mirror of
https://github.com/vale981/bachelor_thesis
synced 2025-03-06 01:51:38 -05:00
602 lines
19 KiB
TeX
602 lines
19 KiB
TeX
\documentclass[10pt, aspectratio=169]{beamer}
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\usepackage{siunitx}
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\usepackage[T1]{fontenc}
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\usepackage{booktabs}
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\usepackage[backend=biber]{biblatex}
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\usepackage{mathtools,amssymb}
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\usepackage{physics}
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\usepackage{slashed}
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\usepackage{tikz}
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\usepackage{tikz-feynman}
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\usepackage[list=true, font=small,
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labelformat=brace, position=top]{subcaption}
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% \setbeameroption{show notes on second screen} %
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\addbibresource{thesis.bib}
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\graphicspath{ {figs/} }
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\usepackage{animate}
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\newfontfamily\DejaSans{DejaVu Sans}
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\usetheme{Antibes}
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% \usepackage{eulerpx}
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\usepackage{ifdraft}
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% \usefonttheme[onlymath]{serif}
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\setbeamertemplate{itemize items}[default]
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\setbeamertemplate{enumerate items}[default]
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\AtBeginSection[]
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{
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\begin{frame}
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\tableofcontents[currentsection]
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\end{frame}
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}
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\setbeamertemplate{footline}[frame number]
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\setbeamertemplate{note page}[plain]
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\setbeamertemplate{bibliography item}{\insertbiblabel} %% Remove book
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%% symbol from references and add
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%% number
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\sisetup{separate-uncertainty = true}
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% Macros
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%% qqgg
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\newcommand{\qqgg}[0]{q\bar{q}\rightarrow\gamma\gamma}
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%% ppgg
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\newcommand{\ppgg}[0]{pp\rightarrow\gamma\gamma}
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%% Momenta and Polarization Vectors convenience
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\DeclareMathOperator{\ps}{\slashed{p}}
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\DeclareMathOperator{\pe}{\varepsilon}
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\DeclareMathOperator{\pes}{\slashed{\pe}}
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\DeclareMathOperator{\pse}{\varepsilon^{*}}
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\DeclareMathOperator{\pses}{\slashed{\pe}^{*}}
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%% Spinor convenience
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\DeclareMathOperator{\us}{u}
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\DeclareMathOperator{\usb}{\bar{u}}
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\DeclareMathOperator{\vs}{v}
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\DeclareMathOperator*{\vsb}{\overline{v}}
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%% Center of Mass energy
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\DeclareMathOperator{\ecm}{E_{\text{CM}}}
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%% area hyperbolicus
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\DeclareMathOperator{\artanh}{artanh}
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\DeclareMathOperator{\arcosh}{arcosh}
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%% Fast Slash
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\let\sl\slashed
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%% Notes on Equations
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\newcommand{\shorteqnote}[1]{ & & \text{\small\llap{#1}}}
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%% Typewriter Macros
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\newcommand{\sherpa}{\texttt{Sherpa}}
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\newcommand{\rivet}{\texttt{Rivet}}
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\newcommand{\vegas}{\texttt{VEGAS}}
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\newcommand{\lhapdf}{\texttt{LHAPDF6}}
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\newcommand{\scipy}{\texttt{scipy}}
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%% Sherpa Versions
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\newcommand{\oldsherpa}{\texttt{2.2.10}}
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\newcommand{\newsherpa}{\texttt{3.0.0} (unreleased)}
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%% Special Names
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\newcommand{\lhc}{\emph{LHC}}
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%% Expected Value and Variance
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\newcommand{\EX}[1]{\operatorname{E}\qty[#1]}
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\newcommand{\VAR}[1]{\operatorname{VAR}\qty[#1]}
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%% Uppercase Rho
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\newcommand{\Rho}{P}
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%% Transverse Momentum
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\newcommand{\pt}[0]{p_\mathrm{T}}
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%% Sign Function
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\DeclareMathOperator{\sign}{sgn}
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%% Stages
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\newcommand{\stone}{\texttt{LO}}
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\newcommand{\sttwo}{\texttt{LO+PS}}
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\newcommand{\stthree}{\texttt{LO+PS+pT}}
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\newcommand{\stfour}{\texttt{LO+PS+pT+Hadr.}}
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\newcommand{\stfive}{\texttt{LO+PS+pT+Hadr.+MI}}
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%% GeV
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\newcommand{\gev}[1]{\SI{#1}{\giga\electronvolt}}
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%% Including plots
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\newcommand{\plot}[2][,]{%
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\includegraphics[draft=false,#1]{./figs/#2.pdf}}
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\newcommand{\rivethist}[2][,]{%
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\includegraphics[draft=false,width=\textwidth,#1]{./figs/rivet/#2.pdf}}
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%% Including Results
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\newcommand{\result}[1]{\input{./results/#1}\!}
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\title{A Study of Monte Carlo Methods and their Application to
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Diphoton Production at the Large Hadron Collider}
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\subtitle{Bachelorvortrag}
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\author{Valentin Boettcher}
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\beamertemplatenavigationsymbolsempty
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\begin{document}
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\hypersetup{pageanchor=false}
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\maketitle
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\hypersetup{pageanchor=true} \pagenumbering{arabic}
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\begin{frame}
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\tableofcontents
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\end{frame}
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\section{Introduction}
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\begin{frame}{Motivation}
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\begin{block}{Monte Carlo Methods}
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\begin{itemize}
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\item (most) important numerical tools (not just) in particle
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physics
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\item crucial interface of theory and experiment
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\item enable precision predictions within and beyond SM
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\end{itemize}
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\end{block}
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\pause
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\begin{block}{Diphoton Process \(\qqgg\)}
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\begin{itemize}
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\item simple QED process, calculable by hand
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\item higgs decay channel: \(H\rightarrow \gamma\gamma\)
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\begin{itemize}
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\item instrumental in its
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discovery~\cite{Aad:2012tfa,Chatrchyan:2012ufa}
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\end{itemize}
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\item dihiggs decay \(HH\rightarrow b\bar{b}\gamma\gamma\)
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\begin{itemize}
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\item process of recent interest~\cite{aaboud2018:sf}
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\end{itemize}
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\end{itemize}
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\end{block}
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\end{frame}
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\section{Calculation of the \(\qqgg\) Cross Section}
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\subsection{Approach}
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\begin{frame}
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\begin{columns}[T]
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\begin{column}{.5\textwidth}
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\begin{figure}[ht]
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\centering
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\begin{subfigure}[c]{.28\textwidth}
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\centering
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\begin{tikzpicture}[scale=.6]
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\begin{feynman}
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\diagram [small,horizontal=i2 to a] { i2
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[particle=\(q\)] -- [fermion, momentum=\(p_2\)] a --
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[fermion, reversed momentum=\(q\)] b, i1
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[particle=\(\bar{q}\)] -- [anti fermion,
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momentum'=\(p_1\)] b, i2 -- [opacity=0] i1, a --
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[photon, momentum=\(p_3\)] f1 [particle=\(\gamma\)], b
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-- [photon, momentum'=\(p_4\)] f2
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[particle=\(\gamma\)], f1 -- [opacity=0] f2, };
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\end{feynman}
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\end{tikzpicture}
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\subcaption{u channel}
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\end{subfigure}
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\begin{subfigure}[c]{.28\textwidth}
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\centering
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\begin{tikzpicture}[scale=.6]
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\begin{feynman}
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\diagram [small,horizontal=i2 to a] { i2
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[particle=\(q\)] -- [fermion, momentum=\(p_2\)] a --
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[fermion, reversed momentum'=\(q\)] b, i1
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[particle=\(\bar{q}\)] -- [anti fermion,
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momentum'=\(p_1\)] b, i2 -- [opacity=0] i1, a --
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[draw=none] f2 [particle=\(\gamma\)], b -- [draw=none]
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f1 [particle=\(\gamma\)], f1 -- [opacity=0] f2, };
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\diagram* { (a) -- [photon] (f1), (b) -- [photon] (f2),
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};
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\end{feynman}
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\end{tikzpicture}
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\subcaption{\label{fig:qqggfeyn2}t channel}
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\end{subfigure}
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%
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\caption{Leading order diagrams for \(\qqgg\).}%
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\label{fig:qqggfeyn}
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\end{figure}
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\end{column}
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\pause
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\begin{column}{.5\textwidth}
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\begin{block}{Task: calculate \(\abs{\mathcal{M}}^2\)}
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\begin{enumerate}[<+->]
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\item translate diagrams to matrix elements
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\item use Casimir's trick to average over spins
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\item use completeness relation to sum over photon
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polarizations
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\item use trace identities to compute the absolute square
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\item simplify with trigonometric identities
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\end{enumerate}
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\end{block}
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\pause Here: Quark masses neglected.
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\end{column}
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\end{columns}
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\end{frame}
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\subsection{Result}
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\begin{frame}
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\begin{equation}
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\label{eq:averagedm_final}
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\langle\abs{\mathcal{M}}^2\rangle = \frac{4}{3}(gZ)^4
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\cdot\frac{1+\cos^2(\theta)}{\sin^2(\theta)} =
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\frac{4}{3}(gZ)^4\cdot(2\cosh(\eta) - 1)
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\end{equation}
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%
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\pause
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\[\overset{\text{Golden Rule}}{\implies}\]
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\pause
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\begin{equation}
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\label{eq:crossec}
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\dv{\sigma}{\Omega} =
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\frac{1}{2}\frac{1}{(8\pi)^2}\cdot\frac{\abs{\mathcal{M}}^2}{\ecm^2}\cdot\frac{\abs{p_f}}{\abs{p_i}}
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= \underbrace{\frac{\alpha^2Z^4}{6\ecm^2}}_{\mathfrak{C}}\frac{1+\cos^2(\theta)}{\sin^2(\theta)}
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\end{equation}
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\pause
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\begin{figure}[ht]
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\centering
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\begin{minipage}[c]{0.3\textwidth}
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\plot[scale=.6]{xs/diff_xs}
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\end{minipage}
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\begin{minipage}[c]{0.3\textwidth}
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\caption{The differential cross section as a function of the
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polar angle \(\theta\).}
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\end{minipage}
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\end{figure}
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\end{frame}
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\begin{frame}{Comparison with \sherpa~\cite{Bothmann:2019yzt}}
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\begin{itemize}
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\item<1-> choose \result{xs/python/eta} and \result{xs/python/ecm}
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and integrate XS
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\begin{equation}
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\label{eq:total-crossec}
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\sigma = {\frac{\pi\alpha^2Z^4}{3\ecm^2}}\cdot\qty[\tanh(\eta_2) - \tanh(\eta_1) + 2(\eta_1
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- \eta_2)]
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\end{equation}
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\item<2-> analytical result: \result{xs/python/xs}
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\item<3-> compatible with \sherpa: \result{xs/python/xs_sherpa}
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\end{itemize}
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\begin{figure}[ht]
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\centering
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\begin{minipage}[c]{0.3\textwidth}
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\plot[scale=.5]{xs/total_xs}
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\end{minipage}
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\begin{minipage}[c]{0.3\textwidth}
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\caption{\label{fig:totxs} The cross section of the process for
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a pseudo-rapidity integrated over \([-\eta, \eta]\).}
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\end{minipage}
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\end{figure}
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\end{frame}
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\section{Monte Carlo Methods}
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\note[itemize]{
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\item Gradually bring in knowledge through distribution. }
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\begin{frame}
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\begin{block}{Basic Ideas}
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\begin{itemize}
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\item<+-> Given some unknown function
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\(f\colon \vb{x}\in\Omega\subset\mathbb{R}^n\mapsto\mathbb{R}\)
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\ldots
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\item<+-> \ldots\ how do we answer questions about \(f\)?
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\end{itemize}
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\;\;\onslide<+->{\(\implies\) Sample it at random points.}
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\end{block}
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\pause
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\begin{block}{Main Applications}
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\begin{enumerate}
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\item<+-> integrate \(f\) over some volume \(\Omega\)
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\item<+-> treat \(f\) as distribution and take random samples
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\end{enumerate}
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\end{block}
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\end{frame}
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\subsection{Integration}
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\note[itemize]{
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\item omitting details (law of big numbers, central limit theorem)
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\item at least three angles of attack
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\item some sort of importance sampling, volume: stratified sampling }
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\begin{frame}
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\begin{itemize}
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\item<+-> we have:
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\(f\colon \vb{x}\in\Omega\subset\mathbb{R}^n\mapsto\mathbb{R}\)
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and \(\rho\colon \vb{x}\in\Omega\mapsto\mathbb{R}_{> 0}\) with
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\(\int_{\Omega}\rho(\vb{x})\dd{\vb{x}} = 1\).
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\item<+-> we seek:
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\begin{equation}
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\label{eq:baseintegral}
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I = \int_\Omega f(\vb{x}) \dd{\vb{x}}
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\onslide<+->{= \int_\Omega
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\qty[\frac{f(\vb{x})}{\rho(\vb{x})}] \rho(\vb{x}) \dd{\vb{x}} = \EX{\frac{F}{\Rho}}}
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\end{equation}
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\item<+-> numeric approximation:
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\begin{equation}
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\label{eq:approxexp}
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\EX{\frac{F}{\Rho}} \approx
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\frac{1}{N}\sum_{i=1}^N\frac{f(\vb{x_i})}{\rho(\vb{x_i})}
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\xrightarrow{N\rightarrow\infty} I
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\end{equation}
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\item<+-> error approximation:
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\begin{align}
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\sigma_I^2 &= \frac{\textcolor<+->{red}{\sigma^2}}{\textcolor<.->{blue}{N}} \\
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\sigma^2 &= \VAR{\frac{F}{\Rho}} = \int_{\textcolor<+(3)->{blue}{\Omega}} \qty[I -
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\frac{f(\vb{x})}{\textcolor<+->{blue}{\rho(\vb{x})}}]^2
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\textcolor<.->{blue}{\rho(\vb{x})} \textcolor<+->{blue}{\dd{\vb{x}}} \approx \frac{1}{N - 1}\sum_i \qty[I -
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\frac{f(\vb{x_i})}{\rho(\vb{x_i})}]^2 \label{eq:varI-approx}
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\end{align}
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\end{itemize}
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\end{frame}
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\begin{frame}{Change of Variables}
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Choose \(\rho(\vb{x}) = \frac{1}{\abs{\Omega}}\)
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\onslide<2->{\(\implies I=\frac{\abs{\Omega}}{N}\sum_{i=1}^N
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f(\vb{x_i})=\abs{\Omega}\cdot\bar{f}\) and
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\(\VAR{\frac{F}{P}}\approx\frac{\abs{\Omega}^2}{N-1}\sum_{i}\qty[f(\vb{x}_i)
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- \bar{f}]^2\)}
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\begin{block}{Results}
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\begin{itemize}
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\item<3-> integrating \(\dv{\sigma}{\theta}\) with target error of
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\(\sigma = \SI{1e-3}{\pico\barn}\) takes
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\result{xs/python/xs_mc_N} samples
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\item<4-> integrating \(\dv{\sigma}{\eta}\) takes just
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\result{xs/python/xs_mc_eta_N} samples
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\end{itemize}
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\end{block}
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\begin{figure}[hb]
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\centering \onslide<3->{
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\begin{subfigure}[c]{.4\textwidth}
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\centering \plot[scale=.6]{xs/xs_integrand}
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\end{subfigure}
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} \onslide<4->{
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\begin{subfigure}[c]{.4\textwidth}
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\centering \plot[scale=.6]{xs/xs_integrand_eta}
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\end{subfigure}
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}
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\end{figure}
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\end{frame}
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\note[itemize]{
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\item proposed by G. Peter Lepage (slac) 1976
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\item own implementation!!!
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}
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\begin{frame}{\vegas\ Algorithm \cite{Lepage:19781an}}
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\begin{columns}
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\begin{column}{.5\textwidth}
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\begin{block}{Idea}
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\begin{enumerate}
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\item subdivide integration volume into grid, take equal
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number of samples in each hypercube \(\iff\) define \(\rho\)
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as step function
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\item iteratively approximate optimal \(\rho = f(\vb{x})/I\)
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with step function
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\item this is quite efficient when \(n\geq 4\)
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\end{enumerate}
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\end{block}
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\pause
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\begin{block}{Result}
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Total function evaluations:
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\result{xs/python/xs_mc_θ_vegas_N}\\
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(for same accuracy as before)
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\end{block}
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\end{column}
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\begin{column}{.5\textwidth}
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\begin{figure}[ht]
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\centering \plot[scale=.6]{xs/xs_integrand_vegas}
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\caption{\(2\pi\dv{\sigma}{\theta}\) scaled to increments
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found by \vegas}
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\end{figure}
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\end{column}
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\end{columns}
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\end{frame}
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\subsection{Sampling}
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\note[itemize]{
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\item prop. to density
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\item generalization to n dim is easy
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\item idea -> cumulative propability the same
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}
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\begin{frame}
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\begin{itemize}[<+->]
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\item we have: \(f\colon x\in\Omega\mapsto\mathbb{R}_{>0}\)
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(choose \(\Omega = [0, 1]\)) and uniformly random samples \(\{x_i\}\)
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\item we seek: a sample \(\{y_i\}\) distributed according to \(f\)
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\end{itemize}
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\begin{block}<+->{Basic Idea}
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\begin{itemize}[<+->]
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\item<.-> let \(x\) be sample of uniform distribution, solve
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\[\int_{0}^{y}f(x')\dd{x'} = x\cdot\int_0^1f(x')\dd{x'} =
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x\cdot A\] for y to obtain sample of \(f/A\)
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\item let \(F\) be the antiderivative of \(f\), then
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\(y=F^{-1}(x\cdot A + F(0))\)
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\begin{itemize}
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\item sometimes analytical form available
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\item otherwise tackle that numerically
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\end{itemize}
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\end{itemize}
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\end{block}
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\end{frame}
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\begin{frame}{Hit or Miss}
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\centering
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\animategraphics[loop,scale=.4,autoplay,palindrome]{5}{pi/pi-}{0}{9}
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\end{frame}
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\begin{frame}{Hit or Miss}
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\begin{block}{Basic Idea}
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\begin{itemize}[<+->]
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\item take samples \({x_i}\) distributed according to \(g/B\),
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where \(B=\int_0^1g(x)\dd{x}\) and
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\(\forall x\in\Omega\colon g(x)\geq f(x)\)
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\item accept each sample with the probability~\(f(x_i)/g(x_i)\)
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(importance sampling)
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\item total probability of accepting a sample: \(\mathfrak{e} =
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A/B < 1\) (efficiency)
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\item simplest choice \(g=\max_{x\in\Omega}f(x)=f_{\text{max}}\)
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\item again: efficiency gain through reduction of variance
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\end{itemize}
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\end{block}
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\begin{block}<+->{Results with \(g=f_{\text{max}}\) }
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\begin{itemize}[<+->]
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\item<.-> sampling \(\dv{\sigma}{\cos\theta}\):
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\result{xs/python/naive_th_samp}
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\item sampling \(\dv{\sigma}{\cos\theta}\):
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\result{xs/python/eta_eff}
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|
\end{itemize}
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|
\end{block}
|
|
\end{frame}
|
|
|
|
\begin{frame}{Hit or Miss}
|
|
\begin{columns}
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|
\begin{column}{.4\textwidth}
|
|
\begin{block}<+->{Results with \(g=a\cdot x^2 + b\)} Modest
|
|
efficiency gain: \result{xs/python/tuned_th_samp}
|
|
\end{block}
|
|
\begin{itemize}
|
|
\item<+-> Of course, we can use \vegas\ to provide a better \(g\).
|
|
\end{itemize}
|
|
\end{column}
|
|
\begin{column}{.6\textwidth}
|
|
\begin{figure}[ht]
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|
\centering \plot[scale=.8]{xs_sampling/upper_bound}
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|
\caption{The distribution \(\dv{\sigma}{\cos\theta}\) and an
|
|
upper bound of the form \(a + b\cdot x^2\).}
|
|
\end{figure}
|
|
\end{column}
|
|
\end{columns}
|
|
\end{frame}
|
|
|
|
\begin{frame}{Stratified Sampling}
|
|
\begin{block}{Basic Idea}
|
|
\begin{itemize}
|
|
\item subdivide sampling volume \(\Omega\) into \(K\) subvolumes
|
|
\(\Omega_i\)
|
|
\item let \(A_i = \int_{\Omega_i}f(x)\dd{x}\)
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|
\item take \(N_i=A_i \cdot N\) samples in each subvolume
|
|
\item efficiency is given by:
|
|
\(\mathfrak{e} = \frac{\sum_i A_i}{\sum_i A_i/\mathfrak{e}_i}\)
|
|
\end{itemize}
|
|
\(\implies\) can optimize in each subvolume independently
|
|
\end{block}
|
|
How do choose the \(\Omega_i\)? \pause {\huge\vegas! :-)}
|
|
\end{frame}
|
|
|
|
\note[itemize]{
|
|
\item no need to know the jacobian ;)
|
|
}
|
|
\begin{frame}{Observables}
|
|
\begin{itemize}
|
|
\item we want: distributions of other observables
|
|
\item turns out: simpliy piping samples \(\{x_i\}\) through a map
|
|
\(\gamma\colon\Omega\mapsto\mathbb{R}\) is enough
|
|
\end{itemize}
|
|
\begin{figure}[p]
|
|
\centering
|
|
|
|
\begin{subfigure}[b]{.49\textwidth}
|
|
\centering \plot[scale=.5]{xs_sampling/histo_sherpa_eta}
|
|
\caption{histogram of the pseudo-rapidity
|
|
(\(\eta\)).}
|
|
\end{subfigure}
|
|
\begin{subfigure}[b]{.49\textwidth}
|
|
\centering \plot[scale=.5]{xs_sampling/histo_sherpa_pt}
|
|
\caption{histogram of the transverse momentum
|
|
(\(\pt\))}
|
|
\end{subfigure}
|
|
\end{figure}
|
|
\end{frame}
|
|
|
|
\begin{frame}[allowframebreaks]
|
|
\frametitle{References}
|
|
\printbibliography
|
|
\end{frame}
|
|
|
|
\appendix
|
|
\section{Appendix}
|
|
|
|
\subsection{More on \vegas}
|
|
|
|
\begin{frame}{\vegas Details}
|
|
\begin{columns}
|
|
\begin{column}{.6\textwidth}
|
|
\begin{block}{Algorithm 1D}
|
|
\begin{enumerate}
|
|
\item start with \(N\) evenly spaced increments
|
|
\(\{[x_i, x_{i+1}]\}_{i\in\overline{1,N}}\)
|
|
\item calculate the integral weights
|
|
\(w_i = \abs{\int_{x_i}^{x_{i+1}}f(x)\dd{x}}\) and define
|
|
\(W=\sum_iw_i\)
|
|
\begin{itemize}
|
|
\item this is done with ordinary MC integration
|
|
\end{itemize}
|
|
\item calculate subdivide the \(i\)-th increment into
|
|
\(K\frac{w_i}{W}\) increments (round up), where
|
|
\(K = \mathcal{O}(1000)\)
|
|
\item amalgamate the new increments into \(N\) groups \(=\)
|
|
new increments
|
|
\end{enumerate}
|
|
\end{block}
|
|
\end{column}
|
|
\pause
|
|
\begin{column}{.4\textwidth}
|
|
\begin{block}{Advantages}
|
|
\begin{itemize}
|
|
\item number of \(f\) evaluations independent of number of
|
|
hypercubes
|
|
\item adaption itself is adaptive
|
|
\item \textcolor{red}{the advantages only show if \(n\)
|
|
``high''.}
|
|
\end{itemize}
|
|
\end{block}
|
|
\end{column}
|
|
\end{columns}
|
|
\end{frame}
|
|
|
|
\begin{frame}
|
|
\begin{figure}[ht]
|
|
\centering \plot[scale=.9]{xs/xs_integrand_vegas}
|
|
\caption{\(2\pi\dv{\sigma}{\theta}\) scaled to increments found by
|
|
\vegas}
|
|
\end{figure}
|
|
\end{frame}
|
|
\begin{frame}{\vegas\ + Hit or Miss}
|
|
\begin{figure}[ht]
|
|
\centering
|
|
\begin{subfigure}{.49\textwidth}
|
|
\centering
|
|
\plot[scale=.8]{xs_sampling/vegas_strat_dist}
|
|
\caption[The distribution for \(\cos\theta\), derived from the
|
|
differential cross-section and the \vegas-weighted
|
|
distribution]{\label{fig:vegasdist} The distribution for
|
|
\(\cos\theta\) and the \vegas-weighted
|
|
distribution.}
|
|
\end{subfigure}
|
|
\begin{subfigure}{.49\textwidth}
|
|
\centering
|
|
\plot[scale=.8]{xs_sampling/vegas_rho}
|
|
\caption[The weighting distribution generated by
|
|
\vegas.]{\label{fig:vegasrho} The weighting distribution generated
|
|
by \vegas. It is clear, that it closely follows the original
|
|
distribution.}
|
|
\end{subfigure}
|
|
\caption{\label{fig:vegas-weighting} \vegas-weighted distribution
|
|
and weighting distribution.}
|
|
\end{figure}
|
|
\end{frame}
|
|
\end{document}
|