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\begin{document}

\pagestyle{empty}
\begin{titlepage}


\noindent
\begin{center}
%{\it {\large version of \today}} \\[.3em] 
\begin{small}
\begin{tabular}{llrr}
%Submitted to & \multicolumn{3}{r}{\footnotesize Electronic Access: {\it http://www-h1.desy.de/h1/www/publications/conf/conf\_list.html}} \\[.2em] \hline 
%Submitted to & \multicolumn{3}{r}{\footnotesize {\it www-h1.desy.de/h1/www/publications/conf/conf\_list.html}} \\[.2em] \hline 
Submitted to & & &
\epsfig{file=/h1/www/images/H1logo_bw_small.epsi
,width=2.cm} \\[.2em] \hline
                & & & \\
\multicolumn{4}{l}{{\bf
                XXII International Symposium on Lepton-Photon Interactions at High Energy},
                June~30,~2005,~Uppsala} \\
                 & Abstract:        & {\bf 389}    &\\
                 & Session: & {\bf QCD and hadron structure}   &\\ \hline
 & \multicolumn{3}{r}{\footnotesize {\it
    www-h1.desy.de/h1/www/publications/conf/conf\_list.html}} \\[.2em]
\end{tabular}
\end{small}
\end{center}
\vspace*{2cm}



\begin{center}
  \Large
{\bf Measurement of the Inclusive DIS Cross Section at Low {\boldmath $Q^2$}
and High {\boldmath $x$} using Events with Initial State Radiation}
  {\bf
  \vspace*{1cm}

  {\Large H1 Collaboration \\
}
}
\end{center}


%%%%%
\begin{abstract}
The inclusive DIS cross section is measured at low $Q^2$ and relatively large
$x$ using events with initial state photon radiation from the 
incoming electron. In this analysis the radiated photon is not explicitly
detected. Instead its energy is inferred from a longitudinal momentum 
imbalance, such that the energy of the interacting electron and the event
kinematics can be reconstructed. The neutral current cross section
is thus measured
for $0.35 \leq Q^2 \leq 0.85 \ {\rm GeV^2}$
and $10^{-4} \ \lapprox \ x \ \lapprox \ 5 \cdot 10^{-3}$.
\end{abstract}

\end{titlepage}
%
\pagestyle{plain}

\section{Introduction}
%
The region of momentum transfer squared $Q^2$ around 1~GeV$^2$
deserves particular attention because it corresponds to the transition
between the non-perturbative and the perturbative domains in deep-inelastic scattering (DIS).  Thus at HERA a special electron-proton scattering run was
performed in which the interaction vertex was shifted by about 70~cm
in the proton beam direction which allows larger electron scattering
angles\footnote{Note that the polar angles $\theta$ are defined with respect 
to the proton beam direction.} $\theta_e$   
and thus lower values of $Q^2$ to be accessed.

The data presented here were taken in August 2000 and correspond to a luminosity
of 0.6\pb . The results represent an extension to our previous measurements \cite{sv00}
of inclusive DIS cross sections at $Q^2 \ \lapprox \ 1 \ {\rm GeV^2}$ based on the same data.
In this analysis, an extension to larger $x$ values
at low $Q^2$ is achieved by making use of events with initial state photon 
radiation (ISR) from the incoming
electron. The photon radiation results in a reduced incoming
electron energy, such that the $ep$ centre of mass energy is reduced and
larger values of $x$ are accessed at fixed $Q^2$ than is the case for
non-radiative events. 
   The kinematic plane and its extension with the new analysis
 is illustrated in Figure~\ref{kinfull}.

The inclusive measurements determine the cross section which at low $Q^2$, 
in the one-photon exchange approximation, can be written in reduced form as
%
\begin{equation}
   \frac{Q^4 x}{2\pi \alpha^2 Y_+}  \cdot \frac{d^2\sigma}{dxdQ^2}  =
 \sigma_r =        F_2(x,Q^2) - \frac{y^2}{Y_+} \cdot F_L(x,Q^2).
       \label{sig}
  \end{equation}
%
Here $y$ is the inelasticity which is related to $x$ and $Q^2$ as
$y=Q^2/sx$. The beam energies determine the centre of mass energy 
squared, $s=4E_eE_p$, and $Y_+$ is defined as
$1+(1-y)^2$.  In the region of inelasticity  below $y = 0.6$ the
contribution of the longitudinal structure function \FL is small
due to the kinematic factor $y^2/Y_+$ and since $F_L \le F_2$. 
Thus the measurement of $\sigma_r$ at lower $y$ directly determines 
\Fc with a small correction for $F_L$.

%Section 2 of this paper describes the experimental methods, i.e.
%the kinematic reconstruction with the H1 detector, the event
%selection, simulation and calibration.  The cross-section
%measurement and the extraction of the structure function \F are
%discussed in Section~3 which is complemented by a study of the rise of
%\F towards low $x$. A brief summary is given in Section~4.
%
%-------------------------------------------------------------
\section{Experimental Methods}
%-------------------------------------------------------------
%
\subsection{Kinematic Reconstruction}
%
In contrast to earlier ISR analyses \cite{ISRold}, this analysis does not
require the observation of the radiated photon. Instead, its presence is
inferred from energy and longitudinal momentum conservation. 
Assuming that the photon is radiated collinear with the electron beam, 
the energy
$E_\gamma$ of the radiated photon is then given by 
$2 E_\gamma = 2 E_e^0 - (E-p_z)_{e^\prime} - (E-p_z)_h$, where 
$E_e^0 = 27.6 \ {\rm GeV}$ is the electron beam energy, 
$(E-p_z)_{e^\prime}$ is the measured difference between the energy and
the longitudinal momentum of the scattered electron and
$(E-p_z)_h$ is the same quantity for the full hadronic final state. The
reduced incoming electron beam energy $E_e$ is then given by 
$E_e = E_e^0 - E_\gamma$.
The reconstruction of $y$ and $Q^2$ proceeds using the 
`$\Sigma$ method', which relies on $\Sigma = (E - p_z)_h$. The reconstruction
of $x$ is then performed in a manner that is independent
of the electron beam energy $E_e^0$. The kinematics are resonstructed using
\begin{eqnarray*}
  y_\Sigma = \frac{\Sigma}{\Sigma + E_{e^\prime} (1 - \cos \theta_e)}
  \hspace*{1.5cm}
  Q^2_\Sigma = \frac{E_{e^\prime}^2 \sin^2 \theta_e}{1 - y_\Sigma}
  \hspace*{1.5cm}
  x_R = \frac{Q^2_\Sigma}{2 \Sigma E_p}
\end{eqnarray*}
where $\theta_e$ is the electron scattering angle. 


 The scattered electron  
energy $E_e'$ is measured in the backward electromagnetic lead 
scintillating fibre calorimeter SPACAL \cite{Spacal}. The polar angle 
$\theta_e$ is measured in the Backward Silicon Tracker (BST) \cite{vova, doris, tomas2}. 
%It is defined with respect to the proton 
%beam direction, defining the $z$ axis. 
%%The BST has a configuration of 8 
%%detector planes placed perpendicularly to the beam axis,
%%each consisting of 16 concentric wafers measuring the radial hit coordinates.
%A scattered positron candidate is defined as a vertex pointing track
%associated to the highest energetic cluster in SPACAL, where the cluster 
%is required to extend by less than two Moli\`{e}re radii in the transverse 
%plane.
The hadronic final state  is reconstructed in the Liquid
Argon calorimeter (LAr) and the SPACAL~\cite{h1detec}. 

%
\subsection{Triggers and Event Selection \label{evsel}}
%
The data are triggered using the local energy sums in the SPACAL calorimeter 
with an energy threshold set to 6~GeV. Low energy deposits can also be caused 
by hadrons and photons from events at very low $Q^2\ \ll\ 1$\gv\, 
which mimic an electron signal in the SPACAL. Part of these photoproduction 
background events is recognised by tagging a scattered electron at very 
small angles in the electron tagger calorimeter upstream the electron beam.

The efficiency of all trigger elements exceeds 98\% and is
controlled by independent tracking triggers to an accuracy of
0.5\%.  From a monitor event sample, defined by a vertex
accurately reconstructed in the central tracker and by a high
energy SPACAL cluster, the BST efficiency is determined and the 
Monte-Carlo simulation correspondingly adjusted.
The hit efficiency of the BST is 97\% on average,
excluding a few malfunctioning sensor modules.

DIS events are required to have a vertex reconstructed from a track
measured in the BST and its intersection with the beam axis. The track
has to be associated to the highest energetic cluster in the SPACAL, where 
the cluster is required to extend by less than two Moli\`{e}re radii in 
the transverse plane. Any energy behind the
electromagnetic cluster measured in the hadronic SPACAL may not exceed
a small fraction of $E'$. 
The criteria of the DIS event selection are summarised in
Table~\ref{tabcut}.
%
\begin{table}[thb] \centering
\begin{tabular}{|l|c|}
\hline
$z$ vertex position     &  $|z-70| < 45$cm           \\
SPACAL cluster radius   &  $< 4$cm                   \\
SPACAL-BST  matching     &  $\delta r < 2$cm         \\
electromagnetic SPACAL energy      & $ >$ 7~GeV      \\
hadronic SPACAL energy      &  $<$ 15\% of $E_{e}'$  \\
\hline
\end{tabular}
\caption{\label{tabcut} {\it Basic criteria to select DIS events.}}
\end{table}
The selection is identical to the selection from the standard analysis \cite{sv00}, with the exception of the $E-p_{z}$ cut, which in the previous analysis was used in order to remove radiative events.

A high statistics simulation of DIS
%and photoproduction
events is performed using the program
%, GEANT~\cite{geant},
DJANGO~\cite{django} with a parameterisation of the parton
distributions (MRST 3,75)~\cite{MRST} extended to very low $Q^2$.
For the extraction of the cross section and comparisons of
experimental with simulated spectra, a recent fit to previous low
$Q^2$ data \cite{thepaper, bpt} was used for reweighting which is
based on the fractal proton structure concept.  This
fit~\cite{tomas} describes the data in the non-perturbative region
and the data in the deep-inelastic domain very well. 
Photoproduction events are
simulated with the program PHOJET~\cite{fojet}. The simulated
events are subject to the same reconstruction and analysis chain
as the real data. In the comparisons of experimental distributions
with the Monte-Carlo spectra, these are normalised to the measured
luminosity.

The luminosity as determined from the cross section of the elastic
bremsstrahlung process is measured with a precision of 1.8\%. 
 
Other analysis details like alignment and calibration of the detector can be found elsewhere \cite{sv00}.
%The luminosity measurement is corrected by a few per cent for the
%occurrence of proton beam satellites which do not enter the selected
%data sample. This leads to an additional uncertainty of 1\% yielding a
%total luminosity measurement error in this analysis of 1.8\%.
%

%
%-------------------------------------------------------------------
%\section{Results}
%\section{The Cross Section and  {\boldmath $ F_2(x,Q^2)$}}
%-------------------------------------------------------------------
%
%
\subsection{Control Plots}
%
Control distributions corresponding to the measured ISR-dominated 
phase space are shown in figure~\ref{one}. 
There is good agreement between the observed uncorrected data distributions
and the expectation based on a DJANGO simulation of inclusive DIS together
with a PHOJET simulation of the photoproduction background. 

% Note that the
%normalisation of the photoproduction simulation was adjusted by about
%20\% based on the electron tagger information. The full size of this 
%renormalisation factor is chosen as the uncertainty of the photoproduction 
%simulation luminosity.
%
\subsection{Systematic Uncertainties}
The total systematic error of the cross section measurement with ISR events is about 10\%.
The main uncertainties and treir contributions to the cross section error are:
\begin{itemize}
\item correlated systematic errors:
\begin{itemize}
\item electron energy uncertainty -- up to 2\% for low energy electrons,
\item error of electron angle measurement with BST -- about 2\% ,
\item hadronic energy uncertainty -- about 4\% ,
\item photoproduction background uncertainty extend to 5\% at low energy ,
\end{itemize}
\item uncorrelated systematic errors:
\begin{itemize}
\item Monte Carlo statistics -- 4\%,
\item BST reconstruction efficiency -- 2\%,
\item radiative corrections -- 2\%,
\item trigger inefficiency -- 0.5\%.
\end{itemize}
\end{itemize}

\section{Results}
\subsection{Cross Section and the Proton Structure Function {\boldmath $ F_2$}}
The kinematic region accessed in this measurement is divided into four 
$Q^2$ intervals.
% in the range $0.35 < Q^2 < 0.85$~\gv. 
The data are also
divided in bins of $y$.  The binning is adapted to the resolution in
the measurement of the kinematic variables.  Bins are accepted if the
purity and stability are bigger than 30\%. Here the purity (stability)
is defined as the number of simulated events which originate from a
bin and which are reconstructed in it, divided by the number of
reconstructed (generated) events in that bin.

Figure~\ref{two} shows
the resulting measurements, expressed in the form of the reduced
cross section.
The data points
cover the region
$0.35 \leq Q^2 \leq 0.85 \ {\rm GeV^2}$
and $10^{-4} \ \lapprox \ x \ \lapprox \ 5 \cdot 10^{-3}$. They are
compared with the extrapolated
predictions of a fit for $F_2$ based on a self-similar
structure to the proton (FRACTAL FIT $F_2$) \cite{tomas},
to which an $F_L$ term has been added, based on a
dipole model \cite{kgbw}. They are also compared with the results from a 
parameterisation (ALLM97 $F_2$) \cite{allm} of inclusive
and exclusive DIS and photoproduction data. All predictions
are in good agreement with the data. 

The extension in
kinematic phase space achieved using the new method is 
illustrated in figure~\ref{three}, where the present data are
compared with the results from the non-radiative shifted
vertex analysis \cite{sv00}. Figure~\ref{four} shows the
overall status of DIS measurements in the low $Q^2$ region,
including in addition data from the ZEUS beampipe 
tracker \cite{bpt}, from a further H1 measurement at low $Q^2$ and
large $x$
using QED-Compton events \cite{qedc} and from NMC \cite{nmc}.
The new data are compatible with the previous measurements.

%
\subsection{Rise of {\boldmath $ F_2(x,Q^2)$} Towards Low  {\boldmath $x$}}
%
Recently the H1 Collaboration has presented~\cite{h1rise} a measurement
of the derivative
%
\begin{equation}
    \left( \frac{\partial \ln F_2(x,Q^2)}{\partial \ln x}\right)_{Q^2}  \equiv
    - \lambda(x,Q^2)
\end{equation}
%
which quantifies the behaviour of the rise of \F towards low $x$ at
fixed $Q^2$. 
The new analysis allows improved extractions of the parameter
$\lambda$. Figure~\ref{five} shows the results for $\lambda$ obtained
by fitting the current data
at fixed $Q^2$ values, together with the previous
shifted vertex measurements \cite{sv00}. Also shown are
H1 results from larger $Q^2$ values \cite{f29697}\cite{h1rise} and
ZEUS results \cite{bpt}. The new measurements
confirm the change in behaviour of $\lambda$ from a logarithmic
dependence on $Q^2$ at large $Q^2$ to a weaker dependence
compatible with reaching a constant consistent with the soft
pomeron intercept as $Q^2 \rightarrow 0$. The change takes
place on distance scales of the order of $0.3 \ {\rm fm}$
and can be interpreted as being related to a transition 
from partonic to hadronic degrees of freedom. 

%
%-------------------------------------------------------------------
\section{Summary}
%-------------------------------------------------------------------
%
A new measurement of the deep-inelastic electron-proton scattering
cross section based on H1 data collected with the interaction vertex shifted by 70 cm is presented.
The analysis introduces a novel method, which allows the previous analysis of this data set to be extended into the higher $x$ region by means of events with initial state photon radiation from the incoming electron. 
The radiated photon is not explicitly detected, but its parameters are inferred from energy and momentum conservation. 
The neutral current cross section
is thus measured
for $0.35 \leq Q^2 \leq 0.85 \ {\rm GeV^2}$
and $10^{-4} \ \lapprox \ x \ \lapprox \ 5 \cdot 10^{-3}$.
In the kinematic region of this
measurement \F still rises towards low $x$ at fixed $Q^2$ like
$F_2(x,Q^2) = c(Q^2) \cdot x^{-\lambda(Q^2)}$.  
The new analysis allows improved extractions of the parameter
$\lambda$ from the H1 data.

%
\vspace{0.5cm}
%
{\bf Acknowledgements}
%=====================
\normalsize
\noindent We are very grateful to the HERA machine group whose
outstanding efforts made this experiment possible. We acknowledge the
support of the DESY technical staff. We appreciate the substantial
effort of the engineers and technicians who constructed and maintain
the detector. We thank the funding agencies for financial support of
this experiment.  We wish to thank the DESY directorate for the
support and hospitality extended to the non-DESY members of the
collaboration.
%
%   References
%
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%
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%
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%
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%
\bibitem{vova}
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%
\bibitem{doris}
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%
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%
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%
%\bibitem{geant}
%R. Brun {\it et al.}, GEANT3 User's Guide, CERN-DD/EE 84-1, Geneva (1987).
%
\bibitem{django}
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%
\bibitem{MRST}
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%
\bibitem{thepaper}
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%
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%
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sigma(tot)(gamma* p): An update, DESY-97-251, [hep-ph/9712415] (1997).
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%
%\bibitem{sasha}
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%
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%
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%

\end{thebibliography}

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\begin{center}
\begin{picture}(150,150)(0,0)
\put(0,0){\epsfig{file=kinplane4b.eps, width=\textwidth}}
\end{picture}
\end{center}
\caption{Kinematic region covered by $F_2$ measurementsd at HERA and fixed targed experiments.
The extension to higher $x$ for low $Q^2$ region provided by the ISR events analysis shown.  }
\label{kinfull}
\end{figure}

\newpage

\begin{figure}[hhh]
\setlength{\unitlength}{1mm}
\begin{center}
\begin{picture}(150,150)(0,0)
\put(0,-20){\epsfig{file=H1prelim-04-042.fig1.eps, width=\textwidth}}
\end{picture}
\end{center}
\caption{Control distributions for the ISR-dominated event selection.
The uncorrected data distributions are compared with the sum of 
the DIS expectation (DJANGO Monte Carlo model) and the expected photoproduction
background (PHOJET Monte Carlo model). The comparisons are shown for the 
effective incoming 
electron energy $E_e$, the electron scattering angle $\theta_e$, 
the sum of the electron and
hadron $E - p_z$ and the reconstructed $y$, $Q^2$ and $x$ as used in the
measurement.}
\label{one}
\end{figure}

\newpage

\begin{figure}[hhh]
\setlength{\unitlength}{1mm}
\begin{center}
\begin{picture}(150,150)(0,0)
\put(0,-30){\epsfig{file=H1prelim-04-042.fig2.eps, width=\textwidth}}
\end{picture}
\end{center}
\caption{Measured reduced cross section from the shifted vertex ISR analysis,
compared with the extrapolated predictions of a `fractal' fit based on
self-similar proton structure, with the sum of the fractal fit result for
$F_2$ and a $F_L$ contribution from a dipole model and with the ALLM97
parameterisation.}
\label{two}
\end{figure}

\newpage

\begin{figure}[hhh]
\setlength{\unitlength}{1mm}
\begin{center}
\begin{picture}(150,150)(0,0)
\put(0,-30){\epsfig{file=H1prelim-04-042.fig3.eps, width=\textwidth}}
\end{picture}
\end{center}
\caption{Measured reduced cross section from both ISR and non-radiative
measurements using shifted vertex data.}
\label{three}
\end{figure}

\newpage

\begin{figure}[hhh]
\setlength{\unitlength}{1mm}
\begin{center}
\begin{picture}(150,150)(0,0)
\put(0,-30){\epsfig{file=H1prelim-04-042.fig4.eps, width=\textwidth}}
\end{picture}
\end{center}
\caption{Compilation of reduced cross section measurements 
with $Q^2 < 1 \ {\rm GeV^2}$ from H1, ZEUS and NMC.}
\label{four}
\end{figure}

\newpage

\begin{figure}[hhh]
\setlength{\unitlength}{1mm}
\begin{center}
\begin{picture}(150,150)(0,0)
\put(0,-30){\epsfig{file=H1prelim-04-042.fig5.eps, width=\textwidth}}
\end{picture}
\end{center}
\caption{Compilation of selected HERA results on the parameter $\lambda$, 
obtained from fits of the form $F_2 = c(Q^2) \cdot x^{- \lambda (Q^2)}$ to
low $x$ data.}
\label{five}
\end{figure}

\end{document}



