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56 lines
2.5 KiB
TeX
56 lines
2.5 KiB
TeX
\section{Parton Density Functions}%
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\label{sec:pdf_basics}
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Parton Density Functions encode, restricting considerations to leading
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order, the probability to encounter a constituent parton of a hadron
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with a certain momentum fraction \(x\) at a certain factorization
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scale \(Q^2\) in a scattering process. PDFs are normalized according
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to \cref{eq:pdf-norm}, where the sum runs over all partons.
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%
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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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%
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More precisely \({f_i}\) denotes a PDF set, which is referred to
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simply as PDF in the following. PDFs can not be derived from first
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principles and have to be determined experimentally for low \(Q^2\)
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and can be evolved to 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 of
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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
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\emph{energy-momentum scale} of the process. If the perturbation
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series behind the process would be expanded to the exact solution, the
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dependence on the factorization scale would vanish. In lower orders,
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one has to choose the scale in a \emph{physically meaningful} way,
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which reflects characteristics of the process~\cite{altarelli:1977af}.
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In the case of \(\qqgg\) the mean of the Mandelstam variables
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\(\hat{t}\) and \(\hat{u}\), which is equal to \(\hat{s}/2\), can be
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used. This choice is lorentz-invariant and reflects the t/u-channel
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nature of the process, although the \(\pt\) of photon would also have
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been a good choice~\cite[18]{buckley:2011ge}.
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The (differential) hadronic cross section for scattering of two
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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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%
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\begin{equation}
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\label{eq:pdf-xs}
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\sigma_{ij} = \int f_i\qty(x_1;Q^2) f_j\qty(x_2;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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%
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Summing \cref{eq:pdf-xs} over all partons in the hadron gives
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the total scattering cross section for the hadron.
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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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