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% Seth Brown, Ph.D.
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% see the macros.tex file for definitions
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% title slide definition
\title{Updates on activities.}
%\subtitle{a bias report}
\author{ Marcin Chrz\k{a}szcz$^{1,2}$ ,  Nicola Serra$^{1}$ }
\institute[UTH, IFJ]
{
 %\begin{tiny}
$ ^1$ University of Zurich , $ ^2$ Institute of Nuclear Physics, Krakow,  
 %\end{tiny}smallsmall
}
  
 
\date{ \today }
%--------------------------------------------------------------------
%                           Introduction
%--------------------------------------------------------------------
 
\begin{document}
 
 
 
 
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%--------------------------------------------------------------------
%                          OUTLINE
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\section[Outline]{}
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\tableofcontents
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%-------------------------------------------------------------------
%                          Introduction
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\title{Update on analysis}
 
 
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\section{$\Ptau \to 3\mu$ many solutions}
\begin{frame}\frametitle{$\PDs \to \eta(\mu\mu\gamma)\mu \nu$}
\only<1>
{
\begin{itemize}
\item Last time I showed you the fits with $\eta$ background. 
\item Now the fits are updated with the $\eta$ calibrated $\PDs \to \eta(\mu\mu\gamma)\mu \nu$ yield.
\item Still everything looks fine.
\end{itemize}
\includegraphics[scale=0.15]{eta/eta_bck.png} \\
}
 
 
	\textref {M.Chrz\k{a}szcz, N.Serra 2014}
\end{frame}
 
\begin{frame}\frametitle{$\PDs \to \eta(\mu\mu\gamma)\mu \nu$}
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\includegraphics[scale=0.09]{figs/exp_with_eta_2011/pid_0p8_1p0001geo0p45_0p61.png} \\
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	\textref {M.Chrz\k{a}szcz, N.Serra 2014}
\end{frame}
 
 
 
 
 
\begin{frame}\frametitle{Expected limit}
\begin{itemize}
\item Note was send to conveners on Monday.
\end{itemize}
\includegraphics[scale=1.4]{hahaha.jpg}
 
\begin{itemize}
\item We decided to give two limits with $\PDs \to \eta(\mu\mu\gamma)\mu \nu$ and with $\eta$ veto.
\end{itemize}
V0 of the note(no systematics in the limit):
\begin{enumerate}
\item $\eta$ veto: $Br(\mu\mu\mu)<4.8\times 10^{-8}$
\item $\eta$: $Br(\mu\mu\mu)<4.7\times 10^{-8}$
\end{enumerate}
Yesterday I evaluated the limits with background systematics. The limits gets around: $5.1\times 10^{-8}$
 
 
 
 
	\textref {M.Chrz\k{a}szcz, N.Serra 2014}
\end{frame}
 
 
 
\section{$\PKstar\mu\mu$}
 
 
\begin{frame}\frametitle{Unfolding for $\PKstar \mu \mu$}
\begin{itemize}
\item Recently every one had statistics problems.
\item I felt alienated that i have none.
\item Thank god that Nico provided some problem :)
\end{itemize}
 
 
 
	\textref {M.Chrz\k{a}szcz, N.Serra 2014}
\end{frame}
 
 
 
 
 
\begin{frame}\frametitle{Nico hypothesis}
We have our PDF:
\begin{multline}
PDF=\dfrac{d^4\Gamma}{dq^2 dcos\theta_k dcos\theta_l d\phi}=\dfrac{9}{32\pi}( J_{1s}sin^2 \theta_k + J_{1c}cos^2 \theta_k + (J_{2s}sin^2 \theta_k + \\ J_{2c}cos^2) cos 2\theta_l  + J_3 sin^2 \theta_k sin^2 \theta_l cos2\phi + J_4 sin2 \theta_k sin \theta_l cos\phi +\\ J_5 sin2 \theta_k sin \theta_l cos \phi +  (J_{6s} sin^2 \theta_k + J_{6c} cos^2 \theta_k) cos \theta_l + \\ J_7 sin 2\theta_k sin \theta_l sin \phi +  J_8 sin 2 \theta_k sin 2 \theta_l sin phi + J_9 sin^2 \theta_k sin^2 \theta_l sin 2 \phi)
\end{multline}
And corresponding moments measured moments: $M^{R}_i$ corresponding to the $i^{th}$ moment.
Nicos hypothesis: 
The true Moments:  $M^{T}_i = A^i_j M^{R}_j$.
But he can't prove it and it looks insane at the first looks. So in the process of proving he is wrong I proved that this is true.
 
 
 
 
	\textref {M.Chrz\k{a}szcz, N.Serra 2014}
\end{frame}
 
\begin{frame}\frametitle{Nico hypothesis, proof}
So the true moments:
 $M^{T}_i = \int PDF f_i = J_i \int f_i^2= J_i \times const$
Now for the measurements you need to have some efficiency: $\epsilon(dcos\theta_k, dcos\theta_l, d\phi)$, we assume it is $C^{\infty}$. So one can Taylor expand this function. \\
The only thing I need to proof now is that the arbitrary element in the Taylor expansion can be write using all $J_i$ in the first order:
$M^{R}_i=\int PDF  f_i \cos\theta^x_k \cos^y\theta_l \phi^z = \sum_j J_j \int f_i f_j  \cos\theta^x_k \cos^y\theta_l \phi^z = \sum_j J_j const_j$
Which ends the proof. I calculated explicit matrix element correspond to $\cos\theta^x_k \cos^y\theta_l \phi^z $, but it's 3 pages long(in the attachment if one likes horrors).
 
 
 
 
 
 
	\textref {M.Chrz\k{a}szcz, N.Serra 2014}
\end{frame}
 
 
 
 
\begin{frame}\frametitle{Back to the unfolding} 
 
The unfolding for the method of moments can(and will) be done with 2 unfolding approaches.
 
\begin{itemize}
\item Unfolding using matrix.
\item Unfolding using event weighting using the same weights as for the fits. 
\item We can check internal consistency.
\end{itemize}
 
 
 
 
 
 
 
	\textref {M.Chrz\k{a}szcz, N.Serra 2014}
\end{frame}
 
 
 
\end{document}