Measurement of sin2θW using μ+μ pairs from pp collisions at √s = 1.96 TeV
18 Sep 2014

W. Sakumoto, A. Bodek, and J.-Y. Han
University of Rochester


Drell-Yan lepton pairs are produced in the process pp → γ*/Z + X, with the subsequent decay of the γ*/Z into lepton pairs. The angular distribution of decay muons provides information on the electroweak-mixing parameter sin2θW via its observable effective-leptonic sin2θW or sin2θefflept. The measurement uses 9.2 fb-1 of CDF Run II data in the μ+μ channel where the pairs are in the mass range above 50 GeV/c2. The value of sin2θefflept is found to be 0.2315 ± 0.0010. [Published in PRD 89,072005 (2014)]


The Drell-Yan process at tree level consists of two parton level diagrams: qq → γ* and Z → μ+μ. Fermions (f) couple to the virtual photon via vector coupling, Qfγμ, where Qf is the fermion charge (in units of e). The fermion coupling to a Z boson has both vector (V) and axial (A) couplings: gVfγμ + gAfγμγ5. The Born couplings are:

where T3f is the third component of the fermion weak isospin. These couplings affect the lepton angular distributions.

The angular distribution of the leptons is analyzed in the Collins-Soper (CS) rest frame of the μμ-pair. The polar angle of the μ is denoted as θ, and the azimuthal angle as φ. In the CS frame, the placement of the z axis is approximately along the direction of the incoming quark. When the ee-pair transverse momentum, PT is zero, the CS and lab coordinates are the same. The angular distribution integrated over φ is

where A0 and A4 are called angular distribution coefficients, and they are ratios of γ*/Z cross sections. A0 appears at NLO with QCD radiation, so it is zero at PT=0.

The A4 coefficient is parity violating, non-zero at PT=0, and induces an asymmetry in the cosθ distribution. It is a consequence of vector and axial current interference, and there are two sources: γ*-Z interference and Z self interference. The γ*-Z interference depends on gA (T3). The Z self interference includes gV, so at tree level, the A4 contribution from it is proportional to gVlgVq

where Ql is the lepton charge and Qq is the quark charge. Loop and vertex electroweak radiative corrections modify the Born-level T3f and sin2θW couplings by a few percent, and what is observed are these effective couplings. As sin2θW ∼ 0.223, A4 is most sensitive to the effective lepton-Z coupling which is denoted as sin2θefflept. A 1% variation of sin2θW results in a 8% change in the lepton-Z factor (containing Ql), and a change of 1.5% in the quark-Z factor (containing Qq) for u-quarks, and of 0.4% for d-quarks.

In this analysis, the forward-backward cosθ asymmetry of the muon pairs is measured as a function of their mass. The asymmetry is defined as:

where σ+ is the total cross section for cosθ≥0, and σ the corresponding cross section for cosθ<0.

This plot on the left illustrates the typical behavior of Afb as a function of the lepton-pair mass M. The vertical line is at M=MZ.
The method used to derive sin2θefflept has two components: the measurement of Afb and calculations of Afb for various input values of sin2θW. The measured Afb is compared to the calculated Afb templates to determine the value of sin2θefflept that best describes the measurement. The calculation of Afb are QCD calculations that use LEP-like implementations of electroweak radiative corrections. These corrections are form factors to the tree-level T3f and sin2θW couplings, and they are 1 to 4% in value.


The Data

The online selection chooses events with high PT muons of PT>18 GeV/c in the |η|<1 region of the CDF detector from 9.2 fb−1 of collisions. The offline selection of muons is more stringent and is designed to identify and deliver high quality muons with PT>20 GeV/c.

Muon candidates are required to have a high quality track in the CDF central charged-particle trackers, and a matching track segment in one of the CDF muon detectors. The central tracker acceptance falls quickly beyond |η| of 1, and vanishes near |η| ∼ 1.5. The CDF detector consists of four muon detectors denoted by CMU, CMP, CMX, and BMU.

  1. The CMU and CMP detectors shadow each other and cover |η| < 0.6.
  2. The CMX detector covers 0.6 < |η| < 1.
  3. The BMU detector covers 1 < |η| < 1.5.
The CDF muon detectors have non-negligible non-fiducial regions, gaps in coverage, and inefficiencies. A less stringent category of muons is allowed for these conditions: the track need not be matched to a muon detector track segment, but the track must still be of high quality and its energy loss in the calorimeters must be consistent with minimum ionization energy loss. The selection criteria for events with muon pairs is summarized below.
  1. Muon kinematics and fiducial regions
  2. Muon pair kinematics
Muon pairs consistent with the passage of a cosmic ray through the CDF detector are rejected. As there is very limited muon pair acceptance beyond |y| of 1, the Afb measurement is for a restricted kinematic region of |y| < 1.

The number of events passing all requirements after background subtraction is 276,623. The backgrounds are from QCD and electroweak (EWK) processes of WW, WZ, ZZ, tt, W+jets, and also Z → τ+τ. The QCD background is primarily from dijets where a particle in a jet has penetrated the hadron absorber in front of the muon detectors. Its level, estimated with same-charge muons pairs, amounts to 0.1%. As there is no Z-peak bump in the same-charge mass distribution, charge misidentification is negligible. The EWK backgrounds are derived from PYTHIA samples with detector simulation, and amount to 0.5%.
Backgrounds are subtracted in the measurement of Afb, which is for M > 50 Gev/c2.

The Simulation

PYTHIA 6.2 simulates the tree level process qq → γ*/Z → μμ followed by parton showering. This is followed by CDF II event and detector simulations. The event simulation includes PHOTOS 2.0 which adds final state QED radiation (QED FSR) to charged particle vertices. The simulated events are selected, reconstructed, as analyzed as the data. The generator-level PT distribution of the γ*/Z boson is adjusted so that the simulated PT distribution is the same as in the data.

The Afb Measurement

The same event selection criteria are applied to both the data and the simulated data. Corrections are applied to both the data and simulated data for the measurement of Afb. Due to the complex nature of the CDF muon detection and reconstuction, direct measurements of muon efficiencies are difficult: There are 10 different muon-pair topologies based on the individual muon categories. The simulation is used to predict the number of events events expected in each muon-pair topology, and they are rescaled to agree with the data. In addition, the simulation of the pp collision vertex is adjusted so that the distribution of the number of vertices per event and the location of the collision along the beamline agrees with the observed distributions. This is important for the muon momentum calibrations.

The muon momentum scale of both the data and simulated data are calibrated using a new technique developed for CMS [A. Bodek, Eur. Phys. J. C 67, 321 (2010)]. It utilizes the peaks in the muon 1/PT and muon-pair mass distributions from the Z-pole. The calibration scale of both the data and simulated data are tied to a common scale: the underlying, generator-level Z-boson peaks of PYTHIA. The calibration is performed in bins of the central-tracker (η, φ), and includes both scale corrections and alignment offset corrections. After the scale calibrations, the simlation muon momentum resolution is adjusted to match the data. The results of the calibrations are presented in the plots below.

The traditional method to measure Afb utilizies corrected cross sections, σ = N/(LεA), where L is the integrated luminosity, ε the reconstruction efficiency, and A the acceptance. The fully corrected forward-backward asymmetry is given by

In this analysis, a new, simpler method to measure the forward-backward asymmetry is used: the event weighting method [A. Bodek et al., Eur. Phys. J. C 72, 2194 (2012)]. The method is equivalent to measurements of Afb in bins of |cosθ| with the assumption that (εA)+ = (εA). The bin measurement is proportional to Afb times an angular weighting factor of |cosθ|/(1+cos2θ + ..). As the angular factor is analogous to an importance sampling weight, the binned measurements are recast in the form of an unbinned measurement with event weights: where the n and d subscripts denote different and separately weighted event sums. The event weights are functions of the boson production kinematics (M, PT) and the angular variables. The method is equivalent to using a maximum likelihood method, and is expected to improve the statistical precision of the measurement by about 20%. The event-weighting method does not compensate the following: These require separate and additional compensation.
Afb(|y|<ymax) slowy increases as ymax increases. The boson kinematic restriction |y| < 1 limits the size of the acceptance correction. The curve labeled PYTHIA shows the (arbitrarily normalized) shape of the underlying rapidity distribution from PYTHIA.
The acceptance and non-uniformity corrections are small.
The raw Afb measurement is shown in the plot on the left, along with the PYTHIA prediction (before QED FSR). The final Afb measurement uses 16 mass bins. The bins shown in this plot are combined for the resolution unfolding.

The event weighted sums N±n and N±d require resolution unfolding in the muon-pair mass and cosθ. The simulation is used to do the unfolding. Bin-to-bin event migrations are tracked by the smearing transfer matrix npr, which tabulates the number of selected events produced in mass bin p that are reconstructed in another mass bin r. The following estimator is used as the resolution unfolding matrix: npr/Nr, where Nr is the total number of events reconstructed in mass bin r. This matrix is also used to estimate the Afb measurement covariance matrix. This method of unfolding requires that the simulated data cosθ distribution match the data. The simulated data cosθ distribution is tuned to match the data. Only adjustments symmetric in cosθ are applied, and the intrinsic asymmetry is unchanged. The default φ distribution is adequately simulated.

The final step of the Afb measurement after the resolution unfolding, is the correction of the small event-weighting biases. They are primarily due to regions of limited kinematic acceptance and detector non-uniformities. The simulation is used to predict the bias, defined as the true value minus the estimate of Afb from the event-weighting method. The true value is the PYTHIA calculation of Afb with the boson rapidity kinematic restriction of |y| < 1. The estimate is the Afb obtained with the simulated data. All corrections applied to the data are applied to the simulated data. The following plots show the bias correction and the fully corrected Afb measurement.

The first bin, denoted as the 'underflow' bin, covers the mass range 50-80 GeV/c2. The last bin, denoted as the 'overflow' bin covers the mass range above 102 GeV/c2. The bias correction for each mass bin is added to the corresponding resolution unfolded Afb measurement to obtain the fully corrected Afb measurement. The uncertainties shown on the Afb measurement are the uncorrelated bin-by-bin unfolding estimate, and not the diagonal elements of the measurement covariance matrix. The vertical line in the measurement plot is the mass of the Z-boson. The PYTHIA prediction is with the default sin2θefflept = 0.232. The ResBos prediction, described in the next section, is calculated with the parameter sin2θW = 0.2233.

QCD Calculations

Electroweak radiative corrections

QCD, QED, and weak radiative corrections can be organized to be separately gauge invariant, and corrections for each can be applied separately and independently. QED radiative corrections (with real photons) are not included in the calculation of Afb. They are included in the simulation physics model, so QED radiative effects are removed from the measurement of Afb.

Weak radiative corrections are based on ZFITTER 6.43's e+e → Z → qq amplitude form factors. The ZFITTER form factors are finite and gauge invariant. Thus photon corrections that involve massive gauge bosons are included in the Z amplitude form factors for gauge invariance. This includes photon propagator W-loop corrections, and photon vertex corrections containing Z propagators. Thus, internal QED radiative corrections to the photon propagator are only from fermion loops, and this correction is another form factor. ZFITTER uses the on-shell renormalization scheme for its form factors. All particle masses are on shell, and sin2θW = 1 − MW2/MZ2 to all orders of perturbation theory. The values of the form factors depend on the input sin2θW and other standard model parameters.

Drell-Yan QCD calculations

The ZFITTER complex-valued form factors are incorporated into QCD calculations for an enhanced Born approximation (EBA) to the electroweak couplings. For the form factors, √s is assigned the mass of the lepton pair. This done for both LO and NLO QCD calculations of the Drell-Yan process. Operationally, the QCD related portion of matrix elements are unchanged, and only the electroweak coupling portions need to be appropriately modified. Two NLO calculations are used in the calculations of Afb: ResBos and Powheg-Box. A simple QCD LO, or tree calculation of the Drell-Yan process is used as a basesline reference for the higher order QCD calculations. The CT10 NLO PDFs are used in all QCD calculations. The EBA form factors modifies the Born gA and gV couplings:

where ρ and κ are s-dependent, complex valued, and fermion charge and weak iso-spin dependent form factors. The combination, κ sin2θW is called an effective sin2θW, and the measured Afb is directly related to this combination, and in particular, most sensitive to the one at the lepton vertex. The κ form factor is different for the l-Z and q-Z vertex. Its reference value, for comparisons with LEP measurements, is the real part at √s = MZ. That is, its value at the Z pole.

The ResBos predictions of Afb for various input values of sin2θW are chosen as the default for comparisons with the measurement. As Powheg-Box has a large and useful variety of calculation options, it is used for the estimation of the systematic uncertainties.

Extraction of sin2θefflept

The fully corrected Afb measurement is compared with predictions using the χ2 statistical measure derived from the measurement covariance matrix. The uncertainties of the bias correction and the predictions are combined in quadrature with the eigenvalues of the covariance matrix in the space of the basis vectors of the covariance matrix. This is also known as regularization in the techniques of matrix singular value decomposition, and its purpose is to regulate the 'simulation noise' terms of the error matrix.

For the EBA-based QCD calculations, the parameter that specifies the electroweak mixing is sin2θW. As the measurement is directly sensitive to sin2θefflept, the best-fit value of sin2θW only represents the corresponding sin2θefflept, which is independent of the standard model parameters used for the EBA form factors. However, the interpretation of the best-fit value of sin2θW only has meaning within the context of the model assumptions used.

The χ2 is calculated for a series of Afb calculations, denoted as scan templates. The series of scan points are fit to a generic χ2 functional form

where the terms with the overlines are fit parameters. The sin2θW parameter is the extracted (best-fit) value of sin2θW, and σ the corresponding measurement uncertainty. The χ2, relative to 16 mass bins, is the χ2 goodness-of-fit for the extracted value.
The χ2 scan over the ResBos NLO templates. The inverted triangles are the scan points, and the solid curve is the fit to those scan points.
The results of the template scans are summarized in the table below.

sin2θefflept sin2θW χ2
ResBos NLO 0.2315±0.0009 0.2233±0.0008 21.1
Powheg-Box NLO 0.2314±0.0009 0.2231±0.0008 21.4
Tree LO 0.2316±0.0008 0.2234±0.0008 24.2
PYTHIA CTEQ5L 0.2311±0.0008 20.8
PYTHIA CTEQ6L1 (*) 0.2314±0.0008 23.6
PYTHIA CTEQ6.6 (*) 0.2314±0.0008 24.0
(CDF A4) 0.2328±0.0010 0.2246±0.0009
(LEP-1+SLD) 0.23152±0.00016

Entries followed by (*): Preliminary. The template scan uncertainties are statistical only. The PYTHIA entry values are from non-EBA calculations using PYTHIA 6.4 with various CTEQ PDFs. All values except the PYTHIA CTEQ6L1 and PYTHIA CTEQ6.6 entries are published in [PRD 89,072005 (2014)]. Also included are two previous measurements: CDF A4 [T. Aaltonen et al. (CDF Collaboration), PRD 88, 072002 (2013), arXiv:1307.0770 [hep-ph]], and LEP-1+SLD [S. Schael et al. (ALEPH, DELPHI, L3, OPAL, and SLD Collaborations), Phys. Rep. 427, 257 (2006)].

Uncertainities of the Extracted sin2θefflept

The uncertainties to sin2θefflept are from the measurement of Afb and the template predictions of Afb. For the measurement, two sources are considered: the muon momentum scale and the backgrounds. For the predictions, three source are considered: QCD mass-factorization and renormalization scales, CT10 PDFs, and the choice of the ResBos calculation as the default prediction.

The momentum scale is well constrained by the precision of the Z-peak within the muon-pair mass distribution. The backgrounds at low muon-pair masses amount to a level of about 5%, and uncertainties in its normalization affect the extracted value of the effective coupling.

All the QCD calculations use the running lepton-pair mass as the factorization and renormalization scales. They are varied from the default running mass by a factor of 0.5 to 2 for the uncertainty. The 26 pairs of CT10 90% CL uncertainty PDFs are used to estimate the uncertainty from PDFs via the standard method and scaled down to 68% CL with the factor 1.645. The extracted value of sin2θefflept using the tree (LO) templates and the Powheg-Box (NLO) templates are slightly different from the ResBos based default. The difference is assigned as a systematic uncertainty for the choice of ResBos as the default calculation.

The uncertainties to sin2θefflept are summarized in the table below. In this table, 'Data' denotes the uncertainties from the Afb measurement, and 'Pred' denotes the prediction uncertainties.

Source Value
Data: Measurement±0.00085 (stat)
Data: Muon momentum scale±0.00005 (syst)
Data: Backgrounds±0.00010 (syst)
Pred: QCD scales±0.00003 (syst)
Pred: QCD PDFs±0.00037 (syst)
Pred: Differences from ResBos±0.00012 (syst)


The results of the extraction are summarized below.

where the first uncertainty is the statistical uncertainty, and the second is the systematic uncertainty. All systematic uncertainties are included and combined in quadrature.

The interpretation of sin2θW and MW only has meaning within the context of the standard model parameters used for the EBA form factors. The context is similar to the environment for standard model fits of Z-pole measurements by LEP-1/SLD [Phys. Rep. 427, 257 (2006)], but with the exception that the Higgs boson mass parameter is set to 125 GeV/c2.

The corresponding sin2θefflept measurements from LEP-1/SLD [Phys. Rep. 427, 257 (2006)] D0 [Phys. Rev. D84, 012007 (2011)], and CDF [T. Aaltonen et al., PRD 88, 072002 (2013), arXiv:1307.0770 [hep-ph]] are: The Z-pole is a standard model analysis of combined LEP-1/SLD Z-pole results and the light quark one only includes hadronic final states with light u, d, and s quarks. The Tevatron Drell-Yan results are produced mainly from light quarks, although the quark mix differs from electron-positron collider production of light quarks. There is also a recent measurement from CMS at the LHC (multivariate analysis of Drell-Yan dimuons) [Phys. Rev. D84, 112002 (2011)]. All results are consistent.

The inferred value of sin2θW is usually expressed as an indirect W-boson mass value. There are other indirect W-boson mass results from LEP-1 and SLD which are from standard model fits to Z-pole measurements with the top quark mass, and an on-shell sin2θW neutrino-nucleon scattering result from NuTeV [Phys. Rev. Lett. 88, 091802, ibid. 90,239902(E)]. There are also direct W-mass measurements from the Tevatron and LEP-2 [Phys. Rev. D88, 052018 (2013)]. Graphical summaries of the result and comparison with other measurements are shown below.

Comparison of sin2θefflept that includes latest LHC results from CMS and ATLAS [ ATL-PHYS-PROC-2013-150; DIS 2013 (Marseilles, Fr) Apr 2013 ].