\graphicspath{{figures/results/}}

\chapter{Results of the Measurement} \label{c:results}

After defining the analysis bins the present chapter shows that the
corrections derived in the last two chapters lead to a level of
agreement between data and simulation in the analysed region of the
phase space that allows for a cross-section determination using the
Monte Carlo method (see section~\ref{c:crosssection}). Subsequently,
all identified uncertainties on the measurement are summarised and
their influence on the cross-section is quantified. The final outcome
of the BDC analysis for the shifted vertex data is then presented: the
double differential inclusive DIS cross-section and the structure
function $F_{2}$. The results are compared to the BST
analysis~\cite{Vargas:2006}, preliminary
results~\cite{Lastovicka:2002, Petrukhin:2004a}, various published
measurements and theoretical predictions.

\section{Definition of the Analysis Bins} \label{c:binning}


For the analysis at hand data and simulated events are binned in the
same way as for the H1 publication~\cite{Adloff:2000qk} to simplify
comparisons. In parts of the phase space the binning is also similar
to that used for the measurement based on the shifted vertex run
from 1995~\cite{Adloff:1997mf}. It is a combination of bins in $x$ and
$Q^{2}$ and in $y$ and $Q^{2}$ (see figure \ref{f:stability} or
\ref{f:purity}). In the bulk of the phase space bins with a rectangular
shape in the \mbox{$xQ^{2}$-plane} are used. The basic grid has five
bins per decade in $x$ and eight bins per decade in $Q^{2}$. Several
bins of the basic grid in $x$ are combined to one final analysis bin
where necessary, according to the criteria given below. The
contribution of the longitudinal structure function $F_{L}$ to the
cross-section is proportional to $y^{2}$. Hence a fine binning in $y$
is preferable at high $y$ to isolate bins with a higher sensitivity to
$F_{L}$. At \mbox{$y = 0.6$} the transition between the two binning
schemes takes place. Bins with a special shape are used in this region
to avoid double counting and losses. The choice of the bin size is
determined by the following considerations:
\begin{itemize}
\item{The bins should be small enough in order to permit a measurement of the double differential cross-section at as many points as possible and large enough to accumulate reasonable statistics. The statistical error in each bin should be smaller than the systematic one.}
\item{The bin size has to be adjusted to the resolution of the kinematic variables. A grid that is too small can lead to large bin-to-bin migrations which can influence the results of the measurement.}
\end{itemize}

\begin{figure}
  \centering
  \includegraphics[width=0.8\textwidth]{binstat_060703_2020_stab.eps}
  \put(-257,541){a)}
  \put(-257,281){b)}
  \caption{Stability $S$ (according to equation~\ref{e:stability}) in per cent for each analysis bin in the \mbox{$xQ^{2}$-plane} for a) the electron and b) the sigma method. Apart from the kinematic limit \mbox{$y=1$} (thick solid lines) the line corresponding to \mbox{$y=0.1$} (dotted lines) and the acceptance limit imposed by the cut $R_{e} > 9\, \mathrm{cm}$ (dashed lines) are indicated for non-radiative events. For each reconstruction method the bins contributing to the cross-section measurement are highlighted.}
  \label{f:stability}
\end{figure}

\begin{figure}
  \centering
  \includegraphics[width=0.8\textwidth]{binstat_060703_2020_puri.eps}
  \put(-257,541){a)}
  \put(-257,281){b)}
  \caption{Purity $P$ (according to equation~\ref{e:purity}) in per cent for each analysis bin in the \mbox{$xQ^{2}$-plane} for a) the electron and b) the sigma method. Apart from the kinematic limit \mbox{$y=1$} (thick solid lines) the line corresponding to \mbox{$y=0.1$} (dotted lines) and the acceptance limit imposed by the cut $R_{e} > 9\, \mathrm{cm}$ (dashed lines) are indicated for non-radiative events. For each reconstruction method the bins contributing to the cross-section measurement are highlighted.}
  \label{f:purity}
\end{figure}

To control the migration effects two quantities are determined in each
bin for the DJANGO Monte Carlo sample, the \emph{stability} $S$ and
the \emph{purity} $P$. They are defined as the ratio of the number of
events generated and reconstructed\footnote{An event is considered as
reconstructed if it passes all selection criteria listed in
table~\ref{t:selcriteria}.} in a bin to the total number of events
generated (reconstructed) in that bin:
\begin{align}
S & = \frac{N_{MC,gen\; \&\&\; rec}^{\square}}{N_{MC,gen}^{\square}} \label{e:stability} \\
P & = \frac{N_{MC,gen\; \&\&\; rec}^{\square}}{N_{MC,rec}^{\square}} \label{e:purity}
\end{align}
The stability specifies the probability that an event generated in a
given bin is also reconstructed in that bin, the purity the
probability that an event reconstructed in a given bin has been
generated in the same one. Hence the stability expresses the migration
out of the given bin while the purity quantifies the migration of
events from adjacent bins into that bin. Both variables can only take
values between 0 and 1 by definition.

Figures \ref{f:stability} and \ref{f:purity} show the stability and
purity for the electron and sigma methods in the \mbox{$xQ^{2}$}
plane. As expected from the discussion of the resolution of the
different reconstruction methods in section~\ref{c:evkindet} the
electron method yields better results at large values of $y$. Values
of $S$ and $P$ above 50\% are typically reached for \mbox{$y>0.1$}. At
lower values of $y$ the sigma method provides larger values of
stability and purity since the resolution of the electron method
degrades as $1/y$ towards low $y$.

Only the sigma method covers the region of the phase space dominated
by ISR events, which are located at lower $Q^{2}$ and larger $x$ than
the acceptance limit of the BDC analysis for non-radiative events
imposed by the requirement $R_{e} > 9\, \mathrm{cm}$ (dashed
lines). In the case of the electron method ISR events are strongly
suppressed by the cut
\mbox{$E - p_{z} > 35\, \mathrm{GeV}$}. However, some of the bins
beyond the acceptance limit are also covered by the electron
method. This is due to the fact that the dashed line corresponds to
the acceptance limit for the average \mbox{$z$-vertex} position of the
shifted vertex run of $\overline{z_{vtx}} = \zvtxmean$. As
\mbox{$z$-positions} of the interaction vertex of up to
$\overline{z_{vtx}} + 30\, \mathrm{cm}$ are accepted for the analysis,
slightly lower values of $Q^{2}$ can also be reached for non-radiative
events.

Bins are only considered for the cross-section measurement if they
have stability and purity larger than 25\%. For bins in which both
reconstruction methods fulfil this requirement the method yielding the
maximum value of
\begin{equation}
W_{i} = \frac{S_{i}}{S_{e} + S_{\Sigma}} + \frac{P_{i}}{P_{e} + P_{\Sigma}}
\end{equation}
is used for the cross-section determination. Here $i$ corresponds
either to the electron or the sigma method. The bins contributing to
the cross-section measurement for each reconstruction method are
highlighted in figures \ref{f:stability} and \ref{f:purity}. The
transition from the usage of the electron method to the sigma method
takes place at \mbox{$y \approx 0.1$}.

\section{Comparison of Data and Monte Carlo Simulation}

\begin{figure}
  \begin{minipage}[c]{0.5\textwidth}
    \includegraphics[width=\textwidth]{controlplots_bdc_inbins_comb_060703_2020_ee.eps}
    \put(-157,161){a)}
  \end{minipage}
  \begin{minipage}[c]{0.5\textwidth}
    \includegraphics[width=\textwidth]{controlplots_bdc_inbins_comb_060703_2020_phie.eps}
    \put(-157,161){b)} 
  \end{minipage} 
  \vskip 0.2cm
  \begin{minipage}[c]{0.5\textwidth}
    \includegraphics[width=\textwidth]{controlplots_bdc_inbins_comb_060703_2020_thetae.eps}
    \put(-157,161){c)} 
  \end{minipage}
  \begin{minipage}[c]{0.5\textwidth}
    \includegraphics[width=\textwidth]{controlplots_bdc_inbins_comb_060703_2020_re.eps}
    \put(-157,161){d)} 
  \end{minipage} 
  \vskip 0.2cm
  \begin{minipage}[c]{0.5\textwidth}
    \includegraphics[width=\textwidth]{controlplots_bdc_inbins_comb_060703_2020_zvtx.eps}
    \put(-157,161){e)} 
  \end{minipage}
  \begin{minipage}[c]{0.5\textwidth}
    \includegraphics[width=\textwidth]{controlplots_bdc_inbins_comb_060703_2020_ptbal.eps}
    \put(-157,161){f)} 
  \end{minipage} 
  \caption{Control distributions of basic observables: a) - d) energy $E_{e}$, azimuthal angle $\phi_{e}$, polar angle $\theta_{e}$ and distance $R_{e}$ of the electron candidate, e) \mbox{$z$-position} of the interaction vertex $z_{vtx}$, f) ratio $p_{t,bal}$ of the transverse momenta of the hadronic final state and the scattered electron. Only events from bins contributing to the cross-section determination (see figures \ref{f:stability} and \ref{f:purity}) are considered.}
  \label{f:controlbasic}
\end{figure}

\begin{figure}
  \begin{minipage}[c]{0.5\textwidth}
    \includegraphics[width=\textwidth]{controlplots_bdc_inbins_elec_060703_2020_empz.eps}
    \put(-157,161){a)}
  \end{minipage}
  \begin{minipage}[c]{0.5\textwidth}
    \includegraphics[width=\textwidth]{controlplots_bdc_inbins_sig_060703_2020_empz.eps}
    \put(-157,161){b)}
  \end{minipage}
  \caption{Distribution of $E-p_{z}$ for events from bins contributing to the cross-section determination covered by a) the electron method and b) the sigma method (see figures \ref{f:stability} and \ref{f:purity}).}
  \label{f:controlempz}
\end{figure}

\begin{figure}
  \begin{minipage}[c]{0.5\textwidth}
    \includegraphics[width=\textwidth]{controlplots_bdc_inbins_elec_060703_2020_q2e.eps}
    \put(-157,161){a)}
  \end{minipage}
  \begin{minipage}[c]{0.5\textwidth}
    \includegraphics[width=\textwidth]{controlplots_bdc_inbins_sig_060703_2020_q2s.eps}
    \put(-157,161){b)}
  \end{minipage}
  \vskip 0.2cm
  \begin{minipage}[c]{0.5\textwidth}
    \includegraphics[width=\textwidth]{controlplots_bdc_inbins_elec_060703_2020_ye.eps}
    \put(-157,161){c)}
  \end{minipage}
  \begin{minipage}[c]{0.5\textwidth}
    \includegraphics[width=\textwidth]{controlplots_bdc_inbins_sig_060703_2020_ys.eps}
    \put(-157,161){d)}
  \end{minipage}
  \vskip 0.2cm
  \begin{minipage}[c]{0.5\textwidth}
    \includegraphics[width=\textwidth]{controlplots_bdc_inbins_elec_060703_2020_xe.eps}
    \put(-157,161){e)}
  \end{minipage}
  \begin{minipage}[c]{0.5\textwidth}
    \includegraphics[width=\textwidth]{controlplots_bdc_inbins_sig_060703_2020_xs.eps}
    \put(-157,161){f)}
  \end{minipage}
  \caption{Control distributions of the kinematic variables $x$, $y$ and $Q^{2}$. In the left (right) column the variables have been reconstructed with the electron (sigma) method and only events are considered from bins which contribute to the cross-section determination and are covered by the electron (sigma) method (see figures \ref{f:stability} and \ref{f:purity}).}
  \label{f:controlkin}
\end{figure}

Figures \ref{f:controlbasic} to \ref{f:controlkin} demonstrate the
agreement between data and Monte Carlo simulation by showing control
distributions for the basic quantities reconstructed in the
analysis. The sum of the predictions of DJANGO for the DIS signal and
PHOJET for the photoproduction background is compared to the data. The
contribution of the former is normalised to the data according to the
luminosities of the DJANGO sample and the data sample. The
normalisation of PHOJET proceeds as described in
section~\ref{c:phojetnorm}. The control distributions for the basic
observables are depicted in figures
\ref{f:controlbasic} and \ref{f:controlempz}. From these the kinematic
variables used for the cross-section determination are derived (see
figure~\ref{f:controlkin}). The distributions in
figure~\ref{f:controlbasic} contain contributions from all analysis
bins used for the cross-section determination (see figure
\ref{f:stability} or \ref{f:purity}). Figure~\ref{f:controlempz} shows 
the $E-p_{z}$ spectrum separately for bins covered a) by the electron
method and b) by the sigma method. The distributions of
figure~\ref{f:controlkin} comprise only analysis bins to which events
have been assigned by the same method that has been used to
reconstruct the displayed variable.

For the basic observables (see figure~\ref{f:controlbasic}) data and
Monte Carlo simulation in general agree well. The distribution of the
azimuthal angle $\phi_{e}$ reflects the complicated fiducial volume of
the analysis. Two bins at $\phi_{e} \approx 60^{\circ}$ and $\phi_{e}
\approx 270^{\circ}$ show an excess and deficit of data compared to
the Monte Carlo prediction, respectively. Single SpaCal cells being
responsible for this could not be identified. The same holds for the
excess of data observed at $R_{e} \approx 12\, \mathrm{cm}$
corresponding to $\theta_{e} \approx 177.2^{\circ}$. The latter does
not occur at the acceptance edge and is reflected in a rather
wide-spread excess of the data over the simulation in the distribution
of $Q_{e}^{2}$ around \mbox{$1.5\, \mathrm{GeV^{2}}$}. Hence it is
likely to be caused by a slightly different behaviour of the structure
functions from that assumed in the simulation and not by a detector
effect.

Figure~\ref{f:controlbasic} f) depicts the distribution of the ratio
$p_{t,bal}$ of the transverse momenta of the hadronic final state and
the electron candidate. The maximum is located near one, which one would
expect from energy and momentum conservation. The large width is due to
the fact that the analysis focuses on events with low values of
$Q^{2}$. Hence the transverse momenta are typically small and small
fluctuations of the measured energies can have large effects on the
ratio. The good agreement between data and Monte Carlo simulation
indicates that the calibrations of the electromagnetic and hadronic
energies are consistent. This is particularly important for the use of
the sigma method.

The spectrum of $E-p_{z}$ for the analysis bins which use the sigma
method extends below the lower limit of $35\, \mathrm{GeV}$ required
in the case of the electron method (see
figure~\ref{f:controlempz}). The ISR events are concentrated in this
region of the distribution. They are accompanied by a large fraction
of photoproduction events since the latter are also characterised by
values of $E-p_{z}$ well below the nominal value of
\mbox{$2E_{e}^{0} = 55.2\, \mathrm{GeV}$}.

For all kinematic variables the Monte Carlo simulation nicely
describes the data as can be seen from figure~\ref{f:controlkin}. Only
non-radiative events contribute to the distributions of $Q_{e}^{2}$,
$y_{e}$ and $x_{e}$ while the spectra of $Q_{\Sigma}^{2}$,
$y_{\Sigma}$ and $x_{\Sigma}$ contain ISR events in addition (see
figures \ref{f:stability} and \ref{f:purity}). This explains why two
separate parts can be distinguished in figures~\ref{f:controlkin} b),
d) and f). The ISR events are located at low $Q_{\Sigma}^{2}$, high
$y_{\Sigma}$ and low $x_{\Sigma}$.


\section{Systematic Uncertainties}

All uncertainties associated with the various aspects of the
measurement which have been identified in the last two chapters are
listed in table~\ref{t:uncertainties}. For comparison the
corresponding uncertainties of the BST analysis are specified in
addition. The references indicate where the uncertainties have been
derived or where additional information about them can be
found. Each uncertainty is the source of a systematic error to be quoted
for the cross-section determination. The calculation of these errors
is the topic of the present section.

\begin{sidewaystable}
\centering
\begin{tabular}{c@{\hspace{2em}}c@{\hspace{2em}}C{3cm}@{\hspace{2em}}C{3cm}@{\hspace{2em}}c}
\toprule
type & source & BST analysis & BDC analysis & reference \\
\midrule
 & scattered electron energy scale & \multicolumn{2}{c}{$1\%$ at $E_{e} = 2\, \mathrm{GeV}$} & section~\ref{c:calibration}, \cite{Vargas:2006}  \\
 & & \multicolumn{2}{c}{$0.2\%$ at $E_{e} = 27.6\, \mathrm{GeV}$} & \\
 & scattered electron polar angle & $0.2\, \mathrm{mrad}$ & $0.5\, \mathrm{mrad}$ & \cite{H1:draft}, section~\ref{c:alignment} \\
 & hadronic energy scale LAr and CT & \multicolumn{2}{c}{$10\%$ at $y = 10^{-3}$} & section~\ref{c:calibration}, \cite{Vargas:2006} \\
\raisebox{1.5ex}[-1.5ex]{correlated} & & \multicolumn{2}{c}{$2\%$ at $y = 10^{-2}$} & \\
 & LAr noise & \multicolumn{2}{c}{$10\%$} & section~\ref{c:calibration}, \cite{Vargas:2006} \\
 & hadronic energy scale SpaCal & \multicolumn{2}{c}{$350\, \mathrm{MeV}$} & section~\ref{c:calibration}, \cite{Vargas:2006} \\
 & PHOJET normalisation & \multicolumn{2}{c}{$15\%$} & section~\ref{c:phojetnorm}, \cite{H1:draft} \\
\cmidrule{1-5}
 & DJANGO statistics & & & \\
uncorrelated & PHOJET statistics & & & \\
 & radiative corrections & \multicolumn{2}{c}{$0.5\%$} & section~\ref{c:radcor}, \cite{Glazov:2005b} \\
\cmidrule{1-5}
 & luminosity & \multicolumn{2}{c}{\lumierr} & \cite{Levonian:2006} \\
 & online event selection & $0.9\%$ & $0.7\%$ & section~\ref{c:onlinesel} \\
normalisation & BST efficiency & $2\%$ & - & \cite{H1:draft} \\
 & selection efficiency & - & $1.5\%$ & section~\ref{c:dissel} \\
 & vertex efficiency correction & - & see table~\ref{t:vtxcoruncer} & section~\ref{c:vertexreco} \\
\bottomrule
\end{tabular}
\caption{Sources of systematic errors for the BST and BDC analyses.}
\label{t:uncertainties}
\end{sidewaystable}

The systematic uncertainties are classified in three different
categories as proposed in~\cite{Adloff:2000qk}: \emph{correlated
systematic errors}, \emph{uncorrelated systematic errors} and global
\emph{normalisation uncertainties} (see
table~\ref{t:uncertainties}). Together with the statistical
uncertainty of the data sample they constitute the total error of the
measurement.

Sources of systematic errors classified as correlated affect the
cross-section measurement in different analysis bins in a correlated
way. The different sources are considered to be uncorrelated among
one another. The distinction of the different error types is important
for fits to the data and comparisons with other measurements. For some
errors the classification is straightforward: the statistical
uncertainty of the Monte Carlo samples is a typical uncorrelated
error, the energy scale of the scattered electron is a typical source
of a correlated error since the latter affects different bins in a
similar way. In contrast, it is rather arbitrary to treat the
uncertainty on the radiative corrections as a source of an
uncorrelated systematic error.

To estimate the correlated systematic errors one proceeds in the
following way: for each of the sources the relevant quantities are
varied by the uncertainty, separately in positive and negative
directions. The reduced cross-sections for the former
($\sigma_{r}^{+}$) and latter case ($\sigma_{r}^{-}$) are redetermined
and the relative systematic error $\delta$ resulting from the source
under consideration is calculated as
\begin{equation}
\delta = \frac{\sigma_{r}^{-} - \sigma_{r}^{+}}{2\, \sigma_{r}^{0}}
\label{e:deltacorr}
\end{equation}
where $\sigma_{r}^{0}$ denotes the reduced cross-section at the
central value. This implies that the whole analysis chain is executed
twice for each correlated error source.

The treatment of the uncorrelated and normalisation uncertainties is
less complex: the errors resulting from the limited DJANGO and PHOJET
statistics are calculated analytically. The quoted uncertainty for the
radiative corrections already corresponds to the systematic error on
the cross-section without further manipulation. The same holds for all
normalisation uncertainties.

The table in appendix~\ref{c:tofresults} provides a detailed overview
of the measurement errors in each analysis bin. All quoted errors are
relative errors. The correlated systematic uncertainties are listed
individually, adopting the sign convention of
equation~\ref{e:deltacorr}, and are added in quadrature to calculate
their cumulated contribution $\delta_{cor}$. The quantity
$\delta_{unc}$ contains all uncorrelated systematic errors as well as
the normalisation uncertainties. Adding $\delta_{cor}$, $\delta_{unc}$
and the statistical error of the data $\delta_{stat}$ in quadrature
yields the total error $\delta_{tot}$.

The error on the cross-section determination is clearly dominated by
the systematic errors. Only for very few bins at low $Q^{2}$ the
statistical error exceeds the systematic one. This is the reason why
so much emphasis has been placed on an accurate determination of the
measurement uncertainties. Among the systematic errors the
uncorrelated and normalisation errors typically exceed the correlated
ones.

For bins with central values of $Q^{2}$ larger than
\mbox{$0.85\, \mathrm{GeV^{2}}$} the total errors vary in general 
between 4\% and 5\%. Only in the bins at lowest $x$ the errors are
larger due to the correlated error $\delta_{\gamma p}$ resulting from
the uncertainty on the PHOJET normalisation. This is expected since
small values of $x$ correspond to large values of $y$ and thus low
energies $E_{e}$, where the PHOJET contribution is large (see
figure~\ref{f:controlbasic}). The largest correlated error in the bins
at higher $x$ is caused by the uncertainty on the polar angle
measurement. At lower values of $Q^{2} < 0.85\, \mathrm{GeV^{2}}$ the
total errors reach 20\%, dominated either by the statistical
error or the uncertainty on the vertex efficiency correction.

The uncertainties of the measurement affect the cross-section obtained
with the electron and the sigma method differently. In order to judge
to which extent the uncertainties are controlled the results of the
two methods are compared before combining
them. Figure~\ref{f:sigmarelecvssigma} shows the reduced cross-section
$\sigma_{r}$ for the electron and the sigma method as a function of
$x$ for various values of $Q^{2}$. It has been determined at the
central values in $x$ and $Q^{2}$ for each bin according to
equation~\ref{e:sigmamcm}. The result is only displayed if stability
and purity exceed 25\% for the respective method. Statistical and
systematic errors are indicated separately. The good agreement between
the two methods shows that the uncertainties of the measurement are
well controlled.

\begin{figure}[t]
  \centering
  \includegraphics[width=\textwidth]{extractf2_060703_2020_sr_binwise_elec_vs_sigma.eps}
  \caption{Comparison of the reduced cross-section $\sigma_{r}$ as a function of $x$ for various values of $Q^{2}$ measured with the electron (full points) and the sigma (open points) method. The inner error bars represent the statistical errors, the outer error bars indicate the statistical and systematic errors added in quadrature.}
  \label{f:sigmarelecvssigma}
\end{figure}


\section{Cross-section Determination}

In order to derive the final cross-section to be quoted for the BDC
analysis the values obtained with the electron and the sigma method
are combined based on stabilities and purities as explained in
section~\ref{c:binning}. The result is depicted in
figure~\ref{f:sigmar} as a function of $x$ in different bins of
$Q^{2}$ between \mbox{$0.2\, \mathrm{GeV^{2}}$} and
\mbox{$3.5\, \mathrm{GeV^{2}}$}. Statistical and total errors are 
indicated separately. The values of the reduced cross-section are also
listed in the table of appendix~\ref{c:tofresults}.

\begin{figure}
  \centering
  \includegraphics[width=\textwidth]{extractf2_060703_2020_sr_binwise.eps}
  \caption{Measurement of the reduced cross-section $\sigma_{r}$ as a function of $x$ for various values of $Q^{2}$. The inner error bars represent the statistical errors, the outer error bars indicate the statistical and systematic errors added in quadrature. The data are compared with a phenomenological prediction for $\sigma_{r}$ based on the ALLM97 parameterisation~\cite{Abramowicz:1997ms} for $F_{2}$ and the saturation model prediction~\cite{Golec-Biernat:1998js} for $F_{L}$ (solid line). The dashed line shows the ALLM97 parameterisation for $F_{2}$ separately.}
  \label{f:sigmar}
\end{figure}

The experimental results are compared to a phenomenological prediction
of $\sigma_{r}$ based on the ALLM97
parameterisation~\cite{Abramowicz:1997ms} for $F_{2}$ and the
saturation model prediction~\cite{Golec-Biernat:1998js} for
$F_{L}$. In addition the expectation for the reduced cross-section from
the ALLM97 parameterisation is displayed for the case that the
longitudinal structure function $F_{L}$ is zero.

As can be seen from the turn-over of the cross-section close to the
kinematic limit (\mbox{$y=1$}) at large values of $Q^{2}$ the data
provide sensitivity to the longitudinal structure function
$F_{L}$. This effect is more pronounced in the data than is predicted
by the saturation model, i.e. there is an indication that the
predicted $F_{L}$ is too small. Apart from this measurement and
prediction agree very well. For the bins with the lowest $Q^{2}$
values the measured cross-section is systematically larger than
is predicted, but the experimental errors are very large in these bins.

Figure~\ref{f:sigmarbdcvsbst} compares $\sigma_{r}$ for the present
analysis and the BST analysis~\cite{Vargas:2006}. The PHOJET
normalisation (see section~\ref{c:phojetnorm}) is treated differently
from the case of~\cite{Vargas:2006}. For the comparison the bin-wise
normalisation is consistently used for both analyses while
in~\cite{Vargas:2006} the global normalisation is applied. For both
analyses statistical and systematic errors are displayed
separately.

\begin{figure}
  \centering
  \includegraphics[width=\textwidth]{extractf2_060703_2020_sr_vs_060703_2021_binwise.eps}
  \caption{Comparison of the reduced cross-section $\sigma_{r}$ as a function of $x$ for various values of $Q^{2}$ between the BDC (full symbols) and the BST (open symbols) analysis. The inner error bars represent the statistical errors, the outer error bars indicate the statistical and systematic errors added in quadrature.}
  \label{f:sigmarbdcvsbst}
\end{figure}

The BST analysis extends to larger $x$ in the whole $Q^{2}$ range as
expected due to the cut \mbox{$y_{\Sigma} > 0.03$} for the BDC
analysis, necessary to ensure a good vertex reconstruction efficiency
(see figure~\ref{f:pscoverage}). In contrast, the BDC analysis
provides some additional data points or data points with a higher
precision in the transition region between the non-radiative and
radiative events at intermediate $Q^{2}$. This follows from the larger
polar angle acceptance of the BDC allowing access to larger values of
$x$ at fixed $Q^{2}$ for non-radiative events compared to the BST (see
figure~\ref{f:pscoverage}). At low $Q^{2}$ the transition region
between non-radiative events and radiative events is not entirely
covered by the cross-section measurement for either of the two
analyses. The reason for this is the finite resolution of
\mbox{$E-p_{z}$} which causes large bin-to-bin migrations and hence
values of stability and purity below the required 25\% in this part of
the kinematic phase space~\cite{Vargas:2006}.

To quantify the consistency of the two analyses is a very difficult
task since the statistical as well as the systematic errors are highly
correlated. The correlation of the former is due to the fact that the
same data set is used and therefore the events passing the DIS
selection criteria in each analysis bin are to a large extent the
same. Table~\ref{t:uncertainties} illustrates the correlation of the
systematic errors. Only the uncertainties on $\theta_{e}$ and the
efficiencies are different. To account for these difficulties the
comparison proceeds in the following way. First the ratio $r$ of the
reduced cross-section of the BDC and the BST analysis is calculated
and its statistical error is determined with the help of a special
method explained in the following. Subsequently it is investigated
whether the statistically significant deviations of the ratio from one
can be explained by systematic errors not correlated between the two
analyses.

To estimate the statistical error of $r$ a special technique is
employed which is applicable for quantities derived from correlated
samples~\cite{Medin:2005}: the DJANGO sample is split into 40
sub-samples with 25000 events each. For every sub-sample the ratio of
the number of events per analysis bin for the BDC and the BST analysis
is calculated. The relative standard deviation of the mean finally
yields the expected statistical uncertainty on $r$ if it is properly
scaled, taking the luminosities of the data and DJANGO sample as well
a the number of sub-samples into account. The validity of the method
is verified in the following way: the event sample of the BDC analysis
is split in half and the ratio of the cross-sections obtained from each
half is calculated. As the two ensembles are statistically totally
uncorrelated the statistical error of the ratio can be calculated from
the statistical error of each sample. The estimation of the
multiple Monte Carlo sample approach reproduces the result almost
perfectly.

Figure~\ref{f:sigmarbdcvsbstrat} shows the ratio $r$ and its
statistical uncertainty in the same bins as in the case of the
reduced cross-section in figure~\ref{f:sigmarbdcvsbst}. For technical
reasons the statistical uncertainty on $r$ could only be calculated
properly for bins in which both analyses use the same reconstruction
method. Accordingly, the ratio is only displayed for these bins. This
requirement excludes only three of the approximately 50 bins available
for the comparison. For each bin in $Q^{2}$ the $\chi^{2}$ per degree
of freedom is indicated for the hypothesis \mbox{$r=1$}, i.e. a
perfect agreement of the two analyses. The dashed lines indicate a
deviation of 5\% from this hypothesis.

\begin{figure}
  \centering
  \includegraphics[width=\textwidth]{extractf2_060703_2020_sr_vs_060703_2021_binwise_comb_ratio.eps}
  \caption{Ratio $r$ of the reduced cross-section for the BDC and BST analysis. The dashed lines indicate a deviation of 5\% from \mbox{$r=1$}. The given values of $\chi^{2}$ per degree of freedom (dof) quantify the level of agreement with \mbox{$r=1$}. The errors displayed and assumed for the calculation of $\chi^{2}$ are the statistical errors of $r$ only, determined with the help of a special method (for details see text).}
  \label{f:sigmarbdcvsbstrat}
\end{figure}

For bins with central values of $Q^{2}$ below \mbox{$1.2\,
\mathrm{GeV^{2}}$} it is almost impossible to draw any statistically
significant conclusion about the agreement of the two analyses since
the statistical uncertainties on $r$ are extremely large. For $Q^{2}$
larger than \mbox{$2\, \mathrm{GeV^{2}}$} the agreement is reasonably
good. For the three intermediate bins the quoted values of $\chi^{2}$
clearly indicate a disagreement of the two analyses. However, one has
to keep in mind that so far only the statistical uncertainty on $r$
estimated from the 40 DJANGO sub-samples has been taken into account.


The following systematic errors, being uncorrelated between the BST and
the BDC analysis, contribute to the uncertainty on $r$: due to the
reduced amount of dead material affecting the track measurement with
the BST compared to the BDC the BST is more efficient in rejecting
photoproduction events (see section~\ref{c:dissel}), i.e. their
contribution to the ensemble of events passing all DIS selection
criteria is significantly smaller. This is reflected in the systematic
error resulting from the uncertainty on the PHOJET normalisation. In
the three intermediate $Q^{2}$ bins it is a factor of two larger for
the BDC than for the BST analysis (compare appendices A
of~\cite{Vargas:2006} and the present thesis). In addition the
different uncertainties on $\theta_{e}$ lead to systematic errors
which are on average roughly a factor of five larger for the BDC than
for the BST analysis. For each of the two sources the difference of
the systematic errors between the two analyses is added quadratically
to the statistical uncertainty on $r$ and the $\chi^{2}$ is
recalculated. This reduces the $\chi^{2}$ by 50\%, 60\% and 30\% for the
$Q^{2}$ bins \mbox{$1.2\, \mathrm{GeV^{2}}$}, \mbox{$1.5\,
\mathrm{GeV^{2}}$} and \mbox{$2\, \mathrm{GeV^{2}}$},
respectively. This is a clear improvement, but not sufficient to claim
agreement between the two analyses in these bins.

A possible explanation why the discrepancy is largest in the
three intermediate bins may be that they are most highly affected by
the lower acceptance limit of the BST analysis for $R_{e}$ as
illustrated in figure~\ref{f:bstacclimit}. The thin solid line shows
the acceptance limit \mbox{$R_{e} = 10\, \mathrm{cm}$} for the BST
analysis for the average \mbox{$z$-position} of the interaction vertex
$\overline{z_{vtx}}$. For larger (smaller) actual \mbox{$z$-vertex}
positions it is located at lower (higher) values of $Q^{2}$. The
dashed lines indicate the change in the acceptance limit in the
\mbox{$xQ^{2}$-plane} for a \mbox{$z$-vertex} position varying by
\mbox{$11\, \mathrm{cm}$} in the positive and negative directions which 
approximately corresponds to the spread of the \mbox{$z$-vertex}
distribution (see figure~\ref{f:vtxstab}). For the analysis, vertices
deviating by up to \mbox{$30\, \mathrm{cm}$} from $\overline{z_{vtx}}$
are accepted. In the region close to the cut the BST acceptance drops
rather quickly with decreasing $R_{e}$, at \mbox{$R_{e} = 10\,
\mathrm{cm}$} it already reaches 50\%~\cite{H1:draft}. For the BDC
analysis the situation is different since the requirement \mbox{$R_{e}
> 9\, \mathrm{cm}$} prevents large migration effects, while the
acceptance at \mbox{$R_{e} = 9\, \mathrm{cm}$} is still close to 100\%
(see figure~\ref{f:bdceff}). Since the lines of constant $R_{e}$
saturate at large values of $x$ a significant number of bins in $x$ is
affected by the BST acceptance limit in the case of the intermediate
bins. It is possible that the assumed uncertainties for $\theta_{e}$
and the BST efficiency do not fully account for such effects in this
region of the kinematic phase space.

\begin{figure}[t]
  \centering
  \includegraphics[width=0.6\textwidth]{bindef_bstacclimit.eps}
  \caption{Illustration of the influence of the lower acceptance limit of the BST analysis for $R_{e}$ on the bins used for the cross-section determination in the \mbox{$xQ^{2}$-plane}. Apart from the kinematic limit $y=1$ (thick solid line) lines representing the acceptance limit \mbox{$R_{e} = 10\, \mathrm{cm}$} for $z$-vertex positions \mbox{$z_{vtx} = \overline{z_{vtx}}$} (thin solid line) and \mbox{$z_{vtx} = \overline{z_{vtx}} \pm 11\, \mathrm{cm}$} (dashed lines) are depicted. The bins with central values of $Q^{2}$ of \mbox{$1.2\, \mathrm{GeV^{2}}$}, \mbox{$1.5\, \mathrm{GeV^{2}}$} and \mbox{$2\, \mathrm{GeV^{2}}$} are highlighted.}
  \label{f:bstacclimit}
\end{figure}


\section{Extraction of the Structure Function $F_{2}$} \label{c:f2extract}

Having measured the reduced cross-section $\sigma_{r}$ the structure
function $F_{2}$ can be determined by correcting $\sigma_{r}$ for the
contribution of $F_{L}$ according to equation~\ref{e:f2mcm}. For this
purpose the prediction of the saturation
model~\cite{Golec-Biernat:1998js} for $F_{L}$ is employed. As has been
discussed in the previous section the sensitivity of $\sigma_{r}$ to
$F_{L}$ at high $y$ (low~$x$) is significant and the prediction of the
saturation model yields too small values for $F_{L}$. Hence to reduce
the dependence of the measurement on $F_{L}$ the $F_{2}$ extraction is
limited to \mbox{$y < 0.6$}. This strategy has been followed already
in~\cite{Adloff:2000qk}. Consequently, in almost all $Q^{2}$ bins no
$F_{2}$ extraction is performed for the bin with the lowest $x$ for
which a measured reduced cross-section is available.

The result of the $F_{2}$ determination is depicted in
figure~\ref{f:f2}, the values are listed in the table of
appendix~\ref{c:tofresults}. The table also contains the values of
$F_{L}$ predicted by the saturation model and assumed for the
extraction.

\begin{figure}
  \centering
  \includegraphics[width=\textwidth]{extractf2_060703_2020_f2_binwise.eps}
  \caption{Measurement of the proton structure function $F_{2}$, plotted as a function of $x$ in bins of $Q^{2}$. The inner error bars represent the statistical errors, the outer error bars indicate the statistical and systematic errors added in quadrature. The data are compared to two phenomenological predictions: the ALLM97 parameterisation~\cite{Abramowicz:1997ms} and the fractal fit~\cite{Lastovicka:2002hw}.}
  \label{f:f2}
\end{figure}

The data are compared to the phenomenological predictions of the
ALLM97 parameterisation~\cite{Abramowicz:1997ms} and the fractal
fit~\cite{Lastovicka:2002hw}. As expected from figure~\ref{f:sigmar}
the ALLM97 parameterisation nicely describes the data. It undershoots
the data only in the lowest $Q^{2}$ bins, where the experimental
errors are large. The fractal fit underestimates the measured $F_{2}$
in all $Q^{2}$ bins.

As one of the motivations for performing the analysis at hand is to
provide an independent cross-check of the preliminary results on
$F_{2}$ for the shifted vertex period figure~\ref{f:f2vsprel} compares
the different analyses. It has already been explained in
section~\ref{c:phasespace} that the preliminary results have been
obtained analysing non-radiative~\cite{Lastovicka:2002} and
ISR~\cite{Petrukhin:2004a} events separately while no separation has
been performed for the present analysis. Since the BST has been
employed for the H1 preliminary results the corresponding data reach
larger values of $x$. The present analysis extends the accessible
phase space towards lower $Q^{2}$ values\footnote{The same holds for
the BST analysis presented in~\cite{Vargas:2006}.}. In the common
region of the phase space, the smaller values of $F_{2}$ for the H1
preliminary analyses compared to the present one constitute the main
difference. This arises due to the fact that for the preliminary
results the wrong satellite corrections have been used for the
luminosity determination (see section~\ref{c:stability}). This causes
the luminosity to be too large by $4\%$ and thus the measured $F_{2}$
to be too small by the same amount. Apart from this the new analysis of
the shifted vertex data benefits from various refinements achieved
in~\cite{Vargas:2006} and the present thesis. Among them are the new
calibration~\cite{Vargas:2006} (section~\ref{c:calibration}), the new
alignment (section~\ref{c:alignment}), the correction of the
implementation of the beam tilt in the Monte Carlo simulation
(section~\ref{c:simulation}) and the adjustment of the subprocess
fractions in DJANGO (section~\ref{c:mcmodels}).

\begin{figure}
  \centering
  \includegraphics[width=\textwidth]{extractf2_060703_2020_f2_binwise_vs_prel.eps}
  \caption{Comparison of the structure function $F_{2}$ for the present analysis  with the preliminary analyses of non-radiative (squares, \cite{Lastovicka:2002}) and ISR (triangles, \cite{Petrukhin:2004a}) events using the BST for the same data taking period. The inner error bars represent the statistical errors, the outer error bars indicate the statistical and systematic errors added in quadrature.}
  \label{f:f2vsprel}
\end{figure}

Finally the structure function $F_{2}$ of the present analysis is
compared to various published HERA measurements relevant in the phase
space covered. These include two analyses performed by the H1
collaboration based on data recorded during the previous shifted
vertex run in 1995~\cite{Adloff:1997mf} and during the years 1996 and
1997~\cite{Adloff:2000qk} with standard running conditions of the HERA
collider. In addition a measurement of the ZEUS collaboration using
data from the year 1997 is shown. It employs a combination of an
electromagnetic calorimeter (Beam Pipe Calorimeter, BPC) and a silicon
tracker (Beam Pipe Tracker, BPT) mounted close to the beam pipe in the
backward region of the ZEUS experiment, designed to reconstruct the
scattered electron at very large large polar angles and thus to access
very low values of $Q^{2}$. For all published measurements the proton
energy of the HERA collider was \mbox{$E_{p}^{0} = 820\,
\mathrm{GeV}$}. The ZEUS collaboration has used a different binning in
$Q^{2}$ compared to H1. Hence for each $Q^{2}$ bin of the present
analysis having a central value $Q_{H1}^{2}$ the bin of the ZEUS
measurement closest in $Q^{2}$ is searched for and the corresponding
values of $F_{2}$ are shifted to $Q_{H1}^{2}$. For this purpose the
$Q^{2}$ dependence of the ALLM97 parameterisation at the central $x$
values of the ZEUS bins is used.

The values of $F_{2}$ for the present analysis are slightly larger
than for the H1 analysis of the years 1996 and 1997 and slightly
smaller than those obtained from the previous shifted vertex run
(see figure~\ref{f:f2vspubl}). In the lowest $Q^{2}$ bins they clearly
overshoot the findings in the ZEUS analysis, however with large
errors.

\begin{figure}
  \centering
  \includegraphics[width=\textwidth]{extractf2_060703_2020_f2_binwise_vs_publ.eps}
  \caption{Comparison of the structure function $F_{2}$ for the present analysis with various published HERA measurements (closed squares~\cite{Adloff:1997mf}, closed triangles~\cite{Adloff:2000qk}, open triangles~\cite{Breitweg:2000yn}). The inner error bars represent the statistical errors, the outer error bars indicate the statistical and systematic errors added in quadrature. The data of~\cite{Breitweg:2000yn} are shifted to the $Q^{2}$ values of the present analysis using the ALLM97 parameterisation.}
  \label{f:f2vspubl}
\end{figure}

The total error of the analysis at hand is typically \mbox{$30-50\%$}
smaller than for the analysis of the previous shifted vertex run. The
statistical error is reduced by almost a factor of two as expected due
to the roughly fourfold increase in luminosity. The most prominent
improvement among the systematic errors is the reduction of the
uncertainty on the PHOJET normalisation from 30\% to 15\%.

To make optimal use of the available precision data for the reduced
cross-section $\sigma_{r}$ and the structure function $F_{2}$ obtained
in this thesis the following steps are in preparation: the results are
going to be combined with those derived for the BST
analysis~\cite{Vargas:2006}. In addition they are complemented by the
data on $\sigma_{r}$ and $F_{2}$ obtained from the minimum bias run in
1999 covering the $Q^{2}$ range \mbox{$0.5\, \mathrm{GeV^{2}} < Q^{2}
< 12\, \mathrm{GeV^{2}}$}~\cite{Eckstein:2001, Lastovicka:2004a}. Also
this analysis has undergone a major revision in the meantime. Based on
the resulting large data set one aims for an extraction of the
longitudinal structure function $F_{L}$ as well as the exponent
$\lambda$ characterising the rise of $F_{2}$ towards low $x$. Finally
new fits to the fractal and saturation model are foreseen. A
comprehensive publication of all results is anticipated for which a
draft already exists~\cite{H1:draft}.


