Impact of heavyflavour production cross sections measured by the LHCb experiment on parton distribution functions at low x
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DOI: 10.1140/epjc/s100520153618z
 Cite this article as:
 Zenaiev, O., Geiser, A., Lipka, K. et al. Eur. Phys. J. C (2015) 75: 396. doi:10.1140/epjc/s100520153618z
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Abstract
The impact of recent measurements of heavyflavour production in deep inelastic ep scattering and in pp collisions on parton distribution functions is studied in a QCD analysis in the fixedflavour number scheme at nexttoleading order. Differential cross sections of charm and beautyhadron production measured by LHCb are used together with inclusive and heavyflavour production cross sections in deep inelastic scattering at HERA. The heavyflavour data of the LHCb experiment impose additional constraints on the gluon and the seaquark distributions at low partonic fractions x of the proton momentum, down to \(x \sim 5 \times 10^{6}\). This kinematic range is currently not covered by other experimental data in perturbative QCD fits.
1 Introduction
Understanding the nucleon structure is one of the fundamental tasks of modern particle physics. In quantum chromodynamics (QCD), the structure of the nucleon is described by parton distribution functions (PDFs), which, in collinear factorisation, represent probability densities to find a parton of longitudinal fraction x of the nucleon momentum at a factorisation scale \(\mu _f\). The scale evolution of the PDFs is uniquely predicted by the renormalisation group equations for factorisation [1, 2, 3, 4, 5, 6, 7, 8, 9]. The xdependence cannot be derived from first principles and must be constrained by experimental measurements. The precision of the PDFs is of key importance for interpreting the measurements in hadronic collisions. In particular, the uncertainty of the proton PDFs must be significantly reduced in order to improve the accuracy of theory predictions for Standard Model (SM) processes at the LHC.
Deep inelastic lepton–proton scattering (DIS) experiments cover a broad range in x and \(\mu _f\). In the perturbative regime, a wide xrange of \(10^{4}<x\lesssim 10^{1}\) is probed by the data of the H1 and ZEUS experiments at the HERA collider [10]. These measurements impose the tightest constraints on the existing PDFs. However, additional measurements are necessary for a better flavour separation and to constrain the kinematic ranges of very small and very high x, where the gluon distribution is poorly known. A better constraint on the highx gluon is needed for an accurate description of the SM backgrounds in searches for newparticle production at high masses or momenta. A significant reduction of the uncertainty of the lowx gluon distribution is important for studies of parton dynamics, nonlinear and saturation effects. Furthermore, the precision of the gluon distribution at low x has implications in physics of atmospheric showers, being crucial for crosssection predictions of highenergy neutrino DIS interactions [11] and for calculations of prompt lepton fluxes in the atmosphere [12].
Heavyflavour production in proton–proton collisions at LHC is dominated by gluon–gluon fusion. Therefore the LHCb measurements of charm [13] and beauty [14] production in the forward region \(2.0 < y < 4.5\) probe the gluon distribution at \(5\times 10^{6} \lesssim x \lesssim 10^{4}\), a region which is not accessible with HERA data. Note that the LHCb data are sensitive to the product of gluon densities in two nonoverlapping low and mediumtohigh x ranges, as illustrated in Fig. 1. Since the medium range is already well constrained by HERA data, which furthermore bridge the gap between the two LHCb ranges, the major impact of the LHCb heavyflavour measurements is expected at \(5\times 10^{6} \lesssim x \lesssim 10^{4}\). The advantage of using heavyflavour data is that the charm and beauty masses provide hard scales for the perturbative QCD (pQCD) expansion all the way down to their production threshold. To estimate the impact of the LHCb measurement of charm and beauty production on the gluon distribution at low x, these data are included in a QCD analysis together with the inclusive DIS [10] and heavyflavour production [15, 16] cross sections measured at HERA.
2 Experimental data used in the QCD analysis
The main objective of the present QCD analysis is to demonstrate the constraining power of the measurements of heavyflavour production in DIS and pp collisions for the determination of the PDFs of the proton. The measurements of charm and beauty production at HERA and LHCb, together with the combined HERA inclusive crosssection measurements, are used in a nexttoleading order (NLO) pQCD analysis.
Neutral current (NC) and charged current (CC) inclusive DIS cross sections in ep scattering are directly sensitive to the valence and seaquark distributions and probe the gluon distribution through scaling violations [1, 2, 3]. HERA measurements of the NC and CC cross sections in DIS at a centreofmass energy \(\sqrt{s}\) = 320 GeV have been combined taking into account systematic correlations [10]. This combined data set contains the complete information on inclusive DIS cross sections published by the H1 and ZEUS Collaborations based on data collected in the years 19942000, and has been used for the determination of the PDF set HERAPDF1.0 [10]. The kinematic range of the NC data is \(6 \times 10^{7} \le x_{Bj} \le 0.65\), \(0.045 \le Q^2 \le 30000\) GeV\(^2\), however, in the original HERAPDF1.0 fit the \(Q^2\) range of the data was restricted to \(Q^2>Q^2_\mathrm {min}\) = 3.5 GeV\(^2\) to ensure the applicability of perturbative calculations [10]. The CC cross sections span the kinematic range of \(1.3 \times 10^{2} \le x_{Bj} \le 0.40\) and \(300 \le Q^2 \le 30000\) GeV\(^2\). These combined NC \(e^\pm p\) and CC \(e^\pm p\) cross sections represent the basis for all PDF determinations.
In ep scattering, charm and beauty quarks are produced predominantly in the photon–gluon fusion process which provides a direct probe of the gluon distribution in the proton. Measurements of opencharm production cross sections in DIS at HERA from the H1 and ZEUS Collaborations have been combined [15]. Cross sections for charm production were obtained in the kinematic range of \(2.5 \le Q^2 \le 2000\) GeV\(^2\) and \( 3 \times 10^{5} \le x_{Bj} \le 5 \times 10^{2}\). The combination method accounts for the correlations of the systematic uncertainties among the different data sets. These combined measurements were used to improve constraints on the gluon distribution and to determine the charmquark mass [15]. The charm reduced cross sections determined as a function of \(Q^2\) and \(x_{Bj}\) are used in the present analysis together with all provided details on the systematic correlations. In addition, cross sections for the production of b quarks in ep scattering, as measured by the ZEUS Collaboration [16] are used in the present analysis. These data correspond to an integrated luminosity of 354 pb\(^{1}\) and cover the kinematic range of \(5 < Q^2 <1000\) GeV\(^2\). The b and cquark content in the events with at least one jet have been extracted using the invariant mass of charged tracks associated with secondary vertices and lifetime information, and the beautyquark mass has been measured [16]. In the present analysis, the beautyquark production data are used mainly to improve constraints on the beautyquark mass.
For additional constraints on the gluon distribution at low x the differential cross sections of charm and beauty production in pp collisions at \(\sqrt{s}=7\) TeV from the LHCb experiment are used for the first time. The measurement of charm production [13] is based on data corresponding to an integrated luminosity of 15 nb\(^{1}\). Charm production is identified through the full reconstruction of decays of the charmed hadrons^{1}\(D^0\), \(D^{+}\), \(D^{*+}\), \(D_s^+\) and \(\Lambda _c^{+}\). The cross sections are measured as a function of the transverse momentum, \(p_T\), and rapidity, y, of the reconstructed hadrons. The LHCb data on Bmeson production in pp collisions [14] correspond to an integrated luminosity of 0.36 fb\(^{1}\). The \(B^+\), \(B^0\) and \(B_s^0\) mesons are reconstructed in exclusive decays mainly involving \(J/\psi \) final states. Correlations between the experimental systematic uncertainties are accounted for as described in the original publications. An uncorrelated systematic uncertainty is obtained for each distribution by subtracting the correlated uncertainties from the total ones. The 3.5 % luminosity uncertainty is treated as correlated between the measurements of charm and beauty production. In the present analysis, the normalised cross sections, \(\frac{\mathrm{d}\sigma }{\mathrm{d}y} / \frac{\mathrm{d}\sigma }{\mathrm{d}y_0}\), for charm and beauty production are calculated from the absolute measurements published by LHCb and are used in the QCD analysis, with \(\frac{\mathrm{d}\sigma }{\mathrm{d}y_0}\) being the cross section in the center bin, \(3 < y < 3.5\), of the measured rapidity range in each \(p_T\) bin. The uncorrelated experimental uncertainty on \(\frac{\mathrm{d}\sigma }{\mathrm{d}y_0}\) is propagated as a correlated uncertainty to the respective complementary rapidity bins. The QCD analysis is performed by using both, absolute or normalised, representations of the LHCb measurements, alternatively.
3 Theoretical predictions for heavyflavour production
In the QCD analysis, the experimental measurements are confronted with corresponding theoretical predictions. Complete fixedorder theoretical predictions for differential production of charm and beauty in both ep and pp collisions only exist at NLO in the fixedflavour number scheme (FFNS). For higher orders, the most comprehensive results for heavyquark production in ep collisions are given in [17], which contain combined approximate nexttonexttoleading order (NNLO) \(O(\alpha _s^3)\) expressions for three kinematic limits: in the limit of high partonic centreofmass energy squared, in the region of the production threshold, and in the highscale region. For heavy quarkpair production in pp collisions, the cross sections in singleparticle kinematics have been calculated at approximate NNLO \(O(\alpha _s^4)\) [18, 19] and at approximate N\(^3\)LO [20] by using methods of threshold resummation beyond the leading logarithmic approximation. Further theory developments are necessary to include the available HERA and LHCb heavyflavour production measurements in a QCD analysis at NNLO.
Consistent with the necessary theory predictions, the presented QCD analysis is performed at NLO in FFNS. This scheme and its applicability to HERA measurements is discussed in detail in Ref. [15] and references therein. Predictions for HERA data are obtained by following the approach of the ABM group at NLO using its implementation in OPENQCDRAD [21, 22, 23] in the framework of HERAFitter [24]. The number of active flavours is set to \(N_f = 3\), and the renormalisation and factorisation scales (pQCD scales) for heavyflavour production are chosen as \(\mu _r = \mu _f=\sqrt{Q^2+4m_Q^2}\), where \(m_Q\) denotes the pole mass of c or b quarks.^{2} For the lightflavour contribution to the inclusive DIS cross sections, the pQCD scales are set to \(\mu _r = \mu _f = Q\).
Theoretical predictions for heavyquark production in pp collisions are obtained using the massive NLO calculations [25, 26, 27] in the FFNS, also available as part of the Mangano–Nason–Ridolfi (MNR) calculations [28]. The pQCD scales are chosen as \(\mu _{r,f} = A_{r,f}^{c,b}\mu _0\), with \(\mu _0=\sqrt{p^2_T+m^2_Q}\) and \(A_{r,f}^{c,b}\) being coefficients for c and b quarks, which are discussed in the following. These predictions were used successfully for beauty production in \(p \bar{p}\) collisions at the \(Sp\bar{p}S\) [29, 30, 31] and the Tevatron^{3} [32]. They are conceptually very similar to the Frixione–Mangano–Nason–Ridolfi (FMNR) predictions [36, 37] employed for heavyflavour photoproduction at HERA [38, 39].
The crosssection predictions for heavyflavoured hadron production not only depend on the kinematics of the heavyflavour production mechanism, but also on the fragmentation of the heavy quark into a particular finalstate hadron. There is no finalstate factorisation scale in the FFNS since collinear logarithms of the heavyquark mass are included in fixedorder perturbation theory. The calculations in [25, 26, 27, 40] describe the production of an onshell heavy quark. Near the kinematic threshold, the transition of the heavy quark into the observed heavyflavoured hadron can be taken into account by multiplying the cross section with the appropriate fragmentation fraction. This leads to an excellent description of B and Dmeson production measurements at the Tevatron and the LHC from \(p_T=0\) up to \(p_T \sim 4 m_Q\) [41, 42]. The scope of these calculations can be extended by convoluting the heavyquark production cross section with a suitable scaleindependent fragmentation function describing the hadronisation of the heavy quark. The implementation of the convolution is not unique once the quark and hadron masses are taken into account, and leads to a potentially \(p_T\)dependent modeling uncertainty which is, however, small compared to the scalechoice uncertainty at NLO. This fragmentation function is used on a purely phenomenological basis, since it does not strictly appear in the context of a factorisation theorem, and therefore it has to be extracted from data. It depends on the order of the perturbation series but is generally assumed to be otherwise universal. Its main effect is to lower the theoretical predictions at large \(p_T\). Typical parametrisations used in the literature are those by Peterson et al. [43] depending on one parameter \(\varepsilon \) and by Kartvelishvili et al. [44] depending on one parameter \(\alpha _K\).
For the HERA measurements, the fragmentation functions and their uncertainties are considered and accounted for in the original publications [15, 16]. The measurements of LHCb are provided as hadronproduction cross sections and the fragmentation functions have to be applied explicitly in order to use these data in the QCD analysis. In addition, fragmentation fractions describing the probability of a quark to fragment into a particular hadron have to be applied. The fragmentation fractions for cflavoured hadrons are taken from [45] and for bflavoured hadrons from [14].
So far, no fragmentation measurements were performed in pp collisions. Because of similarities of the cquarkproduction kinematics at HERA and LHCb, the Kartvelishvili fragmentation function [44] with \(\alpha _K=4.4\pm 1.7\), as obtained from corresponding HERA measurements [46, 47] extracted for the NLO FFNS scheme, is applied for predictions of the LHCb measurements of charmhadron production. The fragmentation is performed in the laboratory frame by rescaling the quark threemomentum, with the energy of the produced hadron being calculated using the hadron mass. This procedure is used for \(D^+\) and \(D^+_s\) mesons, and for \(\Lambda _c^+\) baryons. For \(D^0\) and \(D^+\)meson production, the contribution from \(D^{*+}\) and \(D^{*0}\) mesons is treated as described in [48]. For beauty production, the value \(\alpha _K= 11\pm 4\) is used for all bflavoured hadrons, corresponding to measurements at LEP [49].
The fragmentationfraction uncertainties are assigned to the measurements and are treated as correlated, while the uncertainties arising from the variations of assumptions on the fragmentation functions are treated in the form of variations of the theory predictions in the QCD fit.
4 Details of the QCD analysis
The open source QCD fit framework for PDF determination HERAFitter [24], version 1.0.0, is used. The partons are evolved by using the QCDNUM program [50]. In the presented study, collinear factorisation is assumed throughout the analysis performed using the DGLAP [1, 2, 3, 4, 5, 6, 7, 8, 9] evolution equations. With the available techniques, no direct tests of nonDGLAP models are possible with the data used here. Such tests would require significant developments of the theory and of the corresponding QCD analysis tools, which is beyond the scope of this paper.
The analysisspecific modifications to HERAFitter address the heavyflavour treatment as follows. The massive FFNS [51, 52, 53, 54] with the number of flavours \(N_f=3\) is used for the treatment of heavyflavour contributions. The calculation of oneparticle inclusive heavyquarkproduction cross sections in hadron collisions at NLO according to [27] is implemented by using original routines from the MNR code [55]. The results agree with those obtained with the original MNR code at a level of accuracy better than 1 %.
The 3flavour strong coupling constant in the NLO \(\overline{\mathrm{MS}}\) scheme is set to \(\alpha _S(m_Z)^{N_f=3} =0.1059\pm 0.0005\), which corresponds to the world average value of \(\alpha _S(m_Z)^{N_f=5}= 0.1185\pm 0.0006\), using twoloop evolution equations [50].
The normalisation parameters \(A_{u_{\text {v}}}\), \(A_{d_\text {v}}\), \(A_g\) are determined by the QCD sum rules, the B parameters are responsible for the smallx behaviour of the PDFs, and the parameters C describe the shape of the distribution as \(x \rightarrow 1\). A flexible form for the gluon distribution is adopted with the choice of \(C'_g=25\) motivated by the approach of the MSTW group [56, 57]. The squark distribution is defined through xindependent strangeness fraction, \(f_s\), of the dtype sea, \(x\overline{s} = f_sx\overline{D}\) at \(Q^2_0\), where \(f_s=0.31^{+0.19}_{0.08}\) as in the analysis of [57], including the recent complementary measurement [58]. Additional constraints \(B_{\overline{\text {U}}} = B_{\overline{\text {D}}}\) and \(A_{\overline{\text {U}}} = A_{\overline{\text {D}}}(1  f_s)\) are imposed, with \(x\bar{u} \rightarrow x\bar{d}\) as \(x \rightarrow 0\). The analysis is performed by fitting the remaining 13 free parameters in Eqs. (1–5).
The PDF parameters are determined in HERAFitter by minimisation of a \(\chi ^2\)function taking into account correlated and uncorrelated uncertainties [24] of the measurements. Systematic uncertainties are assumed to be proportional to the central prediction values, whereas statistical uncertainties scale with the square root of the predictions. Correlated uncertainties are treated using a nuisanceparameter representation [24]. To minimise biases arising from the likelihood transition to \(\chi ^2\) when the scaling of the errors is applied, a logarithmic correction is added to the \(\chi ^2\)function [59].
The heavyquark masses are left free in the fit. They are well constrained by the measurements of charm and beauty production in DIS and the fitted values (see Table 2 in the Appendix A) are consistent with the ones obtained in the corresponding HERA analyses [15, 16] within the intrinsic theoretical systematic uncertainty of the polemass definition [60, 61, 62].
The QCD analysis is performed twice using either absolute or normalised differential cross sections of heavyflavour production from LHCb measurements, as defined in Sect. 2. The implementation of the theory calculations [63] as described in Sect. 3 allows the pQCD scales, i.e. the parameters \(A_{r,f}^{c,b}\), and the values for the pole mass of the heavy quarks to be changed at each fit iteration.
In the QCD analysis using the normalised LHCb cross sections, the pQCD scales are fixed to \(A_r=A_f=1\) for the central result. The scale dependence is studied by varying the pQCD scales independently such that \(0.5\le A_r,A_f \le 2\). \(A^c\) and \(A^b\) are always varied simultaneously. The resulting scale dependence is small, since it is largely absorbed by the normalisation, as illustrated in Appendix A.
In the variant of the fit using the absolute LHCb cross sections, the scale dependence of the predicted cross section is the dominant theoretical uncertainty. The same scalechoice and variation procedure, as applied for the variant of the fit using the normalised LHCb cross sections, leads to unacceptably high \(\chi ^2\) values of the respective fits [63]. Therefore, the four scales technically are treated as independent fully correlated systematic uncertainties for the central result. Since the pQCD scales are not physical parameters, the related uncertainties are not obtained from the fit. Instead, the effect of the scale choice on the other fitted parameters is evaluated by an independent variation of \(A_f\) in the range \(0.5< A_f^c=A_f^b < 2\) with \(A_r^c\) and \(A_r^b\) as free parameters, or \(A_r\) in the range \(0.25 < A_r^c = A_r^b < 1\) with \(A_f^c\) and \(A_f^b\) being free parameters. For the variation \(A^c_f=A^b_f=0.5\), a cut \(p_T>2\) GeV is applied for the charm LHCb data to ensure that the factorisation scale is above 1 GeV\({}^2\), since this is technically required in the QCDNUM program. This procedure ensures an acceptable fit quality for all variations [63], as required for a meaningful extraction of the other uncertainties. Because of the unconventional scale treatment the fit using absolute cross sections is considered to be a cross check.
5 PDF uncertainties
The PDF uncertainties are estimated following the approach of HERAPDF1.0 [10] in which experimental, model, and parametrisation uncertainties are taken into account. Experimental uncertainties are evaluated using the Hessian method [24]. A tolerance criterion of \(\Delta \chi ^2 = 1\) is adopted for defining the fit uncertainties that originate from the experimental uncertainties of the measurements included in the analysis.
Model uncertainties arise from the variations in the values assumed for \(Q^2_\mathrm {min}\) imposed on the HERA data, which is varied in the interval \(2.5\le Q^2_\mathrm {min} < 5.0\) GeV\(^2\); the fraction of strange quarks, varied in the range \(0.23<f_s<0.50\) and the value of the strong coupling, varied in the range \(0.1054<\alpha _S(m_Z)^{N_F=3}<0.1064\). The pQCD scales for heavyquark production in DIS are varied simultaneously by a factor of 2 up and down for both, charm and beauty. For the fits with the LHCb data, the model uncertainties include theoretical uncertainties for the cross section predictions for heavyflavoured hadron production, arising from variation of the pQCD scales and of the fragmentation parameters, as described in Sect. 3. Uncertainties, arising from these model variations are referred to as MNR uncertainties in the following.
The parametrisation uncertainty is estimated similarly to the HERAPDF1.0 procedure: for all PDFs, additional parameters are added one by one in the functional form of the parametrisations in Eqs. (1–5), in a similar way as described in [10, 15, 16]. Furthermore, the starting scale is varied to \(Q^2_0=1.9\) GeV\(^2\). The parametrisation uncertainty is constructed as an envelope built from the maximal differences between the PDFs resulting from all the parametrisation variations and the central fit at each x value. The total PDF uncertainty is obtained by adding experimental, model and parametrisation uncertainties in quadrature.
6 Results
The global and partial \(\chi ^2\) values for the data sets used in the analysis of HERA and LHCb measurements
Representation of LHCb data  Absolute  Normalised 

Global \(\chi ^2 / n_\mathrm{dof}\)  1073/1087  958/994 
Global \(\chi ^2\)pvalue  0.61  0.79 
Correlated uncertainties \(\chi ^2\)  73  49 
Logarithmic correction \(\chi ^2\)  \(\)129  48 
Data set  Partial \(\chi ^2 / n_\mathrm{dof}\)  
NC DIS HERA I comb. \(e^p\)  108/145  108/145 
NC DIS HERA I comb. \(e^+p\)  419/379  419/379 
CC DIS HERA I comb. \(e^p\)  26/34  26/34 
CC DIS HERA I comb. \(e^+p\)  39/34  41/34 
\(c\bar{c}\) DIS HERA comb.  78/52  47/52 
\(b\bar{b}\) DIS ZEUS Vertex  16/17  12/17 
LHCb \(D^0\)  68/38  17/30 
LHCb \(D^{+}\)  53/37  18/29 
LHCb \(D^{*+}\)  50/31  19/22 
LHCb \(D_s^{+}\)  24/28  11/20 
LHCb \(\Lambda _c^{+}\)  5.3/6  4.9/3 
LHCb \(B^{+}\)  99/135  81/108 
LHCb \(B^0\)  66/95  35/76 
LHCb \(B_s^0\)  78/75  23/60 
The fitted parameters for the NLO QCD analysis using HERA and LHCb measurements. The value of strong coupling \(\alpha _S(m_Z)^{N_f=3} =0.1059\) is used (which corresponds to \(\alpha _S(m_Z)^{N_f=5}= 0.1185\)). The listed uncertainties correspond to those associated to the experimental measurements used in the fit. Uncertainties are not quoted for parameters that are fixed. The correlation matrix can be made available upon request
Parameter  Absolute  Normalised 

\(B_g\)  \(0.14 \pm 0.07\)  \(0.08 \pm 0.10\) 
\(C_g\)  \(6.83 \pm 0.31\)  \(5.23 \pm 0.34\) 
\(A'_{g}\)  \(1.74 \pm 0.22\)  \(1.29 \pm 0.32\) 
\(B'_{g}\)  \(0.19 \pm 0.04\)  \(0.16 \pm 0.05\) 
\(B_{u_{\text {v}}}\)  \(0.668 \pm 0.020\)  \(0.649 \pm 0.021\) 
\(C_{u_{\text {v}}}\)  \(4.99 \pm 0.23\)  \(4.98 \pm 0.23\) 
\(E_{u_{\text {v}}}\)  \(12.2 \pm 2.4\)  \(13.5 \pm 2.7\) 
\(B_{d_{\text {v}}}\)  \(0.93 \pm 0.09\)  \(0.96 \pm 0.09\) 
\(C_{d_{\text {v}}}\)  \(5.50 \pm 0.56\)  \(5.59 \pm 0.55\) 
\(C_{\overline{\text {U}}}\)  \(1.63 \pm 0.21\)  \(1.63 \pm 0.24\) 
\(A_{\overline{\text {D}}}\)  \(0.173 \pm 0.007\)  \(0.158 \pm 0.007\) 
\(B_{\overline{\text {D}}}\)  \(0.146 \pm 0.006\)  \(0.155 \pm 0.007\) 
\(C_{\overline{\text {D}}}\)  \(10.4 \pm 2.5\)  \(15.1 \pm 4.2\) 
\(m_c\) [GeV]  \(1.709 \pm 0.024\)  \(1.257 \pm 0.014\) 
\(m_b\) [GeV]  \(4.67 \pm 0.08\)  \(4.19 \pm 0.13\) 
\(A_{f}^{c}\)  \(0.659 \pm 0.020\)  1.0 
\(A_{f}^{b}\)  \(0.262 \pm 0.007\)  1.0 
\(A_{r}^{c}\)  \(0.444 \pm 0.021\)  1.0 
\(A_{r}^{b}\)  \(0.335 \pm 0.024\)  1.0 
In case of the variant of the fit based on normalised LHCb cross sections, the uncertainties are reduced by more than a factor of three at \(x \sim 5\times 10^{6}\), which is the edge of the sensitivity of the included measurements (Fig. 1). Consistent results are obtained in the fit using the absolute cross sections, which is considered an important cross check of the selfconsistency of the NLO theory description.
The main differences in the PDF uncertainties between the fits using the absolute and normalised LHCb cross sections are caused by the MNR uncertainties. The variation of the pQCD scales in the prediction of the absolute cross section of heavyflavour production in pp collisions leads to significant changes in the normalisation of the cross section and represents the dominant uncertainty on the PDFs. The variations of the assumptions on the fragmentation parameters result in a negligible uncertainty as compared to those due to the scale variations, see e.g. Fig. 7.13 [63], since changes of the \(p_T\) shape due to variations of the fragmentation function can be compensated by small changes in the scales.
In the case of the PDF fit using the normalised LHCb cross sections, the MNR uncertainty is strongly reduced, since variations of pQCD scales and of the fragmentation parameters do not significantly affect the shape of the y distribution for heavyflavour production. Therefore this is considered to be the primary result of this paper, while the consistency between the absolute and normalised variants is considered to be an important cross check.
7 Conclusions
The sensitivity of heavyflavour production in pp collisions to the lowx gluon distribution was studied in a comprehensive QCD analysis at NLO. The measurements of c and bhadronproduction cross sections at the LHCb experiment are included into a PDF fit together with inclusive and heavyflavourproduction measurements in DIS at HERA. Since the bulk of the heavyflavour data is close to the kinematic threshold, the fixedflavour number scheme at NLO order is used for the predictions of heavyflavour production in ep and pp collisions. A significant reduction of the parametrisation uncertainty of the gluon distribution at very low x is observed, as compared to the result of the PDF fit using only HERA DIS data.
Two ways of using the LHCb measurements in the fit are studied. Although the absolute differential crosssection measurements contain more information, the resulting PDFs suffer from large theoretical uncertainty due to uncalculated higherorder corrections, estimated by the variation of the pQCD scales. By using only the rapidity shape information in the normalised cross sections for the final result, this uncertainty is significantly reduced for the PDF extraction.
The present analysis has illustrated the high potential of the LHCb measurements to constrain the gluon distribution at low x, and global PDF fits clearly can profit from the inclusion of such data. Precise measurements of normalised cross sections of heavyflavour production in the forward kinematic range of the LHC therefore have a great potential to further improve the constraints on the PDFs.
In order to fully exploit the additional constraints from absolute LHC charm and beauty cross sections, a significant reduction of the theoretical uncertainties, e.g. through threshold resummation and/or (partial) NNLO calculations with codes suitable for a usage in QCD analyses, is desirable.
The pole mass is used for consistency with the pp predictions, since \(\overline{\mathrm{MS}}\) running mass predictions are not available for LHCb.
Provided that fragmentation and other uncertainties are properly treated [33]. Note that the NLO+NLL (FONLL) [34] calculations used there, and also used by LHCb [13, 14], slightly reduce the cross sections at high transverse momenta with respect to the pure NLO calculation, while they are identical at low transverse momenta [34]. The Tevatron data are well described by FONLL [33]. This conclusion is also applicable to the NLO calculations [27] used here, since these were used as input for the NLO part of the FONLL calculations. The claim in [32] that the NLO predictions undershoot the data while FONLL describes them must thus be attributed to parametrisations beyond the perturbative part of the calculations, as illustrated, e.g., in Fig. 7 of [35].
Acknowledgments
This work has been supported in part by the Helmholtz Association under contract numbers HGFSO072 and VHHA101. Supported in part by the EUTMR Network Higgstools.
Funding information
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EUTMR Network Higgstools  
Helmholtz Association 

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