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\title{Method of moments for $\PB \to \PKstar \mu \mu$}  
\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{Introduction}
\begin{frame}\frametitle{Plan}
Why method of moments:
\begin{enumerate}
\item Complementary approach in performing the fit.
\item Allows to extract info measuring quantities in event basis depending on the angular distribution.
\item Used in $\PB \to \rho \Plepton \nu$(SLAC-386 UC-414),\\ $\PJpsi \to \PK \PK \gamma$(PRD 71, 032005 (2005) ), etc.
\end{enumerate}


\end{frame}
\section{Method of Moments - Theory}
\begin{frame}\frametitle{Method of moments}
{~}
Let's assume we have our pdf with $k$ unknown parameters :$PDF(x_i, \alpha)$, $dim(\alpha)=k$. One can calculate $k$ moments, which are the functions of $\alpha_i$:
\begin{equation}
\mu_i=f(\alpha_1,..., \alpha_k) = E[W_i]
\end{equation}
If we have $n$ events in our $q^2$ bin, we can estimate:
\begin{equation}
\widehat{\mu}_i=\dfrac{1}{n}\sum_{j=0}^{j=n-1} w_j
\end{equation}
, where $w_j=g(x_i)$

\end{frame}

\begin{frame}\frametitle{Trivial example}
{~}
Lets see how this works in practice:
\begin{equation}
f(x)=\dfrac{x^{a-1}e^{-x/b}} {b^a \Gamma(a)}
\end{equation}
we measure the moments:\\
\begin{center} $m_1=\dfrac{X_1+X_2+...+X_n}{n}$,\\ $m_2=\dfrac{X_1^2+X_2^2+...+X_n^2}{n}$.\\\end{center}
and calculate them analytically:
\begin{center} $m_1=ab$, $m_2=b^2a(a+1)$\end{center}
So one just needs to solve this and get the answer:
\center $a=\dfrac{m_1^2}{m_2-m_1^2}$, $b=\dfrac{m_2-m_1^2}{m_1}$
\end{frame}

\section{Moments of Ss}


\begin{frame}\frametitle{Our PDF}
{~}
The angular terms:
\begin{multline}
\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}
Since we are fitting a PDF we need to ensure it is normalized:
\begin{equation}
\int_{-\pi}^{\pi} d\phi \int_{-1}^{1} d cos\theta_l \int_{-1}^{1} d cos\theta_k \dfrac{d^4\Gamma}{dq^2 dcos\theta_k dcos\theta_l d\phi}=1
\end{equation}


\end{frame}
\begin{frame}\frametitle{Measuring Js}
{~}
From equation 2 we have the following:

\begin{equation}
\dfrac{1}{4}(3 J_{1c} + 6 J_{1s} - J_{2c} - 2 J_{2s}) =1
\end{equation}
For now we will consider the following PDF: 
\begin{equation}
 \dfrac{d^3\Gamma}{\Gamma dcos\theta_k dcos\theta_l d\phi }(cos\theta_k , cos\theta_l, \phi)
\end{equation}

Becouse our PDF is not normalized and we are measuring $\Gamma+\overline{\Gamma}
$ we are effectively fitting the $S_i$ (aka $J_i \to S_i$)
\end{frame}


\begin{frame}\frametitle{Moments for $\PB \to \PKstar \mu \mu$}
{~}
Let's calculate the moments for Ss:
\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}
\begin{frame}\frametitle{Moments for $\PB \to \PKstar \mu \mu$}
{~}
\begin{itemize}
\item The simplest solution one could imagine.
\item We are abusing the fact that the basis is orthogonal.
\item Each of the J doesn't know about other.
\item Only $S_{1s}$, $S_{2s}$, $S_{1c}$, $S_{2c}$ and  $S_{6s}$, $S_{6c}$ are not orthogonal, but to get the answer you just need to solve a linear equation system so it's not a tragedy.
\end{itemize}
\begin{equation}
 \dfrac{d^3\Gamma}{\Gamma dcos\theta_k dcos\theta_l d\phi} sin^2 \theta_k cos \theta_l = 0.1(S_6c+4S_6s)
\end{equation}
\begin{equation}
 \dfrac{d^3\Gamma}{\Gamma dcos\theta_k dcos\theta_l d\phi} cos \theta_l = 0.25(S_{6}c+2S_{6s})
\end{equation}

solution: $S_{6c}=2 (4 M_{S_{6c}} - 5 M_{S_{6s}})$, $S_{6s}= -2 M_{S_{6c}} + 5 M_{S_{6s}}$



\end{frame}



\section{Toy MC study}

\begin{frame}\frametitle{Moments for $\PB \to \PKstar \mu \mu$}
{~}
Lets see if this method actually works. Let's take some random parameters for the PDF and make a toy.
\begin{columns}
\column{2.5in}
\includegraphics[scale=0.25]{images/J5.png}



\column{2.5in}
\includegraphics[scale=0.25]{images/J7.png}


\end{columns}
\begin{itemize}
\item let's take 300 signal events as a working case.
\end{itemize}

\end{frame}




\begin{frame}\frametitle{Moments for $\PB \to \PKstar \mu \mu$}
{~}
\small Lets see if this method actually works. Let's take some random parameters for the PDF and make a toy.
\begin{columns}
\column{2.5in}
\includegraphics[scale=0.25]{images/J3.png}



\column{2.5in}
\includegraphics[scale=0.25]{images/J4.png}


\end{columns}


\begin{itemize}
\item let's take 300 signal events as a working case... we might still change the binning
\end{itemize}

\end{frame}



\begin{frame}\frametitle{Moments for $\PB \to \PKstar \mu \mu$}
{~}
Lets see if this method actually works. Let's take some random parameters for the PDF and make a toy.
\begin{columns}
\column{2.5in}
\includegraphics[scale=0.25]{images/J8.png}



\column{2.5in}
\includegraphics[scale=0.25]{images/J9.png}


\end{columns}

\begin{itemize}
\item let's take 300 signal events as a working case... we might still change the binning
\end{itemize}

\end{frame}

\begin{frame}\frametitle{Error Estimation}
{~}
\begin{itemize}
\item Since moment is the mean of a given distribution the error can be estimated as $mean/RMS$
\item use TOY MC to check this assumption
\end{itemize}

\includegraphics[scale=0.3]{plots/conw.png}\\


\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%5
\begin{frame}\frametitle{Correlation check}
{~}
\begin{itemize}
\item In theory $S_i$ shouldn't be correlated to $S_j$ in the moment calculation.
\item Lets put this to a test.
\end{itemize}

\begin{columns}
\column{2.5in}
\includegraphics[scale=0.2]{plots/J9J4.png}\\


\column{2.5in}
\includegraphics[scale=0.2]{plots/J8J5.png}\\


\end{columns}


\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%5
\begin{frame}\frametitle{Correlation check 2}
{~}
\begin{itemize}
\item Let's now FIX $J_x$ and simulate different $J_y$
\item Again theory would suggest that one J shouldn't know about the other, so $J_x$ shouldn't change with scanning $J_y$ parameter
\end{itemize}

\begin{columns}
\column{2.5in}
\includegraphics[scale=0.24]{plots/J5_vs_J9_300.png}\\


\column{2.5in}
\includegraphics[scale=0.24]{plots/J3_vs_J9_300.png}\\


\end{columns}


\end{frame}



\section{2 am discovery}
\begin{frame}\frametitle{What will happen to our problem with an S-wave?}
{~}
Reminder:
\begin{multline}
\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}
Let's add a very discussing things that keeps us awake at night:
\begin{multline}
W_s =\dfrac{1}{4 \pi} (2 I_{1a} sin^2 \theta_l + 2 I_{1b} sin^2 \theta_l cos \theta_k+ I_4 sin \theta_k sin 2 \theta_lcos \phi \\+ I_5 sin \theta_k sin \theta_l cos \phi + I_7 sin \theta_k sin \theta_l + sin \phi + I_8 sin \theta_k sin 2\theta_l sin\phi)
\end{multline}


\end{frame}

\begin{frame}\frametitle{What will happen to our problem with an S-wave?}
{~}
So now our PDF is sum of eq. 15 and 16. Of coz we need to require normalization:
\begin{equation}
\dfrac{1}{12}(32 I_{1a} + 9 J_{1c} + 18 J_{1s} - 3 J_{2c} - 6 J_{2s}) =1
\end{equation}
No surprises here. If we have a S-wave it has to enter in $\Gamma$.
To build up the preasure, what will happen to our Ss?



\end{frame}


\begin{frame}\frametitle{NOTHING!!!!!!!!}
{~}
We are completely insensitive to S-wave:
\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}

\begin{frame}\frametitle{Thins get better :)}
{~}
We can even measure directly the S-wave:
\begin{equation}
 \dfrac{d^3\Gamma}{\Gamma dcos\theta_k dcos\theta_l d\phi }sin^2 \theta_l cos \theta_k = \dfrac{32 I_{1b} }{45}
\end{equation}
\begin{equation}
 \dfrac{d^3\Gamma}{\Gamma dcos\theta_k dcos\theta_l d\phi } sin \theta_k sin 2 \theta_l cos \phi = \dfrac{16 I_4 }{45}
\end{equation}
\begin{equation}
 \dfrac{d^3\Gamma}{\Gamma dcos\theta_k dcos\theta_l d\phi } sin\theta_k sin\theta_l cos\phi = \dfrac{4 I_5 }{9}
\end{equation}
\begin{equation}
 \dfrac{d^3\Gamma}{\Gamma dcos\theta_k dcos\theta_l d\phi} sin  \theta_k sin 2 \theta_l sin \phi = \dfrac{4 I_7 }{9}
\end{equation}
\begin{equation}
 \dfrac{d^3\Gamma}{\Gamma dcos\theta_k dcos\theta_l d\phi} sin  \theta_k sin2 \theta_l sin \phi = \dfrac{16 S_8 }{45}
\end{equation}



\end{frame}


\begin{frame}\frametitle{Conclusions on S-wave}
{~}
\begin{itemize}
\item S-wave components are transparent to method of moments.
\item If they are orthogonal to others all they toy studies holds for them as well(will reapat for robustness but can bet my house that there is nothing going on there).
\item 
\end{itemize}

\end{frame}



\begin{frame}\frametitle{Conclusions}
{~}
\begin{itemize}
\item Implemented moments method for the K*mm and start testing with toy MC
\item The method converge fast and works for the "simple case", i.e. signal only.
\item Method completely insensitive to S-wave component, thanks to orthogonality. 
\item Complementary one can measure in-depended S-wave component. 
\end{itemize}
TO DO: 
\begin{itemize}
\item  add realism: backgrounds
\item Do the unfolding
\item Study binning
\end{itemize}

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%55
\begin{frame}\frametitle{~}
{~}
\center \Huge BACKUPS

\end{frame}

\begin{frame}
{~}

\begin{columns}
\column{2.5in}
\includegraphics[scale=0.17]{plots/pool_plots/J1c_50.png}\\
\includegraphics[scale=0.17]{plots/pool_plots/J1c_75.png}\\


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\includegraphics[scale=0.17]{plots/pool_plots/J1c_50E.png}\\
\includegraphics[scale=0.17]{plots/pool_plots/J1c_75E.png}


\end{columns}
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{~}

\begin{columns}
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\includegraphics[scale=0.17]{plots/pool_plots/J1c_125.png}\\
\includegraphics[scale=0.17]{plots/pool_plots/J1c_175.png}\\


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\includegraphics[scale=0.17]{plots/pool_plots/J1c_175E.png}


\end{columns}
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\includegraphics[scale=0.17]{plots/pool_plots/J1c_300.png}\\


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\includegraphics[scale=0.17]{plots/pool_plots/J1c_300E.png}


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\includegraphics[scale=0.17]{plots/pool_plots/J1c_400.png}\\



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\includegraphics[scale=0.17]{plots/pool_plots/J1c_400E.png}\\



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{~}

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\includegraphics[scale=0.17]{plots/pool_plots/J1s_175.png}\\


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\includegraphics[scale=0.17]{plots/pool_plots/J1s_175E.png}


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\includegraphics[scale=0.17]{plots/pool_plots/J1s_400.png}\\



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{~}

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\begin{columns}
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\column{2.5in}
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\column{2.5in}
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\includegraphics[scale=0.17]{plots/pool_plots/J4_175.png}\\


\column{2.5in}
\includegraphics[scale=0.17]{plots/pool_plots/J4_125E.png}\\
\includegraphics[scale=0.17]{plots/pool_plots/J4_175E.png}


\end{columns}
\end{frame}

\begin{frame}
{~}

\begin{columns}
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\includegraphics[scale=0.17]{plots/pool_plots/J4_300.png}\\


\column{2.5in}
\includegraphics[scale=0.17]{plots/pool_plots/J4_225E.png}\\
\includegraphics[scale=0.17]{plots/pool_plots/J4_300E.png}


\end{columns}
\end{frame}


\begin{frame}
{~}

\begin{columns}
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\includegraphics[scale=0.17]{plots/pool_plots/J4_500.png}\\


\column{2.5in}
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\includegraphics[scale=0.17]{plots/pool_plots/J4_500E.png}\\


\end{columns}
\end{frame}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
{~}

\begin{columns}
\column{2.5in}
\includegraphics[scale=0.17]{plots/pool_plots/J5_50.png}\\
\includegraphics[scale=0.17]{plots/pool_plots/J5_75.png}\\


\column{2.5in}
\includegraphics[scale=0.17]{plots/pool_plots/J5_50E.png}\\
\includegraphics[scale=0.17]{plots/pool_plots/J5_75E.png}


\end{columns}
\end{frame}

\begin{frame}
{~}

\begin{columns}
\column{2.5in}
\includegraphics[scale=0.17]{plots/pool_plots/J5_125.png}\\
\includegraphics[scale=0.17]{plots/pool_plots/J5_175.png}\\


\column{2.5in}
\includegraphics[scale=0.17]{plots/pool_plots/J5_125E.png}\\
\includegraphics[scale=0.17]{plots/pool_plots/J5_175E.png}


\end{columns}
\end{frame}

\begin{frame}
{~}

\begin{columns}
\column{2.5in}
\includegraphics[scale=0.17]{plots/pool_plots/J5_225.png}\\
\includegraphics[scale=0.17]{plots/pool_plots/J5_300.png}\\


\column{2.5in}
\includegraphics[scale=0.17]{plots/pool_plots/J5_225E.png}\\
\includegraphics[scale=0.17]{plots/pool_plots/J5_300E.png}


\end{columns}
\end{frame}


\begin{frame}
{~}

\begin{columns}
\column{2.5in}
\includegraphics[scale=0.17]{plots/pool_plots/J5_400.png}\\
\includegraphics[scale=0.17]{plots/pool_plots/J5_500.png}\\


\column{2.5in}
\includegraphics[scale=0.17]{plots/pool_plots/J5_400E.png}\\
\includegraphics[scale=0.17]{plots/pool_plots/J5_500E.png}\\


\end{columns}
\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
{~}

\begin{columns}
\column{2.5in}
\includegraphics[scale=0.17]{plots/pool_plots/J7_50.png}\\
\includegraphics[scale=0.17]{plots/pool_plots/J7_75.png}\\


\column{2.5in}
\includegraphics[scale=0.17]{plots/pool_plots/J7_50E.png}\\
\includegraphics[scale=0.17]{plots/pool_plots/J7_75E.png}


\end{columns}
\end{frame}

\begin{frame}
{~}

\begin{columns}
\column{2.5in}
\includegraphics[scale=0.17]{plots/pool_plots/J7_125.png}\\
\includegraphics[scale=0.17]{plots/pool_plots/J7_175.png}\\


\column{2.5in}
\includegraphics[scale=0.17]{plots/pool_plots/J7_125E.png}\\
\includegraphics[scale=0.17]{plots/pool_plots/J7_175E.png}


\end{columns}
\end{frame}

\begin{frame}
{~}

\begin{columns}
\column{2.5in}
\includegraphics[scale=0.17]{plots/pool_plots/J7_225.png}\\
\includegraphics[scale=0.17]{plots/pool_plots/J7_300.png}\\


\column{2.5in}
\includegraphics[scale=0.17]{plots/pool_plots/J7_225E.png}\\
\includegraphics[scale=0.17]{plots/pool_plots/J7_300E.png}


\end{columns}
\end{frame}


\begin{frame}
{~}

\begin{columns}
\column{2.5in}
\includegraphics[scale=0.17]{plots/pool_plots/J7_400.png}\\
\includegraphics[scale=0.17]{plots/pool_plots/J7_500.png}\\


\column{2.5in}
\includegraphics[scale=0.17]{plots/pool_plots/J7_400E.png}\\
\includegraphics[scale=0.17]{plots/pool_plots/J7_500E.png}\\


\end{columns}
\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
{~}

\begin{columns}
\column{2.5in}
\includegraphics[scale=0.17]{plots/pool_plots/J8_50.png}\\
\includegraphics[scale=0.17]{plots/pool_plots/J8_75.png}\\


\column{2.5in}
\includegraphics[scale=0.17]{plots/pool_plots/J8_50E.png}\\
\includegraphics[scale=0.17]{plots/pool_plots/J8_75E.png}


\end{columns}
\end{frame}

\begin{frame}
{~}

\begin{columns}
\column{2.5in}
\includegraphics[scale=0.17]{plots/pool_plots/J8_125.png}\\
\includegraphics[scale=0.17]{plots/pool_plots/J8_175.png}\\


\column{2.5in}
\includegraphics[scale=0.17]{plots/pool_plots/J8_125E.png}\\
\includegraphics[scale=0.17]{plots/pool_plots/J8_175E.png}


\end{columns}
\end{frame}

\begin{frame}
{~}

\begin{columns}
\column{2.5in}
\includegraphics[scale=0.17]{plots/pool_plots/J8_225.png}\\
\includegraphics[scale=0.17]{plots/pool_plots/J8_300.png}\\


\column{2.5in}
\includegraphics[scale=0.17]{plots/pool_plots/J8_225E.png}\\
\includegraphics[scale=0.17]{plots/pool_plots/J8_300E.png}


\end{columns}
\end{frame}


\begin{frame}
{~}

\begin{columns}
\column{2.5in}
\includegraphics[scale=0.17]{plots/pool_plots/J8_400.png}\\
\includegraphics[scale=0.17]{plots/pool_plots/J8_500.png}\\


\column{2.5in}
\includegraphics[scale=0.17]{plots/pool_plots/J8_400E.png}\\
\includegraphics[scale=0.17]{plots/pool_plots/J8_500E.png}\\


\end{columns}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
{~}

\begin{columns}
\column{2.5in}
\includegraphics[scale=0.17]{plots/pool_plots/J9_50.png}\\
\includegraphics[scale=0.17]{plots/pool_plots/J9_75.png}\\


\column{2.5in}
\includegraphics[scale=0.17]{plots/pool_plots/J9_50E.png}\\
\includegraphics[scale=0.17]{plots/pool_plots/J9_75E.png}


\end{columns}
\end{frame}

\begin{frame}
{~}

\begin{columns}
\column{2.5in}
\includegraphics[scale=0.17]{plots/pool_plots/J9_125.png}\\
\includegraphics[scale=0.17]{plots/pool_plots/J9_175.png}\\


\column{2.5in}
\includegraphics[scale=0.17]{plots/pool_plots/J9_125E.png}\\
\includegraphics[scale=0.17]{plots/pool_plots/J9_175E.png}


\end{columns}
\end{frame}

\begin{frame}
{~}

\begin{columns}
\column{2.5in}
\includegraphics[scale=0.17]{plots/pool_plots/J9_225.png}\\
\includegraphics[scale=0.17]{plots/pool_plots/J9_300.png}\\


\column{2.5in}
\includegraphics[scale=0.17]{plots/pool_plots/J9_225E.png}\\
\includegraphics[scale=0.17]{plots/pool_plots/J9_300E.png}


\end{columns}
\end{frame}


\begin{frame}
{~}

\begin{columns}
\column{2.5in}
\includegraphics[scale=0.17]{plots/pool_plots/J9_400.png}\\
\includegraphics[scale=0.17]{plots/pool_plots/J9_500.png}\\


\column{2.5in}
\includegraphics[scale=0.17]{plots/pool_plots/J9_400E.png}\\
\includegraphics[scale=0.17]{plots/pool_plots/J9_500E.png}\\


\end{columns}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
{~}

\begin{columns}
\column{2.5in}
\includegraphics[scale=0.2]{plots/J9J8.png}\\
\includegraphics[scale=0.2]{plots/J9J5.png}\\

\column{2.5in}
\includegraphics[scale=0.2]{plots/J9J7.png}\\
\includegraphics[scale=0.2]{plots/J9J4.png}\\

\end{columns}

\end{frame}

\begin{frame}
{~}

\begin{columns}
\column{2.5in}
\includegraphics[scale=0.2]{plots/J9J3.png}\\
\includegraphics[scale=0.2]{plots/J8J7.png}\\

\column{2.5in}
\includegraphics[scale=0.2]{plots/J8J5.png}\\
\includegraphics[scale=0.2]{plots/J8J4.png}\\

\end{columns}

\end{frame}






\end{document}