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% special definition for this paper
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\begin{titlepage}

\noindent

\noindent
{\bf  H1-prelim-02-015 \hfill 
  \epsfig{file=/h1/www/images/H1logo_bw_small.epsi,width=2.cm} 
} \\
\noindent April 2002 


\vspace*{6cm}

\begin{center}
\begin{Large}

{Elastic electroproduction of \boldmath{$\rho$} mesons \\
   with $8 < Q^2 < 60$ \gevsq\ at HERA}
 
\vspace{2cm}

H1 Collaboration

\end{Large}
\end{center}

\vspace{2cm}

\begin{abstract}

Elastic electroproduction of rho mesons is studied at HERA with the H1 
detector using an integrated luminosity of 42.4 $pb^{-1}$ in the
kinematic range $8 < Q^2 < 60 $ \gevsq , $40 < W < 180$ GeV, 
$0 < |t| < 0.5$ \gevsq. The increase of luminosity compared to previous
measurments gives access to the \qsq\ domain where perturbative QCD is 
expected to apply. Cross sections have been measured as a
function of \qsq , $W$ and $t$ . The $W$ dependence of the $\gamma^{*}p$ cross 
section is observed to increase with \qsq\ from values compatible with a
soft Pomeron exchange at low \qsq\ to hard dependence at large \qsq. 
Spin density matrix elements are measured and their dependence
is compared with a 2 gluon exchange model. 

\end{abstract}

\vspace{1.5cm}

\begin{center}
%To be submitted to Phys. Lett. B
\end{center}

\end{titlepage}

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%\newpage

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

The subject of this paper is the study of the elastic electroproduction 
of \rh\ mesons in $ep$ scattering at high energy:
%
\begin{equation}
e + p \rightarrow e + \rh\ + 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).


The kinematic domain of the measurement is:
%
\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}
%


Three angles are defined to characterise the electroproduction 
of vector mesons 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.




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



\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 both 
electrons and positrons.} 
%===========
were 920 and 27.5 GeV, respectively. 
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-rho,nim}, 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 proton remnant tagger (PRT), a set of 7 scintillator layers
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 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 restricted
to $0.6 < M_{\pi \pi} < 1.1  GeV$.
%
\item
the absence of any energy deposit
larger than 400~MeV in the LAr calorimeter, 
which is not associated 
with the electron or 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 rejecting 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
events with $M_{KK}~<~1.04~{\rm GeV}$ are discarded, 
where $M_{KK}$ is the invariant mass of the two hadron 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 
a cut $\sum(E-p_z) > 45\ {\rm GeV}$ is applied, in order to reduce  
QED radiative corrections. $\sum(E-p_z)$ 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 high energy photons radiated by the electron.
%
\end{itemize}
%
To first approximation, the selected events can be attributed
to elastic scattering.

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

A Monte Carlo simulation based on the DIFFVM program~\cite{diffvm} 
including QED radiation~\cite{heracles} is used to describe   
the elastic production and decay of $\rho$ vector mesons,
% in elastic and proton dissociative scattering, 
and to correct the data for acceptance, smearing and radiative 
effects.

The DIFFVM simulation is also used for the description of the
\rh\ proton dissociative, $\omega$, $\phi$ and \rhoprim\ 
backgrounds.


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

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

In each bin of the kinematical variables, the cross section is 
computed from the numbers of events in the bin, fully corrected for 
backgrounds, acceptance, smearing and radiation effects using the 
Monte Carlo simulations described above.
It is converted into a \gsp\ cross section using the relation:
%
\begin{equation}
\frac{d^2 \sigma_{tot} (ep \rightarrow e \rho p)}{dy \ dQ^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, 
%
\begin{equation}
R\ = \sigma_{L} / \sigma_{T},
                                             \label{eq:Rdef}
\end{equation}
and $\Gamma$ is the flux 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}
$\varepsilon$\ is the polarisation parameter
%
\begin{equation}
\varepsilon = \frac{1 - y}{1-y+y^2/2}.
                                       \label{eq:epsil}
\end{equation}
%
For this experiment, $\varepsilon = 0.996$.
The  \qsq\ and $W$ dependences used for the
computation of the flux factors were taken from the present analysis.

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.

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



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

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

Fig.~\ref{fig:xsq2} presents the \qsq\ dependence of the \gsp\ cross
section, for $W = 95$ GeV.
A fit of the form ${\rm d} \sigma / {\rm d} Q^2
  \propto 1 / (Q^2  + m_\rho^2)$ for the range $8 < Q^2 < 60$~\gevsq\
results in $n = 2.60 \pm 0.04$, with $\chi^2 / ndf = 4.6 / 10$.

%=======================\label{fig:xsq2}===============
\begin{figure}[htbp]
\vspace{-0.cm}
\begin{center}
\epsfig{file=H1prelim-02-015.fig1.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~\protect\cite{h1-rho}, extrapolated to same $W$
 value using the $W$ dependences measured in the present analysis.
% and~\protect\cite{zeus-rho}.
 The inner error bars are statistical,
and the full error bars include the systematic errors added in quadrature. 
 The superimposed line is for
 ${\rm d} \sigma / {\rm d} Q^2 \propto 1 / (Q^2  + m_\rho^2)$ 
 with $n = 2.60$. The fit error include both statistical and uncorrelated
 systematic errors.}
 \label{fig:xsq2}
 \end{figure}
%=======================\end{fig:xsq2} ======================




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

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

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

Fig.~\ref{fig:xsw} presents the $W$ dependence of the \gsp\ cross
section, for four bins in \qsq .
The cross section rise with $W$ is observed to be stronger as \qsq\ 
increases.
The $W$ dependence at fixed \qsq\ being parameterised as 
${\rm d} \sigma / {\rm d} W \propto W^\delta$, the fitted values of
$\delta$ are shown in Fig.~\ref{fig:delta}, together with previous
measurements. 
Values of order 1. are reached for the highest \qsq\ range, 
similar to measurements for \jpsi\ photoproduction~\protect\cite{jpsi}.


%=======================\label{fig:xsw}===============
\begin{figure}[htbp]
\vspace{-0.cm}
\begin{center}
\epsfig{file=H1prelim-02-015.fig2.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 line represents parameterisations of the form
 ${\rm d} \sigma / {\rm d} W \propto W^\delta$.}
 \label{fig:xsw}
 \end{figure}
%=======================\end{fig:xsw} ======================


%=======================\label{fig:delta}===============
\begin{figure}[htbp]
\vspace{-0.cm}
\begin{center}
\epsfig{file=H1prelim-02-015.fig3.eps,width=16.cm}\\ 
\end{center}
 \caption{Results of fits of the form 
 ${\rm d} \sigma / {\rm d} W \propto W^\delta$ 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-rho} 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} ======================


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

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


The \ttra\ dependence of the $e + p \rightarrow e + \rho + p$
cross section has been extracted for four intervals
in \qsq .
The  slopes resulting from exponential fits to the data
are shown as a function of
\qsq\ in fig.~\ref{fig:bq2}, together with previous
measurements. 
As \qsq\ increases, they decrease down to values close to those 
obtained for \jpsi\ photoproduction are obtained~\cite{jpsi}. 

%=======================\label{fig:bq2}===============
\begin{figure}[htbp]
\vspace{-0.cm}
\begin{center}
\epsfig{file=H1prelim-02-015.fig4.eps,width=16.cm}\\ 
\end{center}
 \caption{Results of fits to exponential distributions of the
  to the \protect\ttra\ dependence of the 
 $e + p \rightarrow e + \rho + p$ 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-rho}
 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} ======================


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

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

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 \ttra 
have been fitted to 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 as a function of \qsq , $W$ and \ttra\ in 
fig.~\ref{fig:r400}.
Fig.~\ref{fig:rlt} presents the corresponding value of 
$R = \sigma_{L} / \sigma_{T}$, extracted under the $s$-channel helicity
conservation approximation, together with
previous measurements. 

The \rfour\ 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, \rfour\ is independent of $W$ and \ttra , the latter feature
indicating that, in the measured range, the longitudinal and transverse
\ttra\ distributions have the same behaviour.

%=======================\label{fig:r400}===============
\begin{figure}[htbp]
\vspace{-0.cm} 
\begin{center}
\epsfig{file=H1prelim-02-015.fig5.eps,width=16.cm}\\ 
\end{center}
 \caption{Measurments of \rfour as a function of \qsq , $W$ and \ttra.
  Also shown are measurments \cite{h1-rho,h1-hight,zeus-smde}.
  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:r400}===============
\begin{figure}[htbp]
\vspace{-0.cm} 
\begin{center}
\epsfig{file=H1prelim-02-015.fig6.eps,width=16.cm}\\ 
\end{center}
 \caption{Same as figure \ref{fig:r400} on different scales.}
 \label{fig:r400b}
 \end{figure}
%=======================\end{fig:r400} ======================

%=======================\label{fig:rlt}===============
\begin{figure}[htbp]
\vspace{-0.cm}
\begin{center}
\epsfig{file=H1prelim-02-015.fig7.eps,width=16.cm}\\ 
\end{center}
 \caption{The ratio $R$ of 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-rho}.
  The inner error bars are statistical,
  and the full error bars include the systematic errors added in quadrature.  }
 \label{fig:rlt}
 \end{figure}
%=======================\end{fig:rlt} ======================

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

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

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 \ttra 
have been fitted to the form
\begin{equation}
  \frac {\rm{d} \sigma} {\rm{d} \Phi} \propto 
   1 +
   \sqrt {2 \epsilon (1+\epsilon)} \ \cos {\Phi} \ (\rfivecomb)
   - \epsilon \ \cos {2 \Phi} \ (\ronecomb)
                                \label{eq:Phi}
\end{equation}
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 of 0 and increasing with \ttra\ 
are obtained for the \rfivecomb\ combination, confirming $s$-channel 
helicity non conservation~\cite{h1-rho,h1-hight,zeus-smde}.

%=======================\label{fig:r5}===============
\begin{figure}[htbp]
\vspace{-0.cm}
\begin{center}
\epsfig{file=H1prelim-02-015.fig8.eps,width=16.cm}\\ 
\end{center}
 \caption{Measurments of \rfivecomb as a function of \qsq , $W$ and \ttra.
  Also shown are measurments  \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.
 }
 \label{fig:r5}
 \end{figure}
%=======================\end{fig:r5} ======================
%=======================\label{fig:r5}===============
\begin{figure}[htbp]
\vspace{-0.cm}
\begin{center}
\epsfig{file=H1prelim-02-015.fig9.eps,width=16.cm}\\ 
\end{center}
 \caption{Same as figure \ref{fig:r5} on different scales.}
 \label{fig:r5b}
 \end{figure}
%=======================\end{fig:r5} ======================


%=======================\label{fig:r1}===============
\begin{figure}[htbp]
\vspace{-0.cm}
\begin{center}
\epsfig{file=H1prelim-02-015.fig10.eps,width=16.cm}\\ 
\end{center}
 \caption{Measurments of \ronecomb as a function of \qsq , $W$ and \ttra.
 Also shown are measurments  \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.} 
 \label{fig:r1}
 \end{figure}
%=======================\end{fig:r1} ======================


%=======================\label{fig:r1b}===============
\begin{figure}[htbp]
\vspace{-0.cm}
\begin{center}
\epsfig{file=H1prelim-02-015.fig11.eps,width=16.cm}\\ 
\end{center}
 \caption{Same as fig \ref{fig:r1} on different scales.}
 \label{fig:r1b}
 \end{figure}
%=======================\end{fig:r1b} ======================


%=======================\label{fig:otto}===============
\begin{figure}[htbp]
\vspace{-0.cm}
\begin{center}
\epsfig{file=H1prelim-02-015.fig12a.eps,width=10.cm}\\ 
\epsfig{file=H1prelim-02-015.fig12b.eps,width=10.cm}\\ 
\end{center}
 \caption{$b$ slope as function of $Q^2$ and $Q^2 + M^2$.}
 \label{fig:otto1}
 \end{figure}
%=======================\end{fig:r1} ======================
%=======================\label{fig:otto}===============
\begin{figure}[htbp]
\vspace{-0.cm}
\begin{center}
\epsfig{file=H1prelim-02-015.fig13a.eps,width=10.cm}\\ 
\epsfig{file=H1prelim-02-015.fig13b.eps,width=10.cm}\\ 
\end{center}
 \caption{$\delta$ as function of $Q^2$ and $Q^2 + M^2$.}
 \label{fig:otto1}
 \end{figure}
%=======================\end{fig:r1} ======================
%=======================\label{fig:otto}===============
\begin{figure}[htbp]
\vspace{-0.cm}
\begin{center}
\epsfig{file=H1prelim-02-015.fig14.eps,width=12.cm}\\ 
\end{center}
 \caption{Cross section ratio for $\rho$ and $J/\Psi$.}
 \label{fig:otto1}
 \end{figure}
%=======================\end{fig:r1} ======================


%===========================================
\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\ cross section have been 
measured.
The \qsq\ distribution is well described in the form
${\rm d} \sigma / {\rm d} Q^2
  \propto 1 / (Q^2  + m_\rho^2)$, with $n = 2.60 \pm 0.04$.
The $W$ dependence at fixed \qsq\ of the cross section, when 
parameterised as ${\rm d} \sigma / {\rm d} W \propto W^\delta$, 
indicates a significant increase of the fitted values of
$\delta$ with \qsq , reaching for the highest \qsq\ interval values 
of order 1., similar to \jpsi\ photoproduction measurements.
Fits to exponential distributions of the \ttra\ dependence of 
the cross section give slopes decreasing with increasing \qsq,
reaching for the highest \qsq\ data values close 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\ 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, \rfour\ is independent of $W$ and \ttra , the latter feature
indicating that, in the measured range, the longitudinal and transverse
\ttra\ distributions have the same behaviour.
Values significantly different of 0 and increasing with \ttra\ 
are obtained for the \rfivecomb\ 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.



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\end{document}








