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55 lines
2.3 KiB
TeX
55 lines
2.3 KiB
TeX
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\section{Parton Density Functions}%
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\label{sec:pdf_basics}
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Parton Density Functions encode, restricting the considerations to
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leading order, the probability to ``encounter'' a constituent parton
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(quark or gluon) of a hadron with a certain momentum fraction \(x\) at
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a certain factorization scale \(Q^2\). PDFs are normalized according
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to~\eqref{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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In deep inelastic scattering \(Q^2\) is just the negative over the
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momentum transfer \(-q^2\). In more complicated processes \(Q^2\) has
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to be chosen in a way that reflects the ``energy resolution'' of the
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process. If the perturbation series behind the process would be
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expanded to the exact solution, the dependence on the factorization
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scale vanishes. In leading order, one has to choose the scale in a
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``physically meaningfull'' way, which reflects characteristics of the
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process.
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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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partons in equal hadrons is given in~\eqref{eq:pdf-xs}. Here \(i,j\) are
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the partons participating in a scattering process with the cross
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section \(\sigma_{ij}\). Usually this cross section depends on the
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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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\tilde{\sigma} = \int f_i\qty(x;Q^2) f_j\qty(x;Q^2) \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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Summing~\eqref{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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PDFs can not be derived from first principles and have to be
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determined experimentally for low \(Q^2\) and can be evolved to higher
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\(Q^2\) through the \emph{DGLAP} equations~\cite{altarelli:1977af} at
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different orders of perturbation theory.
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