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\begin{titlepage}

\noindent
\begin{center}
%{\it {\large version of \today}} \\[.3em] 
\begin{small}
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%Submitted to & \multicolumn{3}{r}{\footnotesize Electronic Access: {\it http://www-h1.desy.de/h1/www/publications/conf/conf\_list.html}} \\[.2em] \hline 
%Submitted to & \multicolumn{3}{r}{\footnotesize {\it www-h1.desy.de/h1/www/publications/conf/conf\_list.html}} \\[.2em] \hline 
Submitted to & & &
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,width=2.cm} \\[.2em] \hline
                & & & \\
\multicolumn{4}{l}{{\bf
                International Europhysics Conference 
                on High Energy Physics, EPS03},
                July~17-23,~2003,~Aachen} \\
                (Abstract {\bf 089} & Parallel Session & {\bf 5}) & \\
                & & & \\
\multicolumn{4}{l}{{\bf
                XXI International Symposium on  
                Lepton and Photon Interactions, LP03},
                August~11-16,~2003,~Fermilab} \\ 
                & & & \\
                \hline
 & \multicolumn{3}{r}{\footnotesize {\it
    www-h1.desy.de/h1/www/publications/conf/conf\_list.html}} \\[.2em]
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\vspace*{2cm}

\begin{center}
  \Large
  {\bf 
    Measurement and NLO DGLAP QCD Interpretation \\
    of Diffractive Deep-Inelastic Scattering at HERA \\
  }
  \vspace*{1cm}
  {\Large H1 Collaboration} 
\end{center}

\begin{abstract}

\noindent

A high precision inclusive measurement of the diffractive deep
inelastic scattering (DIS) process $ep \rightarrow eXY$ 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 measurement, based on an integrated luminosity of
$10.6 \rm\ pb^{-1}$, is presented in the form of a
diffractive reduced cross section $\sigma_r^{D(3)}$, measured
in the kinematic range $6.5 \leq Q^2 \leq 120 \rm\ GeV^2$, $0.01 \leq
\beta \leq 0.9$ and $10^{-4}\lapprox\xpom<0.05$.
%
The ratio of the diffractive to the inclusive cross section
$\sigma_r^{D(3)} (x, Q^2, x_\pom) / \sigma_r (x, Q^2)$ is measured and
found to be remarkably flat as a function of $Q^2$ with $x$
and $\xpom$ fixed.
%
The measured cross section is compared with various
models for diffractive DIS.
%
The $\xpom$ dependence of the data is interpreted in terms of a
measurement of the effective pomeron intercept 
%$\alphapom(0)$ with the result 
$\alphapom(0) = 1.173 \ \pm 0.018 \ \mathrm{(stat.)}  \ \pm
0.017 \ \mathrm{(syst.)}  \ ^{+ 0.063}_{-0.035} \ \mathrm{(model)}$.
%which is compared with the same quantity extracted from inclusive DIS.
%
A NLO DGLAP QCD fit is performed to the data,
% and the resulting
%diffractive parton densities are shown, 
together with an assessment of
the experimental and theoretical uncertainties on the resulting
diffractive parton densities.
The diffractive exchange is shown to be dominated by the diffractive
gluon density, which carries an integrated fraction $75\pm15\%$ 
of the exchanged momentum at $Q^2=10 \rm\ GeV^2$ and extends to
large fractional momenta.
%
%The resulting partonic
%decomposition of the diffractive exchange is dominated by the gluon
%density, which extends to large fractional momenta.  The momentum
%fraction of the diffractive exchange carried by gluons is found to be
%$(75\pm15)\%$ at $Q^2=10 \rm\ GeV^2$.  
The parton densities are used
to make updated comparisons with diffractive dijet and open charm
cross sections at HERA and the Tevatron, thus testing the
factorisation properties of hard diffraction.
\end{abstract}

\end{titlepage}

\pagestyle{plain}

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\section{Introduction}

Although
Quantum Chromodynamics (QCD) is the well established gauge
theory of strong interactions, perturbative
QCD calculations are only possible for short distance, hard
partonic interactions, where the strong coupling $\alpha_s$ is small.
On the other hand, hadronic cross sections at high centre-of-mass 
energies 
are dominated by soft interactions, for which
perturbation theory is inapplicable. A large fraction of these soft
interactions are mediated by colour-singlet exchange and are termed
{\em diffractive}. 
The observation of hard sub-processes in diffractive events \cite{UA8}
introduces the exciting possibility of understanding the diffractive
exchange in terms of parton dynamics, which would represent an
important
step towards a deeper understanding of soft strong interactions and
of confinement.
%The elastic scattering at the 
%proton vertex implies the exchange of systems of at least two partons,
%such that new field theory techniques are required. 

Diffractive processes of the type $ep \rightarrow e X p$ have been 
extensively studied in deep-inelastic electron-proton scattering (DIS) at low 
$x$ at the HERA 
collider \cite{obsdiff,h1f2d94,ZEUS:94,f2dhiq2,f2deps,zeus02}.
%In such events, the proton stays intact or dissociates into a low-mass
%leading baryonic system, separated by a large rapidity gap from a
%hadronic final state in the central region.  
In these events, the structure of colour-singlet exchange is probed 
using point-like highly virtual photons.
%, similarly to
%standard DIS probing proton structure.  
%Considerable theoretical effort has been
%invested in understanding diffractive DIS and its relationships to 
%inclusive DIS, diffractive hadron-hadron scattering and 
%the structure of soft strong interactions. 
%description of these
%diffractive DIS processes has become one of the main challenges in the
%development of our understanding of QCD at high energies.
%A breakthrough in the theoretical understanding of diffractive DIS is
%marked by the recent proof of 
A hard scattering QCD factorisation
theorem was recently proven
for a general class of semi-inclusive DIS processes, which
include the process $ep \rightarrow eXp$
\cite{collins}.  This implies that the concept of `diffractive
parton distributions' \cite{facold} can be introduced, expressing 
conditional proton parton
probability distributions under the constraint of a leading
baryonic system of particular 4-momentum. 
This allows diffractive DIS to be tackled with a similar
theoretical description to inclusive DIS, namely the framework of the 
next-to-leading order
(NLO) DGLAP \cite{dglap} evolution equations.

In this paper, a high precision measurement of the diffractive
DIS cross section
is presented, based upon data collected with the H1
detector at HERA.  
This new measurement, first reported in \cite{f2deps},
yields a significant
increase in precision compared with previous H1
\cite{h1f2d94} and ZEUS \cite{ZEUS:94} data.  
%The data are shown in the
%form of a diffractive reduced cross section $\sigma_r^D$.  
%From the
%energy dependence of the cross section, a value for the effective {\em
%  pomeron intercept} $\alpha_\pom(0)$ is determined.  
A NLO DGLAP QCD
fit is performed to the data. For the first time an assessment of
the experimental and theoretical uncertainties 
on the resulting diffractive parton
densities is made.  
To test the validity
of QCD hard scattering factorisation as applied to diffraction, the
parton densities are used for comparisons with diffractive DIS dijet
\cite{h1jets} and open charm \cite{h1dstar} cross sections at HERA as
well as with diffractive dijet production at the Tevatron \cite{CDF:jets}.
%Another topic of interest in this paper is the ratio of the
%diffractive to the total DIS cross section, which is often considered
%in the context of models where partonic fluctuations of the photon,
%represented as colour dipoles, scatter off the proton.  Finally,
%various models for diffractive DIS are confronted with the data.
The ratio of the
diffractive to the total DIS cross section is also investigated
and the data are compared with various theoretical models
for the diffractive exchange.
%, which is often considered
%in the context of models where partonic fluctuations of the photon,
%represented as colour dipoles, scatter off the proton.  Finally,
%various models for diffractive DIS are confronted with the data.

%The kinematics of diffractive DIS are introduced in section
%\ref{sec:kine}, after which the diffractive reduced cross section
%and structure functions are defined in section \ref{sec:sigred}
%and the theoretical picture of QCD in diffraction is
%outlined in section \ref{sec:qcdfac}.

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

\subsection{Kinematics of Diffractive DIS at HERA}
\label{sec:kine}

\begin{figure}[t]
\centering \epsfig{file=H1prelim-02-012.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}

Figure~\ref{diagrams} illustrates the generic diffractive process at
HERA of the type $ep\rightarrow eXY$.  The electron
(with 4-momentum $k$) couples to a virtual photon
($q$) 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}
where $Q^2$ is the photon virtuality, $x$ corresponds
to the longitudinal
momentum fraction of the struck quark with respect to the proton and
$y$ is the inelasticity variable.
The squared invariant masses of the electron-proton and photon-proton systems
$s$ and $W^2$ are given by
\begin{eqnarray}
s =  (k+P)^2 \simeq (300 \ \mathrm{GeV})^2 \ ; \qquad
 W^2 = (q+P)^2 \simeq ys-Q^2 \ .
\label{eq:w2}
\end{eqnarray}
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 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.



\subsection{Diffractive Reduced Cross Section and Structure Functions}
\label{sec:sigred}

The cross section for the diffractive DIS process $ep\rightarrow eXY$ depends
in general on 5 independent variables (neglecting azimuthal
angles).
Here, we work with  
$Q^2$,$x$ (or $\beta$), $x_\pom$, $M_Y$ and $t$.
%In the measurement presented here,
%Diffractive DIS events are selected by the requirement of a large
%rapidity gap and t
The system $Y$ is not measured in this analysis
and the results are integrated
over $|t| < 1.0 \ {\rm GeV^2}$ and $\my < 1.6 \ {\rm GeV}$. They are
expressed in terms of a
{\em reduced diffractive cross section} $\sigma_r^{D(3)}$, defined through
\begin{equation}
\frac{d^3\sigma^D}{\dd x_\pom \ \dd x \ \dd Q^2} = \frac{4\pi
  \alpha^2}{x Q^4}\left ( 1 - y + \frac{y^2}{2} \right )
\sigma_{r}^{D(3)}(\xpom,x,Q^2) \ .
\label{sigmar}
\end{equation}

Neglecting contributions from $Z^0$ exchange, 
%which is a very good
%approximation in the $Q^2$ region under study here,
$\sigma_{r}^{D(3)}$ is
%, analogously to inclusive DIS, 
related to the
diffractive structure functions $F_2^{D(3)}$ and $F_L^{D(3)}$ by
\begin{equation}
\sigma_r^{D(3)} = F_2^{D(3)} - \frac{y^2}{1+(1-y)^2} F_L^{D(3)} \ ,
\label{eq:sigf2fl}
\end{equation}
where $F_L^D$ is the longitudinal diffractive structure function.
The positivity constraint on the cross sections for longitudinally and
transversely polarised photon scattering leads to the relation
$0\leq F_L^D \leq F_2^D$. The contribution
of $F_L^D$ to $\sigrd$ can be sizeable only at large values of
$y$, and in most of the kinematic range the relation $\sigrd \approx F_2^D$
holds to good approximation. In previous measurements of
inclusive diffractive DIS at HERA \cite{h1f2d94,ZEUS:94},
the data were presented in terms of $F_2^D$ and $ F_L^D$ was neglected.

\subsection{Factorisation in Diffraction}
\label{sec:qcdfac}

%A significant step forward in the theoretical description of
%diffractive DIS is represented by t
The proof that QCD hard scattering
factorisation can be applied to diffractive DIS \cite{collins} implies
that in the leading $\log(Q^2)$ approximation, the cross section for
the diffractive process $\gamma^* p \rightarrow XY$ can be written in
terms of convolutions of universal partonic cross sections
$\hat{\sigma}^{\gamma^* i}$ with {\em diffractive parton distributions}
$f_i^D$ \cite{facold}, representing probability distributions for a
parton $i$ in the proton under the constraint that the proton is scattered
with particular values of $t$ and $x_\pom$. Thus, at leading
twist,\footnote{A framework also exists to include higher order 
operators \cite{blumlein}.}
\begin{equation}
\frac{{\rm d^2} \sigma(x,Q^2,x_\pom,t)^{\gamma^*p\rightarrow p'X}}
{{\rm d} x_\pom \ {\rm d} t} \ = \
\sum_i \int_x^{x_\pom}{\rm d}\xi \
\hat{\sigma}^{\gamma^*i}(x,Q^2,\xi) \
f_i^D(\xi,Q^2,x_\pom,t) \ .
\label{equ:diffpdf}
\end{equation}
The factorisation formula is valid for large enough $Q^2$ and
fixed $x_\pom$ and $t$. It also applies to the case of
proton dissociation into a system of fixed mass $M_Y$.
The partonic cross sections
$\hat{\sigma}^{\gamma^* i}$ are the same as those for inclusive DIS and
the diffractive parton distributions $f_i^D$, which are not known from
first principles, should obey the DGLAP \cite{dglap} evolution equations.

In addition to the rigorous theoretical prescription represented
by equation~(\ref{equ:diffpdf}), an additional assumption is often
made, that the shape of the diffractive parton distributions
is independent of $x_\pom$ and $t$ and that
their normalisation is controlled by Regge asymptotics \cite{ingschl}. 
The diffractive parton distributions can then be factorised into a
term depending only on $x_\pom$ and $t$ and a second term
depending only on $x$ (or $\beta$) and $Q^2$:
\begin{equation}
f_i^D(x_\pom,t,x,Q^2) = f_{\pom/p}(x_\pom,t) \cdot 
f_i^{\pom}(\beta=x/x_\pom,Q^2) \ .
\label{reggefac}
\end{equation}
Under this {\em Regge} factorisation assumption, often referred
to in 
the literature as the {\em Ingelman-Schlein} or {\em resolved pomeron} model,
the diffractive exchange can be treated as a
quasi-real object with a partonic structure, given by 
parton distributions $f_i^{\pom}(\beta,Q^2)$. 
The variable $\beta$ then corresponds to the longitudinal
momentum fraction of the diffractive exchange carried by the
struck parton in the {\em pomeron}. The
{\em pomeron flux factor}
$f_{\pom/p}(x_\pom,t)$ 
represents the probability that a pomeron with
particular values of $x_\pom$ and $t$ couples to the proton. 
Although 
equation~(\ref{reggefac}) has no firm basis in QCD,
at the present level of precision it appears to be supported
by data \cite{h1f2d94,ZEUS:94,f2deps}.

%Analogously to inclusive DIS, the diffractive parton
%distributions can be constrained by experimental data by means
%of a DGLAP QCD fit to the inclusive diffractive DIS cross section.
Various fits to diffractive DIS data have been performed under the
assumptions of both QCD hard scattering and Regge factorisation
\cite{h1f2d94,zeus:gpjets,actw,royonbartelsfits}, using the DGLAP
equations to evolve the diffractive parton distributions.
% based on
%measurements of the diffractive structure function $F_2^D$ by
%the H1 and ZEUS collaborations at HERA. 
In \cite{h1f2d94}, previous H1 data for $F_2^{D(3)}$ were fitted.
%At the largest $x_\pom$ studied, a
%contribution from sub-leading reggeon exchange was found to be necessary.
%Parameterizations of pomeron quark
%and gluon distributions and values for the trajectory intercepts
%$\alpha_\pom(0)$ and $\alpha_\reg(0)$ used in the Regge-inspired
%parameterizations of the pomeron and sub-leading exchange flux
%factors were obtained. 
%The pomeron parton densities were found to be
%dominated by gluons, which carry 
$80-90\%$ of the exchanged momentum was found to be carried by gluons.
Two fits, usually referred to as `H1 fit 2' and `H1 fit 3', with
slightly different assumptions for the 
parameterisation of the gluon density at the starting scale,
were presented.
The factorisation properties of diffractive DIS have been
tested by comparing predictions
using these parton distributions with 
diffractive final state observables, such as
jet \cite{h1jets} and heavy quark \cite{h1dstar} cross sections.

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

\section{Experimental Technique}

The data used for the measurement presented in this paper were taken
in 1997 using the H1 detector, when HERA collided protons of energy
$E_p = 820 \ {\rm GeV}$ with positrons\footnote{From now on, the word
  `electron' will be used as a generic term for electrons and
  positrons.}  of energy $E_e = 27.5 \ {\rm GeV}$.  
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
$6.0 < Q^2 < 13.5 \ {\rm GeV^2}$, 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. 

%A brief description
%of the H1 detector is given in section \ref{sec:h1det}, followed by
%discussions of the data selection 
%and kinematic reconstruction 
%(section \ref{sec:evsel}), Monte Carlo
%simulation and cross section measurement (\ref{sec:mc}, \ref{sec:xsect})
%and finally the systematic uncertainties of the measurement (\ref{sec:syst}).

\subsection{The H1 Detector}
\label{sec:h1det}

A full description of the H1 apparatus can be found in \cite{h1det}.
Here, only the parts of the detector relevant for the present analysis
are briefly discussed.  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 $\eta = - \ln \tan
\theta /2$ is referred to as the `forward' direction.

%The interaction region 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
%in the range $-1.5<\eta<1.5$ with a transverse momentum resolution
%of $\sigma(p_T)/p_T \simeq 0.01 \cdot p_T/\mathrm{GeV}$.

%The highly segmented Liquid Argon (LAr) calorimeter, covering the
%range $-1.5<\eta<3.4$, surrounds the tracking chambers and consists of
%electromagnetic and hadronic sections.  The energy resolution for
%hadrons is $\sigma(E)/E \simeq 50\%/\sqrt{E/\mathrm{GeV}}$,
% as obtained from test beam measurements.  The overall
%hadronic energy scale of the LAr is known to $4\%$ for the present
%measurement.

%The backward direction
%($-4.0<\eta<-1.4$) is covered by a lead / scintillating fibre
%calorimeter (SPACAL) with electromagnetic and hadronic sections. The
%energy resolution for electrons is $\sigma(E)/E \simeq
%10\%/\sqrt{E/\mathrm{GeV}}$.  
%For the present analysis, the electromagnetic and hadronic energy scales
%are known to $1.5$ and $7.0\%$, respectively.
%%The energy scale uncertainty is $0.3\%$
%%for electrons with $E'_e=27.5 \rm\ GeV$ and $2.0\%$ at $E'_e=6.5 \rm\
%%GeV$.  The
%%energy scale of the SPACAL is known to $7\%$ for hadrons.  
%In front of the SPACAL, the Backward Drift Chamber (BDC) provides
%track segments of charged particles with a resolution of $\sigma (r)
%=0.4 \rm\ mm$ and $r \sigma (\phi) =0.8 \rm\ mm$, resulting in a
%polar angle measurement accuracy of $0.5 \rm\ mrad$.

%The $ep$ luminosity is
%determined with a precision of $2.0\%$ by comparing the measured event
%rate in a photon tagger calorimeter close to the beam pipe at $z=-103
%\ \mathrm{m}$ with the QED Bremsstrahlung ($ep\rightarrow ep\gamma$)
%cross section.

The interaction region 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
in the range $-1.5<\eta<1.5$.
% with a transverse momentum resolution
%of $\sigma(p_T)/p_T \simeq 0.01 \cdot p_T/\mathrm{GeV}$.
A highly segmented Liquid Argon (LAr) calorimeter, covering the
range $-1.5<\eta<3.4$, surrounds the tracking chambers.
%The energy resolution for
%hadrons is $\sigma(E)/E \simeq 50\%/\sqrt{E/\mathrm{GeV}}$,
% as obtained from test beam measurements.  The overall
%hadronic energy scale of the LAr is known to $4\%$ for the present
%measurement.
The backward direction
($-4.0<\eta<-1.4$) is covered by a lead / scintillating fibre
calorimeter (SPACAL).
Both the LAr and the SPACAL calorimeters contain both 
electromagnetic and hadronic sections. 
%The
%energy resolution for electrons is $\sigma(E)/E \simeq
%10\%/\sqrt{E/\mathrm{GeV}}$.  
%For the present analysis, the electromagnetic and hadronic energy scales
%are known to $1.5$ and $7.0\%$, respectively.
%The energy scale uncertainty is $0.3\%$
%for electrons with $E'_e=27.5 \rm\ GeV$ and $2.0\%$ at $E'_e=6.5 \rm\
%GeV$.  The
%energy scale of the SPACAL is known to $7\%$ for hadrons.  
In front of the SPACAL, the Backward Drift Chamber (BDC) provides
track segments for charged particles.
% with a resolution of $\sigma (r)
%=0.4 \rm\ mm$ and $r \sigma (\phi) =0.8 \rm\ mm$, resulting in a
%polar angle measurement accuracy of $0.5 \rm\ mrad$.
The $ep$ luminosity is
determined 
%with a precision of $2.0\%$ 
by comparing the measured event
rate in a photon tagger calorimeter close to the beam pipe at $z=-103
\ \mathrm{m}$ with the QED Bremsstrahlung ($ep\rightarrow ep\gamma$)
cross section.

To enhance the sensitivity to hadronic activity in the region of the
outgoing proton, the PLUG calorimeter, the Forward Muon Detector 
(FMD) and the Proton Remnant Tagger (PRT) are used.  
The copper-silicon calorimeter PLUG enables energy measurements
to be made in the pseudorapidity range $3.5<\eta<5.5$.
The FMD is located at $z=6.5 \ \mathrm{m}$ 
and covers the range $1.9<\eta<3.7$
directly. Particles produced at larger $\eta$ can also be detected
because of secondary scattering with the beam-pipe. The PRT, a set of
scintillators surrounding the beam pipe at $z=26 \ \mathrm{m}$, 
tags charged particles in the region $6.0 \ \lapprox \ \eta \ \lapprox \ 7.5$.


\subsection{Event Selection and Kinematic Reconstruction}
\label{sec:evsel}

The data were triggered principally on the basis of an energetic
cluster in the electromagnetic section of the 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 scattered electron candidate with energy $E'_e > 6.5 \ {\rm GeV}$
and polar angle in the range $156^\circ < \theta_e < 176.5^\circ$.
%in the range
%\begin{equation}
%E_e^\prime > 6.5 \ {\rm GeV} \ ; \qquad
%156^\circ < \theta_e < 176^\circ \ .
%\end{equation}
To suppress background from photons and hadrons, cuts
%for example 
on the lateral
extension of the cluster forming the electron candidate and its
containment within the electromagnetic part of the SPACAL are applied.
A reconstructed charged track in the BDC,
linked within a transverse distance of $3 \rm\ cm$ to 
the SPACAL cluster is requested.
A reconstructed event vertex within $30 \rm\ cm$ of the nominal
interaction point is required from the Central Tracking Detector.
The hadronic final state 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 \cite{fscomb}. A minimum of two reconstructed 
hadronic final state particles
are required for the analysis.

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: $y$ is reconstructed as
\begin{equation}
  y = y_e^2 + y_d \ (1 - y_d) \ ,
\end{equation}
where  $y_e$ and $y_d$ are obtained from the scattered electron
only (`electron method') or from the angles of the electron and the 
hadronic final state (`double angle method'), respectively.
The reconstruction method limits to the
electron method at high $y$ and the double angle method at low $y$.
$Q^2$ and $x$ are then computed from
\begin{eqnarray}
  Q^2  = \frac{4 E_e^2 \ ( 1 - y)}{\tan^2 (\theta_e / 2)} \ ; \qquad
  x = \frac{Q^2}{ s \ y} \ .
\end{eqnarray}
Events where
%\begin{eqnarray}
$Q^2 > 5.7 \ {\rm GeV^2}$
% \ ; \qquad
and $y > 0.04$
%\end{eqnarray}
enter the final inclusive DIS sample.

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 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 activity above noise thresholds in the PRT, FMD
and
%detectors and a reconstructed energy of less than $3.5 \rm\ GeV$
%in the 
PLUG detectors. 
This selection ensures that the $X$ system is well contained
in the central part of the detector  and is separated by
a large rapidity gap covering at least $3.2 < \eta < 7.5$ from
the $Y$ system.

The mass of the system $X$ is obtained from the hadronic
final state using
\begin{eqnarray}
  M_x^2 = 1.08 \cdot (E^2 - p_x^2 - p_y^2 - p_z^2)_{\rm had} \cdot
\frac{y}{y_h} \ ,
\end{eqnarray}
where the subscript `had' represents a sum over all reconstructed
hadrons
and $y_h=(E-p_z)_{\rm had} / 2 E_e$.
% is the value of $y$
%reconstructed using only hadronic final state information.  
This
method of $\mx$ reconstruction reduces essentially to a measurement of
the total $E + p_z$ of the hadronic final state in the limit of high
$y$, where losses in the backward direction become significant.
The global factor of $1.08$ accounts for residual losses.
The diffractive variables $\beta$ and $x_\pom$ are then reconstructed
using
\begin{eqnarray}
   \beta = \frac{Q^2}{Q^2 + \mx^2 - t} \ ; \qquad
   \xpom = \frac{x}{\beta} \ .
\end{eqnarray}
%To suppress photoproduction
%background and ensure good reconstruction of kinematic variables,
%close agreement is required between $y_e$, $y_h$ and $y_d$:
%\begin{eqnarray}
%|y_e-y_h| < 0.25 \ ; \qquad
%|y_e-y_d| < 0.25 \ .
%\end{eqnarray}
%In addition to the requirements of a rapidity gap in the forward direction
%(see above), a cut
%\begin{eqnarray}
%x_\pom < 0.05 
%\end{eqnarray}
%is applied for the final selection of diffractive DIS events.



\subsection{Monte Carlo Simulation and Backgrounds}
\label{sec:mc}

Corrections for detector inefficiencies, acceptances and migrations between 
measurement intervals are performed 
using a Monte
Carlo simulation which combines several different models. The 
RAPGAP 2.08 \cite{rapgap} model is used 
to simulate colour-singlet exchange processes 
with $\xpom < 0.15$, based on the `resolved pomeron model' QCD fits
in \cite{h1f2d94}, including
contributions from
pomeron and sub-leading 
meson exchange.
%, based on the H1 QCD analysis of previous $F_2^{D(3)}$
%data \cite{h1f2d94}. 
Specifically, the 'fit 2' parton distributions
for the pomeron are used, evolved using $Q^2$ as a scale and
convoluted with leading order QCD matrix elements. Parton
showers \cite{meps} in the leading $\log(Q^2)$ approximation are used to 
approximate higher order QCD effects. Hadronisation is simulated using
the Lund string model in JETSET \cite{lund}.
QED radiative effects are taken into account via an interface to
the HERACLES program \cite{heracles}.


The DIFFVM model \cite{diffvm} is used to
simulate the quasi elastic production of the $\rho$, $\omega$,
$\phi$ and $J/\psi$ vector mesons, which contribute at small
$M_X$ (high $\beta$).
Smearing from the region
$\xpom > 0.15$ is modelled using the DJANGO \cite{django} Monte
Carlo model for standard DIS, based on recent 
structure function data \cite{h1f29697}. 
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. 

%Fig.~\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.

\subsection{Cross Section Measurement and Structure Function Extraction}
\label{sec:xsect}

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 
%\begin{eqnarray}
$\my < 1.6 \ {\rm GeV}$ and 
%\ ; \qquad 
$|t| < 1 \ {\rm GeV^2}$,
%\end{eqnarray}
as was the case for previous H1 data
\cite{h1f2d94}. The correction
factor applied to account for smearing about the $\my$ and $t$
boundaries of the measurement is $-8.7 \pm 8.2 \%$, as evaluated
using the DIFFVM model of elastic and proton dissociative processes.
The final cross sections 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 triple differential reduced cross section $\sigma_r^{D(3)}(x,Q^2,\xpom)$
is extracted according to equation~\ref{sigmar}.
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.

%\begin{eqnarray}
% \sigma_r^{D(3)}(x,Q^2,x_\pom) \ = \frac{x \ Q^4}{4 \pi \alpha^2 \ (1 - y + y^2 / 2)} \ \
%\frac{\dd^3 \sigma_{ep \rightarrow eXY}}{\dd x  \ \dd Q^2 \ \dd \xpom} \ .
%\label{eq:binning3}
%\end{eqnarray}
The measurement is performed using various
binning schemes which are optimised to the study of different aspects 
of the data. To investigate the $\xpom$ dependence for example,
the cross section is measured in many $\xpom$ intervals. By contrast,
the $x,\beta$ and $Q^2$ dependences of $\sigrd$ and their variation
with $\xpom$ can be studied
best in a scheme with fewer $\xpom$ intervals (two per decade) and
an $x$ (or $\beta$) and $Q^2$
binning very similar to that used in inclusive structure function
measurements \cite{h1f29697}. This also leads
naturally to the investigation of the QCD ($x,Q^2$)
structure at fixed $x_\pom$ and to the ratio of diffractive to
inclusive cross sections $\sigma_r^D(x_\pom,x,Q^2) / \sigma_r(x,Q^2)$
at fixed $x_\pom$.

%Three different binning schemes are used in the measurement which
%are optimized in view of different aspects of the data: 
%\begin{itemize}
%\item
%In the first scheme, 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 reduced
%cross section is then extracted using
%\begin{eqnarray}
%  \sigma_r^{D(3)}(\beta,Q^2,x_\pom)\ = \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} \ ,
%\label{eq:binning1}
%\end{eqnarray}
%where $\alpha$ is the fine structure constant.
%\item
%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 five fixed values of $\xpom=0.0003,0.001,0.003,0.01$ and $0.03$. 
%This method allows
%the $\beta$ and $Q^2$ dependence of the data and its variation
%with $\xpom$ to be studied with high precision. 
%The reduced cross section is extracted using
%\begin{eqnarray}
%  \sigma_r^{D(3)}(\beta,Q^2,x_\pom)\ = \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 x_\pom} \ ,
%\label{eq:binning2}
%\end{eqnarray}
%In this scheme, 
%$\sigma_r^{D(3)}$ is evaluated at the same $\beta$ values as for the first
%scheme. 
%\item
%Finally, the third binning scheme is used to extract
%${\rm d^3} \sigma / ({\rm d} x \ {\rm d} Q^2 \ {\rm d} \xpom)$
%at the same five $x_\pom$ values as the second scheme, but using
%an $x=x_\pom*\beta$ binning very similar to the measurement
%of $F_2(x,Q^2)$ by H1 \cite{h1f29697}.
%The reduced cross section is extracted using
%\begin{eqnarray}
% \sigma_r^{D(3)}(x,Q^2,x_\pom) \ = \frac{x \ Q^4}{4 \pi \alpha^2 \ (1 - y + y^2 / 2)} \ \
%\frac{\dd^3 \sigma_{ep \rightarrow eXY}}{\dd x  \ \dd Q^2 \ \dd \xpom} \ .
%\label{eq:binning3}
%\end{eqnarray}
%In this binning scheme, more $\beta$
%bins at fixed $x_\pom$ are measured than has previously been possible.
%In addition, it leads naturally to investigation of the QCD ($x,Q^2$)
%structure at fixed $x_\pom$ and to the ratio of diffractive to
%inclusive cross sections $\sigma_r^D(x_\pom,x,Q^2) / \sigma_r(x,Q^2)$
%at fixed $x_\pom$.
%\end{itemize}

%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\%.
%The data points, quoted at the centre of the measurement intervals,
%cover the kinematic range 
%\begin{eqnarray}
%6.5 \leq Q^2 \leq 120 {\rm\ GeV^2} \ ; \qquad
%0.01 \leq \beta \leq 0.9 \ ; \qquad
%10^{-4} \lapprox x_\pom < 0.05 \ .
%\end{eqnarray}

\subsection{Systematic Uncertainties}
\label{sec:syst}

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. 
%In particular, the following sources of systematic error have been
%taken into account:

%\begin{itemize}
%\item The energy $E'_e$ and polar angle $\theta_e$ of the scattered
%  electron candidate are measured to $1.5 \%$ and $0.5
%  \rm\ mrad$, respectively.
%\item The hadronic energy scales of the LAr and SPACAL calorimeters
%  are for this analysis known to $4\%$ and $7\%$, and 
%  the uncertainty in the energy
%  fraction carried by tracks in the hadronic final state algorithm is $3\%$.
%\item The efficiencies of the trigger and the BDC lead to systematic
%  errors of $1\%$ each;
%  those of the  FMD and PRT detectors are known to $5\%$ and $25\%$,
%  respectively. The correction factor for diffractive events 
%  rejected due to noise in the FMD was varied by $25\%$.
%\item The uncertainty in the luminosity measurement leads to a $2.0\%$
%  normalization error of the measured cross section values.
%\item The model dependence of the acceptance and migration corrections
%  obtained from the Monte Carlo simulations is estimated by varying
%  the shapes of kinematic distributions in the RAPGAP simulation
%  beyond the limits imposed by previous measurements or the present
%  data.  This has been done by reweighting (a) the $x_\pom$
%  distribution by $(1/x_\pom)^{\pm 0.1}$, (b) the $\beta$ distribution
%  by $\beta^{\pm 0.1}$ and $(1-\beta)^{\pm 0.1}$, (c) the $Q^2$
%  distribution by $(\log Q^2)^{\pm 0.2}$ and (d) the $t$ distribution
%  by $e^{\pm 2t}$. Furthermore, the normalizations of the simulation
%  of sub-leading Reggeon exchange (RAPGAP), vector meson production
%  (DIFFVM) and inclusive DIS events (DJANGO) were varied by $\pm
%  25\%$, $\pm 50\%$ and $\pm 100\%$ respectively and the simulation of
%  background due to photoproduction (PHOJET) or QED Compton
%  (COMPTON) events was varied by $25\%$.
%\item The use of different approximations for higher order QCD
%  diagrams (the parton shower (MEPS) model or the colour dipole (CDM)
%  approach) leads to an uncertainty of less than $3\%$ in the cross sections.
%\item There is an uncorrelated error of $3\%$ associated with the
%  bin-centre and QED radiative corrections.
%\item An uncertainty of $8.2\%$ arises from the correction for
%  smearing about the $\my$ limit of the measurement. It is estimated
%  by variations of (a) the ratio of elastic to proton dissociation
%  cross sections between 1:2 and 2:1, (b) the generated $\my$ and $t$
%  distribution in the proton dissociation simulation and (c) the
%  simulated efficiencies of the FMD and PRT detectors.
%\end{itemize}


For this analysis, the energy $E'_e$ and polar angle $\theta_e$
of the scattered
electron candidate are measured to $1.5 \%$ and $0.5
\rm\ mrad$, respectively, the hadronic energy scales of the LAr and 
SPACAL calorimeters are known to $4\%$ and $7\%$ respectively and 
the uncertainty in the energy
fraction carried by tracks in the hadronic final state algorithm is $3\%$.
%The correlations in these uncertainties between different data points
%are fully taken into account.

Normalisation uncertainties arise from the uncertainties in the
efficiencies of the trigger and the BDC ($1\%$ each).
The correction factor for diffractive events 
rejected due to noise in the FMD was varied by $25\%$.
The uncertainty in the luminosity measurement leads to a $2.0\%$
error.

The model dependence of the acceptance and migration corrections
is estimated by varying the shapes of the kinematic distributions in
the RAPGAP simulation beyond the limits imposed by previous
measurements or the present data and by varying the relative
normalisations of the different Monte Carlo models used in the 
acceptance corrections.  
%The $x_\pom$
%distribution is reweighted by $(1/x_\pom)^{\pm 0.1}$, the $\beta$ distribution
%by $\beta^{\pm 0.1}$ and $(1-\beta)^{\pm 0.1}$, the $Q^2$
%distribution by $(\log Q^2)^{\pm 0.2}$ and the $t$ distribution
%by $e^{\pm 2t}$. The normalizations of the simulation
%of sub-leading Reggeon exchange (RAPGAP), vector meson production
%(DIFFVM) and inclusive DIS events (DJANGO) were varied by $\pm
%25\%$, $\pm 50\%$ and $\pm 100\%$ respectively. 
The background subtracted
due to photoproduction (PHOJET) and QED Compton
(COMPTON) events was varied by $25\%$.
The use of different approximations for higher order QCD
diagrams (the parton shower or the colour dipole
approach) leads to an uncertainty of less than $3\%$ in the cross sections.
There is an uncertainty of $3\%$ associated with the
bin-centre and QED radiative corrections.

A normalisation uncertainty of $6.6\%$ and an uncorrelated uncertainty
of $4.9\%$ arises from the correction for
smearing about the $\my$ limit of the measurement. These uncertainties
are estimated
by variations of the ratio of elastic to proton dissociation
cross sections between 1:2 and 2:1, the generated $\my$ and $t$
distribution in the proton dissociation simulation, the
simulated efficiencies of the FMD and PRT detectors by $5\%$ and
$25\%$ respectively and the plug energy scale by 30\%.

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 all figures, the inner error bars on the data points
correspond to the statistical error, the outer error bars to the
statistical and systematic errors added in quadrature. The combined
normalisation error of $6.7\%$ is not shown.

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

\section{The Diffractive Reduced Cross Section}

\subsection{Comparison with Previous Data}

The measured diffractive reduced cross section $\sigrdarg$ 
%using the
%first binning scheme (equation~\ref{eq:binning1}) 
is compared with 
previous
%\footnote{The previous H1
%data were presented in terms of $F_2^{D(3)}$ under the assumption of
%$F_L^{D(3)}=0$, such that $\sigrd=\ftwod$.}
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 $Q^2 \geq 15 \ {\rm GeV^2}$, the statistical precision is
considerably improved. The two measurements are in good agreement with
the exception of the low $\beta$, medium $Q^2$ region,
where the previous data tend to be slightly higher than the new
measurement.

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

\subsection{Dependence on {\boldmath $\xpom$} and Effective Pomeron
Intercept}
%\section{The {\boldmath $\xpom$} dependence of {\boldmath $\sigrd$}}
\label{regge}

The $\xpom$ dependence of the measured diffractive reduced cross section
$\sigrdarg$ for fixed $\beta$ and $Q^2$  (figure~\ref{stamp97}) 
is studied, following a
similar procedure to that adopted in \cite{h1f2d94}.  A fit is
performed to the data using a parameterisation of the form
\begin{equation}
\sigrd(\xpom,\beta,Q^2) = f_\pom(\xpom) A_\pom(\beta,Q^2) +
                           f_\reg(\xpom) A_\reg(\beta,Q^2) \ ,
\end{equation}
which is motivated by 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 \ , 
\label{eq:fluxfac}
\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 \ .
\label{eq:trajectory}
\end{equation}
The values for $\alpha_\pom'$, $B_\pom$, $\alpha_\reg'$ and $B_\reg$
cannot be constrained by the $\sigrd$ data and are taken from
other measurements \cite{fscomb}.  
Furthermore, the secondary reggeon trajectory
intercept $\alpha_\reg(0)$ is not well constrained by the present 
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
parameterisation of the pomeron and reggeon flux factors are given
in table \ref{table:reggepara}.
%$\alpha_\pom'=0.26\pm0.26\rm\ GeV^{-2}$, $B_\pom=4.6^{+3.4}_{-2.6}\rm\ 
%GeV^{-2}$, $\alpha_\reg'=0.90\pm0.10\rm\ GeV^{-2}$,
%$B_\reg=2.0\pm2.0\rm\ GeV^{-2}$ and $\alpha_\reg(0)=0.50\pm0.16$.  
%The longitudinal structure function $F_L^D$ is 
%taken to be zero by default.
%assumed to be zero.

\begin{table}
\centering
\begin{tabular}{|l|l|}
\hline
Parameter & Value \\
\hline
$\alpha_\pom'$   & $0.26\pm0.26\rm\ GeV^{-2}$  \\
$B_\pom$         & $4.6^{+3.4}_{-2.6}\rm\ GeV^{-2}$  \\
$\alpha_\reg'$   & $0.90\pm0.10\rm\ GeV^{-2}$ \\
$B_\reg$         & $2.0\pm2.0\rm\ GeV^{-2}$   \\
$\alpha_\reg(0)$ & $0.50\pm0.16$   \\
\hline
\end{tabular}
\caption{ The parameter values and assumed uncertainties used for the
parameterisation of the pomeron and reggeon flux factors.}
\label{table:reggepara}
\end{table}


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 the unmeasured $F_L^D$, 
which is taken to be zero by default. 
The free parameters in the fit are the 
pomeron intercept parameter $\alpha_\pom(0)$ and 
the coefficients $A_\pom(\beta,Q^2)$
and $A_\reg(\beta,Q^2)$ in each ($\beta$,$Q^2$) bin.  The fit gives a
very good description of the data and has a 
$\chi^2$ per degree of freedom ($\chi^2/{\rm ndf}$) of 0.95.  If the
presence of a sub-leading reggeon exchange contribution is neglected, 
a considerably poorer $\chi^2/{\rm ndf}=1.25$ is obtained.  

The experimental
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 (section \ref{sec:syst}). The fit is also
repeated several times with variations in the theoretical assumptions
and input parameters, in order to evaluate the model dependence
uncertainty.
The possibility of interference between the pomeron and reggeon
exchange contributions is taken into account by repeating the fit with
the assumption of maximal interference, where the interference flux is
parameterised as 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 t
The assumed values for $\alpha_\pom'$,
$B_\pom$, $\alpha_\reg'$, $B_\reg$ and $\alpha_\reg(0)$ are varied within the
limits quoted in table~\ref{table:reggepara}. 
The uncertainty in the size of $F_L^D$ is
conservatively 
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)} \ .
\label{eq:alpha0value}
\end{equation}
The dominant uncertainty
arises from the variation of $F_L^{D(3)}$.
%In the future, the
%prediction for $F_L^D$ as obtained in the NLO QCD fit to the data
%(see section \ref{fits}), can be used to reduce this uncertainty.
%{\bf DO WE WANT TO SAY THIS ???}
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 i
In order to investigate whether
$\alphapom(0)$ has any dependence on $Q^2$, the fits are repeated with
the data divided into
two $Q^2$ intervals covering the ranges $6.5<Q^2<20 \rm\ GeV^2$ 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}
 Within the uncertainties, there is no evidence for a
variation of $\alphapom(0)$ with $Q^2$ in the measured kinematic
range.
These results are shown together with the previous H1 measurement in
figure~\ref{alpha}. The effective pomeron intercept extracted
from the diffractive data is also 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{h1lambda}.
The data suggest that at large $Q^2$, the effective intercept
describing the inclusive data is larger than that from the diffractive
data.


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

\subsection{Dependence on {\boldmath $\beta$ and $Q^2$}}

In order to study the $\beta$ and $Q^2$ dependence of the data with
high precision, the reduced cross section is extracted at 
several fixed values of $\xpom$.
%The diffractive reduced cross
%sections
%in this case not multiplied by $\xpom$, 
%are plotted for all five values of $\xpom$ in
%Figs.~\ref{q2f2d},\ref{xf2d} and \ref{betaf2d} as functions of
%$Q^2$, $x$ and $\beta$ respectively.
In order to illustrate the $\beta$ and $Q^2$ dependences in as much
detail as possible and to compare the results from different $\xpom$ bins, 
the data are presented in figures~\ref{betaflux} and~\ref{q2flux}
in the form $\sigrd / f_\pom(\xpom)$, where $f_\pom(\xpom)$
corresponds to the ``pomeron flux'' used in the Regge and QCD fits
(equation~\ref{eq:fluxfac}), with parameters as
%presented in sections \ref{regge} and \ref{fits}:
%\begin{eqnarray}
%f_{\pom / p} = \left (\frac{1}{\xpom} \right )^{2 \alphapom(0) - 1} 
%\int_{-1 \ {\rm GeV^2}}^{t_{min}} dt \ 
%e^{B_{\pom} t} \ ,
%\label{eq:pomfac}
%\end{eqnarray}
%where the parameters used for the
%pomeron intercept $\alphapom(0)$ and the $\xpom$ dependent slope
%parameter $B_{\pom}$ are as 
explained in section \ref{regge}.\footnote{The results 
%for $\xpom \sigma_r^{D(3)}(\xpom, Q^2, x)$ 
are presented in detail in
in figures~\ref{detailq2}-\ref{betaf2d} in the appendix,
%as functions of $Q^2$, $x$ and $\beta$ at fixed values of $\xpom$.
%Fig.~\ref{detailq2} shows the
%$Q^2$ dependence at fixed values of $\xpom$, $x$ and hence $\beta$.
%Fig.~\ref{detailx} shows the $x$ dependence at fixed $\xpom$ and
%$Q^2$.  Fig.~\ref{detailbeta} shows the $\beta$ dependence at fixed
%$\xpom$ and $Q^2$. 
where H1 preliminary data for
$200 \leq Q^2 \leq 800 \rm\ GeV^2$ \cite{f2dhiq2} are also shown
to be consistent with the observed dependence on 
$Q^2$ (figure~\ref{q2f2d}).}
The data are compared with the
results of the NLO DGLAP QCD fit presented in 
section~\ref{fits}.
In order to avoid
regions which are most likely to be affected by $F_L^D$
or sub-leading reggeon exchange, only data with
$y < 0.6$ and $\xpom \leq 0.01$ are shown. 
%Possible sub-leading exchange contributions are almost negligible for
%the data shown (
%Comparing the 
%data at different values of $\xpom$ in this
%way allows the dependences on $\beta$ and $Q^2$ to be seen over a wide
%range. 
The ``pomeron flux'' is used here primarily as a convenient
parameterisation of the $\xpom$ dependence, though 
the similarity of the normalised reduced cross sections from
different $\xpom$ values in the overlap regions indicates that 
a factorising $\xpom$ dependence of the diffractive
cross section is a good approximation. 
%In an interpretation
%in terms of diffractive parton distributions, this corresponds
%to a situation where the shape of the parton distributions
%is independent of $x_\pom$ (Regge factorization, equation~\ref{reggefac}).
%In Figs.~\ref{betaflux} - \ref{q2flux}, the results of the NLO DGLAP
%fit to the data (section~\ref{fits}) for $F_L^D=0$ are also shown.

%In Figs.~\ref{betaflux} and~\ref{q2flux}, the shapes of the $\beta$
%and $Q^2$ dependences of the cross section are compared for the
%different $x_\pom$ values $0.0003, 0.001, 0.003, 0.01$ by presenting
%the data in the form $\sigrd / f_\pom(\xpom)$, where $f_\pom(\xpom)$
%corresponds to the {\em pomeron flux} used in the Regge and QCD fits
%as defined in equation~\ref{eq:fluxfac},

A striking feature of the data is the scaling
violations with positive $\partial \sigrd / \partial \ln Q^2$ throughout
most of the phase space, becoming negative only at the highest measured
values of $\beta$ (figure~\ref{q2flux}). 
%In a `resolved pomeron'
%model, 
%for which 
%$\beta$ corresponds to the momentum fraction of the struck parton
%relative to the diffractive exchange, 
This behaviour 
%, which are measured with high precision. They remain
%positive over almost the whole $\beta$ or $x$ range of the measurement
%and flatten out or turn negative only at the highest measured values
%of $\beta$. In a picture in which the diffractive exchange is ascribed
%partonic structure, such that $\beta$ corresponds to the momentum
%fraction of the struck parton and thus is the equivalent to $x$ in the
%inclusive case, this behaviour 
is different from that observed for the
scaling violations of $F_2(x,Q^2)$ at fixed $x$, which are negative for 
$x \gapprox 0.1$.  
%The data measured at higher $Q^2$ in \cite{f2dhiq2} are within
%the uncertainties consistent with the observed dependence on
%$\log(Q^2)$ of the cross section at lower $Q^2$, as can be seen in
%Fig.~\ref{detailq2}. 
In a QCD interpretation, the measured $Q^2$
dependences are strongly suggestive of a large gluonic component of
the diffractive exchange.

The $\beta$ or $x$ dependences of the reduced cross section at fixed
$\xpom$ and $Q^2$ (figure~\ref{betaflux}) are relatively flat 
and remain large up to the highest
possible fractional momenta of $\beta=1$ or $x=\xpom$. At the lowest
measured $Q^2$, a rising behaviour of $\sigrd$ is observed in the data
as $\beta\rightarrow 1$, which becomes less pronounced 
with increasing $Q^2$. In
a leading order QCD picture, the reduced cross section can be viewed
as a charge weighted sum over the diffractive quark densities.  The
$\beta$ and $Q^2$
dependences are consistent with DGLAP evolution from high to low
$\beta$ with increasing $Q^2$
%, as expected for DGLAP evolution
%dominated by gluon radiation.
due to gluon radiation.

\subsection{Logarithmic {\boldmath $Q^2$} Derivatives}

For ($x$,$\xpom$) bins in which there are data
points for at least 3 values of $Q^2$, 
the logarithmic $Q^2$ derivative, $B_{\rm D} = \partial
\sigma_r^D / \partial \ln Q^2$ is extracted from fits of the form
\begin{eqnarray}
  \sigma_r^D = A_{\rm D}(x, \xpom) + B_{\rm D}(x, \xpom) \ln Q^2 \ .
\end{eqnarray}
The
logarithmic $Q^2$ derivative is sensitive to the LO diffractive gluon
density convoluted with $\alpha_s$ and the splitting function
$P_{qg}$.  
%Care about kinematic correlations between $\beta$ and $Q^2$
%has to be taken in this interpretation, though the data are all in the
%pQCD region ($Q^2 \geq 6.5 \ {\rm GeV^2}$) and 
The results for $B_{\rm D}$ from these fits are shown in figure~\ref{f2dderiv}
for different $\xpom$ values as a function of $\beta$.
The logarithmic derivatives have a relatively weak dependence on
$\beta$ for $\beta \lapprox 0.6$. Above this value, the behaviour 
changes rapidly, with the derivative changing sign in the region of
%Significant scaling violations with positive $\partial \sigma_r^D /
%\partial \ln Q^2$ persist up to large values of 
$\beta \sim 0.7$. 
%with
%a change in this behaviour for the highest $\beta$ values.  
As expected in Regge factorisation models, the scaling violations at 
fixed $\beta$ are very similar at the different
$\xpom$ values, even though
the data from different $\xpom$ values are sensitive to
different $Q^2$ regions.

\subsection{Comparison With Models For Diffractive DIS}

In this section, several phenomenological models for diffractive DIS
are confronted with the data. Comparisons are made for the diffractive
structure function $\ftwodarg$, which is extracted from the reduced
cross section under the assumption\footnote{Since $F_L^D$ only has a
non-negligible effect on the measured reduced cross section $\sigrd$
at high $y$ values and because of the kinematic range of the present
data, this represents a very good approximation in most of the phase
space of the measurement.} that $F_L^D=0$.

\subsubsection{Semi-classical Model} 

In figure~\ref{semcomp}, the $Q^2$ and $\beta$ dependences of the
diffractive structure function $\ftwodarg$ 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 present measurement, 
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 $M_X < 2 \rm\ GeV$ (corresponding to large $\beta$), is not
expected to be reproduced by the model.

\subsubsection{Saturation Model}

In figure~\ref{satcomp}, the data are compared with 
another colour dipole model by Golec-Biernat and W\"usthoff
\cite{kgb}.  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 $\beta$, corresponding to the longitudinal
$\gamma^*\rightarrow q\bar{q}_L$ photon fluctuation, allowing
comparisons to be made throughout the full measured kinematic region.
Relative to the original predictions in \cite{kgb}, an additional
colour factor of $(4/9)^2$ has been included for the 
$q \bar{q} g$ contribution for the current comparisons \cite{kgb:private}.
%The
%transverse $q \bar{q}$ and $q \bar{q} g$ fluctuations contribute
%mainly at 
%The model gives a reasonable overall description of the data.
%The medium and small $\beta$ regions are dominated by fluctuations
%of transversely poplarised photons into $q \bar{q}$ and
%$q \bar{q} g$ states respectively.
The model gives a reasonable description of the data at high and
medium $\beta$, where the contributions from 
$q \bar{q}$ fluctuations of longitudinally and transversely polarised
photons dominate. 
With the
additional colour factor, the $q \bar{q} g$ 
fluctuations of transversely polarised photons are insufficient
to describe the data in the low $\beta$, high $Q^2$ region.



\subsubsection{Soft Colour Interactions} 

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
previous $F_2^{D(3)}$ data.  The model has been refined \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 were obtained using the LEPTO $6.5.2\beta$
\cite{lepto} Monte Carlo generator. The version of SCI based
on the generalised 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.

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

\section{The Ratio of the Diffractive to the Inclusive Cross Section}

%It is of particular interest and has often been suggested to study 
The
ratio 
\begin{eqnarray}
\left . \frac{\sigma_r^D(\xpom, x, Q^2)}{\sigma_r(x, Q^2)} \right |_{\xpom} 
\label{ratiodef}
\end{eqnarray}
of the diffractive to the inclusive 
DIS reduced cross sections 
%$\sigrd(\xpom, x, Q^2)$ and $\sigma_r(x, Q^2)$
is extracted at fixed $\xpom$ in order to
compare the dynamics of diffractive DIS with those of
inclusive
DIS.
%In particular, {\em dipole models}, which consider the
%interaction of partonic fluctuations of the photon, described as
%(effective) colour dipoles, with the proton, predict the ratio of the
%diffractive to the inclusive cross sections to be flat (e.g.
%\cite{kgb} {\bf NEED MORE REFERENCES}).  It should be noted however that the
%observation of a flatness of the ratio is not a sufficient argument in
%favour of such models since it can arise also in other models in which
%there is no special relation between diffractive and inclusive cross
%sections.  First experimental results on the cross section ratio were
%reported in \cite{andysheffield,ZEUS:94,f2deps,zeus02}.
%In this section, 
%The ratio, defined as
% of $\sigma_r^D$ to the inclusive reduced
%cross section $\sigma_r$, defined as
%at fixed $x_\pom$, is measured. 
The values of $\sigma_r(x, Q^2)$ are
taken from \cite{h1f29697}.  
%The statistical and
%systematic errors for the ratio have been derived under the assumption
%that the uncertainties associated with the inclusive $\sigma_r$
%measurement are negligible with respect to those of the diffractive
%cross section $\sigrd$.
The ratio as a function of $Q^2$, 
shown for all $\xpom$ bins
in figure~\ref{q2rat},\footnote{The results are shown separately
for each $\xpom$ and $x$ bin in 
figure~\ref{q2ratdetail} in the appendix.}
tests the
difference between the scaling violations of $\sigma_r^D$ and $\sigma_r$
when compared at the same $x$. 
At low values of $\beta$ (or $x$), the ratio is remarkably
flat as a function of $Q^2$ for all $\xpom$ values. At the highest $\beta$,
where $x$ approaches $\xpom$,
the ratio falls with increasing $Q^2$. 

In order to quantify the differences between the $Q^2$ dependences of
$\sigma_r^D$ and $\sigma_r$ at fixed $x$ and $\xpom$, the 
logarithmic derivative $B_{\rm R}(x,x_\pom)$ of the ratio
is extracted from fits
of the form
\begin{eqnarray}
\left . \frac{  \sigma_r^D(\xpom,x,Q^2) }{ \sigma_r(x,Q^2) } \right |_{x,\xpom} = A_{\rm R}(x,\xpom) + B_{\rm R}(x,\xpom) \ln Q^2 \ .
\end{eqnarray}
%such that
%\begin{eqnarray}
%  B_{\rm R}(x,\xpom) = \frac{\partial}{\partial \ln Q^2} \ 
%\left . \frac{  \sigma_r^D(\xpom,x,Q^2) }{ \sigma_r(x,Q^2) } 
%\right |_{x,\xpom} \ .
%\end{eqnarray}
%The fits to the data are shown in Figs.~\ref{q2rat} and
%\ref{q2ratdetail}. 
The resulting values of $B_{\rm R}$ are shown in
figure~\ref{ratioderiv} with the ``pomeron flux'' $f_{\pom / p}$
(equation~\ref{eq:fluxfac}) divided out, so that the results at different
$\xpom$ can be compared in normalisation as well as shape.
The 
%are consistent with identical 
$Q^2$ dependences of the diffractive and inclusive
cross sections are consistent with being the same 
away from $x = \xpom$ ($\beta = 1$), suggesting that the ratio of the
diffractive gluon density of the proton to the total gluon density
is approximately constant in this region. This behaviour is expected 
in models in which rapidity gap formation is a purely probabilistic 
mechanism, for example due to multiple soft rearrangements of 
colour configurations \cite{sci}. In dipole models, for which the
diffractive cross
section is proportional to the dipole cross section squared and 
the inclusive cross section depends linearly on the dipole cross
section, this behaviour can
be reproduced provided that the dipole cross section grows with increasing
dipole size \cite{semicl,kgb,mcdermott:strikman}. 
As $\beta \rightarrow 1$, the logarithmic $Q^2$ derivative at fixed
$x$ becomes negative for all $\xpom$. 
This may suggest that the
high $\beta$ region of diffraction is driven by perturbative 2-gluon
exchange \cite{hebecker:teubner}, which is inherently higher twist
and thus leads to a
suppression with increasing $Q^2$. However, this behaviour 
also occurs naturally
in resolved pomeron models, in which gluon radiation shifts the
$\beta$ dependence to lower values as $Q^2$ increases. 
% Elastic vector meson production,
%which contributes in the high $\beta$ region is well described by
%such models.
At fixed $\beta$, there is 
no significant dependence of the logarithmic derivative on $\xpom$.

The ratio as a function of $x$ with $Q^2$ and $\xpom$ fixed is shown
in figure~\ref{xrat}. For each $\xpom$, dashed lines indicate the points in
$x$ at which $\beta = 1$ ($x = \xpom$). Dotted lines indicate the
point where $\beta = 0.1$ ($x = 0.1 \cdot \xpom$). For $\beta > 0.1$,
a complicated structure is observed in the ratio, corresponding to the
high $\beta$ behaviour of $\sigma_r^D$ (see figure~\ref{betaflux}). 
For $\beta < 0.1$, there is a
suggestion that a flatter dependence develops, though there are
limited data in this region.

A different approach to presenting the ratio of the diffractive to the
inclusive cross sections
is to 
extract the quantity $\rho^{D(3)}$, defined as
\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) \ \ \ \
%\rho^{D(3)}(\beta, Q^2, x) \ 
= \ \ \ \ \frac{\mx^2 \ x}{Q^2} \ \cdot \
\frac{\sigrdarg}{\sigma_r (x, Q^2)} \ ,
\end{eqnarray}
This ratio is shown in figure~\ref{epsratio} as 
a function of $W$ in bins of fixed $Q^2$ and $\beta$.
%In terms of virtual photon proton cross sections,
%where the diffractive cross section is integrated over
%$\my < 1.6 \ {\rm GeV}$ and $|t| < 1 \ {\rm GeV^2}$.
$\rho^{D(3)}$ is relatively
flat throughout the full phase space, except at large $\beta$
values (the very low $\mx$ region) and at low $W$ (high $\xpom$), where
sub-leading reggeon exchange becomes important.
When studying the $W$ (or $x$) dependence at
fixed $Q^2$ and $\beta$, $\mx$ is held fixed, as was the case
in \cite{ZEUS:94,zeus02}.  By contrast, in the ratio
defined by equation~(\ref{ratiodef}), $\xpom$ and $Q^2$ are held
fixed as $x$ varies, such that $\mx^2=Q^2(\xpom/x-1)$ also varies with $x$.

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

\section{Next-to-leading Order DGLAP QCD Fit}
\label{fits}

Leading (LO) and next-to-leading order (NLO) QCD fits have been
performed to the diffractive reduced cross section $\sigrdarg$  
to determine diffractive parton distributions (dpdfs)  and to test
QCD hard scattering factorisation.
The data used in the fit are restricted to
%cover the kinematic range $x_\pom<0.05$,
%$0.01\leq\beta\leq0.9$ and 
$M_X>2 \rm\ GeV$
%where the latter cut on
%the diffractive mass $M_X$ is applied 
to justify a leading twist
approach. 
%In the small $\mx$, region, substantial higher-twist
%contributions are expected.  
The NLO fit is performed to the reduced
cross section $\sigma_r^D$.
The effects of $F_L^D$ are considered through its relation to
the NLO gluon density such that no explicit cut on $y$ is required.
%However, due to the limited $y$ range of the
%data there is presently little direct sensitivity to $F_L^D$.
%Fits using leading order (LO) DGLAP have been performed as well for use in
%LO Monte Carlo simulations and for comparisons with previous results.
%In contrast to the NLO fit,
For the LO fit, an additional cut $y<0.45$ is applied to reduce any
possible effect of $F_L^D$ to a negligible level.\footnote{At LO QCD,
  $F_L^D=0$.} 
In addition to the data from the present analysis (284 points),  
%Two data sets are used in the fit: 
%\begin{itemize}
%\item The
%reduced cross section data of the present analysis, covering photon
%virtualities in the 
%range $6.5\leq Q^2\leq
%120 \rm\ GeV^2$ (284 points);
%\item
H1 preliminary data at higher
$Q^2$ \cite{f2dhiq2}, covering $200 \leq Q^2 \leq 800 \rm\ GeV^2$ 
are included (29 points).
%\end{itemize}


\subsection{QCD Fit Technique}

In the fit, the shape of the dpdfs is assumed to be independent of
$x_\pom$ (``Regge factorisation'', equation~\ref{reggefac}), which is
supported by the data (figures~\ref{betaflux},\ref{q2flux}).  
The $x_\pom$ dependence is
parameterised according to equations~(\ref{eq:fluxfac},\ref{eq:trajectory}),
using the values of the parameters given in
table~\ref{table:reggepara}.  The value of $\alpha_\pom(0)=1.173$ is
taken from the result of the fit to the $x_\pom$ dependence of the
data (equation~\ref{eq:alpha0value}).  A sub-leading
exchange contribution is included 
in the fit with parton densities 
taken from a parameterisation of the pion \cite{owens}. 
Since the sub-leading contribution is negligible except
at the
highest $x_\pom>0.01$ values, choosing a
different 
parameterisation \cite{grvpion}, does not significantly affect the results.

The diffractive exchange is modelled in terms of a
light flavour singlet\footnote{$u=d=s=\bar{u}=\bar{d}=\bar{s}$ is
  assumed.} 
\begin{equation}
\Sigma(z) = u(z)+d(z)+s(z)+\bar{u}(z)+\bar{d}(z)+\bar{s}(z) 
\end{equation}
and a gluon distribution $g(z)$ at a starting scale 
$Q_0^2= 3 \rm\ GeV^2$.  Here, $z$ is the momentum
fraction of the parton entering the hard sub-process
with respect to the diffractive
exchange, such that for the lowest-order quark parton model process
$z=\beta$, whereas for higher order processes $0<\beta<z$.  The dpdfs
are parameterised using the form
\begin{equation}
z p_i (z,Q_0^2) = \left [ \sum_{j=1}^n C_j^i P_j(2z-1)   \right ]^2
e^{\frac{a}{z-1}} \ ,
\end{equation}
where, $P_j(\xi)$ is the $j^{\rm th}$ member of a set of Chebychev
polynomials\footnote{$P_1=1$, $P_2=\xi$ and $P_{j+1}(\xi)=2\xi
  P_j(\xi)-P_{j-1}(\xi)$.}. The series is squared to ensure
positivity. 
The additional exponential term is used to guarantee
that in the limit of $z\rightarrow 1$, the dpdfs tend
to zero. The parameter $a$ is set to $0.01$. Leaving out the
exponential term results in slight variations of the shape of the
extracted dpdfs at the highest $z>0.95$, which are well within the
quoted uncertainties.  No momentum sum rule is imposed.  Charm quarks
are treated in the massive scheme (appearing via boson gluon fusion
processes) with
$m_c=1.5\pm0.1 \rm\ GeV$. The strong coupling is set via $\Lambda^{\rm
  \overline{MS}}_{\rm QCD}=200\pm30 \rm\ MeV$.

The LO or
NLO DGLAP equations are used to evolve the dpdfs to $Q^2>Q_0^2$
using the method of
\cite{h1f29697}, extended for diffraction. In the fit, the
statistical and the experimental systematic errors of the data points
and their correlations~\footnote{The systematic errors of the high
$Q^2$ measurement \cite{f2dhiq2} are taken to be fully uncorrelated.}
are propagated to obtain error bands for the
resulting dpdfs \cite{pascaudzomerlal}. The $\chi^2$ is computed as
\begin{equation}
\chi^2 = \sum_{ij} 
\frac{\left[
\sigma^{exp}_{ij} - \sigma^{th}_{ij}
(1-\nu_j\delta_j -\sum_k c_{jk} \delta_{sys,ijk}) 
\right]^2}
{[\sigma^{exp}_{ij}]^2 ( \delta_{stat,ij}^2 + \delta_{unc,ij}^2 ) } + 
\sum_{jk} c_{jk}^2 +
\sum_j \nu_j^2 \ ,
\end{equation}
where the indices correspond to data point $i$ of data set $j$ and $k$
counts the individual sources of correlated error. $\sigma^{exp}_{ij}$
and $\sigma^{th}_{ij}$ are the measured and calculated cross sections,
$\delta_{stat,ij}$ and $\delta_{unc,ij}$ are the relative statistical
and uncorrelated systematic errors of the measurements and
$\delta_{sys,ijk}$ corresponds to the relative correlated systematic
error for a given error source $k$.  The systematics and normalisation
parameters $c_{jk}$ and $\nu_j$, which have zero mean and unit
variance, are determined by the fit.

A theoretical error in the extracted dpdfs
is estimated by variations of $\Lambda_{\rm QCD}$ and $m_c$ within the
limits quoted above and of the parameterisation of the $x_\pom$
dependences by varying the value of $\alpha_\pom(0)$ within its
statistical error and the other parameters in the flux factors within
the uncertainties quoted in table~\ref{table:reggepara}. No
theoretical uncertainty is assigned for the choice of parton
parameterisation, though the results are consistent within the
quoted uncertainties if alternative approaches \cite{MRST}
are used.

The numbers of terms
in the polynomial parameterisations used in the fit have been systematically
optimised to the precision of the data. For the quark singlet as well
as the gluon distributions, the first three terms in the series of
polynomials are used, yielding 3
free parameters ($C_j^\Sigma$ and $C_j^g$)  
for each of the singlet and gluon distributions.
The
normalisation of the sub-leading exchange contribution at high $\xpom$
is also determined by the fit such that the total number of free
parameters is $7$.
The $\chi^2$ for the central NLO fit is $308.7$ for 306 degrees of
freedom.

\subsection{Comparison of the NLO Fit with Data}

The result of the fit is compared with the present data in
%Figs.
%~\ref{q2f2d}-\ref{q2flux} and
figures~\ref{betaflux},~\ref{q2flux} and~\ref{detailq2}-\ref{betaf2d}. A 
comparison with the higher $Q^2$
data \cite{f2dhiq2} is shown in figure~\ref{vancouver}. The fit reproduces the
features of the data very well, in particular the rising behaviour of
$\sigrd$ towards $\beta \rightarrow 1$ 
at low $Q^2$ (figure~\ref{betaflux})
%(Figs.~\ref{detailbeta},\ref{betaf2d} 
%and \ref{betaflux}) 
and the
rising scaling violations which persist up to high $\beta$
values 
%(Figs.~\ref{detailq2},\ref{q2f2d} and 
(figure~\ref{q2flux}).
In figures~\ref{detailq2}-\ref{detailbeta}, the size of the sub-leading
exchange contribution as determined by the fit is also displayed. It
only plays a significant role for $\xpom\geq 0.01$. Compared with previous 
H1 results \cite{h1f2d94}, its normalisation is reduced by
approximately $50\%$.
% Hi Paul, the value+error for the meson normalisation is
%0.44107E-02   0.59995E-03 
% Cheers, Frank

\subsection{Diffractive Parton Distributions}

The dpdfs resulting from the NLO QCD fit
are presented in figures~\ref{partons1} and \ref{partons2}.
The inner error bands correspond to the
experimental error and the outer error bands to the
experimental and theoretical errors added in quadrature. 
%These results are the first determination of
%diffractive parton distributions which include an assessment
%of the experimental and theoretical uncertainties.
The quark singlet as well as 
the gluon distribution extend to large fractional momenta  $z$.
Whereas the singlet distribution is well constrained by the fit,
there is a substantial uncertainty in the gluon distribution at $z>0.5$,
mainly due to the model assumptions. At smaller $z$,
the relative size of the uncertainties is significantly reduced.

The shape of the singlet distribution is a direct consequence of 
the shape of the observed $\beta$ distribution in the reduced
cross section, whereas the gluon distribution is determined mainly 
through the scaling violations. As was the case in previous QCD fits
to diffractive DIS data, the gluon distribution is much bigger
than the quark distribution, 
%consistent with the picture that
showing that
the diffractive exchange is mediated dominantly by gluons.
%To further quantify the fraction of diffractive exchange momentum
%which is carried by gluons, the NLO parton distributions 
%are integrated over the 
%measured kinematic range 
The fraction of the exchanged momentum carried by gluons is quantified
in terms of the ratio
\begin{equation}
\frac
{\int \dd z \ g(z,Q^2)}
{\int \dd z \ \Sigma(z,Q^2) + \int \dd z \ g(z,Q^2)  } \ ,
\nonumber
\end{equation}
integrated over the measured region $0.01<z<1$. This fraction 
is shown as a function of $Q^2$ in figure~\ref{gluefrac}. 
%Again, the
%inner (outer) error bands represent the experimental (total)
%error. 
%In the kinematic range of the measurement, 
The integrated fraction of the exchanged momentum
carried by gluons is $\sim75\pm15\%$ (total error), which is
fully consistent with earlier results \cite{h1f2d94}.

The central values from the LO fit are also shown in 
figures~\ref{partons1} and \ref{partons2}.
The LO gluon density
shows a peak at the highest $z$.  This peak disappears for the
central values when moving to NLO.
%, an effect which is due to the
%different theory rather than the extra high $\beta$ data in the NLO
%fit. 
The LO results are compared with the previous H1 QCD fits
to older $F_2^D$ data \cite{h1f2d94} in figure~\ref{partons3}. 
The
central values of the singlet distributions agree reasonably well
between the previous and the new fits. The shape of the new gluon
distribution is similar to that from `H1 fit 3', 
except that the peak at the highest $z$ is 
significantly 
%smaller\footnote{The functional flexibility of
%the 'fit 2' gluon distribution is limited since it was constrained
%to basically a constant at $Q_0^2$.}. The normalisation is
%different by about $30\%$ at low to medium $z$. 
reduced.\footnote{The functional flexibility of
the 'fit 2' gluon distribution is limited since the parameterisation
allows for no more than a constant dependence on $z$ at 
$Q_0^2$.} The normalisation is
different by about $30\%$ at low to medium $z$. 
%It is however necessary to
%take the uncertainties in these parton distributions into account. 
The uncertainties in the LO dpdfs derived from the present data are
similar to those in the NLO dpdfs (figure~\ref{partons2}).  The errors
on the 'H1 fit 2 / 3' parton distributions 
%have not been determined, but
are significantly larger, especially for the
gluon distribution.
%, since the data used in these fits were much less
%precise.  
Taking these uncertainties into account, 
%it is fair to say that within the errors 
the old and the new parton distributions are in agreement.


\subsection{ The Longitudinal Structure Function $F_L^D$}

At NLO in QCD, the 
leading twist component of the longitudinal diffractive structure
function $F_L^D$ is given by
\begin{equation}
 \frac{F_L^D}{z} \sim \frac{\alpha_s}{2\pi}
\left [ C_q^L \otimes \bar{F}_2^D +
        C_g^L \otimes \sum_i e_i^2 \ g^D
\right ] \ ,
\end{equation}
where $\bar{F}_2^D=\sum_i e_i^2 [ q^D_i + \bar{q}^D_i ]$, $C_q$
and $C_g$
are Wilson coefficients, $e_i$ is the charge of quark species $i$ and 
$\otimes$ represents a convolution integral.
Figure~\ref{fl} shows the prediction for $F_L^D$ from the NLO fit
as a function of 
$\beta$ and $Q^2$ at fixed $\xpom=0.003$. 
The longitudinal structure function 
increases relative to $F_2^D$ towards low $Q^2$ and $\beta$.
The values of $F_L^D$ are relatively large, due to their
relation to the gluon density.

In 
%Figs.~\ref{q2f2d}-\ref{betaf2d}
figures~\ref{detailq2}-\ref{betaf2d}, the 
prediction of the NLO
QCD fit for the diffractive reduced cross section  $\sigrd$ under
the assumption that $F_L^D=0$ is also shown. The effect of a non-zero
$F_L^D$ is a taming of the rise of the cross section towards low
$\beta$ or $x$ (high $y$) at fixed $x_\pom$ and $Q^2$.
%thus corresponding to
%the domain of high $y$. 
%Due to the limited kinematic range in $y$ which
%is accessed by the data, $F_L^D$ 
The effects of $F_L^D$ are constrained to the limits of 
the acceptance of the present measurement, such that the direct
sensitivity is weak.
%. Hence, with the present data
%there is little direct sensitivity on $F_L^D$.



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

\section{ Factorisation Tests With Hadronic Final State Observables}

QCD factorisation in diffraction can be tested by taking the dpdfs
extracted from $\sigma_r^D$ in DIS (see previous section) and predicting
diffractive final state observables 
such as dijet and charm
cross sections at HERA. Both of these processes are highly sensitive to the
diffractive gluon distribution via the boson-gluon fusion process
$\gamma^* g \rightarrow q \bar{q}$. Predictions can also be made
for hard diffractive processes in hadron-hadron interactions, for
example diffractive dijet production at the Tevatron.
In the following, the LO diffractive parton distributions described in
section \ref{fits} are used for these comparisons.

\subsection{Diffractive Jet and Charm Production at HERA}

Predictions for diffractive DIS jet
and charm production cross sections are obtained
using the RAPGAP \cite{rapgap} Monte Carlo program and are made at the
level of stable hadrons. The renormalisation and factorisation scales
are set to $\mu^2=Q^2+p_T^2+m_q^2$.  
In figure~\ref{jetdstarzpomcomp}, comparisons are made between the
new dpdfs and 
H1 measurements of 
diffractive dijet \cite{h1jets} and $D^*$ meson \cite{h1dstar} 
cross sections differential in hadron level estimators of 
$z_\pom$.
%, reconstructed from the dijets or 
%the $D^*$ meson respectively. 
For comparison, predictions
based on the previous H1 QCD fits (`H1 fit 2 and 3' in \cite{h1f2d94}) 
are also shown.
% for dijet as well as $D^*$ production.
The predictions based on the
QCD fit to the present data are generally below those of the
previous fits, due to the reduced normalisation
of the diffractive gluon distribution. This leads 
to an improved description of the magnitude of the $D^*$ cross
section, but a worse description for that of the dijet
cross section. 
The shapes of both $z_\pom$ distributions
are well described by the predictions based on the new dpdfs. 
The uncertainties in the diffractive parton distributions
have not yet been propagated to the final state cross section
predictions,
though they are expected to be substantial. 
Higher order QCD corrections are also not
fully included in the LO simulation. 
%Taking these uncertainties into account,
%the measured cross sections are consistent with the predictions
%based on the new QCD fit.
At the present stage, there is thus no evidence for any breakdown of
%The agreement of the measured cross sections with the predictions
%based on the new dpdfs extracted in the QCD fits to the
%inclusive cross section $\sigrdarg$ lends support
%for the validity of 
QCD hard scattering factorisation
in diffractive DIS \cite{collins}.

Further comparisons with diffractive dijet cross sections
are shown in the appendix in figures~\ref{jetscomp1}-\ref{jetscomp4}.
A comparison with a measurement of 3-jet production in
diffractive DIS is shown in figure~\ref{threejets}.
Additional comparisons with diffractive $D^*$ production
cross sections are shown in figure~\ref{charmcomp}.


\subsection{Diffractive Dijet Production at the Tevatron}

The CDF collaboration 
%at the TEVATRON $\bar{p}p$ collider
has made measurements of
%presented a measurement of 
diffractive dijet production in the process
$p \bar{p} \rightarrow p X$ collisions at
$\surd s = 1800 \ {\rm GeV}$ \cite{CDF:jets}.
The results are presented in terms of an {\em effective diffractive
structure function} $\tilde{F}_{jj}^D$ for dijet events, corresponding
in resolved pomeron models to
\begin{equation}
\tilde{F}_{jj}^D(\beta,\mu^2) \ = \ 
\left\{ \,
\beta \, g(\beta,\mu^2) + 
\frac{4}{9} \, \beta \, q(\beta,\mu^2) \ 
\right\} 
\ \otimes  \ f_{\pom / p}(\xpom)
\ ,
\end{equation}
where $\mu$ is the average scale of the measurement, 
$f_{\pom / p}(\xpom)$ is the pomeron flux
% for a proton losing a
%fraction $\xpom$ of its longitudinal momentum during the interaction
and $\otimes$ represents a convolution of the flux factor with the
effective parton densities 
$g(\beta, \mu^2) + 4/9 \ q(\beta, \mu^2)$.

Figure~\ref{cdf} shows a comparison of the CDF measurement
with a prediction based on the new diffractive parton distributions,
%extracted in section \ref{fits}. It is 
obtained by integrating
the dpdfs over the appropriate range $0.035<\xpom<0.95$, $|t|<1 \rm\
GeV^2$ and taking
$\mu^2 = 75 \rm\ GeV^2$, corresponding to the average squared
transverse jet
energy in the CDF measurement. 
The new prediction is slightly closer to the data than that using the
parton distributions based on the fits to the previous data. However,
a discrepancy at the level of an order of magnitude remains. The discrepancy
appears to be less dependent on $\beta$ when the new dpdfs are used.
The new QCD fit thus confirms the serious breakdown of
factorisation observed
%at hadron-hadron colliders.
when comparing hard diffraction results from $ep$ and $\bar{p} p$ data.
This has often been
interpreted as being due to additional spectator interactions
\cite{gap:survival}.
%(absorptive corrections, ``rapidity gap
%survival''). 

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

\section{Summary}

A high precision measurement of the inclusive cross section for 
the diffractive (DIS) process $ep \rightarrow eXY$ has been presented,
obtained using data taken with the H1 detector at HERA, corresponding to
an integrated luminosity of $\mathcal{L}=10.6 \rm\ pb^{-1}$, a factor
five more than previous measurements.
%The kinematic domain of the measurement is
% $6.5 \leq Q^2 \leq 120 \rm\ GeV^2$, 
%$0.01 \leq \beta \leq 0.9$,
%$10^{-4}\lapprox\xpom<0.05$,
%$\my < 1.6 \rm\ GeV$ and $|t|< 1 \rm\ GeV^2$.
%The measurement is presented in the form of  a
%diffractive reduced cross section $\sigma_r^{D(3)}$ which,
%in the limit of $F_L^D=0$ corresponds to the diffractive
%structure function $\ftwod$.

The $\xpom$ dependence of the data is interpreted in terms of a
measurement of the effective pomeron intercept $\alphapom(0)$. The
resulting value $\alphapom(0) = 1.173 \ \pm 0.018 \ \mathrm{(stat.)}  \ \pm
0.017 \ \mathrm{(syst.)}  \ ^{+ 0.063}_{-0.035} \ \mathrm{(model)}$ 
confirms previous observations that the energy dependence of
diffractive DIS is stronger than that of soft hadronic diffraction. 
There is no significant variation of the $\beta$ and $Q^2$ dependence
of the data when $\xpom$ changes, 
supporting the hypothesis of Regge factorisation. 

The $Q^2$ dependence at fixed $\beta$ and $\xpom$ displays rising
scaling violations for $\beta \lapprox 0.7$, with a clear change
to a falling behaviour with increasing $Q^2$ at the highest $\beta$.
The $\beta$ dependence of the data is relatively flat, though a clear
rise with increasing $\beta$ is observed at the highest $\beta$ and 
low $Q^2$. 

The ratio of the diffractive to the inclusive cross section
$\sigma_r^{D(3)} (x, Q^2, x_\pom) / \sigma_r (x, Q^2)$ is 
found to be remarkably flat as a function of $Q^2$ with $\xpom$ and
$x$ fixed, except at the highest values of $\beta$ ($x$
approaching $\xpom$), where the
diffractive cross section falls faster than the inclusive. 


%The measured cross section is compared with various
%models for diffractive DIS.


A NLO DGLAP QCD fit is performed to the data and an assessment is made
of
the experimental and theoretical uncertainties on the resulting
diffractive parton densities. The results confirm that the diffractive 
exchange is dominated by the gluon
density, which remains large up to high fractional momenta. The 
fraction of the diffractive exchange carried by gluons 
is found be $(75\pm15)\%$ at
$Q^2=10 \rm\ GeV^2$.  

The extracted diffractive parton densities are used to make updated
comparisons with diffractive final state measurements. 
Dijet and open charm cross sections at
HERA are found to be well described, whereas a discrepancy of approximately
an order of magnitude is observed in the predictions of dijet cross
sections from
the Tevatron.
% thus testing QCD hard scattering factorisation
%as applied to diffraction.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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.

% References %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Figures %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% control plots %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\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. 
%NOTE: DO WE NEED AN UPDATE (CHANGED ENERGY CUT) OR NOT SHOW IT AT ALL?
%}
%\label{control}
%\end{figure}

%  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.

% Stamp %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{figure}
\centering
\epsfig{file=H1prelim-02-012.fig2.eps,width=0.875\linewidth}
%\epsfig{file=H1prelim-01-111.fig3.eps,width=0.875\linewidth}
\caption{The measured diffractive reduced cross section {\em (filled
    circles)}, presented as $\xpom \sigma_r^{D(3)}$ 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 normalisation uncertainty
  of $6.7\%$ is not shown.  For comparison, the previous H1
  measurement in \cite{h1f2d94} is also shown
  {\em(triangles)}.  }
\label{stamp97}
\end{figure}

\clearpage

% alphapom summary %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{figure}
\centering
\epsfig{file=H1prelim-02-012.fig3.eps,width=0.85\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{h1lambda} for $x<0.01$. The filled
  circles are the values of $\alphapom(0)$ as obtained from the
  phenomenological Regge fit to the $\sigrd$ 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 triangle is the value of
  $\alphapom(0)$ obtained in \cite{h1f2d94}.  }
\label{alpha}
\end{figure}



% flux normalised sigma_r summary %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{figure}[ht]
\centering
\epsfig{file=H1prelim-02-012.fig4.eps,bbllx=0,bblly=60,bburx=568,bbury=568,
width=1.1\linewidth,clip=}
\caption{$\beta$ dependence of the reduced cross section scaled 
  at each $\xpom$ by the values assumed for the $t$-integrated pomeron
  flux in the QCD fits (equation~\ref{eq:fluxfac}). 
Only data with $y < 0.6$ are shown to minimise the influence of the
longitudinal structure function $F_L^D$. The data
  are compared with the prediction of the NLO QCD fits (experimental
  errors only) under the assumption that $F_L^D=0$ .  
}
\label{betaflux}
\end{figure}

\begin{figure}[ht]
\centering
\epsfig{file=H1prelim-02-012.fig5.eps,bbllx=0,bblly=80,bburx=568,bbury=568,
width=1.1\linewidth,clip=}
\caption{$Q^2$ dependence of the reduced cross section 
  ($y < 0.6$) scaled at each $\xpom$ by the values assumed for the
  $t$-integrated pomeron flux in the QCD fits (equation~\ref{eq:fluxfac}). 
The data are compared
  with the prediction of the NLO QCD fits (experimental errors only) for
$F_L^D=0$.}
\label{q2flux}
\end{figure}

% log derivatives for q2 dep. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{figure}[ht]
\centering
\epsfig{file=H1prelim-02-012.fig6.eps,width=0.85\linewidth}
\caption{Logarithmic $Q^2$ derivatives of $\sigma_r^D$ for different
  $\beta$ and $\xpom=0.001,0.003,0.01$, 
  scaled at each $\xpom$ by the values assumed
  for the $t$-integrated pomeron flux in the QCD fits
(equation~\ref{eq:fluxfac}). 
}
\label{f2dderiv}
\end{figure}


% Model comparisons %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% semi-classical model

\begin{figure}
\centering
\begin{minipage}{0.495\linewidth}
\epsfig{file=H1prelim-02-012.fig7a.eps,width=1.0\linewidth}
\end{minipage}
\begin{minipage}{0.495\linewidth}
\epsfig{file=H1prelim-02-012.fig7b.eps,width=1.0\linewidth}
\vspace{1.5cm}
\end{minipage}
\caption{The diffractive structure function $\xpom\ftwodarg$, 
extracted from the measured reduced cross section
$\sigma_r^{D(3)}$ under the assumption of $F_L^D=0$, 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 $M_X<2 \rm\ GeV$, where the
  model is not expected to be valid.  }
\label{semcomp}
\end{figure}

% saturation model

\begin{figure}
\centering
\begin{minipage}{0.495\linewidth}
%\epsfig{file=H1prelim-01-111.fig14a.eps,width=1.0\linewidth}
%\epsfig{file=satmodel.noevol.corr.beta.xp003.eps,width=1.1\linewidth}
\epsfig{file=H1prelim-02-012.fig8a.eps,width=1.1\linewidth}
\end{minipage}
\begin{minipage}{0.495\linewidth}
%\epsfig{file=H1prelim-01-111.fig14b.eps,width=1.0\linewidth}
%\epsfig{file=satmodel.noevol.corr.q2.xp003.eps,width=1.1\linewidth}
\epsfig{file=H1prelim-02-012.fig8b.eps,width=1.1\linewidth}
%\vspace{1.5cm}
\end{minipage}
\caption{The diffractive reduced cross section
$\sigma_r^{D(3)}$
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 prediction of a
  colour dipole model by Golec-Biernat and W\"usthoff 
  \cite{kgb,kgb:private} {\em
    (solid curves)}. The contributions in the model
from 
$\gamma^* \rightarrow q\bar{q}_T$ ({\em dashed}), 
$\gamma^* \rightarrow q\bar{q}_L$ ({\em dotted}) and
$\gamma^* \rightarrow q\bar{q}g_T$ ({\em dash-dotted}) 
virtual photon fluctuations
are shown separately.
 }
\label{satcomp}
\end{figure}

% SCI

\begin{figure}
\centering
\epsfig{file=H1prelim-02-012.fig9.eps,width=0.9\linewidth}
\caption{The diffractive structure function $\xpom\ftwodarg$, 
extracted from the measured reduced cross section
$\sigma_r^{D(3)}$ under the assumption of $F_L^D=0$,
  plotted as a function of $\xpom$ at fixed values of $Q^2$ and
  $\beta$.  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 generalised 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}

% ratio plots %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\begin{figure}[ht]
\centering
\epsfig{file=H1prelim-02-012.fig10.eps,bbllx=0,bblly=60,bburx=568,bbury=568,
width=1.1\linewidth,clip=}
\caption{The ratio of diffractive to inclusive reduced cross sections,
  shown as a function of $Q^2$. The results of the fit to a
  logarithmic $Q^2$ dependence are overlaid (figure~\ref{ratioderiv}). 
The $Q^2$ dependence of
  the ratio is small for most $\beta$ and varies little with $\xpom$.}
\label{q2rat}
\end{figure}

\clearpage

\begin{figure}[ht]
\centering
\epsfig{file=H1prelim-02-012.fig11.eps,width=0.85\linewidth}
\caption{Logarithmic $Q^2$ derivatives of the ratio $\sigma_r^D / \sigma_r$,
  the $\xpom$ dependence of $\sigma_r^D$ having been divided out using
  a parameterisation of the $x_\pom$ dependence (equation~\ref{eq:fluxfac}). 
The data are consistent
  with identical $Q^2$ dependences of the diffractive and inclusive
  cross sections away from $\beta = 1$, with clear differences from
  this behaviour as $\beta \rightarrow 1$.}
\label{ratioderiv}
\end{figure}

\clearpage

\begin{figure}[ht] 
\centering
\epsfig{file=H1prelim-02-012.fig12.eps,angle=90,width=1.\linewidth,clip=}
\caption{The ratio of diffractive to inclusive reduced cross sections,
  shown as a function of $x$ at fixed $\xpom$ and $Q^2$ values.  Note
  that $\mx^2 = Q^2(\xpom / x - 1)$ varies with $x$ in these plots.
  The dashed (dotted) lines indicate the points at which $\beta = 1$
  ($\beta = 0.1$).  Clear variations in the shape are observed for the
  different $\xpom$ values, arising from the fact that they cover
  different $\beta$ ranges. For the high $\beta$ region, the ratio has
  a complicated shape, since $\sigma_r^D$ has considerable structure
  in this region, whereas $\sigma_r$ is well parameterised
  by $x^{- \lambda}$. For the lowest $\beta$, a flatter dependence of
  the ratio on $x$ is observed.}
\label{xrat}
\end{figure}

\begin{figure}
\centering
\epsfig{file=H1prelim-02-012.fig13.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 fixed according to
$\mx^2 = Q^2 \ (1/\beta - 1)$.}
\label{epsratio}
\end{figure}

\clearpage


% QCD fits %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% pdfs NLO

\begin{figure}[ht] 
\centering
\epsfig{file=H1prelim-02-012.fig14.eps,width=1.0\linewidth}
\caption{Diffractive parton densities obtained from the QCD fit,
  normalised such that the `pomeron
  flux' is unity at $\xpom = 0.003$. The left hand side shows the
  singlet quark distribution ($6 * u$ where we assume $u = d = s =
  \bar{u} = \bar{d} = \bar{s}$). The right hand side shows the gluon
  density.  The red bands show the results of the NLO fits, with inner
  error bands showing the experimental errors (statistical and
  systematic) and the outer errors bands showing the full
  uncertainties, including those arising from theoretical assumptions.
  For comparison, the central values of the parton densities extracted
  from the LO fit are also shown (blue line). 
}
\label{partons1}
\end{figure}

\begin{figure}[ht] 
\centering
\epsfig{file=H1prelim-02-012.fig15.eps,width=1.0\linewidth}
\caption{The same plot as figure~\ref{partons1}, but on a logarithmic
$z$ scale.}
\label{partons2}
\end{figure}

% gluon fraction

\begin{figure}[ht] 
\centering
\epsfig{file=H1prelim-02-012.fig16.eps,width=0.85\linewidth}
\caption{The fraction of the overall diffractive exchange momentum
  carried by gluons in the NLO fits, integrated over the $z$ range
  measured and shown as a function of $Q^2$. The error bands again
  reflect the experimental (inner) and combined experimental and
  theoretical (outer) uncertainties. 
%The result is fully consistent with that quoted in \cite{h1f2d94}.
}
\label{gluefrac}
\end{figure}

% LO fits and 94 pdfs

\begin{figure}[ht] 
\centering
\epsfig{file=H1prelim-02-012.fig17.eps,width=1.0\linewidth}
\caption{Comparison of the central values of the parton densities from the 
  LO fit compared with the results of the two fits including glouns at
  the starting scale made to the 1994 H1 data \cite{h1f2d94}. The
  agreement in the singlet quark density is reasonable for $z < 0.65$
  as used in the fits. The shape of the gluon is fairly similar to
  1994 fit 3 (`peaked' gluon), except that the peak at highest $z$ is
  significantly smaller. The normalisation of the gluon is different
  by about 30\% for low-medium $z$, a difference which would be inside
  the combined errors on the two extractions.}
\label{partons3}
\end{figure}


% FLD

\begin{figure}[ht]
\centering
\epsfig{file=H1prelim-02-012.fig18a.eps,width=0.49\linewidth}
\epsfig{file=H1prelim-02-012.fig18b.eps,width=0.49\linewidth}
\caption{The leading twist component 
of $F_L^D$ as obtained from the
  NLO QCD fit compared with its maximum possible value of $F_2^D$. The
  values of $F_L^D$ are comparatively large, since they are closely
  related to the (dominant) gluon density at NLO.}
\label{fl}
\end{figure}

% Final states comparisons %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% COMBINED JETS AND DSTAR

\begin{figure}[ht]
\centering
\epsfig{file=H1prelim-02-012.fig19a.eps,width=0.49\linewidth}
\epsfig{file=H1prelim-02-012.fig19b.eps,width=0.49\linewidth}
\caption{
H1 measurements of diffractive DIS dijet (\cite{h1jets}, {\em left}) and
$D^*$ meson  (\cite{h1dstar}, {\em right}) hadron level 
cross sections, differential in $z_\pom$, compared with the present
LO QCD fit as described in section \ref{fits} as well
as the fits to the previous data from \cite{h1f2d94} 
('H1 fit 2' and 'H1 fit 3'). The comparisons are obtained
using the RAPGAP program and for the scale  set to $\mu^2=Q^2+p_T^2+m_q^2$.
 }
\label{jetdstarzpomcomp}
\end{figure}


% CDF plot -----------------------------------------------------------------

\begin{figure}[ht]
\centering
\epsfig{file=H1prelim-02-012.fig20.eps,width=0.8\linewidth,clip=}
\caption{The quantity $\tilde{F}_{JJ}^D (\beta)$ as extracted 
  from diffractive dijet data by the CDF collaboration
  \cite{CDF:jets}, compared with the predictions of the new and old
  (\cite{h1f2d94}) LO QCD fits. 
  Assuming Regge and QCD hard
  scattering factorisation, $\tilde{F}_{JJ}^D (\beta)$ corresponds to
  a convolution of the diffractive (and sub-leading) exchange parton
  densities with the appropriate flux factors. 
%The QCD hard scattering
%  factorisation theorem is know to be invalid for diffractive
%  hadron-hadron scattering. 
  The new fits are slightly closer to the
  data at large $\beta$, but a large discrepancy of approximately one
  order of magnitude remains. 
%This is often interpreted as being due
%  to the destruction of rapidity gaps due to spectator interactions
%  (absorptive corrections). 
In contrast to the comparison with the
  fits to 1994 data, the rapidity gap survival probability appears to
  be approximately constant over most of the $\beta$ range. The
  predicted contribution of sub-leading exchanges is approximately
  50\% at low $\beta$, somewhat smaller than that from the old QCD fits.}
\label{cdf}
\end{figure}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
\clearpage
\pagebreak

\appendix
\setcounter{figure}{0}
\renewcommand{\thefigure}{\thesection\arabic{figure}}
 
\section{Additional Figures}

% Detailed plots %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{figure}[ht] 
\centering
\epsfig{file=H1prelim-02-012.figA1a.eps,angle=90,width=0.7\linewidth,clip=}
\epsfig{file=H1prelim-02-012.figA1b.eps,angle=90,width=0.7\linewidth,clip=}
\caption{Reduced diffractive cross section data for fixed 
  $\xpom = 0.0003, 0.001, 0.003, 0.01,0.03$ and fixed $x$ or $\beta$, 
shown as a
  function of $Q^2$ and compared with the results of the NLO QCD fit
  (experimental errors only). The filled data points were included in
  the fit. The open data points ($\mx < 2 \ {\rm GeV}$) were omitted.
  In addition to the present measurement, H1 preliminary data at higher
  $Q^2$ \cite{f2dhiq2} are shown.
  The predictions of the NLO fit for $F_2^D$ (i.e. $\sigma_r^D$ for
  $F_L^D = 0$) are also shown, as well as the contribution to
  the predicted reduced cross section from the leading diffractive exchange
  alone (the remainder being assigned to sub-leading
  meson exchange).
%  Rising scaling violations are observed
%  except at the highest $\beta$. 
}
\label{detailq2}
\end{figure}

%\begin{figure}[ht] 
%\centering
%\epsfig{file=H1prelim-02-012.fig1.eps,angle=0,width=0.9\linewidth,clip=}
%\caption{Reduced diffractive cross section data for fixed 
%  $\xpom = 0.0003, 0.001, 0.003$ and fixed $x$, shown as a function of
%  $Q^2$ and compared with the results of the NLO QCD fit (experimental
%  errors only). The filled data points were included in the fit. The
%  open data points ($\mx < 2 \ {\rm GeV}$) were omitted. The
%  predictions of the NLO fit for $F_2^D$ (i.e. $\sigma_r^D$ for $F_L^D
%  = 0$) are also shown. Rising scaling violations are observed except
%  at the highest $\beta$.}
%\label{detail1}
%\end{figure}

%\begin{figure}[ht] 
%\centering
%\epsfig{file=H1prelim-02-012.fig2.eps,angle=0,width=0.9\linewidth,clip=}
%\caption{Reduced diffractive cross section data for fixed 
%  $\xpom = 0.01, 0.03$ and fixed $x$, shown as a function of $Q^2$ and
%  compared with the results of the NLO QCD fit. The contribution to
%  the predicted reduced cross section from the leading `pomeron' term
%  alone is also shown (the remainder being assigned to sub-leading
%  meson exchange).}
%\label{detail2}
%\end{figure}


\begin{figure}[ht] 
\centering
\epsfig{file=H1prelim-02-012.figA2a.eps,angle=90,width=0.8\linewidth,clip=}
\epsfig{file=H1prelim-02-012.figA2b.eps,angle=90,width=0.8\linewidth,clip=}
\caption{Reduced diffractive cross section data for fixed 
  $\xpom = 0.0003, 0.001, 0.003,0.01,0.03$ and fixed $Q^2$, shown as a function
  of $x$ and compared with the results of the NLO QCD fit. 
  The shape of the distributions at fixed $Q^2$ varies with $\xpom$, since
  different regions of $\beta$ are probed for the different $\xpom$
  values. Note that the cross section is constrained to tend to zero
  as $x \rightarrow \xpom$ ($\beta \rightarrow 1$).}
\label{detailx}
\end{figure}


%\begin{figure}[ht] 
%\centering
%\epsfig{file=H1prelim-02-012.fig3.eps,angle=0,width=0.9\linewidth,clip=}
%\caption{Reduced diffractive cross section data for fixed 
%  $\xpom = 0.0003, 0.001, 0.003$ and fixed $Q^2$, shown as a function
%  of $x$ and compared with the results of the NLO QCD fit. The shape
%  of the distributions at fixed $Q^2$ varies with $\xpom$, since
%  different regions of $\beta$ are probed for the different $\xpom$
%  values. Note that the cross section is constrained to tend to zero
%  as $x \rightarrow \xpom$ ($\beta \rightarrow 1$).}
%\label{detail3}
%\end{figure}

%\begin{figure}[ht] 
%\centering
%\epsfig{file=H1prelim-02-012.fig4.eps,angle=0,width=0.9\linewidth,clip=}
%\caption{Reduced diffractive cross section data for fixed 
%  $\xpom = 0.01, 0.03$ and fixed $Q^2$, shown as a function of $x$ and
%  compared with the results of the NLO QCD fit.}
%\label{detail4}
%\end{figure}


\begin{figure}[ht] 
\centering
\epsfig{file=H1prelim-02-012.figA3a.eps,angle=90,width=0.8\linewidth,clip=}
\epsfig{file=H1prelim-02-012.figA3b.eps,angle=90,width=0.8\linewidth,clip=}
\caption{Reduced diffractive cross section data for fixed 
  $\xpom = 0.0003, 0.001, 0.003, 0.01, 0.03$ 
and fixed $Q^2$, shown as a function
  of $\beta$ and compared with the results of the NLO QCD fit. The
  shape of the distributions at fixed $Q^2$ is similar for different
  $\xpom$ values.}
\label{detailbeta}
\end{figure}

%\begin{figure}[ht] 
%\centering
%\epsfig{file=H1prelim-02-012.fig5.eps,angle=0,width=0.9\linewidth,clip=}
%\caption{Reduced diffractive cross section data for fixed 
%  $\xpom = 0.0003, 0.001, 0.003$ and fixed $Q^2$, shown as a function
%  of $\beta$ and compared with the results of the NLO QCD fit. The
%  shape of the distributions at fixed $Q^2$ is similar for different
%  $\xpom$ values.}
%\label{detail5}
%\end{figure}

%\begin{figure}[ht] 
%\centering
%\epsfig{file=H1prelim-02-012.fig6.eps,angle=0,width=0.9\linewidth,clip=}
%\caption{Reduced diffractive cross section data for fixed 
%  $\xpom = 0.01, 0.03$ and fixed $Q^2$, shown as a function of $\beta$
%  and compared with the results of the NLO QCD fit.}
%\label{detail6}
%\end{figure}

% sigma_r summary %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\begin{figure}[ht]
\centering
\epsfig{file=H1prelim-02-012.figA4.eps,bbllx=0,bblly=20,bburx=568,bbury=568,
width=1.1\linewidth,clip=}
\caption{$Q^2$ dependence of the reduced cross section at fixed
  $\xpom = 0.0003, 0.001, 0.003, 0.01, 0.03$ and fixed $\beta$ or $x$,
  compared with the predictions of the NLO QCD fit.
  The filled data points were included in
  the fit. The open data points ($\mx < 2 \ {\rm GeV}$) were omitted.
  The predictions of the NLO fit for $F_2^D$ (i.e. $\sigma_r^D$ for
  $F_L^D = 0$) are also shown.
}
\label{q2f2d}
\end{figure}

\begin{figure}[ht]
\centering
\epsfig{file=H1prelim-02-012.figA5.eps,bbllx=0,bblly=50,bburx=568,bbury=568,
width=1.1\linewidth,clip=}
\caption{$x$ dependence of the reduced cross section at fixed
  $\xpom = 0.0003, 0.001, 0.003, 0.01, 0.03$ and fixed $Q^2$, compared with the
  predictions of the NLO QCD fit.}
\label{xf2d}
\end{figure}

\begin{figure}[ht]
\centering
%\epsfig{file=H1prelim-02-012.fig8.eps,width=1.0\linewidth,clip=}
\epsfig{file=H1prelim-02-012.figA6.eps,bbllx=0,bblly=50,bburx=568,bbury=568,
width=1.1\linewidth,clip=}
\caption{$\beta$ dependence of the reduced cross section at fixed
  $\xpom = 0.0003, 0.001, 0.003, 0.01, 0.03$ and fixed $Q^2$, compared with the
  predictions of the NLO QCD fit.}
\label{betaf2d}
\end{figure}

% Ratio plot

\begin{figure}[ht] 
\centering
\epsfig{file=H1prelim-02-012.figA7.eps,angle=90,width=1.0\linewidth,clip=}
\caption{The ratio of diffractive to the inclusive reduced cross section,
  shown as a function of $Q^2$ in different $\xpom$ and $x$ bins.  The
  results of the fit to a logarithmic $Q^2$ dependence 
are overlaid (figure~\ref{ratioderiv}).
  The $Q^2$ dependence of the ratio is small except at the highest
  $\beta$ and varies little with $\xpom$.}
\label{q2ratdetail}
\end{figure}

% high q2 data

\begin{figure}[ht] 
\centering
\epsfig{file=H1prelim-02-012.figA8.eps,width=0.8\linewidth}
\caption{High $Q^2$ data on the diffractive reduced cross section $\sigrd$
%as released for the Vancouver ICHEP conference (1998) 
\cite{f2dhiq2} compared with the 
predictions of the NLO QCD fit. The data were included in the fit.}
\label{vancouver}
\end{figure}


% jets

\begin{figure}[ht]
\centering
\epsfig{file=H1prelim-02-012.figA9.eps,width=0.9\linewidth}
\caption{
  H1 measurements of diffractive dijets \cite{h1jets} compared with
  the new and old \cite{h1f2d94} LO QCD fits.
  The
  dijet cross sections differential in $Q^2$, the jet transverse
  momentum in the $\gamma^* p$ centre of mass frame $p_{_{\rm T,jets}}^*$, 
  $\log_{10} \xpom$, $\log_{10} \beta$, $W$ and
  the mean dijet pseudorapidity in the laboratory
  frame $\langle \eta \rangle_{_{\rm jets}}^{^{\rm lab}}$ are shown.
%  Accounting for the uncertainties in the parton densities, for
%  modelling uncertainties such as the choice of renormalisation /
%  factorisation scales and for higher order QCD effects, the overall
%  normalisation difference between data and the new fits is probably
%  not surprising.  The shapes of the distributions remain well
%  described.  
}
\label{jetscomp1}
\end{figure}


\begin{figure}[ht]
\begin{center}
\epsfig{file=H1prelim-02-012.figA10.eps,width=0.55\linewidth}
\end{center}
\caption{H1 measurements \cite{h1jets}
  of diffractive dijet cross sections as a function of $z_\pom^{\rm
    jets}$ in different intervals of $Q^2 + p_{_{\rm T, jets}}^{* \,
    2}$ ({\em left})  and of $\xpom$ ({\em right}), 
compared with the new and old \cite{h1f2d94}
  LO QCD fits. }
\label{jetscomp2}
\end{figure}

\begin{figure}[ht]
\begin{center}
\epsfig{file=H1prelim-02-012.figA11.eps,width=0.55\linewidth}
\end{center}
\caption{H1 measurements \cite{h1jets}
  of diffractive dijet cross sections differential in the fraction
  $x_\gamma^{\rm jets}$ of the virtual photon momentum transferred to
  the dijet system, the energy $E_{\rm rem}^\gamma$ in the photon
  hemisphere reconstructed outside the jets and $\mx$, compared with
  the new and old \cite{h1f2d94} LO QCD fits.}
\label{jetscomp3}
\end{figure}

\begin{figure}[ht]
\begin{center}
\epsfig{file=H1prelim-02-012.figA12.eps,width=0.6\linewidth}
\end{center}
\caption{H1 measurements \cite{h1jets}
  of diffractive dijet cross sections in the restricted region $\xpom
  < 0.01$, compared with the new and old \cite{h1f2d94} LO QCD fits.
  The cross sections are shown differentially in
  $Q^2$, $p_{_{\rm T, jets}}^*$, $z_\pom$ and the transverse momentum
  $p_{\rm T, rem}^{\pom}$ in the `pomeron' hemisphere reconstructed
  outside the jets.}
\label{jetscomp4}
\end{figure}

\begin{figure}[ht]
\begin{center}
\epsfig{file=H1prelim-02-012.figA13.eps,width=0.6\linewidth}
\end{center}
\caption{H1 measurements \cite{h1jets}
  of diffractive three-jet cross sections compared with the new and
  old \cite{h1f2d94} LO QCD fits with the same assumptions for the
  Monte Carlo modelling as in \cite{h1jets}. The cross sections are
  shown differentially in the three jet invariant mass $M_{123}$ and
  in the fraction of the exchanged momentum transferred to the
  three-jet system $z_\pom^{\rm 3 jets}$.}
\label{threejets}
\end{figure}

% dstar ----------------------------------------------------------------


\begin{figure}[ht]
\centering
\epsfig{file=H1prelim-02-012.figA14.eps,width=0.9\linewidth}
\caption{
  H1 measurements \cite{h1dstar} of diffractive $D^*$ production cross
  sections compared with the new and old \cite{h1f2d94} LO QCD fits
  with the same assumptions for the Monte Carlo modelling as in
  \cite{h1dstar}.  The cross section is shown integrated over the full
  measured phase space and differentially in $\log_{10} Q^2$, the
transverse momentum of the $D^*$ in the $\gamma^* p$ system, the
pseudorapidity of the $D^*$, $\xpom$ and $\beta$.
%  overall cross section predicted in the new fits is in very good
%  agreement with the dtaa, though the modelling is subject to the same
%  uncertainties as for the dijet cross sections.  The shapes of the
%  differential distributions are well described. 
 }
\label{charmcomp}
\end{figure}



 

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