\documentclass[xcolor=svgnames]{beamer}
\usepackage[utf8]{inputenc}
\usepackage[english]{babel}
\usepackage{polski}
%\usepackage{amssymb,amsmath}
%\usepackage[latin1]{inputenc}
%\usepackage{amsmath}
%\newcommand\abs[1]{\left|#1\right|}
\usepackage{amsmath}
\newcommand\abs[1]{\left|#1\right|}
\usepackage{hepnicenames}
\usepackage{hepunits}
\usepackage{color}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
\definecolor{mygreen}{cmyk}{0.82,0.11,1,0.25}


\usetheme{Sybila} 

\title[ Comparison of angular methods III]{ Comparison of angular methods III}
\author{Marcin Chrz\k{a}szcz$^{1,2}$, Nicola Serra$^{1}$}
\institute{$^1$~University of Zurich,\\ $^2$~Institute of Nuclear Physics, Krakow}
\date{\today}

\begin{document}
% --------------------------- SLIDE --------------------------------------------
\frame[plain]{\titlepage}
\author{Marcin Chrz\k{a}szcz}
% ------------------------------------------------------------------------------
% --------------------------- SLIDE --------------------------------------------

\begin{frame}\frametitle{Quo vadis $\PBzero \to \PKstar \mu \mu$?}

\center \includegraphics[width=0.8\paperwidth]{diagram.png}\\

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}\frametitle{Quo vadis $\PBzero \to \PKstar \mu \mu$?}

\center \includegraphics[width=0.8\paperwidth]{diagram_com.png}\\

\end{frame}


\section{Introduction}
\begin{frame}\frametitle{How to use Moments}
\begin{columns}
\column{3in}
\begin{enumerate}
\item Method of moments still uses fits.
\item Extended likelihood fit is used to extract number of signal and background events.
\item Calculate raw moments in bck window$~masss>5350$ and signal window$~mass\in[5230,5330]$
\only<1>{\item Use a simple weighted average:}
\only<2>{\item Extract signal moments:}
\end{enumerate}

\column{2.2in}
\includegraphics[width=0.9\textwidth]{mass3.png}

\end{columns}
{~}\\
\only<1>{
\begin{equation}
M_{mix}= \dfrac{n_{sig} M_{sig}+n_{bkg}M_{bkg}}{n_{sig}+n_{bkg}}
\end{equation}
}
\only<2>{
\begin{equation}
M_{sig}=\dfrac{ (n_{sig}+n_{bkg}) M_{mix}}{n_{sig}} - \dfrac{n_{bkg} m_{bkg}}{n_{sig}}
\end{equation}
}

\end{frame}

\begin{frame}\frametitle{How to use fits 1/2}
\begin{columns}
\column{3.0in}
\begin{itemize}
\item Assuming we can decouple in the PDF the angular and mass part:
$PDF(\theta_l, \theta_k, \phi, m)= PDF_1(\theta_l, \theta_k, \phi) \times PDF_2(m)$

\item $PDF_1$ is our PDF defined in previous presentation.
\item $PDF_2$ is double CB as defined in Christoph toy creation. 
\item For background we assume 2nd order Chebyshev polynomials($PDF_{1,bck}$)
\item And single exponential for bkg part($PDF_{2,bck}$)
\end{itemize}

\column{2.2in}
\includegraphics[width=0.6\textwidth]{cosk_tmp.png}\\
\includegraphics[width=0.6\textwidth]{cosl_tmp.png}
\end{columns}

\end{frame}

\begin{frame}\frametitle{How to use fits 2/2}
\begin{columns}
\column{3.2in}
What is left free in the fit:
\begin{itemize}
\item $\lambda$ for exponent.
\item $F_l$, $S_x$ for angular distribution.
\item Relative normalization of signal and bkg.
\end{itemize}
Other remarks:
\begin{enumerate}
\item I am using good old Roofit, not custom developed fitter.
\item No s-wave in the fits, or systematics include in this study.
\item individual toys are now provided by Christoph, so one can compare $1:1$.
\end{enumerate}
\column{2.2in}
\includegraphics[width=0.6\textwidth]{phi_tmp.png}\\
\includegraphics[width=0.6\textwidth]{mass.png}
\end{columns}

\end{frame}


\begin{frame}\frametitle{Small things}
General remark:
\begin{itemize}
\item All pull plots/fits are available: \href{http://nz17-p1.ifj.edu.pl/work_public/LHCb/Kst_mumu/Face2Face_meeting/}{LINK}
\item User: lhcb, password: $2924$.


\end{itemize}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}\frametitle{MM results $1GeV$ binning, Mean bias}


 \tiny{  
    \begin{tabular}{|*{10}{c|}}
	
 \hline
$Q^2$ & \scalebox{0.5}{$F_l$} & \scalebox{0.5}{$S_3$} & \scalebox{0.5}{$S_4$} & \scalebox{0.5}{$S_5$} & \scalebox{0.5}{$S_6$} \\ \hline                                                                      
0 & \scalebox{0.5}{\textcolor{blue}{$-0.431 \pm 0.0184(-23.4)$}} & \scalebox{0.5}{$0.00218 \pm 0.0186(0.118)$} & \scalebox{0.5}{$0.0389 \pm 0.0183(2.13)$} & \scalebox{0.5}{$0.0104 \pm 0.0183(0.564)$} & \scalebox{0.5}{$0.0357 \pm 0.0183(1.96)$}           \\ \hline 
1 & \scalebox{0.5}{$-0.029 \pm 0.0188(-1.56)$} & \scalebox{0.5}{$0.0247 \pm 0.0187(1.32)$} & \scalebox{0.5}{$0.016 \pm 0.0184(0.872)$} & \scalebox{0.5}{$-0.0125 \pm 0.0191(-0.657)$} & \scalebox{0.5}{$-0.00833 \pm 0.0186(-0.449)$}       \\ \hline 
2 & \scalebox{0.5}{$0.0090 \pm 0.0181(0.501)$} & \scalebox{0.5}{$-0.0226 \pm 0.0186(-1.21)$} & \scalebox{0.5}{$-0.00623 \pm 0.018(-0.347)$} & \scalebox{0.5}{$-0.0235 \pm 0.018(-1.3)$} & \scalebox{0.5}{$-0.0201 \pm 0.0183(-1.1)$}        \\ \hline 
3 & \scalebox{0.5}{$-0.046 \pm 0.0183(-2.56)$} & \scalebox{0.5}{$-0.00942 \pm 0.0183(-0.515)$} & \scalebox{0.5}{$-0.00109 \pm 0.0188(-0.0579)$} & \scalebox{0.5}{$0.0108 \pm 0.0188(0.577)$} & \scalebox{0.5}{$-0.0519 \pm 0.0184(-2.83)$}  \\ \hline 
4 & \scalebox{0.5}{$-0.031 \pm 0.0183(-1.71)$} & \scalebox{0.5}{$0.0268 \pm 0.0181(1.48)$} & \scalebox{0.5}{$-0.00687 \pm 0.0182(-0.378)$} & \scalebox{0.5}{$-0.00106 \pm 0.0187(-0.0568)$} & \scalebox{0.5}{$0.00908 \pm 0.0189(0.481)$}   \\ \hline 
5 & \scalebox{0.5}{$-0.034 \pm 0.0178(-1.96)$} & \scalebox{0.5}{$0.0231 \pm 0.0189(1.22)$} & \scalebox{0.5}{$-0.0169 \pm 0.0183(-0.92)$} & \scalebox{0.5}{$-0.0138 \pm 0.0183(-0.75)$} & \scalebox{0.5}{$9.56e-05 \pm 0.0184(0.00521)$}    \\ \hline 
6 & \scalebox{0.5}{$-0.031 \pm 0.0186(-1.69)$} & \scalebox{0.5}{$0.0184 \pm 0.0183(1)$} & \scalebox{0.5}{$-0.000709 \pm 0.0182(-0.039)$} & \scalebox{0.5}{$-0.00728 \pm 0.0189(-0.386)$} & \scalebox{0.5}{$0.0223 \pm 0.0182(1.22)$}       \\ \hline 
7 & \scalebox{0.5}{$0.0042 \pm 0.0188(0.22)$} & \scalebox{0.5}{$0.022 \pm 0.0181(1.22)$} & \scalebox{0.5}{$-0.00412 \pm 0.0188(-0.22)$} & \scalebox{0.5}{$0.0044 \pm 0.0187(0.236)$} & \scalebox{0.5}{$0.0291 \pm 0.0188(1.55)$}           \\ \hline 
8 & \scalebox{0.5}{$0.0061 \pm 0.018(0.34)$} & \scalebox{0.5}{$0.0118 \pm 0.0184(0.643)$} & \scalebox{0.5}{$-0.0169 \pm 0.0185(-0.915)$} & \scalebox{0.5}{$0.0244 \pm 0.0182(1.34)$} & \scalebox{0.5}{$0.00532 \pm 0.0188(0.283)$}         \\ \hline 
9 & \scalebox{0.5}{$0.0056 \pm 0.0184(0.30)$} & \scalebox{0.5}{$-0.00938 \pm 0.0182(-0.514)$} & \scalebox{0.5}{$-0.0405 \pm 0.0182(-2.22)$} & \scalebox{0.5}{$0.00885 \pm 0.0182(0.487)$} & \scalebox{0.5}{$0.0361 \pm 0.0185(1.96)$}      \\ \hline 
10 & \scalebox{0.5}{$-0.0156 \pm 0.0185(-0.84)$} & \scalebox{0.5}{$-0.00611 \pm 0.018(-0.34)$} & \scalebox{0.5}{$-0.0241 \pm 0.0184(-1.31)$} & \scalebox{0.5}{$0.0204 \pm 0.0186(1.09)$} & \scalebox{0.5}{$-0.00378 \pm 0.0189(-0.201)$}   \\ \hline 
11 & \scalebox{0.5}{$-0.0079 \pm 0.018(-0.44)$} & \scalebox{0.5}{$0.0539 \pm 0.0182(2.96)$} & \scalebox{0.5}{$-0.0063 \pm 0.0189(-0.334)$} & \scalebox{0.5}{$-0.0297 \pm 0.019(-1.57)$} & \scalebox{0.5}{$-0.00305 \pm 0.0183(-0.166)$}    \\ \hline 

    \end{tabular}
   
 \begin{tabular}{|*{10}{c|}}                                                                                                                                                                                                                                                                                                                                                                                                                           
 \hline
$Q^2$ &  \scalebox{0.5}{$S_7$} & \scalebox{0.5}{$S_8$} & \scalebox{0.5}{$S_9$} \\ \hline        
0 & \scalebox{0.5}{$-0.0179 \pm 0.018(-0.997)$} & \scalebox{0.5}{$-0.0311 \pm 0.0187(-1.66)$} & \scalebox{0.5}{$0.0274 \pm 0.0184(1.49)$} \\ \hline           
1 & \scalebox{0.5}{$-0.0221 \pm 0.0183(-1.21)$} & \scalebox{0.5}{$-0.0592 \pm 0.0185(-3.2)$} & \scalebox{0.5}{$0.0151 \pm 0.0181(0.836)$} \\ \hline           
2 & \scalebox{0.5}{$-0.00449 \pm 0.0185(-0.243)$} & \scalebox{0.5}{$-0.00996 \pm 0.0188(-0.529)$} & \scalebox{0.5}{$0.00609 \pm 0.0186(0.328)$} \\ \hline     
3 & \scalebox{0.5}{$0.0274 \pm 0.0177(1.55)$} & \scalebox{0.5}{$-0.00217 \pm 0.0186(-0.117)$} & \scalebox{0.5}{$-0.00915 \pm 0.0187(-0.49)$} \\ \hline        
4 & \scalebox{0.5}{$0.0156 \pm 0.0181(0.861)$} & \scalebox{0.5}{$-0.0141 \pm 0.0186(-0.759)$} & \scalebox{0.5}{$0.00498 \pm 0.0178(0.279)$} \\ \hline         
5& \scalebox{0.5}{$0.0158 \pm 0.0186(0.848)$} & \scalebox{0.5}{$0.00727 \pm 0.0191(0.381)$} & \scalebox{0.5}{$0.029 \pm 0.0185(1.57)$} \\ \hline              
6& \scalebox{0.5}{$-0.00232 \pm 0.0186(-0.125)$} & \scalebox{0.5}{$0.00974 \pm 0.0177(0.551)$} & \scalebox{0.5}{$-0.00613 \pm 0.018(-0.341)$}\\ \hline        
7& \scalebox{0.5}{$-0.00853 \pm 0.0184(-0.463)$} & \scalebox{0.5}{$-0.00649 \pm 0.0182(-0.356)$} & \scalebox{0.5}{$0.00402 \pm 0.0183(0.22)$} \\ \hline       
8& \scalebox{0.5}{$-0.0137 \pm 0.0185(-0.739)$} & \scalebox{0.5}{$-0.0167 \pm 0.0185(-0.903)$} & \scalebox{0.5}{$-0.0278 \pm 0.0183(-1.52)$} \\ \hline        
9& \scalebox{0.5}{$-0.0229 \pm 0.019(-1.2)$} & \scalebox{0.5}{$-0.0506 \pm 0.0183(-2.77)$} & \scalebox{0.5}{$-0.0269 \pm 0.0183(-1.46)$} \\ \hline            
10& \scalebox{0.5}{$-0.0155 \pm 0.0185(-0.84)$} & \scalebox{0.5}{$0.0276 \pm 0.0184(1.5)$} & \scalebox{0.5}{$0.0137 \pm 0.0185(0.742)$} \\ \hline              
11& \scalebox{0.5}{$-0.0503 \pm 0.0183(-2.75)$} & \scalebox{0.5}{$-0.0137 \pm 0.0181(-0.758)$} & \scalebox{0.5}{$0.0169 \pm 0.0187(0.902)$} \\ \hline          

    \end{tabular}
    }

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%                                                                                                                                                                                                                                                                                                                                                                                                  
\begin{frame}\frametitle{MM results $1GeV$ binning, Sigma bias}                                                                                                                                                                                                                                                                                                                                                                                             
                                                                                                                                                                                                                                                                                                                                                                                                                                                           
                                                                                                                                                                                                                                                                                                                                                                                                                                                           
 \tiny{                                                                                                                                                                                                                                                                                                                                                                                                                                                    
    \begin{tabular}{|*{10}{c|}}    
           \hline                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
$Q^2$ & \scalebox{0.5}{$F_l$} & \scalebox{0.5}{$S_3$} & \scalebox{0.5}{$S_4$} & \scalebox{0.5}{$S_5$} & \scalebox{0.5}{$S_6$} \\ \hline   
0 & \scalebox{0.5}{$0.969 \pm 0.014(2.24)$} & \scalebox{0.5}{$0.979 \pm 0.0142(1.45)$} & \scalebox{0.5}{$0.958 \pm 0.0136(3.11)$} & \scalebox{0.5}{$0.969 \pm 0.0139(2.21)$} & \scalebox{0.5}{$0.975 \pm 0.014(1.8)$} \\ \hline  
1 & \scalebox{0.5}{$0.998 \pm 0.0154(0.128)$} & \scalebox{0.5}{$0.99 \pm 0.0141(0.684)$} & \scalebox{0.5}{$0.982 \pm 0.014(1.31)$} & \scalebox{0.5}{$1 \pm 0.0142(-0.327)$} & \scalebox{0.5}{$0.988 \pm 0.0152(0.812)$} \\ \hline  
2 & \scalebox{0.5}{$0.957 \pm 0.0134(3.23)$} & \scalebox{0.5}{$0.989 \pm 0.0142(0.807)$} & \scalebox{0.5}{$0.96 \pm 0.0142(2.79)$} & \scalebox{0.5}{$0.965 \pm 0.0135(2.6)$} & \scalebox{0.5}{$0.973 \pm 0.0143(1.9)$} \\ \hline  
3 & \scalebox{0.5}{$0.971 \pm 0.0139(2.11)$} & \scalebox{0.5}{$0.971 \pm 0.0137(2.1)$} & \scalebox{0.5}{$0.996 \pm 0.0152(0.277)$} & \scalebox{0.5}{$0.995 \pm 0.0142(0.374)$} & \scalebox{0.5}{$0.967 \pm 0.014(2.39)$} \\ \hline  
4 & \scalebox{0.5}{$0.971 \pm 0.014(2.09)$} & \scalebox{0.5}{$0.965 \pm 0.0138(2.51)$} & \scalebox{0.5}{$0.966 \pm 0.0137(2.46)$} & \scalebox{0.5}{$0.994 \pm 0.0139(0.44)$} & \scalebox{0.5}{$1 \pm 0.0142(0.00431)$} \\ \hline  
5 & \scalebox{0.5}{$0.947 \pm 0.0136(3.88)$} & \scalebox{0.5}{$1 \pm 0.0138(-0.165)$} & \scalebox{0.5}{$0.967 \pm 0.0138(2.4)$} & \scalebox{0.5}{$0.978 \pm 0.0137(1.63)$} & \scalebox{0.5}{$0.981 \pm 0.0132(1.46)$} \\ \hline  
6 & \scalebox{0.5}{$0.983 \pm 0.0142(1.19)$} & \scalebox{0.5}{$0.981 \pm 0.0139(1.35)$} & \scalebox{0.5}{$0.967 \pm 0.0134(2.46)$} & \scalebox{0.5}{$1 \pm 0.0145(-0.126)$} & \scalebox{0.5}{$0.969 \pm 0.0134(2.29)$} \\ \hline  
7 & \scalebox{0.5}{$1 \pm 0.0144(-0.01)$} & \scalebox{0.5}{$0.968 \pm 0.0142(2.26)$} & \scalebox{0.5}{$0.993 \pm 0.0143(0.474)$} & \scalebox{0.5}{$0.989 \pm 0.0136(0.775)$} & \scalebox{0.5}{$0.991 \pm 0.0138(0.671)$} \\ \hline  
8 & \scalebox{0.5}{$0.956 \pm 0.0133(3.33)$} & \scalebox{0.5}{$0.98 \pm 0.0138(1.43)$} & \scalebox{0.5}{$0.983 \pm 0.0145(1.18)$} & \scalebox{0.5}{$0.966 \pm 0.0136(2.46)$} & \scalebox{0.5}{$0.996 \pm 0.0145(0.309)$}  \\ \hline  
9 & \scalebox{0.5}{$0.959 \pm 0.013(3.14)$} & \scalebox{0.5}{$0.968 \pm 0.0145(2.21)$} & \scalebox{0.5}{$0.954 \pm 0.0128(3.64)$} & \scalebox{0.5}{$0.963 \pm 0.013(2.8)$} & \scalebox{0.5}{$0.985 \pm 0.0142(1.03)$} \\ \hline  
10 & \scalebox{0.5}{$0.972 \pm 0.0131(2.11)$} & \scalebox{0.5}{$0.955 \pm 0.0131(3.46)$} & \scalebox{0.5}{$0.978 \pm 0.0141(1.59)$} & \scalebox{0.5}{$0.991 \pm 0.014(0.635)$} & \scalebox{0.5}{$1.01 \pm 0.0137(-0.502)$} \\ \hline  
11 & \scalebox{0.5}{$0.952 \pm 0.014(3.43)$} & \scalebox{0.5}{$0.969 \pm 0.0141(2.22)$} & \scalebox{0.5}{$0.999 \pm 0.0146(0.0445)$} & \scalebox{0.5}{$1 \pm 0.0144(-0.22)$} & \scalebox{0.5}{$0.979 \pm 0.0143(1.45)$}   \\ \hline  

      
      
   \end{tabular}                                                                                                                                                                                                                                                                                                                                                                                                                                          
  \begin{columns}
  \column{2in}
\begin{tabular}{|c|c|c|c|}                                                                                                                                                                                                                                                                                                                                                                                                                               
 \hline                                                                                                                                                                                                                                                                                                                                                                                                                                                    
$Q^2$ & \scalebox{0.5}{$S_7$} & \scalebox{0.5}{$S_8$} & \scalebox{0.5}{$S_9$} \\ \hline             

0 & \scalebox{0.5}{$0.958 \pm 0.0139(2.98)$} & \scalebox{0.5}{$0.994 \pm 0.0151(0.42)$} & \scalebox{0.5}{$0.972 \pm 0.0145(1.91)$} \\ \hline  
1 & \scalebox{0.5}{$0.969 \pm 0.0138(2.24)$} & \scalebox{0.5}{$0.98 \pm 0.0145(1.4)$} & \scalebox{0.5}{$0.963 \pm 0.0137(2.74)$} \\ \hline  
2 & \scalebox{0.5}{$0.983 \pm 0.0145(1.17)$} & \scalebox{0.5}{$0.998 \pm 0.0146(0.147)$} & \scalebox{0.5}{$0.986 \pm 0.0143(0.98)$} \\ \hline   
3 & \scalebox{0.5}{$0.942 \pm 0.013(4.43)$} & \scalebox{0.5}{$0.987 \pm 0.0132(0.957)$} & \scalebox{0.5}{$0.996 \pm 0.0145(0.295)$} \\ \hline  
4 & \scalebox{0.5}{$0.965 \pm 0.0137(2.58)$} & \scalebox{0.5}{$0.986 \pm 0.0138(1)$} & \scalebox{0.5}{$0.965 \pm 0.0131(2.24)$} \\ \hline       
5 & \scalebox{0.5}{$0.979 \pm 0.0135(1.55)$} & \scalebox{0.5}{$0.978 \pm 0.0149(1.46)$} & \scalebox{0.5}{$0.981 \pm 0.0143(1.33)$} \\ \hline  
6 & \scalebox{0.5}{$0.983 \pm 0.0139(1.22)$} & \scalebox{0.5}{$0.942 \pm 0.013(4.48)$} & \scalebox{0.5}{$0.955 \pm 0.0136(3.33)$} \\ \hline  
7 & \scalebox{0.5}{$0.979 \pm 0.0129(1.63)$} & \scalebox{0.5}{$0.97 \pm 0.0145(2.06)$} & \scalebox{0.5}{$0.972 \pm 0.0135(2.05)$} \\ \hline   
8 & \scalebox{0.5}{$0.979 \pm 0.014(1.5)$} & \scalebox{0.5}{$0.985 \pm 0.0139(1.08)$} & \scalebox{0.5}{$0.971 \pm 0.0136(2.13)$} \\ \hline     
9 & \scalebox{0.5}{$0.995 \pm 0.0139(0.368)$} & \scalebox{0.5}{$0.975 \pm 0.0142(1.73)$} & \scalebox{0.5}{$0.975 \pm 0.0137(1.82)$} \\ \hline  
10 & \scalebox{0.5}{$0.988 \pm 0.0132(0.907)$} & \scalebox{0.5}{$0.977 \pm 0.0142(1.59)$} & \scalebox{0.5}{$0.978 \pm 0.0142(1.56)$} \\ \hline  
11 & \scalebox{0.5}{$0.971 \pm 0.0149(1.97)$} & \scalebox{0.5}{$0.968 \pm 0.0142(2.25)$} & \scalebox{0.5}{$0.988 \pm 0.0137(0.886)$}  \\ \hline           
 \end{tabular} 
  \column{2in}
Conclusions:
\begin{itemize}
\item MM does not bias the mean!
\item There is a trend that it overestimates the errors.
\item From all possible things that could go wrong this is the best. 
\end{itemize}
\end{columns}
  }                                                                                                                                                                                                                                                                                                                                                                                                                                                      
                                                                                                                                                                                                                                                                                                                                                                                                                                                           
\end{frame}               
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%                                                                                                                                                                                                                                                                                                                                                                                                  
\begin{frame}\frametitle{Fit $1GeV$ binning, Mean bias}                                                                                                                                                                                                                                                                                                                                                                                             
\tiny{                                                                                                                                     
   \begin{tabular}{|*{10}{c|}}                                                                                                             
                                                                                                                                           
\hline                                                                                                                                     
$Q^2$ & \scalebox{0.6}{$F_l$} & \scalebox{0.6}{$S_3$} & \scalebox{0.6}{$S_4$} & \scalebox{0.6}{$S_5$} & \scalebox{0.6}{$S_6$} \\ \hline     

0 & \scalebox{0.6}{$-0.54 \pm 0.02091(-25.82)$} & \scalebox{0.6}{$0.01246 \pm 0.0213(0.585)$} & \scalebox{0.6}{$0.01828 \pm 0.02126(0.8598)$} & \scalebox{0.6}{$0.02513 \pm 0.02078(1.209)$} & \scalebox{0.6}{$0.0646 \pm 0.02081(3.104)$} \\ \hline   
1 & \scalebox{0.6}{$-0.27 \pm 0.02232(-12.1)$} & \scalebox{0.6}{$0.03261 \pm 0.02401(1.358)$} & \scalebox{0.6}{$0.08175 \pm 0.02376(3.441)$} & \scalebox{0.6}{$-0.1473 \pm 0.0246(-5.987)$} & \scalebox{0.6}{$-0.244 \pm 0.02276(-10.72)$} \\ \hline
2 & \scalebox{0.6}{$-0.7574 \pm 0.02208(-34.3)$} & \scalebox{0.6}{$-0.0307 \pm 0.02267(-1.354)$} & \scalebox{0.6}{$-0.07512 \pm 0.02344(-3.204)$} & \scalebox{0.6}{$0.03054 \pm 0.02523(1.21)$} & \scalebox{0.6}{$-0.2317 \pm 0.02327(-9.957)$} \\ \hline
3 & \scalebox{0.6}{$-0.7548 \pm 0.02333(-32.35)$} & \scalebox{0.6}{$-0.01416 \pm 0.02193(-0.6458)$} & \scalebox{0.6}{$-0.1786 \pm 0.02354(-7.588)$} & \scalebox{0.6}{$0.2604 \pm 0.02232(11.67)$} & \scalebox{0.6}{$-0.05868 \pm 0.02325(-2.524)$} \\ \hline 
4 & \scalebox{0.6}{$-0.5667 \pm 0.02163(-26.2)$} & \scalebox{0.6}{$0.0153 \pm 0.02265(0.6756)$} & \scalebox{0.6}{$-0.2313 \pm 0.02105(-10.99)$} & \scalebox{0.6}{$0.4372 \pm 0.02207(19.81)$} & \scalebox{0.6}{$0.03735 \pm 0.02385(1.566)$} \\ \hline  
5 & \scalebox{0.6}{$-0.4589 \pm 0.02156(-21.29)$} & \scalebox{0.6}{$0.0305 \pm 0.02265(1.347)$} & \scalebox{0.6}{$-0.2186 \pm 0.02141(-10.21)$} & \scalebox{0.6}{$0.5118 \pm 0.02161(23.69)$} & \scalebox{0.6}{$0.1218 \pm 0.02321(5.249)$} \\ \hline 
6 & \scalebox{0.6}{$-0.3408 \pm 0.02351(-14.5)$} & \scalebox{0.6}{$-0.005133 \pm 0.0235(-0.2184)$} & \scalebox{0.6}{$-0.3205 \pm 0.02116(-15.15)$} & \scalebox{0.6}{$0.628 \pm 0.02207(28.45)$} & \scalebox{0.6}{$0.2972 \pm 0.02256(13.17)$}\\ \hline
7 & \scalebox{0.6}{$-0.2309 \pm 0.02362(-9.778)$} & \scalebox{0.6}{$-0.003081 \pm 0.02186(-0.1409)$} & \scalebox{0.6}{$-0.4115 \pm 0.02268(-18.14)$} & \scalebox{0.6}{$0.6794 \pm 0.02274(29.88)$} & \scalebox{0.6}{$0.4305 \pm 0.02204(19.53)$} \\ \hline     
8 & \scalebox{0.6}{$0.3543 \pm 0.02837(12.49)$} & \scalebox{0.6}{$0.157 \pm 0.02254(6.969)$} & \scalebox{0.6}{$-0.3984 \pm 0.02223(-17.93)$} & \scalebox{0.6}{$0.3877 \pm 0.02171(17.86)$} & \scalebox{0.6}{$0.8208 \pm 0.02474(33.18)$} \\ \hline 
9 & \scalebox{0.6}{$0.2385 \pm 0.02778(8.586)$} & \scalebox{0.6}{$0.1185 \pm 0.02156(5.499)$} & \scalebox{0.6}{$-0.3564 \pm 0.02171(-16.42)$} & \scalebox{0.6}{$0.3164 \pm 0.02264(13.98)$} & \scalebox{0.6}{$0.7744 \pm 0.02437(31.78)$} \\ \hline 
10 & \scalebox{0.6}{$0.2395 \pm 0.0273(8.774)$} & \scalebox{0.6}{$0.2025 \pm 0.02106(9.617)$} & \scalebox{0.6}{$-0.3328 \pm 0.02126(-15.66)$} & \scalebox{0.6}{$0.2583 \pm 0.02247(11.5)$} & \scalebox{0.6}{$0.5419 \pm 0.02453(22.09)$} \\ \hline
11 & \scalebox{0.6}{$0.139 \pm 0.02676(5.194)$} & \scalebox{0.6}{$0.3433 \pm 0.02149(15.98)$} & \scalebox{0.6}{$-0.3043 \pm 0.02151(-14.15)$} & \scalebox{0.6}{$0.1217 \pm 0.02339(5.205)$} & \scalebox{0.6}{$0.3523 \pm 0.02402(14.67)$} \\ \hline


\end{tabular}  

 \begin{columns}
  \column{3in}
 \begin{tabular}{|*{10}{c|}}                                                                                                                                   
 \hline                                                                                                                                                        
$Q^2$ &  \scalebox{0.6}{$S_7$} & \scalebox{0.6}{$S_8$} & \scalebox{0.6}{$S_9$} \\ \hline                                                                       
0 & \scalebox{0.6}{$-0.03252 \pm 0.02039(-1.595)$} & \scalebox{0.6}{$-0.02852 \pm 0.02148(-1.328)$} & \scalebox{0.6}{$0.03152 \pm 0.0207(1.523)$}  \\ \hline          
1 & \scalebox{0.6}{$-0.01799 \pm 0.02456(-0.7325)$} & \scalebox{0.6}{$-0.03414 \pm 0.02435(-1.402)$} & \scalebox{0.6}{$0.04347 \pm 0.02277(1.909)$}  \\ \hline        
2 & \scalebox{0.6}{$-0.1108 \pm 0.02502(-4.429)$} & \scalebox{0.6}{$-0.007344 \pm 0.02559(-0.287)$} & \scalebox{0.6}{$0.04804 \pm 0.02257(2.129)$}  \\ \hline         
3 & \scalebox{0.6}{$-0.009689 \pm 0.02367(-0.4093)$} & \scalebox{0.6}{$0.03343 \pm 0.0234(1.429)$} & \scalebox{0.6}{$-0.01668 \pm 0.0209(-0.7982)$}  \\ \hline        
4 & \scalebox{0.6}{$-0.03027 \pm 0.02361(-1.282)$} & \scalebox{0.6}{$0.05424 \pm 0.02354(2.304)$} & \scalebox{0.6}{$-0.02207 \pm 0.02167(-1.018)$}  \\ \hline         
5 & \scalebox{0.6}{$-0.07159 \pm 0.02324(-3.08)$} & \scalebox{0.6}{$0.02229 \pm 0.02396(0.9304)$} & \scalebox{0.6}{$-0.03396 \pm 0.02147(-1.582)$}  \\ \hline         
6 & \scalebox{0.6}{$-0.06034 \pm 0.02395(-2.52)$} & \scalebox{0.6}{$0.04234 \pm 0.02284(1.854)$} & \scalebox{0.6}{$-0.01762 \pm 0.02174(-0.8102)$}  \\ \hline         
7 & \scalebox{0.6}{$-0.05002 \pm 0.02386(-2.097)$} & \scalebox{0.6}{$0.03382 \pm 0.02289(1.477)$} & \scalebox{0.6}{$0.009742 \pm 0.02147(0.4538)$}  \\ \hline         
8 & \scalebox{0.6}{$-0.05097 \pm 0.0245(-2.08)$} & \scalebox{0.6}{$-0.02328 \pm 0.02359(-0.9872)$} & \scalebox{0.6}{$-0.03285 \pm 0.023(-1.428)$}  \\ \hline          
9 & \scalebox{0.6}{$-0.03206 \pm 0.02543(-1.261)$} & \scalebox{0.6}{$-0.009805 \pm 0.02334(-0.4201)$} & \scalebox{0.6}{$-0.04927 \pm 0.02272(-2.169)$}  \\ \hline     
10 & \scalebox{0.6}{$-0.02589 \pm 0.02487(-1.041)$} & \scalebox{0.6}{$-0.005148 \pm 0.02376(-0.2166)$} & \scalebox{0.6}{$-0.02395 \pm 0.02422(-0.9889)$}  \\ \hline    
11 & \scalebox{0.6}{$-0.04134 \pm 0.02664(-1.552)$} & \scalebox{0.6}{$-0.03522 \pm 0.02517(-1.399)$} & \scalebox{0.6}{$-0.02359 \pm 0.0246(-0.9592)$}  \\ \hline       
                                                                                                                                              
    \end{tabular}                                                                                                                                              
                                                                                                                                               
\column{1.5in}
\begin{itemize}
\item With agreement with Michel previous studies \href{https://indico.cern.ch/event/290857/contribution/3/material/slides/0.pdf}{LINK}
\end{itemize}                                                                                                                                           
                                                                                                                                              

                                                                                                                                                                                                                                                                                                                                                                                                                                                           \end{columns}
}
                                                                                                                                                                                                                                                                                                                                                                                                                                                           
\end{frame}               
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}\frametitle{Conclusion on $1GeV$ binning}
\begin{columns}
\column{2.5in}
\begin{itemize}
\item Fits Highly bias and not usable.
\item MM work fine.
\item Christian developed his own fitter
\item Let's try comparing those(not very fair comparison)!
\end{itemize}
\column{2.5in}
\includegraphics[width=0.8\textwidth]{Q2_4_S8.png}\\
\includegraphics[width=0.8\textwidth]{Q2_5_S5.png}

\end{columns}

\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}\frametitle{MM vs Fit\footnote{Taken form Christoph's presentation}}


\begin{tiny}
\begin{columns}
\column{1in}
 \begin{tabular}{|c|c|c|}
 \hline
 $Q^2$ & $F_l^{MM}$ & $F_l^{Fit}$ \\ \hline
 $0$ & $0.0397$ & $0.025$\\ \hline
 $1$ & $0.0802$ & $0.051$\\ \hline
 $2$ & $0.0869$ & $0.057$\\ \hline
 $3$ & $0.0893$ & $0.050$\\ \hline
 $4$ & $0.0878$ & $0.051$\\ \hline
 $5$ & $0.0848$ & $0.048$\\ \hline
 $6$ & $0.0812$ & $0.047$\\ \hline
 $7$ & $0.0775$ & $0.046$\\ \hline
 $8$ & $0.0657$ & $0.041$\\ \hline
 $9$ & $0.0687$ & $0.044$\\ \hline
 $10$ & $0.0755$ & $0.045$\\ \hline
 $11$ & $0.0967$ & $0.061$\\ \hline
    \end{tabular}                                                                                                                                              
\column{1in}
 \begin{tabular}{|c|c|c|}
 \hline
 $Q^2$ & $S_3^{MM}$ & $S_3^{Fit}$ \\ \hline
 $0$ & $0.0656$ & $0.058$\\ \hline
 $1$ & $0.109$ & $0.100$\\ \hline
 $2$ & $0.116$ & $0.102$\\ \hline
 $3$ & $0.12$ & $0.102$\\ \hline
 $4$ & $0.118$ & $0.101$\\ \hline
 $5$ & $0.115$ & $0.098$\\ \hline
 $6$ & $0.112$ & $0.092$\\ \hline
 $7$ & $0.108$ & $0.087$\\ \hline
 $8$ & $0.0989$ & $0.090$\\ \hline
 $9$ & $0.104$ & $0.098$\\ \hline
 $10$ & $0.114$ & $0.108$\\ \hline
 $11$ & $0.146$ & $0.155$\\ \hline
    \end{tabular}                                                                                                                                              
\column{1in}
 \begin{tabular}{|c|c|c|}
 \hline
 $Q^2$ & $S_4^{MM}$ & $S_4^{Fit}$ \\ \hline
 $0$ & $0.0639$ & $0.054$\\ \hline
 $1$ & $0.123$ & $0.115$\\ \hline
 $2$ & $0.134$ & $0.114$\\ \hline
 $3$ & $0.137$ & $0.123$\\ \hline
 $4$ & $0.134$ & $0.116$\\ \hline
 $5$ & $0.129$ & $0.112$\\ \hline
 $6$ & $0.123$ & $0.104$\\ \hline
 $7$ & $0.117$ & $0.100$\\ \hline
 $8$ & $0.099$ & $0.086$\\ \hline
 $9$ & $0.103$ & $0.098$\\ \hline
 $10$ & $0.112$ & $0.109$\\ \hline
 $11$ & $0.144$ & $0.160$\\ \hline
    \end{tabular}                                                                                                                                              
\column{1in}
 \begin{tabular}{|c|c|c|}
 \hline
 $Q^2$ & $S_5^{MM}$ & $S_5^{Fit}$ \\ \hline
 $0$ & $ 0.0563$ & $0.050$\\ \hline
 $1$ & $0.118$ & $0.114$\\ \hline
 $2$ & $0.131$ & $0.108$\\ \hline
 $3$ & $0.133$ & $0.117$\\ \hline
 $4$ & $0.129$ & $0.105$\\ \hline
 $5$ & $0.122$ & $0.106$\\ \hline
 $6$ & $0.116$ & $0.097$\\ \hline
 $7$ & $0.109$ & $0.094$\\ \hline
 $8$ & $0.0884$ & $0.080$\\ \hline
 $9$ & $0.0925$ & $0.085$\\ \hline
 $10$ & $0.101$ & $0.096$\\ \hline
 $11$ & $0.13$ & $0.144$\\ \hline
    \end{tabular}                                                                                                                                              



    \end{columns}
    
    
    
    
\begin{columns}
\column{1in}
 \begin{tabular}{|c|c|c|}
 \hline
 $Q^2$ &  $S_6^{MM}$ & $S_6^{Fit}$ \\ \hline
 $0$ & $0.0632$ & $0.058$\\ \hline
 $1$ & $0.0936$ & $0.085$\\ \hline
 $2$ & $0.097$ & $0.093$\\ \hline
 $3$ & $0.101$ & $0.085$\\ \hline
 $4$ & $0.101$ & $0.083$\\ \hline
 $5$ & $0.0999$ & $0.085$\\ \hline
 $6$ & $0.0975$ & $0.080$\\ \hline
 $7$ & $0.0941$ & $0.079$\\ \hline
 $8$ & $0.0873$ & $0.073$\\ \hline
 $9$ & $0.0925$ & $0.080$\\ \hline
 $10$ & $0.104$ & $0.100$\\ \hline
 $11$ & $0.137$ & $0.121$\\ \hline
    \end{tabular}                                                                                                                                              
\column{1in}
 \begin{tabular}{|c|c|c|}
 \hline
 $Q^2$ & $S_7^{MM}$ & $S_7^{Fit}$ \\ \hline
 $0$ & $0.0574$ & $0.051$\\ \hline
 $1$ & $0.118$ & $0.100$\\ \hline
 $2$ & $0.131$ & $0.105$\\ \hline
 $3$ & $0.134$ & $0.111$\\ \hline
 $4$ & $0.131$ & $0.111$\\ \hline
 $5$ & $0.125$ & $0.104$\\ \hline
 $6$ & $0.119$ & $0.102$\\ \hline
 $7$ & $0.113$ & $0.097$\\ \hline
 $8$ & $0.0958$ & $0.085$\\ \hline
 $9$ & $0.101$ & $0.091$\\ \hline
 $10$ & $0.112$ & $0.101$\\ \hline
 $11$ & $0.144$ & $0.136$\\ \hline
    \end{tabular}                                                                                                                                              
\column{1in}
 \begin{tabular}{|c|c|c|}
 \hline
 $Q^2$ & $S_8^{MM}$ & $S_8^{Fit}$ \\ \hline
 $0$ & $0.0641$ & $0.058$\\ \hline
 $1$ & $0.123$ & $0.113$\\ \hline
 $2$ & $0.134$ & $0.120$\\ \hline
 $3$ & $0.138$ & $0.119$\\ \hline
 $4$ & $0.135$ & $0.112$\\ \hline
 $5$ & $0.13$ & $0.108$\\ \hline
 $6$ & $0.124$ & $0.103$\\ \hline
 $7$ & $0.118$ & $0.101$\\ \hline
 $8$ & $0.104$ & $0.092$\\ \hline
 $9$ & $0.109$ & $0.095$\\ \hline
 $10$ & $0.121$ & $0.115$\\ \hline
 $11$ & $0.156$ & $0.149$\\ \hline
    \end{tabular}                                                                                                                                              
\column{1in}
 \begin{tabular}{|c|c|c|}
 \hline
 $Q^2$ & $S_9^{MM}$ & $S_9^{Fit}$ \\ \hline
 $0$ & $0.0656$ & $0.057$\\ \hline
 $1$ & $0.109$ & $0.099$\\ \hline
 $2$ & $0.116$ & $0.109$\\ \hline
 $3$ & $0.12$ & $0.104$\\ \hline
 $4$ & $0.119$ & $0.102$\\ \hline
 $5$ & $0.116$ & $0.097$\\ \hline
 $6$ & $0.112$ & $0.094$\\ \hline
 $7$ & $0.108$ & $0.091$\\ \hline
 $8$ & $0.0993$ & $0.079$\\ \hline
 $9$ & $0.105$ & $0.088$\\ \hline
 $10$ & $0.115$ & $0.103$\\ \hline
 $11$ & $0.149$ & $0.143$\\ \hline
    \end{tabular}                                                                                                                                              



    \end{columns}
\end{tiny}
\end{frame}
% ------------------------------------------------------------------------------



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}\frametitle{Conclusion}

\begin{itemize}
\item Method of moments a bit worse($\sim 15\%$) then optimistic fit.
\item MM overestimates the error by fiew \%, fits understimates the error by fiew \%.
\item Fit needs more study to understand why roofit is biased and Christoph fit is not.
\end{itemize}


\end{frame}






\end{document}
