bachelor_thesis/latex/tex/pdf/pdf_basics.tex

\section{Parton Density Functions}%
\label{sec:pdf_basics}

Parton Density Functions encode, restricting considerations to leading
order, the probability to encounter a constituent parton of a hadron
with a certain momentum fraction \(x\) at a certain factorization
scale \(Q^2\) in a scattering process. PDFs are normalized according
to \cref{eq:pdf-norm}, where the sum runs over all partons.
%
\begin{equation}
  \label{eq:pdf-norm}
  \sum_i\int_0^1x\cdot f_i\qty(x;Q^2) \dd{x} = 1
\end{equation}
%
More precisely \({f_i}\) denotes a PDF set, which is referred to
simply as PDF in the following.  PDFs can not be derived from first
principles and have to be determined experimentally for low \(Q^2\)
and can be evolved to higher \(Q^2\) through the \emph{DGLAP}
equations~\cite{altarelli:1977af} at different orders of perturbation
theory.  In deep inelastic scattering \(Q^2\) is just the negative
over the momentum transfer: \(-q^2\). For more complicated processes
\(Q^2\) has to be chosen in a way that reflects the
\emph{energy-momentum scale} of the process. If the perturbation
series behind the process would be expanded to the exact solution, the
dependence on the factorization scale would vanish. In lower orders,
one has to choose the scale in a \emph{physically
  meaningful}\footnote{That means: not in an arbitrary way.} way,
which reflects characteristics of the process~\cite{altarelli:1977af}.

In the case of \(\qqgg\) the mean of the Mandelstam variables
\(\hat{t}\) and \(\hat{u}\), which is equal to \(\hat{s}/2\), can be
used. This choice is lorentz-invariant and reflects the s/u-channel
nature of the process, although the \(\pt\) of photon would also have
been a good choice~\cite[18]{buckley:2011ge}.

The (differential) hadronic cross section for scattering of two
partons in equal hadrons is given in \cref{eq:pdf-xs}. Here \(i,j\)
are the partons participating in a scattering process with the cross
section \(\hat{\sigma}_{ij}\). Usually this cross section depends on
the kinematics and thus the momentum fractions and the factorization
scale\footnote{More appropriately: The factorization scale depends on
  the process. So \(\sigma\qty(Q^2)\) is just a symbol for that
  relation.}.
%
\begin{equation}
  \label{eq:pdf-xs}
  \sigma = \int f_i\qty(x_1;Q^2) f_j\qty(x_2;Q^2) \hat{\sigma}_{ij}\qty(x_1,
  x_2, Q^2)\dd{x_1}\dd{x_2}
\end{equation}
%
Summing \cref{eq:pdf-xs} over all partons in the hadron gives
the total scattering cross section for the hadron.

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