Measurement of the Effective Weak Mixing Angle in $p\bar{p}\rightarrow Z/\gamma^* \rightarrow \ell^+\ell^-$ Events

We present a measurement of the effective weak mixing angle parameter $\sin^2\theta_\text{eff}^{\ell}$, in $p\bar{p}\rightarrow Z/\gamma^* \rightarrow \mu^+\mu^-$ events at a center of mass energy of 1.96 TeV, collected by the D0 detector at the Fermilab Tevatron Collider and corresponding to 8.6 fb$^{-1}$ of integrated luminosity. The measured value of $\sin^2\theta_\text{eff}^{\ell}[\mu\mu]=0.23016 \pm 0.00064$ is further combined with the result from the D0 measurement in $p\bar{p}\rightarrow Z/\gamma^{*}\rightarrow e^{+} e^{-}$ events, resulting in $\sin^2\theta_\text{eff}^{\ell} [\text{comb.}]=0.23095 \pm 0.00040$. This combined result is the most precise measurement from a single experiment at a hadron collider and is the most precise determination using the coupling of the $Z/\gamma^*$ to light quarks.

We present a measurement of the effective weak mixing angle parameter sin 2 θ ℓ eff , in pp → Z/γ * → µ + µ − events at a center of mass energy of 1.96 TeV, collected by the D0 detector at the Fermilab Tevatron Collider and corresponding to 8.6 fb −1 of integrated luminosity. The measured value of sin 2 θ ℓ eff [µµ] = 0.23016 ± 0.00064 is further combined with the result from the D0 measurement in pp → Z/γ * → e + e − events, resulting in sin 2 θ ℓ eff [comb.] = 0.23095 ± 0.00040. This combined result is the most precise measurement from a single experiment at a hadron collider and is the most precise determination using the coupling of the Z/γ * to light quarks. The weak mixing angle θ W is a fundamental parameter of the standard model (SM). It governs the mechanism of spontaneous symmetry breaking of SU (2) × U (1) in which the original vector boson fields W and B 0 are transformed to the physical W ± , Z and γ states. At tree level and in all orders of the on-shell renormalization scheme, the weak mixing angle also relates the W and Z boson masses by sin 2 θ W = 1 − M 2 W /M 2 Z . To include higher order electroweak radiative corrections and allow comparison with experimental measurements, the * with visitors from a Augustana College, Sioux Falls, SD 57197, USA, b  effective weak mixing angle can be defined [1] in terms of the relative strengths of the axial vector and vector couplings, g f A and g f V , of the Z boson to fermions, f : where Q f is the electric charge of the fermions. It is customary to quote the charged lepton effective weak mixing angle parameter sin 2 θ ℓ eff , determined by measurements of observables around the Z boson mass pole (M Z ). The effective mixing angle was precisely measured by the LEP Collaborations and the SLD Collaboration. The combined LEP and SLD result [1] gives a value of sin 2 θ ℓ eff = 0.23153 ± 0.00016 at the energy scale µ = M Z , while the two most precise measurements, the value of 0.23221 ± 0.00029 obtained from the measurement of b-quark forward-backward asymmetry at LEP, and the value of 0.23098 ± 0.00026 obtained from the measurement of the left-right polarization asymmetry at SLD, differ by 3.2 standard deviations. The large difference might indicate a possible bias in one of the measurements and in the world average. A precise independent determination of the effective weak mixing angle is therefore an important test of the SM electroweak breaking mechanism.
At the Tevatron, the weak mixing angle can be measured in the Drell-Yan process pp → Z/γ * → ℓ + ℓ − through a forward-backward charge asymmetry, A F B , defined by A F B = (N F − N B )/(N F + N B ), where N F and N B are the numbers of forward and backward events. Forward (F ) or backward (B) events are defined as those for which cos θ * > 0 or cos θ * < 0, where θ * is the angle between the negatively charged lepton direction and the incoming proton direction in the Collins-Soper frame [2].
For the Z to fermion couplings, both g f A = I f 3 and g f V = I f 3 − 2Q f sin 2 θ W exist, whereas for the photon to fermion couplings there is only a vector coupling. I f 3 is the third component of the weak isospin of the fermion. The parity violation implicit in the forward-backward asymmetry arises from the interference between the vector and axial vector couplings. As the main subprocess for Drell-Yan production is the quark-antiquark annihilation qq → ℓ + ℓ − , A F B depends upon both the couplings to light quarks and the couplings to leptons. The asymmetry can be measured as a function of the invariant mass of the dilepton pair. Since only the vector coupling of the Z boson depends on sin 2 θ W , the information on sin 2 θ W comes from the asymmetry in the vicinity of the Z boson pole. Away from the Z boson mass pole, the asymmetry results from the interference of the axial vector Z coupling and vector photon coupling and depends upon the parton distribution functions (PDFs).
Measurements of sin 2 θ ℓ eff corresponding to the full dataset at the Fermilab Tevatron Collider were performed by the CDF Collaboration using Z/γ * → µ + µ − channel [3] and Z/γ * → e + e − channel [4], and by the D0 Collaboration in the Z/γ * → e + e − channel [5]. The weak mixing angle was also measured at the Large Hadron Collider (LHC) by the ATLAS, CMS and LHCb collaborations [6][7][8]. Because the directions of the initial quarks and anti-quarks in the dominant subprocess qq → Z/γ * → ℓ + ℓ − process are unknown and have to be estimated in pp collisions, the precision of the LHC results is not as good as that of the Tevatron even with higher statistics.
This article reports a measurement of the effective weak mixing angle from the A F B distribution as a function of the dimuon invariant mass using 8.6 fb −1 of data collected by the D0 detector at the Fermilab Tevatron Collider using the Z/γ * → µ + µ − channel. The Z/γ * → µ + µ − measurement is then combined with the D0 Z/γ * → e + e − measurement [5].
The D0 detector comprises a central tracking system, a calorimeter and a muon system [9][10][11]. The central tracking system consists of a silicon microstrip tracker and a scintillating fiber tracker, both located within a 1.9 T superconducting solenoidal magnet and optimized for tracking and vertexing capabilities for detector pseudorapidities of |η det | < 3 [12]. Outside the solenoid, three liquid-argon and uranium calorimeters provide coverage for |η det | < 3.5 for electrons. The muon system is located outside of the calorimeters, providing coverage for |η det | < 2.0. It consists of drift chambers and scintillators and 1.8 T iron toroidal magnets. The solenoid and toroid polarities are reversed every two weeks on average to reduce detector-induced asymmetries. Muons are identified using information from both the tracking system and the muon system. Muon momenta are measured using the tracking system information.
To maximize the event sample, data collected with all triggers are used in this analysis. Events are required to have at least two muon candidates reconstructed in the tracking system and the muon system. Both muon candidates [13] are required to have transverse momentum p T > 15 GeV/c and |η| < 1.8 with at least one muon within |η| < 1.6. The two muon candidates must be isolated from jets in the event by requiring the sum of transverse momenta of tracks in the tracking system or transverse energy in the calorimeter within cones surrounding the muon candidate to be small. Muons must have a track in the tracking system matched with one in the muon system. To suppress backgrounds, the two matched tracks are required to point to the same pp interaction vertex and to have opposite charges. Events with muons nearly back-to-back are removed to reduce cosmic ray background. Events are further required to have a reconstructed dimuon invariant mass 74 < M µµ < 110 GeV/c 2 . The number of events satisfying these requirements is 481,239.
The Monte Carlo (MC) Drell-Yan Z/γ * → µ + µ − sample is generated using leading-order pythia [14] with the NNPDF3.0 [15] PDFs, followed by a geant-based simulation [16] of the D0 detector. Events from randomly selected beam crossings with the same instantaneous luminosity profile as data are overlaid on the simulated events to model detector noise and contributions from the presence of additional pp interactions. The pythia MC samples are used to study the detector's geometric acceptance and the momentum scale and resolution of muons. Separate MC samples are generated for the four different polarity combinations of the solenoid and toroid magnetic fields.
The effective weak mixing angle, which is extracted from A F B as a function of M µµ , depends strongly on the dimuon mass calibration. Therefore, it is critical to have a precise muon momentum measurement and a consistent measured mean value of M µµ for all η, and each muon charge sign q and solenoid polarity S. The D0 muon momentum calibration and resolution smearing procedure [13] is applied to the MC simulation, so as to give agreement of the overall width and peak value of the M µµ distribution with data. However, the muon momentum measurement, especially the scale of the reconstructed muon momentum, still depends on the charge and η of the muons due to imperfect alignment of the detector [17]. Such dependence would translate into a large systematic uncertainty on the A F B measurement. To reduce this dependence, an additional correction, α(q, η, S), to the muon momentum is applied to the data and MC separately. This factor is determined by requiring the mean of the M µµ distribution over the full mass range in each (q, η, S) region to be consistent with the corresponding nominal value obtained from a generator-level MC sample after applying the same kinematic and acceptance cuts as those applied to the data. After the calibration, the mean values of M µµ in data and MC are consistent to within statistical fluctuations. The additional calibration, together with the D0 muon calibration and resolution smearing procedure [13], reduces not only the q-η-S dependence, but also the potential effect from an imperfect modeling on the final state radiation in the pythia generator. The residual difference between data and MC M µµ mean values is propagated to the uncertainty of the weak mixing angle measurement.
Additional corrections and reweightings are applied to the MC simulation to improve the agreement with data. The ratio between the MC and data efficiencies for the muon identification is measured using the tag-andprobe method [13] and applied to the MC distributions as a function of muon η. The simulation is further corrected for higher-order effects not included in pythia by reweighting the MC events at the generator level in two dimensions (p T and rapidity y of the Z boson) to match resbos [18] predictions. In addition, next-to-next-toleading order QCD corrections are applied as a function of Z boson mass [18,19].
The sign of the track matched to the muon is used to determine the charge of the muon and to classify the event as forward or backward. The charge misidentification rate measured in the data is smaller than 0.4%. Since the opposite charge sign requirement is applied in the event selection, the probability of both muons charges to be misidentified, thus transforming a forward event into a backward event or vice versa, is negligibly small.
Background is suppressed by the strict requirements on the muon tracks. The main remaining contribution is from multijet events, in which jets are misidentified as muons, and is estimated from data by selecting events with reversed muon isolation cuts in order to study the shape of the mass distribution of multijet events. The normalization of the multijet background is assumed to be same as that of the selected same sign events after correcting for the presence of the misidentified signal events and the additional background contributions described below. The W +jets background is generated using alpgen [20] interfaced to pythia for showering and hadronization. The Z/γ * → τ τ , di-boson and tt backgrounds, are estimated using pythia. In the dimuon mass range used for the effective weak mixing angle measurement, the multijet background is 0.68% ± 0.68%. An 100% uncertainty is used to safely cover the bias due to corrections for the misidentified signal events. The sum of the W +jets, Z/γ * → τ τ , di-boson (W W and W Z) and tt background is 0.20% ± 0.05%, where the uncertainty is mainly from cross sections of the physics backgrounds.
The effective weak mixing angle is extracted from the background-subtracted A F B spectrum by comparing the data to simulated A F B templates corresponding to different input values of the weak mixing angle. The effective weak mixing angle parameter, here denoted as sin 2 θ p W , corresponds to the input parameter in the calculation from the leading order pythia generator. Higher order corrections are used to convert sin 2 θ p W to sin 2 θ ℓ eff [21]. The templates are obtained by reweighting the twodimensional distribution of the Z boson mass and cos θ * at the generator level to different sin 2 θ p W pythia predictions. The background-subtracted A F B distribution and pythia predictions are shown in Fig. 1. The uncertainties on the fitted sin 2 θ p W , listed in Table I, are dominated by the limited size of the data sample. The systematic uncertainties due to muon momentum calibration and resolution smearing, the estimation of the backgrounds and the efficiency scale factors are themselves also dominated by the limited data samples. The PDF uncertainty is obtained as the standard deviation of the distribution of sin 2 θ p W values given by each of the equal-weighted PDF sets from NNPDF3.0 [15]. The best fit is sin 2 θ p W = 0.22994 ± 0.00059 (stat.) ± 0.00005 (syst.) ± 0.00024 (PDF). Higher order corrections from several sources are introduced. The measured sin 2 θ p W is an average over the leptonic, u-quark and d-quark effective couplings [5]. The mass-scale dependence and complex valued calculation of the weak corrections and the fermion-loop correction to the photon propagator, are not considered in the pythia generator [21]. To obtain the leptonic effective weak mixing angle parameter, we shift the central value by +0.00022, of which +0.00008 is for the u/d-quark correction and +0.00014 is for the complex valued calculation and mass-scale dependence correction [21]. An additional systematic uncertainty of 0.00004 is further introduced [21]. After higher order corrections applied, we get sin 2 θ ℓ eff [µµ] = 0.23016 ± 0.00064. The D0 e + e − measurement [5] and the µ + µ − measurement presented here are used as inputs to a D0 combination result for sin 2 θ ℓ eff . The e + e − measurement in Ref. [5] has been modified for consistency to incorporate the use of additional higher order corrections and the NNPDF3.0 PDFs employed in this letter and in the CDF measurement [4]. The corrected value is sin 2 θ ℓ eff [ee] = 0.23137 ± 0.00047 [21]. The D0 e + e − and µ + µ − measurements agree to within 1.4 standard deviations.
The central values and systematic uncertainties of the e + e − and µ + µ − channels are combined using the inverse of the squares of the statistical uncertainties as weights. The systematic uncertainties are treated as uncorrelated, except the higher order correction uncertainty which is treated as 100% correlated. However, the total combined uncertainty in practice does not depend on whether the systematic uncertainties of the input measurements are taken to be correlated or uncorrelated, because both measurements are dominated by statistical uncertainties. The correlation of the acceptances between e + e − and µ + µ − channels cannot be ignored in treating the PDF uncertainty. Instead of estimating a correlation matrix between sin 2 θ ℓ eff results for these two channels, a combined PDF uncertainty is estimated by first estimating the PDF uncertainty on the average of values for the  e + e − and µ + µ − channels, and then scaling that uncertainty using the linear relation between A F B and sin 2 θ p W calculated using MC.
The combination is: .00007 (syst.) ± 0.00019 (PDF). Table II summarizes the inputs and the results of the combination of the e + e − and µ + µ − measurements. The measured sin 2 θ ℓ eff from D0 and other experiments are compared to the LEP and SLD average in Fig. 2. The D0 combination has an uncertainty close to the precision of the world's best measurements performed by the LEP and SLD Collaborations.
The measurement of sin 2 θ ℓ eff can be used to determine the on-shell value of sin 2 θ W and the mass of the W boson, M W . The relationships between the on-shell sin 2 θ W , sin 2 θ ℓ eff and M W are: where Re[κ e (M 2 Z )] is a form factor whose value at the Z mass pole is 1.037 [22]. The on-shell sin 2 θ W and the indirect determination of M W based on the D0 combined measurement of sin 2 θ ℓ eff are sin 2 θ W = 0.22269 ± 0.00040 M W = 80396 ± 21 MeV/c 2 .
In conclusion, we have measured the effective weak mixing angle parameter from the forward-backward charge asymmetry A F B distribution in the process pp → Z/γ * → µ + µ − at the Fermilab Tevatron Collider. The primary systematic uncertainty arising from muon momentum calibration is reduced by introducing a chargeη-solenoid-dependent calibration. The final result using 8.6 fb −1 of D0 Run II data is sin 2 θ ℓ eff [µµ] = 0.23016 ± 0.00064, which is at the level of the best single channel precision from hadron collider experiments. The D0 combination of the e + e − and µ + µ − measurements is sin 2 θ ℓ eff [comb.] = 0.23095 ± 0.00040, which is the most precise single experiment measurement at hadron colliders and is the most precise result based on the coupling of light quarks to the Z boson.