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\usetheme{Sybila}
\title[Search for Charged Lepton Flavour Violation at LHCb experiment]{Search for Charged Lepton Flavour Violation at LHCb experiment}
\subtitle{Doctoral disertation}
\author[Marcin Chrz\k{a}szcz]{Marcin Chrz\k{a}szcz \\ Supervisor: prof. dr hab. Tadeusz Lesiak \\ Auxiliary supervisor: dr hab. Alberto Lusiani}
\institute[UZH, IFJ]{
Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of Sciences, Cracow, Poland}
\date{\today}
\begin{document}
% --------------------------- SLIDE --------------------------------------------
\frame[plain]{\titlepage}
\author{Marcin Chrz\k{a}szcz}
% ------------------------------------------------------------------------------
% --------------------------- SLIDE --------------------------------------------
\institute{~(IFJ)}
% \begin{frame}\frametitle{Outline}
% \begin{enumerate}
% \item introduction\vspace{.5em}
% \item multivariate technique\vspace{.5em}
% \item normalisation\vspace{.5em}
% % \item backgrounds\vspace{.5em}
% \item expected sensitivity\vspace{.5em}
% \item model dependence\vspace{.5em} data from Reco14Stripping20(r1)
% \end{enumerate}
% Major news wrt.\ the $1~fb^{-1}$ analysis are highlighted in \textcolor{mygreen}{green}
% \end{frame}
\begin{frame}\frametitle{Outline}
\tableofcontents
\end{frame}
\iffalse
\begin{frame}
\frametitle{Errata to thesis}
\begin{itemize}
\item Since submision, the HFAG report has been published, so refence:
\end{itemize}
[53] HFAG Collaboration, M.Chrz\k{a}szcz et. al. Averages of $b$-hadron, $c$-hadron, and $\tau$-lepton properties as of summer 2014, arXiv:1412.7515.
\end{frame}
\fi
\section{Lepton Flavour Violation phenomenon}
\begin{frame}\frametitle{Lepton Flavour/Number Violation}
\begin{small}
Lepton Flavour Violation(LFV):
\end{small}
\begin{footnotesize}
After $\Pmuon$ was discovered (1936) it was natural to think of it as an excited $\Pelectron$.
\begin{columns}
\column{3in}
\begin{itemize}
\item Expected: $B(\mu\to\Pe\gamma) \approx 10^{-4}$
\item Unless there is a nother $\Pnu$.
\end{itemize}
\column{2in}
{~}\includegraphics[width=0.98\textwidth]{rabi.png}
\end{columns}
\begin{columns}
\column{0.5in}
{~}
\column{3in}
\begin{block}{I.I.Rabi:}
"Who ordered that?"
\end{block}
\column{0.3in}{~}
\column{2in}
{~}\includegraphics[scale=0.08]{II_Rabi.jpg}
\end{columns}
\begin{itemize}
\item Up to this day charged LFV is being searched for in various decay modes.
\item LFV was already found in neutrino sector (oscillations).
\end{itemize}
\end{footnotesize}
\begin{footnotesize}
\begin{columns}
\column{3.5in}
\begin{small}
Lepton Number Violation (LNV) %(see J. Harrison \href{https://indico.cern.ch/event/300387/session/17/contribution/74}{\color{blue}talk})
\end{small}
\begin{itemize}
\item Even with LFV, lepton number can be a conserved quantity.
\item Many NP models predict LNV (Majorana neutrinos)
\item LNV searched in s-called neutrinoless double $\beta$ decays.
\end{itemize}
\column{1.5in}
\includegraphics[width=0.73\textwidth]{Double_beta_decay_feynman.png}
\end{columns}
\end{footnotesize}
%Double_beta_decay_feynman.png
% \textref{M.Chrz\k{a}szcz 2014}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Status of searches for $\color{white} \tau \to \mu \mu \mu$}
\begin{columns}
\begin{column}{.62\textwidth}
\includegraphics[width=.95\textwidth]{feymn.png}
{{
\begin{itemize}
\item Charged Lepton Flavour Violation process.
\item The Standard Model contribution: penguin diagram with neutrino oscillation
% \item SM prediction is beyond experimental reach~$O(10^{-40})$.
\end{itemize}
}}
\end{column}
\begin{column}{.45\textwidth}
\begin{alertblock}{Current limits ($ \color{white} 90\,\%$ CL)}
\begin{description}
\item[BaBar] $3.3\times 10^{-8}$
\item[Belle] $2.1\times 10^{-8}$
\end{description}
\end{alertblock}
\begin{alertblock}{Predictions}
\begin{description}
\item[SM] $ O(10^{-40})$
\item[var.\ SUSY] $10^{-10}$
\item[non universal $\PZprime$] $10^{-8}$
\item[mSUGRA+seesaw] $10^{-9}$
\item[and many more...]
\end{description}
\end{alertblock}
\end{column}
\end{columns}
\end{frame}
%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{LHCb detector}
\begin{frame}\frametitle{LHCb detector}
\begin{columns}
\column{3.in}
\begin{center}
\includegraphics[width=0.98\textwidth]{det.jpg}
\end{center}
\column{2.0in}
\begin{footnotesize}
LHCb is a forward spectrometer:
\begin{itemize}
\item Excellent vertex resolution.
\item Efficient trigger.
\item High acceptance for $\Ptau$ and $\PB$.
\item Superb particle identification (PID).
\end{itemize}
\end{footnotesize}
\end{columns}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Selection}
\begin{frame} \frametitle{Strategy}
\begin{enumerate}
\item Data sample: $1 \invfb~7~\TeV$ and $2 \invfb~8\TeV$.
\item Normalization (control) decay channel: $\PDs\to\Pphi(\Pmu\Pmu)\Ppi$.
\item Blind analysis.
\item Event selection:
\begin{itemize}
\item Preselection of three tracks that combine to give a mass close to $m_{\tau}$, with desplaised vertex.
\item Selection based on three classifiers:
\begin{itemize}
\item Geometry and topology ($\mathcal{M}_{3body}$)
\item PID ($\mathcal{M}_{PID}$)
\item Three muon invariant mass ($m_{\mu\mu\mu}$)
\end{itemize}
\end{itemize}
\item Major background contributions:$\PDs \to \eta(\mu\mu\gamma) \mu \nu$ and $\PD \to \PK \Ppi \Ppi$.
\item Evaluation of the upper limit on $\mathcal{B}(\Ptau \to \Pmu \Pmu \Pmu)$ using $CL_s$.
\end{enumerate}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{$\color{white} \tau$ production}
\begin{itemize}
\item $\Ptau$'s in LHCb come from five main sources:
\end{itemize}
\begin{center}
\begin{footnotesize}
\begin{tabular}{| c | c | c | }
\hline
Mode & $7~\TeV$ & $8~\TeV$ \\ \hline
Prompt $\PDs\to\Ptau$ & $71.1\pm3.0\,\%$ & $72.4\pm2.7\,\%$ \\
Prompt $\PDplus\to\Ptau$ & $4.1\pm0.8\,\%$ & $4.2\pm0.7\,\%$ \\
Non-prompt $\PDs\to\Ptau$ & $9.0\pm2.0\,\%$ & $8.5\pm1.7\,\%$ \\
Non-prompt $\PDplus\to\Ptau$ & $0.18\pm0.04\,\%$ & $0.17\pm0.04\,\%$ \\
$X_{\Pbottom}\to\Ptau$ & $15.5\pm2.7\,\%$ & $14.7\pm2.3\,\%$ \\ \hline
\end{tabular}
\end{footnotesize}
\begin{itemize}
\item Pythia produces them in wrong propotions
\item Channels were produced seperatly and added in the given proporitons.
\end{itemize}
\end{center}
\begin{columns}
\column{0.05\textwidth}
{~}
\column{0.9\textwidth}
\begin{exampleblock}{$\mathcal{B}(\PDplus\to\Ptau)$}
\begin{itemize}
\item There is no measurement of $\mathcal{B}(\PDplus\to\Ptau)$.
\item One can calculate it from: $\mathcal{B}(\PDplus\to\Pmu\Pnum)$ + helicity suppression + phase space, \texttt{hep-ex:0604043}.
\item $\mathcal{B}(\PDplus\to\Ptau\Pnut)=(1.0\pm0.1) \times10^{-3}$.
\end{itemize}
\end{exampleblock}
{~}
\column{0.2\textwidth}
{~}
\end{columns}
\end{frame}
\begin{frame}
\frametitle{Triggers at LHCb}
\begin{footnotesize}
\begin{itemize}
\item LHCb uses complex trigger, $\mathcal{O}(100)$ trigger lines. %\footnote{\href{http://arxiv.org/abs/1211.3055}{\color{blue}arXiv:1211.3055}}, $\mathcal{O}(100)$ trigger lines.
\item Lines change with data taking.
\item Optimized choice of triggers based on $\dfrac{s}{\sqrt{b}}$ FOM,
\end{itemize}
\end{footnotesize}
\begin{tiny}
\begin{columns}
\column{0.66\textwidth}
\begin{equation}
\label{eq:efftrig}
\varepsilon(\beta)^\prime_{\text{evt},\text{line}} = \frac{N(\tau~\text{ MC(BKG) events triggered
line, but not by any better line})}{N(\tau\text{ MC(BKG) events triggered by any line})}, \nonumber
\end{equation}
\begin{small}
\begin{itemize}
\item Evaluated different triggers used in 2012 data taking.
\item Found negligible differences in trigger efficiencies.
\end{itemize}
\end{small}
\column{0.32\textwidth}
\begin{equation}
\text{CTFM}=\frac{\sqrt{\sum\limits_\text{trigger lines} \beta^\prime_{\text{evt},\text{line}}}}{\sum\limits_\text{trigger lines} \varepsilon^\prime_{\text{evt},\text{line}}}\nonumber
\end{equation}
\end{columns}
\end{tiny}
\begin{footnotesize}
\end{footnotesize}
\begin{tiny}
\begin{tabular}{|l|c|c|c|}
\hline
name & $\varepsilon^\prime$ & $\beta^\prime$ & CTFM \\\hline
% here
Hlt2TriMuonTauDecision & 0.880708 & 0.736182 & 0.974228\\
Hlt2DiMuonDetachedDecision & 0.0669841 & 0.173396 & 1.00636\\
Hlt2CharmSemilep3bodyD2KMuMuDecision & 0.0206816 & 0.0182935 & 0.99472\\
Hlt2CharmHadD2HHHDecision & 0.00554351 & 0.00666405 & 0.992604\\
Hlt2CharmSemilep3bodyD2KMuMuSSDecision & 0.00195444 & 0.00470404 & 0.993106\\
Hlt2CharmSemilep3bodyD2PiMuMuDecision & 0.00206105 & 0.00679472 & 0.994591\\
Hlt2TopoMu3BodyBBDTDecision & 0.00394442 & 0.0121521 & 0.996937\\
\hline
\end{tabular}
\end{tiny}
\end{frame}
\section{Multivariate technique}
\begin{frame}
\frametitle{Geometric likelihood}
\begin{itemize}
\item As mentioned in LHC we have different production sources of $\Ptau$'s.
\item Each source has different detector response signature.
\item To maximise our performance we trained classifiers for each of the $\Ptau$ sources using:
\begin{itemize}
\item Kinematic properties of $\Ptau$ candidate.
\item Geometric properties of $\Ptau$ candidate, like pointing angle, DOCA, Vertex $\chi^2$, flight distance.
\item Isolations, for vertex and individual tracks.
\end{itemize}
\item After training the individual classifiers one that combines all this information in a single classifier on mixed sample of $\Ptau$'s.
\item This technique is known as Blending or Ensemble learning.
\item Using this approach we gain $6\%$ sensitivity!
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Performance of Blend classifier}
\begin{itemize}
\item Classifier prefers $\Ptau$'s from prompt $\PDs$, the dominant channel.
\end{itemize}
\begin{columns}
\begin{column}{.49\textwidth}
\begin{exampleblock}{MC response for different\newline $\color{white} \tau$ production channels}
\includegraphics[width=.98\textwidth]{./mixing.pdf}
\end{exampleblock}
\end{column}
\begin{column}{.49\textwidth}
\begin{exampleblock}{Response for $\color{white} D_s \rightarrow \phi\pi$\newline data and MC}
\includegraphics[width=.98\textwidth]{./dataMC.pdf}
\end{exampleblock}
\end{column}
\end{columns}
\end{frame}
\begin{frame}
\frametitle{Calibration}
\begin{itemize}
\item Assume all differences between $\Ptau\to\Pmu\Pmu\Pmu$ and $\PDs\to\Pphi\Ppi$ come from kinematics (mass, resonance, decay time), which is correct in MC.
\item Get correction $\PDs\leadsto\Ptau$ from MC.
\item Apply corrections to $\PDs\to\Pphi\Ppi$ on data.
\end{itemize}
\begin{columns}
\begin{column}{.45\textwidth}
\includegraphics[width=.95\textwidth]{m3body_2012.pdf}
\end{column}
\begin{column}{.45\textwidth}
\begin{itemize}
\item $\PDs\to\Pphi\Ppi$ well modelled in MC.
% \item[$\rightarrow$] i.e.\ also badly pointing non-prompt $\PDs$
\end{itemize}
\end{column}
\end{columns}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
% PID
\begin{frame}
\frametitle{Particle Identification (PID)}
\begin{itemize}
\item Classifier trained on inclusive MC sample.
\item Using information from: RICH, Calorimeters, Muon system and tracking.
\item Correct for the MC efficiency using control channel: $\PDs \to \Pphi(\Pmu\Pmu) \Ppi$ and $\PB \to \PJpsi(\Pmu\Pmu) \PK$
\end{itemize}
\begin{columns}
\begin{column}{.45\textwidth}
\includegraphics[width=.95\textwidth]{mPID_2012.pdf}
\end{column}
\end{columns}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
\begin{frame}
\frametitle{Binning optimisation}
\begin{itemize}
\item Events are distributed among $\mathcal{M}_{3body}, \mathcal{M}_{PID}$ plane.
\item In 2D we group the events in groups(bins)
\item Bins are optimised using $CL_s$ method.
\item The lowest bins are rejected, because they do not contribute to the limit sensitivity.
\item In rest of the bins a fit to mass side-bands is performed in order to estimate number of expected background in signal window.
\end{itemize}
\begin{columns}
\column{2.2in}
\center{2011}\\
\includegraphics[width=.99\textwidth]{2D_2011.pdf}
\column{2.2in}
\center{2012}\\
\includegraphics[width=.99\textwidth]{2D_2012.pdf}
\column{0.5in}
{~}
\end{columns}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Mass shape}
\begin{itemize}
\item Double-Gaussian with fixed fraction ($70\,\%$ inner Gaussian).
\item Fix fraction to ease calibration.
\item Correct mass by MC:\newline
$\sigma_{data}^{\Ptau} = \frac{\sigma_{MC}^{\Ptau}}{\sigma_{MC}^{\PDs}}\times\sigma_{data}^{\PDs}$
\end{itemize}
\includegraphics[width=.44\textwidth]{./Ds_data_2011.pdf}
\includegraphics[width=.44\textwidth]{./Ds_data_2012.pdf}
{\footnotesize{
\begin{tabular}{|c|c|c|}
\hline
Calibrated $\Ptau$ Mass shape & 7~TeV & 8~TeV\\
\hline
Mean ($\MeV$) & $1779.1 \pm 0.1$ & $1779.0 \pm 0.1$\\
\hline
$\sigma_1$ ($\MeV$) & $7.7 \pm 0.1$ & $7.6 \pm 0.1$\\
\hline
$\sigma_2$ ($\MeV$) & $12.0 \pm 0.8$ & $11.5 \pm 0.5$\\
\hline
\end{tabular}
}
}
\end{frame}
\section{Normalisation}
\begin{frame}
\frametitle{Relative normalisation}
$\mathcal{B}(\Ptau\to\Pmu\Pmu\Pmu) = \frac{\mathcal{B}(\PDs\to\Pphi\Ppi)}{\mathcal{B}(\PDs\to\Ptau\Pnut)} \times f_{\PDs}^{\Ptau} \times \frac{\varepsilon_\text{norm} }{\varepsilon_\text{sig} } \times \frac{N_\text{sig}}{N_\text{norm}} = \alpha\times N_\text{sig}$
\begin{itemize}
\item where $\varepsilon$ stands for trigger, reconstruction, selection efficiency.
\item $f_{\PDs}^{\Ptau}$ is the fraction of $\Ptau$ coming from $\PDs$.
\item $\text{norm}$ = normalisation channel $\PDs\to\Pphi\Ppi$
\newline i.e.\ $(83\pm3)\,\%$ for 2012.
\end{itemize}
\includegraphics[width=.47\textwidth]{./Ds_data_2011.pdf}
\includegraphics[width=.47\textwidth]{./Ds_data_2012.pdf}
\end{frame}
\section{Backgrounds}
\begin{frame}
\frametitle{Misidentification}
\begin{columns}
\column{3in}
\begin{itemize}
\item Most dominant: $\PDplus\to\PK\Ppi\Ppi$.
\item Also seen $\PDplus\to\Ppi\Ppi\Ppi$ and $\PDs\to\Ppi\Ppi\Ppi$.
\item All contained in the lowest $\mathcal{M}_{PID}$ bin.
% \item Experience from last round: cut away \\low ProbNNmu range
% \item Check remaining data under \\$\PK\Ppi\Ppi$ hypothesis for $\PDplus$ peak
% \item[$\Rightarrow$] misid safely contained in ``trash'' bin
\end{itemize}
\column{2in}
\includegraphics[width=.95\textwidth]{./WMH.pdf}
\end{columns}
\includegraphics[width=.45\textwidth]{./trash.pdf}{~}{~}{~}{~}{~}{~}{~}{~}{~}{~}{~}{~}{~}{~}
\includegraphics[width=.45\textwidth]{./mPID_2012.pdf}
\end{frame}
\begin{frame}
\frametitle{Dangerous backgrounds}
\begin{columns}
\column{3in}
\begin{itemize}
\item $\Pphi\to\Pmu\Pmu + X$: narrow veto on dimuon mass.
\item $\PDs\to\Peta(\Pmu\Pmu\Pphoton)\Pmu\Pnum$: not so easy:
\begin{itemize}
\item Model it
\item \underline{Remove it} with dimuon mass cut:
\begin{itemize}
\item Fits better understood.
\item Sensitivity unchanged when removing veto.
\item Smaller uncertainty on expected background.
\end{itemize}
\end{itemize}
\end{itemize}
\column{2in}
\includegraphics[width=.95\textwidth]{./etaMass.pdf}\\
\includegraphics[width=.95\textwidth]{./etaDalitz.pdf}
\end{columns}
\end{frame}
\begin{frame}
\frametitle{Remaining backgrounds}
\begin{itemize}
\item Fit exponential to invariant mass spectrum in each likelihood bin.
\item Don't use blinded region ( $\pm \unit{30}{\MeV}$ ).
\item[$\rightarrow$] Compatible results blinding only $\pm \unit{20}{\MeV}$\footnote{partially used in classifier development}
\end{itemize}
{\begin{center}
Example of most sensitive regions in 2011 and 2012
\includegraphics[width=0.9\textwidth]{./fits.png}
\end{center}}
\end{frame}
\section{Model dependence}
\begin{frame}
\frametitle{Model dependence}
\begin{itemize}
\item $\Peta$ veto $\Rightarrow$ our limit not constraining to New Physics with small $m_{\APmuon\Pmuon}$.
\item Model description in \href{http://arxiv.org/abs/0707.0988}{\color{blue}\texttt{arXiv:0707.0988}} by S.Turczyk.
\item 5 relevant Dalitz distributions: 2 four-point operators, 1 radiative operator, 2 interference terms.
\end{itemize}
\only<2->{
\begin{itemize}
\item With radiative distribution limit gets worse by a factor of $1.5$ (dominantly from the $\Peta$ veto).
\item The other four Dalitz distributions behave nicely (within $7\,\%$).
\end{itemize}
\begin{center}
\includegraphics[width=.5\textwidth]{./sigDalitz.pdf}
\end{center}
}
\only<1>{
\begin{columns}
\column{0.33\textwidth}
\includegraphics[width=.95\textwidth]{./gammallll2.pdf}\\
\includegraphics[width=.95\textwidth]{./gammarad-llll2.pdf}
\column{0.33\textwidth}
\includegraphics[width=.95\textwidth]{./gammallrr2.pdf}\\
\includegraphics[width=.95\textwidth]{./gammarad-llrr2.pdf}
\column{0.33\textwidth}
\includegraphics[width=.95\textwidth]{./gammarad2.pdf}\\
% \begin{itemize}
% \item Same models as in Z.Was \href{https://indico.cern.ch/event/300387/session/7/contribution/33}{\color{blue}talk}
% \end{itemize}
{~}\\ {~}\\ {~}\\ {~}\\ {~}\\
\end{columns}
}
\end{frame}
% \begin{frame}
% \frametitle{Conclusion}
% \begin{columns}
% \begin{column}{.55\textwidth}
% \begin{itemize}
% \item finally all pieces put together
% \item model (in)dependence of $\Peta$ veto investigated
% \item expected sensitivity computed\newline $5.6\times 10^{-8}$
% \end{itemize}
% \end{column}
% \begin{column}{.45\textwidth}
% \includegraphics[width=\textwidth]{party-music-hd-wallpaper-1920x1200-3850.jpg}
% \end{column}
% \end{columns}
% \end{frame}
\section{Results}
\begin{frame}
\frametitle{Results}
\begin{center}
\includegraphics[width=0.7\textwidth]{banana_line.pdf}
\end{center}
\begin{columns}
\column{0.2in}{~}
\column{2in}
Limits(PHSP):\\
Observed(Expected)\\
$\color{red}4.6~(5.0)\times 10^{-8}$ at $90\%$ CL\\
$\color{pink}5.6~(6.1)\times 10^{-8}$ at $95\%$ CL\\
\column{3in}
\includegraphics[width=0.5\textwidth]{model.png}
\end{columns}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
\section{LFV limit combination}
\begin{frame}\frametitle{Combination of LFV UL 1/2}
\begin{columns}
\column{1.8in}
\begin{itemize}
\item Searches for LFV in $\Ptau$ sector is a domain of B factories.
\item Over last years both BaBar and Belle set very strong limits on branching fractions of several rare $\tau$ decays.
\end{itemize}
\column{3.2in}
\includegraphics[width=0.95\textwidth]{TauLFV_UL_2014001.png}
\end{columns}
\begin{itemize}
\item Since thouse limits are used to constraint NP models, their "official" combination is of paramount importance.
\item Various methods of limit computation used in Belle and BaBar's studies.
\item The HFAQ group recomputed consistently all estimates using the $CL_s$ method and the the same approach was involved in the average evaluation.
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
\begin{frame}\frametitle{Combination of LFV UL 2/2}
\begin{itemize}
\item For each measurement take integrated luminosity ($\mathcal{L}$), cross section ($\sigma_{\tau\tau}$), efficiencies ($\epsilon$), background expected($b$) and all systematics.
\item Calculate number of signal: $s = \mathcal{L} \sigma_{\tau\tau} \epsilon^{tot} \mathcal{B}(\Ptau \to LFV)$.
\item Scan the $CL_s$ wrt. $\mathcal{B}(\Ptau \to LFV)$:
\end{itemize}
\begin{columns}
\column{0.15in}
{~}
\column{2in}
\begin{small}
\begin{align}
CL_s = \dfrac{\prod_{i=1}^{N_{\text{chan}}}\sum_{n=0}^{n_i} \dfrac{e^{-(s_i+b_i)} (s_i+b_i)^{n}}{n!} }{\prod_{i=1}^{n_{\text{chan}}} \sum_{n=0}^{n_i} \dfrac{e^{-b_i} b_i^{n}}{n!}} \nonumber~,
\end{align}
\end{small}
\column{2.7in}
\includegraphics[width=0.95\textwidth]{TauLFV_UL_2014001_averaged.png}
\end{columns}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{"The Rule of Three"}
\begin{columns}
% \column{2.5in}
\begin{column}{1.9in}
\begin{alertblock}{ $\Ptau \to \Pmu \Pmu \Pmu$ limits ($ \color{white} 90\,\%$ CL)}
\begin{description}
\item[BaBar(FC)] $3.3\times 10^{-8}$
\item[Belle(FC)] $2.1\times 10^{-8}$
\item[LHCb(CLs)] $4.6\times 10^{-8}$
\item[HFAG(CLs)] $1.2 \times 10^{-8}$
\end{description}
\end{alertblock}
\end{column}
\begin{column}{2.85in}
\includegraphics[width=1\textwidth]{TauLFV_UL_2014001_averaged_zoom.png}\\
\end{column}
\end{columns}
{~}\\
To conclude:
\begin{columns}
\column{3in}
\begin{small}
\begin{itemize}
% \item LHCb result for $\Ptau \to \Pmu \Pmu \Pmu$ with full data set.
% \item We are getting close to B-factories.
% \item Thanks to 3 experiments we have a world limit: $\mathcal{B}(\Ptau \to \Pmu \Pmu \Pmu)< 1.2 \times 10^{-8}$ at 90\% CL.
\item LHCb is reaching B-factories limits.
\item Many new techniques developed to perform this analysis.
\item Combination of UL within HFAG gave the best sensitivity for $\mathcal{B}(\Ptau \to \Pmu \Pmu \Pmu)< 1.2 \times 10^{-8}$ at 90\% CL.
\end{itemize}
\end{small}
\column{2in}
\includegraphics[width=0.93\textwidth]{banana_tau23mu_hfag.pdf}\\
\end{columns}
\end{frame}
\pagenumbering{gobble}
\begin{frame}\frametitle{Backup}
\includegraphics[width=0.93\textwidth]{ngbbs4f7918ec8ffb8.jpg}\\
\end{frame}
\begin{frame}\frametitle{Prof. J.Ciborowski comments}
\begin{enumerate}
\item Podrozdział (4.10) ten jest de facto zapowiedzią większej pracy. Szkoda, że materiał tu przedstawiony
potraktowany został bardzo skrótowo, co wymusiło na mnie konieczność kilkakrotnego przeczytania tego podrozdziału i utrudniło docenienie wyniku
otrzymanego przez autora w konfrontacji z przewidywaniami teoretycznymi.
\end{enumerate}
\begin{itemize}
\item The theory part of this was presented in detail in 2.3.4. This chapter is just a showing how to reweight the distributions to a given NP model, thats why I tried to keep it short, but I agree I over did it.
\end{itemize}
\end{frame}
\begin{frame}\frametitle{Prof. J.Ciborowski comments}
\begin{enumerate}
\item Jedyne
rzucające się w oczy uchybienie redakcyjne to pomyłki w numerach
rozdziałów, których zawartość wymieniona jest pod koniec Wstępu.
\end{enumerate}
\begin{itemize}
\item Mea Cupla. Completly missed that.
\end{itemize}
\end{frame}
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\end{document}
Marcin Chrząszcz