Search for chargino--neutralino pair production in final states with three leptons and missing transverse momentum in $\sqrt{s} = 13$ TeV $pp$ collisions with the ATLAS detector

A search for chargino$-$neutralino pair production in three-lepton final states with missing transverse momentum is presented. The study is based on a dataset of $\sqrt{s} = 13$ TeV $pp$ collisions recorded with the ATLAS detector at the LHC, corresponding to an integrated luminosity of 139 fb$^{-1}$. No significant excess relative to the Standard Model predictions is found in data. The results are interpreted in simplified models of supersymmetry, and statistically combined with results from a previous ATLAS search for compressed spectra in two-lepton final states. Various scenarios for the production and decay of charginos ($\tilde\chi^\pm_1$) and neutralinos ($\tilde\chi^0_2$) are considered. For pure higgsino $\tilde\chi^\pm_1\tilde\chi^0_2$ pair-production scenarios, exclusion limits at 95% confidence level are set on $\tilde\chi^0_2$ masses up to 210 GeV. Limits are also set for pure wino $\tilde\chi^\pm_1\tilde\chi^0_2$ pair production, on $\tilde\chi^0_2$ masses up to 640 GeV for decays via on-shell $W$ and $Z$ bosons, up to 300 GeV for decays via off-shell $W$ and $Z$ bosons, and up to 190 GeV for decays via $W$ and Standard Model Higgs bosons.


Introduction
Supersymmetry (SUSY) [1][2][3][4][5][6] postulates a symmetry between bosons and fermions, and predicts the existence of new partners for each Standard Model (SM) particle. This extension offers a solution to the hierarchy problem [7][8][9][10][11] and provides a candidate for dark matter as the lightest supersymmetric particle (LSP), which will be stable in the case of conserved R-parity [12]. This paper describes a search for direct production of charginos and neutralinos, mixtures of the SUSY partners of the electroweak gauge and Higgs (h) bosons, decaying to three charged leptons, and significant missing transverse momentum (p miss T , of magnitude E miss T ). The search uses the full Run 2 dataset of proton-proton collisions recorded e-mail: atlas.publications@cern.ch between 2015 and 2018 with the ATLAS detector at the CERN Large Hadron Collider (LHC). Protons were collided at a centre-of-mass energy √ s of 13 TeV and the dataset corresponds to an integrated luminosity of 139 fb −1 [13]. Similar searches at the LHC have been reported by the ATLAS [14][15][16][17][18][19][20] and CMS collaborations [21][22][23][24][25][26][27].
Previous results are extended by analysing the full ATLAS Run 2 dataset, improving the signal selection strategies -particularly for intermediately compressed mass spectra, and exploiting improved particle reconstruction performance. Significant gains in lepton identification and isolation performance follow from updates in the electron reconstruction as well as from the use of a novel multivariate discriminant [28]. Furthermore, the new results are statistically combined with a previous ATLAS search [18] targeting compressed mass spectra and two-lepton final states. Finally, the paper reports updated results for a previous ATLAS search which observed excesses of three-lepton events in the partial, 36 fb −1 , Run 2 dataset [15]. The original analysis using the Recursive Jigsaw Reconstruction (RJR) technique [29,30] is repeated using the full Run 2 dataset, and no significant excesses relative to the SM expectation are observed. A related follow-up search emulating the RJR technique with conventional laboratory-frame variables, also using the full Run 2 dataset, was published in Ref. [16]. The updated RJR results are not included in the combination with the new results, as they are not statistically independent and not competitive with the results of the new search optimised for the full Run 2 dataset. Section 2 introduces the target SUSY scenarios, while a brief overview of the ATLAS detector is presented in Sect. 3, followed by a description of the dataset and Monte Carlo simulation in Sect. 4. After a discussion of the event reconstruction and physics objects used in the analysis in Sects. 5, 6 covers the general analysis strategy, including the definition of signal regions, background estimation techniques, and systematic uncertainties. This is followed by Sect. 7, with details specific to the on-shell W Z selection and the Wh selection, and Sect. 8, with details specific to the offshell W Z selection. Results are presented in Sect. 9, together with the interpretation in the context of relevant SUSY scenarios. Section 10 reports the follow-up RJR analysis, and finally Sect. 11 summarises the main conclusions.

Target scenarios
The bino, the winos, and the higgsinos are respectively the superpartners of the U (1) Y and SU (2) L gauge fields, and the Higgs field. In the minimal supersymmetric extension of the SM (MSSM) [31,32], M 1 , M 2 , and μ are the mass parameters for the bino, wino, and higgsino states, respectively. Through mixing of the superpartners, chargino ( χ ± 1,2 ) and neutralino ( χ 0 1,2,3,4 ) mass eigenstates are formed. These are collectively referred to as electroweakinos, and the subscripts indicate increasing electroweakino mass. If the χ 0 1 is stable, e.g. as the lightest supersymmetric particle (LSP) and with R-parity conservation assumed, it is a viable dark-matter candidate [33,34].
Two physics scenarios are considered in this search. In the first scenario, referred to as the 'wino/bino scenario', mass parameters |M 1 | < |M 2 | |μ| are assumed such that the produced electroweakinos have a wino and/or bino nature, with the χ ± 1 and χ 0 2 being wino dominated, and the χ 0 1 LSP being bino dominated. Such a hierarchy is typically predicted by either a class of models in the framework of gaugino mass unification at the GUT scale (including mSUGRA [35,36] and cMSSM [37]), or a MSSM parameter space where the discrepancy between the measured muon anomalous magnetic moment [38], and its SM predictions [39] can be explained [40][41][42]. When the masssplitting between χ ± 1 and χ 0 1 is 15-30 GeV, this hierarchy is also motivated by the fact that the LSP can naturally be a thermal-relic dark-matter candidate that was depleted in the early universe through co-annihilation processes to match the observed dark-matter density [43][44][45]. These models are poorly constrained by dark-matter direct-detection experiments, and collider searches constitute the only direct probe for |μ| > 800 GeV [46].
The second scenario, referred to as the 'higgsino scenario', considers a triplet of higgsino-like states ( χ ± 1 , χ 0 2 , χ 0 1 ) to be the lightest SUSY particles. This type of scenario is motivated by naturalness arguments [47,48], which suggest that |μ| should be near the weak scale [49][50][51][52], while M 1 and/or M 2 can be larger. The mass-splittings between the light higgsino states are determined by the magnitude of M 1 or M 2 relative to |μ|. For the higgsino scenario this paper considers the regime where the mass-splitting between χ 0 2 and χ 0 1 is about 5-60 GeV, corresponding to cases where the wino and bino states are moderately decoupled (M 1 , M 2 > 0.5 TeV).
Simplified SUSY models [53][54][55] for the two scenarios are considered for optimisation of the selections and interpretation of the results. For the wino/bino scenario, the χ ± 1 and χ 0 2 are assumed to be mass degenerate and purely wino, while the χ 0 1 is purely bino. The product of the two signed neutralino eigenmass parameters m eig (χ 0 2 ) × m eig (χ 0 1 ) can be either positive or negative, 1 and the two cases are referred to as the wino/bino '(+)' or '(−)' scenario, respectively. For the higgsino scenario, the χ ± 1 , χ 0 2 and χ 0 1 are purely higgsino states, and the mass of the χ ± 1 is assumed to be exactly the mean of the χ 0 1 and χ 0 2 masses. In both scenarios, all other SUSY particles are assumed to be heavier, such that they do not affect the production and decay of the χ ± 1 and χ 0 2 . The search targets direct pair production of the lightest chargino and the next-to-lightest neutralino, χ ± 1 χ 0 2 , decaying into a pair of χ 0 1 LSPs via an intermediate state with a W boson and a Z boson (W Z mediated), or a W boson and a SM Higgs boson (Wh mediated). Final states with three lightflavour leptons (electrons or muons, referred to as 'leptons' in the rest of this paper) are explored. One lepton originates from a leptonic decay of a W boson, and two leptons come from the direct decay of a Z boson or the indirect decay of a Higgs boson. The signatures are also characterised by the presence of E miss T originating from the LSPs, and this E miss T component is enhanced when hadronic initial-state radiation (ISR) is present, due to recoil between the χ ± 1 χ 0 2 system and the jets.
The following three simplified model scenarios of χ ± 1 χ 0 2 pair production, as illustrated in Fig. 1, are considered with dedicated selections: • On-shell W Z selection:χ A 100% branching ratio is assumed forχ ± 1 → W ( * )χ 0 1 for all models. Unless otherwise indicated, mass splitting m refers to m(χ 0 2 ,χ 0 1 ) in the rest of this paper. For the considered Wh-mediated scenarios, the Higgs boson has SM properties and branching fractions; and three-lepton final states 1 The mixing matrix used to diagonalise the neutral electroweakino states can be complex, even in the absence of CP violation, but can be made real at the cost of introducing negative mass eigenstates. The sign will affect the couplings and thus the distributions in the decay under consideration. For additional discussion of this, see Ref. [56] and Appendix A of Ref. [57]. For χ ± 1 χ 0 2 pair production with decays via W Z to 3 final states, in the wino/bino (+) scenario, limits were previously set at the LHC for χ ± 1 / χ 0 2 masses up to 500 GeV for massless χ 0 1 , up to 200 GeV for m ∼ m Z , and up to 240 GeV for 50 GeV < m < m Z [21]. Limits for mass splittings m < 50 GeV were set in 2 final states for χ ± 1 / χ 0 2 masses up to 250 GeV [18]. For decays via Wh to 3 final states (including hadronically decaying τ -leptons), limits reached 150 GeV for massless χ 0 1 , and as high as 145 GeV for a χ 0 1 mass of 20 GeV [17]. For the higgsino scenario, the most stringent limits for 5 GeV < m < 55 GeV were set by ATLAS using 2 final states [18] where χ 0 2 masses up to  GeV are excluded depending on m. For m > 55 GeV the best limits were reported by LEP [58][59][60][61][62][63][63][64][65][66][67], excluding χ ± 1 masses up to 103.5 GeV.

ATLAS detector
The ATLAS detector [68] is a general-purpose particle detector with almost 4π solid angle coverage around the interaction point. 2 It consists of an inner tracking system surrounded by a superconducting solenoid, sampling electromagnetic and hadronic calorimeters, and a muon spectrometer encompassing superconducting toroidal magnets.
The inner detector (ID) reconstructs charged-particle tracks in the pseudorapidity range |η| < 2.5, using silicon pixel and microstrip subsystems followed by a transition radiation tracker. For √ s = 13 TeV data-taking an additional innermost layer, the insertable B-layer [69,70], was added to the pixel tracker to improve tracking performance and flavour identification of quark-initiated jets. The ID is immersed in a 2 T axial magnetic field provided by the surrounding thin, superconducting solenoid.
Beyond the ID a high-granularity lead/liquid-argon (LAr) electromagnetic sampling calorimeter (ECAL) and a steel/ scintillator-tile hadronic sampling calorimeter cover |η| < 3.2 and |η| < 1.7 respectively. In the forward regions a copper/LAr endcap calorimeter extends the hadronic coverage to 1.7 < |η| < 3.2, while copper/LAr and tungsten/LAr forward calorimeters are used for electromagnetic and hadronic measurements in the 3.1 < |η| < 4.9 region. The muon spectrometer (MS) surrounds the calorimeters and comprises three layers of trigger and high-precision tracking chambers spanning |η| < 2.4 and |η| < 2.7, respectively. A magnetic field is provided by a system of three superconducting aircore toroidal magnets with eight coils each.
Events of interest are selected using a two-level trigger system [71] consisting of a custom hardware-based first-level (L1) trigger followed by a software-based high-level trigger (HLT). The L1 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 Monte Carlo simulated event samples
This analysis exploits the full Run 2 √ s = 13 TeV pp dataset recorded by the ATLAS experiment during stable beam conditions between 2015 and 2018. The LHC collided protons with bunch-crossing intervals of 25 ns, and the average number of interactions per crossing in data was μ = 34. After applying beam, detector and data-quality requirements [72], the dataset corresponds to a total integrated luminosity of 139 fb −1 [13], with an uncertainty in the integrated luminosity of 1.7%, obtained using the LUCID-2 detector [73] for the primary luminosity measurements.
The expected contributions of SM processes andχ ± 1χ 0 2 SUSY signals are estimated using Monte Carlo (MC) simulation. The MC samples are used in the optimisation of event selection criteria, as well as for yield prediction and the estimation of systematic uncertainties in the yield prediction. The yield prediction for the dominant W Z background is improved by extracting normalisation factors from data in dedicated control regions, as discussed in Sect. 6.2. The background contribution from events with one or more misidentified or non-prompt leptons is estimated using a data-driven method also outlined in Sect. 6.2. For all other processes, the MC-predicted yields are used directly. The samples are produced including an ATLAS detector simulation [74] based on Geant4 [75], or a faster simulation using a parameterised calorimeter response [76] and Geant4 for all other detector systems. Simulated events are reconstructed in the same way as data events. Details of the MC simulation, including the generators used for the matrix element (ME) calculation and the parton shower (PS), hadronisation and underlying event (UE) modelling, the parton distribution function (PDF) sets used in the ME and PS, the set of tuned parameter values used as the UE tune, and the order of the cross-section calculations used for yield normalisation are given in Table 1 and briefly discussed below. The SUSYχ ± 1χ 0 2 → W Z/Wh → 3 signal samples were generated from leading-order (LO) matrix elements with up to two additional partons using MadGraph 2.6 and Pythia 8.2, for both the wino/bino and the higgsino scenarios. MadSpin [125] was used to model off-shell W Z decays. The ME-PS matching was done using the CKKW-L prescription [126,127], with the matching scale set to one quarter of the χ ± 1 / χ 0 2 mass. Samples were generated for χ ± 1 / χ 0 2 masses between 100 GeV and 850 GeV, and mass splittings m between 5 GeV and 850 GeV. Only χ ± 1 / χ 0 2 decays via bosons, which in turn decay leptonically via SM branching fractions, are considered. For the Wh samples, only Higgs boson decays via W W , Z Z and τ τ were generated, with cross section times branching fractions corrected to match the SM Higgs branching fractions [109]. The generated signal events are required to have at least two leptons for the on-shell W Z samples, and at least three leptons for the off-shell W Z samples and the Wh samples; hadronically decaying τ -leptons are not considered in the requirement.
The only difference between the two wino/bino scenarios (positive or negative m eig (χ 0 2 ) × m eig (χ 0 1 )) is the mass lineshape of the Z boson from the χ 0 2 decay, particularly when m < m Z and the Z boson is off-shell. 3 The samples were generated for the (+) scenario and a reweighting in m Z ( * ) , based on an analytic function presented in Ref. [128], was used to simulate the (−) scenario.
Diboson, triboson and Z +jets processes were simulated with the Sherpa 2.2 generator. ME-PS matching and merging is based on Catani-Seymour dipole factorisation [122,129,130], using improved CKKW matching [131,132] extended to NLO accuracy using the MEPS@NLO prescription [130][131][132][133], and including NLO virtual QCD corrections for the ME [134,135]. The diboson samples cover dilepton masses down to 4 GeV for p 1 T , p 2 T > 5 GeV, and down to m > 2m + 250 MeV if p 1 T > 5 GeV and any of m > 4 GeV, p 1 T > 20 GeV, or E miss T > 50 GeV are satisfied. The standard multiboson samples do not include Higgs boson production. An alternative triboson sample including off-shell contributions and leptonically decaying h → V V (with V = W or Z ) contributions is used in the off-shell W Z selection, where W * Z * decays are targeted and off-shell triboson processes are non-negligible in the estimation of the SM background; dilepton masses down to 4 GeV are considered in the sample.
The tt, single-top t W , t-channel, s-channel and tth processes were modelled using Powheg Box 2 + Pythia 8. The h damp parameter 4 was set to 1.5 times the top-quark mass [136]. The samples were generated employing the fiveflavour scheme (four-flavour in case of single-top t-channel), and a diagram removal scheme [137] was used in the case of t W to remove interference and overlap with tt production. Other top-quark processes (tt V , tZ , tW Z, tt V V , tt (t → W b + (γ * /Z → )), 3-top and 4-top) were modelled using MadGraph5_aMC@NLO 2 + Pythia 8. Samples of Higgs boson production via gluon fusion, vector- AZNLO [124] j jet, LO leading order, NLO next-to-leading order, NNLO next-to-next-to-leading order, NNNLO next-to-next-to-next-to-leading order, NLL next-to-leading-log, NNLL next-to-next-to-leading-log, EWK electroweak Eur. Phys. J. C (2021) 81:1118 boson fusion and associated production were generated using Powheg Box 2 + Pythia 8. All background and signal samples make use of Evt-Gen 1.6.0 and 1.2.0 [138] for the modelling of b-and chadrons, except those generated using Sherpa. The effect of additional interactions in the same and neighbouring bunch crossings (pile-up) was included by overlaying simulated minimum-bias interactions onto each hard-scatter process. The simulation was done using Pythia 8.2 with the A3 tune [139] and the NNPDF2.3lo set of PDFs, and the samples were reweighted such that the pile-up distribution matches the one in data.

Event reconstruction and preselection
The strategy for event reconstruction and preselection is defined here, where a common approach has been adopted for all regions in the analysis, unless specified otherwise. Further selection specific to individual regions is discussed in Sects. 6 to 8.
Events are chosen for the Wh and on-shell W Z selections using dilepton triggers and for the off-shell W Z selection using single-lepton, dilepton and trilepton triggers [140,141]. The off-shell W Z selection is complemented at high E miss T with softer-lepton events selected using E miss T triggers [142]. The lepton triggers use various p T thresholds, depending on the lepton type, quality and multiplicity. To ensure trigger efficiencies are well understood in the analysis phase space, tighter quality and p T requirements are applied to fully reconstructed signal leptons, as defined below. Single-electron triggers are not used, to facilitate looser signal-lepton identification criteria. The number of leptons in the event that activate the trigger must be at least as many as the number of leptons required in the trigger, and electrons (muons) activating the trigger must have a fully calibrated p T above 18 GeV (27.3, 14.7 or 6.5 GeV, for increasing trigger-lepton multiplicity). For events selected by a E miss T trigger, an offline requirement of E miss T > 200 GeV is imposed to similarly ensure wellunderstood trigger efficiencies in the analysis phase space.
Events are required to have at least one reconstructed pp interaction vertex [143,144] with a minimum of two associated tracks with p T > 500 MeV. In events with multiple vertices, the primary vertex is defined as the one with the highest p 2 T of associated tracks. The primary objects used in this analysis are electrons, muons and jets. To be considered, reconstructed objects must satisfy 'baseline' loose identification criteria; to be selected for the analysis regions, they must also survive a second, tighter set of 'signal' identification requirements. Additionally, a lepton 'anti-ID' requirement is defined, corresponding to leptons that satisfy the baseline criteria but not the signal criteria. These anti-ID leptons are used in the Z + jets background estimation in Sect. 6.2. Hadronically decaying τ -leptons are not considered in the analysis, and the term 'lepton' always refers to electrons or muons in this document.
Electron candidates are reconstructed from three-dimensional clustered energy deposits in the electromagnetic calorimeter (ECAL), matched to an ID track [145]. Muon candidates are reconstructed by matching MS tracks or track segments to ID tracks [146]. Electron and muon candidates are calibrated in situ [145,146], using Z → ee, J/ψ → ee, Z → μμ and J/ψ → μμ decays. Baseline electrons are required to have p T > 4.5 GeV and fall within the acceptance of the ID (|η| < 2.47). They are further required to satisfy the calorimeter-and tracking-based 'Loose and Blayer likelihood' identification [145]. Baseline muons must have p T > 3 GeV and |η| < 2.5, and satisfy Medium identification criteria [146]. To suppress pile-up, both the baseline electrons and baseline muons are required to have a trajectory consistent with the primary vertex, i.e. |z 0 sin θ | < 0.5 mm. 5 Jet candidates are reconstructed from topological energy clusters in the electromagnetic and hadronic calorimeters [147], grouped using the anti-k t algorithm [148,149] with radius parameter R = 0.4. After subtracting the expected energy contribution from pile-up following the jet area technique [150], the jet energy scale (JES) and resolution (JER) are corrected to particle level using MC simulation, and then calibrated in situ using Z +jets, γ +jets and multijet events [151,152]. Baseline jets must then have p T > 20 GeV, and fall within the full calorimeter acceptance (|η| < 4.5).
Photon candidates are reconstructed from energy clusters in the ECAL provided they have no matched track, or have one or more matched tracks consistent with photon conversion origin. Baseline photons, while not used in the signal regions, are included in the calculation of missing transverse momentum, and used in SM background estimation validation. They are required to have p T > 25 GeV, fall inside the ECAL strip detector acceptance (|η| < 2.37), but outside the ECAL transition region (|η| ∈ [1.37, 1.52]). Candidates must also satisfy Tight identification criteria [145].
Ambiguities may exist between reconstructed objects. To prevent single detector signatures from being identified as multiple objects, the following overlap removal procedure is applied to baseline leptons and jets. First, all electrons sharing an ID track with a muon are discarded to remove bremsstrahlung from muons that is followed by a photon conversion. Second, all jets separated from remaining electrons by less than R = 0.2 are removed. Also, all jets within 5 The transverse impact parameter, d 0 , is defined as the distance of closest approach in the transverse plane between a track and the beam-line. The longitudinal impact parameter, z 0 , corresponds to the z-coordinate distance between the point along the track at which the transverse impact parameter is defined and the primary vertex. R = 0.4 of a muon and associated with fewer than three tracks with p T ≥ 500 MeV are removed. Finally, electrons or muons separated from surviving jets by less than R = 0.4 are discarded to reject non-prompt leptons from decays of band c-hadrons.
The missing transverse momentum is defined as the negative vector sum of the transverse momenta of all baseline objects (electrons, muons, jets, and photons) and an additional soft term [153]. The soft term is constructed from all tracks that pass basic quality requirements and are associated with the primary vertex, but are not associated with any baseline object. In this way, the p miss T is adjusted for the calibration of the contributing objects, while maintaining robustness against pile-up [154]. Additionally, an 'object-based E miss T significance' [155] is defined as |p miss LT )). The p T resolution of the contributing objects, at a given p T and |η|, is determined from parameterised Monte Carlo simulation which well reproduces the resolution measured in data. The quantity σ L denotes the p T resolution of the system, and ρ LT is a correlation factor between the resolutions of the p T components parallel (L) and perpendicular (T) to p miss T . The E miss T significance is used to discriminate events where the E miss T arises from undetected particles in the final state or from events where the E miss T arises from poorly measured particles (and jets). It is also useful in discriminating between signal events with large E miss T and e.g. Z + jets events with medium-to-low E miss T . To ensure high-quality object measurement and selection purity for the analysis regions, leptons and jets must satisfy additional tighter 'signal' criteria and isolation requirements to be selected. Signal jets are selected within |η| < 2.8, and must satisfy Loose quality criteria to reject contamination from non-collision backgrounds or noise bursts [156]. In order to suppress jets originating from pile-up, signal jet candidates with p T < 120 GeV and |η| < 2.5 (within the ID acceptance) are further required to satisfy the Medium working point of the track-based jet vertex tagger (JVT) [150,157]. For jets with |η| < 2.5 a multivariate discriminant -constructed using track impact parameters, information about displaced secondary vertices, and trajectories of b-and chadrons inside the jet [158] -is used for the identification of b-hadron decays, referred to as b-jets. The b-tagging algorithm working point is chosen such that b-jets from simulated tt events are identified with 85% efficiency, with rejection factors of 2.7 for charm-quark jets and 25 for light-quark and gluon jets [158]. Signal electrons must satisfy Medium identification criteria [145]. All signal leptons are then required to be compatible with originating from the primary vertex; the significance of the transverse impact parameter must satisfy |d 0 /σ (d 0 )| < 5 (3) for electrons (muons), where σ (d 0 ) is the track-by-track estimated impact parameter resolution.
Isolation requirements are applied to suppress contributions from conversions, semileptonic decays of heavy-flavour hadrons, or hadrons and jets wrongly identified as leptons, collectively referred as fake or non-prompt (FNP) leptons. The criteria rely on isolation energy variables calculated as p T of tracks or calo-clusters within a certain size of cone around the lepton candidate; the energy of the lepton candidate itself is not considered in this calculation. The isolation working points used in this analysis are based on those described in Refs. [145,146], including updates to improve the performance under the increased pile-up conditions encountered during 2017 and 2018 data-taking. The choice of isolation working points is optimised per selection region and per lepton-flavour to account for different levels of contribution from the FNP lepton background. The Tight working point is used for both electrons and muons in the onshell W Z and Wh selections, while the looser working point Gradient (Loose) is employed for electrons (muons) in the off-shell W Z selection to maintain a reasonable efficiency down to low p T .
To further suppress FNP lepton backgrounds in the offshell W Z selection, a dedicated multivariate discriminant 'non-prompt lepton BDT' [28] is used to tighten the requirements on the lepton with the lowest p T (which is commonly also the most FNP-like lepton of the three), after selecting exactly three baseline leptons in the event. The discriminant uses eight input variables including the isolation information, combined lepton and track quantities, and the b-jet likeliness calculated from the energy deposits and tracks in a cone around the lepton using the DL1mu or RNNIP algorithms [159]. The non-prompt lepton BDT selection is designed to maintain 70-90% efficiency for real leptons, for lepton p T below 20 GeV, with a rejection factor of 2-3 for FNP leptons passing the isolation selection. Figure 2 shows the combined signal lepton selection efficiency (including the reconstruction, identification, isolation, vertex association and nonprompt BDT selection) for the leptons from χ ± 1 χ 0 2 signal events, as well as the differential probability for a Z + jets event to be accompanied by a FNP lepton satisfying the signal lepton selection criteria.
To account for small efficiency differences between simulation and data, simulated events are corrected with scale factors covering lepton reconstruction, identification, isolation and trigger efficiencies, as well as jet pile-up rejection and flavour-tagging efficiencies.
A common preselection is applied for all search regions requiring exactly three signal leptons. Events are also required to have exactly three baseline leptons. This additional baseline requirement ensures orthogonality with other ATLAS SUSY analyses [18,160,161] and facilitates statistical combinations; it also simplifies the FNP lepton background estimation. Muons in the region 2.5 < |η| < 2.7 are exceptionally included in this count if they satisfy all other baseline muon criteria, in order to harmonise with the definition applied in the other analyses. The efficiencies are calculated using simulated samples of wino/bino (+) χ ± 1 / χ 0 2 decays and shown as a function of the generated lepton p T . The associated uncertainties represent the range of efficiencies observed across all signal samples used for the given p T bin. The right panel illustrates the differential probability for a Z + jets event to be accompanied by a FNP lepton satisfying the signal lepton criteria, as a function of the FNP lepton p T . This probability is measured using data events in a region with at least two signal leptons, with the other processes subtracted using the MC samples

Analysis strategy
The selections in this paper -while targeting different simplified model scenarios -all consider final states with exactly three leptons, possible ISR jets, and E miss T . Therefore, a common approach is used throughout most steps of the analyses. The on-shell W Z, off-shell W Z, and Wh selections are optimised independently.
This section describes the general analysis strategy, introducing the common parts of the search region definitions (Sect. 6.1), the background estimation (Sect. 6.2), and the uncertainty treatment (Sect. 6.3). The statistical methods used are outlined in Sect. 6.4. Further details specific to either the on-shell W Z selection and the Wh selection, or the offshell W Z selection, are then discussed in dedicated Sects. 7 and 8.

Search regions
Event selections enriched in signal (signal regions or SRs) are designed independently for the three targeted models, i.e. for the on-shell W Z, off-shell W Z or Wh selections. All the SRs are optimised to the wino/bino (+) scenario, maximising the expected sensitivity using benchmark signal samples. The SRs of the on-shell W Z selection, SR WZ , are optimised for χ ± 1 χ 0 2 signals with W Z-mediated decays and mass splittings near or above the Z -boson mass, m m Z , while the SRs of the off-shell W Z selection, SR offWZ , target W * Z * -mediated decays and mass splittings m < m Z . The SRs of the Wh selection, SR Wh , are optimised for Wh-mediated decays and veto Z -boson candidates.
For SRs targeting W ( * ) Z ( * ) -mediated scenarios, two leptons are assigned to the Z ( * ) -boson candidate by selecting a same-flavour opposite-charge-sign (SFOS) lepton pair in the event, and the remaining lepton is assigned to the W ( * ) boson (labelled W lepton or W ). If more than one SFOS lepton pair is present in the event, the invariant mass m of the SFOS lepton pairs is used to select which pair is assigned to the Z ( * ) -boson candidate. The on-shell W Z selection selects the SFOS lepton pair with m nearest the Zboson mass, m m Z , while the off-shell W Z selection selects the SFOS lepton pair with the smallest m , m min . In the rest of this document, these two types of lepton assignment are referred to as m m Z -based and m min -based lepton assignment, and m refers to m m Z unless otherwise indicated. In Wh-mediated scenarios, the opposite-sign leptons are the indirect product of the Higgs boson decay and can be of either the same or different flavour. Two subsets of SRs are defined depending on lepton flavour composition: the SR Wh SFOS target events with at least one SFOS pair (using m m Z -based lepton assignment), and the SR Wh DFOS target complementary events without a SFOS lepton pair.
For events with at least one SFOS lepton pair the transverse mass, m T , is constructed using the W lepton and the E miss T , and assuming the SM W Z event hypothesis: where φ is the separation in the transverse plane between the lepton and the E miss T . This exploits the difference between SM W Z, which has a Jacobian peak with a sharp cut-off at m T ∼ m W (the Wboson mass), and the targeted signals, which have relatively flat distributions.
For the initial SR segmentation, events with at least one SFOS lepton pair are divided into three m slices: below, in, and above the Z -boson mass window, defined as m ∈ [75,105] GeV. The SR offWZ and SR WZ use the first and second slice, respectively, while the SR Wh SFOS use the first and third slice. The SR WZ are orthogonal to the SR offWZ and the SR Wh SFOS through the m selection. The SR offWZ and the SR Wh SFOS can overlap, but are never used in the same interpretation. The SR Wh DFOS are orthogonal to all other SRs through lepton flavour composition.
For the final selection, a few key discriminating variables are used to further segment and refine the SRs. The SR WZ and SR Wh have a shared binning strategy aside from the m range, while SR offWZ binning focuses on m min and properties of more compressed χ ± 1 χ 0 2 signals. Ultimately, 20, 31, 19, and 2 SR bins are defined for the SR WZ , SR offWZ , SR Wh SFOS , and SR Wh DFOS , respectively. The complete definitions of these nominal SRs are further detailed per selection in subsequent Sects. 7.1 (SR WZ and SR Wh ) and 8.1 (SR offWZ ). The bins within each subset are explicitly disjoint, and are statistically combined when calculating the constraints on the target models. A more detailed overview of the fit configuration is given in Sect. 6.4. Additionally, discovery-oriented inclusive SRs are designed by grouping sets of adjoining nominal-SR bins in order to facilitate quantifying the size of data excesses in a model-independent manner. The inclusive-SR definitions are discussed in Sect. 9.1.

Background estimation
The dominant SM background in most of the SRs in this analysis is from SM W Z events with only leptonic decays, followed in importance by tt and Z + jets processes associated with at least one FNP lepton. In SR Wh DFOS , SM Higgs, triboson and tt production are the dominant processes.
A partially data-driven method is used for the estimation of the W Z background, which produces three real and prompt leptons. The background is predicted using MC simulation samples and normalised to data in dedicated control regions (CRs). This normalisation improves the estimation in the phase space of the selections, and constrains the systematic uncertainties. The CRs are designed to be both orthogonal and similar to the SRs, whilst also having little signal contamination; this is achieved by taking the SR definitions and inverting some of the selection criteria. Dedicated validation regions (VRs) are defined kinematically in between the CRs and SRs, and are used to assess the quality of the background estimation and its extrapolation to the SRs. The final estimation of the yields and uncertainties is performed with a simultaneous fit to the CRs and SRs, as discussed in Sect. 6.4.
The tt background is predicted using MC simulation samples and validated in VRs. It is dominated by decays with a dileptonic final state and an additional lepton from a bor c-hadron decay. As the MC modelling is found to be of good quality, no additional corrections are applied to the MC events. Rare SM processes, including multiboson and Higgs boson production, top-pair production in association with a boson, and single-top production, are estimated from MC simulation in all analysis regions.
The (Z /γ * → )+(jets/γ ) background has two prompt leptons and one FNP lepton from jets or photons. In the rest of this document, 'Z + jets' is used to refer to this set of processes. As there are no invisible particles in these processes at tree level, the observed E miss T is mostly due to mismeasured leptons and/or jets, or due to the E miss T soft term. The FNP leptons originate from a mix of sources, including lightflavour jets faking leptons, electrons from photon conversion, and non-prompt leptons from b-or c-hadron decays. Such FNP leptons often arise from instrumental effects, hadronisation, and the underlying event, all of which are challenging to model reliably in simulation. Therefore a data-driven method, referred to as the 'fake-factor method' [162,163], is used to estimate the Z + jets background. The fake factor (FF) is defined as the ratio of the probability for a given lepton candidate to pass the signal lepton requirements to that to fulfil the anti-ID requirements. This is measured using data in a control region, CRFF, designed to target Z + jets events with FNP leptons whose sources are representative of those expected in the SRs. Exactly three baseline leptons and at least one SFOS lepton pair are required in CRFF. The Z -boson candidate in the event is identified as the SFOS pair yielding the invariant mass closest to the Z -boson mass, and the remaining lepton is tagged as the FNP lepton candidate. The two leptons from the Z -boson candidate must activate the dilepton trigger to ensure there is no selection bias from FNP leptons. The Z + jets prediction in a given region is obtained by applying the FFs to the events in its corresponding 'anti-ID region'. This region is defined by the same selection criteria as used for the nominal region with three signal leptons, except that at least one of the leptons is anti-ID instead of signal. Each event in the anti-ID region is scaled by a weight based on the FF assigned to each anti-ID lepton in the region. The FFs are derived separately per lepton flavour and are parameterised as a function of lepton p T and lepton η or E miss T in the event, depending on the analysis selection. In both the FF measurement and the FF application procedure, contributions from processes other than Z + jets are subtracted using MC simulation samples.
While sharing a common approach, the estimation and validation procedures for the main SM backgrounds were optimised independently for the different selections, which each target a different primary phase-space region with different relative background composition and importance. Details are given in Sect. 7.2 (CR WZ /VR WZ ) and Sect. 8.2 (CR offWZ /VR offWZ ). The analysis considers uncertainties in the predicted yields of signal or background processes due to instrumental systematic uncertaintiesas well as statistical uncertaintiesand theoreticalsystematic uncertaintiesof the MC simulated samples.
Uncertainties are assigned to the yield in each region, except for W Z processes constrained in CRs, in which case they are assigned to the acceptance in each SR relative to that in the CR. The uncertainty treatment is largely common to the onshell W Z, Wh and off-shell W Z selections; exceptions are discussed in Sects. 7.2 (SR WZ and SR Wh ) and 8.2 (SR offWZ ). Relative uncertainties are illustrated in a breakdown per SR in the same sections.
The dominant instrumental uncertainties are the jet energy scale (JES) and resolution (JER). The jet uncertainties are derived as a function of p T and η of the jet, as well as of the pile-up conditions and the jet flavour composition of the selected jet sample. They are determined using a combination of simulated samples and studies in data, such as measurements of the jet p T balance in dijet, Z +jet and γ +jet events [151,152,164]. Another significant instrumental uncertainty is that in the modelling of E miss T , evaluated by propagating the uncertainties in the energy and momentum scale of each of the objects entering the calculation, as well as the uncertainties in the E miss T soft-term resolution and scale [153]. Other instrumental uncertainties concerning the efficiency of the trigger selection, flavour-tagging and JVT, as well as reconstruction, identification, impact parameter selection and isolation for leptons, are found to have minor impact. Each experimental uncertainty is treated as fully correlated across the analysis regions and physics processes considered.
For the processes estimated using the MC simulation, the predicted yield is also affected by different sources of theoretical modelling uncertainty. All theoretical uncertainties are treated as fully correlated across analysis regions, except those related to MC statistics. The uncertaintiesfor the dominant background processes, W Z, Z Z, and tt, are derived using MC simulation samples. For the W Z background, which is normalised to data in CRs, these uncertainties are implemented as transfer factor uncertainties that reflect differences in the SR-to-CR or VR-to-CR ratio of yields, and therefore provide an uncertainty in the assumed shape of MC distributions across analysis regions. The uncertaintiesrelated to the choice of QCD renormalisation and factorisation scales are represented by three Gaussian nuisance parameters in the fit (see Sect. 6.4): the first varies the renormalisation scale up and down, where a one-sigma deviation represents varying that scale up or down by a factor of two, while the factorisation scale is fixed to its nominal value; the second varies the factorisation scale in the same way while fixing the renormalisation scale; and the third nuisance coherently varies both the renormalisation and factorisation scales. There is no nuisance parameter to account for anticorrelated configurations of the renormalisation and factorisation scales, as these are deemed unphysical. For the W Z and Z Z samples, the uncertaintiesdue to the resummation and matching scales between ME and PS as well as the PS recoil scheme are evaluated by varying the corresponding parameters in Sherpa. For tt, modelling uncertainties at ME and PS level are determined by comparing the predictions of nominal and alternative generators, considering Powheg Box versus MadGraph5_aMC@NLO and Pythia 8 versus Herwig 7 [165,166], respectively. Uncertainties in the tt prediction due to ISR and final-state radiation (FSR) uncertainties are evaluated by varying the relevant generator parameters. The uncertainties associated with the choice of PDF set (NNPDF [78,87]) and the uncertainty in the strong coupling constant, α s , are also considered for the major backgrounds. Uncertainties in the cross section of 13%, 12%, 10% and 20% are applied for minor backgrounds tt W , tt Z, tth, and triboson, respectively [109]; for all other rare top processes a conservative uncertainty of 50% is applied.
The data-driven Z + jets estimation is subject to the statistical uncertainty due to the limited data sample size in CRFF or in the anti-ID regions used when applying the FF method, the uncertainty due to varying choice of parameterisation, and the uncertainty in the subtraction of non-Z + jets processes. The uncertainties are evaluated by considering the variations in the FF and propagating the effects to the estimated yields. The prescription applied for the estimation in the off-shell W Z selection is different from that in the on-shell W Z and Wh selections, reflecting the higher presence of Z + jets in SR offWZ . Details are included in Sects. 7.2 and 8.2.
Uncertainties in the expected yields for SUSY signals are estimated by varying by a factor of two the MadGraph5_aMC@NLO parameters corresponding to the renormalisation, factorisation and CKKW-L matching scales, as well as the Pythia8 shower tune parameters. The overall uncertainties in the signal acceptance range from 5% to 20% depending on the analysis region. Uncertainties are smallest in jet-veto regions and slightly larger for higher E miss T and jet-inclusive regions. This uncertainty estimates match the results of a dedicated study using data and MC Z → μμ events in Ref. [18].
In the following results, the uncertainties related to experimental effects are grouped and shown as 'Experimental' uncertainty. This uncertainty is applied for all processes whose yield is estimated from simulation. The 'Modelling' uncertainty groups the uncertainties due to the theoretical uncertainties, including the W Z transfer factor uncertainties. The 'Fakes' group represents the uncertainties for FNP background processes whose yield is estimated from data. 'MC stat' stands for the statistical uncertainties of the simulated event samples. Finally, the 'Normalisation' group describes the uncertainties related to the normalisation factors derived from the CRs.

Statistical analysis
Final background estimates are obtained by performing a profile log-likelihood fit [167], implemented in the Hist-Fitter [168] framework, simultaneously on all CRs and SRs relevant to a given interpretation. The statistical and systematic uncertainties are implemented as nuisance parameters in the likelihood; Poisson constraints are used to estimate the uncertainties arising from limited numbers of events in the MC samples or in the data-driven Z + jets estimation, whilst Gaussian constraints are used for experimental and theoretical systematic uncertainties. Neither the VRs, which solely serve to validate the background estimation in the SRs, nor the CRs used for data-driven Z +jets estimation, are included in any of the fits.
Three types of fit configuration are used to derive the results.
• A 'background-only fit' is performed considering only the CRs and assuming no signal presence. The normalisation of the W Z background is allowed to float and is constrained by the W Z CRs. The normalisation factors and nuisance parameters are adjusted by maximising the likelihood. The background prediction as obtained from this fit is compared with data in the VRs to assess the quality of the background modelling, as well as in the SRs. The significance of the difference between the observed and expected yields is calculated with the profile likelihood method from Ref. [169], adding a minus sign if the yield is below the prediction. • A 'discovery fit' is performed to derive model-independent constraints, setting upper limits on the new-physics cross section. The fit considers the target single-bin SR and the associated CRs, constraining the backgrounds by following the same method as in the background-only fit. Considering only one SR at a time avoids introducing a dependence on the signal model, which may arise from correlations across multiple SR bins. A signal contribution is allowed only in the SR, and a non-negative signal-strength parameter assuming generic beyond-the-SM (BSM) signals is derived. • An 'exclusion fit' is performed to set exclusion limits on the target models. The backgrounds are again constrained by following the same method as in the background-only fit, considering the CRs and the SRs, and the signal contribution to each region participating in the fit is taken into account according to the model predictions.
For each discovery or exclusion fit, the compatibility of the observed data with the signal-plus-background hypotheses is checked using the CL s prescription [170], and limits on the cross section are set at 95% confidence level (CL). Following the independent optimisation of the CRs and SRs, the simultaneous fits are performed separately for the different selections: once for the on-shell W Z and Wh selections combined, and once for the off-shell W Z selection. The results are presented in Sect 9.
The new results of the on-shell and off-shell W Z searches, as well as the results of a previous ATLAS search for electroweak SUSY with compressed mass spectra [18], are statistically combined and interpreted in the simplified models discussed in Sect. 1. Exclusion limits are calculated by statistically combining the results from the signal regions of the contributing searches, which are designed to be orthogonal. The combination is implemented in the pyhf framework [171,172], which was validated against the HistFitter framework [173]. The results are presented in Sect. 9.2.

On-shell WZ and Wh selections
The following subsections discuss the implementation specific to the on-shell W Z selection and the Wh selection, expanding on the general strategy outlined in Sect. 6. The selection is applied on top of the common preselection as defined in Sect. 5, and the SRs are optimised to the wino/bino (+) scenario.

Search regions
The SR WZ and SR Wh selections as introduced in Sect. 6.1 are further refined, taking into consideration differences in signal and background kinematics and composition. Driven by the p T thresholds of the dilepton triggers used in this selection, the leading and sub-leading leptons in the event must satisfy p T > 25, 20 GeV, while the third lepton must satisfy p T > 10 GeV. To reduce SM backgrounds with little to no real E miss T , events are required to have E miss T > 50 GeV. To suppress the contribution of tt events and single-boson production in association with a tt pair, events with at least one b-jet are rejected.
To reduce the contribution from processes with low-mass dilepton resonances, events are vetoed if they contain a SFOS lepton pair with an invariant mass below 12 GeV. Additionally, in events with a SFOS pair, the three-lepton invariant mass m 3 is required to be inconsistent with the mass of a Z boson, |m 3 − m Z | > 15 GeV, in order to suppress contributions from asymmetric photon conversions from the Z + jets process with Z → γ ( * ) and γ ( * ) → , where one of the leptons is out of acceptance.
Events with at least one SFOS lepton pair are divided into three m bins, in order to separate processes that include a Z boson in the decay chain from processes where a SM Higgs boson is involved. The first bin is defined as the Zboson mass window (m ∈ [75,105] GeV), and is used for the SR WZ selection. The second and third bins are defined below and above the Z -boson mass (m ≤ 75 GeV and m ≥ 105 GeV), and are used for the SR Wh SFOS selection. The Z -boson mass window bin is expected to contain a larger irreducible SM background contribution than the other bins. A summary of the common selection criteria is presented in Table 2. The SR WZ and SR Wh regions are further segmented as discussed below, and indexed with '-i'.
Each m bin is further divided into m T and E miss T bins, which enhances the sensitivity to various m scenarios. The m T distribution falls steeply in the region around the W -boson mass, and facilitates discrimination against the background from SM W Z production. Three m T bins, m T < 100, 100 ≤ m T ≤ 160, and m T > 160 GeV, are defined to separate processes with and without a leptonic Wboson decay. The lower and upper bounds on the E miss T bins vary with the m and m T thresholds. The SM background contribution is expected to be higher in low m T and E miss T bins, while the signal populates different m T and E miss T bins, depending on the mass splitting. Signals with smaller m tend to have more events in the lower E miss T and m T range, shifting to higher E miss T and m T bins as the mass difference increases.
Furthermore, events are separated by jet multiplicity, with jet-veto (n jets = 0; SR WZ -1 to 8, SR Wh SFOS -1 to 7 and 17 to 19) and jet-inclusive (n jets > 0; SR WZ -9 to 20, SR Wh SFOS -8 to 16) SRs. The ISR topology is exploited further in the jet-inclusive regions of SR WZ and SR Wh SFOS by categorising the events with at least one jet according to their H T , the scalar p T sum of the jets with p T > 20 GeV. At higher H T , signals with mass splitting m ≈ m Z tend to have more events at high values of E miss T and m T than the SM background, due to the recoil against ISR jets. In the high H T (H T > 200 GeV) regions, softer leptonp T spectra are expected for the signal because of the presence of a massive χ 0 1 , which carries most of the transverse momenta of the boosted χ ± 1 χ 0 2 system. There-fore H lep T , the scalar p T sum of the three selected leptons, is required to be less than 350 GeV. The H T categorisation is applied in regions with m < 105 GeV. Finally, in the high-mass off-peak region (m ≥ 105 GeV), only jet-veto events are considered. The full set of 20 SR WZ and 19 SR Wh SFOS signal regions is summarised in Tables 3 and 4.
In the SR Wh DFOS regions, events are required to have one same-flavour same-charge-sign (SFSS) lepton pair as well as a third lepton which has a different flavour and opposite sign to the SFSS pair, and is referred to as the DFOS lepton. After this selection, tt production dominates the SM background and is minimised by keeping events with low jet multiplicity (n jets < 3). These are then further split into two SR bins, one with n jets = 0 (SR Wh DFOS -1) and the other satisfying n jets ∈ [1, 2] (SR Wh DFOS -2). Due to the presence of the χ 0 1 , signals tend to have higher E miss T significance than the SM background, and therefore the events are required to have E miss T significance > 8. The third lepton in tt production usually arises from a heavy flavour quark decay and is typically lower in p T than the third lepton in the SUSY signal scenarios. To reduce this contribution the lower bound on the third lepton's p T is increased to 15 and 20 GeV in the SR Wh DFOS -1 and SR Wh DFOS -2 regions, respectively. Angular proximity between leptons coming from a Higgs-boson decay is used for further event separation, using the variable R OS,near , defined as the R between the DFOS lepton and the SFSS lepton nearest in φ. The signal is expected to populate the lower range in R OS,near , while the SM background tends to have a flatter distribution. Events in SR Wh DFOS -1 are required to satisfy R OS,near < 1.2. To suppress the higher tt contribution in the SR Wh DFOS -2, a tighter selection on R OS,near is imposed. A complete summary of the selection criteria in SR Wh DFOS is presented in Table 5.

Background estimation
The normalisation of the W Z background is measured in CRs characterised by moderate values of the E miss T and m T variables. The CRs contain only events with at least one SFOS pair with an invariant mass of 75 < m < 105 GeV, targeting on-shell decays. Additional requirements of 50 < E miss T < 100 GeV and 20 < m T < 100 GeV improve the W Z purity, the upper bound on m T at 100 GeV also ensures orthogonality between the W Z CRs and SR WZ . To address the possible mis-modelling of the jet multiplicity in the W Z simulated samples, the cross-section normalisation factor is extracted separately in each jet multiplicity and H T category, using CRWZ WZ 0j , CRWZ WZ low-HT , and CRWZ WZ high-HT . The estimation is cross-checked in kinematically similar, orthogonal VRs: VRWZ WZ 0j , VRWZ WZ low-HT , and VRWZ WZ high-HT . A summary of the selection criteria defining the W Z CRs and VRs is presented in Table 6. The W Z purity is about 80% in all CRs and VRs. The signal contamination is almost negligible in the CRs and increases to 10% in the VRs.
Performing the simultaneous background-only fit for the on-shell W Z and Wh selections, normalisation factors for W Z of 1.07 ± 0.02 (CRWZ WZ 0j ), 0.94 ± 0.03 (CRWZ WZ low-HT ) and 0.85 ± 0.05 (CRWZ WZ high-HT ) are found. A good description of the m T and E miss T distributions in the W Z simulation is crucial in this analysis, especially in the high-m T and high-E miss T tails where new physics may appear. The tail of the m T distribution is a result of, in decreasing order of importance: the use of a wrong pair of leptons to compute the mass of the Z -boson candidate and the m T of the W -boson candidate ('mis-pairing' of the leptons), the E miss T resolution, and the W -boson width. The prediction of lepton mis-pairing in simulation is validated in a control sample in data similar to the one used to calculate the cross-section normalisation factor, but only allowing events with a SFOS pair of different flavour than the W lepton. The Z -boson candidate can then be identified unambiguously, and a mis-paired control sample is obtained using the DFOS pair in the m computation and using the third lepton to calculate m T . Finally, the modelling of the m T and E miss T distributions is validated in a W +γ control sample. The W +γ and W Z processes have very similar m T shapes because their production mechanisms are similar, with the exception that the FSR production diagram of W +γ is much more common than the corresponding diagram in W Z, which is doubly suppressed due to the mass of the Z boson and its weak coupling to leptons. Furthermore, a photon is a good proxy for a leptonically decaying Z boson since photons and leptons are reconstructed with comparable resolutions, and no large extra mismeasurements are expected. The enhancement of the FSR diagram in the W +γ process leads to differences in the m T distribution shapes between W Z and W +γ . When a photon is radiated, leptons lose energy, resulting in a lower m T . In order to use the W +γ m T shape to validate the W Z MC prediction, the FSR contribution in the W +γ control region has to be suppressed. This is done by placing threshold requirements on the p T of the photon, p The tt MC modelling is validated in VRs, enhancing the tt contribution by requiring a DFOS lepton pair and using a moderate E miss T > 50 GeV selection. The main VR, VRtt WZ , requires the presence of one or two b-jets, further increasing the tt contribution. To validate the modelling in the n jets = 0 region as well, an additional VR inclusive in b-jets, VRtt WZ incl , is considered, with a E miss T significance < 8 requirement to ensure orthogonality with the SR Wh DFOS regions. The tt purity is about 80% in the VRtt WZ and 72% in the VRtt WZ incl . The selection requirements for the tt VRs are summarised in Table 7.
The Z + jets estimation uses the FF method as described in Sect. 6.2. For measurement region CRFF WZ , the Z -boson candidate mass must be compatible with the Z -boson mass within 15 GeV, and low E miss T and m T are required to minimise W Z contributions. The typical value of FFs varies from 0.2 to 0.4, depending on the lepton p T and η. The Z + jets estimation is then validated in VRFF WZ , considering the intermediate E miss T range closer to, but orthogonal to, the SRs, and adding a m 3 lower bound to reduce W Z contamination. The selection criteria for CRFF WZ as well as those of VRFF WZ are summarised in Table 7. Figure 3 presents the m T distribution in VRWZ WZ high-HT , and the E miss T distribution in VRtt WZ , showing good agreement between the observed data and the estimated background. The comparisons between the expected and observed yields in the CRWZ WZ and all VR WZ are given in Fig. 4.
The systematic uncertainties considered in the on-shell W Z and the Wh SRs follow the approach discussed in Sect. 6.3. The relative composition of FNP muons is similar    The bottom panel shows the relative difference between the observed data and expected yields for the CRs and the significance of the difference for the VRs, calculated with the profile likelihood method from Ref. [169], adding a minus sign if the yield is below the prediction in the simulation (closure) in various regions of parameter space and is given by the differences between the estimated and simulated yields of events in the given region. In the DFOS region where the triboson contribution becomes dominant, the uncertainties related to the QCD renormalisation and factorisation scales are also evaluated for this background component, in the same way as previously described for diboson and tt. A summary of the considered systematic uncertainties is presented in Fig. 5, with uncertainties grouped as discussed in Sect. 6.3. Bin-to-bin fluctuations in the statistical uncertainty as well as the experimental uncertainty reflect the difference in expected yields in the various search regions, which varies by an order of magnitude. These uncertainties become the dominant ones in SR WZ -3-4, 6-8, 11-12 and 15-16 of the on-shell W Z selection, and SR Wh SFOS -5, SR Wh SFOS -14, and SR Wh SFOS -19 of the Wh selection, due to limited number of MC events at high E miss T and m T . Although the FNP lepton uncertainty is negligible in the majority of the search bins, its relative size reaches 30% in SR Wh DFOS -2, due to the small number of events in the corresponding anti-ID sample.

Off-shell WZ selection
The following subsections discuss the implementation specific to the off-shell W Z selection, expanding on the general strategy outlined in Sect. 6. The selection is applied on top of the common preselection as defined in Sect. 5, and the SRs are optimised to the wino/bino (+) scenario.

Search regions
The SR offWZ selection targets the off-shell W Z region by requiring m min , m max < 75 GeV. The m max is the largest SFOS lepton pair invariant mass in the event, and the double requirement helps to maximally suppress combinatorial backgrounds with an on-shell Z boson. Further variables used in the off-shell W Z selection assume m minbased lepton assignment to the Z * -and W * -boson candidates unless otherwise indicated. The common event selection vetoes events with a b-jet to reduce contamination from tt, requires the three leptons to be well separated in min R 3 = min[ R( i , j ); for all lepton pairs ( i , j )], and requires a lower bound on m min of 1 GeV to remove events with collimated leptons for which FNP lepton background estimation is challenging. Finally, m min mass ranges of [3.0, 3.2] and [9,12] GeV are vetoed to avoid contributions from J/ψ and ϒ resonance backgrounds associated with a FNP lepton, except in the jet-inclusive high E miss T regions (E miss T > 200 GeV) where the contribution is negligible.
Preselected events are further divided into four categories based on the multiplicity of jets with p T > 30 GeV (n 30 GeV jets ), and on E miss T . Jet-veto categories SR offWZ low/ E T -0j and SR offWZ high/ E T -0j reject events containing jets and select low and high E miss T , respectively. Jet-inclusive categories SR offWZ low/ E T -nj and SR offWZ high/ E T -nj require at least one jet and also separate the events with low and high E miss T . As the E miss T is harder in the jet-inclusive categories, due to the recoil between the χ ± 1 χ 0 2 system and the jets, the boundary between the low and high E miss T bins is set at 50 GeV for the jet-veto categories and at 200 GeV for the jet-inclusive categories. The SR offWZ low/ E T -0j, SR offWZ low/ E T -nj and SR offWZ high/ E T -0j primarily target signals with moderate mass splitting ( m ∼ [40,90] GeV), and rely mostly on moderate kinematics and lepton triggers. The SR offWZ high/ E T -nj also target signals with highly compressed mass spectra ( m 40 GeV) -resulting in events with very soft leptons -by exploiting events with large E miss T recoiling against hard hadronic activity. Initial lepton p T requirements are kept as loose as possible, p T > 10 GeV for SR offWZ low/ E T -0j, SR offWZ low/ E T -nj, and SR offWZ high/ E T -0j, and p T > 4.5 (3.0) GeV for e (μ) in SR offWZ high/ E T -nj; however, the selection is restricted by the trigger requirements (Sect. 5) and some further requirements are applied in the bin-by-bin SR optimisation as discussed in the following.
Further common selection criteria are applied to reduce the contamination from Z + jets. First, a lower bound is set to ensure E miss T significance > 1.5 or 3.0, depending on the SR category. For SR offWZ low/ E T , events are then treated separately for different flavours of the lepton from the Wboson decay ( W ), selected using m m Z -based lepton assignment to best capture the SM background topology for rejection. To suppress the contribution from Z (+γ ) → ee caused by bremsstrahlung from prompt electrons and subsequent photon conversions, if W is an electron, the trilepton invariant mass m 3 is required to be off the Zboson peak (|m 3 − m Z | > 20 GeV), and the minimum angular distance between all SFOS lepton pairs must be within min R SFOS ∈ [0.6, 2.4], with min R SFOS defined as min[ R( i , j ); for all SFOS lepton pairs ( i , j )]. The common selection criteria and categorisation are summarised in Table 8.
The primary discriminant in SR offWZ is m min . This variable serves as a proxy for the mass splitting of the targeted signals, and displays a characteristic kinematic edge at their masssplitting value: m min = m, as demonstrated in Fig. 6. A shape fit over the m min spectrum is performed in each SR category. Seven m min bins are defined with boundaries at 1, 12, 15, 20, 30, 40, 60 and 75 GeV, and labelled 'a' to 'g'; the m min bin labels are added to the region names as defined above. Signal regions 'a' are dropped everywhere except in SR offWZ high/ E T -nj, to avoid low-mass resonance backgrounds. A second, similar kinematic edge is present in stransverse mass m T2 [174,175], reflecting the kinematic constraint originating from the χ ± 1 → W * χ 0 1 decay chain. In this selection, m T2 is constructed by assigning the dilepton system providing m min ( 1 2 ) to one visible particle leg, and the remaining lepton ( 3 ) to the other leg: where the transverse mass m T in this m T2 formula is defined by A hypothesised mass m χ is assigned to each invisible particle leg, corresponding to the χ 0 1 mass; m χ is fixed to 100 GeV in this selection. 6 The kinematic edge for signals appears at Table 8 Summary of the common selection criteria applied in the SRs of the off-shell W Z selection. In rows where only one value is given it applies to all regions. '-' indicates no requirement is applied for a given variable/region  Fig. 6. To take advantage of this feature, a sliding cut is applied per m min bin, requiring m 100 T2 to be smaller than the upper m min bin edge + 100 GeV. SM backgrounds can exceed the boundary and are suppressed, while a large fraction of the signal contribution targeted by a given bin is retained. The cut is particularly effective in the lowest m min bins, targeting the smallest mass splittings: e.g. in SR offWZ high/ E T -nja (m min ∈ [1, 12] GeV) the total background is reduced by a factor of signals with finite χ 0 1 mass, where the signal kinematic edges become significantly smeared. three following m 100 T2 < 112 GeV, while the efficiency for m = 10 GeV signals is > 95%. Event selection is tightened further by employing various background rejection criteria, optimised separately for each SR offWZ category and each m min bin. The discriminating variables used and the detailed bin-by-bin cut values are summarised in Table 9.
In order to reduce the FNP lepton background contributions from Z + jets and tt, lepton p T thresholds are raised in SR offWZ low/ E T -0j, SR offWZ low/ E T -nj and SR offWZ high/ E T -0j. In these same three categories, the transverse mass m mllmin T is used to suppress the SM W Z contribution; the m mllmin T variable is con- Table 9 Summary of the selection criteria for SRs for the off-shell W Z selection. SR offWZ common selection criteria are applied (Table 8). '-' indicates no requirement is applied for a given variable/region, while × is marked for regions that aren't considered structed using the W lepton after m min -based lepton assignment and marked with 'mllmin' to distinguish it from the m T variable in the on-shell W Z selection. The SRs target phase space either below or above the SM W -boson peak present at m mllmin T ∼ m W . An upper bound of m mllmin T < 50-70 GeV is applied in low m min bins, while the 'f' and 'g' bins are split into two parts below ('f1', 'g1') and above ('f2', 'g2') the Jacobian peak of SM W Z.
In SR offWZ low/ E T , the selection on min R SFOS is tightened in the low m min bins, exploiting the topology with a relatively boosted Z * in the target signatures, and a lower bound on m 3 is applied for the high m min bins to reject the SM Z → 4 background peaking at m 3 ∼ m Z . The ratio of the magnitude of a vectorial p T sum of the three leptons, |p

Background estimation
The selection criteria for the CRs and the VRs for W Z estimation are summarised in Table 10. An on-shell Z boson (m ∈ [81,101] GeV) is required to ensure orthogonality to the SR offWZ , and an upper bound on E miss T ensures orthogonality to the SR WZ . A lower bound on m T is applied to suppress the Z + jets background. The CRs are further split into two Table 10 Summary of the selection criteria for the CRs and VRs for W Z and tt, for the off-shell W Z selection. In rows where only one value is given it applies to all regions. '-' indicates no requirement is applied for a given variable/region Table 11 Summary of the selection criteria for the CRs and VRs for Z + jets, for the off-shell W Z selection. The corresponding anti-ID regions used for the Z + jets prediction follow the same selection cri-teria, except that at least one of the leptons is anti-ID instead of signal. '-' indicates no requirement is applied for a given variable/region   . 9 Breakdown of the total systematic uncertainties in the background prediction for the SRs of the off-shell W Z selection 75% in the VRs. The contamination from the benchmark signals is negligible in the CRs and below 15% in the VRs. The VRtt offWZ selection criteria are summarised in Table 10. At least one b-jet is required to maintain orthogonality with the SRs, E miss T > 50 GeV is required to suppress the Z + jets contribution, and the low-mass resonance background veto is applied. The tt purity in this region is approximately 65%.
The Z +jets background is estimated using the FF method as described in Sect. 6.2. The FF measurement region for the off-shell W Z selection, CRFF offWZ , is summarised in Table 11. The Z -boson candidate is selected by requiring |m − m Z | < 15 GeV, and E miss T < 40 GeV and m T < 30 GeV are applied to reject contamination from W Z. Additionally m 3 > 105 GeV is applied to suppress Z → 4 . To increase the number of FNP lepton candidates at high p T , the overlap removal procedure described in Sect. 5 is modified for this FF measurement so that muons overlapping with jets are always kept. Finally, a jet veto is applied except for events where the FNP lepton candidate is a muon with p T > 30 GeV, in which case n 30 GeV jets ≤ 1 is required in order to account for the special muon-vs-jet overlap-removal treatment applied to this region.
The FFs are derived separately per lepton flavour of FNP lepton candidates and per signal lepton criterion, i.e. with or without applying the non-prompt BDT, and are parameterised as a function of lepton p T and E miss The contribution of non-Z + jets processes is subtracted using MC simulated samples. A small normalisation correction is applied to the tt events in the simulated anti-ID region to account for the different anti-ID lepton efficiencies in data and MC simulation. Normalisation factors are derived separately depending on the W flavour and the b-jet multiplicity in the event. They are measured using the data events in a tt-enriched control region, CRtt offWZ anti-ID , and are found to be between 0.88 and 0.95. The CRtt offWZ anti-ID selection requires there to be no SFOS lepton pair in the event, as well as p 3 T > 10 GeV and E miss T > 50 GeV to enhance the tt purity.
Two sources of uncertainty specific to the estimation in SR offWZ are considered in addition to those described in Sect. 6.3. The FF parameterisation uncertainty is evaluated from the effect of using a different E miss T binning (E miss T < 50 GeV, 50% larger bin size), or a 3D parameterisation in lepton p T , E miss T and lepton η, additionally taking into account the dependency on lepton η. The impact on the Z + jets background yields in the CRs is ∼ 5%, and 1-7% in the SRs/VRs. The uncertainty from disabling the muon-vsjet overlap removal procedure in the FF measurement region is assessed by comparing those FFs with alternative FFs measured with muon-vs-jet overlap removal applied for events with a FNP muon candidate of p T < 30 GeV. The variation in the estimated Z + jets yields in the SRs/CRs/VRs is found to be 5-15%.
The yields predicted by the FF method are cross-checked in dedicated VRs enriched in FNP lepton backgrounds, as summarised in Table 11. The E miss T significance selection is inverted with respect to the SRs to ensure orthogonality. First, VRFF offWZ 0j and VRFF offWZ nj are designed to validate the yields in SR offWZ low/ E T -0j and SR offWZ low/ E T -nj, respectively, while VRFF offWZ nj-lowpT aims to cross-check the modelling of FNP leptons with p T < 10 GeV specifically. The Z + jets purity is  Fig. 7. Observed and expected yields for all CRs and VRs are summarised in Fig. 8.
The systematic uncertainties considered in the off-shell W Z selection are summarised in Fig. 9, grouped as discussed in Sect. 6.3. As the expected yields can vary by an order of magnitude throughout the regions, bin-to-bin fluctuations are expected in both the statistical and experimental uncertainty; these uncertainties are often dominant in bins with limited MC statistics in the phase space of the selection. The FNP lepton uncertainty is naturally more important in bins with larger FNP lepton background contributions, and can fluc-tuate in bins with few events in the corresponding anti-ID sample, such as SR offWZ high/ E T -0jb and SR offWZ high/ E T -njb. The modelling uncertainty is larger in the presence of ISR jets and at higher values of E miss T ; the fluctuation in SR offWZ originates from the effect of the QCD scale uncertainty on the W Z background.

Results
The observed data in the on-shell W Z, off-shell W Z, and Wh SRs are compared with the background expectation obtained from the background-only fits described in Sect. 6.4. The results are summarised in Tables 12 and 13 as well as visualised in Figs. 10 and 11 for the SR WZ and SR Wh regions, and in Tables 14 and 15 and Fig. 12 for the SR offWZ . Post-fit distributions of key kinematic observables are shown for the  GeV. The bottom panel shows the significance of the difference between the observed and expected yields, calculated with the profile likelihood method from Ref. [169], adding a minus sign if the yield is below the prediction SR WZ and SR Wh regions in Fig. 13 and for the SR offWZ regions in Fig. 14.
To illustrate the sensitivity to various χ ± 1 χ 0 2 signals throughout the regions, representative signal MC predictions are overlaid on the figures. The sensitivity to W Z-mediated models, when the mass difference between theχ  contributing the most. For the W Z-mediated models targeted with the SR offWZ , with mass differences between theχ ± 1 /χ 0 2 andχ 0 1 smaller than the Z -boson mass, the sensitivity to signals with different m depends on the m min range of the bins. The bins with larger (smaller) m min values are sensitive to signals with larger (smaller) mass splittings; for the lowest mass-splitting signals, only SR offWZ high/ E T -nja has sensitivity. No significant deviation from the SM background prediction is found in any of the SRs, and none of the deviations agree with any of the benchmark signal hypotheses. The maximum deviation of the data from the background expectation is in SR offWZ low/ E T -0jd with a 2.3σ data excess, followed by a 2.1σ deficit in SR offWZ high/ E T -0jf2, a 2.0σ excess in SR Wh DFOS -1, and a 2.0σ deficit in SR WZ -5; the significances are computed following the profile likelihood method in Ref. [169].

Model-independent limits on new physics in inclusive regions
Model-independent upper limits and discovery p-values in the SRs are derived by performing the discovery fits as described in Sect. 6.4. The set of single-bin signal regions used in the fits, referred to as 'inclusive SRs', is constructed by logically grouping adjoining, disjoint, nominal SRs of the on-shell W Z, Wh and off-shell W Z selections. Multiple, sometimes overlapping, regions are defined to capture signatures with different unknown m min shapes and jet multiplicities inclusively. Based on the best expected discovery sensitivity and using a number of signal points covering both the W Z-and Wh-mediated scenarios and different  Table 17. For incSR offWZ , contiguous jet-veto regions are merged with jet-inclusive regions, as the m min shape of a signal is assumed to be insensitive to jet multiplicity. The SR offWZ low/ E T and SR offWZ high/ E T regions are kept separate, while the SR offWZ high/ E T -nj regions are considered separately for m min < 20 GeV, as this selection provides the best sensitivity to low-mass-splitting models.
The 95% CL upper limits on the generic BSM cross section are calculated by performing a discovery fit for each target SR and its associated CRs, using pseudo-experiments. Results are reported in Tables 18 and 19 for the on-shell W Z and Wh analysis selections (off-shell W Z selection). The tables list the observed (N obs ) and expected (N exp ) yields in the inclusive SRs, the upper limits on the observed (S 95 obs ) and expected (S 95 exp ) number of BSM events, and the visible cross section (σ 95 vis ) reflecting the product of the production cross section, the acceptance, and the selection efficiency for a BSM process; the p-value and significance (Z ) for the background-only hypothesis are also presented.

Constraints on W Z-and Wh-mediated models
Constraints on the target simplified models are derived using the nominal SRs discussed in Sects. 7.1 and 8.1. The results are statistically combined with the previous results for the electroweakino regions (SR-E) of the two-lepton search targeting compressed mass spectra [18], referred to as the compressed selection. Model-dependent 95% CL exclusion limits are calculated by performing the exclusion fits as GeV. The bottom panel shows the significance of the difference between the observed and expected yields, calculated with the profile likelihood method from Ref. [169], adding a minus sign if the yield is below the prediction described in Sect. 6.4. When performing the combination, common experimental uncertainties are treated as correlated between regions and processes. Theoretical uncertainties of the background and signal are treated as correlated between regions only, while statistical uncertainties are considered uncorrelated between regions and processes.
All regions of the on-shell W Z, off-shell W Z, and compressed selections were explicitly designed to be orthogonal, allowing a statistical combination of the results. The on-shell and off-shell W Z selections are orthogonal due to the m and E miss T requirements, while the off-shell W Z and compressed selections are orthogonal by lepton multiplicity. Results are combined where greater exclusion power is expected over the individual results, ignoring contributions from search regions that do not add sensitivity in a given region of phase space. This approach results in multiple pairwise combinations of the on-shell and off-shell W Z selections, and the off-shell W Z and compressed selections, in bands of the ( m, m(χ 0 2 )) plane. Four separate fits are performed to obtain constraints for the following simplified models: • the wino/bino (+) W Z-mediated model combining the on-shell W Z, off-shell W Z, and compressed selections, • the wino/bino (+) Wh-mediated model using the Wh selection only, • the wino/bino (−) W Z-mediated model combining the off-shell W Z and compressed selections, • the higgsino W Z-mediated model combining the offshell W Z and compressed selections.
For the W Z-mediated model in the wino/bino (+) scenario, only the SR WZ are sensitive for mass splittings m above 100 GeV. Conversely, the SR offWZ dominate the intermediate mass-splitting region, with sensitivity in the m = [5,100] GeV range. In the most compressed region, the SR-E are important, driving the result for m below 10 GeV and adding sensitivity up to m = 50 GeV. Given these contributions, the m range is split into five bands to make optimal use of the different channels, and the combination considers respectively the SR-E only, the SR-E and SR offWZ , the SR offWZ only, the SR offWZ and SR WZ , and the SR WZ only. In the wino/bino (−) an d higgsino scenarios, the on-shell W Z selection is not considered, and only three bands are defined for the combination. The exact m ranges used are illustrated for the different scenarios in Fig. 15.
Expected and observed exclusion contours are reported as a function of the χ 0 1 and χ ± 1 / χ 0 2 masses, and shown in Fig. 16 (W Z-mediated model) and Fig. 17 (Wh-mediated model). The combined results are shown together with the individual contributions. For each mass point, a CL s value is derived to assess the probability of compatibility between the observed data and the signal-plus-background prediction obtained by the exclusion fit. For the W Z-mediated model, the results are obtained by statistically combining the SR WZ , as low as m = 2 GeV for a χ ± 1 / χ 0 2 mass of 100 GeV. Furthermore, constraints are calculated in the bino-wino coannihilation dark-matter scenario by determining the area in the two-dimensional mass plane that yields a thermal darkmatter relic density equal to the observed value [176]. The obtained wino/bino (+) exclusion limits are greatly improved compared to the previous equivalent search presented by the ATLAS experiment using the Run 1, 8 TeV dataset [17] (shown as a light grey shaded area in Fig. 16, top panels), due to a combination of increased production cross section   The obtained results for the wino/bino (−) and higgsino scenarios complement the previous compressed result using two-lepton final states as well. These results from the offshell W Z selection in three-lepton final states make full use of the larger data sample and target a novel phase space in the intermediately compressed m(χ 0 2 ,χ   [58]. The pale blue line in the top right panel represents the mass-splitting range that yields a dark-matter relic density equal to the observed relic density, h 2 = 0.1186 ± 0.0020 [176], when the mass parameters of all the decoupled SUSY partners are set to 5 TeV and tan β is chosen such that the lightest Higgs boson's mass is consistent with the observed value of the SM Higgs [45]. The area above (below) the blue line represents a dark-matter relic density larger (smaller) than the observed Eur. Phys. J. C (2021) 81:1118

Fig. 17
Exclusion limits obtained for the Wh-mediated model in the wino/bino (+) scenario, calculated using the Wh SRs and projected onto the m( χ ± 1 , χ 0 2 ) vs m( χ 0 1 ) plane. The expected 95% CL sensitivity (dashed black line) is shown with ±1σ exp (yellow band) from experimental systematic uncertainties and statistical uncertainties in the data yields, and the observed limit (red solid line) is shown with ±1σ theory (dotted red lines) from signal cross-section uncertainties. The light grey area denotes the constraints obtained by the previous equivalent analysis in ATLAS using the 8 TeV 20.3 fb −1 dataset [17] updated to use the full Run 2 dataset. The SR3 -Low region targets low-mass wino/bino (+) χ ± 1 χ 0 2 production, while the SR3 -ISR region targets wino/bino (+) χ ± 1 χ 0 2 production in association with ISR and mass differences m near the Z -boson mass. The excesses in SR3 -Low and SR3 -ISR observed in the 36 fb −1 result correspond to local significances of 2.1σ and 3.0σ , respectively. The RJR technique endeavours to resolve the ambiguities inherent in reconstructing original particles for event decays including invisible particles, e.g. SUSY particles. By analysing the event starting from the laboratory frame and boosting back to the parent particle's rest frame, assuming given decay chains, the technique can resolve the χ ± 1 and χ 0  ity, in the centre-of-mass frame) are considered. Using the reconstructed leptons, jets, and missing transverse momentum as inputs, the algorithm assigns each particle to a parent sparticle. ISR jets are selected by minimising the invariant mass of the system formed by the candidate jets and the sparticle system, in the centre-of-mass frame. The algorithm then determines the smallest Lorentz-invariant configuration of the particles' four-momenta guaranteeing a non-negative mass parameter for the invisible particles. Finally, object or frame momenta and derived variables can be considered in each of the different frames of each decay tree. The search in the RJR selection regions follows a similar strategy for background estimation, systematic uncertainty treatment, and statistical interpretation to that outlined for the on-shell W Z, off-shell W Z, and Wh selections in Sect. 6. For the search in SR3 -Low (SR3 -ISR), the SM diboson background is taken from MC simulation samples and normalised in a dedicated control region CR3 -VV (CR3 -ISR-VV) and validated in a validation region VR3 -VV (VR3 -ISR-VV). The selection criteria for each of the regions follow the original search [15], except for an additional jet-veto (n jets = 0) in CR3 -VV and VR3 -VV which guarantees the orthogonality between the low-mass and ISR regions. The FNP lepton background component, including tt, t W , W W and Z + jets SM background contributions, is estimated in a data-driven way using the matrix method [177]. The method derives the number of events with one or two FNP leptons by relating the yields for tighter (signal tagged) and looser (baseline tagged) lepton identification criteria. The result is a function of the real-lepton identification efficiencies and the FNP lepton misidentification probabilities. The remaining SM backgrounds, including multiboson and Higgs boson production, and top-pair production in association with a boson, are estimated from MC simulation in all analysis regions. Beyond the treatment of experimental and theoretical systematical uncertainties following the general strategy in Sect. 6.3, uncertainties are assigned to the matrix-method FNP lepton background esti-mation, accounting for limited numbers of events in the measurement region, potentially different compositions (heavy flavour, light flavour, or conversions) between SRs and CRs, and the uncertainty from the subtraction of prompt-lepton contributions using MC simulation samples.
Performing the background-only fit, diboson normalisation factors of 0.92±0.07 (CR3 -VV) and 0.92±0.05 (CR3 -ISR-VV) are determined. Observed and expected yields for all CRs and VRs are summarised in Fig. 18 and a summary of the considered systematic uncertainties is presented in Fig. 19, grouped as discussed in Sect. 6.3.
The observed data in SR3 -Low and SR3 -ISR are compared with the background expectation obtained by the background-only fit. The results are reported in Table 20 and post-fit distributions of key observables for the SRs are shown in Fig. 20. For the low-mass RJR selection, Fig. 20 shows the leading lepton's transverse momentum, p 1 T , and the scalar momentum sum, H PP 3,1 , of the three visible particles (the leptons) and the invisible particles (the LSPs and the neutrino), in the pair-produced parent sparticle-sparticle (PP) frame and assuming the standard decay tree. For the ISR RJR selection, Fig. 20 shows the vector sum of the transverse momenta of all objects, p CM T , and the fraction of the total momentum of the sparticle system carried by the invisible system, R ISR , in the centre-of-mass (CM) frame and assuming the ISR decay tree. Good agreement with the background-only hypothesis is observed in both SRs. The deviations from the SM expectation as found in the 36 fb −1 result are reduced and no longer significant when including the additional 103 fb −1 of data from the 2017-2018 datasets.
Model-independent results for SR3 -Low and SR3 -ISR are shown in Table 21. The 95% CL upper limits on the generic BSM cross section are calculated by performing a discovery fit for each target SR and its associated CR, using pseudo-experiments. The table lists the upper limits on the observed (S 95 obs ) and expected (S 95 exp ) number of BSM events in the inclusive SRs, and the visible cross section (σ 95 vis ) reflecting the product of the production cross section, the acceptance, and the selection efficiency for a BSM process; the pvalue and significance (Z ) for the background-only hypothesis are also presented.

Conclusion
Results of a search for chargino-neutralino pair production decaying via W Z, W * Z * or Wh into three-lepton final states are presented. A dataset of √ s = 13 TeV protonproton collisions corresponding to an integrated luminosity of 139 fb and three selections are developed with a signal region strategy optimised for chargino-neutralino signals decaying via W Z, W * Z * and Wh, respectively. A fourth selection targeting the chargino-neutralino signals decaying via W Z using the Recursive Jigsaw Reconstruction technique is also studied, to follow up on the excesses observed in the previous ATLAS result using the same method and event selection. In all the selections the data are found to be consistent with predictions of the Standard Model. The results are interpreted for simplified models with wino or higgsino production. A statistical combination is performed to include the result of an ATLAS search probing the final state with two soft leptons using the same dataset.
Assuming a simplified model with wino production decaying to a bino LSP, exclusion limits at 95% confidence level are placed on the minimum χ ± 1 / χ 0 2 mass, extending the reach of previous searches [14][15][16][17][18]21]. Limits are set at 640 GeV for the W Z-mediated model signals in the limit of massless χ 0 1 , improving by about 140 GeV; and at 300 GeV for masssplittings between χ ± 1 / χ 0 2 and χ 0 1 close to m Z , improving by about 100 GeV. In the case of a mass splitting of 5-90 GeV, χ ± 1 / χ 0 2 masses up to 200-300 GeV for the W Zmediated model are excluded. The limit extends down to a smallest mass splitting of 2 GeV for a χ ± 1 mass of 100 GeV. The dependency on a model parameter -the sign of the m eig (χ 0 2 )×m eig (χ 0 1 ) product -is also tested, and comparable limits are found for the two scenarios. For the Wh-mediated model signals, the limit on the minimum χ ± 1 / χ 0 2 mass is set at 190 GeV, for χ 0 1 masses below 20 GeV. Limits are also set for simplified models with a higgsino LSP triplet, for the first time including results from threelepton final states, which increases sensitivity to scenarios with moderate mass splittings. Combined with the two-lepton analysis targeting compressed mass spectra, the exclusion limits at 95% confidence level are placed on the minimum χ 0 2 mass up to 210 GeV for W Z-mediated model signals with a mass splitting of 2-60 GeV. In these models, searches in the three-lepton final state enhance the sensitivity in the experimentally challenging region with mass splitting greater than 30 GeV.  [178].

Data Availability Statement
This manuscript has no associated data or the data will not be deposited. [Authors' comment: All ATLAS scientific output is published in journals, and preliminary results are made available in Conference Notes. All are openly available, without restriction on use by external parties beyond copyright law and the standard conditions agreed by CERN. Data associated with journal publications are also made available: tables and data from plots (e.g. cross section values, likelihood profiles, selection efficiencies, cross section limits, ...) are stored in appropriate repositories such as HEPDATA (http:// hepdata.cedar.ac.uk/). ATLAS also strives to make additional material related to the paper available that allows a reinterpretation of the data in the context of new theoretical models. For example, an extended encapsulation of the analysis is often provided for measurements in the framework of RIVET (http://rivet.hepforge.org/).] Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecomm ons.org/licenses/by/4.0/. Funded by SCOAP 3 .