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\title{Probing background with \\ Method of Moments for $\PBzero \to \PKstar \mu \mu$}  
\author{\underline{Marcin Chrzaszcz}$^{1,2}$}
\date{\today} 

\begin{document}

{
\institute{$^1$ University of Zurich, $^2$ Institute of Nuclear Physics}
\setbeamertemplate{footline}{} 
\begin{frame}
\logo{
\vspace{2 mm}
\includegraphics[height=1cm,keepaspectratio]{images/uzh.jpg}~
\includegraphics[height=1cm,keepaspectratio]{images/ifj.png}}

  \titlepage
\end{frame}
}
\institute{UZH,IFJ} 

\section[Outline]{}
\begin{frame}
\tableofcontents
\end{frame}

\section{Reminder}
\begin{frame}\frametitle{Plan}
Method of moments:
\begin{enumerate}
\item Last meeting showed how orthogonality of the does magic for method of moments.
\item Using toy MC (experimental math) checked the errors estimates.
\item Checked that it does not suffer from boundary conditions.
\item Many thanks to Tom for checking all my calculations.
\end{enumerate}
For today:
\begin{enumerate}
\item How this method behaves in terms of background?
\end{enumerate}

\end{frame}
\section{Theory introduction}
\begin{frame}\frametitle{What do we start with}
{~}
Let's assume for simplicity we have our pdf:
\begin{multline}
\dfrac{d^4\Gamma}{ \Gamma dq^2 dcos\theta_k dcos\theta_l d\phi}=\dfrac{9}{32\pi}( \dfrac{3}{4} (1-F_l) \sin^2 \theta_k + F_l\cos^2 \theta_k + ( \dfrac{1}{4}(1-Fl)\sin^2 \theta_k \\ -  F_l\cos^2) cos 2\theta_l  + S_3 \sin^2 \theta_k \sin^2 \theta_l \cos2\phi + S_4 \sin2 \theta_k \sin \theta_l \cos\phi +\\ S_5 \sin2 \theta_k \sin \theta_l \cos \phi +  (S_{6s} \sin^2 \theta_k) \cos \theta_l + \\ S_7 \sin 2\theta_k \sin \theta_l \sin \phi +  S_8 \sin 2 \theta_k \sin 2 \theta_l \sin phi + S_9 sin^2 \theta_k \sin^2 \theta_l \sin 2 \phi)
\end{multline}
What did we assume:
\begin{itemize}
\item  $S_{1x}$,~$J_{2x}$ can be parametrized by $F_l$. 
\item  $S_{6c}=0$.
\item In short what was in the paper.
\end{itemize}
\end{frame}


\begin{frame}\frametitle{Obtained moments 1}
{~}
Lets see how this works in practice:
\begin{equation}
 \dfrac{d^3\Gamma}{\Gamma dcos\theta_k dcos\theta_l d\phi } \sin^2 \theta_k  = \dfrac{2}{5}(2-F_l)
\end{equation}
\begin{equation}
 \dfrac{d^3\Gamma}{\Gamma dcos\theta_k dcos\theta_l d\phi } \cos^2 \theta_k = \dfrac{1}{5}(1+F_l)
\end{equation}
\begin{equation}
 \dfrac{d^3\Gamma}{\Gamma dcos\theta_k dcos\theta_l d\phi } \sin^2 \theta_k \cos 2\theta_l = -\dfrac{2}{25}(2+F_l)
\end{equation}
\begin{equation}
 \dfrac{d^3\Gamma}{\Gamma dcos\theta_k dcos\theta_l d\phi} \cos^2 \theta_k \cos 2\theta_l = -\dfrac{1}{25}(1+8F_l)
\end{equation}
\begin{equation}
 \dfrac{d^3\Gamma}{\Gamma dcos\theta_k dcos\theta_l d\phi} \sin^2 \theta_k \cos \theta_l = \dfrac{2 S_{6s} }{5}
\end{equation}

\end{frame}


\begin{frame}\frametitle{Obtained moments 2}
\begin{equation}
 \dfrac{d^3\Gamma}{\Gamma dcos\theta_k dcos\theta_l d\phi} \cos^2 \theta_k \cos \theta_l = \dfrac{S_{6s} }{10}
\end{equation}
\begin{equation}
 \dfrac{d^3\Gamma}{\Gamma dcos\theta_k dcos\theta_l d\phi }sin^2 \theta_k sin^2 \theta_l cos 2 \phi= \dfrac{8 S_3 }{25}
\end{equation}
\begin{equation}
 \dfrac{d^3\Gamma}{\Gamma dcos\theta_k dcos\theta_l d\phi } sin 2 \theta_k sin 2 \theta_k cos\phi= \dfrac{8 S_4 }{25}
\end{equation}
\begin{equation}
 \dfrac{d^3\Gamma}{\Gamma dcos\theta_k dcos\theta_l d\phi }sin2\theta_k sin\theta_l cos\phi = \dfrac{2 S_5 }{5}
\end{equation}
\begin{equation}
 \dfrac{d^3\Gamma}{\Gamma dcos\theta_k dcos\theta_l d\phi} sin 2 \theta_k sin \theta_l sin \phi = \dfrac{2 S_7 }{5}
\end{equation}
\begin{equation}
 \dfrac{d^3\Gamma}{\Gamma dcos\theta_k dcos\theta_l d\phi} sin 2 \theta_k sin2 \theta_l sin \phi = \dfrac{8 S_8 }{25}
\end{equation}

\end{frame}

\section{Background regions}

\begin{frame}\frametitle{Studied background region}
{~}

\begin{columns}

\column{2.5in}
\begin{itemize}
\item Defined $\PBzero$ mass bins: $1:(5,5,15)\cup 2:(5.15,5.22) \cup 3:(5.35,5.5) \cup 4:(5.5,6)~GeV$
\begin{enumerate}
\item Region 5:$(5.35, 6)$
\item Region 6:$(5, 5.22)$
\end{enumerate}
\item use the old $q^2$ bins:
\begin{itemize}
\item 0:$0, 2$
\item 1:$2,4.3$
\item 2:$4.3, 8,6$
\item 3:$10.1, 12.9$
\item 4:$14.2, 16$
\item 5:$16,19$
\end{itemize}
\end{itemize}

Please remember the numbers, we will need then later on.
\column{2.5in}
\includegraphics[scale=0.22]{plots2/PLOT.png}
\end{columns}


\end{frame}

\section{Results}
\begin{frame}\frametitle{Measurement of background moments}
{~}
\begin{columns}

\column{2.5in}

\includegraphics[scale=0.22]{plots2/Q2_0_S3.png}

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\includegraphics[scale=0.22]{plots2/Q2_1_S3.png}

\end{columns}
\end{frame}

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{~}
\begin{columns}

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\includegraphics[scale=0.22]{plots2/Q2_2_S3.png}

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\includegraphics[scale=0.22]{plots2/Q2_3_S3.png}

\end{columns}
\end{frame}

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\begin{frame}\frametitle{Measurement of background moments}
{~}
\begin{columns}

\column{2.5in}

\includegraphics[scale=0.22]{plots2/Q2_4_S3.png}

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\includegraphics[scale=0.22]{plots2/Q2_5_S3.png}

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\begin{frame}\frametitle{Measurement of background moments}
{~}
\begin{columns}

\column{2.5in}

\includegraphics[scale=0.22]{plots2/Q2_0_S4.png}

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\includegraphics[scale=0.22]{plots2/Q2_1_S4.png}

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\end{frame}

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\begin{frame}\frametitle{Measurement of background moments}
{~}
\begin{columns}

\column{2.5in}

\includegraphics[scale=0.22]{plots2/Q2_2_S4.png}

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\includegraphics[scale=0.22]{plots2/Q2_3_S4.png}

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\begin{columns}

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\includegraphics[scale=0.22]{plots2/Q2_4_S4.png}

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\includegraphics[scale=0.22]{plots2/Q2_5_S4.png}

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{~}
\begin{columns}

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\includegraphics[scale=0.22]{plots2/Q2_0_S5.png}

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\includegraphics[scale=0.22]{plots2/Q2_1_S5.png}

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{~}
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\includegraphics[scale=0.22]{plots2/Q2_2_S5.png}

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\includegraphics[scale=0.22]{plots2/Q2_3_S5.png}

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\includegraphics[scale=0.22]{plots2/Q2_4_S5.png}

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\includegraphics[scale=0.22]{plots2/Q2_5_S5.png}

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\includegraphics[scale=0.22]{plots2/Q2_0_S6s.png}

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\includegraphics[scale=0.22]{plots2/Q2_1_S6s.png}

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\includegraphics[scale=0.22]{plots2/Q2_2_S6s.png}

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\includegraphics[scale=0.22]{plots2/Q2_4_S6s.png}

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\includegraphics[scale=0.22]{plots2/Q2_5_S6s.png}

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{~}
\begin{columns}

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\includegraphics[scale=0.22]{plots2/Q2_0_S6c.png}

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\includegraphics[scale=0.22]{plots2/Q2_1_S6c.png}

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\includegraphics[scale=0.22]{plots2/Q2_2_S6c.png}

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\includegraphics[scale=0.22]{plots2/Q2_3_S6c.png}

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\includegraphics[scale=0.22]{plots2/Q2_4_S6c.png}

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\includegraphics[scale=0.22]{plots2/Q2_4_Fl2.png}

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\includegraphics[scale=0.22]{plots2/Q2_5_Fl2.png}

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{~}
\begin{columns}

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\includegraphics[scale=0.22]{plots2/Q2_0_Fl3.png}

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\includegraphics[scale=0.22]{plots2/Q2_1_Fl3.png}

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{~}
\begin{columns}

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\includegraphics[scale=0.22]{plots2/Q2_2_Fl3.png}

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\includegraphics[scale=0.22]{plots2/Q2_3_Fl3.png}

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{~}
\begin{columns}

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\includegraphics[scale=0.22]{plots2/Q2_4_Fl3.png}

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\includegraphics[scale=0.22]{plots2/Q2_5_Fl3.png}

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\begin{frame}\frametitle{Measurement of background moments}
{~}
\begin{columns}

\column{2.5in}

\includegraphics[scale=0.22]{plots2/Q2_0_Fl4.png}

\column{2.5in}

\includegraphics[scale=0.22]{plots2/Q2_1_Fl4.png}

\end{columns}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}\frametitle{Measurement of background moments}
{~}
\begin{columns}

\column{2.5in}

\includegraphics[scale=0.22]{plots2/Q2_2_Fl4.png}

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\includegraphics[scale=0.22]{plots2/Q2_3_Fl4.png}

\end{columns}
\end{frame}

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\begin{frame}\frametitle{Measurement of background moments}
{~}
\begin{columns}

\column{2.5in}

\includegraphics[scale=0.22]{plots2/Q2_4_Fl4.png}

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\includegraphics[scale=0.22]{plots2/Q2_5_Fl4.png}

\end{columns}
\end{frame}
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\section{Summary}
\begin{frame}\frametitle{Summary}
{~}
\begin{itemize}
\item Background moments are effectivelly 0
\item Apart from $F_l$ and $S_6$
\item $S_6$ is sizeable at the left hand sideband for certain bins, evidence of partially reconstructed semileptonic decays?
\item Because the moments are small, they should have small effect on the final result :)
\end{itemize}

Wish list:
\begin{itemize}
\item Repeat the same with smaller $q^2$ bins.
\item Optimise the binning in $q^2$ taking into account background systematics and error on signal
\item Do unfolding.
\end{itemize}
\end{frame}


\end{document}