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\begin{document}

\newcommand {\pom} {I\!\!P}
\newcommand {\pomsub} {{\scriptscriptstyle \pom}}
\newcommand {\xpom} {x_{\pomsub}}

\title{Diffractive jet and charm production}

\author{N. VLASOV}

\address{Freiburg University \\
Physikal Institute \\
Hermann-Herder st. 3, 79104 Freiburg, Germany \\
E-mail: vlasov@mail.desy.de}

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\maketitle

\abstracts{
The predictions of diffractive models are compared with H1 measurements
of jet production cross sections in diffractive Deep Inelastic Scattering
(DIS) and photoproduction and with H1 and ZEUS measurements of $D^{*\pm}(2010)$
meson production in diffractive DIS.}

\section{Introduction}
\label{sec-int}

The diffractive process at HERA, characterized by a large rapidity gap
in the distribution of the final-state hadrons, is considered
to proceed through the exchange of an object carrying the quantum numbers
of the vacuum, known as the Pomeron ($\pom$). In the
resolved-Pomeron model\cite{pl:b152:256},
the exchanged Pomeron acts as a source of partons, one of which interacts
with the virtual photon. In an alternative view,
the diffractive process at HERA can be described by
the dissociation of the virtual photon into a $q{\bar q}$ or
$q{\bar q}g$ state which interacts with the proton by the exchange
of two gluons or, more generally, a gluon ladder
with the quantum numbers of the vacuum~\cite{pl:b406:171,epj:c11:111,zfp:c74:671}.

Jet and charm production in diffractive Deep Inelastic Scattering (DIS),
which have also been measured by the H1 and
ZEUS collaborations~\cite{epj:c20:29,np:b672:335,pl:b520:191},
allow quantitative tests of the models due to the sensitivity
of jet and charm production to gluon-initiated processes
\cite{proc:hera:1996:691}. Final states containing heavy quarks or high
transverse momentum jets are of particular interest, since the additional hard
scales ensures the applicability of perturbative QCD.

Measurements of inclusive diffractive DIS at HERA have been used to extract
the diffractive parton distributions (diffractive pdfs) of the proton.
If QCD factorisation holds in diffractive DIS, these diffractive pdfs
are universal and may be used to predict the cross sections for exclusive
hard diffractive DIS processes such as jet and heavy flavour production.

\section{Models of Diffractive Charm and Jet Production}
\label{sec-theory}

\subsection{Resolved Pomeron Model}
\label{subsec:res}

In the resolved-Pomeron model, proposed by Ingelman and
Schlein~\cite{pl:b152:256}, the exchanged Pomeron is assumed to be a object
with a partonic structure. The diffractive cross section factorises into a
Pomeron flux factor, describing the probability to find a Pomeron in the
proton,
the Pomeron's parton density functions which specify the probability
to find a given parton in the Pomeron and the interaction cross section
with that parton. Within this model, open charm and jet production are
produced in diffractive DIS via the boson-gluon-fusion (BGF) process,
where the virtual photon interacts with a gluon from the Pomeron.
The HERA measurements of the inclusive diffractive differential cross sections
were found to be consistent with the resolved-Pomeron model
with a Pomeron structure dominated by gluons~\cite{zfp:c74:671,cp:ichep:2002}.

A combined fit of the Pomeron parton densities to the H1 and ZEUS inclusive
diffractive DIS measurements and to the ZEUS data on diffractive dijet
photoproduction has been made by Alvero et al. (ACTW)~\cite{pr:d59:74022}.
The resulting diffractive parton distributions have been compared
to recent ZEUS measurements of $D^*$ meson producton in diffractive DIS
\cite{np:b672:335}.
Leading order (LO) and next-to-leading order (NLO) DGLAP QCD fits have been 
performed by the H1 Collaboration to determine diffractive parton
distributions~\cite{cp:ichep:2002}
which were then used for comparisons with recent H1 measurements of $D^*$ meson
\cite{pl:b520:191} and dijet~\cite{epj:c20:29} production in diffraction.
The fit results have been interfaced to the program HVQDIS~\cite{pr:d57:2806}
to calculate cross sections for diffractive charm production
in DIS~\cite{hep-ph-9806340} and to the program DISENT~\cite{np:b485:29}
to calculate cross sections for diffractive dijet production in DIS, both
leading and next-to-leading order. 

\subsection{Two-gluon-exchange Models}
\label{subsec:tgexc}

The two-gluon-exchange models consider fluctuations of the virtual photon
into $q{\bar q}$ or $q{\bar q}g$ colour dipoles that interact with the proton
via colour-singlet exchange, the simplest form of which is a pair
of gluons~\cite{pr:d12:163,prl:34:1286,pr:d14:246}.
The virtual-photon fluctuations into $c{\bar c}$ and $c{\bar c}g$ states
can lead to diffractive open-charm production.
At high $\xpom$ values, quark exchanges are expected to become significant,
implying that the two-gluon-exchange calculations are expected to be valid
only at low $\xpom$ values ($\xpom<0.01$). In the
calculations~\cite{epj:c11:111,pl:b379:239,BJK:c24:555,pr:d59:014017},
used here, the cross section for two-gluon exchange is related to the square
of the unintegrated gluon distribution of the proton which depends on
the gluon transverse momentum, $k_T$, relative to the proton direction.
In the ``saturation'' model
\cite{pr:d59:014017,proc:ringberg:1999:361n},
the calculation of the $q{\bar q}g$ cross section is performed
under the assumption of strong $k_T$ ordering of the final-state partons,
which corresponds to $k_T^{(g)}\ll k_T^{(q,{\bar q})}$.
The parameters of the model were tuned to describe the total
photon-proton cross section measured at HERA. Alternatively, in the model of
Bartels et al.~\cite{epj:c11:111,pl:b379:239,BJK:c24:555},
configurations without strong $k_T$ ordering are included
in the $q{\bar q}g$ cross-section calculation and the minimum value for
the final-state-gluon transverse momentum, $k_{T,g}^{\rm cut}$, is a free
parameter which has been tuned to describe the H1 dijet cross sections
\cite{epj:c20:29}. The sum of the $q{\bar q}$ and $q{\bar q}g$ contributions
in the saturation model and the model of Bartels et al. are hereafter also
 referred to as SATRAP and BJLW, respectively.

\section{Diffractive Jets and Charm in DIS}
\label{sec-resdis}

In this section the recent H1 and ZEUS measurements of dijets and $D^*$ meson
production in diffractive DIS are compared with the different models.

In diffractive DIS, a photon with virtuality $Q^2$ emitted from the beam
electron interacts with the proton, which loses only a small fraction $\xpom$
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
$z_\pomsub$ (equivalent to $\beta$ at LO). 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 is related to the inelasticity $y$ via $ys = Q^2 + W^2$, where
$s$ is the $ep$ centre-of-mass energy squared. 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 $\sqrt{\hat{s}} = M_{12}$.

\subsection{Diffractive Dijets in DIS}
\label{subsec:jetdis}

The H1 Collaboration has measured differential cross sections for the diffractive
production of dijets in DIS~\cite{epj:c20:29,cp:eps113:2003}. The jets are found
using the CDF cone algorithm with cone radius $R = 1$. The kinematic range of
the measurement is $4 < Q^2 < 80$ GeV$^2$, $0.1 < y < 0.7$, $\xpom < 0.05$,
$|t| < 1.0$ GeV$^2$ and $ M_Y < 1.6$ GeV. The transverse momenta and
pseudorapidities of the two jets, which are searched for in the hadronic
centre-of-mass frame, are required to satisfy $p^*_{T,jet} > 4$ GeV and
$-3 < \eta^*_{jet} < 0$, respectively.

To facilitate a comparison of the cross sections with NLO calculations, the
data were corrected to the kinematic region where the transverse momentum
of the first (second) jet is $p^*_{T,jet} > 5(4)$ GeV, as the next-to-leading
order dijets cross sections are not reliable in regions of phase space where
the two jets have the same transverse momentum~\cite{cp:eps113:2003}.

To calculate diffractive dijet cross sections to NLO in QCD, the DISENT
program was used. Since the calculations refer to
partonic jets, whereas the measurements refer to hadronic jets,
the NLO calculations have been corrected for the hadronisation effects.

In Fig.~\ref{fig-fig1}, the differential cross sections for dijet production
in diffractive DIS are presented as functions of $z_\pomsub^{jets}$, $Q^2$,
$W$, $log_{10}\xpom$ and the average pseudorapidity
$\langle \eta \rangle^{jets}_{lab}$ of the jets. The data are
compared with leading and next-to-leading order calculations using DISENT
interfaced to the H1 diffractive pdf's~\cite{cp:ichep:2002}.
The uncertainty on the NLO calculations was estimated by varying
the renormalisation scale $\mu_R^2 = p_T^2$ by factors of 1/4 and 4
and the hadronisation corrections.
The NLO calculation provides within the theoretical and experimental
uncertainties a reasonable description of the shapes and normalisations of the
measured cross sections.

\begin{figure}[tbhp]
\begin{center}
\vspace *{-0.3cm}
\includegraphics[height=60mm]{vlasov.fig1a.eps}\hspace *{-1cm}
\includegraphics[height=60mm]{vlasov.fig1b.eps}
\end{center}
\vspace *{-0.3cm}
\caption{Differential cross sections for dijet production in diffractive DIS
(dots) as a function of $z_\pomsub^{jets}$, $Q^2$,
$W$, $log_{10}\xpom$ and $\langle \eta \rangle^{jets}_{lab}$. The data are
compared with predictions from NLO calculations.}
\label{fig-fig1}
\end{figure}

In Fig.~\ref{fig-fig2}, the differential dijet cross sections in the
region $\xpom < 0.01$ are compared with the predictions of the Saturation,
BJLW and Resolved Pomeron (Pomeron parametrisation 'h1 fit2'
\cite{pl:b520:191}) models.
The $P^{(\pom)}_{T,rem}$ variable measures the transverse
momentum of all hadronic final state particles in the Pomeron hemisphere of the
$\gamma^* \pom$ centre-of-mass frame which do not belong to the two highest
$p_T^*$ jets.

The saturation model is able to reproduce the shapes of the measured cross
sections, although the overall predicted dijet rate is too low by a factor of
about 2. In the BJLW model, the contribution from $q\bar{q}$ states is
negligible and the dominant contribution comes from $q\bar{q}g$ final states. The
normalisation of the BJLW model for $q\bar{q}g$ production can be controlled by
tuning the lower limit on the transverse momentum of the gluon,
$p^{cut}_{T,g}$, in the calculations. If this limit is set to 1.5 GeV, the
total cross section for dijet production with $\xpom < 0.01$ is in agreement
with the model~\cite{epj:c20:29}.

In Fig.~\ref{fig-fig2}, the Resolved Pomeron model gives a good description of
all observables, including the $P^{(\pom)}_{T,rem}$ distribution. The good
description of the $P^{(\pom)}_{T,rem}$ distribution by both the Resolved
Pomeron and the BJLW models indicates that the present data are not able
to discriminate between models with a soft ''remnant`` and those with a third
high-$p_T$ parton.

\begin{figure}[tbhp]
\begin{center}
\vspace *{-0.3cm}
\includegraphics[height=65mm]{vlasov.fig2.eps}
\end{center}
\vspace *{-0.3cm}
\caption{Differential cross sections for dijet production in diffractive DIS
(dots), shown as functions of $Q^2$, $p_{T,jets}^*$, $z_\pomsub^{jets}$
and $P^{(\pom)}_{T,rem}$. The data are compared with different models.}
\label{fig-fig2}
\end{figure} 

\subsection{Diffractive $D^*$ in DIS}
\label{subsec:difds}

The ZEUS Collaboration has measured charm production in diffractive DIS
\cite{np:b672:335}. Figure~\ref{fig-fig3} shows the differential
cross sections measured in the kinematic range $1.5 < Q^2 < 200$ GeV$^2$,
$0.02 < y < 0.7$, $\beta < 0.8$, $p_T(D^{*}) >1.5$ GeV and
$|\eta(D^*)| < 1.5$ as functions of $\beta$, $\log(\beta)$,
$\log(Q^2)$ and $W$ for $x_\pomsub < 0.035$ and as functions of
$p_T(D^{*})$, $\eta(D^{*})$, $\log(M_X^2)$ and $\beta$
for $x_\pomsub < 0.01$. The data are compared with
the ACTW NLO predictions, calculated with the gluon-dominated fit B,
the SATRAP and the BJLW predictions (only for $x_\pomsub < 0.01$).

The two-gluon-exchange BJLW model predictions, obtained with
the requirement that $k_{T,g}^{\rm cut}=1.5$ GeV, describe the
differential cross sections in the range $x_\pomsub<0.01$ both in shape
and normalisation. The saturation model (SATRAP)
predictions reproduce the shapes and the normalisations of the differential
cross sections measured in both $x_\pomsub$ ranges.
The ACTW NLO predictions, obtained with the gluon-dominated fit B,
describe the data reasonably well in both $x_\pomsub$ ranges.

\begin{figure}[tbhp]
\begin{center}
\vspace *{-0.3cm}
\hspace *{-0.5cm}\includegraphics[height=60mm]{vlasov.fig3a.eps}\hspace *{0.0cm}
\hspace *{-0.5cm}\includegraphics[height=60mm]{vlasov.fig3b.eps}
\end{center}
\vspace *{-0.3cm}
\caption{Differential cross sections for diffractive $D^*$ production
(dots) compared with different models (see text).
The inner error bars indicate the statistical uncertainties,
while the outer ones correspond to the statistical and systematic
uncertainties added in quadrature. The shaded area shows the effect of
varying the charm-quark mass in the ACTW NLO prediction.}
\label{fig-fig3}
\end{figure} 

Fig.~\ref{fig-fig4} shows the quantity $\xpom F_2^{D(3),c{\bar c}}$ measured
by the ZEUS Collaboration~\cite{np:b672:335} where $F_2^{D(3),c{\bar c}}$ is the
open-charm contribution to the diffractive proton structure function.
It is shown as a function of $\log(\beta)$
for different values of $Q^2$ and $\xpom$.
In all cases, $\xpom F_2^{D(3),c{\bar c}}$ rises as $\beta$ decreases.
The curves show the predictions obtained from the
ACTW NLO calculations based on fits B, D and SG~\cite{pr:d59:74022}.
The fit B prediction generally agrees with the data. The fit D (SG)
prediction overestimates (underestimates) the measured
$\xpom F_2^{D(3),c{\bar c}}$ at low $\beta$.

\begin{figure}[tbhp]
\begin{center}
\vspace *{-0.3cm}
\hspace *{-0.5cm}\includegraphics[height=70mm]{vlasov.fig4.eps}
\end{center}
\vspace *{-0.3cm}
\caption{The measured charm contribution to the diffractive structure
function of the proton multiplied by $x_\pomsub$,
 $x_\pomsub F_2^{D(3),c{\bar c}}$, as a function of $\beta$
for different values of $Q^2$ and $x_\pomsub$ (dots).
The inner error bars indicate the statistical uncertainties,
while the outer ones correspond to the statistical and systematic
uncertainties added in quadrature. The shaded area shows the effect of
varying the charm-quark mass in the ACTW NLO prediction.}
\label{fig-fig4}
\end{figure}

Differential cross sections for $D^*$ production in
diffractive DIS have also been measured by H1\cite{pl:b520:191, 
cp:eps113:2003}.
The kinematic range of
the data corresponds to $2 < Q^2 < 100$ GeV$^2$, $0.05 < y < 0.7$,
$\xpom < 0.04$, $|t| < 1$ GeV$^2$, $M_Y < 1.6$ GeV, $p_T(D^{*}) > 2$ GeV
and $|\eta(D^*)| < 1.5$.

Figure~\ref{fig-fig5} shows the differential cross sections as functions of
$\log(Q^2)$, $p_T(D^{*})$, $\eta(D^{*})$ and $x_\pomsub$. The data are
compared with the leading and next-to-leading order calculations from the
diffractive version of HVQDIS, interfaced to the H1 diffractive pdf's
\cite{cp:ichep:2002}. Within the experimental and theoretical uncertainties,
good agreement is observed between the data and NLO
calculations.

\begin{figure}[tbhp]
\vspace *{-0.3cm}
\begin{center}
\includegraphics[height=70mm]{vlasov.fig5.eps}
\end{center}
\vspace *{-0.3cm}
\caption{Differential cross sections for diffractive $D^*$ production
(dots) compared with NLO calculations.
The error bars of the data correspond to the sum of statistical and
systematic uncertainties, added in quadrature. The inner error band
represents the renormalisation scale uncertainity, while the outer
band also includes the uncertainty in the hadronization corrections,
added lineary.}
\label{fig-fig5}
\end{figure} 

\subsection{Dijets in Diffractive Photoproduction}
\label{subsec:jetphp}

As shown in Sec.~\ref{subsec:jetdis}, diffractive dijet production
in DIS is in good agreement with the QCD fits to inclusive diffractive data.
However, applying the diffractive parton densities to predict diffractive jet
production at the Tevatron, leads to an overestimation of the observed rate by one
order of magnitude~\cite{prl:84:2000}. This discrepancy has been attributed to the
presence of the additional beam hadron remnant in $p\bar{p}$ collisions which
leads to secondary interactions and a break-down of factorisation. The suppression
cannot be calculated perturbatively and has been parametrised in various ways
\cite{pl:b438:229,hep-ph9908464,hep-ph0306134}.

The transition from DIS to hadron-hadron scattering has been studied with the H1
detector in the photoproduction regime, where the beam lepton emits a quasi-real
photon which interacts with the proton~\cite{cp:eps087:2003}. Processes in which
the photon is first resolved into partons which then initiate the hard scattering
resemble hadron-hadron scattering. Via the resolved photon process in hard
photoproduction, hadronic final states are accessible, which are present in
the equivalent $p\bar{p}$ collisions but not in DIS.

Dijet events are identified using the inclusive $k_T$ cluster algorithm
\cite{cp:eps087:2003}. Fig.~\ref{fig-fig6} shows the differential
cross sections with respect to $z_\pomsub^{jets}$
and the estimator $x^{jets}_\gamma$ of the fractional photon momentum taking
part in the hard scattering. The cross sections are given at hadron level
and correspond to the kinematic range $Q^2 < 0.01$ GeV$^2$,
$165 < W < 240$ GeV, $\xpom < 0.03$, $E_T^{jet1} > 5$ GeV and
$E_T^{jet2} > 4$ GeV. The H1 2002 fit~\cite{cp:ichep:2002} gives a good
description of the diffractive dijet photoproduction.
The contribution from the direct photon process
($x^{true}_\gamma=1$) is indicated by the hatched histogram. The sum of resolved
and direct photon processes in the model gives a good description of the data both
in normalisation and in shape throughout the $x^{jets}_\gamma$ range. Hence there
is no evidence for any suppression of the diffractive cross section in the
region dominated by resolved photons, as might be expected on the basis of data
from the Tevatron~\cite{prl:84:2000} or phenomenological models
\cite{pl:b438:229,hep-ph9908464,hep-ph0306134}.

\begin{figure}[tbhp]
\begin{center}
\vspace *{-0.3cm}
\includegraphics[height=50mm]{vlasov.fig6a.eps}\hspace *{0.0cm}
\includegraphics[height=50mm]{vlasov.fig6b.eps}
\end{center}
\vspace *{-0.3cm}
\caption{Differential cross sections in $z_\pomsub^{jets}$ and $x^{jets}_\gamma$
in diffractive photoproduction (dots). The inner error bars represent the
statistical errors and the outer error bars the quadratic sum of the statistical
and uncorrelated systematic errors.}
\label{fig-fig6}
\end{figure} 

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\end{document}
