add some monte carlo theory

latex/document.tex

@@ -3,7 +3,7 @@ twoside=false,toc=listof,toc=bibliography,fleqn,leqno,
 captions=nooneline,captions=tableabove,english]{scrbook}
 
 \usepackage{hirostyle}
-%\addbibresource{thesis.bib}
+\addbibresource{thesis.bib}
 
 \title{Title}
 \author{Valentin Boettcher}
@@ -16,15 +16,15 @@ captions=nooneline,captions=tableabove,english]{scrbook}
 \input{./tex/qqgammagamma/calculation.tex}
 \input{./tex/qqgammagamma/comparison.tex}
 
-\listoffigures
-\listoftables
+\input{./tex/monte-carlo.tex}
+\input{./tex/monte-carlo/theory.tex}
+
 
 \appendix
 \input{./tex/appendix.tex}
 
-% Bibliography:
-
 \clearpage
-% \input{./tex/mybibliography.tex}
-
+\listoffigures
+\listoftables
+\printbibliography
 \end{document}

latex/hirostyle.sty

@@ -10,7 +10,7 @@
 \usepackage{graphicx, booktabs, float, scrhack}
 \usepackage{amsmath,amssymb}
 \usepackage[automark]{scrlayer-scrpage}
-%\usepackage[backend=biber,style=verbose,sortcase=false,language=english]{biblatex}
+\usepackage[backend=biber, language=english, style=phys]{biblatex}
 \usepackage{siunitx}
 \usepackage[pdfencoding=auto]{hyperref} % load late
 % \usepackage[activate={true,nocompatibility},final,tracking=true,spacing=true,factor=1100,stretch=10,shrink=10]{microtype}
@@ -57,9 +57,10 @@ labelformat=brace, position=top]{subcaption}
 \captionsetup{justification=centering}
 
 %% Labels
-\labelformat{section}{section #1}
-\labelformat{figure}{figure #1}
-\labelformat{table}{table #1}
+\labelformat{chapter}{chapter~#1}
+\labelformat{section}{section~#1}
+\labelformat{figure}{figure~#1}
+\labelformat{table}{table~#1}
 
 % Macros
 
@@ -93,3 +94,13 @@ labelformat=brace, position=top]{subcaption}
 
 %% Notes on Equations
 \newcommand{\shorteqnote}[1]{ &  & \text{\small\llap{#1}}}
+
+%% Sherpa
+\newcommand{\sherpa}{\texttt{Sherpa}}
+
+%% Expected Value and Variance
+\newcommand{\EX}[1]{\operatorname{E}\qty[#1]}
+\newcommand{\VAR}[1]{\operatorname{VAR}\qty[#1]}
+
+%% Uppercase Rho
+\newcommand{\Rho}{P}

latex/makefile

@@ -7,4 +7,4 @@ thesis: document.tex
 .PHONY: clean
 
 clean:
-	rm -f $(OUTDIR)/*
+	rm -rf $(OUTDIR)/*

latex/tex/monte-carlo.tex

@@ -0,0 +1,12 @@
+%%% Local Variables: ***
+%%% mode: latex ***
+%%% TeX-master: "../document.tex"  ***
+%%% End: ***
+
+\chapter{Survey of Elementary Monte-Carlo Methods}%
+\label{chap:mc}
+
+As monte-carlo methods for multidimensional integration and sampling
+of probability distributions are central tools of modern particle
+physics, their basics shall be studied.  Some simple algorithms will
+be implemented and applied to the results from~\ref{chap:qqgg}.

latex/tex/monte-carlo/theory.tex

@@ -0,0 +1,99 @@
+%%% Local Variables: ***
+%%% mode: latex ***
+%%% TeX-master: "../../document.tex"  ***
+%%% End: ***
+
+\section{Monte-Carlo Integration}
+\label{sec:mctheory}
+
+Consider a function
+\(f: \vb{x}\in\Omega\subset\mathbb{R}^n\mapsto\mathbb{R}\) and a
+probability density on \(\Omega\)
+\(\rho: \vb{x}\in\mathbb{R}^n\mapsto\mathbb{R}_{\geq 0}\) with
+\(\int_{\Omega}\rho(\vb{x})\dd{\vb{x}} = 1\).  By multiplying \(f\)
+with a \(1\) in the fashion of~\eqref{eq:baseintegral}, the Integral
+of \(f\) over \(\Omega\) can be interpreted as the expected value
+\(\EX{F/\Rho}\) of the random variable \(F/\Rho\)
+under the distribution \(\rho\).
+
+\begin{equation}
+  \label{eq:baseintegral}
+  I = \int_\Omega f(\vb{x}) \dd{\vb{x}} = \int_\Omega
+  \qty[\frac{f(\vb{x})}{\rho(\vb{x})}] \rho(\vb{x}) \dd{\vb{x}} = \EX{\frac{F}{\Rho}}
+\end{equation}
+
+The expected value \(\EX{F/\Rho}\) can be approximate by taking the
+mean of \(F/\Rho\) with \(N\) finite samples
+\(\{\vb{x}_i\}_{i\in\overline{1,N}}\sim\rho\) (distributed according to
+\(\rho\)), where \(N\) is usually a very large integer.
+
+\begin{equation}
+  \label{eq:approxexp}
+  \EX{\frac{F}{\Rho}} \approx
+  \frac{1}{N}\sum_{i=1}^N\frac{f(\vb{x_i})}{\rho(\vb{x_i})}
+  \xrightarrow{N\rightarrow\infty} I
+\end{equation}
+
+The convergence of~\eqref{eq:approxexp} is due to the nature of the
+expected value~\eqref{eq:evalue-mean} and
+variance~\eqref{eq:variance-mean} of the mean
+\(\overline{X} = \frac{1}{N}\sum_i X_i\) of \(N\) uncorrelated random
+variables \(\{X_i\}_{i\in\overline{1,N}}\) with the same distribution,
+expected value \(\EX{X_i}=\mathbb{E}\) and variance
+\(\sigma_i^2 = \sigma^2\).
+
+\begin{align}
+  \EX{\overline{X}} = \frac{1}{N}\sum_i\EX{X_i} = \mathbb{E} \label{eq:evalue-mean}\\
+  \sigma^2_{\overline{X}} = \sum_i\frac{\sigma_i^2}{N^2} =
+                            \frac{\sigma^2}{N}  \label{eq:variance-mean}
+\end{algin}
+
+Evidently \(\frac{\sigma^2}{N}\xrightarrow{N\rightarrow\infty} 0\)
+thus the~\eqref{eq:approxexp} really converges to \(I\). For finite
+\(N\) value of~\eqref{eq:approxexp} varies around \(I\) with the
+variance \(\VAR{F/\Rho}\cdot N^{-1}\) as in~\eqref{eq:varI}.
+
+\begin{align}
+  \VAR{\frac{F}{\Rho}} &= \int_\Omega \qty[I -
+  \frac{f(\vb{x})}{\rho(\vb{x})}]^2 \rho({\vb{x}}) \dd{\vb{x}} =
+  \int_\Omega \qty[\qty(\frac{f(\vb{x})}{\rho(\vb{x})})^2 -
+  I^2]\rho({\vb{x}}) \dd{\vb{x}}   \label{eq:varI}
+ \\
+  &\approx \frac{1}{N - 1}\sum_i \qty[I -
+  \frac{f(\vb{x_i})}{\rho(\vb{x_i})}]^2  \label{eq:varI-approx}
+\end{align}
+
+The name of the game is thus to reduce \(\VAR{F/\Rho}\) to speed up
+the convergence of~\eqref{eq:approxexp} and achieve higher accuracy
+with fewer function evaluations.
+
+The simplest choice for \(\rho\) is clearly given
+by~\eqref{eq:simplep}, the uniform distribution.
+
+\begin{equation}
+  \label{eq:simplep}
+  \rho(\vb{x}) = \frac{1}{\int_{\Omega}1\dd{\vb{x'}}} =
+  \frac{1}{\abs{\Omega}}
+\end{equation}
+
+With this distribution~\eqref{eq:approxexp}
+becomes~\eqref{eq:approxexp-uniform}. In other words, \(I\) is just
+the mean of \(f\) in \(\Omega\), henceforth
+called \(\bar{f}\), multiplied with the volume.
+
+\begin{equation}
+  \label{eq:approxexp-uniform}
+  \EX{\frac{F}{\Rho}} \approx
+  \frac{\abs{\Omega}}{N}\sum_{i=1}^N f(\vb{x_i}) = \abs{\Omega}\cdot\bar{f}
+\end{equation}
+
+The variance \(\VAR{I}=\VAR{F/\Rho}\) is now given
+by~\ref{eq:approxvar-I}. Note that the factor \(\abs{\omega}\) gets
+squared when approximating the integral to the sum.
+
+\begin{equation}
+  \label{eq:approxvar-I}
+  \VAR{I} = \abs{\Omega}\int_\Omega f(\vb{x})^2 -
+  I^2 \dd{\vb{x}} \equiv \abs{\Omega}\cdot\sigma_f^2 \approx
+  \frac{\abs{\Omega}^2}{N-1}\sum_{i}\qty[f(\vb{x}_i) - \bar{f}]^2
+\end{equation}

latex/tex/qqgammagamma.tex

@@ -1,6 +1,11 @@
+%%% Local Variables: ***
+%%% mode:latex ***
+%%% TeX-master: "../document.tex"  ***
+%%% End: ***
+
 \chapter{Quark-Antiquark Annihilation into
   two Photons}%
-\label{sec:qqgg}
+\label{chap:qqgg}
 
 Consider the scattering reaction \(\qqgg\). The first order expansion
 of this process is being described by the Feynman diagrams

latex/tex/qqgammagamma/calculation.tex

@@ -1,3 +1,8 @@
+%%% Local Variables: ***
+%%% mode:latex ***
+%%% TeX-master: "../../document.tex"  ***
+%%% End: ***
+
 \section{Calulation of the Cross Section to first Order}%
 \label{sec:qqggcalc}
 

latex/tex/qqgammagamma/comparison.tex

@@ -1,23 +1,30 @@
-\section{Discussion and Comparison with Sherpa}%
+%%% Local Variables: ***
+%%% mode:latex ***
+%%% TeX-master: "../../document.tex"  ***
+%%% End: ***
+
+\section{Discussion and Comparison with \sherpa}%
 \label{sec:compsher}
 
 The result obtained in~\ref{sec:qqggcalc} shall now be verified by
-monte-carlo in \verb|Sherpa|. To facilitate this, an expression for
-the total cross section for a range of \(\theta\) or \(\eta\) has to
-be obtained. Using the golden rule for \(2\rightarrow 2\) processes
-and observing that the initial and final momenta are equal
-(\(p_i=p_f\)) and \(g=\sqrt{4\pi\alpha}\), the
-result~\eqref{eq:crossec} arises.
+monte-carlo in \sherpa{}~\cite{Gleisberg:2008ta}. To facilitate this, an
+expression for the total cross section for a range of \(\theta\) or
+\(\eta\) has to be obtained. Using the golden rule for
+\(2\rightarrow 2\) processes and observing that the initial and final
+momenta are equal (\(p_i=p_f\)) and \(g=\sqrt{4\pi\alpha}\), the
+result~\eqref{eq:crossec} arises. The differential cross section is
+also given in terms of the pseudo-rapidity in~\ref{eq:xs-eta}.
 
 An additional factor of \(\frac{1}{2}\) comes in due to there being
 two identical photons in the final state.
-\begin{equation}
-  \label{eq:crossec}
-  \dv{\sigma}{\Omega} =
-  \frac{1}{2}\frac{1}{(8\pi)^2}\cdot\frac{\abs{\mathcal{M}}^2}{\ecm^2}\cdot\frac{\abs{p_f}}{\abs{p_i}}
-  = \overbrace{\frac{\alpha^2Q^4}{6\ecm^2}}^{\mathfrak{C}}\frac{1+\cos^2(\theta)}{\sin^2(\theta)}
-\end{equation}
-
+\begin{align}
+  \dv{\sigma}{\Omega} &=
+                        \frac{1}{2}\frac{1}{(8\pi)^2}\cdot\frac{\abs{\mathcal{M}}^2}{\ecm^2}\cdot\frac{\abs{p_f}}{\abs{p_i}}
+                        =
+                        \underbrace{\frac{\alpha^2Q^4}{6\ecm^2}}_{\mathfrak{C}}\frac{1+\cos^2(\theta)}{\sin^2(\theta)}\label{eq:crossec}
+  \\
+  \dv{\sigma}{\eta} &= \frac{\alpha^2Q^4}{6\ecm^2}\cdot\qty(\tanh(\eta)^2 + 1)\label{eq:xs-eta}
+\end{align}
 
 \begin{figure}[ht]
   \centering
@@ -43,12 +50,16 @@ also~\ref{fig:diffxs}) is divergent for angles near zero or
 divergence. Because \(m=0\) is the limit for
 \(\ecm\rightarrow\infty\), the cross section would still have strong
 peaks for angles near \(0,\pi\) at high energies so that the result is
-not altogether nonphysical. It is clearly symmetric around
+not altogether nonphysical. The divergence of the cross section itself
+is also not, the problem here, because it can be transformed into a
+form where the divergence does not occur (see~\eqref{eq:xs-eta}).
+
+The differential cross section is clearly symmetric around
 \(\theta=\frac{pi}{2}\) as was to be expected, because the photons are
 indistinguishable. To compare the cross section to experiment and to
 simulation an interval around \(\theta=\frac{\pi}{2}\) has to be
-chosen, where the first order, mass-less approximation may yield
-sensible results.
+chosen, where the first order, mass-less approximation may yield a
+useful result.
 
 The total cross section in such an interval, given by
 integrating~\eqref{eq:crossec} for \(\theta\in [\theta_1, \theta_2]\)
@@ -73,10 +84,12 @@ an interval of \([-\eta, \eta]\), is dominated by the linear
 contributions in~\ref{eq:total-crossec} and would result in an
 infinity if no cut on \(\eta\) would be made.  Choosing
 \(\eta\in [-2.5,2.5]\) 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~\ref{sec:qqggruncard}. This runcard describes the exact same (first
 order) process as the calculated cross section.
 
-Sherpa yields \result{xs/sherpa_xs}. Plugging the same parameters
-into~\eqref{eq:total-crossec} gives \result{xs/xs} which is
-within the uncertainty range of the Sherpa value.
+The monte carlo integration in \sherpa\ yields
+\result{xs/sherpa_xs}. Plugging the same parameters
+into~\eqref{eq:total-crossec} gives \result{xs/xs} which is within the
+uncertainty range of the \sherpa\ value. This verifies the result for
+the total cross section.

latex/thesis.bib

@@ -0,0 +1,38 @@
+@article{Gleisberg:2008ta,
+    author = "Gleisberg, T. and Hoeche, Stefan. and Krauss, F. and Schonherr, M. and Schumann, S. and Siegert, F. and Winter, J.",
+    archivePrefix = "arXiv",
+    doi = "10.1088/1126-6708/2009/02/007",
+    eprint = "0811.4622",
+    journal = "JHEP",
+    pages = "007",
+    primaryClass = "hep-ph",
+    reportNumber = "FERMILAB-PUB-08-477-T, SLAC-PUB-13420, ZU-TH-17-08, DCPT-08-138, IPPP-08-69, EDINBURGH-2008-30, MCNET-08-14",
+    title = "{Event generation with SHERPA 1.1}",
+    volume = "02",
+    year = "2009"
+}
+@article{Krauss:2001iv,
+    author = "Krauss, F. and Kuhn, R. and Soff, G.",
+    archivePrefix = "arXiv",
+    doi = "10.1088/1126-6708/2002/02/044",
+    eprint = "hep-ph/0109036",
+    journal = "JHEP",
+    pages = "044",
+    reportNumber = "CAVENDISH-HEP-01-11",
+    title = "{AMEGIC++ 1.0: A Matrix element generator in C++}",
+    volume = "02",
+    year = "2002"
+}
+@article{Gleisberg:2008fv,
+    author = "Gleisberg, Tanju and Hoeche, Stefan",
+    archivePrefix = "arXiv",
+    doi = "10.1088/1126-6708/2008/12/039",
+    eprint = "0808.3674",
+    journal = "JHEP",
+    pages = "039",
+    primaryClass = "hep-ph",
+    reportNumber = "SLAC-PUB-13232, IPPP-08-31, DCPT-08-62, MCNET-08-08",
+    title = "{Comix, a new matrix element generator}",
+    volume = "12",
+    year = "2008"
+}