#  Introducing scalar leptoquarks into a 3-3-1 model to solve the
$(g-2)_{\mu}$ puzzle

A. Doffb<EMAIL_ADDRESS>C. A. de S. Piresa<EMAIL_ADDRESS>aDepartamento de Física, Universidade Federal da Paraíba, Caixa Postal 5008,
58051-970, João Pessoa, PB, Brazil  bUniversidade Tecnologica Federal do
Parana - UTFPR - DAFIS, R. Doutor Washington Subtil Chueire, 330 - Jardim
Carvalho, 84017-220, Ponta Grossa, PR, Brazil

###### Abstract

In this work we introduce scalar leptoquarks into the 3-3-1 model with right-
handed neutrinos with the aim of solving the $(g-2)_{\mu}$ puzzle. We show
that besides the model supports leptoquarks in the octet, sextet, triplet and
singlet representations, we identified that only one specif leptoquark in the
singlet representation leads to flip of chirality as required to generate
positive and robust contribution to the $(g-2)_{\mu}$. Then we calculate its
contributions to $(g-2)_{\mu}$ and to the decay process $\mu\rightarrow
e\gamma$ and discuss the results.

##  I Introduction

The anomalous magnetic moment of the muon, $a_{\mu}=(\frac{g-2}{2})_{\mu}$ ,
has long stood as a tantalizing puzzle in the particle physicsBrown  _et al._
(2001); Jegerlehner (2017). Experimental measurements of $a_{\mu}$ have
consistently shown a deviation from the predictions of the Standard Model (SM)
of the order of the electroweak contributionBennett  _et al._ (2002),

$\displaystyle a_{\mu}^{Exp}=116592059(22)\times 10^{-11},$ $\displaystyle
a_{\mu}^{SM}=116591802(2)\times 10^{-11},$ $\displaystyle\Delta
a_{\mu}=a_{\mu}^{Exp}-a_{\mu}^{SM}=251(59)\times 10^{-11}.$ (1)

Such result has sparked an intense theoretical and experimental scrutiny to
unravel its underlying causesLindner  _et al._ (2018); Aoyama  _et al._
(2020); Athron  _et al._ (2021).

One intriguing avenue of investigation of $a_{\mu}$ lies in the realm of
leptoquark models. Recent theoretical studies have demonstrated the viability
of chirality flip effects mediated by leptoquarks(LQ) in addressing the muon
anomalous magnetic moment problem, potentially reconciling the observed
anomaly with experimental measurementsAthron  _et al._ (2021); Crivellin and
Hoferichter (2021); Stöckinger and Stöckinger-Kim (2022).

Leptoquarks are hypothetical particles in the form of scalars or vectors that
have gained significant attention in the last decadesDoršner  _et al._ (2016).
These particles combine features of both leptons and quarks and have
fractional electric chargePati and Salam (1974); Georgi and Glashow (1974).
All these features have important impact in flavor physicsDoršner  _et al._
(2016).

Besides the standard model of particle physics does not predict leptoquarks,
they emerge as essential components in various extensions to the Standard
Model, including grand unified theories (GUTs)Pati and Salam (1973),
supersymmetryHewett and Rizzo (1989), and composite models Schrempp and
Schrempp (1985). However the symmetries of the SM support the introduction of
leptoquarks into it in the triplet, doublet and singlet representations. The
last two representations are capables of generating flip of chirality among
the fermions which generate an enhancement of $(g-2)_{\mu}$ by the factor
$\frac{m_{t}}{m\mu}$Coluccio Leskow  _et al._ (2017).

As in the SM case, the $SU(3)_{C}\times SU(3)_{L}\times U(1)_{N}$ (3-3-1)
models face, too, serious difficulities in accommodating the current value of
$a_{\mu}$ at one-loopKelso  _et al._ (2014); De Jesus  _et al._ (2020);
Cárcamo Hernández  _et al._ (2021); Ky  _et al._ (2000); de Jesus  _et al._
(2020); Pinheiro  _et al._ (2021); Hong  _et al._ (2022)111It was showed in
Ref. Pinheiro  _et al._ (2021) that the extension of the model to generate the
inverse seesaw mechanism is not able to explain $\Delta a_{\mu}$ anomaly . In
view of this, we introduce leptoquarks to the original particle content of a
version of the 3-3-1 models that features right-handed neutrinos as third
component of the lepton triplets (331RHN for short)Foot  _et al._ (1994);
Montero  _et al._ (1993). As it is the first work in which leptoquarks are
introduced into this model.222In fact leptoquarks were first introduced in
Doff and de S. Pires (2023) where it was considered the possibility of
including leptoquarks in the minimal 3-3-1 model with the aim of evading the
Landau pole, we first identify the representation content of leptoquarks the
model supports. Next we determine which leptoquark representation leads to
flip of chirality among fermions in the 331RHN. We show that the model allows
leptoquarks in the octet, sextet, triplet and singlet representation but only
one specific leptoquark in the singlet representation is capable of generating
flip of chirality among the fermions. We then focus on this leptoquark and
calculate and discuss its contributions at one loop to $(g-2)_{\mu}$ with the
aim of obtaining which form of enhancement this leptoquark promote. We also
discuss the contribution of such leptoquark to the $\mu\rightarrow e\gamma$
decay and if it discriminata family.

This paper is organized in the following way: in Sec. II we present the main
aspects of the model. In Sec. III we introduce leptoquarks identifying
representation content and Yukawa interactions. In Sec. IV we review the one
loop contributions of the original 331RHN to $(g-2)_{\mu}$ and calculate the
contributions due to the singlet of leptoquark. In Sec. V we calculate the
contribution of the singlet of leptoquark to the $\mu\rightarrow e\gamma$. In
Sec. VI we discuss all the findings and in Sec. VII we present our
conclusions.

##  II The main aspects of the model

In the 331RHN right-handed neutrinos compose the third component of the
leptonic tripletsFoot  _et al._ (1994); Montero  _et al._ (1993),

$L_{a_{L}}=\begin{pmatrix}\nu_{a}\\\ e_{a}\\\ \nu^{c}_{a}\\\
\end{pmatrix}_{L}\sim(1,3,-1/3),\quad e_{a_{R}}\sim(1,1,-1),$ (2)

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

In the hadronic sector anomaly cancellation demands that one family transforms
differently from the other two. Here we chose the first two families of quarks
transforming as anti-triplet while the third one transforming as triplet by
$\text{SU}(3)_{\text{L}}$,

$\displaystyle Q_{i_{L}}=\left(\begin{array}[]{c}d_{i}\\\ -u_{i}\\\
d^{\prime}_{i}\end{array}\right)_{L}\sim(3\,,\,\bar{3}\,,\,0)\,,u_{iR}\,\sim(3,1,2/3),\,\,\,$
(6)
$\displaystyle\,\,d_{iR}\,\sim(3,1,-1/3)\,,\,\,\,\,d^{\prime}_{iR}\,\sim(3,1,-1/3),$
$\displaystyle Q_{3L}=\left(\begin{array}[]{c}u_{3}\\\ d_{3}\\\
u^{\prime}_{3}\end{array}\right)_{L}\sim(3\,,\,3\,,\,1/3),u_{3R}\,\sim(3,1,2/3),$
(10)
$\displaystyle\,\,d_{3R}\,\sim(3,1,-1/3)\,,\,u^{\prime}_{3R}\,\sim(3,1,2/3),$
(11)

where the index $i=1,2$ is restricted to only two generations. The negative
signal in the anti-triplet $Q_{i_{L}}$ is just to standardise the signals of
the charged current interactions with the gauge bosons. The primed quarks are
new heavy quarks with the usual $(+\frac{2}{3},-\frac{1}{3})$ electric
charges.

The original scalar sector of the model is composed by three triplets of
scalars

$\displaystyle\eta=\left(\begin{array}[]{c}\eta^{0}\\\ \eta^{-}\\\
\eta^{\prime 0}\end{array}\right),\,\rho=\left(\begin{array}[]{c}\rho^{+}\\\
\rho^{0}\\\
\rho^{\prime+}\end{array}\right),\,\chi=\left(\begin{array}[]{c}\chi^{0}\\\
\chi^{-}\\\ \chi^{\prime 0}\end{array}\right),$ (21)

with $\eta$ and $\chi$ transforming as $(1\,,\,3\,,\,-1/3)$ and $\rho$ as
$(1\,,\,3\,,\,2/3)$. After spontaneous breaking of the 3-3-1 symmetry
($\langle\eta^{0}\rangle=v_{\eta}\,\,,\,\,\langle\rho^{0}\rangle=v_{\rho}\,\,,\,\,\langle\chi^{\prime
0}\rangle=v_{\chi^{\prime}}$ with $v_{\chi^{\prime}}>>v_{\eta}\,,\,v_{\rho}$ )
the scalar sector333Here we follow Ref.De Jesus  _et al._ (2020), and assume
$v_{\eta}=v_{\rho}=v$ get composed by three CP-even scalars, $h_{1}$, $h_{2}$,
and $H$ where the first recovers the features of the standard Higgs and the
other two are heavy scalars with mass proportional to $v_{\chi^{\prime}}$ with
$m^{2}_{h_{1}}\approx\frac{vv_{\chi^{\prime}}}{2}$ and $m^{2}_{H}\sim
v^{2}_{\chi^{\prime}}$. The model also has in its spectrum of scalars one
pseudoscalar we call $A$, whose mass is given by
$m^{2}_{A}\approx\frac{vv_{\chi^{\prime}}}{2}$ and two singly charged scalars
$h^{+}_{1}$ and $h^{+}_{2}$ with $m^{2}_{h^{+}_{1}}\approx vv_{\chi^{\prime}}$
and $m^{2}_{h^{+}_{2}}\sim v^{2}_{\chi^{\prime}}$. Except by $h_{1}$ that must
recover the 125 GeV standard Higgs, all the other scalars have their masses
determined by $v_{\chi^{\prime}}$ which means that the natural energy scale of
the masses of these scalars is the TeV scale444For a detailed discussion of
the scalar sector, see Refs. Long (1998); Diaz  _et al._ (2004); Pinheiro and
de S. Pires (2023).

Here the most economical Yukawa interactions that generate the correct mass
for all standard quarks and leptons are composed by the terms

$-{\cal L}_{Y}\supset
g_{ia}\bar{Q}_{i_{L}}\eta^{*}d_{a_{R}}+h_{3a}\bar{Q}_{3_{L}}\eta
u_{a_{R}}+g_{3a}\bar{Q}_{3_{L}}\rho
d_{a_{R}}+h_{ia}\bar{Q}_{i_{L}}\rho^{*}u_{a_{R}}+Y\bar{L}\rho
e_{R}+\mbox{H.c.}\,,$ (22)

where $a=1,2,3$ and the parameters $g_{ab}$, $h_{ab}$ and $Y$ are Yukawa
couplings that, for sake of simplification, we consider reals555For a more
comprehensive discussion of the Yukawa sector, see Ref. Doff  _et al._ (2006).

The gauge sector of the model involves nine gauge bosons related to the
electroweak sector, where four of them are the standard gauge bosons
($W^{\pm}$, $Z^{0}$, $\gamma$) and the other five ($W^{\prime\pm}$, $U^{0}$,
$U^{0\dagger}$, $Z^{\prime}$) are the typical 3-3-1 gauge bosons, and eight
gluons $g$ relaterd to the strong force. The standard gauge bosons recover the
physics of the SM while the new gauge bosons have mass belonging to the TeV
scale and interactions that connect the standard particle content with the
specific 3-3-1 particle content666Concerning gauge bosons of the model see
Ref. Long (1996); Cao and Zhang (2016). The mass of $W^{\prime}$ and
$Z^{\prime}$ are given by,

$M_{Z^{\prime}}^{2}\approx\frac{g^{2}c^{2}_{W}}{(3-4s_{w}^{2})}v_{\chi^{\prime}}^{2}\,\,,\,\,\,\,M_{W^{\prime}}^{2}=\frac{g^{2}}{4}\left(\frac{v^{2}}{2}+v_{\chi^{\prime}}^{2}\right).$
(23)

Observe that $v_{\chi^{\prime}}$ practically determine the masses of the
spectrum of new particle characteristic of the 3-3-1.

##  III Leptoquark: representation content and Yukawa interactions

With all this in hand we are ready to determine the possible set of
leptoquarks that the 331RHN supports. Observe that from the quark and lepton
content of the model we can build the following bilinears,

$\displaystyle\bar{L}^{C}_{a_{L}}Q_{3_{L}}\sim(1\,,\,3\,,\,-1/3)\times(3\,,\,3\,,\,1/3)\sim(3\,,\,3^{*}\oplus
6\,,\,0),$
$\displaystyle\bar{L}^{C}_{a_{L}}Q_{i_{L}}\sim(1\,,\,3,-1/3)\times(3\,,\,3^{*}\,,\,0)\sim(3\,,\,1\oplus
8\,,\,-1/3),$
$\displaystyle\bar{L}_{a_{L}}d_{b_{R}}\sim(1\,,\,3^{*},1/3)\times(3\,,\,1\,,\,-1/3)\sim(3\,,\,3^{*}\,,\,0),$
$\displaystyle\bar{L}_{a_{L}}u_{b_{R}}\sim(1\,,\,3^{*},1/3)\times(3\,,\,1\,,\,2/3)\sim(3\,,\,3^{*}\,,\,1),$
$\displaystyle\bar{e}^{C}_{a_{R}}d_{b_{R}}\sim(1\,,\,1,-1)\times(3\,,\,1\,,\,-1/3)\sim(3\,,\,1\,,\,-4/3),$
$\displaystyle\bar{e}^{C}_{a_{R}}u_{b_{R}}\sim(1\,,\,1,-1)\times(3\,,\,1\,,\,2/3)\sim(3\,,\,1\,,\,-1/3),$
(24)

where in the singlet of quarks above we are considering the exotic ones.
According to these bilinears the 331RHN supports the following set of
leptoquarks:

$\displaystyle
S_{8}\sim(3\,,\,8\,,\,1/3),\,\,\,\,\,\,\,\,S_{6}\sim(3\,,\,6\,,\,0),$
$\displaystyle S_{3}\sim(3\,,\
3\,,\,0),\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\tilde{S}_{3}\sim(3\,,3\,,\,1),$
$\displaystyle
S\sim(3\,,\,1\,,\,1/3),\,\,\,\,\,\,\,\,\,\,\tilde{S}\sim(3\,,\,1\,,\,4/3).$
(25)

As we see, we have six categories of leptoquarks with each one of them leading
to a complex phenomenology in regard to flavor physics.

According with the bilinears above, with the leptoquarks $S_{8}$ and $S_{6}$
we compose the following Yukawa interactions,

$\displaystyle{\mathcal{L}}\supset
y^{LQ}_{a3}\bar{L}^{C}_{a_{L}}S_{6}Q_{3_{L}}+y^{LQ}_{ai}\bar{L}^{C}_{a_{L}}S_{8}Q_{i_{L}}+H.c.,$
(26)

where the $S_{8}$ and $S_{6}$ components are given by

$\displaystyle
S_{6}=\left(\begin{array}[]{ccc}\,\tilde{R}_{1}^{-5/3}&R_{2}^{-2/3}&R_{3}^{-5/3}\\\
\newline \\\ R_{2}^{-2/3}&\,\tilde{R}_{2}^{-2/3}&R_{4}^{-5/3}\\\ \newline \\\
R_{3}^{-5/3}&R_{4}^{-5/3}&\,\tilde{R}_{3}^{-1/3}\end{array}\right)\,\,,\,\,S_{8}=\left(\begin{array}[]{ccc}\,\tilde{T}_{1}^{-2/3}&T_{2}^{-5/3}&T_{3}^{-2/3}\\\
\newline \\\ T_{4}^{-2/3}&\,\tilde{T}_{2}^{-5/3}&T_{4}^{-2/3}\\\ \newline \\\
T_{5}^{4/3}&T_{6}^{1/3}&\,\tilde{T}_{3}^{4/3}\end{array}\right).$ (37)

The triplet $S_{3}$ and $\tilde{S}_{3}$ are composed by the following set of
leptoquarks,

$\displaystyle S_{3}=\left(\begin{array}[]{c}S_{3}^{+1/3}\\\ S_{3}^{+2/3}\\\
S_{3}^{\prime+1/3}\end{array}\right)_{L}\,,$
$\displaystyle\tilde{S_{3}}=\left(\begin{array}[]{c}\tilde{S}_{3}^{+2/3}\\\
\tilde{S}_{3}^{+4/3}\\\ \tilde{S}_{3}^{\prime+2/3}\end{array}\right)_{L}$ (44)

The third components of these triplets have $F(S^{\prime
1/3}_{3}\,,\,\tilde{S}_{3}^{\prime 2/3})=-2$ while the other components have
$F=0$ where $F=3B+L$. Their Yukawa interactions are given by

$\displaystyle{\mathcal{L}}\supset\bar{Y}^{LQ}_{a3}\epsilon_{ijk}(\bar{L}^{C}_{a_{L}})_{i}(S_{3})_{j}(Q_{3_{L}})_{k}+\tilde{Y}^{LQ}_{ab}\bar{L}_{a_{L}}\tilde{S}_{3}d_{b_{R}}+H.c.$
(45)

Concerning the allowed singlets of leptoquarks, we have that $\tilde{S}$ has
$F=-2$ and, like the triplets above, do not leads to the flip of chirality.
This means that they do not generate positive and robust contributions to the
$g-2$ of the muon.

The last leptoquark is the singlet $S$. It has $F=-2$ and compose the
following Yukawa interactions with quarks and leptons,

${\cal
L}\supset\tilde{g}^{LQ}_{ia}\bar{Q}^{C}_{i_{L}}L_{a_{L}}S+h^{LQ}_{ab}\bar{u}^{C}_{a_{R}}e_{b_{R}}S+H.c.,$
(46)

with $i=1\,,\,2$ and $a,b=1,2,3$. Opening the first term above we get,

$\bar{Q}^{C}_{i_{L}}L_{a_{L}}=\bar{d}^{C}_{i_{L}}\nu_{a_{L}}-\bar{u}^{C}_{i_{L}}e_{a_{L}}+\bar{\nu}_{a_{R}}d^{\prime}_{i_{L}}.$
(47)

Observe that according with the Yukawa interactions above we identify that $S$
generates flip of chirality. It is due to this fact that from now on we focus
our attention exclusively in developing some physical aspects of this
leptoquark.

For reasons of simplification, and without any loss for the model, we make the
following set of assumptions: charged leptons, neutrinos, right-handed quarks
and the new quarks come in a diagonal basis. With this we obtain the Yukawa
interactions among mass eigenstates of leptons and quarks with the leptoquark
$S$,

${\cal
L}=\bar{\hat{u}}^{C}_{a}(g^{LQ}_{ab}P_{L}+h^{LQ}_{ab}P_{R})e_{b}S+\bar{\hat{d}}^{C}_{a_{L}}g^{\prime
LQ}_{ab}\nu_{a_{L}}S+\bar{\nu}_{a_{R}}\tilde{g}^{LQ}_{ia}d^{\prime}_{i_{L}}S+H.c.,$
(48)

with $g^{LQ}_{ab}=-\tilde{g}^{LQ}_{ib}(V^{u}_{L})_{bi}$ and $g^{\prime
LQ}_{ab}=\tilde{g}^{LQ}_{ib}(V^{d}_{L})_{bi}$. The physical quarks
$\hat{u}_{a}=(u\,,\,c\,,\,t)$ and $\hat{d}_{a}=(d\,,\,s\,,\,b)$ are related to
the symmetric ones by means of the transformations $\hat{u}_{L}=V^{\dagger
u}_{L}u_{L}$ and $\hat{d}_{L}=V^{\dagger d}_{L}d_{L}$. The first term in Eq.
(48) is the one that generates the flip of chirality among the interaction of
$S$ with the charged fermions. The interest in this interaction is because it
provides a positive and robust contribution to the $g-2$ of the muon as we see
below.

As we sad in the introduction, the 331RHN with its original particle content,
and even modified to implement the inverse seesaw mechanism, is not able to
accommodate the current muon $g-2$. But now with the introduction of singlet
of leptoquark $S$ we may cure this deficiency of the model. For this we made a
brief review the contributions of the original particle content of the model
to the muon $g-2$ at one-loop.

##  IV review of the one loop contribution to $\Delta a_{\mu}$ in the 331RHN
and the contribution of the leptoquark S

###  IV.1 331RHN original particle content contribution to $\Delta a_{\mu}$

We review the main contributions at one loop to the muon anomalous magnetic
moment $a_{\mu}=\frac{(g-2)_{\mu}}{2}$ in the 331RHN. For this we follow Ref.
de Jesus  _et al._ (2020) where it is took $v_{\eta}=v_{\rho}=v=174$ GeV. We
are interested in the discrepancy $\Delta a_{\mu}$ produced by the new charged
gauge boson $W^{\prime\pm}$, $Z^{\prime}$, the single charged scalar
$h^{+}_{1}$ , the neutral scalars $h_{2}$ and the pseudo Goldstone $A$777The
charged scalar $h^{+}_{2}$ and the neutral scalar $H$ do not contribute to
$(g-2)_{\mu}$Pinheiro and de S. Pires (2023).

The interactions of interest are given by

${\cal
L}^{CC}_{l}=-\frac{g}{2\sqrt{2}}\left[\overline{\nu^{c}_{R}}\,\gamma^{\mu}(1-\gamma_{5})\bar{l}\,W^{\prime-}_{\mu}\right],$
(49) ${\cal
L}^{NC}=\bar{f}\,\gamma^{\mu}[g_{V}(f)+g_{A}(f)\gamma_{5}]\,f\,Z^{\prime}_{\mu},$
(50) ${\cal
L}^{h^{+}_{1}}_{Y}=\sqrt{2}\frac{m_{l}}{v}\bar{\nu}_{l_{L}}\l_{R}h^{+}_{1},$
(51) ${\cal L}^{h_{2}+A}=\frac{m_{\mu}}{\sqrt{2}v}\bar{\mu}\,\mu
h_{2}+i\frac{m_{\mu}}{v}\bar{\mu}\gamma_{5}\mu A,$ (52)

where $g_{V}$ and $g_{A}$ are the vector and axial coupling constants and are
given by,

$g^{\prime}_{V}(l)=\frac{g}{4c_{W}}\frac{(1-4s_{W}^{2})}{\sqrt{3-4s_{W}^{2}}},\
g^{\prime}_{A}(l)=-\frac{g}{4c_{W}\sqrt{3-4s_{W}^{2}}},$ (53)

The Feymnan diagrams displaying the contributions of these interactions to
$(g-2)_{\mu}$ are showed in FIG. 1.

Figure 1: One-loop contributions to the $g-2$ of the muon in the 331RHN where
$\phi\,\,\mbox{and}\,\,\phi^{-}$ represent the neutral and charged scalar
contributions

The evaluation of these contributions at one-loop to $\Delta a_{\mu}$
generated by these set of particles follow the procedure of Ref. de Jesus  _et
al._ (2020) . To avoid being repetitive we simply quote them in the heavy
mediator regime. With that being said, we have the following contributions of
the 331RHN to the $\Delta a_{\mu}$,

$\displaystyle\Delta
a_{\mu}(\nu,W^{\prime})\simeq\frac{1}{4\pi^{2}}\frac{m_{\mu}^{2}}{M_{W^{\prime}}^{2}}\left|\frac{g}{2\sqrt{2}}\right|^{2}\left(\frac{5}{3}\right),$
(54)

$\displaystyle\Delta
a_{\mu}\left(\mu,Z^{\prime}\right)\simeq\frac{-1}{4\pi^{2}}\frac{m_{\mu}^{2}}{M_{Z^{\prime}}^{2}}\frac{1}{3}\left|-\frac{g}{4c_{W}\sqrt{3-4s_{W}^{2}}}\right|^{2}\left[-\left|1-4s_{W}^{2}\right|^{2}+5\right],$
(55)

$\displaystyle\Delta
a_{\mu}(h^{+}_{1})\simeq\frac{-1}{4\pi^{2}}\frac{m_{\mu}^{2}}{M_{h^{+}_{1}}^{2}}\left|\frac{m_{\mu}\sqrt{2}}{2v_{\eta}}\right|^{2}\frac{1}{6},$
(56)

$\displaystyle\Delta
a_{\mu}(h_{2})\simeq\frac{1}{4\pi^{2}}\frac{m_{\mu}^{2}}{M_{h_{2}}^{2}}\left(\frac{m_{\mu}\sqrt{2}}{2v_{\eta}}\right)^{2}\left[\frac{1}{6}+\left(\frac{3}{4}+\log\left(\frac{m_{\mu}}{M_{h_{2}}}\right)\right)\right].$
(57)

$\displaystyle\Delta
a_{\mu}(A)=-\frac{m_{\mu}^{2}}{8\pi^{2}M_{A}^{2}}\Bigg{(}\frac{gA_{\mu}m_{\mu}}{2M_{W}}\Bigg{)}^{2}H\Bigg{(}\frac{m_{\mu}^{2}}{M_{A}^{2}}\Bigg{)},$
(58)

Figure 2: Overall contribution to $\Delta a_{\mu}$ from Eqs.(54)-(58) to the
331RHN model. The dash-dotted bands in red are delimited by the current
$1\sigma$ bound.

with $H(y)=\int_{0}^{1}\frac{dxx^{3}}{1-x+x^{2}y}$ where
$y=\frac{m^{2}_{\mu}}{M^{2}_{A}}$ . In summary, the 331RHN yields five
contributions to $\Delta a_{\mu}$ with only $W^{\prime}$ and $h_{2}$
contributions being positive. In view of this, according to Ref. De Jesus  _et
al._ (2020), the only way the one-loop contribution would accommodate the
current prediction of the $\Delta a_{\mu}$ was if the positive contributions
of the $W^{\prime}$ and $h_{2}$ could surpass the other ones. This could
happen for $W^{\prime}$ having mass at the electroweak scale which is
completely discarded since current LHC bounds demands the gauge bosons of the
331RHN have mass at the TeV scale. To be precise LHC demands $m_{Z^{\prime}}$
with mass in the range $(3-4)$ TeVsCoutinho  _et al._ (2013), depending if
$Z^{\prime}$ may decay in the new quarks or don´tAlves  _et al._ (2022). This
implies $v_{\chi^{\prime}}$ in the range $10-12$TeVs. In FIG.2 we present the
prediction of the 331RHN to $\Delta a_{\mu}$ in function of
$v_{\chi}^{\prime}$ varying in the range $1-12$ TeVs. As we can see in FIG. 2
one-loop contributions to $\Delta a_{\mu}$ in the 331RHN are not able to
accommodate the current $\Delta a_{\mu}$. Observe that for $v_{\chi^{\prime}}$
around $10^{4}$ GeV the prediction to $\Delta a_{\mu}$ lies three order of
magnitude below the experimental prediction. Thus the one loop contribution to
$\Delta a_{\mu}$ in the 331RHN is practically neglegible. As sad in the
introduction, it is hard to find modifications of the 331RHN that accommodate
the current $\Delta a_{\mu}$ value. A simple route to accommodate $\Delta
a_{\mu}$ into the 331RHN is via the introduction of vectorlike leptons and
inert scalar triplets to the modelDe Jesus  _et al._ (2020). What we are going
to show next is that the addition of only one singlet of leptoquark resolves
the $(g-2)_{\mu}$ anomaly.

###  IV.2 The contribution of $S$ to $\Delta a_{\mu}$ in the 331RHN

Figure 3: One-loop diagrams contributing to $\Delta a_{\mu}$ and $\mu\to
e\gamma$ decay induced by S leptoquarks: To the contributions to $\Delta
a_{\mu}$ in this figure $l_{i}=l_{j}=l_{2}=\mu$, while to muon decay we have
$l_{i}=l_{2}$ and $l_{j}=l_{1}=e$.

The one-loop contributions to $\Delta a_{\mu}$ due to the leptoquark $S$,
depicted in the FIG. 3 and in according to the interactions in Eq. (48) are
given byDoršner  _et al._ (2016); Athron  _et al._ (2021)

$\Delta
a_{\mu}^{LQ}=\frac{m^{2}_{\mu}}{48\pi^{2}m^{2}_{S}}f(m_{\mu})\left(\frac{m_{q}}{m_{\mu}}g^{LQ}_{a2}h^{LQ}_{a2}L_{1}(x_{q})+\frac{(g^{LQ}_{a2})^{2}+(h^{LQ}_{a2})^{2}}{4}L_{2}(x_{q})\right),$
(59)

where $x_{q}=\frac{m^{2}_{q}}{m^{2}_{S}}$ and

$\displaystyle
L_{1}(x)=4F_{F}(x)-F_{C}(x)\,,\,\,\,\,\,L_{2}(x)=2F_{E}(x)-F_{B}(x)$
$\displaystyle
F_{F}(x)=\frac{3(-3+4x-x^{2}-2lnx)}{2(1-x)^{3}}\,,\,\,\,\,\,F_{E}(x)=\frac{2(2+3x-6x^{2}+x^{3}+6xlnx)}{(1-x)^{4}},$
$\displaystyle
F_{C}(x)=\frac{3(1-x^{2}+2xlnx)}{(1-x)^{3}}\,,\,\,\,\,\,F_{B}(x)=\frac{2(1-6x+3x^{2}+2x^{3}-6x^{2}lnx)}{(1-x)^{4}}.$
$\displaystyle f(m_{\mu})=(1+\frac{4\alpha}{\pi}\ln(\frac{m_{\mu}}{m_{S}})).$
(60)

We consider only the maximum contribution given by Eq. (59) which is due to
the top quark ($a=3$ in Eq. (59)). The free parameters involved in such
contributions are the Yukawa couplings $g^{LQ}_{32}$, $h^{LQ}_{32}$ and the
mass of $S$ which we called $m_{S}$. The current LHC bounds demands the lowest
leptoquark to have mass in the range $O(0.98–1.73)TeV$, depending on the spin
of leptoquarks and its couplings to SM fermionsHayrapetyan  _et al._ (2024).
Here we adopt the current assumption $m_{S}=1.8TeV$ and extract constraint on
the Yukawa couplings $g^{LQ}_{32}$ and $h^{LQ}_{32}$.

Our numerical result is displayed in FIG. 4. As we can see there the existence
of a simple singlet of leptoquark $S$ presenting flip of chirality give
positive and robust contributions to $\Delta a_{\mu}$ explaining easily the
observed value of $\Delta a_{\mu}$. That is a very nice result in what concern
the development of the 331RHN. In the figure the dash-dotted line corresponds
to central value $\Delta a_{\mu}=25.1\times 10^{-10}$. The Yellow(Green) bands
corresponds to the current $1\sigma(2\sigma)$ bounds found by requiring
$\Delta a_{\mu}\lesssim 59\times 10^{-11}$. In the next section, we complement
this result with the study of the decay $\mu\rightarrow e\gamma$ once the
contributions are the same being necessary just to take of-shell photon in
FIG. 3.

Figure 4:  In this figure we show in parameter space
($g^{LQ}_{32}-h^{LQ}_{32}$) the allowed values to $\Delta a_{\mu}$. The dash-
dotted line corresponds to central value obtained to $\Delta a_{\mu}$. The
respective bands $1(2)\sigma$ are represented in yellow(green)

##  V Contributions of $S$ leptoquarks to $\mu\to e\gamma$ decay

In the FIG. 3 we show the contributions of $S$ leptoquarks to the muon decay
$\mu\to e\gamma$. These contributions are similar to the corrections of $S$ to
$\Delta a_{\mu}$, and in this Feynman diagrams we consider only the chirality
enhanced terms, given by Eq.(46), that explain the observed $\Delta a_{\mu}$.
To this decay the branching ratio presents the following formJ. Hisano and
Yamaguchi (1996)

$BR(\mu\to
e\gamma)=\frac{m^{5}_{\mu}}{16\pi\Gamma_{\mu}}\left(|A^{L}_{2}|^{2}+|A^{R}_{2}|^{2}\right)$
(61)

where $\Gamma_{\mu}=2.996\times 10^{-19}GeV$ and the structure of the dipole
form factors $A^{2}_{L,R}$ are given by

$\displaystyle
A^{L}_{2}=-\frac{1}{16\pi^{2}}\frac{e}{6m^{2}_{S}}\left(\frac{m_{q}}{m_{\mu}}g^{LQ}_{a1}h^{LQ}_{a2}L_{1}(x_{q})+\frac{1}{4}h^{LQ}_{a1}h^{LQ}_{a2}L_{2}(x_{q})\right),$
$\displaystyle
A^{R}_{2}=-\frac{1}{16\pi^{2}}\frac{e}{6m^{2}_{S}}\left(\frac{m_{q}}{m_{\mu}}h^{LQ}_{a1}g^{LQ}_{a2}L_{1}(x_{q})+\frac{1}{4}g^{LQ}_{a1}g^{LQ}_{a2}L_{2}(x_{q})\right).$
(62)

Following the same line of the previous section, we will consider the maximum
contribution to $\mu\to e\gamma$ decay that is due to the top quark ($a=3$).

Considering the comparison with the experimental results to $\Delta a_{\mu}$,
in the FIG. 4 we show the bounds on products of $g^{LQ}_{32}$ and
$h^{LQ}_{32}$ couplings in parameter space. In this figure the band in Green
restricts the product of these couplings , i. e. the left and handed
leptoquarks couplings of top with the muon. As consequence the $2\sigma$ band
results in the following allowed range of possible couplings

$3.1\times 10^{-3}<g^{LQ}_{32}*h^{LQ}_{32}<9.3\times 10^{-3}.$ (63)

In the Eq.(62) the left and handed leptoquarks couplings of top with the
electron are $g^{LQ}_{31}$ and $h^{LQ}_{31}$. In order to find the allowed
parameter space in the $(g^{LQ}_{31}$ \- $h^{LQ}_{31})$ plane, currently
permitted by experimental boundsBaldini  _et al._ (2020)Baldini  _et al._
(2018), we can consider the above result, combined with the expression to the
branching ratio given by Eq.(61).

Therefore, considering the Eqs.(61), (62) and (63) we can write the following
inequality

Figure 5: Allowed parameter regions for the $\mu\to e\gamma$ decay, assuming
that $\Delta a_{\mu}$ is explained by top quark chirally enhanced term. The
contextualization of the curves behavior are described in the text.

$\left(|h^{LQ}_{32}|^{2}|h^{LQ}_{31}|^{2}+|g^{LQ}_{32}|^{2}|g^{LQ}_{31}|^{2}\right)<3\times
10^{-2}BR(\mu\to e\gamma).$ (64)

In order to generate the parameter region allowed, we will consider the
following possibilities of top couplings

$\displaystyle{\rm(a)Symmetric\,\,leptoquark\,\,coupling:}\,\,h^{LQ}_{32}=g^{LQ}_{32},$
$\displaystyle{\rm(b)Asymmetric\,\,leptoquark\,\,coupling:}\,\,h^{LQ}_{32}>g^{LQ}_{32},$
$\displaystyle{\rm(c)Asymmetric\,\,leptoquark\,\,coupling:}\,\,h^{LQ}_{32}<g^{LQ}_{32}.$
(65)

Assuming the lower bound $(3.1\times 10^{-3}<g^{LQ}_{32}*h^{LQ}_{32})$, we
obtain to Case (a) ($|h^{LQ}_{32}|^{2}=|g^{LQ}_{32}|^{2}>3.1\times 10^{-3}$),
while that to $h^{LQ}_{32}=2$, we have to Case (b) $(g^{LQ}_{32}>1.55\times
10^{-3})$ and similarly to $g^{LQ}_{32}=2$ the Case (c) corresponds to
$(h^{LQ}_{32}>1.55\times 10^{-3})$.

Currently the experimental limits to $BR(\mu\to e\gamma)$ correspond to
$BR(\mu\to e\gamma)=4.2\times 10^{-13}$Baldini  _et al._ (2020) and $BR(\mu\to
e\gamma)=6\times 10^{-14}$Baldini  _et al._ (2018). In FIG. 5 we show in the
plane $(g^{LQ}_{31}$ \- $h^{LQ}_{31})$ the allowed values to
$g^{LQ}_{31}*h^{LQ}_{31}$, assuming the considerations indicated in the
paragraph above.

In FIG. (5a), corresponding to the symmetric case, the region filled in blue
leads to the upper bond to the allowed values of couplings as
$g^{LQ}_{31}*h^{LQ}_{31}<2.056\times 10^{-12}$, while for region filled in
green we have an more restrictive upper bond
$g^{LQ}_{31}*h^{LQ}_{31}<2.93\times 10^{-13}$.

For the asymmetric cases corresponding to FIG. (5b) and (5c), the limits
obtained are smaller compared to case (a), so that they are within the region
covered in this situation. Considering the combination of results presented in
these figures, that illustrate the allowed values to S leptoquarks couplings
permitted by actual experimental bounds, we found that leptoquarks couplings
to electrons must be more than 4 orders of magnitude smaller than the
corresponding couplings to muons.

##  VI discussion

Anomalies cancellation inside 3-3-1 models demand that at least one family of
quarks transforms differently form the other twoFrampton (1992); Ng (1994);
Liu and Ng (1994). This means that for each version of 3-3-1 models three are
three variants as discussed in Ref.Oliveira and de S. Pires (2023); Oliveira
and S. Pires (2023) . Each variant has its proper implications in flavor
physics. However, in adding leptoquarks to the 331RHN, besides the Yukawa
interactions among leptoquarks and fermions seems to discriminate families, as
expressed in Eq. (46), when we go the to mass eigenstates we see that
leptoquarks do not distinguish families, as expressed in Eq. (48). This is so
because multiplet of leptoquarks do not contribute to the masses of the
fermions after spontaneous breaking of the symmetry. Then we can always
redefine the Yukawa couplings by means of the mixing matrix that transform
fermions eigenstate of symmetry into the mass eigenstates. Thus the fact we
have chosen the third family transforming as triplet and the first and second
ones transforming as anti-triplet was a mere question of convenience.

Another point we would like to call the attention is that while the standard
model supports two representaion of leptoquarks inducing flip of
chiralityColuccio Leskow  _et al._ (2017), namely a singlet and a doublet of
leptoquarks, in the 331RHN, besides it supports a more extensive set of
multiplet of leptoquarks (octet, sextet, triplet and singlet), however only a
specific leptoquark transforming as singlet by the 3-3-1 symmetry leads to the
flip of chirality. Nevertheless, such singlet is sufficient to do the job of
explaining $\Delta a_{\mu}$.

Observe that the leptoquark $S$ in eq. (48) enhances the contribution to
$(g-2)_{\mu}$ by a factor $\frac{m_{t}}{m_{\mu}}$. However if we switch on the
VEVs of $\chi^{0}$ and $\eta^{\prime 0}$ we have, as main consequence, a
mixing among the standard quarks with the new ones. As main implication to
$(g-2)_{\mu}$ we will have an enhancement of $\Delta a_{\mu}$ by a factor of
$\frac{m_{u^{\prime}}}{m_{\mu}}$. A $m_{u^{\prime}}$ at TeV scale would give a
very robust contribution to $\Delta a_{\mu}$ rendering constraints in the
mixing among standard and new quarks which translate in constraints on the
VEVs $v_{\chi}$ and $v_{\eta^{\prime}}$.

In FIG. 3 we took $q=t$ which implies an enhancement of the contributions to
$\Delta a_{\mu}$ by a factor of $\frac{m_{t}}{m_{\mu}}$ . This is the reason
why people consider in the standard model only the interaction of leptoquarks
with the third generation of quarks. We can not do this here because the
symmetries do not allow. The leptoquark $S$ has an intriguing set of Yukawa
interactions as we can see in Eq. (46). The interaction with the quark top
arises due to the mixing among the quarks given by the mixing matrix
$V^{u,d}_{L}$.

##  VII conclusion

In this work we introduced leptoquarks into the 3-3-1 model with right-handed
neutrinos with the aim of resolving the problem of the $\Delta a_{\mu}$
anomaly since the original version of the model gives a neglegigle
contribution to $\Delta a_{\mu}$. We saw that the model supports multiplet of
leptoquarks in the representation of octet, sextet, triplet and singlet. We
determined the particle content of these multiplets and presented they Yukawa
interactions with the fermions of the model.

Besides this large spectrum of leptoquarks, only a specific singlet $S$ is
capable of generate a robust contribution to $\Delta a_{\mu}$ since only it
leads to flip of chirality among the fermions. We then developed the main
aspects of the leptoquark $S$ in what concern its contributions to $\Delta
a_{\mu}$ and to the muon decay channel $\mu\rightarrow e\gamma$. Our main
result is that singlet of leptoquark is not only viable but also the most
economical extension of the 3-3-1 model that explain the $\Delta a_{\mu}$
anomaly.

## Acknowledgments

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

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