Search for pair production of third-generation leptoquarks decaying into a
bottom quark and a $\tau$-lepton with the ATLAS detector A search for pair-
produced scalar or vector leptoquarks decaying into a $b$-quark and a
$\tau$-lepton is presented using the full LHC Run 2 (2015–2018) data sample of
139 fb-1 collected with the ATLAS detector in proton–proton collisions at a
centre-of-mass energy of $\sqrt{s}=13$ TeV. Events in which at least one
$\tau$-lepton decays hadronically are considered, and multivariate
discriminants are used to extract the signals. No significant deviations from
the Standard Model expectation are observed and 95% confidence-level upper
limits on the production cross-section are derived as a function of leptoquark
mass and branching ratio into the $\tau$-lepton. For scalar leptoquarks,
masses below 1490 GeV are excluded assuming a 100% branching ratio, while for
vector leptoquarks the corresponding limit is 1690 GeV (1960 GeV) in the
minimal-coupling (Yang–Mills) scenario. EXOT-2021-15 CERN-EP-2022-267

††journal: EPJC

##  1 Introduction

Many extensions of the Standard Model (SM) of particle physics predict
particles known as leptoquarks (LQs) [1, 2, 3, 4, 5, 6, 7]. These particles
provide a connection between the lepton and quark sectors, which are similar
in structure in the SM. LQs can be scalar (spin-0) or vector (spin-1) bosons,
and they carry colour and a fractional electric charge. They also have non-
zero lepton and baryon numbers, and decay into quark–lepton pairs. They can
mediate neutral currents, and therefore can potentially provide an explanation
for hints of violations of lepton universality observed in flavour experiments
[8, 9, 10, 11, 12, 13, 14].

LQs could be produced singly or in pairs in proton–proton ($pp$) collisions at
the LHC, and this analysis targets pair-produced LQs that couple strongly to
the third generation of quarks and leptons. Within the Buchmüller–Rückl–Wyler
(BRW) model [15], which is the benchmark for scalar LQs in this analysis, it
is assumed that these LQs can only interact within the same family via a
Yukawa interaction. This interaction is described by two parameters, a model
parameter $\beta$ and a coupling parameter $\lambda$. In the benchmark models
considered in this paper, the pair-production cross-section is independent of
$\lambda$. This analysis also considers pair-production of vector LQs [16]
corresponding to the $U_{1}$ state in the BRW classification [15]. The
scenarios considered in this model differ by a dimensionless coupling constant
$k$, which is zero for the minimal-coupling scenario and one for the
Yang–Mills scenario. For both scalar and vector LQs, the parameter $\beta$
controls the decay into charged leptons. For these third generation LQs,
results are generally given in terms of the branching ratio (${\cal B}$) and
mass of the LQ ($m_{\text{LQ}}$).

ATLAS and CMS have published searches for LQs coupling to the first, second
and third generations [17, 18, 19, 20, 21, 22, 23, 24, 25, 26]. Each
generation of LQs is split into up-type and down-type LQs with different
electric charges. For instance, for the third generation they are split into
up-type LQs ($\text{LQ}^{\text{u}}_{3}$), which decay into $b\tau$ or $t\nu$,
and down-type LQs ($\text{LQ}^{\text{d}}_{3}$), which decay into $t\tau$ or
$b\nu$. Both types of LQs are currently excluded for masses below 1150 GeV for
the BRW model, for all values of ${\cal B}$.

This paper updates the ATLAS search for an up-type LQ pair decaying into
$b\tau$ [18], shown in Figure 1, using the full Run 2 data sample and an
updated analysis strategy, prioritising high LQ masses that are not yet
excluded for the benchmark models considered. Analysis improvements include
updated analysis-optimisation and background-estimation methods, as well as
updates to several object identification algorithms. The analysis signature is
two jets, at least one of which must be identified as containing a $b$-hadron,
and two $\tau$-leptons. For the $\tau$-leptons, the cases considered are where
both decay hadronically or where one $\tau$-lepton decays into a light lepton
(electron or muon, $\ell$) and neutrinos and the other decays hadronically.
The mass range considered for the LQ is from 300 GeV to 2000 GeV. The
extraction of the signals is performed through a simultaneous likelihood fit
to multivariate discriminants. For the results, both scalar and vector LQs are
considered, with the limits on vector LQs interpreted in the context of two
scenarios, the Yang–Mills scenario and the minimal-coupling scenario [27].

Figure 1: Pair production of a leptoquark (LQ) and its subsequent decay into a
$b$-quark and a $\tau$-lepton.

The paper is structured as follows. After a brief description of the ATLAS
detector, the data sample, simulated backgrounds and simulated signals are
described. This is followed by a description of the event reconstruction, the
object selection, the event selections for the signal regions, and the
multivariate discriminants that are used in the final fit. The next sections
include a description of the data-driven background estimation methods, the
systematic uncertainties, and finally the statistical methods and results.

##  2 ATLAS detector

The ATLAS detector [28] 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 $\Delta R\equiv\sqrt{(\Delta\eta)^{2}+(\Delta\phi)^{2}}$. The inner
tracking detector consists of pixel and microstrip silicon detectors covering
the pseudorapidity region $|\eta|<2.5$, surrounded by a transition radiation
tracker to enhance electron identification in the range of $|\eta|<2.0$. An
additional innermost pixel layer, the insertable B-layer [29, 30], was added
before Run 2 of the LHC. The inner detector (ID) is surrounded by a thin
superconducting solenoid providing a $2\text{\,}\mathrm{T}$ axial magnetic
field, and by a fine-granularity lead/liquid-argon (LAr) electromagnetic (EM)
calorimeter covering $|\eta|<3.2$. Hadronic calorimetry is provided by a
steel/scintillator-tile calorimeter in 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. Three layers of high-precision tracking chambers provide coverage
in the range of $|\eta|<2.7$, while dedicated fast chambers allow triggering
in the region $|\eta|<2.4$. A two-level trigger system [31], consisting of a
hardware-based first-level trigger followed by a software-based high-level
trigger (HLT), is used to select events. An extensive software suite [32] 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 Data and simulation samples

The data used in this search correspond to an integrated luminosity of
$139\,\mbox{fb${}^{-1}$}$ of $pp$ collision data collected by the ATLAS
detector between 2015 and 2018 at a centre-of-mass energy $\sqrt{s}=13$ TeV.
The uncertainty in the combined 2015–2018 integrated luminosity is 1.7% [33],
obtained using the LUCID-2 detector [34] for the primary luminosity
measurements. The presence of additional interactions in the same or
neighbouring bunch crossing, referred to as pile-up, is characterised by the
average number of such interactions, $\langle\mu\rangle$, which was $33.7$ for
the combined data sample. Only events recorded under stable beam conditions
and for which all relevant detector subsystems were known to be in a good
operating condition are used.

Dedicated Monte Carlo (MC) simulated samples are used to model SM processes
and estimate the expected signal yields. All samples were passed through the
full ATLAS detector simulation [35] based on Geant4 [36], except for the
signal samples that use a parameterised fast simulation of the calorimeter
response [37] and Geant4 for the other detector systems. The simulated events
were reconstructed with the same algorithms as used for data. They contain a
realistic modelling of pile-up interactions with pile-up profiles matching the
ones of each data sample between 2015 and 2018, obtained by overlaying
minimum-bias events simulated using the soft QCD processes of Pythia 8.186
[38] with the NNPDF2.3 leading-order (LO) [39] set of parton distribution
functions (PDFs) and the A3 [40] set of tuned parameters (tune). The MC
samples are corrected to account for differences between simulation and data
in terms of the pile-up, the energy and momentum scales, and the
reconstruction and identification efficiencies of physics objects.

Simulated events with pair-produced up-type ($Q=+\frac{2}{3}$) scalar LQs were
generated at next-to-leading order (NLO) in QCD with MadGraph5_aMC@NLO v2.6.0
[41], using the LQ model of Ref. [42], in which fixed-order NLO QCD
calculations [43, 44] are interfaced to Pythia 8.230 [45] for the parton
shower (PS) and hadronisation. Parton luminosities were provided by the five-
flavour scheme NNPDF3.0 NLO [46] PDF set with $\alpha_{s}=0.118$ and the
underlying event (UE) was modelled with the A14 tune [47, 48]. The coupling
parameter $\lambda$ was set to 0.3, resulting in a relative LQ width of
approximately 0.2% and ensuring the LQs decay promptly. In all cases,
$\beta=0.5$ such that the couplings to charged leptons and neutrinos were
equal and the decay products were interfaced to MadSpin [49] to preserve spin
correlations. Different values for ${\cal B}$ were then obtained by
reweighting the simulated events according to the generator information about
their decay following the procedure in Ref. [18]. Signal cross-sections were
obtained from the calculation of the pair production of scalar coloured
particles, such as the hypothesised supersymmetric partner of the top quark,
as these particles have the same production modes and their pair-production
cross-section depends only on their mass. These processes were 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 [50, 51, 52, 53].
The cross-sections do not include contributions from $t$-channel lepton
exchange, which are neglected in Ref. [42] and may lead to corrections at the
percent level [54]. The nominal cross-section and its uncertainty were derived
using the PDF4LHC15_mc PDF set, following the recommendations of Ref. [55].
For LQ masses between 300 GeV and 2000 GeV, the cross-sections range from 10
pb to 0.01 fb.

Simulated events with pair-produced up-type vector LQs were generated at LO in
QCD with MadGraph5_aMC@NLO v2.6.0, using the LQ model of Ref. [16] and the
NNPDF3.0 NLO PDF set with $\alpha_{s}=0.118$. Decays of the LQs were performed
with MadSpin, while PS and hadronisation were simulated using Pythia 8.244
with the A14 tune. The full model includes two additional vector states that
are necessary to obtain a realistic extension of the SM, a colour singlet
$Z^{\prime}$ and a colour octet $G^{\prime}$. However, these are not present
in the MadGraph model and hence do not contribute to the Feynman diagrams
considered for pair production of vector leptoquarks. The samples were
produced with a coupling strength $g_{U}=3.0$, where $g_{U}$ represents the
overall coupling between the LQ and the fermion, motivated by a suppression of
the production cross-section for the additional mediators in the ultraviolet
completion of the model, which might otherwise be in tension with existing LHC
limits. This choice of coupling results in a relative LQ width of around 10%.
In all cases, $\beta=0.5$ and the same reweighting as in the scalar LQ case is
then used to probe different ${\cal B}$ values. As mentioned, the model
introduces two different coupling scenarios the minimal-coupling scenario and
the Yang–Mills scenario. In the latter case the LQ is a massive gauge boson
and has additional couplings to the SM gauge bosons, resulting in enhanced
cross-sections. Since no higher-order cross-sections are available for this
model, the LO MadGraph5_aMC@NLO cross-sections were used and vary between 94
pb (340 pb) and 0.05 fb (0.61 fb) for LQ masses between 300 GeV and 2000 GeV
in the minimal-coupling (Yang–Mills) case. Above 500 GeV, kinematic
differences between the two scenarios are negligible.

Scalar (vector) LQ samples were produced with LQ masses between 300 GeV to
2000 GeV, with a mass interval of 50 GeV in the range of 800–1600 GeV
(1400–1600 GeV) and 100 GeV otherwise.

Table 1: The list of generators used for the simulation of the SM background
processes. Information is given on the matrix element (ME) generator
(including the perturbative QCD order), the PDF set, the parton shower (PS)
and the underlying event (UE). The perturbative order (in QCD unless otherwise
specified) of the cross-section used to normalise the different samples is
also presented. $(\S)$ The $t\bar{t}-Wt$ interference was handled using the
diagram removal scheme. $({\dagger})$ The cross-sections from Sherpa at NLO
were used to normalise the $WW,WZ,ZZ$ and $t\bar{t}W/Z$ events. $({\ddagger})$
The $qq\to ZH$ process was normalised to the NNLO (QCD) + NLO( EW) cross-
section for the $pp\to ZH$ process [56, 57, 58, 59, 60, 61], after subtracting
the $gg\to ZH$ contribution.

Process | ME generator | ME QCD | ME PDF | PS and | UE | Cross-section
---|---|---|---|---|---|---
|  | order |  | hadronisation | tune | order
Top-quark
$t\bar{t}^{(\S)}$ |  Powheg-Box v2 [62] | NLO | NNPDF3.0NLO | Pythia 8.230 | A14 |  NNLO+NNLL [63]
$t$-channel | Powheg-Box v2 | NLO | NNPDF3.0NLO | Pythia 8.230 | A14 |  NLO [64]
$s$-channel | Powheg-Box v2 | NLO | NNPDF3.0NLO | Pythia 8.230 | A14 |  NLO [65]
$Wt^{(\S)}$ | Powheg-Box v2 | NLO | NNPDF3.0NLO | Pythia 8.230 | A14 |  NLO [66]
Top-quark + $W/Z$
$t\bar{t}Z$ |  Sherpa 2.2.1 [67, 68, 69] | NLO | NNPDF3.0NNLO | Sherpa 2.2.1 | Default |  NLO(†)
$t\bar{t}W$ | Sherpa 2.2.8 | NLO | NNPDF3.0NNLO | Sherpa 2.2.8 | Default |  NLO(†)
Vector boson + jets
$W/Z+$jets | Sherpa 2.2.1 |  NLO ($\leq 2$ jets) | NNPDF3.0NNLO | Sherpa 2.2.1 | Default |  NNLO [70]
|  | LO (3,4 jets) |  |  |  |
Diboson
$WW,WZ,ZZ$ | Sherpa 2.2.1 |  NLO ($\leq 1$ jet) | NNPDF3.0NNLO | Sherpa 2.2.1 | Default |  NLO(†)
|  | LO (2,3 jets) |  |  |  |
Higgs boson
ggF | Powheg-Box v2 | NNLO | NNPDF3.0NLO | Pythia 8.212 |  AZNLO [71] |  N3LO(QCD)+NLO(EW) [72, 73, 74, 75, 76]
VBF | Powheg-Box v2 | NLO | NNPDF3.0NLO | Pythia 8.212 | AZNLO |  NNLO(QCD)+NLO(EW) [72, 77, 78, 79]
$qq\to WH$ | Powheg-Box v2 | NLO | NNPDF3.0NLO | Pythia 8.212 | AZNLO |  NNLO(QCD)+NLO(EW) [56, 57, 58, 59, 80, 61]
$qq\to ZH$ | Powheg-Box v2 | NLO | NNPDF3.0NLO | Pythia 8.212 | AZNLO |  NNLO(QCD)+NLO(EW)(‡)
$gg\to ZH$ | Powheg-Box v2 | NLO | NNPDF3.0NLO | Pythia 8.212 | AZNLO |  NLO+NLL [81, 82, 83, 84, 85]
$t\bar{t}H$ | Powheg-Box v2 | NLO | NNPDF3.0NLO | Pythia 8.230 | A14 |  NLO [72]

Background samples were simulated using different MC event generators
depending on the process. All background processes are normalised to the most
accurate available theoretical calculation of their respective cross-sections.
The most relevant event generators, the accuracy of theoretical cross-
sections, the UE parameter tunes, and the PDF sets used in simulating the SM
background processes are summarised in Table 1. For all samples, except those
generated using Sherpa, the EvtGen v1.2.0 [86] program was used to simulate
the properties of the $b$\- and $c$-hadron decays.

##  4 Event reconstruction and object definitions

The LQ signature of interest in this search gives rise to a set of
reconstructed objects that consist primarily of $\tau$-leptons, which may
decay into light leptons or hadronically, and jets from the hadronisation of
quarks, specifically $b$-quarks. In addition, neutrinos produced in the decay
of $\tau$-leptons and the semileptonic decay of $b$-hadrons contribute to the
missing transverse momentum ${\boldsymbol{p}_{\text{T}}^{\text{miss}}}$ of the
event. To be considered for analysis, events are required to have at least one
$pp$ interaction vertex, reconstructed from two or more charged-particle
tracks with $p_{\text{T}}>500$ MeV; the one with the highest summed
$p_{\text{T}}^{2}$ of associated tracks is selected as the primary vertex.

Electron candidates are reconstructed by matching ID tracks to energy clusters
in the EM calorimeter. They must satisfy $p_{\text{T}}>7$ GeV and lie in the
range of $|\eta|<2.47$, excluding the transition region between the barrel and
endcap detectors ($1.37<|\eta|<1.52$). Electrons are further identified using
a likelihood-based method, based on the track quality, the profile of the
shower measured in the EM calorimeter and the consistency between the track
and the energy cluster [87]. Two identification criteria are used to select
electrons in this analysis: ‘veto electrons’ are required to satisfy the
‘loose’ identification working point, while ‘signal electrons’ are required to
satisfy the more stringent ‘tight’ working point.

Muon candidates are reconstructed from tracks in the MS, matched with
compatible tracks in the ID where coverage allows; in regions where the MS is
only partially instrumented ($|\eta|<0.1$) an energy deposit in the
calorimeter compatible with a minimum-ionising particle is combined with a
compatible ID track instead. They must satisfy $p_{\text{T}}>7$ GeV and lie in
the range of $|\eta|<2.7$. Muons are further identified based on the number of
hits in the various ID subdetectors and MS stations, the compatibility between
the measurements in the two detectors and the properties of the resulting
track fit. Two identification criteria [88] are used to select muons: ‘veto
muons’ must satisfy a ‘loose’ identification requirement, while the ‘signal
muons’ are required to satisfy the ‘medium’ (‘high-$p_{\text{T}}$’) working
point if the $p_{\text{T}}$ is less than (greater than) 800 GeV. The more
stringent high-$p_{\text{T}}$ requirements remove around $20\%$ of muons but
improve the $p_{\text{T}}$ resolution by $\approx 30\%$ above 1.5 TeV,
significantly suppressing potential backgrounds [89].

To suppress misidentified leptons or those arising from hadron decays, all
light-lepton candidates must satisfy an isolation criterion that limits the
presence of tracks (calorimeter deposits) in a $p_{\text{T}}$-dependent
(fixed) radius cone. The resulting efficiency is above 99% for both electrons
and muons in the signal regions. Finally, signal leptons must satisfy stricter
requirements on their transverse momenta depending on the data-taking period,
as detailed in Section 5.

Jets are reconstructed from topological energy clusters and charged-particle
tracks, resulting from a particle-flow algorithm [90], using the anti-$k_{t}$
algorithm with a radius parameter of $R=0.4$ [91, 92]. They are required to
satisfy $p_{\text{T}}>20$ GeV and lie in the range of $|\eta|<2.5$. To
suppress jets from pile-up, jets with $p_{\text{T}}<60$ GeV and $|\eta|<2.4$
are required to originate from the primary vertex using a multivariate ‘jet
vertex tagger’ [93]. A multivariate algorithm based on a deep neural network,
known as the ‘DL1r tagger’ [94, 95, 96], is used to identify jets containing
$b$-hadrons ($b$-jets) based on the jet kinematics, the impact parameters of
tracks associated with the jet and the reconstruction of displaced vertices.
This analysis uses a working point with a 77% efficiency for true $b$-jets and
corresponding rejection factors for light-flavour jets, charm jets and
$\tau$-leptons, measured in simulated $t\bar{t}$ events, of 170, 5 and 21,
respectively [97, 98].

Hadronically decaying $\tau$-lepton candidates are seeded by jets, which are
required to have one or three associated tracks (referred to hereafter as
‘one-prong’ or ‘three-prong’ candidates, respectively) with a total charge of
$\pm 1$ [99]. The transverse momentum of the visible decay products
($\tau_{\text{had-vis}}$) must satisfy $p_{\text{T}}>20$ GeV and lie in the
range of $|\eta|<2.47$, excluding the transition region defined above. True
$\tau_{\text{had-vis}}$ candidates are discriminated from quark- and gluon-
initiated jets via a recurrent neural network (RNN) using calorimeter- and
tracking-based variables as input and trained separately on one- and three-
prong candidates [100]. The ‘loose’ working point used has an efficiency of
approximately $85\%$ and $75\%$ for one- and three-prong $\tau_{\text{had-
vis}}$ respectively. A further boosted decision tree (BDT) is used to reject
one-prong $\tau_{\text{had-vis}}$ candidates originating from electrons with
an efficiency of about $95\%$ [101]. For the estimation of the background from
jets misidentified as $\tau_{\text{had-vis}}$ (described in Section 6),
anti-$\tau_{\text{had-vis}}$ candidates are defined in the same way as above
but are required to fail to satisfy the nominal loose RNN working point
requirements and instead satisfy a looser requirement that has an efficiency
of $99\%$ for selecting true $\tau_{\text{had-vis}}$ candidates.

The ${\boldsymbol{p}_{\text{T}}^{\text{miss}}}$ (with magnitude
$E_{\text{T}}^{\text{miss}}$) is computed from the negative vectorial sum of
the selected and calibrated objects described above, along with an extra
track-based ‘soft term’ to account for the energy of particles originating
from the primary vertex but not associated to any of the reconstructed objects
[102, 103].

To resolve ambiguities whereby the same detector signature may be
reconstructed as more than one physics object, a sequential overlap-removal
procedure is applied. First, electron candidates are discarded if they share a
track with a more energetic electron or a muon identified in the MS; if the
muon is identified in the calorimeter it is removed instead. Any
$\tau_{\text{had-vis}}$ candidate within $\Delta R=0.2$ of an electron or a
muon (which must be reconstructed in the MS if the $\tau_{\text{had-vis}}$
$p_{\text{T}}$ is above 50 GeV) is then rejected. Jets are discarded if they
lie within $\Delta R=0.2$ of an electron or have fewer than three associated
tracks and lie within the same distance of a muon. Electron or muon
($\tau_{\text{had-vis}}$) candidates within $\Delta R=0.4$ ($\Delta R=0.2$) of
any remaining jet are then removed. Finally, ambiguities between
anti-$\tau_{\text{had-vis}}$ candidates and jets within $\Delta R=0.2$ are
resolved in favour of the jet if it is $b$-tagged or the
anti-$\tau_{\text{had-vis}}$ otherwise.

##  5 Event selection

The event selection targets a signature consisting of a pair of $\tau$-leptons
and a pair of $b$-quarks. It splits the events into two orthogonal signal
categories based on the $\tau$-lepton decay mode: the
$\tau_{\text{lep}}\tau_{\text{had}}$ channel, which selects events with a
light lepton, an oppositely charged $\tau_{\text{had-vis}}$ and one or two
$b$-jets, and the $\tau_{\text{had}}\tau_{\text{had}}$ channel, which selects
events with two opposite-charge $\tau_{\text{had-vis}}$ and one or two
$b$-jets. Multivariate techniques are used to search for a LQ-pair signal in
the two signal regions (SRs).

###  5.1 Signal regions

Candidate events were recorded using a combination of single-lepton [104, 105]
and single-$\tau_{\text{had-vis}}$ triggers [106]. The single-lepton trigger
used in the $\tau_{\text{lep}}\tau_{\text{had}}$ channel required a
reconstructed light lepton at the HLT, with a minimum $E_{\text{T}}$ threshold
ranging from 24 to 26 GeV for electrons and a minimum $p_{\text{T}}$ threshold
ranging from 20 to 25 GeV for the muons, depending on the data-taking period.
Offline leptons are required to be geometrically matched to the corresponding
trigger object and have a $p_{\text{T}}$ threshold 1–2 GeV above the HLT
threshold in order to operate in the region where the trigger was fully
efficient. The single-$\tau_{\text{had-vis}}$ triggers used in the
$\tau_{\text{had}}\tau_{\text{had}}$ channel required a reconstructed HLT
$\tau_{\text{had-vis}}$ with a period-dependent minimum $p_{\text{T}}$
threshold ranging between 80 GeV and 160 GeV. The corresponding
$p_{\text{T}}$-threshold for the offline $\tau_{\text{had-vis}}$, which is
again required to be geometrically matched to the trigger object, ranges
between 100 GeV and 180 GeV, while the non-trigger-matched $\tau_{\text{had-
vis}}$ is required to have $p_{\text{T}}>20$ GeV.

Following the trigger selection, the $\tau_{\text{lep}}\tau_{\text{had}}$
category requires exactly one ‘signal’ light lepton and an oppositely charged
$\tau_{\text{had-vis}}$, while the $\tau_{\text{had}}\tau_{\text{had}}$
category requires exactly two opposite-charge $\tau_{\text{had-vis}}$ and no
‘veto’ light leptons. Both categories require at least two jets, one or two of
which must be $b$-tagged, with $p_{\text{T}}>45\leavevmode\nobreak\ (20)$ GeV
for the leading (sub-leading) jet.

The invariant mass of the two $\tau$-lepton decay products is an important
variable with which to reject the $Z$+jets background. It is calculated using
the missing mass calculator (MMC) [107], with the light lepton and the
$\tau_{\text{had-vis}}$ (two $\tau_{\text{had-vis}}$) and the
${\boldsymbol{p}_{\text{T}}^{\text{miss}}}$ as input in the
$\tau_{\text{lep}}\tau_{\text{had}}$ ($\tau_{\text{had}}\tau_{\text{had}}$)
category, and it is required to satisfy $m_{\tau\tau}^{\text{MMC}}\notin
40-150$ GeV. Two further selections are applied to target the characteristic
LQ signature while reducing the large multi-jet background. The scalar sum of
the transverse momenta ($s_{\text{T}}$), calculated taking into account the
light lepton, $\tau_{\text{had-vis}}$, two leading jets and the
$E_{\text{T}}^{\text{miss}}$, is a powerful discriminator. It is required to
satisfy $s_{\text{T}}>600$ GeV, while the $E_{\text{T}}^{\text{miss}}$ itself
is required to exceed 100 GeV.

The full event selection is summarised in Table 2 and the resulting acceptance
times efficiency is shown in Figure 2 as a function of $m_{\text{LQ}}$. Since
the analysis prioritises high mass LQs that have not yet been excluded in the
benchmark models under consideration, it is not optimal for low LQ masses.

Table 2: Summary of the event selections for the $\tau_{\text{lep}}\tau_{\text{had}}$ and $\tau_{\text{had}}\tau_{\text{had}}$ categories. Where two objects are required, the thresholds on the sub-leading object are given in parenthesis. Where the selection depends on data-taking period, the different possible threshold values are separated by commas. |  $\tau_{\text{lep}}\tau_{\text{had}}$ channel |  $\tau_{\text{had}}\tau_{\text{had}}$ channel |
---|---|---|---
$e/\mu$ selection |  $=1$ ‘signal’ $e$ or $\mu$ | No ‘veto’ $e$ or $\mu$ |
$p_{\text{T}}^{e}>25,27$ GeV |  |
$p_{\text{T}}^{\mu}>21,27$ GeV |  |
$\tau_{\text{had-vis}}$ selection |  $=1$ $\tau_{\text{had-vis}}$ |  $=2$ $\tau_{\text{had-vis}}$ |
$p_{\text{T}}^{\tau}>100$ GeV |  $p_{\text{T}}^{\tau}>100,140,180\leavevmode\nobreak\ (20)$ GeV |
Jet selection |  $\geq 2$ jets |
$p_{\text{T}}^{\text{jet}}>45\leavevmode\nobreak\ (20)$ GeV |
1 or 2 $b$-jets |
Additional selection | Opposite charge $e,\mu,\tau_{\text{had}}$ and $\tau_{\text{had}}$ |
$m_{\tau\tau}^{\text{MMC}}\notin 40-150$ GeV |
$E_{\text{T}}^{\text{miss}}>100$ GeV |
$s_{\text{T}}>600$ GeV |

Figure 2: The expected acceptance times efficiency for the scalar and vector
LQs, with both the minimal-coupling and the Yang–Mills scenarios, at
$\beta=0.5$ as a function of $m_{\text{LQ}}$ in the (a)
$\tau_{\text{lep}}\tau_{\text{had}}$ and (b)
$\tau_{\text{had}}\tau_{\text{had}}$ channels. The values include the leptonic
and hadronic branching ratios of the tau lepton.

###  5.2 Multivariate signal extraction

Following the event selection, the LQ signal is extracted using a multivariate
discriminant. To obtain near-optimal sensitivity and continuity over the full
range of LQ masses considered, a parameterised neural network (PNN) [108],
parameterised in terms of the generated LQ mass, is chosen. The PNN consists
of three hidden layers, each with 32 nodes, implemented in Keras [109] with
the Tensorflow [110] backend.

The PNN inputs consist of a combination of multiplicity, kinematic and angular
quantities that discriminate between the signal and the dominant background.
In the case of the $b\tau$ invariant mass, the most likely combination of the
$\tau$-lepton and a $b$-jet222In the case of only one $b$-jet, the
highest-$p_{\text{T}}$ non-$b$-jet is taken as the second jet. is chosen based
on a mass-pairing strategy that minimises the mass difference between the two
resulting LQ candidates. The variables, which are similar for both the
$\tau_{\text{lep}}\tau_{\text{had}}$ and $\tau_{\text{had}}\tau_{\text{had}}$
categories, are summarised in Table 3 and defined as follows:

  * •

$\tau_{\text{had-vis}}$ $p_{\text{T}}^{0}$ is the transverse momentum of the
highest-$p_{\text{T}}$ $\tau_{\text{had-vis}}$;

  * •

$s_{\text{T}}$ is the scalar sum of the transverse momenta defined above;

  * •

$N_{b-\text{jets}}$ is the number of $b$-jets;

  * •

$m(\tau,\text{jet})_{0,1}$ are the larger (0) and smaller (1) of the two LQ
masses obtained via the mass-pairing strategy
($\tau_{\text{had}}\tau_{\text{had}}$ channel only);

  * •

$m(\ell,\text{jet})$ and $m(\tau_{\text{had}},\text{jet})$ are the mass of the
light-lepton or $\tau_{\text{had-vis}}$, respectively, combined with its mass-
paired $b$-jet ($\tau_{\text{lep}}\tau_{\text{had}}$ channel only);

  * •

$\Delta R(\ell,\text{jet})$ ($\Delta R(\tau_{\text{had}},\text{jet})$) is the
$\Delta R$ between the light lepton (leading $\tau_{\text{had-vis}}$) and the
mass-paired jet in the $\tau_{\text{lep}}\tau_{\text{had}}$
($\tau_{\text{had}}\tau_{\text{had}}$) category;

  * •

$\Delta\phi(\ell,E_{\text{T}}^{\text{miss}})$ is the azimuthal opening angle
between the lepton and the $E_{\text{T}}^{\text{miss}}$
($\tau_{\text{lep}}\tau_{\text{had}}$ category only);

  * •

$E_{\text{T}}^{\text{miss}}$ $\phi$ centrality quantifies the transverse
direction of the ${\boldsymbol{p}_{\text{T}}^{\text{miss}}}$ relative to the
light lepton and $\tau_{\text{had-vis}}$ (two $\tau_{\text{had-vis}}$) in the
$\tau_{\text{lep}}\tau_{\text{had}}$ ($\tau_{\text{had}}\tau_{\text{had}}$)
category and is defined in Ref. [111].

A selection of representative input distributions, after the background
corrections described in Section 6, are presented in Figures 3 and 4 for the
$\tau_{\text{lep}}\tau_{\text{had}}$ SR and the
$\tau_{\text{had}}\tau_{\text{had}}$ SR, respectively. While the relative
importance of the variables varies with LQ mass, the $s_{\text{T}}$ and mass
variables are generally the most performant. The resulting PNN score
distributions, which peak at higher values for LQ signals than the background
processes, are used as the final analysis discriminant.

Table 3: Summary of variables used as inputs to the PNN in the $\tau_{\text{lep}}\tau_{\text{had}}$ and $\tau_{\text{had}}\tau_{\text{had}}$ categories. The variables are defined in the text. Variable |  $\tau_{\text{lep}}\tau_{\text{had}}$ channel |  $\tau_{\text{had}}\tau_{\text{had}}$ channel
---|---|---
$\tau_{\text{had-vis}}$ $p_{\text{T}}^{0}$ | ✓ | ✓
$s_{\text{T}}$ | ✓ | ✓
$N_{b-\text{jets}}$ | ✓ | ✓
$m({\tau,\text{jet}})_{0,1}$ |  | ✓
$m(\ell,\text{jet})$, $m(\tau_{\text{had}},\text{jet})$ | ✓ |
$\Delta R(\tau,\text{jet})$ | ✓ | ✓
$\Delta\phi(\ell,E_{\text{T}}^{\text{miss}})$ | ✓ |
$E_{\text{T}}^{\text{miss}}$ $\phi$ centrality | ✓ | ✓

Figure 3: Signal (solid lines), post-fit background (filled histograms) and
data (dots with error bars) distributions of representative PNN input
variables in the $\tau_{\text{lep}}\tau_{\text{had}}$ SR: (a) $\Delta
R(\ell,\text{jet})$, (b), $m(\tau_{\text{had}},\text{jet})$ and (c)
$s_{\text{T}}$. The normalisation and shape of the backgrounds are determined
from the background-only likelihood fit to data and the ratios of the data to
the sum of the predicted backgrounds are shown in the lower panels. ‘Other’
refers to the sum of minor backgrounds (vector boson + jets, diboson and Higgs
boson). The hatched band indicates the combined statistical and systematic
uncertainty in the total background prediction. The expected signal for a 1.4
TeV scalar LQ, scaled by the indicated factor for visibility, is overlaid. The
last bin includes the overflow.

Figure 4: Signal (solid lines), post-fit background (filled histograms) and
data (dots with error bars) distributions of representative PNN input
variables in the $\tau_{\text{had}}\tau_{\text{had}}$ SR: (a) $\Delta
R(\tau_{\text{had}}^{0},\text{jet})$ where $\tau_{\text{had}}^{0}$ is the
leading $\tau$-lepton, (b) the larger of the two $\tau$-jet mass combinations
$m(\tau_{\text{had}},\text{jet})_{0}$ and (c) $s_{\text{T}}$. The
normalisation and shape of the backgrounds are determined from the background-
only likelihood fit to data and the ratios of the data to the sum of the
predicted backgrounds are shown in the lower panels. ‘Other’ refers to the sum
of minor backgrounds (vector boson + jets, diboson and Higgs boson). The
hatched band indicates the combined statistical and systematic uncertainty in
the total background prediction. The expected signal for a 1.4 TeV scalar LQ,
scaled by the indicated factor for visibility, is overlaid. The last bin
includes the overflow.

The PNNs are trained on all scalar LQ signal masses simultaneously against the
main $t\bar{t}$ and single-top backgrounds, taking into account both the true
and misidentified $\tau_{\text{had-vis}}$ components with the latter corrected
as described in Section 6. The same PNN training is used for both vector LQ
models since separate trainings were found to provide a negligible improvement
in sensitivity. For the signals, the generated LQ mass is used as the
parameterisation input in addition to the input variables described above,
while in the case of the backgrounds a mock LQ mass is randomly assigned from
the range of signal LQ masses such that the resulting training is independent
of the mass. In all cases, the input variables are standardised by subtracting
the median value and dividing by the interquartile range. The resulting PNN
score distributions are used as the final analysis discriminants.

##  6 Background modelling

The dominant background in the $\tau_{\text{had}}\tau_{\text{had}}$ and
$\tau_{\text{lep}}\tau_{\text{had}}$ channels is top production, including
$t\bar{t}$ and single-top-quark production. A subdominant background is $Z$
boson production in association with heavy-flavour quarks $(bb,bc,cc)$, termed
$Z+\text{HF}$ hereafter. Both top production and $Z+\text{HF}$ are estimated
from simulation to which data-driven corrections are applied. In the
$\tau_{\text{had}}\tau_{\text{had}}$ channel, multi-jet events form a non-
negligible background that is estimated by using data-driven techniques. Small
contributions to the background from all other processes are estimated by
using simulated events. This section describes the background estimation
methods used for top-quark-pair and single-top backgrounds, multi-jet
backgrounds, and the $Z+\text{HF}$ background. The background is validated for
$\tau_{\text{had}}\tau_{\text{had}}$ and $\tau_{\text{lep}}\tau_{\text{had}}$
events in a region with an inverted $s_{\text{T}}$ selection, as well as a
region with a low PNN score and the signal region selection. In addition, the
$\tau_{\text{had}}\tau_{\text{had}}$ multi-jet estimate is validated in a
region where the two $\tau_{\text{had-vis}}$ have the same electric charge.
The potential signal contamination in all regions described in this section is
negligible.

The process of estimating the backgrounds follows several steps. First, an
overall shape correction is determined for the top background, as described in
Section 6.1.1. Then, with this in place, a shape and normalisation correction
is determined for the top backgrounds with jets misidentified as
$\tau_{\text{had-vis}}$, as described in Section 6.1.2. After applying these
corrections, a prediction for the shape and normalisation of multi-jet
backgrounds is determined for the $\tau_{\text{had}}\tau_{\text{had}}$ channel
in Section 6.2. Finally, with all relevant corrections in place, a
normalisation factor is determined for the $Z+\text{HF}$ backgrounds, as
described in Section 6.3. The resulting corrections are only weakly coupled
due to the high purity of each control region, meaning that corrections for a
specific background process do not significantly affect the overall background
in control regions targeting other backgrounds. All of these corrections are
applied in the final SR fit.

###  6.1 Top quark backgrounds

For top-quark-pair and single-top-quark production (top backgrounds), events
are estimated separately based on whether the $\tau_{\text{had-vis}}$
candidate in the event is correctly identified (referred to as a true
$\tau_{\text{had-vis}}$) or whether it is a quark- or gluon-initiated jet
misidentified as a $\tau_{\text{had-vis}}$ (referred to as a fake
$\tau_{\text{had-vis}}$). The small contributions from light leptons that are
misidentified as $\tau_{\text{had-vis}}$ are considered together with the true
$\tau_{\text{had-vis}}$ contribution. Events with a true $\tau_{\text{had-
vis}}$ and a hadronic jet misidentified as a light lepton contribute
negligibly to the $\tau_{\text{lep}}\tau_{\text{had}}$ channel and are not
considered.

These backgrounds are estimated in a multi-step data-driven process that is
applied to simulated events. First, all top backgrounds are scaled by an
$s_{\text{T}}$-dependent reweighting factor (RF), and then simulated
background events with misidentified $\tau_{\text{had-vis}}$ are further
corrected by a scale factor (SF) that is binned in the $\tau_{\text{had-vis}}$
transverse momentum.

####  6.1.1 Overall reweighting of top backgrounds

The motivation for scaling the $t\bar{t}$ and single-top backgrounds arises
from mismodelling of the data by simulation observed in control regions (CRs).
It is seen that this effect becomes more pronounced for events with higher
momentum top quarks, which is where this analysis is primarily focused. This
mismodelling has also been observed in ATLAS measurements of the $t\bar{t}$
differential cross-section, where it is seen that the number of events is
overestimated at high top-quark $p_{\text{T}}$ [112, 113, 114].

For this reason, a CR is defined to determine a binned shape and normalisation
correction of the simulated top quark events to data. Events in this CR are
required to have two $b$-jets with $p_{\text{T}}$ greater than 45 and 20 GeV,
exactly two light leptons with opposite charge,
$E_{\text{T}}^{\text{miss}}>100$ GeV, and a dilepton mass ($m_{\ell\ell}$)
$>110$ GeV. They are also required to have $m_{b\ell}>250$ GeV, where
$m_{b\ell}=\min(\max(m_{b_{0}\ell_{0}},m_{b_{1}\ell_{1}}),\max(m_{b_{0}\ell_{1}},m_{b_{1}\ell_{0}}))$,
where the 0 and 1 indices refer to the leading and sub-leading $b$-tagged jets
and leptons in order of transverse momentum. This region is orthogonal to the
SRs and is over 99% pure in $t\bar{t}$ events.

The RFs are derived by subtracting all non-top backgrounds, as estimated using
simulation, from data. A ratio of the remaining events to the prediction of
$t\bar{t}$ and single-top events in simulation is then calculated. This factor
is binned in $s_{\text{T}}$, with one bin up to 400 GeV, steps of 100 GeV from
400 to 1400 GeV, and then one bin for values greater than 1400 GeV. The values
of the RFs decrease from 0.97 at low $s_{\text{T}}$ to approximately 0.62 in
the highest $s_{\text{T}}$ bin. Even in the highest $s_{\text{T}}$ bin, the
signal contamination remains at the percent level. The largest relative
contribution of single-top events is also at high $s_{\text{T}}$. This
reweighting is applied in both the $\tau_{\text{lep}}\tau_{\text{had}}$ and
$\tau_{\text{had}}\tau_{\text{had}}$ SRs for $t\bar{t}$ and single-top events
with true and misidentified $\tau$-leptons, as well as in all CRs. The
uncertainty in this RF is taken from the statistical uncertainty in the
factor, bin-by-bin in $s_{\text{T}}$, and its impact on the shape and
normalisation of the final PNN score distribution are considered. In addition,
top background modelling uncertainties are propagated through the reweighting
process, so that modified RFs are applied when evaluating such uncertainties
in the final fit.

####  6.1.2 Top backgrounds with jets misidentified as $\tau_{\text{had-vis}}$

In addition to this overall RF, the estimation of top backgrounds with jets
misidentified as $\tau_{\text{had-vis}}$ in the SRs is performed using
simulated events with additional data-driven corrections. A fit is performed
in a $\tau_{\text{lep}}\tau_{\text{had}}$-based CR to simultaneously correct
the overall normalisation of true $\tau_{\text{had-vis}}$ and misidentified
$\tau_{\text{had-vis}}$ events while deriving an SF to be applied to
misidentified $\tau_{\text{had-vis}}$ events in the
$\tau_{\text{lep}}\tau_{\text{had}}$ and $\tau_{\text{had}}\tau_{\text{had}}$
SRs. The RF for top backgrounds is applied to this CR before the fit. The SFs
obtained are then applied in the SRs, in order to correct the
$\tau_{\text{had-vis}}$ misidentification rate in simulation to that observed
in data.

The CR has the same selection as the SR for the
$\tau_{\text{lep}}\tau_{\text{had}}$ channel, except that the
$\tau_{\text{had-vis}}$ $p_{\text{T}}>100$ GeV requirement is removed and
$s_{\text{T}}$ is required to be in a range of 400–600 GeV. This region is 97%
pure in $t\bar{t}$ events, with a mixture of both correctly identified and
misidentified $\tau_{\text{had-vis}}$ that varies with $\tau_{\text{had-vis}}$
$p_{\text{T}}$.

The distribution used for this estimation is the transverse mass of the light
lepton and missing transverse momentum, defined as
$m_{\text{T}}(\ell,E_{\text{T}}^{\text{miss}})=\sqrt{(E_{\text{T}}^{\text{miss}}+p_{\text{T},\ell})^{2}-(E_{\text{T},x}^{\text{miss}}+p_{x,\ell})^{2}-(E_{\text{T},y}^{\text{miss}}+p_{y,\ell})^{2}}$.
The expected shapes for top backgrounds with true and misidentified
$\tau_{\text{had-vis}}$ in this distribution differ significantly, making it
possible to constrain the two background sources. The normalisation of the
true and misidentified $\tau_{\text{had-vis}}$ background is allowed to vary
freely, and SFs for the misidentified $\tau_{\text{had-vis}}$ background are
determined in bins of $\tau_{\text{had-vis}}$ $p_{\text{T}}$. All detector-
related uncertainties and top background modelling uncertainties are included
as nuisance parameters in the fit. An example fit in a single bin of
$p_{\text{T}}$ is shown in Figure 5 for the (a) $\tau_{\text{had-vis}}$ and
(b) anti-$\tau_{\text{had-vis}}$ CRs. Depending on $\tau_{\text{had-vis}}$
$p_{\text{T}}$, the SFs run from 0.90 in the lowest $p_{\text{T}}$ bin down to
$0.56$ in the highest $p_{\text{T}}$ bin.

Figure 5: Post-fit plots for true and misidentified $\tau_{\text{had-vis}}$ in
(a) the $\tau_{\text{had-vis}}$ and (b) the anti-$\tau_{\text{had-vis}}$ CRs,
in a single $p_{\text{T}}$ bin ($\tau_{\text{had-vis}}$ $p_{\text{T}}$ $>$100
GeV). ‘Other’ refers to the sum of minor backgrounds (vector boson + jets,
diboson and Higgs boson). The lower panels show the ratios of the data to the
sum of the predicted backgrounds. The hatched bands indicate the combined
statistical and systematic uncertainty in the total background predictions.
The dashed lines denote the total pre-fit backgrounds for comparison, while
the last bins include the overflow.

For the estimation of top backgrounds with misidentified $\tau_{\text{had-
vis}}$, an uncertainty is considered that arises from the limited number of
events and an additional uncertainty is defined by comparing the nominal SFs
to SFs derived with a more inclusive $s_{\text{T}}$ selection
($s_{\text{T}}<600$ GeV). This last uncertainty is intended to address a
possible $s_{\text{T}}$-dependence in the mismodelling of top backgrounds. The
difference between the central values for SFs measured with these two
$s_{\text{T}}$ selections is taken as the $s_{\text{T}}$-dependence
uncertainty.

###  6.2 Multi-jet backgrounds with jets misidentified as $\tau_{\text{had-
vis}}$

For the $\tau_{\text{had}}\tau_{\text{had}}$ channel, multi-jet processes can
contribute to the SR at non-negligible levels. For this reason, the
$\tau_{\text{had}}\tau_{\text{had}}$ channel uses a data-driven fake-factor
(FF) method to estimate this background. These FFs are measured in a CR with
the same selection as the $\tau_{\text{had}}\tau_{\text{had}}$ SR, except that
the two $\tau_{\text{had-vis}}$ candidates have the same charge and the
$E_{\text{T}}^{\text{miss}}$ requirement is loosened to 80 GeV. The FF is
defined as the ratio of events where both $\tau_{\text{had-vis}}$ are loose to
the number of events where one $\tau_{\text{had-vis}}$ is loose and the other
is an anti-$\tau_{\text{had-vis}}$. These FFs are derived as a function of
transverse momentum and the number of charged-particle tracks of the
$\tau_{\text{had-vis}}$ candidate. The FFs are measured from data after
subtracting all predicted non-multi-jet background contributions. The FFs
range between approximately zero and 0.25.

The non-multi-jet background contributions that are subtracted, however,
suffer from the same mismodelling issues described in the previous two
sections. The top backgrounds are therefore corrected by the RFs and SFs
derived as described in Section 6.1.1 and Section 6.1.2, respectively. Since
the SFs are anticipated to be different for $\tau_{\text{had-vis}}$ and
anti-$\tau_{\text{had-vis}}$, dedicated SFs are measured for this data-driven
estimation. Specifically, the anti-$\tau_{\text{had-vis}}$ region uses SFs
that are derived in a CR as described in Section 6.1.2, except that the
$\tau_{\text{had-vis}}$ identification requirement is changed to that of an
anti-$\tau_{\text{had-vis}}$. An example fit in a single bin of $p_{\text{T}}$
is shown in Figure 5 for the anti-$\tau_{\text{had-vis}}$ CR. Depending on
$\tau_{\text{had-vis}}$ $p_{\text{T}}$, these SFs vary between 0.77 and 0.95.

To construct the background estimate, FFs are applied to a region with the
$\tau_{\text{had}}\tau_{\text{had}}$ SR selection, except that the
$\tau_{\text{had-vis}}$ identification requirement is changed to that of an
anti-$\tau_{\text{had-vis}}$. This provides both a shape and a normalisation
for the multi-jet contribution in the PNN score distribution.

For the estimation of multi-jet backgrounds in
$\tau_{\text{had}}\tau_{\text{had}}$, uncertainties are considered due to the
statistical uncertainty of the SFs, and to the uncertainty in the subtraction
of different backgrounds using simulation. Top events with a correctly
identified $\tau_{\text{had-vis}}$, top events with a misidentified
$\tau_{\text{had-vis}}$, and other small backgrounds are considered
separately. The top events are varied by the overall uncertainty defined by
the procedure to determine the modelling uncertainties, but evaluated in the
anti-$\tau_{\text{had-vis}}$ region. Other backgrounds are varied by 30%. In
addition, a 20% overall uncertainty in the estimate is applied based on checks
of the method in validation regions. The total uncertainty in the multi-jet
background is $-64$% to $+61$%.

###  6.3 $Z+\text{HF}$ background

The normalisation of the $Z+\text{HF}$ background, which is a relatively small
contribution in the SRs, is observed to be in disagreement with the NLO cross-
section in Sherpa (e.g. Ref. [115]). It is therefore determined from data
using a $Z+\text{HF}$ CR that targets events containing a $Z$ boson decaying
into a light-lepton pair and produced in association with two heavy-flavour
jets. The composition of this control region is approximately 60%
$Z+\text{HF}$ events and 40% $t\bar{t}$ events, with less than 1% arising from
backgrounds with misidentified $\tau_{\text{had-vis}}$. Since the contribution
from backgrounds with misidentified $\tau_{\text{had-vis}}$ is negligible,
only the RF for the $t\bar{t}$ shape is included in this CR.

Data for the CR was recorded using a combination of the single-lepton triggers
described above and additional dilepton triggers requiring pairs of same-
flavour leptons. At the analysis level exactly two oppositely-charged same-
flavour leptons, passing the ‘veto’ quality requirements and $p_{\text{T}}$
thresholds based on the corresponding trigger thresholds, are required. The
invariant mass of the resulting lepton pair $m_{\ell\ell}$ is required to lie
between 75 GeV and 110 GeV. In addition, exactly two $b$-jets with
$p_{\text{T}}>20$ GeV are required and their invariant mass $m_{bb}$ is
required to be less than 40 GeV or greater than 150 GeV to avoid the Higgs
boson mass peak. The RFs for top backgrounds derived in Section 6.1.1 are then
applied.

A fit to the $m_{\ell\ell}$ distribution is performed to discriminate between
the $Z+\text{HF}$ and top backgrounds, with the normalisation of both
processes allowed to vary freely and all systematic uncertainties described in
Section 7 included. The resulting $Z+\text{HF}$ normalisation factor is
$1.36\pm 0.11$ and is used to correct the $Z+\text{HF}$ background entering
into the final fit (described in Section 8), which is allowed to vary within
the associated uncertainty.

##  7 Systematic uncertainties

The systematic uncertainties considered include detector-related
uncertainties, modelling and theoretical uncertainties, and uncertainties
derived for the data-driven background estimates, the latter of which have
already been described in Section 6. Uncertainties are evaluated by shifting
the central value upward or downward by one standard deviation, and then
propagating the differences to the PNN score distributions that are used in
the final fit.

Detector-related uncertainties are defined as uncertainties relating to the
detector response, object reconstruction and object identification. There are
systematic uncertainties associated with each of the reconstructed objects
considered, as well as the $E_{\text{T}}^{\text{miss}}$. For light leptons,
$\tau_{\text{had-vis}}$, and jets, uncertainties are considered for energy
scale and resolution, reconstruction and identification, while uncertainties
in isolation are also considered for light leptons. For the
$\tau_{\text{lep}}\tau_{\text{had}}$ and $\tau_{\text{had}}\tau_{\text{had}}$
channel, uncertainties associated with the lepton and $\tau_{\text{had-vis}}$
trigger efficiencies, respectively, are considered. For $b$-jets, additional
uncertainties are considered for the efficiency of (mis)tagging $b$-jets,
$c$-jets, and light-quark-initiated jets. Uncertainties related to energy
scale and resolution, and the inclusion of soft terms, are considered for the
$E_{\text{T}}^{\text{miss}}$. Finally, there is also an uncertainty associated
with shape and normalisation components that arises from uncertainties in the
simulation of pile-up collisions.

Theoretical and modelling uncertainties include uncertainties in the cross-
section calculations of background processes, which have only a normalisation
component, and uncertainties in the acceptance of each process, for which
normalisation and shape components are taken into account. For top
backgrounds, relative acceptance uncertainties are also defined to take into
account normalisation differences for $\tau_{\text{lep}}\tau_{\text{had}}$ and
$\tau_{\text{had}}\tau_{\text{had}}$ SRs.

For $t\bar{t}$ processes, shape and normalisation uncertainties are considered
that arise from changing the matrix element and parton shower simulation
software, and from varying the initial and final state radiation, PDF, and
$\alpha_{s}$. The matrix element uncertainty is determined by comparing the
Powheg+Pythia 8 sample with an aMC@NLO+Pythia 8 sample. The parton shower
uncertainty is determined by comparing the Powheg+Pythia 8 sample with a
Powheg+Herwig 7 [116, 117] sample. The other modelling uncertainties are
evaluated using internal weights in the nominal $t\bar{t}$ sample.

For single-top processes, acceptance uncertainties with shape and
normalisation components are considered. Uncertainties are considered from
changing the matrix element, parton shower, and impacts of diagram
interference. In addition, variations of initial- and final-state radiation
and PDFs are considered. The matrix element uncertainty is determined by
comparing the Powheg+Pythia 8 sample with an aMC@NLO+Pythia 8 sample. The
parton shower uncertainty is determined by comparing the Powheg+Pythia 8
sample with a Powheg+Herwig 7 sample. The diagram interference uncertainty is
evaluated by comparing the nominal single top samples, which use a diagram
removal scheme, with alternative samples that utilise a diagram subtraction
scheme [118]. The other modelling uncertainties are evaluated using internal
weights in the nominal single-top samples.

All $t\bar{t}$ and single-top modelling uncertainties are also propagated
through the top reweighting procedure, such that there is an uncertainty in
the RF corresponding to each modelling uncertainty.

For $Z$+jets processes, uncertainties due to the choice of generator are
evaluated by comparing the nominal Sherpa simulated samples with alternative
samples simulated by MadGraph with LO-accurate matrix elements that contain up
to four final-state partons, using Pythia for parton showering. In addition,
uncertainties are considered by taking an envelope of variations in the
renormalisation and factorisation scales and PDF values using internal weights
in the simulated Sherpa sample. For this process specifically, uncertainties
are also included based on varying the matrix element matching scale and the
resummation scale for soft-gluon emission. All of these uncertainties are
included in the $Z+\text{HF}$ fit described in Section 6.3, and their sum in
quadrature, taking relative acceptance uncertainties into account, is
considered as the uncertainty in the SRs for the final fit.

For signal samples, uncertainties arising from variations of scale, initial-
state radiation, PDF, and $\alpha_{s}$ are considered, using alternative
weights internal to the signal samples. Differences in shape are observed to
be negligibly small in the PNN score distributions, so only variations in
normalisation are included for the final fit.

The relative impact of the different sources of uncertainty on the analysis
varies depending on the LQ model considered and the mass probed. Generally,
the largest impact comes from the statistical uncertainties, which increase
with $m_{\text{LQ}}$. In the scalar LQ case, for example, the statistical
impact on the limit ranges from 60% at the lowest $m_{\text{LQ}}$ evaluated to
80% above 1000 GeV. The main systematic uncertainties come from the $t\bar{t}$
and single-top-quark modelling uncertainties, including their interference,
and normalisation. There is also a significant effect from the signal
acceptance uncertainties, which increases with $m_{\text{LQ}}$, particularly
for the vector LQ models.

##  8 Statistical interpretation and results

The data are compared with the expectation, including the background modelling
corrections outlined in Section 6, by performing simultaneous binned maximum-
likelihood fits to the PNN score distributions, separately for each LQ
hypothesis, in the $\tau_{\text{lep}}\tau_{\text{had}}$ and
$\tau_{\text{had}}\tau_{\text{had}}$ SRs. For each hypothesis, the binning of
the PNN score distributions is chosen separately to maximise the expected
sensitivity, while ensuring sufficient background events in the signal-
enhanced PNN bins and preserving the stability of the fit. In addition to the
relative signal-strength modifier, $\mu$, the top normalisation is free to
float in the fit and is constrained by the background-enhanced PNN bins.

The statistical and systematic uncertainties affecting the signal and
background model, described in Section 7, are represented by deviations from
the nominal model scaled by Gaussian- or Poisson-constrained nuisance
parameters that are profiled in the fit. Common sources of systematic
uncertainty are correlated across the SRs.

The resulting event yields in the $\tau_{\text{lep}}\tau_{\text{had}}$ and
$\tau_{\text{had}}\tau_{\text{had}}$ SRs, based on a background-only fit to
the data, are presented in Table 4. Corresponding post-fit PNN score
distributions for representative LQ signals at masses of 500 GeV, 1.1 TeV and
1.4 TeV are shown in Figure 6 (Figure 7) for the
$\tau_{\text{lep}}\tau_{\text{had}}$ ($\tau_{\text{had}}\tau_{\text{had}}$)
SR. At high values of the PNN score, top backgrounds dominate in the
$\tau_{\text{lep}}\tau_{\text{had}}$ channel, while the
$\tau_{\text{had}}\tau_{\text{had}}$ background consists of a roughly even
mixture of all background sources. Overall, good agreement with the SM
background expectation is observed in all cases, although there is a slight
deficit of data relative to the background prediction in the highest PNN score
bin for the $\tau_{\text{had}}\tau_{\text{had}}$ channel.

Table 4: Post-fit yields for background events, determined from a background-only fit, compared with the observed number of data events in the $\tau_{\text{lep}}\tau_{\text{had}}$ and $\tau_{\text{had}}\tau_{\text{had}}$ SRs. ‘Fake $\tau_{\text{had}}$ (top)’ refers to top backgrounds where a jet is misidentified as the $\tau_{\text{had-vis}}$ of the event, and ‘Other’ refers to the sum of minor backgrounds (vector boson + jets, diboson and Higgs boson). The total background is not identical to the sum of the individual components since the latter are rounded for presentation, while the sum is calculated with the full precision before being subsequently rounded. Systematic uncertainties are included. Due to the large correlations, individual uncertainties can be significantly larger than the total uncertainty.  |  $\tau_{\text{lep}}\tau_{\text{had}}$ channel |  $\tau_{\text{had}}\tau_{\text{had}}$ channel
---|---|---
$t\bar{t}$ |  $2420$$\;\pm\;$ | $90$ |  $93$$\;\pm\;$ | $9$
single-top |  $355$$\;\pm\;$ | $27$ |  $20$$\;\pm\;$ | $4$
Fake $\tau_{\text{had}}$ (top) |  $170$$\;\pm\;$ | $90$ |  $43$$\;\pm\;$ | $18$
Z$\rightarrow\tau\tau$ \+ (bb, bc, cc) |  $13.9$$\;\pm\;$ | $2.4$ |  $10.3$$\;\pm\;$ | $1.4$
Multi-jet |  i– |  $22$$\;\pm\;$ | $11$
Other |  $78$$\;\pm\;$ | $7$ |  $19$$\;\pm\;$ | $5$
Total Background |  $3040$$\;\pm\;$ | $60$ |  $207$$\;\pm\;$ | $13$
Data | 3031 | 211

Figure 6:  The PNN score distributions in the
$\tau_{\text{lep}}\tau_{\text{had}}$ SR for (a) $m_{\text{LQ}}=500$ GeV, (b)
$m_{\text{LQ}}=1.1$ TeV, (c) $m_{\text{LQ}}=1.4$ TeV. The normalisation and
shape of the backgrounds are determined from the background-only likelihood
fit to data and the ratios of the data to the sum of the backgrounds are shown
in the lower panels. ‘Other’ refers to the sum of minor backgrounds (vector
boson + jets, diboson and Higgs boson). The hatched bands indicate the
combined statistical and systematic uncertainty in the total background
predictions. The expected signals for scalar LQs with the corresponding
masses, scaled by the indicated factors for visibility, are overlaid. Since
the PNN score itself is not a physical quantity, it is represented solely by
the bin number.

Figure 7:  The PNN score distributions in the
$\tau_{\text{had}}\tau_{\text{had}}$ SR for (a) $m_{\text{LQ}}=500$ GeV, (b)
$m_{\text{LQ}}=1.1$ TeV, (c) $m_{\text{LQ}}=1.4$ TeV. The normalisation and
shape of the backgrounds are determined from the background-only likelihood
fit to data and the ratios of the data to the sum of the backgrounds are shown
in the lower panels. ‘Other’ refers to the sum of minor backgrounds (vector
boson + jets, diboson and Higgs boson). The hatched bands indicate the
combined statistical and systematic uncertainty in the total background
predictions. The expected signals for scalar LQs with the corresponding
masses, scaled by the indicated factors for visibility, are overlaid. Since
the PNN score itself is not a physical quantity, it is represented solely by
the bin number.

Since no significant excess is observed, upper limits on the scalar and vector
LQ pair production cross-sections for each mass hypothesis are computed based
on the modified frequentist CL${}_{\text{s}}$ method [119], using a profile
likelihood test statistic [120] under the asymptotic approximation. The
resulting observed and expected limits, assuming ${\cal B}=1$, as a function
of $m_{\text{LQ}}$ at 95% confidence level (CL) are shown in Figure 8 for all
LQ models. The expected contributions of the
$\tau_{\text{lep}}\tau_{\text{had}}$ and $\tau_{\text{had}}\tau_{\text{had}}$
channels are approximately equal at high $m_{\text{LQ}}$, while the
$\tau_{\text{had}}\tau_{\text{had}}$ is up to a factor of two more sensitive
at low $m_{\text{LQ}}$. The improvement in the observed limit compared with
the expectation is driven by the data deficit in the highest
$\tau_{\text{had}}\tau_{\text{had}}$ PNN score bin mentioned above and is
larger at high $m_{\text{LQ}}$ since the signal becomes more localised at high
PNN score as $m_{\text{LQ}}$ increases. The theoretical prediction for the
cross-section of scalar or vector LQ pair production is indicated by the solid
line along with its uncertainties.

Table 5: Observed and expected lower limits on the LQ mass at 95% CL for the three different LQ models, assuming ${\cal B}=1$. | Obs. limit [GeV] | Exp. limit [GeV]
---|---|---
Scalar LQ | 1490 | 1410
Vector LQ (minimal-coupling) | 1690 | 1600
Vector LQ (Yang–Mills) | 1960 | 1840

The corresponding expected and observed 95% CL lower limits on the LQ mass for
the three different LQ models are summarised in Table 5, providing an
improvement in mass reach for a scalar LQ of more than 450 GeV compared with
the previous 36 fb-1 result in this channel [18]. They extend the full Run 2
ATLAS reach for third-generation up-type LQs by around 200 GeV in all three
models compared with the search in the $LQLQ\to t\nu t\nu$ decay mode [26].

The results are also expressed as upper limits on the branching ratio to
charged leptons as a function of $m_{\text{LQ}}$ for each LQ model in Figure
9. For all models investigated, constraints on the LQ mass are reduced by no
more than 15% going from ${\cal B}=1$ to ${\cal B}=0.5$, while scalar LQ
masses up to around 850 GeV are excluded for couplings into charged leptons as
low as 0.1; the corresponding ${\cal B}=0.1$ exclusion for vector LQ is around
1100 GeV (1300 GeV) in the minimal-coupling (Yang–Mills) scenario.

Figure 8: The observed (solid line) and expected (dashed line) 95% CL upper
limits on the LQ pair production cross-sections assuming ${\cal B}=1$ as a
function of $m_{\text{LQ}}$ for (a) the scalar LQ case, (b) the vector LQ case
in the minimal-coupling scenario, (c) vector LQs in the Yang–Mills scenario.
The surrounding shaded bands correspond to the $\pm 1$ and $\pm 2$ standard
deviation ($\pm 1\sigma,\pm 2\sigma$) uncertainty in the expected limit. The
theoretical prediction in each model, along with its uncertainty, is shown by
the lines with the hatched bands.

Figure 9: The observed (solid line) and expected (dashed line) 95% CL upper
limits on the branching ratio into charged leptons as a function of
$m_{\text{LQ}}$ for (a) the scalar LQ case, (b) the vector LQ case in the
minimal-coupling scenario, (c) vector LQs in the Yang–Mills scenario. The
observed exclusion region is above the solid line, with the theoretical
uncertainty in the model indicated by the dotted lines around this. The
expected limit is indicated by the dashed line and the surrounding shaded
bands correspond to the $\pm 1$ and $\pm 2$ standard deviation ($\pm
1\sigma,\pm 2\sigma$) uncertainty in the expected limit. No limits are
presented for ${\cal B}<0.1$ due to the lack of expected signal events in this
final state.

##  9 Conclusion

A search for pair-produced scalar or vector leptoquarks decaying into a
$b$-quark and a $\tau$-lepton is presented. The analysis exploits the full
data sample recorded with the ATLAS detector in Run 2 of the LHC,
corresponding to 139 fb-1 of proton–proton collisions at $\sqrt{s}=13$ TeV. No
significant deviations from the Standard Model expectation are observed and
upper limits on the production cross-section are derived as a function of LQ
mass and branching ratio into a charged lepton. Scalar LQs with masses below
1490 GeV are excluded assuming a 100% branching ratio, while for vector LQs
the corresponding limit is 1690 GeV (1960 GeV) in the minimal-coupling
(Yang–Mills) scenario. For branching ratios as low as 10%, scalar LQ masses
below around 850 GeV are excluded; the corresponding mass limits for vector
LQs are 1100 GeV (1300 GeV) in the minimal-coupling (Yang–Mills) scenario.
These results significantly improve the sensitivity compared to previous ATLAS
LQ searches, extending the mass reach for third-generation up-type LQs by more
than 200 GeV in all models and surpassing the previous ATLAS search in this
final state by more than 450 GeV for scalar LQs. In addition to the increased
luminosity, this is due to upgraded hadronic $\tau$-lepton and $b$-jet
identification, improved multivariate techniques and better background
estimation methods.

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