hiro/bachelor_thesis
1
0
Fork 0
mirror of https://github.com/vale981/bachelor_thesis synced 2025-03-05 17:41:41 -05:00
Code Issues Projects Releases Packages Wiki Activity Actions

improve wording in section about pdfs

Browse source
This commit is contained in:
hiro98 2020-05-13 11:05:31 +02:00
parent 7394c322b7
commit a9c24b0804
3 changed files with 29 additions and 27 deletions
Show stats Download patch file Download diff file Expand all files Collapse all files

15
latex/tex/pdf.tex
View file

@ -1,13 +1,14 @@
\chapter{A Simple Model of Proton-Proton Collisions}%
\label{chap:pdf}
Because bare quarks do not occur in nature, one to study hadronic
scattering to obtain experimentally verifiable results. Hadrons are
usually modeled using Parton Density Functions (PDFs). Using a simple
PDF, the cross section for the process \(\ppgg\) (neglecting the
remnants \(x\) and other processes) and event samples of that process
will be obtained. These results will be compared with results from
\sherpa.
Because free quarks do not occur in nature, one has to study the
scattering of hadrons to obtain experimentally verifiable
results. Hadrons are usually modeled as consisting of multiple
\emph{partons} using Parton Density Functions (PDFs). By applying a
simple PDF, the cross section for the process \(\ppgg\) (neglecting
the remnants \(x\) and other processes) and event samples of that
process will be obtained. These results will once again be compared
with results from \sherpa.
%%% Local Variables:
%%% mode: latex

37
latex/tex/pdf/pdf_basics.tex
View file

@ -1,25 +1,31 @@
\section{Parton Density Functions}%
\label{sec:pdf_basics}
Parton Density Functions encode, restricting the considerations to
leading order, the probability to ``encounter'' a constituent parton
(quark or gluon) of a hadron with a certain momentum fraction \(x\) at
a certain factorization scale \(Q^2\). PDFs are normalized according
to \cref{eq:pdf-norm}, where the sum runs over all partons.
Parton Density Functions encode, restricting considerations to leading
order, the probability to \emph{encounter} a constituent parton (quark
or gluon) of a hadron with a certain momentum fraction \(x\) at a
certain factorization scale \(Q^2\). 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}
In deep inelastic scattering \(Q^2\) is just the negative over the
momentum transfer \(-q^2\). In more complicated processes \(Q^2\) has
to be chosen in a way that reflects the ``energy resolution'' of the
process. If the perturbation series behind the process would be
expanded to the exact solution, the dependence on the factorization
scale vanishes. In leading order, one has to choose the scale in a
``physically meaningfull'' way, which reflects characteristics of the
process.
PDFs can not be derived from first principles (at the moment) and have
to be determined experimentally for low \(Q^2\) and are 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{momentum
resolution} of the process. If the perturbation series behind the
process would be expanded to the exact solution, the dependence on the
factorization scale vanishes. 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
@ -44,11 +50,6 @@ scale\footnote{More appropriately: The factorization scale depends on
Summing \cref{eq:pdf-xs} over all partons in the hadron gives
the total scattering cross section for the hadron.
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.
%%% Local Variables:
%%% mode: latex
%%% TeX-master: "../../document"

4
latex/tex/pdf/results.tex
View file

@ -1,7 +1,7 @@
\section{Implementation and Results}%
\label{sec:pdf_results}
The considerations of \cref{sec:pdf_basics} and \cref{sec:lab_xs} can
The considerations of \cref{sec:pdf_basics,sec:lab_xs} can
now be applied to obtain a cross section and histograms of observables
for the scattering of two protons into two photons. Because the PDF is
not available in closed form, event generation is the only viable way
@ -103,7 +103,7 @@ distributions are largely compatible with each other although there
discrepancies arise in regions with low event count (statistics),
which the the ratio plot exaggerators. The sherpa runcard utilized
here and the analysis used to produce the histograms can be found
in \cref{sec:ppruncard} and \cref{sec:ppanalysis}. When
in \cref{sec:ppruncard,sec:ppanalysis}. When
comparing \cref{fig:pdf-eta} with \cref{fig:histeta} it becomes
apparent, that the PDF has substantial influence on the resulting
distribution.