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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}{} 
\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}


%\subsection{From interferometry to particle Physics}
\begin{frame}\frametitle{From interferometry to particle Physics}
\begin{columns}
\column{2.8in}

\includegraphics[height=3.7cm,keepaspectratio]{images/HBT.png}

\column{2.2in}
\begin{small}
Intensity interferometry was discovered in 1950s  by Hanbury-Brown, Twiss (HBT Interferometry) as a method of measuring the angular diameters of radio sources.  \\
It relies on the fact that two photons emitted from the source have to be correlated due to second order interference effect.
\end{small}
\end{columns}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
$C(d)= \dfrac{<I_1><I_2>}{<I_1I_2>} \sim k \theta d$ \\
,where $\theta = R/L$. By changing the $d$ one can measure the diameter of the source.

	
	
\end{frame}

\begin{frame}\frametitle{From interferometry to particle Physics}
\only<1>{
For two identical particles emited from a source we expect a symmetric wave function:\\
$\Psi^s_{1,2} = \dfrac{1}{\sqrt{2}} (\Psi_{11}\Psi_{22}+\Psi_{12}\Psi_{21} )$, where $\Psi_{ij}$ is a wave function of a particle emitted at i and observed at j. 
So the propablity density of observing two bosons with momenta $q_1$ and $q_2$ is:\\
$\abs{\Psi^s_{1,2} }^2 = 1+cos(\Delta \overrightarrow{q} \Delta \overrightarrow{r}	)$, where $\Delta \overrightarrow{q} = q_1-q_2$, $\Delta \overrightarrow{r} = r_1-r_2$
	}
\only<2>
{	
Assuming spherical symmetry of the source: $\mathcal{P}(\overrightarrow{q}) = \int \abs{\rho(r;\overrightarrow{q})}^2 d^3\overrightarrow{r}$, the probability of observing two particles with two momenta is given by:
$\mathcal{P}(\overrightarrow{q_1}, \overrightarrow{q_2}) =\int \abs{\Psi^s_{1,2} }^2 \abs{\rho(\overrightarrow{r_1})}\abs{\rho(\overrightarrow{r_2})}  d^3r_1 d^3r_2$,\\ applying this expression to general $2^{nd}$ correlation function: $C_2(q_1,q_2)=\dfrac{P(q_1,q_2)}{\mathcal{P}(q_1)\mathcal{P}(q_2)}=\dfrac{P(q_1,q_2)}{P(q_1,q_2)^{ref}}$ one gets: \\

	$C_2(q_1,q_2)=1+\dfrac{\int cos[\Delta \overrightarrow{q}(\overrightarrow{r_1} -\overrightarrow{r_2})\abs{\rho(\overrightarrow{r_1}) }^2 \abs{\rho(\overrightarrow{r_2}) }^2 }{\mathcal{P}(q_1)\mathcal{P}(q_2)}$
		
}	

\only<3>
{

Performing a Fourier transform:\\
\begin{columns}
\column{0.3in}{~}
\column{1.6in}
$C(Q)=1+\abs{\widehat{\rho}(Q)}^2$
\column{2.6in}
,where $\widehat{\rho}(Q) = \int e^{-irQ}dr$
\end{columns}
Assuming Gaussian spread of the source: $\rho(r)=R_0 e^{-\frac{r^2}{2R^2}}$, we can simplify the correlation function:\\
\begin{center}
$C(Q)=1+e^{-R^2Q^2}$
\end{center}
This equation is then corrected for the source incoherence, by introducing an free parameter $\lambda$:\\
\begin{equation}
C(Q)=N(1+\lambda e^{-R^2Q^2})
\end{equation}
Eq.(1) is so called Goldhaber parametrization and allows to measure the radius of the source.





}
	
	
\end{frame}



\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 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 Stripping 20.
\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 $TRKChi2<2.$
\item $IP<0.1mm$
\item $IPCHI2 <1.8$
\item $PIDNN(\pi)>0.8$, $PIDNN(K)>0.6$
\item $ghostNN<0.2$
\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}
Enhancement at low $Q^2$ region. We selected $\mathcal{O}(10^8)$ $\pi$ pairs, and $\mathcal{O}(10^6)$ $K$ pairs.
\begin{center}
\includegraphics[scale=.2]{images/q_2011.png}
\end{center}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%





\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/rap.png}
\column{1.6in}
\includegraphics[scale=.2]{images/rap_kaon.png}
\end{columns}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%555

\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{center}
\includegraphics[scale=.2]{images/phi_2011.png}
\end{center}

\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\footnote{Based on Prof. Bialas's talk at cern in July on soft QCD}
	\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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%55
\begin{frame}\frametitle{Dependence R on hadron mass}
	\begin{columns}
	\column{2.6in}
		\includegraphics[scale=.3]{images/mass_dependence.png}

	\column{2.4in}	
\begin{enumerate}
\item Bialas, Zalewski, Phys.Rev. D62 (2000) 114007
\item LHCb can access much higher masses than LEP.
\item Measurement of BEC in charm sector.
\end{enumerate}

	\end{columns}
\end{frame}












%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
\section{Summary}
\begin{frame}\frametitle{Conclusions}
\begin{itemize}
\item Theoretical support from Krakow theorists: prof. Bialas, prof. Zalewski.
\item BEC clearly visible in data.
\item Analysis systematically dominated.
\item Enough events to perform first measurement of 3 body correlations.
\item BEC measurements in charm sector.
\end{itemize}


\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5





\end{document}