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\title{A new Measurement of Exclusive $\rho^0$ Meson Photoproduction at HERA}
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\newcommand{\OurAlphaZero}{1.093 \pm 0.003 (stat.) ^{+0.008}_{-0.007} (syst.)}
\newcommand{\OurAlphaPrime}{0.116 \pm 0.027 (stat.) ^{+0.036}_{-0.046} (syst.)}

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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 
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,width=2.cm} \\[.2em] \hline
\multicolumn{4}{l}{{\bf
                 14th International Workshop on Deep Inelastic Scattering,
                 DIS2006, April 20-24, 2006, Tsukuba, Japan}} \\
%                 & Abstract:        & {\bf xx-xxx}    &\\
%                 & Parallel Session & {\bf x}   &\\ \hline
 & \multicolumn{3}{r}{\footnotesize {\it
    www-h1.desy.de/h1/www/publications/conf/conf\_list.html}} \\[.2em]
 & \multicolumn{3}{r}{H1prelim-06-011} \\[.2em]
\end{tabular}
\end{small}
\end{center}
\vspace*{2cm}

\begin{center}
  \Large
  {\bfseries
    A New Measurement of \\
    Exclusive
    {\boldmath  $\rho^0$} Meson Photoproduction \\
    at HERA\\
  }

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

\begin{abstract}

\noindent
Using a sample of more than 240000 events taken with the H1 detector 
at HERA in 2005, a new measurement of  exclusive
$\rho^0$ photoproduction in the kinematic range $20 < W < 90\,\GeV$
and $|t| < 3\,\GeV^2$ is performed.
From the measurement of 
the $W$ dependence of this process in eight bins of $t$, 
the Pomeron trajectory $\alpha\sub{\pom}\,(t)$ can be extracted
for the first time from the data of a single experiment.
The result is 
$\alpha\sub\pom\,(t) = \OurAlphaZero + (\OurAlphaPrime)\,\GeV^{-2} \cdot t$.
\end{abstract}

  \vspace*{1cm}
  
\end{titlepage}

\pagestyle{plain}

\section{Introduction}
%---------------------

The exclusive photoproduction of $\rho^0$ mesons has 
been studied in great detail over the last 40 years
\cite{bib:rho-fixedtarget,Aston:1982hr,Derrick:1995vq,Aid:1996bs,Derrick:1996vw,Breitweg:1997ed,Breitweg:1999jy}.
%\cite{Crouch:1966,Joos:1976nm,Aleksandrov:1980pt,Apsimon:1991nu,
%Derrick:1995vq,Adams:1997bh,Breitweg:1997ed,Aid:1996bs,Derrick:1996vw,
%Wu:2005wf,Clifft:1976be,Anderson:1976ph,Struczinski:1975ik,Bartalucci:1977cp,
%Erbe:1966,Blechschmidt:1967,Ballam:1971yd,Ballam:1972eq,Egloff:1979mg,
%Aston:1982hr,Atkinson:1984fg,Breitweg:1999jy}.  
This process shows the typical characteristics of
a diffractive reaction, i.e. a weak growth
of the cross section with the photon-proton centre-of-mass 
energy $W$ and an exponential decrease as function
of the modulus of the squared momentum transfer $|t|$.

This behaviour can be understood in terms of the 
vectormeson dominance model (VDM) 
%\cite{Sakurai:1960ju, Gell-Mann:1961tg, Sakurai:1969ss},
\cite{bib:vdm},
where the hadronic part of the photon wavefunction is conceived as a 
superposition of vector--meson states, 
in which the contribution from the $\rho^0$ dominates.
This hadronic component of the photon may then participate
in hadron--hadron interactions,
notably in elastic scattering off the proton.

Thus, an understanding of $\rho^0$ photoproduction
sheds light on the hadronic properties of the photon,
including the total photon--proton cross section,
which is linked to the elastic scattering amplitude
by the optical theorem.

At sufficiently high centre-of-mass energies
$W$,
% ($W > 20\,\GeV$ for $\rho^0$ photoproduction),
elastic hadron-hadron scattering is well described
by the exchange of a single Regge trajectory 
\cite{bib:regge},
%\cite{Regge:1959mz, Regge:1960zc},
the Pomeron,
while at lower values of $W$ 
the exchange of additional meson trajectories
becomes important.
The contribution of the exchange of these so-called subleading trajectories
to the production cross section is often summed up by introducing
an effective Regge trajectory, the Reggeon.

In this high--$W$ region, where Pomeron exchange is dominant,
the energy dependence of elastic 
$\rho^0$ photoproduction at fixed momentum transfer $t$
is in diffractive models directly linked to the Pomeron trajectory 
$\alpha\sub\pom\,(t)$
by
$$
   \frac{\der \sigma\sub{\gamma p}\,(W)}{\der t} 
   \propto 
   \left ( \frac{W}{W\sub{0}} \right ) ^{4 (\alpha\sub\pom\,(t) - 1)}.
$$
Thus, a measurement of the $W$ dependence 
of elastic $\rho^0$ photoproduction in bins of $t$
is paramount to the direct measurement of the 
Pomeron trajectory $\alpha\sub\pom\,(t)$,
which is well approximated by a linear function
$\alpha\sub\pom\,(t) = \alpha\sub{\pom, 0} + \alpha'\sub\pom \cdot t$.

The ZEUS collaboration has measured 
$\rho^0$ photoproduction cross sections
\cite{Breitweg:1999jy}
%\cite {Derrick:1996vw, Breitweg:1997ed, Breitweg:1999jy}
at 
high values of $W \approx 94\,\GeV$,
and has combined its data 
with data taken at lower energies
\cite{Aston:1982hr}
and with earlier HERA data from H1 \cite{Aid:1996bs}
and ZEUS \cite{Derrick:1995vq,Breitweg:1997ed}
in order to extract the $W$ dependence of
$\rho^0$ photoproduction and thus
the Pomeron trajectory.
Their result is
$\alpha\sub\pom\,(t) = (1.096 \pm 0.021) + (0.125 \pm 0.038)\,\GeV^{-2} \cdot t$.
The intercept $\alpha\sub{\pom, 0}$ is consistent 
with the expectations
of Donnachie and Landshoff \cite{Donnachie:1992ny}
($\alpha\sub{\pom, 0} = 1.0808$)
and Cudell \cite{Cudell:1996sh} ($\alpha\sub{\pom, 0} = 1.096^{+0.012}_{-0.009}$),
whereas the measured slope $\alpha'\sub\pom$ is 
significantly lower than the canonical value
of $\alpha'\sub\pom = 0.25\,\GeV^{-2}$
extracted from other diffractive processes \cite{bib:dlslope}.

In the present analysis,
we measure the $W$ dependence of
diffractive $\rho^0$ photoproduction
over an extended $W$ range of $20 < W < 90\,\GeV$
in bins of $t$
within a single experiment.
Thus we can extract the Pomeron trajectory
from the data of one experiment alone,
which reduces the uncertainty from 
different systematic errors of different experiments.
The $\rho^0$ meson is identified by its decay to two charged pions,
which has a branching ratio of $99\,\%$.

\section{Event Selection and Triggering}
%---------------------------------------

The data were taken with the H1 detector in 2005,
when HERA collided $920\,\GeV$ protons with $27.6\,\GeV$ positrons.
The H1 detector has been described in detail elsewhere
%\cite{Abt:1996hi, Abt:1996xv, Appuhn:1996na}. 
\cite{bib:h1detector}. 
A detailed description of the analysis can be found in \cite{bib:rondiss}.

Events were mainly triggered with the Fast Track Trigger (FTT)
\cite{bib:ftt} at the first H1 trigger stage (trigger level 1).
This trigger combines track segments, which are identified in 
four three-wire-groups within the central jet chamber (CJC),
to track candidates within the trigger level 1 decision time of 
$2.3\,\us$. 
Two out of four possible track segments have to be identified
and found to correspond to compatible track parameters (signed curvature and
azimuth) in order to form a track at level 1.
Tracks with transverse momenta as low as $100\,\MeV$ 
can be identified.
%For the data period under consideration, the track finding efficiency
%is approximately 
%\Nonumber{$90\,\%$}
%for single tracks.
%About 
%\Nonumber{$5\,\%$}
%of the tracks are wrongly identified as two separate
%tracks. 
Events were triggered if two or three tracks were found by the FTT with
a total charge of zero or $\pm 1$, and if at least two of the tracks had a
transverse momentum above $160\,\MeV$.

In addition, a trigger signal from the inner proportional chamber (CIP)
trigger, consistent with a low multiplicity interaction in the nominal
interaction region, was demanded. Events with large energy deposits in
the inner forward part of the Liquid Argon (LAr) calorimeter or signals
in a set of scintillators (FTI2) within the forward tracker were
rejected.

The data were taken during a period of three months in 2005,
when the operation parameters of the FTT and the CJC were stable.
Correcting for the trigger prescale,
the data correspond to an integrated luminosity of 
$569.7\,\nb^{-1}$.

% Only data from data taking runs that fulfilled certain quality criteria
% were considered. On one hand, the detector components pertinent to this
% analysis 
% % (CJC, CIP, LAr, FTT, FMD, FTS, ToF and luminosity system)
% had to be operational during the run.
% % In addition, the gain of the jet chambers CJC1 and CJC2, which is
% % constantly monitored by measuring the pulse height spectrum of hits
% % caused by cosmic ray muons, had to be within
% % $4.5\,\%$ ($1.7\,\%$)
% % of its average value for CJC1 (CJC2). 
% % This condition ensures a uniform track trigger efficiency. 
% To ensure a uniform track trigger efficiency,
% runs with unusually high or low gain of the CJC were rejected.
% % Finally, three short run ranges were excluded were the 
% % observed event yield was particularly low; 
% % in all three run cases logbook entries indicated operation problems
% % of the FTT.  

In order to identify events where the $\rho^0$ decays to $\pi^+ \pi^-$,
events were selected by demanding:
\begin{itemize}
\item A reconstructed primary vertex within $25\,\cm$ of the nominal
  interaction point in $z$, 
\item exactly two primary vertex fitted tracks of opposite charge in the central detector,
  with transverse momenta above $200\,\MeV$ and polar angle within the
  range $20^\circ < \theta < 160^\circ$, 
  of good quality,
\item no additional tracks in the central or forward tracker,
\item no scattered electron candidate in the LAr or SpaCal calorimeters,
\item no calorimeter custers with energies above
  $500\,\MeV$ that are not associated with 
  one of the pion tracks. A cluster is 
  considered to be associated with a track when the distance 
  in pseudorapidity $\eta$ and azimuthal angle $\phi$ is less than
  $0.175$.
\end{itemize}
A total of
$271'684$
events passed these selection criteria,
$241'672$
of these events lie in the mass range 
$0.6 < m\sub{\pi \pi} < 1.1\,\GeV$,
where $m\sub{\pi \pi}$ is the invariant mass of the two tracks
under a charged pion mass hypothesis.

The four-vectors of the two decay pions are reconstructed from
their measured track momenta, the $\rho^0$ four vector
is the sum of the two pions momenta.

The momentum transfer $t$ is then calculated from the $\rho^0$ meson's
transverse momentum:
$$
  t = -p\sub{t, \rho}^2
$$
The $\gamma p$ centre-of-mass energy is calculated from
$$
  W = \sqrt{2 E\sub p (E\sub{\rho} - p\sub{z, \rho})},
$$
where $E\sub{p} = 920\,\GeV$ is the proton beam energy.

\section{Monte Carlo Modelling}
%-----------------------------

The elastic and proton-dissociative 
$\rho^0$ production was modeled using the Monte Carlo generator DIFFVM
\cite{List:1998jz}.
The generator models vector meson production 
based on the equivalent photon approximation
%\cite{vonWeizsacker:1934sx,Williams:1934ad,Budnev:1974de},
\cite{bib:epa},
the vector meson dominance model
%\cite{Sakurai:1960ju,Gell-Mann:1961tg,Sakurai:1969ss,Bauer:1977iq},
\cite{bib:vdm},
and pomeron exchange
\cite{Goulianos:1982vk}.

The generator models the diffractive dissociation
of the proton according to a model
where the mass $\MY$ of the dissociative system
is generated according to
$$
  \frac{\der^2 \sigma}{\der t \, \der \MY^2}
  \propto
  \frac{f\,(\MY)}{(\MY^2)^{1 + \epsilon}},
$$
with $\epsilon = 0.0808$, where
$f\,(\MY)$ is a function fitted to experimental data
\cite{Goulianos:1982vk} that parametrizes deviations
in the low mass region $\MY < 1.9\,\GeV$
from a pure $1/(\MY^2)^{1 + \epsilon}$ behaviour.
For this low mass region, the dissociating system
is treated as a $N^*$ resonance that decays according
to the measured branching fractions
\cite{Eidelman:2004wy}. In the higher mass region,
the system is modeled as a quark--diquark system, where
the quark is assumed to be scattered out of the proton,
and the fragmentation is performed using the Lund fragmentation
as implemented in JETSET \cite{bib:jetset}.

The Monte Carlo events were reweighted in the variables $t$, $W$, and 
$\mpipi$ in order to get the best possible description
of the experimentally observed distributions.

In particular, the dipion mass spectrum was reweighted from a 
nonrelativistic Breit--Wigner function (as implemented in DIFFVM)
to a skewed relativistic Breit--Wigner function
according to the model by Ross and Stodolsky \cite{Ross:1965qa},
with the skewing parameter given by
$$
  n\,(t) = 6.1 \cdot \exp\, (t \cdot 1.95\,\GeV^{-2});
$$
this parametrisation was derived from a fit to ZEUS data
\cite{Breitweg:1997ed}.

The $t$ spectrum was generated as a purely exponential
spectrum, and reweighted to the form
$$
  \frac{\der \sigma^{\gamma p}}{\der t} \propto 
  \exp \,\left ( a \ln \left ( 1 - b_0 t/a \right ) \right )
$$
which smoothly interpolates between an exponential decrease 
$\der \sigma^{\gamma p}/\der t \propto\exp(b_0 t + b_0^2 t^2/2 a)$
with $b_0 = 11.4~(6.5)\,\GeV^{-2}$ for $W_0 = 90\,\GeV$ at low values of $|t|$,
and a power--law behaviour 
$\der \sigma^{\gamma p}/\der t \propto |t|^{-a}$
with $a=23~(8.5)$ at large $|t|$
for elastic (proton-dissociative) $\rho^0$ production.
The coefficient $a$ has been determined from fits to large-$|t|$ data 
for the proton-dissociative data, and from fits of the form
$\der \sigma^{\gamma p}/\der t \propto\exp(b_0 t + c_0 t^2)$
\cite{Breitweg:1997ed}
for elastic data.

The DIFFVM generator produces events with all values of $Q^2$
and $\MY$. 
Events were counted as signal events if they fulfilled the cuts
$Q^2 < 4\,\GeV^2$ and
$(\MY^2+Q^2)/(W^2+Q^2) < 0.01$,
otherwise they were treated as background.

\section{Cross Section Determination}
%------------------------------------

\subsection{Corrected Number of $\rho^0$ Mesons}
%-----------------------------------------------

The events are distributed in
 $80$
bins\footnote{Called ``$W/t$ bins'' herafter.} in $W$ and $t$:
In $t$, there are 12 bins
from $|t|=0\,\GeV^2$ to $|t|=3.5\,\GeV^2$.
For $|t|<0.16\,\GeV^2$ ($|t|>0.16\,\GeV^2$),
there are 10 (5) $W$ bins for each $t$ bin.
The $W$ range covered varies with $t$; as $|t|$ rises,
the lower (upper) value of $W$ changes from 
$20\,\GeV$ ($69\,\GeV$) to
$25\,\GeV$ ($90\,\GeV$).

For each $W/t$ bin,
the number of events
is evaluated in dipion mass bins of $25\,\MeV$ width.
This dipion mass spectrum is corrected for acceptance,
reconstruction and trigger efficiency, and fitted.
The overall correction factor is typically $0.25$.
The correction factors that are applied are determined for each
$\mpipi$ bin in each $W/t$ bin individually.

It is well known that due to the large width
of the $\rho$ meson there is a significant interference
between resonant $\rho$ production with a subsequent decay to 
two pions, and nonresonant dipion production.
This interference leads to a distortion of
the observed dipion mass spectrum.
Two models have been put forward to describe 
this effect: 
The ansatz of S\"oding \cite{Soding:1965nh}
describes the distorted spectrum as sum of three terms,
a relativistic Breit--Wigner function for the $\rho$,
a term for the nonresonant dipion production (here taken to be 
constant as function of $\mpipi$), and an interference term:
\begin{equation}
  \frac{\der N}{\der \mpipi} =  
  \frac{ N_0 \mrho\,\Grho\,\mpipi + I (\mrho^2 - \mpipi^2)}
       {(\mrho^2 - \mpipi^2)^2 + \mrho^2 \Grho^2} + B,
  \label{eq:soeding}
\end{equation}
where $N_0$, $I$, and $B$ parametrize the contributions from
resonant $\rho^0$ production, interference, and background,
respectively. Here, $\mrho$ is the nominal $\rho^0$ mass, and
$$
  \Grho = \Grhon \, 
  \left ( \frac{\mpipi^2 - 4 m_\pi^2}{\mrho^2 - 4 m_\pi^2}\right )^{\frac 3 2} 
  \frac{\mrho}{\mpipi}
$$
is the mass dependent width of the $\rho^0$, following the suggestion of Jackson
\cite{Jackson:1964zd} ($\Grhon$ is the
nominal $\rho^0$ width, and $m_\pi$ the charged pion mass).
In the ansatz of Ross and Stodolsky \cite{Ross:1965qa},
a relativistic Breit--Wigner function is multiplied
by a skewing term $(\mpipi/m\sub{\rho})^{-n}$,
where $m\sub{\rho}$ is the nominal $\rho$ meson mass,
and $n$ is the so-called skewing parameter:
\begin{equation}
    \frac{\der N}{\der \mpipi} =  
  N_0 \frac{\mrho\,\Grho\,\mpipi}
       {(\mrho^2 - \mpipi^2)^2 + \mrho^2 \Grho^2}
       \left ( \frac{\mrho}{\mpipi}\right )^n
        + B.
  \label{eq:ross-stodolsky}
\end{equation}

In each of the $W/t$ bins, the corrected dipion 
mass spectrum $\der N\super{corr}/\der \mpipi$
(in $25\,\MeV$ bins)
is fitted in the range $0.6\,\GeV < \mpipi < 1.1\,\GeV$
with a skewed relativistic
Breit--Wigner function as proposed by Ross and Stodolsky,
with the nominal $\rho^0$ mass and width fixed to the
values 
$m\sub{\rho^0} = 766.4\,\MeV$
and
$\Grhon = 145\,\MeV$,
which are the average values of the corresponding fit quantities,
obtained from fits in all 80 $W/t$ bins.
These values are compatible with the PDG values \cite{Eidelman:2004wy} 
$\mrho = 768.5\pm 1.1\,\MeV$
and
$\Grhon = 150\pm 3\,\MeV$.
Fits where the mass and width are
left as free parameters result in compatible values for
$\mrho$ and $\Grhon$ in all
$W/t$ bins.
The average values of \mrho\ and \Grhon\ derived from these
fits are 
$\mrho = 766.4\,\MeV$
and
$\Grhon = 145\,\MeV$.

To extract the number of resonantly produced $\rho^0$ mesons
in each $W/t$ bin, the integral of the unskewed
Breit--Wigner function is evaluated in the region
$2 m\sub{\pi} < \mpipi < \mrho+5 \Grhon$,
with the value of $\mrho+5 \Grhon = 1.52\,\GeV$
taken from the PDG \cite{Eidelman:2004wy}:
$$
  \Ncor = N_0 \int \limits_{\mpipi = 2 m_\pi}^{\mrho+5 \Grhon}
   \frac{\mrho\,\Grho\,\mpipi}
       {(\mrho^2 - \mpipi^2)^2 + \mrho^2 \Grho^2}
  \der \mpipi.
$$

Extracting the number of $\rho^0$ mesons from the corresponding term
in the S\"oding parametrisation leads to 
compatible results.
% a change of the number
% of events of less than 
% \Nonumber{$2\,\% $}
% in all bins with 
% \Nonumber{$|t|<xxx\,\GeV^2$};
% at most, the difference is 
% \Nonumber{$xxx\,\% $}.

There exists a different convention for the
$\rho^0$ cross section due to Spital and Yennie \cite{Spital:1974cx},
used e.\,g. by the Omega collaboration \cite{Aston:1982hr},
who define the cross section by
$$
  \frac{\der \sigma}{\der t} = \frac{\pi \Grhon}{\pi} \,
    \left . \frac{\der^2 \sigma}{\der t \, \der\mpipi} \right | _
    {\mpipi=\mrho}.
$$
For the PDG values of $\mrho$ and $\Grhon$, this definition
leads to a cross section which is larger by a factor of
$1.050$
than the one derived with our definition.


To calculate the photoproduction cross section, 
the $ep$ cross section is divided
by the photon flux, integrated over the respective 
$W$ range, and over $Q^2$ up to $Q^2=4\,\GeV^2$.

The average $Q^2$ value of the events that pass the selection,
as determined from Monte Carlo studies, is $0.010\,\GeV^2$;
it rises from $0.003\,\GeV^2$ for the lowest $t$ bins to $0.4\,\GeV^2$
for the highest $t$ bins. For the $t$ bins that are used to extract
the elastic cross section ($|t|<0.7\,\GeV^2$), the average
$Q^2$ value is $0.0078\,\GeV^2$.

\subsection{Background}
%-----------------------

The following background sources were investigated:
\begin{itemize}
  \item $\phi$ meson production,
  \item $\omega$ meson production,
  \item $\rho(1450)$ and $\rho(1700)$ meson production 
  with subsequent decay to $\pi^+ \pi^- \pi^0 \pi^0$, 
  \item $\rho^0$ meson production outside the signal range
    of $Q^2<4\,\GeV^2$ and $(\MY^2+Q^2)/(W^2+Q^2) < 0.01$.
\end{itemize}

The background from $\phi \to K^+ K^- $ is completely negligible,
as the reconstructed dipion mass is always below $0.6\,\GeV$,
the same holds for $\phi \to K^0\sub{L} K^0\sub{S}$
with subsequent decay $ K^0\sub{S} \to \pi^+ \pi^-$.

The bulk of events from $\omega \to \pi^+ \pi^- \pi^0$ also has
dipion masses below $0.6\,\GeV$,
while the contribution from 
$\omega \to \pi^+ \pi^-$ is small and manifests itself mainly through
the $\rho-\omega$ interference, which has been neglected,
following the procedure of earlier measurements \cite{Aston:1982hr, Breitweg:1997ed}.

The amount of background from $\rho'$ decays ($\rho(1450)$ and
$\rho(1700)$ can be estimated from the 
dipion mass spectrum, which shows an indication for
$\rho' \to \pi^+ \pi^-$ decays in the expected mass region around
$\mpipi \approx 1.6\,\GeV$ at a level of $0.5\,\%$ compared to
$\rho^0 \to \pi^+ \pi^-$.
The Crystal Barrel collaboration has measured \cite{Abele:2001pv}
the ratio
$BR(\rho' \to 4\,\pi)/BR(\rho' \to \pi^+ \pi^-)$
to be $0.37 \pm 0.10$ for $\rho (1450)$ and
$0.16\pm0.04$ for $\rho(1700)$;
however, the channel  $\rho' \to 4\pi$ is expected to be dominated
by $2\pi^+ 2\pi^-$, whereas $\pi^+ \pi^- 2\pi^0$ is significantly
smaller \cite{Clegg:1993mt}.

From these arguments, we estimate the background from $\phi$, $\omega$,
and $\rho'$ production to be smaller than 
$2\,\%$
 and assign a corresponding normalisation uncertainty to the result.
 
Background from $\rho^0$ production outside the
kinematical signal definition used in the present analysis,
i.e. $Q^2 < 4\,\GeV^2$ and 
$(\MY^2+Q^2)/(W^2+Q^2) < 0.01$,
was included in the Monte Carlo simulation and has been corrected for
in the overall reconstruction efficiency.




\subsection{Separation of Elastic and Proton--Dissociative Processes}
%----------------------------------------

To discriminate between events where the proton stays intact (elastic
production) or dissociates diffractively into a state of mass
$\MY$,
signals in additional subdetectors are investigated:
the forward muon detector FMD and the forward tagging system FTS.

The FMD consists of two sets of planar drift chambers in front and
behind a toroidal magnet, covering the polar angle region of 
$3^\circ < \theta < 18^\circ$.
The pre--toroid chambers are very effective in detecting 
secondary particles from interactions of proton remnant particles with
the beampipe, collimators, and other material in the forward direction.
The relevant observable is the total number of hit pairs in the
pre--toroid chambers.

The FTS is a system of 4 sets of scintillator paddles surrounding the
proton beam pipe at distances of $26\,\m$, $28\,\m$, $53\,\m$, and
$92\,\m$ downstream of the interaction point.
Monte Carlo studies indicate that the stations at $26\,\m$ and $28\,\m$
are useful to detect particles from proton dissociation.

Events are considered to be tagged if more than one hit pair in the FMD
is observed (a single hit pair being compatible with noise), or if at
least one scintillator of the $26\,\m$ or $28\,\m$ FTS stations
has detected a particle.

In each $W/t$ bin, the number $N_{el}$ of elastic $\rho^0$ production
events is inferred from the corrected number of $\rho^0$ events
\Ncor\ in this bin, and the fraction of tagged events 
$$
  \ftag = \frac{\Ntag}{\Ntag+\Nuntag}
$$
in this $W/t$ bin, where \Ntag\ (\Nuntag) is the raw number of 
selected tagged (untagged)  events in the mass range $0.6 < \mpipi < 1.1$ in the respective
$W/t$ bin, using the following relation:
$$
  \Nel = \Ncor \frac{\epspd - \ftag}{\epspd - \epsel}.
$$
Here, \epsel\ (\epspd) is the fraction of selected elastic
(proton--dissociative) events that produce a tag, as determined from
Monte Carlo simulations. The fractions \epsel\ and \epspd\ vary with
$t$ and are evaluated for the different $t$ bins separately,
while they are independent of $W$ and are averaged over the dipion mass
range.

\subsection{Differential Cross Sections}
%----------------------------------------

From the corrected number \Ncor\ of $\rho^0$ events per $W/t$ bin,
the differential $\gamma p$ cross section is derived as
$$
  \frac{\der \sigma\super{\gamma p}\,(W)}{\der t}
    = \frac{\Ncor}{\Lint \, \Delta t \, \Delta W \, \Phi\sub{\gamma p}},
$$ 
where $\Delta t$ and $\Delta W$ are the width in $t$ and $W$ of the 
respective bin, \Lint\ is the integrated luminosity, 
and $\Phi\sub{\gamma p}$ is the photon flux factor, integrated up
to $Q^2 = 4\,\GeV^2$.

The separate cross sections for elastic and proton-dissociative $\rho^0$ production
are calculated in the same manner, using the number of elastic (proton-dissociative)
events \Nel\ (\Npd).

The $W$ bin centre is evaluated from the arithmetic mean of
the $W$ of the events detected in the respective bin.
The bin centre in $t$ is calculated according to 
$$
 t_{\textrm{mean}} = t_{\textrm{low}} + \frac{1}{b}\cdot \left ( \log(b\cdot\triangle t) - 
   \log(1-\exp(-b\cdot\triangle t)) \right )
$$
 with $b = 10\,\GeV^{-2}$;
 at this $t_{\textrm{mean}}$, the bin averaged cross section is identical to the 
 differential cross section for the chosen $b$ value.
 
\subsection{\boldmath $W$ Dependency and Pomeron Trajectory}
%----------------------------------------

The differential $\gamma p$ cross sections measured in the same
$t$ range are fitted to the form
\begin{equation}
  \label{eq:wdep}
  \frac{\der \sigma\super{\gamma p}\,(W)}{\der t}
  = \frac{\der \sigma\super{\gamma p}\,(W_0)}{\der t}
    \left ( \frac {W}{W_0} \right ) ^{4 (\alpha - 1)},
\end{equation}
with 
$W_0 = 37\,\GeV$.
The parameter $\alpha$ is, for elastic $\rho^0$ production,
the value of the pomeron trajectory $\alpha = \alpha\sub\pom\,(t)$.

\subsection{Systematic Uncertainties}
%---------------------------------------

The following sources of systematic uncertainties were considered:
\begin{itemize}
  \item The $W$ and $t$ dependence of the $\rho^0$ production cross section.
  \item The $\MY$ dependence of the proton-dissociative $\rho^0$ production cross section
  \item The $\theta$ and $p\sub{t}$ dependence of the 
    FTT single track trigger efficiency
  %\item The CIP trigger efficiency
  \item The tagging efficiencies of the FMD and FTS systems.
\end{itemize}

For each of these systematic uncertainties,
the complete analysis was repeated, and the effect of the corresponding
variation on the individual cross sections per bin, the resulting
$W$ exponent for fixed $t$, and the pomeron trajectory was evaluated.
Thus, all correlations were fully taken into account.

For the $W$ and $t$ dependence of the $\rho^0$ production cross section,
the intercept of the pomeron trajectory was varied by 
$\pm 0.04$,
%(the difference between the values given by Donnachie and Landhoff
%\cite{Donnachie:1992ny} and $0.12$), 
its slope vas varied by 
$\pm 0.25\,\GeV^{-2}$
(corresponding to the extremes of either
twice the slope given by Landhoff and Jaroszkiewicz \cite{bib:dlslope}
and no shrinkage at all).
The $b$ slope parameter was varied by $\pm 10\,\%$ of its value,
i.e. by approximately $\pm 1\,\GeV^{-2}$ for elastic and
$\pm 0.6\,\GeV^{-2}$ for proton-dissociative events.

For the FTT trigger efficiency,
%the $p\sub t$ dependence of the trigger efficiency was varied 
%such that it changed by $-0.4\,\%$ ($+1.2\,\%$) at 
%$p\sub t = 0.2\,\GeV$ ($4\,\GeV$), and such that it changed by
%$-1.0\,\%$ ($-0.4\,\%$).
the $p\sub t$ and $\theta$ dependence was varied such that possible
systematic trends were covered.

For the forward detectors, the efficiency of the FTS and FMD
subdetectors was varied within their uncertainties, 
and proton-dissociative events were reweighted according to
$\MY^{\pm0.3}$ to account for the unknown mass spectrum of the
dissociating proton system.

In addition, the following uncertainties that affect the overall
normalisation were considered:
\begin{itemize}
  \item The uncertainty of the luminosity measurement: $1.5\,\%$;
  \item an uncertainty of $2\,\%$ for the cut on the value of
    $z$ vertex position;
  \item the uncertainty of the track reconstruction efficiency:
    $1.5\,\%$ per track, $3\,\%$ overall;
  \item the overall uncertainty of the track triggering efficiency:
    $1.5\,\%$ per track, $3\,\%$ overall;
  \item the uncertainty of the background contribution:
    $2\,\%$;
  \item an additional uncertainty from radiative corrections, for which
    no corrections were applied: $2\,\%$.
\end{itemize}
Added in quadrature, this results in an overall normalisation
uncertainty of $5.7\,\%$.


\section{Results}
%----------------

Fig.~\ref{fig:dsdt-incl} shows the 
differential photoproduction cross sections $\der \sigma/\der t$ 
for diffractive $\rho^0$ photoproduction
$\gamma p \to \rho^0 Y$
in the kinematic range
$(\MY^2+Q^2)/(W^2+Q^2) < 0.01$
in the different $t$ bins where it was measured.

Fig.~\ref{fig:dsdt-elas-combined-zeus}
shows the same cross section for the elastic process
$\gamma p \to \rho^0 p$,
in comparison with measurements from the ZEUS collaboration
\cite{Breitweg:1997ed, Breitweg:1999jy};
Fig.~\ref{fig:dsdt-elas-combined-9}
shows the same data,
which are additionally compared to data from the
Omega spectrometer 
\cite{Aston:1982hr} and  H1 \cite{Aid:1996bs} collaborations.
For these comparisons, the data points from the ZEUS and H1 
collaborations were
corrected to the closest $t$ bin of the present analysis
using the $t$ dependence as measured by the respective
experiment. 
For the Omega data, the parametrisation of the measured data
as given in \cite[Tab.~3]{Aston:1982hr} that is valid over the range 
$0.06 < |t| < 1\,\GeV^2$ was used.
For this data,
the cross sections were also adjusted (by applying a factor $0.952$) 
to account for the different
definition of the $\rho^0$ cross section that is employed by the 
Omega collaboration,
namely the convention of Spital and Yennie \cite{Spital:1974cx}.
The data of the present analysis
are in good agreement with these previous measurements.

Fig.~\ref{fig:dsdt-pdis} shows
the cross section for the proton-dissociative process
$\gamma p \to \rho^0 Y$,
again in the range $(\MY^2+Q^2)/(W^2+Q^2) < 0.01$.

Fig.~\ref{fig:Wslope-elas-sys}
summarizes the measured values of $\alpha\,(t)$
from a fit of the form given in Eq.~\ref{eq:wdep}.
A straight line fit fit to the observed $\alpha$ values yields
for the Pomeron trajectory
$$
  \alpha\sub\pom\,(t) = \OurAlphaZero + (\OurAlphaPrime)\,\GeV^{-2} \cdot t.
$$
This result is in excellent agreement with the
measurement from the ZEUS collaboration \cite{Breitweg:1999jy},
i.e.
$$
  \alpha\sub\pom\,(t) = 1.096 \pm 0.021
   + (0.125 \pm 0.038)\,\GeV^{-2} \cdot t,
$$
and supports the finding from the ZEUS collaboration
that in the space--like region $t<0$ the Pomeron 
trajectory, as observed in elastic $\rho^0$ photoproduction,
has a significantly smaller slope than the value of
$\alpha'\sub\pom = 0.25\,\GeV^{-2}$ derived
\cite{bib:dlslope} from other 
hadron scattering data.


Figs.~\ref{fig:Wslope-pdis} and \ref{fig:Wslope-incl}
show the effective $\alpha$ values of the $W$ dependency for the 
proton-dissociative and the inclusive diffractive process,
respectively.


%\section{Summary and Conclusions}
%--------------------------------

%\section{References}
%-------------------

\begin{lit}

\bibitem{bib:rho-fixedtarget}
%\bibitem{Crouch:1966} 
  H.~R.~Crouch {\it et al.} [Cambridge Bubble Chamber Group],
  %``Photoproduction on Hydrogen of $\rho^0$ Mesons between threshold  and $6~{\mathrm{BeV}}$,''\\
  Phys.\ Rev.\  {\bf 146} (1966) 994;
  %%CITATION = PRLRA,146,994;%%
\\
%\bibitem{Erbe:1966}
  R.~Erbe {\it et al.} [German Bubble Chamber Collaboration],
  %``Photoproduction of $\omega$, $\phi$, $\eta$ and $X^0$ mesons at energies up to $5.8~{\mathrm{GeV}}$,''\\
  Nuov.\ Cim.\  {\bf 46A} (1966) 795;
  %%CITATION = NUCIA,46A,795;%%
\\ 
%\bibitem{Blechschmidt:1967}
  H.~Blechschmidt {\it et al.},
  %``Photoproduction of $\rho^0$ mesons on hydrogen, carbon and aluminium with photons of known energy,''\\
  Nuov.\ Cim.\  {\bf 52A} (1967) 1348;
  %%CITATION = NUCIA,52A,1348;%%
\\ 
%\bibitem{Ballam:1971yd}
  J.~Ballam {\it et al.},
  %``Bubble chamber study of photoproduction by $2.8~{\mathrm{GeV}}$ and $4.7~{\mathrm{GeV}}$ polarized photons.  1. Cross-section determinations and production of $\rho^0$ and  $\Delta^{++}$ in the reaction $\gamma p \to p \pi^+ \pi^-$,''
  Phys.\ Rev.\ D {\bf 5} (1972) 54;
  % [SLAC-PUB-0941]
  %%CITATION = PHRVA,D5,545;%%
%
%\bibitem{Ballam:1972eq}
%  J.~Ballam {\it et al.},
  %``Vector meson production by polarized photons at $2.8~{\mathrm{GeV}}$, $4.7~{\mathrm{GeV}}$, and $9.3~{\mathrm{GeV}}$,''\\
  %Phys.\ Rev.\ D {\bf 7} (1973) 3150.
  {\it ibid.} {\bf 7} (1973) 3150;
  %%CITATION = PHRVA,D7,3150;%%
\\
%\bibitem{Struczinski:1975ik}
  W.~Struczinski {\it et al.}  [Aachen-Hamburg-Heidelberg-Munich
                  Collaboration],
 %``Study of photoproduction on hydrogen in a streamer chamber with tagged photons for $1.6~{\mathrm{GeV}} < E_\gamma < 6.3~{\mathrm{GeV}}$: Topological and reaction cross-sections,''\\
  Nucl.\ Phys.\ B {\bf 108} (1976) 45;
  %%CITATION = NUPHA,B108,45;%%
\\ 
%\bibitem{Joos:1976nm}
  P.~Joos {\it et al.},
  %``$\rho$ production by virtual photons,''\\
  Nucl.\ Phys.\ B {\bf 113} (1976) 53;
  %%CITATION = NUPHA,B113,53;%%
\\  
%\bibitem{Clifft:1976be}
  R.~W.~Clifft {\it et al.},
  %``Backward photoproduction of $\rho^0$ and $f$ between $2.8~{\mathrm{GeV}}$ and $4.8~{\mathrm{GeV}}$,''\\
  Phys.\ Lett.\ B {\bf 64} (1976) 213;
  %%CITATION = PHLTA,B64,213;%%
\\
%\bibitem{Anderson:1976ph}
  R.~L.~Anderson {\it et al.},
  %``Measurements of exclusive photoproduction processes at large values of $t$ and $u$ from $4~{\mathrm{GeV}}$ to $7.5~{\mathrm{GeV}}$,''
  Phys.\ Rev.\ D {\bf 14} (1976) 679;
  %%CITATION = PHRVA,D14,679;%%
\\
%\bibitem{Bartalucci:1977cp}
  S.~Bartalucci {\it et al.},
  %``Measurement of the photoproduction phases of the $\rho$, $\omega$, and $\phi$ mesons,''\\
  Nuovo Cim.\ A {\bf 44} (1978) 587;
  %%CITATION = NUCIA,A44,587;%%
\\
%\bibitem{Egloff:1979mg}
  R.~M.~Egloff {\it et al.},
  %``Measurements of elastic $\rho$ and $\phi$ meson photoproduction cross-sections on protons from $30~{\mathrm{GeV}}$ to $180~{\mathrm{GeV}}$,''\\
  Phys.\ Rev.\ Lett.\  {\bf 43}, 657 (1979);
  %%CITATION = PRLTA,43,657;%%
\\
%\bibitem{Aleksandrov:1980pt}
  Y.~A.~Aleksandrov {\it et al.},
  %``Photoproduction of $\rho^0$ mesons on hydrogen at gamma quanta energies of $15~{\mathrm{GeV}} - 30~{\mathrm{GeV}}$. (In Russian),''\\
  Yad.\ Fiz.\  {\bf 32} (1980) 651,
  Sov.\ J.\ Nucl.\ Phys.\ {\bf 32} (1980) 334;
  %%CITATION = YAFIA,32,651;%%
  %%CITATION = SJNCA,32,334;%%
\\
%\bibitem{Atkinson:1984fg}
  M.~Atkinson {\it et al.}  [Omega Photon Collaboration],
  %``Inclusive photoproduction of $\rho$ and $\omega$ in the photon energy range $20~{\mathrm{GeV}}$ to $70~{\mathrm{GeV}}$,''\\
  Nucl.\ Phys.\ B {\bf 245} (1984) 189;
  %%CITATION = NUPHA,B245,189;%%
\\
%\bibitem{Apsimon:1991nu}
  R.~J.~Apsimon {\it et al.}  [Omega Photon Collaboration],
  %``Comparison of photon and hadron induced production of $\rho^0$ mesons in the energy range of $65~{\mathrm{GeV}}$ to $175~{\mathrm{GeV}}$,''\\
  Z.\ Phys.\ C {\bf 53} (1992) 581;
  %%CITATION = ZEPYA,C53,581;%%
\\  
%\bibitem{Adams:1997bh}
  M.~R.~Adams {\it et al.}  [E665 Collaboration],
  %``Diffractive production of $\rho^0(770)$ mesons in muon proton interactions at $470~{\mathrm{GeV}}$,''\\
  Z.\ Phys.\ C {\bf 74} (1997) 237;
  %%CITATION = ZEPYA,C74,237;%%
\\  
%\bibitem{Wu:2005wf}
  C.~Wu {\it et al.},
  %``Photoproduction of $\rho^0$ mesons and $\Delta$-baryons in the reaction  $\gamma p \to p \pi^+ \pi^-$ at energies up to $\sqrt{s} = 2.6~{\mathrm{GeV}}$,''\\
  Eur.\ Phys.\ J.\ A {\bf 23} (2005) 317.
  %%CITATION = EPHJA,A23,317;%%
  
\bibitem{Aston:1982hr}
  D.~Aston {\it et al.},
  %``Photoproduction of $\rho^0$ and $\omega$ on hydrogen at photon energies of $20~{\mathrm{GeV}}$ to $70~{\mathrm{GeV}}$,''\\
  Nucl.\ Phys.\ B {\bf 209} (1982) 56.
  %%CITATION = NUPHA,B209,56;%%

\bibitem{Derrick:1995vq}
  M.~Derrick {\it et al.}  [ZEUS Collaboration],
  %``Measurement of elastic $\rho^0$ photoproduction at HERA,''\\
  Z.\ Phys.\ C {\bf 69} (1995) 39.
  %[arXiv:hep-ex/9507011].
  %%CITATION = HEP-EX 9507011;%%

\bibitem{Aid:1996bs}
  S.~Aid {\it et al.}  [H1 Collaboration],
  %``Elastic photoproduction of $\rho^0$ mesons at HERA,''\\
  Nucl.\ Phys.\ B {\bf 463} (1996) 3.
  %[arXiv:hep-ex/9601004].
  %%CITATION = HEP-EX 9601004;%%

\bibitem{Derrick:1996vw}
  M.~Derrick {\it et al.}  [ZEUS Collaboration],
  %``The ZEUS leading proton spectrometer and its use in the measurement of elastic $\rho^0$ photoproduction at HERA,''\\
  Z.\ Phys.\ C {\bf 73} (1997) 253.
  %[arXiv:hep-ex/9609003].
  %%CITATION = HEP-EX 9609003;%%

\bibitem{Breitweg:1997ed}
  J.~Breitweg {\it et al.}  [ZEUS Collaboration],
  %``Elastic and proton-dissociative $\rho^0$ photoproduction at HERA,''\\
  Eur.\ Phys.\ J.\ C {\bf 2} (1998) 247.
  %[arXiv:hep-ex/9712020].
  %%CITATION = HEP-EX 9712020;%%

\bibitem{Breitweg:1999jy}
  J.~Breitweg {\it et al.}  [ZEUS Collaboration],
  %``Measurement of diffractive photoproduction of vector mesons at large momentum transfer at HERA,''\\
  Eur.\ Phys.\ J.\ C {\bf 14} (2000) 213.
  %[arXiv:hep-ex/9910038].
  %%CITATION = HEP-EX 9910038;%%


\bibitem{bib:vdm}
%\bibitem{Sakurai:1960ju}
  J.~J.~Sakurai,
  %``Theory of strong interactions,''\\
  Annals Phys.\  {\bf 11} (1960) 1;
  %%CITATION = APNYA,11,1;%%
%
%\bibitem{Sakurai:1969ss}
  %J.~J.~Sakurai,
  {\it idem},
  %``Vector meson dominance and high-energy electron proton inelastic scattering,''\\
  Phys.\ Rev.\ Lett.\  {\bf 22} (1969) 981;
  %%CITATION = PRLTA,22,981;%%
\\
%\bibitem{Gell-Mann:1961tg}
  M.~Gell-Mann and F.~Zachariasen,
  %``Form-factors and vector mesons,''\\
  Phys.\ Rev.\  {\bf 124} (1961) 953;
  %%CITATION = PHRVA,124,953;%%
\\
%\bibitem{Bauer:1977iq}
  T.~H.~Bauer, R.~D.~Spital, D.~R.~Yennie and F.~M.~Pipkin,
  %``The hadronic properties of the photon in high-energy interactions,''\\
  Rev.\ Mod.\ Phys.\  {\bf 50} (1978) 261
  [Erratum-ibid.\  {\bf 51} (1979) 407].
  %%CITATION = RMPHA,50,261;%%
  
\bibitem{bib:regge}
%\bibitem{Regge:1959mz}
  T.~Regge,
  %``Introduction to complex orbital momenta,''\\
  Nuovo Cim.\  {\bf 14} (1959) 951;
  %%CITATION = NUCIA,14,951;%%
%
%\bibitem{Regge:1960zc}
%  T.~Regge,
  %``Bound states, shadow states and Mandelstam representation,''\\
  %Nuovo Cim.\  {\bf 18}, 947.
  {\it ibid.} {\bf 18} (1960) 947.
  %%CITATION = NUCIA,18,947;%%


\bibitem{Donnachie:1992ny}
  A.~Donnachie and P.~V.~Landshoff,
  %``Total cross-sections,''\\
  Phys.\ Lett.\ B {\bf 296} (1992) 227.
  %[arXiv:hep-ph/9209205].
  %%CITATION = HEP-PH 9209205;%%
  
\bibitem{Cudell:1996sh}
  J.~R.~Cudell, K.~Kang and S.~K.~Kim,
  %``Simple pole fits to $pp$ and $\bar p p$ total cross sections and real parts,''\\
  Phys.\ Lett.\ B {\bf 395} (1997) 311.
  %[arXiv:hep-ph/9601336].
  %%CITATION = HEP-PH 9601336;%%

\bibitem{bib:dlslope}
%\bibitem{Jaroszkiewicz:1974ep}
  G.~A.~Jaroszkiewicz and P.~V.~Landshoff,
  %``Model For Diffraction Excitation,''\\
  Phys.\ Rev.\ D {\bf 10}, 170 (1974);
  %%CITATION = PHRVA,D10,170;%%
\\
%\bibitem{Landshoff:1989ku}
  P.~V.~Landshoff,
  %``Diffractive Processes,''\\
  Nucl.\ Phys.\ Proc.\ Suppl.\  {\bf 12} (1990) 397.
  %%CITATION = NUPHZ,12,397;%%


\bibitem{bib:h1detector}
%\bibitem{Abt:1996hi}
  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{Abt:1996xv}
%  I.~Abt {\it et al.}  [H1 Collaboration],
  %``The Tracking, calorimeter and muon detectors of the H1 experiment at HERA,''
%  Nucl.\ Instrum.\ Meth.\ A {\bf 386} (1997) 348.
  {\it ibid.} 348;
  %%CITATION = NUIMA,A386,348;%%
\\
%\bibitem{Appuhn:1996na}
  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{bib:rondiss}
  R.~Weber,
  ``Diffractive $\rho^0$-photoproduction at H1,''
  PhD thesis ETH Zurich 2006, in preparation.
    
\bibitem{bib:ftt}
%\bibitem{Baird:2001xc}
  A.~Baird {\it et al.},
  %``A fast high resolution track trigger for the H1 experiment,''
  IEEE Trans.\ Nucl.\ Sci.\  {\bf 48} (2001) 1276
  [arXiv:hep-ex/0104010];
  %%CITATION = HEP-EX 0104010;%%
  \\
%\bibitem{Baird:2001sz}
  A.~Baird {\it et al.}  [H1 Collaboration],
  %``A fast track trigger for the H1 Collaboration,''
  Nucl.\ Instrum.\ Meth.\ A {\bf 461} (2001) 461;
  %%CITATION = NUIMA,A461,461;%%
\\
%\bibitem{Schoning:2004jp}
  A.~Sch\"oning  [H1 Collaboration],
  %``A fast track trigger for the H1 collaboration,''
  Nucl.\ Instrum.\ Meth.\ A {\bf 518} (2004) 542;
  %%CITATION = NUIMA,A518,542;%%  
  \\
  N. Berger {\it et al.},
  %(N. Berger, A. Baird, R. Baldinger, S. Baumgartner, M.-O. Boenig, D.P.
  %Brown, D. Dodt, Y.H. Fleming, A.W. Jung,
  %M. Kolander, S. Kolya, K. Krüger, D. Meer, D. Mercer, D. Müller, J. Müller,
  %J. Naumann, P.R. Newman,
  %D.P.C. Sankey, A. Schöning, H.-C. Schultz-Coulon, R. Weber, C. Wissing and
  %W. Yan)
  %
  %First Results from the First Level of the H1 Fast Track Trigger
  Nucl.\ Sc.\ Symp.\ Conf.\ Rec., 2004 IEEE,
  {\bf 3} (2004) 1976.
  % 16-22 Oct. 2004
  
\bibitem{List:1998jz}
  B.~List and A.~Mastroberardino,
  ``DIFFVM: A Monte Carlo generator for diffractive processes in e p
  scattering,''
%\href{http://www.slac.stanford.edu/spires/find/hep/www?irn=4270568}{SPIRES entry}
{\it Prepared for Workshop on Monte Carlo Generators for HERA Physics 
(Plenary Starting Meeting), Hamburg, Germany, 27-30 Apr 1998}
  DESY-PROC-1999-02 (1999) 396.

\bibitem{bib:epa}
%\bibitem{vonWeizsacker:1934sx}
   C.~F.~von~Weizs\"acker,
  %``Ausstrahlung bei St\"o\"sen sehr schneller Elektronen,''\\
  Z.\ Phys.\ {\bf 88} (1934) 612;
  %%CITATION = ZEPYA,88,612;%%
\\
%\bibitem{Williams:1934ad}
  E.~J.~Williams,
  %``Nature of the high energy particles of penetrating radiation and status of ionization and radiation formulae,''\\
  Phys.\ Rev.\  {\bf 45} (1934) 729;
  %%CITATION = PHRVA,45,729;%%
\\
%\bibitem{Budnev:1974de}
  V.~M.~Budnev, I.~F.~Ginzburg, G.~V.~Meledin and V.~G.~Serbo,
  %``The two photon particle production mechanism. Physical problems. Applications. Equivalent photon approximation,''\\
  Phys.\ Rept.\  {\bf 15} (1974) 181.
  %%CITATION = PRPLC,15,181;%%
  
\bibitem{Goulianos:1982vk}
  K.~Goulianos,
  %``Diffractive interactions of hadrons at high energies,''\\
  Phys.\ Rept.\  {\bf 101} (1983) 169.
  %%CITATION = PRPLC,101,169;%%
   
\bibitem{Eidelman:2004wy}
  S.~Eidelman {\it et al.}  [Particle Data Group],
  %``Review of particle physics,''
  Phys.\ Lett.\ B {\bf 592} (2004) 1.
  %%CITATION = PHLTA,B592,1;%%
  
\bibitem{bib:jetset}
%\bibitem{Sjostrand:1985ys}
  T.~Sj\"ostrand,
  %``The Lund Monte Carlo For Jet Fragmentation And E+ E- Physics: Jetset
  %Version 6.2,''
  Comput.\ Phys.\ Commun.\  {\bf 39} (1986) 347;
  %%CITATION = CPHCB,39,347;%%
%    
%\bibitem{Sjostrand:1995iq}
  %T.~Sj\"ostrand,
  {\it idem},
  %``PYTHIA 5.7 and JETSET 7.4: Physics and manual,''
  arXiv:hep-ph/9508391;
  %%CITATION = HEP-PH 9508391;%% 
\\
%\bibitem{Sjostrand:1986hx}
  T.~Sj\"ostrand and M.~Bengtsson,
  %``The Lund Monte Carlo For Jet Fragmentation And E+ E- Physics: Jetset
  %Version 6.3: An Update,''
  Comput.\ Phys.\ Commun.\  {\bf 43} (1987) 367.
  %%CITATION = CPHCB,43,367;%%
  
\bibitem{Ross:1965qa}
  M.~Ross and L.~Stodolsky,
  %``Photon dissociation model for vector meson photoproduction,''\\
  Phys.\ Rev.\  {\bf 149} (1966) 1172.
  %%CITATION = PHRVA,149,1172;%%

\bibitem{Soding:1965nh}
  P.~S\"oding,
  %``On the apparent shift of the $\rho$ meson mass in photoproduction,''\\
  Phys.\ Lett.\  {\bf 19} (1966) 702.
  %%CITATION = PHLTA,19,702;%%
  % \\ \link{pl-19-702.pdf}
  % \\ \spires{J}{PHLTA,19,702}

\bibitem{Jackson:1964zd}
  J.~D.~Jackson,
  %``Remarks on the phenomenological analysis of resonances,''\\
  Nuovo Cim.\  {\bf 34} (1964) 1644.
  %%CITATION = NUCIA,34,1644;%%

\bibitem{Spital:1974cx}
  R.~Spital and D.~R.~Yennie,
  %``$\rho^0$ shape in photoproduction,''\\
  Phys.\ Rev.\ D {\bf 9}, 126 (1974).
  %%CITATION = PHRVA,D9,126;%%

\bibitem{Abele:2001pv}
  A.~Abele {\it et al.}  [CRYSTAL BARREL Collaboration],
  %``$4\pi$ decays of scalar and vector mesons,''\\
  Eur.\ Phys.\ J.\ C {\bf 21} (2001) 261.
  %%CITATION = EPHJA,C21,261;%%

\bibitem{Clegg:1993mt}
  A.~B.~Clegg and A.~Donnachie,
  %``Higher vector meson states produced in electron-positron annihilation,''\\
  Z.\ Phys.\ C {\bf 62} (1994) 455.
  %%CITATION = ZEPYA,C62,455;%%

  
  
\end{lit}

\clearpage
%======================================================================

\begin{figure}[ht]
\center
\setlength{\unitlength}{1cm}
\begin{picture}(12.0,15.0)
\put(0.,0.0)
{\epsfig{file=H1prelim-06-011.fig1.eps,height=15.0cm}}
\end{picture}
\caption{%RhoMassData
  The dipion mass spectrum of the selected candidate events.
  The spectrum is fitted with a skewed relativistic
  Breit-Wigner function according to Eq.~(\ref{eq:ross-stodolsky})
  to guide the eye.
}
\label{fig:RhoMassData}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{figure}[ht]
\center
\setlength{\unitlength}{1cm}
\begin{picture}(12.0,15.0)
\put(0.,0.0)
{\epsfig{file=H1prelim-06-011.fig2.eps,height=15.0cm}}
\end{picture}
\caption{%rhomass-ross-stodolsky
  The dipion mass spectrum of the selected candidate events,
  corrected for acceptance, reconstruction and trigger
  efficiencies,
  with a fit according to Eq.~\ref{eq:ross-stodolsky}.
  The dashed lines show the contributions from
  the background and the unskewed relativistic
  Breit--Wigner distribution.
}
\label{fig:rhomass-ross-stodolsky}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{figure}[ht]
\center
\setlength{\unitlength}{1cm}
\begin{picture}(12.0,15.0)
\put(0.,0.0)
{\epsfig{file=H1prelim-06-011.fig3.eps,height=15.0cm}}
\end{picture}
\caption{%rhomass-soeding  
  The dipion mass spectrum of the selected candidate events,
  corrected for acceptance, reconstruction and trigger
  efficiencies,
  with a fit according to Eq.~\ref{eq:soeding}.
  The dashed lines show the contributions from
  the interference term and the unskewed relativistic
  Breit--Wigner distribution. The background term is set to 0.
}
\label{fig:rhomass-soeding}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{figure}[ht]
\center
\setlength{\unitlength}{1cm}
\begin{picture}(12.0,15.0)
\put(0.,0.0)
{\epsfig{file=H1prelim-06-011.fig4.eps,height=15.0cm}}
\end{picture}
\caption{%W  
  The $W$ spectrum of the selected events, compared to the Monte Carlo
  prediction.
}
\label{fig:W}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{figure}[ht]
\center
\setlength{\unitlength}{1cm}
\begin{picture}(12.0,15.0)
\put(0.,0.0)
{\epsfig{file=H1prelim-06-011.fig5.eps,height=15.0cm}}
\end{picture}
\caption{%t  
  The $t$ spectrum of the selected events, compared to the Monte Carlo
  prediction.
}
\label{fig:t}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{figure}[ht]
\center
\setlength{\unitlength}{1cm}
\begin{picture}(15.0,15.0)
\put(0.,0.0)
{\epsfig{file=H1prelim-06-011.fig6.eps,height=15.0cm}}
\end{picture}
\caption{%dsdt-elas-combined-9
The measured $\gamma p$ cross sections for
elastic  $\rho^0$ photoproduction
$\gamma p \to \rho^0 p$,
for different values of $t$.
The measurent is compared to results from other experiments
\cite{Aston:1982hr,Aid:1996bs,Derrick:1996vw, Breitweg:1997ed, Breitweg:1999jy}.
The lines show the result of a fit 
$\der \sigma / \der t  = a (W/W_0)^{4[\alpha(t)-1]}$ 
to the data of the current analysis, with
$W_0 = 37\,\GeV$; the fit result is extrapolated 
for better comparison with the data from other experiments.
}
\label{fig:dsdt-elas-combined-9}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{figure}[ht]
\center
\setlength{\unitlength}{1cm}
\begin{picture}(15.0,15.0)
\put(0.,0.0)
{\epsfig{file=H1prelim-06-011.fig7.eps,height=15.0cm}}
\end{picture}
\caption{%dsdt-elas-combined-zeus
The measured $\gamma p$ cross sections for
elastic  $\rho^0$ photoproduction
$\gamma p \to \rho^0 p$,
for different values of $t$.
The measurent is compared to results from other experiments
\cite{Derrick:1996vw, Breitweg:1997ed, Breitweg:1999jy}.
}
\label{fig:dsdt-elas-combined-zeus}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{figure}[ht]
\center
\setlength{\unitlength}{1cm}
\begin{picture}(15.0,15.0)
\put(0.,0.0)
{\epsfig{file=H1prelim-06-011.fig8.eps,height=15.0cm}}
\end{picture}
\caption{%dsdt-pdis
The measured $\gamma p$ cross sections for
proton-dissociative  $\rho^0$ photoproduction
$\gamma p \to \rho^0 Y$, 
in the range $(\MY^2+Q^2)/(W^2+Q^2) < 0.01$, 
for different values of $t$.
}
\label{fig:dsdt-pdis}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{figure}[ht]
\center
\setlength{\unitlength}{1cm}
\begin{picture}(15.0,15.0)
\put(0.,0.0)
{\epsfig{file=H1prelim-06-011.fig9.eps,height=15.0cm}}
\end{picture}
\caption{%dsdt-incl
The measured $\gamma p$ cross sections for
diffractive  $\rho^0$ photoproduction
$\gamma p \to \rho^0 Y$, 
in the range $(\MY^2+Q^2)/(W^2+Q^2) < 0.01$, 
for different values of $t$.
}
\label{fig:dsdt-incl}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{figure}[ht]
\center
\setlength{\unitlength}{1cm}
\begin{picture}(14.0,15.0)
\put(0.,0.0)
{\epsfig{file=H1prelim-06-011.fig10.eps,height=15.0cm}}
\end{picture}
\caption{%Wslope-elas-sys
The measured values of the $\alpha$ parameter
for elastic $\rho^0$ photoproduction,
together with a fit to the Pomeron trajectory
of the form
$\alpha\sub{\pom}\,(t) = \alpha\sub{\pom,0} + \alpha'\sub{\pom}\cdot t$,
with 
$\alpha\sub{\pom,0} = \OurAlphaZero$
and
$\alpha'\sub{\pom} = (\OurAlphaPrime)\,\GeV^{-2}$.
The result from the ZEUS collaboration \cite{Breitweg:1999jy} is also shown,
as well as the parametrisation from Donnachie and Landshoff 
\cite{Donnachie:1992ny, bib:dlslope}
}
\label{fig:Wslope-elas-sys}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{figure}[ht]
\center
\setlength{\unitlength}{1cm}
\begin{picture}(14.0,15.0)
\put(0.,0.0)
{\epsfig{file=H1prelim-06-011.fig11.eps,height=15.0cm}}
\end{picture}
\caption{%Wslope-pdis
The measured values of the $\alpha$ parameter of the 
$W$ dependency
for proton-dissociative $\rho^0$ photoproduction.
}
\label{fig:Wslope-pdis}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{figure}[ht]
\center
\setlength{\unitlength}{1cm}
\begin{picture}(14.0,15.0)
\put(0.,0.0)
{\epsfig{file=H1prelim-06-011.fig12.eps,height=15.0cm}}
\end{picture}
\caption{%Wslope-incl
The measured values of the  $\alpha$ parameter
of the $W$ dependency
for diffractive, i.e. elastic and proton-dissociative,
 $\rho^0$ photoproduction.
}
\label{fig:Wslope-incl}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\end{document}
