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19
latex/tex/pdf/pdf_basics.tex
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@ -2,11 +2,10 @@
\label{sec:pdf_basics} \label{sec:pdf_basics}
Parton Density Functions encode, restricting considerations to leading Parton Density Functions encode, restricting considerations to leading
order, the probability to encounter a constituent parton (quark or order, the probability to encounter a constituent parton of a hadron
gluon) of a hadron with a certain momentum fraction \(x\) at a certain with a certain momentum fraction \(x\) at a certain factorization
factorization scale \(Q^2\) in a scattering process. PDFs are scale \(Q^2\) in a scattering process. PDFs are normalized according
normalized according to \cref{eq:pdf-norm}, where the sum runs over to \cref{eq:pdf-norm}, where the sum runs over all partons.
all partons.
% %
\begin{equation} \begin{equation}
\label{eq:pdf-norm} \label{eq:pdf-norm}
@ -16,15 +15,15 @@ all partons.
More precisely \({f_i}\) denotes a PDF set, which is referred to 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 simply as PDF in the following. PDFs can not be derived from first
principles and have to be determined experimentally for low \(Q^2\) principles and have to be determined experimentally for low \(Q^2\)
and are evolved to higher \(Q^2\) through the \emph{DGLAP} and can be evolved to higher \(Q^2\) through the \emph{DGLAP}
equations~\cite{altarelli:1977af} at different orders of perturbation equations~\cite{altarelli:1977af} at different orders of perturbation
theory. In deep inelastic scattering \(Q^2\) is just the negative theory. In deep inelastic scattering \(Q^2\) is just the negative
over the momentum transfer \(-q^2\). For more complicated processes over the momentum transfer: \(-q^2\). For more complicated processes
\(Q^2\) has to be chosen in a way that reflects the \(Q^2\) has to be chosen in a way that reflects the
\emph{energy-momentum scale} of the process. If the perturbation \emph{energy-momentum scale} of the process. If the perturbation
series behind the process would be expanded to the exact solution, the series behind the process would be expanded to the exact solution, the
dependence on the factorization scale vanishes. In lower orders, one dependence on the factorization scale would vanish. In lower orders,
has to choose the scale in a \emph{physically one has to choose the scale in a \emph{physically
meaningful}\footnote{That means: not in an arbitrary way.} way, meaningful}\footnote{That means: not in an arbitrary way.} way,
which reflects characteristics of the process~\cite{altarelli:1977af}. which reflects characteristics of the process~\cite{altarelli:1977af}.
@ -45,7 +44,7 @@ scale\footnote{More appropriately: The factorization scale depends on
% %
\begin{equation} \begin{equation}
\label{eq:pdf-xs} \label{eq:pdf-xs}
\sigma = \int f_i\qty(x;Q^2) f_j\qty(x;Q^2) \hat{\sigma}_{ij}\qty(x_1, \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} x_2, Q^2)\dd{x_1}\dd{x_2}
\end{equation} \end{equation}
% %

6
latex/tex/pdf/results.tex
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@ -10,7 +10,7 @@ experiment, even with this simple leading-order process.
The integrand in \cref{eq:pdf-xs} can be concertized into The integrand in \cref{eq:pdf-xs} can be concertized into
\cref{eq:weighteddist}, where \(q\) runs over all quarks (except the \cref{eq:weighteddist}, where \(q\) runs over all quarks (except the
top quark). The sum has been symmetized, otherwise a double sum with top quark). The sum has been symmetrized, otherwise a double sum with
\(q\) and \(\bar{q}\) would have been necessary. The choice of \(Q^2\) \(q\) and \(\bar{q}\) would have been necessary. The choice of \(Q^2\)
is justified in \cref{sec:pdf_basics} and formulated in is justified in \cref{sec:pdf_basics} and formulated in
\cref{eq:q2-explicit}. \cref{eq:q2-explicit}.
@ -24,7 +24,7 @@ is justified in \cref{sec:pdf_basics} and formulated in
Q^2 = 2x_1x_2E_p^2 Q^2 = 2x_1x_2E_p^2
\end{gather} \end{gather}
% %
The PDF set being used in the following has been fitted (and The PDF set that is being used in the following has been fitted (and
\emph{DGLAP} developed) at leading order and is the central member of \emph{DGLAP} developed) at leading order and is the central member of
the PDF set \verb|NNPDF31_lo_as_0118| provided by \emph{NNPDF} the PDF set \verb|NNPDF31_lo_as_0118| provided by \emph{NNPDF}
collaboration and accessed through the \lhapdf\ collaboration and accessed through the \lhapdf\
@ -253,3 +253,5 @@ imposed alter the distribution quite considerably, cutting of the
%%% mode: latex %%% mode: latex
%%% TeX-master: "../../document" %%% TeX-master: "../../document"
%%% End: %%% End:
% LocalWords: symmetrized