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\begin{document}

\pagestyle{empty}
\begin{titlepage}

\noindent
\begin{center}
\begin{small}
\begin{tabular}{llrr}
Submitted to & & &
\epsfig{file=H1logo_bw_small.epsi,width=2.cm} \\[.2em] \hline
\multicolumn{4}{l}{{\bf
                International Europhysics
                Conference on High Energy Physics},
                July~12,~2001,~Budapest} \\
 {\bf EPS 2001:} 
                 & Abstract:        & {\bf 808}    &\\
                 & Parallel Session & {\bf 2}   &\\[.7em]
%                 & Plenary Session  & {\bf Hebecker, Yoshida}   &\\[.7em]
\multicolumn{4}{l}{{\bf
               XX International Symposium on Lepton and Photon Interactions}, 
               July~23,~2001,~Rome} \\ 
{\bf LP 2001:}  
                 & Abstract:        & {\bf 500} &\\
                 & Plenary Session  & {\bf 8}   &\\ \hline
 & \multicolumn{3}{r}{\footnotesize {\it
    www-h1.desy.de/h1/www/publications/conf/conf\_list.html}} \\[.2em]
\end{tabular}
\end{small}
\end{center}
\vspace*{2cm}

\begin{center}
  \Large
  {\bf 
    Measurement of the Diffractive Structure Function 
    {\boldmath $\ftwodarg$ at HERA} \\ 
  }
  \vspace*{1cm}
    {\Large H1 Collaboration} 
\end{center}

\begin{abstract}

\noindent
A new high precision inclusive measurement of the diffractive deep
inelastic scattering (DIS) process $ep \rightarrow eXY$ 
%using the H1 detector 
is presented, where $Y$ is a proton or a low mass proton
excitation carrying a fraction $1 - \xpom > 0.95$ of the beam
longitudinal momentum and the squared 4-momentum transfer 
at the proton vertex $t > -1 \ {\rm GeV^2}$. 
%The photon dissociation system $X$ 
%is contained within the central part of the detector and 
%is separated from $Y$ by a
%large rapidity gap. 
The measurement is presented in the form of a
diffractive structure function $\ftwodarg$, which is measured in the
kinematic range $6.5<Q^2<120 \rm\ GeV^2$, $0.01<\beta<0.9$ and
$10^{-4}\lapprox\xpom<0.05$. The $\xpom$ dependence of the data is
interpreted in terms of a measurement of the effective pomeron
intercept $\alphapom(0)$, which is compared with the same quantity
extracted from inclusive DIS.   
The $\beta$ and $Q^2$ dependences of
$F_2^D$ are studied at fixed $\xpom$.  
Scaling violations resulting in a rising dependence on $\ln Q^2$ 
are observed up to large values of $\beta$. The $Q^2$ dependence of
$F_2^D$ is compared with the $Q^2$ dependence of $F_2(x,Q^2)$ at
the same value of $x$. 
The data
can be described by the DGLAP evolution equations assuming
QCD hard scattering factorisation for semi-inclusive processes and
an $\xpom$ dependence motivated by Regge theory.
The measured $F_2^D$ is compared with models
for diffractive DIS.
\end{abstract}


\end{titlepage}

\pagestyle{plain}

\section{Introduction}

The description of diffractive Deep-Inelastic Scattering (DIS) has 
become one of the main challenges in the development of our
understanding of Quantum Chromodynamics (QCD). The presence of
large gaps in the rapidity distribution of final state hadrons
implies the exchange of systems of at least two partons in a 
net colour singlet configuration and therefore requires new field
theory techniques for a QCD description.

A hard scattering QCD factorisation theorem has recently been
proven for a general class
of semi-inclusive processes in 
deep-inelastic scattering (DIS), which include the diffractive
dissociation process $ep \rightarrow eXY$ \cite{facold,collins}.
This implies that the
concept of `diffractive parton distributions' 
can be introduced, expressing proton parton probability
distributions under the constraint of a leading baryonic system
at a particular value of $\xpom$, integrated over
a fixed range in $\my$ and $t$.
These diffractive parton distributions evolve with $Q^2$ according to
the DGLAP equations.
The phenomenology of soft hadronic interactions suggests that it may
be possible to extend this factorisation to the idea of a universal  
pomeron exchange with parton densities dependent only on $\beta$ and
$Q^2$ and a flux factor dependent on $\xpom$ and $t$.
Previous H1 inclusive measurements of 
diffractive DIS, presented in the form of a diffractive structure
function $\ftwodarg$ \cite{h1f2d94}, have been found to be consistent with 
both collinear QCD factorisation and pomeron universality. Parton
densities were extracted for the pomeron under these assumptions
and were found to be dominated by a large gluon density with significant
contributions at high momentum fractions $\beta$. These pomeron
parton densities
give a remarkably good description of measurements of the properties
of hadronic final states in diffractive DIS \cite{H1:finalstates},
in particular jet cross sections \cite{H1:jets}, 
which are especially sensitive to the gluon density.

A further challenge in DIS at low $x$ is to relate the 
diffractive dissociation process $ep \rightarrow eXY$ to the
total cross section $ep \rightarrow eX$. Many 
authors (e.g. \cite{semicl,kgb,kgbnew}) have attempted to do
this by considering the scattering from the proton of partonic 
fluctuations of the photon, represented as colour dipoles.

In this paper, we present a new measurement of $\ftwodarg$. The measurement
is used to investigate the factorisation properties of diffractive
DIS and to test QCD models. The new measurement represents a
significant improvement in precision relative to previous HERA
data \cite{h1f2d94,ZEUS:94}.

\subsection{Kinematics of Diffractive DIS at HERA}

Figure~\ref{diagrams} illustrates the generic diffractive process at
HERA of the type $ep\rightarrow eXY$.  The electron
%\footnote{From now
%  on, the word `electron' will be used as a generic term for electrons
%  and positrons.}  
(with 4-momentum $k$) couples to a virtual photon
$\gamma^*$ ($q$) which interacts with the 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=(k+P)^2 \simeq (300 \ \mathrm{GeV})^2 \ ; \qquad
 W^2=(q+P)^2 \simeq ys-Q^2 \ .
\label{eq:w2}
\end{equation}
If the interaction takes place via colour singlet exchange, the photon
and proton dissociate to produce distinct hadronic systems $X$ and
$Y$, 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
$x_\pom$ of the colourless exchange with respect to the incoming
proton and the squared four-momentum transferred at the proton vertex
$t$ are then defined by
\begin{equation}
\xpom = \frac{q \cdot (P - p_Y)}{q \cdot P} = \frac{Q^2+M_X^2-t}{Q^2+W^2-M_p^2}\ ; 
\qquad t=(P-p_Y)^2 \ ,
\end{equation}
where $p_Y$ is the 4-momentum of $Y$. In the analysis presented here,
$t$ and $M_Y$ are not well measured. The results are thus integrated
over $|t| < 1.0 \ {\rm GeV^2}$ and $\my < 1.6 \ {\rm GeV}$.

%\footnote{It is noted that for this analysis $M_Y=M_p$
%  dominantly.}.  
In addition, the quantity $\beta$ is defined as
\begin{equation}
  \beta = \frac{x}{x_\pom} = \frac{Q^2}{2 q \cdot (P - p_Y)} = \frac{Q^2}{Q^2+M_X^2-t}\ .
\end{equation}
In an interpretation in which partonic structure is ascribed to the
colourless exchange, $\beta$ is the longitudinal momentum fraction of
the exchange that is carried by the struck quark, in analogy to $x$ in
the case of inclusive scattering.


\begin{figure}[t]
\centering \epsfig{file=H1prelim-01-111.fig1.eps,width=.5\linewidth}
\caption{ The generic diffractive process at HERA, where the 
  electron (four-momentum $k$) 
couples to a photon ($q$) which interacts with the
  proton ($P$) via net colour singlet exchange, producing two distinct
  final state hadronic systems $X$ and $Y$.  If the masses of $X$ and
  $Y$ are small compared with $W$, the two systems are separated by a
  large gap in rapidity. }
\label{diagrams}
\end{figure}

\section{Data and Analysis Method}

A full description of the H1 apparatus can be found in \cite{h1det}.
The coordinate system used is such that $\theta = 0$ corresponds to
the direction of the outgoing proton beam. The region of low 
$\theta$ and large pseudorapidity $\eta$ is referred to as the 
`forward' direction. 

The data used for the measurement were taken during 1997, when
HERA collided protons of energy $820 \ {\rm GeV}$ with positrons
of energy $27.5 \ {\rm GeV}$. The measurement covers the
kinematic region $Q^2 > 6 \ {\rm GeV^2}$, $y > 0.04$ and 
$\xpom < 0.05$.
For $Q^2 > 13.5 \ {\rm GeV^2}$, a
luminosity of $10.6 \ {\rm pb^{-1}}$ is used in the analysis,
yielding an increase in statistics relative to previous 
measurements \cite{h1f2d94,ZEUS:94} by a 
factor of approximately 5. For the region
$6 < Q^2 < 13.5 \ {\rm GeV^2}$, a sample of $2.0 \ {\rm pb^{-1}}$
is used, taken during a period when the experiment ran with 
dedicated triggers for low $Q^2$ DIS. 

The data were triggered principally on the basis of an energetic 
cluster in the electromagnetic section of the backward `SPACAL' 
calorimeter,
for which the efficiency is close to 100\% throughout the measured
kinematic region. Events are accepted for the analysis if they
contain a positron candidate with energy $E_e^\prime > 8 \ {\rm GeV}$
in the polar angle range $156^\circ < \theta_e < 176^\circ$,
linked to a reconstructed charged
track in the backward drift chambers.
A reconstructed event vertex is required from the Central Tracking
Detector. 

Diffractive events are selected on the basis of a large rapidity
gap separating the leading baryonic system $Y$ from the photon
dissociation system $X$. The large rapidity gap is identified
by the absence of activity in detectors sensitive to forward energy 
flow. The region of the main Liquid Argon Calorimeter with 
$\eta > 3.2$ must show no energy deposits above noise levels. There
must also be no significant activity in the 
`Proton Remnant Tagger' scintillators
surrounding the beam at $z \sim 26 \ {\rm m}$ and sensitive to energy
flow in the region $6.0 < \eta < 7.5$, the 
Forward Muon Detector ($5.0 < \eta < 6.5$) or the
Plug calorimeter ($3.5 < \eta < 5.5$).

The mass $\mx$ of the system $X$ is measured from the hadronic activity
in the SPACAL and Liquid Argon calorimeters and the tracking detectors
using a method that combines tracks and calorimeter deposits without
double counting. A minimum of two hadronic final state objects
are required for the analysis. In order to suppress photoproduction
background and ensure good reconstruction of kinematic variables,
close agreement is required between $y$ as obtained using the
electron only, hadron only and double angle reconstruction methods. 

The reconstruction of kinematic variables is performed using very 
similar techniques to those described in \cite{h1f2d94}. To reconstruct
$y$, $Q^2$ and $x$, a mixed method is used, which reduces to the
electron method at high $y$ and the double angle method at low $y$.
\begin{eqnarray}
  y = y_e^2 + y_d \ (1 - y_d) \ ; \hspace{1.5cm}
  Q^2 = \frac{4 E_e^2 \ ( 1 - y)}{\tan^2 (\theta_e / 2)} \ ; \hspace{1.5cm}
  x = Q^2 / s \cdot y \ ,
\end{eqnarray}
where $y_e$ and $y_d$ are the values of $y$ as reconstructed from the
electron and double angle methods respectively, $E_e$ is the
positron beam energy and $\theta_e$ is the polar angle of the scattered
positron. The mass of the system $X$ is obtained from
\begin{eqnarray}
  M_x^2 = (E^2 - p_x^2 - p_y^2 - p_z^2)_{\rm hadrons} \cdot
\frac{y}{y_h} \ ,
\end{eqnarray}
where $y_h$ is the value of $y$ reconstructed using the hadron only 
method.
This method of $\mx$ reconstruction
reduces essentially to a measurement of the total $E + p_z$
of the hadrons in the limit of high $y$, where losses in the
backward direction become significant. To account for residual
losses, the measured $\mx$ is scaled by $1.08$. The remaining kinematic
variables are reconstructed using
\begin{eqnarray}
   \beta = \frac{Q^2}{Q^2 + \mx^2} \ \ ; \hspace{2cm}
   \xpom = \frac{x}{\beta} \ .
\end{eqnarray}

Corrections for detector inefficiencies, migrations between 
measurement intervals and the acceptance of the selection are performed 
using a Monte
Carlo simulation which combines several different models. The 
RAPGAP \cite{rapgap} model is used 
to simulate dissociative processes with $\xpom < 0.15$, arising from 
pomeron and sub-leading 
meson exchange. The DIFFVM model \cite{diffvm} is used to
simulate the quasi elastic production of the $\rho$, $\omega$,
$\phi$ and $J/\psi$ vector mesons. Smearing from the region
$\xpom > 0.15$ is modelled using the DJANGO \cite{django} Monte
Carlo model. The small QED-Compton background at the largest $\beta$
is subtracted using the
COMPTON \cite{compton} Monte Carlo model. Photoproduction background,
which is negligible except at the highest $y$ values,
is subtracted using the PHOJET \cite{phojet} model. 
Figure~\ref{control} shows comparisons between the uncorrected data
and the full Monte Carlo simulation for several important variables
used in the reconstruction. The simulation gives a good 
overall description
of the data.

The large rapidity gap selection yields a sample dominated by the
single dissociation process $ep \rightarrow eXp$, with a small
admixture of double dissociation events of the type
$ep \rightarrow eXY$ where the proton dissociation system has a 
small mass $\my$. The measurement is corrected to the region
$\my < 1.6 \ {\rm GeV}$ and $t > -1 \ {\rm GeV^2}$ 
as was the case for previous H1 data \cite{h1f2d94}. 
The final structure functions correspond to the case where the 
systems $X$ and $Y$ are separated by the largest gap in the rapidity 
distribution of the final state hadrons. 
The correction
factor applied to account for smearing about the $\my$ and $t$ 
boundaries of the measurement is $-8.7 \pm 8.2 \%$. 

\section{Structure Function Extraction}

Two different binning schemes are used in the measurement. In the first,
the differential cross section
${\rm d^3} \sigma / ( {\rm d} \beta \ {\rm d} Q^2 \ {\rm d} x)$ is measured
at a large number of $x$ values, to allow detailed investigation of the
$\xpom$ (or $x = \xpom \cdot \beta$) dependence of the data. The structure
function is then extracted using
\begin{eqnarray}
  \ftwodarg = \frac{\beta^2 \ Q^4}{4 \pi \alpha^2 \ (1 - y + y^2 / 2)} \ \
\frac{\dd^3 \sigma_{ep \rightarrow eXY}}{\dd \beta  \ \dd Q^2 \ \dd x} \ ,
\end{eqnarray}
where $\alpha$ is the fine structure constant.
In the second binning scheme, the differential cross section
${\rm d^3} \sigma / ({\rm d} \beta \ {\rm d} Q^2 \ {\rm d} \xpom)$ is
extracted at four fixed values of $\xpom$. This second method allows
the $\beta$ and $Q^2$ dependence of the data and its variation
with $\xpom$ to be studied with high precision. The structure
function is extracted using
\begin{eqnarray}
  \ftwodarg = \frac{\beta \ Q^4}{4 \pi \alpha^2 \ (1 - y + y^2 / 2)} \ \
\frac{\dd^3 \sigma_{ep \rightarrow eXY}}{\dd \beta  \ \dd Q^2 \ \dd \xpom} \ .
\end{eqnarray}
For both schemes, $\ftwod$ is extracted under the assumption that the
longitudinal structure function $F_L^{D(3)} = 0$. This assumption
has negligible influence on the extracted $\ftwod$, except at the
largest $y$ values, corresponding to the lowest $\xpom$. 
The effect of a non-zero value of  $F_L^{D(3)}$
is discussed in section \ref{regge}.

For all data points shown, the 
total acceptance exceeds 40\%, the acceptance of the forward
detector selection is greater than 50\% and the bin purities exceed
30\%.
Corrections for initial and final state electromagnetic radiation and
QED virtual loops are performed using the RAPGAP Monte Carlo
model via an interface to HERACLES \cite{heracles}. The final measurements
are quoted at the Born level.

A detailed systematic error analysis has been performed in which the
sensitivity of the measurement to variations in the efficiencies and
energy scales of the detector components and to variations in the details of
the Monte Carlo models used for corrections are evaluated. The resulting
systematic error is in the range 10-15\% for most of
the data points, the largest contribution arising from the correction
to the measured $\my$ and $t$ regions in most cases. 


\section{Results}

The measured $\ftwod$ is compared with previous H1 data \cite{h1f2d94}
in figure~\ref{stamp97}. The improved statistics of the new measurement
allow the structure function to be extracted in an increased number
of $Q^2$ and $\xpom$ bins. In the region $Q^2 > 15 \ {\rm GeV^2}$,
the statistical precision is considerably improved. The two measurements
are in good agreement with the exception of the low $\beta$, medium
$Q^2$ (large $\mx$) region, where the previous data tend to be 
slightly higher than the new measurement. 


\subsection{The {\boldmath $\xpom$} dependence of {\boldmath $F_2^D$}}
\label{regge}

The $\xpom$ dependence of the measured diffractive structure
function $\ftwodarg$ for fixed $\beta$ and $Q^2$
% and $|t|<1.0 \rm\ GeV^2$
is studied, following a similar procedure to that adopted in \cite{h1f2d94}. 
A fit is performed to the data using a parameterization 
of the form
\begin{equation}
F_2^D(\xpom,\beta,Q^2) = f_\pom(\xpom) A_\pom(\beta,Q^2) +
                           f_\reg(\xpom) B_\reg(\beta,Q^2) \ ,
\end{equation}
which is motivated by the ideas of Regge phenomenology.
$f_\pom(\xpom)$ and  $f_\reg(\xpom)$ correspond to
pomeron and sub-leading reggeon {\em flux factors} and are defined as
\begin{equation}
f_{\{\pom,\reg\}}(\xpom) = \int_{t_{cut}}^{t_{min}} \frac{e^{B_{\{\pom,\reg\}}t}}{x_\pom^{2\alpha_{\{\pom,\reg\}}(t)-1}} \ {\rm d}t \ , 
\end{equation}
where $t_{cut}=-1.0 \rm\ GeV^{2}$, $|t_{min}|$ is the 
minimum kinematically allowed value of $|t|$
and the pomeron and reggeon trajectories are assumed to be linear:
\begin{equation}
\alpha_{\{\pom,\reg\}}(t)=\alpha_{\{\pom,\reg\}}(0) + \alpha_{\{\pom,\reg\}}' t \ .
\end{equation}
The values for $\alpha_\pom'$, $B_\pom$, $\alpha_\reg'$ and $B_\reg$ cannot
be constrained by the $F_2^{D(3)}$ data and are taken from other measurements. 
Furthermore, the secondary reggeon trajectory intercept $\alpha_\reg(0)$
is not well constrained by the data due to a lack of high precision
data points at high $x_\pom$ (low $y$) values. 
It is taken from a similar fit to 
previous H1 $F_2^{D(3)}$ data as described in \cite{h1f2d94}.
The values assumed for the parameterization
of the pomeron and reggeon flux factors are 
$\alpha_\pom'=0.26\pm0.26\rm\ GeV^{-2}$,
$B_\pom=4.6^{+3.4}_{-2.6}\rm\ GeV^{-2}$,
$\alpha_\reg'=0.90\pm0.10\rm\ GeV^{-2}$,
$B_\reg=2.0\pm2.0\rm\ GeV^{-2}$ and
$\alpha_\reg(0)=0.50\pm0.16$.
The longitudinal structure function $F_L^D$ is assumed to be zero.
% and $R^{D(5)}=0$.

A fit is performed to all data with $y<0.45$ in every
($Q^2$,$\beta$) bin of the measurement which contains at least four data 
points. The cut in $y$ limits the influence of a non-zero value 
of $F_L^D$. In the fit, the value for the pomeron intercept parameter 
$\alpha_\pom(0)$ is left as a free parameter, as are the
coefficients $A_\pom(\beta,Q^2)$ and  $B_\reg(\beta,Q^2)$ in each 
($Q^2$,$\beta$) bin.
The fit gives a very good description of the data ($\chi^2/{\rm ndf}=0.95$). 
If the presence of a sub-leading reggeon exchange contribution is
neglected in the fit, the description of the data is considerably poorer
and a significantly worse $\chi^2/{\rm ndf}=1.25$ 
is obtained.
The systematic error on $\alpha_\pom(0)$ is obtained by repeating
the fit after shifting the data points according to each individual
source of systematic uncertainty.

The possibility of interference between the pomeron and reggeon
exchange contributions is taken into account by repeating the
fit with the assumption of maximum interference, where the
interference flux is parameterized according to eqs. (14,16) in
\cite{h1f2d94}. Since no significant change in $\chi^2$ is 
observed with or without interference, the results quoted for
the values of  $\alpha_\pom(0)$ correspond to the averages of
the values obtained with or without interference and their 
difference is included in the uncertainty on  $\alpha_\pom(0)$.

Further model dependences of the extracted value of  $\alpha_\pom(0)$
are estimated by varying the assumed values for  $\alpha_\pom'$, 
$B_\pom$, $\alpha_\reg'$, $B_\reg$ and $\alpha_\reg(0)$ within the limits
quoted above. The lack of knowledge on the size of $F_L^D$
is taken into account by repeating the fit under the extreme assumption
of $F_L^{D(3)}=\ftwod$ and taking the difference in the result
as an additional model dependence uncertainty.

The result of the fit is
\begin{equation}
\alphapom(0) = 1.173 \ \pm 0.018 \ \mathrm{(stat.)}
                     \ \pm 0.017 \ \mathrm{(syst.)}
                     \ ^{+ 0.063}_{-0.035} \ \mathrm{(model)} \ .
\end{equation}
The dominant contribution to the model dependence error on  $\alphapom(0)$
originates from the unknown value of $F_L^{D(3)}$.
The fit result is illustrated in figure \ref{stampregge}.
The obtained
value for the pomeron intercept is significantly higher
than the value $\alphapom(0) \simeq 1.08$ for the {\em soft pomeron}
\cite{DL:stot} describing soft hadronic interactions.
The result is compatible with that
obtained from similar fits to previous H1 $F_2^{D(3)}$ data \cite{h1f2d94}.
 
Two further fits are performed in order to investigate whether
$\alphapom(0)$ has any dependence on $Q^2$.
The data are divided 
into two $Q^2$ intervals covering the data with $6.5<Q^2<20 \rm\ GeV^2$ and  
$25<Q^2<120 \rm\ GeV^2$ respectively.
The results are
\begin{equation}
\alphapom(0) = 1.162 \ \pm 0.021 \ \mathrm{(stat.)}
                     \ \pm 0.018 \ \mathrm{(syst.)}
                     \ ^{+ 0.064}_{-0.034} \ \mathrm{(model)} \ \ \ \ 
(\av{Q^2}=10.8 \rm\ GeV^2) \ \ 
\end{equation}
and
\begin{equation}
\alphapom(0) = 1.204 \ \pm 0.034 \ \mathrm{(stat.)}
                     \ \pm 0.027 \ \mathrm{(syst.)}
                     \ ^{+ 0.060}_{-0.037} \ \mathrm{(model)} \ \ \ \
(\av{Q^2}=49.5 \rm\ GeV^2) \ .
\end{equation}
These results are shown together with the previous H1 measurement in
figure \ref{alpha}. Within the uncertainties, there is no evidence 
for a variation of $\alphapom(0)$ with $Q^2$ in the measured
kinematic range.
%The measurement of  $\alphapom(0)$
%in two $Q^2$ intervals shows some indication for a $Q^2$ dependence of 
%the pomeron intercept, which is however not significant within the errors.
In figure \ref{alpha}, the effective pomeron intercept extracted from
the diffractive data is compared with 
$\alphapom(0)=\lambda + 1$ as obtained from fits of the form
$F_2=cx^{-\lambda(Q^2)}$ to
inclusive small $x$ proton structure function data \cite{h1f29697}. 
The data suggest that
at large $Q^2$, the effective intercept describing the inclusive data is 
larger than that from the diffractive data. 

\subsection{The Ratio {\boldmath $F_2^D / F_2$}}

The $\gamma^* p$ centre of mass energy dependences of the diffractive and
inclusive cross sections are further compared in 
figure~\ref{wdepratio}. Here, the quantity 
\begin{eqnarray}
\rho^{D(3)}(\beta, Q^2, x) \ = \ \frac{\mx^2 \ x}{Q^2} \ \cdot \
\frac{\ftwodarg}{F_2 (x, Q^2)}
\end{eqnarray}
is plotted as a function of $W$ in bins of fixed $Q^2$ and 
$\beta$. The $F_2$ data are taken from \cite{h1f29697}.
In order to improve the statistical precision, measurements
at adjacent $Q^2$ values have been combined. In terms of
virtual photon-proton cross sections, 
\begin{eqnarray}
\rho^{D(3)} \ = \ \mx^2 \
\frac{{\rm d} \sigma (\gamma^* p \rightarrow X Y)}{{\rm d} \mx^2}
\ \ / \ \ \sigma (\gamma^* p \rightarrow X) \ ,
\end{eqnarray}
where the diffractive cross section is integrated over 
$\my < 1.6 \ {\rm GeV}$ and $|t| < 1 \ {\rm GeV^2}$.
Up to a factor of $\mx^2 / (Q^2 + \mx^2)$, $\rho^{D(3)}$ is
equal to the quantity $r^{D(3)}$ constructed from previous H1
data \cite{andy:sheffield}. The ratio $\rho^{D(3)}$ is relatively
flat throughout the full phase space, except at large $\beta$
values (the very low $\mx$ region) and at low $W$, where 
sub-leading exchanges play a role in the diffractive data.
The relative flatness of the data is consistent with the
observation by ZEUS in \cite{ZEUS:94}.

\subsection{The {\boldmath $\beta$} and {\boldmath $Q^2$} Dependences
of {\boldmath $\ftwod$}}
\label{qcd}

Figures~\ref{beta1} and~\ref{beta2} show the $\beta$ dependence
of the data at different $Q^2$ values at fixed values of
$\xpom = 0.001$, $0.003$, $0.01$ and $0.03$. Figures~\ref{q2dep1}
and~\ref{q2dep2} show the $Q^2$ dependence at different $\beta$
values for the same four fixed values of $\xpom$. In 
figure~\ref{q2dep2}, preliminary H1 data at larger 
$Q^2$ \cite{vancouver} are also included.

The data show
similar properties to previous $\ftwod$ measurements, but with much
improved precision. The $\beta$ dependence
is relatively flat, with large contributions at high fractional momenta,
in contrast to hadronic structure functions.  
The data exhibit rising scaling violations up to values of 
$\beta$ at least as large as 0.4, falling with $Q^2$ only at the
highest value $\beta = 0.9$.
The $\beta$ dependence
at fixed $\xpom$ thus evolves from high to low $\beta$ with increasing 
$Q^2$, as expected for DGLAP evolution dominated by gluon radiation.

%QCD hard scattering factorisation has recently been proven for
%semi-inclusive processes \cite{collins}, of the type measured.
%This factorisation implies that at fixed $\xpom$ for cross
%sections integrated over a fixed range in $\my$ and $t$, the
%$\beta$ and $Q^2$ dependence of the data can be interpreted as a
%`diffractive structure function' of the proton, the evolution
%in $\beta$ and $Q^2$ being governed by the DGLAP equations.

The data in figures~\ref{beta1}--\ref{q2dep2} are compared with
the results of a fit to the data in which the $\beta$ and $Q^2$
dependences evolve according to the DGLAP equations and the
$\xpom$ dependence follows a Regge behaviour with pomeron ($\pom$) and 
sub-leading meson ($\reg$) exchange contributions, the flux factors
being as described in section~\ref{regge}. In the fit, light singlet
quark ($u,d,s$) and gluon distributions are considered, 
parameterised in terms of non-perturbative input distributions
at the starting scale $Q_0^2 =  2 \ {\rm GeV^2}$ for the QCD evolution.
Only data with $y<0.45$, $\beta\leq0.65$ and $M_X>2 \rm\ GeV$ are
included in the fit.
The fit gives a very good description of the data, including the
kinematic region that was not included in the fit.
The resulting
parton distributions are dominated by a large gluon distribution
extending to large $\beta$ at low scales. 
Sub-leading exchange contributions
are negligible for $\xpom \leq 0.003$ and become significant only
at large $\xpom$ and low $\beta$.

In figures~\ref{q2f2comp1} and~\ref{q2f2comp2}, the $Q^2$ dependence
of the data at fixed $\xpom$ and $\beta$ (and hence fixed $x$) is
compared with the $Q^2$ dependence of $F_2(x, Q^2)$ data 
from \cite{h1f29697} at the same values of $x$. In the region
of low $\beta$, the scaling violations of $\ftwod$ are similar
to those of $F_2$ at the same value of $x$. However, at the largest
values $\beta = 0.9$, $\ftwod$ falls with 
increasing $Q^2$, whereas $F_2$ continues to rise.

\subsection{Comparisons with Models}

In figure~\ref{semcomp}, the $Q^2$ and $\beta$ dependences of the
measured $\ftwodarg$ at fixed $\xpom=0.003$ are compared with the
``semi-classical'' model by Buchm\"uller, Gehrmann and Hebecker
\cite{semicl}. In this model, DIS is considered in terms of the
scattering from the proton of $q \bar{q}$ and $q \bar{q} g$
fluctuations of the virtual photon, modelled as colour dipoles. 
The partonic
fluctuations of the photon scatter 
from a superposition of colour fields of the proton according to a simple
non-perturbative model that averages over all colour field configurations.
All resulting final state configurations contribute to the inclusive proton
structure function $F_2 (x, Q^2)$. Those in which the scattered partons 
emerge in a net colour-singlet state
contribute to the diffractive structure
function $F_2^D$. The model contains only four free parameters,
which are obtained from a combined fit to previous $F_2$ and $F_2^D$
data.
The model reproduces the general features of the data
but lies above the data where $\beta$ and
$Q^2$ are both small.  The behaviour of $F_2^D$ in the region of 
small masses $M_X^2<4 \rm\ GeV^2$ (corresponding to large $\beta$), 
is not expected to be reproduced by the model.

The data are compared 
in figure~\ref{satcomp}
with two versions of another colour dipole 
model by Golec-Biernat and W\"usthoff \cite{kgb,kgbnew}.  In this model, 
the $q \bar{q}$ and $q \bar{q} g$ dipole cross sections are obtained from 
fits to $F_2$ data. The same dipole cross sections are then used to 
predict $\ftwodarg$ under the assumption of two-gluon exchange, 
with only one additional free parameter,  
corresponding to the exponential $t$ dependence of the data, $e^{Bt}$ 
where $B=6 \rm\ GeV^{-2}$. Unlike the semi-classical model, the
saturation model also contains a higher twist contribution at large
$\beta$, allowing comparisons to be made throughout the full
measured kinematic region.
The model in \cite{kgb} gives a good description of the data except
at small $\beta$ and $Q^2$. The model in \cite{kgbnew}, in which
QCD evolution is added,  
underestimates the measured $\ftwodarg$ at high $\beta$ and high $Q^2$.

The $x_\pom$ dependence of the measured $\ftwodarg$
at fixed $\beta$ and $Q^2$ is compared with two versions of the
``Soft Colour Interactions'' (SCI) model \cite{sci,scinew}
in figure~\ref{scicomp}. In these models, the
hard interaction in diffractive DIS is treated identically to that in
inclusive DIS.  Diffraction occurs through soft colour rearrangements
between the outgoing partons, leaving their momentum configuration
unchanged. 
In the original SCI model \cite{sci}, 
diffractive final states are produced using
only one free parameter, the universal colour rearrangement
probability, which is fixed by a fit to $F_2^{D(3)}$.  The model has
been refined recently \cite{scinew} by making the colour rearrangement
probability proportional to the normalised difference in the
generalised areas of the string configurations before and after the
rearrangement. 
The kinematic region shown in figure~\ref{scicomp} is restricted to $M_X>2 \rm\ GeV$, corresponding 
to the region for which the model is intended.
The model predictions have been obtained using the LEPTO $6.5.2\beta$
\cite{lepto}
Monte Carlo generator. The improved version of SCI based
on a generalized area law \cite{scinew} results in a better 
description of $\ftwodarg$ at low $Q^2$ than the original 
version in \cite{sci}, with the exception of the highest $\beta$ region.
%At large $\beta$, the original model gives
%a better description. 
%The large $\beta$ region in which the area law
%model fails corresponds to the regions in figures~\ref{q2f2comp1}
%and~\ref{q2f2comp2} where the scaling violations of $F_2^D$ and
%$F_2$ differ at the same $x$ values.

\section{Conclusions}

A new measurement of the diffractive structure function $\ftwodarg$
has been presented for $6.5<Q^2<120 \rm\ GeV^2$, $0.01<\beta<0.9$ and
$\xpom < 0.05$. In the region $Q^2 \geq 15 \ {\rm GeV^2}$, the 
measurement is significantly more precise than previous data.

The $\xpom$ dependence of the data is well described by a model
motivated by Regge phenomenology, in which a leading ($\pom$) and
a secondary ($\reg$) exchange contribute. The effective pomeron
intercept describing the data is 
\begin{equation}
 \alphapom(0) = 1.173 \ \pm 0.018 \ \mathrm{(stat.)}
                     \ \pm 0.017 \ \mathrm{(syst.)}
                     \ ^{+ 0.063}_{-0.035} \ \mathrm{(model)} \ .
\nonumber
\end{equation}
The results suggest that the effective pomeron intercepts describing 
the total and the diffractive dissociation cross sections for
$\gamma^* p$ interactions 
become different at large $Q^2$.
The ratio of the $t$-integrated diffractive to the total $\gamma^* p$ 
cross section is relatively flat as a function of $\gamma^* p$ centre of 
mass energy.

At fixed $\xpom$ the data show a relatively flat $\beta$ dependence
and a rising dependence on $Q^2$, except at the highest values of
$\beta$. This structure is well described by a fit based on
DGLAP evolution of the $\beta$ and $Q^2$ dependence and a Regge 
motivated $\xpom$ dependence. In this fit, the diffractive
parton distributions of the proton are heavily dominated by a large
gluon density. The scaling violations of $F_2^D$ are similar to
those of $F_2$ when compared at the same $x$ values, except at large
$\beta$ where vector meson and other higher twist contributions
are expected to play a significant role in the diffractive data. 


 


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

\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.

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

\begin{thebibliography}{99}

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\bibitem{kgb}
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\bibitem{compton} A.~Courau {\em et al.}, Proc. of the Workshop on Physics
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%\bibitem{satrap} H.~Kowalski, {\em private communication}.

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\end{thebibliography}

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

\def\contsiz{0.75}

\begin{figure}
\centering
{\sf H1 Diffractive DIS Data} \\
\begin{minipage}{0.495\linewidth}
\centering
\epsfig{bbllx=0,bblly=0,bburx=470,bbury=470,file=H1prelim-01-111.fig2a.eps,width=\contsiz\linewidth}
\end{minipage}
\begin{minipage}{0.495\linewidth}
\centering
\epsfig{bbllx=0,bblly=0,bburx=470,bbury=470,file=H1prelim-01-111.fig2b.eps,width=\contsiz\linewidth}
\end{minipage}

\begin{picture}(0.1,0.1)
\put(-54.0,16.0){\sf (a)}
\put(26.0,16.0){\sf (b)}
\end{picture}

\begin{minipage}{0.495\linewidth}
\centering
\epsfig{bbllx=0,bblly=0,bburx=470,bbury=470,file=H1prelim-01-111.fig2c.eps,width=\contsiz\linewidth}
\end{minipage}
\begin{minipage}{0.495\linewidth}
\centering
\epsfig{bbllx=0,bblly=0,bburx=470,bbury=470,file=H1prelim-01-111.fig2d.eps,width=\contsiz\linewidth}
\end{minipage}

\begin{picture}(0.1,0.1)
\put(-54.0,16.0){\sf (c)}
\put(26.0,16.0){\sf (d)}
\end{picture}

\begin{minipage}{0.495\linewidth}
\centering
\epsfig{bbllx=0,bblly=0,bburx=470,bbury=470,file=H1prelim-01-111.fig2e.eps,width=\contsiz\linewidth}
\end{minipage}
\begin{minipage}{0.495\linewidth}
\centering
\epsfig{bbllx=0,bblly=0,bburx=470,bbury=470,file=H1prelim-01-111.fig2f.eps,width=\contsiz\linewidth}
\end{minipage}

\begin{picture}(0.1,0.1)
\put(-21.0,16.0){\sf (e)}
\put(61.0,16.0){\sf (f)}
\end{picture}

\caption{Observed distributions for {\em (a)} the scattered electron 
  energy and {\em (b)} its polar angle, {\em(c)} the polar angle of
  the hadronic final state, 
  {\em(d)} the maximum pseudorapidity $\eta_{\rm max}$ of all
  hadronic final state objects visible in the detector, {\em (e)} the
  invariant mass of the $X$ system and {\em (f)} $\xpom$, the
  longitudinal momentum fraction of the proton which is transferred
  to the $X$ system. The solid curves correspond to the sum of all
  Monte Carlo simulation contributions as described in the text,
  the dot-dashed curves indicate the sum of the pomeron and reggeon
  exchange contributions as implemented in RAPGAP and the dashed 
  curves correspond to the contributions from vector meson production
  as simulated by DIFFVM. 
%  Additional small contributions
%  from photoproduction background (PHOJET), migrations from high $\xpom$
%  or high $M_Y$ (DJANGO) and from QED-Compton events are not shown 
%  individually.
}
\label{control}
\end{figure}


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

\begin{figure}
\centering
\epsfig{file=H1prelim-01-111.fig3.eps,width=0.875\linewidth}
\caption{The measured diffractive structure function {\em (red filled
circles)}, 
presented as $\xpom\ftwodarg$ and plotted as a function of $\xpom$ 
at fixed values of $\beta$ and $Q^2$. Here and in all other figures,
the inner error bars correspond
to the statistical error, the outer error bars show the statistical
and the systematic error added in quadrature. An additional 
normalization uncertainty of $6.8\%$ is not shown.
For comparison, the 
previous H1 measurement of $F_2^{D(3)}$ 
in \cite{h1f2d94} is also shown {\em(blue triangles)}.
}
\label{stamp97}
\end{figure}

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

\begin{figure}
\centering
\epsfig{file=H1prelim-01-111.fig4.eps,width=0.875\linewidth}
\caption{The measured diffractive structure function {\em (red filled
circles)}, 
presented as $\xpom\ftwodarg$ and plotted as a function of $\xpom$, 
at fixed values of $\beta$ and $Q^2$. The result of the phenomenological
Regge fit to the data as described in the text is shown by the curves, 
where the solid curve corresponds
to the sum of pomeron and reggeon exchange contributions and the dotted
curve indicates the contribution from pomeron exchange only. 
Data points which were excluded from the fit ($y>0.45$ or $N^{\rm{points}}_{(\beta,Q^2)-{\rm bin}}<4$) are indicated by open circles.
}
\label{stampregge}
\end{figure}

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

\begin{figure}
\centering
\epsfig{file=H1prelim-01-111.fig5.eps,width=0.66\linewidth}
\caption{The effective value of $\alphapom(0)$ as a function
of $Q^2$. The squares correspond to 
$\alphapom(0) = 1 + \lambda$ extracted from
a fit $F_2=cx^{-\lambda(Q^2)}$ to inclusive $F_2(x,Q^2)$ data \cite{h1f29697} 
for $x<0.01$. The red filled circles are the values of $\alphapom(0)$ as
obtained from the phenomenological Regge fit to the $F_2^D$ data
as described in the text, for two different $Q^2$ intervals.
The inner error bars correspond to the statistical errors. The middle
error bars show the statistical and experimental systematic errors
added in quadrature. The outer error bars show the full error,
including that arising from model dependence.
The blue triangle is the value of $\alphapom(0)$ which was obtained
in \cite{h1f2d94}.
}
\label{alpha}
\end{figure}

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

\begin{figure}
\centering 
\epsfig{file=H1prelim-01-111.fig6.eps,width=1.0\linewidth}
\caption{Measurements of $\rho^{D(3)}$, 
illustrating the ratio of the diffractive 
to the inclusive cross section. The data points at $\beta = 0.9$ have
been scaled by a factor of 5 for visibility. The filled points 
correspond to the region $\xpom < 0.01$ in the diffractive measurement. 
The open points correspond to $\xpom > 0.01$. At each $\beta$ and $Q^2$
point, the photon dissociation mass is also fixed according to
$\mx^2 = Q^2 \ (1/\beta - 1)$.}
\label{wdepratio}
\end{figure}

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

\begin{figure}
\centering
\begin{minipage}{0.495\linewidth}
\epsfig{file=H1prelim-01-111.fig7a.eps,width=1.0\linewidth}
\vspace{0.9cm}
\end{minipage}
\begin{minipage}{0.495\linewidth}
\epsfig{file=H1prelim-01-111.fig7b.eps,width=1.0\linewidth}
\end{minipage}
\caption{The measured diffractive structure function, plotted as 
  $\xpom\ftwodarg$ {\em (red data points)}, as a function of $\beta$
  for various values of $Q^2$ and at two fixed values of $\xpom=0.001$
  {\em (left)} and $\xpom=0.003$ {\em (right)}. Also shown is the
  result of a QCD fit to the data as described in the text. The solid
  curves correspond to the sum of pomeron and sub-leading reggeon
  exchange contributions, whereas the dotted curves indicate the
  contribution from pomeron exchange only.  }
\label{beta1}
\end{figure}

\begin{figure}
\centering
\begin{minipage}{0.495\linewidth}
\epsfig{file=H1prelim-01-111.fig8a.eps,width=1.0\linewidth}
\end{minipage}
\begin{minipage}{0.495\linewidth}
\vspace{0.6cm}
\epsfig{file=H1prelim-01-111.fig8b.eps,width=1.0\linewidth}
\end{minipage}
\caption{The measured diffractive structure function, plotted as 
  $\xpom\ftwodarg$ {\em (red data points)}, as a function of $\beta$
  for various values of $Q^2$ and at two fixed values of $\xpom=0.01$
  {\em (left)} and $\xpom=0.03$ {\em (right)}. Also shown is the
  result of a QCD fit to the data as described in the text. The solid
  curves correspond to the sum of pomeron and sub-leading reggeon
  exchange contributions, whereas the dotted curves indicate the
  contribution from pomeron exchange only.  }
\label{beta2}
\end{figure}

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

\begin{figure}
\centering
\begin{minipage}{0.495\linewidth}
\vspace{3.8cm}
\epsfig{file=H1prelim-01-111.fig9a.eps,width=1.0\linewidth}
\end{minipage}
\begin{minipage}{0.495\linewidth}
\epsfig{file=H1prelim-01-111.fig9b.eps,width=1.0\linewidth}
\end{minipage}
\caption{The measured diffractive structure function, plotted as 
  $\xpom\ftwodarg$ {\em (red data points)}, as a function of $Q^2$ for
  various values of $\beta$ and at two fixed values of $\xpom=0.001$
  {\em (left)} and $\xpom=0.003$ {\em (right)}. Also shown is the
  result of a QCD fit to the data as described in the text. The solid
  curves correspond to the sum of pomeron and sub-leading reggeon
  exchange contributions, whereas the dotted curves indicate the
  contribution from pomeron exchange only. For each value of $\beta$,
  the corresponding value of $x=\xpom\beta$ is indicated. }
\label{q2dep1}
\end{figure}

\begin{figure}
\centering
\begin{minipage}{0.495\linewidth}
\vspace{2.0cm}
\epsfig{file=H1prelim-01-111.fig10a.eps,width=1.0\linewidth}
\end{minipage}
\begin{minipage}{0.495\linewidth}
\epsfig{file=H1prelim-01-111.fig10b.eps,width=1.0\linewidth}
\end{minipage}
\caption{The measured diffractive structure function, plotted 
  as $\xpom\ftwodarg$ {\em (red filled circles)}, as a function of $Q^2$
  for various values of $\beta$ and at two fixed values of
  $\xpom=0.01$ {\em (left)} and $\xpom=0.03$ {\em (right)}. Also shown
  is the result of a QCD fit to the data as described in the text. The
  solid curves correspond to the sum of pomeron and sub-leading reggeon
  exchange contributions, whereas the dotted curves indicate the
  contribution from pomeron exchange only. For each value of $\beta$,
  the corresponding value of $x=\xpom\beta$ is indicated.  In addition,
  the preliminary H1 data at high $Q^2$ \cite{vancouver} are shown
{\em (blue triangles)}.  }
\label{q2dep2}
\end{figure}

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

\begin{figure}
\centering
\begin{minipage}{0.495\linewidth}
\vspace{3.8cm}
\epsfig{file=H1prelim-01-111.fig11a.eps,width=1.0\linewidth}
\end{minipage}
\begin{minipage}{0.495\linewidth}
\epsfig{file=H1prelim-01-111.fig11b.eps,width=1.0\linewidth}
\end{minipage}
\caption{The measured diffractive structure function $\xpom\ftwodarg$
  {\em (red filled circles)}, plotted as a function of $Q^2$ for various
  values of $\beta$ and at two fixed values of $\xpom=0.001$ {\em
    (left)} and $\xpom=0.003$ {\em (right)}. Also shown are
  $F_2(x,Q^2)$ data from \cite{h1f29697} {\em (squares)}, where
  for each fixed $x=\xpom\beta$ the $F_2$ data points with $x$ values
  closest to those of the $F_2^D$ measurement are plotted.  }
\label{q2f2comp1}
\end{figure}

\begin{figure}
\centering
\begin{minipage}{0.495\linewidth}
\vspace{2.0cm}
\epsfig{file=H1prelim-01-111.fig12a.eps,width=1.0\linewidth}
\end{minipage}
\begin{minipage}{0.495\linewidth}
\epsfig{file=H1prelim-01-111.fig12b.eps,width=1.0\linewidth}
\end{minipage}
\caption{The measured diffractive structure function $\xpom\ftwodarg$
  {\em (red filled circles)}, plotted as a function of $Q^2$ for various
  values of $\beta$ and at two fixed values of $\xpom=0.01$ {\em
    (left)} and $\xpom=0.03$ {\em (right)}. Also shown are
  $F_2(x,Q^2)$ data from \cite{h1f29697} {\em (squares)}, where
  for each fixed $x=\xpom\beta$ the data points with $x$ values
  closest to those of the $F_2^D$ measurement are plotted.  }
\label{q2f2comp2}
\end{figure}

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

\begin{figure}
\centering
\begin{minipage}{0.495\linewidth}
\epsfig{file=H1prelim-01-111.fig13a.eps,width=1.0\linewidth}
\end{minipage}
\begin{minipage}{0.495\linewidth}
\epsfig{file=H1prelim-01-111.fig13b.eps,width=1.0\linewidth}
\vspace{1.5cm}
\end{minipage}
\caption{The measured diffractive structure function $\xpom\ftwodarg$
{\em (red data points)}
at fixed $\xpom=0.003$, plotted
as a function of $\beta$ for various values of $Q^2$ {\em (left)} 
and
as a function of $Q^2$ for various values of $\beta$ {\em (right)}.
The data are compared with the predictions of the semi-classical model 
by Buchm\"uller, Gehrmann and Hebecker \cite{semicl} {\em (solid curves)}.
The dotted curves correspond to an extension of the model into the 
region of small $M^2_X<4 \rm\ GeV^2$,
where the model is not expected to be valid.
}
\label{semcomp}
\end{figure}

\begin{figure}
\centering
\begin{minipage}{0.495\linewidth}
\epsfig{file=H1prelim-01-111.fig14a.eps,width=1.0\linewidth}
\end{minipage}
\begin{minipage}{0.495\linewidth}
\epsfig{file=H1prelim-01-111.fig14b.eps,width=1.0\linewidth}
\vspace{1.5cm}
\end{minipage}
\caption{The measured diffractive structure function $\xpom\ftwodarg$
{\em (red data points)} at fixed $\xpom=0.003$,
plotted as a function of $\beta$ for various values of $Q^2$ {\em (left)}
and
as a function of $Q^2$ for various values of $\beta$ {\em (right)}.
The data are compared with the predictions of two versions of a
colour dipole
model by Golec-Biernat and W\"usthoff \cite{kgb} 
 {\em (dashed curves)}
and \cite{kgbnew}  {\em (solid curves)}.
}
\label{satcomp}
\end{figure}

\begin{figure}
\centering
\epsfig{file=H1prelim-01-111.fig15.eps,width=0.9\linewidth}
\caption{The measured diffractive structure function $\xpom\ftwodarg$,
plotted as a function of $\xpom$ at fixed values of $Q^2$ and $\beta$
{\em(red data points)}.
Only a sub-sample of the data, corresponding to $0.1\leq\beta\leq0.65$
and $Q^2\leq60 \rm\ GeV^2$, are shown. The data are compared with
the predictions of the original Soft Colour Interactions (SCI) model \cite{sci}
{\em (dashed curves)} 
and its refinement based on a generalized area law \cite{scinew}
{\em (solid curves)}. 
The predictions have been obtained using the
LEPTO $6.5.2\beta$ Monte Carlo generator \cite{lepto}.
}
\label{scicomp}
\end{figure}

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

\clearpage
\pagebreak

\section*{Appendix}

\begin{figure}[h]
\centering
\epsfig{file=H1prelim-01-111.fig16.eps,width=0.9\linewidth}
\caption{Summary plot of various H1 measurements of $F_2^{D(3)}$.
}
\label{stampall}
\end{figure}


\begin{figure}[h]
\centering
\epsfig{file=H1prelim-01-111.fig17.eps,width=0.5\linewidth}
\caption{$Q^2$ dependence of $F_2^{D(3)}$ at fixed $\xpom=0.003$, showing
a comparison with the previous H1 measurement.
}
\label{q2qcd9497}
\end{figure}

\begin{figure}[h]
\centering
\begin{minipage}{0.495\linewidth}
\vspace{2.0cm}
\epsfig{file=H1prelim-01-111.fig18a.eps,width=1.0\linewidth}
\end{minipage}
\begin{minipage}{0.495\linewidth}
\epsfig{file=H1prelim-01-111.fig18b.eps,width=1.0\linewidth}
\end{minipage}
\begin{minipage}{0.495\linewidth}
\epsfig{file=H1prelim-01-111.fig18c.eps,width=1.0\linewidth}
\end{minipage}
\begin{minipage}{0.495\linewidth}
\epsfig{file=H1prelim-01-111.fig18d.eps,width=1.0\linewidth}
\end{minipage}
\caption{The $Q^2$ dependence of $F_2^{D(3)}$ as compared with $F_2(x,Q^2)$
at fixed $x=\xpom\beta$,
grouping together available $\xpom$ points for a given fixed $\beta$ value.
}
\label{q2regge}
\end{figure}

\begin{figure}[h]
\centering
\begin{minipage}{0.495\linewidth}
\epsfig{file=H1prelim-01-111.fig19a.eps,width=1.0\linewidth}
\end{minipage}
\begin{minipage}{0.495\linewidth}
\epsfig{file=H1prelim-01-111.fig19b.eps,width=1.0\linewidth}
\end{minipage}
\begin{minipage}{0.495\linewidth}
\epsfig{file=H1prelim-01-111.fig19c.eps,width=1.0\linewidth}
\end{minipage}
\caption{The $\beta$ dependence of $F_2^{D(3)}$ at fixed $Q^2$, combining
all four $\xpom$ bins.
}
\label{betaregge}
\end{figure}





\end{document}
