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+%\usepackage[latin1]{inputenc} +%\usepackage{amsmath} +%\newcommand\abs[1]{\left|#1\right|} +\usepackage{amsmath} +\newcommand\abs[1]{\left|#1\right|} +\usepackage{hepnicenames} +\usepackage{hepunits} +\usepackage{color} +\usepackage{feynmp} +\usepackage{pst-pdf} +\usepackage{hyperref} +\usepackage{xcolor} + +\setbeamertemplate{footline}{\insertframenumber/\inserttotalframenumber} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5 +\definecolor{mygreen}{cmyk}{0.82,0.11,1,0.25} + + + +%\DeclareCaptionFont{uiblack}{\color{uiblack}} +%\DeclareCaptionFont{uipoppy}{\color{uipoppy}} +%\captionsetup{labelfont={uipoppy},textfont=uiblack} + +% see the macros.tex file for definitions + + + +\renewcommand{\PKs}{{\HepParticle{K}{S}{}\xspace}} +\newcommand{\at}{\makeatletter @\makeatother} + +%-------------------------------------------------------------------- +% Introduction +%-------------------------------------------------------------------- + +\usetheme{Sybila} + +\title[ Rare beauty and charm decays at LHCb]{ Rare beauty and charm decays at LHCb} +\author{Marcin Chrz\k{a}szcz$^{1}$ \\ \footnotesize{on behalf of the LHCb collaboration}} +\institute{$^1$~University of Zurich \\{~}\\ Heavy Quarks and Leptons 2014 } +\date{\today} +\begin{document} +% --------------------------- SLIDE -------------------------------------------- +\frame[plain]{\titlepage} +\author{Marcin Chrz\k{a}szcz} +% ------------------------------------------------------------------------------ +% --------------------------- SLIDE -------------------------------------------- + +\institute{~(UZH)} + +%-------------------------------------------------------------------- +% OUTLINE +%-------------------------------------------------------------------- + + +\section[Outline]{} +\begin{frame} +%\tableofcontents +%FIXME! +\begin{enumerate} +\item Rare $\PB$ decays: +\begin{itemize} +\item $\PB \to \PK \Ppi \Ppi \Pphoton$ +\item $\PB \to \mu \mu$. +\item $\Pbeauty \to \Pstrange \Plepton \Plepton$. +\end{itemize} +\item Charm decays: +\begin{itemize} +\item $\PD \to \mu \mu$. +\end{itemize} +\end{enumerate} + +\end{frame} + +%------------------------------------------------------------------- +% Introduction +%------------------------------------------------------------------- +% +% Set the background for the rest of the slides. +% Insert infoline +%\setbeamertemplate{background} +% {\includegraphics[width=\paperwidth,height=\paperheight]{slide_bg}} +%\setbeamertemplate{footline}[bunsentheme] + + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +%\setbeamertemplate{background} +% {\includegraphics[width=\paperwidth,height=\paperheight]{slide_bg}} +%\setbeamertemplate{footline}[bunsentheme] +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%\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 Great Particle ID + % \end{itemize} + + + +%\end{footnotesize} +%\end{columns} + +%\end{frame} + +\section{Introduction} + +\begin{frame}\frametitle{Why rare decays?} +\begin{columns} +\column{4in} +\begin{itemize} +\item CKM structure in SM allows only the charged interactions to change flavour. +\begin{itemize} +\item Other interactions are flavour conserving. +\end{itemize} +\item One can escape the CKM structure and produce $\Pbottom \to \Pstrange$ and $\Pbottom \to \Pdown$ only at loop level. +\begin{itemize} +\item This kind of processes are suppressed in SM $\to$~Rare decays. +\end{itemize} +\end{itemize} +\begin{center} +\includegraphics[scale=0.3]{hql/lupa.png} +\includegraphics[scale=0.3]{hql/example.png} +\end{center} +\column{1.5in} +\includegraphics[width=0.62\textwidth]{hql/couplings.png} +\end{columns} + +\end{frame} + +\begin{frame}\frametitle{Tools} +\begin{itemize} +\item \textbf{Operator Product Expansion and Effective Field Theory} +\end{itemize} +\begin{columns} +\column{0.1in}{~} +\column{3.2in} +\begin{align*} +H_{eff} = - \dfrac{4G_f}{\sqrt{2}} V V^{\ast} \sum_i \left[\underbrace{C_i(\mu)O_i(\mu)}_\text{left-handed} +\underbrace{C'_i(\mu)O'_i(\mu)}_\text{right-handed}\right] +\end{align*} + +\column{2in} +\begin{tiny} +\begin{description} + \item[i=1,2] Tree + \item[i=3-6,8] Gluon penguin + \item[i=7] Photon penguin + \item[i=9.10] EW penguin + \item[i=S] Scalar penguin + \item[i=P] Pseudoscalar penguin + \end{description} + +\end{tiny} +\end{columns} +\begin{center} +\includegraphics[width=0.85\textwidth,height=3cm]{hql/all.png} + +\end{center} + + +\end{frame} + + + +\begin{frame}\frametitle{Radiative decays} + +\begin{columns} +\column{5in} + +\begin{itemize} +\item $\PBzero \to \PKstar \Pphoton$ - first observed penguin! +\begin{itemize} +\item CLEO, [\href{http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.71.674}{\color{blue}PRL, 71 (1993) 674}] +\end{itemize} +\item B-factories probed NP measuring, inclusively/ semi-inclusively $\mathcal{B}(\Pbeauty \to \Pstrange \Pphoton)$ +\end{itemize} +\column{0.1in}{~} + +\end{columns} + + + +\begin{columns} +\column{3in} +\begin{itemize} +\item Is there anyway LHCb can contribute? +\begin{itemize} +\item Measurements of $\mathcal{B}(\Pbeauty \to \Pstrange \Pphoton)$ very difficult. +\item Can probe probe polarization! +\end{itemize} +\end{itemize} +\column{2in} +\includegraphics[width=0.85\textwidth]{hql/btosgamma.png} +\end{columns} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5 + + +\begin{columns} +\column{5in} + +\begin{itemize} +\item In SM, photons form $\Pbeauty \to \Pstrange \Pphoton$ decays are left handed. +\begin{itemize} +\item Charged current interactions: $C_7/C'_7\sim m_{\Pbeauty}/m_{\Pstrange}$ +\end{itemize} +\item Can test $C_7/C'_7$ using: +\begin{itemize} +\item Mixing induced CP violation: \href{http://arxiv.org/abs/hep-ph/9704272}{\color{blue}Atwood et. al. PRL 79 (1997) 185-188} +\item $\PLambdab$ baryons: \href{http://arxiv.org/abs/hep-ph/0108074}{\color{blue}Hiller \& kagan PRD 65 (2002) 074038} +\end{itemize} +\end{itemize} +\column{0.1in}{~} + +\end{columns} + +\end{frame} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5 + +\begin{frame}\frametitle{Photon polarization from $\PB^+ \to \PK^{+} \Ppi^- \Ppi^+ \Pphoton$} + + +\begin{columns} +\column{3.5in} + +\begin{itemize} +\item OR: Study $\PB \to \PK^{\ast \ast} \Pphoton$ decays like $\PBplus \to \PK_1(1270) \Pphoton$ +\begin{itemize} +\item \href{http://arxiv.org/abs/hep-ph/0205065}{\color{blue}Gronau \& Pirjol PRD 66 (2002) 054008} +\end{itemize} +\item The trick is to get the photon polarization from the up-down asymmetry of photon direction in the $\PK \Ppi \Ppi$ rest frame. +\begin{itemize} +\item No asymmetry $\rightarrow$ Unpolarised photons. +\end{itemize} +\item Conceptionally this measurement is similar to the Wu experiment, which first observed parity violation. + + +\end{itemize} +\column{1.5in} + +\includegraphics[width=0.95\textwidth]{hql/polarization.png} + +\end{columns} +\end{frame} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{frame}\frametitle{$\PB^+ \to \PK^{+} \Ppi^- \Ppi^+ \Pphoton$ at LHCb} + + +\begin{columns} +\column{3.in} + +\begin{itemize} +\item LHCb looked at $\PBplus \to \PKplus \Ppiminus \Ppiplus \Pphoton$, using un-converted photons. +\item Got over 13.000 candidates in $3~fb^{-1}$! +\item \href{http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.112.161801}{\color{blue} Phys. Rev. Lett. 112, 161801 } +\item $\PKplus \Ppiminus \Ppiplus$ system has variety of resonances. +\begin{itemize} +\item $\PK \Ppi \Ppi \Ppi$system studied inclusively. +\item Bin the mass and look for polarization there. +\end{itemize} +\end{itemize} +\column{2in} +{~} +\includegraphics[width=0.95\textwidth]{hql/plotspolarization.png} + +\end{columns} +\end{frame} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{frame}%\frametitle{$\color{white} B^+ \to K^{+} \pi^- \pi^+ \gamma$ at LHCb} +\begin{center} +{\color{red} Fit with $\color{red}(C_7'-C_7)/(C_7'+C_7)=0$}, {\color{blue} Best fit} +\includegraphics[width=0.93\textwidth]{hql/photonfit.png} +\end{center} + + +\end{frame} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{frame}\frametitle{Up-down asymmetry} + +\begin{columns} +\column{3.in} + +\begin{itemize} +\item Combining the 4 bins, gives $5.2\sigma$ significance from no photon polarization hypothesis. +\item Unfortunately without understanding the hadron system it is impossible to tell if the photon is left or right -handed. +\end{itemize} + +\column{2in} +{~} +\includegraphics[width=0.95\textwidth]{hql/aud.png} + +\end{columns} +\begin{center} +$\rightarrow$~ First observation of photon polarization in $\Pbeauty \to \Pstrange \Pphoton$! + +\end{center} + + + +\end{frame} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{frame}\frametitle{$\PBzero \rightarrow \Pmu^+ \Pmu^-$} +\begin{columns} +\column{3.2in} + +\begin{itemize} +\item Clean theoretical prediction, GIM and helicity suppressed in the SM: +\begin{itemize} +\item $\mathcal{B}(\PBs \to \Pmuon \APmuon) = (3.65 \pm 0.23)\times 10^{-9}$ +\item $\mathcal{B}(\PBzero \to \Pmuon \APmuon) = (1.06 \pm 0.09)\times 10^{-10}$ +\end{itemize} +\item Sensitive to contributions from scalar and pesudoscalar couplings. +\item Probing: MSSM, higgs sector, etc. +\item In MSSM: $\mathcal{B}(\PBs \to \Pmuon \APmuon) \sim \tan^6 \beta /m_A^4$ +\end{itemize} + +\column{1.5in} +{~} +\includegraphics[width=0.95\textwidth]{hql/bs2mumu1.png}\\ +\includegraphics[width=0.95\textwidth]{hql/bs2mumu2.png}\\ +\includegraphics[width=0.6\textwidth]{hql/higgspen.png} +\end{columns} + + +\end{frame} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{frame}\frametitle{$\PBzero \rightarrow \Pmu^+ \Pmu^-$ searches} + + +\begin{columns} +\column{5in} + +\begin{itemize} +\item Background rejection power is a key feature of rare decays $\rightarrow$ use multivariate classifiers (BDT) and strong PID. +\end{itemize} +\column{0.1in}{~} + +\end{columns} +\begin{columns} +\column{2.5in} +\includegraphics[width=0.95\textwidth]{hql/BDT.png} + +\column{2.5in} + +\includegraphics[width=0.95\textwidth]{hql/mass.png} + +\end{columns} + +\begin{itemize} +\item Normalize the BF to $\PBplus \to \PJpsi(\mu\mu) \PKplus$ and $\PBzero \to \PK \Ppi$. +\end{itemize} +\end{frame} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{frame}\frametitle{$\PBzero \rightarrow \Pmu^+ \Pmu^-$ Results} + + + +\begin{columns} +\column{2.in} +\begin{itemize} +\item Nov. 2012: +\begin{itemize} +\item First evidence $3.5\sigma$ for $B^0 \rightarrow \mu^+ \mu^-$. with $2.1~fb^{-1}$. +\end{itemize} +\item Summer 2013: +\begin{itemize} +\item Full data sample: $3~fb^{-1}$. +\end{itemize} +\end{itemize} +\column{3.0in} + +\includegraphics[width=0.95\textwidth]{hql/mass2.png} + +\end{columns} +\begin{itemize} +\item Measured BF:\\ $\mathcal{B}(\PBs \to \Pmuon \APmuon) =(2.9^{+1.1}_{-1.0}(stat.)^{+0.3}_{-0.1}(syst.))\times 10^{-9}$ +\item $4.0 \sigma$ significance! +\item $\mathcal{B}(\PBzero \to \Pmuon \APmuon) < 7 \times 10^{-10}$ at $95\%$ CL +\item \href{http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.110.021801}{\color{blue} PRL 110 (2013) 021801 } +\item \href{http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.111.101805}{\color{blue} CMS result: PRL 111 (2013) 101805} +\end{itemize} + + +\end{frame} + + + + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{frame}\frametitle{LHCb+CMS Combination} +\begin{Large} + + +\begin{center} +$\mathcal{B}(\PBs \to \Pmuon \APmuon) =(2.9 \pm 0.7 )\times 10^{-9}$\\ +$\mathcal{B}(\PBzero \to \Pmuon \APmuon) =(3.6^{+1.6}_{-1.4} )\times 10^{-10}$ +\end{center} +\end{Large} + +\includegraphics[width=0.95\textwidth]{hql/limits.png} + +Full combination CMS+LHCb with simultaneous fit close to completion! +\begin{itemize} +\item \href{https://cds.cern.ch/record/1564324/files/LHCb-CONF-2013-012.pdf}{\color{blue}LHCb-CONF-2013-012} +\end{itemize} + +\end{frame} + + + + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{frame}\frametitle{$\PBzero \rightarrow \PK^{\ast} \Pmu \Pmu$ angular distributions} +\begin{columns} +\column{2.5in}{~} +\begin{itemize} +\item Can probe photon polarization using virtual photons in $\Pbeauty \to \Pstrange \Plepton \Plepton$. +\item LHCb favourite: $\PBzero \to \PKstar \mu \mu$. +\item Sensitive to lot of new physics models. +\item Decay described by three angles $\theta_l, \theta_K, \phi$ and dimuon invariant mass $q^2$. +\item Analysis is performed in bins of $q^2$. +\end{itemize} +\column{2.5in} +\includegraphics[width=0.95\textwidth]{hql/angles.png} + +\end{columns} +\end{frame} + + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{frame}\frametitle{$\PBzero \rightarrow \PK^{\ast} \Pmu \Pmu$ angular distributions} +\begin{itemize} +\item Angular distributions depends on 11 angular terms: +\includegraphics[width=0.95\textwidth]{hql/eq.png} +\end{itemize} +where the $J_i$ are bilinear combinations of helicity amplitudes. +\begin{itemize} +\item Not enough events in our data sample to fit for 11 parameters $\rightarrow$ need to simplify! +\item Can use symmetries, to reduced the the parameters \\ to 9 $\rightarrow$ still a bit large! +\end{itemize} + +\end{frame} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{frame}\frametitle{$\PBzero \rightarrow \PK^{\ast} \Pmu \Pmu$ Folding} +\begin{itemize} +\item One can simplify the angular distribution by folding: eg. $\phi \to \phi +\pi$ for ($\phi<0$). +\item Cancels terms with $\cos \phi$ and $\sin \phi$. + +\end{itemize} +\includegraphics[width=0.95\textwidth]{hql/eq2.png} +\end{frame} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{frame}\frametitle{$\PBzero \rightarrow \PK^{\ast} \Pmu \Pmu$ angular distributions } +\begin{itemize} +\item Different foldings cancel different angular observables. \href{http://arxiv.org/abs/1308.1707}{\color{blue}[PRL 111 191801 (2013)]} +\includegraphics[width=0.95\textwidth]{hql/ps.png} +\item Observables $P'_{4,5} = S_{4,5}/\sqrt{F_L(1-F_L)}$ +\item Leading form-factor uncertainties cancel. +\item In $1~fb^{-1}$, LHCb observes a local discreapncy of $3.7\sigma$ in $P'_5$. +\item Probability that at least one bin varies by this much is $0.5\%$. +\item SM prediction form: \href{http://arxiv.org/abs/1303.5794}{\color{blue} JHEP 05 (2013) 137} + +\end{itemize} + +\end{frame} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{frame}\frametitle{Understanding the $\PBzero \rightarrow \PK^{\ast} \Pmu \Pmu$ anomaly 1/2} +\begin{columns} +\column{3.5in} +\begin{itemize} +\item Matias, Decotes-Genon \& Virto performed a global fit to the avaible $\Pbeauty \to \Pstrange \Pphoton$ abd $\Pbeauty \to \Pstrange \Plepton \Plepton$. +\item Found $4.5 \sigma$ discrepancy from SM. +\item Fit favours $C_9^{NP}=1.5$ +\item \href{http://arxiv.org/abs/1307.5683}{\color{blue} PRD 88 074002 (2013)} +\end{itemize} +\begin{itemize} +\item Straub \& Altmannshofer performed a global analysis and found discrepancies at the level of $3\sigma$. Data again best describes a modified $C_9$. +\item Data can be explained by introducing a flavour changing $\PZprime$ boson, with mass $\mathcal{O}(10~TeV)$ +\item \href{http://arxiv.org/abs/1212.2263}{\blue{EPJC 73 2646 (2013)}} +\end{itemize} + +\column{1.5in} +\includegraphics[width=0.9\textwidth]{hql/np.png} +\end{columns} +\end{frame} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%\begin{frame}\frametitle{Understanding the $\color{white}B^{0} \rightarrow K^{\ast} \mu \mu$ anomaly 2/2} +%\begin{columns} +%\column{3in} + +%\includegraphics[width=0.99\textwidth]{hql/c9.png} +%\column{2in} +%\begin{itemize} +%\item High $q^2$ differential BF suggests are all below SM. +%\item Better consistency with $C_9^{NP}=-1.5$ +%\end{itemize} + +%\end{columns} + + +%\end{frame} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{frame}\frametitle{Lepton universality} +\begin{columns} +\column{3.0in} +\begin{itemize} +\item If $\PZprime$ is responsible for the $P'_5$ anomaly, does it couple equally to all flavours? +\includegraphics[width=0.9\textwidth]{hql/uni2.png} +\item Challenging analysis. +\item Migration of events modeled by MC. +\item Correct bremsstrahlung. +\item Take double ratio with $\PBplus \to \PJpsi \PKplus$ to cancel systematics. +\item In $3fb^{-1}$, LHCb measures $R_K=0.745^{+0.090}_{-0.074}(stat.)^{+0.036}_{-0.036}(syst.)$ +\item Consistent with SM at $2.6\sigma$. + +\end{itemize} +\column{2.0in} +\includegraphics[width=0.99\textwidth]{hql/universality.png} +\end{columns} +\end{frame} + + + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{frame}\frametitle{FCNC in charm decays} +\begin{columns} +\column{3.0in} +\begin{itemize} +\item GIM cancelation effective in $\Pcharm \to \Pup$ transitions due to small size of $m_b$. +\item SM prediction:~$\mathcal{B}(\PDzero \to \mu \mu) \sim 6 \times 10^{-11}$ + + +\end{itemize} +\column{1.5in} +\includegraphics[width=0.9\textwidth]{hql/ds2mumu.png} +\end{columns} +\includegraphics[width=0.9\textwidth]{hql/ds2mumu2.png} +\begin{small} + +\begin{itemize} +\item Use $\PDstar^{\pm}$ and exploit small $\Delta m$ for background suppression. +\item Limitation is $\pi \to \mu$ mis-id. +\item Limit: $\mathcal{B}(\PDzero \to \mu \mu)<6.2\times 10^{-9}$ at $90\%$ CL +\item \href{http://arxiv.org/abs/1305.5059}{\color{blue} PLB 725 (2013) 15-24} +\end{itemize} + +\end{small} +\end{frame} + + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{frame}\frametitle{Conclusions} +\begin{columns} +\column{3.2in} +\begin{itemize} +\item Rare decays play important role in hutting NP. +\item Can access NP scales beyond reach of GPD. +\item Tension in $\Pbeauty \to \Pstrange \Plepton \Plepton$, theory correct? +\item List of decays presented in this talk is just a tip of iceberg: +\begin{itemize} +\item Please look at ours: isospin, $A_{CP}$. +\item More are on their way. +\end{itemize} + + +\end{itemize} +\column{2in} +\includegraphics[width=0.9\textwidth]{hql/higgs_boring.png} +\end{columns} + +\end{frame} + + +\end{document} diff --git a/DIS2015/Rare_decays/mchrzasz.toc b/DIS2015/Rare_decays/mchrzasz.toc new file mode 100644 index 0000000..8742bca --- /dev/null +++ b/DIS2015/Rare_decays/mchrzasz.toc @@ -0,0 +1,3 @@ +\beamer@endinputifotherversion {3.24pt} +\select@language {english} +\beamer@sectionintoc {2}{Introduction}{3}{0}{1}