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\noindent
\begin{center}
\begin{small}
\begin{tabular}{llrr}
Submitted to & & &
\epsfig{file=H1logo_bw_small.epsi
,width=2.cm} \\[.2em] \hline
\multicolumn{4}{l}{{\bf
                International Europhysics
                Conference on High Energy Physics},
                July~12,~2001,~Budapest} \\
 {\bf EPS 2001:}
             & Abstract:            & {\bf 796}       &\\
                 & Parallel Session & {\bf Soft Interactions, Hadronic
Structure and Diffraction}   &\\ [.7em]
%                 & Plenary Session  & {\bf }   &\\[.7em]
\multicolumn{4}{l}{{\bf
               XX International Symposium on Lepton and Photon
Interactions},
                July~23,~2001,~Rome} \\
{\bf LP 2001:}
                & Abstract:         & {\bf 489 }     &\\
                 & Parallel Session & {\bf s08 }     &\\ \hline
%                 & Plenary Session  & {\bf }     &\\ \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 Measurement of density matrix elements \\
  for \boldmath{$\rho$} meson production as a function 
  of \boldmath{$t$} at HERA}

  \vspace*{1cm}
    {\Large H1 Collaboration}
\end{center}

\begin{abstract}

\noindent
A sample of $e^+ \pi^+ \pi^-$ events with $Q^2 > 2.5$~\gevsq, 
$40 < W < 120$~GeV and $t^\prime = |t| - t_{min}$ 
in the range $0.5 < t^\prime < 3.0$~\gevsq\ is used to
measure the $r^{04}_{00}$ density matrix element and the 
combinations \rfivecomb\ and \ronecomb. 
The \tprim\ dependences are determined.
\end{abstract}


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\pagestyle{plain}

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

The mechanism of diffraction, especially in a QCD approach, can in particular 
be studied through the diffractive electroproduction 
of vector mesons (VM) at high energy:
%
\begin{equation}
e + p \rightarrow e + VM + Y,
                                \label{eq:VM_prod}
\end{equation}
%
where $Y$ is a proton (``elastic'' scattering) or a baryonic system 
of mass $M_Y$ much lower than the invariant mass $W$ of the hadronic 
system: $M_p < M_Y \ll W$ (``proton dissociation'' scattering).

The measurement of the production and decay angle 
distributions of the vector meson provides information on the angular 
momentum structure of the interaction.
Detailed results have been reported on the spin density matrix 
elements for $\rho$ and $\phi$ elastic electroproduction with 
$|t| < 0.5$~\gevsq~\cite{h1-rho,zeus,h1-phi}, $t$ being the square 
of the four-momentum transferred to the proton, and for $\rho$ 
electroproduction with proton dissociation for 
$t^\prime < 1.5 $~\gevsq~\cite{h1-pd}, where $t^\prime$ is defined as 
$t^\prime = |t| - t_{min}$, $t_{min}$ being the minimal 
value of $|t|$ kinematically required to put the vector meson and the
system $Y$ on shell. 
This variable is more relevant than $t$ when $M_Y$ and/or the photon 
virtuality are not negligible with respect to $W$.

Three main features emerge from these measurements~\footnote
%===========
{The following relations~\cite{sch-w} between \mdes\ and helicity 
amplitudes $T_{\lambda_{\rho}\lambda_{\gamma}}$ are
relevant for the present analysis, natural parity exchange 
being assumed to hold 
($T_{-\lambda_{\rho}-\lambda_{\gamma}} = 
 (-1)^{\lambda_{\rho} - \lambda_{\gamma}} \
        T_{\lambda_{\rho}\lambda_{\gamma}}$):
%
\begin{eqnarray}
  r^{04}_{00} \propto \frac {|T_{00}|^2 + |T_{01}|^2} {N}           \nonumber \\
  r^5_{00} \propto \frac {1} {N} {\rm Re} \ (T_{00} \ T_{01}^\dagger) \nonumber \\
  r^5_{11} \propto \frac {1} {N} ({\rm Re} \ (T_{10} T_{11}^\dagger)   
                  - {\rm Re} \ (T_{10} T_{1-1}^\dagger))                \nonumber \\
  r^1_{00} \propto  \frac {-1} {N}  |T_{01}|^2                       \nonumber \\
  r^1_{11} \propto  \frac {1} {N} (T_{1-1} T_{11}^\dagger + T_{11} T_{1-1}^\dagger) \nonumber \\
  {\rm with} \ \  N = \sum_{i,j} |T_{ij}|^2 \ ,  \ \
   i={\lambda}_{\rho},j={\lambda}_{\gamma} \ \ =-1,0,1 
%                            
\end{eqnarray}
}
%===========
%
%===========================================
\begin{itemize}
%%%%\item the dominance of the longitudinal over the transverse helicity 
%%%non-flip amplitude for $Q^2 \gsim 1$ \gevsq\ (\qsq\ is the 
\item the dominance of the longitudinal $T_{00}$ over the transverse
$T_{11}$ helicity non-flip amplitudes for $Q^2 \gsim 1$ \gevsq\ (\qsq\ is the 
negative square of the virtual photon four-momentum), as measured 
by the \rfour\ density matrix element in the 
{\it s}-channel helicity conservation (SCHC) approximation;
\item the presence of a small but significant violation of SCHC, 
revealed by the non-zero value of the \rfive\ matrix element, which 
indicates that 
the dominant helicity flip amplitude is from a transverse photon 
to a longitudinal vector meson ($T_{01} \neq 0$);
\item values compatible with zero for the other amplitudes describing
single or double helicity flip. 
\end{itemize}
%===========================================

These general features are in agreement with perturbative QCD
calculations~\footnote{
%===========
In Ref.~\cite{ivanov}, perturbative QCD calculations are assumed
to be valid for $|t| \leq Q^2 \gamma$, 
where $\gamma$ is the gluon anomalous dimension, depending on \qsq\ and 
the energy. 
Strictly speaking, the predictions are for elastic 
scattering.}
%===========
~\cite{ivanov,niko,royen}.
In addition, the following amplitude hierarchy is predicted for 
moderate \qsq\ and $t$:
$|T_{00}| > |T_{11}| > |T_{01}| > |T_{10}| > |T_{1-1}|$,
the $r^1_{00}$ \mde\ is expected to be slightly negative, 
and the following $t$ dependences are predicted for amplitude ratios:
a constant ratio for helicity conserving amplitudes,
a $\sqrt {|t|}$ dependence for the ratio of the single helicity
flip to the non-flip amplitudes,
and a linear dependence for the ratio of the double flip to
the non-flip amplitudes.

The present paper deals with the measurement, as a function 
of \tprim, of density matrix elements 
for $\rho$ meson diffractive electroproduction:
%
\begin{equation}
e + p \rightarrow e + \rho + Y; \ \ \ \  
   \rho \rightarrow \pi^+ \pi^-,
                                \label{eq:rho_prod}
\end{equation}
%
in the kinematic domain defined as: 
%
\begin{eqnarray}
%\align
 2.5 < Q^2 < 30\ {\rm GeV^2}     \nonumber \\
 40 < W < 120\ {\rm GeV}         \nonumber \\
 0.5 < t' < 3\ {\rm GeV^{2}}     \nonumber \\   
 E-p_z > 52\ {\rm GeV}           \nonumber \\
 0.6 < M_{\rho} < 1.1 \ {\rm GeV}\nonumber \\
 M_Y < 25 \ {\rm GeV},
%\endalign
                                \label{eq:kin_range}
\end{eqnarray}
%
thus extending the \tprim\ range of ref.~\cite{h1-rho,zeus,h1-phi,h1-pd}.

The $\rho$ meson mass $M_{\rho}$ is reconstructed as the invariant 
mass $M_{\pi \pi}$ of the two charged pion candidates, and
the \rh\ momentum is computed as the sum of the pion candidate 
momenta. 
The variable \tprim\ is determined from the scattered electron and charged
pion momentum components transverse to the beam direction, as
%
\begin{equation}
t^\prime = (\bar{p}^{T}_{miss})^2 = 
  (\bar{p}^T_e + \bar{p}^T_{\rho})^2, \ \ \ \ \ 
  \bar{p}^T_{\rho} = 
          \bar{p}^T_{\pi^+} + \bar{p}^T_{\pi^-}.
                                \label{eq:tprim}
\end{equation}
%
The variable $E-p_z$ is the difference of the energies and the 
longitudinal momenta of the scattered electron and the $\rho$ meson candidate.

The \tprim\ distribution in diffractive $\rho$ electroproduction is 
exponentially falling:
$\rm{d} \sigma / \rm{d} \it{t'} \propto 
\it{ e^{- b \cdot t'}}$,
with $b \simeq 7$ \gevsqm\ for elastic scattering~\cite{h1-rho}
and $b \simeq 2.5$ \gevsqm\ for proton dissociative 
scattering~\cite{h1-pd}, in the \qsq\ range relevant for the present 
analysis.
Because of the very small cross section at large \tprim, a major 
difficulty of the analysis is the presence of   
vector mesons produced at low \tprim, but faking $\rho$ production
at higher $t^\prime$.
An example is low \tprim\ electroproduction of \rhoprim\ 
mesons~\footnote
%===========
{The detailed structure~\cite{pdg} of the states previously described as the 
\rhoprim(1600) meson is not relevant for the present study. 
The name \rhoprim\ is thus effectively used for the vector 
meson state combination of mass in the range 1450-1700 MeV and width
of the order of 150 MeV.}
%===========
decaying into two charged pions and two $\pi^0$'s:
%
\begin{equation}
\rho^\prime \rightarrow \rho^+ \pi^- \pi^0, \ \ \ \ \     
  \rho^+ \rightarrow \pi^+ \pi^0       \ \ \ \ \       (+ \ c. c.).
                                \label{eq:rhoprim}
\end{equation}
%
This process mimics channel~(\ref{eq:rho_prod}) when the 
$\pi^+\pi^-$ invariant mass is in the $\rho$ mass range and the decay 
photons of the $\pi^0$'s remain undetected, which happens if they are
associated with the charged pion candidate tracks or do not pass the
detection threshold in the detector.
In such a case, the $p^T$ unbalance of the event due 
to the loss of the $\pi^0$'s is interpreted as $\rho$ 
production at high \tprim\, following eq.~(\ref{eq:tprim}).
In spite of the very small probability for the four decay photons 
to remain undetected, it will be shown below that this channel gives 
a non-negligible background to large \tprim\ \rh\ production.
Similar cases are $\omega$ and $\phi$ production
%
\begin{eqnarray}
\omega \rightarrow \pi^+ \pi^- \pi^0, \nonumber \\    
\phi \rightarrow K^0_S K^0_L, \ \ \ \ \ \phi \rightarrow \pi^+ \pi^- \pi^0,     
                                \label{eq:omegaphi}
\end{eqnarray}
%
when the $\pi^0$ or the $K^0_L$ remains undetected.
Note however that these channels mostly give contributions below the
selected \rh\ mass range.

In the following, any diffractive background of type 
$\pi^+ \pi^- X^0$, where $X^0$ only contains neutral particles, 
will be generically attributed to \rhoprim\ production, unless 
otherwise stated in the case of processes~(\ref{eq:omegaphi}).




\section{Event selection, kinematic variables and simulations}
                               \label{section:selection}
%===========================================
The data used for the present analysis were taken with the H1 detector 
in 1997 (more details for the relevant parts of the detector can be 
found in~\cite{h1-rho}).
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 respectively 820 and 27.5 GeV, 
and the integrated luminosity used for the analysis amounts 
to 6.0 ${\rm pb^{-1}}$.

Events corresponding to channel~(\ref{eq:rho_prod}), in the 
kinematic range defined by relations~(\ref{eq:kin_range}),
were selected by requesting the reconstruction of a cluster 
with energy larger than 17 GeV in the SpaCal backward 
electromagnetic calorimeter~\footnote{
%===========
In the H1 convention, the $z$ axis is defined by the colliding 
beams, the forward direction being that of the outgoing 
proton beam and the backward direction, that of the electron beam.}, 
and the reconstruction in the central tracking detector of the 
trajectories of two and only two charged particles (pion candidates) 
with opposite charges, momenta larger than 0.1 GeV and polar angles 
confined within the interval $20 < \theta < 160^{\rm o}$. 
Reconstructed energy clusters in the electromagnetic (hadronic) 
part of the liquid argon (LAr) calorimeter surrounding the tracker 
were associated with pion candidates when the cluster distance 
to the extrapolated particle direction was smaller than 40 (60) cm.
The event was 
rejected if at least one cluster was reconstructed with energy 
larger than 400 MeV (300 MeV) in the LAr (SpaCal)
calorimeter, and was not associated with a charged pion candidate;
LAr energy clusters in the forward direction, which could be attributed
to products of proton dissociation, were not included in this rejection
criterium.

Events were classified in two categories, corresponding to the 
presence or absence of activity in the forward detectors.
A selected event was classified in the ``notag'' sample when no 
signal was detected neither in the proton remnant tagger (PRT), nor in the 
forward muon detector (FMD), no track was reconstructed with 
polar angle $\theta < 20^{\rm o}$ and no cluster with energy 
larger than 400 MeV was detected in the LAr calorimeter.
Conversely, events were classified in the ``tag'' sample if a 
signal was observed in either the PRT or the FMD, if a track was 
reconstructed with polar angle $\theta < 20^{\rm o}$ or 
if a cluster with energy larger than 400 MeV was reconstructed 
in the forward part of the LAr.  
A pseudorapidity interval of at least 2.2 units was required 
between the edge of the LAr calorimeter or the most backward 
track or cluster attributed to proton dissociation, and the most 
forward charged pion candidate. 

These two samples could be attributed, to first approximation, 
respectively to the elastic and the proton dissociation processes. 
However, elastic events are classified in the tag sample 
if $t$ is large enough that the scattered proton hits the beam pipe walls 
or adjacent material, giving a signal in the forward detectors, 
or in the case of noise in the forward detectors. Conversely, 
proton dissociation events can be classified in the notag 
sample in the case of a small mass $M_Y$ and/or inefficiency 
of the forward detectors.

For the present analysis, it is thus important to recognize 
that proton dissociation and elastic $\rho$ events at large \tprim\
mostly belong to the tag sample, 
whereas low \tprim\ background events, as in 
processes~(\ref{eq:rhoprim},\ref{eq:omegaphi}), belong to 
either the tag or the notag sample, depending on the proton 
dissociation to elastic cross section ratio and on the detector 
response.
The fraction of \rhoprim\ background is thus smaller in the tag 
sample, which will be used for the extraction
of density matrix elements.

The variable \qsq\ is reconstructed (double angle method) as:
%
\begin{equation}
Q^2 = \frac {4 {E_o}^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 ${E_o}$ is the energy of the incoming electron, and
$\theta_e$ and $\theta_{\rho}$ are the scattered electron and \rh\ meson 
polar angles.
The variable $W$ is given by (Jacquet-Blondel method):
%
\begin{equation}
W^2 = y \cdot s - \frac {{p^T_{\rho}}^2} {1-y} , \ \ \ \ \ 
  y= \frac{E_{\rho} - p_{z\rho}} {2 E_o} ,
                                \label{eq:w}
\end{equation}
%
$s$ being the square of the $ep$ centre of mass energy.
For computing  \tprim\ according to eq.~(\ref{eq:tprim}), the electron 
momentum 
is computed using the \rh\ momentum and the electron polar angle.
For $E-p_z$, the electron energy measured in the SpaCal is used.
Fig.~\ref{fig:tfig} presents the \tprim\ distribution for the 
selected tag events.

%=======================\label{fig:tfig}===============
\begin{figure}[htbp]
\vspace{-0.cm}
\begin{center}
\epsfig{file=H1prelim-01-017.fig1.eps,width=7.0cm}
\end{center}
\caption{Uncorrected \tprim\ distribution for the selected tag events in the 
range 0.5-3.0 \gevsq}
\label{fig:tfig}
\end{figure}
%=======================\label{fig:tfig}===============


The angles $\theta^\ast$ and $\Phi$, used for 
the extraction of the \mdes ~\protect\cite{h1-rho,sch-w}, 
are measured using the event kinematics defined by equations
(\ref{eq:qsq}) and (\ref{eq:w}): 
$\theta^\ast$ is the angle between the $\pi^+$ and \rh\ meson directions
in the \rh\ rest frame,
$\Phi$ is the angle between the \rh\ production plane and the electron
scattering plane in the ($\gamma p$) centre of mass system.
%


Monte Carlo simulations based on the DIFFVM 
program~\cite{diffvm} are used to describe the elastic 
and proton dissociation production of $\rho$, $\omega$, 
$\phi$ and \rhoprim\ vector mesons, and to correct the signal 
for acceptance and smearing effects.

For $\rho$ production, the simulations include the angular 
distributions (see definitions in~\cite{h1-rho,sch-w}) 
corresponding to the measurements of the present analysis for
the \rfour\ matrix element (\cost\ distribution) and the
\rfivecomb\ and \ronecomb\ combinations ($\Phi$ distribution).
%%%\rfivecomb\ and \ronecomb\ combinations ($\Phi$ distribution).
Other angular distributions and correlations are taken in the 
{\it s}-channel helicity conservation approximation, the interference
between the longitudinal and transverse amplitudes 
($\cos \delta$ parameter) being taken from the elastic 
scattering measurement~\cite{h1-rho} in the relevant \qsq\ range.
Radiative effects~\cite{heracles} are fully included, and the
measurements below include radiative corrections.
For $\tprim > 0.5$ \gevsq, the \tprim\ slope is taken to be 
$b = 7$~\gevsqm\ for elastic scattering~\cite{h1-rho} and 
$b = 1.7$~\gevsqm\ for proton dissociation scattering.
The \tprim\ integrated cross section ratio for proton dissociation
and elastic scattering is taken as 0.95. 
All parameters in the simulation can be varied.

For $\omega$, $\phi$ and \rhoprim\ production in
channels~(\ref{eq:rhoprim},\ref{eq:omegaphi}), the decay angular distributions 
have not been measured, and are taken isotropic.
The \tprim\ slopes are $b = 6$~\gevsqm\ for elastic scattering and 
$b = 2.5$~\gevsqm\ for proton dissociation.
The ratio of the proton dissociation to elastic 
channels, integrated over \tprim, is taken to be 0.75.
For the present \qsq\ range, the ratio of the $\omega$ to $\rho$ 
and of the $\phi$ to $\rho$ production cross sections, integrated 
over \tprim\ and over the full \rh\ mass range, are taken to be 
0.09~\cite{zeus-omega} and 0.20~\cite{h1-phi}, respectively.
The \rhoprim\ to \rh\ cross section ratio is not measured. It is extracted from
the present data in the mass range
$0.6 < M_{\rho} < 1.1$ \ GeV and for $0.5 < \tprim < 3$ \gevsq. 

\section{Signal extraction and matrix element measurements}
                                  \label{section:extraction}
%===========================================

\subsection{Backgrounds}
%===========================================

To estimate the background contributions, a new variable, $\zeta$,
is introduced:
\begin{equation}
\zeta = \cos {(p^T_{miss}, \ p^T_{\rho})}.
                                \label{eq:y}
\end{equation}
%
For \rhoprim\ production~\footnote{
%===========================================
Here ``\rhoprim'' is generically used for any diffractive system
containing two charged and additional neutral particles.},
%============================================
$p^T_{miss}$ is mostly due to the non-detection of the decay photons of 
the $\pi^0$'s, which induces $p^T$ unbalance.
In the present \qsq\ range, the $\pi^0$'s are mostly emitted in 
the ($\pi^+, \pi^-$) direction, giving a 
$\zeta$ distribution peaking around +1.
In contrast, in the case of high \tprim\ $\rho$ production, 
$p^T_{miss}$ is due to the scattered proton or system $Y$, 
leading to a more symmetric distribution, peaking at +1 and -1. 

The selected events are distributed in 4 subsamples:
tag and notag events, with $\zeta < 0$ or with $\zeta > 0$.
Each of these 4 samples contains 2 contributions, due 
respectively to \rh\ production at large \tprim, and to the \rhoprim\ 
background.

In addition to the conservation of the 
total number of events, 6 constraints are
 obtained from the Monte Carlo simulations:
%===========================================
\begin{itemize}
\item the fractions of \rh\ events with $\zeta < 0$, respectively for the
tag and notag samples, as obtained from the $\zeta$ distributions 
in the \rh\ simulation; 
\item similarly, the fractions of \rhoprim\ events with $\zeta < 0$, as
obtained from the \rhoprim\ simulation; 
\item the fraction of tagged \rhoprim\ events, for $\zeta < 0$ and 
$\zeta > 0$, respectively. 
These fractions are obtained from the detector simulation, under 
the assumption that the ratio of proton dissociation to elastic
\rhoprim\ production cross sections is $0.75$.
% $\pm 0.25$.
\end{itemize}
%===========================================

The \rh\ and \rhoprim\ contributions are then determined through
a global fit of the numbers in the 4~samples under the 6~constraints,
 performed with the MINUIT package~\cite{minuit}.
As expected, the amount of \rhoprim\ background is significantly smaller
in the tag sample.

However, the $\zeta$ distribution in the \rh\ simulation used for the
constraint calculation is related to the $\Phi$ angle distribution,
which in turn depends on the input density matrix elements as:
%
\begin{equation}
  W(\Phi) \propto 
   1 +
   \sqrt {2 \epsilon (1+\epsilon)} \ \cos {\Phi} \ (\rfivecomb)
   - \epsilon \ \cos {2 \Phi} \ (\ronecomb),
                                \label{eq:Phi}
\end{equation}
% 
where the polarisation parameter $\epsilon = 0.99$ in the HERA 
range. 

The measured $\Phi$ distribution for the tag sample, corrected
for the \rhoprim\ background and for acceptance and smearing effects,
is fitted to relation~(\ref{eq:Phi}). The extracted values for the 
\mde\ combinations \rfivecomb\ and \ronecomb
are then fed back into the \rh\ simulation. The new $\zeta$ distribution
provides a modified constraint, and the fit procedure is iterated.
The process converges after a few steps to stable 
background estimates and is independent of the starting value of 
the density matrix elements in the \rh\ Monte Carlo.

Fig.~\ref{fig:mass}a, c, e and g presents the
$\pi^+\pi^-$ mass distribution over the full mass region
$0.3 < M_{\pi \pi} < 1.3$\ GeV for the tag sample, the notag sample,
the $\zeta \ <$ 0 and $\zeta \ >$ 0 sample respectively.  
The background contributions are also shown.
Fits of the background subtracted distributions to relativistic
Breit-Wigner functions with the Ross-Stodolsky skewing 
parameter~\cite{rs} fixed to 1~\cite{h1-rho},
yield \rh\ mass and width values in agreement with 
expectations~\cite{pdg} (see Fig.~\ref{fig:mass}b for the tag sample, and d, f and g for
the other samples).
The data are thus very well described over the full mass range by the 
diffractive \rh\ production at large \tprim\ supplemented by $\omega$,
$\phi$ and \rhoprim\ backgrounds.


%=======================\label{fig:mass}===============
\begin{figure}[htbp]
\vspace{-0.cm}
\begin{center}
\epsfig{file=H1prelim-01-017.fig2ab.col.eps,width=7.5cm}
\epsfig{file=H1prelim-01-017.fig2cd.col.eps,width=7.5cm}
\vspace{-3.cm}

\hspace {0.7cm} a) \hspace {3.0cm} b) \hspace {5.0cm} c) \hspace {3.0cm} d) \\
\vspace{2.5cm}
\vspace{0.5cm}
\epsfig{file=H1prelim-01-017.fig2ef.col.eps,width=7.5cm}
\epsfig{file=H1prelim-01-017.fig2gh.col.eps,width=7.5cm}
\vspace{-3.cm}

\hspace {0.7cm} e) \hspace {3.0cm} f) \hspace {5.0cm} g) \hspace {3.0cm} h) \\
\vspace{2.5cm}
\vspace{0.1cm}
\end{center}
\caption{a-c-e-g) $\pi^+\pi^-$ mass distributions
for the selected events with  $0.3 < M_{\pi \pi} < 1.3$\ GeV;
the dark histograms describe the $\omega$ and $\phi$ background
contributions, while the full histograms 
include the additional \protect\rhoprim\
background ,
b-d-f-h) background subtracted $M_{\pi \pi}$ distributions, 
the curves being the result of fits to a relativistic Breit-Wigner
distribution with the Ross-Stodolsky parameter fixed to 1.
The plots a and b are for the tag events, c and d for the notag events,
e and f for the $\zeta \ <$ 0 sample and g and h for the $\zeta \ >$ 0 sample.} 
\label{fig:mass}
\end{figure}
%=======================\end{fig:mass} ======================


%===========================================
\subsection{(\boldmath{\rfivecomb}) and (\boldmath{\ronecomb})
       determination}
%===========================================

Once the iterative process described above has converged, 
measurements for the \mde\ combinations \rfivecomb\ and \ronecomb\
are obtained from the $\Phi$ distribution in 
two bins in \tprim:
$0.5 < t^\prime < 1.0$~\gevsq\ and $1.0 < t^\prime < 3.0$~\gevsq.
       
Fig.~\ref{fig:final-distr}a-b presents the observed $\Phi$ distributions 
of the tag events for the two bins in \tprim, including the final 
estimates of the background.
Fig.~\ref{fig:final-distr}c-d shows the fully corrected distributions 
used to determine the \mde\ combinations. 

%=======================\label{fig:final-distr}===============
\begin{figure}[htbp]
\vspace{-0.cm}
\begin{center}
\epsfig{file=H1prelim-01-017.fig3ab.eps,width=10.5cm}
\vspace{-5.cm}

\hspace {0.8cm} a) \hspace {4.8cm} b) \\
\vspace{4.5cm}
\vspace{0.1cm}
\epsfig{file=H1prelim-01-017.fig3cd.eps,width=10.5cm}
\vspace{-5.cm}

\hspace {0.8cm} c) \hspace {4.8cm} d) \\
\vspace{4.5cm}
\vspace{0.1cm}
\epsfig{file=H1prelim-01-017.fig3ef.eps,width=10.5cm}
\vspace{-5.cm}

\hspace {0.8cm} e) \hspace {4.8cm} f) \\
\vspace{4.5cm}
\end{center}
\vspace{0.cm}
\caption{a-b) Observed $\Phi$ distributions for the tag
events; the hatched area describes the final estimate of the
backgrounds. 
c-d) Fully corrected $\Phi$ distributions for the tag sample; the
superimposed curves show the results of fits to
relation~(\protect\ref{eq:Phi}).
e-f) Fully corrected \protect\cost\ distributions for the tag sample; the
superimposed curves show the results of fits to
relation~(\protect\ref{eq:cost}).
The plots a-c-e are for $0.5 < t^\prime < 1.0$~\gevsq
and b-d-f are for $1.0 < t^\prime < 3.0$~\gevsq.}
\label{fig:final-distr}
\end{figure}
%=======================\end{fig:final-distr}======================

To compute the errors on the measurement, several uncertainties were 
considered:
%===========================================
\begin{itemize}
\item the uncertainty of the amount of background was estimated by 
 varying, for the fit procedure, the $\zeta$ separation between
 the samples from 0.0 to 
 $\pm 0.2$ and $\pm 0.4$;
\item the fractions of \rhoprim\ events with $\zeta < 0$ 
 were multiplied by factors 2 and 0.5;
\item the proton dissociative to elastic \rhoprim\ production 
 cross section ratio was  changed to 0.5 and to 1.0;
\item the $\Phi$ shape of the background 
 was varied by keeping the total amount 
 fixed, but changing the fraction assigned to 
 the two extreme $\Phi$ bins by $\pm 50$ \%;
\item the energy threshold for the detection of clusters not associated 
 to tracks in the LAR calorimeter was varied between 300~MeV and
 500~MeV;
\item the efficiencies of the PRT and FMD were varied within
 experimental errors;
\item the $M_Y$ spectrum was varied from $1/M_Y^{2.15}$ to 
 $1/M_Y^{1.85}$ and to $1/M_Y^{2.45}$.
\end{itemize}
%===========================================
For each of these variations, the full iterative procedure was
performed. 
The spread of results was quadratically combined with the 
statistical errors from the fit to relation~(\ref{eq:Phi}) to
determine the total errors on the measurements.

The values obtained for the \mde\ combinations \rfivecomb\
and \ronecomb\ are presented in fig.~\ref{fig:mde}a 
and~\ref{fig:mde}b, respectively, together with measurements at lower \tprim\
values in similar energy and \qsq\ ranges 
($Q^2 > 2.5$\ \gevsq\ for ref.~\cite{h1-rho}, 
$Q^2 > 1.5$\ \gevsq\ for ref.~\cite{h1-pd}
and $Q^2 > 3.0$\ \gevsq\ for ref.~\cite{zeus}).

Significant helicity non-conservation is observed for 
the combination \rfivecomb.
Following the results of~\cite{h1-rho,zeus}, this effect is mainly 
attributed to the \rfive\ \mde, which
is proportional to the product of the dominant non-flip amplitude 
$T_{00}$ by the amplitude $T_{01}$, which is 
expected to be the largest helicity flip amplitude.
In contrast, the $r^5_{11}$ \mde\ has a contribution from the 
product of the non-dominant non-flip amplitude $T_{11}$ by the 
non-dominant single flip amplitude $T_{10}$, plus a contribution 
from double flip.
A fit to the measurements of the present analysis, together with 
those of ref.~\cite{h1-rho} which were obtained in the same \qsq\ 
range, is performed assuming a dependence of the type 
$\sqrt {\tprim}$~\cite{ivanov}.
The data are compatible with this dependence, the result of
the fit being 
$\rfivecomb\ = 
  (0.25 \pm 0.03 ) \ 
     \sqrt {\tprim}$, with \tprim\ in \gevsq;
the full errors on the measurements are used to estimate the fit error.
The present analysis is the first measurement of this dependence.


%=======================\label{fig:mde}===============
\begin{figure}[htbp]
\vspace{-0.cm}
\hspace {4.0cm} a) \hspace {8.0cm} b) \\
\epsfig{file=H1prelim-01-017.fig4a.eps,width=7.5cm}
\epsfig{file=H1prelim-01-017.fig4b.eps,width=7.5cm}
\epsfig{file=H1prelim-01-017.fig4c.eps,width=7.5cm}
\hspace{0.8cm}
\epsfig{file=H1prelim-01-017.fig4d.eps,width=7.5cm}
\vspace{-0.2cm}
\hspace {4.0cm} c) \hspace {8.0cm} d) \\
\caption{a) Measurement of \protect\rfivecomb\ as a function of
\protect\tprim, for the present experiment and for
ref.~\protect\cite{h1-rho,zeus,h1-pd};
the curve presents the result of a fit of the form
$r^5_{00} + 2 r^5_{11} = 
  (0.25 \pm 0.03) \ 
     \sqrt {t^\prime \ {\rm [GeV^2]}}$;
b) Measurement of \protect\ronecomb\ as a function of \protect\tprim, 
for the present experiment and for ref.~\protect\cite{h1-rho,zeus};
the curve presents the result of a linear fit with the slope parameter
$-0.13 \pm 0.05 \ \rm{GeV^{-2}} $ ;
c) Measurement of \protect\rfour\ as a function of
\protect\tprim, for the present experiment and for
ref.~\protect\cite{h1-rho,zeus,h1-pd};
the curve presents the result of a constant fit with
$r^{04}_{00} = 0.68 \pm 0.02$;
d) $R = \sigma_L / \sigma_T$ as a function of
\protect\tprim, for the present experiment and for
ref.~\protect\cite{h1-rho,zeus,h1-pd}.
In all plots, the inner error bars are statistical, the full error 
bars include the systematic errors added in quadrature.
The fits are performed to the data of the present
analysis and of ref.~\protect\cite{h1-rho}, which were obtained in the
same $Q^2$ range.}
\label{fig:mde}
\end{figure}
%=======================\end{fig:mde} ======================


In fig.~\ref{fig:mde}b, the values found for \ronecomb\ are shown; they
are different from zero and negative.
The present measurement is the first observation of a significant
non-zero value of the \ronecomb\ \mde\ combination. 
The $r^1_{00}$ element, which is expected to give a
negative contribution~\cite{ivanov}, is proportional to the
square of the single flip amplitude $T_{01}$.
The 
$r^1_{11}$ density matrix element
 is proportional to the product of the non-dominant
non-flip amplitude $T_{11}$ by the double flip amplitude.
The sign of the combination gives interesting information on the
relative strength of the $T_{01}T_{01}^\dagger$ and 
$T_{11}T_{1-1}^\dagger$ products of amplitudes.
It tends to confirm that the $T_{01}$ amplitude is relatively large in
the present kinematical domain, and is presumably the dominant 
single flip amplitude.
A linear fit to the present measurements and those of 
ref.~\cite{h1-rho} gives the slope parameter
$-0.13 \pm 0.05$ \gevsqm.


%===========================================
\subsection{\boldmath{\cost} distribution and \boldmath{\rfour}
       determination}
%===========================================
       
The \cost\ distribution is given by~\cite{sch-w}:
%
\begin{equation}
  W(\cost) \propto 
   1 - \rfour\ + (3 \ \rfour - 1) \cos^2{\theta^{\ast}}.
                                \label{eq:cost}
\end{equation}
%

Fig.~\ref{fig:final-distr}e-f presents the fully 
corrected \cost\ distributions for the two bins in \tprim.
The amount of background was fixed by the iterative procedure described
above, and the shape of this background was taken as flat.
This was obtained in the data itself from the difference in shape 
of the \cost\ experimental distributions for the events with $\zeta < 0$ 
and $\zeta > 0$, which contains different background contributions,
these two samples being predicted by the \rh\ 
simulation to have the same \cost\ distributions.
Systematic uncertainties were taken into account in the same way as
for the $\Phi$ distribution; in the present case, the shape of the
background was varied by supposing that it has the same
distribution as the \rh\ signal.

The \rfour\ element was also extracted using the events with 
$\zeta < 0$, where the background is small. 
The results are compatible with those obtained from the sample of
tag events.
The difference between the two measurements was included in the
systematic errors.

The extracted values of \rfour\ are presented in fig.~\ref{fig:mde}c,
together with measurements at lower \tprim\ values in the same \qsq\ range.
As expected from QCD calculations, no significant variation of 
\rfour\ with \tprim\ is observed.
A fit to the present measurements and to those of 
ref.~\cite{h1-rho} gives 
$r^{04}_{00} = 0.68 \pm 0.02$.

The measurement of \rfour\ yields a determination of
$R = \sigma_L / \sigma_T$ through the relation
%
\begin{equation}
 R = \frac {1} {\epsilon} \frac {\rfour} {1 - \rfour},
                                \label{eq:R}
\end{equation}
%
which is valid in the approximation of
{\it s}-channel helicity conservation (the small SCHC violation
implies an overestimate of the value of {\it R} with a few \% ).
Results are shown in fig.~\ref{fig:mde}d.


\section{Conclusion}
                                     \label{section:concl}
%=============================================

A measurement has been performed of \rh\ meson production in the
\tprim\ range $0.5 < t^\prime < 3.0$~\gevsq, for $Q^2 > 2.5$~\gevsq\
and $40 < W < 120$~GeV.
The \rfour\ \mde\ and the combinations \rfivecomb\ and 
\ronecomb\ have been measured as a function of \tprim.

No significant \tprim\ dependence has been observed for \rfour.

A significant violation of {\it s}-channel helicity conservation is observed
in the \rfivecomb\ combination, which is increasing with \tprim.
The \tprim\ dependence of the combination has
been determined for the first time as 
$\rfivecomb\ = 
   (0.25 \pm 0.03  ) \ 
     \sqrt {\tprim}$, with \tprim\ in \gevsq.
     
The \ronecomb\ element combination appears to be different from zero 
and negative.
This is the first observation of a non-zero value of
this combination. 
The sign gives interesting information on the
relative strength of the $T_{01}T_{01}^\dagger$ and 
$T_{11}T_{1-1}^\dagger$ products of amplitudes.
Together with the \rfivecomb\ measurement, it tends to confirm that 
the $T_{01}$ amplitude is relatively large in the present kinematical 
domain, and is presumably the dominant helicity flip amplitude.

                                     

\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.
%====================================================================

%\section*{References}
\begin{thebibliography}{99}
\input{biblio}
\end{thebibliography}

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


\end{document}





