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 $50~\text {GeV}< \Delta m < m_{Z}$ [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 [\(n_{\mathrm {jets}} < 3\)). 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. [\(p_{\text {T}} >30~\text {GeV}\), 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. [\(WZ\), $Wh$ and off-shell $WZ$ selections. Multiple, sometimes overlap**, 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 $\text {GeV}$.