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\begin{titlepage}

\noindent
\begin{center}
\begin{small}
\begin{tabular}{llrr}
Submitted to & & &
    \includegraphics[width=2.cm]{H1logo_bw_small}
 \\[.2em] \hline
                & & & \\
\multicolumn{4}{l}{{\bf
                International Europhysics Conference 
                on High Energy Physics, EPS05},
                July~21-27,~2003,~Lisboa} \\
                (Abstract {\bf 629} & Parallel Session & {\bf Hard QCD}) & \\
                & & & \\
                \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 
 Inclusive Jet Production in Deep Inelastic Scattering at High $Q^2$
 at HERA
}

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

\begin{abstract}
\noindent
The inclusive jet production in deep inelastic positron-proton 
scattering at HERA is studied.
The data sample corresponds to an integrated 
luminosity of ${\cal{L}}_{int} \sim 61.25$ pb$^{-1}$ and was taken in the 
years 1999-2000 with the H1 detector at a centre-of-mass energy 
$\sqrt{s} = 319$ GeV.
The measured jet cross section is compared to perturbative QCD calculations 
using the NLO program NLOJET++, and the strong coupling 
$\alpha_s$ is extracted.
\end{abstract}
 
\end{titlepage}


\newpage

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

\section{Introduction}

Jet production at HERA in neutral current (NC) deep inelastic
scattering (DIS) provides an important
testing ground for QCD. Jet data are precise at high transverse 
energy, where experimental systematic uncertainties and non-perturbative 
effects are small.
Measurements of jet cross sections allow to test the
understanding of calculational techniques in perturbative QCD, 
to extract the strong coupling
$\alpha_s$ and to constrain the proton and photon parton distribution 
functions (PDFs). Here, we report on a measurement of the inclusive jet 
cross section in NC DIS at high momentum transfers $Q^2$ and 
a determination of the strong coupling constant $\alpha_s(M_Z)$.

\subsection{Deep Inelastic Scattering}
The kinematics of DIS are defined via the four momenta 
of the incoming ($k$) and outgoing electron ($k^{\prime}$), 
incoming proton ($p$) and
exchanged boson ($q = k - k^{\prime}$):

\begin{center}
\begin{tabular}{ll}
$s = (k + p)^2 \simeq 4 E_e E_p$, &
 $Q^2 = -q^2 = -(k - k^{\prime})^2 $, \\
$x = Q^2/(2pq)$, &
$y = Q^2/sx$, \\
\end{tabular}
\end{center}
\noindent
where $s$ is the centre-of-mass energy, $Q^2$ the negative four momentum transfer squared,
$x$ the Bj\o rken scaling variable, and $y$ the inelasticity.
There are several methods to reconstruct the kinematic variables 
in NC DIS events, since there is redundant information on the 
reconstructed level from both, the lepton and the hadronic final state.
Commonly used methods are referred to as the electron method, 
the hadron method, the $\Sigma$ method, the
electron-$\Sigma$ method and the double angle method.
Similarly to a published H1 analysis~\cite{markus}, the present study uses the 
electron-$\Sigma$ method to calculate the kinematic variables. 

\subsection{Jet Production in Deep Inelastic Scattering}

The factorisation property for the calculation of jet production in DIS 
is only given in selected frames~\cite{frame}.
In the present analysis the Breit frame of reference is used.  
By selecting high transverse energy ($E_t$) jets in this frame, 
contributions from the Born level and jets induced by the beam remnant 
are suppressed (Fig.~\ref{fig:breit}).
The lowest order contributions to jet production 
in the Breit frame are of order $\mathcal{O}(\alpha_s)$ and hence directly 
sensitive to QCD radiation.
The scope of the present analysis is on the following differential inclusive jet cross sections:
$\mathrm{d}^2\sigma_{jet}/\mathrm{d}Q^2\mathrm{d}E_t$, $\mathrm{d}\sigma_{\rm jet}/\mathrm{d}Q^2$ and $\mathrm{d}\sigma_{\rm jet}/\mathrm{d}E_t$.


\section{Event and jet selection}

The inclusive jet analysis is based on data taken with the H1
detector~\cite{h1det} during the years 1999 to 2000, corresponding to 
an integrated luminosity of 61.25~pb$^{-1}$ at a centre-of-mass energy 
$\sqrt{s} = 319$ GeV.
High $Q^2$ DIS events are identified by the detection of a scattered 
electron in LAr calorimeter. The main selection cuts are briefly 
listed below.

\begin{itemize}

  \item $150 < Q^2 < 5000$ GeV$^2$ and $0.2 < y < 0.6$, where 
        $Q^2$ and $y$ are obtained using the electron-$\Sigma$ method.
  \item Scattered electron energy $E_e> 11\,\mathrm{GeV}$ and $\theta_\mathrm{e} < 153^o$ \footnote{The polar angle $\theta$ is defined w.r.t\ the z-axis. 
In the H1 coordinate systems the z-axis points in the direction of 
the proton beam.}.
  \item  $45<\sum(E-p_z)<65\,\mathrm{GeV}$, where the sum is over all hadronic final state particles and the electron.
    Further non-ep background is rejected by recognition of characteristic topologies.
  \item Jets
     \begin{itemize} 
       \item Jets are defined by the $k_t$ jet finder in the Breit frame.
       \item Jet selection: $E_{\rm t} > 7$ GeV 
and $-1.0 < \eta^\mathrm{Lab} < 2.5$.
     \end{itemize}

\end{itemize}

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

\section{Correction procedure}
In order to extract the cross sections, the experimental data are corrected for limited detector acceptance
and resolution as well as for QED radiative effects.
The phase space of the cross sections is defined by: 
\begin{center}
 $150 < Q^2 < 5000\,\mathrm{ GeV^2}$, ~~ $0.2 < y < 0.6$, ~~
 $E_{\rm t} > 7$ GeV, ~~ $-1.0 < \eta^\mathrm{Lab} < 2.5$.
\end{center}
The correction factors are determined by the Monte Carlo event generators DjangoH 1.2 
(using the Color Dipole Model with Ariadne) and Rapgap 2.8 (LO matrix elements 
plus parton showers). Both, Rapgap and Django describe the shape of the distributions
from the inclusive jet sample (Figs~\ref{fig:plotjet}~-~\ref{fig:plotet2}).


The bin-by-bin correction 
method is employed to obtain the correction factors.
Two event samples are generated for each generator: the
first sample, which includes QED radiation, is subject to a detector
simulation. The second sample is generated with identical setup but without 
QED radiation and without detector simulation. The correction factor in a bin
is obtained as a ratio of the number of events in this bin
of the second sample to that of in the first sample.
%%%value in the second sample and the value in the first sample.

The arithmetic mean of Rapgap and Django are used as correction factors, 
while half of the difference  is assigned for the model uncertainty.
The correction factors differ by less than 20\% from unity for most of the bins, and the model dependence in most of the bins is below 10\%.

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

\section{Experimental Uncertainties}

Various sources contribute to the systematic uncertainty of the measured 
inclusive jet cross section, such as the
 luminosity measurement, the hadronic and electromagnetic energy scales 
of the calorimeters,
 and the momentum scale and angle of the tracks.
These individual contributions are determined from the data 
and are added in quadrature.
Another important uncertainty is due to the model dependence in the
determination of correction factors.
With respect to correlations, the uncertainties are divided into three classes:

\begin{itemize}
  \item Correlated sources: luminosity measurement uncertainty and 
positron polar angle.
  \item Uncorrelated sources: statistical uncertainty and positron 
azimuthal angle.        
  \item Sources, equally shared between correlated and uncorrelated ones:  positron energy scale, LAr energy scale, track momentum and model dependence.  
\end{itemize}

\noindent
The major experimental uncertainty is due to the the model dependence 
and the LAr energy scale.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{NLO QCD calculation}

The fixed order calculation program NLOJET++~\cite{nlojet} is used to perform the
QCD analysis.

NLOJET++ calculates jet cross sections at
parton level, hence one needs to apply hadronisation correction 
factors to the NLOJET++ prediction in order to compare it with the
measured data points. Here, the hadronisation 
correction is calculated as 
$(1+\delta_\mathrm{had}) = \sigma_\mathrm{had}/\sigma_\mathrm{part}$ for 
Monte Carlo data without QED radiation.
Again, results from Rapgap and Django are averaged, and half of the 
difference is used as uncertainty.
The hadronisation corrections for the inclusive jet 
sample are typically 1.0~to~1.1. Rather large hadronisation corrections 
(up to 1.4) are observed in some bins in the  $d\sigma_{jet}/d\eta$ 
and $d\sigma_{jet}/d\eta_{Lab}$ distributions. 
Hence, these bins are excluded from the analysis.

The NLOJET++ calculation is performed in the $\overline{\mbox{\rm MS}}$ scheme.
 The number of active flavours is set to five, the
renormalisation scale is chosen to be $E_{\rm t}^2$ of jets, and  $Q^2$
is used as factorisation scale.
The strong coupling 
$\alpha_s(\mu_r)$ is evolved at two loop precision and fixed at $\alpha_s(M_Z)$
to 0.118.
The CTEQ5M1 parametrisation of the proton PDFs is used as input to 
~~~~NLOJET++.
The theory uncertainty consists of the uncertainties due to the 
hadronisation correction 
and the renormalisation and factorisation scales.
The latter are obtained by varying the two scales simultaneously
by a factor of 4, i.e.\ $0.25 < \mu_{\rm r}^2/E_{\rm t}^2, 
\mu_{\rm f}^2/Q^2 < 4.0$.
$E_{\rm t}$ as the renormalisation scale offers the benefit of smaller theory uncertainties
compared to the $Q$ scale.

\section{Experimental Results and QCD Analysis}
In the following, the results on the differential jet cross sections are presented, which are used to extract the strong coupling $\alpha_s$.

\subsection{Inclusive jet cross section}

The measured inclusive jet cross sections are shown  as function of $E_{\rm t}$
in four $Q^2$ bins, 
as function of $Q^2$ integrated over $E_{\rm t}$, and of $E_{\rm t}$
integrated over $Q^2$ 
%%%in $150 < Q^2 < 5000\,\mathrm{ GeV^2}$ 
in Figs.~\ref{fig:comet}~-~\ref{fig:comEt} and are compared to the NLO
QCD calculations.
In the lower part of Figs.~\ref{fig:comQ2} and \ref{fig:comEt}, the ratio of 
the measured inclusive jet cross section to the NLOJET++ prediction
is shown. The NLO calculation, corrected for hadronisation effects,
 gives a good description of the data in the full $E_{\rm t}$ and $Q^2$ range. 


\subsection{Determination of $\alpha_s(M_Z)$}

We follow the methods in \cite{markus} and \cite{zeusjet} 
to extract the strong coupling constant $\alpha_s(M_Z)$.
The dependence of the cross section on  $\alpha_s(M_Z)$  is 
determined in each $Q^2$-$E_{\rm t}$ (Fig.~\ref{fig:comet}) and
$E_{\rm t}$ (Fig.~\ref{fig:comEt}) bin
by performing the NLOJET++ calculation using five values of $\alpha_s(M_Z)$
 (0.110, 0.113, 0.116, 0.119 and 0.122).
Using these five calculations per bin, the dependence of the cross 
section on $\alpha_s(M_Z)$ is then approximated by the function

\begin{eqnarray}
  \sigma_i(\alpha_s(M_Z)) = A_i \cdot \alpha_s(M_Z)
                            + B_i \cdot \alpha_s^2(M_Z),
\end{eqnarray}    

\noindent
with the two free parameters determined in the fit.
The function is then used to map the measured jet cross section
$\alpha_s(M_Z)$ for each bin, as it is illustrated in Fig.~\ref{fig:fitting}
for one of the bins. 

\subsection{Results for $\alpha_s(M_Z)$ and $\alpha_s(E_t)$}

The values of $\alpha_s(M_Z)$, determined 
from the inclusive jet cross sections measurements 
(Figs.~\ref{fig:comet},~\ref{fig:comEt})
in the $Q^2$-$E_{\rm t}$ and $E_{\rm t}$ bins,
are shown in Fig.~\ref{fig:alpha0} and Fig.~\ref{fig:alpha}, respectively.
The results on $\alpha_s(M_Z)$ are subsequently evolved
using the two loop solution of the renormalisation group equation
to $\alpha_s(E_{\rm t})$, where $E_{\rm t}$ corresponds to
the mean value of $E_{\rm t}$ in the bin considered.
The results are in good agreement with the world average $\alpha_s(M_Z)=0.1187 \pm 0.0020$~\cite{Eidelman:2004wy}.


\subsection{Averaged $\alpha_s(M_Z)$}

As the determinations of $\alpha_s(M_Z)$ are consistent over the whole $Q^2$ and 
$E_t$ range, they can be combined 
into one average value. Due to correlations of the measured cross sections, 
the individual values of $\alpha_s(M_Z)$ are also correlated. We use the method
presented in  \cite{markus} and \cite{chifit} to
obtain a combined fit to the 15 $\alpha_s(M_Z)$ determinations 
in four $Q^2$ bins (Fig.~\ref{fig:alpha0}).
This procedure gives $\chi^2/\mathrm{ndf} = 20.14/14 = 1.44$, and the 
average $\alpha_s(M_Z)$ value is 

\begin{eqnarray}
\alpha_s(M_Z) = 0.1197 ~\pm 0.0016\,\mathrm{(exp.)}
~ ^{+0.0046}_{-0.0048}\,\mathrm{(th.)}.
\label{etscale}
\end{eqnarray}

\noindent
The largest contribution to the experimental error is due to the
LAr energy scale and the model dependence. The renormalisation and factorisation
scale variations are the major contribution to the theory uncertainty.

The averaged $\alpha_s(M_Z)$ and corresponding $\alpha_s(E_t)$ values
are shown in Fig.~\ref{fig:alpha}.
The averaged $\alpha_s(M_Z)$ value is consistent with the world average
 and with the previous H1
 inclusive jet result~\cite{markus} :
$\alpha_s(M_Z) = 0.1186 ~\pm 0.0030\,\mathrm{(exp.)}
~ ^{+0.0039}_{-0.0045}\,\mathrm{(th.)}$.

%\begin{eqnarray}
%\alpha_s(M_Z) = 0.1186 ~\pm 0.0030\,\mathrm{(exp.)}
%~ ^{+0.0039}_{-0.0045}\,\mathrm{(th.)}.
%\end{eqnarray} 

\noindent


\section{Summary}

Measurements of inclusive jet cross sections in deep inelastic
positron-proton scattering 
in the range of $150 < Q^2 < 5000\,\mathrm{ GeV^2}$
and transverse jet energy $E_{\rm t} > 7$ GeV
have been presented.
Predictions obtained with the NLO QCD
program NLOJET++ provide a good description of the single and double differential
inclusive jet cross sections.
A determination of $\alpha_s(M_Z)$ yields 

\begin{eqnarray}
\alpha_s(M_Z) = 0.1197 ~\pm 0.0016\,\mathrm{(exp.)}
~ ^{+0.0046}_{-0.0048}\,\mathrm{(th.)}.
\end{eqnarray} 

\noindent
This value is consistent with the world average of $\alpha_s(M_Z)$.

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

\begin{thebibliography}{99}

%Markus Wobisch thesis on 95-97 data: jet and published paper
\bibitem{markus}
H1 Collaboration, C. Adloff {\it et al.}, Eur. Phys. J. C19 (2001) 289; \\
Markus Wobisch, Ph. D thesis, PITHA 00/12.

%QCD factorization property in frame
\bibitem{frame}
B. R. Webber, J. Phys. {\bf G19} (1993) 1567.

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

%ZEUS boost code
\bibitem{ktclus}
M. H. Seymour, KTCLUS program.

\bibitem{nlojet}
Z. Nagy and Z. Trocsanyi, Phys. Rev. Lett. 87 (2001) 082001.

\bibitem{zeusjet}
ZEUS Collaboration, S. Chekanov {\it et al.}, Phys. Lett. B547 (2002) 164; \\
H. Raach, Ph. D thesis, DESY-THESIS-2001-046.

\bibitem{chifit}
G. Lobo, C. Pascaud and F. Zomer, H1-01/98-536.

\bibitem{Eidelman:2004wy}
S.~Eidelman {\it et al.}  [Particle Data Group Collaboration],
%``Review of particle physics,''
Phys.\ Lett.\ B {\bf 592} (2004) 1.
%%CITATION = PHLTA,B592,1;%%

\end{thebibliography}

\begin{figure}[ht]
\center
\setlength{\unitlength}{1.0cm}
\begin{picture} (16.0,3.0)
\put (0.5,0.5){\epsfig{file=H1prelim-05-133.fig1a.eps,height=2.2cm}}
\put (9.0,0.5){\epsfig{file=H1prelim-05-133.fig1b.eps,height=2.5cm}}
\put (2.0,0.0) {\bf Born level process}
\put (10.5,0.0) {\bf QCD compton}
\put (13.5,1.6) {\textcolor{red}{\Large $\alpha_s$}}
\end{picture} 
\caption{Deep inelastic scattering in the Breit frame, (a) Born 
level process and (b) QCD Compton process.}
\label{fig:breit}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[ht]
\center
\epsfig{file=H1prelim-05-133.fig2a.eps,width=0.30\textwidth,angle=-90}
\epsfig{file=H1prelim-05-133.fig2b.eps,width=0.30\textwidth,angle=-90}\\[0.5cm]
\epsfig{file=H1prelim-05-133.fig2c.eps,width=0.30\textwidth,angle=-90}
\epsfig{file=H1prelim-05-133.fig2d.eps,width=0.30\textwidth,angle=-90}
\caption{Kinematic variables and jet polar angle of the DIS 
inclusive jet sample, the
 distributions are normalised to one
and are compared to MC model predictions.}
\label{fig:plotjet}
\end{figure}

\begin{figure}[ht]
\center
\epsfig{file=H1prelim-05-133.fig3a.eps,width=0.30\textwidth,angle=-90}
\epsfig{file=H1prelim-05-133.fig3b.eps,width=0.30\textwidth,angle=-90}\\[0.5cm]
\epsfig{file=H1prelim-05-133.fig3c.eps,width=0.30\textwidth,angle=-90}
\epsfig{file=H1prelim-05-133.fig3d.eps,width=0.30\textwidth,angle=-90}
\caption{Jet transverse energy $E_t$ in four $Q^2$ bins,
the distributions are normalised to one 
and are compared to MC model predictions.}
\label{fig:plotet2}
\end{figure}

\begin{figure}[p]
\center
\epsfig{file=H1prelim-05-133.fig4.eps,width=0.650\textwidth,angle=-90}
\caption{Measured $d\sigma/dE_{\rm t}$ in four $Q^2$ bins
compared with NLOJET++ predictions corrected
for hadronisation effects. The bands show the theoretical
uncertainty by varying the renormalisation and factorisation scales by a factor
of two.}
\label{fig:comet}
\end{figure}

\begin{figure}[p]
\center
\epsfig{file=H1prelim-05-133.fig5.eps,width=0.7\textwidth,angle=-90}
\caption{Measured $d\sigma/dQ^2$ 
(integrated over the transverse jet energy $E_{\rm t}$)
compared with NLOJET++ predictions corrected
for hadronisation effects. The band shows the theory uncertainty 
by varying the renormalisation and factorisation scales by a factor of two.
The statistical uncertainty is shown as the inner error bar, while the total error is denoted by the outer error bar.}
\label{fig:comQ2}
\end{figure}

\begin{figure}[p]
\center
\epsfig{file=H1prelim-05-133.fig6.eps,width=0.7\textwidth,angle=-90}
\caption{Measured $d\sigma/dE_{\rm t}$ (integrated over $Q^2$)
compared with NLOJET++ predictions corrected
for hadronisation effects. The bands show the theory uncertainty
by varying the renormalisation and factorisation scales by a factor of two.
The statistical uncertainty is shown as the inner error bar, while the total error is denoted by the outer error bar.}
\label{fig:comEt}
\end{figure}

\clearpage

\begin{figure}[p]
\center
\epsfig{file=H1prelim-05-133.fig7.eps,width=0.7\textwidth,angle=-90}
\caption{Extraction of the strong coupling constant $\alpha_s(M_Z)$ 
from the measured inclusive jet cross section for a given bin
in $Q^2$ and $E_{\rm t}$. The cross section is given in arbitrary units.}
\label{fig:fitting}
\end{figure}
\clearpage

\begin{figure}[p]
\center
\epsfig{file=H1prelim-05-133.fig8.eps,width=0.65\textwidth,angle=-90}
\caption{Results for $\alpha_s(M_Z)$ and $\alpha_s(E_t)$, 
where the $\alpha_s(E_t)$ values have been 
evolved from the measured $\alpha_s(M_Z)$ 
values using the two loop solution of the renormalisation group equation. 
The curves with error bands show values corresponding to
the world average $\alpha_s(M_Z)=0.1187 \pm 0.0020$.
}
\label{fig:alpha0}
\end{figure}
\clearpage

\begin{figure}[p]
\center
\epsfig{file=H1prelim-05-133.fig9.eps,width=0.65\textwidth,angle=-90}
\caption{Results for $\alpha_s(M_Z)$ and $\alpha_s(E_t)$, 
where the $\alpha_s(E_t)$ values have been evolved from 
the measured $\alpha_s(M_Z)$ values for each bin in $E_{\rm t}$
(integrated over $Q^2$)
using the two loop solution of the renormalisation group equation.
The solid curve shows the result of evolving
the averaged (over the 15 $Q^2$ - $E_{\rm t}$ bins) $\alpha_s(M_Z)$,
while the dashed curves
indicate the respective error band.
The horizontal line with the shaded error band corresponds to
the world average $\alpha_s(M_Z)=0.1187 \pm 0.0020$.
}
\label{fig:alpha}
\end{figure}

\end{document}
