\chapter{Cross section Measurement}
\label{Chadisselc}

In this chapter the complete DIS selection is presented and the cross
section measurement for the shifted vertex data is performed.  
The event selection used to identify the scattered electron
with the SpaCal calorimeter and with the BST was already discussed in
chapter \ref{Chdatasample}. The efficiency of the electron
identification was studied using a sample of ISR tagged events,
since it has a minimal background contribution. Nevertheless,
the DIS sample is influenced at low energies of the scattered electron 
by the photoproduction background. Also, the measured distributions
have contribution from processes like elastic QED compton
scattering, which are not included in the MC simulation and have to be
rejected. Thus, it is also important to study the efficiency of the
cuts which define the full DIS selection. In section \ref{SecDISsel}
the complete DIS selection is introduced together with the efficiency
for the most important cuts.  

One important step in the measurement of the DIS cross
section is the identification of the fiducial region, where all
used detector components worked properly, therefore a
summary of the trigger efficiency during the shifted vertex data 
studied in \cite{olaf} and its influence on this analysis is discussed
in section \ref{Sectriggerfiducuts}. 

In section \ref{Secevweight} the $z-$vertex distribution of the event
selection as reconstructed by the BST is 
studied and in section \ref{Subdisplots} the comparison between the
data and the {\small MC} simulation for the most important variables
is presented.

The method used to measure the cross section is explained in section
\ref{Subcrosssection}. The migration of the selected events
between the measured bins is studied in section \ref{Secbinselection}
and the contribution to the cross section from non-tagged radiative
events is discussed in section \ref{Secnotagisr}. 

The cross section measurement and the extraction of the proton
structure function $F_2$ are the subject of section \ref{Subcrosssection}.
As already mentioned in section
\ref{Secshiftedvertex}, this analysis uses the BST as the main
detector to reconstruct the angle of the scattered electron,
nevertheless there is another possibility to reconstruct the angle of
the scattered electron by using the BDC and the CTD.
In this thesis only the BST results are described, the measurement of
the BDC as well as the comparison of 
the cross section measurement determined in this thesis with the BDC
is presented in \cite{olaf}. 

In chapter \ref{Chapspacalib} and \ref{Chaphadfinalstate} it was
explained in detail how the calibration of the SpaCal and the LAr
calorimeters were performed for the shifted vertex data. There, the
scale of the energy measurement of the scattered electron was
derived. Moreover, the deviations between data and MC of the transverse
energy flow were used to determine the uncertainty of the hadronic
energy measurement. These uncertainties together with the results of
other analyses \cite{olaf,tomas} will be used to calculate the
total error of the cross section measurement. These studies are
presented in section \ref{Secsystematic}.

\section{DIS selection}
\label{SecDISsel}

Table \ref{Tabdissel} summarizes the cuts used to
identify DIS events. The event selection is mainly performed by the
SpaCal calorimeter and the BST, a minimal number of cuts depending on
the hadronical final state are considered, this are the $p_{t,bal}$
and the $(E-p_{z})_{tot}$ cuts (see Eq. \ref{Eqptbalvari} and
Eq. \ref{Eqeinc1x} for the definition of both variables).  

The efficiency of a cut is studied using
the number of events that the particular cut has either accepted or
rejected. Nevertheless some background contributions are only present in
the data. For example, events coming from an empty pilot bunch are
rejected from the data, but its efficiency can not be studied with the
{\small MC} since such events are not simulated.
Similarly, the {\small MC} does not simulate events due to the
satellite part of the $z-$vertex distribution, therefore the
$z-$vertex cut can not be excluded. In general, cuts which are applied
to reduce effects which are not considered in the {\small MC}
simulation can not 
be excluded to study the losses due to cut application. These cuts are
marked as {\it fix} in 
Table \ref{Tabdissel} and will allways be applied.

\begin{table}[t]
\begin{center}
\begin{tabular}{l c c c}
\hline
\hline
\multicolumn{4}{c}{DIS selection} \\
\hline
 & Efficiency & Cut & Recontruction  \\
 &  Study   &  & method \\
\hline
SpaCal:& fix & Fiducial volumen & $e,\Sigma$\\
       & fix & $7\,$GeV$< E_{e}< 30\,$GeV & $e,\Sigma$ \\
       & $+$ & $R_{clog}<4\,$cm  & $e,\Sigma$\\ 
       & $+$ & $E_{veto}<1\,$GeV & $e,\Sigma$ \\
       & $+$ & $f_{had}<0.15$ & $e,\Sigma$ \\
\hline
BST: & $+$ & $\Delta R = |R_{bst}-R_{SpaCal}|<1.5\,$cm & $e,\Sigma$ \\ 
%     & fix & $lval=1$ & $e,\Sigma$ \\
     & fix & $|z_{BST} - 70| < 30\,$cm & $e,\Sigma$ \\
     & fix & $R_{bst} > 10\,$cm & $e,\Sigma$ \\
     & $+$ & $N_{BST,hits} < 200$ & $e,\Sigma$ \\
\hline
Hadronical Final State: \\
     & $+$ & $P_{t,bal}>0.3$ & $e,\Sigma$ \\
     & $+$ & $R_{compton}$ & $e,\Sigma$ \\
     & $+$ & $R_{veto}$ & $e,\Sigma$ \\
     & $+$ & $(E-p_{z}) > 35\,$GeV & $e$ \\
\hline
\hline
\end{tabular}
\end{center}
\caption{DIS selection used to measure the inclusive cross section}
\label{Tabdissel}
\end{table}

The efficiency of the cut-i is defined as, 

\begin{equation}
\epsilon(cut-i) = \frac{N_{all}}{N_{all}+N_{rej,cut-i}} 
\label{Eqeffcutpho}
\end{equation}

where $N_{all}$ is the number of events accepted by the DIS
selection and $N_{rej,cut-i}$ is the number of good events rejected by the
$cut-i$. Since the data include a certain amount of background, the {\small PHOJET}
{\small MC} is used to estimate it. Thus,

\begin{equation}
N_{all} = N(data)_{acc} - N(PHOJET)_{acc}  \hspace{1cm}
N_{rej,cut-i} = N(data)_{rej,cut-i} - N(PHOJET)_{rej,cut-i}
\label{Eqcuteffpho}
\end{equation}

with $N(data)_{acc}$ and $N(PHOJET)_{acc}$ being the number of
events accepted in the data and in the {\small PHOJET} {\small MC}
using the whole DIS selection. The {\small PHOJET} contribution is
normalized to the data using the 
normalization factor determined in section \ref{Secphotoproduction}.
In the following sections, the efficiency of the cuts 
cuts marked by a ``plus'' sign in Table \ref{Tabdissel} are studied
and compared with the expectation of the signal {\small MC} {\small DJANGOH}.

\subsection{$R_{clog}$}

The transverse extension of the electron shower over the SpaCal plane
$R_{clog}$ is shown in Fig. \ref{Figdisrclnew}. One can see that the
{\small MC} does not describe the $R_{clog}$ variable (see
Eq. \ref{Eqrclog}), in particular a  
shift between the data and the {\small MC} simulation is observed and
the clusters with 
transverse extension greater than $4\,$cm are concentrated in the
tail of the distribution. The shift of the $R_{clog}$ variable is a
known problem related to the fast shower simulation in {\small MC}
\cite{victor}. 

\begin{wrapfigure}[20]{r}{0.5\textwidth}
%\begin{floatingfigure}[r]{4.5cm}
\begin{center}
\epsfig{figure=./selecdis/figures/rclnew_060515_0020.eps,width=0.4\textwidth}
\begin{minipage}{0.45\textwidth}
\setcaptionwidth{\textwidth}
\caption{$R_{clog}$ distribution of the scattered
  electron. Closed points represent data, while the histogram is the
  sum of {\small DJANGOH}
  and {\small PHOJET} {\small MC}. The background contribution derived
  from PHOJET is shown by the solid area. The line at $4\,$cm
  correspond to the analysis cut.} 
\label{Figdisrclnew}
\end{minipage}
\end{center}
%\end{floatingfigure}
\end{wrapfigure}

The efficiency of the $R_{clog}$ cut as function of the energy of the
  scattered electron $E_e$ for the data and the {\small
  DJANGOH} {\small MC} is presented in 
Fig.\ref{Figcuteffrclnew}(a). The error band of the data corresponds to the
uncertainty asigned to the background substraction $15\%$
\cite{paper}. The limits of the band are obtained varying the phojet
  normalization $15\%$. The efficiency curve is  
not described by the {\small MC} simulation at lower energies. 
%The reason of this discrepancy is related to several mechanisms. First,

The exclusion of the $R_{clog}$ cut leads to more background in the
sample, and the {\small PHOJET} contribution for accepted and rejected
events considerably changes. 
The efficiency was therefore studied using different {\small PHOJET}
contributions for accepted and rejected events. Thus,
the background contribution was determined as in section
\ref{Secphotoproduction} but relaxing the $R_{clog}$ cut.
The normalization of the {\small PHOJET} {\small MC} to the data
increases by almost $70\%$. In contrary to the accepted events, the
energy distribution of the mis-identified scattered electron
$\bf{E_{e}}$  is not described by the {\small MC} simulation for the
{\it rejected} events. 

\begin{figure}[t]
\begin{center}
\epsfig{figure=./selecdis/figures/cuteff_rclnew_060511_1444.eps,width=1.\textwidth}
\put(-390.,50.){(a)}
\put(-180.,50.){(b)}
\caption{Efficiency of the $R_{clog}<4\,$cm cut as function of the
  energy of the scattered electron $E_e$. The open points shows
  the efficiency of the signal {\small MC DJANGOH}, while full points is
  data where the $\gamma p$ is substracted using the {\small MC}
  {\small PHOJET}. (a) The $\gamma p$ background is substracted from
  the data using the same normalization factor for $R_{clog}<4\,$cm
  and $R_{clog}>4\,$cm. (b) Same as Fig. (a), but using different
  normalization factors for accepted and rejected events (see
  text). Thus, for Fig. (b) the $\gamma p$ background is substracted
  considering the relative contribution of 
  $\gamma p$ to the data with $R_{clog}<4\,$cm and $R_{clog}>4\,$cm. }
\label{Figcuteffrclnew}
\end{center}
\end{figure}

The different normalization factors improve the description of the
efficiency curve, as can be seen from
Fig. \ref{Figcuteffrclnew}(b). Nevertheless, discrepancies are still
observed for the interval between $7\,$GeV and $9\,$GeV. This
discrepancy is related to the non-description of the energy ${\bf E_e}$ of
the rejected events by {\small PHOJET} and due to the non description
of the $R_{clog}$ variable itself (see the tail in
Fig. \ref{Figdisrclnew}). In section \ref{Secloweanalysis} it was shown 
that using a sample almost free of background events (the tagged ISR
sample), the efficiency of the $R_{clog}$ cut is in agreement with the
{\small DJANGOH} {\small MC} simulation (see Fig. \ref{Figefflowe}(a)).

\subsection{$f_{had}$, $\Delta R$, $R_{veto}$}

In Fig.\ref{Figcuteffall}(a) and (b) are shown the
efficiency of the hadronic fraction $f_{had}<0.15$ cut and of the SpaCal
BST matching $\Delta R<1.5\,$cm cut, as function of the energy of the scattered
electron $E_e$ (see equations \ref{Eqfhad} and
\ref{EqdeltaRbst}). The $f_{had}$ cut is inefficient 
for low values of the energy of the scattered electron, but its
behaviour is well described by the signal {\small MC} {\small
DJANGOH}. For values of the scattered electron $E_e \approx 7\,$GeV,
the efficiency of the $\Delta R$ cut goes to unphysical values.
This is just a statistical fluctuation coming from the substraction of
{\small PHOJET}. The difference between data and {\small MC} for both cuts
do not exceed $2\%$. 

The $R_{veto}$ cut, rejects events with a wrongly recontructed vertex
by the central tracking devices outside the $z-$vertex measured region
$|z-70|<30\,$cm. From Fig. \ref{Figcuteffall}(d) follows that the
$R_{veto}$ cut is full efficient.

\subsection{$R_{compton}$}

In order to extract correctly the proton structure function from the
measured cross section, it is necessary  to correct for
contributions coming from higher order radiative processes. 
The largest contributions to the corrections arise from QED processes in
first order in $\alpha$. In such processes a photon is radiated
from the incoming electron or from the outgoing electron. 
Experimentally one distinguishes between different event classes in
the detector, the QED Compton (QEDC) being one of those signatures.

\begin{figure}[t]
\begin{center}
\epsfig{figure=./selecdis/figures/cuteff_all_060511_1444.eps,width=1.\textwidth}
\put(-390.,250.){(a)}
\put(-180.,250.){(b)}
\put(-390.,50.){(c)}
\put(-180.,50.){(d)}
\caption{Efficiency of the selection cuts as function of the energy of
  the scattered electron $E_e$ (a) the $f_{had}<0.15\,$ cut (b)
the SpaCal-BST matching $\Delta R<1.5\,$cm (c) the Compton rejection
$R_{compton}$ cut and (d) the
$R_{veto}$ cut. The open points shows the efficiency of the signal
{\small MC DJANGOH}, while full points is data where the $\gamma p$ is
substracted using the {\small MC} {\small PHOJET}.}
\label{Figcuteffall}
\end{center}
\end{figure}

Since the {\small DJANGOH}1.4 {\small MC} includes only the inelastic
component of the QEDC cross section and does not include the elastic
contribution, a rejection of those events has to be performed.  
The QED Compton signature is characterised by two
electromagnetic clusters in the backward region of the H1 detector,
one corresponding to the scattered electron and the another to the
radiated photon. The energy of both clusters can be measured with the
SpaCal calorimeter. The electron and photon cluster should be in a
``back to back'' topology.
% where the difference of the azimuthal angle between the
%cluster should correspond approximately to $180^{\circ}$.

In elastic and most inelastic QEDC events, the
detector is empty except for the electron and the photon, so that no
central tracks are reconstructed. Although the
requirement of a vertex reconstructed with the CTD reduced the
contribution of the QEDC events to the cross section, a small fraction
can still remain if one reconstruct the vertex with the BST
\cite{Arkadov,victor}. 

The cuts used to reject the QED Compton events, follow the
selection criteria developed in \cite{victor}. $R_{compton}$ considers
the following selection to reject the compton events.
The energies of the electron and photon clusters should be greater
than $4\,$GeV 
and the sum of the energy of both clusters should be greater
than $18\,$GeV. Also, the difference of the azimuthal angle between the
electron and photon clusters should correspond approximately to
$\phi \approx 180^{\circ}$. In case the event fulfil those requirement it will be
rejected. 

In Fig. \ref{Figcuteffall}(c) is shown the efficiency of the
$R_{compton}$ cut as function of the energy of the scattered
electron. The rejection due to the $R_{compton}$ cut is localized
mainly between $15\,$GeV and 
$20\,$GeV. The differences of data and {\small MC} do not
exceed $2\%$. 

\subsection{$p_{t,bal}$}
\label{Subsecptbal}

\begin{figure}[t]
\begin{center}
\epsfig{figure=./selecdis/figures/newptbal_wsigma.eps,width=0.955\textwidth}
\put(-260.,180.){(a)}
\put(-60.,180.){(b)}
\caption{(a) $p_{t,bal}$ distribution for the DIS selection, the line
  represent the analysis cut. (b) $W_{\Sigma}$ distribution for
  events which pass the DIS selection and have $p_{t,bal}<0.3$.}
\label{Figcutptbal}
\end{center}
\end{figure}

%As the result of the global LAr calibration , the $P_{bal}$
%distribution for the whole DIS sample is shown in Fig ptbalnocut. 
As was already mentioned in the previous chapter, the acceptance of
the backward and forward part of the detector is limited due to the
beam pipe holes used to store the beams. The hadronic final state can
be produced at small angles escaping the forward calorimeter
acceptance; this cut can create an 
imbalance of the transverse momenta of the hadrons with the
electron. Such non-balanced events are
concentrated mainly at very small $p_{t,bal}$ values, which
is shown in Fig. \ref{Figcutptbal}(a). 

\begin{figure}[t]
\begin{center}
\epsfig{figure=./selecdis/figures/cuteff_ptbal_060511_1444vs060512_1620.eps,width=0.955\textwidth}
\put(-260.,180.){(a)}
\put(-60.,180.){(b)}
\caption{(a) Efficiency of the $p_{t,bal}<0.3$ cut as function of the
energy $E_e$ of the scattered electron. (b) Efficiency of
the $p_{t,bal}< 0.3$ cut having in addition a $E-p_z>35\,$GeV cut. 
The open points show the efficiency of the signal {\small MC
DJANGOH}, while full points correspond to data where the $\gamma p$ is
substracted using {\small PHOJET}}. 
\label{Figcuteffptbal}
\end{center}
\end{figure}


The losses become especially pronounced for high values of $x$ (low
$y$). This region corresponds to the resonance region where $W$ tends
to be small. In Fig. \ref{Figcutptbal}(b) is shown the
$W_{\Sigma}$ distribution for events having $p_{t,bal}<0.3$, it can be
seen that those events are mainly concentrated in the interval
$W_{\Sigma}<50\,$GeV.  

Fig.\ref{Figcutptbal}(a) and (b) demonstrate that the $p_{t,bal}$
cut has to be performed in order to reduce the losses of low
$p_{t,had}$ particles, since the {\small MC} simulation does not
follow the behaviour of the data for values of $p_{t,bal}<0.3$ and
conservatively $W_{\Sigma}<50\,$GeV. 

The efficiency of the $p_{t,bal}<0.3$ cut as function of the energy of
the scattered electron is shown in
Fig. \ref{Figcuteffptbal}(a). The efficiency is not
completely described by the {\small MC} simulation for values of $E_e
\leq 15\,$GeV. Excluding the
$p_{t,bal}<0.3$ cut, more non balanced events enters to the selection and
much of them should be background. In order to minimize the background
contribution, the efficiency for the $p_{t,bal}<0.3$ cut is studied using
in addition a $(E-p_z)_{tot}>35\,$GeV cut, which is used in the analysis for
the region where the electron method is applied. The
$(E-p_z)_{tot}>35\,$GeV cut, will 
guarantee that the longitudinal energy conservation is fulfilled. The
efficiency of the $p_{t,bal}>0.3$ cut subsample is shown in
Fig. \ref{Figcuteffptbal}(b), where a good description of the data
by the {\small MC DJANGOH} simulation is observed. 

\begin{wrapfigure}[20]{r}{0.5\textwidth}
\begin{center}
\epsfig{figure=./selecdis/figures/cuteff_ptbal_060413_0900.eps,width=0.5\textwidth}
\begin{minipage}{0.5\textwidth}
\setcaptionwidth{\textwidth}
\caption{Efficiency of the $p_{t,bal}<0.3$ cut using only events with
$Q^{2}_{\Sigma}>2$GeV$^2$. Open points show the efficiency of the
signal {\small MC DJANGOH}, while full points is data
where the $\gamma p$ is substracted using {\small PHOJET}.}
\label{Figcuteffptbal2}
\end{minipage}
\end{center}
\end{wrapfigure}

However, the $(E-p_z)_{tot}>35\,$GeV cut is not performed in case the
kinematic is 
reconstructed with the $\Sigma$ method. On one hand, one sees that
once the longitudinal energy conservation is fulfill, the $p_{t,bal}$
efficiency is good described, the discrepancy observed in
Fig. \ref{Figcuteffptbal}(a) can be related to background contribution
or to an incomplete implementation of the physics of the MC. 
The {\small DJANGOH} {\small MC} does not generate events 
in the region
where both $x$ and $y$ are small, since $Q^2 \approx syx$, those
events are mainly concentrated for low values of $Q^2$. Thus,
the description of the $p_{t,bal}$ efficiency
for $Q^2>2\,$GeV$^2$ should be better described even not applying the
$(E-p_z)_{tot}$ cut. This efficiency is shown in
Fig. \ref{Figcuteffptbal2}, also a good agreement between data and
{\small MC} is found. 

%In previous analysis the low $W$ region was introduced in
%the {\small MC} simulation using the generator {\small HERWIG}. In order to
%minimize the influence of the losses into the kinematic reconstruction,  
%the cut $p_{t,bal}>0.3$ was introduced. In the present analysis the
%low $W$ regimen is also included, but the generator SOPHIA is used to
%incorporate the events with $W<5\,$GeV.


\subsection{$(E-p_{z})_{tot}$}

The $(E-p_z)_{tot}>35\,$GeV cut is applied only in case the kinematical variables will
be reconstructed with the electron method. This cut is applied to reduce
the contribution of photoproduction background and ISR events.

%The $E-p_z$ distribution for events where the scattered electron have
%energies lower than $15\,$GeV is shown in 
%Fig.\ref{FigcuteffEmpz}(a). The distribution has a maximum at
%$2E_{inc} \approx 55\,$GeV, the events reconstructed at lower $E-p_z$
%have a high contribution of photoproduction events.

The efficiency of the $(E-p_z)_{tot}>35\,$GeV cut is shown in
Fig. \ref{FigcuteffEmpz}, one observes that {\small DJANGOH} {\small
MC} describes the behaviour of the data. For energies of the scattered
electron greater than $17.5\,$GeV, the $(E-p_z)_{tot}$ from the electron is
sufficient to satisfy the cut $(E-p_z)>35\,$GeV cut, so the efficiency
is $100\%$. For energies of the scattered electron lower than
$17.5\,$GeV, the $(E-p_z)_{tot}$ of the electron is not sufficient to satisfy
the $(E-p_z)_{tot}>35\,$GeV cut and therefore the $(E-p_z)_{tot}$ from
the hadrons begin to become important. The 
drop of the efficiency curve can be explained to be due to particles
escaping 
through the beampipe in the backward direction and therefore the measured
$(E-p_z)_{tot}$ will be smaller than $2E_{inc}$ or the event is a ISR one.  

The $(E-p_z)_{tot}$ cut is not applied in case the
kinematical variables are reconstructed with the $\Sigma$
method. However, the kinematical variables are also
correctly reconstructed since, contrary to the electron method, the
centre of mass energy is corrected for the photon emission in this
case (see section \ref{SubSecsigmamethod} for details).

%\begin{wrapfigure}[16]{r}{0.5\textwidth}
\begin{figure}[t]
\begin{center}
\epsfig{figure=./selecdis/figures/cuteff_empz_060529_2357.eps,width=0.45\textwidth}
%\begin{minipage}{0.5\textwidth}
%\setcaptionwidth{\textwidth}
\caption{Efficiency of the $(E-p_z)_{tot}>35\,$GeV cut as function of
  the energy of the scattered electron $E_e$. Open points is the
  efficiency of {\small DJANGOH MC} while closed points are data where the
  $\gamma p$ background is substracted using {\small PHOJET}.}
\label{FigcuteffEmpz}
%\end{minipage}
\end{center}
\end{figure}
%\end{wrapfigure}

\section{Trigger efficiency and fiducial volumen}
\label{Sectriggerfiducuts}

The relevant triggers used in the low $Q^2$ analysis are $S0,S3$ and
$S9$. Table \ref{Tabsubtriggers} summarises the
configuration of $S0,S3,S9$ during the shifted vertex period.

\begin{table}[b]
\begin{center}
\begin{tabular}{| c c c |}
\hline
Subtrigger & L1 & L2  \\
\hline
S0 & {\small{IET>1} || Cen$-$LIET > 2} & $R_{10}$  \\
S3 & {\small{IET>2} || Cen$-$LIET > 3} &  \\
S9 & \small{IET>0} &  \\
\hline
\end{tabular}
\caption{The trigger configuration for the shifted vertex run. The IET
and Cen-IET regions are illustrated in Fig. \ref{Figietregion}.}
\label{Tabsubtriggers}
\end{center}
\end{table}

The basic information comes from the deposited
energy in the SpaCal calorimeter cells. This
energy deposition defines the bits of the Inclusive Electron Trigger
{\small(IET)}, which is the basis of all trigger elements delivered by
SpaCal. {\small IET} considers three different energy thresholds denoted as
{\small IET$>1,2,3$} and discriminates at L1 between different SpaCal
regions. The thresholds of {\small IET$>1,2,3$} during the shifted
vertex period were set to $2,5,10\,$GeV, respectively. The central IET
{\small Cen$-$IET} region cover mainly the inner region of the SpaCal
surface. In Fig. \ref{Figietregion} the coverage of {\small IET} and
{\small Cen$-$IET} is shown. During the shifted vertex period, the
thresholds of {\small Cen$-$IET} were also set to  $2,5,10\,$GeV.


%\begin{wrapfigure}[19]{r}{0.5\textwidth}
\begin{figure}[t]
\begin{center}
\begin{minipage}{0.4\textwidth}
\begin{center}
\epsfig{figure=./selecdis/figures/ietregion.ps,width=\textwidth}
\put(-150.,200.){(a)}
\end{center}
\end{minipage}
\begin{minipage}{0.5\textwidth}
\begin{center}
\epsfig{figure=./selecdis/figures/efficiency_ietcen_060119_2301_effspr10.eps,width=\textwidth}
\put(-150.,200.){(b)}
\end{center}
\end{minipage}
\caption{(a) The inclusive electron trigger structure (IET). The
  Central-IET (CIET) region is illustrated by the shadow area. (b)
  Trigger efficiency of S0 studied in \cite{olaf}}  
\label{Figietregion}
\end{center}
\end{figure}

Due to the acceptance of the BST during the shifted vertex period, only
S0 is used. 
The subtrigger $S0$ has a threshold of $5\,$GeV
and a L2 condition $R_{10}$. The L2 condition
means that only events are considered, which contain at least one
impact outside a circle of $10\,$cm on the SpaCal surface.

The efficiency of $S0$ was studied in \cite{olaf} and is
calculated taking as reference a group of independent subtriggers
which validate the event. 
The reference sample considers events where only one electron
candidate was at L4 reconstructed. In order to minimise
the influence of background events in the reference sample, an
additional electron identification selection has to be applied. Also,
the time of flight of the electron candidate 
should correspond to the time expected of an $ep$ collision. 


%\begin{wrapfigure}[16]{r}{0.5\textwidth}
%\begin{center}
%\epsfig{figure=./selecdis/figures/efficiency_ietcen_060119_2301_effspr10.eps,width=0.5\textwidth}
%\begin{minipage}{0.4\textwidth}
%\setcaptionwidth{\textwidth}
%\caption{Trigger efficiency of s0 studied in \cite{olaf}}
%\label{Figolaftrig}
%\end{minipage}
%\end{center}
%\end{wrapfigure}

Fig \ref{Figietregion}(b) shows the efficiency of $S0$ as function
$R_{box}=max(|x_{cl}|,|y_{cl}|)$ of the scattered electron measured by the
SpaCal calorimeter. Using only S0, the fiducial volumen is restricted
to a SpaCal surface with a box cut of 10 cm. This trigger region together
with the cells which were excluded in chapter 3 constitute the fiducial
volumen on the SpaCal plane used in the BST analysis.

\section{Event Weight}
\label{Secevweight}

\begin{wrapfigure}[22]{r}{0.55\textwidth}
\begin{center}
\begin{minipage}{\textwidth}
\epsfig{figure=./selecdis/figures/zvertexfill_060522_1522.eps,width=0.6\textwidth}
%\put(-260.,180.){(a)}
\put(-60.,180.){(a)}
\put(-60.,40.){(b)}
\end{minipage}
\begin{minipage}{0.5\textwidth}
\setcaptionwidth{\textwidth}
\caption{Position (a) and spread (b) of the interaction point of the ep scattering as
  measured by the BST during the analysed luminosity fills of the
  shifted vertex data period.}
\label{Figzvertexfill}
\end{minipage}
\end{center}
\end{wrapfigure}

As was already mentioned in the previous section, the BST analysis is
performed using only events accepted by the S0 trigger. Since during
the shifted vertex period, S0 had a prescale different of one, each
event had to be corrected with the corresponding prescale factor of
the analysed run. In average the prescale of S0 was $1.15$. 
%Also a L4 weight is applied, having an influence of less than $0.2\%$
%\cite{olaf}. 

For {\small MC} there are two weights which are used, one is related to
the vertex position of the simulated events, and the other
corresponds to the structure function assumption of the {\small MC}. These
two weights are briefly explained in the followings sections.


\subsection{$z-$vertex Reweighting in the MC}
\label{Subvertexreweight}

The determination of the vertex position is important to correctly
reconstruct the kinematic variables, since its precision directly 
influences the measurement of the angle of the scattered electron. In 
Fig \ref{Figzvertexfill} is shown the mean and width of the vertex
position measured by the BST during the shifted vertex data period as
function of the analysed run. The dashed lines indicates 
the luminosity fills. From Fig. \ref{Figzvertexfill}(a) and (b) one can see
that the mean position and the width of the vertex changes during the
luminosity fill \cite{cisserver}. 

Contrary to the data, the vertex position of the {\small MC} is a
gaussian 
distribution with a fixed mean position and width. Since the
distribution of the vertex
position of the data differs from the $z-$vertex simulated, the MC
events are reweighted as a function of the 
$z-$vertex distribution. Such it is possible 
to match the $z-$vertex distribution of the {\small MC} simulation to the
$z-$vertex distribution of the data.

The z-vertex reweighting procedure requires a sample which
ideally is not influenced by additional systematic deviations and be
inside the BST acceptance. 

\begin{figure}[t]
\begin{center}
\epsfig{figure=./selecdis/figures/zvertex_060522_1521.eps,width=0.9\textwidth}
\put(-350.,50.){(a)}
\put(-160.,50.){(b)}
\caption{$z-$vertex distribution measured by the BST in a restricted
  $\theta$ range (see text) using (a) data und (b) {\small MC}. The
  line in (a) and (b) illustrates the fit of the data and MC
  distrubution to the sum of the Gaussian and the polynomial of first
  grad.} 
\label{Figzvertexrew}
\end{center}
\end{figure}

In order to get a $z-$vertex position in the BST acceptance, and to
guarantee that all possible $z-$vertex values are considered, only events
with electron candidates
which are located in the interval $175^{\circ} < \theta <
176.5^{\circ}$ are accepted. This angular interval correspond to the
best radial acceptance of the BST projecting on the SpaCal plane, which
correspond to $12$cm$<R_{SpaCal}<22\,$cm. In addition, the Spacal
selection of Table 
\ref{Tabdissel} is applied, {\it but} the energy of the scattered
electron should be greater than $15\,$GeV in order to reduce the contribution
of photoproduction events. The $z-$vertex 
distribution for data and {\small MC} for the previos selection is
shown in Fig. \ref{Figzvertexrew}. It can be seen, 
that the $z-$vertex distributions are slighlty asymmetric, thus a
sum of a gaussian function and a 
polynomial of first order can describe satisfactorily the tails of the
vertex distributions.

The weight to be used in the analysis is calculated as:

\begin{eqnarray}
weight_{z-vertex} =
\frac{G(z_{v}^{data},P_{i}^{data})}{G(z_{v}^{MC},P_{i}^{MC})} \\
\label{Eqzvertexrew}
\end{eqnarray}

where $G(z_{v}^{data},P_{i}^{data})$, $G(z_{v}^{MC},P_{i}^{MC})$ are
the sum of a gaussian 
function and a polynomial obtained from the fit of the $z-$vertex
distributions of data and {\small MC}, respectively. 

\[
G(z_{v}^{data},P_{i}^{data}) = P1 \exp((z_{v}^{data}-P2)^{2}/2P3^{2})
+ P4 + P5 \,z_{v}^{data}
\]

The fit parameters are shown in
Fig. \ref{Figzvertexrew}(a) and (b), and listed in table \ref{Tabzvertex}.

\begin{table}
\begin{center}
\begin{tabular}{c c c}
\hline
 & data & MC \\
\hline
P0 & $1685.69$ & $10088.98$\\ 
P1 & $73.167$ & $72.274$ \\
P2 & $11.272$ & $11.211$ \\
P3 & $45.858$ & $-60.937$ \\
P4 & $-0.456$ & $-0.297$ \\
\hline
\end{tabular}
\caption{Data and MC parameters of the fit performed to the vertex
  distribution using the function $G$}
\label{Tabzvertex}
\end{center}
\end{table}

%The cross section reweighted technique was used for the monte carlo
%adjustement in order to avoid a reprocessing of the DIS event sample
%comprising millions of events.

\subsection{MC reweight}
\label{Subsecfractalfit}

At generator level, the {\small DJANGOH1.4} simulations were performed
using the parton distribution functions of the MRST(3,75) \cite{mrstv804} group.
These functions were selected due to their reasonable
behaviour over most of the measured kinematic
range. However the MRST functions are only
applicable for $Q^2 \geq 1.25\,$GeV$^2$, and are not expected to
describe completely the data for lower values of $Q^2$. In previous
analyses \cite{tomas}, it was shown that the data can be described
using the fractal model \cite{tomaspaper} and the ALLM97 \cite{allm97} fit, therefore during
the analysis the events were re-weighted according to:

\begin{equation}
weigth_{\sigma} = \frac{\sigma_{red}^{Fit}(Q^2,x)}{\sigma_{red}^{Gen}(Q^2,x)}
\label{Eqweightrcs}
\end{equation}

where $\sigma_{red}^{Gen}$ correspond to the originally generated cross
section and $\sigma_{red}^{Fit}$ is the cross section computed from the
expectation of the fractal and the {\small ALLM97} fit. Since the
fractal fit only describes the low $x$ region, for events having a
generated value of $x$ lower than $0.01$ the fractal fit 
while for the region $x>0.01$ the {\small ALLM97} fit is used.

\section{Control Distributions of the DIS sample}
\label{Subdisplots}

In order to measure the cross section, it is necessary to verify if
the MC simulation can reproduce the behaviour of the data. Therefore
the distributions of the most important variables were
compared with the MC simulation. The MC and the data have to contain
all calibration constants and the weights introduced in the
previous sections.

The final distributions of the main variables used to measure the cross
section and to extract the proton structure function $F_2$
are shown in Fig \ref{Figelectrondist} and in Fig
\ref{Figsigmadist}. The MC is shown as the sum of the signal MC
DJANGOH and the background MC PHOJET. The background contribution in
each distribution is shown in addition as a shaded area. In all
figures presented 
in Fig. \ref{Figelectrondist} and Fig. \ref{Figsigmadist} all cuts
listed in \ref{Tabdissel} are applied. 

The DJANGO MC is reweighted with the procedure explained in section
\ref{Subsecfractalfit} and normalized to the integrated luminosity of
the data which is $450\,$nb$^{-1}$. However the fractal fit used to
reweight the MC was performed using previous H1 measurements with a
wrong luminosity calculation. The difference between the luminosity
used to perform the fractal fit and the value used in this analysis is
$5\%$. The MC simulation is therefore $5\%$ lower than the
data. Nevertheless, the shape of the measured distributions are well
described by the {\small MC} simulation. 

In Fig \ref{Figelectrondist}(a),(b) are shown the energy and the angle
of the scattered electron measured by the SpaCal calorimeter and by the
BST. The vertex distribution is shown in Fig. \ref{Figelectrondist}(c).
The kinematical variables $x,Q^2$ and $y$ reconstructed with the electron
method, are shown in Fig \ref{Figelectrondist} (d),(e),(f). In general
a good agreement between the data and the {\small MC} simulation is
observed. 

In Fig \ref{Figsigmadist}, the $E-p_z$ and $p_{t,had}$ distribution
for the whole DIS selection is presented. The kinematical variables
reconstructed with the $\Sigma-$method are shown in Fig. 
\ref{Figsigmadist}(c),(d) and (e). The $y_{\Sigma}$ distribution is
reasonably described having an excess in $y \approx 10^{-2}$, whose
origin could not be completely explained due to the noise in the LAr.

Since most of the variables are well described, the cross section and
the extraction of $F_2$ can be performed. In the next section the
method used to measure the cross section will be discussed.

\begin{figure}[p]
\begin{center}
\epsfig{figure=./selecdis/figures/electroncontrolplots_060523_1007.eps,width=0.95\textwidth}  
\put(-365.,400.){(a)}
\put(-60.,400.){(b)}
\put(-365.,230.){(c)}
\put(-60.,230.){(d)}
\put(-360.,50.){(e)}
\put(-60.,50.){(f)}
\caption{Control distributions of (a) energy (b) angle of the scattered electron
  (c) vertex position. (d) $Q^2$ (e) $x$ (f) $y$ reconstructed with
  the electron method. The {\small MC} is normalized to the luminosity
  of the data $+5\%$ (see text)}
\label{Figelectrondist}
\end{center}
\end{figure}

\begin{figure}[p]
\begin{center}
\epsfig{figure=./selecdis/figures/sigmacontrolplots_060523_1134.eps,width=0.95\textwidth}
\put(-365.,400.){(a)}
\put(-60.,400.){(b)}
\put(-365.,230.){(c)}
\put(-60.,230.){(d)}
\put(-360.,50.){(e)}
\caption{Control distributions of (a) $E-p_z$ (b) the transvers
  momentum of the hadrons $p_{t,had}$. The kinematical variables (c) $Q^2$
  (d) $x_{\Sigma}$ and (e) $y_{\Sigma}$ reconstructed
  with the $\Sigma$ method. The {\small MC} simulation is normalized to the
  luminosity of the data $+5\%$ (see text)}
\label{Figsigmadist}
\end{center}
\end{figure}


\section{Cross section determination}
\label{Subcrosssection}

The DIS cross section is measured by counting the number of events in
a $x-Q^2$ grid. This number represent the integral of the double
differential cross section over the kinematic interval. The cross
section is given by:

\begin{equation}
\frac{d^{2}\sigma(x,Q^2)}{dxdQ^2} = \frac{N_{data}-N_{background}}{A
  L} \frac{1}{\epsilon} \frac{1}{1+\delta} \, \beta_{BC}
\label{Eqcrosssection}
\end{equation}

where:

\begin{itemize}
\item $N_{data}$ is the number of total events selected in the $x-Q^2$ bin
\item $N_{background}$ represent the background events 
\item $A$ denotes the detector acceptance which is calculated with the
  {\small MC}, as:
\[
A= \frac{N_{rec}}{N_{gen}}
\]
being $N_{rec}$ and $N_{gen}$ the reconstructed and
generated events in the particular bin.    
\item $L$ is the integrated luminosity of the selected runs.
\item $\epsilon$ are additional detector efficiency corrections
  obtained from data which were not included or not correctly
  described in the {\small MC}. In this particular case, this are the
  trigger efficiency of $S0$ \cite{olaf}.
\item $\delta$ represents the radiative corrections, calculated as,
\[
\delta = \frac{\sigma^{full}}{\sigma^{Born}} -1
\]
where $\sigma^{full}$ and $\sigma^{Born}$ denotes the full and Born
cross sections.
\item $\beta_{BC}$ are the bin center corrections which have to be
  applied in order to take the maximun variation of the cross section
  over the bin size and to obtain the cross section at the bin
  central value 
\[
\beta_{BC} = \frac{d^2 \sigma_{Born}}{dxdQ^2} \frac{1}{\sigma_{Born}}
\]
\end{itemize}

The cross section measurement is significantly simplified if the
{\small MC} already contains the radiative corrections. In such case
$N_{gen} = \sigma_{full} L_{{\small MC}}$ and therefore the cross section
formula given in Eq. \ref{Eqcrosssection} can be written as:

\begin{equation}
\frac{d^{2}\sigma(x,Q^2)}{dxdQ^2} =
\frac{N_{data}-N_{background}}{N_{rec}} \frac{L_{MC}}{L} \, \frac{d^{2}\sigma_{Born}}{dxdQ^2}
\frac{1}{\epsilon} 
\label{Eqfinalcrossection}
\end{equation}

since:

\begin{eqnarray*}
\frac{1}{A} \, \frac{1}{1+\delta} \, \beta_{BC} &=&
\left[\frac{N_{gen}}{N_{rec}}\right]
\left[\frac{\sigma_{Born}}{\sigma_{full}}\right]
\left[\frac{d^{2}\sigma_{Born}}{dxdQ^2} \frac{1}{\sigma_{Born}}\right]
\\
& = & \frac{L_{MC}}{N_{rec}} \frac{d^{2}\sigma_{Born}}{dxdQ^2}
\end{eqnarray*}

The Monte Carlo method is applicable just in case the MC already
contains the radiative corrections and the relevant measured
distributions are described by the MC. Thus, it is important to
perform detailed studies on the description between the {\small MC} and the
data for all measured quantities. This involves calibration, alignment and
efficiency corrections. In chapter \ref{Chapspacalib} and
\ref{Chaphadfinalstate} it was in detail 
explained how the calibration of the H1 calorimeter were performed to the
shifted vertex data. The alignment of the H1 detector was studied in
\cite{olaf}, while the BST efficiency were studied in \cite{paper,tomas}. 

\subsection{Bin selection}
\label{Secbinselection}

\begin{wraptable}[9]{r}{0.45\textwidth}
\begin{center}
\begin{tabular}{c c c}
\hline
high & low & bin \\
boundary & boundary & center \\
\hline
0.85 & 0.75 & 0.8 \\
0.75 & 0.6 & 0.675 \\
\hline
\end{tabular}
\label{Tabyq2def}
\setcaptionwidth{0.4\textwidth}
\caption{Boundary definition for the $y-Q^2$ binning}
\end{center}
\end{wraptable}

The binning used in this analysis is the same that was defined to
perform the DIS cross section measurement for the $1995$ data. The
binning is chosen as a non-equidistant grid, mainly in the $x\!-\!Q^2$
plane. Only in case the event has a value of $y$ greater than
$0.6$ a $y\!-\!Q^2$ binning is chosen. This procedure is performed to
profit from the statistic of the high $y$ events.

In Fig.\ref{Figstatistic} is shown the $Q^2\!-\!x$ plane together with
the bin definition. A trapezoidal shape is observed for each $Q^2$ bin
located in the lowest values of $x$. This trapezoidal bins are defined
in a $y$-$Q^2$ grid, with the boundaries defined in Table
\ref{Tabyq2def}. The rectangular grid in $y$, $Q^2$ is transformed to
a trapezoid area in $x$ and $Q^2$.

The statistics accumulated for each bin for the shifted vertex
data is shown in Fig. \ref{Figstatistic}, using either the electron 
or the $\Sigma$ method. From Fig. \ref{Figstatistic}, it follows that
the electron and the $\Sigma$ method have comparable statistics for
``high'' $Q^2$ values and $x$ around $10^{-5}$. For higher values of
$x$ at low $Q^2$, no events are reconstructed with the electron method
due the $E-p_{z}$ cut imposed 
to reduce the influence of radiative events, for which the electron
method leads to a wrong reconstruction. The $E-p_{z}$ cut is not 
necessary for the $\Sigma$ method, allowing therefore to access higher
values of $x$ for low values of $Q^2$. 

\begin{figure}[t]
\begin{center}
\begin{minipage}{0.485\textwidth}
\epsfig{figure=./selecdis/figures/statistic_data_m1_060515_0018.eps,width=\textwidth}
\put(-190.,170.){(a)}
\end{minipage}
\begin{minipage}{0.485\textwidth}
\epsfig{figure=./selecdis/figures/statistic_data_m2_060515_0018.eps,width=\textwidth}
\put(-190.,170.){(b)}
\end{minipage}
\begin{minipage}{0.485\textwidth}
\setcaptionwidth{\textwidth}
\caption{Statistics of each $Q^2-x$ bin during the shifted vertex data
  (a) reconstructed with the electron method and (b) reconstructed
  with the $\Sigma$ method}
\label{Figstatistic}
\end{minipage}
\begin{minipage}{0.485\textwidth}
\epsfig{figure=./selecdis/figures/bincolor.eps,width=\textwidth}
\end{minipage}
\end{center}
\end{figure}

Besides an equally distribution of the statistics, the aim of the
binning is also to have a minimal migration of events between the
bins. A bad 
adjustement of the bin size can cause migrations from bin to bin and
therefore the cross section measurement can be influenced. In order to
control the migration of the events between the bins, the {\it purity}
and {\it stability} value for each bin is studied. 

The purity and stability value are defined as:

\begin{equation}
Purity = \frac{N_{rec\&gen}}{N_{rec}} \hspace{2cm} Stability =  \frac{N_{rec\&gen}}{N_{gen}}  
\label{Eqpurityandstability}
\end{equation}

\begin{figure}[h]
\begin{center}
\begin{minipage}{0.45\textwidth}
\epsfig{figure=./selecdis/figures/purity_m1_060515_0018.eps,width=\textwidth}
\put(-180.,160.){(a)}
\end{minipage}
\begin{minipage}{0.45\textwidth}
\epsfig{figure=./selecdis/figures/purity_m2_060515_0018.eps,width=\textwidth}
\put(-180.,160.){(b)}
\end{minipage}
\begin{minipage}{0.45\textwidth}
\epsfig{figure=./selecdis/figures/stability_m1_060515_0018.eps,width=\textwidth}
\put(-180.,160.){(c)}
\end{minipage}
\begin{minipage}{0.45\textwidth}
\epsfig{figure=./selecdis/figures/stability_m2_060515_0018.eps,width=\textwidth}
\put(-180.,160.){(d)}
\end{minipage}
\begin{minipage}{0.45\textwidth}
\setcaptionwidth{\textwidth}
\caption{Purity and Stability value for each bin during the shifted vertex data
  (a),(c) reconstructed with the electron method and (b),(d) reconstructed
  with the $\Sigma$ method}
\label{Figstabpuri}
\end{minipage}
\begin{minipage}{0.45\textwidth}
\epsfig{figure=./selecdis/figures/bincolorpuristab.eps,width=\textwidth}
\end{minipage}
\end{center}
\end{figure}

where $N_{rec}$ and $N_{gen}$ are the numbers of events reconstructed
and generated in a particular bin. $N_{rec \& gen}$ represents the
number of events which are reconstructed {\it and} generated in
the same bin. 

The purity is a measure for the migration of events coming from
adjacent bins into the studied bin, while the stability
controls the migration of events out of the bin. The values of both
quantities are by definition restricted to the interval 0 to 1. In
Fig. \ref{Figstabpuri}, the purity and stability values in per cent
for each bin are presented. Fig. \ref{Figstabpuri} (a) and (c) show the
purity and stability obtained using the electron method, while in
Fig. \ref{Figstabpuri} (b) and (d) the purity and stability
values obtained with the $\Sigma$ method are presented. From this
plots, one can see 
that the electron method has values of purity and stability between
$40\%$ and $70\%$. The $\Sigma$ method, in contrary has lower values,
between $25\%$ and $50\%$.

The differential cross section is measured in each bin
only in case the values of the purity and stability are greater than
$25\%$. In the kinematical region where the electron and the
$\Sigma$ method overlap, the method having higher values of stability
and purity is chosen.

Although the value of purity and stability for the $\Sigma$ method are
lower than the values of the electron method, the $\Sigma$ method
extends the measurement to higher values of $x$ and lower values of $Q^2$. 
In Fig. \ref{Figstabpuri}, is also shown the acceptance of the BST
when the angle of the scattered electron has its maximal value
($\theta \approx 178^{\circ}$). One can see, that
there are events which are localized below the acceptance line, those
events are ISR events.  
The purity and stability values of the ISR events are greater than $25\%$,
therefore the measurement of the cross section should be possible.
Thus before to present the cross section measurement, the
reconstruction of the incoming energy for the radiative events is in
the next section in detail studied.  

\subsection{Non-tagged ISR analysis}
\label{Secnotagisr}

\begin{wrapfigure}[18]{r}{0.45\textwidth}
\begin{center}
\epsfig{figure=./selecdis/figures/inbinsgen_060523_1134.eps,width=0.4\textwidth}
\begin{minipage}{0.38\textwidth}
\setcaptionwidth{\textwidth}
\caption{Comparison between generated and reconstructed incoming
  energy for ISR events having $Q^2 < 1.2\,$GeV$^2$.}
\label{Figisrgen}
\end{minipage}
\end{center}
\end{wrapfigure}

For the electron method the ISR events are efficiently removed using
the $(E-p_{z})_{tot}> 35\,$GeV cut. Nevertheless, the
$(E-p_z)>35\,$GeV cut is not performed in case the kinematical
variables are reconstructed with the $\Sigma$ method. In this case the
ISR event are used to enlarge the accesible phase space.
%The ISR events are localised below the acceptance of the BST using
%the nominal value of the centre of mass energy $s=4E_{inc}E_{p}$.

In this analysis the radiated photon is {\it not directly
detected} in the photon detector of the luminosity system. The
identification of the initial 
state radiation relies mainly on the recontruction of the $(E-p_z)_{tot}$. One
important check to be performed is the reconstruction of 
the incoming energy $E_{inc}$.

As was already mentioned in Chapter \ref{Chdatasample}, the
$\Sigma$-method uses the 
longitudinal momentum conservation to define the event kinematic, therefore
the incoming energy can be reconstruced as:

\begin{equation}
E_{inc} = \frac{(E-p_{z})_{had} + (E-p_{z})_{el}}{2}
\label{Eqisr}
\end{equation}

\begin{figure}[t]
\begin{center}
\epsfig{figure=./selecdis/figures/inbins_060523_1134.eps,width=0.95\textwidth}
\put(-380.,50.){(a)}
\put(-170.,50.){(b)}
\caption{Control distributions for ISR events having $Q^2 <
  1.2\,$GeV$^2$. (a) The energy of the incoming electron. (b) The
  kinematical variable $y_{\Sigma}$} 
\label{Figisrcontrol}
\end{center}
\end{figure}

For {\small MC} one can verify, if the reconstructed energy of the ISR
events agree with the generated one. In Fig. \ref{Figisrgen} the
distribution of the reconstructed and the generated energy of the
incoming electron are compared. The reconstructed
incoming energy $E_{inc}$ is calculated with
Eq. \ref{Eqisr} and is illustrated with the poins in Fig. \ref{Figisrgen},
while in solid line is shown the incoming energy from
the generator level. Only events with $Q^2<0.85\,$GeV$^2$ and having
values of $x$ in the interval $5 \cdot 10^{-5} < x < 10^{-3}$ are
shown. The selected events are located in bins where the 
values of purity and stability are higher than $25\%$. In
such way the contribution from ISR events 
is high. On generator level the energy of the radiated photon is
substracted from the  
energy of the incoming electron in case the event was explicitly
generated as a ISR event. A good
agreement between the generated and reconstructed energy of the
incoming electron for the main kinematic region is observed. 

A pronounced peak is observed in the incoming energy $E_{inc}$
distribution presented in Fig. \ref{Figisrgen}, those events are
$5.5\%$ of the whole statistic and are non ISR events
reconstructed with small $(E-p_z)_{had}$. However, most of the events are
good described by the MC simulation.
The contribution of such non-radiative events becomes higher for
values of $x<10^{-5}$ and $Q^2 <0.5$GeV$^2$, leading to lower values of
stability and purity (see Fig. \ref{Figstabpuri} (b) and (d)).

%However, the most events are good
%described by the MC simulation.

%For $Q^2<0.85$ and $10^{-6}<x<10^{-5}$ the purity and stability value
%of the bins decreases.Such bins have a higher contribution of events
%which are reconstructed with low value of $(E-p_z)_{had}$. Therefore
%the error of the poin
%bins  and simulates radiative 

In Fig. \ref{Figisrcontrol} (a) and (b) are shown the energy of the
incoming electron and the kinematical variable $y_{\Sigma}$ for data
and {\small MC} for events with $Q^{2}_{\Sigma}<1.2\,$GeV$^2$. The {\small MC}
simulation is in good agreement with the data. 

\begin{figure}[t]
\begin{center}
\epsfig{figure=./selecdis/figures/extractf2_060712_1543_red.eps,width=1.\textwidth}
\caption{Cross section measurement as function of $x$ for different
  $Q^2$ values. Red points were recontructed by the electron method,
  blue poins with the $\Sigma$ method. Only statistical errors are
  shown}
\label{Figcrossection}
\end{center}
\end{figure}

\section{Cross Section Measurement and Extraction of $F_2$}

The cross section is measured following
Eq.\ref{Eqfinalcrossection}. The $full$ cross section from the {\small MC}
uses the proton structure functions $F_2$ from the fractal fit
\cite{tomaspaper} and from the ALLM97 \cite{allm97} parametrizations,
while the $F_L$
contribution is taken from the dipole model \cite{dipoleint}. A
similar Eq. to \ref{Eqfinalcrossection} is obtained for the reduced
cross section (see Eq. \ref{sigmaredred}). 

\begin{figure}[t]
\begin{center}
\epsfig{figure=./selecdis/figures/extractf2_060712_1543_f2.eps,width=1.\textwidth}
\caption{The proton structure function $F_2$ as function of $x$ for different
  $Q^2$ values. The dashed line correspond to the ALLM97 fit \cite{allm97},
  while the full line is the expectation from the Fractal fit \cite{tomaspaper}}
\label{Figf2}
\end{center}
\end{figure}

In Fig. \ref{Figcrossection} is shown the 
reduced cross section as a function of $x$ in different $Q^2$
bins. The points determined by the electron method are presented in
red, while the points measured using the $\Sigma$ method are marked in
blue. The electron method is used up 
to values of $x\! \approx \!10^{-4}$, while the $\Sigma$ method cover the
measurement up to values of $x \approx 10^{-2}$. 

The errors presented in Fig. \ref{Figcrossection} are only statistical
errors. The radiative points have statistical errors which vary between
$10\%-5\%$ for the lowest $Q^2$ bins and $2\% - 3\%$ for the high $Q^2 >
2\,$GeV$^2$ region. The systematics errors are presented in the next
section. 

The proton structure function $F_2$ is extracted from the measured
cross section and the prediction of $F_L$ from the dipole model. 

\begin{equation}
%F_2 = \sigma_{r} \left[1-\frac{y^2}{Y_{+}} \frac{R}{1+R}\right]^{-1}
F_2 = \sigma_{red} + \frac{y^{2}}{Y_{+}} F_{L}
\label{Eqextractf2}
\end{equation}

The measurement of $F_2$ is shown in Fig. \ref{Figf2}, where also
only stastistical errors are presented. 

%For the region of high $x$, $F_2$ will be taken from
%the ALLM97 parametrization, meanwhile for low values of $x$, the {\small MC}
%has to be reweighted with the 


\section{Systematic Errors}
\label{Secsystematic}

The measured cross section and the extraction of $F_2$ is influenced by
systematic effects, some of which were derived in chapter
\ref{Chapspacalib} and \ref{Chaphadfinalstate}. There are two types of
systematic uncertainties, known as uncorrelated and correlated
uncertainties. 

\begin{table}[t]
\begin{center}
\begin{tabular}{c c c}
\hline
\hline 
\multicolumn{3}{c}{Uncorrelated error} \\
Source & value & Reference \\
\hline
\hline
Luminosity & $2.2\%$ & \cite{levonian} \\
Trigger Efficiency & $0.5\%$ & \cite{olaf} \\
Radiative corrections & $0.5\%$ & \cite{paper} \\
BST efficiency & $2\%$ & \cite{tomas,paper} \\
\hline
\hline
\multicolumn{3}{c}{Correlated error} \\
Source & value & Reference \\
\hline
\hline 
angle $\theta_{e}$ & $0.2$mrad & \cite{paper} \\
Electron Energy & $0.02\%$ at $27.5\,$GeV & see chapter \ref{Chapspacalib} \\
& $1\%$ at $2.5$GeV & \\
Hadronic energy & $10\%$ at $y=10^{-3}$ & see chapter \ref{Chaphadfinalstate} \\
& $2\%$ at $y=0.1$ \\
LAr noise & $10\%$ & see chapter \ref{Chaphadfinalstate} \\
Hadronical SpaCal & $350\,$MeV & see chapter \ref{Chaphadfinalstate} \\
$\gamma p$ Background & $15\%$ & \cite{tomas, paper} \\
\hline
\hline
\end{tabular}
\caption{Systematic uncertainties of the DIS cross section measurement}
\label{Tabsyst}
\end{center}
\end{table}

The correlated uncertainties are global normalization uncertainties
which affect as a whole the measurement. This are for example
uncertainties of the
luminosity, the trigger efficiency and radiative corrections.
The uncorrelated uncertainties result from the implemented mechanism
used to reconstruct the kinematical variables, this involves for
example energy
scale uncertainties, noise substraction procedures and angular
measurements. 


Uncorrelated errors influence the measured values
independent of each other. Many systematic errors are partly
correlated and partly uncorrelated. In the following the global
normalization uncertainties will be treated as correlated, while all
other will be treated as fully uncorrelated. In Table 
\ref{Tabsyst} are summarized all the uncorrelated and correlated
sources which will be used to determine the total systematic error.

In order to estimate the influence of the uncorrelated and correlated
uncertainties on the measurement, the values of all relevant quantities were
varied within their uncertainties in positive and negative
direction. After varying the relevant quantities, the kinematical
variables are recalculated and the measurement of the cross section or
$F_2$ extraction is performed again. The deviation of the ``varied''
measurement from the original value is treated as the systematic
uncertainty caused by the particular error source $\delta$. The total
error $\delta_{tot}$ will be computed as the quadratic summation over all error
sources listed in Table \ref{Tabsyst} and the statistical error.

\begin{equation}
\delta_{tot} = \sqrt{\delta_{stat}^{2} +\delta_{corr}^{2} +
  \delta_{uncorr}^{2}} \\
\label{Eqtotalerror}
\end{equation}

where $\delta_{stat},\delta_{corr}$ and $\delta_{uncorr}$ are the
statistical, correlated and uncorrelated errors
respectively. Following the table \ref{Tabsyst} $\delta_{corr}$ and
$\delta_{uncorr}$ are calculated as:  

\[
\delta_{corr} =
  \sqrt{\delta_{ee}^{2}+\delta_{\theta}^{2}+\delta_{LAr}^{2}+\delta_{Noise}^{2}+\delta_{HadSpaCal}^{2}+\delta_{\gamma p}^{2}} 
\]
\[
\delta_{uncorr} =
  \sqrt{\delta_{trigger}^{2}+\delta_{Luminosity}^{2}+\delta_{BST}^{2}+\delta_{rad-corr}^{2}}
\]

\begin{figure}[t]
\begin{center}
\begin{minipage}{1.\textwidth}
\epsfig{figure=./selecdis/figures/syst_q10.eps,width=0.5\textwidth}
\put(-180.,40.){(a)}
\epsfig{figure=./selecdis/figures/crossectionplot_060524_1637.eps,width=0.5\textwidth}
\put(-180.,40.){(b)}
\end{minipage}
\begin{minipage}{\textwidth}
\setcaptionwidth{0.95\textwidth}
\caption{(a) Variation of the $F_2$ points for $Q^{2}=2.5\,$GeV$^2$ in
  case the energy of the scattered electron is moved according its
  uncertainty. (b) Systematic error due to the energy uncertainty
  for $Q^{2}=2.5\,$GeV$^2$.}
\label{Figbinenergy}
\end{minipage}
\end{center}
\end{figure}

Fig. \ref{Figbinenergy}(a) shows an example of the
variations of $F_2$ in case the energy of the scattered electron $E_e$ is
changed according its uncertainty to the positive and negative direction:

\begin{equation*}
E_e^{\pm} = E_e(1+ \Delta^{\pm}) 
\end{equation*}

\noindent where $\Delta^{\pm}$ is defined as:

\begin{equation*}
\Delta^{\pm} = \left[ (\Delta E_{kin})^{\pm} +
            \frac{(\Delta E_{kin})^{\pm}- (\Delta E_{\pi^{0}})^{\pm}}{E_{kin}-E_{\pi^{0}}}(E_e-E_{kin})
            \right]
\end{equation*}

$\Delta E_{kin} = 0.02 \%$ and $\Delta E_{\pi^{0}}=1\%$ are the
uncertainties of the energy measurement at kinematic peak and at
$\pi^{0}$ energy, respectively. 
The systematic error for this particular $Q^2=2.5\,$GeV$^2$ bin due to
the energy measurement uncertainty varies between $0.2\%$ and $1.5\%$ (see
Fig. \ref{Figbinenergy}(b)). 

\begin{figure}[t]
\begin{center}
\epsfig{figure=./selecdis/figures/extractf2_060712_1543_f2sys.eps,width=1.\textwidth}
\caption{The proton structure function $F_2$ as function of $x$ for different
  $Q^2$ values.The inner error bars correspond to the statistical
  uncertainty while the outer error bars show the total error
  obtaining by adding statistical and systematical uncertainties in
  quadrature. The dashed line correspond to the ALLM97 fit,
  while the full line is the expectation from the Fractal fit.}
\label{Figf2syst}
\end{center}
\end{figure}

%In Fig. \ref{Figbinenergy}(c) is shown the systematic error of $F_2$ for
%$Q^2=0.65\,$GeV$^2$ due to the uncertainty of the hadronical
%energy. The systematic error in this case varies from $4\%$ to $6\%$.
In Fig. \ref{Figf2syst} is shown the proton structure function $F_2$
as function of $x$ for different $Q^2$ bins with the total
systematic error. For $Q^2 < 1.5\,$GeV$^2$ the measurement has a total error
which varies between $5\%-15\%$. For $Q^2 > 1.5\,$GeV$^2$ the
total error vary between $2\%-4\%$. The values of the cross section,
$F_2$ and the error sources are summarized in Appendix A. 

The comparison of the measurement performed in this analysis with the
previous analysis is shown in Fig. \ref{Figf2syst2}. One can see that the
measurements agree in shape in the overlap bins. The present analysis
extend the phase space to lower values of $Q^2=0.2$ and
$0.25\,$GeV$^2$. The present measurement is $5\%$ higher than the
presented in \cite{tomas}. This discrepancy can be traced back to a
wrong luminosity value used in \cite{tomas}.

In Fig. \ref{Figf2syst3} is shown the comparison of the measured
points of this analysis with the published data of H1 performed with
shifted vertex data taken in the year 1995. Good agreement in
normalization is found between the analysis here presented and the
published measurement. The systematical errors of this analysis, for
values of $Q^2>2\,$GeV$^2$ are reduced by a factor of two, in
comparison with the previous shifted vertex data. The measurement for
$Q^2<2\,$GeV$^2$ have total errors comparable to
the 1995 shifted vertex measurement.


\begin{figure}[t]
\begin{center}
\epsfig{figure=./selecdis/figures/extractf2_060712_1543_f2sys2.eps,width=1.\textwidth}
\caption{Comparison between the measurement of the proton structure
  function $F_2$ of the present analysis and the preliminary performed
  in \cite{tomas} (open points). The dashed line correspond to the
  ALLM97 fit, while the full line is the expectation from the Fractal fit.}
\label{Figf2syst2}
\end{center}
\end{figure}


\begin{figure}[t]
\begin{center}
\epsfig{figure=./selecdis/figures/extractf2_060712_1543_f2sys_vs95data.eps,width=1.\textwidth}
\caption{Comparison between the measurement of the proton structure
  function $F_2$ of the present analysis and the measurement
  of H1 performed with shifted vertex data in the year $1995$. 
  The dashed line correspond to the
  ALLM97 fit, while the full line is the expectation from the Fractal fit.}
\label{Figf2syst3}
\end{center}
\end{figure}
