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% special definition for this paper
\newcommand{\rfour}{\mbox{$r^{04}_{00}$}}
\newcommand{\rfive}{\mbox{$r^5_{00}$}}
\newcommand{\rfivecomb}{\mbox{$r^5_{00} + 2 r^5_{11}$}}
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\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
     XXII International Symposium on Lepton and Photon Interactions, LP2005},
                June~30,~2005,~Uppsala} \\
                 & Abstract:        & {\bf 399}    &\\
                 & Session & {\bf QCD and hadron structure}   &\\ \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
    Diffractive photoproduction of \boldmath{$\rho$} mesons \linebreak
    with large momentum transfer at HERA}

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

\begin{abstract}

The diffractive photoproduction of $\rho$ mesons with large momentum transfer,\linebreak
$e+p \rightarrow~e+\rho +Y$, is studied at HERA in the kinematic range 
$Q^2 < 0.01~\rm GeV^2$, $75 < W < 95$ GeV, $1.5 < |t| < 10.0~\rm GeV^2$ and 
$M_Y < 5~\rm GeV$. The $t$ dependence of the cross section is measured, as 
well as the spin density matrix elements.  
All results are compared to BFKL model predictions.

\end{abstract}

\noindent

\end{titlepage}

\pagestyle{plain}

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

The subject of this paper is the study of the photoproduction 
of \rh\ mesons at large momentum transfer in high energy $ep$ scattering:
%
\begin{equation}
e + p \rightarrow e + \rh\ + Y; \ \ \ \
\rho \rightarrow \pi^+ + \pi^- ,
    \label{eq:rho_prod}
\end{equation}
%
where the scattered proton is mainly 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).  There is also a 
small contribution from ``proton elastic'' scattering where the proton 
remains intact.

For $|t|$ larger than a few $\rm GeV^2$, $t$ being the square of the four
momentum transferred at the proton vertex, perturbative QCD (pQCD) is
expected to apply and diffractive $\rho$ production is viewed, in the
proton rest frame, as a sequence of three processes well separated in
time: the photon fluctuates into a $q\bar{q}$ pair; the $q\bar{q}$ pair is
involved in a hard interaction with the proton via the exchange, at lowest
order, of two gluons in a colour singlet state and the $q\bar{q}$ pair
recombine to form a bound $\rho$ meson.  In the leading logarithm
approximation the process is represented by the effective exchange of a
gluonic ladder, which, in the low $x$ region of interest, is described by
BFKL evolution.

The spin density matrix elements are defined as bilinear combinations of
the helicity amplitudes $M_{\lambda_{\rho}\lambda_{\gamma}}$, where
$\lambda_{\rho}, \lambda_{\gamma} = -,0,+$ represent the respective
helicities of the $\rho$ meson and the photon~\cite{theory-sdme}. Explicitly 
they are given by: 

\begin{eqnarray}
r_{00}^{04} &=& \frac{\langle |M_{+0}|^2 \rangle}
{\langle |M_{++}|^2 + |M_{+0}|^2 + |M_{+-}|^2 \rangle} \nonumber\\
r_{10}^{04} &=& \frac{1}{2} \frac{\langle M_{++}M_{+0}^{\ast} + M_{+-}M_{-0}^{\ast} \rangle}
{\langle |M_{++}|^2 + |M_{+0}|^2 + |M_{+-}|^2 \rangle}\\
r_{1-1}^{04} &=& \frac{1}{2} \frac{\langle M_{++}M_{+-}^{\ast} + M_{+-}M_{++}^{\ast} \rangle}
{\langle |M_{++}|^2 + |M_{+0}|^2 + |M_{+-}|^2 \rangle} \nonumber
\end{eqnarray}
%
In the
approximation of $s$-channel helicity conservation (SCHC), whereby the
$\rho$ meson retains the helicity of the photon, both the single-flip and
double-flip dependent matrix elements are expected to be zero.  

Measurements of high $|t|$ diffractive vector meson photoproduction
($\rho$, $\phi$ and $J/\Psi$) have been performed by the ZEUS
collaboration~\cite{zeus-hight} and $J/\Psi$ meson photoproduction at high
$|t|$ has also been studied by the H1 Collaboration~\cite{duncan},
extending the kinematic range as far as $\ttra = 30~\rm GeV^{2}$.  The
ZEUS measurements of spin density matrix elements for $\rho$ meson
production indicate violation of $s$-channel helicity conservation.

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

\section{Theoretical Model}
                                  \label{sect:theory}
%===========================================

The process of interest is described by the non-forward BFKL equation for
which a complete analytical solution, in the leading logarithm
approximation, is presented by Poludniowski {\em et al.}
in~\cite{theory-jeff1,theory-jeff2}. Their prescription is to factorise
the meson production from the hard subprocess using a set of meson
wavefunctions expanded on the light-cone.  All expansions are performed up
to twist-3, i.e. next-to-leading twist, which is the lowest order able to
accommodate a non-zero $r^{04}_{10}$ matrix element.  The
constituent, as opposed to current, quark mass is used to enhance the
coupling of the photon to chiral odd $q\bar{q}$ configurations; thereby
producing a strong transverse polarisation as supported by previous
data~\cite{zeus-hight,h1-hight}.

At leading log accuracy there are three free parameters
$\alpha_s^{IF}~(0.21)$ , $\alpha_s^{BFKL}~(0.2)$ and $\gamma~(1.0)$:  where
$\alpha_s^{IF}$ is the coupling of the two gluons to each impact factor,
$\alpha_s^{BFKL}$ is the coupling within the BFKL ladder and $\gamma$
determines the scale $\Lambda^2 = m_{\rho}^2 - \gamma t$.  The numbers in
parenthesis refer to the parameter values chosen to provide the best
description of the previous ZEUS measurements~\cite{zeus-hight} and are
used to describe the results that follow.

The model of Poludniowski {\em et al.} predicts violation of
SCHC with a hierarchy of helicity amplitudes given by: non-flip $(M_{++})$
$>$ double-flip $(M_{+-})$ $>$ single-flip $(M_{+0})$\footnote{There are
also the corresponding amplitudes $M_{--}$, $M_{-+}$ and $M_{-0}$, which
satisfy $M_{--} = M_{++}$, $M_{+-} = M_{-+}$ and $M_{-0} = M_{+0}$.}.
These calculations have subsequently been compared to the ZEUS
data~\cite{zeus-hight} and, while they provide a reasonable description of
the the matrix elements $r^{04}_{00}$ and $r^{04}_{1-1}$, they are unable
to describe the observed small positive value of $\rm{Re}~[r^{04}_{10}]$.

The theoretical challenge is to provide a simultaneous description of both
the $t$ spectra of the vector mesons and the spin density matrix elements;
in particular the largeness of the double-flip dependent matrix element
$r^{04}_{1-1}$ and the smallness of the single-flip dependent matrix
element $r^{04}_{00}$.

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

\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
were 920 and 27.5 GeV, respectively. 
The effective luminosity used for the analysis amounts to 
20.1 ${\rm pb^{-1}}$.  The kinematic domain of the measurement is:
%
\begin{eqnarray}
 &Q^2& < 0.01\ {\rm GeV^2}       \nonumber \\
 75 < &W& < 95\ {\rm GeV}       \nonumber \\
 1.5 < &|t|& < 10\ {\rm GeV^{2}}         
%&M_Y& < 5\ {\rm GeV}.          
    \label{eq:kin_range} \end{eqnarray} 
where $Q^2$ is the modulus
squared of the four momentum carried by the intermediate photon.  The mass
of the proton remnant is limited to 
\begin{eqnarray} &M_Y& < 5\ {\rm GeV}.
    \label{eq:my-range}
\end{eqnarray}

The relevant parts of the detector, for which more details can be 
found in~\cite{nim}, are the central tracking detector, 
the liquid argon (LAr) calorimeter and the 44 m electron tagger, 
a $7 \times 7$ unit array of 22 radiation length crystal calorimeters, 
which detects the scattered positron in the backward 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 
positron beam ($z < 0$).}.
%===========

The events are triggered by a tagged photoproduction trigger based on the
requirement of an energy deposit of greater than 10 GeV in the 44m
electron tagger.  It also has tracking conditions requiring at least one
charged track detected in the central tracker with a transverse momentum larger than
$400$ MeV along with vertexing requirements.  Additionally, there is a
veto on the amount of energy deposited in the forward region of the LAr.
Events corresponding to reaction~(\ref{eq:rho_prod}), in the 
kinematic range defined by relations~(\ref{eq:kin_range}) are 
finally selected by requesting:
%
\begin{itemize}
%
\item
the reconstruction of an energy deposit of more than 15 GeV
in the 44 m electron tagger (the scattered positron candidate);

% 
\item
the reconstruction in the central tracking detector of the 
trajectories of exactly two oppositely charged particles (pion 
candidates) with polar angles confined 
within the interval $20^{\rm o} < \theta < 155^{\rm o}$. 
One track is required to have transverse momentum larger than 
0.15 GeV and the other larger than 0.45 GeV (to fire the 
trigger);

%
\item
the absence of any signal reconstructed with energy 
larger than 400~MeV in the LAr calorimeter, 
which is not associated 
with the two charged pion candidates.
This cut reduces backgrounds due to the production of 
systems decaying into two charged and additional neutral particles.
It also limits the mass of the proton dissociative system to 
$M_Y~\lapprox~5$ GeV;
%
\item
events in the mass range $0.6~<~M_{\pi \pi}~<~1.1~{\rm GeV}$
are selected, while 
events with $M_{KK}~<~1.04~{\rm GeV}$ are discarded, 
where $M_{\pi \pi}$ and $M_{KK}$ are the invariant mass of the two hadron candidates 
when considered as pions or kaons respectively (no direct hadron identification is 
performed for this analysis). The latter cut reduces the 
background due to diffractive production of $\phi$ mesons.

\end{itemize}
%
%

The final sample consists, in total, of 2685 events.

\subsection{Kinematics and angular distributions}
                               \label{sect:kinematics}
%===========================================

The \rh\ three-momentum is computed as the sum of the two charged 
pion candidate momenta.
The variable $W$ is calculated using the Jacquet-Blondel 
method~\cite{jb}:
%
\begin{equation}
W^2 \simeq ys  , {\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, 
$E_0$ is the incoming positron energy and 
$E_{\rho}$, $p_{z, \rho}$ and $p_{t, \rho}$ being the energy, 
longitudinal and
transverse momentum of the \rh\ meson, respectively. 
The variable $|t|$ is determined from the 
\rh\ momentum component transverse to the beam direction as
%
\begin{equation}
|t| \simeq (\vec{p}_{t, \rho})^2 
                                \label{eq:tprim}
\end{equation}
%
The measurement of the production and decay angular distributions provides
access to the spin density matrix elements.  
%Three angles can be defined (see Fig.~\ref{fig:plane}): 
Three angles can be defined: 
$\Phi$ is the angle between the \rh\ production plane 
(defined as the plane containing the virtual photon and the \rh\ 
directions) and the positron
scattering plane in the ($\gamma p$) centre of mass system,
$\theta^*$ and $\phi^*$ are the polar and
the azimuthal angles, respectively, of the positively charged decay 
pion in the \rh\ rest frame, the quantisation axis being 
taken as the direction opposite to that of the scattered proton.
In this paper, only the distributions of the angles $\phi^*$ and 
$\theta^*$ are analysed (the angle $\Phi$ is not accessible in 
photoproduction) giving access to the matrix elements $r^{04}_{00}$
and $r^{04}_{1-1}$.
%
%=======================\label{fig:plane}===============
%\begin{figure}[tbp]
%\vspace{-0cm}
%\begin{center}
%\epsfig{file=pictures/plane_rho.eps,width=16.cm}\\ 
%\end{center}
% \caption{Illustration of the production and decay angles for $\rho$ mesons.}
% \label{fig:plane}
% \end{figure}
%=======================\end{fig:plane} ======================

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

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

A Monte Carlo simulation based on the DIFFVM program~\cite{diffvm} 
is used to describe   
the proton dissociative production and decay of $\rho$ mesons,
and to correct the data for acceptance and smearing 
effects.
The \qsq\ and $W$ dependencies of the cross section 
are taken from the previous measurements~\cite{xavier}, while the $t$ 
dependency is taken according to a power law as measured in the data.
The $W$ dependent effects of the 44 m electron tagger are taken into account
using the luminosity averaged acceptance curve taken from the data. 
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 simulation includes the angular distributions 
corresponding to the measurements of the present analysis for
the \rfour\ matrix element (\cost\ distribution) and for 
$r^{04}_{1-1}$ ($\phi^{\ast}$ distribution), while  the measurement 
from \cite{zeus-hight} is taken for $\rm Re[r^{04}_{10}]$.
Other angular distributions and correlations are taken in the 
{\it s}-channel helicity conservation (SCHC) approximation.

The mass distribution is described by a relativistic Breit-Wigner 
distribution, including skewing effects taken from the current analysis. 
For studies of systematics uncertainties, all simulation parameters have been varied within errors.

The DIFFVM simulation is also used for the description of the
$\omega$, $\phi$ and \rhoprim\ 
backgrounds (see next section).
Here the $t$ distributions are taken to 
have the same parameterisation as for the \rh\ meson, 
and the angular distributions are taken isotropic, except for 
$\phi \rightarrow K^+ K^-$~\cite{h1-phi}.

Fig.~\ref{fig:control} presents the
observed and simulated distributions for several variables of the
selected sample of events.
They include the amounts of background discussed in 
section~\ref{sect:bg}.
Reasonable 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.

%=======================\label{fig:control}===============
\begin{figure}[htbp]
%\vspace{-0.2cm}
\begin{center}
\epsfig{file=H1prelim-05-012.fig1.eps,angle=270,width=14.cm}\\ 
\end{center}
\vspace{-0.2cm}
 \caption{Distribution of the selected events in the kinematic domain
 (\protect\ref{eq:kin_range}) for the transverse momentum (a),
   azimuthal angle of the pion candidates (b), the angular distributions
   \protect\cost\ (c) and $\phi^{\ast}$ (d).
 The crosses present the data and 
 the histograms present the Monte-Carlo predictions, including
 the $\omega$, $\phi$ and 
 \protect\rhoprim\ backgrounds  
 (shaded histogram).}
 \label{fig:control}
 \end{figure}
%=======================\end{fig:control} ======================

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

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

Diffractive photoproduction of $\omega$, $\phi$ and $\rho^\prime$ mesons can 
fake \rh\ production through the decay channels:
%
\begin{eqnarray}
&& \omega \rightarrow \pi^+ \pi^- \pi^0, \ \ \ \ \ 
   \omega \rightarrow \pi^+ \pi^- \nonumber \\    
&& \phi \rightarrow \pi^+ \pi^- \pi^0, \ \ \ \ \ \phi \rightarrow 
         K^0_S K^0_L \nonumber \\ 
&& \rho^\prime \rightarrow \rho^+ \pi^- \pi^0 \nonumber \\
&& \ \ \ \ \ \ \ \ \ \hookrightarrow \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 the cases where the energy of the neutral particle is 
deposited in a crack in the detector, associated to the charged pion 
tracks, or is smaller than the 400 $\rm MeV$ noise threshold.

The $\phi$, $\omega$ and \rhoprim\ cross sections were taken from
measured ratios to the \rh\ cross section, in the \qsq\ range
relevant for the analysis: 
$\phi$~/~$\rho= 0.156$~\cite{zeus-hight}, 
$\omega$~/~$\rho = 0.106$ ~\cite{omega} and
\rhoprim ~/~$\rho= 0.2$. 
In the latter case, the ratio is obtained from the measured ratio
$\rho^{\prime} \rightarrow \pi^+ \pi^- \pi^+ \pi^-~/~\rho \rightarrow \pi^+ \pi^-  = 0.10 \pm 0.05$
~\cite{prime1,prime2} under the assumption
%
\begin{equation}
\frac{(\rho^{\prime} \rightarrow \rho^+ \pi^- \pi^0) + 
(\rho^{\prime} \rightarrow \rho^- \pi^+ \pi^0)}
{\rho \rightarrow \rho \pi^+ \pi^-}  = 2.
\end{equation}

The background contributions in the selected kinematic
domain~(\ref{eq:kin_range}) are estimated with the Monte-Carlo 
simulations, and subtracted from all cross section distributions.
They amount to 0.51~\%, 0.21~\% and 1.30~\% for
the $\omega$, $\phi$ and \rhoprim\ respectively. 

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


\subsection{Systematic errors}
                               \label{sect:syst}
%===========================================

The effect of systematic uncertainties affecting the measurements
are estimated by varying the event selection and the parameters of the 
\rh\ Monte Carlo simulation.

\begin{itemize}
\item {\bf Uncertainties on the acceptance and smearing 
  corrections} \\
The cross section dependence on $t$, used to
compute acceptances and smearing effects, and to propagate the 
measurement to fixed values of the kinematical
variables, is varied by 
$\pm~0.5$ for the power of the $t$ dependence.   
The modelling of the dissociative proton remnant $Y$ is varied
within $(1/M_Y^2)^{\pm 0.3}$.
In the case of the angular distributions, the matrix elements
\rfour\ and $r^{04}_{1-1}$ are varied by the error obtained from 
fits to the current data, which is $\pm~0.05$.
 The matrix element ${\rm Re}[r^{04}_{10}]$
is varied by $\pm~0.02$, in accordance with the error quoted in~\cite{zeus-hight}.

\item {\bf Background distributions} \\
For dissociative \rhoprim\ production, the $t$ dependence is varied by  
$\pm~2.0$ for the power of the $t$ slope.
The ratios of the background cross sections to the \rh\
cross section are varied by:
 $\pm~0.02$ for $\omega$~/~$\rho$, 
 $\pm~0.05$ for $\phi$~/~$\rho$, 
 $\pm~0.1$ for \rhoprim ~/~$\rho$.

\item {\bf Trigger description} \\
The $p_t$ cut of 0.45 GeV, used to restrict the analysis to the
region of high trigger efficiency is varied between 
0.40 GeV and 0.60 GeV. Further errors coming from re-weighting the
trigger efficiency are also taken into account.
  
\item {\bf Tracker description} \\
The cut on the $\theta$ angle of the reconstructed tracks is varied between
$155^\circ$ and $160^\circ$, to estimate the effect of the 
dependence of the correction factors in that region. The variation
of the $p_t$ cut stated above also partially covers the error on the
tracker description.

\item {\bf Electron tagger description} \\ 
The error on the description of the electron tagger is estimated by
varying the acceptance by 3 \% as a function of $y$.

\end{itemize}

The systematic errors are combined quadratically.

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

%\newpage
\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 and smearing effects using the 
Monte Carlo simulations described above. At low $t$, the data are
further corrected for the skewing effect obtained for the present data
from the Ross-Stodolsky~\cite{rs} parametrisation. All cross
sections presented below, as well as the theoretical model predictions,
are normalised to their integrals in the
respective kinematic domain.

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

\subsection{\boldmath{$M_{\pi\pi}$} dependence of the 
            cross section}
    \label{sect:mdepend}

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

The $M_{\pi\pi}$ dependence of the $\gamma + p \rightarrow  \rho + Y$
cross section is shown in fig.~\ref{fig:mass}.
The superimposed curve shows the result of a fit to a relativistic 
Breit-Wigner with Ross-Stodolsky skewing, for which $\chi^2/ndf = 17.12/16$.  The resulting values 
for the \rh\ mass and width, which are $M_{\rho}~=~0.771~\pm~0.007~\rm GeV$
and $\Gamma_{\rho} = 0.180~\pm~0.029~\rm GeV$ 
~respectively, are
compatible with the PDG values~\cite{pdg}.  The overall value of the skewing parameter $n$ is consistent with zero.

%=======================\label{fig:mass}===============
\begin{figure}[htbp]
\vspace{0.2cm}
\begin{center}
\epsfig{file=H1prelim-05-012.fig2.eps,angle=270,width=16.cm}\\ 
\end{center}
 \caption{The \protect{$M_{\pi\pi}$} dependence of the 
 $\gamma + p \rightarrow \rho + Y$ cross section.  The inner error bars
 show the statistical errors, while the outer ones represent
 the sum of the statistical and non-correlated systematic errors added in quadrature.
 The superimposed curve shows the result of a fit to a relativistic Breit-Wigner 
 with Ross-Stodolsky skewing.} 
 \label{fig:mass}
 \end{figure}
%=======================\end{fig:mass} ======================

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

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

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

The $t$ dependence of the $\gamma + p \rightarrow \rho + Y$ cross section is presented in Fig.~\ref{fig:tfit}.  
The dependence is well described by the BFKL model of Poludniowski {\em et al.}
Further, it can be fitted by a power law distribution of the form $d\sigma/dt \propto 
|t|^{-n}$, where $n~=~4.41~\pm~0.07~({\rm stat.})~^{+0.07}_{-0.05}~({\rm syst.})$, over the 
measured range of $t$.

The ZEUS collaboration~\cite{zeus-hight} obtained a power law of
$n = 3.21~\pm~0.04~({\rm stat.})~\pm~0.15~({\rm syst.})$ for the kinematic range 
$Q^2 < 0.02$ \gevsq , $80 < W < 120$ GeV, $1.1 < |t| < 10.0$ \gevsq\ and $M_Y < 25~\rm GeV$, 
which is also well described by theory, and is understood to be shallower
than the value measured here due to the different $M_Y$ range.

%=======================\label{fig:tfit}===============
\begin{figure}[tbp]
\vspace{-0cm}
\begin{center}
\epsfig{file=H1prelim-05-012.fig3.eps,angle=270,width=16.cm}\\ 
\end{center}
 \caption{The $t$ dependence of the $\gamma + p \rightarrow \rho + Y$ 
cross section.  The inner error bars show the statistical 
errors, while the outer ones represent the sum of the statistical and non-correlated systematic 
errors added in quadrature. The dashed line is the result of a fit to a 
power law distribution $|t|^{-n}$, which results in a power 
$n~=~4.41~\pm~0.07~({\rm stat.})~^{+0.07}_{-0.05}~({\rm syst.})$. The full line 
shows the result from the BFKL model of Poludniowski {\em et al.}}
 \label{fig:tfit}
 \end{figure}
%=======================\end{fig:tfit} ======================

%===========================================
%===========================================
\subsection{ \boldmath{\rfour} spin density matrix element}
  \label{sect:r400}

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

Fig.~\ref{fig:ctsfit} presents the \cost\ dependence of the 
$\gamma + p \rightarrow  \rho + Y$ cross section, for three intervals
in \ttra .
The superimposed curves show 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}
along with the expectation from SCHC.
The extracted values of the \rfour\ spin density matrix element
are shown as a function of \ttra\ in fig.~\ref{fig:amps}. Also shown are the results 
of the ZEUS Collaboration~\cite{zeus-hight}, with which there is excellent agreement.
The results are close to zero, 
indicating that the production is dominated by transversely polarised $\rho$ mesons. 
%The small value of $r^{04}_{00}$ agrees well with the BFKL model of Poludinowski {\em et al.}
The BFKL Model of Poludniowski {\em et al.} also correctly describes the small values of $r^{04}_{00}$.

%===========================================
%===========================================
\subsection{ \boldmath{$r^{04}_{1-1}$} spin density matrix element}
  \label{sect:r411}

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

Fig.~\ref{fig:psfit} presents the $\phi^{\ast}$ dependence of the 
$\gamma + p \rightarrow  \rho + Y$ cross section, for three intervals
in \ttra .
The superimposed curves show results of fits of the form
\begin{equation}
  \frac {\rm{d} \sigma} {\rm{d} \phi^{\ast}}  \propto 
   1 - 2 r^{04}_{1-1}\ \cos 2\phi^{\ast}
                                \label{eq:phst}
\end{equation}
along with the expectation from SCHC.
The extracted values of the $r^{04}_{1-1}$ spin density matrix element
are shown as a function of \ttra\ in fig.~\ref{fig:amps}. Also shown are the results 
of the ZEUS Collaboration~\cite{zeus-hight}, with which there is excellent agreement.
Values significantly different from zero are obtained for the $r^{04}_{1-1}$  matrix element, 
confirming $s$-channel helicity non-conservation~\cite{zeus-hight,h1-hight,h1-rho,zeus-smde}.
The prediction arising form the BFKL model of Poludniowski {\em et al.}
for $r^{04}_{1-1}$, although qualitatively able to reproduce the data, is too large at low values
of $|t|$.

%=======================\label{fig:ctsfit}===============
\begin{figure}[htbp]
\vspace{-0.cm}
\begin{center}
\epsfig{file=H1prelim-05-012.fig4.eps,angle=270,width=16.cm}\\ 
\end{center}
 \caption{The \cost\ dependence of the
$\gamma + p \rightarrow \rho + Y$ cross section, for three 
\ttra\ intervals.  The inner error bars
 show the statistical errors, while the outer ones represent
 the sum of the statistical and non-correlated systematic errors added in quadrature.
The superimposed curves show the results of
fits to the relation \ref{eq:cost} (solid) and the predictions of SCHC (dashed)}
 \label{fig:ctsfit}
 \end{figure}
%=======================\end{fig:ctsfit} ======================
%=======================\label{fig:psfit}===============
\begin{figure}[htbp]
\vspace{-0.cm}
\begin{center}
\epsfig{file=H1prelim-05-012.fig5.eps,angle=270,width=16.cm}\\ 
\end{center}
 \caption{The $\phi^{\ast}$ dependence of the
$\gamma + p \rightarrow  \rho + Y$ cross section, for three 
\ttra\ intervals.  The inner error bars
 show the statistical errors, while the outer ones represent
 the sum of the statistical and non-correlated systematic errors added in quadrature.
The superimposed curves show the results of
fits to the relation \ref{eq:phst} (solid) and the predictions of SCHC (dashed)}
 \label{fig:psfit}
 \end{figure}
%=======================\end{fig:psfit} ======================
%=======================\label{fig:r400}===============
\begin{figure}[htbp]
\vspace{-0.cm}
\begin{center}
\epsfig{file=H1prelim-05-012.fig6.eps,angle=270,width=16.cm}\\ 
\end{center}
 \caption{Measurements of the $r^{04}_{00}$ and $r^{04}_{1-1}$
spin density matrix elements 
as a function of $|t|$ 
(full points) together with ZEUS previous measurements \cite{zeus-hight} (open points). 
The inner error bars show the statistical errors, while the outer ones represent 
the sum of the statistical and systematic errors added in quadrature. 
The full lines 
show the results from the BFKL model of Poludniowski {\em et al.}, while the 
dashed lines show the prediction from SCHC}
 \label{fig:amps}
 \end{figure}
%=======================\end{fig:r400} ======================


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

The diffractive  photoproduction of $\rho$ mesons,
$e + p \rightarrow e + \rho + Y$, has been studied using the H1 detector at HERA
in the kinematic range $Q^2 < 0.01 $ \gevsq , $75~<~W~<~95$~GeV, 
 $1.5~<~|t|~<~10$ \gevsq\ and $M_Y~<~5~\rm GeV$. The $t$ dependence of the $\gamma p$ cross section is measured and fitted with
a power law of the form $|t|^{-n}$, which results in \linebreak $n~=~4.41~\pm~0.07~({\rm stat.})~^{+0.07}_{-0.05}~({\rm syst.})$.
Moreover, it is well described by BFKL model predictions. The spin density
matrix elements $r^{04}_{00}$ and $r^{04}_{1-1}$ are measured as a function of $t$. The $r^{04}_{00}$
matrix element is consistent with zero while the $r^{04}_{1-1}$ matrix element differs significantly
from zero and thus confirms violation of SCHC~\cite{zeus-hight,h1-hight,h1-rho,zeus-smde}.  BFKL model predictions describe
$r^{04}_{00}$ well but the prediction for $r^{04}_{1-1}$, although qualitatively able to reproduce the data, 
is too large at low $|t|$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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.
We would also like to thank J.~Forshaw and R.~Enberg for providing
the theoretical model calculations used throughout.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\newpage
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\end{document}
