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\title[$\PB \to \PKstar \Pmu \Pmu$ update ]{$\PB \to \PKstar \Pmu \Pmu$ update }
\author{Marcin Chrz\k{a}szcz$^{1}$}
\institute{$^1$~University of Zurich}
\date{\today}
\begin{document}
% --------------------------- SLIDE --------------------------------------------
\frame[plain]{\titlepage}
\author{Marcin Chrz\k{a}szcz{~}}
\institute{(UZH)}
% ------------------------------------------------------------------------------
% --------------------------- SLIDE --------------------------------------------
\begin{frame}\frametitle{Propagation from $S_x$ to $P_x$}
\begin{itemize}
\item Using MoM we have a direct access to $S_x$.
\item Some theorist (we all know who ;)) like to have instead $P_x$.
\item We need to provide them the $P_x$ or they will do crazy crazy things with the errors.
\item Reminder:
\end{itemize}
\begin{center}
\begin{equation}
P_1=2/(1-F_l),
\end{equation}
\begin{equation}
P_2=\dfrac{2 A_{fb}}{3}/(1-F_l)
\end{equation}
\begin{equation}
P_3=\dfrac{-S_9}{3}/(1-F_l)
\end{equation}
\begin{equation}
P_{4568}=\dfrac{S_{4578}}{3}/\sqrt{F_l(1-F_l)}
\end{equation}
\end{center}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}\frametitle{MC propagation}
Since we have the full information about the $S_x$ with errors(assymetric) and correlations we can make a TOY MC and see how the $P_x$ behaves. To make a toy we need to simulate a correlated multidimensional Gaussian. I didn't know before hand how to do it so here is the recipe if some one will need it for the future:
\begin{itemize}
\item Simulate multidimensional gauss with with 1 and centred at 0.
\item Correlate the dimensions using Cholesky matrix decomposition.
\item Scale the left and the right tail to the right $\sigma_{L,R}$.
\item Now after this scalling you will have a bump at 0 since you have 50:50 events on left and on the right, which is correct only for $\sigma_{L}=\sigma_{R}$. Reject the $\dfrac{\sigma_{SMALLER}}{\sigma_{LARGER}}$ events from smaller $\sigma$.
\item Move the mean of each Gauss to the value of $S_x$ and voila.
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}\frametitle{MC propagation, X-checks 1/2}
\begin{itemize}
\item Since this procedure is not form books I wated to be extra carefull and performed super tests.
\item Check that the correlation between $S_i$ and $S_j$ is in my toys are the same as the one I put in. with 100.000 events the differences that I observed are $\mathcal{O}(0.001)$ so oki doki.
\item Check that the $\sigma_{L,R}$ are the correctly simulate. Fitted left and right separately and compare the results. Oki doki.
\end{itemize}
\begin{columns}
\column{2.5in}
\includegraphics[width=0.9\textwidth]{Kstarmumu_plots/7_110_125_left.pdf}
\column{2.5in}
\includegraphics[width=0.9\textwidth]{Kstarmumu_plots/7_110_125_right.pdf}
\end{columns}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}\frametitle{MC propagation, X-checks 2/2}
\begin{itemize}
\item Since this procedure is not form books I wated to be extra carefull and performed super tests.
\item Check that the correlation between $S_i$ and $S_j$ is in my toys are the same as the one I put in. with 100.000 events the differences that I observed are $\mathcal{O}(0.001)$ so oki doki.
\item Check that the $\sigma_{L,R}$ are the correctly simulate. Fitted left and right separately and compare the results. Oki doki.
\end{itemize}
\begin{columns}
\column{2.5in}
\includegraphics[width=0.9\textwidth]{Kstarmumu_plots/01_110_125.pdf}
\column{2.5in}
\includegraphics[width=0.9\textwidth]{Kstarmumu_plots/10_110_125.pdf}
\end{columns}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}\frametitle{Now how the $P_x$ look like?}
\begin{itemize}
\item Now we can make a $P_x$ histogram and fit it.
\end{itemize}
\begin{columns}
\column{2.5in}
\includegraphics[width=0.9\textwidth]{Kstarmumu_plots/P5_150_170_left.pdf}
\column{2.5in}
\includegraphics[width=0.9\textwidth]{Kstarmumu_plots/P5_60_80_right.pdf}
\end{columns}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}\frametitle{Huston, we have a problem.}
\begin{itemize}
\item The biggest $F_l$ is observed in $1.1-2.5~\GeV^2$.
\item This is clearly unphysical! We need a smarter way to determine the $P_x$ in this bin.
\item Will try bootstraping.
\end{itemize}
\begin{columns}
\column{2.5in}
\includegraphics[width=0.9\textwidth]{Kstarmumu_plots/P5_11_25_left.pdf}
\column{2.5in}
\includegraphics[width=0.9\textwidth]{Kstarmumu_plots/P2_25_40_left.pdf}
\end{columns}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}\frametitle{$\color{white}{F_s}$ pulls, numbers I promised}
\begin{table}
\begin{tabular}{|c|c|c|}
\hline
Bin $[\GeV^2]$ & mean & sigma \\ \hline \hline
$0.1 -0.98$ & $0.038 \pm 0.029$ & $0.918 \pm0.021$ \\ \hline
$1.1 -2 $ & $-0.024 \pm 0.031$ & $0.998 \pm 0.022$ \\ \hline
$2 - 3 $ & $-0.025 \pm 0.029$ & $0.942 \pm 0.021$ \\ \hline
$3 - 4 $ & $-0.069 \pm 0.030$ & $0.971 \pm 0.022$ \\ \hline
$4 - 5 $ & $0.062 \pm 0.030$ & $0.955 \pm 0.021$ \\ \hline
$5 - 6 $ & $0.081 \pm 0.032$ & $0.992 \pm 0.022$ \\ \hline
$6 - 7 $ & $-0.031 \pm 0.031$ & $0.992 \pm 0.022$ \\ \hline
$7 - 8 $ & $-0.054 \pm 0.030$ & $0.962 \pm 0.021$ \\ \hline
$11 - 11.75 $ & $0.002 \pm 0.030$ & $0.947 \pm 0.021$ \\ \hline
$11.75 - 12.5 $ & $-0.027 \pm 0.031$ & $0.979 \pm 0.022$ \\ \hline
$15 - 16 $ & $0.011 \pm 0.029$ & $0.933 \pm 0.021$ \\ \hline
$16 - 17 $ & $0.02 \pm 0.029$ & $0.929 \pm 0.020$ \\ \hline
$17 - 18 $ & $-0.054 \pm 0.030$ & $0.962 \pm 0.021$ \\ \hline
$18 - 19 $ & $ -0.075 \pm 0.030$ & $0.964 \pm 0.021$ \\ \hline
\end{tabular}
\end{table}
\end{frame}
\begin{frame}\frametitle{$\color{white}{F_s}$ conclusions}
\begin{itemize}
\item We are overestimating the error, but we can leave with this.
\item Mean unbiased.
\end{itemize}
\end{frame}
\begin{frame}\frametitle{Correlations between $\color{white}{S_x}$ and $\color{white}{A_x}$, MC Toys}
\begin{small}
\begin{itemize}
\item Toys with acceptance, background (the ones we all like).
\item Plots in backups.
\end{itemize}
\begin{table}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
$q^2$ & $S_3~A_3$ & $S_4~A_4$& $S_5~A_5$& $S_6~A_6$& $S_7~A_7$& $S_8~A_8$& $S_9~A_9$ \\ \hline \hline
$0$ & $0.049$ & $-0.066$ & $-0.001$ & $0.011$ & $-0.034$ & $0.015$ & $0.022$ \\ \hline
$1$ & $0.029$ & $-0.047$ & $-0.003$ & $0.011$ & $-0.003$ & $0.017$ & $0.052$ \\
\hline
$2$ & $0.004$ & $-0.046$ & $0.026$ & $-0.022$ & $-0.010$ & $-0.036$ & $-0.032$ \\
\hline
$3$ & $0.079$ & $0.032$ & $0.035$ & $-0.028$ & $-0.025$ & $0.037$ & $0.027$ \\
\hline
$4$ & $-0.020$ & $0.074$ & $-0.001$ & $0.008$ & $-0.063$ & $-0.015$ & $0.037$ \\
\hline
$5$ & $-0.019$ & $-0.018$ & $-0.041$ & $0.001$ & $0.025$ & $0.015$ & $-0.014$ \\
\hline
$6$ & $0.000$ & $-0.008$ & $-0.021$ & $0.049$ & $-0.014$ & $-0.04$ & $-0.002$ \\ \hline
$7$ & $0.000$ & $-0.008$ & $-0.021$ & $0.049$ & $-0.014$ & $-0.04$ & $-0.002$ \\ \hline
$8$ & $-0.011$ & $-0.009$ & $-0.013$ & $-0.023$ & $0.047$ & $0.006$ & $-0.004$ \\
\hline
$9$ & $-0.009$ & $-0.015$ & $-0.008$ & $-0.009$ & $-0.010$ & $-0.050$ & $-0.029$ \\ \hline
$10$ & $0.030$ & $0.023$ & $0.084$ & $-0.049$ & $0.030$ & $0.002$ & $0.021$ \\ \hline
$11$ & $-0.016 $ & $0.006$ & $0.018$ & $-0.029$ & $0.034$ & $-0.049$ & $-0.039$ \\ \hline
\hline
\end{tabular}
\end{table}
\end{small}
\end{frame}
\begin{frame}
\begin{Huge}
BACKU
\end{Huge}
\end{frame}
\begin{frame} \frametitle{$0.1-0.98~\GeV^2$}
\begin{columns}
\column{2.5in}
\includegraphics[width=0.95\textwidth]{corr2/0_A3_S3.png}
\column{2.5in}
\includegraphics[width=0.95\textwidth]{corr2/0_A4_S4.png}
\end{columns}
\end{frame}
\begin{frame} \frametitle{$0.1-0.98~\GeV^2$}
\begin{columns}
\column{2.5in}
\includegraphics[width=0.95\textwidth]{corr2/0_A5_S5.png}
\column{2.5in}
\includegraphics[width=0.95\textwidth]{corr2/0_A6s_S6s.png}
\end{columns}
\end{frame}
\begin{frame} \frametitle{$0.1-0.98~\GeV^2$}
\begin{columns}
\column{2.5in}
\includegraphics[width=0.95\textwidth]{corr2/0_A7_S7.png}
\column{2.5in}
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\end{frame}
\begin{frame} \frametitle{$0.1-0.98~\GeV^2$}
\begin{columns}
\column{2.5in}
\includegraphics[width=0.95\textwidth]{corr2/0_A9_S9.png}
\column{2.5in}
{~}
\end{columns}
\end{frame}
%%%%%%%%%%%%%%%
\begin{frame} \frametitle{$1.1-2~\GeV^2$}
\begin{columns}
\column{2.5in}
\includegraphics[width=0.95\textwidth]{corr2/1_A3_S3.png}
\column{2.5in}
\includegraphics[width=0.95\textwidth]{corr2/1_A4_S4.png}
\end{columns}
\end{frame}
\begin{frame} \frametitle{$1.1-2~\GeV^2$}
\begin{columns}
\column{2.5in}
\includegraphics[width=0.95\textwidth]{corr2/1_A5_S5.png}
\column{2.5in}
\includegraphics[width=0.95\textwidth]{corr2/1_A6s_S6s.png}
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\end{frame}
\begin{frame} \frametitle{$1.1-2~\GeV^2$}
\begin{columns}
\column{2.5in}
\includegraphics[width=0.95\textwidth]{corr2/1_A7_S7.png}
\column{2.5in}
\includegraphics[width=0.95\textwidth]{corr2/1_A8_S8.png}
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\end{frame}
\begin{frame} \frametitle{$1.1-2~\GeV^2$}
\begin{columns}
\column{2.5in}
\includegraphics[width=0.95\textwidth]{corr2/1_A9_S9.png}
\column{2.5in}
{~}
\end{columns}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
\begin{frame} \frametitle{$2-3~\GeV^2$}
\begin{columns}
\column{2.5in}
\includegraphics[width=0.95\textwidth]{corr2/2_A3_S3.png}
\column{2.5in}
\includegraphics[width=0.95\textwidth]{corr2/2_A4_S4.png}
\end{columns}
\end{frame}
\begin{frame} \frametitle{$2-3~\GeV^2$}
\begin{columns}
\column{2.5in}
\includegraphics[width=0.95\textwidth]{corr2/2_A5_S5.png}
\column{2.5in}
\includegraphics[width=0.95\textwidth]{corr2/2_A6s_S6s.png}
\end{columns}
\end{frame}
\begin{frame} \frametitle{$2-3~\GeV^2$}
\begin{columns}
\column{2.5in}
\includegraphics[width=0.95\textwidth]{corr2/2_A7_S7.png}
\column{2.5in}
\includegraphics[width=0.95\textwidth]{corr2/2_A8_S8.png}
\end{columns}
\end{frame}
\begin{frame} \frametitle{$2-3~\GeV^2$}
\begin{columns}
\column{2.5in}
\includegraphics[width=0.95\textwidth]{corr2/2_A9_S9.png}
\column{2.5in}
{~}
\end{columns}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
\begin{frame} \frametitle{$3-4~\GeV^2$}
\begin{columns}
\column{2.5in}
\includegraphics[width=0.95\textwidth]{corr2/4_A3_S3.png}
\column{2.5in}
\includegraphics[width=0.95\textwidth]{corr2/4_A4_S4.png}
\end{columns}
\end{frame}
\begin{frame} \frametitle{$3-4~\GeV^2$}
\begin{columns}
\column{2.5in}
\includegraphics[width=0.95\textwidth]{corr2/4_A5_S5.png}
\column{2.5in}
\includegraphics[width=0.95\textwidth]{corr2/4_A6s_S6s.png}
\end{columns}
\end{frame}
\begin{frame} \frametitle{$3-4~\GeV^2$}
\begin{columns}
\column{2.5in}
\includegraphics[width=0.95\textwidth]{corr2/4_A7_S7.png}
\column{2.5in}
\includegraphics[width=0.95\textwidth]{corr2/4_A8_S8.png}
\end{columns}
\end{frame}
\begin{frame} \frametitle{$3-4~\GeV^2$}
\begin{columns}
\column{2.5in}
\includegraphics[width=0.95\textwidth]{corr2/4_A9_S9.png}
\column{2.5in}
{~}
\end{columns}
\end{frame}
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\begin{frame} \frametitle{$4-5~\GeV^2$}
\begin{columns}
\column{2.5in}
\includegraphics[width=0.95\textwidth]{corr2/4_A3_S3.png}
\column{2.5in}
\includegraphics[width=0.95\textwidth]{corr2/4_A4_S4.png}
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\begin{frame} \frametitle{$4-5~\GeV^2$}
\begin{columns}
\column{2.5in}
\includegraphics[width=0.95\textwidth]{corr2/4_A5_S5.png}
\column{2.5in}
\includegraphics[width=0.95\textwidth]{corr2/4_A6s_S6s.png}
\end{columns}
\end{frame}
\begin{frame} \frametitle{$4-5~\GeV^2$}
\begin{columns}
\column{2.5in}
\includegraphics[width=0.95\textwidth]{corr2/4_A7_S7.png}
\column{2.5in}
\includegraphics[width=0.95\textwidth]{corr2/4_A8_S8.png}
\end{columns}
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\begin{frame} \frametitle{$4-5~\GeV^2$}
\begin{columns}
\column{2.5in}
\includegraphics[width=0.95\textwidth]{corr2/4_A9_S9.png}
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{~}
\end{columns}
\end{frame}
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\begin{frame} \frametitle{$5-6~\GeV^2$}
\begin{columns}
\column{2.5in}
\includegraphics[width=0.95\textwidth]{corr2/5_A3_S3.png}
\column{2.5in}
\includegraphics[width=0.95\textwidth]{corr2/5_A4_S4.png}
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\begin{frame} \frametitle{$5-6~\GeV^2$}
\begin{columns}
\column{2.5in}
\includegraphics[width=0.95\textwidth]{corr2/5_A5_S5.png}
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\includegraphics[width=0.95\textwidth]{corr2/5_A6s_S6s.png}
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\begin{frame} \frametitle{$5-6~\GeV^2$}
\begin{columns}
\column{2.5in}
\includegraphics[width=0.95\textwidth]{corr2/5_A7_S7.png}
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\begin{frame} \frametitle{$5-6~\GeV^2$}
\begin{columns}
\column{2.5in}
\includegraphics[width=0.95\textwidth]{corr2/5_A9_S9.png}
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{~}
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\begin{frame} \frametitle{$6-7~\GeV^2$}
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\column{2.5in}
\includegraphics[width=0.95\textwidth]{corr2/6_A3_S3.png}
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\includegraphics[width=0.95\textwidth]{corr2/6_A4_S4.png}
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\begin{frame} \frametitle{$6-7~\GeV^2$}
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\includegraphics[width=0.95\textwidth]{corr2/6_A5_S5.png}
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\includegraphics[width=0.95\textwidth]{corr2/6_A6s_S6s.png}
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\begin{columns}
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\begin{frame} \frametitle{$7-8~\GeV^2$}
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\includegraphics[width=0.95\textwidth]{corr2/7_A3_S3.png}
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\begin{frame} \frametitle{$15-16~\GeV^2$}
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\includegraphics[width=0.95\textwidth]{corr2/8_A3_S3.png}
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\includegraphics[width=0.95\textwidth]{corr2/8_A5_S5.png}
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\begin{frame} \frametitle{$15-16~\GeV^2$}
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\begin{frame} \frametitle{$16-17~\GeV^2$}
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\includegraphics[width=0.95\textwidth]{corr2/9_A3_S3.png}
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\begin{frame} \frametitle{$16-17~\GeV^2$}
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\includegraphics[width=0.95\textwidth]{corr2/9_A5_S5.png}
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\begin{frame} \frametitle{$16-17~\GeV^2$}
\begin{columns}
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\includegraphics[width=0.95\textwidth]{corr2/9_A7_S7.png}
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\begin{frame} \frametitle{$16-17~\GeV^2$}
\begin{columns}
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\begin{frame} \frametitle{$17-18~\GeV^2$}
\begin{columns}
\column{2.5in}
\includegraphics[width=0.95\textwidth]{corr2/10_A3_S3.png}
\column{2.5in}
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\begin{frame} \frametitle{$17-18~\GeV^2$}
\begin{columns}
\column{2.5in}
\includegraphics[width=0.95\textwidth]{corr2/10_A5_S5.png}
\column{2.5in}
\includegraphics[width=0.95\textwidth]{corr2/10_A6s_S6s.png}
\end{columns}
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\begin{frame} \frametitle{$17-18~\GeV^2$}
\begin{columns}
\column{2.5in}
\includegraphics[width=0.95\textwidth]{corr2/10_A7_S7.png}
\column{2.5in}
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\begin{frame} \frametitle{$17-18~\GeV^2$}
\begin{columns}
\column{2.5in}
\includegraphics[width=0.95\textwidth]{corr2/10_A9_S9.png}
\column{2.5in}
{~}
\end{columns}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
\begin{frame} \frametitle{$18-19~\GeV^2$}
\begin{columns}
\column{2.5in}
\includegraphics[width=0.95\textwidth]{corr2/11_A3_S3.png}
\column{2.5in}
\includegraphics[width=0.95\textwidth]{corr2/11_A4_S4.png}
\end{columns}
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\begin{frame} \frametitle{$18-19~\GeV^2$}
\begin{columns}
\column{2.5in}
\includegraphics[width=0.95\textwidth]{corr2/11_A5_S5.png}
\column{2.5in}
\includegraphics[width=0.95\textwidth]{corr2/11_A6s_S6s.png}
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\begin{frame} \frametitle{$18-19~\GeV^2$}
\begin{columns}
\column{2.5in}
\includegraphics[width=0.95\textwidth]{corr2/11_A7_S7.png}
\column{2.5in}
\includegraphics[width=0.95\textwidth]{corr2/11_A8_S8.png}
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\begin{frame} \frametitle{$18-19~\GeV^2$}
\begin{columns}
\column{2.5in}
\includegraphics[width=0.95\textwidth]{corr2/11_A9_S9.png}
\column{2.5in}
{~}
\end{columns}
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\end{document}
mchrzasz