Measurement of the inclusive W± and Z/γ* cross sections in the e and μ decay channels in pp collisions at √s = 7 TeV with the ATLAS detector

The production cross sections of the inclusive Drell-Yan processes W (cid:1) ! ‘(cid:1) and Z=(cid:2) (cid:2) ! ‘‘ ( ‘ ¼ e , (cid:3) ) are measured in proton-proton collisions at ﬃﬃﬃ s p ¼ 7 TeV with the ATLAS detector. The cross sections are reported integrated over a ﬁducial kinematic range, extrapolated to the full range, and also evaluated differentially as a function of the W decay lepton pseudorapidity and the Z boson rapidity, respectively. Based on an integrated luminosity of about 35 pb (cid:3) 1 collected in 2010, the precision of these measurements reaches a few percent. The integrated and the differential W (cid:1) and Z=(cid:2) (cid:2) cross sections in the e and (cid:3) channels are combined, and compared with perturbative QCD calculations, based on a number of different parton distribution sets available at next-to-next-to-leading order.

The present measurement determines the cross sections times leptonic branching ratios, W AE Á BRðW ! 'Þ and Z= Ã Á BRðZ= Ã ! ''Þ, of inclusive W and Z production for electron and muon final states, where ' ¼ e, . Compared to the initial measurement by the ATLAS Collaboration [20], the data set is enlarged by 100 and the luminosity uncertainty significantly reduced [21] from 11% to 3.4%. The CMS Collaboration has updated their initial measurement of total W and Z cross sections [22] to include data corresponding to an integrated luminosity similar to that used here [23]. Similar measurements have been performed at the p " p collider Tevatron by the CDF and D0 collaborations [24,25].
The presented cross section values are integrated over the fiducial region of the analysis and also extrapolated to the full kinematic range. The data are also reported differentially, as functions of the lepton pseudorapidity, 3 l , for the W AE cross sections, and of the boson rapidity, y Z , for the Z= Ã cross section. For the ''Z= Ã '' case, which will subsequently often be denoted simply as ''Z,'' all values refer to the dilepton mass window from 66 to 116 GeV. The Z cross section measurement in the electron channel is significantly extended by the inclusion of the forward detector region, which allows the upper limit of the pseudorapidity range for one of the electrons to be increased from 2.47 [20] to 4.9.
The electron and muon W AE and Z cross sections are combined to form a single joint measurement taking into account the systematic error correlations between the various data sets. This also leads to an update of the initial differential measurement of the W charge asymmetry published by ATLAS [26]. Normalized cross sections as a function of the Z boson rapidity and W boson and lepton charge asymmetry measurements have been performed also by the CMS [27,28] and the CDF and D0 collaborations [29][30][31][32][33][34].
The combined W AE and Z cross sections, integrated and differential, are compared with QCD predictions based on recent determinations of the parton distribution functions of the proton. In view of the percent level precision of the measurements, such comparisons are restricted to PDFs obtained to NNLO.
A brief overview of the ATLAS detector, trigger, simulation, and the analysis procedure are presented in Sec. II. The acceptance corrections and their uncertainties are discussed in Sec. III, while Sec. IV presents the selection, the efficiencies, and the backgrounds for both electron and muon channels. The cross section results are first given, in Sec. V, separately for each lepton flavor. In Sec. VI the e and data sets are combined and the results are compared to theoretical predictions. The paper is concluded with a brief summary of the results.

A. ATLAS detector
The ATLAS detector [35] comprises a superconducting solenoid surrounding the inner detector (ID) and a large superconducting toroid magnet system enclosing the calorimeters. The ID system is immersed in a 2 T axial magnetic field and provides tracking information for charged particles in a pseudorapidity range matched by the precision measurements of the electromagnetic calorimeter. The silicon pixel and strip (SCT) tracking detectors cover the pseudorapidity range jj < 2:5. The transition radiation tracker (TRT), which surrounds the silicon detectors, enables tracking up to jj ¼ 2:0 and contributes to electron identification.
The liquid argon (LAr) electromagnetic (EM) calorimeter is divided into one barrel (jj < 1:475) and two end-cap components (1:375 < jj < 3:2, EMEC). It uses an accordion geometry to ensure fast and uniform response and fine segmentation for optimum reconstruction and identification of electrons and photons. The hadronic scintillator tile calorimeter consists of a barrel covering the region jj < 1:0, and two extended barrels in the range 0:8 < jj < 1:7. The LAr hadronic end-cap calorimeter (HEC) (1:5 < jj < 3:2) is located behind the end-cap electromagnetic calorimeter. The forward calorimeter (FCal) covers the range 3:2 < jj < 4:9 and also uses LAr as the active material.
The muon spectrometer (MS) is based on three large superconducting toroids with coils arranged in an eightfold symmetry around the calorimeters, covering a range of jj < 2:7. Over most of the range, precision measurements of the track coordinates in the principal bending direction of the magnetic field are provided by monitored drift tubes (MDTs). At large pseudorapidities (2:0 < jj < 2:7), cathode strip chambers (CSCs) with higher granularity are used in the innermost station. The muon trigger detectors consist of resistive plate chambers (RPCs) in the barrel (jj < 1:05) and thin gap chambers (TGCs) in the end-cap regions (1:05 < jj < 2:4), with a small overlap in the jj ' 1:05 region.
The ATLAS detector has a three-level trigger system consisting of level-1 (L1), level-2 (L2), and the event filter (EF). The L1 trigger rate at design luminosity is approximately 75 kHz. The L2 and EF triggers reduce the event rate to approximately 200 Hz before data transfer to mass storage.

B. Triggers
The analysis uses data taken in the year 2010 with proton beam energies of 3.5 TeV. For the electron channels the luminosity is 36:2 pb À1 . For the muon channels the luminosity is smaller, 32:6 pb À1 , as a fraction of the available data, where the muon trigger conditions varied too rapidly, is not included.
Electrons are triggered in the pseudorapidity range j e j < 2:5, where the electromagnetic calorimeter is finely segmented. A single electron trigger with thresholds in transverse energy of 10 GeV at L1 and 15 GeV at the higher trigger levels is used for the main analysis. Compact electromagnetic energy depositions triggered at L1 are used as the seed for the higher level trigger algorithms, which are designed for identifying electrons based on calorimeter and fast track reconstruction.
The electron trigger efficiency is determined from W ! e and Z ! ee events as the fraction of triggered electrons with respect to the offline reconstructed signal [36]. The efficiency is found to be close to 100%, being constant in both the transverse energy E T and the pseudorapidity e , with a small reduction by about 2% towards the limits of the fiducial region (E T ¼ 20 GeV and j e j ¼ 2:5, see Sec. II D). A systematic uncertainty of 0.4% is assigned to the efficiency determination.
The muon trigger is based at L1 on a coincidence of layers of RPCs in the barrel region and TGCs in the end caps. The parameters of muon candidate tracks are then derived by fast reconstruction algorithms in both the inner detector and muon spectrometer. Events are triggered with a single muon trigger with an EF threshold of transverse momentum p T ¼ 13 GeV.
The muon trigger efficiency is determined from a study of Z ! events. The average efficiency is measured to be 85.1% with a total uncertainty of 0.3%. The lower efficiency of the muon trigger system is due to the reduced geometrical acceptance in the barrel region. reweighted so that the resulting transverse momentum distributions of the W and Z bosons match the data [49,50].
The effect of multiple pp interactions per bunch crossing (''pile-up'') is modeled by overlaying simulated minimum bias events over the original hard-scattering event. MC events are then reweighted so that the reconstructed vertex distribution agrees with the data.
The Monte Carlo simulation is also corrected with respect to the data in the lepton reconstruction and identification efficiencies as well as in the energy (momentum) scale and resolution. Table I summarizes the information on the simulated event samples used for the measurement, including the cross sections used for normalization. The W and Z samples are normalized to the NNLO cross sections from the FEWZ program [20,51]. The uncertainties on those cross sections arise from the choice of PDF, from factorization and renormalization scale dependence, and from the s uncertainty. An uncertainty of ðþ7; À10Þ% is taken for the t " t cross section [52][53][54].

D. Analysis procedure
The integrated and differential W and Z production cross sections are measured in the fiducial volume of the ATLAS detector using the equation where N is the number of candidate events observed in data, B the number of background events, determined using data and simulation, and L int the integrated luminosity corresponding to the run selections and trigger employed. The correction by the efficiency factor C W=Z determines the cross sections fid within the fiducial regions of the measurement. These regions are defined as For the W channels the transverse mass m T is defined as is the azimuthal separation between the directions of the charged lepton and the neutrino.
The main analysis, used to determine the integrated cross sections, is performed for the W and Z electron and muon decay channels for leptons in the central region of the detector of j e j < 2:47 and j j < 2:4, respectively. A complementary analysis of the Z ! ee channel is used in TABLE I. Signal and background Monte Carlo samples as well as the generators used in the simulation. For each sample the production cross section, multiplied by the relevant branching ratios (BR), to which the samples are normalized, is given. The electroweak W and Z cross sections are calculated at NNLO in QCD, t " t at approximate NNLO, and dibosons at NLO in QCD. The inclusive jet and heavy-quark cross sections are given at LO. These samples are generated with requirements on the transverse momentum of the partons involved in the hard-scattering process,p T . No systematic uncertainties are assigned for the jet and heavyquark cross sections, since methods are used to extract their normalization and their systematic uncertainties from data (see text).
The combined efficiency factor C W=Z is calculated from simulation and corrected for differences in reconstruction, identification, and trigger efficiencies between data and simulation (see Sec. IV). Where possible, efficiencies in data and MC are derived from Z ! '' and, in the case of the electron channel, W ! e events [36,55]. The efficiency estimation is performed by triggering and selecting such events with good purity using only one of the two leptons in the Z ! '' case and a significant missing transverse energy in the W ! e case, a procedure often referred to as ''tagging.'' Then the other very loosely identified lepton can be used as a probe to estimate various efficiencies after appropriate background subtraction. The method is therefore often referred to as the ''tag-andprobe'' method.
The total integrated cross sections are measured using the equation where the acceptance A W=Z is used to extrapolate the cross section measured in the fiducial volume fid to the full kinematic region. The acceptance is derived from MC, and the uncertainties on the simulation modeling and on parton distribution functions constitute an additional uncertainty on the total cross section measurement. The total and fiducial cross sections are corrected for QED radiation effects in the final state.
The correction factors C W=Z and A W=Z are obtained as follows: where N MC;rec are sums of weights of events after simulation, reconstruction, and selection; N MC;gen;cut are taken at generator level after fiducial cuts; and N MC;gen;all are sums of weights of all generated MC events (for the Z= Ã channels within 66 < m '' < 116 GeV).
For the measurement of charge-separated W AE cross sections, the C W factor is suitably modified to incorporate a correction for event migration between the W þ and W À samples as  (4) where N MC;recAE and N MC;genAE;cut are sums of weights of events reconstructed or generated as W AE , respectively, without any further charge selection. For example, N MC;recþ includes a small component of charge misidentified events generated as W À , while N MC;genþ;cut contains only events generated as W þ without requirements on the reconstructed charge. This charge misidentification effect is only relevant for the electron channels, and is negligible in the muon channels. Electron and muon integrated measurements are combined after extrapolation to the full phase space available for W and Z production and decay and also to a common fiducial region, chosen to minimize the extrapolation needed to adjust the electron and muon cross sections to a common basis. This kinematic region is defined by extrapolating both channels to j ' j < 2:5 and interpolating the electron measurement over the region 1:37 < j e j < 1:52. The differential cross sections are combined by extrapolating all Z measurements to full phase space in lepton pseudorapidity accessible in Z production and decay and extending the range of the most forward bin of W measurements to 2:18 < j ' j < 2:5. The experimental selections on the transverse momenta of the leptons and on the transverse or invariant mass are retained for the differential cross sections.

III. ACCEPTANCES AND UNCERTAINTIES
The acceptances A W=Z are determined using the MC@NLO Monte Carlo program and the CTEQ 6.6 PDF set. The central values and their systematic uncertainties are listed in Table II, separately for W þ , W À , W AE , and Z= Ã production. The uncertainties due to the finite  (A) and their relative uncertainties (A) in percent for W and Z production in electron and muon channels. The various components of the uncertainty are defined in the text. The total uncertainty (A tot  (i) The uncertainties within one PDF set (A pdf err ). They are derived from the CTEQ 6.6 PDF [10] eigenvector error sets at 90% C.L. (ii) The uncertainties due to differences between PDF sets (A pdf sets ). They are estimated as the maximum difference between the CTEQ 6.6, ABKM095fl [13,14], HERAPDF 1.0 [15], MSTW2008 [12], CT10, CT10W [11], and NNPDF2.1 [18] sets, where samples generated with CTEQ 6.6 are reweighted event by event to other PDFs [56]. (iii) The uncertainties due to the modeling of the hardscattering processes of W and Z production (A hs ). These are derived from comparisons of MC@NLO and POWHEG simulations, using the CTEQ 6.6 PDF set and the parton shower and hadronization models based on the HERWIG simulation. (iv) The uncertainties due to the parton shower and hadronization description (A ps ). These are derived as the difference in the acceptances calculated with POWHEG Monte Carlo, using the CTEQ 6.6 PDF set but different models for parton shower and hadronization descriptions, namely, the HERWIG or PYTHIA programs. In addition, to compute the total cross section ratios (see Sec. VI E), the correlation coefficients between the full W and Z acceptance uncertainties are used. They are 0.80 for W AE À Z, 0.83 for W À À Z, 0.78 for W þ À Z, and 0.67 for W þ À W À .
The corrections, and their uncertainties, to extrapolate the electron and the muon measurements from each lepton fiducial region to the common fiducial region, where they are combined, are calculated with the same approach as described for the acceptances. The extrapolations contribute $3% to the W ! and $7% to the W ! e cross sections. Similarly, the fiducial measurement of the Z cross section is enhanced by $5% in the muon channel and by $12% in the electron channel. The uncertainties on these corrections are found to be on the 0.1% level. The combined fiducial measurements are therefore characterized by negligible theoretical uncertainty due to the extrapolation to the unmeasured phase space.
The differential cross sections for the electron and the muon channels are also combined after extrapolating each measurement to the common fiducial kinematic region. In the case of the W measurements the applied correction is effective only in the highest ' bin and is about 30% in the muon channel and about 9% in the electron channel. The extrapolation factors needed to combine the Z electron and muon measurements, and their systematic uncertainties, are listed in Table III. The uncertainty is of the order of 0.1% in most of the rapidity intervals and increases to 1%-2% near the boundary of the fiducial regions.

IV. EVENT SELECTION, EFFICIENCIES, AND BACKGROUND DETERMINATION
A. Electron channels 1. Event selection: Events are required to have at least one primary vertex formed by at least three tracks. To select W boson events in the electron channel, exactly one well reconstructed electron is required with E T > 20 GeV and jj < 2:47. Electrons in the transition region between the barrel and end-cap calorimeter, 1:37 < jj < 1:52, are excluded, as the reconstruction quality is significantly reduced compared to the rest of the pseudorapidity range. The transverse energy is calculated from calorimeter and tracker information. The electron is required to pass ''medium'' identification criteria [36]. To efficiently reject the QCD background, the electron track must, in addition, have a hit in the innermost layer of the tracking system, the ''pixel b-layer.'' The additional calorimeter energy deposited in a cone of size ÁR 0:3 around the electron cluster is required to be small, where the actual selection is optimized as a function of the electron and p T to have a flat 98% efficiency in the simulation for isolated electrons from the decay of a W or Z boson. The missing transverse energy, E miss T , is determined from all measured and identified physics objects, as well as remaining energy deposits in the calorimeter and tracking information [57]. It is required to be larger than 25 GeV. Further, the transverse mass m T has to be larger than 40 GeV.
The selection as described is also used for the Z boson case with the following modifications: instead of one, two electrons are required to be reconstructed and pass the medium criteria without the additional pixel b-layer and isolation cuts; their charges have to be opposite, and their invariant mass has to be within the interval 66 to 116 GeV.
For the selection of Z events at larger rapidities, a central electron passing ''tight'' [36] criteria, as well as the calorimeter isolation requirement described above for the W channel, is required. A second electron candidate with E T > 20 GeV has to be reconstructed in the forward region, 2:5 jj 4:9, and has to pass ''forward loose'' identification requirements [36]. Its transverse energy is determined from the calorimeter cluster energy and position. As the forward region is not covered by the tracking system, no charge can be measured and the electron identification has to rely on calorimeter cluster shapes only. The invariant mass of the selected pair is required to be between 66 and 116 GeV.

Calibration and efficiencies:
Comprehensive studies of the electron performance are described in [36]. Energy scale and resolution corrections are determined from data as a function of in the central and forward regions, by comparing the measured Z ! ee line shape to the one predicted by the simulation. For the central region, the linearity and resolution are, in addition, cross-checked using J=c ! ee and single electron E=p measurements in W ! e events.
The electron efficiencies are evaluated in two steps called reconstruction and identification. The reconstruction step consists of the loose matching of a good quality track to a high p T calorimeter cluster. Identification summarizes all the further requirements to reduce the background contamination.
The electron reconstruction efficiency in the central region is obtained from the Z tag-and-probe method. The efficiency in data is found to be slightly higher by 1.3% than in MC, and the simulation is adjusted accordingly with an absolute systematic uncertainty of 0.8%.
The identification efficiency for electrons from W or Z decay in the central region is determined using two different tag-and-probe methods, which are performed on selected W and Z data samples, respectively. The W-based determination employs the significant missing transverse energy in those events to obtain an unbiased electron sample. The method benefits from larger statistics but needs more involved procedures for background subtraction, as compared to the Z-related determination. Consistent correction factors to be applied to the simulation are derived from the two methods as a function of the electron rapidity. For the medium identification criteria, the Monte Carlo efficiency is adjusted by about À2:5% on average, with a resulting absolute uncertainty of typically less than 1% on this correction. The quality of the data to MC agreement in the tight identification criteria efficiency is found to depend significantly on electron , and an adjustment by, on average, þ2% with an absolute uncertainty of about 1% is performed. The additional requirements on b-layer hits and calorimeter isolation are found to be very efficient and rather well described in the simulation, resulting in small adjustments and small systematic uncertainties only.
To distinguish W þ from W À events, the charge of the decay electron has to be known. The charge misidentification probability as a function of is determined from a sample of Z ! ee events where both electrons are reconstructed with the same sign. It depends on the identification criteria and, in general, increases at large jj. For electrons passing the medium criteria, about 1% of all electrons are assigned the wrong charge, while for tight electrons this figure is about half. From these measurements, additional uncertainties are derived from the opposite charge requirement on the Z cross section (0.6%) and from migration and charge dependent effects on the W þ and W À cross sections (0.1%).
In the forward region (jj > 2:5), the electron reconstruction is nearly 100% efficient and taken from MC. The identification efficiency is determined using the Z tag-andprobe method in two forward electron rapidity bins, which correspond to the inner part of the EMEC (2:5 < jj < 3:2) and the FCal (3:2 < jj < 4:9), respectively. The simulation overestimates the efficiency by 8.4% and 1.7% in these two bins and is adjusted accordingly, with absolute uncertainties of 5.8% and 8.8%, respectively.
3. Background determination: The largest electroweak background in the W ! e channel is given by the W ! production, mainly from decays involving true electrons, ! e " e . Relative to the number of all W AE candidate events, this contribution is estimated to be 2.6%. The background from t " t events is determined to be 0.4% and further contributions on the 0:1-0:2% level arise from Z ! , Z ! ee, and diboson production. The sum of electroweak and t " t backgrounds are found to be 3.7% in the W À and 3.2% in the W þ channel of the respective numbers of events.
A further significant source of background in the W ! e channel, termed ''QCD background,'' is given by jet production faking electron plus missing transverse energy final states. The QCD background is derived from the data using a template fit of the E miss T distribution in a control sample selected without the E miss T requirement and inverting a subset of the electron identification criteria. The E miss T templates for the signal and the other electroweak and t " t backgrounds are taken from the simulation. The QCD background in the signal region is determined to be 3.4% and 4.8% for the W þ and W À channels, respectively. The statistical uncertainty of this fit is negligible. The background as well as the signal templates are varied to assess the systematic uncertainty on the fraction of QCD background. The relative uncertainty is estimated to be 12% for W þ and 8% for W À , corresponding to a fraction of about 0.5% of the W þ or W À candidates. The fit is performed in each bin of electron pseudorapidity separately to obtain the background for the differential analysis.
The relative background contributions in the central Z ! ee analysis due to electroweak processes, W ! e, Z ! , and W ! , and to t " t production are estimated using the corresponding MC samples to be 0.3% in total. The fraction of candidate events due to diboson decays is 0.2%.
The QCD background in the central Z ! ee analysis is estimated from data by fitting the invariant mass distribution using a background template selected with inverted electron identification cuts and the signal template from MC. This procedure yields a fraction of QCD background of 1.6%. The relative systematic uncertainty on this fraction is dominant and evaluated to be 40% using different background templates and fit ranges, as well as an alternative method based on fitting a sample selected with looser identification criteria. For the differential analysis, the sum of the backgrounds is determined from the global fit, and the relative contributions of each bin are taken from the background template. Differences between templates lead to further relative 25% bin-to-bin uncorrelated uncertainties on the QCD background fraction.
In the forward Z ! ee analysis the main electroweak background comes from W ! e events with an associated jet faking an electron in the forward region. It is estimated to be 1.9%. The QCD background is estimated by fitting the m ee distribution in a similar manner as for the central analysis. Because of the larger level of background the fit can be performed directly in all boson rapidity y Z bins. In total the QCD background is estimated to be 9.4% with relative statistical and systematic uncertainties of 8% and 17%. Differentially, the QCD background fraction varies from 7% to 20% with typical relative total uncertainties of 20% to 40%.

B. Muon channels
1. Event selection: Collision events are selected with the same vertex requirement as for the electron channels. In addition, the vertex with the highest squared transverse momentum sum of associated tracks is selected as the primary vertex for further cuts. To reduce fake collision candidates from cosmic-ray or beam-halo events, the position of the primary vertex along the beam axis is required to be within 20 cm of the nominal position. The efficiency of this requirement is larger than 99.9% in both data and simulation.
Muon track candidates are formed from pairs of standalone tracks in the inner detector and the muon spectrometer, combined using a chi-square matching procedure [58]. W and Z events are selected by requiring at least one or two combined track muons with p T > 20 GeV and jj < 2:4, respectively. The z position of the muon track extrapolated to the beam line has to match the z coordinate of the primary vertex within AE1 cm. A set of ID hit requirements [55] is applied to select high quality tracks also demanding at least one hit in the pixel b-layer.
A track-based isolation criterion is defined by requiring the sum of transverse momenta, P p ID T , of ID tracks with p T > 1 GeV within a cone ÁR < 0:2 around the muon direction, divided by the muon transverse momentum p T , to be less than 0.1. When analyzed after all other selection cuts, this requirement has a high QCD background rejection power, while keeping more than 99% of the signal events in both the W and Z channels.
W ! events are further selected by requiring the missing transverse energy, defined as in the electron analysis, to be larger than 25 GeV and the transverse mass to be larger than 40 GeV. In the Z ! analysis, the two decay muons are required to be of opposite charge, and the invariant mass of the þ À pair to be within the interval 66 to 116 GeV.
2. Calibration and efficiencies: Muon transverse momentum resolution corrections are determined by comparing data and MC as a function of in the barrel and end-cap regions [59]. They are derived by fitting the invariant mass distribution from Z ! events and the curvature difference between inner detector and muon spectrometer tracks weighted by the muon electric charge in Z ! and W ! events. Muon transverse momentum scale corrections are measured by comparing the peak position of the Z ! invariant mass distribution between data and MC and fitting the muon transverse momentum distributions in Z ! events [26,59]. Scale corrections are well below 1% in the central pseudorapidity region, and they increase to about 1% in the high-regions due to residual misalignment effects in the ID and MS.
Muon trigger and identification efficiencies are measured in a sample of Z ! events selected with looser requirements on the second muon and with tighter cuts on the invariant mass window and on the angular correlation between the two muons than in the main analysis in order to reduce the contamination from background events [55]. The efficiencies are measured using a factorized approach: the efficiency of the combined reconstruction is derived with respect to the ID tracks, and the isolation cut is tested relative to combined tracks; finally, the trigger efficiency is measured relative to isolated combined muons. The residual background contamination is measured from data, by fitting the invariant mass spectrum with a signal template plus a background template describing the shape of multijet events measured from a control sample of nonisolated muons. The total background contamination, subtracted from the signal sample, is estimated to be 1.0% in the measurement of the reconstruction efficiency and negligible for other selections. The data-to-Monte Carlo correction factors are all measured to be very close to 1, i.e. 0:993 AE 0:002ðsta:Þ AE 0:002ðsys:Þ for the combined reconstruction, 0:9995 AE 0:0006ðsta:Þ AE 0:0013ðsys:Þ for the isolation, and 1:020 AE 0:003ðsta:Þ AE 0:002ðsys:Þ for the trigger efficiencies. Systematic uncertainties are evaluated by varying the relevant selection cuts within their resolution and the amount of subtracted background within its uncertainty. For the ID reconstruction efficiency, no correction has to be applied.
3. Background determination: The electroweak background in the W ! channel is dominated by the Z ! and the W ! channels. Relative to the number of W AE candidate events, these contributions are determined to be 3.3% and 2.8%, respectively. The contribution from Z ! decay is 0.1% while the t " t contribution is estimated to be 0.4%. Diboson decays contribute 0.1%. Overall, these backgrounds are found to be 6.1% in the W þ and 7.6% in the W À channel, respectively.
The QCD background in the W ! channel is primarily composed of heavy-quark decays, with smaller contributions from pion and kaon decays in flight and hadrons faking muons. Given the uncertainty in the dijet cross section prediction and the difficulty of simulating fake prompt muons, the QCD background is derived from data. The number of expected events is determined by extrapolating from control regions defined by reversing the isolation and missing transverse energy requirements. This analysis yields a fraction of background events of 1.7% in the W þ and of 2.8% in the W À channel, respectively. The systematic uncertainty is dominated by the uncertainty on the extrapolation of the isolation efficiency for QCD events from the control to the signal sample, which is estimated to be about 23% relative to the number of background events.
The relative background contributions in the Z ! channel due to t " t events, Z ! and diboson decays are estimated to be 0.1%, 0.07%, and 0.2%, respectively. The background contaminations from W ! and W ! are found to be negligible.
The QCD background in the Z ! channel is also estimated from data. The number of events is measured in control samples, selected using inverted isolation and m requirements, corrected for the signal and electroweak background contamination, and extrapolated to the signal region. The measured fraction of background events is 0.4%. The systematic uncertainty is evaluated by testing a different isolation definition for the control region, propagating the uncertainties in the electroweak background subtraction, and checking the stability of the method against boundary variations of the control regions. Additional cross-checks of the background estimation are done by comparing with the result of a closure test on simulated events and of an analysis of the invariant mass spectrum based on fit templates, derived from the data and the Monte Carlo. The relative systematic uncertainty amounts to 56% while the relative statistical uncertainty is 40%.
Cosmic-ray muons overlapping in time with a collision event are another potential source of background. From a study of noncolliding bunches this background contribution is found to be negligible.

A. Electron cross sections
1. Control distributions: The understanding of the W and Z measurements can be illustrated by comparing the measured with the simulated distributions. A total of 77 885 W þ and 52 856 W À events are selected in the electron channel. A crucial quantity in the W measurement is the missing transverse energy E miss T , for which the distributions for the two charges are shown in Fig. 1. The requirement E miss T > 25 GeV is seen to suppress a large fraction of the QCD background. Figure 2 shows the distributions of the electron transverse energy E T and the transverse mass m T of the W ! e candidates. The observed agreement between data and MC is good.
A total of 9725 and 3376 candidates are selected by the central and forward Z ! ee analyses, respectively. The invariant mass and boson rapidity distributions are compared to the simulation in Figs. 3 and 4 for the two analyses. The complementarity in the rapidity region covered is easily visible. For the forward Z ! ee analysis the lepton rapidity distributions for the two electrons are shown in Fig. 5. The forward electron reaches  Because of a small number of nonoperational LAr readout channels, the rapidity distributions show an asymmetry, which is well described by the simulation. The overlaps between different calorimeter parts are visible as regions with significantly lower efficiency. Table IV reports the number of candidates, estimated background events, and the C W=Z and A W=Z correction factors used, where the uncertainties on A W=Z are obtained from Table II. The cross sections for all channels are reported in Table V with fiducial and total values and the uncertainties due to data statistics, luminosity, further experimental systematic uncertainties, and the acceptance extrapolation in the case of the total cross sections. Table VI presents the sources of systematic uncertainties in all channels. Excluding the luminosity contribution of 3.4%, the W cross sections are measured with an experimental uncertainty of 1.8% to 2.1%, where the main contributions are due to electron reconstruction and identification as well as missing transverse energy performance related to the hadronic recoil [57].  The Z cross section is measured, apart from the luminosity contribution, with an experimental precision of 2.7%. This is dominated by the uncertainty on the electron reconstruction and identification efficiency.

Results:
The theoretical uncertainties on C W=Z are evaluated by comparisons of MC@NLO and POWHEG Monte Carlo simulations and by testing the effect of different PDF sets, as described in Sec. III for the acceptances. The total theoretical uncertainty is found to be 0.6% for C W and 0.3% for C Z .
The theoretical uncertainty on the extrapolation from the fiducial region to the total phase space for W and Z production is between 1.5% and 2.0%, as mentioned above.
The cross sections measured as a function of the W electron pseudorapidity, for separated charges, and of the Z rapidity are presented in Tables XVI, XVII, XVIII, and XIX. The statistical, bin-correlated, and uncorrelated systematic and total uncertainties are provided. The overall luminosity uncertainty is not included. The statistical uncertainty in each bin is about 1%-2% for the W differential TABLE V. Fiducial and total cross sections times branching ratios for W þ , W À , W AE , and Z= Ã production in the electron decay channel. The electron fiducial regions are defined in Sec. II D. The uncertainties denote the statistical (sta), the experimental systematic (sys), the luminosity (lum), and the extrapolation (acc) uncertainties.  measurements, while the total uncertainty is at the 2.5%-3% level. For the Z rapidity measurement the statistical uncertainty is about 2% for jy Z j < 1:6 and grows to 3%-5% in the more forward bins. The total uncertainty on the Z cross sections is 3%-4% in the central region and up to 10% in the most forward bins. It is mainly driven by the uncertainties on the electron reconstruction and identification efficiencies.

B. Muon cross sections
1. Control distributions: A total of 84 514 W þ , 55 234 W À , and 11 709 Z candidates are selected in the muon channels. A few distributions of these candidate events are compared to the simulation for the signal and the background contributions in the following. Figures 6 and 7 show the distributions of muon transverse momentum and the transverse missing energy of candidate W events for positive and negative charges. The transverse mass distributions are shown in Fig. 8. The invariant mass distribution of muon pairs, selected by the Z analysis, and the boson rapidity distribution are shown in Fig. 9. The agreement between data and Monte Carlo is good in all cases.
3. Results: Table VII reports the number of candidates, the estimated background events, and the C W=Z and A W=Z correction factors used for the different measurements. The fiducial and total cross sections are reported in Table VIII for all channels, with the uncertainties due to data statistics, luminosity, further experimental systematics, and the acceptance extrapolation in the case of the total cross sections.
The breakdown of the systematic uncertainty in all channels is shown in Table IX. Apart from the luminosity contribution of 3.4%, the W ! cross section is measured with an experimental uncertainty of 1.6%. The largest contribution comes from the muon efficiencies (1.1%), followed by several contributions in the 0.3%-0.8% range such as the QCD background, the transverse missing energy scale and resolution uncertainties, and the uncertainty on the momentum scale correction.  The Z ! cross section is measured, apart from the luminosity contribution, with an experimental precision of 0.9%. This is dominated by the uncertainty in the muon reconstruction efficiency (0.6%), with about equal systematic and statistical components due to the limited sample of Z ! events. The uncertainty of   the momentum scale correction has an effect of 0.2%, while the uncertainty from momentum resolution is again found to be negligible. The impact of the QCD background uncertainty is at the level of 3 per mille. The theoretical uncertainties on C W=Z are evaluated as in the electron channels and found to be 0.7%-0.8% for C W and 0.3% for C Z .
The uncertainty on the theoretical extrapolation from the fiducial region to the total phase space for W and Z production is between 1.5% and 2.1%.
The cross sections measured as a function of the W muon pseudorapidity, for separated charges, and of the Z rapidity are shown in Tables XX, XXI, and XXII. The statistical, bin-correlated, and uncorrelated systematic and total uncertainties are provided. The uncertainties on the extrapolation to the common fiducial volume, on electroweak and multijet backgrounds, on the momentum scale and resolution are treated as fully correlated between bins for both W and Z measurements. Other uncertainties are considered as uncorrelated.
The statistical uncertainties on the W differential cross sections are in the range 1%-2%, and the total uncertainties are in the range of 2%-3%.
The differential Z cross section is measured with a statistical uncertainty of about 2% up to jy Z j < 1:6, 2.6% for 1:6 < jy Z j < 2:0, and 4.4% for 2:0 < jy Z j < 2:4. The available number of Z events dominates the total uncertainty, with systematic sources below 1.5% in the whole rapidity range.

A. Data combination
Assuming lepton universality for the W and Z boson e and decays, the measured cross sections in both channels can be combined to decrease the statistical and systematic uncertainties. This combination cannot trivially be applied to the pure fiducial cross sections as somewhat different geometrical acceptances are used for the electron and the muon measurements. This requires the introduction of the common kinematic regions, defined in Sec. II D, where W and Z measurements can be combined.
The method of combination used here is an averaging procedure which has been introduced and described in detail in [60,61]. It distinguishes different sources of systematic errors on the combination of the W and Z cross section measurements, in electron and muon channels.
The sources of uncertainty which are fully correlated between the electron and muon measurements are as follows: the hadronic recoil uncertainty of the E miss T measurement (for W measurements), electroweak backgrounds, pile-up effects, uncertainties of the z-vertex position, the theoretical uncertainties on the acceptance, and extrapolation correction factors.
The sources of uncertainty considered fully correlated bin-to-bin and across data sets are as follows: the extrapolation into noncovered phase space, normalization of the electroweak background, lepton energy or momentum scale and resolution, and systematic effects on reconstruction efficiencies.
In addition, the QCD background systematics are bin-tobin correlated but independent for the e and data sets. The statistical components of the lepton identification efficiencies are largely bin-to-bin uncorrelated but correlated for the W and Z cross sections, whereas the statistical uncertainties of the background and the electron isolation determinations are fully uncorrelated sources. Finally, some sources are considered as fully anticorrelated for W þ and W À production, specifically the PDF uncertainty on C W and the charge misidentification. The luminosity uncertainty is common to all data points, and it is therefore not used in the combination procedure.
In total there are 59 differential cross section measurements entering the combination with 30 sources of correlated systematic uncertainties. The data are combined using the following 2 function [61], which is minimized in the averaging procedure: The sums run over all measurement sets k and points i considered. In case a specific set k contributes a measurement i k to point i, one has w i k ¼ 1; otherwise w i k ¼ 0. The deviations of the combined measurements m i from the original measurements i k are minimized. The correlated error sources j can shift, i.e. b j Þ 0, where b j is expressed in units of standard deviations, and such shifts incur a 2 penalty of b 2 j . The relative statistical and uncorrelated systematic uncertainties of a specific measurement are labeled i sta;k and i unc;k , respectively. The relative correlated systematic uncertainties are given by the matrix i j;k , which quantifies the influence of the correlated systematic error source j on the measurement i in the experimental data set k. In addition, total correlated uncertainty i corr;k can be estimated as a sum in quadrature of i j;k . The combined Z, W À , and W þ differential cross sections are given in Tables XXIII, XXIV, and XXV. The data can be obtained electronically through the HepData repository [62]. The results are quoted with their statistical, uncorrelated, and correlated uncertainties per bin, where the influence of all correlated sources is quantified individually with the matrix i j;k . The data show good compatibility, with the total 2 =dof ¼ 33:9=29. A good level of agreement is also seen if combinations are performed separately for the Z ( 2 =dof ¼ 15:5=9), the W þ ( 2 =dof ¼ 10:2=10), and the W À data ( 2 =dof ¼ 7:0=10).

B. Theoretical calculations
The precision of the current differential and integrated cross section measurements has reached the percent level. Comparisons with QCD predictions therefore are made at next-to-next-to-leading order in perturbation theory using recent NNLO sets of PDFs. The dependence of the cross section predictions on the renormalization ( r ) and factorization ( f ) scales is reduced at NNLO. Varying r and f independently around their central values, taken to be M W or M Z , with the constraint 0:5 < r = f < 2, a maximum effect of about 3% is observed on the NLO cross sections, which is reduced to 0.6% at NNLO, using the MSTW08 PDF sets. The theoretical Z= Ã and W AE predictions, used in the following for a comparison with the data, are obtained with the most recent versions of the programs FEWZ [9,51] and DYNNLO [63,64], which provide NNLO cross sections for vector boson production and decays with full spin correlations and finite width effects. Calculations are performed using the G electroweak parameter scheme and those values of the strong coupling constant s which belong to the original determinations of the PDFs. The predictions obtained with FEWZ and DYNNLO are found to agree to within 0.5% for the total cross sections and to within 1% for the fiducial cross sections when using the same electroweak parameter settings and the standard model predictions for the total and partial widths of the W and Z vector bosons, which also account for higher order electroweak and QCD corrections [65].
The NNLO QCD predictions do not include corrections due to pure weak and interference effects between initial and final state radiation. Both effects have been estimated using the SANC program [66]. The interference effects are below 0.1% for all considered channels. Pure weak effects may change the predicted cross sections by about 0.5%. Shape modifications due to the pure weak corrections are calculated to be at most 10% of the quoted correction values. Since the size of the pure weak corrections is estimated to be of the same order as the level of agreement of the NNLO QCD predictions for the fiducial cross sections, they are not applied for the subsequent comparison of the theory with the data.
For the following comparisons to data, all integrated cross section values, the y Z distributions, and the normalization of the ' distributions are taken from FEWZ. The shapes of the pseudorapidity distributions are taken from DYNNLO which have a higher statistical precision than the differential distributions obtained with FEWZ.

C. Differential cross sections
The differential Z and W AE cross sections are shown in Figs. 10 and 11. The measurements for different channels are seen to be in good agreement with each other. Excluding the overall luminosity normalization uncertainty, the data accuracy reaches about 2% in the central region of the Z rapidity. In the most forward region of the Z cross section measurement, the accuracy is still limited to 6% (10%) at y Z ' 2:6 ð3:2Þ. For the W cross section measurements, a precision of about 2% is obtained in each bin of ' .
The combined differential Z and W AE cross sections are compared in Figs. 12 and 13 with the calculated NNLO predictions using the JR09, ABKM09, HERAPDF1.5, and MSTW08 NNLO PDF sets. The uncertainties of the binwise predictions are a convolution of the PDF uncertainties, considered by the authors of the various PDF sets 4 to correspond to 68% C.L., and a residual numerical uncertainty of below 0.5%. One observes that the measured y Z and ' dependencies are broadly described by the predictions of the PDF sets considered. Some deviations, however, are visible, for example, the lower Z cross section at central rapidities in the case of the JR09 PDF set, or the tendency of the ABKM09 prediction to overshoot the Z and the W cross sections at larger y Z and ' , respectively. It thus can be expected that the differential cross sections presented here will reduce the uncertainties of PDF determinations and also influence the central values.  10 (color online). The combined d=djy Z j cross section, for Z= Ã ! ' þ ' À , compared to measurements obtained separately in the muon and electron (central and forward) channels. The kinematic requirements are 66 < m '' < 116 GeV and p T;' > 20 GeV. For the combined result, the uncorrelated uncertainties are shown as crosses and the total uncertainties as boxes. Only the total uncertainties are shown for uncombined measurements. The luminosity uncertainty is not included. Points are displaced for clarity within each bin.
The combined electron and muon data allow for an update of the measurement of the W charge asymmetry which was previously published [26] by ATLAS based on initial muon measurements alone. The asymmetry values, obtained in the W fiducial region of this analysis, and their uncertainties are listed in Table XXVI. The measurement accuracy ranges between 4% and 8%. The previous and the new measurements are consistent. Since the present measurement is more precise and relies on the same data-taking period, it supersedes the previous result. Figure 14 shows the measured W charge asymmetry together with the NNLO predictions obtained from the DYNNLO program. The ABKM09 and the HERAPDF 1.5 predictions give the best agreement with these results. Some deviations from the measured W þ cross section of ABKM09 (HERAPDF 1.5) observed at larger (smaller) j ' j, however, illustrate that more sensitive information is inherent in the separate W þ and W À cross sections and their correlations rather than in the asymmetry.

D. Integrated cross sections
The combination procedure as outlined above is also used to combine the integrated electron and muon Z and W AE cross sections, separately for the common fiducial and the total cross sections. The integrated fiducial cross sections for the W þ , W À , W AE , and Z channels, listed in Table X with their uncertainties, are all measured to about 1% systematic uncertainty, with significantly smaller uncertainties due to statistics and essentially negligible uncertainties due to the extrapolation to the common fiducial phase space. The luminosity uncertainty of 3.4% is fully correlated between the measurements.
It is instructive to compare the measured integrated cross sections with the theoretical predictions, evaluated in the fiducial region of the measurement. The cross sections are calculated, as described above, to NNLO using the FEWZ program and the four NNLO PDF sets as used also for the differential comparisons. Figure 15 shows the W þ and W À cross sections (left panel) and the (W þ þ W À ) and Z= Ã cross sections (right panel). The outer ellipse is obtained using the correlation coefficients for the total uncertainty, while the inner, much shorter ellipse is obtained excluding the luminosity uncertainty. The numerical values of these  12 (color online). Differential d=djy Z j cross section measurement for Z ! '' compared to NNLO theory predictions using various PDF sets. The kinematic requirements are 66 < m '' < 116 GeV and p T;' > 20 GeV. The ratio of theoretical predictions to data is also shown. Theoretical points are displaced for clarity within each bin. correlation coefficients are given in Table XI. The theoretical ellipses result from the PDF uncertainties, quoted to correspond to about 68% C.L. in their two-dimensional area, 5 and the cross section correlations are obtained from the different error eigenvector sets. The measurement exhibits a sensitivity to differences in the predicted cross sections, which is hindered, however, by the luminosity uncertainty which dominates the error on the integrated cross section measurement.
The predictions rely on the evolution of the PDFs, determined mainly by deep inelastic scattering data from HERA, into the region of the W and Z mass scales. While possible deviations from the measured cross section values are of interest, it is also remarkable, however, to note the overall agreement between theory and experiment. This is evidence that universality of the PDFs and perturbative QCD at high orders continue to work up to the kinematic range probed in W and Z production at the LHC.
The combination and theory comparisons are also performed with the total integrated cross sections, listed in Table XII. The correlation coefficients are given in Table XIII. The pure experimental precision of the total   X. Combined cross sections times leptonic branching ratios for W þ , W À , W AE , and Z= Ã production within the corresponding fiducial regions of the measurements. The uncertainties denote the statistical (sta), the experimental systematic (sys), the luminosity (lum), and the extrapolation (acc) uncertainties. All experimental and theoretical ellipses are defined such that their area corresponds to 68% C.L. This implies that the projections onto the axes correspond to 1.52 times the usual onedimensional uncertainty. Note that this convention differs from the one chosen in [10,12,67], in which the ellipses are narrower to reflect the one-dimensional uncertainties. cross sections is as high as that of the fiducial cross sections. However, the additional extrapolation uncertainty, described in Sec. III, amounts to about 2%, which is larger than the experimental systematic error. The total cross section measurements are thus less able to discriminate details of the PDFs, as may be deduced from comparing Fig. 16 with Fig. 15.
Compared to the first total W, Z cross section measurements by ATLAS [20], the statistical uncertainty is improved by a factor of 10, to 0.2% (0.6%) for W (Z), the systematic uncertainty by a factor of about 5, and the luminosity uncertainty by a factor of 4, to 3.4%.

E. Ratios of cross sections 1. Electron-muon universality
Ratios of electron and muon cross sections can be evaluated in the common kinematic fiducial region. Since the production of the W and Z bosons is independent of the flavor of the decay lepton, the corresponding cross section ratios represent new measurements of the ratios of the e and branching fractions, i.e.   This confirms e À universality in Z decays as well, but the result is much less accurate than the world average  16 (color online). Measured and predicted total cross sections times leptonic branching ratios: W þ vs W À (left panel) and ( W þ þ W À ) vs Z= Ã (right panel). The ellipses illustrate the 68% C.L. coverage for total uncertainties (full) and excluding the luminosity uncertainty (open black). The uncertainties of the theoretical predictions correspond to the PDF uncertainties only.  FIG. 17 (color online). The correlated measurement of the electron-to-muon cross section ratios in the W and the Z channels. The vertical (horizontal) band represents the uncertainty of the corresponding Z (W) branching fractions based on the current world average data. The ellipse illustrates the 68% C.L. for the correlated measurement of R W and R Z , while the error bars correspond to the one-dimensional uncertainties of either R W or R Z , respectively.  value of 0:9991 AE 0:0024 [65]. If one uses this world average as a constraint on the analysis presented here, the correlated systematic uncertainty on R W is reduced, and an improved value R W ¼ 0:999 AE 0:020 is obtained. The correlation of R W and R Z and the comparison with the world average values is illustrated in Fig. 17.

Combined cross section ratios
Ratios of the W AE and Z cross sections are calculated accounting for the correlations between uncertainties. The results obtained in the fiducial region are given in Table XIV.   18 (color online). Measured and predicted fiducial cross section ratios, ð W þ þ W À Þ= Z= Ã (top panel) and W þ = W À (bottom panel). The experimental uncertainty (inner band) includes the experimental systematic errors. The total uncertainty (outer band) includes the statistical uncertainty and the small contribution from the acceptance correction. The uncertainties of the ABKM, JR, and MSTW predictions are given by the PDF uncertainties considered to correspond to 68% C.L., and their correlations are derived from the eigenvector sets. The results for HERAPDF comprise all three sources of uncertainty of that set.  19 (color online). Measured and predicted fiducial cross section ratios, W þ = Z= Ã (left panel) and W À = Z= Ã (right panel). The experimental uncertainty (inner band) includes the experimental systematic errors. The total uncertainty (outer band) includes the statistical uncertainty and the small contribution from the acceptance correction. The uncertainties of the ABKM, JR, and MSTW predictions are given by the PDF uncertainties considered to correspond to 68% C.L., and their correlations are derived from the eigenvector sets. The results for HERAPDF comprise all three sources of uncertainty of that set. TABLE XVI. Differential cross section for the W þ ! e þ process, extrapolated to the common fiducial region. The statistical (sta), uncorrelated systematic ( unc ), correlated systematic ( cor ), and total ( tot ) uncertainties are given in percent of the cross section values. The overall 3.4% luminosity uncertainty is not included.   The precision of these measurements is very high, with a total uncertainty of 0.9% for the W þ =W À ratio and of 1.3% for the W AE =Z ratio.
Ratios for the total cross sections are given in Table XV. The uncertainties of the total cross section ratios are enlarged significantly by the additional acceptance contribution. Compared to the fiducial cross section ratios, the uncertainties are almost doubled, with a value of 1.8% for the W þ =W À ratio and of 1.6% for the W AE =Z ratio.
The cross section ratios, determined in the fiducial regions of the W and Z measurements, are compared in Figs. 18 and 19 with the theoretical predictions, accounting for the correlations inherent in the PDF determinations.
The mean boson rapidity for the data presented here is about zero, and thus on average the Bjorken x values of the incoming partons are equal, x 1 ¼ x 2 ' 0:01. In a rough leading order calculation, neglecting the heavy-quark and Cabibbo disfavored parts of the cross sections and the Ã contribution to the Z cross section, and also assuming the light sea and antiquark distributions to all be the same, xs, the ðW þ þ W À Þ=Z ratio is found to be proportional to ðu v þ d v þ 2sÞ=½ðv 2 u þ a 2 u Þ Â XXIII. Combined differential cross section d=dy Z for the Z ! ' þ ' À process measured for 66 < m '' < 116 GeV and p T;' > 20 GeV. All uncertainties are quoted in % with respect to the cross section values. sta , unc , cor , and tot represent statistical, uncorrelated systematic, correlated systematic, and total uncertainties. 1 À 30 represent diagonalized correlated systematic uncertainties, which are correlated bin to bin and across the W þ , W À , and Z measurements. The overall 3.4% luminosity uncertainty is not included.
Here xu v ðxd v Þ is the up (down) valence-quark momentum distribution and v u;d and a u;d are the vector and axial-vector weak neutral current couplings of the light quarks. As the numerical values for the Z coupling to the up and down quarks, v 2 u;d þ a 2 u;d , are of similar size, the W AE =Z ratio measures a rather PDF insensitive quantity, provided that the sea is flavor symmetric. Since this symmetry assumption, with a small deviation to account for some light sea quark asymmetry near Bjorken x ' 0:1, is inherent in all major PDF fit determinations, there is indeed not much difference observed between the various W AE =Z ratio predictions; see Fig. 18 (top panel). The agreement with the present measurement therefore supports the assumption of a flavor independent light quark sea at high scales, and Bjorken x near 0.01. The predictions for the charge dependent W þ =W À , W þ =Z, and W À =Z ratios, shown in Figs. 18 (bottom panel) and 19, exhibit more significant deviations as they are more sensitive to up-down quark distribution differences. XXIV. Combined differential cross section d=d ' À for the W À ! ' À " process measured for p T;' > 20 GeV, p T; > 25 GeV, and m T > 40 GeV. All uncertainties are quoted in % with respect to the cross section values. sta , unc , cor , and tot represent statistical, uncorrelated systematic, correlated systematic, and total uncertainties. 1 À 30 represent diagonalized correlated systematic uncertainties, which are correlated bin to bin and across the W þ , W À , and Z measurements. The overall 3.4% luminosity uncertainty is not included.

VII. SUMMARY
New measurements are presented of the inclusive cross sections of Drell-Yan W AE and Z= Ã production in the electron and muon decay channels. They are based on the full data sample collected by the ATLAS experiment at the LHC in 2010 at a center-of-mass energy of 7 TeV. With an integrated luminosity of about 35 pb À1 , a total of about 270 000 W boson decays into an electron or muon and the associated neutrino and a total of about 24 000 Z= Ã decays into electron or muon pairs have been observed.
The cross sections are measured in a well-defined kinematic range within the detector acceptance, defined by charged lepton pseudorapidity and charged lepton and neutrino transverse momentum cuts. Integrated cross sections are determined in these fiducial regions and are also extrapolated to the full kinematic range to obtain the total integrated W and Z= Ã cross sections. XXV. Combined differential cross section d=d ' þ for the W þ ! ' þ process measured for p T;' > 20 GeV, p T; > 25 GeV, and m T > 40 GeV. All uncertainties are quoted in % with respect to the cross section values. sta , unc , cor , and tot represent statistical, uncorrelated systematic, correlated systematic, and total uncertainties. 1 À 30 represent diagonalized correlated systematic uncertainties, which are correlated bin to bin and across the W þ , W À , and Z measurements. The overall 3.4% luminosity uncertainty is not included. min À max 0.00-0.21 0. 21-0.42 0.42-0.63 0.63-0.84  The W AE cross sections are measured differentially as a function of the lepton pseudorapidity, extending to j ' j 2:5. The Z= Ã cross section is measured as a function of the boson rapidity jy Z j up to a value of 2.4. An extension to jy Z j 3:6 is obtained through the electron channel measurements, which include the forward detector region and j e j as large as 4.9.
The electron and muon measurements are found to be consistent in the three channels, W þ , W À , and Z= Ã . The data sets are therefore combined using a method which accounts for the different systematic error correlations.
This combination provides the most accurate integrated inclusive W and Z= Ã cross sections so far obtained by the ATLAS Collaboration and the first measurements of rapidity dependent cross sections. An update is also presented of the W charge asymmetry as a function of j ' j.
The precision of the integrated W and Z= Ã cross sections in the fiducial region is $1:2% with an additional uncertainty of 3.4% resulting from the luminosity error. The uncertainties on the total integrated cross sections are about twice as large because of the extrapolation uncertainties in the determination of the acceptance correction. The differential cross sections are determined in the fiducial region with a typical precision of 2%, apart from the most forward part of y Z .
The results are compared with QCD predictions calculated to NNLO in the fiducial regions of the measurements which allows for maximum sensitivity to details of the parton distributions used in these calculations.
The broad agreement of the theory predictions at the few percent level with the data supports the validity of the QCD evolution equations, as the results rely on lower scale parton distribution functions evolved to the W and Z kinematic regions, at the average value of Bjorken x of about 0.01.
Interesting differences between sets of parton distributions are observed, both in the integrated and the differential fiducial cross sections. The results presented in this paper therefore provide a further basis for sensitive tests of perturbative QCD and determinations of the partonic content of the proton.

ACKNOWLEDGMENTS
We thank CERN for the very successful operation of the LHC, as well as the support staff from our institutions without whom ATLAS could not be operated efficiently.