%================================================================
% LaTeX file with preferred layout for the contributed papers to
% the ICHEP Conference 98 in Vancouver
% process with:  latex hep98.tex
%                dvips -D600 hep98
%================================================================

\documentclass[12pt]{article}
\usepackage{epsfig}
\usepackage{amsmath}
\usepackage{hhline}
\usepackage{amssymb}
\usepackage{times}
\usepackage{cite}

\renewcommand{\topfraction}{1.0}
\renewcommand{\bottomfraction}{1.0}
\renewcommand{\textfraction}{0.0}
\newlength{\dinwidth}
\newlength{\dinmargin}
\setlength{\dinwidth}{21.0cm}
\textheight23.5cm \textwidth16.0cm
\setlength{\dinmargin}{\dinwidth}
\setlength{\unitlength}{1mm}
\addtolength{\dinmargin}{-\textwidth}
\setlength{\dinmargin}{0.5\dinmargin}
\oddsidemargin -1.0in
\addtolength{\oddsidemargin}{\dinmargin}
\setlength{\evensidemargin}{\oddsidemargin}
\setlength{\marginparwidth}{0.9\dinmargin}
\marginparsep 8pt \marginparpush 5pt
\topmargin -42pt
\headheight 12pt
\headsep 30pt \footskip 24pt
\parskip 3mm plus 2mm minus 2mm

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

% Some useful tex commands

\newcommand{\GeV}{\rm GeV}
\newcommand{\TeV}{\rm TeV}
\newcommand{\pb}{\rm pb}
\newcommand{\cm}{\rm cm}
\newcommand{\hdick}{\noalign{\hrule height1.4pt}}

% Journal macros
\def\Journal#1#2#3#4{{#1} {\bf #2} (#3) #4}
\def\NCA{\em Nuovo Cim.}
\def\NIM{\em Nucl. Instrum. Methods}
\def\NIMA{{\em Nucl. Instrum. Methods} {\bf A}}
\def\NPB{{\em Nucl. Phys.}   {\bf B}}
\def\PLB{{\em Phys. Lett.}   {\bf B}}
\def\PRL{\em Phys. Rev. Lett.}
\def\PRD{{\em Phys. Rev.}    {\bf D}}
\def\ZPC{{\em Z. Phys.}      {\bf C}}
\def\EJC{{\em Eur. Phys. J.} {\bf C}}
\def\CPC{\em Comp. Phys. Commun.}
\def\PR{{\em Phys. Rev.}}
\def\PL{{\em Phys. Lett.}}
\def\RMP{{\em Rev. Mod. Phys.}}
\newcommand{\etal}{{\em et al.}}        

\newcommand{\pom}{{I\!\!P}}
\newcommand{\reg}{{I\!\!R}}
\newcommand{\xpom}{x_\pom}
\newcommand{\ftwod}{F_2^{D(3)}}
\newcommand{\fiidiii}{F_2^{D(3)}}
\newcommand{\ftwodarg}{F_2^{D(3)} (\beta, Q^2, \xpom)}
\newcommand{\av}[1]{\mbox{$ \langle #1 \rangle $}}
\newcommand{\lapprox}{\stackrel{<}{_{\sim}}}
\newcommand{\alphapom}{\alpha_{_{\rm I\!P}}}
\newcommand{\mx}{M_{_{\rm X}}}
\newcommand{\my}{M_{_{\rm Y}}}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\sigrd}{\sigma_r^{D(3)}}
\newcommand{\sigrdarg}{\sigma_r^{D(3)}(\beta, Q^2, \xpom)}

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

\begin{document}

\pagestyle{empty}
\begin{titlepage}

\noindent
\begin{center}
\begin{small}
\begin{tabular}{llrr}
Submitted to & & &
\epsfig{file=/h1/www/images/H1logo_bw_small.epsi
,width=2.cm} \\[.2em] \hline
\multicolumn{4}{l}{{\bf
                32nd International Conference 
                on High Energy Physics, ICHEP04},
                August~16,~2004,~Beijing} \\
                 & Abstract:        & {\bf 6-0176}    &\\
                 & Parallel Session & {\bf 6}   &\\ \hline
 & \multicolumn{3}{r}{\footnotesize {\it
    www-h1.desy.de/h1/www/publications/conf/conf\_list.html}} \\[.2em]
\end{tabular}
\end{small}
\end{center}
\vspace*{2cm}

\begin{center}
\Large {\bf 
Comparison at NLO between  Predictions from QCD Fits to 
\boldmath{$F_2^D$} and Diffractive Final State Observables at HERA
}

\vspace*{1cm}
{\Large H1 Collaboration} 
\end{center}

\begin{abstract}

\noindent
Diffractive parton distributions, obtained from a next-to-leading
order QCD analysis of recent H1 inclusive diffractive deep-inelastic
scattering data, are convoluted with QCD hard scattering matrix
elements to predict cross sections at next-to-leading order for the
diffractive production of jets and charm quarks off virtual photons.
The predictions are compared with H1 measurements of dijet and $D^*$
meson production cross sections in diffractive DIS. Within the
experimental and theoretical uncertainties, the results are consistent
with QCD factorization in diffractive DIS at NLO.


\end{abstract}

\end{titlepage}

\pagestyle{plain}

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

\section{Introduction}

Understanding the phenomenon of diffraction in hadronic interactions
at high energies, where at least one of the beam hadrons remains
intact (or dissociates to a small mass system $Y$ carrying net proton
quantum numbers) and loses only a small fraction $x_\pom$ of its
incident longitudinal momentum, remains one of the most important
challenges in Quantum Chromodynamics (QCD).  The
$ep$ collider HERA provides a unique possibility to study hard
diffractive interactions in deep inelastic scattering (DIS) at large
values of the photon virtuality $Q^2$. 
 
Measurements of inclusive diffractive DIS at HERA have been used to
extract {\em diffractive parton distributions} (diffractive pdf's) of
the proton. If QCD factorization holds for diffractive DIS
\cite{collins}, these diffractive parton distributions
are universal and may be used to predict the cross sections for
exclusive hard diffractive DIS processes such as jet and heavy flavour
production.

Recently, several new precision measurements of inclusive diffractive
DIS have been presented by the H1 Collaboration at low
\cite{h1f2d99mb}, medium \cite{h1f2d97} and high \cite{h1f2dhiq2}
$Q^2$ values. Using the medium $Q^2$ data, and assuming a factorizing
$x_\pom$ dependence (see section \ref{sec:h1dpdf}), leading order (LO)
and next-to-leading order (NLO) DGLAP QCD fits have been performed to
determine diffractive parton distributions \cite{h1f2d97}, updating
previous results based on earlier data \cite{h1f2d94}.  In this paper,
these pdf's, interfaced to NLO QCD calculations, are used for
comparisons with recent H1 measurements of $D^*$ meson \cite{h1dstar}
and dijet \cite{h1disjets} production cross sections in diffractive
DIS.  The result represents a test of QCD factorization in diffractive
DIS at NLO.

\section{Diffractive Parton Distributions}

\subsection{QCD factorization}

QCD hard scattering factorization in diffractive DIS \cite{collins}
suggests that in the leading $\log(Q^2)$ approximation, the cross
section for the diffractive process $\gamma^* p \rightarrow Xp$ can be
written in terms of convolutions of universal partonic cross sections
$\hat{\sigma}^{\gamma^* i}$ with diffractive parton distributions
$p_i^D$ \cite{facold,semicl,hautmannsoper}, representing probability
distributions for a parton $i$ in the proton under the constraint that
the proton remains intact with particular values of 4-momentum
transfer squared at the proton vertex 
$t$ and $x_\pom$.  Thus, at leading twist,
\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) \
p_i^D(\xi,Q^2,x_\pom,t) \ .
\label{equ:diffpdf}
\end{equation}
The factorization formula is
valid for large enough $Q^2$ and at fixed $x_\pom$ and $t$.
It also applies to the case of proton dissociation
into a system of fixed small mass $M_Y$.  The partonic cross sections
$\hat{\sigma}^{\gamma^* i}$ are the same as for inclusive DIS and the
diffractive parton distributions $p_i^D$, which are not known from
first principles, should obey the DGLAP evolution equations.

Thus, 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.  First
analyses of such kind were performed in
\cite{h1f2d94,actw,royonbartelsfits} based on measurements of the
diffractive structure function $F_2^D$ at HERA. The extracted parton
distributions can then be used to predict diffractive final state
cross sections, such as jet and heavy quark production, which allows
the factorization theorem to be tested.

\subsection{Regge Factorization}

On top of the rigorous theoretical prescription represented by
Eq.~(\ref{equ:diffpdf}), an additional assumption is often made,
namely that the shape of the diffractive parton distributions should
be independent of $x_\pom$ and $t$ and their normalization controlled
by Regge asymptotics \cite{ingschl}.  The diffractive pdf's can then
be factorized into a term depending only on $x_\pom$ and $t$ and a
second term depending only on $x$ (or $\beta=x/x_\pom$) and $Q^2$:
\begin{equation}
p_i^D(x_\pom,t,x,Q^2) = f_{\pom/p}(x_\pom,t) \cdot
p_{i,\pom}(\beta=x/x_\pom,Q^2) \ .
\label{reggefac}
\end{equation}
This so-called {\em Regge} factorization assumption, in the literature
often referred to as the {\em resolved pomeron} model, implies that the
diffractive exchange can be treated as a quasi-real object with a
partonic structure, given by parton distributions
$p_{i,\pom}(\beta,Q^2)$.  The variable $\beta$ then corresponds to the
longitudinal momentum fraction of the diffractive exchange carried by
the struck parton in the {\em pomeron}. The first term
$f_{\pom/p}(x_\pom,t)$ (also called {\em pomeron flux factor})
represents the probability for scattering off a pomeron with
particular values of $x_\pom$ and $t$. It should be stressed that no
proof in QCD exists for the assumption of Eq.~(\ref{reggefac}).
However, at the present level of precision it appears to be consistent
with the data.

\subsection{H1 Diffractive Parton Distributions}
\label{sec:h1dpdf}
 
DGLAP QCD fits were performed in LO and NLO by the H1 Collaboration to
the recent inclusive diffractive DIS data in \cite{h1f2d97}, assuming Regge
factorization. At
high $x_\pom$, a contribution from sub-leading meson (``reggeon")
exchange was taken into account, such that the diffractive pdf's,
integrated over $t$, are given by:
\begin{equation}
p_i^D(\xpom,\beta,Q^2) = f_{\pom/p}(\xpom) p_{i,\pom}(\beta,Q^2) +
                         f_{\reg/p}(\xpom) p_{i,\reg}(\beta,Q^2) \ .
\end{equation}
The pomeron and reggeon flux factors are given by
\begin{equation}
f_{\{\pom,\reg\}/p}(\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 functions of $t$:
\begin{equation}
\alpha_{\{\pom,\reg\}}(t)=\alpha_{\{\pom,\reg\}}(0) +
\alpha_{\{\pom,\reg\}}' t \ .
\label{eq:trajectory}
\end{equation}
The values of the parameters used in Eqs.~\ref{eq:fluxfac} 
and \ref{eq:trajectory} can be found in \cite{h1f2d97}.

The pomeron parton distributions are composed of a light flavour singlet and a
gluon distribution, which are evolved using the DGLAP equations, both in
leading and in next-to-leading order.  The strong coupling
constant $\alpha_s$ was fixed by setting
$\Lambda^{\overline{MS}}_{QCD}=0.2 \rm\ GeV$ for 4 flavours, using the
1(2) loop expression for $\alpha_s$ at LO and NLO respectively, and
the charm mass was set to $m_c=1.5 \rm\ GeV$. The meson pdf's were
parameterised using a pion parton distribution function.  The result
of the fit is shown in Fig.~\ref{fig:h1pdfs}.
The shown uncertainty in the NLO pdf's includes the
experimental uncertainty arising from the errors of the fitted data
as well as the theoretical uncertainty, estimated by variations of
$m_c$, $\Lambda_{QCD}$ and the parameters used in the pomeron
and reggeon flux factors.

 
 
\begin{figure}[t]
\centering
\epsfig{file=H1prelim-03-015.fig1.eps,width=0.7\linewidth}
\caption{Quark singlet (left) and gluon (right) distribution
  functions at LO (lines) and NLO (bands) of the
  diffractive exchange at various values of $Q^2$, obtained from DGLAP
  QCD analyses of inclusive diffractive DIS data, from
  \cite{h1f2d97}.}
\label{fig:h1pdfs}
\end{figure}

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


\section{Diffractive DIS Dijet and \boldmath{$D^*$} 
Cross Section Measurements}

The NLO calculations presented in this paper are compared with recent
H1 measurements of dijet and $D^*$ meson production in diffractive
DIS. An illustration of such processes at leading order QCD is shown
in Fig.~\ref{fig:diagram}.  In diffractive DIS, a photon with
virtuality $Q^2$ emitted from the beam electron interacts with the
proton, which loses only a small fraction $x_\pom$ of its incident
momentum and stays intact (or dissociates into a small mass system $Y$).

The longitudinal momentum fraction of the parton entering the hard
scattering process relative to the diffractive exchange is labelled
$z_\pom$. In the hard scattering process, a pair of high transverse
momentum ($p_T$) jets or heavy quarks is produced. The photon-proton
centre-of-mass energy is $W$, which relates to the inelasticity $y$ by
$ys=Q^2+W^2$, where $s$ is the $ep$ centre-of-mass energy.  The
invariant mass of the diffractively produced system $X$ is $M_X$, and
the invariant mass of the two partons emerging from the hard
sub-process is given by $\sqrt{\hat{s}}=M_{12}$.

\begin{figure}[t]
\centering
\epsfig{file=H1prelim-03-015.fig2.eps,width=0.4\linewidth}\caption{
Diagram for jet or heavy quark pair production in the diffractive
scattering of a virtual photon ($\gamma^*$), emitted by the beam
electron, off the proton ($p$).
}
\label{fig:diagram}
\end{figure}

\subsection{Diffractive Jet Production}

H1 has measured differential cross sections for the diffractive
production of dijets in DIS in \cite{h1disjets}. Jets are defined
using the CDF cone algorithm with cone radius $R=1$.  The kinematic
range of the measurement is $4<Q^2<80 \rm\ GeV^2$, $0.1<y<0.7$,
$x_\pom<0.05$, $|t|<1.0\rm\ GeV^2$ and $M_Y<1.6 \rm\ GeV$. The
transverse momenta and pseudorapidities of the two jets, which are
searched for in the hadronic centre-of-mass frame, are required to be
$p^*_{T,jet} > 4 \rm\ GeV$ and $-3<\eta_{jet}^*<0$, respectively.

\label{sec:asymcorr}

Next-to-leading order dijet cross sections are not reliable in regions
of phase space where the two jets have the same transverse momentum
\cite{frixione}.  To facilitate a comparison of the cross sections
\cite{h1disjets} with NLO calculations, the data were corrected to the
subsample where the transverse momentum of the first (second) jet is
$p^*_{T,jet \ 1(2)} > 5 (4) \rm\ GeV$.  The correction
is performed using leading order Monte Carlo programs, interfaced to
parton showers \cite{meps} to approximate higher order QCD effects.
Partons are fragmented to hadrons using the Lund string model
\cite{lund}.  The correction, defined as
\begin{equation}
c_{asym} = \frac{
\sigma_{dijet}(p^*_{T,jet \ 1(2)} > 5 (4) \ \rm{GeV})
}{
\sigma_{dijet}(p^*_{T,jet \ 1(2)} > 4 (4) \ \rm{GeV}) 
} \ ,
\end{equation}
is calculated using the RAPGAP program \cite{rapgap}, interfaced to the
``H1 fit 2" diffractive parton distributions \cite{h1f2d94},
which yield a very good description of the dijet data in shape
as well as normalization. Alternatively, the correction
factors are evaluated using LEPTO \cite{lepto},
employing the ``generalized area law" \cite{gal} model of soft colour 
interactions \cite{sci}, which provides a poorer
description of the shapes of the measured cross sections.
Despite the different model assumptions, the correction factors
obtained with the two programs are in good agreement. For the final
correction, the average is taken as the central value and half of the
difference is assumed as the uncertainty.  The size of the correction
is on average about $25\%$ and it is found to vary smoothly as a function
of kinematic variables other than $p^*_{T,jet}$.

\subsection{Diffractive \boldmath{$D^*$} Meson Production}

Differential cross sections for the production of $D^*$ mesons in
diffractive DIS were measured by H1 \cite{h1dstar}.  The kinematic
range of the data corresponds to $2<Q^2<100 \rm\ GeV^2$, $0.05<y<0.7$,
$x_\pom<0.04$, $|t|<1.0 \rm\ GeV^2$ and $M_Y<1.6 \rm\ GeV$.  The $D^*$
mesons are identified through their decay $D^{*\pm} \rightarrow
(K^-\pi^+)\pi^+_{slow} \ {\rm (+ c.c.)}$.  The transverse momenta and
pseudo-rapidities of the $D^*$ mesons in the laboratory frame are
required to be $p_T(D^*)> 2 \rm\ GeV$ and $|\eta(D^*)|<1.5$,
respectively.

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

\section{Next-to-leading Order QCD Calculations}

QCD factorization in diffractive DIS \cite{collins} implies that the
hard scattering cross section for the interaction of the
virtual photon with a parton in a diffractive process is identical to the
non-diffractive case.  Therefore, state of the art programs which
calculate fixed order partonic cross sections for dijet or heavy
flavour production in ordinary DIS can be used also in the case of
diffraction.  The available programs calculate the cross section up to
the next-to-leading order ($\mathcal{O}(\alpha_s^2)$) of perturbative
QCD, using DGLAP evolution.

\subsection{Diffractive Jet Production}
\label{sec:dijetnlo}

To calculate diffractive dijet cross sections to NLO in QCD, the DISENT
\cite{disent} program was used, as suggested in \cite{hautmann}.  It
was demonstrated in \cite{nlocomp} that calculations using this
program agree very well with the results from other programs. DISENT
was interfaced to the diffractive parton distributions obtained in
\cite{h1f2d97}.

The cross section at a given fixed value of $x_\pom$ and $t=0$
is calculated by reducing the proton beam energy by a factor
$x_\pom$: $E_p = x_\pom E_{p,nom.}$.  Since the $x_\pom$ and $t$
dependences of the used diffractive pdf's factorize, the proton pdf's
can be replaced by the pdf's of the diffractive exchange
$p_{i,\pom}(z,\mu^2)$, and the calculated cross sections are
multiplied by the $t$-integrated flux factor $f_{\pom/p}(x_\pom)=\int
{\rm d}t f_{\pom/p}(x_\pom,t)$. The partonic configurations are
calculated for $t=0$, such that kinematic effects of a finite value of
$t$ are neglected.  Since the measured cross sections correspond to an
interval in $x_\pom$, the integration over $x_\pom$ is approximated by
summing up the results obtained for a set of suitably chosen $x_\pom$
points (``$x_\pom$ slicing").  The number of $x_\pom$ points
was chosen to ensure the calculation was of sufficient precision.

For the (N)LO calculations, the diffractive parton distributions are
used in their respective version. The strong coupling constant
$\alpha_s$ is set to the value assumed in the QCD fit from which the
pdf's were extracted, using the corresponding 1(2)-loop expression and
taking
$\Lambda_{QCD}^{\overline{MS}}=0.2 \rm\ GeV$ for 4 flavours.  The
renormalization scale is set to $\mu_r^2=p_T^2$, where $p_T^2$
corresponds to the average transverse momentum of the two highest
$p_T$ partons in the Breit frame.  The factorization scale is set to
$\mu_f^2=40 \rm\ GeV^2$, corresponding to the average $p_T^2$ of the
two jets observed in the data, after the correction according to Eq.
(6) has been applied.  The parton configurations resulting from the
calculations are subjected to the same jet algorithm as was used for
the measured cross sections.

Since the calculations refer to jets of partons, whereas the
measurements refer to jets of hadrons, the NLO calculations have to
be corrected for the effects of hadronization. In the case of
diffraction, these also include the definition of the diffractive
kinematics, which at the hadron level are defined on the basis of the
largest gap in rapidity in the hadronic final state, defining two
hadronic systems $X$ and $Y$, from which $x_\pom$, $t$ and $M_Y$ are
calculated.  The hadronization corrections, defined as
\begin{equation}
c_{had} = \frac{\sigma^{hadron}_{dijet}}{\sigma^{parton}_{dijet}} \ ,
\end{equation}
are determined using the leading order Monte Carlo RAPGAP, interfaced
to the ``H1 fit 2'' diffractive pdf's.  Since no alternative model of
hadronization could be used, the correction factors are calculated
using either parton showers (MEPS) or a colour dipole model (CDM) \cite{cdm} 
to take higher order QCD effects into account. The size of
the corrections is of the order of $10\%$ and found to increase
towards low values of $p_T$ or $x_\pom$. For the final corrections,
the average of the values determined using MEPS and CDM is taken, and
for its uncertainty half of the difference is taken with a conservative
minimum uncertainty of $10\%$ being assumed.

\subsection{Diffractive \boldmath{$D^*$} Meson Production}

In \cite{dhvq}, the HVQDIS
\cite{hvqdis} program, based on the NLO calculations
for heavy quark production in DIS from \cite{harris}, was extended 
to diffraction.  The integration over $x_\pom$
and $t$ is performed numerically.  Fragmentation of charm quarks is
modeled using the standard Peterson fragmentation function
\cite{peterson}.

The leading and next-to-leading order cross sections for diffractive
$D^*$ production are calculated using the (N)LO diffractive parton
distributions from \cite{h1f2d97}. The prescription for $\alpha_s$ is
the same as for the dijet calculations, as explained in section
\ref{sec:dijetnlo}.  For the central calculations, both the
renormalization and the factorization scale are set to
$\mu_r^2=\mu_f^2=Q^2+4m_c^2$.  The charm mass is $m_c=1.5\rm\ GeV$,
the hadronization fraction is $f(c\rightarrow D^*)=0.233$ and the
parameter of the Peterson fragmentation function used is
$\epsilon_{Pet.}=0.078$.


\section{Results}

Comparisons at leading and next-to-leading order of
the calculations based on diffractive parton distributions,
obtained from QCD fits to inclusive diffractive DIS data
in \cite{h1f2d97}, and measured dijet production 
cross sections in diffractive DIS \cite{h1disjets} are presented in
Figs.~\ref{fig:jets1}-\ref{fig:jets3}, and for 
$D^*$ production in diffractive DIS \cite{h1dstar}
in Figs.~\ref{fig:dstar1}-\ref{fig:dstar2}.

\subsection{Diffractive Jet Production}

In Fig.~\ref{fig:jets1}, the differential cross section for dijet
production in diffractive DIS is presented as a function of
$z_\pom^{(jets)}$, an estimator for the longitudinal momentum fraction
of the diffractive exchange entering the hard scattering. The data
from \cite{h1disjets} are corrected to asymmetric requirements on the
minimum jet transverse momenta $p^*_{T,jet \ 1(2)} > 5 (4) \rm\ GeV$,
as explained in section \ref{sec:asymcorr}.  The inner error bars of
the data points correspond to the statistical error, whereas the outer
error bars represent the total uncertainty, including the systematic
error as well as the uncertainty of the correction to asymmetric jet
cuts, added in quadrature.

The data are compared with leading and next-to-leading order
calculations using DISENT, interfaced to the diffractive pdf's from
\cite{h1f2d97}, obtained from a QCD analysis of inclusive diffractive
DIS data.  The uncertainty of the next-to-leading order calculation
includes a variation of the renormalization scale $\mu_r^2=p_T^2$
by factors of $1/4$ and $4$ and the uncertainty in the hadronization
corrections, which is added linearly to the renormalization scale
uncertainty, the latter being typically of the order of $20\%$.  

The leading order calculation clearly underestimates the measured
cross section and is also unable to reproduce the observed shape of
the $z_\pom^{(jets)}$ distribution.  The next-to-leading order
corrections to the cross section are found to be large, exceeding a
factor 2 on average. This is considered to be mostly due to the
relatively small average transverse momentum of the jets.  The size of
the NLO correction decreases smoothly with increasing $p_{T,jet}$.
The NLO calculation, corrected for hadronization effects, provides within
the theoretical and experimental uncertainties a reasonable
description of the shape and normalization of the measured cross
section.  The uncertainty of the NLO calculation does not include the
uncertainty in the diffractive pdf's shown in Fig.~\ref{fig:h1pdfs}.
In particular, the uncertainty in the diffractive gluon distribution
is comparatively large for $z>0.5$.

In Fig.~\ref{fig:jets2}, the $z_\pom^{(jets)}$ cross section is presented
in four intervals of $Q^2+p_T^2$ and also in four intervals of
$x_\pom$.  
In Fig.~\ref{fig:jets3} differential dijet cross sections as a
function of $Q^2$, the photon-proton centre-of-mass energy $W$,
$\log_{10} x_\pom$ and the average pseudo-rapidity
$\av{\eta}_{lab}^{jets}$ of the jets are shown. 
Also in these distributions, reasonable agreement is
observed with the NLO calculations, corrected for hadronization, if
the experimental and theoretical uncertainties are taken into account.

\subsection{Diffractive \boldmath{$D^*$} Meson Production}

In Fig.~\ref{fig:dstar1}, the differential cross section for $D^*$
meson production in diffractive DIS, as measured in \cite{h1dstar}, is
shown in bins of $z_\pom$.  The error bars of the data
correspond to the sum of the statistical and systematic uncertainties,
added in quadrature.

The data are compared with leading and next-to-leading order
calculations using the diffractive version of HVQDIS, interfaced to
the diffractive pdf's from \cite{h1f2d97}. The uncertainty of the NLO
calculation corresponds to a variation of the renormalization scale
$\mu_r^2=Q^2+4m_c^2$ by factors of $1/4$ and $4$ and variations of the charm
quark mass within $m_c=1.35 \ldots 1.65 \rm\ GeV$ and of the parameter
of the Peterson fragmentation function within $\epsilon_{Pet.}=0.035
\ldots 0.100$.  Within the experimental and theoretical uncertainties,
good agreement is observed between the data and the NLO calculations.
The size of the NLO corrections is observed to be significantly
smaller than in the case of diffractive dijet production.

The variation of the charm mass by $\pm 0.15 \rm\ GeV$ changes the
cross section by $\pm 12\%$, whereas choosing $\epsilon_{Pet.}=0.035$
($\epsilon_{Pet.}=0.100$) changes the cross section by $+21\%$
($-7\%$).  If the value of $\Lambda_{QCD}$ at 4 flavours is changed by
$\pm 30 \rm\ MeV$, the cross section changes by $\pm5\%$ (not included
in the error bands). If scales other than $\mu_r^2=Q^2+4m_c^2$ are
chosen for the renormalization scale, such as $Q^2+m_c^2$, $Q^2$ or
$m_c^2$, the cross section varies by $10\ldots15\%$.

In Fig.~\ref{fig:dstar2}, differential cross sections are shown in
bins of $\log_{10} Q^2$, the transverse momentum of the $D^*$ in the
hadronic centre-of-mass frame, $p^*_{T,D^*}$, the pseudo-rapidity of
the $D^*$ in the laboratory frame, $\eta_{D^*}$, and of $x_\pom$. Also
for these distributions, good agreement is observed within the
uncertainties.

\section{Conclusions}

Diffractive parton distributions determined by H1 using
inclusive diffractive DIS data have been interfaced with
next-to-leading order QCD calculations for the diffractive production
of jets and heavy quarks in DIS. The calculations have been compared
with recent H1 measurements of diffractive dijet and $D^*$ meson
production in DIS.
Within the experimental and theoretical uncertainties, the
calculations are found to be in good agreement with the data.  The
results are thus consistent with QCD factorization in deep-inelastic
scattering as applied to diffraction at next-to-leading order.

\section*{Acknowledgements}
 
We are grateful to the HERA machine group whose outstanding efforts
have made this experiment possible.  We thank the engineers and
technicians for their work in constructing and maintaining the H1
detector, our funding agencies for financial support, the DESY
technical staff for continual assistance and the DESY directorate for
support and for the hospitality which they extend to the non-DESY
members of the collaboration.

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

\begin{thebibliography}{99}

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

\bibitem{h1f2d99mb} 
H1 Collaboration, ``Measurement of the Diffractive Deep-Inelastic Scattering
Cross Section at low $Q^2$'', paper {\bf 981} 
submitted to the 31st Intl. Conference on
High Energy Physics, ICHEP 2002, Amsterdam (2002). \\
(https://www-h1.desy.de/h1/www/publications/htmlsplit/H1prelim-02-112.long.html)



\bibitem{h1f2d97} 
H1 Collaboration, ``Measurement and NLO DGLAP QCD Interpretation of Diffractive Deep-Inelastic Scattering at HERA'', paper {\bf 980} submitted to the 31st Intl. Conference on
High Energy Physics, ICHEP 2002, Amsterdam (2002). \\
(https://www-h1.desy.de/h1/www/publications/htmlsplit/H1prelim-02-012.long.html)

\bibitem{h1f2dhiq2} 
H1 Collaboration, ``Measurement of the Diffractive Reduced Cross Section
$\sigma_r^{D(3)}$ at high $Q^2$'', paper {\bf 090} submitted to the 
Intl. Europhysics Conference on High Energy Physics, EPS 2003,
Aachen (2003). \\
(https://www-h1.desy.de/h1/www/publications/htmlsplit/H1prelim-03-011.long.html)


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

\bibitem{h1dstar} 
H1 Collaboration, C.~Adloff \etal, \Journal{\PLB}{520}{2001}{191}.

\bibitem{h1disjets} 
H1 Collaboration, C.~Adloff {\em et al.}, \Journal{\EJC}{20}{2001}{29}.

\bibitem{facold} 
L.~Trentadue, G.~Veneziano, \Journal{\PLB}{323}{1994}{201}; \\
A.~Berera, D.~Soper, \Journal{\PRD}{50}{1994}{4328}; \\
M.~Grazzini, L.~Trentadue, G.~Veneziano, \Journal{\NPB}{519}{1998}{394}.

\bibitem{semicl}
W.~Buchm\"{u}ller, T.~Gehrmann, A.~Hebecker, \Journal{\NPB}{537}{1999}{477}.

\bibitem{hautmannsoper}
F.~Hautmann, Z.~Kunszt, D.~E.~Soper, \Journal{\NPB}{563}{1999}{153}; \\
F.~Hautmann, D.~E.~Soper, \Journal{\PRD}{63}{2000}{011501}.

\bibitem{actw}
L.~Alvero, J.~C.~Collins, J.~Terron, J.~J.~Whitmore, 
\Journal{\PRD}{59}{1999}{074022}.

\bibitem{royonbartelsfits}
C.~Royon, L.~Schoeffel, J.~Bartels, H.~Jung, R.~Peschanski,
\Journal{\PRD}{63}{2001}{074004}.

\bibitem{ingschl} 
G.~Ingelman, P.~Schlein, \Journal{\PLB}{152}{1985}{256}.  

\bibitem{frixione}
M.~Klasen, G.~Kramer, \Journal{\PLB}{366}{1996}{385}; \\
S.~Frixione, G.~Ridolfi, \Journal{\NPB}{507}{1997}{315}; \\
B.~Poetter, \Journal{\CPC}{113}{2000}{105}. 

\bibitem{meps} M.~Bengtsson, T.~Sj\"ostrand, \Journal{\ZPC}{37}{1988}{465}.
 
\bibitem{lund} T.~Sj\"ostrand, \Journal{\CPC}{82}{1994}{74}.


\bibitem{rapgap} H.~Jung,  \Journal{\CPC}{86}{1995}{147}. \\
(see also http://www.desy.de/$\sim$jung/rapgap.html)

\bibitem{lepto} A.~Edin, G.~Ingelman, J.~Rathsman,
\Journal{\CPC}{101}{1997}{108}.

\bibitem{gal}  J.~Rathsman, \Journal{\PLB}{452}{1999}{364}.

\bibitem{sci} A.~Edin, G.~Ingelman, J.~Rathsman,
\Journal{\PLB}{366}{1996}{371}; \\
A.~Edin, G.~Ingelman, J.~Rathsman, \Journal{\ZPC}{75}{1997}{57}.

\bibitem{disent} 
S. Catani, M.~H.~Seymour, \Journal{\NPB}{485}{1997}{29}
[erratum-ibid. {\bf B510} (1997) 503].
         
\bibitem{hautmann} 
F.~Hautmann, JHEP {\bf 0210} (2002) 025.

\bibitem{nlocomp}
C.~Duprel, T.~Hadig, N.~Kauer, M.~Wobisch,
``Comparison of next-to-leading order calculations for jet 
cross-sections in deep inelastic scattering" in
A.~Doyle, G.~Grindhammer, G.~Ingelman, H.~Jung (eds.),
Proc. of the Workshop on Monte Carlo Generators for HERA Physics,
DESY-PROC-1999-02 (1999).

\bibitem{cdm}
G.~Gustafson,  \Journal{\PLB}{175}{1986}{453}; \\
G.~Gustafson, U.~Petterson, \Journal{\NPB}{306}{1988}{746}; \\
B.~Andersson, G.~Gustafson, L.~L\"onnblad, U.~Petterson,
    \Journal{\ZPC}{43}{1989}{625}; \\
B.~Andersson, G.~Gustafson, L.~L\"onnblad,
    \Journal{\NPB}{339}{1990}{393}. 

\bibitem{dhvq} L.~Alvero, J.~C.~Collins, J.~J.~Whitmore, 
``Tests of factorization in diffractive charm production and 
double pomeron exchange'', hep-ph/9806340.

\bibitem{hvqdis} 
B.~W.~Harris, J.Smith, \Journal{\PRD}{57}{1998}{2806}.

\bibitem{harris}
B.~W.~Harris, J.Smith, \Journal{\NPB}{452}{1995}{109}; \\
B.~W.~Harris, J.Smith, \Journal{\PLB}{353}{1995}{535}
[erratum-ibid. {\bf B359} (1997) 423].

\bibitem{peterson} C.~Peterson, D.~Schlatter, I.~Schmitt, P.M.~Zerwas,
\Journal{\PRD}{27}{1983}{105}.
 
\end{thebibliography}

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

\newpage

\begin{figure}
\begin{center}
\epsfig{file=H1prelim-03-015.fig3.eps,width=0.7\linewidth}
\end{center}
\caption{
Differential cross section for dijet production in diffractive DIS 
from \cite{h1disjets} (points), corrected
to asymmetric cuts on the jet transverse momentum $p^*_{T,1(2)}>5(4) \rm\ GeV$,
%
shown as a function of $z^{(jets)}_\pom$, an estimator
for the longitudinal momentum fraction of the
diffractive exchange entering the hard process.
%
The data are compared
with predictions based on the (N)LO diffractive parton distributions
from \cite{h1f2d97}, using $\mu_r^2=p_T^2$ and $\mu_f^2=40 \rm\ GeV^2$.
Shown are predictions obtained with
DISENT (using the $x_\pom$ slicing technique, see text) at leading order
QCD (blue, dash-dotted line), at next-to-leading order QCD (black,
dotted line) and at next-to-leading order including
hadronization corrections (central line of error band). The inner error band
represents the renormalization scale uncertainty, while 
the outer band also includes the uncertainty in the hadronization
corrections, added linearly.
}
\label{fig:jets1}
\end{figure}


\begin{figure}
\begin{center}
\epsfig{file=H1prelim-03-015.fig4.eps,width=1.0\linewidth}
\end{center}
\caption{
Differential cross section for dijet production in diffractive DIS 
from \cite{h1disjets} (points), corrected
to asymmetric cuts on the jet transverse momentum $p^*_{T,1(2)}>5(4) \rm\ GeV$,
%
shown as a function of $z^{(jets)}_\pom$, an estimator
for the momentum fraction of the longitudinal momentum fraction of
the diffractive exchange entering the hard process,
for four intervals of $Q^2+p_T^2$ (left) and $x_\pom$ (right),
respectively.
%
The data are compared
with predictions based on the (N)LO diffractive parton distributions
from \cite{h1f2d97}, using $\mu_r^2=p_T^2$ and $\mu_f^2=40 \rm\ GeV^2$.
Shown are predictions obtained with
DISENT (using the $x_\pom$ slicing technique, see text) at leading order
QCD (blue, dash-dotted line), at next-to-leading order QCD (black,
dotted line) and at next-to-leading order including
hadronization corrections (central line of error band). The inner error band
represents the renormalization scale uncertainty, while 
the outer band also includes the uncertainty in the hadronization
corrections, added linearly.
}
\label{fig:jets2}
\end{figure}




\begin{figure}
\begin{center}
\epsfig{file=H1prelim-03-015.fig5.eps,width=1.0\linewidth}
\end{center}
\caption{
Differential cross section for dijet production in diffractive DIS 
from \cite{h1disjets} (points), corrected
to asymmetric cuts on the jet transverse momentum $p^*_{T,1(2)}>5(4) \rm\ GeV$,
%
shown as a function of the photon virtuality $Q^2$, 
the photon-proton centre-of-mass energy $W$, 
the longitudinal proton momentum fraction taken by the diffractive
exchange $x_\pom$ and the average pseudo-rapidity of
the jets $\av{\eta}_{jets}^{lab}$.
%
The data are compared
with predictions based on the (N)LO diffractive parton distributions
from \cite{h1f2d97}, using $\mu_r^2=p_T^2$ and $\mu_f^2=40 \rm\ GeV^2$.
Shown are predictions obtained with
DISENT (using the $x_\pom$ slicing technique, see text) at leading order
QCD (blue, dash-dotted line), at next-to-leading order QCD (black,
dotted line) and at next-to-leading order including
hadronization corrections (central line of error band). The inner error band
represents the renormalization scale uncertainty, while 
the outer band also includes the uncertainty in the hadronization
corrections, added linearly.
}
\label{fig:jets3}
\end{figure}


\begin{figure}
\begin{center}
\epsfig{file=H1prelim-03-015.fig6.eps,width=0.7\linewidth}
\end{center}
\caption{
Differential cross sections for $D^*$
meson production in diffractive DIS from \cite{h1dstar} (points), 
%
shown as a function of $z_\pom$, an estimator
for the longitudinal momentum fraction of the diffractive exchange 
entering the hard process.
%
The data are compared with predictions based on the (N)LO diffractive
parton distributions from \cite{h1f2d97}, using
$\mu_r^2=\mu_f^2=Q^2+4m_c^2$.  Shown are predictions obtained with the
diffractive version of HVQDIS at leading order QCD (blue, dash-dotted
line) and at next-to-leading order QCD (central line of error band).
The inner error band represents the renormalization scale uncertainty,
while the outer error band shows the total uncertainty, which includes
variations of the charm mass in the range $m_c=1.35 \ldots 1.65 \rm\ 
GeV$ and of the parameter of the Peterson fragmentation function
$\epsilon_{Pet.}=0.035 \ldots 0.100$, added in quadrature.  }
\label{fig:dstar1}
\end{figure}

\begin{figure}
\begin{center}
\epsfig{file=H1prelim-03-015.fig7.eps,width=1.0\linewidth}
\end{center}
\caption{
Differential cross sections for $D^*$
meson production in diffractive DIS from \cite{h1dstar} (points), 
%
shown as a function of the photon virtuality $Q^2$,
the transverse momentum of the $D^*$ in the $\gamma^*p$ centre-of-mass frame
$p^*_{T,D^*}$, the pseudo-rapidity of the $D^*$ in the
laboratory frame $\eta_{D^*}$ and 
the longitudinal proton momentum fraction taken by the diffractive
exchange $x_\pom$.
%
The data are compared with predictions based on the (N)LO diffractive
parton distributions from \cite{h1f2d97}, using
$\mu_r^2=\mu_f^2=Q^2+4m_c^2$.  Shown are predictions obtained with the
diffractive version of HVQDIS at leading order QCD (blue, dash-dotted
line) and at next-to-leading order QCD (central line of error band).
The inner error band represents the renormalization scale uncertainty,
while the outer error band shows the total uncertainty, which includes
variations of the charm mass in the range $m_c=1.35 \ldots 1.65 \rm\ 
GeV$ and the parameter of the Peterson fragmentation function
$\epsilon=0.035 \ldots 0.100$, added in quadrature.  
}
\label{fig:dstar2}
\end{figure}

\end{document}

