# Evading the Landau pole in the minimal 3-3-1 model with leptoquarks

A. Doffa, C. A. de S. Piresb a Universidade Tecnológica Federal do Paraná -
UTFPR - DAFIS, 84017-220, Ponta Grossa, PR, Brazil,
b Departamento de Física, Universidade Federal da Paraíba, Caixa Postal 5008,
58051-970, João Pessoa, PB, Brazil,

###### Abstract

In its original version, the minimal 3-3-1 model possess a Landau-pole around
2-6 TeV scale. Current LHC bound on $Z^{\prime}$ implies that the
$SU(3)_{L}\times U(1)_{X}$ symmetry must break spontaneously around 4 TeV
which means that the model may lose its perturbative character even before
symmetry breaking. This is a disaster for the model. Few attention has been
devoted to this problem. Here we investigate the efficiency of scalar
leptoquarks in evading or shifting the Landau pole to a harmless energy scale.

##  I Introduction

The gauge models for the electroweak and strong forces based in the gauge
group $SU(3)_{C}\times SU(3)_{L}\times U(1)_{X}$ (3-3-1 )Singer et al. (1980);
Pisano and Pleitez (1992); Frampton (1992); Foot et al. (1994); Montero et al.
(1993) accommodate two versions determined by the parameter $\beta$ that
appears in the linear combination of the diagonal generators that define the
electric charge operator,

$\frac{Q}{e}=\frac{1}{2}(\lambda_{3}+\beta\lambda_{8})+N.$ (1)

The values $\beta$ can take are $-\frac{1}{\sqrt{3}}$ (case A) and $-\sqrt{3}$
(case B). Each case leads to different models concerning theoretical and
phenomenological aspects111It is interesting to stress that both models
explain family replicationNg (1994) and electric charge quantizationde Sousa
Pires and Ravinez (1998). Here we address a problem that only affect
exclusively case B that is a Landau-like pole that, as showed in previous
studies, manifest already at TeV regimeNg (1994); Dias et al. (2005). The most
popular version of 3-3-1 models related to the case B is the minimal 3-3-1
model.

The landau pole arises in the minimal 3-3-1 model because the couplings
$g_{X}$, associated to $U(1)_{X}$, and $g_{L}$, associated to $SU(3)_{L}$, are
related to the Weinberg angle $\theta_{W}$ in the following wayNg (1994); Dias
et al. (2005)

$\frac{g_{X}}{g_{L}}\sim\frac{\sin^{2}\theta_{W}}{1-4\sin^{2}\theta_{W}}.$ (2)

This dangerous relation generates a Landau-polo,
$g_{X}(\Lambda)\rightarrow\infty$, when
$\sin^{2}\theta_{W}(\Lambda)\rightarrow 1/4$. It was showed in Refs. Ng
(1994); Dias et al. (2005) that this happens already at TeV scale more
precisely in the range $\Lambda\approx 2-6$ TeV depending on the particle
content that contribute to the running of the $\sin^{2}\theta_{W}$. Current
bound on $Z^{\prime}$ demands that the $SU(3)_{L}\times U(1)_{X}$ symmetry
breaks spontaneously around 4 TeVCao and Zhang (2016). This means that, in its
original version, the minimal 3-3-1 model practically lost its power of making
prediction.

It is important to stress that the minimal 3-3-1 model does not accommodate
neutrino massesPires (2014), neither has a candidate for dark matter and faces
difficult in accommodating the recent result of $g-2$ of the muonde Jesus et
al. (2020) and the B anomaliesCárcamo Hernández et al. (2022). In view of all
this the model need to be modified in order to explains such set of points.
Thus, it is natural to look for extensions of the model that provides an
answer to such points and evade or shift the Landau pole to a harmless energy
scale. Following this line of thought, it was showed in Dias (2005) that the
addition of octet of leptons may evade the Landau pole or shift it to a
harmless energy scale.

Here we follow this line of thought and address this issue with the addition
of scalar leptoquarksPati and Salam (1973) to the minimal 3-3-1 model.
Leptoquarks are very well motivated proposal of new physics that may manifest
at TeV scale. It also has an important role in flavor physicsDoršner et al.
(2016). Here, we perform an exhaustive investigation of the impact of scalar
leptoquarks on the landau pole of the model. We show that leptoquarks are as
efficient as octet of leptons to evade or shift the non-pertubative regime of
the minimal 3-3-1 model to a harmless energy scale.

##  II Revisiting the problem

###  II.1 The particle content of the model

In the minimal 3-3-1 model leptons are arranged in triplet representation of
$SU(3)_{L}$,

$f_{aL}=\begin{pmatrix}\nu_{a_{L}}\\\ \ell_{a_{L}}\\\ \ell^{c}_{a_{R}}\\\
\end{pmatrix}\sim(1,3,0),$ (3)

with $a=1,2,3$ representing the three generations of leptons.

In the Hadronic sector, anomaly cancellation requires that one family
transforms differently from the other two. This fact allows three possibility
of arrangements for the quark families. Here we chose the third generation
coming in triplet and the other two coming in anti-triplet representation of
$SU(3)_{L}$,

$\displaystyle Q_{i_{L}}=\left(\begin{array}[]{c}d_{i}\\\ -u_{i}\\\
d^{\prime}_{i}\end{array}\right)_{L}\sim(3\,,\,3^{*}\,,\,-1/3)\,,u_{iR}\,\sim(3,1,2/3),\,\,\,$
(7)
$\displaystyle\,\,d_{iR}\,\sim(3,1,-1/3)\,,\,\,\,\,d^{\prime}_{iR}\,\sim(3,1,-4/3),$
$\displaystyle Q_{3L}=\left(\begin{array}[]{c}u_{3}\\\ d_{3}\\\
u^{\prime}_{3}\end{array}\right)_{L}\sim(3\,,\,3\,,\,2/3),u_{3R}\,\sim(3,1,2/3),$
(11)
$\displaystyle\,\,d_{3R}\,\sim(3,1,-1/3)\,,\,u^{\prime}_{3R}\,\sim(3,1,5/3),$
(12)

where $i=1,2$. The primed quarks are heavy quarks.

The gauge sector is composed by nine gauge bosons where four of them are the
standard ones $A\,\,\,,\,\,\,W^{\pm}\,\,\,,\,\,\,Z^{0}$ and the other five are
the typical 3-3-1 gauge bosons
$U^{\pm\pm}\,\,\,,\,\,W^{\prime\pm}\,\,,\,\,Z^{\prime}$.

The original scalar sector of the model involves three triplets and one sextet
of scalars, namely

$\displaystyle\eta=\left(\begin{array}[]{c}\eta^{0}\\\ \eta^{-}_{1}\\\
\eta^{+}_{2}\end{array}\right)\sim\,\,(1\,,\,3\,,\,0),\,\rho=\left(\begin{array}[]{c}\rho^{+}\\\
\rho^{0}\\\
\rho^{++}\end{array}\right)\sim\,(1\,,\,3\,,\,1)\,\,,\,\chi=\left(\begin{array}[]{c}\chi^{-}\\\
\chi^{--}\\\ \chi^{0}\end{array}\right)\sim(1\,,\,3\,,\,-1).\,$ (22)
$\displaystyle\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,S=\left(\begin{array}[]{ccc}\,\Delta^{0}&\Delta^{-}&\Phi^{+}\\\
\newline \\\ \Delta^{-}&\,\Delta^{--}&\Phi^{0}\\\ \newline \\\
\Phi^{+}&\Phi^{0}&\,H_{2}^{++}\end{array}\right)\sim(1\,,\,6\,,\,0).$ (28)

When $\chi^{0}$ develops vacuum expectation value(VEV) different from zero,
$v_{\chi}$, the $SU(3)_{L}\times U(1)_{X}$ symmetry breaks to the
$SU(2)_{L}\times U(1)_{Y}$ one. When the others neutral scalars develop VEV
different from zero, the standard symmetry is spontaneously broken to the
electromagnetic one. Such scalar content generate masses for all massive
particles of the model, except neutrinos.

After symmetry breaking $Z^{\prime}$ acquires the following mass expressionNg
(1994)

$M^{2}_{Z^{\prime}}\approx\frac{g^{2}\cos^{2}\theta_{W}}{3(1-4\sin^{2}\theta_{W})}v^{2}_{\chi}.$
(29)

The other terms are proportional to the electroweak scale and can be
neglected. Current collider bound on $Z^{\prime}$ imposes $M_{Z^{\prime}}>5$
TeV which implies $v_{\chi}>4.3$ TeVCao and Zhang (2016). As we show below,
the model has a Landau pole that will manifest at some energy scale $\Lambda$.
If $\Lambda<4.3$ TeV than the model is not predictive at all.

###  II.2 Landau-Pole

In order to study the behaviour of the Weinberg angle with energy we, first,
need to know how gauge couplings run with energy. In general the running of
gauge couplings at one-loop is dictated by the relation

$\frac{1}{\alpha(\Lambda)_{i}}=\frac{1}{\alpha(\mu)_{i}}+\frac{1}{2\pi}b_{i}\log(\frac{\mu}{\Lambda}),$
(30)

where $\alpha_{i}=\frac{g^{2}_{i}}{4\pi}$. The renormalization coefficients
for a general $SU(N)$ gauge group are given by

$b_{i}=\frac{2}{3}\sum_{fermions}Tr(F)_{i}+\frac{1}{3}\sum_{scalars}Tr(S)_{i}-\frac{11}{3}C_{2}(G)_{i}.$
(31)

For $SU(N)$ we have $T_{R}(F,S)=1/2$ and $C_{2}(G)=N$. For $U(1)$ we have
$C_{2}(G)=0$. We also use $\sum Tr(F,S)=\sum y^{2}$ for $U(1)_{y}$ with
$y=\frac{Y}{2}$, for the standard model case, and $y=X$ for the 3-3-1 case.

The running of $\sin^{2}\theta_{W}(\Lambda)$ for $\Lambda<\mu_{331}$, where
$\mu_{331}=\langle\chi^{0}\rangle=\frac{v_{\chi}}{\sqrt{2}}$, is given by

$\sin^{2}\theta_{W}(\Lambda)=\frac{1}{1+\frac{\alpha_{2}(\Lambda)}{\alpha_{1}(\Lambda)}}$
(32)

Figure 1:  In this figure we show the running of the electroweak mixing angle
, given by Eq.(32), Fig.(1a), and Eq.(35) that leads to Figs.(1b-1c). In the
figure (1d) we show the $\alpha_{X}(\Lambda)$ running to $\mu_{331}$ = 4TeV
assuming all particle content of the model. The contextualization of the
curves behavior are described in the text.

For we have an idea about the energy scale the Landau pole arises, let us
consider the simplest case in which the scalar sector of the model is composed
only with three triplets of scalars. In this case, when of the
$SU(3)_{L}\times U(1)_{N}$ symmetry breaking, the scalar sector decouples into
the effective two Higgs doublet model (THDM) plus a set of scalars with mass
belonging to the 3-3-1 scale. We, then, consider only the effective THDM which
means the triplet $\chi$ and the singlets $\rho^{++}$ and $\eta_{2}^{+}$ and
the exotic quarks are not active degrees of freedom. In this case we get

$b_{1}=\frac{20}{9}N_{F}+\frac{1}{6}N_{S}=7\,\,\,\,\,\mbox{and}\,\,\,b_{2}=\frac{2}{3}N_{F}+\frac{1}{6}N_{S}-\frac{22}{3}=-3,$
(33)

where $N_{F}$ is the number of fermion families, and $N_{S}=2$, is the number
of scalar doublets. Our results are displayed in FIG. 1.

In this simple case the running of $\sin^{2}\theta_{W}(\Lambda)$ for
$\Lambda<\mu_{331}$ is displayed in Fig.(1a), see dot-dashed line in green,
where the landau pole corresponds to $\Lambda\sim 3.5TeV$. We run the Eqs.
(30) and (32) considering the following values at the $M_{Z}$ energy scale
$\sin^{2}\theta_{W}(M_{Z})=0.2311$ , $\alpha_{1}(M_{Z})=1/128$ ,
$M_{Z}=91.188GeV$ and
$\alpha_{1}(M_{Z})=\alpha_{2}(M_{Z})\tan^{2}\theta_{W}(M_{Z})$.

In the Fig. (1a), dot-dashed line in black, we considered the scalar sextet
S222 This scalar decouples in one triplet with $Y=-2$, one iso-doublet with
$Y=1$ and one doubly charged scalar with $Y=2$, which implies an additional
iso-doublet and a non-hermitian triplet, so that $N_{S}=3$ and the addition
$N_{T}=1$ that leads to

$b_{1}=\frac{20}{9}N_{F}+\frac{1}{6}N_{S}+N_{T}=\frac{49}{6}\,\,\,\,\,\mbox{and}\,\,\,b_{2}=\frac{2}{3}N_{F}+\frac{1}{6}N_{S}+\frac{2}{3}N_{T}-\frac{22}{3}=-\frac{13}{6}.$
(34)

In this case the pole was pushed up a little bit, to the value $\Lambda\sim
6TeV$ which means that the sextet of scalars is sufficient to recover the
pertubative regime of the model concerning current bounds.

Let us now consider that all particle content of the model, the three triples,
the sextet of scalars and the exotic quarks are active degrees of freedom
which means that their masses are below $\mu_{331}$.

To the case of energies above the scale $\mu_{331}$, the running of
$\sin^{2}\theta_{W}(\Lambda)$ for $\Lambda>\mu_{331}$ is

$\sin^{2}\theta_{W}(\Lambda)=\frac{1}{4(1+\frac{\alpha_{L}(\Lambda)}{4\alpha_{X}(\Lambda)})}$
(35)

where now $\alpha_{X}(\Lambda)$ running equation we will be given by

$\frac{1}{\alpha_{X}(\Lambda)}=\left(1-4\sin^{2}\theta_{W}(M_{Z})\right)\frac{1}{\alpha(M_{Z})}+\frac{1}{2\pi}\left(b_{1}-3b_{2}\right)\log(\frac{M_{Z}}{\mu_{331}})+\frac{1}{2\pi}b_{X}\log(\frac{\mu_{331}}{\Lambda}),$
(36)

and $\alpha_{L}(\Lambda=\mu_{331})=\alpha_{2}(\mu_{331})$. In the equation
above $b_{X}$ is the renormalization coefficient for $U(1)_{X}$. When the
degrees of freedom above $\mu_{331}$ are taken into account we have
$b_{X}=20+N_{\rho}+N_{\chi}$. Furthermore, when exotic quarks are omitted, we
can introduce the notation ${b\\!\\!\\!/}_{X}=6+N_{\rho}+N_{\chi}$, where now
${b\\!\\!\\!/}_{X}=8$.

Our results are showed in Fig.(1b). The dot-dashed line in red corresponds to
the case where $\mu_{331}=4TeV$, while the blue line to $\mu_{331}=5TeV$ and
finally the green line to $\mu_{331}=6TeV$. In this figure we considered all
particle content of the minimal 3-3-1 model. This case leads to $b_{X}=22$ and
presents a Landau pole around (2-2.5) TeV which is below 4.3 TeV. This meas
that in this case the model is no predictive at all.

In Fig(1c) contributions of the exotic quarks are omitted ( which gives
${b\\!\\!\\!/}_{X}=8$). We also assumed the same choices for $\mu_{331}$ as in
Fig.(1b). This corresponds to a more restrictive case for $\mu_{331}>4TeV$,
which implies the existence of a Landau pole to $\Lambda<2TeV$. We see here
that the Landau Pole is sensitive to the fact if the exotic quarks are active
degrees of freedom or do not and to the $\mu_{331}$ energy scale.

Finally, in Fig(1d) we present the behavior of the running of
$\alpha_{X}(\Lambda)$ given by Eq.(36) to $\mu_{331}$ = 4TeV assuming all
particle content of the model. The figure let clear the position of the Landau
pole at $\sim 2.5$ TeV and indicates the loss of the perturbative character of
the model already in $\sim 2$ TeV which is very close to the Landau pole. In
what follow we just discuss the Landau pole.

We made here a short review of the problem concerning the Landau pole that
arises in the minimal 3-3-1 model. The results we obtained is in agreement
with the previous one. Conjugating this results with the current bound on the
scale of the 3-3-1 symmetry breaking , given by
$\mu_{331}=\langle\chi^{0}\rangle=\frac{v_{\chi}}{\sqrt{2}}$, which is around
$4.3$ TeV, we conclude that the perturbative regime of the minimal 3-3-1 model
in its original form depends strongly if the exotic quarks are active degrees
of freedom or do not. Even in this case the model is predictive up to the
energy scale of 6 TeV, only, which is very close of $4.3$ TeV.

In what follow we make an exhaustive investigation of the unique proposal
existent in the literature that bring the model to the game by evading the
Landau pole with a particular extension of the particle content of the model.

###  II.3  Evading the pole with octet of leptons

It was proposed in Dias (2005) that we could evade the pole in adding three
octet of leptons to the minimal 3-3-1 model content. The octet is composed by
the following leptons

$\displaystyle\Xi=\left(\begin{array}[]{ccc}\,\frac{1}{\sqrt{2}}t^{0}+\frac{1}{\sqrt{6}}\lambda^{0}&t^{+}&\delta^{-}\\\
\newline \\\
t^{-}&\,-\frac{1}{\sqrt{2}}t^{0}+\frac{1}{\sqrt{6}}\lambda^{0}&\delta^{--}\\\
\newline \\\
\xi^{++}&\xi^{++}&\,-\frac{2}{\sqrt{6}}\lambda^{0}\end{array}\right)\sim(1\,,\,8\,,\,0).$
(42)

Figure 2:  In this figure we show the running of the electroweak mixing angle
, given by Eq.(32), Figs.(2a-2b), and Eq.(35) that leads to Figs.(2c-2d). The
contextualization of the curves behavior are described in the text.

Here we calculate the contribution of the octet of leptons to the running of
$\sin^{2}\theta_{W}(\Lambda)$ given by Eq. (32), that correspond to
$\Lambda<\mu_{331}$, for the cases when we add one, two or three octets of
leptons. In FIG. 2 we show our results. In the Fig. (2a) we have the running
of the minimal 3-3-1 model (dot-dashed line in black) and, the dot-dashed
light-blue curve, we have the running of the minimal 3-3-1 model+1-octet of
leptons(1[8]). Observe that the addition of 1-octet of leptons is sufficient
to shift the landau pole to energies above 50 TeV. In Fig(2b) we have the
running for the cases of two and three octets of leptons.

When the degrees of freedom above $\mu_{331}$ are considered, which means all
particle content of the model is took into account, the running of
$\sin^{2}\theta_{W}(\Lambda)$ is given by Eq. (35). The results for this case
is displayed in Fig(2c) where dot-dashed line in blue corresponds to tree
octets and $\mu_{331}=4TeV$. As we can see in this case the landau pole is
shifted to $\Lambda>O(50)TeV$.

For sake of completeness, in Fig.(2d) we present the running of the Weinberg
angle for three values of $\mu_{331}$ when the contributions of the exotic
quarks are omitted which means ${b\\!\\!\\!/}_{X}=8$, where only in this plot
we consider the addition of one octet. We see that for $\mu_{331}>4$ TeV we
have the absence of the Landau pole just for one octet of leptons. In general
the higher $\mu_{331}$ is, the harmless the pole get. So we conclude that
octet of leptons are efficient in circumventing the landau pole. Next we
analyse other possibilities potentially interesting that evade the landau
pole.

##  III Evading the landau pole with leptoquarks

In this section we consider the contribution of leptoquarks on the running of
the Weinberg angle $\sin^{2}\theta_{W}(\Lambda)$. Leptoquarks are very well
motivated form of new physic that is expect to manifest at TeV scale and then
engenders an interesting flavor physics scenarioDoršner et al. (2016), give
robust contributions to the $g-2$ of the muon and may generate neutrino masses
at 1-loopBabu et al. (2021); Parashar et al. (2022). Moreover, leptoquarks may
be probed at LHCKramer et al. (2005).

Due to the the fact that the Yukawa interactions in 3-3-1 models discriminate
familyNg (1994); Oliveira and S. Pires (2022), i.e., one family of quarks must
transform differently from the other two, then we are going to have a
proliferation of leptoquark multiplets. To see this, observe that from the
quark and lepton content of the minimal 3-3-1 model we can have scalar
leptoquarks in the following representations,

$\displaystyle\bar{L}^{C}_{a_{L}}Q_{3_{L}}\sim(1\,,\,3\,,\,0)\times(3\,,\,3\,,\,2/3)\sim(3\,,\,3^{*}\oplus
6\,,\,2/3),$
$\displaystyle\bar{L}_{a_{L}}Q_{3_{L}}\sim(1\,,\,3^{*}\,,\,0)\times(3\,,\,3\,,\,2/3)\sim(3\,,\,1\oplus
8\,,\,2/3),$
$\displaystyle\bar{L}^{C}_{a_{L}}Q_{i_{L}}\sim(1\,,\,3,0)\times(3\,,\,3^{*}\,,\,-1/3)\sim(3\,,\,1\oplus
8\,,\,-1/3),$
$\displaystyle\bar{L}_{a_{L}}Q_{i_{L}}\sim(1\,,\,3^{*},0)\times(3\,,\,3^{*}\,,\,-1/3)\sim(3\,,\,3\oplus
6^{*}\,,\,-1/3),$
$\displaystyle\bar{L}_{a_{L}}d_{a_{R}}\sim(1\,,\,3^{*},0)\times(3\,,\,1\,,\,-1/3)\sim(3\,,\,3^{*}_{d}\,,\,-1/3),$
$\displaystyle\bar{L}_{a_{L}}u_{a_{R}}\sim(1\,,\,3^{*},0)\times(3\,,\,1\,,\,2/3)\sim(3\,,\,3^{*}_{u}\,,\,2/3)$
(43)

There are also the singlet leptoquarks that we do not consider in this work.
The leptoquarks we are interested are these ones

$\displaystyle\phi^{8}_{a}\sim(3\,,\,8\,,\,2/3),\,\,\,\,\,\,\,\,\,\Phi^{8}_{a}\sim(3\,,\,8\,,\,-1/3),$
$\displaystyle\phi^{6}_{a}\sim(3\,,\,6\,,\,2/3),\,\,\,\,\,\,\,\,\,\,\Phi^{6}_{a}\sim(3\,,\,6^{*}\,,\,-1/3),$
$\displaystyle\phi^{3}_{a}\sim(3\,,\
3^{*}\,,\,2/3),\,\,\,\,\,\,\,\,\,\,\Phi^{3}_{a}\sim(3\,,3\,,\,-1/3),$ (44)

The indice $a$ refer to color. After symmetry breaking, these multiplet
decompose as

$\displaystyle[8]_{X=2/3}=[3]_{Y=4/3}+[2]_{Y=-5/3}+[2^{\prime}]_{Y=13/6}+[1]_{Y=4/3}$
$\displaystyle[8]_{X=-1/3}=[3]_{Y=-2/3}+[2]_{Y=7/3}+[2^{\prime}]_{Y=-11/3}+[1]_{Y=-2/3}$
$\displaystyle[6]_{X=2/3}=[3]_{Y=-2/3}+[2]_{Y=7/3}+[1]_{Y=16/3}$
$\displaystyle[6]_{X=-1/3}=[3]_{Y=-4/3}+[2]_{Y=5/3}+[1]_{Y=14/3}$
$\displaystyle[3_{d}]_{X=2/3}=[2]_{Y=-2/3}+[1]_{Y=7/3}$
$\displaystyle[3_{u}]_{X=-1/3}=[2]_{Y=-5/3}+[1]_{Y=5/3},$ (45)

where $Y$ refers to the hypercharges of the leptoquarks .

Similarly to the previous cases discussed above, here we calculated the
contributions of these leptoquarks to the running of
$\sin^{2}\theta_{W}(\Lambda)$ given by Eqs.(32) as well by Eq.(35) always
having in mind economical scenarios. Our results are displayed in FIG. 3.

Figure 3:  In this figure we show the running of the electroweak mixing angle
, given by Eq.(32), Figs.(3a-3b), and Eq.(35) that leads to Figs.(3c-3d). The
contextualization of the curves behavior are described in the text.

In the Fig. (3a), in the dot-dashed light-blue curve, we assumed the addition
of one octet of leptoquarks ($\phi^{8}_{a}$) to the minimal 3-3-1 model. In
this case the Landau pole is shifted to $\Lambda\sim 50TeV$, while the case
for two and three octet of leptoquarks is depicted in the Fig(3b) where the
dot-dashed orange line correspond to two octets of leptoquarks
($\phi^{8}_{a},\Phi^{8}_{a}$) and the magenta to tree octets, and again the
Landau pole is avoided at $\Lambda<\mu_{331}$. Observe that octet of
leptoquarks have the same effect than the octet of leptons discussed above and
presented in Fig.2a and Fig.2b.

However, when we consider all degrees of freedom above $\mu_{331}$, i.e, when
we consider that all particle content of the model are active, in this case
the running of $\sin^{2}\theta_{W}(\Lambda)$ is given by Eq.(35) and the
results are presented in Fig(3c) with the dot-dashed line in red corresponding
again to the minimal 3-3-1 model at $\mu_{331}=4TeV$ with $b_{X}=22$. In this
case the addition of one octet of leptoquarks( dot-dashed light-blue curve) is
sufficient to push the Landau pole to values above 5 TeV which is a harmless
scale according to current bounds. The other cases, as the addition of one
triplet or one sextet to the octet of leptoquarks
($\phi^{8}_{a},\phi^{(3,6)}_{a}$), are considered(dashed orange and green
curves) and the Landau pole is shifted to $\Lambda\sim O(11-20)TeV$ which is a
harmeless scale, too. In the Fig(3d) we present the most general scenario
involving leptoquarks with hipercharges $(2/3,-1/3)[\phi,\Phi]$.

##  IV Conclusions

In this work we calculated the contributions of leptoquarks to the running of
$\sin^{2}\theta_{W}$ with the aim of obtaining their impact on the non-
perturbative regime of the model. The non-perturnative regime (a Landau-like
pole) of the model may manifest, depending on the particle content that
contribute to the running of $\sin^{2}\theta_{W}$, already at few TeVs.
Current bounds demand the $SU(3)_{L}\times U(1)_{X}$ symmetry breaks around 4
TeVs leaving the model, in its original form, phenomenologicaly unpredictable.
Even in its original form, without the sextet of scalars, the model get non-
perturbative below 4 TeV. The presence of the sextet, which is necessary to
generate lepton masses, push this value to 6 TeV.

We have to resort to extension of the model in order to evade the non-
perturbative regime and then recover its predictability. Thinking in this way,
the addition of octet of leptons to the particle content of the minimal 3-3-1
model can do the job, as we can see in FIG. 2. In general, when we consider
that all the particle content of the model, including the three octet of
leptons, contribute to the running of $\sin^{2}\theta_{W}$ we have that the
model predicts a Landau pole around $\Lambda\sim 100$ TeV. The Landau pole is
completely evaded only when we consider the contributions of three octet of
leptons in an effective 2HDM scenario.

In this work we analyzed the capacity of leptoquarks in evading the Landau
pole. Our results are displayed in FIG. 3. Firstly, the model allows
leptoquarks in the representation of octets, sextets, triplets and singlets.
As main result we showed that leptoquarks are so efficient in evading the
Landau pole as octet of leptons. For example, in adding leptoquarks to the
scenario of 2HDM, one or two octet of leptoquarks are sufficient to shift the
Landau pole to a harmless energy scale while the case of three octet evade
completely the Landau pole. When we consider all particle content active in
the running of $\sin^{2}\theta_{W}$, the most interesting scenario we found
was the case of one octet plus one sextet of leptoquarks which pushed the
Landau pole to an energy scale around 20 TeV. However, if we wish to push up
the Landau pole we may resort to the combinations presented in Fig.3d where
the case of two octets and two sextet are very efficient pushing this energy
scale to $\Lambda\sim 70$ TeV.

In concluding, we have that octet of leptons or multiplets of leptoquarks,
both, are very efficient in evading the Landau pole that arises in in the
minimal 3-3-1 model at TeV scale. The advantage of leptoquarks are in their
phenomenology once they are very attractive concerning flavor physics.

## Acknowledgments

C.A.S.P was supported by the CNPq research grants No. 311936/2021-0 and A.
Doff. was supported by the CNPq research grants No. 310015/2020-0.

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