (95). We see that the charged lepton
masses have contributions which are proportional to the neutrino masses. In
fact, we can rewritte Eq. (96) in terms of the diagonal neutrino mass matrrix
as follows:

$\hat{M}^{l}=V^{l\dagger}_{L}\left(\frac{v_{\eta}}{\sqrt{2}}G^{\eta}+\frac{v_{s_{2}}}{v_{s_{1}}}V^{l*}_{L}V^{*}_{PMNS}\hat{M}^{\nu}V^{\dagger}_{PMNS}V^{l\dagger}_{L}\right)V^{l}_{R},$
(108)

where we have used $V^{\nu}=V^{l}_{L}V_{PMNS}$ and we see that it implies a
fine-tuning. Moreover, if we use the Yukawa coupling in the matrices
$G^{\eta}$ and $G^{S}$ and calculate the $V^{l}_{L,R}$ throuhg the charged
lepton masses and $V^{\nu}$ with neutrino masses, it seems difficult to obtain
at the same time the PMNS matrix [98].

If the scalar sextet is introduced, we have at least four possibilities: i)
The fine tuning is possible and neutrinos are pure Majorana particles in this
model, and no right-handed neutrino at all, at least for this issue. ii)
$v_{s_{1}}=0$ and this condition is stable againts quantum fluctuations the
Zee and/or Zee-Babu mechanims can be implemented and neutrinos gain masses at
1-loop or 2-loop order, respectivaly. Although the Zee mechanism was
considered in Ref. [99] no conditions to maintain $v_{s_{1}}=0$ stable was
given. Hence, in this case, neutrino masses are not calculables in the sense
of Refs. [100, 103, 102, 101]. In order to have calculable neutrino masses, we
must protect $v_{s_{1}}=0$ against quantum effects while maintaining at the
same time lepton number violation interactions in the scalar potential. iii)
The other possibility is to add right-handed neutrinos implementing both
type-I and type II seesaw mechanism. Finally, iv) to include more degrees of
freedom as in Refs. [104, 105, 106, 107].

Which of these mechanisms are used in nature (if any)? In the first and second
one neutrinos are pure Majorana particles and the PMNS matrix is exactly
unitary. In the third case there must exist heavy right-handed neutrinos and
the PMNS matrix is only approximately unitary, moreover the Keung-Senjanovic
mechanism [108] can occur. In the fourth one, the model is not the minimal
version of the m331 since new degrees of freedom different from right-handed
neutrinos have to be added.

Another possible way to generate lepton masses is the following: do not
introduce the sextet or right-handed neutrinos. In this case the neutrinos are
massles at tree level and charged leptons have the mass spectrum: $(0,m,m)$.
However, the Zee mechanism [68] is still implemented by the $\eta^{+}_{2}$
scalar. It solves the problem with the neutrino masses but it is necessary
still to obtain a way to give mass to electron and break out the mass
degeneracy of the muon and tau including the PMNS matriz but using only the
degrees of freedom of the m331. No such mechanism has been appear in the
literature. But, we can wounder ourselves if is it possible a realistic
mechanism for the charged lepton masses analogous to those proposed in Ref.
[103]?

Finally, we would like to comment an interesting possibility that was
implemented in the context of the SM [109] but which may be implemented in
3-3-1 models with $\beta=-1/\sqrt{3}$ and other extensions of the SM. In this
sort of models, the projection into the SM representation content introduce a
vector doublet to the standard model, includes a vector doublet with
$SU(3)_{C}\otimes SU(2)_{L}\otimes U(1)_{Y}$ quantum numbers like the ordinary
lepton doublets,
$V_{\mu}=(V^{0}_{\mu}\,V^{-}_{\mu})^{T}=(\textbf{1};\textbf{2},-1/2)$. This
vector doublet is suitable to address neutrino mass at the 1-loop level
scotogenic mechanism because of the vertex $V^{0}_{\mu}V^{0\mu}H^{0}H^{0}$,
where $H^{0}$ is the SM Higgs. The vector field is also dark matter, and
contributes also to the muon anomalous magnetic moment. The dark matter
candidate is either the lightest of the fermion singlets $N_{1;2;3}$, or one
of the neutral vectors $V_{1};V_{2}$ (mass eigenstates), that are stabilized
by the accidental symmetry $\mathbb{Z}_{2}$. See also [110].

###  VIII.2 Back to the old rare processes

It is interesting that most models with symmetry $SU(3)\otimes U(1)$
considered in Sec. IV were proposed by trying to explain the called “three
muon anomaly” [111, 112],

$\nu_{\mu L}+N\to\mu^{+}\mu^{+}\mu^{-}+X^{-},$ (109)

and also the high-$y$ anomaly, which was an excess at high-$y$ in region,
apparently observed in the process

$\nu_{\mu}(\bar{\nu}_{\mu})+N\to\mu^{-}(\mu^{+})+X,$ (110)

where $y=(E_{\nu}-E_{\mu})/E_{\nu}\equiv E_{h}/E_{\nu}$ and $E_{h}$ is the
hadron energy. Both processes were apparently observed by the Harvard-
Pennsylvania-Wisconsin-Fermilab (HPWF) collaboration [114, 113, 115, 116].

Both processes cannot be explained in the context of the then called
”Weinberg-Salam” electroweak model. New neutral leptons or doubly charged
vector bosons? This was the context in which the old $SU(3)\otimes U(1)$
models considered in Sec. IV, were proposed.

Figure 4: Three muons process at LHC.

However, the Dortmund-Heidelberg-Saclay (CDHS) neutrino experiments at CERN
400 GeV SPS did not see neither the trimuons nor high-$y$ anomaly. Although no
one ever knew what might be wrong with the HPWF experiments, since they were
not confirmed by the CDHS collaboration, both processes were forgotten [117].
Should they?

It happens that the three muon events occur in the m331 model, for instance at
the LHC as can be seen in Fig. 4 and also in supersymmetric models, through
the production of a pair of chargino–neutralino also produce trimuon events in
the final states [118, 119]. See Fig. 5.

Figure 5: Three muons in susy models at LHC [118].

Moreover, in the m331 model three leptons may also occur in $\nu_{\mu}-e$ and
$\mu-N$ collisions as is shown in Fig. 6.

Figure 6: Trileptons in $\nu_{\mu}-e$ and $\mu^{-}-N$ scattering [121].

In 3-3-1 models (independently of the value of the parameter $\beta$)
processes like the trident neutrino production [120, 121], shown in Fig. 7,
may occur.

The process with three muons may occur at LHC also, and assuming the only
source of this decay is the doubly charged vector it could be discovery at the
LHC if its mass is of the order of the 1 TeV [122], see Fig. 8.

Figure 7: Trident events in $\nu_{\mu}-N$ scattering [120].

However the process $\mu\to 3e$ implies, under some reasonable assumptions,
that [124]

$M_{\mathcal{U}}>5,\;\;\textrm{ TeV}$

In this case $U^{++}_{\mu}$ cannot be discovery at the present LHC. However,
this may not be the final word: both results can be made compatible if the
contribution of the scalar fields with double charge were not negligible
or/and there are other realistic solutions for the leptonic mixing matrices.

In fact, charged lepton flavour violation (cLFV) processes are interesting for
searching new physics due to the sensitivity of the next experiments, see
Table 3 and Fig. 9. All models have to be confronted with the future data.
This includes all 3-3-1 models.

Decaimento | Presente | Futuro
---|---|---
$\mu\to e\gamma$ | $4.2\times 10^{-13}$ | $6\times 10^{-14}$
$\mu N\to eN$ | $10^{-12}-10^{-13}$ | $10^{-17}$
$\mu\to ee\bar{e}$ | $10^{-12}$ | $10^{-16}$
Table 3: Lepton Flavour violation in muon decays, from [123].

It is possible to search for the high-$y$ and three muon anomalies, and
similar processes, using the $\nu_{\tau}$ i.e., $\nu_{\tau}+N\to 3\tau+X$
[125].

Figure 8: $pp\to 3\mu+X$ considering only SM+$U^{++}_{\mu}$. This process has
no irreducible background since the SM vertices conserve flavor [122].

###  VIII.3 Neutrino or antineutrino?

In all experiments with neutrinos it has been assumed, whether in the context
of SM or beyond it, that neutrinos always accompany antileptons, while
antineutrinos to leptons. In fact, this is true in the context of the SM where
the only charged interactions are those coupled to the $W^{\pm}_{\mu}$ [4].
Although, that in most extensions of the SM models having other charged
interactions this continue to be true this is not the case in the m331 and
probably in other models. The fact that the identification of the neutrino
flavor cannot be done in a model-independent way, it was noted in Ref. [126].
Although, with the present precision of the experiments this does not seem
necessary to be taken into account, it is conceptually correct.

Figure 9: History of cLFV in the period 1940–2020 [123].

Let us remember that, in the SM, the leptonic transitions are the following:

$\quad W^{+}\to l^{+}_{R}+\nu_{lL},$ (111)

i.e., left-handed neutrinos of flavour $l$ are defined as the neutral particle
emitted together with a right handed positron in the $W^{+}$ decay, while
anti-neutrinos of flavour $l$ are defined as the neutral particle emitted
together with a left-handed electron und a $W^{-}$:

$W^{-}\to l^{-}_{L}+\nu^{c}_{lR}.$ (112)

However, there may be situations where it appears some confusion in the
identification of neutrinos and in some cases there is also a confusion in the
definition of neutrinos and antineutrinos. Let us illustrate this situation
considering the m331 in which there is a second singly charged vector boson.
In this model, besides the transitions in Eqs. (111) and (112), there are
transitions as in Eq. (101),

$V^{-}\to l^{-}_{R}+\nu_{lL},$ (113)

which seems like an antineutrino not a neutrino, unless the chirality of the
charged lepton is measured we cannot say that we are observing the process
(111) or (113). The transition

$V^{+}\to(\nu^{c}_{l})_{R}+l^{+}_{L},$ (114)

also occurs. Hence, a complete identification of neutrino or antineutrinos
needs a measurement of the helicities of both leptons.

Moreover, the vertices involving $V^{\pm}$ have a mixing matrix elements which
are different from those in the usual PMNS matrix:
$\mathbf{V_{LR}\not=V_{PMNS}}$. See Eq. (101). There the mixing matrix is
$\mathbf{V_{LR}=V^{\nu\dagger}_{L}V^{l*}_{R}}$ and, as we already dicussed, in
the m331 it is not possible in any approximation $\mathbf{V^{l}_{R}=1}$. Hence
any phenomenological analysis considering the interactions in Eq. (101) has to
take into account the matrix $\mathbf{V_{LR}}$, otherwise, it cannot be
interpret its result in terms of the m331.

Although the confusion neutrino-anti-neutrino may not have a considerable
effect in active neutrino oscillations, this is not necessarily the case with
rare processes as neutrino$\leftrightarrow$anti-neutrino oscillations. This
sort of oscillations may be viewed as the process
$W^{+}+l^{-}_{L}\to\nu_{L}\to(\nu^{c})_{R}\to l^{+}_{R}+W^{-}$ in which the
intermediate-state neutrino propagates a macroscopic distance [127]. In this
case the intermediate states requires a mass insertion $\nu\to\bar{\nu}$. In
the m331 the respective process we have $l^{-}_{L}+W^{+}\to\nu_{L}\to
l^{+}_{L}+V^{-}$ (no helicity suppressed). Thus, in principle, both mechanism
are distinguishable since the decays of $W^{-}$’s are different from those of
$V^{-}$ and in the latter case there is no helicity suppression. For more
details see Ref. [128].

##  IX The SM non-trivial limit

We know that any model beyond the SM must, in some limit, coincide with the
predictions of that model. This happens in particular with the 3-3-1 models
with arbitrary $\beta$ [129]. Here we use as an example the m331 model. In
this model the masses of the extra particles are proportional to $v_{\chi}$,
the VEV which breaks $SU(3)_{L}\otimes U(1)_{X}\to SU(2)_{L}\otimes U(1)_{Y}$,
if they are fermions or, to $v^{2}_{\chi}$ if they are bosons. On the other
hand, all particles that appear in the SM have masses which are proportional
to a combination of the other VEVs, for instance
$v^{2}_{\eta}+v_{\rho}^{2}+\cdots$ and so on. Hence it is obvious that for a
large $v_{\chi}$, only the degrees of freedom whose masses are not
proportional to $v_{\chi}$, or to $v^{2}_{\chi}$, will be excited at low
energies. For instance, the charged vector masses are given by

$M^{2}_{W}=\frac{g^{2}}{4}v^{2}_{W},\quad
M^{2}_{V}=\frac{g^{2}}{4}(v^{2}_{\rho}+2v^{2}_{s_{2}}+v^{2}_{\chi}),\quad
M^{2}_{U}=\frac{g^{2}}{4}(v^{2}_{\eta}+2v^{2}_{s_{2}}+v^{2}_{\chi})$ (115)

where $v^{2}_{W}=v^{2}_{\eta}+v^{2}_{\rho}+2v^{2}_{s2}$ (neglecting the
$v_{s_{1}}$ in the sextet). All these models have two massive neutral vector
bosons that we will denote $Z_{1}$ and $Z_{2}$ and we will parameterize their
neutral currents as follows [130]:

$\mathcal{L}^{NC}_{331}=-\frac{g_{3L}}{2}\sum_{i}\bar{\psi}_{i}\gamma^{\mu}[(\tilde{g}^{i}_{V}-\gamma_{5}\tilde{g}^{i}_{A})Z_{1\mu}+(f^{i}_{V}-\gamma_{5}f^{i}_{A})Z_{2\mu}]\psi_{i}.$
(116)

Here, we consider only the couplings for leptons; for more details see Ref.
[129]. The couplings $g$’s Eq. (116) are given for leptons by the exact
expressions:

$\displaystyle\tilde{g}^{\nu}_{V}=\tilde{g}^{\nu}_{A}=-\frac{1}{3}N_{1}[1-6A(1+B)-3\bar{v}^{2}_{\rho}+4\bar{v}^{2}_{W}],$
$\displaystyle\tilde{g}^{l}_{V}=-N_{1}(1-\bar{v}^{2}_{\rho}),\;\;\tilde{g}^{l}_{A}=\frac{1}{3}N_{1}[1-6A(1+B)-3\bar{v}^{2}_{W}],$
(117)

where $\bar{v}^{2}_{x}=v^{2}_{x}/v^{2}_{\chi}$ with
$v^{2}_{x}=v^{2}_{\rho},v^{2}_{W}$, and
$v^{2}_{W}=v^{2}_{\eta}+v^{2}_{\rho}+v^{2}_{s_{2}}$, being
$v^{2}_{W}=(246\,\textrm{GeV})^{2}$, and $v_{\eta},v_{\rho},v_{\chi}$ are the
vaccum expectation values for the triplets and $v_{s_{2}}$ for the sextet that
are present in the model. Above we have defined

$\displaystyle
N^{-2}_{1}=(1+4t^{2}_{X})(\bar{v}^{2}_{\rho}-1)^{2}+3\left[2(A(1+B)+\bar{v}^{2}_{\rho}-\frac{4}{3}\bar{v}^{2}_{W}-\frac{1}{3}\right]^{2},$
$\displaystyle
N^{-2}_{2}=(1+4t^{2}_{X})(\bar{v}^{2}_{\rho}-1)^{2}+3\left[2A(1-B)+\bar{v}^{2}_{\rho}-\frac{4}{3}\bar{v}^{2}_{W}-\frac{1}{3}\right]^{2},$
(118)

and $A$ and $B$ are given by

$\displaystyle
3A=1+\bar{v}^{2}_{W}+3\left(1+\bar{v}^{2}_{\rho}\right)t^{2}_{X},$
$\displaystyle
3A^{2}B^{2}=3A^{2}-(1+4t^{2}_{X})\left[(1+\bar{v}^{2}_{\rho})\bar{v}^{2}_{W}-\bar{v}^{4}_{\rho}\right]$
(119)

with $B>0$ and we have defined $t_{X}\equiv\tan\theta_{X}=g_{X}/g_{3L}$.

It was was pointed out in Ref. [129] that, using all the exact expressions for
the $M_{Z_{1}}$ mass in the ratio $M^{2}_{W}/M^{2}_{Z_{1}}c^{2}_{W}=1$ all
neutral current couplings for quarks and leptons $g^{i}_{V,A},i=q,l,\nu$ with
$Z_{1}$ go to the tree level SM values, if we also use the matching condition
$t^{2}_{X}=s^{2}_{W}/(1-4s^{2}_{W})$. For instance, the complicate expressions
in Eq. (117) for neutrinos become $g^{\nu}_{V}=g^{\nu}_{A}=1/2$. Similarly
with the other fermions, the tree level expressions for $g^{f}_{V,A}$ are
obtained and the $f$’s are only functions of $s^{2}_{W}$. For instance,
$f^{\nu}_{V}=f^{\nu}_{A}=f^{l}_{V}=-f^{l}_{A}=-(\sqrt{3}/6)\sqrt{1-4s^{2}_{W}}$.
The same happens with the model with $\beta=-1/\sqrt{3}$ but using
$t^{2}_{X}=s^{2}_{W}/(1-\frac{4}{3}s^{2}_{W})$.

Recently, it was shown that a lower bound on $v_{\chi}$ of 54 GeV can be
obtained using only theoretical aspects of the model. This implies also a
lower bound on the masses of the vector bileptons $M_{U}\geq 24$ GeV,
$M_{V}\geq 78$ GeV and $M_{Z^{\prime}}\geq 78$ GeV [131]. Of course, these are
only theoretical values, phenomenology will say the last word.

A more realistic phenomenology could be done using the exact expressions in
Eq. (117) and similar expressions for $f^{i}_{V,A}$. Then, we may see if more
precise measurements shows how much the $Z_{1}$ could deviates itself from the
predicted couplings of the $Z$ of the SM, including its mass. After all, the
only gauge couplings in the electroweak sector ar $g_{X}$ and $g_{3L}$. The
standard model electroweak mixing angle $s_{W}$ appears only through out a
matching (numerical) condition $t^{2}_{X}=s^{2}_{W}/(1-4s^{2}_{W})$ at a given
energy.

##  X Conclusions

Here we have shown that the parameter space of the 3-3-1 models is much wider
than has usually been considered. In particular, we have stressed that the
3-3-1 models, since they have many parameters, need a more detailed study than
it has been done so far.

We summarize the challenges of the 3-3-1 models as follows:

  1. 1.

Search for exotic quarks $J,J_{i}$ (which are vector-like under the SM
symmetries) at LHC and beyond [132].

  2. 2.

Direct, and indirect, search of the vector bosons $U^{\pm\pm}_{\mu}$ (models
with $\beta=-\sqrt{3}$) or $\mathcal{Z}^{0}_{\mu}\not=\mathcal{Z}^{0*}_{\mu}$
(models with $\beta=-1/\sqrt{3}$).

  3. 3.

Are bounds on the mass of $U^{\pm\pm}_{\mu}$ using cLFV compatible with the
discovery of this vectors at LHC? The answer seems to be yes [133].

  4. 4.

Are right-handed neutrinos needed in the m331 model for generating neutrino
masses? or, just as dark matter [64]? The model does not have a Majoron-like
pseudoscalar, which that is part of the sextet, since $L$ is explicitly
violated in some interactions in the scalar potential. Hence the mechanism of
Ref. [134] could be realized in the model.

  5. 5.

In the model with $\beta=-\sqrt{3}$ and heavy charged leptons $E^{\pm}_{l}$
(which are vector-like under the SM symmetries) the production of four muons
induced by the $Z^{\prime}$ in pp collisions are possible [135]. In the model
with $\beta=-1/\sqrt{3}$ there are neutral vector-like leptons $N_{l}$, and
this sort of process is also produced.

  6. 6.

It is also important to continue the study of the effects of the charged
scalars present in this models at the LHC [136, 137] and other experiments.
The m331 model has a doubly charged scalar field and, as all models with this
sort of scalar, the process $S^{++}\to l^{+}l^{+}$ has signatures in $pp$ and
lepton colliders, in particular in linear colliders [138] and also effects in
the $B$-meson decays [139].

  7. 7.

Study the possibility of these model under the constraints of long-standing
discrepancy of the $A^{b}_{FB}$ asymmetry with the standard model prediction
[140]. For instance, the 3-3-1 models have many scalar multiplets and it is
possible to accomodate the forward-backward anomaly in $Z\to\bar{b}b$ , as it
has been done in the model with two and three Higgs doublets [141].

  8. 8.

Is it possible to have solution for the Cabibbo angle anomaly [142, 143] in
these models? and also, is it possible to have combined solutions to the
$A^{b}_{FB}$ asymmetry, and some anomalous $B$ meson decays as in Ref. [144]
where vector-like quarks were introduced? In fact, models with
$\beta=-1/\sqrt{3}$ allow the mixing $\overline{b_{L}(s)_{L}}\eta^{0*}D_{R}$
where $D_{R}$ is a vector-like quark with $Q=-1/3$ singlet under the SM group.

  9. 9.

The model with heavy leptons [145] deserves a more detailed study. As we said
before, the lightest charged heavy leptons is almost stable since it decays
into leptons only through charged scalars. In this model it is possible that
the Higgs boson decay into long lived particles [146, 147]. Is it possible the
existence of heavy leptonic atoms, $E^{+}e^{-}$?

  10. 10.

The phenomenology of the $Z^{\prime}$ at the energies of the FCC-ee [148]
should be considered, either in flavor physics [149], or in forward-backward
and left-right assymetries in lepton-lepton colliders [150, 152, 151]. Would
this $Z^{\prime}$ may be used to solve some $B$-mesons anomalies [153]. We
must bear in mind that doing the phenomenology of the $Z^{\prime}$ independent
of $\beta$ [154], although it is an interesting step, does not consider, by
principle, the contribution of the scalar fields whose representation content
depends on the value of $\beta$.

  11. 11.

It is interesting to be able to distinguish physical fields in different
representation, as in the scalar sector [155].

  12. 12.

Reliable values of quark and lepton masses and mixing matrix elements at a
given energy scale are important for phenomenological studies at those
energies. For instance, we can recognize the following energy scales (in GeV):
$M_{Z}\sim 91.2$, the LHC $\sim 14\times 10^{3}$, the type I seesaw scale
$\Lambda_{N}\sim 10^{9}$, the Pecci-Quinn $\Lambda_{PQ}\sim 10^{12}$ and the
would-be GUT scale $\Lambda_{GUT}\sim 10^{16}$ [156, 157].

  13. 13.

Are the new hadronic contributions to the $(g-2)_{\mu}$ anomaly important at
least in some regions of the parameter space? The heavy quarks appearing in
all 3-3-1 models could contribute through their vacuum polarisation to this
parameter [158]. Also, non-perturbative effects could be important. Taking
into account only electroweak contributions it may be not enough to explain
the muon $g-2$ anomaly, even if these contributions give a positive result.

  14. 14.

Revisited the decays $b\to sl^{+}l^{-},\;l=e,\mu$ [159] and $c\to
u\nu\bar{\nu}$ [160].

  15. 15.

Finally, we would like to call the attention to the recent CDF II
Collaboration result: the mass of the $W^{+}$ does not agree with the SM
calculation within almost 7$\sigma$ [161]. In the m331 model when all neutral
components of the sextet gain a non zero VEV, the mass matrix of the singly
charged vector bosons are given, in the $(W^{+}\,V^{+})$ basis [162]. Is this
mixing enough to explain the $W$ mass? after all, the scalar triplet extension
of the SM can solve the issue [163]. Or, the model with $\beta=-\sqrt{3}$ and
heavy vector-like (under the SM symmetries) leptons may also solve th $W$ mass
anomaly? [164].

  16. 16.

Production of leptons $l=e,\mu,\tau$ in proton-proton scattering:

$\displaystyle pp\to
l^{-}_{R}l^{+}_{L}+(\nu_{L})^{c}\nu_{L}(\not{E})+jets,\quad(a)$ $\displaystyle
pp\to l^{-}_{R}l^{+}_{L}l^{+}_{R}+\nu_{L}(\not{E})+jets,\quad(b)$
$\displaystyle pp\to
l^{+}_{L}l^{-}_{R}+(\nu_{L})^{c}\nu_{L}(\not{E})+jets,\quad(c)$ $\displaystyle
pp\to l^{-}l^{+}l^{-}l^{+}+jets,\quad(d)$ (120)

may also occur because the resonances involving the exotic quarks are long-
lived, since they are protected by a $\mathbb{Z}_{2}$ symmetry which is broken
only by scalar interactions. For instance, the process (120)(d) occurs through
the subprocess $u\to U^{--}j\to U^{--}U^{++}u\to l^{-}l^{-}l^{+}l^{+}u$, and
so on.

All issues above must be clarify if the model is taken as a well motivated
extension of the SM.

Finally some general remarks. There is one aspect that distinguishes the m331
from the other 3-3-1 models: there are necessarily neutral interactions which
violate flavor via neutral scalars. This is because, as we said before, it is
not possible to assume the mass matrix of the quarks and charged leptons in a
diagonal form from the very start. For this reason, the PMNS matrix is not
determined only by the matrix relating the neutrinos symmetry eigenstates with
the mass eigenstates. In other words, it is not possible to consider neither
$V^{l}_{L}=\mathbf{1}$ in the PMNS matrix, nor $V^{l}_{R}=\mathbf{1}$ in the
interaction with the vector $V^{+}$ in Eq. (101), hence all vertices
necessarily involve the matrix $V_{LR}$ defined in Eq. (102) nor
$\mathbf{\bar{V}_{LR}}$ in Eq. (104). The same happens with the CKM matrix.
This certainly has consequences for other processes. For instance, an analysis
of the possibility of this model to explain, say the muon $g-2$ factor, has to
use the interactions with $V^{+}$ given in Eq. (101). Similarly, with the
interactions with the doubly charged vector, $U^{++}$ in Eq. (103), in which
the vertices include the matrices
$\mathbf{A}=\mathbf{\bar{V}}_{RL}-\mathbf{\bar{V}^{T}}_{RL}$ and
$\mathbf{S}=\mathbf{\bar{V}}_{RL}+\mathbf{\bar{V}^{T}}_{RL}$ with
$\mathbf{\bar{V}}_{RL}$ defined in Eq. (104).

Another point that we must emphasize is the following. The scalar
contributions may be important and cannot be negelected a priori [86, 165].
Last but not least, if a model has new hadronic states, their contributions to
the $(g-2)_{\mu}$ may be important. Moreover, the scalar cotributions cannot
be neglected in all the paraeter space. Even more important is to take into
account the hadronic contributions to the muon factor due to the new quarks in
the models. Hence, without taking into account all the possible contributions
to the $g-2$ we cannot say if a model is dead or alive. From the experimental
point of view, it is not enough to just look for the existence of a certain
type of particle, if it is found, it must be distinguished to which model it
belongs.

We stress once more, a realistic study of 3-3-1 models must take into accountn
the values of the entries of the unitary matrices and also on the mass of the
extra particles in 3-3-1 models. Otherwise, it is like thinking that the SM
could have been tested and confirmed without taking into account the CKM
matrix.

Similar features occurs in model with the gauge symmetries $SU(3)_{C}\otimes
SU(4)_{L}\otimes U(1)_{N}$ [166, 167, 168, 169], or in models with
$SU(3)_{C}\otimes SU(5)_{L}\otimes U(1)_{N}$ [170, 41, 171]. In fact, if
right-handed neutrino do really exist in nature the more interesting model
will be those with 3-4-1 symmetry [166] in which the known leptons and the
right-handed neutrinos are in the fundamental representation of $SU(4)$.

After more than a decade of finding no evidence for new physics at the LHC, we
may wonder if we are studying the various theoretical proposals in the right
way. Both experimental and phenomenological analysis usually are performed in
a model-independent way or with a simplified version of a given model.

From the experimental point of view, the importance of the reinterpretation of
the experimental searches and measurements at the LHC in terms of models for
new physics has been considered recently. In fact, there is no doubt that only
a small subset of the possible theories and parameter combinations have been
tested both, theoretically and experimentally [172].

For instance, the importance of reinterpreting of experimental data can be
appreciated in the case of neutral heavy leptons (HNL), $N$. ATLAS has
considered the trilepton produced by a single HNL which production is possible
via mixing with an electron or muon neutrino produced by an on-shell $W$ boson
decay. The search sets constraints on the mass of $N$ in the range 4.5–50 GeV
and a mixing $|U_{\mu}|^{2}$ of the order of $10^{-6}$ [173]. However,
reinterpreting the ATLAS data by considering two $N$’s mixing with the three
neutrinos and with nearly degenerate masses leads to an exclusion limit on the
total mixing angle which can be up to 3 orders of magnitude weaker than the
limits reported by ATLAS [174]. The model with two heavy neutral leptons is
not just an unnecessary complication of the model with a single $N$: they are
sufficient to explain neutrino masses and oscillations as well as the origin
of the matter-antimatter asymmetry, which are some of the motivation for that
sort of leptons. Besides, a third HNL can play the role of dark matter. Hence,
a model with only a single $N$ and which does not have these properties,
cannot be considered a realistic model.

Another example that the reinterpretation of data seems to be necessary is the
following. For years the MiniBooNE had found more electron neutrinos than has
been expected at low energies. We recall that MinoBooNE, as all experiments
detecting neutrinos, in fact, detect the corresponding charged lepton. Then,
it is worth noting that this result is consistent in energy and magnitude with
the excess of $\bar{\nu}_{e}$ reported by the LSND [175, 176, 177], and the
significance of the combined LSND and MiniBooNE excesses is $6.0\sigma$ [178].
Moreover, these experiments used different detectors, Liquid Scintillator
Neutrino Detector in the LSND, and liquid argon time projection chamber in the
MiniBooNE. However, MiniBooNE does not distinghuishes electron from photons,
and the latter ones could indicate another type of particle. Recently,
MicroBooNE [179, 180] had eliminated the possibility of extra events involving
photons. Then, we can speculate that the difference in the detectors used can
not explain that difference. Would it be an effect of new physics the
explanation of the excess observed in the two experiments? For instance, in
the context of the m331 model, it is possible that MiniBooNE and LSND confused
neutrinos with antineutrinos since in this model the latter would also
accompany electrons, or vice versa. See Eqs. (113) and (114).

We see that is not an exaggeration to say that “the field of particle physics
is at the crossroads” [181]. In fact, it is increasingly evident that we do
not know how to get to observe physics beyond the SM, although we know that it
must exist because of the issues mentioned in the introduction. On one hand,
while we know there is physics beyond the SM, we have no idea where, and how,
it will be manifest itself. On the other hand, we are certain that this model
lost its predictive capacity. Thus, we may wonder ourselves, what are the
hints of the range of couplings and masses to look for that have not been
explored so far? We do not believe that the SM will remain valid up to the
Planck scale. If this were the case, all the issues that the model does not
explain [182] would a priori remain unsolved.

For all that has been discussed above, in the context of the m331 model (but
which may be valid for any other sort of models) we can say with confidence
that until now neither experiments nor phenomenological studies are in a
position to confirm or refute any model. Of course, both the model-dependent
and the model-independent approaches are necessary and must be continued.
Direct search for new particles with as little model bias as possible [183]
seems an illusory strategy. We recall the neutral current discovery. In this
case the role of the theoreticians was crucial [184].

It is necessary that the ceteris paribus condition must be left clear from the
beginning of a work, be it theoretical or experimental.

###### Acknowledgements.

The author would like to thanks FAPESP under the funding Grant No.
2014/19164-6.

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