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% special definition for this paper
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\begin{document}

\pagestyle{empty}
\begin{titlepage}


\noindent
\begin{center}
%{\it {\large version of \today}} \\[.3em] 
\begin{small}
\begin{tabular}{llrr}
%Submitted to & \multicolumn{3}{r}{\footnotesize Electronic Access: {\it http://www-h1.desy.de/h1/www/publications/conf/conf\_list.html}} \\[.2em] \hline 
%Submitted to & \multicolumn{3}{r}{\footnotesize {\it www-h1.desy.de/h1/www/publications/conf/conf\_list.html}} \\[.2em] \hline 
Submitted to & & &
\epsfig{file=/h1/www/images/H1logo_bw_small.epsi
,width=2.cm} \\[.2em] \hline
                & & & \\
\multicolumn{4}{l}{{\bf
                International Europhysics Conference 
                on High Energy Physics, EPS03},
                July~17-23,~2003,~Aachen} \\
                (Abstract {\bf 092} & Parallel Session & {\bf 5}) & \\
                & & & \\
\multicolumn{4}{l}{{\bf
                XXI International Symposium on  
                Lepton and Photon Interactions, LP03},
                August~11-16,~2003,~Fermilab} \\ 
                & & & \\
                \hline
 & \multicolumn{3}{r}{\footnotesize {\it
    www-h1.desy.de/h1/www/publications/conf/conf\_list.html}} \\[.2em]
\end{tabular}
\end{small}
\end{center}
\vspace*{2cm}

\begin{center}
  \Large
  {\bf Elastic Electroproduction of \boldmath{$\rho$} Mesons \\
       at High \boldmath{$Q^2$} at HERA \\} 
%       with $8 < Q^2 < 60$ \gevsq\ at HERA \\}
  \vspace*{1cm}
    {\Large H1 Collaboration} 
\end{center}

\begin{abstract}

\noindent
The elastic electroproduction of $\rho$ mesons is studied at HERA with the H1 
detector 
%using an integrated luminosity of 42.4 ${\rm pb}^{-1}$ 
in the
kinematic range $8 < Q^2 < 60 $ \gevsq , $40 < W < 180$ GeV and 
$|t| < 0.5$ \gevsq. The luminosity 
of 42.4 ${\rm pb}^{-1}$
%compared to previous 
%measurements 
gives access to the \qsq\ domain where perturbative QCD is 
expected to apply. The cross section has been measured as a
function of \qsq , $W$ and $t$. The $W$ dependence of the 
$\gamma^{*}p \rightarrow \rho p$ cross 
section is observed to become stronger with increasing
$Q^2$, changing from a behaviour compatible with 
soft Pomeron exchange at low \qsq\ to a hard dependence at large \qsq. 
Spin density matrix elements are measured as a function of \qsq , $W$ and $t$. 
Significant $t$ dependent violations of $s$-channel helicity conservation
are observed. 


\end{abstract}


\end{titlepage}

\pagestyle{plain}


\section{Introduction}
                               \label{sect:intro}
\noindent
%===========================================

The subject of this paper is the study of the elastic electroproduction 
of $\rho$ mesons in $ep$ scattering at high energy:
%
\begin{equation}
e + p \rightarrow e + \rho + p \ \ ; \ \ \ \
\rho \rightarrow \pi^+ \pi^- .
    \label{eq:rho_prod}
\end{equation}
%
The scattered proton can also be excited into a system $Y$ of mass 
$M_Y$ which is much lower than the photon--proton centre of mass 
energy $W$ (``proton dissociative'' scattering), this process being
the dominant background contribution.

The kinematic domain of the measurement is restricted to:
%
\begin{eqnarray}
 8  < &Q^2& < 60 \ {\rm GeV^2}       \nonumber \\
 40 < &W& < 180  \ {\rm GeV}         \nonumber \\
 0 < &|t|& < 0.5 \ {\rm GeV^{2}},          
    \label{eq:kin_range}
\end{eqnarray}
%
where \qsq\ is the modulus of the intermediate photon four-momentum squared,
$W$ is the hadronic centre of mass energy and $t$
is the square of the four-momentum transfer to the proton.

At low $Q^2$, 
the $\gamma^* p \rightarrow \rho p$ cross section 
is characterized by a ``soft'' energy dependence which can be seen as  
due to pomeron ($\pom$) exchange with 
$ {\rm d}\sigma / {\rm d}t \propto W^{4(\apom(t)-1)}$, where the soft
pomeron trajectory is parametrized~\cite{dl} as
$\apom(t) =\apom(0) - \alpha' \cdot \ttra \simeq 1.08 - 0.25 \cdot \ttra$
(for $t$ in ${\rm GeV^{2}}$). 
For \qsq\ larger than a few \gevsq, perturbative QCD (pQCD) is expected to
apply and diffractive $\rho$ production can be viewed in the proton rest
frame as a sequence of three processes well separated in time : the photon
fluctuation into a \qqbar\ pair, the hard interaction of the \qqbar\ pair
with the proton via the exchange of two gluons in a 
net color-singlet state,
and the \qqbar\ pair recombination into a real $\rho$ meson.
The cross section is then
proportional to the square of the gluon density in the proton,
%$\sigma_{\gamma^{*} p} \propto \alpha^{2}_{s}(\qsq)/Q^6 . |xg(x,\qsq)|^2$,  
which corresponds to a fast increase of the $\gamma^{*} p$ cross section with
the energy (``hard'' behavior)~\cite{ryskin}.

Three angles are defined \cite{sch-w} to characterise the electroproduction 
of vector mesons (VM) decaying into two charged particles: 
$\Phi$ is the angle between the VM production plane 
(defined as the plane containing the virtual photon and the VM 
directions) and the electron
scattering plane in the ($\gamma^{\star} p$) centre of mass system,
$\theta^\ast$ and $\phi$ are the polar and
the azimuthal angles, respectively, of the positively charged decay 
particle in the VM rest frame, the quantisation axis being 
taken as the direction opposite to that of the outgoing 
$Y$ system. In this paper, the distributions of the angles $\Phi$ and
$\theta^*$ are analysed.



%===========================================
%===========================================
%===========================================



\section{Data selection and kinematics }
                                  \label{sect:data_sel}
%===========================================

\subsection{Event selection }
                               \label{sect:selection}
%===========================================

The data used for the present analysis were taken with the H1 
detector in the year 2000. 
The energies of the HERA proton and positron beams~\footnote{
%===========
In the following, the word electron will be used for positrons.} 
%===========
were 920 and 27.5 GeV, respectively. 
The data correspond to an integrated luminosity of 42.4 ${\rm pb^{-1}}$.

%The integrated luminosity used for the analysis amounts to 
%42.4 ${\rm pb^{-1}}$.

The relevant parts of the detector, for which more details can be 
found in~\cite{h1:detector,h1:spacal}, are the central tracking detector, 
the liquid argon (LAr) and the backward electromagnetic (SPACAL) 
calorimeters and the  forward detectors, which are sensitive to 
energy flow close to the outgoing proton direction~\footnote{
%===========
In the H1 convention, the $z$ axis is defined by the colliding
beams, the forward direction being that of the outgoing
proton beam ($z > 0$) and the backward direction that of the 
electron beam ($z < 0$).}.
%===========
The forward detectors consist of
the proton remnant tagger (PRT) made of a set of 7 scintillators 
surrounding the beam pipe 24 m downstream of the $ep$ interaction 
point, and the three planes of the forward muon detector 
(FMD) situated between the interaction point and the solenoidal 
magnet.

Events candidates corresponding to reaction~(\ref{eq:rho_prod}), in the 
kinematic range defined by relations~(\ref{eq:kin_range}) are 
selected by requesting :
%
\begin{itemize}
%
\item
the reconstruction of a cluster 
in the SPACAL calorimeter with energy larger than 17 GeV 
(the scattered electron candidate);
% 
\item
the reconstruction 
in the central tracking detector of the 
trajectories of exactly two charged particles (pion candidates)
with opposite charges, 
transverse momenta larger than 0.1 GeV and polar angles confined 
within the interval $20^{\rm o} < \theta < 160^{\rm o}$;
%
\item
the invariant mass $M_{\pi \pi}$ of the decay pions to be 
$0.6 < M_{\pi \pi} < 1.1$ GeV;
%
\item
the absence of any energy deposit
larger than 400~MeV in the LAr calorimeter
 not associated 
with the two charged pion candidates.
This cut reduces backgrounds due to the diffractive production of 
systems decaying into two charged and additional neutral particles.
It also helps to reject proton dissociative events with large $M_Y$
masses;
%
\item
the absence of activity above noise in the PRT and the FMD, in order
to reject events with proton dissociation;
%
\item
$M_{KK}~>~1.04~{\rm GeV}$, 
where $M_{KK}$ is the invariant mass of the two pions candidates 
when considered as kaons (no direct hadron identification is 
performed for this analysis). This cut reduces the 
background due to diffractive production of $\phi$ mesons;
%
\item 
$\sum(E-p_z) > 45\ {\rm GeV}$, to reduce both the effect of   
QED radiative corrections and background contributions in
which there are unreconstructed particles. 
$\sum(E-p_z)$ is the difference of 
the energies and the longitudinal momenta of the scattered electron 
(measured in the SPACAL) and the pion candidates (measured in the 
central tracking detector). It is expected to be close to twice the 
incident electron beam energy, i.e. 55 GeV, if no other particles 
have been produced except for the forward going system $Y$, in
particular if no high energy photons are radiated by the incoming
or the outgoing electron.
%
\end{itemize}
%
To first approximation, the selected events can be attributed
to elastic scattering.
However, this sample also contains proton dissociative events
%for small masses ($M_Y \ \lapprox \ 1.6$~GeV),
either when the $Y$ system has too small a mass to be detected 
($M_Y \ \lapprox \ 1.6$~GeV),
or due to inefficiencies of the forward detectors.
Conversely, elastic events can be lost due to noise in
the forward detectors or, for the highest \ttra\ values, when
the scattered proton hits the beam pipe walls
or adjacent material, leading to secondary particles which give
a signal in the forward detectors.

To estimate these backgrounds and losses, a ``tag'' sample is defined
comprising the events which pass all of the cuts above, except that a signal is
observed in the forward detectors.
This sample mainly contains proton dissociation events, with
$1.6 \lsim\ M_Y \lsim\ 5$ GeV, the upper bound on $M_Y$ being due to the
request of no energy deposit above noise in the
LAr calorimeter.

The $\pi^+\pi^-$ mass distribution of the selected events is shown
in fig.~\ref{fig:mass}. A clear \rh\ meson signal is visible.
The contributions from $\omega$, $\phi$ and \rhoprim\ backgrounds
extracted from Monte-Carlo simulations (see section~\ref{sect:bg})
are also shown.

\subsection{Kinematics}
                               \label{sect:kinematics}
%===========================================

The three-momentum of the \rh\ is computed as the sum of 
the two charged pion candidate momenta.
The variable \qsq\ is reconstructed using the double angle
method~\cite{da}:
%
\begin{equation}
Q^2 = \frac {4 E_0^2 \ \sin {\theta_{\rho}} \ (1+\cos{\theta_e})}
{\sin{\theta_e} + \sin{\theta_{\rho}} -
\sin{(\theta_e+\theta_{\rho})}},
                                \label{eq:qsq}
\end{equation}
%
where ${E_0}$ is the energy of the incoming electron and
$\theta_e$ and $\theta_{\rho}$ are the scattered electron and \rh\
meson polar angles, respectively.
The $W$ variable is calculated using the Jacquet-Blondel
method~\cite{jb}:
%
\begin{equation}
W^2 = y \cdot s - \frac {{p_{t, \rho}}^2} {1-y} , {\rm with} \
  y= \frac{E_{\rho} - p_{z, \rho}} {2 E_0} ,
                                \label{eq:w}
\end{equation}
%
$s$ being the square of the $ep$ centre of mass energy and
$E_{\rho}$, $p_{z, \rho}$ and $p_{t, \rho}$ being the energy,
the longitudinal and
the transverse momentum of the \rh\ meson, respectively.
The variable $|t|$ is determined
from the scattered electron and
\rh\ momentum components transverse to the beam direction as
%
\begin{equation}
|t| \simeq (\vec{p}_{t, miss})^2 =
  (\vec{p}_{t, e} + \vec{p}_{t, \rho})^2,
                                \label{eq:tprim}
\end{equation}
%
where the electron transverse momentum is computed as
%
\begin{equation}
 p_{t, e} = \frac {2  E_0 - E_{\rho} + p_{z, \rho}}
               {\tan (\theta_e / 2)} \ .
                                \label{eq:pte}
\end{equation}
%



\section{ Monte Carlo simulations and backgrounds}
                                  \label{sect:MC_bg}
%===========================================

\subsection{ Monte Carlo simulations }
                               \label{sect:MC}
%===========================================



A Monte Carlo program based on the DIFFVM program~\cite{diffvm} 
including QED radiation~\cite{heracles} simulates 
the elastic production and the decay of $\rho$ vector meson.
It is used to correct the data for acceptance, smearing and radiative 
effects. 
The \qsq , $W$ and $t$ dependences of the cross section, including
their correlations, are taken from the present measurements (see
figs.~\ref{fig:xsq2}, \ref{fig:delta} and \ref{fig:bq2}).
The simulation includes the angular distributions
corresponding to the measurements of the present analysis for
the \rfour\ matrix element (\cost\ distribution)
and for the \rfivecomb\ and \ronecomb\ combinations ($\Phi$
distribution), combined in the latter case with the
measurements in~\cite{h1-hight}.
Other angular distributions and correlations are taken in the
{\it s}-channel helicity conservation (SCHC) approximation, and the
$\cos \delta$ parameter, which describes the interference
between the longitudinal and transverse amplitudes, is taken
from the measurement from~\cite{h1-rho} in the relevant \qsq\ range.
The mass distribution is described by a relativistic Breit-Wigner
distribution, including skewing effects taken from~\cite{h1-rho}.
For studies of systematic errors all 
the simulation parameters have been varied within errors.


DIFFVM  is also used for the simulation of the
$\rho$ proton dissociative background, and of $\omega$, $\phi$ and \rhoprim\ 
backgrounds. 
For the proton dissociative \rh\ background, the slope of the
exponential $t$ dependence is taken to be 1.7~\gevsqm , a value
determined from the present data using the ``tag'' sample. The
angular distributions are assumed to be the same as for elastic \rh\
case, and the $M_Y$ spectrum is parameterised
as ${\rm d} \sigma / {\rm d} M_Y^2 \propto 1/M_Y^{2.15}$
(see~\cite{goulianos}).
The slope of the $t$ distribution is taken to be 6~\gevsqm\ for
elastic $\omega$, $\phi$ and \rhoprim\ production.
%,
%and  the decay angular distributions are taken isotropic, except for
%$\phi \rightarrow K^+ K^-$~\cite{h1-phi}.

Fig.~\ref{fig:control} shows various
distributions, comparing the data sample with the Monte Carlo simulation.
% for a few variables of the
%selected sample of events.
The background discussed in
section~\ref{sect:bg} are also shown.
Good agreement is observed for all distributions, indicating
that the Monte-Carlo simulations can be reliably used to correct the
data for acceptance and smearing effects.

\subsection{Proton dissociative \boldmath{$\rho$} and
  \boldmath{$\omega$}, \boldmath{$\phi$} and
     \boldmath{\rhoprim } backgrounds}
                               \label{sect:bg}
%===========================================

A proton dissociative background of $11 \pm 4 \%$
is estimated using the number of events in the selected sample and
in the ``tag'' sample, and the probabilities of no signal
in the PRT and the FMD for elastic and proton dissociative \rh\
events, respectively.
The error reflects the statistical precision of the data and the
systematic uncertainties determined by varying the conditions imposed
on the forward detectors, by varying the proton dissociative $t$
slope by $\pm 0.5$~\gevsqm\ and by varying the $M_Y^2$ spectrum
from $1/M_Y^{2.15}$ to $1/M_Y^{1.85}$ and to $1/M_Y^{2.45}$

Diffractive electroproduction of $\omega$ and $\phi$ mesons can
fake \rh\ production through the decay channels:
%
\begin{eqnarray}
&& \omega \rightarrow \pi^+ \pi^- \pi^0, \nonumber \\
&& \phi \rightarrow \pi^+ \pi^- \pi^0, \ \ \ \ \ \phi \rightarrow
         K^0_S K^0_L,  \nonumber \\
&& \rho^\prime \rightarrow \rho^+ \pi^- \pi^0, \ \ \ \ \
   \rho^+ \rightarrow \pi^+ \pi^0       \ \ \ \ \       (+ \ c. c.).
                                \label{eq:bgs}
\end{eqnarray}
%
if the decay photons of the $\pi^0$ or the $K^0_L$ mesons are not
detected.
%This happens in case of deposited energy  associated with the
%charged pion tracks or smaller than 400 MeV. 
Diffractive electroproduction of $\omega$ and $\phi$ mesons also
gives the same topology in the detector as for \rh\ production 
through the decay channels
\begin{eqnarray}
&& \omega \rightarrow \pi^+ \pi^-, \nonumber \\
&& \phi \rightarrow {\rm K}^+ {\rm K}^-. 
                                \label{eq:bgs2}
\end{eqnarray}

The $\omega$, $\phi$ and \rhoprim\ cross sections were taken from
measured ratios to the \rh\ cross section, in the \qsq\ range
relevant for the analysis:
$\omega$~/~$\rho = 0.09$~\cite{zeus-omega},
$\phi$~/~$\rho= 0.20$~\cite{h1-phi} and
\rhoprim ~/~$\rho= 1.00$ (compatible with the analysis
in~\cite{h1-hight}).

The background contributions in the selected kinematic
domain (\ref{eq:kin_range}) and for the selected \rh\ mass
range are estimated with  Monte-Carlo
simulations
The total remaining amount of
background is 
%11 \% for \rh\ proton dissociation,
0.35~\% for $\omega$, 0.25~\% for $\phi$ and 4~\% for \rhoprim .
These contributions are subtracted from all experimental distributions.

\section{Cross sections} \label{sect:xsection}

%===========================================
%===========================================

In each bin of the kinematical variables \qsq\ and $W$, the cross 
section is 
computed from the numbers of events in the bin, fully corrected for 
background, acceptance, smearing and QED radiation effects using the 
Monte Carlo simulations described above.
It is converted into a \gsp\ cross section using the relation:
%
\begin{equation}
\frac{{\rm d}^2 \sigma_{tot} (ep \rightarrow e \rho p)}{{\rm d}y \ 
{\rm d}Q^2} =
\Gamma \ \sigma_{tot} (\gamma ^*p \rightarrow V p) = \
\Gamma \ \sigma_{T} (\gamma ^*p \rightarrow \rho p) \ (1 + \varepsilon \ R),   
                                            \label{eq:sigma}
\end{equation}
%
where $\sigma_{tot}$, $\sigma_{T}$ and $\sigma_{L}$ are the total, 
transverse and longitudinal \gsp\ cross sections and
%
\begin{equation}
R\ = \sigma_{L} / \sigma_{T}.
                                             \label{eq:Rdef}
\end{equation}
The flux $\Gamma$ of transverse virtual photons given by
%
\begin{equation}
\Gamma = \frac {\alpha_{em} \ (1-y+y^2/2)} {\pi\  y \  \qsq};
                                             \label{eq:flux}
\end{equation}
and $\varepsilon$\ is the polarisation parameter
%
\begin{equation}
\varepsilon = \frac{1 - y}{1-y+y^2/2}.
                                       \label{eq:epsil}
\end{equation}
%
For this analysis, $<\varepsilon> = 0.996$ ($0.93<\varepsilon<1$).
The  \qsq\ and $W$ dependences measured in the present analysis are
taken into account in the
computation of the flux factors.

The mass distributions were corrected for the analysis cut, to
the range $2 m_\pi < M_{\pi\pi} < m_\rho + 5 \Gamma_\rho$, where $m_\pi$ and
$m_\rho$ are the pion and \rh\ meson masses, respectively
and $\Gamma_\rho$ is the natural width of the $\rho$ meson. 

%===========================================
%===========================================



\subsection{\boldmath{$Q^2$} dependence of the 
             \boldmath{$\gamma^{*}p$} cross section}
    \label{sect:q2depend} 

%===========================================
%===========================================

Fig.~\ref{fig:xsq2} shows the \qsq\ dependence of the total \gsp\ cross
section, for $W = 95$ GeV.
A fit of the form $\sigma
  \propto 1 / (Q^2  + m_\rho^2)^{n}$ for the range $8 < Q^2 < 60$~\gevsq\
results in $n = 2.60 \pm 0.04$, with $\chi^2 / ndf = 4.6 / 10$.
Previous measurements~\protect\cite{h1-rho,zeus}, extrapolated to 
$W = 95$ GeV, are also shown. Comparing the extracted value of $n$
with the result $n = 2.24 \pm 0.09$ obtained from a sample \cite{h1-rho}
extending to lower $Q^2 > 1 \ {\rm GeV^2}$,
%$1 < Q^2 < 60 \ {\rm GeV^2}$
it is clear that this parameterisation is not sufficient to describe
the sum of transverse and longitudinal photon induced cross sections 
for all $Q^2$.

%===========================================
%===========================================

\subsection{\boldmath{$W$} dependence of the 
             \boldmath{$\gamma^{*}p$} cross section}
    \label{sect:wdepend} 

%===========================================
%===========================================

Fig.~\ref{fig:xsw} shows the $W$ dependence of the \gsp\ cross
section, at four values of \qsq .
The rise of the cross section with $W$ is observed to become 
stronger as \qsq\ 
increases. Parameterising
the $W$ dependence at fixed \qsq\ as
$\sigma (W) \propto W^\delta$, the fitted values of
$\delta$ are shown in fig.~\ref{fig:delta}, together with previous
measurements~\protect\cite{h1-rho,zeus-preli}. 
Values 
similar to those for \jpsi\ photoproduction~\protect\cite{jpsi}
are reached at the highest \qsq\ values, suggesting similarly 
hard production mechanisms for $\rho$ electroproduction at high
$Q^2$ and $J/\psi$ photoproduction. 




%\subsection{\boldmath{$t$} dependence of the \boldmath{$ep$}
%            cross section}
\subsection{\boldmath{$t$} dependence}
    \label{sect:tdepend} 

%===========================================
%===========================================


The $t$ dependence of the 
%$e + p \rightarrow e + \rho + p$
cross section is shown in fig.~\ref{fig:tfit}, for four intervals
in \qsq .
The superimposed curves show the results of fits to exponential
parameterisations
${\rm d} \sigma / {\rm d} t \propto \exp{(b t)}$.
The slopes $b$ are shown as a function of
\qsq\ in fig.~\ref{fig:bq2}, together with previous
measurements~\cite{h1-rho,h1-photoprod,zeus}.
As \qsq\ increases, the $b$ slopes decrease,
reflecting the decrease of the 
transverse size of the \qqbar\ pair into which the photon 
fluctuates as $Q^2$ increases.
At the largest $Q^2$ the $b$ parameter is compatible with that
%to values close to those
obtained in \jpsi\ photoproduction~\cite{jpsi} and is close
to the value expected from the proton form factor.

%===========================================
%===========================================
\subsection{ \boldmath{\rfour} spin density matrix element}
  \label{sect:rfour}

%===========================================
%===========================================

Fig.~\ref{fig:ctsfit} shows the \cost\ dependence of the
%$e + p \rightarrow e + \rho + p$ 
cross section, for three intervals
in \qsq , three intervals in $W$ and three intervals in $t$.
The superimposed curves show the results of fits of the 
form 
\begin{equation}
  \frac {\rm{d} \sigma} {\rm{d} \cost}  \propto
   1 - \rfour\ + (3 \ \rfour - 1) \cos^2{\theta^{\ast}}.
                                \label{eq:cost}
\end{equation}
The extracted values of the \rfour\ spin density matrix element
are shown in fig.~\ref{fig:r400}
as a function of $Q^2$, $W$ and $t$.

The \rfour\ matrix element grows with \qsq , indicating a 
significant increase of the longitudinal to transverse cross section
ratio, up to the highest accessible \qsq\ values.
Fig.~\ref{fig:rlt} shows the corresponding value of
$R = \sigma_{L} / \sigma_{T}$ together with
previous measurements~\cite{h1-rho,h1-photoprod,zeus_phottt,zeus}.
$R$ is detemined under the approximation of $s$-channel helicity
conservation using the relation
\begin{equation}
R = \frac{1}{\varepsilon} \frac{\rfour}{(1-\rfour)}.
\end{equation}
The $Q^2$ dependence of $R$ is well described by the model
of Martin, Ryskin and Teubner \cite{mrt}, based on parton-hadron
duality, using the MRS(R4) \cite{mrsr4}
parton densities.

With the present precision, there is no evidence for any
dependence of \rfour\ on $W$ or $t$.
The lack of $t$ dependence
indicates that, in the measured range, the longitudinal and transverse
cross sections have very similar
$t$ dependences.



%===========================================
\subsection{
 (\boldmath{\rfivecomb}) and (\boldmath{\ronecomb}) combinations}
  \label{sect:combin}

%===========================================
%===========================================

Fig.~\ref{fig:phi2fit} shows the $\Phi$ dependence of the
%$e + p \rightarrow e + \rho + p$ 
cross section, for three intervals
in \qsq , three intervals in $W$ and three intervals in $t$.
The superimposed curves show results of fits of the form
\begin{equation}
  \frac {\rm{d} \sigma} {\rm{d} \Phi} \propto 
   1 +
   \sqrt {2 \varepsilon (1+\varepsilon)} \ \cos {\Phi} \ (\rfivecomb)
   - \varepsilon \ \cos {2 \Phi} \ (\ronecomb),
                                \label{eq:Phi}
\end{equation}
where the spin density matrix element combinations \rfivecomb\
and \ronecomb\ are sensitive to the transition
amplitudes in which the helicity of the $\rho$ meson is different from that 
of the virtual photon \cite{h1-hight}.
 
The extracted combinations of spin density matrix elements 
\rfivecomb\ and \ronecomb\ are shown as a function of \qsq , $W$ 
and \ttra\ in figs.~\ref{fig:r5} and~\ref{fig:r1}, respectively.
Values significantly different from zero and increasing with \ttra\ 
are obtained for the \rfivecomb\ combination, confirming $s$-channel 
helicity non conservation~\cite{h1-rho,h1-hight,zeus-smde} as predicted
by pQCD based models~\cite{ivanov,nikolaev}.
At the largest values of $|t|$ accessed, there is also an indication
that the combination \ronecomb\ becomes negative. There is no evidence
from the present data for any variation of the \rfivecomb\ or
\ronecomb\ combinations with $Q^2$ or $W$.

%\section{Additional Figures}

%A number of additional figures, which compare the new results 
%presented in this paper with other H1 vector meson results can
%be found in the appendix.

%===========================================
\section{Conclusions}
                                     \label{sect:concl}
%=============================================

The elastic electroproduction of $\rho$ mesons,
$e + p \rightarrow e + \rho + p$, has been studied at HERA
in the kinematic range $8 < Q^2 < 60 $ \gevsq , $40 < W < 180$ GeV, 
$0 < |t| < 0.5$ \gevsq .

The \qsq , $W$ and $t$ dependences of the $\gsp \rightarrow \rho p$
 cross section have been measured. 
In the present \qsq\ range, 
the \qsq\ distribution is described by the form
${\rm d} \sigma / {\rm d} Q^2
  \propto 1 / (Q^2  + m_\rho^2)^n$, with $n = 2.60 \pm 0.04$.
This value of $n$ is significantly larger than that obtained when lower
$Q^2$ data are included \cite{h1-rho}. 
The $W$ dependence of the cross section at fixed \qsq, when 
parameterised as $ \sigma (W) \propto W^\delta$, 
confirms a significant increase of the fitted values of
$\delta$ with \qsq . At the highest \qsq\ , $\delta$ reaches values  
similar to that obtained in \jpsi\ photoproduction.
Exponential fits to the \ttra\ dependence of 
the cross section give slope
parameters $b$ decreasing with increasing \qsq.
For the highest \qsq\ data, the slope parameter 
reaches values similar to those obtained for 
\jpsi\ photoproduction. 

The \rfour\ spin density matrix element and the combinations
\rfivecomb\ and \ronecomb\ have been measured as a function of \qsq ,
$W$ and \ttra .
The \rfour\ 
spin density matrix element increases with \qsq , indicating a 
significant increase of the longitudinal to transverse cross section
ratio, up to the highest available \qsq\ values.
In contrast, with the present 
precision, \rfour\ is independent of $W$ and \ttra . The latter feature
indicates that, in the measured range, the longitudinal and transverse
cross sections have similar 
\ttra\ dependences. 
Values significantly different from zero and increasing with \ttra\ 
are obtained for the \rfivecomb\ 
matrix element combination, confirming $s$-channel 
helicity non-conservation~\cite{h1-rho,h1-hight,zeus-smde}.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Acknowledgements}

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



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{thebibliography}{99}


\bibitem{dl} 
    A. Donnachie and P.V. Landshoff, 
    Phys. Lett. {\bf B395} (1997) 311.
    %%CITATION = NONE;%%

\bibitem{ryskin}
    M.G.~Ryskin {\it et al.}, 
    Z. Phys.  {\bf C76} (1997) 231. 

\bibitem{sch-w}
    K.~Schilling and G.~Wolf, 
    %  ``How to analyse vector meson production in inelastic lepton
    %    scattering'' 
    Nucl. \ Phys. \ {\bf B61} (1973) 381. 
    %%CITATION = NONE;%%

\bibitem{h1:detector}
    I.~Abt {\it et al.}  [H1 Collaboration],
    % ``The H1 Detector At Hera'',
    Nucl.\ Instrum.\ Meth.\ A {\bf 386} (1997) 310.
    %%CITATION = NUIMA,A386,310;%%

\bibitem{h1:spacal}
    R.~D.~Appuhn {\it et al.}  [H1 SPACAL Group],
    % ``The H1 lead/scintillating-fibre calorimeter'',
    Nucl.\ Instrum.\ Meth.\ A {\bf 386} (1997) 397.
    %%CITATION = NUIMA,A386,397;%%

\bibitem{da}
    S. Bentvelsen, J. Engelen and P. Kooijman,
    in Proc. of the Workshop on Physics at HERA,
    W. Buchm\"uller and G. Ingelman ed., Hamburg (1992) 23;
    K.C. Hoeger, ibid, p 43. 
    %%CITATION = NONE;%%

\bibitem{jb}
    F. Jacquet and A. Blondel,
    DESY 79-048 (1979) 377.
    %%CITATION = NONE;%%

\bibitem{diffvm}
    B.~List and A. Mastroberardino, in Proc. of the Workshop on 
    Monte Carlo Generators for HERA Physics,
    % `` DIFFVM: A Monte Carlo generator for diffractive processes in 
    %ep scattering ''
    A.T. Doyle et al. ed.,
    DESY-PROC-1999-02 (1999) 396.
    %%CITATION = NONE;%%

\bibitem{heracles}
    A. Kwiatkowski, H.-J. M\"ohring and H. Spiesberger,
    % ``HERACLES: An Event Generator for ep Interactions at HERA
    % energies including radiative processes: Version 1.0''
    Comput. Phys. Commun. {\bf 69} (1992), 155 and Proc. of the Workshop on
    Physics at HERA, W. Buchm\"uller and G. Ingelman ed., Hamburg 
    (1992) 1294. 
    %%CITATION = CPHCB,69,155;%%

\bibitem{h1-hight}
    C.~Adloff {\it et al.}  [H1 Collaboration],
    `` A measurement of the $t$ dependence of the helicity structure 
    of diffractive $\rho$ meson electroproduction at HERA'',
    accepted by Phys Lett. B [hep-ex/0203022].
    %%CITATION = HEP-EX 0203022;%%

\bibitem{h1-rho}
    C.~Adloff {\it et al.}  [H1 Collaboration],
    % `` Elastic Electroproduction of $\rho$ Mesons at HERA''
    Eur. \ Phys. \ J. \ {\bf C13} (2000) 371.
    % [hep-ex/9902019].
    %%CITATION = HEP-EX 9902019;%%

\bibitem{goulianos}
    K. Goulianos,
    Phys. \ Rep. \ {\bf 101} (1983) 169.
    %%CITATION = PRPLC,101,169;%%

\bibitem{h1-phi}
    C.~Adloff {\it et al.}  [H1 Collaboration],
    % ``Measurement of elastic electroproduction of phi mesons 
    %at HERA''
    Phys. \ Lett. \ {\bf B483} (2000) 360. 
    %[hep-ex/0005010].
    %%CITATION = HEP-EX 0005010;%%

\bibitem{zeus-omega}
    J.~Breitweg {\it et al.}  [ZEUS Collaboration],
    %  `` Measurement of the exclusive omega electroproduction
    %     at HERA''
    Phys. \ Lett. \ {\bf B487} (2000) 273.
    %[hep-ex/0006013].
    %%CITATION = HEP-EX 0006013;%%

\bibitem{zeus}
    J.~Breitweg {\it et al.}  [ZEUS Collaboration],
    % `` Exclusive Electroproduction of $\rho^0$ and $J/\Psi$ 
    %Mesons at    HERA''
    Eur. \ Phys. \  J. \ {\bf C6} (1999) 603.
    %[hep-ex/9808020].

\bibitem{zeus-preli}
    ZEUS Collaboration, ``Exclusive electroproduction of $\rho^0$ 
    mesons at HERA'',
    Contrib. pap. 594 to the Eur. Phys. Soc. Conference, Budapest 2001.

\bibitem{jpsi}
    C. Adloff {\it et al.} [H1 Collaboration], 
    Phys. Lett. {\bf B483} (2000) 23.

\bibitem{h1-photoprod}
    S.~Aid {\it et al.}  [H1 Collaboration], 
    Nucl. \ Phys. \ {\bf B463} (1996) 3. 

\bibitem {zeus_phottt}
    J. Breitweg {\it et al.}, [ZEUS Collaboration],
    Eur. \ Phys. \ J. \  {\bf C2} (1998) 247.

\bibitem{mrt}
A. Martin, M. Ryskin, T. Teubner, 
Phys. Rev. {\bf D55} (1997) 4329.

\bibitem{mrsr4} 
A. Martin, R. Roberts, W. Stirling, 
Phys. Lett. {\bf B387} (1986) 419.

\bibitem{zeus-smde} 
    J.~Breitweg {\it et al.}  [ZEUS Collaboration],
    % ``Measurement of the Spin-Density Matrix Elements in Exclusive
    %   Electroproduction of $\rho^0$ Mesons at HERA''
    Eur. \ Phys. \ J. \ {\bf C12} (2000) 393. 
    % [hep-ex/9908026].

\bibitem{ivanov}
    D.Yu.~Ivanov and R.~Kirschner, 
    %  ``Polarization in diffractive electroproduction of light vectors
    %    mesons''
    Phys. Rev. {\bf D58} (1998) 114026.
    %[hep-ph/9807324].
    %%CITATION = HEP-PH 9807324;%%

\bibitem{nikolaev}
    E.V.~Kuraev, N.N.~Nikolaev and B.G.~Zakharov, 
    JETP  Lett.  {\bf68} (1998) 696.
    %[hep-ph/9809539].
    %%CITATION = HEP-PH 9809539;%%



\end{thebibliography}

\clearpage


%=======================\label{fig:mass}===============
\begin{figure}[p]
\vspace{-0.cm}
\begin{center}
\epsfig{file=H1prelim-02-015.fig1.eps,width=16.cm}\\
\end{center}
 \caption{Uncorrected $\pi^+\pi^-$ mass distribution
 for the selected events with  $0.3 < M_{\pi \pi} < 1.5$\ GeV;
 the shaded histograms show the $\omega$ and $\phi$ backgrounds and
 the hatched histograms the \protect\rhoprim\ background.}
 \label{fig:mass}
 \end{figure}
%=======================\end{fig:mass} ======================

%=======================\label{fig:control}===============
\begin{figure}[p]
\vspace{-0.cm}
\begin{center}
\epsfig{file=H1prelim-02-015.fig2.eps,width=16.cm}\\
\end{center}
 \vspace{-0.7cm}
 \setlength{\unitlength}{1.0cm}
 \begin{picture}(16.0,0.1)
    \put(3.00,9.5){(a)}
    \put(7.70,9.5){(b)}
    \put(12.5,9.5){(c)}
    \put(3.00,4.7){(d)}
    \put(7.70,4.7){(e)}
    \put(12.5,4.7){(f)}
 \end{picture}
 \vspace{-0.3cm}
 \caption{Distribution of the selected events in the kinematic domain
 (\protect\ref{eq:kin_range}) and in the mass range
 $0.6 < M_{\pi\pi} < 1.1$ GeV, for 
 the scattered electron polar angle (a), 
 the pion candidate transverse momentum $p_t$ (b),
 \protect\ttra\ (c),
 $W$ for $8 < Q^2 < 12$ \protect\gevsq\ (d),
 $12 < Q^2 < 20$ \protect\gevsq\ (e) and 
 $20 < Q^2 < 60$ \protect\gevsq\ (f).
 The histograms show the Monte-Carlo predictions, including the
 proton dissociative (shaded histograms) and the $\omega$, $\phi$ and
 \protect\rhoprim\ backgrounds (hatched histograms).}
 \label{fig:control}
 \end{figure}
%=======================\end{fig:control} ======================

%=======================\label{fig:xsq2}===============
\begin{figure}[p]
\vspace{-0.cm}
\begin{center}
\epsfig{file=H1prelim-02-015.fig3.eps,width=16.cm}\\ 
\end{center}
 \caption{The $\gamma^* p \rightarrow  \rho p$ cross section as a
 function of \qsq , for $W = 95$ GeV. 
 Also shown are the data from~\protect\cite{h1-rho,zeus},
 extrapolated to the same $W$
 value using the $W$ dependence measured in the present analysis.
% and~\protect\cite{zeus}.
 The inner error bars are statistical,
and the full error bars include the systematic errors added in quadrature. 
 The superimposed line is for
 $\sigma \propto 1 / (Q^2  + m_\rho^2)^{n}$ 
 with $n = 2.60$. The fit error includes both statistical and uncorrelated
 systematic errors.}
 \label{fig:xsq2}
 \end{figure}
%=======================\end{fig:xsq2} ======================

%=======================\label{fig:xsw}===============
\begin{figure}[p]
\vspace{-0.cm}
\begin{center}
\epsfig{file=H1prelim-02-015.fig4.eps,width=16.cm}\\ 
\end{center}
 \caption{The $\gamma^* p \rightarrow  \rho p$ cross section as a
 function of $W$ for several \qsq\ values. The inner error bars are statistical,
 and the full error bars include the systematic errors added in quadrature. 
 The superimposed lines represent parameterisations of the form
 $\sigma (W) \propto W^\delta$.}
 \label{fig:xsw}
 \end{figure}
%=======================\end{fig:xsw} ======================

%=======================\label{fig:delta}===============
\begin{figure}[p]
\vspace{-0.cm}
\begin{center}
\epsfig{file=H1prelim-02-015.fig5.eps,width=16.cm}\\ 
\end{center}
 \caption{Results of fits of the form 
 $\sigma (W) \propto W^\delta$ to the $W$ dependence 
 of the \gsp\ cross section, presented as a function of \qsq . 
 Also shown are the measurements~\protect\cite{h1-rho} 
 and~\protect\cite{zeus-preli} for $\rho$ meson production 
 and~\protect\cite{jpsi} for \jpsi\ photoproduction.
 The errors include both statistical and 
 uncorrelated systematic errors.} 
 \label{fig:delta}
 \end{figure}
%=======================\end{fig:delta} ======================

%=======================\label{fig:tfit}===============
\begin{figure}[htbp]
\vspace{-0.cm}
\begin{center}
\epsfig{file=H1prelim-02-015.fig6.eps,width=16.cm}\\
\end{center}
 \caption{The \protect\ttra\ dependence of the
 cross section, for several
 \protect\qsq\ intervals.
 The superimposed curves show results of fits to exponential
 distributions for $\ttra < 0.5 \ \gevsq$. 
 Only the statistical errors are shown. }
 \label{fig:tfit}
 \end{figure}
%=======================\end{fig:tfit} ================

%=======================\label{fig:bq2}===============
\begin{figure}[p]
\vspace{-0.cm}
\begin{center}
\epsfig{file=H1prelim-02-015.fig7.eps,width=16.cm}\\ 
\end{center}
 \caption{Results of fits to an exponential parameterisation of the
  \protect\ttra\ dependence of the 
 cross section, presented as a 
 function of \qsq .
 Also shown are the measurements~\protect\cite{h1-rho}, 
 \protect\cite{h1-photoprod} and~\protect\cite{zeus}
 for $\rho$ meson production 
 and~\protect\cite{jpsi} for \jpsi\ photoproduction.
 The inner error bars are statistical,
and the full error bars include the systematic errors added in quadrature.  }
 \label{fig:bq2}
 \end{figure}
%=======================\end{fig:bq2} ======================

%=======================\label{fig:ctsfit}===============
\begin{figure}[htbp]
\vspace{-0.cm}
\begin{center}
\epsfig{file=H1prelim-02-015.fig8.eps,width=16.cm}\\
\end{center}
 \caption{The \cost\ dependence of the
cross section, for severals \qsq ,
$W$ and \ttra\ intervals. The superimposed curves show results of
fits to the relation (\ref{eq:cost}). }
 \label{fig:ctsfit}
 \end{figure}
%=======================\end{fig:ctsfit} ======================

%=======================\label{fig:r400}===============
\begin{figure}[p]
\vspace{-0.cm} 
\begin{center}
\epsfig{file=H1prelim-02-015.fig9.eps,width=16.cm}\\
\end{center}
 \caption{Measurements of the \rfour\ spin density matrix element,
  as a function of \qsq , $W$ and \ttra.
  Also shown are the measurements \cite{h1-rho,h1-hight,zeus-smde}.
  The data labelled ``H1 97 diffractive'' correspond to measurements
  without separation of the elastic and the proton dissociative channels.
  The inner error bars are statistical,
  and the full error bars include the systematic errors added in quadrature. }
 \label{fig:r400}
 \end{figure}
%=======================\end{fig:r400} ======================

%=======================\label{fig:rlt}===============
\begin{figure}[p]
\vspace{-0.cm}
\begin{center}
\epsfig{file=H1prelim-02-015.fig10-col.eps,width=16.cm}\\ 
%\epsfig{file=H1prelim-02-015.fig7.eps,width=16.cm}\\ 
\end{center}
 \caption{The ratio $R$ of longitudinal to transverse
  cross sections for elastic \rh\ meson electroproduction 
  by longitudinal and transverse photons, measured in the SCHC approximation 
  and presented as a function of \qsq. The other measurements are from H1
  \cite{h1-rho,h1-photoprod} and ZEUS \cite{zeus_phottt,zeus}.
  The inner error bars are statistical,
  and the full error bars include the 
 systematic errors added in quadrature. 
The line corresponds to the prediction of Martin, Ryskin and
Teubner \cite{mrt}. 
}
 \label{fig:rlt}
 \end{figure}
%=======================\end{fig:rlt} ======================

%=======================\label{fig:phi2fit}===============
\begin{figure}[htbp]
\vspace{-0.cm}
\begin{center}
\epsfig{file=H1prelim-02-015.fig11.eps,width=16.cm}\\
\end{center}
 \caption{The $\Phi$ dependence of the
cross section, for severals \qsq ,
$W$ and \ttra\ intervals. The superimposed curves show results of
fits to the relation (\ref{eq:Phi}).}
 \label{fig:phi2fit}
 \end{figure}
%=======================\end{fig:phi2fit} ======================

%=======================\label{fig:r5}===============
\begin{figure}[p]
\vspace{-0.cm}
\begin{center}
\epsfig{file=H1prelim-02-015.fig12.eps,width=16.cm}\\ 
\end{center}
 \caption{Measurements of the \rfivecomb\ spin density matrix element 
  combination as a function of \qsq , $W$ and \ttra.
  Also shown are the measurements  \cite{h1-rho,h1-hight,zeus-smde}. 
  %The superimposed line is taken from \cite{h1-hight}.
  The inner error bars are statistical,
  and the full error bars include the systematic errors added in quadrature.
  The dashed lines indicate the expected null values in 
  the case of $s$-channel helicity conservation (SCHC).
 }
 \label{fig:r5}
 \end{figure}
%=======================\end{fig:r5} ======================

%=======================\label{fig:r1}===============
\begin{figure}[p]
\vspace{-0.cm}
\begin{center}
\epsfig{file=H1prelim-02-015.fig13.eps,width=16.cm}\\ 
\end{center}
 \caption{Measurements of the \ronecomb spin density matrix element 
  combination as a function of \qsq , $W$ and \ttra.
  Also shown are the measurements  \cite{h1-rho,h1-hight,zeus-smde}. 
  %The superimposed line is taken from \cite{h1-hight}.
  The inner error bars are statistical,
  and the full error bars include the systematic errors added in quadrature.
  The dashed lines indicate the expected null values in 
  the case of $s$-channel helicity conservation (SCHC).
 } 
 \label{fig:r1}
 \end{figure}
%=======================\end{fig:r1} ======================

\clearpage

%\appendix
%\setcounter{figure}{0}
%\renewcommand{\thefigure}{\thesection\arabic{figure}}
 
%\section{Additional Figures}

%\begin{figure}[h]
%\vspace{-0.cm}
%\begin{center}
%\epsfig{file=H1prelim-02-015.fig6.eps,width=16.cm}\\ 
%\end{center}
% \caption{BLAH BLAH BLAH} 
% \end{figure}

%\begin{figure}[p]
%\vspace{-0.cm}
%\begin{center}
%\epsfig{file=H1prelim-02-015.fig7.eps,width=16.cm}\\ 
%\end{center}
% \caption{BLAH BLAH BLAH} 
% \end{figure}

%\begin{figure}[p]
%\vspace{-0.cm}
%\begin{center}
%\epsfig{file=H1prelim-02-015.fig9.eps,width=16.cm}\\ 
%\end{center}
% \caption{BLAH BLAH BLAH} 
% \end{figure}

%\begin{figure}[p]
%\vspace{-0.cm}
%\begin{center}
%\epsfig{file=H1prelim-02-015.fig11.eps,width=16.cm}\\ 
%\end{center}
% \caption{BLAH BLAH BLAH} 
% \end{figure}

%\begin{figure}[p]
%\vspace{-0.cm}
%\begin{center}
%\epsfig{file=H1prelim-02-015.fig12a.eps,width=9.cm}\\ 
%\epsfig{file=H1prelim-02-015.fig12b.eps,width=9.cm}\\ 
%\end{center}
% \caption{BLAH BLAH BLAH} 
% \end{figure}

%\begin{figure}[p]
%\vspace{-0.cm}
%\begin{center}
%\epsfig{file=H1prelim-02-015.fig13a.eps,width=9.cm}\\ 
%\epsfig{file=H1prelim-02-015.fig13b.eps,width=9.cm}\\ 
%\end{center}
% \caption{BLAH BLAH BLAH} 
% \end{figure}

%\begin{figure}[p]
%\vspace{-0.cm}
%\begin{center}
%\epsfig{file=H1prelim-02-015.fig14.eps,width=16.cm}\\ 
%\end{center}
% \caption{BLAH BLAH BLAH} 
% \end{figure}

\end{document}









