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- \title{Update on measurement of Bose-Einstein Correlations}
- \author{\underline{Marcin Chrzaszcz}$^{1,2}$, Marcin Kucharczyk$^{1,3}$,\\Tadeusz Lesiak$^1$}
-
- \date{\today}
-
- \begin{document}
-
- {
- \institute{$^1$ Krakow, $^2$ Zurich, $^3$ Milano}
- \setbeamertemplate{footline}{}
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- \logo{
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- \titlepage
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- \institute{UZH,IFJ}
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- \section[Outline]{}
- \begin{frame}
- \tableofcontents
- \end{frame}
-
- %normal slides
- \section{Theory introduction}
-
- %\begin{bibunit}[apalike]
-
-
-
-
- \subsection{Two particle Correlations}
-
-
-
-
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%55
- \begin{frame}\frametitle{Two particle Correlations}
-
- \begin{itemize}
- \item Let $W(p_1,p_2,x_1,x_2)$ be a Wigner function.
- \item For identical particles observed distributions of momentum takes the form:
- \end{itemize}
- \begin{small}
- \begin{align}\label{eq:one}
- %\begin{equation}
- \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)]
- \end{align}\end{small}
- \begin{itemize}
- \item Space distribution $x_1-x_2$ can be access via $C(p_1,p_2)$.
-
- \end{itemize}
-
-
- \end{frame}
- %\end{bibunit}[apalike]
-
- \begin{frame}\frametitle{Two particle Correlations}
-
- \begin{itemize}
- \item Assuming no correlation in space Wiger function can be factorised:
- \end{itemize}
- \begin{small}
- \begin{equation}
- %\begin{equation}
- 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)
- \end{equation}\end{small}
- \begin{itemize}
- \item This simplifies eq.(\ref{eq:one}): $\Omega(p_1,p_2)=\Omega_0(p_1,p_2)[1+\int dx W(P_12,x)e^{ix(p1-p2)}]$
- \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
- \end{equation}
- \item All LEP experiments measured BEC.
- \end{itemize}
-
- \end{frame}
- %\end{bibunit}[apalike]
-
-
-
- \subsection{Goldhaber parametrisation}
- \begin{frame}\frametitle{Goldhaber parametrisation}
-
- Following Goldhaber\footnote{Goldhaber et. al. Phys. Rev. Lett 3 (1959)} we can parametrize the correlation function:
- \begin{equation}
- C_2(q_1,q_2) = N(1 \pm \lambda e^{-Q^2 R^2})
- \end{equation}
- ,where $Q=q_1-q_2$, $R$ radius of the source, $\lambda$ chaotic parameter, $N$ normalization.
- Second order correlation function is defined:
- \begin{equation}
- 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}}
- \end{equation}
-
-
- \end{frame}
- %\end{bibunit}[apalike]
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \begin{frame}\frametitle{Reference samples}
-
- $\mathcal{P}(q_1,q_2)^{ref}$ can be estimated from reference samples:
- \begin{enumerate}
- \item MC without BEC.
- \begin{itemize}
- \item Absence of Coulomb effects in generator.
- \item Data-MC agreement.
- \end{itemize}
- \item Unlike-sign charge particles
- \begin{itemize}
- \item Resonances contribution
- \item Derived from data
- \end{itemize}
- \item Event-mixing
- \begin{itemize}
- \item Mixing event by events.
- \item PV mixing.
- \end{itemize}
-
-
- \end{enumerate}
-
- \end{frame}
-
-
-
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%55555
- \begin{frame}\frametitle{LCMS}
-
- \begin{columns}
- \column{2.8in}
- \begin{itemize}
- \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).
-
-
- \begin{scriptsize}
- \item $Q^2$ can be written:\\
- $Q^2=1+\lambda e^{-Q_{t,out}^2R_{t,out}^2-Q_{t,side}^2R_{t,side}^2-Q_{t,long}^2R_{t,long}^2} = 1+\lambda e^{-Q_{t,\bot}^2R_{t,bot}^2-Q_{t,\|}^2R_{t,\|}^2}$
- \end{scriptsize}
- \item One can perform 1,2 or 3 dim analysis.
- \end{itemize}
-
- \column{3in}
- \includegraphics[scale=.14]{images/lcms.png}
-
- \end{columns}
-
-
-
- \end{frame}
-
- \begin{frame}\frametitle{LEP and CMS results}
- \only<1>
- {
- \includegraphics[scale=.215]{images/table.png}
- }
- \only<2>
- {
- \includegraphics[scale=.195]{images/table2.png}
-
- }
-
- \end{frame}
- \section{Selection}
- \begin{frame}\frametitle{Preselection}
- \begin{columns}
- \column{3.5in}
- \begin{enumerate}
- \item MiniBias Stripping lines.
- \item 2011 data.
- \item Select all particles that come from PV with cuts:
- \begin{itemize}
- \item $TRKChi2<2.6$
- \item $IP<0.2mm$
- \item $IPCHI2 <2.6$
- \item $PIDNN(\pi, K)>0.25$
- \item $ghostNN<0.3$
- \item $P>0.2GeV$
- \item $Pt>0.1GeV$
- \end{itemize}
- \end{enumerate}
- \column{2.2in}
- \includegraphics[scale=.115]{images/ip.png}\\
- \includegraphics[scale=.115]{images/ipChi2.png}\\
- \includegraphics[scale=.115]{images/ghostNN.png}
-
-
-
- \end{columns}
-
- \end{frame}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \begin{frame}\frametitle{Selection}
- \begin{columns}
- \column{3.5in}
- \begin{enumerate}
- \item MiniBias Stripping lines.
- \item 2011 data.
- \item Select all particles that come from PV with cuts:
- \begin{itemize}
- \item $TRKChi2<2.6$
- \item $IP<0.2mm$
- \item $IPCHI2 <2.6$
- \item $PIDNN(\pi, K)>0.25$
- \item $ghostNN<0.3$
- \item $P>0.2GeV$
- \item $Pt>0.1GeV$
- \end{itemize}
- \end{enumerate}
- \column{2.2in}
-
- \end{columns}
-
- \end{frame}
- \section{Preliminary results}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \begin{frame}\frametitle{Results in 2011 data}
- We can rewrite $Q$ in form:
- \begin{equation}
- Q=\sqrt{-2q_{\bot 1} q_{\bot 2}[cosh(y_1 -y_2)-cos(\phi_1-\phi_2)] }
- \end{equation}
- ,where $y_i$ are the pseudo-rapidity, $\phi_i$ are azimuthal angles.
- We see BEC
- \begin{columns}
- \column{1.6in}
- \includegraphics[scale=.2]{images/phi.png}
- \column{1.6in}
- \includegraphics[scale=.2]{images/rap.png}
- \end{columns}
-
- \end{frame}
-
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \section{Three body correlations}
- \begin{frame}\frametitle{Generalization of two body correlations}
- Assuming no correlations in space the Wigner function can be expressed)analogy to eg.(2):
- \begin{equation}
- 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)
- \end{equation}
- This leads to correlation function:
- \begin{small}
- \begin{align}\label{eq:two}
- %\begin{equation}
- 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 \\
- 2 \mathcal{R}[\widehat{w}(P_{12}, \Delta_{12} ) \widehat{w}(P_{23}, \Delta_{23} )\widehat{w}(P_{31}, \Delta_{31} ) ]
- \end{align}\end{small}
- ,where $\Delta_{ij}=p_i-p_j$, and $\widehat{w}(P_{ij}, \Delta_{ij} )=\int dx_idx_j W(P_{ij}, x)e^{ix\Delta_{ij}}$
-
-
-
-
-
- \end{frame}
-
- \begin{frame}\frametitle{Probing Cluster Model}
- \begin{columns}
- \column{3.2in}
- Let us consider simple ansatz:
- \begin{align}\label{eq:two}
- 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)]
- \end{align}
- ,where $V(x)=\int \phi(x-X)V_c(X)dX$,\\ $V_2=\int V_c(X)\phi(x_1-X)\phi(x_2-X)dX$\\
- \only<2>
- {
- $V_c(X)$ is the distribution of clusters in space.\\
- $\phi(x-X)$ is the shape of the cluster. \\
- $V(x_1)V(x_2)$ emission from two clusters. \\
- $V_2(x_1,x_2)$ emission from single cluster. \\
-
-
- }
-
-
-
- \column{1.6in}
- \includegraphics[scale=.15]{images/clusters.png}
- \end{columns}
- \end{frame}
-
- \begin{frame}\frametitle{Probing Cluster Model}
- \begin{columns}
- \column{3.2in}
- The correlation function for this ansatz takes form:
- \begin{equation}
- 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
- \end{equation}
- where $\widehat{\phi}(\Delta_{12}) = \int dx \phi(x)e^{ix\Delta_12}$
-
- \column{1.6in}
- \includegraphics[scale=.15]{images/clusters.png}
- \end{columns}
- \end{frame}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
- \section{Summary}
- \begin{frame}\frametitle{Conclusions}
- \begin{itemize}
- \item BEC clearly visible in data.
- \item Analysis systematically dominated.
- \item Enought events to perform first measurement of 3 body correlations.
- \end{itemize}
-
-
- \end{frame}
-
-
- \end{document}