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Presentations / lhcb_week_krakow / BEC / BEC_v1.tex
@mchrzasz mchrzasz on 10 Oct 2013 8 KB update before changing laptops
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  33.  
  34.  
  35. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  36. %\beamersetuncovermixins{\opaqueness<1>{25}}{\opaqueness<2->{15}}
  37. \title{Update on measurement of Bose-Einstein Correlations}
  38. \author{\underline{Marcin Chrzaszcz}$^{1,2}$, Marcin Kucharczyk$^{1,3}$,\\Tadeusz Lesiak$^1$}
  39.  
  40. \date{\today}
  41.  
  42. \begin{document}
  43.  
  44. {
  45. \institute{$^1$ Krakow, $^2$ Zurich, $^3$ Milano}
  46. \setbeamertemplate{footline}{}
  47. \begin{frame}
  48. \logo{
  49. \vspace{2 mm}
  50. \includegraphics[height=1cm,keepaspectratio]{images/ifj.png}~
  51. \includegraphics[height=1cm,keepaspectratio]{images/uzh.jpg}}
  52.  
  53. \titlepage
  54. \end{frame}
  55. }
  56.  
  57. \institute{UZH,IFJ}
  58.  
  59.  
  60. \section[Outline]{}
  61. \begin{frame}
  62. \tableofcontents
  63. \end{frame}
  64.  
  65. %normal slides
  66. \section{Theory introduction}
  67.  
  68. %\begin{bibunit}[apalike]
  69.  
  70.  
  71.  
  72.  
  73. \subsection{Two particle Correlations}
  74.  
  75.  
  76.  
  77.  
  78. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%55
  79. \begin{frame}\frametitle{Two particle Correlations}
  80. \begin{itemize}
  81. \item Let $W(p_1,p_2,x_1,x_2)$ be a Wigner function.
  82. \item For identical particles observed distributions of momentum takes the form:
  83. \end{itemize}
  84. \begin{small}
  85. \begin{align}\label{eq:one}
  86. %\begin{equation}
  87. \Omega(p_1,p_2)= \int dx_1 dx_2 (W(p_1,p_2,x_1,x_2)+ e^{(x_1-x_2) (p_1-p_2)} W(P_{12},P_{12},x_1,x_2)) \nonumber \\ \equiv \Omega_0(p_1,p_2)[1+C(p_1,p_2)]
  88. \end{align}\end{small}
  89. \begin{itemize}
  90. \item Space distribution $x_1-x_2$ can be access via $C(p_1,p_2)$.
  91.  
  92. \end{itemize}
  93.  
  94. \end{frame}
  95. %\end{bibunit}[apalike]
  96.  
  97. \begin{frame}\frametitle{Two particle Correlations}
  98. \begin{itemize}
  99. \item Assuming no correlation in space Wiger function can be factorised:
  100. \end{itemize}
  101. \begin{small}
  102. \begin{equation}
  103. %\begin{equation}
  104. W(p_1,p_2,x_1,x_2)= \Omega_0(p_1,p_2)w(p_1,x_1)w(p_2,x_2)
  105. \end{equation}\end{small}
  106. \begin{itemize}
  107. \item This simplifies eq.(\ref{eq:one}): $\Omega(p_1,p_2)=\Omega_0(p_1,p_2)[1+\int dx W(P_12,x)e^{ix(p1-p2)}]$
  108. \item The 2 body correlation can be written as: \begin{equation} C_2(p_1,p_2)=\vert \int dx W(P_{12},x)e^{ix(p1-p2)} \vert^2
  109. \end{equation}
  110. \item All LEP experiments measured BEC.
  111. \end{itemize}
  112. \end{frame}
  113. %\end{bibunit}[apalike]
  114.  
  115.  
  116.  
  117. \subsection{Goldhaber parametrisation}
  118. \begin{frame}\frametitle{Goldhaber parametrisation}
  119. Following Goldhaber\footnote{Goldhaber et. al. Phys. Rev. Lett 3 (1959)} we can parametrize the correlation function:
  120. \begin{equation}
  121. C_2(q_1,q_2) = N(1 \pm \lambda e^{-Q^2 R^2})
  122. \end{equation}
  123. ,where $Q=q_1-q_2$, $R$ radius of the source, $\lambda$ chaotic parameter, $N$ normalization.
  124. Second order correlation function is defined:
  125. \begin{equation}
  126. C_2(q_1,q_2) = \dfrac{\mathcal{P}(q_1,q_2)}{\mathcal{P}(q_1)\mathcal{P}(q_2)} \equiv \dfrac{\mathcal{P}(q_1,q_2)}{\mathcal{P}(q_1,q_2)^{ref}}
  127. \end{equation}
  128.  
  129.  
  130. \end{frame}
  131. %\end{bibunit}[apalike]
  132. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  133. \begin{frame}\frametitle{Reference samples}
  134.  
  135. $\mathcal{P}(q_1,q_2)^{ref}$ can be estimated from reference samples:
  136. \begin{enumerate}
  137. \item MC without BEC.
  138. \begin{itemize}
  139. \item Absence of Coulomb effects in generator.
  140. \item Data-MC agreement.
  141. \end{itemize}
  142. \item Unlike-sign charge particles
  143. \begin{itemize}
  144. \item Resonances contribution
  145. \item Derived from data
  146. \end{itemize}
  147. \item Event-mixing
  148. \begin{itemize}
  149. \item Mixing event by events.
  150. \item PV mixing.
  151. \end{itemize}
  152.  
  153.  
  154. \end{enumerate}
  155.  
  156. \end{frame}
  157.  
  158.  
  159.  
  160. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%55555
  161. \begin{frame}\frametitle{LCMS}
  162.  
  163. \begin{columns}
  164. \column{2.8in}
  165. \begin{itemize}
  166. \item Longitudinal Centre-of-Mass System(LCMS) is defined as a system where sum of 3-momenta $\overrightarrow{q_1} + \overrightarrow{q_2}$ is perpendicular to a reference axis(jet, thrust, z).
  167.  
  168.  
  169. \begin{scriptsize}
  170. \item $Q^2$ can be written:\\
  171. $Q^2=1+\lambda e^{-Q_{t,out}^2R_{t,out}^2-Q_{t,side}^2R_{t,side}^2-Q_{t,long}^2R_{t,long}^2} = 1+\lambda e^{-Q_{t,\bot}^2R_{t,bot}^2-Q_{t,\|}^2R_{t,\|}^2}$
  172. \end{scriptsize}
  173. \item One can perform 1,2 or 3 dim analysis.
  174. \end{itemize}
  175.  
  176. \column{3in}
  177. \includegraphics[scale=.14]{images/lcms.png}
  178.  
  179. \end{columns}
  180.  
  181.  
  182.  
  183. \end{frame}
  184.  
  185. \begin{frame}\frametitle{LEP and CMS results}
  186. \only<1>
  187. {
  188. \includegraphics[scale=.215]{images/table.png}
  189. }
  190. \only<2>
  191. {
  192. \includegraphics[scale=.195]{images/table2.png}
  193.  
  194. }
  195.  
  196. \end{frame}
  197. \section{Selection}
  198. \begin{frame}\frametitle{Preselection}
  199. \begin{columns}
  200. \column{3.5in}
  201. \begin{enumerate}
  202. \item MiniBias Stripping lines.
  203. \item 2011 data.
  204. \item Select all particles that come from PV with cuts:
  205. \begin{itemize}
  206. \item $TRKChi2<2.6$
  207. \item $IP<0.2mm$
  208. \item $IPCHI2 <2.6$
  209. \item $PIDNN(\pi, K)>0.25$
  210. \item $ghostNN<0.3$
  211. \item $P>0.2GeV$
  212. \item $Pt>0.1GeV$
  213. \end{itemize}
  214. \end{enumerate}
  215. \column{2.2in}
  216. \includegraphics[scale=.115]{images/ip.png}\\
  217. \includegraphics[scale=.115]{images/ipChi2.png}\\
  218. \includegraphics[scale=.115]{images/ghostNN.png}
  219.  
  220.  
  221.  
  222. \end{columns}
  223.  
  224. \end{frame}
  225. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  226. \begin{frame}\frametitle{Selection}
  227. \begin{columns}
  228. \column{3.5in}
  229. \begin{enumerate}
  230. \item MiniBias Stripping lines.
  231. \item 2011 data.
  232. \item Select all particles that come from PV with cuts:
  233. \begin{itemize}
  234. \item $TRKChi2<2.6$
  235. \item $IP<0.2mm$
  236. \item $IPCHI2 <2.6$
  237. \item $PIDNN(\pi, K)>0.25$
  238. \item $ghostNN<0.3$
  239. \item $P>0.2GeV$
  240. \item $Pt>0.1GeV$
  241. \end{itemize}
  242. \end{enumerate}
  243. \column{2.2in}
  244.  
  245. \end{columns}
  246.  
  247. \end{frame}
  248. \section{Preliminary results}
  249. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  250. \begin{frame}\frametitle{Results in 2011 data}
  251. We can rewrite $Q$ in form:
  252. \begin{equation}
  253. Q=\sqrt{-2q_{\bot 1} q_{\bot 2}[cosh(y_1 -y_2)-cos(\phi_1-\phi_2)] }
  254. \end{equation}
  255. ,where $y_i$ are the pseudo-rapidity, $\phi_i$ are azimuthal angles.
  256. We see BEC
  257. \begin{columns}
  258. \column{1.6in}
  259. \includegraphics[scale=.2]{images/phi.png}
  260. \column{1.6in}
  261. \includegraphics[scale=.2]{images/rap.png}
  262. \end{columns}
  263.  
  264. \end{frame}
  265.  
  266. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  267. \section{Three body correlations}
  268. \begin{frame}\frametitle{Generalization of two body correlations}
  269. Assuming no correlations in space the Wigner function can be expressed)analogy to eg.(2):
  270. \begin{equation}
  271. W(p_1,p_2,p_3,x_1,x_2,x_3)=\Omega_0(p_1,p_2,p_3)w(p_1,x_1)w(p_2,x_2)w(p_3,x_3)
  272. \end{equation}
  273. This leads to correlation function:
  274. \begin{small}
  275. \begin{align}\label{eq:two}
  276. %\begin{equation}
  277. C_3(p1,p2,p3)=\vert \widehat{w}(P_{12}, \Delta_{12}) \vert^2 + \vert \widehat{w}(P_{23}, \Delta_{23} ) \vert^2+\vert \widehat{w}(P_{31}, \Delta_{31} ) \vert^2 + \nonumber \\
  278. 2 \mathcal{R}[\widehat{w}(P_{12}, \Delta_{12} ) \widehat{w}(P_{23}, \Delta_{23} )\widehat{w}(P_{31}, \Delta_{31} ) ]
  279. \end{align}\end{small}
  280. ,where $\Delta_{ij}=p_i-p_j$, and $\widehat{w}(P_{ij}, \Delta_{ij} )=\int dx_idx_j W(P_{ij}, x)e^{ix\Delta_{ij}}$
  281.  
  282.  
  283.  
  284.  
  285.  
  286. \end{frame}
  287.  
  288. \begin{frame}\frametitle{Probing Cluster Model}
  289. \begin{columns}
  290. \column{3.2in}
  291. Let us consider simple ansatz:
  292. \begin{align}\label{eq:two}
  293. W(p_1,p_2,x_1,x_2)=\Omega_0(p_1,p_2)[V(x_1)V(x_2)\nonumber \\ +\alpha V_2(x_1,x_2)]
  294. \end{align}
  295. ,where $V(x)=\int \phi(x-X)V_c(X)dX$,\\ $V_2=\int V_c(X)\phi(x_1-X)\phi(x_2-X)dX$\\
  296. \only<2>
  297. {
  298. $V_c(X)$ is the distribution of clusters in space.\\
  299. $\phi(x-X)$ is the shape of the cluster. \\
  300. $V(x_1)V(x_2)$ emission from two clusters. \\
  301. $V_2(x_1,x_2)$ emission from single cluster. \\
  302.  
  303.  
  304. }
  305.  
  306.  
  307.  
  308. \column{1.6in}
  309. \includegraphics[scale=.15]{images/clusters.png}
  310. \end{columns}
  311. \end{frame}
  312.  
  313. \begin{frame}\frametitle{Probing Cluster Model}
  314. \begin{columns}
  315. \column{3.2in}
  316. The correlation function for this ansatz takes form:
  317. \begin{equation}
  318. C(p_1,p_2)= \vert \widehat{V_c}(\Delta_{12}) \widehat{\phi}(\Delta_{12}) \vert^2 + \alpha \vert \widehat{\phi}(\Delta_{12}) \vert^2
  319. \end{equation}
  320. where $\widehat{\phi}(\Delta_{12}) = \int dx \phi(x)e^{ix\Delta_12}$
  321.  
  322. \column{1.6in}
  323. \includegraphics[scale=.15]{images/clusters.png}
  324. \end{columns}
  325. \end{frame}
  326. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
  327. \section{Summary}
  328. \begin{frame}\frametitle{Conclusions}
  329. \begin{itemize}
  330. \item BEC clearly visible in data.
  331. \item Analysis systematically dominated.
  332. \item Enought events to perform first measurement of 3 body correlations.
  333. \end{itemize}
  334.  
  335.  
  336. \end{frame}
  337.  
  338.  
  339. \end{document}