mirror of
https://github.com/vale981/bachelor_thesis
synced 2025-03-04 17:11:39 -05:00
1079 lines
34 KiB
TeX
1079 lines
34 KiB
TeX
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labelformat=brace, position=top]{subcaption}
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\usetikzlibrary{arrows,shapes,angles,quotes,arrows.meta,external}
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\graphicspath{ {figs/} }
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\addbibresource{thesis.bib}
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\sisetup{separate-uncertainty = true}
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\usecolortheme{dolphin}
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%\usepackage{newmathpx}
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\institute[TUD] % (optional)
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{
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IKTP, TU Dresden
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}
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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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\AtBeginSubsection[]
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{
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\tableofcontents[currentsubsection]
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}
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\newenvironment<>{results}[1][Results]{%
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\setbeamercolor{block example}{fg=white,bg=red!75!black}%
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\begin{exampleblock}#2{#1}}{\end{exampleblock}}
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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{\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{Bachelor Thesis Presentation}
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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 background for important higgs decay channel:
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\(H\rightarrow \gamma\gamma\)
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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 partial background for 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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\pnote{Why usefult for ev. gen -> later}
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\end{frame}
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\section{The \(\qqgg\) Cross Section}
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\subsection{Approach}
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\begin{frame}
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\begin{figure}[ht]
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\centering
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\begin{subfigure}{.28\textwidth}
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\centering
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\begin{tikzpicture}[scale=1]
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\begin{feynman}
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\diagram [small,horizontal=i2 to a] { i2 [particle=\(q\)] --
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[fermion, momentum=\(p_2\)] a -- [fermion, reversed
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momentum=\(q\)] b, i1 [particle=\(\bar{q}\)] -- [anti
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fermion, 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 [particle=\(\gamma\)], f1
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-- [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}{.28\textwidth}
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\centering
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\begin{tikzpicture}[scale=1]
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\begin{feynman}
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\diagram [small,horizontal=i2 to a] { i2 [particle=\(q\)] --
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[fermion, momentum=\(p_2\)] a -- [fermion, reversed
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momentum'=\(q\)] b, i1 [particle=\(\bar{q}\)] -- [anti
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fermion, momentum'=\(p_1\)] b, i2 -- [opacity=0] i1, a --
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[draw=none] f2 [particle=\(\gamma\)], b -- [draw=none] f1
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[particle=\(\gamma\)], f1 -- [opacity=0] f2, }; \diagram*
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{ (a) -- [photon] (f1), (b) -- [photon] (f2), };
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\end{feynman}
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\end{tikzpicture}
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\subcaption{t channel}
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\end{subfigure}
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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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\begin{center}
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\begin{tikzpicture}[scale=.8]
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\coordinate (origin) at (0,0);
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\draw[-Latex] (origin) -- (-2,0) node[left] {\(p_3\)};
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\draw[-Latex] (origin) -- (2,0) coordinate (p4) node[right]
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{\(p_4\)}; \draw[Latex-,rotate=40] (origin) -- (2,0) coordinate
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(p2) node[right] {\(p_2\)}; \draw[Latex-,rotate=40] (origin) --
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(-2,0) node[left] {\(p_1\)}; \draw[fill=black] (origin) circle
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(.03);
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\draw pic["$\theta$", draw=black, <->, angle eccentricity=1.2,
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angle radius=1cm] {angle=p4--origin--p2};
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\end{tikzpicture}
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\pause
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here: massless limit
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\end{center}
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\end{frame}
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\subsection{Result}
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\begin{frame}{Result}
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\pnote{LEADING ORDER}
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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(\tanh(\eta)^2 + 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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\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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\begin{frame}
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\pnote{
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- Gradually bring in knowledge through distribution. }
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\begin{block}{Basic Idea}
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\begin{center}
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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 \\ \pauses\ldots\ how do we answer questions about
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\(f\)? \\\pause
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\;\;\onslide<+->{\(\implies\) Sample it at random points.}
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\end{center}
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\end{block}
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\pause
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\begin{block}{Concrete Applicationss}
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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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\begin{frame}
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\pnote{
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- WHAT DOES RHO DO
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- omitting details (law of big numbers, central limit theorem)\\
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- at least three angles of attack\\
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- some sort of importance sampling, volume: stratified sampling\\
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- ADVANTAGES OF MC
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- METHOD NAMES
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}
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\begin{itemize}
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\item<+-> we have:
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\(f\colon \vb{x}\in\Omega\subset\mathbb{R}^n\mapsto\mathbb{R}\)\quad
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and\quad \(\rho\colon \vb{x}\in\Omega\mapsto\mathbb{R}_{> 0}\)\quad with\quad
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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 \({\vb{x}_i \sim \rho}\):
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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<+-> integration error:
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\begin{align}
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\sigma_I^2 &= \frac{\textcolor<+->{blue}{\sigma^2}}{\textcolor<.->{red}{N}} \\
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\sigma^2 &= \VAR{\frac{F}{\Rho}} = \int_{\textcolor<+(2)->{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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\pause
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\item<+-> independent of dimensionality :)
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\end{itemize}
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\end{frame}
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\begin{frame}{Naive Integration Change of Variables}
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Choose \(\rho(\vb{x}) = \frac{1}{\abs{\Omega}}\)\\
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\onslide<1->{\quad\(\implies I=\frac{\abs{\Omega}}{N}\sum_{i=1}^N
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f(\vb{x_i})=\abs{\Omega}\cdot\bar{f}\)\quad and\quad
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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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\pause
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\begin{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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\(\SI{1e-3}{\pico\barn}\) 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{results}
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\begin{figure}[hb]
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\centering \onslide<3->{
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\begin{subfigure}[c]{.41\textwidth}
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\centering \plot[width=\columnwidth]{xs/xs_integrand}
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\end{subfigure}
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} \onslide<4->{
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\begin{subfigure}[c]{.41\textwidth}
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\centering \plot[width=\columnwidth]{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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\begin{frame}{\vegas\ Algorithm \cite{Lepage:19781an}}
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\pnote{
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- proposed by G. Peter Lepage (slac) 1976 \\
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- own implementation!!!
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}
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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{results}[Results: Application to \(\dd{\sigma}/\dd{\theta}\)]
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\begin{itemize}
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\item 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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\item without \vegas\: \result{xs/python/xs_mc_N}
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\end{itemize}
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\end{results}
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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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|
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\subsection{Sampling}
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|
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|
\begin{frame}{Why Samples?}
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\begin{itemize}[<+->]
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|
\item same format as experimental data: direct comparison possible
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|
\item easy to generate distributions for other observables
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|
\item events can be ``dressed'' with additional effects
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|
\end{itemize}
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|
\end{frame}
|
|
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|
\begin{frame}
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|
\pnote{
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|
- prop. to density
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|
- generalization to n dim is easy
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|
- idea -> cumulative propability the same
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|
}
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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}
|
|
\begin{columns}
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|
\begin{column}{.5\textwidth}
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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
|
|
\[\int_{0}^{y}f(x')\dd{x'} = x\cdot\int_0^1f(x')\dd{x'} =
|
|
x\cdot A\] for \(y\) to obtain sample of \(f/A\)
|
|
\item let \(F\) be the antiderivative of \(f\), then
|
|
\(y=F^{-1}(x\cdot A + F(0))\)
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|
\begin{itemize}
|
|
\item<.-> sometimes analytical form available
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|
\item<.-> otherwise tackle that numerically
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\end{itemize}
|
|
\end{itemize}
|
|
\end{block}
|
|
\end{column}
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\begin{column}{.5\textwidth}<.(-1)->
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\begin{figure}
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|
\centering
|
|
\includegraphics[width=\columnwidth]{figs/normal_cdf.pdf}
|
|
\caption{CDF of the normal distribution.~\cite{wiki:2020no}}
|
|
\end{figure}
|
|
\end{column}
|
|
\end{columns}
|
|
\end{frame}
|
|
|
|
\begin{frame}{Hit or Miss}
|
|
\begin{block}{Basic Idea}
|
|
\begin{itemize}
|
|
\item take samples \({x_i}\) distributed according to \(g/B\),
|
|
where \(B=\int_0^1g(x)\dd{x}\) and
|
|
\(\forall x\in\Omega\colon g(x)\geq f(x)\)
|
|
\item accept each sample with the probability~\(f(x_i)/g(x_i)\)
|
|
(importance sampling)
|
|
\item total probability of accepting a sample: \(\mathfrak{e} =
|
|
A/B \leq 1\) (efficiency, \(A=\int_0^1f(x)\dd{x}\))
|
|
\item simplest choice \(g=\max_{x\in\Omega}f(x)=f_{\text{max}}\)
|
|
\item again: efficiency gain through reduction of variance
|
|
\end{itemize}
|
|
\end{block}
|
|
\pause
|
|
\begin{results}[Results with \(g=f_{\text{max}}\)]
|
|
\begin{itemize}
|
|
\item<.-> sampling \(\dv{\sigma}{\cos\theta}\):
|
|
\result{xs/python/naive_th_samp}
|
|
\item sampling \(\dv{\sigma}{\eta}\):
|
|
\result{xs/python/eta_eff}
|
|
\end{itemize}
|
|
\end{results}
|
|
\end{frame}
|
|
|
|
\begin{frame}{Hit or Miss}
|
|
\begin{columns}
|
|
\begin{column}{.4\textwidth}
|
|
\begin{results}<+->[Results with \(g=a + b\cdot x^2\)]
|
|
\begin{itemize}
|
|
\item<+-> Modest efficiency gain:
|
|
\result{xs/python/tuned_th_samp}
|
|
|
|
\item<+-> Of course, we can use \vegas\ to provide a better
|
|
\(g\implies\) \result{xs/python/strat_th_samp} \pnote{Has
|
|
problems, not discussing now.}
|
|
\begin{itemize}
|
|
\item<+-> this approach leads to problems when \(f\) is not
|
|
smooth or \emph{very} small over larger regions
|
|
\end{itemize}
|
|
\end{itemize}
|
|
\end{results}
|
|
\end{column}
|
|
\begin{column}{.6\textwidth}
|
|
\begin{figure}[ht]
|
|
\centering \plot[scale=.8]{xs_sampling/upper_bound}
|
|
\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{columns}
|
|
\begin{column}{.6\textwidth}
|
|
\begin{block}{Basic Idea}
|
|
\begin{itemize}
|
|
\item subdivide sampling volume \(\Omega\) into
|
|
subvolumes \(\Omega_i\)
|
|
\item let \(A_i = \int_{\Omega_i}f(x)\dd{x}\)
|
|
\item take \(N_i=\frac{A_i}{\sum_jA_j} \cdot N\) 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}
|
|
\end{column}
|
|
\pause
|
|
\begin{column}{.4\textwidth}
|
|
\begin{center}
|
|
{\LARGE
|
|
How do we choose the \(\Omega_i\)? \pause\\
|
|
\vspace{1em}
|
|
\(\implies\) \vegas! :-) }
|
|
\end{center}
|
|
\end{column}
|
|
\end{columns}
|
|
\end{frame}
|
|
|
|
\begin{frame}{Observables}
|
|
\pnote{
|
|
- no need to know the jacobian ;)
|
|
}
|
|
\begin{itemize}
|
|
\item we want: distributions of other observables \pause
|
|
\item turns out: reconstructing full event (kinematics, ...) from
|
|
sampling variables + calculating other observables from that is enough
|
|
\end{itemize}
|
|
\pause
|
|
\begin{figure}[p]
|
|
\centering
|
|
\begin{subfigure}[b]{.49\textwidth}
|
|
\centering \plot[scale=.7]{xs_sampling/histo_sherpa_eta}
|
|
\end{subfigure}
|
|
\begin{subfigure}[b]{.49\textwidth}
|
|
\centering \plot[scale=.7]{xs_sampling/histo_sherpa_pt}
|
|
\end{subfigure}
|
|
\end{figure}
|
|
\end{frame}
|
|
|
|
\section{A Simple Proton Scattering Event Generator}
|
|
|
|
\subsection{Parton Density Functions}
|
|
\begin{frame}
|
|
\begin{itemize}[<+->]
|
|
\item free quarks are not observed \(\implies\) we have to look at
|
|
hadron collisions
|
|
\item parton density functions (PDFs) are a necessary tool
|
|
\end{itemize}
|
|
\pause
|
|
\begin{block}{Basic Idea (Leading Order)}
|
|
\begin{itemize}
|
|
\item probability density to encounter a parton \(i\) at momentum
|
|
fraction \(x\) and factorization scale \(Q^2\): given by
|
|
\(f_i(x;Q^2)\)
|
|
\item total cross section for a partonic process in the hadron
|
|
collision:
|
|
\begin{equation}
|
|
\label{eq:pdf-xs}
|
|
\sigma_{ij} = \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}
|
|
\end{equation}
|
|
\item have to be obtained experimentally (or through lattice
|
|
QCD\cite{Bhat:2020ktg}) at given \(Q^2\) and evolved with the
|
|
\emph{DGLAP} equations
|
|
\end{itemize}
|
|
\end{block}
|
|
\end{frame}
|
|
|
|
\subsection{Implementation}
|
|
|
|
\begin{frame}
|
|
\pnote{ - took longest time :P }
|
|
\begin{columns}
|
|
\begin{column}{.4\textwidth}
|
|
\begin{block}{What do we need?}
|
|
\begin{itemize}[<+->]
|
|
\item partonic cross section and kinematics in lab frame
|
|
\item \(Q^2\pause = 2x_1x_2E_p^2\) \pause
|
|
\item PDF\pause :
|
|
\texttt{NNPDF31\_lo\_as\_0118}~\cite{NNPDF:2017pd} \pause
|
|
\item beam energies and cuts:\pause
|
|
\begin{itemize}
|
|
\item \result{xs/python/pdf/e_proton}
|
|
\item \result{xs/python/pdf/eta} and
|
|
\result{xs/python/pdf/min_pT}
|
|
\end{itemize}
|
|
\item integration and sampling method: \pause \vegas\ +
|
|
stratified sampling
|
|
\end{itemize}
|
|
\end{block}
|
|
\end{column}
|
|
\begin{column}{.6\textwidth}
|
|
\only<+>{
|
|
\begin{figure}
|
|
\centering \plot[width=\columnwidth]{pdf/dist3d_x2_const}
|
|
\caption{\label{fig:dist-pdf}Differential cross section
|
|
convolved with PDFs for fixed \protect
|
|
\result{xs/python/pdf/second_x} in picobarn.}
|
|
\end{figure}
|
|
} \only<+>{
|
|
\begin{figure}
|
|
\centering \plot[width=\columnwidth]{pdf/dist3d_eta_const}
|
|
\caption{\label{fig:dist-pdf-fixed-eta}Differential cross
|
|
section convolved with PDFs for fixed \protect
|
|
\result{xs/python/pdf/plot_eta} in picobarn.}
|
|
\end{figure}
|
|
}
|
|
\end{column}
|
|
\end{columns}
|
|
\end{frame}
|
|
|
|
\subsection{Results}
|
|
\begin{frame}{Cross Section}
|
|
\begin{center}
|
|
{\huge\result{xs/python/pdf/my_sigma}}
|
|
\end{center}
|
|
\begin{itemize}
|
|
\item compatible with \sherpa: \result{xs/python/pdf/sherpa_sigma}
|
|
\item achieved \result{xs/python/pdf/samp_eff} (\(N=10^7\) samples)
|
|
\item using \result{xs/python/pdf/num_increments} hypercubes
|
|
\end{itemize}
|
|
\end{frame}
|
|
|
|
\begin{frame}[allowframebreaks]{Observables}
|
|
\pnote{effect of the pt cuts}
|
|
\begin{figure}[hp]
|
|
\centering
|
|
\begin{subfigure}{.49\textwidth}
|
|
\centering \plot[width=1\columnwidth]{pdf/eta_hist}
|
|
\end{subfigure}
|
|
\begin{subfigure}{.49\textwidth}
|
|
\centering \plot[width=1\columnwidth]{pdf/cos_theta_hist}
|
|
\end{subfigure}
|
|
\begin{subfigure}{.49\textwidth}
|
|
\centering \plot[width=1\columnwidth]{pdf/pt_hist}
|
|
\end{subfigure}
|
|
\begin{subfigure}{.49\textwidth}
|
|
\centering \plot[width=1\columnwidth]{pdf/inv_m_hist}
|
|
\end{subfigure}
|
|
\end{figure}
|
|
\end{frame}
|
|
|
|
|
|
\section{Phenomenological Studies}
|
|
|
|
\begin{frame}{What is missing?}
|
|
\pause\pnote{of course there's more missing}
|
|
\begin{columns}
|
|
\begin{column}{.5\textwidth}
|
|
\begin{itemize}[<+->]
|
|
\item treatement of the beam remnants
|
|
\item intrinsic \(\pt\)
|
|
\item parton showers \pnote{NLO effects}
|
|
\item hadronization
|
|
\item multiple interactions
|
|
\end{itemize}
|
|
\end{column}
|
|
\begin{column}{.5\textwidth}<+->
|
|
\begin{center}
|
|
{\Huge \sherpa\ can model those effects}
|
|
\end{center}
|
|
\end{column}
|
|
\end{columns}
|
|
\end{frame}
|
|
|
|
|
|
\subsection{Set-Up}
|
|
\begin{frame}
|
|
\pnote{ - cuts and energies same as before\\
|
|
- pun intended\\
|
|
- now discuss impact}
|
|
\begin{itemize}
|
|
\item same phase-space cuts and energies as before
|
|
\item isolation cone cuts
|
|
\end{itemize}
|
|
|
|
\begin{block}{The five Stages}
|
|
\begin{description}
|
|
\item[LO] as before
|
|
\item[LO+PS] parton showers with
|
|
\emph{CSShower}~\cite{schumann2008:ap}
|
|
\item[LO+PS+pT] beam remnants and primordial \(\pt\)
|
|
\item[LO+PS+pT+Hadronization] hadronization with
|
|
\emph{Ahadic}~\cite{Winter2003:tt}.
|
|
\item[LO+PS+pT+Hadronization+MI] Multiple Interactions (MI) with
|
|
\emph{Amisic}~\cite{Bothmann:2019yzt}
|
|
\end{description}
|
|
\end{block}
|
|
\end{frame}
|
|
|
|
\subsection{Results}
|
|
\begin{frame}{Fiducial Cross Sections}
|
|
\pnote{
|
|
- effects of the cuts
|
|
}
|
|
\begin{table}[ht]
|
|
\centering
|
|
\begin{tabular}{l|SSS}
|
|
Stage & {\(\sigma\) [\si{\pico\barn}]}\\
|
|
\toprule
|
|
\stfive & 33.02(7) \\
|
|
\stfour & 34.08(7) \\
|
|
\stthree & 33.97(7) \\
|
|
\sttwo & 34.60(7) \\
|
|
\stone & 38.74(7) \\
|
|
\end{tabular}
|
|
\end{table}
|
|
\end{frame}
|
|
\begin{frame}{Transverse Momentum of the \(\gamma\gamma\) System}
|
|
\begin{columns}
|
|
\begin{column}{.5\textwidth}
|
|
\begin{figure}[ht]
|
|
\rivethist[width=\columnwidth]{pheno/total_pT}
|
|
\end{figure}
|
|
\end{column}
|
|
\begin{column}{.5\textwidth}
|
|
\begin{minipage}[c][.6\textheight][c]{\linewidth}
|
|
\begin{itemize}
|
|
\item photon system acquires recoil momentum
|
|
\item primordial \(\pt\) enhances xs in low momentum regions
|
|
\end{itemize}
|
|
\end{minipage}
|
|
\begin{onlyenv}<-+>
|
|
\begin{figure}
|
|
\centering
|
|
\begin{tikzpicture}[remember picture,overlay]
|
|
\tikzset{yshift=1.2cm}
|
|
\coordinate (origin) at (0,0);
|
|
|
|
\draw[Latex-] (origin) -- (-2,0) node[left] {\(p_1\)};
|
|
\draw[Latex-] (origin) -- (2,0) coordinate (p4) node[right] {\(p_2\)};
|
|
\draw[-Latex] (origin) -- (1.32,1.5) coordinate (p2) node[right] {\(p_3\)};
|
|
\draw[-Latex] (origin) -- (-1.32,-1.5) node[left] {\(p_4\)};
|
|
\draw[fill=black] (origin) circle (.03);
|
|
\end{tikzpicture}
|
|
\end{figure}
|
|
\end{onlyenv}
|
|
\begin{onlyenv}<+>
|
|
\begin{figure}
|
|
\centering
|
|
\begin{tikzpicture}[remember picture,overlay]
|
|
\tikzset{yshift=1.2cm}
|
|
\coordinate (origin) at (0,.1);
|
|
\coordinate (right) at (2,0);
|
|
\coordinate (left) at (-2,0);
|
|
|
|
\draw[Latex-] (origin) -- (left) node[left] {\(p_1\)};
|
|
\draw[Latex-] (origin) -- (right) coordinate (p4) node[right] {\(p_2\)};
|
|
\draw[-Latex] (origin) -- (1.32,1.7) coordinate (p2) node[right] {\(p_3\)};
|
|
\draw[-Latex] (origin) -- (-1.32,-1.3) node[left] {\(p_4\)};
|
|
\draw[fill=black] (origin) circle (.03);
|
|
\draw[dashed] (left) -- (right);
|
|
\end{tikzpicture}
|
|
\end{figure}
|
|
\end{onlyenv}
|
|
\end{column}
|
|
\end{columns}
|
|
\pnote{
|
|
- parton shower: col-linear limit\\
|
|
- others the same
|
|
}
|
|
\end{frame}
|
|
|
|
\begin{frame}{Transverse Momentum of the leading Photon}
|
|
\begin{columns}
|
|
\begin{column}{.5\textwidth}
|
|
\begin{figure}[ht]
|
|
\rivethist[width=\columnwidth]{pheno/pT}
|
|
\end{figure}
|
|
\end{column}
|
|
\begin{column}{.5\textwidth}
|
|
\begin{itemize}
|
|
\item boost to higher \(\pt\)
|
|
\item low \(\pt\lesssim \SI{40}{\giga\electronvolt}\) shape
|
|
known and expected from QCD resummation of multiple emissions
|
|
(parton shower)
|
|
\item all but \stone\ stage largely compatible
|
|
\end{itemize}
|
|
\end{column}
|
|
\end{columns}
|
|
\end{frame}
|
|
|
|
\begin{frame}{Invariant Mass of the \(\gamma\gamma\) System}
|
|
\begin{columns}
|
|
\begin{column}{.5\textwidth}
|
|
\begin{figure}[ht]
|
|
\rivethist[width=\columnwidth]{pheno/inv_m}
|
|
\end{figure}
|
|
\end{column}
|
|
\begin{column}{.5\textwidth}
|
|
\begin{itemize}
|
|
\item some events recoiled so both photons acquire
|
|
\(\pt > \gev{20}\) (very rare)
|
|
\item otherwise shape similar to the \stone\ stage
|
|
\begin{itemize}
|
|
\item largely governed by the PDF
|
|
\end{itemize}
|
|
\end{itemize}
|
|
\end{column}
|
|
\end{columns}
|
|
\end{frame}
|
|
|
|
\begin{frame}{Angular Distributions}
|
|
\begin{figure}[ht]
|
|
\rivethist[width=.49\columnwidth]{pheno/eta}
|
|
\rivethist[width=.49\columnwidth]{pheno/cos_theta}
|
|
\end{figure}
|
|
\end{frame}
|
|
|
|
\begin{frame}{Conclusions}
|
|
\begin{itemize}
|
|
\item parton showering and primordial \(\pt\) have biggest effect on
|
|
shape
|
|
\item hadronization and multiple interactions give rise to isolation
|
|
effects
|
|
\item for angular observables the \stone\ case gives a reasonably
|
|
good qualitative picture
|
|
\end{itemize}
|
|
\pnote{
|
|
- no qed showers\\
|
|
- nlo me
|
|
}
|
|
\end{frame}
|
|
|
|
\section{Summary}
|
|
\begin{frame}
|
|
\begin{columns}
|
|
\begin{column}{.7\textwidth}
|
|
We have...
|
|
\begin{itemize}
|
|
\item calculated the cross section for \(\qqgg\)
|
|
\item studied and implemented Monte Carlo integration and
|
|
sampling
|
|
\begin{itemize}
|
|
\item using in \vegas\ whenever possible :)
|
|
\end{itemize}
|
|
\item built a simple \(\ppgg\) event generator
|
|
\item looked further down the road with \sherpa
|
|
\end{itemize}
|
|
\end{column}
|
|
\pause
|
|
\begin{column}{.3\textwidth}
|
|
\includegraphics[width=\columnwidth]{questions.jpeg}
|
|
\end{column}
|
|
\end{columns}
|
|
\begin{center}
|
|
{\huge Thanks for your attention! Questions: Now!}
|
|
\end{center}
|
|
\end{frame}
|
|
|
|
|
|
\begin{frame}[allowframebreaks]
|
|
\frametitle{References}
|
|
\printbibliography
|
|
\end{frame}
|
|
|
|
\appendix
|
|
\section{Appendix}
|
|
\begin{frame}
|
|
\begin{columns}[T]
|
|
\begin{column}{.5\textwidth}
|
|
\begin{figure}[ht]
|
|
\centering
|
|
\begin{subfigure}[c]{.28\textwidth}
|
|
\centering
|
|
\begin{tikzpicture}[scale=.6]
|
|
\begin{feynman}
|
|
\diagram [small,horizontal=i2 to a] { i2
|
|
[particle=\(q\)] -- [fermion, momentum=\(p_2\)] a --
|
|
[fermion, reversed momentum=\(q\)] b, i1
|
|
[particle=\(\bar{q}\)] -- [anti fermion,
|
|
momentum'=\(p_1\)] b, i2 -- [opacity=0] i1, a --
|
|
[photon, momentum=\(p_3\)] f1 [particle=\(\gamma\)], b
|
|
-- [photon, momentum'=\(p_4\)] f2
|
|
[particle=\(\gamma\)], f1 -- [opacity=0] f2, };
|
|
\end{feynman}
|
|
\end{tikzpicture}
|
|
\subcaption{u channel}
|
|
\end{subfigure}
|
|
\begin{subfigure}[c]{.28\textwidth}
|
|
\centering
|
|
\begin{tikzpicture}[scale=.6]
|
|
\begin{feynman}
|
|
\diagram [small,horizontal=i2 to a] { i2
|
|
[particle=\(q\)] -- [fermion, momentum=\(p_2\)] a --
|
|
[fermion, reversed momentum'=\(q\)] b, i1
|
|
[particle=\(\bar{q}\)] -- [anti fermion,
|
|
momentum'=\(p_1\)] b, i2 -- [opacity=0] i1, a --
|
|
[draw=none] f2 [particle=\(\gamma\)], b -- [draw=none]
|
|
f1 [particle=\(\gamma\)], f1 -- [opacity=0] f2, };
|
|
\diagram* { (a) -- [photon] (f1), (b) -- [photon] (f2),
|
|
};
|
|
\end{feynman}
|
|
\end{tikzpicture}
|
|
\subcaption{\label{fig:qqggfeyn2}t channel}
|
|
\end{subfigure}
|
|
%
|
|
\caption{Leading order diagrams for \(\qqgg\).}%
|
|
\label{fig:qqggfeyn}
|
|
\end{figure}
|
|
\end{column}
|
|
\pause
|
|
\begin{column}{.5\textwidth}
|
|
\begin{block}{Task: calculate \(\abs{\mathcal{M}}^2\)}
|
|
\begin{enumerate}[<+->]
|
|
\item translate diagrams to matrix elements
|
|
\item use Casimir's trick to average over spins
|
|
\item use completeness relation to sum over photon
|
|
polarizations
|
|
\item use trace identities to compute the absolute square
|
|
\item simplify with trigonometric identities
|
|
\end{enumerate}
|
|
\end{block}
|
|
\pause Here: Quark masses neglected.
|
|
\end{column}
|
|
\end{columns}
|
|
\end{frame}
|
|
|
|
\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}
|
|
\begin{frame}{Compatibility of Histograms}
|
|
The compatibility of histograms is tested as described
|
|
in~\cite{porter2008:te}. The test value
|
|
is \[T=\sum_{i=1}^k\frac{(u_i-v_i)^2}{u_i+v_i}\] where \(u_i, v_i\)
|
|
are the number of samples in the \(i\)-th bin of the histograms
|
|
\(u,v\) and \(k\) is the number of bins. This value is \(\chi^2\)
|
|
distributed with \(k\) degrees, when the number of samples in the
|
|
histogram is reasonably high. The mean of this distribution is \(k\)
|
|
and its standard deviation is \(\sqrt{2k}\). The value
|
|
\[P = 1 - \int_0^{T}f(x;k)\dd{x}\] states with which probability the
|
|
\(T\) value would be greater than the obtained one, where \(f\) is the
|
|
probability density of the \(\chi^2\) distribution. Thus
|
|
\(P\in [0,1]\) is a measure of confidence for the compatibility of the
|
|
histograms. These formulas hold, if the total number of events in both
|
|
histograms is the same.
|
|
\end{frame}
|
|
|
|
\begin{frame}{Cut Flow}
|
|
\pnote{
|
|
- 2 kinds of impact: phase space and isolation\\
|
|
- these effects have an impact on fiducial xs\\
|
|
- PS, pT more phase space\\
|
|
- Hadr. and MI isolation
|
|
}
|
|
\begin{table}[ht]
|
|
\centering
|
|
\begin{tabular}{l|SSS}
|
|
&&\multicolumn{2}{c}{events discarded by cuts} \\
|
|
Stage & {\(\sigma\) [\si{\pico\barn}]} & {phase space
|
|
[\si{\percent}]} &
|
|
{isolation
|
|
[\SI{1e-4}{\percent}]} \\
|
|
\toprule
|
|
\stfive & 33.02(7) & 97.63 & 9.56 \\
|
|
\stfour & 34.08(7) & 97.56 & 1.89\\
|
|
\stthree & 33.97(7) & 97.56 & 3.52 \\
|
|
\sttwo & 34.60(7) & 97.52 & 3.63 \\
|
|
\stone & 38.74(7) & 96.77 & 0 \\
|
|
|
|
\end{tabular}
|
|
\caption{\label{tab:xscut}Cross sections and cut statistics.}
|
|
\end{table}
|
|
\end{frame}
|
|
|
|
\begin{frame}
|
|
\begin{figure}[ht]
|
|
\rivethist[width=.49\columnwidth]{pheno/azimuthal_angle}
|
|
\rivethist[width=.49\columnwidth]{pheno/pT_subl}
|
|
\end{figure}
|
|
\end{frame}
|
|
|
|
\begin{frame}
|
|
\begin{figure}[ht]
|
|
\rivethist[width=.49\columnwidth]{pheno/o_angle}
|
|
\rivethist[width=.49\columnwidth]{pheno/o_angle_cs}
|
|
\end{figure}
|
|
\end{frame}
|
|
\end{document}
|