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This behaviour is well known and was observed several times in toys when we done $\PB \to \PKstar \Pmu \Pmu$. +\item In case of small number events LL fits suffer from convergence problems. This behaviour is well known and was observed several times in toys for $\PB \to \PKstar \Pmu \Pmu$. \item LL can exhibit a bias when underlying physics model is not well known, incomplete or mismodeled. -\item The LL have problems converging when parameters of the \pdf are close to their physical boundaries, so-called ''boundary problem'' -\item Accessing uncertainty in LL in some cases requires application of computationally expensive Feldman-Cousins method. +\item The LL have problems converging when parameters of the \pdf are close to their physical boundaries.%, so-called ''boundary problem'' +\item Accessing uncertainty in LL fits sometimes requires application of computationally expensive Feldman-Cousins method. \end{enumerate} @@ -90,7 +90,7 @@ \begin{columns} \column{0.05in}{~} \column{2.2in} -\begin{center} MoM solves the above problems:\end{center} +\begin{center} MoM addresses the above problems:\end{center} \column{2.in} \only<3>{ @@ -118,8 +118,8 @@ %\begin{center} MoM solves the above problems:\end{center} \begin{exampleblock}{Advantages of MoM} \begin{itemize} - \item Each observable can be determined separately from other. - \item Uncertainly follows perfectly $1/\sqrt{N}$ scaling. + \item "For each observable, the mean value can be determined independently from all other observables. + \item Uncertainly follows perfectly $1/\sqrt{N}$ scaling, where N is number of signal events. \end{itemize} \end{exampleblock} } @@ -142,7 +142,7 @@ \begin{frame}\frametitle{Introduction to MoM} Let us a define a probability density function \pdf of a decay: \begin{align} -P(\nuvec, \thvec) \equiv \sum_i S_i(\nuvec) \times f_i(\thvec) + P(\nuvec, \thvec) \equiv \sum_i S_i(\nuvec) \times f_i(\thvec) \end{align} Let's assume further that there exist a dual basis: $\lbrace f_i(\thvec) \rbrace$, $\{\dual{f}_i(\thvec)\}$ that the orthogonality relation is valid: \begin{equation} @@ -189,8 +189,8 @@ \begin{equation} \mathrm{Cov} [S_i, S_j]=\dfrac{1}{N-1} \sum_{k=1}^N [ \widehat{S_i} - \dual{f}_i(x_k) ][ \widehat{S_j} - \dual{f}_j(x_k) ] \end{equation} -\pause -Thanks to the CLT both equations are satisfied. +%\pause +%Thanks to the CLT both equations are satisfied. @@ -230,7 +230,7 @@ \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 muon system of this kind of decays has a fixed angular dependence in terms of $\theta_1$ (lepton helicity angle) and $\theta_3$ (azimuthal angle). \item The hadron system can have arbitrary large angular momentum. \end{itemize} } @@ -269,7 +269,7 @@ \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 To take into account the acceptance effects one needs to simulate the a large sample of MC events.\\ \item Try to figure out the efficiency function. \item Number of possibilities. \item Then you can just weight events: @@ -287,41 +287,28 @@ \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), +P^{\rm{Det}}(x_d) = N \int \int dx_t~ 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)$: +where $N^{-1}=\int \int d x_t~ dx_d~ P^{\rm{Phys}}(x_t) E(x_d \vert x_t)$ and $E(x_d \vert x_t)$ 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 +%\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} +Once we measured the moments $Q$ in data we can invert Eq. 11 and get the $\vec{S}$: $\widehat{\vec{S}}=M^{-1} \widehat{\vec{Q}}.$ + + + + } \end{frame} @@ -380,13 +367,53 @@ \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 MoM viable alternative to LL fits. +\item Allows LHCb 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. +\item Method is universally applicable, as long as an orthonormal + basis for the \pdf exists. \end{enumerate} \end{frame} +\begin{frame} + +\begin{Huge} +BACKUP +\end{Huge} + +\end{frame} + + + + \begin{frame}\frametitle{LHCb detector} +\begin{columns} +\column{3.in} +\begin{center} +\includegraphics[width=0.98\textwidth]{det.jpg} +\end{center} + +\column{2.0in} +\begin{footnotesize} + + + LHCb is a forward spectrometer: + \begin{itemize} + \item Excellent vertex resolution. + \item Efficient trigger. + \item High acceptance for $\Ptau$ and $\PB$. + \item Great Particle ID + \end{itemize} + + + +\end{footnotesize} +\end{columns} + +\end{frame} + + + + \end{document} diff --git a/Kstmumu/Edinburgh/mchrzasz.toc b/Kstmumu/Edinburgh/mchrzasz.toc index d548949..d3419c2 100644 --- a/Kstmumu/Edinburgh/mchrzasz.toc +++ b/Kstmumu/Edinburgh/mchrzasz.toc @@ -2,6 +2,6 @@ \select@language {english} \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} +\beamer@sectionintoc {3}{Systematic uncertainties}{12}{0}{3} +\beamer@sectionintoc {4}{Toy Studies}{17}{0}{4} +\beamer@sectionintoc {5}{Conclusions}{19}{0}{5}