††thanks: E-mail<EMAIL_ADDRESS>E-mail<EMAIL_ADDRESS>
# On the viability of a light scalar spectrum for 3-3-1 models

A. L. Cherchiglia  O. L. G. Peres  Instituto de Física Gleb Wataghin –
Universidade Estadual de Campinas, Campinas-SP, Brasil

###### Abstract

In this work we study an effective version of the 3-3-1 model, in which the
particle content is the same of the 2HDM. We show that the inherited structure
from the $SU(3)_{C}\otimes SU(3)_{L}\otimes U(1)_{X}$ gauge group has a series
of consequences, the most relevant one being the prediction of the masses of
the neutral scalar to be of the order or lower than the mass of the charged
scalar. Given current constraints from collider searches, B-physics, as well
as theoretical constraints such as perturbativity of quartic couplings and
stability of the scalar potential, we find that the new scalars cannot be
lighter than 350 GeV.

##  I Introduction

In the present, the quest for Beyond Standard Model signals is as strong as
ever. From a collider perspective, the LHC has just resumed its operations
aiming to increase the amount of data in a variety of processes, with the hope
to pinpoint deviations from the SM predictions. Although the experimental
program is the one responsible to unveil the path towards BSM, it is of utmost
importance to scrutinize present theoretical proposals aiming to identify the
kind of signals that they can (or not) address. Among these models, an
extension of the SM gauge group is compelling since it may address some of the
open problems of the SM as well as provide a rich phenomenology testable at
the LHC. In particular, we will consider in this contribution the extension of
the gauge group to $SU(3)_{C}\otimes SU(3)_{L}\otimes U(1)_{X}$. We denote
this scenario as 3-3-1 modelPisano and Pleitez (1992); Frampton (1992), which
has been extensively studied in the literature since it can provide a
mechanism for 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. (2016a); 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), allows the inclusion of dark matter candidates
Fregolente and Tonasse (2003); Long and Lan (2003); de S. Pires and Rodrigues
da Silva (2007); Mizukoshi et al. (2011); Ruiz-Alvarez et al. (2012); Profumo
and Queiroz (2014); Dong et al. (2013a, b); Cogollo et al. (2014a); Dong et
al. (2014a, b); Kelso et al. (2014); Dong et al. (2014b); Mambrini et al.
(2016); Dong et al. (2015); de S. Pires et al. (2016); Alves et al. (2017);
Rodrigues da Silva (2016); Carvajal et al. (2017); Dong et al. (2018); Arcadi
et al. (2018); Montero et al. (2018); Huong et al. (2019); Alvarez-Salazar and
Peres (2021); Van Loi et al. (2021); Dutra et al. (2021); Oliveira and de S.
Pires (2022), can address present meson anomalies Cogollo et al. (2012);
Cogollo et al. (2014b); Buras et al. (2014); Buras and De Fazio (2016a);
Queiroz et al. (2016); de Melo et al. (2021); Buras and De Fazio (2016b);
Buras et al. (2021), the strong CP problem Pal (1995); Dias and Pleitez
(2004); Dias et al. (2003); Montero and Sanchez-Vega (2011); Dias et al.
(2018, 2020), among others.

Since we have extended the gauge group of the SM, the 3-3-1 models predicts,
in general, the existence of extra gauge bosons which potentially modifies
many observables. This in turn puts stringent lower bounds in their masses,
given the non-observation of such particles in the LHC. Currently, lower
bounds for the $Z^{\prime}$ mass is around 4 TeV Alves et al. (2022). It
should be emphasized that the mass of these extra gauge bosons is directly
related to the scale in which the 3-3-1 gauge group is broken to the SM one.
Thus, if this breaking occurs at a high scale of hundred of TeVs, for
instance, there is no hope to see these particles in the present and near-
future colliders.

Another consequence of the non-observation of the extra gauge bosons is that
the present discrepancy in the muon magnetic moment measurement Abi et al.
(2021) cannot be addressed by one-loop contributions containing such
particles, due to their high masses de Jesus et al. (2020). However, if the
scalar sector of the 3-3-1 model could be light enough, this anomaly could
potentially be explained in similarity to the 2HDM Broggio et al. (2014); Chun
and Kim (2016); Wang and Han (2015); Abe et al. (2015); Crivellin et al.
(2016); Chun et al. (2015); Han et al. (2016); Ilisie (2015); Cherchiglia et
al. (2018). One of the aims of this contribution is to close this gap.

3-3-1 models have a rich scalar sector, featuring usually three triplets.
Given the bounds on the new gauge bosons masses, there are consequences for at
least some of the scalars predicted. This can be expected given their common
dependence on the vev that breaks the group $SU(3)_{C}\otimes SU(3)_{L}\otimes
U(1)_{X}$ to $SU(3)_{C}\otimes SU(2)_{L}\otimes U(1)_{X}$. However, given that
at least one scalar was discovered at the LHC with mass around the EW scale,
one can wonder if any of the other predicted scalars can have masses below the
TeV region. This decoupled scenario can actually be realized rendering an
effective 3-3-1 model with the same particle content of the 2HDM Okada et al.
(2016b); Fan and Yagyu (2022). In this contribution we will study this
scenario in great detail, applying current constraints coming from collider
searches, b-physics, the W-mass measurement, as well as theoretical
constraints such as perturbativity of quartic couplings and stability of the
scalar potential. Our analysis will complement and extend the ones presented
in Okada et al. (2016b); Fan and Yagyu (2022). In particular the inclusion of
the theoretical constraints will have a major impact on the allowed
enhancement of fermion couplings to the extra scalars. We will also discuss
the impact of the recent W-mass measurement by the CDF collaboration Aaltonen
et al. (2022) for the allowed parameter region.

This work is organized as follows: in section II we review the scalar sector
of the 3-3-1 model, and define its effective version, comparing it to the
general 2HDM. In section III we present the phenomenological constraints we
considered, while in section IV we perform our numerical analysis. Finally, we
conclude in section V.

##  II Scalar sector of 331 models and the 2HDM

In this section we intend to provide a brief review of the scalar sector of
331 models in comparison to the 2HDM. We begin with the scalar potential of
both models, presenting their Yukawa sector afterwards.

###  II.1 Scalar potential

In the version of 331 model we are considering, only three triplets $\eta$,
$\rho$ and $\chi$ are presented. They will be responsible for providing masses
for the gauge bosons as well as all charged fermions through the Higgs
mechanism. As usual, we choose $\chi$ to perform the first breaking of the
gauge group, while both $\eta$, $\rho$ are responsible for the other one.
Schematically,

$SU(3)_{C}\otimes SU(3)_{L}\otimes
U(1)_{X}\xRightarrow{v_{\chi}}SU(3)_{C}\otimes SU(2)_{L}\otimes
U(1)_{X}\xRightarrow{v_{\eta},v_{\rho}}SU(3)_{C}\otimes U(1)_{\text{em}}$

Usually, one adopts

$\eta=\begin{pmatrix}\eta^{0}\\\ \eta^{-}\\\
\eta^{-A}\end{pmatrix},\quad\rho=\begin{pmatrix}\rho^{+}\\\ \rho^{0}\\\
\rho^{-B}\end{pmatrix},\quad\chi=\begin{pmatrix}\chi^{A}\\\ \chi^{B}\\\
\chi^{0}\end{pmatrix},$ (1)

where the charges under $U(1)_{X}$ are

$X_{\eta}=-\frac{1}{2}-\frac{\beta_{Q}}{2\sqrt{3}},\quad
X_{\rho}=\frac{1}{2}-\frac{\beta_{Q}}{2\sqrt{3}},\quad
X_{\chi}=\frac{\beta_{Q}}{\sqrt{3}}.$ (2)

As is well-known, to define a 331 model one must adopt a specific value for
$\beta$, which enters in the charge generator $Q=T_{3}+\beta_{Q}T_{8}+XI$.
From the triplets, one obtains seven physical scalars: three neutral CP-even
($h_{1}$,$h_{2}$,$h_{3}$); one neutral CP-odd ($A_{0}$); three charged scalars
($h^{\pm}$, $h^{\pm A}$, $h^{\pm B}$). In this work we are going to consider
that all couplings in the scalar potential are real (CP conserving potential),
allowing to define CP-even/odd scalars unambiguously. Moreover, the triplets
acquire vevs denoted by $v_{\eta}$, $v_{\rho}$, $v_{\chi}$ where $v_{\chi}$
(which performs the first symmetry breaking) is assumed to be much higher than
the other two. In similarity to 2HDMs, we will define
$\tan\beta=v_{\eta}/v_{\rho}$ and notice that
$v=\sqrt{v^{2}_{\rho}+v^{2}_{\eta}}=246\;\text{GeV}$.

In terms of the triplets, the scalar potential is given by (we adopt the
convention of Costantini et al. (2020))

$\displaystyle V\left(\eta,\rho,\chi\right)$ $\displaystyle=$
$\displaystyle\mu_{1}^{2}\rho^{\dagger}\rho+\mu_{2}^{2}\eta^{\dagger}\eta+\mu_{3}^{2}\chi^{\dagger}\chi+\lambda_{1}\left(\rho^{\dagger}\rho\right)^{2}+\lambda_{2}\left(\eta^{\dagger}\eta\right)^{2}+\lambda_{3}\left(\chi^{\dagger}\chi\right)^{2}$
(3)
$\displaystyle+\lambda_{12}\left(\rho^{\dagger}\rho\right)\left(\eta^{\dagger}\eta\right)+\lambda_{13}\left(\chi^{\dagger}\chi\right)\left(\rho^{\dagger}\rho\right)+\lambda_{23}\left(\eta^{\dagger}\eta\right)\left(\chi^{\dagger}\chi\right)$
$\displaystyle+\zeta_{12}\left(\rho^{\dagger}\eta\right)\left(\eta^{\dagger}\rho\right)+\zeta_{13}\left(\rho^{\dagger}\chi\right)\left(\chi^{\dagger}\rho\right)+\zeta_{23}\left(\eta^{\dagger}\chi\right)\left(\chi^{\dagger}\eta\right)$
$\displaystyle-\sqrt{2}f\epsilon_{ijk}\eta_{i}\rho_{j}\chi_{k}+\textrm{H.c.},$

The above potential is the most general one as long as $\beta_{Q}\neq\pm
1/\sqrt{3}$. The reason is that, for this choice, two of the triplets will
have the same charge under the $U(1)_{X}$ group allowing for terms with an odd
number of the same triplet. This feature can be avoided if, for instance, a
$\mathbb{Z}_{2}$ symmetry is introduced. Since the $\chi$ triplet will be
responsible for the breaking of $SU(3)_{C}\otimes SU(3)_{L}\otimes U(1)_{X}$
to $SU(3)_{C}\otimes SU(2)_{L}\otimes U(1)_{X}$, one usually imposes that only
this field is odd under the $\mathbb{Z}_{2}$ just mentioned. In this
contribution we will, as far as possible, keep the discussion in general terms
which implies that $\beta_{Q}$ will be kept arbitrary. In order to our results
be also applied to the specific choices $\beta_{Q}=\pm 1/\sqrt{3}$, we will
implicitly adopt the $\mathbb{Z}_{2}$ symmetry in this case.

Once the scalar potential is defined, one can study the conditions that the
parameters must fulfill in order to respect stability (boundeness from below),
perturbativity, and unitarity bounds. In Costantini et al. (2020) a
comprehensive study of these conditions were performed. In particular, they
provided expressions which allows one to trade the 13 potential parameters
($\mu_{i}^{2}$, $\lambda_{i}$, $\lambda_{ij}$, $\zeta_{ij}$, $f$) by
$\tan\beta$, $v$, $v_{\chi}$, the masses of the seven scalars ($h_{1}$,
$h_{2}$, $h_{3}$, $A_{0}$, $h^{\pm}$, $h^{\pm A}$, $h^{\pm B}$) plus three
mixing angles that appear when diagonalizing the neutral CP-even mass matrix.
Explicitly,

$\vec{h}=R^{\text{\tiny{331}}}.\vec{H}$ (4)

where
$\vec{H}=(\sqrt{2}\;\Re\;\rho^{0},\sqrt{2}\;\Re\;\eta^{0},\sqrt{2}\;\Re\;\chi^{0})$,
$\vec{h}=(h_{1},h_{2},h_{3})$, and

$R^{\text{\tiny{331}}}=\begin{pmatrix}c_{2}c_{3}&c_{3}s_{1}s_{2}-c_{1}s_{3}&c_{1}c_{3}s_{2}+s_{1}s_{3}\\\
c_{2}s_{3}&s_{3}s_{1}s_{2}+c_{1}c_{3}&c_{1}s_{3}s_{2}-s_{1}c_{3}\\\
-s_{2}&c_{2}s_{1}&c_{1}c_{2}\end{pmatrix}.$ (5)

We are adopting the usual convention $c_{i}=\cos(\alpha_{i})$,
$s_{i}=\sin(\alpha_{i})$. It can be shown (see for instance Okada et al.
(2016b); Fan and Yagyu (2022)) that the limit $v_{\chi}>>v_{\rho},v_{\eta}$
implies $\alpha_{2}\sim\alpha_{1}\sim 0$. Thus, only $h_{1}$ and $h_{2}$ can
mix, in similarity to the 2HDM. Moreover, $h_{3}$, $h^{\pm A}$, and $h^{\pm
B}$ will decouple, rendering an effective scalar potential in
$SU(2)_{L}\otimes U(1)_{X}$

$\displaystyle V_{eff}(\Phi_{\eta},\Phi_{\rho})$ $\displaystyle=$
$\displaystyle\mu^{2}_{\rho}\Phi^{\dagger}_{\rho}\Phi_{\rho}+\mu^{2}_{\eta}\Phi_{\eta}^{\dagger}\Phi_{\eta}+\lambda_{1}(\Phi_{\rho}^{\dagger}\Phi_{\rho})^{2}+\lambda_{2}(\Phi_{\eta}^{\dagger}\Phi_{\eta})^{2}$
(6) $\displaystyle+$
$\displaystyle\lambda_{12}(\Phi_{\rho}^{\dagger}\Phi_{\rho})(\Phi_{\eta}^{\dagger}\Phi_{\eta})+\zeta_{12}(\Phi^{T}_{\eta}\varepsilon\Phi_{\rho})^{\dagger}(\Phi_{\eta}^{T}\varepsilon\Phi_{\rho})$
$\displaystyle+$
$\displaystyle\\{\mu_{\eta\rho}^{2}\Phi^{T}_{\eta}\varepsilon\Phi_{\rho}+H.c.\\}$

where $\varepsilon=i\tau_{2}$ to comply with scalar products in the $SU(2)$
group and the doublets have hypercharge $Y=(-1/2,1/2)$ respectively as below

$\Phi_{\eta}=\left(\begin{array}[]{cc}\eta^{0}\\\
-\eta^{-}\end{array}\right),\quad\Phi_{\rho}=\left(\begin{array}[]{cc}\rho^{+}\\\
\rho^{0}\end{array}\right).$ (7)

It is fruitful to consider the general 2HDM at this point. In this model, one
introduces two doublets ($\Phi_{1}$, $\Phi_{2}$) under $SU(2)_{L}$, with the
same hypercharge under $U(1)_{Y}$ which acquire the vevs $v_{1}$, and $v_{2}$
respectively. Usually, it is defined $\tan\beta=v_{2}/v_{1}$ and we notice
that $v=\sqrt{v^{2}_{1}+v^{2}_{2}}=246\;\text{GeV}$. Explicitly, the doublets
can be expressed as

$\Phi_{i}=\begin{pmatrix}\Phi_{i}^{+}\\\ \Phi_{i}^{0}\end{pmatrix}.$ (8)

In terms of the doublets, the most general scalar potential is Gunion and
Haber (2003)

$\displaystyle V\left(\Phi_{1},\Phi_{2}\right)=$ $\displaystyle
m_{11}^{2}\Phi_{1}^{\dagger}\Phi_{1}+m_{22}^{2}\Phi_{2}^{\dagger}\Phi_{2}+\frac{\Lambda_{1}}{2}\left(\Phi_{1}^{\dagger}\Phi_{1}\right)^{2}+\frac{\Lambda_{2}}{2}\left(\Phi_{2}^{\dagger}\Phi_{2}\right)^{2}$
$\displaystyle+\Lambda_{3}\left(\Phi_{1}^{\dagger}\Phi_{1}\right)\left(\Phi_{2}^{\dagger}\Phi_{2}\right)+\Lambda_{4}\left(\Phi_{1}^{\dagger}\Phi_{2}\right)\left(\Phi_{2}^{\dagger}\Phi_{1}\right)+\big{[}-m_{12}^{2}\Phi_{1}^{\dagger}\Phi_{2}$
$\displaystyle+\frac{\Lambda_{5}}{2}\left(\Phi_{1}^{\dagger}\Phi_{2}\right)^{2}+\Lambda_{6}\left(\Phi_{1}^{\dagger}\Phi_{1}\right)\left(\Phi_{1}^{\dagger}\Phi_{2}\right)+\Lambda_{7}\left(\Phi_{2}^{\dagger}\Phi_{2}\right)\left(\Phi_{1}^{\dagger}\Phi_{2}\right)+\textrm{H.c.}\big{]}$
(9)

A $\mathbb{Z}_{2}$ symmetry can be considered to render
$\Lambda_{6}=\Lambda_{7}=0$, while the term $m_{12}^{2}$ is a soft-breaking
term. This model features four physical scalars: two neutral CP-even ($h$,
$H$); one neutral CP-odd ($A$); one charged scalar ($H^{\pm}$). In Gunion and
Haber (2003) it is shown how to trade seven of the parameters in the potential
($m_{ii}^{2}$, $m_{12}^{2}$, $\lambda_{i}$ where $i=1\cdots 4$) by
$\tan\beta$, $v$, the masses of the four physical scalars and the mixing angle
that appears when diagonalizing the neutral CP-even mass matrix. Explicitly,

$\vec{h}=R^{\text{\tiny{2HDM}}}.\vec{H}$ (10)

where $\vec{H}=(\sqrt{2}\text{Re}\Phi_{1}^{0},\sqrt{2}\text{Re}\Phi_{2}^{0})$,
$\vec{h}=(H,h)$, and

$R^{\text{\tiny{2HDM}}}=\begin{pmatrix}c_{\alpha}&-s_{\alpha}\\\
s_{\alpha}&c_{\alpha}\end{pmatrix}.$ (11)

We emphasize that it is not possible to trade all potential parameters from
$V\left(\Phi_{1},\Phi_{2}\right)$ by physical parameters as we did for
$V\left(\eta,\rho,\chi\right)$. Namely, we retain $\Lambda_{5}$,
$\Lambda_{6}$, and $\Lambda_{7}$. This feature is expected since, by comparing
$V\left(\Phi_{1},\Phi_{2}\right)$ with $V_{eff}(\Phi_{\eta},\Phi_{\rho})$, one
notices that in the latter only four quartic couplings are present. It is a
consequence of the 331 symmetry, that will play an important role in the
phenomenology. To be concrete, even if the usual $\mathbb{Z}_{2}$ symmetry is
assumed in the scalar potential (which renders $\Lambda_{6}=\Lambda_{7}=0$),
in the 2HDM $\Lambda_{5}$ is a free parameter. This implies, for instance,
that the coupling $\lambda_{hAA}$ can always be set to zero by a particular
choice of $\Lambda_{5}$. The same does not occur in the 331 model. Explicitly,
in the alignment limit ($\beta-\alpha=\pi/2$) one obtains for the 2HDM

$\lambda_{hAA}=i\left[2\Lambda_{5}v-\frac{m_{h}^{2}}{v}\right]$ (12)

while the result of the 331 model is obtained by setting $\Lambda_{5}=0$ (we
are ignoring terms $\mathcal{O}(v_{\chi}^{-1})$)111In our numerical analysis
we consider the full expression, showing that the approximation above is
excellent. Another interesting triple scalar couplings is
$\lambda_{hH^{+}H^{-}}$ which, in the alignment limit, is given by

$\displaystyle\lambda_{hH^{+}H^{-}}=\begin{cases}i\left(2\frac{m_{A_{0}}^{2}-m_{h^{\pm}}^{2}}{v}-\frac{m_{h}^{2}}{v}+2\Lambda_{5}v\right),\quad&\mbox{2HDM}\\\
i\left(2\frac{m_{A_{0}}^{2}-m_{h^{\pm}}^{2}}{v}-\frac{m_{h}^{2}}{v}\right),\quad&\mbox{331}\end{cases}$
(13)

Once again, this coupling can be suppressed/enhanced in the 2HDM by a suitable
choice of $\Lambda_{5}$, while for the 331 model the coupling is controlled
only by the splitting among the masses of the charged and CP-odd scalars.

###  II.2 Yukawa sector

We consider now the Yukawa sector of the 331 model in comparison to the 2HDM.
As it is well-known, we have some choices for the leptonic fields, which will
define the quark sector by anomaly cancellation requirements. In a generic way
we define

$L_{i}=\begin{pmatrix}e_{i}\\\ -\nu_{i}\\\ E_{i}\end{pmatrix}_{L}.$ (14)

where $E_{iL}$ can be either a new field or the charge conjugate of the
charged lepton field. As can be seen, we define the leptonic fields to be
antitriplets under 331. The right-handed fields are chosen as singlets. The
electric charge of the $E_{i}$ fields (both left and right) is given by

$Q_{E_{i}}=-\frac{1}{2}+\frac{\sqrt{3}\beta_{Q}}{2}$ (15)

Once the leptonic fields are defined, we must choose how to distribute the
quarks fields. Given the anomaly requirement, one of the quark families must
also be an antitriplet while the other two will be triplets. As usual, we
choose the third family, amounting to

$q_{1}=\begin{pmatrix}u\\\ d\\\ D\end{pmatrix}_{L}\quad\quad
q_{2}=\begin{pmatrix}c\\\ s\\\ S\end{pmatrix}_{L}\quad\quad
q_{3}=\begin{pmatrix}b\\\ -t\\\ T\end{pmatrix}_{L}$ (16)

Once again, the right-handed fields are all singlets. The electric charges of
the $D,S,T$ fields (both left and right-handed) are given by

$Q_{D,S}=\frac{1}{6}-\frac{\sqrt{3}\beta_{Q}}{2}\quad\quad
Q_{T}=\frac{1}{6}+\frac{\sqrt{3}\beta_{Q}}{2}$ (17)

Given the particle content for the fermionic fields, we can write the Yukawa
lagrangian as below

$\displaystyle-\mathcal{L}_{\rm{Yuk}}^{q}$
$\displaystyle=y_{ij}^{u}\bar{q}_{iL}\eta
u_{jR}+y_{3j}^{u}\bar{q}_{3L}\rho^{*}u_{jR}+y_{ij}^{d}\bar{q}_{iL}\rho
d_{jR}+y_{3j}^{d}\bar{q}_{3L}\eta^{*}d_{jR}$
$\displaystyle+y_{i}^{D}\bar{q}_{iL}\chi D_{R}+y_{i}^{S}\bar{q}_{iL}\chi
S_{R}+y^{T}\bar{q}_{3L}\chi^{*}T_{R}+h.c$ (18)
$\displaystyle-\mathcal{L}_{\rm{Yuk}}^{l}$
$\displaystyle=y_{mn}^{e}\bar{L}_{mL}\eta^{*}e_{nR}+y_{mn}^{E}\bar{L}_{mL}\chi^{*}E_{nR}+h.c$
(19)

where $i=\\{1,2\\}$, while the other indexes go from 1 to 3.

For a clearer comparison with the 2HDM, we will rewrite the lagrangian above
as

$\displaystyle-\mathcal{L}_{\rm{Yuk}}^{q}$
$\displaystyle=y_{1j}^{u}\left(\bar{u}_{L}\eta^{0}+\bar{d}_{L}\eta^{-}\right)u_{jR}+y_{2j}^{u}\left(\bar{c}_{L}\eta^{0}+\bar{s}_{L}\eta^{-}\right)u_{jR}+y_{3j}^{u}\left(\bar{b}_{L}\rho^{-}-\bar{t}_{L}\rho^{0}\right)u_{jR}$
$\displaystyle+y_{1j}^{d}\left(\bar{u}_{L}\rho^{+}+\bar{d}_{L}\rho^{0}\right)d_{jR}++y_{2j}^{d}\left(\bar{c}_{L}\rho^{+}+\bar{s}_{L}\rho^{0}\right)d_{jR}+y_{3j}^{d}\left(\bar{b}_{L}\eta^{0}-\bar{t}_{L}\eta^{+}\right)d_{jR}$
$\displaystyle+\mbox{terms with $D$, $S$, $T$}+h.c$ (20)
$\displaystyle-\mathcal{L}_{\rm{Yuk}}^{l}$
$\displaystyle=y_{mn}^{e}\left(\bar{e}_{mL}\eta^{0}-\bar{\nu}_{mL}\eta^{+}\right)e_{nR}+\mbox{terms
with $E_{i}$}+h.c$ (21)

In the format above, it is obvious that $\eta$ will generate the masses for
the charged leptons, the third generation of down-type quarks as well as for
first and second generation of up-quarks, while $\rho$ will provide masses for
all the remaining charged fermions that are present in the SM. We will work in
the limit in which the new fermions do not mix with the SM ones.

We consider that the initial Yukawa lagrangian was written in a basis in which
the charged leptons are diagonal. Thus

$y_{ii}^{e}=\frac{\sqrt{2}m^{e}_{i}}{v}\frac{v}{v_{\eta}}=\frac{\sqrt{2}m^{e}_{i}}{v}\frac{1}{s_{\beta}},$
(22)

where $m^{e}_{i}=\\{m_{e},m_{\mu},m_{\tau}\\}$. For the up and down quarks, we
define the matrices $V$ and $U$ that perform their bi-diagonalization
respectively as below

$\displaystyle\frac{v_{\eta}}{\sqrt{2}}M_{u}\equiv
V^{\dagger}_{u}\begin{pmatrix}y_{11}^{u}&y_{12}^{u}&y_{13}^{u}\\\
y_{21}^{u}&y_{22}^{u}&y_{23}^{u}\\\
\frac{-y_{31}^{u}}{\tan\beta}&\frac{-y_{32}^{u}}{\tan\beta}&\frac{-y_{33}^{u}}{\tan\beta}\end{pmatrix}U_{u}$
$\displaystyle=\begin{pmatrix}m_{u}&0&0\\\ 0&m_{c}&0\\\
0&0&m_{t}\end{pmatrix}$ (23) $\displaystyle
M_{d}\equiv\frac{v_{\eta}}{\sqrt{2}}V^{\dagger}_{d}\begin{pmatrix}\frac{y_{11}^{d}}{\tan\beta}&\frac{y_{12}^{d}}{\tan\beta}&\frac{y_{13}^{d}}{\tan\beta}\\\
\frac{y_{21}^{d}}{\tan\beta}&\frac{y_{22}^{d}}{\tan\beta}&\frac{y_{23}^{d}}{\tan\beta}\\\
y_{31}^{d}&y_{32}^{d}&y_{33}^{d}\end{pmatrix}U_{d}$
$\displaystyle=\begin{pmatrix}m_{d}&0&0\\\ 0&m_{s}&0\\\
0&0&m_{b}\end{pmatrix}$ (24)

As discussed in Fan and Yagyu (2022), the matrix $V_{u}$ can be chosen freely
while the matrix $V_{d}$ will be constructed by the knowledge of the CKM
matrix, $V_{\text{\tiny{CKM}}}=V_{u}^{\dagger}V_{d}$. Under the assumption
that the only source of CP-violation comes from the CKM matrix, we can
parametrize the matrix $V_{u}$ as

$V_{u}^{\dagger}=\begin{pmatrix}1&0&0\\\ 0&c_{\psi}&s_{\psi}\\\
0&-s_{\psi}&c_{\psi}\end{pmatrix}\begin{pmatrix}c_{\theta}&0&s_{\theta}\\\
0&1&0\\\
-s_{\theta}&0&c_{\theta}\end{pmatrix}\begin{pmatrix}c_{\phi}&s_{\phi}&0\\\
-s_{\phi}&c_{\phi}&0\\\ 0&0&1\end{pmatrix}.$ (25)

It can be noticed that, if the angles $\psi$ and $\theta$ are small, the
matrix will be approximately block diagonal. In this scenario, we have

$\sqrt{(y_{31}^{u})^{2}+(y_{32}^{u})^{2}+(y_{33}^{u})^{2}}=\frac{\sqrt{2}m_{t}}{v}\frac{v}{v_{\eta}}\tan\beta=\frac{\sqrt{2}m_{t}}{v}\frac{1}{c_{\beta}}$
(26)

Since $\frac{\sqrt{2}m_{t}}{v}\approx 1$, it is natural to consider the regime
$\tan\beta<1$ to fulfill perturbativity of the Yukawa couplings. As it is
clear from eqs.(23-24), none of the usual 2HDM types (types I, II, X, Y)
Branco et al. (2012) can be mapped in the 331 model. Regarding the effective
331 model considered in this work, one obtains

$\displaystyle-\mathcal{L}_{\rm{Yuk}}^{q}$
$\displaystyle=y_{mi}^{u}\bar{q}_{mL}\phi_{\eta}u_{iR}+y_{3i}^{u}\bar{q}_{3L}\phi_{\rho}^{*}u_{iR}+y_{mi}^{d}\bar{q}_{mL}\phi_{\rho}d_{iR}+y_{3i}^{d}\bar{q}_{3L}\phi_{\eta}^{*}d_{iR}+h.c$
(27) $\displaystyle-\mathcal{L}_{\rm{Yuk}}^{l}$
$\displaystyle=y_{mn}^{e}\bar{L}_{mL}\phi_{\eta}^{*}e_{nR}+h.c$ (28)

where in the equation above only the first two components of $q_{i}$ and
$L_{i}$ are considered. In the mass eigenstate basis and in the alignment
limit, the Yukawa lagrangian is given byFan and Yagyu (2022)

$\displaystyle-\mathcal{L}_{\rm{Yuk}}^{l}$
$\displaystyle=\frac{1}{v}\bar{l}_{i}m^{e}_{i}l_{i}\left(h+\frac{1}{\tan\beta}H+i\frac{1}{\tan\beta}\gamma_{5}A_{0}\right)$
$\displaystyle+\left(\frac{\sqrt{2}}{v}\bar{l}_{i}m^{e}_{i}P_{L}\nu_{i}\frac{1}{\tan\beta}H^{-}+\text{h.c.}\right)$
$\displaystyle-\mathcal{L}_{\rm{Yuk}}^{q}$
$\displaystyle=\frac{1}{v}\sum_{\varphi=h,H,A_{0}}\sum_{q=u,d}p_{\varphi}^{q}\bar{q}\Gamma_{\varphi}^{q}M_{q}^{\rm
diag}P_{R}q\varphi+\text{h.c.}$
$\displaystyle+\frac{\sqrt{2}}{v}\left[\bar{d}\left(M_{L}P_{L}+M_{R}P_{R}\right)u\right]H^{-}+\text{h.c.}$
(29)

where $p_{h}^{q}=p_{H}^{q}=1,\leavevmode\nobreak\ p_{A}^{q}=2iI_{q}$ with
$I_{q}=1/2(-1/2)$ for $q=u(d)$, $P_{L,R}$ are the projection operators and

$\displaystyle M_{L}=-M_{d}\Gamma_{A}^{d}V_{\text{\tiny{CKM}}}^{\dagger},\quad
M_{R}=V_{\text{\tiny{CKM}}}^{\dagger}\Gamma_{A}^{u}M_{u}.$ (30)

The interaction matrices are given by

$\displaystyle\Gamma_{A}^{u}$
$\displaystyle=-\frac{1}{\tan\beta}+V_{u}^{\dagger}\begin{pmatrix}0&0&0\\\
0&0&0\\\ 0&0&\frac{1}{\tan\beta}+\tan\beta\end{pmatrix}V_{u},$
$\displaystyle\Gamma_{A}^{d}$ $\displaystyle=\tan\beta-
V_{\text{\tiny{CKM}}}^{\dagger}V_{u}^{\dagger}\begin{pmatrix}0&0&0\\\ 0&0&0\\\
0&0&\frac{1}{\tan\beta}+\tan\beta\end{pmatrix}V_{u}V_{\text{\tiny{CKM}}},$
(31) $\displaystyle\Gamma_{H}^{q}$
$\displaystyle=-\Gamma_{A}^{q},\quad\Gamma_{h}^{d}=1$ (32)

In the format above, it is clear the angle $\phi$ of $V_{u}$ will not appear,
see eq. (25). It is instructive to write the element (33) of the matrix
$\Gamma_{A}^{u}$ explicitly, since it will be responsible for the coupling
among the neutral scalars $H$, $A_{0}$ and the top quark

$(\Gamma_{A}^{u})_{33}=c_{\psi}^{2}c_{\theta}^{2}\tan\beta-\frac{c_{\psi}^{2}s_{\theta}^{2}+s_{\psi}^{2}}{\tan\beta}$
(33)

As can be seen, if the mixing angles are small, the coupling will be
proportional to $\tan\beta$, being suppressed if $\tan\beta$ is small as we
already discussed after eq. (26). On the other hand, for general choices of
the mixing angles, the coupling will contain a mixture of terms proportional
to $\tan\beta$ and $\cot\beta$. Since the Yukawa coupling related to the top
quark is already of order one, one expect in this case that $\tan\beta\sim 1$.

##  III Phenomenological and theoretical constraints

In this section we describe the constraints we are going to impose in our
model, aiming to illustrate the main differences among the 331 model in the
decoupled regime and the 2HDM.

###  III.1 Theoretical constraints

Since we have a model with multiple scalars, one should consider some
theoretical constraints such as the stability of the scalar potential,
perturbativity of its couplings as well as perturbative unitarity of the
scattering matrix. For 2HDM, this conditions are well-knownManiatis et al.
(2006); Ginzburg and Ivanov (2005). For the 331 model, a detailed study with a
user-friendly implementation in Mathematica was performed in Costantini et al.
(2020). In Sánchez-Vega et al. (2019) the stability conditions for the scalar
potential were also studied. Since they will play a major role in constraining
the scalar masses, we reproduce them below assuming that
$\lambda_{3}\sim\lambda_{i3}\sim\zeta_{i3}\sim 0$ ($i=1,2$)

$\lambda_{1}>0,\quad\lambda_{2}>0,\quad\lambda_{12}+2\sqrt{\lambda_{1}\lambda_{2}}>0,\quad\lambda_{12}+\zeta_{12}+2\sqrt{\lambda_{1}\lambda_{2}}>0,$
(34)

We will show in the numerical section that, in the parameter space considered
in this work the assumption $\lambda_{3}\sim\lambda_{i3}\sim\zeta_{i3}\sim 0$
indeed holds. Moreover, it can be easily seen that the stability conditions of
eq. (34) are exactly the same for the 2HDM under the mapping
$\lambda_{12}\rightarrow\Lambda_{3}$, $\zeta_{12}\rightarrow\Lambda_{4}$, and
$\Lambda_{5}=0$.

Finally, when performing the numerical scan for the 331 model we will use the
full result of Costantini et al. (2020). For the 2HDM we will use an in-house
routine which was checked against the 2HDMC code Eriksson et al. (2010).

###  III.2 $B\rightarrow X_{s}\gamma$

The decay $B\rightarrow X_{s}\gamma$ gives stringent constraints for the
parameter space of 2HDM, in particular for the lower bound of the charged
scalar mass. This is particularly relevant when the Yukawa couplings of the
charged scalar to down-type quarks can be enhanced, which happens in Types II
and Y models in the regime with large $\tan\beta$. In our model, for small
mixing angles in $V_{u}$, a similar pattern is expected to happen since the
couplings to the third family quarks behaves oppositely to each other as in
the Types II and Y. For comparison, we have performed an analysis of this
observable in Types I/X and II/Y using the analytic formulas of Enomoto and
Watanabe (2016). Our results can be seen in Fig 1, where in red we show the
points that are excluded at 95$\%$ C.L. while in green we have the allowed
points. Our results are compatible with Arbey et al. (2018). Since we are
considering in the plots $\tan\beta>1$, the influence of other observables
(for instance $B_{s}\rightarrow\mu\mu$) is very marginal. The main result is
that in Types II/Y there is a lower bound on the mass of the charged scalar
around 600 GeV which is fairly independent of $\tan\beta$.

(a) Type I/X

(b) Type II/Y

Figure 1: Allowed (green) and excluded (red) regions in the plane $\tan\beta$
and $m_{h^{\pm}}$ for the 2HDM.

For the 331 effective model, we considered the Yukawa sector given in eqs.
(27-28). Notice that the third family up-quarks couples with the doublet
$\phi_{1}$ in the notation of eqs. (27-28), which is opposite to the usual
convention in 2HDM. Therefore, in the effective 331 model with small mixing
angles $\psi$ and $\theta$, we obtain in Fig. 2 a result similar to type II/Y
with $\tan\beta$ exchanged by $\cot\beta$.

Figure 2: Allowed (green) and excluded (red) regions in the plane $\cot\beta$
and $m_{h^{\pm}}$ for the effective 331 model.

A interesting feature of the effective 331 model is that there is a upper
limit on $\cot\beta$ if $A$ or $H$ are not degenerate in mass with the charged
scalar $h^{\pm}$. This comes from perturbativity constraints on the quartic
couplings, in particular $\lambda_{2}$ which, in the alignment limit, is given
by

$\lambda_{2}=\frac{m_{h}^{2}}{2v^{2}}-\frac{(\Delta_{Ah^{\pm}}-\Delta_{Hh^{\pm}})(\Delta_{Ah^{\pm}}+\Delta_{Hh^{\pm}}+2m_{h^{\pm}})\cot^{2}\beta}{2v^{2}}$
(35)

where $\Delta_{ab}=m_{a}-m_{b}$. As can be seen from the equation, unless
$\Delta_{Ah^{\pm}}=\Delta_{Hh^{\pm}}$, there will a upper limit for
$\cot\beta$. In fig.2 we depict in black the upper limit for the case
$\Delta_{Ah^{\pm}}=-100\;\mathrm{GeV}$, $\Delta_{Hh^{\pm}}=-50\;\mathrm{GeV}$,
which is an favoured region given the new result for the W boson mass as we
are going to see.

We conclude this subsection commenting about the case with non-vanishing
mixing angles in the matrix $V_{u}$. As noticed in Ref. Fan and Yagyu (2022),
in this scenario one can evade the constraint on $B\rightarrow X_{s}\gamma$
with lower masses for $h^{\pm}$. As a illustrative scenario, we show in fig.3
the allowed regions in the mixing angles plane for
$m_{h^{\pm}}=300\mathrm{GeV}$ and $\tan\beta=1$.

Figure 3:  Allowed (green) and excluded (red) regions in the mixing angles
plane for $m_{h^{\pm}}=300\;\mathrm{GeV}$ and $\tan\beta=1$.

As can be seen, it is necessary to have a strong correlation among the mixing
angles to suppress the bound coming from $B\rightarrow X_{s}\gamma$. Finally,
as we commented earlier, if $\tan\beta$ is not close to unity, it is possible
to have an enhancement on the Yukawa coupling of the neutral scalars to the
top quark, which is disfavored by collider constraints (it would increase the
production rate of these scalars by gluon fusion, for instance). One could
still allow some degree of fine-tuning, since, according to eq. (33), it is
possible to suppress this coupling by particular choices of $\psi$, and
$\theta$. However, for the scenarios we studied with $\tan\beta=2$, the vast
majority of the points that comply with the constraints from $B\rightarrow
X_{s}\gamma$ enforce $(\Gamma_{A}^{u})_{33}>1$. Therefore, in view of this
scenario we will restrict ourselves in this work only to the case with null
mixing angles.

###  III.3 Electroweak constraints

Regarding electroweak constraints, we consider the S, T, and U parameters.
This is particularly relevant given the updated value for the mass of the
W-boson presented by the CDF collaboration Aaltonen et al. (2022). Using the
PDG result for the W mass, one obtains after performing an electroweak fit
(hereafter we consider the case $\Delta U=0$) Zyla et al. (2020)

$\Delta S=0.00\pm 0.07,\quad\Delta T=0.05\pm 0.06,\quad\text{correlation =
}0.92$ (36)

In de Blas et al. (2022); Lu et al. (2022) a electroweak fit taken into
account the CDF result was performed. In de Blas et al. (2022), it is
considered a combination with previous results for the W mass amounting to

$\Delta S=0.086\pm 0.077,\quad\Delta T=0.177\pm 0.070,\quad\text{correlation =
}0.89$ (37)

On the other hand, Lu et al. (2022) considers only the CDF result

$\Delta S=0.15\pm 0.08,\quad\Delta T=0.27\pm 0.06,\quad\text{correlation =
}0.93$ (38)

In the context of the 2HDM, the correction to the oblique parameters is well-
known Grimus et al. (2008). After the release of the CDF result many works
have been performed in the context of 2HDM Kim (2022); Botella et al. (2022);
Benbrik et al. (2022); Ghorbani and Ghorbani (2022); Babu et al. (2022); Song
et al. (2022) with the main conclusion that the degeneracy in mass of the CP-
even and odd scalars with the charged scalar is now disfavoured. We have
confirmed these results by using SPhenoPorod (2003); Porod and Staub (2012)
within the 2HDM, as we are going to show in the numerical section.

For the 331 model, we performed two complementary analysis. First we have only
the effective version of the 331 scalar potential given by eq. (6) with the
Yukawa sector given by eqs. (27-28). In a second step, we used the results
presented in Liu and Ng (1994) for the 331 model. In this reference, it is
considered the contributions coming from the heavy spectrum of the 331 model,
in particular the ones related to the charged scalars $h^{\pm A}$, $h^{\pm B}$
as well as the new charged gauge bosons $X,Y$. We checked in our numerical
section that these contributions are negligible.

###  III.4 Scalar decays

Since we have an extended scalar sector in our model with respect to the SM,
it is natural to consider some extra decays that may be allowed. In
particular, for sufficient light scalars, the Higgs boson may itself decay to
these new particles. In this case, knowledge of triple scalar couplings is
essential and, as we mentioned before, our model in general predicts different
patterns than the ones founded in the 2HDM. For instance, for sufficient light
CP-odd scalar, the decay $h\rightarrow A_{0}A_{0}$ is open, being actually
constant in the alignment regime. Since this decay was not observed, one can
immediately exclude a light CP-odd scalar (mass less than $62.5$ GeV) in the
region of parameter space of the 331 model we are considering in this work.
Notice that the same does not hold true for the 2HDM, since one has an
additional freedom to suppress the coupling by a suitable choice of
$\Lambda_{5}$. A similar reasoning applies for the decay $h\rightarrow HH$. In
order to consider similar analysis in a systematic way we make use of
HiggsBounds and HiggsSignals Bechtle et al. (2014a); Bechtle et al. (2020);
Bechtle et al. (2014b); Bechtle et al. (2021) when performing our numerical
scan.

Finally, we comment on the decay $h\rightarrow\gamma\gamma$ which is only
radiately induced, therefore, influenced by all the extra particles in our
model. Regarding the scalar sector, the extra contributions come from charged
scalars $h^{\pm}$ in the loop222Regarding the heavier charged scalars and
extra gauge bosons, their contributions is negligible in the parameter space
we are interested at Hung et al. (2019). being controlled by the size of the
coupling $\lambda_{hh^{\pm}h^{\pm}}$. As we commented earlier, only if
$\Lambda_{5}=0$ the results in our model and 2HDM will be the same. Therefore,
we expect some differences on the predictions of this decay that we are going
to discuss in the numerical section where we made use of SPheno to obtain the
ratio

$R_{\gamma\gamma}=\frac{\Gamma(h\rightarrow\gamma\gamma)_{331}}{\Gamma(h\rightarrow\gamma\gamma)_{SM}}$
(39)

The ratio above can be analysed in terms of $\kappa$ multipliers, which are
provided by the experimental collaborations. Under the assumption that the
boson $h$ behaves exactly as in the SM Higgs, we obtain the following relation
in terms of the $\kappa_{\gamma}$ multiplier

$k_{\gamma}^{2}=R_{\gamma\gamma}$ (40)

Thus, using the value provided by the Atlas collaboration Aad et al. (2020)

$\kappa_{\gamma}=1.00\pm 0.06$ (41)

we can set the limit

$R_{\gamma\gamma}=1.00\pm 0.12$ (42)

###  III.5 $(g-2)_{\mu}$

As described in the introduction, one of the motivations of this work is to
study if light scalar particles are allowed in the 3-3-1 model, which may
provide a viable explanation to the present $(g-2)_{\mu}$ anomaly ($a_{\mu}$)
in similarity to the 2HDM. As discussed in de Jesus et al. (2020),
contributions coming from new gauge bosons are suppressed given the present
limit on their masses. Using in-house routines based on Heinemeyer et al.
(2004a, b), we have re-derived their analytic formulas, and confirmed that the
contributions are at negligible in the parameter space considered in this
work. Regarding the scalar spectrum, we can see in eq. (II.2) that $h_{3}$
will not couple to the muon line, which prevent their contribution to
$a_{\mu}$ at one-loop. The remaining one-loop contributions come from the
exchange of $h$, $H$, $A_{0}$, $h^{\pm}$. Since in all cases the coupling to
the muon line is proportional to $y_{\mu\mu}^{e}\sim 6\times
10^{-4}/s_{\beta}$, only for very low values of $\tan\beta$ can one expect to
obtain non-negligible contributions. Nevertheless, two-loop diagrams of Barr-
Zee type can surpass this suppression, since in this case the scalar couples
to the muon line only once. Moreover, if one has a fermionic loop with a more
heavy fermion, the coupling may also be less suppressed. For instance, if one
has a loop with the $\tau$ fermion, the coupling will be $y_{\tau\tau}^{e}\sim
10^{-2}/s_{\beta}$. We have performed a general analysis using the results of
Cherchiglia et al. (2017) which were recently implemented in the computational
tool GM2CalcAthron et al. (2022, 2016). We will discuss our findings in the
numerical section. For future reference, we quote the present values for the
experimental Abi et al. (2021) and theoretical prediction in the SM Aoyama et
al. (2020) for $a_{\mu}$

$a_{\mu}^{\text{Exp}}=(11659206.1\pm 4.1)\times 10^{-10},\quad
a_{\mu}^{\text{SM}}=(11659181.0\pm 4.3)\times 10^{-10}$ (43)

which renders a discrepancy by $4.2\sigma$ among the two values. This suggests
a BSM contribution as

$a_{\mu}^{\text{BSM}}=(25.1\pm 5.9)\times 10^{-10}.$ (44)

##  IV Numerical analysis

In this section we perform a focused numerical scan in the parameter space of
the 331 model. This study is complementary to the ones performed in Hung et
al. (2019); Okada et al. (2016b). At this point is it useful to revise our
main assumptions:

  1. 1.

as usual in the literature of 331 models, we are considering the limit
$v_{\chi}>>v_{\rho},v_{\eta}$. This implies that the light spectrum of the 331
model, in addition to the SM particles, features 3 new scalars ($H$, $A_{0}$,
$h^{\pm}$). We identify $h$ as the SM higgs, by considering the alignment
limit ($\beta-\alpha=\pi/2$);

  2. 2.

the scalar potential is CP-conserving (all parameters are real). So, there is
no extra source of CP violation from the scalar sector;

  3. 3.

the exotic fermions do not mix with the SM fermions, and they are decoupled
from the theory.

Regarding our scan strategy, we will adopt the physical basis for the scalar
potential in the 331 model, which is characterized by the 13 parameters

$\\{\tan\beta,v,v_{\chi},m_{h},m_{H},m_{A_{0}},m_{h^{\pm}},m_{h_{3}},m_{h^{\pm
A}},m_{h^{\pm B}},\alpha_{1},\alpha_{2},\alpha_{3}\\}$

Given our assumptions, some of the parameters are chosen as

$\\{v=246\mbox{ GeV},v_{\chi}=100\mbox{
TeV},m_{h}=m_{h_{SM}},\alpha_{1}=\alpha_{2}=0,\alpha_{3}=\alpha=\beta-\pi/2\\}$

leaving us with 7 free parameters whose range we choose as below

$\begin{array}[]{|c|c|c|c|c|c|c|}\hline\cr\tan\beta&m_{H}\,(\mathrm{GeV})&m_{A_{0}}\,(\mathrm{GeV})&m_{h^{\pm}}\,(\mathrm{GeV})&m_{h_{3}}\,(\mathrm{TeV})&m_{h^{\pm
A}}\,(\mathrm{TeV})&m_{h^{\pm B}}\,(\mathrm{TeV})\cr\hline\cr 10^{-2}\div
1&130\div 1000&63\div 1000&600\div 1000&5\div 10&5\div 10&5\div
10\cr\hline\cr\end{array}$ Table 1: Input parameter ranges.

As can be seen, we are adopting $h$ as the lightest CP-even scalar in our
model. Recall that the mass of CP-even or odd scalars cannot be lighter than
half the mass of $h$, leaving just the window (62.5 GeV - 125 GeV). For
$\tan\beta$ we aim to avoid Yukawas larger than unit for the top quark, while
the mass for the charged scalar is constraint by the $B\rightarrow
X_{s}\gamma$ decay.

For comparison, we also performed a numerical scan in the 2HDM with a soft-
broken $\mathbb{Z}_{2}$ symmetry ($\Lambda_{6}=\Lambda_{7}=0$). We adopted the
same ranges for the parameters shared by the 2HDM and 331 model, implying that
the only remaining free parameter is $\Lambda_{5}$ which we vary under the
full perturbative range $|\Lambda_{5}|<4\pi$.

We begin by discussing the influence of the theoretical bounds. It will prove
fruitful to write the quartic couplings for the 331 model in the alignment
limit

$\displaystyle\lambda_{1}$
$\displaystyle=\frac{m_{h}^{2}}{2v^{2}}+\frac{(m_{H}^{2}-m_{A_{0}}^{2})\tan^{2}\beta}{2v^{2}}$
(45) $\displaystyle\lambda_{2}$
$\displaystyle=\frac{m_{h}^{2}}{2v^{2}}+\frac{(m_{H}^{2}-m_{A_{0}}^{2})}{2v^{2}\tan^{2}\beta}$
(46) $\displaystyle\lambda_{12}$
$\displaystyle=\frac{m_{h}^{2}}{2v^{2}}-\frac{(m_{H}^{2}-m_{A_{0}}^{2})}{2v^{2}}$
(47) $\displaystyle\zeta_{12}$
$\displaystyle=\frac{2\left(m_{h^{\pm}}^{2}-m_{A_{0}}^{2}\right)}{v^{2}};$
(48)

Since we require $0<\lambda_{1,2}<4\pi$, in the regime with
$10^{-2}<\tan\beta<1$ the stronger constraint will come from $\lambda_{2}$. In
particular, it will favor the hierarchy $m_{H}>m_{A_{0}}$ and, for small
$\tan\beta$, the two masses will be nearly degenerate. Notice that these
conditions for $\lambda_{2}$ will favor a negative value for $\lambda_{12}$.
Finally, since we have the stability constraint,
$\lambda_{12}+\zeta_{12}>-2\sqrt{\lambda_{1}\lambda_{2}}$, and $\lambda_{12}$
will be mostly negative, $\zeta_{12}$ will need to be mostly positive which
implies that $m_{h^{\pm}}>m_{A_{0}}$. Therefore, for most of the points in the
parameter space we expect the hierarchy $m_{H}>m_{A_{0}}$ and
$m_{h^{\pm}}>m_{A_{0}}$. We emphasize that the same pattern does not occur in
the 2HDM, since we have an additional free parameter, $\Lambda_{5}$, in that
case. This general analysis explain the curves seen in fig. 4. Notice that the
lower limit on $m_{A_{0}}$ strongly depends on the value of $m_{h^{\pm}}$.

Figure 4: In both figures, the yellow points comply with the stabiltiy bound,
the green points with perturbativity and unitarity, while the black points
comply with all theoretical constraints. The dashed red line is for degenerate
masses.

We briefly comment on the remaining quartic couplings which, in the alignment
limit, are given by

$\displaystyle\lambda_{3}\sim\frac{m_{h_{3}}^{2}}{2v_{\chi}^{2}},\quad\lambda_{i3}\sim\frac{m_{A_{0}}^{2}}{v_{\chi}^{2}},\quad\zeta_{13}\sim\frac{2m_{h^{\pm
B}}^{2}}{v_{\chi}^{2}},\quad\zeta_{23}\sim\frac{2m_{h^{\pm
A}}^{2}}{v_{\chi}^{2}}.$ (49)

As can be seen from the equations above, all the remaining quartic couplings
are positive and suppressed given our choice of the parameters. Therefore,
they will not play any important roles into the theoretical bounds.

We consider next the impact of the CDF result for the W mass, which can be
seen in fig. 5, where we show the allowed range at 95$\%$ C.L. for the
difference in mass of the neutral scalars $H$, $A_{0}$ to $h^{\pm}$.

Figure 5:  Difference in mass among the neutral scalars $H$, $A_{0}$ and the
charged scalar $h^{\pm}$ complying at 95$\%$ C.L. with oblique parameters of
equations (36), (37), (38) respectively. In blue we show the points in the
2HDM while in red the points of the 331 model.

In the left plot we show the case where the W mass according to PDG is used
for the prediction of the oblique parameters, eq. (36), while in the middle
plot a combination with the CDF result is performed, eq. (37). For the right
plot we consider only the CDF result, eq. (38). In all plots we show in blue
the points using the 2HDM while the red points correspond to the 331 model.
For the 331 model we checked that the influence of the heavy spectrum is
negligible for the calculation of the S,T parameters. We also checked that, in
all cases, the contribution to the U parameter is negligible. As can be seen
from the plots, the new result from the CDF collaboration favours a split in
mass between the charged scalar and the neutral ones. This finding is
particularly relevant for the 331 effective model, since it affects triple
scalar couplings such as $\lambda_{hh^{+}h^{-}}$, see eq. (13). Moreover,
contrarily to 2HDM, for most of the points only one of the branches is
accessible to the 331 model due to the favoured hierarchy
$m_{h^{\pm}}>m_{A_{0}}$.

In order to illustrate the more predictive behavior of the 331 effective
model, we consider the decay $h\rightarrow\gamma\gamma$ that is directly
affected by the coupling $\lambda_{hh^{+}h^{-}}$. Our result can be seen in
figure 6 where all points comply with theoretical constraints in their
respective model. We use the same color code of figure 5.

(a) $m_{h}^{\pm}=600\mathrm{GeV}$

(b) $m_{h}^{\pm}=800\mathrm{GeV}$

Figure 6: Ratio $R_{\gamma\gamma}$ as a function of the difference in mass
among $A_{0}$ and the charged scalar $h^{\pm}$. Each plot corresponds to a
different choice of $m_{h}^{\pm}$. In all plots, blue points correspond to the
2HDM while red points correspond to the 331 effective model.

Both plots show a similar behavior, where the 331 effective model corresponds
to a very precise prediction to $R_{\gamma\gamma}$ for a given choice of
$m_{A_{0}}-m_{h^{\pm}}$. The lower bound for $m_{A_{0}}$ in the right plot
(fig. 6(b)) is explained by theoretical constraints. Moreover, given the
constraint of eq. (42), the 331 effective model complies with the experimental
value at 1$\sigma$ for all points.

We consider next the influence of a possible light scalar spectrum in
explaining $(g-2)_{\mu}$. In fig. 7 we show in purple the contribution for the
fermionic contribution (2-loop Barr-Zee diagrams) plus the 1 loop contribution
(which are negligible for the parameter space considered). In yellow we show
the absolute value of the bosonic contribution, which is always negative in
the parameter space studied. As can be seen, the value for $(g-2)_{\mu}$
allowed in the effective 331 model is too small, see eq. (44). The behavior in
the left plot is explained for our choice of $m_{H}>m_{h_{SM}}$. As shown in
eq. (35), there is a upper limit on $\cot\beta$ if $m_{H}\neq m_{A_{0}}$.
Since to explain $(g-2)_{\mu}$ an enhancement is required (performed by
choosing higher values of $\cot\beta$), one sees that only when $m_{A_{0}}\sim
m_{H}$ the fermionic contribution to $(g-2)_{\mu}$ can increase. However, it
is still too small. Moreover, our analysis of the oblique parameters shows
that it is not possible to have both $m_{A_{0}}$ and $m_{H}$ far away from
$m_{h^{\pm}}\sim 600\mathrm{GeV}$. Therefore, the region with
$m_{H}<130\mathrm{GeV}$ and $m_{A_{0}}\sim m_{H}$, which was not included in
our scan, is excluded by the oblique parameters.

Figure 7: Values for $(g-2)_{\mu}$ in the effective 331 model. In purple we
show the fermionic contributions (2-loop Barr-Zee diagrams) plus the 1-loop
contributions. In yellow we show the absolute value of the bosonic
contribution (which is always negative in the region scanned). All points
comply with theoretical constraints.

Finally we analyse the constraints coming from searches in colliders. In fig.
8 we show in green the region that is allowed at 95$\%$ C.L. by using the
HiggsBoundsBechtle et al. (2014a); Bechtle et al. (2020) and
HiggSignalsBechtle et al. (2014b); Bechtle et al. (2021) codes, while in red
we show the excluded points. All points comply with theoretical bounds. In
order to obtain two-dimensional plots, we have fixed $m_{h^{\pm}}$ and
$\tan\beta$ in each plot, allowing $m_{A_{0}}$ and $m_{H}$ to vary in the
whole range. As can be seen in fig.8(a), for the choice
$m_{h^{\pm}}=600\mathrm{GeV}$ and $\tan\beta=1$ the region with
$m_{A_{0}}<350\mathrm{GeV}$ is excluded with the exception of a small island
with $m_{H}>700\mathrm{GeV}$ and $m_{A_{0}}<70\mathrm{GeV}$. For
$\tan\beta=0.2$ (fig.8(b)), there are more regions allowed, and notice that
$m_{A_{0}}\sim m_{H}$ as already explained. However, if $\tan\beta$ is
decreased further, all points are excluded. It happens already at
$\tan\beta=0.1$. This is explained by the hierarchy $m_{A_{0}}\sim m_{H}$ as
well as $m_{A_{0}}<m_{h^{\pm}}=600\mathrm{GeV}$, which implies that
$m_{H}<m_{h^{\pm}}=600\mathrm{GeV}$. Since this range of mass is accessible at
the LHC and smaller $\tan\beta$ implies in an enhancement of the Yukawa
couplings to the third family down quarks and leptons, the constraints are
quite strong. For the case of $m_{h^{\pm}}=800\mathrm{GeV}$ and
$\tan\beta>0.2$ the influence of collider searches is very small, after
considering theoretical bounds. Recall that in this case there is a lower
bound on $m_{A_{0}}$ of around $500\mathrm{GeV}$ as can be seen in fig. 4 or
fig.6. For decreasing values of $\tan\beta$ the constraints become stronger,
as can be seen in fig.8(c) where $\tan\beta=0.1$ were chosen. In this case,
masses lower than 720 GeV are excluded.

(a) $m_{h}^{\pm}=600\mathrm{GeV}$, $\tan\beta=1$

(b) $m_{h}^{\pm}=600\mathrm{GeV}$, $\tan\beta=0.2$

(c) $m_{h}^{\pm}=800\mathrm{GeV}$, $\tan\beta=0.1$

Figure 8: Collider constraints for the allowed range of masses for extra
neutral scalars. In red we show the excluded points, while in green the
allowed ones at 95$\%$C.L. Values for $m_{h^{\pm}}$ and $\tan\beta$ are chosen
as shown in the plots. The black dashed line is for degenerate masses. All
points comply with theoretical constraints.

Finally, we show the impact of the maximum value chosen for the pertubative
quartic couplings. In fig. 9 we show in blue the points that comply with all
theoretical constraints using the choice $\lambda_{\max}=4\pi$ while in black
we choose $\lambda_{\max}=\sqrt{4\pi}$. An can be seen, in the second case
there is a much stronger correlation among $m_{A_{0}}$ and $m_{h^{\pm}}$,
implying that $m_{A_{0}}>500\mathrm{GeV}$. The correlation among $m_{A_{0}}$
and $m_{H}$ is also stronger, for instance, if $m_{A_{0}}=500\mathrm{GeV}$,
$m_{H}$ can be at most of order of 700 GeV. In the previous case, $m_{H}$
reached the ceiling of the chosen range for the scan.

Figure 9: In both figures, blue points complies with theoretical constraints
with the choice $\lambda_{\max}=4\pi$, while for the black points we choose
$\lambda_{\max}=\sqrt{4\pi}$. The dashed red line is for degenerate masses.

At this point it is worth to summarize our main findings for a light scalar
spectrum for the effective 331 model considered in this work:

  1. 1.

the constraint from $B\rightarrow X_{s}\gamma$ requires that the mass of the
charged scalar is at least 600 GeV. The same statement holds for Type II/Y
2HDM;

  2. 2.

Theoretical constraints such as perturbativity of quartic couplings and
stability of the scalar potential will favor the hierarchies
$m_{h^{\pm}}>m_{A_{0}}$ and $m_{H}>m_{A_{0}}$. For the 2HDM, there is no
favored hierarchy;

  3. 3.

The CDF result for the W mass disfavors degenerate masses for the scalars.
Given the hierarchy in the 331 effective model, it implies that both $H$ and
$A_{0}$ will be lighter than $h^{\pm}$. For the 2HDM, $H$ and $A_{0}$ can also
be heavier than $h^{\pm}$;

  4. 4.

By considering the CDF result for the W mass, one can predict that the
diphoton decay of the SM-like scalar normalized to the SM value is always less
than unity. For the 2HDM no prediction can be made;

  5. 5.

$(g-2)_{\mu}$ cannot be explained in the effective 331 model, since the region
with the lightest scalar does not allow enhancement due to perturbativity of
the scalar quartic couplings. Moreover, $m_{A_{0}}>63\mathrm{GeV}$. For the
2HDM, not only can $A_{0}$ be lighter (by a suitable choice of $\Lambda_{5}$)
as there is no constraint on the enhancement;

  6. 6.

If the maximum value of the quartic couplings is decreased to $\sqrt{4\pi}$,
one obtains a lower bound on $m_{A_{0}}\sim 500\mathrm{GeV}$. Also, collider
constraints can impose strong constraints for $m_{A_{0}}$, depending on the
value of $\tan\beta$.

We conclude this section with fig.10 which illustrates allowed regions of
$m_{A_{0}}/m_{h^{\pm}}$ and $m_{H}/m_{h^{\pm}}$ for chosen values of
$m_{h^{\pm}}$ and $\tan\beta$, when all constraints are enforced. For each
plot, the different regions correspond to the three values of the electroweak
oblique parameters. The red points correspond to eq. (36), the green points
correspond to eq. (37), and the blue points correspond to eq. (38). We are
adopting the more conservative case of $\lambda_{\max}=4\pi$. As can be seen
in the plot, the mass of the CP-odd neutral scalar cannot be smaller than
50-60$\%$ of $m_{h^{\pm}}$, while for the CP-even scalar this limit increases
to 75-85$\%$. For the case of $m_{h^{\pm}}=600\mathrm{GeV}$, this lower bound
for $m_{A_{0}}$ is enforced by collider constraints while for
$m_{h^{\pm}}=800\mathrm{GeV}$ the theoretical constraints are responsible for
this behavior. For the case of $m_{H}$, the lower bound is due to electroweak
oblique parameters. It is interesting to notice that, regardless of the mass
of $h^{\pm}$, if only the CDF result is considered when computing the
electroweak oblique corrections (blue points), there is an island for the
allowed masses of the neutral scalars which are both lighter than $h^{\pm}$.
For smaller values of $\tan\beta$, the region is even more compressed. This
findings imply that, if a charged scalar is found, for instance, at the LHC,
the effective 331 model predicts the mass region of the other scalars which
are necessarily lighter or at most of the same order of $h^{\pm}$. In this
scenario, the experimental collaboration could focus on the predicted mass
regions and refute (or confirm) the 331 effective model.

(a) $m_{h}^{\pm}=600\mathrm{GeV}$

(b) $m_{h}^{\pm}=800\mathrm{GeV}$

Figure 10: Summary plots when all constraints are enforced. Outside the dashed
lines, the region is excluded by theoretical bounds, while the left region
from the gray line is excluded by collider bounds. The mass of the charged
scalar and $\tan\beta$ are chosen as indicated in the plots.

##  V Conclusion

The 331 model, proposed in the 90’s Pisano and Pleitez (1992); Frampton
(1992), remains an appealing candidate for extending the Standard Model. It
predicts the number of fermion generations to be a multiple of three, which
shed some light on the current pattern observed in nature. It can also explain
the smallness of neutrino masses as well as the strong CP problem. From a more
phenomenological point of view, the current collider constraints set the
masses of all the new gauge bosons to be at the TeV range or higher, the
specific limit depending on the version of the 331 model considered. More
dramatically, since the masses of the extra gauge bosons are directly
connected to the vev of the first gauge symmetry breaking, they can be
completely non-detectable in present and near-future colliders if the symmetry
breaking occurs at a scale of hundred of TeVs, for instance. However, even in
this discouraging scenario, the model allows some of the extra scalars to be
light, at the sub-TeV range, resembling a two-higgs doublet model (2HDM).

In this contribution we have studied in detail this effective realization of
the 331 model, aiming to discuss its differences to a general 2HDM. We found
that there is a lower and upper limit for the mass of the neutral scalars that
strongly depends on the mass of the charged scalar. Thus, if the latter is
found at the LHC, for instance, the effective 331 model predicts the mass
range of the other scalars. This is in clear contrast to the general 2HDM, in
which a upper limit for the masses cannot be enforced. Moreover, the predicted
mass range also depends on the W-mass through the oblique parameters, opening
the possibility to support (or not) the CDF result if new scalars are found
and their masses determined. Finally, we confirmed that the current deviation
for the muon magnetic moment cannot be addressed by 331 models, by explicitly
evaluating its contribution in the effective 331 model. This analysis
complements the one performed in de Jesus et al. (2020).

###### Acknowledgements.

We gratefully acknowledge enlightening discussions with Dominik Stöckinger,
Hyejung Stöckinger-Kim, and Vicente Pleitez. A.C. acknowledges support from
National Council for Scientific and Technological Development – CNPq through
project 166523/2020-8. O.L.G.P. were thankful for the support of FAPESP
funding Grant 2014/19164-6. and CNPq grant 306565/2019-6. This study was
financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível
Superior - Brasil (CAPES) - Finance Code 001.

##  Appendix A Inversion relations for the parameters of the scalar potential

Using the results provided in Costantini et al. (2020) one can obtain
expressions for the parameters of the scalar potential in the effective 331
model. By performing the choices ($\alpha_{2}=\alpha_{1}=0$;
$\alpha_{3}=\alpha$; $m_{h_{2}}=m_{h}$; $m_{h_{1}}=m_{H}$; $m_{A_{0}}=m_{A}$;
$m_{h^{\pm}}=m_{H^{\pm}}$), one obtains in the limit $v_{\chi}\gg
v_{\rho},v_{\eta}$

$\displaystyle\lambda_{1}$
$\displaystyle=\frac{m_{H}^{2}c_{\alpha}^{2}+m_{h}^{2}s_{\alpha}^{2}-m_{A}^{2}s_{\beta}^{2}}{2\;v^{2}c_{\beta}^{2}};$
(50) $\displaystyle\lambda_{2}$
$\displaystyle=\frac{m_{H}^{2}s_{\alpha}^{2}+m_{h}^{2}c_{\alpha}^{2}-m_{A}^{2}c_{\beta}^{2}}{2\;v^{2}s_{\beta}^{2}};$
(51) $\displaystyle\lambda_{12}$
$\displaystyle=\frac{\left(m_{H}^{2}-m_{h}^{2}\right)s_{\alpha}c_{\alpha}+m_{A}^{2}s_{\beta}c_{\beta}}{v^{2}s_{\beta}c_{\beta}};$
(52) $\displaystyle\zeta_{12}$
$\displaystyle={\color[rgb]{1,0,0}-}\frac{2\left(m_{A}^{2}-m_{H^{\pm}}^{2}\right)}{v^{2}};$
(53) $\displaystyle fv_{\chi}$ $\displaystyle=m_{A}^{2}c_{\beta}s_{\beta};$
(54) $\displaystyle\mu_{\rho}^{2}$
$\displaystyle=-\frac{1}{2c_{\beta}}\left(m_{H}^{2}c_{\alpha}c_{\beta-\alpha}-m_{h}^{2}s_{\alpha}s_{\beta-\alpha}\right)+m_{A}^{2}s_{\beta}^{2};$
(55) $\displaystyle\mu_{\eta}^{2}$
$\displaystyle=-\frac{1}{2s_{\beta}}\left(m_{h}^{2}c_{\alpha}s_{\beta-\alpha}+m_{H}^{2}s_{\alpha}c_{\beta-\alpha}\right)+m_{A}^{2}c_{\beta}^{2};$
(56)

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