Observation of gauge boson joint-polarisation states in W ± Z production from pp collisions at

Measurements of joint-polarisation states of W and Z gauge bosons in W ± Z production are presented. The data set used corresponds to an integrated luminosity of 139 fb − 1 of proton–proton collisions at a centre-of-mass energy of 13 TeV recorded by the ATLAS detector at the CERN Large Hadron Collider. The W ± Z candidate events are reconstructed using leptonic decay modes of the gauge bosons into electrons and muons. The simultaneous pair-production of longitudinally polarised vector bosons is measured for the ﬁrst time with a signiﬁcance of 7 . 1 standard deviations. The measured joint helicity fractions integrated over the ﬁducial region are f 00 = 0 . 067 ± 0 . 010, f 0T = 0 . 110 ± 0 . 029, f T0 = 0 . 179 ± 0 . 023 and f TT = 0 . 644 ± 0 . 032, in agreement with the next-to-leading-order Standard Model predictions. Individual helicity fractions of the W and Z bosons are also measured and found to be consistent with joint helicity fractions within the expected amounts of correlation. Both the joint and individual helicity fractions are also measured separately in W + Z and W − Z events. Inclusive and differential cross sections for several kinematic observables sensitive to polarisation are presented. © 2023 Published by Elsevier B.V. This is an open access article under the CC BY license (http:// creativecommons .org /licenses /by /4 .0/). Funded


Introduction
The study of weak boson polarisation provides a way to directly probe the Standard Model (SM) gauge and Higgs sectors and to discriminate between SM and beyond the Standard Model (BSM) signals.The existence of the longitudinally polarised state of weak bosons is a consequence of the non-vanishing mass of the bosons generated by the electroweak (EW) symmetry breaking mechanism.The measurement of the polarisation in diboson production therefore tests both the SM's gauge symmetry structure, through the existence of the triple gauge coupling, and the particular way this symmetry is spontaneously broken, via the longitudinal helicity state.Angular observables can be used to look for new interactions that can lead to different polarisation behaviour than predicted by the SM, to which the  ±  final state would be particularly sensitive [1,2].
At hadron colliders, the polarisation of the  boson was previously measured in the decay of the top quark by the CDF and DØ [3][4][5] Collaborations and by the ATLAS [6] and CMS [7] Collaborations, as well as in single production by ATLAS [8] and CMS [9].Polarisation and several other related angular coefficient measurements of singly produced  bosons were published by the CDF [10], CMS [11] and ATLAS [12] Collaborations.The polarisation of  bosons was also measured in   collisions by the H1 Collaboration [13].Finally, for dibosons, first measurements of the  polarisation were performed by LEP experiments in  pair production in  +  − collisions [14,15] and were used to set limits on anomalous triple gauge couplings in Ref. [16].Correlations between polarisation states of  particles were also probed experimentally at LEP [17][18][19].Helicity fractions of pair-produced vector bosons were measured independently of each other and for the first time in hadronic collisions at a centre-of-mass energy of √  = 13 TeV by the ATLAS Collaboration [20], followed by the CMS Collaboration [21], using integrated luminosities of 36 fb −1 and 137 fb −1 , respectively.Benefiting from the full ATLAS data set available at √  = 13 TeV, combined with novel analysis techniques, observables more difficult to extract from data, such as the correlations between helicity states of the vector bosons, can now be accessed.This Letter reports on an observation and measurement of joint polarisation states of weak bosons in  ±  production, extending the pioneering work of Ref. [20].The  and  bosons are reconstructed using their decay modes into electrons or muons.The   collision data were collected with the ATLAS detector from 2015 to 2018 at a centre-of-mass energy of √  = 13 TeV and correspond to an integrated luminosity of 139 fb −1 .The individual and joint boson polarisation state fractions are measured consistently in the detector fiducial region and their relationships are studied.The reported measurements are compared to the recent fixed-order theory calculation of polarised  and  boson pair production at next-to-leading-order (NLO) accuracy in quantum chromodynamics (QCD) [22,23].Additionally, the  ±  production cross section is measured within a fiducial phase space both inclusively and differentially as a function of several angular variables related to the helicity states of the  ±  system.

Theoretical framework and measurement phase space
A complete description of the polarisation states of the produced  and  bosons is given in terms of the two-particle joint spin density matrix          expressed in the basis of joint-helicity states (    ) where   and   are the helicities of the  and  bosons, respectively.In terms of the  ±  production helicity amplitudes,  (    q )     , the spin density matrix elements are defined by (    q ) *

𝑍
, where  = ∑︁ The initial-state helicity is averaged in the numerator and denominator with sums running over   ,  q = ±1/2.
The quantities  00 ,  TT ,  0T and  T0 are composed of diagonal elements of the full matrix and can be interpreted as probabilities of correlated helicity states of the two  and  bosons.The distinction between the two transverse helicities of the bosons (right-handed, R, and left-handed, L, also indicated with + and −, respectively) is not made.
The single-boson spin density matrix   is derived from the joint spin density matrix          by summing over indices of one of the bosons.It provides information about the helicity of one of the bosons regardless of the state of the other.Diagonal elements of the single-boson spin density matrix are the individual helicity fractions  0 ,  L and  R for the longitudinal, transverse left-handed and transverse right-handed helicity states, respectively.The normalisation is defined such that  0 +  L +  R = 1.Helicity fractions are not Lorentz invariant and hence depend on a chosen reference frame.In diboson production a natural frame choice for single-boson helicity fraction measurement is to fix the reference -axis to the direction of the corresponding boson in the diboson centre-of-mass frame [23].This definition is called the modified helicity coordinate system and is used for this measurement.The back-to-back kinematics of the bosons in the  ±  centre-of-mass frame implies that the polarisation vectors of the two bosons are defined with respect to the same reference axis, but in opposite directions.This choice also maximises the decorrelation between 00 and TT helicity states.
The relative contributions of joint helicity states can be determined from the absolute value of the cosine of the scattering angle of the  or  boson in the   centre-of-mass frame, | cos   |, calculated with respect to the beam axis [24,25].Events with 00 helicity have | cos   | values closer to 0 while events with TT helicity have | cos   | values closer to 1.
The polarisation of an individual gauge boson can be determined from the angular distribution of its decay products.At the Born level, the expected angular distribution for massless fermions in the rest frame of the parent  boson is given by [26][27][28][29] 1 where   is the charge of the  boson and  * ℓ  is defined using the modified helicity frame as the decay angle of the charged lepton in the  rest frame relative to the  direction in the  ±  centre-of-mass frame.All dependencies on the azimuthal angle are integrated over.The expected angular distribution of the lepton decay products of the  boson is described by the generalisation of Equation (1) [26][27][28] 1 where  * ℓ  is defined using the modified helicity frame as the decay angle of the negatively charged lepton in the  rest frame relative to the  direction in the  ±  centre-of-mass frame.The parameter  = (2    )/( 2  +  2  ) = 0.147 is expressed in terms of the vector   = −1 2 + 2 sin 2  eff W and axialvector   = − 1 2 couplings of the  boson to leptons, where the effective value of the Weinberg angle sin 2  eff W = 0.23152 [30] is used.Equation ( 2) also holds for the contribution from  * and its interference with the  boson, with appropriate   and   coefficients.
The reported helicity fractions and cross sections are measured in a fiducial phase space chosen to closely follow the event selection criteria described in Section 5.It is based on the kinematics of particle-level objects as defined in Ref. [31].These are final-state prompt 1 leptons associated with the  and  boson decays.The kinematics of final-state prompt leptons is defined before QED final-state radiation (FSR), i.e.Born leptons.
The Born charged leptons from  and  boson decay must have || < 2.5 and transverse momentum  T above 15 GeV and 20 GeV, respectively; the invariant mass of the two leptons from the  boson decay differs by at most 10 GeV from the world average value of the  boson mass  PDG  = 91.1876GeV [30].
The  transverse mass, defined as where Δ(ℓ, ) is the angle between the lepton and the neutrino in the transverse plane, and  ℓ T and   T are the transverse momenta of the lepton from  boson decay and of the neutrino, respectively, must be greater than 30 GeV.In addition, it is required that the angular distance Δ between the charged leptons from the  and  decay is larger than 0.3, and that Δ between the two leptons from the  decay is larger than 0.2.
The contribution from off-shell effects is minimised by the tight invariant mass window of ±10 GeV around the nominal  boson mass used for the definition of the fiducial phase space.These contributions are estimated from NLO QCD calculations to amount to 2% at the integrated level and in most of the differential distributions [23].However, all the helicity fractions presented here are effective fractions and contain the small non-resonant contributions present in the fiducial phase space, including  * events.Interferences among helicity states are also expected to be small, amounting to 0.6% according to NLO QCD+EW calculations [32].Their effect is included in the measured helicity fractions.
A total phase space is also defined by requiring only the invariant mass of the lepton pair associated with the  boson to be in the range 66 <  ℓℓ < 116 GeV.This serves for the generation of polarised Monte Carlo (MC) event templates described in Section 7 for which no restrictions should be applied to the phase space of the decay leptons from  and  bosons.

ATLAS detector
The ATLAS experiment [33] at the LHC is a multipurpose particle detector with a forward-backward symmetric cylindrical geometry and a near 4 coverage in solid angle. 2 It consists of an inner tracking detector (ID) surrounded by a thin superconducting solenoid providing a 2 T axial magnetic field, electromagnetic and hadron calorimeters and a muon spectrometer (MS).The inner tracking detector covers the pseudorapidity range || < 2.5.It consists of silicon pixel, silicon microstrip and transition radiation tracking detectors.Lead/liquid-argon (LAr) sampling calorimeters provide electromagnetic (EM) energy measurements with high granularity.A steel/scintillator-tile hadron calorimeter covers the central pseudorapidity range (|| < 1.7).The endcap and forward regions are instrumented with LAr calorimeters for both the EM and hadronic energy measurements up to || = 4.9.The muon spectrometer surrounds the calorimeters and is based on three large superconducting air-core toroidal magnets with eight coils each.The field integral of the toroids ranges between 2.0 and 6.0 T m across most of the detector.The muon spectrometer includes a system of precision tracking chambers and fast detectors for triggering.A two-level trigger system is used to select events.The first-level trigger is implemented in hardware and uses a subset of the detector information to accept events at a rate below 100 kHz.This is followed by a software-based trigger that reduces the accepted event rate to 1 kHz on average depending on the data-taking conditions.An extensive software suite [34] is used in the reconstruction and analysis of real and simulated data, in detector operations, and in the trigger and data acquisition systems of the experiment.

Signal and background simulation
MC simulation is used to model signal and most background processes.
An inclusive sample of simulated  ±  events is used to correct the signal yield for detector effects and to compare the measurements with the theoretical predictions.The production of  ±  pairs and the subsequent leptonic decays of the vector bosons were simulated at NLO in QCD and LO in QED using the P -B v2 [35][36][37][38] generator, interfaced to P 8.210 [39] for simulation of parton showering, hadronisation and the underlying event.Final-state radiation resulting from QED interactions was simulated using P 8.210 with the AZNLO [40] set of tuned parameters.The CT10 [41] parton distribution function (PDF) set was used for the hard-scattering process, while the CTEQ6L1 [42] PDF set was used for the parton shower.The sample was generated with dynamic renormalisation and factorisation QCD scales,  r and  f , equal to    /2, where    is the invariant mass of the   system [20].
For systematic uncertainties studies, two alternative inclusive  ±  signal samples were generated at NLO in QCD.The first alternative sample was generated using M G 5_ MC@NLO 2.6.5 [43].Matrix elements containing three leptons, one neutrino and up to two jets in the final state were calculated at NLO QCD and merged with the parton shower from P 8.210 using the FxFx scheme [44].The NNPDF3.0 [45] PDF set was used for the hard-scattering process.The default dynamic renormalisation and factorisation scale set by M G 5_ MC@NLO was used [46].This MC sample is referred to as M G 5_ MC@NLO+P .
2 ATLAS uses a right-handed coordinate system with its origin at the nominal interaction point (IP) in the centre of the detector and the -axis along the beam pipe.The -axis points from the IP to the centre of the LHC ring, and the -axis points upwards.Cylindrical coordinates (, ) are used in the transverse plane,  being the azimuthal angle around the -axis.
The pseudorapidity is defined in terms of the polar angle  as  = − ln tan(/2).Angular distance is measured in units of The second alternative  ±  signal sample was generated using the S 2.2.2 generator [47].Matrix elements contain all diagrams with four electroweak vertices.They were calculated for up to one parton at NLO and up to three partons at LO using Comix [48] and O L [49], and merged with the S parton shower [50] according to the ME+PS@NLO prescription [51].The NNPDF3.0 PDF set was used along with the associated set of tuned parton-shower parameters developed by the S authors.
Polarised  ±  events were generated at LO in QCD using M G 5_ MC@NLO 2.7.3 [52] interfaced to P 8.244 for simulation of parton showering, hadronisation and the underlying event.The generation of polarised events is not possible at the full NLO QCD accuracy.To account for the real part of NLO corrections, events were simulated with either no jets or one jet in the matrix element at LO, and merged with P parton shower using the CKKW-L scheme [53,54].The NNPDF3.0 PDF set was used for the hard-scattering process.Four separate MC samples were generated corresponding to the four joint helicity states 00, 0T, T0 and TT, respectively.An inclusive MC sample was created by adding the four polarised samples.These MC sets are referred to as M G 0,1j@LO.
The background sources in this analysis include processes with two or more electroweak gauge bosons, namely  ,  and  ( = , ); processes with top quarks, such as  t and  t, single top and  ; and processes with gauge bosons associated with jets or photons ( +  and ).Electroweak  ±  production,    −EW, arising at the order  6 EW is also considered as a background and not part of the measured signal.MC simulation is used to estimate the contribution from background processes with three or more prompt leptons.Background processes with at least one misidentified lepton are evaluated using data-driven techniques and simulated events are used to assess the systematic uncertainties of these backgrounds.
The S 2.2.2 event generator was used to simulate both the  q and -initiated   processes, including  →   production, using the NNPDF3.0PDF set.It provides a matrix element calculation accurate to NLO in  s for 0-and 1-jet final states, and LO accuracy for 2-and 3-jet final states.Triboson events were simulated by the S 2.2.2 event generator at NLO accuracy with 0 additional partons and at LO accuracy with one and two additional partons and using the NNPDF3.0PDF set.
Both the  t and  t processes were generated at NLO with the M G 5_ MC@NLO MC generator, using the NNPDF3.0PDF set, and interfaced to the P 8.186 [55] parton shower model.
Finally, the   and    −EW processes were generated at LO by the M G 5_ MC@NLO MC generator interfaced with P 8.244 for the modelling of the parton shower, hadronisation and underlying event.The parton distribution function set was NNPDF3.0 .In the simulation of the parton shower the NNPDF2.3[56] PDF set was used.
All generated MC events were passed through the ATLAS detector simulation [57], based on G 4 [58], and processed using the same reconstruction software as used for the data.The event samples include the simulation of additional   interactions (pile-up) generated with P 8.186 using the NNPDF2.3PDF set and the A3 [59] set of tuned parameters for the underlying event and parton fragmentation.Simulated events were reweighted to match the pile-up conditions observed in the data.Scale factors were applied to simulated events to correct for small differences in the efficiencies observed in data and predicted by MC simulation for the trigger, reconstruction, identification, isolation and impact parameter requirements of electrons and muons [60,61].Furthermore, the electron energy and muon momentum in simulated events were smeared to account for small differences in resolution between data and MC events [60,62].

Event reconstruction and selection
Only data recorded with stable beam conditions and with all relevant detector subsystems operational are considered.Candidate events are selected using triggers [63][64][65] that require at least one electron or muon.These triggers require leptons to pass different transverse momentum threshold and isolation criteria, which depend on the data-taking run period and the instantaneous luminosity.The combined efficiency of these triggers is 99% for  ±  events satisfying the offline selection criteria.Events are required to have a primary vertex compatible with the LHC luminous region inside the ATLAS detector.The primary vertex is defined as the reconstructed vertex with at least two charged-particle tracks that has the largest sum of the  2 T for the associated tracks.Muon candidates are identified by tracks reconstructed in the MS and matched to tracks reconstructed in the ID.Muons are required to pass a 'medium' identification selection, which is based on requirements on the number of hits in the ID and the MS [61,62].The efficiency of this selection averaged over  T and  is larger than 98%.The muon momentum is measured by combining the MS measurement, corrected for the energy deposited in the calorimeters, and the ID measurement.The  T of the muon must be greater than 15 GeV and its pseudorapidity must satisfy || < 2.5.
Electron candidates are reconstructed from energy clusters in the electromagnetic calorimeter matched to ID tracks.Electrons are identified using a discriminant that is the value of a likelihood function constructed with information about the shape of the electromagnetic showers in the calorimeter, the track properties and the quality of the track-to-cluster matching for the candidate [60].Electrons must satisfy a 'medium' likelihood requirement, which provides an overall identification efficiency of 90%.The electron momentum is computed from the cluster energy and the direction of the track.The  T of the electron must be greater than 15 GeV and the pseudorapidity of the cluster must satisfy || < 1.37 or 1.52 < || < 2.47.
Electron and muon candidates are required to originate from the primary vertex and to be isolated from other particles using both calorimeter-cluster and ID-track information.The isolation efficiency is at least 90% for electrons with  T > 25 GeV and muons with  T > 15 GeV and at least 99% for electrons and muons with  T > 60 GeV [60,61].
Jets of hadrons are reconstructed using a particle-flow algorithm based on noise-suppressed positive-energy topological clusters in the calorimeter [66].Energy deposited in the calorimeter by charged particles is subtracted and replaced by the momenta of tracks which are matched to those topological clusters.The jets are clustered using the anti-  algorithm [67,68] with a radius parameter  = 0.4.They are calibrated according to in situ measurements of the jet energy scale [69].All jets must have  T > 25 GeV and be reconstructed in the pseudorapidity range || < 4.5.Jets with  T below 60 GeV and with || < 2.4 have to pass a requirement on the jet vertex tagger [70], a likelihood discriminant that uses a combination of track and vertex information to suppress jets originating from pile-up activity.Jets with || < 2.5 containing a -hadron are identified with a deep-learning neural network (NN) [71][72][73] which uses distinctive features of -hadron decays in terms of the impact parameters of the tracks and the displaced vertices reconstructed in the inner detector.Jets initiated by -quarks are selected by setting the algorithm's output threshold such that an 85% -jet selection efficiency is achieved in simulated  t events, with a rejection factor of 40 against light-flavour jets.
The transverse momentum of the neutrino is estimated from the missing transverse momentum in the event,  miss T , calculated as the negative vector sum of the transverse momentum of all identified hard physics objects (electrons, muons, jets), with a contribution from an additional soft term.This soft term is calculated from ID tracks matched to the primary vertex and not assigned to any of the hard objects [74].
Events are required to contain exactly three lepton candidates satisfying the selection criteria described above.To ensure that the trigger efficiency is well determined, at least one of the candidate leptons is required to have  T > 25 GeV for 2015 and  T > 27 GeV for 2016-2018 data, as well as being geometrically matched to a lepton that was selected by the trigger.No requirement on the number of jets is applied.
In order to suppress background processes with at least four prompt leptons, events with a fourth lepton candidate satisfying looser selection criteria are rejected.For this looser selection, the lepton  T requirement is lowered to  T > 5 GeV, electrons are allowed to be reconstructed in the region 1.37 < || < 1.52 and 'loose' identification requirements [60,61] are used for both the electrons and muons.A less stringent requirement is applied for electron isolation and is based only on ID track information.No dedicated identification algorithm is used to suppress events with electrons and muons originating from the decay of -leptons.
Candidate events are required to have at least one pair of leptons with the same flavour and opposite charge, with an invariant mass that is consistent with  PDG  to within 10 GeV.This pair is considered to be the  boson candidate.If more than one pair can be formed, the pair whose invariant mass is closest to the nominal  boson mass is taken as the  boson candidate.The remaining third lepton is assigned to the  boson decay.The transverse mass of the  candidate, computed using  miss T and the  T of the associated lepton, is required to be greater than 30 GeV.
Backgrounds originating from misidentified leptons are suppressed by requiring the lepton associated with the  boson to satisfy more stringent selection criteria.The transverse momentum of this lepton is therefore required to be greater than 20 GeV.This lepton is also required to pass the 'tight' identification requirements, which results in an efficiency between 90% and 98% for muons and an overall efficiency of 85% for electrons.
The main challenge in the reconstruction of the full kinematics of the selected events arises for the  boson from incomplete knowledge of the neutrino momentum.In the previous ATLAS publication [20] the neutrino longitudinal momentum    was reconstructed analytically using the  mass constraint.In this analysis a neural network regression is developed to reconstruct    .Input variables to the NN are  T and   of the charged lepton from the  boson decay, the  miss T components perpendicular and parallel to the direction of this charged lepton and the    reconstructed analytically.The NN is trained using  ±  P +P MC events at detector level to reconstruct event-by-event the true    value in the MC simulation.The compatibility of the reconstructed  boson mass   with its generated value is ensured by using as the loss function the sum of the root-mean-square (RMS) of the deviations of   and    reconstructed values from their generated values.The regression method provides a reasonable estimate of    for events for which the analytical method fails 3 and reduces the    error RMS by 10% thanks to the improved resolution at high    .This improvement comes from fact that the NN is trained to account for the full width of the generated  boson, whereas the analytical method imposes a single value of the  boson mass, fixed to its pole mass, for all events.Additionally, the regression method improves the   resolution from 41% obtained with the analytical method to 31%.This in turn improves the reconstruction of angular observables in the modified helicity frame.

Background estimation
The background sources are classified into two groups: events where at least one of the candidate leptons is not a prompt lepton (reducible background) and events where all candidates are prompt leptons or are produced in the decay of a -lepton (irreducible background).Candidates that are not prompt leptons are also called 'misidentified' leptons.
Events in the first group mostly originate from  + , ,  t, and  production processes and constitute about 5% of all selected events.This reducible background is estimated with a data-driven method based on the inversion of a matrix containing the efficiencies and misidentification probabilities for prompt and misidentified leptons [20,[75][76][77].The method exploits the classification of the leptons as loose or tight candidates and the probability that a non-prompt lepton is misidentified as a loose or tight lepton.Tight leptons are leptons passing the selection criteria described in Section 5. Loose leptons are leptons that do not meet the isolation and identification criteria for signal leptons but satisfy only looser criteria.The misidentification probabilities for misidentified leptons are determined from data using dedicated control samples each enriched in misidentified leptons from light-or heavy-flavour jets and from photon conversions, respectively.The lepton efficiencies for prompt leptons are determined as detailed in Refs.[60][61][62].The lepton efficiencies and misidentification probabilities are combined with event rates in data samples of  ±  candidate events where at least one and up to three of the leptons are loose.Then, solving a system of linear equations, the number of events with at least one misidentified lepton is obtained, which represents the amount of reducible background in the  ±  sample.About 2% of this background contribution arises from events with two misidentified leptons.The background from events with three misidentified leptons, e.g. from multĳet processes, is negligible.The method allows the shape of any kinematic distribution of reducible background events to be estimated.Another independent method of assessing the reducible background was also considered.This method estimates the amount of reducible background using MC simulations scaled to data by process-dependent factors determined from the data-to-MC comparison in dedicated control regions.Agreement within 20% with the matrix method estimate is obtained in both yield and shape of the distributions of irreducible background events.
The events contributing to the second group of background processes represent ∼ 18% of all selected events.They originate from  ,  t + ,  (where  =  or ) and   events.Events from  ±  production with at least one -lepton decay or from electroweak    −EW production are also considered as backgrounds and not part of the measured signal.The amount of irreducible background is estimated using MC simulations.The dominant contribution in this second group is from   production, where one of the leptons from the   decay falls outside the detector acceptance.It represents about 7.5% of all selected events.The MC-based estimates of the   and  t +  backgrounds are validated by comparing the MC expectations with the event yield and several kinematic distributions of data samples enriched in   and  t +  events, respectively.The   control sample is selected from events with a  candidate which meets all of the analysis selection criteria and which is accompanied by two additional leptons, satisfying the lepton criteria described in Section 5.The   MC expectation is rescaled to match the data by combining the  ±  signal region and a single bin containing all of the events from the   control region in the fit detailed in Section 7 and used to extract the helicity fractions.A scaling factor of 1.13 is obtained from the fit for the   MC contribution, in agreement with the   cross-section measurements performed at √  = 13 TeV [78].The  t +  control sample is selected by requiring selected  ±  events to have at least two reconstructed -jets.The observed data event yield in this validation region is matching the  t +  MC prediction rescaled by a factor of 1.3.This scaling factor relative to predictions is in line with the  t +  cross-section measurements performed at √  = 13 TeV [79].The contribution of  t +  events amounts to ∼ 4% of all selected events.Systematic uncertainties affecting the rescaling of   and  t +  MC predictions are discussed in Section 8. Finally, some  ±  * events produced outside of the total phase space can be selected at detector level.The contribution of such events is estimated from the  ±  P +P MC simulation to amount to 0.5% and is subtracted from data.

Joint boson polarisation
In the Born-level fiducial phase space, the | cos   |,   • cos  * ℓ  and cos  * ℓ  angular observables can be used together to separate the joint helicity states from an inclusive MC sample and determine their respective generated fractions.However, when using the same angular observables at detector level, the separation of the different joint helicity states is not sufficient for the measurement of their respective fractions with an expected precision of 5 .Therefore, a multivariate discriminant exploiting more kinematic observables is used to better separate the four joint helicity states.A deep neural network (DNN) classifier is implemented to exploit kinematic differences between polarisation states in different observables.Keras v2.2.4-tf [80] with the Tensorflow v1.3 [81] backend are used.The DNN is trained and optimised using simulated M G 0,1j@LO polarised MC samples.A total of eight variables are used as inputs to the DNN: the transverse momenta of the three leptons and of the neutrino,  ℓ  T , T and  miss T , angular variables as the absolute difference between the rapidities of the  boson and the lepton from the decay of the  boson, |  −  ℓ  |, the azimuthal angle difference between the two leptons of each  and -boson decay, Δ(ℓ  , ), Δ(ℓ  1 , ℓ  2 ), respectively, and the transverse momentum of the   system    T .The DNN classifier is trained such that the components of the vector output, one for each of the four joint-polarisation states, are combined in a single output node in its last layer, used in the analysis for the separation.The optimisation of the combination is part of the DNN training procedure and a sigmoid activation function is applied to the combination output.The output is therefore a continuous score, labelled 'DNN score,' ranging from 0 to 1 that takes typical values peaking around 0, 0.5, and 1 for TT, mixed 0T and T0, and 00 helicity states, respectively.The DNN classifier achieves integrals of the receiver operating characteristic (ROC) curves between 0.8 and 0.85 for separating the 00 state from each of the three others joint polarisation states.The sensitivity of the analysis in disentangling the mixed helicity states 0T and T0 is achieved by categorising events depending on the cos  * ℓ  and cos  * ℓ  observables.Four categories are defined as 1 and referred to as the 4-category DNN score, is built.
To measure joint helicity fractions a binned maximum-likelihood fit [82] to data of the four helicity states templates is performed.The binned likelihood function used in the fit consists of a product of Poisson probability terms over bins of the 4-category DNN score distribution in the  ±  signal region and a single-bin integrating over the   control region.Each source of systematic uncertainty is implemented in the likelihood function as a nuisance parameter with a Gaussian constraint, except for the NLO QCD modelling systematic uncertainty described in section 8. Independent parameters of the fit are the particle level helicity fractions  00 ,  T0 ,  TT and the integrated fiducial cross section.The fourth joint helicity fraction  0T is obtained using the constraint  00 +  0T +  T0 +  TT = 1.Differences between detector-level and particle-level fractions due to detector efficiencies and QED final-state radiation effects are accounted for in the fit using independent correction factors applied to each helicity fraction.The factors are obtained from simulation and range from 0.95 to 1.01 depending on the helicity state.
As observed in theory calculations [23], a strong relationship exists between NLO QCD corrections and polarisation effects.Therefore, templates for helicity states generated at LO are insufficient.The M G 0,1j@LO MC simulation corrects for the real part of NLO QCD effects but misses virtual corrections that are also important for polarisation measurements.In order to verify that the shapes of the polarised templates are valid, a closure test is performed.Templates are used to fit pseudo-data generated using inclusive MC simulations at NLO QCD accuracy, such as with P +P or M G 5_ MC@NLO+P .The fit is performed on detector-level distributions.Polarisation fractions extracted from the fit are compared with the generated polarisation fractions of the NLO MC samples.Differences from 10% to 50% depending on the polarisation state are observed between extracted and built-in fractions when using polarised templates from the M G 0,1j@LO generation.This demonstrates the need for using polarised templates at NLO QCD accuracy for polarisation measurements.
Templates better approaching the NLO QCD accuracy are built using a DNN-based event-by-event reweighting procedure [83].Four DNNs are trained and each of them is specialised to reweight at particle-level the inclusive M G 0,1j@LO events to one of the four joint polarised states.Input variables of the DNNs are those of the polarisation DNN classifier augmented by the invariant mass of the   system    , and the angular variables cos  * ℓ  , cos  * ℓ  and | cos   |.The four DNNs are then applied to reweight the inclusive P +P MC events, creating four polarised MC templates with NLO-like accuracy in QCD.This method provides the best fit closure, with no bias on the extracted polarisation fractions visible within the statistical precision of the closure test.
A second method is used to create NLO QCD accurate polarised templates, based on the available fixed-order parton-level theory predictions [23].Predicted distributions, including the output score of the DNN classifier, are calculated in the parton-level fiducial phase space [23].Corrections for parton-shower (PS) effects on the shape of the templates are incorporated by binned factors.The correction factors are calculated per helicity state by comparing M G 0,1j@LO predictions at particle and parton level.The templates obtained at particle level are transformed to detector level using a folding procedure to account for detector effects.This method provides a good fit closure, reducing the bias up to ∼ 10%.This template creation method is used to estimate systematic uncertainties in the template shapes.
Interference among polarisation states manifests itself as a shape difference between the sum of polarised templates and the inclusive distribution.This difference is expected to be at most 4% for the DNN score observable [23].The fit procedure adjusts fitted polarisation fractions such that the sum of polarised templates matches the inclusive distribution of data, with increased fit uncertainties.The measured polarisation fractions therefore include interference effects.

Individual boson polarisation
For individual  and  boson polarisation states, the helicity parameters  0 and  L −  R are measured using a fit of templates for the three  0 ,  L and  R helicity states to the   • cos  * ℓ  and cos  * ℓ  distributions.The equation  0 +  R +  L = 1 is used to constrain the independent parameters of the fit to determine  0 ,  L −  R and the integrated fiducial cross section.Templates with NLO QCD accuracy are also required for the   • cos  * ℓ  and cos  * ℓ  distributions.Neither such templates at NLO QCD accuracy nor templates for left-and right-handed helicity states of the vector bosons can be produced by the M G 5_ MC@NLO 2.7.3 MC event generator.The templates for the   • cos  * ℓ  and cos  * ℓ  distributions for each of the three individual helicity states of the  and  bosons are therefore created from the inclusive P +P MC sample using a reweighting technique [20].For each of the gauge bosons the predicted helicity fractions of P +P MC events are determined using the moment method [12,26] in the total phase space.Following the method detailed in Ref. [20], the predicted fractions are used to reweight P +P MC events to represent each of the longitudinal, left-and right-handed states of the  or  bosons.This creates polarised MC templates at detector level.

Differential cross section
Independently of polarisation fractions, the inclusive  ±  production cross section is measured differentially within the fiducial phase space defined at particle level and as a function of the variables the most sensitive to polarisation of the vector bosons.Similarly to measurements in Ref. [20], the differential detector-level distributions of background-subtracted data are corrected for detector resolution, efficiencies and QED final-state radiation effects using simulated signal events and an iterative Bayesian unfolding method [84], as implemented in the RooUnfold toolkit [85].The number of iterations ranges from two to four, depending on the resolution in the unfolded variable and chosen in order to minimise the sum of the expected statistical and unfolding uncertainties in the unfolded distribution.The width of the bins in each distribution is chosen according to the experimental resolution and to the statistical significance of the expected number of events in each bin.The fraction of signal MC events generated in a bin which are reconstructed in the same bin is always greater than 35%.The detector-level distributions are unfolded with a response matrix computed using the P +P MC signal sample that includes all four measured decay channels (, ,  and ) and is divided by four, such that cross sections refer to final states where the  and  bosons decay in a single leptonic channel with muons or electrons.

Systematic uncertainties
Systematic uncertainties affecting the shape and normalisation of the 4-category DNN score distribution for the background contributions, as well as the acceptance of the helicity components and the shape of their templates, are considered.
Uncertainties due to theoretical modelling in the helicity templates for higher order QCD corrections are evaluated by varying in the P +P MC simulation the renormalisation and factorisation scales independently by factors of two and one-half, removing combinations where the variations differ by a factor of four.The uncertainties due to the PDF and the  s value used in the PDF determination are evaluated using the PDF4LHC prescription [86].
For joint polarisation measurements an uncertainty due to the modelling of NLO QCD is estimated by comparing predictions of the 4-category DNN score distributions for each of the joint-helicity states obtained from the parton-level NLO calculation with the DNN reweighting of the P +P MC simulation.The difference between the predicted shapes from the two methods is considered as an uncertainty.The associated nuisance parameter in the fit is parametrised by a flat constraint in the range between the two template shapes from each of the method, with steeply vanishing constraints outside of this range, defined using a double Fermi distribution.A specific uncertainty is attributed to the DNN reweighting method.It is estimated by applying the reweighting method to inclusive M G 0,1j@LO MC events and comparing the obtained distributions with those of polarised M G 0,1j@LO MC events.The difference between the shapes is treated as an uncertainty.
For individual boson polarisation measurements a modelling uncertainty in the helicity templates is estimated by comparing predictions of the   • cos  * ℓ  and cos  * ℓ  distributions from the P +P event generator with those of S .The difference between the predicted shapes from the two generators is considered as an uncertainty.An uncertainty is attributed to the reweighting method used to create templates with individually polarised bosons from inclusive P +P MC events.It is estimated in the same way as for the joint polarisation measurement.
Uncertainties in differential cross-section measurements arising from imperfect description of the data by the P +P MC simulation are evaluated using a data-driven method [87].Each MC differential distribution is corrected to match the data distribution and the resulting weighted MC distribution at detector level is unfolded with the response matrix used in the actual data unfolding.The new unfolded distribution is compared with the weighted MC distribution at particle level and the difference is taken as the systematic uncertainty.An additional uncertainty is derived to account for more subtle differences between the P +P and M G 5_ MC@NLO+P generators (e.g.hadronisation models, additional soft objects, mis-modelling in other kinematic variables).The M G 5_ MC@NLO+P generator is used to unfold the data and deviation from the nominal result is taken as the uncertainty.In order to remove effects already accounted for in the data-driven method, the M G 5_ MC@NLO+P distributions were first reweighed to match P +P distributions at particle level.
Systematic uncertainties affecting the reconstruction and energy calibration of electrons, muons and jets are propagated through the analysis.The uncertainties due to lepton reconstruction, identification, isolation requirements and trigger efficiencies as well as in the lepton momentum scale and resolution are assessed using tag-and-probe methods in  → ℓℓ events [60][61][62].The uncertainties in the jet energy scale and resolution are based on their respective measurements in data [88].The uncertainty in  miss T is estimated by propagating the uncertainties in the transverse momenta of reconstructed objects and by applying momentum scale and resolution uncertainties to the track-based soft term [74].A variation in the pile-up reweighting of MC events is included in order to cover the uncertainty in the ratio of the predicted and measured   inelastic cross sections [89].For the measurements of the  charge-dependent polarisation fractions, an uncertainty arising from the charge misidentification of electrons is also considered.
The uncertainty in the amount of background from misidentified leptons takes into account the limited number of events in the control regions as well as the difference in background composition between the control region used to determine the lepton misidentification rate and the control regions used to estimate the yield in the signal region.This results in an uncertainty of about 25% in the total misidentified-lepton background yield and in the shape of the differential distributions of the reducible background events.
Uncertainties due to the theoretical modelling of   generated events are considered.They arise from higher-order QCD corrections and the PDFs and are evaluated in the same way as for  ±  events.The uncertainty due to irreducible background sources other than   is evaluated by propagating the uncertainty in their MC cross sections.These are 30% for  [90], 25% for    −EW [91] and 15% for   [92] and  t +  [79,93].The contribution of events from  ±  production with at least one -lepton decay is linked in the fit to the measured  ±  production cross section for electron and muon decays.A normalisation uncertainty of 10% is attributed to this contribution to account for a possible mis-modelling in the MC simulation of the efficiency to select  ±  events with a -lepton decay.
The uncertainty in the combined 2015-2018 integrated luminosity is 1.7% [94], obtained using the LUCID-2 detector [95] for the primary luminosity measurements.It is applied to the signal normalisation as well as to all background contributions that are estimated using only MC simulations.
The effect of the systematic uncertainties on the final results after the maximum-likelihood fit is shown in Table 3 where the breakdown of the contributions to the uncertainties in the measured joint polarisation fractions is presented.Data / Pred.

Joint boson polarisation measurements
Figure 1: Post-fit distribution of the 4-category DNN score in  ±  events at detector level in the signal region (SR).The 4-category DNN score is defined for each event as the sum of the DNN score and of the category number ranging from 0 to 3. Respective contributions of the  ±  joint-helicity states 00, 0T, T0 and TT and backgrounds are normalised to the expected number of events after the fit.The sum of background events containing misidentified leptons is labelled 'Misid.leptons'.The sum of ,    −EW and   background events is labelled 'Others'.The uncertainty band around the expectation includes all systematic uncertainties as obtained from the fit.
The distribution of the 4-category DNN score in the  ±  signal region with background normalisations, signal normalisation and nuisance parameters adjusted by the profile-likelihood fit is shown in Figure 1.The corresponding pre-fit and post-fit yields in the signal region and in the   control region are detailed in Table 1.
The measurements of  00 ,  0T ,  T0 and  TT are summarised in Table 2.The hypotheses of having no events in each of the joint helicity states are tested.In  ±  events, the presence of a pair of  and  bosons with Table 1: Expected and observed numbers of events in the  ±  signal region (left) and in the   control region (right).Numbers are presented before and after a fit to the 4-category DNN score distribution in  ±  events.The sum of background events containing misidentified leptons is labelled 'Misid.leptons'.The sum of ,    −EW and   background events is labelled 'Others'.The pre-fit uncertainties quoted are of statistical and systematic origin but do not take into account correlations between parameters of the fit.The post-fit uncertainties quoted are of statistical and systematic origin and take into account the correlations between the various parameters of the fit.For the  ±  signal templates, the correlations arising from the relationship  00 +  0T +  T0 +  TT = 1 explain the larger post-fit uncertainties compared to the pre-fit ones.a simultaneous longitudinal polarisation (  00 ) is observed with a significance of 7.1 , compared to 6.2  expected.The other joint helicity fractions  0T ,  T0 and  TT are also measured with observed (expected) significances of 3.4  (5.4 ), 7.1  (6.6 ) and 11  (9.7 ), respectively.The joint helicity fractions are also measured in  +  and  −  events separately, with observed significances for  00 of 6.9  and 4.1 , respectively.Table 3 shows the main sources of uncertainty in the measurement of the joint helicity fractions.Uncertainties related to data statistics and to QCD effects, namely the QCD scale and modelling uncertainties, make similar contributions to the precision of the measurements.The modelling uncertainty is dominated by the uncertainties in the shape of the distributions of polarised  ±  events due to NLO QCD corrections.Among them, the uncertainty from the DNN reweighting method is dominant, except for  00 where the uncertainty from the template generation method is of equal size.Different correlation schemes between bins and between helicity states are tested for this latter modelling uncertainty.All yield consistent results.Uncertainties in the energy scale and reconstruction of jets are dominant among detector uncertainties, showing how observables for joint polarisation are linked to jet emissions.The values of  00 ,  0T ,  T0 and  TT measured in  ±  events are shown in Figure 2. The global level of agreement between the four measured joint helicity fractions with the fixed-order predictions at NLO QCD [23] is observed to be 1.4 .A similar level of agreement is observed with respect to P +P predictions.The impact of EW corrections on the joint helicity fractions is negligible, smaller than 3% [32].

Individual boson polarisation measurements
The distributions of   • cos  * ℓ  and cos  * ℓ  with background normalisations, signal normalisation and nuisance parameters, adjusted by the profile-likelihood fit, are shown in Figure 3.The post-fit event yields in the  ±  signal region and in the   control region are compatible with those obtained in the fit to the The measurements of  0 and  L −  R are summarised in Table 4 where they are compared with the predictions from P +P and, for  0 , from an NLO QCD fixed-order calculation [23].Good agreement of the measured helicity fractions of both the  and  bosons with the predictions from P +P is observed.Measured  0 values agree within 1 with the prediction, while  L −  R values agree within 1.5 , except for  −  events where the level of agreement deteriorates to 2.3 .Table 5 shows the main sources of uncertainty in the measurement of the individual helicity fractions.The measurements are dominated by statistical uncertainties, except for the  0 measurement of the  boson where the modelling uncertainty is equally important.The values of  0 and  L −  R measured in  ±  events are shown in Figure 4 for the  and  bosons.
The compatibility of the measured joint and individual helicity fractions is tested.Ignoring the interference between polarisations, which should represent only 0.6%-0.8% of the events according to NLO QCD and NLO QCD+EW fixed-order predictions [23,32], the following relationships ensue Using these relationships, joint  00 and individual   0 ,   0 helicity fractions can be measured simultaneously from a fit to the 4-category DNN score observable.The fractions   0 and   0 measured in this procedure agree within less than 1 of their own uncertainties alone with the   0 and   0 values obtained via individual cos  * ℓ  and cos  * ℓ  distributions.This represents a consistency test of the joint helicity measurement procedure.Measured joint helicity fractions  00 ,  0T ,  T0 and  TT of the  and  bosons in  ±  events, compared with NLO QCD fixed-order predictions [23] (upward pointing triangle) and to MC predictions from P +P (downward pointing triangle).The effect of PDF and QCD scale uncertainties on the P +P and NLO QCD fixed-order predictions are of the same size as the respective markers.The full and dashed ellipses around the data points correspond to one and two standard deviations, respectively.
The spin-correlation between longitudinal  and  bosons is tested by comparing joint and individual helicity fractions and measuring the ratio   =  00 /(   0   0 ).In the absence of spin correlations, neglecting interference effects,   is equal to 1.A value of   = 1.3 is predicted by NLO QCD fixed-order calculations, indicating the presence of correlation between polarisation states of the bosons.The ratio   is extracted from data by a simultaneous fit to the 4-category DNN score observable of  00 ,   0 and   0 .It is measured in  ±  events to be   = 1.54 ± 0.35.The observed significance relative to the no spin-correlations hypothesis, i.e.   = 1, is 1.6 .

Differential cross-section measurements
The inclusive cross section of  ±  production in the fiducial region for leptonic decay modes (electrons or muons) is determined simultaneously to the joint helicity fractions.It is measured to be  fid.
±  →ℓ ℓℓ = 64.6 ± 0.5 (stat.)± 1.8 (syst.)± 1.1 (lumi.)fb, where the uncertainties correspond to statistical uncertainties, experimental and modelling systematic uncertainties, and luminosity uncertainties.The corresponding SM NNLO QCD prediction from MATRIX [96] is 64.0 +1.5 −1.3 fb, where the uncertainty corresponds to the QCD scale uncertainty.ℓ  in  ±  events at detector level in the signal region (SR).Respective contributions of the  ±  individual helicity states 0, L and R of the  and  bosons and backgrounds are normalised to the expected number of events after the fit.The sum of background events containing misidentified leptons is labelled 'Misid.leptons'.The sum of ,    −EW and   background events is labelled 'Others'.The uncertainty band around the expectation includes all systematic uncertainties as obtained from the fit.5 are compared with the Born-level predictions at NLO in QCD from the P +P , M G 5_ MC@NLO+P MC generators and to the sum of the polarised M G 0,1j@LO LO predictions.The Born-level DNN score prediction is evaluated using the same DNN as trained at detector level and evaluated using Born-level input observables.The predicted integrated fiducial cross sections of the three MC generators are rescaled to the NNLO cross section from MATRIX [96], as indicated in Figure 5. Good agreement of the shapes of the measured distributions with the NLO QCD predictions is observed.The M G 0,1j@LO LO prediction exhibits larger differences from the measured cross section for the | cos   | distribution.This is expected as the | cos   | distribution is particularly sensitive to NLO QCD corrections [23] and the M G 0,1j@LO LO prediction only includes their real parts.The measured cross sections and the Rivet [97] routine for this measurement, including for the evaluation of the DNN classifier score, are published on HEPData [98]

Conclusion
Measurements of gauge boson polarisation states from  ±  production in √  = 13 TeV   collisions at the LHC are presented.The data analysed were collected with the ATLAS detector during the years 2015-2018 and correspond to an integrated luminosity of 139 fb −1 .The measurements use leptonic decay modes of the gauge bosons to electrons or muons and are performed in a fiducial phase space closely matching the detector acceptance.
Using neural-network-based advanced analysis techniques, joint helicity fractions of pair-produced vector bosons are measured.For the first time, the presence of a  boson and a  boson with a simultaneous longitudinal polarisation is measured with observed and expected significances of 7.1 and 6.2 standard deviations, respectively.Integrated over the fiducial region, the fractions of the  and  bosons both longitudinally or transversely polarised are measured in  ±  events to be  00 = 0.067 ± 0.010 and  TT = 0.644 ± 0.032, respectively, in agreement with the SM predictions at NLO in QCD.The impact of EW corrections is negligible.The measured fractions of mixed polarisation modes are  0T = 0.110 ± 0.029 and  T0 = 0.179 ± 0.023.Joint helicity fractions are also measured separately for  +  and  −  events.
Individual helicity fractions  0 and  L −  R of the  and  bosons are measured.The measured values agree with the SM predictions and are consistent with the measured joint helicity fractions when neglecting interference among polarisation states, which from NLO QCD+EW calculations are expected to be smaller than the uncertainty of the present measurement.
Furthermore, the inclusive cross section of  ±  production in the fiducial region for leptonic decay modes is measured to be  fid. ±  →ℓ ℓℓ = 64.6 ± 2.1 fb, in agreement with the NNLO SM expectation of 64.0 +1.5 −1.3 fb.The  ±  production cross section is measured as a function of several kinematic variables sensitive to polarisation and compared with SM predictions at NLO in QCD from the P +P and M G 5_ MC@NLO+P MC event generators.In particular, the variable related to the output of the deep neural network classifier used for the joint helicity measurement is unfolded in order to remove detector effects.The differential cross-section distributions are well described by the NLO MC predictions.

Figure 2 :
Figure2: Measured joint helicity fractions  00 ,  0T ,  T0 and  TT of the  and  bosons in  ±  events, compared with NLO QCD fixed-order predictions[23]  (upward pointing triangle) and to MC predictions from P +P (downward pointing triangle).The effect of PDF and QCD scale uncertainties on the P +P and NLO QCD fixed-order predictions are of the same size as the respective markers.The full and dashed ellipses around the data points correspond to one and two standard deviations, respectively.

Figure 3 :
Figure 3: Post-fit distributions of (a)   • cos  * ℓ  and (b) cos  *ℓ  in  ±  events at detector level in the signal region (SR).Respective contributions of the  ±  individual helicity states 0, L and R of the  and  bosons and backgrounds are normalised to the expected number of events after the fit.The sum of background events containing misidentified leptons is labelled 'Misid.leptons'.The sum of ,    −EW and   background events is labelled 'Others'.The uncertainty band around the expectation includes all systematic uncertainties as obtained from the fit.

Figure 5 :
Figure 5: The measured  ±  differential cross section in the fiducial phase space as a function of (a)   • cos  * ℓ  , (b) cos  * ℓ  , (c) | cos   | and (d) DNN score.The total cross section integrated over each bin and divided by the bin width is represented.The inner and outer error bars on the data points represent the statistical and total uncertainties, respectively.The measurements are compared with the NLO prediction from P +P (solid line).The dashed band shows how the sum of QCD scale and PDF uncertainties affects the P +P NLO prediction.The predictions from the M G 0,1j@LO and M G 5_ MC@NLO+P MC generators are also indicated by dotted-dashed and dashed lines, respectively.All predictions have been scaled to the NNLO QCD integrated cross section predicted by MATRIX.

Table 2 :
Measured joint helicity fractions  00 ,  0T ,  T0 and  TT in the fiducial phase space, for  ± ,  +  and  −  events.The total uncertainties in the measurements are reported.The measurements are compared with predictions from P +P and from NLO QCD fixed-order calculations[23].The uncertainties in the P +P prediction include statistical, PDF and QCD scale uncertainties; the uncertainties in the NLO QCD fixed-order prediction include QCD scale uncertainties.

Table 3 :
Summary of the relative uncertainties in the joint helicity fractions  00 ,  0T ,  T0 , and  TT measured in  ±  events.The uncertainties are reported as percentages.

Table 4 :
Individual helicity fractions  0 and  L −  R of the  and  bosons measured in the fiducial phase space for  ± ,  +  and  −  events.The total uncertainties in the measurements are reported.

Table 5 :
Summary of the relative uncertainties in the polarisation fractions  0 and  L −  R measured in  ±  events for  and  bosons.The uncertainties are reported as percentages.Individual helicity fractions  0 and  L −  R measured for the (a)  and (b)  bosons in  ±  events, compared with predictions from P +P with sin 2  W = 0.23152 (downward pointing triangle).The effect of PDF and QCD scale uncertainties on the P +P prediction are of the same size as the triangle marker.The full and dashed ellipses around the data points correspond to one and two standard deviations, respectively.The  ±  production cross section is measured as a function of several variables sensitive to the individual and joint boson polarisation:  • cos  * ℓ  , cos  * ℓ  , | cos   |and the DNN score.The measured differential cross sections in Figure