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\noindent
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%Submitted to & \multicolumn{3}{r}{\footnotesize Electronic Access: {\it http://www-h1.desy.de/h1/www/publications/conf/conf\_list.html}} \\[.2em] \hline 
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\multicolumn{4}{l}{{\bf
                International Europhysics
                Conference on High Energy Physics},
                July~12,~2001,~Budapest} \\
 {\bf EPS 2001:} 
                 & Abstract:        & {\bf 813}    &\\
                 & Parallel Session & {\bf 1, 2}   &\\
                 & Plenary Session  & {\bf 1, 2}   &\\[.7em]
\multicolumn{4}{l}{{\bf
               XX International Symposium on Lepton and Photon Interactions}, 
               July~23,~2001,~Rome} \\ 
{\bf LP 2001:}  
                 & Abstract:        & {\bf 505} &\\
                 & Parallel Session & {\bf 7, 8}   &\\
                 & Plenary Session  & {\bf 7, 8}   &\\ \hline
 & \multicolumn{3}{r}{\footnotesize {\it
    www-h1.desy.de/h1/www/publications/conf/conf\_list.html}} \\[.2em]
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\begin{center}
  \Large
  {\bf 
    Measurement of single inclusive high {\boldmath $E_T$} jet cross-sections \\  
    in photoproduction at HERA}

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

\begin{abstract}

\noindent
Inclusive jet cross-sections for the reaction $e^+ p \rightarrow
jet + X$ for  $Q^2 < 1~\gevsq$ and photon-proton energies in
the interval $95 \leq W \leq 285~\gev$ have been measured using 
data collected in the years 1996 - 1997 with the H1 detector at HERA. Jets with transverse energies $E_T^{jet} > 21~\gev$ in the pseudorapidity range $-1 \leq {\eta}^{jet} \leq 2.5$
are searched for  using
the inclusive $k_T$ cluster algorithm in the laboratory frame. 
Single-differential and multi-differential cross-sections as a function of $\eta^{jet}$ and $E_T^{jet}$
%Measurements of $d \sigma / d \eta^{jet}$ and $d\sigma / dE_T^{jet}$ 
are compared with NLO QCD calculations using different sets of photon parton
density functions as input. The cross-sections are in good agreement with NLO QCD up to the highest measured values of $E_T^{jet} = 75 ~\gev$.
\end{abstract}

\end{titlepage}

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\section{Introduction}
At HERA, the collisions of protons with quasi-real photons emitted from the incoming electrons  can result in the production of high transverse momentum jets.
% are dominantly produced in photoproduction, where a quasi-real photon emitted from the incoming lepton collides with the incoming proton. 
In leading order (LO) quantum chromodynamics
(QCD) two types of process are responsible for these jets.
% production at high transverse momentum in $\gamma p$ interactions. 
The photon may interact as a
pointlike particle with a parton in the proton, in so called direct processes. Alternatively, the photon may develop a hadronic structure, such that a parton in the photon interacts with a 
%or it may be considered as a source of partons which scatter off the 
parton in
the proton, in which case these processes are  called resolved. The  hadronic
final state consists of high $E_T$ partons which fragment into jets
with transverse energy $E_T^{jet}$, which acts as a hard scale allowing perturbative
QCD predictions to be made.  Additional energy in the event may come from the
proton and/or the photon remnant, as well as from possible soft remnant interactions.

\noindent
At high
$E_T^{jet}$, where non-perturbative effects are less influential, jet cross-sections reflect the underlying parton-parton dynamics of the photon-proton
collision. Therefore, $E_T^{jet}$ spectra measurements offer a means of testing the
validity of QCD predictions.  The proton parton density functions (PDF) being well
constrained by $F_2^p$ DIS measurements, pseudorapidity\footnote{Pseudorapidity is defined as $\eta \equiv -\ln(\tan~\theta / 2)$ where $\theta$
is the polar angle with respect to the proton direction.} $\eta^{jet}$ spectra are essentially
sensitive to the photon PDF. Measurements of $d\sigma/d\eta^{jet}$ allow the probing of photon structure at higher scales than those reached at $e^+ e^-$ 
colliders and  provide complementary information to that obtained from e.g. 
dijet analysis. 

\noindent
In this paper, measurements of differential jet cross-sections as a function of
the jet transverse energy, $E_T^{jet}$, and pseudorapidity , $\eta^{jet}$, for
inclusive jet production in photoproduction are presented. The measurements are
corrected to the hadron level and compared with   next-to-leading order (NLO)
calculations including  resolved and direct processes and using available NLO 
photon PDFs. 
The data were collected with the H1  detector at the HERA collider in 1996 and 1997, in which positrons of energy $27.5~\gev$ collided with  protons of energy $820~\gev$. The total data sample
corresponds to an integrated luminosity of $24.1 \pm 0.4~\invpb$. The paper is organized as follows. In section~\ref{det} the 
H1 detector is briefly presented. The details of the analysis procedure are
given in section~\ref{analysis} and a discussion of the cross-section
measurements is  presented in section~\ref{results}. The final section provides
a summary of the  results.

\section{Experimental apparatus}
\label{det}
A detailed description of the H1 detector can be found elsewhere~\cite{deth1}.
Here only the components relevant for this measurement are briefly described.
H1 uses a coordinate system whose origin is defined by the nominal interaction 
point. The forward $+z$ direction denotes the direction of the incident proton
beam.

\noindent The tracking system was used to reconstruct the interaction
vertex and to complement the measurement of hadronic energy flow. It consists of a central part  with a polar angle coverage of $25^\circ \leq \theta \leq
155^\circ$, and a forward part with a polar angle coverage of $7^\circ \leq \theta \leq
25^\circ$. The transverse momentum of charged particles was reconstructed from the curvature of tracks due to the homogeneous magnetic field of
$1.15$ Tesla along the beam direction, with a resolution $\sigma (p_T)/p_T \approx 0.6 \% . p_T$/(GeV/$c$) in the central part, and $\sigma (p_T)/p_T \approx 2 \% . p_T$/(GeV/$c$) in the forward part.

%of inner and outer cylindrical jet chambers, $z$-drift chambers and proportional chambers, with a polar coverage of $15^\circ \leq \theta \leq 165^\circ$. The two cylindrical drift chambers, are mounted concentrically around the beam line inside a homogeneous magnetic field of $1.15$ Tesla. 

\noindent Events were triggered using energy deposition in a finely grained
Liquid Argon  (LAr) calorimeter. The LAr calorimeter covers the range in polar angle $4^\circ
\leq \theta \leq 154^\circ$, with full azimuthal acceptance. It consists of an
electromagnetic section with lead as an absorber, and a hadronic section with steel as an  
absorber. The total depth of the calorimeter ranges from $4.5$ to $8$
hadronic interaction lengths. The energy resolution determined in test beam measurements is $\sigma(E)/E \approx 12~\% /
\sqrt{E/\mbox{GeV}} \oplus 1~\%$  for electrons and $\sigma(E)/E \approx 50~\% /
\sqrt{E/\mbox{GeV}} \oplus 2~\%$ for charged pions. The absolute hadronic energy scale is presently known  to $4~\%$.

\noindent The polar region $153^\circ \leq \theta \leq 177.8^\circ$ is covered
by the SPACAL, a lead/scintillating-fibre calorimeter with both electromagnetic
and hadronic sections. The energy resolution for electrons is $\sigma(E)/E \approx 7.5 ~\% /
\sqrt{E/\mbox{GeV}} \oplus 2.5~\%$, the energy resolution for hadrons is $\sigma(E)/E \approx 30 ~\% /
\sqrt{E/\mbox{GeV}}$.

\noindent The luminosity determination is based on the measurement of the
Bethe-Heitler process $e^+ p \rightarrow e^+ \gamma p$, where the positron and the photon are detected in
crystal Cherenkov calorimeters located downstream of the interaction point. The uncertainty on the measured integrated luminosity is $1.5~\%$.

\section{Analysis method}
\label{analysis}
The inclusive jet sample was defined by keeping all events for which at least
one jet with a transverse energy greater than $21~\gev$ in the pseudorapidity
region $-1 \leq \eta^{jet} \leq 2.5$ was reconstructed. The jet search was
performed in the laboratory frame by applying the inclusive $k_T$
cluster  algorithm~\cite{jetalgkt} to all ``objects'' in the final state. Combined objects were defined using calorimeter clusters complemented by tracks measured in the tracking system. 

\noindent
The inelasticity variable $y$ is reconstructed following the method of
Jacquet and Blondel~\cite{jb} from the energies and longitudinal momenta of all objects in the final state.
The event sample is restricted to the kinematic region $0.1 \leq y \leq 0.9$,
corresponding to a photon-proton centre-of-mass energy $W$ ($W = \sqrt{ys}$, where
$s$ is the squared $e^+ p$ centre-of-mass energy) between $95$ and $285~\gev$.

\noindent
The contamination from beam-gas background was reduced to a negligible level by demanding a well-reconstructed vertex in the interaction region.
%The contamination from beam gas background was found to be negligible after demanding that i) exactly one primary vertex is reconstructed, ii) the  vertex position along the beam axis is distant from the nominal interaction point by less than $30~\cm$ and iii) the number of upstream tracks is fewer than the number of tracks pointing to the vertex unless this number is less than four in which case the number of upstream tracks must be less than four.   
Cosmic shower and halo muon backgrounds  were then rejected by using a set of topological
muon finders in addition to the requirement that the missing  transverse
momentum is small compared to the total transverse energy of the  events: 
${P\hspace{-0.1in}/}_T/\sqrt{E_T}\leq(2.5{~\gev})^{\frac {1}{2}}$.
%${P\hspace{-0.1in}/}_T \leq 2.5 \sqrt{E_T . \mbox{GeV}}$.  
At this stage of the selection, the overall
non $ep$ background was estimated to correspond to less than $1~\%$ of the data
sample. 

\noindent
Neutral current deep inelastic scattering background was removed by
rejecting all events with an identified scattered positron in the LAr calorimeter or in the
SPACAL. Inefficiencies of the positron identification algorithm were
compensated for, by further rejection of events with a jet having either a high proportion of electromagnetic energy or a hot core, and containing exactly one track pointing
to a $\phi$ crack in the LAr calorimeter. 

\noindent
The resulting event sample consists of $17\,460$ jets reconstructed in $13\,389$ events. The $\gamma p$ purity of the sample was estimated to be $\sim 99~\%$. The selection limits the virtuality of the incoming photon to the range $Q^2 < 1~\gevsq$.

%With these selections the virtuality of the incoming photon is limited to values below $\sim 1~\gevsq$. The resulting event sample consists of 17460 jets reconstructed in 13389 events. The $\gamma p$ purity of the sample was estimated to be $\sim 99~\%$.

\noindent
Samples of direct and resolved photoproduction events were simulated using the 
PYTHIA 5.7~\cite{pythia} and HERWIG 5.9~\cite{herwig} Monte Carlo generators. The
generated events were passed through a simulation~\cite{geant} of the H1
detector, and reconstructed under the  same conditions as the data. Hard
parton-parton interactions are generated by PYTHIA and HERWIG using LO QCD
matrix elements.  The hard parton-parton cross-section diverges towards low
$\hat{p}_t$ values\footnote{$\hat{p}_t$ corresponds to the transverse momentum of the two outgoing partons in the parton-parton centre-of-mass frame.},  and therefore needs regularisation to normalise to the measured
total cross-section. This is done by the introduction of a lower cut-off
$\hat{p}_t^{min}$. Both generators use initial and final state parton showers
to simulate higher order QCD radiation effects. The hadronisation process is
performed using the Lund string model~\cite{lund} as implemented in 
JETSET~\cite{jetset} in the case of PYTHIA, and using a cluster
model~\cite{cluster} in the case of HERWIG. GRV-LO~\cite{grv} 
parameterisations  of the proton and the photon PDF were used. Multiple
interactions were generated in addition to the primary parton-parton
scattering. Within PYTHIA, these are calculated as LO QCD processes between
partons from the proton and photon remnants and result as  additional final
state partons which are  required to have transverse momenta of at least
$1.2~\gev$.  Soft particles accompanying the hard sub-process are produced in
HERWIG using a soft underlying event (SUE) mechanism which includes
parameterisations of experimental results on soft hadron-hadron collisions. As
shown in figure~\ref{fig:profile_phi}, a reasonable description of the
observed energy flow  around the jet axis is obtained with both PYTHIA and
HERWIG. For the latter, $35 ~\%$ of the resolved interactions were generated
with an additional SUE. This corresponds to an increase of the cross-section of
$\sim 10 ~\%$ at large $\eta^{jet}$ and low $E_T^{jet}$. The effects of SUEs are
negligible at the highest $E_T^{jet}$.
\begin{figure}[hhh]
\center
\epsfig{file=profile_phi.eps,bbllx=13pt,bblly=100pt,bburx=410pt,bbury=536pt,clip=,width=0.65\textwidth}
\caption{\label{fig:profile_phi}
Transverse energy flow integrated over $|\Delta \eta| \leq 1$ around the jet axis as a function of $\Delta \phi $
in different ranges of pseudorapidity and
jet transverse energy.  
}
\end{figure}

\noindent
The HERWIG and PYTHIA Monte Carlo samples have been used to correct the data for the
inefficiencies of the selection criteria and for migrations caused by detector
effects.  
Correction factors were calculated for each bin of each distribution as the ratio
$\mathcal{P}/\mathcal{E}$. The purity $\mathcal{P}$ (resp. efficiency $\mathcal{E}$) of a given bin is defined as the number of jets which enter this bin on both hadron level and reconstructed level (i.e.
after detector smearing and all experimental selections) divided by the number
of jets reconstructed (resp. generated) in this bin.
%The correction factors were calculated as the ratio $\mathcal{P}/\mathcal{E}$ in each bin. Bin purities, $\mathcal{P}$, and efficiencies, $\mathcal{E}$, are  defined as the number of jets that enter the same bin of a distribution on hadron level and on reconstructed level (i.e. after detector smearing and all experimental selections) divided by the number of jets reconstructed, respectively generated in this bin. 
For all measured
cross-sections, bin purities and efficiencies were required to be greater than
$40 ~\%$. The mean values of the correction factors calculated with PYTHIA and
HERWIG were used to correct bin-by-bin the data to the hadron level. The final
correction factors lie between $0.9$ and $1.6$.

\noindent
The following sources of systematic errors have been taken into account :
\begin{itemize}
\item the $4\%$ uncertainty in the absolute hadronic energy scale of the LAr calorimeter 
leads to an uncertainty at the level of $15~\mbox{to}~20~\%$ on the measurements, which is the dominant experimental uncertainty, 
\item the model dependency of the bin-by-bin correction leads to an error of  $\sim
10~\%$ as estimated from the difference between correction factors
calculated with PYTHIA and HERWIG,
\item the trigger efficiency is known with an uncertainty of $5~\%$,
\item the uncertainty in the luminosity determination for this data sample is
$1.5~\%$.
\end{itemize}

\section{Results}
\label{results}
In this section, the measured differential cross-sections are presented and compared with NLO QCD calculations~\cite{nlo} based
on the substraction method. The CTEQ5M~\cite{cteq5m} parameterisation of the
proton PDF was used for the calculations. The value of
$\Lambda_{QCD}$ was chosen to match that of this set of PDF ($\Lambda_{QCD} =
226~\mev$ for five quark flavours, corresponding to $\alpha_s(M_Z) = 0.118$).
GRV-HO~\cite{grv} was chosen as the standard parameterisation of the photon PDF for this analysis. AFG-HO~\cite{afgho} and GSG-HO~\cite{gsgho}
were also used for comparison.  
The  renormalisation and factorisation scales were set to $\mu_R = \mu_F = 1/2 \sum E_T^{parton}$.
%The half sum of the outgoing partons transverse momenta defines the  renormalisation and factorisation scales.
These scales were varied by a factor of 2 up and  down in order to estimate the
uncertainty corresponding to the missing higher-order terms.

\noindent
Since the NLO QCD calculations refer to jets of partons, whereas the 
measurements refer to jets of hadrons, the predicted cross-sections have been
further corrected to the hadron level. These corrections, which do not exceed
$10~\%$, were calculated by taking the average of the ratios of cross-sections
at the hadron and parton levels as given by PYTHIA and HERWIG.

\noindent
The measured differential $e^+ p$ cross-section $d\sigma / dE_T^{jet}$ for
inclusive jet production integrated over $-1 \leq \eta^{jet} \leq 2.5$ in the
kinematic region defined by $Q^2 \leq 1~\gevsq$ and $95 \leq W \leq 285~\gev$
is shown in figure~\ref{fig:xsec_hiet_1}. The cross-section falls off by
three orders of magnitude in the $E_T^{jet}$ range between $21~\gev$ and
$75~\gev$. The data are compared with QCD calculations, both at LO and NLO.
Hadronisation results in a decrease of $\sim 5~\%$ of the calculated parton-level cross-section; this hadronisation correction 
is roughly constant with $E_T^{jet}$. The NLO QCD  prediction including
this effect is shown in the figure together with its theoretical uncertainty, originating from the scale dependence.
The fractional difference between the data and the QCD prediction is shown in figure~\ref{fig:xsec_hiet_2}. The experimental uncertainty due to the
absolute energy scale of the jets, which dominates the systematic errors, is
displayed as a shaded band. As shown in this figure, the calculated cross-section using the GRV-HO photon PDF is typically 5~to~$10~\%$ larger than those
obtained with AFG-HO and GSG-HO. Within the errors, NLO QCD calculations describe
the magnitude and the shape of the measured inclusive $E_T^{jet}$ spectrum very well, up to the highest measured $E_T^{jet}$ values.
 
\noindent
The evolution of $d\sigma / dE_T^{jet}$ with  $y$ is shown in figure ~\ref{fig:xsec_hiet_3}. The two $y$ intervals presented here correspond to photon-proton centre-of-mass energies in
the ranges $95 \leq W \leq 212~\gev$ and $212 \leq W \leq 285~\gev$. The $E_T^{jet}$ spectrum at high  $y$ is harder and extends to higher  $E_T^{jet}$ values than the spectrum at low $y$, as expected from the higher centre-of-mass energy available. Both spectra are well reproduced by  NLO QCD.

\noindent
The measured differential $e^+ p$ cross-section $d\sigma / d\eta^{jet}$ for
inclusive jet production integrated over $21 \leq E_T^{jet} \leq 75~\gev$ in
the kinematic region defined by $Q^2 \leq 1~\gevsq$ and $95 \leq W \leq
285~\gev$ is shown in figure~\ref{fig:xsec_hiet_11} and compared with
theory in figure~\ref{fig:xsec_hiet_12}. The calculated cross-section using 
the GRV-HO photon PDF gives a fair description of the data while calculations using AFG-HO
and GSG-HO lie slightly below the data at low $\eta^{jet}$. In figure~\ref{fig:xsec_hiet_10} the measurement is presented in  three different
intervals of $E_T^{jet}$.  
%As increasing jet transverse energy, the size of the hadronisation corrections decrease from $\sim \pm 10\%$ to $\sim \pm 5\%$. 
The hadronisation corrections 
correspond to an increase (resp. decrease) of the pure partonic prediction in 
the forward (resp. backward) region. They tend to improve the agreement of the NLO calculations with the data, although only in a marginally significant way. Within the errors, the data are well described by the NLO QCD prediction using the GRV-HO photon PDF.  

\noindent
The fraction of the energy of the photon taken by the interacting parton $x_{\gamma}$ can be reconstructed at the parton level as $x_{\gamma} = (\sum_{partons} E_T~e^{- \eta}) / 2 y E_e$. According to this formula, differences in photon PDFs, which result in differences in $x_{\gamma}$ distributions, should reflect mostly in the $\eta^{jet}$ distributions, for restricted ranges of $y$ and $E_T^{jet}$.
The data sample is thus separated into different $y$ and $E_T^{jet}$ regions, to try  to discriminate between the various photon PDFs more efficiently than with the inclusive measurements alone. Measurements of
$d\sigma/d\eta^{jet}$, in two regions of $y$ and three intervals of $E_T^{jet}$, are presented in figure~\ref{fig:xsec_hiet_9}. The maximum of the cross-section is shifted towards
low $\eta^{jet}$ values when increasing $y$. This is expected since, in the
laboratory frame, this corresponds to an increased boost of the events in the 
backward direction. 
%At high $y$, the fall of the cross-sections with increasing $\eta^{jet}$ is all the more steep as $E_T^{jet}$ increases. 
NLO QCD
predictions are in general in good agreement with the measured cross-sections. The only distribution which gives some sensitivity to the different photon PDFs is the one in  the highest $y$ and lowest 
$\eta^{jet}$ region. Here, the GRV-HO photon PDF gives the best description of the data; however, the precision of the experimental measurement (as well as the theoretical uncertainty not presented in figure~\ref{fig:xsec_hiet_9}) does not allow any firm conclusion on which PDF is favoured by data to be drawn.

\noindent
In order to compare with a previous ZEUS measurement~\cite{zeuset}, the kinematic range has been reduced to $0.2 \leq y \leq 0.85$ and $-0.75 \leq \eta^{jet} \leq
2.5$.
Figure~\ref{fig:zeus_h1_combined_Et_kt} shows the differential $e^+ p$ cross-section  $d\sigma / dE_T^{jet}$ measured by H1 for this restricted range, compared with the ZEUS preliminary result.  The inner error bars of the H1 data represent the
statistical errors, the outer show the quadratic sum of statistical and systematic
uncertainties. Within the errors, the two measurements are
in good agreement.

\noindent
To compare with another ZEUS 
measurement~\cite{zeusrp} of the inclusive $\eta^{jet}$ spectrum, the complete analysis chain described in
section~\ref{analysis} has been  re-done using  an iterative cone
algorithm~\cite{cone} with a radius $R=1$ in the $(\eta,\phi)$ plane. The
resulting differential $e^+ p$ cross-sections  $d\sigma / d\eta^{jet}$
integrated for $E_T^{jet}$ above two different thresholds ($E_T^{jet}\geq 21$
and $25~\gev$) are shown
in figure~\ref{fig:zeus_h1_combined_eta_cone}. Again there is no significant difference between the measurements from both experiments.

\section{Summary}
A new measurement of inclusive high $E_T$ jet production cross-sections in quasi-real photoproduction ($Q^2 \leq 1~\gevsq$) $ep$ interactions has been presented, based on ${\cal L} = 24~\invpb$ of $e^+ p$ data collected by H1 in
1996-1997. The kinematic range of the measurement corresponds to photon-proton centre-of-mass energies $95 \leq W \leq 285~\gev$. The jets were found in the pseudorapidity range $-1 \leq {\eta}^{jet} \leq 2.5$ using
the inclusive $k_T$ algorithm in the laboratory frame and were required to have transverse energy
in the interval $ 21 < E_T^{jet} < 75~\gev$. Measured cross-sections were corrected to the hadron level, and compared to NLO QCD calculations, with or without hadronisation corrections. Within the experimental and theoretical uncertainties, the measured distributions are well described, both in normalisation and shape, by NLO QCD calculations using various available photon PDFs, even without hadronisation corrections. The hadronisation corrections to the NLO QCD calculations only slightly improve the agreement with the data. The current precision of the experimental results as well as the theoretical predictions do not allow discrimination between the different photon PDFs. Data are in agreement with previously presented ZEUS results.


\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.
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