Search for new phenomena with top quark pairs in final states with one lepton, jets, and missing transverse momentum in $pp$ collisions at $\sqrt{s}$= 13 TeV with the ATLAS detector

A search for new phenomena with top quark pairs in final states with one isolated electron or muon, multiple jets, and large missing transverse momentum is performed. Signal regions are designed to search for two-, three-, and four-body decays of the directly pair-produced supersymmetric partner of the top quark (stop). Additional signal regions are designed specifically to search for spin-0 mediators that are produced in association with a pair of top quarks and decay into a pair of dark-matter particles. The search is performed using the Large Hadron Collider proton-proton collision dataset at a centre-of-mass energy of $\sqrt{s}$=13 TeV recorded by the ATLAS detector from 2015 to 2018, corresponding to an integrated luminosity of 139 fb$^{-1}$. No significant excess above the Standard Model background is observed, and limits at 95 % confidence level are set in the stop-neutralino mass plane and as a function of the mediator mass or the dark-matter particle mass. Stops are excluded up to 1200 GeV (710 GeV) in the two-body (three-body) decay scenario. In the four-body scenario stops up to 640 GeV are excluded for a stop-neutralino mass difference of 60 GeV. Scalar and pseudoscalar dark-matter mediators are excluded up to 200 GeV when the coupling strengths of the mediator to Standard Model and dark-matter particles are both equal to one and when the mass of the dark-matter particle is 1 GeV.


Introduction
This paper presents a search for new phenomena in events with top quark pairs, in a final state with exactly one isolated charged lepton (electron or muon, 1 henceforth referred to as 'lepton') from the decay of an onor off-shell boson, jets, and a significant amount of missing transverse momentum ( ì miss T ), the magnitude of which is denoted by miss T . This experimental signature may arise in Supersymmetry (SUSY) [1][2][3][4][5][6][7] or in models with a spin-0 mediator produced in association with top quarks [8,9] and subsequently decaying into a pair of dark matter (DM) particles.
SUSY extends the Standard Model (SM) by introducing a supersymmetric partner for each SM particle, the two having identical quantum numbers except for a half-unit difference in spin. Searches for a light supersymmetric partner of the top quark, referred to as the top squark or 'stop', are of particular interest after the discovery of the Higgs boson [10,11] at the Large Hadron Collider (LHC). Stops may largely cancel out divergent loop corrections to the Higgs boson mass [12][13][14][15][16][17][18][19], and thus, supersymmetry may provide an elegant solution to the hierarchy problem [20][21][22][23]. The superpartners of the left-and right-handed top quarks,˜L and˜R, mix to form two mass eigenstates,˜1 and˜2, where˜1 is the lighter of the two. Significant mass splitting between the˜1 and˜2 particles is possible due to the large top quark Yukawa coupling. A generic -parity-conserving 2 minimal supersymmetric extension of the SM (MSSM) [7,12,[24][25][26] predicts pair production of SUSY particles and the existence of a stable lightest supersymmetric particle (LSP). The mass eigenstates from the linear superposition of charged or neutral SUSY partners of the Higgs and electroweak gauge bosons (higgsinos, winos and binos) are called charginos˜± 1,2 and neutralinos 0 1,2,3,4 . The lightest neutralino (˜0 1 ), assumed to be the LSP, may provide a potential dark matter (DM) candidate because it is stable and only interacts weakly with ordinary matter [27,28]. This paper presents a search for direct pair production of˜1 particles, with significant amount of miss T , from the two weakly interacting LSPs that escape detection. Scenarios with on-and off-shell production of bosons and top quarks in the stop decays are considered, leading to two-, three-and four-body decays of the stop.
The search for a spin-0 mediator produced in association with top quarks and subsequently decaying into a pair of DM particles is motivated by SM extensions which respect the principle of minimal flavour violation resulting in the interaction strength between the spin-0 mediator and the SM quarks being proportional to the fermion masses via Yukawa-type couplings.
Dedicated searches for direct˜1 pair production were recently reported by the ATLAS [29-32] and CMS [33][34][35][36][37][38][39][40] Collaborations. Previous ATLAS and CMS searches extend the lower limit on˜1 masses at 95% confidence level to 1.2 TeV in the two-body decay scenario and up to ∼450 GeV in the three-body decay scenario. Searches for spin-0 mediators produced in association with heavy-flavour quarks and decaying into a pair of DM particles have also been reported by the ATLAS [29,41] and CMS [42] Collaborations.

Signal models and search strategy
Two classes of physics models are targeted by this search, the production of˜1 pairs in simplified SUSY models [43][44][45] where the only light sparticles are˜1 and˜0 1 , and simplified benchmark models for DM production that assume the existence of a spin-0 mediator particle that can be produced in association with two top quarks [41,46] and decays into a pair of DM particles¯.
The experimental signatures of stop pair production can vary dramatically, depending on the mass-splitting between˜1 and˜0 1 . Figure 1 illustrates the two-, three-and four-body stop decays considered in this paper. As flavour-changing neutral current processes are not considered, the dominant among the two-, three-or four-body stop decays is assumed to have a 100% branching ratio in a given Δ˜1 The searches for stops presented in this paper use several signal regions dedicated to each of the decay channels˜1 → +˜0 1 ,˜1 →˜0 1 and˜1 → ˜0 1 . For instance, specific signal regions target the so-called compressed region where the stop undergoes a˜1 → +˜0 1 decay but where Δ˜1 The selections are optimised for given benchmark model points, and are binned in key variables to retain sensitivity to the widest possible range of˜1 and˜0 1 masses.  Figure 1: Diagrams illustrating the stop decay modes, which are referred to as (a)˜1 → +˜0 1 , (b)˜1 →˜0 1 and (c)˜1 → ˜0 1 . In these diagrams, the charge-conjugate symbols are omitted for simplicity. All the processes considered involve the production of a squark-antisquark pair. φ/at t g gχ χ Figure 2: A representative Feynman diagram for spin-0 mediator production. The / is the scalar/pseudoscalar mediator, which decays into a pair of dark matter ( ) particles.
The mediator-based DM scenarios consist of simplified models with a DM particle that is a SM singlet and a single spin-0 mediator that couples to SM fermions. Both the scenarios where the mediator is a scalar, , or a pseudoscalar, , are considered, as illustrated in Figure 2. These models have four parameters: the mass of the mediator, med , the DM mass, DM , the DM-mediator coupling, , and the coupling of the mediator to the SM fermions, q . In the models considered, the interaction strength between the mediator and SM particles is proportional to the fermion masses via Yukawa-type couplings, and therefore final states involving top quarks dominate over those involving other fermions. Due to the associated production of top quarks with undetected DM particles in the same event, the mediator-based DM model predicts an excess of¯+ miss T final-state events above the SM expectation. A dedicated signal region common to both the scalar and pseudoscalar models is developed. The signal region is binned in the azimuthal angle Δ ( ì miss T , ℓ) between the missing transverse momentum and the leading lepton, to retain maximum sensitivity to both the scalar and pseudoscalar models and to a large range of mediator and DM particle masses.
The searches presented are based on eight dedicated analyses that target the various scenarios mentioned above. Each of these analyses corresponds to a set of event selection criteria, referred to as a signal region (SR), and is optimised to achieve three standard deviation expected sensitivity to the targeted benchmark model. Two techniques are employed to define the SRs: 'cut-and-count' and 'shape-fit' methods. The former is based on counting events in a single region of phase space, and is employed in the eight analyses. The latter is used in several SRs to improve the exclusion reach if no excess is observed in the cut-and-count signal regions, and employs SRs split into multiple bins in one or two key discriminating kinematic variables. The shape-fit method exploits the varying signal-to-background ratios in different bins to provide sensitivity to a wider range of new-particle masses than can be achieved by a single cut-and-count SR. Including these background-rich regions in the single-bin discovery SRs would significantly reduce the sensitivity to the targeted signatures.
The main background processes after the signal selections include¯,¯+ (→¯), +jets and the associated production of a single top quark and a boson ( ). Backgrounds from these SM processes are estimated by exploiting dedicated control regions (CRs) enriched in these processes. The backgrounds are normalised to data by applying a likelihood fit simultaneously to the SR and associated CRs, making the analysis more robust against potential mis-modelling in simulated events and reducing the uncertainties in the background normalisation. Before looking at the data in the signal regions, the background modelling and the normalisation procedure are tested in a series of validation regions (VRs) by applying the normalisation factors determined by a background-only fit in the CRs. A background-only fit to the CRs and SRs then provides a statistical test that quantifies the existence and extent of a potential excess of events in data in the SRs. In the absence of an excess, exclusion limits are set on the associated model parameters by using the theoretical cross-sections. An overview of the signal regions and the benchmark models for optimisation is presented in Table 1.

ATLAS detector and data collection
The ATLAS detector [47] at the LHC is a multipurpose particle detector with almost 4 coverage in solid angle around the interaction point. 3 It consists of an inner tracking detector (ID) surrounded by a superconducting solenoid providing a 2 T axial magnetic field, electromagnetic and hadronic calorimeters, and a muon spectrometer (MS), which is based on three large air-core toroidal superconducting magnets consisting of eight coils each. The ID provides charged-particle tracking in the range | | < 2.5. During the LHC shutdown between Run 1 (2010-2012) and Run 2 (2015-2018), a new innermost layer of silicon pixels was added [48,49], which improves the track impact parameter resolution, vertex position resolution and -tagging performance [50]. High-granularity electromagnetic and hadronic calorimeters provide energy measurements up to | | = 4.9. The electromagnetic calorimeters, as well as the hadronic calorimeters in the endcap and forward regions, are sampling calorimeters with liquid argon as the active medium and lead, copper, or tungsten absorbers. The hadronic calorimeter in the central region of the detector is a sampling calorimeter with scintillator tiles as the active medium and steel absorbers. The MS surrounds the calorimeters and has three layers of precision tracking chambers with coverage up to | | = 2.7 and fast detectors for triggering in the region | | < 2.4. A two-level trigger system [51] is used to select events. The first-level trigger is hardware-based, followed by a software-based trigger system.
The results in this paper utilise the full Run 2 data sample collected from 2015 to 2018 at a centre-of-mass energy of √ = 13 TeV. The average number of simultaneous interactions per bunch crossing, referred to as 'pile-up', in the recorded data is approximately 34. After the application of beam, detector and data-quality requirements, the total integrated luminosity is 139 fb −1 . The uncertainty in the combined 2015-2018 integrated luminosity is 1.7%. It is derived from the calibration of the luminosity scale using -beam-separation scans, following a methodology similar to that detailed in Ref. [52], and using the LUCID-2 detector for the baseline luminosity measurements [53].
All events were recorded with triggers that accepted events with miss T above a given threshold. The miss T triggers relied on energy measurements in the calorimeter and on several algorithms based on cells, jets or topological clusters in addition to two methods for correcting for the effects of pile-up. The triggers were fully efficient for events passing an offline-reconstruction requirement of miss T > 230 GeV.

Simulated event samples
Samples of Monte Carlo (MC) simulated events are used for the description of the SM background processes and to model the signals. Details of the simulation samples used, including the matrix element (ME) event generator and parton distribution function (PDF) set, the parton shower (PS) and hadronisation model, the set of tuned parameters (tune) for the underlying event (UE) and the order of the cross-section calculation, are summarised in Table 2.  [73,74] The samples produced with M G 5_aMC@NLO [69] and P -B [54, [75][76][77][78] used E G v1.6.0 [79] for the modelling of -hadron decays. The signal samples were all processed with a fast simulation [80], whereas all background samples were processed with the full simulation of the ATLAS detector [80] based on G 4 [81]. All samples were produced with varying numbers of minimum-bias interactions generated by P 8 with the A3 tune [82] and overlaid on the hard-scattering event to simulate the effect of multiple interactions in the same or nearby bunch crossings. The number of interactions per bunch crossing was reweighted to match the distribution in data.
The nominal¯sample and single-top sample cross-sections were calculated at NNLO with the resummation of soft gluon emission at NNLL accuracy and were generated with P -B (at NLO accuracy) interfaced to P 8 for parton showering and hadronisation. Additional¯samples were generated with M G 5_aMC@NLO (at NLO accuracy)+P 8 and P -B +H 7 [83,84] for modelling comparisons and the evaluation of systematic uncertainties [85]. The¯and processes have identical final states and can interfere. Additional¯, and samples were generated as multi-leg processes at LO with M G and used to estimate the systematic uncertainty from the interference modelling. The tN_med and tN_high regions receive significant contributions from both6 and in a phase space where the interference is significant. Techniques used to model the interference such as diagram subtraction (DS) and diagram removal (DR) [86] were shown to provide predictions bracketing the data [87], but can lead to large uncertainties. Both schemes are investigated in this paper, but the DR scheme is ultimately used for the nominal sample.
The +jets and +jets samples were generated with S 2.2.1 [67,88]  The SUSY samples were generated at LO with M G 2.6.2 including up to two extra partons, and interfaced to P 8 for parton showering and hadronisation. For the˜1 → +˜0 1 samples, the stop was decayed in P 8 using only phase-space considerations and not the full ME. Since the decay products in the generated event samples did not preserve spin information, a polarisation reweighting was applied following Refs. [95,96]. A value of cos = 0.553 was assumed, corresponding to a˜1 composed mainly of˜R (∼70%). For the˜1 →˜0 1 and˜1 → ˜0 1 samples the stops were decayed with M S [97], interfaced to P 8 for the parton showering. M S emulates kinematic distributions such as the mass of the ( * ) system to a good approximation without calculating the full ME.
The signal cross-sections for stop pair production were calculated to approximate next-to-next-to-leading order in the strong coupling constant, adding the resummation of soft gluon emission at next-to-nextto-leading-logarithm accuracy (approximate NNLO+NNLL) [72,[98][99][100]]. The nominal cross-section and its uncertainty were derived using the PDF4LHC15_mc PDF set, following the recommendations of Ref. [101]. The stop pair production cross-section varies from approximately 200 fb at˜1 = 600 GeV to about 2 fb at˜1 = 1150 GeV.
Signal events for the spin-0 scalar and pseudoscalar mediator models were generated at LO with up to one additional parton with M G 2.6.2 interfaced to P 8 for parton showering and hadronisation. In the DM sample generation the couplings of the mediator to the DM and SM particles ( and q ) were set to one. When interpreting the experimental results, a single common coupling = = q is always assumed. Coupling values of = 1 as well as < 1 are considered. The kinematics of the mediator decay were found to not depend strongly on the values of the couplings; however, the particle kinematic distributions are sensitive to the scalar or pseudoscalar nature of the mediator and to the mediator and DM particle masses. The cross-sections were computed at NLO [73,74] and decrease significantly when the mediator is produced off-shell. The production cross-section varies from approximately 26 pb to 130 fb over a scalar mediator mass range of 10 to 200 GeV and from approximately 600 fb to 120 fb over a pseudoscalar mediator mass range of 10 to 200 GeV.

Event reconstruction
Events selected in the analysis must satisfy a series of beam, detector and data-quality criteria. The primary vertex, defined as the reconstructed vertex with the highest tracks 2 T , must have at least two associated tracks with T > 500 MeV.
Depending on the quality and kinematic requirements imposed, reconstructed physics objects are labelled as either baseline or signal, where the latter is a subset of the former, with tighter selection criteria applied. Baseline objects are used when classifying overlapping selected objects and to compute the missing transverse momentum. Background contributions from¯and production where both bosons decay leptonically, referred to as dileptonic¯or events, are suppressed by vetoing events with more than one baseline lepton. Signal objects are used to construct kinematic and discriminating variables necessary for the event selection.
Electrons are identified as energy clusters formed in the electromagnetic calorimeter matched to tracks in the ID. Baseline electrons are required to have T > 4.5 GeV and | | < 2.47, and to satisfy 'LooseAndBLayer' likelihood identification criteria that follow the methodology described in Ref. [102]. Furthermore, their longitudinal impact parameter ( 0 ), defined as the distance along the beam direction between the primary vertex and the track's point of closest approach to the beam axis, must satisfy | 0 sin | < 0.5 mm where is the polar angle of the track. Signal electrons must satisfy all the baseline requirements and have a transverse impact parameter ( 0 ) that satisfies | 0 |/ 0 < 5, where 0 is the uncertainty in 0 . Furthermore, signal electrons are required to be isolated. The isolation is defined as the sum of the transverse energy or momentum reconstructed in a cone of size Δ = √︁ (Δ ) 2 + (Δ ) 2 around the electron, excluding the energy of the electron itself. The isolation criteria rely on both track-and calorimeter-based information with a fixed requirement on the isolation energy divided by the electron's T . Electrons that satisfy the signal identification criteria, including the loose isolation, are called loose electrons. In addition, tight electrons must satisfy both a tight electron likelihood identification criterion and a tight isolation criterion.
Muon candidates are reconstructed from combined tracks that are formed from ID and MS tracks, or stand-alone MS tracks. Baseline muons up to | | = 2.7 are used, and are required to have T > 4 GeV, a longitudinal impact parameter | 0 sin | < 0.5 mm, and to satisfy the 'Medium' identification criterion [103]. Signal muons must satisfy all baseline requirements and in addition have a transverse impact parameter satisfying | 0 |/ 0 < 3. Tight signal muons must satisfy tight isolation criteria, similar to those used for tight signal electrons, but with a fixed requirement on track-based isolation energy divided by the muon's T . A category of loose signal muons is also defined, which requires the 'Loose' identification criterion [103] and satisfies a looser isolation criterion.
Dedicated efficiency scale factors are derived from → ℓl and / → ℓl data samples to correct the simulations for minor mis-modelling of electron and muon identification, impact parameter and isolation selections. The T threshold of signal leptons is 25 GeV for electrons and muons in all signal regions except for signal regions dedicated to˜1 → ˜0 1 , where electrons with T > 4.5 GeV and muons with T > 4 GeV are used.
Jet candidates are built from topological clusters [104,105] in the calorimeters using the antialgorithm [106] with a jet radius parameter = 0.4 implemented in the FastJet package [107]. Jets are corrected for contamination from pile-up using the jet area method [108][109][110] and are then calibrated to account for the detector response [111,112]. Jets in data are further calibrated according to in situ measurements of the jet energy scale [112]. Baseline jets are required to have T > 20 GeV. Signal jets are required to have | | < 2.5 and T > 25 GeV in all signal regions, except in the four-body signal regions, where the T threshold of signal jets is 20 GeV. Furthermore, signal jets with T < 120 GeV and | | < 2.5 must satisfy track-based criteria designed to reject jets originating from pile-up [110]. Events containing a signal jet that does not satisfy specific jet-quality requirements ('jet cleaning') are rejected to suppress detector noise and non-collision backgrounds [113,114]. The number of signal jets in an event is denoted jet . In addition to these jet candidates, the same anti-algorithm is used to define larger radius (large-) jets to provide discriminating variables for the reconstruction of top quarks, as described in Section 6.
Jets identified as containing -hadrons are referred to as -tagged jets. Their identification is performed using the MV2c10 -tagging algorithm, which examines quantities such as the impact parameters of associated tracks and characteristics of reconstructed secondary vertices [115,116]. The algorithm is used at a working point that provides a 77% -tagging efficiency in simulated¯events, and corresponds to a rejection factor of about 130 for jets originating from gluons and light-flavour quarks (light jets) and about 6 for jets induced by charm quarks. Corrections derived from data control samples are applied to account for differences between data and simulation in the efficiency and mis-tag rate of the -tagging algorithm. The number of -tagged jets in an event is denoted -jet . Since MV2c10 is only applicable to baseline jets with T > 20 GeV, it is not sensitive to low-T -hadrons. The presence of low-T -hadrons, below 20 GeV, is instead inferred using a soft -tagging algorithm, which does not rely on the presence of a jet in the calorimeter, but requires the presence of secondary vertices [117]. This technique is used to gain sensitivity to the˜1 → ˜0 1 signal in the regime with Δ˜1 ,˜0 1 lower than ∼40 GeV. The number of secondary vertices in an event is denoted SV . Corrections derived from dedicated¯and +jets control regions are applied to the soft -tagging efficiencies to account for differences between data and simulation.
Jets and associated tracks are also used to identify hadronically decaying -leptons using the 'Loose' identification criterion described in Refs. [118,119], which has a 85% (75%) efficiency for reconstructing -leptons decaying into one (three) charged pions. The hadronic -lepton decay candidates are required to have one or three associated tracks, with total electric charge opposite to that of the signal electron or muon, T > 20 GeV, and | | < 2.5. The -lepton candidate T requirement is applied after a dedicated energy calibration [120, 121].
To avoid labelling the same detector signature as more than one object, an overlap removal procedure is applied. Given a set of baseline objects, the procedure checks for overlap based on either a shared track, ghost-matching [109], or a minimum distance Δ between pairs of objects. 4 First, if a baseline lepton and a baseline jet are separated by Δ < 0.2, then the lepton is retained and the jet is discarded. Second, if a baseline jet and a baseline lepton are separated by Δ < 0.4, then the jet is retained and the lepton is discarded, to minimise the contamination from jets misidentified as leptons. For the remainder of the paper, all baseline and signal objects are those that have survived the overlap removal procedure.
The missing transverse momentum ì miss T is reconstructed as the negative vector sum of the transverse momenta of baseline electrons, muons, jets, and a soft term built from high-quality tracks that are associated with the primary vertex but not with the baseline physics objects [122,123]. Photons and hadronically decaying -leptons are not explicitly included but enter either as jets, electrons, or via the soft term.

Discriminating variables
The backgrounds contributing to a final state with one isolated lepton, jets and miss T are primarily semileptonic¯events with one of the bosons decaying leptonically, and +jets events with a leptonic decay of the boson. Both backgrounds can be efficiently reduced by requiring the transverse mass of the event, T , to be significantly larger than the boson mass. The transverse mass is defined as where Δ is the azimuthal angle between the lepton and missing transverse momentum directions and ℓ T is the transverse momentum of the charged lepton. Other discriminating variables used to distinguish signal from several categories of background events are described below.

Dileptonic¯reconstruction
The T2 variable [124] is a generalisation of the transverse mass, applied to signatures where two particles are not directly detected. The variable T2 [125] is a variant of T2 developed to identify and removeē vents where one boson decays into a hadronically decaying -lepton candidate. In this case the ' -jet' is used as the visible particle for one top branch and the observed electron or muon for the other top branch. For¯events where one boson decays leptonically and the other into a hadronically decaying -lepton, T2 has an endpoint at the boson mass. Events with dileptonic decays of¯pairs, where one lepton is not identified, constitute a significant background. The lost lepton can lead to significant missing transverse momentum and T2 above the boson mass. The topness variable [126] quantifies how well an event can be reconstructed under a dileptonic top hypothesis and is defined as the logarithm of the minimum of the following quantity S: when minimised with respect to and with the constraint ì T, + ì T,W = ì miss T . The quantity represents the four-momentum vector of the boson for which the lepton was not reconstructed and is thus completely invisible. The quantities ℓ and are the lepton and neutrino four-momentum vectors from the boson whose lepton was identified. Finally, refer to the two -jets. The sum in the last term runs over the five assumed final-state particles. If the event contains two -tagged jets, the two permutations are tested in the minimisation. If the event has a single -tagged jet, then permutations where the second -jet can be either of the two leading momentum untagged jets are tested during the minimisation. The values of resolution parameters , and CM are constants taken from Ref. [126].

Reconstruction of hadronic top decays
Signal events contain one hadronic top decay →¯ , while such decays are absent from the dileptonicb ackground. Therefore, reconstructing the hadronic top quark decay can provide additional discrimination against dileptonic¯events. A recursive reclustering jet algorithm searches for large-radius jets with radius parameter corresponding to the radius ( T ) = 2 × top / T expected from a hadronic top quark decay →¯ [29]. The algorithm is based on the anti-algorithm using signal jets as inputs and with initial radius parameter 0 = 3.0. If a reclustered large-radius jet is significantly narrower than the radius expected from a hadronic top quark decay of that T , it is discarded. The radius of the remaining reclustered jets is iteratively reduced until the radius approximately matches the radius expected from a hadronic top quark decay. Surviving reclustered jets constitute hadronic top candidates. If more than one hadronic top candidate is found, the candidate whose mass reclustered top is closest to top is retained.
A second hadronic top quark candidate algorithm is employed that fully reconstructs the direction of both the leptonically and the hadronically decaying top quarks, denoted lep and had respectively. This algorithm is applied to events with at least four jets and one -tagged jet. The top variable is defined as the invariant mass of the triplet of signal jets (one of which must be -tagged) most compatible with top , taking into account the jet momentum and energy resolution. The component of the ì miss T perpendicular to lep in ther est frame, miss T,⊥ , is small in semileptonic top quark decays since ì miss T tends to align with the leptonically decaying top quark.

Backgrounds with mismeasured missing momentum
In some signal regions, additional suppression against backgrounds with mismeasured missing momentum, arising from mismeasured jets, is required. This additional rejection is provided by miss is the negative vectorial sum of the momenta of the signal jets and signal lepton [125]. The denominator is computed from the per-event jet energy uncertainties, while the lepton resolution is neglected. The offset parameter is a characteristic scale of the background processes and is fixed at 100 GeV.

Variables for compressed˜1 → +˜0 1
To discriminate stop pair production from SM¯production, in the phase space dominated by the decaỹ 1 → +˜0 1 in the compressed regime Δ˜1 The reconstruction of the event under the semileptonic¯hypothesis starts by searching for the hadronically decaying top quark candidate through the minimisation of the loss function with and top being the experimentally known boson and top quark masses. The boson candidate mass cand is either the mass of a single large anti-jet with radius 1.0 or 1.2 or the invariant mass of two anti-jets with radius 0.4. The hadronically decaying top quark candidate had is either one of the large-jets or the boson candidate plus a -tagged jet. The jet permutation with the minimum loss function is considered as the candidate for the hadronic top. The visible part of the leptonically decaying top quark candidate ( lep vis ) four-momentum vector is determined by adding the four-momentum vectors of the remaining highest-T -tagged jet and the signal lepton.
The reconstruction of the event under the stop hypothesis relies on the collinear approximation [127,128], in which the top quark and the neutralino from the stop decay are collinear. This approximation is valid where the requirement of a high-T initial-state radiation (ISR) jet in the event forces the momentum of the˜1 to be much larger than the sum of the top and 0 1 masses.
With this approximation and a given value of the parameter =˜0 Although the neutrino momentum under the collinear approximation is fully known for a given value of , there is an ambiguity as to how the remaining missing transverse momentum is split between the two neutralinos. To resolve this, the following loss function, which compares the reconstructed leptonic and hadronic˜1 masses with a given˜1 mass hypothesis,˜1, is defined and used in the tN_diag_low SR: A minimisation of this loss function, again under the assumption that = 0.135, is performed with respect to the angles between each neutralino momentum vector and each of the two top quarks. The mass lep 1 , which denotes the leptonic˜1 mass at the minimum of this loss function, takes lower and more peaked values for compressed˜1 → +˜0 1 models than for the SM top quark backgrounds. Finally, the ratio 1 of the hadronic top quark momentum to the parent stop momentum is also used to discriminate between the stop signal and the background. Since it is computed as a projection, 1 can take negative values for background processes, or if the collinear assumption does not hold.

Signal regions
A preselection that exploits the basic characteristics of the signals is applied: the presence of a signal lepton, -tagged jets and missing transverse momentum. The preselection is designed to have very high efficiency for the signal and to remove the most trivial backgrounds. To cover signals with both high-momentum decay products such as in˜1 → +˜0 1 and low-momentum decay products such as in˜1 → ˜0 1 , 'soft-lepton' and 'hard-lepton' preselections are defined and are presented in Table 3. All regions require miss T > 230 GeV to ensure that the trigger was fully efficient. To reject multĳet events with mismeasured jet momenta, a minimum azimuthal angular distance is required between the missing transverse momentum direction and the two leading jets, Δ ( 1,2 , ì miss T ) > 0.4. The signal regions are then optimised using simulated event samples to maximise the expected significance [129, 130] for the benchmark signals. 5 A set of benchmark signal models, selected to cover the various stop and spin-0 mediator models, is used for optimisation. The optimisation is performed using an iterative algorithm, considering all discriminating variables and accounting for statistical and systematic errors in the evaluation of the discovery significance. An overview of the signal regions and the benchmark models for optimisation is presented in Table 1. The SRs are not designed to be orthogonal. The final exclusion limits are obtained by selecting at each point of the model parameter space the SR with the best expected sensitivity.

7.1˜1 → +˜0 1
Two signal regions, tN_med and tN_high, are designed for models with Δ˜1 ,˜0 1 significantly larger than top , and rely on large missing momentum and energetic jets. Selections on T , miss T,sig , miss T,⊥ and topness are dictated by the need to suppress the three main backgrounds, namely +jets,¯, and¯+ . The presence of a hadronic top quark candidate with reclustered top > 150 GeV is required primarily to ensure orthogonality with the control regions. The tN_med and tN_high definitions are given in Table 4. A 5 Significance of observing events for a prediction of ± is defined as common exclusion region is defined by performing a two-variable shape-fit on the tN_med signal region, if no excess is observed in the single-bin discovery signal regions. The binning is designed to maximise the excluded parameter space in the˜1-˜0 1 plane. The two variables chosen for the binning are the two discriminating variables that best distinguish between tN_med and tN_high, namely miss T and T . The resulting six bins are given in Table 4. The kinematics of the decay˜1 → +˜0 1 in the region where Δ˜1 ,˜0 1 ≈ top differ significantly from the two signal regions defined above, and the stop signal is kinematically very similar to the dominant background. This region of parameter space is referred to as the diagonal region. Two dedicated signal regions, tN_diag_low and tN_diag_high, are designed to target scenarios on the diagonal for low-mass and high-mass stops respectively. The sensitivity of the tN_diag_low SR is such that it is expected to be able to exclude scenarios with Δ˜1 ,˜0 1 = top and (˜1) between 200 and 250 GeV. Both the tN_diag_low and tN_diag_high signal regions rely on the presence of a high-T ISR jet, which serves to boost the di-stop system. The signal region definitions are shown in Table 5 and are used both for exclusion and for discovery.
Exclusion technique cut-and-count The signal region for the decay˜1 →˜0 1 is labelled bWN and defined using an optimised two-step machine learning (ML) approach, applied to events preselected according to the hard-lepton preselection criteria and additionally satisfying T > 110 GeV. The background mostly consists of¯, which has strong similarities to the signal in this region of phase space. For this reason the ML technique is selected. The jet multiplicity in signal events varies significantly due to the potential presence of ISR jets and fluctuations in the number of low-energy jets reconstructed from the hadronically decaying boson. To deal with the variable number of signal jets, the first step of the ML procedure is to use a recurrent neural network (RNN) that has the ability to extract information from sequences of variable length [131]. The RNN uses a long short-term memory (LSTM) algorithm [132] and takes the four-momentum vectors of the jets as inputs.
The LSTM output becomes the input of the second step, made up of a shallow neutral network (NN) with a single hidden layer and an output corresponding to the signal probability. The RNN and NN are trained simultaneously in one step. The NN uses the following discriminating variables as input: output of the RNN, miss T , T , the azimuthal angle of ì miss T , the azimuthal angle Δ ( ì miss T , ℓ) between the lepton and ì miss T , the invariant mass ℓb of the lepton and the -tagged jet, the transverse momentum of the -tagged jet, the lepton four-momentum vector, jet and -jet .
Before training, the hard-lepton preselection and the additional selection T > 110 GeV are applied. The size of the training sample is a crucial aspect for the performance of any ML method. Generating fully simulated signal samples with adequate sample sizes after the hard-lepton preselection and T > 110 GeV is computationally expensive. To overcome this difficulty, signal events without detector simulation were used for the training to enhance the number of signal events by two orders of magnitude. Fully simulated SM background events were available in sufficiently large numbers to be used directly for the training. For the signal, the generated events are 'smeared' using a dedicated procedure to emulate the effects of detector simulation and reconstruction. Parameterisations for reconstruction and identification efficiencies are obtained from dedicated ATLAS measurements and applied to jets, leptons and -tagged jet identification.
Particle-level electron, muon and jet four-momentum vectors are smeared according to their respective T , and identification working point. The miss T is recomputed from all smeared objects. The kinematic distributions of all input variables after smearing are found to have fair agreement with distributions after full event reconstruction. The output score of the ML classifier, denoted NN bWN , shows good agreement between smeared samples and fully simulated samples after full event reconstruction. The classifier output also shows a good agreement between simulation and data. The smeared samples are used only for the training, while signal and background predictions are obtained with the samples described in Section 4.
The discovery signal region is defined by selecting events with NN bWN > 0.9. The exclusion limits are obtained by performing a shape-fit using ten bins in NN bWN , with bin boundaries {0.65, 0.7, 0.75, 0.8, 0.82, 0.84, 0.86, 0.88, 0.90, 0.92, 1.0}. The¯background in the first three bins is reduced by applying an additional selection, namely T > 150 GeV. The selections that define the bWN signal region are presented in Table 6. The four-body decay˜1 → ˜0 1 occurs when Δ˜1 ,˜0 1 is smaller than the boson mass. In this scenario, the decay products have low momenta and often fall below the standard jet and lepton reconstruction T thresholds. It is therefore necessary to apply a soft-lepton preselection and require the presence of a high-momentum ISR jet, with T > 200 GeV, to boost the momenta of the final-state particles. A first four-body signal region, labelled as bffN_btag, is optimised by requiring the presence of at least one -tagged jet. The background in the bffN_btag signal region mostly consists of¯events. Because the -tagged jets are required to have T > 20 GeV, bffN_btag is not sensitive to Δ˜1 ,˜0 1 below ∼40 GeV. For this reason a second signal region, labelled as bffN_softb, is defined. This region does not rely on -tagged jets but instead requires a soft -tag identified by the presence of a secondary vertex. The dominant background processes in this region are¯and +jets. The bffN_btag signal region also exploits the correlation between the ISR jet T and miss T by cutting on the T2 variable defined by T2 = min( miss T , ISR T − 25 GeV). The key variable used at the last stage of the selection is the ratio of the lepton's transverse momentum to the missing transverse momentum, ℓ T / miss T , which has small values for the˜1 → ˜0 1 signal and large values for the backgrounds. The exact definitions of the four-body signal regions are given in Table 7. For exclusion limits, the last selection, namely on ℓ T / miss T , is replaced by a shape-fit. In the bffN_softb, the shape-fit is performed in five bins of the variable ℓ T / miss T with bin boundaries {0, 0.015, 0.025, 0.04, 0.06, 0.08}. In the bffN_btag signal region the shape-fit is performed in two variables, namely three bins in ℓ T / miss T with bin boundaries {0, 0.03, 0.06, 0.1} and two bins in Δ ( -jet T , ì miss T ) with bin boundaries {0, 0.8, 1.5}.

Dark matter
The dominant background to the search for spin-0 mediator models is the¯+ process. The optimisation of this signal region favours a selection with at least two -tagged jets and a leading -tagged jet with T > 80 GeV. The distribution of Δ ( ì miss T , ℓ) differentiates the scalar and pseudoscalar models from each other and also from the background. The resulting DM_scalar and DM_pseudoscalar signal region definitions are given in Table 8. In addition to the selection criteria optimised for discovery described above, the exclusion sensitivity is maximised by relying on a shape-fit in the region DM_scalar with the binning in Δ ( ì miss T , ℓ) given in Table 8.

Backgrounds
Data can be used to constrain the normalisation of the most significant background processes. To this end, control regions (CRs) are defined by minimally modifying the SR selections to suppress the signal while enhancing the fraction of the targeted background process. The CRs are then incorporated into a simultaneous likelihood fit to constrain the background process normalisations in the signal region. The ratio of the number of background events of a given process in the SR to those in a CR is estimated in MC background samples but is allowed to deviate from that ratio within dedicated MC modelling systematic uncertainties. Less significant background processes, such as diboson production and +jets, are estimated directly from MC simulation since they typically represent only a few percent of the signal region yields.
CRs are defined to normalise¯(TCR), +jets (WCR), single-top (STCR) and¯+ (TZCR). Whether a control region is defined for a given background and signal region depends on the relative contribution of the process to the SR yield.
To validate the background estimates from the CRs, validation regions (VRs) are introduced for¯(TVR) and +jets (WVR). The VRs are disjoint from both the SRs and CRs. The TZCR is designed to be as close as possible to the signal region in order to obtain the most precise estimate of the large¯+ background, and thus does not leave space between the SR and the CR to introduce a VR for this process. Background normalisations, referred to as normalisation factors (NF), determined in the CRs are applied to the VRs and compared with the data. The VRs are not included in the final simultaneous fit, but provide a statistically independent test of the background estimates in background-dominated regions.
The CRs and VRs are designed to minimise potential contamination from signal processes. The signal contamination is generally well below 10%, but in some TCRs and TVRs, for models close to the previously excluded region of parameter space, it can reach approximately 15%. The signal contributions to the CRs are not included in the background-only fits but are taken into account in the exclusion fits described in Section 11. The CRs and VRs used for each SR are summarised in Table 9. If a process is not normalised via a control region then it is estimated directly from MC simulation and theoretical cross-sections.

Control and validation regions for˜1 → +˜0 1 and spin-0 mediator signals
The dominant background process in the tN_med, tN_high and DM signal regions is¯+ , and therefore each of these SRs uses a dedicated TZCR. The TZCRs aim at capturing¯+ events where the boson decays into two electrons or muons, and thus is kinematically similar to the¯+ background in the signal regions where the boson decays into a pair of neutrinos. This CR is built by selecting events with three leptons (electrons or muons), one pair of which must be of opposite charge and same flavour with an invariant mass within 10 GeV of the boson mass. The exact definitions of the TZCRs follow the definitions of the tN_med, tN_high and DM SRs in terms of the number of jets, -tagged jets and jet T thresholds. A modified missing momentum variable,˜m iss T , is defined, where the leptons associated with the boson decay are considered invisible. The˜m iss T is the magnitude of the vector with components ì miss x,y derived from the , components ì miss x,y of ì miss T introduced in Section 5. The components ì miss x,y are obtained as follows: ì miss x,y = ì miss x,y + ì l2 x,y + ì l3 x,y , where ì l2 x,y and ì l3 x,y are the , components of the momenta of the leptons that make up the boson candidate. The TZCRs require˜m iss T >230 GeV. The remaining SR selections are not applied to the TZCRs, in order to retain a large enough event sample.

Signal Region
Signal Scenario TCR WCR STCR TZCR TVR WVR The +jets and dileptonic¯processes are significant in tN_med and tN_high, and therefore dedicated CRs, WCR and TCR, are employed. The DM signal region also employs a TCR but does not require a WCR due to the smaller size of the +jets background. These CRs have the same requirements on the number of jets, the number of -tagged jets and the jet T thresholds as listed in Tables 4 and 8 for their respective signal regions. Table 10 presents the definitions of the TCRs, WCRs and VRs, by showing which selections differ from the tN_med and tN_high SRs definitions. Neither +jets nor dileptonicp rocesses yield hadronic top decays, so a veto on the presence of a hadronic reclustered top candidate is used to ensure orthogonality with the signal regions. The number of events in TCR and WCR is increased by relaxing several selections compared with the SR selections. The miss T,sig selection is lowered to 10 for both tN_med and tN_high, and to 13 for DM. In addition, miss T,⊥ is lowered to 300 GeV for tN_med, while miss T is lowered to 450 GeV for tN_high. In the DM signal region the requirement on Δ (jet , ì miss T ) is lowered to 0.6.
The topness and T selections are used to differentiate between WCR and TCR. In the WCR, T is required to be in the range 30−−90 GeV, compatible with the presence of a semileptonic decay, but incompatible with dileptonic¯because of the topness selection. In the TCR, the topness selection of the SR is inverted, thus selecting events compatible with dileptonic¯, while T > 120 GeV is required, as larger values are favoured by the presence of two leptonically decaying bosons. The TCR dedicated to the DM signal region has the same T selection as its signal region, T > 180 GeV. The purity of the WCR is further improved by using only positively charged leptons, exploiting the lepton charge asymmetry in +jets events from collisions.
To validate the dileptonic¯background normalisation, a TVR dominated by¯production is designed. The TVRs for tN_med and tN_high have the same selections as the corresponding TCR, except for the veto on the presence of a hadronic reclustered top quark candidate, which is replaced by a selection requiring the presence of such a hadronic top quark, with a mass reclustered top > 150 GeV.
The validation of the +jets background for tN_high is performed with a WVR with the same selection as tN_high but requiring the presence of a hadronic reclustered top candidate with reclustered top > 150 GeV and miss T,sig > 25, in order to be closer to the SR. The WVR for tN_med is defined starting from the WCR selections, but replacing several selections with those used in its SR: miss T > 400 GeV, miss T,sig > 16 and the presence of a hadronic top quark, with a mass reclustered top > 150 GeV.
The DM SR contains only a small fraction of +jets events due to the requirement of two -tagged jets, and therefore only a TVR is considered. It is constructed from the DM SR definition, but with the topness selection inverted, and to increase the number of events and limit signal contamination, the selection on Δ (jet , ì miss T ) is relaxed to 0.6.  Table 14). The hatched area around the total SM prediction and the hatched band in the Data/SM ratio include all statistical and systematic uncertainties. The last (first) bin contains overflows (underflows).  Table 14). The hatched area around the total SM prediction and the hatched band in the Data/SM ratio include all statistical and systematic uncertainties. The last (first) bin contains overflows (underflows).  Table 14). The hatched area around the total SM prediction and the hatched band in the Data/SM ratio include all statistical and systematic uncertainties. The last (first) bin contains overflows (underflows).
The tN_med and tN_high definitions permit the construction of a STCR with enough data events for comparison with the DS-and DR-based MC predictions. The STCR is defined with selections close to those of the WCR, but requires a second -tagged jet, 30 < T < 120 GeV and the distance Δ ( 1 , 2 ) between the two -tagged jets to be larger than 1.4. To ensure orthogonality with the WCR, events with two -tagged jets inside the WCR must have Δ ( 1 , 2 ) < 1.4. It is found that the DS and DR scheme predictions bracket the observed number of events in the STCR data, with a large discrepancy between the two predictions. The largest discrepancy is observed in the STCR associated with the tN_med SR. The data-to-prediction ratio in the STCR is 0.1 +0. 3 −0.1 with the DR scheme and 1.5 ± 1.3 with the DS scheme.
The availability of the STCR allows the normalisation of the single-top background to be constrained from data. The fit to the STCR is performed with both the DS and DR MC schemes, and the resulting two predictions for single-top in the STCR and in the SRs are compatible within uncertainties. Therefore, once the STCR is used to constrain the single-top normalisation, the choice of the DS or DR scheme is found to have a negligible impact on the single-top prediction in the SR. In accordance with Ref.
[87], the DR scheme is used for the default sample.

Figures 3, 4 and 5 compare data and prediction in CRs and
VRs for several variables used in the˜1 → +˜0 1 and DM SRs. Good agreement is observed between data and prediction, within uncertainties.

Control and validation regions for compressed˜1 → +˜0 1
The definitions of the control and validation regions for the compressed two-body decays rely on dedicated discriminating variables introduced in Section 6.4. The dominant background process in both tN_diag_low and tN_diag_high is¯production. Each of these regions has its own dedicated TCR. The TCRs build upon the same jet , -jet , T and hard-lepton preselection as the SRs. In both TCRs the selection on the T of the leading jet is lower than in the SR, and is chosen to be in the range 200-360 GeV for tN_diag_low and in the range 200-440 GeV for tN_diag_high. In the TCR associated with tN_diag_low, Δ T is required to be below zero to avoid signal contamination. In addition, to increase the number of events, the selection on lep 1 is removed and the selection on T is lowered to 110 GeV. The TCR associated with tN_diag_high requires Δ dyn T below 30 GeV to ensure orthogonality with the SR and limit signal contamination. To increase the number of events in the TCR associated with tN_diag_high, the miss T selection is lowered to 350 GeV and selections on dyñ 0 1 and T2 are removed.
The top background normalisation in the two-body compressed region is validated using one VR for each of the regions tN_diag_low and tN_diag_high. The TVR corresponding to tN_diag_low is identical to the TCR except for the leading jet T , required to be above 400 GeV. The TVR associated with tN_diag_high is identical to the TCR, but requires the leading jet T to be larger than 440 GeV. Table 11 summarises the definitions of the TCR and TVR for tN_diag_low and tN_diag_high relative to the SR definitions. Figure 6 compares data and prediction in CRs and VRs for several variables used in the tN_diag_low and tN_diag_high SRs. Good agreement is observed between data and prediction, within uncertainties.  in the tN_diag_high TVR. The distributions shown are post-fit, i.e. each background is scaled by a normalisation factor obtained from a background-only likelihood fit to the CRs (see Table 14). The hatched area around the total SM prediction and the hatched band in the Data/SM ratio include all statistical and systematic uncertainties. The last (first) bin contains overflows (underflows).

Control and validation regions for˜1 →˜0 1
More than 80% of the bWN SR yield consists of dileptonic¯events, while other background components range between 2% and 5% of the SR yield. Therefore, the˜1 →˜0 1 SR requires only a TCR. In addition to the hard-lepton preselection, the bWN SR requires NN bWN > 0.9 and T > 100 GeV. To prevent overlap with the signal region, the TCR requires a lower interval of the output score, namely NN bWN ∈ [0.4, 0.6]. To preserve the same background composition as the signal region, the selection on T is tightened to 150 GeV.
The TVR for the˜1 →˜0 1 signal is designed by selecting NN bWN ∈ [0.6, 0.65], in between the NN bWN ranges used for the CR and the SR. In addition, T > 150 GeV is required to enhance the contribution of semileptonic¯events, representative of the background in the SR. The selections that define the TCR and TVR for bWN are summarised in Table 6. Figure 7 compares data and prediction in the CR and VR associated with bWN. Good agreement is observed between data and prediction, within uncertainties.  Table 14). The hatched area around the total SM prediction and the hatched band in the Data/SM ratio include all statistical and systematic uncertainties. The last (first) bin contains overflows (underflows).

Control and validation regions for˜1 → ˜0 1
In the signal region bffN_softb the largest background is +jets followed by approximately equal amounts of dileptonic and semileptonic¯. A +jets control region, WCR, is built upon the same selections as the SR, but to enhance the fraction of +jets events, ℓ T / miss T is required to be in the interval [0.16, 0.32] while the lepton charge is required to be positive. A¯control region, TCR, is designed to have approximately equal amounts of the two types of¯backgrounds. The TCR builds on the same selections as bffN_softb, but it requires the presence of at least one -tagged jet in order to increase the fraction of¯and to ensure orthogonality with the bffN_softb SR. The TCR also requires ℓ T / miss T to be in the interval [0.12, 0.25] and that the lepton charge be negative for orthogonality with the WCR.
The validation region TVR associated with bffN_softb is defined with the same selections as the TCR, except for the ratio ℓ T / miss T , which is required to be in the interval [0.08, 0.12]. The WVR for bffN_softb has the same selections as the WCR, except ℓ T / miss T is required to be in the interval [0.08, 0.16]. Table 12 shows the selection differences between the bffN_softb SR and the associated CRs and VRs.
The soft -tagging efficiency and mis-tag rate depend on the track multiplicity, kinematics of the -hadrons, and the -hadron fragmentation. Two highly populated soft -tagging regions enriched in¯and +jets are defined, where it is found that the track multiplicity differs between data and MC simulations. In each region a weight is defined as a function of the track multiplicity to reweight the MC simulation to the data. After reweighting, good agreement is found between the data and the simulation in a range of secondary vertex variables such as vertex mass, vertex momentum, vertex distance to the primary vertex and vertex track multiplicity. The 'after-to-before-reweighting' ratios of efficiencies and mis-tag rates define scale factors that are derived separately for S and P 8. The largest discrepancy after reweighting is found in the +jets region and is of the order of 20%. A corresponding 20% systematic uncertainty in the soft -tagging scale factors is introduced. In the signal region bffN_btag, the dominant background process is dileptonic¯, representing almost half of all background events, followed by +jets. To ensure orthogonality with the SR and limit signal contamination, the TCR control region has the same selections as the SR, but requires the leading -tagged jet T to be above 100 GeV, T > 110 GeV and ℓ T / miss T > 0.05. Compared with the SR, the WCR removes the upper bound on the leading -tagged jet T , requires Δ ( The TVR associated with the bffN_btag signal region has the same selections as TCR but the -tagged jet T is required to be in the interval [50,100] GeV, and is thus between the signal region and control region. The WVR has the same selections as the WCR except for the angle Δ (   Table 14). The hatched area around the total SM prediction and the hatched band in the Data/SM ratio include all statistical and systematic uncertainties. The last (first) bin contains overflows (underflows).  Table 14). The hatched area around the total SM prediction and the hatched band in the Data/SM ratio include all statistical and systematic uncertainties. The last (first) bin contains overflows (underflows).

Systematic uncertainties
The systematic uncertainties in the background estimates arise from multiple experimental and theoretical sources and can enter the SR background yield either via direct predictions from theoretical cross-sections or from uncertainties in the extrapolation from CRs to SRs. The sources of systematic uncertainties are grouped into categories whose labels are defined in parentheses in the paragraphs below. Their effect on the background predictions in the SRs is summarised in Table 13. The systematic uncertainties are included as nuisance parameters constrained by Gaussian probability distributions and profiled in the likelihood fits. , as well as the modelling of the -tagging or soft -tagging efficiencies and mis-tag rates [116] ( -tagging experimental). Other experimental uncertainties arise from the modelling of the lepton energy scales, energy resolutions, reconstruction and identification efficiencies (Leptons experimental). There is also an experimental uncertainty arising from the reweighting of the simulation as a function of the number of interactions per bunch crossing in data and the additional cuts applied to jets to ensure they arise from the hard-scatter primary vertex (Pile-up).
Backgrounds such as dibosons and +jets, derived directly from a MC prediction and a theoretical cross-section, have theoretical systematic uncertainties (Theory) arising from theoretical cross-section calculations, including those related to parton distribution functions and factorisation and normalisation scales. The systematic uncertainty on the integrated luminosity is also included in this category. As shown in Table 9, the single-top,¯+ and +jets backgrounds are also predicted directly from MC simulations for some SRs, in which case the theory uncertainties apply also to those processes.
When the yield from a background such as¯, single-top,¯+ or +jets is normalised using a CR, modelling uncertainties affect the extrapolation from the control to the signal region, but not the overall normalisation. In each of these cases, the background has a normalisation systematic uncertainty (Normalisation) from the fit, arising from the statistical power of the CR for the given background and a modelling uncertainty (Modelling) that affects the extrapolation factor from the CR to the SR.
The uncertainties in the modelling of the¯background include effects related to the MC event generator, the hadronisation modelling and the amount of initial-and final-state radiation [85]. The MC generator uncertainty is estimated by taking the full difference in event yields between P -B v2+P 8 and M G 5_aMC@NLO v2.6.0+P 8. Events generated with P -B v2 are showered and subsequently hadronised with either P 8 or H 7.0 in order to estimate the effect from modelling of the hadronisation. The systematic uncertainty from the amount of initial-and final-state radiation is derived by comparing P -B results obtained with different shower radiation, NLO radiation and modified factorisation and renormalisation scales.
The single-top process modelling uncertainty is derived from the size of the interference betweenā nd using the¯, and samples generated with M G . It is obtained by comparing with the difference between and¯. The sample generated with M G is found to be in good agreement with the nominal samples generated with P -B v2+P 8. For the tN_med and tN_high SRs where STCR is used, the modelling uncertainty enters via the ratio of the number of events in the signal region to the number in the STCR. Given the potentially large modelling uncertainty in the interference between¯and , the modelling uncertainty is also evaluated for the DM SR by comparing the predicted single-top yield from with the difference between and¯. The modelling uncertainties considered for¯+ are the renormalisation and factorisation scales, and the amount of initial-and final-state radiation, obtained by considering the variation of the same parameters used for the¯initial-and final-state radiation systematic uncertainties. The +jets modelling uncertainties include generator modelling, derived by considering an alternative +jets sample generated with M G as well as modified factorisation, renormalisation, resummation and parton matching scales.
Most of the SRs are binned in one or two variables in order to enhance sensitivity to a wider range of models for exclusion limits. In this situation the normalisation factors to go from the CR to the SR are rederived specifically for each bin of the SR. The modelling systematic uncertainties are also rederived following the scheme above but applied to the normalisation factor from the CR to each specific bin of the SR.
The SUSY signal cross-section uncertainty is taken from an envelope of cross-section predictions using different PDF sets and factorisation and renormalisation scales as described in Ref. [133]. The uncertainty in the DM production cross-section is derived from the scale variations and PDF choices. Dedicated uncertainties in the SUSY and DM signal acceptance due to the modelling of additional radiation, factorisation, renormalisation and parton matching scales are considered. The total systematic uncertainty for the SUSY models varies between 9% and 35%, increasing at higher stop mass and at lower values of Δ˜1 ,˜0 1 . For the spin-0 mediator signals the total systematic uncertainty is between 15% and 18%.

Results
To determine the SM background yields in the SRs, a background-only likelihood fit is performed for each analysis. The fit does not use the signal region data, but only the dedicated CRs to normalise the backgrounds.
The number of observed events and the predicted number of SM background events from the backgroundonly fits in all VRs and SRs are shown in Figure 10 together with the significance of the observation. The SRs are not mutually exclusive and are therefore not statistically independent. In all SRs, the distributions indicate good agreement between the data and the SM background estimate. The largest excess over the background-only hypothesis is 1.9 observed in the tN_high SR.    luminosity uncertainty are considered. All limits are calculated using the CL s prescription [135]. The NFs are compatible with unity in most cases. One exception is for the single-top NFs in the tN_med and tN_high SRs. The single-top NFs are significantly below unity when using the DR scheme for the treatment of the interference between the and¯processes. When changing to the DS scheme, the NFs become larger than unity but the predicted number of single-top events in the signal regions after the fit does not change significantly. This is explained by the fact that the DS and DR schemes give the same SR to STCR event yield ratio, within uncertainties. The¯NFs in tN_diag_high, bffN_btag and bffN_softb are below unity, which could potentially point to some mismodelling in this extreme region of phase-space. But good agreement is seen in the¯VRs, giving confidence in the¯background estimates. Figures 11 and 12 show comparisons between the observed data and the SM background prediction with all SR selections applied except the requirement on the plotted variable. The expected distributions from representative signal benchmark models are overlaid.

Interpretations
No significant excess is observed, and exclusion limits based on profile-likelihood fits are set for the stop pair production models and the spin-0 mediator models. Exclusion limits at 95% confidence level (CL) are The full event selection in the corresponding signal region is applied, except for the requirement (indicated by an arrow) that is imposed on the variable being plotted. The distributions shown are post-fit, i.e. each background is scaled by a normalisation factor obtained from a background-only likelihood fit to the CRs (see Table 14). In addition to the background prediction, a signal model is shown on each plot. The hatched area around the total SM prediction includes statistical and experimental uncertainties. The last (first) bin contains overflows (underflows).  Table 14). In addition to the background prediction, a signal model is shown on each plot. The hatched area around the total SM prediction includes statistical and experimental uncertainties. The last (first) bin contains overflows (underflows). obtained by selecting the signal region with the lowest expected CL s value for each signal model and the exclusion contours are derived by interpolating in the CL s value. The signal uncertainties and potential signal contributions to all regions are taken into account, and all uncertainties except those in the theoretical signal cross-section are included in the fit. In all exclusion plots, the ±1 exp uncertainty band indicates how much the expected limit is affected by the systematic and statistical uncertainties included in the fit. The ±1 SUSY theory uncertainty lines around the observed limit illustrate the change in the observed limit as the nominal signal cross-section is scaled up and down by the theoretical cross-section uncertainty. Figures 13 and 14 show the expected and observed exclusion contours as a function of the stop mass, the neutralino mass and the mass difference between the stop and the neutralino, for the˜1 → +˜0 1 , 1 →˜0 1 and˜1 → ˜0 1 scenarios. In models with a massless neutralino, stop masses up to 1200 GeV are excluded at 95% CL. In the diagonal region, where the mass difference between the stop and the neutralino coincides with the mass of the top quark, stop masses up to 600 GeV are excluded, which covers the previously unexcluded diagonal region between 210 GeV and 250 GeV in stop mass. In the three-body (four-body) region, stop masses up to 710 GeV (640 GeV) are excluded for a neutralino mass of approximately 580 GeV. The small excess observed in tN_high does not appear because the exclusion limits are obtained from the shape-fit in the tN_med signal region ( Table 4). The miss T,⊥ requirement applied in tN_med but not in tN_high removes most of the excess. The shape-fit is designed to have better expected sensitivity than the single-bin SRs over the whole˜1 → +˜0 1 parameter space. Figure 15 shows the upper limit on the ratio of the production cross-section for the spin-0 mediator model to the theoretical cross-section. Limits are shown under the hypothesis of a scalar or pseudoscalar mediator for a fixed DM candidate mass. Scalar and pseudoscalar mediator masses up to approximately 200 GeV are excluded at 95% CL, assuming a 1 GeV dark-matter particle mass and a common coupling of = 1 to SM and dark-matter particles. With the common coupling reduced to = 0.8, mediator masses up to approximately 100 GeV are excluded. Models with a mediator mass of 10 GeV and a dark-matter particle mass of 1 GeV are excluded down to a coupling of approximately = 0.7.     Figure 15: Upper limit on the ratio of the production cross-section for the spin-0 mediator model to the theoretical cross-section under the hypothesis of (a) a scalar or (b) a pseudoscalar mediator. The limit is shown as a function of the mediator mass for a fixed mass of the DM candidate of 1 GeV. The coupling of the mediator to SM and DM particles is assumed to be = 1.

Conclusion
This paper presents searches for direct stop pair production covering various regions of SUSY phase space and searches for a spin-0 mediator decaying into pair-produced dark-matter particles. The searches use the final state with one isolated lepton, jets, and miss T . The analysis uses 139 fb −1 of collision data collected with the ATLAS detector at the LHC at a centre-of-mass energy of √ = 13 TeV. The largest excess over the background-only hypothesis is 1.9 in the tN_high signal region. As no significant deviation from the Standard Model expectation is observed, exclusion limits at 95% confidence level are derived for the models considered. Stops are excluded up to 1200 GeV (710 GeV) in the two-body (three-body) decay scenario, extended from about 1000 GeV (400-600 GeV) in previous results. The introduction of ML techniques contributes to the significantly improved sensitivity in the challenging three-body decay scenario. In the four-body scenario, the exclusion of stops is extended from about 400 GeV in earlier results to up to 640 GeV for a stop-neutralino mass difference of 60 GeV. The introduction of the soft -tagging algorithm contributes to the significantly improved sensitivity at small mass differences between the stop and the lightest neutralino. Scalar and pseudoscalar dark-matter mediators are excluded up to 200 GeV for a common coupling of = 1 to Standard Model and dark-matter particles, from 100 GeV in earlier results. The introduction of a shape-fit in the DM signal region contributes to this improvement.        The ATLAS Collaboration