\chapter{Data and Monte Carlo sample}
\label{Chdatasample}

In August of $2000$ a special run period was performed by H1 to
measure low values of $Q^2$ below the nominal value of $2\,$GeV$^2$. 
The $ep$ interaction point was shifted by $70\,$cm in the proton
direction, reaching higher acceptance in the backward region of the H1
detector. This
special ``shifted vertex run'' took data over $3$ days, recording an
integrated luminosity of about $600\,$nb$^{-1}$. This amount of luminosity
represents two times more statistic than the previous shifted vertex
run taken in the year $1995$. 

In this chapter will be described the reconstruction methods used to
analyze the shifted vertex data, afterwards the cuts used to
identify the scattered electron are introduced and its
efficiency is studied. At the end of the chapter will be described the
background of DIS events for low values of $Q^2$. 

\section{The Shifted vertex run}
\label{Secshiftedvertex}

In DIS events for values of $Q^2 \leq 100\,$GeV$^2$, the electron is
scattered into the backward region of the H1 detector. 
There are two possibilities to measure the angle of the scattered
electron at low values of $Q^2$: 
One possibility uses the track measured by the backward silicon tracker
(BST), while the other possibility reconstructs the angle with the  
backward drift chamber (BDC) and the central tracking devices  
(see section \ref{Secbst} and \ref{Secbdc}). 

Usually, when the collision between the electron and the proton
beams takes place in the nomi\-nal position, the maximum value of the 
angle of the scattered electron that can be measured by the BST and
the BDC is $176.5^{\circ}$ and $176.8^{\circ}$ respectively. 
During the shifted vertex run period, the interaction point was
moved in the proton direction by $70\,$cm, increasing the 
acceptance of the backwards devices as the BST and the BDC.  

In Fig. \ref{Figbsttheta} the $Q^2-x$ plane is shown, with lines of
constant scattering angles. The solid lines $\theta_{e} = 176.5 
^{\circ}$ and $\theta_{e} = 169 ^{\circ}$, are the maximum and
minimum angle of the scattered electron that can be measured by the
BST at the nominal vertex position ($z=0\,$cm). The
hatched area in Fig. \ref{Figbsttheta} represents the angular
acceptance of the BST in case the vertex 
is shifted by $z=70\,$cm. The solid area in Fig. \ref{Figbsttheta}
illustrates the additional maximal angular acceptance that the BDC and the CTD
have in case the vertex is shifted $70\,$cm from the nominal
position.

From Fig. \ref{Figbsttheta} follows that during the
shifted vertex period the acceptance of the BST and the BDC shifts
towards lower values of $Q^2$, opening the possibility to measure
lower values of $x$ for low values of $Q^2$.  

\begin{figure}[t]
\begin{center}
\includegraphics[width=10.cm]{./resolution/figures/q2xplane.eps}
\caption{Acceptance of the BST during the shifted vertex data period
  $174^{\circ} \leq \theta_e \leq 177.5^{\circ}$ (hatched area)
  compared to the acceptance of the BST if the vertex 
  is localised at the nominal position $z=0.\,$cm ($169 ^{\circ} \leq
  \theta_e \leq 176.5^{\circ}$). The additional maximal angular
  acceptance of the BDC is also shown (solid area $177.5 ^{\circ} \leq
  \theta_e \leq 177.8^{\circ}$ ).} 
\label{Figbsttheta}
\end{center}
\end{figure}

In order to measure the inclusive cross section of the
DIS process $ep \rightarrow eX$ for these low values of $x$ and $Q^2$,
the identification of the scattered electron with the backward
devices is a task of prime importance. In order to profit from the two
detector components used to identify the scattered electron and to
derive the systematic errors of the measurement, two analysis in
parallel were performed. In the analysis developed in
\cite{olaf} the BDC is used as the main detector to reconstruct
together with the CTD and the SpaCal the angle of the scattered
electron. The analysis presented in this thesis uses the BST and
the SpaCal calorimeter to identify the scattered electron.

Several effects like background contributions and detector resolution,
have to be studied in order to correctly identify the scattered
electron, for example:

\begin{itemize}
\item The scattered electron can escape the
  detection of the backward devices and a charged particle coming
  from the hadronical final state can be misidentified as the
  scattered electron. This background events are known as
  photoproduction events and have a contribution of $5\%$ for $y \geq
  0.5$ (see section \ref{Secphotoproduction}). In order to reduce the
  photoproduction background, not only the electron identification is
  important but also the hadronical final state had to be
  studied in detail.
\item The reconstruction of the kinematical variables $x$ and $Q^2$
  can be performed using exclusively the energy and momenta measurements of the
  scattered electron. Although this reconstruction method is the most
  precise for low values of $x \approx 10^{-4}$, it begins to be less
  accurate for values of $x\approx 10^{-3}$. Other possibilities used to
  reconstruct the kinematical variables 
  for high values of $x$ consider also the energy and momenta
  measurement of the hadronical final state.
\end{itemize}

In section \ref{Secrecmethods} the methods used to reconstruct the
kinematical variables $x$ and $Q^2$ are introduced, there the  
resolution of the kinematical variables are studied using a Monte Carlo
simulation (MC) with the characteristics of the shifted vertex data. 
This step is necessary since the measured distributions like energies
and positions are a convolution of pure physics 
characteristic with intrinsic detector resolutions, therefore the Monte Carlo
simulation is used to reproduce the behavior of the data and to
separate as good as possible the pure physic effects from the detector
resolutions. The MC used in this analysis is discussed in section
\ref{Secmontecarlo}. 

In section \ref{Secelecidentification} the electron identification using the
SpaCal calorimeter and the BST is presented, afterwards in section
\ref{Secloweanalysis} the efficiency of the cuts used to identify the
scattered electron are discussed. At the end in section
\ref{Secphotoproduction}, the main source of background to DIS events
is explained and its contribution during the shifted vertex data is
determined.  

\section{Monte Carlo Sample}
\label{Secmontecarlo}

The Monte Carlo (MC) samples are produced in two steps, first a generator is
used and then the detector simulation and the reconstruction of the
event is performed.

The generator has the task to produce pure physic events according to
theory or models, in such way four vectors and cross sections are
obtained. The simulation uses this information to produce the detector
response. In this analysis the generator programs {\small DJANGOH1.4}
\cite{django} 
and {\small PHOJET} \cite{phojet} were used to produce the signal and
background of DIS events. Next, the package {\small H1SIMREC}
\cite{h1sim,h1rec} was used to simulate and produce the detector
response. In the following paragraphs the general characteristic of
the MC generators and the simulation step are described.

\begin{itemize}
\item The {\small DJANGOH1.4} program is based on several packages.
%{\small HERACLES}
%  \cite{heracles} and {\small LEPTO} \cite{lepto}. 
  {\small HERACLES}\cite{heracles} generates the event at parton
  level and allows the integration of the differential cross section
  for $ep \rightarrow e^{\prime}X$ over kinematical regions which can
  be defined in terms of $x,y,Q^2$. {\small HERACLES} can perform a
  separate treatment of the Born term of the cross section and several
  parts of the QED corrections. For NC processes, the QED corrections
  include leptonic 
  initial state radiation (ISR), leptonic final state radiation (FSR),
  a contribution called Compton part and quarkonic radiation. Within
  {\small HERACLES} is possible to 
  choose among a set of parametrizations for input parton
  distributions. In this analysis are used the parton distribution
  parametrization from 
  the {\small MRST} group \cite{mrstv804}. {\small HERACLES} can also
  simulate electroweak processes to first order in $\alpha_s$ and one
  loop corrections. 
  
  In order to have a complete event simulation not only at parton
  level, {\small DJANGOH} call routines from {\small LEPTO}\cite{lepto} and
  {\small JETSET} 
  \cite{jetset}, in such way the complete event simulation including
  fragmentation and hadronization of the scattered quark and the
  proton remmmant are obtained. 

  The program {\small LEPTO} is used to calculate
  the QCD matrix elements and generate the hadronic final state on the
  parton level. Parton 
  cascades are generated by {\small ARIADNE}\cite{ariadne} using the
  Color Dipole Model. While, the fragmentation step is computed using the
  Lund-String model as implemented in {\small JETSET}. 
  In order to simulate the hadronic final state at low $W$, the
  {\small SOPHIA}\cite{sophia} package is used. In this analysis a cut
  on $W=5\,$GeV is performed to define the boundary where {\small
  SOPHIA} and {\small LEPTO+ARIADNE} had to be used.

  Diffractive events are simulated with {\small ARIADNE} using the
  Soft Color Interaction (SCI) model. This implementation does not take
  into account the pomeron quantum numbers. It produces a reasonable
  description of the inclusive energy flow in DIS events, but it
  creates huge backgrounds to exclusive process like
  DVCS\cite{rainer} and QEDC\cite{victor}. A study of the {\small
  DJANGOH1.4} excluding the SCI will be discussed in section
  \ref{Secdiffevents}  

\item {\small PHOJET}\cite{phojet} simulates the total photoproduction cross
  section, incorporating soft and hard hadronic processes. The
  hadronization is performed by {\small JETSET}. Photo-production is
  the main physical background for DIS events at low $Q^2$. The amount
  of photoproduction events in the DIS sample, is studied 
  using part of the luminosity system and will be discussed in
  section \ref{Secphotoproduction} in detail.

\item {\small H1SIMREC}\cite{h1sim,h1rec}: The generated events are fed
  into the detector simulation 
  H1SIM based on GEANT\cite{geant}. H1SIM simulates the H1 detector response,
  taking into account the geometry and acceptance of its subdetectors
  as well as dead material distributed within H1. H1DIGI runs after
  H1SIM and perform a ``digitalisation'' of the simulated hits. In
  this step efficiencies and noise corrections and calibration
  constants obtained from data and test beam studies are
  applied. H1REC reconstruct finally the whole event as it is done
  for real data. 

\end{itemize}

The MC samples used in this analysis have in total 20 millions of events
for both {\small DJANGOH} and {\small PHOJET}.

In the following section, the MC simulation is used to study the
resolution of the reconstructed variables $x$ and $Q^2$. The original
generated variables denoted by the subindex $gen$ are directly
compared with the reconstructed variables having a subindex
$e,\Sigma$. Since the resolution of the kinematical variables is
influenced by several effects, the final deviation of the reconstructed
variables from the generated ones is studied using the DIS
selection cuts introduced in chapter \ref{Chadisselc}.

\section{Reconstruction Methods}
\label{Secrecmethods}

As we have seen in chapter \ref{Ch1detector}, the H1 detector can
measure the energy and the angle of the scattered electron and the
energy depositions and momenta of the particles building the hadronic
final state. The event kinematic is over-constrained and several 
reconstruction methods can be defined using the total final state or
part of it. In the following sections the relevant reconstruction
methods employed are briefly described.

\subsection{Electron Method}
\label{SubSecelecmethod}

The electron method uses {\it purely} the measurement of the four
vectors of the incoming and scattered electron. The event kinematic is
given by: 

\begin{equation}
Q^{2}_{e}= E_{e}^{2} \frac{\sin ^{2} \theta _{e}}{1-y_{e}} \hspace{1.4cm}
y_{e} = 1 - \frac{E_{e}}{E_{inc}^{0}} \sin ^{2} \frac{\theta_{e}}{2}
\hspace{1.4cm}
x_{e} = \frac{Q^{2}_{e}}{sy_{e}}
\label{Eqkinelec}
\end{equation} 

\noindent where $E_{inc}^{0}=27.5\,$GeV is the energy of the incoming
electron. The variables $E_{e}$ and $\theta_{e}$ are the energy and
polar angle of the scattered electron.

The resolution of the event kinematic depends on the
accuracy of the energy measurement of the scattered 
electron $E_e$ and its polar angle $\theta_{e}$, the resolution
is given by:

\begin{eqnarray}
\frac{\delta Q^{2}_{e}}{Q^{2}_{e}} = \frac{\delta E_e}{E_e} \oplus \tan
\left(\frac{\theta_e}{2} \right) \delta \theta_e \\
\label{q2elecres}
\frac{\delta y_{e}}{y_{e}} = \frac{1-y_{e}}{y_{e}} \left(\frac{\delta E_e}{E_{e}} \oplus
\frac{\delta \theta_e}{\tan \theta_e/2} \right)
\label{yelres}
\end{eqnarray}

From Eq. \ref{yelres} it is clear that the electron
method is very precise for high values of $y$, due to the factor
$(1-y)/y$, and begins to be less precise for low values of $y$. 

\begin{figure}[t]
\centering
\begin{minipage}[c]{0.8\textwidth}
\centering
\setlength{\unitlength}{1cm}
\includegraphics[width=\textwidth]{./resolution/figures/xres_elec_060423_2121.eps}
\put(-2.,5.5){\small{(a)}}
\end{minipage}
\centering
\begin{minipage}[c]{0.8\textwidth}
\centering
\setlength{\unitlength}{1cm}
\includegraphics[width=\textwidth]{./resolution/figures/q2res_elec_060423_2121.eps}
\put(-2.,5.5){\small{(b)}}
\end{minipage}
\caption{Resolution of the kinematical variables (a)$x$ and (b)$Q^2$ in the
  case they are reconstructed with the electron method.}
\label{Figresx}
\end{figure}

Fig. \ref{Figresx} show the relative deviation of $x_e$ and $Q_{e}^2$
from the generated variables $x_{gen}$ and $Q^2_{gen}$ for the whole DIS
selection. 

Each point in Fig. \ref{Figresx}(a) and (b) correspond to the mean
position and spread of a Gaussian fit restricted to the interval:

\begin{equation}
-0.3 \leq \frac{x_{e}-x_{gen}}{x_{gen}} \leq 0.3 \hspace{2cm}
-0.5 \leq \frac{Q_{e}^{2}-Q_{gen}^{2}}{Q_{gen}^{2}} \leq 0.5
\label{Eqelectronfit}
\end{equation}

In such way the Gaussian fit converges giving the correct peak
position of the scattered distribution also shown in
Fig. \ref{Figresx}. 

Form Fig. \ref{Figresx}(a) follows that the electron method is very
precise for low values of $x \leq 10^{-4}$ (high $y$) and begins to be
less accurate for higher values of $x_{gen} \approx 10^{-3}$. For such
values $x$ is reconstructed to lower values as it was
generated. The resolution of $x$ degrades from $5\%$ at $x \approx
10^{-5}$ to almost $20\%$ at $x \approx 10^{-3}$. The reconstruction
of $Q^{2}_{e}$ is precise, having 
relative values less than $5\%$ (see Fig. \ref{Figresx}(b)).  

Since the relative resolution of $x$ deviates significantly from the
generated quantities for values of $x>10^{-4}$, the electron method
can be used to measure the cross section up to values of $x\approx
10^{-4}$ (see Fig. \ref{Figresx}(a)). 

The resolution of the electron method depends on the precise
measurement of the angle of the
scattered electron and on a precise energy measurement. This
requires good alignment between the H1 subdetectors and a good
electromagnetic calibration. The alignment of the detector components
was derived in \cite{olaf}, and the calibration of the SpaCal
calorimeter is presented in chapter \ref{Chapspacalib}.

Two important effects influence the reconstruction of events using
the electron method at low $x$ (high $y$); photoproduction ($\gamma
p$) and initial state radiation events (ISR), both phenomena will be
disscussed in section \ref{Secphotoproduction} and \ref{Secnotagisr}
respectively. 

%Photoproduction events generally appears at low $Q^2$ when the
%scattered electron escape the detection of the backwards devices and a
%charged hadron, generally a charged pion is wrongly reconstructed as
%the scattered electron. This background source will be in the section
%\ref{Secphotoproduction} in detail discussed.     

%Initial state radiation events are produced when the incoming
%electron radiates a photon. This results in a decrease of the centre of
%mass energy $s$ of the interaction and thus causes a wrong reconstruction
%of the event kinematic (see Eq. \ref{Eqkinelec}). 
%The ISR contribution is suppressed using the longitudinal energy
%conservation.

\subsection{Hadron or JB Method}
\label{SubSecthadmethod}

This reconstruction method uses the measurement of the four momentum
of all particles which define the hadronical final
state. In practise unfortunately,
some hadrons are undetected due to the uncovered
regions of the H1 detector used to inject the beams. Nevertheless it 
was shown by Jacques and Blondel \cite{hadronmethod} that the event
kinematic can be calculated using the variable:

\begin{equation}
\Sigma = (E_{h}-p_{z,h}) = (E-p_z)_{had}
\label{Eqsigma}
\end{equation}

where $(E-p_z)_{had}$ is the difference between the energy $E_{h}$ and
longitudinal momentum $p_{z,h}$ of all particles which define the hadronical
final state. Particles having low values of $p_{t}$ can escape the
detection, but these particles have a negligible contribution to the
total $(E-p_{z})_{had}$. The event kinematic is given by: 

\begin{equation}
y_{h} = \frac{\Sigma}{2E_e} \hspace{2cm} Q_{h}^{2} =
\frac{p_{t,had}^{2}}{1-y_{h}} \hspace{2cm} x_{h} = \frac{Q_{h}^{2}}{sy_{h}}
\label{hadkin}
\end{equation}

\noindent where $p_{t,had}$ correspond to the total transverse
momentum of the hadronic final state:

\begin{equation}
p_{t,had} = \sqrt{(\sum_{h} p_{x,h})^2 + (\sum_{h} p_{y,h})^{2}}
\label{Eqpthad}
\end{equation}

being $p_{x,h}$ and $p_{y,h}$ the $x$ and $y$ four-momentum components
of each particle which build the hadronical final state (see chapter
\ref{Chaphadfinalstate} for more details).  

The hadron method provides a precise measurement of $y$ for low and
medium values 
$y$($<0.15$). But, the hadron method is not adequate to measure
low $Q^2$ events, since the measurement of low $p_t$ particles is
not precise. 

Using the four momentum of the hadronic final state, the angle of the
hadronical final state $\theta_{had}$ is reconstructed as:

\begin{equation}
\cos \theta_{had} = \frac{\Sigma^2 - p_{t,had}^{2}}{\Sigma^2 +
  p_{t,had}^{2}}
\label{Eqthetahad}
\end{equation}

\subsection{$\Sigma$ Method}
\label{SubSecsigmamethod}

The $\Sigma$ method combines the measurement of the 
scattered electron and of the hadronic final state. This method
has a better precision than the hadron method for the event kinematic
at low values of $Q^2$ and low values of $y$. 

The inelasticity $y$ is reconstructed similar to the hadron method, but the 
energy of the incoming electron $E_{inc}$ is obtained using the longitudinal
energy conservation, thus:

\begin{eqnarray}
2E_{inc} &=& (E-p_{z})_{had} + (E-p_{z})_{el} = (E-p_z)_{tot} \\
\label{Eqeinc1x}
& = & \Sigma + E_{e}(1+\cos \theta_e)
\label{Eqeinc}
\end{eqnarray}

\noindent this substitution ensures a correct reconstruction of the
event kinematic at the hard interaction, even in the case where a
photon from the incoming electron is radiated. The event kinematic is
given by:

%This substitution ensures a correct reconstruction of the
%event kinematic, also for ISR events. The event kinematic is given by: 

\begin{equation}
y_{\Sigma} = \frac{\Sigma}{\Sigma + E_{e}(1+\cos \theta_e)}
\hspace{1.5cm} Q_{\Sigma}^{2} = \frac{E_{e}^{2} \sin ^{2}
  \theta_e}{1-y_{\Sigma}}         
\label{ysigma}
\end{equation}

Due to the correct reconstruction of ISR events, these can be used for
the analysis. Since the emission of the photon of the incoming
electron conduces to a reduced centre of mass energy $s$, the $\Sigma$
method extends the measurement to low values of $y$.
In contrast to the electron method, the ISR events are 
considered with the $\Sigma$ method.  

\begin{figure}[t]
\centering
\centering
\begin{minipage}[c]{0.8\textwidth}
\centering
\setlength{\unitlength}{1cm}
\includegraphics[width=\textwidth]{./resolution/figures/xres_sigma_060424_1809.eps}
\put(-2.,5.5){\small{(a)}}
\end{minipage}
\centering
\begin{minipage}[c]{0.8\textwidth}
\centering
\setlength{\unitlength}{1cm}
\includegraphics[width=\textwidth]{./resolution/figures/q2res_sigma_060424_1809.eps}
\put(-2.,5.5){\small{(b)}}
\end{minipage}
\caption{Resolution of the kinematical variables (a)$x$ and (b)$Q^2$ in 
  case they are reconstructed with the $\Sigma$ method.}
\label{Figxq2sigma}
\end{figure}

The resolution of the $\Sigma$ method is given by,

\begin{equation}
\frac{\delta y}{y} = (1-y) \left(\frac{\delta \Sigma}{\Sigma} \oplus
\frac{\delta E_e}{E_e} \oplus \frac{\delta \theta_e}{\tan \theta_e/2} \right)
\label{Eqsigmares}
\end{equation}

and is dominated by the experimental
error of the term $\delta \Sigma / \Sigma$. At high values of $y$,
the error on $y$ decreases as $(1-y)$ and at low values of $y$ the
error is similar as the resolution of the hadron method.

The relative deviation of $x_{\Sigma}$ and $Q^{2}_{\Sigma}$ from the
generated variables $x_{gen}$, $Q_{gen}^{2}$ is shown in
Fig. \ref{Figxq2sigma}(a) and (b) respectively. Each point in
Fig. \ref{Figxq2sigma} corresponds to a Gaussian fit restricted in the
interval: 

\begin{equation}
-0.3 \leq \frac{x_{\Sigma}-x_{gen}}{x_{gen}} \leq 0.3 \hspace{2cm}
-0.5 \leq \frac{Q_{\Sigma}^{2}-Q_{gen}^{2}}{Q_{gen}^{2}} \leq 0.5
\label{Eqsigmafit}
\end{equation}

From Fig. \ref{Figxq2sigma} follows that the $\Sigma$ method extends 
the measurement to higher values of $x$ above $10^{-3}$ but is less
precise than the electron method for the low $x$ region (compare
Fig. \ref{Figxq2sigma}(a) with Fig. \ref{Figresx}(a)). The
resolution of $x$ using the $\Sigma$ method for values is of the order
of $10\%$ for values of $x \approx 10^{-5}$. The resolution of $Q^2$ with the $\Sigma$
method varies between $10\%$ and $5\%$ for $Q^2<1\,$GeV and
$Q^2>1\,$GeV respectively. 

In the region $ 10^{-4} \leq x_{gen} \leq 10^{-3}$, the
$\Sigma$ method tends to reconstruct $x$ at higher values than it was
generated (see Fig. \ref{Figxq2sigma}(a)). 

The projection of the
relative deviation between $x_{\Sigma}$ and $x_{gen}$ for all values
of $x_{gen}$ is shown in Fig. \ref{Figxgenrec}. There
it can be seen that the relative deviation has a tail over the whole
generated phase space. The tail is created by the substraction of the
identified noise in the LAr calorimeter. The dashed line in
Fig. \ref{Figxgenrec} show the relative deviation of $x_{\Sigma}$ from
$x_{gen}$ in case the noise is not substracted.  
The high $x$ (low $y$) kinematical region is complicated, because it 
is sensitive to noise substraction mechanisms, showering effects and
losses of low $p_{t}$ particles coming from the hadronical final
state.  

\begin{wrapfigure}[18]{r}{0.5\textwidth}
%\begin{figure}[t]
\begin{center}
%\begin{minipage}{\textwidth}
\epsfig{figure=./resolution/figures/compare_res_sigma_0.eps,width=0.5\textwidth}
\begin{minipage}{0.5\textwidth}
\setcaptionwidth{\textwidth}
\caption{Relative deviation of $x_{\Sigma}$ from $x_{gen}$ in the case
the noise in LAr is substracted (solid line) and not substracted
(dashed line)}
\label{Figxgenrec}
\end{minipage}
\end{center}
\end{wrapfigure}

The losses and the identification of the noise are two different
mechanism which can not be easily separated. 
The losses of low $p_{t}$ particles are specially pronounced for low values  
of $W<50\,$GeV. The comparison between $W_{\Sigma}$ and
$W_{gen}$ improves considerably in case the identified noise is
substracted from the measured energy and momenta in the LAr. This
comparison is shown in Fig. \ref{Figwgen}(a) and (b). The losses of
low $p_t$ particles will be studied in section \ref{Subsecptbal}.

%The losses of low $p_{t}$ particles are minimized by requiring that
%the transverse momentum of the hadrons be at least $30\%$ of the
%transverse momentum of the electron. Such balance requirement is
%imposed in both distributions of Fig. \ref{Figwgen}. The noise
%noise identification and its substraction is therefore needed. 

\begin{figure}[t]
\begin{center}
\begin{minipage}{0.45\textwidth}
\setlength{\unitlength}{1cm}
\epsfig{figure=./resolution/figures/wresimple_sigma_060509_1133.eps,width=\textwidth}
\put(-2.,5.){\small{(a)}}
\end{minipage}
\begin{minipage}{0.45\textwidth}
\setlength{\unitlength}{1cm}
\epsfig{figure=./resolution/figures/wresimple_sigma_060508_1823.eps,width=\textwidth}
\put(-2.,5.){\small{(b)}}
\end{minipage}
\caption{Comparison between $W_{gen}$ and $W_{\Sigma}$ not
  substracting the identified noise (a) and substracting the
  identified noise (b)}
\label{Figwgen}
\end{center}
\end{figure}

Although the noise substraction mechanism deteriorate the
resolution in $x$, in chapter \ref{Chaphadfinalstate} it will be shown 
that the LAr noise has to be substracted in order to reproduce
reasonably the behaviour of the data. The noise substraction mechanism
and the identification of the noise sources have to be
studied in detail, therefore such investigations are the subject of chapter
\ref{Chaphadfinalstate}.

In order to guarantie that the reconstructed variable $x_{\Sigma}$ does
not deviate from the generated, a special study of purity and
stability will be performed and discussed in chapter
\ref{Chadisselc}. 


%At low $y$ the resolution of the sigma method is similar as the hadron
%method, since the $\delta \Sigma /Sigma$ is experimentally bigger than
%the terms coming from the electron measurement. At high $y$, the
%precision of the sigma method decreases as $(1-y)$


\subsection{Double angle method}
\label{SubSecdamethod}

The double angle method uses the angle measurement of the scattered
electron $\theta_e$ and hadron systems $\theta_{had}$ to 
calculate the event kinematics. 

\begin{equation}
y_{DA} = \frac{\tan (\theta_{had}/2)}{\tan (\theta_{had}/2) + \tan (\theta_e/2)}
\hspace{1.5cm} Q^{2}_{DA} = 4(E_{inc}^{0})^{2} \frac{\sin
  \theta_{had}(1+\cos \theta_e)}{\sin \theta_{had} + \sin \theta_e -
  \sin(\theta_{had} + \theta_e)}
\label{kinda}
\end{equation}

where $\theta_{had}$ is determined from equation
\ref{Eqthetahad}.
To first order the double angle method is independent of the detector
energy scale and therefore is mainly used for calibration
purposes. For small values of $\theta_e$ or $\theta_{had}$ the
resolution of the double angle method gets worse.

\section{Electron Identification}
\label{Secelecidentification}

As we have seen in the previous sections, the reconstruction of the
event kinematic requires a correct identification of the scattered
electron and of the hadronical final state. 
The electron identification for the low$Q^2$ values studied in this
analysis, start from  
energy clusters measured in the SpaCal calorimeter, while their corresponding
track is reconstructed by the BST.
In previous analysis\cite{tomas} the cuts listed in Table
\ref{Tabcuteffelectron} 
were used to identify the scattered electron using the SpaCal and the
BST. In the following sections these variables will be defined
precisely. 

\begin{table}[h]
\begin{center}
\begin{tabular}{l c l}
\hline
\hline
\multicolumn{3}{c}{Electron validation} \\
\hline
SpaCal: & Fiducial volumen &\\
        & $R_{clog}<4\,$cm  & Eq. \ref{Eqrclog} \\ 
        & $E_{veto}<1\,$GeV & \\
        & $f_{had}<0.15$ & Eq. \ref{Eqfhad}\\
\hline
BST:  & $\Delta R = |R_{bst}-R_{SpaCal}|<1.5\,$cm & Eq. \ref{EqdeltaRbst}\\ 
%      & $lval=1$ \\
      & $|z_{BST} - 70| < 30\,$cm & \\
      & $R_{bst} > 10\,$cm & \\
      & $N_{BST,hits} < 200$ & \\
\hline
\hline
\end{tabular}
\end{center}
\caption{Selection used to identify the scattered electron using the
  SpaCal calorimeter and the BST \cite{tomas}.}
\label{Tabcuteffelectron}
\end{table}

\subsection{Electron identification with the SpaCal}
\label{Subseceidspacal}

The electron is identified as the most energetic
cluster measured in the SpaCal calorimeter. This is justified since
for values of $y<0.5$ the energy of the scattered electron is 
greater than the energy of the hadronic jet. 
For high $y$ this criterium is not appropriate, nevertheless in such
cases, the energy of the hadrons scattered in the backward
region deposit their energy over different cells of the
SpaCal calorimeter. Since the hadronic showers are
in general broader than the electromagnetic ones, two requirements
apart of the maxima energy-cluster criteria are in 
addition applied to identify the electron. 

One requirement is related to the transverse extension of the shower
$R_{clog}$ in the electromagnetic section of the SpaCal
calorimeter. The other selection 
criterium is to have behind the electromagnetic cluster a minimal
deposition of energy ($f_{had}$) in the hadronic section of the
SpaCal. In the following paragraphs both variables will be defined in 
detail.

The energy of the scattered electron is commonly deposited
over more than one cell of the SpaCal calorimeter, in average 
$20$ cells build the electron cluster. The cluster energy
$E_{cl}$ is given by the sum of the energy $E_i$ of all individual
cells which belong to the cluster. The cell with the maximum energy is
known as the ``hottest-cell''. The position of the electron cluster is
defined as 

\begin{equation}
x_{cl} = \sum_{i=1}^{N} x_i w_i \hspace{1cm} y_{cl} = \sum_{i=1}^{N} y_i w_i 
\label{Eqxycluster}
\end{equation}

where $x_i$ and $y_i$ are the geometrical centres of the cells and the
weight of the cell is defined as

\begin{equation}
w_i = \frac{max(0,w_{cut} + ln (E_i/E_{cl}))}{\sum_{j=1}^{N}
  max(0,w_{cut} + ln (E_i/E_{cl}))}
\end{equation}

The parameter $w_{cut}$ decides which cells will be taken into account
in the reconstruction of the cluster position. Through the
variation of the $w_{cut}$ parameter the cluster
position determination was optimized, leading to a value of
$w_{cut}=4.8$ \cite{roman}.   

There are different possibilities to calculate the transverse
extension of the shower. In \cite{SGlazov} was shown that the
logarithmic weighting has the best resolution to distinguish clusters
of electromagnetic and hadronic origin. The transverse extension of
the shower is defined as: 

\begin{equation}
R_{clog} = \sqrt{\sum_i (R_i w_i)^2}
\label{Eqrclog}
\end{equation}

\begin{wrapfigure}[18]{r}{0.5\textwidth}
\begin{center}
\epsfig{figure=./resolution/figures/rclnewplotcalib.eps,width=0.45\textwidth}
%\includegraphics[width=8.cm]{./resolution/figures/tagaccep.eps}
\begin{minipage}{0.5\textwidth}
\setcaptionwidth{\textwidth}
\caption{The transversal extension of the electron cluster $R_{clog}$.}
\label{Figrclogcalib}
\end{minipage}
\end{center}
\end{wrapfigure}

where $R_i$ is the distance of the cell $i$ to the centre of the
cluster. Fig. \ref{Figrclogcalib} shows the transverse
extension of the cluster for 
electrons having energies $20<E_e<30\,$GeV. 

The energy of electromagnetic clusters are mostly contained inside the $27Xo$
electromagnetic section of the SpaCal. The hadronic part of the spacal
was used to sum the energy $E_{had,spa}$ deposited behind the most energetic
cluster in a cylinder having an extension of $15\,$cm of radius. If
the fraction of the energy $E_e$ of the cluster 
to $E_{had,spa}$ is greater than $15\%$ then the cluster is identified
to have an hadronic origin.
The fraction of $E_e$ and $E_{had,spa}$ will be in denoted as $f_{had}$:

\begin{equation}
f_{had} = \frac{E_{had,spa}}{E_{e}}
\label{Eqfhad}
\end{equation}

In case the scattered electron is reconstructed inside the insert
module of the SpaCal calorimeter, a maximum energy of $E_{veto} <
1\,$GeV in the veto cells is required, in order to loose not too much
energy due to shower leakage.

\subsection{Electron identification using the BST}
\label{Subsecelidbst}

The identification of the scattered electron is optimised by
reconstructing in addition its track. This
can be performed using the BST or the BDC and the central tracking
devices.  

The reconstruction of the electron track with the BST was initially
implemented in \cite{Arkadov} where the BST consisted only of 4
planes. Afterwards the reconstruction algorithm was implemented 
in \cite{paper}, it starts with the measured position of the SpaCal
cluster. The three $r-$strip sectors localised
around the azimuthal position $\phi_{spa}$ of the SpaCal cluster are
selected. The $r$ coordinate of each BST hit is projected along the line
which connects the hit and the SpaCal cluster to the outermost BST
plane. The projection of all hits enters into a histogram. The line
which connects the maximum of the histogram with the SpaCal cluster is
used as the starting point for the track finding.

The track finding is performed using an iterative
technique, where all hits of the three selected sectors enters the 
minimisation procedure. The contribution of each hit is weighted
depending on the distance of the hit to the track around an
effective corridor. At the beginning of the track finding this
corridor is equal to the SpaCal resolution, for the next iteration the
corridor is reduced until the BST resolution $5
\sigma_{BST} \approx 250 \mu$m is reached. 
%A reconstructed track inside the BST acceptance, where at least $3$
%$u-$hits are associated with the linked $r-$hits defines the variable
%$lval=1$. 

Once a track is reconstructed with the BST, the additional
requirement:

\begin{equation}
\Delta R=|R_{BST}-R_{SpaCal}|<1.5\,cm
\label{EqdeltaRbst}
\end{equation}

is applied, in order to minimise the influence of background $\gamma
p$ events \cite{tomas}.  

$R_{BST}$ in Eq. \ref{EqdeltaRbst} is the transverse distance of the
impact point of the scattered electron on the SpaCal plane to the
centre of the beam pipe. The subindex $BST$ means that the BST track
was extrapolated to the SpaCal surface in order to determine the
scattered electron position on the SpaCal front plane. In analogy,
$R_{SpaCal}$ is the same transverse distance but measured with the
coordinates of the hottest cluster. 

Due to acceptance problems of the BST during the shifted vertex data
period \cite{paper}, the $R_{BST}>10\,$cm cut is applied. In the
following section, the efficiency of the cuts introduced until now are
studied. 

\section{Efficiency of Electromagnetic cuts}
\label{Secloweanalysis}

The efficiency of the cuts listed in table \ref{Tabcuteffelectron} is
investigated using a sample which is almost free of background
events. This study is performed using data and the Signal MC
{\small DJANGOH}. The efficiency study is important to guaranty that
the applied cut, reject the same fraction of events for data and MC. 

To select a sample almost free of background a so called tagged ISR
sample is used, where the scattered electron is identified as in the
standard selection (table \ref{Tabdissel}) and the initial state
photon is detected using the photon detector.

For normal DIS events the longitudinal momentum has to be conserved and
equal to $2E_{inc}=55\,$GeV, nevertheless this requisite is not
satisfied in case the electron radiates a photon. In such case, from
longitudinal momentum conservation follows: 

\begin{eqnarray}
2E_{inc} - (E-P_{z})_{\gamma} & = & (E-P_z)_{el} + \Sigma \\ \nonumber
& \approx & 2E_{e} + 2 y_{had} E_{inc} \\ \nonumber
%\Rightarrow {for low values of y} \hspace{0.4cm}
E_{inc} & \approx & E_{e}+E_{\gamma}
\label{Eqisr}
\end{eqnarray}

Thus, for ISR events the energy of the incoming electron is
basically distributed between the energy of the photon and the energy
of the scattered electron. Since, the radiated photon can be detected
in the photon-tagger of the luminosity system and the energy of the
electron can be measured in the SpaCal calorimeter, a sample of tagged-ISR
events can be used to study the efficiency of the cuts used to identify
the scattered electron. In order to have a clean sample,
events with {\it only one} cluster in the SpaCal are selected. 


\begin{figure}[t]
\begin{center}
\setlength{\unitlength}{1cm}
\includegraphics[width=15.cm]{./resolution/figures/lowe_bst_ee051110_1630.eps}
\put(-9.5,10.){\small{(a)}}
\put(-2.5,10.){\small{(b)}}
\put(-9.5,4.5){\small{(c)}}
\put(-2.5,4.5){\small{(d)}}
\caption{Efficiency of the electron identification cuts as function of
  the energy of the scattered electron $E_e$. (a) $R_{clog}<4\,$cm (b)
  $\Delta R < 1.5\,$cm (c) $E_{veto}<1\,$GeV and (d) $R_{BST}>10\,$cm
  cut. The efficiency is
  derived using tagged-ISR events. Close points are Data and open
  circles are the signal {\small DJANGOH MC} simulation.}
\label{Figefflowe}
\end{center}
\end{figure}

The selection of tagged-ISR events requires
a minimal photon energy of $1\,$GeV. In addition no activity in the 
electron tagger is required to reduce Bethe Heitler process. Since
ISR events are mainly localised at low $y$, only events with
$y_{had}<0.1$ are considered. 

In order to calculate the efficiency of the cut, the electron
identification selection given in Table \ref{Tabcuteffelectron} in
addition to the tagged-ISR selection is
applied. The efficiency of the cut is calculated as the fraction of
events accepted for the whole sample $N_{all}$ with the events which are
accepted relaxing the cut.

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

In Fig. \ref{Figefflowe} the efficiency is shown of the cuts 
listed in table \ref{Tabcuteffelectron} as function of the energy of the
scattered electron $E_e$.
In Fig. \ref{Figefflowe}(a) is shown the efficiency of the $R_{clog}$
cut. One can see, that the efficiency of the $R_{clog}$ cut is fully
in agreement with the signal MC DJANGOH.
In Fig. \ref{Figefflowe}(b) and (d) are shown the efficiency of the
Spacal-BST  matching $\Delta R$ and the $R_{BST}>10\,$cm cut. 
In Fig. \ref{Figefflowe}(c) is shown the efficiency of the $E_{veto}$
cut. In general, all investigated cuts are well described by the
{\small DJANGOH} {\small MC},  rejecting only a small fraction of
events for data and {\small MC} having differences which do not exceed
$2\%$. 

\section{Photoproduction}
\label{Secphotoproduction}

The cut efficiency previously presented compares the data with the
signal {\small MC} {\small DJANGOH} using a sample which is almost
free of background. But the real DIS selection is influenced 
by the photoproduction background. Therefore it is important to
estimate the influence using the data and the background {\small MC} 
{\small PHOJET}.

%The dominant source of background to DIS events are known as
%photoproduction ($\gamma p$) events. 
In $\gamma p$ events, the interaction
between the electron and the proton occurs at very low $Q^2 <
10^{-2}$GeV$^2$, having thus an $ep$ interaction mediated by an almost 
real photon. 
For small values of $Q^2$, the transverse momentum of the scattered
electron is low, and the probability that the scattered electron
escapes the detection of the backward detectors is high. However a
fraction of $\gamma p$ events can satisfy the electron identification
criteria, when a particle coming from the hadronic final state is
wrongly identified as the scattered electron. 

\begin{wrapfigure}[21]{r}{0.5\textwidth}
\begin{center}
\epsfig{figure=./resolution/figures/tagaccep.eps,width=0.45\textwidth}
%\includegraphics[width=8.cm]{./resolution/figures/tagaccep.eps}
\begin{minipage}{0.5\textwidth}
\setcaptionwidth{\textwidth}
\caption{Acceptance of the ET as function of $y$ for the shifted
  vertex period \cite{gpacceptance}.}
\label{Figtagaaceptance}
\end{minipage}
\end{center}
\end{wrapfigure}

In order to determine how many photoproduction events pass the DIS
selection, a sample of events where the scattered
electron is detected in the ET is used, and at the same time fulfil
the DIS selection listed in table \ref{Tabcuteffelectron}.
Part of the photoproduction events are detected directly by the
electron tagger {\small{ET}}. The subtrigger 115 is used to
select events which posess a shower good contained in the electron
tagger. The energy in the electron tagger is 
required to be greater than $7\,$GeV and the energy deposited in the
photon tagger is required not to exceed $2\,$GeV to exclude Bethe
Heitler events. 

\begin{figure}[t]
\begin{center}
\setlength{\unitlength}{1cm}
\epsfig{file=./resolution/figures/nomphojet_bst_060405_1456.eps,width=0.7\textwidth} 
\put(-9.5,10.5){\small{(a)}}
\put(-4.5,10.5){\small{(b)}}
\put(-9.5,4.5){\small{(c)}}
\put(-4.5,4.5){\small{(d)}}
\caption{Comparison of $\gamma p$ events detected in the ET and passing
  the DIS selection with the contribution given by the {\small{PHOJET
  MC}}. (a) Energy measurement of the scattered electron in the
  ET. (b) Energy of the ``misidentified electron'' (hadron) measured by
  the SpaCal calorimeter. (c),(d) polar angle and $y_{\Sigma}$
  of the misidentified electron}
\label{Figphotoproduction}
\end{center}
\end{figure}

The acceptance of the {\small{ET}} is limited and only a small
fraction of events in a certain kinematic range can be detected. The
acceptance of the ET as function of $y$ is show in
Fig. \ref{Figtagaaceptance} and is calculated using different fills
having the same beam position delivered by HERA. The
acceptance function is used to correct event by event the
photoproduction contribution in the {\small PHOJET MC}. 

One additional background source for the $\gamma p$ events are the
overlap of Bethe-Heitler and normal DIS events. For such
events $E-p_z$ is expected to be greater than $55\,$GeV. The overlap
events are suppressed requiring that the $(E-p_z)_{tot}$ defined as in
Eq. \ref{Eqphotoproduction} be lower than $70\,$GeV. 

%
%For Bethe-Heitler processes Ee=Ee,tag+E_photon,tag
%

\begin{equation}
(E-P_z)_{tot} = \Sigma + (E-P_z)_{el} + 2E_{e,tagger} + 2E_{\gamma,tagger} 
\label{Eqphotoproduction}
\end{equation}

In Fig. \ref{Figphotoproduction} is shown the comparison of data and
Phojet {\small{MC}} for 
events which pass the DIS selection and the tagged-selection.  
The Phojet contribution is normalised to the data using the integral
of the distribution of the energy measured by the wrongly identified
scattered electron by phojet $I_{phojet}$ and in the data
$I_{data}$. The luminosity of phojet is thus obtained as:

\begin{equation}
L_{phojet} = \frac{I_{phojet}}{I_{data}} L_{data}
\label{Eqphojetlumi}
\end{equation}

Fig \ref{Figphotoproduction}(a) shows the energy measured in 
the electron tagger and Fig. \ref{Figphotoproduction}(b) shows the
energy of the hadron that was identified as the scattered electron,
normalized according to Eq. \ref{Eqphojetlumi}.
The photoproduction background is correctly described by {\small
  PHOJET}, hence it can be used to estimate the background
contribution of the final DIS selection. In such way, the
photoproduction contribution will be added to the {\small{DJANGO MC}}
sample of DIS events to reproduce the behaviour of the data.

