Tevatron Run II combination of the effective leptonic electroweak mixing angle

Drell-Yan lepton pairs produced in the process p ¯ p → l þ l − þ X through an intermediate γ (cid:2) =Z boson have an asymmetry in their angular distribution related to the spontaneous symmetry breaking of the electroweak force and the associated mixing of its neutral gauge bosons. The CDF and D0 experiments have measured the effective-leptonic electroweak mixing parameter sin 2 θ lepteff using electron and muon pairs selected from the full Tevatron proton-antiproton data sets collected in 2001-2011, corresponding to 9 – 10 fb − 1 of integrated luminosity. The combination of these measurements yields the most precise result from hadron colliders, sin 2 θ lepteff ¼ 0 . 23148 (cid:3) 0 . 00033 . This result is consistent with, and approaches in precision, the best measurements from electron-positron colliders. The standard model inference of the on-shell electroweak mixing parameter sin 2 θ W , or equivalently the W -boson mass M W , using the ZFITTER software package yields sin 2 θ W ¼ 0 . 22324 (cid:3) 0 . 00033 or equivalently, M W ¼ 80 . 367 (cid:3) 0 . 017 GeV =c 2 .

Drell-Yan lepton pairs produced in the process pp → l þ l − þ X through an intermediate γ Ã =Z boson have an asymmetry in their angular distribution related to the spontaneous symmetry breaking of the electroweak force and the associated mixing of its neutral gauge bosons. The CDF and D0 experiments have measured the effective-leptonic electroweak mixing parameter sin 2 θ lept eff using electron and muon pairs selected from the full Tevatron proton-antiproton data sets collected in 2001-2011, corresponding to 9-10 fb −1 of integrated luminosity. The combination of these measurements yields the most precise result from hadron colliders, sin 2 θ lept eff ¼ 0.23148 AE 0.00033. This result is consistent with, and approaches in precision, the best measurements from electron-positron colliders. The standard model inference of the on-shell electroweak mixing parameter sin 2 θ W , or equivalently the W-boson mass M W , using the ZFITTER software package yields sin 2

I. INTRODUCTION
At the Fermilab Tevatron proton-antiproton (pp) collider, Drell-Yan [1] lepton (l) pairs are produced in the process pp → l þ l − þ X through an intermediate γ Ã =Z boson, where X represents inclusively any other collision products. The forward-backward asymmetry in the polarangle distribution of the l − in the Collins-Soper (CS) frame [2] as a function of the l þ l − -pair invariant mass is directly sensitive to the effective-leptonic electroweak mixing parameter sin 2 θ lept eff . The effective-leptonic parameter is measured using electron and muon pairs (l ¼ e and μ). The electroweak-mixing parameter sin 2 θ W [3] is obtained indirectly in the context of standard model (SM) calculations with the following input parameters: the fine structure constant, the Fermi constant, the strong interaction coupling constant, and the masses of the top quark, Z boson, and Higgs boson. In this SM context, sin 2 θ W and the W-boson mass are related, and a comparison of the Wboson mass inferred from sin 2 θ W to the directly measured mass tests the consistency of the SM. Such tests require precision measurements of sin 2 θ lept eff , and results from hadron colliders such as the Tevatron are complementary to those from electron-positron colliders.
The Drell-Yan process and the production of quark pairs in high-energy e þ e − collisions are analogous processes: qq → l þ l − and e þ e − → qq. processes involving leptons has been investigated at the LEP-1 and SLC [4,5] e þ e − colliders operating on or in the vicinity of the Z-boson pole mass, and at the Tevatron [6][7][8][9][10][11][12][13] and LHC [14][15][16] hadron colliders. Investigations at hadron colliders use Drell-Yan pairs whose range of invariant masses about the Z-boson resonant peak is broad relative to the e þ e − collider investigations. The mixing parameter has been accurately measured at the LEP-1 and SLC colliders, where processes with leptons in the final state are also used. The combined average of six measurements from these lepton colliders [4] yields a value of 0.23149 AE 0.00016 [5]. However, a 3.2 standard-deviation difference exists between the two most precise individual measurements. The combined measurement of the b-quark forward-backward asymmetry ðA 0;b FB Þ with the LEP-1 detectors yields sin 2 θ lept eff ¼ 0.23221 AE 0.00029, while the SLD left-right polarization asymmetry of Z-boson production ðA l Þ yields sin 2 θ lept eff ¼ 0.23098 AE 0.00026. This provides a strong motivation for an accurate determination of sin 2 θ lept eff by the Tevatron experiments.

A. Electroweak couplings
The production of Drell-Yan lepton pairs at the Born level proceeds through two parton-level processes, where the q andq are a quark and antiquark, respectively, that originate from the colliding hadrons. The virtual photon couples the vector currents of the incoming and outgoing fermions (f), and the spacetime structure of the photon-fermion interaction vertex may be represented as hfjQ f γ μ jfi, where Q f , the strength of the coupling, is the fermion charge (in units of e), and jfi is the spinor for fermion f. The interaction vertex of a fermion with a Z boson contains both vector (V) and axial-vector (A) current components, and its structure is hfjg f V γ μ þ g f A γ μ γ 5 jfi. The Born-level coupling strengths are where T f 3 is the third component of the fermion weak isospin, which is T f 3 ¼ 1 2 ð− 1 2 Þ for positively (negatively) charged fermions. Radiative corrections alter the Born-level couplings into effective couplings. At the Born level in the SM, and in all orders of the on-shell renormalization scheme [17], the sin 2 θ W parameter is related to the Wboson mass M W and the Z-boson mass M Z by Since the Z-boson mass is accurately known (to AE0.0021 GeV=c 2 [4,5]), the inference of the on-shell sin 2 θ W is equivalent to an indirect W-boson mass measurement. The angular distributions of l þ l − pairs in the final state of the Drell-Yan process and of l þ l − or qq pairs in the final state of e þ e − collisions are sensitive to the effective sin 2 θ W parameter at the lepton vertex, sin 2 θ lept eff .

B. The forward-backward asymmetry
The rapidity, transverse momentum, and mass of a particle or a system of particles are represented by y, p T , and M, respectively. The energy and momentum of particles are represented as E and ⃗p, respectively. In the laboratory frame, the pp collision axis is the z lab axis, with the positive direction defined to be along the direction of the proton. The transverse component of any vector, such as the momentum vector, is defined relative to that axis. The rapidity is y ¼ 1 2 ln½ðE þ p z Þ=ðE − p z Þ, where p z is the component of the momentum vector along the z lab axis.
The angular kinematic properties of leptons from the Drell-Yan process are defined in the rest frame of the exchanged γ Ã =Z boson. The l − direction is chosen to define the polar and azimuthal angles of the lepton pair, which are denoted as ϑ and φ, respectively. The ideal positive z axis coincides with the direction of the incoming quark so that the definition of ϑ parallels the definition used in e þ e − collisions at LEP [4,5]. This frame is approximated by the CS rest frame [2] for pp collisions, depicted in Fig. 1.
The CS frame angle ϑ is reconstructed using the following laboratory-frame quantities: the lepton energies, the lepton momenta along the beam line, the dilepton invariant mass, M, and the dilepton transverse momentum, p T . The polar angle of the negatively charged lepton is calculated from where l AE ¼ ðE AE p z Þ and the þð−Þ superscript specify that l AE is for the positively (negatively) charged lepton. Similarly, the CS expression for φ in terms of laboratory-frame quantities is given by where ⃗ Δ is the difference between the l − and l þ laboratoryframe momentum vectors;R T is the unit vector along ⃗p p × ⃗p, with ⃗p p being the proton momentum vector and ⃗p the lepton-pair momentum vector; andp T is the unit vector along the transverse component of the lepton-pair momentum vector. At p T ¼ 0, the angular distribution is azimuthally symmetric. The right-hand sides of the definitions of cos ϑ and tan φ are invariant under Lorentz boosts along the z direction in the laboratory frame.
The angular distribution of Drell-Yan lepton pairs is defined as the ratio of the production cross section to the angle-integrated production cross section. Its general structure consists of terms derived from nine helicity cross sections that describe the polarization state of the boson [18,19], The coefficients A 0−7 are functions of kinematic variables of the boson and vanish when the lepton-pair transverse momentum approaches zero, except for A 4 , which contributes to the tree-level amplitude and generates the forward-backward asymmetry in cos ϑ. Thus, at zero transverse momentum, the angular distribution reduces to the tree-level form 1 þ cos 2 ϑ þ A 4 cos ϑ. In the CS frame, the A 0 , A 2 , and A 4 coefficients are large relative to the other coefficients.
The A 4 cos ϑ term violates parity conservation, and is due to the interference of the amplitudes of the vector and axial-vector currents. Its presence induces an asymmetry in the φ-integrated cos ϑ dependence of the cross section. Two sources contribute: the interference between the Z-boson vector and axial-vector amplitudes, and the interference between the photon vector and Z-boson axial-vector amplitudes. The asymmetry component from the γ Ã -Z interference cross section depends on axial-vector couplings g f A to fermions f that are independent of sin 2 θ W . The asymmetry component from Z-boson selfinterference depends on a product of g l V and g q V from the lepton and quark vertices, and thus is related to sin 2 θ W . At the Born level, this product is where l and q denote the lepton and quark, respectively. For the Drell-Yan process, the relevant quarks are predominantly the light quarks u, d, and s. The coupling factor has an enhanced sensitivity to sin 2 θ W at the lepton-Z vertex: for a sin 2 θ W value of 0.223, a 1% variation in sin 2 θ W changes the lepton factor ð1 − 4jQ l j sin 2 θ W Þ by about 8%, and it changes the quark factor ð1 − 4jQ q j sin 2 θ W Þ by about 1.5% (0.4%) for the u (d or s) quark. Electroweak radiative corrections do not significantly alter this Born-level interpretation. Loop and vertex electroweak radiative corrections induce multiplicative form-factor corrections [20][21][22] to the T f 3 and sin 2 θ W terms that change their values by a few percent [7].
The forward-backward asymmetry of the polar-angle distribution is defined as where σ f is the Drell-Yan cross section for the forward (f) orientation of lepton pairs, cos ϑ ≥ 0, and σ b is for the backward (b) orientation of lepton pairs, cos ϑ < 0. Figure 2 shows the typical dependence of the asymmetry The label u þ d denotes the overall asymmetry, and the labels u and d denote the contribution to the overall asymmetry from quarks with charge 2=3 and −1=3, respectively. The asymmetry identified by the u or d label is defined as ðσ q f − σ q b Þ=σ, where q ¼ u or d, σ fðbÞ is the forward (backward) cross section, and σ is the corresponding cross section from quarks of all charges. Thus, the overall asymmetry is the sum of the asymmetries identified by the u and d labels.
as a function of the lepton-pair invariant mass from a Drell-Yan quantum chromodynamics calculation. The offset of A fb from zero at M ¼ M Z is related to sin 2 θ W . At leptonpair masses much smaller or larger than the Z-pole mass, the asymmetry is dominated by the component from γ Ã -Z interference, whose cross section contains the factor ðM 2 − M 2 Z Þ=M 2 from the photon and Z-boson propagators, the factor for the flux of quarks from the parton distribution functions (PDF) of the proton, and the factor for the coupling term 4=9 (1=9) at the quark-photon vertex for the charge 2=3 (−1=3) quark. Consequently, the asymmetry distribution is sensitive to both sin 2 θ W and the PDF of the proton.
C. Extraction of sin 2 θ lept eff The sin 2 θ lept eff parameter is derived from comparisons of the measurement of A fb ðMÞ in bins of mass covering the measurement range, and predictions of the measurement (templates) for various input values of the effective-leptonic parameter. The value that minimizes the χ 2 between the measurement and templates is the extracted value of sin 2 θ lept eff . Detector effects that bias the asymmetry measurement are removed from the measurement by CDF, and incorporated into the templates by D0. In both the CDF and D0 analyses, the forward-backward asymmetries of electron and muon pairs are separately measured.
The CDF and D0 measurements are based on multiple analyses with improvements over time. Different PDFs have been utilized by CDF and D0, as well as slightly different electroweak correction techniques based on the SM in the extraction of the sin 2 θ lept eff parameter. For the combination of electron-and muon-channel results within an experiment, and for the Tevatron combination, a framework with two common components, denoted as the "common framework," is adopted to obtain consistent results for the effective-leptonic parameter. The common components are electroweak-radiative corrections calculated using the SM software package ZFITTER 6.43 [20][21][22], which is used with LEP-1 and SLD measurement inputs for precision tests of the SM at the Z pole [4,5], and NNPDF 3.0 [23][24][25][26][27][28][29] PDFs, which include new HERA and LHC data, and replace the Tevatron W-asymmetry measurements with lepton-asymmetry measurements from the LHC. A summary of the CDF and D0 measurements is presented in Sec. II, along with the strategies used to achieve results consistent with the common-framework agreement. The combination of these CDF and D0 results is presented in Secs. III, and IV provides the summary.

II. INPUT MEASUREMENTS
The Fermilab Tevatron collider provides pp collisions at a center-of-momentum energy of 1.96 TeV. The CDF [30] and D0 [31] detectors are general-purpose detectors that surround the Tevatron collision regions. For both detectors, precision charged-particle tracking detectors (trackers) immersed in solenoidal magnetic fields are located around the beam line. Beyond the trackers are projective-tower electromagnetic and hadronic calorimeters with transverse and longitudinal segmentation, which are used for the identification of electrons, photons, and clusters of outgoing particles (jets), and for measurements of their kinematic properties. The outer regions of each detector consist of hadron absorbers and trackers for the detection and identification of muons.
For particle trajectories, the polar and azimuthal angles of the coordinate systems are denoted by θ lab and ϕ lab respectively. The pseudorapidity of a particle is η ¼ − ln tanðθ lab =2Þ. Detector coordinates are specified as ðη det ; ϕ det Þ, where η det is the pseudorapidity relative to the detector center ðz lab ¼ 0Þ. For particle energies measured in the calorimeters, the transverse energy E T is defined as Both the CDF and D0 measurements utilize electron and muon pairs reconstructed in high-p T electron and muon data samples, respectively. During data taking, events with high-p T electrons and muons are selected online by triggers. Offline selection criteria are applied to these samples to improve the quality of the sample used for the asymmetry measurements.

A. CDF measurement
The CDF measurements for the electron [9] and muon [8] channels use the full Tevatron Run II data set, corresponding to an integrated luminosity of 9 fb −1 of pp collisions. Similar methods are used for both analyses. The A fb measurements are corrected for detector effects. The effects of detector resolution and quantum electrodynamics (QED) final-state radiation (FSR) are removed from the measurements using the CDF simulation of Drell-Yan events. The templates are strictly quantum chromodynamics (QCD) calculations of A fb .
Approximately 485 000 electron pairs and 277 000 muon pairs are utilized in the measurements. Details of the electron-and muon-selection criteria are presented in Refs. [9,8], respectively. Kinematic, fiducial, and leptonidentification criteria are applied. As Drell-Yan leptons are usually produced in isolation from the other activity in the event, isolation criteria that limit event activity around the leptons are also applied. In the electron channel, one electron must be detected in the central-calorimeter region 0.05 < jη det j < 1.05, and its partner electron can be either in the central-calorimeter region or the plug-calorimeter region 1.2 < jη det j < 2.8. For central-central electron pairs, one electron is required to have E T > 25 GeV and the other to have E T > 15 GeV. For central-plug electron pairs, both electrons are required to have E T > 20 GeV. In the muon channel, the selection requires both muons to be predominantly within the central muon-detector region jη det j < 1, and have p T > 20 GeV=c. For the asymmetry measurements, electron and muon pairs are selected to have invariant masses in the range 50-350 GeV=c 2 and 50-1000 GeV=c 2 , respectively. The upper limit for muon pairs is larger to accommodate the significant resolution smearing of the muon momentum at high masses.
The A fb measurement uses a data-driven event-weighting method [32], which is equivalent to performing individual measurements of the asymmetry in j cos ϑj bins and then combining them. The standard expression for the asymmetry in a bin is where N fðbÞ and ðϵAÞ fðbÞ are the signal event counts, and the efficiency (ϵ) and acceptance (A) product, respectively, of forward (backward) lepton pairs. Only leptons reconstructed within the central region, where tracking performance is optimal with small charge misidentification probability, are used to determine the forward (backward) orientation of a pair. For these leptons, the track-finding efficiency is 99% [33]. Within a localized j cos ϑj bin, the forward and backward dependence of the acceptance and efficiency of its lepton pairs cancels out to first order so that The similarity of ðϵAÞ f and ðϵAÞ b for a value of j cos ϑj is related to the proximity of lepton-pair trajectories within the detector to those of the ones with the charges exchanged. With the exchange, the magnitude of cos ϑ is unchanged but the sign is reversed, i.e., forwards and backwards are interchanged. The exchange transforms a lepton with charge Q into one with charge −Q but leaves its laboratory momentum ⃗p unchanged. For a typical lepton with p T ∼ 40 GeV=c, the radius of curvature in the magnetic-field-bend plane is 100 m. For reference, the radial extent of the precision trackers in this plane is 1.3 m. Thus, the trajectories of a lepton and its charge-exchanged configuration through the detector are localized, so to first order, their probabilities of being reconstructed and selected are the same. The expected angular dependencies of the numeratorevent difference and denominator-event sum of Eq. (9) are derived using Eq. (5). The numerator difference is proportional to 2A 4 j cos ϑj, and the denominator sum to 2ð1 þ cos 2 ϑ þ Á Á ÁÞ, where 1 þ cos 2 ϑ þ Á Á Á denotes the cos ϑ symmetric terms from Eq. (5). Together they yield Each bin is an independent measurement of A 4 , with an uncertainty of σ=ξ, where σ is the statistical uncertainty of A fb . When the measurements are combined, each bin has a statistical weight of σ −2 and a sampling weight of ξ 2 . The binned measurements are reformulated into an unbinned-asymmetry expression of the form shown in the right-hand side of Eq. (9) using event weights. Weights for individual events in the numerator and denominator remove the angular dependencies of the event difference and sum, respectively, and provide the sampling weight for the combination of events across j cos ϑj.
Because the event-weighting method needs events for corrections, kinematic regions with few or no events are eliminated from the acceptance region of the measurement and template calculations. Consequently, the kinematicacceptance region of the electron-pair is restricted to jyj < 1.7, and that of the muon-pair to jyj < 1. Small secondary effects not corrected by the event-weighting method are removed with the simulation.
The simulation of Drell-Yan events, described below, uses PYTHIA 6.2 [34] with CTEQ5L [35] PDFs to generate events and includes QED FSR for the decay leptons. To account for QED FSR from promptly decaying hadrons and their decay products, the event generation then uses PHOTOS 2.0 [36][37][38]. This is followed by the detector simulation based on GEANT-3 and GFLASH [39]. Some of the kinematic distributions of the generated γ Ã =Z boson are adjusted with event weights for better agreement between the simulation and the data. For the electron channel, simulation adjustments cover all aspects of boson production because of the extended boson-rapidity coverage of the measurement. The generated boson kinematics are adjusted using the data and the RESBOS [40][41][42][43] calculation with CTEQ6.6 [44] PDFs. The generator-level p T distribution of the boson is adjusted so that the reconstructed p T distribution of simulated events matches the distribution of the data in two rapidity bins, 0 < jyj < 0.8 and jyj ≥ 0.8. The generator-level bosonmass distribution is adjusted with a mass-dependent factor, which is the ratio of the RESBOS boson-mass distribution calculated using CTEQ6.6 PDFs relative to the PYTHIA 6.4 [45] distribution calculated using CTEQ5L PDFs. For the muon channel, the generator-level p T distribution of the boson is adjusted so that the reconstructed p T distribution of simulated events matches the data.
The energy scales of both the data and simulation are calibrated to a common standard following Ref. [46]. The standard consists of idealized calibration distributions derived from the same generated-event samples of the simulation using a detector with perfect calibrations but with the observed resolutions. The energy resolution in the simulation is calibrated to that of the data. In addition to the generator level tuning, other distributions such as the timedependent Tevatron beam-luminosity profile and detector responses near boundaries are tuned. Response adjustments are consistently applied to parity-symmetric variables such as jη det j or j cos ϑj. They are needed for an unbiased detector-resolution unfolding of the asymmetry distribution in mass and cos ϑ for the data. Unfolding matrices are used, and the unfolding statistically removes the effects of resolution smearing and QED FSR. The simulation is also used to derive the error matrix for the A fb measurement.
The backgrounds are from the production of QCD dijets, W þ jets, γ Ã =Z → ττ, diboson (WW, WZ, and ZZ), and tt events. QCD dijet backgrounds are estimated using the data. Other backgrounds listed above are estimated with PYTHIA 6.2 [34]. For the electron channel, the overall background level amounts to 1.1% over the mass range of the asymmetry measurement. For the muon channel, the overall background level amounts to 0.5% over the mass range of the asymmetry measurement. All backgrounds are subtracted from the data.
The CDF templates for A fb which are compliant with the specifications of the common framework are denoted as "common-framework compliant templates". They are calculated using the POWHEG-BOX next-to-leading-order (NLO) implementation [47] of the Drell-Yan process [48] followed by PYTHIA 6.41 [45] parton-showering. The combined implementation has next-to-leading-log resummation accuracy. The NNPDF 3.0 next-to-next-to-leading order (NNLO) PDFs are used for the parton fluxes.
The complex-valued ZFITTER form factors are incorporated into the POWHEG-BOX amplitudes as specified in the Appendix. The QED photon propagator correction from fermion loops is also included. The implementation of these form factors provides an enhanced Born approximation (EBA) to the electroweak couplings. For consistency with the ZFITTER calculations, the NNPDFs selected are derived assuming a value of the strong-interaction coupling of 0.118 at the Z-boson mass. With ZFITTER corrections, the electroweak mixing parameter for the templates is the static on-shell sin 2 θ W . The asymmetry is directly sensitive to the effective mixing terms, which are provided by ZFITTER as κ f sin 2 θ W , where κ f denotes a fermion-flavor (f) dependent form factor. Unlike the directly observable effective mixing terms, sin 2 θ W and κ f are inferred in the context of the SM and their inputs are specified in the Appendix. The effective mixing terms are functions of sin 2 θ W and a mass scale. For comparisons with other measurements, the value of the sin 2 θ lept eff parameter is defined at the Z pole and its value is Re½κ e ðsin 2 θ W ; M 2 Z Þ sin 2 θ W . The NNPDF 3.0 parton distributions consist of an ensemble of 100 equally probable PDFs for which the value of a calculation is the average of the calculations over the ensemble, and the rms about the average is the PDF uncertainty. A measurement can be incorporated into the ensemble of a PDF fit without regenerating the ensemble via various Bayesian methods that reweight each ensemble PDF [49]. With the Giele-Keller (GK) method, the ensemble PDFs, numbered 1 to N, are reweighted with the likelihood of the prediction using the ensemble PDF relative to the measurement, where w k is the weight for PDF number k, and χ 2 k is the χ 2 between the measurement and the prediction using that PDF [50,51].
For the extraction of the sin 2 θ lept eff parameter, commonframework compliant templates with varying values of the effective-leptonic parameter are calculated for each of the ensemble PDFs of NNPDF 3.0, and the GK-weighting method is used to evaluate the ensemble average and rms of the effective-leptonic parameter derived from the measurement and templates. Consequently, additional PDF constraints from the asymmetry measurement are incorporated into the ensemble of PDFs.
The electron-channel measurement of the asymmetry, along with common-framework compliant templates and the error matrix of the measurement, are used to extract the sin 2 θ lept eff parameter. The result for the electron channel is The systematic uncertainty consists of contributions from the energy scale and resolution, the backgrounds, and the QCD scale. The muon-channel measurement of the asymmetry and the error matrix of the measurement used for measurementto-template χ 2 comparisons are presented in Ref. [8]. The published result for the muon channel is The systematic uncertainty consists of contributions from the energy scale and resolution, the backgrounds, and the higher-order terms of QCD. The A fb templates used for the derivation of this result are from previous iterations of the analysis and not compliant with the specifications of the common framework. They are calculated with a modified version of RESBOS using CTEQ6.6 PDFs and the ZFITTER form factors. Systematic uncertainties are estimated using the POWHEG-BOX NLO parton generator with CT10 NLO PDFs [52], and followed by PYTHIA 6.41 parton showering. The PDF uncertainty is derived from the CT10 uncertainty PDFs at 68% C.L. The result shown in Eq. (12) is the standalone result of the muon channel, but the methods used do not facilitate a straightforward combination with the electron-channel analysis. For the combination of the electron-and muon-channel results, the measured asymmetries are directly utilized. Both the electron-and muon-channel measurements are compared against common-framework compliant templates for the extraction of the sin 2 θ lept eff parameter. For comparison purposes, the extracted value of the effective-leptonic parameter from the muon-asymmetry measurement in this framework is 0.23141 AE 0.00086, where the uncertainty is statistical only [9].
The CDF result combining electron and muon channels presented in Ref. [ The systematic uncertainty, consisting of contributions from the energy scale and resolution, the backgrounds, and the QCD scale, are summarized in Table I. As the A fb templates for the electron-and muon-channel asymmetries are both calculated with the same common-framework infrastructure, the electron-and muon-channel comparison χ 2 's between the data and templates are combined into a joint χ 2 for the determination of the best-fit sin 2 θ lept eff . The joint χ 2 takes into account correlations between the electron-and muon-channel asymmetries for each of the ensemble PDFs of NNPDF 3.0. The GK-weighting method using the joint χ 2 incorporates PDF information from both the electron-and muon-channel asymmetry measurements into the PDF ensemble, and thus reduces the PDF uncertainty relative to the default (equal-weight) ensemble.

B. D0 measurement
The D0 measurement of sin 2 θ lept eff is performed in the electron [12] and muon [13] channels. The electron-and muon-channel results use 9.7 fb −1 and 8.6 fb −1 of recorded luminosity, respectively.
The asymmetry A fb is measured in the electron channel using events with at least two electromagnetic (EM) clusters reconstructed in the calorimeter. They are required to be in the central calorimeter (CC) or end calorimeter (EC) with transverse momentum p T > 25 GeV=c. Clusters in the CC must be matched to reconstructed tracks. For events with both clusters in the EC, only one cluster must be trackmatched. Compared to previous D0 results based on 1.1 fb −1 and 5.0 fb −1 of luminosity [10,11], the acceptance is extended from jη det j < 1.0 to jη det j < 1.1 for CC and from 1.5 < jη det j < 2.5 to 1.5 < jη det j < 3.2 for EC, and previously rejected electrons reconstructed near azimuthal CC module boundaries are included. By extending the η det and module boundary acceptance, a 70% increase is achieved in the number of sample events above what would be expected from the increase in luminosity to the full data set, for a total of approximately 560 000 electron pairs in the final sample. Events are categorized as CC-CC, CC-EC, or EC-EC based on the η det regions of the two electron candidates.
Muon-channel events are required to have at least two muon candidates reconstructed in the tracking and muon systems, with transverse momenta p T > 15 GeV=c. Both muon candidates are required to have jη det j < 1.8 with at least one muon within jη det j < 1.6, and they must have tracks matched in the tracking and muon systems. Tracks are required to have opposite curvature. Events with muons nearly back-to-back are removed to reduce cosmic-ray background. The large kinematic acceptance yields a final sample consisting of approximately 481 000 muon pairs. Simulated Drell-Yan events are generated using leading-order PYTHIA 6.23 [34] with the NNPDF 2.3 [24] PDFs for the electron channel and NNPDF 3.0 PDFs for the muon channel, followed by a GEANT-3-based simulation [53] of the D0 detector. The inner-tracker solenoid and muon system toroid polarities are reversed every two weeks on average at D0, enabling cross-checks and cancellations of charge-dependent asymmetries. For example, the muon Drell-Yan samples are generated with different polarities of the solenoid and toroid magnetic fields in the GEANT-3 simulation, and used to model the data corresponding to each combination separately. Data and simulation samples corresponding to different solenoid and toroid polarities are weighted to correspond to equal luminosity exposures for each solenoid-toroid polarity combination. This weighted combination provides cancellation of asymmetries due to variations in detector response. The same reconstruction algorithm is used for the data and simulation.
Relative to previously reported results, new methods of electron energy and muon momentum calibration are developed and applied to both data and simulation. In addition to scale factors, offset parameters are applied to the electron energy as functions of η det and instantaneous luminosity. For muon momentum, a scale factor is introduced that is dependent on charge, η det , and solenoid polarity. With these calibration methods, the systematic uncertainties due to energy and momentum modeling are reduced to negligible levels.
The backgrounds are from the production of multijets, W þ jets, γ Ã =Z → ττ, dibosons (WW and WZ), and tt events. The multijet backgrounds are estimated using the data. The W þ jets events are generated using ALPGEN [54] interfaced to PYTHIA 6.23 for showering and hadronization. The other backgrounds are estimated using the PYTHIA 6.23 simulations. At the Z-boson mass peak, the overall background level is 0.35% (0.88%) with respect to the total number of selected dielectron (dimuon) events in the data. The A fb templates are calculated using PYTHIA 6.23 with NNPDF 2.3 (electron channel) and NNPDF 3.0 (muon channel). They are then reweighted to incorporate higherorder QCD effects. The Z-boson distribution as a function of rapidity and transverse momentum ðy; p T Þ is reweighted in both variables to match that from RESBOS [40][41][42][43] with CTEQ6.6 [44] PDFs. The boson-mass distribution is reweighted with a mass-dependent NNLO K-factor [55]. The events are processed by the D0 detector simulation to yield templates that include detector resolution effects.
The A fb distributions in data are obtained as a function of the dilepton invariant mass. For the electron channel, this is done separately for CC-CC, CC-EC, and EC-EC event categories. The weak mixing parameter is extracted from the background-subtracted A fb spectrum in the regions 75 < M ee < 115 GeV=c 2 for CC-CC and CC-EC events, 81 < M ee < 97 GeV=c 2 for EC-EC events, and 74 < M μμ < 110 GeV=c 2 for muon events, by comparing the data to simulated A fb templates corresponding to different input values of sin 2 θ W .
Combining the weak-mixing-parameter results from the three electron event categories gives the electron-channel result sin 2 θ meas eff ¼ 0.23139 AE 0.00043 ðstatÞ AE 0.00008 ðsystÞ AE 0.00017 ðPDFÞ: The sources of systematic uncertainties include energy calibration, energy resolution smearing, backgrounds, charge misidentification, and electron identification [12]. The largest component is from the electron identification (0.00007). The PDF uncertainty is obtained using the equally probable ensemble PDFs of NNPDF 2.3 following the method prescribed by the NNPDF group [24]. The extraction of the weak mixing parameter from the muon channel A fb distribution in data gives The systematic uncertainties include the following sources: momentum calibration, momentum resolution smearing, backgrounds, and muon identification [13]. The PDF uncertainty is obtained using the 100 equally probable ensemble PDFs of NNPDF 3.0 following the method prescribed by the NNPDF group [23].
Corrections are applied to the values of sin 2 θ meas eff in Eqs. (14) and (15) to make them compliant with the agreed-upon common framework using the CDF EBA electroweak-radiative correction implementation, and NNPDF 3.0. The A fb templates used by D0 for both the electron-and muon-channel analyses are calculated with PYTHIA, which uses the same fixed value for the effective mixing terms sin 2 θ eff of all fermions. The EBA implementation incorporates ZFITTER weak-interaction corrections and the fermion-loop correction to the photon propagator, both of which are complex valued and mass-scale dependent. The effect of using a fixed and constant value for all of the effective mixing terms is investigated by setting all weak form factors to unity so that the effective mixing terms for the templates become sin 2 θ lept eff , where only the real part of the photon propagator correction, the running α em , is retained. This implementation, denoted as nonEBA, is the analog to the PYTHIA calculation. The difference, provides the correction to the values of sin 2 θ lept eff derived using PYTHIA templates, which is needed to convert them to the values derived using ZFITTER form factors.
The difference is calculated using the combination of the CDF electron-and muon-channel A fb measurements. Both nonEBA and EBA template calculations use NNPDF 3.0 NNLO with α s ðM Z Þ ¼ 0.118. Each template contains about 10 9 generated events, and the uncertainty in A fb for the mass bin containing the Z-boson mass is about 5 × 10 −5 . Differences are calculated for 23 ensemble PDFs whose best-fit mixing parameter value from EBA templates is near the average value derived over all ensemble PDFs. The GK-weighted average and rms of Δsin 2 θ lept eff ðZFITTERÞ over the 23 PDFs are 0.00022 and 0.00002, respectively. To accommodate the statistical uncertainty of the effective-leptonic parameter, an additional uncertainty of 0.00003 is assigned, resulting in a total (EBA-nonEBA) uncertainty of AE0.00004. This value of þ0.00022 AE 0.00004 is the correction from the PYTHIA framework, which assumes a single value for all the effective mixing parameters, to the ZFITTER-based one.
The D0 electron-channel A fb template calculations use NNPDF 2.3. Therefore, Eq. (14) must be corrected to comply with the common framework choice of NNPDF 3.0. To calculate the correction, the value of sin 2 θ lept eff is extracted using a default NNPDF 3.0 template and multiple NNPDF 2.3 templates. A PYTHIA template of 5 × 10 8 events is generated, using NNPDF 3.0 with fixed sin 2 θ W input, and taken as pseudodata after applying fast-simulation kinematic requirements. In addition, 40 templates of 3 × 10 8 events each are generated, using NNPDF 2.3 with varying sin 2 θ W inputs and applying the same selection criteria. These templates are used to obtain the best χ 2 fit for the sin 2 θ W value. The difference between the input value for NNPDF 3.0 and the value extracted from NNPDF 2.3, denoted by Δ sin 2 θ lept eff ðPDFÞ, is used to provide a correction to the D0 electron-channel value of sin 2 θ lept eff derived using NNPDF 2.3 to that derived using NNPDF 3.0. The value of Δ sin 2 θ lept eff ðPDFÞ based on the D0 analysis is −0.00024 AE 0.00004, where the uncertainty is statistical. A similar calculation in the CDF framework confirms this value.
The uncorrected central value of the D0 electronchannel measurement is 0.23139 [Eq. (14)]. To update this measurement to one based on templates calculated with NNPDF 3.0 PDFs and ZFITTER-based electroweak radiative corrections, the corrections Δ sin 2 θ lept eff ðPDFÞ, and Δsin 2 θ lept eff ðZFITTERÞ are both applied for a net correction of −0.00002 AE 0.00005, where the uncertainty is denoted as "corrections" in Table II which is the D0 electron-channel value used to combine with the D0 muon-channel and CDF results for the final Tevatron combination. The published D0 electron-channel value of sin 2 θ lept eff in Ref. [12] includes no PDF correction, and only a partial electroweak radiative correction that accounts for differences in the effective couplings to leptons, up-type quarks, and down-type quarks, but not for the complex value and mass-scale dependence of the couplings. Therefore, the fully-corrected Eq. (17) is used for the D0 [13] and Tevatron combinations.
The higher-order weak-interaction radiative correction, Δsin 2 θ lept eff ðZFITTERÞ, is applied to the uncorrected muonchannel result, Eq. (15), giving a þ0.00022 shift to the measured central value and an additional AE0.00004 systematic uncertainty. Applying this correction, the D0 muon-channel final corrected result [13] The D0 combination result [13] for sin 2 θ lept eff is obtained using the corrected electron-and muon-channel results as inputs. The central value and systematic uncertainties are combined using the D0 electron-and muon-channel central values with the inverse of the squares of the measurements' statistical uncertainties as weights. The electron and muon systematic uncertainties are treated as uncorrelated, with the exception of the higher-order radiative correction uncertainty which is treated as 100% correlated. However, the total uncertainty is marginally affected by this choice, because both the electron-and muon-channel total measurement uncertainties are dominated by their statistical uncertainties.
The correlation of PDF uncertainties between the electron-and muon-channel acceptances cannot be ignored. Instead of estimating the correlation matrix between the electron and muon channels, we directly estimate the combined PDF uncertainty. We first estimate the PDF uncertainty on the A fb observable averaged over the electron and muon channels. We then scale that uncertainty using the linear relation between A fb and sin 2 θ W estimated in simulated events.
The D0 combination of electron-and muon-channel results [13]  and a summary of the systematic uncertainties of the combination is presented in Table II.

III. CDF AND D0 COMBINATION
The Tevatron combination of the sin 2 θ lept eff parameter uses a single value from each experiment, the combined result of the electron-and muon-channel analyses. The analyses used to derive these values are described in Sec. II A for CDF, and Sec. II B for D0. Inferences of the sin 2 θ W ðM W Þ parameter corresponding to the Tevatron-combination value of sin 2 θ lept eff and the CDF and D0 input values are obtained using SM calculations in the on-shell renormalization scheme.

A. Tevatron combination
The central values of the combined electron-and muonchannel results from each experiment are where the uncertainties are statistical only. The systematic uncertainties are summarized in Tables I and II. D0 avoids a QCD scale uncertainty by incorporating NNLO effects into the A fb templates, while CDF avoids sensitivity to lepton identification and detector asymmetry uncertainties through the use of the event-weighting method described in Sec. II A.
The PDF uncertainties are treated as 100% correlated. All other systematic uncertainties in Tables I and II are treated as uncorrelated for the CDF and D0 combination. There are some correlations from the common PYTHIA-derived backgrounds, but since the overall contribution from the background is small and the detectors are different, the backgrounds are treated as uncorrelated.
The combination of the two inputs with the "best linear unbiased estimate" (BLUE) method [56] The total uncertainty is AE0.00033, with Table III summarizing the input-source uncertainties and the corresponding combination uncertainties. The combination weights for the CDF and D0 inputs are 0.42 and 0.58, respectively, and the χ 2 probability for the compatibility of the two input measurements is 2.6%.
B. Inference of sin 2 θ W The determination of sin 2 θ lept eff is a direct measurement in that the observed asymmetry is directly sensitive to the effective mixing parameters. In order to obtain sin 2 θ W and its uncertainty, the relationship and the ZFITTER SM calculation of the form factor κ e with a set of input parameters are required. The calculation and parameters specified in the Appendix provide the context for the inference of sin 2 θ W . The calculated value of the form factor, Re½κ e , is 1.0371 for the value of the effective-leptonic mixing parameter specified in Eq. (22) [7]. The choice of the SM-input parameter value for the top-quark mass affects the value of the form factor, and thus the inference of sin 2 θ W . The uncertainty of the inferred value of sin 2 θ W due to the uncertainty from the top-quark mass input, 173.2 AE 0.9 GeV=c 2 [57], is AE0.00008 [9]. This uncertainty is denoted as the "form factor" uncertainty in Table IV. For the CDF measurement, the values of sin 2 θ lept eff , sin 2 θ W , and M W based on the combined electron-and muon-channel results are  where the first contribution to each uncertainty is statistical and the second is systematic. The total systematic uncertainty is the sum in quadrature of all systematic uncertainties listed in Tables III and IV. The form-factor uncertainty is only included in the systematic uncertainty of sin 2 θ W and M W .

C. Result comparisons
The measurements of sin 2 θ lept eff are compared with previous results from the Z-boson pole mass region in Fig. 3. The hadron-collider results are based on A fb measurements. The LEP-1 and SLD results are from the individual asymmetry measurements indicated in the figure.
The W-boson mass inference is compared in Fig. 4 with previous direct and indirect measurements. The direct measurements are from the Tevatron and LEP-2 [58]. The indirect measurements from the Tevatron are derived from the CDF and D0 measurements of A fb , and their combination. The indirect measurement of sin 2 θ W from LEP-1 and SLD, 0.22332 AE 0.00039, is from a SM fit to all Z-pole measurements [4,5] described in Appendix F of Ref. [5]. In that fit, the following input parameters to ZFITTER are varied simultaneously within the constraints of the LEP-1 and SLD data: the Higgs-boson mass m H , the Z-boson mass M Z , the QCD coupling at the Z pole α s ðM 2 Z Þ, and the QED correction Δα ð5Þ em ðM 2 Z Þ. The top-quark mass m t is constrained to the value measured directly at the Tevatron, 173.2 AE 0.9 GeV=c 2 [57]. The precision of the Tevatron indirect measurement almost matches that of the direct measurement combination from the Tevatron and LEP-2.

IV. SUMMARY
The angular distribution of Drell-Yan lepton pairs provides information on the electroweak mixing parameter sin 2 θ W . The effective-leptonic mixing parameter sin 2 θ lept eff is derived from measurements of the forward-backward asymmetry A fb ðMÞ in the polar-angle distribution by the CDF and D0 experiments, where M is the lepton-pair effective mass. The measurements are based on the full Tevatron proton-antiproton data sets collected in 2001-2011. The CDF measurement is derived from electron and muon pairs from a pp collision sample corresponding to 9 fb −1 of FIG. 4. Comparison of experimental determinations of the W-boson mass at high-energy colliders. The horizontal bars represent total uncertainties. The Tevatron combination (this paper) based on CDF and D0 results is denoted as "TeV combined: CDF þ D0". The other indirect measurements are from LEP-1 and SLD [4,5], CDF [8,9], and D0 [12,13]. All indirect measurements use the Tevatron top-quark mass measurement specified in the text [57]. The SM context for the Tevatron inferences is specified in the Appendix, and the SM fit of LEP-1 and SLD is described in Appendix F of Ref. [5]. For the Tevatron inferences of the W-boson mass, the SM Higgs-boson mass parameter is fixed, while in the LEP-1 and SLD SM fit, it is a floating parameter. The direct measurements are from the Tevatron and LEP-2 [58], and the shaded vertical band shows its uncertainty.
FIG. 3. Comparison of experimental measurements of sin 2 θ lept eff in the region of the Z-boson pole mass. The horizontal bars represent total uncertainties. The Tevatron combination (this paper) of CDF and D0 results is denoted as "TeV combined: CDF+D0". The other measurements are from LEP-1 combination [4], SLD [4], CMS [15], ATLAS [14], LHCb [16], CDF [8,9], and D0 [12,13]. The LEP-1 and SLD Z pole result is the combination of their six measurements, and the shaded vertical band shows its uncertainty. integrated luminosity, and the D0 measurement is derived from electron and muon pairs from pp samples corresponding to integrated luminosities of 9.7 fb −1 and 8.6 fb −1 respectively.
The Tevatron combination of the CDF and D0 results yields The combined result is consistent with LEP-1 and SLD measurements at the Z-boson pole. Based on the SM calculations specified in the Appendix, the inferences of sin 2 θ W and the W-boson mass are sin 2 θ W ¼ 0.22324 AE 0.00033; and ð34Þ respectively. Within the context of the SM, sin 2 θ W and the W-boson mass are related. Comparisons of the indirect measurements of the W-boson mass with those from direct measurements provide powerful tests of the self-consistency of the SM. The combined result on the effective sin 2 θ W mixing parameter at the lepton vertex sin 2 θ lept eff is the most precise obtained in hadron collisions. It is consistent with, and approaches in precision, the best measurements from electron-positron colliders. The values of sin 2 θ lept eff from hadron and electron-positron colliders are extracted from a complementary set of processes. At hadron colliders, the partonic processes are qq → e þ e − and μ þ μ − , and the forward-backward asymmetry is sensitive to the vertex couplings of the outgoing leptons and the predominantly light quarks from the hadrons. At electron-positron colliders, the processes are either leptonic, i.e. e þ e − → e þ e − , μ þ μ − , and τ þ τ − , or mixed, i.e. e þ e − → qq. While the asymmetry of a mixed process is analogous to that from hadron collisions, events with b quarks in the final state yield the best experimental precision while those with lighter quarks yield significantly less precision. The result of the Tevatron combination supports the central value of sin 2 θ lept eff derived from the LEP-1 and SLD Z-pole measurements, and the combined values from the Tevatron and from LEP-1 and SLD are nearly identical. quark) [57], and m H ¼ 125 GeV=c 2 (Higgs boson). Form factors and the Z-boson total decay-width Γ Z are calculated. The central values of the parameters provide the context of the ZFITTER SM calculations.
The ZFITTER package uses the on-shell renormalization scheme [17], where particle masses are on-shell and sin 2 θ W ¼ 1 − M 2 W =M 2 Z holds to all orders of perturbation theory by definition. If both G F and m H are specified, sin θ W is not independent, and is related to G F and m H by SM constraints from radiative corrections. To vary the value of the sin θ W (M W ) parameter, the value of G F is not constrained. The M W value is varied in the range 80.0-80.5 GeV=c 2 , and for each value, ZFITTER calculates G F and the form factors. Each set of calculations corresponds to a family of physics models with SM-like couplings where the parameter sin 2 θ W and the G F coupling are defined by the M W parameter. The Higgs-boson mass constraint m H ¼ 125 GeV=c 2 ensures that the form-factor values remain in the vicinity of SM-fit values from LEP-1 and SLD [4,5].
The form factors are calculated in the massless-fermion approximation. Consequently, they only depend on the fermion weak isospin and charge, and are distinguished via three indices, e (electron type), u (up-quark type), and d (down-quark type). For the ee → Z → qq process, the ZFITTER scattering-amplitude ansatz is 4T e 3 T q 3 ρ eq × ½hējγ μ ð1 þ γ 5 Þjeihqjγ μ ð1 þ γ 5 Þjqi − 4jQ e jκ e sin 2 θ W hējγ μ jeihqjγ μ ð1 þ γ 5 Þjqi − 4jQ q jκ q sin 2 θ W hējγ μ ð1 þ γ 5 Þjeihqjγ μ jqi þ 16jQ e Q q jκ eq sin 4 θ W hējγ μ jeihqjγ μ jqi; ðA1Þ where q equals u or d, the terms ρ eq , κ e , κ q , and κ eq are complex-valued form factors, the bilinear γ matrix terms are covariantly contracted, and 1 2 ð1 þ γ 5 Þ is the left-handed helicity projector in the ZFITTER convention. The form factors are functions of the sin 2 θ W parameter and the Mandelstamŝ variable of the e þ e − → Z → ff process. The κ e form factors of the A u and A d amplitudes are not equivalent; however, atŝ ¼ M 2 Z , they are numerically equal. The ρ eq , κ e , and κ q form factors can be incorporated into QCD calculations as corrections to the Born-level g f A and g f V couplings, where f ¼ e or q. The resulting current-current amplitude is similar to A q , but the sin 4 θ W term contains κ e κ q . This difference is eliminated by adding the sin 4 θ W term of A q with the replacement of κ eq with κ eq − κ e κ q to the currentcurrent amplitude. Further details are in Ref. [7].
The products κ f sin 2 θ W , called effective mixing terms, are directly accessible from measurements of the asymmetry in the cos ϑ distribution. However, neither the sin 2 θ W parameter nor theŝ-dependent form factors can be inferred from measurements without assuming the SM. The effective mixing terms are denoted as sin 2 θ eff to distinguish them from the on-shell definition of the sin 2 θ W parameter, sin 2 θ W ¼ 1 − M 2 W =M 2 Z . The Drell-Yan process is most sensitive to the sin 2 θ eff mixing term of the lepton vertex, κ e sin 2 θ W . At the Z-boson pole, κ e is independent of the quark flavor, and the flavor-independent value of κ e sin 2 θ W is commonly denoted as sin 2 θ lept eff . For comparisons with other measurements, the value of sin 2 θ lept eff at the Z-boson pole is taken to be Re½κ e ðsin 2 θ W ; M 2 Z Þ sin 2 θ W .