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\begin{document}
% \eqsec  % uncomment this line to get equations numbered by (sec.num)
\title{EXPERIMENTAL RESULTS ON EXCLUSIVE VECTOR MESONS AND HEAVY QUARKONIUM
%\thanks{Presented at DIS 2002}
}
\author{P. THOMPSON
\address{School of Physics and Astronomy, \\
University of Birmingham, \\
B15 2TT, UK.\\
E-mail: pdt@hep.ph.bham.ac.uk}\\
}
\maketitle
\abstracts{This contribution reviews recent experimental results on the
production of light and heavy vector mesons from the H1 and
ZEUS collaborations during HERA-I data taking.
The data are compared in detail with the predictions
of theoretical models based on perturbative QCD.}
%\PACS{PACS numbers come here}

\section{Introduction}

The study of vector meson production at HERA provides an experimentally 
clean process which allows
the nature of the strong interaction at high energy 
to be investigated. HERA is a unique facility which allows
simultaneous control of the different kinematic scales: the squared 
mass of the vector meson, $M_{\rm VM}^2$, the virtuality of the exchanged
photon, $Q^2$, and the four-momentum transferred at the proton vertex, 
$t$.  
The studies at HERA involve identifying transitions from long range,
or soft, to short distance, or hard, behaviour as a function of 
the various scales, in order to test the applicability of perturbative 
QCD (pQCD).  It is hoped the study of 
hard diffraction processes will lead to a better
understanding of the vacuum-exchange and the strong interaction.


The kinematics for exclusive, or diffractive, vector meson production
$ep \rightarrow e({\rm VM}) Y$
are described in terms of the $ep$ centre-of-mass-energy squared
$s=(k+p)^2$, the virtuality of the photon 
$Q^2=-q^2=-(k-k')^2$,
the square of the centre-of-mass energy of the initial photon-proton system
$W^2 = (q+p)^2$
and the four-momentum transfer squared
$t=(p-p_Y)^2$. Here $k$ ($k'$) is the four-momentum
of the incident (scattered) lepton
and $q$ is the four-momentum of the virtual photon.
The four-momentum of the incident proton is denoted by $p$
and $p_Y$ is the four-momentum of the system $Y$, which
represents either an elastically
scattered proton or a dissociated proton system.


\section{Vector Meson Dominance and Regge Theory}

In the Vector Meson Dominance (VMD) model the 
photon is assumed to fluctuate into a vector meson at a large
distance before the target with the vector meson subsequently undergoing a
soft scattering from the proton target.  In the model, the vector meson 
retains the helicity of the photon and, therefore, s-channel helicity 
conservation (SCHC) is satisfied.  The VMD model can be combined
with Regge phenomenology, which has been successful in parameterising 
soft hadronic elastic and total cross sections. At high energy, in Regge 
theory, the cross section is dominated by the ``soft pomeron'' trajectory 
with an intercept $\alphapom(0) = 1.08$ and a slope 
$\alphapom^{\prime}(t)= 0.25 \ {\rm GeV^2}$~\cite{Donnachie:hf}. 
The intercept describes the experimentally observed 
weak energy dependence $\sim W^{0.2}$ of the total and elastic hadron-hadron
cross sections using $\sigma_{tot} \propto W^{\delta}$ where 
$\delta = 2(\alphapom(0)-1)$.
The soft pomeron is characterised by scattering at small angles and
exhibits an
exponential $t$ dependence $d\sigma / d t \propto e^{b(W)t}$, where $b(W)$ 
is the slope parameter. The foward scattering 
shows a logarithmic shrinkage with increasing $W$ given by
$b = b_0 + 4 \alphapom' \log (W/W_0)$.


\section{The Dipole Picture and QCD}

An increasingly popular approach to describe diffractive interactions 
is to consider  the scattering 
of $q \bar{q}$ fluctuations of the virtual photon, 
as colour dipoles scattering off the proton target in the proton rest frame.
The probability of the photon to fluctuate into a $q \bar{q}$
pair may be parameterised in QCD using the vector meson wave function.
The interaction of the dipole with the proton is described 
at lowest order by
the exchange of two gluons in a colour singlet state. 
In the leading logarithmic (LL) approximation, this process
is described by the effective exchange of a gluonic ladder
and the cross section is 
related to the gluon density within the proton.
In QCD, the transverse separation of the dipole 
is given by $r \sim \frac {1} {(z(1-z)Q^2+m_q^2)^{1/2}}$,
where $m_q$ is the mass of the quark and $z$ is the longitudinal momentum 
fraction of the photon carried by the quark.
Therefore, at large $Q^2$ or large $m_q^2$ the dipole may resolve
short distances within the proton suggesting that perturbative
QCD may be applicable. In contrast, at small $Q^2$ and small $m_q^2$ 
the dipole will be large and the process is likely to contain
a significant non-perturbative component.

There are many different approaches combining the dipole approach 
and pQCD \cite{brodsky,strikman,mrt,mcdermott,fks}. 
However, the majority of the models have common features, 
and include the following. 
A fast rise of the cross section with $W$ ($\sigma \sim W^{0.8}$)
due to the rise of the proton gluon density at low $x$. 
At asymptotically large $Q^2$, the cross section
for longitudinally polarised photons is expected to dominate.
The $Q^2$ dependence
in the longitudinally polarised cross section is expected to be
slower than $1/Q^6$ due to the 
effect of the gluon density.
%The underlying perturbative 2 gluon-dipole scattering means that vector meson
%production is expected to follow an approximate flavour symmetry given by
%$\rho : \omega : \phi : J/\psi = 9 : 1 : 2 : 8$, with slight 
%modifications due to the the dipole to vector meson 
%transition.  
The transverse cross section is expected to have a $1/Q^8$ dependence,
although endpoint effects in the wavefunctions of light vector
mesons  mean that there may be large
non-perturbative contributions.   
At low $|t|$, the $|t|$-dependence is expected to be universal 
and the two gluon form-factor leads to 
${\rm d} \sigma /{\rm d}t \propto e^{-4 |t|}$,
with little or no shrinkage expected.

\begin{figure}[h]
\begin{center}
\epsfig{file=overview10.eps,width=0.49\linewidth}
\end{center}
\caption{The cross section versus $W$ of the elastic photoproduction
of $\rho$, $\omega$, $\phi$, $J/\psi$ and $\Upsilon$ vector mesons.}
\label{overview}
\end{figure}


\section{Elastic Vector Meson Photoproduction and Energy Dependence}

In figure~\ref{overview} the cross section for vector meson photoproduction
is shown as a function of the centre-of-mass energy $W$, integrated over low
values of $|t|$.
The light vector mesons show an energy dependence compatible with the
soft pomeron, whereas the $J/\psi$ shows a steeper behaviour $\sim W^{0.7}$.
The $J/\psi$ energy dependence is qualitatively described by 
pQCD models~\cite{mrt,mcdermott} which
use the mass of the $J/\psi$ as a perturbative scale. 
The $J/\psi$ data may be used to study the $W$ 
dependence at different values of $t$, in order to measure the pomeron
trajectory for the process. The intercept and the slope are 
found to be $\alphapom(0) \sim 1.2$ and 
$\alphapom^{\prime} \sim 0.1 \ {\rm GeV^2}$~\cite{Chekanov:2002xi,h1photopsi}, respectively,
incompatible with the trajectory of the soft pomeron derived from 
hadron-hadron scattering.




\section{Energy Dependence in Elastic Electroproduction }


The variation of the energy dependence with $Q^2$ has been studied with 
increasing accuracy for the $\rho$ meson \cite{rhozeus,rhoh1}. 
The $W$ dependence is found to
become steeper with increasing $Q^2$ and tends toward the same value 
as $J/\psi$
photoproduction, indicating that $Q^2$ is a possible scale for the 
hard process.  As yet, there are no quantitative models to which the data can 
be compared. The variation of the $W$ dependence with $Q^2$ has also been 
studied in $J/\psi$ production~\cite{zeusjpsi}.  
The energy dependence is found to be 
similar at all values of $Q^2$.
The $J/\psi$ data are qualitatively described by pQCD models 
in which the
steep energy dependence arises from the gluon density of 
the proton~\cite{mrt,fks}.
The pomeron trajectory for  $J/\psi$ production was  measured
with increasing precision at $\langle Q^2\rangle = 6.8 \ {\rm GeV^2}$,
and is found to be compatible with that measured in photoproduction.

\begin{figure}[h]
\epsfig{file=rhoq2lt.eps,width=0.49\linewidth}
\epsfig{file=rhor.eps,width=0.49\linewidth}
\caption{(left) The cross sections $\sigma_L$ and $\sigma_T$  
and (right) the ratio $R=\sigma_L/\sigma_T$  
 as a function of $Q^2$ for elastic $\rho$ electroproduction}
\label{rhoq2}
\end{figure}


\section{$Q^2$ Dependence in Elastic Electroproduction}

In pQCD, the cross sections for longitudinally and transversely polarised 
photons are expected to have different $Q^2$ dependences.
The $Q^2$ dependence of the two components has been studied in 
$\rho$ production~\cite{rhozeus,rhoh1}. In figure~\ref{rhoq2}, the data are
parameterised according to $1/(1+Q^2/M^2)^n$.
For the longitudinal cross section a value of $n=1.89\pm 0.03$ is found
which is consistent with the prediction of pQCD.
The parameterisation is found to be inadequate for 
$Q^2 < 5 \ {\rm GeV^2}$, which is thought to be attributable to wave 
function effects.
The ratio $R = \sigma_L/\sigma_T$ is also shown in the figure.
The increase of the ratio with $Q^2$ is compatible with 
calculations~\cite{mrt} based on wave particle 
duality using the details of the wavefunction.  
The $Q^2$ dependence of the cross section in
$J/\psi$ production 
has also been studied~\cite{zeusjpsi}, although the 
determination of $R$, which is
obtained from helicity frame measurements, is not possible due to  
limited statistics. The functional form $(Q^2+M^2)^{-n}$ is found to fit the
$J/\psi$ data, 
although the $\chi^2$ of the fit improves considerably as the minimum 
$Q^2$ of the fit is increased.  Models based on pQCD are able to give a
reasonable description of the $J/\psi$ data.



\section{$t$-dependence at low $t$} 

The $t$-dependence at low $|t|$ is found to be
well described by an exponential form
$\sim e^{bt}$.  
%The $b$-slopes for $\rho$ and 
%$J/\psi$ production as a function of $Q^2$.
The $b$-slope of the $\rho$ decreases with increasing $Q^2$, whereas
the $J/\psi$ $b$-slope is constant with $Q^2$. 
At large $Q^2$ or large $M_{\rm VM}^2$ the $b$-slope
has the value $b \sim 4.5 \ {\rm GeV^2}$, suggestive of an
underlying $t$-dependence which is a feature of pQCD models. 


\section{Helicity Measurements}
The measurement of helicity angles gives information on the 
spin density matrix elements~\cite{Schilling:1973ag}, which in turn
are related to the helicity transition amplitudes.
Precise measurements for $\rho$ and $\phi$ elecroproduction~\cite{h1barb} have established
small violations of SCHC in $r^5_{00}$ indicating that the dominant
single helicity flip amplitude is $T_{01}$.  The data are
well described by a pQCD model~\cite{ivanov} 
based on two gluon exchange in which the 
SCHC violation is attainable if the longitudinal momentum of the
photon is shared asymmetrically by the $q \bar{q}$ pair.
The measurements have been extended~\cite{coll} as a function of 
$|t|$ in events where the final state proton dissociates into a low mass state. 
In figure~\ref{collard}, combinations of spin-density matrix elements
show increasing SCHC violations
with increasing $|t|$, which are well described by the pQCD model. 
The  spin density matrix element dependent on $R$ ($R=\sigma_L/\sigma_T$)
 is observed to be constant in the $t$ range 
studied, indicating that the $b$-slopes for $\sigma_L$ and $\sigma_T$ in 
$\rho$ proton dissociation are similar.  This is suggestive that the 
non-perturbative contribution to $\sigma_T$ is small.
This is further supported by the measurement of elastic 
$\rho$ electroproduction~\cite{rhozeus} in which 
$R$ was found to be constant as 
a function of $W$.


\begin{figure}[h]
\epsfig{file=collard1.eps,width=0.49\linewidth}
\epsfig{file=collard2.eps,width=0.49\linewidth}
\caption{ Measurements of  (left) $r_{00}^5 + 2r^5_{11}$
and (right) $r_{00}^1 + 2r^1_{11}$ as a function of $t$.} 
\label{collard}
\end{figure}

\section{Vector Meson Production at Large $t$}

The diffractive photoproduction of vector mesons
with large negative
momentum transfer squared $t$
at the proton vertex
is a powerful means to probe the parton
dynamics of the diffractive exchange.
The variable $t$ provides a relevant
scale to investigate the application of pQCD.
At sufficiently low values of
Bjorken $x$ (i.e. large values of the centre-of-mass energy $W$),
the gluon ladder
is expected to include contributions from
BFKL evolution~\cite{FKL:1976},
as well as from
standard DGLAP evolution~\cite{Gribov:ri}.


Perturbative QCD models
for the photoproduction of $J\psi$ mesons
have been developed
in the leading logarithmic approximation using either \linebreak
BFKL~\cite{Forshaw:1995ax,Bartels:1996fs,Forshaw:2001pf,Enberg:2002zy}
or DGLAP~\cite{Gotsman:2001ne} evolution.
In the pQCD models,
a non-relativistic approximation~\cite{Ryskin:1992ui} for  the $J/\psi$
wavefunction is used in which
the longitudinal momentum of the vector meson
is shared equally between the quark and the anti-quark.
In this approximation, the vector meson retains the helicity
of the photon such that
$s$-channel helicity conservation (SCHC) is satisfied~\cite{Kuraev:1998ht}.



\begin{figure}[h]
\epsfig{file=zzz.eps,width=0.49\linewidth}
\epsfig{file=h12.eps,width=0.49\linewidth}
\caption{The cross section $d\sigma/dt$ for (left) $\rho$, $\phi$ 
and $J/\psi$ photoproduction and (right) $J/\psi$ photoproduction.}
\label{dsdtzeush1}
\end{figure}


In figure~\ref{dsdtzeush1},  
the large $|t|$ cross sections for $\rho$, $\phi$ 
and $J/\psi$~\cite{Chekanov:2002rm,h1hight} are shown.
The data are described with a power law 
${\rm d} \sigma /{\rm d}t \propto |t|^{-n}$, 
although the value of $n$ in each case increases with the starting 
value of $|t|$ used in the fit.
%n = 3.00 \pm 0.08 \ {\rm (stat.)} \pm 0.05 \ {\rm (syst.)}$
In figure~\ref{dsdtzeush1} (right) 
the data are compared with the predictions from
pQCD calculations in the BFKL leading
logarithmic approximation~\cite{Enberg:2002zy} (solid curve),
including non-leading corrections with fixed $\alpha_s$~\cite{Enberg:2002zy}
(dashed curve)
and including non-leading corrections with
running $\alpha_s$~\cite{Enberg:2002zy} (dotted curve).
The $t$ dependence and normalisation
of the data are well described by the
BFKL LL approximation.
The inclusion of NL corrections with a fixed strong coupling $\alpha_s$
leads to only a small difference with respect to the LL
prediction.  However, with a running $\alpha_s$ the $t$ dependence
becomes steeper and the prediction is unable to describe the data
across the whole $t$ range.
The data are also well described by calculations in the DGLAP LL
approximation~\cite{Gotsman:2001ne} (dashed-dotted curve)
in the region of validity for the model $|t| < M_{J/\psi}^2$.


\begin{figure}[h]
\epsfig{file=h1t.eps,width=0.49\linewidth}
\epsfig{file=zeusdtnew.eps,width=0.49\linewidth}
\caption{The cross section for $J/\psi$ photoproduction (left)
as a function of $W$ in $t$-intervals (right) as a function of $t$ 
at two different values of $W$.}
\label{wintbins}
\end{figure}



In figure~\ref{wintbins}, the cross section is plotted as a function
of $W$ for different intervals of $t$, and also as a function of $t$
in two different $W$ regions.  As can be seen from the plots, 
the $W$ dependence is constant with $t$, and, the $t$ dependence
is constant with $W$. 
The LL BFKL model predicts a stronger rise with $W$ than is observed in the
data, whereas the LL DGLAP model gives a flatter $W$ dependence than 
the data.   
The cross section in $W$ may be used to extract the
pomeron trajectory assuming a
linear form $\alphapom(t) = \alphapom(0) + \alphapom^{\prime}t$,
a fit to the three $\alphapom$ values yields a slope of
$\alphapom^{\prime}= -0.0135 \pm 0.0074 \ {\rm (stat.)}
\pm 0.0051 \ {\rm (syst.)}~{\rm GeV^{-2}}$
with an intercept of
$\alphapom(0) =  1.167 \pm 0.048 \ {\rm (stat.)} \pm 0.024 {\rm (syst.)}$.
The value of the slope parameter $\alpha^{\prime}$
is lower than that observed for the elastic photoproduction of
$J/\psi$ mesons at low $|t|$~\cite{Chekanov:2002xi}.
It is also significantly different from the observations at
low $|t|$ in hadron-hadron scattering.



\begin{figure}[h]
\begin{center}
\epsfig{file=h13.eps,width=0.79\linewidth}
\end{center}
\caption{
The three spin density matrix elements
(top) $r_{1-1}^{04}$,
(middle) $r_{00}^{04}$
and (bottom) ${\rm Re} \{ {r^{04}_{10} } \}$
for the $J/\psi$ as a function of $|t|$. 
The dashed line shows the expectation from SCHC. The results for the
photoproduction of $J/\psi$ and $\rho^0$
mesons \protect\cite{Chekanov:2002rm} are also shown.
}
\label{highthel}
\end{figure}


To obtain information about the helicity structure of the
interaction,
the spin density matrix elements are extracted and shown in 
figure~\ref{highthel} as a function of $|t|$.
In contrast to the $\rho^0$ meson,
the measured spin density matrix elements of the $J/\psi$ meson are all
compatible with zero, within experimental errors, and are thus compatible
with SCHC.
The $J/\psi$ results are therefore
consistent with the longitudinal momentum of the photon being shared
symmetrically
between the heavy quarks.
Hence, the
approximations made in the pQCD
models~\cite{Forshaw:1995ax,Bartels:1996fs,Forshaw:2001pf,Gotsman:2001ne,Enberg:2002zy} for the $J/\psi$ wavefunction
are satisfactory for the present data.



\section{Summary}
The exclusive production of vector mesons at HERA allows the
dynamics of vacuum-exchange processes to be studied.
The transition from soft to hard behaviour is observed with
increasing  $Q^2$ or $M_{\rm VM}^2$.
The analysis of helicity structure provides unique insight into the
structure of the vector meson wavefunctions. The production of vector
mesons at high $|t|$ shows a hard behaviour in $|t|$ and $W$
suggesting that processes beyond DGLAP may be necessary to describe
the data.  In general, pQCD has been able to provide qualitative descriptions
of the experimental data and the second phase of HERA running will continue
to improve our understanding of the strong interaction.




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\end{thebibliography}
\end{document}

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