# 331 Models and Bilepton Searches at LHC

Roberta Calabrese<EMAIL_ADDRESS>Dipartimento di Fisica E. Pancini,
Università di Napoli Federico II
Complesso Universitario di Monte Sant’Angelo, Via Cintia, Napoli (NA), Italy
and INFN, Sezione di Napoli.  Alberto Orso Maria Iorio
<EMAIL_ADDRESS>Dipartimento di Fisica E. Pancini, Università
di Napoli Federico II
Complesso Universitario di Monte Sant’Angelo, Via Cintia, Napoli (NA), Italy
and INFN, Sezione di Napoli.  Stefano Morisi<EMAIL_ADDRESS>Dipartimento
di Fisica E. Pancini, Università di Napoli Federico II
Complesso Universitario di Monte Sant’Angelo, Via Cintia, Napoli (NA), Italy
and INFN, Sezione di Napoli.  Giulia Ricciardi<EMAIL_ADDRESS>Dipartimento di Fisica E. Pancini, Università di Napoli Federico II
Complesso Universitario di Monte Sant’Angelo, Via Cintia, Napoli (NA), Italy
and INFN, Sezione di Napoli.  Natascia Vignaroli
<EMAIL_ADDRESS>Dipartimento di Matematica e Fisica E. De
Giorgi, Università del Salento,
via per Arnesano 73100 Lecce (LE), Italy
and INFN, Sezione di Lecce.  Dipartimento di Fisica E. Pancini, Università di
Napoli Federico II
Complesso Universitario di Monte Sant’Angelo, Via Cintia, Napoli (NA), Italy
and INFN, Sezione di Napoli.

###### Abstract

Despite being remarkable predictive, the Standard Model leaves unanswered
several important issues, which motivate an ongoing search for its extensions.
One fashionable possibility are the so-called 331 models, where the
electroweak gauge group is extended to $SU_{L}(3)\times U(1)$. We discuss
these models and provide a classification. As a second step, we focus on a
minimal extension which includes vector-like quarks (VLQs) and new gauge
bosons. We investigate the phenomenology and perform a consistent analysis of
the production at the LHC of a pair of doubly charged bileptons, including
processes where VLQs contribute, and in particular the associate production
VLQ-bilepton. Finally, we extract the bound on the bilepton mass, $m_{Y}>$
1300 GeV, from a reinterpretation of a recent ATLAS search for doubly-charged
Higgs bosons in multi-lepton final states.

##  1 Introduction

The Standard Model (SM) based on the group $G_{\rm SM}=SU(3)_{c}\times G_{\rm
ew}$, where $G_{\rm ew}=SU(2)_{L}\times U(1)_{X}$, is a remarkably successful
theory, but it has been proven to be incomplete, and needs to be extended.
Since the three gauge couplings, running with energy, are drawn nearer around
$10^{15}$ GeV, an appealing possibility is that $G_{\rm SM}$ arises from the
spontaneously breaking of a larger gauge group, characterized by only one
structure constant which unifies strong and electroweak interactions. Most
Grand Unified Theories (GUTs) are based on the gauge groups $SU(5)$ and
$SO(10)$ Ross (1985). They predict proton decay, which observation in itself
provides a strong motivation. However so far experiments are just putting
stronger and stronger lower limits on proton life-time.

A completely different approach is to consider minimal extensions of $G_{\rm
SM}$ with not-simple gauge groups. There are many possibilities. The simplest
one is to enlarge the electroweak sector to $G_{\rm ew}\times U(1)^{\prime}$
to add a non-standard neutral gauge boson generally named $Z^{\prime}$, see
for instance Ref. Langacker (2009). The next case is inspired to the request
of symmetrise the SM matter assignment, whose complete asymmetry between left
and right-handed fields is probably one of its less appealing feature. In the
so called Left-Right models, $G_{\rm ew}$ is embedded into $SU_{L}(2)\times
SU_{R}(2)\times U(1)$ Mohapatra and Pati (1975); Senjanovic and Mohapatra
(1975). Another interesting possibility is to extend $SU_{L}(2)$ with
$SU_{L}(N)$. In Refs. Schechter and Ueda (1973); Gupta and Mani (1974);
Albright  _et al._ (1975); Georgi and Pais (1979), $G_{\rm ew}$ is extended to
$SU_{L}(3)\times U(1)$. These seminal works focus on the possibility to extend
the GIM mechanism Glashow  _et al._ (1970) to the case of 5 quarks (instead of
4), namely suppressing unwanted flavor changing neutral currents. Then in
Singer  _et al._ (1980) it has been proposed the first $SU_{L}(3)\times U(1)$
model where the three families of fermions are embedded into irreducible
representations of the gauge group. However, the first complete model based on
$SU_{L}(3)\times U(1)$ (hereafter named $331$ where the first 3 refers to
$SU_{c}(3)$) has been proposed by Pisano, Pleitez Pisano and Pleitez (1992)
and Frampton Frampton (1992). In this work we present an analysis to test a
general version of 331 models, with a minimal set of assumptions.

Even if from the group point of view extending $SU_{L}(2)$ to $SU_{L}(3)$
seems a very small modification, from the phenomenological point of view it
has important consequences:

  * •

The matter assignment requires the introduction of extra fermions that can
have exotic charges and can be vector-like. In the minimal model we consider
there are two vector-like quarks with charges $-4/3$ and $5/3$.

  * •

The scalar sector must be extended with extra Higgs fields. For example, in
our case the model contains three scalar $SU_{L}(3)$ triplets.

  * •

The model has 5 more gauge bosons that can have exotic charges. In our case
the extra gauge bosons are $Z^{\prime}$, $W^{\prime\pm}$, $Y^{\pm\pm}$.

The wealth of 331 models of new physics yields to opportunities for searching
at colliders. We provide an inclusive study of the multi-lepton channel
$pp\to\ell^{\pm}\ell^{\pm}\ell^{\mp}\ell^{\mp}+X$ at LHC mediated by 331 new
fields, in particular by the pair production of bileptons $Y$. The multi-
lepton production process cross section is sizable modified by the presence of
331 new physics.

The paper is organized as follows: in section 2 we discuss the basic features
of 331 models, detailing constraints and providing a classification. In
section 3 we identify a minimal model and illustrate its gauge, scalar and
matter content. We analyze its phenomenology in section 4; in particular
(section 5) the bilepton pair production at LHC. Our conclusions are presented
in section 6.

##  2 Basic features of 331 models

In this section, we present the general features of the 331 models. We are
using the same convention of Ref. Buras  _et al._ (2013). We do not discuss
$SU(3)_{C}$ which does not change with respect to the SM. The generators of
the $SU(3)_{\text{L}}$ gauge group are indicated with
$\hat{T}^{1},\cdots,\hat{T}^{8}$. The generators of the fundamental
representation of $SU(3)_{L}$ are $T^{a}=\lambda^{a}/2$, where ${\lambda}^{i}$
are the eight Gell-Mann matrices. For the $\bar{3}$ representation, the
generators are $-(T^{a})^{T}$. The normalisation of the generators is
$\mathrm{Tr}[T^{i}\,T^{j}]=\delta^{ij}/2$, and
${\mathds{1}}=\mathrm{diag}(1,1,1)$ is the identity matrix in the fundamental
space. It is common to define the $U(1)_{X}$ generator as
$T^{9}={\mathds{1}}/\sqrt{6}$. With this normalization
$\mathrm{Tr}[T^{9}\,T^{9}]=1/2$, similarly to the one of $SU(3)_{\text{L}}$
generators.

In analogy with the SM the electric charge is defined as the diagonal operator

$\hat{\mathcal{Q}}=\hat{T}^{3}+\frac{\hat{Y}}{2}$ (1)

where we identify the hypercharge operator ${\hat{Y}}$ as

$\frac{\hat{Y}}{2}=\beta\hat{T}^{8}+X{\mathds{1}}$ (2)

Here $X$ is the quantum number associated with $U(1)_{X}$ and $\beta$ is a
free parameter. Any 331 model is characterized by the pair $(\beta,X)$, whose
values are obviously related to the electric charges of the fields (fermions
and bosons).

In the SM left handed fermions are assigned to doublets of $SU_{L}(2)$.
Similarly in 331 models one assigns left handed fields to fundamental
irreducible representations ($3$ or $\overline{3}$) of $SU_{L}(3)$. We
indicate them generically with $3_{Q,\ell}$ and $\overline{3}_{Q,\ell}$,
distinguishing between quark triplets and anti-triplets ($3_{Q}$ and
$\overline{3}_{Q}$) and lepton ones ($3_{\ell}$ and $\overline{3}_{\ell}$).
Thus, for a generic SM quark-doublet $(u_{L},d_{L})$ we set

$3_{Q}=\begin{pmatrix}u_{L}\\\ d_{L}\\\
\psi_{L}\end{pmatrix}_{i},\qquad\overline{3}_{Q}=\begin{pmatrix}d_{L}\\\
-u_{L}\\\ \Psi_{L}\end{pmatrix}_{j},$ (3)

In a similar way for a generic SM lepton-doublet $(\nu_{L},\ell_{L})$ we set

$3_{\ell}=\begin{pmatrix}\nu_{L}\\\ \ell_{L}\\\
\xi_{L}\end{pmatrix}_{k},\qquad\overline{3}_{\ell}=\begin{pmatrix}\ell_{L}\\\
-\nu_{L}\\\ \zeta_{L}\end{pmatrix}_{r},$ (4)

The field assigned to the third components of these triplets are the left
handed component of a field beyond the SM, namely we have introduced new
fields $\psi_{i},\Psi_{j},\xi_{k},\zeta_{r}$. The indexes $i,j,k,r$ refer to
flavor.

The action of the charge operator $\hat{\mathcal{Q}}$ on $3$ and $\bar{3}$
representations is given by the eigenvalues of the eigenstates
$|3_{Q,\ell}\rangle$, and $|\overline{3}_{Q,\ell}\rangle$ that we indicate,
respectively, as $\mathcal{Q}[3_{Q,\ell}]$ and
$\mathcal{Q}[\overline{3}_{Q,\ell}]$. We have

$\mathcal{Q}[3_{Q,\ell}(\overline{3}_{Q,\ell})]=\left(\begin{array}[]{ccc}\pm\frac{1}{2}\pm\frac{\beta}{2\sqrt{3}}+X[3_{Q,\ell}(\overline{3}_{Q,\ell})]&0&0\\\
0&\mp\frac{1}{2}\pm\frac{\beta}{2\sqrt{3}}+X[3_{Q,\ell}(\overline{3}_{Q,\ell})]&0\\\
0&&\mp\frac{\beta}{\sqrt{3}}+X[3_{Q,\ell}(\overline{3}_{Q,\ell})]\\\
\end{array}\right),$ (5)

The upper (down) signs correspond to the $3$ ($\bar{3}$) representations. The
$X$ values for $3_{Q,\ell}$, and $\overline{3}_{Q,\ell}$ states are indicated,
respectively, as $X[3_{Q,\ell}]$ and $X[\overline{3}_{Q,\ell}]$. We will use a
similar notation $X[f]$ and $X[\overline{f}]$ if, instead of referring to all
fields in an entire multiplet, we refer to the specific field $f$.

The charges of the first two components of (3) and (4) must match the charges
of the corresponding SM fields. This matching fixes uniquely the $X$ values
for the fermions as functions of $\beta$. Let us consider the case of the
quark triplet $3_{Q}$. If we match the up quark charge
$1/2+\beta/(2\sqrt{3})+X[{3}_{Q}]=2/3$, we obtain
$X[{3}_{Q}]=1/6-\beta/(2\sqrt{3})$. The correct down quark charge follows
automatically, and it does not represent an independent constraint. In the
same way we get $X[{3}_{\ell}]=-1/2-\beta/(2\sqrt{3})$ in the lepton sector.
Let us now consider the entries of the matrix (5) for the antitriplets. By
repeating the same reasoning for the quark assignment in (3) we get
$X[\overline{3}_{Q}]=1/6+\beta/(2\sqrt{3})$, whereas for the lepton one in (4)
we obtain $X[\overline{3}_{\ell}]=-1/2+\beta/(2\sqrt{3})$111We are using the
same convention of Ref. Buras  _et al._ (2013), opposite to the one in Ref.
Diaz  _et al._ (2005).. Negative values for $\beta$ can be related to the
positive ones by taking the complex conjugate in the covariant derivative of
each model, which in turn is equivalent to replace $3$ with $\bar{3}$ in the
fermion content of each particular model.

The non-standard fields, $\psi_{i},\Psi_{j},\xi_{k},\zeta_{r}$, have the same
$X$ values of the other members of the respective (anti)triplet, because of
gauge invariance. Thus using eq.(5) we deduce their charges,
$\mathcal{Q}[\psi_{L}]=1/6-\sqrt{3}/2\,\beta$,
$\mathcal{Q}[\Psi_{L}]=1/6+\sqrt{3}/2\,\beta$,
$\mathcal{Q}[\xi_{L}]=-1/2-\sqrt{3}/2\,\beta$ and
$\mathcal{Q}[\zeta_{L}]=-1/2+\sqrt{3}/2\,\beta$.

As far as right handed fields are concerned we assume, in analogy to the SM,
that they are assigned to singlet representations of $SU_{L}(3)$. Since the
action of the charge operator $\hat{\mathcal{Q}}$ on the singlet is given by
$\hat{\mathcal{Q}}|1\rangle=X\,|1\rangle$, the $X$ value of a right-handed
field coincides with its charge. In our notation, given a left-handed fermion
field $f_{L}$, belonging to a triplet or anti-triplet $SU_{L}(3)$
representation, we indicate with $f_{R}$ the corresponding right-handed field
in the $SU_{L}(3)$ singlet representation. Let us observe that the presence of
right-handed quark singlets is demanded for the model to be $SU(3)_{C}$
vector-like, while right-handed neutral lepton singlets are optional.

Table (1) summarises the quantum numbers of the 331 fermions. We highlight
that the $X$ assignments rely on the request of matching the charges of SM
fermions in $3_{Q,\ell}$ and $\overline{3}_{Q,\ell}$ with their SM values.
This assumption does not naturally hold in cases where one considers
additional multiplets including only BSM fermions, which then may have
different $X$ values (see e.g. Ref. Descotes-Genon  _et al._ (2018).)

| $SU_{c}(3)$ | $SU_{L}(3)$ | $U_{X}(1)$ |  | $SU_{c}(3)$ | $SU_{L}(3)$ | $U_{X}(1)$
---|---|---|---|---|---|---|---
$\left(\begin{array}[]{c}u_{L}\\\ d_{L}\\\ \psi_{L}\end{array}\right)_{i}$ | $3$ | $3$ | $\frac{1}{6}-\frac{\beta}{2\sqrt{3}}$ | $\left(\begin{array}[]{c}\nu_{L}\\\ \ell_{L}\\\ \xi_{L}\end{array}\right)_{i}$ | $1$ | $3$ | $-\frac{1}{2}-\frac{\beta}{2\sqrt{3}}$
$\left(\begin{array}[]{c}d_{L}\\\ -u_{L}\\\ \Psi_{L}\end{array}\right)_{i}$ | $3$ | $\overline{3}$ | $\frac{1}{6}+\frac{\beta}{2\sqrt{3}}$ | $\left(\begin{array}[]{c}\ell_{L}\\\ -\nu_{L}\\\ \zeta_{L}\end{array}\right)_{i}$ | $1$ | $\overline{3}$ | $-\frac{1}{2}+\frac{\beta}{2\sqrt{3}}$
$u_{R_{i}}$ | $3$ | $1$ | $\frac{2}{3}$ |  |  |  |
$d_{R_{i}}$ | $3$ | $1$ | $-\frac{1}{3}$ | $\ell_{R_{i}}$ | $1$ | $1$ | $-1$
$\psi_{R_{i}}$ | $3$ | $1$ | $\frac{1}{6}-\frac{\sqrt{3}\beta}{2}$ | $\xi_{R_{i}}$ | $1$ | $1$ | $-\frac{1}{2}-\frac{\sqrt{3}\beta}{2}$
$\Psi_{R_{i}}$ | $3$ | $1$ | $\frac{1}{6}+\frac{\sqrt{3}\beta}{2}$ | $\zeta_{R_{i}}$ | $1$ | $1$ | $-\frac{1}{2}+\frac{\sqrt{3}\beta}{2}$
Table 1:  Assigned representations and $X$ values of matter fields in terms of
the parameter $\beta$ for a generic $331$ model. The subscript $i$ is a
generic flavor index.

Let us now us consider the 331 gauge bosons. Differently from the standard
models we have 8 $W^{a}_{\mu}$ bosons for $SU_{L}(3)$ and one ${B_{X}}_{\mu}$
boson for $U_{X}(1)$. The covariant derivative acting on field representations
are

  * •

triplet:
$D_{\mu}=\partial_{\mu}-igW^{a}_{\mu}T^{a}-ig_{X}X{B_{X}}_{\mu}T^{9}$;

  * •

antitriplet:
$D_{\mu}=\partial_{\mu}+igW^{a}_{\mu}(T^{a})^{T}-ig_{X}X{B_{X}}_{\mu}T^{9}$;

  * •

singlet: $D_{\mu}=\partial_{\mu}-ig_{X}X{B_{X}}_{\mu}T^{9}$;

where the $g$ and $g_{X}$ are gauge couplings and a sum over $a$ is
understood.

We arrange the $SU(3)_{\text{L}}$ gauge bosons $W_{\mu}^{a}$ as

$W_{\mu}=W_{\mu}^{a}T^{a}=\frac{1}{2}\begin{pmatrix}W_{\mu}^{3}+\frac{1}{\sqrt{3}}W_{\mu}^{8}&\sqrt{2}W_{\mu}^{+}&\sqrt{2}Y_{\mu}^{Q_{Y}}\\\
\sqrt{2}W_{\mu}^{-}&-W_{\mu}^{3}+\frac{1}{\sqrt{3}}W_{\mu}^{8}&\sqrt{2}V_{\mu}^{Q_{V}}\\\
\sqrt{2}Y_{\mu}^{-Q_{Y}}&\sqrt{2}V_{\mu}^{-Q_{V}}&-\frac{2}{\sqrt{3}}W_{\mu}^{8}\end{pmatrix}$
(6)

where

$\displaystyle W^{\pm}_{\mu}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(W^{1}_{\mu}\mp iW_{\mu}^{2})$ $\displaystyle
V_{\mu}^{\pm Q_{V}}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(W^{6}_{\mu}\mp iW_{\mu}^{7})$ (7)
$\displaystyle Y_{\mu}^{\pm Q_{Y}}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(W^{4}_{\mu}\mp iW_{\mu}^{5})$

The action of the charge operator $\hat{\mathcal{Q}}$ on gauge bosons is given
by the commutator $\left[\hat{\mathcal{Q}},W_{\mu}\right]$, and it can be
represented by the matrix

$\mathcal{Q}^{\rm
GB}=\left(\begin{array}[]{ccc}0&1&\frac{1}{2}+\beta\frac{\sqrt{3}}{2}\\\
-1&0&-\frac{1}{2}+\beta\frac{\sqrt{3}}{2}\\\
-\frac{1}{2}-\beta\frac{\sqrt{3}}{2}&\frac{1}{2}-\beta\frac{\sqrt{3}}{2}&0\end{array}\right)$
(8)

where each entry represents the charge of the gauge boson in the corresponding
entry of the matrix 6. From eq.(8) we see that there are:

  1. (i)

three neutral gauge bosons, $W^{3}_{\mu}$, $W^{8}_{\mu}$ and ${B_{X}}_{\mu}$.
After the spontaneous symmetry breakings, the neutral gauge bosons mix
originating three mass eigenstates, the SM $\gamma$ and $Z^{0}$ plus an
additional neutral BSM gauge boson, generally indicated with $Z^{\prime}$;

  2. (ii)

two bosons with unit charges, corresponding to $W^{\pm}$;

  3. (iii)

four BSM bosons $Y_{\mu}^{\pm Q_{Y}},V_{\mu}^{\pm Q_{V}}$, with charges
$\pm\mathcal{Q}_{Y}=\pm\frac{1}{2}\pm\beta\frac{\sqrt{3}}{2}$ and
$\pm\mathcal{Q}_{V}=\mp\frac{1}{2}\pm\beta\frac{\sqrt{3}}{2}$, that depend on
the value of $\beta$.

It is useful at this point to summarize the electric charges of BSM fermions
and BSM bosons:

field | $\psi$ | $\Psi$ | $\xi$ | $\zeta$ | $\quad Z^{\prime}\quad$ | $V$ | $Y$
---|---|---|---|---|---|---|---
$\mathcal{Q}$ | $\frac{1}{6}-\frac{\sqrt{3}}{2}\beta$ | $\frac{1}{6}+\frac{\sqrt{3}}{2}\beta$ | $-\frac{1}{2}-\frac{\sqrt{3}}{2}\beta$ | $-\frac{1}{2}+\frac{\sqrt{3}}{2}\beta$ | 0 | $\mp\frac{1}{2}\pm\frac{\sqrt{3}}{2}\beta$ | $\pm\frac{1}{2}\pm\frac{\sqrt{3}}{2}\beta$

Depending on the choice of the parameter $\beta$ the electric charges of the
new fields can be exotic. With exotic we mean charges different from $\pm
1/3,\pm 2/3$ for extra quarks, and different from $\pm 1$ for extra leptons
and gauge bosons. We consider as a minimal requirement that the new gauge
bosons have at least integer charge values, even if there are 331 models that
discard this assumption, see for instance Romero Abad and Ramirez Barreto
(2020).

By imposing that the new gauge bosons $Y_{\mu}^{\pm\mathcal{Q}_{Y}}$ and
$V_{\mu}^{\pm\mathcal{Q}_{V}}$ have integer charge, it follows that the
allowed values for the parameter $\beta$ are

$\beta=\pm\frac{2n+1}{\sqrt{3}}\equiv\frac{r}{\sqrt{3}}$ (9)

where $r$ is an odd integer number (positive or negative). A few examples of
$X$ and $\hat{\mathcal{Q}}$ charges for new fields are reported in Table (2),
corresponding to $r=\pm 1,\pm 3,\pm 5$. We immediately see that only the
choice $r=\pm 1$ guarantees no exotic charges. It is necessary to warn that
another constraint ($|r|\leq 3$) comes from spontaneous symmetry breaking, as
we will see in sect. 3.3. In Table (2) we report the case $\beta=\pm
5/\sqrt{3}$ only to remark that in the minimal assumptions made up to now
there is nothing that prevents $|r|$ from having values larger than $3$. From
Table (2) it is also manifest that the exchange $r\longleftrightarrow-r$ is
equivalent to trade triplets for anti-triplets and viceversa.

$r$ | $+1$ | $-1$ | $+3$ | $-3$ | $+5$ | $-5$
---|---|---|---|---|---|---
$X[\psi]$ | $0$ | $1/3$ | $-1/3$ | $2/3$ | $-2/3$ | $1$
$X[\Psi]$ | $1/3$ | $0$ | $2/3$ | $-1/3$ | $1$ | $-2/3$
$X[\xi]$ | $-2/3$ | $-1/3$ | $-1$ | $0$ | $-4/3$ | $1/3$
$X(\zeta)$ | $-1/3$ | $-2/3$ | $0$ | $-1$ | $1/3$ | $-4/3$
$\mathcal{Q}[\psi]$ | $-1/3$ | $2/3$ | $-4/3$ | $5/3$ | $-7/3$ | $8/3$
$\mathcal{Q}[\Psi]$ | $2/3$ | $-1/3$ | $5/3$ | $-4/3$ | $8/3$ | $-7/3$
$\mathcal{Q}[\xi]$ | $-1$ | $0$ | $-2$ | $1$ | $-3$ | $2$
$\mathcal{Q}[\zeta]$ | $0$ | $-1$ | $1$ | $-2$ | $2$ | $-3$
$|\mathcal{Q}[V]|$ | $0$ | $1$ | $1$ | $2$ | $2$ | $3$
$|\mathcal{Q}[Y]|$ | $1$ | $0$ | $2$ | $1$ | $3$ | $2$
Table 2:  We list the $X$ values and the charges for the third components of
quark triplet $3_{Q}$, lepton triplet $3_{\ell}$, quark anti-triplet
$\overline{3}_{Q}$, lepton anti-triplet $\overline{3}_{\ell}$, and the
absolute values of the charges of BSM gauge bosons for sample values of $r$
($\beta=r/\sqrt{3}$).

The 331 models are fully characterized by the choice of the electric charge
embedding, namely the pair $(\beta,X)$, of its fermionic content and multiplet
assignemnts, and of the scalar sector. There are two major quantum field
theory constraints, the absence of gauge anomalies and the possibility of QCD
asymptotic freedom.

The fermionic structure is restricted by requiring the cancellation of the
triangular anomalies. Both $SU(3)_{L}$ and $U(1)_{X}$ are anomalous, contrary
to the SM, where only $U(1)_{Y}$ is. In the SM, gauge anomalies are cancelled
out for every fermion family. This property is lost in 331 models and
correlation between families is needed. Ref. Diaz  _et al._ (2005) discusses
in detail this topic for an arbitrary value of $\beta$ and number of
generations. Below, we outline anomaly cancellation calculations assuming
$N_{Q(\ell)}^{(3)}$ and $N_{Q(\ell)}^{(\overline{3})}$ quark (lepton)
generations transforming as triplets and anti-triplets with the structure
outlined above. We underline that when we refer to lepton generations, we mean
generations of active leptons, whose left-handed parts are embedded in
triplets or antitriplets. We define the total number of quark (lepton)
generations as

$N_{Q(\ell)}^{F}=N_{Q(\ell)}^{(3)}+N_{Q(\ell)}^{(\overline{3})}$ (10)

The anomaly cancellations with their implications read as

  * A1)

$[SU(3)_{L}]^{3}:$

$N_{c}N_{Q}^{(3)}-N_{c}N_{Q}^{(\overline{3})}+N_{\ell}^{(3)}-N_{\ell}^{(\overline{3})}=0$

$\Rightarrow
N_{Q}^{(3)}=\frac{N_{c}N_{Q}^{F}+N_{\ell}^{F}}{2N_{c}}-\frac{N_{\ell}^{(3)}}{N_{c}}$

  * A2)

$[SU(3)_{L}]^{2}\otimes U(1)_{X}$:

$3N_{c}\left[N_{Q}^{(3)}X[3_{Q}]+N_{Q}^{(\overline{3})}X[\overline{3}_{Q}]\right]+3N_{\ell}^{(3)}X[3_{\ell}]+3N_{\ell}^{(\overline{3})}X[\overline{3}_{\ell}]=0$

$\Rightarrow N_{Q}^{F}=\frac{3}{N_{c}}N_{\ell}^{F}$ (assuming A1))

  * A3)

$[U(1)_{X}]^{3}$

$N_{c}\left[N_{Q}^{(3)}\left(3X[3_{Q}]^{3}-X[\psi_{R}]^{3}\right)+N_{Q}^{(\overline{3})}\left(3X[\overline{3}_{Q}]^{3}-X[\Psi_{R}]^{3}\right)-N_{Q}^{F}\left(X[u_{R}]^{3}+X[d_{R}]^{3}\right)\right]+$
$+N_{\ell}^{(3)}\left(3X[3_{\ell}]^{3}-X[\xi_{R}]^{3}\right)+N_{\ell}^{(\overline{3})}\left(3X[\overline{3}_{\ell}]^{3}-X[\zeta_{R}]^{3}\right)-N_{\ell}^{F}X[\ell_{R}]^{3}=0$

$\Rightarrow$ No condition (always true).

  * A4)

$[SU(3)_{c}]^{2}\otimes U(1)_{X}:$

$N_{c}\left[N_{Q}^{(3)}\left(3X[3_{Q}]-X[\psi_{R}]\right)+N_{Q}^{(\overline{3})}\left(3X[\overline{3}_{Q}]-X[\Psi_{R}]\right)-N_{Q}^{F}\left(X[u_{R}]+X[d_{R}]\right)\right]=0$

$\Rightarrow$ No condition (always true).

  * A5)

$[\text{Grav}]^{2}\otimes U(1)_{X}$

$N_{c}\left[N_{Q}^{(3)}\left(3X[3_{Q}]-X[\psi_{R}]\right)+N_{Q}^{(\overline{3})}\left(3X[\overline{3}_{Q}]-X[\Psi_{R}]\right)-N_{Q}^{F}\left(X[u_{R}]+X[d_{R}]\right)\right]+$
$+N_{\ell}^{(3)}\left(3X[3_{\ell}]-X[\xi_{R}]\right)+N_{\ell}^{(\overline{3})}\left(3X[\overline{3}_{\ell}]-X[\zeta_{R}]\right)-N_{\ell}^{F}X[\ell_{R}]=0$

$\Rightarrow$ No condition (always true).

where $N_{c}$ refers to the color $SU_{3}(N_{c})$, always fixed to $N_{c}=3$.
The minus signs in the sum are due to the different fields chirality. We
remark that these relations hold even under the common assumption
$\zeta_{L}^{a}=(\ell^{a}_{R})^{c}$.

Before drawing conclusions from the anomaly cancellation, let us summarize the
assumptions we have done on an otherwise completely general 331 model:

  * •

all matter multiplets include SM fermions, excluding the possibility of
fermion multiplets including only BSM fields;

  * •

all gauge bosons, including BSM ones, have integer charges;

  * •

all right-handed, $SU(3)$ singlet, fermions have a left-handed counterpart.

We underline that we could have assumed the presence of additional sterile
right-handed or left-handed neutrinos in an arbitrary number without changing
the above A1)-A7) conditions. Nor it would have changed the definition of
$N^{F}_{Q(\ell)}$, which refers to weak interacting fermions.

We found from relation A2)

$\displaystyle N_{\ell}^{F}$ $\displaystyle=$ $\displaystyle N_{Q}^{F}\equiv
N_{F}\,,$ (11)

namely the number of quark and charged lepton families must be the same.
Combining A1) with A2) it follows

$\displaystyle N_{Q}^{(3)}$ $\displaystyle=$
$\displaystyle\frac{1}{3}(2N_{F}-N_{\ell}^{(3)})\,.$ (12)

Both sides of the equality must be integers. This is true if $N_{F}$ is a
multiple of the color $N_{c}=3$, as in minimal models where
$N_{\ell}^{(\bar{3})}=0$. However such proportionality is not strictly
necessary. An example is $N_{F}=4$ with
$N_{\ell}^{(3)}=N_{\ell}^{(\bar{3})}=2$ that gives
$N_{Q}^{(3)}=N_{Q}^{(\bar{3})}=2$ (see also Ref. Diaz  _et al._ (2005)).

Another constraints is given by $SU(3)_{c}$ asymptotic freedom. As in SM QCD,
perturbative calculations limit the number of quark flavours to be equal to or
less than 16. Since each family in 331 models has three flavours, that yields
a constraint on the number of families: $N_{F}\leq 5$.

Summarizing, under the minimal assumptions discussed above, we can classify
331 models by means of $\beta=r/\sqrt{3}$, where $r=\pm 1,\pm 3$ (see for
instance Pleitez (2021)), the family number $N_{F}\leq 5$, and the number of
triplets or antitriplets for either quarks or leptons.

Fixing one of the 4 possible values for $\beta$, and assuming $N_{c}=N_{F}=3$,
we have two possible choices, namely $N_{Q}^{3}=2,1$, that we name case
A[$\beta$] and B[$\beta$], respectively. In Table 3 we report the numbers of
quark and leptons in triplets and antitriplets according to the relations
(10),(11) and (12). Note that this classification works only when triplet and
anti-triplets are used, but it is also possible to assign matter fields to
different representations like sextects, see for instance Ref. Fonseca and
Hirsch (2016).

| $N_{Q}^{(3)}$ | $N_{Q}^{(\bar{3})}$ | $N_{\ell}^{(3)}$ | $N_{\ell}^{(\bar{3})}$
---|---|---|---|---
A[$\beta$] | 2 | 1 | 0 | 3
B[$\beta$] | 1 | 2 | 3 | 0
Table 3:  Number of triplets and anti-triplets in the lepton and quark sectors
in case A[$\beta$] and B[$\beta$] (see text).

Models $A[\beta]$ and $B[-\beta]$ are related by the simultaneous exchange of
fermion triplets $\to$ antitriplets and $\beta\to-\beta$. It implies that
matter and gauge fields possess the same set of charges in both models 222A
complete symmetry between the two models requires additional conditions in the
Higgs sector Hue and Ninh (2019).. Several of these models have been discussed
in literature: $A[1/\sqrt{3}]$ Montero  _et al._ (1993); Buras  _et al._
(2013); Boucenna  _et al._ (2014), $A[-1/\sqrt{3}]$ Cabarcas  _et al._ (2008);
Profumo and Queiroz (2014); Cárcamo Hernández  _et al._ (2016); Alves  _et
al._ (2022), $A[\sqrt{3}]$ Singer  _et al._ (1980); Pisano and Pleitez (1992);
Foot  _et al._ (1993); Montero  _et al._ (1999); Catano  _et al._ (2012);
Frampton (1992); Correia (2018); Corcella  _et al._ (2018). The case
A$[\sqrt{3}]$ is known as the minimal 331 model and will be considered in this
work.

We remark that in order to fully specify a model fixing the matter content as
above is not enough, but it has to be complemented by the assignment of the
Higgs structures and vacuum alignments, see below.

##  3 The model

###  3.1 Matter assignment

In this work, we focus on the minimal 331 model Corcella  _et al._ (2018),
where the first two generations of quarks transform as triplets under
$SU(3)_{L}$, whereas the third generation and all the leptons transform as
anti-triplets (model $A[\sqrt{3}]$). Table (4) contains a summary of the
matter content of the model. As shown explicitly in Table 2, the fermions
$D^{1,2}$ and $T$ in the color triplet are exotic quarks with fractional
charges $-4/3$ and $5/3$ respectively 333These exotic quarks do not have the
same value charge of down type quarks and top quark, but maintain the same
charge sign. With respect to table 1 we have changed the generic BSM fermion
field notation: $\psi\to D,\Psi\to T$., while the fermions $\zeta_{L}^{a}$,
$a$ = 1, 2, 3, which are color singlets, are leptons with positive charge
$+1$. The exotic quarks $D^{1,2}$ and $T$ are assigned total lepton number
$L=-2$ and $L=+2$, respectively. We will see later that this choice allows $L$
to be conserved. Then, since the global symmetry $B-L$ of the SM maintains in
our model, also $B$ is preserved. All three BSM quarks can be assumed to carry
the usual baryonic number for quarks, namely $B=1/3$. In the spirit of
minimally extending the SM, we identify $\zeta^{a}_{L}=(\ell^{a}_{R})^{c}$.

| $SU_{c}(3)$ | $SU_{L}(3)$ | $U_{X}(1)$ |  | $SU_{c}(3)$ | $SU_{L}(3)$ | $U_{X}(1)$
---|---|---|---|---|---|---|---
$\left(\begin{array}[]{c}u^{1,\,2}_{L}\\\ d^{1,\,2}_{L}\\\ D^{1,\,2}_{L}\end{array}\right)$ | $3$ | $3$ | $-\frac{1}{3}$ | $u_{R}^{a}$ | $3$ | $1$ | $\frac{2}{3}$
$d_{R}^{a}$ | $3$ | $1$ | $-\frac{1}{3}$
$D_{R}^{1,\,2}$ | $3$ | $1$ | $-\frac{4}{3}$
$\left(\begin{array}[]{c}b_{L}\\\ -t_{L}\\\ T_{L}\end{array}\right)$ | $3$ | $\overline{3}$ | $\frac{2}{3}$ | $T_{R}$ | $3$ | $1$ | $\frac{5}{3}$
$\left(\begin{array}[]{c}\ell^{a}_{L}\\\ -\nu^{a}_{L}\\\ (\ell^{a}_{R})^{c}\end{array}\right)$ | $1$ | $\overline{3}$ | $0$ | $\chi$ | $1$ | $3$ | $1$
$\rho$ | $1$ | $3$ | $0$
$\eta$ | $1$ | $3$ | $-1$
Table 4:  The fermion and scalar content of the $A[\sqrt{3}]$ model we are
considering.

We underline that $D^{1,2}$ and $T$ are Vector Like Quarks (VLQs). If we
consider the embedding of $SU_{L}(2)$ into $SU_{L}(3)$, we observe that the
generators $T^{i}$ ($i=1,2,3$) do not act on the third component of
$SU_{L}(3)$ triplets and anti-triplets. As a consequence, $D^{1,2}$ and $T$
are singlets of the SM $SU_{L}(2)$ and they do not couple to charged currents
by the covariant derivative. Later on (see Eq. (44)), we explicitly show that
their left and right-handed components behave the same way in the neutral weak
interactions, resulting in a vector electroweak current; hence the epithet
vector-like. Being singlets of the SM $SU_{L}(2)$, VLQs do not require
coupling to $SU(2)_{L}$ scalar Higgs doublets to get mass. Hence they are not
excluded by current measurements of the production rate of the Higgs boson,
which instead exclude the existence of a fourth generation of chiral quarks.
From the general discussion in Sect. 2 it follows that the $D$ and $T$
electric charges are -4/3 and 5/3, respectively.

###  3.2 The scalar sector

As mentioned in the introduction, the spontaneous symmetry breaking (SSB)
pattern of 331 models occurs in two steps: the first one allows to recover the
SM group structure, the second one is identifiable as the electroweak SSB. The
331 models are intrinsically multi-Higgs models, which, as in the SM, are
$SU(3)_{C}$ singlets acquiring non-vanishing vacuum expectation values after
SSB. They are responsible of mass terms for gauge bosons and fermions coupling
to them. The requirement of gauge invariance of the Lagrangian term coupling
one Higgs and two fermions constrains the Higgs representations. Namely, since
the fermions $\psi$ transform either as a $3$ or as a $\bar{3}$ under
$SU(3)_{L}$, we only have the following possibilities Diaz  _et al._ (2004)
for the Higgs field $\Phi$:

  * •

$\bar{\psi}_{L}\psi_{R}\Phi$, with $\Phi\sim 3$;

  * •

$\bar{\psi}_{L}(\psi_{L})^{c}\Phi$, with $\Phi\sim 6$;

  * •

$\bar{\psi}_{R}(\psi_{R})^{c}\Phi$, with $\Phi\sim 1$;

  * •

$\bar{\psi}_{L}(\psi^{\prime}_{L})^{c}\Phi$, with $\Phi\sim\bar{3}$;

The last possibility is allowed only if $\psi^{\prime}_{L}$ are completely
exotic triplets with charged particles which are right handed charge-
conjugated components Descotes-Genon  _et al._ (2018), otherwise is zero.
Singlets scalars are also scalars under $U(1)_{X}$ because of electromagnetic
invariance. Thus, after the two steps of SSB, their vacuum expectation value
will never give rise to a mass term for the gauge bosons or the charged
fermions. Sextets are needed in some models, for instance to give Majorana
masses to neutrinos as in Ref. Foot  _et al._ (1993) or to allow a doubly-
charged scalar to decay into same-sign leptons Corcella  _et al._ (2018). Both
singlets and sextets are not needed for the model and the phenomenology we are
investigating, so we limit to the simpler, and widely used in literature,
assumption of SSB by triplets only. The $X$ values of the triplets, and
therefore also the charges of their components, are constrained by the
requirement that the above Higgs-fermion coupling terms in the Lagrangian are
also invariant under $U(1)_{X}$.

The Higgs triplets are:

$\chi=\begin{pmatrix}\chi^{++}\\\ \chi^{+}\\\ \chi^{0}\end{pmatrix},\hskip
28.45274pt\rho=\begin{pmatrix}\rho^{+}\\\ \rho^{0}\\\
\rho^{-}\end{pmatrix},\,\hskip 28.45274pt\eta=\begin{pmatrix}\eta^{0}\\\
\eta^{-}\\\ \eta^{--}\end{pmatrix}.$ (13)

Their $X$ values are listed in Table 4. The neutral components acquire a
vacuum expectation value, namely $\langle\chi^{0}\rangle\equiv u$,
$\langle\rho^{0}\rangle\equiv v$, $\langle\eta^{0}\rangle\equiv v^{\prime}$.
In the first SSB, triggered by the triplet $\chi$, there are five gauge fields
that acquire mass. In the second SSB, which requires the pair of triplets
$\rho$ and $\eta$, three gauge fields acquire mass, leaving massless the field
associated to the unbroken charge generator, that is the photon.

###  3.3 Gauge sector

In the first stage of SSB and with the choice $\beta=\sqrt{3}$, there emerge
singly and doubly charged gauge bosons, $V_{\mu}^{\pm}$ (also named
$W^{\prime}_{\mu}$) and $Y_{\mu}^{\pm\pm}$. The new charged gauge bosons are
dubbed bileptons because they couple at tree level to SM lepton pairs, but not
to SM quark pairs 444To avoid confusion with old literature, we remind that
they have previously shared the name dilepton with events having two final
state leptons.. Bileptons occur in several models of common SM extensions, as
grand unified theories, technicolor and compositeness scenarios. We can assign
them global quantum numbers, in contrast with the SM where only fermions carry
baryon or lepton numbers, not bosons.

One distinctive feature of our 331 model is that $Y_{\mu}^{\pm\pm}$ bileptons
couple to BSM quarks. One can assign lepton number $L=-2$ to the gauge bosons
$V_{\mu}^{+}$ and $Y_{\mu}^{++}$, and $L=+2$ to $V_{\mu}^{-}$ and
$Y_{\mu}^{--}$. The 331 model does not conserve separate family lepton
numbers, but the total lepton number is conserved once the exotic quarks
$D^{1,2}$ carry $L=-2$ and $T$ carries $L=+2$.

Boson masses stem from the expansion of the covariant derivatives of the
scalar sector

$\mathcal{L}_{S}=(D_{\mu}\chi)(D^{\mu}\chi)^{\dagger}+(D_{\mu}\rho)(D^{\mu}\rho)^{\dagger}+(D_{\mu}\eta)(D^{\mu}\eta)^{\dagger}$
(14)

After both $SU(3)_{L}$ breaking eight gauge bosons acquire a mass. The charged
one are directly diagonal with masses

$\displaystyle m_{W}^{2}$ $\displaystyle=$
$\displaystyle\frac{g^{2}}{4}\left(v^{2}+v^{\prime
2}\right)\equiv\frac{g^{2}}{4}v_{+}^{2}\,,$ (15) $\displaystyle m_{V}^{2}$
$\displaystyle=$
$\displaystyle\frac{g^{2}}{4}\left(u^{2}+v^{\prime\,2}\right)\simeq\frac{g^{2}}{4}u^{2}$
(16) $\displaystyle m_{Y}^{2}$ $\displaystyle=$
$\displaystyle\frac{g^{2}}{4}\left(u^{2}+v^{2}\right)\simeq\frac{g^{2}}{4}u^{2}$
(17)

To correctly restore the Standard Model, we require that $v^{2}_{+}=v^{2}_{\rm
SM}$, where $v_{\rm SM}$ is the Higgs boson vacuum expectation value, and
$u\gg v,v^{\prime}$. These assumptions imply that $m_{W}\ll m_{V}\simeq
m_{Y}$.

The neutral gauge bosons matrix is not diagonal. In the base
$W_{\mu}^{3},\,W_{\mu}^{8},\,B_{X\,\mu}$, it is given by

$M_{0}=\frac{1}{4}\left(\begin{array}[]{ccc}g^{2}v^{2}_{+}&\frac{g^{2}}{\sqrt{3}}v_{-}^{2}&-\frac{gg_{X}}{3\sqrt{2}}\left(v_{+}^{2}\sqrt{3}-v_{-}^{2}\beta\right)\\\
\frac{g^{2}}{\sqrt{3}}v_{-}^{2}&\frac{g^{2}}{3}\left(4u^{2}+v_{+}^{2}\right)&-\frac{gg_{X}}{9\sqrt{2}}\left(u^{2}4\sqrt{3}\beta+v_{+}^{2}\sqrt{3}\beta+3v_{-}^{2}\right)\\\
-\frac{gg_{X}}{3\sqrt{2}}\left(v_{+}^{2}\sqrt{3}-v_{-}^{2}\beta\right)&-\frac{gg_{X}}{9\sqrt{2}}\left(u^{2}4\sqrt{3}\beta+v_{+}^{2}\sqrt{3}\beta+3v_{-}^{2}\right)&\frac{g_{X}^{2}}{18}\left(4u^{2}\beta^{2}+v_{+}^{2}(3+\beta^{2})+v_{-}^{2}2\sqrt{3}\right)\end{array}\right)$
(18)

where $v^{2}_{-}=v^{\prime\,2}-v^{2}$.
Since $u^{2}\gg v_{+}^{2},\,v_{-}^{2}$ we can diagonalize this matrix in two
steps. In the first one $B_{X\,\mu},\,W_{\mu}^{8}$ mix giving rise to
$B^{\prime}_{\mu}$ and $Z^{\prime}_{\mu}$ as

$\displaystyle\left(\begin{array}[]{c}W^{8}_{\mu}\\\
B_{X\,\mu}\end{array}\right)\equiv\left(\begin{array}[]{cc}\cos\theta_{331}&-\sin\theta_{331}\\\
\sin\theta_{331}&\cos\theta_{331}\\\
\end{array}\right)\left(\begin{array}[]{c}B^{\prime}_{\mu}\\\
Z^{\prime}_{\mu}\end{array}\right)\,.$ (25)

With the above definition of the mixing angle, by diagonalizing $M_{0}$ we
obtain

$\displaystyle\sin\theta_{331}\simeq\frac{g}{\sqrt{g^{2}+g_{X}^{2}\beta^{2}/6}}\,.$
(26)

Then at the second step $B^{\prime}_{\mu},Z_{\mu}^{\prime}$ mix both with
$W_{\mu}^{3}$ but the mixing between $Z_{\mu}^{\prime}$ and $W_{\mu}^{3}$ is
small compared to the one between $B^{\prime}_{\mu}$ and $W_{\mu}^{3}$,
therefore the first one can be neglected and we have

$\displaystyle\left(\begin{array}[]{c}W_{\mu}^{3}\\\
B^{\prime}_{X\,\mu}\end{array}\right)\equiv\left(\begin{array}[]{cc}\cos\theta_{W}&\sin\theta_{W}\\\
-\sin\theta_{W}&\cos\theta_{W}\\\
\end{array}\right)\left(\begin{array}[]{c}Z_{\mu}\\\
A_{\mu}\end{array}\right)$ (33)

where $A_{\mu}$ is the Standard Model photon field. The corresponding boson
masses are

$\displaystyle m_{A}$ $\displaystyle=$ $\displaystyle 0\,,$ (34)
$\displaystyle m_{Z}^{2}$ $\displaystyle\simeq$
$\displaystyle\frac{g^{2}}{4\cos^{2}\theta_{W}}v_{\rm SM}^{2}\,$ (35)
$\displaystyle m_{Z^{\prime}}^{2}$ $\displaystyle\simeq$
$\displaystyle\frac{g^{2}u^{2}\cos^{2}\theta_{W}}{3[1-(1+\beta^{2})\sin^{2}\theta_{W}]}\,.$
(36)

We note that $m_{W}^{2}/m_{Z}^{2}=\cos^{2}\theta_{W}$ and therefore we can
identify $\theta_{W}$ with the Weinberg angle. Moreover by comparing Eq. (36)
and Eq. (17), we obtain a useful relation between the masses of these bosons

$\frac{m_{Z^{\prime}}^{2}}{m_{Y}^{2}}\simeq\frac{4\cos^{2}\theta_{W}}{3\left[1-\left(1+\beta^{2}\right)\sin^{2}\theta_{W}\right]}\simeq
9.3\text{ for $\beta=\sqrt{3}$}$ (37)

Another important constraint for 331 models is obtained by imposing a matching
condition for the coupling of $Z$ boson with fermions. In particular,
considering the right-handed leptons $\ell_{R}$ with $X$-charge
$X[\ell_{R}]=-1$, the covariant derivative is given by

$D_{\mu}\ell_{R}=\partial_{\mu}\ell_{R}+ig_{X}\frac{B_{X_{\mu}}}{\sqrt{6}}\,\ell_{R}$
(38)

From Eqs. (25) and (33) we have that

$B_{X_{\mu}}=\cos\theta_{331}\,Z^{\prime}_{\mu}-\sin\theta_{331}\sin\theta_{W}Z_{\mu}+\sin\theta_{331}\cos\theta_{W}A_{\mu}$
(39)

Then by singling out the coupling with $Z$ boson and by matching with the
corresponding Standard Model coupling $g\sin^{2}\theta_{W}/\cos\theta_{W}$ we
get

$\frac{g_{X}}{g}=\frac{\sqrt{6}\sin\theta_{W}}{\cos\theta_{W}\sin\theta_{331}}=\frac{\sqrt{6}\sin\theta_{W}}{\sqrt{1-\sin^{2}\theta_{W}(1+\beta^{2})}}$
(40)

where we have used eq. (26).

We can see that Eq. (40) exhibits a Landau pole when
$\sin^{2}\theta_{W}=1/(1+\beta^{2})$. As in other non asymptotically free
theories, at the Landau pole the coupling constant $g_{X}$ becomes infinite.
The energy scale of the Landau pole depends on the running of
$\sin^{2}\theta_{W}$, which in turn depends on the matter content of the
model. At larger values of the coupling, that is even before the Landau pole
is reached, the models lose their perturbative character Martinez and Ochoa
(2007); Dias  _et al._ (2005); Dias and Pleitez (2009); Dias  _et al._ (2006).

We also note that the left-hand side of eq. (40) is real if (see also Fonseca
and Hirsch (2016))

$|\beta|<\frac{\cos\theta_{W}}{\sin\theta_{W}}\simeq 1.83\,.$ (41)

This upper limit of $|\beta|$ is extremely important for the model building.
As shown above the only allowed values for $|\beta|$ are $r/\sqrt{3}$ with $r$
integer. The constraint coming from eq. (41) yields $n\leq 3$ and therefore
models with $n>3$ are not allowed in this minimal framework.

###  3.4 Yukawa interactions

SM fermions get mass via the Yukawa coupling with the scalars. The gauge
invariant Yukawa Lagrangian reads as

$\begin{split}\mathcal{L}_{Yukawa}=&+\lambda_{i,a}^{d}\overline{Q}_{i}\rho
d_{a,R}+\lambda_{i,a}^{u}\overline{Q}_{i}\eta
u_{a,R}+\lambda_{3,a}^{d}\overline{Q}_{3}\eta^{*}d_{a,R}+\lambda_{3,a}^{u}\overline{Q}_{3}\rho^{*}u_{a,R}+\\\
&+\lambda_{i,j}^{J}\overline{Q}_{i}\chi
D_{j,R}+\lambda_{3,3}^{J}\overline{Q}_{3}\chi^{*}T_{R}\\\
&+\lambda_{a,b}^{\ell}\epsilon_{ijk}\overline{L}_{ai}L_{bj}\rho^{*}_{k}\end{split}$
(42)

where
$[Q^{1},\,Q^{2},\,Q^{3}]\equiv[3_{Q}^{1},\,3_{Q}^{2},\,\overline{3}_{Q}^{3}]$,
$[L^{1},\,L^{2}\,,L^{3}]\equiv[\overline{3}_{\ell}^{1},\,\overline{3}_{\ell}^{2},\,\overline{3}_{\ell}^{3}]$,
$a,b=1,2,3$ and $i,j=1,2$. The first line corresponds to the masses of the
Standard Model quarks, the second to those of the VLQs, and the third pertains
to the charged lepton masses. We underline that the Yukawa interactions
involving quarks are 331-invariant for any $\beta$, whereas the last line,
responsible for the charged lepton masses, is invariant only for
$\beta=\sqrt{3}$.

From the diagonalization of Eq. (42), we obtain the quark mass eigenstates,
$u^{\prime}_{a}$ and $d^{\prime}_{a}$. The relation between left-handed and
right-handed mass and interaction eigenstates reads as

$\left(\begin{array}[]{c}u_{1,}^{\prime}\\\ u_{2}^{\prime}\\\
u_{3}^{\prime}\end{array}\right)_{L,R}={\cal
U}_{L,R}^{-1}\left(\begin{array}[]{c}u_{1}\\\ u_{2}\\\
u_{3}\end{array}\right)_{L,R}\hskip
85.35826pt\left(\begin{array}[]{c}d_{1}^{\prime}\\\ d_{2}^{\prime}\\\
d_{3}^{\prime}\end{array}\right)_{L,R}={\cal
V}_{L,R}^{-1}\left(\begin{array}[]{c}d_{1}\\\ d_{2}\\\
d_{3}\end{array}\right)_{L,R}$ (43)

where ${\cal U}_{L,R}$ and ${\cal V}_{L,R}$ are four unitary matrices. The CKM
mixing matrix $V_{\rm CKM}$ is recovered as the usual product $V_{\rm
CKM}\equiv{\cal U}_{L}^{\dagger}\cdot{\cal V}_{L}$.

We can safely take the new quarks $D$ and $T$ as mass eigenstates and omit
their rotation matrices in the interactions. That is immediately evident for
$T$, which, having charge 5/3, cannot mix with the other quarks. In the case
of $D$ quarks, let us observe that in the interactions a possible rotation
matrix is either cancelled because of unitarity (see Eqs. (44) and (45),) or
it appears together with ${\cal U}_{L}$ or ${\cal V}_{L}$ (see Eqs. (46) and
(47)). In the latter case, we can define new unitary ${\cal U}_{L}$ or ${\cal
V}_{L}$ matrices as the product of the older ${\cal U}_{L}$ or ${\cal V}_{L}$
matrices, whose entries are unknown, and the $D$ rotation matrix, similarly
unknown.

###  3.5 Fermions and gauge boson new couplings

This section illustrates new interactions predicted by the model under
investigation. We start from the interactions involving the SM gauge bosons
and the new fermions. The $W^{\pm}_{\mu}$ bosons do not couple to VLQs, while
the $Z_{\mu}$ boson does. The Lagrangian involving the $Z_{\mu}$ boson reads
as

$i\mathcal{L}_{F}^{Z}\supset+i\frac{g}{c_{W}}Z^{\mu}\Bigg{[}\frac{4}{3}s_{W}^{2}\overline{D}_{L}^{i}\gamma_{\mu}D_{L}^{i}+\frac{4}{3}s_{W}^{2}\overline{D}_{R}^{i}\gamma_{\mu}D_{R}^{i}-\frac{5}{3}s_{W}^{2}\overline{T}_{L}\gamma_{\mu}T_{L}-\frac{5}{3}s_{W}^{2}\overline{T}_{R}\gamma_{\mu}T_{R}\Bigg{]}$
(44)

Here we report the terms involving the $Z^{\prime}$ boson

$\begin{split}i\mathcal{L}_{F}^{Z^{\prime}}=&i\frac{g}{2\sqrt{3}c_{W}\sqrt{1-4\,s_{W}^{2}}}\Biggl{\\{}\left[1-4\,s_{W}^{2}\right](\overline{\nu}_{L}^{a}\gamma_{\mu}\
\nu_{L}^{a}+\overline{\ell}_{L}^{a}\gamma_{\mu}\ell_{L}^{a})-\left[2+4\,s_{W}^{2}\right]\overline{\ell}_{R}^{a}\gamma_{\mu}\ell_{R}^{a}+\\\
&+\left[-1+2\,s_{W}^{2}\right]\overline{u}_{L}^{\prime\,a}\gamma_{\mu}\delta_{ab}u_{L}^{\prime\,b}+2c_{W}^{2}\overline{u}_{L}^{\prime\,a}\gamma_{\mu}u_{L}^{\prime\,b}{{\cal
U}_{L}}^{*}_{3a}{{\cal
U}_{L}}_{3b}+4\,s_{W}^{2}\overline{u}_{R}^{\prime\,a}\gamma_{\mu}u_{R}^{\prime\,b}\delta_{ab}+\\\
&+\left[-1+2\,s_{W}^{2}\right]\overline{d}_{L}^{\prime\,a}\gamma_{\mu}\delta_{ab}d_{L}^{\prime\,b}+2c_{W}^{2}\overline{d}_{L}^{\prime\,a}\gamma_{\mu}d_{L}^{\prime\,b}{{\cal
V}_{L}}^{*}_{3a}{{\cal
V}_{L}}_{3b}-2s_{W}^{2}d_{R}^{\prime\,a}\gamma_{\mu}d_{R}^{\prime\,b}\delta_{ab}+\\\
&+8\,s_{W}^{2}(\overline{D_{L}}^{i}\gamma_{\mu}D_{L}^{j}+\overline{D_{R}}^{i}\gamma_{\mu}D_{R}^{j})+10\,s_{W}^{2}(\overline{T_{L}}\gamma_{\mu}T_{L}+\overline{T_{R}}\gamma_{\mu}T_{R})\Biggl{\\}}\end{split}$
(45)

While in the SM the GIM mechanism guarantees the flavor universality,
$Z^{\prime}_{\mu}$ interactions with quarks violate it, allowing processes
like $e^{+}e^{-}\to\overline{u}c$ or $\overline{u}u\to\overline{u}c$.

The vertices involving $V_{\mu}$ read as

$i\mathcal{L}_{F}^{V}=+i\frac{g}{\sqrt{2}}\,V^{-}_{\mu}\left[\overline{\nu}^{a}_{L}\gamma^{\mu}(\ell_{R}^{a})^{c}+\overline{D}_{L}^{i}\gamma^{\mu}d_{L}^{\prime\,a}{{\cal
V}_{L}}_{ai}+\overline{u}_{L}^{\prime\,a}\gamma^{\mu}T_{L}\,{{\cal
U}_{L}}_{3a}^{*}+{\rm h.c.}\right],$ (46)

whereas the ones with the $Y_{\mu}$ boson are

$i\mathcal{L}_{F}^{Y}=-i\frac{g}{\sqrt{2}}\,Y^{++}_{\mu}\left[\overline{(\ell_{R}^{a})^{c}}\gamma^{\mu}\ell_{L}^{a}+\overline{u}_{L}^{a}\gamma^{\mu}D_{L}^{i}\,{{\cal
U}_{L}}^{*}_{ai}+\overline{T}_{L}\gamma^{\mu}d_{L}^{a}\,{{\cal
V}_{L}}_{3a}+{\rm h.c.}\right]$ (47)

In the appendix, we show the Feynman diagrams of the couplings which are
relevant to our phenomenological analysis.

We observe that the quark couplings to the new gauge bosons depend on the
unknown entries of the rotation matrices ${\cal U}_{L}$ and ${\cal V}_{L}$,
which do not cancel out in the interactions. For simplicity in our analysis we
fix as benchmark case ${\cal U}_{L}=\mathbf{1}$, i.e. the three by three unity
matrix; from the definition of the CKM matrix $V_{CKM}$ it follows that ${\cal
V}_{L}=V_{CKM}$. Under this assumption $D^{i}$ and $T$ couple predominantly to
the $i$-th quark generation and to the third, respectively. For our
phenomenological analysis, it is not worrisome having set ${\cal U}_{L}$ real,
since a possible phase would not affect the diagrams in Fig. 3. It is possible
to connect the entries of the rotation matrices to observables in kaon,
$B_{d}$, and $B_{s}$ systems, yielding absolute values that are generally of
the same order of magnitude than ours Buras  _et al._ (2013).

##  4 Phenomenology of minimal 331

This minimal 331 model is rich of new fields. In the fermion sector we have
two VLQs and in the gauge sector we have one new neutral and two new charged
bosons, namely

$(D^{1,\,2})^{-4/3},\,T^{5/3},\,Z^{\prime\,0},\,V^{\pm},\,Y^{\pm\pm},$ (48)

where we added an apex to indicate the charge of the particle. Several
phenomenological implications of the existence of these exotic particles have
been investigated:

  * •

At colliders, VLQs are typically identified by means of the decay channels
$\rm{VLQs}\to q\,W^{\pm},\,q\,Z,\,q\,H$ Aad  _et al._ (2015); Sirunyan  _et
al._ (2018); Aaboud  _et al._ (2018a); Aad  _et al._ (2023a, b). In 331 models
these decay channels are suppressed and are replaced by the dominant ones
$\rm{VLQs}\to q\,V^{\pm},\,q\,Y^{\pm\pm}$. Therefore the peculiar interactions
of VLQs in 331 theories allow less conventional search strategies, see for
instance Ref. Corcella  _et al._ (2021), where the case of an exotic VLQ with
charge 5/3 has been considered and a lower bound on its mass of about 1.5 TeV
is found by a recasting of the CMS analysis in Ref. Sirunyan  _et al._ (2019).

  * •

The phenomenology of $Z^{\prime}$ bosons in 331 has been studied quite in
detail in the recent literature Oliveira and de S. Pires (2022); Alves  _et
al._ (2022); Addazi  _et al._ (2022); Alves  _et al._ (2023). The $Z^{\prime}$
boson is produced through the Drell-Yan process and subsequently decays into
pairs of SM particles or in exotic channels, like in pairs of VLQs. Note that
these signatures are typical also of $Z^{\prime}$ bosons or, in general, of
heavy resonances, which appear in other BSM theories, as those of composite
Higgs models Contino  _et al._ (2007); Vignaroli (2014, 2015); Bini  _et al._
(2012). Therefore, it would be beneficial to identify, as we will do in this
study, different signatures which are more characteristic of the 331 model.
The dilepton channel gives significant constraints on the new vector, with
bounds of the order of 4 TeV on the $Z^{\prime}$ mass, as extracted from the
ATLAS and CMS searches, already with a luminosity of 139 fb-1 Alves  _et al._
(2022). Note, however, that the $Z^{\prime}$ mass is expected to be
significantly higher than those of $V$ and $Y$ (Eq. 37). While it will
certainly be useful to continue in these searches for the $Z^{\prime}$, it is
essential to explore new strategies for testing the 331 model that can be more
distinctive of the underlying theory and, possibly, even more efficient.

  * •

Differently from the typical LHC phenomenology of $W^{\prime}$ resonances
(such as Sequential-SM $W^{\prime}$ Altarelli  _et al._ (1989), Kaluza-Klein
$W$ from extra-dimension or from composite Higgs theories Contino  _et al._
(2007)), which are dominantly produced via the Drell-Yan process 555The
production via weak-boson-fusion is typically subdominant, except when
considering colliders at very high collision energies, such as FCC-hh Mohan
and Vignaroli (2015), in the case of the $V^{\pm}$ ($W^{\prime}$) from 331
models, other production channels must be considered. Indeed, their Drell-Yan
production is precluded at tree level, given the absence of $V$ couplings with
pairs of SM fermions, see e.g. Ref. Corcella  _et al._ (2021). Nevertheless,
the $V^{\pm}$ could be produced by interactions with a VLQ and a SM quark. It
could be thus interesting to examine less conventional channels, such as the
one in Fig. 2 (bottom) in which the $V$ is produced in association with a VLQ.
We leave this analysis to a future investigation.

In our study, we propose signatures for 331 models, which are also efficient
as search channels for the exotic VLQs. In particular, based on the
considerations above we find particularly interesting to focus on the
inclusive production of a pair of bilepton vector bosons $Y^{\pm\pm}$

$pp\to Y^{++}Y^{--}$ (49)

where each bilepton decays into a pair of same sign charged leptons. Then the
process we look for is

$pp\to\ell^{+}\ell^{+}\,\ell^{-}\ell^{-}+{X}$ (50)

with a multi-lepton final state. We consider the inclusive production,
allowing the presence of extra jets in the final state denoted here as $X$.

Exclusive production of bilepton pairs at colliders, with purely
$Y^{++}Y^{--}$ in the final state, has been already studied, see for instance
Refs. Dion  _et al._ (1999); Corcella  _et al._ (2017, 2018); Nepomuceno and
Meirose (2020). The typical diagram considered (at the parton level) are the
Drell-Yan like processes (Figure 1, left) or with the exchange of an exotic
quark in the $t$-channel (Figure 1, right).

Figure 1: Typical Feynman diagrams for $Y^{++}Y^{--}$ production: Drell-Yan
like processes (left), t-channel mediated by a vector-like anti-quark (right).

Figure 2:  Bilepton $Y$ and $V$ associated production with a VLQ.

Another possibility Dutta and Nandi (1994) (not considered in the previous
searches for exclusive bilepton pair production) is the production from gluon
($g$) interaction, associated to a VLQ, namely $g\,u\to D^{1}Y^{++}$, $g\,c\to
D^{2}Y^{++}$ and $g\,b\to TY^{--}$, as shown in Figure (2). If

$m_{Y}<m_{D},\,m_{T}$ (51)

then the VLQs can decay in bileptons as $D^{1}\to Y^{--}u$, $D^{2}\to Y^{--}c$
and $T\to Y^{++}b$, see Figure (3). The decay widths are reported in appendix.

Figure 3: Feynman diagrams for the dominant contributions to the process
$g\,q\to Y^{++}Y^{--}\,j$ given by the VLQ-Y associated production considered
in this study. The symbols $g$, and $q$ label a gluon and a SM quark,
respectively; $j=q,\,g$ at the parton level.

A contribution to the inclusive process (49) is therefore also given by

$g\,q\to{\rm VLQ}\;Y\to Y^{++}Y^{--}\,q^{\prime},$ (52)

where the VLQ is either a $D$ or a $T$ fermion; $q$ and $q^{\prime}$ are
either an up, a charm or a bottom SM quark. We remark that these processes are
not included in the ones considered in Ref. Corcella  _et al._ (2017), which
focuses on the exclusive final state $Y^{++}Y^{--}\,jj$ with $j=q,\,g$ at the
parton level. Conservatively, in our study we will not include the $D^{2}$
contribution (i.e. we will consider either $D^{2}$ decoupled from the spectrum
or $m_{Y}>m_{D^{2}}$ so that the $D^{2}\to Yc$ decay is off-shell and
suppressed). This is a conservative choice, since an even larger VLQ
contribution to the bilepton pair cross section is expected when $D^{2}$ can
contribute. In agreement with this assumption, we indicate for simplicity
$D^{1}\equiv D$ hereafter and in the previous Fig.s 1,2.

To our knowledge, all previous studies have overlooked the contribution to the
pair bilepton production given by the bilepton associated production with a
vectorlike quark (Figure 2), which subsequently decays into a second bilepton,
see Figure 3.

We find that instead this contribution is significant in a large portion of
the model parameter space. We show in Fig. 4 the cross section for the VLQ-
bilepton associated production, for the cases of $D$ and $T$ VLQ, as a
function of the bilepton mass and for different VLQ mass values. Because of
the smaller PDF for the initial b-quark in the $T-Y$ associated production
process, this latter has a significantly lower cross section compared to the
$D-Y$ associated production. An analogous consideration applies to the case of
the $D^{2}-Y$ associated production, which depends on the $c$-quark PDF.
Nevertheless, identifying these specific associated productions, possibly by
applying a tailored analysis at a high luminosity, could give important
confirmation about the flavor structure of the 331 model. In the case of the
$D$ VLQ, the associated production with a bilepton constitutes up to $\sim$20%
of the total inclusive cross section for the bilepton pair production in a
large portion of the parameter space, in particular when $m_{Y}$ approaches
$m_{D}$ (cfr. Fig. 5). On top of their contribution to the inclusive bilepton
pair production, these channels could be considered for a specific exclusive
analysis. In this case, one could exploit the peculiarities of the VLQ-Y
associated production, as the presence in the final state of a heavy VLQ
(whose invariant mass could be reconstructed), a jet and bileptons with hard
$p_{T}$ and peculiar angular distributions, to select the signal from the
background. We leave this analysis for a future project.

Figure 4: Cross section for the associated production of a bilepton with a
VLQ. We consider the subsequent decay of the VLQ into a bilepton plus a quark,
and the bilepton decay into electron or muon pairs.  Figure 5:  Ratio of the
partial cross section for the bilepton associated production with a $D$ VLQ,
which subsequently decays into a second bilepton, i.e. the $D$-$Y$ $\to YYj$
associated production ($\sigma_{\rm partial}$) over the total $YY+X$ cross
section ($\sigma_{\rm tot}$).

Finally, another contribution to the inclusive bilepton pair production, which
we have included in our study, come from the QCD pair production of extra
VLQs, which both decay into a bilepton plus a jet: $pp\to({\rm VLQ}\to
Y\,q)\;({\rm VLQ}\to Yq)\to YYjj$, with VLQ=$D,T$ and $j=u,b$, see Figure 6.

Figure 6:  QCD pair production of VLQs, leading to a $YYjj$ final state.

##  5 Bilepton pair production at LHC

Given the phenomenological considerations expressed in the previous section,
we find particularly promising the search for bileptons at the LHC. We want to
focus in particular on the inclusive process
$pp\to\ell^{\pm}\ell^{\pm}\ell^{\mp}\ell^{\mp}$, where a pair of $Y$ is
produced on-shell and then decay to same sign leptons, allowing the presence
of extra jets in the final state. We observe that ${\rm
Br}(Y^{\pm\pm}\to\ell^{\pm}\ell^{\pm})\simeq 1$ because this is the only decay
channel at tree level with our mass choice. Our analysis consistently
considers all possible contributions to the inclusive bilepton pair
production, and includes the processes whose diagrams are shown in Figures (1,
3, 6). For our analysis, we have implemented, by means of Feynrules Alloul
_et al._ (2014), the 331 model ($A[\sqrt{3}]$) described in the previous
section in a Universal Feynrules Output model Degrande  _et al._ (2012), which
we use for simulations and numeric calculations in MadGraph5 Alwall  _et al._
(2014).

Figure 7:  Inclusive cross section for the process
$pp\to\ell^{\pm}\ell^{\pm}\ell^{\mp}\ell^{\mp}+X$ as function of the Y mass
and for different values of the VLQ masses, and in the case of decoupled VLQ.

In Figure (7) we provide the inclusive cross section for the process
$pp\to\ell^{\pm}\ell^{\pm}\ell^{\mp}\ell^{\mp}+X$ as function of the $Y$ mass
and for different values of the VLQs masses, and in the case of decoupled
VLQs. It is evident that processes involving VLQs which decay into bileptons
constitute the dominant contribution to the bilepton pair inclusive cross
section. Indeed, a large boost in the cross section, which reaches up to about
two orders of magnitude near the kinematic threshold $m_{VLQ}\simeq m_{Y}$, is
observed when the $VLQ\to Yq$ decays are on-shell. When these decays become
off-shell, the cross section reduces significantly, despite a residual
contribution from VLQ exchanged in the t-channel is still relevant.

A recent ATLAS search for doubly charged Higgs bosons, at the 13 TeV LHC with
139 fb-1, focus on multilepton final states with at least one same-sign lepton
pair Aad  _et al._ (2023c). This analysis is therefore sensitive also to the
bilepton pair production. We apply a recast of this analysis to extract a
lower limit on the $Y$ mass.

###  5.1 Reinterpretation of the ATLAS search for doubly charged Higgs bosons

We summarize in the following paragraphs the main selection criteria
considered in the ATLAS analysis Aad  _et al._ (2023c), also adopting their
notation. The search considers a final state with two (SR2L), three (SR3L) or
four (SR4L) leptons, either electrons or muons, passing triggering
requirements. The leptons must satisfy the transverse momentum $p_{T}$ and
rapidity $\eta$ requirements: $p_{T}>$ 40 GeV and $|\eta_{\mu}|<$2.5,
$|\eta_{e}|<$2.47 (for the electron, the region 1.37$<|\eta_{e}|<$1.52 is
vetoed). Electron candidates are discarded if their angular distance $\Delta
R=\sqrt{\Delta\phi^{2}+\Delta\eta^{2}}$ from a jet (with $p_{T}>$20 GeV and
$|\eta|<$2.5) satisfies 0.2$<\Delta R<$0.4. If a muon and a jet featuring more
than three tracks have a separation $\Delta R<$0.4 and the muon’s $p_{T}$ is
less than half of the jet’s $p_{T}$, the muon is discarded. Signal regions are
optimised to search for one same-charge lepton pair, in the case of two or
three leptons, while four-lepton events are required to feature two same-
charge pairs. The signal selection exploits, as the main variable to
distinguish signal events from the background, the invariant mass of the two
same-charge leptons with the highest $p_{T}$ in the event,
$m(\ell^{\pm},\ell^{\prime\pm})_{\rm lead}$, and applies a cut:

$m(\ell^{\pm},\ell^{\prime\pm})_{\rm lead}\geq 300\;{\rm GeV}\;.$ (53)

Furthermore, for SR2L and SR3L, the vector sum of the two leading leptons’
transverse momenta must exceed 300 GeV:

$p_{T}(\ell^{\pm},\ell^{\prime\pm})_{\rm lead}\geq 300\;{\rm
GeV}\;\;{\rm(SR2L,SR3L)};$ (54)

While, for SR4L, the average invariant mass of the two same-charge lepton
pairs must satisfy:

$\bar{m}=(m_{\ell^{+}\ell^{\prime+}}+m_{\ell^{-}\ell^{\prime-}})/2\geq
300\;{\rm GeV}\;\;{\rm(SR4L)}.$ (55)

On top of these cuts, for SR3L and SR4L, a $Z$-veto condition is required,
excluding the presence of same-charge lepton pairs with invariant mass 71
GeV$<m(\ell,\ell^{\prime})<$111.2 GeV. For SR2L, the requirement on the
separation $\Delta R(\ell^{\pm},\ell^{\prime\pm})_{\rm lead}<3.5$ is
considered. Finally, events containing jets identified as originating from
b-quarks are vetoed.

We apply the above selection to the signal of inclusive bilepton pair
production. Events are generated with Madgraph5 and then passed to Pythia8
Sjöstrand  _et al._ (2015) for showering and hadronization. Jets are clustered
by using Fastjet Cacciari  _et al._ (2012) with an anti-kt algorithm with
radius parameter $R=0.4$. Conservatively, based on the details of lepton
reconstruction procedures described in Aad  _et al._ (2023c), we also consider
the following identification efficiencies for an electron: 0.58, and for a
muon: 0.95.

We obtain the overall signal efficiencies reported in Table 5 for several
signal benchmark points (we also list the values for the separate efficiencies
in the SR4L and SR2L regions). The b-veto applied in the ATLAS analysis
completely suppresses the contribution from the VLQ T to the bilepton pair
production.

$m_{Y}$ (GeV) | 300 | 600 | 1000 | 1300
---|---|---|---|---
$m_{D}$=1.5 TeV | 0.14 ${}^{0.16}_{0.11}$ | 0.37 ${}^{0.37}_{0.11}$ | 0.36 ${}^{0.36}_{0.07}$ | 0.33 ${}^{0.32}_{0.11}$
$m_{D}$=2.0 TeV | 0.14 ${}^{0.13}_{0.10}$ | 0.38 ${}^{0.37}_{0.08}$ | 0.38 ${}^{0.37}_{0.07}$ | 0.37 ${}^{0.36}_{0.14}$
$m_{D}$=3.0 TeV | 0.13 ${}^{0.13}_{0.07}$ | 0.38 ${}^{0.38}_{0.10}$ | 0.38 ${}^{0.37}_{0.09}$ | 0.39 ${}^{0.38}_{0.08}$
$m_{D}\gg m_{Y}$ | 0.13 ${}^{0.13}_{0.06}$ | 0.38 ${}^{0.38}_{0.09}$ | 0.38 ${}^{0.37}_{0.09}$ | 0.39 ${}^{0.38}_{0.08}$
Table 5: Efficiencies of the inclusive bilepton pair production to the ATLAS
selection Aad  _et al._ (2023c) for several benchmark points of the minimal
331 model ($A[\sqrt{3}]$). We report the combined efficiencies for all of the
three signal regions considered in the analysis, together with the specific
efficiencies for the signal regions SR4L (upper values) and SR2L (lower
values). The efficiencies for SR3L are similar to those obtained for SR4L.

For all of the benchmark points analyzed, except the lowest mass point, we
find efficiencies for the bilepton pair process ($\epsilon_{YY}$) which are
comparable to those for doubly-charged Higgs bosons ($\epsilon_{HH}$) in the
SR3L and in the SR4L regions. For the SR2L region, we find in general slightly
lower efficiencies than those obtained for the charged Higgs signal. In order
to correctly compare the cross section for our process $pp\to
YY\to\ell^{\pm}\ell^{\pm}\ell^{\mp}\ell^{\mp}$ with the upper limit band on
the multi-lepton cross section reported in Fig. 8 of the ATLAS search Aad  _et
al._ (2023c), we consider a scaling of our cross section values by a factor
$\epsilon_{YY}/\epsilon_{HH}$. Conservatively, the scaling is not applied when
this factor is greater than 1. We show the result of the comparison in Fig. 8.
We find that in the full mass region covered by the ATLAS analysis, and for
all of the considered values of the mass of extra VLQs, the cross section for
the process of inclusive bilepton pair production lies above the curve
representing the 90% C.L. upper limit, with the lowest values obtained for the
scenario of decoupled VLQs. We can therefore extract, independently of VLQ
masses, the 90% C.L. bound on the bilepton mass:

$m_{Y}>1300\;{\rm GeV}$ (56)

This value supersedes the one obtained in Nepomuceno and Meirose (2020),
$m_{Y}>1200$ GeV, which was based on the results of a previous ATLAS search
for doubly charged Higgs bosons in multi-lepton final states with 36.1 fb-1 at
the 13 TeV LHC Aaboud  _et al._ (2018b). It is clear from Fig. 8 that the
search, with the increasing of the luminosity expected at the High-Luminosity
LHC phase, has a great potential to test even larger bilepton (and VLQs) mass
values, especially when the intertwined contribution of VLQs is taken into
account.

Figure 8: Alongside our prediction for the multi-lepton cross section
generated by the bilepton pair production in the minimal 331 model, we
illustrate the results obtained by ATLAS in Ref. Aad  _et al._ (2023c). In
particular, the solid and dashed lines represent the observed and expected 90%
CL upper limits on the multi-lepton cross section as a function of the
bilepton mass (doubly charged Higgs in the ATLAS analysis). The shaded green
and yellow bands represent the $1\sigma$ and $3\sigma$ uncertainty around the
combined expected limit.

##  6 Conclusions

We have considered a class of models, known as 331 models, characterized by
the electroweak gauge group $SU_{L}(3)\times U(1)$. We discuss their general
classification and detail the passages needed to engineer a 331 model
compatible with the SM, i.e. the choice of the scalar sector and the charge
assignment of the matter fields. Our next step has been to select a minimal
331 model satisfying four general requests: 1) all matter multiplets include
SM fermions, namely there are not multiplets including only BSM fermions; 2)
all gauge bosons have integer charges; 3) all right-handed fermions are
$SU(3)_{L}$ singlet and possess a left-handed counterpart 4) no gauge
anomalies. These assumptions constrain the number of families: one
possibility, although not the only one, is that it is a multiple of the color
number. The model we have chosen has $\beta=\sqrt{3}$, the first two
generations of quarks that transform as triplets under $SU_{L}(3)$, and the
third generation of quark, together with the three lepton generations, that
transform as anti-triplets. In addition to the usual SM fields, this model
accommodates some BSM fields, namely three VLQs, $D^{1,2}$ and $T$, one
neutral gauge boson $Z^{\prime}_{\mu}$ and two bileptons, $V^{\pm}_{\mu}$ and
$Y^{\pm\pm}_{\mu}$. The intertwined phenomenology of bileptons and VLQs looks
particularly promising for the search and characterization of 331 models at
the LHC. In our analysis, we propose bilepton pair signatures for 331 models,
which are also efficient as search channels for VLQs. In particular, we focus
on the inclusive production of a pair of doubly charged bileptons, followed by
the decay of each bilepton into a pair of same sign charged leptons, that is
the process $pp\to Y^{++}Y^{--}\to\ell^{+}\ell^{+}\,\ell^{-}\ell^{-}+{\rm X}$.
The final state of this inclusive process has multi-leptons (with at least one
same-sign lepton pair), and possibly extra jets.

In our analysis, we provide the values of the inclusive cross section for the
process $pp\to\ell^{+}\ell^{+}\,\ell^{-}\ell^{-}+{\rm X}$, as function of the
doubly charged bilepton mass. We include consistently the production of doubly
charged bileptons from gluon interaction, involving VLQs. A highly massive VLQ
can decay into a bilepton and a SM quark $q$, yielding a final state
$Y^{++}Y^{--}q$ or $Y^{++}Y^{--}qq$, which contribute to the inclusive process
under investigation. At the best of our knowledge, the production from gluon-
quark interaction of a bilepton associated to a VLQ has not been taken into
account in previous searches for inclusive or exclusive bilepton pair
production. We find that instead this contribution is significant in a large
portion of the model parameter space (up to $\sim 20\%$ of the total inclusive
cross section for the bilepton pair production).

A recent ATLAS search for doubly charged Higgs bosons focuses on multi-lepton
final states with at least one same-sign lepton pair Aad  _et al._ (2023c). We
apply a recast of this analysis to extract a lower limit on the $Y$ mass. We
found, independently of the VLQs masses, a 90% CL lower limit on the mass of
the doubly charged bilepton:

$m_{Y}>1.3\,\,\mathrm{TeV}.$ (57)

This value supersedes the previous one of $m_{Y}>1200$ GeV Nepomuceno and
Meirose (2020), based on 2017 ATLAS results.

We conclude by remarking that the channel we have investigated remains
relevant with the increase of luminosity expected during the High-Luminosity
LHC phase, due to its potential to test quite large bilepton (and VLQs) mass
values.

## Acknowledgements

SM and GR acknowledge the support of the research projects TAsP (Theoretical
Astroparticle Physics) and ENP (Exploring New Physics), funded by INFN
(Istituto Nazionale di Fisica Nucleare), respectively. The work of NV is
supported by ICSC – Centro Nazionale di Ricerca in High Performance Computing,
Big Data and Quantum Computing, funded by European Union – NextGenerationEU,
reference code CN_00000013.

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## Appendix

Here we report the decay widths of the VLQs introduced in the model:

$\displaystyle\Gamma(T\to Y^{++}d^{a})=\frac{g^{2}|{{\cal
V}_{L}}_{3a}|^{2}}{64\pi}\frac{m^{6}_{T}-3m^{2}_{T}m^{4}_{Y}+2m^{6}_{Y}}{m^{3}_{T}m^{2}_{Y}}\,,$
(58) $\displaystyle\Gamma(D^{i}\to Y^{--}u^{a})=\frac{g^{2}|{{\cal
U}_{L}}_{ia}|^{2}}{64\pi}\frac{m^{6}_{D}-3m^{2}_{D}m^{4}_{Y}+2m^{6}_{Y}}{m^{3}_{D}m^{2}_{Y}}\,,$
(59) $\displaystyle\Gamma(T\to V^{+}u^{a})=\frac{g^{2}|{{\cal
U}_{L}}_{3a}|^{2}}{64\pi}\frac{m^{6}_{T}-3m^{2}_{T}m^{4}_{V}+2m^{6}_{V}}{m^{3}_{T}m^{2}_{V}}\,,$
(60) $\displaystyle\Gamma(D^{i}\to V^{-}d^{a})=\frac{g^{2}|{{\cal
V}_{L}}_{ia}|^{2}}{64\pi}\frac{m^{6}_{D}-3m^{2}_{D}m^{4}_{V}+2m^{6}_{V}}{m^{3}_{D}m^{2}_{V}}\,,$
(61)

where $m_{X}$ stays for the mass of the $X$ field. The bilepton $Y$ decay
width is

$\Gamma(Y\to\ell^{\pm}\ell^{\pm})=\frac{g^{2}}{12\pi}m_{Y}\,.$ (62)

In the following tables we give the relevant vertices of the model:

Gauge boson-fermion interactions
---
Vertex | Coupling | Vertex | Coupling
$V$ boson |  $Y$ boson
| $\frac{g}{\sqrt{2}}\mathcal{U}_{L\,3a}$ |  | $-\frac{g}{\sqrt{2}}\mathcal{V}_{L\,3a}$
| $\frac{g}{\sqrt{2}}\mathcal{V}_{L\,ia}$ |  | $\frac{g}{\sqrt{2}}\mathcal{U}_{L\,ia}$
| $\frac{g}{\sqrt{2}}$ |  | $-\frac{g}{\sqrt{2}}$
$Z$ boson
| $\frac{g}{c_{w}}\left(\frac{1}{6}+\frac{\sqrt{3}}{2}\beta\right)s_{w}^{2}$ |  | $\frac{g}{c_{w}}\left(-\frac{1}{6}+\frac{\sqrt{3}}{2}\beta\right)s_{w}^{2}$
| $\frac{g}{2c_{w}}\left(1-\sqrt{3}\beta\right)s_{w}^{2}$ |  |
$Z^{\prime}$ boson
| $\frac{g\left[1-(1+\sqrt{3}\beta)s_{w}^{2}\right]}{2\sqrt{3}c_{w}\sqrt{1-(1+\beta^{2})s_{w}^{2}}}$ |  | $-\frac{g\beta s_{w}^{2}}{c_{w}\sqrt{1-(1+\beta^{2})s_{w}^{2}}}$
| $-\frac{g\left[2-\left(2-\sqrt{3}\beta(1-\sqrt{3}\beta)\right)s_{w}^{2}\right]}{2\sqrt{3}c_{w}\sqrt{1-(1+\beta^{2})s_{w}^{2}}}$ |  | $-\frac{g(1-\sqrt{3}\beta)\beta s_{w}^{2}}{2c_{w}\sqrt{1-(1+\beta^{2})s_{w}^{2}}}$
| $\frac{g\left[[-1+(\beta/\sqrt{3})s_{w}^{2}]\delta_{ab}+2c_{w}^{2}\mathcal{U}^{*}_{L\,3a}\mathcal{U}_{L\,3b}\right]}{2\sqrt{3}c_{w}\sqrt{1-(1+\beta^{2})s_{w}^{2}}}$ |  | $\frac{2g\beta s_{w}^{2}\delta_{ab}}{3c_{w}\sqrt{1-(1+\beta^{2})s_{w}^{2}}}$
| $\frac{g\left[[-1+(\beta/\sqrt{3})s_{w}^{2}]\delta_{ab}+2c_{w}^{2}\mathcal{V}^{*}_{L\,3a}\mathcal{V}_{L\,3b}\right]}{2\sqrt{3}c_{w}\sqrt{1-(1+\beta^{2})s_{w}^{2}}}$ |  | $-\frac{g\beta s_{w}^{2}\delta_{ab}}{3c_{w}\sqrt{1-(1+\beta^{2})s_{w}^{2}}}$
| $\frac{g\left[2+\left(-2\sqrt{3}+\beta(1-3\sqrt{3}\beta)\right)s_{w}^{2}/\sqrt{3}\right]}{2\sqrt{3}c_{w}\sqrt{1-(1+\beta^{2})s_{w}^{2}}}$ |  | $\frac{g\beta(1-3\sqrt{3}\beta)s_{w}^{2}}{6c_{w}\sqrt{1-(1+\beta^{2})s_{w}^{2}}}$
| $\frac{g\left[-2+\left(2\sqrt{3}+\beta(1-3\sqrt{3}\beta)\right)s_{w}^{2}/\sqrt{3}\right]}{2\sqrt{3}c_{w}\sqrt{1-(1+\beta^{2})s_{w}^{2}}}$ |  | $\frac{g\beta(1+3\sqrt{3}\beta)s_{w}^{2}}{6c_{w}\sqrt{1-(1+\beta^{2})s_{w}^{2}}}$