# Collider signatures of vector-like fermions from a flavor symmetric model

Cesar Bonilla<EMAIL_ADDRESS>Departamento de Física, Universidad
Católica del Norte, Avenida Angamos 0610, Casilla 1280, Antofagasta, Chile  A.
E. Cárcamo Hernández<EMAIL_ADDRESS>Departamento de Física,
Universidad Técnica Federico Santa María
Casilla 110-V, Valparaíso, Chile
Centro Científico-Tecnológico de Valparaíso, Casilla 110-V, Valparaíso, Chile
Millennium Institute for Subatomic Physics at the High-Energy Frontier,
SAPHIR, Chile  João Gonçalves<EMAIL_ADDRESS>Departamento de Física da
Universidade de Aveiro, 3810-183 Aveiro, Portugal
Centre for Research and Development in Mathematics and Applications (CIDMA),
3810-183 Aveiro, Portugal  Felipe F. Freitas<EMAIL_ADDRESS>Departamento
de Física da Universidade de Aveiro, 3810-183 Aveiro, Portugal
Centre for Research and Development in Mathematics and Applications (CIDMA),
3810-183 Aveiro, Portugal  António P. Morais<EMAIL_ADDRESS>Departamento de
Física da Universidade de Aveiro, 3810-183 Aveiro, Portugal
Centre for Research and Development in Mathematics and Applications (CIDMA),
3810-183 Aveiro, Portugal  R. Pasechnik<EMAIL_ADDRESS>Department
of Astronomy and Theoretical Physics, Lund University, Sölvegatan 14A, SE
223-62 Lund, Sweden

(August 28, 2024)

###### Abstract

We propose a model with two Higgs doublets and several $SU(2)$ scalar singlets
with a global non-Abelian flavor symmetry
$\mathcal{Q}_{6}\times\mathcal{Z}_{2}$. This discrete group accounts for the
observed pattern of fermion masses and mixing angles after spontaneous
symmetry breaking. In this scenario only the third generation of fermions get
their masses as in the Standard Model (SM). The masses of the remaining
fermions are generated through a seesaw-like mechanism. To that end, the
matter content of the model is enlarged by introducing electrically charged
vector-like fermions (VLFs), right handed Majorana neutrinos and several SM
scalar singlets. Here we study the processes involving VLFs that are within
the reach of the Large Hadron Collider (LHC). We perform collider studies for
vector-like leptons (VLLs) and vector-like quarks (VLQs), focusing on double
production channels for both cases, while for VLLs single production
topologies are also included. Utilizing genetic algorithms for neural network
optimization, we determine the statistical significance for a hypothetical
discovery at future LHC runs. In particular, we show that we can not safely
exclude VLLs for masses greater than $200\leavevmode\nobreak\ \mathrm{GeV}$.
For VLQ’s in our model, we show that we can probe their masses up to 3.8 TeV,
if we take only into account the high-luminosity phase of the LHC. Considering
Run-III luminosities, we can also exclude VLQs for masses up to
$3.4\leavevmode\nobreak\ \mathrm{TeV}$. We also show how the model with
predicted VLL masses accommodates the muon anomalous magnetic moment.

##  I Introduction

The Standard Model (SM) provides a successful framework to describe three out
of four known fundamental forces of nature, i.e. the electromagnetic, nuclear
strong and weak interactions. However, it does not account for the number of
fermion generations and lacks a natural explanation for the tremendous
hierarchy in the fermion sector, which is extended over a range of 13 orders
of magnitude from the light active neutrino mass scale up to the top quark
mass. Moreover, there is no assertion for the smallness of the quark mixing
angles, which is in contrast with the sizable values of two of the three
leptonic mixing angles. This set of issues is the so called flavor problem
which, among others, motivates the construction of models where the SM
particle content and symmetries are enlarged. One way to tackle the flavor
problem is offered in SM extensions that assume the existence of discrete
flavor symmetries111For more details about flavor symmetry groups see, for
example King and Luhn (2013); Altarelli and Feruglio (2010); Ishimori  _et
al._ (2010); King (2015). These scenarios feature relations in the Yukawa
sector and typically predict correlations between the observed fermion mixing
patterns as well as the fermion mass relations.

With the use of flavor symmetries it can be suggested that the existing number
of fermion families is because they transform as components of a three-
dimensional irreducible representation (irrep) of a non-Abelian discrete
group, such as $A_{4}$. Another option is to have the heaviest fermion
transforming as one-dimensional irrep and the lighter ones being the
components of a doublet under the symmetry group. The smaller non-Abelian
discrete groups, containing one- and two-dimentional irreps, are Ishimori  _et
al._ (2010): $S_{3}$, $Q_{4}$, $D_{4}$ and $Q_{6}$222Pioneer works using these
symmetries to tackle the flavor problem can be found in Gerard (1983);
Frampton and Kephart (1995); Grimus and Lavoura (2003); Kubo  _et al._ (2003);
Kubo (2004); Babu and Kubo (2005); Kajiyama  _et al._ (2006); Lovrekovic
(2012). Both assumptions point to the idea on how to account for the three
generations of quarks and leptons but not for their (very strong) mass
hierarchies.

For this reason, apart of the flavor symmetry, the distribution of the fermion
mass spectrum could suggest the existence of new particles, resulting in
phenomenologically richer setups. One can add more scalars to the SM with non-
zero vacuum expectation values (vevs) whose contributions to the fermion
masses or to different mass matrix elements are restricted by the additional
symmetry. Similar to the role of a $\mathcal{Z}_{2}$ symmetry in a 2-Higgs
doublet model (2HDM) Branco  _et al._ (2012). One fashionable approach to
explain the fermion mass and mixing hierarchies is by using the
Froggatt–Nielsen (FN) mechanism Froggatt and Nielsen (1979), where vector-like
fermions (VLFs) are introduced to the SM and transform under a new
$\mathrm{U(1)_{F}}$ symmetry which is spontaneously broken by the vevs of
$\mathrm{SU(2)}$ scalar singlets (flavons). The new energy scale is much
bigger than the electroweak one as well as the VLFs are heavier than the SM
ones, then all the new fields are effectively integrated out.

Here, we present a framework that somehow gathers the previous ideas. We
consider a multiscalar model with the flavor symmetry
$\mathcal{Q}_{6}\times\mathcal{Z}_{2}$, where the $\mathcal{Z}_{2}$ symmetry
assigns one Higgs to the up-type fermion sector and the other to the down-
type, both scalars are $\mathcal{Q}_{6}$ singlets. In contrast, fermions
transform as ($doublet$+$singlet$) under $\mathcal{Q}_{6}$, preventing the
Yukawa interaction between light fermions and the $\mathrm{SU(2)}$ scalar
doublets. Therefore, one Higgs doublet furnishes the top quark with a no-null
mass, whereas the other one generates the bottom quark and tau mass. In order
to generate the mass of light fermions through a seesaw (or FN) mechanism, we
introduce VLFs and three different flavons with a non-trivial transformation
under the flavor symmetry. We also introduce right-handed (RH) Majorana
neutrinos to generate the small neutrino masses via a type-I seesaw mechanism.

In contrast to the FN mechanism, in our model, the VLFs are not decoupled. For
this reason, we study the processes involving VLFs that are within the reach
of the Large Hadron Collider (LHC). We perform collider studies for vector-
like leptons (VLLs) and vector-like quarks (VLQs), focusing on double
production channels for both cases, while for VLLs the single production
topologies are also included. Furthermore, it has been known that the
experimentally measured muon anomalous magnetic moment deviates from the SM
prediction. The longstanding non-compliance of the muon $(g-2)$ with the SM
was first observed by the Brookhaven E821 experiment at BNL Bennett  _et al._
(2006) and it has been recently confirmed by the Muon $(g-2)$ experiment at
FERMILAB Abi  _et al._ (2021). Hence, we also show how the muon $(g-2)$
anomaly is accommodated within our theory with the predicted VLLs.

The layout of the remainder of the paper is as follows. In Sec. II we describe
the model, i.e. we provide the invariant Yukawa Lagrangian, the scalar
potential and the particle mass spectrum. Afterwards, in Sec. III, the
consequences of our model in the muon anomalous magnetic moment are analyzed.
In Sec. IV we detail the methodology for the collider analysis of the VLFs,
for both the quark and lepton counterparts, with the numerical results being
showcased in the Sec. V. We finalize this paper with Sec. VI, where we take
our conclusions.

##  II Model Description

We propose a model where the SM gauge group is extended with a global flavor
symmetry group, i.e. the complete description is given by the symmetry,
$\mathrm{SU(3)_{C}\times SU(2)_{L}\times
U(1)_{Y}}\times\mathcal{Q}_{6}\times\mathcal{Z}_{2}$. This theory adds to the
SM particle content, a second $\mathrm{SU(2)}$ scalar doublet, three flavon
fields, two RH neutrinos, a flavor doublet of VLQs and flavor doublet of VLLs.
The charge assignments of the particle content under the flavor group are
shown in Tables 1 and 2.

| $H_{1}$ | $H_{2}$ | $Q_{L_{D}}$ | $Q_{L_{3}}$ | $u_{R_{D}}$ | ${u}_{R_{3}}$ | $d_{R_{D}}$ | ${d}_{R_{3}}$ | $L_{L_{D}}$ | $L_{L_{3}}$ | ${\ell}_{R_{D}}$ | $\ell_{R_{3}}$
---|---|---|---|---|---|---|---|---|---|---|---|---
$SU(2)_{L}$ | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1
$U(1)_{Y}$ | 1/2 | 1/2 | 1/6 | 1/6 | 2/3 | 2/3 | -1/3 | -1/3 | -1/2 | -1/2 | -1 | -1
$\mathcal{Q}_{6}$ | $\mathbf{1}_{+-}$ | $\mathbf{1}_{+-}$ | $\mathbf{2}_{2}$ | $\mathbf{1}_{++}$ | $\mathbf{2}_{2}$ | $\mathbf{1_{+-}}$ | $\mathbf{2}_{2}$ | $\mathbf{1}_{+-}$ | $\mathbf{2}_{2}$ | $\mathbf{1}_{++}$ | $\mathbf{2}_{2}$ | $\mathbf{1}_{+-}$
$\mathcal{Z}_{2}$ | $-1$ | $+1$ | $+1$ | $+1$ | $-1$ | $-1$ | $+1$ | $+1$ | $+1$ | $+1$ | $+1$ | $+1$
Table 1: Charge assignments of the SU(2) scalar doublets and SM fermions under the symmetry, $\mathcal{Q}_{6}\times\mathcal{Z}_{2}$. We have arranged the $\mathcal{Q}_{6}$ doublets as, $Q_{L_{D}}\equiv(Q_{L_{1}},Q_{L_{2}})^{T}$, $u_{R_{D}}\equiv(u_{R_{1}},{u}_{R_{2}})^{T}$, $d_{R_{D}}\equiv({d}_{R_{1}},{d}_{R_{2}})^{T}$, $L_{L_{D}}\equiv(L_{L_{1}},L_{L_{2}})^{T}$ and ${\ell}_{R_{D}}\equiv({\ell}_{R_{1}},{\ell}_{R_{2}})^{T}$.  | $\sigma_{1}$ | $\sigma_{2}$ | $\xi$ | $N_{R_{1}}$ | $N_{R_{2}}$ | $T_{L}$ | $T_{R}$ | $B_{L}$ | $B_{R}$ | $E_{L}$ | $E_{R}$
---|---|---|---|---|---|---|---|---|---|---|---
$SU(2)_{L}$ | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1
$U(1)_{Y}$ | 0 | 0 | 0 | 0 | 0 | 2/3 | 2/3 | -1/3 | -1/3 | -1 | -1
$\mathcal{Q}_{6}$ | $\mathbf{1}_{++}$ | $\mathbf{1}_{+-}$ | $\mathbf{2}_{2}$ | $\mathbf{1}_{+-}$ | $\mathbf{1}_{+-}$ | $\mathbf{2}_{1}$ | $\mathbf{2}_{1}$ | $\mathbf{2}_{1}$ | $\mathbf{2}_{1}$ | $\mathbf{2}_{1}$ | $\mathbf{2}_{1}$
$\mathcal{Z}_{2}$ | $-1$ | $-1$ | $-1$ | $+1$ | $+1$ | $+1$ | $-1$ | $-1$ | $+1$ | $-1$ | $+1$
Table 2: Assignments of the singlet scalars and exotic fermions under the
$\mathcal{Q}_{6}$ flavor symmetry irreps. For convenience we have not included
a subindex $D$ for $\mathcal{Q}_{6}$ doublets.

Given the matter content in our theory, the invariant Yukawa Lagrangian is
formed by the contribution from each sector as,

$\mathcal{L}_{Y}=\mathcal{L}_{u}+\mathcal{L}_{d}+\mathcal{L}_{\ell}+\mathcal{L}_{\nu}$
(1)

where

$\mathcal{L}_{u}=y_{u3}\overline{Q}_{L_{3}}u_{R_{3}}\widetilde{H}_{1}+y_{T}\overline{Q}_{D}T_{R}\widetilde{H}_{1}+y_{T1}\overline{T}_{L}T_{R}\sigma_{1}+y_{T2}\overline{T}_{L}u_{R_{D}}\sigma_{2}+y_{T3}\overline{T}_{L}u_{R_{3}}\xi+y_{T4}\overline{T}_{L}T_{R}\xi+\mathrm{H.c.},$
(2)
$\mathcal{L}_{d}=y_{d3}\overline{Q}_{L_{3}}d_{R_{3}}H_{2}+y_{B}\overline{Q}_{D}B_{R}H_{2}+y_{B1}\overline{B}_{L}B_{R}\sigma_{1}+y_{B2}\overline{B}_{L}d_{R_{D}}\sigma_{2}+y_{B3}\overline{B}_{L}d_{R_{3}}\xi+y_{B4}\overline{B}_{L}B_{R}\xi+\mathrm{H.c.},$
(3) $\mathcal{L}_{\ell}=y_{\ell
3}\overline{L}_{L_{3}}\ell_{R_{3}}H_{2}+y_{E}\overline{L}_{L_{D}}E_{R}H_{2}+y_{E1}\overline{E}_{L}E_{R}\sigma_{1}+y_{E2}\overline{E}_{L}\ell_{R_{D}}\sigma_{2}+y_{E3}\overline{E}_{L}\ell_{R_{3}}\xi+y_{E4}\overline{E}_{L}E_{R}\xi+\mathrm{H.c.},$
(4)

and

$\mathcal{L}_{\nu}=\sum_{i=1}^{2}\frac{1}{\Lambda}\left(y_{\nu_{i}}\overline{L}_{L_{3}}N_{R_{i}}\widetilde{H}_{1}\sigma_{1}+y^{\prime}_{\nu_{i}}\overline{L}_{L_{D}}N_{R_{i}}\widetilde{H}_{1}\xi\right)+\sum_{i=1}^{2}M_{R_{i}}N_{R_{i}}\overline{N_{R_{i}}^{C}}+\mathrm{H.c.}$
(5)

with $\tilde{H}_{a}=i\sigma_{2}H^{*}_{a}$. We have defined the
$\mathcal{Q}_{6}$ doublets as, $Q_{L_{D}}\equiv(Q_{L_{1}},Q_{L_{2}})^{T}$,
$u_{R_{D}}\equiv(u_{R_{1}},{u}_{R_{2}})^{T}$,
$d_{R_{D}}\equiv({d}_{R_{1}},{d}_{R_{2}})^{T}$,
$L_{L_{D}}\equiv(L_{L_{1}},L_{L_{2}})^{T}$,
${\ell}_{R_{D}}\equiv({\ell}_{R_{1}},{\ell}_{R_{2}})^{T}$,
$T_{L,R}\equiv(T_{L_{1},R_{1}},T_{L_{2},R_{2}})^{T}$ and
$B_{L,R}\equiv(B_{L_{1},R_{1}},B_{L_{2},R_{2}})^{T}$. Using the multiplication
rules of $\mathcal{Q}_{6}$ given in Appendix A, the above Yukawa interactions
can be rewritten as follows

$\displaystyle\mathcal{L}_{u}$ $\displaystyle=$ $\displaystyle
y_{u3}\overline{Q}_{L_{3}}u_{R_{3}}\widetilde{H}_{1}+y_{T}\left(\overline{Q}_{L_{1}}T_{R_{1}}-\overline{Q}_{L_{2}}T_{R_{2}}\right)\widetilde{H}_{1}+y_{T1}\left(\overline{T}_{L_{1}}T_{R_{2}}-\overline{T}_{L_{2}}T_{R_{1}}\right)\sigma_{1}$
(6) $\displaystyle+$ $\displaystyle
y_{T2}\left(\overline{T}_{L_{1}}u_{R_{1}}-\overline{T}_{L_{2}}u_{R_{2}}\right)\sigma_{2}+y_{T3}\left(\overline{T}_{L_{1}}\xi_{1}-\overline{T}_{L_{2}}\xi_{2}\right)u_{R_{3}}+y_{T4}\left(\overline{T}_{L_{1}}T_{R_{1}}\xi_{2}-\overline{T}_{L2}T_{R_{2}}\xi_{1}\right)$

$\displaystyle\mathcal{L}_{d}$ $\displaystyle=$ $\displaystyle
y_{d3}\overline{Q}_{L_{3}}d_{R_{3}}H_{2}+y_{B}\left(\overline{Q}_{L_{1}}B_{R_{1}}-\overline{Q}_{L_{2}}B_{R_{2}}\right)H_{2}+y_{B1}\left(\overline{B}_{L_{1}}B_{R_{2}}-\overline{B}_{L_{2}}B_{R_{1}}\right)\sigma_{1}$
(7) $\displaystyle+$ $\displaystyle
y_{B2}\left(\overline{B}_{L_{1}}d_{R_{1}}-\overline{B}_{L_{2}}d_{R_{2}}\right)\sigma_{2}+y_{B3}\left(\overline{B}_{L_{1}}\xi_{1}{-}\overline{B}_{L_{2}}\xi_{2}\right)d_{R_{3}}+y_{B4}\left(\overline{B}_{L_{1}}B_{R_{1}}\xi_{2}-\overline{B}_{L_{2}}B_{R_{1}}\xi_{1}\right)+\mathrm{H.c.},$

$\displaystyle\mathcal{L}_{\ell}$ $\displaystyle=$ $\displaystyle y_{\ell
3}\overline{L}_{L_{3}}\ell_{R_{3}}H_{2}+y_{E}\left(\overline{L}_{L_{1}}E_{R_{1}}-\overline{L}_{L_{2}}E_{R_{2}}\right)H_{2}+y_{E1}\left(\overline{E}_{L_{1}}E_{R_{2}}-\overline{E}_{L_{2}}E_{R_{1}}\right)\sigma_{1}$
(8) $\displaystyle+$ $\displaystyle
y_{E2}\left(\overline{E}_{L_{1}}\ell_{R_{1}}-\overline{E}_{L_{2}}\ell_{R_{2}}\right)\sigma_{2}+y_{E3}\left(\overline{E}_{L_{1}}\xi_{1}{-}\overline{E}_{L_{2}}\xi_{2}\right)\ell_{R_{3}}+y_{E4}\left(\overline{E}_{L_{1}}E_{R_{1}}\xi_{2}-\overline{E}_{L_{2}}E_{R_{2}}\xi_{1}\right)+\mathrm{H.c.},$

$\mathcal{L}_{\nu}=\sum_{i=1}^{2}\frac{1}{\Lambda}\left[y_{\nu_{i}}\overline{L}_{L_{3}}N_{R_{i}}\widetilde{H}_{1}\sigma_{1}+y^{\prime}_{\nu_{i}}\left(\overline{L}_{L_{1}}N_{R_{i}}\widetilde{H}_{1}\xi_{1}-\overline{L}_{L_{2}}N_{R_{i}}\widetilde{H}_{1}\xi_{2}\right)\right]+\sum_{i=1}^{2}M_{R_{i}}N_{R_{i}}\overline{N_{R_{i}}^{C}}+\mathrm{H.c.}$
(9)

In addition, the invariant scalar potential reads

$V=V_{\text{2HDM}}+V(\text{$H_{1}$,$H_{2}$,flavons})$ (10)

where the first term corresponds to the 2HDM potential The second term in eq.
(10) has the contributions from the flavons fields, $\sigma_{1}$, $\sigma_{2}$
and $\xi$, i.e. the interactions among them and with the $\mathrm{SU(2)}$
scalar doublets. Explicitly,

$\displaystyle V_{\text{2HDM}}$ $\displaystyle=$
$\displaystyle-\mu_{1}^{2}\left(H_{1}^{\dagger}H_{1}\right)-\mu_{2}^{2}\left(H_{2}^{\dagger}H_{2}\right)+\frac{\lambda_{1}}{2}\left(H_{1}^{\dagger}H_{1}\right)^{2}$
(11) $\displaystyle+$
$\displaystyle\frac{\lambda_{2}}{2}\left(H_{2}^{\dagger}H_{2}\right)^{2}+\lambda_{3}\left(H_{1}^{\dagger}H_{1}\right)\left(H_{2}^{\dagger}H_{2}\right)+\lambda_{4}\left(H_{1}^{\dagger}H_{2}\right)\left(H_{2}^{\dagger}H_{1}\right)+\frac{\lambda_{5}}{2}\left[\left(H_{1}^{\dagger}H_{2}\right)^{2}+\mathrm{H.c.}\right]$

where $H_{i}=\left(\Phi^{+}_{i},\Phi^{0}_{i}\right)^{T}$ with $i=1,2$ and

$\displaystyle V(\text{$H_{1}$,$H_{2}$,flavons})$ $\displaystyle=$
$\displaystyle-\mu_{3}^{2}\sigma_{1}^{*}\sigma_{1}-\mu_{4}^{2}\sigma_{2}^{*}\sigma_{2}-\mu_{5}^{2}\xi_{1}^{*}\xi_{1}-\mu_{6}^{2}\xi_{2}^{*}\xi_{2}-\mu_{7}^{2}(\xi_{1}^{*}\xi_{2}+\mathrm{H.c.})-\mu_{8}\left(\sigma_{1}H_{1}^{\dagger}H_{2}+\mathrm{H.c.}\right)+\lambda_{6}\left(H_{1}^{\dagger}H_{1}\right)(\sigma_{1}^{*}\sigma_{1})$
(12) $\displaystyle+$
$\displaystyle\lambda_{7}\left(H_{2}^{\dagger}H_{2}\right)(\sigma_{1}^{*}\sigma_{1})+\lambda_{8}\left(H_{1}^{\dagger}H_{1}\right)(\sigma_{2}^{*}\sigma_{2})+\lambda_{9}\left(H_{2}^{\dagger}H_{2}\right)(\sigma_{2}^{*}\sigma_{2})+\frac{\lambda_{10}}{2}(\sigma_{1}^{*}\sigma_{1})^{2}+\frac{\lambda_{11}}{2}(\sigma_{2}^{*}\sigma_{2})^{2}$
$\displaystyle+$
$\displaystyle\lambda_{12}(\sigma_{1}^{*}\sigma_{1})(\sigma_{2}^{*}\sigma_{2})+\lambda^{{}^{\prime}}_{12}(\sigma_{1}^{*}\sigma_{2})(\sigma_{2}^{*}\sigma_{1})+\lambda^{{}^{\prime\prime}}_{12}\left[(\sigma_{1}^{*}\sigma_{2})^{2}+\mathrm{H.c.}\right]+\frac{\lambda_{13}}{2}\left(\xi^{*}\xi\right)_{\mathbf{1}_{--}}\left(\xi^{*}\xi\right)_{\mathbf{1}_{--}},$

The terms $\mu_{5,6,7}$ in the last equation break softly the
$\mathcal{Q}_{6}$ symmetry333$\xi$ does not mix with the other scalars because
of the $\mathcal{Q}_{6}$ symmetry. We have, for real $\xi$,
$(\xi^{2})_{++}=\xi_{1}\xi_{2}-\xi_{2}\xi_{1}=0$ and, if complex,
$(\xi^{*}\xi)_{++}+\mathrm{H.c.}=0$. and prevent the appearance of either
Goldstone or tachyonic fields. Since the flavons are real fields, they have no
complex charge assignment, the $\lambda_{12}$, $\lambda^{\prime}_{12}$ and
$\lambda^{\prime\prime}_{12}$ terms are equivalent. Then, eq. (12) can be
rewritten (discarding redundant terms) as follows,

$\displaystyle V(\text{$H_{1}$,$H_{2}$,flavons})$ $\displaystyle=$
$\displaystyle-\mu_{3}^{2}\sigma_{1}^{2}-\mu_{4}^{2}\sigma_{2}^{2}-\mu_{5}^{2}\xi_{1}^{2}-\mu_{6}^{2}\xi_{2}^{2}-\mu_{7}^{2}\xi_{1}\xi_{2}-\mu_{8}\left(\sigma_{1}H_{1}^{\dagger}H_{2}+\mathrm{H.c.}\right)+\lambda_{6}\left(H_{1}^{\dagger}H_{1}\right)\sigma_{1}^{2}+\lambda_{7}\left(H_{2}^{\dagger}H_{2}\right)\sigma_{1}^{2}$
(13) $\displaystyle+$
$\displaystyle\lambda_{8}\left(H_{1}^{\dagger}H_{1}\right)\sigma_{2}^{2}+\lambda_{9}\left(H_{2}^{\dagger}H_{1}\right)\sigma_{2}^{2}+\frac{\lambda_{10}}{2}\sigma_{1}^{4}+\frac{\lambda_{11}}{2}\sigma_{2}^{4}+\lambda_{12}\sigma_{1}^{2}\sigma_{2}^{2}+\frac{\lambda_{13}}{2}\left(2\xi_{1}\xi_{2}\right)^{2},$

All these scalars contribute to the symmetry breaking, they get a no-null vev,
and they are shifted as follows

$\Phi_{i}^{0}=\frac{1}{\sqrt{2}}\left(v_{i}+\varphi_{R_{i}}+i\varphi_{I_{i}}\right),\sigma_{i}=\frac{1}{\sqrt{2}}\left(v_{\sigma_{i}}+\sigma_{R_{i}}\right)\
\ \text{and}\ \
\xi_{i}=\frac{1}{\sqrt{2}}\left(v_{\xi_{i}}+\xi_{R_{i}}\right),$ (14)

where $i=1,2$, the $SU(2)$ scalar vevs satisfy $v^{2}_{1}+v^{2}_{2}=v_{\rm
EW}^{2}$ and $v_{\rm EW}\equiv 246\,\text{GeV}$.

### Fermion mass spectrum

After the spontaneous breaking of the $\mathrm{SU(3)_{C}\times SU(2)_{L}\times
U(1)_{Y}\times\mathcal{Q}_{6}\times\mathcal{Z}_{2}}$ symmetry, using eqs.(2),
(3) and (4), we get $5\times 5$ fermion mass matrices,

$\displaystyle M_{f}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(\begin{array}[]{ccccc}0&0&0&y_{F}v_{H_{1}}&0\\\
0&0&0&0&-y_{F}v_{H_{1}}\\\ 0&0&y_{f3}v_{H_{1}}&0&0\\\
y_{F2}v_{\sigma_{2}}&0&y_{F3}v_{\xi_{1}}&y_{F4}v_{\xi_{2}}&y_{F1}v_{\sigma_{1}}\\\
0&-y_{F2}v_{\sigma_{2}}&-y_{F3}v_{\xi_{2}}&-y_{F1}v_{\sigma_{1}}&-y_{F4}v_{\xi_{1}}\end{array}\right)=\left(\begin{array}[]{cc}C_{f}&A_{f}\\\
B_{f}&M_{F}\end{array}\right),$ (22)

where the sub-indices $f=u,d,\ell$ and $F=T,B,E$. The block matrices, in the
previous equation, are defined as

$C_{f}=\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&0\\\
0&0&M_{f_{33}}\end{array}\right),\ \
A_{f}=\left(\begin{array}[]{cc}M_{f_{14}}&0\\\ 0&M_{f_{25}}\\\
0&0\end{array}\right),\ \
B_{f}=\left(\begin{array}[]{ccc}M_{f_{41}}&0&M_{f_{43}}\\\
0&M_{f_{52}}&M_{f_{53}}\end{array}\right),$ (23)

and

$M_{F}=\left(\begin{array}[]{cc}M_{f_{44}}&M_{f_{45}}\\\
M_{f_{54}}&M_{f_{55}}\end{array}\right)$ (24)

The mass matrices in eq.(22) are diagonalized via the following bi-unitary
transformation:

$\left(U_{L}^{f}\right)^{\dagger}M_{f}U_{R}^{f}=\text{diag}\left(m_{f_{1}},m_{f_{2}},m_{f_{3}},m_{F_{1}},m_{F_{2}}\right),$
(25)

where $f=u,d,\ell$ and $F=T,B,E$.

On can notice, from eq.(23), that only the third fermion family gets the mass
through the Yukawa interaction with one of the two Higgs doublets. That is,
the top quark gets tree-level mass from its Yukawa interaction with $H_{1}$,
whereas the bottom quark and tau lepton obtain their masses from their Yukawa
interactions with the second Higgs doublet $H_{2}$. We will assume that the
symmetry breaking and the masses of the VLFs are around the TeV scale. Thus,
the mass matrices for the SM charged fermions, resulting from a seesaw-like
(or FN-like) mechanism, are given by

$\displaystyle\widetilde{M}_{f}$ $\displaystyle=$ $\displaystyle
C_{f}-A_{f}M_{F}^{-1}B_{f}=\frac{M_{f_{14}}}{M_{f_{45}}^{2}+M_{f_{55}}M_{f_{44}}}\left(\begin{array}[]{ccc}-M_{f_{55}}M_{f_{41}}&-M_{f_{45}}M_{f_{41}}&-M_{f_{55}}M_{f_{43}}-M_{f_{45}}M_{f_{53}}\\\
M_{f_{45}}M_{f_{41}}&-M_{f_{44}}M_{f_{41}}&M_{f_{45}}M_{f_{43}}+M_{f_{44}}M_{f_{53}}\\\
0&0&M_{f_{33}}\end{array}\right)$ (29)

where $f=u,d,\ell$, $F=T,B,E$ and we have used $M_{f_{25}}=-M_{f_{14}}$,
$M_{f_{54}}=-M_{f_{45}}$, $M_{f_{52}}=-M_{f_{41}}$.

Furthermore, from the neutrino Yukawa interactions, we obtain a $5\times 5$
neutrino mass matrix given by

$M_{\nu}=\left(\begin{array}[]{cc}0_{3\times 3}&M_{D_{\nu}}\\\
M_{D_{\nu}}^{T}&M_{R}\end{array}\right),$ (30)

where $M_{D_{\nu}}$ is the Dirac mass matrix and $M_{R}$ is the Majorana mass
matrix for RH neutrinos. These matrices are

$M_{D_{\nu}}=\left(\begin{array}[]{cc}A_{\nu}&C_{\nu}\\\
-\tilde{A}_{\nu}&-\tilde{C}_{\nu}\\\
B_{\nu}&D_{\nu}\end{array}\right),\quad\quad
M_{R}=\left(\begin{array}[]{cc}M_{R_{1}}&0\\\ 0&M_{R_{2}}\end{array}\right),$
(31)

with $A_{\nu}=y^{\prime}_{\nu_{1}}v_{1}v_{\xi_{1}}/2\Lambda$,
$\tilde{A}_{\nu}=y^{\prime}_{\nu_{1}}v_{1}v_{\xi_{2}}/2\Lambda$,
$C_{\nu}=y^{\prime}_{\nu_{2}}v_{1}v_{\xi_{1}}/2\Lambda$,
$\tilde{C}_{\nu}=y^{\prime}_{\nu_{2}}v_{1}v_{\xi_{2}}/2\Lambda$,
$B_{\nu}=y_{\nu_{1}}v_{1}v_{\sigma_{1}}/2\Lambda$ and
$D_{\nu}=y_{\nu_{2}}v_{1}v_{\sigma_{1}}/2\Lambda$. Assuming that the right
handed Majorana neutrinos have masses much larger than the electroweak
symmetry breaking scale $v_{\rm EW}=246$ GeV, the type I seesaw mechanism can
be implemented to generate the tiny masses of the light active neutrinos. The
resulting mass matrix for light active neutrinos takes the form

$\displaystyle\widetilde{M}_{\nu}=M_{D_{\nu}}M_{R}^{-1}M_{D_{\nu}}^{T}$ (32)

Then, the light active neutrino masses are given by

$\displaystyle m_{\nu_{1}}=0,\quad\quad
m_{\nu_{2},\nu_{3}}=\frac{\kappa\pm\kappa^{\prime 2}}{2M_{R_{1}}M_{R_{2}}},$
(33)

where we have defined

$\displaystyle\kappa=\tilde{C}_{\nu}^{2}M_{R_{1}}+C_{\nu}^{2}M_{R_{1}}+D_{\nu}^{2}M_{R_{1}}+\tilde{A}_{\nu}^{2}M_{R_{2}}+A_{\nu}^{2}M_{R_{2}}+B_{\nu}^{2}M_{R_{2}},$
(34) $\displaystyle\kappa^{\prime
2}=-\tilde{C}_{\nu}^{2}M_{R_{1}}-C_{\nu}^{2}M_{R_{1}}-D_{\nu}^{2}M_{R_{1}}-\tilde{A}_{\nu}^{2}M_{R_{2}}-A_{\nu}^{2}M_{R_{2}}-B_{\nu}^{2}M_{R_{2}}^{2})^{2}-$
$\displaystyle\hskip
23.99997pt-4A_{\nu}^{2}\tilde{C}_{\nu}^{2}M_{R_{1}}M_{R_{2}}+4B_{\nu}^{2}\tilde{C}_{\nu}^{2}M_{R_{1}}M_{R_{2}}-8\tilde{A}_{\nu}A_{\nu}\tilde{C}_{\nu}C_{\nu}M_{R_{1}}M_{R_{2}}+$
$\displaystyle\hskip
23.99997pt+4\tilde{A_{\nu}}^{2}C_{\nu}^{2}M_{R_{1}}M_{R_{2}}+4B_{\nu}^{2}C_{\nu}^{2}M_{R_{1}}M_{R_{2}}-8\tilde{A}_{\nu}B_{\nu}\tilde{C}_{\nu}D_{\nu}M_{R_{1}}M_{R_{2}}+$
$\displaystyle\hskip
23.99997pt+8A_{\nu}B_{\nu}C_{\nu}D_{\nu}M_{R_{1}}M_{R_{2}}+4\tilde{A}_{\nu}^{2}D_{\nu}^{2}M_{R_{1}}M_{R_{2}}+A_{\nu}^{2}D_{\nu}^{2}M_{R_{1}}M_{R_{2}}.$

### Scalar mass spectrum

Using eqs. (11) and (13) and solving the tadpole equations, the CP-even
squared mass matrix becomes,

$\displaystyle M^{2}_{\text{CP-
even}}=\left(\begin{array}[]{cc}[M^{2}]_{4\times 4}&0_{4\times 2}\\\
0_{2\times 4}&[\bar{M}^{2}]_{2\times 2}\end{array}\right)$ (37)

where

$\displaystyle
M^{2}=\left(\begin{array}[]{cccc}\lambda_{1}v_{1}^{2}&\left(\lambda_{3}+\lambda_{4}+\lambda_{5}\right)v_{1}v_{2}&\lambda_{6}v_{1}v_{\sigma_{1}}&\lambda_{8}v_{1}v_{\sigma_{2}}\\\
\left(\lambda_{3}+\lambda_{4}+\lambda_{5}\right)v_{1}v_{2}&\lambda_{2}v_{2}^{2}&\lambda_{7}v_{2}v_{\sigma_{1}}&\lambda_{9}v_{2}v_{\sigma_{2}}\\\
\lambda_{6}v_{1}v_{\sigma_{1}}&\lambda_{7}v_{2}v_{\sigma_{1}}&\lambda_{10}v_{\sigma_{1}}^{2}&\lambda_{12}v_{\sigma_{1}}v_{\sigma_{2}}\\\
\lambda_{8}v_{1}v_{\sigma_{2}}&\lambda_{9}v_{2}v_{\sigma_{2}}&\lambda_{12}v_{\sigma_{1}}v_{\sigma_{2}}&\lambda_{11}v_{\sigma_{2}}^{2}\\\
\end{array}\right)$ (42)

and

$\displaystyle\bar{M}^{2}=\left(\begin{array}[]{cc}\frac{\mu_{7}^{2}v_{\xi_{2}}}{2v_{\xi_{1}}}&2\lambda_{13}v_{\xi_{1}}v_{\xi_{2}}-\frac{\mu_{7}^{2}}{2}\\\
2\lambda_{13}v_{\xi_{1}}v_{\xi_{2}}-\frac{\mu_{7}^{2}}{2}&\frac{\mu_{7}^{2}v_{\xi_{1}}}{2v_{\xi_{2}}}\\\
\end{array}\right).$ (45)

From the last equation we find that $\left\langle\xi_{i}\right\rangle\neq 0$.
Notice that the flavon fields get decoupled when
$\left\langle\sigma_{i}\right\rangle\gg v_{\rm EW}$ or simply if one takes
$\lambda_{6,7,8,9}\ll 1$. In this case, the CP-even parts of the
$\mathrm{SU(2)}$ scalar doublets do not mix with the scalar singlets and their
masses are obtained by diagonalizing the matrix

$\displaystyle
M^{2}_{\text{2DHM}}\sim\left(\begin{array}[]{cc}\lambda_{1}v_{1}^{2}&-\mu^{2}_{12}+\left(\lambda_{3}+\lambda_{4}+\lambda_{5}\right)v_{1}v_{2}\\\
-\mu^{2}_{12}+\left(\lambda_{3}+\lambda_{4}+\lambda_{5}\right)v_{1}v_{2}&\lambda_{2}v_{2}^{2}\\\
\end{array}\right)$ (48)

The mass of the CP-odd and charged scalar are, respectively,

$m_{A}^{2}=\frac{\mu_{12}^{2}}{v_{1}v_{2}}-\lambda_{5}\left(v_{1}^{2}+v_{2}^{2}\right),$
(49)

and

$m_{H^{\pm}}^{2}=\frac{\mu_{12}^{2}}{v_{1}v_{2}}-\frac{\left(\lambda_{4}+\lambda_{5}\right)}{2}\left(v_{1}^{2}+v_{2}^{2}\right)$
(50)

Before concluding this section, let us briefly mention that even though the
$SU(2)$ scalar $H_{1}$ is only coupled to the SM up-type sector and $H_{2}$ to
the down-type sector the Weinberg-Glasgow-Paschos theorem Glashow and Weinberg
(1977); Paschos (1977) does not apply in this case, i.e. FCNCs might appear at
tree-level due to the mixing between SM and vector like fermions. However, we
expect these FCNCs to be under control since they will be proportional to the
small mixing angles444 For instance, we have numerically checked that the
mixing angles between SM and vector like quarks are at most of the order of
$10^{-3}$ which are sufficiently small to suppress FCNCs induced by these
mixings. and further suppressed by the square of the heavy non-SM scalar
masses. A thorough analysis on this regard is out of the scope of this paper.

##  III Muon anomalous magnetic moment

In this section we start discussing the implications of our model for the muon
anomalous magnetic moment. It is worth mentioning that the Yukawa
interactions, $-y_{E}\overline{L}_{L_{2}}E_{R_{2}}H_{2}$ and
$-y_{E2}\overline{E}_{L_{2}}\ell_{R_{2}}\sigma_{2}$ in eq. (8) as well as the
quartic scalar interaction
$\lambda_{9}(H_{2}^{\dagger}H_{2})(\sigma_{2}^{\ast}\sigma_{2})$ in eq. (13),
provide the dominant contributions to the muon anomalous magnetic moment.
These contributions to $(g-2)_{\mu}$ are one-loop diagrams which involve the
exchange of a electrically neutral CP-even scalar and the VLL $E_{2}$. To
simplify our analysis, we consider a benchmark close to the decoupling limit
where $\varphi_{R_{2}}$ and $\sigma_{R_{2}}$ are mainly composed of two
orthogonal combinations involving two heavy CP-even $S_{1}^{0}$ and
$S_{2}^{0}$ physical scalar fields. Therefore, the muon anomalous magnetic
moment takes the form:

$\Delta
a_{\mu}\simeq\frac{y_{E}y_{E2}m_{\mu}^{2}}{8\pi^{2}}\left[J\left(m_{E_{2}},m_{S_{1}^{0}}\right)-J\left(m_{E_{2}},m_{S_{2}^{0}}\right)\right]\sin\theta\cos\theta,$
(51)

where $S_{1}^{0}\simeq\cos\theta\sigma_{R_{2}}+\sin\theta\varphi_{R_{2}}$ ,
$S_{2}^{0}\simeq-\sin\theta\sigma_{R_{2}}+\cos\theta\varphi_{R_{2}}$, and
$m_{E_{2}}$ is the mass of the VLL $E_{2}$. Furthermore, the loop
$J\left(m_{E},m_{S}\right)$ function has the following form Diaz  _et al._
(2003); Jegerlehner and Nyffeler (2009); Kelso  _et al._ (2014); Lindner  _et
al._ (2018)

$J\left(m_{E},m_{S}\right)=\int_{0}^{1}dx\frac{x^{2}\left(1-x+\frac{m_{E}}{m_{\mu}}\right)}{m_{\mu}^{2}x^{2}+\left(m_{E}^{2}-m_{\mu}^{2}\right)x+m_{S}^{2}\left(1-x\right)}.$
(52)

It is worth mentioning that in this model there exit other BSM contributions
to the muon anomalous magnetic moment but they turn out to be subleading. For
instance, the loop contributions mediated by heavy neutrinos and the $W$ gauge
boson are strongly suppressed by the quadratic power of both the very tiny
active-sterile neutrino mixing angle and the small effective Dirac neutrino
Yukawa coupling. Notice that the smallness of this coupling is due to the
neutrino Yukawa interactions are dimension-5, Eq. (9). Therefore, Dirac
neutrino Yukawas also suppress one-loop diagrams where neutrinos and
electrically charged Higgs are exchanged. Furthermore, following Refs. Raby
and Trautner (2018); Cárcamo Hernández  _et al._ (2020); Dermisek  _et al._
(2021), one can estimate that the $\left(g-2\right)_{\mu}$ contribution that
involve mediation of the $Z$ gauge boson and heavy vector like leptons turns
out be $\Delta
a^{(Z)}_{\mu}\sim\frac{m_{\mu}m_{E}}{8\pi^{2}m_{Z}^{2}}\theta^{2}G_{loop}$.
Hence, $\Delta a^{(Z)}_{\mu}\sim\mathcal{O}\left(10^{-11}\right)$, for $200$
GeV masses of charged vector like leptons and SM-heavy vector like lepton
mixing angle satisfying $\theta\sim\mathcal{O}\left(10^{-3}\right)$.

Fig. 1 shows the muon anomalous magnetic moment as a function of the VLL mass
$M_{E_{2}}$. The solid horizontal lines correspond to the current upper and
lower experimental bounds for the muon anomalous magnetic moment which are set
by Abi  _et al._ (2021)

$\displaystyle(\Delta a_{\mu})_{\exp}=\left(2.51\pm 0.59\right)\times
10^{-9}.$ (53)

In our numerical analysis we have considered the benchmark point with
$\theta=\pi/4$, $M_{S_{1}}=1.5$ TeV and $M_{S_{2}}=2$ TeV. Furthermore, we
have set $y_{E}=y_{E2}=0.2$ and $y_{E}=y_{E2}=0.3$ for the black and blue
curves, respectively. The mass of the charged exotic lepton $E_{2}$ has been
varied in the range $0.2$ TeV$\leqslant M_{E_{2}}\leqslant$ $2$ TeV. Fig. 1
shows that our model can successfully accommodate the current experimental
anomaly $\Delta a_{\mu}$ within the considered mass range. One can see that
$\Delta a_{\mu}$ requires vector-like states below the TeV scale. For this
reason, in what follows, we will analyze how likely one can observe these VLLs
at the upcoming LHC searches.

Figure 1: Muon anomalous magnetic moment as a function of the charged exotic
lepton mass $M_{E_{2}}$. The black and blue curves correspond to
$y_{E}=y_{E2}=0.2$ and $y_{E}=y_{E2}=0.3$, respectively. The horizontal
magenta and orange lines correspond to the $2\sigma$ upper and lower bounds
for the muon anomalous magnetic moment, respectively.

##  IV Exotic fermionic signatures: Analysis setup

As it was shown in Sect. II, two generations of VLLs and four VLQs are present
in the model. For the VLQs, two are of the up-type whereas the other two are
of the down-type. Such states have masses at the TeV scale, within the range
of future collider runs at the LHC. In this section, we focus on a discussion
of potential signatures characteristic of these particles as well as on the
numerical techniques that we propose to probe them. Such analysis will be
boosted via the implementation of neural networks (NNs), whose hyperparameters
are optimized through the use of genetic algorithms, based on previous work by
some of the authors Freitas  _et al._ (2021). We focus in pair-production
topologies for both VLQs and VLLs, while single-production is considered only
for the VLLs.

The model, at the Lagrangian level, is implemented in SARAH Staub (2014), from
which we generate the relevant UFO Degrande  _et al._ (2012) python codes that
interface with Monte-Carlo simulators. In particular, we employ MadGraph (MG5)
Alwall  _et al._ (2014) for simulation of particle collisions at parton-level
for both signal and background topologies. We add hadronization and showering
effects with Pythia8 Sjöstrand  _et al._ (2015) and Delphes de Favereau  _et
al._ (2014) for fast detector simulation. Angular and kinematic distributions
are extracted from this last step, with the help of ROOT Brun and Rademakers
(1997), which are used as inputs in the NNs, for signal/background separation
and computation of statistical significance. All parton-level events are
generated for proton-proton collisions at the LHC, for a centre-of-mass energy
$\sqrt{s}=14$ TeV and for the parton-distribution function nn23lo1, which
automatically fixes the strong coupling, $\alpha_{s}$, and its evolution. We
generate 250 000 events for each individual topology (background and signal).
We consider the MLM matching scheme Hoeche  _et al._ (2005) for topologies
with at least two jets as final states.

VLLs have long been motivated by various SM extensions and Grand Unified
Theory (GUT) frameworks (see, e.g. Raby and Trautner (2018); Garcia Garcia
_et al._ (2015); Bhattacherjee  _et al._ (2018)), and like it was shown in
Sec. III, are important in addressing the muon $(g-2)$ anomaly, whose
relevance has recently come to the forefront of new physics explorations Abi
_et al._ (2021)555 For further constraints on VLLs we refer the reader to
Crivellin  _et al._ (2020).. Despite the strong theoretical motivations, very
limited collider searches have been performed so far, with the most stringent
constraints coming from CMS Sirunyan  _et al._ (2019a), for doublet VLLs that
strongly couple to the tau lepton. Older searches at LEP Achard  _et al._
(2001) constrain these exotic states to be heavier than
$101.2\leavevmode\nobreak\ \mathrm{GeV}$. Therefore, there is still a plenty
of parameter space left to be explored and as such, phenomenological studies
such as these may help pin-point regions of the model parameter space to look
for at collider experiments.

For the VLLs’ search, we consider topologies identical to some of those
studied in our previous work Freitas  _et al._ (2021). These include pair-
production in the t-channel via vector-boson fusion (VBF) processes (see Fig.
2), characterized by two light jets in the forward region originating from
colliding protons. We also include contributions from pair production via the
exchange of a virtual photon or a $Z^{0}$ boson, which we name as “ZA” in what
follows (see Fig. 3). Both topologies are characterized by having two leptons
and a large missing transverse energy (MET) as final states. The single-
production diagram is characterized by a single lepton and a large MET in the
final state (see Fig. 4), which we dub as the “VLBSM topology”.

(a)  (b)

Figure 2: Leading-order Feynman diagrams for the VBF topologies. Original
quarks from colliding protons are indicated as $q$ and $\bar{q}$, while
$E_{2}$ represents the lightest VLL. Besides the forward jets, we have purely
leptonic final states originating from $W^{\pm}$ decays, with one anti-muon,
$\mu^{+}$, and an electron $e^{-}$, as well as their associated neutrino.
$\nu_{\ell}$ are the SM neutrinos.  Figure 3: Leading-order Feynman diagram
for the ZA topologies. The same nomenclature as in Fig. 2 applies here. Figure
4: Leading-order Feynman diagrams for the VLBSM topologies. The same
nomenclature as in Fig. 2 applies here.

For these processes, we consider the main irreducible backgrounds as follows:

  1. 1.

For ZA topologies, main backgrounds include top quark pair production,
$t\bar{t}$, with two b-jets and fully leptonic decays for the W bosons.
$t\bar{t}$ plus $Z^{0}$ production is also considered, with the $Z^{0}$
decaying in the fully invisible channel (two neutrinos) or into two leptons;

  2. 2.

For VBF topologies, diboson $W^{+}W^{-}$ production is considered, with
subsequent decay into leptons. We also take into account $t\bar{t}$ pair
production plus one or two jets, that is, the top decays into leptons and is
accompanied by one or two light jets;

  3. 3.

For VLBSM topologies, we consider all production channels with a single lepton
in the final state, that is, $pp\rightarrow\ell\nu_{\ell}$, with up to two
light jets.

To maximize the signal region and to reduce the main irreducible backgrounds,
specific kinematic cuts are imposed in ROOT. In particular, we consider

  1. 1.

For VBF and ZA topologies, we require at least two lepton candidates with an
opposite flavour and opposite charge. In particular, one anti-muon originating
from $W^{+}$ decays and one electron candidate originating from $W^{-}$. For
VLBSM, we require only one lepton, an electron.

  2. 2.

Generic to all topologies, we impose kinematic constraints for the final
charged lepton states with $p_{T}>25$ GeV and $|\eta|\leq 2.5$. A minimum MET
is also considered with $\mathrm{MET}>15$ GeV.

  3. 3.

For jet reconstruction, we use the Cambridge/Aachen algorithm CMS (2009) with
cone radius $\Delta R=1.0$, with kinematic constraints $p_{T}>35$ GeV and
$|\eta|\leq 2.5$. For jets originating from bottom quarks, we consider tight-
working points with 90% b-tagging efficiency.

The reconstruction procedure via the use of invisible particles in the final
states follows the same approach as thoroughly described in Freitas  _et al._
(2021). The dimension-full and dimension-less variables are extracted from the
final states to train our Deep Learning models, for different reference
frames, including the laboratory frame, the $W$ bosons frame and
$\bar{E}_{2}E_{2}$ frame. All chosen observables for double production
topologies are shown in Table 3, while in Table 4 we present the variables
used for the VLBSM topology.

| Dimension-full | Dimensionless
---|---|---
Lab. frame |  $p_{T}(e^{-})$, $p_{T}(\mu^{+})$,$p_{T}(E_{2})$ $p_{T}(\bar{E}_{2})$, $M({E_{2}})$, $M(\bar{E}_{2})$ $M_{T}(W^{-})$, $M_{T}(W^{+})$, $p_{T}(W^{+})$, $p_{T}(W^{-})$, MET |  $\cos(\theta_{\bar{\nu}_{e}e})$, $\cos(\theta_{\bar{\nu}_{\mu}\mu^{+}})$, $\cos(\theta_{W^{-}W^{+}})$, $\cos(\Delta\phi)$, $\cos(\Delta\theta)$, $\eta(e^{-})$, $\eta({\mu^{+}})$, $\eta({E_{2}})$, $\eta(\bar{E}_{2})$ $\eta(W^{+})$, $\eta(W^{-})$ |  $\Delta R(e,\bar{\nu_{e}})$, $\Delta R(\mu^{+},\nu_{\mu^{+}})$
$W^{-}$ frame |  $p_{T}(e^{-})$, $p_{T}(E_{2})$ |  $\cos(\theta_{\bar{\nu}_{e}e})$, $\eta(e^{-})$, $\eta({E_{2}})$ |
$W^{+}$ frame |  $p_{T}(\mu^{+})$, $p_{T}(\bar{E}_{2})$ |  $\cos(\theta_{\nu_{\mu}\mu^{+}})$, $\eta(\mu^{+})$, $\eta(\bar{E}_{2})$ |
$E_{2}\bar{E_{2}}$ frame |  |  $\cos(\Delta\Phi)$, $\cos(\Delta\Theta)$ |

Table 3: Angular and kinematic observables selected for study of the pair-production topologies, for different frames of reference, the laboratory frame (in the first row), $W^{+}$ and $W^{-}$ frames (in the second and third row, respectively) and the vector-like frame $E_{2}\bar{E}_{2}$ (last row). $\theta_{i,j}$ denotes the angle between different particles, either in the final state or reconstructed ones. In the $E_{2}\bar{E_{2}}$ frame, the angles $\Delta\Phi$ and $\Delta\Theta$ correspond to the azimuthal and polar angles formed by the decay plane of the two $W$ bosons (see Freitas  _et al._ (2021)). | Dimension-full | Dimensionless
---|---|---
Lab. frame |  $p_{T}(e^{-})$,$M_{T}(W^{-})$, $p_{T}(W^{-})$, MET |  $\cos(\theta_{e^{-}})$, $\cos(\theta_{\bar{\nu}_{e}e^{-}})$, $\cos(\theta_{W^{-}})$, $\eta(e^{-})$, $\eta(W^{-})$, $\phi(e^{-})$
Table 4: Angular and kinematic observables selected to study the single
production channel (VLBSM). We compute observables in the laboratory frame.
The same nomenclature of Table 3 for angles applies here.

A similar analysis is also built for the VLQs. The prediction of VLQs is not
exclusive to the model under consideration. In fact, the existence of such
states has been predicted by a series of distinct models in previous
literature, such as in $\mathrm{E_{6}}$-inspired string and GUT models Hebbar
_et al._ (2016); Hewett and Rizzo (1989) or in other extensions to the SM
Benbrik  _et al._ (2016); Hernández and Schmidt (2021). However, unlike VLLs,
from an experimental point-of-view, there is also a good history of searches,
in particular, at the LHC (see, for example, Aaboud  _et al._ (2019a, b);
Sirunyan  _et al._ (2019b)), where the current constraints on parameter space
constrain the VLQ masses to be between 690 GeV up to 1.85 TeV (for current
constraints, as of March 2021, see the summary plots in Collaboration (2021)).
See also Roy  _et al._ (2020); Araque  _et al._ (2015) where possible
interpretations of current VLQ searches at the LHC are undertaken and Romão
Crispim  _et al._ (2020); Crispim Romão  _et al._ (2021) for a discussion
about Deep-Learning based methods that can be applied in direct VLQ searches.
Naturally, the constraints are heavily dependent on considered assumptions,
mainly when it comes to couplings to the SM states. In this regard, the
majority of searches focus on dominant couplings with the top quark, where the
primary decay channel $\text{VLQ}\rightarrow t(b)W$ is studied, characterized
by a final b-jet and two charged leptons. Of course, such assumption is not
the most general, as there is no reason for mixings with other SM quarks to
not exist. For the purpose of this work, the proposed search topology is
focused on a channel with light jets as final states, as seen in Fig. 5.

Figure 5: Leading-order Feynman diagram for the VLQ pair-production via gluon-
gluon fusion. $T_{1}$ represents the lightest up-type VLQ and $u/\bar{u}$
indicate light up-type quarks (up or charm quarks). This diagram provides a
larger contribution than that with $T_{1}uZ^{0}$ vertices. We refer to
Appendix E for further details.

Identical cuts to the VLL scenario are imposed, with two main differences. The
more obvious one is that we now require at least four lepton candidates (an
anti-muon/muon pair and a positron/electron pair) and at least two light jets.
We also alter the minimum transverse momentum for the jet candidates, with now
$p_{T}>50$ GeV. The reason for such alteration comes from the fact that we
plan to probe higher masses ($m>1.8$ TeV), and therefore, the final jets
emerging from VLQs will be highly boosted and energetic when compared to SM
processes, helping to reduce the number of relevant backgrounds. Without any
missing energy, both VLQs can be more easily reconstructed from the leptons
and light jets. For irreducible backgrounds, we consider the same
$t\bar{t}+Z^{0}$ background as for the ZA topology, with $Z^{0}$ decaying into
two charged leptons. We also include all production channels with the same
final states $pp\rightarrow e^{+}e^{-}\mu^{+}\mu^{-}jj$, where $j$ is a light
jet. Such a process includes the main diboson production backgrounds.
Similarly, dimension-full and dimension-less variables from final and
reconstructed states are used for NNs’ training in three distinct reference
frames: laboratory frame, $j_{1}+\gamma$ frame and $j_{2}+\gamma$ frame, where
we define $j_{1}$ as the leading jet (greatest $p_{T}$), while $j_{2}$ is the
sub-leading jet. All distributions are indicated in Table. 5

| Dimension-full | Dimensionless
---|---|---
Lab. frame |  $M(e^{+},e^{-})$, $M(\mu^{+},\mu^{-})$, $M(e^{-},\mu^{-})$, $M(j_{1},j_{2})$, $p_{T}(e^{-})$, $p_{T}(e^{+})$, $p_{T}(\mu^{+})$, $p_{T}(\mu^{-})$, $p_{T}(j_{n})$, $M(e^{+},e^{-},j_{n})$, $M(\mu^{+},\mu^{-},j_{n})$ |  $\eta(e^{-})$, $\phi(e^{-})$, $\eta(e^{+})$, $\phi(e^{+})$, $\eta(\mu^{-})$, $\phi(\mu^{-})$, $\eta(\mu^{+})$, $\phi(\mu^{-})$, $\eta(j_{n})$, $\phi(j_{n})$, $\cos(\theta_{e^{+}e^{-}})$, $\cos(\theta_{\mu^{+}\mu^{-}})$ $\cos(\theta_{j_{1}j_{2}})$ $\cos(\theta_{e^{-}\mu^{-}})$, $\cos(\theta_{e^{-}\mu^{+}})$, $\cos(\theta_{e^{-}j_{n}})$ $\cos(\theta_{e^{+}j_{n}})$, $\cos(\theta_{\mu^{-}j_{n}})$, $\cos(\theta_{\mu^{+}j_{n}})$, $\Delta\phi(e^{+},e^{-})$, $\Delta\phi(e^{-},j_{n})$, $\Delta\phi(e^{+},j_{n})$, $\Delta\phi(\mu^{+},\mu^{-})$, $\Delta\phi(\mu^{-},j_{n})$, $\Delta\phi(\mu^{+},j_{n})$, $\Delta\phi(e^{-},\mu^{-})$, $\Delta\phi(e^{-},\mu^{+})$, $\Delta\phi(e^{+},\mu^{-})$, $\Delta\phi(e^{+},\mu^{+})$ |  $\Delta R(e^{+},e^{-})$, $\Delta R(e^{-},j_{n})$, $\Delta R(e^{+},j_{n})$ $\Delta R(\mu^{+},\mu^{-})$, $\Delta R(\mu^{-},j_{n})$, $\Delta R(\mu^{+},j_{n})$, $\Delta R(e^{-},\mu^{-})$, $\Delta R(e^{-},\mu^{+})$ , $\Delta R(e^{+},\mu^{-})$, $\Delta R(e^{+},\mu^{+})$
$j_{1}+\gamma$ frame |  |  $\cos(\theta_{\mu^{-}j_{1}})$, $\cos(\theta_{\mu^{+}j_{1}})$, $\cos(\Delta\Phi_{1})$ |
$j_{2}+\gamma$ frame |  |  $\cos(\theta_{\mu^{-}j_{2}})$, $\cos(\theta_{\mu^{+}j_{2}})$, $\cos(\Delta\Phi_{2})$ |

Table 5: Angular and kinematic observables selected for study of VLQ pair-
production, for different frames of reference, the laboratory frame (in the
first row), $j_{1}+\gamma$ and $j_{2}+\gamma$ frames (in the second and third
row, respectively). The same nomenclature of Table 3 for angles applies here.
$\Delta\Phi_{1,2}$ corresponds to the azimuthal angle formed by the decay
planes of the two virtual photons. To simplify notation, it is defined
$j_{n}=j_{1},j_{2}$.

For both VLQs and VLLs, the main objective is to combine the kinematic
distributions into multi-dimensional distributions that are then fed into the
NN, whose job is to solve a classification task, that is, to distinguish
between the background and signal events, so that we can evaluate the
statistical significance. To obtain the most optimal results, one must take
special care in the employed architecture, the number of layers and nodes,
activation functions, etc. The selection of these parameters is often made
based on arbitrary decision-making, by choosing the set of hyperparameters
that have shown to guarantee the best results in earlier analyses. Such a
method is often arbitrary and can not be generalised to other applications.

For the purpose of this work, we implement the same techniques as in Freitas
_et al._ (2021), via a genetic algorithm, that chooses the parameters that
best improve a given significance. A diagrammatic representation of the
algorithm can be seen in Fig. 6. The entire process starts by providing a list
of possible input parameters which the algorithm chooses from. From this list,
the algorithm picks, in a random fashion, a series of parameters from which it
can build an arbitrary number of NNs. Once all the networks have been
constructed, we train them for fixed number of epochs. From the trained
networks, we choose the top networks, that is, the ones that better maximize a
given metric. It is from this point that the evolutionary part of the
algorithm kicks in. The idea is inspired by the process of natural selection.
From the best networks, we create Father-Mother pairs, where 50% of the
father’s traits and 50% of the mother’s traits are used to construct new NNs,
that we dub “daughters”. We also impose a probability of mutation,
$\mathcal{P}(M)$, meaning that after the daughters have been built, we also
add a non-zero probability that each of the traits may change to another,
leading to the creation of “mutated daughters”. Then, we train these new
daughter networks, and the loop repeats for a given number of generations. At
the end, we select the one with the better performance, for a given metric, in
our case, the Asimov significance.

Initiate NN models Top 5 selection: Act as F+M Select best model New NN models
“Daughters” “Mutated” NN models “Daughters” 200 epochsAfter 5 gen.F+M
traits$\mathcal{P}(M)=20\%$traindaughters

Figure 6: Diagram representative of the iterations involved in the evolutive
algorithm as used in this work.

We choose the same set of hyperparameters as in our previous work Freitas  _et
al._ (2021), which we detail as follows:

  * •

Number of hidden layers: 1 to 5;

  * •

Number of nodes per layers: 256, 512, 1024 or 2048;

  * •

Initialisers: ’normal’, ’he normal’ and ’he uniform’;

  * •

L2 regulariser, with penalties $1\times 10^{-3}$, $1\times 10^{-5}$ or
$1\times 10^{-7}$;

  * •

Activation functions: ’ReLU’, ’eLU’, ’tanh’ and ’sigmoid’;

  * •

Optimisers: ’Adam’, ’sgd’, ’AdaMax’ and ’NAdam’.

The networks are built in Keras Chollet (2015) with TensorFlow 2.0 Abadi  _et
al._ (2016) as back-end. Some architectural considerations remain fixed during
the evolution of the genetic algorithm. Namely, the final output layer works
as a prediction layer where we funnel the data into a vector with dimensions
$\text{dim}N=1+N_{b}$, with $N_{b}$ the number of backgrounds. Each entry
corresponds to the probability of being a signal or background. For example,
consider a vector $(S,B1,B2)=(0.98,0.01,0.01)$. The output of this form
indicates that the network considers this event as a signal with probability
of $\mathcal{P}_{S}=0.98$ whereas backgrounds have a probability
$\mathcal{P}_{B_{1,2}}=0.01$. The inputs of the networks are also identical,
with normalized and balanced distributions of kinematic data. The normalized
data are important for training due to potentially high variability in
numerical data structures. By normalizing the datasets, we mitigate numerical
errors when computing gradients during back propagation, which in turn allows
for faster learning and improved performance Sola and Sevilla (1997); LeCun
_et al._ (2012). The datasets are also balanced, as the imposition of cuts in
the kinematics of final states reduces the number of entries for both
background and signal classes. Such unbalanced nature may lead to over-fitting
problems and poor generalization to validation datasets. We use SMOTE Chawla
_et al._ (2002) algorithm to balance the data, by oversampling the minority
classes. We also employ a cyclic learning rate during the training, with the
initial value of 0.01 and the maximum allowed value of 0.1. A fixed batch size
of 32544 is considered.

In this work we are interested in models that provide us with the best
significance. Ss such, we define the Asimov significance that can be modified
to work as the loss function. In our case we define our loss as
$1/(\mathcal{Z_{A}}+\epsilon)$, that we plan to minimize666This methodology
was first proposed by Adam Elwood and Dirk Krücker in Elwood and Krücker
(2018)., with

$\mathcal{Z_{A}}=\Bigg{[}2\Bigg{(}(s+b)\ln\Bigg{(}\frac{(s+b)(b+\sigma_{b}^{2})}{b^{2}+(s+b)\sigma_{b}^{2}}\Bigg{)}-\frac{b^{2}}{\sigma_{b}^{2}}\ln\Bigg{(}1+\frac{\sigma_{b}^{2}s}{b(b+\sigma_{b})}\Bigg{)}\Bigg{)}\Bigg{]}^{1/2},$
(54)

where $s$ is the number of signal events, $b$ the number of backgrounds and
$\sigma_{b}^{2}$ is the variance of backgrounds events. Note that in the limit
of large backgrounds, $\mathcal{Z}_{A}\approx s/\sqrt{b}$.

##  V Exotic fermionic signatures: Results

In this section we discuss the results obtained for collider signatures of
exotic fermions in the context of the $\mathcal{Q}_{6}$ flavored multiscalar
model under consideration. Due to their distinct nature, the mass range that
the LHC can probe for the model’s VLLs differs from that of the VLQs. In
particular, and based on the current exclusion bounds, we focus on the
following two sets,

$m_{E_{2}}\in[200,800]\hphantom{.}\mathrm{GeV},\quad
m_{T_{1}}\in[2.2,4]\hphantom{.}\mathrm{TeV},$ (55)

where the mass range chosen for $m_{E_{2}}$ was based on the $\Delta a_{\mu}$
analysis performed above. Note that both the VLL and VLQ decay widths are
automatically computed by MadGraph for each mass.

In all studied signal events the internal vertices are gauge-interacting,
always involving couplings with the SM vector bosons ($Z^{0}$, $\gamma$,
$W^{\pm}$ or $g$). These do indeed provide the dominant contributions and
their strength is well known. However, the fermion-mixing effects must be
considered and typically provide a suppression factor. This is what happens in
flavor non-diagonal scenarios as is the case of the model that we study in
this article. For the purpose of this work, we assume vector-to-chiral fermion
mixing of order $\mathcal{O}(10^{-2}-10^{-3})$ according to the structures
specified in section II, which in turn corresponds to the benchmark point
$y_{E}=y_{E_{2}}=0.2$ shown in Fig. 1. For example, the lepton mixture is
relevant when probing the $E_{2}\nu_{\ell}W$ vertex via an extended
Pontecorvo-Maki-Nakagawa-Sakata (PMNS) mixing. In particular, we fix the PMNS
matrix in such a way that the block that mixes the charged chiral leptons and
SM-like neutrinos is phenomenologically consistent Zyla  _et al._ (2020). The
same is done in the quark sector with an extended version of the Cabibbo-
Kobayashi-Maskawa (CKM) matrix.

Using the cuts detailed in the previous section we can estimate the production
cross-section for each signal/background topology. For the VLL channel, and
fixing $m_{E_{2}}=200\leavevmode\nobreak\ \mathrm{GeV}$ as well as the mixing
matrix in eq.(25):

$U^{e}_{L}=\begin{bmatrix}-0.999997&0.00122671&-0.00196369&-2.41219\times
10^{-6}&-9.55935\times 10^{-6}\\\
0.00123261&0.999993&-0.00300585&-0.00200382&1.1783\times 10^{-8}\\\
0.00195999&-0.00300821&-0.999994&0.0000294637&9.03987\times 10^{-8}\\\
-1.17835\times 10^{-8}&0.0080039&0.0000234357&0.999997&0.00123275\\\
9.55951\times 10^{-6}&2.4701\times 10^{-6}&-4.27716\times
10^{-8}&0.00123275&-0.999999\end{bmatrix}$ (56)

we have obtained,

$\displaystyle\text{VLBSM signal:}\quad\sigma=1.32\times
10^{-4}\hphantom{.}\mathrm{fb};$ $\displaystyle\text{ZA
signal:}\quad\sigma=6.77\times 10^{-4}\hphantom{.}\mathrm{fb};$
$\displaystyle\text{VBF signal:}\quad\sigma=1.77\times
10^{-4}\hphantom{.}\mathrm{fb};$ $\displaystyle pp\rightarrow
e^{-}\bar{\nu}_{e}\quad\sigma=1.96\times 10^{6}\hphantom{.}\mathrm{fb};$
$\displaystyle pp\rightarrow e^{-}\bar{\nu}_{e}(j,jj)\quad\sigma=8.02\times
10^{5}\hphantom{.}\mathrm{fb};$ $\displaystyle t\bar{t}\quad\sigma=1.21\times
10^{3}\hphantom{.}\mathrm{fb};$ $\displaystyle
t\bar{t}(j,jj)\quad\sigma=2.39\times 10^{3}\hphantom{.}\mathrm{fb};$
$\displaystyle W^{+}W^{-}\quad\sigma=1.63\times
10^{2}\hphantom{.}\mathrm{fb};$ $\displaystyle
t\bar{t}Z^{0}(e^{-}e^{+})\quad\sigma=0.18\hphantom{.}\mathrm{fb};$
$\displaystyle
t\bar{t}Z^{0}(\bar{\nu}_{\ell}\nu_{\ell})\quad\sigma=0.20\hphantom{.}\mathrm{fb}$

and, as one notices, all backgrounds sit well above the expected cross
sections for the signal events. On the other hand, VLQ pair-production reveals
the opposite behaviour. Considering the scenario where $m_{T_{1}}=2.2$ TeV and
the mixing matrix in eq. (25):

$U^{u}_{L}=\begin{bmatrix}-1.&0.000177353&-0.000313111&-4.77474\times
10^{-8}&-9.09229\times 10^{-6}\\\
0.0003595&-0.99997&0.005294&-0.0034525&3.26836\times 10^{-9}\\\
-0.00001530&0.005294&0.999968&0.003161&-1.07557\times 10^{-7}\\\
-9.09229\times 10^{-6}&-9.64642\times 10^{-7}&-6.00768\times
10^{-7}&0.00039356&1.\\\ 3.2693\times
10^{-9}&-0.00026828&-0.0016711&0.999995&-0.0003935\end{bmatrix},$ (57)

the cross sections for this analysis read as,

$\displaystyle\text{VLQ signal:}\quad\sigma=5.09\hphantom{.}\mathrm{fb};$
$\displaystyle pp\rightarrow
e^{-}e^{+}\mu^{+}\mu^{-}jj\quad\sigma=0.16\hphantom{.}\mathrm{fb};$
$\displaystyle t\bar{t}Z^{0}(\mu^{-}\mu^{-})\quad\sigma=1.23\times
10^{-2}\hphantom{.}\mathrm{fb}$

where we note that for the $t\bar{t}Z^{0}(\mu^{-}\mu^{-})$ background, the
electron and positron originate from $W$ decays. As one can clearly see, the
signal production cross-section sits above the main irreducible backgrounds.
This fact remains true up until $m_{T_{1}}\sim 3.8$ TeV, after which the
suppression coming from the VLQ mass and its decay width becomes large enough
yielding an increasingly smaller cross-section. The noticeable difference
between the VLQ and VLL sectors can be mainly attributed to the chosen
collider and couplings present in the production channels. Proton-proton
collisions heavily favour a pair-production of colored particles, which is
further maximized via the strong coupling in the three gluon $ggg$ and
$g\bar{T}_{1}T$ vertices, as opposed to the weak gauge coupling present in all
VLL pair-production processes. We show in Fig. 7 both the VLL and VLQ
production cross-sections in terms of the exotic fermion masses for each of
the studied processes.

(a) VLLs (b) VLQs

Figure 7: The production cross-section as a function of the VLLs mass (left
panel) and the VLQ mass (right panel).

As mentioned in the previous section, the main goal is to compute the
statistical significance of a hypothetical discovery. As such, we need to
extract the relevant kinematic information that helps separating the signal
from background. Let us first dedicate our attention to VLL production
focusing on a scenario where $m_{E_{2}}=200\leavevmode\nobreak\ \mathrm{GeV}$.
We show in Appendix C the distributions in the laboratory frame in Figs. 11,
12 and 13. In these figures, the angular distributions in $\bar{E}_{2}E_{2}$
frame are also presented. The remaining distributions, in the $W$ frame of
reference, are shown in Fig. 14. It is common to all topologies that the
angular distributions are dominated by the imposed backgrounds, since the
signal events do not have major qualitative differences that help separating
different $\cos(\theta)$ distributions. On the other hand, $\Delta R$
distributions are particularly interesting in this regard, as signal
topologies typically involve a peak around $\Delta R\sim 1$, as opposed to the
backgrounds, with a flatter structure and a peak at higher values, $\Delta
R\sim 1-2$. Kinematic information, such as pseudo-rapidity, offers additional
discrimination as the signal events are characterized by a strong peak at
$\eta=0$, while backgrounds typically either possess a double-peak structure
(see $\eta$ distributions for reconstructed particles such as $E_{2}$ and
$W^{+}$ in Fig. 11), or a more uniform distribution over the allowed pseudo-
rapidity range (see $\eta$ plots for the electron and the anti-muon in Fig.
11). This is true for VBF topologies, while for ZA and VLBSM events it is not
(see Figs. 12 and 13). For instance, pseudo-rapidity distributions for ZA and
VLBSM typically follow the same characteristics as the backgrounds. Therefore,
they do not offer the same discriminating power. Transverse momentum
distributions, for both topologies, dot not provide a greater discriminating
power either. However, MET distributions supply some key differences such as
longer tails at large MET values, i.e. $\mathrm{MET}>100\leavevmode\nobreak\
\mathrm{GeV}$. In fact, MET distributions are rather relevant, as this
observable is representative of the signal events that we propose, in
particular for the ZA/VBF cases where the final state contains four neutrinos.
For the case of VLBSM topologies this no longer applies. Even though the final
state contains three neutrinos, MET distributions have the same shape as the
SM backgrounds (see middle panel in the second row of Fig. 13).

Turning our attention to the quark sector, all relevant distributions are
shown in Fig. 10 where we fix the VLQ mass as
$m_{T_{1}}=2.2\leavevmode\nobreak\ \mathrm{TeV}$. For VLQ production, a
substantial amount of variables were used. As such, not all of them, but only
a representative subset is shown. As opposed to the VLLs, $\cos(\theta)$
distributions do offer discriminating power over backgrounds. For the pair of
particles $(e^{+},j_{1})$, $(e^{-},j_{2})$ and $(\mu^{+},j_{2})$, the cosine
distributions preferably peak at $\cos(\theta)=-1$, i.e. at $\theta=\pi$, for
signal events, implying that the outgoing particles are produced back-to-back.
This contrasts with the studied backgrounds where a significant portion of
events feature highly collinear particles, i.e.
$\cos(\theta)=1\Leftrightarrow\theta=0$. Other angular variables such as
$\Delta\phi$ offer further distinction with a typical double-peak structure of
signal events at $|\Delta\phi|\geq 2$, whereas the corresponding backgrounds
tend to populate near $\Delta\phi=0$. The transverse momentum distributions of
the final leptons are also relevant as one notices from the bottom rows of
Fig. 10. In particular, signal events tend to populate regions of phase-space
with larger momentum ($p_{T}>300$ GeV), in stark contrast with the SM
backgrounds, which preferably populate regions of much lower $p_{T}$ values.

A proper analysis requires a combination of the various kinematical variables
into a single multi-dimensional distribution that the NN uses as an input.
This allows one to find the regions of the parameter space that better enhance
the signal region while minimizing the background effects. More importantly,
it also provides us with the ability to compute the statistical significance
as well as determining which mass regions can be excluded within the studied
model. For a complete analysis we calculate the significance focusing on three
distinct statistical metrics, each one being more conservative than the other,
in order to provide the most rigorous and realistic scenarios for further
investigation.

We then consider the following

  * •

$\mathcal{Z}_{A}:$ The Asimov significance, as it is defined in Eq. (54),
assuming 1% systematic errors;

  * •

$\mathcal{Z}(<1\%):$ An adapted version of the Asimov significance, as it is
defined in Eq. (54). For this case we assume a much lower systematic
uncertainty, in particular $10^{-3}$. As such, this is the most lenient metric
and typically offers the highest values for the significance;

  * •

$s/\sqrt{s+b}:$ A more traditional metric, which is the limiting scenario of
the Asimov significance when $s\ll b$.

With this in mind, we apply the genetic algorithm as described in the previous
section and calculate the aforementioned significance metrics in terms of the
score that the NN gives to each event. For the scenarios that have so far been
discussed, $m_{E_{2}}=200$ GeV and $m_{T_{1}}=2.2$ TeV, our results for the
VLLs significance in each channel can be seen in Fig. 15 of Appendix D. Notice
that the significance is calculated under an assumption of the high-luminosity
LHC, with an integrated luminosity of $\mathcal{L}=3000$ $\mathrm{fb^{-1}}$.

Let us first focus on the VLL sector. Provided that each topology is an
independent event we can safely combine their individual significances as

$\sigma_{C}=\sigma_{\mathrm{VBF}}+\sigma_{\mathrm{VLBSM}}+\sigma_{\mathrm{ZA}}.$
(58)

With this in mind, we notice that the highest combined significance is
obtained for the $\mathcal{Z}(<1\%)$ metric, in particular, with
$\sigma_{C}=3.78\sigma$. The dominant contributions to this value are those of
the VLBSM and VBF channels with $\sigma_{\mathrm{VLBSM}}=1.76\sigma$ and
$\sigma_{\mathrm{VBF}}=1.03\sigma$, respectively. Equally interesting is the
$s\sqrt{s+b}$ metric, for which the combined significance is slightly smaller,
$\sigma_{C}=2.56\sigma$, with the VLBSM topology providing the highest
contribution, with $1.13\sigma$. While these results do not permit a confident
exclusion at this mass range, however it is sufficiently large to merit
further inspection if an hypothetical anomaly appears at the LHC experiments.
If this turns out to be the case, we argue here that, if the next generation
of colliders beyond the LHC are still far from becoming operational, a
continuation of the high-luminosity program must be put on the table as we
further discuss below.

As expected, the most conservative metric, $\mathcal{Z}_{A}$, shows the lowest
significance values with $\sigma_{C}=0.032\sigma$. Therefore, within the
context of the model under consideration, we can not safely exclude VLLs up to
masses of $200\leavevmode\nobreak\ \mathrm{GeV}$. In fact, singlet VLLs do not
couple directly to $W$ bosons as it happens for doublet VLLs (see Freitas  _et
al._ (2021) for a previous study.) Instead, such interactions with $W$ bosons
relevant for the three signal processes under consideration, are indirectly
induced via off-diagonal Yukawa interactions becoming suppressed by a factor
of the mass of the VLL itself Kumar and Martin (2015). This implies that the
larger the VLL mass, the smaller the interaction strength with $W$ bosons,
thus, the smaller the cross section and significance.

We show in Fig. 8 the dependence of the significance as a function of the
luminosity for a VLL mass of $200\leavevmode\nobreak\ \mathrm{GeV}$.

(a) $\mathcal{Z}_{A}$ significance (b) $\mathcal{Z}(<1\%)$ significance

(c) $s/\sqrt{s+b}$

Figure 8: Statistical significance as a function of the integrated luminosity
in $\mathrm{fb^{-1}}$ for the three different statistics adopted in the
current analysis and for a fixed VLL mass, $m_{E_{2}}=200$ GeV. The $x$ axis
is on logarithmic scale. In (a) we showcase the Asimov significance, in (b) –
the adapted Asimov significance, and in (c) – the $s/\sqrt{s+b}$ metric. The
colours represent the distinct signal processes under consideration. In
particular, the green curve is representative of VBF events, the red curve
indicates ZA topologies, while the blue curve refers to VLBSM single
production events. The dashed curves indicate that the considered values of
the luminosity are beyond the LHC operation regime.

In particular, we see that by the end of the LHC program, i.e.
$\mathcal{L}=3000\leavevmode\nobreak\ \mathrm{fb^{-1}}$, a combined
significance of $\sigma_{C}=3.78\sigma$ can be achieved, for the
$\mathcal{Z}(<1\%)$ metric. Alternatively, if the standard $s\sqrt{s+b}$
measure is considered, a $\sigma_{C}=2.56\sigma$ anomaly could be observed. In
either scenario, we argue that, if a new generation of colliders that will
succeed the LHC remains decades away from the beginning of operations, such an
anomaly (or any other excess) may justify a continuation of the high-
luminosity runs.

As an example, we show in all panels of Fig. 8 the continuation of the
significance curves for higher luminosities choosing
$\mathcal{L}=6000\leavevmode\nobreak\ \mathrm{fb^{-1}}$ and
$\mathcal{L}=9000\leavevmode\nobreak\ \mathrm{fb^{-1}}$ as merely indicative
values. In particular, we see that a signal confirmation or exclusion would be
realizable with $\sigma_{C}=6.54\sigma$ for the $\mathcal{Z}(<1\%)$ metric, at
$\mathcal{L}=9000\leavevmode\nobreak\ \mathrm{fb^{-1}}$, and
$\sigma_{C}=5.34\sigma$ at $\mathcal{L}=6000\leavevmode\nobreak\
\mathrm{fb^{-1}}$. Note that we use dashed lines in the continuation of the
significance curves for the region beyond
$\mathcal{L}=3000\leavevmode\nobreak\ \mathrm{fb}^{-1}$ to indicate that such
a regime is beyond the planned LHC operation program.

Mass of VLL | $s/\sqrt{s+b}$ | $\mathcal{Z}(<1\%)$ | $\mathcal{Z}_{A}$
---|---|---|---
ZA | VBF | VLBSM | $\sigma_{\mathrm{C}}$ | ZA | VBF | VLBSM | $\sigma_{\mathrm{C}}$ | ZA | VBF | VLBSM | $\sigma_{\mathrm{C}}$
$200$ GeV | $0.70$ | $0.73$ | $1.13$ | 2.56 | $0.99$ | $1.03$ | $1.76$ | 3.78 | $0.02$ | $0.011$ | $0.0022$ | 0.033
$300$ GeV | $0.37$ | $0.38$ | $0.59$ | 1.34 | $0.57$ | $0.54$ | $0.91$ | 2.02 | $0.018$ | $3.08\times 10^{-5}$ | $0.0012$ | 0.019
$400$ GeV | $0.25$ | $0.23$ | $0.38$ | 0.86 | $0.36$ | $0.32$ | $0.55$ | 1.23 | $0.0086$ | $0.0077$ | $0.0022$ | 0.019
$500$ GeV | $0.19$ | $0.15$ | $0.30$ | 0.64 | $0.30$ | $0.21$ | $0.43$ | 0.94 | $0.0079$ | $0.0037$ | $0.0020$ | 0.014
$600$ GeV | $0.15$ | $0.13$ | $0.19$ | 0.47 | $0.20$ | $0.15$ | $0.32$ | 0.67 | $0.00023$ | $0.0022$ | $0.0013$ | 0.0037
$700$ GeV | $0.0024$ | $0.069$ | $0.095$ | 0.17 | $0.091$ | $0.098$ | $0.11$ | 0.30 | $5.40\times 10^{-5}$ | $0.0015$ | $0.0016$ | 0.0032
$800$ GeV | $0.0020$ | $3.8849\times 10^{-6}$ | $0.055$ | 0.057 | $0.0034$ | $0.071$ | $0.077$ | 0.15 | $2.78\times 10^{-5}$ | $2.20\times 10^{-5}$ | $0.0015$ | 0.0015

Table 6: Signal significance for the lightest VLL. The computation follows an
evolutive algorithm that maximizes the Asimov significance metric. All
significances are computed for $\mathcal{L}=3000$ fb-1 with centre-of-mass
energy of $\sqrt{s}=14$ TeV. $\sigma_{C}$ is the combined significance as
defined in (58).

Mass of VLL | $s/\sqrt{s+b}$ | $\mathcal{Z}(<1\%)$ | $\mathcal{Z}_{A}$
---|---|---|---
ZA | VBF | VLBSM | $\sigma_{\mathrm{C}}$ | ZA | VBF | VLBSM | $\sigma_{\mathrm{C}}$ | ZA | VBF | VLBSM | $\sigma_{\mathrm{C}}$
$200$ GeV | $1.22$ | $1.26$ | $1.95$ | 4.43 | $1.71$ | $1.79$ | $3.04$ | 6.54 | $0.034$ | $0.016$ | $0.0032$ | 0.053
$300$ GeV | $0.63$ | $0.66$ | $1.03$ | 2.32 | $0.99$ | $0.93$ | $1.57$ | 3.49 | $0.030$ | $3.73\times 10^{-5}$ | $0.0025$ | 0.033
$400$ GeV | $0.44$ | $0.39$ | $0.65$ | 1.48 | $0.62$ | $0.56$ | $0.95$ | 2.13 | $0.015$ | $0.013$ | $0.0036$ | 0.032
$500$ GeV | $0.34$ | $0.25$ | $0.52$ | 1.11 | $0.35$ | $0.36$ | $0.52$ | 1.13 | $0.0079$ | $0.0065$ | $0.0035$ | 0.0179
$600$ GeV | $0.23$ | $0.20$ | $0.33$ | 0.76 | $0.23$ | $0.29$ | $0.40$ | 0.92 | $0.0039$ | $0.0064$ | $0.0037$ | 0.014
$700$ GeV | $0.10$ | $0.12$ | $0.20$ | 0.42 | $0.13$ | $0.17$ | $0.23$ | 0.53 | $0.00027$ | $0.0026$ | $0.0030$ | 0.00587
$800$ GeV | $0.08$ | $6.73\times 10^{-6}$ | $0.095$ | 0.18 | $0.07$ | $0.12$ | $0.13$ | 0.32 | $0.00015$ | $3.72\times 10^{-5}$ | $0.0028$ | 0.0030

Table 7: Signal significance for the lightest VLL. The computation follows an
evolutive algorithm that maximizes the Asimov significance metric. All
significances are computed for $\mathcal{L}=9000$ fb-1 with centre-of-mass
energy of $\sqrt{s}=14$ TeV. $\sigma_{C}$ is the combined significance as
defined in (58).

Mass of VLL | $s/\sqrt{s+b}$ | $\mathcal{Z}(<1\%)$ | $\mathcal{Z}_{A}$
---|---|---|---
ZA | VBF | VLBSM | $\sigma_{\mathrm{C}}$ | ZA | VBF | VLBSM | $\sigma_{\mathrm{C}}$ | ZA | VBF | VLBSM | $\sigma_{\mathrm{C}}$
$500$ GeV, $\sqrt{s}=28$ TeV | $0.49$ | $0.35$ | $0.36$ | 1.20 | $0.70$ | $0.50$ | $0.72$ | 1.92 | $0.032$ | $0.022$ | $0.0022$ | 0.0562
$500$ GeV, $\sqrt{s}=14$ TeV | $0.19$ | $0.15$ | $0.30$ | 0.86 | $0.36$ | $0.32$ | $0.55$ | 1.23 | $0.0086$ | $0.0077$ | $0.0022$ | 0.019

Table 8: Signal significance for the lightest VLL. The computation follows an
evolutive algorithm that maximizes the Asimov significance metric. All
significances are computed for $\mathcal{L}=3000$ fb-1 for a fixed mass
$m_{E_{2}}=500$ GeV. In the first row, computations are performed for the
centre-of-mass energy of $\sqrt{s}=28$ TeV, while the case of $\sqrt{s}=14$
TeV is in the second row. $\sigma_{C}$ is the combined significance as defined
in (58).

To further complement our analysis, we also perform a scan for different VLL
masses, summarizing our results in Tab. 6 for $3000$ $\mathrm{fb^{-1}}$ of
integrated luminosity. For completeness, in particular, in order to understand
how much we would gain in prolonging the LHC high-luminosity runs, we show in
Tab. 7 the same results but for an integrated luminosity of $9000$
$\mathrm{fb^{-1}}$.

First, in Table 6 we obtain significances of the order of $2\sigma$ up until a
mass of 300 GeV, whereas, in Table 7, the same significance would be achieved
for a mass of 400 GeV. It is also interesting to observe what would be the
effect of a collider at a higher centre-of-mass energy. For this scenario, we
focus our attention at a particular point, $m_{E_{2}}=500$ GeV and a centre-
of-mass energy $\sqrt{s}=28$ TeV. The results are shown in Tab. 8. As
expected, the discovery significance increases with the centre-of-mass energy.
For example, combined significance of the $s/\sqrt{s+b}$ metric increases by a
factor of $1.4$. Similarly, the other metrics also showcase improvements with
an increase by factors of $1.6$ and $0.3$ for the $\mathcal{Z}(<1\%)$ and
$\mathcal{Z}_{A}$ metrics, respectively. We also note that, for the
$s/\sqrt{s+b}$ and $\mathcal{Z}_{A}$ measures of the VLBSM topology, small
variations in statistical significances are obtained, with $\mathcal{Z}_{A}$
seeing no deviation. These small fluctuations might be associated with the
stochastic nature of the tested NNs.

A similar analysis was performed for VLQ searches pair produced via gluon-
gluon fusion. In Fig. 16 of appendix D we show the significance in terms of
the NN score where it a massive significance increase is immediately
noticeable in comparison to the VLL case. This is mostly due to a far superior
cross-section which, for $\mathcal{L}=3000\leavevmode\nobreak\
\mathrm{fb^{-1}}$, promptly results in a large Asimov significance, i.e.
$\mathcal{Z}_{A}=257.53\sigma$, as well as equally large values for the other
two metrics, $s/\sqrt{s+b}=121.71\sigma$ and $\mathcal{Z}(<1\%)=275.76\sigma$.
As we show in Fig. 9, the statistical significance for the VLQ searches is
such that we can probe a large range of masses from $2.2\leavevmode\nobreak\
\mathrm{TeV}$ up to about $4.0\leavevmode\nobreak\ \mathrm{TeV}$.

Figure 9: Statistical significance as a function of the integrated luminosity
for the three statistical metrics under consideration. We fix the the lightest
VLQ mass as $m_{\mathrm{T}}=2.2\leavevmode\nobreak\ \mathrm{TeV}$. The green
curve represents the $\mathcal{Z}(<1\%)$ metric, the red curve indicates the
$s/\sqrt{s+b}$ one while the blue curve shows our results for the Asimov
significance $\mathcal{Z}_{A}$.

The corresponding numerical results can be found in Tab. 9. From this, we note
that for a high-luminosity run, $\mathcal{L}=3000\leavevmode\nobreak\
\mathrm{fb^{-1}}$, one can exclude, or claim a discovery, by over $5$ standard
deviations, of VLQ masses up to $3.8\leavevmode\nobreak\ \mathrm{TeV}$. More
interestingly, the VLQ sector of the model under consideration, can already be
probed at the forthcoming LHC Run-III, $\mathcal{L}=300\leavevmode\nobreak\
\mathrm{fb^{-1}}$, for VLQ masses of around $3.4\leavevmode\nobreak\
\mathrm{TeV}$.

Mass of VLQ |  $300$ $\mathrm{fb^{-1}}$ |  $1000$ $\mathrm{fb^{-1}}$ |  $3000$ $\mathrm{fb^{-1}}$
---|---|---|---
$s/\sqrt{s+b}$ | $\mathcal{Z}(<1\%)$ | $\mathcal{Z}_{A}$ | $s/\sqrt{s+b}$ | $\mathcal{Z}(<1\%)$ | $\mathcal{Z}_{A}$ | $s/\sqrt{s+b}$ | $\mathcal{Z}(<1\%)$ | $\mathcal{Z}_{A}$
$2.2$ TeV | $38.49$ | $87.19$ | $87.91$ | $70.27$ | $159.51$ | $155.98$ | $121.71$ | $275.76$ | $257.53$
$2.4$ TeV | $31.22$ | $67.13$ | $67.96$ | $57.00$ | $122.70$ | $121.85$ | $98.73$ | $212.29$ | $202.30$
$2.6$ TeV | $24.94$ | $50.01$ | $49.84$ | $45.53$ | $91.34$ | $89.28$ | $78.87$ | $158.17$ | $147.40$
$2.8$ TeV | $19.51$ | $36.10$ | $35.94$ | $35.62$ | $65.92$ | $64.62$ | $61.70$ | $114.16$ | $107.57$
$3.0$ TeV | $14.97$ | $25.33$ | $25.22$ | $27.33$ | $27.33$ | $45.48$ | $47.34$ | $80.10$ | $76.23$
$3.2$ TeV | $11.22$ | $17.28$ | $17.21$ | $20.49$ | $31.55$ | $31.11$ | $35.49$ | $54.65$ | $52.42$
$3.4$ TeV | $8.16$ | $11.43$ | $11.39$ | $14.89$ | $20.87$ | $20.62$ | $25.79$ | $36.15$ | $34.88$
$3.6$ TeV | $4.51$ | $4.91$ | $4.98$ | $8.52$ | $8.10$ | $9.11$ | $16.32$ | $17.10$ | $15.43$
$3.8$ TeV | $1.20$ | $3.25$ | $2.93$ | $3.00$ | $7.11$ | $5.02$ | $6.01$ | $8.13$ | $10.05$
$4.0$ TeV | $0.44$ | $0.66$ | $0.47$ | $1.01$ | $1.33$ | $1.28$ | $2.20$ | $2.51$ | $1.91$

Table 9: Signal significance for the lightest VLQ-pair production. The
computation follows a genetic algorithm that maximizes the Asimov significance
metric. All significances are computed for proton-proton collisions at the
centre-of-mass energy of $\sqrt{s}=14$ TeV.

$\mathrm{M_{VLQ}}=2.2\leavevmode\nobreak\ \mathrm{TeV}$ | $5\%$ | $10\%$ | $20\%$ | $40\%$ | $80\%$ | $96\%$
---|---|---|---|---|---|---
$\mathrm{sys}:1\%$ | $58.39\sigma$ | $56.32\sigma$ | $52.02\sigma$ | $42.74\sigma$ | $19.33\sigma$ | $5.21\sigma$
$\mathrm{sys}:10\%$ | $43.14\sigma$ | $41.80\sigma$ | $39.00\sigma$ | $32.83\sigma$ | $16.13\sigma$ | $4.70\sigma$

Table 10: Asimov significance, $\mathcal{Z}_{A}$, for VLQ pair-production. The
last six columns show the Asimov significance assuming a suppression of $5\%$
to $96\%$ in the signal cross section coming from unaccounted effects. In the
first row a systematic uncertainty in $\mathcal{Z_{A}}$ of 1% is considered,
whereas in the second row we assume a systematic uncertainty of 10%. We
consider an integrated luminosity of $\mathcal{L}=139\leavevmode\nobreak\
\mathrm{fb^{-1}}$ corresponding the acquired data after the LHC Run-II.

It is relevant to mention that the obtained significance at low luminosity, in
particular, for Run-II data, may naively suggest that all such scenarios are
excluded. However, one must note that direct searches at the LHC are mostly
focused in VLQ decays to third generation quarks, so far not considering
channels with light jets and di-leptons as we propose in this article, making
a comparison of our results with available data not suitable. With this in
mind, the key point of our analysis relies on the fact that the process in
Fig. 5 results in a cross-section significantly larger than that of the
corresponding irreducible backgrounds, thus yielding a large significance for
this particular channel. In fact, the evolutive algorithm employed in this
work was engineered to find Neural Network models that further enhance such a
discovery (or exclusion) significance. Since the architectures obtained with
this methodology can easily find regions on the feature phase space, in our
case the kinematic and angular observables described in tables 3 and 5, the
separation between signal and background can be maximized as it is shown in
Fig. 16(a). For completeness of information we show in Tab. 10 how the
significance drops with a decreasing cross-section and with increasing
systematic uncertainties. While our calculation has already been subject to
detector effects with Delphes as well as to systematic uncertainties in the
definition of the Asimov metric, one can take a conservative approach and
consider that further unaccounted effects can impose a larger suppression in
the cross-section than the one obtained from the Delphes output. It is
remarkable to note that even for a suppression factor of $96\%$, a
$2.2\leavevmode\nobreak\ \mathrm{TeV}$ VLQ can still be probed with a
statistical significance of o $5.21\sigma$ with current LHC Run-II data. If we
further increase the systematic uncertainties up to $10\%$, we obtain a worst
case scenario lower bound on the discovery (or exclusion) significance of
$4.7$ standard deviations.

##  VI Conclusions

In this paper, we have proposed a novel model where the SM gauge symmetry is
enlarged by the $\mathcal{Q}_{6}\times\mathcal{Z}_{2}$ discrete group. Within
this framework, new exotic VLFs are emergent, both of quark and lepton types,
as well as RH Majorana neutrinos. Furthermore, the scalar sector is enlarged
by the inclusion of a new doublet and singlet scalar fields. We show that
tree-level masses for third-generation fermions (top and tau) due to
interactions with doublet scalars ($H_{1}$ and $H_{2}$) are generated after
the spontaneous breaking of the electroweak gauge and the flavour
$\mathcal{Q}_{6}\times\mathcal{Z}_{2}$ symmetries. The remaining SM charged
fermions gain their masses via a Universal seesaw mechanism mediated by VLFs.
The tiny masses of the light active neutrinos arise from a tree-level type I
seesaw mechanism mediated by heavy right handed Majorana neutrinos.

Due to sizeable couplings between the exotic VLL and the new scalar fields,
contributions to the anomalous magnetic moment of the muon are generated. For
certain benchmark scenarios for the couplings, we demonstrate that the model
can successfully accommodate the measured muon $(g-2)$ anomaly. More
specifically, considering the two benchmark scenarios $y_{E}=y_{E2}=0.2$ and
$y_{E}=y_{E2}=0.3$ we can explain the observed muon $(g-2)$ anomaly within
2$\sigma$ error bars for masses of the $E_{2}$ between 200 GeV and 2 TeV.

Phenomenological studies, in the context of collider physics at the LHC, are
conducted for both VLLs and their quark counterparts. For this purpose, we
employ the genetic algorithms to optimize the construction of neural networks,
whose objective is to maximise the statistical significance of an hypothetical
discovery of these particles at future experiments. For VLLs, we consider
double production channels, either via production of a $Z^{0}$ boson or
virtual photon, or via vector-boson fusion. We also consider the channel for
single production. Using kinematic information of the final states, we
determine that we can not exclude VLLs with more than five standard deviations
for masses above $200\leavevmode\nobreak\ \mathrm{GeV}$, at the high-
luminosity phase of the LHC, $\mathcal{L}=3000$ $\mathrm{fb}^{-1}$. Assuming a
hypothetical extension towards $\mathcal{L}=9000$ $\mathrm{fb}^{-1}$, one can
exclude the lightest VLL with masses up to approximately
$200\leavevmode\nobreak\ \mathrm{GeV}$. We also determine the impact from the
increasing center-of-mass energy at future colliders. For a mass of
$m_{E_{2}}=500$ GeV, we show that the combined significance improves when
moving from $\sqrt{s}=14$ TeV to $\sqrt{s}=28$ TeV. Specifically, the
significance increases from $0.86\sigma$ to $1.20\sigma$ for $s\sqrt{s+b}$,
$1.23\sigma$ to $1.92\sigma$ for $\mathcal{Z}(<1\%)$ and $0.0562\sigma$ to
$0.019\sigma$ for $\mathcal{Z}_{A}$. A similar analysis is made for VLQs,
focusing on double production via strong-interaction channels, which is
characterized by four leptons and two light jets in the final states. We found
that VLQ masses up to $3.8\leavevmode\nobreak\ \mathrm{TeV}$ can be excluded
at a luminosity of $\mathcal{L}=3000$ $\mathrm{fb^{-1}}$ and up to
$3.4\leavevmode\nobreak\ \mathrm{TeV}$ for $\mathcal{L}=300$
$\mathrm{fb^{-1}}$. To the best of our knowledge, the VLQ production channel
proposed in this article has so far not been adopted in direct searches by
experimental collaborations and must be considered both with currently
available as well as with future data to be collected in forthcoming LHC runs.

## Acknowledgments

The authors acknowledge Ulises Saldaña Salazar for fruitful discussions at
early stages of this work. The authors are also very grateful with Martin
Hirsch for providing very useful comments and suggestions. A.E.C.H.
acknowledges support by FONDECYT (Chile) under grant No. 1210378, Milenio-
ANID-ICN2019_044 and ANID PIA/APOYO AFB180002. The work of C.B. was supported
by FONDECYT grant No. 11201240. R.P. is supported in part by the Swedish
Research Council grant, contract number 2016-05996, as well as by the European
Research Council (ERC) under the European Union’s Horizon 2020 research and
innovation programme (grant agreement No 668679). J.G., F.F.F., and A.P.M. are
supported by the Center for Research and Development in Mathematics and
Applications (CIDMA) through the Portuguese Foundation for Science and
Technology (FCT - Fundação para a Ciência e a Tecnologia), references
UIDB/04106/2020 and UIDP/04106/2020. A.P.M., F.F.F. and J.G. are supported by
the project PTDC/FIS-PAR/31000/2017. A.P.M. is also supported by national
funds (OE), through FCT, I.P., in the scope of the framework contract foreseen
in the numbers 4, 5 and 6 of the article 23, of the Decree-Law 57/2016, of
August 29, changed by Law 57/2017, of July 19. The authors also would like to
acknowledge the FCT Advanced Computing Project to provide computational
resources via the project CPCA/A00/7395/2020. This work was partially produced
with the support of INCD funded by FCT and FEDER under the project
01/SAICT/2016 nº 022153. J.G. is also directly funded by FCT through the
doctoral program grant with the reference 2021.04527.BD. C.B. and R.P. would
like to thank the members of the UTFSM particle physics group in Valparaíso
for their hospitality during their visit, where part of this work was made.

##  Appendix A $\mathcal{Q}_{6}$ multiplication rules

The $\mathcal{Q}_{6}$ group contains 12 elements, ${a^{m}b^{n}}$, with
$m=0,1,2,3,4,5$ and $n=0,1$, where the group generators $a$ and $b$ satisfy
$a^{6}=e$, $b^{2}=a^{3}$ and $b^{-1}ab=a^{-1}$. This discrete group has four
1-dimensional irreducible representations, $\mathbf{1}_{++}$,
$\mathbf{1}_{+-}$, $\mathbf{1}_{-+}$, and $\mathbf{1}_{--}$, and two
2-dimensional, $\mathbf{2}_{1}$ and $\mathbf{2}_{2}$. The representation
matrices of $a$ and $b$ for each irreducible representation are given by,

$\displaystyle\begin{array}[]{lllll}{\bf 1}_{++}&:&a=1&&b=1\\\ {\bf
1}_{--}&:&a=1&&b=-1\\\ {\bf 1}_{+-}&:&a=-1&&b=-i\\\ {\bf
1}_{-+}&:&a=-1&&b=i\\\ {\bf 2}_{1}&:&a=\left(\begin{array}[]{cc}\alpha&0\\\
0&\alpha^{-1}\end{array}\right)&&b=\left(\begin{array}[]{cc}0&i\\\
i&0\end{array}\right)\\\ {\bf
2}_{2}&:&a=\left(\begin{array}[]{cc}\alpha^{2}&0\\\
0&\alpha^{-2}\end{array}\right)&&b=\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right)\end{array}$ (73)

with $\alpha=\exp{\left\\{i\frac{2\pi}{6}\right\\}}$. The tensor products for
the $\mathcal{Q}_{6}$ representations are given by

$\displaystyle\left(\begin{array}[]{c}a\\\
b\end{array}\right)_{\mathbf{2}_{2}}\otimes\left(\begin{array}[]{c}c\\\
d\end{array}\right)_{\mathbf{2}_{1}}=\left(ac-
bd\right)_{\mathbf{1}_{+-}}\oplus\left(ac+bd\right)_{\mathbf{1}_{-+}}\oplus\left(\begin{array}[]{c}ad\\\
bc\end{array}\right)_{\mathbf{2}_{1}},$ (80)
$\displaystyle\left(\begin{array}[]{c}a\\\
b\end{array}\right)_{\mathbf{2}_{k}}\otimes\left(\begin{array}[]{c}c\\\
d\end{array}\right)_{\mathbf{2}_{k}}=\left(ad-
bc\right)_{\mathbf{1}_{++}}\oplus\left(ad+bc\right)_{\mathbf{1}_{--}}\oplus\left(\begin{array}[]{c}ac\\\
-bd\end{array}\right)_{\mathbf{2}_{k^{\prime}}},$ (87)

for $k,k^{\prime}=1,2$ and $k^{\prime}\neq k$,

$\displaystyle\left(w\right)_{\mathbf{1}_{++}}\otimes\left(\begin{array}[]{c}a\\\
b\end{array}\right)_{\mathbf{2}_{k}}=\left(\begin{array}[]{c}wa\\\
wb\end{array}\right)_{\mathbf{2}_{k}},\quad\left(w\right)_{\mathbf{1}_{--}}\otimes\left(\begin{array}[]{c}a\\\
b\end{array}\right)_{\mathbf{2}_{k}}=\left(\begin{array}[]{c}wa\\\
-wb\end{array}\right)_{\mathbf{2}_{k}},$ (96)
$\displaystyle\left(w\right)_{\mathbf{1}_{+-}}\otimes\left(\begin{array}[]{c}a\\\
b\end{array}\right)_{\mathbf{2}_{k}}=\left(\begin{array}[]{c}wb\\\
wa\end{array}\right)_{\mathbf{2}_{k}},\quad\left(w\right)_{\mathbf{1}_{-+}}\otimes\left(\begin{array}[]{c}a\\\
b\end{array}\right)_{\mathbf{2}_{k}}=\left(\begin{array}[]{c}wb\\\
-wa\end{array}\right)_{\mathbf{2}_{k}},$ (105)
$\displaystyle\mathbf{1}_{s_{1}s_{2}}\otimes\mathbf{1}_{s^{\prime}_{1}s^{\prime}_{2}}=\mathbf{1}_{s^{\prime\prime}_{1}s^{\prime\prime}_{2}},$
(106)

where $s_{1,2}\in\\{+,-\\}$, $s^{\prime\prime}_{1}=s_{1}s^{\prime}_{1}$ and
$s^{\prime\prime}_{2}=s_{2}s^{\prime}_{2}$.

The invariant Lagrangian of the theory is constructed with the use of eqs.
(73)-(106). To give an example of the field transformation under
$\mathcal{Q}_{6}$, we take the $y_{T}$ term in eq. (2) where

$\overline{Q}_{L_{D}}\sim{\bf
2}_{2}\sim\left(\begin{array}[]{c}\overline{Q}_{L_{1}}\\\
\overline{Q}_{L_{2}}\end{array}\right)\ \ T_{R}\sim{\bf
2}_{1}\sim\left(\begin{array}[]{c}T_{R_{1}}\\\ T_{R_{2}}\end{array}\right)\ \
H_{1}\sim 1_{+-}$ (107)

The generators $a$ and $b$ act in these fields as follows,

$a:\overline{Q}_{L_{D}}\rightarrow
a\overline{Q}_{L_{D}}=\left(\begin{array}[]{c}\alpha^{2}\overline{Q}_{L_{1}}\\\
\alpha^{-2}\overline{Q}_{L_{2}}\end{array}\right),\ \ T_{R}\rightarrow
aT_{R}=\left(\begin{array}[]{c}\alpha T_{R_{1}}\\\
\alpha^{-1}T_{R_{2}}\end{array}\right),\ \ \text{and}\ \
\tilde{H}_{1}\rightarrow aH_{1}=-\tilde{H}_{1}$ (108)

where $\alpha^{3}=\alpha^{-3}-1$ and

$b:Q_{L_{D}}\rightarrow
bQ_{L_{D}}=\left(\begin{array}[]{c}\overline{Q}_{L_{2}}\\\
\overline{Q}_{L_{1}}\end{array}\right),\ \ T_{R}\rightarrow
bT_{R}=i\left(\begin{array}[]{c}T_{R_{2}}\\\ T_{R_{1}}\end{array}\right)\ \
\text{and}\ \ \tilde{H}_{1}\rightarrow b\tilde{H}_{1}=i\tilde{H}_{1}$ (109)

Therefore, the term $y_{T}$ invariant under the SM gauge group and
$\mathcal{Q}_{6}\times\mathcal{Z}_{2}$ is formed with the tensor product
between the representation ${\bf
1}_{+-}=(\overline{Q}_{L_{1}}T_{R_{1}}-\overline{Q}_{L_{2}}T_{R_{2}})\supset{\bf
2}_{2}\otimes{\bf 2}_{1}$ and ${\bf 1}_{+-}\sim\tilde{H}_{1}$, i.e.
$\overline{Q}_{L_{1}}T_{R_{1}}\tilde{H}_{1}-\overline{Q}_{L_{2}}T_{R_{2}}\tilde{H}_{1}$
as in eq. (6). This term is invariant under $a$ and $b$ transformations,

$a:\left(\overline{Q}_{L_{1}}T_{R_{1}}\tilde{H}_{1}-\overline{Q}_{L_{2}}T_{R_{2}}\tilde{H}_{1}\right)\rightarrow(\alpha^{2}\overline{Q}_{L_{1}})(\alpha\overline{T}_{R_{1}})(-\tilde{H}_{1})-(\alpha^{-2}\overline{Q}_{L_{2}})(\alpha^{-1}\overline{T}_{R_{2}})(-\tilde{H}_{1})$
(110)

$b:\left(\overline{Q}_{L_{1}}T_{R_{1}}\tilde{H}_{1}-\overline{Q}_{L_{2}}T_{R_{2}}\tilde{H}_{1}\right)\rightarrow(\overline{Q}_{L_{2}})(i\overline{T}_{R_{2}})(i\tilde{H}_{1})-(\overline{Q}_{L_{1}})(i\overline{T}_{R_{1}})(i\tilde{H}_{1})$
(111)

##  Appendix B $\chi^{2}$-fit in the quark sector

In order to find the best fit point that successfully reproduces the SM quark
masses and CKM parameters, we proceed to minimize the following $\chi^{2}$
function:

$\chi^{2}=\sum_{f}\frac{(m_{f}^{\text{th}}-m_{f}^{\text{exp}})^{2}}{\sigma_{f}^{2}}+\frac{(|\mathbf{V}_{12}^{\text{th}}|-|\mathbf{V}_{12}^{\text{exp}}|)^{2}}{\sigma_{12}^{2}}+\frac{(|\mathbf{V}_{23}^{\text{th}}|-|\mathbf{V}_{23}^{\text{exp}}|)^{2}}{\sigma_{23}^{2}}+\frac{(|\mathbf{V}_{13}^{\text{th}}|-|\mathbf{V}_{13}^{\text{exp}}|)^{2}}{\sigma_{13}^{2}}+\frac{(J_{q}^{\text{th}}-J_{q}^{\text{exp}})^{2}}{\sigma_{J}^{2}}\,\;,$
(112)

where $f=u,c,t,d,s,b$ and $J_{q}$ is the Jarlskog parameter. The experimental
values for the quark masses are given by Xing (2020),

$\displaystyle\begin{split}m_{u}^{\text{exp}}(M_{Z})&=1.24\pm 0.22\text{
MeV}\;,\\\ m_{c}^{\text{exp}}(M_{Z})&=0.626\pm 0.020\text{ GeV}\;,\\\
m_{t}^{\text{exp}}(M_{Z})&=172.9\pm 0.04\text{ GeV}\;,\\\
m_{d}^{\text{exp}}(M_{Z})&=2.69\pm 0.19\text{ MeV}\;,\\\
m_{s}^{\text{exp}}(M_{Z})&=53.5\pm 4.6\text{ MeV}\;,\\\
m_{b}^{\text{exp}}(M_{Z})&=2.86\pm 0.03\text{ GeV}\;,\end{split}$ (113)

and CKM parameters are Zyla  _et al._ (2020)

$\displaystyle\begin{split}|\mathbf{V}_{12}^{\text{exp}}|=0.22452\pm
0.00044\;,\quad&|\mathbf{V}_{23}^{\text{exp}}|=0.04214\pm
0.00076\;,\quad|\mathbf{V}_{13}^{\text{exp}}|=0.00365\pm 0.00012\;,\\\
&J_{q}^{\text{exp}}=(3.18\pm 0.15)\times 10^{-5}\;.\end{split}$ (114)

The parameters in the quark sector are varied considering real up-type quark
Yukawa couplings while the down-type ones are taken as complex. Under this
assumption, we find that the minimization of the $\chi^{2}$ function yields
the following quark mass matrices:

$\displaystyle M_{U}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccccc}0&0&0&13.2667&0\\\
0&0&0&0&-13.2667\\\ 0&0&210.385&0&0\\\ 88.4086&0&-217.233&-847345.&-314.094\\\
0&-88.4086&-1358.2&314.094&-1932.57\\\ \end{array}\right)GeV$ (120)
$\displaystyle M_{D}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccccc}0&0&0&1.97436\,+1.22717i&0\\\
0&0&0&0&-1.97436-1.22717i\\\ 0&0&2.8349&0&0\\\
-7.81061+35.4833i&0&-98.684+14.9212i&18807.\,-19287.6i&4650.77\,-4769.62i\\\
0&7.81061\,-35.4833i&29.0364\,-6.86904i&-4650.77+4769.62i&183.845\,-188.543i\\\
\end{array}\right)$ (126)

It is important to mention that this solution is not unique. Therefore, given
the large amount of free parameters, there are many more viable solutions
solutions that provide consistent values for the SM quark masses and CKM
parameters and that give rise to same values of vector like quark masses used
in our benchmark scenario. Furthermore, let us note that the lepton sector
observables are easily fit using a similar procedure to the one described in
this section.

##  Appendix C Kinematic and angular distributions for the VLLs and VLQs

Figure 10: Set of angular and kinematic observables for VLQ pair-production
with a mass $m_{T_{1}}=2.2$ TeV. All histograms are normalized, with each
having 30 bins. Signal distributions are filled and shown in red, while
background distributions are unfilled and shown in blue ($pp\rightarrow
e^{+}e^{-}\mu^{+}\mu^{-}j_{1}j_{2}$) and green ($t\bar{t}Z$). From top-left to
bottom-right, we have $\Delta\phi$ angles for final states combinations
$(e^{+},\mu^{-})$, $(e^{+},j_{1})$, $(\mu^{+},j_{1})$ and $(\mu^{-},j_{2})$;
$\Delta R$ distributions for the final states combinations $(e^{-},j_{2})$,
$(e^{+},j_{2})$, $(\mu^{+},j_{2})$ and $(\mu^{-},j_{1})$; distributions of the
cosine of the angle within the pair of states $(e^{-},j_{2})$,
$(e^{+},j_{1})$, $(e^{-},\mu^{-})$, $(\mu^{-},j_{2})$ and $(j_{1},j_{2})$;
distributions for the cosine of the angle $\Delta\Phi_{1}$ and
$\Delta\Phi_{2}$; pseudo-rapidity for $e^{-}$, $e^{+}$, $\mu^{-}$ and
$\mu^{+}$; transverse momentum for $e^{-}$, $e^{+}$, $\mu^{-}$, $\mu^{+}$ and
invariant mass distributions for the combinations $(\mu^{+},\mu^{-},j_{1})$,
$(\mu^{+},\mu^{-},j_{2})$, $(e^{+},e^{-},j_{1})$ and $(e^{+},e^{-},j_{2})$.
Figure 11: Set of angular and kinematic observables for VLL pair-production
via the VBF topology with a mass $m_{E_{2}}=200$ GeV. All histograms are
normalized, with each having 30 bins. Signal distributions are filled and
shown in red, while background distributions are unfilled and shown in blue
($W^{+}W^{-}$) and green ($t\bar{t}+(j,jj)$). From top-left to bottom-right,
we have $\Delta R$ distributions for final states combinations $(e^{-},\nu)$
and $(\mu^{+},\nu)$; distributions for cosine of angle $\Delta\Phi$ and
$\Delta\Theta$ in the laboratory and $E_{2}\bar{E}_{2}$ centre-of-mass frames;
distributions of the cosine of the angle between the pairs of particles
$(\nu,e^{-})$, $\nu,\mu^{+}$ and $W^{+}W^{-}$; pseudo-rapidity for the
particles $e^{-}$, $E_{2}$, $\bar{E}_{2}$, $\mu^{+}$, $W^{+}$ and $W^{-}$;
mass distributions for $E_{2}$ and $\bar{E}_{2}$; MET; transverse mass
distributions for $W^{+}$ and $W^{-}$ and transverse momentum distributions
for $e^{-}$, $E_{2}$, $\bar{E}_{2}$, $\mu^{+}$, $W^{+}$ and $W^{-}$. Figure
12: Set of angular and kinematic observables for VLL pair-production via the
ZA topology with a mass $m_{E_{2}}=200$ GeV. All histograms are normalized,
with each having 30 bins. Signal distributions are filled and shown in red,
while background distributions are unfilled and shown in blue
($t\bar{t}^{-}$), green ($W^{+}W^{-}$), yellow ($t\bar{t}+Z(\ell\ell)$) and
pink ($t\bar{t}+Z(\bar{\nu}_{\ell}\nu_{\ell})$). From top-left to bottom-
right, we have $\Delta R$ distributions for final states combinations
$(e^{-},\nu)$ and $(\mu^{+},\nu)$; distributions for cosine of angle
$\Delta\Phi$ and $\Delta\Theta$ in the laboratory and $E_{2}\bar{E}_{2}$
centre-of-mass frames; distributions of the cosine of the angle between the
pairs of particles $(\nu,e^{-})$, $\nu,\mu^{+}$ and $W^{+}W^{-}$; pseudo-
rapidity for the particles $e^{-}$, $E_{2}$, $\bar{E}_{2}$, $\mu^{+}$, $W^{+}$
and $W^{-}$; mass distributions for $E_{2}$ and $\bar{E}_{2}$; MET; transverse
mass distributions for $W^{+}$ and $W^{-}$ and transverse momentum
distributions for $e^{-}$, $E_{2}$, $\bar{E}_{2}$, $\mu^{+}$, $W^{+}$ and
$W^{-}$. Figure 13: Set of angular and kinematic observables for VLL single-
production via the VLBSM topology with a mass $m_{E_{2}}=200$ GeV. All
histograms are normalized, with each having 30 bins. Signal distributions are
filled and shown in red, while background distributions are unfilled and shown
in blue ($pp\rightarrow e^{-}\bar{\nu}_{e}$) and green ($pp\rightarrow
e^{-}\bar{\nu}_{e}+(j,jj)$). From top-left to bottom-right, we have
distributions for cosine of the angle for particles $e^{-}$ and $W^{-}$;
pseudo-rapidity for $e^{-}$ and $W^{-}$; MET; transverse mass of the $W^{-}$
boson, azimuthal angle for $e^{-}$ and transverse momentum for $e^{-}$ and
$W^{-}$. Figure 14: Set of angular and kinematic observables for VLL double-
production for both VBF and ZA topologies with a mass $m_{E_{2}}=200$ GeV. All
histograms are normalized, with each having 30 bins. VBF channel corresponds
to the plots where only two backgrounds are present, whereas ZA plots show
four backgrounds. From top-left to bottom-right, for both signals, we have
pseudo-rapidity distributions for $\bar{E}_{2}$, $E_{2}$, $e^{-}$ and
$\mu^{+}$ and transverse momentum distributions for $E_{2}$, $\bar{E}_{2}$,
$e^{-}$ and $\mu^{+}$.

##  Appendix D Deep Learning significance plots

In this appendix, plots of the statistical significance as a function of the
cut on the classifier score are presented, for both a VLL with mass 200 GeV
(for every topology studied in this paper) and for a 2.2 TeV VLQ. The values
for the cut on the classifier score are assigned by the NN architecture. Such
values are real numbers (often referred to as decision boundary or decision
surface) and can be interpreted as a continuous label that identifies whether
a set of features (kinematic and angular observables) is classified as signal
or as background. As an example, in Fig. 16 (a), the values for the classifier
score around $\sim 0.56$ to $\sim 0.61$ correspond to the region where the
significance is maximized, i.e., where the NN has found a maximal number of
signal events for a non-zero background. Notice that from the definition of
the Asimov significance in Eq. (54), for a region with zero background
$\mathcal{Z_{A}}$ would diverge and therefore is not considered in our
analysis. On the other hand, for scores above $0.61$ the number of background
events increases and the statistical significance quickly drops to zero.

(a) ZA topologies (b) ZA topologies (c) ZA topologies

(d) VBF topologies (e) VBF topologies (f) VBF topologies

(g) VLBSM topologies (h) VLBSM topologies (i) VLBSM topologies

Figure 15: Statistical significance as a function of the classifier score
given by NN for each signal graph, assuming the lightest VLL with mass
$m_{E_{2}}=200\leavevmode\nobreak\ \mathrm{GeV}$ and a luminosity
$\mathcal{L}=3000$ $\mathrm{fb}^{-1}$. All showcased plots are representative
of the best NN found, following a genetic algorithm that maximizes the Asimov
significance. For plots (a), (d) and (g) we present the Asimov significance,
with 1% systematics, for (b), (c) and (h) – the standard significance
$s/\sqrt{s+b}$ and for (c), (f) and (i) – the adapted Asimov significance,
where it is assumed that backgrounds are known with error $10^{-3}$.

(a)  (b)  (c)

Figure 16: Statistical significance as a function of the classifier score
given by NN for each signal graph, assuming the lightest VLQ with mass
$m_{T_{1}}=2.2\leavevmode\nobreak\ \mathrm{TeV}$ and a luminosity
$\mathcal{L}=3000$ $\mathrm{fb}^{-1}$. All showcased plots are representative
of the best NN found, following an genetic algorithm that maximizes the Asimov
significance. For (a) we present the Asimov significance, with 1% systematics,
for (b) – the standard significance $s/\sqrt{s+b}$ and for (c) – the adapted
Asimov significance, where it is assumed that backgrounds are known with error
$10^{-3}$.

##  Appendix E Relevant Feynman Rules for the collider analysis

For the diagrams that we have included in this appendix, we have defined
$\theta_{W}$ as the Weinberg angle, $U^{u}$, $U^{e}$ and $U_{\nu}$ are up-
quark, charged lepton and neutrino mixing matrices, respectively, $g_{2}$ the
SU(2) coupling constant, $g_{1}$ the U(1) gauge coupling constant and
$\gamma_{\mu}$/$\gamma_{5}$ are Dirac gamma matrices. $\delta_{\alpha\beta}$
is a Kronecker delta in color space and $\delta_{ij}$ is a Kronecker delta in
generation space.

The Feynman rule for the quark-antiquark-photon interactions has the form:

$\displaystyle-\frac{i}{6}\delta_{\alpha\beta}\Big{(}U^{u,*}_{L,{j1}}\Big{(}3g_{2}\sin\theta_{W}+g_{1}\cos\theta_{W}\Big{)}U_{L,{i1}}^{u}+U^{u,*}_{L,{j2}}\Big{(}3g_{2}\sin\theta_{W}+g_{1}\cos\theta_{W}\Big{)}U_{L,{i2}}^{u}$
$\displaystyle+g_{1}U^{u,*}_{L,{j3}}\cos\theta_{W}U_{L,{i3}}^{u}+3g_{2}U^{u,*}_{L,{j3}}\sin\theta_{W}U_{{L},{i3}}^{u}+4g_{1}U^{u,*}_{L,{j4}}\cos\theta_{W}U_{L,{i4}}^{u}$
$\displaystyle+4g_{1}U^{u,*}_{L,{j5}}\cos\theta_{W}U_{L,{i5}}^{u}\Big{)}\Big{(}\gamma_{\mu}\cdot\frac{1-\gamma_{5}}{2}\Big{)}-\frac{2i}{3}g_{1}\cos\theta_{W}\delta_{\alpha\beta}\delta_{ij}\Big{(}\gamma_{\mu}\cdot\frac{1+\gamma_{5}}{2}\Big{)},$
(127)

whereas the $Z$-boson quark-antiquark interaction is written as

$\displaystyle-\frac{i}{6}\delta_{\alpha\beta}\Big{(}U^{u,*}_{L,{j1}}\Big{(}3g_{2}\cos\theta_{W}-g_{1}\sin\theta_{W}\Big{)}U_{L,{i1}}^{u}+U^{u,*}_{L,{j2}}\Big{(}3g_{2}\cos\theta_{W}-g_{1}\sin\theta_{W}\Big{)}U_{L,{i2}}^{u}$
$\displaystyle+3g_{2}U^{u,*}_{L,{j3}}\cos\theta_{W}U_{L,{i3}}^{u}-g_{1}U^{u,*}_{L,{j3}}\sin\theta_{W}U_{L,{i3}}^{u}-4g_{1}U^{u,*}_{L,{j4}}\sin\theta_{W}U_{L,{i4}}^{u}$
$\displaystyle-4g_{1}U^{u,*}_{L,{j5}}\sin\theta_{W}U_{L,{i5}}^{u}\Big{)}\Big{(}\gamma_{\mu}\cdot\frac{1-\gamma_{5}}{2}\Big{)}+\,\frac{2i}{3}g_{1}\delta_{\alpha\beta}\sin\theta_{W}\delta_{ij}\Big{(}\gamma_{\mu}\cdot\frac{1+\gamma_{5}}{2}\Big{)}.$
(128)

From the expressions given above, using the unitarity of the $U_{L}^{u}$
matrix and taking into account that $U^{u}_{L,{jn}}\lesssim 10^{-3}$
$(n=4,3)$, (see eq. (57)), it follows that the above given flavor violating
photon interaction is strongly suppressed. Furthermore, it is worth mentioning
that there is also a quark flavor violating $Z$ boson interaction, however its
corresponding amplitude is suppressed by the same unitarity arguments and that
amplitude is lower than the one corresponding to the photon by a factor of
$\sim\tan\theta_{W}$. Because of this reason we do not show in Figure 5 the
subleading contribution arising from $Z$ violating quark interaction to the
VLQ pair-production via gluon fusion mechanism.

$\displaystyle-i\frac{1}{\sqrt{2}}g_{2}\sum_{a=1}^{3}U^{*}_{{\nu},{ja}}U_{L,{ia}}^{e}\Big{(}\gamma_{\mu}\cdot\frac{1-\gamma_{5}}{2}\Big{)}$
(129)

## References

  * King and Luhn (2013) S. F. King and C. Luhn, Rept. Prog. Phys. 76, 056201 (2013), arXiv:1301.1340 [hep-ph] .
  * Altarelli and Feruglio (2010) G. Altarelli and F. Feruglio, Rev. Mod. Phys. 82, 2701 (2010), arXiv:1002.0211 [hep-ph] .
  * Ishimori  _et al._ (2010) H. Ishimori, T. Kobayashi, H. Ohki, Y. Shimizu, H. Okada, and M. Tanimoto, Prog. Theor. Phys. Suppl. 183, 1 (2010), arXiv:1003.3552 [hep-th] .
  * King (2015) S. F. King, J. Phys. G42, 123001 (2015), arXiv:1510.02091 [hep-ph] .
  * Gerard (1983) J. M. Gerard, Z. Phys. C18, 145 (1983).
  * Frampton and Kephart (1995) P. H. Frampton and T. W. Kephart, Int. J. Mod. Phys. A10, 4689 (1995), arXiv:hep-ph/9409330 [hep-ph] .
  * Grimus and Lavoura (2003) W. Grimus and L. Lavoura, Phys. Lett. B572, 189 (2003), arXiv:hep-ph/0305046 [hep-ph] .
  * Kubo  _et al._ (2003) J. Kubo, A. Mondragon, M. Mondragon, and E. Rodriguez-Jauregui, Prog. Theor. Phys. 109, 795 (2003), [Erratum: Prog. Theor. Phys.114,287(2005)], arXiv:hep-ph/0302196 [hep-ph] .
  * Kubo (2004) J. Kubo, Phys. Lett. B578, 156 (2004), [Erratum: Phys. Lett.B619,387(2005)], arXiv:hep-ph/0309167 [hep-ph] .
  * Babu and Kubo (2005) K. S. Babu and J. Kubo, Phys. Rev. D71, 056006 (2005), arXiv:hep-ph/0411226 [hep-ph] .
  * Kajiyama  _et al._ (2006) Y. Kajiyama, E. Itou, and J. Kubo, Nucl. Phys. B743, 74 (2006), arXiv:hep-ph/0511268 [hep-ph] .
  * Lovrekovic (2012) I. Lovrekovic, (2012), arXiv:1212.1145 [hep-ph] .
  * Branco  _et al._ (2012) G. C. Branco, P. M. Ferreira, L. Lavoura, M. N. Rebelo, M. Sher, and J. P. Silva, Phys. Rept. 516, 1 (2012), arXiv:1106.0034 [hep-ph] .
  * Froggatt and Nielsen (1979) C. D. Froggatt and H. B. Nielsen, Nucl. Phys. B 147, 277 (1979).
  * Bennett  _et al._ (2006) G. W. Bennett _et al._ (Muon g-2), Phys. Rev. D73, 072003 (2006), arXiv:hep-ex/0602035 [hep-ex] .
  * Abi  _et al._ (2021) B. Abi _et al._ (Muon g-2), Phys. Rev. Lett. 126, 141801 (2021), arXiv:2104.03281 [hep-ex] .
  * Glashow and Weinberg (1977) S. L. Glashow and S. Weinberg, Phys. Rev. D 15, 1958 (1977).
  * Paschos (1977) E. A. Paschos, Phys. Rev. D 15, 1966 (1977).
  * Diaz  _et al._ (2003) R. A. Diaz, R. Martinez, and J. A. Rodriguez, Phys. Rev. D67, 075011 (2003), arXiv:hep-ph/0208117 [hep-ph] .
  * Jegerlehner and Nyffeler (2009) F. Jegerlehner and A. Nyffeler, Phys. Rept. 477, 1 (2009), arXiv:0902.3360 [hep-ph] .
  * Kelso  _et al._ (2014) C. Kelso, H. N. Long, R. Martinez, and F. S. Queiroz, Phys. Rev. D90, 113011 (2014), arXiv:1408.6203 [hep-ph] .
  * Lindner  _et al._ (2018) M. Lindner, M. Platscher, and F. S. Queiroz, Phys. Rept. 731, 1 (2018), arXiv:1610.06587 [hep-ph] .
  * Raby and Trautner (2018) S. Raby and A. Trautner, Phys. Rev. D97, 095006 (2018), arXiv:1712.09360 [hep-ph] .
  * Cárcamo Hernández  _et al._ (2020) A. E. Cárcamo Hernández, S. F. King, H. Lee, and S. J. Rowley, Phys. Rev. D101, 115016 (2020), arXiv:1910.10734 [hep-ph] .
  * Dermisek  _et al._ (2021) R. Dermisek, K. Hermanek, N. McGinnis, and N. McGinnis, Phys. Rev. Lett. 126, 191801 (2021), arXiv:2011.11812 [hep-ph] .
  * Freitas  _et al._ (2021) F. F. Freitas, J. Gonçalves, A. P. Morais, and R. Pasechnik, JHEP 01, 076 (2021), arXiv:2010.01307 [hep-ph] .
  * Staub (2014) F. Staub, Comput. Phys. Commun. 185, 1773 (2014), arXiv:1309.7223 [hep-ph] .
  * Degrande  _et al._ (2012) C. Degrande, C. Duhr, B. Fuks, D. Grellscheid, O. Mattelaer, and T. Reiter, Comput. Phys. Commun. 183, 1201 (2012), arXiv:1108.2040 [hep-ph] .
  * Alwall  _et al._ (2014) J. Alwall, R. Frederix, S. Frixione, V. Hirschi, F. Maltoni, O. Mattelaer, H. S. Shao, T. Stelzer, P. Torrielli, and M. Zaro, JHEP 07, 079 (2014), arXiv:1405.0301 [hep-ph] .
  * Sjöstrand  _et al._ (2015) T. Sjöstrand, S. Ask, J. R. Christiansen, R. Corke, N. Desai, P. Ilten, S. Mrenna, S. Prestel, C. O. Rasmussen, and P. Z. Skands, Comput. Phys. Commun. 191, 159 (2015), arXiv:1410.3012 [hep-ph] .
  * de Favereau  _et al._ (2014) J. de Favereau, C. Delaere, P. Demin, A. Giammanco, V. Lemaître, A. Mertens, and M. Selvaggi (DELPHES 3), JHEP 02, 057 (2014), arXiv:1307.6346 [hep-ex] .
  * Brun and Rademakers (1997) R. Brun and F. Rademakers, Nucl. Instrum. Meth. A 389, 81 (1997).
  * Hoeche  _et al._ (2005) S. Hoeche, F. Krauss, N. Lavesson, L. Lonnblad, M. Mangano, A. Schalicke, and S. Schumann, in  _HERA and the LHC: A Workshop on the Implications of HERA for LHC Physics: CERN - DESY Workshop 2004/2005 (Midterm Meeting, CERN, 11-13 October 2004; Final Meeting, DESY, 17-21 January 2005)_ (2005) pp. 288–289, arXiv:hep-ph/0602031 .
  * Garcia Garcia  _et al._ (2015) I. Garcia Garcia, K. Howe, and J. March-Russell, JHEP 12, 005 (2015), arXiv:1510.07045 [hep-ph] .
  * Bhattacherjee  _et al._ (2018) B. Bhattacherjee, P. Byakti, A. Kushwaha, and S. K. Vempati, JHEP 05, 090 (2018), arXiv:1702.06417 [hep-ph] .
  * Crivellin  _et al._ (2020) A. Crivellin, F. Kirk, C. A. Manzari, and M. Montull, JHEP 12, 166 (2020), arXiv:2008.01113 [hep-ph] .
  * Sirunyan  _et al._ (2019a) A. M. Sirunyan _et al._ (CMS), Phys. Rev. D 100, 052003 (2019a), arXiv:1905.10853 [hep-ex] .
  * Achard  _et al._ (2001) P. Achard _et al._ (L3), Phys. Lett. B 517, 75 (2001), arXiv:hep-ex/0107015 .
  * CMS (2009) (2009).
  * Hebbar  _et al._ (2016) A. Hebbar, G. K. Leontaris, and Q. Shafi, Phys. Rev. D 93, 111701 (2016), arXiv:1604.08328 [hep-ph] .
  * Hewett and Rizzo (1989) J. L. Hewett and T. G. Rizzo, Physics Reports 183, 193 (1989).
  * Benbrik  _et al._ (2016) R. Benbrik, C.-H. Chen, and T. Nomura, Phys. Rev. D 93, 055034 (2016), arXiv:1512.06028 [hep-ph] .
  * Hernández and Schmidt (2021) A. E. C. Hernández and I. Schmidt, (2021), arXiv:2101.02718 [hep-ph] .
  * Aaboud  _et al._ (2019a) M. Aaboud _et al._ (ATLAS), JHEP 05, 164 (2019a), arXiv:1812.07343 [hep-ex] .
  * Aaboud  _et al._ (2019b) M. Aaboud _et al._ (ATLAS), JHEP 05, 041 (2019b), arXiv:1812.09743 [hep-ex] .
  * Sirunyan  _et al._ (2019b) A. M. Sirunyan _et al._ (CMS), Phys. Rev. D 100, 072001 (2019b), arXiv:1906.11903 [hep-ex] .
  * Collaboration (2021) A. Collaboration, “Atlas exotics searches - 95% cl upper exclusion limits,” https://atlas.web.cern.ch/Atlas/GROUPS/PHYSICS/PUBNOTES/ATL-PHYS-PUB-2021-009/fig_01.png (2021).
  * Roy  _et al._ (2020) A. Roy, N. Nikiforou, N. Castro, and T. Andeen, Phys. Rev. D 101, 115027 (2020), arXiv:2003.00640 [hep-ph] .
  * Araque  _et al._ (2015) J. P. Araque, N. F. Castro, and J. Santiago, JHEP 11, 120 (2015), arXiv:1507.05628 [hep-ph] .
  * Romão Crispim  _et al._ (2020) M. Romão Crispim, N. F. Castro, R. Pedro, and T. Vale, Phys. Rev. D 101, 035042 (2020), arXiv:1912.04220 [hep-ph] .
  * Crispim Romão  _et al._ (2021) M. Crispim Romão, N. F. Castro, and R. Pedro, Eur. Phys. J. C 81, 27 (2021), arXiv:2006.05432 [hep-ph] .
  * Chollet (2015) F. Chollet, “Keras,” https://github.com/fchollet/keras (2015).
  * Abadi  _et al._ (2016) M. Abadi _et al._ , (2016), arXiv:1603.04467 [cs.DC] .
  * Sola and Sevilla (1997) J. Sola and J. Sevilla, IEEE Transactions on Nuclear Science 44, 1464 (1997).
  * LeCun  _et al._ (2012) Y. A. LeCun, L. Bottou, G. B. Orr, and K.-R. Müller, “Efficient backprop,” in  _Neural Networks: Tricks of the Trade: Second Edition_, edited by G. Montavon, G. B. Orr, and K.-R. Müller (Springer Berlin Heidelberg, Berlin, Heidelberg, 2012) pp. 9–48.
  * Chawla  _et al._ (2002) N. V. Chawla, K. W. Bowyer, L. O. Hall, and W. P. Kegelmeyer, Journal of Artificial Intelligence Research 16, 321–357 (2002).
  * Elwood and Krücker (2018) A. Elwood and D. Krücker, (2018), arXiv:1806.00322 [hep-ex] .
  * Zyla  _et al._ (2020) P. A. Zyla _et al._ (Particle Data Group), PTEP 2020, 083C01 (2020).
  * Kumar and Martin (2015) N. Kumar and S. P. Martin, Phys. Rev. D 92, 115018 (2015), arXiv:1510.03456 [hep-ph] .
  * Xing (2020) Z.-z. Xing, Phys. Rept. 854, 1 (2020), arXiv:1909.09610 [hep-ph] .