Measurements of differential Z boson production cross sections in proton-proton collisions at √ s = 13 TeV

Measurements are presented of the differential cross sections for Z bosons produced in proton-proton collisions at √ s = 13 TeV and decaying to muons and electrons. The data analyzed were collected in 2016 with the CMS detector at the LHC and correspond to an integrated luminosity of 35.9 fb−1. The measured fiducial inclusive product of cross section and branching fraction agrees with next-to-next-to-leading order quantum chromodynamics calculations. Differential cross sections of the transverse momentum pT, the optimized angular variable φ∗ η , and the rapidity of lepton pairs are measured. The data are corrected for detector effects and compared to theoretical predictions using fixed order, resummed, and parton shower calculations. The uncertainties of the measured normalized cross sections are smaller than 0.5% for φ∗ η < 0.5 and for pT < 50 GeV. Submitted to the Journal of High Energy Physics c © 2019 CERN for the benefit of the CMS Collaboration. CC-BY-4.0 license ∗See Appendix A for the list of collaboration members ar X iv :1 90 9. 04 13 3v 1 [ he pex ] 9 S ep 2 01 9


Introduction
The measurement of the production of lepton pairs via the Z boson is important for the physics program of the CERN LHC. The large cross section and clean experimental signature allow precision tests of the standard model (SM), as well as constraints on the parton distribution functions (PDFs) of the proton. In addition, a measurement of the Z production process can set stringent constraints on physics beyond the standard model. Moreover, dilepton events are valuable for calibrating the detector and monitoring the LHC luminosity. The Z/γ * → + − process, where is a muon or an electron, is referred to as the Z boson process in this paper.
The Z boson production, identified via its decays into pairs of muons and electrons, can have nonzero transverse momentum, p T , to the beam direction. This is due to the intrinsic p T of the initial-state partons inside the proton, as well as initial-state radiation of gluons and quarks. Measurements of the p T distribution of the Z boson probe various aspects of the strong interaction. In addition, an accurate theoretical prediction of the p T distribution is a key ingredient for a precise measurement of the W boson mass at the Tevatron and LHC.
Theoretical predictions of both the total and the differential Z boson production cross section are available at next-to-next-to-leading order (NNLO) accuracy in perturbative quantum chromodynamics (QCD) [1,2]. Complete NNLO calculations of vector boson production in association with a jet in hadronic collisions have recently become available at O(α 3 S ) accuracy in the strong coupling [3][4][5]. These calculations significantly reduce the factorization (µ F ) and renormalization (µ R ) scale uncertainties, which in turn reduce theoretical uncertainties in the prediction of the p T distribution in the high p T region to the order of one percent. Electroweak corrections are known at next-to-leading order (NLO) and play an important role at high p T [6,7].
However, the fixed-order calculations are unreliable at low p T due to soft and collinear gluon radiation, resulting in large logarithmic corrections [8]. Resummation of the logarithmically divergent terms at next-to-next-to-leading logarithmic (NNLL) accuracy has been matched with the fixed-order predictions to achieve accurate predictions for the entire p T range [9,10]. Fixedorder perturbative calculations can also be combined with parton shower models [11][12][13] to obtain fully exclusive predictions [14][15][16][17]. Transverse momentum dependent (TMD) PDFs [18] can also be used to incorporate resummation and nonperturbative effects.
The Z boson p T and rapidity y Z distributions were previously measured, using e + e − and µ + µ − pairs, by the ATLAS, CMS, and LHCb Collaborations in proton-proton (pp) collisions at √ s = 7, 8, and 13 TeV at the LHC [19][20][21][22][23][24][25][26][27][28][29][30][31][32], and in pp at √ s = 1.8 and 1.96 TeV by the CDF and D0 Collaborations at the Fermilab Tevatron [33][34][35][36][37]. The y Z distribution in pp collisions is strongly correlated with the longitudinal momentum fraction x of the initial partons and provides constraints on the PDFs of proton. The precision of the Z boson p T measurements is limited by the uncertainties in the p T measurements of charged leptons from Z boson decays. The observable φ * η [38,39] is defined by the expression where ∆η and ∆φ are the differences in pseudorapidity and azimuthal angle, respectively, between the two leptons. In the limit of negligible lepton mass rapidity and pseudorapidity are identical. The variable θ * η indicates the scattering angle of the lepton pairs with respect to the beam in the boosted frame where the leptons are aligned. The observable φ * η follows an approximate relationship φ * η ∼ p Z T /m , so the range φ * η ≤ 1 corresponds to p Z T up to about 100 GeV for a lepton pair mass close to the nominal Z boson mass. The measurement resolution of φ * η 4 Event selection and reconstruction The CMS particle-flow event algorithm [49] aims to reconstruct and identify each individual particle in an event, with an optimized combination of all subdetector information. Particles are identified as charged and neutral hadrons, leptons, and photons.
The reconstructed vertex with the largest value of summed physics-object p 2 T is the primary pp interaction vertex. The physics objects are the objects returned by a jet finding algorithm [50,51] applied to all charged particle tracks associated with the vertex plus the corresponding associated missing transverse momentum, which is the negative vector sum of the p T of those jets.
Muons are reconstructed by associating a track reconstructed in the inner silicon detectors with a track in the muon system. The selected muon candidates must satisfy a set of requirements based on the number of spatial measurements in the silicon tracker and in the muon system, and the fit quality of the combined muon track [52,53]. Matching muons to tracks measured in the silicon tracker results in a relative p T resolution of 1% for muons in the barrel and better than 3% in the endcaps, for p T ranging from 20-100 GeV. The p T resolution in the barrel is less than 10% for muons with p T up to 1 TeV.
Electrons are reconstructed by associating a track reconstructed in the inner silicon detectors with a cluster of energy in the ECAL [54]. The selected electron candidates cannot originate from photon conversions in the detector material, and they must satisfy a set of requirements based on the shower shape of the energy deposit in the ECAL. The momentum resolution for electrons from Z → e + e − decays ranges from 1.7% in the barrel region to 4.5% in the endcaps [54].
The lepton candidate tracks are required to be consistent with the primary vertex of the event [55]. This requirement suppresses the background of electron candidates from photon conversion, and lepton candidates originating from in-flight decays of heavy quarks. The lepton candidates are required to be isolated from other particles in the event. The relative isolation for the lepton candidates with transverse momentum p T is defined as where the sums run over the charged and neutral hadrons, and photons, in a cone defined by ∆R ≡ (∆η) 2 + (∆φ) 2 = 0.4 (0.3) around the muon (electron) trajectory. The p PU T denotes the contribution of charged particles from pileup, and the factor 0.5 corresponds to an approximate average ratio of neutral to charged particles [52,54]. Only charged hadrons originating from the primary vertex are included in the sum.
Collision events are collected using single-electron and single-muon triggers that require the presence of an isolated lepton with p T larger than 24 GeV, ensuring a trigger efficiency above 96% for events passing the offline selection. The event selection aims to identify either µ + µ − or e + e − pairs compatible with a Z boson decay. Therefore, the selected Z boson candidates are required to have two oppositely charged same-flavor leptons, muons or electrons, with a reconstructed invariant mass within 15 GeV the nominal Z boson mass [56]. In addition, both leptons are required to have |η| < 2.4 and p T > 25 GeV. To reduce the background from multiboson events with a third lepton, events are rejected if an additional loosely identified lepton is found with p T > 10 GeV.

5 Background estimation
The contribution of background processes in the data sample is small relative to the signal. The background processes can be split into two components, one resonant and the other nonresonant. Resonant multiboson background processes stem from events with genuine Z bosons, e.g., WZ diboson production, and their contributions are estimated from simulation.
Nonresonant background stems from processes without Z bosons, mainly from leptonic decays of W boson in tt, tW, and WW events. Small contributions from single top quark events produced via s-and t-channel processes, and Z → ττ events are also present. The contribution of these nonresonant flavor-symmetric backgrounds is estimated from events with two oppositely charged leptons of different flavor, e ± µ ∓ , that pass all other analysis requirements. The method assumes lepton flavor symmetry in the final states of these processes [57]. Since the W boson leptonic decay branching fractions are well-known, the number of eµ events selected inside the Z boson mass window can be used to predict the nonresonant background in the µµ and ee channels.
A summary of the data, signal, and background yields after the full selection for the dimuon and dielectron final states is shown in Table 1. The contribution of the background processes is below 1%.

Analysis methods
The fiducial region is defined by a common set of kinematic selections applied to both the µ + µ − and e + e − final states at generator level, emulating the selection performed at the reconstruction level. Leptons are required to have p T > 25 GeV and |η| < 2.4, and a dilepton invariant mass |m − 91.1876 GeV| < 15 GeV. The measured distributions, after subtracting the contributions from the background processes, are corrected for detector resolution effects and inefficiencies due to so-called dressed lepton kinematics. The dressed leptons at generator level are defined by combining the four-momentum of each lepton after the final-state photon radiation (FSR) with that of photons found within a cone of ∆R = 0.1 around the lepton. By using this definition, the measured kinematic distributions for Z boson decays to the muon final state and to the electron final state agree to better than 0.1%. The rapidity measurement is restricted to |y Z | < 2.4. The p T and φ * η measurements are restricted to p T < 1500 GeV and φ * η < 50, respectively. There are less than 0.001% of events with p T > 1500 GeV and less than 0.02% with φ * η > 50. The efficiencies for the reconstruction, identification, and isolation requirements on the leptons are obtained in bins of p T and η using the "tag-and-probe" technique [58]. Scale factors are applied as event weights on the simulated samples to correct for the differences in the efficiencies measured in the data and the simulation. The combined scale factor for the reconstruction, identification, and isolation efficiencies for leptons ranges from 0.9 to 1.0, with an uncertainty of about 0.4 (0.7)% for muons (electrons). Momentum scale corrections are applied to the muons and electrons in both data and simulated events [59].
The detector effects are expressed through a response matrix, calculated from the simulated MADGRAPH5 aMC@NLO Z boson sample by associating dressed and reconstructed objects for each observable independently. To account for selection efficiencies and bin migrations, an unfolding procedure based on a least squares minimization with Tikhonov regularization, as implemented in the TUNFOLD framework [60], is applied. The regularization reduces the effect of the statistical fluctuations present in the measured distribution on the high-frequency content of the unfolded spectrum. The regularization strength is chosen to minimize the global correlation coefficient [61].

Systematic uncertainties
The sources of systematic uncertainty in the measurement include the uncertainties in the integrated luminosity, lepton efficiencies (reconstruction, identification, and trigger), unfolding, lepton momentum scale and resolution, and background estimation. A summary of the total uncertainties for the absolute cross section measurements in bins of p Z T , |y Z |, and φ * η is shown in Fig. 1. The uncertainty in the trigger efficiency is included as part of the lepton identification efficiency uncertainty.
Most of the sources of systematic uncertainty are considered fully correlated between bins in all variables. The statistical uncertainties due to the limited size of the data and simulated samples are considered uncorrelated between bins. Some sources of systematic uncertainty have a significant statistical component, such as the statistical uncertainties in the lepton efficiency measurement. This statistical component is considered as uncorrelated between the lepton p T and η bins used for the determination of the lepton efficiencies.
Measurements of the normalized differential cross sections (1/σ)dσ/dp Z T , (1/σ)dσ/d|y Z |, and (1/σ)dσ/dφ * η are also performed. Systematic uncertainties are largely reduced for the normalized cross section measurements. A summary of the total uncertainties for the normalized cross section measurements in bins of p Z T , |y Z |, and φ * η is shown in Fig. 2. The largest source of uncertainty in the inclusive total cross section measurement comes from the measurement of the integrated luminosity and amounts to 2.5% [62]. That uncertainty is relevant only for the absolute cross section measurements. The leading uncertainties for the normalized cross section measurements are related to the momentum scale and the reconstruction efficiency.
A potential bias in the measurement of the reconstruction, identification, and isolation efficiencies with the tag-and-probe technique is estimated by studying the modeling of the background and signal parameterization in the dilepton invariant mass fit. The uncertainty in the modeling of the FSR in the tag-and-probe fits is obtained by weighting the simulation to reflect the difference between a soft-collinear approach [11] and the exact O(α S ) result obtained in PHO-TOS [63]. The tag selection in the tag-and-probe technique can also bias the efficiency measurement. An additional uncertainty is considered by varying the tag selection requirements in the efficiency measurement. The uncertainty in the trigger and lepton reconstruction and selection efficiency is about 0.8 (1.3)% in dimuon (dielectron) final states with a sizable dependence on p Z T , |y Z |, and φ * η . The uncertainty in the dimuon (dielectron) reconstruction efficiency varies between 0.1 (0.2)%        in the central part of the detector and 0.5 (2.5)% at large |y Z | values. The reconstruction efficiency uncertainty also includes the effect of partial mistiming of signals in the forward region in the ECAL endcaps, leading to a one percent reduction in the first-level trigger efficiency. The effect of statistical uncertainties in the measured data-to-simulation scale factors is estimated by varying them within the uncertainties in a series of pseudo-experiments.
The systematic uncertainty due to the choice of the Z boson simulated sample used to determine the response matrices is evaluated by repeating the analysis using POWHEG as the signal sample. The dependence of the measurements on the shapes of p Z T , |y Z |, and φ * η are about 0.3 and 0.5% for the dimuon and dielectron final states, respectively. The uncertainty due to the finite size of the simulated signal sample used for the unfolding reaches about 5% at large p Z T , and the variation with p Z T , |y Z |, and φ * η closely resembles the statistical uncertainty in data. The systematic uncertainties in the absolute cross section measurement arising from the uncertainties in the lepton momentum scale and resolution are at a level of 0.1 (0.5)% for the dimuon (dielectron) final state. The muon and electron momentum scales are corrected for the residual misalignment in the detector and the uncertainty in the magnetic field measurements.
The uncertainty in the nonresonant background contribution is estimated conservatively to be about 5%, leading to an uncertainty in the total cross section measurement below 0.1%. The relative contribution of the nonresonant background processes increases with |y Z | and p T , resulting in an uncertainty of 2% at high p T . The resonant background processes are estimated from simulation and the uncertainties in the background normalization are derived from variations of µ R , µ F , α S , and PDFs [45,48,[64][65][66][67] resulting in uncertainties below 0.1% for the absolute cross section measurement.
When combining the muon and electron channels, the luminosity, background estimation, and modeling uncertainties are treated as correlated parameters, all others are considered as uncorrelated.
Summaries of the uncertainties of the absolute double-differential cross section measurements in p Z T and |y Z | are shown in Figs. 3 and 4. The statistical uncertainties in the data and the systematic uncertainties with a statistical component are large compared to the single-differential cross section measurements. The statistical uncertainty starts to dominate the total uncertainty in the high p Z T regions.

Results
The inclusive fiducial cross section is measured in the dimuon and dielectron final states, using the definition described in Section 6. The combined cross section is obtained by treating the systematic uncertainties, except the uncertainties due to the integrated luminosity and background estimation, as uncorrelated between the two final states. The integrated luminosity and background estimation uncertainties are treated as fully correlated in the combined measurement. The combined cross section is obtained by unfolding simultaneously the dimuon and dielectron final states. The uncertainties are dominated by the uncertainty in the integrated luminosity and the lepton efficiency. A summary of the systematic uncertainties is shown in Table 2. The measured cross sections are shown in Table 3.
The measured differential cross sections corrected for detector effects are compared to various theoretical predictions. The measured absolute cross sections in bins of |y Z | are shown in Fig. 5 for dimuon and dielectron final states, and their combination. The measurement is compared to the predictions using parton shower modeling with both MADGRAPH5 aMC@NLO and POWHEG at NLO accuracy in QCD using the NNPDF 3.0 PDF set. The MADGRAPH5 aMC@NLO prediction includes up to two additional partons at Born level in the matrix element calculations, merged with the parton shower description using the FXFX scheme [73].s A comparison with a fixed order prediction at NNLO accuracy with FEWZ using the NNPDF 3.1 NNLO PDF set is also shown. The MADGRAPH5 aMC@NLO and POWHEG predictions are consistent with the data within the theoretical uncertainties. The FEWZ prediction with the NNPDF 3.1 PDF set is within 5% of the measurement over the entire |y Z | range, which is roughly within the uncertainties. Figure 6 shows the measured absolute cross sections in bins of p Z T for dimuon and dielectron final states, and their combination. The measurement is compared to the predictions using parton shower modeling with both MADGRAPH5 aMC@NLO and POWHEG. A comparison with POWHEG using the MINLO procedure [74] and using the NNPDF 3.1 NLO PDF set is also shown. The predictions are consistent with the measurements within the theoretical uncertainties. The scale uncertainties for the POWHEG-MINLO predictions are evaluated by simultaneously varying µ R and µ F up and down by a factor of two [74]. The POWHEG predictions at high p T , above 100 GeV, disagree with data. The better accuracy of the MADGRAPH5 aMC@NLO and    [75][76][77] and GENEVA [78]. A comparison to the predictions with TMD PDFs obtained [79] from the parton branching method (PB TMD) [80,81] and combined with MADGRAPH5 aMC@NLO at NLO is also shown [82]. The RESBOS predictions are obtained at NNLL accuracy with the CT14 NNLO PDF set and are consistent with the data within the uncertainties at low p T but disagree with the measurements at high p T . The GENEVA predictions include resummation to NNLL accuracy where the resulting parton-level events are further combined with parton showering and hadronization provided by PYTHIA. The GENEVA predictions with the NNPDF 3.1 PDF set and α S (m Z ) = 0.114 are generally consistent with data within the theoretical uncertainties, but disagree with data at p T below 30 GeV. The PB TMD predictions include resummation to NLL accuracy and fixed-order results at NLO, and take into account nonperturbative contributions from TMD parton distributions through fits [79] to precision deep inelastic scattering data. The theoretical uncertainties come from variation of scales and from TMD uncertainties. The PB TMD prediction describes data well at low p T , but deviates from the measurements at high p T because of missing contributions from Z+jets matrix element calculations.
The p Z T distribution for p T > 32 GeV is compared to fixed order predictions, as shown in Fig. 7 (right). A comparison to the MADGRAPH5 aMC@NLO prediction is included as a reference. The data is compared to the FEWZ predictions at NNLO in QCD and to the complete NNLO predictions of vector boson production in association with a jet [4,5]. The comparison is performed for p T > 32 GeV because the Z + 1 jet at NNLO prediction does not exist below that value.
The central values of the µ F and µ R are chosen to be µ F/R = (p Z T ) 2 + m 2 for the FEWZ and Z+1 jet at NNLO predictions. The scale uncertainties are estimated by simultaneously varying the µ F and µ R up and down together by a factor of two. The CT14 [83] NNLO PDF set is used for the Z+1 jet at NNLO predictions. The predictions are consistent with the measurements within the theoretical uncertainties. As can be seen, the Z+1 jet at NNLO calculations significantly reduce the scale uncertainties. The electroweak corrections are important at high p T with expected correction factors of up to 0.9 at p T = 500 GeV and 0.8 at p T = 1000 GeV [6,7]. They are not included in the predictions shown in Fig. 7.

Summary
Measurements are reported of the differential cross sections for Z bosons produced in protonproton collisions at √ s = 13 TeV and decaying to muons and electrons. The data set used corresponds to an integrated luminosity of 35.9 fb −1 . Distributions of the transverse momentum p T , the angular variable φ * , and the rapidity of lepton pairs are measured. The results are corrected for detector effects and compared to various theoretical predictions. The measurements provide sensitive tests of theoretical predictions using fixed-order, resummed, and parton shower calculations. The uncertainties in the normalized cross section measurements are smaller than 0.5% for φ * η < 0.5 and for p Z T < 50 GeV.                         Figure 19: The measured normalized cross sections (left) in bins of p Z T for the 1.6 < |y Z | < 2.4 region. The ratios of the predictions to the data are also shown (right). The shaded bands around the data points (black) correspond to the total experimental uncertainty. The measurement is compared to the predictions with MADGRAPH5 aMC@NLO (square red markers), POWHEG (green triangles), and POWHEG-MINLO (blue circles). The error bands around the predictions correspond to the combined statistical, PDF, and scale uncertainties.

Acknowledgments
We congratulate our colleagues in the CERN accelerator departments for the excellent performance of the LHC and thank the technical and administrative staffs at CERN and at other CMS institutes for their contributions to the success of the CMS effort. 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: [22] ATLAS Collaboration, "Precision measurement and interpretation of inclusive W + , W − and Z/γ * production cross sections with the ATLAS detector", Eur. Phys. J. C 77 (2017) 367, doi:10.1140/epjc/s10052-017-4911-9, arXiv:1612.03016.