\newcommand {\pom}  {I\hspace{-0.2em}P}
\newcommand {\reg}  {I\hspace{-0.2em}R}
\newcommand {\alphapom} {\mbox{$\alpha_{_{\pom}}$}}
\newcommand {\alphareg} {\mbox{$\alpha_{_{\reg}}$}}
\newcommand {\xpom} {\mbox{$x_{_{\pom}}$}}
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\title*{Inclusive Diffraction at HERA\thanks{Presented at
{\it New Trends in HERA Physics 1999}, Ringberg Workshop, June 1999}}
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%\toctitle{Inclusive Diffraction at HERA}
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\titlerunning{Inclusive Diffraction at HERA}
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\author{Henri Kowalski}
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\authorrunning{Henri Kowalski}
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\institute{Deutsches Elektronen Synchrotron DESY, 22603 Hamburg, Germany}

\maketitle              % typesets the title of the contribution

\begin{abstract}
Diffractive phenomena observed at HERA are starting to give us a new
perspective on the deep inelastic scattering. In 
this talk the properties of diffractive events observed 
by the H1 and ZEUS experiments are described.  We give also a theoretical
interpretation of the diffractive results with special emphasis on the
phenomena at the boundary of soft and hard physics. In this region 
perturbative QCD seems to provide a useful guide. 
Simple models can be constructed which 
allow to connect the photoproduction, deep inelastic scattering
 and diffractive phenomena in a quantitative way.
\end{abstract}

\section{Introduction}
%
\begin{figure}[b]
\begin{center}
%\hspace{-3.5cm}
\includegraphics[width=1.0\textwidth]{kowalski.1}
\end{center}
\caption[]{Particle emission characteristics for non-diffractive (left) and 
diffractive events (right).}
\label{kowalski.fig.1}
\end{figure}
One of the most important results of the experiments at HERA is the
 observation 
that a substantial amount $(\sim 10\%)$ of deep-inelastic electron-proton 
scattering (DIS) events is of diffractive origin. 
The usual DIS events are characterized 
by  continuous particle emission between the direction of the struck
quark and the proton remnant. In the $\gamma^*$-proton rest frame this
leads to an almost uniform distribution of emitted particles along the 
$\gamma^*p$ rapidity axis, The rapidity interval in which particles are
radiated has the length  $\ln W^2$, where 
$W$ denotes  the $\gamma^*p$ CMS energy in GeV.  Diffractive processes
are characterized by  particle emission in a considerably shorter rapidity
interval  given by $\ln M_X^2$, where $M_X$ denotes the diffractive mass,
see Fig.~\ref{kowalski.fig.1}.
The  mass $M_X$, which is equal to the invariant mass of all particles emitted 
in the  reaction with exception of the outgoing proton (or the proton 
dissociated system), is in diffractive reactions considerably smaller than 
$W$. Therefore diffractive events exhibit clear rapidity gaps, given by
$\Delta Y = \ln W^2 / \ln M_X^2$,~\cite{kow.feyn}.

The relatively small amount of hadronic radiation emitted in diffractive
reactions is due to the dominance of the colour singlet exchange; this is
in contrast to non-diffractive $\gamma^* p$ interactions where coloured 
gluon exchanges dominate. Diffraction gives  new insight into deep 
inelastic electron proton interactions and gives hope that,
because a hard scale is involved, more can be learned about a possible partonic
mechanism  of diffraction and, eventually,  about the transition between
the perturbative and nonperturbative regimes. 

I will first describe the diffractive measurement
and then discuss a possible theoretical interpretation.
      
\section{Diffractive Measurement}

\subsection{Kinematics}
\begin{figure}[b]
\begin{center}
\includegraphics[width=0.6\textwidth]{kowalski.2}
\end{center}
\caption[]{Diagram showing the diffractive dissociation process in DIS.}
\label{kowalski.fig.2}
\end{figure}
The kinematic quantities used for the description of inclusive DIS,  $e(k) + p(P) \to e(k^{\prime}) + anything$, see  Fig.~\ref{kowalski.fig.2}, are $Q^2 = -q^2 = -(k-k^{\prime})^2$, $x = Q^2/(2P\cdot q)$, $y = (P \cdot q)/(P \cdot k)$ and $W^2 = Q^2 (1-x)/x + m^2_p \approx Q^2/x$ for $x \ll 1$. Here $k, k^{\prime}$ are the four-momenta of the initial and final state positrons; $P$ is the four-momentum of the intial state proton and $y$ is the fractional energy transfer to the proton in its rest 
frame.
 For the range of $Q^2$ and $W$ considered in this talk $W^2 \approx y s$, where $s = 4E_eE_p$ is the square of the $ep$ c.m.s. energy, $\sqrt{s} = 300$ GeV. The scaling variables used to describe DIS diffraction are given by  $\xpom =[(P - N)\cdot q]/(P\cdot q) \approx (M^2_X~+~Q^2)/(W^2~+~Q^2)$ and $\beta = Q^2/[2(P - N)\cdot q] = x/\xpom \approx Q^2/(M^2_X~+~Q^2)$ where $N$ is the four-momentum of the outgoing nucleonic system and $M_X$ is the mass of the system into which the virtual photon dissociated. In models where diffraction is described by the $t$-channel exchange of a system, for example the pomeron, $\xpom$ is the momentum fraction of the proton carried by this system and $\beta$ is the momentum fraction of the struck quark within this system.


\subsection{Diffractive Signature}
%
\begin{figure}[b]
\begin{center}
%\hspace{-3.5cm}
\includegraphics[width=1.05\textwidth]{kowalski.3}
\end{center}
\caption[]{Distribution of $M_X$ in terms of $\ln M_X^2$. 
The solid points show the data.
The solid histograms show
the ARIADNE predictions for non-diffractive reactions.
The dashed histograms show the predictions of RAPGAP for diffractive 
reactions.}
\label{kowalski.fig.3}
\end{figure}
The diffractive and non-diffractive components can be experimentally 
separated by the analysis of the observed $M_X$ 
distribution~\cite{kow.zeus.1,kow.zeus.2}. 
Fig.~\ref{kowalski.fig.3} shows these distributions in terms of $\ln M_X^2$ for
various $W$ intervals as observed in the ZEUS detector. 
The high mass peak is due to non-diffractive events, the plateau-like
structure most notably seen at higher $W$ values, is due to diffractive ones.
The high mass peak has a steep exponential fall-off, $d{\cal N}/d\ln M^2_X \propto \exp (b\ln M^2_X)$, towards smaller $\ln M_X^2$ values. The  position
of the high mass peak changes proportionally to $\ln W^2$, i.e. shows
scaling in $\ln (M^2_X/W^2$) and the slope, $b$, of the exponential
in $\ln M_X^2$ is approximately independent of
$W$ and $Q^2$. These characteristics are properties of events with uniform, random and uncorrelated particle production along the rapidity axis where,
due to the geometrical acceptance of the detector,
 particles are measured in a limited range of rapidity, see Fig.~\ref{kowalski.fig.1} . In models such as the Feynman gas model or ones dominated by longitudinal phase space~\cite{kow.feyn}, the slope $b$ represents the particle multiplicity per unit of rapidity. The exponential in $\ln M^2_X$ and the scaling in $\ln (M^2_X/W^2)$ are directly connected to the exponential suppression of large rapidity gaps by QCD radiation. 

These characteristics are also properties of realistic models for particle production in deep inelastic scattering. ARIADNE~\cite{ariadne}, which gives a good description of particle production by DIS at HERA, also exhibits a pure exponential fall-off with $\ln M^2_X$ and scaling in $\ln M^2_X/W^2$.
The distribution predicted by ARIADNE shown in 
Fig.~\ref{kowalski.fig.3} gives a good account of the high mass peak. 
The plateau-like part of the $\ln M_X^2$ distribution in 
Fig.~\ref{kowalski.fig.3} is well described by the diffractive RAPGAP 
simulation. 

\subsection{Selection of Diffractive Events }
%
\begin{figure}[b]
\begin{center}
%\hspace{-3.5cm}
\includegraphics[width=1.05\textwidth]{kowalski.4}
\end{center}
\caption[]{Distribution of $M_X$ in terms of $\ln M_X^2$. The straight lines 
give the non-diffractive contribution as obtained from the fits.}
\label{kowalski.fig.4}
\end{figure}

In diffractive events, the system $X$ resulting from the dissociation of the virtual photon is almost fully contained in the detector while the outgoing proton or low mass nucleonic system escapes through the forward beam hole. Diffractive dissociation prefers small $M_X$ values and leads to an event distribution of the form $d{\cal N}/dM^2_X \propto 1/(M^2_X)^{(1+n)}$ corresponding to $d{\cal N}/d \ln M^2_X \propto 1/(M^2_X)^n$, approximately independent of $W$. At high energies and for large $M_X$, one expects $n \approx 0$, leading to a constant distribution in $\ln M^2_X$, as discussed below.

In the ZEUS investigation~\cite{kow.zeus.2}  the diffractive contribution 
was identified as the excess
of events at small $M_X$ above the exponential fall-off of the
nondiffractive contribution in $\ln M^2_X$. This method is called the 
$M_X$ method for the determination of the diffractive component and was
developed by H. Kowalski and G. Wolf.
The exponential fall-off
permits the subtraction of the nondiffractive contribution and, therefore,
the extraction of the diffractive contribution without assuming the precise
$M_X$ dependence of the latter. 
The $\ln M^2_X$ distribution for 
both contributions together is expected to be of the form:
\begin{eqnarray} 
\frac{d{\cal N}}{d\,\ln M_X^2} = D +  c\, \exp (b\,\ln {M_X^2}), ~~~~
{\rm for} ~~~ \ln M_X^2 \leq \ln W^2 -\eta_0. 
\label{eq:diffnondiff} 
\end{eqnarray}
Here, $D$ denotes the diffractive contribution and the second term
the nondiffractive contribution. The diffractive term $D$ is corrected for 
the detector distortion effects. The nondiffractive term is not corrected since the detector effects on the exponential fall-off were found to be negligible. The quantity ($\ln W^2 -\eta_0$) specifies the maximum value of $\ln M^2_X$ up to which the exponential behaviour of the nondiffractive part holds. Its value
was determined from the data~\cite{kow.zeus.2}.
The diffractive contribution is not taken from the fit result for $D$ but is determined by subtracting from the observed number of events the nondiffractive contribution found from the fit values of $b$ and $c$, see Fig.~\ref{kowalski.fig.4}.
The $M_X$ method selects diffractive events in which the 
system $N$, emitted close to the direction of the incoming
proton, escapes undetected. Detector acceptance limits the mass
of the system $N$ to $M_N < 5.5$ $GeV$.  

In the H1 investigation~\cite{H1epf2d394}
the selection of diffractive events was performed
by the requirement of a large rapidity gap in the event. This selection
makes use of the information from the central and various forward detectors.
The particles observed for each event are grouped into two systems, called
Y and N,
such that the maximum rapidity gap observed is between Y and N. The system
N, which includes  particles produced in the direction of the incoming
proton is required to have a mass $M_N < 1.6$ $GeV$.
The presence of forward detectors allows H1
to extend the diffractive measurement beyond the region
accessible to the ZEUS $M_X$ measurement.     

The ZEUS $M_X$ and the H1  rapidity gap methods allow only  to 
measure the diffractive cross section integrated over the 
four-momentum transfer~$t$. The $t$ dependence of the diffractive cross section
was measured by the ZEUS leading proton spectrometer (LPS)~\cite{Zeuslps}.
The LPS is designed
to measure small angle ($\sim$~1 mrad) scattered protons which would
otherwise escape undetected inside the beam pipe. Isolation of diffractive
events with the LPS is rather straight forward: 
detection of a proton carrying a
large fraction of the momentum of the incoming proton, 
$x_L = p^{LPS}/p_{p_{beam}} > 0.97$, ensures a large rapidity gap
between the outgoing proton and the system $X$. However, the event
rate is considerably limited by the acceptance of the LPS. 

 
\subsection{Results}

The cross section for the virtual photoproduction process 
$\gamma^* p \to X N$ can be determined 
from corresponding cross sections for  $ep \to e X N$ by: 
\begin{eqnarray}
\frac{d\sigma^{di\!f\!f}_{\gamma^*p \to XN}(M_X,W,Q^2)}{dM_X}  
\approx
\frac{2\pi}{\alpha}\,\frac{Q^2}{ (1-y)^2 +1}
 \,\frac{d\sigma^{di\!f\!f}_{ep \to eXN}(M_X,W,Q^2)}{dM_X d\ln W^2 dQ^2} \; 
\label{eq:sgp} 
\end{eqnarray}
The diffractive cross section determined  by ZEUS is shown in 
Fig.~\ref{kowalski.fig.5} as a function of $W$ for various $M_X$ and $Q^2$ 
values~\cite{kow.zeus.2}.
We observe a rapid rise of the cross section with $W$ at all
$Q^2$ values for the $M_X$ bins up to 7.5 GeV.

\begin{figure}[h]
\begin{center}
\includegraphics[width=0.85\textwidth]{kowalski.5}
\end{center}
\caption[]  { The differential cross sections $d\sigma^{di\!f\!f}
             _{\gamma^* p \to X N}/dM_X, \; M_N < 5.5$ GeV, as a function
             of $W$ at various $M_X$ values. The inner error bars show the 
             statistical errors and the full bars the statistical and
             systematic errors added in quadrature.  
             The solid curves show the result from fitting the diffractive 
             cross section for each ($W,Q^2$) bin separately using 
             the form  $d\sigma^{di\!f\!f}_{\gamma^* p \to X N}/dM_X 
             \propto (W^2)^{a^{diff}}$ where $a^{diff}$ and the normalization
             constants were treated as free parameters. The dashed curves show
             the result from the fit where $a^{diff}$ was assumed to be the
             same for all ($W,Q^2$) bins.
             }
\label{kowalski.fig.5}
\end{figure}
 
The differential $\gamma p$ cross section is related to the diffractive 
structure function, $F^{D(3)}_2(\beta,\xpom,Q^2)$, by
\begin{eqnarray}
\frac{1}{2M_X}\frac{d\sigma^{di\!f\!f}_{\gamma^*p \to XN}(M_X,W,Q^2)}{dM_X} \approx   \frac{4\pi^2\alpha}{Q^2(Q^2 +M^2_X)} \xpom F^{D(3)}_2(\beta,\xpom,Q^2).
\label{eq:f2d3sdiffapprox}
\end{eqnarray}
If $F^{D(3)}_2$ is interpreted in terms of quark densities then it specifies for a diffractive process the probability to find a quark carrying a momentum fraction $x = \beta \xpom$ of the proton momentum, with $x$ being the Bjorken $x$.

The H1 results are shown in Fig.~\ref{kowalski.fig.6}
in terms of diffractive structure function, 
$\xpom F^{D(3)}_2(\beta,\xpom,Q^2)$
as a function of $\xpom$ at various $\beta$ and $Q^2$ 
values~\cite{H1epf2d394}. The quick rise
with $W$ observed in the diffractive cross section measurment corresponds
here to the rise of $\xpom F^{D(3)}_2$ with decreasing $\xpom$. 
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.85\textwidth]{kowalski.6}
\end{center}
\caption[]{The solid points show the measured differential structure function
plotted as $\xpom F^{D(3)}_2(\beta,\xpom,Q^2)$ against $\xpom$ for various
$\beta$ and $Q^2$ values. The inner error bars represent the statistical
errors and the outer error bars represent the statistical and systematic 
error added in quadrature. The solid curves show the results of the Regge fit
described in text.
The dashed curves give the contributions of the pomeron alone and the pomeron 
with the interfence term.  }
\label{kowalski.fig.6}
\end{figure}
 
One of the most important results obtained from the diffractive measurement 
was the observation that the diffractive cross section rises rapidly with 
$W$ at all $Q^2$ values and the measurement of its rate of rise. 
In the ZEUS case the cross section was fitted 
for each ($M_X,Q^2$) bin using the form
\begin{eqnarray}
\frac {d\sigma^{di\!f\!f}_{\gamma^*p \to XN}(M_X,W,Q^2)}{ dM_X } & = h \cdot W^{a^{diff}} \; \; \; , 
\label{eq:gxpint}
\end{eqnarray}
\noindent where $a^{diff}$ and the
normalization constants $h$ were treated as free parameters.
In Regge models, $a^{diff}$ is related to the trajectory of the pomeron $\alphapom (t)$, averaged over $t$: $\overline{\alphapom} = 1+a^{diff}/4$. 
The fit value for $a^{diff}$ leads to $\overline{\alphapom} = 1.127 \pm 0.009 (stat)^{+0.039}_{-0.012}(syst)$.
\begin{figure}[hb]
\begin{center}
\includegraphics[width=0.65\textwidth]{kowalski.7}
\end{center}
\caption{\label{kowalski.fig.7}
        {The diffractive structure function of the proton for 
         $\gamma^* p \to XN, M_N < 5.5$ GeV, multiplied
         by $\xpom$, $\xpom F^{D(3)}_2(\xpom,\beta,Q^2)$, from the $M_X$
         analysis (solid points) compared with the 
         results from ZEUS LPS measurement obtained with an identified
         proton for $\gamma^* p \to Xp$ (stars) 
         and from a subsample of the H1 data (open points)
         for $\gamma^* p \to XN, M_N < 1.6$ GeV.
      }}
\end{figure}


In the H1 case, due to the larger kinematic region accessed by the measurement, $F^{D(3)}_2(\beta,\xpom,Q^2)$ was fitted using a
combined pomeron and reggeon ansatz,
\begin{eqnarray}
F^{D(3)}_2(\beta,\xpom,Q^2) = c_1 (1/\xpom)^{2\alphapom - 1} + 
                              c_2 (1/\xpom)^{2\alphareg - 1}
\label{eq:gxpint2}
\end{eqnarray}



\noindent H1~\cite{H1epf2d394} has given for the intercept of the pomeron trajectory a value of $\alphapom(0) = 1.203 \pm 0.020(stat)\pm 0.013(syst)$$^{+0.030}_{-0.035}(model)$. Averaging over the $t$-distribution gives approximately $\overline{\alphapom} = \alphapom (0) - 0.03$, a value which is in agreement with the ZEUS result.

The $\alphapom$ values of both experiments lie above the $\alphapom$ values deduced from hadron-hadron scattering, where the intercept of the pomeron trajectory was found to be $\alphapom^{soft}~(0)~= 1.08$~\cite{Donlan1}. Averaging over $t$ reduces this value by about 0.02 leading to $\overline{\alphapom}^{soft} = 1.06$~\cite{Donlan1}.

\begin{figure}[h]
\begin{center}
\includegraphics[width=0.5\textwidth]{kowalski.8}
\end{center}
\caption{\label{kowalski.fig.8}
        {$|t|$-dependence of the diffractive cross section as measured by the 
         ZEUS collaboration.
      }}
\end{figure}

In Fig.~\ref{kowalski.fig.7} the data from the ZEUS $M_X$ analysis (solid points) are compared with ZEUS data obtained using the LPS~\cite{Zepf2d3lps94} and with those of  H1~\cite{H1epf2d394}. For ease of comparison the $\xpom F^{D(3)}_2$ values from this analysis were scaled to the ($\beta,Q^2$) values used in the H1 analysis. The LPS data correspond to  events of the type $\gamma^*p \to Xp$ with an identified proton. No correction was applied for the contribution from double dissociation which is present in the ZEUS $M_X$ analysis but not in the LPS data. The correction would increase the LPS data by a factor of $1.45^{+0.40}_{-0.26}$. There is consistency between the $M_X$ analysis and the LPS data. The H1 data correspond to $M_N < 1.6$ GeV while those from the $M_X$ analysis are given for $M_N < 5.5$ GeV. No correction was applied. The data from H1 approximately agree with those from the ZEUS $M_X$ analysis. However, for fixed $\beta$, the H1 values have a tendency to rise faster with $Q^2$ even allowing for an extra scaling factor. 

The dependence of the diffractive cross section 
$d\sigma_{\gamma^*p\rightarrow Xp}/dM_X$ on the square of the four-momentum
transfer $t$ is shown in Fig.~\ref{kowalski.fig.8}
The cross section is steeply falling with $-t$, $d\sigma /dt \propto \exp(bt)$
with $b=7.1\pm 1.(stat)\pm 1.2(syst)$. This shows that small
momentum transfers between incoming and outgoing proton dominate as expected for diffractive scattering.

\section{Theoretical Interpretation}
In the parton model the  DIS process, $\gamma^* p \rightarrow X N$, can be visualized, in the rest frame of the  proton, 
as a fluctuation of the incoming virtual photon into a $q\overline{q}$ pair 
followed by the interaction of this pair 
with the incoming proton~\cite{NikZak0,ForRos}. In a diffractive DIS process the final state 
will be particularly simple: it  consists of a $q\overline{q}$ state 
plus a proton or debris from the dissociation of the proton, 
well separated in rapidity. 
\begin{figure}[hb]
\begin{center}
\includegraphics[width=0.5\textwidth]{kowalski.9}
\end{center}
\caption[]{Feynman diagram describing the fluctuation of $\gamma^* 
\rightarrow  q \overline{q}$ with subsequent scattering by two gluon 
exchange on the proton }
\label{kowalski.fig.9}
\end{figure}

Some thirty years ago, J.D. Bjorken has pointed out that  
 configurations with small $q$ (or $\overline{q}$) 
transverse momenta  relative to the direction of the virtual photon
will lead to a substantial diffractive cross section in DIS~\cite{Bjorken}.
Therefore, since in general one of the quarks of the pair will have a large 
logitudinal momentum,  the so called aligned jet 
configuration will dominate. The cross section for this configuration is 
expected  to scale with $Q^2$ and to have the same $W$ dependence as 
observed for diffraction in hadron-hadron scattering.  

The aligned jet model can be, today, well understood within 
QCD~\cite{BEKW,Buchm}. 
In diffraction,  the interaction of the proton with a quark pair is
mediated by the exchange of a colour singlet gluonic system. Such a system
cannot resolve a  $q \overline{q}$ (neutral) pair when the typical 
wave length of a gluon is larger than the $q \overline{q}$ pair transverse 
separation. Since the colour force which binds the proton has a wave length
of the order of the proton size  only pairs with small transverse momenta 
will interact with the proton. 

The $\beta$ distribution of the aligned quark 
configuration from transverse photons was predicted, by A. Donnachie and 
P.V. Landshoff~\cite{Donnachie87},  to be of the form
\begin{eqnarray}
F^T_{q\overline{q}}\propto \beta(1-\beta).
\label{qqterm}
\end{eqnarray}
N.N. Nikolaev and B.G. Zakharov have shown that
the same $\beta$ dependence was expected in pQCD when the aligned quarks interact with the proton through two-gluon exchange~\cite{Nikzak}. 

\begin{figure}[hb]
\begin{center}
\includegraphics[width=0.5\textwidth]{kowalski.10}
\end{center}
\caption[]{Feynman diagram describing the fluctuation of $\gamma^* 
\rightarrow  q \overline{q} g$ with subsequent scattering by two gluon 
exchange on the proton }
\label{kowalski.fig.10}
\end{figure}
The production of a $q\overline{q}g$ system by transverse photons was 
 found by N.N. Nikolaev and B.G. Zakharov to be also of leading twist and 
was assumed to have a $\beta$ dependence of the type~\cite{Nikzak}
\begin{eqnarray}
F^T_{q\overline{q}g} \propto (1-\beta)^{\gamma}
\label{qqgterm}
\end{eqnarray}
with $\gamma = 2$. A later calculation by M. W\"{u}sthoff~\cite{Wusthoff} 
found $\gamma = 3$.

 The contribution to the production of a $q\overline{q}$ system by longitudinal photons was found to be of higher twist and to have a $\beta$ dependence of the form~\cite{Wusthoff} 
\begin{eqnarray}
F^L_{q\overline{q}} \propto \beta^3(1-2\beta)^2.
\label{longterm}
\end{eqnarray}
In pQCD models the $\xpom$ dependence is expected to be driven by the $x$ dependence of the square of the gluon momentum density of the proton~\cite{Ryskin}, $[x\cdot g(x,\mu^2)]^2$, with $x=\xpom$ and $\mu$  the probing scale.

\subsection{ BEKW Model}
J. Bartels, J. Ellis, H. Kowalski and M. W\"{u}sthoff (BEKW) indentified
the theoretical processes leading to the three terms in eqs.~\ref{qqterm}-~\ref{longterm}  as the major contributors to the diffractive structure function. In~\cite{BEKW} the three contributions were calculated in the perturbative region and extended into the soft region. The $\xpom$ dependence was assumed to be of the form $(1/\xpom)^n$. The power $n$ was allowed to be different for the transverse ($n_T$) and the longitudinal ($n_L$) contributions. 
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.8\textwidth]{kowalski.11}
\end{center}
\caption[]{Comparison of the ZEUS $M_X$ data with the results of the 
BEKW fit (solid curve). The transverse $q\overline{q}$ contribution is shown by the dashed curve, the longitudinal $q\overline{q}$ contribution by the
dashed-dotted curve and the transverse $q\overline{q} g$ contribution by the dotted line.}
\label{kowalski.fig.11}
\end{figure}

The normalizations of the three terms were determined from the data:
\begin{eqnarray}
\xpom F^{D(3)}_2(\beta,\xpom,Q^2) = c_T \cdot F^T_{q\overline{q}} + c_L \cdot F^L_{q\overline{q}} + c_g \cdot F^T_{q\overline{q}g}
\end{eqnarray}
with
\begin{eqnarray}
F^T_{q\overline{q}} & = & (\frac{x_0}{\xpom})^{n_T(Q^2)} \cdot \beta (1-\beta) \\
F^L_{q\overline{q}} & = & (\frac{x_0}{\xpom})^{n_L(Q^2)} \cdot \frac{Q^2_0}{Q^2}\cdot [\ln (\frac{7}{4}+\frac{Q^2}{4\beta Q^2_0})]^2 \cdot \beta^3(1-2\beta)^2 \\
F^T_{q\overline{q}g} & = & (\frac{x_0}{\xpom})^{n_T(Q^2)} \cdot \ln (1+\frac{Q^2}{Q^2_0}) \cdot (1-\beta)^{\gamma} \\
n_{T,L}(Q^2) & = & 0.1 +  n^0_{T,L} \cdot \ln [1+\ln(\frac{Q^2}{Q^2_0})].
\end{eqnarray}

The three terms have different $Q^2$ dependences. $F^T_{q\overline{q}}$  does 
almost not depend on $Q^2$ as a result of the limited quark $p_T$ in the aligned configuration. The term $F^L_{q \overline{q}}$ is higher twist but the power $1/Q^2$ is softened by a logarithmic $Q^2$ factor; $F^T_{q\overline{q}g}$ grows logarithmically with $Q^2$ similar to the proton structure function $F_2$ at low $x$.

\begin{figure}[h]
\begin{center}
\includegraphics[width=0.70\textwidth]{kowalski.12}
\end{center}
\caption[]{Comparison of the $F^{D(2)}_2$  with the results of the BEKW fit.}
\label{kowalski.fig.12}
\end{figure}

The coefficients $c_T, c_L, c_g$ as well as the parameters $n^0_T, n^0_L$ and $x_0, Q^2_0$ were determined from experiment. In the fit the power $\gamma$ was also considered as a free parameter. Assuming $Q^2_0 = 1$ GeV$^2$ and $x_0 = 0.0042$ and treating the other constants as free parameters a good fit ($\chi^2/d.o.f. = 56/47 = 1.2$, statistical errors only) was obtained for the ZEUS $M_X$ data as shown by the solid curves in Fig.~\ref{kowalski.fig.11}. The fit yielded the following parameter values: $n^0_T = 0.13 \pm 0.03, \; n^0_L = 0.32 \pm 0.14, \; \gamma = 3.9 \pm 0.9$~\cite{kow.zeus.2}.
It is instructive to compare the $\beta$  dependence of the three components which build up the diffractive structure function using the results from the BEKW fit as shown in  Fig.~\ref{kowalski.fig.12}.\footnote{ $F^{D(2)}_2$ denotes $F^{D(3)}_2$ at $\xpom=x_0$} 
The ZEUS data suggest that for $\beta > 0.2$ the colourless system couples predominantly to the quarks in the virtual photon. The region $\beta \ge 0.8$ is dominated by the contributions from longitudinal photons. The contribution from coupling of the colourless system to a $q \overline{q} g$ final state becomes important for $\beta < 0.3$.
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.80\textwidth]{kowalski.13}
\end{center}
\caption[]{$F^{D(2)}_2$ obtained from the BEKW fit to the
ZEUS $M_X$ data as a function of $Q^2$}
\label{kowalski.fig.13}
\end{figure}
Figure~\ref{kowalski.fig.13} shows the same quantities as a function of $Q^2$ for $\beta = 0.1, 0.5, 0.9$. One finds within the BEKW model and using ZEUS data, that the gluon term, which dominates at $\beta = 0.1$ rises with $Q^2$ while the quark term, which is important at $\beta = 0.5$ shows almost no evolution with $Q^2$. The contribution from longitudinal photons, which is higher twist and dominates at $\beta = 0.9$, decreases with $Q^2$.

In the BEKW model the $\xpom$-dependence of the quark and gluon contributions for transverse photons is expected to be close to that given by the soft pomeron, $n_T \approx 2(\overline{\alphapom}^{soft} - 1)$. However, perturbative admixtures in the diffractive final state are expected to have a somewhat stronger energy dependence, leading to an effective  $n_T > 2(\overline{\alphapom}^{soft} - 1)$. The $\xpom$ dependence of the longitudinal contribution is driven by the square of the proton's gluon momentum density leading to $n_L > n_T$. The fit results agree with these predictions but the errors are too large for a definitive statement.

Similar fits were performed to H1 data~\cite{BEKW}. In the H1 case 
the data allowed two, from the $\chi^2$ point of view, almost equivalent 
solutions. One with the value of the parameter $\gamma = 0.28 \pm 0.08$ and
another with $\gamma = 8.5 \pm 0.8$. The high $\gamma $ solution leads to 
conclusions which are similar to those obtained from the ZEUS data analysis.
In the low $\gamma $ solution the gluon term is dominant
at large $\beta$ region, which is
in agreement with the H1 analysis~\cite{H1epf2d394} based on a DGLAP NLO fit.

\subsection{Saturation Model}

A.H. Mueller in his DIS 98 talk at Brussel~\cite{Almueller} drew attention 
to the fact that the much stronger rise with $W$ of the diffractive 
cross section in DIS as compared to hadron-hadron scattering combined
with the behaviour of the $F_2$ data suggests the presence of
 saturation in DIS processes.
Motivated by this talk K. Golec-Biernat with M. W\"{u}sthoff developed
a model which connects in a quantitative way the low and higher $Q^2$
total cross section data with the inclusive diffractive
 process~\cite{gbw1,gbw2}. Similar models were also proposed by
J.R. Forshaw et al.~\cite{Forshaw} and W. Buchm\"{u}ller et al.~\cite{Buchm}.

\begin{figure}[h]
\begin{center}
\includegraphics[width=0.6\textwidth]{kowalski.14}
\end{center}
\caption[]{Diagram of the basic saturation model process}
\label{kowalski.fig.14}
\end{figure}

In a frame, where photon and proton are collinear, the total $\gamma p$ 
and the differential diffractive cross 
section can be written as~\cite{NikZak0,ForRos}:
\begin{eqnarray}
\sigma_{T,L}(x,Q^2) = \int d^2 \vec{r} \int dz \, |\Psi_{T,L}(z,\vec{r})|^2
\, \hat{\sigma}(x,r^2) 
\label{satur1}
\end{eqnarray}
\begin{eqnarray}
\frac{d\sigma^{Diff}_{T,L}}{dt} \left|_{t=0} \right.  = \frac{1}{16\pi}
 \int d^2 \vec{r} \int dz \, 
|\Psi_{T,L}(z,\vec{r})|^2 \, |\hat{\sigma}(x,r^2)|^2 
\label{saturD}
\end{eqnarray}

\noindent where $\Psi_{T,L}(z,\vec{r})|$ denotes
the transverse $(T)$ and longitudinally $(L)$  polarized 
photon wave functions, $\hat{\sigma}(x,r^2)$ the dipole cross section of the 
$q\overline{q}$ pair with the proton, $z$  the momentum fraction of the 
photon carried by the quark and $r$ the relative transverse separation 
between the quarks.
The wave functions are solely determined by the coupling
of the photon to the quark and therefore are well known in QCD~\cite{ForRos}.
The cross section of the $q\overline{q}$ pair with the proton
is assumed to be a function of the transverse separation $r$ and the saturation radius $R_0$:  
\begin{eqnarray}
 \hat{\sigma}(x,r^2) = \sigma_0 \, (1 - \exp [-\frac{r^2}{4R^2_0}])
\label{satur2}
\end{eqnarray}

At small $r$, $r \ll 2\, R_0$, the dipole cross section grows quadratically
with $r$, $\hat{\sigma} \sim \sigma_0 r^2/4R^2_0$, at large $r$,
 $r \gg 2\, R_0$, it saturates, $\hat{\sigma}=\sigma_0$.

The wave function, $\Psi(r)$, at small quark transverse separation, $r< 1/Q$, behaves as 
$Q \cdot 1/(Qr)$, whereas for large quark transverse separation, $r > 1/Q$, 
it is 
exponentially suppressed.
When the transverse size of the $q\overline{q}$ pairs is much 
smaller than the saturation radius, $1/Q \ll R_0$, the total cross section 
behaves as:    
        
\begin{eqnarray}
 \sigma_T =  \int dz \int_{0}^{1/Q^2} dr^2 Q^2\, 
\left( \frac{1}{Q^2 r^2} \right) \sigma_0\, \frac{r^2}{4R^2_0}
\sim \frac{1}{Q^2} \frac{\sigma_0}{R^2_0} 
\label{satur3}
\end{eqnarray}

When  the transverse size  of the $q\overline{q}$ 
pairs exceeds the saturation radius, $1/Q \gg R_0$, the integration
over $dr^2$ splits into two parts:

\begin{eqnarray}
\sigma_T =  \int_{0}^{R^2_0} dr^2  
\left( \frac{1}{ r^2} \right) \sigma_0 \, \frac{r^2}{4R^2_0}
+ \int_{R^2_0}^{1/Q^2} dr^2 \left( \frac{1}{ r^2} \right) \sigma_0 
\sim \sigma_0 + \sigma_0\,\ln\left(\frac{1}{Q^2\, R^2_0}\right)
\nonumber
\label{satur4}
\end{eqnarray}

\noindent and $\sigma_T$ becomes almost independent of $Q^2$. From the above approximative arguments it can be seen
 that the model describes the well known scaling behaviour
at higher $Q^2$, $\sigma_T \sim 1/Q^2$, and the break down of scaling
at low $Q^2$, $\sigma_T \sim \sigma_0$. 

In order to describe the $W$ (or $x$) dependence 
of the total DIS cross sections measured at HERA, 
the saturation radius has  to be $x$ dependent. 
\begin{eqnarray}
R^2_0(x) = \frac{1}{Q^2_0}  
\left( \frac{x}{ x_0} \right)^\lambda
\label{satur5}
\end{eqnarray}
\noindent with $Q_0 = 1$ GeV. The eqs.~\ref{satur1}, \ref{saturD}, 
\ref{satur2} and~\ref{satur5} define the model called in the following the 
saturation model. 
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.9\textwidth]{kowalski.15}
\end{center}
\caption[]{The $\gamma^* p$ cross section compared to the results of the
saturation model fit (solid line). The dotted line shows an alternative fit
in which the quark mass was set to zero.}
\label{kowalski.fig.15}
\end{figure}
 
The parameters of the model $\sigma_0$, $x_0$ and $\lambda$ where determined
from the fit to the HERA total cross section data.
 Figure~\ref{kowalski.fig.15} shows the HERA total cross section data
in the $Q^2$ region between 0.1 and 100 $GeV^2$ and compares them to the fit.
Note that a very good description is achieved for the transition between the 
photoproduction-like region at  $Q^2 < 1 GeV^2$ and the scaling 
region at higher $Q^2$. 
The parameters of the model are determined as $\sigma_0 = 23$ (mb), 
$\lambda = 0.29$, $x_0 = 0.0003$.
 The value of $\sigma_0$  is mainly determined  by 
the behaviour of the data in the photoproduction-like region while  $\lambda $
 and $x_0$ is mainly determined by the  DIS region. 
Note that $\sigma_0$, which in the model corresponds to the 
$q \overline{q}$ proton  cross section, is
of the same magnitude as the total $\pi^{\pm} p$ cross section.
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.8\textwidth]{kowalski.16}
\end{center}
\caption[]{Comparison of the ZEUS $M_X$ data with the results of the 
saturation model (solid curve). The transverse $q\overline{q}$ contribution is shown by the dashed curve, the longitudinal $q\overline{q}$ contribution by the
dashed-dotted curve and the transverse $q\overline{q} g$ contribution by the dotted line.}
\label{kowalski.fig.16}
\end{figure}

\begin{figure}[h]
\begin{center}
\includegraphics[width=0.7\textwidth]{kowalski.17}
\end{center}
\caption[]{
The ratio of the diffractive and the total cross sections compared with the
prediction of the saturation model (solid line).}
\label{kowalski.fig.17}
\end{figure}
The results of the saturation model are compared with the data in 
 Fig.~\ref{kowalski.fig.16}. We see a good agreement with data at all
$Q^2$ and $M_X$ values, similar in quality to the BEKW fit as  
seen in Fig.~\ref{kowalski.fig.11}. 
The difference is that it is now an absolute
prediction and not a fit. All parameters of the model were  
fixed by $F_2$ data.

It is interesting to compare the data on the ratio of the  diffractive
 and  total cross sections with the predictions of the
saturation model as shown in Fig.~\ref{kowalski.fig.17}. When this ratio was 
first published the data were difficult to understand since 
the diffractive and total cross sections 
show the same rate of growth with $W$. This is contrary to the na\"{\i}ve 
expectation that according to the optical theorem the diffractive cross
section should grow with a power of $W$ which is twice the power of the 
total one, 
$\sigma_{Diff} \sim W^{2a} $ while $\sigma_{TOT} \sim W^{a}$. This basic 
property is also present in the saturation model;
 note that the dipole cross
section, $\hat{\sigma}$, appears in quadrature in the equation~\ref{saturD} 
while only one power of $\hat{\sigma}$ enters
equation~\ref{satur1}. Therefore, if $\hat{\sigma}$ leads to
a growth of the total cross secton with the power $\lambda$,
$\sigma_T \sim W^\lambda$, (see eqs.~\ref{satur3},~\ref{satur5}) then the 
diffractive cross section would, again,  be expected to grow with the 
power $2\lambda $.
The saturation model leads, however, to a remarkably good description of
the experimentally observed $\sigma_{Diff}/\sigma_{TOT}$ ratio. 
This is due to the fact that the main contribution to diffraction is
coming from $q \overline{q}$ pairs with larger
transverse separation, $r$,  than in the inclusive process.
In the diffractive case, 
a similar  integration as in eq.~\ref{satur3} leads only to
a higher twist contribution. A leading twist contribution can 
be obtained if the $dr^2$ integration is extended beyond the $1/Q^2$ limit.
However, at these large transverse distances  saturation starts to damp 
the growth of the diffractive cross section~\cite{gbw2}.
 
\section{Conclusions}
The first round of experimental data on diffraction at HERA has lead to 
a growing theoretical understanding of relations between the, previously
disconnected, phenomena of photoproduction, deep inelastic scattering and 
diffraction. Perturbative QCD has served as a useful guide into
the transition region between  hard and soft phenomena.   

In the near future  HERA experiments 
can be expected to provide a much larger body of data on diffraction.
These should help to understand diffraction in  deep inelastic
scattering, especially in the transition region between the perturbative and
nonperturbative regimes.

\section{Acknowledgments}
It is a pleasure to thank the organizers, G. Grindhammer, 
B. Kniehl and G. Kramer for a stimulating and enjoyable  workshop.
I have considerably benefited from discussions with 
J. Bartels, E. Levin, A.H. Mueller, G. Wolf and M. W\"{u}sthoff.
I am grateful to G. Wolf for a critical reading of the manuscript and for 
many comments.
  

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\end{thebibliography}

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