Search for pair-produced scalar and vector leptoquarks decaying into third-
generation quarks and first- or second-generation leptons in $pp$ collisions
with the ATLAS detector A search for pair-produced scalar and vector
leptoquarks decaying into quarks and leptons of different generations is
presented. It uses the full LHC Run 2 (2015–2018) data set of
$139\text{\,}\mbox{fb${}^{-1}$}$ collected with the ATLAS detector in
proton–proton collisions at a centre-of-mass energy of
$\sqrt{s}=$13\text{\,}\text{Te\kern-1.00006ptV}$$. Scalar leptoquarks with
charge $-(1/3)e$ as well as scalar and vector leptoquarks with charge
$+(2/3)e$ are considered. All possible decays of the pair-produced leptoquarks
into quarks of the third generation ($t,b$) and charged or neutral leptons of
the first or second generation ($e,\mu,\nu$) with exactly one electron or muon
in the final state are investigated. No significant deviations from the
Standard Model expectation are observed. Upper limits on the production cross-
section are provided for eight models as a function of the leptoquark mass and
the branching ratio of the leptoquark into the charged or neutral lepton. In
addition, lower limits on the leptoquark masses are derived for all models
across a range of branching ratios. Two of these models have the goal of
providing an explanation for the recent $B$-anomalies. In both models, a
vector leptoquark decays into charged and neutral leptons of the second
generation with a similar branching fraction. Lower limits of 1980 GeV and
1710 GeV are set on the leptoquark mass for these two models. EXOT-2019-12
JHEP CERN-EP-2022-145

###### Contents

  1. 1 Introduction
  2. 2 ATLAS detector
  3. 3 Data and simulated event samples
  4. 4 Event reconstruction
  5. 5 Event selection and categorisation
  6. 6 Neural network training
  7. 7 Systematic uncertainties
  8. 8 Statistical interpretation
  9. 9 Results
  10. 10 Conclusion

##  1 Introduction

Recent hints of a potential violation of lepton flavour universality in
various measurements of $B$-meson decays (‘$B$-anomalies’) [1, 2, 3, 4, 5, 6,
7, 8, 9] can be attributed, if confirmed, to the exchange of leptoquarks (LQs)
[10, 11, 12, 13, 14, 15, 16, 17, 18]. In addition, some of these models
introducing LQs aim to simultaneously provide an explanation for the
longstanding discrepancy between the measured and the predicted anomalous
magnetic moment of the muon [19]. LQs have already been discussed for a few
decades, as they are predicted by many extensions of the Standard Model (SM),
e.g. in unified theories [20, 21, 22] and technicolor [23, 24, 25] or
composite models [26, 27, 28]. LQs provide a connection between the quark and
lepton sectors, which exhibit similar structures. They are bosons carrying
colour charge and fractional electrical charge. LQs possess non-zero baryon
and lepton numbers and are assumed to decay into a quark–lepton pair. The
branching ratio into a quark and a charged lepton is denoted by $\mathcal{B}$,
and that into a quark and neutrino by $1-\mathcal{B}$. Leptoquarks can be
scalar or vector bosons and can be produced singly or in pairs in
proton–proton collisions.

The assumption that LQs can only interact with leptons and quarks of the same
generation [29], and are spin-0 particles, has been used in most of the
searches for LQs. Recently, however, searches for LQs with couplings to quarks
and leptons of different generations have aroused interest because these
couplings are required in order to explain the $B$-anomalies mentioned above.
In particular, the theoretical explanations usually require LQs with couplings
to third-generation quarks and second-generation leptons. Similarly, vector
LQs with a charge of 2/3, in units of the elementary charge $e$, have been
identified recently as promising candidates [30, 31].

First results of searches for LQs with couplings to quarks and leptons of
different generations were published recently for pair-produced LQs decaying
into charged leptons, i.e. in dilepton final states. Results from the full LHC
Run 2 data set (2015–2018) of proton–proton collisions at
$\sqrt{s}=$13\text{\,}\mathrm{TeV}$$ collected by ATLAS, corresponding to an
integrated luminosity of $139\text{\,}{\mathrm{fb}}^{-1}$, are available for
pair-produced scalar LQs decaying into a top-quark and an electron or muon
[32] and for the decay into a $b$-, $c$\- or light-quark and an electron or
muon [33]. The CMS Collaboration has published a search for pair-produced LQs
decaying into a top-quark and a muon using a partial
($35.9\text{\,}{\mathrm{fb}}^{-1}$) Run 2 data set [34] where, in addition,
results for vector LQs were derived by scaling the results for scalar LQs to
the larger production cross-sections expected for vector LQs, assuming no
kinematic differences.

Given the good coverage of large branching ratios $\mathcal{B}$ by the
dileptonic measurements described above, the results presented here use a
single-lepton (electron or muon) final state optimised for medium to small
$\mathcal{B}$. In this case, one of the LQs decays into a neutrino and the
other decays into a charged lepton, or both decay into neutrinos and the
charged lepton arises from a leptonically decaying top-quark
($t_{\text{lep}}$). The results are interpreted as searches for pair-produced
LQs with charges of either $\pm(2/3)e$ (up-type) or $\pm(1/3)e$ (down-type).
All possible decays of the pair-produced up-type and down-type LQs into a
quark ($t$, $b$) of the third generation and a lepton ($\ell$, $\nu$) of the
first or second generation are considered, as seen in Figure 1. With flavour
off-diagonal couplings allowed, the model used for up-type
($\mathrm{LQ}_{\text{mix}}^{\textnormal{u}}$) and down-type
($\mathrm{LQ}_{\text{mix}}^{\textnormal{d}}$) scalar LQs is an extension of
that [35] used in previous ATLAS searches [36], where all possible decays of
the pair-produced up-type and down-type scalar LQs into a quark ($t$, $b$) and
a lepton ($\tau$, $\nu$) of the third generation were considered. The present
search for up-type LQs is also optimised for a vector LQ (vLQ) model [37]
designed to provide an explanation for the various $B$-anomalies.

(a)

(b)

Figure 1:  Pair production and decay of 1(a) up-type scalar
($\mathrm{LQ}_{\text{mix}}^{\textnormal{u}}$) and vector
($\mathrm{vLQ}_{\text{mix}}$) LQs and 1(b) down-type scalar
($\mathrm{LQ}_{\text{mix}}^{\textnormal{d}}$) LQs with $\ell=e,\mu$. No
distinction is made between particles and antiparticles.

The analysis strategy is based on a final-state signature with one lepton,
high missing transverse momentum and at least four jets due to a hadronically
decaying top-quark ($t_{\text{had}}$) and a $b\text{-quark}$. Dedicated neural
networks (NNs), trained in a common training region, are used for the
separation of signal and background. This is done separately for scalar and
vector LQ pair-production because they exhibit different kinematic behaviour
for small values of $\mathcal{B}$, i.e. when the charged lepton arises mostly
from the top-quark decay. For each of the models and for various branching
ratios, a signal region (SR) based on the NN output is defined. Control
regions (CRs) are defined so as to be enriched in the various background
processes. They are orthogonal to the SR, and orthogonal to each other. The
statistical interpretation is based on a simultaneous fit to the CRs and the
SR, in which the background normalisations and a possible signal contribution
are determined, while taking into account the experimental and theoretical
systematic uncertainties. The results are presented as limits on the
leptoquark mass as a function of the branching ratio.

##  2 ATLAS detector

The ATLAS experiment [38] at the LHC is a multipurpose particle detector with
a forward–backward symmetric cylindrical geometry and a near $4\pi$ coverage
in solid angle.111ATLAS 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 pipe. 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 $z$-axis. The pseudorapidity is defined in terms of the polar angle
$\theta$ as $\eta=-\ln\tan(\theta/2)$. Angular distance is measured in units
of $\Delta R\equiv\sqrt{(\Delta\eta)^{2}+(\Delta\phi)^{2}}$. It consists of an
inner tracking detector surrounded by a thin superconducting solenoid
providing a $2\text{\,}\mathrm{T}$ axial magnetic field, electromagnetic and
hadron calorimeters, and a muon spectrometer. The inner tracking detector (ID)
covers the pseudorapidity range $|\eta|<2.5$. It consists of silicon pixel,
silicon microstrip, and transition radiation tracking detectors. Lead/liquid-
argon (LAr) sampling calorimeters provide electromagnetic (EM) energy
measurements with high granularity. A steel/scintillator-tile hadron
calorimeter covers the central pseudorapidity range ($|\eta|<1.7$). The endcap
and forward regions are instrumented with LAr calorimeters for both the EM and
hadronic energy measurements up to $|\eta|=4.9$. The muon spectrometer (MS)
surrounds the calorimeters and is based on three large superconducting air-
core toroidal magnets with eight coils each. The field integral of the toroids
ranges between $2.0$ and $6.0\text{\,}\text{Tm}$ across most of the detector.
The muon spectrometer includes a system of precision tracking chambers and
fast detectors for triggering. A two-level trigger system is used to select
events. The first-level trigger is implemented in hardware and uses a subset
of the detector information to accept events at a rate below
$100\text{\,}\mathrm{kHz}$. This is followed by a software-based trigger that
reduces the accepted event rate to $1\text{\,}\mathrm{kHz}$ on average
depending on the data-taking conditions. An extensive software suite [39] is
used 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 Data and simulated event samples

In this search, data from proton–proton collisions at
$\sqrt{s}=$13\text{\,}\mathrm{TeV}$$ corresponding to an integrated luminosity
of $139\text{\,}{\mathrm{fb}}^{-1}$, collected in the years 2015 to 2018 with
the ATLAS detector, are analysed. Data are required to have been collected
during stable beam conditions and with all detector subsystems operational
[40]. The average number of simultaneous $pp$ interactions per bunch crossing,
referred to as pile-up, is approximately 34, averaged over the whole data set.

Monte Carlo (MC) simulated event samples are used to model the signal and
background processes. In all samples except those produced with Sherpa 2.2.1
or Sherpa 2.2.2 [41], decays of heavy-flavour hadrons were modelled with
EvtGen 1.2.0 or EvtGen 1.6.0 [42], depending on the process. Pile-up was
modelled by overlaying minimum-bias events generated with Pythia 8.186 [43]
and the A3 [44] set of tuned parameters (referred to as the ‘tune’) onto the
simulated hard-scatter events. A reweighting procedure was applied in order to
match the pile-up profile of the recorded data. The ATLAS simulation
infrastructure [45] was used to simulate the detector and its response.
Nominal SM background samples were produced with a detailed Geant4 [46]
detector simulation, whereas a faster calorimeter simulation [45] was applied
for the signal samples and systematic variations of the backgrounds. The same
offline reconstruction methods used for data were applied to the simulated
samples. Corrections were applied to the simulated events in order to match
the selection efficiencies and energy and mass scales and resolutions of
reconstructed simulated particles to those measured in data control samples.

Simulated events with pair-produced scalar LQs were generated at next-to-
leading order (NLO) in quantum chromodynamics (QCD) with MadGraph5_aMC@NLO
2.6.0 [47] and the NNPDF3.0nlo parton distribution function (PDF) [48] set
with $\alpha_{\text{s}}=0.118$. An extension of the LQ model of Ref. [35] was
used, allowing flavour off-diagonal couplings. The model is based on previous
fixed-order NLO QCD calculations [49, 50]. To retain information about spin
correlations, the decays of LQs as well as top-quarks were handled with
MadSpin [51]. MadGraph5_aMC@NLO was interfaced with Pythia 8.230 [52] to model
the parton shower (PS), hadronisation, and underlying event (UE). Here and in
the following, Pythia was used with the A14 tune [53] and the NNPDF2.3lo [54]
set of PDFs. The coupling strength $\lambda$ was set to 0.3. The model
parameter $\beta\in[0,1]$ modifies the coupling of LQs to leptons, such that
the coupling to charged leptons is given by $\sqrt{\beta}\lambda$ and to
neutrinos by $\sqrt{1-\beta}\lambda$. It differs from the branching fraction
$\mathcal{B}$ into charged leptons because of phase-space corrections arising
mainly from the large top-quark mass. The parameter $\beta$ was set to 0.5 in
the simulation. Different values for $\mathcal{B}$ were achieved by
reweighting the MC events according to their decay, as described in Ref. [36].
The LQ pair-production cross-sections were obtained from the calculation of
direct top-squark pair production, as this process has the same production
modes, computed at approximate next-to-next-to-leading order (NNLO) in QCD
with resummation of next-to-next-to-leading logarithmic (NNLL) soft gluon
terms [55, 56, 57, 58]. The cross-sections do not include lepton $t$-channel
contributions, which are neglected in Ref. [35] and may lead to corrections at
the percent level [59]. For this analysis, signal samples were produced
separately for electrons and muons and for both up- and down-type LQs with a
mass spacing of $100\text{\,}\mathrm{GeV}$ from $300\text{\,}\mathrm{GeV}$ to
$800\text{\,}\mathrm{GeV}$ and from $1600\text{\,}\mathrm{GeV}$ to
$2500\text{\,}\mathrm{GeV}$, and with a finer spacing of
$50\text{\,}\mathrm{GeV}$ between $800\text{\,}\mathrm{GeV}$ and
$1600\text{\,}\mathrm{GeV}$ to improve the resolution around the expected mass
exclusion limit.

Simulated events with pair-produced up-type vector LQs were generated at
leading order (LO) in QCD with MadGraph5_aMC@NLO 2.8.1 and the NNPDF3.0nlo PDF
set with $\alpha_{\text{s}}=0.118$. Decays of the vLQs and top-quarks were
handled with MadSpin, while the PS and hadronisation were simulated with
Pythia 8.244. The $U_{1}$ vLQ model in Ref. [37] is used for the muon channel,
directly seeking an explanation for the various $B$-anomalies. Couplings to
electrons are assumed to vanish in this model because of existing tight bounds
from low-energy observables, mainly in the lepton-flavour-violating sector.
Nevertheless, an extension of the model [60] was used in this analysis to also
probe the electron channel. The samples were produced with a coupling strength
of $g_{U}=3.0$. The large value of $g_{U}$ is motivated by a suppression of
the production cross-section for additional mediators in an ultraviolet-
complete model, which might otherwise be in tension with existing LHC limits.
The model accommodates both left- and right-handed couplings to fermions,
where in the case of only left-handed couplings the coupling strengths to
charged leptons and neutrinos are equal ($\beta=0.5$). Although only left-
handed couplings were used in producing the MC samples, the analysis uses the
same reweighting of MC events as in the scalar LQ case to also probe different
values of $\mathcal{B}$. The model allows use of either the minimal
($\mathrm{vLQ}_{\text{mix}}^{\text{min}}$) or the Yang–Mills
($\mathrm{vLQ}_{\text{mix}}^{\text{YM}}$) coupling scenario, where in the
latter the vLQ is a heavy gauge boson resulting in enhanced cross-sections.
Kinematic differences between the minimal and the Yang–Mills coupling scenario
were found to be negligible, except for masses below about
$500\text{\,}\mathrm{GeV}$. Samples were produced separately for the two
coupling scenarios and for both muons and electrons, with a mass spacing of
$100\text{\,}\mathrm{GeV}$ from $300\text{\,}\mathrm{GeV}$ to
$1400\text{\,}\mathrm{GeV}$ and from $2300\text{\,}\mathrm{GeV}$ to
$2500\text{\,}\mathrm{GeV}$, and with a finer spacing of
$50\text{\,}\mathrm{GeV}$ between $1400\text{\,}\mathrm{GeV}$ and
$2300\text{\,}\mathrm{GeV}$. No higher-order cross-section computations are
available for this model. Therefore, the cross-sections computed at leading
order by MadGraph5_aMC@NLO are used in the analysis.

Dominant background processes in the search include top-quark pair
($t\bar{t}$), $W\text{+jets}$ and single top-quark production, the last being
mainly associated production of a top-quark and $W$ boson ($tW$). In addition,
$t\bar{t}{+}V$ ($V=W,Z$), diboson, $t\bar{t}{+}H$, and $Z\text{+jets}$
processes are also considered in the analysis. Contributions from multi-jet
background with a jet misidentified as a lepton are negligible in the phase
space of interest.

The production of $t\bar{t}$ events was modelled using the Powheg Box [61, 62,
63, 64] v2 generator at NLO with the NNPDF3.0nlo set of PDFs and the
$h_{\mathrm{damp}}$ parameter222The $h_{\mathrm{damp}}$ parameter is a
resummation damping factor and one of the parameters that controls the
matching of Powheg matrix elements to the PS and thus effectively regulates
the high-$p_{\text{T}}$ radiation against which the $t\bar{t}$ system recoils.
set to $1.5\,m_{t}$ [65]. The cross-section was corrected to the theory
prediction at NNLO including resummation of NNLL soft-gluon terms calculated
using Top++ 2.0 [66]. The events were interfaced to Pythia 8.230 to model the
PS, hadronisation, and UE.

The associated production of a top-quark and a $W$ boson was modelled using
the Powheg Box v2 generator at NLO in QCD using the five-flavour scheme and
the NNPDF3.0nlo set of PDFs. The diagram removal scheme [67] was used to avoid
overlap with $t\bar{t}$ production because of interference. Single-top
$t$-channel ($s$-channel) production was modelled using the Powheg Box v2
generator at NLO in QCD using the four-flavour (five-flavour) scheme and the
corresponding NNPDF3.0nlo PDF set. The events were interfaced with Pythia
8.230 in all cases, except for $tW$ events with large missing transverse
momenta, which were interfaced with Pythia 8.235.

The production of $V\text{+jets}$ was simulated with the Sherpa 2.2.1
generator using NLO-accurate matrix elements for up to two jets, and LO-
accurate matrix elements for up to four jets, calculated with the Comix [68]
and OpenLoops [69, 70] libraries. They were matched with the Sherpa PS [71]
using the MEPS@NLO prescription [72, 73, 74, 75] using the set of tuned
parameters developed by the Sherpa authors. The NNPDF3.0nnlo set of PDFs was
used and the samples were normalised to a NNLO prediction [76].

Samples of diboson final states ($VV$) were simulated with the Sherpa 2.2.1 or
2.2.2 generator, depending on the process, including off-shell effects and
Higgs boson contributions, where appropriate. Fully leptonic final states and
semileptonic final states, where one boson decays leptonically and the other
hadronically, were generated using matrix elements at NLO accuracy in QCD for
up to one additional parton and at LO accuracy for up to three additional
parton emissions. The matrix element calculations were matched and merged with
the Sherpa PS. The NNPDF3.0nnlo set of PDFs was used, along with the internal
Sherpa tune.

The production of $t\bar{t}{+}W$ and $t\bar{t}{+}Z$ events was modelled using
the MadGraph5_aMC@NLO 2.3.3 generator at NLO with the NNPDF3.0nlo PDF set. The
events were interfaced to Pythia 8.210.

The production of $t\bar{t}{+}H$ events was modelled using the Powheg Box v2
generator at NLO with the NNPDF3.0nlo PDF set, interfaced to Pythia 8.230.

An overview of the matrix element (ME) generator, PDF, shower generator and UE
tune for signal and background samples is given in Table 1.

Table 1: List of ME generator and the order of the strong coupling constant in
the perturbative calculation, PDF, shower generator and tune for the different
signal and background processes.

Process | ME generator | ME order | PDF set | PS and hadronisation | UE tune
---|---|---|---|---|---
Scalar LQ |  MadGraph5_aMC@NLO 2.6.0 | NLO | NNPDF3.0nlo |  Pythia 8.230 | A14
Vector LQ |  MadGraph5_aMC@NLO 2.8.1 | LO | NNPDF3.0nlo |  Pythia 8.244 | A14
$t\bar{t}$/single top |  Powheg Box v2 | NLO | NNPDF3.0nlo |  Pythia 8.230/8.235 | A14
$V\text{+jets}$ |  Sherpa 2.2.1 | MEPS@NLO | NNPDF3.0nnlo | Sherpa | internal
Diboson |  Sherpa 2.2.1/2.2.2 | MEPS@NLO | NNPDF3.0nnlo | Sherpa | internal
$t\bar{t}{+}V$ |  MadGraph5_aMC@NLO 2.3.3 | NLO | NNPDF3.0nlo |  Pythia 8.210 | A14
$t\bar{t}{+}H$ |  Powheg Box v2 | NLO | NNPDF3.0nlo |  Pythia 8.230 | A14

##  4 Event reconstruction

Events studied in this analysis are required to have at least one
reconstructed $pp$ interaction vertex with at least two associated tracks with
transverse momentum $p_{\text{T}}>$0.5\text{\,}\mathrm{GeV}$$. The primary
vertex is selected as the one with the largest sum of squared transverse
momenta of tracks associated with the interaction vertex. In the analysis, a
set of reconstructed objects is used, consisting of electrons, muons, and
jets, as well as the missing transverse momentum. When identifying charged
leptons, a staggered approach is used, where so-called baseline leptons fulfil
less stringent requirements than signal leptons. Events are required to have
exactly one signal lepton without any additional baseline leptons.

Electron candidates are reconstructed from energy deposits in the EM
calorimeter matched to charged-particle tracks in the inner detector (ID).
Requirements of $|d_{0}|/\sigma_{d_{0}}<5$ on the transverse impact parameter
$d_{0}$ (with uncertainty $\sigma_{d_{0}}$) and $|\Delta
z_{0}\sin\theta|<$0.5\text{\,}\mathrm{m}\mathrm{m}$$ on the longitudinal track
impact parameter $z_{0}$ ensure matching between track and vertex.
Furthermore, electron candidates are required to lie within a pseudorapidity
range of $|\eta|<2.47$, excluding the EM calorimeter barrel–endcap transition
region $1.37<|\eta|<1.52$. Baseline electrons must have
$p_{\text{T}}>$10\text{\,}\mathrm{GeV}$$ and fulfil loose identification
criteria, using a likelihood-based discriminant that combines information
about tracks in the ID and energy deposits in the calorimeter system [77]. In
addition, baseline electrons are required to have a hit in the innermost layer
of the pixel detector. Isolation requirements in both the calorimeter and the
ID are imposed [77]. Electron candidates are rejected if the scalar sum of
transverse momenta of tracks within a cone of size $\Delta
R=\min\left($10\text{\,}\mathrm{GeV}$/p_{\text{T}},0.2\right)$, excluding the
electron itself, is larger than $15\text{\,}\mathrm{\char 37\relax}$ of the
electron $p_{\text{T}}$. Similarly, an electron is removed if, after
subtracting contributions from pile-up and the electron itself, the transverse
energy deposited in the calorimeter within a cone of size $\Delta R=0.2$
exceeds $20\text{\,}\mathrm{\char 37\relax}$ of the transverse energy of the
electron. To suppress backgrounds due to hadrons misidentified as electrons,
signal electrons must in addition pass the ‘tight’ identification working
point, and have $p_{\text{T}}>$30\text{\,}\mathrm{GeV}$$.

Muon candidates are reconstructed from charged-particle tracks in the ID and
the MS and from energy deposits in the calorimeters. For the reconstruction of
muon candidates, tracks in the ID combined with tracks in the MS are used in
the range $|\eta|<2.5$. In addition, muons in the range $|\eta|<0.1$ are
reconstructed from ID tracks matched to an energy deposit in the calorimeter
compatible with a minimally ionising particle. These muon candidates are
called calorimeter-tagged (CT). Track-to-vertex matching is ensured by
requiring $|d_{0}|/\sigma_{d_{0}}<3$ for the transverse impact parameter and
$|\Delta z_{0}\sin\theta|<$0.5\text{\,}\mathrm{m}\mathrm{m}$$ for the
longitudinal track impact parameter. Baseline muons are required to have
$p_{\text{T}}>$10\text{\,}\mathrm{GeV}$$ and to have compatible individual
measurements in the ID and the MS. Signal muons are required to have
$p_{\text{T}}>$30\text{\,}\mathrm{GeV}$$ and CT muons are not accepted [78].
Additionally, signal muons are rejected if the scalar sum of transverse
momenta of tracks within a cone of size $\Delta
R=\min\left($10\text{\,}\mathrm{GeV}$/p_{\text{T}},0.3\right)$ around the muon
exceeds $6\text{\,}\mathrm{\char 37\relax}$ of its transverse momentum. In
accordance with other searches for pair-produced LQs within ATLAS [33, 32],
signal muons above a $p_{\text{T}}$ threshold of $800\text{\,}\mathrm{GeV}$
must fulfil stricter requirements on the number of hits in the MS to ensure
good momentum resolution.

Small-radius (small-$R$) jet candidates are built from particle-flow objects
[79, 80], using the anti-$k_{t}$ algorithm [81, 82] with a radius parameter of
$R=0.4$. The particle-flow algorithm combines information about ID tracks and
energy deposits in the calorimeters to form the input for jet reconstruction.
Jets with $p_{\text{T}}<$25\text{\,}\mathrm{GeV}$$ or $|\eta|>2.5$ are
rejected. To reduce contributions from pile-up, jet candidates with
$|\eta|<2.4$ and $p_{\text{T}}<$60\text{\,}\mathrm{GeV}$$ are required to
satisfy the ‘tight’ jet vertex tagger criterion [83]. Small-$R$ jets are
categorised as $b\text{-tagged}$ if they satisfy a requirement on the output
of a multivariate algorithm, operating at a tagging efficiency of
$77\text{\,}\mathrm{\char 37\relax}$ as determined with simulated $t\bar{t}$
events [84, 85]. The $R=0.4$ jets are then reclustered with the anti-$k_{t}$
algorithm with $R=1.0$ to obtain large-$R$ jets. Additionally, $R=0.4$ jets
are reclustered iteratively with the recursive method described in Ref. [86]
to reconstruct hadronically decaying top-quark candidates. For this, the
small-$R$ jets are reclustered with an initial radius parameter of $R=3.0$,
which is iteratively reduced to $R(p_{\text{T}})=2m_{\text{top}}/p_{\text{T}}$
to match the jet radius to the top candidate’s transverse momentum. Top
candidates losing large fractions of their $p_{\text{T}}$ in the shrinking
process are discarded. Finally, only the leading-$p_{\text{T}}$ candidate with
a mass larger than $150\text{\,}\mathrm{GeV}$ is kept.

The missing transverse momentum (with magnitude $E_{\text{T}}^{\text{miss}}$)
in an event is defined as the negative vectorial sum of the transverse momenta
of all calibrated objects [87]. It also includes an additional track-based
soft term taking into account energy depositions not associated with any
calibrated object.

An overlap removal procedure is applied to avoid ambiguities when
reconstructing the objects described above, using the baseline lepton
definitions. Electron–muon overlap is handled by removing any calorimeter-
tagged muons sharing a track in the ID with an electron, and then removing any
electrons sharing an ID track with a remaining muon. Subsequently, overlap
between jets and leptons is removed by rejecting any jets within $\Delta
R=0.2$ of an electron and afterwards rejecting any electrons within $\Delta
R=0.4$ of a jet. Similarly, jets are discarded if they have fewer than three
associated tracks and are within $\Delta R=0.2$ of a muon candidate.
Otherwise, the muon is rejected if it lies within $\Delta
R=\text{min}(0.4,0.04+$10\text{\,}\mathrm{GeV}$/p_{\text{T}}(\mu))$ of a jet.

##  5 Event selection and categorisation

Any event considered in this analysis must pass an
$E_{\text{T}}^{\text{miss}}$ trigger [88]. Because the trigger thresholds
varied, a requirement of
$E_{\text{T}}^{\text{miss}}>$250\text{\,}\mathrm{GeV}$$ is imposed in the
offline event selection to ensure full efficiency of all the triggers across
the different data-taking periods. Events are required to contain exactly one
signal lepton. Additionally, a veto is applied on further baseline leptons.
Since the final states of interest contain one hadronically decaying top-quark
and one additional jet, only events with at least four jets are selected. Only
one of those jets needs to be $b\text{-tagged}$ to preserve high efficiency
for the signal. To suppress contributions from fake
$E_{\text{T}}^{\text{miss}}$ caused by mismeasurements of jets or leptons,
events with $m_{\text{T}}(\ell,E_{\text{T}}^{\text{miss}})\leavevmode\nobreak\
{=}\leavevmode\nobreak\
\sqrt{2p_{\text{T}}(\ell)E_{\text{T}}^{\text{miss}}(1-\cos\Delta\phi(p_{\text{T}}(\ell),E_{\text{T}}^{\text{miss}}))}\leavevmode\nobreak\
{<}\leavevmode\nobreak\ $30\text{\,}\mathrm{GeV}$$ or
$\Delta\phi(E_{\text{T}}^{\text{miss}},j_{1,2})<0.4$, where $j_{1,2}$
indicates the highest-$p_{\text{T}}$ and the second-highest-$p_{\text{T}}$
small-$R$ jet, respectively, are rejected.

As the $t\bar{t}$ background is known not to be modelled accurately at high
transverse momenta [89, 90], reweighting factors are derived in bins of the
jet multiplicity as a function of $m_{\text{eff}}$, which is defined as the
scalar sum of the transverse momenta of all reconstructed objects and the
$E_{\text{T}}^{\text{miss}}$. This procedure is referred to as ‘top
reweighting’ in the following. The reweighting factors are determined for the
sum of the $t\bar{t}$ and single-top backgrounds and are parameterised with a
linear function, separately in each of the four jet multiplicity bins (4, 5,
6, $\geq 7$). For this, a dedicated top reweighting region is defined with
$am_{\text{T}2}<$200\text{\,}\mathrm{GeV}$$, where the asymmetric transverse
mass, $am_{\text{T}2}$, is a variant of $m_{\text{T}2}$ [91] and allows the
reconstruction of dileptonic $t\bar{t}$ events in which only one lepton is
reconstructed [92, 93]. The reweighting factor is then applied to single-top
and $t\bar{t}$ events in each of the training and control regions defined in
the following. Figure 2 shows the $m_{\text{eff}}$ and lepton-$p_{\text{T}}$
distributions in the top reweighting region after applying the reweighting
procedure, in addition to the total background expectation before reweighting.
The modelling of kinematic variables is improved by the correction.

(a)

(b)

Figure 2: Distributions of 2(a) $m_{\text{eff}}$ and 2(b) $p_{\text{T}}(\ell)$
in the top reweighting region after applying the top reweighting. The hatched
bands include statistical and systematic uncertainties. The total background
expectation before applying the top reweighting is shown as a dashed line. The
ratios of the observed and expected numbers of background events are shown in
the bottom panels. The last bin contains the overflow.

Control regions (CRs) enriched in the various backgrounds and with negligible
signal contamination are defined. They are orthogonal to the top reweighting
region by requiring $am_{\text{T}2}>$200\text{\,}\mathrm{GeV}$$ and are
orthogonal to each other. The $W\text{+jets}$ CR uses events in a window
around the Jacobian peak, i.e. $$50\text{\,}\mathrm{GeV}$\leq
m_{\text{T}}(\ell,E_{\text{T}}^{\text{miss}})<$120\text{\,}\mathrm{GeV}$$,
with a $b\text{-jet}$ multiplicity of $n_{b}=1$ and no hadronically decaying
top-quark candidate. To increase the purity of selected $W\text{+jets}$
events, only events with a positively charged lepton are considered, because
the cross-section for $W^{+}$ production is larger than that for $W^{-}$
production in $pp$ collisions. For the single-top CR a requirement of
$m_{\text{T}}(\ell,E_{\text{T}}^{\text{miss}})<$120\text{\,}\mathrm{GeV}$$ is
imposed. In order to reduce contributions from $W\text{+jets}$ production,
events must have exactly two $b\text{-tagged}$ jets, with an angular
separation $\Delta R(b_{1},b_{2})>1.2$. Events containing a large-$R$ jet are
vetoed. Distributions of $m_{\text{eff}}$ in the $W\text{+jets}$ CR and the
single-top CR are shown in Figure 3.

(a)

(b)

Figure 3: Distributions of $m_{\text{eff}}$ in 3(a) the $W\text{+jets}$ CR and
3(b) the single-top CR after applying the top reweighting, before the fit to
data in CRs and SR. The hatched bands include statistical and systematic
uncertainties. The total background expectation before applying the top
reweighting is shown as a dashed line. The ratios of the observed and expected
numbers of background events are shown in the bottom panels. The last bin
contains the overflow.

For the training of the NNs, a training region is defined so as to be
orthogonal to the top reweighting region and both control regions by requiring
$am_{\text{T}2}>$200\text{\,}\mathrm{GeV}$$ and
$m_{\text{T}}(\ell,E_{\text{T}}^{\text{miss}})\geq$120\text{\,}\mathrm{GeV}$$.
An overview of the different selection requirements for the training and
control regions is given in Table 2. The product of signal acceptance and
efficiency in the training region is similar for signal hypotheses with
couplings to electrons or muons. For $\mathcal{B}=0.5$, it amounts to around
$17\text{\,}\mathrm{\char 37\relax}$ for up-type scalar LQs and to around
$14\text{\,}\mathrm{\char 37\relax}$ for down-type scalar LQs at
$m_{\text{LQ}}=$1.4\text{\,}\mathrm{TeV}$$. For vector LQs, it reaches
$20\text{\,}\mathrm{\char 37\relax}$ at the same mass for both the Yang–Mills
coupling and the minimal coupling scenario.

Table 2: Overview of event selections applied in the different regions of the
analysis.

Preselection
---
$E_{\text{T}}^{\text{miss}}$ triggers
exactly one signal lepton
veto on additional baseline leptons
$E_{\text{T}}^{\text{miss}}>$250\text{\,}\mathrm{GeV}$$
$\geq 4$ small-$R$ jets
$m_{\text{T}}(\ell,E_{\text{T}}^{\text{miss}})>$30\text{\,}\mathrm{GeV}$$
$\Delta\phi(E_{\text{T}}^{\text{miss}},j_{1,2})>0.4$
Top reweighting region |  $W\text{+jets}$ CR | Single-top CR | Training region
$n_{b}\geq 1$ | $n_{b}=1$ | $n_{b}=2$ | $n_{b}\geq 1$
$m_{\text{T}}(\ell,E_{\text{T}}^{\text{miss}})\geq$120\text{\,}\mathrm{GeV}$$ | $$50\text{\,}\mathrm{GeV}$\leq m_{\text{T}}(\ell,E_{\text{T}}^{\text{miss}})<$120\text{\,}\mathrm{GeV}$$ | $m_{\text{T}}(\ell,E_{\text{T}}^{\text{miss}})<$120\text{\,}\mathrm{GeV}$$ | $m_{\text{T}}(\ell,E_{\text{T}}^{\text{miss}})\geq$120\text{\,}\mathrm{GeV}$$
$am_{\text{T}2}<$200\text{\,}\mathrm{GeV}$$ | $am_{\text{T}2}>$200\text{\,}\mathrm{GeV}$$ | $am_{\text{T}2}>$200\text{\,}\mathrm{GeV}$$ | $am_{\text{T}2}>$200\text{\,}\mathrm{GeV}$$
- |  $t_{\text{had}}$ candidate veto | large-$R$ jet veto | -
- | lepton charge $=+1e$ | - | -
- | - | $\Delta R(b_{1},b_{2})>1.2$ | -

##  6 Neural network training

Simulated signal and background events in the training region are used to
train several NNs for the various signal hypotheses. The NNs are implemented
using the NeuroBayes package [94, 95] which combines a three-layer feed-
forward NN with a complex and robust preprocessing of the input variables
prior to their presentation to the NN. The purpose of the preprocessing is to
facilitate optimal network training by ordering the input variables according
to their ability to discriminate between signal and background, taking
correlations into account, and removing all but the most powerful ones.

NeuroBayes uses Bayesian regularisation techniques for the training process to
improve the generalisation performance and to avoid overtraining. In general,
the network infrastructure consists of one input node for each input variable
plus one bias node, an arbitrary, user-defined number of hidden nodes, and one
output node which gives a continuous NN output score
($N\kern-1.00006ptN_{\textrm{out}}$) in the interval $(0,+1)$, where large
values indicate signal-like events and small values background-like events.
For the NNs of this analysis, 15 nodes are used in the hidden layer and the
ratio of signal to background events in the training is 1:1. The different
background processes are weighted according to their expected number of
events. Only $t\bar{t}$, $W\text{+jets}$, single-top-quark, and $t\bar{t}{+}V$
events are used as background processes in the training. As a check for
potential overtraining, only $80\text{\,}\mathrm{\char 37\relax}$ of the
simulated events serve as input to the training, while the remaining
$20\text{\,}\mathrm{\char 37\relax}$ are used as a test sample. No signs of
overtraining are observed. After the training step, samples of simulated
signal and background events, as well as the observed events, are processed by
the NNs in order to get an $N\kern-1.00006ptN_{\textrm{out}}$ value for each
event. For each NN, the training region is divided into a
low-$N\kern-1.00006ptN_{\textrm{out}}$ control region with
$N\kern-1.00006ptN_{\textrm{out}}$ < 0.5, enriched mainly in $t\bar{t}$
events, and the signal region above 0.5.

The input variables are chosen because of their ability to discriminate
between signal and background. In total, 15 input variables are provided for
the training, including the lepton flavour in order to distinguish between
electrons and muons. This is because a final state with one lepton flavour has
some sensitivity to a LQ model with the other flavour if the lepton stems from
a top-quark decay, i.e. mainly in the low $\mathcal{B}$ region. Table 3 lists
the input variables in order of decreasing ability to discriminate between
signal and background. The order is not absolute, as there is some dependence
on the signal model and $\mathcal{B}$, e.g. the lepton flavour cannot
discriminate at all in the region of low $\mathcal{B}$, but is important
otherwise. The modelling of the input variables in the training region is good
in general, as can be seen for the most important ones in Figure 4. Among the
least well modelled of all input variables is $m_{\text{inv}}(b_{1},\ell)$
shown in Figure 4(c). This is due to the interference between single-top and
$t\bar{t}$ production, which is difficult to describe in MC simulations [96].

Table 3: Input variables for the NN training, approximately sorted in descending ability to discriminate between signal and background. The order is not absolute as there is some dependence on the signal model and $\mathcal{B}$. Variable | Description
---|---
$m_{\text{T}}(\ell,E_{\text{T}}^{\text{miss}})$ | transverse mass of lepton and $E_{\text{T}}^{\text{miss}}$
$m_{\text{eff}}$ | scalar sum of the transverse momenta of leptons, jets, and $E_{\text{T}}^{\text{miss}}$
Lepton flavour | flavour of the signal lepton
$p_{\text{T}}(\ell)$ | transverse momentum of the lepton
$m_{\text{inv}}(b_{1},\ell)$ | invariant mass of the leading-$p_{\text{T}}$ $b\text{-jet}$ and the lepton
$n_{\text{large}}$ | reclustered large-$R$ jet multiplicity
$am_{\text{T}2}$ | asymmetric transverse mass
$E_{\text{T}}^{\text{miss}}$ significance | measure for assessing the compatibility of the observed $E_{\text{T}}^{\text{miss}}$ with zero,
| taking resolutions into account
$m_{\text{T}}(b_{1},E_{\text{T}}^{\text{miss}})$ | transverse mass of leading-$p_{\text{T}}$ $b\text{-jet}$ and $E_{\text{T}}^{\text{miss}}$
$p_{\text{T}}(t_{\text{had}})$ | transverse momentum of $t_{\text{had}}$
$\Delta\phi(E_{\text{T}}^{\text{miss}},b_{2})$ | azimuthal angle separation between $E_{\text{T}}^{\text{miss}}$ and subleading-$p_{\text{T}}$ $b\text{-jet}$
$m_{\text{inv}}(b_{2},\ell)$ | invariant mass of subleading-$p_{\text{T}}$ $b\text{-jet}$ and lepton
$\Delta\phi(E_{\text{T}}^{\text{miss}},b_{1})$ | azimuthal angle separation between $E_{\text{T}}^{\text{miss}}$ and leading-$p_{\text{T}}$ $b\text{-jet}$
$\Delta\phi(t_{\text{had}},\ell)$ | azimuthal angle separation between $t_{\text{had}}$ and lepton
$p_{\text{T}}(b_{1})$ | transverse momentum of leading-$p_{\text{T}}$ $b\text{-jet}$

(a)

(b)

(c)

(d)

Figure 4: Distributions of 4(a) $m_{\text{eff}}$, 4(b) $p_{\text{T}}(\ell)$,
4(c) $m_{\text{inv}}(b_{1},\ell)$, and 4(d)
$m_{\text{T}}(\ell,E_{\text{T}}^{\text{miss}})$ in the training region after
applying the top reweighting, before the fit to data in CRs and SR. The
hatched bands include statistical and systematic uncertainties. Signal
distributions normalised to the total background expectation are overlaid for
up-type scalar LQs with $\mathcal{B}=0.0$ and vector LQs with
$\mathcal{B}=0.0$ and 0.5, each with
$m_{\text{LQ}}=$1.3\text{\,}\mathrm{TeV}$$. The ratios of the observed and
expected numbers of background events are shown in the bottom panels. The last
bin contains the overflow.

Figure 4 also displays three up-type signal hypotheses at a mass of
$1.3\text{\,}\mathrm{TeV}$. It can be seen that the signal shape depends not
only on the value of $\mathcal{B}$ but also on the spin of the LQ. The signal
shape differences due to spin correlations are sizeable for low values of
$\mathcal{B}$ at small values of lepton $p_{\text{T}}$, where the lepton
usually originates from a top-quark decay. The lepton-$p_{\text{T}}$
distribution in the vector LQ model is found to be similar in shape to the
background in contrast to the scalar case as can be seen Figure 4(b).
Therefore, separate NNs are trained for scalar and vector LQs as well as for
various values of $\mathcal{B}$. A total of four NNs per lepton flavour at
$\mathcal{B}=0.0,0.25,0.5$, and $0.9$ are used for up-type scalar and vector
LQs. Since kinematic differences between the Yang–Mills coupling and the
minimal coupling are negligible except for very low masses, the NNs for vector
LQs are only trained on samples with the former coupling and then applied to
the latter as well. In the case of down-type scalar LQs, only one NN is
trained for LQs decaying into muons and another is trained for LQs decaying
into electrons. For both trainings, a branching ratio of 0.5 is assumed, since
only events with one LQ decaying into a charged lepton and the other decaying
into a neutrino contribute significantly to the phase space under
consideration. For scalar LQs, signal samples produced with
$m_{\text{LQ}}=$500\text{\,}\mathrm{GeV}$,$900\text{\,}\mathrm{GeV}$$, and
$1300\text{\,}\mathrm{GeV}$ are combined in each training, as separate
trainings for each signal mass did not yield a significant improvement for
masses above about $600\text{\,}\mathrm{GeV}$. For vector LQs, additional
signal samples with $m_{\text{LQ}}=$1700\text{\,}\mathrm{GeV}$$ are used to
take advantage of their higher production cross-sections and therefore higher
expected mass limits.

##  7 Systematic uncertainties

The largest systematic uncertainties considered in the analysis are related to
the modelling of the background processes. For $t\bar{t}$ production, the
uncertainty due to the method used to match the matrix element to the parton
shower is assessed by comparing a sample produced by MadGraph5_aMC@NLO with
the nominal sample from Powheg Box, using the same parton shower. Conversely,
an estimate of the uncertainties related to the underlying event, parton
shower, and hadronisation is obtained by comparing a sample showered by Herwig
7 [97] with the nominal sample showered by Pythia, using the same ME generator
for both. The effects of uncertainties in the renormalisation and
factorisation scales are estimated by independently varying the scales by a
factor of two. The impact of initial-state radiation (ISR) is estimated by
varying $\alpha_{\text{s}}$ in the A14 tune. Similarly, the uncertainty
related to final-state radiation (FSR) is assessed by varying the
renormalisation scale for final-state parton-shower emissions by a factor of
two. Additionally, an uncertainty related to the choice of value for the
Powheg Box-specific $h_{\mathrm{damp}}$ parameter is evaluated by using a
varied value of $h_{\mathrm{damp}}=3.0\,m_{t}$. PDF uncertainties are obtained
from the PDF4LHC15 PDF set [98].

For single top-quark production, uncertainties in ME-to-PS matching, the
choice of parton shower, renormalisation and factorisation scales, ISR, FSR,
and PDF are evaluated using the same procedures as for $t\bar{t}$ production.
Large uncertainties arise due to interference effects between $t\bar{t}$ and
$tW$ production. They are estimated by comparing the nominal sample based on
the diagram removal scheme with a sample using the diagram subtraction scheme
[67, 99].

In the top reweighting procedure, the parameters of the linear fit are varied
within their $2\sigma$ uncertainty to account for potential non-linearities in
addition to the statistical uncertainty, treating the normalisation and the
shape component in each of the four jet multiplicity bins independently.
Compared to the $1\sigma$ variation, this choice has a negligible impact on
the results.

For $W\text{+jets}$, $Z\text{+jets}$, diboson, $t\bar{t}{+}V$, and
$t\bar{t}{+}H$ processes, renormalisation and factorisation scale variations
are considered, following the same procedures as for $t\bar{t}$. In addition,
an uncertainty of $50\text{\,}\mathrm{\char 37\relax}$ is assigned to the
heavy-flavour component of the $W\text{+jets}$ background to cover differences
in flavour composition between control and signal regions seen in MC studies.

Theoretical systematic uncertainties also include cross-section uncertainties
for the background processes for which no normalisation parameters enter the
profile-likelihood fit. For diboson and $Z\text{+jets}$ production, this
uncertainty is taken to be $6\text{\,}\mathrm{\char 37\relax}$ [100] and
$5\text{\,}\mathrm{\char 37\relax}$ [101], respectively. For $t\bar{t}{+}H$
production, it amounts to $11\text{\,}\mathrm{\char 37\relax}$ [102] and for
$t\bar{t}{+}Z$ production to $15\text{\,}\mathrm{\char 37\relax}$ [102]. For
$t\bar{t}{+}W$ production, the cross-section uncertainty is taken to be
$50\text{\,}\mathrm{\char 37\relax}$ to account for potential differences
between predicted and measured values as reported in Ref. [103].

Systematic uncertainties in the signal prediction arise from acceptance
effects due to renormalisation and factorisation scale, ISR/FSR, and PDF and
$\alpha_{\text{s}}$ variations. They were found to not exceed
$5\text{\,}\mathrm{\char 37\relax}$ in total across the whole mass range, and
this is therefore taken as a conservative estimate.

Additionally, detector-related uncertainties are considered, the dominant ones
being the small-$R$ jet energy scale and resolution uncertainties [80].
Furthermore, systematic uncertainties related to the jet mass scale and
resolution, the lepton identification, isolation, and reconstruction
efficiencies as well as the lepton energy scale and resolution [77, 78], the
$b$-tagging efficiencies [84], and the $E_{\text{T}}^{\text{miss}}$
reconstruction [104] are taken into account. Minor contributions to the total
systematic uncertainty also come from the uncertainty of
$1.7\text{\,}\mathrm{\char 37\relax}$ in the integrated luminosity [105],
obtained using the LUCID-2 detector [106] for the primary luminosity
measurements, and from an uncertainty related to pile-up reweighting.

##  8 Statistical interpretation

The binned distributions of the NN output are used to test for the presence of
a signal. Simultaneous binned profile-likelihood fits are performed for
hypothesis testing, following a modified frequentist method implemented in
RooStats [107] and using the $N\kern-1.00006ptN_{\textrm{out}}$ distribution
in the signal region and the overall number of events in the
low-$N\kern-1.00006ptN_{\textrm{out}}$, $W\text{+jets}$, and single-top
control regions. Systematic uncertainties affecting signal and background
expectations are accounted for by including them in the fit in the form of
nuisance parameters. For uncertainties in the modelling of background
processes for which the normalisation is determined in the likelihood fit,
only shape effects and acceptance differences between the CRs and the SR are
considered, in order to avoid double-counting of normalisation uncertainties.
This procedure is also used for pre-fit uncertainty bands in order to have an
equivalent treatment of systematic uncertainties. A smoothing algorithm is
applied to certain systematic variations in the signal region in order to
reduce statistical fluctuations between bins. To simplify the fitting
procedure, for each region and each process a nuisance parameter is only
considered if the overall effect on the normalisation of the process is larger
than $1\text{\,}\mathrm{\char 37\relax}$. In the case of the signal region,
which has multiple bins, nuisance parameters are also considered if their
effect in any bin within the signal region is above $1\text{\,}\mathrm{\char
37\relax}$.

The binned likelihood function $\mathcal{L}(\mu,\theta)$ is constructed as the
product of Poisson probability terms over all bins considered in the analysis.
It depends on the signal strength parameter $\mu$, a multiplicative factor
applied to the theoretical signal production cross-section, and $\theta$, a
set of nuisance parameters, implemented in the likelihood function as Gaussian
or log-normal priors. The expected number of events in a bin depends on $\mu$
and $\theta$. The nuisance parameters $\theta$ adjust the expectations for
signal and background according to the corresponding systematic uncertainties,
and their fitted values correspond to the amounts that best fit the data.

The test statistic $q_{\mu}$ is defined as the profile likelihood ratio

$\displaystyle
q_{\mu}=-2\ln{\frac{\mathcal{L}(\mu,\hat{\hat{\theta}})}{\mathcal{L}(\hat{\mu},\hat{\theta})}},$

where $\hat{\mu}$ and $\hat{\theta}$ are the values of the parameters that
maximise the likelihood function (with the constraints
$0\leq\hat{\mu}\leq\mu$), and $\hat{\hat{\theta}}$ are the values of the
nuisance parameters that maximise the likelihood function for a given value of
$\mu$. This test statistic is used to determine whether the observed data are
compatible with the background-only hypothesis, i.e. with $\mu=0$.
Furthermore, by using the CL${}_{\text{s}}$ method [108], upper limits on the
signal production cross-section are derived for each of the signal scenarios
considered in this analysis. For a given signal scenario, values of the
production cross-section (parameterised by $\mu$) yielding
CL${}_{\text{s}}<0.05$, where CL${}_{\text{s}}$ is computed using the
asymptotic approximation [109], are excluded at $\geq$95\text{\,}\mathrm{\char
37\relax}$$ confidence level (CL).

##  9 Results

For each NN training, a separate fit to the $N\kern-1.00006ptN_{\textrm{out}}$
distribution in the signal region and the overall number of events in the
low-$N\kern-1.00006ptN_{\textrm{out}}$, $W\text{+jets}$, and single-top
control regions is performed, with free normalisation parameters for the
$t\bar{t}$, single-top, and $W\text{+jets}$ background processes. The
normalisation parameters obtained from fits to data using the background-only
hypothesis are consistent across all trainings. They are always applied in the
following and vary between $1.09\pm 0.22$ and $1.29\pm 0.23$ for $t\bar{t}$,
between $0.84\pm 0.12$ and $0.93\pm 0.12$ for $W\text{+jets}$, and between
$0.46\pm 0.27$ and $0.54\pm 0.26$ for single top. The normalisation parameter
for the single-top process reduces the event yield by approximately a factor
of two; however, the expected yield from the alternative scheme to model the
interference between the $t\bar{t}$ and $tW$ processes leads to an even
smaller yield. Observed and expected event yields after the background-only
fit are listed in Table 4 for one NN training.

Table 4: Observed and expected event yields in the control and signal regions for a training for $\mathrm{vLQ}_{\text{mix}}^{\text{YM}}\rightarrow b\mu/t\nu$ and $\mathcal{B}=0.5$ after the background-only fit. The uncertainties in the background predictions include both the statistical and systematic components. For comparison, expected event yields are shown for a $\mathrm{vLQ}_{\text{mix}}^{\text{YM}}$ signal at a mass point of $1700\text{\,}\mathrm{GeV}$ and $\mathcal{B}=0.5$ including its pre-fit uncertainties. |  $W\text{+jets}$ CR | Single-top CR | Low-$NN_{\text{out}}$ CR | SR
---|---|---|---|---
$t\bar{t}$ | $860\pm 140$ | $186\pm 35$ | $1370\pm 150$ | $53\pm 10$
Single top | $103\pm 87$ | $131\pm 47$ | $200\pm 110$ | $36\pm 14$
$W\text{+jets}$ | $1240\pm 130$ | $101\pm 28$ | $265\pm 55$ | $32.4\pm 6.9$
$t\bar{t}{+}V$ | $11.0\pm 1.8$ | $4.47\pm 0.79$ | $180\pm 28$ | $16.7\pm 2.6$
Diboson | $94.3\pm 9.8$ | $7.6\pm 1.9$ | $94\pm 11$ | $11.1\pm 1.2$
$t\bar{t}{+}H$ | $1.27\pm 0.16$ | $1.00\pm 0.12$ | $14.4\pm 1.7$ | $1.34\pm 0.18$
$Z\text{+jets}$ | $6.46\pm 0.32$ | $2.18\pm 0.11$ | $7.20\pm 0.36$ | $1.315\pm 0.072$
Total background | $2308\pm 48$ | $433\pm 21$ | $2126\pm 46$ | $152\pm 13$
Observed events | 2310 | 430 | 2124 | 157
$\mathrm{vLQ}_{\text{mix}}^{\text{YM}}$ ($1.7\text{\,}\mathrm{TeV}$, $\mathcal{B}=0.5$) | $0.109\pm 0.022$ | $0.097\pm 0.016$ | $1.57\pm 0.10$ | $38.9\pm 2.6$

The $N\kern-1.00006ptN_{\textrm{out}}$ distributions after the background-only
fit are validated with data–MC comparisons in the control regions, as shown in
Figure 5 for one particular NN training. In general, good agreement is found
for all trainings in all control regions, although the single-top CR is
typically more problematic due to the interference effects mentioned above
[96]. NPs corresponding to the systematic uncertainties covering the observed
differences between data and MC simulation are not constrained significantly
because only the overall number of events in each CR enters the fit. The
systematic uncertainties are therefore fully propagated to the SR.

(a)

(b)

(c)

Figure 5: Data and background expectation in 5(a) the $W\text{+jets}$ CR, 5(b)
the single-top CR, and 5(c) the low-$N\kern-0.90005ptN_{\textrm{out}}$ CR
after the simultaneous background-only fit for a training with
$\mathrm{vLQ}_{\text{mix}}^{\text{YM}}\rightarrow b\mu/t\nu$ and
$\mathcal{B}=0.5$. The hatched band indicates the total post-fit uncertainty.
The ratios of data to background expectation are shown in the bottom panels.

A comparison between data and background expectation in the signal region is
shown in Figure 6 after the background-only fit for three different NN
trainings. Good agreement is found for all trainings. The largest
discrepancies at high values of $N\kern-1.00006ptN_{\textrm{out}}$ are
observed for the $\mathrm{LQ}_{\text{mix}}^{\textnormal{u}}$ model with
$\mathcal{B}=0.0$, i.e. for the decay into top-quarks and neutrinos as shown
in Figure 6(c).

(a)

(b)

(c)

Figure 6: Data and background expectation in the signal region after the
simultaneous background-only fit to data for 6(a) a training with
$\mathrm{vLQ}_{\text{mix}}^{\text{YM}}\rightarrow b\mu/t\nu$ and
$\mathcal{B}=0.5$, 6(b) a training with
$\mathrm{LQ}_{\text{mix}}^{\textnormal{d}}\rightarrow te/b\nu$ with
$\mathcal{B}=0.5$, and 6(c) a training with
$\mathrm{LQ}_{\text{mix}}^{\textnormal{u}}\rightarrow t\nu$, i.e.
$\mathcal{B}=0.0$. Minor background contributions from $t\bar{t}{+}H$ and
$Z\text{+jets}$ are combined into ‘other’. Expected pre-fit signal
distributions with $\mathcal{B}$ corresponding to the respective training are
added on top of the background expectation, using a mass of
$1700\text{\,}\mathrm{GeV}$ for vector LQs and $1300\text{\,}\mathrm{GeV}$ for
scalar LQs. The hatched band indicates the total post-fit uncertainty. The
ratios of data to background expectation are shown in the bottom panels.

No significant deviations of the data from the expected SM background are
observed. Upper $95\text{\,}\mathrm{\char 37\relax}$ CL limits on the cross-
sections of pair-produced LQs can be calculated in simultaneous signal-plus-
background fits to the CRs and the SR, in which the background normalisations
and possible signal contributions are determined. The largest uncertainty in
each of the resulting signal strengths is statistical in nature. For the three
signal hypotheses shown in Figure 6, the statistical uncertainty exceeds
$85\text{\,}\mathrm{\char 37\relax}$ of the total uncertainty for the two
scalar LQ models at $m_{\text{LQ}}=$1.3\text{\,}\mathrm{TeV}$$ and rises to
nearly $100\text{\,}\mathrm{\char 37\relax}$ for the vector LQ case at
$m_{\text{LQ}}=$1.7\text{\,}\mathrm{TeV}$$. The resulting limits on the cross-
section for the four scalar LQ models are shown in Figure 7 as a function of
the LQ mass for a fixed $\mathcal{B}=0.5$. Corresponding limits for the four
vector LQ models are shown in Figure 8.

(a)

(b)

(c)

(d)

Figure 7: Expected (dashed black) and observed (solid black)
$95\text{\,}\mathrm{\char 37\relax}$ CL upper limits on the cross-section of
pair-produced scalar LQs, assuming $\mathcal{B}=0.5$. The green (yellow) band
shows the $\pm 1\sigma\,(\pm 2\sigma)$ uncertainty region around the expected
limit. The theoretical prediction and its $\pm 1\sigma$ uncertainty band are
shown in blue. Limits are presented for 7(a) up-type scalar LQs decaying into
muons, 7(b) up-type scalar LQs decaying into electrons, 7(c) down-type scalar
LQs decaying into muons, and 7(d) down-type scalar LQs decaying into
electrons.

(a)

(b)

(c)

(d)

Figure 8: Expected (dashed black) and observed (solid black)
$95\text{\,}\mathrm{\char 37\relax}$ CL upper limits on the cross-section of
pair-produced vector LQs, assuming $\mathcal{B}=0.5$. The green (yellow) band
shows the $\pm 1\sigma\,(\pm 2\sigma)$ uncertainty region around the expected
limit. The theoretical prediction and its $\pm 1\sigma$ uncertainty band are
shown in blue. Limits are presented for 8(a) vector LQs in the Yang–Mills
coupling scenario decaying into muons, 8(b) vector LQs in the Yang–Mills
coupling scenario decaying into electrons, 8(c) vector LQs in the minimal
coupling scenario decaying into muons, and 8(d) vector LQs in the minimal
coupling scenario decaying into electrons.

These cross-section limits are compared with the theoretical cross-section
predictions, shown in blue, resulting in lower limits on the signal mass for
$\mathcal{B}=0.5$. The uncertainty band around the theory prediction includes
PDF, $\alpha_{\text{s}}$, and renormalisation and factorisation scale
uncertainties. The expected and observed limits for $\mathcal{B}=0.5$ are
summarised in Table 5 for the eight LQ models considered in this analysis. The
total impact of systematic uncertainties on the cross-section limits reaches
$15\text{\,}\mathrm{\char 37\relax}$ for LQ masses above
$1\text{\,}\mathrm{TeV}$, corresponding to $20\text{\,}\mathrm{GeV}$ in the
expected mass limit.

Table 5: Expected and observed $95\text{\,}\mathrm{\char 37\relax}$ CL lower limits on the LQ mass at $\mathcal{B}=0.5$ for the eight signal hypotheses considered in this analysis. | Exp. limit [GeV] | Obs. limit [GeV]
---|---|---
$\mathrm{LQ}_{\text{mix}}^{\textnormal{u}}\rightarrow t\nu/b\mu$ | $1440_{-60}^{+60}$ | 1460
$\mathrm{LQ}_{\text{mix}}^{\textnormal{u}}\rightarrow t\nu/be$ | $1440_{-60}^{+60}$ | 1440
$\mathrm{LQ}_{\text{mix}}^{\textnormal{d}}\rightarrow t\mu/b\nu$ | $1380_{-60}^{+50}$ | 1370
$\mathrm{LQ}_{\text{mix}}^{\textnormal{d}}\rightarrow te/b\nu$ | $1410_{-60}^{+60}$ | 1390
$\mathrm{vLQ}_{\text{mix}}^{\text{YM}}\rightarrow t\nu/b\mu$ | $1930_{-60}^{+50}$ | 1980
$\mathrm{vLQ}_{\text{mix}}^{\text{YM}}\rightarrow t\nu/be$ | $1930_{-70}^{+50}$ | 1900
$\mathrm{vLQ}_{\text{mix}}^{\text{min}}\rightarrow t\nu/b\mu$ | $1660_{-50}^{+50}$ | 1710
$\mathrm{vLQ}_{\text{mix}}^{\text{min}}\rightarrow t\nu/be$ | $1650_{-60}^{+50}$ | 1620

Limits on LQ pair-production are also evaluated across a wide range of values
for the branching ratio of LQs into charged leptons. For that, the statistical
interpretation is performed in steps of 0.05 in $\mathcal{B}$ between 0.0 and
0.95 for up-type scalar and vector LQs and between 0.05 and 0.95 for down-type
scalar LQs. For up-type LQs, for which NNs have been trained at four different
values of $\mathcal{B}$, the NN resulting in the best expected cross-section
limit is chosen at each step. The analysis is not sensitive to final states
with zero or two leptons; therefore, $\mathcal{B}=1.0$ is omitted for all LQs
and so is $\mathcal{B}=0.0$ for down-type LQs. The cross-section upper limits
and the mass exclusion curves across the $\mathcal{B}$ plane are shown in
Figure 9 for scalar LQs and in Figure 10 for vector LQs.

(a)

(b)

(c)

(d)

Figure 9:  Expected (solid white, $\pm 1\sigma$ ranges dashed) and observed
(solid red) exclusion limits on the leptoquark mass as a function of the
branching ratio into charged leptons at $95\text{\,}\mathrm{\char 37\relax}$
CL. The observed upper limit on the signal cross-section in each bin is shown
on the z-axis. Limits are presented for 9(a) up-type scalar LQs decaying into
muons, 9(b) up-type scalar LQs decaying into electrons, 9(c) down-type scalar
LQs decaying into muons, and 9(d) down-type scalar LQs decaying into
electrons. For up-type LQs the range in $\mathcal{B}$ is 0–0.95, for down-type
it is 0.05–0.95.

(a)

(b)

(c)

(d)

Figure 10:  Expected (solid white, $\pm 1\sigma$ ranges dashed) and observed
(solid red) exclusion limits on the leptoquark mass as a function of the
branching ratio into charged leptons at $95\text{\,}\mathrm{\char 37\relax}$
CL. The observed upper limit on the signal cross-section in each bin is shown
on the z-axis. Limits are presented for 10(a) vector LQs in the Yang–Mills
coupling scenario decaying into muons, 10(b) vector LQs in the Yang–Mills
coupling scenario decaying into electrons, 10(c) vector LQs in the minimal
coupling scenario decaying into muons, and 10(d) vector LQs in the minimal
coupling scenario decaying into electrons.

Expected and observed limits on the leptoquark mass as a function of
$\mathcal{B}$ agree well everywhere, except for a small deviation for the
$\mathrm{LQ}_{\text{mix}}^{\textnormal{u}}$ model at $\mathcal{B}=0.0$, as
already discussed above in the context of Figure 6. Differences between up-
and down-type LQs can be observed especially for low values of $\mathcal{B}$,
where events with $\text{LQ}\text{LQ}\rightarrow t_{\text{lep}}\nu
t_{\text{had}}\nu$ increase the sensitivity. The shapes of the exclusion
limits for vector LQs with the Yang–Mills and the minimal coupling are very
similar, since no significant kinematic differences exist at these higher
masses. When comparing the limits for up-type scalar LQs with those for vector
LQs, the kinematic differences due to spin correlations in
$\text{LQ}\text{LQ}\rightarrow t_{\text{lep}}\nu t_{\text{had}}\nu$ become
relevant, i.e. when approaching $\mathcal{B}=0.0$, the expected lower limit on
the mass decreases faster for the vector LQ, as expected from Figure 4(b).

##  10 Conclusion

Results of a search for pair-produced scalar and vector leptoquarks decaying
into quarks of the third generation and charged or neutral leptons of the
first or second generation are presented, targeting the single-lepton final
state. The analysis is based on data collected by the ATLAS experiment in
$\sqrt{s}=$13\text{\,}\mathrm{TeV}$$ proton–proton collisions, corresponding
to an integrated luminosity of $139\text{\,}{\mathrm{fb}}^{-1}$. Several
neural networks are trained for various signal hypotheses, covering a wide
range of parameters. No significant deviations from the Standard Model
expectation are observed and upper limits on the production cross-section are
derived for eight models as a function of leptoquark mass and branching ratio
into the charged lepton.

In addition, lower limits on the leptoquark mass are set across a range of
branching ratios for all models. At a branching ratio of 0.5 they reach values
of $1470\text{\,}\mathrm{GeV}$ ($1440\text{\,}\mathrm{GeV}$) for up-type
scalar leptoquarks decaying into muons (electrons) and
$1370\text{\,}\mathrm{GeV}$ ($1390\text{\,}\mathrm{GeV}$) for down-type scalar
leptoquarks decaying into muons (electrons). For the first time, dedicated
neural networks are used for $U_{1}$ vector leptoquarks, directly seeking an
explanation for the various $B$-anomalies in a scenario where the leptoquark
decays with similar branching fractions into either a charged or neutral
lepton of the second generation, assuming couplings to quarks of only the
third generation. The resulting lower limits on the mass are
$1980\text{\,}\mathrm{GeV}$ and $1710\text{\,}\mathrm{GeV}$ for the Yang–Mills
and the minimal coupling scenario, respectively. The decay into electrons is
also probed and limits of $1900\text{\,}\mathrm{GeV}$
($1620\text{\,}\mathrm{GeV}$) for the Yang–Mills (minimal) coupling scenario
are calculated.

## References

  * [1] BaBar Collaboration  “Measurement of an excess of $\bar{B}\to D^{(*)}\tau^{-}\bar{\nu}_{\tau}$ decays and implications for charged Higgs bosons”  In _Phys. Rev. D_ 88.7, 2013, pp. 072012  DOI: 10.1103/PhysRevD.88.072012
  * [2] Belle Collaboration  “Measurement of the branching ratio of $\bar{B}\to D^{(\ast)}\tau^{-}\bar{\nu}_{\tau}$ relative to $\bar{B}\to D^{(\ast)}\ell^{-}\bar{\nu}_{\ell}$ decays with hadronic tagging at Belle”  In _Phys. Rev. D_ 92.7, 2015, pp. 072014  DOI: 10.1103/PhysRevD.92.072014
  * [3] LHCb Collaboration  “Measurement of the Ratio of Branching Fractions $\mathcal{B}(\bar{B}^{0}\to D^{*+}\tau^{-}\bar{\nu}_{\tau})/\mathcal{B}(\bar{B}^{0}\to D^{*+}\mu^{-}\bar{\nu}_{\mu})$”  In _Phys. Rev. Lett._ 115.11, 2015, pp. 111803  DOI: 10.1103/PhysRevLett.115.111803
  * [4] LHCb Collaboration  “Angular analysis of the $B^{0}\to K^{*0}\mu^{+}\mu^{-}$ decay using 3 fb-1 of integrated luminosity”  In _JHEP_ 02, 2016, pp. 104  DOI: 10.1007/JHEP02(2016)104
  * [5] Belle Collaboration  “Measurement of the $\tau$ Lepton Polarization and $R({D}^{*})$ in the Decay $\bar{B}\to D^{*}\tau^{-}\bar{\nu}_{\tau}$”  In _Phys. Rev. Lett._ 118.21, 2017, pp. 211801  DOI: 10.1103/PhysRevLett.118.211801
  * [6] Belle Collaboration  “Lepton-Flavor-Dependent Angular Analysis of $B\to K^{\ast}\ell^{+}\ell^{-}$”  In _Phys. Rev. Lett._ 118.11, 2017, pp. 111801  DOI: 10.1103/PhysRevLett.118.111801
  * [7] LHCb Collaboration  “Test of lepton universality with $B^{0}\rightarrow K^{*0}\ell^{+}\ell^{-}$ decays”  In _JHEP_ 08, 2017, pp. 055  DOI: 10.1007/JHEP08(2017)055
  * [8] LHCb Collaboration  “Measurement of the Ratio of Branching Fractions $\mathcal{B}(B_{c}^{+}\,\to\,J/\psi\tau^{+}\nu_{\tau})$ / $\mathcal{B}(B_{c}^{+}\,\to\,J/\psi\mu^{+}\nu_{\mu})$”  In _Phys. Rev. Lett._ 120.12, 2018, pp. 121801  DOI: 10.1103/PhysRevLett.120.121801
  * [9] LHCb Collaboration  “Test of lepton universality in beauty-quark decays”  In _Nature Phys._ 18.3, 2022, pp. 277–282  DOI: 10.1038/s41567-021-01478-8
  * [10] Gudrun Hiller and Martin Schmaltz  “$R_{K}$ and future $b\to s\ell\ell$ physics beyond the standard model opportunities”  In _Phys. Rev. D_ 90, 2014, pp. 054014  DOI: 10.1103/PhysRevD.90.054014
  * [11] Ben Gripaios, Marco Nardecchia and S.. Renner  “Composite leptoquarks and anomalies in $B$-meson decays”  In _JHEP_ 05, 2015, pp. 006  DOI: 10.1007/JHEP05(2015)006
  * [12] Marat Freytsis, Zoltan Ligeti and Joshua T. Ruderman  “Flavor models for $\bar{B}\to D^{(*)}\tau\bar{\nu}$”  In _Phys. Rev. D_ 92.5, 2015, pp. 054018  DOI: 10.1103/PhysRevD.92.054018
  * [13] Martin Bauer and Matthias Neubert  “Minimal Leptoquark Explanation for the R${}_{D^{(*)}}$ , RK , and $(g-2)_{g}$ Anomalies”  In _Phys. Rev. Lett._ 116.14, 2016, pp. 141802  DOI: 10.1103/PhysRevLett.116.141802
  * [14] Luca Di Luzio and Marco Nardecchia  “What is the scale of new physics behind the $B$-flavour anomalies?”  In _Eur. Phys. J. C_ 77.8, 2017, pp. 536  DOI: 10.1140/epjc/s10052-017-5118-9
  * [15] Dario Buttazzo, Admir Greljo, Gino Isidori and David Marzocca  “B-physics anomalies: a guide to combined explanations”  In _JHEP_ 11, 2017, pp. 044  DOI: 10.1007/JHEP11(2017)044
  * [16] James M. Cline  “$B$ decay anomalies and dark matter from vectorlike confinement”  In _Phys. Rev. D_ 97.1, 2018, pp. 015013  DOI: 10.1103/PhysRevD.97.015013
  * [17] Estefania Coluccio Leskow, Giancarlo D’Ambrosio, Andreas Crivellin and Dario Müller  “$\mathit{(g-2)_{\mu}}$, lepton flavor violation, and $Z$ decays with leptoquarks: Correlations and future prospects”  In _Phys. Rev. D_ 95, 2017, pp. 055018  DOI: 10.1103/PhysRevD.95.055018
  * [18] Debrupa Chakraverty, Debajyoti Choudhury and Anindya Datta  “A nonsupersymmetric resolution of the anomalous muon magnetic moment”  In _Phys. Lett. B_ 506, 2001, pp. 103–108  DOI: 10.1016/s0370-2693(01)00419-1
  * [19] Muon $g-2$ Collaboration  “Measurement of the Positive Muon Anomalous Magnetic Moment to 0.46 ppm”  In _Phys. Rev. Lett._ 126.14, 2021, pp. 141801  DOI: 10.1103/PhysRevLett.126.141801
  * [20] H. Georgi and S.L. Glashow  “Unity of All Elementary Particle Forces”  In _Phys. Rev. Lett._ 32, 1974, pp. 438–441  DOI: 10.1103/PhysRevLett.32.438
  * [21] J.. Pati and A. Salam  “Lepton number as the fourth “color””  In _Phys. Rev. D_ 10, 1974, pp. 275–289  DOI: 10.1103/PhysRevD.10.275
  * [22] W. Buchmüller and D. Wyler  “Constraints on SU(5)-type leptoquarks”  In _Phys. Lett. B_ 177, 1986, pp. 377–382  DOI: 10.1016/0370-2693(86)90771-9
  * [23] S. Dimopoulos and L. Susskind  “Mass without scalars”  In _Nucl. Phys. B_ 155.1, 1979, pp. 237–252  DOI: 10.1016/0550-3213(79)90364-X
  * [24] S. Dimopoulos  “Technicoloured signatures”  In _Nucl. Phys. B_ 168, 1980, pp. 69–92  DOI: 10.1016/0550-3213(80)90277-1
  * [25] E. Eichten and K. Lane  “Dynamical breaking of weak interaction symmetries”  In _Phys. Lett. B_ 90.1,2, 1980, pp. 125–130  DOI: 10.1016/0370-2693(80)90065-9
  * [26] L.. Abbott and E. Farhi  “Are the weak interactions strong?”  In _Phys. Lett. B_ 101, 1981, pp. 69  DOI: 10.1016/0370-2693(81)90492-5
  * [27] L.. Abbott and E. Farhi  “A confining model of the weak interactions”  In _Nucl. Phys. B_ 189, 1981, pp. 547  DOI: 10.1016/0550-3213(81)90580-0
  * [28] Barbara Schrempp and Fridger Schrempp  “Light leptoquarks”  In _Phys. Lett. B_ 153.1, 1985, pp. 101–107  DOI: 10.1016/0370-2693(85)91450-9
  * [29] W. Buchmüller, R. Rückl and D. Wyler  “Leptoquarks in lepton - quark collisions”  In _Phys. Lett. B_ 191, 1987, pp. 442–448  DOI: 10.1016/0370-2693(87)90637-X
  * [30] Rodrigo Alonso, Benjamı́n Grinstein and Jorge Martin Camalich  “Lepton universality violation and lepton flavor conservation in $B$-meson decays”  In _JHEP_ 10, 2015, pp. 184  DOI: 10.1007/JHEP10(2015)184
  * [31] Riccardo Barbieri, Gino Isidori, Andrea Pattori and Fabrizio Senia  “Anomalies in $B$-decays and $U(2)$ flavor symmetry”  In _Eur. Phys. J. C_ 76.2, 2016, pp. 67  DOI: 10.1140/epjc/s10052-016-3905-3
  * [32] ATLAS Collaboration  “Search for pair production of scalar leptoquarks decaying into first- or second-generation leptons and top quarks in proton–proton collisions at $\sqrt{s}=13\,\text{TeV}$ with the ATLAS detector”  In _Eur. Phys. J. C_ 81, 2021, pp. 313  DOI: 10.1140/epjc/s10052-021-09009-8
  * [33] ATLAS Collaboration  “Search for pairs of scalar leptoquarks decaying into quarks and electrons or muons in $\sqrt{s}=13\,\text{TeV}$ $pp$ collisions with the ATLAS detector”  In _JHEP_ 10, 2020, pp. 112  DOI: 10.1007/JHEP10(2020)112
  * [34] CMS Collaboration  “Search for Leptoquarks Coupled to Third-Generation Quarks in Proton–Proton Collisions at $\sqrt{s}=13\,\text{TeV}$”  In _Phys. Rev. Lett._ 121, 2018, pp. 241802  DOI: 10.1103/PhysRevLett.121.241802
  * [35] Tanumoy Mandal, Subhadip Mitra and Satyajit Seth  “Pair production of scalar leptoquarks at the LHC to NLO parton shower accuracy”  In _Phys. Rev. D_ 93.3, 2016, pp. 035018  DOI: 10.1103/PhysRevD.93.035018
  * [36] ATLAS Collaboration  “Searches for third-generation scalar leptoquarks in $\sqrt{s}=13\,\text{TeV}$ $pp$ collisions with the ATLAS detector”  In _JHEP_ 06, 2019, pp. 144  DOI: 10.1007/JHEP06(2019)144
  * [37] Michael J. Baker, Javier Fuentes-Martín, Gino Isidori and Matthias König  “High-$p_{T}$ signatures in vector–leptoquark models”  In _Eur. Phys. J. C_ 79.4 Springer ScienceBusiness Media LLC, 2019, pp. 334  DOI: 10.1140/epjc/s10052-019-6853-x
  * [38] ATLAS Collaboration  “The ATLAS Experiment at the CERN Large Hadron Collider”  In _JINST_ 3, 2008, pp. S08003  DOI: 10.1088/1748-0221/3/08/S08003
  * [39] ATLAS Collaboration  “The ATLAS Collaboration Software and Firmware”, ATL-SOFT-PUB-2021-001, 2021  URL: https://cds.cern.ch/record/2767187
  * [40] ATLAS Collaboration  “ATLAS data quality operations and performance for 2015–2018 data-taking”  In _JINST_ 15, 2020, pp. P04003  DOI: 10.1088/1748-0221/15/04/P04003
  * [41] Enrico Bothmann  “Event generation with Sherpa 2.2”  In _SciPost Phys._ 7.3, 2019, pp. 034  DOI: 10.21468/SciPostPhys.7.3.034
  * [42] D.. Lange  “The EvtGen particle decay simulation package”  In _Proceedings, 7th International Conference on B physics at hadron machines (BEAUTY 2000)_ 462, 2001, pp. 152  DOI: 10.1016/S0168-9002(01)00089-4
  * [43] T. Sjöstrand, S. Mrenna and P. Skands  “A brief introduction to PYTHIA 8.1”  In _Comput. Phys. Commun._ 178, 2008, pp. 852–867  DOI: 10.1016/j.cpc.2008.01.036
  * [44] ATLAS Collaboration  “The Pythia 8 A3 tune description of ATLAS minimum bias and inelastic measurements incorporating the Donnachie–Landshoff diffractive model”, ATL-PHYS-PUB-2016-017, 2016  URL: https://cds.cern.ch/record/2206965
  * [45] ATLAS Collaboration  “The ATLAS Simulation Infrastructure”  In _Eur. Phys. J. C_ 70, 2010, pp. 823  DOI: 10.1140/epjc/s10052-010-1429-9
  * [46] GEANT4 Collaboration and S. Agostinelli  “Geant4 – a simulation toolkit”  In _Nucl. Instrum. Meth. A_ 506, 2003, pp. 250  DOI: 10.1016/S0168-9002(03)01368-8
  * [47] J. Alwall et al.  “The automated computation of tree-level and next-to-leading order differential cross sections, and their matching to parton shower simulations”  In _JHEP_ 07, 2014, pp. 079  DOI: 10.1007/JHEP07(2014)079
  * [48] Richard D. Ball  “Parton distributions for the LHC run II”  In _JHEP_ 04, 2015, pp. 040  DOI: 10.1007/JHEP04(2015)040
  * [49] M. Krämer, T. Plehn, M. Spira and P.. Zerwas  “Pair production of scalar leptoquarks at the CERN LHC”  In _Phys. Rev. D_ 71, 2005, pp. 057503  DOI: 10.1103/PhysRevD.71.057503
  * [50] M. Krämer, T. Plehn, M. Spira and P.. Zerwas  “Pair Production of Scalar Leptoquarks at the Fermilab Tevatron”  In _Phys. Rev. Lett._ 79, 1997, pp. 341  DOI: 10.1103/PhysRevLett.79.341
  * [51] Pierre Artoisenet, Rikkert Frederix, Olivier Mattelaer and Robbert Rietkerk  “Automatic spin-entangled decays of heavy resonances in Monte Carlo simulations”  In _JHEP_ 03, 2013, pp. 015  DOI: 10.1007/JHEP03(2013)015
  * [52] Torbjörn Sjöstrand et al.  “An introduction to PYTHIA 8.2”  In _Comput. Phys. Commun._ 191, 2015, pp. 159  DOI: 10.1016/j.cpc.2015.01.024
  * [53] ATLAS Collaboration  “ATLAS Pythia 8 tunes to $7\leavevmode\nobreak\ \text{TeV}$ data”, ATL-PHYS-PUB-2014-021, 2014  URL: https://cds.cern.ch/record/1966419
  * [54] Richard D. Ball  “Parton distributions with LHC data”  In _Nucl. Phys. B_ 867, 2013, pp. 244  DOI: 10.1016/j.nuclphysb.2012.10.003
  * [55] Wim Beenakker et al.  “NNLL-fast: predictions for coloured supersymmetric particle production at the LHC with threshold and Coulomb resummation”  In _JHEP_ 12, 2016, pp. 133  DOI: 10.1007/JHEP12(2016)133
  * [56] W. Beenakker et al.  “Stop production at hadron colliders”  In _Nucl. Phys. B_ 515, 1998, pp. 3–14  DOI: 10.1016/S0550-3213(98)00014-5
  * [57] Wim Beenakker et al.  “Supersymmetric top and bottom squark production at hadron colliders”  In _JHEP_ 08, 2010, pp. 098  DOI: 10.1007/JHEP08(2010)098
  * [58] Wim Beenakker et al.  “NNLL resummation for stop pair-production at the LHC”  In _JHEP_ 05, 2016, pp. 153  DOI: 10.1007/JHEP05(2016)153
  * [59] Christoph Borschensky, Benjamin Fuks, Anna Kulesza and Daniel Schwartländer  “Scalar leptoquark pair production at hadron colliders”  In _Phys. Rev. D_ 101.11, 2020, pp. 115017  DOI: 10.1103/PhysRevD.101.115017
  * [60] URL: https://feynrules.irmp.ucl.ac.be/wiki/LeptoQuark
  * [61] Stefano Frixione, Giovanni Ridolfi and Paolo Nason  “A positive-weight next-to-leading-order Monte Carlo for heavy flavour hadroproduction”  In _JHEP_ 09, 2007, pp. 126  DOI: 10.1088/1126-6708/2007/09/126
  * [62] Paolo Nason  “A new method for combining NLO QCD with shower Monte Carlo algorithms”  In _JHEP_ 11, 2004, pp. 040  DOI: 10.1088/1126-6708/2004/11/040
  * [63] Stefano Frixione, Paolo Nason and Carlo Oleari  “Matching NLO QCD computations with parton shower simulations: the POWHEG method”  In _JHEP_ 11, 2007, pp. 070  DOI: 10.1088/1126-6708/2007/11/070
  * [64] Simone Alioli, Paolo Nason, Carlo Oleari and Emanuele Re  “A general framework for implementing NLO calculations in shower Monte Carlo programs: the POWHEG BOX”  In _JHEP_ 06, 2010, pp. 043  DOI: 10.1007/JHEP06(2010)043
  * [65] ATLAS Collaboration  “Studies on top-quark Monte Carlo modelling for Top2016”, ATL-PHYS-PUB-2016-020, 2016  URL: https://cds.cern.ch/record/2216168
  * [66] Michal Czakon and Alexander Mitov  “Top++: A program for the calculation of the top-pair cross-section at hadron colliders”  In _Comput. Phys. Commun._ 185, 2014, pp. 2930  DOI: 10.1016/j.cpc.2014.06.021
  * [67] Stefano Frixione et al.  “Single-top hadroproduction in association with a $W$ boson”  In _JHEP_ 07, 2008, pp. 029  DOI: 10.1088/1126-6708/2008/07/029
  * [68] Tanju Gleisberg and Stefan Höche  “Comix, a new matrix element generator”  In _JHEP_ 12, 2008, pp. 039  DOI: 10.1088/1126-6708/2008/12/039
  * [69] Fabio Cascioli, Philipp Maierhöfer and Stefano Pozzorini  “Scattering Amplitudes with Open Loops”  In _Phys. Rev. Lett._ 108, 2012, pp. 111601  DOI: 10.1103/PhysRevLett.108.111601
  * [70] Ansgar Denner, Stefan Dittmaier and Lars Hofer  “Collier: A fortran-based complex one-loop library in extended regularizations”  In _Comput. Phys. Commun._ 212, 2017, pp. 220–238  DOI: 10.1016/j.cpc.2016.10.013
  * [71] Steffen Schumann and Frank Krauss  “A parton shower algorithm based on Catani–Seymour dipole factorisation”  In _JHEP_ 03, 2008, pp. 038  DOI: 10.1088/1126-6708/2008/03/038
  * [72] Stefan Höche, Frank Krauss, Marek Schönherr and Frank Siegert  “A critical appraisal of NLO+PS matching methods”  In _JHEP_ 09, 2012, pp. 049  DOI: 10.1007/JHEP09(2012)049
  * [73] Stefan Höche, Frank Krauss, Marek Schönherr and Frank Siegert  “QCD matrix elements + parton showers. The NLO case”  In _JHEP_ 04, 2013, pp. 027  DOI: 10.1007/JHEP04(2013)027
  * [74] S. Catani, F. Krauss, B.. Webber and R. Kuhn  “QCD Matrix Elements + Parton Showers”  In _JHEP_ 11, 2001, pp. 063  DOI: 10.1088/1126-6708/2001/11/063
  * [75] Stefan Höche, Frank Krauss, Steffen Schumann and Frank Siegert  “QCD matrix elements and truncated showers”  In _JHEP_ 05, 2009, pp. 053  DOI: 10.1088/1126-6708/2009/05/053
  * [76] Charalampos Anastasiou, Lance Dixon, Kirill Melnikov and Frank Petriello  “High-precision QCD at hadron colliders: Electroweak gauge boson rapidity distributions at next-to-next-to leading order”  In _Phys. Rev. D_ 69, 2004, pp. 094008  DOI: 10.1103/PhysRevD.69.094008
  * [77] ATLAS Collaboration  “Electron and photon performance measurements with the ATLAS detector using the 2015–2017 LHC proton–proton collision data”  In _JINST_ 14, 2019, pp. P12006  DOI: 10.1088/1748-0221/14/12/P12006
  * [78] ATLAS Collaboration  “Muon reconstruction and identification efficiency in ATLAS using the full Run 2 $pp$ collision data set at $\sqrt{s}=13\,\text{TeV}$”  In _Eur. Phys. J. C_ 81, 2021, pp. 578  DOI: 10.1140/epjc/s10052-021-09233-2
  * [79] ATLAS Collaboration  “Jet reconstruction and performance using particle flow with the ATLAS Detector”  In _Eur. Phys. J. C_ 77, 2017, pp. 466  DOI: 10.1140/epjc/s10052-017-5031-2
  * [80] ATLAS Collaboration  “Jet energy scale and resolution measured in proton–proton collisions at $\sqrt{s}=13\,\text{TeV}$ with the ATLAS detector”  In _Eur. Phys. J. C_ 81, 2020, pp. 689  DOI: 10.1140/epjc/s10052-021-09402-3
  * [81] Matteo Cacciari, Gavin P. Salam and Gregory Soyez  “The anti-$k_{t}$ jet clustering algorithm”  In _JHEP_ 04, 2008, pp. 063  DOI: 10.1088/1126-6708/2008/04/063
  * [82] Matteo Cacciari, Gavin P. Salam and Gregory Soyez  “FastJet user manual”  In _Eur. Phys. J. C_ 72, 2012, pp. 1896  DOI: 10.1140/epjc/s10052-012-1896-2
  * [83] ATLAS Collaboration  “Tagging and suppression of pileup jets with the ATLAS detector”, ATLAS-CONF-2014-018, 2014  URL: https://cds.cern.ch/record/1700870
  * [84] ATLAS Collaboration  “ATLAS $b$-jet identification performance and efficiency measurement with $t\bar{t}$ events in $pp$ collisions at $\sqrt{s}=13\,\text{TeV}$”  In _Eur. Phys. J. C_ 79, 2019, pp. 970  DOI: 10.1140/epjc/s10052-019-7450-8
  * [85] ATLAS Collaboration  “Optimisation and performance studies of the ATLAS $b$-tagging algorithms for the 2017-18 LHC run”, ATL-PHYS-PUB-2017-013, 2017  URL: https://cds.cern.ch/record/2273281
  * [86] ATLAS Collaboration  “Search for top-squark pair production in final states with one lepton, jets, and missing transverse momentum using $36\,\text{fb}^{-1}$ of $\sqrt{s}=13\,\text{TeV}$ $pp$ collision data with the ATLAS detector”  In _JHEP_ 06, 2018, pp. 108  DOI: 10.1007/JHEP06(2018)108
  * [87] ATLAS Collaboration  “Performance of missing transverse momentum reconstruction with the ATLAS detector using proton–proton collisions at $\sqrt{s}=13\,\text{TeV}$”  In _Eur. Phys. J. C_ 78, 2018, pp. 903  DOI: 10.1140/epjc/s10052-018-6288-9
  * [88] ATLAS Collaboration  “Performance of the missing transverse momentum triggers for the ATLAS detector during Run-2 data taking”  In _JHEP_ 08, 2020, pp. 080  DOI: 10.1007/JHEP08(2020)080
  * [89] ATLAS Collaboration  “Measurement of the $t\bar{t}$ production cross-section and lepton differential distributions in $e\mu$ dilepton events from $pp$ collisions at $\sqrt{s}=13\,\text{TeV}$ with the ATLAS detector”  In _Eur. Phys. J. C_ 80, 2020, pp. 528  DOI: 10.1140/epjc/s10052-020-7907-9
  * [90] ATLAS Collaboration  “Measurements of top-quark pair differential and double-differential cross-sections in the $\ell$+jets channel with $pp$ collisions at $\sqrt{s}=13\,\text{TeV}$ using the ATLAS detector”  In _Eur. Phys. J. C_ 79, 2019, pp. 1028  DOI: 10.1140/epjc/s10052-019-7525-6
  * [91] C.G Lester and D.J Summers  “Measuring masses of semi-invisibly decaying particle pairs produced at hadron colliders”  In _Phys. Lett. B_ 463.1 Elsevier BV, 1999, pp. 99–103  DOI: 10.1016/s0370-2693(99)00945-4
  * [92] Alan J Barr, Ben Gripaios and Christopher G Lester  “Transverse masses and kinematic constraints: from the boundary to the crease”  In _JHEP_ 11.11 Springer ScienceBusiness Media LLC, 2009, pp. 096–096  DOI: 10.1088/1126-6708/2009/11/096
  * [93] Yang Bai, Hsin-Chia Cheng, Jason Gallicchio and Jiayin Gu  “Stop the top background of the stop search”  In _JHEP_ 07.7 Springer ScienceBusiness Media LLC, 2012, pp. 110  DOI: 10.1007/jhep07(2012)110
  * [94] Michael Feindt  “A Neural Bayesian Estimator for Conditional Probability Densities”, 2004  arXiv:physics/0402093 [physics.data-an]
  * [95] M. Feindt and U. Kerzel  “The NeuroBayes Neural Network Package”  In _Nucl. Instrum. Meth. A_ 559, 2006, pp. 190–194  DOI: 10.1016/j.nima.2005.11.166
  * [96] ATLAS Collaboration  “Studies of $t\bar{t}/tW$ interference effects in $b\bar{b}\ell^{+}\ell^{-^{\prime}}\nu\bar{\nu}^{\prime}$ final states with Powheg and MadGraph5_aMC@NLO setups”, ATL-PHYS-PUB-2021-042, 2021  URL: https://cds.cern.ch/record/2792254
  * [97] Johannes Bellm  “Herwig 7.1 Release Note”, 2017  arXiv:1705.06919 [hep-ph]
  * [98] Jon Butterworth  “PDF4LHC recommendations for LHC Run II”  In _J. Phys. G_ 43, 2016, pp. 023001  DOI: 10.1088/0954-3899/43/2/023001
  * [99] Emanuele Re  “Single-top $Wt$-channel production matched with parton showers using the POWHEG method”  In _Eur. Phys. J. C_ 71, 2011, pp. 1547  DOI: 10.1140/epjc/s10052-011-1547-z
  * [100] ATLAS Collaboration  “Multi-boson simulation for $13\leavevmode\nobreak\ \text{TeV}$ ATLAS analyses”, ATL-PHYS-PUB-2016-002, 2016  URL: https://cds.cern.ch/record/2119986
  * [101] ATLAS Collaboration  “Measurement of $W^{\pm}$ and $Z$-boson production cross sections in $pp$ collisions at $\sqrt{s}=13\,\text{TeV}$ with the ATLAS detector”  In _Phys. Lett. B_ 759, 2016, pp. 601  DOI: 10.1016/j.physletb.2016.06.023
  * [102] D. Florian  “Handbook of LHC Higgs Cross Sections: 4. Deciphering the Nature of the Higgs Sector”, 2016  DOI: 10.23731/CYRM-2017-002
  * [103] ATLAS Collaboration  “Measurement of the $t\bar{t}Z$ and $t\bar{t}W$ cross sections in proton–proton collisions at $\sqrt{s}=13\,\text{TeV}$ with the ATLAS detector”  In _Phys. Rev. D_ 99, 2019, pp. 072009  DOI: 10.1103/PhysRevD.99.072009
  * [104] ATLAS Collaboration  “$E_{\text{T}}^{\text{miss}}$ performance in the ATLAS detector using 2015–2016 LHC $pp$ collisions”, ATLAS-CONF-2018-023, 2018  URL: https://cds.cern.ch/record/2625233
  * [105] ATLAS Collaboration  “Luminosity determination in $pp$ collisions at $\sqrt{s}=13\,\text{TeV}$ using the ATLAS detector at the LHC”, ATLAS-CONF-2019-021, 2019  URL: https://cds.cern.ch/record/2677054
  * [106] G. Avoni  “The new LUCID-2 detector for luminosity measurement and monitoring in ATLAS”  In _JINST_ 13.07, 2018, pp. P07017  DOI: 10.1088/1748-0221/13/07/P07017
  * [107] Wouter Verkerke and David Kirkby  “The RooFit toolkit for data modeling”, 2003  arXiv:physics/0306116 [physics.data-an]
  * [108] Alexander L. Read  “Presentation of search results: the $CL_{S}$ technique”  In _J. Phys. G_ 28, 2002, pp. 2693  DOI: 10.1088/0954-3899/28/10/313
  * [109] Glen Cowan, Kyle Cranmer, Eilam Gross and Ofer Vitells  “Asymptotic formulae for likelihood-based tests of new physics”  In _Eur. Phys. J. C_ 71, 2011, pp. 1554  DOI: 10.1140/epjc/s10052-011-1554-0
  * [110] W. Buchmüller, R. Rückl and D. Wyler  In _Phys. Lett. B_ 448, 1999, pp. 320  DOI: 10.1016/S0370-2693(99)00014-3
  * [111] ATLAS Collaboration  In _Eur. Phys. J. C_ 80, 2020, pp. 1092  DOI: 10.1140/epjc/s10052-020-08541-3
  * [112] Glen Cowan, Kyle Cranmer, Eilam Gross and Ofer Vitells  In _Eur. Phys. J. C_ 73, 2013, pp. 2501  DOI: 10.1140/epjc/s10052-013-2501-z