# Vector-Like Fermions and Inert Scalar Solutions to the Muon g-2 Anomaly
and collider probes at the HL-LHC and FCC-hh

Á. S. de Jesusa,b<EMAIL_ADDRESS>F. S. Queiroza,b
<EMAIL_ADDRESS>J. W. F. Vallec<EMAIL_ADDRESS>Y. Villamizard
<EMAIL_ADDRESS>aInternational Institute of Physics, Universidade
Federal do Rio Grande do Norte, Campus Universitário, Lagoa Nova, Natal-RN
59078-970, Brazil  c AHEP Group, Institut de Física Corpuscular –
CSIC/Universitat de València, Parc Científic de Paterna.
C/ Catedrático José Beltrán, 2 E-46980 Paterna (Valencia) - Spain  d Centro de
Ciências Naturais e Humanas, Universidade Federal do ABC, 09210-580, Santo
André, São Paulo, Brasil

###### Abstract

We examine simple models with an inert scalar and vector-like leptons that can
explain the recent $g_{\mu}-2$ measurement reported at FNAL. Prompted by this
exciting result, we explore the viability of a simple interpretation and
determine the required parameters. We also embed these models within a 3-3-1
gauge extension of the Standard Model (SM), showing that the $g_{\mu}-2$
anomaly can be accommodated in agreement with current data. We also show how
our theory can be tested at high-energy colliders such as HL-LHC and FCC-hh.

###### pacs:

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

##  I Introduction

A new measurement of the muon magnetic moment was reported by the Muon
$g_{\mu}-2$ Experiment at Fermilab. In addition to the increase of statistics
using data collected in 2019 (Run-2) and 2020 (Run-3), the collaboration
reduced the systematics by a factor of two. The experiment uses 3.1 GeV
polarized muons produced at the Fermilab Muon Campus, which are then injected
into a storage ring with a radius of 7.1 m. The experiments surpassed the BNL
experiment in many ways Carey  _et al._ (1999); Brown  _et al._ (2000, 2001);
Bennett  _et al._ (2002a, 2004). After much experimental effort, the Muon
$g_{\mu}-2$ Experiment measured the muon’s magnetic moment with unprecedented
precision. We remind the reader that from the Dirac equation, we find the
muon’s magnetic moment to be $\vec{m}=g_{\mu}e/(2m_{\mu})\vec{S}$, where the
gyromagnetic ratio is given by $g=2$ Dirac (1928). The quantum corrections to
this $g$-factor can be expressed through the muon anomalous magnetic moment,
typically described in terms of $a_{\mu}=\nicefrac{{(g_{\mu}-2)}}{{2}}$. The
old BNL result Bennett  _et al._ (2002b, 2006) prompted a lot of attention in
the community, because the discrepancy between theoretical predictions and
experimental results would possibly hint for new physics Czarnecki and
Marciano (2001); Davier and Marciano (2004). There are uncertainties in the
significance of the value of $g_{\mu}-2$ due to theoretical issues stemming
from high-order SM hadronic contributions, particularly the Hadronic Vacuum
Polarization (HVP) and Light-by-Light (HLbL) corrections Aoyama  _et al._
(2020); Jegerlehner (2017); Crivellin  _et al._ (2020); Borsanyi  _et al._
(2021); Blum  _et al._ (2020); Ignatov  _et al._ (2023). A recently conducted
study Borsanyi  _et al._ (2021) has significantly reduced the uncertainty
around the Leading-Order Hadronic Vacuum Polarization (LO-HVP) theoretical
contribution to the muon anomalous magnetic moment ($g_{\mu}-2$) from lattice
QCD calculations, showing a considerable discrepancy between the lattice and
the data-driven approaches. This study combined with other known Standard
Model contributions has reduced the discrepancy in the muon g-2 theory to
around 2$\sigma$. There is also some tension regarding the data-driven
approach. A new result from the CMD-3 collaboration has shown incompatible
results compared with past measurements of the $e^{+}e^{-}\to\pi^{+}\pi^{-}$
channel, showing that it has a larger cross-section than expected. This
process represents the largest contribution to the R-ratio, thus it brings the
theoretical result closer to the experimental limit on $a_{\mu}$, reducing the
anomaly in the data-driven front Ignatov  _et al._ (2023). However, both
results from CMD-3 and BMW collaboration are still under scrutiny, with more
measurements and calculations needed to either confirm or confront these
results. In Table 1 we report the discrepancy and its significance to
highlight that this $g_{\mu}-2$ anomaly is far from being resolved.

Table 1: Summary of the $g_{\mu}-2$ anomaly over the years. Current value to $g_{\mu}-2$ | standard deviation | year | Reference
---|---|---|---
$\Delta a_{\mu}=(261\pm 78)\times 10^{-11}$ | $3.3\sigma$ | 2009 | Prades  _et al._ (2009); Tanabashi  _et al._ (2018)
$\Delta a_{\mu}=(325\pm 80)\times 10^{-11}$ | $4.05\sigma$ | 2012 | Benayoun  _et al._ (2013)
$\Delta a_{\mu}=(287\pm 80)\times 10^{-11}$ | $3.6\sigma$ | 2013 | Blum  _et al._ (2013)
$\Delta a_{\mu}=(377\pm 75)\times 10^{-11}$ | $5.02\sigma$ | 2015 | Benayoun  _et al._ (2015)
$\Delta a_{\mu}=(313\pm 77)\times 10^{-11}$ | $4.1\sigma$ | 2017 | Jegerlehner (2018)
$\Delta a_{\mu}=(270\pm 36)\times 10^{-11}$ | $3.7\sigma$ | 2018 | Keshavarzi  _et al._ (2018)
$\Delta a_{\mu}=(250\pm 48)\times 10^{-11}$ | $5.0\sigma$ | 2023 | Aguillard  _et al._ (2023)

The current value given by the Particle Data Group (PDG) is $\Delta
a_{\mu}=a_{\mu}^{\exp}-a_{\mu}^{\mathrm{SM}}=251(41)(43)\times 10^{-11}$
Workman and Others (2022), but the recent Muon $g_{\mu}-2$ experiment reports
$\Delta a_{\mu}=(250\pm 48)\times 10^{-11}$ Aguillard  _et al._ (2023).

Prompted by the recent Muon $g_{\mu}-2$ result hinting at an exciting
$5.0\sigma$ discrepancy between theory and experiment, we discuss how to
accommodate this anomaly within a simple setup. Several simplified models have
been put forth to account for $g_{\mu}-2$ Padley  _et al._ (2015); Yamaguchi
and Yin (2018); Yin and Yokozaki (2016); Endo and Yin (2019); Endo  _et al._
(2020); Endo and Ueda (2020); Cox  _et al._ (2019); Konar  _et al._ (2018); Un
and Ozdal (2016); An and Wang (2015); Chakraborty and Chakraborty (2016); Jana
_et al._ (2020), see Lindner  _et al._ (2018) for a recent review.

In this work, we are interested in explaining the $g_{\mu}-2$ anomaly with
vector-like fermions and an inert scalar, taking into account existing
restrictions. Vector-like fermions have already been suggested in this context
by several authors Dermisek and Raval (2013); Poh and Raby (2017); Barman  _et
al._ (2019). Here we will be particularly interested in vector-like fermion
interaction terms involving the muon. We will also embed our simplified scheme
into a non-Abelian 3-3-1 gauge extension of the SM in order to show how the
$g_{\mu}-2$ anomaly can be addressed in a comprehensive manner.

Our work is structured as follows: in Section II, we describe the simplified
models featuring inert scalars and vector-like fermions, presenting regions of
parameter space where these models can be used to address the anomaly; in
Section III, we introduce a non-Abelian gauge extension of the SM, outlining
the region of the parameter that can accommodate the $g_{\mu}-2$ anomaly in
agreement with the data from current and future colliders; in Section IV we
draw our conclusions.

##  II  Simplified Models

###  II.1 Inert scalar

In order to illustrate how simplified models can potentially address the
$g_{\mu}-2$ anomaly we first introduce an inert scalar doublet ($\phi$) under
$SU(2)_{L}$ with hypercharge $Y_{\phi}=1$ via the following Lagrangian,

$\mathcal{L}=\lambda_{ab}\bar{L}_{aL}\phi e_{bR}+h.c.$ (1)

where $L_{aL}$ is the SM lepton doublet, $\lambda_{ab}$ is the Yukawa
coupling. With this Lagrangian, there are two contributions to $\Delta
a_{\mu}$ coming from the neutral and charged components of the inert scalar
doublet. However, the main contribution to $g_{\mu}-2$ comes from the neutral
component, and is given by Lindner  _et al._ (2018),

$\Delta a_{\mu}=\frac{\lambda^{\prime
2}}{8\pi^{2}}\int_{0}^{1}\mathrm{\leavevmode\nobreak\
d}x\sum_{b}\frac{\left(g^{s}_{2b}\right)^{2}P^{+}(x)+\left(g^{p}_{2b}\right)^{2}P^{-}(x)}{(1-x)\left(1-x\lambda^{\prime
2}\right)+x\epsilon_{b}^{2}\lambda^{\prime 2}}$ (2)

where $P^{\pm}(x)=x^{2}\left(1-x\pm\epsilon_{b}\right)$,
$\epsilon_{b}=\cfrac{m_{b}}{m_{\mu}}$,
$\lambda^{\prime}=\cfrac{m_{\mu}}{m_{\phi}}$, $g^{s}_{2b}$ and $g^{p}_{2b}$
being the scalar and pseudoscalar couplings of this new scalar with the muon
and other particles in the same vertex, represented by the subscript $b$.

This could explain the $g_{\mu}-2$ anomaly for
$0.0076\,\mbox{GeV}^{-1}<y/m_{\phi}<0.0102\,\mbox{GeV}^{-1}$. However, if one
embeds this scalar doublet within a complete multi-Higgs model, there are a
number of constraints that make this solution rather contrived Wang and Han
(2015); Hektor  _et al._ (2015); Cherchiglia  _et al._ (2018); Wang  _et al._
(2019); Iguro  _et al._ (2019).

Figure 1: Contribution to $\Delta a_{\mu}$ from an inert scalar whose mass is
a free parameter and whose coupling with the muon is $\lambda_{22}=1$ (black)
and $\lambda_{22}=0.1$ (blue).

The inert scalar doublet contribution to $\Delta a_{\mu}$ is calculated by Eq.
2 and shown in Fig. 1. In this figure, we exhibit the region of $\\{\Delta
a_{\mu},m_{\phi}\\}$ parameter space that provides an adequate contribution to
the muon anomalous moment according to the recent data from the $g_{\mu}-2$
experiment at Fermilab. The latter is illustrated by the band between the
horizontal lines in green. We display two different choices for the coupling
between the inert scalar and the muon, given by $\lambda_{22}=1$ and
$\lambda_{22}=0.1$, represented by the solid black and blue lines,
respectively. With this setup, both configurations are capable of addressing
the anomaly for masses of $m_{\phi}=400-550$ GeV ($\lambda_{22}=1$) and
$m_{\phi}=30-40$ GeV ($\lambda_{22}=0.1$).

###  II.2 Vector-like Fermions

#### Vector-like fermion plus a scalar singlet

Vector-like fermions ($E$) represent an interesting class of models to explain
the $g-2$ result Queiroz and Shepherd (2014). Anomaly cancellation requires
vector-like fermions with masses arising from terms of the form
$m_{E}\bar{E_{L}}E_{R}$. If $E$ is a singlet under $SU(2)_{L}$, we can write
down,

$\mathcal{L}=\lambda_{ab}\bar{E}_{aR}\sigma\mu_{R}+h.c$ (3)

where $\sigma$ is now a scalar singlet under $SU(2)_{L}$.

#### Vector-like fermion plus a scalar doublet

Notice that we can also generate a positive contribution to $g_{\mu}-2$
through an exotic charged fermion $(E)$ and inert doublet $(\phi)$ as follows,

$\mathcal{L}=\lambda_{ab}\bar{L}_{aL}\phi E_{bR}+h.c.$ (4)

#### Vector-like fermion doublet plus an inert scalar doublet

Finally, we note that we could also add a fermion doublet under $SU(2)_{L}$ ,
($\psi$) plus an inert scalar doublet $\sigma$, and write down,

$\mathcal{L}=\lambda_{ab}\bar{\psi}_{aL}\sigma\mu_{bR}+h.c.$ (5)

Figure 2: Contributions to $\Delta a_{\mu}$ from inert scalar and vector-like
fermions. The top (lower) panel represents the calculated contribution of a
fixed value for the vector-like fermion (scalar) mass. In contrast, the mass
of the scalar (vector-like fermion) is a free parameter. The horizontal band
between the green lines is the current value.

All of these simplified models offer solutions to the $g_{\mu}-2$ anomaly. We
stress that these exotic charged leptons are required to be vector-like in
order to ensure gauge anomaly cancellation. The contributions to $g-2$ are
determined by Eq.(2), where we introduce another free parameter, namely, the
mass of the vector-like fermion $m_{E}$, which impacts the calculation by
changing $\epsilon_{b}=\epsilon_{E}=m_{E}/m_{\mu}$, while in the inert scalar
case $\epsilon_{b}=\epsilon_{\mu}=1$. The results from this model are
presented in Figs. 2 and 3, where the $g_{\mu}-2$ experiment bounds are
represented in the same way as in Fig. 1, and here we consider three cases:
fixed $m_{E}$ and free $m_{\phi}$, the opposite configuration, and both
$m_{E}$ and $m_{\phi}$ as free parameters.

Figure 3: Parameter space exploration in the $m_{\phi}\times m_{E}$ plane,
assuming $\lambda_{22}=4$ ($g^{s(p)}_{22}=2$). The color bar integrated within
the plot corresponds to the values of the contribution to $\Delta a_{\mu}$
inside the allowed region derived with the new $g_{\mu}-2$ result.

The first two cases are shown in Fig. 2, with the top (lower) panel
representing the first (second) case, both with $\lambda_{22}=3$. Both cases
provide viable solutions to the anomaly, with the top panel representing the
contributions to $\Delta a_{\mu}$ as a function of $m_{\phi}$ for fixed values
of $m_{E}=150$ GeV and $m_{E}=200$ GeV, while the lower panel shows the
contribution to $\Delta a_{\mu}$ versus $m_{E}$ assuming $m_{\phi}=200$ GeV
and $m_{\phi}=300$ GeV.

The third case is illustrated in Fig. 3. In this plot we display the region of
the $\\{m_{E},m_{\phi}\\}$ plane leading to an adequate solution to the
$g_{\mu}-2$ anomaly. We find that the contribution to $\Delta a_{\mu}$ arising
from an inert scalar with vector-like leptons can indeed be in agreement with
the new $g_{\mu}-2$ experiment. In this plot, we assume the muon coupling with
the scalar and vector-like lepton to be $g_{2b}^{s(p)}=2$, implying that
vector-like fermion masses up to $300$ GeV and inert scalar masses up to $450$
GeV are enough to fit the anomaly. This holds for both the inert scalar and
the vector-like lepton model assumed in this work. The horizontal color bar
represents the specific value of $\Delta a_{\mu}$ in units of $10^{-9}$
achieved with each configuration of $m_{\phi}$ and $m_{E}$, limiting the
parameter space to the region inside the new experimental result of $\Delta
a_{\mu}=(250\pm 48)\times 10^{-11}$. An interesting feature of this plot is
the fact that higher values of $m_{\phi}$ imply lower values of $m_{E}$ and
vice-versa. This happens due to the behavior of this channel’s contribution to
$\Delta a_{\mu}$, which depends on the ratio between the masses of the new
particles and the muon mass.

Table 2: Field content and transformation properties  Simplified models
---
Model | Field | SU(3)C | SU(2)L | U(1)Y
Inert Scalar | $\phi$ | 1 | 2 | 1
Vector-like Fermion | E | 1 | 1 | -2
$\phi$ | 1 | 2 | 1
$\psi$ | 1 | 2 | -1
$\sigma$ | 1 | 2 | 1
3-3-1 model
Model | Field | SU(3)C | SU(3)L | U(1)X
3-3-1 | $f$ | 1 | 3 | -1/3
| $N$ | 1 | 1 | 0
| $\phi$ | 1 | 3 | 2/3

In our work, we utilize two local symmetry groups, namely $SU(3)_{C}\times
SU(2)_{L}\times U(1)_{Y}$ and $SU(3)_{C}\times SU(3)_{L}\times U(1)_{X}$. The
new fields and their transformation properties under these symmetry groups are
listed in Table 2. Furthermore, in the upcoming Section III, we will introduce
the 3-3-3 model.

(a) (b) (c)

Figure 4: Feynman diagrams contributing to the muon anomalous magnetic (loop)
in the simplified and 3-3-1 models discussed in this work.

On the other hand, in Fig. 4 we displayed the Feynman diagrams representations
of contributions to the muon’s anomalous magnetic moment analyzed in this
work. Fig. 4(a) represent the $g_{\mu}-2$ contributions involving virtual
fermions ($f$ represent $\ell=e,\,\mu,\,\tau$ or $E$) and a scalar particle
($\phi$ or $\sigma$) that arise in the simplified and the 3-3-1 models (for
instance, see Eqs. 1, 3, 4 and 5). Both scalars are in the simplified models
as detailed in the Table 2, whereas, in the 3-3-1 models, we only have the
representation of the inert scalar as $\phi$. Figs. 4(b) and 4(c) represent
the main Feynman diagrams that contribute to the muon anomalous magnetic
moment (loop) in the 3-3-1 models with a virtual $Z^{\prime}$ and $W^{\prime}$
boson, and fermions $\ell$ and $N$, respectively (see Eqs. 6 and 7).

##  III 3-3-1 Models

The so-called 3-3-1 models form an interesting class of gauge extensions of
the SM based on the $SU(3)_{C}\times SU(3)_{L}\times U(1)_{X}$ symmetry, which
has had by now a very long history Singer  _et al._ (1980); Valle and Singer
(1983); Pisano and Pleitez (1992); Frampton (1992). This new gauge symmetry
was originally introduced to offer a plausible answer to the number of fermion
families in the Standard Model as resulting from anomaly cancellation.
However, this class of electroweak extensions has a broad range of other
possible motivations in Particle Physics. To cite a few, these models have
been extensively studied in the context of flavor physics Cabarcas  _et al._
(2012); Santos and Vasconcelos (2018); Barreto  _et al._ (2018); Wei and
Chong-Xing (2017); Hue  _et al._ (2018), meson oscillations Cárcamo Hernández
_et al._ (2023); Cogollo  _et al._ (2012); Machado  _et al._ (2013); Boucenna
_et al._ (2015a); Kovalenko  _et al._ (2023), charged lepton flavor violation
Boucenna  _et al._ (2015b), collider physics Meirose and Nepomuceno (2011);
Coutinho  _et al._ (2013); Nepomuceno  _et al._ (2016); Queiroz  _et al._
(2016); Nepomuceno and Meirose (2020), neutrino masses Cogollo  _et al._
(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);
Cárcamo Hernández  _et al._ (2019a, b, 2021, 2020), the strong CP problem Dias
_et al._ (2020), unification scenarios Boucenna  _et al._ (2015c); Reig  _et
al._ (2017); Hati  _et al._ (2017); Deppisch  _et al._ (2016); Dong  _et al._
(2018); Hernández  _et al._ (2022), cosmological 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._ (2014); Dong  _et al._ (2014a, b); Alves  _et
al._ (2017); Carvajal  _et al._ (2017); Montero  _et al._ (2018); Huong  _et
al._ (2019); Kang  _et al._ (2019); Leite  _et al._ (2020a); Cárcamo Hernández
_et al._ (2020); Leite  _et al._ (2020b, c); Alvarado  _et al._ (2021), as
well as inflation and leptogenesis Van Dong  _et al._ (2019).

Due to the fact that $SU(2)_{L}$ gauge symmetry is promoted to $SU(3)_{L}$,
the left-handed components of the fermion generations are arranged in the
fundamental (or anti-fundamental) representation of $SU(3)_{L}$, i.e,
triplets. Thus, the structure of the model is now based on triplets, instead
of the usual doublets from the SM. A fundamental consequence of this extension
is the prediction of the existence of other new gauge bosons, such as
$Z^{\prime}$, $W^{\prime}$, as well as other exotics, such as doubly charged
gauge bosons ($U^{++}$). Another change compared to the SM is that more than
one scalar field is necessary to generate masses for all particles. Usually,
3-3-1 models employ three scalar triplets, with one of them being responsible
for the Spontaneous Symmetry Breaking (SSB) of $SU(3)_{L}\times U(1)_{X}\to
SU(2)_{L}\times U(1)_{Y}$ and the others driving electroweak symmetry
breaking. After the $SU(3)_{L}\times U(1)_{X}$ symmetry is spontaneously
broken due to the vacuum expectation value of one of the scalar triplets,
namely $\chi$, a remnant $SU(2)_{L}\times U(1)_{Y}$ survives, recovering the
SM gauge symmetry Borges and Ramos (2016). This change from $SU(2)_{L}$
doublets to $SU(3)_{L}$ triplets implies new fermions contributing to the
gauge anomalies of the extended model. The electric charge operator that
preserves the vacuum is given by,

$\displaystyle\cfrac{Q}{e}$
$\displaystyle=\cfrac{1}{2}(\lambda_{3}+\alpha\lambda_{8})+X\mathcal{I}$
$\displaystyle=\operatorname{diag}\left(\cfrac{1}{2}\left(1+\beta\right)+X,\,\cfrac{1}{2}\left(-1+\beta\right)+X,\,-\beta+X\right),$

where $\lambda_{3}=\operatorname{diag}\left(1,-1,0\right)$,
$\lambda_{8}=\operatorname{diag}\left(1,1,-2\right)/\sqrt{3}$, and
$\mathcal{I}$ are the diagonal generators of $\text{SU(3)}_{\text{L}}$ (Gell-
Mann matrices), with the identity matrix acting as a generator of
$\text{U(1)}_{\text{X}}$, respectively. X and $\alpha$ are free parameters a
priori, with different choices describing distinct models under the 3-3-1
class, and $\beta=\nicefrac{{\alpha}}{{\sqrt{3}}}$.

Since we need to recover the Standard Model spectrum, the first two components
of the lepton triplet should have the same structure as the SM, represented by
a neutrino and a charged lepton. We can fix $\beta=-(2X+1)$ with this choice.
Thus, the third component of the triplet should have an electric charge
quantum number equal to $3X+1$. The choice of X yields different 3-3-1 models,
with the most famous cases being models with electrically charged ($X=0$) or
neutral fields ($X=-1/3$). To cite a few of these models, we have, besides the
original ones Singer  _et al._ (1980); Valle and Singer (1983), the minimal
3-3-1 model Pisano and Pleitez (1992); Frampton (1992), the 3-3-1 model with
exotic charged leptons Ponce  _et al._ (2002a, b); Anderson and Sher (2005);
Cabarcas  _et al._ (2014), the 3-3-1 model with right-handed neutrinos Long
(1996a, b), the economical 3-3-1 model Dong  _et al._ (2006); Dong and Long
(2008); Berenstein  _et al._ (2009); Martinez and Ochoa (2014), and 3-3-1
models with a left-handed neutral lepton Mizukoshi  _et al._ (2011); Catano
_et al._ (2012).

In the context of 3-3-1 models, the main contribution to the muon’s $g-2$
comes from the gauge bosons of the model ($Z^{\prime}$ and $W^{\prime}$),
where the neutral and charged currents are given by

${\cal
L}^{NC}\supset\bar{f}\,\gamma^{\mu}[g_{V}(f)+g_{A}(f)\gamma_{5}]\,f\,Z^{\prime}_{\mu},$
(6) ${\cal
L}^{CC}\supset-\frac{g}{\sqrt{2}}\left[\overline{N_{L}}\,\gamma^{\mu}\bar{l_{L}}\,W^{\prime-}_{\mu}\right],$
(7)

where $f$ is the leptonic triplet representation and $N_{L}$ represents the
third component of the lepton triplet and

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

where $g$ is the weak coupling, $c_{W}=\cos{\theta_{W}}$,
$s_{W}=\sin{\theta_{W}}$ are the cosine and sine of the Weinberg angle,
respectively.

The masses of the $Z^{\prime}$ and $W^{\prime}$ gauge bosons are given by

$\displaystyle M_{Z^{\prime}}^{2}=\frac{g^{2}}{4(3-4s_{W}^{2})}$
$\displaystyle\left(4c_{W}^{2}v_{\chi}^{2}+\frac{v_{\rho}^{2}}{c_{W}^{2}}+\frac{v_{\eta}^{2}(1-2s_{W}^{2})^{2}}{c_{W}^{2}}\right),$
$\displaystyle M_{W^{\prime}}^{2}$
$\displaystyle=\frac{g^{2}}{4}\left(v_{\eta}^{2}+v_{\chi}^{2}\right),$ (8)

where $v_{\rho}$ and $v_{\eta}$ are the vacuum expectation values of the
scalar triplets $\rho$ and $\eta$ driving electroweak symmetry breaking. The
explicit dependence of the gauge boson masses on $v_{\chi}$ can be seen from
Eq. 8. This dependence will be important in what follows, as it will relate
new gauge boson searches at high-energy colliders with the results obtained in
our analysis.

Recently it has been shown that none of the above 3-3-1 models are capable of
accommodating the $g-2$ anomaly, as a result of stringent collider bounds de
Jesus  _et al._ (2020) (see Ky  _et al._ (2000); Kelso  _et al._ (2014); Binh
_et al._ (2015); Cogollo (2017); De Conto and Pleitez (2017) for $g_{\mu}-2$
studies in the context of 3-3-1 models). This result relies on the fact that
the gauge boson masses are proportional to the energy scale that controls
$SU(3)_{L}\times U(1)_{X}\to SU(2)_{L}\times U(1)_{Y}$ spontaneous symmetry
breaking ($v_{\chi}$), which is the main parameter in the analysis. This
dependence on the scale of SSB makes it possible to recast LHC lower bounds on
the mass of the gauge bosons into constraints on the energy scale of 3-3-1
SSB. From LHC searches we have $v_{\chi}>4\,\mbox{TeV}$, which excludes most
of the region where 3-3-1 models can account for the anomaly.

It follows that, by themselves, the simplest 3-3-1 models cannot accommodate
the muon magnetic moment anomaly, requiring the extension we consider here.
The latter consists in embedding the simplified Lagrangians of Sec.II into
3-3-1 models by promoting the inert scalar doublet into a scalar triplet, to
preserve the gauge invariance of the Lagrangian, while the other fields remain
unchanged. The major change is that now the relevant parameter for the
analysis will be the scale of symmetry breaking $v_{\chi}$, instead of the
mass of the inert scalar, as in Figs. 1 and 2. The relationship between the
inert scalar mass and $v_{\chi}$ involves scalar potential mixing terms of the
type $(\phi^{\dagger}\phi)(\chi^{\dagger}\chi)$. After the SSB these generate
a mass to $\phi$ proportional to $v_{\chi}$ 111Mixing with other scalar fields
could also generate mass terms for $\phi$. However, in 3-3-1 models it is
usually assumed that the first SSB takes place at a higher energy scale than
that which characterizes electroweak breaking.. As a result the
$(\phi^{\dagger}\phi)(\chi^{\dagger}\chi)$ term generates the most significant
contribution to the inert scalar mass, which we will assume to be given as
$m_{\phi}\sim\Lambda v_{\chi}$.

Our study of these simplified schemes embedded in 3-3-1 models yields basic
features concerning the interactions involving the inert scalar. We have
meticulously assessed their contribution to the muon’s $g_{\mu}-2$. Our
results are summarized in Figs. 5, 6 and 3. The $5.0\sigma$ deviation from the
experimental value reported by Fermilab Aguillard  _et al._ (2023) is
represented by the region between the solid horizontal green lines in Figs. 5
and 6 and also leads to the colored region in Fig. 3. To get these results we
have first scrutinized the scenario in which the inert scalar couples with the
muon in the loop. Subsequently, we examined the case where it also couples
with the vector-like fermion $E$. By varying the values for the masses of
$\phi$ and E, we have explored which parameter values are most relevant for
the $g_{\mu}-2$ anomaly.

Since our proposed scheme is embedded in a 3-3-1 setup, we can use the bounds
from $Z^{\prime}$ searches in high-energy colliders to constrain $m_{\phi}$.
In order to do this we take advantage of the mutual dependence on $v_{\chi}$,
where the inert scalar mass is given by $m_{\phi}=\Lambda v_{\chi}$ while
$m_{Z^{\prime}}$ is given by Eq. 8.

The high-energy bound from the LHC and the future expected sensitivities are
represented by the vertical dot-dashed lines in Figs. 5 and 6, depicted in
magenta, cyan, and gray color. These give the bounds from the LHC and the
projections for the LHC upgrade to High-Luminosity (HL-LHC), as well as the
Future Circular hadron Collider (FCC-hh), respectively. The bounds from the
LHC were derived using an integrated luminosity of
$\mathcal{L}=139\,\mathrm{fb}^{-1}$ and a center-of-mass energy $\sqrt{s}=13$
TeV. In comparison, simulations for the HL-LHC utilized $\mathcal{L}=$
$3000\,\mathrm{fb}^{-1}$ with $\sqrt{s}=14\,\mathrm{TeV}$. The FCC-hh adopts
the same luminosity level as the HL-LHC, but operates at a significantly
higher center-of-mass energy of $\sqrt{s}=100\,\mathrm{TeV}$, as referenced in
Alves  _et al._ (2022). The limits are set at $4$, $5.6$, and $27$
$\mathrm{TeV}$ for the LHC, HL-LHC, and FCC-hh, respectively.

Fig. 5 represents the contribution to the $\Delta a_{\mu}$ coming from an
inert scalar that couples with the muon as a function of the scale of 3-3-1
symmetry breaking. In this plot, we assumed the Yukawa coupling
$\lambda_{22}=4$, so $g^{s}_{22}=2$ and $g^{p}_{22}=0$, consistent with
perturbativity and unitarity. We took different configurations for the inert
scalar mass and the 3-3-1 scale of symmetry breaking of the 3-3-1 symmetry,
given by $m_{\phi}=0.1v_{\chi}$ and $m_{\phi}=0.05v_{\chi}$, represented by
the blue and black solid lines, respectively. With this configuration, one can
see that the region of $v_{\chi}$ and $m_{\phi}$, where one can accomodate the
5-sigma discrepancy claimed by Fermilab lies beyond the reach of current high-
energy experiments. However, it could possibly be probed by future upgrades of
the LHC or at high-energy lepton colliders such as ILC, CLIC, CEPC or FCC-ee.

Figure 5: Graph illustrating the relationship between the anomalous magnetic
moment ($\Delta a_{\mu}$) and the symmetry-breaking scale ($v_{\chi}$) for the
$g_{\mu}-2$ explanation. We assume $\lambda_{22}=4$ ($g^{s}_{22}=2$) and
$g^{p}_{22}=0$ consistent with perturbative unitarity. The blue and black
lines correspond to two distinct scenarios, characterized by scalar masses of
$m_{\phi}=0.05v_{\chi}$ and $m_{\phi}=0.1v_{\chi}$ respectively. The vertical
magenta, cyan, and gray regions give the current and projected exclusions from
the LHC, HL-LHC, and FCC-hh, as indicated. Figure 6: The plot illustrates the
relationship between the anomalous magnetic moment of the muon ($\Delta
a_{\mu}$) and the scale of symmetry breaking ($v_{\chi}$), providing an
explanation for the $g_{\mu}-2$ anomaly with an exotic charged lepton with a
mass of $m_{E}=270$ GeV. We assume $\lambda_{12}=4$ ($g^{s(p)}_{12}=2$). The
blue and black curves correspond to scalar masses of $m_{\phi}=0.005v_{\chi}$
and $m_{\phi}=0.01v_{\chi}$ respectively. The vertical regions are the current
and projected exclusion areas resulting from high-energy collider searches at
LHC (magenta), HL-LHC (cyan), and FCC-hh (gray).

On the other hand, Fig. 6 illustrates the contribution to the muon anomalous
magnetic moment arising from an inert scalar which is also coupled with the
exotic lepton E. Within this framework, we explore variations in the coupling
constant of this interaction and the mass of E. This way we find the optimal
parameter values that accommodate the current region of $\Delta a_{\mu}$. In
Fig. 6 we display the predicted $\Delta a_{\mu}$ values versus $v_{\chi}$. One
sees that $g_{\mu}-2$ can indeed be understood from first principles with an
exotic charged lepton with a mass of $m_{E}=270$ GeV. Our analysis adopts a
specific value of $\lambda_{12}=4$ ($g^{s(p)}_{12}=2$), consistent with
perturbative unitarity. The depicted blue and black curves give distinct
scenarios corresponding to scalar masses of $m_{\phi}=0.005v_{\chi}$ and
$m_{\phi}=0.01v_{\chi}$ respectively. This way we get further insights on how
this scheme can reproduce the muon’s anomalous magnetic moment. The result is
similar to the case of the inert scalar, and one sees that current data cannot
exclude the region of parameter space that can fit the anomaly. Future
experiments, though, should be able to cover the parameter space that lies
beyond the reach of current LHC searches, and produce a signal in future
colliders, such as the FCC.

##  IV  Conclusions

This work investigated the possibility of inert scalars accounting for the
recent $g_{\mu}-2$ anomaly within simple schemes, as well as in the context of
full-fledged 3-3-1 theories involving vector-like fermions. We considered the
contributions of these particles to the muon anomalous magnetic moment and
examined their viability in light of current experimental constraints. The
simplified schemes with inert scalars and vector-like fermions could provide
solutions to the $g_{\mu}-2$ anomaly (Figs. 1 and 2). The associated regions
of parameter space that could explain the anomaly are given in Fig. 3. We
found that vector-like fermion masses between $250-300$ GeV and inert scalar
masses between $350-450$ GeV are sufficient to accommodate the anomaly.

Accounting for the $g_{\mu}-2$ anomaly in the context of 3-3-1 gauge theories,
however, is subject to limitations that follow mainly due to stringent
collider bounds. These arise from experimental searches for new gauge bosons
at high-energy colliders such as the LHC. These constraints apply to the new
inert scalar triplet via the mass term, which is proportional to the 3-3-1
symmetry breaking scale ($v_{\chi}$) (see Figs. 5 and 6). Detailed analysis
reveals that for the case of an Inert scalar without vector-like fermions one
can accommodate the $g_{\mu}-2$ anomaly while avoiding bounds from current
data. In contrast, this is not possible for the case of the inert scalar plus
vector-like fermions. Indeed, here the constraints from current collider data
exclude a large area of parameter space which can accomodate the anomaly
(magenta region). Projections taking into account the reach of LHC upgrades as
well as those of new high-energy colliders, e.g., HL-LHC and FCC-hh, fully
cover the parameter region compatible with the anomaly. This means that if
this model is indeed the right answer to the anomaly, it should lead to a
signal in the next generation of collider experiments.

## Acknowledgements

FSQ This work was supported by Simons Foundation (Award Number:1023171-RC),
FAPESP Grant 2018/25225-9, 2021/01089-1, 2023/01197-4, ICTP-SAIFR FAPESP
Grants 2021/14335-0, CNPq Grant 307130/2021-5, and ANID-Millennium Science
Initiative Program ICN2019_044 YSV expresses gratitude to São Paulo Research
Foundation (FAPESP) under Grant No. 2018/25225-9 and 2023/01197-4. JWFV is
supported by Spanish grants PID2020-114473GB-I00 (AEI/10.13039/501100011033)
and Prometeo/2021/071 (Generalitat Valenciana). ASJ acknowledges support from
Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) under
Grant No. 88887.497142/2020-00.

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