\documentclass[]{beamer}
\setbeamertemplate{navigation symbols}{}
\usepackage{beamerthemesplit}
\useoutertheme{infolines}
\usecolortheme{dolphin}
%\usetheme{Warsaw}
\usetheme{progressbar} 
\usecolortheme{progressbar}
\usefonttheme{progressbar}
\useoutertheme{progressbar}
\useinnertheme{progressbar}
\usepackage{graphicx}
%\usepackage{amssymb,amsmath}
\usepackage[latin1]{inputenc}
\usepackage{amsmath}

\usepackage{iwona}
\progressbaroptions{imagename=images/lhcb}
%\usetheme{Boadilla}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
\definecolor{mygreen}{cmyk}{0.82,0.11,1,0.25}
\setbeamertemplate{blocks}[rounded][shadow=false]
\addtobeamertemplate{block begin}{\pgfsetfillopacity{0.8}}{\pgfsetfillopacity{1}}
\setbeamercolor{structure}{fg=mygreen}
\setbeamercolor*{block title example}{fg=mygreen!50,
bg= blue!10}
\setbeamercolor*{block body example}{fg= blue,
bg= blue!5}





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\beamersetuncovermixins{\opaqueness<1>{25}}{\opaqueness<2->{15}}
\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}{} 
\begin{frame}
\logo{
\vspace{2 mm}
\includegraphics[height=1cm,keepaspectratio]{images/ifj.png}~
\includegraphics[height=1cm,keepaspectratio]{images/uzh.jpg}}

  \titlepage
\end{frame}
}

\institute{UZH,IFJ} 


\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}