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------------------------------------------------------------------------------ % --------------------------- SLIDE -------------------------------------------- -\tableofcontents +%\tableofcontents +\begin{frame} +\begin{enumerate} +\item Motivation. +\item Method of Moments. +\item Systematic uncertainties. +\item MC toy studies. +\item Conclusions. +\end{enumerate} + +\end{frame} \placelogotrue \section{Motivation} @@ -187,23 +197,196 @@ \end{frame} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Systematic uncertainties} + + +%\placelogofalse + \begin{frame}\frametitle{Partial Waves mismodeling} +\begin{columns} +\column{3in} +\only<1> +{ + +\begin{itemize} +\item Let us consider a decay of $\PB \to P_1 P_2 \Pmuon \APmuon$. +\item In terms of angular \pdf is expressed in terms of partial-wave expansion. +\item For $\PB \to \PK \Ppi \Pmuon \APmuon$ system, S,P,D waves have been studied. +\end{itemize} + +} +\only<2> +{ +\begin{itemize} +\item One can write the \pdf separating the hadronic system: +\end{itemize} +\begin{align} +P(\cos \theta_1, & \cos \theta_2, \theta_3) = \\ &\quad \nonumber \sum _i S_i(\nuvec, \cos \theta_2) f_i (\cos \theta_1, \theta_3) +\end{align} + +} + +\column{2in} +\includegraphics[width=0.9\textwidth]{images/fig-topology.pdf} +\end{columns} +\only<1>{ +\begin{itemize} +\item The muon system of this kind of decays has a fixed angular dependence in terms of $\theta_1$ and $\theta_3$. +\item The hadron system can have arbitrary large angular momentum. +\end{itemize} +} + +\only<2>{ + +\begin{itemize} +\item $S_i(\nuvec, \cos \theta_2)$ can be further expend in terms of Legendre polynomials $p_l^{\vert m \vert }(\cos \theta_2)$: +\end{itemize} +\begin{align} +S_i(\nuvec, \cos \theta_2) = \sum_{l=0}^{\inf} S_{k,l}(\nuvec ) p_l^{\vert m \vert}(\cos \theta_2) +\end{align} + +\begin{small} +\begin{itemize} +\item Experimentally the $ S_{k,l}$ are easily accessible, but there is a theoretical difficulty as one would need to sum over infinite number of partial waves. +\end{itemize}\end{small} +} \end{frame} +\placelogotrue + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{frame}\frametitle{Detector effects} +\begin{columns} + +\column{2.5in} +\begin{itemize} +\item Since our detectors are not a perfect devices the angular distribution observed by them are not the distributions that the physics model creates. +\end{itemize} +\column{2.5in} +\includegraphics[width=0.9\textwidth]{images/Fig2b.pdf} +\end{columns} +\pause +\begin{itemize} +\item To take into account the acceptance effects one needs to simulate the a large sample of MC events.\\ Try to figure out the efficiency function. +\item Try to figure out the efficiency function. +\item Number of possibilities. +\item Then you can just weight events: +\end{itemize} +\begin{align*} +\widehat{E[S_i]}=\dfrac{1}{\sum_{k=1}^N w_k}\sum_{k=1}^{N} w_k \dual{f}(x_k),~w_k=\dfrac{1}{\epsilon(x_k)} +\end{align*} + \end{frame} +\begin{frame}\frametitle{Unfolding matrix} +\only<1>{ +In general one can write the distribution of events after the detector effects: +\begin{align} +P^{\rm{Det}}(x_d) = N \int \int dx_t~ dx_d~ P^{\rm{Phys}}(x_t) E(x_d \vert x_t), +\end{align} +where $N^{-1}=\int \int d x_t~ dx_d~ P^{\rm{Phys}}(x_t) E(x_d \vert x_t)$ and $(x_d \vert x_t)$ denotes the efficiency $\epsilon(x_t)$ and resolution of the detector $ R(x_d\vert x_t)$: +\begin{align} +E(x_d \vert x_t) = \epsilon(x_t) R(x_d\vert x_t) +\end{align} +\pause +One can define the raw moments: +\begin{align} +Q_i^{(m)} = \int \int d x_t~ dx_d~\dual{f}_i(x_d) P^{(m)}(x_t) E(x_d \vert x_t) +\end{align} +The $m$ index corresponds to simulation sample that has $S_0$ and $S_m$ observables set to $\frac{1}{2}$ and rest to zero. +} +\only<2> +{ +Once again we can use MC estimator: +\begin{align} +Q_i^{(m)} \to \widehat{Q}_i^{(m)}= \frac{1}{N_t} \sum_i^{N_d} \dual{f}_i(x_d^{i,m}) +\end{align} +Linearity of the integral ensures that there has to exists a linear transformation: +\begin{align} + \label{eq:eqlin} +\vec{Q}=M\vec{S}, +\end{align} +where $M$ is so-called unfolding matrix, \pause given by the formula: +\begin{align} + M_{ij} = \begin{cases} + 2 Q_i^{(0)} & j = 0\,,\\ + 2\left(Q_i^{(j)} - Q_i^{(0)}\right) & j \neq 0\,,\\ + \end{cases} +\end{align} +Once we measured the moments $Q$ in data we can invert Eq. 11 and get the $\vec{S}$: +\begin{align} +\widehat{\vec{S}}=M^{-1} \widehat{\vec{Q}}, +\end{align} + +} +\end{frame} + + +\section{Toy Studies} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\placelogofalse +\begin{frame}\frametitle{Toy Validation} +\begin{columns} +\column{2.5in} +\begin{itemize} +\item All the statistics properties of MoM have been tested in numbers of TOY MC. +\item As long as you have~$\sim 30$ events your pulls are perfectly gaussian. +\item Uncertainty scales with $\frac{\alpha}{\sqrt{n}}$, $\alpha = \mathcal{O}(1)$. +\item Never observed any boundary problems. +\end{itemize} + +\column{2.5in} +\includegraphics[width=0.9\textwidth]{images/pull-Q2_5_6_S5_200.png}\\ +\includegraphics[width=0.9\textwidth]{images/Q2_1_2_S5.pdf} + +\end{columns} +\end{frame} +\placelogotrue +\begin{frame}\frametitle{Correlation of MoM with Likelihood} +\begin{columns} +\column{2.5in} +\begin{itemize} +\item MoM is highly correlated with LL. +\item Despite the correlation there can be difference of the order of statistical error. +\end{itemize} + +\column{2.5in} +\includegraphics[width=0.9\textwidth]{images/S3_scat.pdf}\\ + + +\end{columns} +\begin{columns} +\column{2.5in} +\includegraphics[width=0.9\textwidth]{images/S5_scat.pdf} + +\column{2.5in} + +\includegraphics[width=0.9\textwidth]{images/S7_scat.pdf} + +\end{columns} +\end{frame} + +\section{Conclusions} +\placelogotrue +\begin{frame}\frametitle{Conclusions} +\begin{enumerate} +\item MoM posses several big advantages with one drawback which is larger statistical uncertainty. +\item Allows us to go smaller $q^2$ bins (get ready for $1~\GeV^2$ soon!). +\item Alternative method of extracting the detector effects. +\item Can be applied to various rare decays. +\end{enumerate} + +\end{frame} \end{document} diff --git a/Kstmumu/Edinburgh/mchrzasz.toc b/Kstmumu/Edinburgh/mchrzasz.toc index d473083..d548949 100644 --- a/Kstmumu/Edinburgh/mchrzasz.toc +++ b/Kstmumu/Edinburgh/mchrzasz.toc @@ -1,5 +1,7 @@ \beamer@endinputifotherversion {3.24pt} \select@language {english} -\beamer@sectionintoc {1}{Motivation}{2}{0}{1} +\beamer@sectionintoc {1}{Motivation}{3}{0}{1} \beamer@sectionintoc {2}{Method of Moments}{7}{0}{2} \beamer@sectionintoc {3}{Systematic uncertainties}{13}{0}{3} +\beamer@sectionintoc {4}{Toy Studies}{19}{0}{4} +\beamer@sectionintoc {5}{Conclusions}{21}{0}{5}