\graphicspath{{figures/basics/}}

\chapter{Basics of the Cross-Section Measurement} \label{c:basics}

The last two chapters have been devoted to the foundations of the
theoretical description and experimental observation of deep inelastic
scattering. The information of both enters the present chapter which
gives a review of the basic concepts of the DIS cross-section
measurement. Most of the technical details are explained in the next
chapter.

For the present analysis a data sample is employed which has been
recorded during a special running period of the HERA collider. To
measure the kinematic variables various methods are available
utilising the information of different detector components and thus
involving several reconstruction algorithms. They are discussed in
detail.

In order to model detector acceptances and efficiencies the recorded
data are compared to event samples obtained from simulations of the
event kinematics and the detector response. The same technique is
applied to determine the contribution of background processes
mimicking the signature of a DIS event. The underlying Monte Carlo
models are introduced.

With these requisites the methods to determine the double differential
cross-section and to extract the proton structure function $F_{2}$ are
presented.

The chapter ends with an overview of the coverage of the kinematic
phase space achieved by the present analysis as well as complementary
analyses using the same data set.

Emphasis is placed in this chapter on the role of radiative
events. Apart from affecting the reconstruction of the event
kinematics they allow for a significant extension of the accessible
kinematic phase space compared to non-radiative events.

\section{Data Sample} \label{c:datasample}

With the scattered electron being detected in the main detector photon
virtualities down to \mbox{$Q^{2} \approx 1.5\, \mathrm{GeV^{2}}$} can
be reached with the H1 experiment for non-radiative events during
standard operation conditions of the HERA collider~\cite{Aid:1996au,
Adloff:2000qk}. To access lower values of $Q^{2}$ a so-called
\emph{shifted vertex run} has been performed during three days in the
year 1995. For this special run the interaction vertex has been
shifted by \mbox{$70\, \mathrm{cm}$} in the direction of the incoming
proton. By this means the acceptance of the existing detector
components is extended towards larger polar angles and hence smaller
values of the photon virtuality\footnote{The correlation between the
polar angle $\theta_{e}$ of the scattered electron and $Q^{2}$ is
discussed in section~\ref{c:evkindet}.}. Values of $Q^{2}$ down to
\mbox{$0.35\, \mathrm{GeV^{2}}$} have been reached in the corresponding
analysis~\cite{Adloff:1997mf}.

A second data sample with the same special running conditions and
approximately the same duration has been collected in August 2000. It
constitutes the basis for the present analysis. With an integrated
luminosity of approximately \mbox{$\lumidat$} the statistics of the
former shifted vertex run is exceeded by a factor of four
corresponding to a reduction of the statistical error by a factor of
two.

In figure~\ref{f:svxevent} a typical event of this data taking period
is depicted: the scattered electron causes a sizeable cluster in the
electromagnetic section of the SpaCal as well as hits in the BST and
BDC. The hadronic final state occupies the forward region of the H1
detector. Its presence is indicated by energy depositions in the
electromagnetic and hadronic part of the LAr calorimeter to which
tracks in the Central Tracker can be attributed\footnote{The tracks in
the Forward Tracker pointing to energy depositions in the forward part
of the LAr calorimeter have been omitted. They are not considered in
the analysis due to general problems with the Forward Tracker
reconstruction.}. In section~\ref{c:evreco} it is explained how the
information of the different detector components contributes to the
determination of the event kinematics.

As can be seen from the displayed tracks the interaction vertex is
sizably shifted in positive \mbox{$z$-direction} w.r.t. the origin
of the H1 coordinate system, i.e. the nominal vertex position for
standard running conditions. To illustrate the effect of this shift
figure~\ref{f:kinshift} compares the region in the
\mbox{$xQ^{2}$-plane} covered by the SpaCal calorimeter for standard
and special running conditions: the two thin solid lines display the
upper acceptance limit of the SpaCal calorimeter concerning the polar
angle $\theta_{e}$ of the scattered electron for a standard
($177.5^{\circ}$) and a shifted ($178.3^{\circ}$) interaction
vertex\footnote{The polar angle values correspond to the geometrical
acceptance limit of the SpaCal as given in section~\ref{c:calorimetry}
for the mean \mbox{$z$-position} of the interaction region. Neither
the sizeable spread of the position of the interaction vertex nor the
fact that the maximal $\theta_{e}$ reached in the analysis is smaller
than the values given are correctly taken into account. Therefore the
following considerations are intended to explain the principal effect
of a shift of the interaction vertex rather than to give an accurate
representation of the kinematic phase space covered by the
measurement.}. The dotted line corresponds to a constant energy of the
scattered electron of \mbox{$E_{e} = 7\, \mathrm{GeV}$} which is the
lower energy limit of the analysis (see section~\ref{c:dissel}). All
events located at larger $x$ than indicated by the line of constant
energy and at larger $Q^{2}$ than indicated by the lines of constant
$\theta_{e}$ can be reconstructed with the help of the SpaCal
calorimeter for the corresponding running conditions. Hence the
highlighted area shows the region that gets accessible due to the
shift of the interaction vertex. The covered phase space is
considerably extended towards lower $Q^{2}$.

\begin{figure}
  \centering
  \includegraphics[width=0.8\textwidth]{278977_19081_onlyCentral_polished_withCross.eps}
  \caption{A typical event of the shifted vertex data taking period. Only tracks reconstructed by the Central Tracker are displayed. The cross indicates the origin of the H1 coordinate system. Run and event number are 278977 and 19081, respectively.}
  \label{f:svxevent}
\end{figure}

\begin{figure}
  \centering
  \includegraphics[width=0.6\textwidth]{svgain.eps}
  \caption{Change of the geometrical acceptance of the SpaCal calorimeter in the \mbox{$xQ^{2}$-plane} due to a shift of the interaction vertex in positive \mbox{$z$-direction} by \mbox{$70\, \mathrm{cm}$}. The thin solid lines depict the acceptance limits for a standard ($177.5^{\circ}$) and a shifted ($178.3^{\circ}$) interaction vertex. Hence the highlighted area represents the gain in acceptance due to the vertex shift. Lines of constant inelasticity $y$ (dotted) and energy (dashed) of the scattered electron $E_{e}$ are displayed in addition.}
  \label{f:kinshift}
\end{figure}


\section{Determination of the Event Kinematics} \label{c:evkindet}


A precise measurement of the DIS cross-section requires an accurate
reconstruction of the kinematic variables describing the scattering
process (see section~\ref{c:kinematics}). Since the H1 experiment is a
collider experiment and the detector covers almost the entire solid
angle the most important part of the hadronic final state particles
can be detected. Therefore the reconstruction of the event kinematics
relies not solely on the scattered electron, i.e. the kinematics is
overconstrained and several reconstruction methods are available. This
redundancy allows for a flexible choice of the reconstruction method
providing the best resolution in each region of the kinematic phase
space. Furthermore it allows for an experimental control of the
systematic errors of the measurement. The different reconstruction
methods are introduced and compared in the following.

In the case of the \emph{electron method} the kinematic variables are
expressed in terms of the energy $E_{e}$ and the polar angle
$\theta_{e}$ of the scattered electron:
\begin{align}
y_{e} & = 1 - \frac{E_{e} (1 - \cos \theta_{e})}{2 E_{e}^{0}} = 1 - \frac{E_{e}}{E_{e}^{0}} \sin^{2} \frac{\theta_{e}}{2} \label{e:ye} \\
Q_{e}^{2} & = 2 E_{e}^{0} E_{e} (1 + \cos \theta_{e}) = \frac{E_{e}^{2} \sin^{2} \theta_{e}}{1 - y_{e}} \label{e:q2e}
\end{align}
The variable $E_{e}^{0}$ corresponds to the HERA electron beam energy
of \mbox{$27.6\, \mathrm{GeV}$}. To determine the Bjorken variable $x$
one makes use of the relation \mbox{$x = Q^{2}/sy$} (see
equation~\ref{e:q2sxy}) as for all other reconstruction methods
presented in the following.

From the given formulae the following conclusion can be drawn: if the
scattered electron is detected in the SpaCal (\mbox{$\theta_{e} >
150^{\circ}$}) the inelasticity $y$ is almost solely defined by its
energy $E_{e}$. The lower the energy the larger values of $y$ are
reached. Furthermore the virtuality $Q^{2}$ depends predominantly on
the polar angle $\theta_{e}$ for low values of $y$. Large $\theta_{e}$
correspond to small values of $Q^{2}$ and vice versa. These
dependencies are reflected in figure~\ref{f:kinshift}: the line of
constant energy is parallel to the lines of constant $y$ and the lines
of constant $\theta_{e}$ get parallel to the abscissa for low $y$ (high $x$).

With the scattered electron being identified all other particles are
attributed to the hadronic final state. Since it is impossible to
build a perfectly hermetic detector particle losses in beam direction
are unavoidable. Due to the large energy imbalance between incoming
electron and proton the losses along the direction of the incoming
proton beam are more important. Variables as insensitive as possible
to losses are chosen for the reconstruction of the event kinematics
based on the hadronic final state. These are the total transverse
momentum
\begin{equation}
p_{t,h} = \sqrt{(\sum_{i} p_{x,i})^{2} + (\sum_{i} p_{y,i})^{2}}
\label{e:pthad}
\end{equation}
and the difference of energy and longitudinal momentum
\begin{equation}
\Sigma = \sum_{i} E_{i} - p_{z,i} = \sum_{i} E_{i} (1 - \cos \theta_{i})
\label{e:sigma}
\end{equation}
In both equations the summation is performed over all particles of the
hadronic final state. The variables $E_{i}$, \mbox{$p_{x,i}$},
\mbox{$p_{y,i}$} and \mbox{$p_{z,i}$} are the components of the four
vector of the hadronic final state particle $i$. The transverse
momentum is rather insensitive to particle losses in the forward and
backward direction while $\Sigma$ is not affected by losses in the
forward direction, but is very sensitive to losses in the backward
direction. Energy and momentum conservation implies the following
relations to hold\footnote{Neglecting particle masses the longitudinal
momentum of the incoming electron and proton is given by
\mbox{$p_{z,p}^{0} = E_{p}^{0}$} and
\mbox{$p_{z,e}^{0} = - E_{e}^{0}$}, respectively. Furthermore both
have only a negligible transverse momentum.}:
\begin{align}
p_{t,e} - p_{t,h} & = E_{e} \sin \theta_{e} - p_{t,h} = 0 \label{e:ptcons} \\
E - p_{z} \equiv (E_{e} - p_{z,e}) + \Sigma & = E_{e} (1 - \cos \theta_{e}) + \Sigma = 2 E_{e}^{0} \label{e:empzcons}
\end{align}
The former conservation rule is employed for the hadronic calibration
(see section~\ref{c:calibration}), the latter proves to be useful in
various aspects, among them the rejection of background processes (see
section~\ref{c:background}).

Finally one defines the so-called \emph{hadron angle}
\begin{equation}
\tan \frac{\theta_{h}}{2} = \frac{\Sigma}{p_{t,h}}
\end{equation}
which corresponds to the direction of the struck quark in the naive
quark parton model.

Using $\Sigma$ and $p_{t,h}$ the equations \ref{e:ye} and \ref{e:q2e}
can be transformed such that the event kinematics is defined based on
the hadronic final state only:
\begin{align}
y_{h} & = \frac{\Sigma}{2 E_{e}^{0}} \label{e:yhad} \\
Q_{h}^{2} & = \frac{p_{t,h}^{2}}{1 - y_{h}} \label{e:q2had}
\end{align}
This approach to determine the kinematics is denoted as \emph{hadron
method}~\cite{Jacqet:1979}. It is the only method applicable for
charged current events as it incorporates no information concerning
the scattered lepton.

The energy of the incoming electron $E_{e}^{0}$ enters the calculation
of the kinematic variables for both, electron and hadron method (see
equations \ref{e:ye} and \ref{e:yhad}). It is assumed to be equal to
the nominal beam energy of \mbox{$27.6\, \mathrm{GeV}$}. However, for
ISR events (see section~\ref{c:radepscat}) the energy available for
the interaction with the proton is lowered due to the photon emission
collinear with the incoming electron. Hence both methods fail to
reconstruct the kinematics correctly in this case. To overcome this
problem $E_{e}^{0}$ is replaced by \mbox{$(E - p_{z})/2$} according to
equation~\ref{e:empzcons} in the case of the \emph{sigma
method}~\cite{Bassler:1994uq} which makes use of information
concerning the scattered electron and the hadronic final state:
\begin{align}
y_{\Sigma} & = \frac{\Sigma}{\Sigma + E_{e} (1 - \cos \theta_{e})} \label{e:ysigma} \\
Q_{\Sigma}^{2} & = \frac{E_{e}^{2} \sin^{2} \theta_{e}}{1 - y_{\Sigma}} \label{e:q2sigma}
\end{align}
In this way the energy loss of the incident electron is implicitly
corrected for.

A fourth method, the so-called \emph{double angle method} defines the
kinematic variables in terms of $\theta_{e}$ and $\theta_{h}$. It is
used for calibration purposes (see section~\ref{c:calibration}).

The resolution of the presented methods in different regions of the
kinematic phase space has been investigated in
detail~\cite{Buchmueller:1992, Bassler:1994uq, Glazov:1998a,
Vargas:2006}. A brief summary of the results follows.

Performing an error propagation for the inelasticity $y$ results in
the following equations:
\begin{align}
\frac{\delta y_{e}}{y_{e}} & = \frac{1 - y_{e}}{y_{e}} \left ( \frac{\delta E_{e}}{E_{e}} \oplus \frac{\delta \theta_{e}}{\tan \frac{\theta_{e}}{2}} \right ) \label{e:yeres} \\
\frac{\delta y_{h}}{y_{h}} & = \frac{\delta \Sigma}{\Sigma} \label{e:yhadres} \\\frac{\delta y_{\Sigma}}{y_{\Sigma}} & = (1 - y_{\Sigma}) \left ( \frac{\delta \Sigma}{\Sigma} \oplus \frac{\delta E_{e}}{E_{e}} \oplus \frac{\delta \theta_{e}}{\tan \frac{\theta_{e}}{2}} \right ) \label{e:ysigmares}
\end{align}
At high $y$ the resolution of the electron method is superior to all
other methods. Towards lower values of $y$ it degrades as $1/y$. Such
a divergence is not present in the case of the hadron and the sigma
method. These two methods show the same behaviour at low $y$ since the
term \mbox{$\delta \Sigma / \Sigma$} dominates the resolution of
$y_{\Sigma}$. For large values of $y$ the resolution of $y_{\Sigma}$
is better than the one of $y_{h}$ due to the suppression by the factor
\mbox{$(1 - y)$}. In this region of the phase space the
contribution from hadrons to \mbox{$E - p_{z}$} becomes comparable to
the one of the electron. Thus the fluctuations of the measured
hadronic energies partly cancel between numerator and denominator of
equation~\ref{e:ysigma}.

For the present analysis the electron and the sigma method are
used. The decision which method to use for a specific region of the
kinematic phase space is taken based on the deviation from the
reconstructed to the generated variables for a Monte Carlo
simulation. As expected the electron method is preferred at high $y$,
the transition to the sigma method takes place at \mbox{$y \approx
0.1$} (see section~\ref{c:binning}). The corresponding regions in
the \mbox{$xQ^{2}$-plane} can be inferred from figure~\ref{f:kinshift}.

\section{Event Reconstruction} \label{c:evreco}

According to the previous section the energy $E_{e}$ and the polar
angle $\theta_{e}$ of the scattered electron as well as the difference
of the energy and the longitudinal momentum $\Sigma$ of the hadronic
final state have to be reconstructed to determine the event kinematics
with the electron or sigma method.

The energy of the scattered electron is measured with the help of the
SpaCal. As the spatial resolution of the BST and BDC is better than
the one of the SpaCal (see sections \ref{c:tracking} and
\ref{c:calorimetry}) either of the two detectors is used for the
reconstruction of the polar angle. Due to the worse resolution
compared to the BST and the long lever arm for the extrapolation to
the interaction region a standalone reconstruction of $\theta_{e}$ is
not possible for the BDC. Therefore its hits are combined with the
vertex position as determined by the Central Tracker. In both cases
the reconstructed position of the scattered electron in the SpaCal is
used as a starting point to search for the corresponding hits in the
backward tracking detector.

It has to be considered that the identification of the scattered
electron is not unambiguous as particles of the hadronic final state
can mimic its typical signature (see
section~\ref{c:background}). Therefore the term \emph{electron
candidate} is used in the following for the particle most likely being
the scattered electron.

For the reconstruction of the hadronic final state energy depositions
in the LAr and SpaCal calorimeter as well as tracks in the Central
Tracker are considered.

\subsection*{Cluster Reconstruction in the SpaCal}


The SpaCal reconstruction starts from the energy depositions in the
individual cells of the electromagnetic section. Adjacent cells with
an energy deposition above a certain threshold are combined to
\emph{clusters} each of them ideally corresponding to a single
particle. The cluster algorithm starts from cells showing a local
maximum of the energy deposition~\cite{Schleif:1996}. Neighbouring
cells are subsequently added to these seeds.

The cluster energy is given by the sum of the energies in the
individual cells belonging to it.

As the SpaCal has no segmentation in \mbox{$z$-direction} apart from
the separation in an electromagnetic and a hadronic section only the
projection of the shower profile to the surface of the SpaCal can be
used for the spatial reconstruction. The centre of gravity $\vec{r}$
of the cluster in the \mbox{$xy$-plane} is calculated via:
\begin{equation}
\vec{r} = \frac{\sum_{i} w_{i} \vec{r}_{i}}{\sum_{i} w_{i}}
\label{e:clustercog}
\end{equation}
Both sums contain all cells in the cluster, $\vec{r}_{i}$ is the
position of the centre of cell $i$ with the energy $E_{i}$. Different
energy dependent weight factors $w_{i}$ have been considered: linear
weighting (\mbox{$w_{i}=w_{i}^{lin}=E_{i}$})~\cite{Drescher:1984rt},
square root weighting (\mbox{$w_{i}=\sqrt{E_{i}}$}) and
finally a logarithmic weight function~\cite{Awes:1992yp} given by:
\begin{equation}
w_{i} = w_{i}^{log} = \max \left( 0, w_{0} + \ln \frac{E_{i}}{\sum_{i} E_{i}} \right)
\label{e:wilog}
\end{equation}
For the latter the weights follow the exponential decrease of the
deposited energy with the transverse distance from the centre of
gravity. The cutoff parameter $w_{0}$ effectively excludes cells with
an energy below a certain threshold from the calculation of
$\vec{r}$. The larger $w_{0}$ is the lower energies are
considered. Detailed studies in~\cite{Poeschl:1996} revealed that the
best results are obtained with the logarithmic weighting. In addition,
the special shape of the cells in the insert module is taken into
account for this analysis as proposed in~\cite{Dirkmann:1997}.

Electrons produce on average more compact showers than
hadrons. Therefore the \emph{cluster radius} which describes the
transverse extension of a cluster is a good cluster shape estimator
to identify the scattered electron. One distinguishes between the
linear cluster radius
\begin{equation}
r_{Cl}^{lin} = \frac{\sum_{i} w_{i}^{lin} \sqrt{(\vec{r}_{i} - \vec{r})^{2}}}{\sum_{i} w_{i}^{lin}}
\label{e:rcluslin}
\end{equation}
and the logarithmic cluster radius
\begin{equation}
r_{Cl} = \sqrt{\frac{\sum_{i} w_{i}^{log} (\vec{r}_{i} - \vec{r})^{2}}{\sum_{i} w_{i}^{log}}}
\label{e:rcluslog}
\end{equation}
It has been shown in~\cite{Glazov:1998a} that $r_{Cl}^{lin}$ strongly
depends on the impact point of the particle. For clusters with a
centre of gravity close to the cell edges the reconstruction
yields systematically larger radii than for those with a centre of
gravity in the middle of the cell. In contrast, the logarithmic
cluster radius $r_{Cl}$ is almost independent of the impact
point. Therefore the latter is preferred for the analysis. The linear
cluster radius is employed for the online event selection on the
fourth trigger level (see section~\ref{c:onlinesel}).


With the centre of gravity $\vec{r}$ the $x$- and
\mbox{$y$-coordinates} of a cluster are known. Since the SpaCal is not
segmented in \mbox{$z$-direction} the \mbox{$z$}-position of the
cluster is derived from a parameterisation~\cite{Shekelyan:1996}:
\begin{equation}
z_{\perp}(E_{Cl})\, [\mathrm{cm}] =  0.001956 \cdot E_{Cl}\, [\mathrm{GeV}] + 0.8529 \cdot \ln (2479 \cdot E_{Cl}\, [\mathrm{GeV}])
\label{e:pendepth}
\end{equation}
The variable $z_{\perp}$ is defined with respect to the SpaCal surface
and corresponds to the penetration depth of a particle impinging on
the SpaCal rectangular to its surface. It is to first approximation
assumed to depend logarithmically on the cluster energy $E_{Cl}$. From
$z_{\perp}$ the $z$-coordinate of the cluster in the H1 coordinate
system can be easily calculated taking the angle of incidence into
account.

\subsection*{Track and Vertex Reconstruction in the Central Tracker}


The track reconstruction of the Central Tracker starts from the hits
in the Central Jet Chambers~\cite{Abt:1996xv}. In the first phase one
searches for short track elements, so called track segments, which are
combinations of three hits detected by wires belonging to the same
angular cell of the CJC (see section~\ref{c:tracking}). These are
combined to track candidates based on the precise drift time
information in the \mbox{$xy$-plane}. Subsequently the trajectories in
the \mbox{$xy$-plane} are determined by subjecting all hits belonging
to a track candidate to a circle fit\footnote{In the magnetic field of
the H1 detector tracks form helices. Thus their projections in the
\mbox{$xy$-plane} are circles. Small deviations from the ideal helix shape 
occur due to multiple scattering in the detector material, energy
losses due to $dE/dx$ and non-uniformities of the magnetic field.}
based on a non-iterative algorithm~\cite{Karimaki:1990zz}. Among the
free parameters of the fit is the azimuthal angle $\phi$ at the
closest distance of the track to the \mbox{$z$-axis} in the
\mbox{$xy$-plane}. In the
\mbox{$Sz$-space}, $S$ being the path length in the \mbox{$xy$-plane},
tracks are straight lines. Thus after the circle fit in the
\mbox{$xy$-plane} a linear fit in the \mbox{$Sz$-space} is performed
allowing to determine the polar angle $\theta$.

For a fixed \mbox{$z$-coordinate} the position of the interaction vertex
in the \mbox{$xy$-plane} varies by only a few hundred $\mathrm{\mu m}$
for a sequence of runs\footnote{This holds only for a fixed
\mbox{$z$-position} as the beams are inclined with respect to the
\mbox{$z$-axis} of the H1 coordinate system (see
section~\ref{c:simulation}).}. Therefore tracks fulfilling certain
quality criteria are used during the online reconstruction on the
fourth trigger level to determine the mean vertex coordinates
$\overline{x_{vtx}}$ and $\overline{y_{vtx}}$ as a function of the
\mbox{$z$-coordinate} $z_{vtx}$~\cite{Kleinwort:2006}.

The average position \mbox{$(\overline{x_{vtx}},\overline{y_{vtx}})$}
at $z_{vtx}=0$ is used as a constraint to improve the parameters of
tracks compatible with this vertex during the next step of the offline
track reconstruction\footnote{During the online track reconstruction
the corresponding parameters of the previous run are used.}. At this
stage corrections for multiple scattering in the detector material and
the non-uniformity of the magnetic field are applied. For all tracks
assigned to the vertex in the $xy$-plane a common fit in the
\mbox{$Sz$-space} is performed to determine an initial approximation
of the \mbox{$z$-position} of the vertex and the polar angles. Knowing
the actual \mbox{$z$-position} the procedure is repeated using the
corresponding values of $\overline{x_{vtx}}$ and
$\overline{y_{vtx}}$~\cite{Kleinwort:2006}.

As the \mbox{$z$-information} of the Central Jet Chambers is not very
precise the resolution of the polar angle and the \mbox{$z$-position}
of the vertex can be significantly improved using the signals of the
two drift chambers CIZ and COZ (see section~\ref{c:tracking}). In the
presence of noise and inefficiencies a stand alone track
reconstruction in the \mbox{$z$-chambers} is rather ineffective given
the relatively small number of sense wires planes. Hence the
individual hits in CIZ and COZ are linked to the tracks reconstructed
in the Central Jet Chambers and new track and vertex parameters are
determined. The assignment of hits and tracks and the track-vertex fit
are parts of a combined procedure. A simple combinatorial approach for
the linking would be very time consuming. Therefore a special
technique, known as the \emph{Method of Deformable Templates} is used
for this purpose~\cite{Ohlsson:1991eh}. A detailed description of its
application for the described task can be found in appendix~B
of~\cite{Glazov:1998a}. In the following the basic concept is shortly
reviewed as the same technique is applied for the track reconstruction
in the BDC and BST.

Since tracks are straight lines in the \mbox{$Sz$-space} the
parameterisation
\begin{equation}
z(S) = a\cdot S + z_{0}
\label{e:szfit}
\end{equation}
is used to fit the hits in the \mbox{$z$-chambers}, $a$ denoting the
slope of the track and $z_{0}$ its \mbox{$z$-coordinate} at \mbox{$S =
0$}. Hence the problem of simultaneously fitting all
\mbox{$z$-chamber} hits can be formulated as the minimisation of the
following functional:
\begin{equation}
\mathcal{L}(a_{j}, z_{0}) = \sum_{k,i,j} W_{ij}^{k} (z_{ij}^{k} -
S_{ij}^{k} \cdot a_{j} - z_{0})
\label{e:szfunctional}
\end{equation}
The sum extends over all wire planes of the \mbox{$z$-chambers} ($k$),
\mbox{$z$-points} ($i$) and CJC tracks ($j$). The number of
\mbox{$z$-points} equals two times the number of hits in CIZ and
COZ. Each hit is considered twice as one cannot distinguish on
which side of a wire a particle has passed the corresponding drift
cell. This \emph{left-right ambiguity} has to be resolved. Furthermore
$z_{ij}^{k}$ and $S_{ij}^{k}$ refer to the \mbox{$z$-coordinate} of
the \mbox{$z$-point} and the path length to reach it,
respectively. The track parameters to be determined are $a_{j}$ and
$z_{0}$. For simplicity is assumed that all tracks stem from the same
vertex\footnote{This simplification is only used for the
explanation. The real implementation handles secondary vertices
correctly as well.}.

The assignment of CJC tracks and $z$-points is governed by the
matrices $W^{k}$, one for each wire plane $k$. The matrix elements
$W_{ij}^{k}$ can take values of 0 or 1. A value of 1 indicates that
track $j$ and \mbox{$z$-point} $i$ of wire plane $k$ are assigned to
each other, if not the value is 0. If a
\mbox{$z$-point} belongs to no CJC track it is most likely caused by
chamber noise, if a track in the acceptance of the \mbox{$z$-chambers}
has no corresponding \mbox{$z$-point} in a wire plane this is a sign
of an inefficiency.

For a given set of weights $W_{ij}^{k}$ the track parameters $a_{j}$
and $z_{0}$ are determined by solving the system of linear equations
belonging to the minimisation of the functional~$\mathcal{L}$. As
already mentioned, a combinatorial search to determine the optimal set
of matrices $W_{ij}^{k}$ is extremely time consuming. Therefore the
given functional is modified according to the Method of Deformable
Templates~\cite{Ohlsson:1991eh}. This implies that weights
$W_{ij}^{k}$ acquire a dependence on an additional parameter, the
temperature $T$.  The combinatorial search among the possible
$W_{ij}^{k}$ is replaced by an iterative minimisation of the new
functional gradually decreasing the temperature to 0. For finite
temperatures the weights $W_{ij}^{k}$ are real numbers between 0 and
1. They can be interpreted as the probability that the
\mbox{$z$-point} $i$ belongs to the track $j$. At \mbox{$T=0$} the new
weights coincide with the old ones.

One expects the \mbox{$z$-points} to have a Gaussian distribution
around the particle trajectory with a width defined by the resolution
$\sigma_{CIZ/COZ}$ of the \mbox{$z$-chambers}. Accordingly, one
assumes the following dependence of the weights on the distance
$D_{ij}^{k}$ of the \mbox{$z$-points} to the tracks\footnote{The
quoted formula shows the basic dependence of the weights on the
distance $D_{ij}^{k}$ and the temperature $T$. The precise expression
for $W_{ij}^{k}$ is significantly more complicated to meet the
boundary condition that the new weights coincide with the old ones for
\mbox{$T=0$}.}:
\begin{equation}
W_{ij}^{k} \sim \exp \left( -\frac{(D_{ij}^{k})^2}{T \sigma_{CIZ/COZ}^{2}} \right)
\label{e:dtmweight}
\end{equation}
With decreasing temperature the weights get smaller for a given
distance $D_{ij}^{k}$, i.e. the corridor around a track from which
\mbox{$z$-points} are assigned to the track gets smaller. Initially
the tracks are defined by the Central Jet Chambers only, i.e. the
initial temperature $T_{ini}$ should satisfy the condition
\begin{equation}
T_{ini} \approx \left( \frac{\sigma_{z,CJC}}{\sigma_{CIZ/COZ}} \right)^{2}
\label{e:tini}
\end{equation}
where $\sigma_{z,CJC}$ corresponds to the \mbox{$z$-resolution} of the
Central Jet Chambers.

The weights depend on the distances of the \mbox{$z$-points} to the
tracks and therefore on the track parameters. This leads to a highly
non-linear functional $\mathcal{L}$. Therefore a two step minimisation
procedure is used for a fixed temperature: at first, given the track
parameters from the previous temperature, the weights $W_{ij}^{k}$ are
calculated. Secondly, knowing the weights, new track parameters
$a_{j}$ and $z_{0}$ are calculated by solving the system of linear
equations implied by minimising $\mathcal{L}$. If the changes of
$W_{ij}^{k}$, $a_{j}$ and $z_{0}$ exceed certain maximal values the
procedure is repeated. Otherwise one continues with a smaller
temperature.

For typical background conditions in total 20 iterations\footnote{Each
iteration comprises one calculation of weights and one minimisation of
$\mathcal{L}$.} are needed for all considered temperatures to perform
the link of CJC tracks and \mbox{$z$-chamber} hits. By this procedure
the resolution of the \mbox{$z$-coordinate} of the vertex is improved
by a factor of 10 from about \mbox{$1\, \mathrm{cm}$} to about
\mbox{$0.1\, \mathrm{cm}$}~\cite{Glazov:1998a}.

\subsection*{Track Reconstruction in the BDC}


To measure the polar angle of the scattered electron with the BDC
SpaCal clusters and BDC information have to be combined. Two
algorithms to do so are discussed in the following.

The first analyses making use of the BDC~\cite{Katzy:1997, Meyer:1997}
have started from standalone SpaCal, vertex and BDC
reconstructions. The former proceed as described above, for the latter
BDC hits are combined to track candidates based on a so-called
Kalman-Filter~\cite{Schwab:1996, Billoir:1989mh}. It turns out that in
a single event a large number of BDC tracks is reconstructed close to
the cluster of the scattered electron~\cite{Katzy:1997}. Only in about
40\% of the events less than five BDC tracks are reconstructed within
a cylinder of \mbox{$3\, \mathrm{cm}$}, up to 100 tracks are
possible. Two effects are responsible for this: for a large fraction
of events (60-70\%) the scattered electron interacts with the dead
material situated between the interaction point and the BDC causing a
shower of secondary particles. Secondly, the BDC provides little
redundancy to resolve drift direction ambiguities of the hit positions
since its eight layers cover a distance on only \mbox{$6.8\,
\mathrm{cm}$} in
\mbox{$z$-direction}. The large number of tracks in the vicinity of 
the SpaCal cluster makes the selection of the corresponding BDC track
difficult. In~\cite{Katzy:1997} a generalised $\chi^{2}$ selection
criterion has been introduced. Among the five tracks with the minimal
$\chi^{2}$ the one with the least distance to the SpaCal cluster
centre of gravity is selected.

In 1998 a new algorithm (\mbox{BDCLEV}\footnote{BDC LEpton Validator})
has been developed to improve the resolution of the polar
angle~\cite{Glazov:1998b}. This uses the line connecting the Central
Tracker vertex and the SpaCal cluster as an initial approximation for
the corresponding track. The azimuthal angle $\phi_{Spa}$ derived from
the SpaCal cluster is used to select the octants of the BDC of which
hits are considered for the track reconstruction\footnote{Depending on
the distance of the cluster to the octant border, one or two octants
are selected in each layer.}. In addition $\phi_{Spa}$ is employed to
transform the radial information derived from the BDC drift time into
a hit position in the \mbox{$xy$-plane}. This is beneficial as the
resolution of $\phi_{Spa}$ is better than the resolution of the
azimuthal angle derived from the stereo information of the BDC.

Hit selection and track fit are performed in a global minimisation
procedure similar to the linking of CJC tracks and \mbox{$z$-chamber}
hits. For the track fit a straight line in the \mbox{$rz$-plane} is
used combining the Central Tracker vertex, the centre of gravity of
the SpaCal cluster and all BDC hits (with a twofold drift direction
ambiguity). Initially the width $\Delta r$ of the corridor around the
track from which hits are assigned to the track is \mbox{$5\,
\mathrm{cm}$} driven by the spatial resolution of the SpaCal. It is
gradually decreased to approximately $5 \sigma_{r,BDC}$, with
$\sigma_{r,BDC} = 300\, \mathrm{\mu m}$~\cite{Glazov:2006} being the
resolution of the BDC for the measurement of the
\mbox{$r$-coordinate}.

For a given SpaCal cluster BDC hits are assigned to a single track
only such that combinatorics is much reduced compared to the linking
of CJC tracks and \mbox{$z$-chamber} hits. If more than one
electromagnetic SpaCal cluster exists the algorithm is run for each
cluster separately.

The conceptual difference between the old and new track reconstruction
is that the assignment of the BDC information to the SpaCal and
Central Tracker information is done on track level for the former
method while on hit level for the latter. This implies that the
imprecise SpaCal information influences the track selection in case
many tracks are present for the old algorithm. For the new algorithm
the SpaCal reconstruction provides the initial approximation while the
resolution at the final stage is defined by the BDC. This results in a
significantly better performance of the new track reconstruction: the
resolution of the polar angle improves from \mbox{$\sigma_{\theta,
BDC} = 0.77\, \mathrm{mrad}$} to \mbox{$\sigma_{\theta, BDC} = 0.57\,
\mathrm{mrad}$} with a simultaneous decrease of the number of
outliers~\cite{Glazov:1998b}.

\subsection*{Track Reconstruction in the BST}


To reconstruct the scattered electron with the help of the BST a
special track finding algorithm has been developed
in~\cite{Arkadov:2000}, thereafter called \emph{BST electron
finder}. It provides apart from the polar angle also the
\mbox{$z$-coordinate} of the interaction vertex.

The BST electron finder can be operated in two modes (see appendix A
of~\cite{Arkadov:2000}): the first one makes use of the Central
tracker vertex. A corridor around the connecting line between SpaCal
cluster and Central Tracker vertex is used for an initial hit
selection. The hits closest to the prediction in each plane are
subjected to a straight line fit in the \mbox{$rz$-plane}. The
resulting parameters are used to define a more narrow corridor for a
second iteration of hit selection and track fit. The alternative
operating mode is completely independent of the Central Tracker
vertex. The track finding proceeds via successively attributing hits to
track candidates based on the position of the SpaCal cluster. The
assignment starts from the BST plane closest to the SpaCal. If more than
one track candidate is found the one with the maximum number of
assigned hits is chosen. Finally, a straight line fit is performed as
for the first operating mode.

For the first analysis making use of the BST to measure the DIS
cross-section the approach independent of the Central Tracker vertex
has been chosen~\cite{Arkadov:2000}. In~\cite{Eckstein:2002}
and~\cite{Lastovicka:2004a}, however, the Central Tracker information
is employed if existing. If no Central Tracker vertex exists different
hypotheses concerning the \mbox{$z$-position} are tested.


For both operation modes it turned out that the BST electron finder is
rather sensitive to hits caused by electronics noise or secondary
particles. This is especially problematic for the shifted vertex
period as the flatter electron track leads to larger showering effects
compared to standard running conditions of the HERA
collider~\cite{Lastovicka:2004b}. The average number of BST hits for
the shifted vertex run is approximately 80 while for the minimum bias
run in 1999\footnote{For this data taking period the average
\mbox{$z$-vertex} position is close to \mbox{$z
\approx 0\, \mathrm{cm}$}.} on average only 20 hits are recorded per
event~\cite{Petrukhin:2006}.

For this reason an improved version of the BST electron finder has
been developed~\cite{H1:draft}: it starts from the azimuthal angle
$\phi_{Spa}$ of the SpaCal cluster to select three sectors of
\mbox{$r$-sensors} in each BST plane to provide hit candidates for the
track fit. Afterwards a small eccentricity correction is applied for
the hits to account for the BST not being centred w.r.t. the beam
axis. Subsequently the \mbox{$r$-coordinate} of each hit is projected
along the line connecting the hit and the SpaCal cluster to the BST
plane closest to the SpaCal and filled into a histogram. The maximum
of this histogram is assumed to indicate the group of hits belonging
to the SpaCal cluster. Therefore the straight line between the maximum
of the histogram and the centre of gravity of the cluster is used as
an initial approximation for the track finding. Hit assignment and
track fit proceed in exactly the same way as in the case of
\mbox{BDCLEV}. The initial width \mbox{$\Delta r = 1.5\,
\mathrm{cm}$} of the corridor is again defined by the SpaCal
resolution and stepwise decreased to \mbox{$\Delta r = 5
\sigma_{r,BST}$} where $\sigma_{r,BST} = 50\, \mathrm{\mu m}$ denotes 
the resolution of the BST for the measurement of the
\mbox{$r$-coordinate}.

\subsection*{Electron Identification}


From the discussion in section~\ref{c:evkindet} it can be inferred
that a correct identification of the scattered electron is essential
to reconstruct the event kinematics since without this also the
hadronic final state is not-well defined. As explained above the
clusters in the backward calorimeter constitute the starting point for
the reconstruction of the scattered electron. For inelasticities
\mbox{$y \gtrsim 0.1$} also hadrons deposit energy in the SpaCal as
the hadron angle $\theta_{h}$ enters its acceptance\footnote{The polar
angle acceptance of the SpaCal for the shifted vertex period is
\mbox{$160.5^{\circ} < \theta_{e} < 178.3^{\circ}$}.} (see
figure~\ref{f:elecid}). Since these energy depositions can mimic the
signature of the scattered electron (see section~\ref{c:background})
its identification is not unambiguous in this region of the kinematic
phase space.

\begin{figure}
  \centering
  \includegraphics[width=0.6\textwidth]{elecid_isoth.eps}
  \caption{Lines of constant inelasticity $y$ (solid), polar angle of the scattered electron $\theta_{e}$  (dotted) and hadron angle $\theta_{h}$ (dashed) in the \mbox{$xQ^{2}$-plane}.}
  \label{f:elecid}
\end{figure}

The situation improves if one restricts the search for the electron
candidate to those clusters having the properties of an
electromagnetic cluster produced by a charged particle. This involves
several selection criteria (see table~\ref{t:selcriteria}) including
the link to a BDC or BST track. Nevertheless a significant fraction of
events remains with more than one cluster passing all the selection
criteria. Therefore an additional requirement has to be found.

If the inelasticity satisfies $y < 0.5$ the polar angle $\theta_{e}$
of the scattered electron is always larger than the polar angle
$\theta_{h}$ of the current jet (see figure~\ref{f:elecid}). The same
holds for the energies. Hence the electromagnetic SpaCal cluster with
the largest energy can be identified with the scattered electron. The
situation is opposite for $y > 0.5$. Nevertheless the maximum cluster
energy remains a reliable criterion since backward going hadronic jets
are typically broad causing several clusters. Hence the cluster of the
scattered electron is likely to remain the one with the highest
energy.

The present analysis considers the SpaCal cluster with the highest
energy passing all selection criteria to be the electron candidate.

%

For a cross-check the cluster with the largest transverse momentum is
used. This criterion is based on the transverse momentum conservation
in the event (see equation~\ref{e:ptcons}). The transverse momentum of
the scattered electron is balanced by the sum of the transverse
momenta of the hadronic final state particles. Hence the particle with
the largest transverse momentum is in most of the cases the scattered
electron. The resulting DIS cross-sections for the two identification
strategies agree well within 0.5\%, differences being only visible at
large inelasticities. This has been checked individually for using the
BST and the BDC to reconstruct the polar angle of the scattered
electron.

A detailed investigation of the electron identification in the
backward region of the H1 detector can be found
in~\cite{Glazov:1998a}.

\subsection*{Reconstruction of the Hadronic Final State}


All particles apart from the electron candidate are attributed to the
hadronic final state. The most important detector component for its
reconstruction is the LAr calorimeter. The hadronic energy resolution
\mbox{$\sigma_{E}/E = 50\% / \sqrt{E/\mathrm{GeV}} \oplus 2\%$} (see
section~\ref{c:calorimetry}) of the LAr deteriorates with decreasing
energy. In contrast, the momentum and thus energy resolution of the
central tracking system improves with decreasing momentum:
\mbox{$\sigma_{p}/p = 0.01\, \mathrm{GeV^{-1}}
p$}~\cite{Abt:1996xv}. Hence the reconstruction of low energetic
particles can be improved by combining the calorimeter information
with Central Tracker tracks.

%

A special algorithm (\mbox{FSCOMB}) has been developed for this
purpose~\cite{Bassler:1995, Adloff:1997mi}: each track with a momentum
\mbox{$p < 10\, \mathrm{GeV}$} fulfilling certain quality criteria is
extrapolated to the surface of the LAr. Subsequently one searches for
energy depositions in a cylinder with a radius of \mbox{$20\,
\mathrm{cm}$} (\mbox{$40\, \mathrm{cm}$}) around the axis defined by
the track direction in the electromagnetic (hadronic) section. If the
summed energy within these cylinders exceeds the track momentum, the
calorimetric information is used for measurement. Otherwise the track
information is employed and the corresponding energy in the LAr is
masked to avoid double counting, i.e. it is not taken into account for
the reconstruction of the hadronic final state. If a track does not
reach the calorimeter surface it is also considered.

Apart from the LAr and the Central Tracker also the hadronic energy
measured in the SpaCal enters the reconstruction of the hadronic final
state. It comprises all energy depositions in the electromagnetic and
hadronic section apart from the ones assigned to the electron
candidate. The SpaCal should contribute only at high $y$ where the
hadronic final state occupies the backward region of the
detector. However, a detailed study in~\cite{Glazov:1998a} has shown
that its contribution is sizeable even at very low values of the
inelasticity. This effect is caused by secondary clusters induced by
the scattered electron which are wrongly identified as hadronic final
state particles. Secondary clusters can be invoked by the scattered
electron due to bremsstrahlung or showering in the material between
interaction point and SpaCal. To correct for this effect 10\% of the
energy of the electron candidate is subtracted from the total hadronic
energy measured in the SpaCal. If the resulting energy is negative the
contribution of the SpaCal is assumed to be equal to zero.

For the reconstruction of the event kinematics with the sigma method
the difference of the energy and longitudinal momentum $\Sigma$ of the
hadronic final state has to be calculated. It has already been
mentioned in section~\ref{c:evkindet} that it is insensitive to losses
in the forward, but very sensitive to losses in the backward region of
the H1 detector. The same holds for additional energy depositions in
the calorimeters not attributed to the genuine hadronic final state,
called `noise' in the following. It leads to a systematic bias of the
event kinematics at low values of $y$ where hadrons are produced in
the forward direction and little energy is deposited in the
calorimeters.

In events recorded with a random trigger\footnote{Among the selected
events there should be nearly no $ep$ interactions.} about 1000
of the 45000 cells of the LAr calorimeter pass a $+2\sigma_{noise}$
noise threshold\footnote{The energy of a LAr cell corresponding to $1
\sigma_{noise}$ varies between \mbox{$15\, \mathrm{MeV}$} and
\mbox{$30\, \mathrm{MeV}$} depending on the calorimeter
region~\cite{Abt:1996xv}.} on average~\cite{Abt:1996xv}. Adding up
this energy for the whole calorimeter yields an average value of
\mbox{$48\, \mathrm{GeV}$}. Apart from pure electronics noise energy
depositions due to beam-induced secondary particles contribute. For
$ep$ events additional energy depositions can be caused by
final state particles reaching the calorimeter via secondary
scattering, e.g. due to interactions with the beam pipe wall.

Various measures are taken during data acquisition and reconstruction
to remove the noise. On top of this \mbox{FSCOMB} performs a
topological noise suppression: LAr cells with an energy below
\mbox{$0.4\, \mathrm{GeV}$} (\mbox{$0.8\, \mathrm{GeV}$}) and
separated from other cells by more than \mbox{$40\, \mathrm{cm}$}
(\mbox{$20\, \mathrm{cm}$}) in the central (forward) region of the LAr
calorimeter are classified as noise and excluded from the
reconstruction of the event kinematics. This procedure does sometimes
exclude isolated clusters from the genuine hadronic final state.

A detailed investigation of the reconstruction of the hadronic final
state including the influence of noise can be found
in~\cite{Vargas:2006}. In section~\ref{c:simulation} the incorporation
of the noise into the simulation of the detector response is
discussed.


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

A number of reactions can imitate $ep$ interactions in the
$Q^{2}$ regime of interest and thus lead to a wrong determination of
the DIS cross-section. Two classes of background processes can be
distinguished:

The term \emph{\mbox{non-$ep$} interactions} is used to summarise
background events induced by non-colliding beam particles, cosmic
muons or beam halo muons. Their contribution can be suppressed by
appropriate selection criteria up to a negligible amount.

The only significant remaining background source are so-called
\emph{photoproduction} events in which the scattered electron escapes
the main detector along the beam pipe and the electron signal is
mimicked by a particle of the hadronic final state.

Both classes of processes are discussed in the following.

\subsection*{Non-ep Background}

Despite the good vacuum in the beam pipe, interactions of the beam
particles with the residual gas molecules or the beam pipe material
create a large background to the $ep$ interactions. Most of them
are already rejected during data taking by requirements concerning the
timing and energy depositions in the SpaCal imposed by the trigger
system (see section~\ref{c:onlinesel}). A further reduction of the
background rate is achieved by the selection criteria of the final
analysis, first of all by the cut on the \mbox{$z$-position} of the
interaction vertex (see section~\ref{c:dissel}). The same holds for
background events caused by cosmic or beam halo muons.

The remaining \mbox{non-$ep$} background is estimated making use of
special non-colliding, i.e. unpaired particle bunches: proton
(electron) \emph{pilot bunches} have an empty electron (proton)
partner bunch. Thus they can interact with residual gas molecules or
beam line elements only. In the case of \emph{empty bunches} electron and
proton bunch carry almost no current. During the time when these
bunches cross in the H1 detector signals can be only caused by random
background processes such as cosmic or beam halo muons.

One determines the number of events passing all selection criteria
(see section~\ref{c:dissel}) for \mbox{$p$-pilot} ($N_{p-pilot}$),
\mbox{$e$-pilot} ($N_{e-pilot}$) and empty bunches ($N_{empty}$) in a
special study. They are used to estimate the residual amount of
background events $N_{non-ep}$ among the number of events $N_{coll}$
passing all selection criteria for the colliding bunches, i.e. among
the total number of events for the cross-section determination. To
account for the different number of colliding ($n_{coll}$) and
unpaired ($n_{p-pilot}$, $n_{e-pilot}$, $n_{empty}$) bunches the
contributions of the different types of unpaired bunches have to be
normalised in the following way:
\begin{equation}
N_{non-ep} = \frac{n_{coll}}{n_{p-pilot}} N_{p-pilot} + \frac{n_{coll}}{n_{e-pilot}} N_{e-pilot} + \frac{n_{coll}}{n_{empty}} N_{empty}
\label{e:nnonep}
\end{equation}
The assumption behind this way of normalising is that all filled
bunches carry almost the same current. Knowing $N_{non-ep}$ the
fraction $f_{non-ep}$ of \mbox{non-$ep$} events in the analysis sample
used for the cross-section measurement can be calculated:
\begin{equation}
f_{non-ep} = \frac{N_{non-ep}}{N_{coll}}
\label{e:fnonep}
\end{equation}

Not a single event passes the event selection for the \mbox{$p$-pilot}
and empty bunches regardless whether the BST or the BDC is used for
the polar angle reconstruction of the scattered electron\footnote{For
the remaining part of this section the statements hold for both
backward tracking detectors. For the figures shown the reconstruction
of $\theta_{e}$ has been performed with the help of the BDC. The
corresponding distributions for the BST look similar.}. Only events
from the \mbox{$e$-pilot} bunches significantly contribute. This is
expected since \mbox{$e$-pilot} bunches should indicate background
events induced by the electron beam. Particles originating from such
events may reach the SpaCal at the same time as particles from
$ep$ interactions. Hence they are not rejected by timing
requirements based on the SpaCal like \mbox{$p$-pilot} or empty bunch
events.

%

Figure~\ref{f:nnonep} shows the estimated contribution of
\mbox{non-$ep$} background to the number of events in the colliding
bunches determined according to equation~\ref{e:nnonep}\footnote{As
the number of bunches for the different bunch types varies with time
the time averaged values over the shifted vertex period are used for
$n_{coll}$, $n_{p-pilot}$, $n_{e-pilot}$ and $n_{empty}$.} as a
function of the \mbox{$z$-position} of the interaction vertex
$z_{vtx}$ and \mbox{$E-p_{z}$}. The fraction $f_{non-ep}$ amounts to
$0.81 \pm 0.09\%$ and $0.81 \pm 0.07\%$ for the BST and the BDC
analysis, respectively. This is already significant for the desired
precision of the analysis.

\begin{figure}
  \begin{minipage}[c]{0.5\textwidth}
    \includegraphics[width=\textwidth]{controlplots_bdc_060616_2304_nonep_zvtx.eps}
    \put(-33,161){a)}
  \end{minipage}
  \begin{minipage}[c]{0.5\textwidth}
    \includegraphics[width=\textwidth]{controlplots_bdc_060616_2304_nonep_empz.eps}
    \put(-33,161){b)}
  \end{minipage}
  \caption{Number of events for colliding and electron pilot bunches passing all selection criteria of the final cross-section analysis as a function of a) $z_{vtx}$ and b) \mbox{$E-p_{z}$}. The distributions of the electron pilot bunches have been renormalised to account for the different number of colliding and electron pilot bunches according to equation~\ref{e:nnonep}.}
  \label{f:nnonep}
\end{figure}

Since the \mbox{$e$-pilot} bunches should indicate background events
one would neither expect that the distribution of $z_{vtx}$ for
\mbox{$e$-pilot} events shows a maximum at the nominal position for
$ep$ interactions of approximately \mbox{$70\, \mathrm{cm}$}
nor that the \mbox{$E-p_{z}$} distribution peaks at
\mbox{$2E_{e}^{0} = 55.2\, \mathrm{GeV}$}, a value expected from
conservation of energy and longitudinal momentum for DIS events (see
equation~\ref{e:empzcons}). Hence there seems to be a large amount of
$ep$ events among the contribution of \mbox{$e$-pilot} bunches.

This is possible if the proton partner bunches of the \mbox{$e$-pilot}
bunches carry a non-negligible current. This is indeed the case for a
fraction of the shifted vertex data taking period as can be seen in
figure~\ref{f:epilotvsrun}~a) which shows the proton beam current for
colliding and \mbox{$e$-pilot} bunches as a function of the luminosity
run. The dashed lines indicate the different luminosity
fills\footnote{Luminosity runs and luminosity fills are well-defined
intervals of the data taking period (see section~\ref{c:runsel}). In
figure~\ref{f:epilotvsrun}~a) a continuous run number is given which
conserves the original sequence of runs.}. For the same luminosity
fills in which the proton partner bunches carry a non-negligible
current the fraction $f_{non-ep}$ is significantly higher than for the
rest of the data taking period (see
figure~\ref{f:epilotvsrun}~b)). Thus a fraction of the
\mbox{$e$-pilot} events cannot be attributed to electron beam induced
background, i.e. the fractions $f_{non-ep}$ given above are overestimated.

\begin{figure}
  \begin{minipage}[c]{0.5\textwidth}
    \includegraphics[width=\textwidth]{showbunches_ip.eps}
    \put(-33,161){a)}
  \end{minipage}
  \begin{minipage}[c]{0.5\textwidth}
    \includegraphics[width=\textwidth]{controlplots_bdc_060616_2304_nonep_lumifill.eps}
    \put(-33,161){b)}
  \end{minipage}
  \caption{a) Proton current $I_{p}$ for colliding (open histogram) and \mbox{$e$-pilot} bunches (hatched histogram) as a function of a continuous run number. The dashed lines separate the different luminosity fills. b) Fraction of \mbox{non-$ep$} background events $f_{non-ep}$ (see equation~\ref{e:fnonep}) as a function of the luminosity fill. Fill 2597 has lasted for only one hour, none of its runs has passed the run selection explained in section~\ref{c:runsel}. The values of $f_{non-ep}$ shown are overestimated (see text).}
  \label{f:epilotvsrun}
\end{figure}

The problem with the not empty proton partner bunches is known and can
be explained in the following way~\cite{Pitzl:2006}: as can be seen
from figure~\ref{f:hera} electrons and protons are accelerated in DESY
and PETRA before they are injected into HERA. DESY has 11 proton
bunches while PETRA and HERA store protons in packets of 10
bunches. The superfluous bunch is supposed to be eliminated by a
so-called kicker magnet. This however does not work completely
reliably such that proton buckets in HERA are filled which are not
supposed to be. Hence an electron bunch is already classified as an
\mbox{$e$-pilot} bunch if the current in the proton partner bunch does
not exceed \mbox{$6\, \mathrm{\mu A}$}.

%

To determine the correct fraction of \mbox{non-$ep$} events the
following model is used~\cite{Krueger:2006}: $N_{coll}$ and
$N_{e-pilot}$ can be split into a fraction of events originating from
$ep$ interactions ($N_{ep,coll}$, $N_{ep,e-pilot}$) and a
fraction of events stemming from electron beam induced background
($N_{e-bgr,coll}$, $N_{e-bgr,e-pilot}$). The contribution from
$ep$ interactions should be proportional to the product of
proton ($I_{p,coll}$, $I_{p,e-pilot}$) and electron current
($I_{e,coll}$, $I_{e,e-pilot}$) carried by the corresponding bunch
type while the number of beam induced background events should rise
linearly with the electron current only. One can therefore specify the
following equations:
\begin{align}
N_{coll} & = N_{ep,coll} + N_{e-bgr,coll} \notag \\
& = a I_{p,coll} I_{e,coll} + b I_{e,coll} \label{e:splitncoll} \\
N_{e-pilot} & = N_{ep,e-pilot} + N_{e-bgr,e-pilot} \notag \\
& = a I_{p,e-pilot} I_{e,e-pilot} + b I_{e,e-pilot} \label{e:splitnepilot}
\end{align}
This system of two linear equations with two unknown coefficients $a$
and $b$ can be easily solved to determine $N_{e-bgr,coll}$. The fraction
\begin{equation}
f_{e-bgr} = \frac{N_{e-bgr,coll}}{N_{coll}}
\label{e:febgr}
\end{equation}
is the sought-after fraction of electron beam induced background
events in the analysis sample. Performing the necessary
steps\footnote{For the currents corresponding to the different bunch
types time averaged values over the shifted vertex period have been
used.} the resulting $f_{e-bgr}$ amounts to \mbox{$0.78\% \pm 0.08\%$}
(\mbox{$0.78\% \pm 0.07\%$}) for the BST (BDC). The quoted errors
denote the statistical uncertainty only, no uncertainty on the
currents has been considered.  The fraction $f_{e-bgr}$ seems to be
again overestimated, i.e. the fraction of $ep$ interactions
contributing to $N_{e-pilot}$ seems to be underestimated.

Two effects can possibly cause this: at first single bunch currents
below \mbox{$1\, \mathrm{\mu A}$} are not recorded in the accessible
database. This is already a significant amount for the proton partners
of the \mbox{$e$-pilot} bunches. Secondly, the proton partner bunches
of $e$-pilot bunches may be more compact than proton colliding bunches
as they contains less protons and therefore the Coulomb repulsion
within the bunches is reduced. For the latter explanation to hold one
expects the \mbox{$z$-vertex} distribution for the \mbox{$e$-pilot}
bunches to be narrower than for the colliding ones\footnote{For
details concerning the dependence of proton bunch size and width of
the \mbox{$z$-vertex} distribution see section~\ref{c:vertexreco}}. As
a matter of fact this is the case (see figure~\ref{f:zvtxepilot}). The
\mbox{$z$-vertex} distribution is sensitive to the longitudinal bunch
size. If one assumes that the transverse size of the proton partner
bunches is also smaller there should be more $ep$ interactions
for a given product of currents than for the colliding bunches. Hence
the coefficient $a$ is not the same in equations \ref{e:splitncoll}
and \ref{e:splitnepilot} and $f_{e-bgr}$ cannot be calculated as
outlined above.

\begin{figure}
\begin{captionbeside}{\mbox{$z$-vertex} distribution for colliding and electron pilot bunches. The distribution for the colliding bunches has been renormalised to contain the same number of events as the distribution for the electron pilot bunches. \vspace{0.5cm}}[i][0.95\textwidth][0.05\textwidth]*
\begin{minipage}[t]{0.5\textwidth}
  \includegraphics[width=\textwidth]{controlplots_bdc_060616_2304_nonep_zvtxlin.eps}
\end{minipage}
\end{captionbeside}
\label{f:zvtxepilot}
\end{figure}

A correct quantitative treatment of the contribution of
\mbox{non-$ep$} background events based on bunch currents and sizes is
beyond the scope of this thesis. The upper limit on the fraction of
electron beam induced background in the \mbox{$e$-pilot} events is
estimated to be 10\% comparing the shape of the \mbox{$E-p_{z}$}
distribution for \mbox{$e$-pilot} and colliding bunches. Given the
estimates for $f_{non-ep}$ this implies that the amount of
\mbox{non-$ep$} background events in the final analysis sample is at
the per mill level and therefore not considered further.

According to~\cite{Pitzl:2006} the upper limit of 10\% is
conservative: special runs with electrons in the machine only have
been performed to study the background conditions for the running
phase II of the HERA collider. In almost none of the events a vertex
has been reconstructed, i.e. the events observed in the present study
should be almost exclusively caused by $ep$ interactions.

\subsection*{Photoproduction Background}


Photoproduction ($\gamma p$) events are $ep$ interactions with
a very small momentum transfer \mbox{$Q^{2} \approx 0$}, i.e. the
exchanged photon is quasi-real. The polar angle of the scattered
electron is so large that it escapes detection in the SpaCal and
leaves the main detector through the beam pipe\footnote{To define
photoproduction events experimentally the lower acceptance limit of
the SpaCal in $Q^{2}$ is used, i.e. for the shifted vertex period
events with $Q^{2} \lesssim 0.5\, \mathrm{GeV^{2}}$ are classified as
photoproduction events (see figure~\ref{f:kinshift}).}. Nevertheless a
photoproduction event can pass the selection criteria of the analysis
if it has a large inelasticity $y$ and a particle of the hadronic
final state mimics the signature of a scattered electron in the
backward detectors. The contribution of this background source is
significant as the inclusive cross-section for
$ep$ interactions is proportional to $1/Q^{4}$ (see
equation~\ref{e:sigmaep}).

Two classes of hadronic final state particles constitute the dominant
sources for spurious electron candidates: neutral $\pi$ mesons and
charged hadrons. The former decay almost exclusively into two
photons. For the cluster energies considered in this analysis
(\mbox{$E_{e} > 7\, \mathrm{GeV}$}) the opening angle of the two decay
photons is so small that they are most likely reconstructed as a
single electromagnetic cluster in the SpaCal. Since photons are
neutral particles one would expect no associated hits in the backward
tracking detectors. However, they convert with a large probability to
$e^{+}e^{-}$ pairs while interacting with the detector material on
their way from the interaction region to the SpaCal. Such conversions
can cause a BST and/or BDC track to be linked to the SpaCal
cluster. Furthermore charged hadrons can imitate the scattered
electron as the corresponding clusters can be classified as
electromagnetic due to fluctuations in the hadronic shower development
or an overlap with a final state photon.

Due to the same mechanisms a final state hadron of a DIS signal event
can mimic the signature of the scattered electron such that the
electron identification is not unambiguous (see
section~\ref{c:evreco}).

Photoproduction events can be rejected to a large extent by applying
selection criteria concerning shower shape, track validation and
\mbox{$E-p_{z}$} (see section~\ref{c:dissel}). The remaining
contribution is statistically subtracted from the data based on a
Monte Carlo simulation (see sections \ref{c:mcmodels} and
\ref{c:crosssection}).


\section{Monte Carlo Models} \label{c:mcmodels}


For the measurement of the DIS cross-section the number of selected
events has to be corrected for the limited detector efficiencies and
acceptances. Furthermore the migration of events due to the finite
resolution or photon radiation from the electron line has to be
considered. It is very difficult to determine the individual
corrections based on the recorded data alone since they are hard to
disentangle. For example, the influence of radiative events may not be
distinguished from inefficiencies (see section~\ref{c:vertexreco}) or
resolution effects. Hence a complicated unfolding procedure would have
to be applied to measure the cross-section.

To avoid this Monte Carlo techniques are employed to perform a
simulation of the $ep$ interactions in the H1 detector. If
detector effects and radiative $ep$ scattering are correctly
implemented the unfolding technique can be replaced by a simple
comparison of the number of events in recorded data and simulation
(see section~\ref{c:crosssection}). This involves a subtraction of the
photoproduction background again performed based on a Monte Carlo
simulation. The simulated event samples are employed further to quantify
some of the sources for systematic errors of the measurement.

The Monte Carlo calculations are divided into two parts: the
simulation of the pure physics process and the simulation of the
detector response. The former is realised by using \emph{Monte Carlo
generators} having specific underlying theoretical models. Two
generators are introduced in the following. The result of the event
generation are the four-momenta and identities of all final state
particles. They are subjected to the detector simulation which is the
topic of section~\ref{c:simulation}. The Monte Carlo events are
reconstructed in the same way as the recorded data.

The generation of events proceeds in the following way: before the
actual event generation the total cross-section $\sigma_{tot}$ for the
$ep$ scattering is calculated via a numerical integration over
the kinematical variables. During the generation a point
\mbox{$(x,Q^{2})$} of the kinematic phase space is chosen randomly for
each event with a probability \mbox{$(d\sigma/dx
dQ^{2})/\sigma_{tot}$}. First the elementary scattering process for
the event is calculated. In a second step additional partons,
so-called \emph{parton cascades} are created. Finally, during the
\emph{hadronisation} the partons are transformed into observable
hadrons.

\subsection*{DJANGO}

DIS events are simulated with the generator DJANGO~\cite{Schuler:1992}
which provides an interface to the programs
HERACLES~\cite{Kwiatkowski:1990es} and LEPTO~\cite{Ingelman:1992}.
For the present analysis the version DJANGOH 1.4 is used.

LEPTO simulates the hadronic final state for the bulk of the kinematic
phase space. This includes the calculation of QCD matrix elements to
first order in $\alpha_{s}$ as well as the generation of parton
cascades. For the latter the Colour Dipole
Model~\cite{Gustafson:1986db, Gustafson:1987rq, Andersson:1988gp} as
implemented in ARIADNE~\cite{Lonnblad:1992tz} is chosen. The
hadronisation is performed by the JETSET
package~\cite{Sjostrand:1993yb} making use of the Lund string
model~\cite{Andersson:1983ia, Andersson:1983jt}.

With previous versions of DJANGO used by the H1 collaboration it was
not possible to generate events with invariant masses of the hadronic
final state $W$ below \mbox{$5\, \mathrm{GeV}$} due to problems with
LEPTO and JETSET in this region of the kinematic phase
space~\cite{Lendermann:2006}. This has led to discrepancies between
simulation and recorded data at low transverse momenta of the hadronic
final state $p_{t,had}$ for former analyses (see
section~\ref{c:dissel}). To overcome this problem the
SOPHIA model~\cite{Mucke:1999yb} has been included into the present
version of DJANGO. It has been developed for astrophysical
applications and provides an accurate description of photon-hadron
interactions reproducing a large set of available data. Depending on
$W$ SOPHIA simulates the production of the major baryon resonances,
direct pion production, multiparticle production based on the Dual
Parton Model~\cite{Capella:1992yb} with subsequent Lund string
fragmentation as well as the diffractive production of the light
vector mesons $\rho$ and $\omega$. For \mbox{$W < 5\, \mathrm{GeV}$}
SOPHIA takes over the simulation of the hadronic final state instead
of LEPTO. The Lund string fragmentation is performed by JETSET.

The complete one-loop electroweak radiative corrections as well as
radiative scattering to first order in $\alpha$ are taken into account
by HERACLES. The influence of radiative events on the cross-section
measurement is discussed in detail in section~\ref{c:radcor}.

The $x$ and $Q^{2}$ dependence of the generated cross-section can be
chosen by selecting a set of proton structure functions available in
the PDFLIB~\cite{Plothow-Besch:2000}. For the present simulation the
parton densities derived in a global QCD analysis by the MRST
group~\cite{Martin:1998sq} are used\footnote{The number of the
corresponding set of proton structure functions in the PDFLIB is 75.},
the longitudinal structure function $F_{L}$ is set to zero. During the
analysis the simulated events are reweighted using the ALLM97
parameterisation~\cite{Abramowicz:1997ms} for $F_{2}$ and a prediction
based on the saturation model~\cite{Golec-Biernat:1998js} for the
longitudinal structure function\footnote{The MRST parton densities
cannot be used for the analysis as they should not be applied below
\mbox{$Q_{min}^{2} = 1.25\,
\mathrm{GeV^{2}}$}~\cite{Plothow-Besch:2000}.} (for details on the two
models see section~\ref{c:phenomenology}).

If the events were generated strictly according to the inclusive $ep$
cross-section only a rather limited amount of events would be
available at large values of $Q^{2}$ leading to sizeable statistical
fluctuations. Therefore only a fraction of the events is generated
below a certain value of $Q^{2}$. In order to restore the original
cross-section weight factors are calculated which have to be applied
during the analysis. This procedure is referred to as
\mbox{\emph{$Q^{2}$ weighting}}.

Ten million DJANGO events have been produced corresponding to an
integrated luminosity of approximately $3\, \mathrm{pb^{-1}}$ which is
more than a factor of five larger than for the data sample. Hence the
systematic error of the cross-section resulting from the limited Monte
Carlo statistics is negligible.

\subsection*{PHOJET}

The photoproduction background is simulated with the help of version
1.10 of the generator PHOJET~\cite{Engel:1995yd}. It aims for a
complete description of the total photoproduction
cross-section\footnote{A comprehensive review on photoproduction
processes at HERA can be found in~\cite{Erdmann:1996}.} based on the
two-component Dual Parton Model~\cite{Capella:1992yb}. Soft and hard
hadronic processes are incorporated. The former are modelled using
Regge phenomenology while the latter are calculated in perturbative
QCD using leading order matrix elements. Like for DJANGO the
hadronisation proceeds according to the Lund String Model as realised
in JETSET.

To exclude double-counting of events the kinematic phase space has to
be divided between DJANGO and PHOJET: for $Q^{2} > 0.1\,
\mathrm{GeV^{2}}$ DJANGO is used, for lower virtualities PHOJET.

The normalisation of the DJANGO sample with respect to the data sample
can be simply performed based on the luminosities of the two
samples. The luminosity for DJANGO can be easily calculated from the
number of events and the total inclusive DIS cross-section taking the
\mbox{$Q^{2}$ weights} into account. For PHOJET the normalisation is
more difficult, section~\ref{c:phojetnorm} is devoted to this
issue. Like in the case of DJANGO a \mbox{$Q^{2}$ weighting} has been
applied and the number of generated events amounts to ten million.

\section{Detector Simulation} \label{c:simulation}


Since the Monte Carlo samples shall be used to model various detector
effects instead of explicitely correcting the data for them it has to
be ensured that they are well described within the simulation. This
includes detector acceptances, efficiencies and resolutions as well as
the influence of noise and dead material. To achieve an accurate
description all generated signal and background events pass a detailed
detector simulation and reconstruction comprising the following steps:

\begin{enumerate}
\item{The passage of the particles through all detector components is
simulated by the H1SIM package based on GEANT3~\cite{Brun:1987}. To
save computing time the shower development in the calorimeters is by
default not simulated in detail. Instead a fast but accurate shower
parameterisation has been developed and implemented into the program
H1FAST~\cite{Grindhammer:1989zg}.}
\item{In the \emph{digitisation} step which is also part of H1SIM 
the simulated subdetector responses are converted into data of the
same form as collected from the real detector components. Basic
efficiency and noise corrections are applied at this stage.}
\item{The fully simulated events are subjected to the same
reconstruction procedure as the recorded data. The software package
H1REC is responsible for this step.}
\end{enumerate}

The version 32800 of H1SIM used for the simulation benefits from the
improved description of the dead material in the backward region of
the H1 detector achieved in~\cite{Lendermann:1999}. This is especially
important for the present analysis as it heavily employs subdetectors
in the backward region.

The crossing angle of the final state particles with the detector
material of the LAr is different between shifted vertex period and
standard running conditions. The shower parameterisation has been
derived for the latter. Therefore the level of detail for the geometry
description and the particle tracking in the LAr has been set to
`fine' instead of `coarse' (default) granularity\footnote{For the fine
granularity the calorimeter stacks are implemented layer by layer
whereas for the coarse granularity a calorimeter stack is realised as
a block of properly mixed homogeneous material without any
substructure.}. This implies simulating the full shower development
instead of employing the fast parameterisation. The change of the
granularity is motivated by the observation of differences in the
distribution of $y_{\Sigma}$ between data and Monte Carlo for the
shifted vertex period~\cite{Vargas:2006}.

Detector simulation and reconstruction are the time consuming part
during the production of Monte Carlo samples, the time needed for the
event generation is negligible compared to this. The complete Monte
Carlo production has been performed on a computer farm at the
University of Dortmund using approximately 50 nodes. Each of the
computers is equipped with two Intel Xeon processors having a clock
rate between \mbox{$2\, \mathrm{GHz}$} and \mbox{$3\,
\mathrm{GHz}$}. On average the simulation of one events takes about
\mbox{$5\, \mathrm{s}$}. As already mentioned, 20 million Monte Carlo
events are used for the cross-section analysis. Dedicated Monte Carlo
samples with special settings used for systematic studies concerning
the reconstruction of the hadronic final state performed
in~\cite{Vargas:2006} amount to approximately the same number of
events. On a single processor of the given type the Monte Carlo
production would have required more than five years.

\subsection*{Noise in the LAr Calorimeter}


To quantify the importance of the different detector components (LAr,
Central Tracker, SpaCal) for the reconstruction of the hadronic final
state, their relative contribution to the inelasticity $y_{h}$
reconstructed with the hadron method (see section~\ref{c:evkindet})
has been determined in~\cite{Vargas:2006}. The result can be seen in
figure~\ref{f:yhaddecomp} as a function of $y_{h}$. For the
calculation of $y_{h}$ the LAr noise identified by the FSCOMB
algorithm (see section~\ref{c:evreco}) is not subtracted, but its
contribution is indicated separately. As expected, the SpaCal
dominates at high $y_{h}$, the Central Tracker contributes at
intermediate $y_{h}$ and at low $y_{h}$ only the LAr can reconstruct
the hadronic final state particles. The identified noise plays an
important role for \mbox{$y \lesssim 0.03$}. Since discrepancies
between data and Monte Carlo simulation have been observed for
\mbox{$y_{\Sigma} < 10^{-2}$}, the implementation of the noise in the
detector simulation has been investigated in detail. The results are
presented in the following.

\begin{figure}
  \centering
  \includegraphics[width=0.7\textwidth]{yhaddecomp_060524_1637.eps}
  \caption{Relative contribution of LAr, Central Tracker and SpaCal to $y_{h}$ as a function of $y_{h}$. The symbols represent the data while the histograms refer to the Monte Carlo simulation~\cite{Vargas:2006}.}
  \label{f:yhaddecomp}
\end{figure}

If the event samples generated by DJANGO and PHOJET were passed to the
detector simulation without any additional measures only energy
depositions originating from $ep$ interactions would be
simulated in the LAr calorimeter. No signals due to noise would be
present. To include noise into the simulation events selected by
random triggers\footnote{Among the selected events there should be
nearly no $ep$ interactions.} are collected in special runs with no
online noise suppression. The total amount of recorded events is
referred to as a \emph{noise file}. During the simulation the signals
in the LAr cells stored in the noise file are superimposed to the
energy depositions caused by the genuine hadronic final state. Noise
files contain typically a few thousand events and are recorded every
few months. While for the simulation of long running periods several
noise files are available, for the shifted vertex period only a single
noise file is employed, referred to as default noise file in the
following. It has been recorded one day after the shifted vertex
run. This implies that at the time the noise events have been
collected the average \mbox{$z$-position} of the interaction vertex
has been already back at its standard value of
\mbox{$\overline{z_{vtx}} \approx 0\, \mathrm{cm}$}.

Due to the different running conditions for analysis sample and noise
file it is desirable to check that the latter is indeed well-suited to
model the noise situation in the LAr during the shifted vertex
run. This is realised in the following way: also during standard data
taking a small fraction of events is selected by random
triggers\footnote{The random trigger rate is typically \mbox{$0.2\,
\mathrm{Hz}$}.}. They are used in the following for comparisons with
the default noise file. It has to be considered that for the random
events recorded during standard data taking an online noise
suppression is performed while for the noise file not.

Three samples of random events recorded before, during and after the
shifted vertex run are investigated. Apart from the default noise file
two additional ones are analysed: one collected roughly two months
before the shifted vertex period and the other during the running
phase II of HERA without any beams in the machine. The latter should
allow to estimate the contribution of beam induced noise. Details
about all samples of random events and noise files can be found in
tables \ref{t:randomsamples} and \ref{t:noisefiles}.

\begin{table}
\centering
\begin{tabular}{c@{\hspace{2em}}c@{\hspace{2em}}c}
\toprule
period & run range & number of events \\
\midrule
before SVX run & 278412 - 278575 & 28644 \\
during SVX run & 278707 - 278891 & 29578 \\
after SVX run & 279053 - 279338 & 25062 \\
\bottomrule
\end{tabular}
\caption{Details about the samples of random events employed for the comparison with the noise files (see table~\ref{t:noisefiles}). The shifted vertex period covers the run numbers 278687 - 278978.}
\label{t:randomsamples}
\end{table}

\begin{table}
\centering
\begin{tabular}{c@{\hspace{2em}}c@{\hspace{2em}}c@{\hspace{2em}}c}
\toprule
name & run & number of events & running conditions \\
\midrule
alternative noise file & 274095 & 3002 & luminosity \\
default noise file & 279046 & 3003 & luminosity \\
noise file w/o beams & 348720 & 15745 & no beams, HERA II \\
\bottomrule
\end{tabular}
\caption{Details about the noise files used for the comparison with the samples of random events (see table~\ref{t:randomsamples}). The shifted vertex period covers the run numbers 278687 - 278978.}
\label{t:noisefiles}
\end{table}

The comparison is realised by subjecting all samples to the module
ASCALE which is the first one in the LAr reconstruction
chain~\cite{Kuester:1991}. It converts the calibrated charges in the
individual cells which are read out during data taking to energies at
a so-called \emph{ideal electromagnetic scale}. These would correspond
to energies of the incident particles if they were all electrons or
photons and no detector imperfections such as dead material would be
present. Cluster formation, correction for dead material and the
reconstruction of energies for hadrons (see
section~\ref{c:calorimetry}) are the task of the later modules of the
reconstruction software.

The energies registered in each calorimeter cell due to pure
electronics noise form a Gaussian distribution~\cite{Borras:1992}. Its
mean value is called pedestal, the corresponding standard deviation is
referred to as $\sigma_{noise}$. After subtraction of the pedestal
which is done online the distribution of the noise signals is
symmetric around zero. Hence the pure physics signal is increased or
decreased depending on the sign of the registered energy.

ASCALE performs a noise suppression on top of the online noise
suppression: first it applies a symmetric cut \mbox{$f \cdot
\sigma_{noise}$} similar to the online noise suppression, i.e. only
LAr cells are taken into account during the further reconstruction for
which the absolute value of the deposited energy after subtraction of
the pedestal exceeds \mbox{$f \cdot \sigma_{noise}$}. This implies
that also cells with negative energies pass the selection which is
important since they partly compensate the positive noise contribution
to the measured signal. The value of the multiplication factor $f$
depends on the calorimeter wheel (see table~\ref{t:fnoise}),
$\sigma_{noise}$ is determined for each LAr cell individually during
special calibration runs. Depending on the calorimeter region the
values of $\sigma_{noise}$ vary between \mbox{$15\, \mathrm{MeV}$} and
\mbox{$30\, \mathrm{MeV}$}~\cite{Abt:1996xv}. This holds also for the
year 2000~\cite{Gayler:2006}.

\begin{table}
\centering
\begin{tabular}{c@{\hspace{2em}}c}
\toprule
calorimeter wheels & f \\
\midrule
CB & 2.0 \\
FB & 2.5 \\
IF, OF & 3.0 \\ 
\bottomrule
\end{tabular}
\caption{Multiplication factor $f$ used for the cell-wise noise cut in ASCALE for the different wheels of the LAr calorimeter. For their notation see figure~\ref{f:larrz}.}
\label{t:fnoise}
\end{table}

In the course of the further reconstruction within ASCALE a
topological noise cut is applied in addition. Cells are only kept for
the reconstruction if their signal or the signal of one of their
neighbouring cells exceeds \mbox{$+4 \sigma_{noise}$} or if their
signal undershoots \mbox{$-4 \sigma_{noise}$}. The cells below
\mbox{$-4 \sigma_{noise}$} are kept in order to compensate for the
noise contribution to the seed cells above \mbox{$+4 \sigma_{noise}$},
but no neighbours are collected around them since they are purely
noise.

Since the cuts against noise applied by ASCALE are at least as
restrictive as the online noise suppression the noise files and
samples of random events can be compared after this reconstruction
step though for the random event samples in contrast to the noise
files an online noise suppression is performed. For the comparison
four quantities are chosen. At first two quantities to investigate the
energy dispersion within the LAr calorimeter for the different
samples: the distribution of the energy $E_{i}^{em}$ reconstructed in
the individual LAr cells and the reconstructed energy per interval of
the polar angle~$\theta$ which is calculated with respect to \mbox{$z
= 0\, \mathrm{cm}$}. To estimate the influence of the noise on the
reconstruction of the hadronic final state two additional kinematic
variables are constructed following equations \ref{e:pthad} and
\ref{e:sigma}:
\begin{align}
p_{t,LAr}^{em} & = \sqrt{(\sum_{i} E_{i}^{em} \sin \theta_{i} \cos{\phi_{i}})^{2} + (\sum_{i} E_{i}^{em} \sin \theta_{i} \sin{\phi_{i}})^{2}} \label{e:pthadnoise} \\
\Sigma_{LAr}^{em} & = \sum_{i} E_{i}^{em} (1 - \cos \theta_{i}) \label{e:sigmanoise}
\end{align}
In both equations the summation is performed over all LAr
cells. Correspondingly, $\theta_{i}$ and $\phi_{i}$ represent the
polar and azimuthal angle of the LAr cell $i$, respectively. The former
is calculated w.r.t. the average \mbox{$z$-position} of the vertex
\mbox{$\overline{z_{vtx}} = \zvtxmean$} for the shifted vertex period.

The noise files and the random event samples are compared among each
other and the default noise file for the simulation of the shifted
vertex period is compared to the corresponding random event
sample. The results are presented in figures \ref{f:noisebasic} and
\ref{f:noisekin}.

\begin{figure}
  \begin{minipage}[c]{0.5\textwidth}
    \includegraphics[width=\textwidth]{noisecomp_274095_348720nobeams_vs_279046_ecell.eps}
    \put(-157,161){a)}
  \end{minipage}
  \begin{minipage}[c]{0.5\textwidth}
    \includegraphics[width=\textwidth]{noisecomp_274095_348720nobeams_vs_279046_theta.eps}
    \put(-157,161){b)}
  \end{minipage}
  \vskip 0.2cm
  \begin{minipage}[c]{0.5\textwidth}
    \includegraphics[width=\textwidth]{noisecomp_nombef_nomaft_vs_svx_ecell.eps}
    \put(-157,161){c)}
  \end{minipage}
  \begin{minipage}[c]{0.5\textwidth}
    \includegraphics[width=\textwidth]{noisecomp_nombef_nomaft_vs_svx_theta.eps}
    \put(-157,161){d)}
  \end{minipage}
  \vskip 0.2cm
  \begin{minipage}[c]{0.5\textwidth}
    \includegraphics[width=\textwidth]{noisecomp_svx_vs_279046_ecell.eps}
    \put(-157,161){e)}
  \end{minipage}
  \begin{minipage}[c]{0.5\textwidth}
    \includegraphics[width=\textwidth]{noisecomp_svx_vs_279046_theta.eps}
    \put(-157,161){f)}
  \end{minipage}
  \caption{Left column: distribution of the energy $E_{i}^{em}$ reconstructed in the individual cells of the LAr calorimeter. Right column: distribution of the energy in the LAr as a function of the polar angle $\theta$. Each distribution is shown for various random event samples (see table~\ref{t:randomsamples}) and noise files (see table~\ref{t:noisefiles}). Each row corresponds to one set of samples.}
  \label{f:noisebasic}
\end{figure}

\begin{figure}
  \begin{minipage}[c]{0.5\textwidth}
    \includegraphics[width=\textwidth]{noisecomp_274095_348720nobeams_vs_279046_empz.eps}
    \put(-157,161){a)}
  \end{minipage}
  \begin{minipage}[c]{0.5\textwidth}
    \includegraphics[width=\textwidth]{noisecomp_274095_348720nobeams_vs_279046_pt.eps}
    \put(-157,161){b)}
  \end{minipage}
  \vskip 0.2cm
  \begin{minipage}[c]{0.5\textwidth}
    \includegraphics[width=\textwidth]{noisecomp_nombef_nomaft_vs_svx_empz.eps}
    \put(-157,161){c)}
  \end{minipage}
  \begin{minipage}[c]{0.5\textwidth}
    \includegraphics[width=\textwidth]{noisecomp_nombef_nomaft_vs_svx_pt.eps}
    \put(-157,161){d)}
  \end{minipage}
  \vskip 0.2cm
  \begin{minipage}[c]{0.5\textwidth}
    \includegraphics[width=\textwidth]{noisecomp_svx_vs_279046_empz.eps}
    \put(-157,161){e)}
  \end{minipage}
  \begin{minipage}[c]{0.5\textwidth}
    \includegraphics[width=\textwidth]{noisecomp_svx_vs_279046_pt.eps}
    \put(-157,161){f)}
  \end{minipage}
  \caption{Left column: distribution of $\Sigma_{LAr}^{em}$ (see equation~\ref{e:sigmanoise}). Right column: distribution of $p_{t,LAr}^{em}$ (see equation~\ref{e:pthadnoise}). Each distribution is shown for various random event samples (see table~\ref{t:randomsamples}) and noise files (see table~\ref{t:noisefiles}). Each row corresponds to one set of samples.}
  \label{f:noisekin}
\end{figure}

The following conclusions can be drawn: the distribution of the
individual cell energies $E_{i}^{em}$ shows almost no entries in the
interval \mbox{$-50\, \mathrm{MeV} < E_{i}^{em} < 50\,
\mathrm{MeV}$}. This gap is expected given the typical values for
$\sigma_{noise}$ and the values of $f$ used by ASCALE (see
table~\ref{t:fnoise}).

The different noise files show differences for the distributions of
the basic quantities as well as the kinematic variables. The
distribution of $E_{i}^{em}$ is symmetric around zero for the noise
file without beams in the machine as one would expect from pure
electronics noise (see figure~\ref{f:noisebasic}~a)). For the two
noise files with beams there is a clear excess at positive
energies. In addition they show a significant rise of the
reconstructed energies towards small $\theta$ in contrast to the noise
file without beams (see figure~\ref{f:noisebasic}~b)). Hence one can
conclude that the majority of the noise is beam-induced. This can be
also clearly seen in figure~\ref{f:noiserz} which shows an
\mbox{$rz$-view} of the average reconstructed energy per event in the
LAr calorimeter for the default noise file and the noise file without
beams. For the default noise file the forward region is occupied by
large positive energy depositions while there are no such depositions
in the case of the noise file without beams. The differences for
$\Sigma_{LAr}^{em}$ and $p_{t,LAr}^{em}$ (see figure~\ref{f:noisekin}
a) and b)) are less significant than for the basic quantities as both
quantities are rather insensitive to losses or additional energy
depositions in the forward region as already explained in
section~\ref{c:evkindet}.

\begin{figure}[t]
\begin{minipage}[c]{1.0\textwidth}
  \centering
  \includegraphics[width=0.7\textwidth]{noisecomp_279046_rz.eps}
  \put(-323,180){a)}
\end{minipage}
\begin{minipage}[c]{1.0\textwidth}
  \centering
  \includegraphics[width=0.7\textwidth]{noisecomp_348720_rz.eps}
  \put(-323,180){b)}
\end{minipage}
\caption{Average energy per event reconstructed in the LAr calorimeter at the ideal electromagnetic scale for the noise files a) 279046 and b) 348720 (see table~\ref{t:noisefiles}) depicted in the \mbox{$rz$-plane}.}
\label{f:noiserz}
\end{figure}

As far as the different random event samples are concerned the sample
recorded before the shifted vertex period shows a difference in the
kinematic distributions compared to the other two (see
figure~\ref{f:noisekin} c) and d)).

The comparison of the default noise file with the random event sample
recorded during the shifted vertex run reveals that the default noise
file is well-suited for the simulation of LAr noise in the shifted
vertex run.

A general outcome of this study is the realisation that the random
events continously recorded during standard data taking are not only
useful to judge the suitability of a noise file, but may be used
instead of the dedicated noise files as an input for the
simulation. Their great advantage is that they are permanently
recorded and hence allow to react on changing running conditions on
short time scales. This option is discussed within the H1
collaboration. Additional studies are necessary to quantify the
influence of the online noise suppression on physics analyses.

To estimate the influence of the LAr noise on the discrepancy between
data and Monte Carlo simulation in the
\mbox{$y_{\Sigma}$-distribution} for $y_{\Sigma} < 10^{-2}$ a number
of dedicated Monte Carlo samples has been produced: for one sample the
alternative noise file has been employed, for a second one a special
noise file containing only half of the events of the default noise
file is considered\footnote{As the noise files contain only three
thousand events the same noise pattern is overlayed to the physics
signal quite often during the simulation. This may create large
fluctuations. If they are the reason for the discrepancy it should be
enhanced when using the special noise file.}. Finally, one Monte Carlo
simulation has been performed without adding any LAr noise at
all. Detailed investigations in~\cite{Vargas:2006} have shown that
none of the samples leads to a significant improvement or worsening of
the discrepancy. With the help of the simulation without noise it has
been demonstrated that a large fraction of the noise identified by
FSCOMB are in fact energy depositions caused by isolated particles of
the genuine hadronic final state.

\subsection*{Beam Tilt}

Due to the changing beam conditions the H1 coordinate system is not
ideally aligned with respect to the beam axis. Correspondingly, the
beams are inclined w.r.t. to the \mbox{$z$-axis}, an effect called
\emph{beam tilt}. It is reflected in the distribution of the
primary vertices as schematically depicted in
figure~\ref{f:beamtilt}~a).

\begin{figure}
  \centering
  \begin{minipage}[t]{0.47\textwidth}
    \includegraphics[width=\textwidth]{beamtilt_para.eps}
    \put(-193,111){a)}
  \end{minipage}
  \hspace{0.01\textwidth}
  \begin{minipage}[t]{0.47\textwidth}
    \includegraphics[width=\textwidth]{beamtilt_corr.eps}
    \put(-193,111){b)}
  \end{minipage}
  \caption{a) Schematic of the beam tilt and the variables used for its parameterisation in the \mbox{$yz$-plane}. The points indicate primary vertices. b) Difference of the polar angle of the scattered electron measured in the H1 coordinate system ($\theta_{e}^{H1}$) and the one describing the $ep$ interaction ($\theta_{e}$) illustrated for a cluster in the SpaCal.}
  \label{f:beamtilt}
\end{figure}

Because of the beam tilt the polar angles of all final state particles
measured in the H1 coordinate system are not identical to the polar
angles describing the scattering process, i.e. all reconstructed polar
angles have to be corrected for the effect of the beam tilt. This holds
especially for the scattered electron as shown in
figure~\ref{f:beamtilt}~b).

To determine the beam tilt the primary vertices reconstructed by the
Central Tracker are subjected to straight line fits in the
$xz$- and \mbox{$yz$-plane}, i.e. the average vertex position
\mbox{$(\overline{x_{vtx}},\overline{y_{vtx}})$} is parameterised in
the following way:
\begin{align}
\overline{x_{vtx}}(z_{vtx}) & = x_{0} + z_{vtx} \tan \delta_{x} \label{e:beamtilt1} \\
\overline{y_{vtx}}(z_{vtx}) & = y_{0} + z_{vtx} \tan \delta_{y} \label{e:beamtilt2}
\end{align}
The free parameters of the fit are the offsets $x_{0}$ and $y_{0}$ at
\mbox{$z_{vtx} = 0\, \mathrm{cm}$} as well as the tilts $\delta_{x}$
and $\delta_{y}$ of the beam. Figure~\ref{f:beamtilt}~a) illustrates
them in the \mbox{$yz$-plane}, their values for the shifted vertex
period as obtained from the data are listed in table~\ref{t:beamtilt}.

\begin{table}
\centering
\begin{tabular}{c@{\hspace{1em}}c@{\hspace{1em}}c@{\hspace{1em}}c}
\toprule
$x_{0}\; [\mathrm{cm}]$ & $y_{0}\; [\mathrm{cm}]$ & $\delta_{x}\; [\mathrm{mrad}]$ & $\delta_{y}\; [\mathrm{mrad}]$ \\
\midrule
-0.27 & 0.20 & 0.2 & 1.0 \\
\bottomrule
\end{tabular}
\caption{Average beam tilts and offsets for the shifted vertex period as defined in equations \ref{e:beamtilt1} and \ref{e:beamtilt2}.}
\label{t:beamtilt}
\end{table}

Since the beam tilt affects $\theta_{e}$ it influences the mapping of
the detector acceptance to the \mbox{$xQ^{2}$-plane} (see
figure~\ref{f:elecid}). Thus the accurate implementation of the beam
tilt in the detector simulation is essential for the method of the
cross-section determination to be valid. Like the simulation of the
finite spread of the \mbox{$z$-vertex} distribution the realisation of the
beam tilt is the task of H1SIM. The beam offsets and tilts determined
from the data sample are used as input parameters for the simulation.

Figure~\ref{f:beamsimdef} shows the distribution of the reconstructed
primary vertices in the \mbox{$yz$-plane} for the Monte Carlo sample
after the detector simulation and after the complete event
reconstruction, respectively. Though only Central Tracker vertices are
used for the analysis Forward Tracker vertices are also contained in
the distributions\footnote{If no primary vertex reconstructed by the
Central Tracker is present for a given event a primary vertex of the
Forward Tracker is searched for.}. The solid lines indicate the
average vertex position $\overline{y_{vtx}}$ observed for the data
calculated from equation~\ref{e:beamtilt2} based on the values listed
in table~\ref{t:beamtilt}. Two problems with the Monte Carlo samples
can be identified: firstly no beam tilt has been taken into account
for parts of the reconstructed vertices. Furthermore the simulated and
reconstructed vertex positions deviate systematically. However, a
significant fraction of the reconstructed vertices agrees with the
findings in the data.

\begin{figure}
  \begin{minipage}[c]{0.5\textwidth}
    \includegraphics[width=\textwidth]{vertexSim_default_yv_vs_zy_sim.eps}
    \put(-33,161){a)}
  \end{minipage}
  \begin{minipage}[c]{0.5\textwidth}
    \includegraphics[width=\textwidth]{vertexSim_default_yv_vs_zy_rec.eps}
    \put(-33,161){b)}
  \end{minipage}
  \caption{Distribution of primary vertices in the \mbox{$yz$-plane} after a) detector simulation and b) reconstruction before correction (see text). The solid line indicates \mbox{$\overline{y_{vtx}}(z_{vtx})$} as observed in the data (according to equation~\ref{e:beamtilt2} and the values from table~\ref{t:beamtilt}).}
  \label{f:beamsimdef}
\end{figure}

Though the observations seem to be contradictory they can be traced
back to an inconsistency concerning the treatment of the vertex
offsets in H1SIM and H1REC: the beam offset $x_{0}$ and $y_{0}$ given
in table~\ref{t:beamtilt} refer to \mbox{$z_{vtx} = 0\,
\mathrm{cm}$}. They are directly used as an input for the
simulation. H1SIM however interprets them as if they would correspond
to the mean \mbox{$z$-vertex} position used for the simulation,
i.e. \mbox{$z_{vtx} \approx 70\, \mathrm{cm}$}. Finally, within H1REC
they are treated as if they would correspond to \mbox{$z_{vtx} = 0\,
\mathrm{cm}$} again. As a consequence the majority of the events seems
to be reconstructed at the correct position though all particles of
the event have the wrong origin in the simulation. The resulting shift
of \mbox{$1\, \mathrm{mm}$} in the \mbox{$y$-direction} is already
significant for the desired precision of the analysis. In contrast,
the effect of the inconsistency in \mbox{$x$-direction} is almost
negligible as the tilt of the beam in the \mbox{$xz$-plane} is a
factor of five smaller than in the \mbox{$yz$-plane}. For standard
running conditions of the HERA collider the problem is almost not
recognisable as the average \mbox{$z$-position} of the vertex is close
to \mbox{$0\, \mathrm{cm}$}.

The vertices which do not show the effect of the tilted beams are all
forward vertices. As they are not used the problem is not relevant for
the present analysis, but has nevertheless been identified: at the
time when the beam tilt has been included into the simulation it has
been forgotten to change the corresponding routine of the Forward
Tracker reconstruction accordingly.

The mentioned problems have been fixed in the following way: the beam
offsets of table~\ref{t:beamtilt} have been converted to the mean
\mbox{$z$-position} of the simulation before using them in the
steering for H1SIM. The routines of H1REC have been changed
accordingly. Finally, the beam tilt has been implemented for the
Forward Tracker. The result of the changes can bee seen in
figure~\ref{f:beamsimcor} which shows that simulated and reconstructed
vertices are in agreement between themselves and with the data.

\begin{figure}
  \begin{minipage}[c]{0.5\textwidth}
    \includegraphics[width=\textwidth]{vertexSim_corrected_yv_vs_zy_sim.eps}
    \put(-33,161){a)}
  \end{minipage}
  \begin{minipage}[c]{0.5\textwidth}
    \includegraphics[width=\textwidth]{vertexSim_corrected_yv_vs_zy_rec.eps}
    \put(-33,161){b)}
  \end{minipage}
  \caption{Distribution of primary vertices in the \mbox{$yz$-plane} after a) detector simulation and b) reconstruction after correction (see text). The solid line indicates \mbox{$\overline{y_{vtx}}(z_{vtx})$} as observed in the data (according to equation~\ref{e:beamtilt2} and the values from table~\ref{t:beamtilt}).}
  \label{f:beamsimcor}
\end{figure}

All simulations employed for the present analysis
have been performed with a corrected version of the software. All
changes have been communicated to the authors of H1SIM and H1REC and
are available in the new versions of the packages.


\section{Radiative Corrections} \label{c:radcor}


The structure functions $F_{2}$ and $F_{L}$ are defined with respect
to the Born cross-section which describes the interaction of electron
and proton in the single photon approximation to the lowest order in
the electromagnetic coupling constant $\alpha$ (see
section~\ref{c:disxsect}). The measured cross-section however contains
contributions from various radiative processes. Apart from a change of
the cross-section they can cause a wrong determination of the event
kinematics depending on the reconstruction method. Their influence can
be sizeable.

Hence the pure Born cross-section needed for the extraction of the
structure functions has to be calculated from the measured
cross-section by applying a correction factor:
\begin{equation}
\frac{d^{2} \sigma}{dx dQ^{2}} = \frac{1}{1 + \delta_{RC}(x,Q^{2})} \left . \frac{d^{2} \sigma}{dx dQ^{2}} \right |_{meas}
\label{e:radcor}
\end{equation}
The variable \mbox{$\delta_{RC}(x,Q^{2})$} quantifies the radiative
corrections and can be estimated from Monte Carlo simulations or
analytical programs. Since radiative processes are included in the
DJANGO generator as already mentioned in the previous section no
explicit correction is needed for the present analysis. It is
performed implicitly by comparing data and Monte Carlo (see
section~\ref{c:crosssection}). The remaining part of this chapter
deals with the relevant radiative processes, their influence on the
measurement and their implementation in DJANGO.

A detailed description of radiative corrections can be found
in~\cite{Spiesberger:1992}. Different contributions are distinguished
therein: \emph{leptonic corrections} are described by diagrams
containing an additional photon attached to the lepton line,
i.e. photon radiation from the lepton line and photonic lepton vertex
corrections. The corresponding processes for the quark line are
referred to as \emph{quarkonic corrections}. In addition the
interference of the two as well as purely weak corrections have to be
considered. For the $Q^{2}$ values considered in this analysis the
dominant contribution stems from the leptonic corrections. Among these
the most important ones are the experimental configurations introduced
in section~\ref{c:radepscat}, i.e. ISR, FSR and QEDC
events\footnote{Bethe-Heitler events do not enter the analysis sample
as final state electron and photon disappear through the beam pipe.}.

Neglecting detector effects all reconstruction methods introduced in
section~\ref{c:evkindet} should give the same results for
non-radiative events. For radiative events this statement does not
longer hold, depending on the method the kinematics is reconstructed
wrongly. This has been already shortly addressed for the case of ISR
events.

While in ISR events the radiated photon escapes from the detector,
final state radiated photons typically end up in the same cluster as
the scattered electron. For only about 0.5\% of the FSR events two
cluster can be resolved~\cite{Meyer:1997}. Hence FSR events are
experimentally almost indistinguishable from non-radiative events and
do not affect the reconstruction of the event kinematics.

In contrast to this the event reconstruction becomes incorrect for ISR
and QEDC events if the electron method is used\footnote{Reference is
made at this point to section~\ref{c:evkindet} where it has been
explained that $y_{e}$ depends predominantly on $E_{e}$ and
$Q_{e}^{2}$ on $\theta_{e}$.}: the energy of either the incoming or
the outgoing electron is lowered due to the radiation. Hence the
energy of the incoming electron at the leptonic vertex is
overestimated or the one of the outgoing underestimated when using
equations \ref{e:ye} and \ref{e:q2e}. Consequently, the reconstructed
$y_{e}$ is larger than the true inelasticity as calculated at the
hadronic vertex. For QEDC events final state electron and photon have
a sizeable transverse momentum. Thus the polar angle of the final
state electron is smaller than the polar angle corresponding to the
scattering process, i.e. the true $Q^{2}$ is smaller than
$Q_{e}^{2}$.

The sigma method reconstructs ISR events correctly as no information
concerning the incoming electron enters the calculation (see equations
\ref{e:ysigma} and \ref{e:q2sigma}). For QEDC events the reconstructed
variables deviate from the true ones as the photon is not
explicitely identified but dealt with as a particle of the hadronic
final state.

Given the explanations above one expects that the radiative
corrections for the electron method are significantly larger than for
the sigma method. This has been shown in various studies (see
e.g.~\cite{Bassler:1994uq}).

DJANGO comprises all radiative processes mentioned above by
interfacing HERACLES apart from elastic QEDC events. Therefore they
are excluded from the data sample (see section~\ref{c:dissel}). For the
simulation at hand photon radiations from the quark line have not been
included as they are estimated to be small for low $x$.

HERACLES allows for a separate treatment of the Born term and several
parts of the QED corrections, i.e. in the present case ISR, FSR and
QEDC processes. The contribution of each subprocess is integrated
first at the initialisation stage and then events are generated
according to the estimated fractions. During the analysis of the
minimum bias run of the year 1999 it has been observed that the
subprocess fractions for the generated events do not agree with the
fractions reported at the initialisation
stage~\cite{Glazov:2005a}. Deviations of up to 5\% occur.

Checking the subprocess fractions for the present analysis revealed
the same problem. While doing so it has been realised that the problem
is connected to the \mbox{$Q^{2}$ weighting}. This has been confirmed
by producing and analysing a sample of generated events without making
use of the \mbox{$Q^{2}$ weighting}. Indeed, the problem does not
occur. Since no \mbox{$Q^{2}$ dependence} of the difference in the
subprocess fractions has been found the problem can be easily
compensated: global weight factors are derived to be applied during
the analysis which allow the restore the original subprocess fractions
from the initialisation step. The problem has been communicated to the
author of DJANGO.

To derive an uncertainty on the simulation of the radiative
corrections it has been checked using semi-analytical calculations
implemented in the HECTOR~\cite{Arbuzov:1995id} package. An agreement
better than 0.5\% is observed in the kinematic phase space relevant
for this analysis~\cite{Glazov:2005b}.


\section{Extraction of the Structure Function $F_{2}$} \label{c:crosssection}


The measurement of the double differential cross-section is performed
in bins of $x$ and $Q^{2}$ or $y$ and $Q^{2}$ depending on the region
in the kinematic phase space (for details see
section~\ref{c:binning}). For the following explanations a bin in
$x$ and $Q^{2}$ denoted by $\square$ is considered\footnote{This is no
restriction as each bin in $y$ and $Q^{2}$ can be mapped to the
\mbox{$xQ^{2}$-plane}.}.

The uncorrected bin-integrated cross-section
$\tilde{\sigma}^{\square}$ is obtained from the number of events in
the bin for the data sample $N_{Data}^{\square}$ that have passed the
DIS selection criteria (see section~\ref{c:dissel}) and the integrated
luminosity $\mathcal{L}_{Data}$ for the selected runs:
\begin{equation}
\tilde{\sigma}^{\square} = \frac{N_{Data}^{\square}}{\mathcal{L}_{Data}}
\label{e:sigmauncor}
\end{equation}
To determine the bin integrated Born cross-section
\begin{align}
\sigma_{Born}^{\square} & = \int_{\square} \frac{d^{2} \sigma}{dx dQ^{2}}\, dx dQ^{2} \\
 & = \frac{N_{Data}^{\square} - N_{\gamma p}^{\square} - N_{non-ep}^{\square}}{A\, \mathcal{L}_{Data}} \, \frac{1}{c_{\epsilon}} \, \frac{1}{1 + \delta_{RC}}
\end{align}
several corrections have to be applied:
\begin{itemize}
\item{$N_{\gamma p}^{\square}$ is the number of photoproduction background events accumulated in the analysis bin which fulfil the DIS selection criteria. It is estimated using the PHOJET Monte Carlo sample (see section~\ref{c:phojetnorm}).}
\item{$N_{non-ep}^{\square}$ is the number of \mbox{non-$ep$} background events passing the DIS selection criteria. According to section~\ref{c:background} their contribution is negligible.}
\item{The detector acceptance $A$ is determined from the Monte Carlo simulation. It is defined as the ratio of the number of reconstructed to the number of generated events in the analysis bin: \mbox{$A = N_{MC}^{\square}/N_{MC,gen}^{\square}$}.}
\item{$c_{\epsilon}$ are corrections for efficiencies not included or not properly described in the Monte Carlo simulation.}
\item{$\delta_{RC}$ denotes the radiative corrections. They are obtained from Monte Carlo simulations or analytic programs and given by \mbox{$\delta_{RC} = \sigma_{rad,MC}^{\square}/\sigma_{Born,MC}^{\square} - 1$}. Here $\sigma_{rad,MC}^{\square}$ and $\sigma_{Born,MC}^{\square}$ refer to the full and Born bin-integrated cross-section, respectively (see section~\ref{c:radcor}).}
\end{itemize}
To derive the differential cross-section at the chosen central values
($x_{c}$,$Q_{c}^{2}$) of the bin from $\sigma_{Born}^{\square}$ the
bin size and the cross-section behaviour inside the bin have to be
taken into account. The bin size correction transforms the integrated
cross-section into a bin averaged differential cross-section. The
so-called \emph{bin centre correction} converts the latter to the
desired cross-section at ($x_{c}$,$Q_{c}^{2}$). Both corrections are
contained in a single factor $c_{BC}$ which is determined from the
Monte Carlo simulation:
\begin{equation}
c_{BC} = \frac{\left . \frac{d^{2} \sigma}{dx dQ^{2}} \right |_{x=x_{c}, Q^{2}=Q^{2}_{c}}^{MC}}{\sigma_{Born,MC}^{\square}} = \frac{\left . \frac{d^{2} \sigma}{dx dQ^{2}} \right |_{x=x_{c}, Q^{2}=Q^{2}_{c}}^{MC}}{\int_{\square} \left . \frac{d^{2} \sigma}{dx dQ^{2}} \right |^{MC}\, dx dQ^{2}}
\label{e:bccor}
\end{equation}
The differential cross-section in the numerator is calculated with the
help of equation~\ref{e:sigmaep} using the predictions for the
structure functions to which the simulated events are reweighted
during the analysis (see below).

Hence the desired differential cross-section is given by:
\begin{equation}
\left . \frac{d^{2} \sigma}{dx dQ^{2}} \right |_{x=x_{c}, Q^{2}=Q^{2}_{c}} = \frac{N_{Data}^{\square} - N_{\gamma p}^{\square} - N_{non-ep}^{\square}}{A\, \mathcal{L}_{Data}} \, \frac{1}{c_{\epsilon}} \, \frac{c_{BC}}{1 + \delta_{RC}}
\label{e:sigmaq2cxc}
\end{equation}
The cross-section determination can be simplified significantly if the
Monte Carlo model employed to simulate DIS events includes radiative
corrections which is the case for DJANGO (see sections \ref{c:mcmodels}
and \ref{c:radcor}). This can be seen in the following way: if
radiative corrections are incorporated in the simulation the relation
\begin{equation}
N_{MC,gen}^{\square} = \mathcal{L}_{MC}\, \sigma_{rad,MC}^{\square}
\end{equation}
holds, $\mathcal{L}_{MC}$ denoting the luminosity corresponding to the
DJANGO sample. Considering this and inserting the definitions of $A$,
$\delta_{RC}$ and $c_{BC}$ into equation~\ref{e:sigmaq2cxc} one
obtains:
\begin{equation}
\left . \frac{d^{2} \sigma}{dx dQ^{2}} \right |_{x=x_{c}, Q^{2}=Q^{2}_{c}} = \frac{N_{Data}^{\square} - N_{\gamma p}^{\square} - N_{non-ep}^{\square}}{N_{MC}^{\square}} \, \frac{\mathcal{L}_{MC}}{\mathcal{L}_{Data}} \, \frac{1}{c_{\epsilon}} \, \left . \frac{d^{2} \sigma}{dx dQ^{2}} \right |_{x=x_{c}, Q^{2}=Q^{2}_{c}}^{MC}
\label{e:sigmamcm}
\end{equation}
To determine the differential cross-section following
equation~\ref{e:sigmamcm} is referred to as the \emph{Monte Carlo
method}. A similar relation to equation~\ref{e:sigmamcm} is valid for
the reduced cross-section. From the latter the structure function
$F_{2}$ can be extracted (see equation~\ref{e:sigmar}):
\begin{equation}
\left . F_{2} (x,Q^{2}) \right |_{x=x_{c}, Q^{2}=Q^{2}_{c}} = \left . \frac{d^{2} \sigma_{r}}{dx dQ^{2}} \right |_{x=x_{c}, Q^{2}=Q^{2}_{c}} + \frac{y^{2}}{Y^{+}} \left . F_{L} (x,Q^{2}) \right |_{x=x_{c}, Q^{2}=Q^{2}_{c}}
\label{e:f2mcm}
\end{equation}
This implies employing external information on the longitudinal
structure function $F_{L}$. For the present analysis a prediction
based on the saturation model is used~\cite{Golec-Biernat:1998js}. It
has been found to be in agreement with the results on $F_{L}$
extracted from the data of the shifted vertex run and other data in
the low $Q^{2}$ regime~\cite{Lobodzinska:2003}. The inelasticity $y$
is calculated from $x_{c}$ and $Q^{2}_{c}$ via equation~\ref{e:q2sxy}
using the nominal centre of mass energy $\sqrt{s}$. This should not
lead to a wrong estimation of the contribution from $F_{L}$ since at
high $y$ and thus low $x$ the fraction of ISR events should be
negligible.

The use of the Monte Carlo method is only justified if all relevant
aspects of the experiment, like subdetector acceptances and
efficiencies, are correctly described in the Monte Carlo
simulation. Therefore the next chapter is devoted to a large extent to
a thorough comparison between data and Monte Carlo
simulation. Remaining differences are absorbed into the extra
correction factor $c_{\epsilon}$ or considered as systematic errors of
the measurement. This is necessary for the vertex reconstruction with
the Central Tracker (see section~\ref{c:vertexreco}) and the polar
angle reconstruction with the BDC (see section~\ref{c:bdceff}).

To calculate the differential cross-section at ($x_{c}$,$Q_{c}^{2}$)
for the Monte Carlo requires an assumption concerning $F_{2}$ and
$F_{L}$. The chosen parameterisation has to be the same as for the
generation of the events. To first order the resulting cross-section
should be independent of the input parameterisations for $F_{2}$ and
$F_{L}$. However, they influence the result via smearing
effects. Therefore the cross-section is determined in an iterative
procedure: for the first iteration arbitrary parameterisations for
$F_{2}$ and $F_{L}$ can be used. The choice of a realistic behaviour
is preferable but not necessary. If the prediction for the
cross-section is too different from the result obtained from
equation~\ref{e:sigmamcm} new parameterisations are derived from model
fits and the cross-section determination is repeated. To avoid a
repetition of the time consuming simulation for each iteration the
Monte Carlo events are reweighted to the parameterisation obtained from
the previous step.

With the ALLM97 parameterisation~\cite{Abramowicz:1997ms} for $F_{2}$
and the saturation model~\cite{Golec-Biernat:1998js} for $F_{L}$
predictions for the proton structure functions are available which
describe the data reasonably well (see~\cite{Lobodzinska:2003} and
section~\ref{c:f2extract}). Therefore the differential cross-section
can be determined in a single step.


\section{Coverage of the Kinematic Phase Space} \label{c:phasespace}

Two options exist to reconstruct the polar angle $\theta_{e}$ of the
scattered electron as has been discussed in section~\ref{c:evreco}:
Either to use the BDC in combination with the Central Tracker or the
BST alone. This already allows for two independent cross-section
determinations for the shifted vertex period. Furthermore it has been
shown in~\cite{Lastovicka:2004a} that the accessible kinematic phase
space can be considerably extended towards larger values of $x$ at low
$Q^{2}$ by employing initial state radiation events in the analysis in
addition to the standard non-radiative events.

The present section is intended to explain how the different detector
components and event types can be utilised to allow for a maximum
possible coverage of the kinematic phase space in the low $Q^{2}$
regime. Special emphasis is placed on the reconstruction of the event
kinematics for ISR events as the reconstruction methods can provide
wrong results for radiative events (see section~\ref{c:radcor}). Since
many people have been involved in the analysis of the shifted vertex
data the aim of the last part of this section is to distinguish the
contributions stemming from the thesis at hand and other people.

Figure~\ref{f:pscoverage} shows the accessible regions in the
\mbox{$xQ^{2}$-plane} defined by the selection criteria explained in
detail in section~\ref{c:dissel}. At first only non-radiative events
are considered: the cut \mbox{$E_{e} > 7\, \mathrm{GeV}$} on the
energy of the electron candidate defines the lowest accessible value
of $x$ for a given value of $Q^{2}$. The minimum value of $Q^{2}$ for
a given $x$ is determined by the maximum polar angle $\theta_{e}$
allowed for the analysis. For the event selection a cut is placed
instead on the distance $R_{e}$ of the electron track to the beam axis
in the plane\footnote{The value $z=-160.5\, \mathrm{cm}$ corresponds
to the average \mbox{$z$-position} of the clusters in the
electromagnetic section of the SpaCal.} $z=-160.5\, \mathrm{cm}$. It
has to be larger than than \mbox{$10\, \mathrm{cm}$} and \mbox{$9\,
\mathrm{cm}$} for the BST and BDC, respectively\footnote{The lines of
constant $R_{e}$ in figure~\ref{f:pscoverage} indicate the acceptance
limit for the average \mbox{$z$-vertex} position of the shifted vertex
run $\overline{z_{vtx}}$. For larger values of $z_{vtx}$ even lower
values of $Q^{2}$ can be accessed.}. For the BST the acceptance at
\mbox{$R_{e} = 10\, \mathrm{cm}$} has already decreased to
approximately $50\%$~\cite{H1:draft} while even at \mbox{$R_{e} = 9\,
\mathrm{cm}$} the BDC acceptance is still close to $100\%$ 
(see figure~\ref{f:bdceff}). Thus the BDC should provide a larger
amount of events at low $Q^{2}$. In return the BDC analysis is
restricted towards large $x$ by the cut \mbox{$y_{\Sigma} > 0.03$}
ensuring the vertex reconstruction efficiency for the Central Tracker
to be larger than 50\%.

\begin{figure}[t]
  \centering
  \includegraphics[width=0.6\textwidth]{pscoverage.eps}
  \caption{Comparison of the coverage in the \mbox{$xQ^{2}$-plane} reached by the BST and the BDC, non-radiative and radiative events. Apart from the kinematic limit \mbox{$y=1$} (thick solid) lines of constant electron energy $E_{e}$ (dotted), distance $R_{e}$ (thin solid) and $y$ (dashed) for non-radiative events are shown. The lowest values of $R_{e}$ accepted for the analysis are \mbox{$10\, \mathrm{cm}$} and \mbox{$9\, \mathrm{cm}$} for the BST and BDC, respectively. The region of the phase space exclusively accessed by ISR events is schematically highlighted.}
  \label{f:pscoverage}
\end{figure}

Typical inclusive analyses concentrate on non-radiative events. Since
ISR events can lead to a wrong determination of the event kinematics
they are excluded by placing a cut on $E - p_{z}$ (see
equation~\ref{e:empzcons}). This is effective since the photon escapes
through the beam pipe undetected such that $E - p_{z}$ is smaller than
$2 E_{e}^{0}$.

Due to the photon radiation an ISR event can be interpreted as a usual
$ep$ scattering event at a lower electron beam energy
$E_{e}^{0}$ resulting in a lower centre of mass energy \mbox{$s = 4
E_{e}^{0} E_{p}^{0}$}. Hence according to equation~\ref{e:q2sxy}
lower values of $Q^{2}$ for a given $x$ or larger values of $x$ for a
given $Q^{2}$ can be reached compared to non-radiative events. ISR
events allow to cover a region of the kinematic phase space at high
$x$ and very low $Q^{2}$ which is inaccessible for non-radiative
events. It is schematically indicated in figure~\ref{f:pscoverage}.

Previous analyses performed by the H1 and ZEUS Collaborations have
explicitely detected the radiated photon in the photon arm of the
luminosity system~\cite{Ahmed:1995cf, Derrick:1995ef, Cole:2004,
Issever:2000}. Assuming a collinear radiation of the photon the
incoming electron energy $E_{e,ISR}^{0}$ can be calculated in this
case from the photon energy $E_{\gamma}$ measured in the photon
detector:
\begin{equation}
E_{e,ISR}^{0} = E_{e}^{0} - E_{\gamma}
\end{equation}
Replacing $E_{e}^{0}$ by $E_{e,ISR}^{0}$ in equations \ref{e:ye} and
\ref{e:q2e} also the electron method yields correct results for ISR
events.

The disadvantage of explicitely detecting the radiated photon is the
contribution of coincidences of photoproduction/DIS and Bethe-Heitler
events to the analysis. They constitute the dominant background
source. While the photoproduction or DIS event causes an electron
signature in the SpaCal the photon of the overlapping Bethe-Heitler
event deposits energy in the photon detector. This source of
background events is significant due to the large rate of
Bethe-Heitler events.

For this reason it has been proposed in~\cite{Lastovicka:2004a} to
incorporate ISR events into the shifted vertex analysis without
explicitely detecting the radiated photon. Apart from the absence of
the mentioned background source this has the advantage of an increased
acceptance for ISR events. The sigma method correctly reconstructs the
kinematic variables $y$ and $Q^{2}$ (see equations \ref{e:ysigma} and
\ref{e:q2sigma}). However, for the determination of $x$ via
equation~\ref{e:q2sxy} one has to take into account that the centre of
mass energy $s$ is lowered in ISR events. This is done by replacing
\mbox{$s = 4 E_{e}^{0} E_{p}^{0}$} by a so-called \emph{reduced centre
of mass energy} $s_{r}$ employing equation~\ref{e:empzcons}:
\begin{equation}
x_{\Sigma} = \frac{Q_{\Sigma}^{2}}{s_{r}\, y_{\Sigma}},\; s_{r} = 2 (E-p_{z}) E_{p}^{0}
\label{e:xsigma}
\end{equation}
The value of $s_{r}$ is close to $s$ for non-radiative events while
smaller for ISR events. Hence the presence of an ISR event is inferred
from energy and longitudinal momentum conservation. In contrast to
\cite{Lastovicka:2004a} no separation between non-radiative and ISR
events is performed for the present analysis. Equation~\ref{e:xsigma}
is consistently applied throughout the entire phase space allowing for
an optimal coverage of the transition region between the parts of the
phase space which are dominated by non-radiative or ISR events.

In case the electron method is used to reconstruct the kinematic
variables a cut on $E-p_{z}$ is applied to avoid a wrong
reconstruction of the kinematics (see section~\ref{c:dissel}). This
implies that for the region in the
\mbox{$xQ^{2}$-plane} with a significant contribution of ISR events
the sigma method is employed by definition.

A first preliminary measurement of the inclusive DIS cross-section for
the shifted vertex data concentrating on non-radiative events was
already presented on conferences in the year
2002~\cite{Lastovicka:2002}. An independent cross-check was considered
to be vital for such a precision analysis. The preliminary results
have been derived based on the BST. It was decided that the
cross-check is to be performed using the BST on the one hand and the
combination of BDC and Central Tracker on the other. The latter should
provide a higher level of independence w.r.t. the preliminary
results. Accordingly, two analyses have been started, referred to as
\emph{BST analysis} and \emph{BDC analysis}. Their results are the
topic of two theses,
\cite{Vargas:2006} and the one at hand. Apart from providing
independent measurements the main focus of the analyses is the
identification and accurate quantification of the systematic errors
affecting the cross-section determination.

Also the first preliminary measurement of the inclusive DIS
cross-section based on ISR events in the shifted vertex data
sample~\cite{Petrukhin:2004a} was presented on conferences in the
course of the work on the two new analyses. Hence the results are
cross-checked as well.

The two analyses have been performed in close collaboration, the work
has been split in the following way: the focus of~\cite{Vargas:2006}
is the BST analysis. The calibrations of the SpaCal and LAr
calorimeters developed therein are applied for the present thesis
which concentrates on the BDC analysis. The BDC analysis also
incorporates ISR events, but covers only a small part of the kinematic
phase space in principle being accessible by ISR events due to the cut
on $y_{\Sigma}$ (see figure~\ref{f:pscoverage}). Therefore details on
this topic beyond the explanations within this chapter can only be
found in~\cite{Vargas:2006}. All results presented in the following
refer to the BDC analysis if not explicitely stated differently. The
selection criteria for the BST analysis are also shortly explained as
a comparison between the two analyses will be presented in the last
chapter of this thesis.

The final outcome of the two new analysis is the proton structure
function $F_{2}$. A first preliminary extraction of the longitudinal
structure function $F_{L}$ already
exists~\cite{Lobodzinska:2003}. More details about the underlying
analyses of the preliminary results can be found
in~\cite{Lastovicka:2004a}.

The data obtained from the shifted vertex run are complemented at
larger values of $Q^{2}$ by the analysis of another data taking period
dedicated to the low $Q^{2}$ regime in 1999, referred to as
\emph{minimum bias run}~\cite{Eckstein:2001, Lastovicka:2004a}. 
The running conditions of the HERA accelerator have been standard, but
the same special trigger setup for the online event selection has been
used as for the shifted vertex run (see section~\ref{c:onlinesel}). A
combination of the results of the shifted vertex and minimum bias run
is anticipated (see section~\ref{c:f2extract}).


