\graphicspath{{figures/selection/}}

\chapter{Data Selection and Treatment} \label{c:selection}

As discussed in the previous chapter, the determination of the deep
inelastic scattering cross-section relies on a comparison of the
number of events for data and a signal Monte Carlo sample in each
analysis bin. For data the background contribution has to be
subtracted beforehand. It is estimated based on a dedicated Monte
Carlo sample. To keep the fraction of background events small an
adequate set of selection criteria is applied.

Employing the described method is only justified if the efficiencies
for the event selection and reconstruction as well as the
uncertainties on the reconstruction are well described by the Monte
Carlo simulation. Investigating and ensuring this is the topic of the
present chapter.

At first problematic periods of the data taking as well as
inefficiencies of the online event selection are identified and
excluded from the analysis. To allow for a precise event
reconstruction the relevant detector components are aligned and
calibrated subsequently. Afterwards the selection criteria used for
the analyses are introduced and it is checked whether their efficiency
agrees between data and Monte Carlo. The remaining part of the chapter
is devoted to corrections for the Monte Carlo samples in order to
ensure a good description of the data. For each step the uncertainty
relevant for the cross-section determination is discussed.

\section{Run Selection} \label{c:runsel}

The operation cycle of the HERA accelerator divides the data
acquisition of the H1 experiment into so-called \emph{luminosity
fills}. A luminosity fill starts as soon as collisions between
electrons and protons have been established and ends if one of the
beams is dumped or lost. To identify periods with nearly constant
experimental conditions the stream of recorded events is further split
into \emph{luminosity runs}. A typical luminosity run lasts for one
hour and contains 100000 events. For each run characteristic
parameters like beam offsets and tilts (see
section~\ref{c:simulation}) are recorded in the H1 database. The
integrated luminosity is stored in the same way.

Before carrying out a detailed selection of DIS events for the final
cross-section measurement a preselection of the data is performed on a
run-by-run basis to ensure stable detector and background
conditions. It is based on the following quality criteria:

\begin{enumerate}
\item{\textbf{Trigger Phase.} Two trigger phases can be distinguished
in the data taking period relevant for this analysis: at the beginning
of the luminosity run the beam currents are high and the beam
parameters are still optimised by the HERA shift crew. To account for
the resulting large beam induced background the H1 trigger system is
operated in phase 1 which is characterised by large prescale factors
(see section~\ref{c:trigger}) especially for the calorimeter
triggers. At this time the tracking detectors have still not reached
their nominal high voltage settings. Once these are achieved trigger
phase 2 is selected and a new run is started. The autoprescale
procedure (see section~\ref{c:trigger}) is used regularly to calculate
new prescale factors to make optimal use of the available bandwidth of
the data acquisition system. Only runs in trigger phase 2 are
considered in this analysis.}
\item{\textbf{Run Quality.} After the complete offline reconstruction
the luminosity runs are classified as \emph{good}, \emph{medium} or
\emph{poor} based on data quality checks. In runs to which the quality
flag poor has been assigned an important detector component (e.g. CJC,
luminosity system or one of the calorimeters) is not operating. Such
runs are excluded from the analysis.}
\item{\textbf{High Voltage and Readout Status.} The high voltage and
readout status of the different H1 detector components is continuously
monitored during data taking. If the high voltage of the detector
components is below its nominal value for a significant amount of time
this could indicate detector instabilities, usually due to a large
rate of background events. Therefore runs are not taken into account
for which the fraction of time with all relevant detector components
being at nominal high voltage is less than 70\%. The same is done for
runs in which one of the important readout branches has not been
operating. The high voltage and readout status is checked for the
subdetectors SpaCal, BDC, BST, LAr, CJC, CIP, COP, CIZ and the
luminosity system. The readout status is verified in addition for the
Central Trigger and the calorimeter trigger system. Within accepted
runs events are rejected in which one of the relevant detector
components is not supplied with the nominal high voltage. The
integrated run luminosity is corrected accordingly.}
\item{\textbf{Trigger Prescale.} To keep the overall trigger rate at a
reasonable level for the data acquisition prescale factors can be
applied to individual subtriggers during data taking (see
section~\ref{c:trigger}). In the course of the analysis the prescale
factors of the relevant subtriggers are used as weights $w_{i}$ for
the events of the data sample in order to obtain the original number
of events (for details see section~\ref{c:onlinesel}). The weights
enter the statistical error of the measurement quadratically:
\mbox{$\sigma_{stat} = \sqrt{\sum_{i} w_{i}^{2}}$}. To reduce the
statistical error of the measurement runs in which one of the
considered subtriggers has a prescale factor larger than 3 are
excluded from the analysis.}
\item{\textbf{Stability.} The stability studies presented in the next
section have shown that the event rate of the luminosity run 278804 is
significantly lower than of all the other runs. The shift crew comment
to the consequent run indicates a loss of the proton beam. It is
likely that the beam conditions have been already unstable in run
278804. Therefore it is not considered for the analysis.}
\end{enumerate}


\section{Stability Studies} \label{c:stability}

The stability of the measurement as a function of time is an important
criteria for the quality of the data used for the cross-section
measurement. Therefore the number of events per $nb^{-1}$ of
integrated luminosity (\emph{event yield}) is investigated per
luminosity run and fill. The former allows to identify problems
occurring on short time scales while the latter provides a
substantially larger statistical significance.

For the stability study the run selection discussed in the last
section is already applied. The events under consideration have passed
all selection criteria of the final cross-section analysis (see
section~\ref{c:dissel}). In addition a special selection is performed
to equalise the detector acceptance for different runs: the mean and
spread of the \mbox{$z$-coordinate} of the interaction vertex changes
from run to run (see figure~\ref{f:vtxstab}). As the detector
acceptance depends on the vertex position this would result in a
varying event yield. Therefore the polar angle of the electron
candidate is restricted to
\mbox{$165.5^{\circ} < \theta_{e} < 175.5^{\circ}$} ensuring a backward 
detector acceptance of 100\% independent of the vertex position (see
section~\ref{c:vertexreco}).

Figure~\ref{f:yieldwrong} shows the obtained event yield as a function
of a continuous run number\footnote{The original sequence of the runs
is conserved.} and the luminosity fill number. For the last luminosity
fill a significant deviation from the mean event yield is
observed. This deviation inspired a complete redetermination of the
luminosity for the shifted vertex run by the luminosity working
group. It revealed that the fraction of the luminosity attributed to
the proton satellite bunches has been wrongly determined beforehand as
is explained in detail in the remaining part of this section.

\begin{figure}
  \begin{minipage}[c]{0.5789\textwidth}
    \includegraphics[width=\textwidth]{controlplots_bdc_060509_0018_yieldrun_wronglumi.eps}
    \put(-193,161){a)}
  \end{minipage}
  \begin{minipage}[c]{0.4211\textwidth}
    \includegraphics[width=\textwidth]{controlplots_bdc_060509_0018_yieldfill_wronglumi.eps}
    \put(-123,161){b)}
  \end{minipage}
  \caption{Event yield per a) continuous run number and b) luminosity fill. The solid lines indicate the average event yield determined by fitting a constant to the data points. The dashed lines in a) separate the different luminosity fills. Fill 2597 has lasted for only one hour, none of its runs has passed the run selection of section~\ref{c:runsel}. The luminosity entering the calculation of the event yield depicted here is based on the wrong estimation of the satellite correction, the corrected event yield is shown in figure~\ref{f:yield} (for details see text).}
  \label{f:yieldwrong}
\end{figure}

From figure~\ref{f:vtxstab} it can be seen that the spread of the
\mbox{$z$-position} of the vertex for the last luminosity fill is
significantly larger than for the others. Figure~\ref{f:satellites}
compares the \mbox{$z$-vertex} distribution of the first and the last
luminosity fill of the shifted vertex run. As expected both
distributions have their main maximum at the nominal \mbox{$z$-vertex}
position of approximately \mbox{$70\, \mathrm{cm}$}. A second maximum
is seen at about \mbox{$140\, \mathrm{cm}$}. The last luminosity fill
has a broader main maximum and more pronounced tails including a larger
second maximum.

\begin{figure}
%\centering
\begin{captionbeside}{\mbox{$z$-vertex} distributions of the first and last luminosity fill of the shifted vertex run normalised w.r.t. each other based on the number of events. The vertical dashed lines indicate the \mbox{$z$-vertex} interval entering the final cross-section analysis. The vertical dashed lines indicate an interval of \mbox{$30\, \mathrm{cm}$} around the nominal vertex position.}[i][0.95\textwidth][0.05\textwidth]*
\begin{minipage}[t]{0.5\textwidth}
\includegraphics[width=\textwidth]{vertexdist_060503_0144_satellites.eps}
\end{minipage}
\end{captionbeside}
\label{f:satellites}
\end{figure}

The additional maximum is caused by the complicated longitudinal
structure of the HERA proton beam: \mbox{$52\, \mathrm{MHz}$} and
\mbox{$208\, \mathrm{MHz}$} radiofrequency systems are installed in
the proton ring for the acceleration. The protons are transferred from
the PETRA pre-accelerator into the \mbox{$52\, \mathrm{MHz}$} buckets
and gradually compressed into the \mbox{$208\, \mathrm{MHz}$} buckets
during the acceleration to decrease the longitudinal size of the
bunches to \mbox{$\sigma_{b} \approx 20\,
\mathrm{cm}$}~\cite{Montag:2000}. This corresponds to a length of the 
luminous region of \mbox{$\sigma_{l} \approx 10\,
\mathrm{cm}$} since the longitudinal size of the electron bunches is a 
factor of 10 smaller and therefore negligible. As the potential wells
created by the radiofrequency systems are not sufficiently deep a
small fraction of the proton current escapes from the main bunches and
forms so-called \emph{satellite bunches}. The satellite bunches are
separated by
\mbox{$\Delta t = \pm \tau_{sat} = \pm 19.2\, \mathrm{ns}$} and
\mbox{$\Delta t = \pm \tau_{sat} = \pm 4.8\, \mathrm{ns}$} from the
main bunch for the \mbox{$52\, \mathrm{MHz}$} and \mbox{$208\,
\mathrm{MHz}$} system, respectively. Most relevant for the analysis
are the satellites closest to the main bunch in the \mbox{$208\,
\mathrm{MHz}$} buckets. They cause additional maxima of the
\mbox{$z$-vertex} distribution separated by 
\mbox{$\Delta z = \mp c \tau_{sat} / 2 = \pm 72\,\mathrm{cm}$} from the 
main one. The maximum separated by \mbox{$\Delta z =
+72\,\mathrm{cm}$} corresponding to the \emph{forward} (or
\emph{early}) satellite is most pronounced since it provides access 
to the lowest values of $Q^{2}$ and thus the largest event rates. It
is well visible in figure~\ref{f:satellites}.

As indicated by the vertical dashed lines only events having a
\mbox{$z$-vertex} position within \mbox{$30\, \mathrm{cm}$} of the
nominal one are accepted for the cross-section analysis (see
section~\ref{c:dissel}). The luminosity system (see
section~\ref{c:lumisystem}) however cannot distinguish Bethe-Heitler
events originating from the main and satellite bunches and thus
measures the luminosity in the entire $z_{vtx}$ range. Therefore a
procedure has been developed to subtract the satellite fraction of the
luminosity based on the information of other
subdetectors~\cite{Levonian:1995, Gogitidze:1996}.

As has been already mentioned the problematic last luminosity fill has
a more clearly developed contribution of the forward satellite to the
\mbox{$z$-vertex} distribution compared to the first fill. Therefore
it has been checked by the luminosity working group whether the
satellite correction has been determined correctly. To derive this
correction data from the proton pickup unit (PPU) located at \mbox{$z
\approx 3\, \mathrm{m}$} are used~\cite{Karstensen:1996,
Wissing:1998}. It measures the voltage induced by the electrically
charged protons in a conductor coil allowing to extract the
longitudinal beam profile. The correction is determined per luminosity
fill.

In order to reach a high luminosity HERA focuses the beams in
transverse direction at the interaction regions of H1 and ZEUS. This
leads to a variation of the transverse beam size with $z$ that has to
be considered when calculating the luminosity corresponding to the
main and satellite bunches from the longitudinal beam profile. The
common measure of the transverse beam size is the
\mbox{$\beta$-function} which is available from the HERA
database. While revising the satellite correction it has been realised
that the wrong \mbox{$\beta$-function} has been applied for the
existing correction~\cite{Levonian:2006}: it has been assumed that the
\mbox{$\beta$-function} for the shifted vertex run is different from
the one for standard running conditions to enhance the interaction
rate at \mbox{$70\, \mathrm{cm}$}. Actually, the
\mbox{$\beta$-function} is the same and the shift of the interaction
vertex has been reached by changing the timing of the beams. In this
way the satellite fraction of the luminosity has been underestimated
by approximately a factor of two.



The new determination of the satellite corrections leads to a decrease
of the total integrated luminosity by 4\%, the measured cross-section
thus increases by the same amount. For the last fill the change is
most significant since it has the largest satellite
bunches. Figure~\ref{f:yield} shows the corrected event yield. The
agreement of the yield of the last fill with the average yield has
improved. Of course the run-wise deviations within this fill are still
present as the satellite correction can only be determined as one global
factor per luminosity fill.

\begin{figure}
  \begin{minipage}[c]{0.5789\textwidth}
    \includegraphics[width=\textwidth]{controlplots_bdc_060509_0018_yieldrun.eps}
    \put(-193,161){a)}
  \end{minipage}
  \begin{minipage}[c]{0.4211\textwidth}
    \includegraphics[width=\textwidth]{controlplots_bdc_060509_0018_yieldfill.eps}
    \put(-123,161){b)}
  \end{minipage}
  \caption{Corrected event yield per a) continuous run number and b) luminosity fill. The solid lines indicate the average event yield determined by fitting a constant to the data points. The dashed lines in a) separate the different luminosity fills.}
  \label{f:yield}
\end{figure}

The uncertainty on the luminosity determination has been estimated to
be $\lumierr$ by the luminosity working group~\cite{Levonian:2006}.


\section{Online Event Selection} \label{c:onlinesel}

In the region of the kinematic phase space covered by this analysis
the scattered electron is detected in the backward region of the H1
detector. Therefore the online event selection is based on energy
depositions releasing the Inclusive Electron Trigger (IET) of the
SpaCal calorimeter (see section~\ref{c:trigger}).

To benefit from the shifted vertex run as much as possible for the
cross-section measurement at low $Q^{2}$ a special \emph{minimum bias}
trigger setup has been used: in comparison to the standard trigger
conditions most of the subtriggers dedicated to high $Q^{2}$ and
exclusive final state analyses have been disabled. Hence a larger
bandwidth of the data acquisition system is available for the
remaining subtriggers allowing them to run with lower prescale factors
and thus reducing the statistical error of the measurement as much as
possible. For the standard trigger setup the central LIET region of
the Inclusive Electron Trigger is usually not taken into account since
it causes a large subtrigger rate.  However, in the case of the
minimum bias setup one can allow to include it for some of the
subtriggers giving access to larger polar angles $\theta$ and thus
lower values of $Q^{2}$ than for the standard subtrigger
definition. The resulting increased trigger rate is partly compensated
by larger energy thresholds.

As described in section~\ref{c:trigger} the online event selection of
the H1 experiment comprises the trigger levels L1, L2 and L4. The
definition of the L1 subtriggers relevant for the analysis and their
validation on L2 is covered in detail in the following with a
subsequent determination of their efficiency. After a discussion of
the event selection on L4 the section closes with an explanation how
to retrieve the original number of events in the data sample by using
event weights to account for the prescale factors applied during data
taking.

\subsection*{Subtrigger Definition}

The L1 subtriggers used for the analysis and their validation on L2
are listed in table~\ref{t:stdef}. Apart from so-called global
options\footnote{Global options are additional veto conditions to
reject beam induced background.} the used subtriggers are solely built
from trigger elements of the Inclusive Electron Trigger (see
section~\ref{c:trigger}). The trigger elements
$\mathtt{SPCLe\_IET>m}\, (\mathtt{m}=\mathtt{0,1,2})$ and
$\mathtt{SPCLe\_IET\_Cen\_n}\, (\mathtt{n}=\mathtt{1,2,3})$ require
that the energy deposition belonging to one cluster bit exceeds a
certain threshold in the outer LIET regions and the central LIET
region, respectively (see table~\ref{t:ietthresh}).

\begin{table}
\centering
\begin{tabular}{c@{\hspace{2em}}c@{\hspace{2em}}c}
\toprule
subtrigger & definition & L2 validation \\
\midrule
S0 & $\mathtt{SPCLe\_IET>1\;\; ||\;\; SPCLe\_IET\_Cen\_2}$ & $\mathtt{SPCL\_R10}$ \\
S3 & $\mathtt{SPCLe\_IET>2\;\; ||\;\; SPCLe\_IET\_Cen\_3}$ & - \\
S9 & $\mathtt{SPCLe\_IET>0}$ & - \\
\bottomrule
\end{tabular}
\caption{L1 subtriggers relevant for this analysis and their validation on L2. Additional veto conditions against beam background are omitted.}
\label{t:stdef}
\end{table}

\begin{table}
\centering
\begin{tabular}{c@{\hspace{2em}}c@{\hspace{2em}}c}
\toprule
\multicolumn{2}{c}{trigger element} & \\
central LIET region & outer LIET regions & \raisebox{1.5ex}[-1.5ex]{threshold} \\
\midrule
$\mathtt{SPCLe\_IET\_Cen\_1}$ & $\mathtt{SPCLe\_IET>0}$ & $2\, \mathrm{GeV}$ \\
$\mathtt{SPCLe\_IET\_Cen\_2}$ & $\mathtt{SPCLe\_IET>1}$ & $5\, \mathrm{GeV}$ \\
$\mathtt{SPCLe\_IET\_Cen\_3}$ & $\mathtt{SPCLe\_IET>2}$ & $10\, \mathrm{GeV}$ \\\bottomrule
\end{tabular}
\caption{Energy thresholds of the Inclusive Electron Trigger. For the standard trigger setup the corresponding thresholds are $0.5\, \mathrm{GeV}$, $2\, \mathrm{GeV}$ and $6\, \mathrm{GeV}$, respectively.}
\label{t:ietthresh}
\end{table}

To keep the trigger rate at a reasonable level the L2TT trigger element
$\mathtt{SPCL\_R10}$ validating subtrigger S0 excludes the innermost
region of the SpaCal~\cite{Bizot:1998}. To be more precise, the 9
cluster bits of the Inclusive Electron Trigger surrounding the beam
pipe do not contribute to the trigger decision leaving a square of
\mbox{$8.1 \times 8.1\,\mathrm{cm^{2}}$} uncovered. This exactly
corresponds to the insert module of the SpaCal.

The subtrigger definition reveals the following trigger strategy for
the shifted vertex analysis: the bulk of the events are selected by
subtrigger S0 as it has a lower energy threshold than S3 and its L2
condition excludes only the innermost region of the SpaCal. The latter
is covered by subtrigger S3 enhancing the statistics for the lowest
$Q^{2}$ values\footnote{As the innermost SpaCal region excluded by
subtrigger S0 is anyhow not covered by the BST acceptance the BST
analysis utilises S0 only.}.

The subtrigger S9 has been defined to access the high $y$ region of
the kinematic phase space with the objective to measure
$F_{L}$. However, it turned out that one of its global options based
on the signals of the proportional chambers CIP and COP and the
Forward Proportional Chambers has a low efficiency, because it has not
been optimised for the shifted interaction vertex. Therefore
subtrigger S9 is not employed for the cross-section measurement, but
it proves to be useful for the determination of the trigger
efficiency.


\subsection*{Trigger Efficiency}

Inefficiencies of the Inclusive Electron Trigger are caused by
individual SpaCal cells. The majority of the cells has an efficiency
close to 100\%, whereas only a few cells show problems. It is
difficult to model the behaviour of the problematic cells in the Monte
Carlo simulation. Therefore the analysis is restricted to regions of
the phase space where the subtrigger efficiency is close 100\%. To be
more precise, the aim is to determine the fiducial volume and the
energy range in which the subtriggers are fully efficient. This
incorporates the identification of problematic SpaCal cells such that
they can be excluded from the subsequent analysis.

The efficency of a specific subtrigger is typically determined by
considering each trigger element separately. The standard procedure to
do so is to use an event sample selected by independent
\emph{reference subtriggers}, i.e. subtriggers which do not contain
any trigger element of the trigger subsystem under
investigation~\cite{Urban:2005}. For the IET trigger elements these
are typically subtriggers based on the trigger subsystems of tracking
detectors or the LAr calorimeter. Due to the reduced number of active
subtriggers in the minimum bias setup however the number of events
selected by the reference subtriggers is not sufficient to determine
the trigger efficiency separately for each SpaCal cell. Therefore the
described method is used as a final cross-check whereas the detailed
studies are performed based on a different approach as outlined below.

Different parts of the Inclusive Electron Trigger electronics cover
the central LIET region and the outer LIET regions. Hence the groups
of trigger elements $\mathtt{SPCLe\_IET>m}$ and
$\mathtt{SPCLe\_IET\_Cen\_n}$ are independent of each other. Therefore
they can be used for a mutual efficiency determination. This proceeds
in the following way: one starts from an ensemble of events which have
been selected by subtriggers containing trigger elements independent
of the trigger element under investigation. From this sample events
are selected which have exactly one cluster in the SpaCal region
covered by the trigger element to be investigated. These events
constitute the so-called \emph{reference sample}. The fraction of
events which are accepted by the trigger element of interest is the
trigger efficiency. The requirement of exactly one cluster in the
region attributed to the trigger element to be investigated is
necessary in order not to overestimate the efficiency due to the
presence of events in which a different SpaCal cluster has fired the
trigger element than has been selected in the reference sample.

To reject background events several selection criteria have been
applied: the \mbox{$z$-po}\-si\-tion of the vertex as determined by the
Central Tracker is to be reconstructed within an interval of $30\,
\mathrm{cm}$ around the nominal position 
$\overline{z_{vtx}} = \zvtxmean$:
\mbox{$|z_{vtx} - \overline{z_{vtx}}| < 30\, \mathrm{cm}$}. The SpaCal
cluster has to match the requirements \mbox{$r_{Cl} < 4\,
\mathrm{cm}$} and \mbox{$f_{had} < 0.15$, $r_{Cl}$} being the
logarithmic cluster radius and $f_{had}$ the fraction of the cluster
energy deposited in the hadronic section of the SpaCal. This ensures
that the cluster is most likely induced by the scattered
electron\footnote{The selection criteria are explained in detail in
section~\ref{c:dissel}.}.

At first the efficiency of the trigger elements
$\mathtt{SPCLe\_IET\_Cen\_2}$ and $\mathtt{SPCLe\_IET\_Cen\_3}$ is
studied by using a sample of events that have been selected by the
subtrigger S9 which covers only the outer LIET regions. Subsequently
the trigger element $\mathtt{SPCLe\_IET>0}$ is checked based on a
reference sample selected by the subtriggers S0 and S3 explicitely
requiring the trigger elements $\mathtt{SPCLe\_IET\_Cen\_2}$ and
$\mathtt{SPCLe\_IET\_Cen\_3}$ to have fired, respectively. To avoid a
possible bias it is ensured that the SpaCal cluster with the highest
energy fulfils the requirements of the reference sample while a
different cluster is used to check the efficiency of the trigger
elements to be investigated. Since the second cluster is most likely
caused by a hadron no cuts on $r_{Cl}$ and $f_{had}$ are placed. Its
energy is so small that it is not sufficient to study the trigger
elements $\mathtt{SPCLe\_IET>1}$ and $\mathtt{SPCLe\_IET>2}$. The
study of $\mathtt{SPCLe\_IET>0}$ solely serves to identify cells with
general problems while the trigger element itself is not relevant for
the subtriggers S0 and S3\footnote{Therefore it should be fine that no
cuts concerning the electron identification are applied.}. Having
identified such cells it is sufficient to restrict the investigation
of the trigger elements $\mathtt{SPCLe\_IET>1}$ and
$\mathtt{SPCLe\_IET>2}$ to their threshold behaviour. The latter is
analysed by again employing the subtrigger S9 to select the reference
sample as its trigger element $\mathtt{SPCLe\_IET>0}$ has a lower
threshold than the ones to be investigated.

The SpaCal cells with the numbers 7, 8, 13, 14, 32 and 33 (see
appendix~\ref{c:spacalcellno}) have been excluded from the Inclusive
Electron Trigger already during data taking. They are located close to
the beam pipe and exhibit an abnormally high trigger rate presumably
induced by synchrotron radiation. In addition the cells 30, 62, 250,
253, 306 and 763 have been identified to show a low trigger
efficiency. All listed cells are excluded from the analysis.

The resulting efficiencies of the L1 trigger elements contributing to
the subtriggers S0 and S3 are depicted in figure~\ref{f:effiet} as a
function of the energy $E_{e}$ of the SpaCal cluster. A Fermi function
is fitted to the data points given by
\begin{equation}
\epsilon(E_{e}) = \frac{\epsilon_{max}}{e^{\frac{E_{t}-E_{e}}{\Delta E}} + 1}
\label{e:fermi}
\end{equation}
Here $\epsilon_{max}$ denotes the efficiency at the plateau, $E_{t}$
the threshold energy and $\Delta E$ the width of the threshold. The
threshold of each trigger element is at its designated position as
specified in table~\ref{t:ietthresh}. The plateau of the efficiency is
reached a few $\mathrm{GeV}$ above the threshold as indicated by the
dashed lines. The cuts for the final analysis are placed at the
corresponding energies.

\begin{figure}
\begin{minipage}[c]{0.5\textwidth}
  \includegraphics[width=\textwidth]{efficiency_ietcen_060119_2301_effcen2.eps}
  \put(-157,161){a)}
\end{minipage}
\begin{minipage}[c]{0.5\textwidth}
  \includegraphics[width=\textwidth]{efficiency_ietcen_060119_2301_effcen3.eps}
  \put(-157,161){b)}
\end{minipage}
\begin{minipage}[c]{0.5\textwidth}
  \includegraphics[width=\textwidth]{threshold_iet_060118_1755_effiet1.eps}
  \put(-157,161){c)}
\end{minipage}
\begin{minipage}[c]{0.5\textwidth}
  \includegraphics[width=\textwidth]{threshold_iet_060118_1755_effiet2.eps}
  \put(-157,161){d)}
\end{minipage}
\caption{Efficiency of the L1 trigger elements of the subtriggers S0 and S3 as a function of the energy of the SpaCal cluster. A Fermi function according to equation~\ref{e:fermi} is fitted to the data (solid line). The dashed line displays the energy cut to be applied for the corresponding trigger element in the final analysis. A deviation of 0.5\% from an efficiency of 100\% is indicated by the horizontal band.}
\label{f:effiet}
\end{figure}

Apart from the efficiency of the IET trigger elements the phase space
covered by the subtrigger S0 is determined by the L2 condition
$\mathtt{SPCL\_R10}$. Its efficiency is analysed using a sample of
events selected by the subtrigger S9 asking for the trigger element
$\mathtt{SPCLe\_IET\_Cen\_2}$ to have fired in addition. The fraction
of these events being accepted by the L2 condition corresponds to the
efficiency. It is depicted in figure~\ref{f:effspr10} as a function of
the quantity \mbox{$R_{box} = \mathrm{max}(|x_{Spa}|,|y_{Spa}|)$}
where $x_{Spa}$ and $y_{Spa}$ are the $x$- and \mbox{$y$-position} of
the SpaCal cluster, respectively. A cut $R_{box}>10\, \mathrm{cm}$
ensures an efficiency of almost 100\% for the L2 condition of S0.

\begin{figure}
\centering
\begin{minipage}[t]{0.5\textwidth}
\includegraphics[width=\textwidth]{efficiency_ietcen_060119_2301_effspr10.eps}
\end{minipage}
\caption{Efficiency of the L2 condition $\mathtt{SPCL\_R10}$ of the subtrigger S0 as a function of \mbox{$R_{box} = \mathrm{max}(|x_{Spa}|,|y_{Spa}|)$}. The cut to be applied for the final analysis is represented by the dashed line. A deviation of 0.5\% from an efficiency of 100\% is indicated by the horizontal band.}
\label{f:effspr10}
\end{figure}

For a final cross-check the efficiencies of the subtriggers S0 and S3
are determined based on independent reference subtriggers as mentioned
at the beginning of this section: the starting point is a sample of
events accepted by subtriggers which do not contain any IET trigger
elements. From this sample events are selected which have a SpaCal
cluster. The fraction of these events for which the raw bit of the
subtrigger to be investigated is set corresponds to the trigger
efficiency.

To reject background events the requirements introduced above for
$z_{vtx}$, $r_{Cl}$ and $f_{had}$ have to be matched. In addition the
time of flight information of the SpaCal has been employed to reject
events which have a SpaCal time $t_{Spa}$ not compatible with an
$ep$ interaction. A cut $t_{Spa} > -20\, \mathrm{ns}$ is
applied. To ensure that this requirement really rejects background
events and not signal events which are hampered by individual SpaCal
cells providing bad timing information the following cross-checks have
been performed: it has been checked that the events rejected by the
timing cut are uniformly distributed over the SpaCal surface within
the limited statistics. In addition it has been verified that the time
measurement of the Central Jet Chambers indicates the presence of a
background interaction for these events\footnote{For the efficiency studies
presented above no cut has been placed on the timing. This is not
necessary, because the subtriggers used to select the reference sample
contain IET trigger elements which can only be released during a time
interval compatible with an $ep$ interaction, i.e. a timing cut
is implicitly present.}.

Figure~\ref{f:effst} shows the resulting trigger efficiencies as a
function of the cluster energy. They show the same threshold behaviour
as the corresponding trigger elements. For the subtrigger S0 only
events satisfying $R_{box}>10\, \mathrm{cm}$ have been used in order
not to mix inefficiencies because of the threshold behaviour and the
L2 condition $\mathtt{SPCL\_R10}$. Especially at large energies one
can see that the statistics for this method is rather limited.

\begin{figure}
\begin{minipage}[c]{0.5\textwidth}
  \includegraphics[width=\textwidth]{efficiency_st_060120_0055_effs0.eps}
  \put(-157,161){a)}
\end{minipage}
\begin{minipage}[c]{0.5\textwidth}
  \includegraphics[width=\textwidth]{efficiency_st_060120_0055_effs3.eps}
  \put(-157,161){b)}
\end{minipage}
\caption{Efficiency of the subtriggers a) S0 and b) S3 as a function of the energy of the SpaCal cluster. A Fermi function according to equation~\ref{e:fermi} is fitted to the data (solid line). The dashed line displays the energy cut to be applied for the corresponding subtrigger in the final analysis. A deviation of 0.5\% from an efficiency of 100\% is indicated by the horizontal band.}
\label{f:effst}
\end{figure}

From the trigger efficiency study different selection criteria arise
for the BST and BDC analyses: since the BST analysis relies solely on
subtrigger S0 only electron candidates fulfilling the requirements
$E_{e}>7\,\mathrm{GeV}$ and $R_{box}>10\, \mathrm{cm}$ are used. The
BDC analysis accepts events selected by S3 in addition and therefore
no cut on $R_{box}$ is necessary for energies above $14\,
\mathrm{GeV}$. For the selection criteria summarised in
table~\ref{t:trcuts} the subtrigger efficiency is 100\% within an
uncertainty of 0.5\% (indicated by the horizontal band in
figures~\ref{f:effiet}~-~\ref{f:effst}).

\begin{table}
\centering
\begin{tabular}{c@{\hspace{2em}}c@{\hspace{2em}}c}
\toprule
analysis & energy range & fiducial volume \\
\midrule
BST & $E_{e}>7\,\mathrm{GeV}$ & $R_{box}>10\, \mathrm{cm}$ \\
\cmidrule{1-3}
    & $E_{e}>14\,\mathrm{GeV}$ & entire SpaCal \\
\raisebox{1.5ex}[-1.5ex]{BDC} & $7 < E_{e} < 14\,\mathrm{GeV}$ & $R_{box}>10\, \mathrm{cm}$ \\
\bottomrule
\end{tabular}
\caption{Selection criteria for the BST and BDC analyses resulting from the trigger efficiency study.}
\label{t:trcuts}
\end{table}

\subsection*{Event Selection on L4}

For the selection of the low $Q^{2}$ events on the fourth trigger
level the L4 finder 18 is responsible. It imposes different selection
criteria on the subtriggers S0/S3 and S9 as specified in
table~\ref{t:l4class18}.

\begin{table}
\centering
\begin{tabular}{c@{\hspace{2em}}c@{\hspace{2em}}c}
\toprule
ST & L4 condition & explanation \\
\midrule
 & $\mathtt{R\_SPAC\_ECRA} < 4\, \mathrm{cm}$ & $r_{Cl}^{lin}$ of cluster with max. $E_{Cl}$ \\
\addlinespace
S0 & & max. $Q_{Cl}^{2}$ of clusters with \\
S3 & \raisebox{1.5ex}[-1.5ex]{$\mathtt{R\_SPAC\_Q2MAX} > 0.2\, \mathrm{GeV^{2}}$} & $E_{Cl} > 4.0\, \mathrm{GeV}$ and $r_{Cl}^{lin} < 4\, \mathrm{cm}$ \\
\addlinespace
 & $\mathtt{R\_SPAC\_EMAXCLUSTER} > 4\, \mathrm{GeV}$ & max. $E_{Cl}$ \\
\cmidrule{1-3}
 & $\mathtt{R\_SPAC\_ECRAMIN} < 5\, \mathrm{cm}$ & min. $r_{Cl}^{lin}$ of clusters with $E_{Cl} > 3.5\, \mathrm{GeV}$ \\
\addlinespace
S9 & $\mathtt{R\_SPAC\_RCLUSTERMAX} > 15\, \mathrm{cm}$ & max. $R_{Cl}$ of clusters with $E_{Cl} > 2\, \mathrm{GeV}$ \\
\addlinespace
 & $\mathtt{R\_SPAC\_EMAXCLUSTER} > 2\, \mathrm{GeV}$ & max. $E_{Cl}$ \\
\bottomrule
\end{tabular}
\caption{Selection criteria on L4. $E_{Cl}$ denotes the cluster energy, $r_{Cl}^{lin}$ the linear cluster radius, $R_{Cl}$ the distance of the cluster to the origin of the SpaCal in the \mbox{$r\phi$-plane} and $Q_{Cl}^{2}$ the photon virtuality as calculated from the cluster properties.}
\label{t:l4class18}
\end{table}

To determine the efficiency of the L4 filter farm events are searched
for which have not been selected by the L4 finder 18, but fulfil all
selection criteria of the final analyses. Events not selected by
finder 18 can pass the online event selection if they are selected by
one of the hard scales, a different finder or if they are among the
fraction of rejected events which are kept for cross-checks (see
section~\ref{c:trigger}). Indeed a few events of the last type have
been found for both analyses. A detailed investigation of the record
of the L4 decision for each of these events has revealed that they
have been classified as non-$ep$ background events, because their
vertex position as reconstructed on L4 is not compatible with an
$ep$ scattering event originating from the interaction region
around $70\, \mathrm{cm}$. They pass the final analysis selection as
the offline vertex reconstruction gives a different result than the
fast vertex reconstruction on the L4 filter farm. The fraction of
rejected events taking into account the downscale factor is listed in
table~\ref{t:l4reject} for the two analyses. The downscaled events are
excluded from the analysis and considered in the uncertainty on the
online event selection. Consequently the latter contains the fraction
of rejected events and the uncertainty on the subtrigger efficiency
added in quadrature. It amounts to 0.9\% for the BST and 0.7\% for the
BDC analysis, respectively.

\begin{table}
\centering
\begin{tabular}{c@{\hspace{2em}}c}
\toprule
analysis & fraction of L4 rejected events \\
\midrule
BST & $0.80\% \pm 0.18\%$ \\
BDC & $0.48\% \pm 0.18\%$ \\
\bottomrule
\end{tabular}
\caption{Fraction of events rejected by the L4 filter farm for the BST
and BDC analyses.}
\label{t:l4reject}
\end{table}

\subsection*{Event Weights}

As prescaling has been employed to account for the changing
interaction rate during a luminosity fill weights have to be applied
to the events of the data sample to obtain the original number of
events.

In the case of the BST analysis only events are accepted which have been
selected by subtrigger S0, i.e. for which the actual bit of subtrigger
S0 is set. Its run-dependent prescale factor is used as a weight for
each event.

The BDC analysis uses the subtrigger S3 in addition. Therefore a more
complicated strategy is necessary to avoid double counting: all events
for which the actual bit of the subtrigger with the lower prescale for
the given run is set are accepted. In addition events are kept if the
actual bit of the subtrigger with the higher prescale is set, but not
the raw bit of the subtrigger with the lower prescale. All remaining
events are rejected. The prescale factor of the subtrigger responsible
for the selection of an event is applied as a weight for this
event. In this way the phase space of the cross-section measurement is
divided into two non-overlapping regions with one subtrigger being
uniquely assigned to each of them. Priority is given to the subtrigger
with the lower prescale factor.


\section{Alignment} \label{c:alignment}

A precise determination of the polar angle of the scattered electron
is crucial for the reconstruction of the event kinematics as well as
the definition of the fiducial volume covered by the analysis. This
demands for an accurate knowledge of the position of the backward
detectors SpaCal, BST and BDC. Therefore the alignment of the SpaCal
and the BDC with respect to the H1 coordinate system is the topic of
the present section, the alignment of the BST is discussed in
~\cite{Lastovicka:2004a}. Offsets in all three dimensions with respect
to the nominal detector position are derived by comparing the angle
measurement of the backward detectors and the Central Tracker with
different methods. The resulting alignment constants for the SpaCal
are cross-checked with QED Compton events.

\subsection*{Alignment with respect to the Central Tracker}

The track reconstruction in the Central Tracker relies on the Central
Jet Chambers CJC1 and CJC2 and the two \mbox{$z$-chambers} CIZ and COZ
as mentioned in section~\ref{c:tracking}. They define the H1
coordinate system and are already aligned with respect to each other
in the data sample used for the analysis. Thus they are well suited
to serve as a reference for the alignment of the SpaCal and the BDC.

A dedicated event sample has been used for the alignment being defined
by the following selection criteria: only events are taken into
account which have one electron candidate and one associated track in
the Central Tracker\footnote{To assign electron candidates and Central
Tracker tracks the distance in the \mbox{$r\phi$-plane} between the
position of the SpaCal cluster and the track extrapolated to the
\mbox{$z$-coordinate} of the cluster is determined. If this
does not exceed \mbox{$10\, \mathrm{cm}$} electron candidate and track are
attributed to each other.}. In addition the same selection criteria
are applied as for the trigger efficiency study in the previous
section. For a further suppression of background events only electron
candidates are considered with an energy \mbox{$E_{e} > 20\,
\mathrm{GeV}$}. To ensure a precise angle measurement in the Central 
Tracker cuts on the estimated error of the \mbox{$z$-position} of the
vertex (\mbox{$\sigma_{z_{vtx}} < 1\, \mathrm{cm}$}) and of the polar
angle of the track ($\sigma_{\theta_{CT}} < 2\, \mathrm{mrad}$) are
placed. The former is more strict than the cut of \mbox{$8\,
\mathrm{cm}$} to be used for the final analysis (see
section~\ref{c:dissel}), the latter ensures that the track has been
linked to at least one of the \mbox{$z$-chambers}. For alignment
methods which use the polar angle of the Central Tracker at least
three of maximal eight linked hits in the \mbox{$z$-chambers} are
required.

The polar and azimuthal angle as reconstructed by the SpaCal and the
BDC are derived from the position of the cluster or track segment and
the Central Tracker vertex assuming that the trajectory of the
electron is a straight line. This is a good approximation for energies
above \mbox{$20\,
\mathrm{GeV}$}. Nevertheless, the azimuthal angle is corrected for the
influence of the magnetic field while its effect on the polar angle
can be neglected~\cite{Merkel:2005}.

As a preparatory study for the alignment the reconstructed polar
angles of Central Tracker, SpaCal and BDC are compared to the
generated angle of the scattered electron for the Monte Carlo
sample. This provides an important check as for the Monte Carlo the
detector components are perfectly aligned by definition. Figure
\ref{f:alithetacor} shows the difference of the reconstructed and
generated polar angle as a function of the reconstructed angle.

\begin{figure}
\centering
\includegraphics[width=0.79\textwidth]{alignspabdc_ct_060202_1427_thetacor.eps}
\caption{Difference of the reconstructed and generated polar angle for Central Tracker, SpaCal and BDC. In the interval \mbox{$2.887 < \theta < 3.005$} (indicated by the vertical dashed lines) a second order polynomial (solid lines) is fitted to the points for the Central Tracker and the SpaCal which is used as a correction function for the alignment (for details see text). The horizontal dashed line indicates the base line.}
\label{f:alithetacor}
\end{figure}

The polar angles as reconstructed by the SpaCal are systematically too
large. The region of large discrepancies at low polar angles
corresponds to the outer acceptance edge of the SpaCal. Here energy
leakage beyond the active detector volume leads to a systematic shift
of the cluster position towards the centre of the SpaCal. Furthermore
the SpaCal possesses no segmentation in the
\mbox{$z$-direction} such that the \mbox{$z$-coordinate} of the
cluster is not measured, but calculated from a parameterisation (see
section \ref{c:evreco}). In the case of the Central Tracker the polar
angles are smaller than the generated ones. The situation improves for
\mbox{$\theta_{CT} < 2.88\, \mathrm{rad}$}, because the electron
enters the acceptance of the COZ. For larger values of $\theta$ the CJC
track is only linked to hits of the CIZ. The BDC reconstruction shows
no significant deviation from the generated quantities.

During the alignment systematic deviations of the polar angle
reconstruction cannot be distinguished from a misalignment of the
detector components. Hence the polar angle range used for the
alignment is restricted to the interval \mbox{$2.887 < \theta < 3.005$}
for which the differences between reconstructed and generated values
are not too large. Furthermore correction functions are derived for
the Central Tracker and the SpaCal by fitting a second order
polynomial to the points in figure \ref{f:alithetacor} for the
specified interval. Both, the restriction of the polar angle range and
the correction function are only applied for alignment methods which
make use of the $\theta$ information.

The selected event sample together with the derived corrections is
used in the following to determine the alignment constants $\Delta x$,
$\Delta y$ and $\Delta z$ for the SpaCal and the BDC. The sign
convention is the following: the alignment constants specify the shift
of the detector component in the corresponding direction with respect
to its nominal position, i.e. they have to be added to the cluster or
track segment coordinates resulting from the reconstruction to
calculate the position in the H1 coordinate system. Three different
methods are employed for their determination:

\begin{itemize}

\item{One possibility to derive alignment constants in $x$- and
  \mbox{$y$-direction} is to measure \mbox{$R_{Spa/BDC} \Delta
  \phi_{Spa/BDC} = R_{Spa/BDC} (\phi_{Spa/BDC}-\phi_{CT})$} as a
  function of $\phi_{CT}$ where $R_{Spa/BDC}$ denotes the distance of
  the cluster or track segment from the origin of the SpaCal or BDC in
  the \mbox{$r\phi$-plane}~\cite{Merkel:2005}. A sinusoidal shape of
  this distribution indicates a shift of the corresponding detector in
  the \mbox{$r\phi$-plane}. To quantify this shift one calculates the
  mean of $R \Delta \phi$ in slices of $\phi_{CT}$ with the help of a
  robust mean estimator~\cite{Yuille:1990} and fits the following
  function to the resulting data points:
  \begin{equation}
    R \Delta \phi = P_{1}^{\Delta \phi} \sin(\phi_{CT} - P_{2}^{\Delta \phi}) + P_{3}^{\Delta \phi}
    \label{e:alirdeltaphi}
  \end{equation}
  The alignment constants are finally obtained from the
  parameters $P_{i}^{\Delta \phi}$ of the fit in the following way:
  \begin{align*}
    \Delta x & = P_{1}^{\Delta \phi} \cos P_{2}^{\Delta \phi} \\
    \Delta y & = P_{1}^{\Delta \phi} \sin P_{2}^{\Delta \phi}
  \end{align*}
  The described alignment method is only used for the SpaCal as
  the BDC electron finder BDCLEV uses the $R$ and $z$ measurement of
  the BDC while $\phi$ stems from the SpaCal, i.e. the values of
  $\phi$ for SpaCal and BDC are the same (for details see
  section~\ref{c:evreco}).}

  \begin{figure}
    \centering
      \begin{minipage}[t]{0.5\textwidth}
	\includegraphics[width=\textwidth]{alignspabdc_ct_060202_1545_rdeltaphispa.eps}
      \end{minipage}
    \caption{Distribution to derive alignment constants in $x$- and \mbox{$y$-direction} for the SpaCal using the Central Tracker as a reference. The accuracy of the Central Tracker information for this alignment method is mainly driven by the hits in the CJCs. To extract the alignment constants a sinus function according to equation~\ref{e:alirdeltaphi} is fitted to the data (solid line).}
    \label{f:alirdeltaphi}
  \end{figure}

\item{An alternative approach to determine the alignment constants in
  $x$- and \mbox{$y$-di}\-rec\-tion  is to plot \mbox{$\Delta R_{Spa/BDC} =
  R_{Spa/BDC} - R_{CT}$} as a function of $\phi_{CT}$. The distance in
  the \mbox{$r\phi$-plane} $R_{CT}$ of the Central Tracker track to
  the origin of the H1 coordinate system is calculated at the
  \mbox{$z$-position} of the SpaCal cluster or the BDC track
  segment. The resulting distribution is parameterised by the following
  function:
  \begin{equation}
    \Delta R = P_{1}^{\Delta R} \cos(\phi_{CT} - P_{2}^{\Delta R}) + P_{3}^{\Delta R}
    \label{e:alideltar}
  \end{equation}
  The alignment constants are calculated with the help of:
  \begin{align*}
    \Delta x & = - P_{1}^{\Delta R} \cos P_{2}^{\Delta R} \\
    \Delta y & = - P_{1}^{\Delta R} \sin P_{2}^{\Delta R}
  \end{align*}
  The two methods used to extract $\Delta x$ and $\Delta y$ 
  are complementary: the difference in the azimuthal angle $\Delta \phi$
  is mainly determined by the CJC as this has a better
  \mbox{$\phi$-resolution}, whereas $\Delta R$ is driven by the
  accurate \mbox{$z$-measurement} of CIZ and COZ. Comparing the
  results of both methods constitutes an important consistency check
  for the alignment.}

  \begin{figure}
    \begin{minipage}[c]{0.5\textwidth}
      \includegraphics[width=\textwidth]{alignspabdc_ct_060202_1545_deltarspa.eps}
      \put(-45,153){a)}
    \end{minipage}
    \begin{minipage}[c]{0.5\textwidth}
      \includegraphics[width=\textwidth]{alignspabdc_ct_060202_1545_deltarbdc.eps}
      \put(-45,153){b)}
    \end{minipage}
    \caption{Distributions to derive alignment constants in $x$- and \mbox{$y$-direction} for a) the SpaCal and b) the BDC using the Central Tracker as a reference. The accuracy of the Central Tracker information for this alignment method is mainly driven by the hits in the \mbox{$z$-chambers}. To extract the alignment constants a cosinus function according to equation~\ref{e:alideltar} is fitted to the data (solid lines).}
    \label{f:alideltar}
  \end{figure}

\item{To obtain the \mbox{$z$-position} of SpaCal and BDC a method
  proposed in~\cite{Merkel:2005} is used: \mbox{$F_{Spa/BDC} \Delta
  \theta_{Spa/BDC} = F_{Spa/BDC} (\theta_{Spa/BDC}-\theta_{CT})$} is
  measured as a function of $\phi_{CT}$, $F_{Spa/BDC}$ being defined
  as:
  \[ F = \frac{z^{2}}{R} \cdot \frac{1}{(\frac{R}{z})^{2} - (\frac{R}{z})^{4} + (\frac{R}{z})^{6} - 1} \]
  The subscript $Spa/BDC$ has been omitted for all variables,
  $z_{Spa/BDC}$ is the \mbox{$z$-coordinate} of the SpaCal cluster or
  BDC track segment. A fit is performed using the function:
  \begin{equation}
    F \Delta \theta = P_{1}^{\Delta \theta} \sin(\phi_{CT} - P_{2}^{\Delta \theta}) + P_{3}^{\Delta \theta}
    \label{e:alifdeltatheta}
  \end{equation}
  The negative of $P_{3}^{\Delta \theta}$ is the sought-after alignment
  constant in \mbox{$z$-direction}.}

  \begin{figure}
    \begin{minipage}[c]{0.5\textwidth}
      \includegraphics[width=\textwidth]{alignspabdc_ct_060202_1545_fdeltathetaspa.eps}
      \put(-45,153){a)}
    \end{minipage}
    \begin{minipage}[c]{0.5\textwidth}
      \includegraphics[width=\textwidth]{alignspabdc_ct_060202_1545_fdeltathetabdc.eps}
      \put(-45,153){b)}
    \end{minipage}
    \caption{Distributions to derive alignment constants in \mbox{$z$-direction} for a) the SpaCal and b) the BDC using the Central Tracker as a reference. To extract the alignment constants a sinus function according to equation~\ref{e:alifdeltatheta} is fitted to the data (solid lines).}
    \label{f:alifdeltatheta}
  \end{figure}

\end{itemize}

The underlying distributions for the different methods are shown in
figures~\ref{f:alirdeltaphi}~-~\ref{f:alifdeltatheta},
table~\ref{t:aliconstctqedc} summarises the alignment constants. The
results are discussed at the end of this section.

\begin{table}
\centering
\begin{tabular}{c@{\hspace{2em}}c@{\hspace{1em}}c@{\hspace{1em}}c@{\hspace{2em}}c@{\hspace{1em}}c@{\hspace{1em}}c}
\toprule
 & \multicolumn{3}{c}{SpaCal} & \multicolumn{3}{c}{BDC} \\
\raisebox{1.5ex}[-1.5ex]{method} & $\Delta x\; [\mathrm{cm}]$ & $\Delta y\; [\mathrm{cm}]$ & $\Delta z\; [\mathrm{cm}]$ & $\Delta x\; [\mathrm{cm}]$ & $\Delta y\; [\mathrm{cm}]$ & $\Delta z\; [\mathrm{cm}]$ \\
\midrule
$R \Delta \phi$ & -0.14 & -0.53 & - & - & - & - \\
$\Delta R$ & -0.25 & -0.26 & - & -0.06 & -0.01 & - \\
$F \Delta \theta$ & - & - & 0.23 & - & - & 0.65 \\
\addlinespace
QEDC & -0.08 & -0.34 & - & - & - & - \\
\bottomrule
\end{tabular}
\caption{Summary of the alignment constants for the SpaCal and the BDC
derived from different methods using the Central Tracker as a reference or QEDC events.}
\label{t:aliconstctqedc}
\end{table}

\subsection*{Alignment with QED Compton Events}

QED Compton events have been introduced in
section~\ref{c:radepscat}. They are characterised by an electron and a
photon in the final state which have sizeable transverse momenta
and are back-to-back in the azimuthal plane. Thus they can be detected
as two electromagnetic clusters in the SpaCal with a difference in the
azimuthal angle $\Delta \phi = |\phi_{1} - \phi_{2}|$ close to
$180^{\circ}$. A useful variable to select QEDC events is the
\emph{acoplanarity} $A$ being defined as
\[ A = |\Delta \phi - 180^{\circ}| \]
To derive alignment constants with the help of QEDC events, as
proposed in \cite{Stamen:1998}, one defines the actual position of the
SpaCal in the H1 coordinate as the position for which the summed
acoplanarity of all QEDC events is minimal. Technically this is done
by performing a $\chi^{2}$ minimisation of the expression
\[ \chi^{2} = \sum_{\mathrm{events}} \frac{A^{2}}{\sigma_{A}^{2}} \]
with
\[ \sigma_{A}^{2} = \frac{y_{1}^{2} \sigma_{x}^{2} + x_{1}^{2} \sigma_{y}^{2}}{(x_{1}^{2} + y_{1}^{2})^{2}} + \frac{y_{2}^{2} \sigma_{x}^{2} + x_{2}^{2} \sigma_{y}^{2}}{(x_{2}^{2} + y_{2}^{2})^{2}} \]
Here $x_{1}$, $y_{1}$ and $x_{2}$, $y_{2}$ are the coordinates of the
electron and the photon cluster, respectively, corrected for the
effects of beam offset and tilt. The average spatial resolution
parameters $\sigma_{x}$ and $\sigma_{y}$ are set to
\mbox{$0.3\, \mathrm{cm}$}. For the alignment with QEDC events the 
$x$- and \mbox{$y$-coordinates} are measured w.r.t. the origin of
the SpaCal which does not coincide with the origin of the H1 coordinate
system in the \mbox{$xy$-plane} due to the misalignment.

For the alignment elastic QEDC events are used as they are
characterised by a smaller acoplanarity than inelastic ones. The
applied selection criteria follow a detailed investigation in
\cite{Lendermann:2001}:

\begin{itemize}
\item{acoplanarity
\[ A < 10^{\circ}\]}
\item{energies $E_{1}$ and $E_{2}$ of the cluster with the highest and second highest energy in the SpaCal
\[ E_{1} > 10\, \mathrm{GeV} \hspace{2em} E_{2} > 4\, \mathrm{GeV} \hspace{2em} 20 \, \mathrm{GeV} < E_{1} + E_{2} < 32\, \mathrm{GeV} \]}
\item{residual energy in the electromagnetic section of the SpaCal
\[ E_{res} = E_{tot} - E_{1} - E_{2} < 1\, \mathrm{GeV} \]
where $E_{tot}$ denotes the total energy in the electromagnetic SpaCal}
\item{logarithmic cluster radius of the two SpaCal clusters
\[ r_{Cl,1} < 4.5\, \mathrm{cm} \hspace{2em} r_{Cl,2} < 4.5\, \mathrm{cm} \]}
\item{rejection of events with a Central Tracker vertex fulfilling
\[ |z_{vtx} - \overline{z_{vtx}}| > 30\, \mathrm{cm} \]
if the vertex accuracy $\sigma_{z_{vtx}}$ is better than \mbox{$8\,
\mathrm{cm}$}}
\end{itemize}

The \mbox{$z$-vertex} cut ensures that events most likely being caused
by beam induced background are excluded from the analysis. Events
having no reconstructed vertex are accepted. This has no negative
influence on the $\chi^{2}$ minimisation as no \mbox{$z$-information}
is needed.

Performing the minimisation for the selected QEDC sample leads to the
alignment constants given in table~\ref{t:aliconstctqedc}. To check the
result one visualises the QEDC events in the following way: the
cluster positions of electron and photon in the SpaCal are connected
by a line in the \mbox{$r\phi$-plane} for all events and the line
density is plotted. The maximum indicates the intersection point of
the beam with the surface of the SpaCal. For a perfectly aligned
SpaCal its $x$- and \mbox{$y$-coordinates} should be
$(0,0)$. Figure~\ref{f:aliqedc} shows the line density before and
after the alignment. Indeed the maximum of the line density indicates
a good alignment of the SpaCal.

\begin{figure}[t]
\begin{minipage}[c]{1.0\textwidth}
  \centering
  \includegraphics[width=0.6\textwidth]{alignspa_qedc_060321_1854_databc.eps}
  \put(-275,180){a)}
\end{minipage}
\vskip 0.3cm
\begin{minipage}[c]{1.0\textwidth}
  \centering
  \includegraphics[width=0.6\textwidth]{alignspa_qedc_060321_2014_databc.eps}
  \put(-275,180){b)}
\end{minipage}
\caption{Density of lines connecting the electron and the photon cluster of QED Compton events in the SpaCal a) before and b) after the alignment with QED Compton events.}
\label{f:aliqedc}
\end{figure}


In principle a similar study can be performed for the BDC as a track
can be reconstructed for QEDC photons with a probability of
approximately 60\%~\cite{Lendermann:2001}. The main reason for this
are photon conversions into electron-positron pairs while passing the
material of the beam pipe or the chamber walls. However, such a study
does not make sense as the \mbox{$\phi$-information} stems from the
SpaCal anyhow.

\subsection*{Discussion and Combination of the Results}

If one considers the alignment constants in table
 \ref{t:aliconstctqedc} obtained from the different methods the
 following conclusions can be drawn: the results for the alignment
 constants $\Delta x$ and $\Delta y$ for the SpaCal do not agree with
 each other, the disagreement being larger for the
\mbox{$y$-direction}. The errors resulting from the fits in the case of
the comparison to the Central Tracker and the $\chi^{2}$ minimisation
for the QEDC events are negligible compared to the differences between
the approaches. Similar discrepancies have been reported before and
have been investigated in detail in~\cite{Merkel:2005}, unfortunately
without a decisive explanation. The difference of the two Central
Tracker results may point to a problem concerning the internal
alignment of the Central Tracker. For the minimum bias run in 1999 a
difference of \mbox{$2\, \mathrm{mm}$} for $\Delta y$ is observed
between the $R \Delta \phi$ and the $\Delta R$
method~\cite{Glazov:2006} which is in agreement with the present
analysis. The alignment with QEDC events provides a value of $\Delta
y$ in between the two Central Tracker results as well.



A number of cross-checks has been performed to test the alignment procedure:

\begin{itemize}
\item{The results of the different approaches for the Monte Carlo
sample have been checked. They are consistent with 0, only the
$F \Delta \theta$ method yields \mbox{$\Delta z = 0.1\, \mathrm{cm}$}
and the QEDC method \mbox{$\Delta y = -0.1\, \mathrm{cm}$} for the
SpaCal.}
\item{A misalignment has been introduced for the Monte Carlo
sample. All methods determine this with high precision apart from the
$\chi^{2}$ minimisation for QED Compton events which gives
\mbox{$\Delta y = -0.2\,
\mathrm{cm}$} though a \mbox{$0.4\, \mathrm{cm}$} shift has been
applied. However, the density line plot indicates a shift of
\mbox{$0.4\, \mathrm{cm}$}.}
\item{To test the stability of the methods the alignment constants
have been applied and the analysis has been repeated. All methods
provide results consistent with zero as expected, only the QEDC method
claims \mbox{$\Delta y = -0.1\, \mathrm{cm}$}.}
\end{itemize}

The $\chi^{2}$ minimisation for the QEDC events seems not to find the
global minimum. Therefore the QEDC alignment is not considered
further.



The alignment constants to be applied to the final analysis are
derived in the following way: in the case of the SpaCal the average of the
$R \Delta \phi$ and the $\Delta R$ method is taken for $\Delta x$ and
$\Delta y$, half of the difference is used as the error. The results
of the $\Delta R$ method for the BDC are compatible with 0. Thus no
alignment in the \mbox{$r\phi$-plane} is needed to improve the
$\theta$ measurement of the BDC. To obtain an error for $\Delta z$ the
alignment is repeated without applying correction functions for
$\theta$ as reconstructed by the Central Tracker and the SpaCal. Half
of the difference to the nominal alignment is used as the error. The
alignment constants to be used for the final analysis are collected in
table~\ref{t:aliconstcomb}.

\begin{table}
\centering
\begin{tabular}{c@{\hspace{2em}}c@{\hspace{1em}}c@{\hspace{1em}}c}
\toprule
detector & $\Delta x\; [\mathrm{cm}]$ & $\Delta y\; [\mathrm{cm}]$ & $\Delta z\; [\mathrm{cm}]$ \\
\midrule
SpaCal & $-0.19 \pm 0.06$ & $-0.39 \pm 0.14$ & $0.23 \pm 0.36$ \\
BDC & - & - & $0.65 \pm 0.15$ \\
\bottomrule
\end{tabular}
\caption{Alignment constants to be applied to the final analysis obtained by combining the different alignment methods (for details see text).}
\label{t:aliconstcomb}
\end{table}


%
%

The uncertainty on the polar angle reconstruction of the BDC is the
figure of merit to be derived from the alignment study for the final
analysis. Therefore the polar angles as reconstructed by the BDC and
the Central Tracker are compared with the alignment constants being
applied. To have a sample of events as close as possible to the one
which is used for the final analysis the cut on the energy of the
scattered electron is relaxed to \mbox{$7\,
\mathrm{GeV}$}. Figure~\ref{f:thetabdcvsctdata} shows the distribution
of \mbox{$\Delta \theta = \theta_{BDC} - \theta_{CT}$}. No residual
systematic shift between the Central Tracker and the BDC is
observed. The tail towards larger values of $\Delta \theta$ is caused
by electrons showering in the dead material behind the Central
Tracker. This can be concluded from comparing the distribution of
$\Delta \theta$ for events with small and large hit multiplicities in
the BDC. A similar effect has been observed in~\cite{Keller:1998} when
comparing the polar angle measurement of the BDC and the BST. The
Monte Carlo sample shows an analogous behaviour. With the error of the
alignment constant $\Delta y$ for the SpaCal (see
table~\ref{t:aliconstcomb}) being the dominant uncertainty on $\theta$
as reconstructed by the BDC the latter has been determined to be
$\sigma_{\theta} = 0.5\, \mathrm{mrad}$.

\begin{figure}
%\centering
\begin{captionbeside}{Comparison of the polar angle reconstruction of the Central Tracker and the BDC after the alignment. The solid line represents the fit of a Gaussian to the distribution.  \vspace{0.5cm}}[i][0.95\textwidth][0.05\textwidth]*
\begin{minipage}[t]{0.5\textwidth}
\includegraphics[width=\textwidth]{alignspabdc_ct_060207_1728_thetabdcvsct_data.eps}
\end{minipage}
\end{captionbeside}
\label{f:thetabdcvsctdata}
\end{figure}


\section{Calibration} \label{c:calibration}


Both the energy of the scattered electron and the hadronic final state
affect the reconstruction of the event kinematics. The former directly
enters the formulas necessary to calculate $y$ and $Q^{2}$ based on
the electron method, the latter affects the quantity $\Sigma$
essential for the sigma method (see
section~\ref{c:evkindet}). Therefore a precise calibration of both
energies is necessary. In the case of the electron energy this implies
a calibration of the electromagnetic section of the SpaCal. For a
precise measurement of the energy of the hadronic final state the
hadronic section of the SpaCal as well as the electromagnetic and
hadronic part of the LAr calorimeter have to be accurately calibrated
in addition. This complex task has been fulfilled
in~\cite{Vargas:2006}, a short overview of which is given in the
chapter at hand\footnote{Unless stated differently all results
presented in this chapter are taken from~\cite{Vargas:2006}.}.

\subsection*{Scattered Electron}

As explained in section~\ref{c:evreco} the energy of the scattered
electron is reconstructed in the electromagnetic section of the
SpaCal. The largest challenge for the calibration of the latter is the
control of the fluctuations of its 1192 photomultipliers. This starts
already during data taking. A special LED system sending well-defined
light pulses to each photomultiplier monitors potential time-dependent
instabilities such that they can be corrected for~\cite{Janoth:1996,
Meyer:1996}. Subsequently the inner cells of the SpaCal are
pre-calibrated with the help of the so-called \emph{kinematic peak}
method~\cite{Janoth:1995}. It makes use of the fact that the energy
distribution of the scattered electron for low $Q^{2}$ DIS events
shows a prominent maximum close to the electron beam energy. Since the
method requires large statistics, the outer part of the SpaCal ($R_{e}
\gtrsim 50\, \mathrm{cm}$) has instead been pre-calibrated with the help of
muons originating from cosmic rays~\cite{Dirkmann:1996} or proton beam
halo events~\cite{Arndt:1995}.

A precise calibration on analysis level is provided by a method
originally proposed in~\cite{Glazov:1998a}. It uses the energy of the
scattered electron reconstructed with the help of the so-called
\emph{double angle method} as a reference. This reconstruction method
expresses the kinematic variables $y$ and $Q^{2}$ in terms of the
polar angle $\theta_{e}$ of the scattered electron, the hadron
angle~$\theta_{h}$ and the electron beam energy $E_{e}^{0}$:
\begin{align}
y_{DA} & = \frac{\tan (\theta_{h}/2)}{\tan (\theta_{e}/2) + \tan (\theta_{h}/2)} \label{e:yda} \\
Q_{DA}^{2} & = 4 (E_{e}^{0})^{2}\, \frac{\cot (\theta_{e}/2)}{\tan (\theta_{e}/2) + \tan (\theta_{h}/2)} \label{e:q2da}
\end{align}
Based on these equations the energy of the scattered electron is calculated to
\begin{equation}
E_{DA} = E_{e}^{0}\, \frac{1 - y_{DA}}{\sin^{2} (\theta_{e}/2)}
\label{e:eda}
\end{equation}
With the double angle energy $E_{DA}$ a reference energy for the
calibration is available which is to first order insensitive to the
energy measurement in the SpaCal and LAr calorimeters. The double
angle method reveals a good resolution for medium values of~$y$
(\mbox{$0.05 < y < 0.2$})~\cite{Bassler:1994uq}. Therefore events are
selected for the calibration which fulfil the requirement
\mbox{$15^{\circ} < \theta_{h} < 80^{\circ}$}. The lower limit
suppresses energy losses in the forward region of the LAr calorimeter,
the upper limit restricts $y$ to maximum values of approximately
$0.15$. Only events with a single electron candidate matching the cuts
$20\, \mathrm{GeV} < E_{e} < 32\,
\mathrm{GeV}$ and $r_{Cl} < 4\, \mathrm{cm}$ are accepted, i.e. 
the calibration sample should contain almost no photoproduction
events. The polar angle of the scattered electron is reconstructed
with the BST or the BDC. If both backward trackers validate the SpaCal
cluster the BST information is preferred. The \mbox{$z$-position} of
the interaction vertex is restricted to an interval of $30\,
\mathrm{cm}$ around the nominal position $\overline{z_{vtx}}$.

The energy of the cluster $E_{Spa}$ as reconstructed by the SpaCal is
the sum of the energies $E_{i}$ in the individual cells belonging to
it:
\begin{equation}
E_{Spa} = \sum_{i} E_{i}
\label{e:espa}
\end{equation}
Based on this the so-called event pull $\delta_{ev}$ is calculated
\begin{equation}
\delta_{ev} = \frac{E_{DA}}{E_{Spa}}
\label{e:deltaev}
\end{equation}
which is investigated as a function of various variables to derive
a sequence of corrections to be applied for a precise energy
measurement.

In a first step calibration constants are derived for each cell of the
SpaCal calorimeter. The event pull $\delta_{ev}$ cannot be attributed
to a single cell since SpaCal clusters typically extend over several
cells. Therefore the following procedure is used: for each SpaCal cell
a histogram is booked capable of storing a distribution of
$\delta_{ev}$. In each event $\delta_{ev}$ is filled to the histograms
of all cells belonging to the cluster with a weight
$E_{i}/E_{Spa}$. The distribution of $\delta_{ev}$ for each SpaCal
cell should have a mean value of one for a perfectly calibrated
calorimeter. Hence the mean pull $\delta_{i}$ is determined for each
SpaCal cell and used as a correction factor. For this purpose the same
robust mean estimator~\cite{Yuille:1990} is employed as for the
alignment.

During the filling of the histograms the same event pull $\delta_{ev}$
is attributed to each cell. In order to separate the influence of the
individual cells several iterations are necessary applying in each
iteration step the results from the previous ones. After eight
iterations the changes are below 0.1\% and the final cell-wise
calibration constants are derived. They correct the energy measurement
by \mbox{$2-3\%$}.

Three additional corrections are derived afterwards having a
significantly smaller influence on the energy measurement: to account
for energy losses at the cell and super-module borders of the SpaCal
$\delta_{ev}$ is investigated as a function of the cluster position
relative to the centre of the cell with the largest energy deposition
(inbox correction) and as a function of \mbox{$R_{box} =
\mathrm{max}(|x_{Spa}|,|y_{Spa}|)$} (crack correction)\footnote{The
variable $R_{box}$ has been introduced in section~\ref{c:onlinesel}}.
Finally losses in the dead material in front of the calorimeter are
compensated by deriving a correction as a function of $R_{e}$ (radial
correction).

The same calibration procedure is carried out for data and Monte Carlo
simulation. It turned out that the resolution for the Monte Carlo
sample is better than for the data sample. Therefore the reconstructed
energy of the Monte Carlo is smeared by 0.2\% to ensure good agreement
with the data. Figure~\ref{f:eleccal} shows the energy distribution of
the scattered electron for the calibration sample a) before the
calibration and b) after applying all derived corrections and smearing
the Monte Carlo. It can be seen that the calibration improves the
agreement between data and Monte Carlo significantly. The uncertainty
on the energy scale of the scattered electron is estimated by
comparing $\delta_{ev}$ as a function of $R_{e}$ between data and
Monte Carlo after applying all corrections. The differences are well
compatible with a systematic uncertainty of 0.2\%.


\begin{figure}
\begin{minipage}[c]{0.5\textwidth}
  \includegraphics[width=\textwidth]{ee_uncalibrated.eps}
  \put(-157,161){a)}
\end{minipage}
\begin{minipage}[c]{0.5\textwidth}
  \includegraphics[width=\textwidth]{ee_calibratedsmeared.eps}
  \put(-157,161){b)}
\end{minipage}
\caption{Energy distribution of the scattered electron as reconstructed by the SpaCal a) before and b) after the calibration~\cite{Vargas:2006}. In the case of b) the energy of the Monte Carlo is additionally smeared by 0.2\%.}
\label{f:eleccal}
\end{figure}

The event sample used for the calibration with the double angle method
is restricted to large energies. To check the applicability of the
calibration at low energies, i.e. the linearity of the SpaCal
response, the invariant mass of $\pi^{0}$ mesons decaying into two
photons is studied\footnote{In contrast to the standard cross-section
analysis the required cluster energy for this study is so small that
for a considerable fraction of the events the opening angle between
the two photons is sufficiently large to detect them as two separate
clusters in the SpaCal.}. Comparing the mass spectrum between data and
Monte Carlo shows that the energy response of the SpaCal in the Monte
Carlo has to be reduced by 2.5\% to achieve a good agreement. The
uncertainty on this correction is determined to 1\% by studying the
dependence of the energy scale on the energy of the photon with the
higher and lower energy, respectively as well as on the distance of
the two photon clusters. These values are attributed to an energy of
\mbox{$2\,
\mathrm{GeV}$} since this is the average energy of the decay photons.

From figure~\ref{f:eleccal} b) it can be seen that no additional
correction to the energy scale on top of the double angle calibration
is necessary at larger energies. The uncertainty on the energy scale
of 0.2\% already identified in the study of $\delta_{ev}$ is
attributed to an energy of \mbox{$27.6\, \mathrm{GeV}$} since the
energy distribution should have its maximum close to the electron beam
energy as explained at the beginning of this section.

Hence the corrections for the energy scale of the scattered electron
and the corresponding uncertainties can be summarised as follows:
\begin{align}
2.5\% \pm 1.0\% & \hspace{2em} \mathrm{at}\; E_{e} = 2\, \mathrm{GeV} \\
0.0\% \pm 0.2\% & \hspace{2em} \mathrm{at}\; E_{e} = 27.6\, \mathrm{GeV}
\end{align}
For correction and uncertainty a linear interpolation is used for intermediate energies.

\subsection*{Hadronic Final State}

The calibration procedure to ensure a precise reconstruction of the
hadronic final state starts with an investigation of the hadronic
energy scale of the SpaCal. This comprises both the electromagnetic
and the hadronic section of the SpaCal since hadron showers occupy
both of them. Several correction factors to the hadronic energy scale
have been derived prior to~\cite{Vargas:2006} and provide the starting
point for the study to be presented in the following. They are listed
in table~\ref{t:spachadcor} separately for the electromagnetic and
hadronic section. The corresponding references are listed in addition,
an overview can be also found in~\cite{Arkadov:2000}. The product of
the correction factors is applied for both data and Monte Carlo to
each cell which does not belong to the electron candidate.

\begin{table}
\centering
\begin{tabular}{c@{\hspace{2em}}c@{\hspace{2em}}c}
\toprule
electromagnetic section & hadronic section & reference \\
\midrule
1.1 & 1.3 & \cite{Zhokin:1995} \\
1.5 & 1.5 & \cite{Meyer:1997, Glazov:1998a} \\
1.07 & 1.07 & \cite{Meyer:1997} \\
\bottomrule
\end{tabular}
\caption{Corrections factors to the hadronic energy scale of the SpaCal calorimeter.}
\label{t:spachadcor}
\end{table}

A final correction is derived using the already calibrated energy of
the scattered electron as a reference. For this purpose events with
\mbox{$y_{e} > 0.5$} are selected. In this region of the phase space $y_{h}$
is almost exclusively determined by the hadrons reconstructed in the
SpaCal (see figure~\ref{f:yhaddecomp}). Therefore a comparison of the
ratio $y_{h}/y_{e}$ reveals differences of the hadronic energy scale
between data and Monte Carlo. Since the amount of photoproduction
background events is large at high $y$ only events with \mbox{$E -
p_{z} > 35\,
\mathrm{GeV}$} are accepted. This cut suppresses the photoproduction
contribution as is explained in the next section. Indeed a shift in
the distribution of $y_{h}/y_{e}$ between data and simulation is
observed. This is accounted for by applying a correction factor of
0.97 to the Monte Carlo. This corresponds to a shift of \mbox{$700\,
\mathrm{MeV}$}. Half of the shift, i.e. \mbox{$350\, \mathrm{MeV}$} is
assumed to be the uncertainty on the hadronic energy scale of the
SpaCal.

The hadronic energy flow in the LAr is distributed over large areas
and thus it is impossible to select individual hadrons for the
calibration. Therefore global quantities have to be employed. An
elegant method to make use of transverse momentum conservation (see
equation~\ref{e:ptcons}) for the calibration is the \emph{Lagrangian
method} developed in~\cite{Arkadov:2000}. The method utilises the
compensation of the transverse momenta of the scattered electron and
the hadronic final state via the minimisation of the following
functional:
\begin{equation}
\mathcal{L} = \sum_{i=1}^{N} \left ( p_{t,e}^{i} - \left ( p_{t,CT}^{i} + p_{t,Spa}^{i} - p_{t,noise}^{i} \right ) - \sum_{j=1}^{M} \alpha_{j} p_{t,LAr}^{ji} \right )^{2}
\label{e:larlagrangian}
\end{equation}
The outer summation is performed over all events, the index $i$ is
omitted for the following discussion. The transverse momentum
$p_{t,e}$ of the scattered electron constitutes the reference. All
other transverse momenta are projected on the electron direction such
that the minimisation can be performed in one dimension. To the
transverse momentum of the hadronic final state $p_{t,h}$ several
subdetectors contribute according to the explanations at the end of
section~\ref{c:evreco}. These are the SpaCal ($p_{t,Spa}$), the
Central Tracker ($p_{t,CT}$) and the LAr calorimeter. Since the latter
is to be calibrated the corresponding transverse momentum is split to
the level of the individual modules. The variable $p_{t,LAr}^{j}$
denotes the transverse momentum of the module $j$ and $\alpha_{j}$ its
calibration constant to be determined by the minimisation of
$\mathcal{L}$. The transverse momentum $p_{t,noise}$ corresponding to
the noise in the LAr calorimeter is subtracted. For the combination of
the LAr and Central Tracker information as well as the noise
identification the FSCOMB algorithm (see section~\ref{c:evreco}) is
employed.

The LAr calorimeter is build of eight wheels with eight octants
each. One of the wheels (BBE) has only one section, all others have
two sections (see figure~\ref{f:larrz}). Hence \mbox{$M = 120$}
calibration constants $\alpha_{j}$ have to be determined. The
minimisation requirement
\begin{equation}
\frac{\partial \mathcal{L}}{\partial \alpha_{j}} = 0 
\label{e:lagrmini}
\end{equation}
defines a set of $M$ linear equations to be solved using matrix techniques.

Figure~\ref{f:hadcal} shows a comparison of the distribution of the quantity
\mbox{$p_{t,bal} = p_{t,had}/p_{t,e}$} between data and Monte Carlo
for two intervals in $y_{\Sigma}$. All calibration constants derived
in this section are applied. For medium values of $y_{\Sigma}$ the
agreement is perfect and the maximum of the distribution is located at
one as expected. At low $y_{\Sigma}$ the distribution is significantly
shifted towards lower values of $p_{t,bal}$ since hadrons escape
detection in the forward region of the LAr calorimeter. In addition a
discrepancy between data and Monte Carlo occurs. This behaviour is
reflected in the uncertainty on the hadronic energy scale of the LAr
calorimeter. It amounts to 10\% at \mbox{$y = 10^{-3}$} and is
linearly interpolated in $\ln y$ to 2\% at \mbox{$y = 10^{-2}$}. For
values above \mbox{$y = 10^{-2}$} it stays at 2\%. The same uncertainty is
assumed for the Central Tracker. It has been derived by varying the
contribution of LAr and Central Tracker and investigating the
agreement between data and Monte Carlo in
figure~\ref{f:yhaddecomp}. In the same way the uncertainty on the
noise contribution in the LAr calorimeter has been determined to be
10\%. Table~\ref{t:haduncer} summarises all uncertainties attributed
to the reconstruction of the hadronic final state.


\begin{figure}
\begin{minipage}[c]{0.5\textwidth}
  \includegraphics[width=\textwidth]{ptbal_lowysigma.eps}
  \put(-37,161){a)}
\end{minipage}
\begin{minipage}[c]{0.5\textwidth}
  \includegraphics[width=\textwidth]{ptbal_mediumysigma.eps}
  \put(-37,161){b)}
\end{minipage}
\caption{Distribution of $p_{t,bal} = p_{t,had}/p_{t,e}$ for two intervals of $y_{\Sigma}$ after application of all correction constants~\cite{Vargas:2006}.}
\label{f:hadcal}
\end{figure}

\begin{table}
\centering
\begin{tabular}{c@{\hspace{2em}}c}
\toprule
source & uncertainty \\
\midrule
hadronic SpaCal & $350\, \mathrm{MeV}$ \\
hadronic energy scale LAr and CT & $10\%$ at $y = 10^{-3}$ \\
 & $2\%$ at $y = 10^{-2}$ \\
 LAr noise & $10\%$ \\
\bottomrule
\end{tabular}
\caption{Uncertainties on the reconstruction of the hadronic final state.}
\label{t:haduncer}
\end{table}


\section{DIS Event Selection} \label{c:dissel}

The aim of the event selection is to provide a sample of events for
the cross-section determination which contains only a small fraction
of background events while keeping the number of selection criteria as
small as possible. The latter requirement should minimise the loss of
signal events and the influence of Monte Carlo reconstruction
efficiencies.

Background contributions to the cross-section measurement arise from
photoproduction events and interactions of the proton or electron beam
particles with the residual gas molecules or the beam pipe walls (see
section~\ref{c:background}). Most of the latter are already rejected
during the online event selection due to the requirements concerning
the timing and energy depositions in the SpaCal.

The selection criteria applied for the BST and BDC analyses (see
table~\ref{t:selcriteria}) are well established as they have been used
in various previous analyses. Therefore the values of the cuts are not
investigated in detail, but their selection and rejection power is
compared between data and Monte Carlo samples for signal and
background events, respectively. Here only the BDC analysis in
covered, a similar study for the BST analysis can be found
in~\cite{Vargas:2006}. An agreement between data and simulation
is essential for the cross-section measurement as explained in the
introduction to this chapter.

\begin{sidewaystable}
\centering
\begin{tabular}{c@{\hspace{2em}}c@{\hspace{2em}}C{4cm}@{\hspace{4em}}C{4cm}}
\toprule
\multicolumn{2}{c}{selection criterion} & \hspace*{2.2em}BST analysis & BDC analysis \\
\midrule
 & position & \multicolumn{2}{c}{$|z_{vtx} - \overline{z_{vtx}}| < 30\, \mathrm{cm}$} \\
vertex & precision & - & $\sigma_{z_{vtx}} < 8\, \mathrm{cm}$ \\
 & type & BST & Central Tracker \\
\cmidrule{1-4}
 & energy & \multicolumn{2}{c}{$E_{e} > 7\, \mathrm{GeV}$} \\
cluster & radius & \multicolumn{2}{c}{$r_{Cl} < 4\, \mathrm{cm}, \hspace{1em} r_{Cl}^{lin} < 3.8\, \mathrm{cm}$} \\
properties & hadronic fraction & \multicolumn{2}{c}{$f_{had} < 0.15$} \\
 & veto cell energy & \multicolumn{2}{c}{$E_{veto} < 1\, \mathrm{GeV}$} \\
\cmidrule{1-4}
 & number of linked hits & $N_{hits} \ge 3$ & $N_{hits} \ge 4$ \\
track & total number of hits & $N_{hits}^{total} < 200$ & - \\
validation & track - cluster match & \multicolumn{2}{c}{$\Delta R < 1.5\, \mathrm{cm}$} \\
 & type & BST & BDC \\
\cmidrule{1-4}
 & & \multicolumn{2}{c}{$E - p_{z} > 35\, \mathrm{GeV}$ (electron method only)} \\
\multicolumn{2}{c}{\raisebox{1.5ex}[-1.5ex]{kinematics}} & $p_{t,bal} > 0.3$ & $y_{\Sigma} > 0.03$ \\
\cmidrule{1-4}
\multicolumn{2}{c}{rejection of elastic QEDC events} & $\checkmark$ & $\checkmark$ \\
\cmidrule{1-4}
 & acceptance & $R_{e} > 10\, \mathrm{cm}$ & $R_{e} > 9\, \mathrm{cm}$ \\
 & trigger & \multicolumn{2}{c}{$R_{box}>10\, \mathrm{cm}$ (BDC: only for $E_{e} < 14\, \mathrm{GeV}$)} \\
 & & \multicolumn{2}{c}{excl. cells: 7,8,13,14,30,32,33,62,250,253,306,763} \\
\raisebox{1.5ex}[-1.5ex]{fiducial cuts} & hot spot & \multicolumn{2}{c}{$R_{e} > 11.5\, \mathrm{cm}$ for $-6.5\, \mathrm{cm} < y < 7.5\, \mathrm{cm}$} \\
 & & \multicolumn{2}{c}{excl. cells: 4,5,6,11,12} \\
 & energy spectrum & \multicolumn{2}{c}{excl. cell: 29} \\
\bottomrule
\end{tabular}
\caption{Selection criteria for the BST and BDC analyses (for details see text). The position of the individual SpaCal cells is shown in appendix~\ref{c:spacalcellno}.}
\label{t:selcriteria}
\end{sidewaystable}

\subsection*{Explanation of the Selection Criteria}

Restricting the \mbox{$z$-coordinate} of the interaction vertex to an
interval of \mbox{$30\, \mathrm{cm}$} around the nominal vertex
position \mbox{$\overline{z_{vtx}} = \zvtxmean$} (see
section~\ref{c:vertexreco}) rejects beam-induced background
events. Contributions from the satellite bunches being separated by
$\pm 72\, \mathrm{cm}$ from the main bunch are removed in
addition. This is of importance as the satellite bunches are not
considered in the Monte Carlo simulations. The uncertainty on the
\mbox{$z$-position} is required to be smaller than \mbox{$8\,
\mathrm{cm}$} and thus the maximal allowed value is significantly 
larger than the resolution of \mbox{$1\, \mathrm{cm}$} quoted in
section~\ref{c:evreco}. This is appropriate since according
to~\cite{Glazov:2006} a resolution of \mbox{$1\, \mathrm{cm}$} can
only be reached for events with a large number of tracks reconstructed
in the Central Tracker. Hence a harsher cut would reject a significant
fraction of events at low $y$ for which only a few particles per event
enter the acceptance of the Central Tracker.

From the clusters validated by the BDC or BST electron finder the one
with the largest energy fulfilling the selection criteria concerning
the cluster properties and track validation is chosen to be the
electron candidate (see section~\ref{c:evreco}).


Photoproduction events can be misinterpreted as signal events if a
particle of the hadronic final state mimics the scattered electron
(see section~\ref{c:background}). Significant contributions stem from
neutral $\pi$ mesons and charged hadrons. The former predominantly
decay into two photons and thus induce electromagnetic showers in the
SpaCal. Moreover, a large fraction of the decay photons converts into
$e^{+} e^{-}$ pairs while interacting with the detector material and
causes signals in the backward trackers. Also purely hadronic showers
can be classified as electromagnetic due to statistical fluctuations
in the shower development. In general, hadron induced clusters are
less compact than electron or photon induced ones.

Thus the contribution of charged hadrons can be reduced by selection
criteria based on the transverse and longitudinal shower
shape. Detailed studies in \cite{Glazov:1998a} and \cite{Eckstein:2002}
have shown that a cut on the logarithmic cluster radius $r_{Cl} < 4\,
\mathrm{cm}$ rejects a significant fraction of the photoproduction
events while accepting electron induced clusters. The rejection
comprises $\pi^{0}$ mesons as for cluster energies considered in this
analysis the opening angle is too small for the two decay photons to
be detected as separate clusters in the SpaCal. Thus both photons are
reconstructed as a single broad cluster.

The SpaCal is designed such that electron induced showers with an
energy up to $30\, \mathrm{GeV}$ are fully contained in the
electromagnetic section. However, for hadrons the electromagnetic
section constitutes only one nuclear interaction length such that
their showers reach the hadronic section of the SpaCal. Therefore
clusters are considered to be hadron induced if the energy fraction
$f_{had}$ deposited in the hadronic section of the SpaCal within a
cylindrical volume of \mbox{$15\, \mathrm{cm}$} radius behind the
cluster identified in the electromagnetic section exceeds 15\% of
the total cluster energy.

A further suppression of photoproduction events is achieved by placing
a cut on the track cluster matching $\Delta R$ which is the distance
of the SpaCal cluster in the \mbox{$r\phi$-plane} to the track of the
BST or BDC extrapolated to the \mbox{$z$-position} of the
cluster. Photons not converted before the BST or BDC cannot fulfil the
selection criterion as they do not produce a track. As photon
conversions in the dead material behind the Central Tracker do only
lead to a signal in the BDC but not in the BST, the cut is more
efficient for the BST analysis in this respect. Charged hadrons may
also fail to match the requirement on $\Delta R$ since the
fluctuations of the energy deposited by them in the electromagnetic
section of the SpaCal is large which may lead to a significantly
misreconstructed cluster position.

At least three (four) hits have to be linked to the track of the
electron candidate for the BST (BDC) analysis. This requirement
ensures a good track resolution and suppresses accidental validations
of SpaCal clusters due to noise hits. The cut \mbox{$N_{hits}^{total}
< 200$} keeps the overall level of noise in the BST at a reasonable
level.

The quantity \mbox{$E-p_{z} = \Sigma + E_{e}(1-\cos \theta_{e})$}
should be equal to \mbox{$2E_{e}^{0} = 55.2\, \mathrm{GeV}$} due to
energy and momentum conservation (see section~\ref{c:evkindet}). For
photoproduction events it is likely to be smaller than $2E_{e}^{0}$ as
the electron is scattered below the SpaCal acceptance and the SpaCal
cluster is a misidentified hadron. However, also in initial state
radiation events \mbox{$E-p_{z}$} has lower values as the photon
escapes through the beam pipe undetected decreasing
\mbox{$E-p_{z}$} by twice its energy. Therefore the cut \mbox{$E -
p_{z} > 35\, \mathrm{GeV}$} is applied if the event kinematics is
reconstructed with the help of the electron method but not if the
sigma method is used. This is due the fact that ISR events are
reconstructed correctly by the sigma method and allow for a
significant extension of the accessible kinematic phase space while
they are reconstructed wrongly by the electron method (see detailed
discussion in the previous chapter).

In earlier analyses a significant discrepancy between data and Monte
Carlo simulation at low values of the ratio \mbox{$p_{t,bal} =
p_{t,had}/p_{t,e}$} of the transverse momenta of the hadronic final
state $p_{t,had}$ and the scattered electron $p_{t,e}$ has been
observed~\cite{Lastovicka:2004a, Arkadov:2000}. Low values of
$p_{t,bal}$ characterise events in which particles of the hadronic
final state are scattered outside the acceptance limits of the H1
detector leading to energy losses and thus low values of
$p_{t,had}$. In the low $Q^{2}$ regime the losses become especially
pronounced at low values of $y$ (large values of $x$) where the
hadronic final state particles are partially scattered below the
forward modules of the LAr calorimeter. In this region of the phase
space the squared centre of mass energy \mbox{$W^{2}$} of the
photon-proton system becomes very small indicating that the resonance
region is reached. The latter has not been included in previous
versions of the DJANGO generator. Thus the fraction of events at low
values of $p_{t,bal}$ is significantly too small in the Monte Carlo
sample. A cut $p_{t,bal} > 0.3$ has been applied to exclude the
problematic region of the kinematic phase space.  For the present
analysis a new version of DJANGO has been used to overcome the
problem, it includes the SOPHIA model allowing to generate nuclear
resonances (see section~\ref{c:mcmodels}). The agreement between data
and Monte Carlo has indeed improved, but is still not
sufficient. Hence the cut has to be sustained~\cite{Vargas:2006}.

However, for the BDC analysis there is no need for a rejection of
events based on $p_{t,bal}$ as the low $y$ region is avoided by the
$y_{\Sigma}$ cut. The latter guarantees a vertex reconstruction
efficiency above 50\% as discussed in the forthcoming section. Another
missing contribution of the DJANGO generator are elastic QEDC events
as already mentioned in section~\ref{c:mcmodels}. Therefore elastic
QEDC events are rejected using similar cuts as for the SpaCal
alignment based on QEDC events described in the previous section.

Several analysis cuts result from the investigation of the online
event selection in section~\ref{c:onlinesel}: for the BST analysis
only events selected by subtrigger S0 are accepted, the BDC analysis
makes use of subtrigger S3 in addition. The energy threshold and the
L2 condition of S0 motivate the rejection of electron candidates with
energies below \mbox{$7\, \mathrm{GeV}$} and the exclusion of a square
with a side length of \mbox{$10\, \mathrm{cm}$} from the inner
SpaCal. In the case of the BDC analysis the latter is only excluded for
energies below $14\, \mathrm{GeV}$ as subtrigger S3 is used in
addition. The cells indicated in table~\ref{t:selcriteria} are not
considered in the analysis either because they have been a priori
excluded from the trigger or reveal bad efficiencies\footnote{The
cells 30 and 62 exhibit a lower efficiency as their threshold is
slightly higher than the nominal one. As they contribute significantly
to the statistics of the measurement they have been only excluded for
energies below the threshold.}. The cut on the linear cluster radius
\mbox{$r_{Cl}^{lin} < 3.8\, \mathrm{cm}$} accounts for the
corresponding selection criterion on trigger level L4 (see
table~\ref{t:l4class18}). The additional rejection of events compared
to the cut on the logarithmic cluster radius is negligible.

Only events are accepted for which the distance of the electron track
to the beam axis $R_{e}$ in the plane $z=-160.5\, \mathrm{cm}$ is
larger than \mbox{$10\, \mathrm{cm}$} and \mbox{$9\, \mathrm{cm}$} for
the BST and BDC analysis, respectively\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.}. In the case of
the BST the acceptance gets lower than 50\% for larger polar
angles~\cite{H1:draft}. For the BDC analysis the cut excludes the
region very close to the beam pipe which exhibits energy leakage
effects and a deterioration of the spatial resolution of BDC and
SpaCal. A cut of \mbox{$8.7\, \mathrm{cm}$} has been used in previous
analyses~\cite{Katzy:1997, Meyer:1997}. The contribution of events
affected by leakage effects is suppressed further by restricting the
energy deposition $E_{veto}$ in the veto cells of the SpaCal to \mbox{$1\,
\mathrm{GeV}$}.

In section~\ref{c:onlinesel} it has been mentioned that the SpaCal
cells 7, 8, 13, 14, 32 and 33 (see appendix~\ref{c:spacalcellno}) have
been excluded from the Inclusive Electron Trigger since they cause an
abnormally high trigger rate. This is presumably caused by synchrotron
radiation, the corresponding region in the SpaCal is referred to as
hot spot. In some of the neighbouring cells a significant excess of
the data over the Monte Carlo prediction has been observed in the
course of the analysis. Therefore the region \mbox{$R_{e} > 11.5\,
\mathrm{cm}$} and
\mbox{$-6.5\, \mathrm{cm} < y < 7.5\, \mathrm{cm}$} is not considered and
the cells 4, 5, 6, 11 and 12 are excluded in addition. Finally, cell
29 is excluded since it shows a suspicious energy distribution for
data. The BST analysis excludes some additional
cells~\cite{Vargas:2006}.

All selection criteria concerning the fiducial volume of the analysis
are applied such that Data and Monte Carlo samples are treated
consistently concerning the kinematic acceptance, i.e. it is assured
that they are identical in the beam coordinate system.

\subsection*{Selection Efficiency}

To verify that the different selection criteria affect data and
simulated events in the same way one defines the fraction
$\epsilon_{i}$ of events accepted by a certain cut $i$:
\[ \epsilon^{i} = \frac{N_{all}}{N_{all \setminus i}} \]
The variable $N_{all}$ ($N_{all \setminus i}$) denotes the number of
events accepted by all cuts including (excluding) cut $i$. For the
selection criteria aiming for a reduction of the photoproduction
background one expects $\epsilon^{i}$ to rise with the electron
energy: for large energies the sample is almost background free and
$\epsilon^{i}$ corresponds to the selection efficiency for signal
events, whereas for lower energies more and more background events get
rejected. To obtain a pure signal selection efficiency one determines
the following quantity:
\begin{equation}
\epsilon^{i}_{s} = \frac{N_{all}^{DIS}}{N_{all \setminus i}^{DIS}}
\label{e:seleff}
\end{equation}
Here $N_{all}^{DIS}$ ($N_{all \setminus i}^{DIS}$) denotes the number
of DIS events accepted by all cuts including (excluding) cut $i$. To
test the agreement of data and simulation one compares
$\epsilon^{i}_{s}$ between data and the DJANGO sample. For the DJANGO
sample the number of selected events for the particular cut scenario
directly corresponds to $N_{all}^{DIS}$ and $N_{all \setminus
i}^{DIS}$ as the DJANGO sample contains only signal events. In the
case of the data sample the photoproduction background has to be
subtracted beforehand, i.e. one determines the quantities
\mbox{$N_{all}^{DIS} = N_{all} - N_{all}^{\gamma p}$} and
\mbox{$N_{all \setminus i}^{DIS} = N_{all
\setminus i} - N_{all \setminus i}^{\gamma p}$}. 
$N_{all}^{\gamma p}$ and $N_{all \setminus i}^{\gamma p}$ are
estimated from the PHOJET sample.

The selection efficiency is determined for all combinations of cuts
listed in table~\ref{t:selefflist}. These are all selection criteria
for the BDC analysis listed in table~\ref{t:selcriteria} apart from
the cuts concerning the \mbox{$z$-vertex} and the fiducial volume. The
latter are excluded as the simulation contains neither satellite
bunches nor are the SpaCal cells excluded from the trigger or the L4
cut implemented correctly. The \mbox{$z$-vertex} is the topic of a
separate study in the next section.

For the determination of the selection efficiency all necessary
corrections to the Monte Carlo simulation which are explained in the
forthcoming sections are already applied. The normalisation of the
PHOJET sample is needed as well. Its determination is the topic of the
last section of this chapter.

\begin{table}
\centering
\begin{tabular}{c@{\hspace{2em}}c}
\toprule
name & definition \\
\midrule
cluster radius & $r_{Cl} < 4\, \mathrm{cm}, \hspace{1em} r_{Cl}^{lin} < 3.8\, \mathrm{cm}$ \\
hadronic energy fraction & $f_{had} < 0.15$ \\
veto cell energy & $E_{veto} < 1\, \mathrm{GeV}$ \\
BDC validation & $N_{hits} \ge 4, \hspace{1em} \Delta R < 1.5\, \mathrm{cm}$ \\
$E-p_{z}$ & $E - p_{z} > 35\, \mathrm{GeV}$ \\
$y_{\Sigma}$ & $y_{\Sigma} > 0.03$ \\
elastic QEDC rejection & see QEDC alignment (section~\ref{c:alignment}) \\
\bottomrule
\end{tabular}
\caption{Combinations of cuts for which the selection efficiency according to equation~\ref{e:seleff} is studied.}
\label{t:selefflist}
\end{table}

The figures \ref{f:seleffee} and \ref{f:seleffre} show the selection
efficiencies for data and Monte Carlo as a function of the energy
$E_{e}$ and the distance $R_{e}$, respectively. Compared to
table~\ref{t:selefflist} the cut on the energy deposition in the veto
cells is not considered as it has turned out that this selection
criterion rejects almost no events. Thus its efficiency is
$\epsilon^{i}_{s} \approx 1$ resulting in data and Monte Carlo
agreeing perfectly by definition. The dominant systematic error for
the comparison between data and Monte Carlo arises from the
uncertainty on the PHOJET normalisation. The shaded band shows this
error assuming an uncertainty on the PHOJET normalisation of 15\% (see
section~\ref{c:phojetnorm}). The individual combinations of cuts are
discussed in the following.

\begin{figure}
  \centering
  \includegraphics[width=\textwidth]{controlplots_bdc_060330_1345_cuteffee.eps}
  \caption{Selection efficiency (see equation~\ref{e:seleff}) as a function of the electron energy $E_{e}$ for the combinations of cuts listed in table~\ref{t:selefflist} apart from the cut concerning the veto cell energy. The shaded band indicates the uncertainty on the PHOJET normalisation.}
  \label{f:seleffee}
\end{figure}

\begin{figure}
  \centering
  \includegraphics[width=\textwidth]{controlplots_bdc_060330_1345_cuteffre.eps}
  \caption{Selection efficiency (see equation~\ref{e:seleff}) as a function of the distance $R_{e}$ for the combinations of cuts listed in table~\ref{t:selefflist} apart from the cut concerning the veto cell energy. The shaded band indicates the uncertainty on the PHOJET normalisation.}
  \label{f:seleffre}
\end{figure}

For the selection criteria concerning the electron identification
(cluster radius, ha\-dro\-nic energy fraction, BDC validation) the
efficiency drops for lower electron energies. This is due to the fact
that for low energies, i.e. large values of $y$, particles of the
hadronic final state reach the SpaCal such that the probability
increases that a cluster of the hadronic final state overlaps with the
cluster of the scattered electron. The radial dependent efficiency of
the BDC validation shows a significant drop around \mbox{$R_{e}
\approx 25\, \mathrm{cm}$}. This is an effect resulting from the
changing cell geometry of the BDC as explained in
section~\ref{c:bdceff}. For energies above \mbox{$17.5\,
\mathrm{GeV}$} the cut on \mbox{$E-p_{z} = \Sigma + E_{e}(1-\cos
\theta_{e})$} does not reject any events as it is fulfilled by the
energy of the scattered electron alone. The efficiency of the cut
against elastic QED Compton events exhibits a dip having its minimum
at \mbox{$E_{e} \approx 15\, \mathrm{GeV}$}. This is in agreement with
the energy distributions for electron and photon clusters of QEDC
events shown in~\cite{Lendermann:2001} having their maxima at half the
initial electron energy.

A large discrepancy between data and Monte Carlo is observed for the
radial dependence of the efficiency of the $y_{\Sigma}$ cut from
\mbox{$30\, \mathrm{cm}$} onwards. In this region the polar angle of
the scattered electron becomes large enough to enter the acceptance of
the Central Jet Chamber CJC1 which has the following implications: 
for distances below \mbox{$30\,
\mathrm{cm}$} the presence of a reconstructed vertex implies that the hadronic final state reaches the acceptance of the Central Tracker and thus that $y_{\Sigma}$ is sufficiently large. In contrast, for distances exceeding \mbox{$30\,
\mathrm{cm}$} a Central Tracker vertex can be reconstructed regardless
of the hadronic final state due to the track of the scattered
electron. This transition may not be well-described in the Monte Carlo
simulation. To verify this explanation the selection efficiency
$y_{\Sigma}$ requirement has been determined again for events
containing at least two tracks in the Central Tracker, i.e. at least
one track in addition to the track corresponding to the scattered
electron. Indeed the discrepancy becomes significantly smaller for
this restricted sample as displayed in
figure~\ref{f:seleffysigmancjkt}. The discrepancy has almost no
relevance for the present analysis as it significant only for
distances $R_{e}$ larger than \mbox{$32\, \mathrm{cm}$}. The
contribution of events from this region to the chosen analysis bins is
negligible even for the largest values of $Q^{2}$ considered. Thus the
discrepancy is not considered further.

\begin{figure}
  \begin{minipage}[c]{0.5\textwidth}
    \includegraphics[width=\textwidth]{controlplots_bdc_060330_1346_cuteff_ysigmaee.eps}
    \put(-32,161){a)}
  \end{minipage}
  \begin{minipage}[c]{0.5\textwidth}
    \includegraphics[width=\textwidth]{controlplots_bdc_060330_1346_cuteff_ysigmare.eps}
    \put(-32,161){b)}
  \end{minipage}
  \caption{Selection efficiency (see equation~\ref{e:seleff}) for the cut \mbox{$y_{\Sigma} > 0.03$} as a function of a) the electron energy $E_{e}$ and b) the distance $R_{e}$ for events containing at least two tracks in the Central Tracker. The shaded band indicates the uncertainty on the PHOJET normalisation.}
  \label{f:seleffysigmancjkt}
\end{figure}

At low energies the significance of the comparison of the selection
efficiency is limited by the uncertainty on the PHOJET contribution
which has been statistically subtracted. Therefore it would be
desirable to perform a similar study with a sample of events
containing scattered electrons down to low energies with almost no
background contribution. This is achieved by selecting Initial State
Radiation events with the radiated photon being detected in the photon
arm of the luminosity system, so-called \emph{tagged} ISR events. For
the cross-section determination the detection of the radiated photon
is not required. This extends the accessible phase space significantly
while at the same time increasing the number of photoproduction
background events. As the latter is to be avoided here a signal in the
photon detector is explicitely required for the selection of the event
sample which proceeds as follows:

\begin{itemize}
\item{energy of the electron candidate
\[ E_{e} < 25\, \mathrm{GeV} \]}
\item{inelasticity $y$ as reconstructed by the hadron method
\[ y_{h} < 0.1 \]}
\item{energy deposition in the photon detector, i.e. energy of the radiated photon
\[ E_{\gamma} > 1\, \mathrm{GeV} \]}
\item{energy deposition in the electron tagger
\[ E_{ET} = 0\, \mathrm{GeV} \]}
\item{sum of electron and photon energy
\[ 22\, \mathrm{GeV} < E_{e} + E_{\gamma} < 32\, \mathrm{GeV} \]}
\end{itemize}

The cut \mbox{$E_{e} < 25\, \mathrm{GeV}$} allows for the radiation of
an ISR photon. By accepting only events with \mbox{$y_{h} < 0.1$} one
selects a region of the phase space where ISR events are present (see
section~\ref{c:phasespace}). The hadron method is used to measure the
inelasticity as it reconstructs the event kinematics correctly
regardless of photon radiation from the lepton line.  As explained in
section~\ref{c:phasespace} Bethe-Heitler events overlapping with DIS
events or photoproduction events constitute the dominant background
sources to the tagged ISR events. A non-negligible fraction of these
events can be identified by detecting the scattered electron of the
Bethe-Heitler or the photoproduction event in the electron tagger of
the luminosity system. Hence restricting the energy deposition in the
electron tagger allows to reduce the influence of this background
source~\cite{Issever:2000}. Apart from resolution effects the sum of
the energies of the scattered electron and the radiated photon cannot
be larger than the initial electron energy in the considered region of
the kinematic phase space. This motivates the upper limit of
\mbox{$32\, \mathrm{GeV}$}. Energies \mbox{$E_{e} + E_{\gamma} < 22\,
\mathrm{GeV}$} are not consistent with inelasticities \mbox{$y_{h} <
0.1$}. In addition to these selection criteria for tagged ISR events
the standard selection cuts for the BDC analysis are applied. Only the
$y_{\Sigma}$ cut is relaxed to
\mbox{$y_{\Sigma} > 0.01$} in order to increase the statistics since
the requirements imposed by the ISR selection and the vertex
reconstruction efficiency leave only a small interval of allowed
inelasticities.

The comparison of the selection efficiencies for tagged ISR events
between data and Monte Carlo simulation (DJANGO) is shown in figures
\ref{f:seleffisree} and \ref{f:seleffisrre} for the cuts concerning
the electron identification and BDC validation. The statistical errors
are significantly larger than the ones obtained for the standard event
selection statistically subtracting the photoproduction background
(compare to figures \ref{f:seleffee} and
\ref{f:seleffre}). Nevertheless, the striking difference is the energy
dependence. The decrease of the efficiency towards lower energies is
by far less developed if not absent for the tagged ISR sample. This is
due the fact that no events with large values of $y$ are present such
that overlaps between the clusters of the scattered electron and a
particle of the hadronic final state are not possible.


At low energies the selection efficiency of the cluster radius cut for
the data is systematically smaller than for the DJANGO sample. A
similar observation has been made using the same method to determine
the selection efficiency for the minimum bias run in 1999. A detailed
investigation has revealed the following
explanation~\cite{Glazov:2005c}: the DJANGO generator does not include
radiative effects beyond leading order, i.e. for ISR events no
additional photons are radiated from the lepton line. Concerning the
event kinematics this should not lead to a discrepancy between data
and simulation since an additional ISR photon should not modify the
event kinematics, because the sigma method resums over multiple
emissions. FSR radiation is also effectively resummed as electron and
FSR photon mostly end up in a single cluster. The contribution of QEDC
radiation is negligible. However, for FSR radiation one expects a
larger cluster radius. Since such events are not present in the DJANGO
sample one expects that more events are rejected by the cluster radius
cut in data than in the simulation, i.e. the selection efficiency
should be smaller for the data sample which is exactly the observed
trend. In ~\cite{Glazov:2005c} the efficiency change due to
photon radiation beyond leading order has been estimated\footnote{The
estimation proceeds in the following way: the efficiency of the
cluster radius cut is compared between non-radiative and FSR events
for the DJANGO sample. Determining the fraction of FSR and
non-radiative events in addition the expected change can be
calculated.} to be 1\% which is in agreement with
figure~\ref{f:seleffisree}.

\begin{figure}[t]
  \centering
  \includegraphics[width=\textwidth]{lowe_bdc_060331_1544_ee.eps}
  \caption{Selection efficiency as a function of the electron energy $E_{e}$ as determined in tagged ISR events for some of the cut combinations listed in table~\ref{t:selefflist}.}
  \label{f:seleffisree}
\end{figure}

\begin{figure}[t]
  \centering
  \includegraphics[width=\textwidth]{lowe_bdc_060331_1544_re.eps}
  \caption{Selection efficiency as a function of the distance $R_{e}$ as determined in tagged ISR events for some of the cut combinations listed in table~\ref{t:selefflist}.}
  \label{f:seleffisrre}
\end{figure}

In general both comparisons of the cut efficiencies between data and
Monte Carlo simulation have shown good agreement apart from the
$y_{\Sigma}$ cut which has been discussed in detail. The remaining
differences are covered by an uncertainty. For its determination the
distance of the DJANGO prediction to the PHOJET uncertainty band
depicted in figures \ref{f:seleffee} and
\ref{f:seleffre} is summed up quadratically for each combination of
cuts. If the DJANGO prediction lies inside the band a distance of zero
is taken. The accumulated difference as a function of $E_{e}$ and
$R_{e}$ is well described by a global uncertainty of 1.5\%. In the case of
$R_{e}$ this holds of course only for the region relevant for the
analysis. Accordingly, a global systematic error of 1.5\% is taken
into account for the cross-section determination.

For the selection efficiencies derived by subtracting the
photoproduction background statistically the $E-p_{z}$ cut has been
included. The study has been repeated excluding it to obtain the
selection efficiencies for the region of the phase space where the
events are reconstructed with the help of the sigma method. The
agreement between data and Monte Carlo is of the same quality.


\section{Vertex Reconstruction} \label{c:vertexreco}

A detailed comparison of the vertex reconstruction in data and Monte
Carlo simulation is performed in the following: at first the fraction
of $ep$ events which should enter the analysis, but are lost due to
the vertex requirement is investigated by determining the \emph{vertex
efficiency}. A difference between data and Monte Carlo simulation
would be reflected directly in a wrong cross-section
determination\footnote{The selection efficiency cannot be determined,
like for most of the other selection criteria, by just relaxing the
vertex requirement as has been explained in the previous
section.}. Secondly, the acceptance of the detector components depends
on the vertex position. Consequently, also the region of the kinematic
phase space corresponding to the fiducial volume of the analysis
depends on the vertex position. Hence the influence of the limited
detector acceptance and the fiducial cuts can be only modelled
correctly in the Monte Carlo simulation if the spatial distribution of
the vertices is the same as for the data sample. To ensure this the
\mbox{$z$-vertex} distribution of the Monte Carlo simulation has to be
reweighted to agree with the one observed in the data.

\subsection*{Vertex Reconstruction Efficiency}

To determine the efficiency of the vertex requirement one selects a
reference sample of events being classified as $ep$ events which
should enter the analysis whereby the vertex is reconstructed
independently of the Central Tracker. The fraction of events for which
a Central Tracker vertex exists is the vertex efficiency:
\begin{equation}
\epsilon_{vtx} = \frac{N_{BST/CIP\, \mathrm{val.}\; \&\&\; \Delta z_{BST/CIP} < 25\, \mathrm{cm} \; \&\&\; \Delta z_{CT} < 30\, \mathrm{cm}\; \&\&\; \sigma_{z_{CT}} < 8\, \mathrm{cm}}}{N_{BST/CIP\, \mathrm{val.}\; \&\&\; \Delta z_{BST/CIP} < 25\, \mathrm{cm}}}
\label{e:vtxeff}
\end{equation}
For the reference sample the vertex is determined with the help of the
BST or the CIP for which the vertex reconstruction solely relies on
the scattered electron. The event is accepted if the electron
candidate is validated by the BST\footnote{The validation by the BST
implies meeting the track validation criteria as given in
table~\ref{t:selcriteria}.} or the CIP and if the \mbox{$z$-vertex}
position fulfils the condition \mbox{$\Delta z = |z_{vtx} -
\overline{z_{vtx}}\, \mathrm{cm}| < 25\, \mathrm{cm}$}. BST and CIP
are used in parallel as the BDC analysis extends into a region of
small polar angles of the scattered electron which exceeds the
acceptance of the BST. In this region the CIP is able to reconstruct
the event vertex. The BST vertex is preferred, only in case it does
not exist the CIP vertex is considered. To enter the reference sample
the event has to pass the selection criteria concerning cluster
properties, QED Compton events and fiducial cuts in addition (see
table~\ref{t:selcriteria}).

The Central Tracker vertex requirement comprises the
\mbox{$z$-coordinate} to lie within \mbox{$30\, \mathrm{cm}$} of the
nominal position and the error of the \mbox{$z$-coordinate} not to
exceed \mbox{$8\, \mathrm{cm}$} (see table~\ref{t:selcriteria}). The
allowed interval of the \mbox{$z$-position} for the BST and CIP is
smaller than the one for the Central Tracker to avoid accounting
resolution effects as inefficiency.

The Central Tracker vertex reconstruction is predominantly based on
tracks caused by hadronic final state particles. Only at sufficiently
small polar angles the scattered electron enters the acceptance of the
CJC. The lower limit of $25^{\circ}$ for the polar angle acceptance of
the Central Tracker (see section~\ref{c:h1overview}) during standard
running conditions corresponds to roughly $45^{\circ}$ for the shifted
vertex period. Therefore one expects Central Tracker vertices to exist
for events with hadron angles larger than this value. Accordingly, the
vertex efficiency should rise steeply as a function of $y$ starting at
\mbox{$y \approx 0.01$} (see figure~\ref{f:ctacceptance}). Hence the
vertex efficiency is investigated as a function of $y$ in intervals of
$Q^{2}$.

\begin{figure}
  \centering
  \includegraphics[width=0.6\textwidth]{ctacceptance.eps}
  \caption{Lines of constant hadron angle $\theta_{h}$ (dashed), inelasticity $y$ (solid) and distance $R_{e}$ (dotted) in the \mbox{$xQ^{2}$-plane}. Hadron angles of $45^{\circ}$ and $165^{\circ}$ roughly correspond to the lower and upper polar angle acceptance limit of the Central Tracker.}
  \label{f:ctacceptance}
\end{figure}

The kinematic variables are determined with the help of the sigma
method. If the electron method was used, $y$ would be reconstructed
systematically too large for ISR events (see
section~\ref{c:radcor}). Events at low $y$ would be misinterpreted as
events at high $y$, such that artificial inefficiencies would emerge
at large inelasticities.

Figure~\ref{f:vtxeffysigma} shows the vertex reconstruction efficiency
for data and Monte Carlo simulation. The intervals in $Q_{\Sigma}^{2}$
have been chosen such that they contain almost the same number of
events. In the case of the Monte Carlo simulation the contributions of
the DJANGO and PHOJET sample have been added according to their
luminosity and normalisation\footnote{The PHOJET normalisation is
discussed in the last section of this chapter.}, respectively. Data
and Monte Carlo simulation show the expected steep rise of
$\epsilon_{vtx}$ with $y_{\Sigma}$. At \mbox{$y_{\Sigma} = 0.03$} the
efficiency exceeds 50\% for all $Q^{2}$ bins. This is the lower limit
of $y_{\Sigma}$ for events to be accepted for the final analysis as
already mentioned in the previous section (see
table~\ref{t:selcriteria}). However, at large values of $y_{\Sigma}$
the efficiency decreases again, the drop being more pronounced for the
data sample. The difference between data and Monte Carlo gets smaller
with increasing $Q_{\Sigma}^{2}$. The efficiency drop is unexpected
since for typical DIS events the region between the current jet and
the proton remnant is filled with particles resulting from colour
forces and gluon radiation, i.e. even if the hadron angle exceeds the
upper acceptance limit\footnote{For standard running conditions the
upper acceptance limit is $155^{\circ}$ (see
section~\ref{c:tracking}).} of the Central Tracker of $165^{\circ}$ a
vertex should be present.

\begin{figure}
  \centering
  \includegraphics[width=\textwidth]{zvertexeffct_bst_060630_1318_effysigma.eps}
  \caption{Vertex reconstruction efficiency according to equation~\ref{e:vtxeff} as a function of $y_{\Sigma}$ in intervals of $Q_{\Sigma}^{2}$. The dashed lines correspond to \mbox{$y_{\Sigma} = 0.03$}.}
  \label{f:vtxeffysigma}
\end{figure}

ISR events should not be the reason for the drop of $\epsilon_{vtx}$
at large values of $y$ as discussed above. The same holds for elasti
QEDC events since they are excluded from the analysis due to the
corresponding cut. In principle they can cause artificial
inefficiencies in this regime: though being characterised by \mbox{$y
\approx 0$}, their reconstructed $y_{\Sigma}$ can be large if the
photon is treated as a particle of the hadronic final state. The
reason for this is that the photon is scattered into the backward
region of the H1 detector causing a large $\Sigma$ and consequently a
large $y_{\Sigma}$ (see equation~\ref{e:ysigma}).


A typical event with a large inelasticity but no reconstructed Central
Tracker Vertex is depicted in figure~\ref{f:evdisnovertex}. It has
passed all selection criteria of the reference sample. Apart from the
scattered electron a hadronic cluster can be identified in the
SpaCal. No activity is observed in the central and forward part of the
detector. Thus the event is characterised by a gap in the polar angle
distribution, or equivalently the pseudorapidity $\eta$ of the final
state particles\footnote{The pseudorapidity is defined as \mbox{$\eta
= -\ln \tan \frac{\theta}{2}$}, positive (negative) values of $\eta$
correspond to small (large) values of $\theta$, i.e. the forward
(backward) region of the detector.}. Such events can be interpreted as
being diffractively produced. Early measurements of H1 and ZEUS have
shown that \emph{diffractive events} constitute about 10\% of all DIS
events~\cite{Derrick:1993xh,Ahmed:1994nw}. The rapidity gap causes the
event to have no reconstructed Central Tracker vertex while at the
same time a large $y_{\Sigma}$ due to the hadron in the SpaCal.

\begin{figure}
  \centering
  \includegraphics[width=0.8\textwidth]{278698_47605_polished.eps}
  \caption{A typical event with \mbox{$y_{\Sigma} > 0.5$} which has no reconstructed Central Tracker vertex. Run and event number are 278698 and 47605, respectively.}
  \label{f:evdisnovertex}
\end{figure}

A difference in the event reconstruction efficiency between data and
Monte Carlo simulation at large values of $y$ has been already
reported in~\cite{Adloff:1997mf}. The reason for this is the absence
of events with large rapidity gaps in the DJANGO Monte Carlo
model. Dedicated simulation studies using generators for diffractive
events have been performed to derive corrections to the predictions of
DJANGO. The most important contribution to this corrections is the
elastic production of low mass vector mesons ($\rho$, $\omega$,
$\phi$). The corrections for the analysis of the previous shifted
vertex run in 1995 covering the \mbox{$Q^{2}$-range} \mbox{$0.35\,
\mathrm{GeV^{2}} < Q^{2} < 3.5\, \mathrm{GeV^{2}}$} can be found in
appendix B of~\cite{Katzy:1997}. They reach up to 7\% increasing with
$y$ and decreasing with $Q^{2}$ and thus are in agreement with the
difference in the vertex efficiency between data and Monte Carlo
observed here. The decrease of the correction with $Q^{2}$ is due to
the fact that the cross-section for the elastic $\rho$ meson
production falls off significantly more rapidly with $Q^{2}$ than the
inclusive DIS cross-section.

To check that the fraction of events with large rapidity gaps is
indeed too small in the DJANGO sample one determines the distribution
of the maximum pseudorapidity $\eta_{max}$ being defined as
\begin{equation}
\eta_{max} = \max_{i}(-\ln \tan \frac{\theta_{i}}{2})
\label{e:etamax}
\end{equation}
The variable $\eta_{max}$ is the pseudorapidity of the cluster in the
LAr or SpaCal with the largest pseudorapidity, not considering the
scattered electron. For non-diffractive events it is expected to lie
near the forward acceptance limit \mbox{$\eta_{max} \approx 3$} of the
LAr calorimeter as the region between the current jet and the proton
remnant is filled with particles. In the case of diffractive events
$\eta_{max}$ can be lower, for the elastic production of vector mesons
it reaches the acceptance region \mbox{$\eta_{max} < -1.8$} of the
SpaCal. Figure~\ref{f:etamax} compares the $\eta_{max}$ distributions
of data and Monte Carlo simulation for \mbox{$y_{\Sigma} > 0.5$}, the
contribution of PHOJET is indicated separately. The majority of the
events is caused by non-diffractive processes. The fraction of
diffractive events in the Monte Carlo sample is in fact too small to
describe the data, supporting the explanation of the inefficiencies in
the previous paragraphs.

\begin{figure}
\begin{captionbeside}{Distribution of $\eta_{max}$ (see equation~\ref{e:etamax}) for events with \mbox{$y_{\Sigma} > 0.5$}. \vspace{1.2cm}}[i][0.95\textwidth][0.05\textwidth]*
\begin{minipage}[t]{0.5\textwidth}
  \includegraphics[width=\textwidth]{zvertexeffct_bst_060630_1318_etamax.eps}
\end{minipage}
\end{captionbeside}
\label{f:etamax}
\end{figure}

To achieve a correct cross-section determination the Monte Carlo
sample has to be corrected for the differences in the vertex
reconstruction efficiency. Therefore the ratio
\begin{equation}
c_{vtx} = \frac{\epsilon_{vtx}^{Data}}{\epsilon_{vtx}^{MC}}
\label{e:vtxcor}
\end{equation}
of the efficiencies of data and Monte Carlo sample is calculated as a
function of $y_{\Sigma}$ in the same $Q_{\Sigma}^{2}$ intervals as for
figure~\ref{f:vtxeffysigma}. Afterwards the obtained data points in
each interval are parameterised with a second order polynomial for the
\mbox{$y_{\Sigma}$-range} relevant for the measurement
(\mbox{$y_{\Sigma} > 0.03$}). The resulting values for each of the
three parameters are in turn plotted as a function of the average
$Q_{\Sigma}^{2}$ in the corresponding interval and also fitted by a
second order polynomial. Thus one ends up up with a continuous
correction function \mbox{$c_{vtx}(Q_{\Sigma}^{2},y_{\Sigma})$} which
has 9 parameters and is applied as an additional weight to each Monte
Carlo event. This is equivalent to the application of an appropriate
correction $c_{\epsilon}$ in equation~\ref{e:sigmamcm}. Using a
function for the reweighting instead of the data points for $c_{vtx}$
directly has the advantage to extrapolate into regions with low
statistics rather than to create statistical fluctuations.

Figure~\ref{f:vtxcorysigma} depicts $c_{vtx}$ as a function of
$y_{\Sigma}$ in the interval \mbox{$1.5\, \mathrm{GeV^{2}} <
Q_{\Sigma}^{2} < 2\, \mathrm{GeV^{2}}$}. Apart from the data points
the original fit to this interval is shown as well as the projection
of the 2-dimensional function $c_{vtx}(Q_{\Sigma}^{2},y_{\Sigma})$,
i.e. \mbox{$c_{vtx}(Q_{\Sigma}^{2},y_{\Sigma})$} for the average
$Q_{\Sigma}^{2}$ of the interval. Original fit and extrapolation agree
to better than 1\% apart from the lowest values of $y_{\Sigma}$ where
the errors are large. Of all $Q_{\Sigma}^{2}$ intervals the chosen one
shows the largest deviation.

\begin{figure}
\centering
\begin{minipage}[t]{0.5\textwidth}
  \includegraphics[width=\textwidth]{zvertexeffct_bst_060630_1318_corysigma.eps}
\end{minipage}
\caption{Ratio $c_{vtx}$ of the vertex reconstruction efficiency of
data and Monte Carlo (see equation~\ref{e:vtxcor}) as a function of
$y_{\Sigma}$ for events with \mbox{$1.5\, \mathrm{GeV^{2}} <
Q_{\Sigma}^{2} < 2\, \mathrm{GeV^{2}}$}. The solid line represents a
second order polynomial fitted to the data points for
\mbox{$y_{\Sigma} > 0.03$}, the dashed line the projection of the
2-dimensional correction function
\mbox{$c_{vtx}(Q_{\Sigma}^{2},y_{\Sigma})$} (for details see
text).}\label{f:vtxcorysigma}
\end{figure}



To estimate the systematic error of the cross-section emerging from
the uncertainty on the vertex efficiency correction the latter is
applied as a function of $E_{e}$. Again intervals in $Q_{\Sigma}^{2}$
are used to take the $Q^{2}$ dependence into account, but no fit is
performed. Instead, the data points for $c_{vtx}$ are used directly
for the reweighting of the Monte Carlo samples. When deriving these
corrections the $y_{\Sigma}$ cut of the final analysis has been
applied.  The resulting cross-section is compared to the one obtained
for the default correction procedure in each final analysis bin. The
differences depend on $Q^{2}$ and are considered as systematic errors
for the cross-section determination as listed in
table~\ref{t:vtxcoruncer}.

\begin{table}
\centering
\begin{tabular}{c@{\hspace{2em}}c}
\toprule
$Q^{2}$ range & systematic error \\
\midrule
$Q^{2} \le 0.35\, \mathrm{GeV^{2}}$ & 10\% \\
$0.5\, \mathrm{GeV^{2}} \le Q^{2} \le 0.85\, \mathrm{GeV^{2}}$ & 5\% \\
$Q^{2} \ge 1.2\, \mathrm{GeV^{2}}$ & 2\% \\
\bottomrule
\end{tabular}
\caption{Systematic error of the cross-section emerging from the uncertainty on the vertex efficiency correction. The $Q^{2}$ values given refer to the central values of the analysis bins for which the cross-section is quoted.}
\label{t:vtxcoruncer}
\end{table}

\subsection*{Vertex Position}

The spatial distribution of the vertices is studied using the standard
selection criteria for the BDC analysis (see
table~\ref{t:selcriteria}) apart from the requirements concerning the
vertex position and the BDC track validation. The track validation is
omitted to avoid a possible bias since the BDC electron finder uses
the vertex position as an input. Therefore the SpaCal is used to
determine the position of the scattered electron. Additional cuts are
applied as explained in the following:
\begin{itemize}
\item{For large \mbox{$z$-coordinates} of the event vertex lower
values of $Q^{2}$ can be accessed than for smaller ones which may lead
to a bias of the vertex distribution.  The latter can be diminished by
restricting the polar angle of the electron candidate to
\mbox{$165.5^{\circ} < \theta_{e} < 175.5^{\circ}$}. For all events
fulfilling \mbox{$\Delta z = |z_{vtx} - \overline{z_{vtx}}\,
\mathrm{cm}| < 40\, \mathrm{cm}$} this cut ensures a SpaCal acceptance
of 100\% for the scattered electron\footnote{To be more precise, the
restriction of the polar angle implies that all scattered electrons
satisfy \mbox{$15\, \mathrm{cm} < R_{e} < 70\, \mathrm{cm}$}. The
upper limit is motivated by the SpaCal acceptance limit, the lower one
covers all fiducial cuts.}.}
\item{The requirement \mbox{$E_{e} > 15\, \mathrm{GeV}$} excludes the
part of the energy spectrum for which the photoproduction background
is large.}
\item{It has been shown in section~\ref{c:dissel} that the Monte Carlo
simulation has problems to describe the vertex reconstruction at large
polar angles where the scattered electron defines the Central Tracker
vertex. In order not to be influenced by this effect only events are
accepted which have at least two reconstructed tracks in the Central
Tracker.}
\end{itemize}
The vertex efficiency correction which has been determined above are
applied for this study.

At first the time dependence of the mean $m(z_{vtx})$ and the spread
$s(z_{vtx})$ of the \mbox{$z$-position} of the event vertices for the
data sample is investigated. For this purpose the \mbox{$z$-vertex}
distribution for each luminosity run is parameterised by a
Gaussian. Figure~\ref{f:vtxstab} displays the extracted fit parameters
as a function of a continuous run number\footnote{The original
sequence of the runs is conserved.}. The dashed lines separate the
different luminosity fills. Mean and spread vary by about \mbox{$4\,
\mathrm{cm}$} between the different runs and show a systematic
development within a luminosity fill. For the last luminosity fill a
particularly large spread $s(z_{vtx})$ is observed.

\begin{figure}
  \begin{minipage}[c]{0.5\textwidth}
    \includegraphics[width=\textwidth]{zvertexrew_ct_060502_1330_zvtxmean.eps}
    \put(-33,161){a)}
  \end{minipage}
  \begin{minipage}[c]{0.5\textwidth}
    \includegraphics[width=\textwidth]{zvertexrew_ct_060502_1330_zvtxspread.eps}
    \put(-33,161){b)}
  \end{minipage}
  \caption{a) Mean $m(z_{vtx})$ and b) spread $s(z_{vtx})$ of the \mbox{$z$-position} of the Central Tracker vertices as a function of a continuous run number. The dashed lines separate the different luminosity fills.}
  \label{f:vtxstab}
\end{figure}

The spread $s(z_{vtx})$ is compatible with the longitudinal size of
the luminous region of approximately \mbox{$10\, \mathrm{cm}$} (see
section~\ref{c:stability}). It increases in the course of each fill as
the proton bunches get larger due to intrabeam scattering and noise of
the radiofrequency being responsible for the acceleration. A
measurement of the evolution of the proton bunch length with time
during a typical HERA operation cycle is presented
in~\cite{Montag:2000}.

Despite the variations of mean position and spread the
\mbox{$z$-vertex} distribution of the entire data sample can be well
parameterised by the sum of a Gaussian and a first order polynomial
(see figure~\ref{f:vtxrew}~a)). The fit shows that the contribution of
the polynomial is almost negligible and provides a mean
\mbox{$z$-position} \mbox{$\overline{z_{vtx}} = \zvtxmean$} of the Central Tracker vertices.

\begin{figure}
  \begin{minipage}[c]{0.5\textwidth}
    \includegraphics[width=\textwidth]{zvertexrew_ct_060502_1330_zvtxdist.eps}
    \put(-33,161){a)}
  \end{minipage}
  \begin{minipage}[c]{0.5\textwidth}
    \includegraphics[width=\textwidth]{controlplots_bdc_060423_2120_zvtxrew.eps}
    \put(-33,161){b)}
  \end{minipage}
  \caption{a) \mbox{$z$-vertex} distribution for data and DJANGO along with a parameterisation by the sum of a Gaussian and a first order polynomial indicated by the solid and dashed line, respectively. The Monte Carlo prediction is scaled by a factor of 0.8 for better visibility. b) Comparison of the \mbox{$z$-vertex} distribution in data and Monte Carlo (DJANGO + PHOJET) with and without applying the weight  $w_{vtx}$ (see equation~\ref{e:vtxrew}) to the Monte Carlo events.}
  \label{f:vtxrew}
\end{figure}

For the cross-section determination it would be ideal to perform the
Monte Carlo simulation with the \mbox{$z$-vertex} distribution
obtained from data. However, in practise this is difficult to
achieve. For example, the final run selection for the data sample is
usually fixed only after the Monte Carlo samples have already been
produced. Therefore the Monte Carlo events are reweighted to bring the
distributions to agreement. For this purpose the \mbox{$z$-vertex}
distribution of the DJANGO sample is subjected to the same fit as the
data (see figure~\ref{f:vtxrew}~a)). The weight $w_{vtx}$ to be applied
to each Monte Carlo event is given by:
\begin{equation}
w_{vtx}(z_{vtx}) = a_{0} \frac{N^{data}(z_{vtx})}{N^{MC}(z_{vtx})}
\label{e:vtxrew}
\end{equation}
Here $N^{data}$ ($N^{MC}$) denotes the number of events at a certain
\mbox{$z$-vertex} position as given by the fit to the data (Monte
Carlo) distribution. Since cuts for a clean electron identification
are applied only the DJANGO sample can be used for the extraction of
the reweighting function. For the final analysis also the PHOJET
sample has to be considered. As this has been simulated with the same
vertex parameters as the DJANGO sample the same reweighting is
applied. Introducing a suitable constant $a_{0}$ ensures that the
total number of events for the DJANGO sample is conserved under the
reweighting. For the PHOJET sample no special care is necessary in
this respect, because the normalisation is performed with the
\mbox{$z$-vertex} reweighting function already applied.

Figure~\ref{f:vtxrew}~b) shows a comparison of the \mbox{$z$-vertex}
distribution of the Monte Carlo simulation with and without
reweighting to the distribution obtained for data for all events
passing the cuts of the final analysis. For both Monte Carlo
distributions the contributions of the DJANGO and PHOJET samples have
been added according to their luminosity and
normalisation\footnote{The PHOJET normalisation is discussed in the
last section of this chapter.}, respectively. Since the agreement of
the distributions in shape is the figure of merit for this comparison
the Monte Carlo distributions have been renormalised to contain the
same number of events as the distribution for the data sample. As
expected, the agreement between data and Monte Carlo is almost perfect
if the reweighting function is applied. It is important that the
difference to the distribution without reweighting is small. Otherwise
large weights would have to be applied implying larger statistical
errors for the Monte Carlo samples.

In the course of the investigations concerning the beam tilt
implementation in the Monte Carlo simulation (see
section~\ref{c:simulation}) it has been checked that the $x$- and
\mbox{$y$-positions} of the vertices for the data sample are well
reproduced by the Monte Carlo simulation.


\section{BDC Efficiency} \label{c:bdceff}

The matching of the SpaCal cluster with hits in the BDC has two
important functions for the analysis: on the one hand it improves the
precision of the polar angle measurement $\theta_{e}$ of the scattered
electron. On the other hand it rejects photoproduction events:
Non-showering neutral particles have no associated track. Furthermore
hadron induced showers exhibit larger fluctuations of the deposited
energy than electron induced ones. Therefore the cut on the track
cluster matching \mbox{$\Delta R < 1.5\, \mathrm{cm}$} should reduce
the probability for a hadronic final state particle to be
misidentified as the scattered electron (see section~\ref{c:dissel}).

Consequently, two aspects of the linking of SpaCal clusters and BDC
tracks have to be studied: the efficiency with which the BDC validates
the scattered electron and the rejection power for background
events. The former is called \emph{BDC efficiency} in the following as
it should directly reflect the operational state of the chamber.

To determine the BDC efficiency a sample of events is selected for
which a misidentification of the scattered electron is almost
impossible: only electron candidates with energies above \mbox{$20\,
\mathrm{GeV}$} are accepted. In addition the standard selection
criteria (see table~\ref{t:selcriteria}) are applied omitting the BDC
validation. This implies that the scattered electron is reconstructed
based on the SpaCal alone. The fraction of events for which the BDC
requirements \mbox{$N_{hits} \ge 4$} and \mbox{$\Delta R < 1.5\,
\mathrm{cm}$} are fulfilled corresponds to the BDC efficiency
$\epsilon_{BDC}$. Vertex efficiency correction and \mbox{$z$-vertex}
reweighting as derived in the last section are applied for this study.

Figure~\ref{f:bdceff} shows $\epsilon_{BDC}$ as a function of the
distance $R_{e}$ of the electron cluster to the beam axis in the SpaCal
plane for data and Monte Carlo simulation. Only the DJANGO sample is
considered as there is no contribution from photoproduction events for
\mbox{$E_{e} > 20\, \mathrm{GeV}$}. Three essential differences between 
data and Monte Carlo are observed which is discussed in the following:
The efficiency of the data sample decreases to 90\% at \mbox{$R_{e}
\approx 25\, \mathrm{cm}$}. This dip is not reproduced in the simulation. 
For small $R_{e}$ the efficiency is significantly overestimated by the
simulation, while for \mbox{$R_{e} > 30\, \mathrm{cm}$} is slightly
underestimated.

\begin{figure}
\begin{captionbeside}{BDC efficiency as a function of the distance $R_{e}$. The dashed lines denote the region affected by the transition cells. \vspace{1.2cm}}[i][0.95\textwidth][0.05\textwidth]*
\begin{minipage}[t]{0.5\textwidth}
  \includegraphics[width=\textwidth]{bdceff_060427_0038_eff.eps}
\end{minipage}
\end{captionbeside}
\label{f:bdceff}
\end{figure}

An efficiency drop or/and a discrepancy between data and Monte Carlo
simulation in the region around \mbox{$R_{e} \approx 25\,
\mathrm{cm}$} has been reported before~\cite{Schwab:1996, Katzy:1997,
Meyer:1997}. In summary two possible explanations have been discussed:
An insufficient description of the dead material in front of the BDC
in the simulation or problems with the track reconstruction in the
region where the BDC cell size changes. For a comparison with the
present study it has to be taken into account that a different track
reconstruction algorithm has been used for the previous analyses and
that the description of the dead material in the backward region of
the H1 detector has meanwhile significantly
improved~\cite{Lendermann:1999}.

For the bulk of the events without a BDC validation in the problematic
region there are less than the required four hits associated to the
electron candidate whereas the requirement concerning $\Delta R$ is
fulfilled. Therefore resolution effects due to interactions of the
scattered electron with dead material seem to be less likely the
reason for the inefficiency.

As mentioned in section~\ref{c:tracking} the radial cell size of the
BDC changes from \mbox{$1\, \mathrm{cm}$} in the inner region to
\mbox{$3\, \mathrm{cm}$} in the outer region. The intermediate region
in each sector is covered by a \emph{transition cell} of \mbox{$2\,
\mathrm{cm}$} size. Taking the chamber geometry into account it can be
derived from figure~\ref{f:bdcrz} that the transition cells of all
sectors affect the interval \mbox{$21\, \mathrm{cm} < R_{BDC} < 27\,
\mathrm{cm}$}. The variable $R_{BDC}$ denotes the distance from the
origin of the BDC in the \mbox{$xy$-plane} at \mbox{$z=-145\,
\mathrm{cm}$}, i.e. at the mean
\mbox{$z$-position} of the BDC. A projection to the SpaCal in
consideration of the possible \mbox{$z$-positions} of the vertex leads
to \mbox{$23\, \mathrm{cm} < R_{e} < 29\, \mathrm{cm}$}. This region
is indicated by dashed lines in figure~\ref{f:bdceff}. A clear
coincidence with the efficiency drop is observed.

The transition cells are asymmetric, i.e. the wire is not centred
w.r.t. to the cell in the radial direction (see
figure~\ref{f:bdcrz}). Furthermore the electric drift field of the
transition cells is simultaneously influenced by the high voltage
supplies of the inner and the outer half-sectors. Hence the transition
cells have a special field configuration. It has been observed that
the drift time spectra after the calibration are of worse quality than
the ones for the standard cells~\cite{Katzy:2006}. If the
reconstruction algorithm is not optimised to account for these effects
they may lead to inefficiencies which are difficult to model in the
Monte Carlo simulation.

The simulation predicts lower efficiencies for the outer half-sectors
than for the inner ones. A similar feature for the data has been
observed in~\cite{Katzy:1997}. Obviously the Monte Carlo simulation has
been optimised for the reconstruction algorithm used therein while the
BDC electron finder BDCLEV does not show such a behaviour.

It is evident that the Monte Carlo events have to be reweighted for
the BDC efficiency of the data sample to be described
adequately. Similar to the vertex reconstruction efficiency the
correction
\begin{equation}
c_{BDC} = \frac{\epsilon_{BDC}^{Data}}{\epsilon_{BDC}^{MC}}
\label{e:bdccor}
\end{equation}
is determined for this purpose. The design of the chamber in sectors
suggests to investigate also the azimuthal variation of the efficiency
in addition to the radial one. Thus the correction $c_{BDC}$ is
determined as a function of $R_{e}$ in four different
\mbox{$\phi$-sectors} as depicted in figure~\ref{f:bdccor}. For
\mbox{$R_{e} < 30\, \mathrm{cm}$} the corrections are derived in
intervals of $1\, \mathrm{cm}$, in particular to model the strongly
changing efficiency in the region of the transition cell. Above
\mbox{$30\, \mathrm{cm}$} larger intervals are used not to be
dominated by statistical fluctuations.

\begin{figure}[t]
  \centering
  \includegraphics[width=\textwidth]{bdceff_060427_0038_cor.eps}
  \caption{Ratio $c_{BDC}$ of the BDC efficiency of data and Monte Carlo (see equation~\ref{e:bdccor}) as a function of the distance $R_{e}$ in quadrants as indicated.}
  \label{f:bdccor}
\end{figure}

All quadrants show the same general trend of the corrections, but
there are significant differences. The most striking one is quadrant
II showing almost no correction at low radii while revealing a
particularly large correction in the region of the transition cell
compared to the other quadrants. As the efficiency
$\epsilon_{BDC}^{MC}$ looks almost the same for all quadrants the
differences can be attributed to the data sample. Therefore it has
been checked whether a correlation to the operational conditions of
the chamber can be found. Figure~\ref{f:bdchv} gives an overview of
the detector status in the year 2000. The highlighted sectors have not
been operated for various reasons~\cite{Bdc:2000}.

\begin{figure}
  \centering
  \includegraphics[width=\textwidth]{deadsectors.eps}
  \caption{Operational status of the BDC in the year 2000~\cite{Bdc:2000}. Sectors not operated are highlighted.}
  \label{f:bdchv}
\end{figure}

A clear correlation between the detector status and $c_{BDC}$ can be
observed if one assumes that the dead sectors are not considered
during the Monte Carlo simulation: in contrast to all others quadrant
II is not affected by a single dead inner sector such that it is
expected to show the smallest correction at lower radii. The
efficiency in the region of the transition cell seems to be most
affected by the status of the outer sectors. Quadrant II is influenced
by two dead outer sectors, quadrants I and III by one each and quadrant
IV by none. Accordingly, quadrant II shows the largest and quadrant IV
the smallest correction in the region of the transition cell.

Using $c_{BDC}$ in quadrants as an additional weight for the Monte
Carlo events ensures that the BDC efficiency is correctly
modelled. This is equivalent to the application of an appropriate
correction $c_{\epsilon}$ in equation~\ref{e:sigmamcm}.

In a second step it has to be checked that the background rejection
power of the BDC selection criteria agrees sufficiently between data
and Monte Carlo simulation for the whole energy range of the final
analysis. This has been done when determining the efficiency of the
selection criteria in section~\ref{c:dissel}. The BDC efficiency
correction has been already applied for that study. Correspondingly,
the uncertainty on the BDC efficiency is already contained in the
uncertainty of the selection efficiency determined to 1.5\%.


\section{PHOJET Normalisation} \label{c:phojetnorm}

If the DIS event selection criteria are applied photoproduction events
are the dominant source of the remaining background. Though their
virtuality $Q^{2}$ is so small that the scattered electron escapes
detection in the backward beam pipe they enter the final event sample
as a particle of the hadronic final state can mimic the scattered
electron (see section~\ref{c:background}).

To determine the DIS cross-section the photoproduction background is
statistically subtracted from the total number of events observed in
the data sample using the PHOJET Monte Carlo model (see
section~\ref{c:crosssection}). For the quantity $N_{\gamma
p}^{\square}$ in equation~\ref{e:sigmamcm} simply to be the number of
PHOJET events passing the selection criteria in
table~\ref{t:selcriteria} the PHOJET sample has to be normalised
w.r.t. the data sample. The straightforward approach for the
normalisation would be to calculate the luminosity of the PHOJET
sample from the photoproduction cross-section and the number of PHOJET
events and to compare it to the luminosity of the data sample. It
requires all event properties on which cuts are placed to be perfectly
described in the simulation which is not the case. This can be seen
from figure~\ref{f:seleffee} which shows that the discrepancy between
data and Monte Carlo simulation for the selection efficiency is
largest at low energies where the PHOJET contribution is sizeable.

Therefore one performs the normalisation by selecting events of the
data sample which can be unambiguously identified as stemming from a
photoproduction process and comparing them to the corresponding events
in the PHOJET sample. In order to reduce the sensitivity of the
normalisation on the selection criteria they are chosen to be as close
as possible to the ones for the final analysis. Two approaches to
identify photoproduction events are discussed: one employs the
electron tagger of the luminosity system, the other the charge of the
electron candidate.

\subsection*{Tagged Photoproduction Events}


As the scattered electron in photoproduction events leaves the main
detector through the backward beam pipe it can be identified with a
certain probability in the electron tagger of the luminosity system
(see section~\ref{c:lumisystem}). Events for which this is the case
are called \emph{tagged events} in the following. On the way from the
interaction region to the electron tagger at \mbox{$z = -33.4\,
\mathrm{m}$} the beam guiding magnets bend the trajectory of the
electrons. This leads to a strong dependence of the tagger acceptance
$A_{ET}$ on the electron energy and thus the inelasticity $y$. The
tagger acceptance has been determined separately for the shifted
vertex period with the help of Bethe-Heitler events (see
section~\ref{c:radepscat}) in which the scattered electron and the
emitted photon are detected in the electron tagger and the photon
detector of the luminosity system,
respectively~\cite{Andreev:1996}\footnote{The
citation~\cite{Andreev:1996} refers to the old electron tagger of the
luminosity system locates at
\mbox{$z = -44\, \mathrm{m}$}. However, the principle of the
acceptance determination is still the same.}. From the result shown in
figure~\ref{f:phonormetag} a) it can be inferred that the scattered
electron can be identified only in a narrow inelasticity band
\mbox{$0.3 < y < 0.6$} with a reasonable probability. This is the
reason why a comparison to a Monte Carlo model is necessary: a direct
estimation of the photoproduction background from tagged events in the
data sample would involve large extrapolation errors due to the
limited tagger acceptance.

\begin{figure}
  \begin{minipage}[c]{0.5\textwidth}
    \includegraphics[width=\textwidth]{etagacc.eps}
    \put(-33,161){a)}
  \end{minipage}
  \begin{minipage}[c]{0.5\textwidth}
    \includegraphics[width=\textwidth]{nomphojet_bdc_060510_1925_eetag.eps}
    \put(-33,161){b)}
  \end{minipage}
  \caption{a) Electron tagger acceptance $A_{ET}$ as a function of the inelasticity $y$. b) Energy spectrum of the scattered electrons reconstructed in the electron tagger. The PHOJET distribution has been renormalised to contain the same number of events as the distribution for data.}
  \label{f:phonormetag}
\end{figure}

The following selection criteria have to be fulfilled for tagged
events: to ensure a reasonable acceptance the reconstructed energy in
the electron tagger is required to lie within an interval of
\mbox{$7\, \mathrm{GeV} < E_{ET} < 16.5\, \mathrm{GeV}$}. All DIS
selection criteria as listed in table~\ref{t:selcriteria} are imposed
apart from the \mbox{$E - p_{z}$} cut. The latter has to be omitted
for a suitable amount of events to remain in the sample of tagged
events given the limited acceptance. Instead cuts are placed on the
quantity
\begin{equation}
(E - p_{z})_{tot} = E - p_{z} + 2 E_{ET} + 2 E_{PD}
\label{e:empztot}
\end{equation}
taking apart from the particles in the main detector (\mbox{$E -
p_{z}$}) also the energy depositions in the electron tagger ($E_{ET}$)
and the photon detector ($E_{PD}$) into account. If no losses occur
\mbox{$(E - p_{z})_{tot}$} should be equal to \mbox{$2E_{e}^{0} =
55.2\, \mathrm{GeV}$} for photoproduction events due to energy
and momentum conservation. Therefore only events with $(E - p_{z})_{tot} >
45\, \mathrm{GeV}$ are accepted for this study.

Also coincidences of DIS and Bethe-Heitler events pass the selection
criteria mentioned so far and would constitute a sizeable background
contribution for the tagged photoproduction sample due to the large
rate of Bethe-Heitler interactions. As the quantity \mbox{$(E -
p_{z})_{tot}$} for overlapping events can reach up to twice the
nominal value they can be efficiently rejected by applying the cut
\mbox{$(E - p_{z})_{tot} < 70\, \mathrm{GeV}$} (for details
see~\cite{Meyer:1997}). Requiring a maximum energy deposition of
\mbox{$2\, \mathrm{GeV}$} in the photon detector further suppresses
these events.

To avoid energy leakage over the detector boundaries a cut is placed
on the \mbox{$x$-co}\-or\-di\-nate of the electron impact point w.r.t. the
centre of the electron tagger for the data sample: \mbox{$|x_{ET}| <
6.5\, \mathrm{cm}$}. Furthermore one asks for the trigger element
115 to be set which requires an energy deposition in the electron
tagger within a certain range. The same cuts have been applied during
the determination of the tagger acceptance.

In the software package H1SIM responsible for the simulation of the H1
detector the electron tagger is not included. This implies that the
latter two selection criteria cannot be applied to the PHOJET sample.
To describe the tagger response the acceptance function $A_{ET}$
determined from data as depicted in figure~\ref{f:phonormetag} a) is
used to reweight the Monte Carlo events according to the generated
inelasticity.


If one applies all mentioned selection criteria as well as the
acceptance reweighting one observes that for about 10\% of the PHOJET
events in the final analysis the scattered electron can be identified
in the electron tagger\footnote{Vertex efficiency correction,
\mbox{$z$-vertex} reweighting and BCD efficiency correction as derived
in the previous sections are applied for this study as
well.}. Figure~\ref{f:phonormetag} shows the energy distribution of the
scattered electron for data and simulation. The PHOJET distribution
has been renormalised to contain the same number of events as the
distribution for data. Both distributions agree reasonably well.

Thus tagged events seem to be well-suited to normalise the number of
PHOJET events entering the cross-section determination w.r.t. the data
sample. To be more precise, the quantity $N_{\gamma p}^{\square}$ in
equation~\ref{e:sigmamcm} is calculated for each analysis bin
$\square$ in $x$ and $Q^{2}$ or $y$ and $Q^{2}$ according
to\footnote{Details concerning the definition of the analysis bins can
be found in section~\ref{c:binning}.}:
\begin{equation}
N_{\gamma p}^{\square} = \frac{N_{Data,tag}}{N_{PHOJET,tag}} N_{PHOJET}^{\square}
\label{e:phonormglo}
\end{equation}
The quantity $N_{PHOJET}^{\square}$ denotes the number of PHOJET
events passing the DIS selection criteria as given in
table~\ref{t:selcriteria} for the analysis bin $\square$, whereas
\mbox{$N_{Data,tag}$} and \mbox{$N_{PHOJET,tag}$} correspond to total the
number of tagged events in the data and PHOJET sample,
respectively. As outlined above the latter have passed the same
selection criteria as $N_{PHOJET}^{\square}$ apart from omitting the
$E-p_{z}$ cut and applying additional cuts to identify them as
tagged. Therefore the normalisation should be almost insensitive to
differences in the distributions of variables on which cuts are placed
(e.g. cluster radius) between data and PHOJET.

Using $N_{\gamma p}^{\square}$ to statistically subtract the photoproduction
background in the described way works correctly only if the PHOJET
sample reproduces the kinematics of photoproduction events in data
reasonably well. This can be checked with the help of tagged
events. Figure~\ref{f:phonormfakee} shows the distributions of the
energy $E_{e}$ and the distance from the beam axis $R_{e}$ of the
hadronic final state particle being misidentified as the scattered
electron. While the simulation almost perfectly describes the energy
distribution it underestimates the number of photoproduction events at
low $R_{e}$ and overestimates it at large $R_{e}$. The discrepancy is
especially large at \mbox{$R_{e} \approx 12\, \mathrm{cm}$}. This
behaviour is reflected in the distributions of the virtuality $Q^{2}$
and Bjorken $x$ and would thus lead to a wrong estimation of the
photoproduction background.

\begin{figure}
  \begin{minipage}[c]{0.5\textwidth}
    \includegraphics[width=\textwidth]{nomphojet_bdc_060510_1925_efakee.eps}
    \put(-33,161){a)}
  \end{minipage}
  \begin{minipage}[c]{0.5\textwidth}
    \includegraphics[width=\textwidth]{nomphojet_bdc_060510_1925_rfakee.eps}
    \put(-33,161){b)}
  \end{minipage}
  \caption{Distributions of a) the energy $E_{e}$ and b) the distance $R_{e}$ for the hadrons misidentified as scattered electrons in tagged events. The PHOJET distributions have been renormalised to contain the same number of events as the distributions for data.}
  \label{f:phonormfakee}
\end{figure}

Therefore the PHOJET normalisation is performed in each analysis bin
separately, i.e. the number of tagged events is counted for each bin
and equation \ref{e:phonormglo} is changed to
\begin{equation}
N_{\gamma p}^{\square} = \frac{N_{Data,tag}^{\square}}{N_{PHOJET,tag}^{\square}} N_{PHOJET}^{\square}
\label{e:phonormbin}
\end{equation}
In this way the local differences between PHOJET and data are
corrected at the price of an increased statistical error compared to
the global normalisation. If an analysis bin contains less than three
tagged events for the data sample the global normalisation factor is
used instead. This imposes no consistency problems on the analysis as
the photoproduction background in these bins is anyhow negligible.

The bin-wise PHOJET normalisation is used only for the final
cross-section measurement. For the determination of the vertex
efficiency correction and the cut efficiencies the global one has been
used for simplicity which should be sufficient. 




The global normalisation factor \mbox{$N_{Data,tag}/N_{PHOJET,tag}$}
for the BST analysis is approximately 8\% larger than for the BDC
analysis. One possible explanation for this is the following: before
being detected in the BDC particles have passed significantly more
dead material than in the case of the BST. If this additional material
including CIP, CIZ and CJC readout electronics and cables is not
correctly modelled in the Monte Carlo simulation the conversion rate
of the decay photons from $\pi^{0}$ mesons is different between data
and Monte Carlo sample. This difference which is absorbed and thus
corrected for in the normalisation factor affects the BDC but not the
BST analysis.

The distribution of $R_{e}$ for the BST analysis shows a better
agreement between data and simulation than the one in
figure~\ref{f:phonormfakee} for the BDC
analysis~\cite{Vargas:2006}. This justifies a global normalisation of
the PHOJET sample. Nevertheless, for consistency reasons the bin-wise
normalisation is used for the comparison of the results of the BST and
BDC analyses.

\subsection*{Normalisation Uncertainty}

For the standard BST analysis only the signals in the
\mbox{$r$-sensors} are employed which allow to measure the distance of
the hits from the \mbox{$z$-axis} in the \mbox{$r\phi$-plane} and
finally the polar angle $\theta_{e}$ of the scattered electron. One of
the 16 BST \mbox{$\phi$-sectors} is equipped with \mbox{$u$-sensors}
in addition (see section~\ref{c:tracking}). By combining the
information of both sensor types it is possible to reconstruct the
\mbox{$3$-dimensional} coordinates of the hits. From these momentum
and charge of the corresponding tracks can be derived.

Knowing the charge of the BST tracks corresponding to the electron
candidates allows to determine the number of photoproduction
background events: in the shifted vertex period positrons have been
accelerated in the HERA ring. Hence, tracks belonging to a scattered
electron should indicate a particle with a positive charge
(\emph{right charge tracks}). Tracks indicating a particle with a
negative charge (\emph{wrong charge tracks}) point to the presence of
a background event. Moreover, assuming a charge symmetry of the
background events the number of events with a wrong charge track gives
an estimate of the remaining background in the correct charge
sample. If a charge asymmetry is present, this has to be taken into
account.

Using this method it is in principle possible to extract the number of
photoproduction events $N_{\gamma p}^{\square}$ for each analysis bin from
the data sample alone. However, this would result in a large
statistical uncertainty on $N_{\gamma p}^{\square}$ as \mbox{$u$-sensors}
have been installed in only one of the 16 \mbox{$\phi$-sectors} of the
BST and thus their acceptance is rather small. Instead the charge
measurement is used to cross-check the PHOJET normalisation for the
BST analysis determined with the help of tagged events and to estimate
its uncertainty.


Such a cross-check has been performed first in~\cite{Eckstein:2002}
for the minimum bias run in 1999. It has been repeated for the same
data set in~\cite{H1:draft}. The results of the latter are summarised
shortly in the following: all events $n_{acc}$ in the acceptance of
the \mbox{$u$-sensors} passing the nominal analysis cuts are
classified as events with right charge ($n_{+}$), wrong charge
($n_{-}$) or events without a linked track from the \mbox{$u$-sensors}
($n_{0}$). The difference of $n_{acc}$ and the number of signal events
$n_{sig}$ is equal to the total amount of background events $n_{\gamma
p}$ in the \mbox{$u$-sensor} acceptance:
\begin{equation}
n_{\gamma p} = n_{acc} - n_{sig} = n_{acc} - \frac{n_{+} - A_{ch} n_{-}}{\epsilon_{u}}
\label{e:phonormusensors}
\end{equation}
In this equation $A_{ch}$ corresponds to the charge asymmetry and
$\epsilon_{u}$ to the efficiency to link a track defined by the
\mbox{$u$-sensors} to the standard track of the BST electron finder
using the information of the \mbox{$r$-sensors}. The charge asymmetry
is defined as \mbox{$A_{ch} = N_{+,\gamma p} / N_{-,\gamma p}$},
i.e. the fraction of the number of background events with a right and
wrong charge track. It has been determined to \mbox{$A_{ch} = 0.90 \pm
0.10$}~\cite{H1:draft}. The different amount of events with positively
and negatively charged tracks is caused by the fact that the
cross-section and released energy for antiproton interactions exceeds
that for proton interactions. Hence the antiproton annihilation leads
to larger energy depositions in the SpaCal than proton interactions
resulting in an asymmetry if the number of events above a certain
threshold is counted~\cite{Adloff:2000qk}.

In order to verify the PHOJET normalisation obtained from the tagged
events the quantity $n_{\gamma p}$ as given by
equation~\ref{e:phonormusensors} is compared between the data and the
PHOJET sample applying the global normalisation factor
\mbox{$N_{Data,tag}/N_{PHOJET,tag}$} to the PHOJET prediction. The
resulting ratio amounts to $R = 1.004 \pm 0.145_{stat}
\pm 0.05_{asym}$, where the first error denotes the statistical
uncertainty while the latter corresponds to the uncertainty of the
charge asymmetry. Hence the ratio indicates an almost perfect
agreement, but is afflicted with a large uncertainty of approximately
15\% dominated the statistical error of $n_{\gamma p}$. Therefore an
uncertainty of 15\% on the PHOJET normalisation is assumed for the
cross-section determination based on the BST. The same uncertainty is
used for the BDC analysis.


