Search for pair production of squarks or gluinos decaying via sleptons or weak bosons in final states with two same-sign or three leptons with the ATLAS detector

A search for pair production of squarks or gluinos decaying via sleptons or weak bosons is reported. The search targets a final state with exactly two leptons with same-sign electric charge or at least three leptons without any charge requirement. The analysed data set corresponds to an integrated luminosity of 139 fb$^{-1}$ of proton$-$proton collisions collected at a centre-of-mass energy of 13 TeV with the ATLAS detector at the LHC. Multiple signal regions are defined, targeting several SUSY simplified models yielding the desired final states. A single control region is used to constrain the normalisation of the $WZ$+jets background. No significant excess of events over the Standard Model expectation is observed. The results are interpreted in the context of several supersymmetric models featuring R-parity conservation or R-parity violation, yielding exclusion limits surpassing those from previous searches. In models considering gluino (squark) pair production, gluino (squark) masses up to 2.2 (1.7) TeV are excluded at 95% confidence level.


Introduction
The ATLAS experiment [1] at the Large Hadron Collider (LHC) [2] probes the electroweak sector of the Standard Model (SM) at the TeV scale. The presence of prompt electrons or muons (collectively referred to as leptons) in reconstructed events provides one of the main experimental signatures to isolate processes mediated by electroweak, scalar or exotic couplings, from the large QCD multĳet background produced in proton-proton ( ) collisions. The production of pairs of leptons with the same electric charge (also referred to as 'same-sign leptons') is particularly rare in the SM, with an inclusive cross section of about 1 pb [3] for centre-of-mass energies around 13 TeV. In contrast, it may occur frequently in scenarios with physics beyond the SM (BSM physics) [4][5][6], and therefore searches for anomalous production of same-sign leptons have been integral parts of the LHC and Tevatron experimental programmes [7][8][9].
Extensions of the SM introducing invariance under supersymmetric transformations [10][11][12][13][14][15] (SUSY) provide many such possibilities. Even minimal realisations such as the Minimal Supersymmetric Standard Model (MSSM) [16,17] contain SUSY partners for all SM fields as well as members of an extended Higgs sector [18] that may decay in complex cascades involving leptons. In the MSSM, SUSY transformations relate each of the Weyl components of a fundamental SM chiral fermion to a new scalar field,˜L or˜R,

Signal models
This analysis considers experimental signatures arising from the production of either gluino pairs or squark-antisquark pairs˜˜ * (with = , , , ). Several gluino and squark decay modes are investigated, and these are summarised in Figure 1: in most cases the first step of the decay chain is to a non-stable neutralino or chargino˜and SM quark(s) of first or second generation. Various˜decay modes may lead to final states featuring two same-sign leptons or three leptons. Amongst those possibilities, this analysis searches especially for the following sources of electrons and muons: •˜decays into SM gauge bosons and˜0 1 LSPs; although the direct decays˜± 1 → ±˜0 1 and˜0 2 →˜0 1 are most efficiently probed with other experimental signatures [32-34], more favourable branching ratios to same-sign leptons are found for cascade decays such as˜± 1 →˜0 2 ± →˜0 1 ± , as illustrated in Figures 1(a) and 1(b), which are thus the focus here.
•˜→˜¯→¯decays via a non-zero RPV coupling , shown in Figure 1(f). The experimental sensitivity to such processes is assessed in simplified models [35][36][37] where only the superpartners directly involved in the process of interest are considered, alternative production and decay modes are ignored, and masses and mixings of superpartners are either varied freely or fixed to chosen values.
For the cascade decays of charginos into pairs of SM bosons (Figures 1(a) and 1(b)), the gluino (or squark) and˜0 1 LSP masses are varied independently to generate different scenarios. The masses of intermediate superpartners are then set halfway in between, i.e. (˜± 1 ) = ( (˜/˜) + (˜0 1 ))/2 and (˜0 2 ) = ( (˜± 1 ) + (˜0 1 ))/2. The decay of˜± 1 and˜0 2 to leptons takes place through real or virtual / bosons, depending on the mass difference from the LSP. Quarks produced in the first step of the gluino decay chain are assumed to be , , or with equal probability. Due to the small Yukawa couplings for the first two quark generations, the decays˜→˜± 1 and˜→˜± 1 are only relevant for a wino-like chargino [38], so only production of left-handed squarks is considered. In minimal models with a neutralino LSP [38], such chargino cascade decays and a non-degenerate mass hierarchy may occur when neutralinos are non-trivial admixtures, but these would compete with several other, generally more favourable, decay modes [19]. In gauge-mediated [39-41] SUSY breaking models, however, this can be a more natural mode in the alternative form˜± 1 → ±˜0 1 → ±˜w here the gravitino˜is the LSP. Previous ATLAS searches using various signatures [25,27,42,43] probed gluino masses up to 2 TeV and squark masses up to 630 GeV with these decay modes.
Similarly, for the decays of charginos and neutralinos into sleptons (Figures 1(c) and 1(d)), gluino (or squark) and˜0 1 LSP masses are varied independently, while the masses of intermediate superpartners are set to (˜± 1 /˜0 2 ) = ( (˜/˜) + (˜0 1 ))/2 and (l/˜) = ( (˜± 1 /˜0 2 ) + (˜0 1 ))/2. Such scenarios arise in the mass spectra of models where sleptons are light, and thus were searched for in early LHC analyses [44][45][46]. Gluinos are chosen to decay only into˜0 2 while squarks are equally likely to decay into either˜± 1 or˜0 2 . In both cases, the neutralino or chargino subsequently decays in an equiprobable way into any of the six SM leptons, together with the left-handed slepton appropriate for charge and lepton number conservation. The latter decays exclusively into its SM partner and the LSP. Previous ATLAS searches [25,27,42,43] probed gluino masses up to 2.2 TeV and squark masses up to 850 GeV. The intermediate charginos or sleptons are also constrained by previous analyses [47][48][49][50], but for these particles the present analysis can probe masses higher than those excluded by the direct-production searches, thanks to their higher production rate in the cascade decays of gluinos or squarks.
Finally, event kinematics in the case of neutralino decays via RPV couplings (Figure 1(e)) are completely determined by the gluino and˜0 1 LSP masses, which again are varied independently to generate different scenarios. Without choosing explicit values, it is assumed that the relevant couplings can be large enough to allow prompt˜0 1 decay into ℓ −¯o r ℓ¯( ℓ = , ), or their charge conjugates, with equal probabilities, while evading low-energy experimental bounds [51]. The˜0 1 natural width also depends on the mass of virtual squarks mediating the decay and is set to 100 MeV. For such a scenario, current experimental sensitivity [52] excludes gluino masses up to 2.2 TeV.
An additional scenario exemplified in Figure 1(f) is considered, in which pair-produced gluinos decay into the lightest top squark (and a SM top quark) which itself decays via a suitable RPV coupling into a pair of quarks, leading in half of the cases to a final state with two same-sign top quarks and up to four jets. Such scenarios were highlighted in Ref. [53] especially, and searches have been performed in thē¯[ 25, 26, 52] and¯¯ [54] final states. In the present analysis, only the former case is addressed, as it yields a clearer experimental signature; for this scenario, Ref.
[52] excluded gluino masses up to 1.8 TeV for top squark masses around 1 TeV. A set of benchmark models is generated by varying the gluino and top squark masses independently, the latter being bounded from below by existing constraints on˜˜ * production [55]. The top squarks are assumed to decay promptly. A new set of search regions allows the sensitivity to be extended beyond that reached in Ref.
[25] with the same data.

Data and samples of simulated events
The results presented here are obtained by analysing proton-proton collision data collected during Run 2 of the LHC at a centre-of-mass energy of 13 TeV. The number of simultaneous inelastic interactions averages to 33.7 for the entire data set [56], but exceeds 70 in a small fraction of the data. Events recorded when parts of the detector were not functional or reserved for detector commissioning or calibration purposes are subsequently ignored, leaving 95.6% of the recorded data [56] available for analysis. The integrated luminosity of this data set amounts to 139 fb −1 , with an uncertainty of 1.7%. The latter is obtained [57] using the LUCID-2 detector [58] for the primary luminosity measurements.
Large samples of simulated events are also employed, mainly to predict contributions from SM processes with prompt 2 leptons to the search regions used in this analysis, as well as those from hypothetical SUSY signal processes. Other usages include validating assumptions employed in data-based background estimation methods and, more generally, assessing systematic uncertainties. Those samples were obtained by simulating individual proton-proton collisions for hard-interaction processes of interest with the different combinations of MC event generators described below. The events were then processed through a detailed simulation of the ATLAS detector [59] based on G 4 [60]. In some cases, notably for BSM signal samples, a faster simulation which relies on a parameterisation of the calorimeter response [61] was used instead. At this stage, additional minimum-bias interactions generated by P 8.186 were simulated separately and overlaid on each simulated hard-interaction event to account for pile-up effects. The response of the detector and its electronic readout chain is then emulated [59], also accounting for effects from interactions in the previous and following bunch-crossings. Reconstructed events are reweighted to reproduce the measured distributions of the number of simultaneous interactions in different data-taking periods and the measured effects of various sources of reconstruction inefficiency, for example in the application of electron identification algorithms. Specific kinematic variables, such as lepton momenta, are smeared to reproduce the measured detector resolution.
The MC generators used to simulate the main SM processes of interest are summarised in Table 1, together with the selected parton shower algorithms, the sets of tuned parameters (tunes), and the sets of parton distribution functions (PDF). When P was used, the decays of bottom and charm hadrons were simulated with the E G program [62]. Diboson processes [63] include all resonant and non-resonant → 3ℓ /4ℓ/ℓ ± ℓ ± processes of order 4 in the fine-structure constant, including Higgs boson contributions, as well as the vector-boson scattering/fusion processes at order 6 . Triboson processes similarly include all the relevant resonant and non-resonant processes with up to six charged leptons in the final state at order 6 . The associated production of¯and an on-shell boson includes a complementary sample generated at leading order (LO) in QCD with the matrix elements of order 3 . Associated production of¯and a pair of same-flavour opposite-sign (SFOS) leptons was generated for dilepton invariant masses as low as 1 GeV. Other processes not identified individually in the table but included in the background estimates comprise associated production of¯and two vector or Higgs bosons, associated production of single top quarks with one or two vector or Higgs bosons, and production of three top quarks. Fast detector simulation was employed for the 4 , , ,¯,¯,¯processes.
The SUSY signal samples were generated with MG5_ MC@NLO 2.6.2 [3] and the NNPDF2.3 PDF, except for the˜→¯˜0 1 and˜→¯¯samples, for which the NNPDF3.0 PDF was used. The pair production of gluinos and squarks was simulated at LO, complemented by matrix elements for up to two extra parton emissions. Superpartners not involved in the model of interest were decoupled by being assigned unreachable masses. The decays of gluinos or squarks were factored out of the hard interaction and simulated with P 8.235 [83], which was also used for the subsequent stages of the event generation with the A14 tune [73]. The matching between matrix elements and parton showers followed the CKKW-L prescription [90], with a matching scale set to one quarter of the gluino or squark mass. For all models, the fast simulation was used to process the generated events. Signal cross sections are calculated to approximate next-to-next-to-leading order in the strong coupling constant, adding the resummation of soft gluon emission at next-to-next-to-leading-logarithm accuracy (approximate NNLO+NNLL) [91][92][93][94][95][96][97][98]. The nominal cross section and its uncertainty are derived using the PDF4LHC15_mc PDF set, following the recommendations of Ref. [99].

Object selection
Charged-particle tracks with | | < 2.5 are reconstructed [100][101][102] in the ID and combined to form primary vertex candidates [103][104][105]. The vertex with the largest track T 2 formed by at least two tracks is taken to be the primary vertex and position of the hard-scattering interaction. The transverse and longitudinal impact parameters of the tracks, denoted by 0 and 0 respectively, are measured at their point of closest approach to the beam line [106]. Requirements are placed on the ratio | sig 0 | of 0 to its estimated uncertainty ( 0 ) and on the value of | 0 sin( )| with respect to the primary vertex are stated below.
Jets with | | < 4.5 are reconstructed with the F J implementation [107] of the anti-algorithm [108], with radius parameter = 0.4, out of particle-flow objects [109,110]  [115] are applied, and pile-up muons are rejected by requiring | 0 sin( )| < 0.5 mm. The candidates satisfying these requirements are referred to as baseline muons. Around 0.1% of events contain a muon from a cosmic-ray shower or with poor expected momentum resolution, and are rejected. Prompt muons are further distinguished from background sources with a requirement | sig 0 | < 3 and an isolation criterion [115] consisting of an upper bound of 6%, relative to the muon T , on the summed | T | of suitable ID tracks 3 within Δ < min(0.3, 10 GeV/ T ( )). They are referred to as signal muons.
Electrons with | | < 2.47 and T > 10 GeV are used. They are reconstructed [117] from clustered energy deposits in the electromagnetic calorimeter matched to an ID track re-fitted to account for bremsstrahlung losses. Their momentum is determined by a calibration procedure [117,118] based on boosted decision trees (BDTs). Electrons within the calorimeter's barrel-to-endcap transition region 1.37 < | | < 1.52 are not considered. Electrons from most background sources are rejected with a likelihood discriminant [117] built from information about the development of the electron shower in the calorimeter, its compatibility with the matched track, and particle identification in the TRT detector. Electrons with | 0 sin( )| > 0.5 mm are rejected. The candidates satisfying these requirements are referred to as baseline electrons. Non-prompt or fake electrons are further suppressed by keeping only electrons within | | < 2.0, imposing the tighter 'MediumLH' identification criteria [117], a combination of track-based and calorimeter-based isolation criteria [117], and requiring | sig 0 | < 5. The first isolation criterion is similar to the one used for muons, while the other consists of an upper bound of 6%, relative to the electron T , on the sum of other calorimeter energy deposits within Δ < 0.2. Electrons very likely to have a wrongly assigned charge are identified and then rejected using the ECIDS discriminant [117], a BDT which is based on the properties of the electron track and which accepts 98% of simulated → decay electrons while rejecting 90% of those with the wrong charge. The remaining candidates are referred to as signal electrons.
An overlap removal procedure is applied to baseline lepton candidates and jets to avoid treating the same detector signals as multiple objects. Jets close to electrons (Δ < 0.2) are removed, unless they are classified as -jets with T < 100 GeV, and so are jets close to muons (Δ < 0.2 or sharing an ID track) if they have less than three associated ID tracks. Leptons close to remaining jets (Δ < min(0.4, 0.1 + 9.6 GeV/ T (ℓ))) are removed, as they are likely to be fake or non-prompt; the Δ bound is smaller at high T to retain leptons from boosted top quark or˜0 1 decays. Electrons close to muons (Δ < 0.01) or to higher-T electrons (Δ < 0.05) are removed. Throughout this procedure, Δ is calculated with the rapidity rather than the pseudorapidity .
The missing transverse momentum p miss T and its magnitude miss T are reconstructed [119] from selected electrons, muons and jets prior to overlap removal, together with reconstructed photons ( T > 25 GeV, | | < 2.37) satisfying 'Tight' identification criteria [117] and a track-based 'soft term' consisting of softer contributions associated with the primary vertex but not included in aforementioned objects. The miss T reconstruction employs its own overlap removal procedure.

Event selection
Events with miss T < 250 GeV were selected using dilepton triggers [120,121]. Since the luminosity increased during Run 2, the lepton T thresholds were raised in steps to a maximum of 24 GeV for triggers requiring two electrons, 22 GeV for triggers requiring two muons, and 17 GeV (14 GeV) for the electron (muon) in different-flavour dilepton triggers. For events with miss T > 250 GeV, a logical OR of these dilepton triggers and miss T triggers [122] was used. Events are preselected by requiring exactly two signal leptons with the same electric charge or at least three signal leptons 4 without any charge requirement.
Multiple signal regions (SRs) were defined with the goal of maximising the sensitivity to the signal models shown in Figure 1. These SRs are not exclusive and can overlap. They are primarily built on requirements placed on the number of signal ( Sig (ℓ)) and/or baseline ( BL (ℓ)) leptons and their relative charges, the number of -jets ( -jets ) with T > 20 GeV, the number of jets ( jets ) with T above 25, 40 or 50 GeV, regardless of their flavour, and also apply independent selections to a series of observables sensitive to kinematic differences between signal and background events. One of the most sensitive observables used across all SRs is the effective mass, eff , which aids in establishing the mass scale of the processes being probed, and is defined as the scalar sum of the T of the jets and leptons and the miss T of the event,

RPC SRs
SRs targeting the RPC models are shown in Tables  the azimuthal separation between the system formed by the two leading leptons and the direction of the missing transverse momentum of the event, Δ (ℓ1ℓ2, p miss T ); and the miss T significance, S( miss T ) [123]. Table 2: Definition of the signal regions used for the RPC model shown in Figure 1(a), where pair production of gluinos leads to cascade decays of charginos into pairs of SM bosons. Table 3: Definition of the signal regions used for the RPC model shown in Figure 1(b), where pair production of squarks leads to cascade decays of charginos into pairs of SM bosons.

SR name
SRs targeting the benchmark models where pair production of gluinos ( Figure 1(c)) and squarks (Figure 1(d)) leads to decays of charginos and neutralinos into sleptons are named SRGGSlep and SRSSSlep, respectively. Multiple SRs are defined for the different mass-splitting scenarios and named as per the convention described previously. An extra SR (SRSSSlep-H (loose)) is defined for the RPC model shown in Figure 1(d), using the same selection criteria as for SRSSSlep-H, but with the eff requirement relaxed to 1 TeV to allow for a binned fit in the model-dependent interpretation (see Section 9). SRs are defined in terms of the variables previously described, with requirements also placed on the T of the leading and sub-leading leptons, denoted by ℓ1 T and ℓ2 T respectively, and on the angular distance between the two leading leptons, Δ (ℓ1, ℓ2). Table 4: Definition of the signal regions used for the RPC model shown in Figure 1(c), where pair production of gluinos leads to decays of charginos and neutralinos into sleptons.

SR name
T > 1200 GeV † : SFOS pairs with 81 < SFOS < 101 GeV are not allowed Table 5: Definition of the signal regions used for the RPC model shown in Figure 1(d), where pair production of squarks leads to decays of charginos and neutralinos into sleptons. Requirements on ℓ T apply to all three leptons.

RPV SRs
SRs targeting the RPV models are shown in Tables 6 and 7. The SRs are named SRLQD and SRUDD, where the former corresponds to the SR defined for the model where the neutralino decays via the RPV coupling of LQD type (Figure 1(e)), and the latter to the SRs defined for the model where gluinos decay via top squarks and the RPV coupling of UDD type (Figure 1(f)). For the latter, the suffix appended to the SR name indicates the requirement on the number of -jets. Those SRs require exactly two leptons with the same electric charge, high jet multiplicity, and high jet T or high eff . No requirement is placed on the miss T as no neutralinos are expected in the final state of RPV model events. Table 6: Definition of the signal region used for the RPV model shown in Figure 1(e), where the neutralino decays via the RPV coupling of LQD type. Table 7: Definition of the signal regions used for the RPV model shown in Figure 1(f), where gluinos decay via top squarks and the RPV coupling of UDD type.

Background estimation
Background contributions to the search regions can be divided into three categories, consisting in SM processes with genuine same-sign leptons in the final state, and other SM processes forming same-sign lepton pairs because of the incorrect reconstruction of an electron's charge or the presence of fake or non-prompt (F/NP) leptons. The following three sub-sections detail the estimation method for each category: processes with genuine same-sign leptons are estimated with MC simulations, aided by a single control region in the data, while for the other two, data events with specific lepton selection criteria are exploited. Comparisons between background predictions and observed data in carefully designed validation regions are presented in a fourth sub-section.

SM processes with prompt same-sign leptons
The largest contribution to the RPC SRs originates from + jets with both bosons decaying into leptons. All these SRs veto events with -jets, which suppresses processes involving top quarks. For + jets, a control region intermediate in jet multiplicity is employed, referred to as CRWZ2j. It is defined with three signal leptons (and no fourth baseline lepton), two of them forming a SFOS pair with invariant mass in the range 81 < SFOS < 101 GeV. The two same-sign leptons of the triplet must have T > 15 GeV, and the sum of the three leptons' T must exceed 130 GeV. Events must contain either two or three jets with T > 25 GeV, with no -jet present. Contributions from hypothetical BSM processes are reduced by requiring 30 < miss T < 150 GeV and eff < 1.5 TeV. This selection results in an estimated purity exceeding 85%, with the remainder dominated by other multiboson processes. The overall normalisation of the + jets MC prediction is then treated as a free parameter and set by fitting the observed data yields in CRWZ2j. This is realised with a maximum likelihood fit formed by the joint probability of expected background contributions to CRWZ2j with their uncertainties. In the case of signal-strength hypothesis testing, the joint probability is expanded to include contributions to the signal regions of interest, as well as the tested signal contribution to all regions. The procedure is fully detailed in Section 9. It is found that a scale factor of 0.84 ± 0.05 relative to the MC prediction using the theoretical inclusive NLO cross section best accommodates the observed data in CRWZ2j. This below-unity value is consistent with past observations [124,125]. In the RPV SRs, no simultaneous CR-SR fit is performed and the + jets normalisation is determined by CRWZ2j alone. This sequential approach does not propagate correlations among the + jets uncertainties for RPV SRs, but this does not affect the results because the + jets background is subdominant in those SRs.
Other SM background processes with prompt same-sign leptons are estimated by normalising the simulated samples to the theoretical cross section. Some weaker sources of same-sign leptons are ignored. Those include radiative top quark decays → ℓ + ℓ + ℓ − in¯events, found to be negligible for the analyses in Refs. [25,126], and multiple parton scattering, for which similar conclusions were reached in Ref. [25].

Electrons with incorrect charge
Electrons and positrons are distinguished by the curvature of their ID tracks, which may be error-prone if they start radiating early in the ID. The probability for signal electrons to exhibit a charge flip is around 0.06% in the validation region with the largest charge-flip contribution. It varies by several orders of magnitude with | | and T as illustrated in Ref. [117]. Contributions of charge-flip background to the various analysis regions are estimated by selecting data events with opposite-sign leptons and weighting them according to the known (| |, T ) values. The latter are measured in simulated¯events, and multiplied by factors (| |, T ) correcting the known mismodelling. The factors , assumed to be process-independent, are determined [117] from the rates of opposite-and same-sign dielectron pairs observed in → decays in data and MC events. They are found to be within 20% of unity.
The dominant uncertainties in the predicted charge-flip yields are those arising from the measurement of the corrections, which is statistically limited and affected by a significant background. The predicted yields suffer from a typical uncertainty of 50%, which, however, does not impact the results of the present analysis since they represent at most 7% of the total background in any of the SRs.

Fake and non-prompt leptons
Non-prompt leptons arising from hadron decays or photon conversions, as well as hadrons misreconstructed as electrons, may survive the identification criteria in Section 5 at a low rate. When combined with a prompt lepton in an event, they may form a same-sign lepton pair. This source of background is estimated primarily with the matrix method summarised in the following paragraphs, and its prediction is then compared with that of a corrected simulation [54,127].
The matrix method [128] exploits the different efficiency of the identification and isolation requirements when applied to F/NP leptons. In a given region of interest, data events are selected by applying lepton selection criteria that are looser than those defining signal leptons, and then categorising those events according to the number of actual signal leptons. A fully determined system of linear equations can then be constructed, relating the numbers of such categorised events to the unknown numbers of events with only prompt leptons, or exactly one F/NP lepton, etc., where the coefficients are functions of the probabilities and for loose prompt leptons or F/NP leptons to also satisfy the nominal criteria. This can be illustrated for events with a single lepton by the matrix equation: The desired estimate signal, F/NP can then be obtained easily. This approach can be readily extended to final states with multiple leptons. The general method is described in detail in Ref. [129], and the present analysis relies upon the software implementation referenced therein. The sample of loosely selected leptons comprises the subset of baseline leptons after overlap removal, which for muons satisfy | sig 0 | < 7, and for electrons satisfy the ECIDS requirement in Section 5 and are restricted to | | < 2.0. The estimated contribution of charge-flip electrons is subtracted from all terms.
The probabilities (| |, T ) are calculated with simulated¯decays to leptons. The probabilities ( T ) are measured in data [25] within the range 10 < T < 75 GeV in regions enriched in¯events with one or two prompt leptons and one F/NP lepton forming a same-sign pair, which is also the leading contributor of F/NP leptons to the SRs. Separate measurements are made depending on the number of -jets in the event (≤ 1 or ≥ 2) and whether the electron satisfies the criteria [120] to trigger the event recording. The undesirable contributions of + jets and¯to these measurement regions, estimated with MC samples, are scaled by correction factors of 0.84 and 1.19 respectively, based on observed data in CRWZ2j and the region VRTTV defined in Table 8. For muons, is found to increase with T from 75% at 10 GeV to 98% at 70 GeV, while varies between 10% and 15% in the most relevant range, 10 < T < 40 GeV, reaching both leptons with T > 25 GeV, one with T > 40 GeV both leptons with T > 25 GeV * : additional baseline leptons are not allowed its lowest values at around 20 GeV. For electrons, also increases with T , from 50% at 10 GeV to 97% at 100 GeV, while varies between 5% and 10% in the aforementioned T range. When the electron satisfies the trigger criteria, values can be up to twice as large. Over the whole measurement range and for both lepton flavours, is always found to be much larger than , a necessary condition for the applicability of the matrix method [129]. The same conservative systematic uncertainties as in Ref.
[25] are used to account for contamination from prompt same-sign lepton processes in the measurement regions and for the assumption that and can be used outside of the regions in which they are measured. In the SRGGWZ-L region, which only selects events containing a third baseline lepton in addition to the pair of same-sign signal leptons, the estimate is calculated with the reasonable assumption that the F/NP lepton is part of the latter pair.
The SR yields predicted with the matrix method were cross-checked against those predicted by MC simulations of¯and / + jets processes. For the latter, simulated events with F/NP leptons are weighted by three correction factors extracted from fits to data in appropriate regions: one for F/NP electrons, one for non-prompt muons from bottom hadron decays, and one for other sources of non-prompt muons. The estimates are compatible within uncertainties with the ones obtained with the matrix method.

Validation of the background estimates
The + jets,¯and¯ℓ + ℓ − background predictions are checked in the validation regions (VRs) defined in Table 8. The purity in the targeted SM process is expected to range from 70% to 80% for the two VRWZ regions, and from 30% to 55% for the four VRTTV/VRTTW regions, while the expected contributions from SUSY signal processes remain small. The VRs are designed to be orthogonal to the SRs, with the exception of VRWZ4j and VRWZ6j, which partially overlap with four of the SRGGWZ and SRSSWZ regions. However, for each of those, the expected number shared SM background events does not exceed 3% of the total number of events in the VR. The background predictions and the observed numbers of events in CRWZ2J and in the VRs are illustrated in Figure 2. The data and the background expectation in all VRs agree within uncertainties. The largest tensions occur in VRTTV1b6j and VRWZ6j: the former is consistent with observations in other¯measurements [130], while for the latter, the level of agreement was found not to become poorer as a function of eff or miss T .

Systematic uncertainties
The predicted background yields in the SRs are affected by several sources of systematic and statistical uncertainty. The systematic uncertainties are grouped into experimental and theoretical ones, as well as those arising from the data-driven methods described above, normalisation, and MC statistics.
The experimental uncertainties come from the possible differences between the data and simulations in elements of this analysis and the uncertainties in the data taken from the operating detectors. The theoretical uncertainties come from the MC modelling of the relevant SM and SUSY processes, including cross sections, choice of scales, the PDF and s . The theoretical uncertainties from the dominant background processes in the signal regions, such as , ± ± , and¯(as well as¯¯for RPV signal regions), are computed in detail. Uncertainties from the choice of renormalisation scales, factorisation scales, resummation scales, merging scales and the recoil schemes, are evaluated by using the varied scales instead of the chosen scales and comparing the results. The impact of the choice of PDF is evaluated by symmetrising the event count variations seen when using the MMHT2014 [139], CT14 [140], and NNPDF [141] PDF sets. The total cross-section uncertainty is not applied to the process because this process is normalised to data in the CRWZ2j control region. For the remaining rarer processes listed in Table 1, an overall 50% uncertainty is assigned.
For the purpose of defining correlations, the different processes are grouped into nine categories: ,¯, ℓℓ,¯¯, other multiboson processes, other processes with top quarks, F/NP leptons, charge-flip electrons, and SUSY signal. Within each group, every source of uncertainty is treated as fully correlated, including across the different regions involved in the simultaneous fit. Experimental uncertainties are also treated as correlated between the different groups, while all other uncertainties are assumed to be uncorrelated. The total uncertainty and the separate contributions from each source are shown in Figure 3 and Figure 4 for CR/VRs and SRs separately. The total uncertainties vary from 1% (CRWZ2j) to 40% (SRUDD-1b). The dominant contributions come from the F/NP-lepton background estimate, and the MC statistics in the SRs. The total uncertainty in CRWZ2j is smaller than the sum in quadrature of the individual components because of the anti-correlation between the normalisation and modelling uncertainties.

Statistical analysis
The expected SM backgrounds are determined with a profile likelihood fit [142], referred to as a background-only fit. The fit strategy differs between the RPC and RPV searches. For the RPC searches, the background-only fit uses the observed event yield in CRWZ2j as a constraint to adjust the normalisation of the + jets background assuming that no signal is present. The inputs to the background-only fit include the number of events observed in CRWZ2j, and the number of events predicted in CRWZ2j, and in the SR(s) of interest, for all background processes. Both the observed and predicted numbers of events are described by Poisson statistics. The systematic uncertainties are included in the fit as nuisance parameters. They are constrained by Gaussian distributions with widths corresponding to the sizes of the uncertainties and are treated as correlated, when appropriate, between the various regions. The product of the various probability density functions forms the likelihood, which the fit maximises by adjusting the normalisation of the + jets background and the nuisance parameters. For the RPV searches, CRWZ2j is not included in the fit, and the likelihood fit is used just to constrain the nuisance parameters associated with the systematic uncertainties. In both cases, the results of the background-only fit are used to test the compatibility of the observed data and the background estimates in the SRs.
In the absence of a significant excess over the SM expectation, two levels of interpretation are provided for BSM physics scenarios: model-independent exclusion limits and model-dependent exclusion limits set on the SUSY benchmark models illustrated in Figure 1. The CL s method [143,144] is used to derive the confidence level (CL) of the exclusion for a particular signal model. A signal model with a CL s value below 0.05 is excluded at 95% CL.
Model-independent exclusion fits are used to set 95% CL upper limits on the possible BSM contributions to the SRs. This fit proceeds in the same way as the background-only fit, with CRWZ2j (for RPC searches) and the SRs both participating in a simultaneous likelihood fit, and the likelihood function including an additional parameter-of-interest to describe the potential signal contribution. Signal contamination in CRWZ2j is assumed to be zero. The hypothesis tests are performed for each of the SRs independently. The limits were evaluated using pseudo-experiments.
Model-dependent exclusion fits are used to set 95% CL exclusion limits on the masses of gluinos and squarks for the SUSY benchmark models considered in this paper. The fit proceeds similarly to the model-independent fit, except that both the signal yield in the SRs and the signal contamination in CRWZ2j expected from the model are taken into account, and the SRs are usually binned. For the models shown in Figures 1(d) and 1(f), only the SRs with the best expected sensitivity are considered. Table 9 shows the SRs used for each benchmark model and the fitted observable in each SR. The observable providing the best sensitivity for each SR is chosen as the fitted variable. The binning of each observable was optimised to provide the best sensitivity for the benchmark model of interest, while keeping enough events in each bin of the fitted SR. This multi-bin approach was found to enhance the sensitivity for all the SUSY scenarios considered in this search. In this model-dependent fit, the CL s value is computed using the asymptotic approximation [145]. For some selected points a comparison with 'toy' experiments was performed, and the exclusion mass limits are overestimated by a maximum of 2%.

Results
The observed number of events in each SR along with the background expectations and uncertainties are summarised in Figure 5, and shown in detail in Tables 10-12. The background prediction corresponds to the estimate after the background-only fit described in Section 9. The overall excess (less than 2 standard deviations) observed in SRSSWZ-ML, SRSSWZ-MH, and SRSSWZ-H, is due to the overlap among these regions, which have three data events in common. Overlap among the remaining SRs is also observed, but in a smaller proportion. The contribution from + jets dominates in the SRs with no -jets, while the production of fake/non-prompt leptons and processes involving the top quark dominate in the SRs where the veto on -jets is not applied. No significant excess of events above the SM prediction is observed in any of the SRs. The highest significance corresponds to SRSSSlep-L with 2.3 standard deviations. The distributions of the most discriminating variable for some of the SRs, with the signal region requirement on the displayed variable removed, are shown in Figure 6.   Figures 1(a) and 1(b). Backgrounds shown with a '−' do not contribute to that region. The displayed yields include all statistical and systematic uncertainties. The individual uncertainties can be correlated or anticorrelated and therefore do not necessarily add in quadrature to equal the total uncertainty. SRGGWZ In the absence of a significant deviation from the SM prediction, the results are interpreted in terms of model-independent upper limits on possible BSM contributions to the SRs, as well as exclusion limits on  the masses of the SUSY particles in the benchmark scenarios shown in Figure 1. The 95% CL upper limits on the number of BSM events, 95 , that may contribute to the SRs are shown in Table 13. Normalising these by the integrated luminosity of the data sample, they can be interpreted as upper limits on the visible BSM cross section ( vis ), defined as vis = prod × A × = 95 / , where prod is the production cross section for an arbitrary BSM signal process, and A and are the corresponding acceptance and reconstruction efficiency for the relevant SR. The probability of the observations being compatible with the SM-only hypothesis is quantified by the -values displayed in the last column of Table 13. For SRs where the data yield is smaller than expected, the -value is capped at 0.50.
Exclusion limits at 95% CL are also set on the masses of the superpartners involved in the SUSY benchmark scenarios considered. Figure 6 shows the fitted variable's distribution for each displayed SR. The bins displayed above the SR requirement correspond to the binning used in the exclusion fit. For illustration purposes, only one SR per benchmark model is shown. Figure 7 shows the exclusion limits obtained for the RPC models shown in Figure 1. For each point of the {˜(˜) ,˜0 1 } parameter space, the SR with the  ), and on the number of signal events given the expected number of background events and its ±1 variations, ( 95 exp ). The last two columns indicate the CL b value, i.e. the confidence level observed for the background-only hypothesis, the discovery -value ( ( = 0)) and its associated significance .      Figure 1. The yellow band shows the ±1 variation of the expected limits. The red dotted lines around the observed limit illustrate the change in the observed limit as the nominal signal cross section is scaled up and down by the theoretical uncertainty. The light blue shaded area indicates the observed limits obtained by previous ATLAS searches.

Conclusion
A search for pair production of squarks or gluinos decaying via sleptons or weak bosons in final states with either two same-sign leptons or at least three leptons is presented. The search uses 139 fb −1 of proton-proton collision data collected by the ATLAS detector at the LHC at a centre-of-mass energy of 13 TeV.
No significant excess of events over the Standard Model expectation is observed. The results are interpreted in the context of several supersymmetric simplified models featuring gluino or squark pair production in R-parity-conserving and R-parity-violating scenarios. Lower limits on particle masses are derived at 95% confidence level for these models, reaching as high as 2.2 TeV for gluinos and 1.7 TeV for squarks, surpassing the exclusion limits from similar previous searches performed by ATLAS. Improved analysis techniques, the inclusion of a control region for the + jets background, and a significantly larger data set in some cases, contributed to this improvement. Model-independent limits on the cross section of a possible signal contribution to the signal regions defined in this search are also set.
[129] ATLAS Collaboration, Tools for estimating fake/non-prompt lepton backgrounds with the ATLAS detector at the LHC,