% proceedings contribution ``Jet Production in DIS''
% Dirk Graudenz
% Ringberg 1999
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\begin{document}

\newcommand{\myabstract}{
The status of next-to-leading-order corrections to jet cross sections
in deeply inelastic scattering is reviewed. 
I give a brief introduction into the techniques which are used to calculate
jet cross sections, and make a few remarks on a recent program comparison.
A third part gives a few results of recent measurements of the strong
coupling constant $\alpha_s(\mu^2)$ and the gluon density 
$f_{g/P}(\xi, \mu_f^2)$.
}

\thispagestyle{empty}
\noindent
\begin{minipage}[t]{10cm}
{\it
\small
Talk given at the Ringberg workshop\\ ``New Trends in HERA Physics 1999''
}
\end{minipage}
\hfill
\hspace*{\fill}
\begin{minipage}[t]{4cm}
\small
PSI--PR/99--25\\
September 1999 
\end{minipage}

\vspace{2cm}

\begin{center}
{
\Large\bf
Jet Production in DIS
}
\end{center}

\vspace{1cm}

\begin{center}
{
Dirk Graudenz\\
\bigskip
Paul Scherrer Institut\\
5232 Villigen PSI\\
Switzerland
}
\end{center}

\vspace{2cm}

\begin{center}
{
\bf
Abstract
}
\end{center}

\myabstract

\newpage
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%\title*{Focusing of a Parallel Beam to Form\protect\newline a Point
%in the Particle Deflection Plane}
\title*{Jet Production in DIS}
%
%
\toctitle{Jet Production in DIS}
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\titlerunning{Jet Production in DIS}
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% to fit in the running head
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\author{Dirk Graudenz}
%
\authorrunning{Dirk Graudenz}
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% please abbreviate author list for running head
%
%
\institute{Paul Scherrer Institut, 5232 Villigen PSI, Switzerland\\
Electronic mail: {\tt Dirk.Graudenz@psi.ch}\\
WWW: {\tt http://www.hep.psi.ch/graudenz/index.html}
}

\maketitle              % typesets the title of the contribution

\begin{abstract}
\myabstract
\end{abstract}


\section{Introduction}
\label{introduction}

The increased integrated luminosity at HERA permits the study of 
hadronic final states of deeply-inelastic processes in great detail 
\cite{H1alphas,ZEUSalphas}. Earlier analyses of this kind of process
suffered mainly from the poorly understood emission of particles
in the forward region. The situation is improved now, because a larger
event sample allows for harder cuts on the photon virtuality $Q$ and the 
transverse momentum
$E_T$ of the jets, the latter cut excluding jets in the forward region.

Besides testing QCD, the measurement of (2+1)-jet final states has two main
applications: the measurement of the strong coupling constant
$\alpha_s(\mu^2)$ and the determination of the proton's gluon density
$f_{g/P}(\xi, \mu_f^2)$. Figure~\ref{lodiagram} shows the basic two 
leading-order processes which contribute to (2+1)-jet final states. Diagram 
(a) is the so-called QCD-Compton process, and (b) is called boson--gluon
fusion\footnote{There are, of course, two additional diagrams: the gluon 
in (a) attached to the outgoing quark, and the diagram with quark and 
antiquark exchanged in (b).}.
The corresponding cross sections 
are of ${\cal O}(\alpha_s)$, and thus the strong
coupling constant can be measured, preferably via the (2+1)-jet rate
$R_{\mbox{\small 2+1}} = \sigma_{\mbox{\small 2+1}}
/\sigma_{\mbox{\small tot}}$, if the parton densities are assumed to be known.
Conversely, the cross section of
process (b) is a convolution of a hard scattering cross section
with the gluon density
$f_{g/P}(\xi, \mu_f^2)$. By 
subtraction of the well-known quark-initiated contribution and a suitable
unfolding procedure, the gluon density can be extracted.

Because of the scale dependence of the strong coupling constant $\alpha_s$ 
and the parton densities $f_{i/P}$, perturbative calculations have to be
done beyond the leading order to be physically meaningful. Virtual corrections
suffer from two problems. Large loop momenta introduce ultraviolet 
singularities, which are removed by renomalisation. Small loop momenta
lead to infrared singularities. These cancel, in principle, against
divergences from the real corrections, but they lead to certain technical 
problems in the calculations. 

\begin{figure}
\unitlength1mm
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\begin{center} 
\begin{picture}(125,90)(0,0)
%\lattice
\SetWidth{0.5}

%% quark diagram

% the proton blob
%\SetColor{\cproton}
\CArc(35,20)(8,0,360)
\Line( 0,20)(27,20)
\Line(43,19)(55,19)
\Line(43,21)(55,21)

% the quark line
%\SetColor{\cquark}
\ArrowLine(35,28)(35,39)
\ArrowLine(35,39)(35,50)
\ArrowLine(35,50)(55,70)

% the photon line
%\SetColor{\cphoton}
\Photon(35,50)(20,65){3}{4}

% the lepton line
%\SetColor{\clepton}
\ArrowLine( 0,65)(20,65)
\ArrowLine(20,65)(40,85)

% the hard gluon
%\SetColor{\cgluon}
\Gluon(35,39)(55,39){2.5}{3}

% vertices
%\SetColor{Black}
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\Vertex(20,65){1.0}
\Vertex(35,39){1.0}

%\Text(10,10)[bl]{perturbative QCD}
%\Text(15, 2)[bl]{$Q^2 \gg \Lambda_{\mbox{\small QCD}}^2$}

\Text(35,20)[]{\Large $f_{q/P}$}

%% gluon diagram

% the proton blob
%\SetColor{\cproton}
\CArc(105,20)(8,0,360)
\Line(70,20)(97,20)
\Line(113,19)(125,19)
\Line(113,21)(125,21)

% the quark line
%\SetColor{\cquark}
\ArrowLine(125,29)(105,39)
\ArrowLine(105,39)(105,50)
\ArrowLine(105,50)(125,60)

% the photon line
%\SetColor{\cphoton}
\Photon(105,50)(90,65){3}{4}

% the lepton line
%\SetColor{\clepton}
\ArrowLine(70,65)(90,65)
\ArrowLine(90,65)(110,85)

% the inc. gluon
%\SetColor{\cgluon}
\Gluon(105,28)(105,39){2.5}{1}

% vertices
%\SetColor{Black}
\Vertex(105,50){1.0}
\Vertex(90,65){1.0}
\Vertex(105,39){1.0}

\Text(105,20)[]{\Large $f_{g/P}$}

%\SetColor{Black}
\Text(35,3)[b]{\Large(a)}
\Text(105,3)[b]{\Large(b)}

\end{picture}  
\end{center}
\caption[]{
The leading-order processes for (2+1)-jet production:
(a) the QCD-Compton diagram, (b) the boson--gluon fusion diagram.
} 
\label{lodiagram}
\end{figure}

The outline of this paper is as follows: in Section~\ref{qcdatwork} I describe 
the cancellation mechanism of infrared divergences in jet cross sections, 
and make a few remarks concerning the techniques used to achieve this.
In Section~\ref{jetcalc}, the available calculation for jet cross sections
in DIS are discussed and compared. Recent results of data analyses of jets
at HERA and a general discussion of scale choices are given in 
Section~\ref{data}.

\section{QCD at Work}
\label{qcdatwork}

QCD is the microscopic theory of the strong interactions. Perturbative QCD, 
the evaluation of the theory at large scales for small values of the strong 
coupling constant, is very successful in describing high-energy scattering
processes. Except for very inclusive observables in e$^+$e$^-$
annihilation, however,
perturbation theory is not
sufficient to arrive at phenomenological predictions, because 
non-perturbative input
is required. Factorization theorems of QCD \cite{CSS89}
guarantee that the non-perturbative
part of the description is process-independent. In DIS, for example, 
parton densities $f_{i/P}(\xi, \mu_f^2)$
are required in order to describe the distribution of partons
in the proton.

\begin{figure}[bht]
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\Line(43,31)(85,31)

% the quark line
%\SetColor{\cquark}
\ArrowLine(35,38)(35,49)
\ArrowLine(35,49)(35,60)
\ArrowLine(35,60)(60,85)

% the photon line
%\SetColor{\cphoton}
\Photon(35,60)(20,75){3}{4}

% the lepton line
%\SetColor{\clepton}
\ArrowLine( 0,75)(20,75)
\ArrowLine(20,75)(45,100)

% the hard gluon
%\SetColor{\cgluon}
\Gluon(35,49)(55,49){2.5}{3}

% vertices
%\SetColor{Black}
\Vertex(35,60){1.0}
\Vertex(20,75){1.0}
\Vertex(35,49){1.0}

% dashed line
%\SetColor{Black}
\DashLine(50,5)(50,120){3}

% the PS part
%\SetColor{\cquark}
\ArrowLine(60,85)(73,98)
\ArrowLine(73,98)(85,110)
\ArrowLine(70,65)(85,80)
\ArrowLine(85,50)(70,65)
%\SetColor{\cgluon}
\Gluon(55,50)(70,65){2.5}{3}
\GlueArc(85,85)(45,-130,-90){2.5}{6}
\Gluon(60,85)(85,85){-2.5}{5}
\Gluon(73,98)(85,98){-2.5}{2}

% more vertices
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\Vertex(73,98){1.0}
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\Vertex(70,65){1.0}

% arcs
\SetWidth{0.2}

\CArc(60,85)(14,12,40)
\CArc(60,85)(15,12,40)

\CArc(73,98)(10,10,40)
\CArc(73,98)(11,10,40)

\CArc(55,50)(14,-11,31)
\CArc(55,50)(15,-11,31)

\CArc(70,65)(10,-30,30)
\CArc(70,65)(11,-30,30)

\SetWidth{0.5}

% the fragmentation box
%\SetColor{Black}
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\Line(90,20)(100,20)
\Line(105,25)(105,115)
\Line(90,120)(100,120)
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\CArc(100,25)(5,-90,0)
\CArc(90,115)(5,90,180)
\CArc(100,115)(5,0,90)

% outgoing particles

\Line(105,35)(120,31)
\Line(105,37)(120,37)
\Line(105,39)(120,43)

\Line(105,80)(120,76)
\Line(105,82)(120,82)
\Line(105,84)(120,88)

\Line(105,100)(120,96)
\Line(105,102)(120,102)
\Line(105,104)(120,108)

\Text(10,10)[bl]{perturbative QCD}
\Text(15, 2)[bl]{$Q^2 \gg \Lambda_{\mbox{\small QCD}}^2$}

\Text(57,10)[bl]{parton cascade \& fragmentation model}
\Text(80, 2)[bl]{$s_{ij} \sim \Lambda_{\mbox{\small QCD}}^2$}

\Text(5,33)[]{\Large\it P}
\Text(5,79)[]{\Large\it l}
\Text(45,105)[]{\Large\it l$\,^\prime$}
\Text(35,30)[]{\Large $f_{i/P}$}

% regions
\SetWidth{0.1}
%\SetColor{Black}

%\CCirc(15,50){5}{Black}{Yellow}
\BCirc(15,50){5}
\Text(15,50)[]{\Large I}

%\CCirc(65,105){5}{Black}{Yellow}
\BCirc(65,105){5}
\Text(65,105)[]{\Large II}

%\CCirc(95,80){5}{Black}{Yellow}
\BCirc(95,80){5}
\Text(95,80)[]{\Large III}

\SetWidth{0.5}

\end{picture}  
\end{center}
\caption[]{
The standard description of processes in QCD phenomenology consists of
three stages.
Processes are modelled by a hard scattering process (I), followed by
a parton cascade (QCD in the collinear / leading logarithmic approximation,
II) and a phenomenological fragmentation model (III). The quantity
$s_{ij}$ is the invariant mass of a typical pair of
partons after the splitting
process.
} 
\label{sdesc}
\end{figure}

The standard description of processes in QCD is illustrated in
Fig.~\ref{sdesc}. A hard scale, in DIS usually the photon virtuality $Q^2 \gg 
\Lambda_{\mbox{\small QCD}}^2$, allows for a perturbative calculation 
in fixed order of perturbation theory (I). In the case of generators, a parton
cascade (II) is appended, which 
simulates QCD in the collinear limit of parton branching processes. Finally, 
in order to obtain final-state hadrons instead of partons, a phenomenological
fragmentation model (III), such as the Lund string model 
\cite{lundmodel}, is used. 
Steps (II) and (III) involve certain choices of parameters, for example 
cut-off parameters for the parton cascade. In contrast, for step (I) there
is, except for the parton density parametrization $f_{i/P}$,
essentially just one parameter determining the process: the strong coupling
constant.
In QCD generators, the matrix elements (I) are usually calculated in 
leading order only --- it is not known how to append a parton cascade
in full generality and 
consistently to a calculation beyond leading order\footnote{
There is a recent proposal for a matching procedure, see Ref.~\cite{FS99}.
}.
The virtue of a generator is that it produces a final state as it is seen in
an experiment, and that hadronization effects are included. The main drawback
is that generator results usually depend on a large number of tuned 
parameters. 

A conceptually cleaner approach is to rely on a fixed-order QCD calculation
with a suitable observable that can be evaluated both for parton and hadron
final states. This is presently the standard procedure for data analysis
of DIS hadronic final states. Generators are used to determine correction
factors from the hadron to the parton level. The spread from various 
fragmentation models gives a contribution to the systematic error of a 
measurement. Observables are chosen according to the criterion of small
hadronization corrections. Another important property for suitable
observables is {\it infrared-safety}. Together with the factorization theorems
of QCD, this property makes sure that expectation values of observables 
calculated in perturbative QCD are finite. Infrared-safe observables behave
well in soft and collinear limits: collinear parton splitting and radiation of
soft partons do not change the value of an observable. Incidentally, this is 
also a property that good experimental observables are supposed to have. 
Calorimetric measurements are insensitive to modifications of the final state
of this kind. For more details, I wish to refer the reader to 
Refs.~\cite{Gra97b,Gra98}.


Cancellation of infrared singularities occurs between real and virtual 
corrections, a mechanism originally discovered in QED. The main problem in 
next-to-leading-order calculations is to perform the calculation 
of the real corrections without 
knowing the specific form of the 
observable to be calculated. There are two basic 
methods to do this: the phase-space-slicing method and the subtraction 
method. Here I do not have enough space to describe these methods in detail, 
descriptions can be found in Refs.~\cite{KS92,Gra97b} and the references cited
therein. I just mention 
that for phase-space slicing a small technical cut-off parameter \ymin{}
has to be introduced. Partons are considered to be technically unresolvable
if a quantity such as their scaled invariant mass is smaller than this cut-off.
Infrared singularities from unresolved configurations of this kind cancel
explicitly. The method is observable-independent, because infrared-safe
observables are insensitive to unresolved partons. Contrary to this, 
the cancellation of IR singularities in the subtraction method is done
locally in phase space. No technical cut-off is required, but the construction
of the required subtraction terms is cumbersome.

\begin{figure}[htb]
\unitlength1mm
\SetScale{2.8346}
\begin{center} \begin{picture}(80,70)(0,0)
%\lattice
\SetWidth{0.5}
%\SetColor{\cquark}
\ArrowLine(10,10)(30,30)
\ArrowLine(30,30)(55,55)
%\SetColor{\cgluon}
\Gluon(30,30)(60,30){-2.5}{5}
%\SetColor{Black}
\Vertex(30,30){1.0}
\CArc(30,30)(25,8,43)
\CArc(30,30)(26,8,43)
\Text(7,7)[]{{\it \Large q}}
\Text(57,57)[]{{\it \Large q}}
\Text(65,28)[]{{\it \Large g}}
%\SetColor{Blue}
\Text(17,24)[]{\Large $p$}
\Text(38,48)[]{\Large $u\cdot p$}
\Text(45,22)[]{\Large $(1-u)\cdot p$}
\SetWidth{0.25}
%\ArrowLine(33,25)(57,25)
%\SetColor{Black}
\Oval(48,48)(17,24)(-30)
\Text(22,60)[]{\Large $\varphi$}
\Text(58,42)[]{\Large $v_{qg}$}
\end{picture}  
\end{center}
\caption[]{
Collinear limit: splitting of a quark into a quark and
a gluon. The incident quark's momentum fraction carried by the outgoing
quark in the collinear limit is $u$, 
and the azimuthal angle of the gluon with respect to a fixed
external axis is $\varphi$.
}
\label{splitting}
\end{figure}

In QCD, there are two distinct types of IR singularities:
\begin{itemize}
\item Two massless partons may be collinear (Fig.~\ref{splitting}). 
The real cross section $\sigma_{\mbox{\small real}}$ 
in the collinear limit may be approximated by the 
expression
\begin{equation}
\label{cfac}
\int\frac{\mbox{d}\varphi}{2\pi}
\sigma_{\mbox{\small real}} \sim \frac{\alpha_s}{2\pi}\,\frac{1}{2p_jp_k}\,
\hat{P}_{kj\leftarrow i}(u)\,\sigma_{\mbox{\small Born}}.
\end{equation}
It is proportional to the Born cross section $\sigma_{\mbox{\small Born}}$
and an unsubtracted splitting function $\hat{P}_{kj\leftarrow i}(u)$.
\item A soft gluon is radiated (Fig.~\ref{soft}). An eikonal approximation
yields an expression of the form
\begin{equation}
\label{sfac}
\sigma_{\mbox{\small real}} \sim \frac{\alpha_s}{2\pi}\,
\sum_{i, j \neq k}\,C_{ijk} \frac{p_i p_j}{(p_i p_k)(p_j p_k)}
\,\sigma_{\mbox{\small Born}}
\end{equation}
with fixed coefficients $C_{ijk}$.
\end{itemize}

\begin{figure}
\unitlength1mm
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\begin{center} 
\begin{picture}(100,90)(0,0)
%\lattice
\SetWidth{0.5}
%\SetColor{\cquark}

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\ArrowLine(25,65)(10,50)
\ArrowLine(10,40)(40,10)

\ArrowLine(90,50)(60,80)
\ArrowLine(75,25)(90,40)
\ArrowLine(60,10)(75,25)

%\SetColor{\cgluon}
\Gluon(25,65)(75,25){2.5}{10}

%\SetColor{Black}
\Vertex(25,65){1.0}
\Vertex(75,25){1.0}

%\SetColor{Blue}
\SetWidth{0.25}
\Line(50,10)(50,40)
\Line(50,50)(50,80)
\CArc(45,10)(5,-90,0)
\CArc(55,80)(5,90,180)

\Text(55,80)[]{{\it \Large i}}
\Text(55,10)[]{{\it \Large j}}
\Text(55,50)[]{{\it \Large k}}

%\SetColor{Black}
\SetWidth{0.25}
\CArc(25,65)(15,-25,37)
\CArc(25,65)(16,-25,37)
\Text(45,65)[]{\Large $v_{ik}$}
\CArc(75,25)(15,150,222)
\CArc(75,25)(16,150,222)
\Text(55,25)[]{\Large $v_{jk}$}
\CArc(95,45)(14,135,225)
\CArc(95,45)(15,135,225)
\Text(77,45)[]{\Large $v_{ij}$}

\Text(15,15)[]{\Large $\overline{\cal M}$}
\Text(85,15)[]{\Large ${\cal M}$}

%\SetColor{Black}
\SetWidth{0.25}
\CArc(0,45)(11.2,-60,60)
\CArc(100,45)(11.2,120,240)

\end{picture}
\end{center}  
\caption[]{ Soft limit: the gluon attaches to two different quark lines~$i$ 
and~$j$ from the 
matrix element ${\cal M}$ and its complex conjugate $\overline{\cal M}$.} 
\label{soft}
\end{figure}

After a separation of overlapping singularities, these expressions are used
to generate an approximation term for the subtraction method. For collinear
singularities in the case of the splitting $g\rightarrow gg$, the subtraction 
term is actually dependent on the angle~$\varphi$.

\section{Calculations of Jet Cross Sections in DIS}
\label{jetcalc}
By now there are several calculations for (2+1)-jet processes
available with corresponding  
weighted Monte-Carlo programs. The two early programs {\tt PROJET} 
\cite{Gra91,Gra94,Gra95}
and {\tt DISJET} \cite{BK92,BM94,BM95} were restricted to the modifed JADE
clustering scheme. Subsequent calculations were universal in the sense
that in principle arbitrary infrared-safe observables could be calculated:
\begin{itemize}
%\item {\tt PROJET} \cite{Gra95}: The jet definition is restricted to 
%the modified JADE jet clustering scheme; 
%the program is based on the calculation published
%in Refs.~\cite{Gra90,Gra91,Gra94}.
%\item {\tt DISJET} \cite{BM94}: Again the jet definition is restricted to 
%the modified JADE scheme; the program is based on the calculation 
%in Refs.~\cite{BK92,BM95}.
\item {\tt MEPJET} \cite{MZ97}: This is a program for the 
calculation of arbitrary
observables which uses the phase-space-slicing method. The 
corresponding calculation 
\cite{MZ96} 
uses the Giele--Glover formalism \cite{GG92} 
for the analytical calculation of the
IR-singular integrals of the real corrections, and the crossing-function
technique \cite{GGK93} 
to handle initial-state singularities. The latter requires
the calculation of ``crossing functions'' for each set of parton densities.
\item {\tt DISENT} \cite{CS97man}: 
This program is based on the subtraction method. 
The subtraction term is defined by means of the dipole 
formalism\footnote{
The subtraction term is written as a sum over dipoles (an ``emitter'' formed
from two of the original partons and a ``spectator'' parton). Besides
the factorization theorems of perturbative QCD, the main ingredient is 
an exact factorization formula for the three-particle phase space, which allows
for a smooth mapping of an arbitrary 3-parton configuration onto the
various singular contributions.
}{}\,~\cite{CS96a,CS97}. 
\item {\tt DISASTER++} \cite{Gra97}: 
This is a {\tt C++} class library\footnote{
The acronym stands for ``Deeply Inelastic Scattering:
All Subtractions Through Evaluated Residues''.
The program is written in {\tt C++}. A {\tt FORTRAN} interface
is available; thus there is no problem to interface the
class library to existing {\tt FORTRAN} code.
}~{}.
The subtraction method
is employed, and the construction of the subtraction term resembles
the method of Ref.~\cite{KS92}, i.e.\ it is obtained by the evaluation
of the residues of the cross section in the soft and collinear limits.
Double counting of soft and collinear singularities
is avoided by means of a general
partial fractions method.
\item {\tt JetViP} \cite{Poett98a}: 
This program implements the calculation of \cite{Poett98b}, which extends the
previous calculations into the photoproduction limit $Q^2\rightarrow 0$.
The calculation has been done by means of the phase space slicing method.
Up to now, the polarization of the virtual photon is restricted to be 
longitudinal or transverse.
\end{itemize}

%The four universal programs {\tt DISENT}, {\tt MEPJET}, {\tt DISASTER++}
%and {\tt JetViP} 

These four programs have been compared in detail during the recent 
workshop on Monte Carlo generators for HERA physics 
\cite{GW99,progcompa,progcompb}, 
both for jet cross sections and event shape variables.
It turns out that {\tt DISENT} and {\tt DISASTER++} agree well for jet cross
sections \cite{progcompa}. 
The results for event shapes are in agreement as well, except 
for the current jet broadening variable~$B$ \cite{progcompb}. 
It is not yet known where this
residual discrepancy comes from; jet broadening is an infrared-safe shape 
variable, but it regularizes the (1+1)-jet singularity poorly. It may well be
that the different results come from a numerical instability in one of the
programs. The {\tt MEPJET} program shows systematic differences with respect 
to {\tt DISASTER++}/{\tt DISENT} for jet cross
sections (no comparison of shape variables has been done) even for
physical setups of integration parameters. The results are stable against
variations of the phase-space slicing parameter. The {\tt JetViP} results 
show deviations from the {\tt DISASTER++}/{\tt DISENT} results as well, 
but the differences are smaller than for {\tt MEPJET}. The 
{\tt JetViP} results depend on the technical phase-space slicing cut-off 
parameter \cite{progcompa}, which is a sign that the final result is not 
completely stable.

\begin{figure}[bht]
\unitlength1mm
\SetScale{2.8346}
\begin{center} 
\begin{picture}(200,60)(0,0)
%\lattice
\SetWidth{0.5}

% the proton
%\SetColor{\cproton}
\Line(10,30)(70,30)
\ArrowLine(10,30)(5,30)

% the photon line
%\SetColor{\cphoton}
\Photon(70,30)(110,30){2}{5}
\ArrowLine(110,30)(115,30)

% vertices
%\SetColor{Black}
\Vertex(70,30){1.0}

% outgoing jets
%\SetColor{\cquark}
\SetWidth{1.0}
\LongArrow(71,32)(85,55)
\LongArrow(71,28)(95,5)
%\SetColor{\cgluon}
\SetWidth{0.3}
\LongArrow(69,29)(65,24)
\SetWidth{0.5}

% E_T marker
%\SetColor{Black}
\DashLine(90,30)(90,55){3}
\LongArrow(90,50)(90,55)
\DashLine(100,30)(100,5){3}
\LongArrow(100,10)(100,5)

%\SetColor{Black}
\Text(95,45)[]{\Large $E_{T,1}$}
\Text(105,15)[]{\Large $E_{T,2}$}
\Text(40,35)[]{\Large\it P}
\Text(110,35)[]{\Large $\gamma^\ast$}
\Text(75,50)[]{\Large jet}
\Text(85,5)[]{\Large jet}
\Text(60,17)[]{\parbox[t]{3cm}
              {\renewcommand{\baselinestretch}{0.7}\Large
               \center soft\\ particle}}

\end{picture}  
\end{center}
\caption[]{
A (2+1)-jet event in the Breit frame. The transverse momentum is balanced.
An additional soft parton of an event close to the transverse momentum
cut-off drives the transverse momentum of one of the jets below threshold.
} 
\label{etbalance}
\end{figure}

A problem with transverse momentum cuts which has recently been observed is 
illustrated in Fig.~\ref{etbalance}. A cut of $E_{T,\mbox{min}}$ on the 
transverse momenta of both jets leads to a quantity which is sensitive to
the emission of soft particles. If the event is close to the threshold, 
the emission of a particle with a very small energy removes it from the event 
sample. This in itself is not critical; after all, this is what cuts are
supposed to do. However, what is affected is the cancellation of
infrared singularities. In a next-to-leading-order calculation, a process 
with two high-$E_T$ partons and a soft gluon contributes to the cancellation 
of IR singularities from the virtual corrections. The singularities 
themselves will of course cancel, but the cross section will become 
very sensitive to other cuts, which is, of course, undesirable. A possible
cure for this is to impose asymmetric cuts on the transverse momenta, i.e.\ 
conditions such as $E_{T,1} > E_{T,\mbox{min}}$, 
$E_{T,2} > E_{T,\mbox{min}}+\Delta$ for $E_{T,1} > E_{T,2}$, 
and $\Delta={\cal O}(\mbox{few GeV})$.


\section{Comparison with Data: $\alpha_s(\mu^2)$ and $f_{g/P}(\xi, \mu_f^2)$}
\label{data}

Both H1 and ZEUS have published measurements of the strong coupling constant 
$\alpha_s(\mu^2)$ from jet rates in DIS \cite{H1alphas,ZEUSalphas}.
The H1 results are shown in Fig.~\ref{h1alphas}. Experimental jet 
cross sections
for the
inclusive $k_\perp$~algorithm
vs.\ 
NLO theory ({\tt DISENT}) for {\tt CTEQ5M} parton densities and
$\alpha_s(m_Z^2) = 0.118$ are displayed in Fig.~\ref{h1alphas}(a).
This impressive agreement suggests that the strong coupling constant 
may be extracted from hadronic final states in DIS. This is illustrated in 
Fig.~\ref{h1alphas}(b). The chosen renormalization scale in this case 
is the transverse momentum $E_T$. The final result of a combined fit of
all data points is $\alpha_s(m_Z^2)=0.1181\pm 0.0030(\mbox{exp.})
^{+0.0039}_{-0.0046}(\mbox{th.})$, 
which is consistent with the world average value. Moreover, the plots 
show the running of the coupling constant. A measurement using 
$Q$ as the renormalization scale (not shown here) leads to the result 
$\alpha_s(m_Z^2)=0.1221\pm 0.0034^{+0.0054}_{-0.0059}$, which is
compatible with the result for $\mu=E_T$. According to 
Ref.~\cite{H1alphas}, the larger theoretical error comes
from the larger renormalization scale dependence of the choice $\mu=Q$.

\begin{figure}[htb]
\unitlength1mm
\SetScale{2.8346}
\begin{center} 
\begin{picture}(200,85)(0,0)
%\lattice
\put(0,12){\includegraphics[width=0.4\textwidth]{graudenz_1.eps}}
\put(50,10){\includegraphics[width=0.6\textwidth]{graudenz_2.eps}}
\put(27,5) {(a)}
\put(87,5) {(b)}
\end{picture}
\end{center}
\caption[]{Preliminary H1 results \cite{H1alphas} 
for jet cross sections and the strong coupling 
constant:
(a) Comparison of data and NLO results ({\tt DISENT}).
(b) Extracted value of $\alpha_s(E_T^2)$ in various bins of $Q^2$.
}
\label{h1alphas}
\end{figure}

The direct extraction of the gluon density by H1 is shown in 
Fig.~\ref{h1gluon} \cite{H1gluon}\footnote{See also Ref.~\cite{Ham97}.}.
The procedure was to do a fit of quark and gluon densities to a combined
data set of the total DIS cross section and the inclusive DIS jet cross
section for a given value of $\alpha_s$; the former constraining the quark 
densities, and the latter constraining the gluon density. The result
for $f_{g/P}(\xi)$ is compatible with the results from global fits.

\begin{figure}[htb]
\unitlength1mm
\SetScale{2.8346}
\begin{center} 
\begin{picture}(200,110)(0,0)
%\lattice
\put(10,0){\includegraphics[width=0.8\textwidth]{graudenz_3.eps}}
\end{picture}
\end{center}
\caption[]{Preliminary H1 result for the direct extraction of the gluon 
density \cite{H1gluon}.
} 
\label{h1gluon}
\end{figure}

There is, of course, the question of the right scale choice.
Scale dependence comes in because the perturbative series is truncated.
Physical results to all orders would be scale-independent. The choice of a
scale essentially corresponds to the choice of a renormalization 
prescription. To a certain degree, perturbation theory takes care of itself:
scale variations of a quantity calculated to order $\alpha_s^n$ in
perturbation theory have a residual scale dependence of order 
$\alpha_s^{n+1}$. This residual scale dependence may be large, which means
that the coefficient of this term is large. 
In practice, 
there are two approaches for the choice of a scale.
The first approach, a procedure employed frequently in experimental 
analyses, is to choose some 
central value $\mu$ equal to one of the energy scales of the process, and
to take the spread of the results in the interval $\mu/2$ to $2\mu$
to be the uncertainty from the theoretical prediction. This is justified
by the reasoning that the scale dependence of the finite-order result is
exactly identical to the scale dependence of the uncalculated higher-order
terms. Thus, the scale dependence reflects the size of these terms, and
consequently is an estimate of the error made. There are two criticisms:
there are 
scale-independent contributions from higher orders, which are not captured
by this procedure, and, in any case,
the variation does not determine the absolute size.
The second approach invokes some higher principle (such as 
BLM \cite{BLM83}, PMS \cite{PMS81} and FAC \cite{SZ79}), 
justified in turn 
either by a physical principle (as for BLM), by an assumption
on the behaviour of the perturbative series (for PMS and FAC), or even
by a fitted parameter $\rho$, with, for example, $\mu=\rho\,Q$.
Not all of these scales can be ``correct'', because they have different
values, and what comes out is (necessarily) a typical scale of the
process multiplied by a numerical factor. Admittedly, this numerical factor
can be different from one. In a certain sense, what is done in the first
approach is to estimate the spread of the scale dependence from various
{\it a priori} 
scenarios.

Typically, there are several 
physical scales which are relevant in a process, in the case at hand
the photon virtuality $Q$ and the transverse momentum $E_T$ of the jets in the
Breit frame. Fixed-order perturbation theory breaks down if these scales 
are very different, because large logarithms of ratios of these scales
lead to large coefficients, which in turn spoil the perturbative 
expansion. The proper way to handle this problem would
be resummation. So far, no resummed calculation for jet cross sections in 
DIS is available. Since none of the available scales is to be preferred, 
the safest procedure at present is to employ all of them in turn, and quote
the resulting differences as a theoretical error.

\section{Summary}
\label{summary}

I have given a brief review of the techniques employed in
next-to-leading-order QCD calculations. In particular, I have discussed
the singularity structure of the real corrections. From a principal
point of view, fixed-order perturbation theory is the cleanest approach 
to theoretical predictions: it depends on the smallest number of input
parameters (essentially the strong coupling constant and parton densities),
and it is not fragmentation-model dependent.

The situation concerning next-to-leading-order calculations for deeply 
inelastic processes has improved.
Several independent calculations for (2+1)-jet-like quantities are
available, with different methods used to extract infrared singularities.
The programs have been compared in some detail in the framework of the
recent workshop on Monte Carlo generators for HERA.
Two of them, {\tt DISENT} and {\tt DISASTER++}, agree well, with the 
exception of the jet broadening shape variable.

Measurements of $\alpha_s$ and $f_{g/P}$ are by now well established. 
Compared to earlier analyses, the present measurements are more reliable,
mainly due to larger event samples and as a consequence the possibility
of stricter cuts on transverse momenta of jets, which reduces the problems
with jets in the forward direction.

\section*{Acknowledgements}
\label{ack}
I would like to thank the organizers of the workshop for the invitation
to Ringberg castle and Doreen Wackeroth for a critical reading of
the manuscript.

\begin{thebibliography}{8.}
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\end{thebibliography}


\end{document}



