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2 changed files with 247 additions and 39 deletions
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@ -344,3 +344,16 @@
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Primaryclass = {hep-ex},
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Reportnumber = {CMS-HIG-12-028, CERN-PH-EP-2012-220},
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}
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@article{Bhat:2020ktg,
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author = "Bhat, Manjunath and Cichy, Krzysztof and
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Constantinou, Martha and Scapellato, Aurora",
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title = "{Parton distribution functions from lattice QCD at
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physical quark masses via the pseudo-distribution
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approach}",
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eprint = "2005.02102",
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archivePrefix ="arXiv",
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primaryClass = "hep-lat",
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month = "5",
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year = "2020"
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}
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273
talk/slides.tex
273
talk/slides.tex
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@ -8,9 +8,10 @@
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\usepackage{slashed}
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\usepackage{tikz}
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\usepackage{tikz-feynman}
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\usepackage{pdfpcnotes}
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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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%\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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@ -30,12 +31,11 @@ labelformat=brace, position=top]{subcaption}
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}
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\AtBeginSubsection[
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{\frame<beamer>{\frametitle{Outline}
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{\frame<beamer>{%
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\tableofcontents[currentsection,currentsubsection]}}%
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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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@ -165,6 +165,7 @@ labelformat=brace, position=top]{subcaption}
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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{Calculation of the \(\qqgg\) Cross Section}
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@ -236,7 +237,7 @@ labelformat=brace, position=top]{subcaption}
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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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\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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@ -289,9 +290,9 @@ labelformat=brace, position=top]{subcaption}
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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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\pnote{
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- Gradually bring in knowledge through distribution. }
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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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@ -311,12 +312,15 @@ labelformat=brace, position=top]{subcaption}
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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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\pnote{
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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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\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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@ -337,7 +341,7 @@ labelformat=brace, position=top]{subcaption}
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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_I^2 &= \frac{\textcolor<+->{blue}{\sigma^2}}{\textcolor<.->{red}{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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@ -346,7 +350,7 @@ labelformat=brace, position=top]{subcaption}
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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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\begin{frame}{Naive Integration 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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@ -374,12 +378,11 @@ labelformat=brace, position=top]{subcaption}
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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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\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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@ -418,13 +421,14 @@ labelformat=brace, position=top]{subcaption}
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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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\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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\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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@ -458,7 +462,7 @@ labelformat=brace, position=top]{subcaption}
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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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A/B \leq 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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@ -468,7 +472,7 @@ labelformat=brace, position=top]{subcaption}
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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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\item sampling \(\dv{\sigma}{\eta}\):
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\result{xs/python/eta_eff}
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\end{itemize}
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\end{block}
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@ -481,7 +485,9 @@ labelformat=brace, position=top]{subcaption}
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efficiency gain: \result{xs/python/tuned_th_samp}
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\end{block}
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\begin{itemize}
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\item<+-> Of course, we can use \vegas\ to provide a better \(g\).
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\item<+-> Of course, we can use \vegas\ to provide a better
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\(g\implies\) \result{xs/python/strat_th_samp}
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\pnote{Has problems, not discussing now.}
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\end{itemize}
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\end{column}
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\begin{column}{.6\textwidth}
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@ -500,7 +506,7 @@ labelformat=brace, position=top]{subcaption}
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\item subdivide sampling volume \(\Omega\) into \(K\) subvolumes
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\(\Omega_i\)
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\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
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\item take \(N_i=\frac{A_i}{\sum_jA_j} \cdot N\) samples in each subvolume
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\item efficiency is given by:
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\(\mathfrak{e} = \frac{\sum_i A_i}{\sum_i A_i/\mathfrak{e}_i}\)
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\end{itemize}
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@ -509,11 +515,11 @@ labelformat=brace, position=top]{subcaption}
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How do choose the \(\Omega_i\)? \pause {\huge\vegas! :-)}
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\end{frame}
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\note[itemize]{
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\item no need to know the jacobian ;)
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}
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\begin{frame}{Observables}
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\begin{itemize}
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\pnote{
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- no need to know the jacobian ;)
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}
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\begin{itemize}
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\item we want: distributions of other observables \pause
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\item turns out: simpliy piping samples \(\{x_i\}\) through a map
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\(\gamma\colon\Omega\mapsto\mathbb{R}\) is enough
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@ -554,15 +560,15 @@ labelformat=brace, position=top]{subcaption}
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\sigma_{ij} = \int f_i\qty(x_1;Q^2) f_j\qty(x_2;Q^2) \hat{\sigma}_{ij}\qty(x_1,
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x_2, Q^2)\dd{x_1}\dd{x_2}
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\end{equation}
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\item have to be obtained experimentally (or through lattice QCD\cite{Bhat:2020ktg})
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\end{itemize}
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\end{block}
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\end{frame}
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\subsection{Implementation}
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\note[itemize]{
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\item took longest time :P
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}
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\begin{frame}
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\pnote{ - took longest time :P }
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\begin{columns}
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\begin{column}{.4\textwidth}
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\begin{block}{What do we need?}
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|
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@ -590,12 +596,11 @@ labelformat=brace, position=top]{subcaption}
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convolved with PDFs for fixed \protect
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\result{xs/python/pdf/second_x} in picobarn.}
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\end{figure}
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}
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\only<+>{
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} \only<+>{
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\begin{figure}
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\centering \plot[width=\columnwidth]{pdf/dist3d_eta_const}
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\caption{\label{fig:dist-pdf-fixed-eta}Differential cross section
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convolved with PDFs for fixed \protect
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\caption{\label{fig:dist-pdf-fixed-eta}Differential cross
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section convolved with PDFs for fixed \protect
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\result{xs/python/pdf/plot_eta} in picobarn.}
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\end{figure}
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}
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@ -633,6 +638,153 @@ labelformat=brace, position=top]{subcaption}
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\end{figure}
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\end{frame}
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\section{Penomenological Studies}
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\begin{frame}{What is missing?}
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\pause
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\begin{itemize}[<+->]
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\item treatement of the beam remnants
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\item intrinsic \(\pt\)
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\item parton showers \pnote{NLO effects}
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\item hadronization
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\item (NLO matrix elements)
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\item multiple interactions
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\end{itemize}
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\pause \(\implies\) \sherpa\ can model those effects
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\end{frame}
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\subsection{Set-Up}
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\begin{frame}
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\pnote{ - cuts and energies same as before\\
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- pun intended\\
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- now discuss impact}
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\begin{itemize}
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\item same phase-space cuts and energies as before
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\item isolation cone cuts
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\end{itemize}
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\begin{block}{The five Stages}
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\begin{description}
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\item[LO] as before
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\item[LO+PS] parton showers with
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\emph{CSShower}~\cite{schumann2008:ap}
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\item[LO+PS+pT] beam remnants and primordial \(\pt\)
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\item[LO+PS+pT+Hadronization] hadronization with
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\emph{Ahadic}~\cite{Winter2003:tt}.
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\item[LO+PS+pT+Hadronization+MI] Multiple Interactions (MI) with
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\emph{Amisic}~\cite{Bothmann:2019yzt}
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\end{description}
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\end{block}
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\end{frame}
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\subsection{Results}
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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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\begin{figure}[ht]
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\rivethist[width=\columnwidth]{pheno/total_pT}
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\end{figure}
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\end{column}
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\begin{column}{.5\textwidth}
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\begin{itemize}
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\item photon system acquires recoil momentum
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\item primordial \(\pt\) enhances xs in low momentum regions
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\end{itemize}
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\end{column}
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\end{columns}
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\pnote{
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- parton shower: col-linear limit\\
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- others the same
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}
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\end{frame}
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\begin{frame}{Transverse Momentum of the leading Photon}
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\begin{columns}
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\begin{column}{.5\textwidth}
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\begin{figure}[ht]
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\rivethist[width=\columnwidth]{pheno/pT}
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\end{figure}
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\end{column}
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\begin{column}{.5\textwidth}
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\begin{itemize}
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\item boost to higher \(\pt\)
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\item all but \stone\ stage largely compatible
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\end{itemize}
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\end{column}
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\end{columns}
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\end{frame}
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\begin{frame}{Invariant Mass of the \(\gamma\gamma\) System}
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\begin{columns}
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\begin{column}{.5\textwidth}
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\begin{figure}[ht]
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\rivethist[width=\columnwidth]{pheno/inv_m}
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\end{figure}
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\end{column}
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\begin{column}{.5\textwidth}
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\begin{itemize}
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\item events can be recoiled past the cuts (very rare)
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\item otherwise shape similar to the \stone\ stage
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\begin{itemize}
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\item largely governed by the PDF
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\end{itemize}
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\end{itemize}
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\end{column}
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\end{columns}
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\end{frame}
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\begin{frame}{Angular Distributions}
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\begin{figure}[ht]
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\rivethist[width=.49\columnwidth]{pheno/eta}
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\rivethist[width=.49\columnwidth]{pheno/cos_theta}
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\end{figure}
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\end{frame}
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\begin{frame}{Conclusions}
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\begin{itemize}
|
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\item parton showering and primordial \(\pt\) have biggest effect on
|
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shape
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\item hadronization and multiple interactions give rise to isolation
|
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effects
|
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\item for angular observables the \stone\ case gives a reasonably
|
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good qualitative picture
|
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\end{itemize}
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\pnote{
|
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- no qed showers\\
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- nlo me
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}
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\end{frame}
|
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|
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\section{Summary}
|
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\begin{frame}
|
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\begin{columns}
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\begin{column}{.7\textwidth}
|
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We have...
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\begin{itemize}
|
||||
\item calculated the cross section for \(\qqgg\)
|
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\item studied and implemented Monte Carlo integration and
|
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sampling
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\begin{itemize}
|
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\item using in \vegas\ whenever possible :)
|
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\end{itemize}
|
||||
\item built a simple \(\ppgg\) event generator
|
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\item looked further down the road with sherpa
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\end{itemize}
|
||||
\end{column}
|
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\pause
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\begin{column}{.3\textwidth}
|
||||
\includegraphics[width=\columnwidth]{questions.jpeg}
|
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\end{column}
|
||||
\end{columns}
|
||||
\begin{center}
|
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{\huge Thanks for your attention! Questions: Now!}
|
||||
\end{center}
|
||||
\end{frame}
|
||||
|
||||
|
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\begin{frame}[allowframebreaks]
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\frametitle{References}
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\printbibliography
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@ -711,4 +863,47 @@ labelformat=brace, position=top]{subcaption}
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and weighting distribution.}
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\end{figure}
|
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\end{frame}
|
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\begin{frame}{Compatibility of Histograms}
|
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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
|
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\(u,v\) 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
|
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\stfive & 33.02(7) & 97.63 & 9.56 \\
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\stfour & 34.08(7) & 97.56 & 1.89\\
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\stthree & 33.97(7) & 97.56 & 3.52 \\
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\sttwo & 34.60(7) & 97.52 & 3.63 \\
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\stone & 38.74(7) & 96.77 & 0 \\
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\end{tabular}
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\caption{\label{tab:xscut}Cross sections and cut statistics.}
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\end{table}
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\end{frame}
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\end{document}
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|
|
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Reference in a new issue