aainstitutetext: Institute of Physics, Bhubaneswar, Sachivalaya Marg, Sainik
School Post, Bhubaneswar 751005, Indiabbinstitutetext: Homi Bhabha National
Institute, Training School Complex, Anushakti Nagar, Mumbai 400094,
Indiaccinstitutetext: Department of Physics and Astrophysics, University of
Delhi, Delhi 110007, India

# Search for exotic leptons in final states with two or three leptons and fat-
jets at 13 TeV LHC

Saiyad Ashanujjaman ID c , Debajyoti Choudhury a,b , Kirtiman Ghosh
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Exotic leptons in large gauge multiplets, appearing in many scenarios beyond
the Standard Model (SM), can be produced at the LHC in pairs or association.
Owing to their large masses, their eventual decay products—SM leptons and
bosons—tend to be highly boosted, with the jets stemming from the SM bosons
more likely to manifest themselves as a single fat-jet rather than two
resolved ones. With the corresponding SM backgrounds being suppressed, final
states with two or three leptons and one or two fat-jets are expected to be
sensitive in probing exotic fermions much heavier than 1 TeV, and we propose
and investigate an appropriate search strategy. To concentrate on the
essential, we consider extensions of the SM by leptonic multiplets of a single
kind (triplets, quadruplets or quintuplets), bearing in mind that such
simplified models typically arise as low-energy limits of more ambitious
scenarios addressing various lacunae of the SM. Performing a systematic and
comprehensive study of nine such scenarios at the 13 TeV LHC, we find that the
corresponding $5\sigma$ discovery reaches a range from 985 GeV to 1650 GeV
(1345 GeV to 2020 GeV) for 300 (3000) fb-1.

###### Keywords:

Doubly charged leptons, Exotic leptons, Fat-jet signatures.

††preprint: IP/BBSR/2021-7

##  1 Introduction

Notwithstanding the Standard Model’s (SM’s) impressive success in describing
the fundamental interactions, it falls short of offering explanations for
quite a few theoretical issues as well as some experimental observations.
Fermion mass and mixing hierarchies, electroweak vacuum instability, the
strong CP problem and the naturalness problem are amongst the pressing
theoretical issues. Experimental inconsistencies include issues like neutrino
masses and mixings, the anomalous magnetic moments of the electron and the
muon, several anomalies in the $b$-quark sector, the baryon asymmetry in the
Universe and the dark matter. Multifarious new physics models going beyond the
SM (BSM) have been proposed to address these shortcomings. A noteworthy
feature of many of these BSM models is the prediction of new particles at the
TeV scale.

A wealth of BSM models addressing the issue of neutrino masses and mixings
such as the canonical see-saw models and their variants Minkowski:1977sc ;
Yanagida:1979as ; Gell-Mann:1979vob ; Glashow:1979nm ; Mohapatra:1979ia ;
Foot:1988aq ; Bajc:2006ia ; FileviezPerez:2007bcw ; FileviezPerez:2007yji ;
Bonnet:2009ej ; Liao:2010ku ; Bonnet:2012kz ; Cepedello:2017lyo ;
Anamiati:2018cuq ; Arbelaez:2019cmj ; Picek:2009is ; Kumericki:2011hf ;
Kumericki:2012bh ; McDonald:2013hsa ; Avnish:2020rhx ; Agarwalla:2018xpc ;
Li:2009mw ; Ashanujjaman:2021jhi ; Ashanujjaman:2020tuv ; Babu:2009aq ;
Delgado:2011iz ; Ma:2013tda ; Ma:2014zda ; Yu:2015pwa ; Ashanujjaman:2021zrh
envisage the presence of new leptonic gauge multiplets. For instance, Ref.
Babu:2009aq  extended the fermion sector by vector-like $SU(2)_{L}$ triplet
leptons with hypercharge $Y=1$ in order to generate neutrino masses via
effective dimension seven operators. Similarly, Refs. Anamiati:2018cuq ;
Ashanujjaman:2020tuv , in generating neutrino masses via effective dimension
nine operators, introduced several fermionic multiplets, namely vector-like
$SU(2)_{L}$ triplets with $Y=1$, vector-like $SU(2)_{L}$ quadruplets with
$Y=1/2$ and chiral $SU(2)_{L}$ quintuplets with $Y=0$. The presence of several
charged exotic leptons within the new weak gauge leptonic multiplets is a
common yet salient feature of most of these neutrino mass models Bonnet:2009ej
; Liao:2010ku ; Bonnet:2012kz ; Kumericki:2012bf ; Cepedello:2017lyo ;
Anamiati:2018cuq ; Arbelaez:2019cmj ; Picek:2009is ; Kumericki:2011hf ;
Kumericki:2012bh ; Yu:2015pwa ; McDonald:2013hsa ; Delgado:2011iz ; Ma:2013tda
; Ma:2014zda ; Ko:2015uma ; Avnish:2020rhx ; Agarwalla:2018xpc ;
KumarAgarwalla:2018nrn ; Kumar:2019tat ; Ashanujjaman:2020tuv ; Babu:2009aq ;
Li:2009mw ; Ashanujjaman:2021jhi ; Kumar:2021umc . Copious production of
different such charged exotic leptons, driven by their electroweak couplings
to the $W$ and $Z$ bosons and the photon, and their eventual decays to the SM
leptons and bosons ($W/Z/h$-boson) offer interesting ways to probe them at the
Large Hadron Collider (LHC). Phenomenology of such exotics leptons (in
particular, the doubly-charged one) at the LHC has been studied extensively in
the literature Alloul:2013raa ; Delgado:2011iz ; Ma:2013tda ; Chen:2013xpa ;
Ding:2014nga ; Ma:2014zda ; Yu:2015pwa ; Leonardi:2014epa ; Biondini:2012ny ;
Kumar:2021umc . However, all of these searches considered multileptons,
missing transverse momentum ($p_{T}^{\rm miss}$) and jets in the final states.
Typically, these final state signatures are beset with considerably large SM
background contributions. Moreover, for signatures with $p_{T}^{\rm miss}$,
multiple leptons and jets with unidentifiable origins, kinematic
reconstructions of the exotic leptons. Consequently, these searches are often
non-optimal in probing exotic leptons with masses beyond 1 TeV. This behoves
us to inspect complementary signatures. For TeV scale exotics, their decay
products (SM leptons and bosons) could be sufficiently boosted that the jets
emanating from them would be collimated. Consequently, the hadronically
decaying bosons are more likely to manifest as a single fat-jet than two
resolved jets. On the contrary, only a small fraction of the SM background
events would give rise to fat-jets in the final state. Not only are signatures
comprising multileptons and fat-jets cleaner than those in the searches
mentioned above Alloul:2013raa ; Delgado:2011iz ; Ma:2013tda ; Chen:2013xpa ;
Ding:2014nga ; Ma:2014zda ; Yu:2015pwa ; Leonardi:2014epa ; Biondini:2012ny ,
but most of these final states also allow for kinematic reconstructions of the
exotic leptons. Consequently, such signatures are expected to be sensitive in
probing exotic leptons with masses beyond 1 TeV.

Adopting the bottom-up approach for new physics, we consider several
simplified models by extending the SM with only one type of new leptonic weak
gauge multiplet at a time. We design a search for the charged exotic leptons
belonging to these simplified models in final states with multilepton and fat-
jets. By performing a systematic and comprehensive collider study, we estimate
the $5\sigma$ discovery reach of this search for nine such simplified models
for both 300 and 3000 fb-1 of the integrated luminosity of proton-proton
collision data at the 13 TeV LHC.

The rest of this work is structured as follows. In Sec. 2, we briefly discuss
some simplified models. Productions and decays of the exotic leptons are
discussed in Sec. 3. We perform a systematic and comprehensive collider study
in Sec. 4 followed by a summary in Sec. 5.

##  2 Simplified models

Understandably, most recent BSM searches by both the CMS and ATLAS
collaborations have been designed to probe specific models that are well-
placed to address particular shortcomings of the SM. Being tuned to a specific
model, such strategies often lose sensitivity when applied to more general
models but with similar final states. This behoves us to design a search
strategy for charged exotic leptons resulting in final states with multilepton
and fat-jets so that the former could be applied to any BSM scenario that
includes such exotic leptons decaying into each other or an SM lepton and a
boson without impinging much on the sensitivity. To this end, we consider
several simplified models111Such simplified models typically arise as low-
energy limits of more ambitious scenarios Anamiati:2018cuq ; Arbelaez:2019cmj
; Picek:2009is ; Kumericki:2011hf ; Kumericki:2012bh ; Yu:2015pwa ;
McDonald:2013hsa ; Delgado:2011iz ; Ma:2013tda ; Ma:2014zda ; Avnish:2020rhx ;
Agarwalla:2018xpc ; Ashanujjaman:2020tuv ; Babu:2009aq  addressing various
shortcomings of the SM. The simplified models, thus, involve a smaller set of
parameters compared to the more complete ones. adopting the bottom-up approach
for new physics.

Keeping the SM gauge group unaltered, we supplement the SM field content with
exotic leptonic states. While a fully general analysis is virtually
impossible, we attempt a generic framework by allowing the new leptonic states
to belong to different $SU(2)_{L}$ representations, namely 3, 4 and 5. As for
the hypercharge assignments, we limit these to (1,2), (1/2,3/2,5/2) and
(0,1,2,3) for the triplet, quadruplet and quintuplet fields, respectively.
This restriction is imposed to ensure that these multiplets have at least one
component of both doubly- and singly-charged fields. Therefore, we have nine
different leptonic multiplets, namely $3^{F}_{1}$, $3^{F}_{2}$, $4^{F}_{1/2}$,
$4^{F}_{3/2}$, $4^{F}_{5/2}$, $5^{F}_{0}$, $5^{F}_{1}$, $5^{F}_{2}$ and
$5^{F}_{3}$, and these, along with their differently charged components, are
listed in Table 1. In our notation, $\chi_{Y}^{Q}$ stands for the $Q$-charged
component of the $SU(2)_{L}$ $\chi$-plet with hypercharge $Y$.

$3^{F}_{1}={\begin{pmatrix}[l]3_{1}^{++}\\\ 3_{1}^{+}\\\
3_{1}^{0}\end{pmatrix}}$, $3^{F}_{2}={\begin{pmatrix}[l]3_{2}^{3+}\\\
3_{2}^{++}\\\ 3_{2}^{+}\end{pmatrix}}$,
$4^{F}_{1/2}={\begin{pmatrix}[l]4_{1/2}^{++}\\\ 4_{1/2}^{+}\\\ 4_{1/2}^{0}\\\
4_{1/2}^{-}\end{pmatrix}}$, $4^{F}_{3/2}={\begin{pmatrix}[l]4_{3/2}^{3+}\\\
4_{3/2}^{++}\\\ 4_{3/2}^{+}\\\ 4_{3/2}^{0}\end{pmatrix}}$,
$4^{F}_{5/2}={\begin{pmatrix}[l]4_{5/2}^{4+}\\\ 4_{5/2}^{3+}\\\
4_{5/2}^{++}\\\ 4_{5/2}^{+}\end{pmatrix}}$,
---
$5^{F}_{0}={\begin{pmatrix}[l]5_{0}^{++}\\\ 5_{0}^{+}\\\ 5_{0}^{0}\\\
5_{0}^{-}\\\ 5_{0}^{--}\end{pmatrix}}$,
$5^{F}_{1}={\begin{pmatrix}[l]5_{1}^{3+}\\\ 5_{1}^{++}\\\ 5_{1}^{+}\\\
5_{1}^{0}\\\ 5_{1}^{-}\end{pmatrix}}$,
$5^{F}_{2}={\begin{pmatrix}[l]5_{2}^{4+}\\\ 5_{2}^{3+}\\\ 5_{2}^{++}\\\
5_{2}^{+}\\\ 5_{2}^{0}\end{pmatrix}}$,
$5^{F}_{3}={\begin{pmatrix}[l]5_{3}^{5+}\\\ 5_{3}^{4+}\\\ 5_{3}^{3+}\\\
5_{3}^{++}\\\ 5_{3}^{+}\end{pmatrix}}$.
Table 1:  Exotic leptonic multiplets and their components.

With the LHC having already constrained such leptons to be heavier than a few
hundred GeV, clearly, such masses cannot be generated through electroweak
symmetry breaking, as this would not only entail uncomfortably large Yukawa
couplings but also generate too large a diphoton branching fraction for the SM
Higgs. Thus, bare mass terms need to be present, and multiplets with non-zero
hypercharges must be vector-like, allowing for a Dirac mass term
$M_{\chi_{Y}}\overline{\chi^{F}_{YL}}\chi^{F}_{YR}$. This would automatically
cancel the $U(1)$ anomaly without introducing additional states. Multiplets
with vanishing hypercharges (and, hence, no contribution to the anomaly) could
very well be chiral, yet allowing for a lepton number violating mass term such
as $M_{\chi_{Y}}\overline{\left(\chi^{F}_{YR}\right)^{c}}\chi^{F}_{YR}$. Here
onwards, we omit the subscript $Y$ from the mass term for brevity. Being gauge
invariant, these masses (whether lepton-number conserving or violating) would
naturally be comparable to the new physics scale. The differently charged
components in a given multiplet are mass-degenerate at the tree-level.
Radiative corrections, dominantly induced by the electroweak gauge bosons,
lift this degeneracy and induce mass-splittings between the differently
charged components $\chi^{Q}_{Y}$ and $\chi^{Q^{\prime}}_{Y}$ Cirelli:2005uq

$\Delta M(\chi^{Q}_{Y},\chi^{Q^{\prime}}_{Y})\approx\frac{\alpha
M_{\chi}}{4\pi}\left[(Q^{2}-Q^{\prime
2})f\left(r_{Z}\right)+\frac{Q-Q^{\prime}}{\sin^{2}\theta_{w}}(Q+Q^{\prime}-2Y)\left\\{f\left(r_{W}\right)-f\left(r_{Z}\right)\right\\}\right]~{},$

where $\theta_{w}$ is the weak-mixing angle, $\alpha$ is the electromagnetic
fine structure constant, $r_{a}\equiv m_{a}/m_{\chi}$, and the function $f(r)$
has the form

$f(r)=\frac{r}{2}\left[2r^{3}\ln
r-2r+\sqrt{r^{2}-4}~{}(r^{2}+2)\ln\left\\{\frac{r^{2}-2-r\sqrt{r^{2}-4}}{2}\right\\}\right]~{}.$

For the mass range of our interest ($M_{\chi}\gg m_{W,Z}$), the function
$f(r)\approx-2\pi r$ with the imaginary part being suppressed by more than an
order of magnitude. Consequently, the mass splittings being completely
determined by the electric- and hyper-charges of the particles involved are
almost independent of the common mass. Most importantly, the components with
larger charges are the heavier ones. Summarised in Table 2, these
splittings222It might be argued that the tree-level mass splittings introduced
by possible Higgs couplings would be larger. Given that we would be including
exotic multiplets of only one type, such couplings are relevant only for the
$3^{F}_{1}$ field. Even there, concerns such as neutrino masses and possible
FCNCs restrict these couplings to be tiny and the splittings generated thereby
to be of little consequence. are inconsequential as far as production cross-
sections at the LHC are concerned. However, these play a crucial role in their
decays; we defer this discussion till Sec. 3.

$\Delta M(3_{1}^{++},3_{1}^{+})\sim$ [860–870] MeV, $\Delta
M(3_{1}^{+},3_{1}^{0})\sim$ [515–525] MeV;
---
$\Delta M(3_{2}^{3+},3_{2}^{++})\sim$ [1545–1570] MeV, $\Delta
M(3_{2}^{++},3_{2}^{+})\sim$ [1200–1220] MeV;
$\Delta M(4_{1/2}^{++},4_{1/2}^{+})\sim$ [685–695] MeV, $\Delta
M(4_{1/2}^{+},4_{1/2}^{0})\sim$ [340–350] MeV, $\Delta
M(4_{1/2}^{0},4_{1/2}^{-})\sim$ [-5–2] MeV;
$\Delta M(4_{3/2}^{3+},4_{3/2}^{++})\sim$ [1370–1390] MeV, $\Delta
M(4_{3/2}^{++},4_{3/2}^{+})\sim$ [1025–1045] MeV, $\Delta
M(4_{3/2}^{+},4_{3/2}^{0})\sim$ [685–700] MeV;
$\Delta M(4_{5/2}^{4+},4_{5/2}^{3+})\sim$ [2060–2090] MeV, $\Delta
M(4_{5/2}^{3+},4_{5/2}^{++})\sim$ [1715–1745] MeV, $\Delta
M(4_{5/2}^{++},4_{5/2}^{+})\sim$ [1370–1400] MeV;
$\Delta M(5_{0}^{\pm\pm},5_{0}^{\pm})\sim$ [515–520] MeV, $\Delta
M(5_{0}^{\pm},5_{0}^{0})\sim$ 170 MeV;
$\Delta M(5_{1}^{3+},5_{1}^{++})\sim$ [1200–1220] MeV, $\Delta
M(5_{1}^{++},5_{1}^{+})\sim$ [855–870] MeV, $\Delta
M(5_{1}^{+},5_{1}^{0})\sim$ [515–525] MeV, $\Delta M(5_{1}^{0},5_{1}^{-})\sim$
170 MeV;
$\Delta M(5_{2}^{4+},5_{2}^{3+})\sim$ [1890–1915] MeV, $\Delta
M(5_{2}^{3+},5_{2}^{++})\sim$ [1545–1570] MeV, $\Delta
M(5_{2}^{++},5_{2}^{+})\sim$ [1200–1220] MeV, $\Delta
M(5_{2}^{+},5_{2}^{0})\sim$ [855–875] MeV;
$\Delta M(5_{3}^{5+},5_{3}^{4+})\sim$ [2580–2615] MeV; $\Delta
M(5_{3}^{4+},5_{3}^{3+})\sim$ [2230–2265] MeV, $\Delta
M(5_{3}^{3+},5_{3}^{++})\sim$ [1885–1920] MeV, $\Delta
M(5_{3}^{++},5_{3}^{+})\sim$ [1540–1570] MeV;

Table 2:  Mass-splittings between the differently charged components of the
exotics multiplets.

In this work, we extend the SM with a single copy of only one of the
aforementioned multiplets at a time. In other words, we consider nine simple
scenarios, each of which we denote as Model-$\chi^{F}_{Y}$ depending on the
identity of the exotic multiplets (see Table 1). Such simplified models can
often be viewed as low-energy limits of specific well-motivated BSM models
when some of the fields decouple and/or some of the interactions cease to
exist (see Table 3). For instance, Model-$3^{F}_{1}$ is a limit of the
neutrino mass model presented in Ref. Babu:2009aq  when the scalar quadruplet
with $Y=3/2$ decouples. Likewise, Model-$3^{F}_{1}$ (Model-$4^{F}_{1/2}$)
[Model-$5^{F}_{0}$] could be thought of as a limit of another neutrino mass
model in Ref. Ashanujjaman:2020tuv  when the $4^{F}_{1/2}$ and $5^{F}_{0}$
($3^{F}_{1}$ and $5^{F}_{0}$) [$3^{F}_{1}$ and $4^{F}_{1/2}$] multiplets get
decoupled. Understandably though, such a correspondence is not unique.
Moreover, a given scenario, say Model-$5^{F}_{3}$, may not be easily relatable
to any popular theory.

Simplified model | Corresponding more ambitious scenario
---|---
Refs. | Motivation
Model-$3^{F}_{1}$ | Babu:2009aq  | neutrino mass
Delgado:2011iz ; Ma:2013tda ; Ashanujjaman:2020tuv  | neutrino mass
Model-$4^{F}_{1/2}$ | McDonald:2013hsa  | neutrino mass
Ashanujjaman:2020tuv  | neutrino mass
Model-$5^{F}_{0}$ | Cai:2011qr ; Kumar:2021umc  | neutrino mass and dark matter
Kumericki:2012bf ; Kumericki:2012bh ; Yu:2015pwa ; Kumar:2019tat ; Ashanujjaman:2020tuv  | neutrino mass
Cirelli:2009uv ; Ko:2015uma ; KumarAgarwalla:2018nrn  | dark matter
Model-$5^{F}_{1}$ | Picek:2009is ; Kumericki:2011hf  | neutrino mass
KumarAgarwalla:2018nrn ; Ko:2015uma  | dark matter
Model-$5^{F}_{2}$ | KumarAgarwalla:2018nrn ; Ko:2015uma  | dark matter

Table 3: Some simplified models and corresponding more ambitious scenarios
studied in the literature.

Irrespective of such correspondence, each of these simplified models is UV-
complete in its own right. And while some of the exotics can play a direct
role in generating neutrino masses Yu:2015pwa ; McDonald:2013hsa ;
Delgado:2011iz ; Ma:2013tda ; Ma:2014zda ; Avnish:2020rhx ; Agarwalla:2018xpc
; Picek:2009is ; Kumericki:2011hf ; Kumericki:2012bh ; Babu:2009aq ;
Anamiati:2018cuq ; Arbelaez:2019cmj ; Ashanujjaman:2020tuv  and/or addressing
other concerns like the anomalous magnetic moments of the muon and the
electron or even the possible evidence of lepton flavour violation in
$B$-decays, the others may not play a role in facilitating these. In
particular, in order to generate three non-zero masses for the light
neutrinos, most of the neutrino mass models Bonnet:2009ej ; Liao:2010ku ;
Bonnet:2012kz ; Cepedello:2017lyo ; Anamiati:2018cuq ; Arbelaez:2019cmj ;
Picek:2009is ; Kumericki:2011hf ; Kumericki:2012bh ; Yu:2015pwa ;
McDonald:2013hsa ; Delgado:2011iz ; Ma:2013tda ; Ma:2014zda ; Avnish:2020rhx ;
Agarwalla:2018xpc ; Ashanujjaman:2020tuv ; Babu:2009aq ; Li:2009mw ;
Ashanujjaman:2021jhi  require three copies of the exotic multiplets. While it
might be argued that, in all strictness, as of now, only two of the light
neutrino states need to be massive, and thus only two of the heavy generations
are called for, it should be realised that were more sensitive experiments to
conclude that all the light neutrino species are indeed massive, one would
need three exotic species. However, most extant experimental searches
ATLAS:2018ghc ; CMS:2019lwf ; ATLAS:2020wop ; ATLAS:2021xxb ; CMS:2021zkl ;
ATLAS:2021eyc  consider one such species, and thus an easy comparison with
such studies is facilitated by the inclusion of just one species rather than
two or three.

All the simplified models considered in this work accommodate large
$SU(2)_{L}$ multiplet. The introduction of such multiplet changes the running
of the $U(1)_{Y}$ and $SU(2)_{L}$ gauge couplings ($g_{1}$ and $g_{2}$). In
Table 10 (in Appendix A), we list the one-loop $\beta$-functions describing
the evolution of $g_{1}$ and $g_{2}$ in these models. Also tabulated are the
energy scales at which the gauge couplings blow up (the Landau poles in
$g_{1}$ and/or $g_{2}$), scales that vary strongly across these models. The
said energy scale is well below the grand unification scale for some of these
models. For instance, the Landau pole in $g_{2}$ appear at a scale of
$10^{10}$ GeV for all the vector-like quintuplet models. For the
Model-$5^{F}_{3}$, the Landau pole in $g_{1}$ appears at a scale of $10^{6}$
GeV. In other words, these specific models need to be superseded well below
such scales.

##  3 Production and decays of the exotics

The exotic leptons are pair produced aplenty at the LHC by quark-antiquark
annihilation through $s$-channel $\gamma/Z$ and $W^{\pm}$ exchanges:

$q\bar{q^{\prime}}\to W^{*}\to\chi^{Q}\chi^{-Q\pm 1}\quad{\rm and}\quad
q\bar{q}\to\gamma^{*}/Z^{*}\to\chi^{Q}\chi^{-Q}.$

They can also be pair produced via the $t/u$-channel photon-photon fusion
process,333They can also be pair produced via vector boson-fusion ($\gamma
Z,\gamma W^{\pm},ZZ,ZW^{\pm}$) processes, with one or two associated forward
jets at the LHC. However, such processes are rather sub-dominant, and can be
safely neglected. namely

$\gamma\gamma\to\chi^{Q}\chi^{-Q}\quad(Q\neq 0).$

Despite their very small (as compared to the other partons) density, photon
initiated pair production becomes significant for large masses of the multi-
charged exotics Babu:2016rcr ; Ghosh:2017jbw , with the bulk of the
contribution emanating from subprocesses wherein one of the photons originates
from a resolved proton. And while single production (in association with an SM
lepton) is, in principle, possible, such processes are highly suppressed on
account of the small mixing.

We evaluate the leading order (LO) production cross-sections of the exotics at
the 13 TeV LHC using the UFO modules generated from SARAH Staub:2013tta ;
Staub:2015kfa  in MadGraph Alwall:2011uj ; Alwall:2014hca  with the
LUXqed17-plus-PDF4LHC15-nnlo-100 parton distribution function (PDF)
Manohar:2016nzj ; Manohar:2017eqh ; Butterworth:2015oua .444The LUXqed17 PDF
Manohar:2016nzj ; Manohar:2017eqh , the photon PDF inside the proton
determined in a model-independent manner using electron-proton scattering
data, combined with QCD partons from the PDF4LHC15 PDF Butterworth:2015oua
results in the LUXqed17-plus-PDF4LHC15-nnlo-100 PDF.’555In order to estimate
‘true’ LO production cross-section, one should, in principle, convolute the LO
partonic cross-section with the LO partonic distributions only Basu:2002uu ,
and thus the extensively used NNPDF23-LO-as-0130-qed PDF should have been an
obvious choice. However, as pointed out in Refs. Ghosh:2018drw ; Fuks:2019clu
, in the case of this PDF, the predicted production cross-sections for the
photon fusion initiated processes suffer from substantial uncertainties due to
large uncertainties in the photon density inside the proton. On the contrary,
the LUXqed17-plus-PDF4LHC15-nnlo-100 PDF has much smaller errors in
determining the photon density inside the proton. Moreover, for the mass range
of our interest, both the PDFs predicts almost the same Drell-Yan production
cross-sections—the difference being well within a few per cent. In view of
these, we chose the latter PDF over the former one.

The plots in Fig. 1 show different pair and associated LO production cross-
sections for the exotic leptons at the 13 TeV LHC as a function of their tree-
level mass for the different scenarios considered in this work. The left
(right) plot in the top panel shows the pair production cross-sections for the
doubly- (singly-) charged exotics, while the left plot in the bottom panel
shows the associated production cross-sections for the doubly- and singly-
charged exotics.666In models with higher-charged leptons as well, those too
would be produced, and we return to this issue later.’777Some of the
simplified models considered in this work also contain neutral leptons (see
Table 1). However, we do not consider them, partly because their pair
production cross-sections are smaller than those for the charged ones (see the
right plot in the bottom panel of Fig. 1). Though in some of these models,
their productions in association with the singly-charged ones are comparable
to those of the other processes, with signal regions in the present search
being designed for the charged exotic leptons, the inclusion of the neutral
ones would not entail significant added contributions. For instance, trilepton
events from charged exotics production would demonstrate a peak in the
invariant mass distribution of a particular lepton—distinguishable from the
others—and the fat-jet, whereas this is not the case for those from neutral
production. In view of these and the existing search paradigm (that the
charged ones are likely to be discovered prior to the neutral ones) adopted by
the ATLAS/CMS collaborations, we only consider the charged exotic
leptons.Refs. Fuks:2012qx ; Fuks:2013vua ; Ruiz:2015zca  have estimated the
QCD corrections to the production of heavy $SU(2)_{L}$ triplet leptons at
hadron colliders. The resulting next-to-leading order (NLO) $K$-factors varies
from 1.1 to 1.4 for both charged current and neutral current processes over a
triplet mass range of 100 GeV to 2 TeV. Further, Ref. Ruiz:2015zca  shows that
the NLO differential $K$-factors for heavy lepton kinematics are substantially
flat, suggesting that naive scaling by an overall $K$-factor is a very good
approximation. And while the corrections have not been calculated for the
higher multiplets, it is obvious that the said corrections pertain to the
hadronic ends of the Feynman diagrams, and these are universal for all the
cases under discussion. Guided by this, we apply an overall QCD $K$-factor of
1.25 to the LO cross-section.

Figure 1: LO production cross-sections of the exotics in different simplified
models: doubly-charged pair production (top left), singly-charged pair
production (top right), doubly- and singly-charged associated production
(bottom left), and neutral pair and neutral-charged productions (bottom
right).

The decays of the exotics often depend on the details of the UV-completion,
for example, through their effective mixings with the SM counterparts, either
through direct Yukawa couplings or through a combination of couplings
involving yet other exotics. In broad terms, the leading decays can be
categorised into three classes:

  * •

Category-I: The aforementioned mixing can induce two-body decays into a SM
lepton and a boson. The rates are, understandably, dependent on the elusive
parameters.

  * •

Category-II: The radiative mass-splittings between the differently charged
components within a multiplet induces the heavier exotics to decay into the
lighter ones in association with $\ell^{+}\nu$, $\pi^{+}$, $K^{+}$,
$\pi^{+}\pi^{0}/\rho(770)$, etc. if kinematically accessible; we refer to
these decays as heavy state transitions. Driven by gauge interactions, these
are governed only by mass splittings.

  * •

Category-III: It could go to into a $W$-boson and an off-shell (exotic)
partner which goes into an SM lepton and an SM boson. Thus, in effect, we have
a 3-body decay into two bosons and an SM lepton.

The dominance of one category of decays over the others depends, naturally, on
the parameter space. For example, being suppressed by both the mixing and the
exotic lepton mass in the propagator, the rates for Category-III are,
typically, much smaller Kumericki:2011hf  than the Category-II decays and can
be safely neglected. Similarly, for the singly- and doubly-charged exotic
leptons, the corresponding mass-splittings are not large enough ($\sim$ a few
hundred MeV – 1 GeV) so that the Category-I decays dominate over the Category-
II ones. On the contrary, for the multi-charged (with $|Q|\geq 3$) exotic
leptons, Category-I decays are not allowed at all. Consequently, these undergo
Category-II decays. In summary, the dominant decay modes for the exotic
leptons are

$\chi^{\pm\pm}\to\ell^{\pm}W^{\pm}\quad{\rm
and}\quad\chi^{\pm}\to\ell^{\pm}Z,\,\ell^{\pm}h,\,\nu W^{\pm}\quad{\rm
with}\quad\ell=e,\mu,\tau~{},$

while for $n\geq 3$,

$\chi^{n\pm}\to\chi^{(n-1)\pm}+\hat{X}\,\quad{\rm
with}\quad\hat{X}\in\pi^{\pm},\,K^{\pm},\,\pi^{\pm}\pi^{0},\,\rho^{\pm}(770),\,\ell^{\pm}\nu.$

Although the relevant mass-splittings (see Table 2) are small, these are
substantial enough to ensure that some of the heavy state transitions are
prompt. However, the hadrons/leptons stemming from such decays are very soft,
and thus can not be easily identified (as the resultant of a primary decay) in
the LHC environment owing to the inherent large hadronic activity. Therefore,
multi-charged exotics’ pair and associated productions serve only to enhance
the doubly-charged pair production effectively. For instance, the effective
pair production cross-section of doubly-charged exotics in Model-$3^{F}_{2}$
is given by

$\sigma^{\rm
eff}(3^{\pm\pm}_{2}3^{\mp\mp}_{2})=\sigma(3^{\pm\pm}_{2}3^{\mp\mp}_{2})+\sigma(3^{3\pm}_{2}3^{3\mp}_{2})+\sigma(3^{3\pm}_{2}3^{\mp\mp}_{2})~{}.$

We plot the LO effective cross-sections for doubly-charged pair production as
a function of their tree-level mass in different simplified models in Fig. 2.
We see that all the models, particularly those with multiplet lying in the
higher representation of $SU(2)_{L}$, yield sizeable cross-sections at the 13
TeV LHC.

Figure 2: Effective cross-section of doubly-charged pair production at LO in
different simplified models.

For simplicity, we assume equal branching fractions888This assumption does not
have a substantial impact on our final results. For one, both the $W$ and $Z$
boson have similar hadronic branching fractions; second, the analysis to be
presented here does not distinguish between the fat-jets emanating from them.
for the decays $\chi^{\pm}\to\ell^{\pm}Z$ and $\chi^{\pm}\to\ell^{\pm}h$.
Further, to facilitate an easy comparison with existing search paradigms
adopted by the ATLAS/CMS collaborations ATLAS:2018ghc ; CMS:2019lwf ;
ATLAS:2020wop ; ATLAS:2021xxb , we assume that their branching fractions are
identical across all lepton flavours (flavour-democratic scenario).

##  4 Collider phenomenology

The exotic leptons are copiously produced via neutral or charged current
Drell-Yan-like processes at the LHC. As mentioned earlier, the multi-charged
exotic leptons decay promptly into the doubly-charged exotic partner in
association with mesons/leptons, which are very soft, and thus are not likely
to pass the reconstruction criteria of the LHC detectors. Therefore, the
production of the multi-charged exotics effectively enhances the cross-
sections for the doubly-charged ones. Subsequent prompt decays to SM particles
lead to various final state signatures at the LHC. Listed in Table 4, these
include smoking gun signatures like mono-Higgs, di-Higgs, di-gauge boson, etc.
in association with multiple leptons. Further decays of the SM bosons lead to
various final states, some of which allow for kinematic reconstructions of the
exotics. The exotics are expected to be in the TeV regime, so their decay
products (SM leptons and bosons) would be highly boosted. Consequently, the
jets emanating from the decaying SM bosons would tend to be collimated, with
such a boson more likely to manifest itself as a single fat-jet (with a larger
radius) rather than two resolved jets. Though many of the final states are
interesting in their own right, we emphasise only those with multilepton and
fat-jets.

| $\bm{\chi^{++}\to\ell^{{}^{\prime}+}W^{+}}$ | $\bm{\chi^{+}\to\nu W^{+}}$ | $\bm{\chi^{+}\to\ell^{{}^{\prime}+}Z}$ | $\bm{\chi^{+}\to\ell^{{}^{\prime}+}h}$
---|---|---|---|---
$\bm{\chi^{--}\to\ell^{-}W^{-}}$ | $\ell^{-}\ell^{{}^{\prime}+}W^{-}W^{+}$ | $\ell^{-}\nu W^{-}W^{+}$ | $\ell^{-}\ell^{{}^{\prime}+}W^{-}Z$ | $\ell^{-}\ell^{{}^{\prime}+}W^{-}h$
$\bm{\chi^{-}\to\nu W^{-}}$ | $\nu\ell^{{}^{\prime}+}W^{-}W^{+}$ | $\nu\nu W^{-}W^{+}$ | $\nu\ell^{{}^{\prime}+}W^{-}Z$ | $\nu\ell^{{}^{\prime}+}W^{-}h$
$\bm{\chi^{-}\to\ell^{-}Z}$ | $\ell^{-}\ell^{{}^{\prime}+}ZW^{+}$ | $\ell^{-}\nu ZW^{+}$ | $\ell^{-}\ell^{{}^{\prime}+}ZZ$ | $\ell^{-}\ell^{{}^{\prime}+}Zh$
$\bm{\chi^{-}\to\ell^{-}h}$ | $\ell^{-}\ell^{{}^{\prime}+}hW^{+}$ | $\ell^{-}\nu hW^{+}$ | $\ell^{-}\ell^{{}^{\prime}+}hZ$ | $\ell^{-}\ell^{{}^{\prime}+}hh$

Table 4: Possible final state signatures of pair/associated productions of
charged exotic leptons.

In the following, we briefly review the processes contributing to the final
state signatures of our interest.

  * •

Trilepton and fat-jet: Both pair and associated production of doubly-charged
exotics contribute to this final state as follows:

    1. (a)

$\chi^{++}\chi^{--}\to\ell^{+}W^{+}\ell^{{}^{\prime}-}W^{-}\to\ell^{+}\ell^{{}^{\prime}-}\ell^{{}^{\prime\prime}\pm}\nu
J$,

    2. (b)

$\chi^{\pm\pm}\chi^{\mp}\to\ell^{\pm}W^{\pm}\ell^{{}^{\prime}\mp}Z/h\to\ell^{\pm}\ell^{{}^{\prime}\pm}\ell^{{}^{\prime\prime}\mp}\nu
J$,

where $J$ denotes a fat-jet emanating from the hadronically decaying SM boson.
Here, (one of) the $W$-boson(s) from the $\chi^{\pm\pm}$ decays leptonically
and the other $W(Z/h)$ from the $\chi^{\pm\pm}(\chi^{\pm})$ decays
hadronically. Hence, two same-charge (sc) leptons result from the
$\chi^{\pm\pm}$ decay, and the other exotic fermion results in an opposite-
charge (oc) lepton and a hadronically decaying boosted boson. Thus, not only
would the mass of the fat-jet peak at $M_{W/Z/h}$, but the mass of
$\chi^{\pm\pm}$ can also be reconstructed from the peak of the invariant mass
distribution of the opposite-charge lepton and the fat-jet.

  * •

Same-charge dilepton and fat-jet: Only associated production of doubly-charged
exotics contribute to this final state:

    1. $\chi^{\pm\pm}\chi^{\mp}\to\ell^{\pm}W^{\pm}\nu W^{\mp}\to\ell^{\pm}\ell^{{}^{\prime}\pm}\nu\nu J$.

In this case, the $W$-boson from the $\chi^{\pm\pm}$ decays leptonically while
that from the $\chi^{\pm}$ decays hadronically. Since neither of
$\chi^{\pm\pm}$ nor $\chi^{\pm}$ decays to entirely visible final states, and
since there are two neutrinos in the final state, none of two masses can be
reconstructed.

  * •

Opposite-charge dilepton and fat-jets: All possible production of doubly- and
singly-charged exotics contribute to this final state999Note that the
opposite-charge dilepton final states usually have huge SM backgrounds.
However, the signal rates are also large because of contributions from all
possible production channels. Hoping to reduce the background contributions to
a manageable level using appropriate kinematic cuts, we consider these final
states. as follows:

    1. (a)

$\chi^{\pm\pm}\chi^{\mp}\to\ell^{\pm}W^{\pm}\nu
W^{\mp}\to\ell^{\pm}\ell^{{}^{\prime}\mp}\nu\nu J$,

    2. (b)

$\chi^{+}\chi^{-}\to\ell^{\pm}Z/h\;\nu
W^{\mp}\to\ell^{\pm}\ell^{{}^{\prime}\mp}\nu\nu J$,

    3. (c)

$\chi^{++}\chi^{--}\to\ell^{+}W^{+}\ell^{{}^{\prime}-}W^{-}\to\ell^{{}^{\prime}+}\ell^{{}^{\prime}-}JJ$,

    4. (d)

$\chi^{\pm\pm}\chi^{\mp}\to\ell^{\pm}W^{\pm}\ell^{{}^{\prime}\mp}Z/h\to\ell^{\pm}\ell^{{}^{\prime}\mp}JJ$,

    5. (e)

$\chi^{+}\chi^{-}\to\ell^{+}Z/h\ell^{{}^{\prime}-}Z/h\to\ell^{+}\ell^{{}^{\prime}-}JJ$.

For processes (a) and (b), the $W$($Z/h$)-boson from the
$\chi^{\pm\pm}$($\chi^{\pm}$) decays hadronically, and the other $W$ from the
$\chi^{+}$ decays leptonically. The lepton from the decay of
$\chi^{\pm\pm}/\chi^{\pm}$ usually has a larger $p_{T}$ than that coming from
the $W$-decay. Therefore, the peak of the invariant mass distribution of the
leading lepton and the fat-jet system is expected to yield $M_{\chi^{\pm\pm}}$
(or $M_{\chi^{\pm}}$, as the case may be). For processes (c), (d) and (e),
both the $W/Z/h$-bosons from $\chi^{\pm\pm}/\chi^{\pm}$ decay hadronically so
that the final state contains two fat-jets. For such events, the
reconstruction of both the exotic fermions is possible, although there are two
independent pairs of invariant masses: $(i)~{}M_{J_{0}\ell_{0}}$ and
$M_{J_{1}\ell_{1}}$, and $(ii)~{}M_{J_{1}\ell_{0}}$ and $M_{J_{0}\ell_{1}}$,
where $\ell_{0}$ and $\ell_{1}$ ($J_{0}$ and $J_{1}$) denote, respectively,
the leading and subleading leptons (fat-jets). Since the mass difference
between dissimilar exotics is expected to be small (see Table 2), we may
safely assume the pairing with the smaller mass difference to be the correct
one.

In what follows, we briefly describe the reconstruction and selection of
various physics objects, event selection, and the classification of selected
events into mutually exclusive signal regions (SRs).

###  4.1 Object reconstruction and selection

We use Delphes deFavereau:2013fsa  to reconstruct different physics objects,
namely photons, electrons, muons and jets. For reasons to be elaborated later
(in Sec. 4.2), for each event, we attempt jet reconstruction with two
different values for the distance parameter ($R$), namely $0.8$ (applicable
for fat-jets) and $0.4$ (for ordinary jets). The jet constituents are
clustered using the anti-kT algorithm Cacciari:2008gp  as implemented in
FastJet Cacciari:2011ma , with a winner-take-all axis being used for the fat-
jets. At the reconstruction level, fat-jets and ordinary-jets (denoted by $J$
and $j$, respectively) are not dissociated objects; rather a pair of ordinary-
jets may coalesce into a single fat-jet on increasing $R$, and, conversely, a
single fat-jet may resolve into two ordinary jets if a small $R$ is
used.101010Obviously, both kinds of jets are not used for a given event. The
jet pruning algorithm Ellis:2009su ; Ellis:2009me  is used to remove the soft
yet wide-angle QCD emissions from the fat-jets,. Following Ref. Ellis:2009su ;
Ellis:2009me , we choose the default values for the parameters of the pruning
algorithm, namely $z_{cut}=0.1$ and $R_{cut}=0.5$.111111The parameters
$z_{cut}$ and $R_{cut}$ determine the level of pruning—the larger (smaller)
the $z_{cut}$ ($R_{cut}$), the more aggressive is the pruning. Pruning with a
smaller $R_{cut}$ (larger $z_{cut}$) degrades the putative jet mass resolution
by aggressively pruning the QCD shower of the daughter partons. For a larger
$R_{cut}$ (smaller $z_{cut}$), the algorithm merges uncorrelated soft
radiations from the underlying event and pile-up with the putative jet,
thereby augmenting the jet mass. Ref. Ellis:2009su ; Ellis:2009me
demonstrated that the choice $z_{cut}=0.1$ and $R_{cut}=0.5$ yields close to
optimal results of pruning, and the result is largely insensitive to small
changes in the values of the cuts. Further, $N$-subjettiness121212An inclusive
jet shape variable, N-subjettiness is a good measure of the number of putative
subjets a jet resolves into. It is defined as
$\tau_{N}=\frac{1}{d_{0}}\sum_{k}p_{T,k}{\rm min}\left\\{\Delta R_{1,k},\Delta
R_{2,k},...,\Delta R_{N,k}\right\\}$, where $N$ is the number of subjets a jet
is presumably composed of, $k$ runs over the constituent particles in a given
jet $j$ with $p_{T,k}$ being their transverse momenta, and $\Delta R_{j,k}$
the distance in the rapidity-azimuth plane between a candidate subjet $j$ and
a constituent particle $k$. Here, $d_{0}=\sum_{k}p_{T,k}R_{0}$ is the
normalisation factor with $R_{0}$ being the characteristic jet radius used in
the original jet clustering algorithm. The ratio $\tau_{N}/\tau_{N-1}$ is an
useful discriminant between events with $N$\- and $(N-1)$-prong jets
Thaler:2010tr ; Thaler:2011gf . Thaler:2010tr ; Thaler:2011gf  is used to
divulge the multi-prong nature of the fat-jets. We choose one-pass $k_{T}$
axes for the minimal axes with a thrust parameter $\beta=1$. Reconstructed
jets are required to be central (pseudorapidity range $|\eta|<2.5$) and have a
transverse momentum $p_{T}>30$ GeV. Lepton candidates (electrons and muons)
are considered for further analysis only if these satisfy both $p_{T}>10$ GeV
and $|\eta|<2.5$. Lepton isolation is ensured by demanding that the scalar sum
of the $p_{T}$s of photons and hadrons within a cone of $\Delta R=0.5$ around
the lepton be smaller than 12%(15%) of the electron (muon) $p_{T}$. By
truncating the hadronic activity inside the isolation cone, this requirement
subdues the reducible backgrounds such as $Z+$jets and $tt+$jets, where
additional leptons emanate from heavy quark decays or jet misidentification.
Finally, the missing transverse momentum vector $\vec{p}_{T}^{\rm~{}miss}$
(with magnitude $p_{T}^{\rm miss}$) is estimated using all reconstructed
particle-flow objects in an event.

To circumvent possible ambiguities in assigning particles to jets, we require
the latter to be separated by $\Delta R(j,j)>0.4$. Since the jet
reconstruction algorithm may also reconstruct leptons as jets, any jet within
a cone of $\Delta R<0.4$ of a selected lepton is discarded to avoid double
counting. Similarly, all selected electrons within a cone of $\Delta R<0.05$
of a selected muon are thrown away as these could arise due to the muon
bremsstrahlung interactions with the inner detector material. To mitigate the
effect of jet momenta mismeasurement contributing to missing transverse
momenta, the leading fat-jet $J_{0}$ (or the leading and sub-leading jets
$j_{0}$ and $j_{1}$) should satisfy $|\Delta\phi(J_{0},p_{T}^{\rm miss})|>0.2$
($|\Delta\phi(j_{0,1},p_{T}^{\rm miss})|>0.2$). It needs to be realised that
some of the jets, escorted by loosely isolated leptons or hadrons that reach
the muon detectors traversing the hadronic calorimeter or hadronic showers
that deposit a significant fraction of energy in the electromagnetic
calorimeter, could be misidentified as leptons. Though the composition of the
misidentified lepton contribution varies appreciably among the analysis
channels, without going into the intricacy of modelling the misidentified
lepton contributions, we adopt a conservative approach, assuming the
probability for a jet to be misidentified as a lepton to be in the 0.1 – 0.3%
range ATLAS:2016iqc .131313As delineated in Ref. ATLAS:2016iqc , which we
follow, this probability varies linearly from 0.3% to 0.1% for $p_{T}$ ranging
from 20 to 50 GeV, saturating at 0.1% for $p_{T}>50$ GeV.

Furthermore, bremsstrahlung interactions of an electron with the inner
detector material could lead to deflections and kinks in its tracks which, in
turn, lead to incorrect measurement of the track curvature and thus to a
charge misidentification. Similarly, bremsstrahlung photons with a high enough
energy may subsequently convert to $e^{-}e^{+}$ pairs within the detector
material and thus give rise to additional tracks near the primary electron
trajectory leading to the ambiguous association of an electromagnetic
calorimeter cluster to a track. We adopt the charge misidentification
probability from Ref. ATLAS:2017xqs : $P(p_{T},\eta)=\sigma(p_{T})\times
f(\eta)$, where $\sigma(p_{T})$ and $f(\eta)$ range, respectively, in the
intervals $[0.02,0.1]$ and $[0.03,1]$ such that $P(p_{T},\eta)$ ranges from
0.02% to 10%.141414$\sigma(p_{T})$ is found to vary linearly from 0.02 to 0.1
for $p_{T}$ ranging from 40 to 300 GeV saturating at 0.02 (0.1) for $p_{T}<40$
($p_{T}>300$) GeV. Similarly, $f(\eta)$ varies from 0.03 to 1.0 for
$0.5<\eta<2.5$, while $f(\eta)\approx 0.03$ for $\eta<0.5$.Note that higher-
energy electrons are more likely to suffer charge-misidentification owing to
the smaller curvature of their tracks. Similarly, electrons coming out with a
larger $|\eta|$ have a larger charge misidentification probability as they
traverse through a higher amount of inner detector material and, thus, suffer
more bremsstrahlung interactions.

###  4.2  Event preselection and signal regions selection

After object reconstruction and selection, only events with one of same-sign
dileptons (SSD) or opposite-sign dileptons (OSD) or trileptons (3L) are
considered for further analysis, which proceeds in two steps—the preselection
and the signal region (SR) selection. Henceforth, we restrict our discussion
to only those objects (leptons, jets, etc.) that pass the reconstruction and
selection criteria described in Sec. 4.1. The preselection requirements are as
follows. Events are selected only if the absolute value of the sum of charges
of the energetic and isolated leptons is two, zero and one for the SSD, OSD
and 3L events, respectively. To subdue background contributions from
low-$\Delta R$ final-state radiations as well as low-mass
resonances—$\gamma^{(*)}$ and quarkonia (such as $J/\psi,\Upsilon$)—events
containing a lepton pair with $\Delta R(\ell,\ell)<0.4$ or an opposite sign
same-flavour lepton pair with invariant mass below 10 GeV are discarded. In
addition, events containing an opposite sign same-flavour lepton pair with an
invariant mass within the nominal Z-boson mass window, i.e. $M_{Z}\pm 10$ GeV
are vetoed. This not only suppresses background contributions from processes
like $Z\to\ell\ell$ but also from processes like
$Z\to\ell\ell+\gamma^{(*)}\to\ell\ell\ell^{\prime}\ell^{\prime}$ where (one
or) two of the leptons fall(s) outside the detector coverage or (is) are too
soft to pass the object reconstruction and selection criteria or get(s)
misidentified by the detector. Furthermore, the $Z$-boson invariant mass veto
is also applied to the same-sign dielectron events to subjugate the background
contribution of electron charge misidentification.151515This is of little
concern for muons, owing to the much superior charge identification.

Events satisfying the above-mentioned preselection criteria (denoted,
henceforth, by S0) are categorised into the following primary SRs —

  1. (i)

3L-1J: 3L events with at least one fat-jet,

  2. (ii)

SSD-1J: SSD events with at least one fat-jet,

  3. (iii)

OSD-1J: OSD events with at least one fat-jet and high-$p_{T}^{\rm miss}(>100$
GeV),

  4. (iv)

OSD-2J: OSD events with at least two fat-jets.161616At this stage, the OSD-1J
and OSD-2J SRs are not mutually exclusive. However, they become mutually
exclusive after all the selection cuts (see Table 6) are imposed.

As we will see in Sec. 4.4, not all the events in the above-mentioned SRs
would pass the SR specific selection criteria (Table 6), in particular, the
twin criteria for a fat-jet to be identified as a ${W/Z/h}$-fat-jet (denoted
as $J_{W/Z/h}$), and defined by the requirements dubbed as Selection I-2 in
Sec. 4.4.171717Typically, only 35%–50% of the fat-jets would satisfy the
$J_{W/Z/h}$ selection criteria. Firstly, often some of the constituents of a
putative fat-jet may go missing in the jet clustering algorithm. Second, other
jets or hadrons not stemming from $W/Z/h$ could superfluously be clustered
with a putative fat-jet. Further, the jet momenta mismeasurement makes the
state of affairs worse. Therefore, to supplement the sensitivity of this
search, in addition to the above-mentioned SRs, we consider the following two
secondary SRs corresponding to 3L-1J and SSD-1J SRs:

  1. (v)

3L-2j: 3L events with at least two ordinary-jets, and

  2. (vi)

SSD-2j: SSD events with at least two ordinary-jets.

The only difference between the last two SRs ((v) and (vi)) and their fat-jet
counterparts ((i) and (ii)) is that two or more ordinary-jets are demanded
instead of at least one fat-jet. In order to make these two SRs orthogonal to
their fat-jet counterparts, only the events which fail to pass all the
selection cuts for the 3L-1J and SSD-1J SRs are considered in the 3L-2j and
SSD-2j SRs, respectively. The sole purpose of defining these two secondary SRs
is, by and large, to supplement the sensitivity of this search. Note that we
do not define any ordinary-jet counterpart for the OSD-1J (OSD-2J) SRs, as for
such ordinary-jet SRs, the relevant backgrounds (signals) are comparatively
large (small), and, thus, they are of little consequence.

###  4.3 SM Backgrounds

Having defined all the SRs of our interest, we now briefly mention the
relevant backgrounds. These include diboson ($VV$ with $V=W,Z/\gamma^{*}$),
triboson ($VVV$) and tetraboson ($VVVV$) production, Higgsstrahlung processes
($Zh$, $Wh$, $t\bar{t}h$, $ZZh$, $WWh$), multi-top production ($t\bar{t}$,
$t\bar{t}t$, $t\bar{t}t\bar{t}$), top-pair in association with gauge bosons
($t\bar{t}V$, $t\bar{t}VV$), single top production ($tb,tW,tj$), and Drell-Yan
processes ($W,Z/\gamma^{*}$). In order to meticulously account for the jet
multiplicity in the final states, all the background samples are generated in
association with up to two jets using MadGraph Alwall:2011uj ; Alwall:2014hca
at the LO using the five flavour scheme followed by MLM matching in PYTHIA
Sjostrand:2014zea . The background event samples are normalised to NLO or
higher Campbell:1999ah ; Ciccolini:2003jy ; Brein:2003wg ; Catani:2007vq ;
Campanario:2008yg ; Balossini:2009sa ; Bredenstein:2009aj ; Catani:2009sm ;
Kidonakis:2010ux ; Campbell:2011bn ; Brein:2011vx ; Bevilacqua:2012em ;
Garzelli:2012bn ; Brein:2012ne ; Altenkamp:2012sx ; Nhung:2013jta ;
Kidonakis:2013zqa ; Denner:2014cla ; Harlander:2014wda ; Kidonakis:2015nna ;
Muselli:2015kba ; Shen:2015cwj ; Frederix:2017wme .

The relevant backgrounds can be classified into two classes—reducible and
irreducible. The leptons stemming from the SM bosons’ decays, commonly
referred to as prompt leptons, are usually indiscernible in momentum and
isolation from those produced in signal events. Hence, all SM processes giving
rise to two or more isolated leptons, such as $VV$, $VVV$, $VVVV$, $t\bar{t}$,
$t\bar{t}V$ and Higgsstrahlung processes, constitute the irreducible
backgrounds in this analysis. On the contrary, the reducible
backgrounds—mainly from the SM processes like $Z/\gamma^{*}+$jets and
$t\bar{t}+$jets, where a jet is misidentified as lepton or additional leptons
originates from heavy quark decays—are considerably subdued by applying
stringent lepton isolation requirements.

###  4.4  SR-specific event selection

In this section, we discuss the SR-specific selection criteria that would be
effective in subjugating the relevant SM backgrounds. We use the distributions
of various kinematic variables as guiding a premise to choose the selection
cuts to ameliorate the signal-to-background ratio.

####  4.4.1 The 3L-1J signal

In Fig. 3, we display different normalised kinematic distributions for a
specific benchmark point BP1: $M_{\chi}=1$ TeV and BR$(\chi^{\pm}\to\nu
W^{\pm})\approx 50\%$ in Model-$3^{F}_{1}$. The left plot in the top panel
shows the missing transverse momentum $p_{T}^{\rm miss}$ distribution, whereas
the middle and right plots show, respectively, the $p_{T}$ distributions of
the opposite-charge and leading same-charge leptons, $p_{T}(\ell^{\rm oc})$
and $p_{T}(\ell^{\rm sc}_{0})$. In the bottom panel, we plot the $L_{T}$ and
$M_{\rm eff}=L_{T}+H_{T}+p_{T}^{\rm miss}$ distributions, with $L_{T}(H_{T})$
being the scalar sum of transverse momenta of all the leptons (jets)
satisfying the reconstruction and selection requirements in an event. We see,
in Fig. 3, that the kinematic distributions for the signal are much harder
than those for the background. Therefore, reasonably strong cuts on
$p_{T}^{\rm miss}$, $p_{T}(\ell^{\rm oc})$, $p_{T}(\ell^{\rm sc}_{0})$,
$L_{T}$ and $M_{\rm eff}$ would appreciably curtail the background while
keeping the signal relatively unharmed. Led the way by the kinematic
distributions in Fig. 3, we impose the following selection cuts (in GeV):

$\textbf{Selection~{}I-1~{}({SI-1}):}~{}~{}p_{T}^{\rm
miss}>100,\,\,p_{T}(\ell^{oc})>100,\,\,p_{T}(\ell_{0}^{sc})>100,\,\,L_{T}>500\,\,{\rm
and}\,\,m_{\rm eff}>800.$

Figure 3: Normalised kinematic distributions for 3L-1J events for BP1:
$M_{\chi}=1$ TeV and BR$(\chi^{\pm}\to\nu W^{\pm})\approx 50\%$ in
Model-$3^{F}_{1}$. Top panel: $p_{T}^{\rm miss}$ (left), $p_{T}$ of the
opposite-charge lepton (middle) and $p_{T}$ of the leading same-charge lepton
(right); bottom panel: $L_{T}$ (left) and $M_{\rm eff}$ (right).

In Fig. 4 are shown a few normalised kinematic distributions pertaining to the
leading fat-jet $J_{0}$ in such events. The left and middle panels show,
respectively, the distributions of $p_{T}$ and mass of $J_{0}$, namely
$p_{T}(J_{0})$ and $M_{J_{0}}$, for events satisfying SI-1 cuts. The
$p_{T}(J_{0})$ distribution for the background events peak at $\sim 100\,$GeV,
whereas the signal events peak at a substantially larger value. However, a
strong cut on $p_{T}(J_{0})$ clearly impinges on the signal strength as well.
As for the $M_{J_{0}}$ distribution for the background events, it is almost a
monotonically falling one, barring the sharp fall at very low $M_{J_{0}}$ that
is occasioned by the requirements on the jet energies. On the other hand, the
signal, boasts a peak around 100 GeV emanating from gauge boson decay. Taking
advantage of this, we select only those events in which at least one of the
fat-jets not only has a sufficient $p_{T}$ but would also be identified as
$W/Z/h$-fat-jet ($J_{W/Z/h}$), namely

$\textbf{Selection~{}I-2~{}({SI-2}):}~{}~{}p_{T}(J)>200{\rm~{}GeV}\,\,{\rm
and}\,\,M_{J}\in[70,140]{\rm~{}GeV}.$

Also displayed in the right panel is the distribution of the ratio of two
subjettiness parameters, viz. $\tau_{21}\equiv\tau_{2}/\tau_{1}$ Thaler:2010tr
of $J_{0}$ for events satisfying SI-2. A robust discriminant, $\tau_{21}$ can
distinguish between genuine two-prong events (such as those from a boosted
$W/Z/h$), which are expected to have a much lower value than pure QCD events
(one-prong) or those arising from (non-boosted) top-decays. Taking inspiration
from the figure, we impose a modest (and conventional) $\tau_{21}$ cut:

$\textbf{Selection~{}I-3~{}({SI-3}):}~{}~{}\tau_{21}(J)<0.5.$

Figure 4: Normalised kinematic distributions of $J_{0}$ for 3L-1J events for
BP1 in Model-$3^{F}_{1}$: $p_{T}(J_{0})$ (left) and $M_{J_{0}}$ (middle)
distributions for events satisfying SI-1; $\tau_{21}$ (right) distribution for
events satisfying SI-2.

####  4.4.2 The 3L-2j signal

These are events very similar to the previous (3L-1J) class, but for the fact
that the jets from the bosons are not collimated enough for them to merge to a
single fat-jet. Consequently, as far as the variables of Fig. 3 are concerned,
there is relatively little difference between the corresponding distributions.
Consequently, we use the same selection cut as in the 3L-1J case:

$\textbf{Selection~{}V-1~{}({SV-1}):}~{}~{}p_{T}^{\rm
miss}>100,\,\,p_{T}(\ell^{oc})>100,\,\,p_{T}(\ell_{0}^{sc})>100,\,\,L_{T}>500\,\,{\rm
and}\,\,m_{\rm eff}>800.$

Figure 5: Normalised kinematic distributions for 3L-2j events for BP1 in
Model-$3^{F}_{1}$: $p_{T}(jj)$ (left) and $M_{jj}$ (right) distributions for
events satisfying SI-1.

In Fig. 5, we plot the distributions for the transverse momentum (left) and
invariant mass (right) of the closest two ordinary-jets system,181818As
mentioned earlier, the leptons and the SM bosons coming from the decays of TeV
scale exotic leptons would be inordinately boosted. Thus, the jets emanating
from the decaying SM bosons would be in close proximity with each other.
$p_{T}(jj)$ and $M_{jj}$, for the 3L-2j events satisfying SI-1 cuts for BP1 in
Model-$3^{F}_{1}$. Because of similar structures for the background and signal
distributions, it would not be possible to subjugate the background by
applying selection cuts on $p_{T}(jj)$ and $M_{jj}$ without harming much the
signal strength. However, to enhance the signal-to-background ratio by
selecting only those events in which at least one pair of the ordinary-jets
emanating from gauge boson decay has a sufficient $p_{T}$ such that the signal
would demonstrate a characteristic peak in the invariant mass distribution of
a particular lepton–distinguishable from the others–and the ordinary-jets pair
(see the second left plot in the top panel of Fig. 7), we apply the following
selection cuts:191919A conundrum needs to be addressed here. The selection cut
SV-2 improves the signal-to-noise ratio marginally but at the cost of signal
strength. Therefore, whether to apply this cut or not is seemingly a dilemma.
Note that, without this cut, the signal would not have a characteristic peak
in the invariant mass distribution of a particular lepton and the ordinary-
jets pair, thereby making the reconstruction of the exotics very challenging.

$\textbf{Selection~{}V-2~{}({SV-2}):}~{}~{}p_{T}(jj)>200{\rm~{}GeV}\,\,{\rm
and}\,\,M_{jj}\in[70,140]{\rm~{}GeV}.$

Event sample |  3L-1J SR |  3L-2j SR
---|---|---
S0 |  SI-1 |  SI-2 |  SI-3 | S0 |  SV-1 |  SV-2
$t\bar{t}$ | $\sim 1.1\times 10^{5}$ |  177 |  41 |  11 | $\sim 6.8\times 10^{4}$ |  110 |  13
$WZ$ | $\sim 2.1\times 10^{4}$ |  117 |  14 |  7 | 8451 |  49 |  6
$t\bar{t}W$ | 2676 |  36 |  7 |  6 | 2264 |  20 |  4
$t\bar{t}Z$ | 1897 |  18 |  5 |  3 | 1795 |  11 |  2
Other backgrounds | $\sim 5.3\times 10^{4}$ |  88 |  15 |  8 | $\sim 2.1\times 10^{4}$ |  37 |  6
Total background | $\sim 1.9\times 10^{5}$ |  436 |  82 |  35 | $\sim 10^{5}$ |  227 |  31
Signal | 146 |  108 |  50 |  40 | 64 |  29 |  8

Table 5:  The numbers of expected background and signal events in 3L-1J and
3L-2j SRs after passing various selection cuts for BP1 in Model-$3^{F}_{1}$
for 1000 fb-1 of luminosity data at the 13 TeV LHC.

Table 5 displays the progression of the number of expected background and
signal events (for BP1 in Model-$3^{F}_{1}$) in the 3L-1J and 3L-2j SRs as
subsequent selection cuts are imposed for 1000 fb-1 of luminosity data at the
13 TeV LHC. For the 3L-1J SR, the SI-1 cut, as expected from Fig. 3, turns out
to be efficacious in vanquishing the background without impinging much on the
signal strength while the subsequent cuts SI-2 and SI-3, in particular SI-2,
enhance(s) the signal-to-background ratio appreciably. Similarly, for the
3L-2j SR, the SV-1 cut subdues the background while keeping almost half of the
signal events, and the SV-2 cut further improves the signal-to-background
ratio.

####  4.4.3 The other signal profiles

As for the remaining signals discussed in Sec. 4.1, the distributions (and,
hence, the discussions) are very similar. We do not discuss these in detail
for brevity’s sake but to summarise all the SR-specific selection cuts in
Table 6. The cut flows of expected background and signal events in SSD-1J and
SSD-2j SRs are shown in Table 7, while those for OSD-1J and OSD-2J SRs are
shown in Table 8.

3L-1J SR
---
SI-1 |  $p_{T}^{\rm miss}>100{\rm~{}GeV},\,\,p_{T}(\ell^{\rm oc})>100{\rm~{}GeV},\,\,p_{T}(\ell_{0}^{\rm sc})>100{\rm~{}GeV},\,\,L_{T}>500{\rm~{}GeV}\,\,{\rm and}\,\,m_{\rm eff}>800{\rm~{}GeV}$
SI-2 and SI-3 |  at least one fat-jet with $p_{T}(J)>200{\rm~{}GeV},M_{J}\in[70,140]{\rm~{}GeV}$ and $\tau_{21}(J)<0.5$
SSD-1J SR
SII-1 |  $p_{T}^{\rm miss}>100{\rm~{}GeV},\,\,p_{T}(\ell_{0})>250{\rm~{}GeV},\,\,L_{T}>400{\rm~{}GeV}\,\,{\rm and}\,\,m_{\rm eff}>700{\rm~{}GeV}$
SII-2 and SII-3 |  at least one fat-jet with $p_{T}(J)>200{\rm~{}GeV},M_{J}\in[70,140]{\rm~{}GeV}$ and $\tau_{21}(J)<0.5$
OSD-1J SR ($p_{T}^{\rm miss}>100{\rm~{}GeV}$)
SIII-1 |  $M_{\ell\ell}>100{\rm~{}GeV},\,\,p_{T}(\ell_{0})>300{\rm~{}GeV},\,\,p_{T}(\ell_{1})>100{\rm~{}GeV},\,\,L_{T}>500{\rm~{}GeV},\,\,H_{T}>250{\rm~{}GeV}\,\,{\rm and}\,\,m_{\rm eff}>800{\rm~{}GeV}$
SIII-2 and SIII-3 |  exactly one fat-jet with $p_{T}(J)>200{\rm~{}GeV},M_{J}\in[70,140]{\rm~{}GeV}$ and $\tau_{21}(J)<0.5$
OSD-2J SR
SIV-1 |  $M_{\ell\ell}>100{\rm~{}GeV},\,\,p_{T}(\ell_{0})>300{\rm~{}GeV},\,\,p_{T}(\ell_{1})>100{\rm~{}GeV},\,\,L_{T}>500{\rm~{}GeV},\,\,H_{T}>250{\rm~{}GeV}\,\,{\rm and}\,\,m_{\rm eff}>800{\rm~{}GeV}$
SIV-2 and SIV-3 |  exactly two fat-jets with $p_{T}(J)>200{\rm~{}GeV},M_{J}\in[70,140]{\rm~{}GeV}$ and $\tau_{21}(J)<0.5$
3L-2j SR
SV-1 |  $p_{T}^{\rm miss}>100{\rm~{}GeV},\,\,p_{T}(\ell^{\rm oc})>100{\rm~{}GeV},\,\,p_{T}(\ell_{0}^{\rm sc})>100{\rm~{}GeV},\,\,L_{T}>500{\rm~{}GeV}\,\,{\rm and}\,\,m_{\rm eff}>800{\rm~{}GeV}$
SV-2 |  at least one ordinary dijet with $p_{T}(jj)>200{\rm~{}GeV}\,\,{\rm and}\,\,M_{jj}\in[70,140]{\rm~{}GeV}$
SSD-2j SR
SVI-1 |  $p_{T}^{\rm miss}>100{\rm~{}GeV},\,\,p_{T}(\ell_{0})>250{\rm~{}GeV},\,\,L_{T}>500{\rm~{}GeV}\,\,{\rm and}\,\,m_{\rm eff}>700{\rm~{}GeV}$
SVI-2 |  at least one ordinary dijet with $p_{T}(jj)>200{\rm~{}GeV}\,\,{\rm and}\,\,M_{jj}\in[70,140]{\rm~{}GeV}$

Table 6:  Summary of the SR-specific selection cuts. The OSD-1J and OSD-2J SRs
become mutually exclusive after all the selection cuts are imposed.

Event sample |  SSD-1J SR |  SSD-2j SR
---|---|---
S0 |  SII-1 |  SII-2 |  SII-3 | S0 |  SVI-1 |  SVI-2
$t\bar{t}$ | $\sim 1.2\times 10^{5}$ |  271 |  104 |  63 | $\sim 8.9\times 10^{4}$ |  97 |  13
$t\bar{t}W$ | $\sim 1.5\times 10^{4}$ |  138 |  48 |  30 | $\sim 1.5\times 10^{4}$ |  64 |  13
$WWW$ | $\sim 1.2\times 10^{4}$ |  93 |  40 |  28 | $\sim 10^{4}$ |  31 |  4
$WZ$ | $\sim 5.6\times 10^{4}$ |  160 |  52 |  27 | $\sim 2.7\times 10^{4}$ |  69 |  14
Other backgrounds | $\sim 8.4\times 10^{4}$ |  145 |  52 |  31 | $\sim 5.2\times 10^{4}$ |  62 |  9
Total background | $\sim 2.9\times 10^{5}$ |  807 |  296 |  179 | $\sim 1.9\times 10^{5}$ |  323 |  53
Signal | 141 |  90 |  51 |  41 | 79 |  38 |  9

Table 7:  The numbers of expected background and signal events in SSD-1J and
SSD-2j SRs after passing various selection cuts for BP1 in Model-$3^{F}_{1}$
for 1000 fb-1 of luminosity data at the 13 TeV LHC.

Event sample |  OSD-1J SR |  OSD-2J SR
---|---|---
S0 |  SIII-1 |  SIII-2 |  SIII-3 | S0 | SIV-1 |  SIV-2 |  SIV-3
$t\bar{t}$ | $\sim 3.1\times 10^{6}$ |  2653 |  719 |  265 | $\sim 10^{7}$ | 4317 |  42 |  3
$WW$ | $\sim 1.1\times 10^{6}$ |  738 |  240 |  89 | $\sim 4.2\times 10^{6}$ | 1070 |  5 |  1
$\gamma^{*}/Z$ | $\sim 1.6\times 10^{7}$ |  503 |  168 |  64 | $\sim 5.1\times 10^{8}$ | 6427 |  76 |  17
$WZ$ | 7781 |  89 |  34 |  14 | $\sim 5.4\times 10^{4}$ | 393 |  10 |  1
Other backgrounds | $\sim 3.2\times 10^{4}$ |  199 |  96 |  50 | $\sim 10^{5}$ | 400 |  15 |  6
Total background | $\sim 2.0\times 10^{7}$ |  4182 |  1257 |  482 | $\sim 5.3\times 10^{8}$ | $\sim 1.3\times 10^{4}$ |  148 |  28
Signal | 339 |  194 |  149 |  92 | 382 | 269 |  63 |  39

Table 8:  The numbers of expected background and signal events in OSD-1J and
OSD-2J SRs after passing various selection cuts for BP1 in Model-$3^{F}_{1}$
for 1000 fb-1 of luminosity data at the 13 TeV LHC.

To enhance the sensitivity of this search, the signal regions are further
classified into several independent search bins using a primary kinematic
discriminant between the signal and background:

  * •

The 3L-1J (3L-2j) SR uses the invariant mass of the opposite-charge lepton and
fat-jet (ordinary dijet) system, $M_{J\ell^{oc}}$ ($M_{jj\ell^{oc}}$), as the
primary kinematic discriminant. For the signal events, this invariant mass
distribution would peak at the mass of the exotic lepton.

  * •

For the SSD-1J and SSD-2j SRs, the exotic lepton mass can not be reconstructed
from any invarint mass distribution and, thus, we use $L_{T}+p_{T}^{\rm miss}$
as primary kinematic discriminant.

  * •

For the OSD-1J SR, we use $M_{J\ell_{0}}$ as primary kinematic discriminant.

  * •

For the OSD-2J SR, the kinematic reconstruction of both the exotics is
possible. Since the mass difference between dissimilar exotics is small (see
Table 2), we may safely assume the pairing with the smaller mass difference to
be the correct one. For the signal events, this mass difference is expected to
be dominated by the uncertainties inherent in jet reconstruction, while for
the background events, large additional contributions accrue from the
underlying hard scattering process itself. This difference is exemplified by
the distributions depicted in Fig. 6, and their very shapes induce us to
impose the final selection cut, namely restrict this difference to be less
than 100 GeV. As the final kinematic discriminant, we use the average of the
two lepton-fat-jet invariant masses computed from the leading particles that
correspond to the smaller difference. In other words, if
$|M_{J_{0}\ell_{0}}-M_{J_{1}\ell_{1}}|<|M_{J_{1}\ell_{0}}-M_{J_{0}\ell_{1}}|$,
then the discriminant is $M_{J\ell}^{\rm
avg}=(M_{J_{0}\ell_{0}}+M_{J_{1}\ell_{1}})/2$, else it is $M_{J\ell}^{\rm
avg}=(M_{J_{0}\ell_{1}}+M_{J_{1}\ell_{0}})/2$.

Figure 6: Distributions of the difference between the two lepton-fat-jet
invariant masses for the OSD-2J SR for BP1: $M_{\chi}=1$ TeV and
BR$(\chi^{\pm}\to\nu W^{\pm})\approx 50\%$ in Model-$3^{F}_{1}$. The events
are weighted for 1000 fb-1 at the 13 TeV LHC.

Clearly, except in the second (SSD-1J and SSD-2j) case, these invariant mass
distributions could be used to reconstruct the exotic leptons kinematically.
Each SR is divided into ten independent search bins in the [0:2000] GeV
range202020The range [0:2000] GeV is chosen to maximise the sensitivity of the
present search for the exotics of 1-2 TeV masses. However, for the exotics
heavier than 2 TeV, a broader range (say [1000:3000] GeV) would be beneficent
in reconstructing the exotics. for the corresponding discriminant. The width
of the bins is chosen to ensure smooth and monotonic behaviour of the expected
background while retaining sensitivity to the simplified models. The overflow
events are contained in the last bin for each signal region. This binning
scheme yields a total of 60 statistically independent search bins (see Table
9).

Label | Kinematic discriminant | Range (GeV) | Number of bins
---|---|---|---
3L-1J | $M_{J\ell^{oc}}$ | $[0,2000]$ | 10
3L-2j | $M_{jj\ell^{oc}}$ | $[0,2000]$ | 10
SSD-1J |  $L_{T}$+$p_{T}^{\rm miss}$ | $[0,2000]$ | 10
SSD-2j |  $L_{T}$+$p_{T}^{\rm miss}$ | $[0,2000]$ | 10
OSD-1J | $M_{J\ell_{0}}$ | $[0,2000]$ | 10
OSD-2J | $M_{J\ell}^{\rm avg}$ | $[0,2000]$ | 10
Table 9: Binning scheme of different signal regions.

In Fig. 7, we show the distributions of the primary kinematic discriminants
for both signal and background events passing all the selection cuts for all
the SRs for BP1: $M_{\chi}=1$ TeV and BR$(\chi^{\pm}\to\nu W^{\pm})\approx
50\%$ in Model-$3^{F}_{1}$. The events are weighted for 1000 fb-1 luminosity
at the 13 TeV LHC. It is evident from Fig. 7 that the exotics can be
kinematically reconstructed from the distributions of the primary
discriminants in the 3L-1J, 3L-2j, OSD-1J and OSD-2J SRs. As expected, these
invariant mass distributions peak near 1000 GeV and thereby reconstruct the
exotic mass in the Model-$3^{F}_{1}$.

Figure 7: Distributions of the primary kinematic discriminants for all the SRs
for BP1: $M_{\chi}=1$ TeV and BR$(\chi^{\pm}\to\nu W^{\pm})\approx 50\%$ in
Model-$3^{F}_{1}$. The events are weighted for 1000 fb-1 at the 13 TeV LHC.

###  4.5 Discovery reach for the simplified models

To estimate the discovery reach of the search strategies presented above, we
use a hypothesis tester named Profile Likelihood Number Counting Combination,
which uses a library of C++ classes RooFit Verkerke:2003ir  in the ROOT
Brun:1997pa  environment. Treating all the bins in different signal regions as
independent channels for both signal and background events, the uncertainties
are included via the Profile Likelihood Ratio. Without going into the
intricacy of estimating the background uncertainties, we assume an overall 20%
total uncertainty on the estimated background.212121The SM backgrounds usually
suffer from substantial uncertainties—both experimental and theoretical. The
experimental uncertainties arise from several sources such as the
reconstruction, identification, isolation, trigger efficiency, energy scale
and resolution of different physics objects, the luminosity measurements, the
pile-up modelling, and the data-driven estimation of the misidentified lepton
and charge-misidentified electron backgrounds. The theoretical uncertainties
arise from the normalisation of some of the dominant irreducible backgrounds,
the parton-shower modelling, the higher-order QCD corrections
(renormalisation, factorisation and resummation scales), the PDF sets, and the
strong coupling constant. Further, the limited statistics in the data and
simulation samples entail significant uncertainty to the total background. For
the typical LHC searches ATLAS:2018ghc ; CMS:2019lwf ; ATLAS:2020wop ;
ATLAS:2021xxb ; CMS:2021zkl ; ATLAS:2021eyc , both the theoretical and
experimental uncertainties are $\mathcal{O}(10)\%$ yielding a total
uncertainty of less than 20%.Further, to ensure robustness in statistical
interpretations, we replace the less than one per-bin expected background
yield at 3000 fb-1 with one background yield.

Figure 8: Required Luminosity for the $5\sigma$ discovery for the
Model-$3^{F}_{1}$ with BR$(\chi^{\pm}\to\nu W^{\pm})\approx 50\%$ in different
SRs.

In Fig. 8, projections of the required luminosities—in individual channels—for
a $5\sigma$ discovery of the exotics in the Model-$3^{F}_{1}$ with
BR$(\chi^{\pm}\to\nu W^{\pm})\approx 50\%$ are shown as a function of the
mass. Since all the SRs considered in this search are mutually exclusive, it
is reasonable to combine them together, and this too is depicted in the same
figure. That the OSD-1J, SSD-2j and 3L-2j SRs are the least sensitive among
all, is understandable. For the OSD-1J SR, the SM background is rather large,
whereas for the SSD-2j and 3L-2j SRs, the signal strengths are small.222222For
the OSD-1J, SSD-2j and 3L-2j SRs, the curves in Fig. 8 have sharp turns. For
one, the signals are beset with comparatively large SM backgrounds for those
SRs, and second, sizeable uncertainty (which we assumed to be flat 20%) on the
latter that even increasing luminosity does not help much. With the increasing
amount of data at the LHC (and thus better insight into the detector
response), the experimental uncertainties are expected to be reduced
significantly. However, a precise estimation of the same is beyond the realm
of this work.Consequently, the $5\sigma$ discovery reach for the $3^{F}_{1}$
exotics extends up to 910 GeV in these SRs. On the contrary, the OSD-2J SR is
the most promising one with the discovery reach of 1310 GeV because of its
small background and ability to reconstruct the mass of the exotic leptons
without any ambiguity. Indeed, this contributes the maximum to the total
sensitivity reach of approximately 1385 GeV reached by combining all the
channels.

Proceeding to a comparison of the different scenarios, we present, in Fig. 9,
the $5\sigma$ discovery reach, as defined by combining all the channels. The
left and right plots correspond, respectively, to 300 and 3000 fb-1 luminosity
of data at 13 TeV LHC. A key input here is the BR($\chi^{\pm}\to\nu W^{\pm}$).
While this branching fraction is trivially calculable if the multiplet
represented the only BSM physics operative at these scales, it would depend on
other parameters in a more generic situation. Hence, rather than use a
particular value, we parametrize our ignorance by allowing it to vary over the
entire physical range.

For 300(3000) fb-1 luminosity, the discovery reach for Model-$3^{F}_{1}$
ranges from 985 GeV to 1140 GeV (1345 GeV to 1485 GeV) with the search
sensitivity increasing as the branching fraction of the singly-charged exotics
into the neutrino modes decreases. For Model-$5^{F}_{3}$, the discovery reach
ranges from 1625 GeV to 1650 GeV (1985 GeV to 2020 GeV) for 300(3000) fb-1
luminosity. This substantiates that a significant mass range of the exotics in
these simplified models can be probed for discovery at the 13 TeV LHC. Note
that, for some of the simplified models such as the Model-$4^{F}_{1/2}$, the
discovery reach depends moderately on BR($\chi^{\pm}\to\nu W^{\pm}$), while
the reach is almost independent for some other models like Model-$5^{F}_{3}$.
This is because, for the former, the associated production of the doubly- and
singly-charged exotics contributes significantly to the signal, while for the
latter, the signal is dominated by the effective pair production of the
doubly-charged exotics.

Figure 9: $5\sigma$ discovery reach of exotic leptons belonging to different
simplified models for different values of BR($\chi^{\pm}\to\nu W^{\pm}$) for
300 (left) and 3000 (right) fb-1 luminosity of data at 13 TeV LHC.

##  5 Summary and outlook

The presence of several charged exotic leptons within the new weak gauge
fermionic multiplets is a common yet salient feature of many new physics
models Bonnet:2009ej ; Liao:2010ku ; Bonnet:2012kz ; Kumericki:2012bf ;
Cepedello:2017lyo ; Anamiati:2018cuq ; Arbelaez:2019cmj ; Picek:2009is ;
Kumericki:2011hf ; Kumericki:2012bh ; Yu:2015pwa ; McDonald:2013hsa ;
Delgado:2011iz ; Ma:2013tda ; Ma:2014zda ; Ko:2015uma ; Avnish:2020rhx ;
Agarwalla:2018xpc ; KumarAgarwalla:2018nrn ; Kumar:2019tat ;
Ashanujjaman:2020tuv ; Babu:2009aq ; Li:2009mw ; Ashanujjaman:2021jhi ;
Kumar:2021umc  which are well-placed to address various shortcomings of the SM
(see Table 3 for an incomplete list of such models and their motivations).
Aplenty production of these exotic leptons and their decays to the SM
particles offers an up-and-coming way to probe them at the LHC. Owing to their
large masses, their eventual decay products—SM leptons and bosons—tend to be
highly boosted, with the jets stemming from the SM bosons more likely to
manifest themselves as a single fat-jet rather than two resolved ones. Not
only are signatures comprising two or three leptons and one or two fat-jets
cleaner than the usual LHC searches, but most of these final states also allow
for kinematic reconstructions of the exotic leptons. Consequently, such
signatures are expected to be sensitive in probing exotic leptons with large
masses. In this work, we propose and investigate an appropriate search
strategy. Adopting the bottom-up approach for new physics, we consider several
simplified models by extending the SM with only one type of new leptonic weak
gauge multiplet (triplet, quadruplet or quintuplet) at a time bearing in mind
that such simplified models typically arise as low-energy limits of more
ambitious scenarios addressing various lacunae of the SM when some of the
fields decouple and/or some of the interactions cease to exist. We have
performed a systematic and comprehensive collider study for nine such
scenarios and estimated the $5\sigma$ discovery reach for each of these for
both 300 and 3000 fb-1 luminosity of proton-proton collision data at the 13
TeV LHC. The $5\sigma$ discovery reach extends from 985 GeV to 1650 GeV (1345
GeV to 2020 GeV) for 300 (3000) fb-1 luminosity for different simplified
models. The analysis presented in this work appears to carry through for a
large class of simplified models with exotic charged leptons within a new
$SU(2)_{L}$ multiplet. Thus, collider searches, such as the one presented in
this work, are anticipated to play a notable role in constraining or testing
the new physics scenarios.

##  Appendix A  Renormalization Group flows in the different models

In all the simplified models considered in this work, the SM field content is
supplemented by new leptonic weak gauge multiplets. The introduction of such
multiplets changes the running of the gauge couplings of the $U(1)_{Y}$ and
$SU(2)_{L}$ groups. At the 1-loop order, the renormalization group equations
(RGEs) for the gauge couplings $g_{1},g_{2}$ and $g_{3}$ of the
$U(1)_{Y},SU(2)_{L}$ and $SU(3)_{C}$ groups can be written as:

$\frac{dg_{i}}{dt}=b_{i}\frac{g_{i}^{3}}{(4\pi)^{2}}$

where $t=\ln\mu$ with $\mu$ being the renormalization point. Within the SM,
the coefficients are
$(b_{1},b_{2},b_{3})=\left(\frac{41}{10},-\frac{19}{6},-7\right)$. Table 10
shows the corresponding coefficients in the respective models as well as the
energy scales at which the gauge couplings blow up. The existence of a Landau
pole below the Planck scale signals that the theory under consideration cannot
be an UV-complete one, and must be subsumed in a larger theory that
ameliorates the growth of the couplings. Note that Table 10 includes the
effect of only a single pair of the vector-like multiplet (a single chiral
field in case of $5^{F}_{0}$) and that the inclusion of multiple copies would
only bring the scale down. While the inclusion of higher-order terms in the
$\beta$-functions would change the location of the Landau pole to an extent,
the difference is only quantitative and not qualitative.

Model | $(b_{1},b_{2})$ |  $\mu_{1}$ (GeV) |  $\mu_{2}$ (GeV)
---|---|---|---
Model-$3^{F}_{1}$ | $\left(\frac{13}{2},\frac{-1}{2}\right)$ | $6.9\times 10^{26}$ | –
Model-$3^{F}_{2}$ | $\left(\frac{137}{10},\frac{-1}{2}\right)$ | $8.1\times 10^{13}$ | –
Model-$4^{F}_{1/2}$ | $\left(\frac{49}{10},\frac{7}{2}\right)$ | $7.5\times 10^{34}$ | $3.5\times 10^{25}$
Model-$4^{F}_{3/2}$ | $\left(\frac{113}{10},\frac{7}{2}\right)$ | $2.5\times 10^{16}$ | $3.5\times 10^{25}$
Model-$4^{F}_{5/2}$ | $\left(\frac{241}{10},\frac{7}{2}\right)$ | $7.5\times 10^{8}$ | $3.5\times 10^{25}$
Model-$5^{F}_{0}$ | $\left(\frac{41}{10},\frac{7}{2}\right)$ | $1.7\times 10^{41}$ | $3.5\times 10^{25}$
Model-$5^{F}_{1}$ | $\left(\frac{81}{10},\frac{61}{6}\right)$ | $9.6\times 10^{21}$ | $1.8\times 10^{10}$
Model-$5^{F}_{2}$ | $\left(\frac{201}{10},\frac{61}{6}\right)$ | $1.6\times 10^{10}$ | $1.8\times 10^{10}$
Model-$5^{F}_{3}$ | $\left(\frac{401}{10},\frac{61}{6}\right)$ | $1.7\times 10^{6}$ | $1.8\times 10^{10}$

Table 10:  The coefficients in the $\beta$-functions and the energy scales
$\mu_{1}$ and $\mu_{2}$ at which the Landau poles appear in, respectively,
$g_{1}$ and $g_{2}$ for the different models. For the onset of the RG running,
we take $g_{1}=0.3587$ and $g_{2}=0.6483$ at $\mu=m_{t}$, where $m_{t}=173.35$
GeV Buttazzo:2013uya .

###### Acknowledgements.

KG acknowledges support from the DST INSPIRE Research Grant
[DST/INSPIRE/04/2014/002158] and SERB Core Research Grant [CRG/2019/006831].
The simulations were supported in part by the SAMKHYA: High Performance
Computing Facility provided by Institute of Physics, Bhubaneswar.

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