\graphicspath{{figures/dis/}}

\chapter{Deep Inelastic Scattering} \label{c:dis}

The present chapter provides the theoretical basis for the measurement
of the proton structure functions $F_{2}$ and $F_{L}$ in deep
inelastic electron\footnote{Unless otherwise stated, the term
`electron' is used in this thesis to generically denote electrons or
positrons.}-proton scattering at low values of $Q^{2}$. After the
basic variables describing the event kinematics are introduced the
parameterisation of the inclusive electron-proton scattering
cross-section in terms of proton structure functions is
discussed. Subsequently, predictions of their behaviour by the quark
parton model, perturbative Quantum Chromodynamics and various
phenomenological models are presented. It turns out that also a class
of processes referred to as \emph{radiative electron-proton
scattering} is relevant for the measurement. Therefore the
corresponding experimental configurations are introduced.

The chapter can only provide a short overview of the individual
topics. A detailed discussion can be found in \cite{Devenish:2004} and
\cite{Brock:1993sz}.

\section{Event Kinematics} \label{c:kinematics}

The interaction of electron~($e$) and proton~($p$) can be described in
lowest order perturbation theory via the exchange of virtual gauge
bosons. If a neutral gauge boson is exchanged, i.e. a photon~($\gamma$)
or a $Z^{0}$, the electron is scattered off the proton and a hadronic
final state~$X$ is produced. In case the gauge boson carries a
charge~($W^{\pm}$) the final state consists of a neutrino and the
hadronic system~$X$. Figure~\ref{f:feyndis} shows the corresponding
Feynman diagram.

\begin{figure}[t]
  \centering
  \begin{fmffile}{feynmf/neutralCurrent}
    \begin{fmfgraph*}(165,120)
      \fmfleft{ip,il}
      \fmflabel{$p$}{ip}
      \fmflabel{$e$}{il}
      \fmfright{oq1,oq2,oq3,d1,d2,d3,ol}
      \fmflabel{$X$}{oq2}
      \fmf{fermion,tension=2,label=${\mathbf P}$,label.side=left,label.dist=10.}{ip,vp}
      \fmf{fermion}{vp,oq1}
      \fmf{fermion}{vp,oq2}
      \fmf{fermion}{vp,oq3}
      \fmf{photon,tension=0.7,label=$\gamma (Z^{0},,W^{\pm})$,label.side=left}{vl,vp}
      \fmf{photon,tension=0.7,label=${\mathbf q}$,label.side=right}{vl,vp}
      \fmf{fermion,tension=2,label=${\mathbf k}$,label.side=left}{il,vl}
      \fmf{fermion,label=${\mathbf k'}$,label.side=left}{vl,ol}
      \fmfv{decor.shape=circle,decor.filled=empty,decor.size=.1w}{vp}
      \fmfdot{vl}
      \fmffreeze
      \fmfi{plain}{vpath (__ip,__vp) shifted (thick*(0,2))}
      \fmfi{plain}{vpath (__ip,__vp) shifted (thick*(1,-1.6))}
    \end{fmfgraph*}
  \end{fmffile}
  \vspace{0.3cm}
  \caption{Lowest order Feynman graph for the electron-proton scattering process.}
  \label{f:feyndis}
\end{figure}

At a fixed centre of mass energy $\sqrt{s}$ the event kinematics are
fully determined by two of the three Lorentz invariant variables
$Q^{2}$, $x$ and $y$. They are defined by the four-momenta of the
particles involved in the scattering process: the four-momenta of the
incoming electron and proton are denoted by $\mathbf k$ and $\mathbf
P$, respectively, $\mathbf k'$ corresponds to the scattered
electron. Finally, $\mathbf q$ denotes the four-momentum of the
exchanged boson. Neglecting the electron and the proton mass the squared
centre of mass energy is determined by the energies of the incoming
electron $E_{e}^{0}$ and the incoming proton $E_{p}^{0}$:
\begin{equation}
s = (\mathbf{P} + \mathbf{k})^{2} \approx 4 E_{e}^{0} E_{p}^{0}
\label{e:cmse}
\end{equation}
The negative four momentum transfer squared at the electron vertex is
defined as
\begin{equation}
Q^{2} = -\mathbf{q}^{2} = -(\mathbf{k} - \mathbf{k'})^{2}
\label{e:q2}
\end{equation}
It is also referred to as the virtuality of the exchanged boson.

Furthermore, the Bjorken scale variable $x$ and the inelasticity $y$
are used to describe the event kinematics:
\begin{equation}
x = \frac{Q^{2}}{2\, \mathbf{P} \cdot \mathbf{q}}, \hspace{2em} y = \frac{\mathbf{P} \cdot \mathbf{q}}{\mathbf{P} \cdot \mathbf{k}}
\label{e:xandy}
\end{equation}
In the quark-parton model $x$ is interpreted as the fraction of the
proton's momentum carried by the struck quark (see
section~\ref{c:qpm}). The inelasticity $y$ corresponds to the relative
energy transfer of the incoming electron to the proton in the rest
frame of the latter. Correspondingly, both variables are dimensionless
and limited to the range \mbox{$0 \leq x,y \leq 1$}.

Neglecting the particle masses the previously defined quantities are
related by:
\begin{equation}
Q^{2} \approx xys
\label{e:q2sxy}
\end{equation}
Finally the invariant mass of the hadronic final state is given by
\begin{equation}
W^{2} = (\mathbf{P} + \mathbf{q})^{2} = \frac{1-x}{x}\, Q^{2} +
m_{p}^{2} 
\label{e:w2}
\end{equation}
with $m_{p}$ denoting the proton mass.

The values of $Q^{2}$ relevant to this analysis are small compared to
the large masses of the $Z^{0}$ and $W^{\pm}$ bosons (\mbox{$m_{W}
\approx 80\, \mathrm{GeV}$}, \mbox{$m_{Z^{0}} \approx 91\,
\mathrm{GeV}$}). Therefore $Z^{0}$ and $W^{\pm}$ exchange can be
neglected and only the photon exchange is considered in the following.
In the limit \mbox{$Q^{2} \rightarrow 0\, \mathrm{GeV^{2}}$} the $ep$
scattering process can be interpreted as an interaction of a real
photon with the proton. The corresponding events are called
\emph{photoproduction events} in contrast to DIS events.

\section{DIS Cross-Section} \label{c:disxsect}

At lowest order in electroweak theory (Born approximation) the DIS
cross-section can be expressed in terms of a tensor product:
\begin{equation}
d\sigma \sim L_{\mu \nu} W^{\mu \nu}
\label{e:sigmatensor}
\end{equation}
The leptonic tensor $L_{\mu \nu}$ describes the interaction of the
electron with the exchanged virtual gauge boson and can be precisely
calculated in the electroweak theory treating the electron as a
point-like particle. Correspondingly, the interaction of the gauge
boson and the proton is represented by the hadronic tensor $W^{\mu
\nu}$. Since the internal structure of the proton is unknown the
precise form of $W^{\mu \nu}$ cannot be specified. Instead, the most
general Lorentz invariant form is assumed. Considering the photon
exchange only and employing current conservation at the hadron vertex
$W^{\mu \nu}$ can be expressed in terms of two unknown functions.

Hence the inclusive differential $ep$ cross-section at low $Q^{2}$ can
be expressed in terms of two independent proton structure functions
$F_{2}$ and $F_{L}$:
\begin{equation}
\frac{d^{2} \sigma}{dx dQ^{2}} = \frac{2 \pi \alpha^{2}}{Q^{4}x} Y_{+} \left ( F_{2}(x,Q^{2}) - \frac{y^{2}}{Y_{+}} \cdot F_{L}(x,Q^{2}) \right ) \hspace{1em} \mathrm{with}\; Y_{+} = 1 + (1-y)^{2}
\label{e:sigmaep}
\end{equation}
A simultaneous measurement of both structure functions is only
possible if the inelasticity $y$ is varied for fixed values of $x$ and
$Q^{2}$. According to equation~\ref{e:q2sxy} this requires a variation
of the centre of mass energy $\sqrt{s}$. The latter can be achieved either by
changing the beam energies of the HERA collider or by analysing events
for which the energy of the electron available for the interaction
with the proton is reduced due to photon radiation (see
section~\ref{c:radepscat}). However, at sufficiently low values of $y$
the contribution of $F_{L}$ to the cross-section is strongly
suppressed due the factor $y^{2}$. Since this is the case for the bulk
of the phase space accessible at HERA the structure function $F_{2}$
can be directly determined from the differential DIS cross-section
even at fixed $s$.

It turns out to be convenient to define the so-called \emph{reduced
cross-section} $\sigma_{r}$ derived from equation~\ref{e:sigmaep} by
absorbing the common factor of both structure functions:
\begin{equation}
\sigma_{r} = F_{2}(x,Q^{2}) - \frac{y^{2}}{Y_{+}} \cdot F_{L}(x,Q^{2})
\label{e:sigmar}
\end{equation}

The $ep$ scattering process at low $Q^{2}$ can be interpreted as the
interaction of a flux of virtual photons with the proton. Due to their
effective mass virtual photons can have a longitudinal as well as a
transverse polarisation. Hence the differential $ep$ cross-section in
the Born approximation can be decomposed accordingly:
\begin{equation}
\frac{d^{2} \sigma}{dx dQ^{2}} = \Gamma(y,Q^{2}) \left ( \sigma_{T}(x,Q^{2}) + \epsilon(y) \sigma_{L}(x,Q^{2}) \right )
\label{e:sigmaegammap}
\end{equation}
Here $\sigma_{T}$ and $\sigma_{L}$ denote the cross-section for the
absorption of transversely and longitudinally polarised photons by the
proton, respectively, $\Gamma(y,Q^{2}) = \alpha Y_{+}/(2 \pi Q^{2} x)$
describes the photon flux and $\epsilon(y) = 2 (1-y) / Y_{+}$ is the
photon polarisation\footnote{It should be emphasised that there is a
certain amount of freedom in the definition of the flux factor since
the only constraint is the correct description of the real photon flux
in the photoproduction limit (\mbox{$Q^{2} \rightarrow 0\,
\mathrm{GeV^{2}}$}). Here the convention of Hand is
used~\cite{Hand:1963bb}. The quoted formulae for $\Gamma(y,Q^{2})$ and
$\epsilon(y)$ are approximations valid for $W^{2} \gg m_{p}^{2}$.}.

Comparing equations \ref{e:sigmaep} and \ref{e:sigmaegammap} one obtains:
\begin{align}
F_{2}(x,Q^{2}) & = \frac{Q^{2}}{4\pi^{2}\alpha} \left ( \sigma_{T}(x,Q^{2}) + \sigma_{L}(x,Q^{2}) \right ) \label{e:f2gammap} \\
F_{L}(x,Q^{2}) & = \frac{Q^{2}}{4\pi^{2}\alpha}\, \sigma_{L}(x,Q^{2}) \label{e:flgammap}
\end{align}
Due to its relation to the cross-section for longitudinally polarised
photons $F_{L}$ is called \emph{longitudinal structure function}. The
condition that both the longitudinal and the transverse cross-section have
to be positive imposes the following constraint on the proton
structure functions:
\begin{equation}
0 \le F_{L}(x,Q^{2}) \le F_{2}(x,Q^{2})
\end{equation}

\section{Quark Parton Model} \label{c:qpm}

The quark parton model has been the first approach to provide a
reasonable interpretation of the SLAC measurements mentioned in the
preface. In the quark parton model the proton is assumed to consist of
point-like \mbox{spin-$\frac{1}{2}$} particles (partons) that
can be identified with the three quarks necessary to explain the
quantum numbers of the proton. The interaction of electron and proton
is viewed in the \emph{infinite momentum frame} in which the
transverse momenta of the quarks are neglected and \mbox{$Q^{2} \gg
m_{p}^{2}$}. In this frame relativistic time dilatation slows down the
rate at which the partons interact with each other, i.e. during the
short time the virtual photon interacts with the proton the quarks are
essentially free. Hence deep inelastic $ep$ scattering can be viewed
as an incoherent sum of electron-quark scattering processes as
displayed in figure~\ref{f:feynqpm}.

\begin{figure}
  \centering
  \begin{fmffile}{feynmf/qpm}
    \begin{fmfgraph*}(165,120)
      \fmfleft{ip,il}
      \fmflabel{$p$}{ip}
      \fmflabel{$e$}{il}
      \fmfright{oq1,oq2,d1,oq3,d2,d3,ol}
      \fmflabel{struck quark}{oq3}
      \fmflabel{proton remnant}{oq2}
      \fmflabel{$e'$}{ol}
      \fmf{fermion}{ip,vp}
      \fmf{fermion,label=$\xi \mathbf{P}$,label.side=left}{vp,vq}
      \fmf{fermion}{vq,oq3}
      \fmf{phantom,label=$\mathbf{P}$,label.side=left,label.dist=10.}{ip,vp}
      \fmf{fermion}{vp,oq1}
      \fmf{fermion}{vp,oq2}
      \fmf{photon}{vl,vq}
      \fmf{fermion}{il,vl,ol}
      \fmf{phantom}{il,vl}
      \fmfv{decor.shape=circle,decor.filled=empty,decor.size=.1w}{vp}
      \fmfdot{vq,vl}
      \fmffreeze
      \fmfi{plain}{vpath (__ip,__vp) shifted (thick*(0,2))}
      \fmfi{plain}{vpath (__ip,__vp) shifted (thick*(1,-2))}
    \end{fmfgraph*}
  \end{fmffile}
  \vspace{0.3cm}
  \caption{Deep inelastic scattering in the quark parton model.}
  \label{f:feynqpm}
\end{figure}

Accordingly, the differential $ep$ cross-section is expressed as a sum
of elastic $eq$ scattering cross-sections
\begin{equation}
\frac{d^{2} \sigma}{dx dQ^{2}} = \sum_{i}\int_{0}^{1} d \xi q_{i}(\xi) \left ( \frac{d^{2} \sigma}{dx dQ^{2}} \right )_{eq_{i} \rightarrow eq_{i}}
\label{e:sigmaqpm}
\end{equation}
where the \emph{parton density function} $q_{i}(\xi)$ is the probability
to find a quark $i$ carrying the fraction $\xi$ of the proton's
momentum before entering the scattering process. The cross-section for
electron-quark elastic scattering can be calculated in QED to
\begin{equation}
\left ( \frac{d^{2} \sigma}{dx dQ^{2}} \right )_{eq_{i} \rightarrow eq_{i}} = \frac{4\pi\alpha^{2}}{Q^{4}} e_{i}^{2} \frac{1}{2}(1+(1-y)^{2}) \delta(x-\xi)
\label{e:sigmaeq}
\end{equation}
The result implies that the momentum fraction $\xi$ of the struck
quark is identical to the dimensionless variable Bjorken $x$
introduced in section~\ref{c:kinematics}. Equation~\ref{e:sigmaqpm}
illustrates the basic concept of the quark parton model, namely the
splitting of the DIS cross-section into an calculable hard process and
a parton density function to be determined by the experiment.

Inserting equation~\ref{e:sigmaeq} in~\ref{e:sigmaqpm} and comparing the result to~\ref{e:sigmaep} yields:
\begin{align}
F_{2}(x,Q^{2}) & = \sum_{i} e_{i}^{2}xq_{i}(x) \label{e:f2qpm} \\
F_{L}(x,Q^{2}) & = 0 \label{e:flqpm}
\end{align}
From these equations the following predictions of the quark parton
model can be derived: the structure function $F_{2}$ is scale
invariant, i.e. it does not depend on $Q^{2}$ but only on $x$. This
behaviour is called \emph{scaling} and was observed in the first DIS
experiments at SLAC for $x \approx 0.25$ in 1969~\cite{Bloom:1969kc,
Breidenbach:1969kd}. The longitudinal structure function $F_{L}$ is
predicted to vanish. Equation~\ref{e:flqpm} is also known as
Callan-Gross relation. It follows from the helicity conservation for
massless \mbox{spin-$\frac{1}{2}$} partons~\cite{Callan:1969uq}. In
the same year when it was derived (1969) it was confirmed in a
combined analysis of SLAC and DESY~\cite{Albrecht:1969, Taylor:1969}
giving strong evidence for the \mbox{spin-$\frac{1}{2}$} structure of
the quarks.

\section{Quantum Chromodynamics} \label{c:qcd}


As already pointed out in the preface the experimental observation
that the quarks carry only about 50\% of the proton's momentum cannot
be explained within the quark parton model. Nor can it account for
free quarks not being observed in the final state of any particle
reaction while in DIS experiments quarks appear to be
quasi-free. Further problems occurred in the classification of the
hadron spectra: to explain the quantum numbers of the $\Delta^{++}$
resonance it is assumed to consist out of three $u$ quarks. Since
quarks are \mbox{spin-$\frac{1}{2}$} particles the Fermi statistics
seems to be violated.

In first instance to overcome the latter caveat a new quantum number,
\emph{colour}, was introduced~\cite{Han:1965pf,
Greenberg:1964pe}. Each quark carries one of the colours red, green or
blue. The idea of extending the colour model to a gauge theory obeying
SU(3) symmetry led to the development of \emph{Quantum Chromodynamics}
(QCD), the theory of the strong interaction~\cite{Fritzsch:1973pi,
Gross:1973ju, Weinberg:1973un}. The gauge bosons of QCD are eight
massless \emph{gluons} mediating the interaction between the
quarks. They carry the missing fraction of the proton's momentum. Due
to the non-Abelian structure of the SU(3) group they have themselves a
colour and an anticolour allowing for gluon self interactions. The
colour plays the same role for the strong interaction as the charge
for QED. However, the Abelian structure of the U(1) symmetry group
allows no photon self couplings in QED.

The vacuum polarisation in QED leads to a screening of a charge to be
probed and the coupling constant $\alpha$ increases with
decreasing distance (increasing $Q^{2}$). On the contrary, in QCD the
self coupling of the gluons causes an antiscreening effect leading to
a decrease of the strong coupling constant $\alpha_{s}$ with increasing
$Q^{2}$. This behaviour is referred to as \emph{running of
$\alpha_{s}$} and quantified in the renormalisation group equation
given in leading order by
\begin{equation}
\alpha_{s}(Q^{2}) = \frac{12 \pi}{(33 - 2n_{f}) \ln \frac{Q^{2}}{\Lambda_{QCD}^{2}}}
\end{equation}
where $n_{f}$ denotes the number of active quark flavours, i.e. those
with \mbox{$m_{q}^{2} < Q^{2}$}. The parameter $\Lambda_{QCD}$
determines the scale at which $\alpha_{s}$ becomes large. For $Q^{2}$
values much larger than $\Lambda_{QCD}^{2}$ the coupling constant
$\alpha_{s}$ becomes small and the quarks inside the hadrons are
quasi-free. The discovery of this property of QCD~\cite{Gross:1973id,
Politzer:1973fx}, called \emph{asymptotic freedom}, allows to conduct
perturbative calculations. Approaching $\Lambda_{QCD}^{2}$ with
$Q^{2}$ implies $\alpha_{s}$ getting large. Hence the quarks start to
be tightly bound in the hadrons and perturbative methods cannot be
used. Either $\alpha_{s}(Q^{2})$ or $\Lambda_{QCD}$ has to be
determined from experimental data. The latter is of the order of a few
hundred MeV.

One of the achievements of perturbative QCD (pQCD), applicable in the
domain of asymptotic freedom, is the field theoretical justification
of the parton model. This is realised by the \emph{factorisation
theorem} of deep inelastic scattering~\cite{Collins:1989}. It states
that for the general case of lepton-hadron scattering the interaction
of the exchanged vector boson with the hadron can be divided into two
independent parts, a
\emph{short distance} part which is perturbatively calculable and a
\emph{long distance} part to be determined experimentally. It is a
direct transfer of the basic assumption of the quark parton model to
separate the DIS cross-section into a hard scattering process and a
parton density function. Formally, factorisation implies to express
the structure function $F_{2}^{V,h}$ as convolution of a
\emph{coefficient function} $C_{2}^{V,i}$ and a parton density
$f_{i/h}$ summing over all partons $i$ (quarks~$q$,
antiquarks~$\bar{q}$, gluons~$g$):
\begin{equation}
F_{2}^{V,h}(x,Q^{2}) = \sum_{i = q,\bar{q},g} \int_{x}^{1} dz \; C_{2}^{V,i} \left ( \frac{x}{z},\frac{Q^{2}}{\mu_{r}^{2}},\frac{\mu_{f}^{2}}{\mu_{r}^{2}},\alpha_{s}(\mu_{r}^{2}) \right ) f_{i/h}(z,\mu_{r}^{2},\mu_{f}^{2})
\label{e:f2fact}
\end{equation}
The coefficient function $C_{2}^{V,i}$ describes the interaction of
the exchanged vector boson $V$ with a parton $i$. It is independent of
the long distance effects and in particular independent of the
identity of the hadron $h$. For a particular exchanged boson $V$ it
can be expanded in a power series of $\alpha_{s}$ and perturbatively
calculated. The parton density $f_{i/h}(z)$ is a direct generalisation
of the parton model quark distribution, i.e. it corresponds to the
probability to find a parton $i$ carrying a fraction $z$ of the
hadron's momentum. It is specific to the hadron $h$, but
\emph{universal}, that is, independent of the particular hard
scattering process. Two scales are present: the factorisation scale
parameter $\mu_{f}$ defines the boundary between the short and long
distance part and $\mu_{r}$ is the so-called renormalisation scale
parameter used to absorb divergent parts of the perturbation
theory. Figure~\ref{f:feynfact} shows an illustration of the
factorisation theorem for deep inelastic $ep$ scattering\footnote{As
this thesis is concerned with $ep$ scattering in a region of the
kinematic phase space in which the photon exchange dominates the
indices $V$ and $h$ are omitted in the following implicitly assuming
that they correspond to $\gamma$ and $p$.}.

\begin{figure}
  \centering
  \begin{fmffile}{feynmf/factorisation}
    \begin{fmfgraph*}(165,120)
      \fmfleft{ip,il}
      \fmflabel{$p$}{ip}
      \fmflabel{$e$}{il}
      \fmfright{oq1,oq2,oq3,d1,d2,oq4,oq5,oq6,d3,d4,d5,d6,ol}
      \fmflabel{$e'$}{ol}
      \fmf{fermion,tension=5.,label=$\mathbf{P}$,label.side=left,label.dist=10.}{ip,vp}
      \fmf{fermion,label=$z \mathbf{P}$,label.side=left}{vp,vq}
      \fmf{phantom,label=$i$,label.side=right}{vp,vq}
      \fmf{fermion}{vp,oq1}
      \fmf{fermion}{vp,oq2}
      \fmf{fermion}{vp,oq3}
      \fmf{fermion}{vq,oq4}
      \fmf{fermion}{vq,oq5}
      \fmf{fermion}{vq,oq6}
      \fmf{photon,tension=2.}{vl,vq}
      \fmf{fermion,tension=2.}{il,vl,ol}
      \fmf{phantom}{il,vl}
      \fmfv{label=$f_{i}$,label.dist=0,decor.shape=circle,decor.filled=empty,decor.size=.17w}{vp}
      \fmfv{label=$C^{V,,i}$,label.dist=0,decor.shape=circle, decor.filled=empty, decor.size=.17w}{vq}
      \fmfdot{vl}
      \fmffreeze
      \fmfi{plain}{vpath (__ip,__vp) shifted (thick*(0,2))}
      \fmfi{plain}{vpath (__ip,__vp) shifted (thick*(1,-2))}
    \end{fmfgraph*}
  \end{fmffile}
  \vspace{0.3cm}
  \caption{Illustration of the factorisation theorem for deep inelastic $ep$ scattering.}
  \label{f:feynfact}
\end{figure}

The separation between the coefficient functions and parton density
functions is not unique. To eliminate this ambiguity a specific
\emph{factorisation scheme} has to be applied. The two most commonly
used schemes are the DIS~\cite{Altarelli:1979ub} and the modified
minimal subtraction scheme
$\mathrm{\overline{MS}}$~\cite{Bardeen:1978yd} scheme. In the DIS
scheme the structure function $F_{2}$ is given by the quark parton
model equation~\ref{e:f2qpm} to all orders in perturbation theory,
just that the quark densities are scale dependent. Choosing $\mu_{r} =
\mu_{f} = Q$ one obtains:
\begin{equation}
F_{2}(x,Q^{2}) = \sum_{i=q,\bar{q}} e_{i}^{2}xf_{i}(x,Q^{2}) \label{e:f2dis}
\end{equation}
In case of the $\mathrm{\overline{MS}}$ scheme the parton density
functions are defined directly in terms of hadronic matrix
elements. The latter scheme is more useful for theoretical
calculations while the former is the natural choice for DIS
experiments.

A remarkable consequence of factorisation is that measuring the parton
density functions at one scale $\mu$ allows their prediction for any
other scale $\mu'$ given that both scales are large enough for
$\alpha_{s}$ at each of them to be small. This possibility increases
the predictive power of pQCD enormously and is referred to as
\emph{evolution} of the parton densities. The underlying evolution
equations can be derived from equation~\ref{e:f2fact} by requiring
that the structure functions as measurable quantities should be
independent of the factorisation and renormalisation scale. Choosing
again $\mu_{r} = \mu_{f} = Q$ one obtains the DGLAP evolution equations
named after Dokshitzer, Gribov, Lipatov, Altarelli and
Parsisi~\cite{Gribov:1972ri, Dokshitzer:1977sg, Altarelli:1977zs}:
\begin{align}
\frac{dq_{i}(x,Q^{2})}{d \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} \left ( \frac{x}{z} \right ) + g(z,Q^{2}) P_{ig} \left ( \frac{x}{z} \right ) \right ] \label{e:dglapq} \\
\frac{dg(x,Q^{2})}{d \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} \left ( \frac{x}{z} \right ) + g(z,Q^{2}) P_{gg} \left ( \frac{x}{z} \right ) \right ] \label{e:dglapg}
\end{align}
Here $q_{j}(z,Q^{2})$ and $g(x,Q^{2})$ denote the quark and gluon
density functions, respectively. Like the coefficient functions the
so-called \emph{splitting functions} $P_{\alpha \beta}$ are calculable
in pQCD expanding them into a power series in $\alpha_{s}$:
\begin{equation}
P_{\alpha \beta}(z,\alpha_{s}(Q^{2})) = \frac{\alpha_{s}}{2 \pi} P_{\alpha \beta}^{(0)}(z) + \left ( \frac{\alpha_{s}}{2 \pi} \right )^{2} P_{\alpha \beta}^{(1)}(z) + \dots
\end{equation}
The leading order splitting functions $P_{\alpha \beta}^{(0)}(x/z)$
have a simple physical interpretation: they correspond to the probability
that a parton $\alpha$ carrying a fraction $x$ of the proton's
momentum originates from a parton $\beta$ with a momentum fraction
$z$. Figure~\ref{f:splitfunc} shows the graphs for all leading order
splitting functions.

\begin{figure}
  \centering
  \begin{minipage}[c]{0.18\textwidth}
  \begin{fmffile}{feynmf/splitPqq}
    \begin{fmfgraph*}(75,56)
      \fmfleft{iq}
      \fmfright{og,oq}
      \fmflabel{$j$}{iq}
      \fmflabel{$i$}{oq}
      \fmf{fermion,tension=1.5,label=$z$,label.side=left,label.dist=7.5}{iq,v}
      \fmf{fermion,label=$x$,label.side=left}{v,oq}
      \fmf{gluon}{v,og}
    \end{fmfgraph*}
  \end{fmffile}
  \put(-80,70){{\small $P_{ij}^{(0)} \left ( \frac{x}{z} \right )$}}
  \end{minipage}
  \hspace{0.03\textwidth}
  \begin{minipage}[c]{0.18\textwidth}
  \begin{fmffile}{feynmf/splitPqg}
    \begin{fmfgraph*}(75,56)
      \fmfleft{ig}
      \fmfright{oq1,oq2}
      \fmflabel{$g$}{ig}
      \fmflabel{$i$}{oq2}
      \fmf{gluon,tension=1.5,label=$z$,label.side=left,label.dist=7.5}{ig,v}
      \fmf{fermion,label=$x$,label.side=left}{v,oq2}
      \fmf{fermion}{v,oq1}
    \end{fmfgraph*}
  \end{fmffile}
  \put(-80,70){{\small $P_{ig}^{(0)} \left ( \frac{x}{z} \right )$}}
  \end{minipage}
  \hspace{0.03\textwidth}
  \begin{minipage}[c]{0.18\textwidth}
  \begin{fmffile}{feynmf/splitPgq}
    \begin{fmfgraph*}(75,56)
      \fmfleft{iq}
      \fmfright{oq,og}
      \fmflabel{$j$}{iq}
      \fmflabel{$g$}{og}
      \fmf{fermion,tension=1.5,label=$z$,label.side=left,label.dist=7.5}{iq,v}
      \fmf{gluon,label=$x$,label.side=left}{v,og}
      \fmf{fermion}{v,oq}
    \end{fmfgraph*}
  \end{fmffile}
  \put(-80,70){{\small $P_{gj}^{(0)} \left ( \frac{x}{z} \right )$}}
  \end{minipage}
  \hspace{0.03\textwidth}
  \begin{minipage}[c]{0.18\textwidth}
  \begin{fmffile}{feynmf/splitPgg}
    \begin{fmfgraph*}(75,56)
      \fmfleft{ig}
      \fmfright{og1,og2}
      \fmflabel{$g$}{ig}
      \fmflabel{$g$}{og2}
      \fmf{gluon,tension=1.5,label=$z$,label.side=left,label.dist=7.5}{ig,v}
      \fmf{gluon,label=$x$,label.side=left}{v,og2}
      \fmf{gluon}{v,og1}
    \end{fmfgraph*}
  \end{fmffile}
  \put(-80,70){{\small $P_{gg}^{(0)} \left ( \frac{x}{z} \right )$}}
  \end{minipage}
  \caption{Diagrams of the leading order DGLAP slitting functions. $P_{\alpha \beta}^{(0)}(x/z)$ denotes the probability that a parton $\alpha$ carrying a fraction $x$ of the proton's momentum originates from a parton $\beta$ with a momentum fraction $z$.}
  \label{f:splitfunc}
\end{figure}

Since the momentum fraction of the radiated parton is always smaller
or equal to that of the parent parton, the integration in equations
\ref{e:dglapq} and \ref{e:dglapg} is restricted to the interval $x \le
z \le 1$. Hence the evolution of the parton density functions can be
predicted without having to know them to arbitrary small values of
$x$.

This allows for a determination of the parton density functions from
experimental data according to the following procedure: the parton
density functions are parameterised by smooth analytical functions at a
low starting scale $Q_{0}^{2}$ as a function of $x$ with a certain
number of free parameters. They are evolved in $Q^{2}$ using the
DGLAP equations. Afterwards predictions for structure functions and
cross-sections are calculated. The free parameters are determined by
performing a $\chi^{2}$ fit to the data. Several constraints are
imposed during this procedure.

For an accurate determination of the parton density functions the
coefficient and splitting functions have to be precisely known as they
govern the calculation of the structure functions (see
equation~\ref{e:f2fact}) and the evolution (see equations
\ref{e:dglapq} and \ref{e:dglapg}), respectively. Since 2005 all of
them are known to next-to-next-to-leading order
(NNLO)~\cite{Moch:2004pa, Vogt:2004mw, Vermaseren:2005qc}.

Global QCD analyses applying the fitting procedure outlined above to
the data of different experiments have been carried out by several
theoretical groups, resulting in a large set of available
parameterisations for the parton density functions. Among them are the
ones from Gl\"uck, Reya and Vogt (GRV)~\cite{Gluck:1998xa}, from
Martin, Roberts, Stirling and Thorne (MRST)~\cite{Martin:2001es,
Martin:2002aw} and the CTEQ collaboration~\cite{Pumplin:2002vw}.

Also the H1 collaboration has performed a QCD fit (H1 PDF 2000) based
on positron-proton neutral and charged current events recorded with
the H1 detector in the years 1994 to 2000~\cite{Adloff:2003uh}. The
parton densities have been evolved according to the DGLAP equations in
next to leading order. Figure~\ref{f:h1qcdana} compares the H1PDF 2000
fit to the data used for its derivation as well as to data from the
ZEUS collaboration and the fixed target experiments BCDMS and NMC. The
structure function $F_{2}$ is displayed as a function of $Q^{2}$ for
various values of $x$. Good agreement between the data and the fit is
observed except for the BCDMS data at $x=0.65$. This implies that pQCD
successfully describes the data over almost five orders of magnitude
in $x$ and $Q^{2}$ including the scaling violation of $F_{2}$
explained in the following.

\begin{figure}
  \centering
  \includegraphics[width=0.9\textwidth]{fig_f2_col_zeus.eps}
  \caption{Proton structure function $F_{2}$ as a function of $Q^{2}$ for various values of $x$ measured by the H1 and ZEUS collaborations as well as the fixed target experiments BCDMS and NMC (figure taken from~\cite{Newman:2003vh}). The results are compared to the H1 PDF 2000 fit represented by the error bands. The width of the latter contains the experimental uncertainty as derived from the fit and the model uncertainty added in quadrature. In case of the data points the inner error bars indicate the statistical errors, the outer ones the statistical and systematic errors added in quadrature. For a better visibility the values of the structure function $F_{2}$ are multiplied by $2^{i}$ with $i$ depending on the particular value of $x$.}
  \label{f:h1qcdana}
\end{figure}

The structure function $F_{2}$ rises (decreases) with $Q^{2}$ at small
(large) values of $x$, i.e. the scaling behaviour predicted by the
quark parton model and observed in the first DIS experiments at
SLAC~\cite{Bloom:1969kc, Breidenbach:1969kd} is violated for the bulk
of the phase space. Only in the region of
\mbox{$x \approx 0.25$} where the SLAC measurements have been
performed scaling is still found. The scaling violations are caused by
gluon radiation: by radiating a gluon with a momentum fraction $x''$
the original momentum fraction $x$ of the quark is reduced to
\mbox{$x' < x$} because of \mbox{$x = x' + x''$}. In addition gluons
can split into quark antiquarks pairs\footnote{The quarks and
antiquarks originating from gluon splittings are called \emph{sea
quarks} in contrast to the three \emph{valence quarks} necessary to
explain the quantum numbers of the proton.} carrying a relatively
small momentum fraction. With increasing resolution, i.e. larger
values of~$Q^{2}$, more of such processes can be resolved by the
photon. Therefore one expects the quark density functions to rise with
$Q^{2}$ at low $x$ and to decrease with it at high $x$. Since the
structure function $F_{2}$ is the charged weighted sum of the quark
densities (see formula~\ref{e:f2dis}) this behaviour directly leads to
the observed scaling violations.


\section{The Rise of $F_{2}$ towards Low $x$} \label{c:f2rise}


It has been already mentioned in the preface that one of the first
observations at HERA was a substantial rise of $F_{2}$ with decreasing
$x$ at fixed $Q^{2}$~\cite{Abt:1993cb, Derrick:1993ft}. Such a
behaviour is expected in pQCD: the DGLAP equations can be solved in
the so-called double leading log approximation predicting $F_{2}$ to
rise approximately as a power of $x$ towards low
$x$~\cite{DeRujula:1974rf, Ball:1994du}. The rise is a reflection of
the gluon density and the sea quark densities getting larger with
decreasing $x$. However, for sufficiently small $x$ the power-like
growth of $F_{2}$ causes the cross-section to violate unitarity. Hence
one expects the rise of $F_{2}$ to be tamed as $x \rightarrow 0$, an
effect referred to as \emph{saturation}. One possible explanation for
saturation are gluon recombinations in the
nucleon~\cite{Gribov:1981ac, Gribov:1984tu}. Concerning the event
kinematics it is most promising to search for saturation effects in
the low $Q^{2}$ regime\footnote{At very low values of $Q^{2}$ gluon
self-interactions cannot be used as an explanation for the taming of
the rise of $F_{2}$ since individual partons are not resolved any
more.} since in this region the lowest values of $x$ are reached
according to equation~\ref{e:q2sxy}.

It has been suggested~\cite{Navelet:1994qp} to measure the derivative
\begin{equation}
\lambda(x,Q^{2}) = - \left . \frac{\partial F_{2}(x,Q^{2})}{\partial \ln x} \right |_{Q^{2}}
\label{e:lambda}
\end{equation}
in order to investigate the behaviour of $F_{2}$ at low $x$. The H1
collaboration has determined this quantity for various data
sets~\cite{Adloff:2001rw, Lastovicka:2002, Petrukhin:2004a}. It was
found that $\lambda(x,Q^{2})$ is independent of $x$ at fixed $Q^{2}$
for \mbox{$x \lesssim 0.01$} within the experimental accuracy. This
implies that the structure function $F_{2}$ at low $x$ can be
parameterised as
\begin{equation}
F_{2} = c(Q^{2}) \cdot x^{-\lambda(Q^{2})} 
\label{e:f2lambda}
\end{equation}
Hence the $x$ dependence of $F_{2}$ at low $x$ is indeed consistent
with a power law as predicted by pQCD and there is no evidence for a
taming of this rise. Both H1 and ZEUS have subjected their data to
fits of the function~\ref{e:f2lambda} for \mbox{$x <
0.01$}~\cite{Breitweg:1998dz, Adloff:2001rw, Lastovicka:2002}. The
results for $\lambda(Q^{2})$ are shown in figure~\ref{f:lambda}. Two
distinct regions can be distinguished: for \mbox{$Q^{2} \gtrsim 3\,
\mathrm{GeV^{2}}$} where partons are the relevant degrees of freedom,
$\lambda$ depends logarithmically on $Q^{2}$. This behaviour is well
reproduced by DGLAP based QCD fits. In contrast, for \mbox{$Q^{2}
\lesssim 1\, \mathrm{GeV^{2}}$}, $\lambda(Q^{2})$ deviates from the
logarithmic dependence on $Q^{2}$ and tends to the value of $0.08$
known from fits to hadron-hadron and photoproduction total
cross-sections based on Regge phenomenology (see next section). In
this region $\alpha_{s}$ is large and due to the confinement hadrons
are the relevant degrees of freedom. Since perturbative methods are
not applicable in this regime a reasonable description of the
transition between quasi-real photoproduction and DIS remains one of
the biggest challenges in QCD. Presently the best description is
provided by phenomenological models. Three of them are discussed in
the coming section.

\begin{figure}[t]
  \centering
  \includegraphics[width=0.65\textwidth]{lambda.eps}
  \caption{Exponent $\lambda(Q^{2})$ obtained from fits of the function~\ref{e:f2lambda} to H1 and ZEUS structure function data for \mbox{$x < 0.01$}~\cite{Breitweg:1998dz, Adloff:2001rw, Lastovicka:2002} (figure taken from~\cite{Newman:2003vh}). The inner error bars illustrate the statistical errors, the outer error bars indicate the statistical and systematic errors added in quadrature. The straight line represents a fit of the form \mbox{$\lambda(Q^{2}) = a \ln(Q^{2}/\Lambda^{2})$} using data for \mbox{$Q^{2} \ge 3.5\, \mathrm{GeV^{2}}$}, the dashed line shows an extrapolation of this fit to lower values of $Q^{2}$.}
  \label{f:lambda}
\end{figure}

\section{Phenomenological Models at Low $Q^{2}$} \label{c:phenomenology}

A large fraction of the phenomenological models describing the
transition from high to low values of $Q^{2}$ make use of the fact
that the $ep$ scattering process can be interpreted as the interaction
of a virtual photon with the proton (see
section~\ref{c:disxsect}). They provide a smooth fit of the structure
functions or equally the photoabsorption cross-sections ensuring that
in the limit $Q^{2} \rightarrow 0$ the corresponding photoproduction
values are reached. Since real photons can have no longitudinal
polarisation this implies for $\sigma_{T}$ and $\sigma_{L}$:
\begin{equation}
\sigma_{T} \rightarrow \sigma_{\gamma p}, \hspace{2em} \sigma_{L} \rightarrow 0
\label{e:gplim}
\end{equation}
According to equations \ref{e:f2gammap} and \ref{e:flgammap} this
implies that the structure functions $F_{2}$ and $F_{L}$ have to
vanish in the photoproduction limit.

At very low values of $Q^{2}$ the exchanged photon becomes almost real
and it can fluctuate into a virtual quark-antiquark pair. The lifetime
of this state can become large and comparable to the photon-proton
interaction time since the virtuality is small. Thus it is possible to
interpret the photon-proton scattering as a hadron-hadron interaction.

\subsection*{ALLM Parameterisation}


The ALLM parameterisation is inspired by the Regge theory which has
been developed well in advance of QCD to describe hadron-hadron
interactions~\cite{Regge:1959mz, Regge:1960zc, Chew:1961yz}. Due to
the hadronic nature of quasi real photons mentioned above it can be
used to describe $ep$ scattering at small values of $Q^{2}$. In Regge
theory the interaction of two hadrons is described by the exchange of
collective states of colour neutral particles, all of them having the
same quantum numbers apart from the spin. The spin depends
approximately linearly on the particle mass squared $t$. The
corresponding straight line is called
\emph{Regge trajectory} and parameterised as follows
\begin{equation}
\alpha(t) = \alpha(0) + \alpha' \cdot t
\end{equation}
with $\alpha(0)$ being the intercept and $\alpha'$ the slope of the
trajectory.

Using this approach Donnachie and Landshoff~\cite{Donnachie:1992ny}
have fitted $pp$ and $p\bar{p}$ total cross-sections by the form
\begin{equation}
\sigma_{i}^{tot} = A_{i}^{\pom} s^{\alpha_{\pom}(0)-1} + A_{i}^{\reg} s^{\alpha_{\reg}(0)-1}\label{e:sigmadl}
\end{equation}
The coefficients $A_{i}^{\pom}$ and $A_{i}^{\reg}$ are process
dependent constants whereas the intercepts $\alpha_{\pom}(0)$ and
$\alpha_{\reg}(0)$ are universal and have been determined to be
\mbox{$\alpha_{\pom}(0) = 1.0808$} and \mbox{$\alpha_{\reg}(0) =
0.5475$}. The trajectory $\reg$ is an effective trajectory comprising
several trajectories of known mesons (reggeon trajectories). It
describes the decrease of the cross-section as a function of $s$ at
low $s$. To account for the observed increase of $\sigma_{i}^{tot}$ at
larger values of $s$, the so-called pomeron trajectory $\pom$ has been
introduced which has the quantum numbers of the vacuum.

It has been shown that equation~\ref{e:sigmadl} also describes other
hadronic total cross-section data including that of real
photoproduction on protons. In addition it directly implies $F_{2}$ to
rise as \mbox{$F_{2} \sim x^{1-\alpha_{\pom}(0)}$} or \mbox{$F_{2}
\sim x^{-0.08}$} towards small values of $x$~\cite{Donnachie:1993it}. 
This is in agreement with the observations in the last section.

In contrast, the given parameterisation fails to to reproduce DIS
data. To achieve a description of the entire $Q^{2}$ range the ansatz
of Donnachie and Landshoff has been extended by several authors. In
the parameterisation of Abramowicz, Levin, Levy and Moar
(ALLM)~\cite{Abramowicz:1991xz} the structure function $F_{2}$ is
composed of a pomeron ($F_{2}^{\pom}$) and a reggeon ($F_{2}^{\reg}$)
contribution:
\begin{equation}
F_{2}(x,Q^{2}) = \frac{Q^{2}}{Q^{2} + m_{0}^{2}} \left ( F_{2}^{\pom}(x,Q^{2}) + F_{2}^{\reg}(x,Q^{2}) \right )
\end{equation}
The basic difference between this model and equation~\ref{e:sigmadl}
is the introduction of $Q^{2}$ dependent intercepts. The version of
the parameterisation employed for the present analysis has been
obtained in 1997 by fitting its 23 free parameters to DIS data of HERA
and fixed target experiments together with total $pp$ and $\gamma p$
cross-sections~\cite{Abramowicz:1997ms}.

\subsection*{Saturation Model}


The saturation model of Golec-Biernat and W\"usthoff
~\cite{Golec-Biernat:1998js, Golec-Biernat:1999qd} is intended to
provide a description of the transition between the high and low
$Q^{2}$ regime in inclusive $ep$ scattering at low $x$. It is based on
the colour dipole model. The latter has a physical interpretation in
the proton rest frame (see figure~\ref{f:dipolemodel} a)): the virtual
photon $\gamma^{*}$ splits into a quark-antiquark pair, i.e. a colour
dipole, long before the interaction with the proton since the
formation time,
\mbox{$\tau_{q\bar{q}} \sim 1/(xm_{p})$}, is large. Subsequently the
$q\bar{q}$ pair scatters coherently off the proton in a time which is
short compared to $\tau_{q\bar{q}}$. The dissociation of the photon
into the $q\bar{q}$ pair is described by QED while the strong
interaction is contained in the cross-section for the dipole-proton
interaction. It is assumed that the transverse separation $r$ of quark
and antiquark remains constant during the interaction. Quark and
antiquark carry the fraction $z$ and \mbox{$1-z$} of the photon's
momentum, respectively.

\begin{figure}
  \hspace{0.01\textwidth}
  \begin{minipage}[c]{0.38\textwidth}
    \vspace{-0.3cm}
    \includegraphics[width=\textwidth]{feydiag1.eps}
    \put(-150,93){a)}
  \end{minipage}
  \hspace{0.02\textwidth}
  \begin{minipage}[c]{0.55\textwidth}
    \includegraphics[width=\textwidth]{dipcs.eps}
    \put(-195,120){b)}
  \end{minipage}
  \hspace{0.01\textwidth}
  \caption{a) Illustration of the photon-proton scattering process within the dipole model. b) Dipole cross-section $\hat{\sigma}(x,r^{2})$ as a function of the transverse separation $r$ of the $q\bar{q}$ pair according to equation~\ref{e:sigmadpm} for different values of $x$ (both figures taken from~\cite{Golec-Biernat:1999qd}).}
  \label{f:dipolemodel}
\end{figure}

The idea of the dipole model is expressed formally by factorising the
absorption cross-sections for transversely and longitudinally
polarised photons in the following way:
\begin{equation}
\sigma_{T,L}(x,Q^{2}) = \int d^{2} \mathbf{r} \int_{0}^{1} dz \, |\Psi_{T,L}(z,\mathbf{r})|^{2} \, \hat{\sigma}(x,r^{2})
\label{e:dpmfact}
\end{equation}
The photon wave functions $\Psi_{T}$ and $\Psi_{L}$ describe the
photon dissociation in a $q\bar{q}$ pair and are calculable in QED. In
contrast, the dipole cross-section $\hat{\sigma}(x,r^{2})$ is
substantially influenced by non-perturbative contributions and needs
to be modelled.

In the saturation model~\cite{Golec-Biernat:1998js,
Golec-Biernat:1999qd} the dipole cross-section is assumed to have the
form
\begin{equation}
\hat{\sigma}(x,r^{2}) = \sigma_{0} \left [ 1 - \exp \left (-\frac{r^{2}}{4R_{0}^{2}(x)} \right ) \right ]
\label{e:sigmadpm}
\end{equation}
where the $x$ dependent \emph{saturation radius} $R_{0}(x)$ is given by
\begin{equation}
R_{0}(x) = \frac{1}{Q_{0}} \left ( \frac{x}{x_{0}} \right )^{\frac{\lambda}{2}}
\label{e:r0dpm}
\end{equation}
To ensure a smooth transition to the photoproduction limit ($Q^{2} \rightarrow 0$) $x$ is modified to be
\begin{equation}
x = x_{B} \left ( 1 + \frac{4 m_{q}^{2}}{Q^{2}} \right )
\label{e:xbjdpm}
\end{equation}
with $x_{B}$ denoting Bjorken $x$ in its original definition according
to equation~\ref{e:xandy} and $m_{q}$ being an effective quark mass.

Fixing $Q_{0}$ to $1\, \mathrm{GeV}$ and $m_{q}$ to \mbox{$140\,
\mathrm{MeV}$} three model parameters remain which have been
determined from fits to DIS data for \mbox{$x <
10^{-2}$}~\cite{Golec-Biernat:1998js}:
\begin{equation}
\sigma_{0} = 23\, \mathrm{mb}, \hspace{1em} \lambda = 0.29, \hspace{1em} x_{0} = 3 \cdot 10^{-4}
\label{e:paradpm}
\end{equation}
The resulting dipole cross-section $\hat{\sigma}(x,r^{2})$ as a
function of the transverse separation $r$ is depicted in
figure~\ref{f:dipolemodel} b) for different values of $x$. For small
values of $r$ the dipole cross-section rises like $r^{2}$ while for
large values of $r$ it saturates. Saturation sets in at \mbox{$r \sim
2R_{0}$}, i.e. it is governed by $R_{0}$. The characteristic size of
the dipoles is given by \mbox{$r \sim 1/\sqrt{Q^{2}}$}. Hence
$\hat{\sigma}(x,r^{2})$ rises with decreasing values of $Q^{2}$ and
finally saturates if low values of $Q^{2}$ are approached\footnote{The
saturation of the dipole cross-section as a function of $Q^{2}$ has to
be distinguished from the saturation as a function of $x$ which is
described below and leads to the small $x$ saturation of $F_{2}$
introduced in section~\ref{c:f2rise}.}. This feature of the dipole
model can explain the change in the behaviour of $\lambda$ at very low
values of $Q^{2}$ (see section~\ref{c:f2rise}). The dipole model
predicts a pomeron intercept of \mbox{$\alpha_{\pom}(0) \approx 1.08$}
if the parameterisation obtained from fits to DIS data is extrapolated
to the photoproduction regime. This result is in perfect agreement
with the model of Donnachie and Landshoff~\cite{Donnachie:1992ny} as
well as the experimental results for $\lambda$ obtained at HERA (see
figure~\ref{f:lambda}).

Due to the $x$ dependence of the saturation radius $R_{0}$ given by
equation~\ref{e:r0dpm} the dipole cross-section
$\hat{\sigma}(x,r^{2})$ saturates at smaller values of $r$ and hence
larger values of $Q^{2}$ as $x$ decreases. For a fixed value of $r$,
i.e. $Q^{2}$, it saturates as a function of $x$ when approaching low
values of $x$. In this way the saturation model incorporates the
saturation of $F_{2}$ expected at small $x$. The saturation sets in at
larger values of $x$ when $Q^{2}$ decreases.

While being very successful in describing the experimental
observations at low $Q^{2}$ the saturation model undershoots the
measured structure function $F_{2}$ for \mbox{$Q^{2} > 20\,
\mathrm{GeV^{2}}$}. This is caused by the fact that it does not 
include any scaling violations. Therefore the saturation model has
been extended by incorporating leading order DGLAP
evolution~\cite{Bartels:2002cj}. In order to preserve the good
agreement with the data at low $Q^{2}$ the dipole cross-section is
only modified at small values of $r$ while at large $r$ is still shows
saturation:
\begin{equation}
\hat{\sigma}(x,r^{2}) = \sigma_{0} \left [ 1 - \exp \left ( -\frac{\pi^{2}\, r^{2}\, \alpha_{s}(\mu^{2})\, x g(x,\mu^{2})}{3\, \sigma_{0}} \right ) \right ]
\label{e:sigmadpmdglap}
\end{equation}
The scale $\mu^{2}$ is assumed to have the form $\mu^{2} = \mu_{0}^{2}
+ C/r^{2}$ with $\mu_{0}^{2}$ and $C$ being parameters to be
determined from a fit to the data. Since the model is intended to
describe the data at low $x$, only the gluon density $g(x,Q^{2})$ is
included in the leading order DGLAP evolution, because it is the
dominating parton density in this regime. The modified saturation
model with four additional parameters describes the $Q^{2}$ dependence
of $\lambda$ as observed at HERA (see figure~\ref{f:lambda})
remarkably well over the whole $Q^{2}$ range~\cite{Bartels:2002cj}.

\subsection*{Fractal Fit}


At small values of $x$ the proton structure function $F_{2}$ is almost
solely determined by the sea quarks. In QCD their behaviour is driven
by gluon emissions and splittings. The larger values of $Q^{2}$ are
reached the more of these emissions and splittings can be
observed. In~\cite{Lastovicka:2002hw} it has therefore been attempted
to parametrise $F_{2}$ assuming a self-similar structure of the
proton, i.e. expecting that the proton reveals a fractal nature at low
$x$. Two magnification scales are used: $1/x$ and \mbox{$1 +
Q^{2}/Q_{0}^{2}$}. Including $Q_{0}^{2}$ the parameterisation contains
five free parameters. They have been determined using results from the
H1 and ZEUS experiments at \mbox{$0.045 \le Q^{2} \le 120\,
\mathrm{GeV^{2}}$} and \mbox{$6.2 \cdot 10^{-7} \le x \le 0.01$}. The
parameterisation, which is referred to as \emph{fractal fit} in the
following, provides a good description of the data in the kinematic
region given.

\section{Radiative $ep$ Scattering} \label{c:radepscat}


\begin{figure}
  \begin{minipage}[c]{0.48\textwidth}
  \centering
  \begin{fmffile}{feynmf/ncIsr}
    \begin{fmfgraph*}(160,110)
      \fmfstraight
      \fmftop{d1,og,ol}
      \fmfcurved
      \fmfleft{ip,d2,d3,il,d4}
      \fmfright{oq1,oq2,oq3,d5,d6,d7,ol}
      \fmf{fermion,tension=3,label=${\mathbf P}$,label.side=left,label.dist=10.}{ip,vp}
      \fmf{fermion}{vp,oq1}
      \fmf{fermion}{vp,oq2}
      \fmf{fermion}{vp,oq3}
      \fmf{photon,tension=0.8,label=${\mathbf q}$,label.side=left}{vl,vp}
      \fmf{fermion,tension=1.7,label=${\mathbf k}$,label.side=left}{il,vg}
      \fmf{fermion,tension=1.7,label=${\mathbf q'}$,label.side=right}{vg,vl}
      \fmf{fermion,label=${\mathbf k'}$, label.side=left}{vl,ol}
      \fmfv{decor.shape=circle,decor.filled=empty,decor.size=.1w}{vp}
      \fmfdot{vl}
      \fmffreeze
      \fmf{photon,label=${\mathbf l}$,label.side=left}{vg,og}
      \fmfi{plain}{vpath (__ip,__vp) shifted (thick*(0,2))}
      \fmfi{plain}{vpath (__ip,__vp) shifted (thick*(1,-1.6))}
    \end{fmfgraph*}
  \end{fmffile}
  \end{minipage}
  \begin{minipage}[c]{0.48\textwidth}
  \centering
  \begin{fmffile}{feynmf/ncFsr}
    \begin{fmfgraph*}(160,110)
      \fmfstraight
      \fmftop{d1,d2,ol}
      \fmfcurved
      \fmfleft{ip,d2,d3,il,d4}
      \fmfright{oq1,oq2,oq3,d5,og,d7,ol}
      \fmf{fermion,tension=3,label=${\mathbf P}$,label.side=left,label.dist=10.}{ip,vp}
      \fmf{fermion}{vp,oq1}
      \fmf{fermion}{vp,oq2}
      \fmf{fermion}{vp,oq3}
      \fmf{photon,tension=0.8,label=${\mathbf q}$,label.side=left}{vl,vp}
      \fmf{fermion,label=${\mathbf k}$,label.side=left}{il,vl}
      \fmf{fermion,tension=1.7,label=${\mathbf q''}$, label.side=left}{vl,vg}
      \fmf{fermion,tension=1.7,label=${\mathbf k'}$, label.side=left}{vg,ol}
      \fmfv{decor.shape=circle,decor.filled=empty,decor.size=.1w}{vp}
      \fmfdot{vl}
      \fmffreeze
      \fmf{photon,label=${\mathbf l}$,label.side=right}{vg,og}
      \fmfi{plain}{vpath (__ip,__vp) shifted (thick*(0,2))}
      \fmfi{plain}{vpath (__ip,__vp) shifted (thick*(1,-1.6))}
    \end{fmfgraph*}
  \end{fmffile}
  \end{minipage}
  \vspace{0.3cm}
  \caption{Lowest order Feynman diagrams for the process \mbox{$ep \rightarrow e \gamma X$} with photon emission from the electron line.}
  \label{f:radepscat}
\end{figure}

Figure~\ref{f:feyndis} depicts the $ep$ scattering process to lowest
order perturbation theory (Born approximation). The corresponding Born
cross-section is given in equation~\ref{e:sigmaep} taking only the
photon exchange into account. At higher orders various radiative
processes have to be considered which are discussed in
section~\ref{c:radcor}. The present section focuses solely on their
dominant contribution to the cross-section at low $Q^{2}$, the
radiation of a real photon from the incoming or outgoing electron. For
these process class several configurations can be experimentally
observed and are important for the analysis as outlined in the
following.

The lowest order Feynman diagrams for the photon emission from the
lepton line are shown in figure~\ref{f:radepscat}. Also the notation
for the particle four-momenta to be used in the following can be
inferred from there. The amplitudes of both diagrams and the
interference term between them contribute to the scattering
cross-section. Hence it is impossible to attribute a single event to
either the one or the other diagram. The amplitudes for the first and
second diagram are inversely proportional to \mbox{$({\mathbf{q}'}^{2}
- m_{e}^{2})\, \mathbf{q}^{2}$} and \mbox{$({\mathbf{q}''}^{2} -
m_{e}^{2})\, \mathbf{q}^{2}$}, respectively. Correspondingly, the
differential cross-section exhibits several maxima when one or both of
these term approach zero. These peaks correspond to the following
experimental configurations:

\begin{itemize}
\item{
\emph{Radiative DIS events} correspond to the maxima of the cross-section reached for \mbox{${\mathbf{q}'}^{2} \simeq 0$} or \mbox{${\mathbf{q}''}^{2} \simeq 0$}, but $\mathbf{q}^{2}$ still being finite. This implies that the radiated photon is emitted collinear either with the incoming (\mbox{${\mathbf{q}'}^{2} \simeq 0$}) or the scattered (\mbox{${\mathbf{q}''}^{2} \simeq 0$}) electron while the virtuality \mbox{$Q^{2} = -\mathbf{q}^{2}$} of the exchanged photon is large enough for the scattered electron to be detected in the main detector.

In the first case the Feynman diagram shown in
figure~\ref{f:radepscat} a) dominates the cross-section. Such events
are called \emph{Initial State Radiation} (ISR) events. Due to the
photon radiation the energy of the electron available for the
interaction with the proton is lowered and hence an ISR event can be
interpreted as a non-radiative $ep$ scattering event at a
reduced centre of mass energy $\sqrt{s}$. According to
equation~\ref{e:q2sxy} this allows on the one hand to extract the
structure function $F_{2}$ at lower values of $Q^{2}$ for a given $x$
or at larger values $x$ for a given $Q^{2}$~\cite{Ahmed:1995cf,
Derrick:1995ef, Cole:2004, Issever:2000}. In addition the lowered
centre of mass energy in ISR events can be exploited to measure the
proton structure function $F_{L}$~\cite{Cole:2004, Issever:2000} (see
section~\ref{c:disxsect}). Also the present analysis makes use of ISR
events to extend the accessible kinematic phase space for the $F_{2}$
measurement, but in contrast to the previous measurements the radiated
photon is not explicitely detected.

In the second case the main contribution stems from the Feynman
diagram of figure~\ref{f:radepscat} b), the corresponding events are
referred to as \emph{Final State Radiation} (FSR) events. For them the
opening angle between the scattered electron and the radiated photon
is typically too small for the two particles to be separately
reconstructed in the calorimeter. Therefore they usually cannot be
distinguished from non-radiative events.
}

\item{
A further peak of the radiative cross-section is reached for
\mbox{${\mathbf{q}}^{2} \simeq 0$}, but ${\mathbf{q}'}^{2}$ and 
${\mathbf{q}''}^{2}$ finite. For this configuration electron and
photon have a sizeable transverse momentum and are almost back-to-back
in azimuth. Since the virtuality of the exchanged photon is small the
process can be interpreted as the scattering of a quasi-real photon
off an electron. For this reason the process is called \emph{QED
Compton Scattering} (QEDC). Due to the sizeable transverse momenta of
the outgoing particles QEDC events can be rather easily identified
experimentally. Therefore they are ideally suited to measure the
proton structure function $F_{2}$ at small values of
$Q^{2}$~\cite{Aktas:2004ek, Lendermann:2001}. Furthermore they can be
employed for the detector alignment by making use of their property to
be back-to-back in azimuth (see section~\ref{c:alignment}).  }

\item{
Due to the maximum of the cross-section in the limit
\mbox{${\mathbf{q}}^{2} \simeq 0$}, \mbox{${\mathbf{q}'}^{2} \simeq 0$} 
and \mbox{${\mathbf{q}''}^{2} \simeq 0$} most of the radiative events
are characterised by both electron and radiated photon being scattered
at very small angles. Hence they leave the main detector through the
beam pipe without being detected. The majority of these events are
elastic scattering processes called
\emph{Bethe-Heitler events}~\cite{Bethe:1934za}. Their cross-section can 
be precisely calculated in QED and is insensitive to the internal
proton structure. For these reasons the Bethe-Heitler process is used as
reference for the luminosity measurement of the H1 experiment (see
section~\ref{c:lumisystem}).  }

\end{itemize}


