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\noindent
% {\bf H-UM \& JS: version of \today} \\[.3em] 
Submitted to the 30th International Conference on 
High-Energy Physics ICHEP2000, \\ 
Osaka, Japan, July 2000


\vspace*{3cm}

\begin{center}
  \Large
  {\bf 
           QCD Interpretation of \\
     Inclusive Deep-Inelastic Scattering and \\
        Jet Production Data at HERA} 

  \vspace*{1cm}
    {\Large H1 Collaboration} 
\end{center}

\begin{abstract}

\noindent
The gluon density in the proton is extracted using the information from
scaling violations of the inclusive cross sections and from the direct
contribution to the dijet cross section. The dijet data provides
additional information in the region of high $x$ ($0.01 < x < 0.1$)
where a significant fraction of the gluon momentum is found. The fit
uses massless next-to-leading order QCD predictions folded in Mellin
space with parameterized parton densities evolved from a starting scale
to the corresponding scale. The gluon density from scaling violations
only and from the combined fit are compared and demonstrate the
consistency and influence of the data sets.
\end{abstract}


\vfill
\begin{flushleft}
  {\bf Abstract: 992 } \\
  {\bf Parallel session: 3} \\
  {\bf Plenary talk: 11 } 
\end{flushleft}

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\section{Introduction}

Deep-inelastic $ep$ scattering can be described by the scattering of a
parton inside the proton with the beam electron. Parton density
functions describe the non-perturbative part, while for the hard parton
photon scattering a perturbative expansion in the strong coupling
strength $\alpha_s$ is performed.

The graph in zeroth order in $\alpha_s$ arises from a quark or
anti-quark of the proton scattering off the photon radiated by the
electron. This diagram may lead to a one jet event and contributes to
the lowest order of the inclusive cross section.

The simplest QCD process that directly depends on the gluon density in
the proton is given by the Boson-Gluon-Fusion graph. The matrix element
is of order one in $\alpha_s.$ In addition, the gluon contributes to the
observed scaling violations and, therefore, to the dependence of the
total cross section or, equivalently, the structure functions on the
scale. This allows the gluon density to be extract indirectly.
Equivalent diagrams appear in calculations of the next-to-leading order
inclusive cross section for deep-inelastic scattering and in the lowest
order dijet cross section.

The extractions of the gluon density in the proton performed by the H1
collaboration currently either use the indirect scaling violation
or the direct dijet information. In this fit, the gluon density is
extracted using both informations simultaneously.

\section{Data sets}

The inclusive cross section $\sigma$ has been measured by the H1
collaboration in the kinematic region $20\GeV^2 \le Q^2 \le
5000\GeV^2$ for the data taking period of 1995 till
1997\cite{h1hq,slaclowq}. For these data sets together with additional
fixed target data, extractions of the gluon density from scaling
violations have been performed in the cited papers.

The hadronic final state of the events in the same data set is studied
with the help of the  the longitudinal boost-invariant $k_t$ algorithm\cite{inclkt}. The
double differential dijet cross section 
$\displaystyle{d^2\sigma_2\over dQ^2 dx}$
for photon virtualities between $200\GeV^2 \le Q^2 \le 5000\GeV^2$ is
determined\cite{ICHEP}. 

\section {Fit method}

The fit is performed by using the MINUIT package\cite{minuit} where the
quality parameter $\chi^2$ is defined by comparing the theoretical
calculation with the corresponding measured value taking into account
the statistical and systematic errors.

While the evolution of the parton densities is defined by the
perturbative expansion, the densities have to be parameterized at a
starting scale of $Q_0^2 =4\;\GeV^2.$ The parameterizations used for the
gluon, the up valence quark and the combined sea quark densities are of
the form
\begin{equation}xf(x,Q_0^2) = A_f x^{\alpha_f} (1-x)^{\beta_f}.\end{equation}
Leaving additional freedom by introducing a polynom of the form
$(1 +\gamma_f x + \epsilon_f \sqrt{x})$ or $(1 + \gamma_f x^{\epsilon_f})$
does neither change the gluon distribution more than allowed by the error
band nor improves the quality of the fit significantly.

The down valence quark density is assumed to be correlated to the up valence 
quark density by $d(x,Q_0^2) = R (1-x)^2 u(x,Q_0^2),$ where $R$
is a constant given by the valence counting rules. Studies on the
dependence of the fit on this assumption showed that, while the quarks
densities vary drastically, the gluon density does not change
significantly and, thus, is independent of this input.

The sea is taken to have the same up and down contribution. The
strange contribution is suppressed by a factor of two while the heavy
quark contributions are negligible at the input scale. Repeating the fit
with a flavor symmetric light sea does not change the results.

The insensitivity of the gluon density of the exact composition of
the quark densities of the individual flavors is understood by the
fact that for both data sets the quark distributions to a good
approximation appear in a sum over all flavors weighted by the quark
charge squared.

The parton densities are evolved using Mellin space evolution. Both,
Mellin space and $x$ space evolutions, are expansions of the evolution
equations up to next-to-leading order in $\alpha_s$ but differ in the
treatment of the higher order contributions. Therefore, the comparison
of fits using Mellin and $x$ space evolutions can give an estimate on
the size of higher order contributions.

The theoretical calculations in this fit have been performed in
next-to-leading order. For the dijet data, the DISENT\cite{disent}
program is used in order to take into account the definition of the jet
algorithms and the cuts. For all calculations, the renormalization and
the factorization scales are set to $Q^2$ and the dependence on the
scales are estimated by varying the values independently between
${1\over 4}Q^2$ to $4Q^2.$

The value of the theory prediction also strongly depends on the value of
the strong coupling strength. Since this value has been measured very
accurately in various processes and is much better known than the
gluon density, the current world average of $\alpha_s(M_Z^2)=0.119$
enters the fit. The influence of the uncertainty on the strong coupling
is taken into account by repeating the fit with values of $0.116$ and
$0.122$ which is displayed by superimposing the resulting error bands.

\section {Gluon density}

The dependence in performing the evolution in Mellin space and in $x$
space is evaluated by performing fits using the inclusive data set only
with both fit programs. In figure~\ref{fig1} both fits are shown for
data starting at $Q_{\mbox{\small min}}^2=3.5\;\GeV^2.$ The curves agree
well in a large region of the momentum fraction $x.$ For low
virtualities, mass effects become important. Therefore, a fit
including massive partons is shown as a separate error band.

%{{{}}}
%{{{  figure
\begin{figure}[tb]
\includegraphics[width=\hsize]{H1prelim-00-033.fig1.eps}
\caption{
\label{fig1}
 Result of the fits to the inclusive data. Dijet data are not included.
 The full line shows a fit using evolution in Mellin space and the
 dashed line a fit with evolution in $x$ space. For comparison the $x$
 space fit with massive partons is plotted including error band as
 hashed area.
} 
\end{figure}
%}}}

While the inclusive data dominate the low $x$ region, the dijet data
contribute at high $x$ ($0.01 \le x \le 0.1$) where data from other
processes are sparse. The combined fit is shown in figure~\ref{fig2} in
the medium $x$ region. The thick line surrounds the error band of the
combined fit. Two fits using single data sets allow for a comparison of
the influence of the individual data sets. The gluon extracted from
dijet data (dashed line) alone is consistent but has considerably larger
uncertainties. The thin line gives the error band of a fit of the
inclusive data only. The size of the uncertainty of this fit is only
slightly larger than the one of the combined fit. This demonstrates that
with the current HERA data the scaling violations provide valuable
information on the gluon density which are confirmed by the dijet data.
The dijet data also improve the fit stability.

%{{{  figure
\begin{figure}[tb]
\includegraphics[width=.95\hsize]{H1prelim-00-033.fig2.eps}
\caption{
\label{fig2}
Result of fit at $Q^2=200\GeV^2.$ The gluon density is plotted together
with corresponding error band (thick, full line). For comparison the
gluon densities of a fit using the dijet data (dashed line) or the
inclusive data only (thin, full line) is shown. }
\end{figure}
%}}}

In figure~\ref{fig3}, the measured dijet cross section is shown together
with the NLO theory prediction by DISENT using the parton densities
extracted in the fit. A good agreement is found.

%{{{  figure
\begin{figure}[p]
\includegraphics[width=.95\hsize]{H1prelim-00-033.fig3.eps}
\caption{
\label{fig3}
Dijet cross section data and NLO theory prediction using the parton
densities as extracted in the fit.}
\end{figure}
%}}}

\section{Conclusions}

The gluon density in the proton is extracted in this article. For the
first time, the fit uses both, the size of the scaling violations of the
inclusive cross section and the direct information from the dijet cross
section. Both data sets, taken from measurements of the H1 experiment
only, are well described. The inclusion of the dijet cross section
results improves the fit stability compared to an extraction using the
inclusive cross section only.

The result is compatible with the latest fits\cite{slaclowq} despite of
the different evolution codes indicating that the uncertainties in the
treatment of the higher orders in the evolution codes is smaller than
the uncertainty on the gluon density.

%
%   References 
%
\begin{thebibliography}{99}

\bibitem{h1hq}
  H1 Collaboration. Eur.\ Phys.\ J.\  {\bf C13} (2000) 609-639.

\bibitem{slaclowq}
  M Klein. {\em Proceedings, Lepton Photon Conference,
  Stanford, USA} 1999.

\bibitem{inclkt}
  S.\ Catani, Yu.L.\ Dokshitzer, and B.R.\ Webber.\ Phys.\ Let.\ {\bf B} 
  285:291, 1992.

\bibitem{ICHEP}
  H1 Collaboration. 
  {\em Conf.\ Paper 520, 29th Intern.\ Conf.\ on High-Energy Physics, 
  Vancouver, Canada} 1998

\bibitem{minuit}
F.~James and M.~Roos. {\em Comput. Phys. Commun.}, 10:343, 1975.

\bibitem{zomer}
G.~Lobo, C.~Pascaud, and F.~Zomer.
On parton density error band calculation in {QCD} analysis of proton
structure. Technical Report 98-536, DESY, 1998.

\bibitem{disent}
S.~Catani and M.~H. Seymour. {\em Nucl. Phys.}, B485:291--419, 1997.



\end{thebibliography}

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