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%  pdftitle    = {Measurement of the Proton Structure Function F2
%  at low Q2 in QED Compton Scattering at HERA},
%  pdfauthor   = {H1 Collaboration},
%  pdfkeywords = {QEDC, proton structure function},
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\begin{document}

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\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 
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Submitted to & & &
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                & & & \\
\multicolumn{4}{l}{{\bfseries
                International Europhysics Conference 
                on High Energy Physics, EPS03},
                July~17-23,~2003,~Aachen} \\
                (Abstract {\bfseries 084} & Parallel Session & {\bfseries 4}) & \\
                & & & \\
\multicolumn{4}{l}{{\bfseries
                XXI International Symposium on  
                Lepton and Photon Interactions, LP03},
                August~11-16,~2003,~Fermilab} \\ 
                & & & \\
                \hline
 & \multicolumn{3}{r}{\footnotesize {\itshape
    www-h1.desy.de/h1/www/publications/conf/conf\_list.html}} \\[.2em]
\end{tabular}
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\end{center}
\vspace*{2cm}

\begin{center}
  \Large
  {\bf 
    Measurement of the Proton Structure Function {\boldmath $F_2$} \\ 
    at low {\boldmath $Q^2$} in QED Compton Scattering at HERA}\\

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

\begin{abstract}
\noindent
The proton structure function $F_2$ is measured in QED Compton scattering
at HERA using data collected in 1997 by the H1 experiment. QED Compton events
give access to the kinematic range of very low $Q^2$, down to $0.1\,\GeV^2$,
and medium Bjorken $x$, a region which is not covered by inclusive DIS 
measurements at HERA. The results are in agreement with the measurements
from fixed target experiments wherever they overlap.
\end{abstract}

\end{titlepage}

\pagestyle{plain}


\section{Introduction} \label{s:intro}

Measurements of deep-inelastic lepton-proton scattering (DIS) provide
information that is crucial to our understanding of proton
structure.  Since the fixed target experiments have discovered
scaling violations~\cite{Fox:1974ry,deGroot:1978hr},
much progress has been made in extending the kinematic range of measurement
in terms of the Bjorken variable $x$ and the four-momentum transfer $Q^{2}$.
%This holds especially for the HERA $ep$ experiments which,
% with their wealth of data,
Especially the HERA $ep$ scattering experiments have shown that the
$Q^{2}$ evolution of the proton structure function $F_2(x,Q^2)$ is well 
described by perturbative Quantum Chromodynamics (pQCD) throughout a 
wide range in $x$ and $Q^{2}$\,%
\cite{Adloff:2000qk,Aid:1996au,Derrick:1996hn,Adloff:1997yz}.
However, at small $Q^2$ deviations from pQCD predictions are
observed~\cite{Adloff:1997yz,Breitweg:2000yn}, indicating the transition
into a regime in which non-perturbative effects dominate and the data
can only be described by phenomenological models such as
those derived from the Regge approach~\cite{Collins:1977jy}.

In order to study this non-perturbative regime, the structure function
$F_2$ has been measured at very low values of $Q^{2}$ and $x$, which
are accessible at HERA via special devices mounted close to the
outgoing electron beam direction~\cite{Breitweg:2000yn} thus
facilitating measurements of the scattered electron at very low angles.
These devices, however, do not cover the transition region at
\mbox{$Q^{2}$ $\sim 1\,$GeV$^{2}$}, which up to now has only been
investigated using ``shifted vertex''
data~\cite{Derrick:1996ef,Adloff:1997mf}.
In this paper preliminary results of a new measurement of $F_2$
in this kinematic domain are presented, which
utilises $ep$ data with wide angle hard photon radiation,
so called QED Compton (QEDC) events.

\begin{figure}[bh]
\centerline{\epsfig{figure=H1prelim-03-042.fig1.eps,width=.7\textwidth}}
\caption{Lowest order Feynman diagrams for the radiative process $ep 
\rightarrow e{\gamma}X$ with photon emission from the electron line.
$l$, $P$ represent the four-momenta of the incoming electron and the 
incoming proton, while $l^{\prime}$, $k$ and $X$ are the momenta of 
the scattered electron, the radiated photon and the hadronic final 
state, respectively.}
% $\hat{s}$ and $\hat{t}$ denote the squared 
%four-momenta of the virtual lepton.} 
\label{f:feynman_rad}
\end{figure}


\section{QED Compton Scattering Cross Section} \label{s:theory}

Radiative processes in $ep$ scattering, as depicted in
fig.\,\ref{f:feynman_rad}, may be split into three different
classes~\cite{Courau:1992ht,Ahmed:1995cf} with (i) the bremsstrahlung
or Bethe-Heitler process corresponding to small masses of both the
virtual electron and the virtual photon, (ii) the QED Compton process
with a low virtual photon and a large virtual electron mass and
finally (iii) the radiative DIS process where the photon is collinear 
either with the incoming (Initial State Radiation, ISR) or the
outgoing (Final State Radiation, FSR) electron.  
All three classes correspond to distinct experimental signatures.
For the QEDC scattering process the final
state topology is given by an azimuthal back-to-back configuration of the
outgoing electron and photon detected under rather large scattering angles. 
In this configuration their transverse momenta balance such that very
low values of the exchanged photon virtuality $Q^{2}$ are experimentally
accessible.

To correctly describe the process $ep\rightarrow e\gamma X$ the
standard kinematic variables $x$ and $Q^{2}$, used to describe inclusive
deep-inelastic scattering (DIS), have to be redefined in order to
account for the additional photon in the final state
%
\begin{equation}
Q^2 = -q^{2} = -(l - l^{\prime} - k)^2 \,\,, \,\,\,
x = \frac{Q^2}{2 P \cdot (l - l^{\prime} - k)} \,\,, \,\,\,
y = \frac{Q^2}{x s} \,\,.
\label{eq=kin_inv}
\end{equation}
%
Here $l$ and $P$ are the four-momenta of the incoming electron and the
incoming proton, and $l^{\prime}$ and $k$ represent the momenta of the
scattered electron and the radiated photon, respectively
(fig.\,\ref{f:feynman_rad}).  Three further independent variables are needed
for a full description of the differential QEDC scattering cross section. 
In the formalism presented in~\cite{Courau:1992ht} the Lorentz
invariant scale variable $x_{\gamma} = q \cdot l / P \cdot l$ and the
scattering solid angle $\Omega^{*}$ defined in the centre-of-mass
frame of the virtual Compton process %and encapsulating two degrees of freedom 
are employed. The cross section is then given by~\cite{Courau:1992ht}
%
\begin{equation}
    \label{e:QEDCxsec}
    \frac{d^{4}\sigma^{ep\rightarrow e\gamma
    X}}{d{x}dx_{\gamma}d{Q}^{2}d\Omega^{*}} =
    f^{T}_{\gamma^{*}/p}({x},x_{\gamma},{Q}^{2}) \left[
    \frac{d\sigma}{d\Omega^{*}} \right]^{T} +
    f^{L}_{\gamma^{*}/p}({x},x_{\gamma},{Q}^{2}) \left[
    \frac{d\sigma}{d\Omega^{*}} \right]^{L} \,\,, 
\end{equation}
%
where $[ d\sigma/d\Omega^{*} ]^{T,L}$ are the differential cross sections
of the process $e\gamma^{*} \rightarrow e\gamma$ for transverse and 
longitudinal polarised photons, fully calculable in the framework of 
QED~\cite{Courau:1992ht}, and $f_{\gamma^{*}/p}^{T,L}$ represent the 
corresponding virtual photon spectra, which may be expressed in terms
of the photo-absorption cross sections $\sigma_{\gamma^{*}p}^{T,L}$.
In order to specify $\sigma_{\gamma^{*}p}^{T,L}$ one
has to consider three separate contributions depending on the value
of the invariant mass $W$ of the outgoing hadronic final state:

\begin{enumerate}
%
    \item {\rm Elastic scattering}, for which the proton stays intact
    ($W = m_{p}$). This channel is well measured, and the cross section
    is given by the electric and magnetic form factors $G_E$ and~$G_M$;
%
    \item {\rm Resonance production}, where the total mass of the
    hadronic final state $X$ lies in the range $m_p + m_{\pi}
    \lesssim W \lesssim 2$\,GeV;
%
    \item {\rm Continuum inelastic scattering} at $W \gtrsim 2$\,GeV. 
    In this region the $\gamma^{*}p$ cross section is
    defined through the proton structure functions $F_2$ and $F_L$.
%
\end{enumerate}

The above cross section expression~(\ref{e:QEDCxsec}) is implemented
in the COMPTON Monte Carlo event generator~\cite{Carli:1991yn}. The program
also calculates higher order corrections for Initial State Radiation
in the peaking approximation~\cite{Etim:1967,Pancheri:1969}. However,
as this generator was primarily written for an application in analyses 
of elastic QEDC events, in the original version a rather basic approach 
is employed to describe the resonance region  and only simple scale 
invariant $F_2$ parameterisations are used to model the continuum 
inelastic domain.
Furthermore, no hadronisation of the final state $X$ is performed.

Since this analysis aims at the investigation of inelastic QEDC events a new
version of the COMPTON generator was developed~\cite{Lendermann:2002} which
includes detailed parameterisations for the resonance~\cite{Brasse:1976bf}
and the continuum~\cite{Abramowicz:1997ms} regions. In addition,
several packages for a complete simulation of the hadronic final 
state have been implemented into the program.
For the present study the SOPHIA package~\cite{Mucke:2000yb} 
is used in the range of low $Q^2$  and low masses, $W$, of the hadronic 
final state, while the Quark Parton Model with subsequent Lund string 
fragmentation \cite{Sjostrand:2000wi} is employed at high $W$ and high $Q^2$.

The SOPHIA model which is employed in most of the analysed
phase space provides a minimum bias description of 
photoproduction processes reproducing a large set of available data.
To be specific, the production of major baryon resonances,
direct pion production, multiparticle production based on the
Dual Parton Model~\cite{Capella:1994yb} with subsequent Lund string 
fragmentation, as well as 
the diffractive production of light vector mesons $\rho$ and $\omega$ 
are simulated.
%Decays of unstable particles are performed by the routine DECSIB from the
%SIBYLL package~\cite{Fletcher:1994bd}.


\section{Experimental Technique} \label{s:experiment}

The outgoing electron and photon in QEDC events are selected 
in the H1 detector~\cite{Abt:1997hi}
by requiring two energy depositions (clusters) in the electromagnetic part
of the backward\footnote{The $z$ axis of the right-handed coordinate 
system used by H1 is defined to lie along the direction of the 
incident proton beam and the origin to be at the nominal $ep$ 
interaction vertex. The backward direction is thus defined through 
$z < 0$.} lead-fibre calorimeter SpaCal~\cite{Appuhn:1996vf} with a polar
acceptance of $153^{\circ} < \theta < 177^{\circ}$,
an electromagnetic energy resolution of
%$\sigma / E = (7.1 \pm 0.2)\% / \sqrt{E/\GeV} \oplus (1.0 \pm 0.1)\%$
$\sigma / E = 7\% / \sqrt{E/\GeV} \oplus 1\%$
and the spatial resolution of
%$\sigma = (6.3 \pm 0.4)\,\mm / \sqrt{E/\GeV} \oplus (1.7 \pm 0.1)\,\mm$.
$\sigma = 6.3\,\mm / \sqrt{E/\GeV} \oplus 1.7\,\mm$.
The Backward Drift Chamber (BDC) \cite{Schwab:1996} situated in front of 
the SpaCal and providing an average resolution of $0.57$\,mrad for the $\theta$
measurement is employed as a pre-shower detector to increase the precision
of the cluster position measurement for electrons and converted photons.
In events with the electron scattered in the inner part of the SpaCal,
the Backward Silicon Tracker (BST)~\cite{Arkadov:2000} 
with an angular coverage of $171.5^{\circ} < \theta < 176.5^{\circ}$
and resolution of $0.3$\,mrad is used to measure the electron polar angle
and to reconstruct the interaction vertex position.
In events in which the electron is scattered out of the BST acceptance,
the Central Inner Proportional Chamber (CIP)~\cite{Abt:1997hi} in conjunction
with BDC and SpaCal is employed to determine the vertex coordinates.
The vertex reconstruction efficiencies are measured using inclusive DIS data 
and found to be 91\% on average for the BST and above $99\%$ for the CIP.
The hadronic final state is measured in the Liquid Argon Calorimeter
(LAr)~\cite{Abt:1997hi} covering the angular range 
$4^{\circ} < \theta < 153^{\circ}$. It has an energy resolution of
$12\% / \sqrt{E} \oplus 1\%$ for the electromagnetic and
$50\% / \sqrt{E} \oplus 2\%$ for the hadronic section.

The analysed events were collected using a dedicated QEDC trigger chain
ivolving three trigger levels. The chain is designed to select events
with two clusters in the electromagnetic SpaCal with the azimuthal back-to-back
topology. Unfortunately,
this trigger chain was built to keep only elastic QEDC events, 
which are extensively used for detector alignment, calibration
and luminosity cross checks. A significant number of events with
hadronic activity in the central region of the detector were rejected
by additional conditions based on the tracks in the Central Jet Chamber.
The phase space of the measurement is therefore restricted to the range
where the hadronic final state is detected in the forward part of the detector.
The trigger efficiency
is determined as a function of the largest polar angle $\theta_{\rm LAr}$
of a cluster in the LAr calorimeter with the energy above the noise
threshold of $0.5\,\GeV$ and falls down from $99\%$ at minimum
$\theta_{\rm LAr}$ to $79\%$ at $\theta_{\rm LAr} = 30^\circ$.
The efficiencies of all other trigger conditions are $\gtrsim 99\%$.


\section{Background Processes} \label{s:background}

A prominent background to inelastic QEDC scattering is created by inclusive
DIS events in which one particle from the hadronic final state
(typically a $\pi^0$) fakes the outgoing photon producing a cluster
in the electromagnetic SpaCal.
At high $y$, where the hadronic final state lies mostly in the hemisphere
of the scattered electron and photon, this process dominates the QEDC signal
making a clear QEDC event selection impossible.
For this reason the measurement is restricted to relatively low $y$ values:
$y \lesssim 0.006$.
%
Inclusive DIS events are modelled using the DJANGO MC
generator~\cite{Schuler:1991} which includes LEPTO~\cite{Ingelman:1991} and
ARIADNE~\cite{Lonnblad:1992tz} for the hard interaction and higher order QCD
effects as well as HERACLES~\cite{Kwiatkowski:1992es} for the calculation of
leading order radiative corrections.
DJANGO events with hard photons emitted from the lepton side in the analysed
phase space are excluded from the analysis to avoid double counting
with COMPTON events.
It however cannot be a priori expected that the specific non-perturbative 
process of single $\pi^0$ production far from the other final state hadrons
is correctly simulated by an inclusive event generator.
%The DIS background sample is therefore renormalised after extensive
%dedicated studies, in which, in addition to the scattered electron,
%$\pi^0$'s decaying in two photons, which produce two separate energy
%depositions in SpaCal, were explicitely selected 
%by cutting on their invariant mass.
%For events with no addional activity
%in the LAr calorimeter DJANGO is found to largely overestimate the
%real background contribution (at least by a factor of $20$). This was
%cross-checked by studying the shapes of SpaCal showers and estimating the
%number of events in which the two photons produce a common energy cluster.
%The result is also in agreement with other similar HERA 
%analyses~\cite{Adloff:2001cn}. However, for events with an additional
%hadronic activity in the LAr calorimeter, which make the dominant
%contribution for this analysis,
%DJANGO appears to provide a much more reasonable description.
%This contribution is also scaled down, but only by a factor of $1.8$.
%%The systematic uncertainty for the DIS background is estimated to be 100\%.
The DIS background sample is therefore renormalised after extensive
dedicated studies, which resulted into scaling down 
the DJANGO contribution by a factor of $1.8$.
The DIS background is thus estimated to contribute up to $3\%$ of the total
cross section with a systematic uncertainty of $100\%$.

Another source of significant background is Deeply Virtual Compton Scattering
(DVCS), in which the final state photon is radiated from the proton side.
DVCS and QEDC are indistinguishable experimentally,
and differ only in the kinematic distributions of the outgoing electron
and photon. The interference between the two processes does
not influence the energy and polar angle distributions of the final state
particles in the leading twist approximation, thus allowing for separate
MC simulations of both processes. Elastic DVCS events were simulated
by the TINTIN generator~\cite{Stamen:2001} and the cross section was
normalised to the H1 results~\cite{Adloff:2001cn}.

At present there is no theoretical prediction and no measurement for the
inelastic DVCS cross section. The ratio of the inelastic to elastic scattering
cross section was therefore estimated experimentally to be of a similar size
as for diffractive vector meson electroproduction. From this
inelastic DVCS is derived to contribute $5.5\%$ of the measured cross section
with a $100\%$ uncertainty.

Further background sources considered are typically of the order
$\lesssim 1\%$. These are:
%
\begin{itemize}
%
\item elastic and inelastic dilepton production, modelled using the
GRAPE event generator~\cite{Abe:1999}. 
%The uncertainty of the cross section calculation is 1\% and 10\%
%for the elastic and inelastic channels respectively.
%The contribution to the total cross section is $\lesssim 1\%$;
%
\item inclusive photoproduction, simulated by the PHOJET program
\cite{Engel:1996yd} which in particular includes production of light vector
meson $\rho$, $\omega$ and $\phi$; %After applying all selection cuts, 
%the $\gamma p$ contribution is $\lesssim 0.5\%$; % with 20\% uncertainty;
%
\item diffractive electroproduction of light vector mesons as well as
diffractive $J/\psi$ photo- and electroproduction, all being simulated
by the DIFFVM MC generator~\cite{List:1999} 
for both elastic and proton-dissociative cases;
%The generated samples are rescaled to the measured cross sections;
%
\item two photon resonance and possible odderon production. The
contributions of these processes were estimated analytically 
employing results from \cite{Kilian:1998ew,Berger:1999ca} and found negligible;
%
\item beam--gas/beam--wall interactions, estimated using pilot bunch events.
%and found to be below $0.5\%$.
\end{itemize}

The detector response for all generated events is simulated in detail
by a program based on GEANT~\cite{Geant:1994}.
The simulated events are subject to the same reconstruction procedure
as the data.


\section{Data Analysis} \label{s:analysis}

The measurement is performed on $9.25\,\pb^{-1}$ of $e^+p$ data
with the centre-of-mass energy $\sqrt{s} = 301\,\GeV$, which were collected
with the H1 detector at HERA in 1997.
%
For the event selection the following requirements are imposed:
%
%\renewcommand{\labelitemii}{--}
\begin{itemize}
\item General selection criteria for the QED Compton process
  \begin{itemize}
\item 
Energies $E_1$, $E_2$ of the highest and the second highest energy cluster
in the electromagnetic SpaCal
  \begin{equation} \label{eq:e_cut}
%E_1 > 10\,{\rm GeV}\,, \hspace{2em} 
E_2 > 4\,{\rm GeV}\,, \hspace{2em}
20\,{\rm GeV} < E_1 + E_2 < 31\,{\rm GeV}\,,
  \end{equation}
%
\item 
$e\gamma$ acoplanarity
  \begin{equation} \label{eq:aco_cut}
A \equiv \pi - \Delta\phi < 45^\circ\,,
  \end{equation}
where $\Delta\phi$ is the azimuthal angle between
the electron and photon directions;
  \end{itemize}
%
\item Additional criteria for inelastic QED Compton process selection
  \begin{itemize}
\item
At least one cluster with the energy above the noise threshold
of 0.5\,GeV is found in the LAr calorimeter;
\item $e\gamma$ acoplanarity
  \begin{equation} \label{eq:aco_incut}
A > 2^\circ\,
  \end{equation}
which suppresses the contamination of elastic QEDC events
remaining due to the noise in the LAr calorimeter;
  \end{itemize}
%
\item Background suppression requirements
  \begin{itemize}
\item 
Maximum polar angle of LAr clusters with the energy above the noise threshold
  \begin{equation}  \label{eq:thlar_cut_mb97}
\theta_{\rm LAr} < 30^\circ\,,
  \end{equation}
which restricts the phase space to the range where the inclusive DIS
background is relatively low, and also where the trigger efficiency is high;
%
\item 
Residual energy in the electromagnetic SpaCal 
  \begin{equation}  \label{eq:eres_cut}
  E_{\rm res} < 1\,{\rm GeV}
  \end{equation}
defined as \mbox{$E_{\rm res} = E_{\rm tot} - E_1 - E_2$},
where $E_{\rm tot}$ is the total energy in the electromagnetic SpaCal.
This cut suppresses DIS, $\gamma p$, dilepton and vector meson backgrounds;
%
\item Cuts on the lateral shower profiles of the two SpaCal clusters,
allowing a good separation of electrons and photons from hadrons;
%
\item Selection of events with the identified electron and photon.
The electron identification in BST and CIP involves also 
the vertex reconstruction;
%
\end{itemize}
%
%\item Limits necessary for correct detector acceptance description
\item Acceptance limits
  \begin{itemize}
\item 
Vertex $z$ coordinate
\begin{equation}
  -30\,{\rm cm} < z_{\rm vtx} < 30\,{\rm cm}\,,
\end{equation}
which also reduces the beam induced background,
%
\item Fiducial cuts on the BST and CIP acceptance.
  \end{itemize}
\end{itemize}
%
For the kinematic range in question, the reconstruction of the variables
$x$ and $y$ cannot be performed using only the four-momenta of the outgoing
electron and photon,
as the resolution deteriorates with $1/y$ thus becoming inapplicable
at low values of the inelasticity $y$.
Therefore these variables have to be reconstructed using in addition the
kinematics of the final state hadrons.
This is done using the Sigma method~\cite{Bassler:1995uq}, which is based on 
the measurement of $\sum_{i} (E - p_{z})_{i}$ summing 
over all objects $i$ of the hadronic final state.
Hence one of the main challenges for the analysis is a good understanding
of the hadronic final state and the acceptance corrections in this region.
The quality of the description employed in the measurement,
combining both the hadron distribution by the SOPHIA package and
the subsequent simulation of the H1 detector, is illustrated in 
fig.~\ref{f:control1}, in which the ratio of the total transverse momentum of 
measured hadrons $p_{t,{\rm had}}$ to the total transverse momentum of 
the $e\gamma$ system $p_{t,e\gamma}$ is plotted. A very good agreement
between data and the simulation is visible%
\footnote{The peaks in the background distributions are artefacts
of the limited MC statistics available at the moment for the inclusive
DIS background estimation.}.
Due to losses in the very forward region,
beyond the edge of the LAr calorimeter, the distribution peaks at values
smaller than one. One should note, however, that for the calculation of
kinematic variables $y$ and $x$ with the Sigma method the total $E - p_z$
of hadrons is used which is much less sensitive to the losses in the beam
pipe than the transverse momentum.
The acceptance limit defining
the lowest measurable $y$ is given by the struck quark direction and
is approximately the same as used in inclusive DIS measurements 
performed by H1.

The control distribution of $y$ shown in fig.~\ref{f:control2} demonstrates 
the good quality of both the hadronic simulation and the cross section
description given by the COMPTON program down to the lowest $y$ values,
even beyond the range used for the measurement.

The distributions depicted in fig.~\ref{f:control3} illustrate a good
description of the electron--photon final state provided by the simulation.
This was studied additionally with higher
precision using all (elastic and inelastic) QED Compton events as shown in
fig.~\ref{f:control4}. For these plots, all analysis cuts except for
inelastic QEDC selection, are imposed, thus obtaining an event sample
with higher statistics and almost insensitive to hadronisation effects%
\footnote{Only due to the cut $\theta_{\rm LAr} < 30^\circ$ there is
a small influence of hadronisation corrections on this event sample.}.


\section{Measurement Uncertainties}  \label{s:errors}

The following systematic errors are estimated:
%
\begin{itemize}
\item %A $10\%$ uncertainty of the noise description in the LAr calorimeter
  %and of backscattering effects in the forward region
  A 10\%  uncertainty on the energy attributed to noise or backscattered
  particles in the LAr calorimeter
  is estimated studying the energy of the particles rejected by the noise cut.
  It contributes typically a $4 - 6\%$ error to the $F_2$ measurement;
%
\item The uncertainty of the hadronic measurement, which
  takes into account the errors of the hadronic final state simulation,
  detector acceptance and LAr calorimeter energy scale, is estimated
  from the $p_t$ balance between the reconstructed hadron and electron-photon
  final states. The systematic error is of the order $1 - 2\%$ rising up to
  $6\%$ at lowest $y$ values;
%
\item The uncertainty of the SpaCal energy scale, affecting
  simultaneously the energies of both clusters, amounts to $1\%$ at 
  $E > 17$\,GeV and increases linearly at lower energies 
  up to 2.7\% at $E = 4$\,GeV. This source contributes $1 - 2\%$ error
  to the measurement;
%
\item The uncertainties of the measurement due to polar angle and
$z_{\rm vtx}$ reconstruction are typically within $1\%$;
%
\item The total efficiency of the electron/photon identification and vertex
reconstruction is studied using inclusive DIS and elastic QEDC events
and the corresponding error of the measurement is estimated to be $3\%$;
%
\item Out of the errors of the background estimation, as described 
in sect.~\ref{s:background}, main contributions are $5.5\%$ by inelastic DVCS
and up to $3\%$ by inclusive DIS. Other errors are below $1\%$;
%
\item The luminosity measurement contributes a $1.5\%$ error;
%
\item The uncertainties of several trigger conditions stay each within $0.5\%$;
%
\item The error due to radiative corrections amounts to $2\%$.
\end{itemize}
%
Both the statistical and the systematic errors lie in the range $8 - 13\%$
making thus the total error to be $12 - 18\%$.


\section{Results of the Measurement} \label{s:results}

In order to 
extract the structure function $F_2$ the data sample is divided into 
subsamples corresponding to a grid in $y$ and $Q^{2}$. The bin sizes are
adapted to the resolution in the measured kinematic quantities such that
purity and stability in all bins shown are greater than $30\%$. Here, 
the purity (stability) is defined as the ratio of the number of simulated
QEDC events originating from and reconstructed in a specific bin to the
number of reconstructed (generated) events in the same bin.

The $F_2$ values measured in QED Compton scattering are depicted
in fig.\,\ref{f:f2qedc} as a function of $x$ at fixed $Q^2$ and compared
to other HERA~\cite{H1mb99,H1sv00,Breitweg:2000yn} and fixed target~%
\cite{Whitlow:1992uw,Arneodo:1997qe,Adams:1996gu} data.
The present analysis extends the kinematic range of HERA at very low $Q^2$
towards higher $x$ values thus complementing standard inclusive
and shifted vertex measurements. For inclusive DIS at nominal vertex position
the outgoing electron is not detected in the H1 apparatus for such low values
of $Q^2$ as it is scattered at small angles escaping through the beam pipe
unseen.
On the other hand shifted vertex data cannot extend the low $Q^2$ measurements
to such high $x$ due to stronger acceptance limitations in the forward region.
The region covered mostly overlaps with the domain of fixed target 
experimets. A good agreement with their results is observed.

%The region covered mostly overlaps with
%the domain of fixed target data, and at highest $x$ and lowest $Q^2$ it
%reaches the range not yet measured in the neutral current lepton--proton
%scattering. A good agreement with the fixed target data is observed.


\section{Summary}  \label{s:conclusion}

Preliminary results of a measurement of the proton structure function
$F_2$ in QED Compton scattering at HERA are presented. The data analysis
is performed at very low $Q^2$ down to $0.1\,\GeV^2$, in the transition region
from DIS to photoproduction. The measurement extends the kinematic domain
of HERA complementing the analyses of standard inclusive low $Q^2$ and
shifted vertex data in the medium $x$ region. The results are
in a good agreement with data of fixed target experiments.

%
%   References for QED Compton paper
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\clearpage

\begin{figure}[!tb]
\centerline{\epsfig{figure=H1prelim-03-042.fig2.eps,width=.9\textwidth}}
\caption{Ratio of the total measured transverse momentum of hadrons to the
total transverse momentum of the $e\gamma$ system.
%Only events used for the measurement are plotted.
H1 data are depicted by the closed circles. 
The solid histogram represents the sum of COMPTON MC events with all
background contributions added. The hatched histogram denotes the sum of
all simulated background events}
\label{f:control1}
\end{figure}

\begin{figure}[!tb]
\centerline{\epsfig{figure=H1prelim-03-042.fig3.eps,width=.9\textwidth}}
\caption{$y_\Sigma$ distribution after applying all selection cuts. 
The solid histogram represents the sum of COMPTON MC events with all
background contributions added. The hatched histogram denotes the sum of
all simulated background events}
\label{f:control2}
\end{figure}

\begin{figure}[!tb]
\centerline{\epsfig{figure=H1prelim-03-042.fig4.eps,width=\textwidth}}
\caption{Control distributions for the measured electron and photon
in events used for the measurement.
Shown are: a) electron energy, b) photon energy, c) sum of both energies,
d) invariant mass of the $e\gamma$ system, e) electron scattering angle and
f) photon scattering angle}
\label{f:control3}
\end{figure}

\begin{figure}[!tb]
\centerline{\epsfig{figure=H1prelim-03-042.fig5.eps,width=\textwidth}}
\caption{Control distributions for the measured electron and photon
in all (elastic and inelastic) QED Compton events. All selection cuts
are applied except those demanding explicitly inelastic events.
Shown are: a) electron energy, b) photon energy, c) sum of both energies,
d) invariant mass of the $e\gamma$ system, e) electron scattering angle and
f) photon scattering angle}
\label{f:control4}
\end{figure}

\begin{figure}[!p]
\centerline{\large H1 preliminary}
\centerline{\epsfig{figure=H1prelim-03-042.fig6.eps,width=\textwidth}}
\caption{Results of $F_2$ measurement in QED Compton scattering
(closed circles) compared with other measurements at 
HERA (closed squares~\cite{Breitweg:2000yn}, open diamonds~\cite{H1mb99}
and open circles~\cite{H1sv00}) and fixed target experiments
(open squares~\cite{Whitlow:1992uw}, open stars~\cite{Arneodo:1997qe}
and open crosses~\cite{Adams:1996gu}).
The solid line depicts the ALLM97 parameterisation~\cite{Abramowicz:1997ms},
while the dashed line represents the Fractal fit~\cite{Lastovicka:2002hw},
which is plotted in the range $y > 0.003$.}
\label{f:f2qedc}
\end{figure}

\end{document}

