2020-05-04 19:56:04 +02:00
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\section{Parton Density Functions}%
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\label{sec:pdf_basics}
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2020-05-13 11:05:31 +02:00
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Parton Density Functions encode, restricting considerations to leading
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order, the probability to \emph{encounter} a constituent parton (quark
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or gluon) of a hadron with a certain momentum fraction \(x\) at a
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certain factorization scale \(Q^2\). PDFs are normalized according to
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\cref{eq:pdf-norm}, where the sum runs over all partons.
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\begin{equation}
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\label{eq:pdf-norm}
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\sum_i\int_0^1x\cdot f_i\qty(x;Q^2) \dd{x} = 1
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\end{equation}
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2020-05-13 11:05:31 +02:00
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PDFs can not be derived from first principles (at the moment) and have
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to be determined experimentally for low \(Q^2\) and are evolved to
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higher \(Q^2\) through the \emph{DGLAP}
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equations~\cite{altarelli:1977af} at different orders of perturbation
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theory. In deep inelastic scattering \(Q^2\) is just the negative
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over the momentum transfer \(-q^2\). For more complicated processes
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\(Q^2\) has to be chosen in a way that reflects the \emph{momentum
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resolution} of the process. If the perturbation series behind the
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process would be expanded to the exact solution, the dependence on the
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factorization scale vanishes. In lower orders, one has to choose the
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scale in a \emph{physically meaningful}\footnote{That means: not in
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an arbitrary way.} way, which reflects characteristics of the
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process~\cite{altarelli:1977af}.
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2020-05-04 19:56:04 +02:00
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In the case of \(\qqgg\) the mean of the Mandelstam variables \(\hat{t}\)
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and \(\hat{u}\), which is equal to \(\hat{s}/2\), can be used. This
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choice is lorentz-invariant and reflects the s/u-channel nature of the
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process.
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The (differential) hadronic cross section for scattering of two
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2020-05-18 09:55:57 +02:00
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partons in equal hadrons is given in \cref{eq:pdf-xs}. Here \(i,j\)
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are the partons participating in a scattering process with the cross
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section \(\hat{\sigma}_{ij}\). Usually this cross section depends on
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the kinematics and thus the momentum fractions and the factorization
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scale\footnote{More appropriately: The factorization scale depends on
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the process. So \(\sigma\qty(Q^2)\) is just a symbol for that
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relation.}.
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\begin{equation}
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\label{eq:pdf-xs}
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\sigma = \int f_i\qty(x;Q^2) f_j\qty(x;Q^2) \hat{\sigma}_{ij}\qty(x_1,
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x_2, Q^2)\dd{x_1}\dd{x_2}
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\end{equation}
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2020-05-13 10:37:16 +02:00
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Summing \cref{eq:pdf-xs} over all partons in the hadron gives
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2020-05-04 19:56:04 +02:00
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the total scattering cross section for the hadron.
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2020-05-05 18:59:40 +02:00
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2020-05-13 10:37:16 +02:00
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%%% Local Variables:
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%%% mode: latex
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%%% TeX-master: "../../document"
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%%% End:
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