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 electrons is used to measure the electroweak-mixing sin^{2}θ_{W}. The measurement uses 2.1 fb^{-1} of CDF Run II data in the e^{+}e^{−} channel, where the pairs are in the mass range of 66—116 GeV/c^{2}. In the standard model, the Drell-Yan process at tree level consists of two parton level diagrams: qq → γ^{*} or Z → e^{+}e^{−}. 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 electrons is analyzed in the Collins-Soper (CS) rest frame of the ee-pair. The distribution depends on the polarization state of the γ^{*}/Z boson. QCD radiation from the colliding quarks changes the polarization state. The polar angle of the e^{−} 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. Thus, the polar angle θ is similar to the one used in e^{+}e^{−} collisions. The angular distribution integrated over φ is
The method used to extract sin^{2}θ_{W} has two components: the measurement of A_{4} and calculations of A_{4} for various input values of sin^{2}θ_{W}. This analysis measures A_{4} for electron pairs over the mass range of 66‐116 GeV. There are no restrictions on the pair rapidity or transverse momentum. The measurement is denoted as A_{4}. The calculation of A_{4} uses LEP-like implementation of electroweak radiative corrections.
The A_{4} measurement is derived from our 2.1 fb^{-1} measurement of the electron angular distribution coefficients as a function of the ee-pair transverse momentum [Phys. Rev. Lett. 106, 241801 (2011)]. Analysis details are available on the CDF EWK web page (under W/Z Cross Sections and Asymmetries). The 2.1 fb^{-1} dataset is understood well and the simulation of the data is tuned well. The simulation modeling is data driven. The modeling starts with the Pythia 6.2 simulation of pp → γ^{*}/Z (→ ee) + X (at tree level) followed by parton showering. PHOTOS 2.0 adds final state QED radiation to charged particle vertices. This is then followed by the full CDF detector simulation. All data efficiencies, both global and η_{det} dependent, as well as time-dependent, are measured and incorporated into the simulation. Both the physics and detector models are tuned (iteratively) so that the simulated, reconstructed event distributions match the data. The measured P_{T} dependent angular distibution coefficients are incorporated in the physics model for the A_{4} analysis.
The Data
The dataset is identical to the angular distribution
coefficient measurement, but is summarized below.
The electron candidates used in this analysis can be in
either the CDF central (C) or the forward end plug (P)
calorimeters. The central and plug calorimeters
cover the range, |η_{det}|<1.1 and
1.1<|η_{det}|<3.5 respectively. Each
electron is required to have an associated track and pass
standard CDF electron selection and fiducial cuts. Three
electron-pair topology categories are used with the following
kinematic selections.
The determination of cosθ requires the identification of the e^{−} and this done with the track associated with the electrons. Charge misidentification is very small for central electrons. However, it is large for plug electrons at large |η_{eta}|. So for the PP topology, |cosθ|, which is indepenent of charge, is used.
This plot shows the tuned data-vs-simulation cosθ distribution for the combined CC+CP topologies. The black crosses are the data and the blue are the simulated data. |
This plot shows the tuned data-vs-simulation cosθ distribution for the PP topology. The black crosses are the data and the blue are the simulated data. |
The QED FSR added by PHOTOS and PYTHIA are checked with the ZGrad2 Drell-Yan simulation which has an exact order α_{em} calculation. PHOTOS and PYTHIA provides a good model for QED FSR in regions that are important for electron reconstruction. The cosθ resolution estimated by the simulation has negligible bias and a narrow rms of 0.01 but with non-Gaussian tails. Charge misidentification in the CC and CP topologies has a negligible bias.
PHOTOS simulates QED FSR to α_{em}^{2} in leading-logarithmic accuracy using a fragmentation function approach. To compare QED FSR added by PHOTOS and PYTHIA against the ZGrad2 calculation, the ee-γ system is first boosted to its CM frame to mimimize distortions due to QCD radiation. ZGrad2 only produces 0 or 1 real photons. As PHOTOS and PYTHIA adds multiple photons, these photons are clustered into a single object by summing up their momentum vectors prior to their comparison with ZGrad2. For PYTHIA+PHOTOS to ZGrad2 comparisons, the minimum photon energy is 0.5 GeV (ZGrad2 threshold).
Photon energy comparison between ZGrad2 and PHOTOS based on the absolute rates. The PHOTOS plus PYTHIA total cross section is normalized to the ZGrad2 total cross section. |
cosβ separation between the photon (cluster) and the nearest e^{+} or e^{−}. The ZGrad2 and PHOTOS total cross section is normalized over 0<cosβ<0.8, so this is a shape comparison for 0<cosβ<1. |
The
A_{4}
Measurement
The tuned simulation physics model which includes the previously
measured angular distribution coefficients, is used to
calculate
A_{4}.
This is a fully corrected cross section weighted average.
The blue simulated data histograms in the cosθ plots
above is the result of the tuned simulation physics model.
As a cross check,
A_{4}
is also determined with a second method. The cosθ
data distributions shown above are fit with same technique
used in the P_{T} binned measurement of
A_{4}.
The result is consistent with the physics model calculation.
The second method is used for uncertainty estimates:
statistical fit uncertainty, and systematic uncertainties
from the background and electron efficiency measurments.
The result is
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_{4}.
They are included in the simulation physics model, so QED
radiative effects are removed from the measurement of
A_{4}.
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 (and so potentially measureable). 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. All ZFITTER form factors, including the photon propagator form factor, are complex valued, and at least functions of the s Mandelstam variable.
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. As the Z mass is well known, sin^{2}θ_{W} and M_{W} are equivalent, ie, the determination of one implies the other. ZFITTER is used to calculate tables of form factors in sin^{2}θ_{W} and the Mandalstam variable s (ie, M_{ee}). The total decay width of the Z is also calculated for the given sin^{2}θ_{W}. The Z-boson propagator is the LEP standard, [s − (M_{Z}^{2} − i sΓ_{Z}/M_{Z})].
The ZFITTER interface that calculates e^{+}e^{−} cross sections and asymmetries implement complex-valued form factors, including the one for the photon correction. The real part of the photon correction is often absorbed into α_{em} and this is called the running α_{em}. The imaginary part of the photon correction is not negligible. ZGrad2 also implements weak form factors but with a different implementation that ZFITTER. The effective sin^{2}θ_{W} from both calculations are very similar. However, ZGrad2 only uses the real valued running α_{em} for the photon amplitude. This is an issue as it affects γ^{*}-Z interference, which is significant away from the Z pole.
LO calculation of A_{fb}(M), where M=√s (ee-pair mass). The vertical line is at M=M_{Z}, and this is where γ^{*}-Z interference is zero. The intercept with the A_{fb} curve is the level of the sin^{2}θ_{W} "signal" from Z self interference. |
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 ee-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_{4}
for input values of sin^{2}θ_{W}.
ZFITTER form factor EBA calculations of A_{4} for various values of input sin^{2}θ_{W}. |
R_{4} = A_{4}(NLO) / A_{4}(tree), where NLO are the ResBos and Powheg-Box calculations, and tree is the simple LO calculation. The Powheg-Box LO matrix element calculation with parton showering is also shown. |
The ResBos predictions of A_{4} for various input values of sin^{2}θ_{W} are chosen as the default prediction for use with the measurement. It is about 0.7% below the tree predictions. The Powheg-Box calculations are similar to ResBos, but there are differences. For these differences, a systematic uncertainty of ±1% is assigned to each predicted value of A_{4} from ResBos.
The Powheg-Box has a large and useful variety of calculation options, so it is used for the estimation of the following systematic uncertainties: QCD mass factorization and renormalization scale uncertainties, and uncertainties inherent in the CT10 PDFs. All the QCD calculations use the running ee-pair mass as the factorization and renormalization scales. They are varied from the default running ee-pair mass by a factor of 0.5 to 2 for the uncertainty; this gives a QCD scale uncertainty of ΔA_{4} = ±0.0004. Update, 17 Oct 2013: For consistency, the PDF uncertainties are now calculated with the same method used for the CDF W-mass measurement [Phys. Rev. D 77, 112001 (2008)]. The 26 pairs of CT10 90% CL uncertainty PDFs are used to estimate the uncertainty from PDFs to A_{4}, and this value is scaled down to the 68% CL by a factor of 1.645 to give the PDF uncertainty of ΔA_{4} = ±0.0026.Overall, the QCD prediction uncertainties are small compared to the A_{4} measurement uncertainty.
The uncertainites from the measurement (Data) and predictions (Pred) are summarized below.
Source | Value |
Data: Measurement | ±0.0079 (stat) |
Data: Background normalization | ±0.0003 (syst) |
Data: Electron efficiencies | Negligible (syst) |
Pred: QCD scales | ±0.0004 (syst) |
Pred: QCD PDFs | ±0.0026 (syst) |
Pred: Differences from ResBos | ±0.01 A_{4} (syst) |
The inferred value of sin^{2}θ_{W} (or M_{W}) is obtained by taking the measurement, A_{4} = 0.1100 ± 0.0079 (stat) ± 0.0004 (syst), and extracting the sin^{2}θ_{W} point corresponding to the measurement in the ResBos prediction. However, the A_{4} measurement is directly and most sensitive to the effective sin^{2}θ_{W} at the lepton vertex, which is denoted as sin^{2}θ_{eff}^{lept}. This is the reference value of the ZFITTER based effective leptonic sin^{2}θ_{W}, or κ_{e}(s) sin^{2}θ_{W} at √s = M_{Z}. Thus, the more direct measurement is
The inferred value of sin^{2}θ_{W} is usually expressed as an indirect W-boson mass value.
Comparison with direct W-mass measurements from the Tevatron and LEP2, and the indirect W-mass from LEP1/SLD SM fits at the Z pole. |