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%     Investigation of the Proton Structure                      %
%     Using Radiative Events at HERA                             %
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%                                             H1 Collaboration   % 
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% February 2001                                                  %
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% Editors: C. Issever/H.-C. Schultz-Coulon                       %
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\noindent
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Submitted to & & &
\epsfig{file=/h1/www/images/H1logo_bw_small.epsi
,width=2.cm} \\[.2em] \hline
\multicolumn{4}{l}{{\bf
                31st International Conference 
                on High Energy Physics, ICHEP02},
                July~24,~2002,~Amsterdam} \\
                 & Abstract:        & {\bf 976}    &\\
                 & Parallel Session & {\bf 5}   &\\ \hline
 & \multicolumn{3}{r}{\footnotesize {\it
    www-h1.desy.de/h1/www/publications/conf/conf\_list.html}} \\[.2em]
\end{tabular}
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\vspace*{2cm}

\begin{center}
\begin{Large}

%
% Original Title
% {\bf Measurement of Radiative Deep Inelastic Scattering with the H1 Detector at HERA}
% 
% Final Title
% {\bf Investigation of the Proton Structure \\ Using Radiative Events at HERA}
% 
% Bekl... Title
{\bf Measurement of the Proton Structure Function\\ Using Radiative Events at HERA}

\vspace*{1cm}

H1 Collaboration

\end{Large}
\end{center}

\begin{abstract}
\noindent Deep-inelastic electron-proton scattering (DIS) with initial 
state photon radiation is investigated using the H1 detector at HERA. 
The proton structure function $F_2(x,Q^2)$ is measured for Bjorken-$x$ 
values of $2 \cdot 10^{-5} \leq x \leq 2 \cdot 10^{-2}$ over a $Q^2$ 
range extending from $20$\;GeV$^2$, where perturbative QCD 
calculations are clearly valid, down to $0.35$\;GeV$^2$, where the 
influence of non-perturbative effects is large and only 
phenomenological models can describe the data.
% The 
% measurements thus help to close the gap between data on real and 
% virtual photoproduction and provide information on the transition 
% between DIS and real photon interactions.
% %  Deep-inelastic electron-proton scattering data with initial state photon 
% %  radiation (ISR) are investigated.  With these data the proton 
% %  structure function $F_{2}$ is measured, extending the kinematic range 
% %  of the H1 experiment for intermediate Bjorken-$x$ of $x\approx 
% %  10^{-4}$ down to four-momentum transfers of $Q^{2} = 
% %  0.35$\;GeV$^{2}$.  The measurements based on a coherent set of 
% %  radiative events span from a region ($Q^{2}=20$\;GeV$^{2}$) in which 
% %  perturbative calculations are clearly valid, to a phase space regime 
% %  ($Q^{2}<1$\;GeV$^{2}$) where only phenomenological models can 
% %  describe the data due to large influence of non-perturbative effects.
% % %  In addition, the special feature of the ISR data resulting in an 
% % %  effective reduction of the electron-proton center-of-mass energy is 
% % %  used to extract information on the longitudinal structure function 
% % %  $F_{L}$.
% %  The data thus help to close the gap between measurements of real and 
% %  virtual photoproduction and provide further information on the 
% %  transition between deep-inelastic scattering (DIS) and real photon 
% %  interactions at $Q^{2}\approx 0$.
\end{abstract}

\vspace{1.5cm}

% \begin{center}
% To be submitted to {\em Physics Letters B}
% \end{center}

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

The study of deep-inelastic lepton-proton scattering (DIS) provides 
important information that is crucial to our understanding of 
proton structure.  Since the first 
observation by the HERA $ep$ 
experiments~\cite{Abt:1993cb,Derrick:1993ft} of the strong rise of the 
structure function $F_{2}$ towards low values of the Bjorken-variable $x$, 
much progress has been made in investigating this kinematical 
regime~\cite{Adloff:2000qk, Aid:1996au, Derrick:1996hn, Adloff:1997mf, 
Adloff:1997yz}.  While the evolution of the structure function with 
the four-momentum transfer $Q^{2}$ is well described by perturbative 
Quantum Chromo Dynamics (pQCD) for $Q^{2}>1.5$\;GeV$^{2}$, deviations 
from pQCD predictions are observed~\cite{Breitweg:2000yn} if $Q^{2}$ 
is small, indicating a transition into a regime where 
non-perturbative effects dominate and the data can only be described 
by phe\-no\-meno\-lo\-gical models such as those derived from Regge 
theory~\cite{Collins:1977jy}.

% Measurements of deep-inelastic lepton-nucleon scattering (DIS) are 
% important for the understanding of the proton structure and the 
% theories which describe it.  Since the first observation by the HERA 
% $ep$ experiments~\cite{Abt:1993cb,Derrick:1993ft} of the strong rise 
% of the structure function $F_{2}$ for low values of the 
% Bjorken-variable $x$, there has been much progress in investigating 
% the structure of the proton in many detail~\cite{Adloff:2000qk, 
% Aid:1996au, Derrick:1996hn, Adloff:1997mf, Adloff:1997yz}.  It has
% been shown that, while the evolution of the structure  
% function with the four-momentum transfer $Q^{2}$ is well described by 
% perturbative Quantum Chromo Dynamics (pQCD) for 
% $Q^{2}>1.5$\;GeV$^{2}$, the observed shape deviates~\cite{Breitweg:2000yn} 
% from the prediction by pQCD if $Q^{2}$ is  
% small, indicating a transition into a regime where non-perturbative 
% effects dominate and the data can only be described by 
% phe\-no\-meno\-lo\-gical models as those derived from Regge 
% theory~\cite{Collins:1977jy}.

In order to study this non-perturbative regime, the structure function 
$F_{2}$ has been measured at very low values of $Q^{2}$, which at HERA 
are only accessible via special devices mounted close to the 
outgoing electron\footnote{During the 1997 data taking period in which 
the data for this analysis were recorded, HERA 
provided $e^{+}p$ collisions.  In this paper the incident and 
scattered lepton are always referred to as ``electron''.} beam 
direction~\cite{Breitweg:2000yn} thus facilitating measurements of the 
scattered electron at very low angles.  These low $Q^{2}$ data, 
however, do not cover the transition region at $Q^{2}$ $\sim 
1$\;GeV$^{2}$ which up to now has only been examined using ``shifted 
vertex'' data taken in 1995~\cite{Adloff:1997mf}.

In this paper the HERA kinematic domain is extended towards low values 
of $Q^{2}$ by making use of DIS events in which a photon is radiated 
collinear to the incident electron (ISR events).  These events, which 
can be identified by detection of the radiated photon in the small 
angle photon detector of the luminosity system, effectively have a
%photon detection system of the luminosity system, effectively have a 
lower initial electron beam energy and thus, for a given angular 
acceptance for the scattered electron, allow access to the low-$Q^{2}$ 
regime.
% In addition, the ISR events are used to 
% extract information on the longitudinal structure function $F_{L}$.

Previous investigations of radiative events at HERA are reported 
in~\cite{Ahmed:1995cf,Aid:1996au,Favart:1998fa}.  The present 
analysis takes advantage of both improved instrumentation in the 
backward region in H1~\cite{Appuhn:1997na}, i.e. in the electron direction, and the larger 
integrated luminosity recorded during the 1997 running period.
% In line with 
% recent H1 publications studying deep-inelastic $ep$ scattering based 
% on data recorded during the 1996 and 1997 running 
% periods~\cite{Adloff:2000qk} the present analysis takes --- apart from 
% the higher statistics recorded during these years --- advantage of the 
% improved instrumentation in the backward region.  
The presented 
$F_{2}$-measurement covers a kinematic range of $0.35 \leq Q^{2} 
\leq 20$\;GeV$^{2}$ and $2 \cdot 10^{-5} \leq x \leq 2 \cdot 
10^{-2}$.
% Following the 
% suggestions of~\cite{krasny,favart} the data set is also used to 
% extract information on the longitudinal proton structure function 
% $F_{L}$ at $<Q^2>=3.7$\;GeV$^{2}$ and $<x>=0.25\cdot 10^{-3}$.
Except for the very low $Q^{2}$ regime these data supersede an earlier 
preliminary analysis based on 1996 data~\cite{Favart:1998fa}. 


\section{The H1 Detector}
\label{sec:detector}

The H1 detector~\cite{Abt:1997hi} consists of a number of subdetectors 
designed to provide both complementary and redundant measurements of 
various aspects of high energy electron-proton collisions.  Central to 
this analysis are the backward calorimeter, 
SpaCal~\cite{Appuhn:1997na}, and the H1 luminosity system.  The latter 
consists of two main components, a photon detection system and an 
electron detector (electron tagger) and is used to determine the 
luminosity by measuring the cross-section of the Bethe-Heitler 
reaction $ep \rightarrow ep\gamma$.  Within the framework of this 
analysis, it serves to identify ISR events by detecting an energy deposition in 
the photon detector; the accuracy of the energy measurement provided 
by the system is $\Delta E_{\gamma}/E_{\gamma} = (28\pm 
3)\%/\sqrt{E/{\rm GeV}} \oplus 1.0\%$.  The scattered electrons in ISR 
events are detected in the electromagnetic part of the SpaCal 
calorimeter with an energy resolution of $7\%/\sqrt{E/{\rm GeV}} 
\oplus 1.0\%$.  The electron identification is improved by a track 
requirement based on the information of the backward drift chamber 
(BDC), which in addition provides a polar angle 
measurement\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.} accurate to $0.5$\;mrad.  The hadronic 
final state is reconstructed with the LAr 
calorimeter~\cite{Andrieu:1993kh}, the tracking detectors and the 
SpaCal.  The total depth of the LAr calorimeter varies between 4 and 
8.5 interactions lengths, depending on the polar angle, and provides a 
hadronic energy resolution of about $50\%/\sqrt{E/{\rm GeV}} \oplus 
2.0\%$~\cite{Andrieu:1994yn}.  In addition, this analysis makes use of 
the track detectors for vertex reconstruction.


\section{Radiative Processes}
\label{sec:rad. processes}

Radiative processes in $ep$ scattering, as depicted in 
Figure~\ref{fig=isr_diag}, 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 at sizeable 
$Q^{2}$ where the photon is collinear either with the 
incoming (ISR process) or the outgoing (FSR process) electron line.

A set of kinematic invariants taking into account the energy 
loss from the radiated photon and suitable for describing the 
process $ep\rightarrow e\gamma X$ is given 
by~\cite{Krasny:1991wq,Anlauf:1999re}
\begin{equation}
\hat{Q}^{2}=-(p-p^{\prime}-k)^{2} \;\;, \;\;\;
\hat{x}=\frac{\hat{Q}^{2}}{2P\cdot(p-p^{\prime}-k)} \;\;,\;\;\;
\hat{y}=\frac{P\cdot(p-p^{\prime}-k)}{P\cdot(p-k)} \;\;,
\label{eq=kin_inv}
\end{equation}
where $p$ and $P$ are the four-momenta of the incoming electron and 
the incoming proton, and $p^{\prime}$ and $k$ represent the momenta 
of the scattered electron and the radiated photon, respectively 
(Fig.~\ref{fig=isr_diag}).  Events in which the photon is seen in the
photon detector of the luminosity system,  
i.e.\ is radiated at very small angles $\theta_{\gamma}$ with respect 
to the incident electron beam direction, are
dominated by the ISR process.  The kinematic quantities defined 
in~(\ref{eq=kin_inv}) are then related to the $ep$ center-of-mass energy 
squared $s \approx 2Pp$ and the energy fraction $z=(E_{0}-E_{\gamma})/E_{0}$ 
of the electron after the initial photon radiation via
\begin{equation}
        \hat{Q}^{2}=z\hat{x}\hat{y}s\,.
\end{equation}
Here, $E_{0}$ represents the electron beam energy and $E_{\gamma}$ 
the energy of the radiated photon deposited in the photon detector of 
the luminosity system.

\begin{figure}[t]
  \vspace*{0.4cm} 
  \centerline{\epsfig{figure=H1prelim-01-042.fig1.eps,width=.8\textwidth}}
  \vspace*{0.3cm} 
  \caption{Schematic Feynman diagrams for the 
  radiative process $ep\rightarrow e\gamma X$ with (a) initial (ISR) 
  and (b) final state radiation (FSR); $p$, $P$ represent the 
  four-momenta of the incoming electron and the incoming proton, while 
  $p^{\prime}$, $k$ and $P^{\prime}$ are the momenta of the scattered 
  electron, the radiated photon and the hadronic final state $X$, 
  respectively.}
   \label{fig=isr_diag}
\end{figure}

For the experimental determination of $\hat{x}$, $\hat{y}$ and 
$\hat{Q}^{2}$ the kinematics of the scattered electron together with 
the energy of the radiated photon are used (electron method) with
\begin{equation}
        \hat{y}_{e}=1-\frac{E_{e}}{z E_{0}}\sin^{2}\frac{\theta_{e}}{2}\;\;,\;\;\;\;\;
        \hat{Q}_{e}^{2}=4zE_{0}E_{e}\cos^{2}\frac{\theta_{e}}{2}\;\;,\;\;\;\;\;
        \hat{x}_{e}=\hat{Q}_{e}^{2}\hat{y}_{e}/zs\;\;,
\end{equation}
where $E_{e}$ and $\theta_{e}$ are the energy and scattering angle of 
the scattered electron. At low values of $\hat{y}$ the resolution of 
the measurement can be improved by using the hadronic final state 
kinematics for the determination of $\hat{y}$.  This is done with the 
so-called $\Sigma$-method~\cite{Bassler:1995uq}, which also reduces the 
influence of higher order radiative effects. Using 
$\Sigma_{h}=\sum_{i} E_{i}-p_{i,z}$, where one sums over all hadronic 
final state objects, and $\Sigma_{e}=E_{e}(1-\cos\theta_{e})$ this 
method yields
\begin{equation}
        \hat{y}_{\Sigma}=\frac{\Sigma_{h}}{\Sigma_{h}+\Sigma_{e}}\;\;,\;\;\;\;\;
        \hat{Q}_{\Sigma}^{2}=\frac{E_{e}^{2}\sin^{2}\theta_{e}}{1-\hat{y}_{\Sigma}}\;\;,\;\;\;\;\;
        \hat{x}_{\Sigma}=\hat{Q}_{\Sigma}^{2}\hat{y}_{\Sigma}/zs\;\;.
\end{equation}

Since they are complementary with respect to resolution and their 
sensitivity to radiative corrections both reconstruction methods 
presented above were used.  The final results are determined using the method 
which yields the better experimental accuracy in a particular region 
of phase space.

In terms of the kinematic variables described above, the differential 
cross section for the ISR process $ep \rightarrow e\gamma X$ 
integrated over the photon emission angle 
$0<\theta_{\gamma}<\Theta_{a}$ and the energy fraction $z$ 
reads~\cite{Krasny:1991wq}
\begin{eqnarray}
\frac{d^{2}\sigma}{d\hat{x}d\hat{Q}^{2}} & = &
    \int_{0}^{z_{\rm
        max}}{dz\frac{d^{3}\sigma}{d\hat{x}d\hat{Q}^{2}dz}} \nonumber 
    \label{eq=xsec1}\\ 
    & = &
    \int_{0}^{z_{\rm max}}{
       dz\frac{\alpha^{3}}{\hat{x}\hat{Q}^{4}}P(z)
       \left[ 2(1-\hat{y}) + \frac{\hat{y}^{2}}{1+R} \right]
       F_{2}(\hat{x},\hat{Q}^{2})(1+\delta_{\rm RC})
    }\label{eq=xsec2} 
\end{eqnarray}
where
\begin{equation}
   P(z) \;\; = \;\; \frac{1+z^{2}}{1-z} \ln 
     \left[\frac{E_{0}^{2}\Theta_{a}^{2}}{m^2_e}\right] - \frac{2z}{1-z} 
     \;\;\;\raisebox{-10pt}{,}
     \hspace*{4.5cm}
     \label{eq=pz}
\end{equation}
$R = F_{L}/(F_{2}-F_{L})$ is related to $F_{2}$ and the 
longitudinal structure function $F_{L}$, and $m_{e}$ represents the mass 
of the electron.  In the derivation of 
equations~(\ref{eq=xsec1})--(\ref{eq=pz}) all infrared and collinear 
finite terms have been neglected~\cite{Krasny:1991wq}.  Furthermore, 
terms of order ${\cal O}(m_{e}^{2})$ are not taken into account and 
equation~(\ref{eq=pz}) is valid only for angle cutoffs 
$\Theta_{a} \gg m_{e}/E_{0}$, given by the photon detector acceptance 
$\Theta_{a}\approx 0.45$\;mrad.  Finally, since contributions from 
$Z^{0}$ boson exchange are small for the $Q^{2}$ range relevant to 
this analysis, only pure photon exchange is considered.  
Radiative corrections of ${\cal O}(\alpha^{4})$ are represented by the
term $1+\delta_{\rm RC}$. 


\section{Event Simulation and Higher Order Corrections}
\label{sec:MC}

Acceptance corrections and background contributions are studied with 
both the data and Monte Carlo simulations.  Radiative and non-radiative DIS 
events were generated using DJANGO~\cite{Schuler:1991yg}.  This 
program uses HERACLES~\cite{Kwiatkowski:1992es} for 
simulation of the electroweak interactions and
LEP\-TO~\cite{Ingelman:1997mq} to simulate the hadronic final states 
using the colour dipole model (CDM) as implemented in 
ARIADNE~\cite{Lonnblad:1992tz}.  The hadronisation is performed with 
JETSET, which is based on the LUND string 
model~\cite{Bengtsson:1987kr,Sjostrand:1987hx}.

Since HERACLES includes all corrections to 
deep-inelastic $ep$ scattering up to ${\cal O}(\alpha^3)$, the DJANGO 
program is well suited to generate ISR signal events.  Radiative 
corrections of higher order are, however, not included and were
estimated using recent calculations~\cite{Anlauf:1999re, 
Anlauf:1998fg, Anlauf:1999ns} especially performed for the case of 
collinear photon radiation off the incident lepton.
% These
% corrections take into account the contributions of those terms at
% ${\cal O}(\alpha)$ that involve leading and next-to-leading large
% logarithms.  
The calculations 
were adapted to the specific selection criteria of this analysis and 
allow the computation of bin-by-bin correction factors for both 
the electron and $\Sigma$ reconstruction methods.  The resulting correction 
factors depend strongly on the method used and reach 
values of $5\%$ ($20\%$) for the $\Sigma$--method (electron method).

One of the main difficulties in correctly simulating the ISR signal is 
the modelling of events in which the measured photon energy is 
artificially raised by an additional photon from the coincidental 
overlap of a
Bethe-Heitler reaction.  For this purpose Bethe-Heitler events were 
simulated and randomly added to the ISR Monte Carlo according to an 
overlap probability extracted from the experimental
data~\cite{Issever:2000ci}.

The overlap of Bethe-Heitler events with other $ep$ reactions 
leads to the two main background sources. These are (i) the overlap 
of bremsstrahlung events with a non-radiative DIS process faking an 
ISR signature (DISBH) and (ii) $\gamma p$ interactions with an 
additional Bethe-Heitler overlap (GPBH) where part of the hadronic 
final state mimics the electron.  Photoproduction (GPBH) 
background is simulated using the PHOJET~\cite{Engel:1996yd} generator 
by overlaying real Bethe-Heitler data and the generated 
sample~\cite{Issever:2000ci}.  For the DISBH background the same 
method is applied using generated DIS events.  For both, an 
average overlap probability for Bethe-Heitler events with 
$E_{\gamma}>5$\;GeV of $4.21 \pm 0.07$\% was determined.

The detector response of the Monte Carlo events was simulated in 
detail with a program based on GEANT~\cite{Brun:1987fy}.  After this 
step the events were subjected to the same reconstruction and analysis 
chain as the real data.


\section{Event Selection}
\label{sec:selection}

\begin{table}[t]
\begin{center}
\begin{tabular}{l@{\hspace{3cm}}l}
  {\bf Item} & {\bf Cut value} \\ 
  \hline 
  \hline
  Photon ID               & \rule[-2mm]{0mm}{6mm} $E_{\gamma}>8$\;GeV          \\ 
  \hline 
  Electron ID             & \rule[-2mm]{0mm}{6mm} $E_{e}>5$\;GeV               \\ 
                          & \rule[-2mm]{0mm}{6mm} $E_{\rm had}<0.5$\;GeV       \\
                          & \rule[-2mm]{0mm}{6mm} $R_{\rm clu}<3.5$\;cm        \\
  \hline
  Background              & \rule[-2mm]{0mm}{6mm} $|z|<35$\;cm, $W>30$\;GeV    \\ 
  rejection               & \rule[-2mm]{0mm}{6mm} $E_e+E_{\gamma}<33$\;GeV     \\
                          & \rule[-2mm]{0mm}{6mm} $|(E_{\gamma}-E_{\rm miss})/E_{\gamma}|<0.5$ \\ 
                          & \rule[-2mm]{0mm}{6mm} $\sum E_{i}-p_{z,i}<61$\;GeV \\ 
                          & \rule[-2mm]{0mm}{6mm} $E_{\it ET}<2$\;GeV          \\
  \hline
  \hline
\end{tabular} 
\caption{Summary of ISR selection criteria as described in text.}
\label{tab=selection}
\end{center}
\end{table}


% \begin{table}
% \begin{center}
% \begin{tabular}{llll}
%   selection cut & 1996 & 1997 \\ 
%   \hline 
%   \hline 
%   photon energy & $>7$\,GeV & $>8$\,GeV \\ 
%   SpaCal energy & $>6$\,GeV & $>5$\,GeV \\ 
%   radius of cluster & $<3.5$\,cm & $<3.5$\,cm \\
%   energy behind cluster in hadronic SpaCal 
%                   & $<0.5$\,GeV & $<0.5$\,GeV \\
%   $E_e$+$E_{\gamma}$  & (?)  & $<33$\,GeV \\
%   z vertex position & $|z|<35$\,cm & $|z|<35$\,cm \\
%   $W^2$ & -- & $<0.01\cdot s$ \\ 
%   $|(E_{\gamma}-E_{\rm miss})/E_{\gamma}|$ & $<0.5$ &$<0.5$ \\ 
%   $E-p_z$ & $<61$\,GeV (?)  & $<61$\,GeV \\ 
%   $E_{\it ET}$ & $<2$\,GeV & $<2$\,GeV \\
% \end{tabular} 
% \caption{ISR selection criteria applied for the 1996
% and 1997 data  periods.}
% \label{tab=selection}
% \end{center}
% \end{table}

The presented results are derived from data which were recorded 
during the 1997 data taking period and correspond to an integrated 
luminosity of $11.1$\;pb$^{-1}$.
% Due to special trigger conditions not available during 
% 1997 the 1996 data sample especially allows exploration of the lowest 
% $Q^{2}$ range ($Q^{2}<0.5$\;GeV$^{2}$), while it only adds little (?)  to 
% the significance of most of the 1997 results at $Q^{2}$ above 
% $0.5$\;GeV$^{2}$.
%
% The two data samples undergo slightly different selection criteria 
% taylored to the running conditions of the individual years.  
% Naturally, the analysis strategy for both samples is, however, the 
% same: ...
%
The analysis strategy is based on the detection of a photon in the H1 
luminosity system and the requirement of an electron signature in the 
backward calorimeter, SpaCal.  In addition, the selection demands a 
vertex in the vicinity of the nominal interaction point and places 
several constraints derived from momentum/energy conservation in 
order to reduce background due to the overlap of Bethe-Heitler reactions 
with non-radiative events.
The selection criteria are given in Table~\ref{tab=selection} and 
summarized in the following:

\begin{itemize}
        
        \item {\bf Photon identification:} An energy deposition 
        $E_{\gamma}$ above $8$~GeV is required inside the photon 
        detector well separated, i.e\ more than one Moliere radius away, 
        from the edges of the detector.
        
        \item {\bf Electron identification:} The scattered electron is 
        assumed to be the cluster of maximum energy $E_{e}$ in the 
        electromagnetic part of the backward calorimeter. Within a 
        radius of $15$~cm of the electromagnetic shower centre 
        there must be no further energy deposition $E_{\rm had}$ of 
        more then $0.5$~GeV detected in the hadronic SpaCal.  In 
        addition, to guarantee a selection of clean electromagnetic 
        showers, a cut of $3.5$~cm is used on the energy weighted 
        cluster radius
        \begin{equation} 
             R_{\rm clu} = \frac{\sum_i E_i \sqrt{(x_{\rm clu}-x_i)^2 + 
                                  (y_{\rm clu}-y_i)^2}}{E_{\rm clu}}\,, 
        \end{equation}  
        where the sum runs over all cells $i$ of a cluster with energies
        $E_i$ and coordinates $x_i$, $y_i$; $x_{\rm clu}$, $y_{\rm clu}$ 
        represent the cluster coordinates and $E_{\rm clu}$ is the cluster energy.  
	Finally, an electron  
        candidate must be associated with a track segment in the 
        BDC~\cite{Adloff:2000qk}; thus only clusters within the acceptance of 
        the BDC are subject to the analysis: $156^{\circ} < 
        \Theta_{\rm BDC}< 175^{\circ}$.  A minimum energy requirement 
        of $E_{e}>5$\;GeV is applied.
        
        \item{\bf Background rejection:} Only events with a 
        reconstructed vertex with $|z_{\rm vtx}|<35$~cm are 
        considered; this substantially reduces contributions from 
        beam-gas and beam-wall interactions.  To obtain a reasonably 
        high vertex reconstruction efficiency of more than $90$\% a cut 
        on the $\gamma^* p$ center-of-mass energy of $W>30$~GeV is 
        applied to guarantee sufficient hadronic activity in the main 
        detector. 
	This only affects the high-$x$ region and is taken into account
	in the acceptance calculation. 
        DIS background with overlapping Bethe-Heitler 
        reactions is reduced by requiring $E_{e}+E_{\gamma}<33$\;GeV 
        as expected from energy conservation.  Since it is required 
        that the scattered 
        electron be seen in the main detector, no energy 
        $E_{\rm ET}$ should be observed in the electron tagger of the 
        luminosity system.  This is ensured by requiring $E_{\rm 
        ET}<2$\;GeV. Furthermore, the missing energy in the central detector, $E_{\rm miss}$, 
        must match the energy deposition measured in the photon system 
        such that
        \begin{equation}
              -0.5 < (E_{\gamma}-E_{\rm miss})/E_{\gamma} < 0.5\;\;,
        \end{equation}
        where $E_{\rm miss}$ is given by $E_{\rm 
        miss}=E_{0}-\frac{1}{2}\sum_{i} (E_{i}-p_{z,i})$ with the sum 
        running over all particles seen in the main detector components.  
        Longitudinal momentum conservation implies that $\sum_{i} 
        E_{i}-p_{z,i} = 2 E_0 \approx 55$\;GeV, if summed over all 
        final state particles including the radiated photon.  An 
        additional reduction of DISBH and GPBH background is thus 
        achieved by demanding $\sum_{i} E_{i}-p_{z,i}<61$\;GeV.
    
\end{itemize}

This selection was designed to have a high detection efficiency 
for radiative DIS events.  Within the kinematic range studied the total 
efficiency is better than $95$\%.  After the final selection $13\,701$ data 
events remain.


\section{Calibration and Background Estimate}
\label{sec:calib}

Since the selection and kinematic reconstruction rely strongly on 
the energy measurement in the backward calorimeter, SpaCal, and the 
photon detector of the luminosity system, the accurate calibration of these 
detectors is essential for the analysis.

The energy scale of the SpaCal has been carefully determined with QED 
Compton events using the double angle method for kinematic 
reconstruction of the energies of the photon and the scattered 
electron~\cite{Lendermann:2000ci}. The remaining uncertainty 
is $0.5$\% for energies $E_{e}$ above $12$~GeV and less than $4$\% for
$E_{e}<12$~GeV.   
The results are in good agreement 
with the calibration used in the most recent $F_{2}$ 
analysis~\cite{Adloff:2000qk}.  
%Comparison of data and Monte Carlo 
%reveal a remaining uncertainty of 0.5\% for energies $E_{e}$ above 
%12~GeV and less than 4\% for $E_{e}<12$~GeV. 
The photon detector is calibrated using Bethe-Heitler events in which  
both the photon and the electron were detected in the luminosity 
system.  By fitting the theoretically well known Bethe-Heitler 
spectrum~\cite{Kotkin:1988dx} to the observed photon energy 
distribution, the absolute energy scale of the photon detector is 
determined to better than $1$\%~\cite{Issever:2000ci}.  
For the selected event sample, Figure~\ref{fig=control} shows several energy and
angular distributions for the scattered electron, the tagged photon and the 
hadronic final state, demonstrating adequate agreement between the data 
and the Monte Carlo simulation.  Concerning the hadronic final state, the 
calibration is done using the $p_{t}$ balance between the scattered 
electron and the hadronic jets, as described in~\cite{Adloff:2000qk, 
Wallny:2000rw,Glazov:1998ag}.  The remaining differences observed are 
covered by the experimental systematic errors described below.

\begin{figure}[t]
  \centerline{\epsfig{figure=H1prelim-01-042.fig2.eps,width=0.8\textwidth}}
  \caption{Experimental (points) and Monte Carlo (histogram) 
  distributions of (a) the energy $E_{e}$ of the scattered electron 
  and (b) the corresponding scattering angle, (c) the energy 
  $E_{\gamma}$ of the ISR photon, (d) $E_{e}+E_{\gamma}$, (e) the 
  longitudinal momentum balance $\Sigma_{H} = \sum_{i} E_{i} - 
  p_{z,i}$, where the sum runs over all particles of the hadronic 
  final state, and (f)~the polar angle $\Theta_{H}$ of the hadronic system.  The 
  open histograms show the total Monte
  Carlo prediction which, in addition to the ISR signal, includes 
  contributions from Bethe-Heitler reactions overlayed with 
  non-radiative DIS processes or photoproduction events.  The shaded 
  band indicates the statistical accuracy of the Monte Carlo 
  simulation.}
  \label{fig=control}
\end{figure}

\begin{figure}[tb]
  \centerline{\epsfig{figure=H1prelim-01-042.fig3.eps,width=\textwidth}}
  \caption{Photon energy distribution for background events with an 
  energy deposition in the electron tagger of the luminosity system 
  for three different energy ranges of the electron candidate seen in 
  the SpaCal: (a) $22<E_{e}<30$\;GeV (b) $12<E_{e}<22$\;GeV (c) $ 
  5<E_{e}<12$\;GeV. Data are shown as points, the Monte Carlo 
  prediction for the different background contributions as histograms.  
  As described in section~\ref{sec:MC}, the two main background 
  sources arise from overlap of bremsstrahlung events with the 
  non-radiative DIS process (DISBH) and $\gamma p$ interactions 
  (GPBH).  In addition, the ISRBH contribution shown represents signal events with an 
  coincidentally overlapping Bethe-Heitler process in which the electron was seen in 
  the electron tagger.  It should be noted that the observed 
  differences between the three $E_{\gamma}$ spectra arise from the 
  requirement that the energy sum $E_{e}+E_{\gamma}<33$\;GeV, which 
  restricts the photon energy $E_{\gamma}$ for higher values of 
  $E_{e}$.}
  \label{fig=background_check}
\end{figure}

The second important issue is the understanding of the different 
background contributions due to coincidental overlap of DIS and $\gamma p$ 
reactions with Bethe-Heitler events, as mentioned in 
section~\ref{sec:MC}.  In order to test the modelling of the 
Bethe-Heitler overlap, and in particular to control the normalization 
of the background Monte Carlo, a clean sample of background events had 
to be selected.  This was possible since $30$\% of such events can be 
directly identified by detecting the electron from the Bethe-Heitler reaction. 
This was done by requiring an energy deposition $E_{\rm ET}$ above 
$2$\;GeV in the 
electron tagger of the luminosity system.  
Figure~\ref{fig=background_check} shows the $E_{\gamma}$ distribution 
for these events 
for three ranges of the energy $E_{e}$ of the electron candidate seen 
in the SpaCal; good agreement between data and Monte Carlo is 
observed.  Since the background sample with $22<E_{e}<30$~GeV is 
dominated by DIS events with Bethe-Heitler overlap, it can be 
used to check the normalization of the DISBH background.  For 
lower values of $E_{e}$ there is an additional contribution from GPBH 
events due to parts of the hadronic final state faking an electron 
signature.  A fit of the $E_{\gamma}$ Monte Carlo distributions to the 
data leads to normalization factors of $0.82$ for the DISBH and $1.04$ for 
the GPBH simulation which are applied in Figure~\ref{fig=background_check}.
The deviation of these normalization factors 
from one is included in the systematic error of the measurement.


\section{Results}
\label{sec:results}

In order to extract the structure function $F_{2}$ in terms of $x$ and 
$Q^{2}$ the selected data sample is divided into subsamples 
corresponding to a grid in $x$ and $Q^2$ as illustrated in 
Figure~\ref{fig=binning}.  The bin sizes are adapted to the resolution 
in the measured kinematic quantities.  They are increased in regions 
where the number of events is low. Bins are accepted if the purity and 
stability are greater than $30$\% with typical values being $50$\%, where 
the purity (stability) is defined as the ratio of the number of 
simulated events originating from and reconstructed in a specific bin 
to the number of reconstructed (generated) events in the same bin.  
The proton structure function $F_2$ was then extracted using a 
bin-by-bin unfolding method.

With the number of events $N$ ($N_{\rm MC}$) reconstructed in the data 
(Monte Carlo) and the background contribution, $N_{\rm BG}$, estimated 
from the DISBH and GPBH Monte Carlo, the numerical value of the 
structure function at a given point ($x$,$Q^{2}$) is obtained from the 
ratio ($N-N_{\rm BG})/N_{\rm MC}$ times the corresponding $F_{2}^{{\rm 
MC}}$ value of the parameterization used in the simulation:
\begin{equation}
   F_{2}(x,Q^{2}) = \frac{N-N_{\rm BG}}{N_{\rm MC}} F_{2}^{{\rm MC}}(x,Q^{2}) 
   \cdot \left( \frac{{\cal L}_{\rm MC}}{{\cal L}} \right)
   \frac{1\;\;}{1+\delta_{\rm RC}}\;\;\; \raisebox{-10pt}{.}
   \label{eq=F2}
\end{equation}
The additional factor ${\cal L}_{\rm MC}/{\cal L}$ takes account of 
the different luminosities for data and Monte Carlo, and $\delta_{\rm 
RC}$ corrects for higher order QED effects not included in the simulation 
(see section~\ref{sec:MC}).  Application of~(\ref{eq=F2}) implies a correct 
modelling of the detector acceptance, selection efficiencies and 
migrations, which have been carefully checked~\cite{Issever:2000ci}.  
% In the simulation the ALLM97\cite{Abramowicz:1997ms} structure 
% function parameterization is used to reweight events at low 
% $Q^{2}<0.5$\;GeV$^{2}$, while for $Q^{2}>0.5$\;GeV$^{2}$ the most 
% recent H1-fit to $F_{2}$-data~\cite{Adloff:2000qk} is taken as 
% input
%\footnote{It has been shown that the final result does not depend 
%strongly on the input parameterization of the structure function if it 
%only roughly reproduces the general trend of the data.  If starting 
%with a completely uncorrelated $F_{2}$ parameterization one in 
%principle would have to use an iterative procedure as described 
%in~\cite{Adloff:1997mf} which finally would, however, lead to the same 
%results.}.  
The influence of the longitudinal structure function is assumed to be 
negligible within the kinematic range studied; any impact on the 
measurement due to this assumption is estimated using the Monte Carlo and 
included into the systematic error.

\begin{figure}[t]
\begin{minipage}{0.59\textwidth}
  \centerline{\epsfig{figure=H1prelim-01-042.fig4.eps,height=\textwidth}}
\end{minipage}
\raisebox{-1.0cm}{
\begin{minipage}{0.39\textwidth}
  \caption{  
  \label{fig=binning}
  Division of the ($x$,$Q^{2}$)--plane for the measurement of 
  the proton structure function $F_{2}$ from radiative DIS
  events.   
  The figure shows the distribution of the selected ISR events as
  expected from simulation.  The shaded bins are those accepted by the
  quality criteria described in the text.}
\end{minipage}}
\end{figure}

Systematic uncertainties have been determined by studying the 
stability of the results under variation of energy scales and detector 
resolutions within the accuracies quoted below, variations of selection 
efficiencies, changes of the background normalization and modification 
of the theoretical input:

\begin{itemize}

   \item The electron energy scale is varied by $0.5$\% for 
   $E_e>12$\;GeV and $2.5$\% at $E_e=4$\;GeV, where its uncertainty is assumed to  
   decrease linearly in the energy range $4<E_e<12$\;GeV. This 
   variation leads to an average uncertainty on the $F_2$ measurement 
   of $3$\%.

   \item The uncertainty of $0.5$\;mrad on the measurement of the polar 
   scattering angle of the electron leads to an uncertainty of about
   $1$\% on $F_2$.
  
   \item The uncertainty on the energy scale of the photon detector 
   reaches values up to $1$\% and leads to an average uncertainty on $F_2$ 
   of $2.5$\%.

   \item The energy resolution of the photon detector is varied within its 
   uncertainty resulting in a $3$\% systematic error contribution 
   on $F_{2}$.
   
   \item The accuracy of the hadronic energy scale is influenced by the 
   uncertainties of the energy measurement in the LAr calorimeter 
   ($\pm 4$\%), the energy measurement of the SpaCal ($\pm 5$\%) and 
   the track momenta measured with the tracking detectors ($\pm 2$\%).  
   In total, the hadronic energy scale uncertainty effects $F_{2}$ by 
   about $4$\%.

   \item The background normalization is known to within 
   $18$\% and leads to an uncertainty of less than $1$\% on $F_{2}$.  

   \item The modelling of the Bethe-Heitler photon spectrum overlayed 
   with the ISR signal Monte Carlo (see section~\ref{sec:MC}) is done 
   from a minimum photon energy of $E^{\rm 
   min}_{\gamma}=0.15$\;GeV, needed to penetrate an active filter
   placed in  
   front of the photon detector to protect it from synchrotron 
   radiation.  The uncertainty on this limit is $\pm 0.02$~GeV and 
   adds about $5$\% to the systematic error on the final result.
   
   \item The vertex reconstruction efficiency is known 
   to about $4$ to $5$\% and
   leads to a systematic error contribution of $5$\% to the $F_{2}$ 
   measurement.
   
   \item Changing $F_{L}$ in the acceptance calculation from zero to
   the value resulting from the  
   most recent H1 NLO QCD-fit~\cite{Adloff:2000qk} leads to a 
   variation of the structure function $F_{2}$ of $0.8$\%.
      
   \item The estimated uncertainty of the higher order 
   corrections~\cite{Anlauf:2001pc} leads to an additional systematic 
   error of 1\% on the $F_2$ measurement.
%    For the higher order corrections an absolute uncertainty of 
%    $\pm 1$\% was assumed, which affects $F_{2}$ directly.
%    and thus leads to an additional contribution of $1$\% to the total systematic 
%    error of the measurement.
   
\end{itemize}

The total systematic error is determined by summing the individual 
contributions quoted in quadrature.  Finally a $1.5$\% normalization uncertainty of 
the luminosity measurement has to be considered, which is not included 
in the systematic error of the measurement shown in the 
figures.

The result of the measurement is given in Figure~\ref{fig=F2_in_x_q2} 
which shows $F_{2}$ in bins of $x$ and $Q^{2}$ for $2 \cdot 10^{-5} 
\leq x \leq 2 \cdot 10^{-2}$ and $0.35 \leq Q^{2} \leq 20$\;GeV$^{2}$.  
The data points are compared to predictions from the ALLM97 
parameterization~\cite{Abramowicz:1997ms}, a model based on Regge phenomenology, 
and the recent H1 QCD-fit~\cite{Adloff:2000qk} to H1 and BCDMS 
non-radiative $F_{2}$ data with $Q^{2}>3.5$\;GeV$^{2}$.  
At larger $Q^{2}$ the data are well 
described by the pQCD based H1 parameterization.
% thus indicating good 
%agreement of this new measurement with the most recent H1 precision 
%$F_{2}$ results.  
For $Q^{2}$ values below $1$\;GeV$^{2}$ perturbative 
QCD however fails
\footnote{The presented H1 parameterization using 
$Q^{2}_{\rm min}=3.5$\;GeV$^{2}$ as starting scale shows deviations 
from the data at $Q^{2}<3$\;GeV$^{2}$.  It has however been shown that 
$F_{2}$ can successfully be described down to $Q^{2}=1.5$\;GeV$^{2}$ 
when $Q^{2}_{\rm min}$ is chosen accordingly\cite{Adloff:2000qk}.} 
and only non-perturbative models, e.g. those which incorporate Regge 
phenomenology, can describe the measurement.

In order to allow comparison with earlier $F_{2}$ measurements, 
Figure~\ref{fig=F2_in_q2_fixedW} shows the $Q^{2}$ dependence of the 
structure function from various analyses at fixed values of the 
photon-proton centre-of-mass energy $W$ between $25$~GeV and $230$~GeV. 
The $F_{2}$ data shown in Figure~\ref{fig=F2_in_x_q2} are transformed 
to the $W$ values given using the 
ALLM97 parameterization.  The results based on the analysis of 
radiative events span a region which reaches from 
$Q^{2} \geq 1.5$\;GeV$^{2}$ where pQCD calculations are applicable to the 
non-perturbative regime, thus probing the transition between 
DIS and photoproduction within a single data 
set.  The new data are in good agreement with earlier $F_{2}$ results 
both for the low $Q^{2}$ domain~\cite{Breitweg:2000yn} and the pQCD 
regime~\cite{Adloff:2000qk}.  In the transition region they agree with 
the earlier H1 result~\cite{Adloff:1997mf}.  Data from an earlier
preliminary  
analysis~\cite{Favart:1998fa} using radiative events from the year 
1996 are also shown where they extend this measurement to lower 
$Q^{2}$.
% due to a different detector configuration during that year.


\section{Conclusion}
\label{sec:Conclusion}

Deep-inelastic scattering data with initial state photon radiation are 
used to extract the proton structure function $F_{2}$ for 
four-momentum transfers $0.35 \leq Q^{2} \leq 20$\;GeV$^{2}$ and 
Bjorken-$x$ values of $2 \cdot 10^{-5} \leq x \leq 2 \cdot 10^{-2}$.  
The results provide a 
link between the deep-inelastic scattering (DIS) regime and 
photoproduction and thus aid the development of an understanding of
the transition from 
regions where pQCD calculations hold into the domain in which 
non-perturbative effects become large.  The measurements are in 
agreement with previous structure function measurements performed at 
HERA and with predictions based on a phenomenological approach to the 
transition region between DIS and photoproduction.


\section*{Acknowledgements}

We are grateful to the HERA machine group whose outstanding efforts 
have made and continue to make this experiment possible.  We thank the 
engineers and technicians for their work in constructing and now 
maintaining the H1 detector, our funding agencies for financial 
support, the DESY technical staff for continual assistance, and the 
DESY directorate for the hospitality which they extend to the non DESY 
members of the collaboration.  We also would like to thank H.\ Anlauf 
for providing the code to estimate the higher order corrections and 
for many valuable discussions.

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Z.\ Phys.\ C {\bf 39} (1988) 61.
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%
%\cite{Glazov:1998ag}
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A.~A.~Glazov,
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%\cite{Wallny:2000rw}
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R.~Wallny,
Dissertation, ETH Z\"urich, in preparation.
%\cite{Anlauf:2001pc} 
\bibitem{Anlauf:2001pc} 
H.~Anlauf, privat communication.
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\end{thebibliography}

\cleardoublepage

\begin{figure}[t]
  \centerline{\epsfig{figure=H1prelim-01-042.fig5.eps,height=.8\textheight}}
  \caption{The structure function $F_2$ as determined from ISR events 
  as a function of $x$ for fixed values of $Q^2$.  The inner error 
  bars represent the statistical and the outer error bars the total 
  uncertainty of the measurement.  The data are compared to the ALLM97 
  (solid curve) parameterization and the most recent H1 NLO QCD fit 
  to non-radiative $F_2$ data (dashed curve) starting at $Q^2_{\rm 
  MIN} = 3.5$\;GeV$^{2}$~\cite{Adloff:2000qk}.}
  \label{fig=F2_in_x_q2}
\end{figure}

\begin{figure}[t]
  \centerline{\epsfig{figure=H1prelim-01-042.fig6.eps,height=.8\textheight}}
  \caption{The structure function $F_2$ as a function of $Q^2$ for 
  fixed values of $W$.  ISR measurements (closed circles: this 
  analysis; open circles: H1 radiative ¹96~\cite{Favart:1998fa}) are shown 
  together with the ¹96/97 non­radiative $F_2$ results of 
  H1~\cite{Adloff:2000qk} (triangles), the $F_2$ shifted vertex ¹95 
  data~\cite{Adloff:1997mf} (diamonds) and the low $Q^{2}$ results 
  from ZEUS~\cite{Breitweg:2000yn} (open crosses).  
  The measurements are compared with the ZEUS Regge fit (dashed curve) 
  and with the H1 NLO QCD fit to the non-radiative $F_2$ 
  data (solid curve) starting at $Q^2_{\rm MIN} = 
  3.5$\;GeV$^{2}$~\cite{Adloff:2000qk}.}
  \label{fig=F2_in_q2_fixedW}
\end{figure}


\end{document}