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latex/tex/abstract.tex
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@ -2,7 +2,7 @@
The analytical cross section for the parton-level \(\qqgg\) diphoton The analytical cross section for the parton-level \(\qqgg\) diphoton
process is calculated. With the application to this in mind, some process is calculated. With the application to this in mind, some
monte carlo integration and sampling methods are studied, implemented Monte Carlo integration and sampling methods are studied, implemented
and compared. Proton-Proton scattering is simulated on the partonic and compared. Proton-Proton scattering is simulated on the partonic
level through the use of parton density functions. Throughout, level through the use of parton density functions. Throughout,
obtained results are verified through comparison with the \sherpa\ obtained results are verified through comparison with the \sherpa\
@ -15,7 +15,7 @@ including parton showers, hadronization and multiple interactions at
Der analytische wirkungsquerschnitt fuer den \(\qqgg\) diphoton prozess Der analytische wirkungsquerschnitt fuer den \(\qqgg\) diphoton prozess
wird berechnet. Mit hinblick auf die Anwendung auf diesen Prozess wird berechnet. Mit hinblick auf die Anwendung auf diesen Prozess
werden monte carlo Methoden untersucht, implementiert und werden Monte Carlo Methoden untersucht, implementiert und
verglichen. Proton-Proton Streuung wird auf der partonischen ebene verglichen. Proton-Proton Streuung wird auf der partonischen ebene
mithilfe von parton-density Funktionen simuliert. Gewonnene resultate mithilfe von parton-density Funktionen simuliert. Gewonnene resultate
werden durchweg mit dem \sherpa\ event generator verifiziert. Zuletzt werden durchweg mit dem \sherpa\ event generator verifiziert. Zuletzt

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latex/tex/introduction.tex
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@ -4,11 +4,11 @@
Monte carlo methods have been and still are one of the most important Monte carlo methods have been and still are one of the most important
tools for theoretical calculations in particle physics. Be it for tools for theoretical calculations in particle physics. Be it for
validating the well established standard model or for making validating the well established standard model or for making
predictions about new theories, monte carlo simulations are the predictions about new theories, Monte Carlo simulations are the
crucial interface of theory and experimental data, making them crucial interface of theory and experimental data, making them
directly comparable. Furthermore horizontal scaling is almost trivial directly comparable. Furthermore horizontal scaling is almost trivial
to implement in monte carlo algorithms, making them well adapted to to implement in Monte Carlo algorithms, making them well adapted to
modern parallel computing. In this the thesis, the use of monte carlo modern parallel computing. In this the thesis, the use of Monte Carlo
methods will be traced through from simple integration to the methods will be traced through from simple integration to the
simulation of proton-proton scattering. simulation of proton-proton scattering.
@ -22,11 +22,11 @@ scope of this thesis. The differential and total cross section of this
process is being calculated in leading order in \cref{chap:qqgg} and process is being calculated in leading order in \cref{chap:qqgg} and
the obtained result is compared to the total cross section obtained the obtained result is compared to the total cross section obtained
with the \sherpa~\cite{Gleisberg:2008ta} event generator, used as with the \sherpa~\cite{Gleisberg:2008ta} event generator, used as
matrix element integrator. In \cref{chap:mc} some simple monte carlo matrix element integrator. In \cref{chap:mc} some simple Monte Carlo
methods are discussed, implemented and their results compared. First methods are discussed, implemented and their results compared. First
monte carlo integration is studies and the \vegas\ Monte Carlo integration is studies and the \vegas\
algorithm~\cite{Lepage:19781an} is implemented and algorithm~\cite{Lepage:19781an} is implemented and
evaluated. Subsequently monte carlo sampling methods are explored and evaluated. Subsequently Monte Carlo sampling methods are explored and
the output of \vegas\ is used to improve the sampling the output of \vegas\ is used to improve the sampling
efficiency. Histograms of observables are generated and compared to efficiency. Histograms of observables are generated and compared to
histograms from \sherpa using the \rivet~\cite{Bierlich:2019rhm} histograms from \sherpa using the \rivet~\cite{Bierlich:2019rhm}

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latex/tex/monte-carlo/integration.tex
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@ -9,7 +9,7 @@ probability density on
with a \(1\) in the fashion of \cref{eq:baseintegral}, the Integral with a \(1\) in the fashion of \cref{eq:baseintegral}, the Integral
of \(f\) over \(\Omega\) can be interpreted as the expected value of \(f\) over \(\Omega\) can be interpreted as the expected value
\(\EX{F/\Rho}\) of the random variable \(F/\Rho\) under the \(\EX{F/\Rho}\) of the random variable \(F/\Rho\) under the
distribution \(\rho\). This is the key to most monte carlo methods. distribution \(\rho\). This is the key to most Monte Carlo methods.
\begin{equation} \begin{equation}
\label{eq:baseintegral} \label{eq:baseintegral}
@ -77,9 +77,9 @@ The convergence of \cref{eq:approxexp} is not dependent on the
dimensionality of the integration volume as opposed to many other dimensionality of the integration volume as opposed to many other
numerical integration algorithms (trapezoid rule, Simpsons rule) that numerical integration algorithms (trapezoid rule, Simpsons rule) that
usually converge like \(N^{-\frac{k}{n}}\) with \(k\in\mathbb{N}\) as usually converge like \(N^{-\frac{k}{n}}\) with \(k\in\mathbb{N}\) as
opposed to \(N^{-\frac{k}{n}}\) with monte carlo. Because integrals in opposed to \(N^{-\frac{k}{n}}\) with Monte Carlo. Because integrals in
particle physics usually have a high dimensionality, monte carlo particle physics usually have a high dimensionality, Monte Carlo
integration is suitable there. When implementing monte carlo methods, integration is suitable there. When implementing Monte Carlo methods,
the random samples can be obtained through hardware or software random the random samples can be obtained through hardware or software random
number generators (RNGs). Most implementations utilize software RNGs number generators (RNGs). Most implementations utilize software RNGs
because supply pseudo-random numbers in a reproducible way, which because supply pseudo-random numbers in a reproducible way, which

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latex/tex/pdf/results.tex
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@ -97,7 +97,7 @@ stratified sampling (as discussed in \cref{sec:stratsamp-real}) and
the hit-or-miss method. Because the stratified sampling requires very the hit-or-miss method. Because the stratified sampling requires very
accurate upper bounds, they have been overestimated by accurate upper bounds, they have been overestimated by
\result{xs/python/pdf/overesimate}, which lowers the efficiency \result{xs/python/pdf/overesimate}, which lowers the efficiency
slightly but reduces bias. The monte carlo integrator was used to slightly but reduces bias. The Monte Carlo integrator was used to
estimate the location of the maximum in each hypercube and then this estimate the location of the maximum in each hypercube and then this
estimate was improved by gradient ascend\footnote{Which becomes estimate was improved by gradient ascend\footnote{Which becomes
problematic, when performed close to a cut.}. problematic, when performed close to a cut.}.

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latex/tex/qqgammagamma/comparison.tex
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@ -2,7 +2,7 @@
\label{sec:compsher} \label{sec:compsher}
The result obtained in \cref{sec:qqggcalc} shall now be verified by the The result obtained in \cref{sec:qqggcalc} shall now be verified by the
monte carlo event generator \sherpa{}~\cite{Gleisberg:2008ta}. To Monte Carlo event generator \sherpa{}~\cite{Gleisberg:2008ta}. To
facilitate this, an expression for the total cross section for a range facilitate this, an expression for the total cross section for a range
of \(\theta\) or \(\eta\) has to be obtained. Using the golden rule of \(\theta\) or \(\eta\) has to be obtained. Using the golden rule
for \(2\rightarrow 2\) processes and observing that the initial and for \(2\rightarrow 2\) processes and observing that the initial and
@ -86,7 +86,7 @@ an interval of \([-\eta, \eta]\), is dominated by the linear
contributions in \cref{eq:total-crossec} and would result in an contributions in \cref{eq:total-crossec} and would result in an
infinity if no cut on \(\eta\) would be made. Choosing infinity if no cut on \(\eta\) would be made. Choosing
\(\eta\in [-2.5,2.5]\) and \(\ecm=\SI{100}{\giga\electronvolt}\) the \(\eta\in [-2.5,2.5]\) and \(\ecm=\SI{100}{\giga\electronvolt}\) the
process was monte carlo integrated in \sherpa\ using the runcard process was Monte Carlo integrated in \sherpa\ using the runcard
in \cref{sec:qqggruncard}. This runcard describes the exact same (first in \cref{sec:qqggruncard}. This runcard describes the exact same (first
order) process as the calculated cross section. order) process as the calculated cross section.