# Dead or Alive? Implications of the Muon Anomalous Magnetic Moment for 3-3-1
Models

Alvaro S. de Jesusa<EMAIL_ADDRESS>Sergey Kovalenkob
<EMAIL_ADDRESS>C. A. de S. Piresc<EMAIL_ADDRESS>Farinaldo S.
Queiroza<EMAIL_ADDRESS>Yoxara S. Villamizara
<EMAIL_ADDRESS>aInternational Institute of Physics, Universidade Federal
do Rio Grande do Norte, Campus Universitário, Lagoa Nova, Natal-RN 59078-970,
Brazil
b Departamento de Ciencias Físicas, Universidad Andres Bello, Sazié 2212,
Santiago, Chile
cDepartamento de Física, Universidade Federal da Paraíba, Caixa Postal 5008,
58051-970, Joao Pessoa, PB, Brazil

###### Abstract

We have witnessed a persistent puzzling anomaly in the muon magnetic moment
that cannot be accounted for in the Standard Model even considering the large
hadronic uncertainties. A new measurement is forthcoming, and it might give
rise to a $5\sigma$ claim for physics beyond the Standard Model. Motivated by
it, we explore the implications of this new result to five models based on the
$SU(3)_{C}\times SU(3)_{L}\times U(1)_{N}$ gauge symmetry and put our
conclusions into perspective with LHC bounds. We show that previous
conclusions found in the context of such models change if there are more than
one heavy particle running in the loop. Moreover, having in mind the projected
precision aimed by the g-2 experiment at FERMILAB, we place lower mass bounds
on the particles that contribute to muon anomalous magnetic moment assuming
the anomaly is resolved otherwise. Lastly, we discuss how these models could
accommodate such anomaly in agreement with existing bounds.

###### pacs:

95.35.+d, 14.60.Pq, 98.80.Cq, 12.60.Fr

††preprint: IIPDM-2020

##  I Introduction

The Standard Model (SM) offers an excellent description of the strong and
electroweak interactions in nature. Its theoretical predictions, calculated
beyond tree level, are compatible with experimental measurements with
unprecedented accuracy. However, there are many open questions, which cannot
be solved within the SM. The long-standing muon anomalous magnetic moment
anomaly, if experimentally confirmed, is one of them. The deviation of the
magnetic moment of any charged fermion from its Dirac prediction, $g/2=1$, is
quantified by the aforementioned anomalous magnetic moment $(g-2)/2$. For the
case of muon, the SM predicts

$a_{\mu}^{SM}=\frac{g-2}{2}=116591802(2)(42)(26)\times 10^{-11}.$ (1)

This value takes into account the electromagnetic, weak and hadronic
corrections (See Lindner  _et al._ (2018) for a recent review). The
theoretical errors shown in Eq. (1) correspond to the electroweak, lowest-
order hadronic, and higher-order hadronic contributions, respectively. There
are several diagrams that contribute to $a^{SM}_{\mu}$ and they have been
computed beyond one-loop.

Experiments have measured the muon anomalous magnetic moment using the
principle of Larmor precession, whose frequency is proportional to the
magnetic field which the charged particle is subject to. As the theoretical
and experimental errors shrank, a discrepancy, quantified by $\Delta
a_{\mu}=a_{\mu}^{exp}-a_{\mu}^{SM}$, was observed between the Standard Model
prediction and the experimental measurements. Comparing the SM prediction with
the measurements from Brookhaven National Lab Bennett  _et al._ (2002, 2006),
we get,

$\displaystyle\Delta a_{\mu}$ $\displaystyle=$ $\displaystyle(261\pm 78)\times
10^{-11}\,\,(3.3\sigma)\,\,\text{ \cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Prades:2009tw,Tanabashi:2018oca}{\@@citephrase{(}}{\@@citephrase{)}}}
- (2009)}$ $\displaystyle\Delta a_{\mu}$ $\displaystyle=$
$\displaystyle(325\pm 80)\times 10^{-11}\,\,(4.05\sigma)\,\,\text{
\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Benayoun:2012wc}{\@@citephrase{(}}{\@@citephrase{)}}} -
(2012)}$ $\displaystyle\Delta a_{\mu}$ $\displaystyle=$ $\displaystyle(287\pm
80)\times 10^{-11}\,\,(3.6\sigma)\,\,\text{ \cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Blum:2013xva}{\@@citephrase{(}}{\@@citephrase{)}}} -
(2013)}$ $\displaystyle\Delta a_{\mu}$ $\displaystyle=$ $\displaystyle(377\pm
75)\times 10^{-11}\,\,(5.02\sigma)\,\,\text{ \cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Benayoun:2015gxa}{\@@citephrase{(}}{\@@citephrase{)}}} -
(2015)}$ $\displaystyle\Delta a_{\mu}$ $\displaystyle=$ $\displaystyle(313\pm
77)\times 10^{-11}\,\,(4.1\sigma)\,\,\text{ \cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Jegerlehner:2017lbd}{\@@citephrase{(}}{\@@citephrase{)}}}-
(2017)}$ $\displaystyle\Delta a_{\mu}$ $\displaystyle=$ $\displaystyle(270\pm
36)\times 10^{-11}\,\,(3.7\sigma)\,\,\text{ \cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Keshavarzi:2018mgv}{\@@citephrase{(}}{\@@citephrase{)}}} -
(2018)}$ (2)

The different values quoted in Eq.(2) refer to different studies where the
overall SM contribution was reassessed based on different calculations of the
hadronic contribution. According to the Particle Data Group (PDG), the current
discrepancy reads $3.3\sigma$, but the PDG review already acknowledges recent
studies where the significance approaches $4\sigma$. It is clear that such
hadronic contribution blurs the significance of this anomaly, but it seems a
clear sign of new physics. We will be conservative in our study and adopt
$\Delta a_{\mu}=(261\pm 78)\times 10^{-11}$ Tanabashi  _et al._ (2018).

Fortunately, there are two experiments (g-2 at FERMILAB Grange  _et al._
(2015) and Muon g-2 at J-PARC Abe  _et al._ (2019)) that will be able to push
down the error bar and increase the discrepancy if the central value remains
the same. The goal is to bring the error down by a factor of four. In
particular, the g-2 experiment is about to announce new results. Keeping the
central value of the previous measurement, the g-2 experiment has the
potential to claim the first $5\sigma$ signal after the Higgs boson discovery,
and with the expected theoretical improvements this significance could
increase to nearly $8\sigma$ Al-Binni  _et al._ (2013). Such observation will
have profound implications for particle physics. If the anomaly is not
confirmed though, we will place conservative $1\sigma$ bounds by enforcing the
overall contribution of a given model to g-2 to be at most $78\times
10^{-11}$. Furthermore, having in the mind the sensitive aimed by the g-2
collaboration and the expected reduction of the theoretical uncertainties of
the hadronic contribution Carey  _et al._ (2009), we can further impose a
projected $1\sigma$ bound by requiring $\Delta a_{\mu}<34\times 10^{-11}$
Carey  _et al._ (2009). In summary, we are arguably on the climax of the muon
magnetic moment history and for this reason, our investigation is timely
important.

Instead of exploring g-2 in a simplified model where there are few new physics
contributions to g-2 and the experimental constraints are easier to bypass, we
investigate g-2 in Ultra-Violet complete models based on the $SU(3)_{C}\times
SU(3)_{L}\times U(1)_{X}$ (3-3-1) gauge symmetry. Such models are well-
motivated for several reasons. They are capable of addressing the number of
fermion generations due to anomaly cancellations and QCD asymptotic freedom
requirement. Such models are anomaly free if there is an equal number of
triplets and antitriplets fermion multiplets. The anomaly cancellation does
not occur generation by generation, it happens when all three fermion
generations are taken into account. In this way, we need three fermion
generations to have an anomaly free model. On the other side, asymptotic
freedom of QCD requires less than 17 quarks. Thus, the models with three
fermion generations stand as the simplest non-trivial anomaly free
representation of the $SU(3)_{L}\times U(1)_{N}$ gauge group Foot  _et al._
(1993).

These models can explain neutrino masses Montero  _et al._ (2001); Tully and
Joshi (2001); Montero  _et al._ (2002); Cortez and Tonasse (2005); Cogollo
_et al._ (2009, 2010, 2008); Okada  _et al._ (2016); Vien  _et al._ (2019);
Cárcamo Hernández  _et al._ (2018); Nguyen  _et al._ (2018); de Sousa Pires
_et al._ (2019); Carcamo Hernandez  _et al._ (2019a, b); Cárcamo Hernández
_et al._ (2020) and dark matter Fregolente and Tonasse (2003); Long and Lan
(2003); de S. Pires and Rodrigues da Silva (2007); Mizukoshi  _et al._ (2011);
Profumo and Queiroz (2014); Dong  _et al._ (2013a, b); Cogollo  _et al._
(2014a); Dong  _et al._ (2014a, b); Carvajal  _et al._ (2017); Montero  _et
al._ (2018); Huong  _et al._ (2019). Due to the enlarged scalar sector these
models are entitled to a rich phenomenology that can be explored in the
context of meson oscillations, colliders, flavor violation Cabarcas  _et al._
(2012); Hue  _et al._ (2018), among others Montero and Sanchez-Vega (2011);
Santos and Vasconcelos (2018); Barreto  _et al._ (2018); De Conto  _et al._
(2018).

Having said that, there are models based on this gauge symmetry that have
become quite popular. They are know as Minimal 3-3-1 Pisano and Pleitez
(1992), 3-3-1 with right-handed neutrinos (3-3-1 r.h.n), Long (1996a, b),
3-3-1 with neutral lepton (3-3-1 LHN) Mizukoshi  _et al._ (2011); Catano  _et
al._ (2012), Economical 3-3-1 Dong  _et al._ (2006); Dong and Long (2008);
Martinez and Ochoa (2014), and 3-3-1 with exotic leptons Ponce  _et al._
(2002a, b); Anderson and Sher (2005); Cabarcas  _et al._ (2014). Each model
gives rise to several new contributions to $a_{\mu}$ and these corrections
come either from gauge bosons, or scalar fields, or new fermions. Some of them
induce a negative contribution to $a_{\mu}$. As we plan to draw solid
conclusions, we exploit the fact all these new particles have masses
proportional to the energy scale at which the $SU(3)_{L}\times U(1)_{N}$
symmetry is broken down to $SU(2)_{L}\times U(1)_{Y}$. Therefore, we can
compute up to some constants the overall contribution to g-2 stemming from
each of these five models in terms of the scale of symmetry breaking. In this
way, we can connect our findings to existing bounds in the literature rising
from collider physics. We believe that our findings will represent a new
direction concerning 3-3-1 model building endeavors if the g-2 anomaly is
confirmed.

There were studies of the muon magnetic moment in the context of 3-3-1 models
in the past Ky  _et al._ (2000); Kelso  _et al._ (2014a); Binh  _et al._
(2015); Cogollo (2017); De Conto and Pleitez (2017); Carcamo Hernandez  _et
al._ (2020). We extend them by investigating five different models
simultaneously and putting them into perspective with existing bounds.
Moreover, we show that previous conclusions found in the literature are not
valid in the presence of two heavy new fields in the loop. At loop level,
there can be more than one heavy particle in the loop that contributes to the
muon magnetic moment. This is a common feature in 3-3-1 models as they have a
rich particle spectrum. We show how our conclusions change depending on the
masses of those particles and draw robust conclusions. We provide a
Mathematica notebook, available at Queiroz and Shepherd (2013).

##  II 3-3-1 Models

It is important to highlight some key features of the models based on the
$SU(3)_{C}\times SU(3)_{L}\times\times U(1)_{N}$ gauge symmetry before
discussing each model individually. Enlarging the $SU(2)_{L}$ symmetry to
$SU(3)_{L}$ implies that the fermion generations are now triplets or
antitriplets under $SU(3)_{L}$. After the $SU(3)_{L}\times U(1)_{N}$ symmetry
is spontaneously broken, a remnant $SU(2)_{L}\times U(1)_{Y}$ is observed
Borges and Ramos (2016). The fermionic and bosonic contents are dictated by
the electric charge operator which is generally written as a combination of
the diagonal generators of the group as follows,

$\frac{Q}{e}=\frac{1}{2}(\lambda_{3}+\alpha\lambda_{8})+NI=\begin{pmatrix}1/2(1+\frac{\alpha}{\sqrt{3}})+N\\\
1/2(-1+\frac{\alpha}{\sqrt{3}})+N\\\ -\frac{\alpha}{\sqrt{3}}+N\end{pmatrix},$
(3)

where $\lambda_{3,8}$ and $I$ are the generators of $SU(3)_{L}$ and
$U(1)_{N}$, respectively.

As we want to reproduce the Standard Model spectrum, the first two components
of the triplet should be a neutrino and a charged lepton. From this
requirement, we get $\alpha/\sqrt{3}=-(2N+1)$, which implies that the third
component should have a $3N+1$ quantum number under $U(1)_{N}$. For instance,
taking $N=-1/3$, the third component would be a right-handed neutrino
$\nu_{R}^{c}$, or simply a neutral fermion, $N$. These choices lead to the
3-3-1 r.h.n., and 3-3-1 LHN. If we took N=0, i.e. $\alpha=-\sqrt{3}$, then the
third component would be a positively charged lepton, either $l^{c}$, or $E$,
where $l^{c}$ is simply the charge-conjugate of the Standard Model lepton $l$,
while $E$ is an exotic charged lepton. The last two choices lead to the
Minimal 3-3-1 model and 3-3-1 model with exotic leptons, respectively. These
3-3-1 models usually feature three scalar triplets in the scalar sector, but
if there are only two, then, it is called Economical 3-3-1. There are other
possible ways to extend these models by introducing extra singlet fermions
under $SU(3)_{L}$, additional scalar multiplets, etc. The initial motivation
behind all these models is the possibility to solve the number of fermion
generations. Later on, it was realized that many of them are also capable of
explaining neutrino masses, dark matter, and some flavor anomalies Wei and
Chong-Xing (2017).

We now move on to a brief description of each model. We emphasize that we will
not discuss the models in detail, we will focus rather on the aspects that are
relevant to our phenomenology.

##  III MINIMAL 3-3-1

In the Minimal 3-3-1 model the leptotonic triplet is arranged as Pisano and
Pleitez (1992); Montero  _et al._ (1993),

$f^{a}_{L}=\begin{pmatrix}\nu^{a}\\\ l^{a}\\\ (l^{c})^{a}\end{pmatrix}$ (4)

where the $a=1,2,3$ is the generation index. As explained above this model is
a consequence of taking $\alpha=-\sqrt{3}$. It is well-known that an
$SU(3)_{L}\times U(1)_{N}$ gauge has 9 gauge bosons. Four of them are
identified as the $W^{\pm}$, $Z$ and the photon. There are other five massive
gauge bosons known as $W^{\prime\pm}$, $U^{\pm\pm}$ and $Z^{\prime}$. The
$W^{\prime}$ gauge boson experiences a charge current similar to the W boson
of the Standard Model. The $U^{\pm\pm}$ is a doubly charged gauge boson. The
dynamics of the gauge boson interactions with leptons are governed by the
charged (${\cal L}^{CC}$) and neutral (${\cal L}^{NC}$) currents Lagrangian
Pisano and Pleitez (1992),

${\cal
L}^{CC}_{l}\supset-\frac{g}{2\sqrt{2}}\left[\bar{\nu}\gamma^{\mu}(1-\gamma_{5})C\bar{l}^{T}W^{\prime-}_{\mu}-\bar{l}\gamma^{\mu}\gamma_{5}C\bar{l}^{T}U^{--}_{\mu}\right],$
(5) ${\cal
L}^{NC}\supset\bar{f}\,\gamma^{\mu}[g_{V}(f)+g_{A}(f)\gamma_{5}]\,f\,Z^{\prime}_{\mu},$
(6)

where $g_{V}$ and $g_{A}$ are the vector and axial coupling constants, which
for charged leptons read,

$\displaystyle g_{A}(l)$
$\displaystyle=\frac{g}{2c_{W}}\frac{\sqrt{3}\sqrt{1-4s_{W}^{2}}}{6},\
g_{V}(l)=3g_{A}(l),$ (7)

with

$g^{\prime}=g\frac{s_{W}}{\sqrt{1-\frac{4}{3}s_{W}^{2}}},$

where, $s_{W}=sen(\theta_{W})$, $c_{W}=cos(\theta_{W})$, and $\theta_{W}$ is
the Weinberg angle. All these gauge bosons will contribute to $a_{\mu}$. We
left out the hadronic sector as it is irrelevant to our g-2 study.

The masses of all fermions and gauge bosons are obtained via the introduction
of three scalar triplets and one scalar sextet,

$\chi=\begin{pmatrix}\chi^{-}\\\ \chi^{--}\\\
\chi^{0}\end{pmatrix},\rho=\begin{pmatrix}\rho^{+}\\\ \rho^{0}\\\
\rho^{++}\end{pmatrix},\eta=\begin{pmatrix}\eta^{0}\\\ \eta_{1}^{+}\\\
\eta_{2}^{+}\end{pmatrix},$ (8)

$S=\left(\begin{array}[]{ccc}\sigma_{1}^{0}&h_{2}^{-}&h_{1}^{+}\\\
h_{2}^{-}&H_{1}^{--}&\sigma_{2}^{0}\\\
h_{1}^{+}&\sigma_{2}^{0}&H_{1}^{++}\end{array}\right),$ (9)

where the vacuum expectation value ($vev$) for every one of the neutral
components of scalars are $\left<\eta^{0}\right>=v_{\eta}$,
$\left<\rho^{0}\right>=v_{\rho}$, $\left<\chi^{0}\right>=v_{\chi}$,
$\left<\sigma^{0}_{2}\right>=v_{\sigma_{2}}$ and
$\left<\sigma^{0}_{1}\right>=v_{\sigma_{1}}$. One may notice that after
spontaneous symmetry breaking mechanism scalar sextet breaks down to a scalar
triplet, doublet and a singlet field Montero  _et al._ (2001). This scalar
sextet is important to generate neutrino masses via a type II seesaw mechanism
Montero  _et al._ (2001); Tully and Joshi (2001); Queiroz  _et al._ (2010);
Ferreira  _et al._ (2019). Similarly to what occurs in the type II seesaw
mechanism, the combination
$v_{\eta}^{2}+v_{\rho}^{2}+v_{\sigma_{2}}^{2}+2v^{2}_{\sigma_{1}}$ contributes
to the $W$ mass, this allows we recognize
$v_{\eta}^{2}+v_{\rho}^{2}+v_{\sigma_{2}}^{2}=v^{2}$, where $v\approx 246$GeV.
In this work we consider $v_{\sigma_{1}}$ to be sufficiently small as required
by the $\rho$ parameter Camargo  _et al._ (2019). In this way we can take
$v_{\eta}=v_{\rho}=v_{\sigma_{2}}\approx v/\sqrt{3}$. We highlight that we did
not include the scalar sextet in our calculations of $a_{\mu}$. The scalar
sextet gives rise to a negative contribution to g-2 and small when compared to
the doubly charged gauge boson. Therefore, its inclusion in our discussion is
simply a matter of completeness.

As a result, the masses of the new gauge bosons are given by Pisano and
Pleitez (1992),

$\displaystyle
M_{W^{\prime}}^{2}=\frac{g^{2}}{4}\left(v_{\eta}^{2}+v_{\chi}^{2}+v_{\sigma_{2}}^{2}+2v^{2}_{\sigma_{1}}\right),$
$\displaystyle
M_{U}^{2}=\frac{g^{2}}{4}\left(v_{\rho}^{2}+v_{\chi}^{2}+4v_{\sigma_{2}}^{2}\right),$
$\displaystyle M_{Z^{\prime}}^{2}\approx\left(\frac{g^{2}+\frac{g^{\prime
2}}{3}}{3}\right)v_{\chi}^{2}.$ (10)

It is important to mention that the relevant interactions involving scalar
fields as far as the muon magnetic moment is concerned are, Foot  _et al._
(1993),

${\cal L}\supset
G_{l}\,\left[\overline{l_{R}}\,\nu_{L}\eta_{1}^{-}+\overline{l_{R}^{c}}\,\nu_{L}h_{1}^{+}+\overline{l_{R}}\,\nu_{L}h_{2}^{+}+\overline{l_{R}}l_{L}R_{\sigma_{2}}\right]+h.c,$
(11)

with $G_{l}=m_{l}\sqrt{2}/v_{\eta}$ and $R_{\sigma_{2}}$ the real component of
the $\sigma_{2}^{0}$, where the masses for $\eta^{+}_{1}$, $h^{+}_{1}$,
$h^{+}_{2}$ and $R_{\sigma_{2}}$ are given by Tonasse (1996),

$\displaystyle M_{\eta_{1}^{+}}^{2}$ $\displaystyle\sim$ $\displaystyle
fv_{\chi},$ $\displaystyle M_{h_{1}^{+},h_{2}^{+}}$ $\displaystyle\sim
v_{\chi},$ $\displaystyle M_{R_{\sigma_{2}}}$ $\displaystyle\sim v_{\chi},$
(12)

where $f$ is an energy parameter whose value must lie around $v_{\chi}$Tonasse
(1996). These scalars interact with leptons through the Yukawa Lagrangian in
Eq.(11) meaning that they couple to leptons proportionally to their masses.
Hence, their contribution to $a_{\mu}$ will be suppressed. The neutral scalar
contribution arises via the Feynman diagram in FIG. 1(a), the charged scalar
via FIG. 1(c), the doubly charged scalar through FIGS. 1(e)-1(f), the
$Z^{\prime}$ via FIG. 1(g), the $W^{\prime}$ via FIG. 1(i) and the
$U^{\pm\pm}$ through FIG. 1(l)-1(m). The doubly charged gauge boson
contribution gives rise to the largest modification of $a_{\mu}$. If we had
included the scalar sextet in our calculations our conclusions would not have
changed.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

Figure 1: Feynmann diagrams that contribute to the muon anomalous magnetic
moment in the 3-3-1 models investigated in this work.

##  IV 3-3-1 r.h.n

The 3-3-1 model that features right-handed neutrinos is called (3-3-1 r.h.n),
the third component of the leptonic triplet is replaced by a right-handed
neutrino. For this reason, the model has the following leptonic sector Long
(1996b, a),

$f^{a}_{L}=\begin{pmatrix}\nu^{a}\\\ l^{a}\\\
(\nu^{c})^{a}\end{pmatrix};\,\,l^{a}_{R}.$ (13)

In this case the five new gauge bosons are the $W^{\prime\pm}$, $Z^{\prime}$,
$X^{0}$ and $X^{0\dagger}$. As the third component of the fermion triplet is a
neutral fermion, there is no doubly charged boson and the $X^{0}$ and
$X^{0\dagger}$ bosons are neutral. Following the notation in Eq.(6), the
vector and axial-vector couplings of the neutral current are found to be Long
(1996b, a),

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

while the charged current takes the form,

$\displaystyle{\cal
L}\supset-\frac{g}{2\sqrt{2}}\left[\overline{\nu^{c}_{R}}\,\gamma^{\mu}(1-\gamma_{5})\bar{l}\,W^{\prime-}_{\mu}\right],$
(14)

which is similar to the Minimal 3-3-1 model. It is important to highlight that
the $X^{0}$ and $X^{0\dagger}$ gauge bosons do not contribute to the neutral
current.

The scalar of the model also features three scalar triplet as follows,

$\chi=\begin{pmatrix}\chi^{0}\\\ \chi^{-}\\\
\chi^{0\prime}\end{pmatrix},\rho=\begin{pmatrix}\rho^{+}\\\ \rho^{0}\\\
\rho^{+\prime}\end{pmatrix},\eta=\begin{pmatrix}\eta^{0}\\\ \eta^{-}\\\
\eta^{0\prime}\end{pmatrix}.$ (15)

The spontaneous symmetry breaking mechanism leads to the gauge boson masses,

$\displaystyle
M_{Z^{\prime}}^{2}=\frac{g^{2}}{4(3-4s_{w}^{2})}\left(4c_{W}^{2}v_{\chi}^{2}+\frac{v_{\rho}^{2}}{c_{W}^{2}}+\frac{v_{\eta}^{2}(1-2s_{W}^{2})^{2}}{c_{W}^{2}}\right),$

and

$\displaystyle
M_{W^{\prime}}^{2}=M_{X^{0}}^{2}=\frac{g^{2}}{4}\left(v_{\eta}^{2}+v_{\chi}^{2}\right).$
(17)

For simplicity we take $\left<\chi^{0\prime}\right>=v_{\chi}$,
$\left<\rho^{0}\right>=v_{\rho}$, $\left<\eta^{0}\right>=v_{\eta}$, with the
other neutral scalar not developing a $vev$ different from zero. We also
assume that the $SU(3)_{L}\times U(1)_{N}$ symmetry breaking energy scale,
$v_{\chi}$, occurs at energies much higher than the weak scale. As before the
condition $v_{\eta}^{2}+v_{\rho}^{2}=v^{2}$ is obeyed with $v\sim
246\,\mbox{GeV}$.

The Yukawa Lagrangian involving the charged scalars is essentially the same as
in Eq.(11) and for the same reason the contributions to $a_{\mu}$ stemming
from charged scalars are small. As far as the scalar sector is concerned,
there is a small difference which is due to the presence of a neutral scalar
that interacts with the muon through the Yukawa lagrangian,

${\cal L}\supset G_{ab}\bar{f_{a}}_{L}\rho e_{b_{R}},$ (18)

which leads to

${\cal L}\supset G_{s}\bar{\mu}\,\mu S_{2},$ (19)

where $G_{s}=m_{\mu}\sqrt{2}/(2v)$.

As the scalar couples to muons proportionally to the muon mass, we conclude
that their corrections to $a_{\mu}$ are dwindled. We included all these
contributions in our computations anyway. The masses of these scalars are
given by,

$\displaystyle M^{2}_{S_{2}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(v_{\chi}^{2}+2v^{2}(2\lambda_{2}-\lambda_{6}))\,$
(20) $\displaystyle M^{2}_{h^{+}}$ $\displaystyle=$
$\displaystyle\frac{\lambda_{8}+\frac{1}{2}}{2}(v^{2}+v_{\chi}^{2})$

where the constants $\lambda_{2}$, $\lambda_{6}$ and $\lambda_{8}$ are
coupling constants of the scalar potential (see Eq.6 of Mizukoshi  _et al._
(2011).). In order to obtain analytical expressions for the masses of the
scalars simplifying assumptions were made concerning some couplings. We
emphasize that they do not affect our conclusions.

Summarizing, the contributions of the 3-3-1 r.h.n model to $a_{\mu}$ rise from
the neutral (FIG. 1(g)) and singly-charged gauge bosons (FIG. 1(i)), the
neutral (FIG. 1(a)) and charged scalars (FIG. 1(c)). The main difference
between this model and the Minimal 3-3-1 model is the absence of a doubly
charged gauge boson which was the main player in the Minimal 3-3-1 model.

##  V 3-3-1 LHN

The 3-3-1 model with heavy neutral lepton (3-3-1 LNH) Mizukoshi  _et al._
(2011); Catano  _et al._ (2012) differs from the 3-3-1 r.h.n because now the
right-handed neutrino is replaced by a heavy neutral lepton ($N$) whose mass
is governed by the scale of symmetry breaking of the 3-3-1 symmetry. The
lepton generations are arranged as follows,

$f^{a}_{L}=\begin{pmatrix}\nu^{a}\\\ l^{a}\\\
N^{a}\end{pmatrix};\,\,l^{a}_{R},N^{a}_{R}.$ (21)

The same scalar triplets defined in Eq.15 appear in this model. Consequently,
the $Z^{\prime}$ and $W^{\prime}$ masses as precisely the same as Eqs.(IV) and
(17) respectively. The neutral current is also the same. There is a subtle but
important difference that resides in the charged current,

$\displaystyle{\cal
L}\supset-\frac{g}{\sqrt{2}}\left[\overline{N_{L}}\,\gamma^{\mu}\bar{l_{L}}\,W^{\prime-}_{\mu}\right].$
(22)

The vector and vector-axial couplings in the charged current that appears in
Eq.22 are easily extracted. Similarly to the 3-3-1 r.h.n this model will have
interactions involving scalars of the form,

$\displaystyle{\cal L}\supset$ $\displaystyle
G_{l}\,\overline{l_{R}}\,N_{L}h_{1}^{-}+G_{l}\,\overline{l_{R}}\,\nu_{L}h_{2}^{+}+G_{s}\bar{\mu}\,\mu
S_{2},$ (23)

where again $G_{s}=m_{\mu}\sqrt{2}/(2v)$, and $G_{l}=m_{l}\sqrt{2}/v_{\eta}$.
We point out that the scalar fields $h_{1}^{-}$ and $h_{2}^{-}$ are the mass
eingenstates that arise of the diagonalization procedure of the
$(\chi^{-},\rho^{\prime-})$ and $(\eta^{-},\rho^{-})$ bases and $S_{2}$ is a
combination of $R_{\eta}$ and $R_{\rho}$Mizukoshi  _et al._ (2011). Their
masses are found to be Mizukoshi  _et al._ (2011); Catano  _et al._ (2012),

$\displaystyle M^{2}_{h^{-}_{1}}$ $\displaystyle=$
$\displaystyle\frac{\lambda_{8}+\frac{1}{2}}{2}(v^{2}+v_{\chi}^{2})\,,$
$\displaystyle M^{2}_{h^{-}_{2}}$ $\displaystyle=$
$\displaystyle\frac{v_{\chi}^{2}}{2}+\lambda_{9}v^{2}\,,$ $\displaystyle
M^{2}_{S_{2}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(v_{\chi}^{2}+2v^{2}(2\lambda_{2}-\lambda_{6})),$
(24)

where $\lambda_{2}$, $\lambda_{8}$, $\lambda_{9}$ are the coupling constants
of the scalar potential in this model (See Eq.(6) of Mizukoshi  _et al._
(2011)). As always, $v_{\chi}$ is assumed to be much larger than $v_{\rho}$
and $v_{\eta}$, as $v_{\rho}^{2}+v_{\eta}^{2}=246^{2}$ GeV2. For simplicity we
take $v_{\rho}=v_{\eta}=v/\sqrt{2}$. We highlight that this assumption does
not interfere in our conclusions as the scalar contributions are suppressed
anyway in comparison with the ones stemming from gauge bosons.

In summary, in the 3-3-1 LHN we have a contributing coming from the $S_{2}$ (
FIG. 1(a) ), $h_{2}^{-}$ ( FIG. 1(c) ), $h_{1}^{-}$ ( FIG. 1(d) ),
$Z^{\prime}$ (FIG. 1(g) ) and $W^{\prime}$ ( FIG. 1(j) ).

##  VI Economical 3-3-1 Model

The Economical 3-3-1 model Dong  _et al._ (2006); Dong and Long (2008);
Berenstein  _et al._ (2009); Martinez and Ochoa (2014) is built with the same
leptonic triplet of the 3-3-1 r.n.h model, but now with scalar sector reduced
to incorporate two scalar triplet , only:

$\chi=\begin{pmatrix}\chi_{1}^{0}\\\ \chi_{2}^{-}\\\
\chi_{3}^{0}\end{pmatrix},\eta=\begin{pmatrix}\eta_{1}^{+}\\\ \eta_{2}^{0}\\\
\eta_{3}^{+}\end{pmatrix},$ (25)

where the $vev$ of neutral fields takes the form
$\left<\eta^{0}_{2}\right>=v_{\eta^{0}_{2}}=v/\sqrt{2}$,
$\left<\chi^{0}_{1}\right>=v_{\chi_{1}}=u/\sqrt{2}$,
$\left<\chi^{0}_{3}\right>=v_{\chi^{0}_{3}}=v_{\chi}/\sqrt{2}$. Once again we
assume that $u,v\ll v_{\chi}$.

The corrections for $a_{\mu}$ arise from neutral and charged scalars via
Yukawa Lagrangian Dong  _et al._ (2006),

${\cal L}\supset
G^{\ell}_{ij}\overline{f}_{iL}\eta\ell_{jR}+G^{\epsilon}_{ij}\epsilon_{pmn}(\overline{f}^{c}_{iL})_{p}(f_{iL})_{m}(\eta)_{n}+h.c.$
(26)

From this Lagrangian we will obtain terms that go with
$G_{l}\bar{l_{R}}\nu_{L}\eta_{1}^{+}$ and $G_{s}\bar{\mu}\mu S_{2}$, where,

$\displaystyle
M_{\eta_{1}^{+}}^{2}=\frac{\lambda_{4}}{2}\left(u^{2}+v^{2}+v_{\chi}^{2}\right),\
M_{S_{2}}^{2}=2\lambda_{1}v_{\chi}^{2}.$ (27)

The neutral and charged currents are the same as the 3-3-1 r.h.n, but the
masses of the gauge bosons take different forms because we now have only two
scalar triplets,

$M_{Z^{\prime}}^{2}\approx\frac{g^{2}c^{2}_{W}v_{\chi}^{2}}{3-4s_{W}^{2}},\
\,\ M_{W^{\prime}}^{2}=\frac{g^{2}}{4}\left(v_{\eta}^{2}+v_{\chi}^{2}\right).$
(28)

The Economical 3-3-1 features basically the same new physics contributions to
$a_{\mu}$ as the 3-3-1 r.h.n. The main differences appear in the expressions
for the $W^{\prime}$ and $Z^{\prime}$ masses. As we are going to plot the
overall contribution of each 3-3-1 model to $a_{\mu}$ as a function of the
$v_{\chi}$, as we will see, the results will differ.

##  VII 3-3-1 Model with exotic Leptons

3-3-1 models are known for being versatile, with many possible combinations of
fermionic and scalar multiplets that can be constructed in an anomaly-free
way. However, the price that usually has to be paid is that the quarks
generations cannot be represented equally under $SU(3)_{L}$. This happens in
the 3-3-1 model with exotic leptons. A consequence of this necessity is that
$Z^{\prime}$ interactions are no longer universal, leading to flavor changing
neutral currents at tree level Cogollo  _et al._ (2014b). In this section, the
goal is to investigate models that have generations with different
representations under $SU(3)_{L}$ in the leptonic sector. Here we are going to
consider the following leptonic sector Ponce  _et al._ (2002a)

$f_{1L}=\begin{pmatrix}\nu_{1}\\\ l_{1}\\\
E_{1}^{-}\end{pmatrix};\,\,l_{1}^{c};f_{2,3L}=\begin{pmatrix}\nu_{2,3}\\\
l_{2,3}\\\ N_{2,3}\end{pmatrix};\,\,l_{2,3}^{c};$ (29)
$f_{4L}=\begin{pmatrix}E_{2}^{-}\\\ N_{3}\\\
N_{4}\end{pmatrix};\,\,E_{2}^{c};\\\ f_{5L}=\begin{pmatrix}N_{5}\\\
E_{3}^{+}\\\ l_{3}^{+}\end{pmatrix};\,\,E_{3}^{c};$ (30)

where $N$ and $E$ are the exotic neutral and charged leptons, respectively.

We label the new gauge bosons as $K^{\pm}$, $K^{0}$ and $Z^{\prime}$. The
scalar sector is formed by three triplets, given by Ponce  _et al._ (2002a,
b); Anderson and Sher (2005); Cabarcas  _et al._ (2014),

$\chi_{i}=\begin{pmatrix}\chi_{i}^{-}\\\ \chi_{i}^{0}\\\
\chi_{i}^{0\prime}\end{pmatrix},\chi_{3}=\begin{pmatrix}\chi_{3}^{0}\\\
\chi_{3}^{+}\\\ \chi_{3}^{\prime+}\end{pmatrix}.$ (31)

with $i=1,2$, $\left<\chi_{1}\right>=\left(0,\ 0,\ v_{\chi}\right)^{T}$,
$\left<\chi_{2}\right>=\left(0,\ v/\sqrt{2},\ 0\right)^{T}$ and
$\left<\phi_{3}\right>=\left(v^{\prime}/\sqrt{2},\ 0,\ 0\right)^{T}$, where
$v_{\chi}\gg v,v^{\prime}$, with $v^{\prime}\sim v$.

The relevant interactions to $a_{\mu}$ are Cabarcas  _et al._ (2014),

$\displaystyle{\cal L}\supset$
$\displaystyle\frac{g^{\prime}}{2\sqrt{3}s_{W}c_{W}}\bar{\mu}\gamma_{\mu}\left(g_{V}+g_{A}\right)\mu\,Z^{\prime}$
(32)
$\displaystyle-\frac{g}{\sqrt{2}}\left(\overline{N_{1L}}\,\gamma_{\mu}\mu_{L}+\bar{\mu}_{L}\gamma_{\mu}N_{4L}\right)K^{+}_{\mu}$
$\displaystyle-\frac{g}{\sqrt{2}}\left(\bar{\mu}_{L}\gamma_{\mu}E_{L}\right)K^{0}_{\mu}+h_{1}\bar{\mu}(1-\gamma_{5})N\chi^{+}$
$\displaystyle+h_{2}\bar{\mu}E^{-}\chi^{0}+h_{3}\bar{\mu}E_{2}^{-}\chi^{0}+\mbox{H.c.},$

where $\chi^{+}$ and $\chi^{0}$ are scalars coming from the scalar triplets.
The vector and vector-axial couplings of the $Z^{\prime}$ to charged leptons
are,

$g_{V}=\frac{-c_{2W}+2s_{W}^{2}}{2},\,g_{A}=\frac{c_{2W}+2s_{W}^{2}}{2},$ (33)

and the masses of the gauge bosons are found to be Cabarcas  _et al._ (2014),

$\displaystyle M_{Z^{\prime}}^{2}$ $\displaystyle=$
$\displaystyle\frac{2}{9}\left(3g^{2}+g^{\prime 2}\right)v_{\chi}^{2},$
$\displaystyle M_{K^{+}}^{2}$ $\displaystyle=M_{K^{0}}^{2}=$
$\displaystyle\frac{g^{2}}{4}\left(2v_{\chi}^{2}+v^{2}\right),$ $\displaystyle
g^{\prime}$ $\displaystyle=$
$\displaystyle\frac{g\tan_{W}}{\sqrt{1-\tan_{W}^{2}/3}}.$ (34)

As highlighted previously, the corrections to $a_{\mu}$ coming from the
scalars are suppressed by the lepton masses, so they will not be considered in
this case. Hence, the main corrections to $a_{\mu}$ rise from the gauge bosons
$Z^{\prime}$ ( FIG. 1(g)), $K^{0}$ ( FIG. 1(h)) and $K^{-}$ ( FIG. 1(k)). The
Feynman diagrams induced by the presence of $K^{-}$ and $K^{0}$ also involve
an exotic charged lepton $E$, whose mass can be very large.

##  VIII Results

In this section, we will present the main results of our work and put them
into perspective with collider bounds. In all models discussed here, the
leading corrections to $a_{\mu}$ stem from the new gauge bosons. As we are
dealing with a $SU(3)_{L}$ gauge group there are several gauge bosons that
play a role in $a_{\mu}$. We have computed all individual corrections to
$a_{\mu}$ and expressed our results in terms of the scale of symmetry
breaking, $v_{\chi}$. For completeness we display the individual contributions
in the Appendix. As the gauge bosons and scalar fields have masses governed by
$v_{\chi}$ we can sum up all the individual contributions and draw conclusions
in terms of the total $\Delta a_{\mu}$ for each model. The main results are
summarized in the Table 1. We start discussing our findings in the context of
the Minimal 3-3-1 model.

###  VIII.1 Minimal 3-3-1

In FIG. 2 we show the total contribution to $\Delta a_{\mu}$ as a function of
$v_{\chi}$ for the Minimal 3-3-1 model with a black curve. There we also
display a green band at which the $a_{\mu}$ anomaly is addressed. Moreover, we
exhibit the current and projected $1\sigma$ bounds considering the precision
aimed by the g-2 experiment at FERMILAB. It is important to mention that the
individual contributions were calculated numerically considering
$v_{\eta}=v_{\rho}=147$ GeV. The $Z^{\prime}$ corrections to $a_{\mu}$ is
small. The largest corrections to $a_{\mu}$ stem from the singly
($W^{\prime}$) and doubly ($U^{++}$) charged gauge bosons. The latter gives
rise to a negative contribution to $a_{\mu}$. This is an important point that
has been overlooked in Ky  _et al._ (2000); Kelso  _et al._ (2014b). The
vector current vanishes for a doubly charged gauge boson Lindner  _et al._
(2018). The main contributions come from the $W^{\prime}$ and $U^{++}$ gauge
bosons in the Minimal 3-3-1 model, which are positive and negative
respectively. The overall correction to $\Delta a_{\mu}$ is small and
negative. Looking at FIG. 2 one can easily conclude that the $v_{\chi}>1.8$
TeV, and using the projected g-2 sensitivity we will be able to impose
$v_{\chi}>2.8$ TeV. This model cannot explain $\Delta a_{\mu}$ as it yields a
negative correction to $a_{\mu}$. Moreover, one should keep in mind the
existing collider bounds which were placed on the mass of the new gauge bosons
(See Table 1). One may use the relations $M_{Z^{\prime}}=0.395v_{\chi}$, and
$M_{W^{\prime}}=M_{U^{\pm\pm}}=0.33v_{\chi}$ to obtain constraints on the
scale of symmetry breaking. Currently LHC data imposes $M_{Z^{\prime}}>3.7$
TeV which means $v_{\chi}>9.3$ TeV. Having in mind the Landau pole found at
5TeV, the Minimal 3-3-1 model has been excluded by LHC searches. There are
ways to extend the Minimal 3-3-1 model and add extra decay channels to the
$Z^{\prime}$ gauge boson Machado  _et al._ (2019), without altering the g-2
predictions. Such new decay modes may weaken the LHC bound. Therefore it is
worthwhile to determine the contributions to $a_{\mu}$ as it gives rise to an
orthogonal and complementary bound on the model.

We point out that each contribution to $a_{\mu}$ depends on the mass of the
particles running in the loop. However, we can describe each correction to
$a_{\mu}$ in terms of the scale of symmetry breaking, $v_{\chi}$, as the
particle masses depend on it. In this way, we obtain the lower mass bounds on
the masses of the gauge bosons using FIG. 2. We summarize these bounds in
Table 1.

In summary, if the $a_{\mu}$ anomaly is confirmed by the g-2 experiment, the
Minimal 3-3-1 model cannot offer an answer. New interactions with charged
leptons must arise.

Figure 2: Overall contribution to $\Delta a_{\mu}$ from the Minimal 3-3-1
model. The green bands are delimited by $\Delta a_{\mu}=(261\pm 78)\times
10^{-11}$. The current $1\sigma$ bound is found by requiring $\Delta
a_{\mu}<78\times 10^{-11}$ while the projected bound is obtained for $\Delta
a_{\mu}<34\times 10^{-11}$. We used $M_{Z^{\prime}}=0.395v_{\chi}$,
$M_{W^{\prime}}=M_{U^{\pm\pm}}=0.33v_{\chi}$.  Figure 3: Overall contribution
to $\Delta a_{\mu}$ from the 3-3-1 r.h.n model. We used
$M_{Z^{\prime}}=0.395v_{\chi}$, $M_{W^{\prime}}=M_{U^{\pm\pm}}=0.33v_{\chi}$.

Figure 4: Overall contribution to $\Delta a_{\mu}$ from the 3-3-1 LHN model
for $m_{N}=1$ GeV (left-panel) and $m_{N}=100$ GeV (right-panel). One can
clearly from the plots that our conclusions concerning the 3-3-1 LHN heavily
depend on mass used for the neutral lepton. For $m_{N}=100$ GeV we can place a
projected limit of $v_{\chi}>6.4$ TeV, whereas for $m_{N}\leq 240$ GeV no
limit on the scale of symmetry breaking can be found because the corrections
to $\Delta a_{\mu}$ is too small. We used
$M_{Z^{\prime}}=0.395v_{\chi}$,$M_{W^{\prime}}=M_{U^{\pm\pm}}=0.33v_{\chi}$.
Figure 5: Overall contribution to $\Delta a_{\mu}$ from the Economical 3-3-1
model. We used
$M_{Z^{\prime}}=0.395v_{\chi}$,$M_{W^{\prime}}=M_{U^{\pm\pm}}=0.33v_{\chi}$.

Figure 6: The overall $\Delta a_{\mu}$ with contribution from the 3-3-1 model
with exotic leptons assuming $m_{N}=10$ GeV, $m_{E}=150$ GeV (left-panel) and
$m_{N}=100$ GeV, $m_{E}=500$ GeV (right-panel). We used
$M_{Z^{\prime}}=0.55v_{\chi}$, $M_{K^{\prime}}=M_{K^{0}}=0.46v_{\chi}$.

###  VIII.2 3-3-1 r.h.n

In FIG.3 we show $\Delta a_{\mu}$ prediction from the 3-3-1 r.h.n model. The
individual contributions are shown in the Appendix. The $W^{\prime}$ and
$Z^{\prime}$ gauge bosons give rise to the largest corrections to $a_{\mu}$.
As we explained earlier we can express each correction to $a_{\mu}$ in terms
of the scale of symmetry breaking as the particle masses depend on it. This
feature allowed us to plot $\Delta a_{\mu}$ as a function of $v_{\chi}$ in
FIG.3. We conclude from FIG. 3 that $v_{\chi}<200$ GeV is needed to reproduce
the measured $\Delta a_{\mu}$. However, such a small value has been ruled by
collider searches for $Z^{\prime}$ bosons. Collider bounds impose
$M_{Z^{\prime}}>4$ TeV Lindner  _et al._ (2018), which implies that
$v_{\chi}>12$ TeV. We hilight that such lower mass bound on the $Z^{\prime}$
mass is not robust because it assumed that the $Z^{\prime}$ field decays only
into charged leptons, which may not be true in this model as the $Z^{\prime}$
can also decay into exotic quarks. The presence of such decay modes certainly
will weaken the LHC limit. We have implemented the model in in Calchep Belyaev
_et al._ (2013) via Lanhep Semenov (2016) and found that indeed the
$Z^{\prime}$ branching ratio into charged leptons can be suppressed up to
$34\%$ when all three exotic quarks are sufficiently light for the
$Z^{\prime}$ decay into. Therefore, conservatively speaking the lower mass
bound on the $Z^{\prime}$ boson should read $2.64$ TeV, implying that
$v_{\chi}>6.68$ TeV. Hence, regardless of the inclusion of new exotic decays,
collider bounds still forbid the 3-3-1 r.h.n model to provide a solution to
the muon anomalous magnetic moment anomaly. That said, we obtain lower mass
bounds requirement the 3-3-1 r.h.n contribution to $a_{\mu}$ to be within the
$1\sigma$ error bars. The bounds are summarized in Table 1.

###  VIII.3 3-3-1 LHN

The 3-3-1 model that features the presence of neutral leptons instead of
right-handed neutrinos is known as 3-3-1 LHN. This ingredient is sufficient to
change out conclusions. It is a fact unexplored in the literature. Again we
show for completeness the individual contributions in the APPENDIX. In Fig. 4
we show $\Delta a_{\mu}$ when the contribution from the 3-3-1 LHN model are
accounted for, assuming $M_{N}=1$ GeV. At first look it appears that the 3-3-1
LHN may explain the anomaly for $1\,{\rm TeV}<v_{\chi}<2$ TeV. However, the
collider bounds we discussed previously in the 3-3-1 r.h.n model are in
principle also applicable here. Thus, one could imagine that $v_{\chi}$ should
be also larger than $12$ TeV to be consistent with current LHC bounds on the
$Z^{\prime}$ mass. As aforementioned, such collider bound can be weakened by
the presence of $Z^{\prime}$ decays into exotic quarks. In this model, there
are additional decay models in N’s pairs. We have implemented the model in
Calchep Belyaev  _et al._ (2013) via Lanhep Semenov (2016) and assessed the
impact of these new decay in the $Z^{\prime}$ branching ratio into charged
leptons. We concluded that branching ratio into charged leptons may be
diminished up to 60%, when also three exotic quarks and three neutral leptons,
N, are light enough for the $Z^{\prime}$ to decay into. Conservatively
speaking, it means that the lower mass bound should also be weakened by 50%,
which implies that $M_{Z^{\prime}}>2$ TeVs and that $v_{\chi}>5.06$ TeV. Thus,
even considering new exotic decays, the 3-3-1 LHN is not capable of addressing
the muon anomaly because the scale of symmetry breaking is too small to be
consistent with LHC limits. The conclusion would not change if we had adopted
different values for $M_{N}$. Enforcing the $\Delta a_{\mu}$ to be smaller
than $78\times 10^{-11}$ and $34\times 10^{-11}$ we get $v_{\chi}>407$ GeV and
$v_{\chi}>722$ GeV that yield $M_{Z^{\prime}}>160$ GeV and
$M_{Z^{\prime}}>285$ GeV, respectively. A similar logic applies to the
$W^{\prime}$ boson. Our lower mass bounds are shown in Table 1 even for the
case when $M_{N}=100$ GeV.

###  VIII.4 Economical 3-3-1

The total correction to $a_{\mu}$ from the Economical 3-3-1 model is displayed
in FIG. 5. This model is rather similar to the 3-3-1 r.h.n model. Thus it
cannot accommodate the $a_{\mu}$ anomaly either. The main difference appear in
the mass of the gauge bosons which have a different relation with $v_{\chi}$.
This results into different lower mass bounds summarized in Table 1. Looking
at Table 1 can easily conclude that the constraints derived from $a_{\mu}$ on
this model are quite weak compared to those stemming from collider searches.

Model | LHC-13TeV | g-2 current | g-2 projected
---|---|---|---
Minimal 3-3-1 |  $M_{Z^{\prime}}>3.7$ TeV Nepomuceno and Meirose (2020) |  $M_{Z^{\prime}}>434.5$ GeV |  $M_{Z^{\prime}}>632$ GeV
|  $M_{W^{\prime}}>3.2$ TeV Nepomuceno and Meirose (2020) |  $M_{W^{\prime}}>646$ GeV |  $M_{W^{\prime}}>996.1$ GeV
3-3-1 r.h.n |  ${}^{*}M_{Z^{\prime}}>2.64$ TeV Lindner  _et al._ (2018) |  $M_{Z^{\prime}}>158$ GeV |  $M_{Z^{\prime}}>276.5$ GeV
| ——- |  $M_{W^{\prime}}>133$ GeV |  $M_{W^{\prime}}>239$ GeV
3-3-1 LHN |  ${}^{*}M_{Z^{\prime}}>2$ TeV Lindner  _et al._ (2018) |  $M_{Z^{\prime}}>160$ GeV |  $M_{Z^{\prime}}>285$ GeV
for $M_{N}=1$ GeV | —- |  $M_{W^{\prime}}>134.3$ GeV |  $M_{W^{\prime}}>238.3$ GeV
3-3-1 LHN |  ${}^{*}M_{Z^{\prime}}>2$ TeV Lindner  _et al._ (2018) |  $M_{Z^{\prime}}>136.7$ GeV |  $M_{Z^{\prime}}>276.5$ GeV
for $M_{N}=100$ GeV | —— |  $M_{W^{\prime}}>114.2$ GeV |  $M_{W^{\prime}}>231$ GeV
Economical |  ${}^{*}M_{Z^{\prime}}>2.64$ TeV Lindner  _et al._ (2018) |  $M_{Z^{\prime}}>59.3$ GeV |  $M_{Z^{\prime}}>271.4$ GeV
3-3-1 | —— |  $M_{W^{\prime}}>49.5$ GeV |  $M_{W^{\prime}}>226.7$ GeV
3-3-1 exotic leptons |  ${}^{*}M_{Z^{\prime}}>2.91$ TeV Salazar  _et al._ (2015) |  $M_{Z^{\prime}}>429$ GeV |  $M_{Z^{\prime}}>693$ GeV
for $m_{N}=10$ GeV, $m_{E}=150$ GeV | —— |  $M_{W^{\prime}}>359$ GeV |  $M_{W^{\prime}}>579.6$ GeV
3-3-1 exotic leptons |  ${}^{*}M_{Z^{\prime}}>2.91$ TeV Salazar  _et al._ (2015) |  $M_{Z^{\prime}}>369$ GeV |  $M_{Z^{\prime}}>600$ GeV
for $m_{N}=10$ GeV, $m_{E}=150$ GeV | —— |  $M_{W^{\prime}}>309.1$ GeV |  $M_{W^{\prime}}>501.4$ GeV
Table 1: Summary of the lower bounds based on our calculations. For comparison
we include the LHC bounds at 13TeV center-of-mass energy. These LHC bounds are
based on either $36fb^{-1}$ or $139fb^{-1}$ of data. The lower mass bounds on
the $Z^{\prime}$ and $W^{\prime}$ bosons from the 3-3-1 r.h.n model are also
applicable to the Economical 3-3-1 model as they have the same interactions.
We emphasize that the limits quoted in Lindner  _et al._ (2018) neglected
$Z^{\prime}$ decays into exotic quarks and heavy leptons which can weaken the
lower mass bounds up to 50%. The same is true for the bounds obtained in
Salazar  _et al._ (2015). The effect of $Z^{\prime}$ exotic decays were
considered in the Minimal 3-3-1 model in Nepomuceno and Meirose (2020).
Conversely, the lower mass constraints we found are robust.

###  VIII.5 3-3-1 with exotic leptons

The 3-3-1 model with exotic leptons features gauge bosons, $K^{\pm}$ and
$K^{0}$, that induce the largest corrections to $a_{\mu}$. The interactions
that these bosons experience are absent in the previous models, because they
couple to exotic charged leptons, E, which can be heavy. The values obtained
for $\Delta a_{\mu}$ are shown in FIGs. 6 for $M_{N}=10$ GeV, $M_{E}=150$ GeV
(left-panel) and $M_{N}=100$ GeV, $M_{E}=500$ GeV (right-panel). We considered
these two cases to assess the dependence of our finding on the masses of these
particles. We remind that the individual contributions are also exhibited in
the Appendix. The total contribution to $a_{\mu}$ is rather small because
there more than one heavy field running in the loop. Consequently, the scale
of symmetry breaking needed to reproduce the measured $\Delta a_{\mu}$ value
is too small to be consistent with collider bounds. Therefore, once again, we
can simply derive $1\sigma$ lower bound on scale of symmetry breaking which
can be translated into lower mass limits on the gauge boson masses as shown in
Table 1.

##  IX Discussions

We have shown that all five models studied cannot accommodate the muon
anomalous magnetic moment anomaly in agreement with existing bounds. Our
conclusion holds even if one departs from the $v_{\eta}=v_{\rho}$ assumption
adopted throughout. These choices for $v_{\eta}$ and $v_{\rho}$ affect more
the scalar contributions which are already suppressed by the muon mass.
However, one can think of ways of extending such models that give rise to
sizeable contribution to $a_{\mu}$ while being consistent with collider
bounds. One way could be via the addition of an inert scalar triplet under
$SU(3)_{L}$ Carcamo Hernandez  _et al._ (2020), inert singlet scalar, vector-
like leptons, etc., which represent viable possibilities to accommodate this
anomaly Queiroz and Shepherd (2014). However, each of these avenues constitute
a new model, with either own phenomenological implications that we plan to
explore in the near future. For concreteness, we have added an inert scalar
triplet, $\phi$, into the 3-3-1 LHN model and computed the overall
contribution to $a_{\mu}$. Such scalar triplet which gets a mass from the
quartic coupling in the scalar potential,
$\lambda\phi^{\dagger}\phi\chi^{\dagger}\chi$, after the scalar triplet $\chi$
acquires a $vev$. This mass goes as $M_{\phi}\sim\lambda v_{\chi}$. This inert
triplet scalar allow us to include $\mathcal{L}\supset y_{ab}\bar{f_{a}}\phi
e_{bR}$. Taking $y_{22}$ equal to unit we get FIG. 7. There we display the
total contribution of the 3-3-1 LHN model augmented by this inert scalar
triplet. We can now thrivingly explain the muon anomalous magnetic moment
anomaly for $v_{\chi}\sim 10$ TeV, while being consistent with LHC constraint
that rules out the region with $v_{\chi}<5.06$ TeV. Therefore, we have
conclusively presented a solution to the muon anomalous magnetic moment in the
context of 3-3-1 models.

Figure 7: Overall contribution of the 3-3-1 LHN model augmented by an inert
scalar triplet $\phi$. The LHC bound is represented by a vertical line which
excluded the region of parameter space to the left. One can notice that this
extended version of the 3-3-1 LHN model now successfully accommodate the
$a_{\mu}$ anomaly for $v_{\chi}\sim 10$ TeV, while being consistent with LHC
constraint.

##  X Conclusions

We have revisited the contribution to the muon anomalous magnetic moment
stemming from five different models based on the $SU(3)_{C}\times
SU(3)_{L}\times U(1)_{N}$ gauge symmetry. We have assessed the impact of
changing the masses of the exotic leptons present in such models and shown
that our quantitative conclusions do change depending on the value assumed for
their masses. A fact unexplored before. Moreover, we corrected previous
estimations in the literature such as the contribution stemming from the
doubly charged gauge boson, as it does not have a vectorial current to muons.
Moreover, we have drawn our conclusions in perspective with collider bounds
and concluded that none of the five models investigated here are capable of
accommodating the anomaly. Consequently, we derived robust and complementary
$1\sigma$ lower mass bounds on the masses of the new gauge bosons, namely the
$Z^{\prime}$ and $W^{\prime}$ bosons. In summary, if the anomaly observed in
the muon anomalous magnetic moment is confirmed by the g-2 experiment at
FERMILAB these models must be extended. For concreteness, we presented a
plausible extension to the 3-3-1 LHN model, which features an inert scalar
triplet. This extension can accommodate the anomaly for $v_{\chi}\sim 10$ TeV,
while being consistent with LHC limits. We make our Mathematica numerical
codes available at Queiroz (2020) to allow the reader to double check our
findings and apply our tool to other studies of the muon anomalous magnetic
moment.

## Acknowledgement

The authors are grateful to Antonio Santos for discussions. SK acknowledges
support from CONICYT-Chile Fondecyt No. 1190845 and ANID-Chile PIA/APOYO
AFB180002. ASJ and YSV acknowledge support from CAPES. CASP is supported by
the CNPq research grants No. 304423/2017-3. FSQ thanks CNPq grants
303817/2018-6 and 421952/2018-0, and ICTP-SAIFR FAPESP grant 2016/01343-7 for
the financial support. We thank the High Performance Computing Center (NPAD)
at UFRN for providing computational resources.

##  XI Appendix

In this appendix we present for completeness the individual contributions of
the particles to $a_{\mu}.$

Figure 8: Overall $\Delta a_{\mu}$ taking into account each individual
contribution from the Minimal 3-3-1 model and 3-3-1 r.h.n. Figure 9: Overall
$\Delta a_{\mu}$ taking into account each individual contribution from the
3-3-1 LHN model for $m_{N}=1$ GeV

Figure 10: Overall $\Delta a_{\mu}$ taking into account each individual
contribution from the 3-3-1 LHN model for $m_{N}=100$ GeV (left-panel) and
$m_{N}=240$ GeV (right-panel). Figure 11: Overall $\Delta a_{\mu}$ taking into
account each individual contribution from the Economical 3-3-1 model.

Figure 12: Overall $\Delta a_{\mu}$ taking into account each individual
contribution from the 3-3-1 with exotic leptons, for $m_{N}=10$GeV,
$m_{E}=150$ GeV (left-panel) and $m_{N}=100$GeV, $m_{E}=500$ GeV (right-
panel).

## References

  * Lindner  _et al._ (2018) M. Lindner, M. Platscher, and F. S. Queiroz, Phys. Rept. 731, 1 (2018), arXiv:1610.06587 [hep-ph] .
  * Bennett  _et al._ (2002) G. W. Bennett _et al._ (Muon g-2), Phys. Rev. Lett. 89, 101804 (2002), [Erratum: Phys. Rev. Lett.89,129903(2002)], arXiv:hep-ex/0208001 [hep-ex] .
  * Bennett  _et al._ (2006) G. W. Bennett _et al._ (Muon g-2), Phys. Rev. D73, 072003 (2006), arXiv:hep-ex/0602035 [hep-ex] .
  * Prades  _et al._ (2009) J. Prades, E. de Rafael, and A. Vainshtein, Adv. Ser. Direct. High Energy Phys. 20, 303 (2009), arXiv:0901.0306 [hep-ph] .
  * Tanabashi  _et al._ (2018) M. Tanabashi _et al._ (Particle Data Group), Phys. Rev. D98, 030001 (2018).
  * Benayoun  _et al._ (2013) M. Benayoun, P. David, L. DelBuono, and F. Jegerlehner, Eur. Phys. J. C73, 2453 (2013), arXiv:1210.7184 [hep-ph] .
  * Blum  _et al._ (2013) T. Blum, A. Denig, I. Logashenko, E. de Rafael, B. L. Roberts, T. Teubner, and G. Venanzoni, (2013), arXiv:1311.2198 [hep-ph] .
  * Benayoun  _et al._ (2015) M. Benayoun, P. David, L. DelBuono, and F. Jegerlehner, Eur. Phys. J. C75, 613 (2015), arXiv:1507.02943 [hep-ph] .
  * Jegerlehner (2018) F. Jegerlehner,  _Proceedings, KLOE-2 Workshop on $e^{+}e^{-}$ Collision Physics at 1 GeV: Frascati, Italy, October $26-28$, 2016_, EPJ Web Conf. 166, 00022 (2018), arXiv:1705.00263 [hep-ph] .
  * Keshavarzi  _et al._ (2018) A. Keshavarzi, D. Nomura, and T. Teubner, Phys. Rev. D97, 114025 (2018), arXiv:1802.02995 [hep-ph] .
  * Grange  _et al._ (2015) J. Grange _et al._ (Muon g-2), (2015), arXiv:1501.06858 [physics.ins-det] .
  * Abe  _et al._ (2019) M. Abe _et al._ , PTEP 2019, 053C02 (2019), arXiv:1901.03047 [physics.ins-det] .
  * Al-Binni  _et al._ (2013) U. Al-Binni _et al._ , in  _Proceedings, 2013 Community Summer Study on the Future of U.S. Particle Physics: Snowmass on the Mississippi (CSS2013): Minneapolis, MN, USA, July 29-August 6, 2013_, edited by A. S. Kronfeld and R. S. Tschirhart (2013) arXiv:1306.5009 [hep-ex] .
  * Carey  _et al._ (2009) R. M. Carey _et al._ ,  (2009), 10.2172/952029.
  * Foot  _et al._ (1993) R. Foot, O. F. Hernandez, F. Pisano, and V. Pleitez, Phys. Rev. D47, 4158 (1993), arXiv:hep-ph/9207264 [hep-ph] .
  * Montero  _et al._ (2001) J. C. Montero, C. A. de S. Pires, and V. Pleitez, Phys. Lett. B502, 167 (2001), arXiv:hep-ph/0011296 [hep-ph] .
  * Tully and Joshi (2001) M. B. Tully and G. C. Joshi, Phys. Rev. D64, 011301 (2001), arXiv:hep-ph/0011172 [hep-ph] .
  * Montero  _et al._ (2002) J. C. Montero, C. A. De S. Pires, and V. Pleitez, Phys. Rev. D65, 095001 (2002), arXiv:hep-ph/0112246 [hep-ph] .
  * Cortez and Tonasse (2005) N. V. Cortez and M. D. Tonasse, Phys. Rev. D72, 073005 (2005), arXiv:hep-ph/0510143 [hep-ph] .
  * Cogollo  _et al._ (2009) D. Cogollo, H. Diniz, and C. A. de S. Pires, Phys. Lett. B677, 338 (2009), arXiv:0903.0370 [hep-ph] .
  * Cogollo  _et al._ (2010) D. Cogollo, H. Diniz, and C. A. de S. Pires, Phys. Lett. B687, 400 (2010), arXiv:1002.1944 [hep-ph] .
  * Cogollo  _et al._ (2008) D. Cogollo, H. Diniz, C. A. de S. Pires, and P. S. Rodrigues da Silva, Eur. Phys. J. C58, 455 (2008), arXiv:0806.3087 [hep-ph] .
  * Okada  _et al._ (2016) H. Okada, N. Okada, and Y. Orikasa, Phys. Rev. D93, 073006 (2016), arXiv:1504.01204 [hep-ph] .
  * Vien  _et al._ (2019) V. V. Vien, H. N. Long, and A. E. Carcamo Hernandez, Mod. Phys. Lett. A34, 1950005 (2019), arXiv:1812.07263 [hep-ph] .
  * Cárcamo Hernández  _et al._ (2018) A. E. Cárcamo Hernández, H. N. Long, and V. V. Vien, Eur. Phys. J. C78, 804 (2018), arXiv:1803.01636 [hep-ph] .
  * Nguyen  _et al._ (2018) T. P. Nguyen, T. T. Le, T. T. Hong, and L. T. Hue, Phys. Rev. D97, 073003 (2018), arXiv:1802.00429 [hep-ph] .
  * de Sousa Pires  _et al._ (2019) C. A. de Sousa Pires, F. Ferreira De Freitas, J. Shu, L. Huang, and P. Wagner Vasconcelos Olegário, Phys. Lett. B797, 134827 (2019), arXiv:1812.10570 [hep-ph] .
  * Carcamo Hernandez  _et al._ (2019a) A. E. Carcamo Hernandez, N. A. Perez-Julve, and Y. Hidalgo Velasquez, Phys. Rev. D100, 095025 (2019a), arXiv:1907.13083 [hep-ph] .
  * Carcamo Hernandez  _et al._ (2019b) A. E. Carcamo Hernandez, Y. Hidalgo Velasquez, and N. A. Perez-Julve, Eur. Phys. J. C79, 828 (2019b), arXiv:1905.02323 [hep-ph] .
  * Cárcamo Hernández  _et al._ (2020) A. E. Cárcamo Hernández, L. T. Hue, S. Kovalenko, and H. N. Long, (2020), arXiv:2001.01748 [hep-ph] .
  * Fregolente and Tonasse (2003) D. Fregolente and M. D. Tonasse, Phys. Lett. B555, 7 (2003), arXiv:hep-ph/0209119 [hep-ph] .
  * Long and Lan (2003) H. N. Long and N. Q. Lan, Europhys. Lett. 64, 571 (2003), arXiv:hep-ph/0309038 [hep-ph] .
  * de S. Pires and Rodrigues da Silva (2007) C. A. de S. Pires and P. S. Rodrigues da Silva, JCAP 0712, 012 (2007), arXiv:0710.2104 [hep-ph] .
  * Mizukoshi  _et al._ (2011) J. K. Mizukoshi, C. A. de S. Pires, F. S. Queiroz, and P. S. Rodrigues da Silva, Phys. Rev. D83, 065024 (2011), arXiv:1010.4097 [hep-ph] .
  * Profumo and Queiroz (2014) S. Profumo and F. S. Queiroz, Eur. Phys. J. C74, 2960 (2014), arXiv:1307.7802 [hep-ph] .
  * Dong  _et al._ (2013a) P. V. Dong, T. P. Nguyen, and D. V. Soa, Phys. Rev. D88, 095014 (2013a), arXiv:1308.4097 [hep-ph] .
  * Dong  _et al._ (2013b) P. V. Dong, H. T. Hung, and T. D. Tham, Phys. Rev. D87, 115003 (2013b), arXiv:1305.0369 [hep-ph] .
  * Cogollo  _et al._ (2014a) D. Cogollo, A. X. Gonzalez-Morales, F. S. Queiroz, and P. R. Teles, JCAP 1411, 002 (2014a), arXiv:1402.3271 [hep-ph] .
  * Dong  _et al._ (2014a) P. V. Dong, D. T. Huong, F. S. Queiroz, and N. T. Thuy, Phys. Rev. D90, 075021 (2014a), arXiv:1405.2591 [hep-ph] .
  * Dong  _et al._ (2014b) P. V. Dong, N. T. K. Ngan, and D. V. Soa, Phys. Rev. D90, 075019 (2014b), arXiv:1407.3839 [hep-ph] .
  * Carvajal  _et al._ (2017) C. D. R. Carvajal, B. L. Sanchez-Vega, and O. Zapata, Phys. Rev. D96, 115035 (2017), arXiv:1704.08340 [hep-ph] .
  * Montero  _et al._ (2018) J. C. Montero, A. Romero, and B. L. Sanchez-Vega, Phys. Rev. D97, 063015 (2018), arXiv:1709.04535 [hep-ph] .
  * Huong  _et al._ (2019) D. T. Huong, D. N. Dinh, L. D. Thien, and P. Van Dong, JHEP 08, 051 (2019), arXiv:1906.05240 [hep-ph] .
  * Cabarcas  _et al._ (2012) J. M. Cabarcas, J. Duarte, and J. A. Rodriguez,  _Proceedings on 11th International Conference on Heavy Quarks and Leptons (HQL 2012): Prague, Czech Republic, Jun. 11-15, 2012_ , PoS HQL2012, 072 (2012), arXiv:1212.3586 [hep-ph] .
  * Hue  _et al._ (2018) L. T. Hue, L. D. Ninh, T. T. Thuc, and N. T. T. Dat, Eur. Phys. J. C78, 128 (2018), arXiv:1708.09723 [hep-ph] .
  * Montero and Sanchez-Vega (2011) J. C. Montero and B. L. Sanchez-Vega, Phys. Rev. D84, 055019 (2011), arXiv:1102.5374 [hep-ph] .
  * Santos and Vasconcelos (2018) A. C. O. Santos and P. Vasconcelos, Adv. High Energy Phys. 2018, 9132381 (2018), arXiv:1708.03955 [hep-ph] .
  * Barreto  _et al._ (2018) E. R. Barreto, A. G. Dias, J. Leite, C. C. Nishi, R. L. N. Oliveira, and W. C. Vieira, Phys. Rev. D97, 055047 (2018), arXiv:1709.09946 [hep-ph] .
  * De Conto  _et al._ (2018) G. De Conto, A. C. B. Machado, and J. P. B. C. de Melo, Phys. Lett. B784, 255 (2018), arXiv:1711.06718 [hep-ph] .
  * Pisano and Pleitez (1992) F. Pisano and V. Pleitez, Phys. Rev. D46, 410 (1992), arXiv:hep-ph/9206242 [hep-ph] .
  * Long (1996a) H. N. Long, Phys. Rev. D54, 4691 (1996a), arXiv:hep-ph/9607439 [hep-ph] .
  * Long (1996b) H. N. Long, Phys. Rev. D53, 437 (1996b), arXiv:hep-ph/9504274 [hep-ph] .
  * Catano  _et al._ (2012) M. E. Catano, R. Martinez, and F. Ochoa, Phys. Rev. D86, 073015 (2012), arXiv:1206.1966 [hep-ph] .
  * Dong  _et al._ (2006) P. V. Dong, H. N. Long, D. T. Nhung, and D. V. Soa, Phys. Rev. D73, 035004 (2006), arXiv:hep-ph/0601046 [hep-ph] .
  * Dong and Long (2008) P. V. Dong and H. N. Long, Adv. High Energy Phys. 2008, 739492 (2008), arXiv:0804.3239 [hep-ph] .
  * Martinez and Ochoa (2014) R. Martinez and F. Ochoa, Phys. Rev. D90, 015028 (2014), arXiv:1405.4566 [hep-ph] .
  * Ponce  _et al._ (2002a) W. A. Ponce, J. B. Florez, and L. A. Sanchez, Int. J. Mod. Phys. A17, 643 (2002a), arXiv:hep-ph/0103100 [hep-ph] .
  * Ponce  _et al._ (2002b) W. A. Ponce, Y. Giraldo, and L. A. Sanchez,  _Proceedings, 8th Mexican Workshop on Particles and Fields: Zacatecas, Mexico, November 14-20, 2001_ , AIP Conf. Proc. 623, 341 (2002b), arXiv:hep-ph/0201133 [hep-ph] .
  * Anderson and Sher (2005) D. L. Anderson and M. Sher, Phys. Rev. D72, 095014 (2005), arXiv:hep-ph/0509200 [hep-ph] .
  * Cabarcas  _et al._ (2014) J. M. Cabarcas, J. Duarte, and J. A. Rodriguez, Int. J. Mod. Phys. A29, 1450015 (2014), arXiv:1310.1407 [hep-ph] .
  * Ky  _et al._ (2000) N. A. Ky, H. N. Long, and D. Van Soa, Phys. Lett. B486, 140 (2000), arXiv:hep-ph/0007010 [hep-ph] .
  * Kelso  _et al._ (2014a) C. Kelso, P. R. D. Pinheiro, F. S. Queiroz, and W. Shepherd, Eur. Phys. J. C74, 2808 (2014a), arXiv:1312.0051 [hep-ph] .
  * Binh  _et al._ (2015) D. T. Binh, D. T. Huong, and H. N. Long, Zh. Eksp. Teor. Fiz. 148, 1115 (2015), [J. Exp. Theor. Phys.121,no.6,976(2015)], arXiv:1504.03510 [hep-ph] .
  * Cogollo (2017) D. Cogollo, (2017), arXiv:1706.00397 [hep-ph] .
  * De Conto and Pleitez (2017) G. De Conto and V. Pleitez, JHEP 05, 104 (2017), arXiv:1603.09691 [hep-ph] .
  * Carcamo Hernandez  _et al._ (2020) A. E. Carcamo Hernandez, Y. H. Velasquez, S. Kovalenko, H. N. Long, N. A. Perez-Julve, and V. V. Vien, (2020), arXiv:2002.07347 [hep-ph] .
  * Queiroz and Shepherd (2013) F. S. Queiroz and W. Shepherd, “Numerical Code for g-2,” https://drive.google.com/open?id=1lBKcCKfvBtNJHKjUaGBq3IX506seRqK0 (2013), [Online; accessed 18-February-2020].
  * Borges and Ramos (2016) J. S. Borges and R. O. Ramos, Eur. Phys. J. C76, 344 (2016), arXiv:1602.08165 [hep-ph] .
  * Wei and Chong-Xing (2017) M. Wei and Y. Chong-Xing, Phys. Rev. D95, 035040 (2017), arXiv:1702.01255 [hep-ph] .
  * Montero  _et al._ (1993) J. C. Montero, F. Pisano, and V. Pleitez, Phys. Rev. D47, 2918 (1993), arXiv:hep-ph/9212271 [hep-ph] .
  * Queiroz  _et al._ (2010) F. Queiroz, C. A. de S. Pires, and P. S. R. da Silva, Phys. Rev. D82, 065018 (2010), arXiv:1003.1270 [hep-ph] .
  * Ferreira  _et al._ (2019) M. M. Ferreira, T. B. de Melo, S. Kovalenko, P. R. D. Pinheiro, and F. S. Queiroz, (2019), arXiv:1903.07634 [hep-ph] .
  * Camargo  _et al._ (2019) D. A. Camargo, A. G. Dias, T. B. de Melo, and F. S. Queiroz, JHEP 04, 129 (2019), arXiv:1811.05488 [hep-ph] .
  * Tonasse (1996) M. D. Tonasse, Phys. Lett. B381, 191 (1996), arXiv:hep-ph/9605230 [hep-ph] .
  * Berenstein  _et al._ (2009) D. Berenstein, R. Martinez, F. Ochoa, and S. Pinansky, Phys. Rev. D79, 095005 (2009), arXiv:0807.1126 [hep-ph] .
  * Cogollo  _et al._ (2014b) D. Cogollo, F. S. Queiroz, and P. Vasconcelos, Mod. Phys. Lett. A29, 1450173 (2014b), arXiv:1312.0304 [hep-ph] .
  * Kelso  _et al._ (2014b) C. Kelso, H. N. Long, R. Martinez, and F. S. Queiroz, Phys. Rev. D90, 113011 (2014b), arXiv:1408.6203 [hep-ph] .
  * Machado  _et al._ (2019) A. C. B. Machado, J. Montaño, and V. Pleitez, J. Phys. G46, 115005 (2019), arXiv:1604.08539 [hep-ph] .
  * Belyaev  _et al._ (2013) A. Belyaev, N. D. Christensen, and A. Pukhov, Comput. Phys. Commun. 184, 1729 (2013), arXiv:1207.6082 [hep-ph] .
  * Semenov (2016) A. Semenov, Comput. Phys. Commun. 201, 167 (2016), arXiv:1412.5016 [physics.comp-ph] .
  * Nepomuceno and Meirose (2020) A. Nepomuceno and B. Meirose, Phys. Rev. D101, 035017 (2020), arXiv:1911.12783 [hep-ph] .
  * Salazar  _et al._ (2015) C. Salazar, R. H. Benavides, W. A. Ponce, and E. Rojas, JHEP 07, 096 (2015), arXiv:1503.03519 [hep-ph] .
  * Queiroz and Shepherd (2014) F. S. Queiroz and W. Shepherd, Phys. Rev. D89, 095024 (2014), arXiv:1403.2309 [hep-ph] .
  * Queiroz (2020) F. S. Queiroz,  _Mathematica numerical codes of the Muon Anomalous Magnetic Moment to 3-3-1 Models_ (Accessed in 2020), https://bit.ly/2vFZLnG.


# Why is superconductivity absent in bulk Infinite-layer Nickelates?

Akariti Sharmaa, Bharathiganesh Devanarayanan a,b,∗, Pratik D.Patela,
Navinder Singha
aTheoretical Physics Division, Physical Research Laboratory, Navrangpura
Ahmedabad, India - 380009
bIndian Institute of Technology, Gandhinagar, Palaj, Gujarat, India - 382355

∗Corresponding author: Bharathiganesh Devanarayanan<EMAIL_ADDRESS>
###### Abstract

We answer in affirmative the absence of superconductivity in bulk prepared
samples of doped infinite-layer Nickelates through a simple magnetostatic
model. In an independent approach we also employ Density Functional Theory
(DFT) calculations for the same investigation. Our investigations reveal that
it is the formation of Nickel clusters and Nickel deficient regions in bulk
prepared samples (as opposed to thin films) that leads to the absence of
superconductivity in bulk prepared samples.

Ever since the discovery of high temperature (Tc) superconductivity in doped
Cuprates[1], there have been several layers of mysteries in understanding the
mechanism of superconductivity in such materials. One approach to possibly
understand the mechanism was to synthesise other families of high Tc
superconductors and then compare their properties with that of Cuprates. Ni+1
with d9 configuration similar to the configuration of Copper in Cuprates made
Nickelates a leading candidate for such a family. Several decades of search
for such a superconducting cousin finally found success when Li, Danfeng et
al.[2] synthesised superconducting thin films of NdNiO2 doped with Sr
(Nd1-xSrxNiO2).
Contrary to just helping our understanding of superconducting mechanism in
Cuprates, Nickelate superconductors came with their own additional layers of
mysteries[3]. Li, Qing et al.[4] had synthesised bulk samples of Nd1-xSrxNiO2
and have reported absence of superconductivity in the same. Several other
groups had synthesised bulk samples of infinite layer Nickelates and
superconductivity has not been seen in any of them[5] but it is observed in
thin films of several nanometer thickness [6]. It was thought that the
substrate on which the thin film is mounted is playing a key role in emergence
of superconductivity. This also lead to prepositions that superconductivity in
the thin films could be an interface phenomena in the interface of the
material Nd1-xSrxNiO2 and the substrate SrTiO3.
Here we emphasise that one should clearly understand that the substrate can
either play an indirect role by providing strain modulation and crystal
stability to the thin film sample or directly participate in the
superconducting mechanism. It should be made clear that only the latter makes
superconductivity in Nickelates an interfacial phenomena and not the former
one. Experiments on Meissner effects confirms that the substrate does not
directly participate in superconducting mechanism ruling out that it is an
interface phenomena[7].

Figure 1: Schematic diagram of Ni-clusters in NdNiO2 bulk sample.

It has been reported that by Li, Qing et al.[4] that in the bulk samples of
infinite layer Nickelates there are Ni deficiencies and these Ni atoms form
clusters and remain together with the grains of Nd1-xSrxNiO2(fig.1). It has
also been mentioned that the size of such clusters can go up to a few $\mu$ms
[private communication]. Since Ni metal is a ferromagnet these Nickel clusters
of micro and sub-micro meter size can also be considered as tiny ferromagnets
present within the sample. It should also be noted that the first critical
field ($H_{c_{1}}$) of Nd1-xSrxNiO2 in thin film samples is reported to be
0.79 mT.
In our simple model we estimate the magnetic fields produced by these Nickel
clusters and notice that it can be greater than the first critical field
$H_{c_{1}}$. The magnetic fields produced by the Nickel clusters can be
calculated either from the magnetic dipole model (point model) or the Ampérian
loop model (extended model). In the first model the Ni cluster is modeled to
be magnetic dipole with a moment equal to the sum of individual moments from
each Nickel atom. This is very well justified because the temperatures are
well below the ordering temperature of Nickel and the moments are definitely
aligned. The magnetic field due to a magnetic dipole is given by the formula:

$\mathbf{B}(\mathbf{r})=\frac{\mu_{0}}{4\pi}\left(\frac{3\mathbf{\hat{r}}\left(\mathbf{\hat{r}.m}\right)-\mathbf{m}}{|\mathbf{r}|^{3}}\right)$
(1)

where $\mathbf{m}$ is the total magnetic dipole moment of the cluster and
$\mathbf{r}$ is the vector pointing from the center of the magnetic dipole to
the point where the magnetic field is measured.
In the Ampérian loop model the nickel clusters are modelled as tiny current
carrying loops with the radius being the size of the Nickel cluster and the
magnetic moment the sum of magnetic moments all Nickel atoms in the cluster
(see fig.2). The current in the fictitious loop can then be straight forwardly
calculated as:

$\mathbf{m}=\mathbf{IA},$ (2)

Here $\mathbf{I}$ is the current in the loop and A is the area of the loop.

Figure 2: Nickel cluster modeled as a current carrying loop of same radius in
the Ampérian model.

To get a rough estimate of the total magnetic moment of the Nickel cluster we
need to approximately calculate the number of Nickel atoms in each cluster.
Suppose the radius of a Nickel cluster is r (1$\mu$m). Given the lattice
constant of Ni metal is $a$ ( $3.5\times 10^{-10}$ m) and Nickel crystallises
in a Face centered Cubic (FCC) lattice, the number of Nickel atoms in each
cluster is approximately:

$N_{c}=\frac{4\frac{4\pi}{3}(r)^{3}}{(a)^{3}}=\frac{4\frac{4\pi}{3}(10^{-6})^{3}}{(3.5\times
10^{-10})^{3}}=1.19\times 10^{12},$ (3)

Since each Nickel atom contributes 3 Bohr magneton($\mu_{B}$) of magnetic
moment, the total magnetic moment of Nickel cluster is:

$\mathbf{m}=3\,N_{C}\,\mathbf{\mu_{B}}.$ (4)

From the value of the magnetic moment calculated above the magnetic field can
be directly calculated from eqn.(1) in the magnetic dipole model. For the
Ampérian loop model we will have to calculate current from the total magnetic
moment. The magnetic field at a point in the axis of a current carrying loop
is given by the following formula:

$\mathbf{B}_{axial}(\mathbf{x})=\frac{\mu_{0}I}{2}\frac{r^{2}}{(r^{2}+x^{2})^{3/2}}\mathbf{\hat{z}}=\mu_{0}\mu_{B}\left(\frac{2r}{a\sqrt{r^{2}+x^{2}}}\right)^{3}\mathbf{\hat{z}},$
(5)

where $r$ is the radius of the current loop, $x$ is the distance from the
centre of the loop to a point in the axis where magnetic field is calculated,
$I$ is the current in the loop and $\mathbf{\hat{z}}$ is an unit vector along
the axis and pointing away from the loop.

Radius of the cluster ($\mu$ms) | Distance from the cluster ($\mu$ms) | Magnetic field (mT)
---|---|---
1 | 1 | 768.47
5 | 16.39
10 | 2.14
5 | 5 | 768.47
10 | 194.41
20 | 31.01
Table 1: Magnetic field produced by the Nickel clusters of various sizes at
various distances from the Nickel clusters.

Our results for the magnetic field at various distances for various size of
clusters have been tabulated in Table 1. The value for magnetic field obtained
from both the models are of same order. We can observe the general trend that
for distance up to an order higher than the radius of the cluster the field is
of the order of mT. We have already noted that the size of these clusters are
of the order of 1-6 $\mu$ms and their average inter cluster separation is of
the order of 10 $\mu$ms.
This leads to the conclusion that a magnetic field of an order of mT pervades
the entire sample.This magnetic field is of an order of magnitude higher than
the $H_{c_{1}}(=79\,$ Oe) of Nickelates and is enough to destroy the
superconductivity.

Figure 3: Orbital resolved PDOS for non Ni-deficient Nd1-xSrxNiO2 (0.25) in
panel (a), one and two Ni-atom deficient supercells in panel (b) and (c),
respectively.

Another argument (from Nickel deficient regions) can be made in the following
way. From our DFT calculations we infer that Nickel deficient regions modify
the electronic structure. The DFT calculations have been performed for the
system Nd0.75Sr0.25NiO2 and the results are shown in fig.3. Nickel deficiency
leads to the reduced weight of the critical Ni 3d${}_{x^{2}-y^{2}}$ orbital at
the Fermi level which is detrimental to superconductivity.
Therefore we conclude that the lack of superconductivity can be understood in
the following way: The Nickel deficiency in the Nd1-xSrxNiO2 grains leads to
decrease in the weight of Ni 3d${}_{x^{2}-y^{2}}$ orbital which in turn
affects superconductivity. Further the Nickel atoms form disconnected clusters
between these grains. These clusters produce a magnetic field of an order of
at least a milliTesla (mT) killing superconductivity if it exists. We have
given two possible explanations for ”Why is superconductivity absent in bulk
Infinite-layer Nickelates?” that have been synthesised with the currently
available state of the art chemical methods. It would be interesting to see in
future the possibility of existence of superconductivity in bulk prepared
Nickelate samples, if the two differences of Nickel deficiency and Nickel
clusters between the film and bulk samples are resolved with more advanced
chemical methods of synthesis.

Acknowledgment

We thank Dr Qing Li for his correspondence regarding size of Nickel clusters.
Computation work related to these investigations were performed using the HPC
resources (Vikram-100 HPC) project at Physical Research Laboratory (PRL).

## References

  * [1] M. K. Wu, J. R. Ashburn, C. J. Torng, P. H. Hor, R. L. Meng, L. Gao, Z. J. Huang, Y. Q. Wang, and C. W. Chu.  Superconductivity at 93 k in a new mixed-phase y-ba-cu-o compound system at ambient pressure.  Phys. Rev. Lett., 58:908–910, Mar 1987.
  * [2] Danfeng Li, Kyuho Lee, Bai Yang Wang, Motoki Osada, Samuel Crossley, Hye Ryoung Lee, Yi Cui, Yasuyuki Hikita, and Harold Y Hwang.  Superconductivity in an infinite-layer nickelate.  Nature, 572(7771):624–627, 2019.
  * [3] Navinder Singh.  A” road-map” of nickelate superconductivity.  arXiv preprint arXiv:1909.07688, 2019.
  * [4] Qing Li, Chengping He, Jin Si, Xiyu Zhu, Yue Zhang, and Hai-Hu Wen.  Absence of superconductivity in bulk nd1- xsrxnio2.  Communications Materials, 1(1):1–8, 2020.
  * [5] Bi-Xia Wang, Hong Zheng, E. Krivyakina, O. Chmaissem, Pietro Papa Lopes, J. W. Lynn, Leighanne C. Gallington, Y. Ren, S. Rosenkranz, J. F. Mitchell, and D. Phelan.  Synthesis and characterization of bulk ${\mathrm{nd}}_{1-x}{\mathrm{sr}}_{x}\mathrm{Ni}{\mathrm{o}}_{2}$ and ${\mathrm{nd}}_{1-x}{\mathrm{sr}}_{x}\mathrm{Ni}{\mathrm{o}}_{3}$.  Phys. Rev. Materials, 4:084409, Aug 2020.
  * [6] Kyuho Lee, Berit H Goodge, Danfeng Li, Motoki Osada, Bai Yang Wang, Yi Cui, Lena F Kourkoutis, and Harold Y Hwang.  Aspects of the synthesis of thin film superconducting infinite-layer nickelates.  Apl Materials, 8(4):041107, 2020.
  * [7] L. E. Chow and A. Ariando.  Infinite-layer nickelate superconductors: A current experimental perspective of the crystal and electronic structures.  Frontiers in Physics, 10, 2022.