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\begin{titlepage}

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\noindent
Submitted to the 30th International Conference on High-Energy Physics 
ICHEP2000, \\
Osaka, Japan, July 2000

% INTERNAL %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\vspace*{1cm}
%\noindent
%\begin{center}
%\begin{tabular}{|ll|}
%\hline
%{\bf Date:}     &  \today \\
%{\bf Version:}  &  0.8.1 \\
%{\bf Editors:}   &  F.-P.~Schilling ({\em fpschill@mail.desy.de}) , \\
%                &   P.~R.~Newman ({\em prn@hep.ph.bham.ac.uk}) \\
%{\bf Referees:} &  H.~Jung ({\em jung@mail.desy.de}) , \\
%                &  B.~List ({\em blist@mail.desy.de}) \\
%\hline
%\end{tabular}
%\end{center}
%\vspace*{1cm}

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

\vspace*{3cm}

\begin{center}
\begin{Large}

%{\bf  Diffractive Dijet and 3-Jet Electroproduction \\ at HERA}
%{\bf  Diffractive Jet-Production \\ in Deep-inelastic Scattering at HERA}
{\bf  Diffractive Jet-Production \\ in Deep-Inelastic $\bf e^+p$ Collisions
at HERA}
%$\bf \sqrt{s}=300 \bf\ GeV$}

\vspace*{1cm}

H1 Collaboration

\end{Large}
\end{center}

\begin{abstract}
\noindent
A measurement is presented of dijet and 3-jet cross sections in low-$|t|$ 
diffractive  deep-inelastic scattering interactions of the type 
$ep \rightarrow eXY$, where the photon dissociation system $X$ is 
separated by a large rapidity gap from a leading low-mass baryonic system $Y$.
Data taken in 1996 to 1997 with the H1 detector at HERA, corresponding to 
%$\mathcal{L}_{int}=18.0 \rm\ pb^{-1}$, 
an integrated luminosity of $18.0 \rm\ pb^{-1}$, 
are used to measure a comprehensive set
of single and double differential cross sections in the kinematic
range $4<Q^2<80 \rm\ GeV^2$, $x_\pom<0.05$ and $p_{T,jet}>4 \rm\ GeV$.
The nature of the energy flow not attributed to the dijet
system is also investigated. 
%The data exhibit a clear 
%dominance of $q\overline{q}g$ states over $q\overline{q}$ states. 
%In a partonic interpretation, 
Viewed in terms of the diffractive scattering of parton fluctuations
of the photon, the data require the 
dominance of $q\overline{q}g$ over $q\overline{q}$ states. 
They constrain the 
diffractive gluon distribution and require a large 
fraction  of the colorless exchange momentum to be carried by gluons. 
The data are consistent with factorization 
%in $x_\pom$, 
of the $x_\pom$ dependence, a Pomeron 
intercept value of $\alpha_\pom(0)=1.2$ and a sizeable fraction of events
where the virtual photon is resolved. Soft color neutralization models
can reproduce the shapes of the differential distributions but  
underestimate 
the cross section. A perturbative QCD calculation based on 2-gluon
exchange is in agreement with the data at low $x_\pom$ values.
The 3-jet cross sections are in excess of the model predictions.
\end{abstract}

\vfill

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\noindent
\begin{flushleft}
%\begin{tabular}{ll}
{\bf Abstract:          960 }  \\
{\bf Parallel session:   2  } \\
{\bf Plenary talk:       12 }
%\\[0.2cm] 
%{\bf Electronic Access:} & 
%                 http://www-h1.desy.de/publications/H1\_sci\_results.shtml \\
%\end{tabular}
\end{flushleft}

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%\begin{center}
%To be submitted to {\em ICHEP2000} / \EJC
%\end{center}

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\end{titlepage}

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%       AT THE TIME OF THE T0-TALK INTO YOUR AREA
%
% from /h1/iww/ipublications/h1auts.tex 

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%  \input{h1auts}
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\section{Introduction}
\label{chapter0}

The observation of deep-inelastic scattering (DIS) events at HERA containing
a large gap in the rapidity distribution of the final state hadrons
has generated considerable renewed interest in understanding the
phenomenon of diffractive scattering or color singlet exchange.
Traditionally, such interactions have been described within
Regge phenomenology of high energy scattering. With the advent of the 
electron-proton collider HERA, it has become possible to study the dynamics 
of diffractive scattering using a point-like, highly virtual photon emitted
from the beam electron as a probe. This offers the chance to illuminate
the underlying dynamics of diffractive scattering in terms of
quantum chromodynamics (QCD).

Apart from measurements of inclusive diffractive scattering  
at HERA \cite{H1:F2d93,H1:F2d94,f2dzeus}, it is particularly 
interesting to focus on those hadronic final states where additional hard 
scales $\mu^2$ are introduced, 
%such as longitudinal vector mesons 
%($\mu^2=m_V^2$), heavy quarks ($\mu^2=m_q^2$) or jets ($\mu^2=p_T^2$). 
for example due to the presence of heavy quarks or high transverse
momentum ($p_T$) jets. 
%This allows to apply the techniques of perturbative QCD.
%Diffractive vector meson production at HERA has been studied in detail 
%\cite{vmesons:h1,vmesons:zeus}. 
The production of high $p_T$ jet final states
in diffractive scattering was observed in $p\overline{p}$ collisions 
\cite{difjetsUA8,difjetsCDF,difjetsD0} as well as at HERA 
\cite{H1:d2j94,djzeus}. The advantage of diffractive dijet production 
%with respect to inclusive measurements 
is the direct sensitivity to the gluon
component of the diffractive exchange, which 
can only be inferred indirectly from scaling violations
in the case of inclusive structure
function measurements.

In this article, a high statistics measurement of dijet and 3-jet
production in color singlet exchange events is presented which was performed
with the H1 detector at HERA. Deep-inelastic scattering events 
($Q^2>4 \rm\ GeV^2$) are selected
where the proton (or a low-mass excitation) looses only a small
fraction of its incoming momentum and escapes undetected through the beam pipe.
Separated from it by a large rapidity gap devoid of hadronic activity, 
the photon dissociation system $X$ is well contained within the central part
of the detector. The selection of events implies that 
%both the squared 
%4-momentum transfer $|t|$ at the proton vertex and
the longitudinal momentum fraction of the proton transferred
to the $X$ system ($\xpom$) is small. In the analysis, events with
either at least two or exactly three high $p_T$ jets contained in $X$ are 
considered. 

This article is organized as follows. The kinematics of diffractive
scattering at HERA are introduced in section
\ref{chapter1}. In section 
\ref{chapter2}, an overview of phenomenological models
relevant for diffractive jet production is given 
and the simulation of diffractive events using Monte Carlo generators is
described. In section 
\ref{chapter3}, the data selection, the measurement of the
jet cross sections and the determination of the systematic uncertainties
are presented. The results, expressed in terms of 
hadron level single and 
double differential cross sections, are presented and discussed
in section \ref{chapter4}.
The article ends with a summary and final remarks in section \ref{chapter5}.

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

\section{Kinematics of Diffractive Scattering at HERA}
\label{chapter1}

\subsection{Inclusive Diffractive Scattering}

\begin{figure}[t]
\centering
\epsfig{file=fig1.eps,width=.5\linewidth}
\caption{The generic diffractive process at HERA, where the electron ($k$) 
emits a photon ($q$) which 
interacts with a proton ($P$) via net color singlet exchange,
producing two distinct final state hadronic systems $X$ and $Y$. 
Where the masses of $X$ and $Y$ are small, the two systems are separated by
a large gap in rapidity. }
\label{diagrams}
\end{figure}

Fig.~\ref{diagrams} illustrates the generic diffractive process at HERA of the
type $ep\rightarrow eXY$. 
%HERA collides $E_e=27.5 \ \mathrm{GeV}$ electrons 
%or positrons\footnote{The word ``electron'' will be used as a generic term for
%electrons and positrons throughout the article.} with protons\footnote{The 
%proton beam energy was increased from $820$ to $920 \rm\ GeV$ in 1998.}
%of $E_p=820 \ (920) \ \mathrm{GeV}$. 
At the time when the data presented here were taken, 
HERA collided $E_e=27.5 \rm\ GeV$ 
positrons\footnote{The word ``electron'' will be used as a generic term for
electrons and positrons throughout the article.} with protons
of $E_p=820 \ \mathrm{GeV}$. 
In deep-inelastic scattering, the 
incoming electron (with 4-momentum $k$) emits a virtual photon $\gamma^*$ 
($q$) which interacts with a proton
($P$). The usual DIS kinematic variables are defined as
\begin{equation}
Q^2=-q^2 \ ; \qquad y=\frac{P\cdot q}{P \cdot k} \ ; \qquad x=\frac{-q^2}{2 P\cdot q} \ .
\end{equation}
The squared invariant masses of the electron-proton and photon-proton systems
$s$ and $W^2$ are given by
\begin{equation}
s=\frac{Q^2}{xy} \approx 4 E_e E_p \approx (300 \ \mathrm{GeV})^2 \ ; \qquad W^2=(q+P)^2 \approx ys-Q^2 \ .
\label{eq:w2}
\end{equation}
If the interaction takes place via color singlet exchange, two distinct 
hadronic systems of the dissociating photon and proton, $X$ and $Y$, are 
produced, with invariant masses $M_X$ and $M_Y$ respectively. 
In the case where $M_X$ and $M_Y$ are small compared with $W$, the two systems
are separated by a large rapidity gap. The longitudinal momentum fraction of
the colorless exchange with respect to the incoming proton $x_\pom$ and the 
squared four-momentum transferred at the proton vertex $t$ are then defined by
\begin{equation}
  \xpom = \frac{q \cdot (P - p_{_{\rm Y}})}{q \cdot P} \ ; \qquad t=(P-p_Y)^2 \ ,
\end{equation}
where $p_Y$ is the 4-momentum of $Y$.
In addition, the quantity $\beta$ is defined as
\begin{equation}
  \beta = \frac{x}{x_\pom} = \frac{Q^2}{2 q \cdot (P - p_{_{\rm Y}})} \ .
\end{equation}
In a partonic interpretation, $\beta$ is the longitudinal
momentum fraction of the colorless exchange carried by the struck quark,
in analogy to $x$ in the case of inclusive scattering.
In the analysis presented
here, the system $Y$ escapes undetected through
the proton beam-pipe. Therefore, $t$ and $M_Y$ are not measured and thus 
integrated over implicitly\footnote{It is noted that for this analysis 
$M_Y=M_p$ dominantly.}. 

%The cross section for the process
%$ep \rightarrow eXY$ depends on five independent 
%variables\footnote{Dependencies on azimuthal angles are not considered here.},
%usually chosen to be $\xpom$, $\beta$, $Q^2$, $M_Y$ and $t$.

%variables\footnote{Dependencies on azimuthal angles are not considered here.}
%and can be expressed in terms of a diffractive structure function $F_2^{D(5)}$:
%\begin{equation}
%\frac{d^5\sigma _{ep \rightarrow eXY}}{d\xpom d\beta dQ^2 dM_Y dt} =  \frac{4\pi \alpha_{em}^2}{\beta Q^4} \left ( 1-y+\frac{y^2}{2(1+R^{D(5)})} \right )  \ F_2^{D(5)}(\beta,Q^2,\xpom,M_Y,t) \ .
%\end{equation}
%Here $\alpha_{em}$ denotes the electro-magnetic coupling. For the range of $y$ 
%discussed in this article, the value of $R^{D(5)}=\sigma_L/\sigma_T$, 
%denoting the ratio of the longitudinal to the transverse photon cross 
%sections, is assumed to be negligible. Moreover, in the analysis presented
%here, the dissociating proton system $Y$ is not detected and escapes through
%the proton beam-pipe. Therefore, $t$ and $M_Y$ are not measured and thus 
%integrated over implicitly\footnote{It is noted that for this analysis 
%$M_Y=M_p$ dominantly.}. The inclusive diffractive cross section can then be 
%expressed as
%\begin{equation}
%\frac{d^3\sigma _{ep \rightarrow eXY}}{d\xpom d\beta dQ^2 } = \frac{4\pi \alpha
% _{em}^2}{\beta Q^4} \left ( 1-y+\frac{y^2}{2} \right )  \ \fiidiii (\beta,Q^2,\xpom) \ .
%\end{equation}

\subsection{Diffractive Dijet Production}

\begin{figure}[t]
\centering
\epsfig{file=fig2.eps,width=0.66\linewidth}
\caption{
Diffractive scattering, viewed in the proton rest frame ({\em left}) and the
infinite momentum frame ({\em right}). ({\em a}) In the proton rest
frame, the virtual photon dissociates into a $q\overline{q}$ state,
scattering off the proton by color singlet exchange (diffractive 
quark scattering). ({\em b}) The emission of an additional gluon forms a
$q\overline{q}g$ state (diffractive gluon scattering or Boson-Gluon-Fusion).
}
\label{diagrams2}
\end{figure}

Viewed in the proton rest frame, the virtual photon 
dissociates into a $q\overline{q}$ pair well before the interaction with
the proton (Fig.~\ref{diagrams2}a). The $q\overline{q}$ state 
is coupled to the elastically scattered proton by a color singlet exchange
(diffractive quark scattering). It is expected that for larger 
values of $M_X$, corresponding to low values of $\beta$, this picture is 
not sufficient and additional contributions like the 
radiation of an extra gluon become important, producing 
%an effective dipole
an incoming
$q\overline{q}g$ system, which is often also modelled as a colour dipole 
(Fig.~\ref{diagrams2}b).
Small size dipole configurations
with high transverse momenta are calculable within perturbative QCD, whereas
large size, low $p_T$ configurations are similar to soft hadron-hadron 
scattering. 

The diffractive quark scattering process (Fig.~\ref{diagrams2}a)
%$\gamma^*q\rightarrow q'$ 
represents the lowest order, i.e. $O(\alpha_s^0)$, 
diagram. High $p_T$ final states can be produced
in leading order QCD, i.e. $O(\alpha_s)$,
via the Boson-Gluon-Fusion (BGF) and QCD-Compton (QCDC) processes.
Figure (Fig.~\ref{diagrams2}b) corresponds to
diffractive Boson-Gluon-Fusion in the proton infinite momentum frame.
Because of the non-zero invariant mass
$\sqrt{\hat{s}}=M_{12}$ of the two final state partons, a new variable $z_\pom$
is introduced:
\begin{equation}
   z_\pom = \beta \cdot \left (1+\frac{\hat{s}}{Q^2} \right ) \ .
\end{equation}
The interpretation of $z_\pom$ is equivalent to the one of $\beta$ in the case
of the $O(\alpha_s^0)$ diagram. In models where the colorless exchange is
attributed an internal structure, it corresponds to the longitudinal
momentum fraction of the exchange which takes part in the hard interaction.
In such models, diffractive jet-production
is directly sensitive to the gluon distribution
$g^D(z,\mu^2) $ of the diffractive exchange.
%because of the Boson-Gluon-Diagram, in contrast to inclusive 
%measurements of $F_2^{D(3)}$.

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

\section{Phenomenological Models and Monte Carlo Simulation}
\label{chapter2}

In this section, several phenomenological approaches and QCD
calculations, attempting to describe diffractive DIS and 
especially diffractive jet production, will be discussed. The focus will be 
on those which are compared with the data in 
section \ref{chapter4}.

\subsection{Diffractive Parton Distributions}
\label{section:difpdf}

In the leading $\log(Q^2)$ approximation, the cross section for the 
diffractive process
$\gamma^*p\rightarrow p'X$ can be written in terms of convolutions of partonic
cross sections $\hat{\sigma}^{\gamma^*i}$ with diffractive parton 
distributions $f_i^D$, representing probability distributions for a parton in
the proton under the additional constraint that the proton remains
intact with particular values of $x_\pom$ and $t$:
\begin{equation}
\frac{d\sigma(x,Q^2,x_\pom,t)^{\gamma^*p\rightarrow p'X}}{dx_\pom dt}= \sum_i \int_x^{x_\pom}dy \ \hat{\sigma}(x,Q^2,y)^{\gamma^*i}\left(\frac{df_i^D(y,x_\pom,t)}{dx_\pom dt}\right) \ .
\label{equ:diffpdf}
\end{equation}
This factorization formula holds for fixed values of $x$, $x_\pom$ and $t$
and is exactly valid in the limit $Q^2\rightarrow\infty$, where higher twist
contributions become negligible. 
This ansatz was introduced in \cite{trentadue} and applied to hard 
diffraction in \cite{berera}. The proof of Eq.~(\ref{equ:diffpdf}) 
for inclusive diffractive lepton-hadron scattering was given in 
\cite{facproof} in the framework of a scalar model and in \cite{collins} for
full QCD. The partonic cross sections are identical to those in 
non-diffractive scattering. The diffractive parton distributions
however, which should obey the DGLAP \cite{dglap} evolution equations, are not
known from first principles. Recently, there have been approaches
to calculate the distributions at the starting scale $\mu_0^2$ of 
the QCD evolution under certain assumptions. In \cite{hautmann},
the case of a $q\overline{q}$-pair made of heavy quarks coupling to a 
small-size hadron
has been studied, which is perturbatively calculable.
A different approach is the Semiclassical model by Buchm\"uller,
Gehrmann and Hebecker \cite{semicl}, based on the
opposite extreme of a very large hadron. In spite of the different
assumptions, both approaches give rather
similar results for the diffractive parton distributions: The parton 
distributions follow the same general behavior at the endpoints $z=0$ and
$z=1$ and the gluon distribution dominates.

\subsection{Resolved Pomeron Model and Pomeron Parton Distributions}
\label{chapter:rapgapmodel}

The application of Regge phenomenology for soft hadronic high energy 
interactions to the concept of diffractive parton distributions
leads naturally to the Ingelman-Schlein model of a resolved Pomeron
with a partonic structure \cite{IngSchl} invariant under changes in $\xpom$
and $t$. The diffractive parton
distributions then factorize into a flux factor $f_{\pom/p}$ and 
Pomeron parton distributions $f_i^{\pom}$:
\begin{equation}
\frac{df_i^D(x,Q^2,x_\pom,t)}{dx dQ^2 dx_\pom dt}= f_{\pom /p}(x_\pom ,t) \ f_i^{\pom }(\beta=x/x_\pom ,Q^2) \ .
\end{equation}
The universal flux factor describes the probability to find a
colorless state in the proton, the ``Pomeron'', as a function of 
$x_\pom$ and $t$.
%The diffractive structure function $F_2^{D}$ factorizes:
%\begin{equation}
%F_{2}^{D(4)}(x_\pom ,t, \beta , Q^2) = f_{\pom /p}(x_\pom ,t) \ F_2^{\pom }(\beta,Q^2) \ .
%\end{equation}
%Here $F_2^{\pom}(\beta,Q^2)$ denotes the Pomeron structure function which
%is, in analogy to $F_2(x,Q^2)$, related to the Pomeron parton distributions 
%and obeys the usual DGLAP QCD evolution equations.

%The H1 collaboration has interpreted their measurements of $F_2^{D(3)}$ in
%terms of such a model \cite{H1:F2d94}. For the range of $x_\pom$ under study,
%it turned out to be necessary to consider more
%generally contributions from sub-leading Reggeon exchanges as well:
%\begin{equation}
%F_{2}^{D(4)} =  f_{\pom /p} \  F_2^{\pom } \ + \  f_{\reg /p} \  F_2^{\reg } \ .
%\end{equation}
The H1 collaboration has interpreted their measurements of inclusive 
diffraction
(presented in the form of a diffractive structure function $F_2^{D(3)}$) in
terms of such a model \cite{H1:F2d94}. 
For the range of $x_\pom$ under study,
it turned out to be necessary to consider more
generally contributions from sub-leading Reggeon exchanges as well
as the pomeron:
\begin{equation}
F_{2}^{D(3)} (\beta, Q^2 , \xpom) =  
\int d {\rm t} \ f_{\pom /p} (\xpom, t) \cdot F_2^{\pom } (\beta, Q^2) \ + \  
\int d {\rm t} \ f_{\reg /p} (\xpom, t) \cdot F_2^{\reg } (\beta, Q^2) \ .
\label{h1flux}
\end{equation}
$F_2^{\pom}(\beta,Q^2)$ and $F_2^{\reg}(\beta,Q^2)$ are the Pomeron and
Reggeon structure functions. 
The flux factors for the Pomeron and Reggeon exchanges were parameterized
in a Regge-inspired form:
\begin{equation}
f_{\{\pom,\reg\}/p}(\xpom , t) = C_{\{\pom,\reg\}} \ {\xpom}^{1-2\alpha_{\{\pom,\reg\}}(t)} \ e^{b_{\{\pom,\reg\}} t} \ .
\end{equation}
From mixed Regge and QCD 
fits to the measured data, a value for the Pomeron intercept
$\alpha_\pom (0)$ and
%and from a QCD analysis of the scaling violations of $F_2^{\pom}(\beta,Q^2)$, 
parameterizations of the Pomeron parton densities
were obtained. The resulting value of 
$\alpha_\pom (0)=1.203\pm0.020(stat.)\pm0.013(syst.)\pm0.030(model)$
is significantly higher than that obtained from soft hadronic interactions,
where $\alpha_\pom(0) \sim 1.08$ \cite{DL:stot}. The 
parton densities extracted for the Pomeron are
dominated by gluons, which carry $80\ldots 90\%$ of the Pomeron
momentum.

% FOLLOWING LOOKS LIKE REPETITION - ALREADY SAID EARLIER
%For dijet events, the dominating 
%leading order QCD processes is then Boson-Gluon-Fusion (BGF).
%Thus, measuring diffractive dijet production directly constrains the Pomeron
%gluon distribution. In contrast, when measuring the inclusive diffractive
%structure function $F_2^{D(3)}$, the gluon distribution can only be inferred
%indirectly from scaling violations and is subject to a substantial
%uncertainty. Especially in the region of large $z_\pom$ or $\beta$, the
%gluon density is not well constrained because data with $\beta>0.65$ were
%excluded from the H1 QCD fits because of possible
%non-negligible higher twist contributions.

\subsection{Colour Dipole and 2-gluon Exchange Models}

In the proton rest frame, the virtual photon splits into a $q\overline{q}$
or $q\overline{q}g$ system well in advance of the target. These states
are then scattered off the proton by net color singlet exchange. Many 
properties of the diffractive final state can be deduced from a knowledge of
the partonic wave functions of the photon alone. The $\beta$
distribution for example is largely determined by the photon wave functions
and depends only weakly on the way these are coupled to the proton. 
%In a recent analysis \cite{bekw} it has been shown that 
In a recent QCD motivated parameterisation \cite{bekw}
longitudinally and transversely
polarized $q\overline{q}$ states contribute at high and medium values
of $\beta$ respectively, whereas the $q\overline{q}g$ state originating
from a transversely polarized photon is dominant at low $\beta$ (i.e. high 
$M_X$). 
%The coupling of these states to the proton is undefined
The diffractive coupling of these states to the proton is not known
a priori. However, the simplest realization of a net color singlet 
exchange at the parton level is a pair of gluons with opposite color 
charges \cite{lownussinov}.
%An essential feature of 2-gluon exchange models is 
The cross section is then closely related to the square of the unintegrated
gluon distribution of  the proton $\mathcal{F}(x,k_i^2)$.
We will focus here on two recent models based on 
the ideas of diffractive scattering
of partonic fluctuations of the photon and two gluon exchange. Other approaches
can be found in \cite{gg}, for example.

%In impact parameter space, the condition of $k_T$-factorization is equivalent
%to the factorization of the $\gamma^*p$ cross section into an effective photon
%dipole wave function $|\Psi|^2$ and the dipole cross section 
%$\hat{\sigma}$. The photon-proton cross section for the $q\overline{q}$ dipole
%can be expressed as \cite{nikzak90}:
%\begin{equation}
%\sigma^{\gamma^*p}_{T,L}(x,Q^2) \sim \int d^2r \int_0^1 d\alpha |\Psi_{T,L}(\alpha,r)|^2 \ \hat{\sigma}(x,r^2) \ .
%\end{equation}
%Here, the subscripts $T$ and $L$ denote transverse and longitudinal
%polarization, $r$ is the relative transverse separation between
%the quarks and $\alpha$ is the momentum fraction of the
%quark. In the case of diffraction,
%the square of the dipole cross section enters in the formula.
%In the Saturation model by 
%Golec-Biernat and W\"usthoff \cite{sat}, an Ansatz for the
%dipole cross section is made which interpolates between the perturbative
%and non-perturbative regions of $\sigma^{\gamma^*p}$.
%It is motivated by the ideas of color 
%transparency, $\log 1/x$ summation and 
%low-$x$ saturation. 

In impact parameter space, the condition of $k_T$-factorization is equivalent
to the factorization of the $\gamma^*p$ cross section into an effective photon
dipole wave function $|\Psi|^2$ and the dipole cross section 
$\hat{\sigma}$ \cite{nikzak90}. 
%The photon-proton cross section for the $q\overline{q}$ dipole
%can be expressed as \cite{nikzak90}:
%\begin{equation}
%\sigma^{\gamma^*p}_{T,L}(x,Q^2) \sim \int d^2r \int_0^1 d\alpha |\Psi_{T,L}(\alpha,r)|^2 \ \hat{\sigma}(x,r^2) \ .
%\end{equation}
%Here, the subscripts $T$ and $L$ denote transverse and longitudinal
%polarization, $r$ is the relative transverse separation between
%the quarks and $\alpha$ is the momentum fraction of the
%quark. 
The diffractive cross section depends on the
square of the dipole cross section.
Investigating diffractive final states with varying 
transverse momenta, for example by studying jets,
probes the dipole cross section as a function of
the dipole size in the transition region between Regge phenomenology and
perturbative QCD.
In the `Saturation' model by 
Golec-Biernat and W\"usthoff \cite{sat}, an Ansatz for the
dipole cross section is made which interpolates between the perturbative
and non-perturbative regions of $\sigma^{\gamma^*p}$.
This model is able to give
a reasonable description of $F_2(x,Q^2)$ at low $x$,
which determines the free parameters of the model (only 4).
The diffractive structure function $F_2^{D(3)}$ is 
then predicted and also described reasonably well. The model
predicts a constant ratio of the diffractive over the inclusive cross sections.
The calculation of the $q\overline{q}g$ cross 
section for medium $\beta$ is made under the assumption of strong $k_T$ 
ordering of the final state partons, corresponding to 
$k^{(g)}_T \ll k^{(q,\overline{q})}_T$.

Cross sections for diffractive $q\overline{q}$ and $q\overline{q}g$ production 
by 2-gluon exchange have been calculated by Bartels, Lotter and 
W\"usthoff ($q\overline{q}$) \cite{bartelsqq} and by Bartels, Jung and 
W\"usthoff ($q\overline{q}g$) \cite{bartelsqqg}.
The calculation of the $q\overline{q}g$ final state is performed 
in the low-$\beta$ or ``triple Regge'' limit, taking also
configurations without strong $k_T$ ordering into account. The calculations 
require high $p_T$ of all outgoing partons. Thus, this
model is particularly suited for diffractive jet production.

\subsection{Soft Color Neutralization Models}

An alternative approach to diffractive DIS, not based on special
concepts for diffraction but closely related to inclusive
scattering, is given by soft color neutralization models. 
These models naturally lead to very similar
properties of inclusive and diffractive DIS final states.

One example is Soft Color Interaction (SCI) model by Edin, Ingelman and
Rathsman \cite{sci}. 
In its original version, the hard interaction in diffractive DIS was treated 
identically to that in inclusive DIS.
%based on the QCD-improved parton model formulated
%in the infinite momentum frame. 
Diffraction enters through
soft color rearrangements between the outgoing partons, leaving
their momentum configuration unchanged. If two color
singlet systems are produced by such a mechanism, the hadronic final state
can exhibit a visible rapidity gap. 
There is only one additional free parameter,
namely the universal probability for color rearrangements to occur, which
is fixed by a fit to $F_2^{D(3)}$.
The model has been refined recently by 
making the color rearrangement probability proportional to the
difference in the generalized areas of the string configurations
before and after the rearrangement \cite{scinew}.

Another approach is the Semiclassical model \cite{semicl}, a non-perturbative
model which was already mentioned in section \ref{section:difpdf}. Viewed in
the proton rest frame, $q\overline{q}$
and $q\overline{q}g$ configurations scatter off a superposition of
soft color fields originating from the proton. Those configurations which
emerge in a net color singlet configuration contribute to the diffractive
cross section. A parameterization of diffractive parton distributions
is derived from a combined 4 parameter fit to the inclusive and diffractive 
structure functions $F_2$ and $F_2^D$ at low $x$.

\subsection{Monte Carlo Simulation}
\label{section:mc}

Monte Carlo simulations are used to determine the corrections
to be applied to the measured data to compensate for the 
limited efficiencies, acceptances and resolutions of the detector. 
The generated Monte Carlo events are subjected to a detailed simulation of
the H1 detector and passed through the same reconstruction and analysis chain
as the measured data. 

The main Monte Carlo generator used for this purpose is RAPGAP 2.08 
\cite{rapgap}. Events are generated according to a resolved (partonic) Pomeron
model. Contributions from Pomeron and sub-leading meson exchanges are included.
The parameterizations of the Pomeron and meson flux factors and parton
distributions are taken from the H1 analysis
of $F_2^{D(3)}$ \cite{H1:F2d94} (see Eq.~\ref{h1flux}).
The Pomeron and meson trajectories and 
slope parameters are
$\alpha_\pom(t)=1.20+0.26t$, $b_\pom=4.6 \ \mathrm{GeV}^{-2}$  and
$\alpha_\reg(t)=0.50+0.90t$, $b_\reg=2.0 \ \mathrm{GeV}^{-2}$ respectively.
The Pomeron parton distributions are the ``flat gluon'' (or ``fit 2'') 
parameterizations extracted from the leading order QCD fits
to $F_2^\pom(\beta,Q^2)$. The meson structure function is taken from
a parameterization of the pion \cite{owens}. The renormalization and factorization
scales are set to $\mu^2=Q^2+p_T^2$, where $p_T$ is the transverse momentum
of the partons emerging from the hard scattering. 
The parton distributions are convoluted 
with hard scattering matrix elements 
to leading order in QCD. Transverse momenta of the incoming partons are
not included in the calculation of the cross sections. Outgoing 
charm quarks are produced in the
massive scheme via Boson-Gluon-Fusion. For the production of light 
quarks, a lower cutoff in $p_T^2$ is introduced
in the ${\cal O} (\alpha_s)$ QCD matrix elements   
to avoid divergences in the calculation.
Higher order QCD diagrams are approximated with parton showers in the
leading $\log(\mu)$ approximation (MEPS) \cite{meps}.
Hadronisation is simulated using the Lund string model \cite{lund}.

In RAPGAP, a contribution of events where the virtual
photon $\gamma^*$ is resolved and assigned an internal partonic structure 
can also be simulated.
The parton densities for the virtual photon are taken from the SAS-2D 
\cite{sas} parameterization, which has been found to give a reasonable
description of non-diffractive dijet production at low $Q^2$ in a previous 
H1 measurement \cite{h1:virtgam}.

Monte Carlo generators are also used to compare the measured hadron level
cross sections to the predictions of the phenomenological models and QCD
calculations presented in the previous sections. RAPGAP is used to
obtain the 
predictions of the resolved Pomeron model with different Pomeron intercept 
values and parton distributions. It also contains
implementations of the Saturation model \cite{mckowalski} and the 
Semiclassical model \cite{mcfpschill} as well as the 2-gluon exchange 
model by Bartels et al. Both versions of the Soft Color Interaction 
(SCI) model are  implemented in the LEPTO $6.5.2\beta$ generator \cite{lepto},
which was used for the comparisons with these models.

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

\section{Experimental Procedure}
\label{chapter3}

The analysis presented in this article is based on H1 data 
corresponding to an integrated luminosity of $18.0 \rm\ pb^{-1}$. The data 
were taken in the years 1996 and 1997, when HERA collided 820 GeV protons with
27.5 GeV positrons. A detailed description of the measurement
can be found in \cite{fpschill}.

\subsection{H1 Detector}

The H1 detector is described in detail elsewhere \cite{H1:det}.
Here, we will give a brief description of the detector components most
relevant for the analysis. The $z$-axis of the H1 coordinate system
corresponds to the nominal beam axis such that positive $z$ values 
indicate the direction of the outgoing proton beam which is often
referred to as the ``forward'' direction\footnote{This direction
corresponds to positive values of the pseudorapidity  
$\eta=-\ln\tan \theta/2$.}.

The beam pipe is surrounded by the tracking system. Two large concentric
drift chambers (CJC), located within a solenoidal magnetic field 
of $1.15 \rm\ T$,  measure the trajectories of charged particles and their
momenta in the range $-1.5<\eta<1.5$ with a precision of 
$\sigma(p)/p< 0.01 \cdot p/\mathrm{GeV}$.
%
Energies of final state particles are measured in a highly segmented Liquid
Argon  (LAr) calorimeter covering the range $-1.5<\eta<3.4$, surrounding the
tracking detectors. The energy resolution
is $\sigma(E)/E \approx 11\%/\sqrt{E}$ for electro-magnetic showers and
$\sigma(E)/E \approx 50\%/\sqrt{E}$ for hadrons. The systematic uncertainty
on the hadronic energy measurement in the LAr is $4\%$. The ``backward'', i.e.
outgoing electron beam, direction ($-4.0<\eta<-1.4$) is covered by
a lead / scintillating fibre calorimeter (SPACAL) with electro-magnetic
and hadronic sections. In this measurement, the scattered DIS electron
is identified in the SPACAL with an energy resolution between $0.3\%$ for
kinematic peak electrons ($E_{e'}=27.5 \rm\ GeV$) and $2.0\%$ at low
energies ($E_{e'}=8 \rm\ GeV$).
The energy resolution of the hadronic part of the SPACAL is $7\%$.
In front of the SPACAL, the Backward Drift Chamber (BDC) provides
track segments of charged particles with a resolution of
$\sigma_{dr}=0.4 \rm\ mm$ and $\sigma_{rd\phi}=0.8 \rm\ mm$.

Beam induced backgrounds are suppressed using a time-of-flight scintillator
system (TOF). The $ep$ luminosity is determined by comparison of the QED
cross section for the bremsstrahlung reaction $ep\rightarrow ep\gamma$
with the measured event rate in a photon tagger calorimeter close
to the beam pipe at $z=-103 \ \mathrm{m}$ with a precision of $2\%$.

To enlarge the coverage for hadronic activity up to pseudorapidities
of $\eta=7.5$ in the region of the outgoing proton, the Forward Muon Detector
(FMD) and the Proton Remnant Tagger (PRT) are used for the analysis.
The FMD is located at $z=6.5 \ \mathrm{m}$ and covers the pseudorapidity
range $1.9<\eta<3.7$ directly. It also has sensitivity to larger $\eta$ values 
because of secondary scattering. The Proton Remnant Tagger (PRT), a set of 
scintillators surrounding the beam pipe at $z=26 \ \mathrm{m}$, can tag
hadrons in the region $6<\eta<8$.

\subsection{Data Selection}

Deep-inelastic scattering events are triggered by an electro-magnetic energy 
cluster in the SPACAL with $E_{e'}>8 \ \mathrm{GeV}$ in coincidence
with a reconstructed CJC track associated to the event vertex. Due to the 
track requirement, the trigger efficiency varies for the selected events 
between 80 and 
$90\%$, depending on the kinematics. DIS electron candidates are selected in 
the angular range $156^o<\theta_{e'}<176^o$. Various 
cuts are applied on these
candidates in order to select electrons and reject background originating
from photons and hadrons. Among these are requirements on the width of 
the shower, the containment
within the electro-magnetic part of the SPACAL and the existence of a 
reconstructed track segment in the BDC pointing from the vertex to the 
electron candidate. The $z$ coordinate of the reconstructed vertex is
required to lie within $\pm35 \ \mathrm{cm}$ ($\pm \sim 3\sigma$) 
of the nominal
interaction point. To suppress events with
initial state QED radiation, the summed $E-p_Z$ of all reconstructed 
final state particles including the electron\footnote{For DIS events 
fully contained in the detector,
the total $E-p_Z$ satisfies $55 \rm\ GeV$.} has to be greater than
$35 \ \mathrm{GeV}$. 
The DIS kinematic variables are calculated from
the polar angle and energy measurements of the scattered electron:
\begin{equation}
Q^2=4E_eE_{e'}\cos^2\frac{\theta_{e'}}{2} \ ; \qquad y=1-\frac{E_{e'}}{E_e}\sin^2\frac{\theta_{e'}}{2} \ .
\end{equation}
Events which fulfill
\begin{equation}
4<Q^2<80 \ \mathrm{GeV}^2 \ ; \qquad 0.1<y<0.7 
\end{equation}
are selected.

The selection of diffractive events is based on requirements on the
absence of hadronic activity in the outgoing proton region. No signal
above noise levels is allowed in the FMD and PRT detectors. The most forward
part ($\eta>3.2$) of the LAr calorimeter has to be devoid of hadronic energy clusters with energies $E>400 \ \mathrm{MeV}$. This selection ensures
that the photon dissociation system $X$ is well contained within the 
central part of the H1 detector and is separated by a large rapidity gap
covering at least $3.2 < \eta < 7.5$ from the $Y$ system, which
escapes undetected through the beam pipe. This imposes the constraint
$M_Y<1.6 \ \mathrm{GeV}$ and $|t|<1.0 \ \mathrm{GeV}^2$.

The $X$ system, measured in the LAr and SPACAL
calorimeters together with the CJC, is reconstructed using a 
method that combines calorimeter clusters and tracks carefully avoiding
double counting \cite{fscomb}. Its mass is then calculated according to
\begin{equation}
M_X^2=(\textstyle\sum_{i} \displaystyle E_i)^2 - (\textstyle \sum_{i} \displaystyle {\bf p}_i)^2 \ ,
\end{equation}
were the sum runs over all reconstructed objects except for the
scattered electron. $W^2$ is calculated according to Eq.~(\ref{eq:w2}). 
$x_\pom$ and $\beta$ are then computed by
\begin{equation}
x_\pom=\frac{Q^2+M_X^2}{Q^2+W^2} \ ; \qquad \beta=\frac{Q^2}{Q^2+M_X^2} \ .
\end{equation}
The correlation between the hadron level and detector level values of
$x_\pom$, as obtained from the Monte Carlo simulations, 
is shown in Fig.~\ref{fig11}a. The resolution in $\log x_\pom$ is $8\%$. A cut 
\begin{equation}
 x_\pom<0.05
\end{equation}
is applied to suppress contributions from non-diffractive scattering
and secondary exchanges.

The 4-vectors of the hadronic final state particles associated to the $X$ 
system are then Lorentz-transformed to the $\gamma^*p$ center-of-mass 
frame\footnote{This frame is also called 
the ``hadronic center-of-mass frame''.},
where they are subjected to the CDF cone jet algorithm \cite{cdfcone} with
a cone radius of $R_{cone}=\sqrt{\Delta\eta^2+\Delta\phi^2}=1.0$ . Transverse
energies and momenta are calculated with respect to the $\gamma^*p$ axis.
%which is not collinear with the $ep$ beam axis. 
Events with either
at least two or exactly three jets 
with transverse momenta 
$p^*_{T,jet}>4 \ \mathrm{GeV}$
are selected for the dijet and 3-jet
samples respectively. The jets are required to lie within the region
of $-1.0<\eta^{lab}_{jet}<2.2$ to ensure a good containment within the LAr 
calorimeter. In Fig.~\ref{fig11}b, the correlation between the hadron
level and the reconstructed values of $p_{T,jets}^*$, the mean dijet 
transverse momentum, is shown. The resolution is $14\%$. The final event 
selection yields approx. 2.500 dijet and 130 3-jet events.

\subsection{Cross Section Measurement}

\begin{table}[!t]
\begin{center}
\begin{tabular}{|c|} \hline
Cross Section Definition \\ \hline  \hline
$4 < Q^2 < 80 \ {\rm GeV^2}$ \\ 
$0.1 < y < 0.7$ \\ \hline 
$\xpom < 0.05$ \\ 
$M_Y < 1.6 \ {\rm GeV}$ \\ 
$|t| < 1.0 \ {\rm GeV^2}$ \\ \hline
$N_{\rm jets} \geq 2 \ \ \mathrm{or} \ \  N_{\rm jets} = 3 $ \\ 
$p^*_{T,jet} > 4 \ {\rm GeV}$  \\
$-3 < \eta^*_{jet} < 0 $ \\ \hline
\end{tabular}
\end{center}
\caption{The kinematic range in which the cross sections are measured. }
\label{kinrange}
\end{table}

The measured distributions are corrected for detector acceptances, 
efficiencies and resolutions using the RAPGAP program (see section 
\ref{section:mc}), interfaced to HERACLES \cite{heracles} to take QED 
corrections into account. The simulations give a very good description of all
relevant kinematic distributions of the selected dijet and 3-jet events. 
Smearing in  $x_\pom$ is taken into account up to $x_\pom=0.2$ in RAPGAP.
Migrations from $\xpom > 0.2$ or from large values of $M_Y>5 \ \mathrm{GeV}$
are covered by a RAPGAP simulation of non-diffractive DIS. This 
contribution is at the level of $5\%$ averaged over all bins
and is concentrated at large
$x_\pom$. An additional factor of $-2.3\pm5.0\%$ is applied
to account for the net smearing about the $M_Y=1.6 \rm\ GeV$ boundary
which is not covered by RAPGAP because it only generates elastically scattered 
protons. The factor has been determined using the DIFFVM 
\cite{diffvm} simulation of proton dissociation.
Furthermore, a correction of $7.8\pm1.9\%$ takes into account 
diffractive events rejected due to fluctuations in the
amount of noise in the FMD detector. The correction has been determined using
randomly triggered events. 

The cross sections are corrected to the Born level. 
QED corrections are small for 
most of the data points, typically at the level of $5\%$.
The data are corrected using a bin-to-bin correction method.
The bin purities and stabilities are typically of the order of $50$ to $60\%$
and it is ensured that they exceed $30\%$ in every bin of the
resulting cross sections.

The corrected hadron level cross sections are defined in terms of a model
independent definition of rapidity gap events. Two
systems $X$ and $Y$ are defined by searching for 
%the largest gap in pseudorapidity in the event (Fig.~\ref{diagrams}). 
the largest gap in rapidity among the hadrons in 
the $\gamma^* p$ center-of-mass
frame (Fig.~\ref{diagrams}). 
%From these, the 
%quantities $x_\pom$, $M_Y$ and $|t|$ are
%calculated. 
No $\eta_{\rm\max}$ or similar cuts
are imposed in the definition of the measured cross sections. 
The full definition of the hadronic cross
sections is given in Tab.~\ref{kinrange}.
The cross section definition in terms of jet pseudorapidities in the hadronic 
center-of-mass frame $-3<\eta_{jet}^*<0$ approximately matches the 
$-1<\eta^{lab}_{jet}<2.2$ cut for the selected events.

Fig.~\ref{fig9} shows the transverse energy flow
around the jet axes for the dijet
sample. For the jet profiles in $\eta$ and $\phi$, only transverse energies
within one unit in azimuth and pseudorapidity are included in the plots 
respectively.  The jet profiles for backward and forward jets are shown
separately in Figs.~\ref{fig9}a,c and b,d respectively. The data exhibit a
clear back-to-back dijet structure in azimuth. The energy flow is well 
described by the RAPGAP simulation.

\subsection{Analysis of Systematic Uncertainties}

The following sources of systematic uncertainties contribute to
the total systematic error on the measured cross sections.
The experimental uncertainties taken into account are:

\begin{enumerate}

\item The uncertainties on the hadronic calibrations of the LAr and SPACAL
calorimeters are $\pm 4\%$ and $\pm7\%$ respectively.
Both influence the measured values of $p^*_{T,jet}$ and
$x_\pom$ and result in uncertainties in the measured cross sections
of up to $10\%$ (with a mean value of $5\%$) for the LAr and 
$0.5\%$ for the SPACAL.

\item The uncertainties on $E_{e'}$ and $\theta_{e'}$ propagate
into the reconstruction of $Q^2$, $y$ and $W$ and the definition of the
$\gamma^*p$ axis for the boost into the $\gamma^*p$ frame.
The error on the polar angle measurement of the scattered positron in 
the SPACAL is $\pm 1 \rm\ mrad$, leading to a systematic error of $1\%$ to
$2\%$. The uncertainty on the electron energy measurement, which is $0.3\%$
at the kinematic peak ($E_{e'}=27.5 \rm\ GeV $)
and increases to $2\%$ at $E_{e'}=8 \rm\ GeV $, results in a systematic
error between $\pm1\%$ and $\pm5\%$, depending on the kinematics. 

\item The uncertainty on the fraction of energy of the reconstructed hadronic
objects carried by tracks is $\pm 3\%$, leading to a systematic error in the
range $1\%$ to $5\%$.

\item The uncertainties on the determinations of the trigger efficiency 
and the $ep$ luminosity affect the total normalization by $5\%$ and 
$2\%$ respectively.

\item There is an uncertainty of $25\%$ on the fraction of rejected events
due to noise in the FMD detector, which translates into a $2\%$ error on
the measured cross sections.

\end{enumerate}

\noindent
The corrections applied to the
measured data are affected by the following uncertainties:

\begin{enumerate}
\setcounter{enumi}{5}

\item The uncertainty on the number of background events migrating into the 
sample from $x_\pom>0.2$ or $M_Y>5 \rm\ GeV$ is estimated as
$\pm 25\%$, leading to a $1\%$ to $3\%$ systematic error, with the 
biggest values at large $x_\pom$.

\item A $\pm5\%$ uncertainty arises from the $M_Y$ 
smearing correction. It is estimated by:
\begin{enumerate}
\item Variation of the ratio of proton elastic to proton dissociation cross
sections to either 1:2 or 2:1
\item Variation of the generated $M_Y$ distribution in DIFFVM by 
$1/M_Y^{2.0\pm0.3}$
\item Variation of the $t$ dependencies in the simulations by changing the 
slope parameter by $\pm1 \rm\ GeV^{-2}$ and
$\pm2 \rm\ GeV^{-2}$ in the proton dissociation and proton elastic
simulations respectively  
\item Variation of the efficiencies of the forward detectors FMD and PRT 
by $\pm4\%$ and $\pm25\%$ respectively 
\end{enumerate}

\item There is an uncertainty on the calculation of the
QED radiative corrections of $\pm 5\%$, originating from the limited  
statistics of the Monte Carlo event samples.


\item The use of different approximations for higher order QCD diagrams,
the parton shower model (MEPS) or the color dipole model (CDM) as 
implemented in the ARIADNE program \cite{ariadne}, 
for the determination of the correction
factors leads to a $\pm3\%$ uncertainty on the resulting cross sections.

\item The model dependence of the acceptance and migration
corrections was estimated by varying the
shapes of kinematic distributions in the simulations:
\begin{enumerate}
\item Variation of the $z_\pom$ distribution by $z_\pom^{\pm0.2}$ and
$(1-z_\pom)^{\pm0.2}$ 
\item Variation of the $p_T$ distribution by $(1/p_T)^{\pm0.5}$ 
\item Variation of the $x_\pom$ distribution by $(1/x_\pom)^{\pm0.2}$
\item Variation of the $t$ distribution by $e^{\pm2t}$ 
\item Reweighting of the $\eta_{jet}$ distribution 
to that observed in the data
\end{enumerate}
The resulting systematic uncertainties range between $\pm6\%$ and $\pm13\%$,
where the largest contributions originate from the 
assumed shape of the $x_\pom$ and
$\eta$ distributions in the simulation.

\item The lower $p_T$-cutoff 
chosen to avoid collinear divergences in the leading order QCD matrix 
elements in RAPGAP is relatively high ($p_T^2>9 \ GeV^2$)
with respect to the experimental cut of $p^{* 2}_{T,jet}>16 \rm\ GeV^2$, 
%because of the dominance of gluon-induced processes. 
%Below the cutoff parton showers are used. 
Studying the dependence of the cross sections on
the cutoff value results in an additional uncertainty of $\pm5\%$.
\end{enumerate}

The total systematic error has been evaluated in each bin of the cross 
sections which are presented by adding all individual systematic
errors in quadrature. The systematic error dominates the total
uncertainty on the dijet cross sections. In the case of 3-jet 
production, the statistical errors are more important.

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

\section{Results}
\label{chapter4}

This section presents the obtained differential cross sections for
dijet and 3-jet production in diffractive DIS in the 
kinematic region specified in  Tab.~\ref{kinrange} . 
The obtained cross sections are shown in
Figs.~\ref{fig1a}-\ref{fig8} . The inner error bars 
correspond to the statistical error, the outer error bars represent the
quadratic sum of the statistical and systematic errors.

\subsection{General Properties of the Dijet Data}

In Fig.~\ref{fig10}a, the uncorrected average transverse energy flow per event
for the dijet sample is shown in the $\gamma^*\pom$ center-of-mass 
frame\footnote{This frame is equivalent to the rest frame of $X$.} 
as a function of the pseudorapidity $\eta^+$. Positive values of $\eta^+$ 
correspond to the Pomeron\footnote
{The term ``Pomeron'' is used synonymously for the colorless exchange here.} 
hemisphere, negative values to the photon hemisphere. Comparing the total 
energy flow to that where only particles not belonging to
the two highest $p_T$ jets contribute, the data exhibit considerable
additional hadronic energy not associated with the jets.
This additional energy is distributed in both hemispheres with a certain
preference for the Pomeron hemisphere.
In order to examine the sharing of energy within the $X$ system on an
event-by-event basis, Fig.~\ref{fig10}b shows the uncorrected correlation
between the dijet invariant mass squared $M_{12}^2$ and the total 
diffractive mass 
squared $M_X^2$. Except for a small subset of the events at low $M_X$, 
only a fraction
of the available energy of the $X$ system is contained in the dijet
system. Typically, a significant amount of additional energy is present
which is not associated with the jets, leading to $M_{12}^2<M_X^2$.
Even taking into account the
effects of hadronisation and detector resolution,
this observation suggests the dominance of $q\overline{q}g$ states 
over $q\overline{q}$ states alone.

Figs.~\ref{fig1a} and \ref{fig1b} present differential dijet cross sections as
functions of the following observables: The photon virtuality $Q^2$, the mean
dijet transverse momentum $p_{T,jets}^*$, defined as
\begin{equation}
p_{T,jets}^* = \textstyle \frac{1}{2}\displaystyle \left (p_{T,1}^*+p_{T,2}^*\right) \ ,
\end{equation}
the $\gamma^*p$ invariant mass $W$, the mean dijet pseudorapidity in the
laboratory frame $\av{\eta}^{lab}_{jets}$, defined as
\begin{equation}
\av{\eta}^{lab}_{jets} = \textstyle \frac{1}{2}\displaystyle \left(\eta_1^{lab}+\eta_2^{lab}\right) \ ,
\end{equation}
and the $x_\pom$ and $\beta$ variables. 
The $Q^2$ and $p_{T,jets}^*$ distributions are steeply falling.
Due to the selection of events with $Q^2>4 \ \mathrm{GeV}^2$ and 
$p_{T,jets}^{* 2}>16 \ \mathrm{GeV}^2$, the relation $p_T^2>Q^2$ holds for the
bulk of the data. The $W$ range is $90<W<260 \rm\ GeV$. 
%The $\av{\eta}^{lab}_{jets}$ distribution demonstrates
%that, due to the rapidity gap selection
%and the cuts in $y$ and $x_\pom$, the jets
%are located in the central part of the detector. 
The $x_\pom$
distribution shows a rising behavior from the lowest accessible values
of 0.003 up to the cut value of 0.05 . For kinematic reasons, the
dijet measurement is restricted to larger $x_\pom$ values compared to
inclusive measurements. 
%The observed $x_\pom$ dependence is opposite to the
%energy dependence of the Pomeron trajectory because $ep$ cross sections
%are presented here. The photon flux increases towards lower values of
%$y$, which corresponds to larger values of $x_\pom$. 
The $\beta$ range covered
by the measurement extends down to almost $10^{-3}$, 
lower than accessed by the measurements of $F_2^{D(3)}$.
The measured cross sections are generally 
well described by the RAPGAP simulation
used to correct the data, except for the $\av{\eta}^{lab}_{jets}$ 
distribution, which indicates that the measured jets typically have
slightly larger pseudorapidities compared to the simulations.

Fig.~\ref{fig3} presents the cross section as a function of $z^{(jets)}_\pom$,
calculated from $M_X$, $Q^2$ and the invariant dijet mass $M_{12}$:
\begin{equation}
z_\pom^{(jets)}=\frac{Q^2+M^2_{12}}{Q^2+M_X^2} \ .
\end{equation}
$M_{12}$ is calculated from the massless jet 4-vectors.
Monte Carlo studies show that the resolution in $z_\pom$ is $25\%$ 
(see Fig.~\ref{fig11}c) and that a good correlation between 
$z_\pom^{(jets)}$ and the true value of $z_\pom$ exists. In loose terms, 
this observable measures the fraction of the hadronic final state energy of
the $X$ system which is contained in the two jets. 
Exclusive $q\overline{q}$ final states, which at the parton level
satisfy $z_\pom=1.0$, can be smeared down to $z_\pom^{(jets)}$
values of around 0.6 because of fragmentation and jet resolution effects.
The shape of the measured $z_\pom^{(jets)}$ distribution thus confirms the 
observation that 
$q\overline{q}g$ states dominate over $\overline{q}q$ states.

\subsection{Interpretation within a Partonic Pomeron Model}
\label{section:resultsrespom}

A model which has been applied quite frequently in the past to diffractive 
scattering data from HERA is the Ingelman-Schlein model as 
described in section \ref{chapter:rapgapmodel}.
It has previously been found that
Pomeron parton densities dominated by gluons
proved successful to describe not only inclusive measurements of 
the diffractive structure function \cite{H1:F2d93,H1:F2d94,f2dzeus}
but also more exclusive hadronic final state analyses performed by H1 and
ZEUS \cite{H1:diffhfs,diffhfs-zeus}, as well as first results
on dijet production reported by H1 \cite{H1:d2j94}. 
The free parameters of the model to which dijet production is most sensitive
are
the Pomeron intercept $\alpha_\pom(0)$ and the Pomeron gluon distribution
$g_\pom(z,\mu^2)$. The sub-leading Reggeon contribution and the Pomeron
quark distribution are well constrained 
by the inclusive $F_2^{D(3)}$ measurement.

The measured distribution of $z^{(jets)}_\pom$ (Fig.~\ref{fig3}),
corresponding to the momentum fraction of the colorless exchange
transferred to the dijet system, is shown together with predictions based on
the two gluon distributions obtained from the scaling violation
analysis of  $F_2^\pom(\beta,Q^2)$ in \cite{H1:F2d94}. 
The gluon distributions shown are evaluated at a scale 
$\mu^2=Q^2+p_T^2=42 \ \mathrm{GeV}^2$,
representing the mean value of the selected events. The dijet data
have a strong sensitivity to the shape of the gluon distribution.
Especially in the region of large $z_\pom$ or $\beta$,
it is not much constrained by the inclusive measurements, where data with 
$\beta>0.65$ were excluded from the QCD analysis.
If the ``flat gluon'' or ``fit 2'' gluon density is used, a very good 
agreement with the data is achieved. 
The  ``peaked gluon''  or ``fit 3'' parameterization leads to an 
overestimate of the dijet cross section at high values of  $z^{(jets)}_\pom$.
%Having determined the shape and
%normalization of the gluon distribution, 
The model is also able to give
a very good description of the other differential distributions.
This can be interpreted as support for a Pomeron structure
strongly dominated by gluons.
%To investigate how the data follow the evolution of the parton 
%densities according to the DGLAP equations used in the resolved Pomeron model,
Fig.~\ref{fig4}a shows the $z^{(jets)}_\pom$ cross section in four bins of
the factorization scale $\mu^2=Q^2+p_T^2$. Even in this double differential 
view, the resolved Pomeron model with parton densities evolving according to
the DGLAP equations gives a very good description of the data. 
The ``peaked 
gluon'' solution of the H1 QCD fits overestimates the cross section at high
$z^{(jets)}_\pom$ in all regions of $\mu^2$.

%The second main free parameter of the resolved Pomeron model apart from
%the gluon distribution is
%the value of t
The Pomeron intercept  $\alpha_\pom(0)$, controls
the energy or $x_\pom$ dependence of the cross section. In the predictions
of the resolved Pomeron model shown in
Figs.~\ref{fig1a}-\ref{fig4}, a value of  $\alpha_\pom(0)=1.2$ is used,
taken from the H1 analysis of $F_2^{D(3)}$ \cite{H1:F2d94}.
Since the value
of $\alpha_\pom(0)$ has been shown to be different at high $Q^2$ than
in soft interactions, it is interesting to investigate whether further
variation takes place with the additional hard scale introduced in the
dijet sample. In Fig.~\ref{fig1b}a, the effect on the predicted dijet
cross section is shown if
$\alpha_\pom(0)$ is changed to the ``soft Pomeron'' value of 1.08 or
to 1.4.
%the so-called ``hard'' or ``BFKL Pomeron''.
%Because the normalization of the cross 
%section in this model is a free parameter to be determined from the data, 
The normalization of the Pomeron cross section with a
changed value of $\alpha_\pom(0)$ is chosen such that the visible dijet
cross section is kept fixed. If the intercept is set to 1.08, 
%the predicted $x_\pom$
%cross section gets steeper towards higher $x_\pom$. 
the data are underestimated at low $x_\pom$, whereas at high $x_\pom$ the model
is above the data.
%The $\chi^2$ increases from 0.9 to 4.7.
If the intercept is increased from 1.2 to 1.4,
the opposite effect is observed.
% and the $\chi^2$ increases even further
%to 29.0 .
%It can thus be concluded that the dijet data
%are consistent with a value for $\alpha_\pom(0)$ of around 1.2 .
The dijet data therefore suggest a value for $\alpha_\pom(0)$ that
is close to that measured from inclusive diffraction in a similar $Q^2$
region.

In Fig.~\ref{fig4}b, the data are used to test Regge factorization,
i.e. the factorization of the cross section into a Pomeron probability 
distribution in the proton and a cross section for the interaction
between the Pomeron and the electron.
%Pomeron parton densities:
%\begin{equation}
%\sigma_{diffr.}=f_{\pom/p}(x_\pom) \cdot p_{i/\pom}(z_\pom,\mu^2) \ .
%\end{equation}
The  $z^{(jets)}_\pom$ cross section is measured in four bins of $x_\pom$.
A substantial dependence of the shape of the $z^{(jets)}_\pom$ distribution
on $x_\pom$ is observed, which is dominantly a kinematic effect 
because $x_\pom$ and $z^{(jets)}_\pom$ are connected via the relation
\begin{equation}
x_p^{(jets)} = x_\pom \cdot z_\pom^{(jets)} \ ,
\end{equation}
where
$x_p^{(jets)} $ denotes the proton momentum fraction which enters the hard 
process. 
%For a certain range of $x_p^{(jets)}$, given by the acceptance and 
%the kinematic cuts, the $z_\pom^{(jets)} $ distribution has to become softer
%(harder) for larger (smaller) values of $x_\pom$. 
It is thus only possible to draw 
conclusions from a comparison of the cross sections with Monte Carlo models 
based either on the assumption of Regge factorization or not. 
Again, the factorizing resolved Pomeron model describes the distributions 
well. Also the Saturation model, 
%not implying Regge factorization,
in which Regge factorisation is not incorporated by construction,
is able to roughly reproduce the data, if a constant scale
factor of 2.4 is applied. Thus, at the present level of precision 
firm conclusions are difficult to draw. However, interpreting Fig.~\ref{fig4}b 
in terms of the Pomeron intercept and the gluon distribution within
the resolved Pomeron model, it is clear
that there is little freedom to change $\alpha_\pom(0)$ and 
accommodate this by adjusting the gluon distribution or vice versa.

\subsection{Resolved Virtual Photons and non $k_T$-ordered Contributions} 
\label{section:xgam}

Figs.~\ref{fig1a},\ref{fig1b} and \ref{fig3} indicate that the description
of the dijet data can be further improved
if an additional contribution from resolved virtual photons is added. Here,
the virtual photon has an internal partonic structure. For the parton densities
of the photon, the SaS-2D \cite{sas} parameterization was chosen, which lead
to a good  description of an earlier measurement of inclusive
dijet production at low $Q^2$ by H1 \cite{h1:virtgam}.

Fig.~\ref{fig2} presents cross sections for two observables which are
particularly suited to interpret the data in terms of direct and resolved 
photon contributions. Similar to real photoproduction analyses 
(see e.g. \cite{gammap}), a quantity
$x_\gamma$ is defined as the momentum fraction of the photon which enters
the hard scattering interaction. If the 4-vector of the parton from the
photon is labeled $u$, $x_\gamma$ is defined as
\begin{equation}
x_\gamma=\frac{P\cdot u}{P\cdot q} \ .
\end{equation}
Direct photon events satisfy $x_\gamma=1$ by definition. Events where
the photon is resolved have $x_\gamma<1$. At the hadron level, an observable 
$x_\gamma^{(jets)}$ can be constructed by measuring the ratio of the summed 
$E-p_Z$ of the two jets over the total $E-p_Z$:
% which is correlated to the true $x_\gamma$ value:
\begin{equation}
x_\gamma^{(jets)}=\frac{\sum_{jets} E-p_Z}{\sum_{all} E-p_Z} \ .
\end{equation}
$x_\gamma^{(jets)}$ is reconstructed with a resolution of $12\%$ (see 
Fig.~\ref{fig11}d). The $x_\gamma^{(jets)}$ distribution of the data, shown in
Fig.~\ref{fig2}a, is peaked at values around 1 but also shows a sizeable
fraction of the events at lower values. The resolved Pomeron model 
including direct photon contributions alone describes
only the high $x_\gamma^{(jets)}$ region but is significantly lower than
the data at low values of $x_\gamma^{(jets)}$. It is not zero however
because of migrations from  the true value of $x_\gamma$ to the hadron
level quantity $x_\gamma^{(jets)}$. If the contribution from resolved photons
is included, which increases the total dijet cross section by $17\%$, 
a much improved agreement with the data is archived.

It is also possible to look into the part of the hadronic final state
not associated to the two highest $p_T$ jets.
In the $\gamma^*\pom$ center-of-mass frame, the $\eta^+=0$ plane defines
two hemispheres associated with the outgoing photon and Pomeron
directions. In addition to QCD radiation and the possible presence of a third
jet, 
%below the threshold of $p_{T,jet}^*=4 \rm\ GeV$, 
hadronic final state 
particle production in the two hemispheres can originate from possible
photon and Pomeron remnants. In order to further investigate
the possible presence of a photon remnant, a new observable 
$E_{rem}^{(\gamma)}$ is constructed. It is defined as the energy sum
of all final state hadrons in the photon hemisphere ($\eta^+<0$) in the
$\gamma^*\pom$ CMS which are not associated to the two highest $p_T$ jets, 
i.e. lie outside the two jet cones in the $(\eta,\phi)$ plane.
The cross section is shown differentially in
$E_{rem}^{(\gamma)}$ in Fig.~\ref{fig2}b.
The distribution is dominated by small values,
indicating the dominance of direct photon scattering. The description at
higher $E_{rem}^{(\gamma)}$ values (corresponding to $x_\gamma<1$)
is again much improved by adding the resolved $\gamma^*$ 
contribution. 

Similar observations are also made for the jet profiles in Fig.~\ref{fig9}
and the distribution of the 
transverse energy flow not associated to the jets shown in 
Fig.~\ref{fig10}. An increased energy 
flow behind the backward jet, corresponding to the photon direction, and 
in the $\eta^+<0$ hemisphere of the $\gamma^*\pom$ system can 
be better described
if the resolved photon contribution is added, which would not be
possible just by adjusting the gluon distribution.
The picture of a resolved virtual photon can be viewed as 
an approximation to
next-to-leading order QCD diagrams and/or contributions without
strong $k_T$ ordering. The presence of such contributions will be further
investigated in section \ref{2gluon} .

\subsection{Soft Color Neutralization}

Soft Color Interactions (SCI)
and recently the Semiclassical model have both been able to
give a reasonably good description of inclusive diffraction at HERA
with a small number of free parameters. 
In Fig.~\ref{fig6}, they are compared with the dijet cross sections 
as functions of $p^*_{T,jets}$, $M_X$, $x_\pom$ and $z_\pom^{(jets)}$.
The original version 
of SCI \cite{sci}, which gave an acceptable 
description of $F_2^{D(3)}$ with a single free parameter fixed by the data,
also gives a reasonable description of the shapes of the differential 
distributions of the dijet data, but is too low in normalization. 
The semi-classical model \cite{semicl} gives similar results.
%It is 
%interesting to see that the Semiclassical model \cite{semicl} gives rather 
%similar results. 
Both models underestimate the cross sections by a factor 
of about 2. This is the case even in the region of 
low $x_\pom<0.01$, where secondary 
exchanges are negligible.
%, the models are underestimating the cross sections.

The refined version of the 
SCI model \cite{scinew}, based on a generalized area law for
string rearrangements, is also compared to the data.
It has been shown to give a better description of $F_2^{D(3)}$ at
low $Q^2$. The new version of SCI reproduces 
the normalization of the dijet cross sections much better. However,
the shapes of the differential distributions, such 
as $\log x_\pom$ or $z_\pom^{(jets)}$, are not described.

Soft color neutralization models predict the shape and normalizations
of the dijet cross sections
in leading order QCD approaches based on 
fits to $F_2^{D(3)}$. The observed disagreement with the measured cross 
sections may be reduced if next-to-leading order (NLO) calculations were
available. Definite conclusions about the validity of these
approaches are thus difficult to draw at this stage.

\subsection{Colour Dipole and 2-Gluon Exchange Models}
\label{2gluon}

In this section, models based on the ideas of dipole cross sections and
two-gluon exchange are 
compared with the dijet data: the Saturation model by Golec-Biernat 
and W\"usthoff and the calculations by Bartels et al. For this purpose, a 
restricted data sample with the additional cut 
\begin{equation}
x_\pom<0.01
\end{equation}
is studied. The calculations were
carried out under the assumption of low $x_\pom$ 
to avoid the valence quark region in the proton and
contributions from
%secondary Reggeon exchanges, which set in at large $x_\pom$, are not included.
secondary Reggeon exchanges.
Applying this additional restriction reduces the number of events in the data
sample by a factor of approx. 4 .

The resolved Pomeron model implies the presence of a soft
Pomeron remnant. The same is true for $q\overline{q}g$ production
within the Saturation model, because of the $k_T$-ordering condition 
imposed for the calculations. By contrast, the $q\overline{q}g$ calculation of
Bartels et al. (``BJLW'') relies on high transverse momenta of all particles
and is not restricted to
$k_T$-ordered configurations. 
Any `remnant' system in this model is expected to have larger $p_T$. 
To gain more insight into the properties of the
part of the hadronic final state not belonging to the jets,
a new observable $p_{T,rem}^{(\pom)}$ is introduced.
Similarly to the definition of $E_{rem}^{(\gamma)}$, which was introduced
in section \ref{section:xgam}, this variable measures the transverse
momentum of the summed hadronic final state particles in the Pomeron
hemisphere of the $\gamma^*\pom$-CMS not belonging to the two
highest $p_T$ jets.

Dijet cross sections for the region $x_\pom<0.01$ differential in
$Q^2$, $p_{T,jets}^*$, $z^{(jets)}_\pom$ and $p_{T,rem}^{(\pom)}$
are shown in Fig.~\ref{fig7} . The Saturation model is not able to reproduce 
the absolute normalization of the data\footnote{It is possible that a 
fraction of the observed difference between
the Saturation and the BJLW models is due to the choices of
unintegrated gluon structure functions $\mathcal{F}(x,k_i^2)$ used in the
models. The BJLW model uses the NLO parameterization of GRV \cite{GRVfgluon},
whereas in the Saturation model $\mathcal{F}$ is parameterized from the fit to
the $F_2(x,Q^2)$ data.}, falling short by a factor 
of approx. 2 except at the highest
measured values of $Q^2$ and $p_T^*$.
It also does not describe the observed shapes of the
distributions, e.g. $z^{(jets)}_\pom$.
For the BJLW model, the contribution from $q\overline{q}$
states alone is shown scaled by a factor of 5.
It is negligibly small except at large values of $z_\pom$. As 
expected for large values of $M_X$ (low $\beta$), the $q\overline{q}g$
contribution is much more significant. 
The normalization of the BJLW model for $q\overline{q}g$
production is principally controlled by the lower cutoff on the transverse 
momentum of the gluon in the calculations. If
this cutoff is set to $p^2_{T,g}>1.0 \ \mathrm{GeV^2}$, there is  
a reasonable description of the overall dijet cross section for $x_\pom<0.01$.
The description of the shapes of the distributions
is also reasonable given that there are only two free parameters. 
The  $p_{T,rem}^{(\pom)}$  distribution is particularly well described.
Lowering the gluon transverse momentum cutoff to 
$p^2_{T,g}>0.5 \ \mathrm{GeV^2}$
leads to a cross section significantly above the measured data,
notably at low $p_{T,rem}^{(\pom)}$.
This behavior and the fact that the Saturation model
underestimates the cross section
considerably, is suggestive of
non-$k_T$-ordered contributions in the data, as discussed in
section \ref{section:xgam}.
If the BJLW model is compared to the bulk of the data with
$0.01<x_\pom<0.05$,
it substantially underestimates the cross sections.


%The BJLW model for $q\overline{q}g$
%production requires a lower cutoff on the transverse momentum of the gluon. If
%this cutoff is set to $p^2_{T,g}>1.0 \ \mathrm{GeV^2}$, there is  
%a reasonable description of the dijet data for $x_\pom<0.01$.
%As far as the differential distributions are concerned, the description
%is satisfactory, taking into account 
%the small number of free parameters ($p_{T,g}$ and an internal cutoff
%in the matrix elements in order to avoid divergences).
%The  $p_{T,rem}^{(\pom)}$  distribution is well described.
%Lowering the gluon transverse momentum cutoff to 
%$p^2_{T,g}>0.5 \ \mathrm{GeV^2}$
%leads to a cross section significantly above the measured data. The inclusion
%of lower gluon transverse momenta becomes clearly visible at low
%values of  $p_{T,rem}^{(\pom)}$, where the cross section is considerably
%enhanced. This behavior and the fact that the Saturation model with
%strong $k_T$ ordering is underestimating the cross section
%considerably, can be interpreted as a further manifestation of
%non-$k_T$-ordered contributions, in addition to the interpretation of the
%data in terms of resolved photon events in section \ref{section:xgam} .
%It is also possible that a fraction of the observed discrepancy between
%the Saturation and the BJLW models is due to differences in the 
%unintegrated gluon structure functions $\mathcal{F}(x,k_i^2)$ used in the
%models. The BJLW model uses the NLO parameterization of GRV \cite{GRVfgluon},
%whereas in the Saturation model $\mathcal{F}$ is parameterized from the fit to
%the $F_2(x,Q^2)$ data.
%If the BJLW model is compared to the bulk of the data at larger $x_\pom$ 
%values, it is substantially underestimating the cross sections in the region 
%$0.01<x_\pom<0.05$.

\subsection{3-Jet Production}

The diffractive production of 3 high-$p_T$ jets as components of the $X$ 
system has been investigated. 
%The process under study is:
%\begin{equation}
%\nonumber
%e p\longrightarrow eXY ; \qquad X = jet + jet + jet + X' \ .
%\end{equation}
Except for the requirement on the number of jets, the analysis is identical
to the dijet analysis. 
%It has become possible due to the very much 
%enlarged amount of data analyzed with respect to \cite{H1:d2j94}. 
%It is
Again, {\em inclusive} jet-production is
studied, not requiring the absence of hadronic activity beyond the jets.
%Since three-parton final states are suppressed by an
%additional power of $\alpha_s$, the observed number of 3-jet events in DIS
%diffraction is small for the integrated luminosity analyzed here. 
Approximately 130 events are observed for $p_{T,jet}^*>4 \ \mathrm{GeV}$. 
The statistical precision of the measurement is thus
much poorer than for the dijet analysis. 

In Fig.~\ref{fig8}, the measured 3-jet cross sections are presented as
functions of two observables, the 3-jet invariant mass $M_{123}$ and
\begin{equation}
z_\pom^{(3 \ jets)}=\frac{Q^2+M_{123}^2}{Q^2+M_X^2} \ ,
\end{equation}
which is, in analogy to the 
definition of $z_\pom^{(jets)}$ for dijet events,
a measure of the fraction of the energy of the $X$ system which is contained in
the jets. 
%Again, in a partonic Pomeron model $z_\pom^{(3 \ jets)}$ corresponds 
%to the fraction of the Pomeron momentum  fraction taking part in the hard scattering 
%process.
The measured cross sections are generally above the predictions from the
resolved partonic Pomeron model based on the H1 QCD fits to $F_2^{D(3)}$.
The ``flat gluon'' parton distributions, evaluated at a scale
$\mu^2=Q^2+p_T^2$, are used. 
Direct and resolved $\gamma^*$ contributions are included.
Because the leading order for 3-parton final states is $O(\alpha_s^2)$,
two different approximations for higher order QCD diagrams are considered 
here, 
the Parton Shower model (MEPS) and the Color Dipole model (CDM).
The difference between the data and the models may be explained
by the lack of a full next-to-leading order treatment of the
three-parton final states.
The BJLW model
with $p^2_{T,g}= 1.0 \ \mathrm{GeV^2}$
is not able to accommodate the observed rate of 3-jet
events. 
However, for kinematic reasons, the 3-jet
events have large values of $x_\pom>0.01$, where contributions from
the proton valence region can no longer be neglected.

The cross section differential in
$z_\pom^{(3 \ jets)}$ demonstrates that
additional hadronic activity beyond the jets is typically present
even in the 3-jet sample. An improvement in the description of the
cross section by dipole models may come through the inclusion of 
higher multiplicity 
states such as $q\overline{q}gg$, which have not yet been calculated. 

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

\section{Summary and Conclusions}
\label{chapter5}

An analysis of the production of jets as components of the dissociating photon
system $X$ in the diffractive deep-inelastic scattering reaction 
$ep\rightarrow eXY$ was presented in the kinematic region 
$4<Q^2<80 \ \mathrm{GeV^2}$, $x_\pom<0.05$ and $p^*_T>4 \ \mathrm{GeV}$. The
values of $M_Y$ and $|t|$ satisfy $M_Y<1.6 \ \mathrm{GeV}$ and 
$|t|<1.0 \ \mathrm{GeV^2}$. The  
kinematic range has been extended compared to previous
diffractive dijet measurements 
\cite{H1:d2j94}\footnote{The principal differences are the 
lowered $Q^2$ and $p^*_{T,jet}$ cuts.} and the statistical precision is
much improved. The production of three 
high transverse momentum jets has also 
been studied for the first time in diffraction. 
%The analysis of H1 data corresponding to 
%$\mathcal{L}_{int}=18.0 \rm\ pb^{-1}$ yields approx. 2.500 dijet and 
%130 3-jet events.

The observed dijet events typically exhibit a structure where
the $X$ system contains
additional hadronic energy with transverse momentum below the
jet scale, in addition to the reconstructed jets. The dijet invariant 
mass is thus generally smaller than $M_X$.
The additional energy is distributed in both hemispheres of the
rest frame of $X$, with a certain preference for the Pomeron hemisphere.
This can be interpreted in terms of a dominance of higher 
multiplicity parton level
states (e.g. $q\overline{q}g$) over the simple $q\overline{q}$ configuration.

In a resolved partonic Pomeron model, 
%the Pomeron
%quark distribution and the small sub-leading Reggeon contribution are 
%determined by inclusive $F_2^{D(3)}$ measurements. 
the dijet data give highly competitive constraints on
the diffractive gluon distribution and are also sensitive to
the Pomeron intercept $\alpha_\pom(0)$.
The data require a large fraction of the Pomeron momentum to be 
carried by a gluon distribution which is 
comparatively flat in $z_{\pom}$, compatible with the ``flat
gluon'' (or ``fit 2'') parameterization in \cite{H1:F2d94}. 
The data are compatible
with a factorisable $x_\pom$ dependence and a value for the
Pomeron intercept of $\alpha_\pom(0) \sim 1.2$, significantly higher than the 
soft Pomeron value of $\alpha_\pom(0)=1.08$.
The dijet cross sections
are best described when a contribution from resolved virtual photons
of about $17\%$ of the measured cross sections is added. The need
for this contribution is confirmed by 
the observation of enhanced hadronic activity in the photon hemisphere.
This contribution can also be considered as a first approximation to 
NLO QCD terms or to non $k_T$-ordered contributions.

The Soft Color Interactions model (SCI)
and the similarly motivated 
Semiclassical model are not able to
reproduce the dijet cross sections in shape and normalization at the same time.
The Semiclassical model and the original SCI model give 
reasonable descriptions of the shapes of the
differential distributions, but underestimate the overall
cross sections by a factor of around 2. 
%Because this discrepancy may
%originate from missing NLO contributions, final judgments on these
%models cannot yet be made. 
The area-law-improved version of SCI is better in normalization, but
fails to describe the differential distributions.

Models based on colour dipole cross sections and
two-gluon exchange have been compared with the dijet data in
the restricted kinematic region of $x_\pom<0.01$.
%, where the valence quark
%distribution in the proton and contributions from Reggeon exchanges
%are negligible. 
The Saturation model of Golec-Biernat and W\"usthoff, taking only $k_T$ 
ordered configurations into account, predicts jet 
cross sections too low in normalization by a factor of approx. 2 . 
The model of Bartels et al., 
%exploring a larger
%region of phase-space for $q\overline{q}g$ production without a 
in which strong $k_T$ ordering is not imposed, 
%in the region of large $M_X$ (low $\beta$),
%where the dijet production cross sections are measured, the contribution 
%from $q\overline{q}$
%states alone is tiny and  
is more successful. In this model, 
$q\overline{q}g$ or higher multiplicity states are
dominant at the relatively large $M_X$ values of the present data. 
With a cutoff for the gluon transverse momentum of 
$p^2_{T,g}> 1.0 \ \mathrm{GeV^2}$, 
a reasonable description of the dijet cross sections is obtained
in the region $x_\pom<0.01$. Lower values of this cutoff are 
disfavored
by the data.
% consistent with a picture that contributions from small $k_T$
%gluons are suppressed.
%For the bulk of the dijet data with $0.01<x_\pom<0.05$ however,
%the calculations are clearly underestimating the cross
%section.

For the 3-jet production cross sections, strong conclusions cannot yet 
be drawn, because of the limited statistical accuracy
and the kinematic restriction to large $\xpom$ implied by the requirement 
of 3 high $p_T$ jets. Nonetheless, the 3-jet cross sections are 
found to be significantly above the predictions based on 
H1 fits to $F_2^{D(3)}$.
The 2-gluon model by Bartels et al.
%, lacking higher multiplicity states
%such as $q\overline{q}gg$, is not 
is also unable to reproduce the rate of observed 3-jet events.

Diffractive jet production has been shown to be a powerful tool
to gain insight into the underlying QCD dynamics of diffraction,
in particular the role of gluons. 
%Compared
%to inclusive measurements of the diffractive structure function 
%$F_2^{D(3)}$,
%it has the advantage of being directly sensitive to the gluonic
%structure of the colorless exchange, whereas there is only an indirect
%sensitivity in inclusive measurements.
%The dijet data constrain well the diffractive gluon distribution and
%give a strong support of a Pomeron structure heavily dominated by gluons.
The jet cross sections are sensitive to differences between 
phenomenological models which all give a reasonable description 
of $F_2^{D(3)}$. For the first time, a calculation
based on 2-gluon exchange has been able to successfully predict low-$|t|$
diffractive dissociation cross sections at HERA in terms of perturbative QCD.

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

\section*{Acknowledgements}

We are grateful to the HERA machine group whose outstanding efforts have made
and continue to make this experiment possible. We thank the engineers and
technicians for their work in constructing and now maintaining the H1 
detector, our funding agencies for financial support, the DESY technical staff
for continual assistance, and the DESY directorate for the hospitality which
they extend to the non DESY members of the collaboration.
% - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
%We have benefited from discussions with J.~Bartels and A.~Hebecker.

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

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%\\ {\em Saturation model}

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%\\ {\em Bartels {\em et al.} qqg model }

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%\\ {\em  Rapgap }

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%\\{\em MEPS}

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%\\{\em Lund string model}

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%\\ {\em h1 virtual photon structure paper}

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\end{thebibliography}

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

%\newpage

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

\begin{figure}[b]
\centering
\epsfig{file=fig3.eps,width=.8\linewidth}
\caption{Correlations between the generated hadron level (labelled ``had'') 
and the reconstructed level (labeled ``rec'') values of 
{\em(a)} $\log_{10} x_\pom $, {\em(b)} $p_{T,jets}^*$, 
{\em(c)} $z_\pom^{(jets)}$ and {\em(d)} $x_\gamma^{(jets)}$ for the simulated
sample of RAPGAP events as described in section \ref{section:mc}. }
\label{fig11}
\end{figure}

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

\begin{figure}
\centering
\epsfig{file=fig4.eps,width=1.\linewidth}
\caption{Observed distributions of the average transverse energy flow
 per event around the jet axes in the diffractive dijet sample. 
$\Delta\eta^*$ and $\Delta\phi^*$ are
the distances from the jet axes in pseudorapidity and azimuthal angle in the
hadronic center-of-mass frame. 
The jet profiles in $\eta$ and $\phi$ are integrated over  $\pm 1$ unit in 
$\phi$ and $\eta$ respectively. {\em(a)} and {\em(c)} show the distributions
for the backward jet in the laboratory frame, whereas {\em(b)} and {\em(d)} 
show those for the forward jet. For comparison, the distributions for the
simulated sample of RAPGAP events are also shown. Here, the contribution from 
direct photons only and the sum of direct and resolved photon contributions
are shown.}
\label{fig9}
\end{figure}

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

\begin{figure}
\centering
\epsfig{file=fig5.eps,width=1.\linewidth}
\caption{{\em(a)} The uncorrected distribution of the average transverse 
energy per event in the $\gamma^*\pom$ center-of-mass frame as a function of
the pseudorapidity $\eta^+$ for the diffractive dijet sample. 
Shown are the distributions for all final state particles {\em(solid points)}
and for only those particles which do not belong to the two highest $p_T$ 
jets {\em (open points)}. The prediction of the RAPGAP simulations for either 
direct or direct plus resolved virtual photon contributions are also shown. 
%
{\em(b)} The uncorrected correlation between the invariant mass of the $X$ 
system squared $M_X^2$ and the dijet invariant mass squared $M^2_{12}$
for the diffractive dijet sample. The dotted line corresponds to 
$M_X^2=M_{12}^2$. }
\label{fig10}
\end{figure}

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

\begin{figure}
\centering
\epsfig{file=fig6.eps,width=1.\linewidth}
\caption{ Diffractive dijet cross sections as a function of {\em(a)} the
photon virtuality $Q^2$, {\em(b)} the mean transverse jet momentum 
$p^*_{T,jets}$, {\em(c)} the $\gamma^*p$ invariant mass $W$ and {\em(d)} the 
mean jet pseudorapidity in the lab frame $\av{\eta}^{lab}_{jets}$.
Also shown are the predictions from a resolved (partonic) Pomeron
model with gluon dominated Pomeron parton distributions as obtained from 
the QCD analysis of $F_2^{D(3)}$ by H1 \cite{H1:F2d94}.
Here, the ``fit 2'' parton distributions for the Pomeron were used, 
evolved to a scale $\mu^2=Q^2+p_T^2$. The dotted line corresponds to 
the direct virtual photon cross section, whereas for the solid line direct and
resolved $\gamma^*$ contributions according to the SaS-2D parameterization 
have been added.}
\label{fig1a}
\end{figure}

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

\begin{figure}
\centering
\epsfig{file=fig7.eps,width=1.\linewidth}
\caption{Differential diffractive dijet cross sections as a function of
{\em(a)} the longitudinal proton momentum fraction of the colorless exchange
$x_\pom$ and {\em(b)} $\beta$, the longitudinal momentum fraction of the 
exchange carried by the quark coupling to the photon.
The solid curves represent the predictions of the resolved
Pomeron model as described in the text with direct and 
resolved photon contributions.
For the $\log_{10} x_\pom$ distribution, the contribution from
sub-leading Reggeon exchange within a resolved Pomeron model, based on the H1
fits to $F_2^{D(3)}$ in \cite{H1:F2d94}, is indicated by the
hatched area. The dashed and dashed-dotted curves correspond to the
cross section predictions where the value of the Pomeron intercept
 $\alpha_\pom(0)$ in the model was changed from the default value of 1.20 to
1.08 and 1.40 respectively while the normalization was kept fixed.
For the $\beta$ distribution, the contribution from direct photons only is
also shown and the range covered by the inclusive H1 
measurement of $F_2^{D(3)}$ is indicated. }
\label{fig1b}
\end{figure}

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

\begin{figure}
\centering 
\epsfig{file=fig8.eps,width=0.5\linewidth}
\caption{The diffractive dijet cross section as a function of 
$z^{(jets)}_\pom$, corresponding within a partonic Pomeron model to the
Pomeron momentum fraction which enters the hard scattering process.
The predictions of the resolved Pomeron model for two 
different parameterizations of the Pomeron gluon density,
obtained from the H1 QCD fits to $F_2^{D(3)}$ \cite{H1:F2d94},
are also shown, namely the ``flat gluon'' (labeled ``H1 fit 2'') and the 
``peaked gluon''  (labeled ``H1 fit 3'') distributions. These gluon 
distributions, evolved to a scale $\mu^2=Q^2+p_T^2=42 \rm\ GeV^2$, 
corresponding to the mean value of the event sample, are also
shown. The model predictions are shown for the sum of direct and resolved 
photon contributions. The size of the direct photon contribution alone 
is indicated by the dotted line.}
\label{fig3}
\end{figure}

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

\begin{figure}
\centering
\epsfig{file=fig9.eps,width=1.\linewidth}
\caption{ {\em (a)} Diffractive dijet cross sections as a function of
$z^{(jets)}_\pom$, shown in four bins of the scale $\mu^2=Q^2+p_T^2$.
The data are compared to the resolved Pomeron model based on the
two fits to  $F_2^{D(3)}$  from H1, adding direct and resolved $\gamma^*$
contributions.
{\em (b)} The $z^{(jets)}_\pom$ cross section in four bins 
of $\log_{10}x_\pom$. Also shown are the predictions from the resolved Pomeron
model, where the ``H1 fit 2'' parameterization was used and direct and 
resolved $\gamma^*$ contributions are added, and the Saturation model, scaled
by a constant factor of $2.4$.}
\label{fig4}
\end{figure}

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

\begin{figure}
\centering
\epsfig{file=fig10.eps,width=1.\linewidth}
\caption{ Differential diffractive dijet cross sections as a function of 
{\em (a)} $x_\gamma^{(jets)}$, an estimator for the photon momentum fraction
entering the hard scattering process, and {\em (b)} $E_{rem}^{(\gamma)}$, the
summed hadronic final state energy, not belonging to the two highest $p_T$ jets,
in the photon hemisphere of the $\gamma^*\pom$-CMS.
The data are compared
to the resolved Pomeron model with and without an 
additional contribution from resolved virtual photons,
parameterized according to the SaS-2D photon parton distributions.}
\label{fig2}
\end{figure}

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

\begin{figure}
\centering
\epsfig{file=fig11.eps,width=1.\linewidth}
\caption{ Differential dijet cross sections as functions of {\em (a)} 
$p^*_{T,jets}$,
{\em (b)} $M_X$, {\em (c)} $\log_{10}x_\pom$ and {\em (d)} $z_\pom^{(jets)}$. 
The data are compared to the original version of the Soft Color
Interaction (SCI) model, labeled ``SCI (original)'',
the prediction of the improved SCI version based on a generalized area law 
for string reconnections, labeled ``SCI (area law)'', and
to the Semiclassical model.}
\label{fig6}
\end{figure}

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

\begin{figure}
\centering
\epsfig{file=fig12.eps,width=1.\linewidth}
\caption{ Diffractive dijet cross sections in the restricted kinematic range
$x_\pom<0.01$, shown as functions of {\em (a)} $Q^2$, {\em (b)}  
$p^*_{T,jets}$, {\em (c)} $z_\pom^{(jets)}$ and {\em (d)}  
$p_{T,rem}^{(\pom)}$, denoting the summed transverse 
momentum of the final state particles not belonging to the two highest $p_T$
jets and located in the Pomeron hemisphere of the $\gamma^*\pom$-CMS.
The data are compared to the Saturation model and to the calculations
by Bartels et al. (labeled ``BJLW''). There, the contribution 
from $q\overline{q}$ states alone, scaled by a factor of 5, and the sum
of the $q\overline{q}$ and $q\overline{q}g$ contributions for two different
values of the $p_T$-cutoff for the gluon in the case 
of $q\overline{q}g$ production are shown.}
\label{fig7}
\end{figure}

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

\begin{figure}
\centering
\epsfig{file=fig13.eps,width=1.\linewidth}
\caption{ Differential cross sections for diffractive 3-jet production
as functions of {\em (a)} the 3-jet invariant mass $M_{123}$ and
{\em (b)} the corresponding $z_\pom$-variable
$z_\pom^{(3 \ jets)}$, measuring the colorless exchange momentum fraction 
which enters the hard interaction. The data are compared to the resolved 
Pomeron model with two different
approaches for higher order QCD diagrams, the Parton Shower
model (labeled ``MEPS'') and the Color Dipole model (labeled ``CDM''). 
The ``H1 fit 2'' parameterization is used and direct and resolved
virtual photon contributions are added. Also shown is the 2-gluon
exchange model by Bartels et al. (labeled ``BJLW''), where $q\overline{q}$ and 
$q\overline{q}g$ contributions are added and the cutoff for the gluon 
$p_T$ is set to $p^2_{T,g}>1.0 \rm\ GeV^2$.}
\label{fig8}
\end{figure}

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

\end{document}
