Published for SISSA by

Springer

Received: December 19, 2023
Accepted: January 30, 2024
Published: February 21, 2024

A novel search strategy for right-handed charged gaugebosons at the Large Hadron Collider,b Adil Jueid,c Stefano Morettid,ea

Department of Physics, Concordia University,
7141 Sherbrooke St. West, Montreal, Quebec H4B 1R6, CanadabLaboratoire de Physique Théorique et Hautes Énergies (LPTHE), UMR 7589,
Sorbonne Université & CNRS, 4 place Jussieu, 75252 Paris Cedex 05, FrancecParticle Theory and Cosmology Group, Center for Theoretical Physics of the Universe,
Institute for Basic Science (IBS),
Daejeon, 34126, Republic of KoreadSchool of Physics & Astronomy, University of Southampton,
Highfield, Southampton SO17 1BJ, U.K.
e
Department of Physics & Astronomy, Uppsala University,
Box 516, 75120 Uppsala, Sweden

E-mail: mariana.frank@concordia.ca, fuks@lpthe.jussieu.fr,
adiljueid@ibs.re.kr, stefano.moretti@cern.ch, ozer.ozdal@soton.ac.uk
Abstract: We explore the potential of the Large Hadron Collider (LHC) in detecting
a signal originating from the production of a heavy SU(2)R charged gauge boson thatthen decays into a top-bottom quark pair via the mediation of a right-handed neutrino,
pp → WR → NR ℓ → (ℓ′ tb)ℓ. Such a channel, that we study in the context of the minimal
Left-Right Symmetric Model, contrasts with conventional smoking-gun signatures targeted
experimentally and phenomenologically in which only light quarks are involved. We propose aselection strategy aimed at extracting such a top-bottom signal and we estimate the resulting
sensitivity of the LHC to the model. Our results demonstrate the potential impact of such asearch and we therefore urge the experimental collaborations to carry out a similar analysis
in the light of present and future data.
Keywords: Left-Right Models, Jets and Jet Substructure, Higher-Order Perturbative
Calculations
ArXiv ePrint: 2312.08521

Open Access, © The Authors.
Article funded by SCOAP3 .

https://doi.org/10.1007/JHEP02(2024)150

JHEP02(2024)150

Mariana Frank ,a Benjamin Fuksand Özer Özdal a,dContents
1

2 The model and current bounds
2.1 The model
2.2 Limits on WR properties from collider and low-energy experiments
2.3 Production of WR bosons at the LHC
2.4 Searches for WR bosons in events with two leptons and two jets

3
3
4
5
7

3 Sensitivity of ℓℓtb probes to left-right models
3.1 Benchmark scenarios and backgrounds
3.2 Description of the analysis
3.3 Results

11
11
13
15

4 Conclusions

19

A Reinterpretation of CMS searches for WR bosons with leptons and jets
A.1 Object definitions and detector modeling
A.2 CMS-EXO-17-011
A.3 CMS-EXO-20-002

20
21
22
24

1

Introduction

The Standard Model (SM) of particle physics is unable to address fundamental questions in
Nature despite being in excellent agreement with all measurements carried out at experiments
so far. For instance, the origin of parity violation in electroweak (EW) interactions remains
a mystery to which the SM has no answer. Left-Right Symmetric Models (LRSMs) are
one of the most attractive solutions to such a problem [1–4]. Their minimal incarnationis dubbed the minimal LRSM (mLRSM). It is based on the SU(2)L ⊗ SU(2)R ⊗ U(1)B−L
gauge group, and it predicts a particle spectrum comprising several extra states, including
new charged and neutral gauge bosons, extra (pseudo)scalars (i.e. neutral, singly-charged
and doubly-charged bosons), and three generations of right-handed neutrinos. Furthermore,this minimal model accommodates naturally small neutrino masses through the seesaw
mechanism [5–7], it elegantly addresses the problem of parity violation, and it has a richphenomenology both at colliders and relative to low-energy experiments. The presenceof extra right-handed charged-currents and new (pseudo)scalar interactions indeed impact
lepton flavour violation in charged lepton decays and e → µ conversions [8], EW precisionobservables [9–11], Charge-Parity (CP) violations in meson decays and meson-antimeson
oscillations [12–19], electric dipole moments of various particle types [20–24], and nuclear βdecays [25, 26]. In particular, a recent comprehensive global analysis of the mLRSM by meansof low-energy observables has been carried in ref. [27], while the consequences on its leptonic
and (pseudo)scalar sectors have been analysed in refs. [28–34] and [35–38] respectively. In–1–

JHEP02(2024)150

1 Introduction

1

These bounds can be relaxed in cases where the WR boson couples in a generic way to leptons and quarks,
i.e. when gL ̸= gR (see e.g. refs. [53, 54]).

–2–

JHEP02(2024)150

contrast to other models featuring multi-Higgs doublets, the extra (pseudo)scalar states ofthe mLRSM originate from an SU(2)L ⊗ SU(2)R bi-doublet, and it has been shown that theyneed to be as heavy as O(10) TeV to avoid large flavour-changing neutral-current transitions.
The mLRSM also predicts lepton number violation that contribute to neutrinoless doublebeta (0νββ) decay rates. Such contributions are highly correlated with the neutrino mass
parameters and the SU(2)R gauge boson masses, while being amenable to discovery even forsmall neutrino masses (see refs. [39–42] for recent analyses). The mLRSM correspondingly
features high-energy analogues of the usual low-energy 0νββ processes. One of them consistsof the so-called Keung-Senjanovic mechanism that is expected to occur at high-energy hadroncolliders [43]. The idea is that on-shell and off-shell SU(2)R gauge bosons, denoted in what
follows as WR , can decay into a lepton and right-handed neutrino NR that itself further
decays into two light quarks and another (charged) lepton. If the two charged leptons havethe same electric charge, then this process breaks lepton number conservation by two units,providing therefore a direct correlation with the rates of 0νββ decays. Furthermore, theproduction of opposite-sign charged leptons is also interesting on its own [44–47], which hasconsequently motivated the ATLAS and CMS collaborations to carry out several searches fora WR state at the Large Hadron Collider (LHC), both in the same-sign and opposite-sign
dileptonic channels [48–52]. Depending on the NR mass and the lepton flavour structure, WR boson masses up to 4.8–5.0 TeV are excluded today [52].1 Furthermore, at large momentum
transfer the same 0νββ matrix elements imply several enhancements in the production rates
of same-sign lepton pairs of possibly different flavours via vector-boson scattering [55–58].
Constraints on heavy neutrinos with masses in the 50 GeV — 20 TeV range can here be
imposed, hence even above the LHC energy scale.
In the light of these efforts, an interesting question, that was also raised in ref. [59],arises: what if we use heavy quarks instead of light ones in searching for WR and NR states?
In this case the produced NR particle undergoes a three-body decay into a charged lepton,
a bottom quark, and a top quark, leading thus to a very rich final state. This consists ofan intriguing signature, as it is widely believed that the top quark can play an importantrole in probing new physics Beyond the SM (BSM).
The aim of this study is to propose a novel search strategy for the production of an SU(2)R
charged gauge boson WR and a neutrino NR at the LHC, relying on the WR → NR ℓ → tbℓℓ
decay chain with the WR emerging from charged current Drell-Yan (DY) production. The
top quark, being produced from a heavy neutrino decay, is highly boosted in most of thecases. Jet substructure methods and top taggers have therefore the potential to efficiently
reduce the SM background, together with specific kinematics variables exploiting the richnessof the final state. The use of this channel can thus be crucial not only for discovery purposes
but also for diagnostics as a probe of the properties of the mLRSM as a whole.
The rest of this paper is organised as follows. In section 2 we briefly discuss the mLRSM,
its field content and the associated Lagrangian, and we assess the constraints emerging
from recent searches for WR bosons and NR neutrinos at colliders (with some technicaldetails on the reinterpretation of the LHC results being reported in appendix A). Next westudy WR and NR production and decay in the ℓℓtb channel in section 3, where we alsopresent our signal and background analysis, and its results. The latter include an estimationof the sensitivity of the LHC to the mLRSM signal considered, and we demonstrate thepotential usefulness of designing corresponding searches in real LHC data. We conclude
and summarise our work in section 4.

2

The model and current bounds

2.1

The model

qL ≡

uL
dL

!

,

qR ≡

(2,1, 13 )

uR
dR

!

νL
eL

ℓL ≡

,
(1,2, 13 )

!

,

NR
eR

ℓR ≡

(2,1,−1)!

,(1,2,−1)(2.1)
in which the subscripts refer to the representation of the fermion fields under SU(2)L ⊗SU(2)R ⊗ U(1)B−L . In addition, we denote the three gauge couplings associated with thisgauge symmetry by gL , gR , and gB−L . Parity conservation at high scales dictates that thegauge interactions be invariant under(2.2){WL , qL , ℓL } ←→ {WR , qR , ℓR }.

An immediate consequence of this symmetry is that gL and gR are equal, i.e. gL ≡ gR ≡ g.
Thus EW symmetry breaking is minimally achieved through the vacuum expectation values
acquired by the neutral components of three scalar multiplets: a bi-doublet (Φ) and two
triplets (∆L,R ) represented as:


ϕ01

Φ≡
ϕ−
1

ϕ+
2



ϕ02



 +

δ
√L
2

∆L ≡
δL0

,
(2,2,0)

δL++



δ+



− √L2

 +

δ
√R
2

∆R ≡
0
δR

,

(3,1,2)

++
δR



δ+



− √R2

.

(1,3,2)(2.3)

The model’s Lagrangian L is given by/ L + iq̄R Dq/ R + iℓ̄L Dℓ/ L + iℓ̄R Dℓ/ R
L = Lgauge + iq̄L Dq
h

i

h

i

h

+ Tr (Dµ Φ)† (Dµ Φ) + Tr (Dµ ∆L )† (Dµ ∆L ) + Tr (Dµ ∆R )† (Dµ ∆R )
i− q̄L Yq Φ+ Ỹq Φ̃ qR + ℓ̄L Yℓ Φ+ Ỹℓ Φ̃ ℓR + ℓ̄cL iσ2 ∆L YL ℓL + ℓ̄cR iσ2 ∆R YR ℓR + H.c.


− V (Φ, ∆L , ∆R ),



(2.4)

where Lgauge includes kinetic terms for all gauge bosons, V (Φ, ∆L , ∆R ) is the scalar potential(which exact form is irrelevant for our study), and all Yukawa couplings Y are 3 × 3 matrices
in the flavour space. Moreover, Φ̃ is the dual bi-doublet Higgs field, and Dµ stands forthe covariant derivative operator
gB−L
II
Dµ = ∂µ − igTLI WL,µ− igTRI WR,µ−i
QB−L Bµ ,
(2.5)
2

–3–

JHEP02(2024)150

In this section, we briefly describe the mLRSM, its particle content, and the interaction
Lagrangian relevant for our analysis (more technical details can be found in refs. [60–64]).
The fermion sector of the mLRSM includes the following fields:

with TLI and TRI being the generators of SU(2)L and SU(2)R taken in the relevant representation, and QB−L is the associated U(1)B−L charge. EW symmetry breaking proceeds
in two steps. First, the vacuum expectation value acquired by the triplet ∆R breaks theSU(2)R ⊗ U(1)B−L symmetry down to the hypercharge group U(1)Y . Second, the bi-doublet
Φ and triplet ∆L Higgs fields choose a configuration breaking the SU(2)L ⊗ U(1)Y symmetry
down to electromagnetism. The full corresponding vacuum configuration is hence given by
⟨Φ⟩ =

v1
0
0 −v2 e−iα!

,⟨∆L,R ⟩ ≡00vL,R 0

!

.

(2.6)

1
2
MW
≈ g2v2,
L
2
2.2

2
2
MW
≈ g 2 vR
.
R

(2.7)

Limits on WR properties from collider and low-energy experiments

Several searches have focused on testing for the existence of WR bosons associated with
the SU(2)L × SU(2)R × U(1)B−L gauge symmetry. While at tree-level the mixing between
the two electrically charged bosons WL and WR results in a shift of the WL mass from its
SM value MW , data indicate that the corresponding mixing angle must be smaller than
10−2 [65]. This is consistent with our model configuration in which this mixing is negligible,
and in which the WR boson couples essentially to right-handed fermions. The structure ofsuch a coupling in the flavour space is further dictated by the values of the elements of a
Cabibbo-Kobayashi-Maskawa (CKM) matrix relating right-handed fermions, which generally
needs not be proportional to the known CKM matrix (that relates left-handed fields).
At hadron colliders searches for signatures of a WR boson have concentrated on resonant
production. The most commonly looked for signal is made up of high-momentum electrons
or muons accompanied by a large amount of missing transverse energy, originating fromthe process
pp → WR X → ℓNR X

withℓ = e, µ, τ.

(2.8)

Searches in this channel assume that the narrow width approximation is valid (with the
WR boson width-over-mass ratio ΓWR /MWR ≤ 7%),2 and that the heavy neutrino NR islighter than the WR boson and escapes detection. Relying on the Sequential Standard Model(SSM) as a benchmark new physics setup, the ATLAS and CMS collaborations have set
stringent limits on the WR boson by making use of 139 fb−1 of data at a centre-of-mass√energy s = 13 TeV. In the electronic channel, they constrained the WR boson mass tosatisfy MWR > 6 TeV, while in the muonic channel the bounds are reduced to 5.6 TeV [68, 69].
In contrast, limits only reach MWR > 5 TeV for a final state comprising tau leptons. By
virtue of the Keung-Senjanovic mechanism, the right-handed neutrino NR produced throughprocess (2.8) could also decay, through a virtual WR boson exchange, into an eejj, µµjj or
2

The effects of the WR -boson width on the signal have been explored in refs. [66, 67].

–4–

JHEP02(2024)150

We further define the mixing angle β such that v = v1 sin β = v2 cos β, and we enforce
the hierarchy vL ≪ v ≪ vR to get agreement with neutrino data (vL being small) andconstraints on additional gauge bosons (vR being large). In this setup, the masses of thecharged gauge bosons readContents

Contents

ℓ

q
WR
q̄

WR

ℓ
NR
WR

ℓ

q

j

q̄

ℓ
NR
WR

j

t
b

–1–

–1–

τ τ jj system. Such a new physics signature was explored by both collaborations, and crosssection limits as functions of the NR and WR masses have been determined from 13 TeV LHC
data [49, 52, 70]. This is further addressed in section 2.4.
Searches for di-jet resonances can also be used to set bounds on the signal originatingfrom the process pp → WR → q q̄ ′ . Within the SSM, limits on the WR boson mass of 4 TeV
have been obtained [52, 71]. On the other hand, The ATLAS collaboration also excluded
MWR lighter than 3.25 TeV from pp → WR → tb̄ + h.c. using 36 fb−1 of integrated luminosity[72]. Furthermore, both collaborations have searched for the signal that would emerge fromthe processes pp → WR → ZWL and pp → WR → HWL in all possible final states (leptonic,
semi-leptonic and hadronic), and derived that MWR ≥ 3.9 TeV [73, 74].
On different grounds, mass limits on WR bosons can also be obtained indirectly fromlow-energy constraints, especially from box diagrams contributing to kaon mixing. In thecase where the CKM matrices in the right-handed and left-handed quark sectors are thesame, we get MWR > 2.9 TeV [75]. Parity violation effects to be observed in polarised muon
decays additionally impose that MWR > 600 GeV [76], and combined limits on MWR and
MNR can be additionally derived [28].
2.3

Production of WR bosons at the LHC

We consider the DY-like production of a (possibly off-shell) WR boson in pp collisions, followedby its decay via the Keung-Senjanovic mechanism into two charged leptons and either twojets or a top-bottom system,
pp → WR → ℓNR → ℓℓjj

orpp → WR → ℓNR → ℓℓtb.

(2.9)

In our notation, j represents a light jet resulting from the fragmentation of quarks or
antiquarks from the first or second generation, and ℓ stands for either an electron or a muon.
Moreover, among all possible flavour and electric-charge assignments for the two leptons, we
restrict our analysis to the case of a Same-Sign (SS) and Opposite-Sign (OS) lepton pairof the Same Flavour (SF). Representative parton-level Feynman diagrams illustrating theprocesses (2.9) are shown in figure 1. In the following, we consider an mLRSM scenario in
which the right-handed neutrino is lighter than the WR gauge boson (MNR ≤ MWR ), andwe set the two CKM matrices to the 3 × 3 identity matrix.
We begin with a calculation of the cross section for the first of the processes (2.9),
pp → ℓNR → ℓℓjj, both at Leading Order (LO) and Next-to-Leading Order (NLO) in–5–

JHEP02(2024)150

Figure 1. Representative parton-level Feynman diagrams illustrating the production of two leptons
and either a pair of jets (left) or a top-bottom system (right) from the decay of a heavy right-handed
neutrino NR . The latter originates from the decay of a possibly off-shell SU(2)R charged gauge boson
WR emerging from DY production.

MNR (GeV)LO
NNPDF40_lo_as_01180
103

5000

NLO
NNPDF40_nlo_as_01180
103

101

4000

100
10−1

3000

2000

10

−2

10

−3

10

−4

102

σ(pp → WR → ``jj) (fb)

102

101

4000

100
10−1

3000
10−2
10−3

2000

σ(pp → WR → ``jj) (fb)

MNR (GeV)

5000

10−4

10−5

10−5

1000
1000

2000

3000

4000

5000

1000

2000

3000

MNR (GeV)

MWR (GeV)

4000

5000

MWR (GeV)

5000

3.6
3.3

2.7
2.4

3000

2.1
1.8

K ≡ σNLO/σLO

3.0

4000

1.5

2000

1.2
0.9

1000
1000

2000

3000

4000

5000

MWR (GeV)

Figure 2. Production cross sections related to the process pp → WR → ℓNR → ℓℓjj at LO (top√row, left) and NLO (top row, right), for a centre-of-mass energy s = 13 TeV. We also display the
K-factor defined as in (2.10) (bottom row).

QCD. We rely on the narrow-width approximation so that we can factorise the full process
into a ‘production part’ (pp → ℓNR ) and a ‘decay part’ (NR → ℓjj). The generationof the corresponding ‘production’ LO and NLO matrix elements is achieved by means of
MadGraph5_aMC@NLO version 3.4.2 [77] and a UFO [78, 79] implementation of therelevant sectors of the mLRSM. The latter is obtained by means of FeynRules [80, 81]
and NLOCT [82], and an in-house effective implementation of the Lagrangian (2.4) thatconsiders only the WR -boson interactions with fermions and the SM sector (on the samegrounds as in the parametrisation proposed in ref. [83]). The ‘decay’ matrix elements are
instead always evaluated at LO thanks to MadSpin [84], which allows us to keep track of all
spin correlations inherent to the fermionic nature of the right-handed neutrino NR and its
decay products. Hadronic cross sections are next obtained by convoluting the resulting ‘full’matrix elements with the LO (NNPDF40_lo_as_01180) and NLO (NNPDF40_nlo_as_01180)
sets of NNPDF 4.0 Parton Distribution Functions (PDFs) [85] for LO and NLO calculations,
respectively, and this relies on a central scale choice in which both the factorisation and–6–

JHEP02(2024)150

1000

renormalisation scales are fixed to the WR -boson mass, µR = µF = MWR . To visualise thesize of the NLO corrections, we define a global K-factor by the ratio of the LO to NLO rates
σLO ≡ σ(pp → ℓℓjj)LO and σNLO ≡ σ(pp → ℓℓjj)NLO ,
K≡

σNLO
.
σLO

(2.10)

2.4

Searches for WR bosons in events with two leptons and two jets

In the previous sections, we mentioned the potential bounds that could be extracted fromthe first of the processes shown in eq. (2.9) and the associated searches at the LHC. In
this section, we study the consequences of such searches on the model. To this aim, wereinterpret the results of the CMS-EXO-20-002 search targeting final states comprising two
leptons (i.e. electrons or muons) and two (light) jets, using data collected between 2016 and2018 and corresponding to an integrated luminosity of 138 fb−1 [52]. However, instead ofextracting constraints directly from the published experimental results, we undertake ourown interpretation study. The reason is that in the initial analysis, the CMS collaboration
simulated the signal at LO, used global K-factors (as computed in section 2.3), and followed
the prescription of refs. [59, 87]. Instead, we perform a more precise signal simulation at NLO
in QCD, including real and virtual contributions to O(αs ) at the fully differential level, andwe additionally rely on more recent PDF sets. Our reinterpretation study however ignores
any potential correlation between the different signal regions of the analysis, as relevant
information has not been released by the CMS collaboration. In light of our findings andthe experimental results published, we however do not expect that this issue would yield
any significant change in our conclusions.
–7–

JHEP02(2024)150

Production cross sections for the ‘full’ process pp → ℓNR → ℓℓjj are displayed in figure 2
at LO (top row, left) and NLO (top row, right), together with the associated K-factors as
defined in (2.10) (bottom row). They are found to vary between 100–1000 fb for a relatively
light WR boson with a mass between 1 and 2 TeV, to around 10−3 fb for a very heavy WR
boson with a mass around 5 TeV, i.e. in the vicinity of the exclusion limits relevant for asequential extra charged gauge boson (the precise reinterpretation of these bounds in themLRSM being addressed in section 2.4). Furthermore, for a fixed value of the WR -boson
mass, the cross section value depends on the mass splitting between the WR and NR states.
It hence decreases with decreasing values of ∆ ≡ MWR − MNR , as expected from the reduced
phase space available for the decay when the two masses are similar. We also recall that crosssections for the electron and muon channels are identical, as predicted by the universality of
the SU(2)R gauge coupling in the Lagrangian of eq. (2.4). Finally, K-factors are modest formost of the mass regime considered (i.e. for MWR < 5 TeV), and they lie between 1.1 and1.5. The lowest values correspond to lighter WR boson mass configurations while the larger
ones correspond to the heaviest setups considered. Such a result is typical of the DY-likeproduction of extra gauge bosons with masses in the TeV range, recalling that the behaviourof the K-factor is mostly independent of how the central scale is chosen [83, 86]. The K-factor
however further increases to up to 3.3 for heavier WR bosons with masses above 5 TeV, that
also correspond to a regime in which PDF uncertainties are much larger and yield larger
differences between LO predictions (involving a LO set of parton densities often associatedwith a poorer fit to data) and NLO predictions (involving an NLO set of parton densities).

Electron channel

Muon channel

Data

Background

Data

Background]800, 1000] GeV

1106.01103.5 ± 26.607

1639.01670.7 ± 39.774]1000, 1200] GeV

646.0

631.51 ± 16.968

946.0

925.99 ± 23.917]1200, 1400] GeV

332.0

323.23 ± 10.736

518.0

500.33 ± 14.869]1400, 1600] GeV

170.0

169.69 ± 6.8418

268.0

263.88 ± 9.3498]1600, 2000] GeV

143.0

157.55 ± 9.505

216.0

215.18 ± 8.2146]2000, 2400] GeV

62.0

52.327 ± 3.9676

80.0

73.482 ± 4.4654]2400, 2800] GeV

25.0

19.567 ± 1.5493

30.0

25.943 ± 2.3125]2800, 3200] GeV

10.0

8.9907 ± 1.209

13.0

9.7557 ± 1.1603]3200, 8000] GeV

13.0

6.2463 ± 0.77892

11.0

7.8119 ± 0.84286

Table 1. Definition of the 18 signal regions of the CMS-EXO-20-002 search for WR -boson production
and decay in the ℓℓjj channel [52]. After a common pre-selection (see the description in the text), thedifferent regions are defined according to the lepton flavour (electron or muon) and the value of thereconstructed WR -boson mass Mℓℓjj . For each region, we show the number of observed events andthe associated SM expectation.

The CMS-EXO-20-002 search is dedicated to the analysis of events featuring exactly two
isolated leptons (electrons or muons, regardless of their electric charge), and at least two jets
reconstructed by means of the anti-kT algorithm [88, 89] with a radius parameter fixed to
R = 0.4. For events exhibiting more than two jets, the two leading jets in terms of transverse
momentum are considered to originate from the decay of a WR boson, together with thetwo leptons. The leading and sub-leading charged leptons are required to have transverse
momenta pT > 60 and 53 GeV respectively, and to be within the detector acceptance (i.e. witha pseudo-rapidity |η| < 2.4, excluding electrons and positrons with a pseudo-rapidity lying in
the interval 1.44 < |η| < 1.57 that corresponds to the transition region between the barrel
and endcap of the CMS electromagnetic calorimeter). Furthermore, selected jets must have atransverse momentum pT > 40 GeV and pseudo-rapidity |η| < 2.4, the invariant mass of thereconstructed WR -boson candidate is enforced to verify Mℓℓjj > 800 GeV, and the invariant
mass of the lepton pair is constrained to fulfill Mℓℓ > 400 GeV. After this pre-selection, theanalysis defines 18 signal regions, 9 in the eejj channel and 9 in the µµjj channel, the various
regions corresponding in different bins in Mℓℓjj . We report the exact definition of these bins,
together with their background expectations and the associated observations, in table 1.
In order to determine the constraints that could be imposed on the mLRSM from LHC
searches for WR signatures in the ℓℓjj channel, we have implemented the CMS-EXO-20-002
search in the MadAnalysis 5 framework [90–92]. Our implementation relies on the SFS
module for the simulation of the detector effects via a parametrisation through transfer
functions [93, 94], and it has been thoroughly validated by ensuring an excellent agreementwith predictions provided by the CMS collaboration on HepData [95]. Details on our
implementation and its validation can be found in appendix A, as well as on the Public
Analysis Database (PAD) [96] and the dataverse [97] of MadAnalysis 5. In addition, we also
provide in this appendix information on the implementation and validation of the superseded–8–

JHEP02(2024)150

min , M max ]]Mℓℓjjℓℓjj

Resolved
1.0

0.8

4000

0.6

3000

0.4

5000

L = 138 fb−1
Resolvedµµ channelσ(pp → µµjj)LO

1.0

4000

0.8

0.6

3000

CL95%
s

MNR (GeV)

L = 138 fb−1
ee channelσ(pp → eejj)LO

CL95%
s

MNR (GeV)

5000

0.4

0.2
0.2

2000

2000

0.0

0.0

1000
2000

3000

4000

5000

1000

2000

3000

4000

L = 138 fb−1
Resolved
ee channelσ(pp → eejj)NLO

MNR (GeV)

5000

1.0

4000

0.6

0.4

5000

L = 138 fb−1
Resolvedµµ channelσ(pp → µµjj)NLO

1.0

4000

0.8

3000

5000

MWR (GeV)

CL95%s

MNR (GeV)

MWR (GeV)

0.8

0.6

3000

CL95%
s

1000

0.4

0.2
0.2

2000

2000

0.0

0.0

1000

1000
1000

2000

3000

4000

5000

1000

MWR (GeV)

2000

3000

4000

5000

MWR (GeV)

Figure 3. Exclusion contours extracted from the reinterpretation of the results of the CMS-EXO-20002 analysis, projected on the plane (MWR , MNR ) in the electron (left) and muon (right) channel,
when using matrix elements calculated at LO (top row) and NLO (bottom row). The black solidline corresponds to a CLs value of 1 − CLs = 0.95, all mass points localised on its lower left being
thus excluded.

CMS-EXO-17-011 search dedicated to the same signal, but in which only a partial LHC
Run 2 dataset is analysed. This older search had been initially used for the present work,
until the more recent CMS-EXO-20-002 results appeared.
The resulting constraints on the model are derived from a scan of the two-dimensional
plane defined by the masses MWR and MNR , which we vary in the range [800, 5300] GeV.
We analysed 120 signal configurations with MNR ≤ MWR , generating 500,000 hard-scattering
events for each. Subsequently, we utilised Pythia version 8.309 [98] for simulating parton
showering and hadronisation. The resulting hadron-level events were then analysed with
MadAnalysis 5, employing the CLs method [99] to assess the signal’s viability with respect
to data. We report the obtained CLs values in figure 3, both for the electron (left) and muon(right) channels and using LO (top row) and NLO (bottom row) simulations. We obtain in
general results in the same ballpark as those presented officially in the CMS analysis [52],demonstrating hence by different means the validity of our implementation of the CMS
analysis in our tool chain.
–9–

JHEP02(2024)150

1000

ee channelσ(pp → eejj)NLO

L = 138 fb−1
L = 3000 fb−1(syst)

MNR (GeV)MNR (GeV)

5000

5000

µµ channelσ(pp → µµjj)NLO

L = 3000 fb−1 (stat)
4000

3000

3000

2000

2000

3000

4000

5000

1000

MWR (GeV)

2000

3000

4000

5000

MWR (GeV)

Figure 4. Exclusion contours in the plane (MWR , MNR ) as obtained from the recasting of the
CMS-EXO-20-002 search in the eejj (left) and µµjj (right) channel, from NLO signal simulations.
We compare current exclusions (solid red) to expectations at the high-luminosity LHC with 3000 fb −1 ,assuming that the background uncertainties are dominated by the statistical (dotted green) or
systematic (dashed green) components.

In the electron channel, we exclude WR bosons with masses smaller than 4.3 TeV, giventhat the mass splitting ∆ between the WR boson and the right-handed neutrino NR issignificant. In the most extreme situation featuring a WR boson of 4–5 TeV, NR masses
ranging up to 3–3.2 TeV can be reached. The CMS official bounds are, in comparison,slightly stronger by about 1-2σ. This difference is mitigated when comparing our NLO
exclusions to the CMS official ones, which are based on LO simulations including an NLO
K-factor. In this case, the remaining difference is less than 10% when considering the mass
limit values, as expected due to the entirely different nature of the detector simulations
employed in this work (the SFS framework) and in the CMS analysis (the CMS simulation
software based on Geant 4).
Results in the muonic channel are more intriguing. Similar to the electronic case, ourLO limits are in the same ballpark as the CMS official ones, with the latter being slightly
stronger. However, with NLO simulations we observe a decrease in sensitivity predicted by
MadAnalysis 5. It is important to note that the limits found lie close to the regime where
the K-factors displayed in figure 2 are larger than 2, in contradiction with naive expectationsof a DY-like process. This behaviour can be traced back to the impact of the differences
between the poor LO and the better NLO NNPDF 4.0 parton density fits, which therefore
significantly modify not only total rates but also event kinematics. As NLO PDF fits are
better than LO fits, we rely on NLO simulations only from now on.
We close this section by estimating the change in the exclusion contours when the LHC
luminosity is scaled to L = 3000 fb−1 , which corresponds to the high-luminosity LHC phase.
We assume the same search strategy as in [52], and naively re-scale the exclusion contours
following the methods presented in ref. [100]. We consider that the expected number ofbackground events for each signal region, shown in table 1, scales proportionally to theluminosity, and that the scaling of the uncertainties on the background is achieved in two
different complementary ways depending on whether background errors are dominated bystatistical uncertainties or by systematic ones. We consider the two cases, and derive expected– 10 –JHEP02(2024)150

2000

L = 3000 fb−1 (syst)
L = 3000 fb−1 (stat)

4000

1000

L = 138 fb−1

limits by assuming observations in agreement with the SM background. The results are
displayed in figure 4. They show that WR -boson masses between 5 and 6 TeV can be reached,
together with right-handed neutrino masses ranging up to 4 TeV.

3

Sensitivity of ℓℓtb probes to left-right models

3.1

Benchmark scenarios and backgrounds

In order to build our analysis, we define Benchmark Points (BPs) that fulfill all constraints
from searches for WR bosons in the ℓℓjj channel presented in the previous section. We consider
two mass values for a WR boson decaying in the di-electron channel (WR → eetb), namely
MWR = 4800 and 5500 GeV, and two mass values for the di-muon channel (WR → µµtb),
namely MWR = 5100 and 5500 GeV. For each of these mass values, we then define three
scenarios differing by the choice of the right-handed neutrino mass,

MN R ∈

MW R MW R,, MWR − 400 GeV .
52Such a choice allows us to span most of the possible kinematic configurations in terms of the
splitting ∆ ≡ MWR − MNR . The definitions of these different BPs are collected in table 2 forboth the electron channel (upper part of the table) and the muon channel (lower part of thetable), and we also indicate some of the properties of these scenarios relevant for the signal.
In order to extract the mLRSM signal from the SM background, we focus on hadronic
top quark decays so that the final state considered comprises exactly two charged leptons,
and several jets whose hardness and multiplicity depend on the kinematic properties of thesignal. For relatively small top quark transverse momenta (typically smaller than 500 GeV),
events generally feature at least four small-radius jets, while for larger top quark transverse
momenta we should have at least one large-R jet and at least one small-R jet. The lowest
NR mass chosen for the scenarios explored in our study being fixed to 960 GeV (BPe1), theproduced top quarks have typically very high transverse momenta in most of the targetedcases. mLRSM spectra yielding softer top quarks are indeed already challenged by existingdata and thus hard to justify phenomenologically. The backgrounds associated to our signal
process thus usually emerge from the production of high-pT top quarks, with a much smaller
contribution originating from the production of two massive gauge bosons in association withjets that we therefore do not consider. The list of all background contributions is given intable 3, together with the related LO and NLO cross sections.

– 11 –JHEP02(2024)150

In section 2, we focused on existing LHC searches for mLRSM signals in which a WR boson
decays into a lepton pair and a di-jet system. However, for heavier spectra, decays into a
lepton pair and a top-bottom system could as well contribute. We therefore focus in thissection on the second process of eq. (2.9), illustrated by the second diagram in figure 1. Inorder to assess the potential of the LHC to such a signal, we consider a set of benchmark
scenarios that satisfy all current constraints, and then design a dedicated phenomenologicalanalysis that could potentially seed future experimental searches in proton-proton collisions√
at a centre-of-mass energy of s = 13.6 TeV.

BPBP
MWR [GeV]MNR [GeV]
σ(pp → ℓℓtb)LO [fb]σ(pp → ℓℓtb)NLO [fb]
ΓWR [GeV]ΓNR [GeV]BP
MWR [GeV]MNR [GeV]
σ(pp → ℓℓtb) [fb]σ(pp → ℓℓtb)NLO [fb]
ΓWR [GeV]ΓNR [GeV]BP
MWR [GeV]MNR [GeV]
σ(pp → ℓℓtb)LO [fb]σ(pp → ℓℓtb)NLO [fb]
ΓWR [GeV]ΓNR [GeV]

BPe2

BPe3

4800
960
1.22 × 10−1
1.73 × 10−1
134
2.47 × 10−5

4800
2400
7.77 × 10−2
1.13 × 10−11302.96 × 10−3

4800
4400
4.27 × 10−3
6.43 × 10−3122
1.21 × 10−1

BPe4

BPe5

BPe6

5500
1100
3.39 × 10−2
5.50 × 10−2
153
2.88 × 10−5

5500
2750
1.73 × 10−2
3.13 × 10−2
149
3.41 × 10−3

5500
5100
6.87 × 10−4
1.34 × 10−3
139
1.52 × 10−1

BPµ1

BPµ2

BPµ3

5100
1020
6.99 × 10−2
1.04 × 10−1142
2.65 × 10−5

5100
2550
4.09 × 10−2
6.45 × 10−2
138
3.15 × 10−3

5100
4700
1.54 × 10−3
3.25 × 10−31291.34 × 10−1

BPµ4

BPµ5

BPµ6

5500
1100
3.39 × 10−2
5.50 × 10−2
153
2.88 × 10−5

5500
2750
1.73 × 10−2
3.13 × 10−2
149
3.41 × 10−3

5500
5100
6.87 × 10−4
1.34 × 10−3
139
1.52 × 10−1

Table 2. Definition of the 12 BPs used in our analysis, for WR decays into a eetb system (BPe1 toBPe6) and µµtb system (BPµ1 to BPµ6). These benchmarks satisfy current constraints originatingfrom WR production and decay into an ℓℓjj final state, and they feature a variety of split and morecompressed spectra. For each scenario, we additionally report LO and NLO signal cross sections at√
s = 13.6 TeV, as well as the total decay widths of the WR and NR states.

– 12 –JHEP02(2024)150

MWR [GeV]MNR [GeV]
σ(pp → ℓℓtb)LO [fb]σ(pp → ℓℓtb)NLO [fb]
ΓWR [GeV]ΓNR [GeV]

BPe1

ProcessσLO [fb]σNLO [fb]K ≡ σNLO /σLO

Nevents

pp → t t̄ H

345.5

503.8

1.45

1.6 × 106

pp → t t̄ Z

519.1

838.9

1.61

1.6 × 106

pp → t t̄ W ±

434.0

670.2

1.54

1.6 × 106

pp → t Zj + c.c. [101]

821.4

903.5

1.10

2.2 × 106

pp → t W − b̄ + c.c. [102]976.4

1331.51.36

4.0 × 106

Mℓℓ
Mℓℓtb
]1200, ∞)]1400, ∞)]1600, ∞)]1800, ∞)]2000, ∞)]2500, ∞)]3000, ∞)]400, ∞)

]600, ∞)

]800, ∞)

SRa1
SRa2
SRa3
SRa4
SRa5
SRa6
SRa7

SRb1
SRb2
SRb3
SRb4
SRb5
SRb6
SRb7

SRc1
SRc2
SRc3
SRc4
SRc5
SRc6
SRc7

Table 4. Definitions of the SRs of our analysis in terms of bins in the invariant mass of thereconstructed WR boson Mℓℓtb and the invariant mass of the lepton pair Mℓℓ .

Background event generation and cross section calculations have been achieved withthe same tool chain as the one described in section 2.3, with the exception of the NLO
total cross sections relevant to single top production in association with a Z boson or a W
boson that we extracted from refs. [101] and [102] respectively. Detector simulation relieson the SFS framework with a detector parametrisation matching the one that we used forthe implementation of the CMS-EXO-20-002 search in MadAnalysis 5 (see appendix A
for more details), with the exception of the b-tagging efficiency that we take equal to 70%.
Event reconstruction relies on the definition of two jet collections. The first one includes
small-R ‘AK04’ jets clustered using the anti-kT algorithm and a jet radius parameter of
R = 0.4, while the second one comprises large-R ‘CA15’ jets clustered using the CambridgeAachen algorithm [103, 104] with a radius parameter of R = 1.5. Finally, boosted top quark
candidates are reconstructed by means of HepTopTagger [105, 106], as integrated within
the jet substructure module of MadAnalysis 5 [107].
3.2

Description of the analysis

Events are selected if they contain exactly two isolated electrons or muons with pT > 25 GeV
for electrons and pT > 33 GeV for muons, and |η| < 2.4 in both cases. Moreover, we– 13 –JHEP02(2024)150√
Table 3. Total cross sections at s = 13.6 TeV for all background processes relevant to our study,
shown both at LO and NLO, along with the corresponding K-factor and the number of generated events.
Here, σtZj ≡ σ(pp → tZj) × BR(t → bjj) × BR(Z → ℓ+ ℓ− ) and σtW b ≡ σ(pp → tW b) × BR(t →bℓν) × BR(W → ℓν). Total rate predictions have been obtained with MadGraph5_aMC@NLO,
unless indicated otherwise through a reference.

BPµ1
BPµ2
BPµ3
Backgrounds

10

1/σ dσ/Mtb``

1/σ dσ/M``

10−3

10−3

−4

10−5

10−4

10−5

1000

2000

3000

4000

5000

1000

M`` (GeV)

2000

3000

4000

5000

6000

7000

Mtb`` (GeV)

×10−5

Backgrounds

Signal

1.75
1200

1.50

1000

1.25

5000

×10−7
2.0

1.5
M`` (GeV)

M`` (GeV)

4000

1.00

800

0.75

3000

1.0
2000

600

0.50
400

0.5
1000

0.25
1000

2000

3000

4000

0.00

1000 2000 3000 4000 5000 6000 7000

Mtb`` (GeV)

0.0

Mtb`` (GeV)

Figure 5. Normalised distributions in the invariant mass of the di-lepton system Mℓℓ (top left) andof the reconstructed WR boson Mℓℓtb (top right) after the selection described in the text, both forthe background (solid black) and the three representative signal scenarios BPµ1 (blue), BPµ2 (red)and BPµ3 (orange). We additionally display the correlations between these two variables, both forthe background (bottom left) and the BPµ2 signal (bottom right). The colour code refers to a total
number of entries in the maps, normalised to unity.

remind that lepton isolation is encoded directly in our SFS detector parametrisation. The
leading and sub-leading charged leptons are next enforced to satisfy pT > 60 GeV andpT > 53 GeV respectively, and we veto events featuring hadronically decaying tau-leptons
with pT > 30 GeV and |η| < 2.5. We then select events that contain at least one AK04
b-tagged jet with pT > 30 GeV and |η| < 2.5, and at least one top-tagged CA15 jet withpT > 200 GeV, |η| < 2, and an invariant mass MHTT ∈ ]145, 210[ GeV. Finally, we requirethat the invariant mass of the two charged leptons satisfies Mℓℓ > 200 GeV. The resulting
selection efficiency is 6.67 × 10−4 and 1.54 × 10−3 for the t + X and tt̄ + X components ofthe background, respectively. On the other hand, for the signal processes the accumulated
efficiency varies between 2.9% and 5.4% for the electron channel and between 14% and 18% for
the muon channel, the higher efficiencies corresponding to scenarios with larger right-handed– 14 –JHEP02(2024)150

10−6

10−6

3.3

Results

In figure 6 we display the number of events populating the different signal regions of ouranalysis, both for the signal process in the case of the 12 benchmark points considered andthe different contributions to the background. We collectively group the latter into a single
top component (orange) and tt̄ pair component (kaki), and we show results both for theelectron channel (upper panel) and muon channel (lower panel). As already mentioned in
section 3.1, events are normalised to their total rates at NLO in the strong coupling. It is
immediate to see that there is a high discovery potential for the signal, specifically in themuon channel. This originates from the fact that our analysis comprises some SRs thatare essentially background-free, and that turn out to be the SRs dedicated to scenarioswith heavy WR bosons.
In order to assess the sensitivity of the LHC to the signal process considered as a functionof the integrated luminosity, we extract the CLs exclusion and associated 1σ and 2σ bands
for the best expected signal region (SRb7), by means of the package Pyhf [108]. Whereas
more aggressive estimates could be obtained by combining the different signal regions of
the analysis, we refrain from doing so in order to get predictions as conservative as possible.
Our goal is indeed to demonstrate that for scenarios not excluded by analyses of the ℓℓjj
signature of WR -boson production and decay, there is a potential gain in studying thecorresponding third-generation signal ℓℓtb. Our calculation relies on several assumptions.
First, we conservatively assume that there is at least one background event regardless of theluminosity. Consequently, we always fix the number of background event to nb = 1 for signal
regions in which our simulations yield nb < 1. Next, we assume that there is a systematic
uncertainty of 20% on the background yields. Finally, we estimate that the number of events
to be observed is always equal to those predicted in the context of the SM.
The results are shown in figure 7 for the electron (upper panel) and muon (lower panel)
channels. For the electron channel, it is clear that a large luminosity is required in order toprobe the model. A few ab−1 are indeed necessary to exploit the small signal cross sections– 15 –JHEP02(2024)150

neutrino masses. Moreover, the higher prospects for the muon channel are directly related tothe higher reconstructed efficiencies for high-pT muons than for corresponding electrons.
To improve signal significance we define several signal regions (SRs) targeting different
bins in the invariant masses of the di-lepton system (Mℓℓ ) and of the reconstructed WRboson candidate (Mℓℓtb ). To better understand this choice of variables, we display in figure 5(top row) the associated normalised distributions for the background and a selection of
three illustrative signal benchmark points (BPµ1, BPµ2 and BPµ3), after imposing theselection Mℓℓ > 200 GeV. The resonant contributions specific to the signal leads to very hard
distributions exhibiting a very broad plateau (or peak), deep in the multi-TeV regime. This
contrasts with the SM background, that exhibits a steeply falling spectrum with very few
events expected for invariant masses larger than 1 TeV for the Mℓℓ spectrum or 2–3 TeV for
the Mℓℓtb one. The same information is displayed jointly through the correlations between
these two variables in the bottom row of the figure, in which we focus on the background(bottom left) and the illustrative BPµ2 scenario (bottom right). We subsequently define 21
SRs in each of the electron and muon channels as shown in table 4.

Electron channel

Number of events

10

BPe1
BPe2
BPe3

2

BPe4
BPe5
BPe6

t+X
tt̄ + X

101

100

SRc7

SRc6

SRc5

SRc4

SRc3

SRc2

SRc1

SRb7

SRb6

SRb5

SRb4

SRb3

SRb2

SRa7

SRb1

SRa6

SRa5

SRa4

SRa3

SRa2

SRa1

Muon channel

BPµ1
BPµ2
BPµ3

Number of events

102

BPµ4
BPµ5
BPµ6

t+X
tt̄ + X

101

SRc7

SRc6

SRc5

SRc4

SRc3

SRc2

SRc1

SRb7

SRb6

SRb5

SRb4

SRb3

SRb2

SRa7

SRb1

SRa6

SRa5

SRa4

SRa3

SRa2

SRa1

100

Figure 6. Number of background and signal events populating the different SRs in the electron(upper) and muon (lower) channels, for L = 3000 fb−1 . Our predictions are normalised to NLO
in QCD, and we collect the background contributions into a single top contribution (orange) andtop-antitop contribution (kaki). Signal predictions are made for the 12 BPs considered (solid lines).

(see table 2). Consequently, only benchmarks for which the WR boson is not too heavy andin which the new physics spectrum is not too compressed could yield some sensitivity. We
find that it is indeed the case, as shown by the predictions made in the upper panel of the
figure. Only the two first scenarios, BPe1 and BPe2, have the potential to be excluded at 95%
confidence level, and this can only be achieved around the ultimate end of the high-luminosity
LHC runs. In contrast, results in the muon channel are more promising, by virtue of thelarger reconstruction efficiencies for high-pT muons than for electrons. More of our selectedbenchmarks are found reachable, and this for a smaller integrated luminosity (of about
1 ab−1 ) than in the electron channel. All muonic scenarios with the exception of BPµ6 arehence potentially observable, and it becomes clear that charged gauge boson masses ranging– 16 –JHEP02(2024)150

10−1

Electron channel
100
10−1
10−2

BPe3

10−3

BPe1

1 − CL95%
s

10−5

BPe2

100

10−1
10−2

BPe6 (×10)

10−3
10

−4

BPe4

10−5

BPe5

10−6
102

103

102

103

Integrated luminosity (fb−1)

102

103

Muon channel
100

10−1

1 − CL95%
s

BPµ1
10

BPµ2

BPµ3

−2

101
100

10−1
10−2

BPµ6 (×10)

10−3

BPµ4

10−4

102

BPµ5
103

102

103

Integrated luminosity (fb−1)

102

103

Figure 7. Evolution of the exclusion CLs as a function of the luminosity for the electron (upper) andmuon (lower) channels, for each of the BPs considered. The central values are depicted through a solid
black line, together with the associated 1σ (lime) and 2σ (yellow) bands. The horizontal blue dashedline corresponds to CLs = 0.95, above which the parameter point is excluded at 95% confidence level.

– 17 –JHEP02(2024)150

10−4Electron channel

S

BPe1
BPe2
BPe3

BPe4
BPe5
BPe6

Muon channel

101

S=5

S=5

S

101

S=2

S=2

100

100

10−1
10

BPµ1
BPµ2
BPµ3

−1

BPµ4
BPµ5
BPµ6

10−2

103

102

Integrated luminosity (fb−1)

103

Integrated luminosity (fb−1)

Figure 8. Signal significance S for the electron (left) and muon (right) channels. Results are shownfor the scenarios BPℓ1 (green), BPℓ2 (red), BPℓ3 (yellow), BPℓ4 (cyan), BPℓ5 (purple) and BPℓ6(gray) and for two choices of background uncertainties: the ideal case in which there is no error on the
background (solid lines), and a more realistic situation embedding a level of systematics equal to 20%.
The horizontal dashed and solid black lines correspond to the exclusion and discovery limits for which
S = 1 and S = 5 respectively.

up to at least WR = 5.5 TeV could be probed during the high-luminosity phase of the LHC,
provided that the mass splitting between the WR boson and the right-handed heavy neutrino
NR is not too small. The ℓℓtb channel has thus a strong potential as a probe to left-rightmodels, similar to the ℓℓjj channel (see figure 4).
In figure 8, we illustrate these results in a complementary way through the calculationof the signal significance S as a function of the integrated luminosity, both for the electron(left panel) and muon (right panel) channel. Using the formulas introduced in ref. [109],
we make use of
S=

√

"

2 (ns + nb ) log(ns + nb )(nb + δb2 )n2b + (ns + nb )δb2!

n2
δb2 ns− 2b log 1 +δbnb (nb + δb2 )

!#1/2

,

(3.1)

where ns and nb are the number of signal and background events, and where the systematic
error on the background δb = xnb . We use the same error configuration as above, and we thus
assume that x = 0.20 and that nb is of at least 1 (dashed lines). For informative purposes,
we additionally consider the ideal case of x = 0 (solid lines). This consistently confirmsthat the two electron scenarios BPe1 and BPe2 can be reached at the high-luminosity LHC,
and this for luminosities around 2.5 ab−1 , and that the first five muonic BPs can be probedwith luminosities of O(1) ab−1 .
We close this section by discussing briefly the capability of our search strategy todisentangle between the different BPs, i.e. by assessing the value of the right-handed neutrino
mass. In our analysis we did not make use of the reconstructed NR mass, defined as theinvariant mass of the tbℓ system, for the reason that we wanted to optimise the signal-tobackground ratio with minimal model assumptions. However, the determination of the NR
mass can always be achieved post-discovery. In the production channel considered, thecharged lepton that is emerging from the decay of the right-handed neutrino NR , along with– 18 –JHEP02(2024)150

102

MWR = 5100 GeV, MNR = 2550 GeV

1.4

×10−7

6000

6000

1.2

5000

1.0

4000

0.8

3000

0.6

2000

0.4

1000

0.2

MWR = 5100 GeV, MNR = 4700 GeV

2000

4000
6000
Mtb`` (GeV)

8000

1.75

6
4000

5
4

3000

5000

1.50

4000

1.25
1.00

3000

0.75

3
2000

2

0.0

6000

7

5000

1000

1
2000

4000
6000
Mtb`` (GeV)

8000

0

×10−7
2.00

8

Mtb` (GeV)

×10−6

Mtb` (GeV)

Mtb` (GeV)

MWR = 5100 GeV, MNR = 1020 GeV

2000

0.50
0.25

1000
2000

4000
6000
Mtb`` (GeV)

8000

0.00

the top and bottom jets, can be tagged using a ∆R requirement: the NR candidate comprises
the lepton with the minimal separation from the top-bottom system. As an example, we
display in figure 9 correlation maps relating the invariant mass of the WR system (Mℓℓtb )and the one of the NR candidate (Mℓtb ), for the three most optimistic scenarios BPµ1 (leftpanel), BPµ2 (central panel) and BPµ3 (right panel). These scenarios all feature the same
WR boson mass, MWR = 5.1 TeV, but they differ with the chosen value of the right-handed
neutrino mass so that different level of spectrum compression are considered. Predictions
are presented for the signal region SRb7, that is essentially background-free and dedicatedto scenarios with very large WR -boson mass. This region corresponds to a mass window
reasonably populated, which ensures good statistics for the signal (ns ≈ 10–40 events). Our
results therefore demonstrate that our simple analysis has an intrinsic ability to identify
the underlying mLRSM scenario.

4

Conclusions

We have proposed a novel search channel for SU(2)R charged gauge bosons typical of left-rightsymmetric models, which exploits the fact that such bosons often decay, via right-handed
neutrinos, into a final state made of a highly-boosted top quark, a b-jet and two charged
leptons. This new avenue supplements traditional analyses that exploit the ℓℓjj signature,
where j represents a light jet. After discussing in detail the effects of existing searcheson the viability of scenarios featuring a WR boson and a right-handed neutrino NR in aminimal left-right configuration, we have defined a few phenomenologically-viable scenariosthat could be tested at future runs of the LHC. We have analysed the properties of WR
and NR production and decay in these benchmark scenarios, and we have then built a new
analysis strategy relying on the richness of the ℓℓtb final state. We have in particular made useof two kinematic variables to enhance the signal-to-background ratio, namely, the invariant
masses of the di-lepton system (Mℓℓ ) and that of the WR -boson candidate (Mℓℓtb ). Thesevariables have allowed for the definition of different SRs targeting different mLRSM spectrain terms of mass scale and regime of compression, guaranteeing hence in a very significant
way a potential observation of the signal above the background.

– 19 –JHEP02(2024)150

Figure 9. 2D correlations between Mℓℓtb and Mℓtb for the signal for BPµ1 (left upper panel), BPµ2
(right upper panel) and BPµ3 (lower panel). The colour code corresponds to the number of entries
normalised to unit integral.

Acknowledgments
The work of AJ is supported by the Institute for Basic Science (IBS) under project code
IBS-R018-D1. SM is supported in part through the NExT Institute and Science & Technology
Facilities Council (STFC) Consolidated Grant No. ST/L000296/1. The work of MF has been
partly supported by NSERC through the grant number SAP105354. BF acknowledges support
from Grant ANR-21-CE31-0013, Project DMwithLLPatLHC, from the Agence Nationale
de la Recherche (ANR), France.

A

Reinterpretation of CMS searches for WR bosons with leptons and jets

In this section, we present details about predictions obtained with in-house implementations,in the MadAnalysis 5 framework [90–92], of two CMS searches for SU(2)R gauge bosons
and right-handed neutrinos in final states comprising two charged leptons (electrons or√muons) and at least two jets. We consider LHC run 2 searches at s = 13 TeV that focus
on integrated luminosities of 35.9 fb−1 [50] and 138 fb−1 [52]. We have validated these– 20 –JHEP02(2024)150

Our results demonstrated that analyses of the ℓℓtb channel could probe left-right models
as well as analyses relying on the ℓℓjj channel. We have shown that, in the electron mode
(eetb), scenarios featuring a WR -boson with mass ranging up to 4.8 TeV and a right-handed
neutrino NR with mass lying between MWR /5 and MWR /2, have the potential to be probed
during the high-luminosity phase of the LHC. Furthermore, in the muon channel (µµtb),
WR -boson masses ranging up to 5.5 TeV can be reached for NR masses lying between MWR /5and MWR /2, although MNR can also sometimes be even larger. In addition, our approach isbeneficial when it comes to profiling the underlining new physics scenario, as NR reconstructionis also possible. This in turn enables new physics characterisation through the exploitation of
all information available from the rich final state inherent to ℓℓtb production via WR and NR
exchanges, by virtue of the combination of WR and NR reconstruction.
The method that we proposed in this work can be improved further. First, better
electron and muon identification efficiencies could increase fiducial signal rates, which would
subsequently increase the sensitivity reach. This is especially crucial for the electron channel
where lower reconstruction and identification efficiencies are currently in order. Second, one
could use more sophisticated jet substructure techniques that will allow for the definition of
the whole NR candidate as a single large R-jet with some specific substructure properties.
Consequently, heavy-neutrino jets could be distinguished from more regular top-quark, W boson, Z-boson or Higgs-boson jets. Such methods targeting jets with more than threeprongs belong however to an uncharted territory, as far as we know, and they will need to bestudied comprehensively. Third, we expect that machine-learning techniques could impact
the signal significance very positively, and improve both large-R jet identification and cut-flow
optimisation. Finally, we remark that a simple search strategy, similar to the one that we
proposed in this work, could also be used to probe lepton-flavour-violating or lepton-numberviolating WR -boson decays. All these items should however preferably be addressed in thecontext of an experimental search. We therefore urge the experimental collaborations at the
LHC to carry out a search for WR bosons and right-handed neutrinos NR as we proposed.

implementations by comparing cut-flows predicted with our tool chain to official results
provided by the CMS collaboration for well-defined benchmark scenarios. The corresponding
codes and the validation material is public, integrated in MadAnalysis 5, and can be foundon the software dataverse [97, 110]. In the following, we first discuss generalities common
to the two implemented analyses in section A.1, and we next detail the validation of thetwo implementations in sections A.2 and A.3.
A.1

Object definitions and detector modeling

We model detector effects with the SFS module of MadAnalysis 5 [93, 94]. Smearing
and reconstruction efficiencies related to the different objects are tuned according to the
CMS analyses considered, that we implement through approximate semi-analytical formulas.
For energy and momentum smearing, we use the parametrisation included in the default
CMS card shipped with MadAnalysis 5 (see also refs. [113–116]). We then implement thefollowing identification efficiencies for jets (Ej ), muons (Eµ ) and electrons (Ee ):



0.925



if |η| ≤ 1.5,



0.80

if 2.5 < |η|.

Ej = 0.875


(A.1)

if 1.5 < |η| ≤ 2.5,

– 21 –JHEP02(2024)150

The signal topology is such that the two produced final-state leptons carry a large fraction of
the SU(2)R gauge boson energy. The two CMS analyses considered exploit this, and theyconsequently rely on triggers on leptons with very high transverse momentum to optimise signal
efficiency. In CMS-EXO-17-011, the electron channel relies on a double-electron trigger that
requires that the event final state features at least two electrons with a transverse momentum
pT > 33 GeV and associated to important deposits in the electromagnetic calorimeter. For themuon channel, a single-muon trigger is used instead, which requires the presence of at least one
muon with pT > 50 GeV. In its more recent CMS-EXO-20-002 search, the CMS collaboration
improves electron triggering through a combination of three triggers. The event final statemust exhibit the presence of at least either an isolated electron with pT > 27 (32) GeV, anelectron with pT > 115 GeV, or a photon with pT > 175 (200) GeV for the 2016 (2018) dataset.
As in all CMS searches, event reconstruction is performed by means of the particle-flow
algorithm [111]. We briefly summarise below object identification requirements used in the
two CMS searches considered, and we refer interested readers to refs. [50, 52] for more details.
• Electron candidates are identified by associating charged-particle tracks from theprimary vertex with energy deposit clusters in the electromagnetic calorimeter.
• Muon candidates are identified by combining tracker and muon chamber information.
• Charged hadrons reconstruction involves the matching of tracks with calorimeter cells,
together with the absence of any associated activity in the muon chamber.
• Neutral hadrons arises from the presence of clusters in both the electromagnetic andhadronic calorimeters and the absence of any associated charged particle track.
• Jets are defined from the clustering of reconstructed particles by means of the anti-kT
algorithm [88, 89] with a jet radius R = 0.4. Charged and neutral hadrons originating from pile-up interactions are removed through a dedicated pileup subtraction
method [111] and residual average area-based corrections [112].

Cut

Definition

Initial

Initial number of events corresponding to 35.9 fb−1

Njets ≥ 2

At least two jets with pT > 25 GeV and |η| < 2.4

Nℓ ≥ 2At least two isolated charged leptons

pjT > 40 GeV
pℓT1 > 60 GeV
pℓT2 > 53 GeV

The pT of the two leading jets should be larger than 40 GeV∆Rℓj > 0.4

Leptons and jets should be separated by at least ∆R > 0.4

Mℓℓ > 200 GeV

The di-lepton invariant mass should be larger than 200 GeV

The pT of the leading lepton should be larger than 60 GeV
The pT of the sub-leading lepton should be larger than 53 GeV

Table 5. Definition of the different cuts used in the CMS–EXO–17–011 search [50].

Ee =

Eµ =

A.2




0.00






0.73






0.95






for pT ≤ 0.1 GeV,



0.50






0.83






0.83




0.00

for 1.5 < |η| ≤ 2.5 and pT ∈ ]0.1, 1] GeV,

0.83

for |η| ≤ 1.5 and pT ∈ ]0.1, 1] GeV,
for |η| ≤ 1.5 and pT ∈ ]1, 100] GeV,
for |η| ≤ 1.5 and pT > 100 GeV,

(A.2)

for 1.5 < |η| ≤ 2.5 and pT ∈ ]1, 100] GeV,
for 1.5 < |η| ≤ 2.5 and pT > 100 GeV,
for |η| > 2.5,




0.00






0.75






0.99






0.99×exp{0.5−5×10−4 pT }



0.70






0.98






0.98×exp{0.5−5×10−4 pT }




0.00

for pT ≤ 0.1 GeV,
for |η| ≤ 1.5 and pT ∈ ]0.1, 1] GeV,
for |η| ≤ 1.5 and pT ∈ ]1, 100] GeV,
for |η| ≤ 1.5 and pT > 100 GeV,
for 1.5 < |η| ≤ 2.5 and pT ∈ ]0.1, 1] GeV,

(A.3)

for 1.5 < |η| ≤ 2.5 and pT ∈ ]1, 100] GeV,
for 1.5 < |η| ≤ 2.5 and pT > 100 GeV,
for |η| > 2.5,

CMS-EXO-17-011

The CMS-EXO-17-011 analysis is dedicated to events with at least two high-pT leptons andtwo jets. The two leptons, together with the two jets with the largest pT , are considered tooriginate from the decay of a WR boson. The selection, that is described in table 5, first
imposes that the leading and sub-leading leptons have transverse momentum pT > 60 and
53 GeV respectively, and that they both lie within the detector acceptance (|η| < 2.4, withelectrons being rejected if 1.444 < |η| < 1.566). Muon isolation is enforced by imposing thatthe charged-track activity in a cone of ∆R = 0.3 centred on the muon is of at most 10% of– 22 –JHEP02(2024)150

Mℓℓjj > 600 GeV The reconstructed WR invariant mass should be larger than 600 GeV

MWR = 2200 GeV MWR = 2800 GeV MWR = 3600 GeV
Electron channel

EventsεEventsεEventsεInitial1507.8

−

366.2

−

58.9

−

1505.4 ± 0.6 0.998 365.5 ± 0.2 0.998 58.9 ± 0.0 0.999

Nℓ ≥ 2752.1 ± 5.3 0.500 167.7 ± 1.2 0.459 24.6 ± 0.2 0.418

pT (jets) > 40 GeV

751.0 ± 5.3 0.999 167.5 ± 1.2 0.999 24.6 ± 0.2 0.999

pℓT1 > 60 GeV, pℓT2 > 53 GeV

735.1 ± 5.3 0.979 164.8 ± 1.2 0.984 24.4 ± 0.2 0.992∆Rℓj > 0.4

735.1 ± 5.3 1.000 164.8 ± 1.2 1.000 24.4 ± 0.2 1.000

Mℓℓ > 200 GeV

717.4 ± 5.2 0.976 162.5 ± 1.2 0.986 24.2 ± 0.2 0.992

Mℓℓjj > 600 GeV

717.0 ± 5.2 0.999 162.5 ± 1.2 1.000 24.2 ± 0.2 1.000

Muon channel

EventsεEventsεEventsεInitial1507.8

-

366.2

-

58.9

-

Njets ≥ 2

1403.2 ± 3.7 0.931 346.7 ± 0.8 0.947 56.1 ± 0.1 0.953

Nℓ ≥ 21213.0 ± 5.4 0.864 300.8 ± 1.3 0.867 46.3 ± 0.2 0.824

pT (jets) > 40 GeV

1145.3 ± 5.6 0.944 285.7 ± 1.3 0.950 44.5 ± 0.2 0.963

pℓT1 > 60 GeV, pℓT2 > 53 GeV

1121.7 ± 5.7 0.979 282.5 ± 1.4 0.989 44.2 ± 0.2 0.992∆Rℓj > 0.4

1121.7 ± 5.7 1.000 282.5 ± 1.4 1.000 44.2 ± 0.2 1.000

Mℓℓ > 200 GeV

1103.7 ± 5.7 0.984 279.5 ± 1.4 0.989 43.9 ± 0.2 0.994

Mℓℓjj > 600 GeV

1103.7 ± 5.7 1.000 279.5 ± 1.4 1.000 43.9 ± 0.2 1.000

Table 6. Cut-flow table relevant to the CMS-EXO-17-011 analysis, for the electron (upper) andmuon (lower) channels. Three scenarios are considered, with MWR = 2200, 2800, and 3600 GeV andMNR = MWR /2. Cut efficiencies are defined in eq. (A.4).

the muon pT , while electron isolation imposes that the same variable is smaller than 5 GeV.
On the other hand, the two jet candidates must each satisfy pT > 40 GeV and |η| < 2.4. In
addition, objects overlapping within ∆R > 0.4 are removed. Signal region definition further
constrains that the two leptons have the same flavour, that the invariant mass of the di-lepton
system is above 200 GeV (to avoid contamination from the Drell-Yan background), and thatthe invariant mass of the reconstructed WR boson Mℓℓjj is greater than 600 GeV.
Cut-flow tables for selected benchmark scenarios are given in table 6 for both the electron
and the muon channels. We consider three WR -boson masses of 2200, 2800 and 3600 GeV, withthe right-handed neutrino mass being fixed in each case to MWR /2. We present the number of
events surviving each cut, normalised to an integrated luminosity of 35.9 fb−1 and a production
cross section evaluated at LO. Moreover, we define the efficiency εi after the ith cut by
εi =

Ni
,
Ni−1

(A.4)

where Nk corresponds to the number of events surviving the k th cut. In table 7, we compare
the predictions obtained with MadAnalysis 5 for the benchmarks considered with the official
results released by the CMS collaboration for different bins in the reconstructed WR -boson– 23 –JHEP02(2024)150

Njets ≥ 2

min , M max ])MWR ([Mℓℓjjℓℓjj

MadAnalysis 5

CMS

δ [%]

2200 ([1950, 2810])

464.8 ± 3.8

474.0 ± 3.7 ± 44.7

1.942800 ([2530, 3840])

117.2 ± 0.9

114.1 ± 0.9 ± 10.6

2.713600 ([3250, 5170])

17.7 ± 0.1

19.2 ± 0.2 ± 1.8

7.82

2200 ([1860, 2800])

885.9 ± 5.7

744.0 ± 4.7 ± 47.5

16.02

2800 ([2430, 3930])

211.2 ± 1.4

177.0 ± 1.1 ± 13.1

16.193600 ([3190, 5500])

30.2 ± 0.2

29.2 ± 0.2 ± 2.6

3.31

Electron Channel

Muon Channel

mass. To better quantify the agreement we compute the quantity δ defined byδ = 100 × 1 −

nMA5
,
nCMS

(A.5)

where nMA5 (nCMS ) refers to the number of events surviving all cuts and with Mℓℓjj ∈min , M max ] as obtained with MadAnalysis 5 (reported by the CMS collaboration). We[Mℓℓjjℓℓjj
observe that our implementation leads to an agreement of about 2%–20%, and can thus
be considered as validated.
A.3

CMS-EXO-20-002

The CMS-EXO-17-011 analysis described in section A.2 has been recently superseded by the
CMS-EXO-20-002 search exploring similar signs of new physics, but in 138 fb−1 of LHC
run 2 data. This recent search for WR -boson production and decay in an ℓℓjj system via aheavy neutrino includes two classes of search regions. A first one is dedicated to the resolved
regime where (at least) four well-separated final-state objects are identified, while a second
one focuses on a boosted situation in which (at least) only two well-separated objects are
identified. We only consider the former, as it is sufficient for the present study. Report on
the recasting of the boosted-regime analysis is left for future work.
The resolved analysis is similar the CMS-EXO-17-011 search, although it embeds a few
differences. Events are selected provided that their final state includes exactly two isolated
leptons, and at least two small-radius jets (with R = 0.4) with pT > 40 GeV and |η| < 0.4.
Among all jets, those with the highest transverse momenta are assumed to originate from theWR -boson decay, together with the leading and sub-leading charged leptons that are required
to satisfy pT > 60 GeV and 53 GeV respectively, and to be within |η| < 2.4 (and not to satisfy
1.44 < |η| < 1.57 for electrons). Lepton isolation is enforced through a dedicated variable
Iℓ ≡X
i∈tracks– 24 –

piT ,

(A.6)

JHEP02(2024)150

Table 7. Comparison between the number of events surviving all cuts as predicted by MadAnalysis 5,
and those reported by the CMS collaboration. We consider three scenarios respectively featuring
MWR = 2200, 2800 and 3600 GeV, and MNR = MWR /2. For each scenario, a different bin in theinvariant mass of the ℓℓjj system is considered.

CMS

MadAnalysis

EventsεEventsεδ [%]Initial1175.4-1174.4--Nℓ ≥ 2 with pℓT > 60 (53) GeV

379.5

0.323

363.5 ± 3.0

0.309

4.1

≥ 2 AK4 jets with pT > 40 GeV

363.1

0.957

363.0 ± 3.0

0.999

4.4

∆R > 0.4 between all pairs of objects

355.4

0.979

363.0 ± 3.0

1.000

2.2

mℓℓ > 200 GeV

335.6

0.944

358.3 ± 3.0

0.987

4.5

mℓℓjj > 800 GeV

335.6

1.000

357.1 ± 3.0

0.997

0.3

mℓℓ > 400 GeV

324.1

0.966

339.5 ± 2.9

0.951

1.6

{MWR , MNR } = (4000, 2000) GeV

EventsεEventsεδ [%]Initial

115.2

-

115.2

-

-

Nℓ ≥ 2 with pℓT > 60 (53) GeV

37.2

0.322

32.9 ± 0.3

0.285

11.5

≥ 2 AK4 jets with pT > 40 GeV

36.0

0.969

32.7 ± 0.3

0.996

2.8

∆R > 0.4 between all pairs of objects

35.1

0.974

32.7 ± 0.3

1.000

2.6

mℓℓ > 200 GeV

33.2

0.947

32.5 ± 0.3

0.993

4.8

mℓℓjj > 800 GeV

33.2

1.000

32.5 ± 0.3

0.999

0.1

mℓℓ > 400 GeV

32.5

0.979

31.4 ± 0.3

0.966

1.3

{MWR , MNR } = (5000, 3000) GeV

EventsεEventsεδ [%]Initial

11.8

-

11.8

-

-

Nℓ ≥ 2 with pℓT > 60 (53) GeV

3.8

0.317

3.1 ± 0.0

0.264

16.9

≥ 2 AK4 jets with pT > 40 GeV

3.7

0.987

3.1 ± 0.0

0.999

1.3

∆R > 0.4 between all pairs of objects

3.6

0.968

3.1 ± 0.0

1.000

3.4

mℓℓ > 200 GeV

3.4

0.947

3.1 ± 0.0

0.997

5.3

mℓℓjj > 800 GeV

3.4

1.000

3.1 ± 0.0

1.000

0.0

mℓℓ > 400 GeV

3.4

0.988

3.0 ± 0.0

0.974

1.4

Table 8. Cut-flow tables as predicted with MadAnalysis 5 and as reported by the CMS collaboration,
for the analysis of the eejj final state. Three scenarios are considered, with {MWR , MNR } =(3000, 1400), (4000, 2000) and (5000, 3000) GeV, for which we provide number of events normalised to
138 fb−1 and cut-by-cut efficiencies (ε) as defined in eq. (A.4). We also display the level of agreement
between the CMS and MadAnalysis 5 results, as quantified by the δ variabble introduced in eq. (A.5).

where the sum runs over all tracks separated by ∆R < 0.3 from the lepton direction. Lepton
definition then requires that Iµ < 0.1pµT , and that Ie < 5 GeV. Furthermore, all finalstate object candidates are imposed not to overlap, and to satisfy ∆R > 0.4. In addition,a good background rejection is guaranteed by a first selection on the di-lepton invariant
mass, Mℓℓ > 400 GeV, and by a second selection on the invariant mass of the reconstructed
WR -boson candidate (i.e. the invariant mass of the ℓℓjj system), Mℓℓjj > 800 GeV.

– 25 –JHEP02(2024)150{MWR , MNR } = (3000, 1400) GeVCMSMadAnalysis

EventsεEventsεδ [%]Initial

586.6

-

587.2

-

-

Nℓ ≥ 2 with pℓT > 60 (53) GeV

494.5

0.843

498.6 ± 1.9

0.849

0.7

≥ 2 AK4 jets with pT > 40 GeV

473.2

0.957

450.6 ± 2.2

0.904

5.6

∆R > 0.4 between all pairs of objects

443.1

0.936

450.6 ± 2.2

1.000

6.8

mℓℓ > 200 GeV

420.8

0.950

447.3 ± 2.2

0.993

4.5

mℓℓjj > 800 GeV

420.7

1.000

447.2 ± 2.2

1.000

0.0

mℓℓ > 400 GeV

407.3

0.968

431.6 ± 2.2

0.965

0.3

{MWR , MNR } = (4000, 2000) GeV

EventsεEventsεδ [%]Initial

57.6

-

57.6

-

-

Nℓ ≥ 2 with pℓT > 60 (53) GeV

49.8

0.864

45.2 ± 0.2

0.786

9.1

≥ 2 AK4 jets with pT > 40 GeV

48.4

0.971

41.8 ± 0.2

0.925

4.8

∆R > 0.4 between all pairs of objects

45.4

0.940

41.8 ± 0.2

1.000

6.4

mℓℓ > 200 GeV

43.5

0.957

41.7 ± 0.2

0.997

4.2

mℓℓjj > 800 GeV

43.5

1.000

41.7 ± 0.2

1.000

0.0

mℓℓ > 400 GeV

42.8

0.985

40.7 ± 0.2

0.977

0.8

{MWR , MNR } = (5000, 3000) GeV

EventsεEventsεδ [%]Initial

5.9

-

5.9

-

-

Nℓ ≥ 2 with pℓT > 60 (53) GeV

5.1

0.858

4.4 ± 0.0

0.749

12.7

≥ 2 AK4 jets with pT > 40 GeV

5.0

0.984

4.1 ± 0.0

0.923

6.2

∆R > 0.4 between all pairs of objects

4.7

0.940

4.1 ± 0.0

1.000

6.4

mℓℓ > 200 GeV

4.5

0.960

4.1 ± 0.0

0.997

3.9

mℓℓjj > 800 GeV

4.5

1.000

4.1 ± 0.0

1.000

0.0

mℓℓ > 400 GeV

4.5

0.991

4.0 ± 0.0

0.982

0.9

Table 9. Same as in table 8 but for the µµjj channel.

A comparison between the results obtained with our implementation of the CMS-EXO-20002 analysis in MadAnalysis 5 and the official results as released by the CMS collaboration
are shown in tables 8 (for the electron channel) and 9 (for the muon channel). In thesetables, we consider three scenarios defined by {MWR , MNR } = (3000, 1400), (4000, 2000) and(5000, 3000) GeV. To quantify the agreement between our predictions with MadAnalysis 5
and the official results provided by the CMS collaboration, we make use of the δ variable
introduced in eq. (A.5). An amazingly good agreement between the two is found, which
validates our implementation.

– 26 –JHEP02(2024)150{MWR , MNR } = (3000, 1400) GeV

Open Access. This article is distributed under the terms of the Creative Commons
Attribution License (CC-BY4.0), which permits any use, distribution and reproduction inany medium, provided the original author(s) and source are credited.

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