\chapter{Hadronic Final State}
\label{Chaphadfinalstate}

The calibration of the LAr calorimeter and the understanding of the
hadronic final state (HFS) requires special attention for the
$x \geq 10^{-4}$ region, where the electron method begins to be less
accurate and the kinematical variables can be better reconstructed 
with the $\Sigma$ method. 
Thus, for values of $x \approx 10^{-4}$, the measurement of the inclusive DIS
cross section depends not only on the reconstruction of the 
scattered electron but also depends on the reconstruction of the HFS.

In order to reconstruct as good as possible the kinematical variables
$x$ and $Q^2$ with the $\Sigma$ method, detailed studies on the energy
flow of the HFS have to  
be performed. This studies essentially involve the calibration
of the main calorimeters of the H1 detector. The calibration is
performed using the longitudinal and the transverse momentum of the
particles which form the initial and the final state.
Measuring with a perfect detector, energy and momentum conservation
requires that the total $(E-p_z)_{tot}$ and $p_{t,tot}$ of the
scattered electron and of the HFS be equal to:

\begin{equation}
(E-p_z)_{tot} = 2E_{inc} \approx 55\,GeV \hspace{1cm}
\vec{p}_{t,tot} = \vec{p}_{t}^{el} + \vec{p}_{t}^{had} =0
\label{Eqglobal}
\end{equation}

However, real experiments suffer from acceptance losses and resolution
effects which lead to deviations of the expected values. In this
chapter the HFS is studied in detail using the global quantities
written in Eq. \ref{Eqglobal} and those already discussed in section
\ref{SubSecthadmethod} and \ref{SubSecsigmamethod}. 

In section \ref{Sechadspacalib} the response of the SpaCal calorimeter
to hadrons is studied using the longitudinal 
energy conservation $(E-p_{z})_{tot} \approx 55\,$GeV. The
reconstruction of the HFS is explained 
in section \ref{Secrechfs}. The calibration of the LAr calorimeter is
described in section \ref{SecLarcalib}, where the transverse momentum
of the electron is used as the energy scale reference. Part of the
energy measured in the LAr calorimeter is influenced by noise 
signals originating either from the electronics of the LAr calorimeter
readout or it is beam induced. Additional noise due to secondary
scattering of  
final state particles which deposit its energy into the calorimeter
can influence the energy measurement. The influence of the noise
on the kinematic reconstruction is explained in section
\ref{Seclarnoi}. At the end of the chapter the
uncertainties assigned to the description of the HFS are summarized. 

\section{Hadronic SpaCal}
\label{Sechadspacalib}

The hadronic section of the SpaCal calorimeter is  
non-compensating, thus its response is 
different for hadronic and electromagnetic showers.
The combined response of the SpaCal calorimeter for hadrons was
investigated in \cite{episephad} using data taken at the CERN proton
Synchroton. Two parts of the electromagnetic and hadronic sections
of the SpaCal where installed at the CERN PS T9 beam line,
where electrons and $\pi^{-}$ having incident energies between $1$ and
$7\,$GeV were used to investigate the hadronic response and the $e/\pi$
separation.  

The combined response of the SpaCal to hadrons was studied using
$\pi^{-}$, the energy
measurement in the electromagnetic and hadronic section were added in
case the $\pi^{-}$ interacted at least in one of the two sections of
the SpaCal. An optimal constant scale factor 
$1.94$ was found for the electromagnetic section while for the hadronic
section the cali\-bration factor varies from 1.29 at $3\,$GeV to $1.46$
at $7\,$GeV \cite{episephad}.

\begin{table}[t]
\begin{center}
\begin{tabular}{c c c}
\hline
\hline
\multicolumn{3}{c}{SpaCal Calibration factor for hadrons} \\
Reference & Electromagnetic section & Hadronic section \\
\hline
\cite{Minutes} & $1.1$ & $1.3$ \\
\cite{SGlazov,AMeyer} & $1.5$ & $1.5$ \\
\cite{AMeyer} & $1.07$ & $1.07$ \\
\hline
\hline
\end{tabular}
\label{Tabhadspa}
\caption{Calibration constants for the hadronic response of the SpaCal
  calorimeter.}
\end{center}
\end{table}
 
The hadronic energy corrections for the SpaCal calorimeter 
were also estimated in a Monte Carlo study obtaining values of
$1.1$ for the electromagnetic part and $1.3$ for the
hadronic part \cite{Minutes}. These corrections were applied for both
data and {\small MC} for all cells which do not belong to the electron
cluster.  

Using $ep$ collisions the longitudinal energy conservation was used in
several analyses in order to study the response to hadrons using the
SpaCal. Such investigations exploited the combined 
energy measurement in the electromagnetic and hadronic section of
the SpaCal, obtaining cali\-bration factors which are summarized in table
\ref{Tabhadspa}. The product of the correction factors represent the
starting scale of the SpaCal hadronic energy in the data sample which 
is investigated here.

% An additional correction factor of $1.5$ had to be
%applied to the energy of the hadronic final state measured in the
%SpaCal for both data and {\small MC}. The same correction were derived
%independently in \cite{SGlazov} which used the comparison of the hadronic
%energy with the momentum measurement of charged particles using the
%central jet chambers. 
%Further analysis were performed in \cite{AMeyer} to get a reasonable
%agreement between the data and the {\small MC}.
%Two additional corrections are applied to the data
%in the following way: the energies in the cells of the hadronic
%SpaCal are multiplied with a correction factor of 0.7 and in addition,
%the hadronic energies in both electromagnetic and hadronic SpaCal are
%multiplied by a factor of 1.07. 

\begin{figure}[t]
\begin{center}
\setlength{\unitlength}{1cm}
\begin{minipage}{0.45\textwidth}
\epsfig{figure=./hadron/figures/yhadoverye_060316_1936.eps,width=\textwidth}
\put(-2.,6.2){\small{(a)}}
\end{minipage}
\begin{minipage}{0.45\textwidth}
\epsfig{figure=./hadron/figures/yhadoverye_060326_1607.eps,width=\textwidth}
\put(-2.,6.2){\small{(c)}}
\end{minipage}
\begin{minipage}{0.9\textwidth}
\epsfig{figure=./hadron/figures/empzhadspa_060316_1936.eps,width=\textwidth}
\put(-3.,6.2){\small{(b)}}
\end{minipage}
\caption{Comparison between $y_{had}/y_{e}$ for events with
  $y_{e}>0.5$ and $(E-p_z)_{tot}>35\,$GeV using the initial hadronic
  scale. (b) Mean value of 
  $(E-p_z)_{had}$ of the hadrons with the $(E-p_z)_{el}$ of the
  electron for events having $y_{e}>0.5$ and $(E-p_z)_{tot}>35\,$GeV. (c)
  Comparison between $y_{had}/y_{e}$ with the same selection of Fig.(a)
  but correcting the $(E-p_z)_{had}$ of the hadrons in the {\small MC}
  by $3\%$ (see text).} 
\label{Figyhadspa}
\end{center}
\end{figure}

The energy scale of the hadronic section of the
SpaCal can be studied with DIS events \cite{Arkadov}. 
For high values of $y$, the hadronic final state is scattered mainly
in the backward region of the H1 detector, thus the measurement of the
HFS for high values of $y$ is mainly performed by the 
SpaCal calorimeter. Therefore in that region, the
energy of the scattered electron can be used as the reference scale to
study the hadronic energy scale. The inelasticity variable $y$
reconstructed with the electron $y_e$ (see equation \ref{Eqkinelec}) and the
hadron $y_{had}$ (see equation \ref{hadkin}) method
are the variables used to study the scale.

Since for high values of $y>0.5$ the photoproduction background
contributes strongly (see Fig. \ref{Figphotoproduction}(d)),hence only
events having a $(E-p_z)_{tot}>35\,$GeV are 
selected to study the hadronic energy scale of the SpaCal
calorimeter. In Fig \ref{Figyhadspa}(a) 
the ratio $y_{had}/y_e$ is shown for events where $y_e > 0.5$ and
$(E-p_{z})_{tot}>35\,$GeV. A clear shift between the data and the
{\small MC} is observed. 

The energy shift is studied analyzing the behaviour of the measured
$(E-p_z)_{had}$ from the hadrons taking as reference the
$(E-p_{z})_{el}$ of the scattered electron. Due to 
longitudinal energy conservation the total $(E-p_z)_{tot}$ has to be
conserved and equal to $55\,$GeV. Thus in average the $(E-p_z)_{had}$ with
the $(E-p_z)_{el}$ of the electron should be directly correlated. 
In Fig. \ref{Figyhadspa}(b) is shown the dependence of the
$(E-p_z)_{had}$ with $(E-p_z)_{el}$ for data and {\small MC}. The
deviation between data and {\small MC} reflect the shift observed in the
$y_{had}/y_e$ distribution. The shift can be obtained using the ratio or the
difference between the data and the {\small MC} points shown in
Fig. \ref{Figyhadspa}(b). 

The $(E-p_z)_{had}$ of the hadrons in the {\small MC} had to be
corrected with a factor $0.97$ ($3\%$). This correction correspond to
a shift of $700\,$MeV. In Fig. \ref{Figyhadspa}(c)
is shown the $y_{had}/y_e$ distribution after applying the correction
leading to a good agreement. The
uncertainty of the hadronic SpaCal is quoted as half of the
needed correction which is $350\,$MeV.
 
\section{Reconstruction of the Hadronic final State}
\label{Secrechfs}

In contrast to the identification of the scattered electron,
the identification of the hadronic final state (HFS) is even more
complex, since the HFS can have different topologies inside the H1
detector depending on which values of $Q^2$ and $x$ the DIS
event occur. This involves several energy depositions which can be
measured using the subdetector components of the H1 detector. The aim
of the reconstruction of the HFS is to provide a four vector which
contains all measured energy depositions.
  
%The reconstruction of the particles belonging to
%the HFS rely basically in the momentum measurement from the central tracker
%devices, the energy deposition in the LAr calorimeter cells and the
%energy deposition in the cells belonging to the hadronic part of
%the SpaCal calorimeter. 
%The aim is to construct a four vector which contain all
%measured energy depositions.

%The HFS is reconstructed with an elaborated algorithm known as {\small FSCOMB},
%which combines the energy and momentum measurements of the
%particles which are detected in the several subdetector components of the
%H1 detector. 

The reconstruction of the particles belonging to the HFS rely
basically on the momentum measurement of the central tracking devices,
the energy deposition in the LAr calorimeter cells and the energy
deposition in the cells belonging to the hadronic activity in
the SpaCal calorimeter. The HFS is reconstructed with the
elaborated algorithm {\small FSCOMB} \cite{FSCOMB0,FSCOMB1,FSCOMB2},
which combines the measurements of the subdector components previously
mentioned. 

In order to improve the kinematic reconstruction, the {\small{FSCOMB}}
algorithm optimizes the momentum measurement according to the resolution 
of the subdetector components. For example for low energetic
particles the central tracking devices are used, since their resolution
is propotional to the energy $E$ of the particle, whereas the
calorimeter resolution is $50\%/\sqrt{E}$ leading to better
performance at high energies.

Using the measured hits in the tracking devices, the curvature
$\kappa$ and the spacial position of the track is determined. The
information of each track is used to determine the vertex position of
the event. Thus, using the vertex and the track parameters 
a four momentum $P^{Track}$ of all measured tracks by the tracker
devices is constructed. In the same way, the four momentum $P^{LAr}$
and $P^{HadSpa}$  is constructed using the  vertex position and the
spacial coordinates of the cells where energy was deposited.

{\small FSCOMB} combines the calorimeter and the track information in the
following way. Tracks with momenta less than
$2\,$GeV are selected and extrapolated to the LAr surface where they
are linked to clusters. 
The extrapolated track with the
calorimeter cluster is linked, under the assumption that the particle
yielding the track is a pion \cite{FSCOMB0,h1phan}. All clusters 
inside a cylinder with a radius of
$20\,$cm in the electromagnetic section and $40\,$cm in the
hadronic section of the LAr are considered.
A track is also used if its extrapolation does
not reach the LAr surface. In order to avoid double counting of the
measured four momentum, those calorimeter cells, where the energy
deposition is replaced by the track measurement, are masked and are
not used to build the LAr four momentum $P^{LAr}$. In general the four momentum
of the hadronic final state can be expressed as the following vectorial
sum: 

\begin{equation}
\vec{P}^{had} = \vec{P}^{Track} + \vec{P}^{LAr} + \vec{P}^{HadSpa}
\label{Eqpthad}
\end{equation}

The combination of the track with the calorimeter measurement uses
primary vertex constrained tracks. In the course of this analysis it was
found that the vertex position determined by the central tracking
devices and the BST differ considerably in case the tracks
reconstructed by forward trackers are used to determine the vertex
position. Thus, the link of tracks and calorimeter four momentum is
performed using the primary vertex only in case reconstructed tracks
by the CJC are found. In case no tracks are in the CJC found, then the
default vertex position $z=70\,$cm is used. At analysis level the
four momentum is again calculated using the vertex position of the BST.

The contribution of the four momentum of the SpaCal $P^{HadSpa}$
contains the energy deposited in the electromagnetic section, which
does not belong to the electron cluster and the energy which is
deposited in the hadronic section.  

For low values of $y$, the determination of $(E-p_z)_{had}$ is
affected by extra activity in 
the calorimeter. The scattered electron can be rescattered off the
detector material behind the central tracker with secondary
particles. These particles can be wrongly identified as taking part of
the HFS originating from the primary 
interaction. This extra activity is reduced by substracting $10\%$ of the
$(E-p_z)_{el}$ of the electron from the total hadronic
contribution measured in the SpaCal, if the result is
negative a zero contribution for the SpaCal is taken \cite{SGlazov}.

\subsection{Noise identification using FSCOMB}
\label{Subsecfscombnoise}

During the reconstruction of the four momentum in the LAr, the {\small
  FSCOMB} algorithm also identifies noise. This is neccesary, since
during the $ep$ 
collision, the energy measured by the LAr calorimeter is overlayed
with noise signals produced by example from the
electronics of the LAr cells which allow to measure the energy
depositions in the LAr \cite{Larelectronicnoise}. Thus, not all
measured energy in the LAr calorimeter 
belongs to the energy of the HFS of the $ep$ collision. Therefore it is
necessary to investigate the noise.
%This noise signals do not belong to the HFS,
%thus its contribution has to be studied. 

The strategy is to look for isolated energy depositions in
the cells of the LAr, thus all cells which have an energy deposition
are investigated. Two different energy thresholds depending on the
angular position of the cell are considered. 

The angular position of the cell $\theta_{cell}$ is measured with
respect the origin of coordinate system of the H1 detector. In case a cell
is localised at $\theta_{cell}<15^{\circ}$ and its energy energy is
below $800\,$MeV, the energy of the neighbouring
cells localised in a transversal cone of $40\,$cm is
summed. In case the summed energy of all neighbouring cells is also
below $800\,$MeV, all those cells are masked as
``noise-cells''. Similarly a energy threshold of $400\,$MeV is used if
the cell is localised at $\theta_{cell}>15^{\circ}$, but the
transversal extension of the neightbouring cells is only $20\,$cm. For
$\theta_{cell}<15^{\circ}$ the cell is located in the IF1, IF2 wheels
(see Fig. \ref{Figlar}).

Following this procedure, the four momentum measured by the LAr can be
expressed as the sum of the identified Noise and the Signal:

\begin{equation}
\vec{P}^{LAr} = \vec{P}^{LAr}_{Signal} + \vec{P}^{LAr}_{Noise}
\label{Eqlar4momentum}
\end{equation}

where $ P^{LAr}_{Noise}$ is constructed with the cells masked as
noise-cells. 

\section{Calibration of the LAr calorimeter}
\label{SecLarcalib}

In contrast to the electromagnetic SpaCal calibration the energy flow
in LAr does usually not consist of a single cluster, but it is
distributed over collimated jets or it is
just dispersed into the H1 detector. Therefore
it is very difficult to select single hadrons and use them to 
calibrate. 

Since in neutral current DIS events, the transverse momentum of the
electron $p_{t}^{el}$ and of the hadronic final state $p_{t,had}$
have to be balanced, it is possible to use the well measured transverse
momentum of the electron to achieve the calibration of the LAr
calorimeter. 

The trasversal balance of the event $p_{t,bal}$ defined as 

\begin{equation}
p_{t,bal} = \frac{p_{t,had}}{p^{el}_{t}}
\label{Eqptbalvari}
\end{equation}

is the most used variable to derive the calibration of the
LAr calorimeter. The transverse momentum of the total HFS $p_{t,had}$
is calculated using: 

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

where the contributions of the LAr signal, tracks and hadrons measured
in the SpaCal are considered in the vectorial sum of $p_{x,h}$ and
$p_{y,h}$: 

\begin{eqnarray*}
p_{x,h} = p_{x,Signal}^{LAr}+ p_{x}^{Track} + p_{x}^{HadSpa} \\
p_{y,h} = p_{y,Signal}^{LAr}+ p_{y}^{Track} + p_{y}^{HadSpa}
\end{eqnarray*}

\subsection{Calibration procedure}
\label{SubSeccalmethod}

In this analysis the Lagrangian method introduced in \cite{Arkadov} is
used to derive 
the calibration constants of the LAr calorimeter. The Lagrangian method
minimizes the difference between the transverse momenta of the 
scattered electron and of the hadronic final state particles. The
function used has the form:

\begin{equation}
L = \sum_{i=1}^{N_{events}} \frac{1}{\sigma_{i}^{2}} \left(p_{t,i}^{el} -
\left[p_{t,i}^{Track}+p_{t,i}^{HadSpa}+ p_{t,i,LAr}^{Noise}\right] -
\sum_{j=1}^{M=120} \alpha_{j} p_{t,i,LAr}^{j}  \right)^{2}
\label{Eqlagrangian}
\end{equation}

where $p_{t}^{el}$ is the transverse momentum of the electron, measured
in the electromagnetic section of the SpaCal calorimeter. The variable
$\sigma_{i}$ is the weight of each event, which is taken as $1$ in the
following calculation.
 
$p_{t}^{Track}$ and $p_{t}^{HadSpa}$ are the transverse momenta of
the hadrons measured with the central tracking devices and with 
the SpaCal calorimeter, respectively. 

The contribution of the LAr is considered by the last term in
Eq.\ref{Eqlagrangian}, where the index $j$ corresponds to a continues
number asigned to the different wheels and octants of the LAr
calorimeter. Since the LAr calorimeter is build of $8$ wheels
subdivided in $8$ octants and $7$ wheels have one
electromagnetic and one hadronic section, in total $M=7*8*2+8=120$
contributions of the transverse impulse of the LAr $p_{t,LAr}$ are
considered. The variable $\alpha_{j}$ are the  calibration constants
to be determined by minimizing the function $L$. 

The term $p_{t}^{Noise}$ in Eq. \ref{Eqlagrangian} represents the
transverse momentum of the cells which 
are identified by the {\small FSCOMB} algorithm as noise. This noise
can be substracted globally from the LAr contribution or for 
each module. 

The mimimization procedure is performed such that the transverse
momentum of the hadrons
is projected onto the electron direction, in such way only the $p_{t}$
component parallel (or antiparallel) to the $p_{t}^{el}$ is
considered. The minimization requirement

\begin{equation}
\frac{\partial L}{\partial \alpha_{j}} = 0
\end{equation}

defines a system of $M$ equations corresponding to the $M$ considered
modules and octants: 

\begin{equation}
\sum_{i=1}^{N_{events}} p_{t,LAr}^{i,k} \cdot \left(p_{t}^{el}
-\left[p_{t}^{Track} + p_{t}^{HadSpa} - p_{t,LAr}^{Noise} \right] - \alpha_{K}
\sum_{j=1}^{M} p_{t,LAr}^{i,j} \right) = 0
\label{Eqminimize}
\end{equation}

The results of the minimization determine the energy scale factors
$\alpha_{k}$ for the $M=120$ modules and octants of the LAr
calorimeter. The solution of Eq. \ref{Eqminimize} is obtained using
numerical matrix techniques. 

Since the Lagrangian method relies on the balance of the transverse
momentum of the electron and the hadrons to derive the calibration
constants of the LAr calorimeter 
modules, it is important to consider a {\bf calibration sample} almost free
of background events. 

A precise determination of $p_{t}^{el}$ is ensured by requiring a BST track
validation of the electron candidate cluster. Photoproduction
background is considerably reduced by requiring the energy of
the scattered electron to be $E_{e}>20\,$GeV (see
Fig. \ref{Figphotoproduction}(b)). 

In addition, the calibration sample uses events where the hadronic angle
lies in the LAr acceptance. In order to suppress events where the
hadronic final state escapes undetected through the beam pipe, the
angular coverage is restricted to $15^{\circ} \leq \gamma_{h} \leq
150^{\circ}$. Furthermore the transverse momentum of the hadrons
$p_{t,had}$ has to have at least $30\%$ of the electron momentum. This
cut also reduces the losses in the forward direction where $p_{t,had}$ 
tends to be zero.  


\begin{figure}[t]
\begin{center}
\epsfig{figure=./hadron/figures/hadcalib060227_1948vs060227_1951.eps,width=0.9\textwidth}
%\includegraphics[width=16.0cm]{./hadron/figures/hadcalib060227_1948vs060227_1951.eps}
\caption{Calibration constants for the LAr calorimeter for data of the
  Shifted vertex period (closed points) and {\small DJANGOH} {\small MC} (open
  points). Each wheel is labelled according its position
  with respect the nominal interaction point (central (CB), forward (IF,OF),
  Backward (B) see also Fig. \ref{Figlar}). The lines indicate the
  octants of the corresponding wheels of the LAr calorimeter.}
\label{Figcalibcons}
\end{center}
\end{figure}

\begin{figure}[h]
\begin{center}
\setlength{\unitlength}{1cm}
\epsfig{figure=./hadron/figures/ptbalcalib.eps,width=0.9\textwidth}
\put(-9.0,6.0){\small{(a)}}
\put(-2.4,6.0){\small{(b)}}
\caption{Distribution of $p_{t,bal}$ for the calibration sample
  before (a) and after (b) the calibration constants presented in
  Fig. \ref{Figcalibcons} are applied.}
\label{Figptbalcalib}
\end{center}
\end{figure}

In Fig. \ref{Figcalibcons} are shown the calibration constants of the
LAr calorimeter for data of the shifted vertex period and the signal
{\small MC} {\small DJANGOH}. The calorimeter response is well
described by the {\small MC} simulation only for the electromagnetic
part, but differences are observed in the hadronic modules,
specially for the forward module IF2.

The $p_{t,bal}$ distribution for the {\bf calibration sample},
before and after the calibration constants are applied is
shown in Fig. \ref{Figptbalcalib} (a) and (b), respectively. A good
description between the data  and the {\small MC} simulation is found after
applying the calibration constants. Nevertheless, the mean of the
distribution is not located at the expected 
value $p_{t,bal}=1$. The values of $p_{t,bal}$ vary according to the phase
space in which the event occur. Therefore the dependence of the $p_{t,bal}$
distribution on the angle of the hadrons and on the kinematic
variables has to be studied in more detail for the final DIS analysis
sample (for a detailed description of the DIS sample see table
\ref{Tabdissel}). This studies are presented in the next section.

\subsection{Description of the HFS in the DIS analysis sample}
\label{Secdescriptionhfs}

\begin{figure}[t]
\begin{center}
\begin{minipage}{\textwidth}
\begin{center}
\setlength{\unitlength}{1cm}
\epsfig{figure=./hadron/figures/ptbal060308_1009rebin.eps,width=0.8\textwidth}
\put(-7.8,5.4){\small{(a)}}
\put(-2.,5.4){\small{(b)}}
\end{center}
\end{minipage}
\begin{minipage}{\textwidth}
\begin{center}
\setlength{\unitlength}{1cm}
\epsfig{figure=./hadron/figures/ptbal060302_1448rebin.eps,width=0.8\textwidth}
\put(-7.8,5.4){\small{(c)}}
\put(-2.,5.4){\small{(d)}}
\end{center}
\end{minipage}
\caption{(a) Distribution of $p_{t,bal}$ for the whole DIS selection
  before applying the LAr calibration constants, (b) the ratio of data
  over MC for the uncalibrated $p_{t,bal}$ distribution. (c)
  Calibrated $p_{t,bal}$ distribution (d) ratio of data over MC for
  the calibrated $p_{t,bal}$ distribution.}
\label{Figptcalibdis}
\end{center}
\end{figure}

As already mentioned at the beginnig of this chapter the HFS is a
complex system. The $p_{t,bal}$ and the $(E-p_z)_{tot}$
distributions of the DIS sample are the unique quantities used
to study the total energy flow of the HFS. On the one hand, the $p_{t}$
of the hadrons should be balanced with the $p_{t}^{el}$ of the
electron, but on the other hand the event sample should also conserve its
longitudinal energy. 

%During the calibration procedure the transverse impulse of the
%electron $p_{t}^{el}$ 
%serves as the reference to calibrate the LAr. Nevertheless for low values
%of $y$ the resolution of the $E-p_z$ of the electron degrates (see
%Fig. \ref{Figresx}(a)), 
%thus in that kinematical region the $E-p_z$ used to reconstruct the variable
%$y_{\Sigma}=E-p_z/2E_{inc}$ does
%not have a direct reference scale. Therefore it is important that 
%data and {\small MC} show the same behaviour over the measured phase
%space. The deviation of the {\small MC} from the data of the $p_{t,bal}$
%distribution over the different accessed kinematical phase space regions,
%will give a measure of the hadronic energy uncertainty.
%In order to derive the uncertainty of the energy flow of the HFS, the
%$p_{t,bal}$ and the $y_{\Sigma}$ distributions for the final DIS
%selection are in the following paragraphs in detail studied.

In Fig. \ref{Figptcalibdis}(a) and (c) are shown the $p_{t,bal}$
distributions for the whole DIS event selection before and after
applying the calibration constants. An improvement of the
description between the data and  the {\small MC} is observed for the
whole DIS sample. Apart from the 
normalization problem which will be discussed in the next chapter, a
slight imbalance of the order of $3\%$ is observed for values of
$p_{t,bal}<0.6$ (see Fig. \ref{Figptcalibdis}(c)). The deviation
between the data and the {\small MC} simulation is quantified by its ratio, 
which is shown in Fig. \ref{Figptcalibdis} (b) and (d). 

As in the {\bf calibration sample}, the $p_{t,bal}$ distribution is
good described by the {\small MC} simulation, but it does not peak at
the expected 
value $1$. In order to investigate how the balance of the transverse
momentum is distributed over the accessed kinematical region, the
dependence of $p_{t,bal}$ on the angle of the hadrons and on the
kinematic variables is studied.

Using the transversal and longitudinal momentum of the HFS, the polar
angle of the hadrons $\theta_{had}$ can be calculated (see
Eq. \ref{Eqthetahad}). The polar angular dependence of the $p_{t,bal}$
distribution is shown in Fig. \ref{Figptbalvsangle}(a). The mean
value of the $p_{t,bal}$ distribution deviates from 
$1$ depending on which hadronic angle $\theta_{had}$ is accessed. For
low and high values of $\theta_{had}$ the ratio deviates from $1$ by
$5\%$. This behaviour leads to a
distribution of $p_{t,bal}$ with a mean value of $0.85$. The deviation
between the data and the {\small MC} is shown in
Fig. \ref{Figptbalvsangle}(b), which does not exceed $2\%$ for
$\theta_{had}>20^{\circ}$.  

\begin{figure}[h]
\begin{center}
\begin{minipage}{\textwidth}
\begin{center}
\setlength{\unitlength}{1cm}
\epsfig{figure=./hadron/figures/ptbalvsthetahad060302_1448.eps,width=0.85\textwidth}
\put(-8.2,2.0){\small{(a)}}
\put(-2.,2.0){\small{(b)}}
\end{center}
\end{minipage}
\begin{minipage}{\textwidth}
\begin{center}
\setlength{\unitlength}{1cm}
\epsfig{figure=./hadron/figures/ptbalvspte060302_1448.eps,width=0.85\textwidth}
\put(-8.2,2.0){\small{(c)}}
\put(-2.,2.0){\small{(d)}}
\end{center}
\end{minipage}
\caption{Comparison between data and {\small MC} for the $p_{t,bal}$
  distribution as function of (a) the hadronic angle $\theta_{had}$, (c) the
  transverse momentum of the electron $p_{t,el}$. The ratio of data
  over MC of Fig (a) and (c) are shown in Fig. (b) and (d). The ratio
  of data and MC has a drawn band corresponding to a deviation of $2\%$.}
\label{Figptbalvsangle}
\end{center}
\end{figure}

The dependence of $p_{t,bal}$ on the transversal momentum $p_{t}^{el}$
of the electron is shown in Fig. \ref{Figptbalvsangle}(c). In this
case, the difference between data and MC is concentrated at low values
of $p_{t}^{el}$. The
{\small MC} can describe the behaviour of the data even at low values
of $p_{t}^{el}$. For values of $p_{t}^{el} >1\,$GeV, the mean value of 
$p_{tbal}$ approximates to $1$. Hence the quality of the $p_{t,bal}$
reconstruction varies with the kinematical regions characterized by
$\theta_{had}$ and $p_{t}^{el}$. 

In order to investigate how the transverse energy flow is related
with the longitudinal energy flow, the dependence of the $p_{t,bal}$
distribution for different values of $y$ is analysed. In such way a
``measure'' of the transverse and longitudinal energy flow can be
determined.  

The kinematical variable $y$ characterizes the fraction of the
longitudinal energy 
that the HFS takes after the $ep$ collision. For high values of $y$
the HFS is located mainly in the backward region of the H1 detector,
while for low values of $y$ the HFS is scattered in the forward
region of the H1 detector. Using this naive characteristic of $y$, one
can understand that depending on which kinematical region the event
occurs, the different subdetector components of the H1 detector 
play a relevant role.

\begin{figure}[t]
\begin{center}
\setlength{\unitlength}{1cm}
\epsfig{figure=./hadron/figures/ptbalysigma060302_1448.eps,width=\textwidth}
\put(-9.5,15.){\small{(a)}}
\put(-2.,15.){\small{(b)}}
\put(-9.5,9.5){\small{(c)}}
\put(-2.,9.5){\small{(d)}}
\put(-9.5,4.3){\small{(e)}}
\put(-2.,4.3){\small{(f)}}
\caption{$p_{t,bal}$ distributions for different values of the
  inelasticity $y_{\Sigma}$. Closed points are data, while the
  histogram correspond to the sum of the {\small DJANGOH} and {\small
  PHOJET MC}. The $\gamma p$ background contribution is illustrated by
  the full area.} 
\label{Figptbalsig}
\end{center}
\end{figure}

For low values of $y$ the transverse momentum of the hadrons tends to
be reconstructed to lower values than the transverse momentum of the
electron $p_{t}^{el}$. This can be seen in Fig. \ref{Figptbalsig} where
the $p_{t,bal}$ distribution is shown for different intervals of the
inelasticity $y_{\Sigma}$. From Fig. \ref{Figptbalsig}(a) and (b)
follows that for low values of $y_{\Sigma}< 10^{-2}$, the $p_{t,bal}$
is lower than $1$. The opposite behaviour is observed at $y>10^{-1}$
where $p_{t,had}$ is reconstructed to higher values of $p_{t}^{el}$
(see Fig. \ref{Figptbalsig} (e) and (f)). Nevertheless for values of
$y_{\Sigma}>10^{-2}$ the balance between the electron and the hadrons
is well described by the {\small MC} simulation, the difference between
data and {\small MC} do not exceed $2\%$ for this $y_{\Sigma}$ region.

The $p_{t,bal}$ distribution is used to investigate the energy flow
of the HFS in the transverse direction, and it is observed that for
values of $y_{\Sigma} \geq 10^{-2}$ is good described by the {\small
  MC} simulation (see Figures \ref{Figptbalsig}(c) to
\ref{Figptbalsig}(d)).
However the kinematic variable $y_{\Sigma}$ is not well described in
the whole sample as can be seen from Fig. \ref{Figysigmaall}. The {\small MC}
simulation can not reproduce the behaviour of the data for low values
of $y_{\Sigma}$, spefically an excess is observed around $y_{\Sigma} \approx
10^{-2}$. Since $y_{\Sigma}$ is one of the most important kinematical
variables used to measure $F_2$, and $p_{t,bal}$ is used as a reference
for the hadronic energy scale, the disagreement between data and
{\small MC} for low values of $y$
had to be investigated in more detail. The reason of the discrepancy
might be related to an incomplete description of the physics which is
implemented in the {\small MC} simulation. That would mean that the
discrepancy comes from the 
proton structure function assumptions, or it might be that the
discrepancy is a detector effect. In the following paragraphs this
last option is investigated in detail. 

\begin{figure}[t]
\begin{center}
\setlength{\unitlength}{1cm}
\epsfig{figure=./hadron/figures/ysigma060302_1448def.eps,width=0.85\textwidth}
\put(-11.5,4.4){\small{(a)}}
\put(-5.5,4.4){\small{(b)}}
\caption{(a) The inelasticity variable $y_{\Sigma}$ for the whole DIS
  sample,(b) the ratio of data over MC of $y_{\Sigma}$.}
\label{Figysigmaall}
\end{center}
\end{figure}

\begin{figure}[t]
\begin{center}
\epsfig{figure=./hadron/figures/yhadcompcontrolplots_bst_060524_1637.eps,width=0.85\textwidth}
\caption{Contributions to $y_{had}$ from the different detector
  components. The closed points represent the data while the histogram
  correspond to the MC simulation.}
\label{Figyhadcomp}
\end{center}
\end{figure}

The reconstruction of $y$ with the hadron method allows to identify
which subdetector component has the main impact in a
specific kinematical region. The contribution of a subdetector
component $i-had$ to the kinematical variable $y_{had}$ is
calculated through the fraction of the $(E-p_{z})_{i-had}$ measured by the
subdetector component $i-had$ to the total $(E-p_z)_{had}$ of the hadrons. For
example, in order to calculate the fraction of $y$ coming from the
track measurement, each event is weighted with

\begin{equation}
f_{y}^{Track} = \frac{y_{Track}}{y_{had}} =
\frac{(E-p_z)^{Track}}{(E-p_z)_{had}} 
\label{Eqydecomposition}
\end{equation}

where

\begin{equation}
(E-p_z)_{had} = (E-p_z)_{LAr} + (E-p_z)_{Track} + (E-p_{z})_{HadSpa}
\end{equation}

$(E-p_z)_{had}$ considers the contributions from the LAr, tracks and the
hadrons measured with the SpaCal calorimeter. The decomposition of
$y_{had}$ is shown in Fig. \ref{Figyhadcomp}, where the whole measured
energy is taken into account. At high $y_{had}$ the contribution of
the SpaCal calorimeter dominates, while the tracks contribute mainly
at medium values of $y_{had}$. The LAr calorimeter contributes over
the whole kinematic region, and the 
identified LAr-{\small FSCOMB} noise plays a major role in the low
$y_{had}$ region. From Fig. \ref{Figyhadcomp} it is clear that in the
region $y\approx 10^{-2}$, where the ``$y_{\Sigma}-$bump'' occurs (see
Fig. \ref{Figysigmaall}(a)), the identified LAr-{\small FSCOMB} noise has
a large influence on the reconstruction of the kinematics from the HFS.

In addition to the noise identification performed by the {\small{FSCOMB}}
algorithm there are several noise suppression mechanisms implemented
in the reconstruction of the energy measured by the LAr calorimeter. 
The first noise rejection is performed during the recording of
raw data. Further noise suppression are done by the algorithms which
merge the energy deposition into clusters \cite{cigdemdiplom,cigdem}.

In order to study the influence of the noise in the DIS selection,
different {\small MC} simulations were performed varying the noise
input. The procedure used to include the noise in the {\small MC}
simulation of the H1 calorimeter and how the noise is treated during
the reconstruction is explained in the next section.  

\section{LAr Noise}
\label{Seclarnoi}

\subsection{Noise in the MC simulation}

The energy deposited in the LAr calorimeter is overlayed with noise
signals. Basically, two contributions not originating from the genuine
$ep$ DIS event are present during the energy
measurement. One noise source is related to the capacitance of the LAr
cells and of the amplifiers which
process the signals, while another noise source is beam
induced \cite{Larelectronicnoise,olaf}. This two noise sources are
indistingible from the real LAr signal during the data taking phase.

In contrast to the data, the energy depositions obtained over the
whole detector with the {\small MC} simulation
arise exclusively from the $ep$ collision. Thus,
in order to compare the energy measured in the LAr with 
the {\small MC} simulation, the noise present in the data also has to be
considered. Two strategies can be followed, one can either try
to simulate the noise or one can add the noise to the {\small MC}
extracted from real data. The 
noise simulation is difficult, since it changes
with the run period and the beam conditions and its shape is not a simple
gaussian distribution which could be added to the signal of the
calorimeter cells. Therefore in H1 the LAr noise is added simply to
the {\small MC} simulation using its sprectra recorded with real data.


The noise is readout in special runs known as {\it ``noise
files''}. The aim of the {\it noise-file} is to record the sprectrum of the
noise of each LAr cell.  
The {\it noise-file} is taken with random triggers. The task of
the random trigger is to get empty 
events without any physics signal originating from $ep$ event. Thus,
in the ideal case pure noise 
should be collected. The {\it noise-files} are runs qualified as poor and
usually are taken with $ep$ collisions and contains typically only 1000
events. The noise file recorded for the shifted vertex
period will be in the following section called {\it``default noise-file''}.


Each event of the {\small MC} simulation, will contain in addition to the
energy depositions of the $ep$ collision the
energy deposited of one event of the {\it noise-file}. This means that
each $N-$th events \footnote{where N is the length of the noise file},
the same noise pattern is fed into the {\small MC} simulation. 
After the noise from the {\it noise file} is incorporated in the
{\small MC} simulation, the
noise suppresion algorithms will be performed in the same way as
applied to the data. 

For events coming from the normal data taking, the signal of the LAr 
cells is readout only in case, its deposited charge is greater than
the noise cut $\sigma$. The value of $\sigma$ depends on the position
of the module, having a value which varies
between $10\,$MeV and $30\,$MeV \cite{Abt:1996xv,gayler}. This noise
suppression is applied 
online for all data. However, this cut is {\it not} applied if a
{\it noise-file} is taken, since the aim of the
{\it noise-file} is to readout the whole 
noise spectrum. Thus, the first noise suppression which is performed in
the {\small MC}, is the application of the online $\sigma$ cut.

The noise suppression is performed to all LAr calorimeter cells, and it
can occur that the investigated 
cell has a energy which is below the $\sigma$ or pedestrial value.
In order not to overestimate the
signal, the cells which have negative energy below the $\sigma$ value
will not be excluded. Thus, the 
noise substraction is performed using the absolute value of the energy
threshold $n \cdot \sigma$ (the value of $n$ depend on the octant in
which the cell is located). 

During the reconstruction of the energy measurement a so called
``topological noise suppresion'' is also performed. One cell
will be considered in the cluster algorithm if its energy or the energy
of its neighbor is higher than $4\sigma$ \cite{cigdem}. The noise suppression 
mechanism explained in the previous paragraphs together with the
cluster algorithm is done in the same way for data and {\small MC}.  

\subsection{Investigation of the noise in the MC}

In order to investigate the influence of the {\it noise file}
and the noise which is identified by the {\small FSCOMB}
algorithm, different {\small MC} simulations were produced 
varying the input {\it noise-file}. In order
not to change the physics coming from the $ep$ DIS event
the {\small MC} simulations must have the same generated files as the ones
disccussed in section \ref{Secmontecarlo}. 

The following {\small MC} samples were produced:

\begin{enumerate}
\item {\bf Half-Noise MC sample.} \\
As already mentioned the noise-files posess only 1000 events, thus
its statistic could create a bias in the final measured distributions,
since each $1000$ event the same pattern of noise is used. To study
the influence of this possible bias a
{\small MC} with half of the statistics of the used noise-file was produced.

\item {\bf Alternative noise file MC sample.} \\
In order to investigate how the final measured distribution are
influenced by the added noise, an alternative {\it noise-file} was
used to produce the {\small MC} simulation. The {\it noise-file} used
until now, were recorded one day after the shifted vertex run period. 
The alternative {\it noise-file} was recorded three month before the
shifted vertex period \footnote{The {\it noise-file} for the shifted
  vertex data was taken at 21.08.00 in the lumi fill 2601 while
  the alternative {\it noise-file} was taken in the lumi fill 2462 the
  11.06.00}.  

\begin{figure}[t]
\begin{center}
\includegraphics[height=6.0cm]{./hadron/figures/noisecomptheta.eps}
\caption{Comparison of the angular energy deposition in the {\it
    noise-files} (plot from \cite{olaf}).}
\label{Figthetanoisefiles}
\end{center}
\end{figure}

The amount of added energy on the {\it default} and {\it alternative
  noise file} after applying the online cut as function of 
$\theta_{cell}$ is shown in Fig. \ref{Figthetanoisefiles}. One can see
that the {\it default noise-file} of the shifted vertex data has in average
more deposited energy than the {\it alternative} one. How much this
  additional noise changes the hadronic distributions will be
  discussed in the next section. 

\item {\bf No-Added noise MC sample} \\
A {\small MC} where only the energy deposition in the LAr comes from the
real $ep$ scattering is obtained by dropping the contribution derived from
the {\it noise-file}. In order to compare with the data the
pedestal or $\sigma$ cut value is performed. This {\small MC} can
be used to derive an upper limit of the error due to a
``wrong'' identification of the noise. 
\end{enumerate}

In the following subsections, the HFS distributions
for each of the special {\small MC} simulations together with the
default {\small MC} simulation are presented. The data sample of the
three special 
{\small MC} simulations were calibrated with the 
Lagrangian method explained in section \ref{SecLarcalib}. 


\begin{figure}[t]
\begin{center}
\begin{minipage}{1.\textwidth}
\begin{center}
\setlength{\unitlength}{1cm}
\includegraphics[width=14.0cm]{./hadron/figures/hadcalib_defvsonoi.eps}
\put(-2.4,4.8){\small{(a)}}
\end{center}
\end{minipage}
\begin{minipage}{1.\textwidth}
\begin{center}
\setlength{\unitlength}{1cm}
\includegraphics[width=14.0cm]{./hadron/figures/hadcalib_defvsnonoi.eps}
\put(-2.4,4.8){\small{(b)}}
\end{center}
\end{minipage}
\caption{Comparison of the calibration constants for the LAr
  calorimeter for {\small DJANGOH} {\small MC}, open points represent the default
  simulation, closed circles correspond to the {\small MC} simulation with
  (a) a {\it noise-file} recorded before Shifted vertex period and (b) no
  electronic noise applied to the simulation. The percent values
  correspond to the deviation of the constants from the default
  simulation}
\label{Figcalibhad}
\end{center}
\end{figure}

\subsubsection{Half-Noise MC}

The calibration constants with half of the statistic of the 
default {\it noise-file} do not considerably change from the calibration
constants compared to the whole statistic. The main distributions
as $p_{t,bal}$ and $y_{\Sigma}$ show the same behaviour as the
distributions presented in section \ref{Secdescriptionhfs}. Therefore
this result does confirm that the statistics of the {\it noise-file}
does not affect the measured final distributions.  

\begin{figure}[t]
\setlength{\unitlength}{1cm}
\begin{center}
\begin{minipage}{0.65\textwidth}
\epsfig{figure=./hadron/figures/ysigallnoi.eps,width=\textwidth}
\put(-2.,2.0){\small{(a)}}
\end{minipage}
\begin{minipage}{0.65\textwidth}
\epsfig{figure=./hadron/figures/yfscomballnoi.eps,width=\textwidth}
\put(-2.,2.0){\small{(b)}}
\end{minipage}
\caption{(a) Comparison between data and the different MC simulations
  of the $y_{\Sigma}$ distributions of the DIS sample. (b) Comparison
  between data and MC for the mean value in $y$ of the noise
  identified by the {\small FSCOMB} algorithm.}
\label{Figysigma}
\end{center}
\end{figure}

\subsubsection{Alternative-Noise and No-Added-Noise MC}

Variations of the calibration constants are observed in the
simulation when the {\it alternative noise-file} is used and when
no-noise is added. In Fig. \ref{Figcalibhad} is shown the comparison of the
calibration constants between the default {\small MC} simulation (closed
points) and the alternative {\it noise-file} (a) and the simulation without
addition of noise (b).  The calibration constants for the {\it
alternative-noise} {\small MC} differ in average by $2\%$ from
the calibration constants of 
the default {\it noise-file}. In case the noise is not added in the {\small MC}
simulation, the calibration constants differ $3\%$ and $6\%$ from
the default {\small MC}, as 
follows from Fig. \ref{Figcalibhad} (b). The calibration constants of
the {\small CBH1} modules vary strongly due to the limited statistic.

The comparisons between the data and the different {\small MC}
simulations for the $y_{\Sigma}$ distributions are shown in
Fig. \ref{Figysigma}(a). The excess of data over {\small MC} at
$y_{\Sigma} \approx 10^{-2}$ is
always present and seems to be independent of the choice of the {\it
  noise-file}. 

Fig. \ref{Figysigma}(b) shows the mean value in $y_{had}$ of the noise
identified by the  {\small FSCOMB} algorithm (see
\ref{Subsecfscombnoise}). The noise identified by the 
{\small FSCOMB} algorithm is localised in average at the same values
of $y_{had}$ for each MC simulation.
In fact, also the simulation where {\bf no noise} was added, shows the same
behaviour. For the no-noise {\small MC} the average value of $y_{had}$ is
found towards lower values than the simulations where the noise was
explicity added using the {\it noise-files} (see 
Fig.\ref{Figysigma}(b)). This is expected since, no 
additional noise is explicitly added.
Comparing the distributions of the position in $y_{had}$ of the noise in
Fig. \ref{Figysigma}(b), it is clear that the {\small FSCOMB}
algorithm identifies more noise than acctually is added.
The isolated clusters which the {\small FSCOMB} algorithm
identifies as noise do not always correspond to the noise added with the
{\it noise-file} in the {\small MC} simulation. {\bf Part} of those isolated
cluster can be real signals coming from the $ep$ scattering. 

\begin{figure}[t]
\begin{center}
\setlength{\unitlength}{1cm}
%\epsfig{figure=./hadron/figures/ysigmanoi_060421_1147.eps,width=0.5\textwidth}
\epsfig{figure=./hadron/figures/ysigma060421_1147def.eps,width=0.85\textwidth}
\put(-11.5,4.4){\small{(a)}}
\put(-5.5,4.4){\small{(b)}}
\caption{The inelasticity variable $y_{\Sigma}$ not substracting the
  {\small FSCOMB} noise.}
\label{Figysigmawithnoise}
\end{center}
\end{figure}

\begin{figure}[p]
\setlength{\unitlength}{1cm}
\begin{center}
\begin{minipage}{0.85\textwidth}
\epsfig{figure=./hadron/figures/yhadecom2noi.eps,width=\textwidth}
\put(-2.,8.5){\small{(a)}}
\end{minipage}
\begin{minipage}{0.85\textwidth}
\epsfig{figure=./hadron/figures/ydecompnonoi.eps,width=\textwidth}
\put(-2.,8.5){\small{(b)}}
\end{minipage}
%\includegraphics[height=10.0cm]{./hadron/figures/ydecompnonoi.eps}
\caption{Contribution to $y_{had}$ from the different subdetector
  components for data (closed points) and MC (a) 
  using the {\it official} and
  {\it alternative} noise file. The bands represent the variation
  between the two MC simulation, their width does not exceed $2\%$.
  (b) Contribution to $y_{had}$ of data (closed points) and the
  No-noise {\small MC} simulation (histogram).}
\label{Figydecompnonoi}
\end{center}
\end{figure}

The isolated clusters which are identified by {\small FSCOMB} as noise, are
located mainly in the forward region 
and are isotropically distributed in azimuth. However, the
substraction of the {\small FSCOMB} noise is necessary 
since its incorporation into the reconstruction of the kinematical
variables conduces to a huge disagreement between the data and the
{\small MC} simulation of more than $10\%$.
Fig. \ref{Figysigmawithnoise} show the distributions of
$y_{\Sigma}$ in case the {\small FSCOMB} noise is not substracted. The
$y_{\Sigma}$ distribution is better described if the {\small FSCOMB}
noise is substracted (compare Fig. \ref{Figysigmaall}(a) with
Fig. \ref{Figysigmawithnoise}(a).) 

Another source of noise can influence the measurement of the
$(E-p_z)_{had}$ of the HFS. These energy depositions are coming from back scattered
particles in the forward region. This rescattering contribution is included 
already in the {\small MC} simulation.
% such scattering the {\small FSCOMB} algorithm minimizes \ref{FSCOMB2}. 

From this results one concludes that:

\begin{itemize}

 \item The addition of the {\it noise-files} does not considerably 
   change the measured distributions relevant for the $F_2$
   measurement. 
   
   The relative contribution of the LAr calorimeter differ from the
   {\it official} and the {\it alternative noise file} by less than $2\%$.
   Fig. \ref{Figydecompnonoi}(a) shows the $y_{had}$ decomposition of
   the data and the comparison between the {\small MC} simulations
   using different noise files. The drawn band in
   Fig. \ref{Figydecompnonoi}(a) 
   represent the variations of the hadronic response using the
   default {\small MC} and the {\small MC} with the {\it alternative
   noise-file}. The width of the band is less than $2\%$ over all
   accessed values of $y_{had}$

 \item The noise identified by the {\small FSCOMB} algorithm has the
   biggest influence on the reconstruction of $y_{had}$ and
   $y_{\Sigma}$ for values of $O(10^{-2})$. 
 
\end{itemize}

The excess of the data in comparison with the MC remains.
Nevertheless the uncertainty of the noise identification performed by
{\small FSCOMB} has to be taken into account. The uncertainty will be
calculated 
using the variation between data and {\small MC} of the {\small FSCOMB} noise
over the whole $y_{had}$ range. Since the {\small MC} without {\it
  noise-file} is not able to describe the data, the 
uncertainty is derived using the {\small MC} with the {\it default
  noise-file}. In Fig. \ref{Figydecompnonoi}(b)
is shown the $y_{had}$ decomposition distribution for the no noise {\small MC}
simulation. For this {\small MC} simulation, the LAr contribution and
of course the LAr-noise contribution 
completely disagree with the measured data. 

\begin{figure}[p]
\begin{center}
\setlength{\unitlength}{1cm}
\begin{minipage}{0.75\textwidth}
\epsfig{figure=./hadron/figures/yhaddecomp_4_060524_1637.eps,width=\textwidth}
\put(-2.,8.0){\small{(a)}}
\end{minipage}
\begin{minipage}{0.75\textwidth}
\epsfig{figure=./hadron/figures/ynoise_4_060524_1637.eps,width=\textwidth}
\put(-2.,2.8){\small{(b)}}
\end{minipage}
\begin{minipage}{0.75\textwidth}
\epsfig{figure=./hadron/figures/yhaddecomp_3_060524_1637.eps,width=\textwidth}
\put(-2.,8.0){\small{(c)}}
\end{minipage}
\caption{(a)Contributions to $y_{had}$ from different detector components 
  in case the {\small FSCOMB} noise is varied by $10\%$. (b) The ratio
  of data and the {\it official} {\small MC} of the {\small FSCOMB}
  noise fraction. (c) Contributions to $y_{had}$ from different
  detector components in case the LAr signal is varied linearly by
  $10\%$ at $y=10^{-3}$ and $2\%$ at $y=10^{-1}$.}
\label{Figyhadsyst}
\end{center}
\end{figure}

The uncertainty due to the noise substraction procedure is taken to be
$10\%$. In Fig. \ref{Figyhadsyst}(a) 
the $y_{had}$ decomposition distribution is shown again. The band
corresponds to the 
variation of the LAr and the {\small FSCOMB} noise, in case the {\small
FSCOMB} noise is varied by $10\%$. In Fig. \ref{Figyhadsyst}(b) is
shown the ratio of the noise fraction between data and MC taken from
the default MC. From this distribution follows that the assigned $10\%$
uncertainty is conservative over the whole measured phase space. 

Since at $y=10^{-3}$ the noise is significant, the uncertainty of
the {\it LAr signal} is taken to be $10\%$ at $y = 10^{-3}$ and is
extrapolated linearly to $2\%$ at $y=10^{-1}$.
Fig. \ref{Figyhadsyst}(c) shows the decomposition of $y_{had}$ for
data and the default MC in case the LAr signal is varied by $2\%$. The
drawn band correspond to the asigned uncertainty of the {\it LAr
  signal}. 

The uncertainties of the {\it LAr signal} and {\it Noise} will be
taken into account for the determination of the systematic errors of
the $F_2$ measurement. 

One additional remark concerning the official {\it noise-file} used
for the shifter vertex data period is necessary: it was recorded one day
after the shifted vertex period ended. Thus, it was not clear if the
noise conditions of the {\it noise-file} are appropiate for the
shifted vertex data sample. One 
possibility to verify it is using so called
``random-events''. These data are recorded always during the data taking
with a constant rate. The main difference between the random-events and the
noise-files is, that the random-runs posess the online noise
suppression cut of $2\sigma$ \footnote{The online noise suppresion cut
is different for each module of the LAr.}. Therefore not all cells are 
read out and recorded. While, the {\it noise-files} contains the whole noise
spectrum for all cells. However, this random events can be used to
check the noise behaviour of the data period by comparing the noise
contribution above the online noise $\sigma$ cut. The comparison of the 
{\it noise-file} used for the shifted vertex {\small MC} with the
random runs was performed in \cite{olaf}. Good agreeement was found
between the official {\it noise-file} and the random events of the
shifted vertex period. Thus, the official {\it noise-file} is well
suited to be used for the {\small MC} of the shifted vertex period. 

\section{Diffractive Events}
\label{Secdiffevents}

\begin{figure}[t]
\centering
\begin{minipage}{0.48\textwidth}
\setlength{\unitlength}{1cm}
\epsfig{figure=./hadron/figures/ysigpomnopom.eps,width=\textwidth}
\put(-5.8,3.4){\small{(a)}}
\end{minipage}
\begin{minipage}{0.48\textwidth}
\setlength{\unitlength}{1cm}
\epsfig{figure=./hadron/figures/ptbalpomnopom.eps,width=\textwidth}
\put(-6.6,3.4){\small{(b)}}
\end{minipage}
\caption{(a) $y_{\Sigma}$ and (b) $p_{t,bal}$ distributions for the {\small MC}
simulation where no diffractive events were generated.}
\label{Figypomnopom}
\end{figure}

As was already pointed in section \ref{Secdescriptionhfs}, the {\small
  DJANGOH} {\small MC} can 
reasonably describe the behaviour of the inclusive data. Nevertheless,
in previous analysis it was already observed that {\small DJANGOH} is not
appropiate to study exclusive processes like DVCS\cite{rainer} and QEDC
\cite{victor}. 

One possible explanation of the $y_{\Sigma}$ discrepancy between data
and MC could
come from the diffractive component of the cross section.
Diffractive events constitute $6\%$ of the inclusive DIS cross
section \cite{diffractive}. The {\small DJANGOH} {\small MC} used in
this analysis, includes diffractive 
events through the model of soft color interactions.
In order to investigate if this component
has an influence on the $y_{\Sigma}$ distribution and the energy
flow, a {\small MC} sample using the same version of {\small DJANGOH}
without the soft color interaction model was produced. 

In Fig. \ref{Figypomnopom} the $y_{\Sigma}$ and the $p_{t,bal}$ 
distributions are shown. The data are compared to the {\small MC}
simulation where no 
diffractive events were generated. The diffractive contribution of
the {\small DJANGOH} simulation is not localized in low $y$ region. In
addition the exclusion of the diffractive events creates an
imbalance of the transversal energy as follows from
Fig. \ref{Figypomnopom}(b). Therefore the diffractive component can
not explain the differences observed between 
the data and the {\small MC} simulation in the low $y$ region. 
 
\section{Hadronic energy scale uncertainty}

In this chapter, the HFS was studied using the $p_{t,bal}$ and $(E-p_z)_{tot}$
distributions. The deviations of the {\small MC}
simulation to the data were used to derive the systematic
uncertainties which are listed in table \ref{Tabhadun}. 

\begin{table}[h]
\begin{center}
\begin{tabular}{c c c}
\hline
Source & Uncertainty & reference \\
\hline
Noise identification & $10\%$ & Fig\ref{Figyhadsyst}(b) \\
Hadronic energy & $10\%$ for $y=10^{-3}$ & Fig \ref{Figyhadsyst}(c) \\
                     & $2\%$ for $y=0.1$ \\
Hadronic SpaCal & $350\,$MeV & Fig. \ref{Figyhadspa}(b) \\
\hline
\end{tabular}
\caption{List of uncertainties on the hadronic measurement.}
\label{Tabhadun}
\end{center}
\end{table}


