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

Abstract

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 $\hbox {fb}^{-1}$. 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.

1 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 ($\mathbf{p }_{\mathrm {T}}^{\mathrm {miss}}$, of magnitude $E_{\text {T}}^{\text {miss}}$). The search uses the full Run 2  dataset of proton–proton collisions recorded between 2015 and 2018 with the ATLAS detector at the CERN Large Hadron Collider (LHC). Protons were collided at a centre-of-mass energy $\sqrt{s}$ of $13~\text {TeV}$ 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 $WZ$ selection and the $Wh$ selection, and Sect. 8, with details specific to the off-shell $WZ$ 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.

2 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 $\mu$ are the mass parameters for the bino, wino, and higgsino states, respectively. Through mixing of the superpartners, chargino (${\widetilde{\chi }}^\pm _{1,2}$) and neutralino (${\widetilde{\chi }}^{0}_{1,2,3,4}$) mass eigenstates are formed. These are collectively referred to as electroweakinos, and the subscripts indicate increasing electroweakino mass. If the ${\widetilde{\chi }}^{0}_{1}$ 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 $|M_1| < |M_2| \ll |\mu |$ are assumed such that the produced electroweakinos have a wino and/or bino nature, with the ${\widetilde{\chi }}^{\pm }_{1}$ and ${\widetilde{\chi }}^{0}_{2}$ being wino dominated, and the ${\widetilde{\chi }}^{0}_{1}$ 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 mass-splitting between ${\widetilde{\chi }}^{\pm }_{1}$ and ${\widetilde{\chi }}^{0}_{1}$ is 15–30 $\text {GeV}$, 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 $|\mu | > 800~\text {GeV}$ [46].

The second scenario, referred to as the ‘higgsino scenario’, considers a triplet of higgsino-like states (${\widetilde{\chi }}^{\pm }_{1}$, ${\widetilde{\chi }}^{0}_{2}$, ${\widetilde{\chi }}^{0}_{1}$) to be the lightest SUSY particles. This type of scenario is motivated by naturalness arguments [47, 48], which suggest that $|\mu |$ 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 $|\mu |$. For the higgsino scenario this paper considers the regime where the mass-splitting between ${\widetilde{\chi }}^{0}_{2}$ and ${\widetilde{\chi }}^{0}_{1}$ is about 5–60 $\text {GeV}$, corresponding to cases where the wino and bino states are moderately decoupled ($M_1,M_2 > 0.5~\text {TeV}$).

Fig. 1

Diagrams of the targeted simplified models: pair production with subsequent decays into two , via leptonically decaying W, Z and SM Higgs bosons, three leptons and a neutrino. Diagrams are shown for (left) intermediate $WZ$ ($W\smash {^{*}}Z\smash {^{*}}$) as well as (right) intermediate $Wh$, with the Higgs boson decaying indirectly into leptons+X (where X denotes additional decay products) via $WW$, $ZZ$, or $\tau \tau$

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 ${\widetilde{\chi }}^{\pm }_{1}$ and ${\widetilde{\chi }}^{0}_{2}$ are assumed to be mass degenerate and purely wino, while the ${\widetilde{\chi }}^{0}_{1}$ is purely bino. The product of the two signed neutralino eigenmass parameters 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 ${\widetilde{\chi }}^{\pm }_{1}$, ${\widetilde{\chi }}^{0}_{2}$ and ${\widetilde{\chi }}^{0}_{1}$ are purely higgsino states, and the mass of the ${\widetilde{\chi }}^{\pm }_{1}$ is assumed to be exactly the mean of the ${\widetilde{\chi }}^{0}_{1}$ and ${\widetilde{\chi }}^{0}_{2}$ 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 ${\widetilde{\chi }}^{\pm }_{1}$ and ${\widetilde{\chi }}^{0}_{2}$.

The search targets direct pair production of the lightest chargino and the next-to-lightest neutralino, ${\widetilde{\chi }}^{\pm }_{1}{\widetilde{\chi }}^{0}_{2}$, decaying into a pair of ${\widetilde{\chi }}^{0}_{1}$ LSPs via an intermediate state with a W boson and a Z boson ($WZ$ mediated), or a W boson and a SM Higgs boson ($Wh$ mediated). Final states with three light-flavour 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 $E_{\text {T}}^{\text {miss}}$ originating from the LSPs, and this $E_{\text {T}}^{\text {miss}}$ component is enhanced when hadronic initial-state radiation (ISR) is present, due to recoil between the ${\widetilde{\chi }}^{\pm }_{1}{\widetilde{\chi }}^{0}_{2}$ system and the jets.

The following three simplified model scenarios of ${\widetilde{\chi }}^{\pm }_{1}{\widetilde{\chi }}^{0}_{2}$ pair production, as illustrated in Fig. 1, are considered with dedicated selections:

• On-shell $WZ$ selection: with 100% branching ratio, where , for the wino/bino (+) scenario.

• Off-shell $WZ$ selection: with 100% branching ratio, where , for the wino/bino (+), the wino/bino (−), and the higgsino scenarios.

• $Wh$ selection: with 100% branching ratio, where , for the wino/bino (+) scenario.

A 100% branching ratio is assumed for for all models. Unless otherwise indicated, mass splitting $\Delta m$ refers to  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 are expected with one lepton coming from the W boson and the remaining two from Higgs boson decays via $WW$, $ZZ$ or $\tau \tau$.

For ${\widetilde{\chi }}^{\pm }_{1}{\widetilde{\chi }}^{0}_{2}$ pair production with decays via $WZ$ to $3\ell$ final states, in the wino/bino (+) scenario, limits were previously set at the LHC for ${\widetilde{\chi }}^{\pm }_{1}$/${\widetilde{\chi }}^{0}_{2}$ masses up to 500 $\text {GeV}$ for massless ${\widetilde{\chi }}^{0}_{1}$, up to 200 $\text {GeV}$ for $\Delta m \sim m_{Z}$, and up to 240 $\text {GeV}$ for $50~\text {GeV}< \Delta m < m_{Z}$ [21]. Limits for mass splittings $\Delta m < 50~\text {GeV}$ were set in $2\ell$ final states for ${\widetilde{\chi }}^{\pm }_{1}$/${\widetilde{\chi }}^{0}_{2}$ masses up to 250 $\text {GeV}$[18]. For decays via $Wh$ to $3\ell$ final states (including hadronically decaying $\tau$-leptons), limits reached 150 $\text {GeV}$ for massless ${\widetilde{\chi }}^{0}_{1}$, and as high as 145 $\text {GeV}$ for a ${\widetilde{\chi }}^{0}_{1}$ mass of 20 $\text {GeV}$[17].

For the higgsino scenario, the most stringent limits for $5~\text {GeV}< \Delta m < 55~\text {GeV}$ were set by ATLAS using $2\ell$ final states [18] where ${\widetilde{\chi }}^{0}_{2}$ masses up to 130–190 $\text {GeV}$ are excluded depending on $\Delta m$. For $\Delta m > 55~\text {GeV}$ the best limits were reported by LEP [58,59,60,61,62,63, 63,64,65,66,67], excluding ${\widetilde{\chi }}^{\pm }_{1}$ masses up to $103.5~\text {GeV}$.

3 ATLAS detector

The ATLAS detector [68] is a general-purpose particle detector with almost $4\pi$ 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 $|\eta | < 2.5$, using silicon pixel and microstrip subsystems followed by a transition radiation tracker. For $\sqrt{s}= 13$ $\text {TeV}$ 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 $|\eta | < 3.2$ and $|\eta | < 1.7$ respectively. In the forward regions a copper/LAr endcap calorimeter extends the hadronic coverage to $1.7< |\eta | < 3.2$, while copper/LAr and tungsten/LAr forward calorimeters are used for electromagnetic and hadronic measurements in the $3.1< |\eta | < 4.9$ region. The muon spectrometer (MS) surrounds the calorimeters and comprises three layers of trigger and high-precision tracking chambers spanning $|\eta | < 2.4$ and $|\eta | < 2.7$, respectively. A magnetic field is provided by a system of three superconducting air-core 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.

4 Data and Monte Carlo simulated event samples

This analysis exploits the full Run 2  $\sqrt{s}=13~\text {TeV}$ 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 $\langle \mu \rangle = 34$. 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 and 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 $WZ$ 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.

Table 1 Monte Carlo simulation details by physics process. The table lists the event generators used for ME and PS calculations, the accuracy of the ME calculation, the PDF sets and UE parameter tunes used, and the order in $\alpha _\text {s}$ of cross-section calculations used for yield normalisation (‘–’ if the cross section is taken directly from MC simulation)

The SUSY 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 $WZ$ decays. The ME–PS matching was done using the CKKW-L prescription [126, 127], with the matching scale set to one quarter of the ${\widetilde{\chi }}^{\pm }_{1}$/${\widetilde{\chi }}^{0}_{2}$ mass. Samples were generated for ${\widetilde{\chi }}^{\pm }_{1}$/${\widetilde{\chi }}^{0}_{2}$ masses between 100 $\text {GeV}$ and 850 $\text {GeV}$, and mass splittings $\Delta m$ between 5 $\text {GeV}$ and 850 $\text {GeV}$. Only ${\widetilde{\chi }}^{\pm }_{1}$/${\widetilde{\chi }}^{0}_{2}$ decays via bosons, which in turn decay leptonically via SM branching fractions, are considered. For the $Wh$ samples, only Higgs boson decays via WW, ZZ and $\tau \tau$ 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 $WZ$ samples, and at least three leptons for the off-shell $WZ$ samples and the $Wh$ samples; hadronically decaying $\tau$-leptons are not considered in the requirement.

The only difference between the two wino/bino scenarios (positive or negative ) is the mass lineshape of the Z boson from the ${\widetilde{\chi }}^{0}_{2}$ decay, particularly when $\Delta m <m_{Z}$ 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.

Inclusive production cross sections are computed at next-to-leading order (NLO) plus next-to-leading-log (NLL) precision [79,80,81,82,83,84]. For wino production the computation is performed in the limit of mass-degenerate ${\widetilde{\chi }}^{\pm }_{1}$ and ${\widetilde{\chi }}^{0}_{2}$, and with light ${\widetilde{\chi }}^{0}_{1}$, while for higgsino production a partially degenerate case is considered, with the ${\widetilde{\chi }}^{\pm }_{1}$ mass equal to the mean of the ${\widetilde{\chi }}^{0}_{1}$ and ${\widetilde{\chi }}^{0}_{2}$ masses; all the other supersymmetric particles (sparticles) are assumed to be heavy and decoupled. For production at a centre-of-mass energy of $\sqrt{s} = 13~\text {TeV}$, the wino (higgsino) ${\widetilde{\chi }}^{\pm }_{1}{\widetilde{\chi }}^{0}_{2}$ cross section ranges between $22.67 \pm 0.97$ pb ($12.22 \pm 0.26$ pb, $\Delta m =80$ $\text {GeV}$) and $3.42 \pm 0.41$ fb ($87.2 \pm 3.2$ fb, $\Delta m =20$ $\text {GeV}$) for ${\widetilde{\chi }}^{0}_{2}$ masses between 100 $\text {GeV}$ and 850 (320) $\text {GeV}$, with the higgsino cross section depending additionally on $\Delta m$.

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 $4~\text {GeV}$ for $p_{\text {T}}^{\ell _{1}},p_{\text {T}}^{\ell _{2}} > 5~\text {GeV}$, and down to $m_{\ell \ell } > 2 m_{\ell }+250~\text {MeV}$ if $p_{\text {T}}^{\ell _{1}} > 5~\text {GeV}$ and any of $m_{\ell \ell } >4~\text {GeV}$, $p_{\text {T}}^{\ell _{1}} >20~\text {GeV}$, or $E_{\text {T}}^{\text {miss}} >50~\text {GeV}$ are satisfied. The standard multiboson samples do not include Higgs boson production. An alternative triboson sample including off-shell contributions and leptonically decaying $h\rightarrow VV$ (with V = W or Z) contributions is used in the off-shell $WZ$ selection, where $W\smash {^{*}}Z\smash {^{*}}$ decays are targeted and off-shell triboson processes are non-negligible in the estimation of the SM background; dilepton masses down to 4 $\text {GeV}$ are considered in the sample.

The $t\bar{t}$, single-top $t{}W$, t-channel, s-channel and $t\bar{t} h$ processes were modelled using Powheg Box 2 + Pythia 8. The $h_\mathrm {damp}$ parameter⁴ was set to 1.5 times the top-quark mass [136]. The samples were generated employing the five-flavour 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 $t\bar{t}$ production. Other top-quark processes ($t\bar{t} V$, tZ, t$WZ$, $t\bar{t} VV$, $t\bar{t} \ell \ell$ ($t\rightarrow W{}b{}+(\gamma ^{*}\!/Z\rightarrow \ell \ell ))$, 3-top and 4-top) were modelled using MadGraph5_aMC@NLO 2 + Pythia 8. Samples of Higgs boson production via gluon fusion, vector-boson fusion and associated production were generated using Powheg Box 2 + Pythia 8.

All background and signal samples make use of EvtGen 1.6.0 and 1.2.0 [138] for the modelling of b- and c-hadrons, 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.

5 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 $WZ$ selections using dilepton triggers and for the off-shell $WZ$ selection using single-lepton, dilepton and trilepton triggers [140, 141]. The off-shell $WZ$ selection is complemented at high $E_{\text {T}}^{\text {miss}}$ with softer-lepton events selected using $E_{\text {T}}^{\text {miss}}$ triggers [142]. The lepton triggers use various $p_{\text {T}}$ thresholds, depending on the lepton type, quality and multiplicity. To ensure trigger efficiencies are well understood in the analysis phase space, tighter quality and $p_{\text {T}}$ 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 $p_{\text {T}}$ above 18 $\text {GeV}$ (27.3, 14.7 or 6.5 $\text {GeV}$, for increasing trigger-lepton multiplicity). For events selected by a $E_{\text {T}}^{\text {miss}}$ trigger, an offline requirement of $E_{\text {T}}^{\text {miss}} > 200~\text {GeV}$ is imposed to similarly ensure well-understood 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 $p_{\text {T}} > 500~\text {MeV}$. In events with multiple vertices, the primary vertex is defined as the one with the highest $\sum p_{\text {T}} ^2$ 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 $Z+\text {jets}$ background estimation in Sect. 6.2. Hadronically decaying $\tau$-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 $Z\rightarrow ee$, $J/\psi \rightarrow ee$, $Z\rightarrow \mu \mu$ and $J/\psi \rightarrow \mu \mu$ decays. Baseline electrons are required to have $p_{\text {T}} > 4.5~\text {GeV}$ and fall within the acceptance of the ID ($|\eta | < 2.47$). They are further required to satisfy the calorimeter- and tracking-based ‘Loose and B-layer likelihood’ identification [145]. Baseline muons must have $p_{\text {T}} > 3~\text {GeV}$ and $|\eta |<2.5$, 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. $|z_0\sin \theta |<0.5$ 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, $\gamma$+jets and multijet events [151, 152]. Baseline jets must then have $p_{\text {T}} > 20~\text {GeV}$, and fall within the full calorimeter acceptance ($|\eta |<4.5$).

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 $p_{\text {T}} >25~\text {GeV}$, fall inside the ECAL strip detector acceptance ($|\eta |<2.37$), but outside the ECAL transition region ($|\eta |\in [1.37,1.52]$). 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 $\Delta R = 0.2$ are removed. Also, all jets within $\Delta R = 0.4$ of a muon and associated with fewer than three tracks with $p_{\text {T}} \ge 500~\text {MeV}$ are removed. Finally, electrons or muons separated from surviving jets by less than $\Delta R = 0.4$ are discarded to reject non-prompt leptons from decays of b- and 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 $\mathbf{p }_{\mathrm {T}}^{\mathrm {miss}}$ is adjusted for the calibration of the contributing objects, while maintaining robustness against pile-up [154]. Additionally, an ‘object-based $E_{\text {T}}^{\text {miss}}$ significance’ [155] is defined as . The $p_{\text {T}}$ resolution of the contributing objects, at a given $p_{\text {T}}$ and $|\eta |$, is determined from parameterised Monte Carlo simulation which well reproduces the resolution measured in data. The quantity $\sigma _{\text {L}}$ denotes the $p_{\text {T}}$ resolution of the system, and $\rho _{\text {LT}}$ is a correlation factor between the resolutions of the $p_{\text {T}}$ components parallel (L) and perpendicular (T) to $\mathbf{p }_{\mathrm {T}}^{\mathrm {miss}}$. The $E_{\text {T}}^{\text {miss}}$ significance is used to discriminate events where the $E_{\text {T}}^{\text {miss}}$ arises from undetected particles in the final state or from events where the $E_{\text {T}}^{\text {miss}}$ arises from poorly measured particles (and jets). It is also useful in discriminating between signal events with large $E_{\text {T}}^{\text {miss}}$ and e.g. $Z+\text {jets}$ events with medium-to-low $E_{\text {T}}^{\text {miss}}$.

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 $|\eta |<2.8$, 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 $p_{\text {T}} < 120~\text {GeV}$ and $|\eta | < 2.5$ (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 $|\eta |<2.5$ a multivariate discriminant – constructed using track impact parameters, information about displaced secondary vertices, and trajectories of b- and c-hadrons 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 $t\bar{t}$ 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 $|d_0/\sigma (d_0)| < 5~(3)$ for electrons (muons), where $\sigma (d_0)$ 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 $\sum p_{\text {T}}$ 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 on-shell $WZ$ and $Wh$ selections, while the looser working point Gradient (Loose) is employed for electrons (muons) in the off-shell $WZ$ selection to maintain a reasonable efficiency down to low $p_{\text {T}}$.

To further suppress FNP lepton backgrounds in the off-shell $WZ$ selection, a dedicated multivariate discriminant ‘non-prompt lepton BDT’ [28] is used to tighten the requirements on the lepton with the lowest $p_{\text {T}}$ (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 $p_{\text {T}}$ below 20 $\text {GeV}$, 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 non-prompt BDT selection) for the leptons from ${\widetilde{\chi }}^{\pm }_{1}{\widetilde{\chi }}^{0}_{2}$  signal events, as well as the differential probability for a $Z+\text {jets}$ event to be accompanied by a FNP lepton satisfying the signal lepton selection criteria.

Fig. 2

The left panel shows the combined lepton selection efficiencies for the signal electron/muon requirements applied to the lowest-$p_{\text {T}}$ lepton after selecting three baseline leptons in the off-shell $WZ$ selection. The efficiencies are calculated using simulated samples of wino/bino (+) ${\widetilde{\chi }}^{\pm }_{1}/{\widetilde{\chi }}^{0}_{2}$ decays and shown as a function of the generated lepton $p_{\text {T}}$. The associated uncertainties represent the range of efficiencies observed across all signal samples used for the given $p_{\text {T}}$ bin. The right panel illustrates the differential probability for a $Z+\text {jets}$ event to be accompanied by a FNP lepton satisfying the signal lepton criteria, as a function of the FNP lepton $p_{\text {T}}$. 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

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 $2.5< |\eta | < 2.7$ 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.

6 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 $E_{\text {T}}^{\text {miss}}$. Therefore, a common approach is used throughout most steps of the analyses. The on-shell $WZ$, off-shell $WZ$, 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 $WZ$ selection and the $Wh$ selection, or the off-shell $WZ$ selection, are then discussed in dedicated Sects. 7 and 8.

6.1 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 $WZ$, off-shell $WZ$ 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 $WZ$ selection, , are optimised for ${\widetilde{\chi }}^{\pm }_{1}{\widetilde{\chi }}^{0}_{2}$ signals with $WZ$-mediated decays and mass splittings near or above the Z-boson mass, $\Delta m \gtrsim m_{Z}$, while the SRs of the off-shell $WZ$ selection, , target $W\smash {^{*}}Z\smash {^{*}}$-mediated decays and mass splittings $\Delta m < m_{Z}$. The SRs of the $Wh$ selection, , 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 $\smash {\ell _{\mathrm {W}}}$). If more than one SFOS lepton pair is present in the event, the invariant mass $m_{\ell \ell }$ of the SFOS lepton pairs is used to select which pair is assigned to the $Z^{(*)}$-boson candidate. The on-shell $WZ$ selection selects the SFOS lepton pair with $m_{\ell \ell }$ nearest the Z-boson mass, $\smash {m_{\ell \ell }^{m_{Z}}}$, while the off-shell $WZ$ selection selects the SFOS lepton pair with the smallest $m_{\ell \ell }$, $m_{\ell \ell } ^{\mathrm {min}}$. In the rest of this document, these two types of lepton assignment are referred to as $\smash {m_{\ell \ell }^{m_{Z}}}$-based and $m_{\ell \ell } ^{\mathrm {min}}$-based lepton assignment, and $m_{\ell \ell }$ refers to $\smash {m_{\ell \ell }^{m_{Z}}}$ 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 target events with at least one SFOS pair (using $\smash {m_{\ell \ell }^{m_{Z}}}$-based lepton assignment), and the target complementary events without a SFOS lepton pair.

For events with at least one SFOS lepton pair the transverse mass, $m_\mathrm {T}$, is constructed using the W lepton and the $E_{\text {T}}^{\text {miss}}$, and assuming the SM $WZ$ event hypothesis: ${m_\mathrm {T} = {\sqrt{2{p_{\text {T}} ^{{\ell _{\mathrm {W}}}}}E_{\text {T}}^{\text {miss}} (1-\cos {(\Delta \phi )})}}}$, where $\Delta \phi$ is the separation in the transverse plane between the lepton and the $E_{\text {T}}^{\text {miss}}$. This exploits the difference between SM $WZ$, which has a Jacobian peak with a sharp cut-off at $m_\mathrm {T} \sim m_W$ (the W-boson 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 $m_{\ell \ell }$ slices: below, in, and above the Z-boson mass window, defined as $m_{\ell \ell } \in [75,105]~\text {GeV}$. The and use the first and second slice, respectively, while the use the first and third slice. The are orthogonal to the and the through the $m_{\ell \ell }$ selection. The and the can overlap, but are never used in the same interpretation. The 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 and have a shared binning strategy aside from the $m_{\ell \ell }$ range, while binning focuses on $m_{\ell \ell } ^{\mathrm {min}}$ and properties of more compressed ${\widetilde{\chi }}^{\pm }_{1}{\widetilde{\chi }}^{0}_{2}$  signals. Ultimately, 20, 31, 19, and 2 SR bins are defined for the , , , and , respectively. The complete definitions of these nominal SRs are further detailed per selection in subsequent Sects. 7.1 ( and ) and 8.1 (). 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.

6.2 Background estimation

The dominant SM background in most of the SRs in this analysis is from SM $WZ$ events with only leptonic decays, followed in importance by $t\bar{t}$ and $Z+\text {jets}$ processes associated with at least one FNP lepton. In , SM Higgs, triboson and $t\bar{t}$ production are the dominant processes.

A partially data-driven method is used for the estimation of the $WZ$ 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 $t\bar{t}$ 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 b- or 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 $(Z/\gamma ^{*} \rightarrow \ell \ell ) + (\text {jets}/\gamma )$ background has two prompt leptons and one FNP lepton from jets or photons. In the rest of this document, ‘$Z+\text {jets}$ ’ is used to refer to this set of processes. As there are no invisible particles in these processes at tree level, the observed $E_{\text {T}}^{\text {miss}}$ is mostly due to mismeasured leptons and/or jets, or due to the $E_{\text {T}}^{\text {miss}}$ soft term. The FNP leptons originate from a mix of sources, including light-flavour 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 $Z+\text {jets}$ 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, , designed to target $Z+\text {jets}$ 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 . 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 $Z+\text {jets}$ 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 $p_{\text {T}}$ and lepton $\eta$ or $E_{\text {T}}^{\text {miss}}$ in the event, depending on the analysis selection. In both the FF measurement and the FF application procedure, contributions from processes other than $Z+\text {jets}$ 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 (/) and Sect. 8.2 (/).

6.3 Systematic uncertainties

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 WZ 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 on-shell $WZ$, $Wh$ and off-shell $WZ$ selections; exceptions are discussed in Sects. 7.2 ( and ) and 8.2 (). 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 $p_{\text {T}}$ and $\eta$ 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 $p_{\text {T}}$ balance in dijet, Z+jet and $\gamma$+jet events [151, 152, 164]. Another significant instrumental uncertainty is that in the modelling of $E_{\text {T}}^{\text {miss}}$, 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 $E_{\text {T}}^{\text {miss}}$ 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, $WZ$, $ZZ$, and $t\bar{t}$, are derived using MC simulation samples. For the $WZ$ 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 anti-correlated configurations of the renormalisation and factorisation scales, as these are deemed unphysical. For the $WZ$ and $ZZ$ 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 $t\bar{t}$, 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 $t\bar{t}$ 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, $\alpha _\mathrm {s}$, are also considered for the major backgrounds. Uncertainties in the cross section of 13%, 12%, 10% and 20% are applied for minor backgrounds $t\bar{t} W$, $t\bar{t} Z$, $t\bar{t} h$, and triboson, respectively [109]; for all other rare top processes a conservative uncertainty of 50% is applied.

The data-driven $Z+\text {jets}$ estimation is subject to the statistical uncertainty due to the limited data sample size in 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-$Z+\text {jets}$ 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 $WZ$ selection is different from that in the on-shell $WZ$ and $Wh$ selections, reflecting the higher presence of $Z+\text {jets}$ in . 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 $E_{\text {T}}^{\text {miss}}$ and jet-inclusive regions. This uncertainty estimates match the results of a dedicated study using data and MC $Z\rightarrow \mu \mu$ 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 $WZ$ 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.

6.4 Statistical analysis

Final background estimates are obtained by performing a profile log-likelihood fit [167], implemented in the HistFitter [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 $Z+\text {jets}$ 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 $Z+\text {jets}$ 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 $WZ$ background is allowed to float and is constrained by the $WZ$ 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 $\hbox {CL}_{\text {s}}$ 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 $WZ$ and $Wh$ selections combined, and once for the off-shell $WZ$ selection. The results are presented in Sect 9.

The new results of the on-shell and off-shell $WZ$ 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.

7 On-shell $WZ$ and $Wh$ selections

The following subsections discuss the implementation specific to the on-shell $WZ$ 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.

7.1 Search regions

The and selections as introduced in Sect. 6.1 are further refined, taking into consideration differences in signal and background kinematics and composition. Driven by the $p_{\text {T}}$ thresholds of the dilepton triggers used in this selection, the leading and sub-leading leptons in the event must satisfy $p_{\text {T}}>$ 25, 20 $\text {GeV}$, while the third lepton must satisfy $p_{\text {T}} > 10$ $\text {GeV}$. To reduce SM backgrounds with little to no real $E_{\text {T}}^{\text {miss}}$, events are required to have $E_{\text {T}}^{\text {miss}} > 50$ $\text {GeV}$. To suppress the contribution of $t\bar{t}$ events and single-boson production in association with a $t\bar{t}$ pair, events with at least one $b\text {-jet}$ 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 $\text {GeV}$. Additionally, in events with a SFOS pair, the three-lepton invariant mass $m_{3\ell }$ is required to be inconsistent with the mass of a Z boson, $|m_{3\ell }-m_{Z} | > 15$ $\text {GeV}$, in order to suppress contributions from asymmetric photon conversions from the $Z+\text {jets}$ process with $Z\rightarrow \ell \ell \gamma ^{(*)}$  and $\gamma ^{(*)}\rightarrow \ell \ell$, where one of the leptons is out of acceptance.

Table 2 Summary of the common selection criteria applied in the SRs of the on-shell $WZ$ and $Wh$ selections. In rows where only one value is given it applies to all regions. ‘–’ indicates no requirement is applied for a given variable/region

Events with at least one SFOS lepton pair are divided into three $m_{\ell \ell }$ 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 Z-boson mass window ($m_{\ell \ell } \in [75,105]~\text {GeV}$), and is used for the selection. The second and third bins are defined below and above the Z-boson mass ($m_{\ell \ell } \le 75$ $\text {GeV}$ and $m_{\ell \ell } \ge 105$ $\text {GeV}$), and are used for the 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 and regions are further segmented as discussed below, and indexed with .

Each $m_{\ell \ell }$ bin is further divided into $m_\mathrm {T}$ and $E_{\text {T}}^{\text {miss}}$ bins, which enhances the sensitivity to various $\Delta m$ scenarios. The $m_\mathrm {T}$ distribution falls steeply in the region around the W-boson mass, and facilitates discrimination against the background from SM $WZ$ production. Three $m_\mathrm {T}$ bins, $m_\mathrm {T} < 100$, $100 \le m_\mathrm {T} \le 160$, and $m_\mathrm {T} > 160$ $\text {GeV}$, are defined to separate processes with and without a leptonic W-boson decay. The lower and upper bounds on the $E_{\text {T}}^{\text {miss}}$ bins vary with the $m_{\ell \ell }$ and $m_\mathrm {T}$ thresholds. The SM background contribution is expected to be higher in low $m_\mathrm {T}$ and $E_{\text {T}}^{\text {miss}}$ bins, while the signal populates different $m_\mathrm {T}$ and $E_{\text {T}}^{\text {miss}}$ bins, depending on the mass splitting. Signals with smaller $\Delta m$ tend to have more events in the lower $E_{\text {T}}^{\text {miss}}$ and $m_\mathrm {T}$ range, shifting to higher $E_{\text {T}}^{\text {miss}}$ and $m_\mathrm {T}$ bins as the mass difference increases.

Furthermore, events are separated by jet multiplicity, with jet-veto ($n_{\mathrm {jets}} = 0$; to , to and ) and jet-inclusive ($n_{\mathrm {jets}} > 0$; to , to ) SRs. The ISR topology is exploited further in the jet-inclusive regions of and by categorising the events with at least one jet according to their $H_{\text {T}}$, the scalar $p_{\text {T}}$ sum of the jets with $p_{\text {T}} > 20$ $\text {GeV}$. At higher $H_{\text {T}}$, signals with mass splitting $\Delta m \approx m_\mathrm {Z}$ tend to have more events at high values of $E_{\text {T}}^{\text {miss}}$ and $m_\mathrm {T}$ than the SM background, due to the recoil against ISR jets. In the high $H_{\text {T}}$ ($H_{\text {T}} > 200~\text {GeV}$) regions, softer lepton-$p_{\text {T}}$ spectra are expected for the signal because of the presence of a massive ${\widetilde{\chi }}^{0}_{1}$, which carries most of the transverse momenta of the boosted ${\widetilde{\chi }}^{\pm }_{1}{\widetilde{\chi }}^{0}_{2}$ system. Therefore $H_\mathrm {T}^{\mathrm {lep}}$, the scalar $p_{\text {T}}$ sum of the three selected leptons, is required to be less than 350 $\text {GeV}$. The $H_{\text {T}}$ categorisation is applied in regions with $m_{\ell \ell } < 105$ $\text {GeV}$. Finally, in the high-mass off-peak region ($m_{\ell \ell } \ge 105$ $\text {GeV}$), only jet-veto events are considered. The full set of 20 and 19 signal regions is summarised in Tables 3 and 4.

Table 3 Summary of the selection criteria for the SRs targeting events with at least one SFOS lepton pair and $m_{\ell \ell } \in [75,105]~\text {GeV}$, for the on-shell $WZ$ search regions. Region selections are binned by $m_\mathrm {T}$ (rows) and $E_{\text {T}}^{\text {miss}}$ for the two sets of regions, where each set has different $n_{\mathrm {jets}}$ and $H_{\text {T}}$ requirements. common selection criteria are applied (Table 2)

Table 4 Summary of the selection criteria for the SRs targeting events with at least one SFOS lepton pair and $m_{\ell \ell } \notin [75,105]~\text {GeV}$, for the $Wh$ search regions. Region selections are binned by $m_\mathrm {T}$ (rows) and $E_{\text {T}}^{\text {miss}}$ for the three sets of regions, where each set has different $m_{\ell \ell }$, $n_{\mathrm {jets}}$, and $H_{\text {T}}$ requirements. common selection criteria are applied (Table 2)

In the 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, $t\bar{t}$ production dominates the SM background and is minimised by keeping events with low jet multiplicity ($n_{\mathrm {jets}} < 3$). These are then further split into two SR bins, one with $n_{\mathrm {jets}} = 0$ () and the other satisfying $n_{\mathrm {jets}} \in [1, 2]$ (). Due to the presence of the ${\widetilde{\chi }}^{0}_{1}$, signals tend to have higher $E_{\text {T}}^{\text {miss}}$  significance than the SM background, and therefore the events are required to have $E_{\text {T}}^{\text {miss}} ~\text{ significance } > 8$. The third lepton in $t\bar{t}$ production usually arises from a heavy flavour quark decay and is typically lower in $p_{\text {T}}$ than the third lepton in the SUSY signal scenarios. To reduce this contribution the lower bound on the third lepton’s $p_{\text {T}}$ is increased to 15 and 20 $\text {GeV}$ in the and regions, respectively. Angular proximity between leptons coming from a Higgs-boson decay is used for further event separation, using the variable ${\Delta }R_{\text {OS,near}}$, defined as the $\Delta R$ between the DFOS lepton and the SFSS lepton nearest in $\phi$. The signal is expected to populate the lower range in ${\Delta }R_{\text {OS,near}}$, while the SM background tends to have a flatter distribution. Events in are required to satisfy ${\Delta }R_{\text {OS,near}} < 1.2$. To suppress the higher $t\bar{t}$ contribution in the , a tighter selection on ${\Delta }R_{\text {OS,near}}$ is imposed. A complete summary of the selection criteria in is presented in Table 5.

For the $WZ$-mediated ${\widetilde{\chi }}^{\pm }_{1}{\widetilde{\chi }}^{0}_{2}$ signal sample with NLSP mass of 600 $\text {GeV}$ and massless ${\widetilde{\chi }}^{0}_{1}$, the and regions have selection acceptance times efficiency values of $2.0\times 10^{-3}$ and $3.0\times 10^{-3}$, respectively. For the $Wh$-mediated ${\widetilde{\chi }}^{\pm }_{1}{\widetilde{\chi }}^{0}_{2}$ signal sample with NLSP mass of 200 $\text {GeV}$ and massless ${\widetilde{\chi }}^{0}_{1}$, the , , and regions have selection acceptance times efficiency values of $9.1\times 10^{-5}$, $1.0\times 10^{-4}$, and $3.7\times 10^{-5}$, respectively.

Table 5 Summary of the selection criteria for the SRs targeting events with a DFOS lepton pair, for the $Wh$ selection. common selection criteria are applied (Table 2)

7.2 Background estimation

The normalisation of the $WZ$ background is measured in CRs characterised by moderate values of the $E_{\text {T}}^{\text {miss}}$ and $m_\mathrm {T}$ variables. The CRs contain only events with at least one SFOS pair with an invariant mass of $75< m_{\ell \ell } < 105$ $\text {GeV}$, targeting on-shell decays. Additional requirements of $50< E_{\text {T}}^{\text {miss}} < 100~\text {GeV}$ and $20< m_\mathrm {T} < 100~\text {GeV}$ improve the $WZ$ purity, the upper bound on $m_\mathrm {T}$ at 100 $\text {GeV}$ also ensures orthogonality between the $WZ$ CRs and . To address the possible mis-modelling of the jet multiplicity in the $WZ$ simulated samples, the cross-section normalisation factor is extracted separately in each jet multiplicity and $H_{\text {T}}$ category, using , , and . The estimation is cross-checked in kinematically similar, orthogonal VRs: , , and . A summary of the selection criteria defining the $WZ$ CRs and VRs is presented in Table 6. The $WZ$ 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.

Table 6 Summary of the selection criteria for the CRs and VRs for $WZ$, for the on-shell $WZ$ and $Wh$ selections. In rows where only one value is given it applies to all regions. ‘–’ indicates no requirement is applied for a given variable/region

Performing the simultaneous background-only fit for the on-shell $WZ$ and $Wh$ selections, normalisation factors for $WZ$ of $1.07\pm 0.02$ (), $0.94\pm 0.03$ () and $0.85\pm 0.05$ () are found.

A good description of the $m_\mathrm {T}$ and $E_{\text {T}}^{\text {miss}}$ distributions in the $WZ$ simulation is crucial in this analysis, especially in the high-$m_\mathrm {T}$ and high-$E_{\text {T}}^{\text {miss}}$ tails where new physics may appear. The tail of the $m_\mathrm {T}$ 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 $m_\mathrm {T}$ of the W-boson candidate (‘mis-pairing’ of the leptons), the $E_{\text {T}}^{\text {miss}}$ 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 $m_{\ell \ell }$ computation and using the third lepton to calculate $m_\mathrm {T}$. Finally, the modelling of the $m_\mathrm {T}$ and $E_{\text {T}}^{\text {miss}}$ distributions is validated in a $W{+}\gamma$ control sample. The $W{+}\gamma$ and $WZ$ processes have very similar $m_\mathrm {T}$ shapes because their production mechanisms are similar, with the exception that the FSR production diagram of $W{+}\gamma$ is much more common than the corresponding diagram in $WZ$, 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 $W{+}\gamma$ process leads to differences in the $m_\mathrm {T}$ distribution shapes between $WZ$ and $W{+}\gamma$. When a photon is radiated, leptons lose energy, resulting in a lower $m_\mathrm {T}$. In order to use the $W{+}\gamma$ $m_\mathrm {T}$ shape to validate the $WZ$ MC prediction, the FSR contribution in the $W{+}\gamma$ control region has to be suppressed. This is done by placing threshold requirements on the $p_{\text {T}}$ of the photon, $p_{\text {T}} ^{\gamma } > 50$ $\text {GeV}$, and the separation between the lepton and the photon, $\Delta R(\ell ,\gamma ) > 0.4$, in $W{+}\gamma$ events, as FSR photons are expected to be close to the lepton radiating them and also tend to have low $p_{\text {T}}$. The distribution shapes of $m_\mathrm {T}$ and $E_{\text {T}}^{\text {miss}}$, as well as other kinematic variables, are compared in data and MC events in the $W{+}\gamma$ region. The $m_\mathrm {T}$ distribution in the validation region with mis-paired leptons and the $W{+}\gamma$ validation region are shown in Fig. 3. Good agreement in both control samples is observed and no extra corrections or scale factors are applied to correct the $m_\mathrm {T}$ distribution for the $WZ$ background.

The $t\bar{t}$ MC modelling is validated in VRs, enhancing the $t\bar{t}$ contribution by requiring a DFOS lepton pair and using a moderate $E_{\text {T}}^{\text {miss}} > 50~\text {GeV}$ selection. The main VR, , requires the presence of one or two b-jets, further increasing the $t\bar{t}$ contribution. To validate the modelling in the $n_{\mathrm {jets}} = 0$ region as well, an additional VR inclusive in b-jets, , is considered, with a $E_{\text {T}}^{\text {miss}} ~\text{ significance } < 8$ requirement to ensure orthogonality with the regions. The $t\bar{t}$ purity is about 80% in the and 72% in the . The selection requirements for the $t\bar{t}$ VRs are summarised in Table 7.

Fig. 3

Distributions of $m_{\text {T}}$ showing the data and the pre-fit expected background in (top left) the mis-paired lepton validation region and (top right) the $W{+}\gamma$ validation region, used to validate the $WZ$ background. Distributions of (bottom left) $m_\mathrm {T}$ in and (bottom right) $E_{\text {T}}^{\text {miss}}$ in , showing the data and the post-fit expected background in each region. The last bin includes overflow. The ‘Others’ category contains backgrounds from single-top, $WW$, triboson, Higgs and rare top processes. The bottom panel shows the ratio of the observed data to the predicted yields. The hatched bands indicate the combined theoretical, experimental, and MC statistical uncertainties

Table 7 Summary of the selection criteria for the CRs and VRs for $t\bar{t}$ and $Z+\text {jets}$, for the on-shell $WZ$ and $Wh$ selections. The corresponding anti-ID regions used for the $Z+\text {jets}$ prediction follow the same selection criteria, except that at least one of the leptons is anti-ID instead of signal. ‘–’ indicates no requirement is applied for a given variable/region

The $Z+\text {jets}$ estimation uses the FF method as described in Sect. 6.2. For measurement region , the Z-boson candidate mass must be compatible with the Z-boson mass within 15 $\text {GeV}$, and low $E_{\text {T}}^{\text {miss}}$ and $m_\mathrm {T}$ are required to minimise $WZ$ contributions. The typical value of FFs varies from 0.2 to 0.4, depending on the lepton $p_{\text {T}}$ and $\eta$. The $Z+\text {jets}$ estimation is then validated in , considering the intermediate $E_{\text {T}}^{\text {miss}}$ range closer to, but orthogonal to, the SRs, and adding a $m_{3\ell }$ lower bound to reduce $WZ$ contamination. The selection criteria for as well as those of are summarised in Table 7.

Figure 3 presents the $m_\mathrm {T}$ distribution in , and the $E_{\text {T}}^{\text {miss}}$ distribution in , showing good agreement between the observed data and the estimated background. The comparisons between the expected and observed yields in the and all are given in Fig. 4.

Fig. 4

Comparison of the observed data and expected SM background yields in the CRs (pre-fit) and VRs (post-fit) of the on-shell $WZ$ and $Wh$ selections. The ‘Others’ category contains the single-top, $WW$, triboson, Higgs and rare top processes. The hatched band indicates the combined theoretical, experimental, and MC statistical uncertainties. 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

The systematic uncertainties considered in the on-shell $WZ$ and the $Wh$ SRs follow the approach discussed in Sect. 6.3. The relative composition of FNP muons is similar between the and , whereas for FNP electrons the main source in the is photon conversions, while in the the heavy-flavour decay contribution dominates. An additional source of uncertainty that is considered accounts for the different FNP lepton compositions in the and . This uncertainty arises from the method’s performance 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 $t\bar{t}$. 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 and 15–16 of the on-shell $WZ$ selection, and , , and of the $Wh$ selection, due to limited number of MC events at high $E_{\text {T}}^{\text {miss}}$ and $m_\mathrm {T}$. Although the FNP lepton uncertainty is negligible in the majority of the search bins, its relative size reaches 30% in , due to the small number of events in the corresponding anti-ID sample.

Fig. 5

Breakdown of the total systematic uncertainties in the background prediction for the SRs of (left) the on-shell $WZ$ selection and (right) the $Wh$ selection

8 Off-shell $WZ$ selection

The following subsections discuss the implementation specific to the off-shell $WZ$ 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.

8.1 Search regions

The selection targets the off-shell $WZ$ region by requiring $m_{\ell \ell } ^{\mathrm {min}}, m_{\ell \ell } ^{\mathrm {max}} < 75~\text {GeV}$. The $m_{\ell \ell } ^{\mathrm {max}}$ 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 $WZ$ selection assume $m_{\ell \ell } ^{\mathrm {min}}$-based lepton assignment to the $Z^{*}$- and $W^{*}$-boson candidates unless otherwise indicated. The common event selection vetoes events with a $b\text {-jet}$ to reduce contamination from $t\bar{t}$, requires the three leptons to be well separated in , and requires a lower bound on $m_{\ell \ell } ^{\mathrm {min}}$ of 1 $\text {GeV}$ to remove events with collimated leptons for which FNP lepton background estimation is challenging. Finally, $m_{\ell \ell } ^{\mathrm {min}}$ mass ranges of [3.0, 3.2] and [9, 12] $\text {GeV}$ are vetoed to avoid contributions from $J/\psi$ and $\Upsilon$ resonance backgrounds associated with a FNP lepton, except in the jet-inclusive high $E_{\text {T}}^{\text {miss}}$ regions ($E_{\text {T}}^{\text {miss}} > 200~\text {GeV}$) where the contribution is negligible.

Preselected events are further divided into four categories based on the multiplicity of jets with $p_{\text {T}} >30~\text {GeV}$ ($n_{\text {jets}}^{30~\text {GeV}}$), and on $E_{\text {T}}^{\text {miss}}$. Jet-veto categories and reject events containing jets and select low and high $E_{\text {T}}^{\text {miss}}$, respectively. Jet-inclusive categories and require at least one jet and also separate the events with low and high $E_{\text {T}}^{\text {miss}}$. As the $E_{\text {T}}^{\text {miss}}$ is harder in the jet-inclusive categories, due to the recoil between the ${\widetilde{\chi }}^{\pm }_{1}{\widetilde{\chi }}^{0}_{2}$ system and the jets, the boundary between the low and high $E_{\text {T}}^{\text {miss}}$ bins is set at 50 $\text {GeV}$ for the jet-veto categories and at 200 $\text {GeV}$ for the jet-inclusive categories. The , and primarily target signals with moderate mass splitting ($\Delta m \sim [40,90]~\text {GeV}$), and rely mostly on moderate kinematics and lepton triggers. The also target signals with highly compressed mass spectra ($\Delta m \lesssim 40~\text {GeV}$) – resulting in events with very soft leptons – by exploiting events with large $E_{\text {T}}^{\text {miss}}$ recoiling against hard hadronic activity. Initial lepton $p_{\text {T}}$ requirements are kept as loose as possible, $p_{\text {T}} >10~\text {GeV}$ for , , and , and $p_{\text {T}} >4.5\,(3.0)~\text {GeV}$ for e ($\mu$) in ; 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.

Table 8 Summary of the common selection criteria applied in the SRs of the off-shell $WZ$ selection. In rows where only one value is given it applies to all regions. ‘-’ indicates no requirement is applied for a given variable/region

Further common selection criteria are applied to reduce the contamination from $Z+\text {jets}$. First, a lower bound is set to ensure $E_{\text {T}}^{\text {miss}} ~\text{ significance } >1.5$ or 3.0, depending on the SR category. For , events are then treated separately for different flavours of the lepton from the W-boson decay ($\smash {\ell _{\mathrm {W}}}$), selected using $\smash {m_{\ell \ell }^{m_{Z}}}$-based lepton assignment to best capture the SM background topology for rejection. To suppress the contribution from $Z(+\gamma ) \rightarrow \ell \ell ee$ caused by bremsstrahlung from prompt electrons and subsequent photon conversions, if $\smash {\ell _{\mathrm {W}}}$ is an electron, the trilepton invariant mass $m_{3\ell }$ is required to be off the Z-boson peak ($|m_{3\ell }-m_{Z} |>20~\text {GeV}$), and the minimum angular distance between all SFOS lepton pairs must be within $\text {min}\,\Delta R _{\mathrm {SFOS}} \in [0.6,2.4]$, with $\text {min}\,\Delta R _{\mathrm {SFOS}}$ defined as . The common selection criteria and categorisation are summarised in Table 8.

Fig. 6

Distributions of (left) $m_{\ell \ell } ^{\mathrm {min}}$ and (right) $m_{\text {T2}}^{100}$ showing the expected SM background as well as signals with various mass splittings (), for a selection of exactly three baseline and signal leptons. The distributions are normalised to unity. Signals demonstrate a cut-off in both variables matching the mass splitting, while backgrounds do not. The dominant background in this selection is $WZ$, with the Z-boson mass peak visible in both distributions

The primary discriminant in is $m_{\ell \ell } ^{\mathrm {min}}$. This variable serves as a proxy for the mass splitting of the targeted signals, and displays a characteristic kinematic edge at their mass-splitting value: $m_{\ell \ell } ^{\mathrm {min}} = \Delta m$, as demonstrated in Fig. 6. A shape fit over the $m_{\ell \ell } ^{\mathrm {min}}$ spectrum is performed in each SR category. Seven $m_{\ell \ell } ^{\mathrm {min}}$ bins are defined with boundaries at 1, 12, 15, 20, 30, 40, 60 and 75 $\text {GeV}$, and labelled to ; the $m_{\ell \ell } ^{\mathrm {min}}$ bin labels are added to the region names as defined above. Signal regions  are dropped everywhere except in , to avoid low-mass resonance backgrounds.

A second, similar kinematic edge is present in stransverse mass $m_\mathrm {T2}$ [174, 175], reflecting the kinematic constraint originating from the ${\widetilde{\chi }}^{\pm }_{1} \rightarrow W\smash {^{*}} {\widetilde{\chi }}^{0}_{1}$ decay chain. In this selection, $m_\mathrm {T2}$ is constructed by assigning the dilepton system providing $m_{\ell \ell } ^{\mathrm {min}}$ ($\ell _1\ell _2$) to one visible particle leg, and the remaining lepton ($\ell _3$) to the other leg:

$$\begin{aligned}&m_\mathrm {T2} ^{m_\chi } \left( \mathbf{p }_\mathrm{T}^{\ell _1\ell _2},\mathbf{p }_\mathrm{T}^{\ell _3},\mathbf{p }_\mathrm{T}^\mathrm {miss}\right) \\&\quad = \underset{\mathbf{q }_\mathrm{T}}{\min }\left( \max \left[ m_\mathrm {T} \left( \mathbf{p }_\mathrm{T}^{\ell _1\ell _2},\mathbf{q }_\mathrm{T},m_\chi \right) , m_\mathrm {T} \left( \mathbf{p }_\mathrm{T}^{\ell _3},\mathbf{p }_{\mathrm {T}}^{\mathrm {miss}}-\mathbf{q }_\mathrm{T},m_\chi \right) \right] \right) , \end{aligned}$$

where the transverse mass $m_{\text {T}}$ in this $m_\mathrm {T2}$ formula is defined by

$$\begin{aligned}&m_\mathrm {T} \left( \mathbf{p }_\mathrm{T}^{\ell },\mathbf{q }_\mathrm{T},m_\chi \right) \\&\quad = \sqrt{ m_\ell ^2 + m_\chi ^2 + 2\left( \sqrt{(p_{\text {T}} ^{\ell })^2 + m_\ell ^2} \sqrt{q_\mathrm {T}^2 + m_\chi ^2} - \mathbf{p }_\mathrm{T}^{\ell } \cdot \mathbf{q }_\mathrm{T}\right) }. \end{aligned}$$

A hypothesised mass $m_\chi$ is assigned to each invisible particle leg, corresponding to the ${\widetilde{\chi }}^{0}_{1}$ mass; $m_\chi$ is fixed to 100 $\text {GeV}$ in this selection.⁶ The kinematic edge for signals appears at $m_{\text {T2}}^{100} = \Delta m ({\widetilde{\chi }}^{0}_{2},{\widetilde{\chi }}^{0}_{1}) + 100~\text {GeV}$ as illustrated in Fig. 6. To take advantage of this feature, a sliding cut is applied per $m_{\ell \ell } ^{\mathrm {min}}$ bin, requiring $m_{\text {T2}}^{100}$ to be smaller than the upper $m_{\ell \ell } ^{\mathrm {min}}$ bin edge + 100 $\text {GeV}$. 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 $m_{\ell \ell } ^{\mathrm {min}}$ bins, targeting the smallest mass splittings: e.g. in ($m_{\ell \ell } ^{\mathrm {min}} \in [1,12]~\text {GeV}$) the total background is reduced by a factor of three following $m_{\text {T2}}^{100} <112~\text {GeV}$, while the efficiency for $\Delta m =10~\text {GeV}$ signals is $>95\%$.

Event selection is tightened further by employing various background rejection criteria, optimised separately for each category and each $m_{\ell \ell } ^{\mathrm {min}}$ bin. The discriminating variables used and the detailed bin-by-bin cut values are summarised in Table 9.

Table 9 Summary of the selection criteria for SRs for the off-shell $WZ$ selection. common selection criteria are applied (Table 8). ‘–’ indicates no requirement is applied for a given variable/region, while $\times$ is marked for regions that aren’t considered

In order to reduce the FNP lepton background contributions from $Z+\text {jets}$ and $t\bar{t}$, lepton $p_{\text {T}}$ thresholds are raised in , and . In these same three categories, the transverse mass $m_{\text {T}}^{\text {mllmin}}$ is used to suppress the SM $WZ$ contribution; the $m_{\text {T}}^{\text {mllmin}}$ variable is constructed using the W lepton after $m_{\ell \ell } ^{\mathrm {min}}$-based lepton assignment and marked with ‘’ to distinguish it from the $m_{\text {T}}$ variable in the on-shell $WZ$ selection. The SRs target phase space either below or above the SM W-boson peak present at $m_{\text {T}}^{\text {mllmin}} \sim m_W$. An upper bound of $m_{\text {T}}^{\text {mllmin}}<$ 50–70 $\text {GeV}$ is applied in low $m_{\ell \ell } ^{\mathrm {min}}$ bins, while the and bins are split into two parts below () and above () the Jacobian peak of SM $WZ$.

In , the selection on $\text {min}\,\Delta R _{\mathrm {SFOS}}$ is tightened in the low $m_{\ell \ell } ^{\mathrm {min}}$ bins, exploiting the topology with a relatively boosted $Z^{*}$ in the target signatures, and a lower bound on $m_{3\ell }$ is applied for the high $m_{\ell \ell } ^{\mathrm {min}}$ bins to reject the SM $Z\rightarrow 4\ell$ background peaking at $m_{3\ell } \sim m_Z$. The ratio of the magnitude of a vectorial $p_{\text {T}}$ sum of the three leptons, $|\mathbf{p }_{\mathrm {T}}^{\mathrm {lep}}|$, to $E_{\text {T}}^{\text {miss}}$, is labelled $|\mathbf{p }_{\mathrm {T}}^{\mathrm {lep}}|/E_{\text {T}}^{\text {miss}}$ and represents the extent to which the transverse momentum of the hard-scatter ${\widetilde{\chi }}^{\pm }_{1} {\widetilde{\chi }}^{0}_{2}$ system, recoiling against ISR jets, is converted into leptons as opposed to $E_{\text {T}}^{\text {miss}}$. Due to the presence of a massive ${\widetilde{\chi }}^{0}_{1}$, contributing to the $E_{\text {T}}^{\text {miss}}$, signals tend to populate lower parts of the $|\mathbf{p }_{\mathrm {T}}^{\mathrm {lep}}|/E_{\text {T}}^{\text {miss}}$ spectrum than SM backgrounds, particularly for the compressed signals in the high $E_{\text {T}}^{\text {miss}}$ regions where the $E_{\text {T}}^{\text {miss}}$ is almost fully generated by the ISR jets. A tight upper bound $|\mathbf{p }_{\mathrm {T}}^{\mathrm {lep}}|/E_{\text {T}}^{\text {miss}}$ is therefore imposed in the low $m_{\ell \ell } ^{\mathrm {min}}$ bins of .

Table 10 Summary of the selection criteria for the CRs and VRs for $WZ$ and $t\bar{t}$, for the off-shell $WZ$ 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 $Z+\text {jets}$, for the off-shell $WZ$ selection. The corresponding anti-ID regions used for the $Z+\text {jets}$ prediction follow the same selection criteria, except that at least one of the leptons is anti-ID instead of signal. ‘–’ indicates no requirement is applied for a given variable/region

After applying the selection criteria, for the wino/bino (+) model  signal sample with NLSP masses of 200 $\text {GeV}$ and a mass splitting of $\Delta m =20$ $\text {GeV}$, the , , , and regions (taking the union of the bins inside each region) have acceptance times efficiency values of $2.2\times 10^{-5}$, $1.1\times 10^{-5}$, $3.4\times 10^{-6}$, and $6.0\times 10^{-5}$, respectively. Similarly, for a mass splitting of $\Delta m =60$ $\text {GeV}$, values of $1.6\times 10^{-4}$, $1.7\times 10^{-4}$, $2.8\times 10^{-4}$, and $7.9\times 10^{-5}$ are found. The acceptance times efficiency values for the wino/bino (−) and higgsino model signal samples are typically 15–55% and 20–60% lower, depending on the region.

Fig. 7

Example kinematic distributions after the background-only fit, showing the data and the post-fit expected background, in regions of the off-shell $WZ$ selection. The figure shows (top left) the $m_{\ell \ell } ^{\mathrm {min}}$ distribution in , (top right) the $|\mathbf{p }_{\mathrm {T}}^{\mathrm {lep}}|/E_{\text {T}}^{\text {miss}}$ distribution in , (bottom left) the $E_{\text {T}}^{\text {miss}}$ distribution in , and (bottom right) the $m_{\ell \ell } ^{\mathrm {min}}$ distribution in . The last bin includes overflow. The ‘Others’ category contains backgrounds from single-top, $WW$, triboson, Higgs and rare top processes. The bottom panel shows the ratio of the observed data to the predicted yields. The hatched bands indicate the combined theoretical, experimental, and MC statistical uncertainties. The slope change in the bottom left $E_{\text {T}}^{\text {miss}}$ distribution illustrates the selection extension with $E_{\text {T}}^{\text {miss}}$ triggered events, which start contributing at $E_{\text {T}}^{\text {miss}} \gtrsim 200~\text {GeV}$

Fig. 8

Comparison of the observed data and expected SM background yields in the CRs and VRs of the off-shell $WZ$ selection. The SM prediction is taken from the background-only fit. The ‘Others’ category contains the single-top, $WW$, triboson, Higgs and rare top processes. The hatched band indicates the combined theoretical, experimental, and MC statistical uncertainties. 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

8.2 Background estimation

The selection criteria for the CRs and the VRs for $WZ$ estimation are summarised in Table 10. An on-shell Z boson ($m_{\ell \ell } \in [81,101]~\text {GeV}$) is required to ensure orthogonality to the , and an upper bound on $E_{\text {T}}^{\text {miss}}$ ensures orthogonality to the . A lower bound on $m_{\text {T}}$ is applied to suppress the $Z+\text {jets}$ background. The CRs are further split into two bins ( and ), based on the absence or presence of jets, to constrain $WZ$ events without or with hard ISR jets separately with individual normalisation factors.

Fig. 9

Breakdown of the total systematic uncertainties in the background prediction for the SRs of the off-shell $WZ$ selection

Table 12 Observed and expected yields after the background-only fit in the SRs for the on-shell $WZ$ selection. The normalisation factors of the $WZ$ sample are extracted separately for the , and regions, and are treated separately in the combined fit. The ‘Others’ category contains the single-top, $WW$, triboson, Higgs and rare top processes. Combined statistical and systematic uncertainties are presented

Table 13 Observed and expected yields after the background-only fit in the SRs for the $Wh$ selection. The normalisation factors of the $WZ$ sample are extracted separately for the , and regions, and are treated separately in the combined fit. The ‘Others’ category contains the single-top, $WW$, $t\bar{t} {+}X$ and rare top processes. Combined statistical and systematic uncertainties are presented

Three validation regions are defined in the region with $m_{\ell \ell } ^{\mathrm {min}},m_{\ell \ell } <75~\text {GeV}$, similar to . First, and are designed to validate the $WZ$ estimation in the phase space. A window in $m_\mathrm {T}$ around the Jacobian peak ($m_{\text {T}} \in [60,90]~\text {GeV}$) is selected to enhance $WZ$, as well as to ensure the orthogonality with respect to the SRs. Further kinematic selection criteria similar to those in are applied. Two additional variables are employed in the to suppress the signal contamination in the region. The W-boson mass, $m_{\mathrm {W}}^{\text {reco}\,\text {W}\text {Z}}$, is reconstructed assuming the $WZ$ topology and balanced longitudinal momenta of the W and Z bosons, and $\Delta R(\ell _{\mathrm {W}},E_{\text {T}}^{\text {miss}})$ is defined by where leptons are assigned according to the $m_{\ell \ell } ^{\mathrm {min}}$ approach, and $\ell _{\mathrm {W}}$ is the lepton associated with the W boson. Since $m_{\mathrm {W}}^{\text {reco}\,\text {W}\text {Z}}$ peaks around $m_{W}$ with a long tail to higher masses for $WZ$ background, while signals tend to have a flatter distribution, $m_{\mathrm {W}}^{\text {reco}\,\text {W}\text {Z}} > 75~\text {GeV}$ is found to effectively reduce signal contamination.

In the very low $m_{\ell \ell } ^{\mathrm {min}}$ region, is used to validate the $WZ$ estimation in the phase space. This region has the low-mass resonance veto applied and a lower bound on $|\mathbf{p }_{\mathrm {T}}^{\mathrm {lep}}|/E_{\text {T}}^{\text {miss}}$ to ensure orthogonality with the SRs. Other kinematic cuts are loosened relative to , or removed entirely, to increase the number of data events in the region. The $WZ$ purity is 85–90% in the CRs and 70–75% in the VRs. The contamination from the benchmark signals is negligible in the CRs and below 15% in the VRs.

The selection criteria are summarised in Table 10. At least one b-jet is required to maintain orthogonality with the SRs, $E_{\text {T}}^{\text {miss}} >50~\text {GeV}$ is required to suppress the $Z+\text {jets}$ contribution, and the low-mass resonance background veto is applied. The $t\bar{t}$ purity in this region is approximately 65%.

The $Z+\text {jets}$ background is estimated using the FF method as described in Sect. 6.2. The FF measurement region for the off-shell $WZ$ selection, , is summarised in Table 11. The Z-boson candidate is selected by requiring $|m_{\ell \ell }-m_{Z} |<15~\text {GeV}$, and $E_{\text {T}}^{\text {miss}} <40~\text {GeV}$ and $m_\mathrm {T} <30~\text {GeV}$ are applied to reject contamination from $WZ$. Additionally $m_{3\ell } >105~\text {GeV}$ is applied to suppress $Z\rightarrow 4\ell$. To increase the number of FNP lepton candidates at high $p_{\text {T}}$, 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 $p_{\text {T}} >30~\text {GeV}$, in which case $n_{\text {jets}}^{30~\text {GeV}} \le 1$ 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 $p_{\text {T}}$ and $E_{\text {T}}^{\text {miss}}$ in the event. Typical FF values are 0.2–0.4 (0.2–0.6) without the BDT applied, and 0.05–0.2 (0.07–0.2) when applying the BDT, for electrons (muons) in a $p_{\text {T}}$ range of 4.5–30 (3.0–30) $\text {GeV}$. The parameterisation in $E_{\text {T}}^{\text {miss}}$ is used to reflect the variation of FNP lepton source with $E_{\text {T}}^{\text {miss}}$, which is required in order to model the shape of fake $E_{\text {T}}^{\text {miss}}$ correctly. Typically the fraction of FNP leptons originating from heavy-flavour decays varies with $E_{\text {T}}^{\text {miss}}$, in the range 20–30% (60–70%) for electrons (muons), because of the neutrinos from the leptonic b-/c-decays.

The contribution of non-$Z+\text {jets}$ processes is subtracted using MC simulated samples. A small normalisation correction is applied to the $t\bar{t}$ 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 $\smash {\ell _{\mathrm {W}}}$ flavour and the b-jet multiplicity in the event. They are measured using the data events in a $t\bar{t}$-enriched control region, , and are found to be between 0.88 and 0.95. The selection requires there to be no SFOS lepton pair in the event, as well as $p_{\text {T}}^{\ell _{3}} >10~\text {GeV}$ and $E_{\text {T}}^{\text {miss}} >50~\text {GeV}$ to enhance the $t\bar{t}$ purity.

Two sources of uncertainty specific to the estimation in are considered in addition to those described in Sect. 6.3. The FF parameterisation uncertainty is evaluated from the effect of using a different $E_{\text {T}}^{\text {miss}}$ binning ($E_{\text {T}}^{\text {miss}}$ $<50$ $\text {GeV}$, 50% larger bin size), or a 3D parameterisation in lepton $p_{\text {T}}$, $E_{\text {T}}^{\text {miss}}$ and lepton $\eta$, additionally taking into account the dependency on lepton $\eta$. The impact on the $Z+\text {jets}$ background yields in the CRs is $\sim 5\%$, and 1–7% in the SRs/VRs. The uncertainty from disabling the muon-vs-jet 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 $p_{\text {T}} <30~\text {GeV}$. The variation in the estimated $Z+\text {jets}$ 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 $E_{\text {T}}^{\text {miss}}$  significance selection is inverted with respect to the SRs to ensure orthogonality. First, and are designed to validate the yields in and , respectively, while aims to cross-check the modelling of FNP leptons with $p_{\text {T}} <10~\text {GeV}$ specifically. The $Z+\text {jets}$ purity is in the VRs is 50–80%, while the contamination from signals is negligible.

Performing the background-only fit, $WZ$ normalisation factors of $1.06\pm 0.03$ () and $0.93\pm 0.03$ () are determined. Examples of kinematic distributions in the CRs and VRs, demonstrating good agreement, are presented in Fig. 7. Observed and expected yields for all CRs and VRs are summarised in Fig. 8.

The systematic uncertainties considered in the off-shell $WZ$ 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 fluctuate in bins with few events in the corresponding anti-ID sample, such as and . The modelling uncertainty is larger in the presence of ISR jets and at higher values of $E_{\text {T}}^{\text {miss}}$; the fluctuation in originates from the effect of the QCD scale uncertainty on the $WZ$ background.

Fig. 10

Comparison of the observed data and expected SM background yields in the SRs of the on-shell $WZ$ selection. The SM prediction is taken from the background-only fit. The ‘Others’ category contains the single-top, $WW$, triboson, Higgs and rare top processes. The hatched band indicates the combined theoretical, experimental, and MC statistical uncertainties. Distributions for wino/bino (+) ${\widetilde{\chi }}^{\pm }_{1}{\widetilde{\chi }}^{0}_{2}$ $\rightarrow$ $WZ$  signals are overlaid, with mass values given as  $\text {GeV}$. 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

Fig. 11

Comparison of the observed data and expected SM background yields in the SRs of the $Wh$ selection. The SM prediction is taken from the background-only fit. The ‘Others’ category contains the single-top, $WW$, $t\bar{t} {+}X$ and rare top processes. The hatched band indicates the combined theoretical, experimental, and MC statistical uncertainties. Distributions for wino/bino (+) ${\widetilde{\chi }}^{\pm }_{1}{\widetilde{\chi }}^{0}_{2}$ $\rightarrow$ $Wh$  signals are overlaid, with mass values given as  $\text {GeV}$. 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

Table 14 Observed and expected yields after the background-only fit in . The normalisation factors of the $WZ$ sample are extracted separately for 0j and nj, and are treated separately in the combined fit. The ‘Others’ category contains the single-top, $WW$, triboson, Higgs and rare top processes. Combined statistical and systematic uncertainties are presented

Table 15 Observed and expected yields after the background-only fit in . The normalisation factors of the $WZ$ sample are extracted separately for 0j and nj, and are treated separately in the combined fit. The ‘Others’ category contains the single-top, $WW$, triboson, Higgs and rare top processes. Combined statistical and systematic uncertainties are presented

Fig. 12

Comparison of the observed data and expected SM background yields in the SRs of the off-shell $WZ$ selection. The SM prediction is taken from the background-only fit. The ‘Others’ category contains the single-top, $WW$, triboson, Higgs and rare top processes. The hatched band indicates the combined theoretical, experimental, and MC statistical uncertainties. Distributions for wino/bino (+) ${\widetilde{\chi }}^{\pm }_{1}{\widetilde{\chi }}^{0}_{2}$ $\rightarrow$ $W\smash {^{*}}Z\smash {^{*}}$  signals are overlaid, with mass values given as  $\text {GeV}$. 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

9 Results

The observed data in the on-shell $WZ$, off-shell $WZ$, 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 and regions, and in Tables 14 and 15 and Fig. 12 for the . Post-fit distributions of key kinematic observables are shown for the and regions in Fig. 13 and for the regions in Fig. 14.

To illustrate the sensitivity to various ${\widetilde{\chi }}^{\pm }_{1}{\widetilde{\chi }}^{0}_{2}$ signals throughout the regions, representative signal MC predictions are overlaid on the figures. The sensitivity to $WZ$-mediated models, when the mass difference between the / and is large, is driven by the with large $m_\mathrm {T}$ and $E_{\text {T}}^{\text {miss}}$ values. On the other hand, when the mass splitting is close to the Z-boson mass, the sensitivity is dominated by the high $H_{\text {T}}$ region and moderate $m_\mathrm {T}$ and $E_{\text {T}}^{\text {miss}}$ bins of the $n_{\mathrm {jets}} = 0$ and low $H_{\text {T}}$ regions. For the $Wh$-mediated scenarios the sensitivity is driven by and regions, with contributing the most.

For the $WZ$-mediated models targeted with the , with mass differences between the and smaller than the Z-boson mass, the sensitivity to signals with different $\Delta m$ depends on the $m_{\ell \ell } ^{\mathrm {min}}$ range of the bins. The bins with larger (smaller) $m_{\ell \ell } ^{\mathrm {min}}$ values are sensitive to signals with larger (smaller) mass splittings; for the lowest mass-splitting signals, only 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 with a $2.3\sigma$ data excess, followed by a $2.1\sigma$ deficit in , a $2.0\sigma$ excess in , and a $2.0\sigma$ deficit in ; the significances are computed following the profile likelihood method in Ref. [169].

9.1 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 $WZ$, $Wh$ and off-shell $WZ$ selections. Multiple, sometimes overlapping, regions are defined to capture signatures with different unknown $m_{\ell \ell } ^{\mathrm {min}}$ shapes and jet multiplicities inclusively. Based on the best expected discovery sensitivity and using a number of signal points covering both the $WZ$- and $Wh$-mediated scenarios and different mass splittings, 12 inclusive SRs are formed by merging and regions, creating and , respectively. They are summarised in Table 16. Similarly, 17 inclusive SRs are formed by merging regions, creating ; their definitions are summarised in Table 17. For , contiguous jet-veto regions are merged with jet-inclusive regions, as the $m_{\ell \ell } ^{\mathrm {min}}$ shape of a signal is assumed to be insensitive to jet multiplicity. The and regions are kept separate, while the regions are considered separately for $m_{\ell \ell } ^{\mathrm {min}} < 20~\text {GeV}$, 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 $WZ$ and $Wh$ analysis selections (off-shell $WZ$ selection). The tables list the observed ($N_{\text {obs}}$) and expected ($N_{\text {exp}}$) yields in the inclusive SRs, the upper limits on the observed ($S^{95}_{\mathrm {obs}}$) and expected ($S^{95}_{\mathrm {exp}}$) number of BSM events, and the visible cross section ($\sigma ^{95}_{\mathrm {vis}}$) 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.

9.2 Constraints on $WZ$- 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 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 $WZ$, off-shell $WZ$, and compressed selections were explicitly designed to be orthogonal, allowing a statistical combination of the results. The on-shell and off-shell $WZ$ selections are orthogonal due to the $m_{\ell \ell }$ and $E_{\text {T}}^{\text {miss}}$ requirements, while the off-shell $WZ$ 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 $WZ$ selections, and the off-shell $WZ$ and compressed selections, in bands of the plane.

Four separate fits are performed to obtain constraints for the following simplified models:

• the wino/bino (+) $WZ$-mediated model combining the on-shell $WZ$, off-shell $WZ$, and compressed selections,

• the wino/bino (+) $Wh$-mediated model using the $Wh$ selection only,

• the wino/bino (−) $WZ$-mediated model combining the off-shell $WZ$ and compressed selections,

• the higgsino $WZ$-mediated model combining the off-shell $WZ$ and compressed selections.

For the $WZ$-mediated model in the wino/bino (+) scenario, only the are sensitive for mass splittings $\Delta m$ above $100~\text {GeV}$. Conversely, the dominate the intermediate mass-splitting region, with sensitivity in the $\Delta m = [5,100]~\text {GeV}$ range. In the most compressed region, the SR-E are important, driving the result for $\Delta m$ below $10~\text {GeV}$ and adding sensitivity up to $\Delta m = 50~\text {GeV}$. Given these contributions, the $\Delta m$ 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 , the only, the and , and the only. In the wino/bino (−) an

d higgsino scenarios, the on-shell $WZ$ selection is not considered, and only three bands are defined for the combination. The exact $\Delta m$ ranges used are illustrated for the different scenarios in Fig. 15.

Fig. 13

Kinematic distributions after the background-only fit showing the data and the post-fit expected background, in SRs of the on-shell $WZ$ and $Wh$ selections. The figure shows (top left) the ${\Delta }R_{\text {OS,near}}$ distribution in , (top right) the 3rd leading lepton’s $p_{\text {T}}$ in , and the (bottom left) $E_{\text {T}}^{\text {miss}}$ and (bottom right) $m_{\text {T}}$ distributions in (with all SR-i bins of summed). The SR selections are applied for each distribution, except for the variable shown, for which the selection is indicated by a black arrow. The last bin includes overflow. The ‘Others’ category contains backgrounds from single-top, $WW$, triboson, Higgs and rare top processes, except in the top panels, where triboson and Higgs production contributions are shown separately, and $t\bar{t} {+}X$ is merged into Others. Distributions for wino/bino (+) ${\widetilde{\chi }}^{\pm }_{1}{\widetilde{\chi }}^{0}_{2}$ $\rightarrow$ $WZ$/$Wh$  signals are overlaid, with mass values given as  $\text {GeV}$. The bottom panel shows the ratio of the observed data to the predicted yields. Ratio values outside the graph range are indicated by a red arrow. The hatched bands indicate the combined theoretical, experimental, and MC statistical uncertainties

Fig. 14

Kinematic distributions after the background-only fit showing the data and the post-fit expected background, in SRs of the off-shell $WZ$ selection. The figure shows the $m_{\text {T}}^{\text {mllmin}}$ distribution in (top left) , (top right) and (bottom left) , and the $|\mathbf{p }_{\mathrm {T}}^{\mathrm {lep}}|/E_{\text {T}}^{\text {miss}}$ distribution in (bottom right) . The contributing $m_{\ell \ell } ^{\mathrm {min}}$ mass bins within each category are summed. The SR selections are applied for each distribution, except for the variable shown, for which the selection is indicated by a black arrow. The last bin includes overflow. The ‘Others’ category contains backgrounds from single-top, $WW$, triboson, Higgs and rare top processes. Distributions for wino/bino (+) ${\widetilde{\chi }}^{\pm }_{1}{\widetilde{\chi }}^{0}_{2}$ $\rightarrow$ $WZ$  signals are overlaid, with mass values given as  $\text {GeV}$. The bottom panel shows the ratio of the observed data to the predicted yields. The hatched bands indicate the combined theoretical, experimental, and MC statistical uncertainties

Expected and observed exclusion contours are reported as a function of the ${\widetilde{\chi }}^{0}_{1}$ and ${\widetilde{\chi }}^{\pm }_{1}$/${\widetilde{\chi }}^{0}_{2}$ masses, and shown in Fig. 16 ($WZ$-mediated model) and Fig. 17 ($Wh$-mediated model). The combined results are shown together with the individual contributions. For each mass point, a $\hbox {CL}_{\text {s}}$ 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 $WZ$-mediated model, the results are obtained by statistically combining the , and SR-E contributions, following the prescription outlined above. For the $Wh$-mediated model, the results are taken from a simultaneous fit of the 19 bins of .

For the wino/bino (+) $WZ$-mediated model, shown in Fig.  16 (top panels), observed (expected) lower limits for equal-mass ${\widetilde{\chi }}^{\pm }_{1}$/${\widetilde{\chi }}^{0}_{2}$ are set at 640 (660) $\text {GeV}$ for massless ${\widetilde{\chi }}^{0}_{1}$, and up to 300 (300) $\text {GeV}$ for scenarios with mass splittings $\Delta m$ near $m_{Z}$, driven by the on-shell $WZ$ selection. The exclusion for the scenarios with $\Delta m < m_Z$ is driven by the off-shell $WZ$ selection. For ${\widetilde{\chi }}^{\pm }_{1}$ and ${\widetilde{\chi }}^{0}_{2}$ decaying via off-shell $WZ$ bosons, observed and expected limits are set at values up to 300 $\text {GeV}$ for $\Delta m > 35~\text {GeV}$, and up to 210–300 $\text {GeV}$ for $\Delta m = 20$–35 $\text {GeV}$. Below $\Delta m = 15~\text {GeV}$ the observed and expected limits are extended by the combination with the compressed selection, up to 240 $\text {GeV}$ for $\Delta m = 10~\text {GeV}$, and down to as low as $\Delta m = 2~\text {GeV}$ for a ${\widetilde{\chi }}^{\pm }_{1}$/${\widetilde{\chi }}^{0}_{2}$ mass of 100 $\text {GeV}$. Furthermore, constraints are calculated in the bino–wino co-annihilation dark-matter scenario by determining the area in the two-dimensional mass plane that yields a thermal dark-matter relic density equal to the observed value [176]. Figure 16 (top right) shows this area in blue, with the over- and under-abundant regions marked above and below; ${\widetilde{\chi }}^{\pm }_{1}$/${\widetilde{\chi }}^{0}_{2}$ (${\widetilde{\chi }}^{0}_{1}$) masses are excluded in this dark-matter scenario up to 210 (195) $\text {GeV}$.

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 $\text {TeV}$ dataset [17] (shown as a light grey shaded area in Fig. 16, top panels), due to a combination of increased production cross section at the increased collision centre-of-mass energy, larger data sample, and improved analysis techniques.

Table 16 Summary of the selection criteria for the inclusive SRs in the on-shell $WZ$ and $Wh$ selections

Table 17 Summary of the selection criteria for the inclusive SRs in the off-shell $WZ$ selection

Table 18 Observed ($N_{\text {obs}}$) yields after the discovery fit and expected ($N_{\text {exp}}$) after the background-only fit, for the inclusive SRs of the on-shell $WZ$ and $Wh$ selections. The third and fourth columns list the 95% CL upper limits on the visible cross section ($\sigma _{\text {vis}}^{95}$) and on the number of signal events ($S_\text {obs}^{95}$). The fifth column ($S_\text {exp}^{95}$) shows the 95% CL upper limit on the number of signal events, given the expected number (and $\pm 1\sigma$ excursions of the expectation) of background events. The last two columns indicate the $\hbox {CL}_{\text {b}}$ value, i.e. the confidence level observed for the background-only hypothesis, and the discovery p-value ($p(s = 0)$). If the observed yield is below the expected yield, the p value is capped at 0.5

Table 19 Observed ($N_{\text {obs}}$) yields after the discovery fit and expected ($N_{\text {exp}}$) after the background-only fit, for the inclusive SRs of the off-shell $WZ$ selection. The third and fourth columns list the 95% CL upper limits on the visible cross section ($\sigma _{\text {vis}}^{95}$) and on the number of signal events ($S_\text {obs}^{95}$). The fifth column ($S_\text {exp}^{95}$) shows the 95% CL upper limit on the number of signal events, given the expected number (and $\pm 1\sigma$ excursions of the expectation) of background events. The last two columns indicate the $\hbox {CL}_{\text {b}}$ value, i.e. the confidence level observed for the background-only hypothesis, and the discovery p value (p$(s = 0)$). If the observed yield is below the expected yield, the p-value is capped at 0.5

Expected and observed exclusion contours are also derived for the $WZ$-mediated model in the wino/bino (−) and higgsino scenarios, shown in Fig. 16 (bottom panels) as a function of the ${\widetilde{\chi }}^{0}_{1}$ and ${\widetilde{\chi }}^{0}_{2}$ masses. The results are obtained by statistically combining the and SR-E contributions, following the prescription outlined above.

In the wino/bino (−) scenario, shown in Fig. 16 (bottom left), observed (expected) lower limits for equal-mass ${\widetilde{\chi }}^{\pm }_{1}$/${\widetilde{\chi }}^{0}_{2}$ are set at values up to 310 (300) $\text {GeV}$ for mass splittings $\Delta m$ around 80 $\text {GeV}$, and up to 250 (250) $\text {GeV}$ for $\Delta m$ around 40 $\text {GeV}$. For $\Delta m$ of 10–20 $\text {GeV}$, the impact of the combination of the off-shell $WZ$ and compressed results is the largest, and raises the expected limit to ${\widetilde{\chi }}^{\pm }_{1}$/${\widetilde{\chi }}^{0}_{2}$ masses of 270 $\text {GeV}$, with the observed limit still showing a mild deficit similar to that visible in the compressed contribution. At a ${\widetilde{\chi }}^{\pm }_{1}$/${\widetilde{\chi }}^{0}_{2}$ mass of 100 $\text {GeV}$, the observed (expected) exclusion extends down to $\Delta m = 1~(1.5)$ $\text {GeV}$.

In the higgsino scenario, shown in Fig. 16 (bottom right), with the ${\widetilde{\chi }}^{\pm }_{1}$ mass between that of the ${\widetilde{\chi }}^{0}_{1}$ and ${\widetilde{\chi }}^{0}_{2}$, limits are set for mass splittings $\Delta m$ up to 60 $\text {GeV}$. For $\Delta m$ between 30 and 60 $\text {GeV}$, observed (expected) limits extend to around 150–210 (160–215) $\text {GeV}$. The impact of the combination of the off-shell $WZ$ and compressed results is largest in the $\Delta m = 15$–30 $\text {GeV}$ range, improving on the individual result by up to 15 $\text {GeV}$. Below $\Delta m = 20~\text {GeV}$, the result is dominated by the compressed contribution, and limits extend down to $\Delta m = 2~\text {GeV}$.

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 off-shell $WZ$ selection in three-lepton final states make full use of the larger data sample and target a novel phase space in the intermediately compressed region. The new results extend the exclusion by up to 100 $\text {GeV}$ in ${\widetilde{\chi }}^{0}_{2}$ mass.

For the wino/bino (+) $Wh$-mediated model, observed (expected) lower limits for equal-mass ${\widetilde{\chi }}^{\pm }_{1}$/${\widetilde{\chi }}^{0}_{2}$ are set at values up to 190 (240) $\text {GeV}$ for ${\widetilde{\chi }}^{0}_{1}$ masses below 20 $\text {GeV}$, as shown in Fig. 17. The observed exclusion is weaker than the expected exclusion, which is explained by the mild excess found in ; the limits are, however, compatible within $2\sigma$. The obtained observed (expected) limits show an improvement of up to 40 (80) $\text {GeV}$ compared to the previous Run 1, 8 $\text {TeV}$, ATLAS search [17].

Fig. 15

Illustration of the selections considered for the combined result for each scenario, dependent on $\Delta m$

Fig. 16

Exclusion limits obtained for the $WZ$-mediated models in the (top left and right) wino/bino (+) scenario, (bottom left) the wino/bino (−) scenario, and (bottom right) the higgsino scenario. The expected 95% CL sensitivity (dashed black line) is shown with $\pm 1\sigma _{\text {exp}}$ (yellow band) from experimental systematic uncertainties and statistical uncertainties in the data yields, and the observed limit (red solid line) is shown with $\pm 1\sigma _{\text {theory}}$ (dotted red lines) from signal cross-section uncertainties. The statistical combination of the on-shell $WZ$, off-shell $WZ$, and compressed results is shown as the main contour, while the observed (expected) limits for each individual selection are overlaid in green, blue, and orange solid (dashed) lines, respectively. The exclusion is shown projected (top left) onto the $m({\widetilde{\chi }}^{\pm }_{1},\,{\widetilde{\chi }}^{0}_{2})$ vs $m({\widetilde{\chi }}^{0}_{1})$ plane or (top right and bottom) onto the $m({\widetilde{\chi }}^{0}_{2})$ vs $\Delta m$ plane. The light grey area denotes (top) the constraints obtained by the previous equivalent analysis in ATLAS using the 8 $\text {TeV}$ 20.3 fb$^{-1}$ dataset [17], and (bottom right) the LEP lower ${\widetilde{\chi }}^{\pm }_{1}$ mass limit [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, $\Omega h^2=0.1186\pm 0.0020$ [176], when the mass parameters of all the decoupled SUSY partners are set to $5~\text {TeV}$ and $\tan \beta$ 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

Fig. 17

Exclusion limits obtained for the $Wh$-mediated model in the wino/bino (+) scenario, calculated using the $Wh$ SRs and projected onto the $m({\widetilde{\chi }}^{\pm }_{1},\,{\widetilde{\chi }}^{0}_{2})$ vs $m({\widetilde{\chi }}^{0}_{1})$ plane. The expected 95% CL sensitivity (dashed black line) is shown with $\pm 1\sigma _{\text {exp}}$ (yellow band) from experimental systematic uncertainties and statistical uncertainties in the data yields, and the observed limit (red solid line) is shown with $\pm 1\sigma _{\text {theory}}$ (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 $\text {TeV}$ 20.3 fb$^{-1}$ dataset [17]

Fig. 18

Comparison of the observed data and expected SM background yields in the CRs and VRs of the RJR selection. The SM prediction is taken from the background-only fit. The ‘FNP leptons’ category contains backgrounds from $t\bar{t}$, tW, WW and $Z+\text {jets}$  processes. The ‘Others’ category contains backgrounds from Higgs and rare top processes. The hatched band indicates the combined theoretical, experimental, and MC statistical uncertainties. 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

10 Recursive Jigsaw Reconstruction selection and results

To follow up on an earlier ATLAS search performed using the Recursive Jigsaw Reconstruction  (RJR) technique with the 2015–2016, 36 fb$^{-1}$ dataset [15], the search in this paper includes two signal regions in which the original search observed excesses of three-lepton events. The original search in the two regions is repeated following the same methods, updated to use the full Run 2 dataset. The region targets low-mass wino/bino (+) ${\widetilde{\chi }}^{\pm }_{1}{\widetilde{\chi }}^{0}_{2}$ production, while the region targets wino/bino (+) ${\widetilde{\chi }}^{\pm }_{1}{\widetilde{\chi }}^{0}_{2}$ production in association with ISR and mass differences $\Delta m$ near the Z-boson mass. The excesses in and observed in the 36 fb$^{-1}$ result correspond to local significances of 2.1$\sigma$ and 3.0$\sigma$, respectively.

Fig. 19

Breakdown of the total systematic uncertainties in the background prediction for the SRs of the RJR selection

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 ${\widetilde{\chi }}^{\pm }_{1}$ and ${\widetilde{\chi }}^{0}_{2}$ particles. For this search, both the standard decay tree applied to a three-lepton final state (representing the decay of pair-produced sparticles into a final state with two invisible objects and three leptons, in the laboratory frame) and the ISR decay tree (representing the decay of an intermediate sparticle into a visible and an invisible component, recoiling from ISR activity, 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.

Table 20 Observed and expected yields after the background-only fit in the SRs for the RJR selection. The ‘FNP leptons’ category contains backgrounds from $t\bar{t}$, tW, WW and $Z+\text {jets}$  processes. The ‘Others’ category contains backgrounds from Higgs and rare top processes. Combined statistical and systematic uncertainties are presented

Fig. 20

Example of kinematic distributions after the background-only fit, showing the data and the post-fit expected background, in regions of the RJR selection. The figure shows the (top left) $H^{\text {PP}}_{\text {3,1}}$  and (top right) $p_{\text {T}}^{\ell _{1}}$   distributions in , and the (bottom left) $p^{\text {CM}}_{\text {T,ISR}}$  and (bottom right) $R_{\text {ISR}}$  distributions in . The last bin includes overflow. The ‘FNP leptons’ category contains backgrounds from $t\bar{t}$, tW, WW and $Z+\text {jets}$  processes. The ‘Others’ category contains backgrounds from Higgs and rare top processes. Distributions for wino/bino (+) ${\widetilde{\chi }}^{\pm }_{1}{\widetilde{\chi }}^{0}_{2}$ $\rightarrow$ $WZ$  signals are overlaid, with mass values given as  $\text {GeV}$. The bottom panel shows the ratio of the observed data to the predicted yields. The hatched bands indicate the combined theoretical, experimental, and MC statistical uncertainties

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 $WZ$, off-shell $WZ$, and $Wh$ selections in Sect. 6. For the search in (), the SM diboson background is taken from MC simulation samples and normalised in a dedicated control region () and validated in a validation region (). The selection criteria for each of the regions follow the original search [15], except for an additional jet-veto ($n_{\text {jets}}=0$) in and which guarantees the orthogonality between the low-mass and ISR regions. The FNP lepton background component, including $t\bar{t}$, tW, WW and $Z+\text {jets}$ 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 estimation, 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 () and 0.92±0.05 () 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 and 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, $p_{\text {T}}^{\ell _{1}}$, and the scalar momentum sum, $H^{\text {PP}}_{\text {3,1}}$, 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, $p^{\text {CM}}_{\text {T}}$, and the fraction of the total momentum of the sparticle system carried by the invisible system, $R_{\text {ISR}}$, 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 and 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 ($S^{95}_{\mathrm {obs}}$) and expected ($S^{95}_{\mathrm {exp}}$) number of BSM events in the inclusive SRs, and the visible cross section ($\sigma ^{95}_{\mathrm {vis}}$) 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.

11 Conclusion

Results of a search for chargino–neutralino pair production decaying via $WZ$, $W^{*}Z^{*}$ or $Wh$ into three-lepton final states are presented. A dataset of $\sqrt{s}=13~\text {TeV}$ proton–proton collisions corresponding to an integrated luminosity of 139 fb$^{-1}$, collected by the ATLAS experiment at the CERN LHC, is used. Events with three light-flavour charged leptons and missing transverse momentum are preselected, and three selections are developed with a signal region strategy optimised for chargino–neutralino signals decaying via $WZ$, $W^{*}Z^{*}$ and $Wh$, respectively. A fourth selection targeting the chargino–neutralino signals decaying via $WZ$ 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.

Table 21 Results of the discovery fit for the SRs of the RJR selection, calculated using pseudo-experiments. The first and second column list the 95% CL upper limits on the visible cross section ($\sigma _{\text {vis}}^{95}$) and on the number of signal events ($S_\text {obs}^{95}$). The third column ($S_\text {exp}^{95}$) shows the 95% CL upper limit on the number of signal events, given the expected number (and $\pm 1\sigma$ excursions of the expectation) of background events. The last two columns indicate the $\hbox {CL}_{\text {b}}$ value, i.e. the confidence level observed for the background-only hypothesis, and the discovery p value ($p(s = 0)$). If the observed yield is below the expected yield, the p-value is capped at 0.5

Assuming a simplified model with wino production decaying to a bino LSP, exclusion limits at 95% confidence level are placed on the minimum ${\widetilde{\chi }}^{\pm }_{1}$/${\widetilde{\chi }}^{0}_{2}$ mass, extending the reach of previous searches [14,15,16,17,18, 21]. Limits are set at 640 $\text {GeV}$ for the $WZ$-mediated model signals in the limit of massless ${\widetilde{\chi }}^{0}_{1}$, improving by about 140 $\text {GeV}$; and at 300 $\text {GeV}$ for mass-splittings between ${\widetilde{\chi }}^{\pm }_{1}$/${\widetilde{\chi }}^{0}_{2}$ and ${\widetilde{\chi }}^{0}_{1}$ close to $m_{Z}$, improving by about 100 $\text {GeV}$. In the case of a mass splitting of 5–90 $\text {GeV}$, ${\widetilde{\chi }}^{\pm }_{1}$/${\widetilde{\chi }}^{0}_{2}$ masses up to 200–300 $\text {GeV}$ for the $WZ$-mediated model are excluded. The limit extends down to a smallest mass splitting of 2 $\text {GeV}$ for a ${\widetilde{\chi }}^{\pm }_{1}$ mass of 100 $\text {GeV}$. The dependency on a model parameter – the sign of the product – is also tested, and comparable limits are found for the two scenarios. For the $Wh$-mediated model signals, the limit on the minimum ${\widetilde{\chi }}^{\pm }_{1}$/${\widetilde{\chi }}^{0}_{2}$ mass is set at 190 $\text {GeV}$, for ${\widetilde{\chi }}^{0}_{1}$ masses below 20 $\text {GeV}$.

Limits are also set for simplified models with a higgsino LSP triplet, for the first time including results from three-lepton 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 ${\widetilde{\chi }}^{0}_{2}$ mass up to 210 $\text {GeV}$ for $WZ$-mediated model signals with a mass splitting of 2–60 $\text {GeV}$. In these models, searches in the three-lepton final state enhance the sensitivity in the experimentally challenging region with mass splitting greater than 30 $\text {GeV}$.