$\sin^2\theta^{\rm lept}_{\rm eff}$ and $M_W$(indirect) extracted from 9 fb$^{-1}$ $\mu^+\mu^-$ event sample at CDF

We report on the extraction of $\sin^2\theta^{\rm lept}_{\rm eff}$ and indirect measurement of the mass of the W boson from the forward-backward asymmetry of $\mu^+\mu^-$ events in the $Z$ boson mass region. The data sample collected by the CDF detector corresponds to the full 9 fb$^{-1}$ run II sample. We measure $\sin^2 \theta^{\rm lept}_{\rm eff} = 0.2315 \pm 0.0010$, $ \sin^2 \theta_W = 0.2233 \pm 0.0009$ and $M_W ({\rm indirect}) = 80.365 \pm 0.047 \;{\rm GeV}/c^2$, where each uncertainty includes both statistical and systematic contributions. Comparison with the results of the D0 collaboration are presented.


Introduction
Now that the Higgs mass is known, the Standard Model is over constrained. Therefore, any inconsistency between precise measurements of SM parameters would be indicative of new physics. The parameter that needs to be measured more precisely is M W (with errors <15 MeV), or equivalently sin 2 θ W = 1 − M 2 W /M 2 Z (with errors <0.0003). Similarly, in order to help resolve the long standing 3σ difference in sin 2 θ • A new technique [1] for calibrating the muon energy scale as a function of detector η and φ (and sign), thus greatly reducing systematic errors from the energy scale. A similar method can also used for electrons.
• A new event weighting technique [3]. With this technique all experimental uncertainties in acceptance and efficiencies cancel (by measuring the cos θ coefficient A 4 and using the relation A FB = 8A 4 /3). Similarly, additional weights can be included for antiquark dilution, which makes the analysis independent of the acceptance in dilepton rapidity.
• The implementation [2] of Z fitter Effective Born Approximation (EBA) electroweak radiative corrections into the theory predictions of POWHEG and RESBOS which allows for a measurement of both sin 2 θ lept eff (M Z ) and

Momentum-energy scale corrections
This new technique [1] is used in CDF (for both muons and electrons) and also in CMS. In CMS it is used to get a precise measurement of the Higgs mass in the four lepton channel. The technique relies on the fact that the Z boson mass is well known as follows: • Any correlation between the scales of the two leptons is removed by getting an initial calibration using Z events. It is done by requiring that the mean 1/P T of each lepton in bins of detector η, φ and charge is equal to the expected value for generated Z events, smeared by the momentum/energy resolution.
• The Z mass is is used as a second order correction. The measured Z mass as a function of detector η, φ and charge of the lepton is required to be equal to the value for generated Z events (smeared by the momentum/energy resolution).
• Another check is the measured J/ψ mass as a function of η of the lepton.
The momentum/scale corrections are determined for both data events and reconstructed hit level Monte Carlo events. After corrections, the reconstructed Z mass as a function η, φ and charge for both the data and hit level MC agree with the generator level Monte Carlo (smeared by resolution, and with experimental acceptance cuts). All charge bias is removed. For muons each bin in η and φ the following calibration constants are extracted.
• A multiplicative calibration correction in the quantity 1/P T which accounts for possible miscalibration of the magnetic field.
• A calibration correction which is additive in 1/P T which accounts for tracker mis-alignments.
• For very low energy muons, the J/ψ mass and Υ mass are used to determine a small additional calibration constant to tune the dE/dx energy loss in the amount of material in the tracker as a function of detector η.
When the technique is used for electrons, the multiplicative correction accounts for tower mis-calibration and there is no additive correction since the tracker is not used in the reconstruction of the electron energy.

The event weighting technique
The forward-backward A FB asymmetry of leptons measured with this technique [3] is insensitive to the acceptance and lepton detection efficiency. Therefore, the raw A FB which is measured using this technique is automatically corrected for efficiency and acceptance. The only corrections that need to be made are corrections for momentum/energy resolution which lead to event migration between different bins in dilepton mass. All experiment dependent systematic errors cancel to first order.
The event weighting technique utilizes two kinds of weights. Angular weights are used to remove the sensitivity to acceptance and lepton detection efficiency as a function of cos θ. In the CDF analysis, only angular weights are used. For proton-proton collisions at the LHC, one can add weights which correct for the rapidity dependent dilution and therefore removes the sensitivity to the acceptance in Boson rapidity.

Effective Born approximation (EBA) electroweak
radiative corrections These radiative corrections are derived from the approach adopted at LEP [2,5]. The Z-amplitude form factors are calculated by ZFITTER 6.43 [5] which is used with LEP-1 and SLD measurement inputs for precision tests of the standard model [6].
A f b in the region of the mass of the Z boson is sensitive to the effective weak mixing angle sin 2 θ eff (M, f lavor), where M is the dilepton mass. Here, sin 2 θ eff is related to the on-shell [7] electroweak mixing angle sin 2 θ W = 1 − M 2 W /M 2 Z via complex mass and flavor dependent electroweak radiative corrections form factors.
The parameter which is measured at LEP and SLD is sin 2 θ lept eff (M Z ). Previous extraction of sin 2 θ lept eff (M Z ) from Drell-Yan A f b neglected the dependent of sin 2 θ eff on flavor and dilepton mass. The input to the theory predictions is then one number sin 2 θ eff which is assumed to be independent of mass or flavor and therefore is interpreted as sin 2 θ lept eff (M Z ). When the full EBA EW radiative corrections are included, the input to the theory is sin 2 θ W = 1 − M 2 W /M 2 Z , which when compared to the data yields a measurement of the best fit value of sin 2 θ W . From that value of sin 2 θ W , and the full complex EBA radiative corrections form factors one also gets the corresponding sin 2 θ lept eff (M Z ). We find that sin 2 θ lept eff (M Z ) ≈ 1.037 sin 2 θ W .

ZGRAD type EW radiative corrections
An approximate way to correct for the flavor dependence of sin 2 θ eff from EW radiative corrections is used by the D0 collaboration. This is done by making the following corrections (proposed by Baur and collaborators [8]): We will refer to these EW corrections as ZGRAD type corrections. The D0 collaboration reports [14] that the difference between sin 2 θ lept eff (M Z ) extracted using resbos (with CTEQ 6.6 -NLO PDFs) including ZGRAD type radiative corrections and sin 2 θ  larger than the value of sin 2 θ lept eff (M Z ) extracted using pythia 6.323 [9] with NNPDF2.3-NLO PDFs [10]) and no EW radiative corrections. Note that pythia matrix elements are QCD leading order as compared to resbos matrix elements which are NLO.
The above procedure partially corrects for the flavor dependence of sin 2 θ eff , but does not account for the mass dependence of sin 2 θ eff . However, since the data are dominated by events in the region of the Z boson mass, the average sin 2 θ eff is interpreted as sin 2 θ lept eff (M Z ). Note that this kind of analysis cannot not yield a measurement of sin 2 2. Analysis of CDF µ + µ − full 9 fb −1 run II sample We report on the published analysis of the full 9 fb −1 run II µ + µ − data sample [4] collected by the CDF detector. [4] After applying the calibrations and muon scale corrections to the experimental and simulated data, A fb is measured in bins of µ + µ − invariant mass using the event-weighting method. This measurement is denoted as the raw A fb measurement because the eventweighting method provides a first-order acceptance correction, but does not include resolution unfolding and With the event weighting technique, the events near cos θ=0 are assigned zero weight, Therefore, the migration of events between positive and negative cos θ is negligible. Resolution smearing and FSR primarily transfer events between bins in invariant mass.
The raw A fb in bins of dimuon invariant mass is unfolded [4] for resolution smearing and FSR using a transfer matrix which is obtained from the Monte Carlo simulation. The unfolded A fb is shown in the right side of Fig. 1.
The electroweak (EWK) mixing parameters sin 2 θ lept eff and sin 2 θ W are extracted from the fully unfolded A fb measurements using A fb templates calculated with different values of sin 2 θ W . Three QCD calculations are used: LO (tree), resbos NLO, and powheg-box NLO. The calculations were modified to include EWK radiative correction [2] using the Effective Born Approximation (EBA). For the EBA electroweak form-factor cal-culations, the EW parameter is sin 2 θ W . The A fb measurement is directly sensitive to the effective-mixing parameters sin 2 θ lept eff which are combinations of the form-factors and sin 2 θ W . Most of the sensitivity to sin 2 θ lept eff comes from the Drell-Yan A fb near the Z pole, where A fb is small. In contrast, A fb at higher mass values where A fb is large, is mostly sensitive to the axial coupling, which is known.
While the extracted values of the effective-mixing parameter sin 2 θ lept eff are independent of the details of the EBA model, the interpretation of the best-fit value of sin 2 θ W and its corresponding form factors depend on the details of the EBA model.
Calculations of A fb (M) with different values of the electroweak-mixing parameter are compared with the measurement to determine the value of the parameter that best describes the data. The calculations include both quantum chromodynamic and EBA electroweak radiative corrections.
The measurement and templates are compared using the χ 2 statistic evaluated with the A fb measurement error matrix. Each template provides a scan point for the χ 2 function (sin 2 θ W , χ 2 (sin 2 θ W )). The scan points are fit to a parabolic χ 2 functional form. The χ 2 distribution of the scan over templates from the resbos NLO calculation (with CT10 PDFs) is shown in Fig. 2. The EBA-based resbos calculations of A fb are used to ex-tract the central value of sin 2 θ W . The other calculations are used to estimate the systematic error from the electroweak radiative corrections and QCD NLO radiation.
3. Systematic errors in the extraction of sin 2 θ lept eff from the full 9 fb −1 run II sample In all QCD calculations, the mass-factorization and renormalization scales are set to the muon-pair invariant mass. To evaluate the effects of different scales, the running scales are varied independently by a factor ranging from 0.5 to 2 in the calculations. The largest observed deviation of the best-fit value of sin 2 θ W from the default value is considered to be the QCD-scale uncertainty. This uncertainty is ∆ sin 2 θ W (QCD scale) = ±0.00003.
The CT10 PDFs are derived from a global analysis of experimental data that utilizes 26 fit parameters and the associated error matrix. In addition to the best global-fit PDFs, PDFs representing the uncertainty along the eigenvectors of the error matrix are also derived. For each eigenvector i, a pair of PDFs are derived using 90% C.L. excursions from the best-fit parameters along its positive and negative directions. The difference between the best-fit sin 2 θ W values obtained from the positive (negative) direction excursion PDF and the global best-fit PDF is denoted as δ +(−) i . The 90% C.L. uncertainty for sin 2 θ W is given by the expres- where the sum i runs over the 26 eigenvectors. This value is scaled down by a factor of 1.645 for the 68.3% C.L. (one standard-deviation) uncertainty yielding ∆ sin 2 θ W (PDF) = ±0.00036. The PDF error is expected to be a factor of 2 smaller with more modern PDFs.
The resbos A fb templates are the default templates for the extraction of sin 2 θ lept eff . The scan with the powhegbox or the tree templates yields slightly different values for sin 2 θ W . The difference, denoted as the EBA uncertainty, is ∆ sin 2 θ W (EBA) = ±0.00012. Although the resbos and powheg-box predictions are fixed-order NLO QCD calculations at large boson P T , they are all-orders resummation calculations in the low-to-moderate P T region, which provides most of the total cross section. The EBA uncertainty is a combination of differences between the resummation calculations and the derived value of sin 2 θ W with and without QCD radiation.
In summary, the total systematic uncertainties on sin 2 θ W from the QCD mass-factorization and renormalization scales, and from the CT10 PDFs is ±0.00036. All component uncertainties (shown in Table 1)  Each uncertainty includes statistical errors, and various sources of systematic errors combined in quadrature.
The results for sin 2 θ lept eff (M Z ) are consistent with other measurements at the Z-boson pole, as shown on the left panel of Fig, 3. The results for M W are consistent with other direct and indirect measurements of M W as shown on the right panel of Fig. 3.
Because of the larger angular acceptance for electrons, the error in sin 2 θ lept eff for the 9 fb − 1 e + e − sample are expected to be smaller by a factor of two (about ±0.0005). Both the statistical errors and systematic errors such as PDFs are smaller for events with large cos θ. The corresponding expected error in the CDF extracted value of M indirect W (± 24 MeV) will be competitive with the direct measurements of M W . The results from the CDF full 9 fb −1 run II e + e − sample are expected by end of 2014.

Comparison of CDF and D0 results
Also shown in Figure 3 and Table 2 is the most recent (Aug. 2014) value [14] of sin 2 θ lept eff (M Z ) extracted from the full 9.7 fb −1 run II e + e − sample in D0 [14] (0.23146 ± 0.00047).
In order to make a more direct comparison with the D0 results we have done preliminary extractions of sin 2 θ lept eff from the CDF data using pythia 6.4 [13] with no EW radiative corrections. The values extracted with CTEQ6.6 NLO PDFs are the same as the values extracted with CTEQ6L1-LO PDFs as shown in Table 2. In contrast the values extracted using the older CTEQ5L-LO PDFs are 0.0003 lower.
When the CDF extraction of sin 2 θ lept eff (M Z ) from the full 9 fb −1 run II e + e − data sample is completed, the uncertainty in the average of both CDF and D0 9 fb −1 measurements of sin 2 θ lept eff in the e + e − channel will be competitive with LEP and SLC.