Measurement of the forward $Z$ boson production cross-section in $pp$ collisions at $\sqrt{s}$ = 7 TeV

A measurement of the production cross-section for $Z$ bosons that decay to muons is presented. The data were recorded by the LHCb detector during $pp$ collisions at a centre-of-mass energy of 7 TeV, and correspond to an integrated luminosity of 1.0 fb$^{-1}$. The cross-section is measured for muons in the pseudorapidity range $2.0<\eta<4.5$ with transverse momenta $p_{T}>20$ GeV/c. The dimuon mass is restricted to $60<M_{\mu^{+}\mu^{-}}<120$ GeV/c$^{2}$. The measured cross-section is $$\sigma_{Z\rightarrow\mu^{+}\mu^{-}} = (76.0 \pm 0.3 \pm 0.5 \pm 1.0 \pm 1.3) \, \text{pb}$$ where the uncertainties are due to the sample size, systematic effects, the beam energy and the luminosity. This result is in good agreement with theoretical predictions at next-to-next-to-leading order in perturbative quantum chromodynamics. The cross-section is also measured differentially as a function of kinematic variables of the $Z$ boson. Ratios of the production cross-sections of electroweak bosons are presented using updated LHCb measurements of $W$ boson production. A precise test of the Standard Model is provided by the measurement of the ratio $$\frac{\sigma_{W^{+}\rightarrow\mu^{+}\nu_{\mu}} + \sigma_{W^{-}\rightarrow\mu^{-}\bar{\nu}_{\mu}}}{\sigma_{Z\rightarrow\mu^{+}\mu^{-}}} = 20.63\pm0.09\pm0.12\pm0.05,$$ where the uncertainty due to luminosity cancels.


Introduction
Measurements of the total and differential cross-sections for the production of Z bosons in pp collisions test the Standard Model (SM) and provide constraints on parton density functions (PDFs) of the proton. 1 Theoretical predictions for these cross-sections are available at nextto-next-to-leading order (NNLO) in perturbative quantum chromodynamics (pQCD) [1][2][3][4][5].
The dominant uncertainty on these predictions reflects the uncertainties on the PDFs, which vary as functions of the kinematic variables studied. The forward acceptance of the LHCb detector allows the PDFs to be constrained at Bjorken-x values down to 10 −4 [6]. Ratios of the W and Z cross-sections provide precise tests of the SM as the sensitivity to the PDFs in the theoretical calculations is reduced and many of the experimental uncertainties cancel.
LHCb has measured the Z boson production cross-section at √ s= 7 TeV using decays to muon pairs in a data set corresponding to 37 pb −1 [7], and using electron [8] and tau lepton [9] pairs in a data set of 1.0 fb −1 . Production cross-sections of W bosons and the W + /W − cross-section ratio have been measured in the muon channel [10] with the 1.0 fb −1 data set. Similar measurements have also been performed by the ATLAS [11] and CMS [12] collaborations.
The analysis described here is an update of the one described in Ref. [7], using a total integrated luminosity of about 1.0 fb −1 . This increases statistical precision and allows better control of systematic uncertainties, with the result that the total uncertainties on the measurements are significantly reduced. Measurements are performed for muons with transverse momentum p T > 20 GeV/c and pseudorapidity in the range 2.0 < η < 4.5. In the case of Z boson measurements, the invariant mass of the two muons is required to be in the range 60 < M µ + µ − < 120 GeV/c 2 . These kinematic requirements define the fiducial region of the measurement and in this article are referred to as the fiducial requirements. Total cross-sections are presented as well as differential cross-sections as functions of the Z boson rapidity y Z , p T,Z and φ * Z . Here φ * Z is defined as [13] φ * Z ≡ tan (φ acop /2) cosh (∆η/2) .
The angle φ acop = π − |∆φ| depends on the difference ∆φ in azimuthal angle between the two muon momenta, while the difference between their pseudorapidities is denoted by ∆η. The W boson cross-sections given in Ref. [10] are re-evaluated using a more precise determination of the event trigger efficiency. The cross-sections are presented as a function of the η of the muon from the W boson decay. The values presented here supersede those of Ref. [10]. This paper is organised as follows: Section 2 describes the LHCb detector; Sections 3 and 4 detail the selection of Z boson candidates, the Z boson cross-section definition and relevant sources of systematic uncertainty; Section 5 presents the results and Section 6 concludes the paper. Appendices A and B provide tables of differential cross-sections and correlations between these measurements.
shower. Parton showering is performed using Herwig [42,43], with different values for the root mean-square-deviation of the intrinsic k T , and Herwiri [46][47][48], which is based on infrared-improved [49] DGLAP-CS [50][51][52][53][54][55] theory. All calculations are performed with the renormalisation and factorisation scales set to the electroweak boson mass. Scale uncertainties are estimated by varying these scales by factors of two around the boson mass [56]. Total uncertainties correspond to the PDF and α s uncertainties at 68.3% confidence level and scale uncertainties, added in quadrature.

Event selection
Events considered in this analysis are selected by the muon trigger. At the hardware stage, this trigger requires a muon with p T > 1.5 GeV/c and imposes an upper limit of 600 hits in the SPD sub-detector. At the software stage, a muon with p T > 10 GeV/c is required. The muon track must also satisfy additional track quality criteria. Candidate events are selected by requiring a pair of well-reconstructed particles of opposite charge, identified as muons, that also pass the fiducial requirements [7]. In total, 58 466 Z boson candidates are selected and their invariant mass distribution is shown in Fig. 1.  The background contamination in the candidate sample is low. Five background sources are investigated: decays of heavy flavour hadrons, hadron misidentification, Z → τ + τ − decays, tt and W + W − production. Unlike muons from signal, muons arising from decays of heavy flavour hadrons are neither directly produced at the primary interaction vertex, nor are they isolated particles. Using the techniques from Ref. [7], the contribution from this background is estimated from the data to be 227 ± 32 events, which amounts to 0.4% of the candidate sample. The contribution from hadrons that decay in flight or have sufficient energy to traverse the calorimeters and be detected in the muon stations is studied in randomly triggered data, as described in Ref. [7], and determined to be 116 ± 45 events, which is 0.2% of the candidate sample. Other electroweak and QCD backgrounds are estimated using Pythia simulation [25] and normalised to the measured total cross-sections for Z → τ + τ − decays [57,58], tt [59,60] and W + W − [61,62] production. The estimate for these sources is 66 ± 6 events, or 0.1% of the candidate sample. In total, the background is estimated to be 409 ± 56 events, or 0.7% of the candidate sample.
The purity, defined to be the ratio of signal to total candidate events, is ρ=0.993±0.002. It is assumed to be constant as a function of y Z , p T,Z and φ * Z . A systematic uncertainty, discussed later, is assigned to allow for possible inaccuracies in this assumption.

Cross-section measurement
Cross-sections are quoted in the kinematic range defined by the measurement and are corrected for quantum electrodynamic (QED) final-state radiation (FSR) in order to provide a consistent comparison with NLO and NNLO QCD predictions. No corrections are applied for initial-state radiation, electroweak effects, nor their interplay with QED effects. The cross-section in a given bin i of y Z , p T,Z or φ * Z , with both final-state muons inside the fiducial region, is measured as The indices i and j run over the bins of the variable under study. The index k runs over the candidates contributing to bin j. The total muon reconstruction efficiency for an event is given by ε(η µ + k , η µ − k ), which is dependent on the pseudorapidity of the two muons. The matrix U corrects the data for bin migrations due to detector resolution effects. It is determined using an unfolding procedure, which is described in Section 4.4. The efficiency of the requirement on the number of SPD hits in the hardware trigger is denoted by ε GEC . The correction factors for QED final-state radiation are denoted by f FSR (i) and are determined for each bin. The integrated luminosity is denoted by L. Though not entering the expression for the cross-section, an uncertainty due to the beam energy is assigned to all cross-sections. More detail on these individual components is given below. Once the binned cross-sections are determined, they are summed to give the total cross-section The most precise estimate of the total cross-section is obtained by summing the crosssections as a function of rapidity, where uncertainties due to data unfolding are negligible.

Muon reconstruction efficiencies
The data are corrected for efficiency losses due to track reconstruction, muon identification, and trigger requirements. All efficiencies are determined from data using the techniques detailed in Refs. [7,8], where the track reconstruction, muon identification, and muon trigger efficiencies are obtained using tag-and-probe methods on the Z resonance. The tag and probe tracks are required to satisfy the fiducial requirements. The tag must be identified as a muon and be consistent with triggering the event, while the probe is defined so that it is unbiased by the requirement for which the efficiency is being measured. The efficiency is studied as a function of several variables, which describe both the muon kinematics and the detector occupancy. In this analysis the efficiency as a function of muon η is used. The efficiency in each bin of η is defined as the fraction of tag-and-probe candidate events where the probe satisfies a track reconstruction, identification or trigger requirement. The tracking efficiency is determined using probe tracks that are reconstructed by combining hits from the muon stations and the large-area silicon-strip detector. The efficiency depends on η and varies between 89.5% and 98.5% with uncertainties between 0.4% and 1.9%.
The muon identification efficiency is determined using probe tracks that are reconstructed without using the muon system. The efficiency depends on η and varies between 91.3% and 99.2% with uncertainties between 0.1% and 0.9%.
The single-muon trigger efficiency is determined using reconstructed muons as probes. The efficiency depends on η and varies between 71.6% and 82.0% with uncertainties between 0.5% and 1.2%. Since only one muon candidate is required for the event to pass the trigger requirements, the overall trigger efficiency for the analysis is about 95%.
The efficiency to reconstruct any given event is taken to be the product of the three individual efficiencies and determined on an event-by-event basis as a function of muon η, In Equation 4, the efficiency ε is written explicitly in terms of the muon tracking (ε trk ), identification (ε id ) and trigger (ε trg ) efficiencies. The average reconstruction efficiency for the analysis is about 85%. Effects that correlate the efficiency of the two muons are considered, but these are negligible at the current level of precision.

GEC efficiency
The GEC efficiency is the efficiency of the SPD multiplicity limit at 600 hits in the muon trigger. This efficiency is evaluated from data using two independent methods. The first exploits the fact that the SPD multiplicities of single pp interactions are always below the 600 hit threshold. The expected SPD multiplicity distribution of signal events is constructed by adding the multiplicities of signal events in single pp interactions to the multiplicities of randomly triggered events, as in Ref. [7]. The convolution of the distributions extends to values above 600 hits, and the fraction of events that the trigger rejects can be determined. The second method consists of fitting the SPD multiplicity distribution and extrapolating the fit function to determine the fraction of events that are rejected, as in Ref. [8]. Both methods give consistent results and ε GEC = (94.0 ± 0.2)% is used in this analysis. The central value is the estimate from the first method, while the difference between the two estimates contributes to the uncertainty. This efficiency depends linearly on y Z with about 2% variation across the full range. A weaker dependence on both the p T,Z and φ * Z is also observed. Corrections for these effects are made.

Final-state radiation
The FSR correction is taken to be the mean of the corrections calculated with Herwig++ [63] and Pythia8 [25]. The corrections are tabulated in Appendix A and are on average about 2.5%.

Unfolding detector response
To correct for detector resolution effects, an unfolding is performed (matrix U of Equation 2) using LHCb simulation and the RooUnfold [64] software package. The momentum resolution in the simulation is calibrated to the data. The data are then unfolded using the iterative Bayesian approach proposed in Ref. [65]. Other unfolding techniques [66,67] give similar results. Additionally, all unfolding methods are tested for model dependence using underlying distributions from leading order Pythia [24,25], leading order Herwig++ [63], as well as NLO Powheg [41,68,69] showered with both Pythia and Herwig using the Powheg matching scheme. The correction is on average about 2% as a function of p T,Z , while it is significantly less as a function of φ * Z . Only the p T,Z and φ * Z distributions are unfolded. Since y Z is well measured, no unfolding is performed and U is the identity matrix in this case.

Systematic uncertainties
Sources of systematic uncertainty and their effect on the total cross-section measurement are summarised in Table 1. 2 The measured cross-sections as a function of p T,Z and φ * Z have additional systematic uncertainties due to unfolding.
The systematic uncertainty associated with the trigger, identification and tracking efficiencies is determined by re-evaluating all cross-sections with the values of the individual efficiencies increased or decreased by one standard deviation. The full covariance matrix of the differential cross-section measurements is evaluated in this way for each source of uncertainty separately. The covariance matrices for each source are added and the diagonal elements of the result determine the total systematic uncertainty due to reconstruction efficiencies. These vary between 0.5 and 2.0% on the differential cross-section measurements. The systematic uncertainty on the FSR correction is the quadratic sum of two components. The first is due to the statistical precision of the Pythia and Herwig++ estimates and the second is half of the difference between their central values. The latter dominates, with the uncertainties on the differential cross-sections varying between 0.3 and 3%.
The systematic uncertainty on the purity is determined from the number of candidate and background events. In addition, an uncertainty based on the assumption that the purity is the same for all variables and bins of the analysis is evaluated by comparing to cross-section measurements using a binned purity, rather than a global one. The total uncertainties on the differential cross-section measurements due to variations in purity are typically less than 1%.
The GEC efficiency is determined in each bin of y Z , p T,Z and φ * Z . The systematic uncertainty is the sum in quadrature of a component due to the available sample size in each bin and a component due to the 0.2% uncertainty on the integrated number, as determined in Section 4.2. This varies between 0.4 and 4% across the differential measurements.
The systematic uncertainty due to unfolding is estimated by the differences between the differential cross-sections using Bayesian and matrix inversion unfolding techniques. The typical size is 1.5%.
The measurement is specified at centre-of-mass energy √ s = 7 TeV. The beam energy, and consequently the centre-of-mass energy, is known to 0.65% [70]. The sensitivity of the cross-section to the centre-of-mass energy is studied using DYNNLO [71] and a systematic uncertainty of 1.25% is assigned.

Z boson production cross-section
The measured rapidity distribution as shown in Fig. 2 is compared to the prediction from Fewz [30,31] with six different PDF sets. To compare the shapes of the differential cross-sections, measurements and predictions are normalised to the total fiducial crosssection. The normalised differential cross-sections are shown in Figs. 2, 3 and 4. The measurements are compared to the predictions from ResBos [38][39][40] and Powheg [41] where events are interfaced with a parton shower that is simulated using Herwig [42,43]. The p T,Z and φ * Z distributions are well described by ResBos and Powheg, with the central values overestimating the data slightly at low φ * Z and underestimating slightly at high φ * Z . Comparisons to MC@NLO + Herwiri and MC@NLO + Herwig are shown in Figs. 5 and 6. Here Herwig is configured with the root mean-square-deviation of the intrinsic k T distribution set to 0 GeV/c in one instance and 2.2 GeV/c in another. The predictions straddle the measurement at low p T,Z and φ * Z . The high p T,Z and φ * Z tails are underestimated.
The total inclusive cross-section for Z → µ + µ − production for muons with p T > 20 GeV/c in the pseudorapidity region 2.0 < η < 4.5 and the dimuon invariant mass range 60 < M µ + µ − < 120 GeV/c 2 is measured to be where the first uncertainty is statistical, the second systematic, the third is due to the beam energy and the fourth is due to the luminosity. The upper plot of Fig. 7 shows agreement between this measurement and NNLO predictions given by Fewz configured with various PDF sets. The measurement also agrees with the measurements of the Z boson production cross-section performed in the electron [8] and tau lepton [9] channels but with a significantly smaller uncertainty. All binned cross-sections are detailed in Tables 3, 4 and 5 of Appendix A.

Ratios of electroweak boson production cross-sections
The cross-section ratios are defined for muons with p T > 20 GeV/c, 2.0 < η < 4.5 and, in the case of the Z boson cross-section, a dimuon invariant mass between 60 and 120 GeV/c 2 . The ratio of W boson to Z boson production is defined as The separate ratios of W + and W − to Z boson production cross-sections are defined as while the W boson cross-section ratio is defined as Many sources of systematic uncertainty cancel or are reduced in the ratios. As the data sets are identical, the largest single source of uncertainty on the individual cross-sections, due to the luminosity determination, is removed. The trigger used to select both samples is identical and most of the uncertainty on the determination of the trigger efficiency cancels. In particular, the GEC is common to both the W and Z boson analyses and it is expected that the size of the efficiency correction is similar for W and Z events. Crosschecks in data and simulation support this assumption with a precision of approximately 0.3%, which is included as a systematic uncertainty. The GEC efficiency was determined for the previous W boson measurement [10] to be (95.9 ± 1.1)%, whereas an improved precision of (94.0 ± 0.2)% is obtained in the current analysis. Consequently, the W boson cross-section results are updated to benefit from the more precise value. These results are listed in Tables 6, 7 and 8 of Appendix A, along with the muon charge ratios and asymmetries, and supersede those in Ref. [10]. The uncertainties on the tracking and muon identification partially cancel in the ratios of W and Z bosons. The uncertainty on the W + (W − ) cross-section due to beam energy is 1.06(0.91)% and most of this uncertainty also cancels in the ratios. The uncertainties on the purities of the W and Z boson selections are uncorrelated. The FSR uncertainties are taken to be uncorrelated. The sources of uncertainty contributing to the determination of the ratios are summarised in Table 2. The dominant uncertainties on the ratios are due to the purity and the size of the sample. The correlation coefficients used in the uncertainty calculations are tabulated in Appendix B.
The updated W + boson cross-section is σ W + →µ + νµ = (878.0 ± 2.1 ± 6.7 ± 9.3 ± 15.0) pb, where the uncertainties are due to the sample size, systematic effects, the beam energy and the luminosity determination. The updated W − boson cross-section is These measurements are in good agreement with the predictions of NNLO pQCD, as shown in Fig. 7. Using the Z boson cross-section from Section 5.1, electroweak boson cross-section measurements and theoretical predictions, with different parameterisations of the PDFs, are compared in Fig. 8, with contours corresponding to the 68.3% confidence level.
The W to Z boson cross-section ratio is measured as where the first uncertainty is statistical, the second is systematic and the third is due to the beam energy. The charged W to Z boson cross-section ratios are measured as R W + Z = 11.56 ± 0.06 ± 0.08 ± 0.02, R W − Z = 9.07 ± 0.05 ± 0.07 ± 0.03, while the W boson cross-section ratio is measured as R W = 1.274 ± 0.005 ± 0.009 ± 0.002.
These measurements, as well as their predictions, are displayed in Fig. 9. For R WZ and R W + Z , the data are well described by HERA1.5 and JR09, while the values from CT10, MSTW08, NNPDF3.0 and ABM12 are larger than those measured. All PDF sets show good agreement for R W − Z . As previously reported [10], all PDF sets except ABM12 show good agreement for R W . The R WZ and R W ratios are measured with a fractional uncertainty of 0.8%, which is similar both to the precision due to the PDFs on the individual theoretical predictions and to the spread between the predictions. Considering the spread in the different predictions, the experimental measurements are in good agreement with SM predictions and can be used to improve the determination of the PDFs.

Conclusions
A measurement of the forward Z boson production cross-section at √ s = 7 TeV is presented, where the Z bosons are reconstructed in the decay Z → µ + µ − . The total cross-section in the fiducial range of the selection is in agreement with NNLO pQCD calculations. Normalised differential cross-sections as a function of y Z , φ * Z and p T,Z are compared to the predictions of various generators. The increased precision on the determination of the event trigger efficiency motivates a re-evaluation of the recently measured W boson production cross-section. These are presented here and supersede the values given in Ref. [10]. Combining the Z boson cross-section with updated W boson cross-sections measured in a similar fiducial volume allows for precision measurements of electroweak boson cross-section ratios. In particular, the W to Z boson ratio is determined with a relative precision of 0.8%. The measured ratios are consistent with SM predictions but are sensitive to the particular choice of PDF. Consequently, these results are expected to provide significant constraints on PDFs.              Figure 9: Ratios of electroweak boson production R W + Z , R W − Z , R WZ , R W , compared to various theoretical predictions. The shaded (yellow) bands indicate the statistical and total uncertainties on the measurements, which are symmetric about the central value. 3.000 -3.125 7.783 ± 0.099 ± 0.059 ± 0.097 ± 0.134 1.026±0.003 3.125 -3.250 7.094 ± 0.096 ± 0.058 ± 0.089 ± 0.122 1.028±0.004 3.250 -3.375 5.894 ± 0.087 ± 0.049 ± 0.074 ± 0.101 1.026±0.004 3.375 -3.500 4.160 ± 0.073 ± 0.041 ± 0.052 ± 0.072 1.027±0.005 3.500 -3.625 2.896 ± 0.061 ± 0.030 ± 0.036 ± 0.050 1.026±0.005 3.625 -3.750 1.741 ± 0.047 ± 0.023 ± 0.022 ± 0.030 1.021±0.007 3.750 -3.875 0.825 ± 0.032 ± 0.014 ± 0.010 ± 0.014 1.025±0.010 3.875 -4.000 0.321 ± 0.020 ± 0.008 ± 0.004 ± 0.006 1.011±0.015 4.000 -4.250 0.115 ± 0.013 ± 0.006 ± 0.001 ± 0.002 1.018±0.033 4.250 -4.500 − − Table 4: Inclusive differential cross-sections for Z boson production as a function of p T,Z . Uncertainties are due to the sample size, systematic effects, the beam energy and the luminosity.  Table 5: Inclusive differential cross-sections for Z boson production as a function of φ * Z . Uncertainties are due to the sample size, systematic effects, the beam energy and the luminosity.