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- \title{Update on measurement of Bose-Einstein Correlations}
- \author{\underline{Marcin Kucharczyk$^{1,2}$}}
-
- \date{\today}
-
- \begin{document}
-
- {
- \institute{$^1$ Krakow, $^2$ Milano}
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- \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}
-
- \begin{frame}\frametitle{Work in progress}
- \
- \begin{enumerate}
- \item We observed a difference between Magnet Polarity as reported by P. Koppenburg LHCb-INT-2013-047
-
- \end{enumerate}
-
- \center \includegraphics[scale=.25]{images/patrick.png}
-
- \end{frame}
-
-
- \begin{frame}\frametitle{Work in progress}
- \
- \begin{itemize}
- \item We are preparing TOY-MC study to see the impact of this for our measurements.
- \item Under consideration: Do the analysis for two polarities separate, and then combine.
- \end{itemize}
- \center \includegraphics[scale=.25]{images/patrick.png}
-
- \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
-
-
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-
-
- \end{document}