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\begin{document}

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\begin{titlepage}

%\noindent
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%Submitted to & & &
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%\multicolumn{4}{l}{{\bf
%                10. International Workshop on Deep Inelastic Scattering
%                Conference on High Energy Physics},
%                April 29 - May 5,~2001,~Krakow} \\
% {\bf DIS 2002:} 
%                 & Abstract:        & {\bf 808}    &\\
%                 & Parallel Session & {\bf 2}   &\\[.7em]
%                 & Plenary Session  & {\bf Hebecker, Yoshida}   &\\[.7em]
%\multicolumn{4}{l}{{\bf
%               XX International Symposium on Lepton and Photon Interactions}, 
%               July~23,~2001,~Rome} \\ 
%{\bf LP 2001:}  
%                 & Abstract:        & {\bf 500} &\\
%                 & Plenary Session  & {\bf 8}   &\\ \hline
% & \multicolumn{3}{r}{\footnotesize {\it
%    www-h1.desy.de/h1/www/publications/conf/conf\_list.html}} \\[.2em]
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\begin{center}
{\em Perpared for DIS 2002, Krakow}
\end{center}



\vspace*{1cm}

\begin{center}
\Large
{\bf 
Measurement and NLO DGLAP QCD interpretation of \\
%the Diffractive Structure Function  {\boldmath $\ftwodarg$} \\
Diffractive Deep Inelastic Scattering \\
%at HERA \\ 
}
\vspace*{1cm}
{\Large H1 Collaboration} 
\end{center}
\vspace*{1.5cm}

\begin{abstract}

\noindent

A measurement and DGLAP QCD analysis of the inclusive cross section for
diffractive deep-inelastic scattering
is presented, This corresponds to an extension of the measurement of
$F_2^{D(3)} (\xpom, Q^2, \beta)$ 
presented in H1prelim-01-111 and uses the same data. The data are 
now presented
in the form of a reduced diffractive cross section
$\sigma_r^{D(3)} (\xpom, Q^2, \beta)$. A new binning scheme is used
with fixed values of $\xpom$, $Q^2$ and $x = \beta \cdot \xpom$. These
changes facilitate measurements of the ratio of the diffractive to the
inclusive DIS cross sections 
$\sigma_r^{D(3)} (x, Q^2, \xpom) / \sigma_r (x, Q^2)$,
which is found to be remarkably flat as a function of $Q^2$ with $\xpom$
fixed. A NLO DGLAP QCD fit is performed to the diffractive data and
the resulting diffractive parton densities are shown, together with an
assessment of the experimental and theoretical uncertainties. The resulting
partonic decomposition of the diffractive exchange is 
dominated by the gluon density, which extends to large
fractional momenta. The parton densities are used to make updated 
comparisons with diffractive dijet and open charm cross sections at HERA and 
the Tevatron, thus testing QCD hard scattering factorisation as applied
to diffraction.

\end{abstract}


\end{titlepage}

\pagestyle{plain}

% Basic QCD fit related plots %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Introduction}

This document should be taken together with previous H1 results on the
inclusive diffractive DIS cross section
released for the Budapest conference 
(H1prelim-01-111) \cite{H1:F2Dprel}. The measurements presented use the
same data and cover
essentially the same kinematic range ($Q^2 \geq 6.5 \ {\rm GeV^2}$,
$\beta > 0.01$, $\xpom < 0.05$). There have been several minor adjustments
to the underlying measurement, the only visible changes arising from a 
modified
scattered electron energy cut from $E_e^\prime > 8 \ {\rm GeV}$ to
$E_e^\prime > 6.5 \ {\rm GeV}$.

The presentation has now changed in several ways. Firstly, we now
work with the `reduced diffractive cross section' $\sigma_r^D$ rather
than $F_2^D$, where
\begin{eqnarray*}
  \sigma_r^D = F_2^D - \frac{y^2}{1 + (1 - y)^2} \ F_L^D \ .
\end{eqnarray*}
This is essentially a cosmetic change, since our previous extractions
of $F_2^D$ always assumed that the longitudinal diffractive structure
function $F_L^D = 0$.
However since $F_L^D$ has not yet directly
been constrained by data, it is more 
correct to quote the measurement as a 
reduced cross section. The NLO QCD fits that follow use $\sigma_r^D$
as their input.

A new binning scheme is now used, where we work with fixed $\xpom$ bins
(2 per decade) and then study the ($x, Q^2$) dependence in a binning
scheme very similar to that used for the H1 1996-7 low $Q^2$ $F_2$ 
measurement \cite{H1:F2}. Since 
$\beta = x / \xpom$, $\beta$ is also always fixed when $\xpom$ and
$x$ are fixed. This binning scheme yields more
$\beta$ bins at fixed $x$ and $\xpom$ than has previously been possible. 
It also leads naturally 
to investigation of the QCD ($x, Q^2$) or ($\beta, Q^2$) structure at
fixed $\xpom$ and to measurements of the ratio of diffractive to inclusive 
cross sections.

The data are subjected to a DGLAP QCD analysis in order to extract
diffractive parton densities for comparisons with hadronic 
final state measurements. This exercise tests the QCD hard scattering
factorisation theorem for semi-inclusive DIS \cite{collins}. With the
present precision and kinematic range of the data, it is not yet possible
to perform these fits at fixed $\xpom$ and $t$ as would strictly be required 
by the proof in \cite{collins}. Instead we assume a 
`Regge factorisation' between the ($\xpom, t$) and ($\beta, Q^2$) variables,
as tested in \cite{H1:F2Dprel} and use all available data.

\section{The Reduced Diffractive Cross Section}

Measurements of the 
recuced diffractive cross section $\sigma_r^{D(3)}(\xpom, Q^2, x)$,
multiplied by $\xpom$,
are presented in detail in 
figures~\ref{detail1}-~\ref{vancouver} using the new binning scheme.
Figures~\ref{detail1} and~\ref{detail2} show the $Q^2$ dependence at
fixed values of $\xpom$, $x$ and hence $\beta$. 
Figures~\ref{detail3} and~\ref{detail4} show the $x$ dependence at
fixed $\xpom$ and $Q^2$.
Figures~\ref{detail5} and~\ref{detail6} show the $\beta$ dependence at
fixed $\xpom$ and $Q^2$. In each case, the data are compared with the
results of the NLO DGLAP QCD fit presented in section~\ref{fits}. For
completeness, measurements of $\sigma_r^{D(3)}$ at high $Q^2$ \cite{vancf2d},
which are also used in the QCD fits, are shown in figure~\ref{vancouver}.

\section{Summary Plots}

To compare the data in a single plot, the diffractive reduced cross 
section (no additional $\xpom$ factor) is plotted in fig~\ref{betaf2d}
as a function of $\beta$ for the $\xpom = 0.001, 0.003, 0.01$ bins.
Figure~\ref{xf2d} shows the same thing as a function
of $x$. In figure~\ref{q2f2d}, the $Q^2$ dependence is shown, including
data points from all five $\xpom$ bins. 
In figs~\ref{betaflux} and~\ref{q2flux}, the data points from each
$\xpom$ bin are scaled by the assumed values of the `pomeron flux' in
the QCD fits, 
\begin{eqnarray*}
f_{\pom / p} = (1 / \xpom)^{2 \alphapom(0) - 1} 
\int_{-1 \ {\rm GeV^2}}^{t_{min}} dt \ 
e^{B_{\pom} t} \ ,
\end{eqnarray*}
where the parameters used for the
pomeron intercept $\alphapom(0)$ and the $\xpom$ dependent slope
parameter $B_{\pom}$ are as explained in \cite{H1:F2Dprel}. 
No account is taken of possible sub-leading exchange contributions at
large $\xpom$ and small $\beta$, though these contributions are small
for the data shown ($\xpom \leq 0.01$). In order to avoid
regions which are most likely to be affected by $F_L^D$, only data with
$y < 0.6$ are shown. At LO in QCD, the reduced cross section can then be 
interpreted in the usual way as a charge weighted sum over the diffractive
quark densities. 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 similarity of the flux-normalised reduced cross sections from
different $\xpom$ values in the overlap regions indicates that the Regge
factorisation assumption is a good approximation. 
%(factorisation between the
%$\xpom$ dependent flux factors and the ($\beta, Q^2$) dependence is assumed
%in the fits). 
In all figures~\ref{betaf2d}-~\ref{q2flux}, the results of the NLO DGLAP
fit to the data (section~\ref{fits}) are also shown.

For ($x, \xpom$) bins in which
there are at least 3 data points, the logarithmic $Q^2$ derivative,
$B = \partial \sigma_r^D / \partial \ln Q^2$ is extracted from fits
of the form
\begin{eqnarray}
  \sigma_r^D = A(x, \xpom) + B(x, \xpom) \ln Q^2 \ .
\end{eqnarray}
The results for $B$ from 
these fits are shown in fig~\ref{f2dderiv}. 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. The logarithmic $Q^2$ derivative is sensitive
to the diffractive gluon density, which is required by the data
to remain large up
to high values of $\beta$. The behaviour is once again similar at the
different values of $\xpom$.

\section{Diffractive to Inclusive Cross Section Ratio}

The ratio of $\sigma_r^D$ to the inclusive reduced cross section
$\sigma_r$ is formed by dividing $\sigma_r^D(\xpom, x, Q^2)$ in each $\xpom$
bin by the measured $\sigma_r(x, Q^2)$, as obtained in \cite{H1:F2}.
%the H1 publication of $F_2$ on 1996-7 data (DESY-00-181). 
The ratio as a function of $Q^2$ tests the
difference between the scaling violations of $\sigma_r^D$ and $\sigma_r$
when compared at the same $x$. 
This ratio is shown separately for each $\xpom$ and $x$ bin in 
figure~\ref{q2ratdetail} and in a single plot for all $\xpom$ bins
in figure~\ref{q2rat}. At low values of $\beta$, the ratio is remarkably
flat as a function of $Q^2$ for all $\xpom$ values. At the highest $\beta$,
the ratio falls with increasing $Q^2$. 

In order to quantify the differences between the $Q^2$ dependences
of $\sigma_r^D$ and $\sigma_r$, the logarithmic derivative of the ratio
is obtained, corresponding to parameter $B$ in fits of the form
\begin{eqnarray*}
  \sigma_r^D / \sigma_r = A + B \ln Q^2 \ ,
\end{eqnarray*}
such that
\begin{eqnarray}
  B = \frac{\partial}{\partial \ln Q^2} \ \left( \sigma_r^D / \sigma_r 
\right) \ .
\end{eqnarray}
The fits to the data are shown in figures~\ref{q2ratdetail} 
and~\ref{q2rat}. The resulting values of $B$
are shown in fig~\ref{ratioderiv}. Once again, the `pomeron
flux' at fixed $\xpom$ is divided out, so that the results at different
$\xpom$ can be compared in normalisation as well as shape.

The ratio as a function of $x$ with $Q^2$ and $\xpom$ fixed 
is shown in fig~\ref{xrat}. In each figure, dashed lines indicate the
point 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 
figs~\ref{detail5},~\ref{detail6} and~\ref{betaf2d}). For $\beta < 0.1$,
there is a suggestion that a flatter dependence develops, though there are 
limited data in this region.

It should be noted that the diffractive to inclusive ratios as presented
here
must be interpreted slightly differently to those shown by H1 in 
\cite{H1:F2Dprel} 
as\footnote{Note that $\rho^{D(3)}$ was 
found to vary
little with $W$ (or $x$) with $Q^2$ and $\beta$ (hence $\mx$) 
held fixed in \cite{H1:F2Dprel}.} $\rho^{D(3)}$ and 
by ZEUS \cite{zeus}. In the two former cases, $\beta$ and
$Q^2$ (hence also $\mx$) are held fixed as $x$ (or $W$)
varies. In the present work, $\xpom$ and $Q^2$ are held fixed as $x$
varies, such that $\mx$ also varies with $x$. 

\section{NLO DGLAP QCD Fits}
\label{fits}

Both LO and NLO QCD fits have been performed to $\sigma_r^D$ in the
($\beta, Q^2, \xpom$) binning 
scheme (see \cite{H1:F2Dprel}). It is assumed that the diffractive
parton densities vary only in normalisation with $\xpom$ and $t$ and the
dependence on these variables is
parameterised using the Regge flux factor $f_{\pom / p}$ as in the
`Regge' fits from \cite{H1:F2Dprel}, with $\alphapom(0) = 1.173$.
The massive scheme is used for heavy quarks.
The LO fit is subject to the restrictions $y<0.45$ and $\mx>2 \ {\rm GeV}$
in order to avoid regions that are most likely to be influenced
by $F_L^D$ or where significant higher twist contributions may be present.
For the NLO fit, where the effects of $F_L^D$ are considered through its
relation to the NLO gluon density, the only cut is $\mx > 2 \ {\rm GeV}$. 
No explicit $\beta$ cut is made in either fit.

Using a modified version of the fitting code used for $F_2$ in 
\cite{H1:F2}
(Pascaud, Zomer), it has been possible for the first time to asign
experimental errors to the extracted parton densities, including
a full propagation of all systematic uncertainties considered in
the measurement. In addition, theoretical uncertainties corresponding
to $\Lambda = 200 \pm 30 \ {\rm MeV}$, $m_c = 1.5 \pm 0.1 \ {\rm GeV}$
and all assumed parameters for the pomeron and meson flux 
(uncertainties as in \cite{H1:F2Dprel}) have been evaluated.
High Q2 H1 data points \cite{vancf2d}
are also included in the fits, their systematic uncertainties
taken to be fully uncorrelated. 
%Full details of the fits (H1 internal)
%have been shown by F. Schilling at various H1 meetings. 

The data in the new ($x, Q^2, \xpom$) binning scheme are
compared with the predictions of the NLO QCD fit in 
figs~\ref{detail1}-~\ref{q2flux}. The extracted 
NLO singlet-quark and gluon parton densities, together with their
uncertainties are shown in figures~\ref{partons1}
and~\ref{partons2}. For comparison, the
central result from the LO fit is also shown. In these figures, the variable
$z$ corresponds to the momentum fraction carried by an individual parton
($z = \beta$ for quarks, $z > \beta$ for gluons).
In figure~\ref{partons3}, the results are compared with the results of
a previous H1 LO fit using earlier data \cite{h1f2d94}. 
Figure~\ref{gluefrac} shows the fraction of the total exchanged longitudinal
momentum that is carried by gluons, integrated over the measured kinematic
range $0.01 < z < 1$. This fraction is approximately
$75 \pm 15 \%$, relatively constant with $Q^2$. Figure~\ref{fl} shows the
leading twist component of $F_L^D$ as predicted from the fits through its
relation to the NLO gluon density.
%associated plots are shown in figs~\ref{partons1}
%-~\ref{fl}. 


\section{Comparisons with hadronic Final State Observables}

The partons extracted from the LO QCD fits have
been implemented in the RAPGAP Monte Carlo generator and used to make
predictions for various hadronic final state observables. The results
are shown for the H1 diffractive jets measurements \cite{H1:jets} in
figs~\ref{dijet1}-~\ref{dijetend} and for the H1 diffractive $D^*$
measurements \cite{H1:charm} in figs~\ref{dstar1}-~\ref{dstarend}. 
The new predictions based on $\sigma_r^D$ are in better agreement
with the charm data was previously the case based on the partons 
from \cite{h1f2d94}. The agreement with the 
dijet data is not as good as that using the partons from \cite{h1f2d94}.
Taking account of the considerable uncertainties in the parton densities,
final state modelling uncertainties and undetermined higher order QCD
effects, the data remain in agreement with hard scattering factorisation
as proven for semi-inclusive DIS \cite{collins}.

A comparison of predictions using the 
extracted partons densities with 
the quantity $\tilde{F}_{JJ}^D (\beta)$ as measured
by the CDF collaboration \cite{CDF:jets} is shown
in figure~\ref{cdf}. The new prediction is slightly closer to the data than
that using the partons extracted in \cite{h1f2d94}, though a discrepancy
at the level of an order of magnitude remains. This has often been
interpreted as evidence for absorptive corrections in diffractive
hadron-hadron scattering (rapidity gap survival probabilities). The
comparison with the newly obtained parton densities suggests that the
gap survival probability may be less dependent on $\beta$ than was previously
thought.

\section{Summary}

The inclusive cross section for diffractive DIS has been presented in a
modified way, allowing closer inspection of the $\beta$ and $Q^2$ dependences
at fixed values of the fractional loss of longitudinal momentum of the
proton during the interaction, $1 - \xpom$. 
The data are presented in ways that are directly sensitive to the shapes
of the diffractive singlet-quark and gluon densities. These illustrations
confirm the need for a dominant gluon density in diffration, extending to
large fractional momenta. 

The $x$ and $Q^2$ dependences of the ratio of the diffractive to the 
inclusive DIS cross sections has been presented for fixed $\xpom$
values. The $Q^2$ dependence of this ratio is remarkably small over a
large kinematic region.

New parton densities describing diffractive DIS have been extracted from
NLO DGLAP QCD fits to the data. For the first time, it has been
possible to ascribe experimental and theoretical errors to these parton
densities. The results are broadly in line with a previous result from
H1, though the magnitude of the gluon density is somewhat smaller in the
new version. Approximately 75\% of the total exchanged momentum is
carried by gluons. 

The parton densities have been used to make updated
comparisons with hadronic final state data assuming QCD hard scattering
factorisation. The predictions for lepton
proton scattering remain consistent with the final state measurements, 
whilst large additional gap survival probabilities continue to be necessary
to obtain a good description of $p \bar{p}$ data from the Tevatron.

\begin{thebibliography}{9}

\bibitem{H1:F2Dprel} H1 Collaboration, {\em 'Measurement of the diffractive 
structure function $F_2^{D(3)}$'}. Paper submitted to International 
Europhysics Conference on High Energy Physics EPS01,
Budapest, July 2001 (abstract 808).
Available from http://www-h1.desy.de/h1/www/publications/htmlsplit/H1prelim-01-111.long.html

\bibitem{H1:F2}
H1 Collaboration, C.~Adloff {\em et al.}, \Journal{\EJC}{21}{2001}{33}.

\bibitem{collins} 
J.~Collins, \Journal{\PRD}{57}{1998}{3051}
and erratum-ibid. {\bf D 61} (2000) 019902.

\bibitem{vancf2d} H1 Collaboration, {\em Measurement and 
Interpretation of the Diffractive Structure Function 
$F_2^{D(3)}$ at HERA}. Paper submitted to International Conference on High
Energy Physics ICHEP98, Vancouver, July 1998 (abstract 571).

\bibitem{zeus} ZEUS Collaboration, J.~Breitweg {\em et al.},
\Journal{\EJC}{6}{1999}{43}. \\
ZEUS Collaboration, S. Chekanov et al., 
DESY-02-029, submitted to Eur. Phys. Journal C.

\bibitem{h1f2d94} H1 Collaboration, C.~Adloff {\em et al.}, 
\Journal{\ZPC}{76}{1997}{613}.

\bibitem{H1:jets} 
H1 Collaboration, C.~Adloff {\em et al.}, \Journal{\EJC}{20}{2001}{29}.

\bibitem{H1:charm}
H1 Collaboration, C.~Adloff {\em et al.}, \Journal{\PLB}{520}{2001}{191}.

\bibitem{CDF:jets} 
CDF Collaboration,  T.~Affolder {\em et al.}, \Journal{\PRL}{84}{2000}{5043}.

\end{thebibliography}

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

% Data compared with fit %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{figure}[h] 
\begin{center}
%\epsfig{file=qcdfit.q2dep.loxpom.eps,angle=0,width=0.9\linewidth,clip=}
\epsfig{file=H1prelim-02-012.fig1.eps,angle=0,width=0.9\linewidth,clip=}
\end{center}
\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}[h] 
\begin{center}
%\epsfig{file=qcdfit.q2dep.hixpom.eps,angle=0,width=0.9\linewidth,clip=}
\epsfig{file=H1prelim-02-012.fig2.eps,angle=0,width=0.9\linewidth,clip=}
\end{center}
\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 asigned to sub-leading meson exchange).}
\label{detail2}
\end{figure}

\begin{figure}[h] 
\begin{center}
%\epsfig{file=qcdfit.xbjdep.loxpom.eps,angle=0,width=0.9\linewidth,clip=}
\epsfig{file=H1prelim-02-012.fig3.eps,angle=0,width=0.9\linewidth,clip=}
\end{center}
\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}[h] 
\begin{center}
%\epsfig{file=qcdfit.xbjdep.hixpom.eps,angle=0,width=0.9\linewidth,clip=}
\epsfig{file=H1prelim-02-012.fig4.eps,angle=0,width=0.9\linewidth,clip=}
\end{center}
\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}[h] 
\begin{center}
%\epsfig{file=qcdfit.betadep.loxpom.eps,angle=0,width=0.9\linewidth,clip=}
\epsfig{file=H1prelim-02-012.fig5.eps,angle=0,width=0.9\linewidth,clip=}
\end{center}
\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}[h] 
\begin{center}
%\epsfig{file=qcdfit.betadep.hixpom.eps,angle=0,width=0.9\linewidth,clip=}
\epsfig{file=H1prelim-02-012.fig6.eps,angle=0,width=0.9\linewidth,clip=}
\end{center}
\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}

\begin{figure}[h] 
\begin{center}
%\epsfig{file=H1prelim-02-012.fig5.eps,width=0.6\linewidth}
\epsfig{file=H1prelim-02-012.fig7.eps,width=0.6\linewidth}
\end{center}
\caption{High $Q^2$ data as released for the Vancouver ICHEP conference 
(1998) \cite{vancf2d} compared with the predictions of the QCD fit.}
\label{vancouver}
\end{figure}

\clearpage
\pagebreak

\begin{figure}[h]
\begin{center}
%\epsfig{file=H1prelim-02-012.fig17.eps,width=1.0\linewidth,clip=}
%\epsfig{file=allinone.sigma.beta.zoom.eps,width=1.0\linewidth,clip=}
\epsfig{file=H1prelim-02-012.fig8.eps,width=1.0\linewidth,clip=}
\end{center}
\caption{$\beta$ dependence of the reduced cross section at fixed
$\xpom = 0.001, 0.003, 0.01$ and fixed $Q^2$, compared with the predictions
of the NLO QCD fit.}
\label{betaf2d}
\end{figure}

 
\clearpage
\pagebreak

\begin{figure}[h]
\begin{center}
%\epsfig{file=H1prelim-02-012.fig19.eps,width=1.0\linewidth,clip=}
%\epsfig{file=allinone.sigma.xbj.zoom.eps,width=1.0\linewidth,clip=}
\epsfig{file=H1prelim-02-012.fig9.eps,width=1.0\linewidth,clip=}
\end{center}
\caption{$x$ dependence of the reduced cross section at fixed
$\xpom = 0.001, 0.003, 0.01$ and fixed $Q^2$, compared with the predictions
of the NLO QCD fit.}
\label{xf2d}
\end{figure}

\clearpage
\pagebreak

\begin{figure}[h]
\begin{center}
%\epsfig{file=H1prelim-02-012.fig20.eps,width=1.0\linewidth,clip=}
%\epsfig{file=allinone.sigma.q2.eps,width=1.0\linewidth,clip=}
\epsfig{file=H1prelim-02-012.fig10.eps,width=1.0\linewidth,clip=}
\end{center}
\caption{$Q^2$ dependence of the reduced cross section at fixed
$\xpom = 0.0003, 0.001, 0.003, 0.01, 0.03$ and fixed $x$, compared with 
the predictions
of the NLO QCD fit.}
\label{q2f2d}
\end{figure}

\begin{figure}[h]
\begin{center}
%\epsfig{file=allinone.sigma.beta.flux.liny.eps,width=1.0\linewidth,clip=}
%\epsfig{file=allinone.sigma.beta.flux.eps,width=1.0\linewidth,clip=}
\epsfig{file=H1prelim-02-012.fig11.eps,width=1.0\linewidth,clip=}
\end{center}
\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. Only
data with $y < 0.6$ are shown. The data are compared
with the prediction of the NLO QCD fits (experimental errors only). 
Since the different $\xpom$ bins are sensitive to different 
regions of $\beta$, comparing $\xpom$ bins in this way 
shows the shape of the singlet
density extracted in the fits directly from the data. Although the
pomeron flux factor is used here primarily as a convenient parameterisation
of the $\xpom$ dependence, the level of agreement between the data and
the QCD fit also indicates the extent to which the Regge factorisation
hypothesis is valid.}
\label{betaflux}
\end{figure}

\begin{figure}[h]
\begin{center}
%\epsfig{file=allinone.sigma.q2.flux.liny.eps,width=1.0\linewidth,clip=}
%\epsfig{file=allinone.sigma.q2.flux.eps,width=1.0\linewidth,clip=}
\epsfig{file=H1prelim-02-012.fig12.eps,width=1.0\linewidth,clip=}
\end{center}
\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. The data are compared
with the prediction of the NLO QCD fits (experimental errors only).}
\label{q2flux}
\end{figure}

\begin{figure}[h]
\begin{center}
%\epsfig{file=f2d.q2derivs.final.beta.eps,width=0.8\linewidth}
\epsfig{file=H1prelim-02-012.fig13.eps,width=0.8\linewidth}
\end{center}
\caption{Logarithmic $Q^2$ derivatives of $\sigma_r^D$ for different
$\beta$ and $\xpom$, scaled 
at each $\xpom$ by the
values assumed for the 
$t$-integrated pomeron flux in the QCD fits. The plot shows that the
scaling violations of $\sigma_r^D$ remain positive up to high values of
$\beta \sim 0.7$, but become negative at the highest $\beta$. The scaling
violations at fixed $\beta$ are very similar at the different $\xpom$ values,
as expected in Regge factorisation models. The logarithmic $Q^2$ dependence
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 there is no evidence for
changes in the results with $\xpom$, which might be expected since the
difference $\xpom$ bins cover different $Q^2$ regions.}
\label{f2dderiv}
\end{figure}

\clearpage
\pagebreak

% ratios %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\begin{figure}[h] 
%\begin{center}
%\epsfig{file=ratio.betadep.eps,angle=0,width=0.9\linewidth,clip=}
%\end{center}
%\caption{The ratio of diffractive to inclusive reduced cross sections,
%shown as a function of $\beta$. }
%\label{betarat}
%\end{figure}

\begin{figure}[h] 
\begin{center}
%\epsfig{file=ratio.q2dep.eps,angle=0,width=0.8\linewidth,clip=}
\epsfig{file=H1prelim-02-012.fig14.eps,angle=0,width=0.8\linewidth,clip=}
\end{center}
\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. The $Q^2$ dependence of the ratio is
small except at the highest $\beta$ and varies little with $\xpom$.}
\label{q2ratdetail}
\end{figure}

\begin{figure}[h]
\begin{center}
%\epsfig{file=H1prelim-02-012.fig21.eps,width=1.0\linewidth,clip=}
%\epsfig{file=allinone.ratio.q2.eps,width=1.0\linewidth,clip=}
\epsfig{file=H1prelim-02-012.fig15.eps,width=1.0\linewidth,clip=}
\end{center}
\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. The $Q^2$ dependence of the ratio is
small for most $\beta$ and varies little with $\xpom$.}
\label{q2rat}
\end{figure}


\begin{figure}[h]
\begin{center}
%\epsfig{file=ratio.q2derivs.final.x.eps,width=0.8\linewidth}
\epsfig{file=H1prelim-02-012.fig16.eps,width=0.8\linewidth}
\end{center}
\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 `pomeron' flux. 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}

\begin{figure}[h] 
\begin{center}
%\epsfig{file=ratio.xbjdep.eps,angle=0,width=0.725\linewidth,clip=}
\epsfig{file=H1prelim-02-012.fig17.eps,angle=0,width=0.725\linewidth,clip=}
\end{center}
\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 $\beta = 1$
($\beta = 0.1$).
%Due to this simultaneous
%variation of $\mx$, the ratio as a function of $x$ or $\beta$ is not
%easily interpreted. 
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 the structure
of $\sigma_r^D$ is rather complicated 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}

% QCD fits

\clearpage
\pagebreak

\begin{figure}[h] 
\begin{center}
%\epsfig{file=H1prelim-02-012.fig1.eps,width=1.0\linewidth}
\epsfig{file=H1prelim-02-012.fig18.eps,width=1.0\linewidth}
\end{center}
\caption{Diffractive parton densities 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).
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 uncertainties
in this region remain large. At smaller $z$, the relative size of the 
uncertainties is significantly reduced.}
\label{partons1}
\end{figure}

\clearpage
\pagebreak
 

\begin{figure}[h] 
\begin{center}
%\epsfig{file=H1prelim-02-012.fig2.eps,width=1.0\linewidth}
\epsfig{file=H1prelim-02-012.fig19.eps,width=1.0\linewidth}
\end{center}
\caption{The same as figure \protect \ref{partons1} \protect, but
on a logarithmic $z$ scale.}
\label{partons2}
\end{figure}

\clearpage
\pagebreak
 

\begin{figure}[h] 
\begin{center}
%\epsfig{file=H1prelim-02-012.fig3.eps,width=1.0\linewidth}
\epsfig{file=H1prelim-02-012.fig20.eps,width=1.0\linewidth}
\end{center}
\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 significnatly 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}

\clearpage
\pagebreak

\begin{figure}[h] 
\begin{center}
%\epsfig{file=H1prelim-02-012.fig4.eps,width=0.6\linewidth}
\epsfig{file=H1prelim-02-012.fig21.eps,width=0.6\linewidth}
\end{center}
\caption{The fraction of the overall exchanged (`pomeron') 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}

\begin{figure}[h]
\begin{center}
%\epsfig{file=H1prelim-02-012.fig6a.eps,width=0.49\linewidth}
%\epsfig{file=H1prelim-02-012.fig6b.eps,width=0.49\linewidth}
\epsfig{file=H1prelim-02-012.fig22a.eps,width=0.49\linewidth}
\epsfig{file=H1prelim-02-012.fig22b.eps,width=0.49\linewidth}
\end{center}
\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}

\clearpage
\pagebreak



 
% jets comparisons %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{figure}[h]
\begin{center}
%\epsfig{file=H1prelim-02-012.fig7.eps,width=0.7\linewidth} \\
\epsfig{file=H1prelim-02-012.fig23.eps,width=0.7\linewidth} \\
\end{center}
\caption{H1 measurements of diffractive dijets \cite{H1:jets}
compared with the
new and old (\cite{h1f2d94})
LO QCD fits with the same assumptions for the Monte Carlo
modelling as in \cite{H1:jets}. 
The dijet cross sections differential in $Q^2$, the jet transverse momentum
in the $\gamma^* p$ centre of mass frame ($p_{_{\rm T, jets}}^*$), 
$W$ and the jet 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{dijet1}
\end{figure}

 


\begin{figure}[h]
\begin{center}
%\epsfig{file=H1prelim-02-012.fig8.eps,width=0.7\linewidth}
\epsfig{file=H1prelim-02-012.fig24.eps,width=0.7\linewidth}
\end{center}
\caption{H1 measurements \cite{H1:jets} of diffractive dijet cross sections
differential in the diffractive variables $\log \xpom$,
$\log \beta$ and the fraction $z_\pom^{\rm jets}$ of the exchanged momentum
transferred to the dijet system, compared with the
new and old (\cite{h1f2d94})
LO QCD fits with the same assumptions for the Monte Carlo
modelling as in \cite{H1:jets}.}
\end{figure}

 
\clearpage
\pagebreak


\begin{figure}[h]
\begin{center}
%\epsfig{file=H1prelim-02-012.fig9.eps,width=0.5\linewidth}
\epsfig{file=H1prelim-02-012.fig25.eps,width=0.5\linewidth}
\end{center}
\caption{H1 measurements \cite{H1:jets}
of diffractive dijet cross sections as a function of
$z_\pom^{\rm jets}$ in different intervals of 
$Q^2 + p_{_{\rm T, jets}}^{* \, 2}$ and of $\xpom$,
compared with the
new and old (\cite{h1f2d94})
LO QCD fits with the same assumptions for the Monte Carlo
modelling as in \cite{H1:jets}. }
\end{figure}

 


\begin{figure}[h]
\begin{center}
%\epsfig{file=H1prelim-02-012.fig10.eps,width=0.5\linewidth}
\epsfig{file=H1prelim-02-012.fig26.eps,width=0.5\linewidth}
\end{center}
\caption{H1 measurements \cite{H1:jets}
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 with the same assumptions for the Monte Carlo
modelling as in \cite{H1:jets}.}
\end{figure}


 \clearpage
\pagebreak



\begin{figure}[h]
\begin{center}
%\epsfig{file=H1prelim-02-012.fig11.eps,width=0.5\linewidth}
\epsfig{file=H1prelim-02-012.fig27.eps,width=0.5\linewidth}
\end{center}
\caption{H1 measurements \cite{H1:jets}
of diffractive dijet cross sections in the restricted region $\xpom < 0.01$, 
compared with the
new and old (\cite{h1f2d94})
LO QCD fits with the same assumptions for the Monte Carlo
modelling as in \cite{H1:jets}. The cross sections are shown differentially
in $Q^2$, $p_{_{\rm T, jets}}^*$, $\zpomj$ and the
transverse momentum $p_{\rm T, rem}^{\pom}$ in the `pomeron' hemisphere
reconstructed outside the jets.}
\end{figure}


\begin{figure}[h]
\begin{center}
%\epsfig{file=H1prelim-02-012.fig12.eps,width=0.5\linewidth}
\epsfig{file=H1prelim-02-012.fig28.eps,width=0.5\linewidth}
\end{center}
\caption{H1 measurements \cite{H1:jets}
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 DESY-00-174. 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 $\zpom^{\rm 3 jets}$.}
\label{dijetend}
\end{figure}

 
\clearpage
\pagebreak


% charm comparisons %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{figure}[h]
\begin{center}
%\epsfig{file=H1prelim-02-012.fig13.eps,width=0.5\linewidth}
\epsfig{file=H1prelim-02-012.fig29.eps,width=0.5\linewidth}
\end{center}
\caption{H1 measurements \cite{H1:charm}
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{H1:charm}. 
The cross section is shown integrated over the full measured phase space and
differentially in $\log Q^2$, $\xpom$ and an estimator of the incoming gluon
momentum fraction, $\zpom$. 
The 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{dstar1}
\end{figure}

 


\begin{figure}[h]
\begin{center}
%\epsfig{file=H1prelim-02-012.fig14.eps,width=0.5\linewidth}
\epsfig{file=H1prelim-02-012.fig30.eps,width=0.5\linewidth}
\end{center}
\caption{H1 measurements \cite{H1:charm} of diffractive $D^*$ 
cross sections differential in $\log \beta$, 
the $D^*$ transverse momentum
in the $\gamma^* p$ centre of mass frame ($p_{_{\rm T, D^*}}^*$)
and the $D^*$ pseudorapidity in the laboratory frame $\eta_{D^*}$.
The results are compared with the new and old (\cite{h1f2d94})
LO QCD fits with the same assumptions 
for the Monte Carlo modelling as in \cite{H1:charm}.}
\label{dstarend}
\end{figure}

 
\clearpage
\pagebreak


% cdf plot %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{figure}[h]
\begin{center}
%\epsfig{file=H1prelim-02-012.fig15.eps,width=0.7\linewidth,clip=}
\epsfig{file=H1prelim-02-012.fig31.eps,width=0.7\linewidth,clip=}
\end{center}
\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 `pomeron'
(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 1994 fits.}
\label{cdf}
\end{figure}

 
\clearpage
\pagebreak




% combined plots %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\begin{figure}[h]
%\begin{center}
%\epsfig{file=H1prelim-02-012.fig16.eps,width=1.0\linewidth,clip=}
%\end{center}
%\caption{Beta dependence with QCD fit.}
%\end{figure}

 
%\clearpage
%\pagebreak

%\begin{figure}[h]
%\begin{center}
%\epsfig{file=H1prelim-02-012.fig18.eps,width=1.0\linewidth,clip=}
%\end{center}
%\caption{x dependence with QCD fit.}
%\end{figure}

%\clearpage
%\pagebreak





\end{document}
