Measurement of associated production of a \mathrm{W} boson and a charm quark in proton-proton collisions at \sqrt{s}=13\,\text{TeV}
###### Abstract

Measurements are presented of associated production of a boson and a charm quark () in proton-proton collisions at a center-of-mass energy of 13. The data correspond to an integrated luminosity of 35.7 collected by the CMS experiment at the CERN LHC. The bosons are identified by their decay into a muon and a neutrino. The charm quarks are tagged via the full reconstruction of mesons that decay via . A cross section is measured in the fiducial region defined by the muon transverse momentum , muon pseudorapidity , and charm quark transverse momentum . The inclusive cross section for this kinematic range is . The cross section is also measured differentially as a function of the pseudorapidity of the muon from the boson decay. These measurements are compared with theoretical predictions and are used to probe the strange quark content of the proton.

EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

CERN-EP-2018-282 2019/\two@digits7/\two@digits19

CMS-SMP-17-014

Measurement of associated production of a boson and a charm quark in proton-proton collisions at

The CMS Collaboration111See Appendix A for the list of collaboration members

Abstract

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Submitted to the European Physical Journal C

© 2019 CERN for the benefit of the CMS Collaboration. CC-BY-4.0 license

## 1 Introduction

Precise knowledge of the structure of the proton, expressed in terms of parton distribution functions (PDFs), is important for interpreting results obtained in proton-proton () collisions at the CERN LHC. The PDFs are determined by comparing theoretical predictions obtained at a particular order in perturbative quantum chromodynamics (pQCD) to experimental measurements. The precision of the PDFs, which affects the accuracy of the theoretical predictions for cross sections at the LHC, is determined by the uncertainties of the experimental measurements used, and by the limitations of the available theoretical calculations. The flavor composition of the light quark sea in the proton and, in particular, the understanding of the strange quark distribution is important for the measurement of the boson mass at the LHC [1]. Therefore, it is of great interest to determine the strange quark distribution with improved precision.

Before the start of LHC data taking, information on the strange quark content of the nucleon was obtained primarily from charm production in (anti)neutrino-iron deep inelastic scattering (DIS) by the NuTeV [2], CCFR [3], and NOMAD [4] experiments. In addition, a direct measurement of inclusive charm production in nuclear emulsions was performed by the CHORUS experiment [5]. At the LHC, the production of or bosons, inclusive or associated with charm quarks, provides an important input for tests of the earlier determinations of the strange quark distribution. The measurements of inclusive or boson production at the LHC, which are indirectly sensitive to the strange quark distribution, were used in a QCD analysis by the ATLAS experiment, and an enhancement of the strange quark distribution with respect to other measurements was observed [6].

The associated production of bosons and charm quarks in pp collisions at the LHC probes the strange quark content of the proton directly through the leading order (LO) processes and , as shown in Fig. 1. The contribution of the Cabibbo-suppressed processes and amounts to only a few percent of the total cross section.

Therefore, measurements of associated production in pp collisions provide valuable insights into the strange quark distribution of the proton. Furthermore, these measurements allow important cross-checks of the results obtained in the global PDF fits using the DIS data and measurements of inclusive and boson production at the LHC.

Production of in hadron collisions was first investigated at the Tevatron [7, 8, 9]. The first measurement of the cross section of production in collisions at the LHC was performed by the CMS Collaboration at a center-of-mass energy of with an integrated luminosity of 5 [10]. This measurement was used for the first direct determination of the strange quark distribution in the proton at a hadron collider [11]. The extracted strangeness suppression with respect to and quark densities was found to be in agreement with measurements in neutrino scattering experiments. The cross section for production was also measured by the ATLAS experiment at  [12] and used in a QCD analysis, which supported the enhanced strange quark content in the proton suggested by the earlier ATLAS analysis [6]. A subsequent joint QCD analysis [13] of all available data that were sensitive to the strange quark distribution demonstrated consistency between the measurements by the ATLAS and CMS Collaborations. In Ref. [13] possible reasons for the observed strangeness enhancement were discussed. Recent results of an ATLAS QCD analysis [14], including measurements of inclusive and boson production at , indicated an even stronger strangeness enhancement in disagreement with all global PDFs. In Ref. [15], possible reasons for this observation were attributed to the limitations of the parameterization used in this ATLAS analysis [14].

In this paper, the cross section for production is measured in pp collisions at the LHC at using data collected by the CMS experiment in 2016 corresponding to an integrated luminosity of 35.7. The bosons are selected via their decay into a muon and a neutrino. The charm quarks are tagged by a full reconstruction of the charmed hadrons in the process , which has a clear experimental signature. The pion originating from the decay receives very little energy because of the small mass difference between and (1865) and is therefore denoted a “slow” pion . Cross sections for production are measured within a selected fiducial phase space. The cross sections are compared with theoretical predictions at next-to-leading order (NLO) QCD, which are obtained with mcfm 6.8 [16, 17, 18], and are used to extract the strange quark content of the proton.

This paper is organized as follows. The CMS detector is briefly described in Section 2. The data and the simulated samples are described in Section 3. The event selection is presented in Section 4. The measurement of the cross sections and the evaluation of systematic uncertainties are discussed in Section 5. The details of the QCD analysis are described in Section 6. Section 7 summarizes the results.

## 2 The CMS detector

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 (ECAL), and a brass and scintillator hadron calorimeter (HCAL), each composed of a barrel and two endcap sections. Forward calorimeters extend the pseudorapidity coverage provided by the barrel and endcap detectors. Muons are detected in gas-ionization chambers embedded in the steel flux-return yoke outside the solenoid.

The silicon tracker measures charged particles within the pseudorapidity range . It consists of 1440 silicon pixel and 15 148 silicon strip detector modules. For nonisolated particles of and , the track resolutions are typically 1.5% in and 25–90 (45–150) in the transverse (longitudinal) impact parameter [19]. The reconstructed vertex with the largest value of summed physics-object is taken to be the primary interaction vertex. The physics objects are the jets, clustered using the jet finding algorithm [20, 21] with the tracks assigned to the vertex as inputs, and the associated missing transverse momentum, taken as the negative vector sum of the of those jets. Muons are measured in the pseudorapidity range , with detection planes made using three technologies: drift tubes, cathode strip chambers, and resistive-plate chambers. The single muon trigger efficiency exceeds 90% over the full range, and the efficiency to reconstruct and identify muons is greater than 96%. Matching muons to tracks measured in the silicon tracker results in a relative transverse momentum resolution, for muons with up to 100, of 1% in the barrel and 3% in the endcaps. The resolution in the barrel is better than 7% for muons with up to 1 [22]. A more detailed description of the CMS detector, together with a definition of the coordinate system used and the relevant kinematic variables, can be found in Ref. [23].

## 3 Data and Monte Carlo samples and signal definition

Candidate events for the muon decay channel of the boson are selected by a muon trigger [24] that requires a reconstructed muon with . The presence of a high- neutrino is implied by the missing transverse momentum, , which is defined as the negative vector sum of the transverse momenta of the reconstructed particles.

Muon candidates and are reconstructed using the particle-flow (PF) algorithm [25], which reconstructs and identifies each individual particle with an optimized combination of information from the various elements of the CMS detector. The energy of photons is obtained directly from the ECAL measurement. The energy of electrons is determined from a combination of the electron momentum at the primary interaction vertex determined by the tracking detector, the energy of the corresponding ECAL cluster, and the energy sum of all bremsstrahlung photons spatially compatible with originating from the electron track. The muon momentum is obtained from the track curvature in both the tracker and the muon system, and identified by hits in multiple stations of the flux-return yoke. The energy of charged hadrons is determined from a combination of their momentum measured in the tracker and the matching ECAL and HCAL energy deposits, corrected for both zero-suppression effects and the response function of the calorimeters to hadronic showers. Finally, the energy of neutral hadrons is obtained from the corresponding corrected ECAL and HCAL energy.

The meson candidates are reconstructed from tracks formed by combining the measurements in the silicon pixel and strip detectors through the CMS combinatorial track finder [19].

The signal and background processes are simulated using Monte Carlo (MC) generators to estimate the acceptance and efficiency of the CMS detector. The corresponding MC events are passed through a detailed Geant4 [26] simulation of the CMS detector and reconstructed using the same software as the real data. The presence of multiple pp interactions in the same or adjacent bunch crossing (pileup) is incorporated by simulating additional interactions (both in-time and out-of-time with respect to the hard interaction) with a vertex multiplicity that matches the distribution observed in data. The simulated samples are normalized to the integrated luminosity of the data using the generated cross sections. To simulate the signal, inclusive +jets events are generated with MadGraph5_amc@nlo (v2.2.2) [27] using the NLO matrix elements, interfaced with pythia8 (8.2.12) [28] for parton showering and hadronization. A matching scale of 10 is chosen, and the FxFx technique [29] is applied for matching and merging. The factorization and renormalization scales, and , are set to . The proton structure is described by the NNPDF3.0nlo [30] PDF set. To enrich the sample with simulated events, an event filter that requires at least one muon with and , as well as at least one meson, is applied at the generator level.

Several background contributions are considered, which are described in the following. An inclusive +jets event sample is generated using the same settings as the signal events, but without the event filter, to simulate background contributions from  events that do not contain mesons. Events originating from Drell–Yan (DY) with associated jets are simulated with MadGraph5_amc@nlo (v2.2.2) with and set to . Events originating from top quark-antiquark pair () production are simulated using powheg (v2.0) [31], whereas single top quark events are simulated using powheg (v2.0) [32, 33] or powheg (v1.0) [34], depending on the production channel. Inclusive production of , , and bosons and contributions from the inclusive QCD events are generated using pythia8. The CUETP8M1 [35] underlying event tune is used in pythia8 for all, except for the sample, where the
CUETP8M2T4 [36] tune is applied.

The dominant background originates from processes like or , with quarks produced in gluon splitting. In the signal events the charges of the boson and the charm quark have opposite signs. In gluon splitting, an additional quark is produced with the same charge as the boson. At the generator level, an event is considered as a event if it contains at least one charm quark in the final state. In the case of an odd number of quarks, the quark with the highest and a charge opposite to that of the boson is considered as originating from a process, whereas the other quarks in the event are labeled as originating from gluon splitting. In the case of an even number of quarks, all are considered to come from gluon splitting. Events containing both and quarks are considered to be  events, since quarks are of higher priority in this analysis, regardless of their momentum or production mechanism. Events containing no quark and at least one quark are classified as . Otherwise, an event is assigned to the category.

The contribution from gluon splitting can be significantly reduced using data. Events with the same charge sign for both the boson and charm quark, which correlates to the charge sign of the meson, are background, which is due to gluon splitting. Since the gluon splitting background for opposite charge pairs is identical, it can be removed by subtracting the same-sign distribution from the signal.

For validation and tuning of MC event generators using a Rivet plugin [37], the measurement is performed. This requires a particle-level definition without constraints on the origin of mesons. Therefore, any contributions from meson decays and other hadrons, though only a few pb, are included as signal for this part of the measurement.

## 4 Event selection

The associated production of bosons and charm quarks is investigated using events, where and the quarks hadronize into a meson. The reconstruction of the muons from the boson decays and of the candidates is described in detail in the following.

### 4.1 Selection of W boson candidates

Events containing a boson decay are identified by the presence of a high- isolated muon and . The muon candidates are reconstructed by combining the tracking information from the muon system and from the inner tracking system [22], using the CMS particle-flow algorithm. Muon candidates are required to have , , and must fulfill the CMS “tight identification” criteria. To suppress contamination from muons contained in jets, an isolation requirement is imposed: {linenomath}

 (1)

where the sum of PF candidates for charged hadrons (CH), neutral hadrons (NH), photons (EM) and charged particles from pileup (PU) inside a cone of radius is used, and the factor 0.5 corresponds to the typical ratio of neutral to charged particles, as measured in jet production [25].

Events in which more than one muon candidate fulfills all the above criteria are rejected in order to suppress background from DY processes. Corrections are applied to the simulated samples to adjust the trigger, isolation, identification, and tracking efficiencies to the observed data. These correction factors are determined through dedicated tag-and-probe studies.

The presence of a neutrino in an event is assured by imposing a requirement on the transverse mass, which is defined as the combination of and :

 mT≡√2pμT→pmissT(1−cos(ϕμ−ϕ→pmissT)). (2)

In this analysis, is required, which results in a significant reduction of background.

### 4.2 Selection of D∗(2010)± candidates

The mesons are identified by their decays using the reconstructed tracks of the decay products. The branching fraction for this channel is  [38].

The candidates are constructed by combining two oppositely charged tracks with transverse momenta , assuming the and masses. The candidates are further combined with a track of opposite charge to the kaon candidate, assuming the mass, following the well-established procedure of Ref. [39]. The invariant mass of the combination is required to be , where  [38]. The candidate and tracks must originate at a fitted secondary vertex [40] that is displaced by not more than 0.1 cm in both the -plane and -coordinate from the third track, which is presumed to be the candidate. The latter is required to have and to be in a cone of around the direction of the candidate momentum. The combinatorial background is reduced by requiring the transverse momentum and by applying an isolation criterion . Here is the sum of transverse momenta of tracks in a cone of 0.4 around the direction of the momentum. The contribution of mesons produced in pileup events is suppressed by rejecting candidates with a between the muon and the .

The meson candidates are identified using the mass difference method [39] via a peak in the distribution. Wrong-charge combinations with pairs in the accepted mass range mimic the background originating from light-flavor hadrons. By subtracting the wrong-charge combinations, the combinatorial background in the distribution is mostly removed. The presence of nonresonant charm production in the right-charge combinations introduces a small normalization difference of distributions for right- and wrong-charge combinations, which is corrected utilizing fits to the ratio of both distributions.

### 4.3 Selection of W+c candidates

An event is selected as a signal if it contains a boson and a candidate fulfilling all selection criteria. The candidate events are split into two categories: with combinations falling into the same sign (SS) category, and combinations falling into the opposite sign (OS) category. The signal events consist of only OS combinations, whereas the and background processes produce the same number of OS and SS candidates. Therefore, subtracting the SS events from the OS events removes the background contributions from gluon splitting. The contributions from other background sources, such as and single top quark production, are negligible.

The number of events corresponds to the number of mesons after the subtraction of light-flavor and gluon splitting backgrounds. The invariant mass of candidates, which are selected in a window of 1, is shown in Fig. 2, along with the observed reconstructed mass difference . A clear peak at the expected mass and a clear peak around the expected value of [38] are observed. The remaining background is negligible, and the number of mesons is determined by counting the number of candidates in a window of . Alternately, two different functions are fit to the distributions, and their integral over the same mass window is used to estimate the systematic uncertainties associated with the method chosen.

## 5 Measurement of the fiducial W+c cross section

The fiducial cross section is measured in a kinematic region defined by requirements on the transverse momentum and the pseudorapidity of the muon and the transverse momentum of the charm quark. The simulated signal is used to extrapolate from the fiducial region of the meson to the fiducial region of the charm quark. Since the kinematics is integrated over at the generator level, the only kinematic constraint on the corresponding charm quark arises from the requirement on the transverse momentum of meson. The correlation of the kinematics of charm quarks and mesons is investigated using simulation, and the requirement of translates into . The distributions of and in the simulation are shown to reproduce very well the fixed order prediction at NLO obtained, using mcfm 6.8 [16, 17, 18] calculation. The kinematic range of the measured fiducial cross section corresponds to , , and .

The fiducial cross section is determined as:

 σ(W+c)=NselSLintBC, (3)

where is the number of selected events in the distribution and is the signal fraction. The latter is defined as the ratio of the number of reconstructed candidates originating from to the number of all reconstructed . It is determined from the MC simulation, includes the background contributions, and varies between 0.95 and 0.99. The integrated luminosity is denoted by . The combined branching fraction for the channels under study is a product of 0.0049 [41] and = 0.0266 0.0003 [38]. The correction factor accounts for the acceptance and efficiency of the detector. The latter is determined using the MC simulation and is defined as the ratio of the number of reconstructed candidates to the number of generated originating from events that fulfill the fiducial requirements.

The measurement of the cross section relies to a large extent on the MC simulation and requires extrapolation to unmeasured phase space. To reduce the extrapolation and the corresponding uncertainty, the cross section for production is also determined in the fiducial phase space of the detector-level measurement, , , and , in a similar way by modifying Eq. (3) as follows: only the branching fraction is considered and the factor is defined as the ratio between the numbers of reconstructed and of generated candidates in the fiducial phase space after subtraction.

The cross sections are determined inclusively and also in five bins of the absolute pseudorapidity of the muon originating from the boson decay. The number of signal () events in each range of is shown in Fig. 3. Good agreement between the data and MC simulation within the statistical uncertainties is observed.

### 5.1 Systematic uncertainties

The efficiencies and the assumptions relevant for the measurement are varied within their uncertainties to estimate the systematic uncertainty in the cross section measurement. The resulting shift of the cross section with respect to the central result is taken as the corresponding uncertainty contribution. The various sources of the systematic uncertainties in the production cross section are listed in Table 2 for both the inclusive and the differential measurements.

• Uncertainties associated with the integrated luminosity measurement are estimated as 2.5% [42].

• The uncertainty in the tracking efficiency is 2.3% for the 2016 data. It is determined using the same method described in Ref. [43], which exploits the ratio of branching fractions between the four-body and two-body decays of the neutral charm meson to all-charged final states.

• The uncertainty in the branching fraction of the is 2.4% [41].

• The muon systematic uncertainties are 1% each for for the muon identification and isolation, and 0.5% for the trigger and tracking corrections. These are added in quadrature and the resulting uncertainty for muons is 1.2%, which is referred to as the ’muon uncertainty’.

• The uncertainty in the determination of is 1.5%, which is estimated from the difference in using a Gaussian or Crystal Ball fit [44].

• Uncertainties in the modeling of kinematic observables of the generated meson are estimated by reweighting the simulated and distributions to the shape observed in data. The respective uncertainty in the inclusive cross section measurement is 0.5%.

• The uncertainty in the difference of the normalization of the distributions for and combinations (’background normalization’) is 0.5%.

• Uncertainties in the measured are estimated in Ref. [45] and result in an overall uncertainty of 0.9% for this analysis.

• Uncertainties due to the modeling of pileup are estimated by varying the total inelastic cross section used in the simulation of pileup events by 5%. The corresponding uncertainty in the cross section is 2%.

• The uncertainty related to the requirement of a valid secondary vertex, fitted from the tracks associated with a candidate, is determined by calculating the reconstruction efficiency in data and MC simulation for events with and without applying this selection criterion.

• The number of reconstructed candidates after the SS event subtraction is compared for events with or without a valid secondary vertex along with the proximity requirement (, ).

• The difference in efficiency between data and MC simulation is calculated and an uncertainty in the inclusive cross section of % is obtained. Since this variation is not symmetric, the uncertainty is one-sided.

• The PDF uncertainties are determined according to the prescription of the PDF group [30]. These are added in quadrature to the uncertainty related to the variation of in the PDF, resulting in an uncertainty of 1.2% in the inclusive cross section.

• The uncertainty associated with the fragmentation of the quark into a meson is determined through variations of the function describing the fragmentation parameter . The investigation of this uncertainty is inspired by a dedicated measurement of the fragmentation function in electron-proton collisions [46], in which the fragmentation parameters in various phenomenological models were determined with an uncertainty of 10%. In the pythia MC event generator, the fragmentation is described by the phenomenological Bowler–Lund function [47, 48], in the form

 f(z)=1zrcbm2q(1−z)aexp(−bm2⊥/z)c,

with controlled by the two parameters and . In the case of charm quarks, = 1 and are the pythia standard settings in the CUETP8M1 tune, whereas the value of is related to the average transverse momentum of generated in the MC sample. The parameters , and are determined in a fit to the simulated distribution of , where is needed for the normalization. Since the presence of a jet is not required in the analysis, the charm quark transverse momentum is approximated by summing up the transverse momenta of tracks in a cone of 0.4 around the axis of the candidate. The free parameters are determined as and . To estimate the uncertainty, the parameters and are varied within 10% around their central values, following the precision achieved for the fragmentation parameters in [46]. An additional constraint on the upper boundary on the parameter in pythia is consistent with this 10% variation. The resulting uncertainty in the cross section is 3.9%.

### 5.2 Cross section results

The numbers of signal events and the inclusive fiducial cross sections with their uncertainties are listed in Table 4 together with the ratio of . For the differential measurement of the cross section, the numbers of signal events are summarized in Table 6 together with the corrections derived using MC simulations in each bin. The results are presented for , as well as for and for .

The measured inclusive and differential fiducial cross sections of are compared to predictions at NLO () that are obtained using mcfm 6.8. Similarly to the earlier analysis [11], the mass of the charm quark is chosen to be , and the factorization and the renormalization scales are set to the value of the boson mass. The calculation is performed for , , and . In Fig. 4, the measurements of the inclusive cross section and the charge ratio are compared to the NLO predictions calculated using the ABMP16nlo [49], ATLASepWZ16nnlo [14], CT14nlo [50], MMHT14nlo [51], NNPDF3.0nlo [30], and NNPDF3.1nlo [52] PDF sets. The values of the strong coupling constant are set to those used in the evaluation of a particular PDF. The details of the experimental data, used for constraining the strange quark content of the proton in the global PDFs, are given in Refs. [53, 14, 50, 51, 30]. In these references, the treatment of the sea quark distributions in different PDF sets is discussed, and the comparison of the PDFs is presented. The ABMP16nlo PDF includes the most recent data on charm quark production in charged-current neutrino-nucleon DIS collected by the NOMAD and CHORUS experiments in order to improve the constraints on the strange quark distribution and to perform a detailed study of the isospin asymmetry of the light quarks in the proton sea [54]. Despite differences in the data used in the individual global PDF fits, the strangeness suppression distributions in ABMP16nlo, NNPDF3.1nlo, CT14nlo and MMHT14nlo are in a good agreement among each other and disagree with the ATLASepWZ16nnlo result [14].

The predicted inclusive cross sections are summarized in Table 8. The PDF uncertainties are calculated using prescriptions from each PDF group. For the ATLASepWZ16nnlo PDFs no respective NLO set is available and only Hessian uncertainties are considered in this paper. For other PDFs the variation of is taken into account as well. The uncertainties due to missing higher-order corrections are estimated by varying and simultaneously by a factor of 2 up and down, and the resulting variation of the cross section is referred to as the scale uncertainty, . Good agreement between NLO predictions and the measurements is observed, except for the prediction using ATLASepWZ16nnlo. For the cross section ratio /, all theoretical predictions are in good agreement with the measured value. In Table 10, the theoretical predictions for using different PDF sets are summarized. In Fig. 5, the measurements of differential and cross sections are compared with the mcfm NLO calculations and with the signal MC prediction, respectively. Good agreement between the measured cross section and NLO calculations is observed except for the prediction using the ATLASepWZ16nnlo PDF set. The signal MC prediction using NNPDF3.0nlo is presented with the PDF and uncertainties and accounts for simultaneous variations of and in the matrix element by a factor of 2. The cross section is described well by the simulation.

## 6 Impact on the strange quark distribution in the proton

The associated production at 13 probes the strange quark distribution directly in the kinematic range of at the scale of . The first measurement of a fiducial cross section in collisions was performed by the CMS experiment at a center-of-mass energy with a total integrated luminosity of 5 [11]. The results were used in a QCD analysis [10] together with measurements of neutral- and charged-current cross sections of DIS at HERA [55] and of the lepton charge asymmetry in production at at the LHC [11].

The present measurement of the production cross section at 13, determined as a function of the absolute pseudorapidity of the muon from the boson decay and , is used in an NLO QCD analysis. This analysis also includes an updated combination of the inclusive DIS cross sections [56] and the available CMS measurements of the lepton charge asymmetry in boson production at  [11] and at  [57]. These latter measurements probe the valence quark distributions in the kinematic range and have indirect sensitivity to the strange quark distribution. The earlier CMS measurement [10] of production at is also used to exploit the strange quark sensitive measurements at CMS in a joint QCD analysis. The correlations of the experimental uncertainties within each individual data set are taken into account, whereas the CMS data sets are treated as uncorrelated to each other. In particular, the measurements of production at 7 and 13 are treated as uncorrelated because of the different methods of charm tagging and the differences in reconstruction and event selection in the two data sets.

The theoretical predictions for the muon charge asymmetry and for production are calculated at NLO using the mcfm program [16, 17], which is interfaced to applgrid 1.4.56 [58].

Version 2.0.0 of the open-source QCD fit framework for PDF determination xFitter [59, 60] is used with the parton distributions evolved using the Dokshitzer–Gribov–Lipatov–Altarelli–Parisi equations [61, 62, 63, 64, 65, 66] at NLO, as implemented in the qcdnum 17-00/06 program [67].

The Thorne–Roberts [68, 69] general mass variable flavor number scheme at NLO is used for the treatment of heavy quark contributions with heavy quark masses and , which correspond to the values used in the signal MC simulation in the cross section measurements. The renormalization and factorization scales are set to , which denotes the four-momentum transfer for the case of the DIS data and the mass of the boson for the case of the muon charge asymmetry and the measurement. The strong coupling constant is set to . The range of HERA data is restricted to to ensure the applicability of pQCD over the kinematic range of the fit. The procedure for the determination of the PDFs follows the approach used in the earlier CMS analyses [11, 57]. In the following, a similar PDF parameterization is used as in the most recent CMS QCD analysis [57] of inclusive boson production.

The parameterized PDFs are the gluon distribution, , the valence quark distributions, , , the -type, , and -type anti-quark distributions, with () denoting the strange (anti-)quark distribution. The initial scale of the QCD evolution is chosen as . At this scale, the parton distributions are parameterized as:

 xuv(x) =Auv xBuv (1−x)Cuv (1+Euvx2), (4) xdv(x) =Adv xBdv (1−x)Cdv, (5) x¯¯¯u(x) =A¯¯¯u xB¯u (1−x)C¯u (1+E¯¯¯ux2), (6) x¯¯¯d(x) =A¯¯¯d xB¯d (1−x)C¯d, (7) x¯s(x) =A¯s xB¯s (1−x)C¯s, (8) xg(x) =Ag xBg (1−x)Cg. (9)

The normalization parameters , , are determined by the QCD sum rules, the parameter is responsible for small- behavior of the PDFs, and the parameter describes the shape of the distribution as . The strangeness fraction is a free parameter in the fit.

The strange quark distribution is determined by fitting the free parameters in Eqs. (4)–(9). The constraint ensures the same normalization for and densities at . It is assumed that .

In the earlier CMS analysis [11], the assumption was applied. An alternative assumption led to a significant change in the result, which was included in the parameterization uncertainty. In the present analysis, the parameters of the light sea quarks are independent from each other,