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https://github.com/vale981/bachelor_thesis
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shorten the first slide and some tweaks
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1 changed files with 120 additions and 61 deletions
181
talk/slides.tex
181
talk/slides.tex
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@ -185,63 +185,45 @@ labelformat=brace, position=top]{subcaption}
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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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\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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\pause
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\begin{center}
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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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@ -329,11 +311,13 @@ labelformat=brace, position=top]{subcaption}
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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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- METHOD NAMES
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}
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\begin{itemize}
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\item<+-> we have:
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\(f\colon \vb{x}\in\Omega\subset\mathbb{R}^n\mapsto\mathbb{R}\)
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@ -346,11 +330,11 @@ labelformat=brace, position=top]{subcaption}
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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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\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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\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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@ -715,7 +699,23 @@ labelformat=brace, position=top]{subcaption}
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\end{frame}
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\subsection{Results}
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\begin{frame}{Cross Sections}
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\pnote{
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- effects of the cuts
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}
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\begin{table}[ht]
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\centering
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\begin{tabular}{l|SSS}
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Stage & {\(\sigma\) [\si{\pico\barn}]}\\
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\toprule
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\stfive & 33.02(7) \\
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\stfour & 34.08(7) \\
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\stthree & 33.97(7) \\
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\sttwo & 34.60(7) \\
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\stone & 38.74(7) \\
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\end{tabular}
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\end{table}
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\end{frame}
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\begin{frame}{Transverse Momentum of the \(\gamma\gamma\) System}
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\begin{columns}
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\begin{column}{.5\textwidth}
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@ -827,6 +827,65 @@ labelformat=brace, position=top]{subcaption}
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\appendix
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\section{Appendix}
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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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\begin{frame}{\vegas\ Details}
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\begin{columns}
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