Combination and QCD analysis of charm and beauty production crosssection measurements in deep inelastic ep scattering at HERA
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Abstract
Measurements of open charm and beauty production cross sections in deep inelastic ep scattering at HERA from the H1 and ZEUS Collaborations are combined. Reduced cross sections are obtained in the kinematic range of negative fourmomentum transfer squared of the photon \(2.5~\hbox {GeV}^2\le Q^2 \le 2000\, \hbox {GeV}^2\) and Bjorken scaling variable \(3 \cdot 10^{5} \le x_\mathrm{Bj} \le 5 \cdot 10^{2}\). The combination method accounts for the correlations of the statistical and systematic uncertainties among the different datasets. Perturbative QCD calculations are compared to the combined data. A nexttoleading order QCD analysis is performed using these data together with the combined inclusive deep inelastic scattering cross sections from HERA. The running charm and beautyquark masses are determined as \(m_c(m_c) = 1.290^{+0.046}_{0.041} \mathrm{(exp/fit)}\) \({}^{+0.062}_{0.014} \mathrm{(model)}\) \({}^{+0.003}_{0.031} \mathrm{(parameterisation)}\) GeV and \(m_b(m_b) = 4.049^{+0.104}_{0.109} \mathrm{(exp/fit)}\) \({}^{+0.090}_{0.032} \mathrm{(model)}\) \({}^{+0.001}_{0.031} \mathrm{(parameterisation)}~\mathrm{GeV}\).
1 Introduction
Measurements of open charm and beauty production in neutral current (NC) deep inelastic electron^{1}–proton scattering (DIS) at HERA provide important input for tests of the theory of strong interactions, quantum chromodynamics (QCD). Measurements at HERA [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24] have shown that heavyflavour production in DIS proceeds predominantly via the bosongluonfusion process, \(\gamma g\rightarrow {\mathrm{Q}}\overline{\mathrm{Q}}\), where Q is the heavy quark. The cross section therefore depends strongly on the gluon distribution in the proton and the heavyquark mass. This mass provides a hard scale for the applicability of perturbative QCD (pQCD). However, other hard scales are also present in this process: the transverse momenta of the outgoing quarks and the negative four momentum squared, \(Q^2\), of the exchanged photon. The presence of several hard scales complicates the calculation of heavyflavour production in pQCD. Different approaches have been developed to cope with the multiple scale problem inherent in this process. In this paper, the massive fixedflavournumber scheme (FFNS) [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36] and different implementations of the variableflavournumber scheme (VFNS) [37, 38, 39, 40, 41] are considered.
At HERA, different flavour tagging methods are applied for charm and beauty crosssection measurements: the full reconstruction of D or \(D^{*\pm }\) mesons [1, 2, 4, 5, 6, 10, 11, 12, 15, 17, 20, 21, 22], which is almost exclusively sensitive to charm production; the lifetime of heavyflavoured hadrons [7, 8, 9, 14, 23] and their semileptonic decays [13, 16, 19], both enabling the measurement of the charm and beauty cross section simultaneously. In general, the different methods explore different regions of the heavyquark phase space and show different dependencies on sources of systematic uncertainties. Therefore, by using different tagging techniques a more complete picture of heavyflavour production is obtained.
In this paper, a simultaneous combination of charm and beauty production crosssection measurements is presented. This analysis is an extension of the previous H1 and ZEUS combination of charm measurements in DIS [42], including new charm and beauty data [13, 14, 16, 19, 21, 22, 23] and extracting combined beauty cross sections for the first time. As a result, a single consistent dataset from HERA of reduced charm and beauty cross sections in DIS is obtained, including all correlations. This dataset covers the kinematic range of photon virtuality 2.5 GeV\(^2 \le Q^2 \le 2000\) GeV\(^2\) and Bjorken scaling variable \(3 \times 10^{5} \le x_\mathrm{Bj} \le 5 \times 10^{2}\).
The procedure follows the method used previously [42, 43, 44, 45, 46, 47]. The correlated systematic uncertainties and the normalisation of the different measurements are accounted for such that one consistent dataset is obtained. Since different experimental techniques of charm and beauty tagging have been employed using different detectors and methods of kinematic reconstruction, this combination leads to a significant reduction of statistical and systematic uncertainties with respect to the individual measurements. The simultaneous combination of charm and beauty crosssection measurements reduces the correlations between them and hence also the uncertainties. The combined reduced charm cross sections of the previous analysis [42] are superseded by the new results presented in this paper.
The combined data are compared to theoretical predictions obtained in the FFNS at nexttoleading order (NLO, \(O(\alpha _s^2)\)) QCD using HERAPDF2.0 [48], ABKM09 [29, 30] and ABMP16 [32] parton distribution functions (PDFs), and to approximate nexttonexttoleading order (NNLO, \(O(\alpha _s^3)\)) using ABMP16 [32] PDFs. In addition, QCD calculations in the RTOPT [37] VFNS at NLO and approximate NNLO are compared with the data. The NLO calculations are at \(O(\alpha _s^2)\) except for the massless parts of the coefficient functions, which are at \(O(\alpha _s)\); the NNLO calculations are one order of \(\alpha _s\) higher. A comparison is also made to predictions of two variants of the FONLLC scheme [38, 39, 40] (\(O(\alpha _s^3)\) (NNLO) in the PDF evolution, \(O(\alpha _s^2)\) in all coefficient functions): the default scheme, which includes nexttoleadinglog (NLL) resummation of quasicollinear final state gluon radiation, and a variant which includes NLL lowx resummation in the PDFs and the matrix elements (NLLsx) [41] in addition.
The new data are subjected to a QCD analysis together with the final inclusive DIS crosssection data from HERA [48] allowing for the determination at NLO of the running charm and beautyquark masses, as defined from the QCD Lagrangian in the modified minimumsubtraction (\(\overline{\mathrm{MS}}\)) scheme.
Datasets used in the combination. For each dataset, the tagging method, the \(Q^2\) range, integrated luminosity (\(\mathcal{L}\)), centreofmass energy (\(\sqrt{s}\)) and the numbers of charm (\(N_c\)) and beauty (\(N_b\)) measurements are given. The tagging method VTX denotes inclusive measurements based on lifetime information using a silicon vertex detector. Charge conjugates are always implied for the particles given in the column ’Tagging’
Dataset  Tagging  \(Q^2\) range (\(\hbox {GeV}^2\))  \(\mathcal{L}\) (\(\hbox {pb}^{1}\))  \({\sqrt{s}}\) (GeV)  \(N_c\)  \(N_b\)  

1  H1 VTX [14]  VTX  5–2000  245  318  29  12 
2  H1 \(D^{*\pm }\) HERAI [10]  \(D^{*+}\)  2–100  47  318  17  
3  H1 \(D^{*\pm }\) HERAII (medium \(Q^2\)) [20]  \(D^{*+} \)  5–100  348  318  25  
4  H1 \(D^{*\pm }\) HERAII (high \(Q^2\)) [15]  \(D^{*+}\)  100–1000  351  318  6  
5  ZEUS \(D^{*+}\) 9697 [4]  \(D^{*+}\)  1–200  37  300  21  
6  ZEUS \(D^{*+}\) 9800 [6]  \(D^{*+}\)  1.5–1000  82  318  31  
7  ZEUS \(D^0\) 2005 [12]  \(D^{0}\)  5–1000  134  318  9  
8  ZEUS \(\mu \) 2005 [13]  \(\mu \)  20–10000  126  318  8  8 
9  ZEUS \(D^{+}\) HERAII [21]  \(D^+\)  5–1000  354  318  14  
10  ZEUS \(D^{*+}\) HERAII [22]  \(D^{*+}\)  5–1000  363  318  31  
11  ZEUS VTX HERAII [23]  VTX  5–1000  354  318  18  17 
12  ZEUS e HERAII [19]  e  10–1000  363  318  9  
13  ZEUS \(\mu + {\mathrm{jet}}\) HERAI [16]  \(\mu \)  2–3000  114  318  11 
Reduced cross section for charm production, \(\sigma _{\mathrm{red}}^{c\overline{c}}\, \), obtained by the combination of H1 and ZEUS measurements. The crosssection values are given together with the statistical \((\delta _{\mathrm{stat}})\) and the uncorrelated \((\delta _{\mathrm{uncor}})\) and correlated \((\delta _{\mathrm{cor}})\) systematic uncertainties. The total uncertainties \((\delta _{\mathrm{tot}})\) are obtained through adding the statistical, uncorrelated and correlated systematic uncertainties in quadrature
#  \(Q^2\) (GeV\(^2\))  \(x_{\mathrm{Bj}}\)  \(\sigma _{\mathrm{red}}^{c\overline{c}}\, \)  \(\delta _{\mathrm{stat}} (\%)\)  \(\delta _{\mathrm{uncor}} (\%)\)  \(\delta _{\mathrm{cor} }(\%)\)  \(\delta _{\mathrm{tot}} (\%)\) 

1  2.5  0.00003  0.1142  8.9  10.7  9.4  16.9 
2  2.5  0.00007  0.1105  5.8  6.7  8.2  12.1 
3  2.5  0.00013  0.0911  7.1  6.2  7.9  12.3 
4  2.5  0.00018  0.0917  4.8  9.6  7.2  12.9 
5  2.5  0.00035  0.0544  5.3  8.2  6.9  12.0 
6  5.0  0.00007  0.1532  11.6  9.6  8.2  17.1 
7  5.0  0.00018  0.1539  5.3  3.4  7.8  10.0 
8  5.0  0.00035  0.1164  5.2  5.3  5.7  9.3 
9  5.0  0.00100  0.0776  4.8  8.7  5.6  11.4 
10  7.0  0.00013  0.2249  4.3  3.3  6.7  8.6 
11  7.0  0.00018  0.2023  6.8  5.7  7.2  11.4 
12  7.0  0.00030  0.1767  2.3  2.4  5.4  6.4 
13  7.0  0.00050  0.1616  2.5  1.8  5.2  6.0 
14  7.0  0.00080  0.1199  4.6  4.0  4.9  7.8 
15  7.0  0.00160  0.0902  4.1  3.9  5.2  7.7 
16  12.0  0.00022  0.3161  4.9  2.9  5.7  8.0 
17  12.0  0.00032  0.2904  2.9  1.5  6.3  7.1 
18  12.0  0.00050  0.2410  2.4  1.3  4.6  5.3 
19  12.0  0.00080  0.1813  2.1  1.4  4.5  5.1 
20  12.0  0.00150  0.1476  3.2  1.5  5.1  6.2 
21  12.0  0.00300  0.1010  4.4  4.0  5.1  7.8 
22  18.0  0.00035  0.3198  5.2  3.3  5.2  8.1 
23  18.0  0.00050  0.2905  2.6  1.4  6.4  7.0 
24  18.0  0.00080  0.2554  2.2  1.2  4.2  4.9 
25  18.0  0.00135  0.2016  2.0  1.1  4.1  4.7 
26  18.0  0.00250  0.1630  1.9  1.3  4.2  4.7 
27  18.0  0.00450  0.1137  5.5  4.1  5.4  8.7 
28  32.0  0.00060  0.3885  8.5  9.3  5.8  13.9 
29  32.0  0.00080  0.3756  2.3  1.4  4.4  5.2 
30  32.0  0.00140  0.2807  2.0  1.1  3.4  4.1 
31  32.0  0.00240  0.2190  2.3  1.4  3.9  4.7 
32  32.0  0.00320  0.2015  3.6  1.6  5.4  6.6 
33  32.0  0.00550  0.1553  4.2  3.0  4.1  6.6 
34  32.0  0.00800  0.0940  8.7  5.4  6.0  11.9 
35  60.0  0.00140  0.3254  3.2  1.4  4.8  5.9 
36  60.0  0.00200  0.3289  2.3  1.2  4.1  4.9 
37  60.0  0.00320  0.2576  2.2  1.2  3.6  4.4 
38  60.0  0.00500  0.1925  2.3  1.6  4.1  5.0 
39  60.0  0.00800  0.1596  4.8  3.1  3.4  6.7 
40  60.0  0.01500  0.0946  8.1  6.5  4.9  11.5 
41  120.0  0.00200  0.3766  3.3  2.6  5.0  6.5 
42  120.0  0.00320  0.2274  14.6  13.7  2.7  20.2 
43  120.0  0.00550  0.2173  3.3  1.6  5.4  6.5 
44  120.0  0.01000  0.1519  3.9  2.3  5.2  6.9 
45  120.0  0.02500  0.0702  13.6  12.6  4.4  19.1 
46  200.0  0.00500  0.2389  3.1  2.4  4.5  6.0 
47  200.0  0.01300  0.1704  3.4  2.3  5.0  6.5 
48  350.0  0.01000  0.2230  5.1  3.0  6.4  8.7 
49  350.0  0.02500  0.1065  6.1  2.9  7.4  10.0 
50  650.0  0.01300  0.2026  5.4  3.7  9.1  11.2 
51  650.0  0.03200  0.0885  7.8  3.8  12.8  15.4 
52  2000.0  0.05000  0.0603  16.0  6.7  26.4  31.6 
2 Heavyflavour production in DIS
Various theoretical approaches can be used to describe heavyflavour production in DIS. At values of \(Q^2\) not very much larger than the heavyquark mass, \(m_{\mathrm{Q}}\), heavy flavours are predominantly produced dynamically by the photongluonfusion process. The creation of a \({\mathrm{Q}}\overline{Q}\) pair sets a lower limit of \(2m_{\mathrm{Q}}\) to the mass of the hadronic final state. This low mass cutoff affects the kinematics and the higher order corrections in the phase space accessible at HERA. Therefore, a careful theoretical treatment of the heavyflavour masses is mandatory for the pQCD analysis of heavyflavour production as well as for the determination of the PDFs of the proton from data including heavy flavours.
In this paper, the FFNS is used for pQCD calculations for the corrections of measurements to the full phase space and in the QCD fits. In this scheme, heavy quarks are always treated as massive and therefore are not considered as partons in the proton. The number of (light) active flavours in the PDFs, \(n_f\), is set to three and heavy quarks are produced only in the hardscattering process. The leadingorder (LO) contribution to heavyflavour production (\(O(\alpha _s)\) in the coefficient functions) is the photongluonfusion process. The NLO massive coefficient functions using onshell mass renormalisation (pole masses) [25, 26, 27, 28] were adopted by many global QCD analysis groups [31, 33, 34, 35], providing PDFs derived from this scheme. They were extended to the \(\overline{\mathrm{MS}}\) scheme [30], using scale dependent (running) heavyquark masses. The advantages of performing heavyflavour calculations in the \(\overline{\mathrm{MS}}\) scheme are reduced scale uncertainties and improved theoretical precision of the mass definition [24, 36]. In all FFNS heavyflavour calculations presented in this paper, the default renormalisation scale \(\mu _r\) and factorisation scale \(\mu _f\) are set to \(\mu _r=\mu _f=\sqrt{Q^2+4m_\mathrm{Q}^2}\), where \(m_\mathrm{Q}\) is the appropriate pole or running mass.
For the extraction of the combined reduced cross sections of charm and beauty production, it is necessary to predict inclusive cross sections as well as exclusive cross sections with certain phasespace restrictions applied. For this purpose, the FFNS at NLO is used to calculate inclusive [25, 26, 27, 28] and exclusive [50] quantities in the polemass scheme. This is currently the only scheme for which exclusive NLO calculations are available.
The QCD analysis at nexttoleading order^{2} including the extraction of the heavyquark running masses is performed in the FFNS with the OPENQCDRAD programme [31, 51, 52] in the xFitter (former HERAFitter) framework [53]. In OPENQCDRAD, heavyquark production is calculated either using the \( {\overline{\mathrm{MS}}}\) or the polemass scheme of heavyquark masses. In this paper, the \({\overline{\mathrm{MS}}}\) scheme is adopted.
Reduced cross section for beauty production, \(\sigma _{\mathrm{red}}^{b\overline{b}}\, \), obtained by the combination of H1 and ZEUS measurements. The crosssection values are given together with the statistical \((\delta _\mathrm{stat})\) and the uncorrelated \((\delta _\mathrm{uncor})\) and correlated \((\delta _\mathrm{cor})\) systematic uncertainties. The total uncertainties \((\delta _\mathrm{tot})\) are obtained through adding the statistical, uncorrelated and correlated systematic uncertainties in quadrature
#  \(Q^2\) (GeV\(^2\))  \(x_\mathrm{Bj}\)  \(\sigma _{\mathrm{red}}^{b\overline{b}}\, \)  \(\delta _\mathrm{stat} (\%)\)  \(\delta _\mathrm{uncor}\) (%)  \(\delta _\mathrm{cor}\) (%)  \(\delta _\mathrm{tot} (\%)\) 

1  2.5  0.00013  0.0018  28.4  22.4  11.4  37.9 
2  5.0  0.00018  0.0048  10.5  7.1  19.8  23.5 
3  7.0  0.00013  0.0059  8.8  11.2  12.7  19.1 
4  7.0  0.00030  0.0040  8.5  10.3  15.2  20.2 
5  12.0  0.00032  0.0072  4.9  5.8  10.5  13.0 
6  12.0  0.00080  0.0041  4.6  6.9  11.1  13.9 
7  12.0  0.00150  0.0014  32.2  26.9  3.6  42.1 
8  18.0  0.00080  0.0082  4.8  5.0  12.8  14.5 
9  32.0  0.00060  0.0207  8.9  7.8  8.9  14.8 
10  32.0  0.00080  0.0152  5.8  6.1  10.0  13.1 
11  32.0  0.00140  0.0113  3.9  5.3  9.0  11.2 
12  32.0  0.00240  0.0082  9.0  9.5  12.9  18.4 
13  32.0  0.00320  0.0046  32.2  41.9  3.0  52.9 
14  32.0  0.00550  0.0058  39.8  20.4  57.4  72.8 
15  60.0  0.00140  0.0260  4.8  6.9  8.8  12.2 
16  60.0  0.00200  0.0167  7.5  6.5  10.5  14.4 
17  60.0  0.00320  0.0097  10.7  7.7  14.4  19.5 
18  60.0  0.00500  0.0129  5.4  4.2  14.7  16.2 
19  120.0  0.00200  0.0288  6.3  5.4  9.0  12.2 
20  120.0  0.00550  0.0127  21.2  14.9  10.9  28.1 
21  120.0  0.01000  0.0149  20.5  20.6  23.6  37.5 
22  200.0  0.00500  0.0274  3.8  3.7  6.9  8.7 
23  200.0  0.01300  0.0123  9.5  4.8  19.5  22.2 
24  350.0  0.02500  0.0138  20.4  26.2  35.0  48.2 
25  650.0  0.01300  0.0164  8.1  7.5  13.1  17.1 
26  650.0  0.03200  0.0103  8.1  8.7  14.6  18.8 
27  2000.0  0.05000  0.0052  30.6  15.2  47.6  58.6 
The \(\chi ^2\), p values and number of data points of the charm and beauty data with respect to the NLO and approximate NNLO calculations using various PDFs as described in the text. The measurements at \(Q^2=2.5~\hbox {GeV}^2\) are excluded in the calculations of the \(\chi ^2\) values for the NNPDF3.1sx predictions, by which the number of data points is reduced to 47, as detailed in the caption of Fig. 12
Dataset  PDF (scheme)  \(\chi ^2\) (p value)  

charm [42]  (\(\mathrm{N}_\mathrm{data}\) = 52)  HERAPDF20_NLO_FF3A (FFNS)  59 [0.23] 
ABKM09 (FFNS)  59 [0.23]  
ABMP16_3_nlo (FFNS)  61 [0.18]  
ABMP16_3_nnlo (FFNS)  70 [0.05]  
HERAPDF20_NLO_EIG (RTOPT)  71 [0.04]  
HERAPDF20_NNLO_EIG (RTOPT)  66 [0.09]  
(\(\mathrm{N}_\mathrm{data}\) = 47)  NNPDF31sx NNLO (FONLLC)  106 \([1.5\cdot 10^{6}]\)  
NNPDF31sx NNLO+NLLX (FONLLC)  71 [0.013]  
charm, this analysis  (\(\mathrm{N}_\mathrm{data}\) = 52)  HERAPDF20_NLO_FF3A (FFNS)  86 [0.002] 
ABKM09 (FFNS)  82 [0.005]  
ABMP16_3_nlo (FFNS)  90 [0.0008]  
ABMP16_3_nnlo (FFNS)  109 \({[6\cdot 10^{6}]}\)  
HERAPDF20_NLO_EIG (RTOPT)  99 \({[9\cdot 10^{5}]}\)  
HERAPDF20_NNLO_EIG (RTOPT)  102 \({[4\cdot 10^{5}]}\)  
(\(\mathrm{N}_\mathrm{data}\) = 47)  NNPDF31sx NNLO (FONLLC)  140 \([1.5\cdot 10^{11}]\)  
NNPDF31sx NNLO+NLLX (FONLLC)  114 \([5\cdot 10^{7}]\)  
beauty, this analysis  (\(\mathrm{N}_\mathrm{data}\) = 27)  HERAPDF20_NLO_FF3A (FFNS)  33[0.20] 
ABMP16_3_nlo (FFNS)  37 [0.10]  
ABMP16_3_nnlo (FFNS)  41 [0.04]  
HERAPDF20_NLO_EIG (RTOPT)  33 [0.20]  
HERAPDF20_NNLO_EIG (RTOPT)  45 [0.016] 
3 Combination of H1 and ZEUS measurements
The different charm and beautytagging methods exploited at HERA enable a comprehensive study of heavyflavour production in NC DIS.
Using fully reconstructed D or \(D^{*\pm }\) mesons gives the best signaltobackground ratio for measurements of the charm production process. Although the branching ratios of beauty hadrons to D and \(D^{*\pm }\) mesons are large, the contributions from beauty production to the observed D or \(D^{*\pm }\) meson samples are small for several reasons. Firstly, beauty production in ep collisions is suppressed relative to charm production by a factor 1 / 4 due to the quark’s electric charge coupling to the photon. Secondly, the photongluonfusion cross section depends on the invariant mass of the outgoing partons, \(\hat{s}\), which has a threshold value of \(4m_\mathrm{Q}^2\). Because the beautyquark mass, \(m_b\), is about three times the charmquark mass, \(m_c\), beauty production is significantly suppressed. Thirdly, in beauty production D and \(D^{*\pm }\) mesons originate from the fragmentation of charm quarks that are produced by the weak decay of B mesons. Therefore the momentum fraction of the beauty quark carried by the D or \(D^{*\pm }\) meson is small, so that the mesons often remain undetected.
Fully inclusive analyses based on the lifetime of the heavyflavoured mesons are sensitive to both charm and beauty production. Although the first two reasons given above for the suppression of beauty production relative to charm production also hold in this case, sensitivity to beauty production can be enhanced by several means. The proper lifetime of B mesons is on average a factor of 2 to 3 that of D mesons [59]. Therefore, the charm and beauty contributions can be disentangled by using observables directly sensitive to the lifetime of the decaying heavyflavoured hadrons. The separation can be further improved by the simultaneous use of observables sensitive to the mass of the heavyflavoured hadron: the relative transverse momentum, \(p_\mathrm{T}^\mathrm{rel}\), of the particle with respect to the flight direction of the decaying heavyflavoured hadron; the number of tracks with lifetime information; the invariant mass calculated from the charged particles attached to a secondaryvertex candidate.
The analysis of lepton production is sensitive to semileptonic decays of both charm and beauty hadrons. When taking into account the fragmentation fractions of the heavy quarks as well as the fact that in beauty production leptons may originate both from the \(b\rightarrow c\) and the \(c\rightarrow s\) transitions, the semileptonic branching fraction of B mesons is about twice that of D mesons [59]. Because of the large masses of B mesons and the harder fragmentation of beauty quarks compared to charm quarks, leptons originating directly from the B decays have on average higher momenta than those produced in D meson decays. Therefore, the experimentally observed fraction of beautyinduced leptons is enhanced relative to the observed charminduced fraction. Similar methods as outlined in the previous paragraph are then used to further facilitate the separation of the charm and beauty contributions on a statistical basis.
List of uncertainties for the charm and beautyquark mass determination. The PDF parameterisation uncertainties not shown have no effect on \(m_c(m_c)\) and \(m_b(m_b)\)
Parameter  Variation  \(m_c(m_c)\) uncertainty (GeV)  \(m_b(m_b)\) uncertainty (GeV) 

Experimental / Fit uncertainty  
Total  \(\Delta \chi ^2=1\)  \(^{+0.046}_{0.041}\)  \(^{+0.104}_{0.109}\) 
Model uncertainty  
\(f_s\)  \(0.4^{+0.1}_{0.1}\)  \(^{0.003}_{+0.004}\)  \(^{0.001}_{+0.001}\) 
\(Q^2_{\text {min}}\)  \(3.5^{+1.5}_{1.0}\) GeV\(^2\)  \(^{0.001}_{+0.007}\)  \(^{0.005}_{+0.007}\) 
\(\mu _{r,f}\)  \({\mu _{r,f}}^{\times 2.0}_{\times 0.5}\)  \(^{+0.030}_{+0.060}\)  \(^{0.032}_{+0.090}\) 
\(\alpha _s^{n_f=3}(M_Z)\)  \(0.1060^{+0.0015}_{0.0015}\)  \(^{0.014}_{+0.011}\)  \(^{+0.002}_{0.005}\) 
Total  \(^{+0.062}_{0.014}\)  \(^{+0.090}_{0.032}\)  
PDF parameterisation uncertainty  
\(\mu _\mathrm{f,0}^2\)  \(1.9 \pm 0.3\) GeV\(^2\)  \(^{+0.003}_{0.001}\)  \(^{0.001}_{+0.001}\) 
\(E_{u_v}\)  set to 0  \(0.031\)  \(0.031\) 
Total  \(^{+0.003}_{0.031}\)  \(^{+0.001}_{0.031}\) 
3.1 Data samples
The H1 [60, 61, 62] and ZEUS [63] detectors were general purpose instruments which consisted of tracking systems surrounded by electromagnetic and hadronic calorimeters and muon detectors, ensuring close to \(4\pi \) coverage of the ep interaction region. Both detectors were equipped with highresolution silicon vertex detectors [64, 65].
The datasets included in the combination are listed in Table 1. The data have been obtained from both the HERA I (in the years 1992–2000) and HERA II (in the years 2003–2007) datataking periods. The combination includes measurements using different tagging techniques: the reconstruction of particular decays of D mesons [4, 6, 10, 12, 15, 20, 21, 22] (datasets \(27, 9, 10\)), the inclusive analysis of tracks exploiting lifetime information [14, 23] (datasets 1, 11) and the reconstruction of electrons and muons from heavyflavour semileptonic decays [13, 16, 19] (datasets 8, 12, 13).
The datasets 1–8 have already been used in the previous combination [42] of charm crosssection measurements, while the datasets 9–13 are included for the first time in this analysis. Dataset 9 of the current analysis supersedes one dataset of the previous charm combination (dataset 8 in Table 1 of [42]), because the earlier analysis was based on a subset of only about \(30\,\%\) of the final statistics collected during the HERA II running period.
3.2 Extrapolation of visible cross sections to \({\varvec{\sigma }}_{\mathrm{red}}^{{{\rm Q}\overline{\rm Q}}}\)
In pQCD, \(\sigma _\mathrm{red}^\mathrm{th}\) can be written as a convolution integral of proton PDFs with hard matrix elements. For the identification of heavyflavour production, however, specific particles used for tagging have to be measured in the hadronic final state. This requires that in the calculation of \(\sigma _\mathrm{vis}^\mathrm{th}\), the convolution includes the proton PDFs, the hard matrix elements and the fragmentation functions. In the case of the HVQDIS programme, nonperturbative fragmentation functions are used. The different forms of the convolution integrals for \(\sigma _\mathrm{red}^\mathrm{th}\) and \(\sigma _\mathrm{vis}^\mathrm{th}\) necessitate the consideration of different sets of theory parameters.

The renormalisation and factorisation scales are taken as \(\mu _\mathrm{r}=\mu _\mathrm{f}=\sqrt{Q^2+4m_\mathrm{Q}^2}\). The scales are varied simultaneously up or down by a factor of two.

The pole masses of the charm and beauty quarks are set to \(m_c=1.50 \pm 0.15\) GeV, \(m_b=4.50 \pm 0.25\) GeV, respectively. These variations also affect the values of the renormalisation and factorisation scales.

For the strong coupling constant, the value \(\alpha _s^{n_f=3}(M_Z) = 0.105 \pm 0.002\) is chosen, which corresponds to \(\alpha _s^{n_f=5}(M_Z) = 0.116 \pm 0.002\).

The proton PDFs are described by a series of FFNS variants of the HERAPDF1.0 set [42, 45] at NLO determined within the xFitter framework. No heavyflavour measurements were included in the determination of these PDF sets. These PDF sets are those used in the previous combination [42] which were calculated for \(m_c=1.5\pm 0.15\) GeV, \(\alpha _s^{n_f=3}(M_Z) = 0.105 \pm 0.002\) and simultaneous variations of the renormalisation and factorisation scales up or down by a factor two. For the determination of the PDFs, the beautyquark mass was fixed at \(m_b=4.50\) GeV. The renormalisation and factorisation scales were set to \(\mu _r=\mu _f=Q\) for the light flavours and to \(\mu _r=\mu _f=\sqrt{Q^2+4m_\mathrm{Q}^2}\) for the heavy flavours. For all parameter settings considered, the respective HERAPDF1.0 set is used. As a cross check of the extrapolation procedure, the cross sections are also evaluated with the 3flavour NLO versions of the HERAPDF2.0 set (FF3A) [48]; the differences are found to be smaller than the PDFrelated crosssection uncertainties.

The charm fragmentation function is described by the Kartvelishvili function [67] controlled by a single parameter \(\alpha _K\) to describe the longitudinal fraction of the charmquark momentum transferred to the D or \(D^{*\pm }\) meson. Depending on the invariant mass \(\hat{s}\) of the outgoing parton system, different values of \(\alpha _K\) and their uncertainties are used as measured at HERA [68, 69] for \(D^{*\pm }\) mesons. The variation of \(\alpha _K\) as a function of \(\hat{s}\) observed in \(D^{*\pm }\) measurements has been adapted to the longitudinalfragmentation function of ground state D mesons not originating from \(D^{*\pm }\) decays [42]. Transverse fragmentation is modelled by assigning to the charmed hadron a transverse momentum \(k_T\) with respect to the direction of the charmed quark with an average value of \(\langle k_T\rangle =0.35\pm 0.15\) GeV [42].

The charm fragmentation fractions of a charm quark into a specific charmed hadron and their uncertainties are taken from [70].

The beauty fragmentation function is parameterised according to Peterson et al. [71] with \(\epsilon _{b}=0.0035 \pm 0.0020\) [72].

The branching ratios of D and \( D^{*\pm }\) mesons into the specific decay channels analysed and their uncertainties are taken from [59].

The branching fractions of semileptonic decays of heavy quarks to a muon or electron and their uncertainties are taken from [59].

The decay spectra of leptons originating from charmed hadrons are modelled according to [73].

The decay spectra for beauty hadrons into leptons are taken from the PYTHIA[74] Monte Carlo (MC) programme, mixing direct semileptonic decays and cascade decays through charm according to the measured branching ratios [59]. It is checked that the MC describes BELLE and BABAR data [75, 76] well.

When necessary for the extrapolation procedure, partonlevel jets are reconstructed using the same clustering algorithms as used on detector level, and the cross sections are corrected for jethadronisation effects using corrections derived in the original papers [16, 23].^{4}
3.3 Combination method
The quantities to be combined are the reduced charm and beauty cross sections, \(\sigma _{\mathrm{red}}^{c\overline{c}}\, \) and \(\sigma _{\mathrm{red}}^{b\overline{b}}\, \), respectively. The combined cross sections are determined at common (\(x_\mathrm{Bj} , Q^2\)) grid points. For \(\sigma _{\mathrm{red}}^{c\overline{c}}\, \), the grid is chosen to be the same as in [42]. The results are given for a centreofmass energy of \(\sqrt{s}=318\) GeV. When needed, the measurements are transformed to the common grid \((x_\mathrm{Bj} ,Q^2)\) points using inclusive NLO FFNS calculations [25, 26, 27, 28]. The uncertainties on the resulting scaling factors are found to be negligible.
4 Combined cross sections
The values of the combined cross sections \(\sigma _{\mathrm{red}}^{c\overline{c}}\, \)and \(\sigma _{\mathrm{red}}^{b\overline{b}}\, \), together with the statistical, the uncorrelated and correlated systematic and the total uncertainties, are listed in Tables 2 and 3. A total of 209 charm and 57 beauty data points are combined simultaneously to obtain 52 reduced charm and 27 reduced beauty crosssection measurements. A \(\chi ^2\) value of 149 for 187 degrees of freedom (d.o.f.) is obtained in the combination, indicating good consistency of the input datasets. The distribution of pulls of the 266 input data points with respect to the combined cross sections is presented in Fig. 1. It is consistent with a Gaussian around zero without any significant outliers. The observed width of the pull distribution is smaller than unity which indicates a conservative estimate of the systematic uncertainties.
There are 167 sources of correlated uncertainties in total. These are 71 experimental systematic sources, 16 sources due to the extrapolation procedure (including the uncertainties on the fragmentation fractions and branching ratios) and 80 statistical charm and beauty correlations. The sources of correlated systematic and extrapolation uncertainties are listed in the “Appendix”, together with the crosssection shifts induced by the sources and the reduction factors of the uncertainties, obtained as a result of the combination. Both quantities are given in units of \(\sigma \) of the original uncertainties. All shifts of the systematic sources with respect to their nominal values are smaller than \(1.5\sigma \). Several systematic uncertainties are reduced significantly – by up to factors of two or more. The reductions are due to the different heavyflavour tagging methods applied and to the fact that for a given process (charm or beauty production), an unique cross section is probed by the different measurements at a given \((x_\mathrm{Bj},Q^2)\) point. Those uncertainties for which large reductions have been observed already in the previous analysis [42] are reduced to at least the same level in the current combination, some are further significantly reduced due to the inclusion of new precise data [21, 22, 23]. The shifts and reductions obtained for the 80 statistical correlations between charm and beauty cross sections are not shown. Only small reductions in the range of \(10\%\) are observed and these reductions are independent of \(x_\mathrm{Bj} \) and \(Q^2\). The crosssection tables of the combined data together with the full information on the uncertainties can be found elsewhere [77].
The combined reduced cross sections \(\sigma _{\mathrm{red}}^{c\overline{c}}\, \)and \(\sigma _{\mathrm{red}}^{b\overline{b}}\, \)are shown as a function of \(x_\mathrm{Bj} \) in bins of \(Q^2\) together with the input H1 and ZEUS data in Figs. 2 and 3, respectively. The combined cross sections are significantly more precise than any of the individual input datasets for charm as well as for beauty production. This is illustrated in Fig. 4, where the charm and beauty measurements for \(Q^2 = 32~\hbox {GeV}^2\) are shown. The uncertainty of the combined reduced charm cross section is \(9\%\) on average and reaches values of about \(5\%\) or better in the region \(12~\hbox {GeV}^2\le Q^2\le 60~\hbox {GeV}^2\). The uncertainty of the combined reduced beauty cross section is about \(25\%\) on average and reaches about \(15\%\) at small \(x_\mathrm{Bj} \) and \(12~\hbox {GeV}^2\le Q^2\le 200~\hbox {GeV}^2\).
In Fig. 5, the new combined reduced charm cross sections are compared to the results of the previously published combination [42]. Good consistency between the different combinations can be observed. A detailed analysis of the crosssection measurements reveals a relative improvement in precision of about \(20\,\%\) on average with respect to the previous measurements. The improvement reaches about \(30\,\%\) in the range \(7~\hbox {GeV}^2\le Q^2\le 60~\hbox {GeV}^2\), where the newly added datasets (datasets 9–11 in Table 1) contribute with high precision.
5 Comparison with theory predictions
The combined heavyflavour data are compared with calculations using various schemes and PDF sets. Predictions using the FFNS [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35] and the VFNS [37, 38, 39, 40, 41] are considered, focussing on results using HERAPDF2.0 PDF sets. The data are also compared to FFNS predictions based on different variants of PDF sets at NLO and approximate NNLO provided by the ABM group [29, 32]. In the case of the VFNS, recent calculations of the NNPDF group based on the NNPDF3.1sx PDF set [41] at NNLO, which specifically aim for a better description of the DIS structure functions at small \(x_\mathrm{Bj} \) and \(Q^2\), are also confronted with the measurements. The calculations in the FFNS based on the HERAPDF2.0 FF3A PDF set will be considered as reference calculations in the subsequent parts of the paper.
5.1 FFNS predictions
In Figs. 6 and 7, theoretical predictions of the FFNS in the \(\overline{\mathrm{MS}}\) running mass scheme are compared to the combined reduced cross sections \(\sigma _{\mathrm{red}}^{c\overline{c}}\, \)and \(\sigma _{\mathrm{red}}^{b\overline{b}}\, \), respectively. The theoretical predictions are obtained within the opensource QCD fit framework for PDF determination xFitter [53], which uses the OPENQCDRAD programme [31, 51, 52] for the crosssection calculations. The running heavyflavour masses are set to the world average values [59] of \(m_c(m_c)=1.27\pm 0.03\) GeV and \(m_b(m_b)=4.18\pm 0.03\) GeV. The predicted reduced cross sections are calculated using the HERAPDF2.0 FF3A [48] and ABMP16 [32] NLO PDF sets using NLO \((O(\alpha _s^2))\) coefficient functions and the ABMP16 [32] NNLO PDF set using approximate NNLO coefficient functions. The charm data are also compared to NLO predictions based on the ABKM09 [29] NLO PDF set used in the previous analysis [42] of combined charm data. This PDF set was determined using a charmquark mass of \(m_c(m_c)=1.18\) GeV. The PDF sets considered were extracted without explicitly using heavyflavour data from HERA with the exception of the ABMP16 set, in which the HERA charm data from the previous combination [42] and some of the beauty data [14, 23] have been included. For the predictions based on the HERAPDF2.0 FF3A set, theory uncertainties are given which are calculated by adding in quadrature the uncertainties from the PDF set, simultaneous variations of \(\mu _r\) and \(\mu _f\) by a factor of two up or down and the variation of the quark masses within the quoted uncertainties.
The FFNS calculations reasonably describe the charm data (Fig. 6) although in the kinematic range where the data are very precise, the data show a \(x_\mathrm{Bj}\) dependence somewhat steeper than predicted by the calculations. For the different PDF sets and QCD orders considered, the predictions are quite similar at larger \(Q^2\) while some differences can be observed at smaller \(Q^2\) or \(x_\mathrm{Bj}\). For beauty production (Fig. 7) the predictions are in good agreement with the data within the considerably larger experimental uncertainties.
The description of the charmproduction data is illustrated further in Fig. 8, which shows the ratios of the reduced cross sections for data, ABKM09 and ABMP16 at NLO and approximate NNLO with respect to the NLO reduced cross sections predicted in the FFNS using the HERAPDF2.0 FF3A set. For \(Q^2\ge 18~\hbox {GeV}^2\), the theoretical predictions are similar to each other in the kinematic region accessible at HERA. In this region, the predictions based on the different PDF sets and orders are well within the theoretical uncertainties obtained for the HERAPDF2.0 FF3A set. Towards smaller \(Q^2\) and \(x_\mathrm{Bj}\), some differences in the predictions become evident. In the region of \(7~\hbox {GeV}^2\le Q^2\le 120~\hbox {GeV}^2\), the theory tends to be below the data at small \(x_\mathrm{Bj}\) and above the data at large \(x_\mathrm{Bj}\), independent of the PDF set and order used.
In Fig. 9, the corresponding ratios are shown for the beauty data. In the kinematic region accessible at HERA, the predictions are very similar to each other. Within the experimental uncertainties, the data are well described by all calculations.
5.2 VFNS predictions
In Fig. 10, predictions of the RTOPT [37] NLO and approximate NNLO VFNS using the corresponding NLO and NNLO HERAPDF2.0 PDF sets are compared to the charm measurements. As in Fig. 8, the ratio of data and theory predictions to the reference calculations are shown. While the NLO VFNS predictions are in general consistent with both the data cross sections and the reference calculations, the approximate NNLO cross sections show somewhat larger differences, about \(10\%\) smaller than the reference cross sections in the region \(12~\hbox {GeV}^2\le Q^2\le 120~\hbox {GeV}^2\). On the other hand, at \(Q^2\le 7~\hbox {GeV}^2\) the \(x_\mathrm{Bj}\) slopes of the NNLO VFNS predictions tend to describe the data somewhat better than the reference calculations. Overall, the NLO and approximate NNLO VFNS predictions describe the data about equally well, but not better than the reference FFNS calculations.
In Fig. 11, the same ratios as in the preceding paragraph are shown for beauty production. In the kinematic region accessible in DIS beauty production at HERA, the differences between the different calculations are small in comparison to the experimental uncertainties of the measurements.
The calculations considered so far generally show some tension in describing the \(x_\mathrm{Bj}\) slopes of the measured charm data over a large range in \(Q^2\). Therefore the charm data are compared in Fig. 12 to recent calculations[41, 78] in the FONLLC scheme with (NNLO+NLLsx) and without (NNLO) lowx resummation in both \(O(\alpha _s^2)\) matrix elements and \(O(\alpha _s^3)\) PDF evolution, using the NNPDF3.1sx framework, which aim for a better description of the proton structure functions at low \(x_\mathrm{Bj} \) and \(Q^2\). The charm data from the previous combination have already been used for the determination of the NNPDF3.1sx PDFs. Both calculations provide a better description of the \(x_\mathrm{Bj}\) shape of the measured charm cross sections for \(Q^2< 32~\hbox {GeV}^2\). However, the predictions lie significantly below the data in most of the phase space. This is especially the case for the NNLO+NLLsx calculations. Overall, the description is not improved with respect to the FFNS reference calculations.
5.3 Summary of the comparison to theoretical predictions
The comparison to data of the different predictions considered is summarised in Table 4 in which the agreement with data is expressed in terms of \(\chi ^2\) and the corresponding fit probabilities (p values). The table also includes a comparison to the previous combined charm data [42]. The agreement of the various predictions with the charm crosssection measurements of the current analysis is poorer than with the results of the previous combination, for which consistency between theory and data within the experimental uncertainties is observed for most of the calculations. As shown in Sect. 4, the charm cross sections of the current analysis agree well with the previous measurements but have considerably smaller uncertainties. The observed changes in the \(\chi ^2\) values are consistent with the improvement in data precision if the predictions do not fully describe reality. The tension observed between the central theory predictions and the charm data ranges from \(\sim 3\sigma \) to more than \(6\sigma \), depending on the prediction. Among the calculations considered, the NLO FFNS calculations provide the best description of the charm data. For the beauty cross sections, good agreement of theory and data is observed within the larger experimental uncertainties. In all cases, the effect of the PDF uncertainties on the \(\chi ^2\) values is negligible.
6 QCD analysis
The combined charm and beauty data are used together with the combined HERA inclusive DIS data [48] to perform a QCD analysis in the FFNS using the \(\overline{\mathrm{MS}}\) massrenormalisation scheme at NLO. The main focus of this analysis is the simultaneous determination of the running heavyquark masses \(m_c(m_c)\) and \(m_b(m_b)\). The theory description of the \(x_\mathrm{Bj}\) dependence of the reduced charm cross section is also investigated.
6.1 Theoretical formalism and settings
The analysis is performed with the xFitter [53] programme, in which the scale evolution of partons is calculated through DGLAP equations [79, 80, 81, 82, 83, 84, 85, 86] at NLO, as implemented in the QCDNUM programme [87]. The theoretical FFNS predictions for the HERA data are obtained using the OPENQCDRAD programme [31, 51, 52] interfaced in the xFitter framework. The number of active flavours is set to \(n_f = 3\) at all scales. For the heavyflavour contributions the scales are set to \(\mu _r = \mu _f=\sqrt{Q^2+4m_\mathrm{Q}^2}\). The heavyquark masses are left free in the fit unless stated otherwise. For the lightflavour contributions to the inclusive DIS cross sections, the pQCD scales are set to \(\mu _r = \mu _f = Q\). The massless contribution to the longitudinal structure function \(F_\mathrm{L}\) is calculated to \(O(\alpha _s)\). The strong coupling strength is set to \(\alpha _s^{n_f=3}(M_Z) = 0.106\), corresponding to \(\alpha _s^{n_f=5}(M_Z) = 0.118\). In order to perform the analysis in the kinematic region where pQCD is assumed to be applicable, the \(Q^2\) range of the inclusive HERA data is restricted to \(Q^2 \ge Q^2_\mathrm{min} = 3.5~\hbox {GeV}^2\). No such cut is applied to the charm and beauty data, since the relevant scales \(\mu _r^2 = \mu _f^2=Q^2+4m_\mathrm{Q}^2\) are above \(3.5~\hbox {GeV}^2\) for all measurements.
This theory setup is slightly different from that used for the original extraction [48] of HERAPDF2.0 FF3A. In contrast to the analysis presented here, HERAPDF2.0 FF3A was determined using the onshell mass (polemass) scheme for the calculation of heavyquark production and \(F_\mathrm{L}\) was calculated to \(O(\alpha ^2_s)\).
Perturbative QCD predictions were fit to the data using the same \(\chi ^2\) definition as for the fits to the inclusive DIS data (equation (32) in reference [48]). It includes an additional logarithmic term that is relevant when the estimated statistical and uncorrelated systematic uncertainties in the data are rescaled during the fit [88]. The correlated systematic uncertainties are treated through nuisance parameters.
The selection of parameters in Eq. (6) from the general form, Eq. (5), is made by first fitting with all D and E parameters set to zero, and then including them one at a time in the fit. The improvement in the \(\chi ^2\) of the fit is monitored. If \(\chi ^2\) improves significantly, the parameter is added and the procedure is repeated until no further significant improvement is observed. This leads to the same 14 free PDF parameters as in the inclusive HERAPDF2.0 analysis [48].
The PDF uncertainties are estimated according to the general approach of HERAPDF2.0 [48], in which the experimental, model, and parameterisation uncertainties are taken into account. The experimental uncertainties are determined from the fit using the tolerance criterion of \(\Delta \chi ^2 =1\). Model uncertainties arise from the variations of the strong coupling constant \(\alpha _s^{n_f=3}(M_Z) = 0.1060 \pm 0.0015\), simultaneous variations of the factorisation and renormalisation scales up or down by a factor of two, the variation of the strangeness fraction \(0.3 \le f_{s} \le 0.5\), and the value of \(2.5~\hbox {GeV}^2\le Q^2_{\text {min}}\le 5.0~\hbox {GeV}^2\) imposed on the inclusive HERA data. The total model uncertainties are obtained by adding the individual contributions in quadrature. The parameterisation uncertainty is estimated by extending the functional form in Eq. (6) of all parton density functions with additional parameters D and E added one at a time. An additional parameterisation uncertainty is considered by using the functional form in Eq. (6) with \(E_{u_v} = 0\). The \(\chi ^2\) in this variant of the fit is only 5 units worse than that with the released \(E_{u_v}\) parameter; changing this parameter noticeably affects the mass determination. In addition, \(\mu _\mathrm{f,0}^2\) is varied within \(1.6~\hbox {GeV}^2< \mu _\mathrm{f, 0}^2 < 2.2~\hbox {GeV}^2\). The parameterisation uncertainty is determined at each \(x_\mathrm{Bj} \) value from the maximal differences between the PDFs resulting from the central fit and all parameterisation variations. The total uncertainty is obtained by adding the fit, model and parameterisation uncertainties in quadrature. The values of the input parameters for the fit and their variations considered, to evaluate model and parameterisation uncertainties, are given in Table 5.
6.2 QCD fit and determination of the running heavyquark masses
In the QCD fit, the running heavyquark masses are fitted simultaneously with the PDF parameters in Eq. (6). The fit yields a total \(\chi ^2=1435\) for 1208 degrees of freedom. The ratio \(\chi ^2/\mathrm{d.o.f.}=1.19\) is similar in size to the values obtained in the analysis of the HERA combined inclusive data [48]. The resulting PDF set is termed HERAPDFHQMASS. The central values of the fitted parameters are given in the “Appendix”.
A cross check is performed using the Monte Carlo method [90, 91]. It is based on analysing a large number of pseudo datasets called replicas. For this cross check, 500 replicas are created by taking the combined data and fluctuating the values of the reduced cross sections randomly within their statistical and systematic uncertainties taking into account correlations. All uncertainties are assumed to follow a Gaussian distribution. The central values for the fitted parameters and their uncertainties are estimated using the mean and RMS values over the replicas. The obtained heavyquark masses and their experimental/fit uncertainties are in agreement with those quoted in Eq. (7).
In order to study the influence of the inclusive data on the mass determination, fits to the combined inclusive data only are also tried. In this case, the fit results are very sensitive to the choice of the PDF parameterisation. When using the default 14 parameters, the masses are determined to be \(m_c(m_c) = 1.80^{+0.14}_{0.13} \mathrm{(exp/fit)}~\mathrm{GeV}\), \(m_b(m_b) = 8.45^{+2.28}_{1.81} \mathrm{(exp/fit)}~\mathrm{GeV}\), where only the experimental/fit uncertainties are quoted. In the variant of the fit using the inclusive data only and the reduced parameterisation with \(E_{u_v} = 0\), the central fitted values for the heavyquark masses are: \(m_c(m_c) = 1.45~\mathrm{GeV}\), \(m_b(m_b) = 4.00~\mathrm{GeV}\). The sensitivity to the PDF parameterisation and the large experimental/fit uncertainties for a given parameterisation demonstrate that attempts to extract heavyquark masses from inclusive HERA data alone are not reasonable in this framework. The large effect on the fitted masses observed here, when setting \(E_{u_v} = 0\), motivates the \(E_{u_v}\) variation in the HERAPDFHQMASS fit.
The NLO FFNS predictions based on HERAPDFHQMASS are compared to the combined charm and beauty cross sections in Figs. 14 and 15, respectively. The predictions based on the HERAPDF2.0 set are included in the figures. Only minor differences between the different predictions can be observed. This is to be expected because of the similarities of the PDFs, in particular that of the gluon and the values of the heavyquark masses. The description of the data is similar to that observed for the predictions based on the HERAPDF2.0 FF3A set.
In Fig. 16, the ratios of data and predictions based HERAPDFHQMASS to the predictions based on HERAPDF2.0 FF3A are shown for charm production. The description of the data is almost identical for both calculations. The data show a steeper \(x_\mathrm{Bj}\) dependence than expected in NLO FFNS. The partial \(\chi ^2\) value of 116 for the heavyflavour data^{7} (d.o.f. \(=79\)) in the fit presented is somewhat large. It corresponds to a p value^{8} of 0.004, which is equivalent to \(2.9\sigma \). A similar behaviour can be observed already for the charm cross sections from the previous combination [42], albeit at lower significance due to the larger uncertainties.
In Fig. 17, the same ratios as in Fig. 16 are shown for beauty production. Agreement is observed between theory and data within the large uncertainties of the measurements.
6.3 Reduced heavyflavour cross sections as a function of the partonic \(\varvec{x}\)
6.4 Increasing the impact of the charm data on the gluon density
While inclusive DIS cross sections constrain the gluon density indirectly via scaling violations, and directly only through higher order corrections, heavyflavour production probes the gluon directly already at leading order. Contributions to heavyflavour production from lightflavour PDFs are small. For charm production they amount to five to eight per cent, varying only slightly with \(x_\mathrm{Bj}\) or \(Q^2\) [49]. Because of the high precision of \(\sigma _{\mathrm{red}}^{c\overline{c}}\, \) reached in this analysis, a study is performed to enhance the impact of the charm measurement on the gluon determination in the QCD fit.
To reduce the impact of the inclusive data in the determination of the gluon density function, a series of fits is performed by requiring a minimum \(x_\mathrm{Bj} \ge x_\mathrm{Bj,min}\) for the inclusive data included in the fit, with \(x_\mathrm{Bj,min}\) varying from \(2\cdot 10^{4}\) to 0.1. No such cut is applied to the heavyflavour data. The \(\chi ^2/\mathrm{d.o.f.}\) values for the inclusive plus heavyflavour data and the partial \(\chi ^2/\mathrm{d.o.f.}\) for the heavyflavour data only are presented in Fig. 19 as a function of \(x_{\mathrm{Bj,min}}\). The partial \(\chi ^2/\mathrm{d.o.f.}\) for the heavyflavour data improves significantly with rising \(x_{\mathrm{Bj,min}}\) cut reaching a minimum at \(x_{\mathrm{Bj,min}}\approx 0.04\), while the \(\chi ^2/\mathrm{d.o.f.}\) for the inclusive plus heavyflavour data sample is slightly larger than that obtained without a cut in \(x_\mathrm{Bj}\). For further studies \(x_{\mathrm{Bj,min}}=0.01\) is chosen. The total \(\chi ^2\) is 822 for 651 degrees of freedom. The partial \(\chi ^2\) of the heavyflavour data improves to 98 for 79 degrees of freedom (corresponding to a p value of 0.07 or \(1.8\sigma \)). The resulting gluon density function, shown in Fig. 20 at the scale \(\mu _\mathrm{f}^2=1.9~\hbox {GeV}^2\), is significantly steeper than the gluon density function determined when including all inclusive measurements in the fit. The other parton density functions are consistent with the result of the default fit.
In Fig. 21, a comparison is presented of the ratios of the combined reduced charm cross section and the cross section as calculated from the alternative fit, in which the inclusive data are subject to the cut \(x_\mathrm{Bj}\ge 0.01\), to the reference cross sections based on HERAPDF2.0 FF3A. The predictions from HERAPDFHQMASS are also shown. As expected, the charm cross sections fitted with the \(x_\mathrm{Bj}\) cut imposed on the inclusive data rise more strongly towards small \(x_\mathrm{Bj}\) and describe the data better than the other predictions. In general, the predictions from the fit with \(x_\mathrm{Bj}\) cut follow nicely the charm data. A similar study for beauty is also made but no significant improvement in the description of the beauty data is observed. The heavyquark masses extracted from the fit with \(x_{\mathrm{Bj}} \ge 0.01\) are consistent with those quoted in Eq. (7).
Crosssection predictions based on the three PDF sets, discussed in the previous paragraph, are calculated for inclusive DIS. In Fig. 22, these predictions are compared to the inclusive reduced cross sections [48] for NC \(e^+p\) DIS. The predictions based on HERAPDF2.0 FF3A and on HERAPDFHQMASS agree with the inclusive measurement. The calculations based on the PDF set determined by requiring \(x_\mathrm{Bj}\ge 0.01\) for the inclusive data predict significantly larger inclusive reduced cross sections at small \(x_\mathrm{Bj}\).
This study shows that a better description of the charm data can be achieved by excluding the low\(x_\mathrm{Bj}\) inclusive data in the fit. However, the calculations then fail to describe the inclusive data at low \(x_\mathrm{Bj}\). In the theoretical framework used in this analysis, it seems impossible to resolve the \(2.9\sigma \) difference in describing simultaneously the inclusive and charm measurements from HERA, using this simple approach of changing the gluon density. The comparison of various theory predictions to the charm data in section 5 suggests that the situation is unlikely to improve at NNLO because the NNLO predictions presented provide a poorer description of the charm data than that observed at NLO. The combined inclusive analysis [48] already revealed some tensions in the theory description of the inclusive DIS data. The current analysis reveals some additional tensions in describing simultaneously the combined charm data and the combined inclusive data.
7 Summary
Measurements of charm and beauty production cross sections in deep inelastic ep scattering by the H1 and ZEUS experiments are combined at the level of reduced cross sections, accounting for their statistical and systematic correlations. The beauty cross sections are combined for the first time. The datasets are found to be consistent and the combined data have significantly reduced uncertainties. The combined charm cross sections presented in this paper are significantly more precise than those previously published.
Nexttoleading and approximate nexttonexttoleadingorder QCD predictions of different schemes are compared to the data. The calculations are found to be in fair agreement with the charm data. The nexttoleadingorder calculations in the fixedflavournumber scheme provide the best description of the heavyflavour data. The beauty data, which have larger experimental uncertainties, are well described by all QCD predictions.
The QCD analysis reveals some tensions, at the level of \(3 \sigma \), in describing simultaneously the inclusive and the heavyflavour HERA DIS data. The measured reduced charm cross sections show a stronger \(x_\mathrm{Bj}\) dependence than obtained in the combined QCD fit of charm and inclusive data, in which the PDFs are dominated by the fit of the inclusive data. A study in which inclusive data with \(x_\mathrm{Bj}<0.01\) are excluded from the fit is carried out. A better description of the charm data can be achieved this way. However, the resulting PDFs fail to describe the inclusive data in the excluded \(x_\mathrm{Bj}\) region. Alternative nexttoleadingorder and nexttonextleadingorder QCD calculations considered, including those with lowx resummation, do not provide a better description of the combined heavyflavour data.
Footnotes
 1.
In this paper the term ‘electron’ denotes both electron and positron .
 2.
 3.
The pseudorapidity is defined as \(\eta =\ln \tan \frac{\Theta }{2}\), where the polar angle \(\Theta \) is defined with respect to the proton direction in the laboratory frame.
 4.
Since no such corrections are provided, an uncertainty of \(5\%\) is assigned to cover the untreated hadronisation effects [16].
 5.
In the FFNS this scale is decoupled from the charmquark mass.
 6.
The previous analysis did not consider scale variations and a less flexible PDF parameterisation was used.
 7.
It is not possible to quote the charm and the beauty contribution to this \(\chi ^2\) value separately because of the correlations between the combined charm and beauty measurements.
 8.
The \(\chi ^2\) and the p value given here do not correspond exactly to the statistical definition of \(\chi ^2\) or p value because the data have been used in the fit to adjust theoretical uncertainties. Therefore the theory is somewhat shifted towards the measurements. However this bias is expected to be small because the predictions are mainly constrained by the much larger and more precise inclusive data sample.
Notes
Acknowledgements
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.
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