Search for new physics in the lepton plus missing transverse momentum final state in proton-proton collisions at $\sqrt{s} =$ 13 TeV

A search for physics beyond the standard model (SM) in final states with an electron or muon and missing transverse momentum is presented. The analysis uses data from proton-proton collisions at a centre-of-mass energy of 13 TeV, collected with the CMS detector at the LHC in 2016-2018 and corresponding to an integrated luminosity of 138 fb$^{-1}$. No significant deviation from the SM prediction is observed. Model-independent limits are set on the production cross section of W' bosons decaying into lepton-plus-neutrino final states. Within the framework of the sequential standard model, with the combined results from the electron and muon decay channels a W' boson with mass less than 5.7 TeV is excluded at 95% confidence level. Results on a SM precision test, the determination of the oblique electroweak $W$ parameter, are presented using LHC data for the first time. These results together with those from the direct W' resonance search are used to extend existing constraints on composite Higgs scenarios. This is the first experimental exclusion on compositeness parameters using results from LHC data other than Higgs boson measurements.


Introduction
Experimental evidence from the last half-century has established the standard model (SM) as the foundational theory of particle physics. However, several fundamental aspects of Nature cannot be explained by the SM [1]. To address a variety of open issues, many models beyondthe-SM (BSM) have been proposed. To resolve the hierarchy problem [2,3], some of these models propose additional heavy gauge bosons as remnants of electroweak (EW) symmetry breaking. To achieve gauge coupling unification, other models invoke extended gauge sectors [4] that can give rise to extra spatial dimensions [5,6].
This paper focuses on a search for a new heavy charged gauge boson (W ) decaying into an energetic charged lepton (electron or muon) and a neutrino. As neutrinos cannot be detected directly with the CMS detector, their presence is inferred from an imbalance in the transverse momentum, p T , in the event; this imbalance is referred to as missing transverse momentum p miss T [7], and is defined as the negative vector sum of the p T of all reconstructed particles in an event. The transverse mass (M T ) of the charged lepton ( ) and the p miss T is used as the main discriminating variable of the search, and is given by where p T is the magnitude of the lepton p T , p miss T is the magnitude of p miss T , and ∆φ( p T , p miss T ) is the azimuthal opening angle between the directions of the lepton and p miss T .
In this search, we look for the presence of a resonant signal in the high mass tail of the M T distribution, where the contributions from background processes are small. The shape of the M T distribution is studied using a binned likelihood method. This approach is especially powerful as the examined models predict different M T distributions for different hypotheses, and allow BSM signals to be detected through significant deviations from SM predictions in their kinematic distributions. Results are interpreted in the context of several BSM scenarios, involving the production and decay of a W boson in the sequential standard model (SSM) [8] or of a Kaluza-Klein (KK) excitation of the W boson in a model with split universal extra dimensions (split-UED) [9, 10]. Additionally, we consider the production and flavour violating decay of a slepton mediator in the R parity violating supersymmetric (RPV SUSY) model [11,12].
Searches for a W boson at the CERN LHC have been reported by the ATLAS and CMS Collaborations with data collected at 7 TeV [13,14], 8 TeV [15,16], and 13 TeV [17][18][19]. ATLAS reported the exclusion of an SSM W below 6.0 TeV in the combined electron and muon channels at 95% confidence level (CL) [20] with data collected from 2015 to 2018. So-called "Universal" effects from new physics are also studied using effective field theory (EFT) interpretations. This EFT approach quantifies potential deviations from the SM expectations through the oblique EW W parameter [21]. Constraints on this parameter are derived in this paper, establishing a lower limit on the scale for universal new physics effects. This is the first time using LHC data that constraints have been set on such a parameter in the +p miss those of the SM W boson. In this analysis, the tb decay channel is allowed, while diboson decays are considered to be suppressed. These assumptions yield a predicted branching fraction (B) of about 8.5% for each of the leptonic channels [8]. The expected intrinsic width of a 1 TeV W boson is about 33 GeV. The leading order Feynman diagram showing the production of a charged W boson in a quark-antiquark (qq ) interaction is depicted in Fig. 1 (left).
The signal events for the SSM W → e/µ+ν channels are generated using PYTHIA v8. 230 [32] at leading order (LO) accuracy in quantum chromodynamics (QCD) with the NNPDF3.1 NNLO [33] parton distribution function (PDF) set. The masses of the W boson (M W ) are varied in steps of 0.2 TeV from 0.2 to 6.4 TeV. Regarding the shape of the M T distribution, as M W increases, the fraction of off-shell production increases in the low M T region as predicted by the PDFs. A mass-dependent K factor [34], which is defined as the ratio of the next-to-nextto-LO (NNLO) to the LO cross section, is calculated using FEWZ v3.1 [35] with the NNPDF3.1 NNLO PDF set, and varies from 1.1 (M W = 0.2 and 6.0 TeV) to 1.3 (M W = 3.0 TeV). This K factor for the signal is used to achieve QCD NNLO precision in production cross sections. . The coupling strength, g W , is allowed to vary. In RPV SUSY, a τ slepton ( τ ) could also act as a mediator (right) with the corresponding coupling strength, λ, for the decay. This coupling strength is allowed to be different between the two final states, denoted by λ 231 and λ 132 for the electron and muon final states, respectively. Here, C.C. stands for charge conjugation.

Varying coupling strength in the Sequential Standard Model
The W boson coupling strength, g W , can be given in terms of the SM weak coupling strength g W . The coupling ratio, g W /g W , is set to unity in the SSM model; however, various coupling strengths could be possible. Variations in the W boson's coupling will affect its width and consequently its M T distribution. Signal samples for a range of coupling ratios were generated using MADGRAPH5 aMC@NLO [36] at LO with the NNPDF2.3 LO [37] PDF set for the electron and muon channels. The coupling ratios range from 10 −2 to 3. These signals exhibit different widths as well as different cross sections.
The generated distributions of PYTHIA samples are reweighted to include the decay width dependence and provide the appropriate M T distributions. For a coupling ratio of unity, the theoretical LO cross sections apply. For all other coupling ratios, the theoretical cross sections scale with the square of the coupling strength, g W .

Split-UED model
Other models in which a mediator may decay into a charged lepton and a neutrino are characterized by the existence of universal extra dimensions. In particular, we consider the split-UED model [6,9,10]. This model extends the conventional notion of space-time to include an additional spatial dimension of radius R. Here, all SM particles have a corresponding Kaluza where M W is the W boson mass. The split-UED model requires a bulk mass parameter µ for the fermion field in five dimensions. The special case of minimal UED corresponds to the limit of vanishing bulk mass. The W (n) KK couplings to SM fermions depend on µ. Additionally, the product of the cross section for W (n) KK boson production and its branching fraction to SM fermions goes to zero as µ goes to zero. One of the important features of this model is the conservation of KK-parity. The n = odd (even) KK modes must have odd (even) KK-parity, and the lightest odd KK-parity particle cannot directly couple to the even modes. Therefore, the conservation of KK-parity forbids coupling of KK-odd modes of the W (n) KK boson to SM fermions (zero mode). For the n = 2 mode, the W (2) KK boson decay to leptons is kinematically similar to the SSM W boson decay ( Fig. 1 left). This allows the limits obtained from the W → eν and W → µν searches to be reinterpreted directly in terms of the M(W

R parity violating SUSY with a slepton mediator
It is possible for a SUSY slepton ( ) by virtue of R parity violation to undergo a lepton flavour violating decay to a charged lepton and a neutrino [11,12]. The analysis focuses on a τ slepton ( τ ) decaying to eν µ or µν e channels as illustrated in Fig. 1 (right). The coupling at the production vertex is always a function of the hadronic-leptonic RPV coupling λ 3ij (i, j = 1, 2, 3). This is the coupling strength related to the third generation, specifically the τ . The coupling at the decay vertex is governed by λ 231 (λ 132 ) for the decay to eν µ (µν e ).

Universal new physics through constraints on the W parameter
One possible way to incorporate effects from new physics is to consider dimension-6 operators involving only H, W, and B fields. Here, H is the Higgs boson doublet, W is the weak isospin triplet, and B is the hypercharge field, which can be expressed as the mixture of photon and Z boson fields. This BSM extension would involve new physics effects related to EW symmetry breaking, without the introduction of either new strong interactions or flavour violations up to the scale of new physics currently considered. In this context, there are four parameters, S, T, W, and Y [21], quantifying the universal deviations. They correspond to operators that modify the propagators of the SM EW vector bosons both on pole (S and T) and off-pole (W and Y). The terms accompanying W and Y grow with the centre-of-mass energy of the process under study, thus LHC is particularly suited to study observables like W and Y, in the tail of relevant distributions. These parameters are constrained by the EW precision measurements, mainly from the CERN LEP and LHC colliders [38]. These parameters are further constrained by this analysis.
In the specific case of four-fermion contact interactions with difermion final states, LHC data is sensitive to the oblique W and Y parameters through the interpretation of the dilepton mass distribution. In the SM assumption, the values of the W and Y parameters are zero. In the case of nonvanishing W and Y parameter values, and in the ν channel, deviations (either an excess or a deficit) in the M T distribution would be manifested, especially in the high-M T region. This expected effect is implemented in the signal events via a reweighting technique (details in Section 7.6) applied on the events from the simulated SM background samples. These predictions are then compared to data to place constraints on the W parameter. This procedure assumes that the new resonance lies beyond the current kinematic limit. Deviations from the SM predictions in the mass region between the SM W boson and the hypothetical new resonance could, however, still be observed.

Composite Higgs boson models
A reinterpretation of the results from the W boson search is done in the framework of composite Higgs boson models. These BSM models predict the Higgs boson to be a composite particle associated with strong interaction dynamics at a higher energy scale [24]. Two parameters characterize the dynamics: m * and g * , the mass scale of the new spectrum of composite resonances and the coupling associated to the new interaction, respectively. The input for this reinterpretation comes in two complementary ways. The first uses the limits on the W coupling, g W , presented in Section 3.2 and with values later shown in Section 7.3, obtained from the direct W search. The second uses the constraints on the oblique W parameter presented in Section 3.5 and with values shown in Section 7.6.
In the first case, the W boson is assumed to be a state composed of more fundamental particles. The gauge coupling to the new constituents, g * , is screened by the presence of the resonance so that it can be seen at low energy in the form of g 2 /g * [25], i.e.
where g is the SU(2) L SM gauge coupling and g W is the W boson coupling strength to fermions, present at both the resonance production and decay vertices.
In the second case, the value we obtain for the W parameter in this analysis is used to quantify deviations from the SM using the relation [38] Using a third approach, additional constraints are placed in the m * -g * plane when applying current measurements of the Higgs boson cross section from a combination of production mode and decay channels [26]. In this case, the new interactions would modify the SM predictions for Higgs production and decay, generating an extra contribution to the Higgs kinetic term in the effective Lagrangian [24]. The corresponding modification would scale as where µ H is the measured Higgs boson cross section relative to that of its SM cross section and v is the EW symmetry breaking scale.
Diboson decay channels are allowed in this model, following the W description in the Heavy Vector Triplet (HVT) model [39], in contrast to the physics models described in Sections 3.1 to 3.5, where they are assumed to be suppressed.

Background simulation
Simulated event samples are used to estimate the event yield of the SM background processes. In the following we describe the simulation of the background samples used to analyze the data obtained in 2017-2018.  [41,42] with the NNPDF3.1 nnlo as0118 luxqed PDF set [43,44]. The QCD NNLO corrections are computed with the FEWZ v3.1 package [35]. These corrections are combined with an additive method [45]. The K factor applied is largest at low mass, with a value of 1.20 ± 0.04 at M W = 0.2 TeV, falling to 0.89 ± 0.19 at high mass (M W = 6 TeV).
For the low-mass region (M W < 0.1 TeV), the inclusive on-shell W+jets sample and the highp T W+jets sample are used. Both samples are generated at LO with MADGRAPH5 aMC@NLO v2.4.2 [36,46] and the simulated jets are matched to the matrix element and parton shower produced by PYTHIA, following the MLM approach [47]. The on-shell sample is used in the low-H T (< 0.1 TeV) region and the high-p T sample is used at low masses and in the high-H T (> 0.1 TeV) region, where H T is defined as the scalar p T sum of all jets in an event. The onshell W sample is scaled to QCD NLO precision using MADGRAPH5 aMC@NLO, following the FxFx scheme [46], and the high-p T W sample is corrected with the W boson p T -dependent NLO QCD+NLO EW K factors, using MADGRAPH5 aMC@NLO [48].
The inclusive WZ and ZZ samples are generated with PYTHIA and scaled to QCD NLO precision [55,56] in comparing the samples with data. The multijet background is estimated using simulated PYTHIA samples generated with sufficient numbers of events at high lepton p T values, and normalized to data as described in Section 5.
In the electron channel, γ+jets samples are generated at LO using MADGRAPH5 aMC@NLO with MLM matching. These samples are used to estimate the effects of photons misidenti-fied as electrons. In the muon channel, several background samples that give nonnegligible contributions are taken into account in the low-M T (<0.4 TeV) region. These backgrounds are Z/γ * +jets (Z/γ * → νν), generated at QCD NLO with POWHEG, and Wγ (Wγ → νγ with = e, µ, and τ) samples generated at LO using MADGRAPH5 aMC@NLO. Their contributions are negligible in the electron channel because of the much larger trigger thresholds for electrons used during 2017-2018 data-taking, as described in Section 5.
The simulation of the CMS detector response is performed using the GEANT4 [57,58] framework. All simulated samples used in the analysis are normalized to each total integrated luminosity of the three years and combined together. The average number of additional pp collisions in the same or nearby bunch crossings (pileup) was 32 in 2017-2018, corresponding to a total inelastic pp cross section of 69.2 mb [59]. All simulated samples are weighted to have the same pileup multiplicity distributions as measured in data.

Event reconstruction and particle identification
The reconstruction algorithms are similar to those used in an earlier analysis of 13 TeV data [19]. The signature of interest is characterized by a single high-p T lepton and significant missing transverse momentum.
The candidate vertex with the largest value of summed physics-object p 2 T is taken to be the primary pp interaction vertex (PV). The physics objects used for this determination are the jets, clustered using the jet finding algorithm [60,61] with the tracks assigned to candidate vertices as inputs, and the associated p miss T , taken as the negative vector p T sum of those jets [7]. Information from all components of the CMS detector is used by a particle-flow (PF) technique [62] to reconstruct and identify each individual particle in an event. Photon energy is obtained from the ECAL measurement. The electron energy is determined from a combination of the electron momentum at the PV as determined by the tracker, the energy of the corresponding ECAL cluster, and the sum of energy from all photon bremsstrahlung processes spatially compatible with originating from the electron track. The energy of muons is obtained from the curvature of the corresponding track. The charged hadron energy is determined from a combination of the momentum measured in the tracker and the matching ECAL and HCAL energy deposits, corrected for the response function of the calorimeters to hadronic showers. Finally, the energy of neutral hadrons is obtained from the associated corrected ECAL and HCAL energies.
Electrons are reconstructed from clusters of energy deposits in the ECAL. These are matched to hits in the tracker and the candidates are required to pass a set of identification and isolation criteria [63][64][65] optimized for high energy values. Electrons with transverse energy (E T ) larger than 35 GeV and |η| < 1.40 (1.57 < |η| < 2.50) in the barrel (endcap) are required and the measured energy is taken directly from the ECAL cluster without using track information. High-E T electrons can saturate the ECAL readout electronics, biasing the determination of showershape. Therefore, subdetector-based isolation is used instead of PF isolation. The electron is required to be isolated in a cone of radius ∆R = (∆η) 2 +(∆φ) 2 = 0.3 in both the calorimeters and the tracker. In the calorimeters, the energy sum is required to be less than 3% of the electron's p T . The HCAL has two readout depths in the endcap. Only the first longitudinal depth is used for the HCAL isolation because the second depth can be affected by detector noise. In the tracker, the p T sum must be less than 5 GeV, including only well-measured tracks consistent with originating from the same vertex as the electron. Energy deposits in the HCAL around the direction of the electron, corrected for noise, pileup, and leakage of electrons through the inter-module gaps, are required to be less than 5% of the reconstructed electron energy, in order to reject jets. For high energy electrons, the last contribution dominates. Other requirements are applied on the number of lost hits in the inner tracker layers and on the impact parameter relative to the centre of the luminous region in the transverse plane, d xy . The former must have no more than one missing hit and the latter is required to be less than 0.2 (0.5) mm for the barrel (endcap). Taken together, these requirements suppress background sources such as electrons originating from photon conversion, hadronic activity misidentified as electrons, and electrons from semileptonic decays of b or c quarks.
Muons are reconstructed by matching tracks from the inner tracker with hits, or segments, in the muon system. The information from the tracker and muon system is combined in such a way as to ensure good p T resolution and low sensitivity to energy losses due to muon bremstrahlung. The muon p T is determined using a specific combination (TuneP) of different algorithms optimized for p T > 200 GeV [66]. Muon candidates with |η| < 2.4 are required to satisfy the track fit and matching quality requirements to ensure a good p T measurement and suppression of potential muon misidentification from any hadron shower remnants reaching the muon system. For a reliable p T measurement, we require that σ p T /p T < 0.3, where σ p T is the uncertainty in p T from the TuneP reconstruction. For good track quality, the muon track must have at least one hit in the pixel detector, and at least six tracker layers with hits, to suppress muons from nonprompt meson decays. The muon track is also required to have segments with hits in at least two consecutive muon detector planes, to reduce contamination from hadronic punch-through. To reduce the background due to cosmic ray muons, we require that d xy and the longitudinal impact parameter, d z , be less than 0.02 and 0.5 mm, respectively. These parameters are defined relative to the PV. The muon isolation in the tracker requires the scalar p T sum of all tracks originating from the interaction vertex within a cone of ∆R = 0.3 around the muon direction, excluding the muon itself, to be less than 10% of the muon p T .
Jets are clustered from PF candidates using the anti-k T algorithm [60,61] with a ∆R of 0.4. Jet momentum is determined as the vector sum of all particle momenta in the jet, and is found from simulation to be, on average, within 5-10% of the true momentum over the whole p T spectrum and detector acceptance. Pileup can contribute additional tracks and calorimetric energy depositions, increasing the apparent jet momentum. To mitigate this effect, tracks identified to be originating from pileup vertices are discarded and an offset correction is applied to correct for remaining contributions. Jet energy corrections are derived from simulation studies so that the average measured energy of jets becomes identical to that of particle level jets. In situ measurements of the momentum balance in dijet, γ+jet, Z+jet, and multijet events are used to determine any residual differences between the jet energy scale in data and in simulation, and appropriate corrections are made [67]. Additional selection criteria are applied to each jet to remove jets mismeasured because of instrumental effects or reconstruction failures. Jets originating from the hadronization of b quarks are identified by a secondary vertex algorithm, DeepCSV [68]. This b tagging algorithm is applied to the leading jet with a tight working point that has an identification efficiency of about 50% for b quark jets and misidentification probabilities of 2.0% for c quark jets and 0.1% for light-flavour or gluon jets. To correct the simulated events, these efficiencies are measured as a function of jet p T and η in data samples enriched in tt and multijet events.
The vector p miss T is computed as the negative vector p T sum of all the PF candidates in an event, and its magnitude is denoted as p miss T [7]. The p miss T must satisfy reconstruction quality criteria chosen to reduce detector noise effects. It is modified to account for the corrections to the jet energy scale [67,69] in the event. In both the electron and muon channels, further corrections are applied to account for possible differences in the lepton p T relative to that obtained from the PF algorithm.
The lepton reconstruction and identification efficiencies for both channels are measured using the "tag-and-probe" method [70] with samples of high mass Drell-Yan pairs. Scale factors (SFs) are applied to the simulated events to account for any differences in the selection efficiencies between observed and simulated data. The SFs for the electron identification, isolation, and reconstruction efficiencies are 0.967 (0.973) for barrel (endcap) in 2017, and 0.969 (0.984) in 2018 [64]. The SFs for the muon identification efficiencies are in the range 0.974-0.994, as a function of η, for both 2017 and 2018 [66]. The dependence on η reflects the geometry of the muon detectors system and the effect of radiative processes associated with muon interactions in the material of the detector at very high p T . The muon isolation is well reproduced in simulated samples and the SF is consistent with unity, within systematic uncertainties, for both 2017 and 2018 data.

Event selection
Events are selected in the high-level trigger by requiring the presence of one isolated highp T lepton. Electron events require a single electromagnetic cluster having E T > 200 GeV and |η| < 2.5. Muon events require a single muon with p T > 50 GeV and |η| < 2.4.
In addition to satisfying the tight offline identification criteria described in Section 4, electrons (muons) are required to have p T > 240 (53) GeV, so that they are on the efficiency plateau of the trigger. The trigger efficiency for the electron channel is measured in simulation and in an independent data sample recorded with single-muon triggers for each data-taking year. The SF is determined to be 0.995 (0.994) for barrel (endcap) in 2017, and 0.997 (0.998) for barrel (endcap) in 2018. In addition, the SFs for the level-1 trigger timing drift [28] are applied for the electron channel in 2017. This timing drift caused a partial loss of events in the trigger, as the reconstructed clusters in the ECAL could be wrongly assigned to the previous bunch crossing, an occurrence referred to as "trigger prefiring". The prefiring mostly impacts events with large electromagnetic activity at large |η| values (2.0-3.0). The trigger SF values for the muon channel, applied as a function of η, range 0.911-1.011 in 2017-2018 [29,66].
To reduce the Drell-Yan, tt, and multiboson backgrounds, events are required not to have a second lepton, either electron or muon, with p T > 25 GeV. For a high efficiency, this second lepton is identified with a set of loose selection criteria. The backgrounds from tt and single top quark events in both semileptonic and dileptonic decay modes are sources of p miss T and high-p T prompt leptons. In the muon channel, the contribution from tt background is further reduced by suppressing events containing more than five jets (p T > 25 GeV, |η| < 2.5 and ∆R(jet, ) > 0.5) and events with the leading jet consistent with originating from a b quark [68]. This requirement is not needed in the electron channel since the higher electron p T threshold applied in the selection already significantly reduces the tt contribution. The agreement of data with simulation lies within uncertainties for the jet multiplicity distributions for up to 6 jets. For higher jet multiplicities, an additional 10% uncertainty in the normalization of the tt sample is included.
The multijet background has the largest cross section among all background processes, but it is efficiently reduced thanks to the lepton high-p T cut and the isolation criteria. The shape of the lepton p T distribution of the multijet background is taken from the simulated samples and normalized to data using a control region enriched in events with energetic leptons from QCD production of multijets. This region is obtained by applying the same selection criteria as in the analysis, but inverting the lepton isolation requirement and limiting the amount of p miss T in the event.
The lepton p T and p miss T are expected to be nearly back-to-back in the transverse plane and balanced in p T . To incorporate these characteristics in the analysis, the ratio of the lepton p T to p miss T and the angle between the directions of the lepton and p miss T provide additional kinematic criteria to select events. The ratio p T /p miss T is required to be between 0.4 and 1.5, and the angular difference must be ∆φ( p T , p miss T ) > 2.5.
For simulated events passing all of the selection criteria, the signal efficiency for an SSM W boson with no requirement on the reconstructed M T in the event reaches a maximum value of 0.70 (0.77) in the electron (muon) decay channel, for an M W range of 1.5 to 2.4 TeV. This value decreases gradually to about 0.4 (0.6) in the electron (muon) channel for high masses (M W ∼ 6 TeV). The difference in the efficiencies between the channels is due to the difference in the p T thresholds applied. Off-shell production of the W boson resonance will manifest itself in the lower M T region. The effect increases as the W mass increases. The lower p T threshold of the muon trigger gives rise to a higher efficiency in the muon channel than the electron channel, for low and very high masses. The signal efficiency as a function of M W is shown in Fig. 2 for both channels.  The resulting M T distributions of the electron (left) and muon (right) channels are shown in Fig. 4. The minimum value chosen for M T depends on the trigger threshold. It is 250 GeV (2016) and 500 GeV (2017-2018) for the electron and 120 GeV for the muon channels, respectively. The M T distribution, which is used as an input to set limits, is binned according to the resolution.
The observed and expected numbers of events for a selected set of M T thresholds are shown in Table 1 for the electron and muon channels, from the combination of the 2016-2018 data KK , and RPV SUSY τ signals at the masses of 3.8 and 5.6 TeV are also indicated. The lower panels show the ratios of data to the SM prediction, and the shaded bands represent the systematic uncertainty.
sets. The highest M T values of events observed in the electron and muon channels are 3.1 and 2.9 TeV, respectively.

Systematic uncertainties
Several sources of uncertainty affect the shape, or normalization, of the M T distribution of the lepton plus p miss T system, impacting the background and signal description. Unless otherwise specified, to estimate the systematic uncertainty on the overall normalization and on the shape of the M T distribution, the full analysis is repeated, with the values of the parameters associated with that source shifted up and down by one standard deviation.
For the dominant W boson background, the theoretical uncertainties in the PDF are estimated for each M T bin of the differential distribution. The uncertainties are obtained using the PDF4-LHC [45] procedure accounting for the NNPDF3.1 NNLO [33] PDF set of replicas. The effect on the background event yield due to this PDF uncertainty increases gradually to 4, 40, and 120% at M W = 2, 4, and 6 TeV in this analysis. The uncertainties in the W boson cross section at high mass related to the QCD+EW higher-order corrections are taken to be 4-19%, depending on the M W . These uncertainties get larger with increasing M W .
The global uncertainties in the integrated luminosity measurement are estimated with a precision of 2.3 and 2.5% [71][72][73] for the 2017 and 2018 data-taking periods, respectively. The uncertainty related to pileup modelling is evaluated for variations of 4.6% in the total inelastic pp cross section relative to the nominal value of 69.2 mb [59]. The resulting shift in weights is propagated through the analysis and the corresponding M T spectra are used as inputs in the statistical treatment of the data. The variation of the yields induced by this procedure is less than 0.5%.
The uncertainties related to the lepton energy or momentum measurement and identification efficiency for high-p T leptons in data and simulation are estimated from dedicated measurements performed with Z bosons with a large transverse boost. For the electron channel, the uncertainty associated with the electron energy scale and resolution corrections is taken as 0.05-0.10 (0.1-0.3)% in the ECAL barrel (endcap) [64]. For the muon channel, the uncertainty in the muon p T scale and resolution is estimated by studying the curvature of the muon tracks in different regions of η and φ, using energetic muons. These tracks are taken from cosmic ray data and from dimuon events from high-p T Z boson decays in the collision data, together with the corresponding simulation samples [66]. Based on this study, the scale correction for high-p T muons is assigned an uncertainty for p T > 200 GeV, which varies as a function of η. This uncertainty amounts to a potential p T shift of 5-10% for p T > 2 TeV in the endcaps (|η| > 1.2), and around 5% in the barrel (|η| < 1.2), during 2017. During 2018, these values were reduced to 2% for p T > 2 TeV, independent of muon η. For muons with p T > 100 GeV, the uncertainty associated with p T resolution is negligible in the barrel region and taken to be 1% in the endcaps. These uncertainties are propagated to the p miss T and M T distributions.
The uncertainty in the trigger SF in the electron channel is determined to be 0.5 (1.1)% for the barrel (endcap). In addition, the uncertainty derived from the ECAL endcap level-1 trigger timing drift effect [28] is taken to be a maximum of 20% (prefiring probability) and the statistical uncertainty associated with the particular bin. The uncertainty in the identification and isolation SF for electrons in the barrel is 1% for p T < 90 GeV. This uncertainty increases linearly from 1 to 3% for p T in the range of 90-1000 GeV and corresponds to 3% for p T > 1 TeV. In the endcap the SF uncertainty is 2% for p T < 90 GeV. This uncertainty increases linearly from 2 to 5% for p T in the range of 90-300 GeV, and is 5% for p T > 300 GeV.
In the muon channel, the uncertainty in the trigger SF is estimated to be 0.5-8.0%, depending on p T and η. The corresponding prefiring probability is less than 0.4% and its effect on the trigger efficiency is included in the systematic uncertainty. The SF of the muon isolation is assigned an uncertainty of 0.1% for both 2017 and 2018 data. For the high-p T muon identification, the uncertainty in the SF is 0.4% for |η| < 2.0 and 2.1% for |η| > 2.0.
The uncertainty in the determination of p miss T is derived from the individual uncertainties assigned to the PF objects [7,67]. The PF objects considered are jets, e, µ, τ, γ, and unclustered energy. The TuneP muon p T determination described in Section 4 is taken into account.
For energetic electrons and muons the procedure has been explained above. The effect on the p miss T measurement of the uncertainty in the jet energy scale is estimated by shifting the jet energy by ±1 standard deviation in simulation. The resulting uncertainty is 2-5%, which depends on p T and η [67]. In addition, an uncertainty of 10% in the p T is used for the unclustered energy. The uncertainty associated with p miss T is derived by summing quadratically the uncertainties in the separate contributions to the PF calculation of this quantity.
The dominant uncertainties in the electron and muon channels are related to the description of the SM background used to compare to the data. On the experimental side, the most important uncertainty is associated with the measurement of the lepton energy or momentum, arising from the detector resolution and the energy scale calibration. From the theoretical side, the main uncertainties come from the choice of the PDF set and the determination of K factors covering higher-order corrections for high mass off-shell W boson cross sections.
All systematic uncertainties are taken to be correlated between the three data-taking years. The exceptions to this are the object-related (leptons, jet, p miss T ) uncertainties, which derive from statistically independent sources, and integrated luminosity uncertainties. These exceptions are treated as partially correlated between different years, because they include some components that are fully correlated year-to-year and other components that are uncorrelated year-to-year.

Results
The M T distributions for each channel observed in data and predicted by the SM simulation are compared in Fig. 4 and Table 1. They agree within the statistical and systematic uncertainties. Thus, no significant deviation from the SM expectation is observed. These observations are interpreted in the context of the models described in Section 3.
The M T distributions shown in Fig. 4 are used as the basis for interpreting the results in the framework of the SSM and the split-UED model. We perform a binned shape fit using a maximum likelihood to the M T spectrum. Upper limits at 95% CL on the product of the resonance production cross section and the branching fraction to a charged lepton and a neutrino as a function of M W are calculated using a Bayesian method [1] with a uniform positive prior probability distribution for the signal cross section. Each source of systematic uncertainty described in Section 6 is considered as a nuisance parameter in the expected signal and background yields. The rate uncertainties (e.g. luminosity) that affect only the normalization of the simulation are modelled with log-normal distributions, and the uncertainties affecting the shapes are modelled using a template-morphing method with Gaussian distributions [74]. The statistical uncertainties from the events in the simulated signal and the SM background are considered by incorporating them into the fit as nuisance parameters using the Barlow-Beeston "lite" approach [74,75].

Limits on the SSM W boson and on the W
(2) KK boson in split-UED models are calculated using a binned maximum-likelihood fit method for the electron and muon channels. These limits are set using the M T distributions shown in Fig. 4 above 400 (120) GeV for electrons (muons) obtained after the complete event selection. A model-independent (MI) limit is determined using a single bin of all events above a threshold M min T summed together. The MI upper limit at 95% CL on σ W B(W → ν)A ( = e or µ) is set as a function of the M min T . If the background yield is low enough, this method provides comparable sensitivity to that of the binned likelihood fit method. The resulting MI limit can be reinterpreted using other models having the same final states. An example for the RPV SUSY model is given in Section 7.5. Additionally, constraints on the W parameter and on the composite Higgs sector are computed.  As shown in Fig. 5, M W below 5.4 TeV (5.3 TeV expected) and 5.6 TeV (5.5 TeV expected) are excluded in the electron (upper left) and muon (upper right) channels, respectively. For the very low M W region, the muon channel becomes more sensitive because of the much higher trigger threshold in the electron channel as described in Section 5. The lower exclusion limit for the combination of the two channels increases to 5.7 TeV (5.6 TeV expected), as shown in Fig. 5 (lower). The one and two standard deviation bands include both the statistical and systematic uncertainties described in Section 6. The results are summarized in Table 2.

Exclusion limit on the model-independent cross section
The model-independent limits on σBA for a new particle decaying to a lepton and a neutrino are shown in Fig. 6 for the electron (left), muon (right), and the combined channels (lower). The limits depend strongly on the M min T threshold. For the combination of the two channels, the MI limit ranges from about 10 fb at a M min T of 500 GeV to about 0.02 fb at M min T of 3.6 TeV.
The exclusion limit for any specific physics model that predicts a massive charged mediator with the same final states can be determined using the MI limit in Fig. 6. For a specific physics model, one first finds the model-dependent efficiency for the signal, namely the product of the acceptance and the efficiency, A , which includes the effects of the kinematic selections, the geometrical acceptance, and the trigger threshold. The effect of the threshold M min T on the signal is expressed by the fraction f M T (M min T ) of generated signal events that pass the requirement M T > M min T , where M T is calculated at generator level. A limit on the production cross section times the branching fraction for the specific physics model in the fiducial phase space of the measurement, [σBA ] excl , can be obtained as a function of M min T . This is done by dividing the excluded cross section of the MI limit [σBA ] MI (M min T ) given in Fig. 6 by the calculated fraction f M T (M min T ): To interpret a new signal model using the MI limit, first, a (σB) excl is determined as a minimum value of the [σBA ] excl (M min T ) by scanning M min T . Then, a limit on the model parameters can be set by comparing (σB) excl with the theoretical cross section (σB) theo . Models where (σB) theo > (σB) excl can be excluded.

Exclusion limits on the W boson coupling strength
The cross section exclusion limit depends on the width and the mass range of the potential signal. A limit can be derived on the ratio of coupling strengths, g W /g W , from the relationship between the coupling strength of a W and the resonance decay width.
Two steps are required to derive a coupling ratio limit for every M W . The first one consists in setting, for each M W point, an upper limit on σ W B as a function of a coupling ratio. The second step is to find an intersection point of the σ W B limit line with the theoretical cross section for every coupling ratio. The theoretical cross sections used are calculated to LO in QCD precision for the M W in this model. Coupling ratios where the limit is less than the theoretical prediction are excluded. This procedure is repeated for every M W , and the corresponding intersection points provide the input for the result. The resulting exclusion limit on the coupling strength ratio, as a function of the M W , is shown in Fig. 7 for the electron (left) and muon (right) channels. No combination of channels is performed in this case since there is no assumption that the couplings have to be equal for both decay channels. The area above the limit line is excluded. For the M W ≈ 1 TeV, coupling ratios above 2.7 × 10 −2 (3.0 × 10 −2 expected) and 2.7 × 10 −2 (2.9 × 10 −2 expected) are excluded in the electron and muon channels, respectively.  Figure 7: The observed (solid line) and expected (dashed line) upper limits at 95% CL on the coupling strength ratio, g W /g W as functions of the mass of the W boson. These are shown for the electron (left) and muon (right) channels. The one and two standard deviation uncertainty bands for the expected limits are shown. The area above the limit curve is excluded. The dotted line represents the case of SSM couplings, g W , being equal to g W , the SM coupling.

Exclusion limits on the W boson in the split-UED model
To place limits on the split-UED model, the SSM W boson signal is reinterpreted using the extra dimension radius, R, and the bulk mass parameter of the five-dimensional fermion field, µ. For the lowest KK-even mode, n = 2, the decay of the W (2) KK to leptons is kinematically similar to the SSM W boson decay, and the differences in the widths of the resonances have little impact on the results. The impact is at most 2% for µ < 100 GeV, and is negligible for higher values of µ. Taking into account the similar signal shapes and efficiencies allows the lower mass limits on the SSM W boson to be reinterpreted as lower limits on 1/R, for a given value of µ. Separate limits on the electron and muon channels, and on their combination are shown in Fig. 8. For the given value µ = 2 TeV, the observed (expected) lower limits on 1/R are 2.7 (2.6) TeV for electrons, 2.7 (2.7) TeV for muons, and 2.8 (2.7) TeV for the combination of the two channels.

Exclusion limits on R parity violating SUSY
The RPV SUSY interpretation assumes a τ mediator with distinct RPV SUSY couplings at the production and decay vertices. By using the MI limit in Fig. 6, the generated M T distribution and the corresponding f M T fractions, the (σB) excl value was calculated as described in Section 7.2. The cross sections calculated by MADGRAPH5 aMC@NLO at LO for the different coupling values were used to convert (σB) excl into limits on RPV SUSY couplings. While λ 3ij is common to both decay channels, the coupling at the decay vertex is either λ 231 (for the eν µ channel) or λ 132 (for the µν e channel). Upper exclusion limits on λ 231 and λ 132 , as a function of the τ mediator mass, M τ , are shown in Fig. 9  M τ = 1 TeV and λ 3ij = 0.5, values of λ 231 above 3.7 × 10 −3 (4.6 × 10 −3 expected) and λ 132 above 4.7 × 10 −3 (4.7 × 10 −3 expected) are excluded.

Universal new physics through constraints on the W parameter
The comparison between the charged lepton plus a neutrino M T spectra in data and the expected SM background contributions allows the oblique W and Y parameters [38] to be constrained. For nonzero W and Y values, the modification of the M T distributions of the ν system at generator level, relative to that of the SM, can be obtained via event reweighting. The weight is defined as where P W is the expression for the propagator of the qq → W → ν process that depends on the W and Y parameters, P W is the SM W boson propagator (W = Y = 0), t is the tangent of the SM weak mixing angle (t 2 ≈ 0.3), and q 2 is the square of the invariant mass of the ν system at the hard scattering level.
For high M T values, in practice, the weight depends solely on the oblique W parameter and is independent of the oblique Y parameter. This region is where most of the signal sensitivity occurs. Thus, the ν analysis allows the determination of the W parameter value.
As mentioned in the Section 1, the 2016 data set has not been reanalyzed. The combined electron and muon channel data taken during 2017-2018, corresponding to an integrated luminosity of 101 fb −1 , are used to constrain the W parameter.
To find the oblique W parameter value that best reproduces data, a binned negative log likelihood fit has been implemented using MINUIT [76]. The distribution used as input in the fit is the binned M T distribution for data and for the SM prediction as previously presented, where the W → ν contribution is reweighted as explained. The oblique W parameter appears in the fit through the weight of each event in the SM W boson background sample. Systematic uncertainties are included as nuisance parameters in the fit with log-normal distributions. The best-fit value of the parameter and an approximate 68% CL confidence interval are extracted following the procedure described in Section 3.5 of Ref. [77].
We assume that only the SM W → ν process is modified by new physics at high energy scales, disregarding any potential effect on the Drell-Yan Z → background process, which contributes only around 4% of the selected W → ν events. In addition, and according to Ref. [38], the sensitivity of Z boson processes to the W parameter is approximately one half that of W boson processes for similar integrated luminosity and √ s values. This effect is statistically translated to having an excess (or deficit) of 1% of the W boson sample, which is included as a systematic uncertainty in the fit.
By combining the electron and muon channel distributions from the 2017-2018 data sets, a fit value of W = −1.2 +0.5 −0.6 × 10 −4 is found. The uncertainties reflect both statistical and systematic components, and they are in agreement with sensitivity studies performed using pseudodata instead of the experimental data themselves. The outcome of the fit and the region allowed (band at 95% CL) by these results in the oblique (W, Y) parameter space together with the allowed region derived [38] from LEP results [22] at 95% CL are shown in Fig. 10. The SM value of W = 0 lies within the 2 sigma region constrained at 95% CL by the current analysis. This result improves by an order of magnitude on previous limits for the value of the oblique W parameter.

Constraints on the composite Higgs boson model
Results obtained in this search are reinterpreted in the context of composite Higgs boson models. Constraints are placed in the m * -g * plane, where m * is the mass scale of new composite resonances generated in these models and g * the coupling of the new interaction, respectively.
The coupling strength ratio of the W boson to the SM W boson, g W /g W , is related to the g * coupling through Eq. (2). The limits derived in Section 7.3 yield the constraints shown in Fig. 11 (light blue shaded area) where the equivalence between M W and m * is assumed. For clarity, only the observed limit is shown, the expected one being very similar as it is deduced from the expected limit on the coupling strength ratio in Fig. 7. Data sets from 2016-2018 are used, combining the electron and muon channels. The shaded region is excluded, given the assumption of an SSM W , with only couplings to fermions used in the extraction of the coupling ratio limit.  Figure 11: Regions in the m * -g * plane excluded at 95% CL by the results derived in the framework of the different models considered in the current analysis. Constraints from the limit on the oblique W parameter are shown in red. Constraints from the limit on the W boson coupling strength are shown in light blue for the SSM W hypothesis (only coupling to fermions), and in dark blue for the HVT W hypothesis (coupling to bosons as well). In addition, the exclusion coming from current CMS constraints on the Higgs boson cross section is shown as a orange shaded area.
In order to accommodate couplings to bosons and fermions (Heavy Vector Triplet model, HVT [39]), a reweighting procedure has been introduced. It implements the ratio of Breit-Wigner distributions of corresponding fermionic plus bosonic widths relative to that of the fermionic width, and takes into account the different cross section values. The region excluded in the m * -g * plane, achieved by combining the results from the electron and muon channels for the 2017-2018 data sets, is shown by the dark blue area in Fig. 11. This region is less restrictive than the light blue one because of the smaller branching fraction being tested with the current final state and the reduced amount of data. However, the interpretation is more general in this HVT framework.
Being quantities extracted from a direct search, these constraints are kinematically limited by √ s. The EFT approach followed to set constraints on the oblique W parameter at 95% CL extends beyond the limit established by the direct search. The constraint from the oblique W parameter, derived from the combination of the electron and muon channels, is also used to set limits in the m * -g * plane, following Eq. (3). Numerically, this implies g * < (4770 GeV)/m * . Figure 11 shows the region (red shaded) excluded by these results at 95% CL. This approach extends the constraints up to high resonance masses, since the EFT approach remains valid in this region.
In addition, current CMS constraints on the deviation (∆µ H ) of the Higgs boson production cross section from the SM value also set limits in the m * -g * plane of the composite Higgs boson sector. Considering the value measured by CMS in a combination of several Higgs bo-son decay modes [26] of ∆µ H < 0.20 at 95% CL, and following Eq. (4), the exclusion region g * > (0.00182 GeV −1 )m * is established. The excluded region coming from the Higgs boson cross section constraint, together with the regions already presented, is shown in Fig. 11 (orange triangular area in the upper-left part of the plot). Taking all exclusion regions into account, an energy scale for Higgs boson compositeness can be excluded below m * = 3 TeV under the SSM W boson assumption, or alternatively, below 1 TeV for an HVT interpretation of the W resonance.
All limits obtained in this paper are summarized in Table 3. Composite Higgs Regions in m * -g * plane in SSM e + µ m * > 3 TeV Regions in m * -g * plane in HVT e + µ m * > 1 TeV

Summary
A search for a deviation relative to standard model (SM) expectations in events with a final state consisting of a lepton (electron or muon) and missing transverse momentum in proton-proton collisions at a centre-of-mass energy of 13 TeV has been performed. The analysis strategy is similar to that of the previous study using data corresponding to an integrated luminosity of 36 fb −1 [19]. This search uses data collected by the CMS detector in 2016-2018 corresponding to 138 fb −1 of total integrated luminosity. In addition to employing four times larger data set, this search considers two new interpretations.
No evidence for new physics is observed when examining the transverse mass distributions. These observations are interpreted as limits on the parameters of several models. The exclusion limits at 95% confidence level (CL) on the mass of a sequential standard model (SSM) W boson are found to be 5.4 and 5.6 TeV for the electron and muon channels, respectively. The 95% CL exclusion limit from the combination of both channels is 5.7 TeV. Variations in the coupling strength of the SSM W boson are also examined. Models for which the ratio of the coupling strength of the W boson to the SM W boson is at the level of 2 × 10 −2 are excluded for W masses up to 0.5 TeV. For higher masses the constraint on the coupling weakens, approaching g W /g W = 1 at the value of M W corresponding to the exclusion limit obtained in the SSM analysis. The result is also interpreted in the split universal extra dimensions model. The inverse radius of the extra dimension, 1/R, is constrained to be larger than 2.8 TeV if the bulk mass parameter of the five-dimensional fermion field, µ = 2 TeV.
Model-independent limits are also provided. These can be used to constrain parameters of several models through reinterpretations. As an example, the limits have been interpreted in the context of an R-parity violating supersymmetric model. This interpretation provides limits on the coupling strengths at the decay vertex as a function of the mediator τ mass, for various coupling values, λ 3ij , at the production vertex.
In addition to these results, new interpretations of an ν final state analysis using LHC data are considered for the first time. One is the measurement of the oblique W parameter, using the combined electron and muon channels. The oblique W parameter fitted value is W = −1.2 +0.5 −0.6 × 10 −4 , in agreement within uncertainties with the SM value. This is the most precise measurement to date of the oblique W parameter, reducing its allowed range by more than an order of magnitude relative to previous bounds derived from LEP results.
Another new interpretation is made within the framework of composite Higgs boson models. The 95% CL exclusion limit for the combined channels is set in the m * -g * plane, where m * indicates the mass scale of compositeness and g * the coupling strength of the new composite sector. In this case, interpretations of several of the presented results have been used in a complementary way, as different inputs for setting the limit. Each of the inputs is sensitive to a different region in the m * -g * plane. Constraints coming from the limit on the W coupling strength dominate the low-m * region and are kinematically limited by the collider energy. The high-m * region is probed by the indirect oblique W parameter interpretation. Finally, we set a lower limit of 3 (1) TeV for the mass scale of such a composite Higgs boson, when an SSM (Heavy Vector Triplet) W boson is assumed, extending existing constraints from previous measurements.     [13] ATLAS Collaboration, "ATLAS search for a heavy gauge boson decaying to a charged lepton and a neutrino in pp collisions at √ s = 7 TeV", Eur. Phys. J. C 72 (2012) 2241, doi:10.1140/epjc/s10052-012-2241-5, arXiv:1209.4446.