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Measurement of W AE γ differential cross sections in proton-proton collisions at ffiffi s p = 13 TeV and effective field theory constraints Differential cross section measurements of W AE γ production in proton-proton collisions at ffiffi ffi s p ¼ 13 TeV are presented. The data set used in this study was collected with the CMS detector at the CERN LHC in 2016-2018 with an integrated luminosity of 138 fb −1 . Candidate events containing an electron or muon, a photon, and missing transverse momentum are selected. The measurements are compared with standard model predictions computed at next-to-leading and next-to-next-to-leading orders in perturbative quantum chromodynamics. Constraints on the presence of TeV-scale new physics affecting the WWγ vertex are determined within an effective field theory framework, focusing on the O 3W operator. A simultaneous measurement of the photon transverse momentum and the azimuthal angle of the charged lepton in a special reference frame is performed. This two-dimensional approach provides up to a factor of ten more sensitivity to the interference between the standard model and the O 3W contribution than using the transverse momentum alone. DOI: 10.1103/PhysRevD.105.052003

I. INTRODUCTION
The measurement of the properties of vector boson pair production is an important test of the electroweak sector of the standard model (SM). The production cross sections of these processes are sensitive to the WWV triple gauge couplings (TGCs), where V ¼ γ or Z. Because the strengths of these TGCs are predicted precisely in the SM, the measurement of an anomalous value would indicate the presence of physics beyond the SM (BSM).
This paper reports an analysis of W AE γ production in proton-proton (pp) collisions at ffiffi ffi s p ¼ 13 TeV using data recorded with the CMS detector at the CERN LHC in 2016-2018 with an integrated luminosity of 138 fb −1 . This process has previously been studied by the CDF and D0 Collaborations [1][2][3]  All of these studies found the measured inclusive cross sections to be compatible with the SM predictions and set limits on the presence of anomalous TGCs.
In the present analysis, W AE γ events are selected with one charged lepton (l), which is either an electron or muon, one neutrino (ν), and one photon (γ) in the final state. A few leading order (LO) Feynman diagrams are shown in Fig. 1. The photon can be produced as a result of either initial-or final-state radiation, in addition to the contribution involving a TGC vertex.
Differential cross sections are measured for several observables, including the transverse momentum p γ T and pseudorapidity η γ of the photon, and the transverse mass of the lνγ system, denoted m cluster T . The cross section is also measured as a function of the number of additional jets. The measured values are compared with the SM predictions at next-to-leading order (NLO) and next-to-next-to-leading order (NNLO) in perturbative quantum chromodynamics (QCD). Interference between the LO W AE γ production diagrams results in a cross section that vanishes in specific phase space regions. This effect is known as a radiation amplitude zero (RAZ) [7][8][9][10][11]. A differential cross section measurement of the pseudorapidity difference between the lepton and the photon, Δηðl; γÞ, is used to explore this effect.
Constraints on BSM contributions to the WWγ vertex are determined in an effective field theory (EFT) framework. For the first time in the study of this process, both p γ T and the angular properties of the final-state particles are exploited to increase the sensitivity to the interference between the SM and BSM amplitudes. This novel approach is referred to as "interference resurrection" [12,13].
The paper is organized as follows. The interference resurrection technique is described in Sec. II. The CMS detector, data samples, and event simulation are summarized in Secs. III and IV. The object reconstruction and event selection are described in Secs. V and VI. The estimation of the main backgrounds is given in Sec. VII. The systematic uncertainties are discussed in Sec. VIII, the results presented in Sec. IX, and the paper summarized in Sec. X.

II. INTERFERENCE RESURRECTION
An EFT approach can be used to study how new physics entering at an energy scale Λ, assumed to be much larger than the electroweak scale, leads to deviations from the SM at an energy regime accessible at the LHC. The SM EFT is constructed by the addition of higher-dimensional operators, O i , to the SM Lagrangian, where i enumerates the set of operators under consideration, C ðDÞ i are Wilson coefficients that scale as Λ 4−D and D denotes the operator dimension. The leading deviations from the SM are generally expected to occur at D ¼ 6, since D ¼ 5 operators violate lepton number conservation [14]. Examples of the relationship between the operators affecting diboson production and specific BSM scenarios are described in Refs. [15,16].
The dimension-six operator of interest in this analysis, O 3W , is a CP-even modification of the WWV TGC defined in the EFT basis of Ref. [17] as where W iν μ is the weak isospin field strength tensor and ϵ ijk is the totally antisymmetric tensor with ϵ 123 ¼ 1. The cross section in the presence of this operator can be expressed as where C 3W is the Wilson coefficient, σ int is the contribution from the interference between the SM and O 3W , and σ BSM is the pure BSM component.
However, it has been demonstrated [12,13] that in the high-energy limit, E > m W , the 2 → 2 amplitudes for transverse vector boson production, ff → W T V T , have different final-state helicity configurations for the SM (AE ∓) and BSM (AEAE) components. This means the effect of the interference is typically not detectable when considering observables inclusive over the decay angles, for example, the p T of the photon or W AE boson. This narrows our sensitivity in such observables to just the pure BSM contribution at order C 2 3W , which scales as Λ −4 . In this scenario, the validity of any derived constraints can be limited by the unknown effect of the leading dimensioneight contributions, which also enter at order Λ −4 . Therefore, increasing the sensitivity to the interference, which scales as Λ −2 , is important for improving the validity of the constraints in any global EFT interpretation [18].
A method has been proposed [12,19] that gives sensitivity to the SM-BSM interference by measuring the decay angles of the final-state fermions. A special coordinate system, illustrated in Fig. 2, is defined event-by-event by a Lorentz boost to the center-of-mass frame of the W AE γ system, where the boost direction is denotedr. Since the longitudinal component of the neutrino momentum is not measurable, additional constraints are required in the calculation of the W AE γ four-momentum, described in FIG. 2. Scheme of the special coordinate system for W AE γ production, defined by a Lorentz boost to the center-of-mass frame along the directionr. The z axis is chosen as the W AE boson direction in this frame, and y is given byẑ ×r. The W AE boson decay plane is indicated in blue, where the labels f þ and f − refer to positive and negative helicity final-state fermions. The angles ϕ and θ are the azimuthal and polar angles of f þ .
FIG. 1. LO Feynman diagrams for W þ γ production showing initial-state (left) and final-state (center) radiation of the photon, and the WWγ TGC process (right). A. TUMASYAN et al. PHYS. REV. D 105, 052003 (2022) Sec. VI A. In the boosted frame the boson momenta are back-to-back, and the z axis is taken as the direction of the W AE boson, the y axis direction is given byẑ ×r, and the x axis given byŷ ×ẑ. The final-state fermions from the W AE boson decay are labeled as f þ and f − , referring to the positive and negative fermion helicity states, respectively. In this frame the angle θ is the polar angle of the W AE boson decay in its rest frame, taken as the angle between the three-momenta of the W AE boson and fermion f þ . The angle ϕ ∈ ½−π; π is the azimuthal angle of f þ : A measurement of ϕ is sensitive to the interference of the O 3W term with the SM contribution. Figure 3 (upper) shows the particle-level distribution in ϕ in the LO 2 → 2 scattering process for varying values of C 3W , with only the inclusion of the interference term. A clear modulation is present, which grows with increasing C 3W . Figure 3 (lower) shows that the magnitude of this modulation is reduced with the inclusion of additional jets in the matrix element calculations, with MLM merging [20] used in the matching to the parton shower. However, a veto on the presence of additional jets in the event with p T > 30 GeV and jηj < 2.5 is shown to substantially restore the effect.

III. THE CMS DETECTOR
The central feature of the CMS apparatus is a superconducting solenoid of 6 m internal diameter, providing a magnetic field of 3.8 T. Within the solenoid volume are a silicon pixel and strip tracker, a lead tungstate crystal electromagnetic calorimeter (ECAL), and a brass and scintillator hadron calorimeter (HCAL), each composed of a barrel and two endcap sections. Forward calorimeters extend the pseudorapidity coverage provided by the barrel and endcap detectors. Muons are detected in gas-ionization chambers embedded in the steel flux-return yoke outside the solenoid.
In the barrel section of the ECAL, an energy resolution of about 1% is achieved for unconverted or late-converting photons in the tens of GeV energy range. The remaining barrel photons have a resolution of about 1.3% up to jηj ¼ 1, rising to about 2.5% at jηj ¼ 1.4. In the endcaps, the resolution of unconverted or late-converting photons is about 2.5%, while the remaining endcap photons have a resolution of 3%-4% [21].
The electron momentum is estimated by combining the energy measurement in the ECAL with the momentum measurement in the tracker. The momentum resolution for electrons with p T ≈ 45 GeV from Z → ee decays ranges from 1.7% to 4.5%. It is generally better in the barrel region than in the endcaps, and also depends on the bremsstrahlung energy emitted by the electron as it traverses the material in front of the ECAL [22].
Muons are measured in the range jηj < 2.4, with detection planes made using three technologies: drift tubes, cathode strip chambers, and resistive plate chambers. Matching muons to tracks measured in the silicon tracker results in a relative p T resolution of 1% in the barrel and 3% in the endcaps for muons with p T up to 100 GeV. The p T resolution in the barrel is better than 7% for muons with p T up to 1 TeV [23].
Events of interest are selected using a two-tiered trigger system. The first level (L1), composed of custom hardware processors, uses information from the calorimeters and muon detectors to select events at a rate of around 100 kHz within a fixed latency of about 4 μs [24]. The second level, known as the high-level trigger, consists of a farm of   (2022) processors running a version of the full event reconstruction software optimized for fast processing, and reduces the event rate to around 1 kHz before data storage [25]. A more detailed description of the CMS detector, together with a definition of the coordinate system used and the relevant kinematical variables, can be found in Ref. [26].

IV. DATA SAMPLES AND EVENT SIMULATION
This analysis uses pp collision data collected by the CMS experiment between 2016 and 2018 at ffiffi ffi s p ¼ 13 TeV with an integrated luminosity of 138 fb −1 . Simulated event samples are used to model both the W AE γ signal and several backgrounds. The W AE γ signal, with up to one additional jet, and the Z þ jets, Zγ, γγ, single-t þ γ, WW, WZ, and ZZ background samples are generated with MadGraph5_aMC@NLO [27] v2.3.3-2.6.5 at NLO in QCD. Backgrounds from tt þ jets and single-t production are simulated with POWHEG v2.0 [28][29][30][31][32][33]. The ttγ background is simulated with MadGraph5_aMC@NLO v2.6.0 at LO in QCD. The effect of EFT operators on W AE γ distributions is modeled using a dedicated sample generated with MadGraph5_aMC@NLO v2.6.7 at LO with up to two additional jets. Event weights for varying values of C 3W are embedded, utilizing LO matrix element reweighting [34] and the SMEFTSIM UFO model [35,36].  [43]. Additional pp interactions occurring in both the same and adjacent bunch crossings (pileup) are simulated as a set of minimum bias interactions that are mixed with the hard scattering event. In the analysis of each datataking year the simulated events are weighted based on the number of pileup events to match the distribution measured in data.

V. EVENT AND OBJECT RECONSTRUCTION
The pp collision vertices in an event are reconstructed by grouping tracks consistent with originating at a common point in the luminous region. The candidate vertex with the largest value of summed physics-object p 2 T is taken to be the primary pp interaction vertex. The physics objects are the jets, clustered using the anti-k T jet finding algorithm [44,45] with all the tracks assigned to candidate vertices as inputs, and the associated missing transverse momentum, taken as the negative vector p T sum of those jets.
The full event reconstruction uses the particle-flow (PF) algorithm [46], which aims to reconstruct and identify each individual particle in an event, with an optimized combination of information from the various elements of the CMS detector. The energy of photons is obtained from the ECAL measurement. The energy of electrons is determined from a combination of the electron momentum at the primary interaction vertex as determined by the tracker, the energy of the corresponding ECAL cluster, and the energy sum of all bremsstrahlung photons spatially compatible with originating from the electron track. The energy of muons is obtained from the curvature of the corresponding track. The energy of charged hadrons is determined from a combination of their 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 corresponding corrected ECAL and HCAL energies.
For each event, hadronic jets are clustered from these reconstructed particles using the infrared-and collinearsafe anti-k T algorithm [44,45] with a distance parameter of 0.4. The 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 entire p T spectrum and detector acceptance. Pileup interactions can contribute additional tracks and calorimetric energy depositions to the jet momentum. To mitigate this effect, charged particles identified to be originating from pileup vertices are discarded, and an offset correction is applied to subtract remaining contributions. Jet energy corrections are derived from simulation to bring the measured response of jets to that of particle level jets on average. In situ measurements of the momentum balance in dijet, photon þ jet, Z þ jet, and multijet events are used to correct any residual differences in the jet energy scale between data and simulation [47].
Events of interest in this analysis are characterized by the presence of a single isolated charged lepton, either an electron or a muon; an isolated photon; and missing transverse momentum. Identification and isolation criteria are used to improve the purity of events containing prompt leptons or photons, while rejecting candidates that originate from jets, pileup interactions, or misreconstruction.
The identification criteria for photons [48] include selections on the ratio of hadronic to electromagnetic energy deposited in calorimeters (H=E), a variable quantifying the lateral extension of the shower (σ ηη ), and a requirement that the photon ECAL cluster is not matched to an electron track that has hits in the innermost tracker layers. Electron candidates are required to pass identification criteria related to the impact parameters of the track with respect to the primary vertex, the matching between the track and ECAL cluster, the shower shape in the ECAL, and the same H=E variable as for photons. A veto on A. TUMASYAN et al. PHYS. REV. D 105, 052003 (2022) electrons originating from photon conversions is applied [21]. Muon candidates are reconstructed by a simultaneous track fit to hits in the tracker and in the muon chambers. Identification criteria are applied to the impact parameters with respect to the primary vertex, to the number of spatial measurements in the silicon tracker and the muon system, and to the fit quality of the combined muon track. Both lepton and photon candidates are required to be isolated from any other activity in the event. For electrons and muons, a relative isolation variable is defined as where the sums are over the PF candidates of the indicated type that are within ΔR < 0.4 of the lepton, with ΔR ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ðΔηÞ 2 þ ðΔϕÞ 2 p . To suppress the effect of pileup, only charged hadrons compatible with originating at the primary vertex are included. For the neutral hadron and photon sums, an estimate of the expected pileup contribution is subtracted [23,48]. Requirements of I μ rel < 0.15 and I e rel < 0.05-0.09 are applied, where the latter selection varies with p e T and jη e j. For photons, isolation requirements are applied separately to each of the three components in Eq. (4). In addition, a selection is applied on the maximal charged particle isolation sum I max ch computed with respect to any vertex. A requirement of I max ch < minð0.05p γ T ; 6 GeVÞ is applied. This more than halves the rate of photons from pileup jets, which constitute around 60% of misidentified photons at low p γ T . For leptons and photons, several corrections are applied to the simulation to improve the modeling of the data. Correction factors are applied as event weights for the efficiencies of electrons and muons to be reconstructed, pass their respective identification and isolation criteria, and pass the single-lepton trigger requirements. These are determined using the "tag-and-probe" (T&P) technique [49] with Z → ll data. Similarly, scale factors for the efficiencies of photons to pass identification and isolation criteria are applied. These are measured with the T&P technique in Z → ee events, where the probe electron is reconstructed as a photon. The muon momentum scale in data and simulation is corrected using a calibration derived in Z → μμ events [23,50]. In simulation this includes an additional resolution smearing to match the performance in data. Similarly, scale and resolution corrections are applied to simulated electrons and photons to better match the data resolution.
The missing transverse momentum 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 [51]. The ⃗p miss T vector is modified to account for corrections to the energy scale of the reconstructed jets. The pileup-per-particle identification algorithm [52] is applied to reduce the pileup dependence of the ⃗p miss T observable. In the sum over PF candidates, each p T is weighted by the probability for that particle to originate from the primary interaction vertex.
Anomalous high-p miss T events can be due to a variety of reconstruction failures, detector malfunctions, or noncollision backgrounds. Such events are rejected by event filters that are designed to identify more than 85% of the spurious high-p miss T events with a mistagging rate of less than 0.1% [51].

VI. EVENT SELECTION
Single-lepton triggers are used to select data events. These require that candidates pass loose identification and isolation criteria and have a p T above thresholds of 27-32 GeV for electrons and 24-27 GeV for muons, with the exact values depending on the data-taking year.
The offline event selection requires the presence either of an electron with p T > 35 GeV and jηj < 2.5 (electron channel) or a muon with p T > 30 GeV and jηj < 2.4 (muon channel). The lepton must pass the identification criteria outlined in Sec. V and be spatially matched to the object that fired the trigger. Electron candidates falling in the transition region between the ECAL barrel and endcap calorimeters, 1.44 < jηj < 1.57, are rejected. Events are vetoed if they contain additional leptons with p T > 10 GeV that pass looser versions of the identification criteria. The same photon selection is used for both channels. The photon is required to have p T > 30 GeV and jηj < 2.5, not fall in the ECAL transition region, pass the identification and isolation requirements and be separated from the selected lepton by ΔR > 0.7. Events with more than one photon passing these requirements are rejected. Hadronic jets in the event are counted if having p T > 30 GeV, jηj < 2.5, and a separation from both the lepton and photon of ΔR > 0.4.
In both channels, events are required to have p miss T > 40 GeV. This reduces a large source of background from processes that do not contain any intrinsic p miss T , for example Z → ll. Events with m lγ , the invariant mass of the lepton-photon pair, close to m Z are rejected. In the electron channel this veto is on events with 70 GeV < m eγ < 110 GeV, which reduces the remaining Z → ee contribution. In the muon channel the range is 70 GeV < m μγ < 100 GeV, which rejects events where the photon originates as final-state radiation, Z → μμ → μμγ, and takes a large fraction of the original muon momentum.
This baseline event selection results in the selection of 72,798 electron channel and 109,669 muon channel events, of which W AE γ production is expected to account for 53% D 105, 052003 (2022) and 58%, respectively. Figure 4 shows the p l T , p γ T , m T ðl; p miss T Þ, and m cluster T distributions in data and simulation, summed over the electron and muon channels. The m cluster T observable is the transverse mass of the lνγ system, defined as where p T;lγ is the transverse momentum of the lepton and photon system, and p T;lγmiss is the transverse momentum of the lepton, photon, and p miss T system. The baseline selection is used for the majority of the differential cross section measurements reported in Sec. IX. For the EFT analysis a tighter selection, the EFT selection, with p l T > 80 GeV and p γ T > 150 GeV is applied, and events containing hadronic jets, as defined above, are vetoed.

A. Reconstruction of ϕ
In the EFT analysis the neutrino four-momentum is required to construct the W AE γ center-of-mass frame and determine the azimuthal angle ϕ. The reconstruction procedure described in Ref. [12] is followed. The ⃗p miss T vector in the event is assumed to be the transverse momentum of the neutrino, ⃗p ν T . Then by requiring that the invariant mass of the charged lepton and neutrino system equals the W AE boson pole mass m W , it is possible to form an equation for η ν , Stat.+Syst. Uncertainty

CMS
η l is the pseudorapidity of the lepton, and m T is the transverse mass of the lν system. Of the two possible solutions for η ν , only one will correspond to the unknown true value. One of the two solutions is picked at random event-by-event. In the limit of high W AE boson momentum, it can be demonstrated that the two solutions for ϕ, ϕ þ , and ϕ − , are related by ϕ þ ¼ π − ϕ − , modulo 2π. This intrinsic ambiguity does not, however, prevent the observation of the interference effect. This is illustrated in Fig. 3, where the deviation in the ϕ distribution is unaffected by a transformation ϕ → π − ϕ.
One additional complication to this reconstruction is in cases where m T > m W , either because the W AE boson was produced off shell, or because of mismeasured ⃗p miss T in the reconstruction. In this case Eq. (6) has no real solutions and instead η ν is assumed to equal η l , which gives a leptonneutrino mass as close as possible to m W . In these events ϕ is biased towards AEπ=2. Figure 5 shows the two-dimensional (2D) distribution of ϕ gen versus ϕ true , where ϕ gen is the reconstructed ϕ angle using particle-level quantities.
Given this ambiguity in ϕ, the variable ϕ f is used in the subsequent analysis, where Furthermore, since the distributions of ϕ f shown in Fig. 3 are symmetric about zero, the magnitude jϕ f j will be used.

VII. BACKGROUND ESTIMATION
The background from processes containing prompt leptons and photons is estimated from simulation. This includes Z=γ Ã ðllÞγ, ttγ, WVγ, and single-t þ γ production, where V ¼ W, Z.
An important background in the electron channel is from events with two prompt electrons, predominantly Z → ee, in which one is misidentified as the photon candidate. The rate at which electrons, having passed the photon identification criteria described in Sec. V, are subsequently not rejected by the specific electron veto is measured in both data and simulation using the T&P method. The probability in data is typically 1%-5%, depending on the p T and η of the electron. The resulting correction factors are applied in all simulated events where the photon candidate is matched to a prompt electron at the generator level.
The remaining backgrounds originate from other processes where one or both of the lepton and photon are misidentified, and these are discussed in the following sections.

A. Jets misidentified as photons
After the baseline selection, the main background in both channels comes from W þ jets events in which a jet is misidentified as a photon. These jets tend to have high fractions of energy deposited in the ECAL, and may contain genuine photons, for example from π 0 → γγ decays. Since the probability for jets to pass the photon identification criteria is not guaranteed to be well modeled in simulation, this background is estimated using data.
Independent control regions, with a high purity in misidentified jets, are defined by inverting some of the photon identification requirements. Two variables are used: the charged isolation sum defined in Eq. (4) denoted I ch , and σ ηη . This exploits the fact that misidentified jets tend to be less well isolated and have broader showers than prompt photons. For a given selection and observable bin, the events in the inverted-σ ηη region are used to predict the number of misidentified jet events in the nominal region, N misid . Each inverted-σ ηη event is assigned an extrapolation factor, f γ , corresponding to the expected inverted-tonominal ratio given the p T and jηj of the photon candidate. This gives where the sum runs over all data events in the inverted-σ ηη region, and N prompt-γ sim is the estimated prompt photon contamination, determined by applying the same extrapolation factors to simulated events in this region. The f γ FIG. 5. The two-dimensional distribution of ϕ gen versus ϕ true , where the former is reconstructed using the particle-level lepton and photon momenta and ⃗p miss T . The off-diagonal components correspond to events where the incorrect solution for η ν is chosen, as described in the text.
MEASUREMENT OF W AE γ DIFFERENTIAL CROSS … PHYS. REV. D 105, 052003 (2022) value for each p T and jηj bin is determined from a region where I ch is inverted. The σ ηη extrapolation factor, calculated after the subtraction of the expected prompt photon contribution, varies linearly with I ch . Therefore, a linear fit to the ratio is performed, with the result extrapolated to the nominal I ch region to give f γ .
The statistical uncertainty in f γ is 2%-35%, depending on the p T and jηj bin. Systematic uncertainties come from the normalization of the prompt photon contribution that is subtracted, and the bias in the extrapolated value of f γ , compared with the true value, when the method is validated in simulation. The former is typically 10%, and the latter is up to 15% in a given bin. The total statistical and systematic uncertainty, which ranges from 10% to 45%, is propagated through all subsequent results.
In the EFT selection, because of the limited number of control region events for p γ T > 150 GeV, this method is modified. In the inverted-σ ηη region, the nominal requirement on I ch is retained, whereas all other photon identification criteria are removed and replaced with only a loose requirement of H=E < 0.15. The total uncertainty in f γ is 20%-60% under this selection, and the statistical component is typically dominant.

B. Misidentified leptons
Another contribution comes from events with a prompt photon but a nonprompt or misidentified lepton, e.g., within a jet containing a semileptonic meson decay. The main identifying feature of such events is that the leptons will generally be less well isolated. The contribution is estimated by applying an inverted lepton isolation requirement, and applying a per-event extrapolation factor f l , binned in p T and jηj, to predict the number of events in the nominal isolated selection.
The f l factors are measured in a control region in data designed to increase the purity of events with misidentified leptons. This control region is obtained by applying the baseline selection in both channels, but vetoing the presence of an identified photon and instead requiring the presence of a jet with p T > 30 GeV and jηj < 2.5. A requirement of m T ðl; p miss T Þ < 30 GeV is applied to reject the background from W þ jets events. The remaining contamination from W þ jets and other prompt-lepton sources is estimated using simulation and subtracted. The normalization uncertainty in this prompt subtraction dominates the uncertainties in the f l factors, which are typically 10%-40%.
Both the misidentified lepton and misidentified photon background estimates predict the contribution from events where both objects are misidentified. To avoid doublecounting these events, their expected contribution is subtracted from the misidentified lepton background estimate. This is estimated using events in a region where both the lepton isolation and photon σ ηη requirements are inverted, where the extrapolation factors for both methods are applied.

VIII. SYSTEMATIC UNCERTAINTIES
The effects of systematic uncertainties in the signal and background expectations are incorporated for all measurements in this analysis. The majority of the uncertainties affect both the shape and normalization of the predictions, whereas several affect only the normalization. The shape variations are modeled by a smooth interpolation between the nominal and AE1 standard deviation predictions in each bin and a linear extrapolation beyond this. The systematic uncertainties and their typical effect on the expected yields are summarized in Table I. The experimental sources are as follows: (a) Integrated luminosity: The 2016, 2017, and 2018 data set uncertainties are 1.2%, 2.3%, and 2.5% [53-55], respectively, applied to all processes estimated purely with simulation. The total integrated luminosity has an uncertainty of 1.6%, the improvement in precision reflecting the (uncorrelated) time evolution of some systematic effects. (b) Pileup modeling: The uncertainty in the weighting of the pileup distribution in simulation is derived by varying the total inelastic cross section by AE5% in the estimate of the distribution in data [56]. (c) L1 trigger: During the 2016 and 2017 data-taking periods, a gradual shift in the timing of the inputs of the ECAL L1 trigger in the jηj > 2.4 region led to a specific inefficiency. A correction amounting to 2%-3% is applied to the simulation along with the corresponding uncertainty in the inefficiency measurement. (d) Lepton ID and trigger efficiencies: The uncertainties in the measured correction factors are applied to the simulation, which include both statistical and systematic sources. The latter dominate at low p T and the former in the higher p T range probed in this analysis. (e) Photon ID efficiency: The uncertainties in the identification correction factors, derived in Z → ee events, are applied to the simulation. The highest p T measurement bin (p γ T > 200 GeV) is dominated by the statistical uncertainty and so is uncorrelated from the uncertainty at lower p T . An uncertainty in the extrapolation of the efficiencies to the maximum p γ T considered, 1.5 TeV, is also included. calculation. An additional uncertainty is included for the energy scale of PF candidates not clustered in jets.
(i) Electrons misidentified as photons: An uncertainty in the correction of the electron to photon misidentification rate, due to the subtraction of genuine photon events in the T&P measurement, is derived. (j) Jets misidentified as photons: For jets misidentified as photons, the uncertainties from the method based on control samples in data described in Sec. VII A are applied, independently for each p γ T and jη γ j bin that f γ is measured in. These uncertainties are correlated between the electron and muon channels. (k) Misidentified leptons: The dominant source of uncertainty in the f l measurement, described in Sec. VII B, comes primarily from the normalization of the subtracted prompt lepton contamination in the measurement region.
(l) Simulated sample size: The statistical uncertainty due to the finite number of simulated events per bin is applied following the Barlow-Beeston method [57]. Theoretical uncertainties in the W AE γ signal prediction because of missing higher orders in the perturbative cross section calculation are evaluated by varying the renormalization and factorization scales, denoted μ R and μ F , respectively, in simulated events by factors of 0.5 and 2. The maximum variation among all possible pairings of μ R and μ F is considered, with the exception of the (0.5,2) and (2,0.5) combinations. For constraints on C 3W , the full uncertainty from this procedure is applied, which amounts to an 8%-11% normalization effect depending on the bin. For the cross section measurements, the effect on the predicted cross sections is neglected, whereas the effect on the reconstruction-level acceptance is retained. These values typically vary from 0.5% to 2% and are treated as independent for each measured cross section bin. Conversely, for the contamination of signal events from TABLE I. Summary of the systematic uncertainties affecting the signal and background predictions. The table notes whether each uncertainty affects the shape of the measured observable or just the normalization (Affects shape), and whether the effect is correlated between the data-taking years (Correlated years). The normalization effect on the expected yield of the applicable processes is also given. For some shape uncertainties the values vary significantly across the observable distribution. In these cases the typical range and maximum values are given, where the former is the central 68% interval considering all bins.

Uncertainty
Affects shape Correlated years Relative effect on expected yield  (2022) outside the fiducial region, the effect on both cross section and acceptance is included. Uncertainties from the NNPDF3.1 PDF set eigenvectors are also evaluated following the PDF4LHC prescription [58], and affect the acceptance by up to 2%. For measurements that include a jet veto, an additional migration uncertainty between the 0-jet and ≥ 1-jet regions is applied, following the procedure described in Ref. [59]. This results in an 8.6% uncertainty in the 0-jet cross section. Normalization uncertainties are assigned for each background process estimated with simulation to account for missing higher-order corrections and the PDF choice. These are treated as uncorrelated between different processes, and are calculated following the same procedures as for the signal. The missing higher-order correction uncertainties typically range from 2% to 3.5%, and the PDF uncertainties are up to 4.8%, depending on process.

IX. RESULTS
The results described in this section are obtained using statistical procedures developed by the ATLAS and CMS Collaborations, detailed in Refs. [60,61] and implemented in the ROOFIT [62] and ROOSTATS [63] frameworks.
The parameters of interest ⃗ α for a particular result are estimated with their corresponding confidence intervals using a profile likelihood ratio test statistic qð⃗ αÞ [64], in which experimental and theoretical uncertainties are incorporated via nuisance parameters (NPs) ⃗ θ: The asymptotic approximation, whereby qð⃗ αÞ is assumed to follow a χ 2 distribution, is used in the determination of the confidence intervals. The likelihood functions in the numerator and denominator of Eq. (10) are constructed using products of binned signal and background probability density functions, as well as constraint terms for the NPs.
The quantities⃗α and⃗ θ denote the unconditional maximum likelihood estimates of the parameter values, while⃗ θ ⃗ α denotes the conditional maximum likelihood estimate for fixed values of the parameters of interest ⃗ α. The binned likelihood function is expressed as where n i is the number of observed events in each bin, s and b are the expected numbers of signal and background events, and⃗ θ represents the nominal values of the NPs, determined by external measurements.
For the measurement of a set of fiducial cross sections, ⃗ α ≡ σ j , the signal expectation s is constructed as where j runs over the set of W AE γ fiducial cross sections being measured; k runs over a set of unmeasured fiducial cross section bins, referred to as out-of-acceptance (OOA) and discussed further below; L is the integrated luminosity of a given data set; and R ij is the response matrix, parameterized by ⃗ θ, giving the probability for an event in fiducial bin j to be reconstructed in analysis bin i.
In this measurement a significant OOA contribution can come from events that are within the fiducial acceptance with the sole exception of the p miss T requirement, because p miss T is measured with poor resolution compared with the transverse momentum of the leptons and photons. This component is therefore split into the same set of fiducial bins j but with a modified p miss T selection.
Since it is not possible to measure these OOA cross sections σ miss j simultaneously, they are related to the σ j using the predicted ratios from the W AE γ simulation: σ miss j ¼ ðσ miss;MC j =σ MC j Þσ j . The remaining OOA contribution σ OOA is small and is fixed to the value predicted by the simulation. This includes events in which the electron or muon originates from a tau lepton decay. Equation (12) above can then be expressed as In addition to measuring the absolute σ j , it is useful to present the results as fractional cross sections, f j ¼ σ j =σ tot , where σ tot is treated as a free parameter in the likelihood fit and P f j is constrained to unity. This facilitates comparisons between different predictions purely on the basis of observable shape, and reduces the impact of systematic uncertainties that are uniform across the distribution, for example from the integrated luminosity.
Tabulated versions of all the results in this section are provided in the HEPData record for this analysis [65].

A. Differential cross sections
The differential cross sections σ j ðpp → W AE γ → l AE νγÞ, where l denotes all three lepton flavors, are measured in the following fiducial region: (a) p l T > 30 GeV, jη l j < 2.5,   These selections are chosen to match as closely as possible with the reconstruction-level baseline described in Sec. VI, with two exceptions: in the electron channel p e T > 35 GeV is required because of the higher trigger thresholds, and in the muon channel jη μ j < 2.4 is applied to reflect the acceptance of the muon system.

CMS
The lepton momenta are completed by adding the fourmomenta of any photons with ΔRðl; γÞ < 0.1 to the fourmomentum of the lepton. A smooth-cone photon isolation selection is also applied. This is necessary to ensure an infrared-safe cross section calculation in perturbative QCD. It requires that for every cone in η-ϕ around the photon direction with radius δ < δ 0 the total scalar transverse energy sum E T must be smaller than E max T ðδÞ [66] where and the values of the tunable parameters are set as δ 0 ¼ 0.4, ϵ ¼ 1.0, and n ¼ 1. Differential cross sections are measured for p γ T , η γ , ΔRðl; γÞ, Δηðl; γÞ, and m cluster T under the baseline selection. The jϕ f j observable is measured only under the higher-p T EFT selection, given in Sec. IX D, where the resolution is sufficiently improved compared to the baseline selection.
The signal model is constructed as in Eq. (13), where the OOA contributions σ miss j are defined for events with 0 < p miss T < 40 GeV, but otherwise within the fiducial region for j. A simultaneous fit is performed to the reconstructed distributions of the observables in each year and for each of the electron and muon channels, where A. TUMASYAN et al. PHYS. REV. D 105, 052003 (2022) lepton flavor universality in the W boson decay is assumed. These distributions use the same bin boundaries as the fiducial-level cross section bins. The measurements are compared with the predictions from the NLO MadGraph5_aMC@NLO+PYTHIA simulation (MG5_aMC+PY8) and NNLO predictions from the MATRIX [67,68], MCFM [69,70], and GENEVA [71][72][73][74] frameworks. The MATRIX calculation implements a dedicated q Tsubtraction formalism [75,76], where q T is the total transverse momentum of the colorless final state particles. The framework utilizes amplitudes up to one-loop level from OPENLOOPS [77,78], as well as dedicated two-loop calculations [79]. The MCFM prediction uses the N-jettiness slicing method, with all one-and two-loop amplitudes [79] calculated using analytic formulae. It is given with and without the inclusion of NLO electroweak corrections, which have been shown [69,80] to modify W AE γ differential distributions by Oð10%Þ. The NLO electroweak correction to the qq-initiated process is applied multiplicatively to the NNLO QCD cross section, which is then summed with the contribution from the distinct photon-induced qγ channel. The GENEVA framework combines the fixed-order NNLO result with a next-to-next-to-leading logarithmic resummed calculation, in this case using soft-collinear effective theory to resum the 0-jettiness variable. The output events are interfaced with PYTHIA8 [39] for parton showering, hadronization, and underlying event simulation. Figures 6-8 show the resulting measurements for each observable. For p γ T in Fig. 6, the uncertainty composition and correlation matrices are also given.
The absolute cross section, as a function of p γ T , is measured with a precision of 4%-5% at low p T , dominated  (2022) by experimental systematic uncertainties, to 40% in the highest bin, dominated by the statistical uncertainty. The fractional cross section uncertainty falls to 2%-3% at low p T , owing to a cancellation of systematic uncertainties. Both measurements show a tendency towards the lower values of the NNLO predictions at high p T . Although the NLO electroweak correction is predicted to reduce the cross section in this region by up to 30%, the current precision is not sufficient to distinguish this effect. The correlation matrices illustrate how large positive correlations between bins in the absolute measurement are reduced significantly when using the fractional parametrization. The ΔRðl; γÞ measurement is in agreement with all the predictions in the bulk of the distribution, but favors the higher NNLO predictions in the ΔR > 4.0 region. The Δηðl; γÞ measurement shows significant enhancement in the highest jΔηj bins compared with the MG5_aMC+PY8 prediction. The NNLO predictions feature a similar, though less pronounced effect. The m cluster T distribution is broadly in agreement with the predictions, however with a moderate disagreement in the shape of the peak in the 100-300 GeV region.

B. Cross section as a function of the jet multiplicity
The cross section is measured as a function of the jet multiplicity using the same fiducial region as the previous results. The fiducial selection for the jets is chosen to match the experimental selection. Jets are clustered from final state particles, excluding neutrinos, with the anti-k T algorithm and a distance parameter of 0.4. They are required to have p T > 30 GeV, jηj < 2.5, and be separated from both the lepton and photon by ΔR > 0.4. The measured cross sections and their correlation matrix are shown in Fig. 9, and the cross section values are listed in Table II. The measured 0-jet cross section is 5%-15% smaller than the various predictions. A similar trend is observed in the 1-jet measurement, whereas the ≥ 2-jet measurement is higher than the predictions. Each measurement is positively correlated with the other two, with the strongest correlation of þ0.86 between the 0-jet and 1-jet bins.

C. The radiation amplitude zero effect
In pp collisions the RAZ effect is observed [4] in the rapidity difference between the charged lepton and the   (2022) photon; it manifests itself as a suppression of the cross section near Δηðl; γÞ ¼ 0. This region is of interest because new physics effects could make the minimum in this region less pronounced. Several models with resonances that would exhibit this feature are discussed in Ref. [81]. In the EFT framework this enhancement is generated by C 3W , however, the Δηðl; γÞ observable is significantly less sensitive than the approach pursued in this paper, as described in Sec. II.
To observe the RAZ effect in data it is necessary to add further requirements to the baseline selection. A veto on the presence of any jet with p T > 30 GeV and jηj < 2.5 is applied to preferentially select events in the Born configuration, and a requirement that m cluster T > 150 GeV. Figure 10 shows the differential cross section measurements, where a pronounced dip at Δηðl; γÞ ¼ 0 is observed. The jet veto in this selection gives a similar difference in normalization between measurement and predictions as for the 0-jet cross section in the previous section. In addition, the dip at Δηðl; γÞ ¼ 0 is larger than the predictions, though closer to the NNLO calculations, and the enhancement in the high jΔηðl; γÞj region is similar to that observed under the baseline selection in Fig. 8.

D. EFT constraints
In this section, constraints on the C 3W coefficient are derived via a parametrization of the fiducial cross section in p γ T and jϕ f j, under the EFT selection described in Sec. VI. The bin boundaries are [150,200,300,500,800,1500] GeV and [0, π=6, π=3, π=2], respectively. For the corresponding 2D differential cross section measurement the two highest p T bins are merged, owing to the small number of events expected in the highest bin. As for the one-dimensional (1D) differential measurements the signal model is constructed following Eq. (13), where j runs over all p γ T and jϕ f j bins.  T × jϕ f j bins. Values are given relative to the SM prediction: μ int ¼ σ int =σ SM and μ BSM ¼ σ BSM =σ SM , with the total relative normalization defined as  (2022) Constraints on C 3W are determined by parametrizing the fiducial cross section in each bin according to Eq. (3). Table III shows the corresponding values of the σ int and σ BSM terms, relative to σ SM . Figure 11 shows the expected and observed distributions, summed over the electron and muon channels, before the maximum likelihood fit is performed. The change in the expected distribution when C 3W is set to either −0.2 or 0.2 TeV −2 is also shown, considering only the SM and interference terms in Eq. (3).
The 95% confidence level (CL) intervals on C 3W are calculated using the asymptotic properties of the profile likelihood ratio of Eq. (10). Figure 12 shows scans of this ratio as a function of C 3W . The intervals are ½−0.062; 0.052 TeV −2 with the inclusion of the pure BSM term in Eq. (3) and ½−0.38; 0.17 TeV −2 with only the SM and interference terms. The observation is therefore compatible with the SM prediction of C 3W ¼ 0. Constraints are also determined as a function of the maximum p γ T bin that is included in the fit, shown in Fig. 13 (upper) and Table IV. Such a presentation is useful for interpretations where the highest bins may be beyond the validity of the EFT or specific BSM model being tested [82]. These constraints are also determined with and without the inclusion of the pure BSM term. This shows that with the current amount of data the pure BSM term dominates for all values of the cutoff. However, the 200 400 600 800 1000 1200 1400 0.5 π 200 400 600 800 1000 1200 1400 A. TUMASYAN et al. PHYS. REV. D 105, 052003 (2022) sensitivity to the interference term is comparable at lower values of p γ T . The constraint including the pure BSM term is similar in value to that of Ref.
[6], after accounting for a difference in the normalization of C 3W due to the use of an alternative EFT basis. However, the main feature of this result is the ability to constrain C 3W when only the interference term is considered. To demonstrate this, the constraints are also determined without the binning in jϕ f j applied. The same set of p γ T bins are used, but the three bins in jϕ f j are replaced by a single bin integrating over the full jϕ f j range. The result is shown in Fig. 13 (lower), without the inclusion of the pure BSM term. This demonstrates the significant improvement, by up to a factor of ten, that the jϕ f j binning gives to this constraint. Best-fit values of C 3W and the corresponding 95% CL confidence intervals as a function of the maximum p γ T bin included in the fit (upper). Measurements with and without the pure BSM term are given by the black and red lines, respectively. The limits without the pure BSM term given with and without the binning in jϕ f j are also shown (lower), with black and blue lines, respectively. Please note the different vertical scales; the black lines in both figures correspond to the same limits.
TABLE IV. Best fit values of C 3W and corresponding 95% CL confidence intervals as a function of the maximum p γ T bin included in the fit.  FIG. 14. Response matrix for the differential p γ T × jϕ f j cross section measurement. The entry in each bin gives the probability for an event of a given particle-level fiducial bin to be reconstructed in one of the corresponding reconstruction-level bins. The inner labels give the jϕ f j bin and the outer labels indicate the p γ T bin.
MEASUREMENT OF W AE γ DIFFERENTIAL CROSS … PHYS. REV. D 105, 052003 (2022) The interference-only confidence intervals on C 3W can be compared with measurements of other processes that are sensitive to the O 3W operator. An ATLAS measurement of WW production [83], in which the coefficient is denoted c W , uses an alternative technique to improve sensitivity to the interference. The presence of an additional high-p T jet is required, which partially mitigates the helicity suppression effect [13], and gives an interference-only sensitivity that is around a factor of eight lower. The ATLAS measurement of electroweak Z boson production in association with two jets [84] gives comparable sensitivity to our result. It exploits the distribution of the azimuthal angle between the jets, which is sensitive to the interference contribution.
The response matrix R ij for the 2D differential cross section measurement of p γ T and jϕ f j is shown in Fig. 14. Although the migration between p γ T bins is small, the migration between jϕ f j bins is larger, owing both to the limited p miss T resolution and the fundamental limitations of the method used to reconstruct the neutrino four-momentum via the m W pole mass constraint.
The resulting cross section measurements are shown in Fig. 15. The measured values are compared with the prediction from the NLO MG5_aMC+PY8 simulation. The correlation matrix is presented in Fig. 16. Unlike the 1D p γ T cross section, the correlations between different p γ T bins are relatively small, since these measurements at high p γ T are much more dominated by the statistical uncertainties. For a given p γ T bin the (anti)correlation between jϕ f j bins is larger, owing to the migration in the response matrix discussed previously.

X. SUMMARY
This paper has presented an analysis of W AE γ production in ffiffi ffi s p ¼ 13 TeV proton-proton collisions using data recorded with the CMS detector at the LHC, corresponding to an integrated luminosity of 138 fb −1 . Differential cross    (2022) sections have been measured for several observables and compared with standard model (SM) predictions computed at next-to-leading and next-to-next-to-leading orders in perturbative quantum chromodynamics. The radiation amplitude zero effect, caused by interference between W AE γ production diagrams, has been studied via a measurement of the pseudorapidity difference between the lepton and the photon. Constraints on the presence of TeV-scale new physics affecting the WWγ vertex have been determined using an effective field theory framework. Confidence intervals on the Wilson coefficient C 3W , determined at the 95% CL, are ½−0.062; 0.052 TeV −2 with the inclusion of the interference and pure beyond the SM contributions, and ½−0.38; 0.17 TeV −2 when only the interference is considered. A novel two-dimensional approach is used with the simultaneous measurement of the photon transverse momentum and the azimuthal angle of the charged lepton in a special reference frame. With this method, the sensitivity to the interference between the SM and the O 3W operator is enhanced by up to a factor of ten compared to a measurement using the transverse momentum alone. This improves the validity of the result, as the dependence on the missing higher-order contributions in the EFT expansion is reduced. The technique will also be valuable in the future when sufficiently small values of C 3W are probed such that the interference contribution will be dominant.

ACKNOWLEDGMENTS
We congratulate our colleagues in the CERN accelerator departments for the excellent performance of the LHC and thank the technical and administrative staffs at CERN and at other CMS institutes for their contributions to the success of the CMS effort. In addition, we gratefully acknowledge the computing centers and personnel of the Worldwide LHC Computing Grid and other centers for delivering so effectively the computing infrastructure essential to our analyses. Finally, we acknowledge the enduring support for the construction and operation of the LHC, the CMS detector, and the supporting computing infrastructure provided by the following funding agencies: BMBWF and FWF ( 2020 Grant, Contracts No. 675440, No. 724704, No. 752730, No. 758316, No. 765710, No. 824093, and No. 884104, and COST Action CA16108 (European Union); the Leventis Foundation; the Alfred P. Sloan  research Grants No. 123842, No. 123959, No. 124845, No. 124850, No. 125105, No. 128713, No. 128786, and No. 129058 (Hungary) and identification with the CMS detector in proton-proton collisions at ffiffi ffi s p ¼ 8 TeV, J. Instrum. 10, P08010 (2015).