Angular coeﬃcients of Z bosons produced in pp collisions at √ s = 8 TeV and decaying to μ + μ − as a function of transverse momentum and rapidity

Measurements of the ﬁve most signiﬁcant angular coeﬃcients, A 0 through A 4 , for Z bosons produced in pp collisions at √ s = 8 TeV and decaying to μ + μ − are presented as a function of the transverse momentum and rapidity of the Z boson. The integrated luminosity of the dataset collected with the CMS detector at the LHC corresponds to 19 . 7 fb − 1 . These measurements provide comprehensive information about the Z boson production mechanisms, and are compared to the QCD predictions at leading order, next-to-leading order, and next-to-next-to-leading order in perturbation theory.

d cos θ * dφ * ∝ (1 + cos 2 θ * ) + A 0 1 2 (1 − 3 cos 2 θ * ) + A 1 sin(2θ * ) cos φ * + A 2 1 2 sin 2 θ * cos(2φ * ) + A 3 sin θ * cos φ * + A 4 cos θ * + A 5 sin 2 θ * sin(2φ * ) + A 6 sin(2θ * ) sin φ * + A 7 sin θ * sin φ * ]. (1) Here, θ * and φ * are the polar and azimuthal angles of the negatively charged lepton in the rest frame of the lepton pair. In this analysis we choose the Collins-Soper (CS) frame [1] to measure the angular coefficients A i , considering the momentum of the beam proton closest in rapidity to the Z boson as the "target momentum" in [1]. The parameters A 0 , A 1 , and A 2 are related to the polarization of the Z boson, whilst A 3 and A 4 are also sensitive to the V-A structure of the couplings of the muons. All angular coeffi-E-mail address: cms-publication-committee-chair@cern.ch. cients vanish as the Z boson transverse momentum q T approaches zero except for A 4 , which is the electroweak parity violation term.
The only previous measurement of four of the angular coefficients was performed by the CDF Collaboration in pp interactions for q T up to 55 GeV [2]. The angular coefficients in pp collisions are expected to differ from those in pp collisions for several reasons. For pp collisions, the Z boson production occurs predominantly via the qq annihilation process, whilst the contribution of the qg Compton process is larger in pp collisions than pp collisions. Using the powheg estimation [3][4][5][6] the fraction of qg process in pp collisions at the LHC is 47%; it is 35% near q T = 0 and increases to ∼80% at q T > 100 GeV. For the qq process in the CS frame, [7][8][9][10], where M Z is the Z boson mass. For the qg Compton process, A 0 = A 2 ≈ 5q 2 T /(M 2 Z + 5q 2 T ) [11]. The relation A 0 = A 2 is known as the Lam-Tung relation [12], reflecting the full transverse polarization of vector boson coupling to quarks, as well as rotational invariance [13]. Processes containing nonplanar configurations (e.g., from higher order multi-gluon emission) smear the transverse polarization, leading to A 2 < A 0 [14]. In contrast to what happens at the Tevatron, the average handedness of Z bosons is nonzero at the LHC, as for the W boson [15][16][17].
The angular coefficients of Z bosons produced in pp collisions at √ s = 8 TeV and decaying to μ + μ − are measured as a function of q T and rapidity y. The data, taken with the CMS detector at the LHC, corresponds to an integrated luminosity of 19.7 fb −1 . The large Z boson event sample collected by the CMS experiment allows precision measurements of the angular distribution for q T < 200 GeV and |y| < 2.1. The coefficients, measured as a function of q T and |y|, are compared with three perturbative QCD predictions by fewz at next-to-next-to-leading order (NNLO) [18], powheg at next-to-leading order (NLO) [3][4][5][6], and MadGraph at leading order (LO) [19].
The central feature of the CMS apparatus is a superconducting solenoid of 6 m internal diameter, providing a magnetic field of 3.8 T. Within the solenoid volume are a silicon pixel and strip tracker, a lead tungstate crystal electromagnetic calorimeter, and a brass and plastic scintillator hadron calorimeter, each composed of a barrel and two endcap sections. Muons are measured in gasionization detectors embedded in the steel flux-return yoke outside the solenoid. Extensive forward calorimetry complements the coverage provided by the barrel and endcap detectors. Muons are measured in the pseudorapidity range |η| < 2.4, with detection planes made using three technologies: drift tubes, cathode strip chambers, and resistive plate chambers. A more detailed description of the CMS detector, together with a definition of the coordinate system and the relevant kinematic variables, can be found in Ref. [20].
Matching muons to tracks measured in the silicon tracker results in a relative p T resolution for muons with 20 < p T <  [24]. The matching between the matrix element calculation and the parton shower is performed with the k T -MLM algorithm [25]. The CTEQ6L1 [26] parton distribution functions (PDF) are used for the event generation. Multiple-parton interactions are simulated by pythia. The powheg generator [3][4][5][6] interfaced with pythia (same version used for MadGraph) and the CT10 PDF set [27] are used as an alternate to test any model dependence in the shapes of the angular distributions.
Each muon candidate is required to be reconstructed in the muon detectors and in the inner tracker, and the global track fit is required to have a reduced χ 2 < 10. The vertex with the highest sum of p 2 T for associated tracks is defined as the primary vertex. The distance of the muon candidate trajectories with respect to the primary vertex must be smaller than 2 mm in the transverse plane and 5 mm along the beam axis. The leading (subleading) muon is required to have p T > 25 (10) GeV and |η| < 2.1 (2.4). In order to suppress background events, the muons are required to be isolated from nearby particles. The relative isolation is calculated as the ratio of the scalar sum of p T of all PF candidates from the same primary vertex, within a cone of R = ( η) 2 + ( φ) 2 < 0.4, and the p T of the muon. For the leading (subleading) muon in p T , the relative isolation must be less than 0.12 (0.5). Oppositely charged muon pairs with an invariant mass in the range 81-101 GeV are selected. In the rare case that more than two muons are selected, the muon pair with invariant mass closest to the Z boson mass is chosen. The muon pair must satisfy |y| < 2.1 since at higher |y| the acceptance varies rapidly. After the event selection, 4.3 × 10 6 events with Z boson candidates remain for |y| < 1.0 and 2.5 × 10 6 events for 1.0 < |y| < 2.1.
A "tag-and-probe" method [33] is used to measure the efficiencies for track reconstruction, trigger, muon isolation, and muon identification in data and simulation. Efficiency corrections are applied as multiplicative scale factors to the simulation values. The efficiency for track reconstruction is measured in bins of η since the p T dependence is weak. The trigger efficiency is determined in bins of p T and η, separately for μ + and μ − . The identification efficiency is measured in bins of p T and η. Since the subleading muon can point in the direction of the hadronic activity, a looser isolation requirement is used and its efficiency is measured as a function of q T , cos θ * , and φ * . The efficiency of the isolation requirement for the leading muon is measured as a function of p T and η of the muon, as detector effects relate to these variables more directly than to the Z boson q T and y.
After event selection, the background contribution ranges from ∼0.1% at low q T to ∼1.5% at high q T . The yields of the backgrounds from tt, τ τ , WW, tW, and W + jets production are estimated from data using lepton flavor universality. Most of these backgrounds typically have two prompt leptons, which may have the same flavor. The W + jets background is flavor asymmetric, but its contribution is small. We assume that the ratio of the number of oppositely charged background μμ and eμ events is the same in data and simulation. We use the ratio of the eμ yields in data and simulation after applying muon and electron selection criteria [33,34] to normalize the simulation to data. The acceptance and the efficiency at the event level vary in cos θ * and φ * , and strongly with q T and y. In order to avoid a bias in the acceptance due to the modeling of the Z boson kinematics, the simulation is reweighted in fine bins of q T and y to match the background-subtracted data distribution. The weights are determined at the reconstruction level and applied at the generator level. The weighting is iterated four times, with negligible change between the second and fourth iteration. The angular coefficients are measured in eight bins of q T and two bins of |y|, by fitting the two-dimensional (cos θ * , φ * ) distribution in data with a linear combination of templates. These templates are built for each coefficient A i by reweighting the simulation at generator level to the corresponding angular distribution, as given in Eq. (1). The templates are based on reconstructed muons, and thereby incorporate the effects of resolution, efficiency and acceptance. A template is also built for the term (1 + cos 2 θ * ) of Eq. (1). An additional template, with shape and normalization fixed, is developed for fitting the backgrounds. A binned maximum-likelihood method with Poisson uncertainties is employed for the fit. The angular coefficients A 5 , A 6 , and A 7 are predicted to be very small; they are set to zero and excluded from the fit. Since A 0 through A 4 are sign invariant in φ * , the absolute value |φ * | is used. The fit is made in 12 × 12 equidistant bins in cos θ * and |φ * |. The statistical uncertainties from the fit are confirmed by comparison with pseudo-experiments.
To test the robustness of the result with respect to the analysis method and trigger effect, the angular coefficients A 0 , A 2 , A 3 , and A 4 are also measured by an independent analysis similar to that reported in Ref. [2], where one-dimensional (1D) templates produced using powheg are fitted to the distributions in cos θ * and |φ * |. The 1D fit analysis is performed iteratively, so as to be unbiased with respect to the assumed templates and to possible correlations between cos θ * and |φ * |. The analysis differs in the triggers, estimation of backgrounds, simulation, and selection criteria. The 1D fit analysis uses a sample that requires a dimuon trigger with asymmetric muon p T thresholds of 17 and 8 GeV. Both results are consistent within their total systematic uncertainties, excluding uncertainties common to both analyses. Some examples of the measured cos θ * and |φ * | distributions from the 1D analysis are given in Fig. 1. The measured and simulated distributions are shown together using the best fit values of the angular coefficients. The shape of the cos θ * distribution changes with q T and |y| because the acceptance and efficiency in cos θ * depend strongly on these two variables. For |φ * |, the shape of the distribution changes moderately with q T , and is almost insensitive to |y|. The comparison of data and simulation shown in Fig. 1 gives confidence that the acceptance and efficiency dependences are correctly modeled in the simulation.
Several sources of systematic uncertainties are taken into account. The most significant source is the muon efficiency that includes the trigger, track reconstruction, isolation, and identification. The statistical uncertainties of the measured efficiency scale factors are taken into account by simulating 500 pseudoexperiments in which the templates are reformed, each time varying the scale factors randomly within the given uncertainty. The systematic uncertainties in the extraction of the efficiency (e.g., background estimates) are also included. Another significant uncertainty stems from the statistical precision of the templates, which is estimated using pseudo-experiments. The pileup uncertainty is estimated by varying the cross section of the minimum bias events by ±5%. The muon momentum bias is measured in data and simulation, and corresponding corrections are applied [35]. The statistical uncertainties in the muon momentum correction factors are propagated to a systematic uncertainty using pseudo-experiments. In addition, a systematic uncertainty is assessed to take into account possible global offsets from the peak position of the Z boson mass. The systematic uncertainties for the background are estimated by varying the normalization scale factor of the eμ sample by 10% and the yields of WZ and ZZ events by 50%. The sta-tistical precision of the iterative reweighting is determined using pseudo-experiments. The difference between the last two iterations is assigned as additional systematic uncertainty. The effect of final-state radiation is taken into account by adding the energy of photons within a cone of radius 0.1 around the muon direction [36]. Weights are applied to the simulation to reflect the difference between a soft-collinear approach and the exact O(α QED ) result and the reconstructed template is rebuilt using the weighted simulation. The difference between templates is used to estimate the systematic uncertainty from final-state radiation. Finally, the acceptance uncertainty, related to the values of A i assumed in the simulation, is estimated by reweighting with the fitted values of A i , and the difference in results is included as a systematic uncertainty. Generally, the statistical uncertainties dominate in the highest bins in q T , whilst the systematic uncertainty in the efficiency tends to be the most important elsewhere.
The results of the q T and |y| dependent measurements of the angular coefficients A 0 to A 4 as well as the difference A 0 − A 2 are presented along with MadGraph, powheg, and fewz (at NNLO) calculations in Figs. 2 and 3. The various systematic uncertainties of the five angular coefficients A 0 to A 4 are presented in Fig. 4. The values and uncertainties of the coefficients are provided in Tables 1 and 2. The PDF sets used in the calculations are CTEQ6L for MadGraph and CT10 for powheg (at NLO) and fewz (at NNLO). The MadGraph predictions for A 4 are systematically higher than those of powheg and fewz because MadGraph uses a weak mixing angle calculated without considering radiative corrections. The measured A 0 and A 2 coefficients agree better with the prediction of MadGraph than with those of powheg and fewz, especially at high q T . At q T = 0, the powheg prediction for A 0 is negative, which is unphysical and has been traced to approximations in  the shower matching algorithm. The fewz prediction is shown for q T > 20 GeV, where the calculations are considered reliable. We find that A 0 (q T ) and A 2 (q T ) are larger in pp collisions than those measured in pp collisions at the Tevatron. The larger contribution from the qg process in pp collisions at the LHC is responsible for this difference. We observe the violation of the Lam-Tung relation ( A 0 = A 2 ) anticipated by QCD calculations beyond leading order [37]. We find that A 0 > A 2 , especially for high q T . In addition, we measure nonzero values of A 1 and A 3 . The comparison of the results for |y| < 1 and 1 < |y| < 2.1 is shown in Fig. 5.
In summary, we presented the five major angular coefficients, A 0 through A 4 , for the production of the Z boson decaying to muon pairs as a function of q T and |y| in pp collisions. These results play an important role in future high-precision measurements, such as the measurement of the mass of the W boson and of the electroweak mixing angle. Some theoretical predictions deviate from the measurements in q T . Further refinements of the theory are needed to achieve a better agreement with the experimental results.  The vertical bars represent the statistical uncertainties and the boxes the systematic uncertainties of the measurement.

Table 1
The five angular coefficients A 0 to A 4 and A 0 − A 2 in bins of q T for |y| < 1.

q T [GeV]
A  Table 2 The five angular coefficients A 0 to A 4 and A 0 − A 2 in bins of q T for 1 < |y| < 2. In addition, we gratefully acknowledge the computing centers and personnel of the Worldwide LHC Computing Grid for delivering so effectively the computing infrastructure essential to our analyses. Finally, we acknowledge the enduring support for the construction and operation of the LHC and the CMS detector provided by the following funding agencies: