Extraction of the proton parton density functions using a NLO-QCD fit of the combined H1 and ZEUS inclusive DIS cross sections
The combined HERA-I data set, of neutral and charged current inclusive cross-sections for and scattering, is used as the sole input for a next-to-leading order (NLO) QCD parton distribution function (PDF) fit. The consistent treatment of systematic uncertainties in the joint data set ensures that experimental uncertainties on the PDFs can be calculated without need for an increased tolerance. This results in PDFs with greatly reduced experimental uncertainties compared to the separate analyses of the ZEUS and H1 experiments. Model uncertainties, including those arising from parametrization dependence, are also carefully considered. The resulting HERAPDFs have impressive precision compared to the global fits.
The kinematics of lepton hadron scattering is described in terms of the variables , the invariant mass of the exchanged vector boson, Bjorken , the fraction of the momentum of the incoming nucleon taken by the struck quark (in the quark-parton model), and which measures the energy transfer between the lepton and hadron systems. The differential cross-section for the neutral current (NC) process is given in terms of the structure functions by
where . The structure functions and are directly related to quark distributions, and their dependence, or scaling violation, is predicted by perturbative QCD. For low , , is sea quark dominated, but its evolution is controlled by the gluon contribution, such that HERA data provide crucial information on low- sea-quark and gluon distributions. At high , the structure function becomes increasingly important, and gives information on valence quark distributions. The charged current (CC) interactions also enable us to separate the flavour of the valence distributions at high-, since their (LO) cross-sections are given by,
Parton Density Function (PDF) determinations are usually obtained from global NLO QCD fits [2, 3, 4], which use fixed target DIS data as well as HERA data. In such analyses, the high statistics HERA NC data have determined the low- sea and gluon distributions, whereas the fixed target data have determined the valence distributions. Now that high- HERA data on NC and CC and inclusive double differential cross-sections are available, PDF fits can be made to HERA data alone, since the HERA high cross-section data can be used to determine the valence distributions. This has the advantage that it eliminates the need for heavy target corrections, which must be applied to the -Fe and fixed target data. Furthermore there is no need to assume isospin symmetry, i.e. that in the proton is the same as in the neutron, since the distribution can be obtained directly from CC data.
The H1 and ZEUS collaborations have both used their data to make PDF fits , . Both of these data sets have very small statistical uncertainties, so that the contribution of systematic uncertainties becomes dominant and consideration of point to point correlations between systematic uncertainties is essential. The ZEUS analysis takes account of correlated experimental systematic errors by the Offset Method, whereas H1 uses the Hessian method . Whereas the resulting ZEUS and H1 PDFs are compatible, the gluon PDFs do have rather different shapes, see Fig. 1, and the uncertainty bands spanned by these analyses are comparable to those of the global fits.
It is possible to improve on this situation since ZEUS and H1 are measuring the same physics in the same kinematic region. These data have been combined them using a ’theory-free’ Hessian fit in which the only assumption is that there is a true value of the cross-section, for each process, at each point , . The resulting systematic uncertainties on each of the combined data points are significantly smaller than the statistical errors. In the present paper this combined data set is used as the input to a NLO QCD PDF fit. The consistency of the input data set and its small systematic uncertainties enable us to calculate the experimental uncertainties on the PDFs using the tolerance, . This represents a further advantage compared to those global fit analyses where increased tolerances of are used to account for data inconsistencies.
For the present HERAPDF0.1 fit, the role of correlated systematic uncertainties is no longer crucial since these uncertainties are relatively small. This ensures that similar results are obtained using either Offset or Hessian methods, or by simply combining statistical and systematic uncertainties in quadrature. For our central fit we have chosen to combine the 43 systematic uncertainties which result from the separate ZEUS and H1 data sets in quadrature, and to Offset the 4 sources of uncertainty which result from the combination procedure. This results in the most conservative uncertainty estimates on the resulting PDFs.
Despite our conservative procedure the experimental uncertainties on the resulting PDFs are impressively small and a thorough consideration of further uncertainties due to model assumptions is necessary. In section 2 we describe the NLO QCD analysis and model assumptions. In section 3 we give results and in section 4 we give a summary.
The QCD predictions for the structure functions are obtained by solving the DGLAP evolution equations at NLO in the MSbar scheme with the renormalisation and factorization scales chosen to be . The DGLAP equations yield the PDFs at all values of provided they are input as functions of at some input scale . This scale has been chosen to be GeV and variation of this choice is considered as one of the model uncertainties. The resulting PDFs are then convoluted with NLO coefficient functions to give the structure functions which enter into the expressions for the cross-sections. The choice of the heavy quark masses is, GeV, and variation of these choices is included in the model uncertainties. For this preliminary analysis, the heavy quark coefficient functions have been caluclated in the zero-mass variable flavour number scheme. The strong coupling constant was fixed to , and variations in this value of have also been considered.
The fit is made at leading twist. The HERA data have of the invariant mass of the hadronic system,, of GeV and maximum , , such that they are in a kinematic region where there is no sensitivity to target mass and large- higher twist contributions. However a minimum cut is imposed to remain in the kinematic region where perturbative QCD should be applicable. This has been chosen such that GeV. Variation of this cut is included as one of the model uncertainties.
A further model uncertainty is the choice of the initial parametrization at . The PDFs are parametrized by the generic form
and the number of parameters is chosen by ’saturation of the ’, such that parameters are only varied if this brings significant improvement to the . Otherwise they are set to zero.
For our central fit, the PDFs which are parametrized are , , and , . The normalisation parameters, , for the and valence are constrained to impose the number sum-rules and the normalisation parameter for the gluon is constrained to impose the momentum sum-rule. The parameters which constrain the low- behaviour of the and valence distributions are set equal, and the parameters are also set equal for and , such that there is a single parameter for the valence and another different single parameter for the sea distributions. Assuming that the strange and charm quark distributions can be expressed as independent fractions, and , of the and type sea, gives the further constraint . The value of has been chosen to be consistent with determinations of this fraction using neutrino induced di-muon production. This value has been varied to evaluate model uncertainties. The charm fraction has been set to be consistent with dynamic generation of charm from the start point of in a zero-mass-variable-flavour-number scheme. A small variation of the value of is included in the model uncertainties. Saturation of the leads us to set the parameters , for all partons except for which only .
The results are presented using this parametrization, including six sources of model uncertainty due to variation of: . Comparison is made to three other classes of parametrization, one based on the ZEUS-JETs parametrization , one based on the H1 parametrization  and one based on the current parametrization but allowing for the gluon. Comparison is also made to results obtained by varying , see reference  for details. Our central choice has less model dependence than the ZEUS-Style parametrization because it has fewer asumptions concerning , and it has less model dependence than the H1-style parametrization in that it does not assume equality of all parameters. Furthermore, although all types of parametrization give acceptable values, the central parametrization has the best and it gives the most conservative experimental errors.
The total uncertainty of the PDFs obtained from the HERA combined data set is much reduced compared to the PDFs extracted from the analyses of the separate H1 and ZEUS data sets, as can be seen from the summary plots in Fig. 1, where these new HERAPDF0.1 PDFs are compared to the ZEUS-JETS and H1PDF2000 PDFs.
Now that high- HERA data on NC and CC and inclusive double differential cross-sections are available, PDF fits can be made to HERA data alone, since the HERA high cross-section data can be used to determine the valence distributions and HERA low cross-section data can be used to determine the Sea and gluon distributions. The combined HERA-I data set, of neutral and charged current inclusive cross-sections for and scattering, has been used as the sole input for a NLO QCD PDF fit in the DGLAP formalism. The consistent treatment of systematic uncertainties in the joint data set ensures that experimental uncertainties on the PDFs can be calculated without need for an increased tolerance. This results in PDFs with greatly reduced experimental uncertainties compared to the separate analyses of the ZEUS and H1 experiments. Model uncertainties, including those arising from parametrization dependence, have also been carefully considered. The resulting HERAPDFs have impressive precision compared to the global fits.
-  A.D. Martin et al., Eur. Phys.J C23, 73 (2002)
-  J. Pumplin et al., JHEP 0207, 012 (2002)
-  ZEUS Coll., S. Chekanov et al., Phys. Rev D 67, 012007 (2003)
-  ZEUS Coll., S. Chekanov et al., Eur.Phys.J C 42, 1 (2005)
-  H1 Coll., C.Adloff et al., Eur.Phys.J C 30, 32 (2003)
-  A.M. Cooper-Sarkar, J.Phys G 28, 2669 (2002)
-  ZEUS and H1 Collaborations, ZEUS-prel-07-036, H1prelim-07-007
-  J Feltesse, Proceedings of DIS 2008
-  W-M Yoo et al, J.Phys G 33, 1 (2006)
-  G Watt, Proceedings of DIS 2008