Measurements of the W boson rapidity, helicity, double-differential cross sections, and charge asymmetry in pp collisions at $\sqrt{s} =$ 13 TeV

The differential cross section and charge asymmetry for inclusive W boson production at $\sqrt{s}=$ 13 TeV is measured for the two transverse polarization states as a function of the W boson absolute rapidity. The measurement uses events in which a W boson decays to a neutrino and either a muon or an electron. The data sample of proton-proton collisions recorded with the CMS detector at the LHC in 2016 corresponds to an integrated luminosity of 35.9 fb$^{-1}$. The differential cross section and its value normalized to the total inclusive W boson production cross section are measured over the rapidity range $|y_\mathrm{W}| \lt $ 2.5. In addition to the total fiducial cross section, the W boson double-differential cross section, d$^2\sigma$/d$p^\ell_\mathrm{T}$d$|\eta|$ and the charge asymmetry are measured as functions of the charged lepton transverse momentum and pseudorapidity. The precision of these measurements is used to constrain the parton distribution functions of the proton using the next-to-leading order NNPDF3.0 set.


Introduction
The standard model (SM) of particle physics provides a description of nature in terms of fundamental particles and their interactions mediated by vector bosons. The electromagnetic and weak interactions are described by a unified gauge theory based on the SU(2) L ×U(1) Y symmetry group, where the photon, the W boson, and the Z boson act as mediators of the unified electroweak interaction [1][2][3].
Measurements of the kinematic properties of W bosons produced at hadron colliders provide stringent tests of perturbative quantum chromodynamics (QCD) calculations and probe the nature of the electroweak interaction. In particular, the measurement of the polarization of the W boson is fundamental in determining its production mechanism.
At leading order (LO) in QCD, W bosons are produced at a hadron collider with small transverse momentum (p T ) through the annihilation of a quark and an antiquark: ud for the W + and ud for the W − . At the CERN LHC, W bosons with large rapidity (|y W |) are produced predominantly with momentum in the same direction as the momentum of the quark that participates in the hard scattering. This is because the parton distribution functions (PDFs) of the proton favor the quark to carry a larger fraction (x) of the proton momentum rather than the antiquark [4].
Because of the V−A coupling of the W boson to fermions in the SM, the spin of the W boson is aligned with that of the quark, i.e., purely left-handed, and thus aligned opposite to the direction of the momenta of both the W boson and the quark. With smaller |y W |, the W bosons produced at the LHC become a mixture of left-, and right-handed polarization states at LO in QCD, and the rates of the two polarizations become equal at |y W | = 0. With increasing W boson p T (p W T ), next-to-leading order (NLO) amplitudes contribute in its production, and longitudinally polarized W bosons arise. The relative fractions of the three polarization states depend on the relative size of the amplitudes of the three main production processes: ud → W + g, ug → W + d, and gd → W + u, and are determined by the PDFs at high values of x. Overall, left-handed W bosons are favored at the LHC over right-handed and longitudinally polarized W bosons. The relative fraction of positively (negatively) charged left-handed W bosons is around 65 (60)%, of right-handed W bosons around 28 (33)%, and of longitudinally polarized W bosons around 7 (7)% of the total production cross section. The fraction of longitudinally polarized W bosons increases with p W T in the kinematic region that contains the bulk of the W boson production cross section with p W T < 50 GeV. At the LHC, W bosons are produced in large quantities and it is easy to trigger on their leptonic decays (W → ν) with high purity. Since the escaping neutrino means the momentum of the W boson is not known, the direct measurement of the fully differential cross section of the W boson is not possible. In particular, the polarization and rapidity distributions of the W boson must be inferred by using the PDFs. Uncertainties stemming from the imperfect knowledge of these PDFs contribute a large fraction of the overall uncertainties in recent measurements of the mass of the W boson [5], and in other high-precision measurements at the LHC [6].
Constraints on the PDFs and their uncertainties are possible through many different measurements. Recently, the ATLAS and CMS Collaborations published PDF constraints from doubledifferential measurements of Z boson production and the accurate measurement of sin 2 θ W [7-9]. Studies of W bosons have been used by the ATLAS and CMS Collaborations to set constraints on PDFs through the measurement of charge asymmetries, in particular as a function of the charged lepton pseudorapidity η [10-18]. Measurements of associated production of a W boson and a charm quark by the ATLAS, CMS, and LHCb Collaborations at the LHC [19-itself. In addition, a requirement on the relative isolation of the reconstructed muon is applied to suppress muons from background processes, such as leptonic heavy-flavor decays. This isolation variable is defined as the pileup-corrected ratio of the sum of the p T of all charged hadrons, neutral hadrons, and photons, divided by the p T of the muon itself [37]. It is calculated for a cone around the muon of ∆R = √ (∆φ) 2 + (∆η) 2 < 0.4, where φ is the azimuthal angle, and it is required to be smaller than 15%.
Electron candidates are formed from energy clusters in the ECAL (called superclusters) that are matched to tracks in the silicon tracker. Their p T is required to exceed 30 GeV and they are selected within the volume of the CMS tracking system up to |η e | < 2.5. Electrons reconstructed in the transition region between the barrel and the endcap sections, within |η e | > 1.4442 and |η e | < 1.566, are rejected.
Electron identification is based on observables sensitive to bremsstrahlung along the electron trajectory and geometrical and momentum-energy matching between the electron trajectory and the associated supercluster, as well as ECAL shower-shape observables and variables that allow the rejection of the background arising from random associations of a track and a supercluster in the ECAL. Energetic photons produced in pp collision may interact with the detector material and convert into electron-positron pairs. The electrons or positrons originating from such photon conversions are suppressed by requiring that there is no more than one missing tracker hit between the primary vertex and the first hit on the reconstructed track matched to the electron; candidates are also rejected if they form a pair with a nearby track that is consistent with a conversion. Additional details of electron reconstruction and identification can be found in Refs. [38,39].
A relative isolation variable similar to that for muons is constructed for electrons, in a cone of ∆R < 0.3 around their momenta. This variable is required to be less than a value that varies from around 20% in the barrel part of the detector to 8% in the endcap part. The values used are driven by similar requirements in the HLT reconstruction.
Offline selection criteria are generally equal to or tighter than the ones applied at the HLT. Despite this, differences in the definition of the identification variables defined in the online system and offline selection create differences between data and simulation that need dedicated corrections, as described in Section 4.1.
The analysis is carried out separately for W + and W − bosons, and aims to measure the charge asymmetry in W boson production, so any charge misidentification has to be reduced to a minimum. Thus, the offline electron selection also employs a tight requirement for the charge assignment, which reduces the charge misidentification to 0.02 (0.20)% in the barrel region (endcap sections) in the p T range of interest [40].
Events coming from W → ν decays are expected to contain one charged lepton (muon or electron) and significant p miss T resulting from the neutrino. The missing transverse momentum vector p miss T is computed as the negative vector sum of the transverse momenta of all the PF candidates in an event, and its magnitude is denoted as p miss T [41]. No direct requirement on p miss T is applied, but a requirement is placed on the transverse mass, defined as m T = √ 2p T p miss T (1 − cos ∆φ), where ∆φ is the angle in the transverse plane between the directions of the lepton p T and the p miss T . Events are selected with m T > 40 GeV. This requirement rejects a large fraction of QCD multijet backgrounds.
Events from background processes that are expected to produce multiple leptons, mainly Z → , tt, and diboson production, are suppressed by a veto on the presence of additional electrons

Efficiency corrections
or muons in the event. To maximize the rejection efficiency, these events are rejected if additional leptons, selected with looser identification and isolation criteria than the selected lepton, have p T > 10 GeV.

Efficiency corrections
The measurement of differential cross sections relies crucially on the estimation of the lepton selection efficiencies both in the collision data and in the MC because these are among the dominant contributions to the uncertainty. For the total absolute cross sections, the uncertainties are dominated by the integrated luminosity uncertainty. For normalized differential cross sections the correlation of the luminosity uncertainty between the inclusive and differential measurements is such that it mostly cancels out in their ratio. Thus, the dominant uncertainties are the ones related to the lepton efficiency that are not fully correlated through the lepton kinematics phase space.
The lepton efficiency is determined separately for three different steps in the event selection: the trigger (L1+HLT), the offline reconstruction, and the offline selection, which includes identification and isolation criteria. The lepton efficiency for each step is determined with respect to the previous one.
A technique called tag-and-probe is used, in which the efficiency for each step is measured for MC simulation and collision data using samples of Z → events with very high purity [42]. The sample is defined by selecting events with exactly two leptons. One lepton candidate, denoted as the tag, satisfies tight identification and isolation requirements. The other lepton candidate, denoted as the probe, is selected with the selection criteria that depend on the efficiency of the above steps being measured. The number of probes passing and failing the selection is determined from fits to the invariant mass distribution, with Z → signal and background components. The backgrounds in these fits stem largely from QCD multijet events and are at the percent level. In certain regions of phase space, especially in the sample of failing probes, these backgrounds contribute significantly, requiring an accurate modeling of the background components. The nominal efficiency in collision data is estimated by fitting the Z signal using a binned template derived from simulation, convolved with a Gaussian function with floating scale and width to describe the effect of the detector resolution. An exponential function is used for the background. The nominal efficiency in MC simulation is derived from a simple ratio of the number of passing probes over all probes.
For each step, the tag-and-probe method is applied to data and to simulated samples, and the efficiency is computed as a function of lepton p T and η. The ratio of efficiencies in data and simulation is computed together with the associated statistical and systematic uncertainties, and is used to weight the simulated W boson events. The uncertainties in the efficiencies are propagated as a systematic uncertainty in the cross section measurements. The analysis strategy demands a very high granularity in the lepton kinematics. Therefore, the efficiencies are computed in slices of ∆η = 0.1 and steps of p T ranging from 1.5 to 5.0 GeV. A smoothing is applied as a function of lepton p T for each slice in η, modeled by an error function. Systematic uncertainties associated with this method are propagated to the measurement and are discussed in Section 7.1.3. These include a correlated component across η and an uncorrelated component related to the statistical uncertainty in each of the slices in η .

Background estimation
The selection requirements described in Section 4 result in a data sample of 114 (51)×10 6 W + and 88 (42)×10 6 W − candidate events in the muon (electron) final state with small background. A summary of the inclusive background-to-signal ratios is shown in Table 1. The most significant residual background is QCD multijet production, where the selected nonprompt leptons stem from either semileptonic decays of heavy-flavor hadrons or are the product of misidentified jets (usually from light quarks). The former is the principal source of QCD background in the muon channel; the latter dominates the background in the electron channel, along with the production of electron-positron pairs from photon conversions.
The nonprompt-lepton background is estimated directly from data. A control sample (the application sample) is defined by one lepton candidate that fails the standard lepton selection criteria, but passes a looser selection. The efficiency, pass , for such a loose lepton object to pass the standard selection is determined using another independent sample (the QCDenriched sample) dominated by events with nonprompt leptons from QCD multijet processes. This QCD-enriched sample, which is disjoint to the signal sample by means of the requirement m T < 40 GeV, is defined by one loosely identified lepton and a jet with p T > 45 GeV recoiling against it. The measured efficiency for the leptons in this sample, parameterized as a function of p T and η of the lepton, is used to weight the events in the application sample by pass /(1 − pass ) to obtain the estimated contribution from the nonprompt-lepton background in the signal region. The efficiency pass is computed with granularity of ∆η = 0.1, and in each η bin it is parameterized as a linear function of p T .
A small fraction of the events passing the selection criteria are due to other electroweak processes, and this contribution is estimated from simulation. Drell-Yan (DY) events that produce a pair of muons or electrons and one of the two leptons falls outside the detector acceptance mimic the signature of W boson events rather closely. A smaller effect from DY production stems from Z → ττ decays where one τ lepton decays leptonically and the other hadronically. Additionally, events from W → τν decays are treated as background in this analysis. The light leptons from the τ decays typically exhibit lower p T than that in signal events, and are strongly suppressed by the minimum p T requirements. Other backgrounds arise from tt and single top production, with one of the top quarks producing a W boson that subsequently decays leptonically. There are small contributions to the background from diboson (WW, WZ, ZZ) production. Finally, for the electron channel only, the background from W → eν, where the lepton is reconstructed with the wrong charge, is estimated. This background is completely negligible for the muon final state.

Template construction and fitting procedure
The measurement strategy is to fit 2D templates in the charged-lepton kinematic observables of p T and η to the observed 2D distribution in data. Whereas each of the background processes results in a single template, the simulated W boson signal is divided into its three helicity states as well as into slices of the W boson rapidity, |y W |. The procedure of constructing these helicityand rapidity-binned signal templates is described below.

Construction of helicity and rapidity signal templates
The inclusive W boson production cross section at a hadron collider, with its subsequent leptonic decay, neglecting the small terms which are exclusively NLO in QCD, is given by [43]: where θ * and φ * are the polar and azimuthal decay angles of the lepton in the Collins-Soper frame of reference [44], where the lepton refers to the charged lepton in the case of W − , and the neutrino in the case of W + . The angular coefficients A 0 to A 4 in Eq. (1) depend on the W boson charge, p W T , and y W , and receive contributions from QCD at leading and higher orders. When integrating Eq. (1) over φ * , the cross section is written as: dN d cos θ * ∝ (1 + cos 2 θ * ) This expression can equivalently be written as a function of the helicity amplitudes [45]: where the coefficients f i are the helicity fractions, and the upper (lower) sign corresponds to W + (W − ) boson, respectively. Thus, the fractions of left-handed, right-handed, and longitudinal W bosons ( f L , f R and f 0 , respectively) are related to the coefficients A i of Eq. (2), with The generated leptons are considered before any final-state radiation ("pre-FSR leptons") and are called pre-FSR leptons.
Since there is no helicity information in the simulated MC signal sample, the reweighting procedure is implemented based on the production kinematics of the W boson, and the kinematics of the leptonic decay of the W boson.
The coefficients f i depend strongly on the production kinematics of the W boson, namely p W T , |y W |, and its charge. Therefore, a reweighting procedure is devised in which the cos θ * distribution is fitted in bins of p W T and |y W |, separately for each charge, to extract the predicted f i . These spectra of the decay angle are constructed in the full phase space of the W boson production. Each simulated event is reweighted three separate times to obtain pure samples of left-handed, right-handed, and longitudinally polarized W bosons. The results of this procedure are illustrated in Fig. 1, where the simulated signal is split into the three helicity states by reweighting by the extracted helicity fractions f i . Distributions of p W T and |y W | are shown for both charges of W bosons, along with the resulting distribution of the charged lepton η.
The distributions of p W T and |y W | are substantially different for the three helicity components. Whereas the left-handed W bosons (W L ) and the right-handed W bosons (W R ) behave in the same way as a function of p T , their behavior in |y W | is significantly different. Their production cross sections are equal at |y W | = 0, but that of the W L component increases up to a maximum at |y W | between 3.0 and 3.5, whereas the W R component decreases monotonically with |y W |. The longitudinally polarized W bosons (W 0 ) have an overall much lower production cross section, which is relatively flat in |y W | and increases as a function of p T , as expected in the Collins-Soper reference frame. The different distributions in |y W | of the W R and W L components, paired with the preferential decay direction of the charged lepton for these two helicity states, results in distinctly different η distributions. For positively charged W bosons at a given |y W |, the W L component causes the charged lepton to have values of η closer to zero. In contrast, the positively charged W R component tends to have larger values of |η |. The opposite is true for negatively charged W bosons, i.e., the charged lepton |η | will tend to be large for left-handed W − bosons, whereas right-handed W − bosons lead to leptons observed mostly at small |η |.

Fitting strategy for the rapidity-helicity measurement
The characteristic behavior of the lepton kinematics for different polarizations of the W boson can be exploited to measure the cross section for W boson production differentially in |y W | and separately for the three helicity components. This is done by splitting each of the three helicity states into bins of |y W | and constructing the charged lepton p T versus η templates for each of the helicity and charge components from the MC as described above. Example 2D templates are shown in Fig. 2, where three different templates are shown for W + bosons. The blue template is obtained from events with a W R produced from 0.00 to 0.25 in |y W |, the red template from events with a W R produced between 0.50 and 0.75 in |y W |, and the green template from events with a W L produced between 2.00 and 2.25 in |y W |. The behavior described above is clearly seen. Another important aspect of the underlying physics may also be understood from Fig. 2: while the W bosons are produced in orthogonal regions of phase space, the resulting templates for the observable leptons overlap considerably for the different helicity and rapidity bins. This overlap is most striking for adjacent bins in |y W | in a given helicity state. In Fig. 2, the two distributions for the right-handed W boson and the distribution for the left-handed W boson show sizeable overlap, albeit with contrasting shapes as a function of the observable lepton kinematics. A consequence of the large overlaps in general, and in neighboring bins in rapidity in particular, are large (anti-)correlations in the fitted differential cross sections in helicity and rapidity.
The 2D templates in the observable lepton kinematics extend from the minimum p T requirement of 26 (30) GeV for muons (electrons) to a maximum value of 45 GeV in bins with width of 1 GeV. In the observable η , the width of the bins is 0.1, extending from −2.4 (−2.5) to 2.4 (2.5) for muons (electrons).
To extract the differential cross sections in W boson rapidity for the three helicity states, the full sample of simulated W boson events is divided using the method described earlier into the three helicity components and 10 bins of |y W | of width 0.25 up to |y W | = 2.5. These separate signal processes are left freely floating in a maximum likelihood (ML) fit to the observed 2D distribution for p T versus η . All events above the threshold |y W | = 2.5 are fixed to the prediction from simulation and are treated as background because of the rapid loss in acceptance for certain charge and helicity combinations at higher rapidity. Additionally, the longitudinally polarized states are fixed to the MC prediction. This results in 40 freely floating cross sections in the fit, corresponding to the 10 bins in W boson rapidity for each charge, and for the leftand right-handed polarizations.

Fitting strategy for the double-differential W boson cross section
The double-differential W boson production cross sections, as functions of p T and |η |, are measured with an analogous technique. The double-differential cross section for each charge of the W boson is denoted by and can be measured in very fine bins of η and p T . Current theoretical calculations predict these cross sections with next-to-NLO (NNLO) accuracy in perturbative QCD, and such a measurement is a more rigorous test of these calculations than the previous studies performed by the CDF and D0 Collaborations at the Fermilab Tevatron pp collider [10, 11], or by the ATLAS, CMS, and LHCb Collaborations at the LHC [12-18], which all measured the cross section as a function of reconstructed η only. The CDF Collaboration has also inferred the charge asymmetry as a function of |y W | in Ref.
[10]. When integrating either over the |η | or in the p T dimension, the usual differential cross section measurement can be recovered.
This measurement is performed by fitting the same 2D distributions of p T versus η , with different signal processes. Instead of constructing each signal template from an underlying |y W | and helicity state, each signal process in this measurement corresponds to the underlying generated lepton p T and lepton |η| bin. The generated leptons in this measurement are obtained by a so-called "dressing" procedure, where electroweak radiation is added back to the chargedlepton momentum within a cone of ∆R < 0.1. The unfolding corrects for bin-by-bin differences in generated versus reconstructed p T and η . The resulting number of underlying signal processes increases from the 40 processes in the helicity/rapidity fit to a total of 324, corresponding to 18 bins in the p T times 18 bins in |η |. The generated p T ranges from 26 to 56 GeV in bins of width between 1.5 and 2 GeV, where the bins at low and high values of p T are wider than around the Jacobian peak at around half the mass of the W boson. The bin width in |η | is 0.1 up to |η | = 1.3, followed by 4 bins of width 0.2, and a final bin ranging from |η | = 2.1 to 2.4. Events in which the generated leptons are outside of the reconstructed acceptances are treated as a background component in this fit. The treatment of the backgrounds and the systematic uncertainties remains the same as for the rapidity/helicity fit.

Likelihood construction and fitting
An ML fit is performed to extract the parameters of interest. The construction and calculation of the likelihood, as well as the minimization are implemented using the TENSORFLOW software package originally developed for machine learning applications [46]. The benefit of such an implementation is that the gradients required for minimization are computed automatically by back-propagation, which is both faster and more numerically accurate and stable than finite difference approaches used in existing tools. The calculation of the likelihood, and the additional linear algebra associated with the minimization algorithm can also be parallelized on vector processing units and/or multiple threads, as well as using graphics processing units, for a further improvement in the speed of the fit. The implementation is also optimized to keep memory usage acceptable given the large number of measurement bins and parameters, with a sparse tensor representation used where appropriate.
The negative log-likelihood function can be written as with where: n obs i is the observed number of events in each bin, assumed to be independently Poissondistributed; n exp i,p is the expected yield per bin per process; µ p is the freely floating signal strength multiplier per signal process, fixed to unity for background processes; θ k are the nuisance parameters associated with each systematic uncertainty; and κ i,p,k is the size of the systematic effect per bin, per process, and per nuisance parameter. The systematic uncertainties are implemented with a unit Gaussian constraint on the nuisance parameter θ k such that the factor κ θ k i,p,k multiplying the yield corresponds to a log-normal distribution with the mean equal to 0 and the width equal to ln κ i,p,k . This parameterization corresponds to the one used by the LHC Higgs Combination Working Group [47].
The signal strength modifiers and nuisance parameters are extracted directly from the ML fit, with the corresponding covariance matrix computed from the Hessian of the likelihood at the minimum, which can also be calculated to high numerical accuracy using back-propagation. The unfolded cross sections are extracted simultaneously in the ML fit by including the dependence of the predicted cross section on the nuisance parameters associated with the theoretical uncertainties. The cross sections and corresponding covariance matrix are extracted based on the post-fit values of the signal strength modifiers and nuisance parameters and their covariance.
While the cross section vectors σ are left freely floating when fitting for the rapidity/helicity or the double-differential cross sections, it is also possible to fix these parameters to their expected values. Performing the fit in such a way allows for the direct measurement of the constraints set by the data on every nuisance parameter. This is especially interesting for the case of the PDF uncertainties, as the large and quite pure selected sample of W bosons can place strong constraints on the PDF uncertainties by using the charged lepton kinematics.

Measurement of the charge asymmetry and unpolarized cross sections
The fit to the data is performed simultaneously for the two charge categories and to the three helicity states. Therefore, the minimization can yield combinations of the measured cross sections with the proper propagation of the uncertainties through the fit covariance matrix, either differentially in rapidity or double-differentially in p T and |η |.
One of the additional quantities considered is the polarized W boson charge asymmetry, defined as: where pol represents the W polarization state. The charge asymmetry, as a function of |y W | as extracted from the ML fit, differentially in the three polarizations, provides a more direct constraint on the PDF than the previous measurements at the LHC, which are performed differentially in the reconstructed lepton pseudorapidity [12,16]. In the CDF Collaboration measurement [10], the W boson charge asymmetry was extracted as a function of |y W |, but not separately in the W boson helicity state.
The charge asymmetry of W bosons, which is also determined from the double-differential cross section measurement, is written as: When the distribution is integrated over p T , the results may be compared directly with previous measurements of A(|η |) at hadron colliders. Similarly, when integrating over |η |, A(p T ) is obtained. These one-dimensional (1D) distributions as functions of p T and η are obtained by integrating over the other variable after performing the fully differential 2D fit. Associated uncertainties are included properly from the full 2D covariance matrix of the fit.

Systematic uncertainties
This section describes the treatment of systematic uncertainties from experimental sources, as well as from modeling and theoretical uncertainties. In general, systematic uncertainties are divided into two types: those affecting only the normalization of the templates and those affecting their shape.
Normalization uncertainties are treated as log-normal nuisance parameters acting on a given source of background or signal. They change the overall normalization of the process by the given value, while retaining the relative contributions of the process in each of the p T and η bins.
Shape uncertainties do the exact opposite. While the integral of a background or signal component is kept constant at the central value, the relative shape of the 2D template is allowed to float. This necessitates both an up and down variation of each shape nuisance parameter. These uncertainties are incorporated by means of vertical interpolation of the event count in each bin of the template.
Uncertainties can also be a combination of the two, i.e., change the normalization as well as the shape of the 2D templates simultaneously.

QCD multijet background
The QCD multijet background is estimated from data sidebands in the lepton identification and isolation variables, as described in Section 5.
The uncertainty in the method itself is estimated from closure tests in a background-dominated region, obtained by inverting the m T requirement, i.e., m T < 40 (30) GeV for the µ (e) channel. The level of agreement in this background-dominated region is an estimate of the uncertainty in the normalization of this process. The agreement in the 2D (p T , η )plane is rather good for both muons and electrons, and varies with lepton η and p T . In the case of electrons, where this background is larger than in the muon case, the central value of the QCD background is also rescaled by the values derived in this closure test.
The nonclosure amounts to about 5% in the muon final state for all the |η | bins, and 0.5 to 5.0% in the electron final state, with larger uncertainties at higher |η |. The smaller uncertainty for electrons is related to the increased size of the misidentified-lepton dominated control sample used for closure. Each of these normalization uncertainties is treated as uncorrelated with the others.
A systematic uncertainty in the normalization of the QCD multijet background is also estimated by a closure test in the background-dominated region in bins of p T 3 (5) GeV wide. The uncertainties range from 30 to 15% (10 to 20%), depending on the p T region for the muon (electron) final state. Although the uncertainty is related to differences in the composition of misidentified leptons in the control and signal regions, which are common across the whole p T range, the fraction of real leptons from jets and random combinations of tracks and ECAL deposits within jets might change across the phase space. Thus, conservatively, these normalization uncertainties are also considered uncorrelated among each other.
The closure test is also evaluated for the two charges separately, weighting the events with the charge-independent pass misidentification efficiency. The two estimates are consistent within the uncertainties, with a similar dependency on p T and η . A further check was carried out by computing a charge-dependent ± pass . Based on these checks, an additional chargedependent uncertainty of 2% is introduced in the muon case, in the same coarse bins of |η |, to include possible charge asymmetries in the production of true muons from decays in flight of heavy quarks. No additional uncertainty for electrons is added, since the dominating source of misidentified electrons is random geometric association of energy deposits in the ECAL with tracks within jets, which is charge-symmetric.
The uncertainty in the extraction of the QCD multijet efficiency pass is evaluated as follows. This lepton misidentification rate, pass , is extracted through a linear fit to p T , which has an uncertainty associated with it. While a variation of the offset parameter of this fit is absorbed by the normalization uncertainty, the linear parameter of the fit is varied, which therefore varies the QCD multijet background as a function of p T . This uncertainty is applied in the same uncorrelated bins of |η | as the normalization uncertainty.
In total, 46 (55) nuisance parameters that affect the QCD multijet background estimation are considered for each charge of the muon (electron) final state. The larger number of parameters for the electrons is due to a more granular binning and the larger acceptance in η .

Lepton momentum scale
The lepton momentum scales are calibrated and corrected using events from Z boson decays. Closure tests are performed by fitting the invariant mass spectrum in data and simulation with a Breit-Wigner lineshape, convolved with a Crystal Ball function. The data-to-MC difference in the fitted mass of the Z boson is taken as the nonclosure. This nonclosure is of the order of 10 −4 in the muon case. For such a precision, a detailed nuisance model was implemented to cover residual effects [48] that can remain after the calibration procedure is applied.
Systematic uncertainties in the derivation of the muon momentum scale corrections are included. These uncertainties are related to: the modeling of p Z T , electroweak effects on the Z boson line shape, and the effect of the acceptance on the dimuon invariant mass. Hence, they are finely grained in muon η and p T . Furthermore, the uncertainty in the limited data and sim-ulated Z sample is estimated from 100 statistical replicas of the two data sets. Every replica is constructed from a subset of the total event ensemble via a bootstrap procedure (case resampling with replacement is used) [49]. Each of them is also finely binned in muon η and p T . The 99 independent statistical uncertainties are diagonalized with the procedure of Ref. [50] and their independent contributions are included as shape nuisance effects.
For electron candidates, the observed residual differences in the energy scales for the data and the simulated Z sample are of the order of 10 −3 . A procedure similar to that used for the muon momentum scale is adopted. Two systematic effects are included in fine bins of η e and p e T . The first is the difference in the Z boson mass value obtained by fitting the mass peak for Z → e + e − events in two different ways. The first fit uses an MC template convolved with a Gaussian resolution function and the second with a functional form consisting of a Breit-Wigner lineshape for a Z boson, convolved with a Crystal Ball function, with floating mean and width parameters [51,52]. The effect is the main contribution to the systematic uncertainty, and ranges from 0.1 to 0.2% for p e T < 45 GeV and 0.2-0.3% at higher values. The second, smaller systematic effect comes from the modeling of p Z T . In the muon case, the limited size of the samples used to derive the energy scale corrections is accounted for by the means of 100 replicas of the data and MC samples, diagonalized to get 99 independent nuisance parameters.
For both lepton flavors the precision in the estimate of the momentum scale decreases when increasing |η |. The W boson sample with a lepton in the more forward regions of the detector still has sufficient statistical power to allow the fit to constrain the momentum scale nuisance parameters. If the systematic effect related to the momentum scale is fully correlated across the full η acceptance, then its constraint in the profiling procedure, driven by the large effect on the templates at high |η |, may result in an unphysical constraint in the central region. This is avoided by decorrelating the nuisance parameters related to the various momentum scale systematics in wide bins of η , for both muons and electrons. In contrast, the parameters relating to the statistical part of this uncertainty are kept fully correlated across η .
Since the systematic uncertainty in the momentum scale of the leptons allows the p T of a lepton to be changed and therefore for bin-to-bin migration, it is applied as a shape uncertainty.

Lepton efficiency scale factors
Data-to-simulation efficiency scale factors are derived through the tag-and-probe method, also using Z → events. Two systematic uncertainties are considered. The first uncertainty comes from the scale factor and depends on the functional forms used to describe the background and signal components when fitting the efficiencies in each bin of η as a function of p T of the probe lepton. Alternative fits are performed by using different models for the dilepton invariant mass line shape for either the Z boson events or for the combinatorial background events, resulting in different efficiencies. The alternative signal shape is a Breit-Wigner function with the nominal Z boson mass and width, convolved with an asymmetric resolution function (Crystal Ball function) with floating parameters. The alternative background description is done with a function modeling the invariant mass of random combinations of two leptons satisfying the minimum p T criteria. This systematic change is assumed to be correlated among all bins in η . The size of this uncertainty ranges from a few per mill at low |η |, to around 1-2% in the very forward region. Another uncertainty in the lepton efficiency scale factors that is uncorrelated between η bins arises from the statistical uncertainties in the event count in each η bin. These uncertainties are derived by varying the parameters of the error function that is used to interpolate between the measured efficiency values as a function of η , described in Section 4.1, by their uncertainties. This introduces three nuisance parameters for each bin in η , resulting in a total of 144 (150) nuisance parameters per charge in the muon (electron) final state. The larger number of parameters for the electrons is due to the larger acceptance in η . These systematic uncertainties are considered uncorrelated for the two charges, since they are measured independently, and the statistical uncertainty of the data and MC sample in each bin is large. An additional uncertainty in the trigger efficiency is included for events with electrons in the endcap sections of the detector. This uncertainty is due to a radiation-induced shift in the ECAL timing in the 2016 data-taking period, which led to early event readout (referred to as pre-firing) in the L1 trigger and a resulting reduction in the efficiency for events with significant energy deposits in the ECAL endcap sections. The correction is estimated using a set of the Z → e + e − events collected in collisions where, because of L1 trigger rules, the event is saved regardless of the L1 trigger decision for the in-time bunch crossing (BX). This sample is composed of events where the L1 decision is positive for the third BX before the in-time BX: this records only about 0.1% of the total Z → e + e − events, and is thus statistically limited. The uncertainty ranges from 0.5% for |η| ≈ 1.5 to 10% at |η| ≈ 2.5 for electrons from W boson decays.

Extra lepton veto
The efficiency for the requirement that only one lepton be present in the event, which is especially effective in reducing the Z → background, is also affected by differences in the efficiencies in data and in simulation. As more background survives the selection at higher |η |, where the uncertainty in the efficiency is larger, a normalization uncertainty is applied, equal to 2 (3)% for the muon (electron) channel. In the electron channel, an additional uncertainty is included to account for the L1 trigger pre-firing effect, described previously in Section 7.1.3, in Z → e + e − events in which one electron is in one of the ECAL endcap sections. This uncertainty ranges from 2% at low electron p T to 10% in the highest |η | and p T bins.

Charge misidentification
The probability of mistakenly assigning the incorrect charge to a muon in the p T range considered is negligible (10 −5 ) [53], thus no uncertainty is introduced to describe this effect. For the electrons, the statistical uncertainty in the estimate of wrong charge assignment in Z → e + e − events reconstructed with same-sign or opposite-sign events is used. It is dominated by the limited sample of same-sign events in the 2016 dataset. The uncertainty assigned to this small background component, in the electron channel only, is 30% [40].

Integrated luminosity
A normalization uncertainty is assigned to the imperfect knowledge of the integrated luminosity. This is applied as an overall normalization uncertainty in all processes estimated from MC simulation and is set at a value of 2.5% [54].

Modeling and theoretical uncertainties 7.2.1 Missing higher orders in QCD
Theoretical uncertainties resulting from missing higher orders in the QCD calculations are implemented in the following way. Renormalization and factorization scales, µ R and µ F , respectively, are changed to half and twice their original value. This change is propagated to the resulting weight for each simulated event in three variations: the uncorrelated ones in which either µ R or µ F is varied, and the correlated one in which both are varied simultaneously, but in the same direction, i.e., both up or down by a factor of two. This uncertainty is applied to all signal processes, as well as to the simulated Z → background. For the signal processes, these variations lead to a normalization shift that is largely independent of η . The impact on the shape of the p T distribution is within 0.5% up to p T < 35 GeV; however, for p T > 35 GeV a significant modification of the predicted p T distribution is seen. These uncertainties change both the normalization and the shape of the overall 2D templates. In the case of the signal, they are split into several components, as described below.

p W T modeling
Imperfect knowledge of the p W T spectrum results in an uncertainty that affects the p T spectrum. It is most important in the region of low p W T , where fixed-order perturbative calculations lead to divergent cross sections as p W T approaches zero, which can be fixed by using resummation. The nominal templates are evaluated from the MADGRAPH5 aMC@NLO simulated sample, with the p W T spectrum reweighted by the measured data versus MC corrections in the p Z T distribution obtained in data, as described in Section 3. Uncertainties are associated with the p W T modeling in such a way as to reduce the sensitivity to the theoretical prediction, at the cost of increasing the statistical uncertainty of the results. The uncertainties in µ R and µ F , described above, are divided into ten bins of p W T : [0.0, 2.9, 4.7, 6.7, 9.0, 11.8, 15.3, 20.1, 27.2, 40.2, 13 000] GeV. These nuisance parameters are uncorrelated for each charge. In the case of the polarized cross section measurement, an uncorrelated uncertainty is used for each helicity state to account for the different production mechanisms of the longitudinally, left, and right polarized W bosons. The µ R and µ F uncertainties in the W → τν process are binned in the same p W T bins, albeit integrated in polarization, and so are uncorrelated with the signal processes.

Parton distribution functions
Event weights in MC simulation derived from 100 variations of the NNPDF3.0 PDF set, referred to as replica sets, are used to evaluate the PDF uncertainty in the predictions. These 100 replicas are transformed to a Hessian representation to facilitate the treatment of PDF uncertainties in the analysis via the procedure described in Ref. [50], with 60 eigenvectors and a starting scale of 1 GeV. Because the PDFs determine the kinematics and the differential polarization of the W boson, variations of the PDFs alter the relative contribution of the W boson helicity states in p W T and |y W |. Thus, the alternative weighting of the signal templates described in Section 6.1 is repeated independently for each of the 60 Hessian variations. Each signal process is reweighted once for each of the 60 independent variations as the up variation, corresponding to one positive standard deviation. The corresponding down variation is obtained by mirroring the up variation with respect to the nominal template. Since the underlying PDF uncertainties also affect the DY and W → τν backgrounds, the same procedure is applied to the simulated events for these backgrounds, and the uncertainties are treated as fully correlated between the signal and these two background processes. This procedure changes the overall normalization of the templates as well as their shapes. The magnitudes of the Hessian variations are 1% or lower for the normalization, but show significantly different behavior in the p T versus η plane, from which a constraint on these PDF uncertainties is expected.

Choice of α S value
The 100 PDF replicas of the NNPDF3.0 set are accompanied by two variations of the strong coupling. The central value of α S at the mass of the Z boson of 0.1180 is varied from 0.1195 to 0.1165. Both normalization and shape are affected by this variation.

Simulated background cross sections
The backgrounds derived from simulation, namely DY, diboson, and W → τν production, and all top quark backgrounds are subject to an overall normalization-only uncertainty. The main contributions to the theoretical uncertainty in the Z and W boson production cross section arise from the PDF uncertainties, α S , and µ R and µ F . These are included as shape nuisance parameters affecting the templates of such processes, and they are fully correlated with the same parameters affecting the signal. For the W → τν process, a further 4% normalization uncertainty is assigned, to address the residual uncertainty because of the much lower p T of the decay lepton.
For the top quark and diboson backgrounds, the kinematic distributions are well modeled by the higher-order MC generators. The uncertainties assigned to the normalization are 6 and 16% respectively, motivated by the large theoretical cross section uncertainty for each of the contributing processes. Because these processes make a small contribution to the selected sample of events, the effect of these relatively large uncertainties is small.

Choice of the m W value
Events are reweighted to two alternative values of m W , with values ±50 MeV with respect to the default m W value in the generator of 80.419 GeV, using a Breit-Wigner assumption for the invariant mass distribution at the generator level. Since the central value of m W does not significantly influence the W boson cross sections, the impact of this uncertainty is very small.

Modeling of QED radiation
The simulation of the signal processes models the lepton FSR through the quantum electrodynamic (QED) showering in PYTHIA within the MADGRAPH5 aMC@NLO MC generator. An uncertainty in this modeling is assessed by considering an alternative showering program, PHOTOS 3.56 [55]. A large sample of W → ν ( = e + , e − , µ + , µ − separately) events is produced at the generator level only at NLO in QCD, and is interfaced to either PYTHIA or PHOTOS. The variable sensitive to FSR, which accounts for the different radiation rate and, in case of radiation, for the harder FSR photon spectrum produced by PHOTOS with respect to PYTHIA, is the ratio r FSR = p dress T /p bare T between the dressed lepton p T and the bare lepton p T (after radiation). Alternative templates are built by reweighting the nominal MADGRAPH5 aMC@NLO events by the ratio between PHOTOS and PYTHIA, as a function of r FSR .
The effect of QED FSR is largely different for the two lepton flavors, because of the differences in the lepton masses and the estimate of the lepton momentum. For the muons, only the track is used and there is no explicit recovery of the FSR. For these reasons, the nuisance parameters related to this effect are kept uncorrelated between the two lepton flavors. For the electrons the effect is derived from a combination of the measurements using the track and the ECAL supercluster. The latter dominates the estimate for the energy range exploited in this analysis, and its reconstruction algorithm, optimized to gather the bremsstrahlung photons, also efficiently collects the FSR photons.

Statistical uncertainty in the W simulation
An uncertainty is assigned to reflect the limited size of the MC sample used to build the signal templates. The sample size, when considering the negative weights of the NLO corrections, corresponds to approximatively one fifth of the data sample. This is included in the likelihood with the Barlow-Beeston Lite approach [56] and represents one of the dominant contributions to the systematic uncertainty. Table 2: Systematic uncertainties for each source and process. Quoted numbers correspond to the size of log-normal nuisance parameters applied in the fit, while a "yes" in a given cell corresponds to the given systematic uncertainty being applied as a shape variation over the full 2D template space. A summary of the systematic uncertainties is shown in Table 2. They amount to 1176 nuisance parameters for the helicity fit.

Impact of uncertainties in the measured quantities
The effects of the systematic uncertainties on the measured quantities (signal strength modifiers for one process, µ p in Eq. (5), absolute cross sections σ p , or normalized cross sections σ p /σ tot ) are presented as the impact of an uncertainty in the parameter of interest. The impact on a given measured parameter µ p from a single nuisance parameter, θ k in Eq. (5), is defined as C pk /σ(θ k ), where C pk is the covariance for the nuisance parameter and the parameter of interest, and σ(θ k ) is the post-fit uncertainty on the nuisance parameter. In the limit of Gaussian uncertainties, this is equivalent to the shift that is induced as the nuisance parameter θ k is fixed and brought to its +1σ or −1σ post-fit values, with all other parameters profiled as normal. The procedure is generalized to groups of uncertainties, gathered such that each group includes conceptually related and/or strongly correlated sources. Groups are defined for: • luminosity: uncertainty in integrated luminosity, • efficiency stat.: uncorrelated part (in η ) of the lepton efficiency systematics, • efficiency syst.: correlated part (in η ) of the lepton efficiency systematics (coming from the tag-and-probe method), L1 pre-firing uncertainty for the signal electron or the second electron from Z → e + e − events, • QCD bkg.: includes both the normalization and shape uncertainties related to the misidentified lepton background from QCD multijet events, • lepton scale: uncertainty in the lepton momentum scale, • other experimental: systematic uncertainties estimated from simulation and the extralepton veto, • other bkg: normalization uncertainties for all backgrounds, except for the nonprompt background, • PDFs ⊕ α S : 60 Hessian variations of the NNPDF3.0 PDF set and α S , • µ F , µ R , µ F+R : separate µ R and µ F variations, plus the correlated variation of both µ R and µ F , • FSR: modeling of final state radiation, • MC sample size: statistical uncertainty per bin of the template for all the samples, • statistical: the statistical uncertainty in the data sample.
The impact of each group is the effect of the combined variation of all the parameters included in it. It is evaluated as is (the transpose of) the matrix of the correlations between the measured parameter and the nuisance parameters within the group, and C is is the subset of the covariance matrix corresponding to the nuisances parameters in the group. This is equivalent to computing the combined impact of the eigenvectors for the postfit nuisances within a group. These groups cover all the nuisance parameters included in the likelihood and are mutually exclusive. Figure 3 summarizes the relative impact of groups of systematic uncertainties for two illustrative measurements: the normalized cross sections and the charge asymmetry for W L , for the combination of the muon and electron final states. The total uncertainty is not expected to be exactly equal to the sum in quadrature of the impacts due to remaining correlations between groups. The impact of uncertainties that are strongly correlated among all the rapidity bins mostly cancel when considering either the cross section normalized to the total cross section or in the charge asymmetry. In these plots the groups of subleading uncertainties with respect to the ones shown are suppressed for simplicity.
In a similar manner, the effect of the statistical and systematic uncertainties is shown for the normalized double-differential cross section, and for its charge asymmetry. For simplicity, the distribution is integrated over p T and it is shown as a function of |η | in Fig. 4.
The two most dominant sources of uncertainties are the uncertainty in the integrated luminosity and the uncertainty due to the limited size of the MC sample compared with the size of the recorded data set. The latter dominates for all normalized quantities, while the former is the largest contribution to the total uncertainty in most regions of the phase space for absolute quantities.

Results and interpretations
The template fit to the (p T , η )distribution is performed on the four independent channels: The observed events as a function of lepton η and p T are shown in Figs. 5 (6) for the muon final state and Figs. 7 (8) for the electron final state, for the positive (negative) charge. These distributions represent the 2D templates, unrolled into one dimension, such that bin unrolled = 1 + bin η + 48(50)bin p T , with bin η ∈ [0, 48(50)] and bin p T ∈ [0, 18(14)] for the muon (electron) channel, along with the 1D projections in each of the two variables. In the projections, the sum in quadrature of the uncertainties in the 2D distribution is shown, neglecting any correlations. Therefore, these uncertainties are for illustration purpose only.

Cross section measurements
The W ± → ν cross section measurements are performed in both the muon and electron channels, by using the negative log likelihood minimization in Eq. (5). This provides a cross-check of experimental consistency of the two decay modes, and provides a method of reducing the impact of the statistical and systematic uncertainties when combining the measurements in the two channels and accounting for correlated and uncorrelated uncertainties.

Combination procedure
Measurements in different channels are combined by simultaneously minimizing the likelihood across channels, with common signal strengths and nuisance parameters as appropriate. Uncertainties that are correlated among channels are those corresponding to the integrated luminosity, the knowledge of specific process cross sections in the background normalizations when the process is estimated from simulation, and effects that are common to multiple processes. Uncertainties related to the estimate of the QCD background are considered uncorrelated between muon and electron channels, since they originate from the closure test of the estimate in the background-dominated regions, which are independent of each other. The estimate of the lepton misidentification probability pass is also performed independently. The systematic uncertainty on pass is 100% correlated between the two charges for each lepton flavor.
The statistical uncertainties in the efficiency correction factors are assumed as uncorrelated among positive and negative charges, and among the channels, since they are derived from independent samples. The fully correlated part of the systematic uncertainty in the efficiency within a channel is assumed uncorrelated between muons and electrons, since the dominant effects from the Z → line-shape and the background sources are very different.
Most of the theoretical uncertainties are assumed 100% correlated among the four channels. They are uncertainties in the boson p T spectrum modeling because of µ F and µ R uncertainties and the uncertainty in the knowledge of α S . Another large group of nuisance parameters that are correlated among all the channels represent the effects of the PDF variations within the NNPDF3.0 set used on both the shape of the templates used and their normalization. The 60 nuisance parameters associated with the Hessian representation of the 100 PDF replicas as well as the uncertainty in α S are 100% correlated among all the four lepton flavor and charge channels. These 60+1 systematic uncertainties are also fully correlated with the respective uncertainties considered for the Z and W → τν processes.

Differential cross sections in |y W |
The measured |y W |-dependent cross section, for the left-and right-handed polarizations, is extracted from the fit in 10 bins of |y W |, with a constant width of ∆y W = 0.25 in a range |y W | < 2.5. The cross sections in the two additional bins, 2.5 < |y W | < 2.75 and 2.75 < |y W | < 10, that integrate over the kinematic region in which the detector acceptance is small, are fixed to the expectation from MADGRAPH5 aMC@NLO with a large 30% normalization uncertainty. To achieve a partial cancellation of uncertainties that are largely correlated among all |y W | bins, the cross sections are normalized to the fitted total W boson cross section integrating over all the rapidity bins within the acceptance. As stated before, the longitudinally polarized component is fixed to the MADGRAPH5 aMC@NLO prediction with a 30% normalization uncertainty. Therefore, it is not a freely floating parameter in the fit, and hence only the W L and W R components are shown in the following.
The measured W boson production cross sections, split into the left-and right-handed helicity states, for the combination of the muon and the electron channels, are presented in Fig. 9, normalized to the total cross section in the whole rapidity range. The experimental distributions are compared with the theoretical prediction from MADGRAPH5 aMC@NLO. The central value from the MADGRAPH5 aMC@NLO prediction, where the p W T spectrum in simulation is weighted by the ratio of measured and predicted spectrum for DY production as described in Section 3, is also shown as a line within the error bands, and denoted as MAD-GRAPH5 aMC@NLO * . It is evident that this weighting has a small impact on the rapidity spectrum, and the alternative expected distributions are well within the other theoretical uncertainties. The uncertainty shown in the theoretical prediction includes the contribution from the PDFs (NNPDF3.0 set), the envelope of the µ F and µ R variations, and the α S .
The main systematic uncertainty in the signal cross section, the 2.5% uncertainty in the integrated luminosity [54], is fully correlated across all the rapidity bins, thus it cancels out when taking the ratio to the total W cross section. The ratio of the expected normalized cross section using the nominal MADGRAPH5 aMC@NLO simulation to the measured one in data is also presented. As described in Section 6.5, the fitted |y W |-dependent cross sections is used to simultaneously derive the differential charge asymmetry. This is presented, differentially in |y W | and polarization, in Fig. 10.
There are significant correlated uncertainties between neighboring W boson rapidity bins. The correlations arising only from the overlap of the signal templates in the (p T , η ) plane, i.e., of purely statistical nature, are in the range 50-80% for adjacent W boson rapidity bins (∆|y W | = 1), raising with |y W |, about 20% for ∆|y W | = 2, about 10% or less for ∆|y W | = 3 and negligible otherwise. An overall correlation sums up to these statistical correlations, originating from systematic uncertainties common to all the signal processes, such as the uncertainty in the integrated luminosity.
The cross section results differential in W boson rapidity are tested for statistical compatibility with a smooth functional shape, taking these correlations into account. Monte Carlo pseudoexperiments show that the results are quantitatively consistent with smooth third-order polynomial functions of |y W |. This test is performed simultaneously in both helicity states, both charges, and all |y W | bins, taking into account the full covariance matrix of the fit.
Results are also shown as an unpolarized normalized cross section, i.e., by summing over all helicity states as a function of |y W |, in Fig. 11. The unpolarized charge asymmetry as a function of |y W | is shown in Fig. 12.
In addition to these normalized and unpolarized cross sections, the results of the fits are also presented as absolute cross sections in Fig. 13, where the absolute, unpolarized cross sections are shown for the combined flavor fit. Generally good agreement is observed in the shape of the measured distribution with respect to the expectation, albeit with an offset of the order of a few percent.
After the fit with floating cross sections is performed, only few nuisance parameters are significantly constrained. Mainly the nuisance parameters related to the normalization of the nonprompt-lepton background and its shape in η and p T are constrained by the fit. Because of the large data sample, this effect is expected. 8.1.3 Double-differential cross sections in p T and |η | Double-differential cross sections in p T and |η | are measured from a fit to the observed data in the (p T , η )plane. The underlying generated templates are unfolded to the dressed lepton definition in 18 bins of p T and 18 bins of |η |, as described in Section 6.3. These cross sections are shown in Fig. 14, normalized to the total cross section. These results come from the combination of the muon and electron final states, divided into two categories of the lepton charge. From the measured cross sections, the double-differential charge asymmetry is computed, where the uncertainty is computed from the full covariance matrix from the fit, and it is shown in Fig. 15.
The agreement of the measured normalized W boson cross sections and charge asymmetry with the prediction of MADGRAPH5 aMC@NLO is at the level of 1% in the central part of the lepton acceptance (|η | < 1). In the outer endcap sections of the detector, especially for lower p T , the agreement with the prediction becomes worse.
Although these normalized cross sections of the combined flavor fit represent the result with the smallest total uncertainty because of the cancellation of the fully correlated components, the absolute cross sections are also of interest. In particular, the agreement of the absolute cross sections between the flavor channels highlights the understanding of the systematic uncertainties. These plots are displayed in Fig. 16, where the measured absolute cross sections are shown separately for the muon, electron, and combined fits. Good agreement is found within the uncertainties in the regions with sufficient event count. Uncertainties become large in the high-|η | region for the electron-only fit, rendering a precise comparison difficult.
From the results of this fit, the single-differential cross section is measured by integrating in one of the two dimensions, as a function of the other variable. Along with these cross sections, the charge asymmetry differential in one dimension is extracted. This approach has the added value, with respect to a single-differential measurement, that it is independent of the modeling of the lepton kinematics in the variable that is integrated over. The resulting absolute cross sections for the combination of the two lepton flavors is shown as a function of η for both W + and W − in Fig. 17. The corresponding W charge asymmetry is shown in Fig. 18. This result can be directly compared with previous measurements of W boson differential cross section and charge asymmetry as functions of η performed at 7 and 8 TeV by the CMS and ATLAS Collaborations [12, 53].
As a further summary of this fit, the total W boson production cross section, integrated over the fiducial region, 26 < p T < 56 GeV and |η | < 2.4, is measured. The fiducial, charge-integrated cross section is 8.47 ± 0.10 nb, which agrees well with the NLO prediction. The values for each charge, and their ratio to the theoretical prediction, are also shown in Fig. 19, as well as the ratio of the two charges to the prediction from MADGRAPH5 aMC@NLO.

Constraining the PDF nuisances through likelihood profiling
When the cross section parameters in the likelihood function of Eq. (5) are fixed to their expected values (µ p = 1) within their uncertainties, the fit has the statistical power to constrain the PDF nuisance parameters. This procedure corresponds to the PDF profiling method described in Ref. [12], with associated caveats about the interpretation of constraints far from the initial predictions. The constraints in this case are derived directly from the detector-level measurements rather than passing through an intermediate step of unfolded cross sections.
The input PDF and MC predictions are both accurate to NLO in QCD, with the MC prediction implicitly including resummation corrections through the parton shower. The theoretical uncertainties included in this procedure for missing higher orders in QCD correspond to the full model used for the measurement as described in Section 7.2. This is in contrast to typical global PDF fits or QCD analyses that are performed at NNLO accuracy, though at fixed order without resummation, and with the inclusion of missing higher order uncertainties only in dedicated studies at NLO so far [57,58].
For each variation, the fit input value (pre-fit) is trivially represented by a parameter with mean zero and width one. The expected post-fit values of these parameters all have mean zero, but a reduced uncertainty after the likelihood profiling procedure, i.e., width smaller than unity. Finally, the points representing the observed post-fit values of the parameters may have mean different from zero, indicating a pull of the associated systematic uncertainty, and width smaller than one.
Such a result can be obtained in both the helicity and the double-differential cross sections fit, and they indeed provide a consistent set of PDF nuisance parameter values. The ones reported in this Section, shown in Fig. 20, come from the former fit. These parameters correspond to the 60 orthogonalized Hessian PDF variations corresponding to the NNPDF3.0 replicas, plus α S . All of the variants, i.e., pre-fit, post-fit expected, and post-fit observed, are shown. Post-fit constraints of 70% of the pre-fit values are observed in some of the PDF nuisance parameters, Whereas the mean constraint is closer to 90%. The post-fit nuisance parameter values with respect to the prefit values and uncertainties give a χ 2 value of 117 for 61 degrees of freedom. This suggests that the PDF set used here at NLO QCD plus parton shower accuracy may not be sufficient to describe the data. It is possible that NNLO QCD accuracy combined with additional developments in fitting methodology incorporated in more recent PDF fits may improve the situation, and this can be studied in detail on the basis of the unfolded cross sections measured here.

Additional plots
Additional plots on the helicity and rapidity analysis are presented in Appendix A.1, and additional plots on the two-dimensional cross sections are presented in Appendix A.2.          and bin unrolled (lower) for W + → µ + ν events for observed data superimposed on signal plus background events. The signal and background processes are normalized to the result of the template fit. The cyan band over the data-to-prediction ratio represents the uncertainty in the total yield in each bin after the profiling process.      and bin unrolled (lower) for W + → e + ν events for observed data superimposed on signal plus background events. The signal and background processes are normalized to the result of the template fit. The cyan band over the data-to-prediction ratio represents the uncertainty in the total yield in each bin after the profiling process.     Figure 9: Measured normalized W + → + ν (left plot) or W − → − ν (right plot) cross section as functions of |y W | for the left-handed and right-handed helicity states from the combination of the muon and electron channels, normalized to the total cross section. Also shown is the ratio of the prediction from MADGRAPH5 aMC@NLO to the data. The MADGRAPH5 aMC@NLO * spectrum stands for the prediction with the p W T weighting applied. The lightly-filled band corresponds to the expected uncertainty from the PDF variations, µ F and µ R scales, and α S .   Figure 11: Measured normalized W + → + ν (left plot) and W − → − ν (right plot) cross section as a function of |y W | from the combination of the muon and electron channels, normalized to the total cross section, integrated over the W polarization states. Also shown is the ratio of the prediction from MADGRAPH5 aMC@NLO to the data. The MADGRAPH5 aMC@NLO * spectrum stands for the prediction with the p W T weighting applied. The lightly-filled band corresponds to the expected uncertainty from the PDF variations, µ F and µ R scales, and α S .  Figure 12: Measured W charge asymmetry as a function of |y W | from the combination of the muon and electron channels, integrated over the W polarization states. Also shown is the ratio of the prediction from MADGRAPH5 aMC@NLO to the data. The MADGRAPH5 aMC@NLO * spectrum stands for the prediction with the p W T weighting applied. The lightly-filled band corresponds to the expected uncertainty from the PDF variations, µ F and µ R scales, and α S .  Figure 13: Measured absolute W + → + ν (left plot) or W − → − ν (right plot) cross section as functions of |y W | from the combined flavor fit. The ratio of the prediction from MAD-GRAPH5 aMC@NLO to the data is also shown. The MADGRAPH5 aMC@NLO * spectrum stands for the prediction with the p W T weighting applied. The lightly-filled band corresponds to the expected uncertainty from the PDF variations, µ F and µ R scales, and α S .               Figure 19: Ratio of the measured over predicted absolute inclusive cross section in the fiducial region 26 < p T < 56 GeV and |η | < 2.5, charge-integrated, charge-dependent, and the ratio for W + and W − . The measurement is the result of the combination of the muon and electron channels. The colored bands represent the prediction from MADGRAPH5 aMC@NLO with the expected uncertainty from the quadrature sum of the PDF⊕α S variations (blue) and the µ F and µ R scales (bordeaux).

Summary
The differential W boson cross sections as functions of the W boson rapidity, |y W |, and for the two charges separately, W + → + ν and W − → − ν , are measured in the W boson helicity states. Double-differential cross sections of the W boson are measured as a function of the charged-lepton transverse momentum p T and absolute pseudorapidity |η |. For both W + and W − bosons, the differential charge asymmetry is also extracted.
The measurement is based on data taken in proton-proton collisions at the LHC at a center-ofmass energy of √ s = 13 TeV, corresponding to an integrated luminosity of 35.9 fb −1 . Differential cross sections are presented, both absolute and normalized to the total production cross section within a given acceptance. For the helicity measurement, the range |y W | < 2.5 is presented, whereas for the double-differential cross section the range |η | < 2.4 and 26 < p T < 56 GeV is used. The measurement is performed using both the muon and electron channels, combined together considering all sources of correlated and uncorrelated uncertainties.
The precision in the measurement as a function of |y W |, using a combination of the two channels, is about 2% in central |y W | bins, and 5 to 20%, depending on the charge-polarization combination, in the outermost acceptance bins. The precision of the double-differential cross section, relative to the total, is about 1% in the central part of the detector, |η | < 1, and better than 2.5% up to |η | < 2 for each of the two W boson charges.
Charge asymmetries are also measured, differentially in |y W | and polarization, as well as in p T and |η |. The uncertainties in these asymmetries range from 0.1% in high-acceptance bins to roughly 2.5% in regions of phase space with lower detector acceptance. Furthermore, fiducial cross sections are presented by integrating the two-dimensional differential cross sections over the full acceptance of the analysis.
The measurement of the W boson polarized cross sections as functions of |y W | is used to constrain the parameters related to parton distribution functions in a simultaneous fit of the two channels and the two W boson charges. The constraints are derived at the detector level on 60 uncorrelated eigenvalues of the NNPDF3.0 set of PDFs within the MADGRAPH5 aMC@NLO event generator, and show a total constraint down to 70% of the pre-fit uncertainties for certain variations of the PDF nuisance parameters.

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 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 and the CMS detector provided by the following funding agencies: the Austrian Federal Ministry of Education, Science and Research and the Austrian Science Fund; the Belgian    [11] D0 Collaboration, "Measurement of the muon charge asymmetry in pp → W+X → µν + X events at √ s = 1.96 TeV", Phys. Rev. D 88 (2013) 091102, doi:10.1103/PhysRevD.88.091102, arXiv:1309.2591.

A Additional material
This Appendix includes the plots and figures that shall be provided as additional material in addition to those already featured in the body of the text. The comparison of the measured unpolarized W boson charge asymmetry as a function of |y W | with the prediction from another matrix-element generator, FEWZ 2.0 [59], is shown in Fig. A.4. The calculation is coupled with either the NNPDF3.1 NNLO PDF set or the CT18 [60] NNLO PDF set.  Measured absolute W + → + ν cross section as a function of |y W | from three distinct fits: the combination of muon and electron channels (green), the muon-only fit (blue), and the electron-only fit (red). The ratio of the prediction from MADGRAPH5 aMC@NLO to the data is also shown. The lightly-filled band corresponds to the expected uncertainty from the PDF variations, µ F and µ R scales, and α S . of the W boson. The numbering corresponds to the bins in |y W | of width 0.25 starting at zero. It is worthwhile to note here that the correlations of neighboring bins in rapidity are large, especially for each helicity. There are also nontrivial correlations across the helicity states.        pdf3  pdf4  pdf5  pdf6  pdf7  pdf8  pdf9  pdf10  pdf11  pdf12  pdf13  pdf14  pdf15  pdf16  pdf17  pdf18  pdf19  pdf20  pdf21  pdf22  pdf23  pdf24  pdf25  pdf26  pdf27  pdf28  pdf29  pdf30  pdf31  pdf32  pdf33  pdf34  pdf35  pdf36  pdf37  pdf38  pdf39  pdf40  pdf41  pdf42  pdf43  pdf44  pdf45  pdf46  pdf47  pdf48  pdf49  pdf50  pdf51  pdf52  pdf53  pdf54  pdf55  pdf56  pdf57  pdf58  pdf59

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Pre-fit Post-fit expected Post-fit observed       A.2 2D differential cross section Figure A.17 shows the absolute cross sections for the combined muon and electron channel fit unrolled along p T , in bins of |η | for both charges of the W boson. Figure A.18 and A.20 shows these same absolute cross sections and charge asymmetry, but integrated over all the bins in |η |. Figures A.19 and A.21 show the normalized differential cross sections for both charges of the W boson as a function of p T and |η |, respectively. The charge asymmetry as a function of p T is also shown in Fig. A.19. 25 show the remaining impacts of the 2D differential cross sections analysis, which were omitted in the main paper: the impacts on the normalized W cross sections as a function of |η | for W bosons with negative charge in Fig. A.22; the impacts on the absolute cross sections for both charges as a function of p T in Fig. A.23; the impacts on the normalized W boson production cross sections as a function of p T for both charges, along with the impacts on the charge asymmetry, in Fig. A.24; and, finally, the impacts on the absolute W boson production cross sections as a function of p T , for both charges of the W boson in Fig. A.25.