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 leptons provides information on the electroweak-mixing parameter sin^{2}θ_{W} via the observable effective-leptonic sin^{2}θ_{W} (sin^{2}θ_{eff}^{lept}) mixing parameter. Measurements of sin^{2}θ_{eff}^{lept} based on the observed the lepton asymmetry (A_{fb}) as a function of lepton-pair mass from the following measurements are combined:
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The D0 and CDF combined result for sin^{2}θ_{eff}^{lept} are derived from the following references.
D0 result summary
Result type | sin^{2}θ_{eff}^{lept} |
Raw | 0.23139 ± 0.00043 (stat.) ± 0.00008 (syst.) ± 0.00017 (PDF) |
Final | 0.23147 ± 0.00043 (stat.) ± 0.00008 (syst.) ± 0.00017 (PDF) |
The raw result is obtained directly from the A_{fb} distributions of electron pairs. Both the central (CC) and end cap (EC) calorimeters are used, and there are three pair topoplgies, CC-CC, CC-EC, and EC-EC, which are combined for the results above. The mixing angle sin^{2}θ_{eff}^{lept} is extracted from background subtracted A_{fb} distributions. The A_{fb} templates used for the extraction of sin^{2}θ_{eff}^{lept} are based on PYTHIA and NNPDF 2.3 (NLO) PDFs. They include D0 detector simulation. The PDF uncertainty is obtained using the 100 equally probable ensemble PDFs of NNPDF 2.3.
The A_{fb} templates based on PYTHIA use the same real value for the effective couplings at the lepton, u-quark, and d-quark vertices. However, radiative corrections to the Born level couplings from ZFITTER used by CDF, and ZGRAD used by D0 are complex valued and differ for the different fermions. The real part of the effective couplings for the u- and d-quark vertices are shifted relative to the leptonic vertex by −0.0001 and −0.0002, respectively. These real shifts are incorporated into a version of RESBOS with CTEQ6.6 PDFs. Relative to this modified version of RESBOS, the sin^{2}θ_{eff}^{lept} from PYTHIA is shifted by −0.00008 with no change in uncertainties. Applying this correction to the raw value gives the final measurement central value without any change of uncertainties.
The uncertainties of sin^{2}θ_{eff}^{lept} are summarized in the following table.
Category | Uncertainty |
Statistical | 0.00043 |
Total systematic | 0.00008 |
Energy calibration | 0.00001 |
Energy smearing | 0.00002 |
Background | 0.00001 |
Charge misidentification | 0.00003 |
Electron identification | 0.00007 |
Fiducial asymmetry | 0.00001 |
NNPDF 2.3 | 0.00017 |
CDF result summary
Result type | sin^{2}θ_{eff}^{lept} |
Muon A_{fb} | 0.2315 ± 0.0009 (stat.) ± 0.0002 (syst.) ± 0.0004 (PDF) |
Electron A_{fb} | 0.23248 ± 0.00049 (stat.) ± 0.00004 (syst.) ± 0.00019 (PDF) |
Combined | 0.23221 ± 0.00043 (stat.) ± 0.00007 (syst.) ± 0.00016 (PDF) |
A_{fb} is measured using the event-weighting method [Eur. Phys. J. C 67, 321 (2010)], and is fully corrected. The effects of detector resolution and QED FSR are removed using the simulation. For the simulation, PYTHIA generates the Drell-Yan event, PHOTOS 2.0 applies QED FSR, and CDF detector simulation is applied. The A_{fb} templates for the extraction of sin^{2}θ_{eff}^{lept} are calculated with the POWHEG-BOX NLO framework for Drell-Yan events, and the NNPDF 3.0 ensemble PDFs for NNLO and α_{s}(M_{Z}^{2}) = 0.118. PYTHIA 6.4 parton showering follows the event generation. The ensemble consists of 100 equally probable PDFs, which are used to predict the value of observables and their uncertainties. Standard model electroweak radiative corrections calculated by ZFITTER are implemented in the template calculations. This includes the photon propagator correction, whose real part is usually called the running α_{em}.
The muon A_{fb} result in the table above uses templates calculated with RESBOS with CTEQ6.6 PDFs, and with uncertainties estimated with POWHEG-BOX and CT10 NLO PDFs. For the combination of the muon- and electron-channel results, the muon A_{fb} measurement is unchanged but the templates are calculated with the POWHEG-BOX framework with NNPDF 3.0 PDFs described above. The corresponding muon-channel result with templates calculated using POWHEG-BOX and NNPDF 3.0 PDFs is sin^{2}θ_{eff}^{lept} = 0.23141±0.00086, where the uncertainty is statistical only.
The uncertainties of sin^{2}θ_{eff}^{lept} for the combination of the muon- and electron-channels are summarized in the following table.
Category | Uncertainty |
Statistical | 0.00043 |
Total systematic | 0.00007 |
Energy scale and resolution | 0.00002 |
Background | 0.00003 |
QCD scale | 0.00006 |
NNPDF 3.0 | 0.00016 |
Standardization corrections
The CDF and D0 implementations of A_{fb} templates differ in these aspects: the PDFs and a slight variance in the method of radiative corrections. Adjustments that standardize results to NNPDF 3.0 PDFs and ZFITTER radiative corrections are calculated and implemented. The NNPDF 3.0 PDF set is preferred as it includes LHC data and the PDF implementation is more robust. The ZFITTER implementation of electroweak radiative corrections is a precision package used for standard model fits by LEP-1 and SLD.
The CDF results already use the standardization framework of NNPDF 3.0 PDFs and ZFITTER electroweak radiative corrections. The D0 results are corrected for consistency with this framework. There are two standardization corrections
For the radiative correction standardization offset, the CDF ZFITTER based framework is used. The CDF A_{fb} templates include ZFITTER radiative correction form factors, and this implementation is denoted as EBA (Enhanced Born Approximation). The form factors to the Born level vertex couplings are
To update the D0 central value of sin^{2}θ_{eff}^{lept} (0.23147) to one based on templates calculated with NNPDF 3.0 PDFs and ZFITTER-based radiatiive corrections, the two additive adjustments are applied:
Combination of sin^{2}θ_{eff}^{lept} results
The input values of the CDF and D0 results, and their combination via the Best Linear Unbiased Estimator (BLUE) are
Category | Uncertainty |
Statistical | ±0.00030 |
Uncorrelated | ±0.00005 |
Correction | ±0.00003 |
NNPDF PDF | ±0.00017 |
Inference of sin^{2}θ_{W}
The observed asymmetry is directly sensitive to the effective couplings κ_{f}sin^{2}θ_{W}. The standard model is used to infer the value of sin^{2}θ_{W} and values of κ_{f} that correspond to the effective couplings. The effective leptonic weak mixing angle in terms of the ZFITTER lepton-vertex form factor κ_{e} [PRD 93 112016 (2016)] is
The inference of sin^{2}θ_{W} using ZFITTER implies the inference of M_{W} because of its use of the on-shell renormalization scheme where sin^{2}θ_{W} = 1 − (M_{W}/M_{Z})^{2} holds to all orders. The inferences for sin^{2}θ_{W} (M_{W}) based on the updated D0 value of sin^{2}θ_{eff}^{lept} and the combination of CDF and D0 values are
Comparisons of sin^{2}θ_{eff}^{lept} that includes latest LHC results from CMS [ Phys. Rev. D84 112002, 2011 ], ATLAS [ J. High Energy Phys. 09 (2015) 049 ], and LHCb [ J. High Energy Phys. 11 (2015) 190 ]. LEP-1+SLD [ Phys. Rept. 428, 257 (2006), Phys. Rept. 532, 119 (2013) ]: Z-pole entry is the standard model analysis using all Z-pole measurements, A_{FB}^{0,b} is the b-quark asymmetry based measurement, and A_{l} measurement corresponds to pure leptonic couplings. The entries with D0 are based on the updated results of the previous section, and are preliminary. |
The inferred value of sin^{2}θ_{W} from on-shell renormalization scheme frameworks is also expressed as an indirect W-boson mass. There are other indirect W-boson mass results, including those from LEP-1 and SLD which are from standard model fits to Z-pole measurements with the top quark mass, and there are direct W-mass measurements from the Tevatron and LEP-2 [Phys. Rev. D86, 010001 (2012): PDG W mass review, 2015 update].
All measurements except for 'TeV and LEP-2' are indirect W-mass measurements that use the standard model (on-shell scheme). The LEP-1 and SLD (m_{t}) result, except for the top mass constraint, only uses LEP-1 and SLD data to constrain the ZFITTER standard model input parameters. NuTeV is the neutrino neutral current measurement [ PRL 88, 091802 (2002); PRL 90, 239902(E) (2003) ] from the Tevatron. The entries with D0 are based on the updated results of the previous section, and are preliminary. |