%================================================================
% LaTeX file with preferred layout for the contributed papers to
% the ICHEP Conference 98 in Vancouver
% process with:  latex hep98.tex
%                dvips -D600 hep98
%================================================================
\documentclass[12pt]{article}
\usepackage{epsfig}
\usepackage{amsmath}
\usepackage{hhline}
\usepackage{amssymb}
\usepackage{times}
\usepackage{rotating}

\renewcommand{\topfraction}{1.0}
\renewcommand{\bottomfraction}{1.0}
\renewcommand{\textfraction}{0.0}
\newlength{\dinwidth}
\newlength{\dinmargin}
\setlength{\dinwidth}{21.0cm}
\textheight23.5cm \textwidth16.0cm
\setlength{\dinmargin}{\dinwidth}
\setlength{\unitlength}{1mm}
\addtolength{\dinmargin}{-\textwidth}
\setlength{\dinmargin}{0.5\dinmargin}
\oddsidemargin -1.0in
\addtolength{\oddsidemargin}{\dinmargin}
\setlength{\evensidemargin}{\oddsidemargin}
\setlength{\marginparwidth}{0.9\dinmargin}
\marginparsep 8pt \marginparpush 5pt
\topmargin -42pt
\headheight 12pt
\headsep 30pt \footskip 24pt
\parskip 3mm plus 2mm minus 2mm

%===============================title page=============================

% Some useful tex commands
%
%\newcommand{\GeV}{\rm GeV}
%\newcommand{\TeV}{\rm TeV}
%\newcommand{\pb}{\rm pb}
%\newcommand{\cm}{\rm cm}
%\newcommand{\hdick}{\noalign{\hrule height1.4pt}}

\newcommand{\xpom}{x_{\mbox{\scriptsize{$I\!P$}}}}
\newcommand{\kev}{\,\mbox{keV}}
\newcommand{\mev}{\,\mbox{MeV}}
\newcommand{\gev}{\,\mbox{GeV}}
\newcommand{\mub}{\,\mu\mbox{b}}
\newcommand{\mrad}{\,\mbox{mrad}}
\newcommand{\meter}{\,\mbox{m}}
\newcommand{\mum}{\mu\mbox{m}}
\newcommand{\cm}{\,\mbox{cm}}
\newcommand{\mm}{\,\mbox{mm}}
\newcommand{\mb}{\,\mbox{mb}}
\newcommand{\nb}{\,\mbox{nb}}
\newcommand{\der}{\mbox{d}}
\newcommand{\pttrk}{p_T^{\mathrm{trk}}}
\newcommand{\zvtx}{z_{\mathrm{Vtx}}}
\newcommand{\invpb}{\,\mbox{pb}^{-1}}
\newcommand{\ftwolpthree}{F_2^{LP(3)}}
\newcommand{\ftwovdm}{F_2^{VDM}}
\newcommand{\ftwopart}{F_2^{part}}
\newcommand{\cvm}{C_{VM}}
\newcommand{\qsqvm}{Q^2_{VM}}
\newcommand{\cvmlp}{C_{VM}^{LP}}
\newcommand{\pom}{I\!\!P}
\newcommand{\reg}{I\!\!R}

\begin{document}

\pagestyle{empty}
\begin{titlepage}

\noindent
Submitted to the 30th International Conference on 
High-Energy Physics ICHEP2000, \\ 
Osaka, Japan, July 2000


\vspace*{3cm}

\begin{center}
  \Large
  {\bf Measurement of the Photoproduction Cross Section with a 
Leading Proton 
    at HERA}

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

\begin{abstract}

\noindent
\end{abstract}
The total cross-section for the semi-inclusive photoproduction process with a 
leading proton in the final state has been measured at centre-of-mass energies
between 91 and 231~GeV. The measured cross-sections refer to the kinematic 
range with    
transverse momenta of the scattered proton restricted to
$p_T \leq 0.2 \gev$ and $0.66 \leq z \leq 0.90$,
where $z = E_p'/E_p$ is the 
scattered proton energy normalized to the beam energy. That is a region, 
in which diffractive processes are suppressed
relative to processes with pion and reggeon exchange. The cross-section 
is measured to be about $7.7 \mub$ per unit $z$. It is
found to be independent of the photon-proton centre-of-mass energy and the
scattered proton momentum. The measured cross-sections are compared with 
semi-inclusive 
deep inelastic scattering data with a leading proton in the final state, which
is subject to the same kinematic restrictions as in the case of photoproduction. 
This allows to
study the transition from finite photon virtuality $Q^2$ to $Q^2=0$
in a model in which the proton structure function at low $Q^2$ is split into
a nonperturbative vector meson dominated contribution 
and a partonic contribution. The VDM contribution determined in the framework
of this model in case of the semi-inclusive
process is only $23\%$ of the amount found in the inclusive reaction.


\vfill
\begin{flushleft}
  {\bf Abstract: 967 } \\
  {\bf Parallel session: PA-02} \\
  {\bf Plenary talk: PL-12 } 
\end{flushleft}

\end{titlepage}

\pagestyle{plain}
% ====================
\section{Introduction}
% ====================
The HERA electron proton collider allows to reconsider an old problem
in deep inelastic lepton scattering (DIS) concerning the transition of
photoproduction with virtual photons of virtuality $Q^2$ to photoproduction
with real photons at $Q^2 = 0$. Only now sufficient data become available
at the same centre-of-mass energy for both reactions, which are recorded
within the same experimental set-up so that the bulk of systematic errors
are identical and will have no impact on the analysis. Such type of
investigations have been performed by the HERA experiment 
H1~\cite{h1f2fits} 
for inclusive DIS and photoproduction final states. The present
paper deals with the transition in semi-inclusive reactions with a 
forward scattered proton in the final state.

The transition from finite $Q^2$ reactions to the limiting case of $Q^2=0$ is
conveniently described by introducing a total virtual photon proton
cross-section\footnote{The exact formula used is $
\sigma^{tot}_{\gamma^{*} p}(W,Q^2) = 
(\frac{4\pi^2 \alpha}{Q^4})(4M^2_p x^2_p+Q^2)/
(1-x) \cdot F_2(x,Q^2)$, $M_p$ being the proton mass.}

\begin{equation}
\label{formsigmatotincl}
\sigma^{tot}_{\gamma^{(*)} p}(W,Q^2) \approx 
\frac{4\pi^2 \alpha}{Q^2} F_2(x,Q^2)
\end{equation}

where $x = Q^2/(2p\cdot q)$ is the Bjorken scaling variable with $p$ and
$q$ being the four-momenta of the incident proton and photon respectively.
$\alpha$ is the fine-structure constant and $F_2$
the proton structure function. $W$ denotes the total $\gamma^{\ast} p$
centre-of-mass energy, which for the present application of small values of the 
scaling variable $x$ can be approximated by $W^2 \approx Q^2/x$.

The transition  $Q^2 \to 0$ has to obey boundary
conditions imposed by conservation laws and by measurements.
Due to the conservation of the 
electromagnetic current, $F_2$ has to vanish as $Q^2$ tends to 0. The other
boundary condition which has to be obeyed is based on an experimental
observation. Measurements in DIS reactions show that 
the total $\gamma^{*} p$ cross-section at low $x$ 
follows a power law behaviour~\cite{h1f2fits}
$$
\sigma^{tot}_{\gamma^{(*)} p} \propto (W^2)^{\lambda}.
$$
 The values of
the parameter $\lambda$ increase with increasing $Q^2$ and range between
0.15 at \linebreak
$Q^2= 0.9 \gev^2$ and 0.2 at $Q^2= 3\gev^2$. In the case of
photoproduction $\lambda$ approaches the so called soft pomeron limit of
0.08~\cite{dola}.

It is interesting to compare this transition for inclusive processes with
semi-inclusive DIS processes where the proton remains intact leaving the
interaction with a fractional energy\footnote{In some publications, 
$\xpom=1-z$ is used instead.} \linebreak
$z = E_p'/E_p$, where $E_p'$ and $E_p$ 
are the energies of the incident and scattered protons respectively.
The process is sketched in Figure~\ref{picfeyndiag}.
In a previous publication~\cite{lbproduction}
it has been demonstrated that in the kinematic range of this experiment
$0.66\leq z \leq 0.90$ the semi-inclusive reaction is dominated by
processes where the virtual photon interacts with colorless constituents of
the proton. In Regge terminology, this corresponds to a hadronic exchange 
object ${\cal R}$, a reggeon or a virtual pion in the kinematic range of $z$
under study, colliding with the photon at fixed energy $(1-z) E_p'$.
The variables describing this process are  $\beta=x/(1-z)$ 
instead of $x$ and the invariant mass of the $\gamma{\cal R}$ 
system $M_X=W\sqrt{1-z}$. Note that for each value of $z$,
the choice of $W$ fixes $M_X$. 

\begin{figure}[h]
\begin{center}
  \setlength{\unitlength}{1mm}
  \begin{picture}(200,75)(-15,-10)
  \put(5,0){
    \psfig{file=/afs/desy.de/user/m/mahlke/private/diss/fdg/gammap1.ps,%
       width=6cm}}
        \put(70,30){\mbox{$\left. \rule{0mm}{0.7cm}\right\}$}}
        \put(77,30){\mbox{\small $M_X$}}
        \put(20,28){\mbox{\begin{rotate}{-30}
                  $\left\{ \rule{0mm}{1.9cm}\right.$\end{rotate}}}
        \put(14,30){\mbox{\small $W$}}
        \qbezier[20](23,56)(29,63)(37,57)
        \put(28,63){\mbox{\small $Q^2$}}
        \qbezier[20](24,7)(29,-1)(37,5)
        \put(29,-3){\mbox{\small $t$}}
        \put(40,15){\mbox{ $(\pom, \reg, \pi)$}}
        \put(48,20){\mbox{$\!{\cal R}$}}
        \put(70,60){\mbox{\small$y=1-E_e'/E_e$}}
        \put(70,3){\mbox{\small$z=E_p'/E_p=1-\xpom$}}
  \end{picture}
  \begin{minipage}{13cm}{
  \caption[]{
  \sl The process $e p \to e'p'X$. At the proton vertex, 
  $z=E_p'/E_p$ is the fractional momentum kept by the proton.
\label{picfeyndiag}
  } 
}
\end{minipage}
\end{center}
\end{figure}

A comparison of semi-inclusive data with a leading proton in the final state
at finite values of $Q^2$ with data at $Q^2=0$ thus investigates the limit
$Q^2 \to 0$ for $\gamma\reg$ or $\gamma \pi$ reactions.
In order to avoid  
uncertainties due to flux factors describing the probability to hit ${\cal R}$
(${\cal R}=\reg,\pi$)
in a proton, the measurement need not be converted into 
a $\gamma\reg$ or $\gamma \pi$ cross section. 
Instead the flux factors can be considered as a
common normalisation parameter for photoproduction and DIS.

In the present paper measurements of the semi-inclusive 
photoproduction process \linebreak  $\gamma p \to p'X$
will be described, which are then combined with the corresponding DIS data 
from the same experiment published in~\cite{lbproduction}, where the proton 
is subject to the same kinematic restrictions in both cases.  

% =================
\section{Apparatus}
% =================
The H1 detector is described in detail elsewhere~\cite{theh1detathera}.
In this paper we make use of 
the H1 coordinate system, originating at the nominal interaction
point. It is defined as follows: the direction
of the incoming proton beam is taken as the $+z$ axis,
the $x$ axis points towards the centre of the storage rings, and the
$y$ axis points upwards w.r.t.~the plane of the HERA electron ring.

The main H1 components used for this analysis are:
\begin{itemize}

\item The {\bf central tracking chambres} CJC1 and CJC2 and the
{\bf central proportional chambres} CIP and COP for trigger purposes. They are 
of cylindrical shape, enclosing the beam pipe along the beam line.
Charged particle momenta are calculated from
the curvature introduced by the 1.2~Tesla field of the H1 solenoid. 
The momentum resolution is 
$\sigma_p/p^2\approx  10^{-2}\gev^{-2}$ at momenta relevant for this
analysis.
The track triggers are efficient for tracks with
a transverse momentum $\pttrk> 450\mev$. 

\item The {\bf luminosity system}, consisting of electron 
calorimeters and a photon detector, is
located downstream of the H1 detector. It is capable of
measuring the luminosity online with a precision
of better than $1\%$ by selecting events for the
Bethe-Heitler process \linebreak $ep\to e'p'\gamma$.

In 1996, two electron calorimeters,
ETAG33 and ETAG44, were operated at $z=-33\meter$
and $z=-44\meter$ from the nominal H1 interaction point.
The system is 
used to identify photoproduction events where a quasi-real
photon is emitted by the incident lepton and the
scattered lepton being observed in one
of the small angle electron calorimeters. Due to the
acceptance of the electron detectors, the virtuality
of the photon is restricted to $Q^2<0.01\gev^2$ for tagged
photoproduction events. 

\item The {\bf Forward Proton Spectrometer 
(FPS)}~\cite{fpsnimpaper} is used to
measure the trajectory of protons which are scattered
under small angles at the interaction point.
In 1996, two stations were operated at distances of $81\meter$ 
and $90\meter$ from the H1 interaction region.
The FPS stations consist of  detectors  housed in
movable plunger vessels, so called Roman Pots, which are moved into the 
HERA machine
vacuum when stable beam conditions are \linebreak established. 
A track measurement is performed
by four fiber hodoscopes per station.
The hodoscopes are sandwiched between scintillator
planes for triggering purposes.
 
Scattered proton momenta are reconstructed from the deviation of their
trajectories at the position of the Roman Pots from the actual stored beam
orbit. Thus in 
a calibration procedure, the
position  of the circulating beam with
respect to the nominal beam  is determined for
each proton fill. The track segments of each station 
are combined into a local vector tangential to a proton trajectory 
at $85\meter$ from the collision point. Making use of 
the known properties of the HERA beam optics, the scattering angle
at the interaction point and the energy of the scattered 
proton can be inferred from the parameters of the measured local 
trajectory vector. This procedure results in an error on the
energy measurement of less than $8\gev$.
The detector size, the distance of the detector stations from the interaction
point, and the proton beam size restrict the
acceptance in the proton's transverse momentum to
$p_T \leq 0.2 \gev$ and in the two projections  $x$ and $y$ to values
below $|\theta_x|,|\theta_y|<0.5\mrad$. 
Proton energies $ 500 \gev \leq  E_p' \leq 780
\gev$ are accepted by the spectrometer. For the present analysis the 
energy range was further restricted to $0.66 \leq z=E_p'/E_p \leq 0.90$.
\end{itemize}
% ======================
\section{Data Selection}
% ======================
The analysis is based on a data set from the 1996 running 
period, where HERA collided $820\gev$ protons with
$27.5\gev$ positrons.
Data were analysed where the FPS was in a stable 
position close to the circulating beam and all relevant components of 
the H1 detector
were fully operational. The corresponding integrated
luminosity amounts to $3.3\invpb$.

In order to study the reaction $\gamma p \to p'X$, events were
selected with a reconstructed track through both FPS stations at 
$z=81\meter$ and $z=90\meter$,
a positron candidate in one
of the small-angle electron detectors and at least one reconstructed track with 
$\pttrk>500\mev$ in the central H1 detector.

The distributions of emission angle projections
as a function of the energy of the scattered 
proton are shown in Figure~\ref{figdatafidvol}.

 
For the analysis, the data are grouped into five $40\gev$ wide bins
in the energy range
$540\gev \leq E_p' \leq 740\gev$. The measured cross-sections correspond 
to an integral over $p_T<200\mev$
in the scattered proton's transverse
momentum. In order to be accepted for the analysis,
the kinematic quantities of the scattered proton were required to lie within a 
fiducial volume defined as limits on the projected angles as a function of 
the proton's energy. This results in
maximum values for the scattering angles of 
{$|\theta_x|,|\theta_y|<0.3\mrad$}. 

\begin{figure}[h]
\begin{center}
  \setlength{\unitlength}{1mm}
  \begin{picture}(200,65)(-10,0)
  \put(-10,0){
   \put(0,-3){\epsfig{file=H1prelim-00-113.fig1.eps,width=17cm}}
   \put(8,50){\mbox{\large \boldmath $\Theta_x$[mrad]}}
   \put(85,50){\mbox{\large \boldmath $\Theta_y$[mrad]}}
   \put(60,0){\mbox{\large \boldmath $E_p'$[GeV]}}
   \put(137,0){\mbox{\large \boldmath $E_p'$[GeV]}}
  }
  \end{picture}
  \begin{minipage}{13cm}{
  \caption[]{
  \sl The projection of the proton's polar scattering angle onto the
  $x$ and $y$ axis in dependence on the proton's energy for the selected
  data sample. The borders of the fiducial volume explained in the text
  are indicated.
   \small
\label{figdatafidvol}
  } 
}
\end{minipage}
\end{center}
\end{figure}

The positron candidates in ETAG33/ETAG44 were accepted 
if there was an energy deposition between 
$8.3\gev$ and $19.3\gev$ in ETAG33 or larger than
$20\gev$ in ETAG44 and if a cut on the energy deposition in the
photon detector rejecting Bethe-Heitler overlay
events was fulfilled.
The data sample was binned according to the value of
the inelasticy $y=1-E_e'/E_e$, where $E_e'$ is equal to the
energy measured in the tagger calorimeters and $E_e$ is
the positron beam energy. 
The ETAG33 acceptance varies for the considered $y$~range of 
$0.3\leq y \leq 0.7$ between $8\%$ and $66\%$.
ETAG44 accepts events within $0.04 \leq y \leq 0.2$.
A mean acceptance of 29\% was determined 
for events with an energy deposition in ETAG44. 
%A fraction of $0.6\%$ had positron candidates
%in both electron taggers. These events were dealt with
%using several additional criteria to find the correct
%bin. 
All events of the ETAG44 sample form one
bin. The ETAG33 sample resulted in two bins. The
values for the mean $\gamma p$ centre-of-mass
energy $W$, calculated as $W=\sqrt{ys}$, for the
three bins are: $W_{ETAG44}=91\gev$, $W_{ETAG33,1}=187\gev$,
$W_{ETAG33,2}=231\gev$.

At least one track in the H1 detector was required to have a
transverse momentum of \linebreak $\pttrk>500\mev$,  
and a polar angle $\theta$
of $20^\circ < \theta < 160^\circ$ in order to guarantee a reasonable
trigger efficiency. The correction due to this cut
for the full kinematic range
was performed using a Monte Carlo simulation.

For this purpose the Monte
Carlo generator PHOJET~\cite{phojetman} has been used, since it reproduces the 
measured transverse momentum spectra and charged particle multiplicities
sufficiently well so that it can be employed for correcting experimental data.

In order to reject background from beam-gas interactions,
the $z$ coordinate of the vertex was restricted to 
$|\zvtx| <35 \cm$. 
Further background rejection conditions were imposed, which require  the 
timing of the event to match the bunch crossing time.

Imposing the cuts described in the previous sections, 
23072 events met all criteria.

% --------------
\section{Cross Section Measurement}
% --------------
The aim is to measure the differential cross section
$\der\sigma_{\gamma p \to Xp'}(ys, E_p') / \der z $. It can be
determined using the relation
$$
\frac{\der\sigma_{e p \to eXp'}(y,z)}{\der y\der z \der Q^2}
= {\cal F}_{\gamma /e}(y,Q^2)
\frac{\der\sigma^{\gamma p \to Xp'}(ys,z)}{\der z},
$$
where ${\cal F}_{\gamma /e}(y,Q^2)$ is the photon flux
in the Equivalent Photon Approximation~\cite{equivphotonapprox}.

The data have to be corrected for acceptances and efficiencies.
Weights are introduced for the FPS trigger efficiency,
the FPS track reconstruction efficiency, the track multiplicity dependent
trigger efficiency, 
and the electron tagger acceptances. The latter
includes a correction for cases where events are lost due to an 
energy deposition in the photon detector by 
a Bethe Heitler process occuring simultaneously with a 
photoproduction reaction, so called overlay events. Binwise
correction factors are used to correct for
track reconstruction efficiency in the H1 tracking
chambers and to compensate the
migration between proton energy bins and the limitation due 
to the finite FPS acceptance. The values for the
migration corrections vary between $0.911$ and
$1.125$, the acceptance corrections range from
$0.269$ to $0.903$.

\subsection{Background}
% ========================
The following sources of background were studied:
\begin{itemize}
\item Fake proton: A proton in the FPS can be
faked by showers due to interactions of the
proton beam with the beam pipe wall or residual
gas. These particles do not originate from the nominal 
interaction point and fail to match a valid trajectory coming from the
interaction point.
The remaining background is negligibly small.

\item Tracks in the central H1 detector produced
by beam gas interactions: A handle on this kind of
background is given by the $z$ coordinate of the
event vertex. The quantity $\zvtx$ is approximately
Gaussian distributed in $z$ around a mean value
close to the nominal interaction point, while 
beam gas interactions show no correlation.
The fraction of background events entering the
data sample, where $|\zvtx|<35\cm$ is required,
can be estimated from the tails
of the $\zvtx$ distribution and has been
determined to vary between $2.2\%$ (lowest $y$ bin)
and $3.4\%$ (highest $y$ bin).  These numbers contribute
to the systematic error.

\item Bethe Heitler events $ep\to e'p'\gamma$: They are not a 
significant source of background since the acceptance
for the photons is large and with the cut imposed on
the maximum energy deposited in the photon detector
these events are suppressed to a level of $0.3\%$. 

%\item Background from fake  signals in 
%one of the taggers has been evaluated using non-colliding positron bunches,
%so called pilot bunches. The
%resulting contribution to the data sample
%is $0.8\%$, which is included
%into the systematic error.
\end{itemize}

%
\subsection{Systematic errors}
% ===============================
Three types of systematic errors are distinguished:
errors common for events in the same \linebreak $y$ interval,
errors related to the proton energy (or $z$) interval,
and global errors.

%
\paragraph{Errors depending on \boldmath $y$.}
% ++++++++++++++++++++++++++++++++++++++++++++
The following sources contribute to this type of
error: background from proton beam gas interactions
in the central region of the H1 detector, 
errors on the acceptance of the electron taggers, errors on the
selection efficiency for  tracks in the H1
tracking chambers, and in case of ETAG33 an 
uncertainty of the positron energy measurement resulting in 
an imprecise determination of the number of events
belonging to one of the two $y$ bins.
The individual contributions are listed in Table~\ref{tabsyserrybin}.
\begin{table}[h]
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline 
                          & ETAG44 & \multicolumn{2}{c|}{ETAG33} \\
			  &        & 
		                     $y^{\mbox{\footnotesize ETAG}} < 0.5$ 
		          & $y^{\mbox{\footnotesize ETAG}}> 0.5$ \\
\hline 
proton beam bas background & 2.2\% &  2.3\%  & 3.4\% \\
error on energy meas. (ETAG33) %
                          &          & $0.8\%$ & $5.0\%$ \\
central track selection   & $4.2\%$  & \multicolumn{2}{c|}{3.3\%} \\
ETAG acceptances          & 6\%    & \multicolumn{2}{c|}{5\%}\\
\hline 
total             & 7.6\% & 6.5\% & 8.5\% \\
\hline
\end{tabular}
\caption{\sl Systematic errors common for events in the same inelasticity bin.}
\label{tabsyserrybin}
\end{center}
\end{table}

%
\paragraph{Errors depending on \boldmath $z$.}
% ++++++++++++++++++++++++++++++++++++++++++++
Uncertainties on the acceptance and migration corrections
for each $z$ bin are grouped together in 
Table~\ref{tabsyserrzbin}.

For the acceptance correction, the biggest effect is 
related to the distribution of the proton's transverse
momentum. The effect was evaluated using the PHOJET~\cite{phojetman}
and POMPYT~\cite{pompytman} Monte Carlo generators. These two generators
produce events with different $p_T$ spectra of the scattered proton, while the
description of the remaining hadronic final state is satisfactory in both
cases. Because the transverse momentum distributions are not yet  measured,
the two programs have been used to get an estimate of the uncertainty for the
correction.

The migration between $z$ bins is greatly
influenced by the stability of the FPS calibration procedure.
To estimate this effect, the calibration was performed
on subsamples of the Monte Carlo data set.
\begin{table}[h!]
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline 
$z$ interval & migration & acceptance & total \\
\hline 
$0.66\ldots 0.71$ & 7.8\% &11.8\% &14.1\% \\
$0.71\ldots 0.76$ & 3.0\% & 3.1\% & 4.3\% \\
$0.76\ldots 0.80$ & 1.2\% & 1.3\% & 1.8\% \\
$0.80\ldots 0.85$ & 2.0\% & 1.6\% & 2.6\% \\
$0.85\ldots 0.90$ & 6.7\% & 5.2\% & 8.5\% \\
\hline 
\end{tabular}
\caption{\sl 
Systematic errors common for all events in the same $z$ bin.
\small The error on the migration correction is the result of
Monte Carlo studies 
investigating the influence of errors on 
the calibration constants. The error on the acceptance correction
is estimated using two different Monte Carlo models.
\label{tabsyserrzbin}
}
\end{center}
\end{table}

% 
\paragraph{Global contributions.}
% +++++++++++++++++++++++++++++++
These are listed in Table~\ref{tabsyserrglob}
and are given by 
the uncertainty on the luminosity measurement, a value
for FPS systematic errors such as alignment errors, 
uncertainty on the hodoscope
efficiencies, and other effects not covered by the simulation.
Additionally systematic errors on the vertex cut efficiency and 
positron beam related background are included.
% --------------
% fehler auf null dregg ereignisse? is schon drin (4.2%, 3.3%)!! :-)
% --------------
\begin{table}[h!]
\begin{center}
\begin{tabular}{|c|c|}
\hline 
Source &  \\
\hline 
track reconstruction probability FPS & 5\% \\
Bethe Heitler background & 0.3\% \\
vertex cut & 0.4\% \\
luminosity & 1.77\% \\
\hline
normalization uncertainty & 5.3\%\\
\hline 
\end{tabular}
\caption{\sl Systematic errors common to all bins.
\label{tabsyserrglob}
}
\end{center}
\end{table}


%
\subsection{Results}
% ================
The cross section $\der\sigma_{\gamma p \to p'X}(W)/\der z$
for $W=91$, $187$, and $231\gev$ and five values of $z$ for 
$0.66 \leq z \leq 0.90$ is shown in Figure~\ref{sigmaplot}. The errors 
shown are the sqare root of the quadratic sum of the systematic errors
as described above and the statistical error taking into account the
weights of the events. The total errors  vary between $9.2\%$ and $21.0\%$.

It is observed that in all $z$ bins the measured cross-sections for different
values of the $\gamma p$ centre-of-mass energy $W$ agree with each other within 
the experimental errors. Furthermore the $z$ dependence is only weak without a 
clear trend so that the present data may be represented by one average
cross-section value of $7.7\mub$ per unit of $z$ (the measured values 
ranging between $7.4$ and $9.8$) which corresponds to
$\der\sigma_{\gamma p \to p'X}(W)/\der E_p' \sim 9.4 \nb/\gev$
for the kinematic range covered by this experiment. In particular the
restriction in the transverse momentum of the final state proton to 
$p_T\leq 0.2\gev$
should be kept in mind, which is equivalent to the statement that the measured
cross-section represents only about $24\%$ of the total semi-inclusive 
photoproduction cross-section if a slope of $b=6.8\gev^{-2}$ in 
$\der \sigma/\der{p_T^2} \sim e^{-b p_T^2}$ is assumed as measured by the
ZEUS collaboration \cite{ZEUSgamp}. 

The present data can be used to extend a measurement by the ZEUS 
collaboration~\cite{ZEUStampere} of the fraction of events with a
leading proton with $0.60 < z < 0.97$ and $p_T^2<0.5\gev^2$
compared to inclusive DIS events for $0.1\gev^2 < Q^2 < 260\gev^2$.
The ZEUS experiment observes that this ratio is largely independent of the DIS 
scaling variables $x$ and $y$ over the $Q^2$~range covered by the experiment
in~\cite{ZEUStampere}.
At $Q^2=0$, the data have to be extrapolated to the kinematic range of the
ZEUS experiment assuming $b=6.8\pm 0.8\gev^{-2}$ and a flat $z$~dependence
in accordance with published data. It is found that at $Q^2=0$ this ratio
is $r=0.07 \pm 0.01$, within the errors compatible with the values measured 
at finite virtuality.

The observed $z$ independence is in accord with 
the assumption of factorisation in a Regge description of the reaction.

From the fact that the cross-section is independent of $z$ and $W$ we
infer that it is also independent of $M_X$, the mass of the hadronic
final state apart from the leading proton.
This observation is in contrast to diffractive processes,
where a $1/M_X^2$ dependence is a good approximation to the data. 
The present measurement is reminiscent of 
data from semi-inclusive proton-proton scattering $p + p \to p + X$
at lower energy centre-of-mass energies squared 
$s < 3900 \gev^2$~\cite{albrow}, where a flattening
of the cross-section $\der \sigma/ \der M_X^2$ is observed for 
normalized masses above $M_X/\sqrt s \sim 0.2$, which is the same 
regime as covered by the present experiment.


\begin{figure}[h]
\begin{center}
  \setlength{\unitlength}{1mm}
  \begin{picture}(200,80)(-10,0)
  \put(-5,0){
    \epsfig{file=H1prelim-00-113.fig2.eps,width=15cm}
  }
  \end{picture}
  \begin{minipage}{13cm}{
  \caption[]{
  \sl The cross section $\der \sigma_{\gamma p\to p'X}/\der z$ as a
      function of $z$ for three values of $W$. All data points in a box belong
     to the same $z$ value \linebreak
      ($z=0.682, 0.732, 0.780, 0.829, 0.878$)
      and only for a better distinction they are drawn apart.
\label{sigmaplot}
  } 
}
\end{minipage}
\end{center}
\end{figure}

% ============================================================
\section{Leading Proton Production in DIS and Photoproduction}
% ============================================================
\subsection{Cross Section Measurement for Real and Virtual Photons}
% +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Applying Equation~\ref{formsigmatotincl}, inclusive deep
inelastic scattering and inclusive photoproduction data from the reaction
$\gamma^{(*)}p \to X$ have been combined in order to study
the transition between the two kinematic regimes in~\cite{h1f2fits}. 
In the present  paper, the same investigation is performed for the
process $\gamma^{(*)}p \to p'X$.

Data points on deep inelastic scattering with a leading proton
observed in the FPS with the same set-up as used for
the photoproduction investigation described in the previous
chapters are published in~\cite{lbproduction}. 
Radiative corrections have been applied to the DIS data. They amount to
not more than $6\%$ for all bins in $(x,Q^2)$.

The kinematic
range covered by the DIS data sample is $0.707 \leq z \leq 0.902$, 
divided into four
same-size intervals, $2\gev^2 \leq Q^2 \leq 50\gev^2$ in five bins, and 
$6\cdot 10^{-5} \leq x \leq 6\cdot 10^{-3}$ in four bins.
A structure function $\ftwolpthree$ was defined as:
$$
\frac{\der^3\sigma}{\der x \der Q^2 \der z} = 
    \frac{4\pi\alpha^2}{xQ^4}
    \left( 1-y+\frac{y^2}{2[1+R(x,Q^2,z)]} \ftwolpthree(x,Q^2,z)\right),
$$
where $R(x,Q^2,z)$ is the ratio of the longitudinal and the
transverse DIS cross-sections with a leading proton. The impact of $ R(x,Q^2,z)$
on the leading proton structure function \linebreak
$\ftwolpthree(x,Q^2,z)$ is small.
In the worst case a change of 9\% is introduced when $R=\infty$ is
assumed instead of $R=0$. For the present analysis $R$ is set to zero. 
This structure function is related to the leading proton
photoproduction data as follows:
$$
\frac{\der\sigma_{\gamma^{(*)} p \to p' X}(W)}{\der z} =
    \frac{4\pi^2\alpha}{Q^2}\ftwolpthree(x,Q^2,z).
$$
\begin{figure}[h]
\begin{center}
  \setlength{\unitlength}{1mm}
  \begin{picture}(200,155)(-10,0)
  \put(-5,0){
    \epsfig{file=H1prelim-00-113.fig3.eps,width=15cm}
  }
  \end{picture}
  \begin{minipage}{13cm}{
  \caption[]{
  \sl The cross section $\der \sigma_{\gamma^{(*)}p\to p'X}/\der z$ for
      four bins of $z$ and four bins of $M_X$ as a function of $Q^2$.
   \small
\label{plotsigmatotlp}
  } 
}
\end{minipage}
\end{center}
\end{figure}

\subsection{Model}
% ++++++++++++++++
In~\cite{h1f2fits},
the data have been contrasted with a model where the proton
structure is assumed to consist of two components: $\ftwovdm$ dominant
at low values of the photon virtuality describing the contribution
of vector mesons coupling to the virtual photon, and a partonic
structure function $\ftwopart$, governing the region 
$Q^2>1\gev^2$~\cite{bkpaper}. In~\cite{h1f2fits}, deep inelastic
scattering data were fitted to the functional form
$$
F_2(x,Q^2) = \cvm \ftwovdm(x,Q^2) 
   + \frac{Q^2}{Q^2+\qsqvm} F_2^{H1QCD}(\bar x,Q^2 + \qsqvm).
$$ 
with $\bar x = (Q^2+\qsqvm)/(W^2+Q^2+\qsqvm)$. The VDM part of the
structure function is given by 
$\ftwovdm(x,Q^2) = Q^2/(4\pi)\sum_V 
(M_V^4\sigma_{Vp}(W))/(\gamma_V^2(Q^2+M_V^2)^2)$, the sum running over the
low mass vector mesons. The parameters $\gamma_V$ and $M_V$ represent the
vector meson's coupling to the photon and its mass, respectively. The cross
section for the $Vp$ scattering is given by~$\sigma_{Vp}(W)$. 
The contribution for $\ftwopart$ was taken from the structure function
measurement.
The fit parameter $\qsqvm$ was found to be $0.45\gev^2$. 
The relative normalization $\cvm$ of the vector meson term is not
part of the model ~\cite{bkpaper} but was introduced as a free parameter
in order to match the photoproduction data. The fit
resulted in $\cvm=0.77$. 

The model of \cite{bkpaper} has not been developed to
cover reactions with a leading proton, 
which according to the Feynman diagram of Figure~\ref{picfeyndiag}
corresponds to a
situation where the real or virtual photon interacts with a colourless
constituent of the proton and not the proton itself. An argument, however, 
could be made that also the exchange object has a partonic content
and the VDM part describes a vector meson-reggeon or pion
scattering process so that an equivalent ansatz as in the inclusive case could
be justified. Therefore, making use of the observation that also the deep
inelastic semi-inclusive cross-section is nearly independent of $W$ and $z$,  
two additional parameters are introduced: 
a common scale factor $F$ including the flux factor for 
the emission of particles ${\cal R}$ by the proton and a
compensation for the reduction in phase space
due to the limited transverse momentum of the scattered protons, 
and a normalization factor $\cvmlp$ for the 
vector meson contribution. The variable comparable to $x$ of the inclusive
case is $\beta=x/(1-z)$ here. The function is modelled as follows:
$$
\frac{\ftwolpthree(x,Q^2,z)}{F} 
= \left( 0.77 \cdot \cvmlp \ftwovdm(\beta,Q^2) 
   + \frac{Q^2}{Q^2+0.45} F_2^{H1QCD}(\bar \beta,Q^2 + 0.45)
   \right).
$$ 

\pagebreak
Since no data in the genuine transition region is
available, the contributions to $F_2^{LP}$ are fixed
independently. The global factor $F$ is
determined by the DIS data alone, resulting in
$F=0.12$, independent of the $z$  and $M_X$.  
With $F$ fixed, a common fit is made to the semi-inclusive DIS and 
photoproduction data. In this fit the parameter $\cvmlp$ is determined by the 
the photoproduction data alone with negligible impact on $F_2^{H1QCD}$. 
The set of values obtained for $\cvmlp$ ranges from
0.13 to 0.30 with errors such that a averaging process is justified.
A mean value of $\cvmlp=0.23\pm 0.10$ is obtained.

It is thus observed that the VDM term is suppressed by a factor of ~0.23
with respect to the inclusive case. This is significantly
smaller than the factor of~0.65 between the total
cross sections for pion proton and proton proton scattering,
which one would expect to be applicable 
if cross-sections were related to the number of quarks
involved in a particular reaction.

The fits are displayed in Figure~\ref{plotsigmatotlp}
together with the DIS and photoproduction data points. 
Like in the case of inclusive reactions, a transition region
is observed near $Q^2\approx 1\gev^2$, where the DIS cross-section turns over
into a non-scaling regime and approaches the photoproduction value.

Unfortunately the semi-inclusive data are not precise enough and do
not cover a sufficiently wide range in $M_X$ to study the
transition as a function of $M_X$ in all $Q^2$ bins.

% ======================
\section{Summary}
Photoproduction reactions with a final state proton of $p_T \leq 0.2 \gev$
observed in the H1 Forward Proton Spectrometer have
been analysed in the kinematic range $0.66 \leq z \leq 0.90$
at $\gamma$p centre-of-mass energies $W=91$, $187$ and $231\gev$. The cross
section $\der \sigma(W) /\der z$ was determined to
be on the average $7.7\mub$ per unit of $z$. It is independent of $z$
and $W$ within the experimental errors.

The fraction of events with a leading proton extrapolated to the kinematic
range $0.60 < z < 0.97$ and $p_T^2 < 0.5\gev^2$ compared with the total
$\gamma p$ cross section is $0.07\pm 0.01$. This result is compatible with
values measured by the ZEUS collaboration for 
$0.1\gev^2 < Q^2 < 260\gev^2$.

The combination of data from photoproduction and
deep inelastic scattering with a final state proton
observed in the FPS, $\gamma^{(*)} p\to p'X$,
in the kinematic range $0.71 \leq z \leq 0.90$
shows that a qualitatively similar behaviour as observed when 
going from virtual to real photons in the inclusive reaction
$\gamma^{(*)} p \to X$ is expected.
A model where the contributions to the cross section are divided
into a VDM and a partonic component shows
a fair description of the data, provided the overall
normalization and the relative normalization between
the two components are adjusted accordingly. However, the
relative size of the vector meson fraction w.r.t.~the partonic  
contribution within the model
is $23\%$ of the value found for the inclusive reaction.
\pagebreak
 
% ========================
\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 constructing and maintaining
the H1 detector, our funding agencies for financial support,
the DESY technical staff for continuous assistance, and the DESY directorate
for the hospitality which they extend to the non-DESY members of the
collaboration. The forward proton spectrometer was supported by the 
INTAS93-43 project and the NATO contract PST.CLG.975100.

%
%   References for LP photoproduction paper
%
\begin{thebibliography}{99}

\bibitem{h1f2fits} H1 Collab., C. Adloff {\em et al.},
               Nucl. Phys. B~497 (1997) 3--30
\bibitem{dola} A. Donnachie, P.V. Landshoff,
               Phys. Lett. B~296 (1992) 227--232       
\bibitem{lbproduction} H1 Collaboration, C. Adloff {\em et al.},
               Eur. Phys. J. C~6 (1999) 587-602
\bibitem{theh1detathera} H1 Collaboration, I. Abt {\em et al.},
               Nucl. Instrum. Meth. A~386 (1997) 310--347
\bibitem{fpsnimpaper} P. Van Esch {\em et al.},
               Nucl. Instrum. Meth. A~446 (2000) 409--425
\bibitem{equivphotonapprox} S. Frixione {\em et al.},
               Phys. Lett. B~319 (1993) 339--345
\bibitem{schildknecht} J.J. Sakurai, D. Schildknecht,
               Phys. Lett. B~40, Nr. 1 (1972) 121--126
\bibitem{ZEUSgamp} ZEUS preliminary in ''Proceedings of ICHEP98 Conference,
               Vancouver'', p. 924, World Scientific A. Astbury et al. ed.
\bibitem{albrow} M.G. Albrow {\em et al.},
               Nucl. Phys. B~108 (1967) 1--29
\bibitem{ZEUStampere} A. Garfagnini for the ZEUS collaboration:
               ZEUS preliminary in ''Proceedings of EPS HEP Conference,
               Tampere, Finland'', in press
\bibitem{phojetman} R. Engel, ''PHOJET manual (Program version 1.05c, June
               96)'', University of Siegen preprint 95-05 (1995) 
	       (revised Feb. 96)
\bibitem{pompytman} P. Bruni, A. Edin, G. Ingelman,
               ''POMPYT version 2.6 -- A Monte Carlo 
               Program to Simulate Diffractive Hard Scattering Processes'',
	       unpublished program manual (1996), see 
               {\tt http://www3.tsl.uu.se/thep/pompyt}
\bibitem{bkpaper} B. Bade{\l}ek, J. Kwieci\'nski,
               Phys. Lett. B~295 (1992) 263--268
\end{thebibliography}

\end{document}

