This paper presents an analysis at next-to-next-to-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\documentclass[12pt]{minimal}
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\begin{document}$$\sqrt{s} = 7$$\end{document}, 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±\documentclass[12pt]{minimal}
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\begin{document}$$Z/\gamma ^*$$\end{document} boson production, W±\documentclass[12pt]{minimal}
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\begin{document}$$t\bar{t}$$\end{document} production, 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.
publisher-imprint-nameSpringervolume-issue-count12issue-article-count51issue-toc-levels0issue-pricelist-year2022issue-copyright-holderThe Author(s)issue-copyright-year2022article-contains-esmNoarticle-numbering-styleContentOnlyarticle-registration-date-year2022article-registration-date-month3article-registration-date-day14article-toc-levels0toc-levels0volume-typeRegularjournal-productNonStandardArchiveJournalnumbering-styleContentOnlyarticle-grants-typeOpenChoicemetadata-grantOpenAccessabstract-grantOpenAccessbodypdf-grantOpenAccessbodyhtml-grantOpenAccessbibliography-grantOpenAccessesm-grantOpenAccessonline-firstfalsepdf-file-referenceBodyRef/PDF/10052_2022_Article_10217.pdfpdf-typeTypesettarget-typeOnlinePDFissue-typeRegulararticle-typeOriginalPaperjournal-subject-primaryPhysicsjournal-subject-secondaryElementary Particles, Quantum Field Theoryjournal-subject-secondaryNuclear Physics, Heavy Ions, Hadronsjournal-subject-secondaryQuantum Field Theories, String Theoryjournal-subject-secondaryMeasurement Science and Instrumentationjournal-subject-secondaryAstronomy, Astrophysics and Cosmologyjournal-subject-secondaryNuclear Energyjournal-subject-collectionPhysics and Astronomyopen-accesstrueIntroduction
Precise knowledge of the content of protons, the parton distribution functions (PDFs), is a necessary ingredient for accurate predictions of both Standard Model (SM) and beyond-the-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\documentclass[12pt]{minimal}
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\begin{document}$$Q^2$$\end{document}, 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 by measurement of different processes to constrain different regions of phase space, which span the kinematic region: 10-5≲x≲1\documentclass[12pt]{minimal}
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\begin{document}$$1 \lesssim Q^2 \lesssim 10^6~\text {GeV}{}^2$$\end{document}.
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 Q2\documentclass[12pt]{minimal}
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\begin{document}$$Q^2$$\end{document} and x. The PDF set HERAPDF2.0 was determined from HERA data alone [5] using information from e±p\documentclass[12pt]{minimal}
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\begin{document}$$e^{\pm } p$$\end{document} neutral-current (NC) and charged-current (CC) processes. The lower-Q2\documentclass[12pt]{minimal}
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\begin{document}$$Q^2$$\end{document} NC data constrain the low-x sea-quark distribution but are not able to distinguish between quark flavours in the sea at low x, or between the down-type quarks, d¯\documentclass[12pt]{minimal}
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\begin{document}$$Q^2$$\end{document}, together with the high-Q2\documentclass[12pt]{minimal}
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\begin{document}$$Q^2$$\end{document} CC data, constrains the valence distributions. The Q2\documentclass[12pt]{minimal}
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\begin{document}$$Q^2$$\end{document} dependence 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\documentclass[12pt]{minimal}
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\begin{document}$$\sqrt{s}$$\end{document}) 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±\documentclass[12pt]{minimal}
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\begin{document}$$\text {TeV}$$\end{document} 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\documentclass[12pt]{minimal}
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\begin{document}$$x<0.1$$\end{document} was improved. The effect of adding differential tt¯\documentclass[12pt]{minimal}
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\begin{document}$$\text {TeV}$$\end{document}, to the HERA data and the inclusive W,Z/γ∗\documentclass[12pt]{minimal}
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\begin{document}$$W,Z/\gamma ^*$$\end{document} data was studied in Ref. [10]. These tt¯\documentclass[12pt]{minimal}
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\begin{document}$$t\bar{t}$$\end{document} data are complementary to the W,Z/γ∗\documentclass[12pt]{minimal}
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\begin{document}$$W,Z/\gamma ^*$$\end{document} data in their power to constrain PDFs because they are sensitive to the high-x gluon distribution. The resulting PDF set was called ATLASepWZtop18. In Ref. [11], data on the production of W and Z bosons in association with jets (V+ jets) were added to the HERA data and inclusive W, Z/γ∗\documentclass[12pt]{minimal}
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\begin{document}$$Z/\gamma ^*$$\end{document} data, resulting in the ATLASepWZVjets20 PDF set. The V+ jets data are sensitive to partons at higher x than can be accessed by inclusive W and Z/γ∗\documentclass[12pt]{minimal}
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\begin{document}$$Z/\gamma ^*$$\end{document} data and, in particular, they constrain the d¯\documentclass[12pt]{minimal}
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\begin{document}$$\bar{s}$$\end{document} quarks at higher x.
It would clearly be advantageous to combine ATLAS W,Z/γ∗\documentclass[12pt]{minimal}
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\begin{document}$$t\bar{t}$$\end{document} data 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 described in the following. Firstly, the data on W [12] production and Z/γ∗\documentclass[12pt]{minimal}
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\begin{document}$$\text {TeV}$$\end{document} are added, providing further constraints on the valence PDFs and on the composition of the light-quark sea. Secondly, the direct photon production differential cross sections with 20.2 fb-1\documentclass[12pt]{minimal}
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\begin{document}$$\text {TeV}$$\end{document} are added in the form of their ratios [14], whereby many systematic uncertainties cancel out. Thirdly, the tt¯\documentclass[12pt]{minimal}
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\begin{document}$$\text {TeV}$$\end{document} [18] are considered. These direct photon data, tt¯\documentclass[12pt]{minimal}
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\begin{document}$$t\bar{t}$$\end{document} 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 uncertainties deriving from jet measurements, since these 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 2 presents the input data sets. Section 3 describes the theoretical framework of the fit. Section 4 presents the fitting methodology, including the definition of the fit χ2\documentclass[12pt]{minimal}
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\begin{document}$$\chi ^2$$\end{document} and details of the parameterisation. Section 5 presents 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 6 presents a study of the χ2\documentclass[12pt]{minimal}
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\begin{document}$$\chi ^2$$\end{document} tolerance and a comparison with other modern PDF sets. Section 7 gives the summary and conclusions. Appendix A gives a more detailed exposition of the impact of scale uncertainties for the inclusive W,Z/γ∗\documentclass[12pt]{minimal}
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\begin{document}$$W,Z/\gamma ^*$$\end{document} data. The impacts of inclusive jet data at different centre-of-mass energies are compared in Appendix B. Appendices C and D compare the ATLASpdf21 fit predictions with the ATLAS input data sets and with data sets from other experiments, respectively.
Input data setsDescription of data sets
The combined e±p\documentclass[12pt]{minimal}
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\begin{document}$$t\bar{t}$$\end{document} cross 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 tt¯\documentclass[12pt]{minimal}
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\begin{document}$$p_{\mathrm {T}}^t$$\end{document}. 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%\documentclass[12pt]{minimal}
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\begin{document}$$|y_{t\bar{t}}$$\end{document}|. 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].
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\begin{document}$$\text {TeV}$$\end{document}, 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 reaches higher absolute rapidity of the tt¯\documentclass[12pt]{minimal}
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\begin{document}$$y^\mathrm {b}_{t\bar{t}}$$\end{document}. 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 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 pTt\documentclass[12pt]{minimal}
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\begin{document}$$\text {TeV}$$\end{document}. 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 η\documentclass[12pt]{minimal}
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All the ATLAS input data sets for the QCD fit are summarised in Table 1.
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, the Z + jets and the inclusive jet spectra at 8 TeV\documentclass[12pt]{minimal}
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\begin{document}$$\chi ^2$$\end{document} of the fits to V+ jets, but has little impact on the fitted PDFs. Similarly, the parton shower systematic uncertainty is decorrelated between the pTt\documentclass[12pt]{minimal}
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\begin{document}$$\chi ^2$$\end{document} to acceptable levels. This decorrelation 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\documentclass[12pt]{minimal}
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\begin{document}$$\text {TeV}$$\end{document} 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’,5 and 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 pTjet\documentclass[12pt]{minimal}
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\begin{document}$$R=0.6$$\end{document} 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].6 Alternative decorrelation scenarios are also considered in Sect. 5.3.1.
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\begin{document}$$\text {TeV}$$\end{document}). Thus, correlations of lepton uncertainties between these inclusive W and Z measurements 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∗\documentclass[12pt]{minimal}
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\begin{document}$$\text {TeV}$$\end{document} 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∗\documentclass[12pt]{minimal}
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\begin{document}$$\text {TeV}$$\end{document}, is very similar to the impact of these data in Ref. [11], where uncombined 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 jet-measurement uncertainties are considered. The W+ jets data and the Z + jets data at 8 TeV\documentclass[12pt]{minimal}
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\begin{document}$$100\%$$\end{document} 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 tt¯\documentclass[12pt]{minimal}
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\begin{document}$$\text {TeV}$$\end{document} in the dilepton channel and other data sets cannot be applied because the correlations for the dilepton channel are supplied as a covariance matrix and the separate individual sources of uncertainty cannot 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 tt¯\documentclass[12pt]{minimal}
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\begin{document}$$t\bar{t}$$\end{document} data 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 rapidity and pT\documentclass[12pt]{minimal}
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\begin{document}$$p_\mathrm {T}$$\end{document} [17]. Consequently, there are no longer corresponding systematic sources in the V+ jets data and tt¯\documentclass[12pt]{minimal}
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\begin{document}$$t\bar{t}$$\end{document} data. 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.
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\begin{document}$$100\%$$\end{document} correlation and no correlation, yielding little difference between the resultant PDFs. For the central fit a correlation of 100%\documentclass[12pt]{minimal}
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\begin{document}$$100\%$$\end{document} is used.
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\begin{document}$$R=0.6$$\end{document}, 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 ∗\documentclass[12pt]{minimal}
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The measurement of the direct-photon production ratio already considered correlations between the data at 8 TeV\documentclass[12pt]{minimal}
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\begin{document}$$\text {TeV}$$\end{document} data sets.
Theoretical frameworkNNLO 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-flavour-number scheme of Refs. [38, 39], known as the optimised TRVFN scheme.
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 NLO in QCD and LO in EW in the form of fast interpolation grids and are corrected to NNLO QCD and NLO EW by K-factors. For the tt¯\documentclass[12pt]{minimal}
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\begin{document}$$t\bar{t}$$\end{document} lepton + jets channel the grids are calculated at NNLO in QCD directly, so no entry appears in the column ‘NLO QCD code’
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In the present analysis, the photon PDF within the proton is not accounted for and thus data sets with substantial sensitivity to the photon PDF, such as very high-mass Drell–Yan data, are excluded. It is estimated [40] that the photon takes only ∼0.3%\documentclass[12pt]{minimal}
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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 tt¯\documentclass[12pt]{minimal}
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\begin{document}$$t\bar{t}$$\end{document} production 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 K-factors 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 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 W and Z inclusive cross sections at 7 TeV\documentclass[12pt]{minimal}
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\begin{document}$$\text {TeV}$$\end{document} 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 DYNNLO 1.5 [41, 42] and FEWZ 3.1.b2 [43, 44] are compared. A small difference between the NNLO predictions of FEWZ and DYNNLO of up to ∼1%\documentclass[12pt]{minimal}
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\begin{document}$$Z/\gamma ^*$$\end{document} 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)%\documentclass[12pt]{minimal}
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\begin{document}$${\sim }3\%$$\end{document} 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 Gμ\documentclass[12pt]{minimal}
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\begin{document}$$\text {TeV}$$\end{document} were calculated by the authors of Ref. [52]. The renormalisation and factorisation scales are set to μr=μf=12mℓℓ2+pT,ℓℓ2+ΣpT,partons\documentclass[12pt]{minimal}
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\begin{document}$$10\%$$\end{document} 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.
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\begin{document}$$\text {TeV}$$\end{document} 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 mtt¯\documentclass[12pt]{minimal}
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\begin{document}$$\text {GeV}$$\end{document} is the pole mass. Non-perturbative corrections and their uncertainties are supplied with the data and these uncertainties are applied as correlated systematic uncertainties. For the dilepton channel, APPLgrid is interfaced to MCFMto 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\documentclass[12pt]{minimal}
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\begin{document}$$y_{t\bar{t}}$$\end{document}. 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]. For the 8 TeV\documentclass[12pt]{minimal}
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\begin{document}$$\text {TeV}$$\end{document} have been calculated by the authors of Ref. [53] in the form of fast interpolation grids [45, 54, 55] for the data in the lepton + jets channel [57]. The predictions for mtt¯\documentclass[12pt]{minimal}
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\begin{document}$$t\bar{t}$$\end{document} data is studied as part of the model uncertainties and found to be negligible (see Sect. 5.3.1). NLO EW corrections for the spectra are also considered, using K-factors taken from Ref. [56]. For the 13 TeV\documentclass[12pt]{minimal}
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\begin{document}$$p_{\mathrm {T}}^t$$\end{document}. Non-perturbative corrections and their uncertainties are again supplied with the data and these uncertainties are applied as correlated systematic uncertainties.
The NLO QCD predictions for direct photon production were taken from MCFM interfaced to APPLgrid, and K-factors from Ref. [58] were applied to obtain NNLO QCD predictions. Resummation of electroweak effects is also applied as in Ref. [59]. The corrections from these K-factors are 2%–15% and 0.5%–10% for the separate 8 and 13 TeV\documentclass[12pt]{minimal}
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\begin{document}$$\mu _{\mathrm {r}} =\mu _{\mathrm {f}} = p_{\mathrm {T}}^{\mathrm {jet}}$$\end{document}, as discussed in Sect. 5.3.1. Table 3 details the code used for NNLO/NLO QCD corrections and NLO/LO EW corrections to ATLAS data in the QCD fit, for ease of reference.
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\begin{document}$$\chi ^2$$\end{document} calculation in the same way as experimental systematic uncertainties (see Sect. 4). These scale uncertainties are evaluated as follows. The K-factors are evaluated for separate changes of the renormalisation and factorisation scales by factors of 2 and 0.5. The magnitude of the K-factor difference is symmetrised as (K[μr(2)]-K[μr(0.5)])/2\documentclass[12pt]{minimal}
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\begin{document}$$\mu _{\mathrm {f}} $$\end{document} makes the K-factor increase and negative if it makes the K-factor decrease. For these renormalisation and factorisation scale changes, the ratios of the signed K-factor changes to the K-factors for the nominal scales are calculated to obtain the fractional scale uncertainties. These can then be used in the fit like any other systematic uncertainty, as suggested in Ref. [65].
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\begin{document}$$\text {TeV}$$\end{document} for the central fit. Studies of alternative approaches for the scale uncertainties are presented in Appendix A.
Scale uncertainties are accounted for in the central fit only for the W and Z inclusive data sets, because the scale uncertainties are comparable to the experimental uncertainties only in this case. For the other ATLAS data sets the experimental uncertainties are larger and scale uncertainties are treated by repeating the fit with varied scales as discussed in Sect. 5.3.1.
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\begin{document}$${\sim }1\%$$\end{document} between the NNLO predictions of FEWZ and DYNNLO. A fractional uncertainty can be defined as the ratio of the signed difference between the FEWZ and DYNNLO predictions to the FEWZ prediction, and applied similarly to the scale uncertainties. The inclusion of such a theoretical uncertainty makes no further difference to the fits, provided the scale uncertainties are applied.
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\begin{document}$$^{3}$$\end{document}LO) calculations for the total cross section of the Drell–Yan process have become available [66]. It is notable that the N3\documentclass[12pt]{minimal}
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Fit methodology
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\begin{document}$$\begin{aligned} \begin{aligned} \chi ^2&= \sum _{i,k}\left( D_i - T_i(1-\sum _{j}\gamma _{ij}b_{j})\right) \\&\, \times C^{-1}_{\text {stat,uncor}, ik} (D_i,D_k)\left( D_k \!-\! T_k(1\!-\!\sum _{j}\gamma _{kj}b_{j})\right) \\&\, + \sum _{i} \log \frac{\delta ^{2}_{i,\text {uncor}}T_i^2 + \delta ^{2}_{i,\text {stat}}D_i T_i}{\delta ^{2}_{i,\text {uncor}}D_i^2 + \delta ^{2}_{i,\text {stat}}D_i^2}\\&\, + \sum _{j} b_j^2 \end{aligned} \end{aligned}$$\end{document}where Di\documentclass[12pt]{minimal}
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\begin{document}$$\delta _{i,\text {stat}}$$\end{document} are the uncorrelated systematic uncertainties and the statistical uncertainties of Di\documentclass[12pt]{minimal}
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\begin{document}$$C_{\text {stat,uncor}, ik}$$\end{document} is a covariance matrix for both the statistical and uncorrelated systematic uncertainties. Summations over i and k run over all data points and summation over j runs over all sources of correlated systematic uncertainty. For each data set, the first term gives the main contribution to the partial χ2\documentclass[12pt]{minimal}
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\begin{document}$$\chi ^2$$\end{document} of the data set and the second term is a small bias correction term, referred to as the log penalty, which arises because the diagonal term of the matrix, C, is given by Cii=δi,uncor2Ti2+δi,stat2DiTi\documentclass[12pt]{minimal}
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\begin{document}$$C_{ii} =\delta ^{2}_{i,\text {uncor}}T_i^2 + \delta ^{2}_{i,\text {stat}}D_i T_i$$\end{document}, with different weighting for statistical uncertainties and uncorrelated systematic uncertainties. This form of the χ2\documentclass[12pt]{minimal}
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\begin{document}$$\chi ^2$$\end{document} is used as standard in HERA and ATLAS PDF fits [5, 9, 10]. In reporting the χ2\documentclass[12pt]{minimal}
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\begin{document}$$\chi ^2$$\end{document} arising from the penalty for the nuisance parameters describing correlations of systematic uncertainties within and between data sets. The experimental uncertainties of the fit are first set using the usual tolerance, T=1\documentclass[12pt]{minimal}
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\begin{document}$$T^2 = \Delta \chi ^2 =1$$\end{document}. The use of an enhanced tolerance is considered in Sect. 6.
Results
In this section the ATLASpdf21 PDF set is presented. The impact of variations of the central choice of fit settings and parameterisation is discussed in Sect. 5.3. Table 4 gives the total χ2\documentclass[12pt]{minimal}
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\begin{document}$$\chi ^2/\mathrm {NDP}$$\end{document} for NDP data points, of each data set. The correlated terms (term 3) are shown separated into groups with common systematic correlations. In order to evaluate the separate contributions of the data sets to this correlated term, the fit is run with its final parameters fixed for each data set separately. These values follow the total correlated terms in brackets, in order of their appearance in the table. The quality of the fit to the HERA data is χ2/NDP=1.14\documentclass[12pt]{minimal}
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\begin{document}$$\chi ^2/\mathrm {NDP} =1.14$$\end{document}, comparable to that of HERAPDF2.0, so that there is no tension between the ATLAS data and HERA data. The quality of the fit to the ATLAS W, Z data is χ2/NDP=1.44\documentclass[12pt]{minimal}
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\begin{document}$$\chi ^2/\mathrm {NDP} =1.44$$\end{document}, and to the ATLAS V+ jets, tt¯\documentclass[12pt]{minimal}
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\begin{document}$$\chi ^2$$\end{document} values are comparable to those obtained by the global PDF fits for similar data sets, but indicate a need to consider the appropriate χ2\documentclass[12pt]{minimal}
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\begin{document}$$\chi ^2$$\end{document} tolerance of the fit. Both of these points are discussed further in Sect. 6.
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\begin{document}$$\chi ^2$$\end{document} contributions for the all data sets entering the PDF fit. The partialχ2\documentclass[12pt]{minimal}
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\begin{document}$$\chi ^2$$\end{document} for the individual data sets are given with respect to the number of data points (NDP). They represent the addition of terms 1 and 2 in Eq. (1). The correlated terms (term 3) are shown separated into groups with common systematic correlations. The total value of the correlated term for these groups is also split into the addition of the separate contributions in the order in which they are given in the table
Comparison of PDFs with and without inclusion of correlations between data sets
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\begin{document}$$^2$$\end{document} (the starting scale) are shown in Figs. 1 and 2. Only experimental uncertainties with tolerance T=1\documentclass[12pt]{minimal}
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\begin{document}$$T=1$$\end{document} are shown for the comparisons throughout this section. Full uncertainties including model and parameterisation variations are considered for the ATLASpdf21 fit in Sect. 5.3.
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\begin{document}$$x\bar{d}$$\end{document}. The lower panels show the comparison as a ratio to the default ATLASpdf21 PDF
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\begin{document}$$t\bar{t}$$\end{document} (lepton + jets) and inclusive jets data sets are not applied, with the exception of the luminosities for the data at the same centre-of-mass energy.9 The systematic uncertainties which are correlated are specified in Table 2 and their treatment for the central fit is discussed in Sect. 2.2.
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\begin{document}$$O(1\%)$$\end{document} accuracy is to be a realistic goal for PDFs [73].
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These plots also allow a comparison of the size of the current experimental uncertainty of the PDFs with and without inclusion of inter-data-set correlations. The sizes are generally similar but the uncertainty in the high-xd¯\documentclass[12pt]{minimal}
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\begin{document}$$\bar{d}$$\end{document}, strange and gluon PDFs is larger if such correlations are not taken into account.
Impact of each data set
In this section the impact of each data set is considered. Only experimental uncertainties with tolerance T=1\documentclass[12pt]{minimal}
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Impact of <italic>W</italic>, <italic>Z</italic> inclusive data
Figure 5 shows the ratio Rs\documentclass[12pt]{minimal}
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\begin{document}$$\text {TeV}$$\end{document} still add information.
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However, the W, Z data taken at 8 TeV\documentclass[12pt]{minimal}
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\begin{document}$$x\bar{d}$$\end{document} distribution, as also shown in Fig. 7.
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\begin{document}$$\text {TeV}$$\end{document} 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. N3\documentclass[12pt]{minimal}
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\begin{document}$$^{3}$$\end{document}LO 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.
Impact of <italic>V</italic>+ jets data
The impact of the V+ jets data is shown in Figs. 8 and 9. There are significant changes in the xd¯\documentclass[12pt]{minimal}
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\begin{document}$$x(\bar{d}-\bar{u})$$\end{document} 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 Rs\documentclass[12pt]{minimal}
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\begin{document}$$x\bar{s}$$\end{document} at high x is 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.
These V+ jets data also effect a modest change in the xdv\documentclass[12pt]{minimal}
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\begin{document}$$\text {TeV}$$\end{document} and other data sets in the fit, since all data are fitted well at the minimum chosen by these V+ jets data.
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The impact of the tt¯\documentclass[12pt]{minimal}
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\begin{document}$$t\bar{t}$$\end{document} data 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-x gluon distribution, which are reduced. There is no significant tension between the tt¯\documentclass[12pt]{minimal}
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\begin{document}$$\text {TeV}$$\end{document} (left) or only the tt¯\documentclass[12pt]{minimal}
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\begin{document}$$\text {TeV}$$\end{document} (right). It is clear that the data at 8 TeV\documentclass[12pt]{minimal}
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\begin{document}$$\text {TeV}$$\end{document} have the stronger impact on the shape of the xg PDF but both data sets contribute to a modest reduction in the uncertainties.
Impact of photon data and inclusive jet data
ATLASpdf21 xg PDF compared with xg for fits not including various data sets. Only experimental uncertainties are shown, evaluated with tolerance T=1\documentclass[12pt]{minimal}
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There is little impact from the addition of the direct-photon production ratio data apart from a marginal softening of the high-x gluon 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-x gluon 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\documentclass[12pt]{minimal}
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\begin{document}$$R=0.4$$\end{document}. The impact of using inclusive jet data at 7 TeV and at 13 TeV, instead of at 8 TeV, is explored in Appendix B.
Model, theoretical and parameterisation uncertainties
The consideration of additional uncertainties affecting the PDFs is presented in this section. These are classified and labeled here as either model, theoretical or parameterisation uncertainties.
Model and theoretical uncertainties
The class of model uncertainties includes effects due to variations of the heavy-quark masses input to the TRVFN heavy-quark-mass scheme for the inclusive DIS calculations, the minimum Q2\documentclass[12pt]{minimal}
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\begin{document}$$\text {GeV}$$\end{document} [71]. The variations of mc\documentclass[12pt]{minimal}
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An additional model uncertainty comes from the assumed value of the top-quark mass. The interpolation grids for the NNLO predictions for the tt¯\documentclass[12pt]{minimal}
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\begin{document}$$\text {TeV}$$\end{document} data set are input to the final model uncertainty.
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\documentclass[12pt]{minimal}
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\begin{document}$$\chi ^2/\mathrm {NDF}$$\end{document} for the total fit and the contribution χ2/NDP\documentclass[12pt]{minimal}
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\begin{document}$$\chi ^2/\mathrm {NDP}$$\end{document} 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
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 Q02\documentclass[12pt]{minimal}
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\begin{document}$$Q^2_0$$\end{document}. 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 the 8 TeV\documentclass[12pt]{minimal}
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\begin{document}$$\text {TeV}$$\end{document} jets is shown in Fig. 12. The only significant change is in the gluon PDF. This small difference between the PDFs extracted using jet data at 8 TeV\documentclass[12pt]{minimal}
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\begin{document}$$R = 0.6$$\end{document} 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 tt¯\documentclass[12pt]{minimal}
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\begin{document}$$t\bar{t}$$\end{document} data from the lepton + jets channel and V+ jets data, and for these data sets only R=0.4\documentclass[12pt]{minimal}
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\begin{document}$$R = 0.4$$\end{document} jets are available.
The effect of using jet production data at different centre-of-mass energies of 7, 8 and 13 TeV\documentclass[12pt]{minimal}
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\begin{document}$$\text {TeV}$$\end{document} 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.
All model uncertainties are added in quadrature to form a total model uncertainty.
Theoretical uncertainties include the scale uncertainties of the predictions. The largest of these are the scale uncertainties for the inclusive W and Z data which are considered in detail in Sect. 3.2. Since these are treated as correlated systematic uncertainties in the fit χ2\documentclass[12pt]{minimal}
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\begin{document}$$\chi ^2$$\end{document} they are actually already included in what has been labelled as the experimental uncertainty of the fit.
Impact of adding Fuv\documentclass[12pt]{minimal}
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\begin{document}$$D_{\bar{d}}$$\end{document} as 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\documentclass[12pt]{minimal}
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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 tt¯\documentclass[12pt]{minimal}
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\begin{document}$$100\%$$\end{document} correlation of the scale uncertainties is assumed between the 8 and 13 TeV\documentclass[12pt]{minimal}
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\begin{document}$$\text {TeV}$$\end{document} 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\documentclass[12pt]{minimal}
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\begin{document}$$\text {TeV}$$\end{document} requires further consideration. These K-factors were supplied for two choices of scale: μr=μf=pTmax\documentclass[12pt]{minimal}
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\begin{document}$$p_{\mathrm {T}}^{\mathrm {jet}}$$\end{document} is favoured over pTmax\documentclass[12pt]{minimal}
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\begin{document}$$p_{\mathrm {T}}^{\mathrm {max}}$$\end{document} [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 χ2\documentclass[12pt]{minimal}
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\begin{document}$$\chi ^2$$\end{document} values for these fits are given in Table 6. The only significant change in χ2\documentclass[12pt]{minimal}
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\begin{document}$$\chi ^2$$\end{document} comes from the inclusive jets, not from other data sets.
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\begin{document}$$\mu _{\mathrm {r}} =\mu _{\mathrm {f}} = p_{\mathrm {T}}^{\mathrm {jet}}/2$$\end{document} are considered. The effect on the gluon PDF is shown in Fig. 13 (right). The χ2\documentclass[12pt]{minimal}
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\begin{document}$$\chi ^2$$\end{document} values for these fits are given in Table 6. Again, the only significant change in χ2\documentclass[12pt]{minimal}
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\begin{document}$$\chi ^2$$\end{document} comes from the inclusive jets, not from other data sets.
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\begin{document}$$\chi ^2$$\end{document}/NDF for each model and parameterisation variation considered for the ATLASpdf21 PDF fit. The notation ‘(sym)’ indicates that the upward and downward model variations have been symmetrised
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 χ2\documentclass[12pt]{minimal}
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\begin{document}$$\chi ^2$$\end{document} value for this fit is also given in Table 6. It is lower than that for smoothed K-factors because the statistical uncertainties of the unsmoothed K-factors are ∼1%\documentclass[12pt]{minimal}
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\begin{document}$${\sim }1\%$$\end{document}. However, the resulting PDFs are very similar to those obtained using smoothed K-factors. 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)\documentclass[12pt]{minimal}
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\begin{document}$$\alpha _{\text {s}} (m_{Z})$$\end{document}, which affects the shape of the gluon PDF. The correlation between αs(mZ)\documentclass[12pt]{minimal}
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\begin{document}$$\alpha _{\text {s}} (m_{Z})$$\end{document} is beyond the scope of the current paper, since a correct NNLO treatment requires variation of the K-factor calculations with αs(mZ)\documentclass[12pt]{minimal}
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\begin{document}$$\alpha _{\text {s}} (m_{Z}) = 0.118$$\end{document} is used, in line with the value used by the global fitting groups, CT [76], MSHT [77] and NNPDF [78].
Parameterisation uncertainties
The optimal number of parameters was determined by ‘saturation’ of the χ2\documentclass[12pt]{minimal}
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\begin{document}$$\chi ^2$$\end{document} of the fit, are summarised in Table 7.
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\begin{document}$$^2$$\end{document}. This removes 82 points from the fit: 63 from the inclusive jets data, 6 from the V+ jets data, 11 from the direct-photon ratio data, and one each from the tt¯\documentclass[12pt]{minimal}
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\begin{document}$$\chi ^2$$\end{document}/NDF value of the fit improves from 2010/1620 to 1870/1538, with the largest improvement coming from the inclusive jets, for which the χ2\documentclass[12pt]{minimal}
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Combination of uncertainties
The PDFs are now presented with experimental uncertainties from the fit and with additional model and parameterisation uncertainties as discussed in this section. These model and parameterisation uncertainties are added in quadrature to the experimental uncertainty. It should be noted that the scale uncertainties for the inclusive W, Z data appear as part of the experimental uncertainty, whereas the scale uncertainties for other data sets are negligible. Figure 16 shows the valence distributions and the xu¯\documentclass[12pt]{minimal}
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\begin{document}$$x\bar{d}$$\end{document} affects both xd¯\documentclass[12pt]{minimal}
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\begin{document}$$x=0.013$$\end{document}, compared with other PDFs. For the ATLASpdf21 fit, the value of Rs\documentclass[12pt]{minimal}
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\begin{document}$$R_s$$\end{document} at this low scale has decreased relative to previous ATLAS fits. The ATLASpdf21 central value of Rs∼0.8\documentclass[12pt]{minimal}
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\begin{document}$$R_s \sim 0.8$$\end{document} is due firstly to the input of the V+ jets data, as seen in the change from ATLASepWZ16 (Rs∼1.15\documentclass[12pt]{minimal}
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\begin{document}$$R_s \sim 1.15$$\end{document}) to ATLASepWZVjets20 (Rs∼1.0\documentclass[12pt]{minimal}
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\begin{document}$$R_s \sim 1.0$$\end{document}), secondly to the new freedom in the low-x parameterisation, whereby the ATLASpdf21 central parameterisation corresponds to the low end of the uncertainty band of the ATLASepWZVjets20 result (Rs∼0.85\documentclass[12pt]{minimal}
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\begin{document}$$R_s \sim 0.85$$\end{document}), and thirdly to the input of the W, Z data at 8 TeV\documentclass[12pt]{minimal}
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\begin{document}$$\text {TeV}$$\end{document}. On the other hand, Rs\documentclass[12pt]{minimal}
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\begin{document}$$R_s$$\end{document} has increased for some of the other PDFs, e.g. MSHT20 [77] compared with MMHT14 [79], and CT18A [76] compared with CT14 [80]. This increase is due to the input of the ATLAS precision inclusive W, Z data at 7 TeV\documentclass[12pt]{minimal}
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\begin{document}$$\text {TeV}$$\end{document}. Thus, the recent evaluations of this ratio of strangeness to light quarks are now in fair agreement. However, the strangeness is still substantially larger at low x than the older values which estimated Rs∼0.5\documentclass[12pt]{minimal}
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\begin{document}$$R_s\sim 0.5$$\end{document}. This illustration of the ratio Rs\documentclass[12pt]{minimal}
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\begin{document}$$x=0.023$$\end{document} should be supplemented by a comparison of the full shape of the Rs\documentclass[12pt]{minimal}
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\begin{document}$$R_s$$\end{document} distribution in Sect. 6.
Consideration of <inline-formula id="IEq996"><alternatives><mml:math><mml:msup><mml:mi>χ</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math><tex-math id="IEq996_TeX">\documentclass[12pt]{minimal}
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\begin{document}$$\chi ^2$$\end{document}</tex-math><inline-graphic xlink:href="10052_2022_10217_Article_IEq996.gif"/></alternatives></inline-formula> tolerance and comparison with global fits<inline-formula id="IEq997"><alternatives><mml:math><mml:msup><mml:mi>χ</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math><tex-math id="IEq997_TeX">\documentclass[12pt]{minimal}
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\begin{document}$$\chi ^2$$\end{document}</tex-math><inline-graphic xlink:href="10052_2022_10217_Article_IEq997.gif"/></alternatives></inline-formula> tolerance
The experimental uncertainties presented for the fitted PDFs were evaluated by the usual criterion of parameter setting, Δχ2=1\documentclass[12pt]{minimal}
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\begin{document}$$\Delta \chi ^2 = 1$$\end{document}. However, global PDF fitting groups such as CT and MSHT use enhanced χ2\documentclass[12pt]{minimal}
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\begin{document}$$T^2$$\end{document} is a dynamic tolerance ranging from 10 to 100.10 The justification for this comes from consideration of the consistency, or relative inconsistency, of input data sets [84]. In the present case, whereas the HERA data is a single consistent set of data with common correlated uncertainties, the ATLAS data are more disparate and a study to find an appropriate tolerance was performed.
One way to estimate a suitable tolerance for a fit including the ATLAS data is to consider the dynamic tolerance procedure of the MSHT group, as first introduced in the MSTW paper [84]. Briefly restated, the method is as follows. The inverse of the Hessian matrix of the fit is diagonalised to give the eigenvector combinations of the parameters of the fit. The eigenvalues are the squares of the uncertainties in these eigenvector combinations of the parameters. The parameter displacements from the global minimum can then be expanded in a basis of rescaled eigenvectors, eik=λkvik\documentclass[12pt]{minimal}
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\begin{document}$$v_{ik}$$\end{document} is the i-th component of the k-th orthonormal eigenvector. Pairs of eigenvector PDFs have parameters given by ai(k±)=ai0±teik\documentclass[12pt]{minimal}
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\begin{document}$$T^2 = \Delta \chi ^2=1$$\end{document}. In the quadratic approximation, t=T\documentclass[12pt]{minimal}
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\begin{document}$$t=T$$\end{document}, and this is reasonably well obeyed for the present ATLASpdf21 21-parameter fit, at least up to t=5\documentclass[12pt]{minimal}
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\begin{document}$$\chi ^2_n$$\end{document} goes above this limit, this sets the maximum value of t, and hence of T, that can be tolerated for this eigenvector in this direction, for this data set. Each eigenvector will have a most constraining data set, for each direction, and in principle the values of the tolerances T for the most constraining data set can be different for each eigenvector and each direction, hence the term ‘dynamic’.
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This method is made more sophisticated in a number of ways. Firstly, the hypothesis-testing criterion chosen is not exactly the simple χn2<N+2N\documentclass[12pt]{minimal}
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\begin{document}$$\chi ^2_{n,0}$$\end{document} for the data set may not be perfect, where perfect would imply χn,02=ξ50\documentclass[12pt]{minimal}
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\begin{document}$$\text {TeV}$$\end{document}). The tolerance values for all the eigenvectors, in both directions, are all in the range 2.4–4.2. These values are all fairly similar, so a single value can be used and, since the tolerance should not be set too large whereby a data set is far outside its 68%\documentclass[12pt]{minimal}
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\begin{document}$$T = 3$$\end{document}.
In the above procedure to evaluate the dynamic tolerance, the fact that the data sets are not fitted perfectly was taken into account. One may estimate how much of the increased tolerance comes from this part of the procedure by increasing the uncertainties of each data set by χ2/NDP\documentclass[12pt]{minimal}
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\begin{document}$$\chi ^2/\mathrm {NDP}\sim 1$$\end{document} is achieved in the fit. This will clearly increase the uncertainties of the output PDFs. However, a naive application of this procedure would increase all the experimental uncertainties and this seems somewhat unnecessary for the statistical uncertainties, which are well known. Hence, only the systematic uncertainties are increased. Starting with the χ2/NDP\documentclass[12pt]{minimal}
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It is non-trivial that for these reweightings of the data the PDF shapes hardly differ from those of an unweighted fit. This shows that the fit is stable for modest reweighting of the input data sets. The uncertainties of the PDFs have increased, but only by a factor which corresponds to a tolerance of up to T∼1.5\documentclass[12pt]{minimal}
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\begin{document}$$T \sim 1.5$$\end{document}. Thus the greater part of the dynamic tolerance estimate of T=3\documentclass[12pt]{minimal}
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The tolerance T=3\documentclass[12pt]{minimal}
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\begin{document}$$T=3$$\end{document} is applied for the final estimation of PDF uncertainty; however, ATLASpdf21 PDFs with both choices of tolerance, T=1\documentclass[12pt]{minimal}
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\begin{document}$$T=3$$\end{document}, will be made available on the LHAPDF PDF repository [85]. Having fixed the tolerance for the experimental uncertainties of the PDFs, the full uncertainties, including experimental, model and parameterisation uncertainties are compared for the choices T=3\documentclass[12pt]{minimal}
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\begin{document}$$T=1$$\end{document} in Fig. 20. When an enhanced tolerance such as this is applied to the experimental uncertainties, the model and parameterisation uncertainties are far less significant.
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\begin{document}$$T=3$$\end{document} and full uncertainties, are shown in Appendix C.
Comparison of ATLASpdf21 with global PDFs
In Figs. 21, 22 and 23, the ATLASpdf21 PDFs are compared with the HERAPDF2.0 [5], CT18, CT18A [76], NNPDF3.1 [78], MSHT20 [77] and ABMP16 [81] PDFs. All figures in this section show ATLASpdf21 PDFs with full uncertainties, such that experimental uncertainties are evaluated with T=3\documentclass[12pt]{minimal}
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\begin{document}$$T=3$$\end{document} and model and parametrisation uncertainties are also included. It is observed that the addition of the ATLAS data sets to the HERA data brings the PDFs much closer to those of the global fits than to HERAPDF2.0, particularly for xdv\documentclass[12pt]{minimal}
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\begin{document}$$xd_v$$\end{document} distributions of the ATLASpdf21 fit are in good agreement with CT18, CT18A, MSHT20 and ABMP16, whereas NNPDF3.1 is an outlier. The xu¯\documentclass[12pt]{minimal}
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\begin{document}$$x\bar{d}$$\end{document} distribution is in fair agreement with CT, MSHT and NNPDF, with ABMP presenting as somewhat of an outlier.11 However, it should be noted that xd¯\documentclass[12pt]{minimal}
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\begin{document}$$x\bar{d}$$\end{document} from the ATLASpdf21 fit is somewhat higher at high x. This is further illustrated by the x(d¯-u¯)\documentclass[12pt]{minimal}
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\begin{document}$$x(\bar{d}-\bar{u})$$\end{document} distribution, which changes from negative at high x when fitting only HERA data in HERAPDF2.0, to positive at high x for ATLASpdf21, in agreement with the global PDFs. However, the ATLASpdf21 x(d¯-u¯)\documentclass[12pt]{minimal}
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\begin{document}$$x >0.3$$\end{document}, 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 xs¯\documentclass[12pt]{minimal}
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\begin{document}$$x\bar{s}$$\end{document} distribution is larger than those in NNPDF3.1 and CT18 for x∼0.02\documentclass[12pt]{minimal}
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\begin{document}$$x\sim 0.02$$\end{document}, 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 Rs\documentclass[12pt]{minimal}
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\begin{document}$$R_s$$\end{document} for 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 inclusive W, Z data at 7 TeV.13 The uncertainties of ABMP16 are smaller for the usual reason that they use tolerance T=1\documentclass[12pt]{minimal}
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\begin{document}$$T=1$$\end{document}. The input of ATLAS inclusive W, Z data at 7 TeV is the only difference between the CT18 and CT18A analyses. After input of these data the CT18A Rs\documentclass[12pt]{minimal}
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\begin{document}$$R_s$$\end{document} PDF ratio has smaller uncertainties and a larger central value of Rs\documentclass[12pt]{minimal}
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\begin{document}$$R_s$$\end{document} at low x than CT18. CT18A is in good agreement with ATLASpdf21, for both central value and uncertainty, over the full x range illustrated. The PDF analyses of MSHT20 and NNPDF3.1 also use ATLAS inclusive W, Z data at 7 TeV and have a similar size of uncertainty and level of agreement with ATLASpdf21. In addition, the MSHT20 analysis uses ATLAS inclusive W, Z data at 8 TeV, just as the ATLASpdf21 analysis does. Note that NNPDF have updated their study of strangeness recently to NNPDF3.1_strange [83], which has a somewhat larger strangeness at low x than NNPDF3.1. The modern PDFs which use ATLAS inclusive W, Z data, 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 fixed-target 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 χ2\documentclass[12pt]{minimal}
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\begin{document}$$\chi ^2$$\end{document} values for the data sets included in the ATLASpdf21 fit are HERAPDF2.0: 2262, CT18: 2135, CT18A: 2133, MSHT20: 2218, NNPDF3.1: 2109, compared with ATLASpdf21: 2010. Although the global PDFs from CT, MSHT and NNPDF have more flexible parameterisations, the χ2\documentclass[12pt]{minimal}
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\begin{document}$$\chi ^{2}$$\end{document} value of the ATLASpdf21 fit is better for the data sets considered. This is a further indication that the ATLASpdf21 parameterisation is sufficiently flexible.
ATLASpdf21 fit with full uncertainties (experimental T=3\documentclass[12pt]{minimal}
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\begin{document}$$T=3$$\end{document}, model, parameterisation) compared with CT18A, MSHT20 shown as a ratio with each distribution centred on unity. Top left: xuv\documentclass[12pt]{minimal}
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Figure 24 shows the uncertainties of the ATLASpdf21 fit, with the enhanced tolerance T=3\documentclass[12pt]{minimal}
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\begin{document}$$x< 2\times 10^{-3}$$\end{document} is very similar to that of both CT18A and MSHT20, but the high x gluon uncertainty is somewhat larger for the ATLASpdf21 fit, reflecting the absence of Tevatron jet data. In the valence sector, the xuv\documentclass[12pt]{minimal}
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\begin{document}$$xu_v$$\end{document} uncertainties of the ATLASpdf21 fit are comparable to those of CT18A at low x, but larger at high x, reflecting the absence of Tevatron W, Z data. For xdv\documentclass[12pt]{minimal}
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\begin{document}$$x\sim 0.003$$\end{document}–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.
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\begin{document}$${\sim }2\sigma $$\end{document}.
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 W and Z data to HERA data allows a fit to be made without any constraints imposed on the relationship between xu¯\documentclass[12pt]{minimal}
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\begin{document}$$x\bar{d}\sim x\bar{u}$$\end{document} is observed at low x. The strangeness at low x is 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, Z data. This increase in the low x strange 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, Z data 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 x gluon distribution. The tt¯\documentclass[12pt]{minimal}
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\begin{document}$$t\bar{t}$$\end{document} data serve to moderately reduce the uncertainty in the high x gluon 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 x and they reduce its uncertainty considerably.
The role of scale uncertainties is considered for all data sets and these theoretical uncertainties are implemented in the fit for the inclusive W, Z data, where they are most impactful. The effects of such scale uncertainties are small.
A study is made of excluding high-scale data (with a scale above 500 GeV\documentclass[12pt]{minimal}
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\begin{document}$$\text {GeV}$$\end{document}), 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 and NNPDF in future fits. The effects of these correlations are relatively small, O(∼1%\documentclass[12pt]{minimal}
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\begin{document}$$O({\sim }1\%$$\end{document}), at the scale Q2∼10,000\documentclass[12pt]{minimal}
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\begin{document}$$^2$$\end{document} considered for precision physics at the LHC, but large enough to be considered in future precision data analyses such as the measurement of mW\documentclass[12pt]{minimal}
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\begin{document}$$\sin ^2\!\theta _{\text {W}}$$\end{document}, 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; BMWFW and FWF, 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; MES of Russia and NRC KI, Russian Federation; 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
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 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].]
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In this appendix, different choices for the treatment of the scale uncertainties in inclusive W, Z data are considered. For these data sets the experimental uncertainties are comparable to the scale uncertainties of NNLO predictions (∼0.5%\documentclass[12pt]{minimal}
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\begin{document}$${\sim }0.5\%$$\end{document}). 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 W and Z data and to be correlated between the 7 and 8 TeV\documentclass[12pt]{minimal}
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Comparison showing the ratio of the ATLASpdf21 gluon PDF and the Rs\documentclass[12pt]{minimal}
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\begin{document}$$^2$$\end{document}, 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\documentclass[12pt]{minimal}
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\begin{document}$$\text {TeV}$$\end{document} inclusive W, Z data sets is shown by the green line in these figures and it can be seen that it is generally a smaller effect.
Comparison of the impact of inclusive jet data at different centre-of-mass energies
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Goodness of the fit: comparison with data sets included in the fit
Figures 28, 29, 30, 31, 32, 33, 34, 35, 36 and 37 show comparisons 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.
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The ATLASpdf21 fit does not include data from the Tevatron or from fixed-target DY data. In this appendix the predictions of the ATLASpdf21 fit for some of these data sets, which were found to be most impactful in the global fits, are explored and found to be satisfactory.
The differential cross-section measurements 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 Z boson 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 bj\documentclass[12pt]{minimal}
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In Figs. 38 and 39 the ATLASpdf21 fit is shown in comparison with Tevatron W and Z data.
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The definitions of the scale for the DIS and inclusive vector boson processes 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.
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\begin{document}$$p_{\text {T}}$$\end{document} balance and the ‘Pile-up Rho Topology’ takes into account the uncertainty in the density ρ\documentclass[12pt]{minimal}
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\begin{document}$$\rho $$\end{document} of pile-up activity in a given event.
The decorrelations used at next-to-leading order (NLO) in Ref. [17], include decorrelations of systematic uncertainties due to scale choice. These are not applied in the present NNLO analysis because scale uncertainties are much smaller. NNLO scale uncertainties for the inclusive jets are studied in Sect. 5.3.1.
Note that the ATLAS data are all at much higher scales and the charm and bottom quarks are treated in the 5-flavour zero-mass variable flavour number scheme, for which the top is an additional heavy quark.
For the other processes considered in this section the statistical precision of the K-factors is much higher and further consideration of this source of uncertainty is not necessary.
The W+ jets and Z + jets data sets are always considered as a single data set with correlations applied and these correlations are maintained in the present exercise. Consideration of the impact of correlations between the W+ jets and Z + jets data sets was studied in Ref. [11].
NNPDF uses a completely different method for estimation of PDF uncertainties, but the magnitude of their uncertainties is comparable to those of MSHT and CT.
Note that ABMP has much smaller uncertainties due to their use of tolerance T=1\documentclass[12pt]{minimal}
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\begin{document}$$T=1$$\end{document}.
The ABMP16 gluon PDF corresponds to αs(mZ)=0.1145\documentclass[12pt]{minimal}
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\begin{document}$$\alpha _{\text {s}} (m_{Z})=0.1145$$\end{document}, a lower value than in the other PDFs illustrated, which all use αs(mZ)=0.118\documentclass[12pt]{minimal}
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\begin{document}$$\alpha _{\text {s}} (m_{Z})=0.118$$\end{document}.
In the case of HERAPDF2.0 the uncertainty in the strange PDF is not measured but estimated.