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%       Measurements in Photoproduction}              %
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\noindent
Submitted to the 30th International Conference on High-Energy Physics 
ICHEP2000, \\
Osaka, Japan, July 2000

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%\begin{flushleft}
%%  DESY-??-??? \hfill ISSN ????????\\
%  \noindent
%Date:          \today
%Version:  OSAKA write-up        \\
%Editors:   P. Bate, S. Caron, A. Mehta \\
%Referees:  S.  Schmitt,  S. Maxfield   \\
%%Comments by Thursday 24/6/99  12:00

%\end{flushleft}

\vspace{2cm}

\begin{center}
\begin{Large}
  
  {\bf Measurements of 2-Jet and 3-Jet Cross Sections at High Transverse Momentum in Photoproduction}

  
  \vspace{1cm}
  
  H1 Collaboration

\end{Large}
\end{center}

\vspace{1cm}

\begin{abstract}
\noindent
Single differential 2-jet and 3-jet cross sections as a function of
jet transverse momentum and invariant mass are measured in
photoproduction using the H1 detector at HERA. The data sample
comprises all available $e^+p$ data collected from 1994-97. Jets
are found using the longitudinal boost-invariant \kt algorithm run in
the laboratory frame with a minimum transverse momentum of the highest
transverse momentum jet of greater than $25~\gev$. When the cross
sections for two and three jet production are compared with QCD
predictions, no significant deviations from expectation are found.
\end{abstract}

\vfill

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\noindent
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{\bf Abstract:          977/978 }  \\
{\bf Parallel session:   3,11  } \\
{\bf Plenary talk:       11,7b }
%\\[0.2cm] 
%{\bf Electronic Access:} & 
%                 http://www-h1.desy.de/publications/H1\_sci\_results.shtml \\
%\end{tabular}
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%============================The Paper=======================
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\section{Introduction and Motivation}
%
At HERA high transverse momentum jets are preferentially produced in
photoproduction where the photon virtuality, $\qsq \approx 0~\gevsq$.
In this paper single differential photoproduction 2-jet and 3-jet $ep$
cross sections in jet transverse momentum \ptj up to $\sim 75\gev$ and
invariant mass up to $\sim 180 \gev$ are presented. These jet cross
sections are sensitive primarily to the dynamics of the hard partonic
scatter, allowing the validity of Quantum Chromodynamics (QCD) to be
tested up to the highest transverse momenta and masses available at
HERA. The understanding of QCD jet production is essential if the
sensitivity to hadronic decays of massive particles is to be
exploited.

A data sample requiring at least two hard jets of which one must have
transverse momentum greater than $25~\gev$ is defined.  A subsample,
requiring a third hard jet, is then constructed. From these samples
differential 2-jet and 3-jet $ep$ cross sections in jet transverse
momentum and jet invariant mass are evaluated.

The paper is organized as follows. In section \ref{det} the H1
detector is presented while Section \ref{models} reviews the
theoretical models. The jet algorithm used and the phase space
considered in the analysis are defined in section \ref{jets}.  Section
\ref{analysis} details the analysis method. A discussion of the cross
section measurements is presented in section \ref{results}. The final
section provides a summary of the results and their interpretation. A
full description of the analysis may be found in \cite{pb_thesis}.

\section{The H1 Detector}
\label{det}
%
The H1 detector is described in detail in~\cite{h1det}. Only those
components relevant to the analysis are described here. The Liquid
Argon (LAC )~\cite{Lar} and SpaCal~\cite{SpaCal} calorimeters are used
to trigger events, reconstruct the hadronic energy of the final state
and anti-tag positrons. The central tracking detector (CTD) is used to
reconstruct the interaction vertex and to supplement the measurement
of hadronic energy flow. A coordinate system in which the nominal
interaction point is at the origin and the incident proton beam
defines the $+z$ direction is used.


The LAC covers the polar angle range $4^\circ < \theta <
154^\circ$ with full azimuthal acceptance.  It consists of an
electromagnetic section and a hadronic section. The total depth varies
between 4.5 and 8 hadronic interaction lengths. The angular region
\mbox{$153^\circ < \theta < 177.8^\circ$} is covered by the SpaCal, a
lead/scintillating-fibre calorimeter. It also contains an
electromagnetic and a hadronic section. The hadronic energy scale of
the LAC is known to $4\%$ and of the SpaCal to $7\%$.

The CTD consists of two concentric cylindrical drift chambers, coaxial
with the beam-line, with a polar angle coverage of $15^\circ <
\theta < 165^\circ$. The entire CTD is immersed in a 1.15 T magnetic
field. The luminosity determination is based on measurement of the $ep
\rightarrow ep \gamma$ Bethe-Heitler process, where the positron and
photon are detected in crystal Cherenkov calorimeters located
downstream of the interaction point.

\section{Theoretical Models and Simulations}
\label{models}

In this paper QCD-inspired Monte Carlo event generators are used to
unfold the data and, along with next to leading order (NLO)  QCD
calculations, are compared to the corrected data cross sections. A
brief description of the generators and calculations used for these
purposes are described below.

To simulate the direct and resolved QCD photoproduction of jets the
PYTHIA 5.7~\cite{pythia} and HERWIG 5.9~\cite{herwig} event generators
are used. Both programs contain the Born level QCD hard scattering
matrix elements, regulated by a minimum cutoff, \ptmin. Leading
logarithmic parton showers simulate higher order QCD radiation.
GRV-LO~\cite{Gluck:1992jc} parton distribution functions (PDFs) for
the proton and photon are applied. The Lund String model is used by
PYTHIA to hadronize the outgoing partons, while HERWIG uses the
Cluster Hadronization approach. Multiple interactions between the
proton and resolved photon are dealt with in PYTHIA by extending the
perturbative parton-parton scattering below \pthat, the transverse
momentum of the main partonic scatter, to a new limit, \ptmia. In
HERWIG (in order to model the multiple interactions), 20$\%$ of events
were simulated with the proton and resolved photon remnants
interacting, producing soft particles. Table \ref{param_table}
summarizes the major parameters used in these two event generators.
The EPVEC~\cite{epvec} program is used to model the contribution to
the cross sections arising from real $W$ production. A full detector
simulation~\cite{geant} is applied to all Monte Carlo events.

\begin{table}
\input{param_table.tex}
\caption{Description of event generators used to simulate
photoproduction processes.}
\label{param_table} 
\end{table}

The JetViP \cite{jetvip} program, in which hard scattering matrix
elements up to \Oaaa are implemented, is also used to compare QCD
predictions to the data. The VEGAS~\cite{vegas} package is used to
perform the integration over the allowed phase space with five
iterations to ensure stability.  Divergences in the real and virtual
corrections are handled by JetViP using the phase-space slicing
approach. In calculating NLO 2-jet cross sections, 2-loop $\alpha_s$
with 5 flavours is used in the \msbar scheme and
GRV-HO~\cite{Gluck:1992jc} PDFs chosen for the proton and photon,
whilst for the (LO) 3-jet cross sections, 1-loop $\alpha_s$ with 5
flavours and GRV-LO PDFs were used. In all cases, \lqcd is taken from
the proton PDF.

For estimating the Neutral Current Deep Inelastic Scattering (NC DIS)
background, events with \qsq $>$ 4 \gevsq were generated using RAPGAP
2.3~\cite{rapgap} with CTEQ4M~\cite{cteq4m} parameterizations for the
proton (a structureless photon is assumed) and parton showering.
%
\section{Jet Selection}
\label{jets}
The longitudinally boost-invariant \kt algorithm as proposed in
\cite{invkt} is used in this analysis to define jets. This algorithm
avoids the ambiguities that occur in cone-type
algorithms~\cite{seyphen}. The algorithm is implemented using the
KTCLUS~\cite{ktclus} package and run in the inclusive mode. Details of
the implementation of this algorithm can be found in \cite{subjets}.
The $\pt$-weighted scheme has been used throughout this analysis:

\begin{equation}
\etj = \sum_{i} E_{T, i} , \hskip8mm \etaj = \frac{\sum_i E_{T,i} \;
\eta_i}{\sum_{i} E_{T,i}} , \hskip8mm \phij = \frac{\sum_i E_{T,i} \;
\phi_i}{\sum_{i} E_{T,i}} ,
\label{recomb}
\end{equation}

where the index, $i$ runs over all final state objects. Since the jets
are defined as massless, the rapidity and pseudorapidity of the jets are
equivalent, as are the transverse momentum, \ptj and transverse
energy, \etj.

Cuts on jet transverse momentum are made such that $\ptji>25$~GeV,
$\ptjii>15$~GeV and for the 3-jet sample, $\ptjiii >10$~GeV.  The
pseudorapidity of each jet, \etaj is restricted to $-0.5 < \etaj <
2.5$. All jets are then well contained in the LAC. Asymmetric cuts on
the \ptj of the top two jets are made to avoid NLO uncertainties due
to infrared-sensitive regions of phase space.  The measured kinematic
region is restricted to $y < 0.9$ and $\qsq < 4 \gevsq $.
%
\section{Cross Section Measurement Procedure}
\label{analysis}

\subsection{Event Selection}

The data sample was collected with the H1 detector from 1994-97, when
820 \gev protons collided with 27.6 \gev positrons, resulting in a
centre-of-mass energy of 300~GeV. The integrated luminosity obtained
is 35.7~$\rm pb ^{-1}$. Energy deposits in the calorimeters and tracks
in the CTD were combined to reconstruct the hadronic energy of
events~\cite{mx_paper}. The $z$ coordinate of the event vertex,
reconstructed from hit pattens in the CTD, is limited to -36 $<$
$z_{vertex}$ $<$ 34 cm. A cut on the missing transverse momentum
\ptm$<$ 20 \gev is imposed to remove Charged Current  and non-$ep$
events.  Jet selections as described in section~\ref{jets} are
applied. The measured jet energies are calibrated to their true
energies using a large sample of single jet NC DIS events. In these
events the jet momentum may be precisely estimated from kinematic
constrains~\cite{pb_thesis}.
 
The only significant background in the di-jet sample arises from NC
DIS events. These are suppressed by removing events with a electron
identified in the LAC or SpaCal with energy above 5 GeV and by
requiring $y<0.9$, with $y$ reconstructed using hadronic
variables~\cite{yJB}.  Further cuts detailed in \cite{pb_thesis}
reduce the contamination from NC DIS interactions to less than $4\%$
in all bins.

Kinematic and jet distributions are well described by the Monte Carlo
predictions.
%
\subsection{Correction of the Data for Detector Effects} 
%
The data cross sections are obtained from the measured event rates by
correcting for acceptance losses and inefficiencies bin-by-bin, using
the cross-section weighted sum of events generated by PYTHIA and
EPVEC. The background from NC DIS processes is estimated using RAPGAP
and is subtracted on a statistical basis. The bin purity, defined as
the number of events generated and reconstructed in a bin divided by
the number reconstructed in the bin, is greater than $35\%$ for
each bin. The ratio of bin contents before and after detector
simulation is close to one. Bin centre corrections, radiative
corrections and hadronization corrections are not applied.
\subsection{Systematic Uncertainties}
The systematic error on the measurement arises from the following
sources:
\begin{itemize}
\item A $4$ \% uncertainty on the jet energy scale including a $2$ \%
  uncertainty that is correlated bin to bin. The jet energy scale
  uncertainty results in an error of $15$--$20$ \% on the cross sections.
\item The uncertainty in the acceptance correction was estimated by
  taking the difference between the corrections obtained from HERWIG
  and PYTHIA. The resulting error on the cross sections is $\sim
  10$\%.
\item The uncertainty in the trigger efficiency results in a error of
  $\sim 4$\% on the cross section.
\item The uncertainty on the background from DIS events is evaluated by
  varying the Monte Carlo estimate by $\pm 100\%$. The resulting error
  on the cross sections is $\sim 1\%$.
\item The uncertainty in the integrated luminosity results in an
  overall normalisation error of $1.5$\% on the cross section.
\end{itemize}  

All systematic errors are added in quadrature.

\section{Results}
\label{results}
%


Cross sections for the 2-jet sample as functions of the transverse
momentum of the leading jet and the average of the two leading jets 
are shown in
figure \ref{fig:2jets}. Cross sections for the 3-jet sample as
functions of the transverse momentum of the third highest jet, the
invariant mass of the two leading jets
$M^{12}$ and of all three jets $M^{123}$ are shown in figure
\ref{fig:3jets}. In the measured ranges the cross
sections fall by three orders of magnitude. The total uncertainty,
evaluated by adding systematic and statistical errors in quadrature,
is typically $15\%$ on the 2-jet cross sections and $20\%$ on the
3-jet cross sections.



The shape of the data are well described by the implementations of
PYTHIA 5.7 and HERWIG 5.9. 
%Both models underestimate the total cross
%section, with scale factors of 1.13 (2.1) and 1.19 (4.3) being applied
%to PYTHIA (HERWIG) for the 2-jet and 3-jet cross sections respectively
%in figures~\ref{fig:2jets} and \ref{fig:3jets}.
Hadronization corrections are studied with PYTHIA and HERWIG by
comparing cross sections before and after hadronization of the
outgoing partons (after parton showering). The correction is $\sim 10
\%$ or lower in all bins and is not applied to the data, nor to any of
the QCD models. The cross sections are also compare to QCD
calculations as implemented in the JetViP program. The calculations
agree well in shape and normalisation with the data. 

The kinematic configuration of massive particle decay would correspond
in the 3-jet sample to the case where all three jets have large
transverse momenta and are well separated from each other such that
invariant masses of jet pairs are large.  While no significant
deviation is seen in figure~\ref{fig:3jets} it is noted that the QCD
prediction yields a considerable number of events also at large $\ptj$
and invariant mass in the inclusive distributions presented here. In
the highest bin of the invariant mass of the top three jets, the EPVEC
generation of $W$ production ($ep \rightarrow eq W$, $W \rightarrow q
q$) accounts for $7\%$ of the cross section prediction.

%
\section{Conclusions}

Photoproduction of jets with high transverse momentum is investigated.
Cross sections are measured for two or more jets and three or more
jets as a function of transverse momentum up to $75\gev$ and as a
function of invariant mass up to $180\gev$. The measured cross sections
are compatible within
errors with QCD predictions for photoproduction processes based on
Monte Carlo event generator models and NLO QCD calculations.
%The invariant mass cross sections are particularly
%interesting in view of their sensitivity to massive particle decay.
%
\section*{Acknowledgements}
%
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.  We wish to thank B. P\"{o}tter and M.
Seymour for many helpful discussions.

% the bibliography
\begin{thebibliography}{99}

\bibitem{pb_thesis}
Paul Bate; PhD thesis, Univ. of Manchester, England : ``High Transverse Momentum 2-Jet and 3-Jet Cross Section Measurements in Photoproduction''.

\bibitem{h1det} H1 Collaboration, I.\ Abt et al., Nucl. Instr. and
  Meth.  {\bf A386} (1997) 310 and 348.

\bibitem{Lar} H1 Calorimeter Group, B.\ Andrieu et al., Nucl. Instr.
  and Meth.  {\bf A336} (1993) 460.

\bibitem{SpaCal} H1 SpaCal Group, R.\ D.\ Appuhn et al., Nucl. Instr.
  and Meth.  {\bf A386} (1997) 397.

\bibitem{pythia} T. Sjostrand, CERN-TH-6488 (1992), Comput. Phys.
  Commun. {\bf 82} (1994) 74.

%
%\bibitem{Adloff:1998aw} H1 Collab., C.~Adloff et al.;
%``Observation of events with an isolated high energy lepton and missing  transverse momentum at HERA,''
%Eur.\ Phys.\ J.\ {\bf C5}, 575 (1998).%, hep-ex/9806009.
%%CITATION = EPHJA,C5,575;%%

  
\bibitem{herwig} G.\ Marchesini et al., Comp. Phys. Comm. {\bf 67}
  (1992) 465.
  
\bibitem{Gluck:1992jc} M.~Gl\"uck, E.~Reya and A.~Vogt, Phys. Rev.
  {\bf D46} (1992) 1973.


\bibitem{epvec}
U.~Baur,~J.A.M.~Vermaseren and D.~Zeppenfeld, 
%\Journal{\NPB}{375}{1992}{3}.
Nucl.\ Phys.\ {\bf B375} (1992) 3.

\bibitem{geant}
R. Brun et al., GEANT3 User's Guide,
CERN-DD/EE-84-1 (1987). 


\bibitem{jetvip} B.~Potter,
%``JetViP 1.1: Calculating one- and two-jet cross sections with virtual  photons
%  in NLO {QCD},''
Comput.\ Phys.\ Commun.\ {\bf 119} (1999) 45.%, hep-ph/9806437.
%
\bibitem{vegas} G.P. Lepage, J. Comp. Phys. {\bf 27} (1978) 192.

%
%\bibitem{tbeam} H1 Calorimeter Group, B.\ Andrieu et al., Nucl. Instr.
%  and Meth.
%  {\bf A350} (1994) 57; \\
%  H1 Calorimeter Group, B.\ Andrieu et al., Nucl. Instr. and Meth.
%  {\bf A336} (1993) 499.


\bibitem{rapgap} H.\ Jung, Comp. Phys. Comm. {\bf 86} (1995) 147.
  

\bibitem{cteq4m} H. L. Lai et al., Phys. Rev. {\bf D55} (1997), 1280.


\bibitem{invkt} {S.D. Ellis and D.E. Soper}{,} \newblock Phys. Rev.
  {\bf D48} (1993) 3160.

\bibitem{seyphen} {M.H. Seymour}{,} \newblock in: Les Rencontres de la
  Vall\'ee d'Aoste, La Thuile, France (1997) and Nucl. Phys. {\bf
    B513} (1998) 269.

\bibitem{ktclus} {S. Catani, Yu.L. Dokshitzer, M.H. Seymour and B.R.
    Webber}{,} \newblock Nucl. Phys. {\bf B406} (1993) 187.
  
  
\bibitem{subjets} H1 Collab., C.~Adloff et al.,
%``Measurement of internal jet structure in dijet production in  deep-inelastic scattering at HERA,''
  Nucl.\ Phys.\ {\bf B545} (1999) 3.
%, hep-ex/9901010.

%\bibitem{django} G. A. Schuler and H. Spiesberger, Proceedings of the
%  Workshop ``Physics at HERA'', vol. 3, eds. W. Buchm\"uller, G.
%  Ingelman, DESY (1992) 1419.
  
%\bibitem{ariadne} L. L\"onnblad, Comp. Phys. Comm. {\bf 71} (1992) 15.
  
%\bibitem{heracles} A. Kwiatkowski, H. Spiesberger and H.-J. M\"ohring,
%  Comp. Phys. Comm. {\bf 69} (1992) 155.

\bibitem{mx_paper} H1 Collab., C. Adloff et al., Z. Phys. {\bf C74}
  (1997) 221.



\bibitem{yJB} A. Blondel and F. Jacquet, Proceedings of the Study of
  an $ep$
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\bibitem{pozo} {L.A. del Pozo}{,} \newblock PhD Thesis University of
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%\bibitem{seyjets} M.H.~Seymour, presented at 10th Topical Workshop on
%  Proton-Antiproton Collider Physics, Batavia, IL, 9-13 May 1995.
%, hep-ph/9506421. 
%``Jets in QCD,''

\end{thebibliography}
\newpage
%
%
%
%
%
% Figures and Tables

%%   \begin{figure}[p] \unitlength 1cm \begin{center}
%%      \begin{picture}(16.0,9.0)(-3,0.)  \resizebox{!}{9cm}
%%      {\includegraphics*[1pt,1pt][536pt,502pt]{./epsfiles/etaprof.eps}}
%%      \end{picture} \caption{{ The pseudorapidity jet profile,
%%          split into regions of \ptj and \etajns. The vertical axis
%%          shows the summed scalar transverse momentum, $\Delta P_T$ in
%%          each bin
%%      divided by the summed scalar transverse momentum, $\Sigma P_T$ of all hadronic final state objects.}} \label{etaprof} %\end{center}
%\end{center}

%\begin{center}
%   \begin{figure}[htb] \unitlength 1cm \begin{center}
%%      \begin{picture}(16.0,10.0)(-3.,0.)  \resizebox{!}{9cm}
%%      {\includegraphics*[1pt,1pt][536pt,502pt]{./epsfiles/phiprof.eps}}
%%      \end{picture} \caption{{
%%          The azimuthal angle jet profile, split into regions of \ptj
%%          and \etajns. The vertical axis shows the summed scalar
%%          transverse momentum, $\Delta P_T$ in each bin divided by the
%%          summed scalar transverse momentum, $\Sigma P_T$ of all
%%          hadronic final state objects.}}
%%      \label{phiprof}
%%    \end{center}
%%      \end{figure}
%\end{center}
%
\begin{figure}[p]
\begin{center}
%  \center \epsfig{file=epsfiles/anal_thesis_comb_ptl.eps,width=7.5cm}
%  \epsfig{file=epsfiles/anal_thesis_comb_ptav.eps,width=7.5cm}
%  \epsfig{file=epsfiles/anal_thesis_comb_m12_2jet.eps,width=7.5cm}
\center  \epsfig{file=./H1prelim-00-052.fig1.eps,width=18cm}
\caption{Hadronic cross sections for 2-jet production as function of
  the leading jet transverse momentum, $P_{T}^{Jet1}$ (left) and the average
  transverse momentum of the top two jets, $(P_{T}^{Jet1}+P_{T}^{Jet2})/2$ 
  (right) . The inner
  error bars denotes the statistical, the outer error bars the
  quadratic sum of all statistical and systematic errors of the data.
  The predictions given by the PYTHIA 5.7
  using GRV-LO structure functions for the proton and photon are
  shown, as are the NLO parton cross sections implemented with the
  JetViP numerical integration program using GRV-HO
  structure functions respectively.}
\label{fig:2jets} 
\end{center}
\end{figure}


%%\begin{figure}[t]
%%  \center \epsfig{file=epsfiles/anal_thesis_comb_ptl.eps,width=7.5cm}
%%  \epsfig{file=epsfiles/anal_thesis_comb_ptav.eps,width=7.5cm}
%%  \epsfig{file=epsfiles/anal_thesis_comb_m12_2jet.eps,width=7.5cm}
%% \center  \epsfig{file=epsfiles/anal_am_comb_2jet.eps,width=\textwidth}
%%\caption{Hadronic cross sections for 2-jet production as function of
%%  the leading jet transverse momentum, \ptji (top left), the average
%%  transverse momentum of the top two jets, \ptjav (top right) and the
%%  invariant mass of the two leading jets $M_{12}$ (bottom). The inner
%%  error bars denotes the statistical, the outer error bars the
%%  quadratic sum of all statistical and systematic errors of the data.
%%  The predictions given by the PYTHIA 5.7 and HERWIG 5.9 programs,
%%  using GRV LO structure functions for the proton and photon are
%% shown, as are the NLO parton cross sections implemented with the
%%  JetViP 1.1 numerical integration program using GRV LO and HO
%%  structure functions respectively. The HERWIG and PYTHIA cross
%%  sections have been scaled to the data cross section.}
%%\label{fig:2jets} 
%%\end{figure}

%\begin{figure}[p]
%  \center
%  \epsfig{file=epsfiles/anal_thesis_comb_pt3.eps,width=17cm}
%
%\caption{Hadronic cross sections for 3-jet production as function of
%  the jet transverse momentum of the third highest jet. The inner error bars denotes the statistical, the outer
%  error bars the quadratic sum of all statistical and systematic
%  errors of the data. The predictions given by the PYTHIA 5.7 and
%  HERWIG 5.9 programs, using GRV LO structure functions for the proton
%  and photon are shown, as is the LO parton cross sections implemented
%  with the JetViP 1.1 numerical integration program using GRV LO
%  structure functions. The HERWIG and PYTHIA cross sections have been
%  scaled to the data cross section.}
%\label{fig:3jet_masses_pt3} 
%\end{figure}

%%\begin{figure}[p]
%  \center
%  \epsfig{file=epsfiles/anal_thesis_comb_m12_3jet.eps,width=7.5cm}
%  \epsfig{file=epsfiles/anal_thesis_comb_m13.eps,width=7.5cm}
%  \epsfig{file=epsfiles/anal_thesis_comb_m23.eps,width=7.5cm}
%  \epsfig{file=epsfiles/anal_thesis_comb_m123.eps,width=7.5cm}
%  \vspace{-3cm} \center
%  \epsfig{file=epsfiles/anal_am_comb_3jet.eps,width=\textwidth}
%\caption{Hadronic cross sections for 3-jet production s function of
%  the jet transverse momentum of the third highest jet \ptjiii (top
%  left); as function of the invariant mass of the two leading jets
%  $M_{12}$ (top right), the first and third jet $M_{13}$ (middle
%  left), the second and third jet $M_{23}$ (middle right) and all
%  three jets $M_{123}$ (lower left). The inner error bars denotes the
%  statistical, the outer error bars the quadratic sum of all
%  statistical and systematic errors of the data. The predictions given
%  by the PYTHIA 5.7 and HERWIG 5.9 programs, using GRV LO structure
%  functions for the proton and photon are shown, as is the LO parton
%  cross sections implemented with the JetViP 1.1 numerical integration
%  program using GRV LO structure functions. The HERWIG and PYTHIA
%  cross sections have been scaled to the data cross section.}
%\label{fig:3jets} 
%\end{figure}

\begin{figure}[p]
%\vspace{-2cm}
\begin{center}
%  \center
%  \epsfig{file=epsfiles/anal_thesis_comb_m12_3jet.eps,width=7.5cm}
%  \epsfig{file=epsfiles/anal_thesis_comb_m13.eps,width=7.5cm}
%  \epsfig{file=epsfiles/anal_thesis_comb_m23.eps,width=7.5cm}
%  \epsfig{file=epsfiles/anal_thesis_comb_m123.eps,width=7.5cm}
  \vspace{-3cm} \center
  \epsfig{file=./H1prelim-00-052.fig2.eps,width=18cm}
\caption{Hadronic cross sections for 3-jet production as function of
  the jet transverse momentum of the third highest jet $P_{T}^{Jet3}$ (top
  left); as function of the invariant mass of the two leading jets
  $M^{12}$  (top right) and as function of all
  three jets $M^{123}$ (lower left). The inner error bars denotes the
  statistical, the outer error bars the quadratic sum of all
  statistical and systematic errors of the data. The predictions given
  by the PYTHIA 5.7, using GRV-LO structure
  functions for the proton and photon are shown, as is the LO parton
  cross sections implemented with the JetViP numerical integration
  program using GRV-LO structure functions.}
\label{fig:3jets} 
\end{center}
\end{figure}





%      \begin{picture}(15.,11) \put(-1.0,0){\psfig{
%      figure=./psfiles/ktconenewest.ps,height=15cm,bburx=548,bbury=637,bbllx=7,bblly=160,clip=}}
%      \put(1.6,10.){\bf\Large{{$a)$}}}
%      \put(9.9,10.){\bf\Large{{$b)$}}}
%      \put(1.6,2.6){\bf\Large{{$c)$}}}
%      \put(9.9,2.6){\bf\Large{{$d)$}}} \end{picture} \caption{{\it The
%      Hadron Level cross sections for a) the leading jet \pt, b) the
%      third jet \pt, c) the 2-jet invariant mass of the highest two \pt jets and
%      d) the 3-jet invariant mass of the highest three \pt jets. The
%      superscripts `$2-JET$' and `$3-JET$' are used for cross sections
%      where two and three jets are required respectively. Error bars
%      are not shown.}}  \label{cl_cone_fig}  \end{center} \end{figure}
%\end{center}


%
%
%
%\begin{table}
%\begin{center}
%\input{table_two.tex}
%\end{center}
%\caption{The triple-differential dijet cross-section 
%\triple for $0.1 < y < 0.7$ in ranges of $Q^2$, $\etbarsq$ and $\xgjets$. The cross-section in pb
%is given together with the statistical, positive systematic and negative
%systematic errors.
%The table shows measurements for 
%$Q^2\leq 8.0~{\rm GeV}^2$. The higher $Q^2$ region is given in
%table~\ref{Tab2b}.}
%\label{Tab2a}
%\end{table}
%

\end{document}

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