Determination of the parton distribution functions of the proton using diverse ATLAS data from pp collisions at root s=7, 8 and 13 TeV

Eur. Phys. J. C (2022) 82:438 https://doi.org/10.1140/epjc/s10052-022-10217-z Regular Article - Experimental Physics Determination of the parton distribution functions of the proton using diverse ATLAS data from pp collisions at √s=7, 8 and 13 TeV ATLAS Collaboration CERN, 1211 Geneva 23, Switzerland Received: 22 December 2021 / Accepted: 12 March 2022 / Published online: 13 May 2022 © CERN for the benefit of the ATLAS collaboration 2022 Abstract This paper presents an analysis at next-to-nextto-leading order in the theory of quantum chromodynamics for the determination of a new set of proton parton distribution functions using diverse measurements in pp collisions at √s=7, 8 and 13 TeV, performed by the ATLAS experiment at the Large Hadron Collider, together with deep inelastic scattering data from ep collisions at the HERA collider. The ATLAS data sets considered are differential cross-section measurements of inclusive W±and Z/γ ∗boson production, W±and Zboson production in association with jets, t¯ tproduction, inclusive jet production and direct photon production. In the analysis, particular attention is paid to the correlation of systematic uncertainties within and between the various ATLAS data sets and to the impact of model, theoretical and parameterisation uncertainties. The resulting set of parton distribution functions is called ATLASpdf21. Contents 1 Introduction ..................... 1 2 Input data sets .................... 2 2.1 Description of data sets ............. 2 2.2 Correlationsofuncertaintieswithinandbetween data sets ..................... 5 3 Theoretical framework ................ 8 3.1 NNLO QCD and NLO electroweak predictions for each data set ................. 8 3.2 Scale uncertainties and sensitivity to NNLO code 10 4 Fit methodology ...................11 5 Results ........................12 5.1 Comparison of PDFs with and without inclusion of correlations between data sets ........12 5.2 Impact of each data set .............16 5.3 Model, theoretical and parameterisation uncertainties ......................21 e-mail: [email protected] 5.4 Combination of uncertainties ..........26 6 Consideration of χ2tolerance and comparison with global fits .......................28 6.1 χ2tolerance ...................28 6.2 Comparison of ATLASpdf21 with global PDFs 30 7 Conclusion ......................38 Appendix ........................39 A Scale uncertainties in W,Zinclusive data ......39 B Comparison of the impact of inclusive jet data at different centre-of-mass energies ............41 C Goodness of the fit: comparison with data sets included in the fit ...................41 D Comparison with extra data sets not included in the fit 51 References ........................54 1 Introduction Preciseknowledgeofthecontent ofprotons, thepartondistribution functions (PDFs), is a necessary ingredient for accurate predictions of both Standard Model (SM) and beyondthe-SM (BSM) cross sections at the Large Hadron Collider (LHC). Searching for BSM effects in the deviations of SM parameters from their predicted SM values [1,2] requires better knowledge of PDFs in the kinematic regions where they are already best known [3,4]. This means that many effects which were previously considered negligible must now be examined seriously. In this paper, particular attention is paid to some of these effects, including theoretical scale uncertainties and the correlations of experimental systematic uncertainties between, as well as within, data sets. With present theoretical knowledge, PDFs can be determined at up to next-to-next-leading order (NNLO) in perturbative quantum chromodynamics (QCD). In order to determine the PDFs to the required precision, a wide coverage of the scale, Q2, and x, the fraction of the proton’s longitudinal momentum carried by the parton participating in the initial interaction, is required. This is facilitated by combining new inputs with older data from various experiments and 123 438 Page 2 of 70 Eur. Phys. J. C (2022) 82 :438 by measurement of different processes to constrain different regions of phase space, which span the kinematic region: 10−5x1 and 1 Q2106GeV2. Knowledge of the PDFs of the proton, in the kinematic region relevant for the LHC, comes mainly from the precision deep inelastic scattering (DIS) data from ep collisions at the HERA collider, which covers a broad range of Q2and x. ThePDFsetHERAPDF2.0wasdeterminedfromHERAdata alone [5] using information from e±pneutral-current (NC) and charged-current (CC) processes. The lower-Q2NC data constrain the low-xsea-quark distribution but are not able to distinguish between quark flavours in the sea at low x, or between the down-type quarks, ¯ dand ¯sat any x.The difference between the NC e+pand e−pcross sections at high Q2, together with the high-Q2CC data, constrains the valence distributions. The Q2dependence measured in the data constrains the gluon distribution. Additional information about the gluon comes from using HERA cross sections measured at various centre-of-mass energies (√s) through the contribution of the longitudinal structure function [6–8]. Diverse ATLAS data sets can be used in addition to the HERA data to constrain the PDFs better. In Ref. [9], high precision measurements of the inclusive differential W± and Z/γ ∗boson cross sections at 7 TeV were added to the HERA data, resulting in the PDF set ATLASepWZ16, which improved on the HERAPDF2.0 set in various respects. Firstly, the strange content of the sea was determined rather than assumed to be a fixed fraction of the light sea. Indeed, compared to previous determinations, the strange sea was found to be enhanced at low x. Secondly, the accuracy of the valence quark distributions for x<0.1 was improved. The effect of adding differential t¯ tdistributions, in the lepton+jets and dilepton channels at 8 TeV, to the HERA data and the inclusive W,Z/γ ∗data was studied in Ref. [10]. Theset¯ tdataarecomplementary tothe W,Z/γ ∗data in their power to constrain PDFs because they are sensitive to the high-xgluon distribution. The resulting PDF set was called ATLASepWZtop18. In Ref. [11], data on the production of Wand Zbosons in association with jets (V+jets) were added to the HERA data and inclusive W,Z/γ ∗data, resulting in the ATLASepWZVjets20 PDF set. The V+jets data are sensitive to partons at higher xthan can be accessed by inclusive Wand Z/γ ∗data and, in particular, they constrain the ¯ dand ¯squarks at higher x. It would clearly be advantageous to combine ATLAS W,Z/γ ∗data, t¯ tdata and V+jets data in a single QCD fit. This is done in the present paper, and additional ATLAS data sets are also included. This paper presents the first comprehensive and comparative NNLO perturbative QCD analysis of a number of ATLAS data sets with potential sensitivity to parton distributions. The additional ATLAS data sets are describedinthefollowing. Firstly, thedata on W[12]production and Z/γ ∗[13] production with 20.2 fb−1at 8 TeV are added, providing further constraints on thevalence PDFs and onthecompositionofthelight-quarksea.Secondly,thedirect photon production differential cross sections with 20.2 fb−1 at 8 TeV and 3.2 fb−1at 13 TeV are added in the form of their ratios [14], whereby many systematic uncertainties cancel out. Thirdly, the t¯ tdifferential cross sections in the lepton+jets channel from 3.2 fb−1at 13 TeV [15] are added and, fourthly, inclusive jet production cross sections with 4.5 fb−1at 7 TeV [16], 3.2 fb−1at 8 TeV [17] and 3.2 fb−1 at 13 TeV [18] are considered. These direct photon data, t¯ t data and inclusive jet data all have impact on the gluon PDF at medium to high x. All ATLAS data sets considered in this study have full information about correlated systematic uncertainties. The analysis considers systematic uncertainty correlations between, as well as within, data sets. This is important now that several of the input data sets have systematic uncertaintiesderivingfromjetmeasurements,sincethese uncertainties are larger than those from the lepton measurements. Theoretical uncertainties, such as scale uncertainties, are also considered, including their correlations between data sets. The resultant PDF set is called ATLASpdf21. The structure of the paper is as follows. Section 2presents the input data sets. Section 3describes the theoretical framework of the fit. Section 4presents the fitting methodology, including the definition of the fit χ2and details of the parameterisation. Section 5presents ATLASpdf21 PDFs and then compares them with a fit in which correlations between data sets are not implemented. The impact of each data set is considered and then model uncertainties and further theoretical uncertainties, including scale uncertainties, are considered. Section 6presents a study of the χ2tolerance and a comparisonwithothermodernPDFsets.Section7givesthesummary and conclusions. Appendix Agives a more detailed exposition of the impact of scale uncertainties for the inclusive W,Z/γ ∗data. The impacts of inclusive jet data at different centre-of-mass energies are compared in Appendix B. Appendices Cand Dcompare the ATLASpdf21 fit predictions with the ATLAS input data sets and with data sets from other experiments, respectively. 2 Input data sets 2.1 Description of data sets The combined e±pNC cross-section measurements of H1 and ZEUS [5] cover a kinematic range of Q2, defined as the negative four-momentum transfer squared, from 0.045 GeV2 to 50,000 GeV2and of Bjorken-x,xBj, which is equal to the fraction of the proton’s longitudinal momentum carried by the struck parton at leading order (LO) in QCD, from 6×10−7to 0.65. The CC cross-section measurements cover 123 Eur. Phys. J. C (2022) 82 :438 Page 3 of 70 438 a kinematic region Q2∼300 GeV2to beyond 104GeV2 and of xfrom ∼0.65 down to ∼10−2.Low-xdata, below xBj =10−5, are excluded from this analysis by requiring Q2>10 GeV2, motivated by the poorer fit observed in this region compared to the rest of HERA data [5], which may reflectthe needforresummationcorrections atlow x[19,20]. Full information about correlated systematic uncertainties is provided. There are 169 sources of correlated systematic uncertainty. Total uncertainties are below 1.5% over the Q2 range of 3 <Q2<500 GeV2and remain below 3% up to Q2=3000 GeV2. The ATLAS W,Z/γ ∗differential cross-section measurements at √s=7 TeV based on an integrated luminosity of 4.6fb −1are used [9]. A combination of the electron and muon decay channels is used. They access a kinematic range whereby the scale is identified with the measured boson mass range, and the xrange is determined by the scale, the proton beam energy and the measured rapidity (y)ranges, such that Q2=m2and x=(Q/√s)e±y, which gives an xrange 0.001 x0.1.1The W±differential cross sections were measured as a function of the W decay lepton pseudorapidity, η, with an experimental precision of 0.6%–1.0%. The Z/γ ∗boson rapidity distribution, yZ,wasmeasuredinthreemassranges:46 <m <66GeV; 66 <m <116 GeV and 116 <m <136 GeV, with the two decay leptons in central-central (c-c) and centralforward (c-f) rapidity ranges,2and experimental uncertainties as low as 0.4% for c-c rapidity and 2.3% for c-f rapidity. There are 131 sources of correlated systematic uncertainty. The low-mass off-peak Z/γ ∗data are not used in the present analysis because they are subject to much larger corrections for non-fixed order, parton shower pT-resummation effects than the other W,Z/γ ∗data [21]. These low-mass data had little impact on the ATLASepWZ16 PDFs. ATLAS has published differential cross sections and forward-backward asymmetries for Z/γ ∗production at √s=8 TeV, based on 20.2fb −1of data [13]. The measurement is presented in terms of three variables for both the c-c and c-f regions of the dilepton rapidity, y.Forthec-c region, there are 12 bins of absolute rapidity ranging from 0.0 to 2.4, in intervals of 0.2, divided into 7 bins of the dilepton mass m ranging from 46 to 200 GeV and 6 bins of the polar angle, θ, of the decay lepton in the Collins–Soper frame [22]. For the c-f region, there are 5 bins of absolute 1Thedefinitionsof the scaleforthe DIS andinclusivevectorbosonprocesses considered so far are conventional. They have been defined here in order to give an idea of the kinematic coverage of the fit. For the other data sets considered, the scale definitions are more conveniently given together with the description of the theoretical predictions in Sect. 3. 2The central-central region requires both decay leptons to have absolute rapidity less than 2.5 and the central-forward region requires one lepton with absolute rapidity less than 2.5 and the other in the range 2.5–4.9. rapidity ranging from 1.2to3.6 and 5 mass bins ranging from 66 to 150 GeV in the same 6 bins of the Collins– Soper angle. The dilepton data were extracted in both the dielectron and dimuon channels up to |y|=2.4, which is extended to |y|=3.6 in the dielectron channel. The dimuon and dielectron data are combined for the c-c region. Measurements are used as cross sections when the acceptance of the lepton fiducial cuts with respect to the (m, |y|, cos θ∗) bin is greater than 95%, such that the leptons are constructed with high accuracy and the cross sections are known to ∼0.5% accuracy. There are 184 cross-section data points which fulfil these requirements and all of them lie in the c-c region of dilepton rapidity. These are used for the central fit. The selection of the bins in m,y and θensures that the NNLO predictions for these high acceptance bins are not highly sensitive to fiducial cuts applied to leptons [23]. The uncertainties of the high-|y|data range from 2% to 33% and the combined central rapidity data reaches a precision ranging from 0.4%, for the Z-mass-peak bins at |y|<1.0, to 1.8% for the high-mass bins at |y|=2.4. There are 330 sources of correlated systematic uncertainty. The remaining lower-acceptance data are used only in cross-checks. They are converted to 188 data points for forward-backward asymmetries across the Collins–Soper angle in order to minimise sensitivity to acceptance cuts. These data lie in both the c-c and c-f dilepton rapidity regions. TheWinclusivedifferentialcrosssections,from20.2fb−1 of data at √s=8TeV[12], are measured as a function of muon pseudorapidity, ημ. Data are provided for W+and W− cross sections separately as well as for the W-asymmetry. The separate W+and W−cross sections are used for the fit, rather than the W-asymmetry. The precision of these crosssectionmeasurements varies from0.8%to1.5%as afunction of ημ. The pseudorapidity covers the same central range as the √s=7 TeV inclusive Wdata. There are 41 sources of correlated systematic uncertainty. The ATLAS differential cross-section measurements of Wproduction in association with jets (W+jets) are based on data recorded during pp collisions at √s=8 TeV, and a total integrated luminosity of 20.2 fb−1, in the electron decay channel only. Each event contains at least one jet with transverse momentum pT>30 GeV and rapidity |y|<4.4. The data are split into W+and W−cross sections. There are 51 sources of correlated systematic uncertainty. As explained inRef. [11] the differential spectra of the transversemomenta of the W+and W−bosons pW Tare fitted in the range 25 < pW T<800 GeV. The data precision ranges from 10 to 25% from low to high pW T[24]. The ATLAS differential cross-section measurements of Zproduction in association with jets [25] are based on data recordedduring pp collisionsat√s=8TeV,andatotalintegrated luminosity of 20.2 fb−1, in the electron decay channel only. The measurement was performed as a function of the 123 438 Page 4 of 70 Eur. Phys. J. C (2022) 82 :438 absoluterapidityofinclusivejets,|yjet|,forseveralbinsofthe transverse momentum within 25 GeV <pjet T<1050 GeV. There are 42 sources of correlated systematic uncertainty. The data precision ranges from ∼10% to ∼30% from low |yjet|and pjet Tto high |yjet|and pjet T. The t¯ tdifferential cross sections were measured at √s= 8 TeV using 20.2fb −1of data in the lepton+jets [26] and dilepton [27] decay channels. The t¯ tcross sections are provided as both normalised and absolute spectra. The absolute spectra are used in the present study since they carry extra information about the normalisation of the cross-sections, which is not input anywhere else in the fit. As detailed in Ref. [10], in the lepton+jets channel the differential spectra that are used are the mass of the t¯ tpair, mt¯ t, and the average top-quark transverse momentum, pt T. Each of these spectra has full information about systematic and statistical bin-to-bin correlations, including statistical correlations between the spectra. There are 55 sources of correlated systematic uncertainty in common between the different spectra. The precision of these lepton+jets data is ∼15% for pt T and ranges from ∼10% to ∼20% from low to high mt¯ t.In the dilepton channel, the spectrum for the rapidity of the t¯ t pair, yt¯ t, is used. The precision of these data ranges from ∼5to∼15% from low to high |yt¯ t|. For these dilepton channel data the correlations are provided as a total covariance matrix. Further studies of these top-quark data are presented in Ref. [10]. The t¯ tdifferential cross sections were measured at √s= 13 TeV using 36 fb−1of data in the lepton+jets [15] channel. Similarly to the t¯ tdata at 8 TeV, the absolute spectra are used. For the t¯ tdata at 13 TeV, two complementary topologies are measured; the ‘resolved’ and ‘boosted’ topologies, which differ in that the decay products of the hadronically decaying top quark are well separated for the resolved case and collimated for the boosted case. The ‘boosted’ topology reacheshigherabsolute rapidityofthe t¯ tpair.Thedifferential spectra used for the PDF fit are the mass of the t¯ tpair, mt¯ t, the average top-quark transverse momentum, pt T, the average top-quark rapidity, yt, and the boosted rapidity of the t¯ tpair, yb t¯ t. These spectra were found to be the most sensitive to the gluon PDF in Ref. [28]. Each of these spectra has full information about systematic and statistical bin-to-bin correlations, including statistical correlations between the spectra.3 3The final two data points, for which pt T>360 GeV, are excluded from the fitted data set. This is because the full statistical correlation matrix could not be inverted when these bins are included. A crosscheck is made in which the fit uses only two of the t¯ t1-D spectra pt T and mt¯ tat 13 TeV, for which the statistical matrix could be inverted, both including and excluding the final two bins of the pt Tspectrum. The resulting PDFs were compared to the PDFs from a fit to the 2Dpt T,mt¯ tspectra. The PDFs of these three fits are almost identical, thus the exclusion of the high pt Tbins does not affect the PDF shapes significantly. There are 88 sources of correlated systematic uncertainty in common between the different spectra and statistical correlations between the spectra are implemented. Double differential spectra in pt Tand mt¯ twere also studied. The precision of these lepton+jets data ranges from 15% to 30% for pt Tand from 15% to 22% for mt¯ t, with the largest uncertainties in the lowest and highest bins. For both of the rapidity spectra the total uncertainty is ∼15%. The ratio of the cross sections for inclusive isolatedphoton production at √s=8 and 13 TeV comes from integrated luminosities of 20.2fb −1and 3.2fb −1respectively. The data are presented as a function of photon transverse energy, Eγ T,forEγ T>125 GeV in bins of photon pseudorapidity, ηγ, for the central region |ηγ|<2.37. Details of the isolation criteria are given in Ref. [14]. Evaluation of experimental uncertainties in the ratio data takes into account the correlations between the data at the two different centreof-mass energies. The dominant source of correlated uncertainty is the photon energy scale. The uncertainty due to this source is much lower when considering the ratio of cross sections. Nevertheless, 25 correlated uncertainty components from this source remain, and there are 21 further uncertainty components which are treated as correlated or uncorrelated asspecified inRef. [14]. Itshould benoted that forthe present fits the combined luminosity uncertainty at 8 and 13 TeV is not used, so the separate luminosity uncertainties may be correlated with those of other data at these centre-of-mass energies. The experimental precision of the ratios ranges from ∼4% at low Eγ Tand |ηγ|to ∼10% at the largest Eγ Tand |ηγ|. The ATLAS measurements of inclusive jet cross sections at √s=7TeV[16], 8 TeV [17] and 13 TeV [18] were considered. The data are presented as a function of jet pjet T,in six bins of absolute rapidity from |yjet|=0.0to|yjet|=3.0. Jets were reconstructed using the anti-ktalgorithm with jet radius parameter R=0.4 and R=0.6 for data at 7 and 8 TeV, and with R=0.4 for data at 13 TeV. Statistical correlation matrices between bins are provided for all jet data sets. However, it is not possible to fit inclusive jet production data at different centre-of-mass energies simultaneously, because the full experimental systematic uncertaintycorrelations betweenthesedata setshavenotbeen fully specified. The data at 8 TeV are selected for the central fit, firstly because a better understanding of correlated systematic uncertainties has been achieved for these data than for the data at 7 TeV, and secondly because they are available for R=0.6, unlike for the data at 13 TeV, and a larger R value is considered more reliable theoretically [29]. Details of the data set are therefore given only for the jet data at 8 TeV. The data at 7 and 13 TeV have similar features, the main difference being in the pjet Trange probed. These data sets, including the alternative choice of jet radius, are used 123 Eur. Phys. J. C (2022) 82 :438 Page 5 of 70 438 Table 1 Summary of all the ATLAS input data sets considered in the QCD fit. It should be noted that: (i) the inclusive Wcross-section data at7and8TeVareusedasW+and W−data separately, (ii) the isolated photon data are used as the ratio of the 13 TeV to 8 TeV cross sections and (iii) the inclusive jet production data at 7 and 13 TeV are used only for a cross-check Data set √s(TeV) Luminosity (fb−1) Decay channel Observables entering the fit Inclusive W,Z/γ ∗[9]7 4.6 e,μcombined η(W), yZ(Z) Inclusive Z/γ ∗[13] 8 20.2 e,μcombined cos θ∗in bins of y,m Inclusive W[12] 8 20.2 μη μ W±+jets[24] 8 20.2 ep W T Z+jets[25] 8 20.2 ep jet Tin bins of |yjet| t¯ t[26,27] 8 20.2 lepton+jets, dilepton mt¯ t,pt T,yt¯ t t¯ t[15] 13 36 lepton+jets mt¯ t,pt T,yt,yb t¯ t Inclusive isolated γ[14] 8, 13 20.2, 3.2 – Eγ Tin bins of ηγ Inclusive jets [16–18] 7, 8, 13 4.5, 20.2, 3.2 – pjet Tin bins of |yjet| in cross-checks. At 8 TeV there are 171 jet data points from 20.2 fb−1of integrated luminosity. The fitted pjet Trange is from 85 GeV to 2.5 TeV. The dominant uncertainties come from the estimation of the jet energy scale and jet energy resolution. Extensive work [17] has been done to understand correlated sources of uncertainty. There are 320 sources of correlated systematic uncertainty, in addition to the luminosity uncertainty, and the jet energy scale’s ηintercalibration uncertainty is split into 250 sources. The total uncertainty in the central pjet Trange, 300 <pjet T<600 GeV, is ∼5% for |yjet|<0.5, rising to ∼10% for |yjet|in the range 2.5–3.0. For low pjet T,85<pjet T<300 GeV, the uncertainty is ∼15% and for high pjet T, 600 <pjet T<2000 GeV, it is ∼50%. All the ATLAS input data sets for the QCD fit are summarised in Table 1. 2.2 Correlations of uncertainties within and between data sets The correlated systematic uncertainties are applied within each data set with the following small number of exceptions. Firstly, for the W+jets,theZ+jets and the inclusive jet spectra at 8 TeV, the systematic uncertainty due to the unfolding procedure is treated as being uncorrelated, both within and between these spectra.4As shown in Ref. [11], this affects the χ2of the fits to V+jets, but has little impact on the fitted PDFs. Similarly, the parton shower systematic uncertainty is decorrelated between the pt Tand mt¯ tspectra in the t¯ tlepton+jets channel, as done in Ref. [10]. It was established that this decorrelation has a minimal effect on the PDFs, while reducing the fit χ2to acceptable levels. This decorrelation 4The two systematic uncertainties in each of the W+jetsand Z+jets spectra related to unfolding (one related to the MC modelling and one to the size of the data samples) are fully decorrelated between spectra and bins within a single spectrum as they contain a large statistical component in both data sets owing to MC simulation statistics. and the aforementioned decorrelation of the unfolding systematic uncertainty in V+jets and inclusive jet data, can be justified because the systematic uncertainties concerned are evaluated from the difference of two Monte Carlo estimates, and thus do not represent well-behaved Gaussian uncertainties. Similar conclusions were reached in a recent study in the MMHT framework [30]. Thirdly, in the inclusive jet data at 8 TeV further decorrelations of such systematic uncertainties, derived from the difference of two Monte Carlo estimates, are considered following Ref. [17]. The experimental systematic uncertainties for the jet energy scale (JES) such as the ‘Flavour Response’, ‘Multi-Jet Balance Fragmentation’, ‘Pile-up Rho Topology’,5and the ‘Non-Perturbative Correction’ uncertainty, are not considered completely correlated between all rapidity bins. Instead, they are split into two or three components as a function of rapidity and pjet T as specified in the various splitting options described in the Appendix of Ref. [17]. For the central fit, the preferred set of splitting options for R=0.6 is used, in which the JES ‘Flavour Response’ is split into three components (see Table 6 of Ref. [17]). In the present paper, this is called ‘Decorrelation Scenario 2’ and it is chosen because it is one of the two preferred options as determined in the analysis in Ref. [17].6Alternative decorrelation scenarios are also considered in Sect. 5.3.1. 5The ‘Flavour Response’ is the systematic uncertainty due to the response difference between quarkand gluon-induced jets, the ‘MultiJet Balance Fragmentation’ represents the jet fragmentation uncertainty in the multi-jet pTbalance and the ‘Pile-up Rho Topology’ takes into account the uncertainty in the density ρof pile-up activity in a given event. 6The decorrelations used at next-to-leading order (NLO) in Ref. [17], include decorrelations of systematic uncertainties due to scale choice. Thesearenotappliedinthe present NNLO analysisbecausescaleuncertainties are much smaller. NNLO scale uncertainties for the inclusive jets are studied in Sect. 5.3.1. 123 438 Page 6 of 70 Eur. Phys. J. C (2022) 82 :438 Correlations of systematic uncertainties between data sets are explained below. The luminosity uncertainties are considered fully correlated for all data sets at the same centreof-mass energy. For the data sets considered in this analysis, systematic uncertainties involving electron and muon measurements are small (<1%), whereas systematic uncertainties involving the jet measurements can be much larger (O(10%)). Moreover, the high-precision inclusive W,Z/γ ∗ differential cross-section measurements at 7 TeV and the inclusive Z/γ ∗triple differential cross-section measurements at 8 TeV both had the electron and muon channel data combined and thus the identities of the systematic uncertainty sources are lost in the combination procedure such that the analysis cannot correlate them with muon and electron uncertainties in other data sets (or between the inclusive Z/γ ∗measurements at 7 and 8 TeV). Thus, correlations of lepton uncertainties between these inclusive Wand Zmeasurements and the other data sets are neglected. This is not a significant limitation in the study of the effect of correlations between data sets since the lepton uncertainties are small compared to the jet uncertainties, as shown in the recent study of the impact of V+jets data on PDF fits [11]. This study reverted to the use of the separate electron and muon channel data for the inclusive W,Z/γ ∗data at 7 TeV in order to correlate the electron systematic uncertainties with those of the W+jets data and the Z+jets data at 8 TeV in the electron channel. These studies produce PDFs which are barely different from those in which the electron/muon combined data are used and lepton systematic uncertainty correlations are not applied between the data sets. Indeed, in Sect. 5.2 of the present analysis, it is shown that the impact of the V+jets data within the present analysis, which uses combined electron/muon data for both inclusive W,Z/γ ∗data at 7 TeV and inclusive Z/γ ∗data at 8 TeV, is very similar to the impact of thesedata in Ref. [11], whereuncombined electron and muon data are used. Thus, the use of the more accurate combined electron and muon data is preferred. In this paper, the correlations of the much larger jetmeasurement uncertainties are considered. The W+jets data and the Z+jets data at 8 TeV have common sources of jet systematic uncertainties, which are listed in Table 2. These sources are considered 100% correlated. Since the t¯ tcross sectionsat both8and 13TeVconsiderdata inthelepton+jets channel,therearesomecommonsourcesofsystematicuncertainty, due to the jet measurements, between these data and the V+jets data. These are also listed in Table 2and are considered 100% correlated between these data sets. This table also lists correlations of some sources of smaller systematic uncertainties, in the muon measurements and diboson, single-top and Z+jets backgrounds, which can be correlated between the t¯ tmeasurements. It should be noted that detailed correlationsbetween the t¯ tdataat 8 TeV in the dilepton channel and other data sets cannot be applied because the correlations for the dilepton channel are supplied as a covariance matrixand theseparate individual sourcesof uncertaintycannot be separated. This does not have a significant effect on the fit since the dilepton data itself has only a small impact on the fit, as seen in Ref. [10]. Correlations of jet systematic uncertainties between the inclusive jet data and the V+jets data and t¯ tdata in the lepton+jets channel have been identified and are listed in Table 2. There is a choice to be made in the implementation of these inter-data-set correlations. The JES ‘Flavour Response’ and ‘Pile-up Rho Topology’ are part of the Jet Decorrelation Scenario 2 chosen for the central fit, and thus they have been split into three components according to jet rapidityand pT[17]. Consequently, thereareno longercorresponding systematic sources in the V+jets data and t¯ tdata. The choice made for the central fit is to correlate only the remaining six sources (in the right-hand column of Table 2) between the data sets. The alternative choice of correlating these two systematic sources with the other data sets but not including them in the jet decorrelation scenario was also investigated. The effect on the resulting PDFs is negligible. There is a further caveat on how the correlations between the inclusive jet data set and the V+jets and t¯ tdata sets are applied, namely that the last two data sets have radius R=0.4 jets, and hence the systematic uncertainties may not be fully correlated. Checks were made using 100% correlation and no correlation, yielding little difference between the resultant PDFs. For the central fit a correlation of 100% is used. The systematic uncertainties of the inclusive jet data at different beam energies are correlated with each other, but understanding these correlations in detail is non-trivial. In the present study, these data sets are fitted separately and results are compared. As already stated the data at 8 TeV are used for the central fit. The measurement of the direct-photon production ratio already considered correlations between the data at 8 TeV and 13 TeV. The photon energy scale is the largest correlated systematic uncertainty between the two measurements. There are no further important correlations with the other data sets. The luminosity uncertainties of the data at 8 TeV and 13 TeV are not combined for the present study. Instead, the8 TeV luminosityiscorrelated with thatof the other8TeV data sets and the 13 TeV luminosity is correlated with that of the other 13 TeV data sets. 123 Eur. Phys. J. C (2022) 82 :438 Page 7 of 70 438 Table 2 Systematic uncertainties that are correlated between the W+jets data at 8 TeV, Z+jets data at 8 TeV, t¯ tlepton+jets data at 8 TeV, t¯ tlepton+jets data at 13 TeV and inclusive jets data at 8 and 13 TeV are listed. The names of the systematic uncertainties are those found in the HEPData entries [31]. Entries in the same row are taken as 100% correlated for the V+jets and t¯ t lepton+jets data, which all have jet radius R=0.4. Different degrees of correlation are considered for the inclusive jet data at R=0.6, because of the differing choice of jet radius. Where entries are omitted, that systematic uncertainty does not exist for that data set (denoted by ‘–’). The luminosity uncertainty of data sets at the same centre-of-mass energy are also fully correlated. The JES ‘Flavour Response’ and JES ‘Pile-up Rho topology’ are considered fully correlated with other data sets only for cross-checks. They are not correlated for the central fit because they are part of the Decorrelation Scenario 2 which is applied to the inclusive jet measurements, as explained in the text. For this reason they are marked with the symbol ∗ Systematic uncertainty 8 TeV W+jets 8TeV Z+jets 8TeVt¯ tlepton+jets 13 TeV t¯ tlepton+jets 8 TeV inclusive jets Jet flavour response JetScaleFlav2 Flavor Response flavres-jes JET29NP JET Flavour Response syst JES Flavour Response∗ Jet flavour composition JetScaleFlav1Known Flavor Comp flavcomp-jes JET29NP JET Flavour Composition syst JES Flavour Comp Jet punchthrough JetScalepunchT Punch Through punch-jes – syst JES PunchThrough MC15 Jet scale JetScalePileup2 PU OffsetMu pileoffmu-jes – syst JES Pileup MuOffset – PU Rho pileoffrho-jes JET29NP JET Pileup RhoTopology syst JES Pileup Rho topology∗ JetScalePileup1 PU OffsetNPV pileoffnpv-jes JET29NP JET Pileup OffsetNPV syst JES Pileup NPVOffset – PU PtTerm pileoffpt-jes JET29NP JET Pileup PtTerm syst JES Pileup Pt term Jet JVF selection JetJVFcut JVF jetvxfrac – syst JES Zjets JVF B-tagged jet scale – btag-jes JET29NP JET BJES Response – – Jet resolution – jeten-res JET JER SINGLE NP – – Muon scale – – mup-scale MUON SCALE – Muon resolution – – muonms-res MUON MS – Muon identification – – muid-res MUON ID – Diboson cross section – – dibos-xsec Diboson xsec – Z+jets cross section – – zjet-xsec Zjets xsec – Single-tcross section – – singletop-xsec st xsec – 123 438 Page 8 of 70 Eur. Phys. J. C (2022) 82 :438 3 Theoretical framework 3.1 NNLO QCD and NLO electroweak predictions for each data set The present analysis uses the xFitter framework [32–34]. This program interfaces to theoretical calculations directly or uses fast interpolation grids to make theoretical predictions for the considered processes. The program MINUIT [35]is used for the minimisation. Each step is cross-checked with an independent fit program [36]. This section describes how these predictions are obtained for each data set. For the DIS processes the light-quark coefficient functions are calculated to NNLO in QCD theory as implemented in QCDNUM [37]. The contributions of charm and bottom quarks are calculated in the general-mass variable-flavournumber scheme of Refs. [38,39], known as the optimised TRVFN scheme. The renormalisation and factorisation scales for the DIS processes are taken to be the conventional choices, μr= μf=Q2, where Q2is the negative four-momentum transfer squared as already stated. For the DIS data electroweak (EW) effects are already unfolded to leading order, such that LO-EW corrections need not be applied, apart from the running of α. NLO-EW corrections are not well defined and no such corrections are applied to the DIS data. Inthepresentanalysis,thephotonPDFwithintheprotonis not accounted for and thus data sets with substantial sensitivitytothephotonPDF,suchasveryhigh-massDrell–Yandata, are excluded. It is estimated [40] that the photon takes only ∼ 0.3% of the momentum of the proton in our kinematic range. The DIS processes are the only ones for which fast NNLO QCD predictions can be made analytically, such that they can be used in an iterative fit. For the LHC processes, NNLO calculations are too time-consuming and thus fast interpolation grids are used. However, currently only the t¯ tproduction processes in the lepton+jets channel have grids available for NNLO calculations. For the remainder of the data sets the grids are available only at NLO and K-factors are used to correct the QCD predictions for the differential cross sections from NLO to NNLO. These K-factors are calculated from the ratio of the NNLO to NLO cross sections, with the same cuts as the data, using a fixed input PDF. These Kfactors have very little dependence on the choice of input PDF for this calculation, and iteration using the output PDF of the fit is not necessary. The calculations used to construct the interpolation grids are usually to leading order (LO) in the EW part of the calculation. Correction from LO to NLO in EW predictions is also applied by a K-factor technique. For some data sets, this is applied together with the QCD predictions’ correction from NLO to NNLO while for others it is a separate calculation. Details for each data set are given below and summarised in Table 3. For data sets measuring Table 3 Summary of code used for NLO, NNLO QCD and LO, NLO EW predictions for ATLAS data as applied in the QCD fit. For most data sets, predictions are provided at NLOinQCDand LO in EW in the form of fast interpolation grids and are corrected to NNLO QCD and NLO EW by K-factors. For the t¯ tlepton+jets channel the grids are calculated at NNLO in QCD directly, so no entry appears in the column ‘NLO QCD code’ Data set NLO QCD code LO EW code NNLO QCD code NLO EW code Inclusive W,Z/γ ∗[9]MCFM MCFM DYNNLO1.5, FEWZ3.1.b2 DYNNLO1.5, FEWZ3.1.b2 Inclusive Z/γ ∗[13]MCFM MCFM NNLOjet NNLOjet Inclusive W[12]MG5_aMC@NLO 2.6.4 MG5_aMC@NLO 2.6.4 DYNNLO1.5 DYNNLO1.5 W±+jets[24]N jetti Njetti Njetti Sherpa Z+jets[25]Ref.[52]Ref.[52]Ref.[52]Sherpa t¯ t(lepton+jets) [26]– Ref.[53]Ref.[53]Ref.[56] t¯ t(dilepton) [27]MCFM MCFM Ref. [28]Ref.[56] t¯ t[15]– Ref.[53]Ref.[53]Ref.[56] Inclusive isolated γ[14]MCFM MCFM Ref. [58]Ref.[59] Inclusive jets [16–18]NLOjet++ NLOjet++ NNLOjet Ref. [64] 123 Eur. Phys. J. C (2022) 82 :438 Page 9 of 70 438 final-state jets, there are also non-perturbative corrections to account for hadronisation. These are also applied using a K-factor technique as specified below.7 The Wand Zinclusive cross sections at 7 TeV are calculated at fixed order, to NNLO in QCD theory and to NLO in EW theory, as described in Ref. [9]. The results obtained from DYNNLO1.5 [41,42] and FEWZ3.1.b2 [43,44]are compared. A small difference between the NNLO predictions of FEWZ and DYNNLO of up to ∼1% is observed for the Wand Z-peak data. The sensitivity of NNLO programs to the fiducial cuts applied to the data was highlighted again recently [23]. This is considered further in Sect. 3.2.The scales for the Drell–Yan (DY) processes are taken to be the decay dilepton invariant mass appropriate to the centre of the mass bin in the NC case and the W-boson mass in the CC case.ThexFitterpackage usesthe APPLgrid code[45]interfaced to the MCFM program [46,47] for fast calculation of the differential Wand Z/γ ∗boson cross sections at NLO in QCD theory and LO in EW theory, and a K-factor technique is used to correct the NLO QCD predictions to NNLO and the LO EW predictions to NLO. These K-factors are within 1–2% of unity for the W±and Z/γ ∗in the c-c dilepton rapidity region and within ∼4% of unity in the c-f dilepton rapidity region. The high-mass sideband of Z/γ ∗production is also subject to background from photon-induced dilepton production, which was estimated using the MRST2004qed photon PDF [48] and subtracted from the data. Compared to the signal, the size of this background is ∼(1.5±0.5)%. The QCD predictions for the triple differential distributions of Z/γ ∗production at 8 TeV are made to NLO by MCFM interfaced to APPLgrid, and K-factors are applied for NNLO QCD and NLO EW effects using NNLOjet. These K-factors are within ∼3% of unity. The renormalisation and factorisation scales for these data are taken to be the decay dilepton invariant mass appropriate to the centre of the mass bin. For the present study, Particle Data Group (PDG) values [49] are used for the electroweak parameters in the Gμscheme, in particular the value of sin2θWis sin2θW=0.23127. The QCD predictions for the Wcross sections at 8 TeV are calculated to NLO and the EW predictions to LO using MG5_aMC@NLO 2.6.4 interfaced to APPLgrid via aMCfast 1.3.0, and the NNLO QCD + NLO EW K-factors are calculated using DYNNLO1.5. These K-factors are within 1%–2%ofunity.Therenormalisationandfactorisationscales for these data are taken to be the W-boson mass. The predictions for W+jets production at 8 TeV are obtained at fixed order, up to NNLO in QCD theory and at LO in EW theory, using the Njetti program [50]. Out7Note that the ATLAS data are all at much higher scales and the charm andbottomquarksaretreated in the5-flavour zero-massvariable flavour number scheme, for which the top is an additional heavy quark. puts from the APPLgrid code are used for fast calculation at NLO in QCD theory, and K-factors from the aforementioned Njetti prediction are used to correct this calculation to NNLO. The renormalisation and factorisation scales are set tom2 W+(pjet T)2,wherethesecondterminthesquareroot is a sum over the transverse momentum of each jet. Additionally, all non-perturbative corrections, such as for hadronisation effects, are included in the K-factors. Further K-factors are applied for NLO EW corrections as computed by the authors of Sherpa [51]. As in the previous ATLAS analysis of V+jets data [11], the lowest pjet Tbin, pjet T<25 GeV, is not used, since the calculation is effectively only at NLO for such low pT. The NNLO QCD predictions for Z+jets production at 8 TeV were calculated by the authors of Ref. [52]. The renormalisation and factorisation scales are set to μr=μf= 1 2m2  +p2 T, +pT,partonswhere m is the invariant massoftheelectronpair, pT, isthetransversemomentumof the electron pair and pT,partons is the sum of the transverse momenta of the outgoing partons. Four sets of K-factors are provided: the first corrects the NLO QCD predictions to NNLO, the second gives corrections for non-perturbative effects, the third corrects from LO to NLO in QED and the fourth applies NLO EW corrections (excluding QED radiation)as computed by Sherpa 2.2.10. The K-factorsfor both W+jets and Z+jets production are typically within 10% of unity. Uncertainties in the non-perturbative corrections are supplied for the Z+jets predictions and these are applied as correlated systematic uncertainties. Their impact is very small. Such uncertainties are not supplied for the W+jets predictions. The NNLO QCD predictions for t¯ tproduction at 8 TeV were calculated by the authors of Ref. [53] and are available in the form of fast interpolation grids, APPLgrid [45] or fastNLO [54,55], for the data in the lepton+jets channel. The predictions for mt¯ tare given for renormalisation and factorisation scales equal to HT/4, where HT= m2 t+(pt T)2+m2 t+(p¯ t T)2,and the predictionsfor pt Tare given for scales equal to mT/2, where mT=m2 t+(pt T)2 and mt=173.3 GeV is the pole mass. Non-perturbative corrections and their uncertainties are supplied with the data and theseuncertaintiesareappliedascorrelatedsystematicuncertainties. For the dilepton channel, APPLgrid is interfaced to MCFM to produce NLO grids, and a K-factor technique is used to correct NLO predictions to NNLO, using K-factors from Ref. [28]. The scales used for the predictions are equal to HT/4. The NNLO/NLO K-factors are ∼7% above unity and constant for yt¯ t. The calculations are for zero width of the top mass. NLO EW corrections for the spectra are also considered, using the additional K-factors given in Ref. [56]. 123 438 Page 16 of 70 Eur. Phys. J. C (2022) 82 :438 x 3− 10 2− 10 1− 10 ref ) 2 sQ,x( )/x 2 sQ,x( x 0.8 1 1.2 1.4 1.6 1.8 Q2= 10000 GeV2 ATLASpdf21, T=1 No uncertainty correlation between data sets ATLAS x 3− 10 2− 10 1− 10 ref ) 2 )/xg(x,Q 2 xg(x,Q 0.9 0.95 1 1.05 1.1 Q2= 10000 GeV2 ATLASpdf21, T=1 No uncertainty correlation between data sets ATLAS Fig. 4 Ratios of ATLASpdf21 PDFs extracted from a fit including correlations of systematic uncertainties between data sets to those extracted from a fit in which only the luminosity uncertainties for each centre-of-mass energy are correlated between data sets, at scale Q2=10,000 GeV2. Only experimental uncertainties are shown, evaluated with tolerance T=1. Left: xs. Right: xg 5.2 Impact of each data set In this section the impact of each data set is considered. Only experimental uncertainties with tolerance T=1are shown for these comparisons. Full uncertainties including model and parameterisation variations are considered for the ATLASpdf21 fit in Sect. 5.3. 5.2.1 Impact of W,Z inclusive data Figure 5shows the ratio Rsfor the ATLASpdf21 fit and compared with a fit in which the inclusive W,Zdata at 7 and 8 TeV are removed (left-hand plot), as well as to a fit in which only W,Zdata at 8 TeV are removed (right-hand plot). It is clear that without W,Zinclusive data the ratio Rscannot be x 3− 10 2− 10 1− 10 ) 2 dQ,x() s(x /) x(s+ 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 ATLAS No 7, 8 TeV W, Z ATLASpdf21, T=1 Q2= 1.9GeV2 x 3− 10 2− 10 1− 10 ) 2 dQ,x()u+s) (x/ x(s+ 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Q2= 1.9 GeV2 No 8 TeV W, Z ATLAS ATLASpdf21, T=1 u+ Fig. 5 The PDF ratio Rs=x(s+¯s)/x(¯u+¯ d)from ATLASpdf21 compared with Rsfor fits not including some of the W,Zdata sets. Only experimental uncertainties are shown, evaluated with tolerance T=1. Left: not including W,Zdata at both 7 and 8 TeV. Right: not including W,Zdata at 8 TeV 123 Eur. Phys. J. C (2022) 82 :438 Page 17 of 70 438 x 3− 10 2− 10 1− 10 V /xd V xdδ 0.8 0.9 1 1.1 1.2 Q2= 1.9 GeV2 No 7, 8 TeV W, Z ATLAS ATLASpdf21, T=1 x 3− 10 2− 10 1− 10 V /xd V xdδ 0.8 0.9 1 1.1 1.2 ATLAS Q2= 1.9 GeV2 ATLASpdf21, T=1 No 8 TeV W, Z x 3− 10 2− 10 1− 10 xg/xg δ 0.8 0.9 1 1.1 1.2 ATLAS ATLASpdf21, T=1 No 7, 8 TeV W, Z Q2= 1.9 GeV2 x 3− 10 2− 10 1− 10 xg/xg δ 0.8 0.9 1 1.1 1.2 ATLAS ATLASpdf21, T=1 No 8 TeV W, Z Q2= 1.9 GeV2 Fig. 6 Relative uncertainties in ATLASpdf21 xdvand xg compared with fits not including some of the W,Zdata sets. Only experimental uncertainties are shown, evaluated with tolerance T=1. Top: xdv uncertainties, (left) not including inclusive W,Zdata at both 7 and 8 TeV, (right) not including inclusive W,Zdata at 8 TeV. Bottom: xg uncertainties, (left) not including inclusive W,Zdata at both 7 and 8 TeV, (right) not including inclusive W,Zdata at 8 TeV determined reliably. Once W,Zdata at 7 TeV are input the determination improves considerably, but the inclusive W,Z data at 8 TeV still add information. Incontrast,thevalenceandgluonPDFsarestillreasonably well determined without any W,Zdata but the input of these data decreases their uncertainties significantly, as illustrated for the xdvand xg PDFs on the left-hand side of Fig. 6. On the right-hand side of Fig. 6the decrease in the uncertainties of the xdvand xg PDFs from removing only the W,Zdata taken at 8 TeV is illustrated, showing that the major decrease comes from retaining the W,Zdata taken at 7TeV. However, the W,Zdata taken at 8 TeV have a major role to play in ensuring that x¯u∼x¯ dholds at low x, even though this constraint is not imposed. Without them, one observes x¯ d<x¯uat low x, as seen in Fig. 7. These data also somewhat reduce the low-xstrange distribution and harden the high-xstrange distribution, while softening the high-xx ¯ d distribution, as also shown in Fig. 7. There is mild tension between the W,Zdata at 8 TeV and the W,Zdata at 7 TeV. The partial χ2/NDP for the W,Zdata at 7 TeV decreases from 68/55 to 50/55 if the W,Zdata at 8 TeV are excluded from the fit, and the partial χ2/NDP for the W,Zdata at 8 TeV decreases from 239/206 123 438 Page 18 of 70 Eur. Phys. J. C (2022) 82 :438 x 3− 10 2− 10 1− 10 ) 2 uQ ,x( x 0 0.1 0.2 0.3 0.4 0.5 ATLAS ATLASpdf21, T=1 No 8 TeV W, Z Q2= 1.9 GeV2 x 3− 10 2− 10 1− 10 ) 2 dQ,x( x 0 0.1 0.2 0.3 0.4 0.5 ATLAS ATLASpdf21, T=1 No 8 TeV W, Z Q2= 1.9 GeV2 x 3− 10 2− 10 1− 10 ) 2 sQ ,x( x 0 0.1 0.2 0.3 0.4 0.5 ATLAS ATLASpdf21, T=1 No 8 TeV W, Z Q2= 1.9 GeV2 Fig. 7 ATLASpdf21 PDFs compared with those from a fit not including the W,Zdata at 8 TeV. Only experimental uncertainties are shown, evaluated with tolerance T=1. Top left: x¯u. Top right: x¯ d. Bottom: x¯s to 222/206 if the W,Zdata at 7 TeV are excluded from the fit. These increases in χ2are most pronounced for the 7 TeV c-c data around the Zmass-peak (66–116 GeV) and for the mass bins around the Zpeak in 8 TeV data. As already remarked, theoretical scale uncertainties for W,Zdata at both 7 and 8 TeV are added to the fit uncertainties. If these uncertainties are not added the tension between W,Zdata at 7 and 8 TeV increases. The partial χ2/NDP for W,Z data at 7 TeV increases to 80/55 and the partial χ2/NDP for W,Zdata at 8 TeV increases to 268/206 if both data sets are included in the fit and scale uncertainties are not applied. The differences between the PDFs are not large, whether or not scale uncertainties are applied, compared to the current experimental precision. However, consideration of such theoretical uncertainties is important if accuracy to 1% is the ultimate goal for the PDFs. N3LO calculations [74] indicate that higher-order corrections are likely to be larger than our current estimate of scale uncertainties. A further study of scale uncertainties is given in Appendix A. 5.2.2 Impact of V + jets data The impact of the V+jets data is shown in Figs. 8and 9. There are significant changes in the x¯ dand x¯sPDF shapes such that the high-xx¯sand x¯ dPDFs become softer and harder, respectively, with the input of V+jets data. Because of the change in the x¯ dshape, the difference x(¯ d−¯u)is also strongly affected. This is shown in Fig. 8. This figure also shows that, without the V+jets data, there is little information about the ratio Rsat high x. The changes in Fig. 8are large because the V+jets data resolve a double minimum in 123 Eur. Phys. J. C (2022) 82 :438 Page 19 of 70 438 x 3− 10 2− 10 1− 10 ) 2 dQ,x( x 0 0.1 0.2 0.3 0.4 0.5 ATLAS ATLASpdf21, T=1 No 8 TeV V+jets Q2= 1.9 GeV2 x 3− 10 2− 10 1− 10 ) 2 sQ ,x( x 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 ATLAS ATLASpdf21, T=1 No 8 TeV V+jets Q2= 1.9 GeV2 x 3− 10 2− 10 1− 10 ) 2 dQ,x() u+ s(x/) x(s+ 0 0.5 1 1.5 2 2.5 3 3.5 ATLAS ATLASpdf21, T=1 No 8 TeV V+jets Q2= 1.9 GeV2 x 3− 10 2− 10 1− 10 ) 2 uQ,x()dx( 0.04− 0.02− 0 0.02 0.04 0.06 0.08 0.1 ATLAS ATLASpdf21, T=1 No 8 TeV V+jets Q2= 1.9 GeV2 Fig. 8 ATLASpdf21 PDFs compared with those from a fit not including V+jets data. Only experimental uncertainties are shown, evaluated with tolerance T=1. Top left: x¯ d. Top right: x¯s.Bottomleft:Rs. Bottom right: x(¯ d−¯u) parameter space such that the fit now prefers a hard x¯ dand soft x¯sat high x, whereas it previously had an additional minimum with x¯ d∼x¯sat high x, which was marginally preferred. These results are similar to those already seen and fully explained in the ATLASepWZVjets20 PDF analysis [11] and the double minimum may be seen in Figure 5 of that paper. A fit with only HERA data, or with HERA plus ATLAS W,Zdata at 7 TeV (ATLASepWZ20), prefers the minimum with x¯ d∼x¯sat high x, but once ATLAS V+jets data at 8 TeV are added to the fit the double minimum with a hard x¯ dand soft x¯sat high xis preferred and the previous minimum disappears. In Figure 3 of Ref. [11], it can also be seen that for ATLASepWZ20 (the fit without V+jets data) the full uncertainties, including model and parameterisation variations, are very large because they cover both minima, whereas the full uncertainties of the ATLASepWZVjets20 fit (including V+ jets data) are much smaller because there is no double minimum. There is thus no strong inconsistency between the fits with or without the V+jets data because of the large uncertainties of the fit without the V+ jets data. This also applies in the present analysis, but in Fig. 8, only experimental uncertainties are shown. The full uncertainties for the fit without V+ jets data in our present analysis are of little interest and are not presented. Model and parametrisation uncertainties for the ATLASpdf21 fit will be discussed in Sect. 5.3. 123 438 Page 20 of 70 Eur. Phys. J. C (2022) 82 :438 x 3− 10 2− 10 1− 10 ) 2 (x,Q V xd 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 ATLAS ATLASpdf21, T=1 No 8 TeV V+jets Q2= 1.9 GeV2 x 3− 10 2− 10 1− 10 ref ) 2 )/xg(x,Q 2 xg(x,Q 0.8 0.9 1 1.1 1.2 ATLAS ATLASpdf21, T=1 No 8 TeV V+jets Q2= 1.9 GeV2 Fig. 9 ATLASpdf21 PDFs compared with those from a fit not including V+jets data. Only experimental uncertainties are shown, evaluated with tolerance T=1. Left: xdv. Right: the ratio of xg for the two fits These V+jets data also effect a modest change in the xdv shape and the gluon PDF shape as shown in Fig. 9.Theutype quarks are not strongly affected by these data and thus they are not shown. There is no tension between the V+jets data at 8 TeV and other data sets in the fit, since all data are fitted well at the minimum chosen by these V+jets data. 5.2.3 Impact of t ¯ t data Theimpact of the t¯ tdatais showninthe top half of Fig.10. The high-xgluon distribution is mildly softened when the t¯ tdata are added to the fit. This effect is opposite to the one observed in the ATLASepWZtop18 fit. This is because more data which harden the gluon PDF, in particular the V+jets and inclusive jet data, are included in the present fit. The more significant effect is in the uncertainties of the high-xgluon distribution, which are reduced. There is no significant tension between the t¯ tdata and the other data in the fit. Figure 10 (bottom half) also shows the impact of removing only the t¯ tdata at 13 TeV (left) or only the t¯ tdata at 8 TeV (right). It is clear that the data at 8 TeV have the stronger impact on the shape of the xg PDF but both data sets contribute to a modest reduction in the uncertainties. 123 Eur. Phys. J. C (2022) 82 :438 Page 21 of 70 438 x 3− 10 2− 10 1− 10 ref ) 2 )/xg(x,Q 2 xg(x,Q 0.95 1 1.05 ATLAS ATLASpdf21, T=1 No8, 13 TeV top Q2= 1.9 GeV2 x 3− 10 2− 10 1− 10 xg/xg δ 0.95 1 1.05 ATLAS ATLASpdf21, T=1 No 8, 13 TeV top Q2= 1.9 GeV2 x 3− 10 2− 10 1− 10 ref ) 2 )/xg(x,Q 2 xg(x,Q 0.95 1 1.05 ATLAS ATLASpdf21, T=1 No13 TeV top Q2= 1.9 GeV2 x 3− 10 2− 10 1− 10 ref ) 2 )/xg(x,Q 2 xg(x,Q 0.95 1 1.05 ATLAS ATLASpdf21, T=1 No8 TeV top Q2= 1.9 GeV2 Fig. 10 ATLASpdf21 xg PDF compared with xgfor a fit not including various t¯ tdata sets. Only experimental uncertainties are shown, evaluated with tolerance T=1. Top left: the shape ratio to a fit not including t¯ tdata at 8 and 13 TeV. Top right: the ratio to a fit not including t¯ t data at 8 and 13 TeV for which both distributions are centred on unity. Bottom left: the shape ratio to a fit not including t¯ tdata at 13 TeV. Bottom right: the shape ratio to a fit not including t¯ tdata at 8 TeV 5.2.4 Impact of photon data and inclusive jet data There is little impact from the addition of the direct-photon production ratio data apart from a marginal softening of the high-xgluon distribution as shown in Fig. 11 (left). However, it is notable that these data can now be well fitted at NNLO in QCD, given that they have been excluded from PDF fits for the last 20 years because of poor fits to lower-energy data [60,75]. There is minimal tension with other data sets. The principal impact of the inclusive jet data is on the gluon PDF. The main effect is a considerable decrease in high-xgluon uncertainties, with a mild hardening of the gluon PDF at high x, as shown in Fig. 11 (right). There is minimal tension with other data sets. As explained earlier in Sect. 2, the central ATLASpdf21 fit includes only the inclusive jet data at 8 TeV, with R=0.6. The full uncertainties of the fit, see Sect. 5.3.1, include the difference between the choices R=0.6 and R=0.4. The impact of using inclusive jet data at 7 TeV and at 13 TeV, instead of at 8 TeV, is explored in Appendix B. 5.3 Model, theoretical and parameterisation uncertainties The consideration of additional uncertainties affecting the PDFs is presented in this section. These are classified and 123 438 Page 22 of 70 Eur. Phys. J. C (2022) 82 :438 x 3− 10 2− 10 1− 10 ref ) 2 )/xg(x,Q 2 xg(x,Q 0.95 1 1.05 ATLAS ATLASpdf21, T=1 No photon Q2= 1.9 GeV2 x 3− 10 2− 10 1− 10 ref ) 2 )/xg(x,Q 2 xg(x,Q 0.95 1 1.05 ATLAS ATLASpdf21, T=1 No jets Q2= 1.9 GeV2 Fig. 11 ATLASpdf21 xg PDF compared with xg for fits not including various data sets. Only experimental uncertainties are shown, evaluated with tolerance T=1. Left: not including the direct-photon production ratio data taken at 13 and 8 TeV. Right: not including inclusive jet data at 8TeV labeled here as either model, theoretical or parameterisation uncertainties. 5.3.1 Model and theoretical uncertainties Theclass of modeluncertaintiesincludes effectsdueto variations of the heavy-quark masses input to the TRVFN heavyquark-mass scheme for the inclusive DIS calculations, the minimum Q2cut on the HERA inclusive DIS data and the value of the starting scale for evolution. The minimum Q2 cut was varied in the range 7.5<Q2 min <12.5GeV 2and the starting scale was varied in the range 1.6<Q2 0<2.2GeV 2. In the inclusive DIS calculations, the heavy-quark masses were varied in the ranges 1.37 <mc<1.45 GeV and 4.1<mb<4.3GeV[71]. The variations of mcand Q2 0 are coupled since the requirement Q2 0<m2 cmust be met. For this reason the upward variation of mcand the downward variation of Q2 0are symmetrised. Figure 12 illustrates the effect of these variations on the gluon distribution since this is the PDF most sensitive to these changes. The impact of the choice of mcand mbis modest. The effect of the variation of the Q2 min cut is larger but still within the experimental uncertainties. The largest of these uncertainties is due to the choice of Q2 0, which gives a gluon PDF differing from the central fit by ∼2σfor x∼0.1. An additional model uncertainty comes from the assumed value of the top-quark mass. The interpolation grids for the NNLO predictions for the t¯ tdata at 8 TeV are available for pole masses, mt=172.5,173.3,175.0 GeV. The smaller and larger values are used to estimate an asymmetric model uncertainty around the central value, shown in Fig. 12.For the t¯ t1-D distributions at 13 TeV the grids are only available for mt=172.5 GeV. The effect of this change in central value is negligible, as illustrated in Fig. 12. A cross-check was performed using the double differential pt Tand mt¯ tdistributions at 13 TeV, for which predictions are available for mt=171.0,172.5,174.0 GeV and the effect was found to be negligible. The 13 TeV t¯ tdata have a smaller impact in thefitthanthe8TeVt¯ tdata, so only the mtvariations of the 8 TeV data set are input to the final model uncertainty. A further potential model uncertainty comes from the treatment of the jet systematic uncertainties. Alternative decorrelation scenarios are considered as follows: the alternative option in which the JES Flavour Response is split into only two components rather than three, called “Decorrelation scenario 1”; complete decorrelation of the Jet Flavour Response between rapidity bins, called “FR decorrelated”; and no decorrelation, called “Fully correlated”. Table 5gives the total χ2/NDP for the jets (including all three terms of Eq. (1)) for alternative correlation scenarios, showing that the difference between full correlations and various choices of decorrelation can have a significant effect on the χ2. However, the difference between the ATLASpdf21 gluon PDFs obtained using jet data with these different correlation scenarios is relatively small compared to the model uncertainties considered above, e.g. the variation of Q2 0. There are also no changes in the PDF uncertainties as a result of using different correlation scenarios. Hence, these variations are not considered as a source of significant uncertainty. A further source of uncertainty for the inclusive jet data comes from the choice of jet radius R. The difference between the PDFs due to a different choice of jet radius for the8TeVjets isshowninFig.12.Theonlysignificant change is in the gluon PDF. This small difference between the PDFs 123 Eur. Phys. J. C (2022) 82 :438 Page 23 of 70 438 x 3− 10 2− 10 1− 10 ref ) 2 )/xg(x,Q 2 xg(x,Q 0.95 1 1.05 1.1 2 ATLAS ATLASpdf21, T=1 Q = 1.9 GeV mbup mcsymmetric mbdown 2 x 3− 10 2− 10 1− 10 ref ) 2 )/xg(x,Q 2 xg(x,Q 0.95 1 1.05 1.1 2 ATLAS ATLASpdf21, T=1 Q = 1.9 GeV Qmin up Qmin down Q0symmetric 2 2 2 2 x 3− 10 2− 10 1− 10 ref ) 2 )/xg(x,Q 2 xg(x,Q 0.95 1 1.05 1.1 ATLAS Q2= 1.9 GeV2 ATLASpdf21, T=1 mtup mtdown 8 TeV jets R = 0.4 Fig. 12 Impact of model assumption variations on the ATLASpdf21 gluon PDF illustrated in their ratio to the xg distribution for the central choice. Uncertainties of the central fit are experimental, evaluated with tolerance T=1. Top left: variation of mcand mbvalues. The mc upward variation is shown and this is symmetrised. Top right: impact of variations of Q2 min and Q2 0on the ATLASpdf21 gluon PDF. The Q2 0 downward variation is shown and it is symmetrised. Bottom: impact of the variation of mtand of the choice of jet radius for the inclusive jets on the ATLASpdf21 gluon PDF Table 5 χ2contributions for the inclusive jet data set at 8 TeV with R= 0.6, for different correlation scenarios, as explained in the text. The χ2 values given here represent the addition of all terms in Eq. (1) Jets 8 TeV R= 0.6 Fully correlated FR decorrelated Decorrelation scenario 1 Decorrelation scenario 2 (default) χ2/NDP 289/171 227/171 250/171 248/171 extracted using jet data at 8 TeV for R=0.4 and R=0.6 is considered as an extra uncertainty because, although it is clear that the choice R= 0.6 is theoretically favoured [29], this paper also inputs t¯ tdata from the lepton+jets channel and V+jets data, and for these data sets only R=0.4jets are available. The effect of using jet production data at different centreof-mass energies of 7, 8 and 13 TeV is shown in Appendix B. This is not considered as an additional uncertainty but is presented as a cross-check that the choice of centre-of-mass energy for the jet production data does not have a significant influence on the PDFs extracted. 123 438 Page 24 of 70 Eur. Phys. J. C (2022) 82 :438 x 3− 10 2− 10 1− 10 ref ) 2 )/xg(x,Q 2 xg(x,Q 0.95 1 1.05 22 ATLAS Q = 1.9 GeV ATLASpdf21, T=1 KFpT max x 3− 10 2− 10 1− 10 ref ) 2 )/xg(x,Q 2 xg(x,Q 0.95 1 1.05 ATLAS Q2= 1.9 GeV2 ATLASpdf21, T=1 μ r ,μ f μ r ,μ f up down Fig. 13 Ratios of the ATLASpdf21 gluon PDFs for different inclusive jet scale choices. Left: pmax Tover pjet T(default). Right: 2pjet Tand pjet T/2 over pjet T(default). Uncertainties of the central fit are experimental, evaluated with tolerance T=1 Table 6 χ2/NDF for the total fit and the contribution χ2/NDP for the inclusive jet data, for various scale choices, and treatment of K-factors and their uncertainties. The first row represents the default values used in the central ATLASpdf21 fit Total χ2/NDF χ2/NDP for jets Treatment of K-factors Scale choice 2010/1620 248/171 Smoothed pjet Tscale 2019/1620 257/171 Smoothed pmax Tscale 2032/1620 272/171 Smoothed 2pjet Tscale 1991/1620 228/171 Smoothed pjet T/2 scale 1983/1620 223/171 Unsmoothed pjet Tscale All model uncertainties are added in quadrature to form a total model uncertainty. Theoreticaluncertaintiesincludethe scale uncertainties of thepredictions.Thelargestofthesearethescaleuncertainties for the inclusive Wand Zdata which are considered in detail in Sect. 3.2. Since these are treated as correlated systematic uncertainties in the fit χ2they are actually already included in what has been labelled as the experimental uncertainty of the fit. The effect of scale uncertainties for the V+jets data was studied in Ref. [11] and was found to be negligible. The effect of scale uncertainties for the t¯ tlepton+jets data was studied for the present paper and was also found to be negligible. The direct-photon production ratio data are relatively insensitive to scale variations because 100% correlation of the scale uncertainties is assumed between the 8 and 13 TeV data. Since the fit is relatively insensitive to these data, alternative assumptions for these correlations were not pursued. The scale uncertainty for the inclusive jets at 8 TeV requires further consideration. These K-factors were supplied for two choices of scale: μr=μf=pmax Tor μr=μf= pjet T. A scale related to pjet Tis favoured over pmax T[29], as already mentioned. However, this difference in scale choice is now investigated. A comparison of the gluon PDFs for these two scales is shown in Fig. 13 (left), since this is the PDF most sensitive to this change. It can be seen that there is no significant difference between the resulting gluon PDFs. Changes in the nuisance parameter values for the correlated systematic uncertainties absorb the change in the predictions for the two scales. The χ2values for these fits are given in Table 6. The only significant change in χ2comes from the inclusive jets, not from other data sets. Next, scale variations μr=μf=2pjet Tand μr=μf= pjet T/2 are considered. The effect on the gluon PDF is shown in Fig. 13 (right). The χ2values for these fits are given in Table 6. Again, the only significant change in χ2comes from the inclusive jets, not from other data sets. Finally, a cross-check is performed in which the K-factors are not smoothed but their statistical uncertainties are used as an additional uncorrelated uncertainty. The χ2value for this fit is also given in Table 6. It is lower than that for smoothed K-factors because the statistical uncertainties of 123 Eur. Phys. J. C (2022) 82 :438 Page 25 of 70 438 the unsmoothed K-factors are ∼1%. However, the resulting PDFs are very similar to those obtained using smoothed Kfactors. Since none of the scale variation effects produced significant changes in the PDFs, no further theoretical uncertainty is added for this source. Thus the theoretical uncertainties considered so far are either already included in the experimental uncertainties of the fit, or they are negligible. It should be noted that the data are also sensitive to the value of αs(mZ), which affects the shape of the gluon PDF. The correlation between αs(mZ)and the gluon PDF shape is specified by the DGLAP formalism. A determination of αs(mZ)is beyond the scope of the current paper, since a correct NNLO treatment requires variation of the K-factor calculations with αs(mZ), or direct NNLO grids for all processes. This is left for future work. The conventional value αs(mZ)=0.118 is used, in line with the value used by the global fitting groups, CT [76], MSHT [77] and NNPDF [78]. 5.3.2 Parameterisation uncertainties The optimal number of parameters was determined by ‘saturation’ of the χ2as explained in Sect. 4. However, the effect on the PDFs of adding extra parameters is investigated. Although there is no significant further decrease in χ2, some small shape changes are observed when adding an Fuvterm to the u-valence PDF and/or a D¯ dterm to the x¯ d PDF. Figure 14 shows the impact of adding both of these as free parameters on the xuv,xdv,x¯uand x¯ dPDFs, which are the PDFs which show the largest variations. The total x 3− 10 2− 10 1− 10 ref ) 2 (x,Q V )/xu 2 (x,Q V xu 0.85 0.9 0.95 1 1.05 1.1 1.15 ATLAS Q2= 1.9 GeV2 ATLASpdf21, T=1 Add parameters Fu, Dd _ v x 3− 10 2− 10 1− 10 ref ) 2 (x,Q V )/xd 2 (x,Q V xd 0.85 0.9 0.95 1 1.05 1.1 1.15 ATLAS Q2= 1.9 GeV2 v ATLASpdf21, T=1 Add parameters Fu, Dd _ x 3− 10 2− 10 1− 10 ref ) 2 uQ,x( )/x 2 uQ,x( x 0.85 0.9 0.95 1 1.05 1.1 1.15 ATLAS Q2= 1.9 GeV2 v ATLASpdf21, T=1 Add parameters Fu, Dd _ x 3− 10 2− 10 1− 10 ref ) 2 dQ,x()/x 2 dQ,x ( x 0.85 0.9 0.95 1 1.05 1.1 1.15 ATLAS Q2= 1.9 GeV2 v ATLASpdf21, T=1 Add parameters Fu, Dd _ Fig. 14 Impact of adding Fuvand D¯ das free parameters on the valence PDFs in comparison with the central ATLASpdf21 21-parameter fit. Uncertainties of the central fit are experimental, evaluated with tolerance T=1. Top left: xuv. Top right: xdv.Bottomleft:x¯u. Bottom right: x¯ d 123 438 Page 32 of 70 Eur. Phys. J. C (2022) 82 :438 x 3− 10 2− 10 1− 10 V xu (x,Q 2 ) 0 0.2 0.4 0.6 0.8 1 ATLASATLASATLASATLAS CT18 CT18A HERAPDF2.0 ATLASpdf21, full uncertainties Q Q2= 1.9 GeV2 x 3− 10 2− 10 1− 10 0 0.2 0.4 0.6 0.8 1 ATLASATLASATLASATLAS MSHT20 NNPDF3.1 ABMP16 ATLASpdf21, full uncertainties Q2= 1.9 GeV2 V xu (x,Q 2 ) x 3− 10 2− 10 1 − 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 ATLASATLASATLASATLAS CT18 CT18A HERAPDF2.0 ATLASpdf21, full uncertainties Q2= 1.9 GeV2 V xd (x,Q 2 ) x 3− 10 2− 10 1 − 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 ATLASATLASATLASATLAS MSHT20 NNPDF3.1 ABMP16 ATLASpdf21, full uncertainties Q2= 1.9 GeV2 xd V (x,Q 2 ) Fig. 21 ATLASpdf21 xuvand xdvdistributions with full uncertainties (experimental T=3, model, parameterisation) compared with other PDFs. Left: CT18, CT18A, HERAPDF2.0. Right: MSHT20, NNPDF3.1, ABMP16 x(¯ d−¯u)distributionismorepositivethantheglobalPDFsfor x>0.3, in better agreement with the newer E906 Drell–Yan data [86], rather than the E866 data which is in the global fits. This is explored further in Appendix D. The central value of the x¯sdistribution is larger than those in NNPDF3.1 and CT18 for x∼0.02, but is in good agreement with CT18A and MSHT20. The gluon distribution agrees best with NNPDF3.1, but is in reasonable agreement with the other global PDFs.12 The distributions of Rsfor all the PDFs illustrated are now in broad agreement but there are differences in detail. The uncertainties of HERAPDF2.0 and CT18 are larger than the others because they do not include ATLAS 12 The ABMP16 gluon PDF corresponds to αs(mZ)=0.1145, a lower value than in the other PDFs illustrated, which all use αs(mZ)=0.118. inclusive W,Zdataat7TeV.13 TheuncertaintiesofABMP16 aresmaller fortheusualreason thattheyuse tolerance T=1. The input of ATLAS inclusive W,Zdata at 7 TeV is the only difference between the CT18 and CT18A analyses. After input of these data the CT18A RsPDF ratio has smaller uncertainties and a larger central value of Rsat low xthan CT18. CT18A is in good agreement with ATLASpdf21, for both central value and uncertainty, over the full xrange illustrated. The PDF analyses of MSHT20 and NNPDF3.1 also use ATLAS inclusive W,Zdata at 7 TeV and have a similar sizeofuncertaintyandlevelofagreementwithATLASpdf21. In addition, the MSHT20 analysis uses ATLAS inclusive W,Zdata at 8 TeV, just as the ATLASpdf21 analysis does. Note that NNPDF have updated their study of strangeness 13 In the case of HERAPDF2.0 the uncertainty in the strange PDF is not measured but estimated. 123 Eur. Phys. J. C (2022) 82 :438 Page 33 of 70 438 x 3 − 10 2 − 10 1− 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 ATLASATLASATLASATLAS CT18 CT18A HERAPDF2.0 ATLASpdf21, full uncertainties Q Q2= 1.9 GeV2 xu(x,Q2) x 3 − 10 2 − 10 1− 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 MSHT20 NNPDF3.1 ABMP16 ATLASpdf21, full uncertainties S A LTASALT ASA L TASALTA Q2= 1.9 GeV2 xu(x,Q2) x 3− 10 2− 10 1− 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 CT18 CT18A HERAPDF2.0 ATLASpdf21, full uncertainties SA LTA SA L TASAL TAS A L TA Q2=1.9 GeV2 xd(x,Q2) x 3− 10 2− 10 1− 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 MSHT20 NNPDF3.1 ABMP16 ATLASpdf21, full uncertainties SALTASALTASALTASALTA Q2= 1.9 GeV2 xd(x,Q2) x 3 − 10 2 − 10 1− 10 0.15− 0.1− 0.05 − 0 0.05 0.1 0.15 0.2 CT18 CT18A HERAPDF2.0 ATLASpdf21, full uncertainties ATLAS Q2= 1.9 GeV2 x(d -u) (x,Q2) x 3 − 10 2 − 10 1− 10 0.15− 0.1− 0.05 − 0 0.05 0.1 0.15 0.2 ATLAS Q2= 1.9 GeV2 MSHT20 NNPDF3.1 ABMP16 ATLASpdf21, full uncertainties x(d - u) (x,Q2) Fig. 22 ATLASpdf21 x¯u,x¯ dand x(¯ d−¯u)distributions with full uncertainties (experimental T=3, model, parameterisation) compared with other PDFs. Left: CT18, CT18A, HERAPDF2.0. Right: MSHT20, NNPDF3.1, ABMP16 123 438 Page 34 of 70 Eur. Phys. J. C (2022) 82 :438 x 3 10 2 10 1 10 sx 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 ATLASATLAS CT18 CT18A HERAPDF2.0 ATLASpdf21, full uncertainties Q2= 1.9 GeV2 (x,Q2) x 3 − 10 2− 10 1− 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 ATLASATLAS MSHT20 NNPDF3.1 ABMP16 ATLASpdf21, full uncertainties Q2= 1.9 GeV2 xs(x,Q2) x 3− 10 2− 10 1− 10 xg 0 0.5 1 1.5 2 2.5 3 3.5 4ATLASATLAS CT18 CT18A HERAPDF2.0 ATLASpdf21, full uncertainties Q2= 1.9 GeV2 (x,Q2) x 3− 10 2− 10 1− 10 0 0.5 1 1.5 2 2.5 3 3.5 MSHT20 NNPDF3.1 ABMP16 ATLASpdf21, full uncertainties Q2=1.9 GeV2 ATLAS 4 xg (x,Q2) 3− 10 2− 10 s R 0 0.5 1 1.5 2 CT18 CT18A HERAPDF2.0 ATLASpdf21, full uncertainties ATLAS Q2= 1.9 GeV2 10−1xx 3 10 2 10 1 10 s R 0 0.5 1 1.5 2 ATLAS Q2= 1.9 GeV2 MSHT20 NNPDF3.1 ABMP16 ATLASpdf21, full uncertainties Fig. 23 ATLASpdf21 x¯s,xg and Rsdistributions with full uncertainties (experimental T=3, model, parameterisation) compared with other PDFs. Left: CT18, CT18A, HERAPDF2.0. Right: MSHT20, NNPDF3.1, ABMP16 123 Eur. Phys. J. C (2022) 82 :438 Page 35 of 70 438 recently to NNPDF3.1_strange [83], which has a somewhat larger strangeness at low xthan NNPDF3.1. The modern PDFs which use ATLAS inclusive W,Zdata, including ATLASpdf21 itself, all agree that strangeness is not suppressed strongly at low x, but there is substantial suppression at high x. In summary, the ATLASpdf21 fit now agrees with the global fits as well as they agree with each other. Thus ATLAS data seem able to replicate most of the features that the fixedtarget DIS and DY data plus the Tevatron data brought to the global PDFs, see Appendix D. Using only the HERA and ATLAS data allows a more rigorous treatment of correlated systematic uncertainties and, in particular, of correlations between data sets. For the global fits considered here, the χ2values for the data sets included in the ATLASpdf21 fit are HERAPDF2.0: 2262, CT18: 2135, CT18A: 2133, MSHT20: 2218,NNPDF3.1:2109,comparedwithATLASpdf21:2010. Although the global PDFs from CT, MSHT and NNPDF have more flexible parameterisations, the χ2value of the ATLASpdf21 fit is better for the data sets considered. This is a further indication that the ATLASpdf21 parameterisation is sufficiently flexible. Figure 24 shows the uncertainties of the ATLASpdf21 fit, with the enhanced tolerance T=3, compared with CT18A and MSHT20 PDFs. These two PDFs are chosen because theyboth usethe ATLAS W,Zinclusive cross-sectiondata at 7 TeV and they both evaluate uncertainties in a similar way to the present analysis. In general, the uncertainties of CT18A are somewhat larger than those of MSHT20, even though both represent 68% CL uncertainties. For the x¯uand x¯ dsea distributions at low x, the uncertainties of ATLASpdf21 lie between those of CT18A and MSHT20, becoming larger at high x,x>0.1, reflecting the absence of fixed-target DY data.Forthe x¯ssea distribution there is a similar pattern, with the low xuncertainties of the ATLAS PDF being comparable to those of CT18A. The uncertainty of the ATLASpdf21 gluon distribution at x<2×10−3is very similar to that of both CT18A and MSHT20, but the high xgluon uncertainty is somewhat larger for the ATLASpdf21 fit, reflecting the absence of Tevatron jet data. In the valence sector, the xuvuncertainties of the ATLASpdf21 fit are comparable to those of CT18A at low x, but larger at high x, reflecting the absence of Tevatron W,Zdata. For xdv, the uncertainties are larger for all xexcept x∼0.003–0.008, reflecting the fact that there is less information about the d-quark than the u-quark in the absence of deuterium target data. Appendix D shows comparisons between the ATLASpdf21 predictions and some of these older data sets which, while not fitted, are generally well described. In order to focus more on the high xregime, which is important in searches for new physics, the comparisons between the ATLASpdf21 fit, with the enhanced tolerance of T=3, and the CT18A and MSHT20 PDFs, are repeated with a linear-xand log-yscaleinFig.25. Some discrepancies, not evident before, appear for x≥0.7. In this high x region there is no data to determine any of the PDFs. Nevertheless, there is agreement within ∼2σ. 123 438 Page 36 of 70 Eur. Phys. J. C (2022) 82 :438 x 3 − 10 2− 10 1− 10 V /xu V xuδ 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 SALTA Q Q2= 1.9GeV2 MSHT20 CT18A ATLASpdf21, full uncertainties x 3 − 10 2− 10 1− 10 V /xd V xdδ 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 ATLAS Q2= 1.9 GeV2 MSHT20 CT18A ATLASpdf21, full uncertainties x 3− 10 2− 10 1 − 10 u/xuxδ 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 SALTA Q2=1.9 GeV2 MSHT20 CT18A ATLASpdf21, full uncertainties x 3− 10 2− 10 1 − 10 d/xdxδ 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 SALTA Q2= 1.9 GeV2 MSHT20 CT18A ATLASpdf21, full uncertainties x 3 − 10 2− 10 1− 10 xg/xgδ 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 SALTA Q2= 1.9GeV2 MSHT20 CT18A ATLASpdf21, full uncertainties x 3 − 10 2− 10 1− 10 s/xsxδ 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 ATLAS MSHT20 CT18A ATLASpdf21, full uncertainties Q2=1.9 GeV2 Fig. 24 ATLASpdf21 fit with full uncertainties (experimental T=3, model, parameterisation) compared with CT18A, MSHT20 shown as a ratio with each distribution centred on unity. Top left: xuv. Top right: xdv. Middle left: x¯u. Middle right: x¯ d. Bottom left: xg. Bottom right: x¯s 123 Eur. Phys. J. C (2022) 82 :438 Page 37 of 70 438 x 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 V xu 3− 10 2− 10 1− 10 1 10 SALTA Q Q 2 = 1.9 GeV 2 (x,Q 2 ) MSHT20 CT18A ATLASpdf21, full uncertainties x 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 V xd 3− 10 2− 10 1− 10 1 10 SALTA Q 2 = 1.9GeV 2 (x,Q 2 ) MSHT20 CT18A ATLASpdf21, full uncertainties x 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 ux 3− 10 2 − 10 1− 10 1 10 SALTA Q 2 =1.9 GeV 2 (x,Q 2 ) MSHT20 CT18A ATLASpdf21, full uncertainties x 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 dx 3− 10 2 − 10 1− 10 1 10 ATLAS Q 2 = 1.9GeV 2 (x,Q 2 ) MSHT20 CT18A ATLASpdf21, full uncertainties x 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 sx 3 − 10 2− 10 1− 10 1 10 SALTA Q 2 = 1.9 GeV 2 (x,Q 2 ) MSHT20 CT18A ATLASpdf21, full uncertainties x 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 xg 3 − 10 2− 10 1− 10 1 10 SALTA Q 2 = 1.9 GeV 2 (x,Q 2 ) MSHT20 CT18A ATLASpdf21, full uncertainties Fig. 25 ATLASpdf21 fit with full uncertainties (experimental T=3, model, parameterisation) compared with global PDFs: CT18A and MSHT20, focusing on high xbehaviour. Top left: xuv. Top right: xdv. Middle left: x¯u. Middle right: x¯ d. Bottom left: x¯s. Bottom right: xg 123 438 Page 38 of 70 Eur. Phys. J. C (2022) 82 :438 7 Conclusion A PDF fit is presented using selected proton–proton collision data sets recorded by the ATLAS experiment at the LHC together with the final combined inclusive HERA data in order to assess the impact of the ATLAS data with a minimum of other input data. The resulting PDF set is called ATLASpdf21. The addition of ATLAS inclusive Wand Z data to HERA data allows a fit to be made without any constraints imposed on the relationship between x¯u,x¯ dand x¯s, nevertheless, x¯ d∼x¯uis observed at low x. The strangeness at low xis found not to be as suppressed as found in earlier PDFs such as CT14, MMHT14 and NNPDF3.0, but to be in agreement with modern PDFs such as CT18A, MSHT20 and NNPDF3.1_strange which use these ATLAS inclusive W,Zdata. This increase in the low xstrange was first seen in the ATLASepWZ12 and ATLASepWZ16 PDF analyses, but is now somewhat moderated by further ATLAS data, and the freedom of the parametrisation at low x. These inclusive W,Zdata also exhibit sensitivity to the valence quark distributions and the gluon distribution, considerably reducing their uncertainty. The ATLAS V+jets data constrain the light-quark sea at higher x, such that strangeness is strongly suppressed at high x, and impact the high xgluon distribution.The t¯ tdataserveto moderatelyreduce theuncertaintyin the high xgluon distribution. Ratios of direct photon production at different proton–proton collision energies have only a mild impact on the gluon PDF, but it is notable that they may now be reliably fitted to NNLO in QCD. The jet production data are sensitive to the gluon distribution at medium to high xand they reduce its uncertainty considerably. The role of scale uncertainties is considered for all data setsandthesetheoreticaluncertaintiesareimplementedinthe fitfor theinclusive W,Zdata,where theyaremostimpactful. The effects of such scale uncertainties are small. A study is made of excluding high-scale data (with a scale above 500 GeV), which may contain subtle effects of physics beyond the Standard Model, from the fit. The resulting PDFs are not strongly affected by this restriction of the fitted kinematic region. It is observed that the addition of the ATLAS data sets to the HERA data brings the PDFs much closer to the global PDFs of MSHT, CT and NNPDF than to HERAPDF2.0. The ATLASpdf21 PDFs agree with these global fits as well as they agree with each other. Thus, ATLAS data seem to be able to replicate many of the features that the fixed-target deep inelastic scattering and Drell–Yan data plus the Tevatron Drell–Yan data bring to the global PDFs. Using only the HERA and ATLAS data allows a detailed treatment of correlated systematic uncertainties. Correlations of systematic uncertainties within and between ATLAS data sets are considered, with particular emphasis on the larger systematic uncertainties appertaining to the jet energy scale. Information about correlations between data sets is provided such that the major correlations could be considered by the global fitting groups CT, MSHT andNNPDFinfuturefits.Theeffectsofthesecorrelationsare relatively small, O(∼1%), at the scale Q2∼10,000 GeV2 consideredforprecisionphysicsattheLHC,butlargeenough to be considered in future precision data analyses such as the measurement of mWand sin2θW, given that such a level of accuracy is now desired to resolve effects of physics beyond the Standard Model in the measurements of Standard Model parameters. Acknowledgements We thank CERN for the very successful operation of the LHC, as well as the support staff from our institutions without whom ATLAS could not be operated efficiently. We acknowledge the support of ANPCyT, Argentina; YerPhI, Armenia; ARC, Australia; BMWFWandFWF,Austria;ANAS,Azerbaijan;SSTC,Belarus;CNPq and FAPESP, Brazil; NSERC, NRC and CFI, Canada; CERN; ANID, Chile; CAS, MOST and NSFC, China; Minciencias, Colombia; MSMT CR, MPO CR and VSC CR, Czech Republic; DNRF and DNSRC, Denmark; IN2P3-CNRS and CEA-DRF/IRFU, France; SRNSFG, Georgia; BMBF, HGF and MPG, Germany; GSRI, Greece; RGC and Hong Kong SAR, China; ISF and Benoziyo Center, Israel; INFN, Italy; MEXT and JSPS, Japan; CNRST, Morocco; NWO, Netherlands; RCN, Norway; MEiN,Poland;FCT,Portugal;MNE/IFA,Romania;JINR;MESofRussiaandNRC KI,RussianFederation;MESTD,Serbia;MSSR,Slovakia; ARRS and MIZŠ, Slovenia; DSI/NRF, South Africa; MICINN, Spain; SRC and Wallenberg Foundation, Sweden; SERI, SNSF and Cantons of Bern and Geneva, Switzerland; MOST, Taiwan; TAEK, Turkey; STFC, United Kingdom; DOE and NSF, United States of America. In addition, individual groups and members have received support from BCKDF, CANARIE, Compute Canada and CRC, Canada; COST, ERC, ERDF, Horizon 2020 and Marie Skłodowska-Curie Actions, European Union; Investissements d’Avenir Labex, Investissements d’Avenir Idex and ANR, France; DFG and AvH Foundation, Germany; Herakleitos, Thales and Aristeia programmes co-financed by EU-ESF and the Greek NSRF, Greece; BSF-NSF and GIF, Israel; Norwegian Financial Mechanism 2014–2021, Norway; NCN and NAWA, Poland; La Caixa Banking Foundation, CERCA Programme Generalitat de Catalunya and PROMETEO and GenT Programmes Generalitat Valenciana, Spain; Göran Gustafssons Stiftelse, Sweden; The Royal Society and Leverhulme Trust, United Kingdom. The crucial computing support from all WLCG partners is acknowledged gratefully, in particular from CERN, the ATLAS Tier-1 facilities at TRIUMF (Canada), NDGF (Denmark,Norway,Sweden),CC-IN2P3(France),KIT/GridKA(Germany), INFN-CNAF (Italy), NL-T1 (Netherlands), PIC (Spain), ASGC (Taiwan), RAL (UK) and BNL (USA), the Tier-2 facilities worldwide and large non-WLCG resource providers. Major contributors of computing resources are listed in Ref. [87]. Data Availability Statement This manuscript has no associated data or the data will not be deposited. [Authors’ comment: All ATLAS scientific output is published in journals, and preliminary results are made available in Conference Notes. All are openly available, without restriction on use by external parties beyond copyright law and the standard conditions agreed by CERN. Data associated with journal publications 123 Eur. Phys. J. C (2022) 82 :438 Page 39 of 70 438 are also made available: tables and data from plots (e.g. cross section values, likelihood profiles, selection efficiencies, cross section limits, ...) are stored in appropriate repositories such as HEPDATA (http:// hepdata.cedar.ac.uk/). ATLAS also strives to make additional material related to the paper available that allows a reinterpretation of the data in the context of new theoretical models. For example, an extended encapsulation of the analysis is often provided for measurements in the framework of RIVET (http://rivet.hepforge.org/). This information is taken from the ATLAS Data Access Policy, which is a public document that can be downloaded from http://opendata.cern.ch/record/413 [opendata.cern.ch].] Open Access This article is licensed under a Creative Commons Attribution4.0InternationalLicense,whichpermitsuse,sharing,adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecomm ons.org/licenses/by/4.0/. Funded by SCOAP3. Appendix A Scale uncertainties in W,Zinclusive data In this appendix, different choices for the treatment of the scale uncertainties in inclusive W,Zdata are considered. For these data sets the experimental uncertainties are comparable to the scale uncertainties of NNLO predictions (∼0.5%). These theoretical uncertainties can therefore have some significant impact on the PDF fit. For the central fit, these theoretical uncertainties are applied in the same way as correlated systematic uncertainties. Changes in renormalisation and factorisation scales are taken to be correlated between Wand Zdata and to be correlated between the 7 and 8 TeV data sets. Two alternatives are considered here: (i) the scale uncertainties are not correlated between the 7 and 8 TeV data sets and (ii) scale uncertainties are not applied at all. The results of fits for these two cases, compared with the central fit, are shown as a ratio at a scale Q2=10,000 GeV2, relevant for LHC physics, in Fig. 26. If should be noted that the central fit is shown with just experimental uncertainties, evaluated with tolerance T=1, so that its uncertainties are directly comparable with those of the fit without scale uncertainties. The uncertainties are very similar in size. The differences between the shapes of the PDFs are not large, but they can be important if the desired accuracy of the PDFs is O(∼1%). The difference between the cases where the scale uncertainties are applied as being correlated or uncorrelated between the 7 and 8 TeV inclusive W,Zdata sets is shown by the green line in these figures and it can be seen that it is generally a smaller effect. 123 438 Page 40 of 70 Eur. Phys. J. C (2022) 82 :438 x 3− 10 2− 10 1− 10 ref ) 2 (x,Q V )/xu 2 (x,Q V xu 0.9 0.95 1 1.05 1.1 1.15 ATLAS Q 2 = 10000 GeV 2 ATLASpdf21, T = 1 No scale uncertainties Scale uncertainties uncorrelated x 3− 10 2− 10 1− 10 ref ) 2 (x,Q V )/xd 2 (x,Q V xd 0.9 0.95 1 1.05 1.1 1.15 ATLAS Q 2 = 10000 GeV 2 ATLASpdf21, T = 1 No scale uncertainties Scale uncertainties uncorrelated x 3− 10 2− 10 1− 10 ref ) 2 uQ,x()/x 2 uQ,x( x 0.9 0.95 1 1.05 1.1 1.15 ATLAS Q 2 = 10000 GeV 2 ATLASpdf21, T = 1 No scale uncertainties Scale uncertainties uncorrelated x 3− 10 2− 10 1− 10 ref ) 2 dQ ,x ( )/x 2 dQ , x ( x 0.9 0.95 1 1.05 1.1 1.15 ATLAS Q 2 = 10000 GeV 2 ATLASpdf21, T = 1 No scale uncertainties Scale uncertainties uncorrelated x 3− 10 2− 10 1− 10 ref ) 2 sQ,x()/x 2 sQ,x( x 0.9 0.95 1 1.05 1.1 1.15 ATLAS Q 2 = 10000 GeV 2 ATLASpdf21, T = 1 No scale uncertainties Scale uncertainties uncorrelated x 3− 10 2− 10 1− 10 ref ) 2 )/xg(x,Q 2 xg(x,Q 0.9 0.95 1 1.05 1.1 1.15 ATLAS Q 2 = 10000 GeV 2 ATLASpdf21, T = 1 No scale uncertainties Scale uncertainties uncorrelated Fig. 26 ATLASpdf21, showing the ratios of a fit not including theoretical scale uncertainties in the inclusive W,Zdata to the central fit which does include these uncertainties, at the scale Q2=10,000 GeV2.Also shown is the result of a fit in which the scale uncertainties are applied but not correlated between the 7 and 8 TeV data. All three fits are shown with just experimental uncertainties, evaluated with tolerance T=1. Top left: xuv. Top right: xdv. Middle left: x¯u. Middle right: x¯ d.Bottom left: xs. Bottom right: xg 123 Eur. Phys. J. C (2022) 82 :438 Page 41 of 70 438 x 3− 10 2− 10 1− 10 re f ) 2 )/xg(x,Q 2 xg(x,Q 0.8 0.9 1 1.1 1.2 ATLAS Q2= 1.9 GeV2 ATLASpdf21, full uncertainties 8 TeV jets R =0.4 13 TeV jets R =0.4 7 TeV jets R =0.6 x 3− 10 2− 10 1− 10 re f ) 2 dQ , x()u+s(x/))/x(s+ 2 dQ,x( ) s(x/) x(s+ 0.5 1 1.5 8 TeV jets R =0.4 13 TeV jets R =0.4 7 TeV jets R =0.6 ATLAS Q2= 1.9 GeV2 ATLASpdf21, full uncertainties u+ Fig. 27 Comparison showing the ratio of the ATLASpdf21 gluon PDF and the RsPDF ratio, using inclusive jet data at 8 TeV with R=0.6, to the gluon PDF and RsPDF ratio for fits using various different jet production data sets at 7, 8 and 13 TeV, with differing choices of jet radius. The data sets in these plots are for scale choice pjet T. Uncertainties of the central fit are full uncertainties: experimental, evaluated with tolerance T=3, plus model and parameterisation uncertainties B Comparison of the impact of inclusive jet data at different centre-of-mass energies It is not possible to fit inclusive jet production data at different centre-of-mass energies simultaneously, because the full experimental systematic uncertainty correlations between these data sets are not known. The inclusive jet production data at 8 TeV with R=0.6 were selected for input to the central fit. In this appendix, fits using the inclusive jet production data at 7 or 13 TeV instead of the data at 8 TeV are compared with the central fit. The data at 7 TeV shown here were extracted for R=0.6 whereas the data at 7 TeV were extracted only for R=0.4. The gluon PDF and RsPDF ratio using these jet production data sets at 7 and 13 TeV are shown in their ratio to the central fit results in Fig. 27. The scale choice was pjet Tfor all the jet production data sets included in this figure. Since the effect of using R=0.4or R=0.6 is very similar, as illustrated for the jet production data at 8 TeV, the differences between the PDFs are dominated by the change in centre-of-mass energy. The PDFs using the jet production data at 8 TeV lie between those of the jet production data at 7 TeV and the jet production data at 13 TeV. However, these differences are not significant compared to the full uncertainties of the PDFs evaluated with T=3. C Goodness of the fit: comparison with data sets included in the fit Figures28,29,30,31,32,33,34,35,36 and37 showcomparisons of the various ATLAS differential cross-section measurements used in the ATLASpdf21 fit, together with the predictions of this fit. Further details are provided in the figure captions. 123 438 Page 48 of 70 Eur. Phys. J. C (2022) 82 :438 0 100 200 300 [pb/GeV] t T /dpσ d 2− 10 1− 10 1 10 -1 = 13 TeV, 36.1 fbst, t→pp ATLAS Data uncorrelatedδ totalδ + shifts ATLASpdf21, full uncertainties [GeV] t T p 0 100 200 300 Theory/Data 0.6 0.8 1 1.2 1.4 ATLAS 400 500 1000 [pb/GeV] tt /dm σ d 3− 10 2− 10 1− 10 1 10 -1 = 13 TeV, 36.1 fbst, t→pp ATLAS Data uncorrelatedδ totalδ + shifts ATLASpdf21, full uncertainties [GeV] tt m 300 400 500 1000 2000 Theory/Data 0.8 0.9 1 1.1 1.2 ATLAS 012 | [pb] t /d|yσd 2 10 3 10 -1 = 13 TeV, 36.1 fbst, t→pp ATLAS Data uncorrelatedδ totalδ + shifts ATLASpdf21, full uncertainties | t |y 012 Theory/Data 0.8 0.9 1 1.1 1.2 ATLAS 012 | [pb] b t t /d|yσd 2 10 3 10 -1 = 13 TeV, 36.1 fbst, t→pp ATLAS Data uncorrelatedδ totalδ + shifts ATLASpdf21, full uncertainties | b t t |y 012 Theory/Data 0.8 0.9 1 1.1 1.2 ATLAS Fig. 35 The differential cross-section measurements of t¯ tat 13 TeV in Ref. [15] (black points) as functions of the (top left) average top transverse momentum pt T, (top right) invariant mass of the t¯ tpair mt¯ t, (bottom left) average top absolute rapidity |yt|and (bottom right) absolute boosted rapidity of the t¯ tpair, |yb t¯ t|. The bin-to-bin uncorrelated part of the data uncertainties is shown as black error bars, while the total uncertainties are shown as a yellow band. The cross sections are compared with the predictions computed with the PDFs resulting from the ATLASpdf21 fit. The solid line shows the predictions without shifts of the systematic uncertainties, while for the dashed line the bjparameters associated with the experimental systematic uncertainties as shown in Eq. (1) are allowed to vary to minimise the χ2.Theredbandrepresents the full uncertainty (experimental (evaluated with T=3) + model + parameterisation) of the fit prediction 123 Eur. Phys. J. C (2022) 82 :438 Page 49 of 70 438 γ T /dE 8 TeV σ/d γ T /dE 13 TeV σ d 20− 10− 0 10 20 30 40 | < 0.6 γ η, | -1 /20.2 fb -1 8 TeV, 3.2 fbγ 13 TeV/γ ATLAS Data uncorrelatedδ totalδ + shifts ATLASpdf21, full uncertainties [GeV] γ T E 500 1000 1500 Theory/Data 0.5 1 1.5 ATLAS 500 1000 γ T /dE 8 TeV σ /d γ T /dE 13 TeV σ d 5− 0 5 10 15 | < 1.37 γ η, 0.6 < | -1 /20.2 fb -1 8 TeV, 3.2 fbγ 13 TeV/γ ATLAS Data uncorrelatedδ totalδ + shifts ATLASpdf21, full uncertainties [GeV] γ T E 500 1000 Theory/Data 0.6 0.8 1 1.2 1.4 ATLAS 200 400 600 γ T /dE 8 TeV σ/d γ T /dE 13 TeV σ d 10− 5− 0 5 10 15 20 | < 1.81 γ η, 1.56 < | -1 /20.2 fb -1 8 TeV, 3.2 fbγ 13 TeV/γ ATLAS Data uncorrelatedδ totalδ + shifts ATLASpdf21, full uncertainties [GeV] γ T E 200 400 600 Theory/Data 0.8 1 1.2 ATLAS 200 400 600 γ T /dE 8 TeV σ /d γ T /dE 13 TeV σ d 30− 20− 10− 0 10 20 30 40 50 60 | < 2.37 γ η, 1.81 < | -1 /20.2 fb -1 8 TeV, 3.2 fbγ 13 TeV/γ ATLAS Data uncorrelatedδ totalδ + shifts ATLASpdf21, full uncertainties [GeV] γ T E 200 400 600 Theory/Data 0.8 1 1.2 ATLAS Fig. 36 The ratios of the cross sections for inclusive isolated-photon production at 8 and 13 TeV in Ref. [14] (black points) as functions of the photon transverse energy, Eγ T,forEγ T>125 GeV in bins of photon absolute pseudorapidity, |ηγ|.Topleft:|ηγ|<0.6. Top right: 0.6<|ηγ|<1.37. Bottom left: 1.56 <|ηγ|<1.81. Bottom right: 1.81 <|ηγ|<2.37. The bin-to-bin uncorrelated part of the data uncertainties is shown as black error bars, while the total uncertainties are shown as a yellow band. The cross sections are compared with the predictions computed with the PDFs resulting from the ATLASpdf21 fit. The solid line shows the predictions without shifts of the systematic uncertainties, while for the dashed line the bjparameters associated with the experimental systematic uncertainties as shown in Eq. (1)are allowed to vary to minimise the χ2. The red band represents the full uncertainty (experimental (evaluated with T=3) + model + parameterisation) of the fit prediction 123 438 Page 50 of 70 Eur. Phys. J. C (2022) 82 :438 Fig. 37 The differential cross-section measurements of inclusive jet production at 8 TeV in Ref. [17] (black points), for R=0.6, as a function of the jet pjet T, in six bins of absolute rapidity, |yjet|.Topleft: |yjet|<0.5. Top right: 0.5<|yjet|<1.0. Middle left: 1.0<|yjet|<1.5. Middle right: 1.5<|yjet|<2.0. Bottom left: 2.0<|yjet|<2.5. Bottom right: 2.5<|yjet|<3.0. The bin-to-bin uncorrelated part of the data uncertainties is shown as black error bars, while the total uncertainties are shown as ayellowband.Thecross sections are compared with the predictions computed for the scale choice pjet T, with the PDFs resulting from the ATLASpdf21 fit. The solid line shows the predictions without shifts of the systematic uncertainties, while for the dashed line the bj parameters associated with the experimental systematic uncertainties as shown in Eq. (1) are allowed to vary to minimise the χ2.Theredbandrepresents the full uncertainty (experimental (evaluated with T=3) + model + parameterisation) of the fit prediction 60 100 200 1000 2000 [pb/GeV] jet T |dp jet /d|y σ d 15− 10 13− 10 11− 10 9− 10 7− 10 5− 10 3− 10 1− 10 10 3 10 5 10 7 10 9 10 11 10 | < 0.5 jet , |y -1 = 8 TeV, 20.2 fbs j, →pp ATLAS Data uncorrelatedδ totalδ + shifts ATLASpdf21, full uncertainties [GeV] jet T p 60 100 200 1000 2000 Theory/Data 0.6 0.8 1 1.2 1.4 ATLAS 60 100 200 1000 2000 [pb/GeV] jet T |dp jet /d|yσ d 15− 10 13− 10 11− 10 9− 10 7− 10 5− 10 3− 10 1− 10 10 3 10 5 10 7 10 9 10 11 10 | < 1.0 jet |y≤, 0.5 -1 = 8 TeV, 20.2 fbs j, →pp ATLAS Data uncorrelatedδ totalδ + shifts ATLASpdf21, full uncertainties [GeV] jet T p 60 100 200 1000 2000 Theory/Data 0.6 0.8 1 1.2 1.4 ATLAS 60 100 200 300 1000 2000 [pb/GeV] jet T |dp jet /d|y σ d 15− 10 13− 10 11− 10 9− 10 7− 10 5− 10 3− 10 1− 10 10 3 10 5 10 7 10 9 10 11 10 | < 1.5 jet |y≤, 1.0 -1 = 8 TeV, 20.2 fbs j, →pp ATLAS Data uncorrelatedδ totalδ + shifts ATLASpdf21, full uncertainties [GeV] jet T p 60 100 200 300 1000 2000 Theory/Data 0.6 0.8 1 1.2 1.4 ATLAS 70 100 200 300 1000 [pb/GeV] jet T |dp jet /d|yσ d 15− 10 13− 10 11− 10 9− 10 7− 10 5− 10 3− 10 1− 10 10 3 10 5 10 7 10 9 10 11 10 | < 2.0 jet |y≤, 1.5 -1 = 8 TeV, 20.2 fbs j, →pp ATLAS Data uncorrelatedδ totalδ + shifts ATLASpdf21, full uncertainties [GeV] jet T p 70 100 200 300 1000 Theory/Data 0.6 0.8 1 1.2 1.4 ATLAS 70 100 200 300 [pb/GeV] jet T |dp jet /d|y σ d 15− 10 13− 10 11− 10 9− 10 7− 10 5− 10 3− 10 1− 10 10 3 10 5 10 7 10 9 10 11 10 | < 2.5 jet |y≤, 2.0 -1 = 8 TeV, 20.2 fbs j, →pp ATLAS Data uncorrelatedδ totalδ + shifts ATLASpdf21, full uncertainties [GeV] jet T p 70 100 200 300 Theory/Data 0.6 0.8 1 1.2 1.4 ATLAS 70 100 200 300 400 [pb/GeV] jet T |dp jet /d|yσ d 15− 10 13− 10 11− 10 9− 10 7− 10 5− 10 3− 10 1− 10 10 3 10 5 10 7 10 9 10 11 10 | < 3.0 jet |y≤, 2.5 -1 = 8 TeV, 20.2 fbs j, →pp ATLAS Data uncorrelatedδ totalδ + shifts ATLASpdf21, full uncertainties [GeV] jet T p 70 100 200 300 400 Theory/Data 0.6 0.8 1 1.2 1.4 ATLAS 123 Eur. Phys. J. C (2022) 82 :438 Page 51 of 70 438 0123 | [pb] Z /d|yσ d 60− 40− 20− 0 20 40 60 80 100 120 -1 = 1.96 TeV: 2.1 fbs Z, →pp CDF Data uncorrelatedδ totalδ + shifts ATLASpdf21, full uncertainties | Z |y 0123 Theory/Data 0.8 1 1.2 ATLAS 0123 W y A 0.6− 0.4− 0.2− 0 0.2 0.4 0.6 0.8 1 -1 = 1.96 TeV, 1.0 fbs W, → p p CDF Data uncorrelatedδ totalδ + shifts ATLASpdf21, full uncertainties | W |y 0123 Theory/Data 0.5 1 ATLAS Fig. 38 The differential cross-section measurements of (left) Zand (right) Wbosons in Refs. [88,89] (black points) as a function of the absolute rapidity of the boson, |yZ|or |yW|. The bin-to-bin uncorrelated part of the data uncertainties is shown as black error bars, while the total uncertainties are shown as a yellow band. The cross sections are compared with the predictions computed with the PDFs resulting from the ATLASpdf21 fit. The solid line shows the predictions without shifts of the systematic uncertainties, while for the dashed line the bj parameters associated with the experimental systematic uncertainties as shown in Eq. (1) are allowed to vary to minimise the χ2.Theredband represents the full uncertainty (experimental (evaluated with T=3) + model + parameterisation) of the fit prediction D Comparison with extra data sets not included in the fit The ATLASpdf21 fit does not include data from the Tevatron or from fixed-target DY data. In this appendix the predictions oftheATLASpdf21fitforsomeofthesedatasets,whichwere found to be most impactful in the global fits, are explored and found to be satisfactory. In Figs. 38 and 39 the ATLASpdf21 fit is shown in comparison with Tevatron Wand Zdata. The χ2/NDF values for the CDF data are 31/28 for the Z data and 35/13 for the W-asymmetry data. The χ2/NDF values for the D0 data are 23/28 for the Z data, 25/13 for the W-electron asymmetry data and 13/10 for the W-muon asymmetry data. The ATLASpdf21 fit therefore provides a fair description, χ2/NDF =126/92, of these Tevatron data, which mostly influence the high-xvalence quarks. 123 438 Page 52 of 70 Eur. Phys. J. C (2022) 82 :438 012 | Z /d|yσ d⋅σ 1/ 0.2− 0.1− 0 0.1 0.2 0.3 0.4 -1 = 1.96 TeV, 0.4 fbs Z, → pp D0 Data uncorrelatedδ totalδ + shifts ATLASpdf21, full uncertainties | Z |y 012 Theory/Data 0.5 1 1.5 ATLAS 0123 [%] e η A 120− 100− 80− 60− 40− 20− 0 20 40 60 -1 = 1.96 TeV, 9.7 fbs,ν e→ W → pp > 25 GeV e T D0 Data p uncorrelatedδ totalδ + shifts ATLASpdf21, full uncertainties | e η | 0123 Theory/Data 0 2 4 ATLAS 0 0.5 1 1.5 2 [%] μ η A 15− 10− 5− 0 5 10 15 20 25 -1 = 1.96 TeV, 7.3 fbs,μν→ W → pp > 25 GeV μ T D0 Data p uncorrelatedδ totalδ + shifts ATLASpdf21, full uncertainties | μ η | 0 0.5 1 1.5 2 Theory/Data 0.5 1 ATLAS Fig. 39 The differential cross-sectionmeasurements of (left) Z, (right) W(electron decay channel) and (bottom) W(muon decay channel) in Refs. [90–92] (black points) as a function of the absolute rapidity of the Zboson or absolute pseudorapidity of the decay lepton. The bin-to-bin uncorrelated part of the data uncertainties is shown as black error bars, while the total uncertainties are shown as a yellow band. The cross sections are compared with the predictions computed with the PDFs resulting from the ATLASpdf21 fit. The solid line shows the predictions without shifts of the systematic uncertainties, while for the dashed line the bjparameters associated with the experimental systematic uncertainties as shown in Eq. (1) are allowed to vary to minimise the χ2.The red band represents the full uncertainty (experimental (evaluated with T=3) + model + parameterisation) of the fit prediction Itisalsointerestingtoconsiderthedescriptionofthefixedtarget Drell–Yan data from E866 and E906, since these are uniquely able to constrain the difference x(¯ d−¯u)at high x. Figure 40 compares the predictions of the ATLASpdf21 fit with the E866 pD/pp data [93] in three mass regions. The χ2/NDF values are 9.6/10, 14/14 and 21/15 for the low-, intermediateand high-mass regions, respectively. Thus, the description of the E866 data, which mostly give information about high-xsea quarks, is good. However, the high-mass region, which covers larger Bjorken-x, is not as well fitted as the lower-mass regions. The ATLASpdf21 fit is in better agreementwith thenewdata from E906[86]in thehigh-mass region, as seen in Fig. 41, which compares ATLASpdf21 and other PDFs with the x¯ d/x¯uratios extracted from E866 and E906. 123 Eur. Phys. J. C (2022) 82 :438 Page 53 of 70 438 0.05 0.1 0.15 pp σ/2 pd σ 0.4 0.6 0.8 1 1.2 1.4 1.6 →μμpp, pd < 8.8 GeV μμ E866 Data 4.0 < m uncorrelatedδ totalδ + shifts ATLASpdf21, full uncertainties 2 x 0.05 0.1 0.15 Theory/Data 0.8 1 1.2 ATLAS 0.1 0.2 0.3 pp σ /2 pd σ 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 pp, pd→μμ μμ < 8.8 GeV, 10.8 GeV < m μμ E866 Data 4.3 < m uncorrelatedδ totalδ + shifts ATLASpdf21, full uncertainties 2 x 0.1 0.2 0.3 Theory/Data 0.8 1 1.2 ATLAS 00.10.20.3 pp σ/2 pd σ 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 pp, pd→μμ μμ < 9.0 GeV, 10.7 GeV < m μμ E866 Data 4.5 < m uncorrelatedδ totalδ + shifts ATLASpdf21, full uncertainties 2 x 00.10.20.3 Theory/Data 0.6 0.8 1 1.2 1.4 ATLAS Fig. 40 σpD/2σpp from Ref. [93] (black points) in the (top left) lowmass, (top right) intermediate-mass and (bottom) high-mass regions as a function of x2. The bin-to-bin uncorrelated part of the data uncertainties is shown as black error bars, while the total uncertainties are shown as a yellow band. The cross sections are compared with the predictions computed with the PDFs resulting from the ATLASpdf21 fit. The solid line shows the predictions without shifts of the systematic uncertainties, while for the dashed line the bjparameters associated with the experimental systematic uncertainties as shown in Eq. (1) are allowed to vary to minimise the χ2. The red band represents the full uncertainty (experimental (evaluated with T=3) + model + parameterisation) of the fit prediction 123 438 Page 54 of 70 Eur. Phys. J. C (2022) 82 :438 x 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 u/xdx 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 ATLAS ATLASpdf21 (T=1) CT18A MSHT20 NNPDF3.1 NuSea (E866) SeaQuest (E906) ATLASpdf21 (T=1) total unc. 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Zwalinski35 1Department of Physics, University of Adelaide, Adelaide, Australia 2Department of Physics, University of Alberta, Edmonton, AB, Canada 3(a)Department of Physics, Ankara University, Ankara, Turkey; (b)Application and Research Center for Advanced Studies, Istanbul Aydin University, Istanbul, Turkey; (c)Division of Physics, TOBB University of Economics and Technology, Ankara, Turkey 4LAPP, Univ. Savoie Mont Blanc, CNRS/IN2P3, Annecy , France 5High Energy Physics Division, Argonne National Laboratory, Argonne, IL, USA 6Department of Physics, University of Arizona, Tucson, AZ, USA 7Department of Physics, University of Texas at Arlington, Arlington, TX, USA 8Physics Department, National and Kapodistrian University of Athens, Athens, Greece 9Physics Department, National Technical University of Athens, Zografou, Greece 10 Department of Physics, University of Texas at Austin, Austin, TX, USA 11 (a)Faculty of Engineering and Natural Sciences, Bahcesehir University, Istanbul, Turkey; (b)Faculty of Engineering and Natural Sciences, Istanbul Bilgi University, Istanbul, Turkey; (c)Department of Physics, Bogazici University, Istanbul, Turkey; (d)Department of Physics Engineering, Gaziantep University, Gaziantep, Turkey 12 Institute of Physics, Azerbaijan Academy of Sciences, Baku, Azerbaijan 13 Institut de Física d’Altes Energies (IFAE), Barcelona Institute of Science and Technology, Barcelona, Spain 14 (a)Institute of High Energy Physics, Chinese Academy of Sciences, Beijing, China; (b)Physics Department, Tsinghua University, Beijing, China; (c)Department of Physics, Nanjing University, Nanjing, China; (d)University of Chinese Academy of Science (UCAS), Beijing, China 15 Institute of Physics, University of Belgrade, Belgrade, Serbia 16 Department for Physics and Technology, University of Bergen, Bergen, Norway 17 Physics Division, Lawrence Berkeley National Laboratory and University of California, Berkeley, CA, USA 18 Institut für Physik, Humboldt Universität zu Berlin, Berlin, Germany 123 438 Page 66 of 70 Eur. Phys. J. C (2022) 82 :438 19 Albert Einstein Center for Fundamental Physics and Laboratory for High Energy Physics, University of Bern, Bern, Switzerland 20 School of Physics and Astronomy, University of Birmingham, Birmingham, UK 21 (a)Facultad de Ciencias y Centro de Investigaciónes, Universidad Antonio Nariño, Bogotá, Colombia; (b)Departamento de Física, Universidad Nacional de Colombia, Bogotá, Colombia 22 (a)Dipartimento di Fisica e Astronomia A. Righi, Università di Bologna, Bologna, Italy; (b)INFN Sezione di Bologna, Bologna, Italy 23 Physikalisches Institut, Universität Bonn, Bonn, Germany 24 Department of Physics, Boston University, Boston, MA, USA 25 Department of Physics, Brandeis University, Waltham, MA, USA 26 (a)Transilvania University of Brasov, Brasov, Romania; (b)Horia Hulubei National Institute of Physics and Nuclear Engineering, Bucharest, Romania; (c)Department of Physics, Alexandru Ioan Cuza University of Iasi, Iasi, Romania; (d)Physics Department, National Institute for Research and Development of Isotopic and Molecular Technologies, Cluj-Napoca, Romania; (e)University Politehnica Bucharest, Bucharest, Romania; (f)West University in Timisoara, Timisoara, Romania 27 (a)Faculty of Mathematics, Physics and Informatics, Comenius University, Bratislava, Slovak Republic; (b)Department of Subnuclear Physics, Institute of Experimental Physics of the Slovak Academy of Sciences, Kosice, Slovak Republic 28 Physics Department, Brookhaven National Laboratory, Upton, NY, USA 29 Departamento de Física (FCEN) and IFIBA, Universidad de Buenos Aires and CONICET, Buenos Aires, Argentina 30 California State University, Long Beach, CA, USA 31 Cavendish Laboratory, University of Cambridge, Cambridge, UK 32 (a)Department of Physics, University of Cape Town, Cape Town, South Africa; (b)iThemba Labs, Western Cape, South Africa; (c)Department of Mechanical Engineering Science, University of Johannesburg, Johannesburg, South Africa; (d)National Institute of Physics, University of the Philippines, Diliman, Philippines; (e)Department of Physics, University of South Africa, Pretoria, South Africa; (f)School of Physics, University of the Witwatersrand, Johannesburg, South Africa 33 Department of Physics, Carleton University, Ottawa, ON, Canada 34 (a)Faculté des Sciences Ain Chock, Réseau Universitaire de Physique des Hautes Energies-Université Hassan II, Casablanca, Morocco; (b)Faculté des Sciences, Université Ibn-Tofail, Kénitra, Morocco; (c)Faculté des Sciences Semlalia, Université Cadi Ayyad, LPHEA-Marrakech, Morocco; (d)LPMR, Faculté des Sciences, Université Mohamed Premier, Oujda, Morocco; (e)Faculté des sciences, Université Mohammed V, Rabat, Morocco; (f)Mohammed VI Polytechnic University, Ben Guerir, Morocco 35 CERN, Geneva, Switzerland 36 Enrico Fermi Institute, University of Chicago, Chicago, IL, USA 37 LPC, Université Clermont Auvergne, CNRS/IN2P3, Clermont-Ferrand, France 38 Nevis Laboratory, Columbia University, Irvington, NY, USA 39 Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark 40 (a)Dipartimento di Fisica, Università della Calabria, Rende, Italy; (b)INFN Gruppo Collegato di Cosenza, Laboratori Nazionali di Frascati, Frascati, Italy 41 Physics Department, Southern Methodist University, Dallas, TX, USA 42 Physics Department, University of Texas at Dallas, Richardson, TX, USA 43 National Centre for Scientific Research “Demokritos”, Agia Paraskevi, Greece 44 (a)Department of Physics, Stockholm University, Stockholm, Sweden; (b)Oskar Klein Centre, Stockholm, Sweden 45 Deutsches Elektronen-Synchrotron DESY, Hamburg and Zeuthen, Germany 46 Fakultät Physik , Technische Universität Dortmund, Dortmund, Germany 47 Institut für Kernund Teilchenphysik, Technische Universität Dresden, Dresden, Germany 48 Department of Physics, Duke University, Durham, NC, USA 49 SUPA-School of Physics and Astronomy, University of Edinburgh, Edinburgh, UK 50 INFN e Laboratori Nazionali di Frascati, Frascati, Italy 51 Physikalisches Institut, Albert-Ludwigs-Universität Freiburg, Freiburg, Germany 52 II. Physikalisches Institut, Georg-August-Universität Göttingen, Göttingen, Germany 53 Département de Physique Nucléaire et Corpusculaire, Université de Genève, Geneva, Switzerland 54 (a)Dipartimento di Fisica, Università di Genova, Genoa, Italy; (b)INFN Sezione di Genova, Genoa, Italy 123 Eur. Phys. J. C (2022) 82 :438 Page 67 of 70 438 55 II. Physikalisches Institut, Justus-Liebig-Universität Giessen, Giessen, Germany 56 SUPA-School of Physics and Astronomy, University of Glasgow, Glasgow, UK 57 LPSC, Université Grenoble Alpes, CNRS/IN2P3, Grenoble INP, Grenoble, France 58 Laboratory for Particle Physics and Cosmology, Harvard University, Cambridge, MA, USA 59 (a)Department of Modern Physics and State Key Laboratory of Particle Detection and Electronics, University of Science and Technology of China, Hefei, China; (b)Institute of Frontier and Interdisciplinary Science and Key Laboratory of Particle Physics and Particle Irradiation (MOE), Shandong University, Qingdao, China; (c)School of Physics and Astronomy, Shanghai Jiao Tong University, Key Laboratory for Particle Astrophysics and Cosmology (MOE), SKLPPC, Shanghai, China; (d)Tsung-Dao Lee Institute, Shanghai, China 60 (a)Kirchhoff-Institut für Physik, Ruprecht-Karls-Universität Heidelberg, Heidelberg, Germany; (b)Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg, Germany 61 (a)Department of Physics, Chinese University of Hong Kong, Shatin, N.T., Hong Kong; (b)Department of Physics, University of Hong Kong, Hong Kong, China; (c)Department of Physics and Institute for Advanced Study, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong 62 Department of Physics, National Tsing Hua University, Hsinchu, Taiwan 63 IJCLab, Université Paris-Saclay, CNRS/IN2P3, 91405 Orsay, France 64 Department of Physics, Indiana University, Bloomington, IN, USA 65 (a)INFN Gruppo Collegato di Udine, Sezione di Trieste, Udine, Italy; (b)ICTP, Trieste, Italy; (c)Dipartimento Politecnico di Ingegneria e Architettura, Università di Udine, Udine, Italy 66 (a)INFN Sezione di Lecce, Lecce, Italy; (b)Dipartimento di Matematica e Fisica, Università del Salento, Lecce, Italy 67 (a)INFN Sezione di Milano, Milan, Italy; (b)Dipartimento di Fisica, Università di Milano, Milan, Italy 68 (a)INFN Sezione di Napoli, Naples, Italy; (b)Dipartimento di Fisica, Università di Napoli, Naples, Italy 69 (a)INFN Sezione di Pavia, Pavia, Italy; (b)Dipartimento di Fisica, Università di Pavia, Pavia, Italy 70 (a)INFN Sezione di Pisa, Pisa, Italy; (b)Dipartimento di Fisica E. Fermi, Università di Pisa, Pisa, Italy 71 (a)INFN Sezione di Roma, Rome, Italy; (b)Dipartimento di Fisica, Sapienza Università di Roma, Rome, Italy 72 (a)INFN Sezione di Roma Tor Vergata, Rome, Italy; (b)Dipartimento di Fisica, Università di Roma Tor Vergata, Rome, Italy 73 (a)INFN Sezione di Roma Tre, Rome, Italy; (b)Dipartimento di Matematica e Fisica, Università Roma Tre, Rome, Italy 74 (a)INFN-TIFPA, Trento, Italy; (b)Università degli Studi di Trento, Trento, Italy 75 Institut für Astround Teilchenphysik, Leopold-Franzens-Universität, Innsbruck, Austria 76 University of Iowa, Iowa City, IA, USA 77 Department of Physics and Astronomy, Iowa State University, Ames, IA, USA 78 Joint Institute for Nuclear Research, Dubna, Russia 79 (a)Departamento de Engenharia Elétrica, Universidade Federal de Juiz de Fora (UFJF), Juiz de Fora, Brazil; (b)Universidade Federal do Rio De Janeiro COPPE/EE/IF, Rio de Janeiro, Brazil; (c)Instituto de Física, Universidade de São Paulo, São Paulo, Brazil 80 KEK, High Energy Accelerator Research Organization, Tsukuba, Japan 81 Graduate School of Science, Kobe University, Kobe, Japan 82 (a)Faculty of Physics and Applied Computer Science, AGH University of Science and Technology, Krakow, Poland; (b)Marian Smoluchowski Institute of Physics, Jagiellonian University, Krakow, Poland 83 Institute of Nuclear Physics Polish Academy of Sciences, Krakow, Poland 84 Faculty of Science, Kyoto University, Kyoto, Japan 85 Kyoto University of Education, Kyoto, Japan 86 Research Center for Advanced Particle Physics and Department of Physics, Kyushu University, Fukuoka , Japan 87 Instituto de Física La Plata, Universidad Nacional de La Plata and CONICET, La Plata, Argentina 88 Physics Department, Lancaster University, Lancaster, UK 89 Oliver Lodge Laboratory, University of Liverpool, Liverpool, UK 90 Department of Experimental Particle Physics, Jožef Stefan Institute and Department of Physics, University of Ljubljana, Ljubljana, Slovenia 91 School of Physics and Astronomy, Queen Mary University of London, London, UK 92 Department of Physics, Royal Holloway University of London, Egham, UK 93 Department of Physics and Astronomy, University College London, London, UK 94 Louisiana Tech University, Ruston, LA, USA 123 438 Page 68 of 70 Eur. Phys. J. C (2022) 82 :438 95 Fysiska institutionen, Lunds universitet, Lund, Sweden 96 Departamento de Física Teorica C-15 and CIAFF, Universidad Autónoma de Madrid, Madrid, Spain 97 Institut für Physik, Universität Mainz, Mainz, Germany 98 School of Physics and Astronomy, University of Manchester, Manchester, UK 99 CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France 100 Department of Physics, University of Massachusetts, Amherst, MA, USA 101 Department of Physics, McGill University, Montreal, QC, Canada 102 School of Physics, University of Melbourne, Victoria, Australia 103 Department of Physics, University of Michigan, Ann Arbor, MI, USA 104 Department of Physics and Astronomy, Michigan State University, East Lansing, MI, USA 105 B.I. Stepanov Institute of Physics, National Academy of Sciences of Belarus, Minsk, Belarus 106 Research Institute for Nuclear Problems of Byelorussian State University, Minsk, Belarus 107 Group of Particle Physics, University of Montreal, Montreal, QC, Canada 108 P.N. Lebedev Physical Institute of the Russian Academy of Sciences, Moscow, Russia 109 National Research Nuclear University MEPhI, Moscow, Russia 110 D.V. Skobeltsyn Institute of Nuclear Physics, M.V. Lomonosov Moscow State University, Moscow, Russia 111 Fakultät für Physik, Ludwig-Maximilians-Universität München, Munich, Germany 112 Max-Planck-Institut für Physik (Werner-Heisenberg-Institut), Munich, Germany 113 Graduate School of Science and Kobayashi-Maskawa Institute, Nagoya University, Nagoya, Japan 114 Department of Physics and Astronomy, University of New Mexico, Albuquerque, NM, USA 115 Institute for Mathematics, Astrophysics and Particle Physics, Radboud University/Nikhef, Nijmegen, The Netherlands 116 Nikhef National Institute for Subatomic Physics and University of Amsterdam, Amsterdam, The Netherlands 117 Department of Physics, Northern Illinois University, DeKalb, IL, USA 118 (a)Budker Institute of Nuclear Physics and NSU, SB RAS, Novosibirsk, Russia; (b)Novosibirsk State University Novosibirsk, Novosibirsk, Russia 119 Institute for High Energy Physics of the National Research Centre Kurchatov Institute, Protvino, Russia 120 Institute for Theoretical and Experimental Physics named by A.I. Alikhanov of National Research Centre “Kurchatov Institute”, Moscow, Russia 121 (a)New York University Abu Dhabi, Abu Dhabi, United Arab Emirates; (b)United Arab Emirates University, Al Ain, United Arab Emirates; (c)University of Sharjah, Sharjah, United Arab Emirates 122 Department of Physics, New York University, New York, NY, USA 123 Ochanomizu University, Otsuka, Bunkyo-ku, Tokyo, Japan 124 Ohio State University, Columbus, OH, USA 125 Homer L. Dodge Department of Physics and Astronomy, University of Oklahoma, Norman, OK, USA 126 Department of Physics, Oklahoma State University, Stillwater, OK, USA 127 Palacký University, Joint Laboratory of Optics, Olomouc, Czech Republic 128 Institute for Fundamental Science, University of Oregon, Eugene, OR, USA 129 Graduate School of Science, Osaka University, Osaka, Japan 130 Department of Physics, University of Oslo, Oslo, Norway 131 Department of Physics, Oxford University, Oxford, UK 132 LPNHE, Sorbonne Université, Université de Paris, CNRS/IN2P3, Paris, France 133 Department of Physics, University of Pennsylvania, Philadelphia, PA, USA 134 Konstantinov Nuclear Physics Institute of National Research Centre “Kurchatov Institute”, PNPI, St. Petersburg, Russia 135 Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, PA, USA 136 (a)Laboratório de Instrumentação e Física Experimental de Partículas-LIP, Lisbon, Portugal; (b)Departamento de Física, Faculdade de Ciências, Universidade de Lisboa, Lisbon, Portugal; (c)Departamento de Física, Universidade de Coimbra, Coimbra, Portugal; (d)Centro de Física Nuclear da Universidade de Lisboa, Lisbon, Portugal; (e)Departamento de Física, Universidade do Minho, Braga, Portugal; (f)Departamento de Física Teórica y del Cosmos, Universidad de Granada, Granada, Spain; (g)Instituto Superior Técnico, Universidade de Lisboa, Lisbon, Portugal 137 Institute of Physics of the Czech Academy of Sciences, Prague, Czech Republic 138 Czech Technical University in Prague, Prague, Czech Republic 139 Charles University, Faculty of Mathematics and Physics, Prague, Czech Republic 140 Particle Physics Department, Rutherford Appleton Laboratory, Didcot, UK 123 Eur. Phys. J. C (2022) 82 :438 Page 69 of 70 438 141 IRFU, CEA, Université Paris-Saclay, Gif-sur-Yvette, France 142 Santa Cruz Institute for Particle Physics, University of California Santa Cruz, Santa Cruz, CA, USA 143 (a)Departamento de Física, Pontificia Universidad Católica de Chile, Santiago, Chile; (b)Instituto de Investigación Multidisciplinario en Ciencia y Tecnología y Departamento de Física, Universidad de La Serena, La Serena, Chile; (c)Universidad Andres Bello, Department of Physics, Santiago, Chile; (d)Instituto de Alta Investigación, Universidad de Tarapacá, Arica, Chile; (e)Departamento de Física, Universidad Técnica Federico Santa María, Valparaíso, Chile 144 Universidade Federal de São João del Rei (UFSJ), São João del Rei, Brazil 145 Department of Physics, University of Washington, Seattle, WA, USA 146 Department of Physics and Astronomy, University of Sheffield, Sheffield, UK 147 Department of Physics, Shinshu University, Nagano, Japan 148 Department Physik, Universität Siegen, Siegen, Germany 149 Department of Physics, Simon Fraser University, Burnaby, BC, Canada 150 SLAC National Accelerator Laboratory, Stanford, CA, USA 151 Department of Physics, Royal Institute of Technology, Stockholm, Sweden 152 Departments of Physics and Astronomy, Stony Brook University, Stony Brook, NY, USA 153 Department of Physics and Astronomy, University of Sussex, Brighton, UK 154 School of Physics, University of Sydney, Sydney, Australia 155 Institute of Physics, Academia Sinica, Taipei, Taiwan 156 (a)E. Andronikashvili Institute of Physics, Iv. Javakhishvili Tbilisi State University, Tbilisi, Georgia; (b)High Energy Physics Institute, Tbilisi State University, Tbilisi, Georgia 157 Department of Physics, Technion, Israel Institute of Technology, Haifa, Israel 158 Raymond and Beverly Sackler School of Physics and Astronomy, Tel Aviv University, Tel Aviv, Israel 159 Department of Physics, Aristotle University of Thessaloniki, Thessaloniki, Greece 160 International Center for Elementary Particle Physics and Department of Physics, University of Tokyo, Tokyo, Japan 161 Department of Physics, Tokyo Institute of Technology, Tokyo, Japan 162 Tomsk State University, Tomsk, Russia 163 Department of Physics, University of Toronto, Toronto, ON, Canada 164 (a)TRIUMF, Vancouver, BC, Canada; (b)Department of Physics and Astronomy, York University, Toronto, ON, Canada 165 Division of Physics and Tomonaga Center for the History of the Universe, Faculty of Pure and Applied Sciences, University of Tsukuba, Tsukuba, Japan 166 Department of Physics and Astronomy, Tufts University, Medford, MA, USA 167 Department of Physics and Astronomy, University of California Irvine, Irvine, CA, USA 168 Department of Physics and Astronomy, University of Uppsala, Uppsala, Sweden 169 Department of Physics, University of Illinois, Urbana, IL, USA 170 Instituto de Física Corpuscular (IFIC), Centro Mixto Universidad de Valencia-CSIC, Valencia, Spain 171 Department of Physics, University of British Columbia, Vancouver, BC, Canada 172 Department of Physics and Astronomy, University of Victoria, Victoria, BC, Canada 173 Fakultät für Physik und Astronomie, Julius-Maximilians-Universität Würzburg, Würzburg, Germany 174 Department of Physics, University of Warwick, Coventry, UK 175 Waseda University, Tokyo, Japan 176 Department of Particle Physics and Astrophysics, Weizmann Institute of Science, Rehovot, Israel 177 Department of Physics, University of Wisconsin, Madison, WI, USA 178 Fakultät für Mathematik und Naturwissenschaften, Fachgruppe Physik, Bergische Universität Wuppertal, Wuppertal, Germany 179 Department of Physics, Yale University, New Haven, CT, USA aAlso at Borough of Manhattan Community College, City University of New York, New York, NY, USA bAlso at Bruno Kessler Foundation, Trento, Italy cAlso at Center for High Energy Physics, Peking University, Beijing, China dAlso at Centro Studi e Ricerche Enrico Fermi, Rome, Italy eAlso at CERN, Geneva, Switzerland fAlso at Département de Physique Nucléaire et Corpusculaire, Université de Genève, Geneva, Switzerland 123 438 Page 70 of 70 Eur. Phys. J. C (2022) 82 :438 gAlso at Departament de Fisica de la Universitat Autonoma de Barcelona, Barcelona, Spain hAlso at Department of Financial and Management Engineering, University of the Aegean, Chios, Greece iAlso at Department of Physics and Astronomy, Michigan State University, East Lansing, MI, USA jAlso at Department of Physics and Astronomy, University of Louisville, Louisville, KY, USA kAlso at Department of Physics, Ben Gurion University of the Negev, Beer Sheva, Israel lAlso at Department of Physics, California State University, East Bay, USA mAlso at Department of Physics, California State University, Sacramento, USA nAlso at Department of Physics, King’s College London, London, UK oAlso at Department of Physics, St. Petersburg State Polytechnical University, St. Petersburg, Russia pAlso at Department of Physics, University of Fribourg, Fribourg, Switzerland qAlso at Faculty of Physics, M.V. Lomonosov Moscow State University, Moscow, Russia rAlso at Graduate School of Science, Osaka University, Osaka, Japan sAlso at Hellenic Open University, Patras, Greece tAlso at Institucio Catalana de Recerca i Estudis Avancats, ICREA, Barcelona, Spain uAlso at Institut für Experimentalphysik, Universität Hamburg, Hamburg, Germany vAlso at Institute of Particle Physics (IPP), Victoria, Canada wAlso at Institute of Theoretical Physics, Ilia State University, Tbilisi, Georgia xAlso at Instituto de Fisica Teorica, IFT-UAM/CSIC, Madrid, Spain yAlso at Dept. of Physics, Istanbul University, Istanbul, Turkey zAlso at Istinye University, Istanbul, Turkey aa Also at Joint Institute for Nuclear Research, Dubna, Russia ab Also at Moscow Institute of Physics and Technology State University, Dolgoprudny, Russia ac Also at National Research Nuclear University MEPhI, Moscow, Russia ad Also at Physics Department, An-Najah National University, Nablus, Palestine ae Also at Physikalisches Institut, Albert-Ludwigs-Universität Freiburg, Freiburg, Germany af Also at The City College of New York, New York, NY, USA ag Also at TRIUMF, Vancouver, BC, Canada ah Also at Universita di Napoli Parthenope, Naples, Italy ai Also at University of Chinese Academy of Sciences (UCAS), Beijing, China aj Also at Physics Department, Yeditepe University, Istanbul, Turkey ∗Deceased 123