Search for new phenomena in multi-body invariant masses in events with at
least one isolated lepton and two jets using $\sqrt{s}=13$ TeV proton–proton
collision data collected by the ATLAS detector CERN-EP-2022-179 A search for
resonances in events with at least one isolated lepton ($e$ or $\mu$) and two
jets is performed using $139\leavevmode\nobreak\ {\text{fb}}^{-1}$ of
$\sqrt{s}=13$ TeV proton–proton collision data recorded by the ATLAS detector
at the LHC. Deviations from a smoothly falling background hypothesis are
tested in three- and four-body invariant mass distributions constructed from
leptons and jets, including jets identified as originating from bottom quarks.
Model-independent limits on generic resonances characterised by cascade decays
of particles leading to multiple jets and leptons in the final state are
presented. The limits are calculated using Gaussian shapes with different
widths for the invariant masses. The multi-body invariant masses are also used
to set 95% confidence level upper limits on the cross-section times branching
ratios for the production and subsequent decay of resonances predicted by
several new physics scenarios. EXOT-2020-15 JHEP

##  1 Introduction

Searches for resonances in dijet invariant mass distributions provide a means
to investigate a wide range of theories beyond the Standard Model (BSM
theories). Such searches are sensitive to heavy particles that decay into two
partons (quarks or gluons) which, following hadronisation, form jets (i.e.
dijets). Previous searches for heavy resonances in dijet mass distributions
performed by ATLAS [1, 2, 3] and CMS [4, 5, 6] have set limits on the
production of heavy resonances over a range of masses for a variety of BSM
models.

Searches in dijet mass distributions were also performed in events triggered
by leptons [7]. In contrast to the searches in inclusive dijet events, an
exclusive selection requiring events with a light lepton (electron $e$ or muon
$\mu$, referred to simply as a ‘lepton’ herein) in addition to dijets in the
final state increases the sensitivity to various exotic models. Using lepton
triggers to select these events provides a means to study low-mass regions
that are difficult to access in inclusive dijet events, where the large jet
production rate imposes high threshold requirements on jet triggers. In
addition, requiring an additional lepton reduces the QCD multi-jet background
rate and thus can be favourable for BSM model searches. A recent ATLAS search
[8] similarly looked for resonances in complementary final states with $Z\to
ll+X$ events.

The searches [7] for a heavy dijet resonance in two-jet events triggered by
leptons can be extended into a search for a resonance in a final state which
includes two jets ($jj$) and a lepton ($\ell$). Searches for narrow peaks in
three-body decays (such as invariant masses of two jets and a lepton,
$m_{jj\ell}$) are sensitive to a new class of models with cascade decays of
heavy particles which have not been studied extensively at the LHC. Such
studies in multi-body decays can be performed in exactly the same way as
searches for resonances in dijet mass ($m_{jj}$) distributions using a data-
driven technique for background determination [9]. This technique mitigates
problems arising from insufficient theoretical knowledge of the Standard Model
(SM) background in Monte Carlo (MC) simulations.

There are a number of theoretical models [10, 11, 12, 13, 14, 15] that, for a
certain choices of parameters, can exhibit resonance-like structures in multi-
body invariant masses. For example [9], the invariant mass $m_{jj\ell}$ for
decays such as $X\to Y\>W\to jj\ell\nu$ (with a $W$ boson decaying to
$\ell\nu$) can be used in searches for a heavy unknown particle $X$ in
scenarios where the mass-splitting between $X$ and a second unknown particle
$Y$ is not large (a few hundred GeV). This leads to a relatively small
transverse momentum for the neutrino. Therefore, the inclusion of missing
transverse momentum may not improve the reconstruction of $X$ masses.

More generally, a generic $s$-channel process, $X\rightarrow Y\>C$, can be
considered, where $X$ is an unknown heavy particle. The particles $Y$ and $C$
are SM or BSM particles that produce quarks, gluons, leptons or other
particles via two-body decay kinematics. As a result, an enhancement in multi-
body invariant masses is expected near the mass of the particle $X$, which can
be used in searches. However, two-body invariant masses may not be the best
tool for identifying such events if $Y$ and $C$ are broad resonances. The
experimental limitations due to large widths, $\Gamma/m>0.15$ (where $\Gamma$
is the width and $m$ is the mass of a resonance), of dijet resonances are
discussed in the previous dijet searches [1, 2, 3, 7].

This study reports model-independent searches for new phenomena in three- and
four-body invariant mass distributions in events with at least one isolated
lepton. Such searches can lead to the discovery of heavy resonances with
cascade decays in situations where two-body decays are broad, or affected by a
large SM background. Three- and four-body masses have not been explored
extensively at the LHC using data-driven techniques for background
determination. These invariant masses are also used here to study a number of
specific BSM models, such as dark-matter models with an axial-vector mediator
[10], the Sequential Standard Model with $W^{\prime}$ and $Z^{\prime}$ bosons
[11], composite resonances breaking lepton-flavour universality [13], and
radion models [14, 15]. The search uses an integrated luminosity of
$139\leavevmode\nobreak\ \text{fb}^{-1}$ of $\sqrt{s}=13$ TeV proton–proton
collision data recorded by the ATLAS detector.

##  2 ATLAS detector

The ATLAS detector [16, 17, 18] consists of an inner tracking detector (ID),
surrounded by a superconducting solenoid which provides a 2 T magnetic field,
electromagnetic and hadronic calorimeters, and a muon spectrometer (MS). The
ID provides tracking in the pseudorapidity111ATLAS uses a right-handed
coordinate system with its origin at the nominal interaction point (IP) in the
centre of the detector and the $z$-axis along the beam line. The $x$-axis
points from the IP to the centre of the LHC ring, and the $y$-axis points
upwards. Cylindrical coordinates $(r,\phi)$ are used in the transverse plane,
$\phi$ being the azimuthal angle around the beam line. Transverse momentum and
energy are defined as $p_{\text{T}}=p\sin\theta$ and
$E_{\text{T}}=E\sin\theta$, respectively. The pseudorapidity is defined in
terms of the polar angle $\theta$ as $\eta=-\ln\tan(\theta/2)$. The angular
separation between two objects in $\eta$–$\phi$ space is defined as $\Delta
R=\sqrt{(\Delta\eta)^{2}+(\Delta\phi)^{2}}$. region $|\eta|\leq 2.5$ and
consists of silicon pixel and microstrip detectors surrounded by a transition
radiation tracker, which also provides information for electron
identification. Each tracking detector consists of a central barrel and two
endcap sections.

The electromagnetic calorimeter is a sampling device made of lead absorbers
with liquid argon (LAr) as active medium. It comprises a barrel ($|\eta|\leq
1.475$) and two endcaps ($1.375\leq|\eta|\leq 3.2$). To facilitate corrections
for energy losses upstream of the calorimeter, the cryostat is equipped with a
presampler layer in the region $|\eta|\leq 1.8$. Hadronic sampling calorimetry
is provided by a steel and scintillator-tile calorimeter in the region
$|\eta|\leq 1.7$, complemented by a copper/LAr system in the region
$1.5\leq|\eta|\leq 3.2$. The forward region ($3.1\leq|\eta|\leq 4.9$) is
equipped with both electromagnetic and hadronic calorimeters composed of
copper/LAr and tungsten/LAr, respectively.

The muon spectrometer is the outermost ATLAS subsystem. It detects muons in
the pseudorapidity region up to $|\eta|=2.7$, with triggering capability up to
$|\eta|=2.4$. The spectrometer consists of a barrel ($|\eta|\leq 1.05$) and
two endcap sections ($1.05\leq|\eta|\leq 2.7$). A system of three large
superconducting air-core toroidal magnets, each with eight coils, provides a
magnetic field with a bending integral of about 2.5 Tm in the barrel and up to
6 Tm in the endcaps.

The trigger system [19] consists of a first-level trigger implemented in
hardware using a subset of the detector information to accept events from the
40 MHz bunch crossings at a rate of 100 kHz, followed by a software-based
trigger implemented in a large computer farm, which reduces the acceptance
rate so that events are recorded at about 1 kHz. An extensive software suite
[20] is used in data simulation, in the reconstruction and analysis of real
and simulated data, in detector operations, and in the trigger and data
acquisition systems of the experiment.

##  3 Monte Carlo simulations

Monte Carlo (MC) simulations of various SM background processes are used to
estimate their contributions to the multi-body mass distributions and to
validate some aspects of the analysis procedure. The considered sources of SM
background are the QCD multi-jet, $t\bar{t}$, single-top and $W$+jets
processes. The QCD multi-jet event sample was created at leading order in QCD
with parton showering using the Pythia 8 [21] generator. It uses the NNPDF2.3
[22] set of parton distribution functions (PDF) and a set of tuned parameters
called the A14 tune [23]. The $t\bar{t}$, single-top ($t$\- and $Wt$-channels)
and $W$+jets events were produced at next-to-leading order in QCD using the
Powheg Box v2 [24, 25, 26, 27] generator interfaced with Pythia 8\. This
simulation uses the CT10nlo PDF set [28] and the AZNLO tune [29].

For the model-dependent searches, MC simulations are also used to predict the
expected signal shapes for the BSM models mentioned earlier: (1)
$W^{\prime}\to WZ^{\prime}$ in the Sequential Standard Model (SSM), (2) a
simplified dark-matter (DM) model with an axial-vector mediator $Z^{\prime}$,
(3) a Kaluza–Klein (KK) gauge boson model with a SM $W$ boson and a radion
that decays into two gluons, (4) a model with composite SU(2)${}_{\text{L}}$
fermion doublets that breaks lepton-flavour universality (‘composite lepton’).
Representative Feynman diagrams for these models are shown in Figure 1.

((a))  ((b))  ((c))  ((d))

Figure 1:  Representative Feynman diagrams for the models considered in this
analysis: (a) $W^{\prime}\to Z^{\prime}\>W$ production in the SSM, (b) the
simplified DM model, (c) the radion model, (d) the composite-lepton model. The
text provides more detailed descriptions.

Simulations for the SSM [11] were performed at leading order in QCD using
Pythia 8 with the CT10nlo PDF set. Event-level simulations for all other BSM
models were performed with MadGraph5_aMC@NLO [30] at leading order in QCD. The
model assumes the heavy $Z^{\prime}$ boson decays into $q$ and $\bar{q}$,
which give rise to two high-energy jets, while the $W$ boson decays
leptonically. The $W^{\prime}\to WZ^{\prime}$ branching fraction was chosen to
be 0.5 and the mass difference between the $W^{\prime}$ and $Z^{\prime}$
bosons was set to 250 GeV. The latter setting yields the largest predicted
cross-section for the desired final state. This model is also used to estimate
systematic uncertainties for the limits when using generic signals
approximated by Gaussian functions.

A benchmark simplified DM model [10] includes an axial-vector mediator
$Z^{\prime}$ which decays as $q\bar{q}\rightarrow Z^{\prime}\rightarrow
q\bar{q}$, and one of the quarks radiates a $W$ or $Z$ boson which then decays
via $W/Z\rightarrow\ell\nu/\ell\ell$. This model assumes the leptophobic
couplings $g_{q}=0.25$, $g_{\ell}=0$, and $g_{\text{DM}}=1$, following the
recommendations of the LHC Dark Matter Working Group [10]. Here $g_{q}$,
$g_{\ell}$ and $g_{\text{DM}}$ are the couplings of the mediator to quarks,
leptons and the DM particle, respectively.

A warped-extra-dimensions model assumes KK excitations of gauge bosons and
their decay into a radion and a SM gauge boson, e.g. $q\bar{q}\rightarrow
W_{\text{KK}}\rightarrow\phi W$, followed by $W\rightarrow\ell\nu$ and the
radion $\phi$ decaying into a pair of gluons [14, 15]. A mass difference
between KK bosons and radions of 250 GeV was chosen for all samples in order
to minimise signal peak widths. A larger mass difference leads to the
production of higher-$p_{\text{T}}$ neutrinos that need to be taken into
account in the $W_{\text{KK}}$ mass reconstruction.

The composite-lepton model [13] predicts a $Z^{\prime}$-like vector boson that
decays into a new vector-like composite lepton $E$ along with a SM lepton,
$Z^{\prime}\rightarrow E\ell$. The composite lepton $E$ can then decay as
$E\rightarrow Z/h\>\ell$, with the $Z/h$ boson eventually decaying into two
jets. The two leptons produced in this process have sufficient transverse
momenta to give higher acceptances in the $m_{jj\ell\ell}$ channel.

The parton showering and hadronisation processes for all the used BSM models
were performed using Pythia 8 with the A14 tune. All SM background MC
processes were passed through the full ATLAS detector simulation [31] based on
Geant4 [32]; samples for the BSM signal processes were simulated using the
ATLAS fast simulation framework, ATLFAST-II, which uses parameterisations of
electromagnetic and hadronic showers in the calorimeters. Additional simulated
$pp$ collisions generated using Pythia 8, with the A3 set of tuned parameters
[33] and the NNPDF2.3 PDF set, were overlaid to simulate the effects of pile-
up in a manner that matches the multiplicity distribution of additional
collisions in the data. Simulated MC events are reconstructed and analysed
with the same algorithms as used for data. All simulated events are corrected
so that the object identification, reconstruction and trigger efficiencies,
energy scales and energy resolution match those determined from data control
samples. After these corrections, additional systematic uncertainties are
applied to cover differences between the data and MC simulations as described
in Section 6.

##  4 Object definitions and event selection

The analysis presented in this paper is based on data collected with the ATLAS
detector during the 2015–2018 data-taking period, referred to as Run 2. The
data sample corresponds to an integrated luminosity of $139$ fb-1 with an
uncertainty of 1.7%. The uncertainty was derived from a calibration of the
luminosity scale using $x$–$y$ beam-separation scans, following a methodology
similar to that detailed in Ref. [34] and using data from the LUCID-2 detector
[35] for the baseline measurement.

Candidate events were accepted by either single-muon or single-electron
triggers [19, 36, 37] with various thresholds for the transverse momentum
$p_{\text{T}}$ of muons or transverse energy $E_{\text{T}}$ of electrons, as
well as data quality and lepton isolation requirements. The lowest
$p_{\text{T}}$ ($E_{\text{T}}$) threshold without trigger prescaling is 24
(26) GeV and includes a lepton isolation requirement that is not applied for
triggers with higher thresholds. A trigger matching requirement [19] is
applied to ensure that the reconstructed lepton lies in the vicinity of the
corresponding trigger-level object.

Muons reconstructed from a combination of hits in the ID and MS are required
to have $p_{\text{T}}\geq 7$ GeV and $|\eta|\leq 2.5$. They must satisfy
‘medium’ quality criteria [38]. Identification requirements based on the
number of hits in the ID and MS subsystems, as well as the significance of the
difference $|q/p_{\text{MS}}-q/p_{\text{ID}}|$, where $q$ is the charge and
$p_{\text{MS}}$ ($p_{\text{ID}}$) is the momentum as measured in the MS (ID),
are applied to the combined tracks. Muon tracks are required to satisfy
$|d_{0}|/\sigma(d_{0})\leq 3$ and $|z_{0}\sin(\theta)|\leq 0.5$ mm, where
$d_{0}$ is the transverse impact parameter relative to the beam line,
$\sigma(d_{0})$ is its uncertainty, $z_{0}$ is the distance along the beam
line to the primary vertex from the point where $d_{0}$ is measured, and
$\theta$ is the polar angle of the track. The primary vertex is chosen as the
vertex with the highest $\sum{p_{\text{T}}^{2}}$, where the sum is over tracks
associated with that vertex and having $p_{\text{T}}>500$ MeV; at least two
such tracks are required.

Electrons are identified as energy clusters in the electromagnetic calorimeter
[39] matched to tracks in the ID, with requirements of $E_{\text{T}}>7$ GeV
and $|\eta|\leq 2.47$. Candidate electrons must have a track satisfying
$|d_{0}|/\sigma(d_{0})\leq 5$ and $|z_{0}\sin(\theta)|\leq 0.5$ mm and must
meet ‘tight’ quality criteria [39]. Electrons within the barrel–endcap
transition region of the electromagnetic calorimeter, $1.37\leq|\eta|\leq
1.52$, or which share a track with an identified muon, are discarded.

Leptons are also required to be isolated from other objects in the event,
using $p_{\text{T}}$-dependent criteria based on calorimeter and tracking
information. The isolation parameters of the FCTight isolation working point
[36] were tuned to provide a constant efficiency as a function of transverse
momentum, and the highest background rejection below 60 GeV. The lepton
isolation and $p_{\text{T}}$ requirements allow a consistent definition of
lepton candidates when considering data-taking periods in which different
trigger configurations were used. Lepton misidentification rates for the
chosen isolation requirements are discussed in Refs. [38, 39]. The trigger
efficiencies for electrons and muons are above 90% [40, 41].

Jets are reconstructed using the anti-$k_{t}$ algorithm [42] with a radius
parameter of $R=0.4$, as implemented in the FastJet package [43], using
particle-flow objects [44] as inputs. These jets are corrected for
contributions arising from additional collisions in the same or neighbouring
bunch crossings (pile-up) [45], and calibrated to the particle energy scale
(i.e. before interaction with the detector) [46]. Jet candidates must have
$p_{\text{T}}\geq 30$ GeV and be within $|\eta|\leq 2.4$. To suppress jets
arising from pile-up, a jet-vertex-tagging technique [47, 48] is applied,
using the "Tight" working point for central jets with $|\eta|<2.5$. Jets
originating from bottom quarks (‘$b$-jets’) are identified using the DL1r
algorithm [49, 50] with an offline working point providing 77% $b$-jet tagging
efficiency [51].

An overlap-removal procedure is applied to remove ambiguities which arise when
the same object is reconstructed by different algorithms. Candidate jets with
fewer than three associated tracks are discarded if they lie within a cone of
$\Delta R=0.2$ around a muon candidate. Jets with any number of associated
tracks are discarded if they are within $\Delta R=0.2$ of an electron.
Electron and muon candidates are discarded if they are within a cone of
$\Delta R=0.4$ around the axis of any remaining jets.

Following the lepton and jet selection, a signal region is defined by
requiring at least one isolated lepton ($e$ or $\mu$) with
$p_{\text{T}}^{\ell}\geq 60$ GeV and at least two jets. The invariant masses
are constructed by combining the two jets having the highest $p_{\text{T}}$
with one ($m_{jj\ell}$) or two ($m_{jj\ell\ell}$) leptons with the highest
$p_{\text{T}}^{\ell}$. No requirements on the charges of leptons for
$m_{jj\ell\ell}$ are applied, as this keeps the selection as model-independent
as possible. Invariant masses with at least one $b$-jet ($m_{jb\ell}$) or two
$b$-jets ($m_{bb\ell}$) are also reconstructed. These invariant masses are
sensitive to various BSM scenarios including a few supersymmetric models with
R-parity-violating interactions [52, 53].

For all distributions, only events with invariant masses above $0.4$ TeV are
considered; this minimum value is chosen in order to avoid the low-mass region
where the event rate does not decrease monotonically with increasing mass, due
to a kinematic bias in jet $p_{\text{T}}$ from the
minimum-$p_{\text{T}}^{\ell}$ requirement imposed on leptons.

##  5 Analysis procedure

A new resonant state $X$ with a mass ${m_{X}}$ can decay into partons and
leptons, either directly or via complex cascade decays. Such decays could lead
to an observable excess of events in multi-body invariant masses, such as
$m_{jj\ell}$ and $m_{jj\ell\ell}$, near ${m_{X}}$ on an otherwise smooth and
monotonically decreasing background distribution. This analysis presents a
search for such an excess in the mass range above $0.4$ TeV for $m_{jj\ell}$,
$m_{jj\ell\ell}$, $m_{jb\ell}$ and $m_{bb\ell}$. These signal regions are not
orthogonal, with $m_{jj\ell}$ being the most inclusive region, and all other
regions being subsets.

The bin widths of the multi-body distributions are chosen to be larger than
the resolution for the four-body mass $m_{jj\ell\ell}$, which is fully
dominated by the two-jet mass resolution at a given mass. The bin widths are
increased from 19 GeV to 148 GeV over the range of 0.4–8 TeV to reflect the
changes in the mass resolution. The following fit function is used to model
the shape of the estimated background [54, 3, 2, 7]:

$f(x)=p_{1}(1-x)^{p_{2}}x^{p_{3}+p_{4}\ln x+p_{5}\ln^{2}x},$ (1)

where $x\equiv m/\sqrt{s}$ and the $p_{i}$ ($i=1,\cdots 5$) are five free
parameters. In the following, this function is referred to as a $5p$ function.

According to MC simulations, about 90% of events contributing to $m_{jj\ell}$
originate from the multi-jet background where a jet is misidentified as an
electron. The study of such events is limited by the available MC statistics.
To investigate the ability of Eq. (1) to accurately describe the background in
the signal region, a likelihood fit is performed on a background-enhanced
sample obtained from a ‘loose electron’ control region (LE-CR) defined in
data. This control region is dominated by multi-jet events where jets are
misidentified as electrons. The LE-CR is populated by selecting events that
contain at least two jets and one electron, where the electron satisfies only
a set of loose identification criteria and not the more stringent
identification criteria that events must satisfy to enter the signal region,
thus ensuring orthogonality. The main purpose of this control region is to
verify that the data do not show structures that could be interpreted as
signals when using the analytic fits. The number of events in the LE-CR is a
factor of 10 smaller in the $m_{jj\ell}$ channel than in the signal region,
and a factor of 40 smaller in the $m_{jj\ell\ell}$ channel. The $5p$ fit
describes the LE-CR well, without structures in the pulls of the fit that
could be interpreted as signals.

Due to the over-representation of QCD multi-jet events in the LE-CR, $W+$jets,
$t\bar{t}$ and single-top MC events were added to the LE-CR using the
predicted cross-sections. The normalisations of the simulated distributions
were set to the expected rate in each signal region to create a hybrid control
region (MC+LE-CR). The combined contribution from $W$ boson and top quark
processes in the MC simulations varies from 10% to 95% as a function of
invariant mass and the $b$-jet selection. The fit hypothesis of Eq. (1) was
validated again for all channels on this MC+LE-CR, and was found to provide a
good description for invariant masses above 0.4 TeV, giving
$\chi^{2}/\text{ndf}\simeq 1.37,0.95,1.43,1.28$ for the $m_{jj\ell}$,
$m_{jj\ell\ell}$, $m_{jb\ell}$ and $m_{bb\ell}$ distributions respectively,
where ndf is the number of degrees of freedom. The distributions of the fit
residuals are also consistent with normal distributions with means of zero.
The fit parameters are strongly constrained by the low-mass region, below 1
TeV, which has the most events. Functions with the same form as Eq. (1) but
with fewer than five parameters fail to describe $m_{jj\ell}$ adequately in
the LE-CR and MC+LE-CR regions.

An alternative five-parameter function for the description of the background
was investigated. This alternative function, proposed in Ref. [7], differs
from Eq. (1) by replacing $p_{5}\ln^{2}x$ with $p_{5}/\sqrt{x}$. Similarly to
the nominal function, applying the alternative function to the LE-CR and
MC+LE-CR regions leads to residuals distributed according to a normal
distribution with a mean consistent with zero. Differences between fits of the
$m_{jj\ell}$ and $m_{jj\ell\ell}$ distributions with the nominal and
alternative functions were statistically insignificant. As the two functions
were statistically equivalent when applied to the MC+LE-CR, Eq. (1) was chosen
as the nominal function in order to remain consistent with Ref. [7].

To investigate potential biases in the description of the MC+LE-CR by Eq. (1),
‘signal-injection’ and ‘spurious-signal’ tests are performed. For the signal-
injection test, signal events modelled according to Gaussian distributions are
added to the expected background distribution to assess whether or not the
correct numbers of events can be extracted using signal-plus-background fits
with unconstrained fit parameters, assuming the known Gaussian signal shape.
Such tests define signal-injection efficiency correction factors to be applied
to the limits. The signal-injection tests are performed for the $m_{jj\ell}$,
$m_{jj\ell\ell}$, $m_{jb\ell}$ and $m_{bb\ell}$ distributions, assuming
injected signals with various widths from 3% to 15% of the mean mass. The
correction factors range from 0.96 to 1.0.

For the spurious-signal test, signal-plus-background fits are run on the
background-only spectra of the control region for different signal masses, and
the extracted signal yield is taken as an estimate of a potentially false
signal. The extracted signal yield in the spurious-signal tests is less than
50% of the statistical uncertainty in each mass bin. This event rate for the
signal region is considered as a source of systematic uncertainty in the
estimation of cross-section limits (discussed in Section 6).

Based on the studies of the control regions, the background-only hypothesis
for the signal region is constructed using Eq. (1), over the mass range
starting from 0.4 TeV.

To determine whether the data deviate significantly from the background-only
hypothesis, the BumpHunter (BH) algorithm [55] is used. This test calculates
the significance of any excess found in mass intervals at all possible
locations in the binned invariant mass. The width of the search window varies
from a minimum of two bins to half of the full width of the invariant mass
distribution. For each of the invariant mass intervals, a local $p$-value is
calculated from a hypothesis-test statistic. The look-elsewhere effect [56] is
taken into account by combining the separate hypothesis tests to form a new
hypothesis test, and calculating the minimum $p$-value amongst all tests. A
global $p$-value is then calculated and transformed into a significance by
assuming that bin-by-bin fluctuations of the data follow a Poisson
distribution. Pseudo-experiments are then used to determine the most
significant local excess and, finally, a global significance is calculated.
Signal-injection tests show a lower signal-extraction efficiency for wide
signals than for narrow signals.

If no statistically significant deviations are observed according to the BH
test, upper limits are set at the 95% confidence level (CL) on the production
cross-section times branching ratio using the Bayesian method [57] for generic
resonances, as well as for the benchmark BSM models discussed in Section 3.
Systematic uncertainties are included as nuisance parameters, assuming
Gaussian priors. A constant prior is used for the signal normalisation. The
$0.95$ quantile in the signal contribution is calculated; this is taken as the
upper limit on the number of possible signal events in the data, for each
resonance mass and width hypothesis. This value, divided by the integrated
luminosity, provides a measure of the upper limit on the cross-section times
acceptance times efficiency times branching ratio for a resonance with that
mass and width. This method is also used to determine the expected limits,
along with the corresponding one-standard-deviation ($1\sigma$) and two-
standard-deviation ($2\sigma$) uncertainty bands, using 100 pseudo-
experiments.

For the BSM models, the limits calculated using the reconstructed events are
corrected for the acceptance and efficiency as a function of the mass. The
acceptance is constrained by the $p_{\text{T}}$ and $\eta$ requirements placed
on the leptons and jets, and by the minimum dijet invariant mass used in this
analysis. The acceptance is typically 50% for the lowest mass point, and
increases to 60%–90% for the highest mass, depending on the model. The
efficiency correction accounts for various instrumental effects, such as the
trigger, lepton identification, and lepton reconstruction inefficiencies. A
typical efficiency, averaged over the two lepton flavours, is 65%–75%,
depending on the particle mass and the type of BSM model.

##  6 Systematic uncertainties

A number of experimental systematic uncertainties are considered for the limit
calculations. They are divided into signal-shape-changing uncertainties that
affect the limit calculations, uncertainties in the background shape, and
uncertainties in efficiencies (and acceptances) that affect the overall
normalisation of the final limits for the BSM benchmark models. The first two
categories of uncertainties are common to the limit calculations for Gaussian
signals and BSM models. Systematic uncertainties affecting the shapes of
invariant mass distributions are estimated from the BSM MC event samples. They
are included as nuisance parameters in the calculation of the Bayesian limits.
The uncertainty of 1.7% in the integrated luminosity is used for all limits.

The effects of jet energy scale (JES) and jet energy resolution (JER)
uncertainties [46] are estimated using signal model MC events. The combined
effect of all the systematic uncertainties is to raise the cross-section upper
limits relative to the limits without uncertainties. This effect has a typical
size comparable to the width of the $1\sigma$ statistical uncertainty band
around the expected limits. Uncertainties related to $b$-jet tagging are
included in the $m_{jb\ell}$ and $m_{bb\ell}$ invariant masses. The JES
uncertainty accounts for more than 50% of the total systematic uncertainty,
while the JER uncertainty makes the second-largest contribution. The effect of
lepton energy scale and resolution uncertainties is found to be negligible.
The effects of lepton trigger, identification and reconstruction
uncertainties, contributing to differences between data and MC events, are
taken into account by assigning a combined uncertainty of 1%.

For the generic Gaussian signals, JES and JER systematic uncertainties from
the $W^{\prime}\to WZ^{\prime}\to\ell\nu\>q\bar{q}$ MC events are used,
parameterised as a function of mass to generate uncertainties for mass values
that were not used in generating the $W^{\prime}\to WZ^{\prime}$ samples. As a
cross-check, other models predicting different signal widths are also used,
but no significant differences are observed.

The uncertainties arising from imperfect knowledge of the background shapes
are investigated by using the alternative background fit and the spurious-
signal test (discussed previously) using the nominal $5p$ fit to the data. The
uncertainty due to the choice of background function is estimated by
performing the background fit with the alternative fit function. The
difference between the results of this fit and the nominal one, averaged
across a set of pseudo-data, is considered as a systematic uncertainty. Both
uncertainties are found to have a negligible effect on the limits.

Uncertainties in the efficiency and acceptance correction factors related to
JES, JER, $b$-jet and lepton identification (and reconstruction) are found to
be smaller than $1\%$. An additional uncertainty in the limits is associated
with the PDF choice for the acceptance calculation [58]. This is accounted for
by assigning a 1% PDF uncertainty to the calculated limits.

##  7 Results

Figures 2, 3 and 4 show the invariant mass distributions discussed above,
together with the results of the likelihood fits using the $5p$ function from
Eq. (1). The quality of the fit in terms of the $\chi^{2}/$ndf is indicated.
In order to establish the presence or absence of a signal in the $m_{jj\ell}$
distribution, the BH algorithm is used. It is also used to calculate local and
global significances of the largest deviation from the background hypothesis.

Figure 2: Invariant $2j+\ell$ mass distribution for $m_{jj\ell}>0.4$ TeV
together with the fit with the $5p$ background function. The lower panel shows
the bin-by-bin significances of deviations from the background hypothesis. The
largest deviation reported by the BumpHunter algorithm is shown by the
vertical dashed lines. The global $p$-value of this deviation in the range
1.27–1.38 TeV is indicated. Figure 3: Invariant $2j+2\ell$ mass distribution
for $m_{jj\ell\ell}>0.4$ TeV together with the fit with the $5p$ background
function. The lower panel shows the bin-by-bin significances of deviations
from the background hypothesis. The largest deviation reported by the
BumpHunter algorithm is shown by the vertical dashed lines. The global
$p$-value of this deviation in the range 3.59–4.41 TeV is indicated.

((a)) $m_{jb\ell}$ invariant mass ((b)) $m_{bb\ell}$ invariant mass

Figure 4: Invariant mass distributions (a) $m_{jb\ell}$ and (b) $m_{bb\ell}$
with the $5p$ background fits. The lower panels show the bin-by-bin
significances of deviations from the background hypothesis. The largest
deviations reported by the BumpHunter algorithm are indicated by the vertical
dashed lines. The global $p$-values of these deviations in the range 4.78–5.07
TeV (for $m_{jb\ell}$) and 5.07–5.27 TeV (for $m_{bb\ell}$) are indicated.

Figure 2 shows the invariant mass distribution $m_{jj\ell}$ obtained from the
selected $2j+\ell$ events. The data are described by the $5p$ fit function,
with $\chi^{2}=149$ and $\text{ndf}=102$. The lower panel shows the
significances [59] of deviations from the background hypothesis, which can be
approximated by $(d_{i}-f_{i})/\delta_{i}$, where $d_{i}$ is the number of
events in the bin $i$, $f_{i}$ is the fit value, and $\delta_{i}$ is an
uncertainty. This uncertainty includes the statistical and systematic
uncertainties of the fit result $f_{i}$.

The largest deviation of the $m_{jj\ell}$ data from the background-only
hypothesis reported by the BH is near 1.3 TeV, with a local $p$-value of
$2\times 10^{-5}$, corresponding to a significance of $3.5$ standard
deviations. Accounting for the look-elsewhere effect for the $m_{jj\ell}$
distribution, the global $p$-value for the largest deviation using the nominal
fit is $0.06$, leading to a global significance of $1.5\sigma$. When using the
alternative background fit, this deviation has a lower local significance of
$2.8\sigma$. This deviation from the background hypothesis is thus consistent
with a statistical fluctuation.

The deviation near 1.3 TeV is equally present in the electron and muon
channels, but its statistical significance is smaller than for the combined
channel. For the separate channels, the BH reports that the largest deviations
are in the region near 5 TeV. The global $p$-values for the deviations near 5
TeV are 0.13 (for muons) and 0.38 (for electrons). Thus, such deviations from
the $5p$ background are also consistent with statistical fluctuations.

Figure 3 shows the $m_{jj\ell\ell}$ mass distribution for the combined muon
and electron channels. The global $p$-value for the largest deviation, near 4
TeV, is $0.69$, indicating no significant deviations from the $5p$ background.
The separate muon and electron channels do not show significant deviations.

Figure 4 shows the distributions for three-body masses with one or two
$b$-jets. The global $p$-values for the largest deviations are larger than
0.6. Thus the distributions do not show significant deviations from the
background.

###  7.1 Model-independent limits

In the absence of any statistically significant deviations from the $5p$
background function, the Bayesian method is used to set 95% CL upper limits on
the cross-section for new processes that have a signature of a new particle
with decays producing two jets and leptons. The limits are obtained for
hypothetical signals with a Gaussian shape.222Using the Gaussian limits
requires some care in treating the realistic shapes from specific models. For
example, the appendix of Ref. [60] gives a recipe for how the Gaussian limits
should be used for dijet studies.

Figure 5:  The 95% CL observed upper limits, in the $m_{jj\ell}$ channel, on
the cross-section times acceptance ($A$), efficiency ($\epsilon$) and
branching ratio ($B$) for a Gaussian signal with various signal widths. The
expected limit and the corresponding $\pm 1\sigma$ and $\pm 2\sigma$ bands are
shown for ${\sigma_{X}}/{m_{X}}=0$ signals. The limit for a width of 15% is
shown starting from 0.75 TeV in order to avoid the region close to the minimum
value of the $m_{jj\ell}$ distribution.  Figure 6:  The 95% CL observed upper
limits, in the $m_{jj\ell\ell}$ channel, on the cross-section times acceptance
($A$), efficiency ($\epsilon$) and branching ratio ($B$) for a Gaussian signal
with various signal widths. The expected limit and the corresponding $\pm
1\sigma$ and $\pm 2\sigma$ bands are shown for ${\sigma_{X}}/{m_{X}}=0$
signals. The limit for a width of 15% is shown starting from 0.75 TeV in order
to avoid the region close to the minimum value of the $m_{jj\ell\ell}$
distribution.

Figure 5 shows the $95\%$ CL expected and observed limits on the cross-section
times acceptance ($A$), signal efficiency ($\epsilon$) and branching ratio
($B$) of a BSM particle decaying into two jets and a lepton. The observed
limits are shown for different widths, ranging from a narrow width –
comparable to the experimental resolution of about 3% – up to 15% of the mean
mass. The expected limit and the corresponding $\pm 1\sigma$ and $\pm 2\sigma$
bands are shown for narrow width (${\sigma_{X}}/{m_{X}}=0$) resonances. As
expected, the limits are weaker for broader resonances that span more
$m_{jj\ell}$ bins. Signals making Gaussian contributions to the $m_{jj\ell}$
distribution are excluded by the obtained limits as a function of the mean
mass. The expected limit and the corresponding $\pm 1\sigma$ and $\pm 2\sigma$
bands are shown for the narrowest Gaussian width. The observed exclusion
limits range from $20$ fb to $0.1$ fb in the mass range 0.5–8 TeV.

Figure 6 shows the observed limits for four-body invariant masses,
$m_{jj\ell\ell}$. The limits are obtained for Gaussian signals with different
widths, from a narrow width – compatible with the experimental resolution – up
to 15% of the mean mass. Figure 7(a) shows similar Gaussian limits for the
$m_{jb\ell}$ distribution. Figure 7(b) shows the observed limits for the
$m_{bb\ell}$ invariant mass. The obtained limits range from about 10 fb to 0.1
fb for resonance masses between 0.5 TeV and 4–5 TeV.

((a)) Limits for $m_{jb\ell}$ ((b)) Limits for $m_{bb\ell}$

Figure 7:  The 95% CL observed upper limits, in the $m_{jb\ell}$ and
$m_{bb\ell}$ channels, on the cross-section times acceptance ($A$), efficiency
($\epsilon$) and branching ratio ($B$) for a Gaussian signal with various
signal widths. The expected limits and the corresponding $\pm 1\sigma$ and
$\pm 2\sigma$ bands are shown for ${\sigma_{X}}/{m_{X}}=0$ signals. The limits
for a width of 15% are shown starting from 0.75 TeV in order to avoid the
region close to the minimum value of the invariant mass distributions.

###  7.2 Limits on BSM models

To calculate the exclusion limits for the benchmark models, the exact shapes
of the $m_{jj\ell}$ and $m_{jj\ell\ell}$ distributions of the signal MC events
after simulation and reconstruction are used. The Bayesian limits are
calculated using the $m_{jj\ell}$ and $m_{jj\ell\ell}$ distributions with the
background description discussed earlier, and then the acceptance and
efficiency corrections are applied. The limits on the composite-lepton model
are calculated using $m_{jj\ell\ell}$ in the muon channel for two masses, 250
GeV and 500 GeV, of a composite lepton $E$. The muon channel was chosen for
this model as the $Z^{\prime}$ preferentially couples to muons over electrons.
The calculated limits include the systematic uncertainties discussed in
Section 6.

Figures 8 and 9 show the observed (filled circles) and expected (dotted line
with uncertainty bands) 95% CL upper limits on the cross-section times
branching ratios. These results exclude contributions from the $W^{\prime}\to
WZ^{\prime}$ process in the SSM for masses of the $W^{\prime}$ boson (decaying
into jet pairs) below 2.5 TeV and a mass difference of 250 GeV between the
$W^{\prime}$ and $Z^{\prime}$ bosons, which maximises the cross-section for
this process. This is consistent with the previous result for the same model
using dijet events, where a $Z^{\prime}$ boson (decaying into jet pairs) with
a mass below 2 TeV was excluded [7]. For the radion model, in which the mass-
splitting between a radion $\phi$ and $W_{\text{KK}}$ is small (250 GeV), the
data exclude $\phi$ with a mass below 1 TeV, as shown in Figure 8(b). For the
studied mass range, the current analysis cannot exclude the DM model with a
$Z^{\prime}$ axial-vector mediator, as seen in Figure 8(c). Figure 9 shows the
limits for the composite-lepton model with a $Z^{\prime}$ boson. The data
exclude a $Z^{\prime}$ boson with a mass below 1.3 TeV for a composite lepton
mass of 500 GeV.

((a)) SSM model ((b)) $W_{\text{KK}}$ \+ $\phi$ model ((c)) $Z^{\prime}W$ (DM)
model

Figure 8:  Observed (filled circles) and expected (dotted line with
uncertainty bands) 95% CL upper limits on the cross-section ($\sigma$) times
branching ratio ($B$) for (a) the SSM $W^{\prime}$ boson, (b) the
$W_{\text{KK}}+\phi$ radion model with $m_{W_{\text{KK}}}-m_{\phi}=250$ GeV,
and (c) the $Z^{\prime}W$ simplified DM model. The presented limits were
obtained using the $m_{jj\ell}$ ($\ell=\mu,e$) distribution in events with at
least one isolated lepton with $p_{\text{T}}^{\ell}>60$ GeV. Branching ratios
of 100% to the final states of interest were assumed in the signal generation
and computation of theory cross-sections. The black lines shown for the
observed limits are provided to guide the eye.

((a)) Composite-lepton model with $m_{E}=250$ GeV ((b)) Composite-lepton model
with $m_{E}=500$ GeV

Figure 9:  Observed (filled circles) and expected (dotted line with
uncertainty bands) 95% CL upper limits on the cross-section ($\sigma$) times
branching ratio ($B$) for $Z^{\prime}\to E\mu$ where the composite lepton $E$
has a mass of (a) $250$ GeV or (b) $500$ GeV. The presented limits were
obtained using the $m_{jj\ell\ell}$ ($\ell=\mu$) distribution in events with
at least one isolated muon with $p_{\text{T}}^{\ell}>60$ GeV. Branching ratios
of 100% to the final states of interest were assumed in the signal generation
and computation of theory cross-sections. The black lines shown for the
observed limits are provided to guide the eye.

##  8 Conclusion

A model-independent search for new resonances in invariant masses constructed
from jets and leptons with $p_{\text{T}}^{\ell}>60$ GeV using an integrated
luminosity of 139 fb-1 of proton–proton collisions with a centre-of-mass
energy of $\sqrt{s}=13$ TeV recorded by the ATLAS detector at the LHC is
presented. Several three- and four-body invariant masses were studied:
$m_{jj\ell}$, $m_{jj\ell\ell}$, $m_{jb\ell}$ and $m_{bb\ell}$ in the mass
range from 0.4 to 8 TeV. For the first time, searches in these invariant
masses are performed with a data-driven technique, assuming a smoothly falling
background described by a fit function, without relying on statistically
limited MC predictions.

The multi-body invariant masses do not exhibit significant excesses above a
data-derived estimate of the smoothly falling background. The most significant
deviation from the data-derived estimate of the SM background in the combined
electron and muon channel is observed around $m_{jj\ell}=1.3$ TeV. This excess
has a local statistical significance of 3.5 standard deviations. After
accounting for the look-elsewhere effect, this deviation corresponds to a
global statistical significance of $1.5\sigma$. A check with an alternative
description leads to a lower local significance of 2.8$\sigma$. Other multi-
body invariant-mass combinations show smaller local deviations. Thus the data
are consistent with the background-only hypothesis.

In the absence of signals indicating the presence of new phenomena, this
analysis sets 95% CL upper limits on the cross-section times acceptance,
efficiency and branching fraction for new processes that can produce a
Gaussian contribution to the invariant mass distributions of jets and leptons.
The limits are calculated for a Gaussian width ranging from that determined by
the detector resolution up to 15% of the resonance mass. These results exclude
generic BSM models predicting resonances that decay to produce jets and
leptons with cross-sections larger than the limits reported in this paper.
This study extends the search range by 2 TeV beyond that of previous searches
which did not include leptons in the invariant masses.

Model-dependent limits are also set on a variety of BSM models, without the
use of additional selection criteria tailored to the specific final states
investigated. These results exclude contributions from the $W^{\prime}\to
WZ^{\prime}$ process in the SSM for masses of a $W^{\prime}$ boson below 2.5
TeV, assuming the mass difference between the $W^{\prime}$ and $Z^{\prime}$
bosons is 250 GeV to maximise the cross-section for this process. For the
radion model, in which the mass-splitting between a radion $\phi$ and
$W_{\text{KK}}$ is small (250 GeV), the data exclude $\phi$ at masses below 1
TeV. For the composite-lepton model with a $Z^{\prime}$ boson, the data
exclude a $Z^{\prime}$ boson with a mass below 1.3 TeV for a composite lepton
mass of 500 GeV.

Altarelli:1989ff,Altarelli:1989ff:erratum AltarelliMele AltarelliMele

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