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\usetheme{Sybila} 

\title[Method of moments for $\PBzero \to \PKstar \mu \mu$]{Method of moments for $\PBzero \to  \PKstar \mu \mu$}
\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_mm.png}\\

\end{frame}


\section{Introduction}
\begin{frame}\frametitle{Introduction}
Why method of moments:
\begin{enumerate}
\item Complementary approach to LL fits.
\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}
For $n$ events, 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{small}
\begin{multline}
PDF(\cos \theta_k ,\cos \theta_l, \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-F_l)\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 \sin 2 \theta_k \sin \theta_l \cos \phi +  (S_{6s} \sin^2 \theta_k + S_{6c} \cos^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}
\end{small}
\only<1>{
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}
}
\only<2>
{
\begin{small}
For further use let's introduce a notation:
\begin{multline}
PDF(\cos \theta_k ,\cos \theta_l, \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-F_l)\sin^2 \theta_k -F_l\cos^2) \cos 2\theta_l  + \sum_{x=3}^{9} S_x f_x(\cos \theta_k ,\cos \theta_l, \phi)
\end{multline}
\end{small}
}

\end{frame}

\begin{frame}\frametitle{Moments for $\PB \to \PKstar \mu \mu$ 1/2}
{~}
\begin{footnotesize}



Let's calculate the moments(means of the given distribution):
\begin{equation}
\int_{-1}^{1}d\cos \theta_l \int_{-1}^{1}d\cos \theta_k \int_{-\pi}^{\pi}d\phi PDF(\cos \theta_k ,\cos \theta_l, \phi) \sin^2 \theta_k =\frac{2}{5} (2-F_l)
\end{equation}
\begin{equation}
\int_{-1}^{1}d\cos \theta_l \int_{-1}^{1}d\cos \theta_k \int_{-\pi}^{\pi}d\phi PDF(\cos \theta_k ,\cos \theta_l, \phi) \cos^2 \theta_k =\frac{1}{5} (2F_l+1)
\end{equation}
\begin{equation}
\int_{-1}^{1}d\cos \theta_l \int_{-1}^{1}d\cos \theta_k \int_{-\pi}^{\pi}d\phi PDF(\cos \theta_k ,\cos \theta_l, \phi) \cos^2 \theta_k =-\dfrac{2}{25}(2 + F_l)
\end{equation}
\begin{equation}
\int_{-1}^{1}d\cos \theta_l \int_{-1}^{1}d\cos \theta_k \int_{-\pi}^{\pi}d\phi PDF(\cos \theta_k ,\cos \theta_l, \phi) \sin^2 \theta_k =-\dfrac{1}{25}(1+8F_l)
\end{equation}
\end{footnotesize}

\end{frame}


\begin{frame}\frametitle{Moments for $\PB \to \PKstar \mu \mu$ 2/2}
{~}


\begin{small}

Let's calculate the moments(means of the given distribution):
\begin{equation}
\int_{-1}^{1}d\cos \theta_l \int_{-1}^{1}d\cos \theta_k \int_{-\pi}^{\pi}d\phi PDF(\cos \theta_k ,\cos \theta_l, \phi) f_{S_x}= \frac{8}{25}S_x,
\end{equation}
for $x=3,4,8,9$, and:

%%
\begin{equation}
\int_{-1}^{1}d\cos \theta_l \int_{-1}^{1}d\cos \theta_k \int_{-\pi}^{\pi}d\phi PDF(\cos \theta_k ,\cos \theta_l, \phi) f_{S_x}= \frac{2}{5}S_x,
\end{equation}
for $x=5,6,7$.\\
New physics apparently as we like orthogonal world:
\begin{equation}
\int_{-1}^{1}d\cos \theta_l \int_{-1}^{1}d\cos \theta_k \int_{-\pi}^{\pi}d\phi~( f_{S_x} \times f_{S_y})		= \alpha_{xy} \delta_{x~y}
\end{equation}

\end{small}
\end{frame}



\begin{frame}\frametitle{Moments for $\PB \to \PKstar \mu \mu$}
{~}
\begin{itemize}
\item We are abusing the fact that the basis is orthogonal and moments do not mix.
\item Makes live easier and reduces the systematics.
\item Each of the S does not know about other.
\item In case of full PDF, $S_{1s}$, $S_{2s}$, $S_{1c}$, $S_{2c}$ $S_{6s}$, $S_{6c}$ are not orthogonal.
\item Still we can get them solving equation system:
\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}

\small 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/J3.png}

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


\end{columns}

\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
\item Do not worry, detail description an numbers will come in other presentation.
\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{S-wave}



\begin{frame}\frametitle{S-wave pollution}
{~}
\begin{columns}
\column{3in}
\begin{itemize}
\item Unfortunately in our perfect orthogonal world lives an imposter.
\item This imposter is $\PBzero \to (\PK \Ppi)_{S-wave}~\mu \mu$
\item This "ghost" dilutes our NP! Like dark matter the universe.
\item We need something to bust this ghost away
\end{itemize}

\column{2in}
\includegraphics[width=0.95\textwidth]{P5.png}\\

\includegraphics[width=0.5\textwidth]{gb.jpg}
\end{columns}


\end{frame}





%%%%%%%%%%%%%%%%%%%5
\begin{frame}\frametitle{S-wave hunting}
{~}
Our PDF with the S-wave will look as follows:
\begin{multline}
PDF_{full}(\cos \theta_k ,\cos \theta_l, \phi) =\dfrac{9}{32\pi}(  \textcolor{red}{(1-F_s)}(\dfrac{3}{4}(1-F_l)\sin^2 \theta_k + F_l \cos^2 \theta_k + \\ (\dfrac{1}{4}(1-F_l)\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 \sin 2 \theta_k \sin \theta_l \cos \phi +  \\ (S_{6s} \sin^2 \theta_k + S_{6c} \cos^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) + \\
\textcolor{red}{\dfrac{2}{3} F_s \sin^2 \theta_l + \frac{4}{3} A_s \sin^2 \theta_l \cos \theta_k + I_4 \sin \theta_k \sin 2 \theta_l \cos \phi} \\ \textcolor{red}{+ 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}

\small In this form we ensure normalization.
\end{frame}

\begin{frame}\frametitle{How does the dilution work? 1/2}
{~}
\begin{small}
\begin{equation}
\int_{-1}^{1}d\cos \theta_l \int_{-1}^{1}d\cos \theta_k \int_{-\pi}^{\pi}d\phi PDF_{full}(\cos \theta_k ,\cos \theta_l, \phi) f_{S_x}= \frac{8}{25}S_x\textcolor{red}{(1-F_s)},
\end{equation}
for $x=3,4,8,9$, and:

%%
\begin{equation}
\int_{-1}^{1}d\cos \theta_l \int_{-1}^{1}d\cos \theta_k \int_{-\pi}^{\pi}d\phi PDF_{full}(\cos \theta_k ,\cos \theta_l, \phi) f_{S_x}= \frac{2}{5}S_x\textcolor{red}{(1-F_s)},
\end{equation}
for $x=5,6,7$.\\
Not much harm and easy to control.
\end{small}

\end{frame}

\begin{frame}\frametitle{How does the dilution work? 2/2}
{~}
\begin{small}
Unfortunately $F_l$ and $F_s$ will mix with each other:
\begin{multline}
\int_{-1}^{1}d\cos \theta_l \int_{-1}^{1}d\cos \theta_k \int_{-\pi}^{\pi}d\phi PDF_{full}(\cos \theta_k ,\cos \theta_l, \phi) \sin^2 \theta_k= \\ \frac{2}{15} (6 + 3 F_l (F_s-1) - F_s)=M_{F_l}
\end{multline}
\begin{multline}
\int_{-1}^{1}d\cos \theta_l \int_{-1}^{1}d\cos \theta_k \int_{-\pi}^{\pi}d\phi PDF_{full}(\cos \theta_k ,\cos \theta_l, \phi) \sin^2 \theta_l= \\ \frac{1}{5} (3 + F_l + F_s - F_l F_s)=M_{F_s}
\end{multline}

They can solve this system:

$ \begin{cases} F_s = \frac{15}{4} (M_{F_l} + 2 M_{F_s}) \\ F_l = \frac{(15 M_{F_l} + 10 M_{F_s}-18)}{(15 M_{F_l} + 30 M_{F_s}-34)}  \end{cases}$


\end{small}

\end{frame}


\begin{frame}\frametitle{S-wave moments}
{~}
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}
{~}
\begin{itemize}
\item Method of moments very suitable for $\PBzero \to \PKstar \mu \mu$.
\item The method converge fast and works for the "simple case", i.e. signal only.
\item Method very insensitive to S-wave component, thanks to orthogonality. 
\item Complementary one can measure in-depended S-wave component.
\item No problem with boundary problems.

\end{itemize}
What comes in the next talks(stay tuned):
\begin{itemize}
\item This method reduces the error on unfolding.
\item No problem with convergence.
\item Systematics easy accessible.
\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}\\


\column{2.5in}
\includegraphics[scale=0.17]{plots/pool_plots/J1c_50E.png}\\
\includegraphics[scale=0.17]{plots/pool_plots/J1c_75E.png}


\end{columns}
\end{frame}

\begin{frame}
{~}

\begin{columns}
\column{2.5in}
\includegraphics[scale=0.17]{plots/pool_plots/J1c_125.png}\\
\includegraphics[scale=0.17]{plots/pool_plots/J1c_175.png}\\


\column{2.5in}
\includegraphics[scale=0.17]{plots/pool_plots/J1c_125E.png}\\
\includegraphics[scale=0.17]{plots/pool_plots/J1c_175E.png}


\end{columns}
\end{frame}

\begin{frame}
{~}

\begin{columns}
\column{2.5in}
\includegraphics[scale=0.17]{plots/pool_plots/J1c_225.png}\\
\includegraphics[scale=0.17]{plots/pool_plots/J1c_300.png}\\


\column{2.5in}
\includegraphics[scale=0.17]{plots/pool_plots/J1c_225E.png}\\
\includegraphics[scale=0.17]{plots/pool_plots/J1c_300E.png}


\end{columns}
\end{frame}


\begin{frame}
{~}

\begin{columns}
\column{2.5in}
\includegraphics[scale=0.17]{plots/pool_plots/J1c_400.png}\\



\column{2.5in}
\includegraphics[scale=0.17]{plots/pool_plots/J1c_400E.png}\\



\end{columns}
\end{frame}

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\begin{frame}
{~}

\begin{columns}
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\includegraphics[scale=0.17]{plots/pool_plots/J1s_50.png}\\
\includegraphics[scale=0.17]{plots/pool_plots/J1s_75.png}\\


\column{2.5in}
\includegraphics[scale=0.17]{plots/pool_plots/J1s_50E.png}\\
\includegraphics[scale=0.17]{plots/pool_plots/J1s_75E.png}


\end{columns}
\end{frame}

\begin{frame}
{~}

\begin{columns}
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\includegraphics[scale=0.17]{plots/pool_plots/J1s_125.png}\\
\includegraphics[scale=0.17]{plots/pool_plots/J1s_175.png}\\


\column{2.5in}
\includegraphics[scale=0.17]{plots/pool_plots/J1s_125E.png}\\
\includegraphics[scale=0.17]{plots/pool_plots/J1s_175E.png}


\end{columns}
\end{frame}

\begin{frame}
{~}

\begin{columns}
\column{2.5in}
\includegraphics[scale=0.17]{plots/pool_plots/J1s_225.png}\\
\includegraphics[scale=0.17]{plots/pool_plots/J1s_300.png}\\


\column{2.5in}
\includegraphics[scale=0.17]{plots/pool_plots/J1s_225E.png}\\
\includegraphics[scale=0.17]{plots/pool_plots/J1s_300E.png}


\end{columns}
\end{frame}


\begin{frame}
{~}

\begin{columns}
\column{2.5in}
\includegraphics[scale=0.17]{plots/pool_plots/J1s_400.png}\\



\column{2.5in}
\includegraphics[scale=0.17]{plots/pool_plots/J1s_400E.png}\\



\end{columns}
\end{frame}

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\begin{frame}
{~}

\begin{columns}
\column{2.5in}
\includegraphics[scale=0.17]{plots/pool_plots/J2c_50.png}\\
\includegraphics[scale=0.17]{plots/pool_plots/J2c_75.png}\\


\column{2.5in}
\includegraphics[scale=0.17]{plots/pool_plots/J2c_50E.png}\\
\includegraphics[scale=0.17]{plots/pool_plots/J2c_75E.png}


\end{columns}
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\begin{frame}
{~}

\begin{columns}
\column{2.5in}
\includegraphics[scale=0.17]{plots/pool_plots/J2c_125.png}\\
\includegraphics[scale=0.17]{plots/pool_plots/J2c_175.png}\\


\column{2.5in}
\includegraphics[scale=0.17]{plots/pool_plots/J2c_125E.png}\\
\includegraphics[scale=0.17]{plots/pool_plots/J2c_175E.png}


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\begin{columns}
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\includegraphics[scale=0.17]{plots/pool_plots/J2c_225.png}\\
\includegraphics[scale=0.17]{plots/pool_plots/J2c_300.png}\\


\column{2.5in}
\includegraphics[scale=0.17]{plots/pool_plots/J2c_225E.png}\\
\includegraphics[scale=0.17]{plots/pool_plots/J2c_300E.png}


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\begin{frame}
{~}

\begin{columns}
\column{2.5in}
\includegraphics[scale=0.17]{plots/pool_plots/J2c_400.png}\\
\includegraphics[scale=0.17]{plots/pool_plots/J2c_500.png}\\


\column{2.5in}
\includegraphics[scale=0.17]{plots/pool_plots/J2c_400E.png}\\
\includegraphics[scale=0.17]{plots/pool_plots/J2c_500E.png}\\


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{~}

\begin{columns}
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\includegraphics[scale=0.17]{plots/pool_plots/J3_50.png}\\
\includegraphics[scale=0.17]{plots/pool_plots/J3_75.png}\\


\column{2.5in}
\includegraphics[scale=0.17]{plots/pool_plots/J3_50E.png}\\
\includegraphics[scale=0.17]{plots/pool_plots/J3_75E.png}


\end{columns}
\end{frame}

\begin{frame}
{~}

\begin{columns}
\column{2.5in}
\includegraphics[scale=0.17]{plots/pool_plots/J3_125.png}\\
\includegraphics[scale=0.17]{plots/pool_plots/J3_175.png}\\


\column{2.5in}
\includegraphics[scale=0.17]{plots/pool_plots/J3_125E.png}\\
\includegraphics[scale=0.17]{plots/pool_plots/J3_175E.png}


\end{columns}
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{~}

\begin{columns}
\column{2.5in}
\includegraphics[scale=0.17]{plots/pool_plots/J3_225.png}\\
\includegraphics[scale=0.17]{plots/pool_plots/J3_300.png}\\


\column{2.5in}
\includegraphics[scale=0.17]{plots/pool_plots/J3_225E.png}\\
\includegraphics[scale=0.17]{plots/pool_plots/J3_300E.png}


\end{columns}
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\begin{frame}
{~}

\begin{columns}
\column{2.5in}
\includegraphics[scale=0.17]{plots/pool_plots/J3_400.png}\\
\includegraphics[scale=0.17]{plots/pool_plots/J3_500.png}\\


\column{2.5in}
\includegraphics[scale=0.17]{plots/pool_plots/J3_400E.png}\\
\includegraphics[scale=0.17]{plots/pool_plots/J3_500E.png}\\


\end{columns}
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\begin{frame}
{~}

\begin{columns}
\column{2.5in}
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\includegraphics[scale=0.17]{plots/pool_plots/J4_75.png}\\


\column{2.5in}
\includegraphics[scale=0.17]{plots/pool_plots/J4_50E.png}\\
\includegraphics[scale=0.17]{plots/pool_plots/J4_75E.png}


\end{columns}
\end{frame}

\begin{frame}
{~}

\begin{columns}
\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}
\column{2.5in}
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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}
\column{2.5in}
\includegraphics[scale=0.17]{plots/pool_plots/J4_400.png}\\
\includegraphics[scale=0.17]{plots/pool_plots/J4_500.png}\\


\column{2.5in}
\includegraphics[scale=0.17]{plots/pool_plots/J4_400E.png}\\
\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}