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%\begin{document}
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\title*{Measurements of Jet Production\\ 
in Deep Inelastic Scattering}
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\toctitle{Measurements of Jet Production in 
Deep Inelastic Scattering}
%\protect\newline in 
%\protect\newline Deep Inelastic Scattering}
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\titlerunning{Measurements of Jet Production in Deep Inelastic Scattering}
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\author{Leif J\"onsson}
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\authorrunning{Leif J\"onsson}
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\institute{Phys. Dept., Lund University, Box 118, 
221 00 Lund, Sweden}

\maketitle              % typesets the title of the contribution

\begin{abstract}
Some recent results based on jet production at the HERA ep-collider are
presented. Inclusive jet cross sections have proven to be in good
agreement with predictions from QCD Monte Carlo models up to very high 
momentum transfer, whereas
di-jet cross sections are less well described. Recent 
next-to-leading order calculations 
turn out to reproduce the di-jet data very well which enables a direct
determination in next-to-leading order of the gluon density in the 
proton and an improved measurement of the strong coupling constant.

\end{abstract}

\section{Introduction}

Considerable progress in our understanding of how matter is constructed
has been made over the past decades mainly thanks to results from
deep inelastic scattering (DIS) experiments.
The basic idea behind these experiments dates as far back as
to Rutherford's famous experiment from 1911 in which it was proven that the 
atom contains a massive nucleus.
In modern experiments leptons are used to probe the structure of matter, 
the advantage being that the leptons, as far as we know, are pointlike 
particles and 
that their interaction with matter can be described by a well founded
theory, the electroweak theory.
The basic DIS process (zeroth order in QCD) is a purely electroweak 
interaction between a highly virtual
photon emitted by the incoming lepton and a parton in the proton
(Fig.~\ref{Feynman}a).


The HERA storage ring, which collides 27.5 GeV positrons (electrons) with
820 GeV protons, has provided a great increase in resolution by
extending the accessible kinematic range by several
orders of magnitude in the transverse momentum squared, $Q^2$, transferred 
by the virtual photon and in the Bjorken scaling variables, $x$ and $y$.
 
The scattered electron from a DIS process carries information about 
the parton density in
the proton as does also the particle jet from the scattered quark,
which is usually called the current jet. Thus an inclusive measurement 
of either the scattered electron or the current jet will 
determine the proton structure function, $F_2$. Results from HERA have
revealed a steep rise of the structure function as the fractional
parton momentum decreases. This can be explained by a model where
the emitted
gluons split up into quark-antiquark pairs and thus create a large
number of sea-quarks. The more partons sharing the proton
momentum the less fraction each of them will carry. Since the gluons
are driving the strong rise via the sea-quarks, the gluon density can
be determined from the proton structure function assuming that the
QCD splitting functions describing the parton emission are correct.

A more direct determination of the gluon density would be to
measure the rate of boson-gluon fusion (BGF) processes, which is of first
order in the strong coupling constant, $\alpha_S$. The BGF process 
is an interaction between the virtual photon and a gluon which has 
fragmented into a quark-antiquark pair (Fig.~\ref{Feynman}c), and consequently 
the final state will contain two hard jets. Another first order $\alpha_S$
process is QCD-Compton scattering, where a gluon is emitted from the
scattered quark (Fig.~\ref{Feynman}b). Being of first order in $\alpha_S$, both 
processes are thus sensitive to the strong coupling constant and can
be used to determine its value.
Even though the di-jet cross section is proportional to $\alpha_S$,
leading order (LO) calculations alone suffer from large scale
dependence. Thus higher order corrections are necessary to reduce 
the dependence of the cross section on the renormalization ($\mu_r$) and 
factorization ($\mu_f$) scales. It turns out that already NLO corrections
leads to much smaller dependences on $\mu_r$ and $\mu_f$ which
make reliable cross section calculations possible. In recent years
several Monte Carlo programs, like MEPJET ~\cite{MEPJET},
DISENT ~\cite{DISENT1} and DISASTER++ ~\cite{DISASTER},
have been developed, providing
cross section calculations on the parton level and allowing for
the application of various kinematic cuts and 
choices of jet algorithms.
In order to cancel infrared divergencies which appear in the analytic
calculations due to soft or collinear gluon emissions two methods have
been used. The MEPJET program uses the phase space slicing method
~\cite{slicing}
which requires the matrix elements for the real emission process to be
approximated in the regions close to the singularities. These regions
are defined by the introduction of non-physical cut-off parameters. 
The DISENT program is based on the subtraction method ~\cite{subtraction}
which uses the
exact matrix element expression in the whole phase space region.
In this method the singularities due to soft and collinear gluon emission 
will explicitly cancel against the poles in the virtual diagrams.
     
Since the cross section for di-jet production at HERA energies is
in the range 100 pb to few nb, enough statistics will be available 
for various precision tests of QCD.
 
\begin{figure}
\begin{center}
\rotatebox{-90}{\scalebox{0.40}{\includegraphics{joensson.1}}}
\end{center}
\caption[]{Generic Feynman diagrams for zeroth and first order
$\alpha_S$ processes.}
\label{Feynman}
\end{figure}

The kinematics of a deep inelastic scattering event can be determined
from two independent Lorentz variables, which can be any of the two
Bjorken scaling variables $x_{Bj}$ or $y$, the square of the momentum, $Q^2$, 
transferred by the exchanged boson, and the invariant mass squared of 
the hadronic system, $W^2$. These variables are defined in terms of the 
4-momenta of the incoming proton, $P$, the incoming and scattered electron,
$p_e$ and $p'_e$, and of the virtual boson, $q$.

\begin{eqnarray}
Q^2 \equiv -q^2 = -(p_e-p'_e)^2 \\
y \equiv \frac{P \cdot q}{P \cdot p_e} \\
x_{Bj} \equiv \frac{-q^2}{2P \cdot q} = \frac{Q^2}{ys} \\
W^2 \equiv (q+P)^2 = Q^2 \left( \frac{1-x_{Bj}}{x_{Bj}} \right) + m_p,
\end{eqnarray}

where $s$ is the centre-of-mass energy squared and $m_p$ is the mass of 
the proton.

Other variables which are useful in describing the kinematic properties 
of di-jet events are the 
Lorentz invariant partonic scaling variables $x_p$ and $z_p$.

\begin{eqnarray}
x_p = \frac{Q^2}{2p \cdot q} = \frac{Q^2}{2 \xi P \cdot q} = \frac{x_{Bj}}{\xi}
\approx \frac{Q^2}{Q^2+\hat{s}} \\
z_p = \frac{P \cdot p_{jet}}{P \cdot q} 
\approx \frac{E_{jet}(1-cos\theta_{jet})}{\sum_i E_i(1-cos\theta_i)},
\end{eqnarray}

where $\xi$ is the fraction of the proton momentum carried by the 
interacting parton
and $\hat{s}$ is the invariant mass squared of the hard subsystem. 
$E_{jet}$ and $\theta_{jet}$ are the energy and polar angle of the jet,
whereas $E_i$ and $\theta_i$ are the energy and polar angle of the
individual hadrons in the hard subsystem. The variable $x$ in 
Fig. ~\ref{Feynman2} represents the fraction of the longitudinal
proton momentum taken by the parton. For LO processes $x$
is equivalent to $\xi$ but for higher orders it becomes a non-measurable
quantity which can only be defined within a specific theoretical frame.
$M^2_{jj}$ is the invariant mass squared of the two jets produced in 
the hard scattering and is thus equivalent to $\hat{s}$.    

\begin{figure}
\begin{center}
\rotatebox{0}{\scalebox{0.60}{\includegraphics{joensson.1_a}}}
\end{center}
\caption[]{Generic Feynman diagram illustrating the meaning of some
kinematic variables.}
\label{Feynman2}
\end{figure}

The first order parton level process must be corrected for higher order 
perturbative QCD effects causing the emission of additional partons on
a short space-time scale before hadronisation. For example the LEPTO 
~\cite{LEPTO1} and HERWIG ~\cite{HERWIG1}
event generators calculate processes up to the first order in 
$\alpha_S$ by using exact QCD matrix elements convulated with the parton 
densities in the proton. Additional parton emission is generated using 
the DGLAP ~\cite{DGLAP1} equations in a so called parton shower evolution. 
In the ARIADNE ~\cite{ARIADNE1} Monte Carlo program gluons
are emitted according to the colour dipole model ~\cite{cdm}
from independently radiating colour dipoles which are 
stretched between the partons.

Due to the bremsstrahlung nature of this radiation, most partons will
be soft or collinear with the emitting parton and will thus not produce
separately observable jets, but rather cause softening of the hadron 
momenta and broadening of the jet. The reconstruction of a jet must
therefore account correctly for both the parton emission process and
the hadronisation process.

There are different approaches to describe the fragmentation of 
partons into hadrons. The most naive picture is to assume that
the different partons fragment independently 
(\emph{independent fragmentation} ~\cite{indep}).
This model is, however, connected with several problems like 
non-conservation of the total flavour, energy and momentum, 
and it is also not Lorentz invariant.
In the \emph{string fragmentation} ~\cite{string} picture a colour flux 
tube is streched
between a quark and an antiquark. If the transverse dimension of the
tube is assumed to be uniform along  its length, this automatically leads
to confinement with a linearly rising potential. In case a gluon is
emitted the colour string will be streched from the quark via the gluon
to the antiquark and, if several gluons are emitted, the string picture
becomes correspondingly more complicated. As the partons move apart the
energy stored in the string will increase and eventually it
will break up creating new quark antiquark pairs.
The Lund string fragmentation is for example
used in the LEPTO and ARIADNE programs.
The concept of \emph{cluster fragmentation} ~\cite{cluster} contains 
three components. The
generation of clusters by parton shower evolution, the fragmentation
of large clusters into smaller ones, and the decay of clusters into 
hadrons. The clusters are assumed to be the basic units from which
hadrons are produced. The decay is assumed to be isotropic in the 
rest frame of the cluster which gives a compact description with few 
parameters. This fragmentation model is used by the HERWIG generator.

The intuitive picture of a jet as a collimated flow of particles implies
that a minimum energy must be available for the jet formation. In
particular, the particle momenta along the initial parton direction
must be much larger than the transverse momentum fluctuations induced
in the hadronisation process. As the parton momentum increases, the jets
tend to be more collimated which simplifies their identification.
Nevertheless, the definition of a jet is not unambigous.   

A number of jet-finding algorithms have been developed to provide ways
of reconstructing jets but since the definition of a jet is not unique,
the algorithms all include a parameter which specifies the desired resolution. 
The input to a jet algorithm are 4-vectors representing particles
and/or energy clusters in the detector and as output the 4-momenta of the
jets are given. If the resolution depends on the kinematic region
in which the algorithm is applied, a suitable scaling variable has
to be found to account for this kinematic dependence. 

There are two general types of jet algorithms. 
The \emph{cone algorithms} ~\cite{cone} are based on the 
definition of a cone with its tip in the interaction point and with
a predefined opening angle, normally defined through 
$R = \sqrt{\eta^2+\phi^2}$, where $\eta$ is the pseudorapidity and
$\phi$ the azimuthal angle. 
The cone is swept over the solid angle covered by
the detector and a jet is defined by maximizing the total energy 
inside the cone.
The \emph{cluster algorithms} calculate the distances between 
pairs of particles or energy clusters using some distance measure
which is specific to each algorithm. In an iterative process 
pairs may be merged until the distance measure exceeds a predefined
cut-off value, specifying the resolution of the algorithm. 
The jet algorithms which will be mentioned here are two versions of
the $k_T$ algorithm and the modified Durham algorithm, which are
all cluster algorithms.
The factorizable $k_T$ algorithm ~\cite{factkt} uses two distance 
measures to determine whether
a particle should be merged with another particle or with the proton
remnant which is represented by an infinite momentum particle.
The distance measures are defined as 
$k_{T,ij}^2 = 2min(E_i^2, E_j^2)(1-cos\theta_{ij})$ and 
$k_{T,ir}^2 = 2E_i^2(1-cos\theta_{ir})$,
where $i$ and $j$ represent particles and $r$ the proton remnant.
The longitudinally invariant $k_T$ algorithm ~\cite{longkt} uses the same 
procedure but with cone like distance measures;
$k_{T,ij}^2 = 2min(E_{T,i}^2, E_{T,j}^2)(\Delta\eta_{ij}^2+\Delta\phi_{ij}^2)$
and $k_{T,ir}^2 = 2E_{T,i}^2$
The modified Durham algorithm is a modification of the JADE
algorithm ~\cite{JADE} with one distance measure, 
$k_{T,ij}^2 = 2min(E_i^2, E_j^2)(1-cos\theta_{ij})$ and the cut-off parameter 
scaling with $W^2$.        

\section{Inclusive jet cross sections}

The production of multi-jet events at high $Q^2$ can be used to 
test the predictions by perturbative QCD. The ZEUS collaboration
has used a data sample corresponding to 42.5 ~pb$^{-1}$ to study
inclusive jet production. The jets were reconstructed in the
laboratory system using the 
longitudinally invariant $k_T$ algorithm on clusters in the 
calorimeter. Reconstructed jets were required 
to have a transverse momentum greater 14 GeV and only jets within
the pseudorapidity range $-1 < \eta_{LAB} < 2$ were accepted, since 
this is the region where the calorimeter offers good acceptance.
As an example of the comparisons between data and QCD models
the differential jet cross section $d\sigma/dQ^2$ is shown in
Fig~\ref{inclusive}a and b. It is seen that ARIADNE describes the data very
well up to the highest $Q^2$ bin where a slight deviation can be
observed. This is, however, probably due to the fact that the
version of ARIADNE which was used doesn't give a proper description
of the jet cross sections at high $Q^2$. It should also be kept in
mind that since the jet reconstruction has been performed in the 
laboratory frame the tranverse energies of the jets are strongly 
influenced by event kinematics. Furthermore the event sample is
dominated by zeroth order $\alpha_S$ events and therefore the
sensitivity to QCD effects is suppressed.  

\begin{figure}
\begin{center}
\rotatebox{0}{\scalebox{0.40}{\includegraphics{joensson.2}}}
\put(-120,50){\Large $a)$}
\rotatebox{0}{\scalebox{0.40}{\includegraphics{joensson.3}}}
\put(-120,50){\Large $b)$}
\end{center}
\caption[]{The differential inclusive jet cross section 
$d\sigma/dQ^2$ compared to the predictions by ARIADNE.}
\label{inclusive}
\end{figure}

\section{A direct determination of the gluon density in NLO}

A direct determination of the gluon density is not only motivated 
by its own interest, but a comparison with indirect
measurements constitutes an important test of QCD.  
The lack of cross section calculations
to next-to-leading order restricted an earlier analysis to a LO
determination ~\cite{gluon}. Since then several NLO programs, to
which various jet algorithms and also experimental cuts can be
applied,
have become available. Although the QCD-Compton process has the same
event topology as the BGF process and thus constitutes a background, 
the BGF process dominates in the kinematic region investigated. 

A recent H1 analysis was based on an integrated luminosity of 36 ~pb$^{-1}$
collected in the years 1994-97. It was performed in a kinematic region 
which covers a wide range in the transverse momentum transfer squared, 
$10 < Q^2 < 5000$ ~GeV$^2$, and
was restricted to $0.2 < y < 0.6$ in order to get a good determination 
of $y$ from the measurement of the scattered electron and to suppress 
backgrounds from photoproduction processes. The jet reconstruction 
was performed in the Breit frame using the longitudinally invariant 
$k_T$ algorithm, since this turns out to give the smallest
hadronisation corrections, as determined from Monte Carlo
models. The jets were required to have a transverse energy sum,
$E_{T,jet1}+E_{T,jet2}$, greater than 17 GeV in the Breit frame and 
simultaneously the transverse energy of each jet had to be greater 
than 5 GeV. With the additional cut $-1 < \eta_{jet} < 2.5$ in the
laboratory system, it was ensured that the jets were well inside the
hadronic calorimeter. The data were corrected for detector effects
and initial and final state electroweak radiation by performing
detailed simulations using DJANGO ~\cite{DJANGO}, which provides an 
interface between the ARIADNE and LEPTO events generators, repectively
and HERACLES ~\cite{HERACLES}.  

Since hadronisation corrections are small, the data have been compared 
directly to a NLO calculation
performed with the DISENT program, using the 
CTEQ5M ~\cite{CTEQ5M} parton density
parametrization and its corresponding $\alpha_S(M_Z^2)$ value of 0.116.
The factorization scale was chosen to be $\mu_f^2 = 200$ ~GeV$^2$, 
which is approximately equal to the average transverse energy 
squared of the jets in the Breit frame. $Q^2$ was taken as the
renormalization scale squared, $\mu^2_r$. Fig.~\ref{gluon23}a shows the 
corrected differential cross section as a function of the jet
scattering angle, defined
as $\eta' = \frac{\eta _{1,Breit} - \eta _{2,Breit}}{2}$,
for three regions in $E_{T,Breit}$ of the jets.  
The agreement between the data and the NLO prediction is good 
and it can be observed that the jet production at larger values of
pseudorapidity, $\eta'$,
is suppressed as $E_T$ increases. In Fig.~\ref{gluon23}b the differential 
cross section is plotted against the fractional parton momentum, $\xi$,
in four different $Q^2$ bins for comparisons of the data with both the
NLO calculation and QCD model predictions. Again the NLO prediction
reproduces the data, whereas at low $Q^2$ the LEPTO predictions fall
below the data and at high $Q^2$ ARIADNE underestimates the cross
section. HERWIG with a one loop $\alpha_S$ calculation, similar to 
LEPTO and ARIADNE, is unable to describe the data everywhere.
It should also be noted that the data tend to overshoot the
NLO prediction in the lowest $\xi$ bin, which may be an indication
that the gluon density has been underestimated in the calculation.

\begin{figure}
\begin{center}
\rotatebox{0}{\scalebox{0.5}{\includegraphics{joensson.4}}}
\put(-20,50){\Large $a)$}
\rotatebox{0}{\scalebox{0.5}{\includegraphics{joensson.5}}}
\put(-20,50){\Large $b)$}
\end{center}
\caption[]{The doubly differential di-jet cross section 
a) $d\sigma/dE_td\eta '$ and b) $d\sigma/dQ^2d\xi$ compared to the predictions
by a NLO calculation and QCD models.}
\label{gluon23}
\end{figure}

The jet cross section can be written as a convolution of perturbative
coefficients with the parton densities in the proton:        

\begin{equation}
\sigma_{jet} = \sum_{n} \alpha_S^n(\mu_r^2)
               \int_0^1 dx [c_{g,n}(\mu_r^2,\mu_f^2) g(x,\mu_f^2) +  
               \sum_{f} c_{f,n} (\mu_r^2,\mu_f^2) q(x,\mu_f^2)],
\end{equation}

where the cross section is calculated to all orders in $\alpha_S$ and
for all quark flavours $f$. The perturbative coefficients are denoted
$c_{g,n}$ for the gluon and $c_{f,n}$ for the quarks and the parton density 
functions are written $g(x,\mu_f^2)$ and $q(x,\mu_f^2)$ for the
gluon and the quarks, respectively. The formula illustrates the 
correlation between the strong coupling constant and the parton densities,
which prevents an unambigous determination of the gluon density from
a deep inelastic measurement alone. In order to circumvent this 
problem the $\alpha_S(M_Z^2)$ value of $0.119 \pm 0.005$ from measurements
at LEP ~\cite{alphaslep} was used.

The perturbative coefficients are not all independent but can be reduced 
to three independent coefficients by linear combinations of the single
flavour coefficients.

\begin{eqnarray}
c_G = c_g , \nonumber \\
c_{\Delta} = 3(c_u-c_d) , \nonumber \\
c_{\Sigma} = 1/3(4c_d-c_u).
\setcounter{eqsubcnt}{0}
\end{eqnarray} 

This results in three independent parton densities:

\begin{eqnarray}
\Delta(x,\mu_f) = \sum_f e_f^2(q_f(x,\mu_f)+\overline{q}(x,\mu_f)), \nonumber \\
G(x,\mu_f) = g(x,\mu_f), \nonumber \\
\Sigma(x,\mu_f) = \sum_f (q_f(x,\mu_f)+\overline{q}(x,\mu_f)). 
\setcounter{eqsubcnt}{0}
\end{eqnarray}

Table~\ref{table1} summarizes which parton densities contribute to which order
in $\alpha_S$ for the inclusive, $\sigma^{incl}$, and the di-jet,
$\sigma^{dijet}$, cross sections, respectively.

\begin{table}
\caption{LO and NLO contributions of the various parton densities to
the inclusive and di-jet cross sections, respectively.}
\begin{center}
\renewcommand{\arraystretch}{1.4}
\setlength\tabcolsep{15pt}
\begin{tabular}{@{}llp{1.8cm}l}
\hline\noalign{\smallskip}
 & LO & NLO \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
$\sigma^{incl}$ & $\Delta(x)$ & $\Delta(x),G(x)$ \\
$ \sigma^{dijet}$ & $G(x),\Delta(x)$ & $G(x),\Delta(x),\Sigma(x)$ \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
\end{tabular}
\end{center}
\label{table1}
\end{table}

With the value of $\alpha_S(M_Z^2)$ fixed to $0.119 \pm 0.005$ a
simultaneous fit of the gluon and quark densities in the proton
was performed using data both on inclusive DIS and on di-jet 
production. The $Q^2$ region for the inclusive data was restricted
to $200 < Q^2 < 650$ ~GeV$^2$, which is approximately equal to
the jet transverse energy squared in the di-jet data. 
This also motivates a fixed value of 200 GeV$^2$ for the 
factorization scale, $\mu^2_f$. In this kinematic 
region the inclusive data constrain the quark density strongly,
whereas the di-jet data is mainly sensitive to the gluon density.
The di-jet events were required to have $Q^2 > 200$ ~GeV$^2$ in order
to avoid large NLO corrections and hadronisation effects.
For the di-jet data fits were made to the double differential 
cross sections $d\sigma/dQ^2d\xi$ and $d\sigma/dQ^2dx_{Bj}$,
and for the inclusive data to $d\sigma/dQ^2dx_{Bj}$, since they
are the most sensitive ones to the $x$-dependence of the parton densities.

In Fig.~\ref{gluon1} the result of the gluon density fit is compared to global
fits using parton density parametrizations according to CTEQ4M,
MRST and GRV94HO, and to a QCD analysis of the $F_2$ structure
function as measured by the H1 collaboration. The direct measurement 
indicates a somewhat steeper rise with decreasing $x$ than the
global fits although the results are compatible within the error
band. On the other hand, the direct measurement is in good agreement
with the H1 QCD analysis of $F_2$. The errors are dominated by the
uncertainty in the energy scales of the electromagnetic and hadronic
calorimeters, and in the determination of the luminosity.

\begin{figure}
\begin{center}
\rotatebox{-0}{\scalebox{0.70}{\includegraphics{joensson.6}}}
\end{center}
\caption[]{The gluon density in the proton. The result of the direct 
determination is compared to global fits with various parton density 
parametrizations and to a QCD analysis of the proton structure 
function $F_2$.}
\label{gluon1}
\end{figure}

\section{A study of differential di-jet rates}

In an H1 analysis differential di-jet rates have been used for
comparisons with various QCD models and to a NLO prediction.
The differential di-jet rate is defined as the number of di-jet events
which is obtained with 
a predefined cut-off value for the resolution parameter, $y_{cut}$,
in the jet algorithm, divided by the total number of DIS events. 
In an iterative process the jet candidates with
the smallest invariant masses are combined until exactly two hard
jets remain. The maximum value of the resolution parameter at which
two hard jets are resolved is denoted $y_2$.

The data used for the analysis by H1 were collected in the years 1995-97
and correspond to an integrated luminosity of 35 pb$^{-1}$.
The events were restricted to the kinematic range $Q^2 > 150$ ~GeV$^2$
and $0.1 > y > 0.7$ which results in a data sample of 60000 events.
The statistics allow a subdivision into three different ranges in
$Q^2$.

Two jet algorithms, the factorizable $k_T$-algorithm and
the modified Durham algorithm, have been used in the study. The 
$k_T$-algorithm was applied in the Breit frame with a cut-off value
of $y_2>0.8$, which approximately corresponds to a mean 
transverse jet energy of 10 GeV in the Breit frame. The modified
Durham algorithm was used in the laboratory system. The 
resolution parameter was given by $y_2 = k_T^2/W^2$, and the cut-off
value was chosen to be $y_2 = 0.005$, which again corresponds to a mean
transverse jet energy of 10 GeV in the Breit frame. For both 
algorithms the jets were restricted to an angular region 
$10^o < \theta_{LAB} < 140^o$ to ensure good detector acceptance.
In addition, jets reconstructed below $7^o$ by the Durham algorithm
were rejected.

The di-jet distributions presented have been corrected for detector
effects and the influence of QED radiation. The major systematic
error comes from the uncertainties in the energy scales of the 
electromagnetic and hadronic calorimeters.

Data have been compared to the LEPTO and ARIADNE models and to 
a NLO calculation performed with the DISENT program using $Q^2$
as renormalization scale. The hadronisation effects have been 
estimated from simulations with the QCD models and were found 
to be between $10\%$ and $20\%$.

Fig.~\ref{tobien1}a shows that the $y_2$ distributions are in good 
agreement with both
Monte Carlo models and the NLO prediction in all three $Q^2$ bins. 
The spectra get harder as $Q^2$ increases. The $z_p$ distributions,
however, are not so well described by the models which tend to
overshoot the data as seen in Fig.~\ref{tobien1}b. Again the NLO calculation
reproduces the data. In Fig.~\ref{tobien3} the distribution of the 
polar angle
for the most forward going jet is plotted and compared to the NLO prediction
where also the BGF fraction is indicated. At high $Q^2$, where the
BGF fraction is of the order of $30\%$, the agreement is good,
whereas in the low $Q^2$ region, which contains $70\%$ BGF, the
NLO calculation seems to fall below the data point in the lowest
$\theta$ bin. This might indicate an underestimation of the gluon 
density in the NLO calculation.   

\begin{figure}
\begin{center}
\rotatebox{-90}{\scalebox{0.70}{\includegraphics{joensson.7}}}
\put(-230,-280){\Large $a)$}
\put(-40,-280){\Large $b)$}
\end{center}
\caption[]{The differential di-jet rates as a function of a) $y_2$
and b) $z_p$ compared to a NLO calculation and QCD models.}
\label{tobien1}
\end{figure}

\begin{figure}
\begin{center}
\rotatebox{0}{\scalebox{0.40}{\includegraphics{joensson.8}}}
\end{center}
\caption[]{The distribution in polar angle for the most forward
going jet compared to a NLO calculation. The contribution from the BGF 
process is indicated.}
\label{tobien3}
\end{figure}

\section{Measurement of the strong coupling constant \protect\newline from  
di-jet rates}

The ZEUS collaboration has used data collected in the years 1996-97
to investigate di-jet rates in deep inelastic scattering
with the aim of determining the strong coupling constant, $\alpha_S$.
The data correspond to an integrated luminosity of 38 pb$^{-1}$. 
In order to avoid experimental and theoretical difficulties as far as
possible the analysis was performed in the high $Q^2$ region, 
$470 < Q^2 < 20000$ ~GeV$^2$. 
The event kinematics was extracted using the double angle method 
~\cite{2-angle} 
which is the method least sensitive to errors in the absolute energy 
scale of the calorimeter. It was also proven to be the most accurate 
one in the $Q^2$ region investigated.

The jet reconstruction was performed in the Breit frame using the
longitudinally invariant $k_T$-algorithm. Energy clusters in the 
calorimeter were used as jet seeds, and the jet reconstruction
procedure followed the Snowmass scheme.
Events with exactly two jets were selected, where both jets were 
required to have energies above 5 GeV and their energy sum to be 
above 17~GeV. This cut scenario avoids problems with divergencies, 
related to soft and collinear gluon emissions, in fixed order
calculations.
In order to ensure that the jets were well inside the acceptance 
of the detector they were restricted to the rapidity range
$-1 < \eta_{LAB} < 2$.

Monte Carlo simulations have been used to correct the data for
detector inefficiences and to take effects from QED radiation and
hadronisation into account. Further, contributions from $Z^o$ exchange
had to be included since the $Q^2$ range extends as far up as 20000 $GeV^2$.
Comparisons of the data have been made with two versions of the 
DJANGO program based on the LEPTO and ARIADNE Monte Carlo generators.   
Also the HERWIG program was selected for comparisons since it uses a 
different fragmentation algorithm to describe the hadronisation.

For calulations of NLO cross sections the DISENT program was used 
and it was checked to be consistent with the results of the DISASTER++
program to within a few percent.

Various control plots were produced in order to see how well the data
were reproduced by Monte Carlo models and the NLO prediction. 
Fig. ~\ref{alphas2} 
shows the differential di-jet cross section as a function of the 
the transverse energies, $E_t$, and rapidities, $\eta$,
of the two jets in the event. The uncertainty due to the hadronic 
energy scale of the calorimeter is the dominant systematic error.
The error bars contain contributions from all other systematic
errors and the statistical errors added in quadrature. 
The NLO calculations have been performed by DISENT in the 
$\overline{MS}$
renormalization scheme using the parton density parametrization 
according to CTEQ4M ~\cite{CTEQ4M}
and choosing $\mu_R^2$ = $\mu_F^2$ = $Q^2$ as 
renormalization and factorization scales. 
A closer look at the renormalization scale dependence of the NLO 
calculation motivates the lower $Q^2$ cut of 470 GeV$^2$ above which
the dependence is very small. Also the dependence on the parton
density functions is small in this region.
An excellent agreement between data and the NLO calculation is observed
in all plots, which is a necessary condition for a reliable extraction
of the strong coupling constant.

\begin{figure}
\begin{center}
\rotatebox{0}{\scalebox{0.70}{\includegraphics{joensson.9}}}
\put(-230,160){\Large $a)$}
\put(-90,160){\Large $b)$}
\put(-230,30){\Large $c)$}
\put(-90,30){\Large $d)$}
\end{center}
\caption[]{The di-jet cross sections a) $d\sigma/dE_{t1}$,
b) $d\sigma/d\eta_1)$, c) $d\sigma/dE_{t2}$ and d) $d\sigma/d\eta_2$
compared to the predictions by the NLO calculation.}
\label{alphas2}
\end{figure}

In each $Q^2$-bin the ratio of di-jet events, 
$R_{2+1} = \sigma_{2+1}/\sigma_{tot}$, 
was calculated with the DISENT program. Five different values of the 
strong coupling constant, ranging from 0.110 to 0.122, were used in 
the CTEQ4M parametrization of the
parton density functions. The MRST parametrization was also
used for a consistency check. The di-jet rates obtained for the five
$\alpha_S$ values were fitted with a straight line in each $Q^2$-bin
to allow for a continuous variation of $\alpha_S$ with the di-jet rate.
The $\alpha_S$ value was then extracted through a $\chi^2$ fit of the
NLO prediction to the measured di-jet ratio according to:

\begin{equation}
\chi^2=\sum_{i=1}^{5} \frac{(R_{2+1,meas}^{i} - R_{2+1,theor}^{i})^2}
{\sigma_{2+1}^{2,i}}
\end{equation}    

The value obtained is:

\begin{equation}
\alpha_S(M_Z^2) = 0.120 \pm 0.003 (stat)
\end{equation}

A summary of the systematic errors are given in Fig.~\ref{alphas1}. As has been
mentioned previously, and which is clearly illustrated in the figure,
the by far dominant systematic error comes from the uncertainty
in the jet energy scale. Another contribution to the experimental 
systematic error is due to the model dependence. The uncertainty due to a
variation of the renormalization scale between $1/2Q^2$ and $2Q^2$, and
the dependence on the parton density functions and the NLO program 
used, define the theoretical systematic error. The final result including
all errors is:

\begin{equation}
\alpha_S(M_Z^2) = 0.120 \pm 0.003 (stat) \pm_{0.006}^{0.005} (exp) 
\pm_{0.003}^{0.002} (th) 
\end{equation}

\begin{figure}
\begin{center}
\rotatebox{0}{\scalebox{0.40}{\includegraphics{joensson.10}}}
\end{center}
\caption[]{Contributions to the systematic errors on the measured
$\alpha_S$ value.}
\label{alphas1}
\end{figure}

\subsection{Summary}

Jet production in DIS provides an interesting testing ground of
perturbative QCD, which in many respects offers higher sensitivity 
and contains more information about the proton structure than the 
inclusive measurements of the proton structure function. 
Over the past few years quite some progress has been made in the 
description of jet production in deep inelastic scattering. Several
Monte Carlo programs for calculations of 
next-to-leading order $\alpha_S$ jet cross sections have been developed,
which allow different jet algorithms to be applied and experimental
cuts to be defined. Although the new programs offer quantitative
agreement at large $Q^2$, deviations were observed in the low $Q^2$ 
region. Only recently an explanation of these data could be achieved
both by a Monte Carlo model ~\cite{rapgap} and by a NLO calculation
~\cite{NLOres} by, in addition to interactions with a pointlike 
photon, also including contributions where the interaction takes place 
with the parton content of the photon.

\subsection{Acknowledgement}

This presentation was given on behalf of the H1 and ZEUS collaborations.
I would like to thank S. Schlenstedt, L. Sinclair, E. Tasso, M. Weber,
N. Tobien and M. Wobisch for providing me with the relevant material.

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