## Tevatron Run II combination of the effective leptonic electroweak mixing angles

### CDF and D0 Collaborations

Phys. Rev. D publication: Phys. Rev. D97, 112007
arXiv: arXiv:1801.06283

### Abstract

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,

$\sin^2 \theta^{\rm lept}_{\rm eff} = 0.23148 \pm 0.00033$.

This result is consistent with, and approaches in precision, the best measurements from electron-positron colliders. The standard model inference of the on-shell electroweak mixing parameter $\sin^2\theta_W$, or equivalently the $W$-boson mass $M_W$, using the ZFITTER software package yields $\sin^2 \theta_W = 0.22324 \pm 0.00033$ or equivalently, $M_W = 80.367 \pm 0.017 \;{\rm GeV}/c^2$.

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### Measurements and the combination strategy

The forward-backward asymmetry Afb is measured in bins of electron- and muon-pair mass in a range about the Z-boson resonant peak mass. The sin2θefflept parameter is derived from comparisons of the asymmetry measurement with predictions of the measurement (templates) for various input values of sin2θefflept.

The CDF and D0 measurements of sin2θefflept 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 sin2θefflept, 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 (gVf) and axial-vector (gAf) vertex couplings,

• gVf → √ρf T3f [ 1 − 4|Qf| κf sin2θW ]
• gAf → √ρf T3f
where ρ and κ are form factors which depend on the fermion flavor (f) at the vertex and a mass scale (Mff). The photon propagator correction from fermion loops is utilized.

### Summary of CDF and D0 sin2θefflept results used for the combination

The documentation of measurements of the asymmetry and the extraction of sin2θefflept from them are

The measurements of Afb utilize the full Run II data sets for CDF and D0. For CDF, both the electron- and muon-channels have an integrated luminosities of 9 fb−1. For D0, the electron-channel measurement uses 9.7 fb−1 of integrated luminosity, and the muon-channel measurement uses 8.6 fb−1 of integrated luminosity.

The corresponding public web pages are

CDF result summary

 Result type sin2θefflept Electron Afb 0.23248 ± 0.00049 (stat.) ± 0.00004 (syst.) ± 0.00019 (PDF) Muon Afb 0.2315 ± 0.0009 (stat.) ± 0.0002 (syst.) ± 0.0004 (PDF) Combined 0.23221 ± 0.00043 (stat.) ± 0.00007 (syst.) ± 0.00016 (PDF)

Afb 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 Afb templates for the extraction of sin2θefflept are calculated with the POWHEG-BOX NLO framework for Drell-Yan events, and the NNPDF 3.0 ensemble PDFs for NNLO and αs(MZ2) = 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 sin2θefflept 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 Afb 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 sin2θefflept = 0.23141±0.00086, where the uncertainty is statistical only.

The χ2 between the measurement and templates for both the electron and muon channel are combined into joint χ2 distributions, which are used to extract the combined value of sin2θefflept. The uncertainties of sin2θefflept 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 sin2θefflept 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 Afb 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 sin2θefflept is extracted from background subtracted Afb distributions. The Afb templates used for the extraction of sin2θefflept 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 Afb distributions of muon pairs. Muons have PT > 15 GeV/c. Both muons have |η| < 1.8, with one having |η| < 1.6. The mixing angle sin2θefflept is extracted from background subtracted Afb distributions. The Afb templates used for the extraction of sin2θefflept 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

• Δsin2θefflept(PDF) = −0.00024±0.00004: NNPDF 2.3 → NNPDF 3.0 for the electron channel, and
• Δsin2θefflept(RadCor) = +0.00022 ± 0.00004: PYTHIA → ZFITTER radiative corrections.

The electron-channel determination of sin2θefflept requires a PDF correction, which is determined as follows. Pseudodata Afb distributions are calculated using the default PDFs of NNPDF 3.0 and NNPDF 2.3.
• NNPDF 3.0: a single Afb distribution with a fixed value of sin2θefflept.
• NNPDF 2.3: 40 templates with varying values of sin2θefflept.
The NNPDF 3.0 Afb distribution is fit to the templates to obtain the best-fit value of sin2θefflept for NNPDF 2.3. The difference between the fixed input value of sin2θefflept of the NNPDF 3.0 Afb distribution and the best fit value is Δsin2θefflept(PDF), and its value is −0.00024±0.00004, where the uncertainty is statistical. The muon-channel PDF is NNPDF 3.0, so it does not require Δsin2θefflept(PDF).

For Δsin2θefflept(RadCor), the CDF ZFITTER based framework is utilized, where Afb 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 αem part of the photon propagator correction. In this case, "sin2θW" is equivalent to the single value sin2θefflept, and this implementation is denoted by nonEBA. The difference between the EBA template and nonEBA (or PYTHIA equivalent) template fits to Afb for sin2θefflept gives Δsin2θefflept(RadCor) = +0.00022 ± 0.00004. This is calculated using the sin2θefflept 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 sin2θefflept is the "Raw ee" value with the Δsin2θefflept(RadCor) and Δsin2θefflept(PDF) corrections additively included. The published value of sin2θefflept [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 sin2θefflept, only the Δsin2θefflept(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 sin2θefflept 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 CDF and D0 results

Combination of sin2θefflept results

The input values of the CDF and D0 results, and their combination via the Best Linear Unbiased Estimator (BLUE) are

• sin2θefflept = 0.23221 ± 0.00043 (stat.) ← CDF, and
• sin2θefflept = 0.23095 ± 0.00035 (stat.) ← D0,
• sin2θefflept = 0.23148 ± 0.00027 (stat.) ± 0.00005 (syst.) ± 0.00018 (PDF) ← Combined.
The uncertainties for the combination are listed below, and they are derived using the individual CDF and D0 uncertainties listed in the tables of the previous section. For the combination, three categories of uncertainties are used: statistical, PDF, and uncorrelated systematics. The input PDF uncertainties are treated as 100% correlated. All other input systematic uncertainties are uncorrelated, and combined into a single combination category denoted as "uncorrelated". The following table gives the combination uncertainties from BLUE.

 Category Uncertainty Statistical ±0.00027 Uncorrelated ±0.00005 NNPDF PDF ±0.00018

Inference of sin2θW

The sin2θefflept parameter in terms of the ZFITTER lepton-vertex form factor κe [PRD 93 112016 (2016)] is

• sin2θefflept = Re[κe(sin2θW,MZ2)] sin2θW.
Consequently, SM calculations of Re[κe(sin2θW,MZ2)] are required to infer sin2θW from the observable sin2θefflept. The calculation of κe depends on the context of SM inputs to ZFITTER calculations. The top-quark mass input of 173.2 ± 0.9 GeV/c2 [Phys. Rev. D86, 092003 (2012)] affects the inference. Its measurement uncertainty propagates to an uncertainty of ±0.00008 for the inferred value of sin2θW. This uncertainty, denoted as the "form factor" uncertainty, only applies to sin2θW.

The inference of sin2θW using ZFITTER implies the inference of MW because of its use of the on-shell renormalization scheme where sin2θW = 1 − (MW/MZ)2 holds to all orders. The inference for sin2θW (MW) based on the combined sin2θefflept is

• sin2θW = 0.22324±0.00026±0.00019 (80.367±0.014±0.010 GeV/c2),
where the first uncertainty is statistical, and the second is the systematic uncertainty which includes the PDF and form factor uncertainties.

### Comparisons with other measurments

 Comparisons of sin2θefflept 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, AFB0,b is the b-quark asymmetry based measurement, and Al measurement corresponds to pure leptonic couplings.

The inferred value of sin2θW 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/c2), 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 (mt) result, except for the Tevatron top-mass constraint, only uses LEP-1 and SLD data to constrain the ZFITTER SM input parameters.