Combination of measurements of inclusive deep inelastic \({e^{\pm }p}\) scattering cross sections and QCD analysis of HERA data
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Abstract
A combination is presented of all inclusive deep inelastic cross sections previously published by the H1 and ZEUS collaborations at HERA for neutral and charged current \(e^{\pm }p\) scattering for zero beam polarisation. The data were taken at proton beam energies of 920, 820, 575 and 460 GeV and an electron beam energy of 27.5 GeV. The data correspond to an integrated luminosity of about 1 fb\(^{1}\) and span six orders of magnitude in negative fourmomentumtransfer squared, \(Q^2\), and Bjorken x. The correlations of the systematic uncertainties were evaluated and taken into account for the combination. The combined cross sections were input to QCD analyses at leading order, nexttoleading order and at nexttonexttoleading order, providing a new set of parton distribution functions, called HERAPDF2.0. In addition to the experimental uncertainties, model and parameterisation uncertainties were assessed for these parton distribution functions. Variants of HERAPDF2.0 with an alternative gluon parameterisation, HERAPDF2.0AG, and using fixedflavournumber schemes, HERAPDF2.0FF, are presented. The analysis was extended by including HERA data on charm and jet production, resulting in the variant HERAPDF2.0Jets. The inclusion of jetproduction cross sections made a simultaneous determination of these parton distributions and the strong coupling constant possible, resulting in \(\alpha _s(M_Z^2)=0.1183 \pm 0.0009 \mathrm{(exp)} \pm 0.0005\mathrm{(model/parameterisation)} \pm 0.0012\mathrm{(hadronisation)} ^{+0.0037}_{0.0030}\mathrm{(scale)}\). An extraction of \(xF_3^{\gamma Z}\) and results on electroweak unification and scaling violations are also presented.
1 Introduction
Deep inelastic scattering (DIS) of electrons^{1} on protons at centreofmass energies of up to \(\sqrt{s} \simeq 320\,\)GeV at HERA has been central to the exploration of proton structure and quark–gluon dynamics as described by perturbative Quantum Chromo Dynamics (pQCD) [1]. The two collaborations, H1 and ZEUS, have explored a large phase space in Bjorken x, \(x_\mathrm{Bj}\), and negative fourmomentumtransfer squared, \(Q^2\). Cross sections for neutral current (NC) interactions have been published for \(0.045 \le Q^2 \le 50{,}000 \) GeV\(^2\) and \(6 \times 10^{7} \le x_\mathrm{Bj} \le 0.65\) at values of the inelasticity, \(y = Q^2/(sx_\mathrm{Bj})\), between 0.005 and 0.95. Cross sections for charged current (CC) interactions have been published for \(200 \le Q^2 \le 50{,}000 \) GeV\(^2\) and \(1.3 \times 10^{2} \le x_\mathrm{Bj} \le 0.40\) at values of y between 0.037 and 0.76.
HERA was operated in two phases: HERA I, from 1992 to 2000, and HERA II, from 2002 to 2007. From 1994 onwards, and for all data used here, HERA operated with an electron beam energy of \(E_e \simeq 27.5\) GeV. For most of HERA I and II, the proton beam energy was \(E_p = 920\) GeV, resulting in the highest centreofmass energy of \(\sqrt{s} \simeq 320\,\)GeV. During the HERA I period, each experiment collected about 100 pb\(^{1}\) of \(e^+p\) and 15 pb\(^{1}\) of \(e^p\) data. These HERA I data were the basis of a combination and pQCD analysis published previously [2]. During the HERA II period, each experiment added about 150 pb\(^{1}\) of \(e^+p\) and 235 pb\(^{1}\) of \(e^p\) data. As a result, the H1 and ZEUS collaborations collected total integrated luminosities of approximately 500 pb\(^{1}\) each, divided about equally between \(e^+p\) and \(e^p\) scattering. The paper presented here is based on the combination of all published H1 [3, 4, 5, 6, 7, 8, 9, 10] and ZEUS [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24] measurements from both HERA I and II on inclusive DIS in NC and CC reactions. This includes data taken with proton beam energies of \(E_p = 920\), 820, 575 and 460 GeV corresponding to \(\sqrt{s}\simeq \) 320, 300, 251 and 225 GeV. During the HERA II period, the electron beam was longitudinally polarised. The data considered in this paper are cross sections corrected to zero beam polarisation as published by the collaborations.
The combination of the data and the pQCD analysis were performed using the packages HERAverager [25] and HERAFitter [26, 27]. The method [3, 28] also allowed a modelindependent demonstration of the consistency of the data. The correlated systematic uncertainties and global normalisations were treated such that one coherent data set was obtained. Since H1 and ZEUS employed different experimental techniques, using different detectors and methods of kinematic reconstruction, the combination also led to a significantly reduced systematic uncertainty.
Within the framework of pQCD, the proton is described in terms of parton density functions, f(x), which provide the probability to find a parton, either gluon or quark, with a fraction x of the proton’s momentum. This probability is predicted to depend on the scale at which the proton is probed, called the factorisation scale, \(\mu _\mathrm{f}^2\), which for inclusive DIS is usually taken as \(Q^2\). These functions are usually presented as parton momentum distributions, xf(x), and are called parton distribution functions (PDFs). The PDFs are convoluted with the fundamental pointlike scattering cross sections for partons to calculate cross sections. Perturbative QCD provides the framework to evolve the PDFs to other scales once they are provided at a starting scale. However, pQCD does not predict the PDFs at the starting scale. They must be determined by fits to data using ad hoc parameterisations.
The name HERAPDF stands for a pQCD analysis within the DGLAP [29, 30, 31, 32, 33] formalism. The \(x_\mathrm{Bj}\) and \(Q^2\) dependences of the NC and CC DIS cross sections from both the H1 and ZEUS collaborations are used to determine sets of quark and gluon momentum distributions in the proton. The set of PDFs denoted as HERAPDF1.0 [2] was based on the combination of all inclusive DIS scattering cross sections obtained from HERA I data. A preliminary set of PDFs, HERAPDF1.5 [34], was obtained using HERA I and selected HERA II data, some of which were still preliminary. In this paper, a new set of PDFs, HERAPDF2.0, is presented, based on combined inclusive DIS cross sections from all of HERA I and HERA II.
Several groups JR [35], MSTW/MMHT [36, 37], CTEQ/CT [38, 39], ABM [40, 41, 42] and NNPDF [43, 44], provide PDF sets using HERA, fixedtarget and hadroncollider data. The strength of the HERAPDF approach is that a single coherent highprecision data set containing NC and CC cross sections is used as input. The new combined data used for the HERAPDF2.0 analysis span four orders of magnitude in \(Q^2\) and \(x_\mathrm{Bj}\). The availability of precision NC and CC cross sections over this large phase space allows HERAPDF to use only ep scattering data and thus makes HERAPDF independent of any heavy nuclear (or deuterium) corrections. The difference between the NC \(e^+p\) and \(e^p\) cross sections at high \(Q^2\), together with the high\(Q^2\) CC data, constrain the valencequark distributions. The CC \(e^+p\) data especially constrain the valence downquark distribution in the proton without assuming strong isospin symmetry as done in the analysis of deuterium data. The lower\(Q^2\) NC data constrain the lowx seaquark distributions and through their precisely measured \(Q^2\) variations they also constrain the gluon distribution. A further constraint on the gluon distribution comes from the inclusion of NC data at different beam energies such that the longitudinal structure function is probed through the y dependence of the cross sections [45].
The consistency of the input data allowed the determination of the experimental uncertainties of the HERAPDF2.0 parton distributions using rigorous statistical methods. The uncertainties resulting from model assumptions and from the choice of the parameterisation of the PDFs were considered separately.
Both H1 and ZEUS also published charm production cross sections, some of which were combined and analysed previously [46], and jet production cross sections [47, 48, 49, 50, 51]. These data were included to obtain the variant HERAPDF2.0Jets. The inclusion of jet cross sections allowed for a simultaneous determination of the PDFs and the strong coupling constant.
The paper is structured as follows. Section 2 gives an introduction to the connection between cross sections and the partonic structure of the proton. Section 3 introduces the data used in the analyses presented here. Section 4 describes the combination of data while Sect. 5 presents the results of the combination. Section 6 describes the pQCD analysis to extract PDFs from the combined inclusive cross sections. The PDF set HERAPDF2.0 and its variants are presented in Sect. 7. In Sect. 8, results on electroweak unification as well as scaling violations and the extraction of \(xF_3^{\gamma Z}\) are presented. The paper closes with a summary.
2 Cross sections and parton distributions
The relations within the QPM illustrate in a simple way which data contribute which information. However, the parton distributions are determined by a fit to the \(x_\mathrm{Bj}\) and \(Q^2\) dependence of the new combined data using the linear DGLAP equations [29, 30, 31, 32, 33] at leading order (LO), nexttoleading order (NLO) and nexttonexttoleading order (NNLO) in pQCD. These are convoluted with coefficient functions (matrix elements) at the appropriate order [54, 55]. Already at LO, the gluon PDF enters the equations giving rise to logarithmic scaling violations which make the parton distributions depend on the scale of the process. This factorisation scale, \(\mu _\mathrm{f}^2\), is taken as \(Q^2\) and the experimentally measured scaling violations determine the gluon distribution.^{2}
3 Measurements
3.1 Detectors
The H1 [56, 57, 58] and ZEUS [59, 60, 61, 62] detectors were both multipurpose detectors with an almost \(4\pi \) hermetic coverage.^{3} They were built following similar physics considerations but the collaborations opted for different technical solutions resulting in slightly different capabilities [63]. The discussion here focuses on general ideas; details of the construction and performance are not discussed.
In both detectors, the calorimeters had an inner part to measure electromagnetic energy and identify electrons and an outer, lesssegmented, part to measure hadronic energy and determine missing energy. Both main calorimeters were divided into barrel and forward sections. The H1 collaboration chose a liquidargon calorimeter while the ZEUS collaboration opted for a uranium–scintillator device. These choices are somewhat complementary. The liquidargon technology allowed a finer segmentation and thus the identification of electrons down to lower energies. The uranium–scintillator calorimeter was intrinsically “compensating” making jet studies easier. In the backward region, ZEUS also opted for a uranium–scintillator device. The H1 collaboration chose a lead–scintillating fibre or socalled “spaghetti” calorimeter. The backward region is particularly important to identify electrons in events with \(Q^2 < 100\) GeV\(^2\).
Both detectors were operated with a solenoidal magnetic field. The field strength was 1.16 and 1.43 T within the tracking volumes of the H1 and ZEUS detectors, respectively. The main tracking devices were in both cases cylindrical drift chambers. The H1 device consisted of two concentric drift chambers while ZEUS featured one large chamber. Both tracking systems were augmented with special devices in the forward and backward region. Over time, both collaborations upgraded their tracking systems by installing silicon microvertex detectors to enhance the capability to identify events with heavyquark production. In the backward direction, the vertex detectors were also important to identify the electrons in low\(Q^2\) events.
During the HERA I running period, special devices to measure very backward electrons were operated and events with very low \(Q^2\) were reconstructed. This became impossible after the luminosity upgrade for HERA II due to the placement of finalfocus magnets further inside the detectors. This also required some significant changes in both main detectors. Detector elements had to be retracted, and as a result the acceptance for low\(Q^2\) events in the main detectors was reduced.
Both experiments measured the luminosity using the Bethe–Heitler reaction \( ep \rightarrow e\gamma p\). In HERA I, H1 and ZEUS both had photon taggers positioned about 100 m down the electron beam line. For the higher luminosity of the HERA II period, both H1 [8, 64, 65] and ZEUS [66, 67, 68] had to upgrade their luminosity detectors and analysis methods. The uncertainties on the integrated luminosities were typically about 2 %.
3.2 Reconstruction of kinematics
The usage of different reconstruction techniques, due to differences in the strengths of the detector components of the two experiments, contributes to the reduction of systematic uncertainties when combining data sets. The choice of the most appropriate kinematic reconstruction method for a given phasespace region and experiment is based on resolution, possible biases of the measurements and effects due to initial or finalstate radiation. The different methods are described in the following.
The deep inelastic ep scattering cross sections of the inclusive neutral and charged current reactions depend on the centreofmass energy, \(\sqrt{s}\), and on the two kinematic variables \(Q^2\) and \(x_\mathrm{Bj}\). The variable \(x_\mathrm{Bj}\) is related to y, \(Q^2\) and s through the relationship \(x_\mathrm{Bj}=Q^2/(sy)\). The HERA collider experiments were able to determine the NC event kinematics from the scattered electron, e, or from the hadronic final state, h, or from a combination of the two.
3.3 Inclusive data samples
The 41 data sets from H1 and ZEUS used for the combination. The marker [2] in the column “Data Set” indicates that the data are treated as two data sets in the analysis. The markers \(^{1.5p}\) and \(^{1.5}\) in the column “Data Set” indicate that the data were already used for HERAPDF1.5, see Appendix A. The p in \(^{1.5p}\) denotes that the crosssections measurements were preliminary at that time. The markers \(^{*y.5}\) and \(^{*y}\) in the column “Data Set” are explained in Sect. 4.1. The marker \(^1\) for [8] indicates that published cross section were scaled by a factor of 1.018 [65]. Integrated luminosities are quoted as given by the collaborations. The equations used for the reconstruction of \(x_\mathrm{Bj}\) and \(Q^2\) are given in Sect. 3.2
Data set  \(x_\mathrm{Bj}\) grid  \(Q^2 (\)GeV\(^2\)) grid  \(\mathcal{L}\) (pb\(^{1}\))  \(e^+/e^\)  \(\sqrt{s}\) (GeV)  \(x_\mathrm{Bj}\),\(Q^2\) from equations  References  

From  To  From  To  
HERA I \(E_p=820\) GeV and \(E_p=920\) GeV data sets  
H1 svxmb [2]  9500  0.000 005  0.02  0.2  12  2.1  \(e^+p\)  301, 319  [3]  
H1 low \(Q^2\) [2]  9600  0.000 2  0.1  12  150  22  \(e^+p\)  301, 319  [4]  
H1 NC  9497  0.0032  0.65  150  30, 000  35.6  \(e^+p\)  301  [5]  
H1 CC  9497  0.013  0.40  300  15, 000  35.6  \(e^+p\)  301  [5]  
H1 NC  9899  0.0032  0.65  150  30, 000  16.4  \(e^p\)  319  [6]  
H1 CC  9899  0.013  0.40  300  15, 000  16.4  \(e^p\)  319  [6]  
H1 NC HY  9899  0.0013  0.01  100  800  16.4  \(e^p\)  319  [7]  
H1 NC  9900  0.0013  0.65  100  30, 000  65.2  \(e^+p\)  319  [7]  
H1 CC  9900  0.013  0.40  300  15, 000  65.2  \(e^+p\)  319  [7]  
ZEUS BPC  95  0.000 002  0.000 06  0.11  0.65  1.65  \(e^+p\)  300  [11]  
ZEUS BPT  97  0.000000 6  0.001  0.045  0.65  3.9  \(e^+p\)  300  [12]  
ZEUS SVX  95  0.000 012  0.0019  0.6  17  0.2  \(e^+p\)  300  [13]  
ZEUS NC [2] high/low \(Q^2 \)  9697  0.000 06  0.65  2.7  30, 000  30.0  \(e^+p\)  300  [14]  
ZEUS CC  9497  0.015  0.42  280  17, 000  47.7  \(e^+p\)  300  [15]  
ZEUS NC  9899  0.005  0.65  200  30, 000  15.9  \(e^p\)  318  [16]  
ZEUS CC  9899  0.015  0.42  280  30, 000  16.4  \(e^p\)  318  [17]  
ZEUS NC  9900  0.005  0.65  200  30, 000  63.2  \(e^+p\)  318  [18]  
ZEUS CC  9900  0.008  0.42  280  17, 000  60.9  \(e^+p\)  318  [19]  
HERA II \(E_p=920\) GeV data sets  
H1 NC \(^{1.5p}\)  0307  0.0008  0.65  60  30, 000  182  \(e^+p\)  319  [8]\(^1\)  
H1 CC \(^{1.5p}\)  0307  0.008  0.40  300  15, 000  182  \(e^+p\)  319  [8]\(^1\)  
H1 NC \(^{1.5p}\)  0307  0.0008  0.65  60  50, 000  151.7  \(e^p\)  319  [8]\(^1\)  
H1 CC \(^{1.5p}\)  0307  0.008  0.40  300  30, 000  151.7  \(e^p\)  319  [8]\(^1\)  
H1 NC med \(Q^2\) \(^{*y.5}\)  0307  0.000 0986  0.005  8.5  90  97.6  \(e^+p\)  319  [10]  
H1 NC low \(Q^2 \) \(^{*y.5}\)  0307  0.000 029  0.000 32  2.5  12  5.9  \(e^+p\)  319  [10]  
ZEUS NC  0607  0.005  0.65  200  30, 000  135.5  \(e^+p\)  318  [22]  
ZEUS CC \(^{1.5p}\)  0607  0.0078  0.42  280  30, 000  132  \(e^+p\)  318  [23]  
ZEUS NC \(^{1.5}\)  0506  0.005  0.65  200  30, 000  169.9  \(e^p\)  318  [20]  
ZEUS CC \(^{1.5}\)  0406  0.015  0.65  280  30, 000  175  \(e^p\)  318  [21]  
ZEUS NC nominal \(^{*y}\)  0607  0.000092  0.008343  7  110  44.5  \(e^+p\)  318  [24]  
ZEUS NC satellite \(^{*y}\)  0607  0.000071  0.008343  5  110  44.5  \(e^+p\)  318  [24]  
HERA II \(E_p=575\) GeV data sets  
H1 NC high \(Q^2 \)  07  0.00065  0.65  35  800  5.4  \(e^+p\)  252  [9]  
H1 NC low \(Q^2 \)  07  0.0000279  0.0148  1.5  90  5.9  \(e^+p\)  252  [10]  
ZEUS NC nominal  07  0.000147  0.013349  7  110  7.1  \(e^+p\)  251  [24]  
ZEUS NC satellite  07  0.000125  0.013349  5  110  7.1  \(e^+p\)  251  [24]  
HERA II \(E_p=460\) GeV data sets  
H1 NC high \(Q^2 \)  07  0.00081  0.65  35  800  11.8  \(e^+p\)  225  [9]  
H1 NC low \(Q^2 \)  07  0.0000348  0.0148  1.5  90  12.2  \(e^+p\)  225  [10]  
ZEUS NC nominal  07  0.000184  0.016686  7  110  13.9  \(e^+p\)  225  [24]  
ZEUS NC satellite  07  0.000143  0.016686  5  110  13.9  \(e^+p\)  225  [24] 
The very low\(Q^2\) region is covered by data from both experiments taken during the HERA I period. The lowest, \(Q^2 \ge 0.045\) GeV\(^2\), data come from measurements with the ZEUS detector using special tagging devices. They are named ZEUS BPT in Table 1. During the course of this analysis, it was discovered that in the HERA I analysis [2], values given for \(F_2\) were erroneously treated as reduced cross sections. This was corrected for the analysis presented in this paper. All other individual data sets from HERA I were used in the new combination exactly as in the previously published combination [2].
The \(Q^2\) range from 0.2 to 1.5 GeV\(^2\) was covered using special HERA I runs, in which the interaction vertex position was shifted forward, bringing backward scattered electrons with small scattering angles into the acceptance of the detectors [3, 13, 74]. The lowest\(Q^2\) values for these shiftedvertex data were reached using events in which the electron energy was reduced by initialstate radiation [3].
The \(Q^2 \ge 1.5\) GeV\(^2\) range was covered by HERA I and HERA II data in various configurations. The highstatistics HERA II data sets increase the accuracy at high \(Q^2\), particularly for \(e^p\) scattering, for which the integrated luminosity for HERA I was very limited.
The 2007 running periods with lowered proton energies [9, 10, 24] were included in the combination and provide data with reduced \(\sqrt{s}\) and \(Q^2\) up to 800 GeV\(^2\). These data were originally taken to measure \(F_\mathrm{L}\).
3.4 Data on charm, beauty and jet production
The QCD analyses presented in Sect. 6 also used selected results on heavyquark and jet production.
The charm production cross sections were taken from a publication [46] in which data from nine data sets published by H1 and ZEUS, covering both the HERA I and II periods, were combined. The beauty production cross sections were taken from two publications, one from ZEUS [75] and one from H1 [76]. The heavyquark events form small subsets of the inclusive data. Correlations between the charm and the inclusive data are small and were not taken into account.
The data on jet production cross sections were taken from selected publications: ZEUS inclusivejet production data from HERA I [47], ZEUS dijet production data from HERA II [48], H1 inclusivejet production data at low \(Q^2\) [49] and high \(Q^2\) from HERA I [50] and HERA II [51]. The HERA II H1 publication provides inclusivejet, dijet and trijet cross sections normalised to the inclusive NC DIS cross sections in the respective \(Q^2\) range. This largely reduces the correlations with the H1 inclusive DIS reduced cross sections. The HERA I H1 high\(Q^2\) jet data are similarly normalised. The other ZEUS and H1 jet data sets are small subsamples of the respective inclusive sample; correlations are small and are thus ignored.
For the heavyquark and jet data sets used, the statistical, uncorrelated systematic and correlated systematic uncertainties were used as published.
4 Combination of the inclusive cross sections
In order to combine the published cross sections from the 41 data sets listed in Table 1, they were translated onto common grids and averaged.
4.1 Common \({\sqrt{s}}\) values, common \({(x_\mathrm{Bj},Q^2)}\) grids and translation of data
The data were taken with several \(E_p\) values and the doubledifferential cross sections were published by the two experiments for different reference \(\sqrt{s}\) and \((x_\mathrm{Bj},Q^2)\) grids. In order to average a set of data points, the points had to be translated to common \(\sqrt{s}_{\mathrm{com}}\) values and common \((x_\mathrm{Bj,grid},Q^2_\mathrm{grid})\) grids. The following choices were made.
In most of the phase space, separate measurements from the same data set were not translated to the same grid point. Only 9 out of 1307 grid points accumulated two and in one case three points from the same data set. Up to 10 data sets were available for a given process. The vast majority of grid points accumulated data from both H1 and ZEUS measurements; the typical case is six measurements from six different data sets. However, 22 % of all grid points have only one measurement, predominantly at low \(Q^2\). For \(Q^2\) above 3.5 GeV\(^2\), only 13 % of the grid points have only one measurement.
For the translation of the crosssection values, predictions for the ratios of the doubledifferential cross section at the \((x_\mathrm{Bj},Q^2)\) and \(\sqrt{s}\) where the measurements took place, and the \((x_\mathrm{Bj,grid},Q^2_\mathrm{grid})\) to which they were translated, were needed. These predictions, \(T_\mathrm{grid}\), were obtained from the data themselves by performing fits to the data using the HERAFitter [26, 27] tool. For \(Q^2 \ge 3\) GeV\(^2\), a nexttoleadingorder QCD fit using the DGLAP formalism was performed.^{4} In addition, a fit using the fractal model^{5} [3, 77] was performed for \(Q^2 \le 4.9\) GeV\(^2\). For \(Q^2 < 3\) GeV\(^2\), the fit to the fractal model was used^{6} to obtain factors \(T_\mathrm{grid,FM}\). For \(Q^2 > 4.9\) GeV\(^2\), the QCD fit was used to provide \(T_\mathrm{grid,QCD}\). For \(3 \le Q^2 \le 4.9\) GeV\(^2\), the factors were averaged as \(T_\mathrm{grid} = T_\mathrm{grid,FM} (1(Q^23)/1.9) + T_\mathrm{grid,QCD} (Q^23)/1.9 \) where \(Q^2\) is in GeV\(^2\). The upper edge of the application of the fractal fit was varied between 3 GeV\(^2\) and 5 GeV\(^2\); the effect was negligible.
4.2 Averaging cross sections
The original doubledifferential crosssection measurements were published with their statistical and systematic uncertainties. The systematic uncertainties were classified as either pointtopoint correlated or pointtopoint uncorrelated. For each data set, all uncorrelated systematic uncertainties were added in quadrature before averaging. Correlated systematic uncertainties were kept separately. Some of the systematic uncertainties were originally reported as asymmetric. They were symmetrised by the collaborations before entering the averaging procedure.
The leading systematic uncertainties on the crosssection measurements used for the combination arose from the uncertainties on the acceptance corrections and luminosity determinations. Thus, both the correlated and uncorrelated systematic uncertainties are multiplicative in nature, i.e. they increase proportionally to the central values. In Eq. 22, the multiplicative nature of these uncertainties is taken into account by multiplying the relative errors \(\gamma ^{i,ds}_j\) and \(\delta _{i,ds,\mathrm{uncor}}\) by the estimate \(m^i\). The denominator in the first righthandside term in Eq. 22 contains an estimate of the squared statistical uncertainty of the crosssection measurement, \(\delta _{i,ds,stat}^2 \mu ^{i,ds} (m^{i}  \sum _j \gamma ^{i,ds}_j m^i b_j)\), which is assumed^{7} to scale with the expected number of events in bin i, as calculated from \(m^{i}\). Corrections due to the shifts to accommodate the correlated systematic uncertainties are introduced through the term \(\sum _{j} \gamma ^{i,ds}_j m^i b_j\).
The ratio of \(\chi ^2_\mathrm{min}\) and the number of degrees of freedom, \(\chi ^2_\mathrm{min}/\mathrm{d.o.f.}\), is a measure of the consistency of the data sets. The number \(\mathrm{d.o.f.}\) is the difference between the total number of measurements and the number of averaged points \(N_M\).
Some systematic uncertainties \(\gamma _j^i\), which were treated as having pointtopoint correlations, may be common for several data sets. A full table of the correlations of the systematic uncertainties across the data sets can be found elsewhere [79]. The systematic uncertainties were in general treated as independent between H1 and ZEUS. However, an overall normalisation uncertainty of \(0.5\,\%\), due to uncertainties on higherorder corrections to the Bethe–Heitler crosssection calculations, was assumed for all data sets which were normalised with data from the luminosity monitors.
All the NC and CC crosssection data from H1 and ZEUS are combined in one simultaneous minimisation. Therefore, the resulting shifts of the correlated systematic uncertainties propagate coherently to both NC and CC data. Even in cases where there are data only from a single data set, the procedure can still produce shifts with respect to the original measurement due to the correlation of systematic uncertainties.
4.3 Combination procedure
4.4 Consistency of the data
4.5 Procedural uncertainties
Procedural uncertainties are introduced by the choices made for the combination. Three kinds of such uncertainties were considered.
4.5.1 Multiplicative versus additive treatment of systematic uncertainties
The \(\chi ^2\) definition from Eq. 22 treats all systematic uncertainties as multiplicative, i.e. their size is expected to be proportional to the “true” values \(\varvec{m}\). While this is a good assumption for normalisation uncertainties, this might not be the case for other uncertainties. Therefore an alternative combination was performed, in which only the normalisation uncertainties were taken as multiplicative, while all other uncertainties were treated as additive. The differences between this alternative combination and the nominal combination were defined as correlated procedural uncertainties \(\delta _\mathrm{rel}\). This is a conservative approach but still yields quite small uncertainties. The typical values of \(\delta _\mathrm{rel}\) for the \(\sqrt{s}_\mathrm{com,1}=318\) GeV (\(\sqrt{s}_\mathrm{com,2/3}\)) combination were below 0.5 % (1 %) for medium\(Q^2\) data, increasing to a few percent for low and high\(Q^2\) data.
4.5.2 Correlations between systematic uncertainties on different data sets
Similar methods were often used to calibrate different data sets obtained by one or by both collaborations. In addition, the same Monte Carlo simulation packages were used to analyse different data sets. These similar approaches might have led to correlations between data sets from one or both collaborations. This was investigated in depth for the combination of HERA I data [2]. The important correlations for this period were found to be related to the background from photoproduction and the hadronic energy scales. The correlations for the HERA I period were taken into account as before [2].
The correlations between the experiments for the HERA II period were considered much less important, because both experiments developed different methods to address calibration and normalisation. In the case of H1, some potential correlations between the data from the HERA I and HERA II periods were identified. In the case of ZEUS, no such correlations were found; this is due to significant changes in the detector and in the data processing.
The differences between the nominal combination and the combinations, in which systematic sources for the photoproduction background and hadronic energy scale were taken as correlated across data sets, were defined as additional signed procedural uncertainties \(\delta _{\gamma p}\) and \(\delta _\mathrm{had}\). Typical values of \(\delta _{\gamma p}\) and \(\delta _\mathrm{had}\) are below \(1\,\%\) (0.5 %) for NC (CC) scattering. For the data at low \(Q^2\), they can reach a few percent.
4.5.3 Pull distribution of correlated systematic uncertainties
The distribution of pulls shown in Fig. 3 is not Gaussian; it has a rootmeansquare value of 1.34. Out of the 162 pointtopoint correlated uncertainties, 40 were identified with \( \mathrm{p}_j > 1.3\). This might indicate that these uncertainties were either underestimated or do not fulfil the implicit assumptions of the linear procedure applied. Scaling these 40 uncertainties by a factor of two would reduce the rootmeansquare value to 1.03 and the \(\chi ^2_\mathrm{min}\) of the combination would be reduced from 1687 to 1614 for the 1620 degrees of freedom.
 1.
very low\(Q^2\) data from HERA I (14 uncertainties);
 2.
low\(Q^2\) data from HERA II with lowered proton beam energies (10 uncertainties);
 3.
medium and high\(Q^2\) data from HERA I and II (11 uncertainties);
 4.
normalisation issues from HERA I and II (5 uncertainties).
5 Combined inclusive \(\varvec{e^{\pm }p}\) cross sections
The total uncertainties are below 1.5 % over the \(Q^2\) range of \(3 \le Q^2 \le 500\) GeV\(^2\) and below 3 % up to \(Q^2 = 3000\) GeV\(^2\). Cross sections are provided for values of \(Q^2\) between \(Q^2=0.045\) GeV\(^2\) and \(Q^2=50{,}000\) GeV\(^2\) and values of \(x_\mathrm{Bj}\) between \(x_\mathrm{Bj}=6\times 10^{7}\) and \(x_\mathrm{Bj}=0.65\). The events have a minimum invariant mass of the hadronic system, W, of 15 GeV.
In Fig. 4, the individual and the combined reduced cross sections for NC \(e^+p\) DIS scattering are shown as a function of \(Q^2\) for selected values of \(x_\mathrm{Bj}\). The improvement due to combination is clearly visible. In Fig. 5, a comparison between the new combination and the combination of HERA I data alone is shown. The improvement is especially significant at high \(Q^2\). The results for NC \(e^p\) scattering are depicted in Figs. 6 and 7. As the integrated luminosity for \(e^p\) scattering was very limited for the HERA I period, the improvements due to the new combination are even more substantial than for \(e^+p\) scattering.
The results of the combination of the data with lower proton beam energies are shown in Figs. 8 and 9 as a function of \(x_\mathrm{Bj}\) in selected bins of \(Q^2\). These data augment the data with standard proton energy to provide increased sensitivity to the gluon density in the proton.
The combined NC \(e^+p\) data for very low \(Q^2\) with proton beam energies of 920 and 820 GeV are shown in Figs. 10 and 11. These data were taken during the HERA I period, but due to the systematic shifts introduced by the combination with HERA II data, the numbers are not always the same as in the old HERA I combination.
The highprecision DIS cross sections provided here form a coherent set spanning six orders of magnitude, both in \(Q^2\) and \(x_\mathrm{Bj}\). They are a major legacy of HERA.
6 QCD analysis
In this section, the pQCD analysis of the combined data resulting in the PDF set HERAPDF2.0 and its released variants is presented. The framework established for HERAPDF1.0 [2] was followed in this analysis. A breakdown of pQCD is expected for \(Q^2\) approaching 1 GeV\(^2\). To safely remain in the kinematic region where pQCD is expected to be applicable, only cross sections for \(Q^2\) starting from \(Q^2_\mathrm{min} = 3.5\) GeV\(^2\) were used in the analysis. In this kinematic region, targetmass corrections are expected to be negligible. Since the centreofmass energy at the \(\gamma p\) vertex W is above 15 GeV for all the data, large\(x_\mathrm{Bj}\) highertwist corrections are also expected to be negligible. The \(Q^2\) range of the cross sections entering the fit is \(3.5 \le Q^2 \le 50{,}000\,\)GeV\(^2\). The corresponding \(x_\mathrm{Bj}\) range is \(0.651 \times 10^{4} \le x_\mathrm{Bj} \le 0.65 \).
In addition to experimental uncertainties, model and parameterisation uncertainties were also considered. The latter were evaluated by variations of the values of various input settings at the starting scale and the form of the parameterisation.
6.1 Theoretical formalism and settings
Predictions from pQCD are fitted to data. These predictions were obtained by solving the DGLAP evolution equations [29, 30, 31, 32, 33] at LO, NLO and NNLO in the \(\overline{\mathrm{MS}}\ \) scheme [80]. This was done using the programme QCDNUM [81] within the HERAFitter framework [26, 27] and an independent programme, which was already used to analyse the combined HERA I data [2]. The results obtained by the two programmes were in excellent agreement, well within fit uncertainties. The numbers on fit quality and resulting parameters given in this paper were obtained using HERAFitter.
The value of \(M_c\) was chosen after performing \(\chi ^2\) scans of NLO and NNLO pQCD fits to the combined inclusive data from the analysis presented here and the HERA combined charm data [46]. The procedure is described in detail in the context of the combination of the reduced charm crosssection measurements [46]. All correlations of the inclusive and of the charm data were considered in the fits. Figure 16 shows the \(\Delta \chi ^2 = \chi ^2 \chi ^2_\mathrm{min}\), where \(\chi ^2_\mathrm{min}\) is the minimum \( \chi ^2\) obtained, of these fits versus \(M_c\) at NLO and NNLO. As a result, the value of \(M_c\) was chosen as \(M_c=1.47\,\)GeV at NLO and \(M_c=1.43\,\)GeV at NNLO. The settings for LO were chosen as for NLO unless otherwise stated.
The value of the beautyquark mass parameter \(M_b\) was chosen after performing \(\chi ^2\) scans of NLO and NNLO pQCD fits using the combined inclusive data and data on beauty production from ZEUS [75] and H1 [76]. The \(\chi ^2\) scans are shown in Fig. 17. The value of \(M_b\) was chosen to be \(M_b=4.5\) GeV at LO, NLO and NNLO. The value of the topquark mass parameter was chosen to be 173 GeV [52] at all orders.
The value of the strong coupling constant was chosen to be \(\alpha _s(M_Z^2)= 0.118\) [52] at both NLO and NNLO and \(\alpha _s(M_Z^2)= 0.130\) [38] for the LO fit.
6.2 Parameterisation
The normalisation parameters, \(A_{u_v}, A_{d_v}, A_g\), are constrained by the quarknumber sum rules and the momentum sum rule. The B parameters \(B_{\bar{U}}\) and \(B_{\bar{D}}\) were set as equal, \(B_{\bar{U}}=B_{\bar{D}}\), such that there is a single B parameter for the sea distributions. The strangequark distribution is expressed as an xindependent fraction, \(f_s\), of the dtype sea, \(x\bar{s}= f_s x\bar{D}\) at \(\mu ^2_\mathrm{f_{0}}\). The value \(f_s=0.4\) was chosen as a compromise between the determination of a suppressed strange sea from neutrinoinduced dimuon production [36, 85] and a recent determination of an unsuppressed strange sea, published by the ATLAS collaboration [86]. A further constraint was applied by setting \(A_{\bar{U}}=A_{\bar{D}} (1f_s)\). This, together with the requirement \(B_{\bar{U}}=B_{\bar{D}}\), ensures that \(x\bar{u} \rightarrow x\bar{d}\) as \(x \rightarrow 0\).
The parameters appearing in Eqs. 27–31 were selected by first fitting with all D and E parameters and \(A_g'\) set to zero. This left 10 free parameters. The other parameters were then included in the fit one at a time. The improvement of the \(\chi ^2\) of the fits was monitored and the procedure was ended when no further improvement in \(\chi ^2\) was observed. This led to a 15parameter fit at NLO and a 14parameter fit at NNLO. A common parameterisation with 14parameters was chosen as “central”, both at NLO and at NNLO, such that any differences between these fits reflect only the change in order. The central fits satisfy the criterion that all the PDFs are positive in the measured region. The 15parameter NLO fit was used as a parameterisation variation, see Sect. 6.5.
6.3 Definition of \(\varvec{\chi ^2}\)
Correlated systematic uncertainties were treated as for the combination of data, see Sect. 4.2. For the combined inclusive data, the correlated systematic uncertainties are smaller or comparable to the statistical and uncorrelated uncertainties. Nevertheless, the remaining correlations are significant and thus the 162 systematic uncertainties present for the H1 and ZEUS data sets plus the seven sources of procedural uncertainty which resulted from the combination procedure, see Sect. 4.5, were all individually treated as correlated uncertainties.
6.4 Experimental uncertainties
Input parameters for HERAPDF2.0 fits and the variations considered to evaluate model and parameterisation (\(\mu _{f_{0}}\)) uncertainties
Variation  Standard value  Lower limit  Upper limit 

\(Q^2_\mathrm{min}\) (GeV\(^2\))  3.5  2.5  5.0 
\(Q^2_\mathrm{min}\) (GeV\(^2\)) HiQ2  10.0  7.5  12.5 
\(M_c\) (NLO) (GeV)  1.47  1.41  1.53 
\(M_c\) (NNLO) (GeV)  1.43  1.37  1.49 
\(M_b\) (GeV)  4.5  4.25  4.75 
\(f_s\)  0.4  0.3  0.5 
\(\alpha _s(M_Z^2)\)  0.118  –  – 
\(\mu _{f_{0}}\) (GeV)  1.9  1.6  2.2 
A cross check was performed using the Monte Carlo method [87, 88]. It is based on analysing a large number of pseudo data sets called replicas. For this cross check, 1000 replicas were created by taking the combined data and fluctuating the values of the reduced cross sections randomly within their given statistical and systematic uncertainties taking into account correlations. All uncertainties were assumed to follow Gaussian distributions. The PDF central values and uncertainties were estimated using the mean and RMS values over the replicas.
The uncertainties obtained by the Monte Carlo method and the Hessian method were consistent within the kinematic reach of HERA. This is demonstrated in Fig. 18 where experimental uncertainties obtained for HERAPDF2.0 NNLO by the Hessian and Monte Carlo methods are compared for the valence, the gluon and the total seaquark distributions. The RMS values taken as Monte Carlo uncertainties tend to be slightly larger than the standard deviations obtained in the Hessian approach.
6.5 Model and parameterisation uncertainties
For the NLO and NNLO PDFs, the uncertainties on HERAPDF2.0 due to the choice of model settings and the form of the parameterisation were evaluated by varying the assumptions. A summary of the variations on model parameters is given in Table 2. The variations of \(M_c\) and \(M_b\) were chosen in accordance with the \(\chi ^2\) scans related to the heavyquark mass parameters as shown in Figs. 16 and 17. The data on heavyquark production from HERA II led to a considerably reduced uncertainty on the heavyquark mass parameters compared to the HERAPDF1.0 and HERAPDF1.5 analyses, see Appendix A.
The variation of \(f_s\) was chosen to span the ranges between a suppressed strange sea [36, 85] and an unsuppressed strange sea [86]. In addition to this, two more variations of the assumptions about the strange sea were made. Instead of assuming that the strange contribution is a fixed fraction of the dtype sea, an xdependent shape, \(x\bar{s}= f_s' \, 0.5 \tanh (20(x0.07))\, x\bar{D}\), was used in which highx strangeness is highly suppressed. This was suggested by measurements published by the HERMES collaboration [89, 90]. The normalisation of \(f_s'\) was also varied between \(f_s'=0.3\) and \(f_s'=0.5\).
Input parameters for HERAPDF2.0FF fits. All other parameters were set as for the standard HERAPDF2.0 NLO fit
Scheme  \(\alpha _s(M_Z^2)\)  \(F_\mathrm{L}\)  \(m_c\) (GeV)  \(m_b\) (GeV) 

FF3A  \(\alpha _s^{N_F=3} = 0.106375\)  \(\mathcal{O}(\alpha _s^2)\)  \(m_c^\mathrm{pole} = 1.44\)  \(m_b^\mathrm{pole} = 4.5\) 
FF3B  \(\alpha _s^{N_F=5} = 0.118\)  \(\mathcal{O}(\alpha _s)\)  \(m_c(m_c) = 1.26\)  \(m_b(m_b) = 4.07\) 
The values of \(\chi ^2\) per degree of freedom for HERAPDF2.0 and its variants
HERAPDF  \(Q^2_\mathrm{min}\) (GeV\(^2\))  \(\chi ^2\)  dof  \(\chi ^2\)/dof 

2.0 NLO  3.5  1357  1131  1.200 
2.0HiQ2 NLO  10.0  1156  1002  1.154 
2.0 NNLO  3.5  1363  1131  1.205 
2.0HiQ2 NNLO  10.0  1146  1002  1.144 
2.0 AG NLO  3.5  1359  1132  1.201 
2.0HiQ2 AG NLO  10.0  1161  1003  1.158 
2.0 AG NNLO  3.5  1385  1132  1.223 
2.0HiQ2 AG NNLO  10.0  1175  1003  1.171 
2.0 NLO FF3A  3.5  1351  1131  1.195 
2.0 NLO FF3B  3.5  1315  1131  1.163 
2.0Jets \(\alpha _s(M_Z^2)\) fixed  3.5  1568  1340  1.170 
2.0Jets \(\alpha _s(M_Z^2)\) free  3.5  1568  1339  1.171 
Two kinds of parameterisation uncertainties were considered, the variation in \(\mu ^2_\mathrm{f_{0}}\) and the addition of parameters D and E, see Eq. 26. The variation in \(\mu ^2_\mathrm{f_{0}}\) mostly increased the PDF uncertainties of the sea and gluon at small x. The parameters D and E were added separately for each PDF. The only significant difference from the 14parameter central fit came from the 15parameter fit, for which \(D_{u_v}\) was non zero. This affected the shape of the Utype sea as well as the shape of \(u_v\). The final parameterisation uncertainty for a given quantity is taken as the largest of the uncertainties. This uncertainty is valid in the xrange covered by the QCD fits to HERA data.
6.6 Total uncertainties
The total PDF uncertainty is obtained by adding in quadrature the experimental, the model and the parameterisation uncertainties described in Sects. 6.4 and 6.5. Differences arising from using alternative values of \(\alpha _s(M_Z^2)\), alternative forms of parameterisations, different heavyflavour schemes or a very different \(Q^2_\mathrm{min}\) are not included in these uncertainties. Such changes result in the different variants of the PDFs to be discussed in the subsequent sections.
6.7 Alternative values of \(\varvec{\alpha _s(M_Z^2)}\)
6.8 Alternative forms of parameterisation
An “alternative gluon parameterisation”, AG, was considered at NNLO and NLO. The value of \(A_g'\) in Eq. 27 was set to zero and a polynominal term for xg(x) as in Eq. 26 was substituted. This potentially resulted in a different 14parameter fit. However, in practice a 13parameter fit with a nonzero \(D_g\) was sufficient for the AG parameterisation, since there was no improvement in \(\chi ^2\) for a nonzero \(E_g\). Note that AG was the only parameterisation considered at LO.
The standard parameterisation fits the HERA data better; however, especially at NNLO, it produces a negative gluon distribution for very low x, i.e. \(x < 10^{4}\). This is outside the kinematic region of the fit, but may cause problems if the PDFs are used at very low x within the conventional formalism. Therefore, a variant HERAPDF2.0AG using the alternative gluon parameterisation is provided for predictions of cross sections at very low x, such as very highenergy neutrino cross sections.
HERAPDF has a certain ansatz for the parameterisation of the PDFs, see Sect. 6.2. Different ways of using the polynomial form, such as parameterising xg, \(xu_v\), \(xd_v\), \(x\bar{d}+x\bar{u}\) and \(x\bar{d}x\bar{u}\) or xg, xU, xD, \(x\bar{U}\) and \(x\bar{D}\) were investigated. The resulting PDFs agreed with the standard PDFs within uncertainties and no improvement of fit quality resulted. Therefore, these alternative parameterisations were not pursued further.
6.9 Alternative heavyflavour schemes
The standard choice of heavyflavour scheme for HERAPDF2.0 is the variableflavournumber scheme RTOPT [84]. Investigations using other heavyflavour schemes were also carried out.
Two other variableflavournumber schemes, FONLL [91, 92] and ACOT [93], were considered, as implemented in HERAFitter at the time of the analysis. The FONLL scheme is implemented via an interface to the APFEL program [94] and was used at NLO and NNLO. The ACOT scheme is implemented using kfactors for the NLO corrections. The three heavyflavour schemes differ in the order at which \(F_\mathrm{L}\) is evaluated. At NLO, the massless contribution to \(F_\mathrm{L}\) is evaluated to \(\mathcal{O} (\alpha _s^2)\) for RTOPT and to \(\mathcal{O}(\alpha _s)\) for FONNLB and ACOT. At NNLO, the massless contribution to \(F_\mathrm{L}\) is evaluated to \(\mathcal{O} (\alpha _s^3)\) for RTOPT and to \(\mathcal{O}(\alpha _s^2)\) for FONNLC. Fixedflavournumber schemes were also investigated. In such schemes, the number of (massless) light flavours in the PDFs remains fixed across “flavour thresholds” and (massive) heavy flavours only occur in the matrix elements.
For some calculations, e.g. charm production at HERA, the availability of fixedflavour variants of the PDFs is useful or even mandatory. Many PDF groups provide either fixedflavour fits only, or variableflavour fits only, with a fixedflavour variant calculated from the variableflavour parton distributions at the starting scale using theory. For HERAPDF2.0, fixedflavour variants are provided which were actually fitted to the data.
 scheme FF3A:

Threeflavour running of \(\alpha _s\);

\(F_\mathrm{L}\) calculated to \(\mathcal{O}(\alpha _s^2)\);

pole masses for charm, \(m_c^\mathrm{pole}\), and beauty, \(m_b^\mathrm{pole}\);

 scheme FF3B:

Variableflavour running of \(\alpha _s\) [95]. This is sometimes called the “mixed scheme” [81];

massless (light flavour) part of the \(F_\mathrm{L}\) contribution calculated to \(\mathcal{O}(\alpha _s)\);

\(\overline{\mathrm{MS}}\) [80] running masses for charm, \(m_c(m_c)\), and beauty \(m_b(m_b)\).

The fits providing the variants HERAPDF2.0FF3A and HERAPDF2.0FF3B were obtained using the OPENQCDRAD [96] package as implemented in HERAFitter, partially interfaced to QCDNUM. This was proven to be consistent with the standalone version of OPENQCDRAD and, in the case of the A variant, with the FFNS definition used by the ABM [40, 41, 42] fitting group. The HERAFitter implementation allows an external steering of the order of \(\alpha _s\) in \(F_\mathrm{L}\), as listed in Table 3.
6.10 Adding data on charm production to the HERAPDF2.0 fit
The data on charm production described in Sect. 3.4 were used to find the optimal value of \(M_c\) for the HERAPDF2.0 fits as described in Sect. 6.1.
The impact of adding charm data to inclusive data as input to NLO QCD fits has been extensively discussed in a previous publication [46]. This previous analysis was based on the HERA I combined inclusive data and combined charm data. It was established that the main impact of the charm data on the PDF fits is a reduction of the uncertainty on \(M_c\). It was also established that the optimal value of \(M_c\) can differ according to the particular generalmass variableflavournumber scheme chosen for the fit. The fits for all schemes considered were of similar quality.
Central values of the HERAPDF2.0 parameters at NLO
A  B  C  D  E  \(A'\)  \(B'\)  

xg  4.34  \(\)0.015  9.11  1.048  \(\)0.167  
\(xu_v\)  4.07  0.714  4.84  13.4  
\(xd_v\)  3.15  0.806  4.08  
\(x\bar{U}\)  0.105  \(\)0.172  8.06  11.9  
\(x\bar{D}\)  0.176  \(\)0.172  4.88 
Central values of the HERAPDF2.0 parameters at NNLO
A  B  C  D  E  \(A'\)  \(B'\)  

xg  2.27  \(\)0.062  5.56  0.167  \(\)0.383  
\(xu_v\)  5.55  0.811  4.82  9.92  
\(xd_v\)  6.29  1.03  4.85  
\(x\bar{U}\)  0.161  \(\)0.127  7.09  1.58  
\(x\bar{D}\)  0.269  \(\)0.127  9.58 
The inclusion of the charm data had little influence on the result of the fit. This was not unexpected, since the main effect of the charm data, i.e. to constrain \(M_c\), was already used for the fit to the inclusive data. The charm data were proven to be consistent with the inclusive data, but only a marginal reduction in the uncertainty on the lowx gluon PDF was obtained. The situation is similar at NNLO. Therefore no HERAPDF2.0 variants with only the addition of data on charm production are released.
6.11 Adding data on jet production to the HERAPDF2.0 fit
The jet data were included in the fits at NLO by calculating predictions for the jet cross sections with NLOjet++ [97, 98], which was interfaced to FastNLO [99, 100, 101] in order to achieve the speed necessary for iterative PDF fits. The predictions were multiplied by corrections for hadronisation and \(Z^0\) exchange before they were used to fit the data [47, 48, 49, 50, 51]. A running electromagnetic \(\alpha \) as implemented in the 2012 version of the programme EPRC [102] was used for the treatment of jet cross sections when they were included in the PDF fits. The factorisation scale was chosen as \(\mu _\mathrm{f}^2 = Q^2\), while the renormalisation scale was linked to the transverse momenta, \(p_T\), of the jets by \(\mu _\mathrm{r}^2 = (Q^2 + p_{T}^2)/2\). Jet data could not be included at NNLO for the analysis presented here, because the matrix elements were not available at the time of writing.
Fits including these jet data and including the combined charm data were performed with \(\alpha _s(M_Z^2)=0.118\) fixed and with \(\alpha _s(M_Z^2)\) as a free parameter in the fit. The resulting HERAPDF variant with free \(\alpha _s(M_Z^2)\) is called HERAPDF2.0Jets. A full uncertainty analysis was performed for the HERAPDF2.0Jets variant, including model and parameterisation uncertainties and additional hadronisation uncertainties on the jet data as evaluated for the original publications [47, 48, 49, 50, 51].
6.12 The \(\chi ^2\) values of the HERAPDF2.0 fits and alternative \(Q^2_\mathrm{min}\)
The \(\chi ^2/\mathrm{d.o.f.}\) of the fits for HERAPDF2.0 and its variants are listed in Table 4. These values are somewhat large, typically around 1.2. The dependence of \(\chi ^2\) on \(Q^2_\mathrm{min}\) was investigated in detail. Figure 19 shows the \(\chi ^2/\mathrm{d.o.f.}\) values for the LO, NLO and NNLO fits versus \(Q^2_\mathrm{min}\). The \(\chi ^2/\mathrm{d.o.f.}\) drop steadily until \(Q^2_\mathrm{min} \approx 10\,\)GeV\(^2\). Also shown are \(\chi ^2\) values obtained for an NLO fit to HERA I data only. These values are substantially closer to one, but they show the same trend as seen for HERAPDF2.0.
The influence of the choice of heavyflavour scheme, and the order at which the massless contribution to \(F_\mathrm{L}\) is evaluated, on the \(\chi ^2/\mathrm{d.o.f.}\) behaviour was also investigated. Scans at NLO and NNLO of the \(\chi ^2/\mathrm{d.o.f.}\) versus \(Q^2_\mathrm{min}\) for fits done with the heavyflavour schemes described in Sect. 6.9 are illustrated in Fig. 20. The decrease of the \(\chi ^2/\mathrm{d.o.f.}\) with increasing \(Q^2_\mathrm{min}\) is observed for every scheme. At NLO and low \(Q^2_\mathrm{min}\), all fits using schemes for which the \(F_\mathrm{L}\) contributions are calculated using matrix elements of the order of \(\alpha _s\) result in slightly lower \(\chi ^2/\mathrm{d.o.f.}\) than fits for schemes using matrix elements of the order of \(\alpha _s^2\). The increase of \(\chi ^2/\mathrm{d.o.f.}\) for lower \(Q^2_\mathrm{min}\) is also less pronounced for fits using the “\(\mathcal{O}(\alpha _s)\)schemes”. However, at NNLO, the trend reverses and RTOPT, which uses matrix elements of order \(\alpha _s^3\) in the calculation of \(F_\mathrm{L}\), results in lower \(\chi ^2/\mathrm{d.o.f.}\) than the FONNL scheme, for which matrix elements of order \(\alpha _s^2\) are used. The \(\chi ^2/\mathrm{d.o.f.}\) values for fits with the RTOPT scheme are quite similar at NLO and NNLO.
The two fixedflavournumber schemes considered, see Sect. 6.9, also differ in using lightflavour matrix elements of order \(\alpha _s\) (FF3B) and \(\alpha _s^2\) (FF3A). The FF3A fit variant results in \(\chi ^2/\mathrm{d.o.f.}\) values very similar to the values from the standard fit using RTOPT while the values for the FF3B variant closely follow the results for fits using the FONNL scheme. This suggests that the determining factor for the \(\chi ^2\) of the fits is the order of \(\alpha _s\) of the matrix elements used to calculate the massless \(F_\mathrm{L}\) contribution. Other differences between FF3A and FF3B as well as differences [103] between different variableflavournumber schemes, and differences between fixedflavournumber and variableflavournumber schemes, seem to have less influence on \(\chi ^2\).
At HERA, the low\(Q^2\) data are also dominantly at low \(x_\mathrm{Bj}\). Some of the poor \(\chi ^2\) values in this kinematic region could be due to low\(x_\mathrm{Bj}\) physics not accounted for in the current framework [1, 104]. This could mean that the inclusion of low\(x_\mathrm{Bj}\), low\(Q^2\) data into the fits introduces bias. To study this, NLO and NNLO fits with \(Q^2_\mathrm{min} = 10\,\)GeV\(^2\) were also fully evaluated. This variant is called HERAPDF2.0HiQ2. As part of the evaluation, the settings were reexamined. No significant changes for the optimal parameterisation or for the optimal value of \(M_c\) or \(M_b\) were observed. Model and parameterisation variations were also performed in order to better assess possible bias. For the NLO fits, the \(\chi ^2/\mathrm{d.o.f.}\) of 1156 / 1002 for the \(Q^2_\mathrm{min} = 10\,\)GeV\(^2\) fit can be compared to the 1357 / 1131 for the \(Q^2_\mathrm{min} = 3.5\,\)GeV\(^2\) fit. This is a significant improvement, but still larger than observed for HERAPDF1.0. The values are similar at NNLO, see Table 4. In particular, the NNLO fit does not fit the lower\(Q^2\) data better than the NLO fit, see Fig. 19, just as, at NLO, the higherorder evaluation of \(F_\mathrm{L}\) does not fit these data better, see Fig. 20.
Fits were also performed with the alternative gluon parameterisation and \(Q^2_\mathrm{min} = 10\,\)GeV\(^2\). The \(\chi ^2/\mathrm{d.o.f.}\) was always worse than for the standard parameterisation, see Table 4.
The \(\chi ^2/\mathrm{d.o.f.}\) values obtained for HERAPDF2.0Jets, both for fixed and for free \(\alpha _s(M_Z^2)\) are better than the value for the standard HERAPDF2.0 NLO fit, see Table 4. The partial \(\chi ^2\) for the jet data is 161 for 162 data points, while it is 41 for 47 data points for the charm data. The partial \(\chi ^2\) for the inclusive data remains practically the same as for HERAPDF2.0 NLO. This demonstrates the compatibility of the data on charm and jet production with the inclusive data.
7 HERAPDF2.0

the data include four different processes, NC and CC for \(e^+p\) and \(e^p\) scattering, such that there is sufficient information to extract the \(xd_v\), \(xu_v\), \(x\bar{U}\) and \(x\bar{D}\) PDFs, and the gluon PDF from the scaling violations;

the NC \(e^+p\) data include data at centreofmass energies sufficiently different to access different values of y at the same \(x_\mathrm{Bj}\) and \(Q^2\); this makes the data sensitive to \(F_\mathrm{L}\) and thus gives further information on the lowx gluon distribution;

it is based on a consistent data set with small correlated systematic uncertainties;

the experimental uncertainties are Hessian uncertainties;

the uncertainties introduced both by model assumptions and by assumptions about the form of the parameterisation are provided;

no heavytarget corrections were needed as all data are on ep scattering; the assumption of \(u_\mathrm{neutron}=d_\mathrm{proton}\) was not needed.
7.1 HERAPDF2.0 NLO, NNLO and 2.0AG
A summary of HERAPDF2.0 NLO is shown in Fig. 21 at the scale \(\mu ^2_\mathrm{f}=10\) GeV\(^2\). The experimental, model and parameterisation uncertainties, see Sects. 6.4 and 6.5, are shown separately. The model and parameterisation uncertainties are asymmetric. The uncertainties arising from the variation of \(\mu ^2_\mathrm{f_{0}}=1.9\) GeV\(^2\) and \(Q^2_\mathrm{min}=3.5\) GeV\(^2\) affect predominantly the lowx region of the sea and gluon distributions. The parameterisation uncertainty from adding the \(D_{u_{v}}\) parameter is important for the valence distributions for all x.
The gluon distribution of HERAPDF2.0 NLO does not become negative within the fitted kinematic region. The distributions of HERAPDF2.0AG with the alternative gluon parameterisation as described in Sect. 6.2 and discussed in Sect. 6.8 are shown superimposed on the standard PDFs.
A summary of HERAPDF2.0 NNLO is shown in Fig. 23 at the scale \(\mu ^2_\mathrm{f}=10\) GeV\(^2\). At NNLO, the gluon distribution of HERAPDF2.0 ceases to rise at low x. Consequently, xg from HERAPDF2.0AG NNLO deviates significantly. As at NLO, the uncertainties arising from the variation of \(\mu ^2_\mathrm{f_{0}}\) and \(Q^2_\mathrm{min}\) affect predominantly the lowx region of the sea and gluon distributions. The parameterisation uncertainty from adding the \(D_{u_v}\) parameter is not important for the NNLO fit, since there was no significant improvement in \(\chi ^2\) from the addition of the 15th parameter. The parameters of the NNLO fit are listed in Table 6.
The flavour breakdown of the sea into \(x\bar{u}\), \(x\bar{d}\), \(x\bar{c}\) and \(x\bar{s}\) for HERAPDF2.0 NNLO is shown in Fig. 24. The uncertainties are also shown as fractional uncertainties. They are dominated by model uncertainties and derive from the same sources as already described at NLO. The parameterisation uncertainties are less important at NNLO than at NLO.
A comparison between HERAPDF2.0 NNLO and NLO is shown in Fig. 25 with total uncertainties, using both linear and logarithmic x scales. The main difference is the different shapes of the gluon distributions as expected from the differing evolution at NLO and NNLO.
At leading order, HERAPDF2.0 is only available as HERAPDF2.0AG LO with the alternative gluon parameterisation. It has thus to be compared to HERAPDF2.0AG NLO. HERAPDF2.0AG LO was determined with experimental uncertainties only, because its main purpose is to be used in LO Monte Carlo programmes. A comparison between the distributions of HERAPDF2.0AG LO and HERAPDF2.0AG NLO is shown in Fig. 26. The gluon distribution at LO rises much faster than at NLO, as expected from the different evolution. The \(xu_v\) distribution is softer at LO than at NLO.
7.1.1 Comparisons to inclusive HERA data
The data with the proton beam energy of 920 GeV (\(\sqrt{s}=318\,\)GeV) are the most precise data due to the large integrated luminosity, see Table 1. HERAPDF2.0 predictions are compared at NNLO, NLO and LO to these highprecision data.
The predictions of HERAPDF2.0 NNLO, NLO and AG LO are compared to the high\(Q^2\) NC \(e^+p\) data in Figs. 27, 28 and 29. The data are well described by the predictions at all orders. Figure 30 shows the cross sections already shown in Fig. 5 together with the predictions of HERAPDF2.0 NNLO and NLO. The predictions at NNLO and NLO are very similar.
For \(e^+p\) scattering, data at low \(Q^2\) are available. Figures 34, 35, and 36 show comparisons between the predictions of HERAPDF2.0 NNLO, NLO and AG LO and these low\(Q^2\) data. The description of the data is generally good and for the predictions at NNLO and NLO, it remains so even for \(Q^2\) below the fitted kinematic region. However, at low \(x_\mathrm{Bj}\) and low \(Q^2\), the turnover in the cross sections related to \(F_\mathrm{L}\) is not well described, and HERAPDF2.0 NNLO does not describe these data better than HERAPDF2.0 NLO. The HERAPDF2.0AG LO predictions show a clear turnover, but the prediction is significantly too high at all \(x_\mathrm{Bj}\) for the lowest \(Q^2\).
The predictions of the NNLO and NLO fits are compared to the CC \(e^+p\) data with \(\sqrt{s} = 318\,\)GeV in Figs. 37 and 38 and to CC \(e^p\) data in Figs. 39 and 40. The precise predictions describe the CC cross sections well. The CC data are in general less precise than the NC data.
The predictions of HERAPDF2.0 NLO compared to low\(Q^2\) and high\(Q^2\) NC \(e^+p\) data for \(\sqrt{s} = 300\,\)GeV are shown in Figs. 41 and 42. Equivalent comparisons for \(\sqrt{s} = 251\,\)GeV and \(\sqrt{s} = 225\,\)GeV are shown in Figs. 43 and 44, and Figs. 45 and 46, respectively. The data with reduced proton beam energy are also reasonably well described.
7.1.2 Comparisons to HERAPDF1.0 and 1.5
A comparison between HERAPDF2.0 NNLO and HERAPDF1.5 NNLO is provided in Fig. 49. As in the case of the NLO PDFs, a reduction of the uncertainty at high x has been achieved by including further high\(x_\mathrm{Bj}\) data. There is also a reduction of uncertainties at low x. This is mostly due to the better stability of the fit under the variation of \(Q^2_\mathrm{min}\), which is part of the model uncertainties. The shapes of the HERAPDF1.5 and HERAPDF2.0 at NNLO are rather similar, but the gluon distribution at high x has moved to the lower end of its previous uncertainty band.
7.1.3 Comparisons to other sets of PDFs
The PDFs of HERAPDF2.0 NLO and NNLO can be directly compared to the PDFs of MMHT 2014 [37], for which the same heavyflavour scheme, i.e. RTOPT, was used. Comparisons are also made to the PDFs of CT10 [39, 105], for which a heavyflavourscheme based on ACOT was used, and NNPDF3.0 [44], for which the FONLL scheme was used. The results are shown in Figs. 50 and 51 for NLO and NNLO, respectively. For the PDFs themselves, the uncertainties are only shown for HERAPDF2.0. All uncertainties are shown when the ratios of the other PDFs with respect to HERAPDF2.0 are illustrated. Taking the full uncertainties into account, all PDFs are compatible. The largest relative discrepancy (\(\approx \)2.5\(\sigma \)) is found in the shape of the \(xu_v\) distribution at \(x\approx 0.4\) for both NLO and NNLO PDFs. In addition, at NLO, the gluon distribution of HERAPDF2.0 at high x is softer than that of the other PDFs, whereas at NNLO it is close to their \(68\,\%\) uncertainty bands.
7.2 HERAPDF2.0HiQ2
Figures 52 and 53 show summaries for HERAPDF2.0 NLO and NNLO as already shown in Figs. 21 and 23 together with the equivalent plots for HERAPDF2.0HiQ2. The only difference is that HERAPDF2.0 has \(Q^2_\mathrm{min} = 3.5\,\)GeV\(^2\) while HERAPDF2.0HiQ2 has \(Q^2_\mathrm{min} = 10\,\)GeV\(^2\). At NLO, the gluon distributions of HERAPDF2.0 and HERAPDF2.0HiQ2 are compatible within uncertainties. At NNLO, the two gluon distributions differ significantly. Using the higher \(Q^2_\mathrm{min}\) at NNLO causes the gluon distribution to turn over significantly at low x. The distributions of HERAPDF2.0AG are also shown in Figs. 52 and 53. They are not very different for the two \(Q^2_\mathrm{min}\) values. At NNLO, this causes the gluon distribution of HERAPDF2.0AG to be completely different than that of the standard parameterisation for \(x < 10^{3}\).
7.2.1 Comparison of HERADPF2.0HiQ2 to HERAPDF2.0
A comparison of the NLO PDFs of HERAPDF2.0 to HERAPDF2.0HiQ2 at the scale \(\mu _\mathrm{f}^2=10\,\)GeV\(^2\) is shown in Fig. 54. The different shapes of the gluon distribution at low x are compatible within uncertainties. In Sect. 6.12, the question arose whether including data from the kinematic region of low \(x_\mathrm{Bj}\) and low \(Q^2\), i.e. below 10 GeV\(^2\), in the PDF fits would introduce a bias on predictions for high \(x_\mathrm{Bj}\) and high \(Q^2\). Figure 55 demonstrates that at the high scale of \(\mu _\mathrm{f}^2=10{,}000\,\)GeV\(^2\), the PDFs resulting from the two fits are very similar. This confirms that the value of \(Q^2_\mathrm{min} = 3.5\,\)GeV\(^2\) is a safe value for pQCD fits to HERA data and no bias is introduced for applications at higher scales like crosssection predictions for LHC.
A comparison of the NNLO PDFs of HERAPDF2.0 to those of HERAPDF2.0HiQ2 at the scale \(\mu _\mathrm{f}^2=10\,\)GeV\(^2\) is shown in Fig. 56. The differences in the gluon distributions are pronounced. The gluon distribution of HERAPDF2.0HiQ2 NNLO turns over for \(x < 10^{3}\). The valence distributions at NNLO also differ between HERAPDF2.0HiQ2 and HERAPDF2.0, but they are compatible within uncertainties. At the high scale of \(\mu _\mathrm{f}^2=10{,}000\,\)GeV\(^2\), the PDFs resulting from the two fits are, as at NLO, very similar, see Fig. 57. This demonstrates that again no bias is introduced at higher scales when low\(x_\mathrm{B_j}\) and low\(Q^2\) data are included in the fit at NNLO .
7.2.2 Comparison of HERAPDF2.0HiQ2 to data
7.3 HERAPDF2.0FF
Summaries of HERAPDF2.0FF3A and HERAPDF2.0FF3B as introduced in Sect. 6.9 are shown in Fig. 60. The experimental, model and parameterisation uncertainties were evaluated as for the standard HERAPDF2.0 NLO, see Sects. 6.4 and 6.5, and are shown separately.
A comparison of the PDFs of HERAPDF2.0FF3A and HERAPDF2.0FF3B to the standard HERAPDF2.0 NLO using the RTOPT heavyflavour scheme is shown in Fig. 61. This comparison is presented at the starting scale \(\mu _\mathrm{f_0}\), because a meaningful comparison can only be done at scales below the charm mass. There are differences in the valence and in the gluon distributions. The latter originate mainly from the different \(\mathcal{O}(\alpha _s)\) at which the massless contribution to \(F_\mathrm{L}\) is calculated and on the \(\alpha _s\) evolution scheme. A comparison of the predictions from HERAPDF2.0FF3B and HERAPDF2.0 NLO to selected data as already used for Fig. 30 is shown in Fig. 62. The predictions are very similar. However, at low \(x_\mathrm{Bj}\) and low \(Q^2\), the \(Q^2\) dependence predicted from HERAPDF2.0FF3B is a bit less steep than the prediction from HERAPDF2.0 NLO. The predictions of HERAPDF2.0FF3A are also very similar. The \(Q^2\) dependence predicted from HERAPDF2.0FF3A is however slightly steeper than the prediction from HERAPDF2.0 NLO at low \(x_\mathrm{Bj}\) and low \(Q^2\).
A comparison of the PDFs of HERAPDF2.0FF3B to the PDFs of NNPDF3.0 FF(3N) [44] is shown in Fig. 64. These two sets of PDFs can be directly compared at the starting scale due to their equivalent treatment of the \(F_\mathrm{L}\) contribution and of the \(\alpha _s\) evolution.^{12} The gluon distributions are quite similar. Some differences are observed in the \(xu_v\) and \(xd_v\) valence distributions.
7.4 HERAPDF2.0Jets
Data on jet production were included in the analysis as described in Sect. 6.11. This inclusion was first used to validate the choice of \(\alpha _s(M_Z^2)=0.118\) for HERAPDF by investigating the dependence of the \(\chi ^2\)s of the HERAPDF pQCD fits on \(\alpha _s(M_Z^2)\). Three \(\chi ^2\) scans vs. the value of \(\alpha _s(M_Z^2)\) were performed at NLO for three values of \(Q^2_\mathrm{min}\). The result is depicted in the top panel of Fig. 65. A distinct minimum at \(\alpha _s(M_Z^2)\approx 0.118\) is observed, which is basically independent of \(Q^2_\mathrm{min}\). This validates the choice of \(\alpha _s(M_Z^2)= 0.118\) for HERAPDF2.0 NLO. Scans at NLO and NNLO were also performed for fits to inclusive data only. The middle and bottom panels of Fig. 65 show that these scans yielded similar shallow \(\chi ^2\) dependences and the minima were strongly dependent on the \(Q^2_\mathrm{min}\). This demonstrates that the inclusive data alone cannot constrain \(\alpha _s(M_Z^2)\) reasonably.
7.4.1 PDFs and measurement of \({\alpha _s(M_Z^2)}\)
The PDFs from the HERAPDF2.0Jets fit with \(\alpha _s(M_Z^2)=0.118\) fixed are also very similar to the standard PDFs from HERAPDF2.0 NLO. This is demonstrated in Fig. 67. This is again the result of the choice of \(\alpha _s(M_Z^2)=0.118\) for HERAPDF2.0 which is also the preferred value for HERAPDF2.0Jets. Consequently, there is only a small reduction of the uncertainty on the gluon distribution observed for HERAPDF2.0Jets.
7.4.2 Comparison of HERAPDF2.0Jets to data
The predictions of HERAPDF2.0Jets with free \(\alpha _s(M_Z^2)\) are shown together with the charm input data [46] in Fig. 68. The description of the data is excellent.
Comparisons of the predictions of HERAPDF2.0Jets to the data on jet production used as input are shown in Figs. 69, 70, 71, 72 and 73. All analyses were performed using the assumption of massless jets, i.e. the transverse energy, \(E_T\), and the transverse momentum of a jet, \(p_T\), are equivalent. For inclusive jet analyses, each jet is entered separately with its \(p_T\). For dijet and trijet analyses, the average of the transverse momenta is used as \(p_T\). These different definitions of \(p_T\) were also used to set the renormalisation scale to \(\mu _\mathrm{r}^2 = (Q^2 + p_{T}^2)/2\) for calculating predictions. The factorisation scale was chosen as \(\mu _\mathrm{f}^2 = Q^2\). Scale uncertainties were not considered for the comparisons to data.
8 Electroweak effects and scaling violations
The precise data and the predictions from HERAPDF2.0 were used to examine both electroweak effects and scaling violations.
8.1 Electroweak unification
8.2 The structure function \(xF_3^{\gamma Z}\)
Figures 75 and 76 show the reduced cross sections for both \(e^+p\) and \(e^p\) inclusive NC scattering and predictions from HERAPDF2.0 at NLO and NNLO as a function of \(Q^2\) for selected values of \(x_\mathrm{Bj}\). The differences in the cross sections at high \(Q^2\) are clearly visible and well described by HERAPDF2.0, both at NLO and at NNLO. The predictions at NNLO have slightly lower uncertainties than at NLO. As described in Sect. 2, the structure function \(xF_{3}^{\gamma Z}\) can be extracted by subtracting the NC \(e^+p\) from the NC \(e^p\) cross sections. This directly probes the valence structure of the proton. Equations 2 and 7 were used to obtain \(xF_{3}^{\gamma Z}\) for \(Q^2 \ge 1000\) GeV\(^2\). The result is shown in Fig. 77 in bins of \(Q^2\) together with the predictions of HERAPDF2.0 NLO. The values are listed in Table 7. The subtraction yields precise results above \(Q^2\) of 3000 GeV\(^2\).
The valencequark distributions and hence \(xF_{3}^{\gamma Z}\) depend only minimally on the scale, i.e. only small corrections are needed to translate all values of \(xF_{3}^{\gamma Z}\) to a common scale of \(1000\,\)GeV\(^{2}\). This was done using HERAPDF2.0 NLO. The translation factors were close to unity for most points. The largest factors of up to 1.6 were obtained for points at the highest \(Q^2\) and \(x_\mathrm{Bj}\) where \(xF_{3}^{\gamma Z}\) is very small.
The translated \(xF_{3}^{\gamma Z}\) values were averaged using the method described in Sect. 4. A full covariance matrix was built using the information on the individual sources of uncertainty. The averaging of the \(xF_{3}^{\gamma Z}\) values has a \(\chi ^2 /\mathrm{d.o.f.} = 58.8 / 57\) demonstrating the consistency of the data for different values of \(Q^2\). The result is presented in Fig. 78 together with the prediction of HERAPDF2.0 NLO. The values are listed in Table 8. The data are well described by the HERAPDF2.0 NLO prediction.
8.3 Helicity effects in CC interactions
Figures 79 and 80 present the reduced cross sections for CC inclusive \(e^+p\) and \(e^p\) scattering. The \(e^+p\) cross sections are affected strongly by the helicity factor \((1y)^2\), see Eq. 12. Therefore, the contribution of the valence quarks is supressed at high y which translates to high \(Q^2\) for fixed \(x_\mathrm{Bj}\). The \(e^p\) cross section is almost unaffected, because the helicity factor applies to the antiquarks which as part of the sea are already supressed at high \(x_\mathrm{Bj}\).
8.4 Scaling violations
Scaling violations, i.e. the dependence of the structure functions on \(Q^2\) at fixed \(x_\mathrm{Bj}\), are a consequence of the strong interactions between the partons in the nucleon. The larger the kinematic range, the more clearly these violations are demonstrated. They have been used to extract the gluon content of the proton.
Figures 81 and 82 show the inclusive NC \(e^+p\) and \(e^p\) HERA data together with fixedtarget data [107, 108] and the predictions of HERAPDF2.0 NLO and NNLO, respectively. The data presented span more than four orders of magnitude, both in \(Q^2\) and \(x_\mathrm{Bj}\). The scaling violations are clearly visible and are well described by HERAPDF2.0, both at NLO and NNLO. The scaling violations were also already clearly visible in Fig. 30, in which a closeup for a particular kinematic range was presented.
The function \(\tilde{F_2}\) rises toward low \(x_\mathrm{Bj}\) at fixed \(Q^2\). The scaling violations manifest themselves by the rise becoming steeper as \(Q^2\) increases. In the conventional framework of pQCD, this implies an increasing gluon density. The predictions of HERAPDF2.0 NLO describe the data well.
9 Summary and conclusions
The H1 and ZEUS collaborations measured inclusive \(e^{\pm }p\) scattering cross sections at HERA from 1994 to 2007, collecting a total integrated luminosity of about 1 fb\(^{1}\). The data were taken in two different beam configurations, called HERA I and HERA II, at four different centreofmass energies and with two different detectors changing and improving over time. All inclusive data were combined to create one consistent set of NC and CC crosssection measurements for unpolarised \(e^{\pm }p\) scattering, spanning six orders of magnitude in both negative fourmomentumtransfer squared, \(Q^2\), and Bjorken x. The data from many measurements made independently by the two collaborations proved to be consistent with a \(\chi ^2\) per degree of freedom being 1.04 for the combination. Combined cross sections are provided for values of \(Q^2\) between \(Q^2=0.045\) GeV\(^2\) and \(Q^2=50{,}000\) GeV\(^2\) and values of \(x_\mathrm{Bj}\) between \(x_\mathrm{Bj}=6\times 10^{7}\) and \(x_\mathrm{Bj}=0.65\). They are the most precise measurements ever published for ep scattering over such a large kinematic range and have been used to illustrate scaling violation. The precision of the data has also been exploited to illustrate electroweak unification and extract \(xF_3^{\gamma Z}\) above \(Q^2 = 1000\,\)GeV\(^2\).
The inclusive cross sections were used as input to a QCD analysis within the DGLAP formalism. In order to constrain the heavyquark mass parameters, additional information from data on charm and beauty production at HERA was used. The resulting parton distribution functions are denoted HERAPDF2.0 and are available at LO, NLO and NNLO. They were calculated for a series of fixed values of \(\alpha _s(M_Z^2)\) around the central value of 0.118. HERAPDF2.0 has small experimental uncertainties due to the high precision and coherence of the input data. Parameterisation and model uncertainties have also been estimated. HERAPDF2.0 makes precise predictions which describe the input data well.
The heavyflavour scheme used for HERAPDF2.0 is RTOPT, a variableflavour number scheme. Two variants HERAPDF2.0 FF3A and FF3B, using fixedflavour number schemes, are also available at NLO.
The perturbative QCD fits yielding HERAPDF2.0 are based on data with \(Q^2\) above 3.5 GeV\(^2\). Their \(\chi ^2/\mathrm{d.o.f.}\) values are around 1.2. An extensive investigation included fits with different \(Q^2_\mathrm{min}\), below which data were excluded. For \(Q^2_\mathrm{min} = 10\,\)GeV\(^2\), a full set of PDFs named HERAPDF2.0HiQ2 is also released. These fits have an improved \(\chi ^2/\mathrm{d.o.f.}\) of about 1.15. However, the resulting PDFs do not describe the data in the excluded low\(Q^2\) region well. HERAPDF2.0 shows tensions between data and fit, independent of the heavyflavour scheme used, at low \(Q^2\), i.e. below \(Q^2 = 15\,\)GeV\(^2\), and at high \(Q^2\), i.e. above \(Q^2 = 150\,\)GeV\(^2\). Comparisons between the behaviour of the fits with different \(Q^2_\mathrm{min}\) values indicate that the NLO theory evolves faster than the data towards lower \(Q^2\) and x. Fits at NNLO do not improve the agreement. HERAPDF2.0 NNLO and NLO have a similar fit quality.
A measurement of \(\alpha _s(M_Z^2)\) was made using a perturbative QCD fit for which the inclusive cross sections were augmented with selected jet and charmproduction cross sections as measured by both the H1 and ZEUS collaborations. The value obtained is \(\alpha _s(M_Z^2)=0.1183 \pm 0.0009 \mathrm{(exp)} \pm 0.0005\mathrm{(model/parameterisation)} \pm 0.0012\mathrm{(hadronisation)} ^{+0.0037}_{0.0030}\mathrm{(scale)}\). This value is in excellent agreement with the value of the world average \(\alpha _s(M_Z^2)= 0.1185\) [109]. The set of PDFs obtained from the analysis with free \(\alpha _s(M_Z^2)\) is released as HERAPDF2.0Jets.
Structure function \(xF_3^{\gamma Z}\) for different values of \(Q^2\) and \(x_\mathrm{Bj}\); \(\delta _\mathrm{stat}\), \(\delta _\mathrm{syst}\) and \(\delta _\mathrm{tot}\) represent the statistical, systematic and total uncertainties, respectively
\(Q^2\) (\(\mathrm{GeV}^2\))  \(x_\mathrm{Bj}\)  \(xF_3^{\gamma Z}\)  \(\delta _{\mathrm{stat}}\)  \(\delta _{\mathrm{syst}}\)  \(\delta _{\mathrm{tot}}\) 

1000  0.013  0.293  0.227  0.144  0.269 
1000  0.020  0.378  0.254  0.141  0.290 
1000  0.032  0.619  0.357  0.214  0.416 
1000  0.050  \(\)0.472  \(\)0.500  \(\)0.341  \(\)0.606 
1000  0.080  \(\)0.342  \(\)0.760  \(\)0.396  \(\)0.857 
1000  0.130  0.567  1.256  0.650  1.415 
1000  0.180  3.669  1.622  0.903  1.853 
1000  0.250  4.189  2.044  1.265  2.404 
1000  0.400  0.657  2.477  1.886  3.113 
1200  0.014  0.497  0.142  0.107  0.178 
1200  0.020  0.362  0.137  0.087  0.162 
1200  0.032  0.089  0.178  0.107  0.208 
1200  0.050  0.826  0.227  0.139  0.266 
1200  0.080  0.763  0.329  0.192  0.382 
1200  0.130  0.919  0.509  0.261  0.573 
1200  0.180  \(\)0.709  \(\)1.288  \(\)0.618  \(\)1.429 
1200  0.250  \(\)0.574  \(\)0.763  \(\)0.377  \(\)0.851 
1200  0.400  \(\)1.128  \(\)0.996  \(\)0.767  \(\)1.258 
1500  0.020  0.511  0.121  0.081  0.146 
1500  0.032  0.487  0.149  0.070  0.164 
1500  0.050  0.009  0.193  0.100  0.218 
1500  0.080  0.852  0.268  0.135  0.300 
1500  0.130  0.897  0.443  0.187  0.481 
1500  0.180  \(\)0.001  \(\)1.013  \(\)0.407  \(\)1.092 
1500  0.250  0.855  0.725  0.300  0.785 
1500  0.400  0.444  0.871  0.583  1.048 
1500  0.650  0.042  0.456  0.267  0.528 
2000  0.022  0.630  0.234  0.103  0.255 
2000  0.032  0.340  0.103  0.055  0.116 
2000  0.050  0.426  0.134  0.055  0.145 
2000  0.080  0.211  0.180  0.078  0.196 
2000  0.130  0.181  0.296  0.110  0.315 
2000  0.180  0.335  0.374  0.142  0.400 
2000  0.250  0.316  0.483  0.179  0.515 
2000  0.400  \(\)0.371  \(\)0.542  \(\)0.236  \(\)0.591 
2000  0.650  \(\)0.739  \(\)0.296  \(\)0.166  \(\)0.340 
3000  0.032  0.347  0.096  0.049  0.108 
3000  0.050  0.303  0.068  0.033  0.075 
3000  0.080  0.463  0.095  0.041  0.104 
3000  0.130  0.440  0.150  0.059  0.161 
3000  0.180  0.279  0.194  0.073  0.208 
3000  0.250  0.723  0.241  0.102  0.262 
3000  0.400  0.227  0.268  0.128  0.297 
3000  0.650  \(\)0.022  \(\)0.106  \(\)0.053  \(\)0.118 
5000  0.055  0.320  0.078  0.033  0.084 
5000  0.080  0.333  0.041  0.019  0.045 
5000  0.130  0.548  0.072  0.027  0.077 
5000  0.180  0.500  0.087  0.030  0.092 
5000  0.250  0.207  0.115  0.035  0.120 
5000  0.400  0.132  0.124  0.046  0.132 
5000  0.650  0.096  0.055  0.027  0.062 
8000  0.087  0.425  0.084  0.029  0.089 
8000  0.130  0.493  0.043  0.016  0.046 
8000  0.180  0.415  0.056  0.018  0.059 
8000  0.250  0.321  0.070  0.022  0.074 
8000  0.400  0.120  0.072  0.025  0.077 
8000  0.650  \(\)0.004  \(\)0.031  \(\)0.013  \(\)0.034 
12,000  0.130  0.637  0.125  0.038  0.131 
12,000  0.180  0.385  0.040  0.013  0.042 
12,000  0.250  0.379  0.049  0.013  0.050 
12,000  0.400  0.272  0.056  0.019  0.059 
12,000  0.650  \(\)0.012  \(\)0.027  \(\)0.009  \(\)0.028 
20,000  0.250  0.388  0.040  0.013  0.042 
20,000  0.400  0.218  0.040  0.012  0.041 
20,000  0.650  0.016  0.019  0.009  0.021 
30,000  0.400  0.178  0.036  0.008  0.037 
30,000  0.650  0.060  0.025  0.009  0.026 
Structure function \(xF_3^{\gamma Z}\) averaged over \(Q^2 \ge 1000\) GeV\(^2\) at the scale 1000 GeV\(^2\); \(\delta _\mathrm{stat}\), \(\delta _\mathrm{syst}\) and \(\delta _\mathrm{tot}\) represent the statistical, systematic and total uncertainties, respectively
\(Q^2\) (\(\mathrm{GeV}^2\))  \(x_\mathrm{Bj}\)  \(xF_3^{\gamma Z}\)  \(\delta _{\mathrm{stat}}\)  \(\delta _{\mathrm{syst}}\)  \(\delta _{\mathrm{tot}}\) 

1000  0.014  0.422  0.120  0.082  0.146 
1000  0.020  0.443  0.080  0.051  0.094 
1000  0.032  0.334  0.058  0.034  0.067 
1000  0.050  0.312  0.045  0.023  0.050 
1000  0.080  0.365  0.033  0.016  0.037 
1000  0.130  0.523  0.035  0.014  0.038 
1000  0.180  0.423  0.032  0.011  0.034 
1000  0.250  0.407  0.031  0.011  0.032 
1000  0.400  0.245  0.028  0.009  0.029 
1000  0.650  0.026  0.017  0.007  0.018 
Settings for HERAPDF2.0 and HERAPDF1.5
HERAPDF2.0  HERAPDF1.5  

NNLO  NLO  NNLO  NLO  
Data as in Table 1  Combination  Preliminary combination  
Uncertainties  
Experimental  Hessian  Hessian  
Procedural  7  3  
Parameterisation  
Number of parameters  14  14  14\(^{**}\)  10\(^{*}\) 
Variations  15 [\(D_{u_{v}}\)]  15 [\(D_{u_{v}}\)]  None  11 [\(D_{u_{v}}\)], 12 [\(D_{\bar{U}}\)] 
\(\mu ^2_\mathrm{f_{0}}\) (GeV\(^2\))  1.9  1.9  1.9  1.9 
Variations  1.6, 2.2\(^\mathrm{a}\)  1.6, 2.2\(^\mathrm{b}\)  1.5, 2.5\(^\mathrm{c}\)  1.5\(^\mathrm{d}\), 2.5\(^\mathrm{c}\) 
\(M_c\) (GeV)  1.43  1.47  1.4  1.4 
Variations  1.37\(^\mathrm{e}\), 1.49  1.41, 1.53  1.35\(^\mathrm{f}\), 1.65  1.35\(^\mathrm{f}\), 1.65\(^\mathrm{a}\) 
\(M_b\) (GeV)  4.5  4.5  4.75  4.75 
Variations  4.25, 4.75  4.25, 4.75  4.30, 5.00  4.30, 5.00 
\(f_s\) (GeV)  0.40  0.40  0.31  0.31 
Variations  0.30, 0.50  0.30, 0.50  0.23, 0.38  0.23, 0.38 
\(Q^2_\mathrm{min}\) (GeV\(^2\)) of data  3.5  3.5  3.5  3.5 
Variations  2.5, 5.0  2.5, 5.0  2.5, 5.0  2.5, 5.0 
Fixed \(\alpha _s\)  0.118  0.118  0.1176  0.1176 
Footnotes
 1.
In this paper, the word “electron” refers to both electrons and positrons, unless otherwise stated.
 2.
The definition of what is meant by LO can differ; it can be taken to mean \(\mathcal O(1)\) in \(\alpha _s\), or it can be taken to mean the first nonzero order. For example, the longitudinal structure function \(F_\mathrm{L}\) is zero at \(\mathcal O(1)\) such that its first nonzero order is \(\mathcal O(\alpha _s\)). This is what is meant by LO here unless otherwise stated. Higher orders follow suit such that at NLO, \(F_2\) has coefficient functions calculated up to \(\mathcal O(\alpha _s)\), whereas \(F_\mathrm{L}\) has coefficient functions calculated up to \(\mathcal O(\alpha _s^2)\).
 3.
Both experiments used a righthanded Cartesian coordinate system, with the Z axis pointing in the proton beam direction, referred to as the “forward direction”, and the X axis pointing towards the centre of HERA. The coordinate origins were at the nominal interaction points. The polar angle, \(\theta \), was measured with respect to the proton beam direction.
 4.
As a cross check, predictions using HERAPDF1.0 were used instead. The induced changes were negligible.
 5.
The ansatz of the fractal model is based on the selfsimilar properties in \(x_\mathrm{Bj}\) and \(Q^2\) of the proton structure function at low \(x_\mathrm{Bj}\). They are represented by two continuous, variable and correlated fractal dimensions.
 6.
A cross check was performed using the colour dipole model [78] as implemented in HERAFitter. The results did not change significantly.
 7.
For the DIS crosssection measurements, the background contributions were small and thus it is justified to take the square root of the number of events as the statistical uncertainty.
 8.
For the first iteration, terms are modified as \(\gamma ^{i,ds}_j m^i \rightarrow \gamma ^{i,ds}_j \mu ^{i,ds},~~ \delta _{i,ds,\mathrm{uncor}}\, m^i \rightarrow \delta _{i,ds,\mathrm{uncor}}\, \mu ^{i,ds}\) and \(\delta ^2_{i,ds,\mathrm{stat}}\, \mu ^{i,ds}\Big (m^i  \sum _j \gamma ^{i,ds}_j m^i b_j\Big ) \rightarrow (\delta _{i,ds,\mathrm{stat}}\, \mu ^{i,ds})^2\), respectively.
 9.
For subsequent iterations, terms are modified as \(\gamma ^{i,ds}_j m^i \rightarrow \gamma ^{i,ds}_j \mu ^{i},~~ \delta _{i,ds,\mathrm{uncor}}\, m^i \rightarrow \delta _{i,ds,\mathrm{uncor}}\, \mu ^{i}\) and \(\delta ^2_{i,ds,\mathrm{stat}}\, \mu ^{i,ds}\Big (m^i  \sum _j \gamma ^{i,ds}_j m^i b_j\Big ) \rightarrow \delta ^2_{i,ds,\mathrm{stat}}\, \mu ^{i,ds}\Big (\mu ^i  \sum _j \gamma ^{i,ds}_j \mu ^i b_j\Big )\), respectively.
 10.
The full information about correlations between crosssection measurements is available elsewhere [79].
 11.
In the analysis presented here, \(C_g'\) is fixed to \(C_g' = 25\) [36]. The fits are not sensitive to the exact value of \(C_g'\) once \(C_g' \gg C_g\), such that the term does not contribute at large x.
 12.
The NNPDF3.0FF(3N) is based on a fixed number of flavours, NF=3, evolution, but it is calculated from the FONLLB fit, which is based on a variable NF evolution [44, 106]. Thus, close to the starting scale, the PDFs of NNPDF3.0FF(3N) can be directly compared to the PDFs of HERAPDF2.0FF3B, which are also based on a variable NF evolution.
Notes
Acknowledgments
We are grateful to the HERA machine group whose outstanding efforts have made these experiments possible. We appreciate the contributions to the construction, maintenance and operation of the H1 and ZEUS detectors of many people who are not listed as authors. We thank our funding agencies for financial support, the DESY technical staff for continuous assistance and the DESY directorate for their support and for the hospitality they extended to the nonDESY members of the collaborations. We would like to give credit to all partners contributing to the EGI computing infrastructure for their support.