Search for new phenomena in events with two opposite-charge leptons, jets and missing transverse momentum in $pp$ collisions at $\sqrt{s} = 13$ TeV with the ATLAS detector

The results of a search for direct pair production of top squarks and for dark matter in events with two opposite-charge leptons (electrons or muons), jets and missing transverse momentum are reported, using 139 fb$^{-1}$ of integrated luminosity from proton-proton collisions at $\sqrt{s} = 13$ TeV, collected by the ATLAS detector at the Large Hadron Collider during Run 2 (2015-2018). This search considers the pair production of top squarks and is sensitive across a wide range of mass differences between the top squark and the lightest neutralino. Additionally, spin-0 mediator dark-matter models are considered, in which the mediator is produced in association with a pair of top quarks. The mediator subsequently decays to a pair of dark-matter particles. No significant excess of events is observed above the Standard Model background, and limits are set at 95% confidence level. The results exclude top squark masses up to about 1 TeV, and masses of the lightest neutralino up to about 500 GeV. Limits on dark-matter production are set for scalar (pseudoscalar) mediator masses up to about 250 (300) GeV.


Introduction
The Standard Model (SM) of particle physics is extremely successful in describing the phenomena of elementary particles and their interactions. Its predictive power has been proven with high precision by a wide range of experiments. However, despite its success, several important questions remain unanswered within the SM. One particularly striking omission is that it does not provide any explanation for dark matter (DM) [1,2]. This is a non-baryonic, non-luminous matter component of the universe, for which there is strong evidence from a range of astrophysical observations. A weakly interacting dark-matter candidate particle can be produced at the Large Hadron Collider (LHC) [3] in a variety of ways, as described, for example, by supersymmetry (SUSY) [4][5][6][7][8][9] or DM models. At the LHC, one of the most promising modes is the production of DM particle pairs in association with on-or off-shell top quarks. Previous searches for DM candidates in association with a top quark pair have been performed by the ATLAS [10][11][12][13][14][15][16] and CMS [17][18][19][20][21][22][23][24][25][26] collaborations. However, those previous searches were statistically limited, or sensitive only up to limited particle masses. They also suffered from significant regions in which no limit could be placed because the kinematics of the decays made the signal events particularly difficult to identify. This paper aims to extend the sensitivity beyond that of the previous searches to higher masses, and to cover the regions in which the previous ATLAS results had no sensitivity [27,28]. It achieves this in part by exploiting a larger dataset, corresponding to 139 fb −1 of proton-proton collision data collected by the ATLAS experiment during Run 2 of the LHC (2015-2018) at a centre-of-mass energy √ = 13 TeV. Further improvements in sensitivity are obtained by using a new discriminating variable, the 'object-based miss T significance' [29], lowering the lepton T thresholds, and optimising a dedicated selection to target signal models in the most difficult kinematic regions.

Signal models and kinematic regions
For DM production, the simplified benchmark models [30][31][32] assume the existence of a mediator particle which couples both to the SM and to the dark sector [33][34][35]. The couplings of the mediator to the SM fermions are then severely restricted by precision flavour measurements. An ansatz that automatically relaxes these constraints is Minimal Flavour Violation [36]. This assumption implies that the interaction between any new neutral spin-0 state and SM matter is proportional to the fermion masses via Yukawa-type couplings. 1 It follows that colour-neutral mediators would be produced mainly through loop-induced gluon fusion or in association with heavy-flavour quarks. Here, the DM particles are assumed to be pair produced through the exchange of a spin-0 mediator, which can be a colour-neutral scalar or pseudoscalar particle (denoted by or , respectively), in association with a top quark pair: →¯¯ (Figure 1(a)).
Alternatively, dark-matter particles are also predicted in supersymmetry, a space-time symmetry that for each SM particle postulates the existence of a partner particle whose spin differs by one-half unit. To avoid violation of baryon number ( ) and lepton number ( ) conservation, a multiplicative quantum number -parity [37], defined as = (−1) 3( − )+2 , is assumed to be conserved. SUSY particles are then produced in pairs, and the lightest supersymmetric particle (LSP) is stable and, if only weakly interacting, a candidate for dark matter [38,39]. In the framework of a generic -parity-conserving Minimal Supersymmetric Standard Model (MSSM) [40,41], the supersymmetric scalar partners of right-handed and left-handed quarks (squarks),˜R and˜L, can mix to form two mass eigenstates,˜1 and˜2, with˜1 defined  Figure 1: Diagrams representing the signal models targeted by the searches: (a) the spin-0 mediator models, where the mediator decays into a pair of dark-matter particles and is produced in association with a pair of top quarks ( →¯¯), (b) the three-body˜1 decay mode into an on-shell boson, a -quark and the lightest neutralino (˜1 →˜0 1 ), (c) the four-body˜1 decay mode (˜1 →l˜0 1 ) wherel and are a anti-lepton with its neutrino and (d) the two-body˜1 decay into an on-shell top quark and the lightest neutralino (˜1 →˜0 1 ). For all the diagrams (a-d) the distinction between particle and anti-particle is omitted.
to be the lighter one. In the case of the supersymmetric partner of the top quark,˜, large mixing effects can lead to one of the top squark mass eigenstates,˜1, being significantly lighter than the other squarks. The charginos and neutralinos are mixtures of the bino, winos and Higgsinos that are superpartners of the U(1) and SU(2) gauge bosons and the Higgs bosons, respectively. Their mass eigenstates are referred to as˜± ( = 1, 2) and˜0 ( = 1, 2, 3, 4) in order of increasing mass. In a large variety of models, the LSP, which is the DM candidate, is the lightest neutralino˜0 1 . Searches for direct pair production of the top squark and DM particles can be performed in final states with two leptons (electrons or muons) of opposite electric charge, jets and missing transverse momentum (Figures 1(b)-1(d)). Depending on the mass difference between the top squark and the lighter SUSY particles, different decay modes are relevant. For ( ) + ( ) < (˜1) − (˜0 1 ) < ( ), the three-body decay˜1 →˜0 1 occurs through an off-shell top quark (Figure 1(b)). For smaller mass differences, i.e. (˜1) − (˜0 1 ) < ( ) + ( ), the four-body decay channel˜→ ˜0 1 , where and are two fermions from the off-shell ( * ) decay, is assumed to occur (Figure 1(c)). In this search, and are a charged lepton and its associated anti-neutrino (or vice versa). For each of these two decay modes a dedicated event selection is performed to maximise the sensitivity. These selections are referred to as three-body and four-body selections in this paper. Direct pair production of top squarks which decay into an on-shell top quark and the lightest neutralino˜1 →˜0 1 , will occur when (˜1) − (˜0 1 ) > ( ) (Figure 1(d)). The signature of the¯+DM process is similar to that of the simplified model shown in Figure 1(a), so the same selection is also used to constrain the˜1 →˜0 1 model and it is referred to as the two-body selection.
The paper proceeds as follows; after a description of the ATLAS detector in Section 2, the data and simulated Monte Carlo (MC) samples used in the analysis are detailed in Section 3 and the object identification is documented in Section 4. The search strategy, the SM background estimations, and the systematic uncertainties are discussed in Sections 5, 6 and 7. The results and their statistical interpretations are presented in Sections 8 and 9. Finally, Section 10 presents the conclusions.

ATLAS detector
The ATLAS detector [42] at the LHC covers nearly the entire solid angle around the collision point. 2 It consists of an inner tracking detector surrounded by a thin superconducting solenoid, electromagnetic and hadronic calorimeters, and a muon spectrometer with three large superconducting toroidal magnets.
The inner-detector system (ID) is immersed in a 2 T axial magnetic field and provides charged-particle tracking in the range | | < 2.5. The high-granularity silicon pixel detector covers the vertex region and typically provides four measurements per track, the first hit normally being in the insertable B-layer installed before Run 2 [43,44]. It is followed by the silicon microstrip tracker, which usually provides eight measurements per track. These silicon detectors are complemented by the transition radiation tracker (TRT), which enables radially extended track reconstruction up to | | = 2.0. The TRT also provides electron identification information based on the fraction of hits (typically 30 in total) above a higher energy-deposit threshold corresponding to transition radiation.
The calorimeter system covers the pseudorapidity range | | < 4.9. Within the region | | < 3.2, electromagnetic calorimetry is provided by barrel and endcap high-granularity lead/liquid-argon (LAr) calorimeters, with an additional thin LAr presampler covering | | < 1.8 to correct for energy loss in material upstream of the calorimeters. Hadronic calorimetry is provided by the steel/scintillating-tile calorimeter, segmented into three barrel structures within | | < 1.7, and two copper/LAr hadronic endcap calorimeters. The solid angle coverage is completed with forward copper/LAr and tungsten/LAr calorimeter modules optimised for electromagnetic and hadronic measurements respectively.
The muon spectrometer (MS) comprises separate trigger and high-precision tracking chambers measuring the deflection of muons in a magnetic field generated by the superconducting air-core toroids. The field integral of the toroids ranges between 2.0 and 6.0 T m across most of the detector. A set of precision chambers covers the region | | < 2.7 with three layers of monitored drift tubes, complemented by cathode-strip chambers in the forward region, where the background is highest. The muon trigger system covers the range | | < 2.4 with resistive-plate chambers in the barrel, and thin-gap chambers in the endcap regions.
Interesting events are selected to be recorded by the first-level trigger system implemented in custom hardware, followed by selections made by algorithms implemented in software in the high-level trigger [45]. The first-level trigger accepts events from the 40 MHz bunch crossings at a rate below 100 kHz, which the high-level trigger reduces in order to record events to disk at about 1 kHz.

Data and simulated event samples
The data used in this analysis were collected by the ATLAS detector during collisions at a centre-of-mass energy of √ = 13 TeV from 2015 to 2018. The average number of interactions per bunch crossing (pile-up) varies from 14 during 2015 to 38 during 2017-2018. Only events taken in stable beam conditions, 2 ATLAS uses a right-handed coordinate system with its origin at the nominal interaction point (IP) in the centre of the detector and the -axis along the beam pipe. The -axis points from the IP to the centre of the LHC ring, and the -axis points upwards. Cylindrical coordinates ( , ) are used in the transverse plane, being the azimuthal angle around the -axis. The pseudorapidity is defined in terms of the polar angle as = − ln tan( /2), and the rapidity in terms of energy and momentum as = 0.
A vector energy ì is defined by combining the energy deposited in the calorimeter with its deposit direction. and for which all relevant detector systems were operational, are considered in this analysis. After data-quality requirements the data sample amounts to a total integrated luminosity of 139 fb −1 . The uncertainty in the combined 2015-2018 integrated luminosity is 1.7% [46], obtained using the LUCID-2 detector [47].
The two-body and three-body selections use events accepted by a trigger that requires a minimum of two electrons, two muons, or an electron and a muon [45]. Different trigger-level thresholds for the transverse momentum of the leptons were used in different data-taking periods, ranging between 8 and 22 GeV. Tighter thresholds are applied in the lepton offline selection, to ensure that the trigger efficiency is 'on plateau' in all of the relevant kinematic region. Missing transverse momentum triggers [48] are used in the four-body selection to increase the acceptance of low-T leptons. The missing transverse momentum trigger threshold varied depending on data-taking conditions in the four years: 70 GeV for data collected during 2015; in the range 90-110 GeV for data collected during 2016, and 110 GeV for data collected during 2017 and 2018. Tighter offline requirements on the missing transverse momentum are defined accordingly to ensure event selection on the plateau region of the trigger efficiency curve.
Simulated event samples are used for SM background estimations and to model the signal samples. Standard Model MC samples were processed through a full G 4 [49] simulation of the ATLAS detector, while a fast simulation based on parameterisation of the calorimeter response and G 4 simulation for all the other detector components [50] is used for the SUSY and DM signal samples. MC events are reconstructed using the same algorithms used for the data. To compensate for small residual differences between data and simulation in the lepton reconstruction efficiency, energy scale, energy resolution, trigger modelling, and -tagging efficiency, the simulated events are reweighted using correction factors derived from data [51][52][53].
The events targeted by this analysis are characterised by two leptons with opposite electric charge, jets and missing transverse momentum. The main SM background contributions are expected to come from top quark pair production (¯), associated production of a boson and a top quark pair (¯), single-top decay in the production channel ( ), / * + jets production and diboson processes ( with = , ).
Matrix element and showering generators used for the SM backgrounds and signals are listed in Table 1 along with the relevant parton distribution function (PDF) sets, the configuration of underlying-event and hadronisation parameters (tunes), and the cross-section order in s used to normalise the event yields. Additional MC samples are used to estimate systematic uncertainties, as detailed in Section 7.
The SUSY top squark pair signal samples were generated from leading-order (LO) matrix elements with up to two extra partons using M G 5_aMC@NLO 2.6.2 [54]. M G 5_aMC@NLO was interfaced to P 8.212 + M S [55,56] for the signal samples used in the three-body and four-body selections, while it was interfaced to P 8.212 for the SUSY signal samples used for the interpretation of the two-body selection results. Signal cross-sections were calculated to next-to-next-to-leading order (NNLO) in s , adding the resummation of soft gluon emission at next-to-next-to-leading-logarithm accuracy (NNLO+NNLL) [57][58][59][60][61][62][63][64]. The nominal cross section and the uncertainty are derived using the PDF4LHC15 PDF set, following the recommendations presented in Ref. [65]. Jet-parton matching was performed following the CKKW-L prescription [66]. The A14 tune [67] was used for the modelling of parton showering, hadronisation and the underlying event. Parton luminosities were provided by the NNPDF2.3LO PDF set [68].
The dark-matter signal samples were also generated from leading-order matrix elements, with up to one extra parton, using M G 5_aMC@NLO 2.6.2 interfaced to P 8.212. In the DM samples generation the couplings of the scalar and pseudoscalar mediators to the SM and DM particles ( and ) are set to one. The kinematics of the mediator decay are not strongly dependent on the values of the couplings; however, the particle kinematic distributions are sensitive to the nature of the mediator and to the mediator and DM particle masses. The cross-sections were computed at NLO [69,70].

Inelastic
interactions were generated and overlaid onto the hard-scattering process to simulate the effect of multiple proton-proton interactions occurring during the same (in-time) or a nearby (out-of-time) bunch crossing. These were produced using P 8.186 [71] and EvtGen [72] with the NNPDF2.3LO set of PDFs [68] and the A3 tune [73]. The MC samples were reweighted so that the distribution of the average number of interactions per bunch crossing reproduces the observed distribution in the data.

Object identification
Candidate events are required to have a reconstructed vertex with at least two associated tracks, each with T > 500 MeV and originating from the beam collision region in the -plane. The primary vertex in the event is the vertex with the highest scalar sum of the squared transverse momenta of associated tracks.
The leptons selected for analysis are classified as baseline or signal leptons depending on an increasingly stringent set of reconstruction quality criteria and kinematic selections, so that signal leptons are a subset of the baseline leptons. Baseline leptons are used in the calculation of missing transverse momentum (p miss T ), to resolve ambiguities between the analysis objects in the event, as described later, and for the fake/non-prompt (FNP) lepton background estimation described in Section 6. Signal leptons are used for the final event selection.
Baseline electron candidates are reconstructed from three-dimensional clusters of energy deposition in the electromagnetic calorimeter matched to ID tracks. These electron candidates are required to have pseudorapidity | | < 2.47, T > 4.5 GeV, and to pass a Loose likelihood-based identification requirement [51] with an additional condition on the number of hits in the B-layer. The tracks associated with electron candidates are required to have a longitudinal impact parameter 3 relative to the primary vertex | 0 sin | < 0.5 mm, where is the track's polar angle.
Baseline muon candidates are reconstructed by matching ID tracks, in the pseudorapidity region | | < 2.4 for the two-body and three-body selections and | | < 2.7 for the four-body selection, with MS tracks or energy deposits in the calorimeter compatible with a minimum-ionising particle (calo-tagged muon). The resulting tracks are required to have a T > 4 GeV and a | 0 sin | < 0.5 mm from the primary vertex. Muon candidates are required to satisfy the Medium identification requirement, defined in Ref. [52], based on the numbers of hits in the different ID and MS subsystems, and on the significance of the charge-to-momentum ratio / .
Additional tighter selections are applied to the baseline lepton candidates to select the signal electrons or muons. Signal electrons are required to satisfy a Medium likelihood-based identification requirement [51] and the track associated with a signal electron is required to have a significance | 0 |/ ( 0 ) < 5, where 0 is the transverse impact parameter relative to the reconstructed primary vertex and ( 0 ) is its uncertainty. Isolation criteria are applied to electrons by placing an upper limit on the sum of the transverse energy of the calorimeter energy clusters in a cone of size Δ = √︁ (Δ ) 2 + (Δ ) 2 = 0.2 around the electron (excluding the deposit from the electron itself) and the scalar sum of the T of tracks within a cone of Δ = 0.2 around the electron (excluding its own track). The isolation criteria are optimised such that the isolation selection efficiency is uniform across . This varies from 90% for T = 25 GeV to 99% for T = 60 GeV in events with a boson decaying into pair of electrons [51].
For signal muons a significance in the transverse impact parameter | 0 |/ ( 0 ) < 3 is required. Isolation criteria applied to muons require the scalar sum of the T of tracks inside a cone of Δ = 0.3 around the muon (excluding its own track) to be less than 15% of the muon T . In addition, the sum of the transverse energy of the calorimeter energy clusters in a cone of Δ = 0.2 around the muon (excluding the energy from the lepton itself) must be less than 30% of the muon T [52].
Jets are reconstructed from three-dimensional clusters of energy in the calorimeter [92] using the antijet clustering algorithm [93] as implemented in the FastJet package [94], with a radius parameter = 0.4. The reconstructed jets are then calibrated by the application of a jet energy scale derived from 13 TeV data and simulation [95]. Only jet candidates with T > 20 GeV and | | < 2.8 are considered. 4 To reduce the effects of pile-up, for jets with | | ≤ 2.5 and T < 120 GeV a significant fraction of the tracks associated with each jet are required to have an origin compatible with the primary vertex, as defined by the jet vertex tagger (JVT) [96]. This requirement reduces the fraction of jets from pile-up to 1%, with an efficiency for pure hard-scatter jets of about 90%. Finally, in order to remove events impacted by detector noise and non-collision backgrounds, specific jet-quality requirements [97,98] are applied, designed to provide an efficiency of selecting jets from proton-proton collisions above 99.5% (99.9%) for T > 20 (100) GeV.
The MV2C10 boosted decision tree algorithm [53] identifies jets containing -hadrons (' -jets') by using quantities such as the impact parameters of associated tracks, and well-reconstructed secondary vertices. A selection that provides 77% efficiency for tagging -jets in simulated¯events is used. The corresponding rejection factors against jets originating from -quarks, from -leptons, and from light quarks and gluons in the same sample at this working point are 4.9, 15 and 110, respectively.
To avoid reconstruction ambiguities and double counting of analysis objects, an overlap removal procedure is applied to the baseline leptons and jets in the order which follows. First, the calo-tagged muons are removed if sharing the track with electrons and, next, all electrons sharing an ID track with a muon are removed. Jets which are not -tagged (with the tagging parameters corresponding to an efficiency of 85%) and which lie within a cone of Δ = √︁ (Δ ) 2 + (Δ ) 2 = 0.2 around an electron candidate are removed. All jets lying within Δ = 0.2 of an electron are removed if the electron has T > 100 GeV. Finally, any lepton candidate is removed in favour of a jet candidate if it lies a distance Δ < min(0.4, 0.04 + 10/ T (ℓ)) from the jet, where T (ℓ) is the T of the lepton.
The missing transverse momentum (p miss T ), with magnitude miss T , is defined as the negative vector sum of the transverse momenta for all baseline electrons, photons, muons and jets. Low-momentum tracks from the primary vertex that are not associated with reconstructed analysis objects are also included in the calculation. The miss T value is adjusted for the calibration of the selected physics objects [99]. Linked to the miss T value is the 'object-based miss T significance', called simply ' miss T significance' in this paper. This quantity measures the significance of miss T based upon the transverse momentum resolution of all objects used in the calculation of the p miss T . It is defined as where L is the (longitudinal) component parallel to the p miss T of the total transverse momentum resolution for all objects in the event and the quantity LT is the correlation factor between the parallel and perpendicular components of the transverse momentum resolution for each object. On an event-by-event basis, given the full event composition, miss T significance evaluates the -value that the observed miss T is consistent with the null hypothesis of zero real miss T , as further detailed in Ref. [29]. In this way miss T significance helps to separate events with true miss T , arising from weakly interacting particles such as dark matter or neutralinos, from those where miss T is consistent with particle mismeasurement, resolution or identification inefficiencies, thus providing better background rejection.

Event selection
Different event selections are inspired by previous published strategies [27,28] reoptimised to fully exploit the larger available dataset. For all selections, an improvement in the sensitivity is obtained with the introduction of the miss T significance variable, which enables further optimisation of the selection variables. The four-body sensitivity also benefits from a reduction in the lepton T threshold in the region with small mass differences Δ (˜1,˜0 1 ) between˜1 and˜0 1 . The threshold for the muon (electron) T was lowered from 7 GeV to 4 GeV (4.5 GeV).
Events are required to have exactly two signal leptons (two electrons, two muons, or one electron and one muon) with opposite electric charge. In the two-body and three-body selections, an invariant mass ℓℓ greater than 20 GeV condition is applied to remove leptons from Drell-Yan and low-mass resonances, while in the four-body selection, given the softer T spectrum of the leptons, ℓℓ is required to be higher than 10 GeV. Events with same flavour (SF) lepton pairs ( ± ∓ and ± ∓ ) with ℓℓ between 71.2 and 111.2 GeV are rejected to reduce the boson background, except for the four-body selection. No additional ℓℓ selection is imposed on the different flavour (DF) lepton pairs ( ± ∓ ). Different jet ( -jet) multiplicities, labelled as jets ( −jets ), are required in the three selections, as detailed below.

Discriminators and kinematic variables
Final event selections are obtained by separating signal from SM background using different kinematic variables. Two variables are constructed from the miss T and the T of the leading leptons and jets: 2ℓ = miss T /( T (ℓ 1 ) + T (ℓ 2 )) and 2ℓ4 = miss where T (ℓ 1 ) and T (ℓ 2 ) are the leading and sub-leading lepton transverse momenta respectively and T ( =1,..., ≤4 ) are the transverse momenta of the up to four leading jets, in decreasing order. For some backgrounds, e.g. / * + jets, the variable 2ℓ has a distribution that peaks at lower values than the signal, and it is thus used to reject those backgrounds. Similarly, 2ℓ4 is employed for its high rejection power against multi-jet events.
Another variable employed is p ℓℓ T,boost , which is defined as the vectorial sum of p miss T and the leptons' transverse momentum vectors p T (ℓ 1 ) and p T (ℓ 2 ). Its magnitude, ℓℓ T,boost , can be interpreted as the magnitude of the vector sum of all the transverse hadronic activity in the event. The azimuthal angle between the p miss T vector and the p ℓℓ T,boost vector is defined as Δ boost . This variable is useful for selecting events where the non hadronic component ( , , and or˜0 1 ) is collimated.
The lepton-based stransverse mass [100, 101] is a kinematic variable used to bound the masses of a pair of identical particles which have each decayed into a visible and an invisible particle. This quantity is defined as where T indicates the transverse mass, 5 p T,1 and p T,2 are the transverse momentum vectors of two visible particles, and q T,1 and q T,2 are transverse momentum vectors with p miss T = q T,1 + q T,2 . The minimisation is performed over all the possible decompositions of p miss T . In this paper, p T,1 and p T,2 are the transverse momentum vectors of the two leptons and T2 (p T (ℓ 1 ), p T (ℓ 2 ), p miss T ) is referred to simply as ℓℓ T2 . For the ℓℓ T2 calculation, the invisible particles are assumed to be massless. The ℓℓ T2 distribution is expected to have an endpoint corresponding to the mass for backgrounds such as¯while it is expected to reach higher values in the case of SUSY events, due to the presence of the neutralinos [102,103].
The three-body selection uses a number of 'super-razor' variables [104], which are derived with a series of assumptions made in order to approximate the centre-of-mass energy frame (Razor Frame) of two parent particles (i.e. top squarks) and the decay frames. Each parent particle is assumed to decay into a set of visible (only leptons are considered in this case) and invisible particles (i.e. neutrinos and neutralinos). These variables are T , the Lorentz factor R+1 , the azimuthal angle Δ R and R Δ . The first variable is T as the vector sum of the transverse momenta of the visible particles and the missing transverse momentum, and

√ˆR
as an estimate of the system's energy in the razor frame , defined as the frame in which the two visible leptons have equal and opposite longitudinal momentum ( z ). The value of | ì T | vanishes for events where leptons are the only visible particles, such as diboson events, leading to T values that tend toward zero. Instead, in events that contain additional activity, such as¯, this variable tends towards unity. The Lorentz factor, R+1 , is associated with the boost from the razor frame to the approximation of the two decay frames of the parent particles and is expected to have values tending towards unity for back-to-back visible particles or when they have different momenta. Lower values of R+1 are otherwise expected when the two visible particles are collinear and have comparable momentum. The azimuthal angle Δ R is defined between the razor boost from the laboratory to the frame and the sum of the visible momenta as evaluated in the frame. It is a good discriminator when used in searches for signals from models with small mass differences between the massive pair-produced particle and the invisible particle produced in the decay. Finally, the last variable is R Δ = √ˆR / R+1 , which is particularly powerful in discriminating between signal events and¯and diboson background, since it has a kinematic end-point that is proportional to the mass-splitting between the parent particle and the invisible particle.

Two-body event selection
This selection targets the dark-matter signal model that assumes the production of a pair of dark-matter particles through the exchange of a spin-0 mediator, in association with a pair of top quarks (Figure 1(a)). It is also used for a search for top squarks decaying into an on-shell top and neutralino ( Figure 1(d)).
For each event, the leading lepton, ℓ 1 , is required to have T (ℓ 1 ) > 25 GeV, while for the sub-leading lepton, ℓ 2 , the requirement is T (ℓ 2 ) > 20 GeV. The event selection also requires at least one reconstructed -jet, Δ boost lower than 1.5 and miss T significance greater than 12, and finally ℓℓ T2 greater than 110 GeV. Following the classification of the events, two sets of signal regions (SRs) are defined: a set of exclusive SRs binned in the ℓℓ T2 variable, to maximise model-dependent search sensitivity, and a set of inclusive SRs, to be used for model-independent results. For the binned SRs, events are separated according to the lepton flavours, different flavour or same flavour, and by the range ) . For the inclusive signal regions, referred to as SR

2−body
[ ,∞) with being the lower bound placed on the ℓℓ T2 variable, DF and SF events are combined. The common definition of these two sets of signal regions is shown in Table 2.

Three-body event selection
The three-body decay mode of the top squark shown in Figure 1 The signal kinematics in this region resemble that of production when Δ (˜,˜0 1 ) ∼ ( ) and that of¯production when Δ (˜,˜0 1 ) ∼ ( ). The signal selection was optimised to reject these dominant backgrounds while not degrading signal efficiency. The -jet multiplicity is highly dependent on the mass-splitting between the top squark and the neutralino, Δ (˜1,˜0 1 ) = (˜1) − (˜0 1 ), since for lower Δ (˜1,˜0 1 ) the -jets have lower momentum and cannot be reconstructed efficiently. Accordingly, two orthogonal signal regions were defined: SR  Table 3.

Four-body event selection
In the kinematic region defined by (˜1) < (˜0 1 ) + ( ) + ( ) and (˜1) > (˜0 1 ) + ( ), the top squarks are assumed to decay via a four-body process through an off-shell top quark and boson as shown in Figure 1(c). In this region the final-state leptons from the virtual boson decay are expected to have lower momentum and can be efficiently selected when imposing both a lower and upper bound on the T of the leptons. A transverse momentum lower bound of 4.5 GeV (4 GeV) is applied for electrons (muons), together with an upper bound, which is optimised separately for the leading and the sub-leading leptons. Two separate signal regions are defined to cover different Δ (˜1,˜0 1 ) ranges: the first one, SR 4−body Small Δ , targets small values of Δ (˜1,˜0 1 ) and requires T (ℓ 1 ) < 25 GeV and T (ℓ 2 ) < 10 GeV; the second one, SR 4−body Large Δ , targets larger values of Δ (˜1,˜0 1 ) and instead requires T (ℓ 2 ) > 10 GeV. This condition also ensures orthogonality between the two SRs. The presence of an energetic initial-state radiation (ISR) jet recoiling against the system of the two top squarks is required, introducing an imbalance in the event kinematics with an enhanced value of miss T that allows signal events to be distinguished from SM processes. For this reason, for each event, the leading jet 1 is considered to be a jet from ISR and required to have T > 150 GeV. A further reduction of the SM background is achieved with selections on miss T significance, ℓℓ T,boost , 2ℓ and 2ℓ4 variables. An additional requirement is applied to improve the sub-leading lepton isolation, using the following isolation variable: where '[jets]' contains all the jets in the event. This reduces the probability of lepton misidentification or selecting a lepton originating from heavy-flavour or / decays in jets. The definitions of these regions are summarised in Table 4.

Background estimation
The MC predictions for the dominant SM background processes are improved using a data-driven normalisation procedure, while non-dominant processes are estimated directly using MC simulation. A simultaneous profile likelihood fit [105] is used to constrain the MC yields with the observed data in dedicated background control regions (CRs). The fit is performed using standard minimisation software [106,107] where the normalisations of the targeted backgrounds are allowed to float, while the MC simulation is used to describe the shape of kinematic variables. Systematic uncertainties that could affect the expected yields in the different regions are taken into account in the fit through nuisance parameters. Each uncertainty source is described by a single nuisance parameter, and correlations between nuisance parameters, background processes and selections are taken into account. A list of the systematic uncertainties considered in the fits is provided in Section 7. The SM background thus modelled is validated in dedicated validation regions (VRs) which are disjoint from both the control and signal regions.
Important sources of reducible background are events with jets which are misidentified as leptons. The fake/non-prompt (FNP) lepton background comes from / and heavy-flavour hadron decays and photon conversions. This is particularly important for the low-T leptons targeted by the four-body selection. The FNP background is mainly suppressed by the lepton isolation requirements described in Section 4, but a non-negligible residual contribution is expected. This is estimated from data using the 'fake factor' method [108][109][110][111] which uses two orthogonal lepton definitions, labelled as 'Id' and 'anti-Id', to define a control data sample enriched in fake leptons. The Id lepton corresponds to the signal lepton identification criteria used in this analysis. Anti-Id electrons fail either the signal identification or isolation requirement, while anti-Id muons fail the isolation requirement. The sample used for the fake-factor computation is enriched in +jets events. Events with three leptons are selected, with the two same-flavour leptons of opposite electric charge (SFOS leptons) identified as the boson decay products (ℓ 1 and ℓ 2 , in order of decreasing T ) satisfying the Id requirements, and the third unpaired lepton, called the probe lepton (ℓ probe ), satisfying either the Id or anti-Id criteria. The fake factor is defined as the ratio of the Id lepton yield to the anti-Id probe lepton yield. Residual contributions from processes producing prompt leptons are subtracted using the MC predictions. Fake factors are measured separately for electrons and muons and as a function of the lepton T and . These are derived in the CR FNP region whose selection is summarised in Table 5. The FNP estimates in each analysis region are derived by applying the fake factors to events satisfying that region's criteria but replacing at least one of the signal leptons by an anti-Id one.
Additional requirements T (ℓ probe ) < 16 GeV or miss T

< 50 GeV
The three selections in this paper use different sets of CRs and VRs, specifically designed to be kinematically similar to the respective SRs. The definitions of the regions used in each analysis and the results of the fits are described in the following subsections.

Estimation of the backgrounds in the two-body selection
The main background sources for the two-body selection are¯and¯with invisible decay of the boson. These processes are normalised to data in dedicated CRs: CR 2−bodȳ and CR¯. The¯normalisation factor is extracted from different-flavour dilepton events. In order to test the reliability of the¯background prediction, two validation regions VR 2−bodȳ ,DF and VR 2−bodȳ ,SF are defined. The¯production events with invisible decay of the boson are expected to dominate the tail of the ℓℓ T2 distribution in the SRs and are normalised in the dedicated control region CR¯. Given the difficulty in achieving sufficient purity for this SM process because of the high contamination from¯events, a strategy based on a three-lepton final state is adopted. Events are selected if characterised by three charged leptons including at least one pair of SFOS leptons having invariant mass consistent with that of the boson (| ℓℓ − | < 20 GeV). If more than one pair is identified, the one with ℓℓ closest to the boson mass is chosen. Events are further required to have a jet multiplicity, jets , greater than or equal to three with at least two -tagged jets. These selections target¯production with the boson decaying into two leptons and¯decaying in the semileptonic channel. In order to select¯events whose kinematics, regardless of subsequentā nd decays, emulate the kinematics of this background in the SRs, the momenta of the two leptons of the SFOS pair (p(ℓ Z 1 ), p(ℓ Z 2 )) are vectorially added to the p miss T , effectively treating them like the neutrino pair from the boson decay. A variable called miss T,corr = p miss T and the momenta of the remaining two leptons. The definition of the control and validation regions used in the two-body selection is summarised in Table 6. The expected signal contamination in the CRs is generally below ∼ 1%. The signal contamination in the VRs is less than 15% (7%) for a DM signal model with scalar (pseudoscalar) mediator mass of 100 GeV and DM mass of 1 GeV.  The results of the fit are reported in Table 7 for the two-body CRs and VRs. The normalisations for fitted backgrounds are found to be consistent with the theoretical predictions when uncertainties are considered: the normalisation factors obtained from the fit for¯and¯are 0.88 ± 0.08 and 1.07 ± 0.14 respectively. ,¯,¯¯,¯,¯,¯,¯, and . The hatched bands represent the total statistical and detector-related systematic uncertainty. The rightmost bin of (b) includes overflow events. In the upper panels, red arrows indicate the control region selection criteria. The bottom panels show the ratio of the observed data to the total SM background prediction, with hatched bands representing the total uncertainty in the background prediction; red arrows show data outside the vertical-axis range.
Good agreement, within one standard deviation of the SM background prediction, is observed in the VRs (see Figure 3).

Estimation of the backgrounds in the three-body selection
The dominant SM backgrounds in the three-body signal regions are diboson,¯and¯production. Dedicated CRs were defined, labelled as CR 3−body and CR 3−bodȳ , which are kinematically close to the SRs and which have good purity in diboson and¯events respectively. The orthogonality between CRs and SRs is mainly ensured by the inversion of the Δ R cut. The normalisation of the¯background is extracted using the same control region CR¯defined for the two-body selection in Section 6.1. Dedicated validation regions were defined to test the modelling of these processes: VR  Table 8. The expected signal contamination is below 2% in the CRs and reaches a maximum of 10% in the VRs for a top squark mass of ∼ 430 GeV. Table 9 shows the expected and observed numbers of events in each of the control and validation regions after the background fit. The normalisation factors extracted from the fit of the backgrounds for the diboson, and¯production processes are 0.92 ± 0.28, 0.96 ± 0.09 and 1.06 ± 0.15 respectively. The total number of fitted background events in the validation regions is in agreement with the observed number of data events. Figure 4 shows the distributions of Δ R for the CR   Table 8: Three-body selection. Control and validation regions definitions. The common selection defined in Section 5 also applies to all regions. A further control region CR¯was defined previously in Table 7.
> 20 > 20 > 20 > 20 > 20     ,¯,¯¯,¯,¯,¯,¯, and processes. The hatched bands represent the total statistical and detector-related systematic uncertainty. The bottom panels show the ratio of the observed data to the total SM background prediction, with hatched bands representing the total uncertainty in the background prediction.

Estimation of the backgrounds in the four-body selection
The dominant irreducible SM background sources for the four-body selection are¯and diboson: these backgrounds are normalised in two dedicated background-enriched control regions labelled as CR T and ℓ,corr is defined as the ratio of miss ,1ℓ,corr to the sum of the transverse momenta of two remaining OS leptons. The invariant mass of the remaining two leptons, called ℓℓ,corr , is also used. The definition of the control and validation regions used in the four-body selection is summarised in Table 10. In thec ontrol region the signal contamination is ∼ 1% or less. In CR 4−body , the typical signal contamination is about ∼ 1 − −2%, but reaches a maximum value of ∼ 5% for a top squark mass of ∼ 400 GeV and lightest-neutralino mass of ∼ 310 GeV at the boundary of the region excluded by the previous analysis. Signal contamination in the validation regions is below 10%. Table 11 shows the expected and observed numbers of events in each of the control and validation regions after the background fit. The normalisation factors extracted by the fit for the diboson and¯production processes are 1.00 ± 0.25 and 0.90 ± 0.12 respectively. The distributions of miss   ,¯,¯¯,¯,¯,¯,¯, and processes. The hatched bands represent the total statistical and detector-related systematic uncertainty. The rightmost bin of each plot includes overflow events. In the upper panels, red arrows indicate the control region selection criteria. The bottom panels show the ratio of the observed data to the total SM background prediction, with hatched bands representing the total uncertainty in the background prediction. ,¯,¯¯,¯,¯,¯,¯, and processes. The hatched bands represent the total statistical and detector-related systematic uncertainty. The rightmost bin of each plot includes overflow events. In the upper panels, red arrows indicate the validation region selection criteria. The bottom panels show the ratio of the observed data to the total SM background prediction, with hatched bands representing the total uncertainty in the background prediction.

Systematic uncertainties
Systematic uncertainties are evaluated for the signal and for the background predictions. The main experimental uncertainties in the yields of the reconstructed objects, the theoretical uncertainties in the processes' yields, and the uncertainties related to the MC modelling of the SM backgrounds are described in this section. The statistical uncertainties in the simulated event samples are also taken into account.
The main sources of experimental uncertainty are related to the jet energy scale (JES) and the jet energy resolution (JER). The JES and JER uncertainties are derived as a function of the T and of the jet, as well as of the pile-up conditions and the jet-flavour composition of the selected jet sample [112]. Uncertainties associated with the modelling of the -tagging efficiencies for -jets, -jets and light-flavour jets [113,114] are also considered. The systematic uncertainties related to the modelling of miss T in the simulation are estimated by propagating the uncertainties in the energy and momentum scales of electrons, muons and jets, as well as the uncertainties in the resolution and scale of the soft term [115]. Other detector-related systematic uncertainties, including those arising from lepton reconstruction efficiency, energy scale, energy resolution and in the modelling of the trigger efficiency [45,51,52,116,117], or the ones due to the pile-up reweighting and JVT are found to have a small impact on the results.
Systematic uncertainties in the theoretical modelling of the observed final states can be broadly divided into uncertainties in the description of the parton-level final states (uncertainties in the proton PDF, cross-section, and strong coupling constant) and further uncertainties arising from the parton showering and hadronisation processes that convert partons into the hadronic final states. The uncertainties in the modelling of theb ackground are estimated by varying the renormalisation and factorisation scales, as well as the amount of initial-and final-state radiation produced when generating the samples [118,119]. Comparison between the yields obtained with P and M G 5_aMC@NLO [118] is used to estimate uncertainties from the event generator choice. For¯production, in the two-body and three-body selections, the effects of QCD scale uncertainties are evaluated using seven-point variations of the factorisation and renormalisation scales [120]. Uncertainties for additional radiation contributions (ISR, FSR) are evaluated by comparing the nominal sample with one obtained with a P tune enhancing the radiation [55]. In the four-body selection, since the¯background contribution is minor, a total theoretical error of 14%, coming from the cross-section uncertainty [121], is applied instead. For¯and¯production, the parton showering and hadronisation uncertainties are covered by the difference between samples obtained using the two different showering models implemented in P and in H . Single top quark production via the -channel is a minor background in all the selections. An uncertainty in the acceptance due to the interference between¯and production is assigned by comparing dedicated samples produced with P and P using the diagram removal (DR) and the diagram subtraction (DS) approaches [122]. The modelling uncertainties for the diboson background are estimated using the seven-point variations of the renormalisation and factorisation scales. Additional uncertainties in the resummation (QSF) and matching (CKKM) scales between the matrix element generator and parton shower are computed by varying the scale parameters in S [90]. For the other background processes which make minor contributions a conservative uncertainty is applied. These minor backgrounds are mainly¯and processes. A 30% uncertainty, driven by the DR versus DS difference for the¯[123] process, is applied in the two-body and three-body selections. For the four-body selection a 22% uncertainty is applied for the uncertainty in the¯cross-section [121]. For all the processes mentioned above the PDF uncertainties [65] were evaluated and found to be negligible.
Systematic uncertainties in the data-driven FNP background estimate are expected due to potential differences in the FNP composition (heavy flavour, light flavour or photon conversions) between the regions defined in Section 6 and the CR FNP used to extract the fake factor. A FNP systematic error is evaluated in each of the regions by varying the FNP composition in the CR FNP to match that of the considered analysis region. The statistical error is also included by propagating the statistical uncertainty in the ratio used to compute the fake factor. For the four-body selection, where the FNP lepton background is dominant, a FNP closure uncertainty is also evaluated from the full difference between the data and the FNP predictions as observed in a validation region with two same-charge leptons with kinematics similar to the four-body selection. The closure uncertainty ranges between 13% and 33% in the regions where the FNP background is important. A 1.7% uncertainty in the luminosity measurement is considered for all signal and background estimates that are derived directly from MC simulations [46].
Tables 12, 13 and 14 summarise the contributions from the different sources of systematic uncertainty in the total SM background predictions for the two-body, three-body and four-body signal regions. The total systematic uncertainty ranges between 14% and 26%, with the dominant sources being the MC statistical error, the JES and JER, the uncertainty in the background normalisation and the theoretical uncertainties.
The SUSY signal cross-section uncertainty is evaluated from an envelope of the cross-section predictions using different PDF sets and factorisation and renormalisation scales as described in Ref. [124]. The uncertainty in the DM production cross-section is derived from the scale variations and the PDF choices. The SUSY and DM theory signal uncertainties are computed from the variation of the radiation, renormalisation, factorisation and merging scales. These uncertainties are most relevant for the four-body selection, where the largest theory uncertainties are those resulting from radiation and are in the range 10% to 24% depending on the mass difference (˜1) − (˜0 1 ). For the DM signals the total systematic uncertainty is between 5% and 20%.

Results
A set of simultaneous likelihood fits is performed, for each one of the three different selections, using standard minimisation software packages, HistFitter and pyhf [106,107]. For the normalisation of the semi-data-driven backgrounds, only the CRs are considered in the background fit, while for the computation of the exclusion limits both the CRs and SRs are included as constraining channels. The likelihood is a product of Poisson probability density functions (pdf), describing the observed number of events in each CR/SR, and Gaussian pdf distributions that describe the nuisance parameters associated with all the systematic uncertainties. Systematic uncertainties that are correlated between different samples are accounted for in the fit configuration by using the same nuisance parameter. The uncertainties are applied in each of the CRs and SRs and their effect is correlated for events across all regions in the fit.
The results of the background fit are shown in Figures 8-10 for each of the three analysis selections. In general, good agreement, within about one standard deviation, is observed in all the SRs and VRs except in SR-DF 3−body where the data fluctuates well below the fit.

Two-body selection results
The estimated SM yields in the binned and inclusive SRs defined in the two-body selection are obtained with a background fit which simultaneously determines the normalisations of the background contributions from¯and¯. Figure 11 shows the ℓℓ T2 distribution for events satisfying all the selection criteria of the SR 2−body 110,∞ (SF and DF) signal regions, after the background fit. Each bin corresponds to one of the binned SRs. No significant excess over the SM prediction is observed, as can be seen from results shown in Tables 15 and 16 for the binned SRs. ,¯,¯¯,¯,¯,¯,¯, and processes. The hatched bands represent the total statistical and systematic uncertainty. The rightmost bin of each plot includes overflow events. Reference dark-matter signal models are overlayed for comparison. Red arrows in the upper panels indicate the signal region selection criteria. The bottom panels show the ratio of the observed data to the total SM background prediction, with hatched bands representing the total uncertainty in the background prediction.

Three-body selection results
The dominant background processes in the three-body selection are diboson,¯and¯production, and the yields are determined with a simultaneous fit. Figure 12 shows the distributions of R Δ in SR 3−body (top) and in SR 3−body (bottom), for events satisfying all the selection criteria except the one for the presented variable, after the background fit. Table 17 shows the observed events in each signal region and the SM background estimates. No excess over the SM prediction is observed while a fluctuation of about −2 is observed in SR-DF 3−body and is also visible in Figure 12

Four-body selection results
The estimated SM yields in SR 4−body Small Δ and SR

4−body
Large Δ are determined with a background fit that provides the normalisation factors for¯and diboson production. Figure 13 shows the distributions of (a) miss T in SR 4−body Small Δ and (b) 2ℓ4 in SR 4−body Large Δ for events satisfying the selection criteria of the given SR, except the one for the presented variable, after the background fit. The background fit results are shown in Table 18. The observed yield in the SR is within one standard deviation of the background prediction. The bottom panels show the ratio of the observed data to the total SM background prediction, with hatched bands representing the total uncertainty in the background prediction; red arrows show data outside the vertical-axis range.

Interpretation
No excess is observed in the data relative to the expected background. The analysis results are therefore interpreted in terms of model-independent upper limits on the visible cross-section ( vis ) of new physics, defined as the 95% confidence level (CL) upper limit on the number of signal events ( 95 ) divided by the integrated luminosity, and in terms of exclusion limits in the plane of the masses parameters of our simplified models. For the two-body selection the upper limits are derived using the inclusive SRs.
The upper limits on vis are derived, in each SR, by performing a model-independent hypothesis test, which introduces a free signal as an additional process to be constrained by the observed yield. The CL s method [126] is used to derive all the exclusion confidence levels. Model-independent upper limits are presented in Table 19. These limits assume negligible signal contamination in the CRs, resulting in a more conservative result than from the model-dependent limits, where a small signal contamination is allowed in the CRs. Large Δ for the four-body selection. Limits for simplified models in which pair-produced˜1 decay with 100% branching ratio into a top quark and˜0 1 are shown in the˜1-˜0 1 mass plane in Figure 14(a) and in the (˜1)-Δ (˜1,˜0 1 ) plane in Figure 14(b). The exclusion contour is the envelope of the exclusion regions obtained separately for the three selections. Top squark masses up to 1 TeV are excluded for a massless lightest neutralino. Neutralino masses up to 500 GeV are excluded for (˜1) above the top quark production kinematic limit. In the three-body decay region, top squark masses are excluded up to 600 GeV for Δ (˜1,˜0 1 ) = 120 GeV, up to 550 GeV for Δ (˜1,˜0 1 ) close to the top quark mass and up to 430 GeV for Δ (˜1,˜0 1 ) close to the boson mass. In the four-body decay region, top squark masses are excluded up to 540 GeV for Δ (˜1,˜0 1 ) = 40 GeV. Top squark decay around the boson production kinematic limit is not fully excluded for (˜1) above 400 GeV because there the four-body and three-body decay exclusion regions do not overlap. The four-body selection loses sensitivity for Δ (˜,˜0 1 ) ( ) due to the upper bound of the sub-leading lepton T while, for the three body selection, the R Δ requirement suppresses the sensitivity for Δ (˜,˜0 1 ) ( ) because of the smaller mass splitting. The three-body and two-body overlap in the sensitivity provides exclusion coverage around the top quark production kinematic limit up to (˜1) of 540 GeV.  For the DM mediator models, Figure 15 shows upper limits at 95% CL on the observed signal cross-section scaled to the theoretical signal cross-section for a coupling = = = 1, denoted by obs / Th ( = 1.0). These limits are obtained as a function of the mediator mass, assuming a specific DM particle mass of 1 GeV. Both the scalar and pseudoscalar mediator cases are considered. The sensitivity is approximately constant for mediator masses below 100 GeV and the models are excluded for scalar (pseudoscalar) mediator masses up to 250 (300) GeV when assuming = 1.   Compared to previous searches a significant improvement in sensitivity is obtained by using additional integrated luminosity and a new discriminating variable, the object-based miss T significance. Moreover, in the small-Δ (˜1,˜0 1 ) region, an important gain in sensitivity is also achieved by lowering the T threshold for lepton selection.
The data are found to be consistent with the Standard Model predictions. Assuming direct˜1 pair production with both top squarks decaying in either the two-body channel˜1 →˜0 1 , the three-body channel the minimum˜1 and˜0 1 masses up to about 1 TeV and 500 GeV respectively. The results improve on the previous ATLAS limits obtained in a two-lepton final state and provide unique sensitivity among the ATLAS searches in the mass region where the decay˜1 →˜0 1 becomes kinematically allowed. For the dark-matter model, assuming spin-0 mediator production in association with a pair of top quarks and decay with 100% branching ratio into a pair of dark-matter particles, scalar (pseudoscalar) mediator masses up to about 250 (300) GeV are excluded at 95% confidence level for mediator couplings = = 1 to Standard Model and dark-matter particles.