aainstitutetext: Department of Physics, Concordia University, 7141 Sherbrooke
St. West, Montreal, Quebec H4B 1R6, Canadabbinstitutetext: Laboratoire de
Physique Théorique et Hautes Énergies (LPTHE), UMR 7589, Sorbonne Université
et CNRS, 4 place Jussieu, 75252 Paris Cedex 05, France

# Collider imprint of vector-like leptons in light of anomalous magnetic
moment and neutrino data

Parham Dehghani a , Mariana Frank b  and Benjamin Fuks
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We investigate the impact of incorporating anomaly-free representations of
vector-like leptons into the Standard Model, aiming to address longstanding
puzzles related to the anomalous magnetic moments of the muon and electron
while maintaining consistency with neutrino masses and mixings. We find that
among the various representations of vector-like leptons permitted by the
Standard Model gauge symmetry, only weak doublets and singlets offer
satisfactory solutions, all associated with a significantly constrained
parameter space. Our analysis delves into the associated parameter space,
identifying representative benchmark scenarios suitable for collider studies.
These setups yield a distinctive six-lepton signature whose associated signals
can easily be distinguished from the Standard Model background, providing a
clear signal indicative of new physics models featuring vector-like leptons.
Our work hence sheds light on the potential implications of vector-like
leptons in resolving discrepancies inherent to the Standard Model, while also
offering insights into experimental avenues for further exploration.

##  1 Introduction

While the Standard Model (SM) has achieved remarkable success in describing
fundamental particles and their interactions, its formulation includes
important theoretical shortcomings. These have prompted extensive exploration
of models Beyond the Standard Model (BSM), which often involve enlarging the
particle content and/or gauge structure of the SM. Such extensions aim to
address specific challenges inherent to the SM, including the existence of
dark matter, the origin of neutrino masses, and the asymmetry between matter
and antimatter in the universe. On the other hand, discrepancies between SM
predictions and experimental measurements provide strong motivation for the
development of new physics theories.

One notable example of this tension is evident in the precise measurement of
the muon’s anomalous magnetic moment, denoted $a_{\mu}$, and the associated
theoretical predictions. Combining data from experiments at Fermilab (FNAL)
and Brookhaven National Laboratory (BNL), we get an experimental value given
by Muong-2:2021ojo ; Muong-2:2006rrc ; Muong-2:2023cdq ,

$a_{\mu}^{\rm exp}\,(\text{FNAL+BNL})=116~{}592~{}059\,(22)\times
10^{-11}\,(0.19\,\rm ppm)\,.$ (1)

This measured value exhibits a significant deviation from the SM prediction,

$a_{\mu}^{\rm SM}=116~{}591~{}810\,(43)\times 10^{-11}\,,$ (2)

as evaluated by the Muon $g-2$ Theory Initiative in 2020 Aoyama:2020ynm .
However, the recent evaluation of the leading-order hadronic vacuum
polarisation contribution to $a_{\mu}$ using lattice simulations suggests a
reconciliation between the theoretical and experimental values for $a_{\mu}$
Borsanyi:2020mff ; Ce:2022kxy ; ExtendedTwistedMass:2022jpw ;
FermilabLatticeHPQCD:2023jof . These lattice results nevertheless lead to
discrepancies with the measurements of properties of
$e^{+}e^{-}\to\pi^{+}\pi^{-}$ scattering Wittig:2023pcl , leaving hence the
possibility open for new physics contributions. Thus, the $(g-2)_{\mu}$ puzzle
continues to motivate the exploration of BSM avenues for restoring agreement
between theory and data, as showcased for instance in Jegerlehner:2009ry .

On the other hand, there also exist tensions between measurements and SM
predictions for the electron anomalous magnetic moment $a_{e}$. In the SM,
$a_{e}$ computations include QED corrections up to the tenth order, and they
rely on the measurement of the fine-structure constant using the recoil
velocity and frequency of atoms that absorb a photon Aoyama:2017uqe ,

$a_{e}^{\rm SM}=1159652181.643\times 10^{-12}\,.$ (3)

The most recent combined experimental measurements yield Fan:2022eto

$a_{e}^{\rm exp}=115965218(12)\times 10^{-11}\,,$ (4)

although there is currently a $5.5\sigma$ discrepancy between data obtained
using Rubidium-87 Morel:2020dww  and Cesium-133 Parker:2018  atoms,

$\Delta
a_{e}\equiv\,a^{\text{exp}}_{e}-a^{\text{SM}}_{e}=\left\\{\begin{array}[]{cc}48(30)\times
10^{-14}&{\rm Rb}\\\ -88(36)\times 10^{-14}&{\rm Cs}\end{array}\right.~{}.$
(5)

As in the case of the anomalous magnetic moment of the muon, this puzzle could
be indicative of the presence of physics beyond the SM.

Several BSM extensions propose the existence of new exotic heavy leptons,
notably vector-like leptons (VLLs) with identical left-handed and right-handed
couplings akin to those originating from composite paradigms Panico:2015jxa ;
Cacciapaglia:2020kgq ; Cacciapaglia:2022zwt  or present in several neutrino
mass models Cai:2017mow . Equipping particle physics theories with VLLs has
moreover shown promises in addressing various challenges and anomalies. For
instance, BSM setups with VLLs can encompass a dark matter candidate
Schwaller:2013hqa ; Bahrami:2016has , and they may contribute to resolving
discrepancies in the muon’s anomalous magnetic moment, hence potentially
restoring agreement between theory and data Poh:2017tfo ; Chun:2020uzw ;
Lee:2022nqz . Additionally, VLLs could predict deviations from lepton
universality in $B$-decays Poh:2017tfo , offer an explanation for the Cabibbo
angle anomaly Crivellin:2020ebi , and align predictions of the $W$-boson mass
with recent experimental measurements from the CDF collaboration Lee:2022nqz .

Experimental constraints on VLLs remain relatively weak and highly dependent
on the specific model under consideration. These constraints primarily depend
on the VLL’s representation under the electroweak symmetry group and their
couplings to SM leptons. For example, weak VLL doublets coupling to taus are
constrained to be heavier than approximately 1 TeV CMS:2019hsm ; ATLAS:2023sbu
, while the limit drops to the range of 100-200 GeV in case of VLL singlets
CMS:2022nty . Additionally, bounds from previous experiments at LEP are always
valid, imposing a lower limit on the VLL mass of about 100 GeV, the exact
value depending on the specific model details ALEPH:1996akm ; DELPHI:2004ili ;
L3:2001xsz ; OPAL:2002wkp .

One crucial motivation for including VLLs in theoretical frameworks is their
potential to extend the SM neutrino sector to accommodate a variety of
experimental data. Neutrinos, among the most mysterious particles in the
universe, play essential roles in understanding weak interactions, as well as
cosmology, nuclear physics and astrophysics. While expected to be massless in
the SM, neutrinos have been shown to oscillate between flavours, indicating
that they have non-zero masses. While the neutrino absolute masses are not
known, neutrino oscillations measured mass-square differences, found to be
$m_{\nu_{2}}^{2}-m_{\nu_{1}}^{2}=7.5\times 10^{-5}$ eV2 and
$m_{\nu_{3}}^{2}-m_{\nu_{1}}^{2}=\pm 2.5\times 10^{-3}$ eV2 deGouvea:2016qpx .
Additionally, cosmic surveys indirectly constrain the sum of neutrino masses
to be at the sub-eV level, $\sum_{j}m_{\nu_{j}}<0.23$ eV at the 95% confidence
level Loureiro:2018pdz . The discovery of non-zero neutrino masses is probably
among the most important particle physics results of the last two decades, as
it suggests that the SM is incomplete. Thus any consistent model of BSM
physics should be able to provide a mechanism and an explanation for neutrino
masses and mixings.

In this work, we conduct a comprehensive analysis of models incorporating VLLs
irrespective of their representation under the SM gauge symmetry, with a main
focus on anomaly-free setups and their consequences at colliders. We explore
various BSM constructions with VLL singlets, doublets and triplets, and we
probe their potential role in addressing fundamental physics puzzles such as
dark matter, the muon and electron anomalous magnetic moments, and neutrino
masses (all of these corresponding to limitations of the SM that could be
related to the existence of an extended leptonic sector). Despite introducing
only one or multiple VLL representations, we find no common explanation that
could account for all these phenomena. While the idea of dark matter being
associated with a vector-like neutrino has been extensively discussed in prior
literature Jegerlehner:2009ry , our emphasis lies on options that provide
explanations for discrepancies between theoretical predictions and
experimental measurements in the context of the lepton anomalous magnetic
moments, and that are suitable with respect to neutrino mass and mixing
constraints. Additionally, we ensure that our solutions are compatible with
electroweak precision data, and finally we investigate their implications at
collider experiments.

The rest of this study is organised as follows. In section 2 we introduce our
theoretical framework and write down the VLL Lagrangian that we consider, we
present the experimental constraints included in our analysis, and we
introduce the software tools that we use to conduct a comprehensive parameter
space scan. Section 3 is dedicated to a presentation of our results, with a
special focus on the anomalous magnetic moments of the electron and muon in
section 3.1, on neutrino data in section 3.2, and on electroweak precision
tests in section 3.3. In section 4 we select representative benchmark
scenarios and determine their collider signatures. Finally, we summarise our
findings and conclude in section 5.

##  2 Theoretical Framework

Unlike SM-like chiral fermions, whose left-handed and right-handed components
transform differently under the SM gauge symmetry $SU(3)_{C}\otimes
SU(2)_{L}\otimes U(1)_{Y}$, vector-like fermions exhibit identical properties
for both their chiralities. However, the specific choice of representation for
these fermions is not unique. In models featuring VLLs, building a consistent,
renormalisable, and anomaly-free extension of the SM imposes constraints on
the possible choices for their $SU(2)_{L}$ and hypercharge quantum numbers.

The list of acceptable choices, along with our notation for the fields and
their weak multiplet components, is provided in Table 1. The first two singlet
possibilities, ${\cal S}_{1}$ and ${\cal S}_{2}$, correspond to VLLs with
quantum numbers matching those of a right-handed charged lepton $E$ or
neutrino $N$ (considered as a Dirac fermion in the following), respectively.
In addition, VLL doublets can be either SM-like or possess different quantum
numbers (${\cal D}_{1}$, and ${\cal D}_{2}$). In the cases of the ${\cal
D}_{1}$ state, the doublet component fields correspond to a neutral and
singly-charged lepton, while for ${\cal D}_{2}$ they include a singly-charged
and a doubly-charged state. Similarly, the first triplet option, ${\cal
X}_{1}$, includes only neutral and singly-charged VLLs and is considered as a
Dirac fermion in the following, whereas the second possibility, ${\cal
X}_{2}$, is more exotic and features a neutral, a singly-charged and a doubly-
charged component.

| ${\cal S}_{1}$ | ${\cal S}_{2}$ | ${\cal D}_{1}$ | ${\cal D}_{2}$ | ${\cal X}_{1}$ | ${\cal X}_{2}$
---|---|---|---|---|---|---
Multiplets | $E^{-}$ | $N^{0}$ | $\begin{pmatrix}{\cal D}_{1}^{0}\\\ {\cal D}_{1}^{-}\end{pmatrix}$ | $\begin{pmatrix}{\cal D}_{2}^{-}\\\ {\cal D}_{2}^{--}\end{pmatrix}$ | $\begin{pmatrix}\frac{1}{\sqrt{2}}{\cal X}_{1}^{0}&{\cal X}_{1}^{+}\\\ {\cal X}_{1}^{-}&-\frac{1}{\sqrt{2}}{\cal X}_{1}^{0}\end{pmatrix}$ | $\begin{pmatrix}\frac{1}{\sqrt{2}}{\cal X}_{2}^{-}&{\cal X}_{2}^{0}\\\ {\cal X}_{2}^{--}&-\frac{1}{\sqrt{2}}{\cal X}_{2}^{-}\end{pmatrix}$
$SU(2)_{L}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{2}$ | $\mathbf{2}$ | $\mathbf{3}$ | $\mathbf{3}$
$U(1)_{Y}$ | $-1$ | 0 | $-1/2$ | $-3/2$ | $0$ | $-1$
Table 1:  Possible VLL representations leading to an anomaly-free and
renormalisable BSM theory. We consider weak singlets (${\cal S}_{1}$ and
${\cal S}_{2}$), doublets (${\cal D}_{1}$, ${\cal D}_{2}$), and triplets
(${\cal X}_{1}$ and ${\cal X}_{2}$) with different hypercharge quantum number
assignments. In the second line of the table, we provide the components of the
various weak multiplets considered (with their electric charge explicitly
indicated), while the $SU(2)_{L}\times U(1)_{Y}$ quantum numbers are given in
the last lines of the table.

To address the anomalous magnetic moment of both the electron and the muon as
well as other puzzles of the SM such as neutrino masses and mixings, we
construct a theoretical framework that incorporates one or more of the
considered VLL representations, each endowed with a generation index.
Furthermore, we enforce that a given VLL species only interacts with the
corresponding generation of ordinary leptons. This approach, distinct from
introducing VLL fields with inter-generational couplings, allows us to analyse
the impact of the model on the anomalous magnetic moments of the electron and
muon separately, in a way that avoids inducing lepton-violating decays. In our
new physics setup, the VLL collider signatures and their corresponding rates
(including production cross sections and relevant branching ratios) depend on
how the different VLLs mix with ordinary leptons, which stems from their
Yukawa interaction with the Higgs field $\Phi$, as well as their mass terms.
The corresponding Lagrangian contributions, with generational indices
understood, are given by

$\begin{split}&{\cal
L}_{\mathrm{Y}}\\!=-\\!\bigg{[}\mathbf{Y_{e}}\,{\bar{L}}\Phi
e+\mathbf{Y_{\nu}}\,({\bar{L}}\cdot\Phi^{\dagger})\nu+\mathbf{\hat{Y}_{\mathcal{S}_{1}}}\,{\bar{L}}\Phi{\cal
S}_{1}+\mathbf{\hat{Y}_{\mathcal{S}_{2}}}\,({\bar{L}}\\!\cdot\\!\Phi^{\dagger}){\cal
S}_{2}+\mathbf{\hat{Y}_{\mathcal{D}_{1}}}\,{\bar{\cal D}}_{1}\Phi
e+\mathbf{\hat{Y}_{\mathcal{D}_{2}}}\,({\bar{\cal
D}}_{2}\\!\cdot\\!\Phi^{\dagger})e\\\
&\quad+\mathbf{\hat{Y}_{\mathcal{X}_{1}}}\,{\bar{L}}{\cal
X}_{1}\Phi+\mathbf{\hat{Y}_{\mathcal{X}_{2}}}\,\Phi\\!\cdot\\!({\cal
X}_{2}L)+\mathrm{H.c.}\bigg{]}-\sum_{i=1}^{2}\bigg{[}M_{\mathcal{S}_{i}}\,{\bar{\cal
S}}_{i}{\cal S}_{i}+M_{\mathcal{D}_{i}}\,{\bar{\cal D}}_{i}{\cal
D}_{i}+M_{\mathcal{X}_{i}}\,{\bar{\cal X}}_{i}{\cal
X}_{i}\bigg{]}\,.\end{split}$ (6)

Here, $L$, $e$ and $\nu$ represent the SM doublet and singlet of ordinary
leptons and neutrinos111While strictly speaking there is no right-handed
neutrino $\nu\equiv\nu_{R}$ in the SM, we consider a SM sector containing such
a state, that is thus differently tagged from the VLL representations
studied., respectively, and the explicit invariant products appearing in the
Lagrangian denote $SU(2)$-invariant products of two fields lying in the
fundamental or antifundamental representation of $SU(2)_{L}$. The different
parameters appearing in the Lagrangian ${\cal L}_{\mathrm{Y}}$ include the
$3\times 3$ SM Yukawa matrices in the flavour space $\mathbf{Y_{e}}$,
$\mathbf{Y_{\nu}}$, $\mathbf{\hat{Y}_{\mathcal{S}_{1}}}$,
$\mathbf{\hat{Y}_{\mathcal{S}_{2}}}$, $\mathbf{\hat{Y}_{\mathcal{D}_{1}}}$,
$\mathbf{\hat{Y}_{\mathcal{D}_{2}}}$, $\mathbf{\hat{Y}_{\mathcal{X}_{1}}}$ and
$\mathbf{\hat{Y}_{\mathcal{X}_{2}}}$, along with the mass parameters
associated with the VLLs $M_{\mathcal{S}_{1}}$, $M_{\mathcal{S}_{2}}$,
$M_{\mathcal{D}_{1}}$, $M_{\mathcal{D}_{2}}$, $M_{\mathcal{X}_{1}}$ and
$M_{\mathcal{X}_{2}}$. All these matrices are imposed to be flavour diagonal,
which ensures that a vector-like representation of a given generation only
interacts with the SM lepton of the same generation.

Constructing a new physics framework suitable for explaining dark matter with
the desired properties singles out the singlet representation ${\cal S}_{2}$.
This representation is the only one that includes an electrically neutral
state that can be stabilised through an additional $\mathbb{Z}_{2}$ parity,
once we enforce that all SM states are $\mathbb{Z}_{2}$-even and $N^{0}$ is
$\mathbb{Z}_{2}$-odd. However, this configuration prevents neutrinos from
mixing and acquiring mass, and has no impact on predictions for the anomalous
magnetic moments of the electron and the muon. Since the focus of the present
work does not encompass dark matter, we do not consider this possibility.
Instead, we utilise the possibility of supplementing the SM with an
electrically-neutral vector-like singlet ${\cal S}_{2}$ along with the SM-like
doublet ${\cal D}_{1}$ to generate mixing between all neutral leptons, account
for the observed neutrino masses, and satisfy all experimental bounds based on
neutrino oscillations. Furthermore, our numerical analysis indicates that
meaningful and acceptable contributions to the anomalous magnetic moments of
the electron and the muon can additionally be obtained by including both the
SM-like doublet ${\cal D}_{1}$ and the charged vector-like singlet ${\cal
S}_{1}$ in the model, which mix with the SM doublets of ordinary leptons $L$.

While embedding the remaining representations shown in Table 1 into the model
could also yield acceptable solutions for neutrino mass generation and resolve
the puzzles related to the anomalous magnetic moments of the muon and the
electron, this option quickly becomes highly non-minimal and escalates in a
non-necessary complexity for an initial exploration of such BSM models.
Therefore, we limit our study to an extension of the field content of the SM
in which we incorporate three generations of ${\cal S}_{1}$, ${\cal S}_{2}$
and ${\cal D}_{1}$ VLLs, deferring the analysis of other options to future
work. The incorporation of the three sets of VLL species ${\cal S}_{1}$,
${\cal S}_{2}$ and ${\cal D}_{1}$ into the model necessitates supplementing
the Lagrangian by extra contributions involving several of the VLL species
considered. The terms allowed by the SM gauge symmetry read

$\begin{split}{\cal L}_{\mathrm{VLL}}=&\ \bigg{[}\big{(}{\bar{\cal
D}}_{1}\Phi\big{)}\big{(}\lambda_{L}^{(1)}P_{L}+\lambda^{(1)}_{R}P_{R}\big{)}{\cal
S}_{1}+\big{(}{\bar{\cal
D}}_{1}\cdot\Phi^{\dagger}\big{)}\big{(}\lambda_{L}^{(2)}P_{L}+\lambda^{(2)}_{R}P_{R}\big{)}{\cal
S}_{2}+\mathrm{H.c.}\bigg{]}\,.\end{split}$ (7)

In this expression, $P_{L}$ and $P_{R}$ represent the left-handed and right-
handed chirality projectors, respectively, while the couplings
$\lambda_{L,R}^{(1,2)}$ are matrices in the flavour space. Once again, these
matrices are imposed to be diagonal, which automatically ensures that each
additional VLL is connected to a single generation.

Observable | Constraints |  |  | Observable | Constraints
---|---|---|---|---|---
$m_{h}$ | [122, 128] GeV |  |  | BR$(B^{0}_{s}\rightarrow\mu^{+}\mu^{-})$ | $[1.1,6.4]\times 10^{-9}$
BR$(h\rightarrow\mu^{+}\mu^{-})$ (strength) | $[0.51,1.87]$ |  |  | $\frac{\mathrm{BR}(B\rightarrow\tau\nu_{\tau})}{\mathrm{BR}_{{\rm SM}}(B\rightarrow\tau\nu_{\tau})}$ | [0.15, 2.41]
BR$(H\rightarrow\gamma\gamma)$ (strength) | $[0.96,1.24]$ |  |  | $\mathrm{BR}(B^{0}\rightarrow\chi_{s}\gamma)$ | $[2.99,3.87]\times 10^{-4}$
$m_{\nu_{1}}+m_{\nu_{2}}+m_{\nu_{3}}$ |  $\leq 0.26$ eV |  |  | Neutrino masses | See section 3.2
Table 2: Experimental bounds imposed on the SM sector during our parameter
space scanning procedure.

Subsequently, we proceed to demonstrate, by means of a scan over the model’s
free parameters, that there exist benchmark scenarios allowing a broad range
of low-energy constraints to be satisfied. Moreover, we show that these
choices fulfil requirements from neutrino oscillation data and measurements of
the anomalous magnetic moment of the electron and the muon. More precisely, we
include constraints from the results of recent LHC searches for VLLs, which
require us to impose lower bounds on the mass of any charged VLL state
CMS:2019hsm ; CMS:2022nty ; ATLAS:2023sbu . Additionally, we ensure that any
scenario retained in the scan features a Higgs boson with mass and decay
properties consistent with observational data ATLAS:2020fzp ; ATLAS:2023oaq ;
CMS:2020xrn ; CMS:2020xwi ; Workman:2022ynf , up to potential parametric
uncertainties arising from the new physics sector. Furthermore, we impose
constraints from rare $B$-hadron decays HFLAV:2022pwe ; LHCb:2013ghj ;
LHCb:2012skj  and neutrino cosmological data Loureiro:2018pdz . These
constraints are compiled in Table 2.

Parameter | Scanned range |  |  | Parameter | Scanned range
---|---|---|---|---|---
$|\mathbf{\hat{Y}_{\mathcal{D}_{1}}}|$ | $[10^{-5},1]$ |  |  | $|\mathbf{\hat{Y}_{\mathcal{S}_{1}}}|$ | $[10^{-5},1]$
$|\mathbf{\hat{Y}_{\mathcal{S}_{2}}}|$ | $[10^{-7},10^{-3}]$ |  |  | $|\mathbf{Y_{\nu}}|$ | $[10^{-15},10^{-10}]$
$|\lambda_{L}^{(1)}|\,,\ |\lambda_{R}^{(1)}|$ | $[10^{-5},1]$ |  |  | $|\lambda_{L}^{(2)}|\,,\ |\lambda_{R}^{(2)}|$ | $[10^{-7},10^{-3}]$
$M_{\mathcal{S}_{1}}$ | [100, 500] GeV |  |  | $M_{\mathcal{S}_{2}}$ | [100, 1000]
$M_{\mathcal{D}_{1}}$ | [100, 300] GeV |  |  |  |
Table 3: Ranges over which the model’s free parameters are allowed to vary.
These ranges are uniformly applied to each component of the diagonal Yukawa
coupling and mass matrices in the flavour space.

To explore the model’s parameter space, we make use of the SARAH package
(version 4.15.1) Staub:2013tta ; Goodsell:2017pdq  along with an in-house
implementation of the model described above. Our parameter space scan involves
a randomised sampling over the model parameters within the ranges specified in
Table 3. To ensure a thorough probe of diverse regions across the parameter
space, we employ logarithmic samplings for all Yukawa couplings, as the lower
part of the ranges are typically of greater interest and more likely to
satisfy existing experimental constraints. Moreover, we bias our results
towards light or moderately light VLL states, a choice that enhances the
prospect of detecting these particles at the LHC or future high-energy
colliders, and therefore testing our BSM construction in the near future. To
evaluate the implications of each parameter set, we utilise the SPheno package
(version 4.0.5) Porod:2011nf  to compute the associated mass spectrum, decay
tables, and relevant information for the experimental observables under
investigation.

We next proceed to evaluate, throughout the scan, how each benchmark scenario
selected could potentially address discrepancies between theoretical
predictions and experimental measurements regarding the anomalous magnetic
moment of the muon $a_{\mu}$. Specifically, we assess the extent to which each
scenario predicts a value of $a_{\mu}$ with up to a $3\sigma$ deviation from
the combined result of experiments at BNL and FNAL Muong-2:2023cdq ,

$\Delta
a_{\mu}\,\equiv\,a^{\text{exp}}_{\mu}-a^{\text{SM}}_{\mu}\,=\,[\,105,\,393\,]\times
10^{-11}\,.$ (8)

Moreover, we also assess whether the selected scenarios can yield large
deviations from the SM regarding the anomalous magnetic moment of the
electron. In this case, we consider two distinct sets of measurements
involving Cs and Rb atoms Fan:2022eto , and allow our predictions to deviate
by 3$\sigma$ from Cs measurements and 2$\sigma$ from Rb measurements as

$\Delta
a^{\text{Cs}}_{e}\,\equiv\,a^{\text{exp}}_{e}-a^{\text{SM}}_{e}\,=\,[\,-168,\,-108\,]\times
10^{-14}\,,\qquad\Delta
a^{\text{Rb}}_{e}\,\equiv\,a^{\text{exp}}_{e}-a^{\text{SM}}_{e}\,=\,[\,2,\,66\,]\times
10^{-14}\quad\,.$ (9)

##  3 Scan Results

After conducting a random scan across the parameter space of the model as
defined in Table 3, we have identified 328 solutions that satisfy all bounds
listed in Table 2 along with restrictions on VLL masses and milder constraints
originating from the anomalous magnetic moments of the muon and the electron
to orient the scan towards the physics case studied. Many of these solutions
hence exhibit significant deviations from the SM predictions for the anomalous
magnetic moments of both the muon and the electron, potentially offering a
pathway to reconciling theory with data. Specifically, these solutions feature
$\Delta{a}_{\mu}$ values reaching up to $3\sigma$, as defined by Eq. (8), and
$\Delta{a}^{\rm Rb}_{e}$ values showing positive deviations of up to
$2\sigma$, aligning with the results obtained from measurements carried out
from Rubidium atoms and presented in Eq. (9). However, none of the obtained
solutions predicts a negative deviation for $a_{e}$, demonstrating that our
model cannot account for the value of $a_{e}$ derived from measurements
carried out on Cesium atoms. In the following, we investigate these findings
in detail, first relative to the anomalous magnetic moments of the muon and
the electron in section 3.1, and next in the context of neutrino data in
section 3.2. Finally, we verify in section 3.3 that adding multiplets of VLLs
to the theory does not impact electroweak precision tests.

###  3.1 Anomalous magnetic moments of the muon and the electron

Figure 1: Predictions for the new physics contributions to the anomalous
magnetic moment of the muon and their dependence on the second generation
vector-like mass $M^{22}_{\mathcal{D}_{1}}$ (top left), the Yukawa coupling of
the second-generation VLL doublet with the ordinary lepton singlet
$|\mathbf{\hat{Y}_{\mathcal{D}_{1}}}^{22}|$ (top right) and the masses of the
lightest vector-like charged lepton (bottom left) and neutrino (bottom right).
Each point represents a scenario compliant with all constraints imposed during
the scan, and the colour code represents either the sum of the masses of the
three lightest neutrinos (top row) or the ratio of the second-generation
Yukawa couplings of VLL doublet and singlet (bottom row).

In figure 1 we present various properties of the 328 scenarios identified in
our scan. We display the dependence of the new physics contributions
$\Delta{a}_{\mu}$ to the anomalous magnetic moment of the muon on the second-
generation vector-like mass parameter $M^{22}_{\mathcal{D}_{1}}$ (top left
panel), on the absolute value of the second-generation lepton-VLL Yukawa
coupling $|\mathbf{\hat{Y}_{\mathcal{D}_{1}}}^{22}|$ (top right panel), as
well as on the mass of the lightest charged VLL state (bottom left panel) and
neutral VLL state (bottom right panel). We observe that the constraints
enforced lead to relatively large values for the second-generation Yukawa
coupling $|\mathbf{\hat{Y}_{\mathcal{D}_{1}}}^{22}|$, typically ranging from
0.01 and 0.5. Additionally, the colour coding in the bottom row indicates that
the Yukawa coupling $|\mathbf{\hat{Y}_{\mathcal{S}_{1}}}^{22}|$ relating the
second-generation VLL singlet $\mathcal{S}_{1}$ and the SM weak doublet $L$
must be even larger (by a factor of a few). On the other hand, the dependence
of $\Delta{a}_{\mu}$ on the doublet VLL mass $M^{22}_{\mathcal{D}_{1}}$ is
moderate, with values spanning the entire scanned range, like the one on the
masses of the lightest vector-like states.

Figure 2: Predictions for the new physics contributions to the anomalous
magnetic moment of the electron and their dependence on the first-generation
vector-like mass parameter $M^{11}_{\mathcal{D}_{1}}$ (top left), the Yukawa
coupling of the first-generation VLL doublet with the ordinary electron
singlet $|\mathbf{\hat{Y}_{\mathcal{D}_{1}}}^{11}|$ (top right), the masses of
the lightest vector-like charged lepton (bottom left) and neutrino (bottom
right). Each point represents a scenario compliant with all constraints
imposed during the scan, and the colour code either represents how scenarios
can simultaneously provide an explanation for the two anomalous magnetic
moments considered (top row), or it highlights the ratio of the first-
generation Yukawa couplings of VLL doublet and singlet (bottom row).

The observed pattern of large Yukawa couplings results in a complex mixing
between the VLL doublet state $\mathcal{D}_{1}$, the VLL singlet state
$\mathcal{S}_{1}$, and the left-handed and right-handed muon fields, while not
spoiling observations and ensuring large values for new physics contributions
to the anomalous magnetic moment of the muon. Subsequently, this provides
sufficient flexibility to adjust the other input parameters of the model to
achieve agreement with data in the neutrino sector or feature deviations in
the anomalous magnetic moment of the electron that are significant enough to
provide an explanation for its observed value. This is exemplified below for
the anomalous magnetic moment of the electron, as well as in figure 1 by the
values obtained for the sum of the masses of the three lighter neutrinos
(represented through the colour code in the top row of the figure) and in
section 3.2. Upon examination of the bottom row of figure 1, it becomes
evident that a substantial portion of the solutions featuring a $3\sigma$
deviation from the SM predictions for $a_{\mu}$ also feature a VLL spectrum
where the lightest state is predominantly singlet-like (as
$|\mathbf{\hat{Y}_{\mathcal{D}_{1}}}^{22}|<|\mathbf{\hat{Y}_{\mathcal{S}_{1}}}^{22}|$).
Nonetheless, we have verified that introducing both VLL singlets and doublets
and enforcing them to mix is necessary to obtain scenarios providing an
explanation for the deviation between the SM predictions for $a_{\mu}$ and the
associated measured value. It is worth noting, however, that several of these
scenarios still exhibit VLLs with masses smaller than the existing collider
limits, and are thus excluded phenomenologically.

Next, we present predictions for the VLL contributions to $\Delta a_{e}$ in
the considered scenarios in figure 2, investigating their dependence on
various model parameters and properties while comparing them with experimental
measurements conducted on Rb atoms Fan:2022eto . A noticeable contrast emerges
relative to the muon case. In the top-left and top-right panels of the figure,
we analyse the dependence of $\Delta a_{e}$ on the first-generation vector-
like doublet mass parameter $M^{11}_{\mathcal{D}_{1}}$ and on the lepton-VLL
doublet Yukawa coupling $|\mathbf{\hat{Y}_{\mathcal{D}_{1}}}^{11}|$,
respectively. We observe that scenarios in which the new physics contributions
to $a_{e}$ align with data predictions exist within the entire scanned range
for these parameters. Furthermore, as indicated by the colour coding, many of
these solutions also offer a potential explanation for the observed
discrepancies in the anomalous magnetic moment of the muon, as expected given
the different parametric dependence of $a_{\mu}$ and $a_{e}$. In the bottom
row of figure 2, we further demonstrate the existence of solutions spanning a
range of masses of electrically charged and neutral VLLs coupled to first-
generation ordinary leptons, with a significant portion being excluded by
collider bounds. Furthermore, acceptable scenarios feature diverse VLL
composition, as indicated by the associated
$|\mathbf{\hat{Y}_{\mathcal{D}_{1}}}^{11}|/|\mathbf{\hat{Y}_{\mathcal{S}_{1}}}^{11}|$
ratios that can be either large or small. However, the doublet Yukawa coupling
is again found to be always smaller than the singlet one.

Figure 3: Predictions for the new physics contributions to the anomalous
magnetic moment of the electron (left) and the muon (right) and their
dependence on the relevant couplings
($|\mathbf{\hat{Y}_{\mathcal{S}_{1}}}^{11}|$ and
$|\mathbf{\hat{Y}_{\mathcal{S}_{1}}}^{22}|$ respectively) of the additional
VLL singlet with the ordinary SM charged lepton singlet. The colour coding
depicts how likely the selected scenarios can provide a solution to both the
$a_{e}$ and $a_{\mu}$ puzzles.

To complement our exploration of the significance of incorporating charged VLL
singlets $\mathcal{S}_{1}$ in the model, figure 3 illustrates the dependence
of the new physics contributions to $a_{e}$ (left panel) and $a_{\mu}$ (right
panel) and their correlation with the magnitude of the deviation relative to
measurements conducted at Northwestern on Rb atoms and the combination of FNAL
and BNL data, respectively. Our analysis reaffirms previous conclusions,
indicating that the Yukawa coupling linking a vector-like singlet of a given
generation to the corresponding SM lepton doublet must be large in the case of
the muon, with $|\mathbf{\hat{Y}_{\mathcal{S}_{1}}}^{22}|>0.01$. However, for
$a_{e}$, $|\mathbf{\hat{Y}_{\mathcal{S}_{1}}}^{11}|$ can span the entire
considered set of coupling values, ranging from $10^{-5}$ to $1$. Furthermore,
our findings demonstrate the feasibility of constructing scenarios that
achieve agreement between theory and data for both the electron and the muon
anomalous magnetic moments.

Figure 4: Heat map based on the pairwise Pearson correlation coefficients
between the quantities $\Delta a_{e}$, $M^{11}_{\mathcal{D}_{1}}$,
$M^{11}_{\mathcal{S}_{1}}$, $\mathbf{\hat{Y}_{\mathcal{D}_{1}}}^{11}$ and
$\mathbf{\hat{Y}_{\mathcal{S}_{1}}}^{11}$ (left), as well as that computed
from $\Delta a_{\mu}$, $M^{22}_{\mathcal{D}_{1}}$, $M^{22}_{\mathcal{S}_{1}}$,
$\mathbf{\hat{Y}_{\mathcal{D}_{1}}}^{22}$ and
$\mathbf{\hat{Y}_{\mathcal{S}_{1}}}^{22}$ (right).

To further explore the impact of all model’s free parameters on $\Delta
a_{\mu}$ and $\Delta a_{e}$, we now examine how they correlate with
predictions for the value of the electron and muon anomalous magnetic moments.
Figure 4 shows heat maps based on the pairwise Pearson correlation
coefficients JamesISL ; Lista:2017jsy  between these parameters, and $\Delta
a_{e}$ (left) and $\Delta a_{\mu}$ (right). Pearson correlation coefficients
measure the linear relationship between two quantities, by calculating the
ratio of their covariance to the product of their standard deviations.
According to this definition, the coefficient lies within the range $[-1,1]$,
where a value equal to $1$ suggests a perfect positive linear relationship
(the two quantities exhibiting a similar behaviour), and $-1$ a perfect
negative linear relationship (the two quantities exhibiting an opposite
behaviour), while $0$ represents no linear relationship. However, such
coefficients ignore other types of (non-linear) relationships between
quantities. The corresponding formula for calculating the Pearson coefficients
between two parameters $x$ and $y$ across a given dataset is given by

$\displaystyle\rho\,(x,y)\,=\,\frac{\text{cov}(x,y)}{\sigma_{x}\,\sigma_{y}}\,,$
(10)

where $\mathrm{cov}(x,y)$ denotes the covariance between the two parameters
across the dataset, and $\sigma_{x}$ and $\sigma_{y}$ the standard deviation
obtained for the parameters $x$ and $y$ respectively.

The results depicted in figure 4 highlight the significance of the parameter
$M^{11}_{\mathcal{D}_{1}}$ in determining the VLL contributions to the
anomalous magnetic moment of the electron. Across all 328 solutions identified
in our scan, we observe a Pearson correlation of $-0.29$ between
$M^{11}_{\mathcal{D}_{1}}$ and $\Delta a_{e}$. This negative correlation
suggests that increasing the mass parameter associated with the first-
generation VLL doublet leads to a decrease in the new physics contribution to
the electron’s anomalous magnetic moment. To reconcile this with experimental
data, a spectrum featuring lighter VLL mass eigenstates may be required, as
other coupling parameters play no role. This is achievable through lepton-VLL
mixings. Conversely, regions within the parameter space characterised by lower
$M^{11}_{\mathcal{D}_{1}}$ values are favoured in light of considerations
regarding $a_{e}$, which is consistent with findings from prior studies (as
e.g. in Dermisek:2013gta ). In the right panel of figure 4, we demonstrate
that a similar correlation exists for $\Delta a_{\mu}$ and the second-
generation parameter $M^{22}_{\mathcal{D}_{1}}$.

A notable feature from figure 4 is the absence of a clear hierarchy between
the Yukawa interactions linking VLL doublets and singlets to the SM ordinary
leptons when predictions for the anomalous magnetic moment are enforced to
match experimental data. The Yukawa couplings
$\mathbf{\hat{Y}_{\mathcal{S}_{1}}}^{11}$ and
$\mathbf{\hat{Y}_{\mathcal{D}_{1}}}^{11}$
($\mathbf{\hat{Y}_{\mathcal{S}_{1}}}^{22}$ and
$\mathbf{\hat{Y}_{\mathcal{D}_{1}}}^{22}$) exhibit no discernible linear
correlation with $a_{e}$ ($a_{\mu}$), suggesting that any hierarchy among them
may be suitable. This can also be seen from figures 1, 2 and 3 where we note
that the Yukawa couplings of the VLL singlets to the SM leptons consistently
exceed those of the VLL doublets.

###  3.2 Properties of the neutrino sector

The introduction of VLLs has implications for the neutrino sector, providing a
mechanism to generating neutrino masses. While compelling evidence from
oscillation experiments indicates that their mass is non-zero SNO:2002tuh ;
Super-Kamiokande:2005mbp , the actual mass spectrum is unresolved. It can
nevertheless be constrained thanks to cosmology. Indeed, the abundance of
cosmic neutrinos and their contribution to the universe’s total energy density
during its early stages enables the study of their characteristics.
Specifically, the sum of the neutrino masses $\sum m_{\nu}$ influences the
cosmic microwave background, supernovæ features, large scale structure, and
big bang nucleosynthesis Loureiro:2018pdz . Present data has yielded an
approximate upper limit of $\sum m_{\nu}<0.26$ eV, a constraint that we have
incorporated in our parameter space scans (see section 2). In our analysis of
the neutrino sector, we have considered scenarios with the normal ordering
(NO) and inverted ordering (IO) of the Majorana neutrino masses, and we have
explored all mixing possibilities of the SM neutrinos with the neutral
components of the VLL doublets and neutral singlets. The constraints imposed
on the neutrino sector during the scan are detailed in Table 4.

Cosmological constraints | Neutrino masses (IO) | Neutrino masses (NO)
---|---|---
$\Sigma\,m_{\nu}\,\leq\,$0.26 eV |  $m_{\nu_{1}}\in[8.37,8.85]\times 10^{-3}$ eV |  $m_{\nu_{2}}\in[8.46,8.88]\times 10^{-3}$ eV
$m_{\nu_{2}}\in[4.93,5.04]\times 10^{-2}$ eV |  $m_{\nu_{3}}\in[4.88,5.02]\times 10^{-2}$ eV
Table 4: Constraints imposed on the neutrino sector during our parameter space
scan. We have allowed a $2\sigma$ uncertainty on the neutrino masses, based on
the experimental values determined for the neutrino squared mass differences
$\Delta m_{12}^{2}$ and $\Delta m_{31}^{2}$.

Figure 5: Predictions for the masses of the heavier neutrinos at one loop, for
scenarios featuring a normal ordering of the neutrino masses (top row) and an
inverted ordering of the neutrino masses (bottom row). The results are shown
as a function of the neutrino Yukawa couplings
$\mathbf{Y}_{\mathbf{\nu}}^{22}$ (left) and $\mathbf{Y}_{\mathbf{\nu}}^{33}$
(right), while the dotted lines correspond to tree-level predictions. The
narrow red and green bands depict the range of masses consistent with neutrino
oscillation data within $2\sigma$.

Figure 6: Heat map based on the pairwise Pearson correlation coefficients
between the mass of the third neutrino $m_{\nu_{3}}$ and the parameters
entering its calculation at one loop (left). The effects of the most impactful
parameter $\mathbf{Y}_{\mathbf{\nu}}^{33}$ on the corrections to
$m_{\nu_{3}}$, denoted by $\Delta m_{\nu_{3}}$, is additionally shown (right).

In figure 5, we analyse the results of our parameter space scan in relation to
their impact on neutrino masses. We classify the solutions into those with
normal (top row) and inverted (bottom row) neutrino mass ordering, and we
display the dependence of the masses of the heavier neutrinos on the neutrino
Yukawa couplings to ordinary leptons $\mathbf{Y}_{\mathbf{\nu}}^{22}$ (left
panels) and $\mathbf{Y}_{\mathbf{\nu}}^{33}$ (right panels). We recall that
for normal-ordering scenarios, the lightest neutrino is $\nu_{1}$, while for
inverted-ordering scenarios, it is $\nu_{3}$. Additionally, we use colour
coding to highlight how each solution aligns with observed deviations in the
anomalous magnetic moments of the muon and the electron. Many solutions
exhibit a direct proportionality to the Yukawa coupling
$\mathbf{Y}_{\mathbf{\nu}}$, as indicated by the dashed lines shown in the
different figures. In such cases, loop corrections to neutrino masses, that
involve the VLL states of the model, are negligibly small. However, this is
not systematic, and when loop corrections are not negligible, results show a
slight dependence on the magnitude of the neutrino Yukawa couplings. We also
represent solutions compatible with the constraints on the neutrino masses
listed in Table 4 by using narrow green and red bands. Our analysis reveals
that solutions explaining observed deviations observed in $a_{\mu}$ and
$a_{e}$, as well as generating neutrino masses in agreement with data, exist
for both normal and inverted neutrino mass orderings.

In order to assess which model parameters have the most significant impact on
the neutrino sector, we present a heat map in the left panel of figure 6 based
on the pairwise Pearson correlation coefficients between the mass of the
neutrino $\nu_{3}$ (chosen for illustrative purposes) and various model
parameters relevant for neutrino masses at one loop. It turns out that the
parameter $\mathbf{Y}_{\mathbf{\nu}}^{33}$, governing the Yukawa interactions
between ordinary leptons and the right-handed neutrino field, has the largest
effect. Turning our attention to the value of the one-loop corrections $\Delta
m_{\nu_{3}}$ to the tree-level mass, we observe, in the right panel of the
figure, that the value of $\mathbf{Y}_{\mathbf{\nu}}^{33}$ plays a minimal
role on the neutrino mass for $\mathbf{Y}_{\mathbf{\nu}}^{33}\lesssim
10^{-12}$. However, for larger values, these effects become more pronounced as
one-loop contributions $\Delta m_{\nu_{3}}$ must compensate for a too large
tree-level mass. This easily leads to predictions that conflict with neutrino
oscillation data, as visible from the figure. Therefore, achieving agreement
with experimental constraints necessitates a very weak interaction term
between the right-handed neutrino field $\nu$ and the SM-like lepton doublet
$L$.

###  3.3 Electroweak precision tests

Figure 7: New physics contributions to the $S$ and $T$ parameters, and
comparison with current limits at $2\sigma$, shown in red and green
respectively. Colour coding indicates the consistency of the solutions chosen
in the scan with respect to their potential to explain deviation in the
anomalous magnetic moments of the muon and the electron.

Given that our theoretical framework includes a non-trivial $SU(2)_{L}$
multiplet, we perform an additional verification to ensure that the selected
scenarios in our scan comply with electroweak precision data. We undertake
this verification by calculating the new physics contributions to the Peskin-
Takeuchi parameters Peskin:1991sw . In light of recent bounds on these
parameters ($S=-0.01\pm 0.07$, $T=0.04\pm 0.06$, and $U=0$ Workman:2022ynf  ),
we show in figure 7 that the majority of scenarios compatible with neutrino
oscillation data and providing an explanation for the anomalous magnetic
moment of the muon and the electron do not violate electroweak precision data.

##  4 Collider phenomenology

In this section, our focus is on analysing the phenomenological implications
at high-energy hadronic colliders of scenarios that meet all previously
imposed constraints. We examine two representative benchmark setups and
demonstrate that they yield discernible signals amidst the overwhelming
background of the SM. For the first scenario (see section 4.1), we exploit a
signature characterised by the production of a significant number of charged
leptons, while for the second one (see section 4.2), we leverage the
production of substantial missing energy. In general, studying these two
signals is sufficient to obtain excellent collider handles on model scenarios
consistent with all constraints examined in section 3.

To conduct our study, we generate a UFO model Darme:2023jdn  with the SARAH
package (version 4.15.1) Staub:2013tta ; Goodsell:2017pdq , starting from the
model implementation discussed in section 2. This allows us to utilise
MadGraph5_aMC@NLO Alwall:2014hca  (version 3.2.0) to generate both signal and
background events relevant to a potential proton-proton collider operating at
a centre-of-mass energy of 100 TeV, and aiming to collect an integrated
luminosity of 3 ab-1. We obtain hard-scattering events by convoluting leading-
order matrix elements with the next-to-leading-order set of NNPDF 2.3 parton
densities Ball:2012cx . Subsequently, these events are matched with parton
showering using Pythia 8.3 Bierlich:2022pfr , which we also use to simulate
hadronisation. Our simulations match matrix elements including up to one
additional parton at the Born level, using the MLM matching scheme
Mangano:2006rw ; Alwall:2008qv , and we re-weight events to incorporate
higher-order corrections in QCD into the relevant total rates Alwall:2014hca ;
Gauld:2021ule ; AH:2023hft . Finally, hadron-level events are processed
through the fast detector simulator SFS Araz:2020lnp  equipping the
MadAnalysis 5 package Conte:2012fm ; Conte:2014zja ; Conte:2018vmg  (version
1.10.12). This internally relies on FastJet Cacciari:2011ma  (version 3.3.3)
for event reconstruction, together with its implementation of the anti-$k_{T}$
jet algorithm Cacciari:2008gp  with a radius parameter $R=0.4$. To emulate
realistic detector resolution and particle identification, we employ the ATLAS
configuration card provided with the SFS module of MadAnalysis 5, which we
also use for implementing our collider analysis.

###  4.1 A signature involving multiple charged leptons

$\mathbf{\hat{Y}_{\mathcal{D}_{1}}}^{11}$ | $1.34\times 10^{-1}$ | $\mathbf{\hat{Y}_{\mathcal{S}_{1}}}^{11}$ | $-1.15\times 10^{-3}$ | $M^{11}_{\mathcal{S}_{2}}$ |  $400.47$ GeV
---|---|---|---|---|---
$\mathbf{\hat{Y}_{\mathcal{D}_{1}}}^{22}$ | $-3.18\times 10^{-2}$ | $\mathbf{\hat{Y}_{\mathcal{S}_{1}}}^{22}$ | $6.43\times 10^{-1}$ | $M^{22}_{\mathcal{S}_{2}}$ |  $435.47$ GeV
$\mathbf{\hat{Y}_{\mathcal{D}_{1}}}^{33}$ | $-3.06\times 10^{-1}$ | $\mathbf{\hat{Y}_{\mathcal{S}_{1}}}^{33}$ | $-1.75\times 10^{-5}$ | $M^{33}_{\mathcal{S}_{2}}$ |  $156.56$ GeV
$(\lambda_{L}^{(1)})^{11}$ | $-1.38\times 10^{-4}$ | $(\lambda_{R}^{(1)})^{11}$ | $-8.50\times 10^{-2}$ | $M^{11}_{\mathcal{D}_{1}}$ |  $131.39$ GeV
$(\lambda_{L}^{(1)})^{22}$ | $-2.86\times 10^{-1}$ | $(\lambda_{R}^{(1)})^{22}$ | $6.55\times 10^{-4}$ | $M^{22}_{\mathcal{D}_{1}}$ |  $125.97$ GeV
$(\lambda_{L}^{(1)})^{33}$ | $-3.29\times 10^{-5}$ | $(\lambda_{R}^{(1)})^{33}$ | $-5.29\times 10^{-1}$ | $M^{33}_{\mathcal{D}_{1}}$ |  $178.34$ GeV
$\mathbf{\hat{Y}_{\mathcal{S}_{2}}}^{11}$ | $-7.62\times 10^{-5}$ | $\mathbf{Y}_{\mathbf{\nu}}^{11}$ | $-2.81\times 10^{-13}$ | $M^{11}_{\mathcal{S}_{1}}$ |  $391.12$ GeV
$\mathbf{\hat{Y}_{\mathcal{S}_{2}}}^{22}$ | $9.62\times 10^{-7}$ | $\mathbf{Y}_{\mathbf{\nu}}^{22}$ | $3.53\times 10^{-15}$ | $M^{22}_{\mathcal{S}_{1}}$ |  $431.56$ GeV
$\mathbf{\hat{Y}_{\mathcal{S}_{3}}}^{11}$ | $-1.17\times 10^{-7}$ | $\mathbf{Y}_{\mathbf{\nu}}^{33}$ | $1.33\times 10^{-13}$ | $M^{33}_{\mathcal{S}_{1}}$ |  $177.65$ GeV
$(\lambda_{L,R}^{(2)})^{11}$ | $2.15\times 10^{-7}$ | $(\lambda_{L,R}^{(2)})^{22}$ | $-1.93\times 10^{-5}$ | $(\lambda_{L,R}^{(2)})^{33}$ | $-4.10\times 10^{-7}$
$M_{h}$ | 127.66 GeV | $m_{e^{-}_{4}}$ | 126.12 GeV | $m_{e^{-}_{5}}$ | 134.23 GeV
$m_{e^{-}_{6}}$ | 144.01 GeV | $m_{e^{-}_{9}}$ | 448.21 GeV | $m_{\nu_{1}}$ |  $6.18\times 10^{-4}$ eV
$m_{\nu_{2}}$ |  $4.51\times 10^{-3}$ eV | $m_{\nu_{3}}$ |  $4.94\times 10^{-2}$ eV | $S$ | $2.33\times 10^{-2}$
$T$ | $3.81\times 10^{-2}$ | $\Delta a_{e}$ | $3.74\times 10^{-13}$ | $\Delta a_{\mu}$ | $1.44\times 10^{-9}$
Table 5: Values of the free parameters of the model for the benchmark scenario
BM I, together with predictions for the most relevant observables used in our
parameter space exploration.

As a first representative benchmark scenario (BM I) satisfying all constraints
imposed, we consider a choice of parameters leading to a new physics spectrum
featuring heavy VLLs with masses ranging in about $[125,450]$ GeV with a
collider signature leading to enriched muon production. The corresponding
model parameters are given in Table 5, together with the numerical values for
the observables relevant for the constraints that we imposed during the scan.
In particular, we observe that predictions for the oblique parameters are
compliant with the latest observations, that the sum of the masses of the
three lightest neutrinos, for normal ordering, satisfies cosmological bounds,
and that the BM I scenario provides an explanation for observed deviations in
the anomalous magnetic moments of both the electron and the muon at $2\sigma$
and $1\sigma$ respectively. While many of the six electrically-charged VLL
eigenstates of the spectrum have masses conducive to significant production
rates at colliders, only a select few leave discernible signatures once decay
rates are accounted for. Among these, the heaviest VLL, denoted as $e^{-}_{9}$
(as the heaviest of all nine charged leptons, with $e_{4}^{-}$ being, in this
notation, the lightest of all electrically-charged BSM leptonic states),
emerges as the leading candidate.

Among the various production and decay processes involving VLLs, the most
promising collider signature involves multiple leptons. This signature stems
from the electroweak production of a pair of VLLs, each decaying into a
Standard Model lepton and a weak boson. We choose to focus on the leptonic
decay channel of the weak bosons to maximise the number of leptons in the
final state, since final states with four or more leptons typically exhibit
minimal or negligible SM background. By considering predictions for the VLL
electroweak couplings, masses, and branching ratios within the framework of
benchmark scenario BM I, we identify the production of a pair of $e_{9}$
states decaying into a muon and a $Z$-boson as the most promising channel,

$p\,\,p\ \ \rightarrow\ \ e^{-}_{9}\,\,e^{+}_{9}+X\ \ \rightarrow\ \
\mu^{-}\,Z\,\mu^{+}\,\,Z+X\ \
(\rightarrow\mu^{-}\ell^{+}\ell^{-}\,\mu^{-}\ell^{\prime+}\ell^{\prime-}+X)\,.$
(11)

This process is associated with a production rate at next-to-leading order
(NLO) and includes soft-gluon resummation at the next-to-next-to-leading-
logarithmic (NNLL) accuracy of 26.4 fb AH:2023hft , and with a VLL branching
ratio $\mathrm{BR}(e_{9}\to\mu\,Z)=47.7\,$%. This indicates a significant
probability of producing events with six isolated leptons after accounting for
leptonic $Z$-boson decays, which is largely free from SM background.
Additionally, the presence of two hard muons originating from the decay of
$e_{9}$ VLLs with masses in the several hundred GeV range provides an
additional discriminating feature for the signal. In the subsequent analysis,
we provide quantitative evidence that investigating such a process is
sufficient for a potential observation of a VLL signal at a future proton-
proton collider operating at a centre-of-mass energy
$\sqrt{s}=100\,\text{TeV}$, when the properties of this VLL state are such
that they provide an explanation for several leptonic anomalies. For this, we
perform signal simulation as described above, with a merging scale set to
$m_{e_{9}}/4$.

While we anticipate a small SM background, we nonetheless perform its detailed
phenomenological analysis, including event generation for all relevant SM
processes using the previously introduced toolchain. These processes consist
of:

$p\,\,p\ \rightarrow\ h\,\,Z+X\,,\quad p\,\,p\ \rightarrow\
Z\,\,Z\,\,Z+X\quad\text{and}\quad p\,\,p\ \rightarrow\
t\,\,\bar{t}\,\,Z\,\,Z+X\,,$ (12)

which all lead to a six-lepton final state once $Z$-boson and Higgs-boson
leptonic decays are accounted for. The generated events are re-weighted to
ensure that the corresponding hard-scattering cross sections, including the
branching ratios related to leptonic Higgs and $Z$ boson decays, are accurate
to next-to-next-to-leading order (NNLO) for the first process Gauld:2021ule ,
and NLO for the last two processes Alwall:2014hca . This yields cross section
values of $\sigma_{Zh}=0.455~{}\text{ab}$, $\sigma_{ZZZ}=0.0602~{}\text{fb}$
and $\sigma_{t\bar{t}ZZ}=0.371~{}\text{ab}$, that we determine by multiplying
the leading-order predictions returned by MadGraph5_aMC@NLO by constant
$K$-factors extracted from the literature.

Figure 8: Key observables for distinguishing the BM I signal from the SM
background. We display the distribution of the number of reconstructed leptons
$N_{\ell}$ (left), and of the total transverse energy $E_{T}^{\mathrm{Total}}$
(right). The signal is plotted in solid lines, the background in dashed lines.
Histograms are normalised to a luminosity of $3\,\text{ab}^{-1}$ of $pp$
collisions at $\sqrt{s}=100\,\text{TeV}$.

In our analysis, we begin by defining a lepton collection comprising all
electrons and muons well-separated from any jet candidate by a transverse
distance $\Delta R>0.4$, with a pseudo-rapidity $|\eta^{\ell}|\leq 2.5$, and a
transverse momentum $p_{T}^{\ell}>15$ GeV. Additionally, we consider as jet
candidates all reconstructed jets with a pseudo-rapidity $|\eta^{j}|\leq 2.5$
and a transverse momentum $p_{T}^{j}>15$ GeV. We then select events containing
$N_{\ell}\geq 6$ reconstructed leptons and a total transverse energy
$E_{T}^{\mathrm{Total}}$ defined as the sum of the $p_{T}$ of all
reconstructed objects of at least 1.4 TeV. Figure 8 illustrates the potential
of these criteria for a high signal efficiency while effectively rejecting
background events. After applying these criteria, we select $n_{s}$ signal
events and $n_{b}$ background events, where $n_{b}$ is determined by the sum
of the number of surviving $Zh$, $ZZZ$ and $t\bar{t}ZZ$ events ($n_{Zh}$,
$n_{ZZZ}$ and $n_{t\bar{t}ZZ}$, respectively). Specifically, we find

$n_{s}=28.43\qquad\text{and}\qquad n_{b}=9.24\quad\text{with}\ \ n_{Zh}\simeq
0,\ \ n_{ZZZ}=6.23\ \ \text{and}\ \ n_{t\bar{t}ZZ}=2.99\,.$ (13)

We estimate the sensitivity to the signal using two metrics Cowan:2010js ,

$\begin{split}s=&\ \frac{n_{s}}{\sqrt{n_{b}+\Delta_{b}^{2}}}\,,\\\ Z_{A}=&\
\sqrt{2\left\\{(n_{s}+n_{b})\ln\left[\frac{(n_{s}+n_{b})(n_{s}+\Delta_{b}^{2})}{n_{b}^{2}+(n_{s}+n_{b})\Delta_{b}^{2}}\right]-\frac{n_{b}^{2}}{\Delta_{b}^{2}}\ln\left[1+\frac{\Delta_{b}^{2}n_{s}}{n_{b}(n_{b}+\Delta_{b}^{2})}\right]\right\\}}\
,\end{split}$ (14)

where the systematics on the background are conservatively assumed to be
$\Delta_{b}=0.2n_{b}$. We obtain

$s\,=\,8\,\sigma\qquad\text{and}\qquad Z_{A}\,=\,10\,\sigma\,,$ (15)

indicating a clear possibility of observing the signal at a future collider
aiming to operate at a centre-of-mass energy of 100 TeV.

###  4.2 A signature with significant missing transverse energy

$\mathbf{\hat{Y}_{\mathcal{D}_{1}}}^{11}$ | $-3.87\times 10^{-2}$ | $\mathbf{\hat{Y}_{\mathcal{S}_{1}}}^{11}$ | $3.13\times 10^{-4}$ | $M^{11}_{\mathcal{S}_{2}}$ |  $152.84$ GeV
---|---|---|---|---|---
$\mathbf{\hat{Y}_{\mathcal{D}_{1}}}^{22}$ | $-2.27\times 10^{-2}$ | $\mathbf{\hat{Y}_{\mathcal{S}_{1}}}^{22}$ | $-4.33\times 10^{-1}$ | $M^{22}_{\mathcal{S}_{2}}$ |  $231.35$ GeV
$\mathbf{\hat{Y}_{\mathcal{D}_{1}}}^{33}$ | $2.14\times 10^{-4}$ | $\mathbf{\hat{Y}_{\mathcal{S}_{1}}}^{33}$ | $-5.60\times 10^{-3}$ | $M^{33}_{\mathcal{S}_{2}}$ |  $141.43$ GeV
$(\lambda_{L}^{(1)})^{11}$ | $-1.81\times 10^{-1}$ | $(\lambda_{R}^{(1)})^{11}$ | $-2.39\times 10^{-5}$ | $M^{11}_{\mathcal{D}_{1}}$ |  $137.82$ GeV
$(\lambda_{L}^{(1)})^{22}$ | $9.32\times 10^{-1}$ | $(\lambda_{R}^{(1)})^{22}$ | $-9.40\times 10^{-3}$ | $M^{22}_{\mathcal{D}_{1}}$ |  $125.82$ GeV
$(\lambda_{L}^{(1)})^{33}$ | $-1.76\times 10^{-3}$ | $(\lambda_{R}^{(1)})^{33}$ | $-6.34\times 10^{-2}$ | $M^{33}_{\mathcal{D}_{1}}$ |  $296.30$ GeV
$\mathbf{\hat{Y}_{\mathcal{S}_{2}}}^{11}$ | $-1.46\times 10^{-4}$ | $\mathbf{Y}_{\mathbf{\nu}}^{11}$ | $-2.79\times 10^{-13}$ | $M^{11}_{\mathcal{S}_{1}}$ |  $622.87$ GeV
$\mathbf{\hat{Y}_{\mathcal{S}_{2}}}^{22}$ | $-6.28\times 10^{-5}$ | $\mathbf{Y}_{\mathbf{\nu}}^{22}$ | $1.01\times 10^{-15}$ | $M^{22}_{\mathcal{S}_{1}}$ |  $883.75$ GeV
$\mathbf{\hat{Y}_{\mathcal{S}_{2}}}^{33}$ | $-3.17\times 10^{-4}$ | $\mathbf{Y}_{\mathbf{\nu}}^{33}$ | $6.60\times 10^{-14}$ | $M^{33}_{\mathcal{S}_{1}}$ |  $892.46$ GeV
$(\lambda_{L,R}^{(2)})^{11}$ | $-5.87\times 10^{-5}$ | $(\lambda_{L,R}^{(2)})^{22}$ | $4.53\times 10^{-4}$ | $(\lambda_{L,R}^{(2)})^{33}$ | $-1.30\times 10^{-5}$
$M_{h}$ | 122.57 GeV | $m_{e^{-}_{4}}$ | 110.94 GeV | $m_{e^{-}_{5}}$ | 118.70 GeV
$m_{e^{-}_{6}}$ | 148.66 GeV | $m_{e^{-}_{7}}$ | 189.18 GeV | $m_{\nu_{1}}$ |  $1.78\times 10^{-4}$ eV
$m_{\nu_{2}}$ |  $4.92\times 10^{-3}$ eV | $m_{\nu_{3}}$ |  $4.86\times 10^{-2}$ eV | $S$ | $1.02\times 10^{-2}$
$T$ | $6.38\times 10^{-2}$ | $\Delta a_{e}$ | $2.06\times 10^{-14}$ | $\Delta a_{\mu}$ | $1.97\times 10^{-9}$
Table 6: Same as Table 5 but for scenario BM II.

For our second representative benchmark scenario, denoted as BM II, we
consider a new physics setup characterised by light VLLs that can be copiously
produced at hadronic colliders and that predominantly decay into tau leptons.
This scenario has been carefully selected to satisfy all constraints imposed
during the parameter scan, the corresponding parameters being provided in
Table 6 alongside the values of relevant observables. One of the distinctive
collider signatures associated with BM II arises from the pair production and
subsequent decay of the $e_{7}^{-}$ state. The corresponding signature
manifests itself through final states comprising multiple leptons, including
at least two tau leptons given that $\mathrm{BR}(e_{7}\to\tau Z)=28.71\%$,

$p\,\,p\ \ \rightarrow\ \ e^{-}_{7}\,\,e^{+}_{7}+X\ \ \rightarrow\ \
\tau^{-}\,Z\,\tau^{+}\,\,Z+X\ \
(\rightarrow\tau^{-}\ell^{+}\ell^{-}\,\tau^{-}\ell^{\prime+}\ell^{\prime-}+X)\,.$
(16)

In our analysis, we consider leptonic tau decays together with $Z$-boson
decays in electrons and muons. This yields a signal rate of 2.8 fb at
NLO+NNLL, as well as a final state exhibiting missing transverse energy. As
shown below, the presence of these specific characteristics provides valuable
handles on the model.

Figure 9: Key observables aimed at distinguishing the BM II signal from the SM
background. We display the distribution of the number of reconstructed leptons
$N_{\ell}$ (left), and of the total missing transverse energy $\not{E}_{T}$
(right). Histograms are normalised to a luminosity of $3\,\text{ab}^{-1}$ of
$pp$ collisions at $\sqrt{s}=100\,\text{TeV}$.

Backgrounds are similar to those depicted in section 4.1, with cross sections
given by $\sigma_{Zh}=1.52~{}\text{ab}$, $\sigma_{ZZZ}=0.2~{}\text{fb}$ and
$\sigma_{t\bar{t}ZZ}=0.17~{}\text{fb}$ after including relevant weak boson and
tau lepton branching ratios. To extract the signal, events with at least six
electrons or muons produced in association with missing transverse energy
$\not{E}_{T}>100$ GeV are selected, with lepton and jet collections defined
like in section 4.1. Such choices are illustrated by the signal and background
distributions presented in figure 9. These basic criteria yield the following
number of signal events and background events,

$n_{s}=53.3\qquad\text{and}\qquad n_{b}=23.8\quad\text{with}\ \ n_{Zh}\simeq
0,\ \ n_{ZZZ}=2.3\ \ \text{and}\ \ n_{t\bar{t}ZZ}=21.5\,,$ (17)

and the corresponding sensitivities, considering a 20% level of systematics on
the background, are found to be

$s\,=\,7.8\,\sigma\qquad\text{and}\qquad Z_{A}\,=\,10.3\,\sigma\,.$ (18)

Once again, these results demonstrate the clear potential of our setup to
distinguish the signal from the background, exploiting a second collider
characteristics of the model.

##  5 Summary and conclusions

In the absence of direct signals of new physics at colliders, indirect
constraints play a crucial role in constructing relevant and motivated BSM
theories. This often involves reconciling discrepancies between SM predictions
and experimental data, such as those observed in the anomalous magnetic
moments of the muon and electron that can be further linked to neutrino
physics. Our study exploits precisely this. We examined a simplified extension
of the SM to which we added three generations of vector-like leptons, each
associated with a specific SM family.

After exhaustively listing anomaly-free representations for VLLs and their
interaction Lagrangian, we identified parameter space regions compatible with
neutrino data and consistent with measurements of the anomalous magnetic
moment of the electron and the muon. We determined that this requires,
minimally, the inclusion of two weak VLL singlets (one with the quantum
numbers of a right-handed electron and the other with the quantum numbers of a
right-handed neutrino), along with a weak VLL doublet with hypercharge $-1/2$.
The compatible parameter space regions imply distinct collider signatures,
prominently featuring six hard leptons accompanied by either significant total
transverse energy or missing transverse energy. We performed a detailed
phenomenological analysis to demonstrate the sensitivity of future proton-
proton colliders operating at a centre-of-mass energy of 100 TeV to these
signatures, focusing on two representative benchmark scenarios with VLL masses
in the range of $100-400$ GeV. Our findings underscore that if new physics
involves VLLs with properties explaining neutrino data and reconciling
discrepancies in anomalous magnetic moments of the light-generation leptons,
it will yield signals that will be unmistakable at future proton-proton
colliders currently under discussion within the high-energy physics community.

###### Acknowledgements.

We thank Kivanc Cingiloglu for discussions in the earlier stages of this
project, as well as Mark Goodsell, Olivier Mattelaer and Stephen Mrenna for
their respective help with the packages SARAH, MadGraph5_aMC@NLO and Pythia.
Our parameter space scan has been conducted on the Béluga cluster of the
Digital Research Alliance of Canada. The work of MF has been partly supported
by NSERC through grant number SAP105354, and that of BF by the grant
ANR-21-CE31-0013 (project DMwithLLPatLHC) from the French _Agence Nationale de
la Recherche_. For the purpose of open access, a CC-BY public copyright
license has been applied by the authors to the present document, and will be
applied to all subsequent versions up to the one accepted for a publication
arising from this submission.

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