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\begin{document}

\newcommand{\jpsi}{J/\psi}
\newcommand{\psip}{\psi^\prime({\mbox{2S}})}
\newcommand{\as}{\alpha_{\rm S}}
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\newcommand{\aPp}{\alpha_{\rm P}^\prime}
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%\begin{document}
%
\title*{Heavy Quarkonium Production at HERA}
%
%
\toctitle{Heavy Quarkonium Production at HERA}
% allows explicit linebreak for the table of content
%
%
\titlerunning{Heavy Quarkonium Production}
% allows abbreviation of title, if the full title is too long
% to fit in the running head
%
\author{Christian Kiesling}
%
\authorrunning{C. Kiesling}
% if there are more than two authors,
% please abbreviate author list for running head
%
%
\institute{Max-Planck-Institut f\"ur Physik
  (Werner-Heisenberg-Institut), \protect\newline F\"ohringer Ring 6,
  D-80805 M\"unchen }  
%
\maketitle              % typesets the title of the contribution

\begin{abstract}
In this article, recent results on electro- and photoproduction of
heavy vector mesons at HERA are reviewed. Various models for the
inelastic and elastic production mechanisms are 
discussed and confronted with the measurements. 
%Inelastic heavy vector meson production at HERA is dominated by the
%(virtual) photon-gluon fusion process, giving direct access to the gluon 
%density in the nucleon. 
Elastic production allows  
comparison of two distinct theoretical concepts, i.e. the Regge 
ansatz and models using perturbative QCD (pQCD), where the colorless
exchange is modeled by multiple gluon exchange. While production of
light vector mesons is well described in the Regge picture, the heavy
vector meson data show clear deviations from the universal ``soft''  
Pomeron prediction. Perturbative QCD, however, is able to reproduce
the trend of the data, as well as a more general Regge fit using an
additional ``hard'' Pomeron.
\end{abstract}
%
%=========================================================================
\section{Introduction}
%
One of the most striking early results from the HERA collider was the
detection of the strong rise \cite{h1f2lowx,zeusf2lowx} of the proton
structure function $F_2$ towards 
low values of the Bjorken variable $x$. These findings could be
interpreted by a dramatic increase of the gluon density within the proton 
and provided strong motivation to numerous subsequent measurements of
the electron-protron interaction at low $x$ and low momentum transfer
$Q^2$. In the course of these measurements a new class of events were
(re-)discovered \cite{h1rap,zeusrap}, namely the diffractive
production of hadrons, both in  
electroproduction and photoproduction (at HERA the latter are defined
as events with an undetected electron in the final state, emitted at
very small angles  
and exhibiting very small $Q^2$). A surprisingly large fraction of the
produced events showed a large rapidity gap between the produced secondary
hadrons and the outgoing nucleon, indicative for a color singlet exchange
between the two hadronic systems. These events led to a revival of
interest in Regge theory, more precisely in Pomeron exchange, which in 
the past could successfully describe ``soft'' diffractive processes in
hadron-hadron and photon-hadron collisions. The correct prediction of
the total cross section for quasi-real photons ($\gamma^*$)
colliding with protons ($p$) at HERA, which entered an entirely new
kinematic domain, was a clear triumph for the Pomeron
exchange interpretation: Pomeron exchange is characterized by a universal
trajectory, independent of the composition of the initial hadronic state,
and predicts a slow rise of the total cross section with increasing
hadronic center-of-mass energy $W=\Wgsp$. Other data, such as 
the slowly rising cross sections for exclusive light vector meson
($\rho, \omega, \phi$)
production, contributed to the strengthening of the ``soft'' Pomeron
interpretation.    

A severe blow against this consistent Pomeron picture came through the
measurement of $\jpsi$ photoproduction at HERA which, in 
contrast to $\rho$ photoproduction and the soft Pomeron expectation,
was observed to exhibit a much steeper rise\cite{Zeusjpsigp,H1jpsigp}
of the total cross section
towards increasing $W$. The two above-mentioned phenomena,
i.e. strong rise of $F_2$ towards low $x$ and strong rise of the cross
section for diffractive processes with increasing $W$, can
however be understood within perturbative QCD (pQCD): The rise is
caused by an increased gluon density inside the proton as a
consequence of the $Q^2$ evolution predicted in pQCD.  

An even more direct path towards the gluon density within the proton
is offered by inelastic heavy vector meson electroproduction: At HERA, 
this process is dominated by photon-gluon fusion, which is tagged as
such, e.g., through the presence of a $\jpsi$ meson, originating from the 
fluctuation of the virtual photon emitted from the electron into a
charm-anticharm quark pair. The color singlet nature of the $\jpsi$ in 
the $\gamma g$ subprocess is ensured by the emission of a hard gluon
in the final state (inelastic reaction) or by emission of a soft gluon 
re-absorbed by the partonic constituents of the proton. 

All these processes, involving heavy vector meson production at HERA,
are extremely useful in the investigation of the detailed gluonic
structure of the nucleon and complement nicely other analyses such as
jet production or the $Q^2$ development of the structure function
$F_2$. 

%
%======================================================================
\section{Kinematics and Experimental Procedures}
%
Figure \ref{kiesling.fig1} shows the lowest order Feynman diagram
for inelastic $\jpsi$ production, where also the kinematics of the
reaction can be defined.
%
\begin{figure}[b]
  \begin{minipage}[t]{0.40\textwidth}
      \makebox[0cm][t]{}        % this box is only there to align text
      \begin{center}
        \includegraphics[width=0.95\textwidth]{kiesling.1}
      \end{center}
  \end{minipage}
  \hfill
  \begin{minipage}[t]{0.59\textwidth}
    \vspace{0.1cm}
    \caption[]{Feynman diagram for inelastic vector meson (e.g. $\jpsi$)
      production as a boson gluon fusion process. The following
four-vectors define the kinematics: incoming proton $p_p$,
incoming electron $k$, scattered electron $k^\prime$, exchanged photon
$q = k-k^\prime$, outgoing vector meson 
$p_V$. The hadronic center-of-mass energy $W$ is given by the virtual
photon, emitted from the electron, and the incoming proton:
$W^2=(p_p+q)^2$}.  
    \label{kiesling.fig1}
  \end{minipage}
\end{figure}
%
The process is photon-gluon fusion, where the (virtual) photon
$\gamma^*$ with four-momentum $q$ is
emitted from the incoming electron (or positron) of four-momentum $k$,
and the gluon is radiated off the constituents of the proton, the
four-momentum of which is $p_p$. Besides the mass squared of the produced
vector meson $M^2_V$, two other scales exist 
in the process, relevant for the subsequent discussions: The photon
virtuality $Q^2 = -q^2$ and the momentum  transfer squared $t=(p_p -
p_X)^2$ at the lower (``proton'') vertex. 
 
The energy $y$ of the virtual photon, in units of the electron energy
in the proton rest system, is given by    
%
\begin{equation}
y = \frac{q \cdot p_p}{k \cdot p_p} \; .
\end{equation}
%
The produced (heavy) vector meson $V$ is emitted with energy fraction
$z$ (``elasticity'') relative to the incident photon energy fraction
$y$ in the proton rest system, which is given by  
%
\begin{equation}
z = \frac{p_p \cdot p_V}{p_p \cdot q}=\frac{E_V}{E_\gamma} \; .
\label{zdef}
\end{equation}
%
The variable $z$ can be used to separate photon-gluon fusion from
resolved photon processes, which dominate for $z<0.2$
\cite{hannes}. 
For elastic reactions, no hard gluon is radiated and the hadronic
system $X$ is just the proton. In this limit the vector meson's
elasticity $z$ approaches unity: $z \approx 1$. 
%

In order to measure the reaction $\gamma^* p \rightarrow V X$, both H1 
and ZEUS require the heavy vector meson $V$ to be detected essentially 
in the central detector (H1 also uses the backward electromagnetic
calorimeter to detect $\jpsi$ mesons decaying into
electron-positron pairs). One can
distinguish the following reactions producing a heavy vector meson in
the final state: 
\begin{itemize}
\item {\it inelastic production}: The vector meson is produced accompanied by
additional hadrons from the hard gluon (see
fig.~\ref{kiesling.fig1}). The initial proton at the lower vertex is transformed
to a general hadronic state $X$. Typically, the elasticity for the
$\jpsi$ is small, $z \le 0.9$ 
but larger than about 0.2 in order to exclude background where the
meson $V$ is produced in the fragmentation chain of a quark- or
gluon-initiated jet.  
\item {\it dissociative production}: The vector meson is produced
``elastically'' (no hard gluon in the final state), but the proton
dissociates into a state $X$ with $M_X > M_p$ and $z \le 1$.
\item {\it elastic (exclusive) production}: The vector meson is produced
truly elastically and the proton stays intact, $z \approx 1$. 
\end{itemize} 
%

%======================================================================
\section{Theoretical Concepts of Heavy Quarkonium Production} 
%
 Electro- and photoproduction of vector mesons $V (=\rho, \omega,
\phi, J/\psi, \Upsilon)$, i.e. $ep \rightarrow e X 
V$, provides a wealth of information on the properties of the strong
interaction. As indicated schematically in fig.~\ref{kiesling.fig1}, one of the
motivations for the study of inelastic production of vector mesons is
the possibility to directly probe the gluon content in the
nucleon. Processes such as exclusive vector meson production ($ep
\rightarrow ep V$), on the other hand, may be used as sensitive tools
to study soft and hard diffractive processes and the hadronic
structure of the virtual photon. While perturbative QCD is expected to
describe inelastic vector-meson production - given a
sufficiently large scale in the scattering process - there are two
major theoretical concepts for the description of the diffractive
reactions, i.e. Regge phenomenology and, again, perturbative QCD.  

%------------------------------------------------------------------------
\subsection{Regge Models}
%
High energy hadron-hadron and photon-hadron scattering display many
features characteristic for optical diffraction, among them total
cross sections slowly rising with the hadronic center of mass energy
$W$ and a strong (exponential) drop of the elastic differential cross
section $d \sigma /dt$ away from the forward direction. Donnachie and
Landshoff\cite{DL} have found an extremely simple parameterization of
the energy dependence of all the total and differential cross sections
at high energies, based on the exchange of a particular Regge
trajectory\cite{regge1}, the so-called Pomeron (P) trajectory $\alpha_{\rm
P}$, defined by two parameters:
%
\begin{equation}
\aP(t) = \aPz +\aPp \; t \; .
\end{equation}
%
The differential cross section $d \sigma /dt$ for elastic scattering
processes at low values of $\tbb$ 
is well described by a single exponential in $t$ with a slope parameter,
varying in a characteristic manner with the energy $W$:
%
\begin{equation}
\frac{d\sigma}{dt} \propto  e^{-b |t|} \quad ,\quad  b =
b_0 + 2\aPp \ln\left(\frac{W^2}{W_0^2}\right) \; .
\label{bdef}
\end{equation}
%
Here, $b_0$ and $W_0$ are process-dependent constants.
%Note here the dependence of the slope on a mass scale, which is the
%mass of the dissociated proton ($M_X$).
%
Using the optical theorem, the total cross section is then given by
%
\begin{equation}
\sigma(W) \propto \frac {W^{4(\aPz-1)}}{b} \sim W^{0.22}\; ,
\label{wdep}
\end{equation}
%
where the numerical value in the exponent of $W$ results from
$\aPz=1.08$ and $\aPp=0.25$GeV$^{-2}$, which have been determined from 
fits to all available total cross section data\cite{DL}.
%
\begin{figure}[htb]
  \begin{minipage}[t]{0.40\textwidth}
      \makebox[0cm][t]{}        % this box is only there to align text
      \begin{center}
        \includegraphics[width=0.95\textwidth]{kiesling.2}
      \end{center}
  \end{minipage}
  \hfill
  \begin{minipage}[t]{0.59\textwidth}
    \vspace{0.1cm}
    \caption[]{Feynman diagram for elastic vector meson (e.g. $\jpsi$)
      production within the framework of Regge phenomenology,
exhibiting vector-meson dominance and 
Pomeron exchange. In electroproduction, the hadronic center of mass
energy $W$ is given by the virtual photon, emitted from the
electron, and the incoming proton.} 
    \label{kiesling.fig2}
  \end{minipage}
\end{figure}
%

In the framework of Regge phenomenology, electroproduction of vector-mesons
is viewed as a two-step process as indicated in fig. 2:
First, the (virtual) photon emitted
from the electron fluctuates into a vector meson, described by the
Vector-Meson Dominance model (VDM\cite{VDM}), and the vector meson in turn 
scatters elastically off the proton, exchanging Reggeons like ``ordinary''
hadron-hadron interactions. Based on this picture, the following
predictions are made: The  
total cross section for $\gamma^* p \rightarrow pV$ at high virtual-photon
proton energy $W$ rises slowly with $W$, no additional
dependence on the photon virtuality $Q^2$ is expected concerning the
rise of cross section with $W$. The differential cross section is falling 
exponentially with $t$, the slope parameter $b$ increasing
logarithmically with $W$, driven by $\aPp = 0.25$
GeV$^{-2}$\cite{DL}. This feature of the differential cross 
section is called ``shrinkage'': due to the increasing $b$ the
differential cross section 
steepens with energy, the diffractive forward peak becomes narrower,
or ``shrinks''. A further prediction of the Regge model is the
complete transfer of the photon helicity to the final state vector
meson ($s$-channel helicity conservation, SCHC). SCHC can be tested
analyzing the angular distribution of the decay particles in the rest
frame of the vector meson. 

It is well-known by now that the HERA data on $\jpsi$ photo- and
electroproduction are not described by the simple (``soft'') Regge
model. A natural extension within the Regge phenomenology was to
introduce in addition to the soft Pomeron trajectory $\aP(t)$ a
``hard'' Pomeron trajectory $\aPP(t)$ with suitable parameters
$\aPPz$ and $\aPPp$, so that the data can, again, be correctly
described. Such an approach was recently proposed by Donnachie and
Landshoff\cite{twopom1}.   
%

%------------------------------------------------------------------------
\subsection{Perturbative QCD}
%
Given a sufficiently large scale
such as the photon virtuality, the momentum transfer $t$, or the
mass of the produced vector 
meson, diffractive processes can be viewed as mediated, to lowest
order, by an exchange of two gluons (in a color singlet state) between a
quark-antiquark 
pair (emitted from the photon) and the constituents inside the proton
(see fig.~\ref{kiesling.fig3}).
%
\begin{figure}[htb]
  \begin{minipage}[t]{0.40\textwidth}
      \makebox[0cm][t]{}        % this box is only there to align text
        \begin{center}
          \includegraphics[width=0.95\textwidth]{kiesling.3}
        \end{center}
  \end{minipage}
  \hfill
  \begin{minipage}[t]{0.59\textwidth}
    \vspace{0.1cm}
%
    \caption[]{Feynman diagram for elastic $\jpsi$ production within the
      framework of perturbative QCD: The virtual photon couples to a
$c \bar{c}$ pair and two gluons in a color singlet state are exchanged,
in lowest order, between the quark pair and the proton. The $c
\bar{c}$ pair forms a $\jpsi$ with a probability derived from the
bound-state wave function.}  
    \label{kiesling.fig3}
  \end{minipage}
\end{figure}
%
In models based on pQCD the reaction $\gamma^* p \rightarrow V p$ (the
virtual photon $\gamma^*$ being radiated off an incoming electron or 
positron) is viewed as a temporal sequence of three
sub-processes\cite{BFGMS94}:  
First the virtual photon $\gamma^*$ fluctuates into a $q \bar{q}$
pair, which then scatters off the proton target exchanging, in lowest
order, a color singlet two-gluon system or, more generally, a gluon
ladder. Finally the $q \bar{q}$ 
pair turns into a 
vector meson. Elastic (heavy) vector meson production may thus be used 
as an alternative way to probe the gluon content of the
proton\cite{Ryskin93}. While in \cite{BFGMS94} only the case of high $Q^2$
is considered, where the longitudinal components of the virtual photon
and the produced vector meson are supposed to dominate, several
authors\cite{FKS96,FKS98,RRML97} have generalized this model to
include also transversely polarized photons and vector mesons, and
heavy vector meson photoproduction, the calculations being carried out
in leading order (LO) $\as \ln(Q^2/\lqcd^2)$. The calculations of
Frankfurt et 
al.\cite{FKS98} yield the following prediction for the forward 
differential cross section (the Ryskin formula\cite{RRML97} is similar):  
%
\begin{eqnarray}
\frac{d \sigma^{\gamma^* p}}{d t}\bigg|_{t=0} =
\frac{12 \pi^3 \Gamma_{ll}M_V^3}{\alpha_{\rm em}(Q^2+4m^2_c)^4}
 & \cdot & \left| \as(\tilde{Q}^2)\left(1+\frac{i \pi}{2}
\frac{d}{d \ln x}\right) 
xg(x,\tilde{Q}^2)\right|^2 \nonumber 
\\ 
&\cdot& \left( 1+\epsilon  
\frac{Q^2}{M^2_V}\right){\cal C}(Q^2)  \; .
\label{dsqcd}
\end{eqnarray}
%
In this expression, $\epsilon$ is the flux ratio of longitudinally to
transversely polarized incoming virtual photons, given by 
\begin{equation}
\epsilon=\frac{\Gamma_L}{\Gamma_T} = \frac{1-y}{1-y+y^2/2} \; .
\end{equation}
At HERA, $\epsilon$ is very close to unity. The function ${\cal
C}(Q^2)$ is related to the vector meson's wave 
function and approaches unity from below for $Q^2 \rightarrow
\infty$. The momentum fraction $x_g$ carried by
the gluon is given by
%
\begin{equation}
x_g \approx \frac{Q^2 + M_V^2}{W^2} \; . 
\end{equation}
%

The main uncertainties in these calculations for $\jpsi$ production come
from relativistic effects in the vector meson wave function (Fermi
motion of the charmed quarks) and from the uncertainty of the
charm quark mass. In the model of Ryskin the effective scale is
$\tilde{Q}^2=(Q^2+M_V^2)/4$, while it is roughly a factor of 2 larger 
in the Frankfurt et al. model.  

As can be seen from eq.~\ref{dsqcd}, pQCD makes a prediction for the
forward elastic differential cross section. In order to predict the
total elastic cross section, the slope parameter $b$ (see
eq.~\ref{bdef}) needs to be known, for which no QCD prediction
exists. The slope has therefore to be taken from experiment and the total
cross section is determined by
\begin{equation}
\sigma^{\gamma^* p} = \frac{1}{b} \frac{d \sigma^{\gamma^*
p}}{dt}\biggr|_{t=0} \quad .
\end{equation}

Concerning inelastic heavy vector meson production, 
substantial progress has been made during the past years with the
introduction of an effective field-theory framework of nonrelativistic
QCD (NRQCD)\cite{BBL1}. In this approach the NRQCD matrix
elements factorize into short-distance coefficients describing the
production of a heavy quark-antiquark system, calculated using
ordinary perturbative techniques, and effects from long-distance
physics describing the hadronization of the quark-antiquark
pair.
The long-distance effects are parameterized in terms of color
multiplet matrix elements (see, e.g., \cite{FleMeh}), which can be
determined from experiment or, at least in principle, from
non-perturbative calculus such as lattice QCD.  In contrast to elastic
vector meson production, the inelastic cross section depends linearly
on the gluon density in the proton.  
 
Compared to previous
work on heavy quarkonium production employing the color singlet model
(see, e.g. \cite{Schuler} for an extensive review), the NRQCD
formalism allows for the possibility that a $c \bar{c}$ pair produced
in a color octet state may evolve non-perturbatively into a $\jpsi$ or
a $\psi^\prime$. Such color octet contributions were in fact
successful in explaining the large $\jpsi$ and $\psi^\prime$
production rates observed by the CDF and D0
collaborations\cite{jpsitev} at the Tevatron. 
  
%
%======================================================================
\section{Heavy Quarkonium Production \protect\newline
in Deep Inelastic Scattering}
%
The subsequent discussion of the experimental data on heavy quarkonium
production will concentrate mainly on the $\jpsi$. We will start with
inclusive production and then turn to the inelastic and exclusive
channels. In order to make meaningful comparisons of the various models 
introduced above with the experimental data, the separation of elastic
and inelastic production of vector mesons is a major experimental issue;   
the distinction between photoproduction and electroproduction is
straightforward through the observation of the scattered electron in
the final state.  

%----------------------------------------------------------------------
\subsection{Inelastic Production of $\jpsi$ Mesons}
%
In general, the ``diffractive'' and ``inelastic'' regimes are
separated by a cut in the variable $z$ (see eq.\ref{zdef}). The
diffractive component is expected at values of $z$ close to unity.
However, large non-diffractive contributions, most importantly color
octet contributions, are expected\cite{FleMeh}  
at high $z$ as well, motivating a model independent analysis without
such a cut. H1 has recently published data on inclusive $\jpsi$
production\cite{Hjpsiep99} (see also \cite{meyerdiss}). The $\jpsi$
was observed in its leptonic decay modes and the kinematic range
covered was $2 < Q^2 < 80$ GeV$^2$ and $40 < \W < 180$ GeV.   
%
\begin{figure}[htb]
%  \begin{minipage}[t]{0.65\textwidth}
%      \makebox[0cm][t]{}        % this box is only there to align text
      \begin{center}
        \includegraphics[width=0.95\textwidth]{kiesling.4}
      \end{center}
%  \end{minipage}
%  \hfill
%  \begin{minipage}[t]{0.33\textwidth}
%      \makebox[0cm][t]{}        % this box is only there to align text
%    \vspace{0.4cm}
    \caption[]{Di-lepton mass spectra for events from inclusive (points) 
and inelastic (histogram) electroproduction of $\jpsi$ mesons as
measured in the H1 experiment\protect\cite{Hjpsiep99}: a) 
$0.2 < z < 0.6$, b) $0.6 < z$. The curves are from fits to the signal
region, convoluting a Gaussian with an exponential tail to account for 
radiation loss in the case of electronic decay, and an exponentially
falling non-resonant background.}
    \label{kiesling.fig4a}
%  \end{minipage}
\end{figure}
%
The di-lepton mass spectra for two different regions of $z$ are shown in
fig.~\ref{kiesling.fig4a}, where the inelastic component of the events was
identified by a cut in the energy deposited close to the proton
(forward) direction ($E_{\rm fwd} > 5$ GeV). This cut excludes the low 
mass excitations $M_X$ of the nucleon and with it the elastic and most of
the quasi-elastic diffractive events. The selection is, however,
believed to retain the color octet contributions. It was verified by
Monte-Carlo methods that this cut is equivalent to a cut $M_X > 10$
GeV. As expected, the inelastic component dominates at lower values of 
$z$ (fig.~\ref{kiesling.fig4a}a)), while the elastic and quasi-elastic component
dominates at high $z$ (see fig.~\ref{kiesling.fig4a}b)).

%
\begin{figure}[htb]
  \begin{minipage}[t]{0.70\textwidth}
      \makebox[0cm][t]{}        % this box is only there to align text
      \begin{center}
        \includegraphics[width=0.95\textwidth]{kiesling.5}
      \end{center}
  \end{minipage}
  \hfill
  \begin{minipage}[t]{0.29\textwidth}
      \makebox[0cm][t]{}        % this box is only there to align text
    \vspace{0.4cm}
    \caption[]{Differential cross sections for the inclusive (open
points) and inelastic ($M_X > 10$ GeV, full symbols) process $ep
\rightarrow e \jpsi X$ for $z>0.2$ from
H1\protect\cite{Hjpsiep99}. The inner error bars are statistical, 
the outer bars combined statistical and systematic errors. The curves
are leading order predictions for inelastic $\jpsi$ production within
the NRQCD factorization approach\protect\cite{FleMeh}. The histogram
is from a Monte-Carlo simulation incorporating soft color
interactions\protect\cite{IngelRath}.}
    \label{kiesling.fig4}      
  \end{minipage}
\end{figure}
%
The corrected differential cross sections with respect to several
kinematic quantities ($Q^2$, transverse momentum squared
$p^2_{t,\psi}$ of the $\jpsi$, $z$, $y^*$ - the rapidity of the
$\jpsi$ in the $\gamma^* p$ center of mass system - and $W$) are
shown in fig.~\ref{kiesling.fig4} for the inclusive as 
well as the inelastic data. For comparison, the results of a NRQCD
calculation\cite{FleMeh} are given, which should describe the inelastic 
part of the cross sections. The predictions   
include contributions from the color octet states $^3P_0$ and $^1S_0$, 
which are of order ${\cal O}(\as)$, and from the color singlet state
$^3S_1$, which is of order ${\cal O}(\ass)$. Note that both
contributions are in leading
order; the color singlet contribution is shown separately. The
color octet contributions are  
found to dominate for all values of $Q^2$, the leading order color
singlet alone falling below the data by factors 2-3. The observed
differences in magnitude between 
predicted and measured cross sections may be indicative of the necessity
to adjust  
the color multiplet matrix elements, determined so far from the Tevatron
data alone. The differences in shape (see e.g. the $y^*$ distribution),
might be influenced by a relative adjustment of the matrix
elements. Note also that a full prediction of $d\sigma/dz$ is not
possible at present\cite{FleMeh} and is therefore not
shown. 

One might be led to the conclusion that, like for the Tevatron
data, color octet contributions seem required by the $ep$ data, the
color singlet contribution alone not being sufficient to saturate the
cross sections. Still, since the calculations are only in leading
order, one cannot yet claim the necessity for color octet
contributions in the HERA data.


%---------------------------------------------------------------------------
\subsection{Exclusive Production of $\jpsi$ Mesons}
%
Both ZEUS\cite{Zjpsirhoepex99} and H1\cite{Hjpsiep99} have recently
published data on exclusive electroproduction of $\jpsi$ mesons, $ep
\rightarrow e V p$. The $Q^2$ ($W$) ranges covered are $2 < Q^2 <
80$ GeV$^2$ ($25 < W  < 180$ GeV) for H1 and $2 < Q^2 < 40$
GeV$^2$ ($50 < W < 150$ GeV) for ZEUS. Exclusive production of
$\jpsi$ mesons is ensured in both experiments by requiring exactly two
oppositely charged leptons in the main detector and by suppressing
quasi-elastic (proton dissociation) events by requiring no activity in the
forward (proton direction) parts of the detector. Residual backgrounds 
in the $\jpsi$ mass region (coming mainly from low mass proton
dissociation and from the Bethe-Heitler process $ep \rightarrow l^+
l^- p$) are of order few percent and have been subtracted using
Monte-Carlo simulations. The electroproduction cross section
is converted to a virtual photoproduction cross section using 
\begin{equation}
\frac{d^2 \sigma(ep \rightarrow e \jpsi p)}{dy dQ^2} = \Gamma
\sigma(\gamma^* p \rightarrow \jpsi p) \; ,
\label{virtsig}
\end{equation}
where the function $\Gamma$ is the flux of virtual photons\cite{handconv},
defined by
\begin{equation}
\Gamma
= \frac{\alpha_{\rm em}}{2 \pi y Q^2} \left(1 + (1-y)^2\right) \; .
\end{equation}

%
\begin{figure}[htb]
  \begin{minipage}[t]{0.65\textwidth}
      \makebox[0cm][t]{}        % this box is only there to align text
      \begin{center}
        \includegraphics[width=0.95\textwidth]{kiesling.6}
      \end{center}
  \end{minipage}
  \hfill
  \begin{minipage}[t]{0.34\textwidth}
   \vspace{0.4cm}
    \caption[]{Exclusive electroproduction of 
$\jpsi$ mesons from H1\protect\cite{Hjpsiep99} and ZEUS\protect\cite
{Zjpsirhoepex99},
converted to virtual photoproduction (see eq.~\protect\ref{virtsig})
together with real photoproduction 
data (see discussion below). The full error bars correspond to
statistical and systematic errors added in quadrature. The
dashed-dotted lines result from fits of the 
form $W^\delta$. Typical values are $\delta \approx 0.8-0.9$, with no
significant variation with $Q^2$. The data are
compared to various theoretical models (see text).} 
    \label{kiesling.fig5}
  \end{minipage}
\end{figure}
%
$W$ $dependence$: Figure~\ref{kiesling.fig5} shows the resulting cross section
measurements for $\gamma^* p
\rightarrow \jpsi \; p$ as a function of $W$ for various values of
the momentum transfer $Q^2$ in the range $2 < Q^2 < 80$ GeV$^2$. For
comparison, also the photoproduction ($Q^2 = 0$)
cross sections are shown (see also below). All cross sections exhibit a
steep rise with $W$, in contrast to the corresponding data on
exclusive light vector meson production (e.g. $\rho$
electroproduction\cite{Zjpsirhoepex99,Hrhoepel99,Proskur99}) 
in a similar (low) $Q^2$ range. Fits of the form $\sigma \sim W^\delta$ for
given $Q^2$ yield values $\delta \approx 0.8-0.9$ with typical
errors of about 0.2, clearly incompatible with the ``soft Pomeron'' 
slope of about 0.22 (see eq.~\ref{wdep}). The dash-dotted lines give the
results of these fits. Although in the $Q^2=10.1$ GeV$^2$ bin a
somewhat large slope 
($\delta=1.3 \pm 0.4$) is measured, no significant variation of the
slope with $Q^2$ is observed within the present precision.
In addition to the phenomenological description,  the full and
dashed lines in the figure are results of calculations based on
pQCD\cite{FKS98} using two different parameterizations for the gluon
density inside the proton(GRV\protect\cite{GRV95},
MRSR2\protect\cite{MRS96}). The absolute normalization of these
predictions is sensitive to the mass $m_c$ of the charm quark (see
eq.~\ref{dsqcd}). This is 
indicated in fig.~\ref{kiesling.fig5} by the length of 
the little arrows denoting the change in normalization at the point
$W=200$ GeV, when $m_c$ is changed from 1.4 to 1.5 GeV.
%
%\begin{equation}
%\frac{d \sigma^{\gamma p}}{d t} \propto e^{-b|t|}
%\left(\frac{W^2}{W^2_0}\right)^{2(\alpha(t)-1)} 
%\end{equation}

$Q^2$ $dependence$: The $Q^2$ dependence of the cross section for
$\gamma^* p \rightarrow 
\jpsi \; p$ in the $W$ range between 40 and 160 GeV (mean value at $W =
90$ GeV) is shown in fig.~\ref{kiesling.fig6}. At large $Q^2$, the naive
expectation from QCD (see eq.~\ref{dsqcd}) would suggest a $Q^6$
dependence of the cross section. However, one has to take into account
the $Q^2$ 
evolution of the gluon distribution which is known to increase
strongly as $Q^2$ grows, and the function ${\cal C}(Q^2)$, which both
reduce the $Q^2$ dependence. Parameterizing the $Q^2$ dependence in
the form $\propto (Q^2 + M_V^2)^{-n(V)}$, the $\jpsi$ data are well
described\cite{Hjpsiep99} by the fit with $n(\jpsi) = 2.38 \pm 0.11$. 
The dotted line in fig.~\ref{kiesling.fig6} is from the pQCD model of Frankfurt
et al.\cite{FKS98}  
with the MRSR2 gluon distribution, which was able to describe the $W$
dependence of the total cross section (see fig.~\ref{kiesling.fig5}). This model 
also reasonably well describes the $Q^2$ dependence. For comparison,
the cross section for $V = \rho$ is also shown in fig.~\ref{kiesling.fig6}, at
somewhat lower $W$. Fitting the $\rho$ data with the same $Q^2$
dependence yields similar values as for the $\jpsi$: $n(\rho) = 2.24
\pm 0.09$ from H1\cite{Hrhoepel99}, in good agreement with
ZEUS\cite{Zjpsirhoepex99} ($n(\rho) = 2.32 \pm 0.10$). While the
$\rho$ dominates the cross section at low $Q^2$, the $\jpsi$
approaches the $\rho$ with increasing $Q^2$. At  sufficiently large
$Q^2$ ($Q^2 \gg M_V^2$), the ratio of cross sections should, according 
to the additive quark model, only
depend on the electric charges of the $q \bar{q}$  state forming the
vector meson, $\rho : \omega : \phi : \jpsi = 9 : 1 : 2: 8$. This
limit seems to be reached for $Q^2 > 20$ GeV$^2$. At very large $Q^2$
Frankfurt et al.\cite{FKS96} predict an enhancement of the
$\jpsi$ cross section, relative to the asymptotic quark model limit,
by a factor of about 3.5. Accordingly, the $\jpsi$
cross section would overtake the $\rho$ cross
section by such an amount. The presently accessible $Q^2$ range (up to 
50 GeV$^2$) is obviously not yet high enough to test this prediction.  

%
\begin{figure}[htb]
  \begin{minipage}[t]{0.65\textwidth}
      \makebox[0cm][t]{}        % this box is only there to align text
      \begin{center}
        \includegraphics[width=0.95\textwidth]{kiesling.7}
      \end{center}
  \end{minipage}
  \hfill
  \begin{minipage}[t]{0.34\textwidth}
      \vspace{0.2cm}
      \caption[]{Exclusive virtual-photoproduction cross sections for
        $\jpsi$ ($\langle W \rangle =90$ GeV) and $\rho$ mesons
($\langle W \rangle = 75$ GeV) as  
        function of the momentum transfer $Q^2$. The full lines 
are from dipole-like fits (see text) to the measurements. At large $Q^2$,
the cross sections for $\rho$ and $\jpsi$ approach the SU(4) limit of
9:8. The curves are from fits of the form $(Q^2 + M_V^2)^{-n(V)}$,
resulting in $n(\rho)= 2.24 \pm 0.09$\cite{Hrhoepel99} and $n(\jpsi)=
2.38 \pm 0.11$\protect\cite{Hjpsiep99} } 
      \label{kiesling.fig6}
  \end{minipage}
\end{figure}
%

$t$ $Distribution$: Since for electroproduction events the scattered
electron is observed in the final state, the momentum transfer $t$ can
be determined unambiguously . For $\tbb$ values below about 1 GeV$^2$ the
differential cross section for elastic vector meson production can be
described by a single exponential fall-off away from $\tbb = 0$,
characterized by the slope parameter $b$ (see eq.~\ref{bdef}).
After correcting the experimental $\tbb$ distributions for residual
backgrounds (mainly from proton 
dissociation and the Bethe-Heitler process) H1\cite{Hjpsiep99} obtain
$b = 4.1 \pm 0.3 ({\rm stat.}) \pm 0.4 ({\rm syst.})$ GeV$^{-2}$ for
the kinematic range $2 < Q^2 < 80$ GeV$^2$, $ 40 < W < 160$ GeV, in
good agreement with the ZEUS\cite{Zjpsirhoepex99} value of $b = 5.1 \pm
1.1 ({\rm stat.}) \pm 0.7 ({\rm syst.})$ GeV$^{-2}$ for the range  $2
< Q^2 < 40$ GeV$^2$, $55 < W < 125$ GeV. The fits, carried out in the 
$\tbb$ region below 1 GeV$^2$, are consistent with a single exponential
for the elastic $t$ distribution in both experiments. The statistical
accuracy of the $\jpsi$ electroproduction data do not yet allow for a
determination of $b$ as function of $W$ to test shrinkage. 

Note that the slope parameter $b(\jpsi)$ for elastic $\jpsi$
production (average $Q^2 \approx 8$ GeV$^2$) is substantially smaller
than the one for light vector mesons: For elastic $\rho$ 
production\cite{Zjpsirhoepex99,Hrhoepel99} at a similar $Q^2$, e.g., the
value $b(\rho) \approx 8$ GeV$^{-2}$ is found. This behavior is 
qualitatively explained in the optical picture of scattering, where
the $b$ parameter can be viewed as the sum of the scattering
partner's radii squared: $b \approx R_p^2 + R_V^2$. Due to the
large mass of the charm quarks their velocities in a $\jpsi$ bound
state is small as is, as a consequence of the uncertainty principle,
the $\jpsi$ radius. The slope $b$ is therefore given mainly by the
(larger) radius of the proton, which is composed of light quarks. On
the basis of this it is expected that the $b$ slope for elastic
$\Upsilon$ production will be similar to the one for the $\jpsi$.  

Compared to elastic production, the $b$ slopes for $\jpsi$ production
with proton dissociation are smaller\cite{meyerdiss,Hrhoeldiss99} by
typically a factor of 2, 
both for heavy and light vector mesons. Again the ``optical'' picture
provides a qualitative understanding: The elastic scattering is a
coherent effect involving the entire proton, while the 
dissociative process is thought to proceed as quasi-elastic scattering
off the (``smaller'') constituent quarks in the proton\cite{MPP80}.


%\begin{figure}[b]
%  \begin{minipage}[t]{0.55\textwidth}
%      \makebox[0cm][t]{}        % this box is only there to align text
%      \begin{center}
%        \includegraphics[width=0.95\textwidth]{figures/brun99p14b.eps}
%      \end{center}
%  \end{minipage}
%  \hfill
%  \begin{minipage}[t]{0.44\textwidth}
%       \vspace{0.7cm}
%    \caption[]{Ratio of production cross sections of $\jpsi$ and $\rho$
%      mesons \protect\cite{Zjpsirhoepex99}. The asymptotic $SU(4)$ ratio
%      is also shown.} 
%    \label{fig7}
%  \end{minipage}
%\end{figure}
%
$Helicity$ $analysis$: In order to disentangle the longitudinal and
transverse virtual photon components in exclusive vector meson
production, the helicity structure of the interaction can be
investigated. This is usually done in the $s$-channel helicity frame,
where the meson emission direction in the $\gamma^*  p$ rest system
serves as the 
quantization axis. At given values for $W$ and $Q^2$ the exclusive
production and decay of a meson  $V \rightarrow l^+ l^-$ is described
by three angles: $\theta^*$ and $\phi$, the polar and azimuthal
angles of the positively charged lepton in the $V$ rest system, and
the angle $\Phi$  between the production plane of the meson $V$ and
the electron scattering plane. The probability distribution of these
three angles can be expressed by the spin density matrix elements or
linear combinations of these (see, e.g., \cite{SchilWolf73}). Assuming 
$s$-channel helicity conservation (SCHC), the number of independent angles
reduces to two, where the polarization angle $\Psi = \phi - \Phi$
absorbs one degree of freedom. The two remaining angular distributions 
depend on the matrix elements (bilinear combinations of the helicity
amplitudes) $r_{00}^{04}$ and $r_{1,-1}^1$, where the lower indices
denote the helicities of the vector meson in the bilinear combination
and the upper indices are conventional\cite{SchilWolf73} and refer to 
the photon helicity producing the vector meson. Integrating over the
other respective angle, the following distributions are obtained:  
%
\begin{eqnarray}
\frac{d\sigma}{d \cos\theta^*} &\propto& 1 + r_{00}^{04} + \left( 1 - 3
r_{00}^{04}\right) \cos^2 \theta^*  \\
\frac{d \sigma}{d\Psi} &\propto& 1 - \epsilon \, r_{1 -1}^1 \cos 2
\Psi \; .
\end{eqnarray}
%
\begin{figure}[tb]
  \begin{minipage}[t]{0.70\textwidth}
      \makebox[0cm][t]{}        % this box is only there to align text
      \begin{center}
        \includegraphics[width=0.95\textwidth]{kiesling.8}
      \end{center}
  \end{minipage}
  \hfill
  \begin{minipage}[t]{0.29\textwidth}
    \vspace{0.9cm}
    \caption[]{Distributions of $\cos\theta^*$ (a) and
b)) and of $\Psi$ (c) and d)) for the positive lepton from 
$\jpsi$ decay from the quasi-elastic (elastic plus proton-dissociative)
process $e p \rightarrow e \jpsi X$  in the range $40 < W < 160$ GeV,
for two intervals in $Q^2$ from H1\protect\cite{Hjpsiep99}. The inner
error bars indicate statistical, 
the outer error bars statistical and systematic errors added in
quadrature. The lines are the results of fits described in the text.}
    \label{kiesling.fig8a}
  \end{minipage}
\end{figure}
%
The acceptance corrected distributions from recent H1
measurements\cite{Hjpsiep99} of $\cos\theta^*$ and $\Psi$
are shown in fig.~\ref{kiesling.fig8a} for two ranges of $Q^2$. The
$\cos\theta^*$ distribution nicely supports the expectation of
a dominating longitudinal photon polarization as $Q^2$ is
increased: Fits to the data from H1\cite{Hjpsiep99} yield the values
$r_{00}^{04} = 0.15 \pm 
0.11$ at $\langle Q^2 \rangle = 4$ GeV$^2$ and $r_{00}^{04} = 0.48 \pm
0.15$ at $\langle Q^2 \rangle = 16$ GeV$^2$, suggesting that the
longitudinal component of the 
$\jpsi$ indeed increases with $Q^2$. The $\Psi$ distributions provide
measurements of $r_{1 -1}^1=0.50 \pm 0.08$ at $\langle Q^2 \rangle  =
4$ GeV$^2$ and $r_{1 -1}^1=0.39 \pm 0.13$ at $\langle Q^2 \rangle =
16$ GeV$^2$. Assuming natural 
parity exchange (NPE), the two matrix elements are related by $r_{1
-1}^1 = \frac{1}{2}(1 - r_{00}^{04})$. Using this relation and the
measured values for $r_{00}^{04}$ one obtains agreement within one
standard deviation with the direct measurement of $r_{1 -1}^1$ from
the $\Psi$ distributions. Thus both SCHC and NPE are
hypotheses supported by the elastic $\jpsi$ production data at HERA.

Assuming SCHC to hold, the measurements of $r_{00}^{04}$ can be used
to determine the ratio $R$ of longitudinal to transverse virtual
photon cross sections: 
%
\begin{equation}
R = \frac{\sigma_L}{\sigma_T} =
\frac{1}{\epsilon}\frac{r_{00}^{04}}{1-r_{00}^{04}} \; .
\end{equation}
%
For the two $Q^2$ regions H1 obtain the results $R = 0.18 {{+0.18} \atop
{-0.14}}$ at $\langle Q^2 \rangle  = 4$ GeV$^2$ and  $R = 0.94
{{+0.79} \atop {-0.43}}$ at $\langle Q^2 \rangle = 16$ GeV$^2$,
compatible with the ZEUS 
result\cite{Zjpsirhoepex99} of $R = 0.41 {{+0.45} \atop
{-0.52}}$ at $\langle Q^2 \rangle = 5.9$ GeV$^2$. Together with the
photoproduction 
data (see below) the measurements suggest a rise of $R$, i.e. an increase of
the fraction of longitudinally polarized virtual photons, with
$Q^2$. 

Within the above-mentioned pQCD models, $R$ is approximately
proportional to $Q^2/M_V^2$. This relation can be tested
comparing $\rho$ with $\jpsi$ production with the result that the
value for $R(\rho)$ from elastic $\rho$ production is found 
to be significantly larger\cite{Zjpsirhoepex99,Hrhoepel99} than
$R(\jpsi)$ from elastic $\jpsi$ production. But $R(\rho)$ and
$R(\jpsi)$ are of the same order, when the value of $Q^2$ is
reduced appropriately in elastic $\rho$ production to match the values
of $Q^2/M_V^2$ in both reactions.  

\subsection{Electroproduction of $\psip$ Mesons}
%
%\begin{equation}
%R = \frac{\sigma^{\gamma^* p}(\psi^\prime)}{\sigma^{\gamma^* p}(J/\psi)}
%\end{equation}
H1\cite{Hjpsiep99} has observed quasi-elastic production of
$\psi(2S)$ in deep inelastic scattering using the decay channel
$\psi(2S) \rightarrow \jpsi \; \pi^+ \pi^-$. No separation of elastic and 
proton dissociation was possible due to reasons of lacking
statistics. Qualitatively, the cross section ratio
$\sigma(\psi(2S))/\sigma(\jpsi)$ is expected to be 
significantly smaller than unity due to the smaller wave function at
the origin for the $\psi(2S)$ particle ($1^{\rm st}$ radial excitation)
compared to the $\jpsi$. Within the $Q^2$ range from 1 to 80 GeV$^2$
the cross section ratio $\sigma(\psi(2S))/\sigma(\jpsi)$ was
determined and compared to pQCD models\cite{FKS96,Nemch94}, which
predict a rise of this ratio from $\approx 0.2$ (photoproduction, see
below) to $\approx 0.4$ for large $Q^2$ ($Q^2 \gg M_V^2$), in
agreement with the measurements.

%
%\begin{figure}[b]
%  \begin{minipage}[t]{0.50\textwidth}
%      \makebox[0cm][t]{}        % this box is only there to align text
%      \begin{center}
%        \includegraphics[width=0.95\textwidth]{figures/brun99p22b.eps}
%      \end{center}
%  \end{minipage}
%  \hfill
%  \begin{minipage}[t]{0.49\textwidth}
%       \vspace{0.2cm}
%    \caption[]{Ratio of production cross sections for $\jpsi$ and $\psip$
%      mesons as function of the photon virtuality $Q^2$.}
%    \label{fig8}
%  \end{minipage}
%\end{figure}
%


%-----------------------------------------------------------------------
%\subsection{Electroproduction of $\phi$ Mesons}
%
%\begin{figure}[b]
%  \begin{minipage}[t]{0.65\textwidth}
%      \makebox[0cm][t]{}        % this box is only there to align text
%      \begin{center}
%        \includegraphics[width=0.95\textwidth]{figures/brun99p11.eps}
%      \end{center}
%  \end{minipage}
%  \hfill
%  \begin{minipage}[t]{0.34\textwidth}
%    \vspace{0.2cm}
%    \caption[]{Dependence of energy slope parameter $\delta$ (see text)
%      for the production cross section for $\phi$  mesons as function of
%      the photon virtuality $Q^2$.} 
%    \label{fig9}
%  \end{minipage}
%\end{figure}
%

%======================================================================
\section{Photoproduction of Heavy Quarkonia}
%

Diffractive photoproduction is one of the 
classical domains for the successful application of Regge theory.
Perturbative QCD 
calculations, on the other hand, are deprived of their ``natural''
scale $Q^2$ and cannot be applied here in the usual way, since in
photoproduction the photon virtuality does not provide a hard
scale. However, 
due to the $Q^2$ effect in the emission of photons from the incoming
electron or positron suppressing high virtualities,
photoproduction events are more abundant at HERA compared to deep
inelastically produced events, allowing for selection of data
exhibiting an alternative hard scale. In vector meson photoproduction,
the mass of a produced heavy quarkonium state and/or the momentum
transfer at the proton vertex may serve as such scales. For inelastic 
reactions, mediated dominantly by photon-gluon fusion, Regge theory
cannot be applied but full next-to-leading order (NLO) QCD calculations
exist\cite{Kraemer95}.
 
\subsection{Inelastic Photoproduction of Charmonium} 
%
Both H1\cite{Hjpsiin99} and ZEUS\cite{Zjpsiin98} have recently
presented analyses on inelastic production of $\jpsi$ in 
the photoproduction limit. The inelastic component was extracted by a
cut in $z$: ZEUS use $0.55 < z < 0.9$ with a $W$ range of $50 < W <
180$ GeV, while H1 selects $0.3 < z < 0.9$ and $ 60 < W < 180$
GeV. The data are corrected for the lower $z$ cuts using Monte Carlo
techniques. The differential cross sections $d \sigma/dz$ for both
experiments (not shown) are well
described by the color singlet model in NLO, but are not compatible with 
a leading order color octet model, as found in the ZEUS analysis. It
has been pointed out\cite{kniehl97}, however, that the theoretical
calculations contain considerable uncertainties, so that the color
octet model based on the framework on NRQCD\cite{BBL1} cannot be ruled
out by the HERA data. This should be compared with the conclusions
from the electroproduction  data (see above), where the leading order
color singlet model was not sufficient to describe the measurements,
but, due to missing NLO predictions, octet contributions could still
not be claimed. 

The ratio of the inelastic $\psi(2S)$ over $\jpsi$ photoproduction
cross section was remeasured recently by H1\cite{Hjpsiin99} and
ZEUS\cite{Zinelcharmon99} at photon proton center of mass energies $60 
< W < 180$ GeV (H1) and $50 < W < 180$ GeV (ZEUS). H1 finds the cross
section ratio  
to be $0.21 \pm 0.06$ (the error contains statistical and systematic
uncertainties), in good agreement with the ZEUS value of $0.242 \pm
0.065 {\rm (stat.)} {+0 \atop -0.019}{\rm (syst.)}$. Comparing to
similar measurements from fixed target experiments at typical energies 
$W \approx 10$ GeV,  no dependence of this ratio on $W$ can be
observed within the large uncertainties. 

%-----------------------------------------------------------------------
\subsection{Exclusive Photoproduction of $\jpsi$ Mesons}
%
The total cross section for elastic photoproduction of light vector
mesons, $\gamma p \rightarrow V  p$, $V = \rho, \omega, \phi$, is well
described within the Regge model by soft Pomeron exchange (see,
e.g. \cite{Proskur99}), predicting a universal, weak dependence
$\sigma \sim W^{0.22}$ . This model, however, cannot accommodate heavy 
vector meson photoproduction ($V=\jpsi, \psi(2S), \Upsilon$), as clearly
established with the HERA data
\cite{Hjpsigpel96,Zjpsigpel97,HjpsigpJer97,ZjpsiJer97,Hpsipgp98,Zjpsigpin97}
shown in   
fig.~\ref{kiesling.fig10}. Fitting the $\jpsi$ cross section to the form
$\sigma \sim W^\delta$ yields $\delta \approx 0.80$, resulting from a
much steeper rise of the cross section with increasing $W$ compared 
to the universal expectation. 
%
\begin{figure}[bt]
  \begin{minipage}[t]{0.65\textwidth}
      \makebox[0cm][t]{}        % this box is only there to align text
      \begin{center}
        \includegraphics[width=0.95\textwidth]{kiesling.9}
      \end{center}
  \end{minipage}
  \hfill
  \begin{minipage}[t]{0.34\textwidth}
    \vspace{0.1cm}
    \caption[]{Data on the total photoproduction cross section and
cross sections for elastic vector meson production as function of the 
photon-proton center of mass energy $W$, for fixed target experiments
and from the HERA collider. The cross sections for light vector mesons
$V=\rho, \omega, \phi$ are well described by the soft Pomeron model, while
$\jpsi$ cross sections are rising much steeper with $W$. Statistics
for $\Upsilon$ production is not yet sufficient to determine the $W$
dependence of the cross section at HERA.}
    \label{kiesling.fig10}
  \end{minipage}
\end{figure}
%

As is discussed in more detail
elsewhere\cite{Proskur99}, the electroproduction cross 
sections for the light vector mesons also start to rise faster with
$W$ than the soft expectation , when the photon virtuality $Q^2$
approaches a ``hard'' scale, typically 10 GeV$^2$. The fact that the
total cross section of elastic photoproduction of $\jpsi$ mesons shows 
already ``hard'' behavior suggests that the hard scale is provided by
the mass of the heavy vector meson itself ($M_{\jpsi}^2 \sim 10$ GeV$^2$):
Perturbative QCD can be applied to the deep inelastic and to the
photoproduction processes, 
since the relevant scale $\tilde{Q}^2$ for these processes can be
written\cite{Ryskin93} as   
\begin{equation}
\tilde{Q}^2 = \frac{Q^2 + M^2_V}{4} \; ,
\end{equation}
%
which is much larger than $\lqcd$, even for photoproduction ($Q^2=0$)
of heavy vector mesons such as the $\jpsi$.

Figure \ref{kiesling.fig11} shows the cross section $\gamma p \rightarrow \jpsi 
\; p$ from the fixed target experiments\cite{E40182,E51684} and the
results from ZEUS\cite{Zjpsigpel97} and
H1\cite{Hjpsigpel96,HjpsigpJer97} 
together with the leading order pQCD predictions of Frankfurt et 
al.\cite{FKS98}, using various parameterizations of the gluon 
distribution inside the proton, such as 
GRV(HO)\cite{GRV95}, MRSR2\cite{MRS96}, and CTEQ4M\cite{CTEQ97}. Note the
large $W$ range of the preliminary H1 data\cite{HjpsigpJer97}, reaching
up to $W=285$ GeV, close to the kinematic limit of HERA (300
GeV). These measurements have become possible through the use of a
novel hardware trigger system at level 2, based on the neural network
architecture\cite{l2syst}. 
%
\begin{figure}[htb]
  \begin{minipage}[t]{0.65\textwidth}
      \makebox[0cm][t]{}        % this box is only there to align text
      \begin{center}
        \includegraphics[width=0.95\textwidth]{kiesling.10}
      \end{center}
  \end{minipage}
  \hfill
  \begin{minipage}[t]{0.34\textwidth}
    \vspace{0.2cm}
    \caption{Total cross section for elastic photoproduction of
$\jpsi$ mesons from
HERA\protect\cite{Hjpsigpel96,Zjpsigpel97,HjpsigpJer97} and fixed
target experiments\protect\cite{E40182,E51684}. The power law fit
shown (full line) is for the HERA data alone. The curves are from
a pQCD model\protect\cite{FKS98} using various assumptions of the gluon
density (see text). The absolute normalization of the pQCD predictions
strongly depends on the mass of the charmed quark. Note that these
models predict a ``convex'' shape in this $\log\sigma$ vs $\log W$
representation.}    
      \label{kiesling.fig11} 
  \end{minipage}
\end{figure}
%

While there is a large uncertainty in the absolute value of the prediction,
depending strongly on the charm quark mass (see also
fig.~\ref{kiesling.fig5}), the $W$ dependence is a ``hard'' prediction from
pQCD: The slope of the data is best described using MRSR2 or CTEQ4,
while GRV(HO) is slightly too steep. Fitting a power law ($\sigma \sim 
W^\delta$) to the combined HERA and fixed target data
yields\cite{Hdiffjpsi99} $\delta = 0.83 \pm 0.05$, much steeper than
the prediction of the Donnachie and Landshoff model using the soft
Pomeron ($\delta \approx 0.22 - 0.32$). A phenomenological
solution to this problem was offered by Donnachie and
Landshoff\cite{twopom1}, introducing a second, ``hard'', Pomeron
trajectory  $\alpha_{\rm P^\prime}(t) = 1.4 + 0.1 t$. This trajectory
exhibits a much stronger $W$ rise (the intercept $\alpha_{\rm
P^\prime}(0) = 1.4$, as opposed to 1.08 from the soft Pomeron), and a
much reduced $W$ dependence of the $b$ parameter 
($\alpha^\prime_{\rm P^\prime} = 0.1$ instead of 0.25 GeV$^{-2}$ for the
soft Pomeron). A two-Pomeron fit, adjusting the
relative contributions of the two Pomerons, is able to describe
the data\cite{Hdiffjpsi99}. A distinctive feature of the two-Pomeron
model as opposed to the pQCD prediction is its $W$ dependence: While
the pQCD model suggests a ``convex'' shape in the representation $\log
\sigma$ vs $\log W$ (see fig.~\ref{kiesling.fig11}), the two-Pomeron model
predicts a ``concave'' shape with an accelerated rise at the end of
the HERA kinematic range. Within the present experimental accuracy,
neither of the two shapes can be excluded.

$t$ $distribution$: For photoproduction, the momentum transfer $t$ at
the proton vertex can be approximated by the transverse momentum squared, 
$p_T^2$, of the $\jpsi$. The $p_T^2$-distribution can then be used to fit 
an exponential exp$(-b \tbb)$ for low values of $\tbb$ (typically $\tbb < 
1$ GeV$^2$).  The resulting $b$ parameters are very close to the ones
obtained from electroproduction: H1\cite{Hdiffjpsi99}, e.g., find a
value of $b = 4.2 \pm 0.2 ({\rm stat.}) \pm 0.6 ({\rm syst.})$, in 
good agreement with the result from ZEUS\cite{Zjpsiph97}, $b = 4.6 \pm
0.4 ({\rm stat.}) {{+0.4} \atop {-0.6}}({\rm syst.})$.  
Statistics in photoproduction is larger than in electroproduction so
that the $W$ dependence of the slope parameter can be
studied\cite{Hdiffjpsi99}: In the 
range $40 < W < 150$ GeV five $W$ bins were created
and $b(W)$ was parameterized according to eq.~\ref{bdef}, where the
shrinkage parameter $\aPp$ was allowed to freely vary in the
fit. The result of the fit to the entire $W$ range is
\begin{equation}
b = (4.5 \pm 0.2) + 4 \cdot (0.18 \pm 0.09)\cdot \ln \left(\frac{W}{90 
{\rm GeV}}\right) \left[{\rm GeV}^{-2}\right] \; .
\end{equation}
Shrinkage due to the soft Pomeron model demands $\aPp = 0.25$,
compared to $0.18 \pm 0.09$ from the fit. While the data are clearly
compatible with being independent of $W$, shrinkage cannot
be excluded within the present uncertainties.


%\begin{figure}
%  \begin{minipage}[t]{0.65\textwidth}
%      \makebox[0cm][t]{}        % this box is only there to align text
%      \begin{center}
%        \includegraphics[width=0.95\textwidth]{figures/brun99p19.eps}
%      \end{center}
%  \end{minipage}
%  \hfill
%  \begin{minipage}[t]{0.34\textwidth}
%    \vspace{0.6cm}
%    \caption{Slope parameter as function of the $\gamma p$
%      center-of-mass energy}
%    \label{fig16}
%  \end{minipage}
%\end{figure}
%



%\begin{equation}
%\sigma^{\rm eff}_{\gamma p} \approx \frac{1}{Q^2} F_2(x,Q^2) \approx
%\frac{x^{-\lambda}}{Q^2} \approx W^{2 \lambda} = W^\delta
%\end{equation}
%
%
%
%----------------------------------------------------------------------
%\subsection{Inelastic Photoproduction of $\psi(2S) Mesons}

%----------------------------------------------------------------------
\subsection{Photoproduction of $\Upsilon$ Mesons}
%
%\begin{figure}
%  \begin{minipage}[t]{0.65\textwidth}
%      \makebox[0cm][t]{}        % this box is only there to align text
%      \begin{center}
%        \includegraphics[width=0.95\textwidth]{figures/brun99p23a.eps}
%      \end{center}
%  \end{minipage}
%  \hfill
%  \begin{minipage}[t]{0.34\textwidth}
%    \vspace{0.3cm}
%    \caption{Mass spectra of $\psip$ mesons from the ZEUS experiment}
%    \label{fig12}
%  \end{minipage}
%\end{figure}
%
%\begin{figure}
%  \begin{minipage}[t]{0.65\textwidth}
%      \makebox[0cm][t]{}        % this box is only there to align text
%      \begin{center}
%        \includegraphics[width=0.95\textwidth]{figures/brun99p23b.eps}
%      \end{center}
%  \end{minipage}
%  \hfill
%  \begin{minipage}[t]{0.34\textwidth}
%    \vspace{0.2cm}
%    \caption{Mass spectra of $\psip$ mesons from the H1 experiment}
%    \label{fig13}
%  \end{minipage}
%\end{figure}
An even larger scale in the process $\gamma p \rightarrow V p$ is
provided when $\Upsilon$ mesons are produced ($V = \Upsilon$). The
naive expectation would suggest a rise of the cross
section with $W$, even steeper than for the $\jpsi$, driven by the
gluon density in the proton which 
becomes larger as $x_g$ decreases and the relevant scale $\tilde{Q}$
(here the mass of the vector meson $V$) increases.  
Both ZEUS\cite{ZelYph98} and H1\cite{HelYph98} have observed
photoproduction of $\Upsilon$ mesons, decaying into $\mu$ pairs. Due
to limited mass resolution, the three radial excitations
$\Upsilon(1S),\Upsilon(2S)$ and $\Upsilon(3S)$ cannot be resolved. The 
main source of background is the non-resonant Bethe-Heitler process,
which is subtracted statistically. Experimentally, the sum of the
products of the elastic $\Upsilon(iS)$ photoproduction cross sections
and their respective branching ratios $B_i$ into $\mu^+ \mu^-$ is
determined. ZEUS obtain $13.3 \pm 6.0 ({\rm stat.}) {{+2.7} \atop
{-2.3}} ({\rm syst.})$ pb at a mean center of mass energy of 120 GeV,
the preliminary value of H1 is $16 \pm 8 ({\rm stat.}) \pm 4 ({\rm
syst.})$ pb at a mean center of mass energy of 160 GeV. Both values
are in mutual agreement and are well described by a QCD model of Martin 
et al.\cite{MRT99}, using parton-hadron duality. Since $\Upsilon$
photoproduction is reduced by a factor of about 400 relative to the
$\jpsi$ (suppressed by $(M_{\jpsi}/M_\Upsilon)^5$, see
eq.~\ref{dsqcd}), it is not surprising that statistics is not yet
sufficient to study the $W$ dependence of the $\Upsilon$ cross section.

%----------------------------------------------------------------------
\subsection{Photoproduction at High Momentum Transfers}
%
Diffractive processes are characterized by an exponential $t$
dependence, $\sigma/d t \sim e^{-b \tbb}$. Most of the above discussions 
concentrated on elastic processes, exhibiting a steeply falling $t$
behavior and therefore limiting the $\tbb$ reach to below $\sim 1$
GeV$^2$. When scattering to larger $\tbb$ values is considered, an
additional hard scale is introduced and improved QCD calculations become
possible\cite{Forsh95,BFLW96}. Calculations concentrating on high
momentum transfers can be applied also
to the lighter vector mesons. 

When $\tbb$ is large, the diffractive
proton dissociation process is expected to dominate, where a gluon ladder
described by the
BFKL formalism\cite{bfklpom} can be exchanged and predictions can be
made for both the $t$ and $W$ dependences\cite{Forsh95,BFLW96}.
%
\begin{figure}[htb]
  \begin{minipage}[t]{0.65\textwidth}
      \makebox[0cm][t]{}        % this box is only there to align text
      \begin{center}
        \includegraphics[width=0.95\textwidth]{kiesling.11}
      \end{center}
  \end{minipage}
  \hfill
  \begin{minipage}[t]{0.34\textwidth}
    \vspace{0.2cm}
    \caption{Differential cross section for $\jpsi$ photoproduction
with proton dissociation,
as measured by H1\protect\cite{Hjpsibigt97} (full symbols). The open
points are taken from \cite{H1jpsigp}, where $Q^2 < 4$ GeV$^2$. The
solid line represents the combination of a pQCD
calculation\protect\cite{BFLW96} for $\tbb > 1$ GeV$^2$ and a
non-perturbative model when  $\tbb < 1$ GeV$^2$. The normalization was
determined using the measured cross section for $\tbb > 1$ GeV$^2$. The 
dashed line represents the prediction from the non-perturbative model
when the full kinematic range is used.}  
    \label{kiesling.fig14}
  \end{minipage}
\end{figure}
%

Both H1\cite{Hjpsibigt97} and ZEUS\cite{Zvectbigt98} have studied
quasi-elastic $\jpsi$ photoproduction at large $\tbb$. As in the elastic
case, the $\tbb$ distribution is obtained from the measured $p_T^2$ of
the final state $\jpsi$, where $t = -p_T^2$ holds to a very good
approximation. As an example, the differential
cross section $d\sigma/dt$ for $\jpsi$ photoproduction with proton
dissociation from H1 is shown in fig.~\ref{kiesling.fig14}(full symbols), the
open points are from elastic photoproduction\cite{H1jpsigp}. Both sets
of data are restricted to the kinematic range $30 < W < 150$ GeV. A
clear decrease of the slope with increasing $p_T^2$ is observed. The
solid line in the figure is the result of a 
pQCD calculation\cite{BFLW96} in the region $\tbb > 1$ GeV$^2$, using a
non-perturbative model for $\tbb < 1$ GeV$^2$. The prediction
agrees well with the data, implying that the pQCD calculation describes
correctly the large $\tbb$ behavior. 

According to the expectations from
perturbative QCD, the diffractive photoproduction of vector mesons at a hard
scale should be, to a good approximation, flavor-independent and
proportional to the electromagnetic coupling of the photon to the
mesonic $q \bar{q}$ state. 
In a ZEUS analysis\cite{Zvectbigt98} the large $\tbb$ behavior in
photoproduction also of the light vector mesons ($\rho, \phi$) has
been studied. It is found that the cross section ratios
$\sigma(\jpsi)/\sigma(\rho)$ and $\sigma(\phi)/\sigma(\rho)$ approach
their quark model limits 8:9 and 2:9, respectively, as $\tbb$ increases. 
These results also suggest that $\tbb$, just like $Q^2$ and $M_V$, can
play the role of a hard scale, enabling perturbative QCD
calculations. A combination of all the three relevant scales may
provide an interesting hard scale $\tilde{Q}^2$: It has in fact been
suggested\cite{Ryskin93} earlier, although limited to heavy quarkonia
 ($\jpsi$) production, to use the combined scale $\tilde{Q}^2 =
(Q^2 + M_V^2 + \tbb)/4$. From the present HERA data it seems that such a
restriction to heavy quarkonia may not be necessary.

%----------------------------------------------------------------------
%\subsection{The $t$-Slope Parameter}
%
%
%
%\begin{figure}
%  \begin{minipage}[t]{0.65\textwidth}
%      \makebox[0cm][t]{}        % this box is only there to align text
%      \begin{center}
%        \includegraphics[width=0.95\textwidth]{figures/brun99p17b.eps}
%      \end{center}
%  \end{minipage}
%  \hfill
%  \begin{minipage}[t]{0.34\textwidth}
%    \vspace{0.2cm}
%    \caption{Ratio of production cross section for the various vector
%      mesons as function of the vector meson mass}
%    \label{fig15}
%  \end{minipage}
%\end{figure}
%


%======================================================================
\section{Conclusions}
%
HERA has provided a wealth of precise data on electro- and
photoproduction of heavy quarkonia, giving access to the rich field of 
diffractive phenomena in $ep$ collisions. While the diffractive processes
involving light vector mesons ($\rho, \omega, \phi$) show the
typical ``soft'' behavior, i.e. weak dependence of the cross section
on $W$, well described by Pomeron
exchange, the corresponding cross sections for heavy quarkonia
($\jpsi$) exhibit a steep rise with $W$, incompatible with the soft
picture. Another prediction of the soft Pomeron model , i.e. shrinkage
of the differential cross section, cannot be seriously tested with
heavy quarkonia at present due to lack of statistics.
 
The $W$ dependence of the cross sections $\sigma(\gamma^{(*)} p
\rightarrow V p, V=\jpsi, \Upsilon)$, however, is explained within
pQCD by the steep rise of the gluon density in the proton with
decreasing $x$ (= increasing $W$). Also in contrast to the soft
Pomeron model, pQCD does not expect shrinkage of the differential
cross section. 
The hard scale necessary to justify pQCD is apparently provided by the
mass of the heavy vector meson. This view is beautifully supported by
diffractive photoproduction of $\jpsi$, where pQCD can successfully
describe the observed cross sections, the $\jpsi$ mass being the only
hard scale in the reaction. 

While the soft, single Pomeron model is unable to account for the
cross section rise in heavy quarkonium production, models with an
additional ``hard'' Pomeron (two new parameters which need adjustment, 
plus the relative strength of the two Pomeron components) 
are able to describe the HERA data again. Also the question of shrinkage,
thought to be a distinctive feature, loses its importance: The hard
Pomeron parameters suggest a largely reduced shrinkage effect. 
As a consequence many of the salient  
features distinguishing Regge models from pQCD predictions need largely
improved statistics, such as shrinkage, the exact shape of the
cross section as function of $W$ (``convex'' or ``concave''), or the
open question of SCHC. Future running at HERA with high luminosity
will significantly improve the precision of the data. 

For future comparisons of theoretical models
with, hopefully, statistically improved data one should keep in mind
that many of the pQCD predictions for 
heavy quarkonium production at present are available only in
leading order, thus precluding serious confrontation with
experiment. One such example is the inelastic production of $\jpsi$ 
mesons, where the size of possible color octet contributions,
apparently necessary for the Tevatron data, cannot yet be
quantified in $ep$ collisions: For photoproduction, the NLO color
singlet calculation is able to describe the differential cross
section $d \sigma / d z$. Additional leading order color octet
contributions, especially at high $z$, would largely exceed the
data. But a suppression of such additional color octet contributions
in higher order, then being in agreement with the HERA data, is not
excluded. On the other hand for 
electroproduction only the LO color singlet and octet calculations are
available. Here, the color singlet 
prediction alone clearly undershoots the data. Due to a lacking
NLO calculation, however, one cannot use this observed discrepancy to
claim the necessity of additional octet contributions. 

To reach a more complete theoretical understanding along with the
expected improvement of the experimental results would be highly
desirable: One of the main motivations for studying heavy vector meson
production in $ep$ collisions is the possibility to directly extract
the gluon density in the proton. This important additional information
from the heavy vector meson sector has yet to be supplied and would
nicely complement other analyses on the gluonic structure of the
nucleon.     

\section*{Acknowledgments}
I am indebted to the organizers for this very pleasant meeting. I
would like to thank my colleague Beate Naroska for a critical reading
of the manuscript and many helpful suggestions.
 

%======================================================================
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ZEUS Collaboration:  J. Breitweg et al.,
%     Measurement of Elastic $\Upsilon$ Photoproduction at HERA
%     DESY 98-089 (July 1998)
     Phys. Lett. B437 (1998) 432.
%
\bibitem{HelYph98}
H1 Collaboration: Observation of $\Upsilon$ Production at HERA,
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%
\bibitem{MRT99}A.D. Martin, M.G. Ryskin and T. Teubner,
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%
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%
\bibitem{Zvectbigt98}ZEUS Collaboration: Study of Vector Meson
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%
\end{thebibliography}


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