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\APmuon$]{Jacobian for $\PBzero \to \PKstar \Pmuon \APmuon$ \\ proposed solution} +\author{Marcin Chrz\k{a}szcz$^{1}$} +\institute{$^1$~University of Zurich} +\date{\today} + +\begin{document} +% --------------------------- SLIDE -------------------------------------------- +\frame[plain]{\titlepage} +\author{Marcin Chrz\k{a}szcz{~}} +\institute{(UZH)} +% ------------------------------------------------------------------------------ +% --------------------------- SLIDE -------------------------------------------- +\begin{frame}\frametitle{Introduction} + +\begin{itemize} +\item Lets down explicit on what we all agree ( I hope at least ;) ). +\begin{itemize} +\item Measurement of $\overrightarrow{S}=(F_l,~S_x)$ is unbiased. +\item Error is also correctly estimated ensuring the correct coverage. +\end{itemize} +\item The questions what I am answering: what is the corresponding confidence and probability distribution in a new space: $\overrightarrow{P}=(F_l,~P_x)$. +\item To put it a bit more simple: I want to map one space on the other one. +\end{itemize} + +\end{frame} + +\begin{frame}\frametitle{Some mathematical theorems assumptions 1} +\begin{itemize} +\item We have our standard transformation of ($\overrightarrow{S} \to \overrightarrow{P}$): +\begin{footnotesize} +\begin{align*} +F_l &\leftarrow F_l\\ +P_1 &\leftarrow 2\frac{S_3}{1-F_{\rm L}}\\ +P_2 &\leftarrow \frac{1}{2}\frac{S_6^s}{1-F_{\rm L}} = \frac{2}{3}\frac{A_{\rm FB}}{1-F_{\rm L}}\\ +P_3 &\leftarrow -\frac{S_9}{1-F_{\rm L}}\\ +P_4^\prime &\leftarrow \frac{S_4}{\sqrt{F_{\rm L}(1-F_{\rm L})}}\\ +P_5^\prime &\leftarrow \frac{S_5}{\sqrt{F_{\rm L}(1-F_{\rm L})}}\\ +P_6^\prime &\leftarrow \frac{S_7}{\sqrt{F_{\rm L}(1-F_{\rm L})}}\\ +P_8^\prime &\leftarrow \frac{S_8}{\sqrt{F_{\rm L}(1-F_{\rm L})}}. +\end{align*} +\end{footnotesize} +\end{itemize} +\end{frame} + + + \begin{frame}\frametitle{Some mathematical theorems assumptions 2} +\begin{itemize} +\item We know about this transformation: +\begin{itemize} +\item The parameter space is bounded domain ($D$) \checkmark +\item The angular PDF is smooth function in the domain \checkmark +\item There exists 1:1 transformation between $\overrightarrow{S}$ and $\overrightarrow{P}$ \checkmark +\item Inside the domain the Jacobian is non-zero. ($J \neq 0$) \checkmark +\end{itemize} +\item Next slide you will know why those assumptions are needed. +\end{itemize} +\end{frame} + + + +\begin{frame}\frametitle{Some mathematical theorems assumptions 3} +\begin{itemize} +\item Now since there is 1:1 correspondence the central point in the $\overrightarrow{P}$ should be derived from the central point of the $\overrightarrow{S}$ basis. +\item Now the confidence belt. In the $\overrightarrow{S}$ a $68\%$ confidence belt ($D$) is: +\begin{align*} +\int_D f(\overrightarrow{S}) d \overrightarrow{S} = 0.68 +\end{align*} +\item In this equation our $D$ is effectively the errors that we quote. +\item Now form analysis thats to previous slide we can write : +\begin{align*} +\int_D \underbrace{f(\overrightarrow{S})}_{\rm{What~we~simulate/bootstrap}} d \overrightarrow{S} = \int_{\Delta} \underbrace{f'(\overrightarrow{P})}_{\rm{What~we~get~in~P}} \times \vert J \vert d\overrightarrow{P} +\end{align*} + +\end{itemize} +\end{frame} + + \begin{frame}\frametitle{Toys} +\begin{itemize} +\item So to get the integral(coverage) correct we need to take the Jacobian into account. +\item Let's make a toy example calculating $P_2$. Values used (mean $\pm$ error): $F_l= 0.7679 \pm 0.2$, $A_{FB} = -0.329 \pm 0.13$. +\end{itemize} +\includegraphics[width=0.85\textwidth]{images/Fl_AFb.png} +\end{frame} + + \begin{frame}\frametitle{Toys} +\begin{itemize} +\item I am rejecting the non-physical regions: + +\end{itemize} +\includegraphics[width=0.85\textwidth]{images/Fl_boundry.png} +\end{frame} + + +\begin{frame}\frametitle{Toys} +\begin{itemize} +\item Now how does the new space look like. +\item Important to take into account the boundary as without all my theorems fall down. +\end{itemize} +\includegraphics[width=0.85\textwidth]{images/2D.png} +\end{frame} + + + +\begin{frame}\frametitle{Toys ZOOM} +\begin{itemize} +\item Now how does the new space look like. +\item Important to take into account the boundary as without all my theorems fall down. +\end{itemize} +\includegraphics[width=0.85\textwidth]{images/2DZ.png} +\end{frame} + + + \begin{frame}\frametitle{Toys Conclusions} +\begin{itemize} +\item One needs to take into account the Jacobian! +\item The most probable value agrees perfectly. +\item The confidence belt can be transformed analytically even :) +\item There is one hack here: one cannot look at $P_2$ stand alone any more: + +\end{itemize} +\begin{columns} +\column{2.5in} +\includegraphics[width=0.85\textwidth]{images/P2.png} +\column{2.5in} +\includegraphics[width=0.85\textwidth]{images/cast.png} +\end{columns} + + + +\end{frame} + + + + + + + + + + +\end{document} diff --git a/Kstmumu/Jakobian/mchrzasz.toc b/Kstmumu/Jakobian/mchrzasz.toc new file mode 100644 index 0000000..ab3aa12 --- /dev/null +++ b/Kstmumu/Jakobian/mchrzasz.toc @@ -0,0 +1,2 @@ +\beamer@endinputifotherversion {3.24pt} +\select@language {english} diff --git a/Kstmumu/Ref_meeting/beamerthemeSybila.sty b/Kstmumu/Ref_meeting/beamerthemeSybila.sty deleted file mode 100644 index 633f89e..0000000 --- a/Kstmumu/Ref_meeting/beamerthemeSybila.sty +++ /dev/null @@ -1,164 +0,0 @@ -\mode - 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- - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5 -\definecolor{mygreen}{cmyk}{0.82,0.11,1,0.25} - - -\usetheme{Sybila} - -\title[Method of Moments for $\PBzero \to \PKstar \Pmuon \APmuon$]{Method of Moments for $\PBzero \to \PKstar \Pmuon \APmuon$ \\ status report} -\author{Marcin Chrz\k{a}szcz$^{1}$, Nicola Serra$^{1}$} -\institute{$^1$~University of Zurich} -\date{\today} - -\begin{document} -% --------------------------- SLIDE -------------------------------------------- -\frame[plain]{\titlepage} -\author{Marcin Chrz\k{a}szcz{~}} -\institute{(UZH)} -% ------------------------------------------------------------------------------ -% --------------------------- SLIDE -------------------------------------------- -\begin{frame}\frametitle{What was done for MoM} - -\begin{itemize} -\item Statistical error is evaluated using bootstraps. -\begin{itemize} -\item The weighted variance is not super well known in the literature. -\item We check and both errors are in perfect agreement -\item Also in the bootstraps we get the correlation for free. -\end{itemize} -\item We finished all the systematics which are already in the analysis note. -\item We added the measurement $A_{\rm{CP}}$. -\end{itemize} - -\end{frame} - -\begin{frame}\frametitle{Bootstraps} - -\begin{columns} -\column{2.5in} -\begin{itemize} -\item Observed perfect agreement with previous error determination. -\item The agreement was tested both on data and toy MC. -\item For the paper we produced 1M bootstraps peer $q^2$ bin. -\item The distributions look super nice. -\end{itemize} - - - - -\end{columns} - -\begin{itemize} -\item Statistical error is evaluated using bootstraps. -\begin{itemize} -\item The weighted variance is not super well known in the literature. -\item We check and both errors are in perfect agreement -\item Also in the bootstraps we get the correlation for free. -\end{itemize} -\item We finished all the systematics which are already in the analysis note. -\item We added the measurement $A_{\rm{CP}}$. -\end{itemize} - -\end{frame} - - -\end{document} diff --git a/Kstmumu/Ref_meeting/mchrzasz.toc b/Kstmumu/Ref_meeting/mchrzasz.toc deleted file mode 100644 index ab3aa12..0000000 --- a/Kstmumu/Ref_meeting/mchrzasz.toc +++ /dev/null @@ -1,2 +0,0 @@ -\beamer@endinputifotherversion {3.24pt} -\select@language {english}