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\begin{document}
%
\title*{Light Vector Meson Production at HERA}
%
%
\toctitle{Light Vector Meson Production at HERA}
% allows explicit linebreak for the table of content
%
%
\titlerunning{Light Vector Meson Production}
% allows abbreviation of title, if the full title is too long
% to fit in the running head
%
\author{Alexander Proskuryakov for the H1 and ZEUS collaborations}
%
\authorrunning{Alexander Proskuryakov}
% if there are more than two authors,
% please abbreviate author list for running head
%
%
\institute{Moscow State University, 119899 Moscow, Russia}

\maketitle              % typesets the title of the contribution

\begin{abstract}
Recent results from HERA
on exclusive vector meson production are reviewed.
The diffractive reaction 
$\gamma^{\star} p \to V p$ ($V=\rho^0,\omega.\phi$)
has been measured at photon-proton centre-of-mass
energies $20<W<160$ GeV, for photon virtualities
$Q^2<60$ GeV$^2$ and for $|t|$ up to 11 GeV$^2$, where $t$
is the square of the four-momentum transferred at the proton
vertex. The $\gamma^{\star} p \to V p$ cross section as a
function of $W$, $Q^2$ and $t$ is presented. The polarisation
properties of the $\rho^0$ produced at low and high $Q^2$ are
discussed. The data are compared with perturbative QCD calculations.

\end{abstract}

% INTRODUCTION
\section{Introduction}
A detailed description of results on the diffractive production
of the vector mesons by real and virtual photons is
available in~\cite{bauer,jim}.
%
\subsection{Kinematics}
%
Figure 1 shows a schematic diagram for exclusive vector meson
production in the reaction $e p \to e p V$, where $V$ stands for
a vector meson ($\rho^0$, $\omega$, $\phi$, $J/\psi$...).
\begin{figure}[h]
\begin{center}
\includegraphics[bb=80 80 402 402,width=.5\textwidth]{proskuryakov.1}
\caption[]{Schematic diagram for exclusive vector meson production}
\label{Fig. 1}
\end{center}
\end{figure}
%
\par\medskip\noindent
The kinematics of this reaction is described by the following
variables:
\begin{itemize}
\item
the negative of the four-momentum squared of the exchanged
photon, $Q^2=-(k-k')^2$, where $k(k')$ is the four-momentum of
the incident (scattered) electron;
\item
the centre-of-mass energy of the photon-proton system, 
$W=\sqrt{(q+P)^2}$, where $q$ and $P$ are the four-momenta of the
exchanged photon and the incident proton respectively; 
\item
the Bjorken variable $y=pq / pk$;
\item
the four-momentum transfer squared at the proton vertex, 
$t=(P-P')^2$, where $P'$ is the four-momentum of the scattered
proton;
\item
the angle between the vector meson production plane and the electron
scattering plane, $\Phi$.
\end{itemize}
The kinematics of the vector meson decay (in the following only
two particle decay is considered) is described by the polar angle
$\theta$ and the azimuthal angle $\phi$ in the $s$-channel helicity
frame~\cite{angle}, in which
the quantisation axis is defined as the direction of the
momentum of the final-state proton in the $V$ rest frame.
%
%  Cross section
\subsection{Cross section}
The cross section of exclusive vector meson production from virtual
photons has contributions from longitudinal (helicity 0 state) and
transverse (helicity $\pm1$ state) photons.
%
The differential electron-proton cross section is related to the virtual 
longitudinal
and transverse photon-proton cross sections $\sigma^{\gamma^{\star}p}_L$ and
$\sigma^{\gamma^{\star}p}_T$ as:
\begin{equation}
\frac{d\sigma^{ep}}{dydQ^2}= \Gamma_T(y,Q^2)(\sigma^{\gamma^{\star}p}_T+\epsilon\sigma^{\gamma^{\star}p}_L),
\end{equation}
with the flux of transverse photons given by
\begin{equation}
\Gamma_T = \frac{\alpha_{em}}{2\pi} \frac{1+(1-y)^2-2(1-y)Q^2_{min}/Q^2}{yQ^2},
\end{equation} 
and the polarisation parameter $\epsilon$ defined as
\begin{equation}
\epsilon=\frac{2(1-y)}{1+(1-y)^2-2(1-y)Q^2_{min}/Q^2},
\end{equation} 
where $Q^2_{min} \simeq m^2_e y^2/(1-y)$ is the minimum kinematically
allowed value of $Q^2$. In the kinematic region covered by the HERA
data the value of the polarisation parameter $\epsilon$ is close to unity.
The virtual photon-proton cross section, 
$\sigma^{\gamma^{\star}p}=\sigma^{\gamma^{\star}p}_T+\epsilon\sigma^{\gamma^{\star}p}_L$,
can be related to the total cross section,
$\sigma^{\gamma^{\star}p}_{TOT}=\sigma^{\gamma^{\star}p}_T+\sigma^{\gamma^{\star}p}_L$,
through the relation:
\begin{equation}
\sigma^{\gamma^{\star}p}_{TOT}=\frac{1+R}{1+\epsilon R} \sigma^{\gamma^{\star}p},
\end{equation}
where $R=\sigma^{\gamma^{\star}p}_L/\sigma^{\gamma^{\star}p}_T$ is the ratio of the
longitudinal and the transverse cross sections. 
%
% Angles
\subsection{Decay angular distributions}
The decay angular distributions for the reaction 
$ep \to e\rho^0p$ can be written as~\cite{angle}:
\begin{eqnarray}
W(\cos{\theta},\phi,\Phi)=
\frac{3}{4\pi}
\biggl[\frac{1}{2}(1-r^{04}_{00})+\frac{1}{2}(3r^{04}_{00}-1)\cos^2{\theta} \nonumber \\
%
 -\sqrt{2}{\rm Re}\{r^{04}_{10}\}\sin{2\theta}\cos{\phi}
-r^{04}_{1-1}\sin^2{\theta}\cos{2\phi} \nonumber \\
%
 -\epsilon\cos{2\Phi}(r^1_{11}\sin^2{\theta}
+r^1_{00}\cos^2{\theta}-\sqrt{2}{\rm Re}\{r^1_{10}\}\sin{2\theta}\cos{\phi}-
r^1_{1-1}\sin^2{\theta}\cos{2\phi}) \nonumber \\
%
 -\epsilon\sin{2\Phi}(\sqrt{2}{\rm Im}\{r^2_{10}\}\sin{2\theta}\sin{\phi}
+{\rm Im}\{r^2_{1-1}\}\sin^2{\theta}\sin{2\phi}) \nonumber \\
%
 +\sqrt{2\epsilon (1+\epsilon)}\cos{\Phi}
(r^5_{11}\sin^2{\theta}+r^5_{00}\cos^2{\theta} \nonumber \\
%
-\sqrt{2}{\rm Re}\{r^5_{10}\}\sin{2\theta}\cos{\phi}
-r^5_{1-1}\sin^2{\theta}\cos{2\phi}) \nonumber \\
%
 +\sqrt{2\epsilon (1+\epsilon)}\sin{\Phi}
(\sqrt{2}{\rm Im}\{r^6_{10}\}\sin{2\theta}\sin{\phi}
+{\rm Im}\{r^6_{1-1}\}\sin^2{\theta}\sin{2\phi})\biggr].
\setcounter{eqsubcnt}{0}
\end{eqnarray}
The spin-density matrix elements $r^{04}_{ij}$,
$r^{\alpha}_{ij}$ are the combinations of the helicity amplitudes
$T_{\lambda_{V}\lambda_{\gamma}}$ (the nucleons helicities are omitted),
where $\lambda_{V}$ and $\lambda_{\gamma}$ are
the helicities of the vector meson and the photon respectively.
\par\medskip\noindent
The assumption of helicity conservation in the photon-proton 
centre-of-mass frame (SCHC) reads:
\begin{equation}
T_{\lambda_{V}\lambda_{\gamma}}=
T_{\lambda_{V}\lambda_{\gamma}}
\delta_{\lambda_{V}\lambda_{\gamma}}.
\end{equation}
Under this assumption
all matrix elements except $r^{04}_{00}$, $r^1_{1-1}$,
${\rm Im}\{r^2_{1-1}\}$, ${\rm Re}\{r^5_{10}\}$ and ${\rm Im}\{r^6_{10}\}$ are
zero. The remaining matrix elements are related via:
\begin{equation}
r^1_{1-1}+{\rm Im}\{r^2_{1-1}\}=0,
\end{equation}
\begin{equation}
{\rm Re}\{r^5_{10}\}+{\rm Im}\{r^6_{10}\}=0.
\end{equation}
The angular distribution in the case of SCHC depends on two angles,
$\theta$ and $\psi = \phi-\Phi$:
\begin{eqnarray}
W(\cos{\theta},\psi)=
\frac{3}{4\pi}
\biggl[\frac{1}{2}(1-r^{04}_{00})+\frac{1}{2}(3r^{04}_{00}-1)\cos^2{\theta} \nonumber \\
 +\epsilon r^1_{1-1} \sin^2{\theta} \cos{2\psi}
-2\sqrt{2\epsilon (1+\epsilon)} {\rm Re}\{r^5_{10}\} 
\sin{2\theta} \cos{\psi}\biggr].
\setcounter{eqsubcnt}{0}
\end{eqnarray}
The SCHC hypothesis allows the direct determination of the ratio
$R=\sigma_{L}^{\gamma^{\star}p}/\sigma_{T}^{\gamma^{\star}p}$
from the value of $r^{04}_{00}$:
\begin{equation}
R = \frac{r^{04}_{00}}{\epsilon (1-r^{04}_{00})}.
\end{equation}

% Theory
\subsection{Theoretical models}
Exclusive production of light vector mesons has been studied in a wide
range of $W$ and $Q^2$ in fixed target experiments~\cite{bauer,vmfixed} and
at HERA~\cite{vmhera,vmhera15}. For $W > 10$ GeV and at low photon virtuality
these reactions have the characteristics of a soft diffractive
process: a cross section rising weakly with the centre-of-mass energy,
a sharp forward diffractive peak in the differential $t$ distribution
and $s$-channel helicity conservation (SCHC). These features are well described
within the framework of Regge phenomenology~\cite{regge} and the
Vector Dominance Model (VDM)~\cite{vdm,vdm2}.
In this approach
the energy dependence of exclusive vector meson production at high
energies is determined by the exchange of a Pomeron trajectory:
%
\begin{equation}
\frac{d\sigma}{d|t|} \propto e^{-b_{0}|t|}\biggl(\frac{W^2}{W^2_0}\biggr)^{2(\alpha(t)-1)},
\end{equation}
where the Pomeron trajectory is parameterised as
$\alpha(t)=\alpha(0)+\alpha' t = 1.08+0.25t$~\cite{dl}.
The $W$ dependence of cross section in the HERA kinematic region can
approximately be parameterised by a simple power behaviour, $W^{\delta}$,
where $\delta \simeq 4(\alpha(0)-\alpha'/b-1)$.
The slope of the differential $t$ distribution depends on the energy as
$b=b_0 + 2\alpha' ln(W^2/W^2_0)$.
\par\medskip\noindent
The same approach fails to describe the energy dependence of the cross
section for elastic $J/\psi$ photoproduction measured at HERA~\cite{psihera}.
The rapid rise with energy of the elastic $J/\psi$ photoproduction 
cross section
is consistent with a perturbative QCD calculation~\cite{psiqcd}. The hard
scale for this reaction is given by the charm quark mass.
\par\medskip\noindent
The hard scale can also be provided by the photon virtuality $Q^2$. 
Exclusive electroproduction at high values of $Q^2$ has been the subject
of many theoretical investigations (see e.g.~\cite{qcd1,qcd2,qcd3,qcd4,qcd5,qcd6}). 
The main prediction of
the perturbative QCD models is a fast increase of the cross section with $W$,
connected with the $Q^2$ evolution of the gluon density of the proton
$xg(x,Q^2)$. The $Q^2$ dependence for the longitudinal
vector meson production, which dominates at high $Q^2$, is
$\sigma^{\gamma^{\star}p}_L \sim (xg(x,Q^2))^2 /Q^6$. The QCD models
predict also universal, flavor and energy independent, $t$ dependence
for this exclusive diffractive process.
\par\medskip\noindent
Recent calculations of the helicity amplitudes for the reaction
$\gamma^{\star}p \to \rho^0 p$, performed in the framework of
perturbative QCD~\cite{hqcd1,hqcd2,hqcd3} predict small violations
of SCHC. The largest helicity non-conserving amplitude is expected
to be $T_{01}$, which describes the production of
longitudinally polarised vector mesons from
transverse photons.
% results
\section{Results}

\subsection{The $W$ dependence of the 
$\gamma^{\star}p \to Vp$ cross sections}
Figure 2 shows the energy dependence of elastic $\rho^0$,
$\omega$, $\phi$ and $J/\psi$ photoproduction ($Q^2 \simeq 0$ GeV$^2$) cross sections
measured at HERA and fixed target experiments. The $W$ dependence
of the cross section for light vector meson photoproduction 
is well parameterised 
at high energy as 
$\sigma^{\gamma p} \propto W^{\delta}$
with $\delta \simeq 0.22$.
This weak energy dependence is in agreement with the 
'soft' Pomeron parameterisation of eq. (11).
% Cross section vs W , php
\begin{figure}[h]
\begin{center}
\includegraphics[width=.8\textwidth,height=.75\textwidth]{proskuryakov.2}
\caption[]{The energy dependence of the total photoproduction and
vector meson photoproduction cross sections for fixed target and
HERA experiments.}
\label{Fig. 2}
\end{center}
\end{figure}
% Cross section vs W, ar different Q2
\vspace{1.0cm}
%
\begin{figure}[h]
\begin{minipage}[t]{0.5\linewidth}
\centering
%
\includegraphics[width=0.99\textwidth,totalheight=1.05\textwidth]{proskuryakov.3}
\end{minipage}%
%
\begin{minipage}[t]{0.5\linewidth}
\centering
\includegraphics[width=1.0\textwidth,totalheight=1.08\textwidth]{proskuryakov.4}
\end{minipage}
\caption[]{The energy dependence of exclusive $\rho^0$ production
cross sections for different values of $Q^2$. The lines represent
the function $\sigma \propto W^{\delta}$ fitted to the data.}
\caption[]{The values of $\delta$ as a function of $Q^2$ for 
exclusive production
of $\rho^0$ and $\phi$ mesons.}
\end{figure}
%
\clearpage
\noindent
In contrast, the rapid rise with energy of the elastic $J/\psi$
photoproduction cross section is consistent with a perturbative
QCD calculation~\cite{psiqcd}.
\par\medskip\noindent
The $W$ dependence of the exclusive $\rho^0$ production
cross section at different values of $Q^2$ is shown in
Fig. 3. The lines represent the result of a fit to the data
with a function of type $\sigma \propto W^{\delta}$. The results
of the fit are summarised in Fig. 4. As can be seen from 
Fig. 4, the $W$ dependence of the cross section for the reaction
$\gamma^{\star}p \to Vp$ is consistent with the 'soft' Pomeron
approach at low values of $Q^2$. In the region of high $Q^2$,
where QCD based models predict a steeper cross section dependence
on $W$, there is an indication that the rise of the cross section
becomes faster.

%\clearpage
% cross section Q2 dependence
\subsection{The $Q^2$ dependence of the 
$\gamma^{\star}p \to Vp$ cross sections}
The cross section for exclusive $\rho^0$ production as a function
of $Q^2$ is shown in Fig. 5. The $Q^2$ dependence of the
$\gamma^{\star}p \to \rho^0 p$ cross section is well described
in a wide range of $Q^2$ by the function
$d\sigma/dQ^2 \propto 1/(1+Q^2/M^{2}_{\rho})^n$, 
with $n=2.28\pm0.06$. The curves shown in Fig. 5 represent the predictions
of QCD~\cite{qcd5,qcd6} and Generalized VDM~\cite{vdm2}.
\par\medskip\noindent
Figure 6 shows the ratio of the $\phi$ and $\omega$ production
cross sections to that for the $\rho^0$ meson as a function of $Q^2$.
The ratios in the high $Q^2$ region are consistent with the prediction
% $9:1:2$ for $\rho^0 : \omega : \phi$
of QCD models.


\begin{figure}[h]
\begin{minipage}[t]{0.5\linewidth}
\centering
\includegraphics[width=0.99\textwidth,height=1.25\textwidth]{proskuryakov.5}
\end{minipage}%
%
\begin{minipage}[t]{0.5\linewidth}
\centering
% ratio VS Q2
\includegraphics[width=0.99\textwidth,height=1.25\textwidth]{proskuryakov.6}
\end{minipage}
\caption[]{The $Q^2$ dependence of the exclusive $\rho^0$ production
cross section. The curves represent the predictions of the QCD 
models~\cite{qcd5,qcd6} and Generalized VDM~\cite{vdm2}.}
\caption[]{Ratios of $\phi$ and $\omega$ cross sections to that
for the $\rho^0$ meson as a functions of $Q^2$.}
\label{Fig. 5}

\end{figure}

% t dependence
\subsection{The $t$ dependence of the 
$\gamma^{\star}p \to Vp$ cross sections}
The exponential $t$ dependence of the 
$\gamma^{\star}p \to Vp$ cross section is a typical
characteristic of diffractive processes. The slope parameter
$b$, which is related in optical models to the sizes of the
interacting particles, is expected to depend on the photon virtuality.
Figure 7 shows the slope parameter for exclusive $\rho$ production
as a function of $Q^2$. Despite large statistical errors of the
experimental data these results indicate that the slope $b$
decreases with $Q^2$, approaching the value of $b \simeq 4-5$ GeV$^{-2}$.
% slope VS Q2
\begin{figure}[h]
\begin{center}
\includegraphics[width=.7\textwidth]{proskuryakov.7}
\caption[]{The slope parameter $b$ as a function of $Q^2$ for exclusive 
$\rho^0$ production.}
\label{Fig. 7}
\end{center}
\end{figure}
%
\par\medskip\noindent
The slope parameter also depends on the vector meson mass.
The value of $b$ is~\cite{vmhera,psihera}
$9.8\pm0.8({\rm stat})\pm1.1({\rm syst})$ GeV$^{-2}$,
$7.3\pm1.0({\rm stat})\pm0.8({\rm syst})$ GeV$^{-2}$ and
$4.0\pm0.4({\rm stat})^{+0.6}_{-0.7}({\rm syst})$ GeV$^{-2}$,
for photoproduction of $\rho^0$,
$\phi$ and $J/\psi$ mesons respectively.
\par\medskip\noindent
The differential $t$ distribution at low $|t|$  is
well described by Regge phenomenology assuming that the
exchange of the Pomeron trajectory dominates at high enough
energies. The parameters of the Pomeron trajectory can be
measured by studying the energy dependence of $d\sigma/dt$
at different $t$ values (see eq. 11).
% pomeron trajectory
\begin{figure}[h]
\begin{minipage}[t]{0.5\linewidth}
\centering
\includegraphics*[bb=63 180 499 682,width=0.99\textwidth,height=1.4\textwidth]{proskuryakov.8}
\end{minipage}%
%
\begin{minipage}[t]{0.5\linewidth}
\centering
% 
\includegraphics*[bb=65 154 487 717,width=0.99\textwidth,height=1.40\textwidth]{proskuryakov.9}
\end{minipage}
\caption[]{Cross section for elastic $\rho^0$ photoproduction
as a function of $W$ at different values of $t$. The $W$
dependence is fitted by 
$d\sigma/dt \propto (W^2)^{2\alpha(t)-2}$.}
%
\caption[]{The Pomeron trajectory determined from the reaction
$\gamma p \to \rho^0 p$. The dots are the values of the trajectory
at a given $t$ as determined from figure 8. The full line is the
result of a linear fit to these values. The Pomeron trajectory
given in~\cite{dl} is shown as the dashed line.}
\end{figure}
%
Figure 8 shows the $W$ dependence for the elastic $\rho^0$
photoproduction cross section at fixed $t$ values.
The resulting Pomeron trajectory, shown in Fig. 9, is
$\alpha(t)=(1.096\pm0.021)+(0.125\pm0.038)t$. 



% Decay angular distributions
\subsection{Decay angular distributions}
The analysis of  vector meson decay distribution
provides important information about helicity
properties of the reaction $\gamma^{\star}p \to Vp$.
In the case of unpolarised beams, the angular distributions
are determined by 15 spin density matrix elements (see eq. 5), which
are the combinations of the helicity amplitudes
$T_{\lambda_V,\lambda_{\gamma}}$.
The full set of spin density matrix elements has been measured for
exclusive $\rho^0$ production.
The most significant deviation from the expectation of SCHC
is observed for the matrix element $r^5_{00}$. This matrix
element is proportional to the interference between
the helicity non-flip amplitude $T_{00}$ and the helicity
single-flip amplitude $T_{01}$. The size of this effect
can be quantified by the ratio of the helicity single-flip
amplitude to the helicity conserving amplitudes
$|T_{01}|/\sqrt{|T_{00}|^2+|T_{11}|^2}$, which gives
$(8\pm3)\%$~\cite{vmhera15}.
\par\medskip\noindent
Figure 10 shows the 15 spin density matrix elements as a function
of $Q^2$.
The results of the QCD based calculations~\cite{hqcd2,hqcd3}
shown in Fig. 10 are in good agreement with the data. 
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.999\textwidth,height=1.35\textwidth]{proskuryakov.10}
\caption[]{The 15 spin density matrix elements as a function of $Q^2$.
The full curves are the prediction of Ref.~\cite{hqcd2}, the dotted curves 
the prediction of Ref.~\cite{hqcd3}. The dashed lines indicate the
expectation of SCHC.}
\label{Fig. 10}
\end{center}
\end{figure}
%
\clearpage
%\par\medskip\noindent
The ratio $R=\sigma^{\gamma^{\star}p}_L/\sigma^{\gamma^{\star}p}_T$, 
calculated according to eq. 10 (neglecting small violation of SCHC) ,
is displayed in Fig. 11 as a function of $Q^2$ for exclusive 
$\rho^0$ production. The curves represent the predictions of
the QCD~\cite{qcd5,qcd6} and  Generalized VDM~\cite{vdm2}.
Fig. 11 shows that the longitudinal photon-proton cross section
dominates at large $Q^2$ and that the
non-linear rise of $R$ can be reproduced by the theoretical
models.
\begin{figure}[h]
\begin{center}
\includegraphics[width=.7\textwidth]{proskuryakov.11}
\caption[]{The ratio of the longitudinal and transverse photon-proton 
cross sections as a function of $Q^2$.
The curves represent the predictions of the 
QCD~\cite{qcd5,qcd6} and Generalized VDM~\cite{vdm2}.}
\label{Fig. 11}
\end{center}
\end{figure}
%

%\clearpage
% conclusion
\section{Conclusions}

\begin{itemize}
\item
The energy dependence of the light vector meson production
cross section exhibits a slow rise with $W$ at low values of
$Q^2$. There is an indication that in the region of high $Q^2$
this rise becomes faster, in agreement with predictions by
QCD models. The cross section for exclusive $J/\psi$ production
shows a steep rise with $W$ already a low photon virtualities.
\item
The $Q^2$ dependence of the
$\gamma^{\star}p \to \rho^0 p$ cross section is well described 
(for $Q^2 > 1$ GeV$^2$)
by the
function $d\sigma/dQ^2 \propto 1/(1+Q^2/M_{\rho})^n$, with
$n=2.24\pm0.09$.
The ratios of the $\phi$ and $\omega$ production
cross sections to that for the $\rho^0$ meson
are consistent in the high $Q^2$ region with the predictions
of QCD models.
\item
The $t$ distribution for the exclusive vector meson production
is well parameterised by an exponential dependence $e^{bt}$.
The slope parameter $b$ decreases with $Q^2$.
The Pomeron trajectory, determined from the reaction of
elastic $\rho^0$ photoproduction, is 
$\alpha(t)=(1.096\pm0.021)+(0.125\pm0.038)t$. 
\item
The full set of spin density matrix elements measured for exclusive 
$\rho^0$ productions shows a small deviation from the SCHC hypothesis.
The ratio of the longitudinal and transverse photon-proton cross sections
exhibits a non-linear rise with $Q^2$. The theoretical models reproduce
this behaviour.
\end{itemize}


\begin{thebibliography}{8.}
\addcontentsline{toc}{section}{References}

\bibitem{bauer} T.H. Bauer et al., Rev. Mod. Phys. 50 (1978) 261.

\bibitem{jim} J.A. Crittenden, Exclusive Production of Neutral Vector
Mesons at the Electron-Proton Collider HERA, Springer Tracts in
Modern Physics, Volume 140 (Springer, Berlin Heidelberg, 1997).

\bibitem{angle} K. Schilling and G. Wolf, Nucl. Phys. B61~(1973)~381.

\bibitem{vmfixed} R.M. Egloff et al., Phys. Rev. Lett. 43 (1979) 657; \\
R.M. Egloff et al., Phys. Rev. Lett. 43 (1979) 1545; \\
J. Busenits et al., Phys. Rev. D40 (1989) 1; \\
EMC Collab., J.J. Aubert et al., Phys. Lett. B161 (1985) 203; \\
NMC Collab., M. Arneodo et al., Nucl. Phys. B429 (1994) 503; \\
M.R. Adams et al., Z. Phys., C74 (1997) 237.

\bibitem{vmhera} ZEUS Collab., M Derrick et al., Z. Phys. C69 (1995) 39; \\
H1 Collab., S. Aid et al., Nucl. Phys. B463 (1996) 3; \\
ZEUS Collab., M. Derrick et al., Phys. Lett. B377 (1996) 259; \\
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