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 sin^{2}θ_{W} via its observable effective-leptonic sin^{2}θ_{W} or sin^{2}θ_{eff}^{lept}. 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/c^{2}. The value of sin^{2}θ_{eff}^{lept} 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, Q_{f}γ_{μ}, where Q_{f} is the fermion charge (in units of e). The fermion coupling to a Z boson has both vector (V) and axial (A) couplings: g_{V}^{f}γ_{μ} + g_{A}^{f}γ_{μ}γ_{5}. The Born couplings are:
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, P_{T} is zero, the CS and lab coordinates are the same. The angular distribution integrated over φ is
The A_{4} coefficient is parity violating, non-zero at P_{T}=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 g_{A} (T_{3}). The Z self interference includes g_{V}, so at tree level, the A_{4} contribution from it is proportional to g_{V}^{l}g_{V}^{q}
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:
This plot on the left illustrates the typical behavior of A_{fb} as a function of the lepton-pair mass M. The vertical line is at M=M_{Z}. |
The Data
The online selection chooses events with high P_{T} muons
of P_{T}>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
P_{T}>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.
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
P_{T} distribution of the γ^{*}/Z
boson is adjusted so that the simulated P_{T}
distribution is the same as in the data.
The A_{fb} 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 A_{fb}. 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/P_{T} 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 A_{fb} 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
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 n_{pr}, 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: n_{pr}/N_{r}, where N_{r} is the total number of events reconstructed in mass bin r. This matrix is also used to estimate the A_{fb} 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 A_{fb} 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 A_{fb} from the event-weighting method. The true value is the PYTHIA calculation of A_{fb} with the boson rapidity kinematic restriction of |y| < 1. The estimate is the A_{fb} 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 A_{fb} measurement.
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
A_{fb}. They are included in the simulation physics model,
so QED radiative effects are removed from the measurement of
A_{fb}.
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 sin^{2}θ_{W} = 1 − M_{W}^{2}/M_{Z}^{2} to all orders of perturbation theory. The values of the form factors depend on the input sin^{2}θ_{W} 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 A_{fb}:
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 g_{A} and g_{V}
couplings:
The ResBos predictions of A_{fb} for various input values of sin^{2}θ_{W} 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.
The fully corrected A_{fb} measurement is compared with predictions using the χ^{2} 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 sin^{2}θ_{W}. As the measurement is directly sensitive to sin^{2}θ_{eff}^{lept}, the best-fit value of sin^{2}θ_{W} only represents the corresponding sin^{2}θ_{eff}^{lept}, which is independent of the standard model parameters used for the EBA form factors. However, the interpretation of the best-fit value of sin^{2}θ_{W} only has meaning within the context of the model assumptions used.
The χ^{2} is calculated for a series of A_{fb} calculations, denoted as scan templates. The series of scan points are fit to a generic χ^{2} functional form
The χ^{2} scan over the ResBos NLO templates. The inverted triangles are the scan points, and the solid curve is the fit to those scan points. |
Template (Measurement) |
sin^{2}θ_{eff}^{lept} | sin^{2}θ_{W} | χ^{2} |
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 A_{4}) | 0.2328±0.0010 | 0.2246±0.0009 | |
(LEP-1+SLD) | 0.23152±0.00016 |
The uncertainties to sin^{2}θ_{eff}^{lept} are from the measurement of A_{fb} and the template predictions of A_{fb}. 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 sin^{2}θ_{eff}^{lept} 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
sin^{2}θ_{eff}^{lept}
are summarized in the table below.
In this table, 'Data' denotes the uncertainties from the
A_{fb} 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.
The interpretation of sin^{2}θ_{W} and M_{W} 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/c^{2}.
The corresponding sin^{2}θ_{eff}^{lept} 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 inferred value of sin^{2}θ_{W} 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 sin^{2}θ_{W}
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 sin^{2}θ_{eff}^{lept} that includes latest LHC results from CMS and ATLAS [ ATL-PHYS-PROC-2013-150; DIS 2013 (Marseilles, Fr) Apr 2013 ]. |