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\begin{titlepage}

\noindent
Submitted to the XXXth International Conference on High Energy Physics
ICHEP2000, \\
Osaka, Japan, July 2000

\vspace{2cm}

\begin{center}
\begin{Large}

{\bf A measurement of Dijet Rates in Deep Inelastic Scattering  at HERA
}

\vspace{2cm}

H1 Collaboration

\end{Large}
\end{center}

\vspace{2cm}

\begin{abstract}
\noindent
Dijet event rates, $R_2$, have been measured for deep inelastic $ep$
scattering using data collected with the H1 detector in the years
1996-1997 corresponding to an integrated luminosity of 22~${\rm
  \,pb}^{-1}$.  $R_2$ is shown as a function of $x_B$ and $Q^2$ in the
kinematic region $10^{-4}<x_B<10^{-2}$ and ${\rm} 5<Q^{2}<100 {\rm
  \,GeV}^{2}$.  The data are confronted with next-to-leading order QCD
calculations using two different choices of the renormalization scale
$\mu^{2}_{r}$ and assuming a point-like structure of the virtual
photon.  While for $\mu^{2}_r=Q^2$ the NLO predictions match the
measured dijet rate within the rather large scale uncertainties, the
choice $\mu^{2}_r=Q^2+E^{2}_t$ leads to a substantially lower but
theoretically more reliable prediction demanding the inclusion of
higher order processes. Therefore the data are compared to NLO QCD
calculations which include a resolved structure of the virtual photon.
In addition, the dependence of the measured dijet rates on the
transverse energy requirement for the jets is studied in bins of 
$x_B$ and $Q^2$ providing a
highly differential measurement of $R_2$ for comparison to current
fixed order and future resummed calculations.
%%%shown and compared to theoay.

%%%Dijet event rates, $R_2$, have been measured for deep inelastic scattering
%%%using data collected with the H1-Detector in the years 1996-1997.
%%%The analysis covers the kinematic domain $10^{-4}<x_B<10^{-2}$ and
%%%${\rm} 5<Q^{2}<100 {\rm \,GeV}^{2}$. The integrated luminostiy of $\approx$
%%%22~${\rm \,pb}^{-1}$ allows to measure $R_2$ as a function of both
%%%$x_B$ and $Q^{2}$ herewith extending a single differential analysis
%%%recently published by H1 with 1994 data. It opens the possibility
%%%of a more detailed study of jet production at low $x_B$ in a regime where the
%%%squared four momentum transfer $Q^{2}$ is comparable to the squared jet $p_t$. 
%%%The data are confronted with next-to-leading order QCD calculations
%%%including pointlike and non-pointlike structure of the virtual photon.
\end{abstract}

\vspace{1.5cm}

\begin{flushleft}
{\bf Abstract: 997} \\
{\bf Parallel session: 3} \\
{\bf Plenary Talk: 11  } \\ 
\vspace*{0.2cm}
{\bf Electronic Access: http://www-h1.desy.de/h1/www/publications/H1\_sci\_results.shtml}
\end{flushleft}

\end{titlepage}

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\newpage

\section{Introduction}

\noindent

%This paper is divided into 5 sections. In section 2 the data and
%event selection are described. The calibration procedure
%is given in section 3, and the cross-section measurements and
%comparisons are presented in section 4. In section 5 the paper is
%summarised.

The study of jets in deep-inelastic $ep$ scattering (DIS) at HERA, 
colliding $27.5~{\rm GeV}$ electrons on $820~{\rm GeV}$ protons,
provides a large phase space for testing perturbative QCD. 

If QCD in next-to-leading order (NLO) is employed to confront 
theory with measurement, the dependence of the
theoretical prediction on the choice of the renormalization scale $\mu^{2}_r$ is
expected to be small. However, the correct choice of $\mu^{2}_r$
remains a matter of concern especially at low $x_B$ and low $Q^2$
where higher orders in the perturbation series might be of more importance.
For the study of dijet events in DIS at least two scales are suggested by the
nature of the process, the first of which is  the
four-momentum transfer $Q^2$. The other
possible scale for jet production is the transverse energy of the
outgoing jets. The comparison of the H1 jet data with NLO predictions
for these different scales is addressed by this paper. The H1
collaboration has measured the fraction $R_2$ of dijet events in all DIS events as a function of the
Bjorken variable $x_B$ in different regions of the four-momentum
transfer $Q^2$. 


\section{The H1 Detector}

A detailed description of the H1 apparatus is given elsewhere\,\cite{h1}. Those
components essential for this analysis are briefly described
in the following. The interaction area is surrounded by a tracking
system of interleaved multi-wire proportional chambers and drift chambers which in total
cover the polar angle\footnote{Polar angles are defined with respect 
to the positive $z$-axis,  the proton 
beam direction} between $7^{o}<\theta<155^{o}$. They 
provide the primary measurement of charged particles. A highly segmented liquid argon
calorimeter, covering the polar angle $4^{o}<\theta<154^{o}$, 
surrounds the tracking system, providing the measurement of both,
electromagnetic and hadronic interacting particles.  
In the direction of large polar angles a lead scintillating-fiber
calorimeter (SpaCal)\,\cite{spa}  and a drift chamber (BDC), both covering the polar angle
$153^{o}<\theta<178^{o}$, enable a precise measurement of the scattered electron.

\section{Data selection}

The data used in the presented analysis were taken in the years
1996/97 with the H1 detector and correspond to an
integrated luminosity of 22~${\rm \,pb}^{-1}$. DIS events relevant
for this study are characterized by the electron being scattered into
the SpaCal. An energy deposit  of at least  $6 {\rm \,GeV}$ in the SpaCal and a loose 
track requirement is demanded in the trigger. The most energetic
cluster in the SpaCal defines the electron
candidate and events with a scattered electron of at least $9 {\rm
  \,GeV}$ are accepted. The cluster position has to be between
$156^o<\theta<175^o$ such that it is well measured within
the acceptance of the SpaCal and the BDC, which
is used to measure the polar angle of the electron candidate. The event kinematics
are then reconstructed from the energy and the polar angle of the
scattered electron alone.  

The hadronic final state is reconstructed from all remaining 
energy depositions in the SpaCal and the liquid argon calorimeter 
as well as from track momenta measured in the tracking system. While all
clusters in the calorimeters are considered, the four momentum 
of a track is only allowed to
contribute up to a momentum of $350 {\rm \,MeV}$. The inclusion of
tracks into the hadronic final state reconstruction substantially
improves the energy measurement of the identified jets (see below).

Background to the DIS events arises from photoproduction, where
the electron leaves the detector undetected and a particle from the
hadronic final state is misidentified as the scattered electron, as
well as from beam-gas and beam-wall interactions.  Therefore
additional cuts are applied to minimize the contamination by
these background sources and the influence of QED radiation:

\begin{itemize}
\item The event must have a reconstructed vertex within $\pm 35 {\rm \,cm}$ 
of its nominal position.

\item The quantity $\sum (E-p_z)$ is calculated from the
energy and the longitudinal momenta of scattered
electron and the objects associated with the hadronic final state.
The value of $\sum (E-p_z)$  is expected to be equal to twice the
electron beam energy of $27.5~{\rm GeV}$. 
An undetected electron in a photoproduction event or initial state photon
radiation will decrease the value of this observable. For this
analysis $35<\sum E-p_z<70 {\rm \,GeV}$ is required.
\end{itemize}

The phase space of the analysis is finally fixed by the following requirements:

\begin{itemize}
\item The four-momentum transfer $Q^2$ must be in the range $5<Q^2<100
  {\rm \,GeV^2}$ and the Bjorken scaling variable $x_B$ must be in the
  range $10^{-4}<x_B<10^{-2}$.

\item The inelasticity $y$ must be greater than
  0.1. The requirement of a minimum energy of the scattered
  electron of $9 {\rm \,GeV}$ corresponds to an upper cut on $y$ of
  0.7 leading to a $y$-range of $0.1<y<0.7$.


\end{itemize} 


In order to distinguish dijet events from the bulk of the DIS events,
the objects of the hadronic final state described above are input to
a jet algorithm. The jet search is performed in the photon-proton center of mass
system using the inclusive $k_t$ algorithm\,\cite{es}. Dijet events are
defined to have at least two jets fulfilling the following criteria:
The transverse energy $E_{t,jet}$ of each jet has to be larger than $5 {\rm \,GeV
}$ and their pseudo-rapidity $\eta$ as measured in the laboratory
frame has to be between $-$1 and 2.5.


\section{Correction of the data and systematic errors}

The data are corrected for detector effects and  effects of QED
radiation using events simulated with the
DJANGO/ARIADNE\,\cite{scs,ll} and RAPGAP\,\cite{hj} Monte-Carlo
generators. Both give a satisfactory description of the event
characteristics\,\cite{rp}. The mean of the predictions of these
models was used for the bin-wise correction of the
data. The difference between the correction factors obtained by the 
models and the mean is considered as a systematic error of
the final result leading to an uncertainty well below 10\%.

Additional systematic errors result from uncertainties of the hadronic
energy scale of $\pm 4\%$ which leads to a typical error
of 10\% on the final result and constitutes the dominant error source.
Other systematic uncertainties like that on the
electromagnetic energy scale of the SpaCal or the measurement 
of the polar angle of the scattered electron are much smaller.  


\section{Comparison between data and theory}
\subsection{Comparison to NLO QCD calculations}


The dijet rates are compared to NLO calculations using the
DISENT\,\cite{cs} program which calculates the cross section for $ep$
scattering to order ${\cal O}(\alpha^{2}_{s})$ assuming the virtual
photon to be point-like. To avoid infrared sensitivity of the
predictions\,\cite{kk,fr}, the transverse energies of the jets are
required to be \mbox{$E_{t,jet1}>7 {\rm \,GeV}$}, \mbox{$E_{t,jet2}>5
{\rm \,GeV}$}. The corresponding dijet rates are shown in figure
\ref{fig:r2nlodir}.  The dijet rates increase for decreasing $x_B$ 
as expected, since the available phase space increases with
the invariant mass of the hadronic final state $W^{2}\approx\frac{Q^2}{x_B}$.


In figure \ref{fig:r2nlodir} the NLO QCD calculations 
are shown for two choices of the renormalization scale, namely $\mu^{2}_r=Q^2$ and
$\mu^{2}_{r}=Q^{2}+E^{2}_{t}$. The factorization scale is set to
$\mu^{2}_f=Q^2$ and the CTEQ4M\,\cite{cte} parton densities are used for
the proton. The calculations have been corrected for hadronization
effects which lower the prediction by approximately 10\%.

The theoretical predictions describe the data for $\mu^{2}_r=Q^2$
with scale uncertainties up to 30\%, if $\mu^{2}_r$ is varied by
a factor of two. If $\mu^{2}_r=Q^2+E^{2}_t$
(where $E_{t}=(E_{t,jet1}+E_{t,jet2})/2$ ) is chosen, the NLO QCD
calculations describe the data for large $x_B$ and $Q^2$ only and 
clearly lie below the data for low $x_B$ and low $Q^2$. In this case, however,
the scale uncertainties are considerably smaller. The calculations leave room
for introducing additional contributions to jet production,
for example processes where the virtual photon becomes resolved and
acts as a source of partons.


\subsection{Contributions from a virtual photon structure?}

In order to investigate a possible resolved contribution of the photon
to dijet production, the data are compared to NLO QCD calculations  using
the program JETVIP\,\cite{bp}. For these studies the renormalization and
factorization scale are set to $\mu^{2}_r=\mu^{2}_f=Q^2+E^{2}_t$
and the SAS-1D\cite{sas} parton densities for the virtual photon are used.
As shown in figure \ref{fig:r2jv} the full calculation  
including all resolved and direct contributions 
leads to a better agreement with the data than a JETVIP
calculation which only considers a point-like photon. 


\subsection{Dependence of $R_2$ on the transverse energy of the jets}

%%%To study the dijet rate as a function of the
%%%transverse energy of the jets the cut on $E_{t,jet1}>(5+\Delta) {\rm \,GeV}$ is
%%%varied by introducing the parameter $\Delta\in[0,7~{\rm\,GeV}]$ which
%%%can be interpreted as the threshold difference between the transverse
%%%energy of the hardest jet and the minimum value of the transverse
%%%energy as required in the jet selection.
To study the dependence of the dijet rate on
the transverse energy of the jets, $R_2$ is measured as a function of the
parameter $\Delta$ where the highest $E_{t,jet}$  satisfies $E_{t,jet1} > 
(5+\Delta )$~GeV . Note that as $\Delta \rightarrow 0$ the cuts on the
jets become symmetric and the NLO calculations become infrared sensitive.
The measured $R_2$ as a function of $\Delta$ is presented in figure
\ref{fig:r2del} for two bins in $Q^2$ and $x_B$. 


At large $x_B$ and large $Q^2$ and
$\Delta\geq1$~GeV, when staying away from the infrared sensitive
region of the calculation, the NLO QCD calculations using DISENT 
describe the data well for both choices of $\mu^{2}_{r}$ .  
In the low $x_B$,$Q^2$ regime the predictions only
describe the data when choosing $\mu^{2}_{r}=Q^2$. For
$\mu^{2}_{r}=Q^2+E^{2}_{t}$ a reasonable agreement between the
prediction and the measurement is only observed for $\Delta=7$~GeV
which is equivalent to the requirement of large transverse energies of
the jets.  Finally, the NLO calculations show a
turnover for $\Delta$ below about $1$~GeV whereas the data
continue to increase towards $\Delta=0$ showing the need for
resummed calculations.

The two bins shown in figure~\ref{fig:r2del} are part of a detailed scan of the
phase space which is presented in figure \ref{fig:r2com}. It is evident
that for fixed $Q^2$ and decreasing $x_B$ the NLO description fails
to describe the data for $\mu^{2}_{r}=Q^2+E^{2}_{t}$ and that
$\mu^{2}_{r}=Q^2$ is favored by the data.

\section{Summary}

A measurement of the dijet rate $R_2$ in the  phase space region
$10^{-4}<x_B<10^{-2}$ and ${\rm} 5<Q^{2}<100 {\rm \,GeV}^{2}$
has been presented. The dijet rate is described by NLO calculations if $\mu^{2}_{r}=Q^2$
is chosen as the renormalization scale albeit at the cost of large scale
uncertainties. If, however, $\mu^{2}_{r}=Q^2+E^{2}_{t}$
is chosen --which considerable reduces the scale uncertainties--
a substantial contribution from other sources of dijet
production is needed. When studying the
contribution from resolved virtual photons it was
found that its inclusion helps to describe the data. 

The dijet rate has also been studied as a function of the transverse energy of the
jets. As before when choosing $\mu^{2}_{r}=Q^2+E^{2}_{t}$ as
renormalization scale the data at low  $Q^2$ and $x_B$ are only described 
for large transverse energies of the jets.
For large values of $Q^2$ and $x_B$, however, the theory successfully matches the
data and the choice of the scale variable is less important.
The measurement is extended to a region of phase space where
resummation is required in the calculations and provides an important 
reference for improved theoretical predictions.

%%%\section*{Acknowledgments}
%%%We are grateful to the HERA machine group whose outstanding efforts
%%%have made and continue to make this experiment possible.  We thank the
%%%engineers and technicians for their work in constructing and now
%%%maintaining the H1 detector, our funding agencies for financial
%%%support, the DESY technical staff for continual assistance, and the
%%%DESY directorate for the hospitality which they extend to the non-DESY
%%%members of the collaboration.

\begin{thebibliography}{99}
\bibitem{h1} H1 collaboration \Journal{\NIM}{386}{310}{1996}.

\bibitem{spa} H1 SpaCal Group \Journal{\NIM}{374}{149}{1996}.
 
\bibitem{es}S.D. Ellis and D.E Soper, \Journal{\PRD}{48}{3160}{1993}.

\bibitem{scs} G.A. Schuler and H. Spiesberger, {\em Proceedings of the
workshop on Physics at HERA}, eds. W. Buchm\"uller and G. Ingelmann,
Hamburg (1992) 1419

\bibitem{ll}L. L\"onnblad, \Journal{\em Comp. Phys. Comm.}{71}{15}{1992}.

\bibitem{hj}H. Jung, \Journal{\em Comp. Phys. Comm.}{86}{147}{1995}.

\bibitem{rp}R. P\"oschl, {\em PhD thesis}, in preparation.

\bibitem{cs}S. Catani and M.H. Seymour \Journal{\NPB}{485}{291}{1997}.

\bibitem{kk}M. Klasen and G. Kramer \Journal{\PLB}{366}{385}{1996}.

\bibitem{cte} H.L. Lai et al. , \Journal{\PRD}{55}{1280}{1997}.

\bibitem{fr}S. Frixione and G. Ridolfi \Journal{\NPB}{507}{315}{1997}.

\bibitem{bp}B. P\"otter, \Journal{\em Comp. Phys. Comm.}{119}{45}{1999}.

\bibitem{sas}G.A. Schuler and T. Sj\"ostrand, \Journal{\ZPC}{68}{607}{1995}

\end{thebibliography}
\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[hb]
\begin{center}
%%\rule{5cm}{0.2mm}\hfill\rule{5cm}{0.2mm}
%%%\vskip 2.5cm
%%%\rule{5cm}{0.2mm}\hfill\rule{5cm}{0.2mm}
\psfig{figure=H1prelim-00-031.fig1.eps,height=20cm}
\caption{Dijet rate $R_2$ as a function of $x_B$ in different bins of
$Q^2$. The inner error bars of the data points indicate
the statistical error only, the total error bars include also the
statistical error added in quadrature. 
The data are compared to NLO QCD calculations which have been
corrected for hadronization effects. The full bands indicate the
uncertainty from the hadronization corrections and the hatched band
indicate the error from the hadronization corrections and from the
renormalization scale uncertainty added in quadrature.  \label{fig:r2nlodir} }
\end{center}
\end{figure}


\begin{figure}
\begin{center}
%%%\rule{5cm}{0.2mm}\hfill\rule{5cm}{0.2mm}
%%%\vskip 2.5cm
%%%\rule{5cm}{0.2mm}\hfill\rule{5cm}{0.2mm}
\psfig{figure=H1prelim-00-031.fig2.eps,width=15cm}
\caption{Dijet rate $R_2$ as a function of $x_B$ in two different bins
of $Q^2$. The data are compared to NLO QCD predictions assuming a direct
coupling of the virtual photon and for predictions which include a
virtual photon structure. The NLO QCD predictions have been corrected
for hadronization effects. \label{fig:r2jv} }
\end{center}
\end{figure}

\begin{figure}
%%\rule{5cm}{0.2mm}\hfill\rule{5cm}{0.2mm}
%%%\vskip 2.5cm
%%%\rule{5cm}{0.2mm}\hfill\rule{5cm}{0.2mm}
\begin{center}
%%%\psfig{figure=stamp.dist.eps,width=15cm}
\psfig{figure=H1prelim-00-031.fig3.eps,height=8.2cm}
\caption{Dijet rate $R_2$ as a function of $\Delta$ in two
extreme regions of $Q^2$ and $x_B$ for this analysis. The variable $\Delta$ 
is defined by requiring $E_{t,jet}>(5+\Delta)$~GeV for the jet with the
largest transverse energy. The data are compared to NLO predictions
for two choices of the renormalization scale $\mu^{2}_{r}$. The NLO
predictions have been corrected for hadronization effects. The
innermost band indicates the the error from the hadronization
corrections and the full error band includes the scale uncertainty
added in quadrature. \label{fig:r2del} }
\end{center}
\end{figure}


\begin{figure}
%%\rule{5cm}{0.2mm}\hfill\rule{5cm}{0.2mm}
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\begin{center}
%%%\psfig{figure=stamp.test.eps,width=18cm}
\psfig{figure=H1prelim-00-031.fig4.eps,height=21cm}
\caption{Dijet rate $R_2$ as a function of $\Delta$ in various 
regions of $Q^2$ and $x_B$. The two bins shown in
figure~\ref{fig:r2del} are included in this figure. The
innermost band indicates the the error from the hadronization
corrections and the full error band includes the scale uncertainty
added in quadrature. \label{fig:r2com} }
\end{center}
\end{figure}




\end{document}

