Drell-Yan lepton pairs produced in the process $p \bar{p} \rightarrow \ell^+\ell^- + X$ through an intermediate $\gamma^*/Z$ 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 $\sin^2\theta^{\rm lept}_{\rm eff}$ 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,
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The forward-backward asymmetry A_{fb} is measured in bins of electron- and muon-pair mass in a range about the Z-boson resonant peak mass. The sin^{2}θ_{eff}^{lept} parameter is derived from comparisons of the asymmetry measurement with predictions of the measurement (templates) for various input values of sin^{2}θ_{eff}^{lept}.
The CDF and D0 measurements of sin^{2}θ_{eff}^{lept} are based on multiple analyses with improvements over time. Slightly different electroweak correction techniques and different parton distribution functions (PDFs) have been used. To obtain consistent results for combinations of sin^{2}θ_{eff}^{lept}, a framework with two common components, denoted as the "common framework", is adopted. Both CDF and D0 combine their electron- and muon-channel results with techniques consistent with the agreed upon common framework. The Tevatron combination utilizes these combined electron- and muon-channel results.
The common components are a) the NNPDF 3.0 set of PDFs, and b) electroweak radiative corrections calculated with the ZFITTER standard model (SM) package. The NNPDF 3.0 PDF set includes LHC data and the PDF implementation is more robust. NNPDFs are ensembles of 100 equally probable PDFs, for which the value of a calculation is the average over the ensemble, and the rms about the average is the PDF uncertainty. The ZFITTER implementation of electroweak radiative corrections is a precision package used for SM fits by LEP-1 and SLD. ZFITTER provides complex-valued form factors corrections to the Born-level vector (g_{V}^{f}) and axial-vector (g_{A}^{f}) vertex couplings,
The documentation of measurements of the asymmetry and the extraction of sin^{2}θ_{eff}^{lept} from them are
The corresponding public web pages are
CDF result summary
Result type | sin^{2}θ_{eff}^{lept} |
Electron A_{fb} | 0.23248 ± 0.00049 (stat.) ± 0.00004 (syst.) ± 0.00019 (PDF) |
Muon A_{fb} | 0.2315 ± 0.0009 (stat.) ± 0.0002 (syst.) ± 0.0004 (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 along with 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. ZFITTER electroweak radiative corrections are implemented in the template calculations and this implementation is denoted as EBA (Enhanced Born Approximation). This includes the photon propagator correction.
The muon-channel result above uses templates calculated with RESBOS with CTEQ6.6 PDFs, and with uncertainties estimated with POWHEG-BOX and CT10 NLO PDFs. While this is the standalone result for sin^{2}θ_{eff}^{lept} of the muon channel, it is not compliant with the common framework agreement. For the combination of the electron- and muon-channel results, the muon A_{fb} measurement is used directly with templates calculated using the POWHEG-BOX framework with NNPDF 3.0 PDFs described above. The corresponding muon-channel result with these templates is sin^{2}θ_{eff}^{lept} = 0.23141±0.00086, where the uncertainty is statistical only.
The χ^{2} between the measurement and templates
for both the electron and muon channel are combined into joint
χ^{2} distributions, which are used to extract the
combined value of sin^{2}θ_{eff}^{lept}.
The uncertainties of
sin^{2}θ_{eff}^{lept} for the combination
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 |
D0 result summary
Result type | sin^{2}θ_{eff}^{lept} |
Raw ee | 0.23139 ± 0.00043 (stat.) ± 0.00008 (syst.) ± 0.00017 (PDF) |
Final ee | 0.23137 ± 0.00043 (stat.) ± 0.00009 (syst.) ± 0.00017 (PDF) |
Raw μμ | 0.22994 ± 0.00059 (stat.) ± 0.00005 (syst.) ± 0.00024 (PDF) |
Final μμ | 0.23016 ± 0.00059 (stat.) ± 0.00006 (syst.) ± 0.00024 (PDF) |
Final combined | 0.23095 ± 0.00035 (stat.) ± 0.00007 (syst.) ± 0.00019 (PDF) |
The raw ee 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 ensemble PDFs of NNPDF 2.3.
The raw μμ result is obtained from the A_{fb} distributions of muon pairs. Muons have P_{T} > 15 GeV/c. Both muons have |η| < 1.8, with one having |η| < 1.6. 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 3.0 (NLO) PDFs. They include D0 detector simulation. The PDF uncertainty is obtained using the ensemble PDFs of NNPDF 3.0.
To achieve results compliant with the common framework agreement, corrections are calculated and applied to the "raw" results. There are two corrections
For Δsin^{2}θ_{eff}^{lept}(RadCor), the CDF ZFITTER based framework is utilized, where A_{fb} templates use ZFITTER radiative corrections. The analog to the D0 PYTHIA templates with a single real-valued weak-mixing angle parameter is obtained by setting all form factors to unity, and retaining only the running α_{em} part of the photon propagator correction. In this case, "sin^{2}θ_{W}" is equivalent to the single value sin^{2}θ_{eff}^{lept}, and this implementation is denoted by nonEBA. The difference between the EBA template and nonEBA (or PYTHIA equivalent) template fits to A_{fb} for sin^{2}θ_{eff}^{lept} gives Δsin^{2}θ_{eff}^{lept}(RadCor) = +0.00022 ± 0.00004. This is calculated using the sin^{2}θ_{eff}^{lept} derived from fits to the combination of the CDF electron- and muon-channel asymmetries for 23 ensemble PDFs of NNPDF 3.0. The fit values from the EBA templates are near the average value derived over all ensemble PDFs.
The "Final ee" value of sin^{2}θ_{eff}^{lept}
is the "Raw ee" value with the
Δsin^{2}θ_{eff}^{lept}(RadCor) and
Δsin^{2}θ_{eff}^{lept}(PDF)
corrections additively included.
The published value of sin^{2}θ_{eff}^{lept}
[PRL 115 041801 (2015)]
includes no PDF correction, and electroweak radiative
corrections that partially accounts for ZFITTER based corrections.
For the "Final μμ" value of
sin^{2}θ_{eff}^{lept}, only the
Δsin^{2}θ_{eff}^{lept}(RadCor) correction
is additively applied to the "Raw μμ" value.
The Final ee and μμ values are combined to yield the
"Final combined" value in the table above. The uncertainties of
sin^{2}θ_{eff}^{lept} for the combination
of the electron and muon channels are summarized in the following table.
Category | Uncertainty |
Statistical | 0.00035 |
Total systematic | 0.00007 |
Energy calibration | 0.00001 |
Energy smearing | 0.00003 |
Background | 0.00002 |
Charge misidentification | 0.00003 |
Lepton identification | 0.00005 |
Fiducial asymmetry | 0.00001 |
NNPDF 2.3/3.0 | 0.00019 |
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 | ±0.00027 |
Uncorrelated | ±0.00005 |
NNPDF PDF | ±0.00018 |
Inference of sin^{2}θ_{W}
The sin^{2}θ_{eff}^{lept} parameter 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 inference for sin^{2}θ_{W} (M_{W}) based on the combined sin^{2}θ_{eff}^{lept} is
Comparisons of sin^{2}θ_{eff}^{lept} that includes 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 inferred value of sin^{2}θ_{W} from the on-shell renormalization scheme framework 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 SM fits to Z-pole measurements with the Tevatron constraint to the top quark mass (173.2 ± 0.9 GeV/c^{2}), and there are direct W-mass measurements from the Tevatron and LEP-2 [Phys. Rev. D88, 052018 (2013)].
All measurements except for 'TeV and LEP-2' and 'ATLAS' are indirect W-mass measurements that use the standard model (on-shell scheme). The LEP-1 and SLD (m_{t}) result, except for the Tevatron top-mass constraint, only uses LEP-1 and SLD data to constrain the ZFITTER SM input parameters. |