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\begin{document}

\pagestyle{empty}
\begin{titlepage}

\noindent
\begin{center}
%{\it {\large version of \today}} \\[.3em] 
\begin{small}
\begin{tabular}{llrr}
%Submitted to & \multicolumn{3}{r}{\footnotesize Electronic Access: {\it http://www-h1.desy.de/h1/www/publications/conf/conf\_list.html}} \\[.2em] \hline 
%Submitted to & \multicolumn{3}{r}{\footnotesize {\it www-h1.desy.de/h1/www/publications/conf/conf\_list.html}} \\[.2em] \hline 
Submitted to & & &
\epsfig{file=/h1/www/images/H1logo_bw_small.epsi
,width=2.cm} \\[.2em] \hline
\multicolumn{4}{l}{{\bf
                International Europhysics
                Conference on High Energy Physics},
                July~12,~2001,~Budapest} \\
 {\bf EPS 2001:} 
                 & Abstract:        & {\bf 798}    &\\
                 & Parallel Session & {\bf 1}   &\\
                 & Plenary Session  & {\bf Hard QCD, Structure Functions}   &\\[.7em]
\multicolumn{4}{l}{{\bf
               XX International Symposium on Lepton and Photon Interactions}, 
               July~23,~2001,~Rome} \\ 
{\bf LP 2001:}  
                 & Abstract:        & {\bf 491} &\\
                 & Plenary Session  & {\bf S7,S8}   &\\ \hline
 & \multicolumn{3}{r}{\footnotesize {\it
    www-h1.desy.de/h1/www/publications/conf/conf\_list.html}} \\[.2em]
\end{tabular}
\end{small}
\end{center}
\vspace*{2cm}

\begin{center}
  \Large
{\bf Measurement of Dijet Cross Sections in Photoproduction\\}
 \vspace*{1cm}
    {\Large H1 Collaboration} 
\end{center}


\begin{abstract}
\noindent
Dijet cross sections as functions of
various jet observables are measured in
photoproduction using the H1 detector at HERA. 
The data sample
comprises $e^+p$ data collected from 1995-97. 
Jets
are found using the inclusive $k_{\bot}$ algorithm 
with a minimum transverse momentum of the highest
transverse momentum jet of $25~\gev$. 
The results are compared to the predictions of next-to-leading order
perturbative QCD.

\end{abstract}

\end{titlepage}

\pagestyle{plain}

\section{Introduction}
\noindent
The production of high transverse momentum jets in $\gamma p$-reactions 
is, to leading order, described by the pointlike
interaction of real photons with the quarks (QCD compton effect)
and
the gluons (boson gluon fusion) of the proton. Higher order
effects can be approximated by the notion of photonic parton
densities which, analogously to proton parton densities, depend
on a factorization scale $\mu_\gamma$ and $x_\gamma$, the longitudinal momentum
fraction of the photon taken by the parton. The limiting case
of pointlike interactions is thus given by $x_\gamma=1$. At HERA
these photoproduction reactions can be investigated in inelastic electron
(positron) proton reactions at very small squared four-momentum transfers
$Q^2$. Starting from the first investigation of this sort at HERA
\cite{berger}
the comparison of the results with the predictions
of Quantum Chromo Dynamics (QCD) has been a central topic of
interest \cite{gPpaper}.

High transverse momentum jets naturally provide
a hard scale which makes perturbative QCD calculations reliable and renders
the results less sensitive to non-perturbative physics.
A measurement of jet cross sections at these high transverse momenta can
therefore be used to test the current predictions of next-to-leading
order (NLO) perturbative QCD -- including the parameterizations
of  photon and proton parton densities at
large scales -- with high precision.
The photonic quark densities have been extracted
up to $x_{\gamma}$ of $0.6$ at scales up  to $500 \gev^2$
investigating the photon structure function $F_{2,\gamma}$
in experiments at  $e^+ e^-$-colliders\cite{nisius}.
This analysis extends the $x_{\gamma}$ range up to $1$  at
scales between $600$ and $6000 \gev^2$, where  the
quark density parameterizations of the photon
are presently only partially  constrained by measurements.
The Photoproduction of jets  is, in contrast to the
$F_{2,\gamma}$-experiments
directly sensitive to the gluon density of the photon, which by now
is also only insufficiently known.
Photoproduction data can be used to constrain
the parton density functions in regions where only few 
measurements are available.
In addition, the data is sensitive to the parton densities of the proton at
$x_P$ values up to $0.6$. In this kinematical regime, the quark densities
are well known from deep-inelastic scattering data, whilst the gluon
density has uncertainties of the order 10 to 50 percent\cite{gluon}.

This paper
is based on an $e^+p$-data sample collected with the H1 detector
from 1995-1997 with an integrated luminosity of 34.9 pb $^{-1}$.
It  presents dijet cross
sections in a range of observables for a sample of jet events with
mean jet transverse energies $20 < E_{T} < 80 \gev$,
observed $x_{\gamma}$ values $0.1 < x_{\gamma} < 1$
and observed values of the fractional energies of the
partons out of the proton $0.05 < x_P < 0.6$.

\section{Jets in Photoproduction}

\subsection{Phenomenology}
The cross section for the photoproduction of hard jets in electron-proton
collisions can be calculated from the photon-proton scattering result
using the
well known factorization ansatz, where $y$ is the longitudinal
momentum fraction of
the incoming photon and $f_{ \gamma,e}$ is the Weizs\"acker-Williams photon flux\cite{wwa,budlev,frixww}:
\begin{equation}
  \sigma_{eP \rightarrow eX} =\int dy f_{ \gamma ,e}(y) \sigma_{\gamma P} (y)
\end{equation}
The hadronic photon-proton jet cross section can be calculated in
perturbative QCD according to the factorization theorem.
The cross section $\sigma_{\gamma P}$ is calculated  as the convolution
of the partonic cross sections
with the parton momentum distributions of the proton $f_{i/P}$ and the
photon $f_{j/ \gamma}$ .
It is usually divided into a sum of two components, the
pointlike part $\sigma_{\gamma P}^{\mbox{\tiny{pointlike}}}$, where the photon
directly interacts with a parton out of the proton and the resolved
part $\sigma_{\gamma P}^{\mbox{\tiny{resolved}}}$, where the photon first
splits into partons which then interact with a parton out of the proton.
This distinction is only unambiguously defined in leading order and depends
on the photon factorization scale $\mu_{\gamma}$. These components
can be written as:

\begin{eqnarray}
  \sigma_{\gamma P}^{resolved} & = &\sum_{j,i} \int dx_{ \gamma} f_{j/ \gamma} 
(x_{ \gamma},\mu_{\gamma}) dx_P f_{i/P}(x_P,\mu_{P}) \hat{\sigma}_{ij} (\hat{s},\mu_{\gamma},
\mu_P,\alpha_s(\mu_r),\mu_r)  \\
 \sigma_{\gamma P}^{pointlike} & = &\sum_{i} \int dx_P f_{i/P}(x_P,\mu_{P}) \hat{\sigma}_{i\gamma}
 (\hat{s},\mu_{\gamma},\mu_P,\alpha_s(\mu_r),\mu_r)  
\end{eqnarray}

The squared center-of-mass energy of the hard subprocess is $\hat{s}=x_p x_{\gamma} y s$,
where
$\sqrt{s}$ is the total center of mass energy in the $ep$-system, i.e. 300
GeV at HERA.
The proton factorization scale is $\mu_P$, the renormalization scale is
$\mu_r$.
The partonic cross sections $\hat{\sigma}$ can be expanded as a perturbative series
in powers of $\alpha_s$.
These jet cross sections have been calculated up to the next-to-leading
order in QCD\cite{frix,nlo}.

As can be seen from these equations the total cross section is
obtained by integrating over $y$, $x_p$ and $x_\gamma$. The partonic
cross sections $\hat{\sigma}_{ij}$ and $\hat{\sigma}_{i\gamma}$
contain an integration e.g. over $\cos \theta^*$ the scattering angle
in the center-of-mass system of the partonic two body reaction.
In order to avoid mass singularities in the partonic cross section
a minimum cut in $\theta^*$ or in the transverse energy of the
outgoing partons has to be applied.


Much more detailed information on the reaction dynamics can
be obtained by measuring differential cross sections in the kinematical
observables.
The two scaled longitudinal parton momenta $x_{\gamma}$ and $x_{P}$ can be calculated
from the jets produced in the hard subprocess, using:
\begin{eqnarray}
 x_{\gamma} & = & \frac{1}{2 E_e y} (E_{T,1} e^{- \eta_1} + E_{T,2} e^{- \eta_2}) \\
 x_{P} & = & \frac{1}{2 E_P} (E_{T,1} e^{ \eta_1} + E_{T,2} e^{ \eta_2})
\end{eqnarray}
In this formula $E_{T,1}$ and $E_{T,2}$ are the transverse momenta of the two
jets of the hard subprocess, $\eta_1$ and $\eta_2$ are their pseudorapidities and
$E_e$ and $E_P$ are the energies of the electron and proton beams.
The transverse momenta of the jets are equal for exclusive two jet production. 
The pseudorapidities of the jets give also access to $\theta^*$
via
\begin{equation}
\cos \theta^* = | \tanh ((\eta_1-\eta_2)/2) | \enspace .
\end{equation}

Events at high $y$ increase sensitivity to the low $x_{\gamma}$
region, these at low $y$ to high $x_{\gamma}$.
Similarly, high $E_T$ is correlated with high $x_{\gamma}$ and $x_P$.
These different kinematic regions in $y$ and $E_{T}$
are therefore sensitive to the photon and proton structure
at different $x_{\gamma}$ and $x_P$.

In principle one has to measure the dependence of the 
fourfold differential cross section on all
four variables. 
This however requires a careful study of the resolution
effects and a very large statistics.
In this analysis  therefore,
more inclusive quantities like
the distribution of the invariant mass of the two leading jets, $M_{JJ}$, 
and the transverse momentum distribution 
of the highest transverse momentum jet, $E_{T,max}$, are presented first.
The cross section differential in
the average value of the pseudorapidities
$\overline{\eta}=(\eta_1+\eta_2)/2$ is particularly sensitive to
QCD calculations. It is thus presented  for different
photon-proton center-of-mass energies ($y$ regions) and different scales
($E_{T,max}$ regions).

Single differential cross sections in  $x_{\gamma}$ and $x_{P}$,
are measured
in different scale regions (different $E_{T,max}$ regions)
and for different $x_{\gamma}$ or $x_P$
cut-off values.
The angle $\theta^*$ is sensitive to the kinematics of jet production and
the differential cross section  is therefore presented
for different
$x_{\gamma}$ regions for all $M_{JJ}$ and in addition with a cut-off
in $M_{JJ}$.
These angular distribution are also sensitive to the production
of $W$ or $Z^0$ bosons and would differ from QCD expectation
if significant $W$ or $Z^0$ production were occuring.

In this analysis, jets are always defined using the inclusive $k_{\bot}$ algorithm
as proposed in \cite{invkt,invkt2}.
This works with a definition of jets in which not
all particles are assigned to hard jets.
The algorithm is applied in the laboratory frame and a separation parameter
of 1 is used.
An $E_T$ weighted recombination scheme is used.
The use of this algorithm has become standard in jet analyses at HERA \cite{markus}.

\subsection{QCD predictions and Models}
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 cut-off. Leading
logarithmic parton showers simulate higher order QCD radiation.
GRV-LO~\cite{Gluck:1992jc} parton distribution functions (PDFs) for
the proton and photon are used. 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 adding 
soft processes within the same event.
These processes are calculated by extending the 
perturbative parton-parton scattering to a low $E_T$ cut-off. 
In HERWIG (in order to model the multiple interactions), 35$\%$ of events
were simulated to contain so called {\it soft underlying events}.
In order to obtain a realistic description of the data 
a full detector
simulation~\cite{geant} is applied to all Monte Carlo events.
\\
The measured cross sections are compared to perturbative QCD
calculations.
The LO and NLO dijet cross sections are calculated using a program based
on the subtraction method \cite{frix,frix2}
for the analytic cancellation of infrared singularities. In calculating
LO and NLO cross sections a 2-loop $\alpha_s$ is taken with 5 flavours.
$\Lambda_{QCD}$ is set to $0.226$ GeV, which is the value used
in the proton parton density functions.
CTEQ5M\cite{cteq5m} parton density functions were chosen for the proton.
MRST99\cite{mrst} parton density functions were used to test the 
dependence of the NLO cross sections on the proton PDF uncertainties.
For the photon we have chosen GRV-HO\cite{grv} as a main setting and the
parameterizations of GS-HO\cite{gsg} and AFG-HO\cite{afg} for comparisons.
The  renormalization scale $\mu_r$ and the factorization scales,
$\mu_p$ and $\mu_\gamma$ have been set to the sum of the transverse
momenta of the outgoing partons divided by two.
The QCD program allows the variation of this scale.
It was varied
from 0.5 to 2 times the default scale to estimate
the scale uncertainty of the NLO calculation.
This uncertainty is approximately $\pm 10$ to $20 \%$.

\section{Experimental Technique}
\subsection{H1 Detector}
%
The H1 detector is described in detail in~\cite{h1det}. Only those
components relevant to the analysis are briefly 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 consists of an
electromagnetic and a hadronic section. 

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.



\subsection{Event Selection}
The data sample was collected with the H1 detector from 1995-97, when
820 GeV protons collided with 27.6 GeV positrons, in HERA, resulting in a
centre-of-mass energy of 300~GeV. 
The obtained integrated luminosity
was 34.9~$\rm pb ^{-1}$.
The events were triggered by a standard set of 
electromagnetic energy and missing energy triggers.
The trigger efficiencies are above $94$ percent for the 
event sample described in this analysis.
Energy deposits in the calorimeters and tracks
in the CTD were combined to reconstruct the hadronic energy of
events~\cite{mx_paper}. It was required that an event vertex 
was reconstructed within 
$35$ cm of the nominal $z$ position. 

A cut on the missing transverse momentum
$E_{T,miss}<$ 20 GeV is imposed to remove Charged Current  and non-$ep$
events. A special cosmic finder is used to remove remaining cosmic 
events.
The measured jet energies are calibrated to their true
energies. The calibration of the jet energies is tested 
by the jet-electron balance with deep inelastic scattering 
events and the jet-jet balance for this photoproduction 
sample in different kinematic regions. 
The Monte Carlo description of the jet energy calibration was found
to be better than $3$ percent.
 
The most significant background in the data sample arises from 
neutral current deep-inelastic scattering events.
Deep-inelastic scattering dijet data and the ARIADNE \cite{ariadne}
Monte Carlo interfaced with DJANGO \cite{django} are used to estimate these backgrounds, which
are suppressed by removing events with an electron
identified in the LAC or SpaCal and by
requiring $y<0.9$, with $y$ reconstructed using hadronic
variables~\cite{yJB}. 
Further electron finding algorithms reduce this 
background to less than $1 \%$ for the total sample.
In the bins with the highest $y$ at low $\overline{\eta}$ this
background is estimated to be $5 \%$ at maximum and 
was subtracted on a statistical basis.

Asymmetric cuts on
the $E_T$ of the highest transverse momentum jets are needed in order 
to avoid NLO uncertainties due
to infrared-sensitive regions of phase space \cite{frix}.
The jet selection criteria require that the $E_T$ of the highest 
transverse momentum jet $E_{T,max}>25$~GeV,
and the transverse momentum of the second highest transverse momentum jet $E_{T,second}>15$~GeV.
%For the 3-jet sample a third jet is demanded to have $E_{T,third}>10$~GeV.
The
pseudorapidity of each jet, $\eta$ is restricted to $-0.5 < \eta_{Jet} <
2.5$. All jets are then well contained in the LAC. 
The measured kinematic
region is restricted to $0.1< y < 0.9$ and $Q^2 < 1 \gev^2$.

The total number of events found was then 4732.

\subsection{Correction of the Data for Detector effects}
The data are corrected for detector effects such as limited resolution
and ineffiencies. 
To determine these effects the HERWIG and PYTHIA Monte Carlo
samples are used. 
Both programs describe all tested jet distributions well.
The bin sizes of all distributions are chosen to give an
efficiency and purity above 30-50 percent.
The correction is done by using the bin-to-bin method.
This is possible due to the good 
detector resolution and the obtained bin purities.
The mean values of the two Monte Carlo generators are taken 
for the correction.
The correction functions of the two models are in good agreement
and differ on average by 5 $\%$ and at most by 20 $\%$.
All correction factors lie typically between
$0.9$ and $1.2$ and reach in some cases $1.5$.


\subsection{Systematic Uncertainties}

The following sources of systematic error were considered:
\begin{itemize}
\item An $4$\% uncertainty on the LAr energy scale
  which results in an uncertainty with a mean value of 
  $20$\% on the jet cross sections.
\item An $8$\% uncertainty of the hadronic Spacal energy scale 
  which results in an uncertainty of $1$ \% for all bins
  in the jet cross sections.
\item 
  Half the difference between HERWIG and PYTHIA
  is taken as the uncertainty of the detector correction.
  The resulting uncertainty on the jet 
  cross sections is in the range of $5$ to $10$\%.
\item The uncertainty in the trigger efficiency results in an error of
  $\sim 3$\% on the cross section.
\item The uncertainty on the background substraction results in
  an error
  on the cross sections of $\sim 2\%$.
\item The uncertainty in the integrated luminosity results in an
  overall normalization error of $1.5$\% on the cross section.
\end{itemize}  


All systematic errors are added in quadrature.
The resulting systematic uncertainty ranges from $20$ to $30$ percent 
and is larger than the statistical uncertainty.

The effect of multiple scattering is tested by comparing the
data to HERWIG with and without some fraction $P$ of soft underlying 
events.
If $P$ is $\sim 30$ to $35 \%$ the jet-profiles
are well described for all regions of $x_{\gamma}$.
The difference of HERWIG 
for the obtained cross sections with and without $35$
percent of soft underlying events 
is below 5-10 percent for $x_{\gamma}$ between 0.3-0.8 and
10-20 percent for $x_{\gamma} < 0.3$.
For $x_{\gamma}>0.8$ the difference is negligible.  
Neither for these effects nor for the underlying event 
energies corrections are applied.
 

\section{Results}

The measured cross sections for inclusive dijet production
in the reaction $e+p \rightarrow e+p+\mbox{jets}$ are presented 
as single differential cross sections for all distributions.
The data are always corrected for detector effects and presented for
the phase space region defined in section $3.2$.
The inner
error bars of the data points denote the statistical, 
the outer error bars the
quadratic sum of all statistical and systematic uncertainties.
All results are compared to next-to-leading order (NLO) QCD 
predictions.
The NLO QCD prediction is always 
obtained with our standard setting (described in section $2$)
if not quoted.
In addition the predictions of NLO QCD  
corrected for hadronization effects are shown. 
The hadronization effects are obtained with the 
PYTHIA and HERWIG Monte Carlos and the mean value of the 
two predictions is used for corrections. 
Here the difference between the two Monte Carlo models 
is generally very small and 
at maximum 10 percent in some bins of the measured cross sections.

In figure $1$ the dijet cross section is shown as a function of
the invariant mass $M_{JJ}$ of the dijet system.
The data is presented for $M_{JJ}$ values between $45$ and  
$180$ $\gev$. The measured cross section falls by more than
3 orders of magnitude. Hadronization corrections are less than 5
percent for all bins. 
NLO QCD describes the measured data for the
whole mass range, while the calculation using LO matrix
elements fails to describe the low 
$M_{JJ}$ region. This is partly due to the fact that the 
low $M_{JJ}$ region is populated by events which are influenced
by the asymmetric cuts on the jet transverse momenta. 
Events in which the second jet has an $E_T$ less than 25 $\gev$
contribute mainly in this region and are only present in
dijet calculations beyond the leading order.
These observations agree with the fact that the scale uncertainties 
of the QCD predictions are largest at low $M_{JJ}$ values.

The dijet cross section as a function of the transverse momentum
of the highest transverse momentum jet $E_{T,max}$
is shown in figure $2$.
The distribution demonstrates that the data is well
described by
NLO QCD up to the highest $E_{T,max}$ bin.
The cross sections are little affected by 
hadronization corrections which are around 5 
percent for all bins.

The differential cross section $d\sigma / d\overline{\eta}$
is displayed for
two different $y$ regions and two $E_{T,max}$ bins  
in figure $3$.
An overall agreement between data and NLO QCD is observed
taking into account the uncertainties of 
the calculations and data points.
At low $\overline{\eta}$, where direct interactions 
are dominant, the QCD predictions lie above the data, 
but hadronization corrections are largest in this region and 
these corrections lower the partonic predictions.
Nevertheless the predictions tend to lie slightly above 
the data at low $\overline{\eta}$.
At high $\overline{\eta}$, where resolved interactions are 
dominant and hadronization corrections are small, 
the NLO QCD predictions agree well with the
measured data taking into account the experimental and 
theoretical uncertainties.
 
The upper plots in figure $4$ show the dijet cross section
$d \sigma / d x_{\gamma}$ 
as a function of
$x_{\gamma}$
for two different $x_{P}$ regimes.
The data lie below the calculations only for 
$x_{\gamma}>0.85$. Again, hadronization
corrections are largest in this region and reduce this discrepancy.
Different proton PDF sets (MRST99 1-3,CTEQ5M) result in differences
of less than $5$ percent for the predicted cross section
for $x_P < 0.1$ and up to $10$ percent for $x_P > 0.1$. 
Therefore, the differences in the proton PDFs are
less than the scale uncertainties for $x_P < 0.1$ and
of the same order for $x_P > 0.1$.
These findings are corroborated by the lower part of figure $4$ where
the cross section $d\sigma / d x_P$ as a function of $x_P$ is shown
for two different $x_{\gamma}$ regions.
Even at the highest $x_P$ the measured cross section
agrees well with the QCD predictions.

Figure $5$ displays the dijet cross sections $d \sigma / d x_{\gamma}$
as a function of $x_{\gamma}$ for
two regions of $E_{T,max}$, representing
different factorization scales for the photon and proton PDFs.
The data is compared to NLO calculations
with three different parameterizations of the photon structure.
All these predictions are NLO QCD calculations corrected for
hadronization effects. 
The predictions vary only little with the photon PDFs used.
In contrast the NLO scale uncertainties produce a significant
effect as can be inferred from figure $6$.
For high values of $x_{\gamma}$ hadronization
correction have a sizeable influence.
Again for the direct dominated part at high $x_{\gamma}$ 
the data is slightly below 
all predictions.

A more sensitive comparison between data and theory
is obtained by plotting the relative 
difference as shown in figure $7$.
The symbol $\sigma$ stands for $d \sigma / d x_{\gamma}$.
In contrast to the other figures the error bars of the data 
only contain the uncorrelated systematic errors, the correlated
errors due to the uncertainty of the calometric energy are 
shown as a hatched band.
The figure clearly demonstrates that these uncertainties and
the NLO scale uncertainties are much more important than 
the differences induced by using different photon PDFs.
All NLO predictions shown in the figure include hadronization
corrections.

Finally, the dijet cross section $d \sigma / d \cos \theta^*$
is plotted in 
figure $8$ for $x_{\gamma}>0.8$ and $x_{\gamma}<0.8$ (upper two plots).
Again, the data are well described by NLO QCD for low $x_{\gamma}$, whereas 
at higher $x_{\gamma}$ the predictions overshoot the data for
small values of $\cos \theta^*$.
In addition these cross sections are shown with a cut on
the invariant mass $M_{JJ}$ of the dijet system (lower two plots).
Although this cut changes the shape of 
the distribution dramatically, the QCD calculations
reproduce this behavior nicely. 

 
\section{Conclusions}
New measurements of dijet events in photoproduction at high 
transverse momentum are presented for various
jet kinematic observables. The measurements cover invariant dijet masses 
up to $160 \gev$ and transverse momenta up to $75 \gev$.
In this kinematic domain non-perturbative effects like 
multiple scattering and hadronization are found to be
small, which allows a direct comparison 
to QCD calculations to be made. 
The results again demonstrate the power of perturbative
QCD in predicting the measured cross sections
in a wide kinematical range.
Even though the photon PDFs have been obtained
from measurements at lower scales,
their QCD evolution correctly reproduces the 
data at high scales.
A further improvement in testing the theory
can only be reached by reducing the 
theoretical scale uncertainties and the systematic 
uncertainties of the data.

                                                
                              


 
\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 
S. Frixione and B. P\"{o}tter for many helpful discussions.
 





 



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\begin{figure}[hhh]
\center
\epsfig{file=H1prelim-01-031.fig1.ps,width=0.5\textwidth}
\caption{Differential $eP$ cross sections for dijet production as a function of
  the invariant dijet mass $M_{JJ}$ of the two highest $E_T$ jets. The inner
  error bars denote the statistical, the outer error bars the
  quadratic sum of all statistical and systematic errors of the data.
  The LO predictions 
  using CTEQ5M structure functions for the proton and
  GRV-HO structure functions for the photon are
  shown as a dotted line.
  NLO predictions with the same structure functions are shown
  as a dashed line.
  NLO predictions with hadronization corrections are shown as 
  a full line, the grey band
  indicates the renormalization and factorization
  scale uncertainties of the NLO prediction.}

\epsfig{file=H1prelim-01-031.fig2.ps,width=0.5\textwidth}
\caption{Differential $eP$ cross sections for dijet production as a function of
  the $E_T$ of the highest $E_T$ jet $E_{T,max}$. The inner
  error bars denote the statistical, the outer error bars the
  quadratic sum of all statistical and systematic errors of the data.
  The LO predictions 
  using CTEQ5M structure functions for the proton and
  GRV-HO structure functions for the photon are
  shown as a dotted line.
  NLO predictions with the same structure functions are shown
  as a dashed line.
  NLO predictions with hadronization corrections are shown as 
  a full line, the grey band
  indicates the renormalization and factorization
  scale uncertainties of the NLO prediction.}


\end{figure}
%\begin{figure}[hhh]
%\center
%\epsfig{file=/x01/usr/scaron/plots/h2jets_m2.ps,width=0.5\textwidth}
%\epsfig{file=/x01/usr/scaron/plots/h2jets_pt.ps,width=0.5\textwidth}
%\end{figure}
%\begin{figure}[hhh]
%\center
%\epsfig{file=/x01/usr/scaron/plots/3.2jets_m2.ps,width=0.5\textwidth}
%\epsfig{file=/x01/usr/scaron/plots/3.2jets_pt.ps,width=0.5\textwidth}
%\end{figure}




\begin{figure}[hhh]
\center
\epsfig{file=H1prelim-01-031.fig3.ps,width=\textwidth}
\caption{Differential $eP$ cross sections for dijet production as a function of
  the $\overline{\eta}$ of the two highest $E_T$ jets.
  The upper plots show the low $y$ and the lower plots the high
  $y$ region. The left plots show the low $E_{T,max}$ and the
  right plots the high $E_{T,max}$ region.
  The inner
  error bars denote the statistical, the outer error bars the
  quadratic sum of all statistical and systematic errors of the data.
  NLO predictions 
  using CTEQ5M structure functions for the proton and
  GRV-HO structure functions for the photon are
  shown as a dashed line.
  NLO predictions with hadronization corrections are shown as 
  a full line, the grey band
  indicates the renormalization and factorization
  scale uncertainties of the NLO prediction.}

  
\end{figure}
%\begin{figure}[hhh]
%\center
%\epsfig{file=/x01/usr/scaron/plots/hetam_ybin.ps,width=\textwidth}
%\end{figure}
%\begin{figure}[hhh]
%\center
%\epsfig{file=/x01/usr/scaron/plots/3etam_ybin.ps,width=\textwidth}
%\end{figure}


\begin{figure}[hhh]
\center
\epsfig{file=H1prelim-01-031.fig4.ps,width=\textwidth}
\epsfig{file=H1prelim-01-031.fig4b.ps,width=\textwidth}
\caption{Differential $eP$ cross sections for dijet production as a function of
  $x_{\gamma}$ (upper plots) and $x_P$ (lower plots).
  The upper left plot shows the cross sections as a function of 
  $x_{\gamma}$ for the lower $x_P$ region, the upper
  right plot for the higher $x_P$ region. 
  The lower right plot shows the cross sections as a function of 
  $x_P$ for the higher $x_{\gamma}$ region, the lower 
  left plot for the lower $x_{\gamma}$ region. 
  The inner
  error bars denote the statistical, the outer error bars the
  quadratic sum of all statistical and systematic errors of the data.
  The NLO predictions 
  using CTEQ5M structure functions for the proton and
  GRV-HO structure functions for the photon are
  shown as a dashed line.
  NLO predictions with hadronization corrections are shown as 
  a full line, the grey band
  indicates the renormalization and factorization
  scale uncertainties of the NLO prediction.}

\end{figure}

%\begin{figure}[hhh]
%\center
%\epsfig{file=/x01/usr/scaron/plots/hxgam_xp.ps,width=\textwidth}
%\end{figure}
%\begin{figure}[hhh]
%\center
%\epsfig{file=/x01/usr/scaron/plots/3xgam_xp.ps,width=\textwidth}
%\end{figure}

%\begin{figure}[hhh]
%\center
%\epsfig{file=/x01/usr/scaron/plots/2jets_xgampt.ps,width=\textwidth}
%\end{figure}

%\begin{figure}[hhh]
%\center
%\epsfig{file=/x01/usr/scaron/plots/h2jets_xgampt.ps,width=\textwidth}
%\end{figure}

\begin{figure}[hhh]
\center
\epsfig{file=H1prelim-01-031.fig5.ps,width=\textwidth}
\caption{Differential $eP$ cross sections for dijet production as a function of
  $x_{\gamma}$.
  The left plot shows the cross sections 
  for the lower $E_{T,max}$ region, the right 
  plot for the higher $E_{T,max}$ region. 
  The inner
  error bars denote the statistical, the outer error bars the
  quadratic sum of all statistical and systematic errors of the data.
  The NLO predictions 
  using CTEQ5M structure functions for the proton and
  GRV-HO structure functions for the photon and
  including hadronization corrections
  are shown
  as a full line. NLO predictions using AFG-HO and GSG-HO parametrizations
  of the photon structure functions and including hadronization 
  corrections are shown as dashed (AFG) and dotted (GSG) line. }
\end{figure}

\begin{figure}[hhh]
\center
\epsfig{file=H1prelim-01-031.fig6.ps,width=\textwidth}
\caption{Differential $eP$ cross sections for dijet production as a function of
  $x_{\gamma}$.
  The left plot shows the cross sections 
  for the lower $E_{T,max}$ region, the right 
  plot for the higher $E_{T,max}$ region. 
  The inner
  error bars denote the statistical, the outer error bars the
  quadratic sum of all statistical and systematic errors of the data.
  The NLO predictions 
  using CTEQ5M structure functions for the proton and
  GRV-HO structure functions for the photon are
  shown as a dashed line.
  NLO predictions with hadronization corrections are shown as 
  a full line, the grey band
  indicates the renormalization and factorization
  scale uncertainties of the NLO prediction.}

\end{figure}

\begin{figure}[hhh]
\center
\epsfig{file=H1prelim-01-031.fig7.ps,width=\textwidth}

\caption{The relative difference of the dijet cross sections as a
 function of $x_{\gamma}$ to
the NLO prediction with hadronization corrections applied
 using CTEQ5M structure functions for the proton and
  GRV-HO structure functions for the photon.
  The data is shown as points.
  The inner
  error bars denote the statistical, the outer error bars the
  quadratic sum of all statistical and of the uncorrelated 
  systematic errors of the data.
  The correlated systematic errors are shown in the middle two plots 
  as a shaded band.
  The grey band (two lower plots) 
  shows the renormalization and factorization
  scale uncertainties of this NLO prediction.}

\end{figure}



\begin{figure}[hhh]
\center
\epsfig{file=H1prelim-01-031.fig8.ps,width=\textwidth}
\caption{Differential $eP$ cross sections for dijet production as a function of
  $\cos \theta^*$.
  The left plots shows the cross sections 
  for the higher $x_{\gamma}$ region, the right 
  plots for the lower $x_{\gamma}$ region. 
  The lower two plots show the cross sections for an additional cut on
  the invariant mass of the dijet system. 
  The inner
  error bars denote the statistical, the outer error bars the
  quadratic sum of all statistical and systematic errors of the data.
    The NLO predictions 
  using CTEQ5M structure functions for the proton and
  GRV-HO structure functions for the photon are
  shown as a dashed line.
  NLO predictions with hadronization corrections are shown as 
  a full line, the grey band
  indicates the renormalization and factorization
  scale uncertainties of the NLO prediction.}

\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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

\bibitem{berger}
H1 Collab., T. Ahmed et al., Phys. Lett. {\bf B 297} (1992) 205
\bibitem{gPpaper}
H1 Collab., C. Adloff et al., Phys. Lett. {\bf B 483} (2000) 36;\\
H1 Collab., C. Adloff et al., Eur. Phys. J. {\bf C 1} (1998) 97;\\ 
ZEUS Collab., J. Breitweg et al., Eur. Phys. J. {\bf C 1} (1998) 109;\\
ZEUS Collab., J. Breitweg et al., Eur. Phys. J. {\bf C 11} (1999) 35;
 
%\bibitem{h1a} 
%H1 Collab., I. Abt et al., Phys. Lett. {\bf B 314} (1993) 436

\bibitem{nisius} R. Nisius, Phys. Reports {\bf 332} (2000) 165


\bibitem{gluon}
J. Huston et al, Phys. Rev. {\bf D 58} (1998) 114034 

\bibitem{wwa}
P. Kessler, Il Nuovo Cimento {\bf 17} (1960) 809
\bibitem{budlev}
V.M. Budnev et al., Phys. Rep. {\bf C 15} (1974) 181
\bibitem{frixww}
S. Frixione et al., Phys. Lett. {\bf B 319} (1993) 339

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

\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{geant}
R. Brun et al., GEANT3 User's Guide,
CERN-DD/EE-84-1 (1987). 
\bibitem{invkt} {S.D. Ellis and D.E. Soper}{,} \newblock Phys. Rev.
  {\bf D48} (1993) 3160

\bibitem{invkt2} {S. Catani, Yu.L. Dokshitzer, M.H. Seymour and B.R. Webber }{,} Nucl. Phys.
  {\bf B 406} (1993) 187

\bibitem{markus} H1 Collab., C. Adloff et al., Eur. Phys. J. {\bf C 19} (2001) 289 

\bibitem{yJB} A. Blondel and F. Jacquet, Proceedings of the Study of
  an $ep$
  Facility for Europe\\
  ed. U. Amaldi, DESY 79/48 (1979) 391.
\bibitem{frix} {S. Frixione and G. Ridolfi},  Nucl. Phys. {\bf B507} (1997) 315 

\bibitem{frix2} {S. Frixione}, Nucl. Phys. {\bf B 507} (1997) 295

\bibitem{nlo} {M. Klasen, G. Kramer}, Z. Phys. {\bf C 76} (1997) 67 ; \\
              {B.W. Harris, J.F. Owen}, Phys. Rev. {\bf D 56} (1997) 4007;\\
              {P. Aurenche et al.}, Eur. Phys. J. {\bf C 17} (2000) 413; 



\bibitem{grv}  M.~Gl\"uck, E.~Reya and A.~Vogt , Phys. Rev. {\bf D45} 
 (1992) 3986

\bibitem{afg}  P.~Aurenche, J.P.~Guillet and M.~Fontannaz,  
Z.Phys. {\bf C64} (1994) 621 

\bibitem{gsg}  L.E.~Gordon, J.K.~Storrow, Nucl.Phys. {\bf B489} (1997) 405 

\bibitem{h1det} H1 Collaboration, I. Abt et al., Nucl. Instr. Meth. 
{\bf A 386} (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{mx_paper} H1 Collab., C. Adloff et al., Z. Phys. {\bf C74}
  (1997) 221.

\bibitem{ariadne} L. L\"onnblad, Comp. Phys. Comm. 71 (1992) 15
\bibitem{django} K. Charchula et al., Comp. Phys. Comm. 81 (1994) 381

\bibitem{cteq5m} CTEQ Collaboration, H.L.~Lai et al., Phys. Rev. {\bf D55}
, (1997), 1280
\bibitem{mrst} A. D. Martin, R. G. Roberts, W. J. Stirling, R. S. Thorne, 
Eur. Phys. J. {\bf C 14} (2000), 133

\end{thebibliography}

\end{document}
