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Presentations / Kstmumu / Update_Ps / mchrzasz_kstmumu.tex
@mchrzasz mchrzasz on 6 May 2015 31 KB big commit of everything
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  1. \documentclass[xcolor=svgnames]{beamer}
  2. \usepackage[utf8]{inputenc}
  3. \usepackage[english]{babel}
  4. \usepackage{polski}
  5. %\usepackage{amssymb,amsmath}
  6. %\usepackage[latin1]{inputenc}
  7. %\usepackage{amsmath}
  8. %\newcommand\abs[1]{\left|#1\right|}
  9. \usepackage{amsmath}
  10. \newcommand\abs[1]{\left|#1\right|}
  11. \usepackage{hepnicenames}
  12. \usepackage{hepunits}
  13. \usepackage{color}
  14.  
  15.  
  16.  
  17. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
  18. \definecolor{mygreen}{cmyk}{0.82,0.11,1,0.25}
  19.  
  20.  
  21. \usetheme{Sybila} 
  22.  
  23. \title[$\PB \to \PKstar \Pmu \Pmu$ update ]{$\PB \to \PKstar \Pmu \Pmu$ update }
  24. \author{Marcin Chrz\k{a}szcz$^{1}$, Nicola Serra$^{1}$}
  25. \institute{$^1$~University of Zurich}
  26. \date{\today}
  27.  
  28. \begin{document}
  29. % --------------------------- SLIDE --------------------------------------------
  30. \frame[plain]{\titlepage}
  31. \author{M.Chrz\k{a}szcz{~}, N.Serra}
  32. \institute{~(UZH)}
  33. % ------------------------------------------------------------------------------
  34. % --------------------------- SLIDE --------------------------------------------
  35. \begin{frame}\frametitle{Propagation from $S_x$ to $P_x$}
  36.  
  37. \begin{itemize}
  38. \item Using MoM we have a direct access to $S_x$.
  39. \item Some theorist (we all know who ;)) like to have instead $P_x$.
  40. \item We need to provide them the $P_x$ or they will do crazy crazy things with the errors.
  41. \item Reminder:
  42. \end{itemize}
  43. \begin{center}
  44. \begin{equation}
  45. P_1=2/(1-F_l),
  46. \end{equation}
  47. \begin{equation}
  48. P_2=\dfrac{2 A_{fb}}{3}/(1-F_l)
  49. \end{equation}
  50. \begin{equation}
  51. P_3=\dfrac{-S_9}{3}/(1-F_l)
  52. \end{equation}
  53. \begin{equation}
  54. P_{4568}=\dfrac{S_{4578}}{3}/\sqrt{F_l(1-F_l)}
  55. \end{equation}
  56. \end{center}
  57.  
  58. \end{frame}
  59.  
  60. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  61. \begin{frame}\frametitle{MC propagation}
  62.  
  63. 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:
  64. \begin{itemize}
  65. \item Simulate multidimensional gauss with with 1 and centred at 0.
  66. \item Correlate the dimensions using Cholesky matrix decomposition.
  67. \item Scale the left and the right tail to the right $\sigma_{L,R}$.
  68. \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 $\sigma$.
  69. \item Move the mean of each Gauss to the value of $S_x$ and voila.
  70.  
  71.  
  72. \end{itemize}
  73.  
  74. \end{frame}
  75.  
  76. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  77. \begin{frame}\frametitle{MC propagation, X-checks 1/2}
  78. \begin{itemize}
  79. \item Since this procedure is not form books I wated to be extra carefull and performed super tests.
  80. \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.
  81. \item Check that the $\sigma_{L,R}$ are the correctly simulate. Fitted left and right separately and compare the results. Oki doki.
  82. \end{itemize}
  83. \begin{columns}
  84. \column{2.5in}
  85. \includegraphics[width=0.9\textwidth]{Kstarmumu_plots/7_110_125_left.pdf}
  86. \column{2.5in}
  87. \includegraphics[width=0.9\textwidth]{Kstarmumu_plots/7_110_125_right.pdf}
  88. \end{columns}
  89.  
  90. \end{frame}
  91.  
  92. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  93. \begin{frame}\frametitle{MC propagation, X-checks 2/2}
  94. \begin{itemize}
  95. \item Since this procedure is not form books I wated to be extra carefull and performed super tests.
  96. \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.
  97. \item Check that the $\sigma_{L,R}$ are the correctly simulate. Fitted left and right separately and compare the results. Oki doki.
  98. \end{itemize}
  99. \begin{columns}
  100. \column{2.5in}
  101. \includegraphics[width=0.9\textwidth]{Kstarmumu_plots/01_110_125.pdf}
  102. \column{2.5in}
  103. \includegraphics[width=0.9\textwidth]{Kstarmumu_plots/10_110_125.pdf}
  104. \end{columns}
  105.  
  106. \end{frame}
  107.  
  108. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  109. \begin{frame}\frametitle{Now how the $P_x$ look like?}
  110. \begin{itemize}
  111. \item Now we can make a $P_x$ histogram and fit it.
  112. \end{itemize}
  113. \begin{columns}
  114. \column{2.5in}
  115. \includegraphics[width=0.9\textwidth]{Kstarmumu_plots/P5_150_170_left.pdf}
  116. \column{2.5in}
  117. \includegraphics[width=0.9\textwidth]{Kstarmumu_plots/P5_60_80_right.pdf}
  118. \end{columns}
  119.  
  120. \end{frame}
  121.  
  122. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  123. \begin{frame}\frametitle{Huston, we have a problem.}
  124. \begin{itemize}
  125. \item The biggest $F_l$ is observed in $1.1-2.5~\GeV^2$.
  126. \item This is clearly unphysical! We need a smarter way to determine the $P_x$ in this bin. 
  127. \item Will try bootstraping.
  128. \end{itemize}
  129. \begin{columns}
  130. \column{2.5in}
  131. \includegraphics[width=0.9\textwidth]{Kstarmumu_plots/P5_11_25_left.pdf}
  132. \column{2.5in}
  133. \includegraphics[width=0.9\textwidth]{Kstarmumu_plots/P2_25_40_left.pdf}
  134. \end{columns}
  135.  
  136. \end{frame}
  137. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  138. \begin{frame}\frametitle{$\color{white}{F_s}$ pulls, numbers I promised}
  139.  
  140. \begin{table}
  141. \begin{tabular}{|c|c|c|}
  142. \hline
  143. Bin $[\GeV^2]$ & mean & sigma \\ \hline \hline
  144. $0.1 -0.98$ & $0.038 \pm 0.029$ & $0.918 \pm0.021$ \\ \hline
  145. $1.1 -2 $ & $-0.024 \pm 0.031$ & $0.998 \pm 0.022$ \\ \hline
  146. $2 - 3 $ & $-0.025 \pm 0.029$ & $0.942 \pm 0.021$ \\ \hline
  147. $3 - 4 $ & $-0.069 \pm 0.030$ & $0.971 \pm 0.022$ \\ \hline
  148. $4 - 5 $ & $0.062 \pm 0.030$ & $0.955 \pm 0.021$ \\ \hline
  149. $5 - 6 $ & $0.081 \pm 0.032$ & $0.992 \pm 0.022$ \\ \hline
  150. $6 - 7 $ & $-0.031 \pm 0.031$ & $0.992 \pm 0.022$ \\ \hline
  151. $7 - 8 $ & $-0.054 \pm 0.030$ & $0.962 \pm 0.021$ \\ \hline
  152.  
  153. $11 - 11.75 $ & $0.002 \pm 0.030$ & $0.947 \pm 0.021$ \\ \hline
  154. $11.75 - 12.5 $ & $-0.027 \pm 0.031$ & $0.979 \pm 0.022$ \\ \hline
  155. $15 - 16 $ & $0.011 \pm 0.029$ & $0.933 \pm 0.021$ \\ \hline
  156. $16 - 17 $ & $0.02 \pm 0.029$ & $0.929 \pm 0.020$ \\ \hline
  157. $17 - 18 $ & $-0.054 \pm 0.030$ & $0.962 \pm 0.021$ \\ \hline
  158. $18 - 19 $ & $ -0.075 \pm 0.030$ & $0.964 \pm 0.021$ \\ \hline
  159.  
  160. \end{tabular}
  161. \end{table}
  162.  
  163.  
  164. \end{frame}
  165. \begin{frame}\frametitle{$\color{white}{F_s}$ conclusions}
  166.  
  167. \begin{itemize}
  168. \item We are overestimating the error, but we can leave with this.
  169. \item Mean unbiased.
  170. \end{itemize}
  171.  
  172. \end{frame}
  173.  
  174. \begin{frame}\frametitle{Correlations between $\color{white}{S_x}$ and $\color{white}{A_x}$, MC Toys}
  175. \begin{small}
  176.  
  177.  
  178. \begin{itemize}
  179. \item Toys with acceptance, background (the ones we all like).
  180. \item Plots in backups.
  181. \end{itemize}
  182. \begin{table}
  183. \begin{tabular}{|c|c|c|c|c|c|c|c|}
  184. \hline 
  185. $q^2$ & $S_3~A_3$ & $S_4~A_4$& $S_5~A_5$& $S_6~A_6$& $S_7~A_7$& $S_8~A_8$&  $S_9~A_9$ \\ \hline \hline
  186. $0$ & $0.049$ & $-0.066$ & $-0.001$ & $0.011$ & $-0.034$ & $0.015$ & $0.022$ \\ \hline
  187. $1$ & $0.029$ & $-0.047$ & $-0.003$ & $0.011$ & $-0.003$ & $0.017$ & $0.052$ \\
  188. \hline
  189. $2$ & $0.004$ & $-0.046$ & $0.026$ & $-0.022$ & $-0.010$ & $-0.036$ & $-0.032$ \\
  190. \hline
  191. $3$ & $0.079$ & $0.032$ & $0.035$ & $-0.028$ & $-0.025$ & $0.037$ & $0.027$ \\
  192. \hline
  193. $4$ & $-0.020$ & $0.074$ & $-0.001$ & $0.008$ & $-0.063$ & $-0.015$ & $0.037$ \\
  194. \hline
  195. $5$ & $-0.019$ & $-0.018$ & $-0.041$ & $0.001$ & $0.025$ & $0.015$ & $-0.014$ \\ 
  196. \hline
  197. $6$ & $0.000$ & $-0.008$ & $-0.021$ & $0.049$ & $-0.014$ & $-0.04$ & $-0.002$ \\ \hline
  198. $7$ & $0.000$ & $-0.008$ & $-0.021$ & $0.049$ & $-0.014$ & $-0.04$ & $-0.002$ \\ \hline
  199. $8$ & $-0.011$ & $-0.009$ & $-0.013$ & $-0.023$ & $0.047$ & $0.006$ & $-0.004$ \\
  200. \hline
  201. $9$ & $-0.009$ & $-0.015$ & $-0.008$ & $-0.009$ & $-0.010$ & $-0.050$ & $-0.029$ \\ \hline
  202. $10$ & $0.030$ & $0.023$ & $0.084$ & $-0.049$ & $0.030$ & $0.002$ & $0.021$ \\ \hline
  203. $11$ & $-0.016 $ & $0.006$ & $0.018$ & $-0.029$ & $0.034$ & $-0.049$ & $-0.039$ \\ \hline
  204.  
  205.  
  206. \hline
  207.  
  208.  
  209.  
  210. \end{tabular}
  211. \end{table}
  212.  
  213. \end{small}
  214. \end{frame}
  215.  
  216.  
  217.  
  218. \begin{frame}\frametitle{Things to do for me}
  219. \begin{itemize}
  220. \item Finish systematics (already started).
  221. \item Study combination of $F_s$ from $m_{K\pi}$ fit and BW phase shift.
  222. \item There was a discussion with Tom about the errors that we quote: aka do we use Bootstrapping or just weighted error? Need a decision today.
  223.  
  224. \end{itemize}
  225.  
  226.  
  227.  
  228. \end{frame}
  229.  
  230.  
  231.  
  232.  
  233.  
  234.     \begin{frame}
  235.  
  236. \begin{Huge}
  237. BACKU
  238. \end{Huge}
  239.  
  240.   \end{frame}
  241.  
  242.     \begin{frame}       \frametitle{$0.1-0.98~\GeV^2$}
  243.     \begin{columns}
  244.     \column{2.5in}
  245.      \includegraphics[width=0.95\textwidth]{corr2/0_A3_S3.png}
  246.     \column{2.5in}
  247.  \includegraphics[width=0.95\textwidth]{corr2/0_A4_S4.png}
  248. \end{columns}
  249.  
  250.  
  251.   \end{frame}
  252.  
  253.      \begin{frame}       \frametitle{$0.1-0.98~\GeV^2$}
  254.     \begin{columns}
  255.     \column{2.5in}
  256.      \includegraphics[width=0.95\textwidth]{corr2/0_A5_S5.png}
  257.     \column{2.5in}
  258.  \includegraphics[width=0.95\textwidth]{corr2/0_A6s_S6s.png}
  259. \end{columns}
  260.  
  261.  
  262.   \end{frame}
  263.  
  264.        \begin{frame}       \frametitle{$0.1-0.98~\GeV^2$}
  265.     \begin{columns}
  266.     \column{2.5in}
  267.      \includegraphics[width=0.95\textwidth]{corr2/0_A7_S7.png}
  268.     \column{2.5in}
  269.  \includegraphics[width=0.95\textwidth]{corr2/0_A8_S8.png}
  270. \end{columns}
  271.  
  272.  
  273.   \end{frame}
  274.  
  275.  
  276.          \begin{frame}       \frametitle{$0.1-0.98~\GeV^2$}
  277.     \begin{columns}
  278.     \column{2.5in}
  279.      \includegraphics[width=0.95\textwidth]{corr2/0_A9_S9.png}
  280.     \column{2.5in}
  281. {~}
  282. \end{columns}
  283.  
  284.  
  285.   \end{frame}
  286.  
  287.  
  288. %%%%%%%%%%%%%%%
  289.  
  290.   \begin{frame}       \frametitle{$1.1-2~\GeV^2$}
  291.     \begin{columns}
  292.     \column{2.5in}
  293.      \includegraphics[width=0.95\textwidth]{corr2/1_A3_S3.png}
  294.     \column{2.5in}
  295.  \includegraphics[width=0.95\textwidth]{corr2/1_A4_S4.png}
  296. \end{columns}
  297.   \end{frame}
  298.      \begin{frame}       \frametitle{$1.1-2~\GeV^2$}
  299.     \begin{columns}
  300.     \column{2.5in}
  301.      \includegraphics[width=0.95\textwidth]{corr2/1_A5_S5.png}
  302.     \column{2.5in}
  303.  \includegraphics[width=0.95\textwidth]{corr2/1_A6s_S6s.png}
  304. \end{columns}
  305.   \end{frame}
  306.  
  307.        \begin{frame}       \frametitle{$1.1-2~\GeV^2$}
  308.     \begin{columns}
  309.     \column{2.5in}
  310.      \includegraphics[width=0.95\textwidth]{corr2/1_A7_S7.png}
  311.     \column{2.5in}
  312.  \includegraphics[width=0.95\textwidth]{corr2/1_A8_S8.png}
  313. \end{columns}
  314.   \end{frame}
  315.          \begin{frame}       \frametitle{$1.1-2~\GeV^2$}
  316.     \begin{columns}
  317.     \column{2.5in}
  318.      \includegraphics[width=0.95\textwidth]{corr2/1_A9_S9.png}
  319.     \column{2.5in}
  320. {~}
  321. \end{columns}
  322.   \end{frame}
  323. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
  324.   \begin{frame}       \frametitle{$2-3~\GeV^2$}            
  325.     \begin{columns}                                          
  326.     \column{2.5in}                                           
  327.      \includegraphics[width=0.95\textwidth]{corr2/2_A3_S3.png}
  328.     \column{2.5in}                                           
  329.  \includegraphics[width=0.95\textwidth]{corr2/2_A4_S4.png}    
  330. \end{columns}                                                
  331.   \end{frame}                                                
  332.      \begin{frame}       \frametitle{$2-3~\GeV^2$}         
  333.     \begin{columns}                                          
  334.     \column{2.5in}                                           
  335.      \includegraphics[width=0.95\textwidth]{corr2/2_A5_S5.png}
  336.     \column{2.5in}                                           
  337.  \includegraphics[width=0.95\textwidth]{corr2/2_A6s_S6s.png}  
  338. \end{columns}                                                
  339.   \end{frame}                                                
  340.                                                              
  341.        \begin{frame}       \frametitle{$2-3~\GeV^2$}       
  342.     \begin{columns}                                          
  343.     \column{2.5in}                                           
  344.      \includegraphics[width=0.95\textwidth]{corr2/2_A7_S7.png}
  345.     \column{2.5in}                                           
  346.  \includegraphics[width=0.95\textwidth]{corr2/2_A8_S8.png}    
  347. \end{columns}                                                
  348.   \end{frame}                                                
  349.          \begin{frame}       \frametitle{$2-3~\GeV^2$}     
  350.     \begin{columns}                                          
  351.     \column{2.5in}                                           
  352.      \includegraphics[width=0.95\textwidth]{corr2/2_A9_S9.png}
  353.     \column{2.5in}                                           
  354. {~}                                                          
  355. \end{columns}                                                
  356.   \end{frame}                                                
  357. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
  358.   \begin{frame}       \frametitle{$3-4~\GeV^2$}              
  359.     \begin{columns}                                          
  360.     \column{2.5in}                                           
  361.      \includegraphics[width=0.95\textwidth]{corr2/4_A3_S3.png}
  362.     \column{2.5in}                                           
  363.  \includegraphics[width=0.95\textwidth]{corr2/4_A4_S4.png}    
  364. \end{columns}                                                
  365.   \end{frame}                                                
  366.      \begin{frame}       \frametitle{$3-4~\GeV^2$}           
  367.     \begin{columns}                                          
  368.     \column{2.5in}                                           
  369.      \includegraphics[width=0.95\textwidth]{corr2/4_A5_S5.png}
  370.     \column{2.5in}                                           
  371.  \includegraphics[width=0.95\textwidth]{corr2/4_A6s_S6s.png}  
  372. \end{columns}                                                
  373.   \end{frame}                                                
  374.                                                              
  375.        \begin{frame}       \frametitle{$3-4~\GeV^2$}         
  376.     \begin{columns}                                          
  377.     \column{2.5in}                                           
  378.      \includegraphics[width=0.95\textwidth]{corr2/4_A7_S7.png}
  379.     \column{2.5in}                                           
  380.  \includegraphics[width=0.95\textwidth]{corr2/4_A8_S8.png}    
  381. \end{columns}                                                
  382.   \end{frame}                                                
  383.          \begin{frame}       \frametitle{$3-4~\GeV^2$}       
  384.     \begin{columns}                                          
  385.     \column{2.5in}                                           
  386.      \includegraphics[width=0.95\textwidth]{corr2/4_A9_S9.png}
  387.     \column{2.5in}                                           
  388. {~}                                                          
  389. \end{columns}                                                
  390.   \end{frame}                                                
  391. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5                  
  392.   \begin{frame}       \frametitle{$4-5~\GeV^2$}               
  393.     \begin{columns}                                           
  394.     \column{2.5in}                                            
  395.      \includegraphics[width=0.95\textwidth]{corr2/4_A3_S3.png} 
  396.     \column{2.5in}                                            
  397.  \includegraphics[width=0.95\textwidth]{corr2/4_A4_S4.png}     
  398. \end{columns}                                                 
  399.   \end{frame}                                                 
  400.      \begin{frame}       \frametitle{$4-5~\GeV^2$}            
  401.     \begin{columns}                                           
  402.     \column{2.5in}                                            
  403.      \includegraphics[width=0.95\textwidth]{corr2/4_A5_S5.png} 
  404.     \column{2.5in}                                            
  405.  \includegraphics[width=0.95\textwidth]{corr2/4_A6s_S6s.png}   
  406. \end{columns}                                                 
  407.   \end{frame}                                                 
  408.                                                               
  409.        \begin{frame}       \frametitle{$4-5~\GeV^2$}          
  410.     \begin{columns}                                           
  411.     \column{2.5in}                                            
  412.      \includegraphics[width=0.95\textwidth]{corr2/4_A7_S7.png} 
  413.     \column{2.5in}                                            
  414.  \includegraphics[width=0.95\textwidth]{corr2/4_A8_S8.png}     
  415. \end{columns}                                                 
  416.   \end{frame}                                                 
  417.          \begin{frame}       \frametitle{$4-5~\GeV^2$}        
  418.     \begin{columns}                                           
  419.     \column{2.5in}                                            
  420.      \includegraphics[width=0.95\textwidth]{corr2/4_A9_S9.png} 
  421.     \column{2.5in}                                            
  422. {~}                                                           
  423. \end{columns}                                                 
  424.   \end{frame}                                                 
  425. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5                        
  426.   \begin{frame}       \frametitle{$5-6~\GeV^2$}                     
  427.     \begin{columns}                                                 
  428.     \column{2.5in}                                                  
  429.      \includegraphics[width=0.95\textwidth]{corr2/5_A3_S3.png}       
  430.     \column{2.5in}                                                  
  431.  \includegraphics[width=0.95\textwidth]{corr2/5_A4_S4.png}           
  432. \end{columns}                                                       
  433.   \end{frame}                                                       
  434.      \begin{frame}       \frametitle{$5-6~\GeV^2$}                  
  435.     \begin{columns}                                                 
  436.     \column{2.5in}                                                  
  437.      \includegraphics[width=0.95\textwidth]{corr2/5_A5_S5.png}       
  438.     \column{2.5in}                                                  
  439.  \includegraphics[width=0.95\textwidth]{corr2/5_A6s_S6s.png}         
  440. \end{columns}                                                       
  441.   \end{frame}                                                       
  442.                                                                     
  443.        \begin{frame}       \frametitle{$5-6~\GeV^2$}                
  444.     \begin{columns}                                                 
  445.     \column{2.5in}                                                  
  446.      \includegraphics[width=0.95\textwidth]{corr2/5_A7_S7.png}       
  447.     \column{2.5in}                                                  
  448.  \includegraphics[width=0.95\textwidth]{corr2/5_A8_S8.png}           
  449. \end{columns}                                                       
  450.   \end{frame}                                                       
  451.          \begin{frame}       \frametitle{$5-6~\GeV^2$}              
  452.     \begin{columns}                                                 
  453.     \column{2.5in}                                                  
  454.      \includegraphics[width=0.95\textwidth]{corr2/5_A9_S9.png}       
  455.     \column{2.5in}                                                  
  456. {~}                                                                 
  457. \end{columns}                                                       
  458.   \end{frame}                                      
  459. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5                 
  460.   \begin{frame}       \frametitle{$6-7~\GeV^2$}              
  461.     \begin{columns}                                          
  462.     \column{2.5in}                                           
  463.      \includegraphics[width=0.95\textwidth]{corr2/6_A3_S3.png}
  464.     \column{2.5in}                                           
  465.  \includegraphics[width=0.95\textwidth]{corr2/6_A4_S4.png}    
  466. \end{columns}                                                
  467.   \end{frame}                                                
  468.      \begin{frame}       \frametitle{$6-7~\GeV^2$}           
  469.     \begin{columns}                                          
  470.     \column{2.5in}                                           
  471.      \includegraphics[width=0.95\textwidth]{corr2/6_A5_S5.png}
  472.     \column{2.5in}                                           
  473.  \includegraphics[width=0.95\textwidth]{corr2/6_A6s_S6s.png}  
  474. \end{columns}                                                
  475.   \end{frame}                                                
  476.                                                              
  477.        \begin{frame}       \frametitle{$6-7~\GeV^2$}         
  478.     \begin{columns}                                          
  479.     \column{2.5in}                                           
  480.      \includegraphics[width=0.95\textwidth]{corr2/6_A7_S7.png}
  481.     \column{2.5in}                                           
  482.  \includegraphics[width=0.95\textwidth]{corr2/6_A8_S8.png}    
  483. \end{columns}                                                
  484.   \end{frame}                                                
  485.          \begin{frame}       \frametitle{$6-7~\GeV^2$}       
  486.     \begin{columns}                                          
  487.     \column{2.5in}                                           
  488.      \includegraphics[width=0.95\textwidth]{corr2/6_A9_S9.png}
  489.     \column{2.5in}                                           
  490. {~}                                                          
  491. \end{columns}                                                
  492.   \end{frame}                                                
  493. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5                 
  494.   \begin{frame}       \frametitle{$7-8~\GeV^2$}              
  495.     \begin{columns}                                          
  496.     \column{2.5in}                                           
  497.      \includegraphics[width=0.95\textwidth]{corr2/7_A3_S3.png}
  498.     \column{2.5in}                                           
  499.  \includegraphics[width=0.95\textwidth]{corr2/7_A4_S4.png}    
  500. \end{columns}                                                
  501.   \end{frame}                                                
  502.      \begin{frame}       \frametitle{$7-8~\GeV^2$}           
  503.     \begin{columns}                                          
  504.     \column{2.5in}                                           
  505.      \includegraphics[width=0.95\textwidth]{corr2/7_A5_S5.png}
  506.     \column{2.5in}                                           
  507.  \includegraphics[width=0.95\textwidth]{corr2/7_A6s_S6s.png}  
  508. \end{columns}                                                
  509.   \end{frame}                                                
  510.                                                              
  511.        \begin{frame}       \frametitle{$7-8~\GeV^2$}         
  512.     \begin{columns}                                          
  513.     \column{2.5in}                                           
  514.      \includegraphics[width=0.95\textwidth]{corr2/7_A7_S7.png}
  515.     \column{2.5in}                                           
  516.  \includegraphics[width=0.95\textwidth]{corr2/7_A8_S8.png}    
  517. \end{columns}                                                
  518.   \end{frame}                                                
  519.          \begin{frame}       \frametitle{$7-8~\GeV^2$}       
  520.     \begin{columns}                                          
  521.     \column{2.5in}                                           
  522.      \includegraphics[width=0.95\textwidth]{corr2/7_A9_S9.png}
  523.     \column{2.5in}                                           
  524. {~}                                                          
  525. \end{columns}                                                
  526.   \end{frame}                                                
  527. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5                 
  528.   \begin{frame}       \frametitle{$15-16~\GeV^2$}              
  529.     \begin{columns}                                          
  530.     \column{2.5in}                                           
  531.      \includegraphics[width=0.95\textwidth]{corr2/8_A3_S3.png}
  532.     \column{2.5in}                                           
  533.  \includegraphics[width=0.95\textwidth]{corr2/8_A4_S4.png}    
  534. \end{columns}                                                
  535.   \end{frame}                                                
  536.      \begin{frame}       \frametitle{$15-16~\GeV^2$}           
  537.     \begin{columns}                                          
  538.     \column{2.5in}                                           
  539.      \includegraphics[width=0.95\textwidth]{corr2/8_A5_S5.png}
  540.     \column{2.5in}                                           
  541.  \includegraphics[width=0.95\textwidth]{corr2/8_A6s_S6s.png}  
  542. \end{columns}                                                
  543.   \end{frame}                                                
  544.                                                              
  545.        \begin{frame}       \frametitle{$15-16~\GeV^2$}         
  546.     \begin{columns}                                          
  547.     \column{2.5in}                                           
  548.      \includegraphics[width=0.95\textwidth]{corr2/8_A7_S7.png}
  549.     \column{2.5in}                                           
  550.  \includegraphics[width=0.95\textwidth]{corr2/8_A8_S8.png}    
  551. \end{columns}                                                
  552.   \end{frame}                                                
  553.          \begin{frame}       \frametitle{$15-16~\GeV^2$}       
  554.     \begin{columns}                                          
  555.     \column{2.5in}                                           
  556.      \includegraphics[width=0.95\textwidth]{corr2/8_A9_S9.png}
  557.     \column{2.5in}                                           
  558. {~}                                                          
  559. \end{columns}                                                
  560.   \end{frame}                                                                                                      
  561. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5                 
  562.   \begin{frame}       \frametitle{$16-17~\GeV^2$}            
  563.     \begin{columns}                                          
  564.     \column{2.5in}                                           
  565.      \includegraphics[width=0.95\textwidth]{corr2/9_A3_S3.png}
  566.     \column{2.5in}                                           
  567.  \includegraphics[width=0.95\textwidth]{corr2/9_A4_S4.png}    
  568. \end{columns}                                                
  569.   \end{frame}                                                
  570.      \begin{frame}       \frametitle{$16-17~\GeV^2$}         
  571.     \begin{columns}                                          
  572.     \column{2.5in}                                           
  573.      \includegraphics[width=0.95\textwidth]{corr2/9_A5_S5.png}
  574.     \column{2.5in}                                           
  575.  \includegraphics[width=0.95\textwidth]{corr2/9_A6s_S6s.png}  
  576. \end{columns}                                                
  577.   \end{frame}                                                
  578.                                                              
  579.        \begin{frame}       \frametitle{$16-17~\GeV^2$}       
  580.     \begin{columns}                                          
  581.     \column{2.5in}                                           
  582.      \includegraphics[width=0.95\textwidth]{corr2/9_A7_S7.png}
  583.     \column{2.5in}                                           
  584.  \includegraphics[width=0.95\textwidth]{corr2/9_A8_S8.png}    
  585. \end{columns}                                                
  586.   \end{frame}                                                
  587.          \begin{frame}       \frametitle{$16-17~\GeV^2$}     
  588.     \begin{columns}                                          
  589.     \column{2.5in}                                           
  590.      \includegraphics[width=0.95\textwidth]{corr2/9_A9_S9.png}
  591.     \column{2.5in}                                           
  592. {~}                                                          
  593. \end{columns}                                                
  594.   \end{frame}                                      
  595. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5                 
  596.   \begin{frame}       \frametitle{$17-18~\GeV^2$}            
  597.     \begin{columns}                                          
  598.     \column{2.5in}                                           
  599.      \includegraphics[width=0.95\textwidth]{corr2/10_A3_S3.png}
  600.     \column{2.5in}                                           
  601.  \includegraphics[width=0.95\textwidth]{corr2/10_A4_S4.png}    
  602. \end{columns}                                                
  603.   \end{frame}                                                
  604.      \begin{frame}       \frametitle{$17-18~\GeV^2$}         
  605.     \begin{columns}                                          
  606.     \column{2.5in}                                           
  607.      \includegraphics[width=0.95\textwidth]{corr2/10_A5_S5.png}
  608.     \column{2.5in}                                           
  609.  \includegraphics[width=0.95\textwidth]{corr2/10_A6s_S6s.png}  
  610. \end{columns}                                                
  611.   \end{frame}                                                
  612.                                                              
  613.        \begin{frame}       \frametitle{$17-18~\GeV^2$}       
  614.     \begin{columns}                                          
  615.     \column{2.5in}                                           
  616.      \includegraphics[width=0.95\textwidth]{corr2/10_A7_S7.png}
  617.     \column{2.5in}                                           
  618.  \includegraphics[width=0.95\textwidth]{corr2/10_A8_S8.png}    
  619. \end{columns}                                                
  620.   \end{frame}                                                
  621.          \begin{frame}       \frametitle{$17-18~\GeV^2$}     
  622.     \begin{columns}                                          
  623.     \column{2.5in}                                           
  624.      \includegraphics[width=0.95\textwidth]{corr2/10_A9_S9.png}
  625.     \column{2.5in}                                           
  626. {~}                                                          
  627. \end{columns}                                                
  628.   \end{frame}                                                
  629. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5                    
  630.   \begin{frame}       \frametitle{$18-19~\GeV^2$}               
  631.     \begin{columns}                                             
  632.     \column{2.5in}                                              
  633.      \includegraphics[width=0.95\textwidth]{corr2/11_A3_S3.png}  
  634.     \column{2.5in}                                              
  635.  \includegraphics[width=0.95\textwidth]{corr2/11_A4_S4.png}      
  636. \end{columns}                                                   
  637.   \end{frame}                                                   
  638.      \begin{frame}       \frametitle{$18-19~\GeV^2$}            
  639.     \begin{columns}                                             
  640.     \column{2.5in}                                              
  641.      \includegraphics[width=0.95\textwidth]{corr2/11_A5_S5.png}  
  642.     \column{2.5in}                                              
  643.  \includegraphics[width=0.95\textwidth]{corr2/11_A6s_S6s.png}    
  644. \end{columns}                                                   
  645.   \end{frame}                                                   
  646.                                                                 
  647.        \begin{frame}       \frametitle{$18-19~\GeV^2$}          
  648.     \begin{columns}                                             
  649.     \column{2.5in}                                              
  650.      \includegraphics[width=0.95\textwidth]{corr2/11_A7_S7.png} 
  651.     \column{2.5in}                                              
  652.  \includegraphics[width=0.95\textwidth]{corr2/11_A8_S8.png}     
  653. \end{columns}                                                   
  654.   \end{frame}                                                   
  655.          \begin{frame}       \frametitle{$18-19~\GeV^2$}        
  656.     \begin{columns}                                             
  657.     \column{2.5in}                                              
  658.      \includegraphics[width=0.95\textwidth]{corr2/11_A9_S9.png} 
  659.     \column{2.5in}                                              
  660. {~}                                                             
  661. \end{columns}                                                   
  662.   \end{frame}                                                   
  663.  
  664.  
  665.  
  666.  
  667.  
  668.  
  669.  
  670. \end{document}