\chapter{Introduction}

One of the most amazing questions in physics is how matter is composed
and created. One method used to investigate the structure of the
matter, uses the scattering of elementary particles.
Since electrons seem to be elemental they have been 
used to probe the structure of the matter. For example one needs
electrons of few keV of energy to resolve the size of the atom
better than 1A ($10^{-10}$m). In order to discover the structure of the 
nucleus one has to resolve distances of approximately $1\,$fermi. These
distances can be studied using the scattering of nucleus with
electrons having an energy of a few hundred of MeV.
%and for that is necessary to scatter electrons having energies of a
%few hundred MeV.

Similar to the nucleus structure one can probe the structure of the
proton using electrons with a few GeV of energy. In the electron energy
spectrum of such $ep$ collisions there is an elastic peak in
which the proton recoils as a whole and secondary peaks appear due
to the excitation of the proton into various higher-mass $N^{*}$
resonant states. After the resonance region a continuum distribution
is observed and this corresponds to those electrons which have been
scattered by the constituents of the proton, now known as {\it quarks}. 

The suggestion that the protons and neutrons could be composed of more
basic objects called ``quarks'' was made in 1964 by Gell-Mann
and Zweig \cite{GellMann}. Experiments to probe the structure of the proton by
scattering electrons began at Stanford in the late 1960 and there
it was revealed that protons contain point-like constituents. It was
Feynman \cite{Feynmann} who called them ``partons'' and only after the
identification 
of the ``partons'' with ``quarks'' it was possible to establish
the first two of the three families which summaries the
standard model of particles.  

Deep inelastic scattering has thus played an important role in the
development of the standard model of particles. In this thesis the
structure of the proton is studied using also the deep inelastic
scattering of electrons with protons. In order to understand the
kinematic region explored, the most important variables and experimental
results of DIS will be summarized in the following sections.

\section{Deep inelastic scattering}
 
The generic diagram of the electron proton scattering is shown in
Fig. \ref{Figgendis}, where $k$ and $p$ represent the four momemtum of
the incoming electron and proton, respectively. The four
momentum of the outcoming lepton is denoted by $k^{\prime}$ while
$P_{x}$ represent the four momentum of the hadronic final
state. Depending on which vector boson  $V^{*}$ is exchanged, the
electron proton scattering can be divided in Neutral Current (NC) and
Charge Current processes (CC). 

\begin{figure}[h]
\begin{center}
\epsfig{figure=./introduction/texfig/genericdis.eps,width=0.55\textwidth}
%\includegraphics{./introduction/texfig/genericdis.eps}
\caption{Schematic diagram of lepton proton scattering via
  vector-boson exchange}
\label{Figgendis}
\end{center}
\end{figure}

%In charge current processes a charged boson is exchanged, therefore
%the produced scattered lepton is a neutrino.
%neutrino is produced as the scattered lepton

%In charge current processes a charged boson $W^{\pm}$ is
%exchanged, leading a neutrino as the scattered lepton.

In case a CC process occurs a charged boson $W^{\pm}$ mediates the
electron proton interaction, producing a neutrino as the outcoming
lepton. In NC events a neutral boson ($\gamma$ or $Z^{0}$) is
exchanged, and then an scattered lepton is 
observed in the final state. While CC events are produced purely
by weak interaction, the NC events will be dominated by the
electromagnetic interaction for low values of the virtuality
$Q=\sqrt{-(k-k^{\prime})^2}$. 

The kinematic of the electron-proton reaction is usually
described by the following  Lorentz-invariant quantities:

\begin{itemize}
\item The center of mass energy of the electron proton interaction,

\begin{equation}
s = (p+k)^2 
\label{sform}
\end{equation}

\item The invariant mass squared of the virtual exchanged boson,

\begin{equation}
Q^{2} = -q^{2}
\label{q2form} 
\end{equation}

\item The Bjorken $x$ variable, which is interpreted in the lowest
  order of QCD, as the fraction of the momentum of the incoming proton
  taken by the struck quark, 

\begin{equation}
x = \frac{Q^2}{2p \cdot q} 
\label{xbjk}
\end{equation}

\item The inelasticity $y$, which gives a measure of the amount of
  energy transferred between the electron and the proton. 

\begin{equation}
y = \frac{p \cdot q}{p \cdot k}
\label{yvar}
\end{equation}

\end{itemize}

Due its definition $x$ and $y$ vary in the interval $[0,1]$. Ignoring
the electron and proton masses, from Eqs.\ref{q2form}, \ref{xbjk},
\ref{yvar} follows:

\begin{equation}
Q^2 \approx sxy
\label{qxy}
\end{equation} 

\noindent therefore only two of these quantities are
independent, and then the $ep$ scattering can be described in terms 
of $s,Q^2$ and $x$. 

An additional variable used to describe the
hadronic final state is the center of mass energy of the $\gamma p$
system:

\begin{eqnarray*}
W^2 &=& (q+p)^2 = P_{x}^{2} \\
&=& Q^2 \frac{1-x}{x} + m_{p}^{2} 
\end{eqnarray*}

One speaks of deep inelastic scattering of electron proton collisions, 
if the virtuality $\sqrt{Q^2}$ of the exchanded boson is above the proton mass and 
if the center of mass energy of the $\gamma p$ system goes beyond the
resonance domain $W^2 >> m_{p}^2$.

If we concentrate on NC processes in the region where the photon
exchange dominates ($Q^2 < 100\,$GeV$^{2}$) then the amplitude of the
$ep \rightarrow eX$ reaction is proportional to  

\[
\mathcal{A} \propto \frac{1}{q^2} L_{\mu \nu} W^{\mu \nu}
\]

\noindent where $L_{\mu \nu}$ is the leptonic tensor which describes
the interaction between the electron and the photon and is calculable
throught QED. $W^{\mu \nu}$ is the hadronic tensor associated to the
$\gamma p$ vertex.

In contrast to the leptonic tensor $L_{\mu \nu}$, the exact form of
the current in the hadronic vertex can not be calculated. The hadronic
tensor $W^{\mu \nu}$ has to be parametrized 
in terms of so called proton structure functions. The proton structure
functions will describe the $\gamma p \rightarrow X$ transition.
For NC processes
only two arbitrary functions $F_{2}$, $F_{1}$ depending on
$x$ and $Q^2$ are relevant \cite{A.D.Martin,rdevenish}.

The differential cross section for $ep \rightarrow eX$ is given by:

\begin{equation}
\frac{d^{2} \sigma}{dx dQ^2} = k \left[F_{2}(x,Q^2) -
  \frac{y^2}{Y_{+}} F_{L}(x,Q^2) \right] 
\label{sigmared}
\end{equation} 

\noindent where

\[
Y_{+} = 1 + (1-y)^{2} \hspace{1.cm} k = \frac{2 \pi \alpha^{2}}{Q^{4}x} Y_{+} \]
\[
F_{L} = F_{2}-2xF_{1}
\]

Thus the proton structure functions can be obtained from the
experiment if the cross section, gets measured differentially in $Q^2$
and $x$. One useful quantity is obtained from Eq. \ref{sigmared},
absorving the kinematic factor $k$, and is known as the reduced cross
section $\sigma_{red}$. 


\begin{equation}
\sigma_{red} = F_{2}(x,Q^2) - \frac{y^2}{Y_{+}} F_{L}(x,Q^2)
\label{sigmaredred}
\end{equation} 

\subsection{The Naive Quark Parton Model}

The quark parton model (QPM) introduced by Feynman in 1969
\cite{Feynmann} provides a simple physical interpretation of the
electron proton scattering. This 
model considers that at large values of $Q^2$ the $\gamma p$
interaction can be expressed as the sum of incoherent scattering from
point like quark constituents which behaves as free inside 
the proton during the interaction. This picture can only be correct if 
the scattering process occurs over a short time scale during which the
photon sees a frozen state of non interacting quarks \cite{Paschos}.

Let $f_i$ be the probability to find a quark-$i$ in the
proton with a fraction $\chi$ of the proton momentum. If the quarks do
not interact with each other, and assuming partons of spin
$0$, then the total $ep$ cross
section can be expressed as:

\begin{eqnarray}
\frac{d^{2} \sigma}{dx dQ^2} &=& \sum_{i} \int d\chi f_{i}(\chi)
\left(\frac{d^2 \sigma_i}{dx dQ^2}\right)_{eq_{i} \rightarrow
  eq_{i}} \nonumber \\ 
 & = & \sum_{i} \int d\chi f_{i}(\chi) \frac{4 \pi
  \alpha^{2}}{Q^4}\frac{e_{i}^{2}}{2} [1+(1-y)^2] \delta(x-\chi)
\nonumber \\ 
&=& \sum_{i} \frac{2 \pi \alpha^{2}}{Q^4}e_{i}^{2} Y_{+} f_{i}(x)
\label{formqpmtot1}
\end{eqnarray}

\noindent where in Eq. \ref{formqpmtot1} the elastic $eq_{i}
\rightarrow eq_{i}$ cross section have been introduced. 

Comparing the two $ep$ cross section formulae in 
Eqs. \ref{formqpmtot1}, \ref{sigmared} it follows that: 

\begin{equation}
F_{2}(x,Q^2) = \sum_{i} f_{i}(x) x e_{qi}^{2} 
\label{callangross}
\end{equation}

\begin{figure}[t]
\begin{center}
\epsfig{file=./introduction/figures/slacdata.eps,angle=90,width=0.8\textwidth}
\caption{The $\nu F_2 = F_2$ structure function as function of $Q^2$
  at fixed $x=0.25$, as it was measured by the SLAC-MIT group \cite{slacmit}}
\label{Figqpm}
\end{center}
\end{figure}

The QPM predicts that $F_{2}$ is a function of $x$ only, a phenomenon
known as {\it scaling}. Fig. \ref{Figqpm} shows the measurement of
$F_2$ as function of $Q^2$ performed by the SLAC group. The relation
$F_2(x)=2xF_{1}(x)$ was derived by Callan-Gross \cite{CallanGross},
representing a direct consequence of the spin $1/2$ of the quarks.

Soon after the SLAC experiments were performed the violation of the
scale was observed in muon nucleon \cite{muondis} scattering and later
confirmed by neutrino nucleon scattering experiments
\cite{neutrinodis0,neutrinodis1}. It was not 
clear how to reconcile the picture of the free partons into the
experimental observations. 

\section{The QCD improved parton model}

Altough the QPM provides a simple picture of the electron proton
scattering and can describe the scaling behaviour of the SLAC and
Gargamelle data \cite{gargamelle},
it was not able to explain the scaling violation behavior observed
afterwards.  
Also it can not explain the fact that quarks are not detected as free
particles but always build bound systems as observed in 
hadron spectra experiments, a phenomenon called ``confinement''. 
It turns out that the QPM is the lowest order approximation of a more
general theory known as Quantum Chromodynamics (QCD).  

\begin{figure}[h]
\begin{center}
\includegraphics[height=4cm]{./introduction/figures/allloopsorig2.ps}
\caption{Loop diagrams illustrating the interaction between quarks and
gluons.}
\label{Figloops}
\end{center}
\end{figure}

QCD is a non-abelian theory based on the symmetry group $SU(3)_{c}$,
there quarks besides electric charge, also carry ``color charge''
which can be of three types  
``red, green, blue''. The interaction between quarks is mediated through the
exchange of one of the eight-gluons which generate the symmetry group. The
gluons play a equivalent role as the
photons in QED do, but contrary to the photons, the interaction
between gluons is possible. 
Therefore within QCD, the interaction between quarks considers diagrams as
shown in Fig. \ref{Figloops}. The contributions of the loop diagrams
to any amplitude diverge but can be resummed introducing a $Q^2$
dependence of the
coupling between the quarks and gluons $\alpha_s$. Formally this
behaviour is known as the ``running'' of the strong coupling constant, 
which in the leading order logarithmic approximation is given by: 

\begin{equation}
\alpha_{s}(Q^2) = \frac{12 \pi}{(33-2N_{f}) \ln (Q^2/ \Lambda ^{2})}
\label{alphas}
\end{equation}

where $N_{f}$ is the number of quarks flavours with the mass
$m_{q}^{2} < Q^{2}$, and $\Lambda \approx 100 MeV$ determines the
energy scale at which $\alpha_s$ becomes large.

Two important limits are deduced from Eq. \ref{alphas}.
Since momentum is related to distance due to the uncertainty principle,
at low momentum the coupling between quarks is leading to quark
{\it confinement}.  
An opposite behaviour is seen at high $Q^2$, since $\alpha_s$ becomes small
meaning that the coupling between quarks is weak and therefore can be 
treated as nearly free particles \cite{gw73,pol73}. The running of the strong
coupling constant $\alpha_s$ is able to describe at the same time 
the experimental results of confinement related to the existence of
hadron spectra and asymptotic freedom seen in
DIS experiments. Note however that confinement has
not been proven exactly to follow from Eq. \ref{alphas}.

%
% Explain that pQCD is applicable up values of q2> 3.5GeV^2
%

The introduction of gluons in QCD changed the picture of the proton
and the interpretation of the $ep$ interaction as follows. 
The proton is composed of three ``valence quarks'' which interact
throught the exchange of gluons. The gluons
can produce quark-antiquark pairs also known as ``sea quarks'' and
radiation of gluons due to their self-interaction is possible. 


\begin{figure}[t]
\begin{center}
\includegraphics[height=10cm]{./introduction/figures/fsplitt2.eps}
\caption{Splitting functions}
\label{Figsplitfunctions}
\end{center}
\end{figure}

At lowest order, QCD reproduces the QPM picture of the
interaction, but considering the first order of the $\alpha_s$
expansion (LO), the calculations of the $ep$ cross section involve
integrals which are 
divergent and have to be resummed (renormalized) as in the case of
$\alpha_s$. This resummation introduces a ``mass factorization scale''
which separates the short distance partonic effects from the
long distance hadronic effects. The choise of the separation is
arbitrary and in case the DIS scheme is used, then: 

\begin{equation}
F_2(x,Q^2) = \sum_{i} e_{i}^{2} x q_{i}(x,Q^2)
\label{f2qcd} 
\end{equation}

\noindent and contrary to the QPM, $F_2$ depends on $x$ and $Q^2$.
The parton distribution $q_i(x,Q^2)$ in Eq. \ref{f2qcd} can not be
predicted. But the requirement that they should not depend on the
``factorization scale'' leads to the prediction of its evolution in
$Q^2$, which in leading order can be written as:

\begin{eqnarray}
\frac{\partial q_{i}(x,Q^2)}{\partial \ln Q^2} = \frac{\alpha_{S}(Q^2)}{2 \pi}
\int_{x}^{1} \frac{dz}{z} \left[\sum_{j}
  q_{j}(z,Q^2)P_{ij}^{(0)}\left(\frac{x}{z}\right) +
  g(z,Q^2)P_{ig}^{(0)}\left(\frac{x}{z}\right) \right]
%\hspace{1cm} \\
\label{dglapquark}
\end{eqnarray}

\noindent where the functions $P_{ij}$ and $P_{ig}$ are known as splitting
functions. In LO the splitting functions have a natural physical
meaning e.g., $P_{ij}$ is the probability that a quark $i$ with
momentum $z$ radiates a gluon with $1-z$ of its momentum. 
Since quarks are coupled to gluons, the evolution of the gluon density
function is also required and is given by:

\begin{eqnarray}
\frac{\partial g(x,Q^2)}{\partial \ln Q^2} = \frac{\alpha_{S}(Q^2)}{2 \pi}
\int_{x}^{1} \frac{dz}{z} \left[\sum_{j}
  q_{j}(z,Q^2)P_{gj}^{(0)}\left(\frac{x}{z}\right) +
  g(z,Q^2)P_{gg}^{(0)}\left(\frac{x}{z}\right) \right] \hspace{1cm}
\label{dglapgluon}
\end{eqnarray}

where $P_{gj}$, $P_{gg}$ are the probability that a gluon splits in a
quark antiquark pair or split to 2 gluons (see Fig \ref{Figsplitfunctions}). 
The set of equations \ref{dglapquark}, \ref{dglapgluon} are known as
Altarelli Parisi equations \cite{Altarelli,gl72}.
The evolution of the parton distribution function on $x$ is not
predicted, but for a given scale $Q^2$ its dependence on $x$ can be
parametrized using the counting rules. The free parameters of the
parametrization will be determined using the measured data over
several order of magnitude of $x$ and $Q^2$. 
There are several parametrizations used, by example Glueck, Reya and Vogt
(GRV) \cite{GRV}, Martin,Ryskin,Sterling and Thorne MRST \cite{MRST}
and die CTEQ group \cite{CTEQ}. 

\begin{figure}[t]
\begin{center}
\epsfig{figure=./introduction/figures/f2vsQ2_allx_h19400_col.eps,width=0.9\textwidth}
\caption{Scaling violations measured in lepton proton scattering
  experiments \cite{h1fitfractal,H1F2-2001b,H1F2-2000,ZEUSF1-a,ZEUSF2-b,BCDMS,E665,NMC}.}
\label{Figf2vsq2x}
\end{center}
\end{figure}

Using this picture of the proton, the phenomenon of 
scaling violations observed at several experiments 
is one of the most important successes of QCD. Fig \ref{Figf2vsq2x}
shows the measurement of $F_2$ as function of $Q^2$ for different
values of $x$. A good agreement is found between the measured data
which cover almost five orders of magnitude in $Q^2$ with the
predictions of pQCD. As it can be seen from Fig. \ref{Figf2vsq2x} the
lower values of $x$ are accessed mainly at low values of $Q^2$. This
kinematical region will be discussed in the next sections.

\section{The behavoiur of the cross section at low $Q^2$}

\begin{figure}[t]
\begin{center}
\includegraphics[height=5cm]{./introduction/texfig/regge2.eps}
\caption{Contributions to the reaction $A+B \rightarrow B+C$ in the t-channel from
  a Reggeon or Pomeron trajectory $\alpha(t)$}
\label{Figregge2}
\end{center}
\end{figure}

At low values of $Q^2\!<\!1\,$GeV$^{2}$, pQCD can not predict the behaviour
of the electron proton scatte\-ring since the strong coupling constant
$\alpha_s$ is large. This kinematic domain can only be investigated in
terms of phenomenological models and hadron-hadron descriptions like
Regge theory. 

A powefull method used to investigate the behaviour of $F_2$ at very low
$Q^2$ proceeds via its connection to the total $\gamma^{*}p$ cross
section. 
%Considering the electron as a source of virtual photons,
%the $\gamma^{*} p$ total cross section is given by:

\begin{eqnarray}
\sigma_{tot}^{\gamma^{*}p} = \frac{4\pi^{2} \alpha}{Q^2} F_{2}(W^{2},Q^{2})
\label{sigmagammaptot}
\end{eqnarray}

thus the electron can be seen as a source of quasi real photons which
interact with the proton. This is almost a hadron hadron interaction
which can be studied in terms of Regge theory.

\subsection{Introduction to Regge theory}

Regge theory investigates the dynamics of hadrons by studying the two
particle scattering $A+B \rightarrow C+D$ (see Fig. \ref{Figregge2}) \cite{collins}. 
The physical idea is to
relate the high energy behaviour of the $s-$channel amplitude to the
quantum numbers that are exchanged in the $t-$channel. Instead of
considering the exchange of single particles as QCD does, Regge theory
considers the exchange of a complete trajectory of particles
$\alpha(t)$ having the {\it same internal quantum number} (strangeness,
isospin, bayon number, etc) but {\it different spin}. The particles lying
on a Regge-trayectory relate their spin ($J$) and mass ($m$) such that 
in good aproximation $m^{2}= J$.

In order to describe the total hadron-hadron cross section
($\sigma_{tot}$) measurements, 
two universal trajectories known as Pomeron 
($\alpha_{P}$) and Reggeon ($\alpha_{R}$) are needed. A Reggeon
exchange usually is related with a exchange of quantum numbers,
while a pomeron exchange involves the exchange of the vacuum
quantum numbers. 
Donnachie and Landshoff (1992) \cite{donnlan92} performed a global fit
to all existing hadron-hadron total cross section
data and discovered that all Reggeon
intercepts can be represented by one effective intercept having a
value of $\alpha_{R}(0) \approx 0.5$. In order to reproduce the rising
behaviour of the hadron-hadron cross section at high energies, DL
parametrized the Pomeron intercept as 
$\alpha_{P}(0)=1+\epsilon$ obtaining a value of $\epsilon$ varying
between $[0.08,0.09]$. Following this ideas, the total cross section of any
hadron-hadron interaction can be written as:

\begin{equation}
\sigma _{i}^{tot} (s) = A_{i}^{P} s ^{\alpha_{P}(0)-1} +  
A_{i}^{R} s ^{\alpha_{R}(0)-1}
\label{regeecs}
\end{equation}

where the coefficients $A_{i},B_{i}$ dependend on the process under
study and on $Q^2$. Donnachie and Landshoff (1993) \cite{donnlan93} showed that the
Pomeron plus Reggeon exchange model could as well describe successfully the
existing data on $\sigma_{tot}^{\gamma^{*}p}$ up to values of
$Q^2 \leq 10\;$GeV$^2$. The measurement at high energies of the total
photoproduction cross section at $Q^2 \approx 0\,$GeV$^2$ ,was
performed by HERA in 1992 \cite{h1gammap}, and the 
measurement was fully in agreement with the  Regge theory prediction.  

This result has an important impact on the proton structure
function $F_2$ at low $x$, since $F_2$ is related to the total
$\gamma^{*}p$ cross section (see Eq. \ref{sigmagammaptot}) and then,
the Regge description of DL predicts a rise of $F_2$ at low $x$ as:

\[ 
F_{2} \propto x^{-0.08} = x^{-\lambda}
\]

\begin{figure}[t]
\begin{center}
%\includegraphics[height=10cm]{./introduction/texfig/lambda_isr_notag_modified.eps}
\includegraphics[height=10cm]{./introduction/figures/lammergez.eps}
\caption{Fits of the rise $\lambda$ of $F_2$ parametrized at low $x$
  as $c x^{-\lambda}$ \cite{thelambda}}
\label{Figlambdaplot}
\end{center}
\end{figure}

H1 and ZEUS collaborations have measured the proton
structure function 
$F_2(x,Q^2)$ over a large range of values of $x,Q^2$, and therefore
the rise $\lambda$ 
can be derived from the derivative of $F_2$ for differents values of
$Q^2$ and for low $x$ ($x<0.02$). 

\[
\left[\frac{\partial \ln F_2(x,Q^2)}{\partial \ln x}\right]_{Q^2} = -
\lambda(x,Q^2) 
\]

The measurements of the rise $\lambda$ are summarized in Fig. 
\ref{Figlambdaplot} \cite{thelambda}.
One can clearly see that the predicted value of $\alpha _{P}(0)
\approx 0.08$ of
Regge theory is compatible with the data for the region of low
$Q^2<0.5\,$GeV$^2$. On the other hand,
the rise of $F_2$ is well described by pQCD for values of
$Q^2>1.5\,$GeV$^2$. 

\subsection{The behaviour of $F_2$ in the transition region}

The model of Donnachie and Landshof is in agreement with the data for
low values of $Q^2$, but can not describe the transition of
photoproduction to deep inelastic scattering. In order to improve the
description of
the data over the whole phase space, the model of Donnachie and
Landshoff has been extended. The model of Abramowicz, Levin, Levy and
Maor (ALLM97 \cite{allm97}) makes the assumption that the intercept
$\alpha_{P}(0)$ 
depends on $Q^2$. Using the data of deep inelastic scattering and of
hadron and hadron scattering, the free parameters are fixed. This
parametrization is used in this thesis.

%As already mentioned in the last section, at low values of
%$Q^2<1\,$GeV$^2$, the strong coupling constant $\alpha_s$ increases
%and pQCD can not predict the behaviour of the cross section of the
%electron proton scattering.
In addition to the ALLM97 parametrisation based on Regge theory,
there are other phenomelogical models which have been developed in
order to describe  
the transition region $Q^2 \approx 1\;$GeV$^2$ of the electron proton 
scattering. Some of such models involves dipole
interactions \cite{dipoleint}, vector meson dominance models
\cite{vmd}, and the fractal structure of the proton
\cite{tomas,tomaspaper}.  

In QCD the behaviour of the sea quark densities is described by gluon
emissions and splitting. The higher the resolution power $Q^2$ is, the deeper
the proton structure is probed and thus more gluon-gluon interactions
can be ``observed''. Taking this interpretation in analogy to fractals,
the proton could be treated of fractal nature only at low values of
$x$, in the region where the sea quark densities dominates. Two
magnification factors are used to simulate the fractal structure 
of the proton, which are defined in terms of $Q^2$ and $x$ as
$1+Q^2/Q_{0}^{2}$ and $1/x$ \cite{tomaspaper}. Using this two magnification factors, the
proton structure $F_2$ is predicted to be described by

\begin{equation}
F_{2}(x,Q^2) = \frac{e^{D_{0}Q_{0}^{2}x^{-D_{2}+1}}}{1+D_{3}-D_{1}log(x)}
\left[x^{-D_{1}log(1+Q^2/Q_{0}^{2})}\left(1+\frac{Q^2}{Q_{0}^{2}}\right)^{D_{3}+1}-1\right] 
\label{Eqf2fractal}
\end{equation}

where $D_{i}$ and $Q_{0}^{2}$ are the five parameters
that the fractal model needs to parametrize the behavior of
$F_2$ as function of $x$ and $Q^2$. The value of the parameters were
determined using data from the HERA experiments H1 \cite{h1fitfractal} and
ZEUS \cite{zeusfitfractal} in the range $x<0.01$ and $1.5 \leq Q^2 \leq
120$GeV$^2$, $0.045 \leq Q^2 \leq 0.65\,$GeV$^2$, respectively. In
addition to ALLM97 this parametrization will be compared to the data in
this analysis.


\section{Motivation of this Analysis}

This analysis is concentrated on the behaviour of the proton structure
function $F_2$ in the transition region of non-perturbative to 
perturbative QCD. The data set which is analysed here are
$ep$ scattering data recorded by the H1 detector at the HERA collider. In
order to measure the cross section at low values of $Q^2$, a special
data sample is analysed where the interaction point was moved from its 
nominal position. 

%Two main detector components can be used to measure independently the
%inclusive cross section.

The measurement of the inclusive DIS cross section $ep \rightarrow e +
X$ and the extraction of the proton structure function $F_2$ at low
values of $Q^2$ is dominated by systematic errors than by
statistical uncertanties. The first analysis of the data sample which
is investigated in this thesis was performed in \cite{tomas} obtaining 
preliminary results \cite{preliminary}. This
thesis together with the analysis developed in \cite{olaf} performs an
independent cross section measurement using the same data
sample, but analysing in more detail the sources of systematic errors. 

The inclusive cross section measurement can be obtained 
using {\it two independent} main detector components. Therefore two
independent analysis are performed to determine the cross section
measurement. In order to identify and quantify 
the systematic errors obtained from both analysis, detailed studies
concerning the vertex reconstruction and the energy measurement of the
particles building the final state are required. The investigations
performed in \cite{olaf} concentrates on the uncertainties due to the
reconstruction of the interaction point of the $ep$ collision. The
cross section measurements of the {\it two analysis}, the one
presented in this thesis and the one from \cite{olaf} are also
compared there.

This analysis includes detailed studies on the systematic
errors resulting from the uncertainty of the energy measurement of the
scattered electron 
and of the energy flow of the particles building the hadronic final
state. Therefore, special emphasis is given to the influence of the
noise identification algorithms on the reconstruction of the
kinematical variables $x$ and $Q^2$, which are essential for the
measurement of the proton structure function $F_2$. 

%The studies presented in this thesis consist an independent analysis of
%\cite{tomas}, where the proton structure function was measured 
%with the same data sample.
The thesis is organized as follows. In chapter \ref{Ch1detector} the
H1 detector is introduced, special emphasis 
will be given to the subdetector components which are used to measure
the inclusive cross section for low values of $Q^2$. In chapter
\ref{Chdatasample} the data sample and the methods
used to identify the scattered electron coming from the $ep$
collision will be presented. The methods used to reconstruct the
kinematical variables 
$x$ and $Q^2$ are also subject of chapter \ref{Chdatasample}.
The techniques used to calibrate the main calorimeters of
the H1 detector will be in detail discussed in 
chapter \ref{Chapspacalib} and \ref{Chaphadfinalstate}. In chapter
\ref{Chadisselc} the cross section measurement and the extraction of
the proton structure function $F_2$ is presented. 

