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\author{ {\fontspec{Trebuchet MS}Marcin Chrz\k{a}szcz} (Universit\"{a}t Z\"{u}rich, IFJ PAN)}
\institute{UZH, IFJ PAN}
\title[Anomalies in Flavour physics]{Anomalies in Flavour physics}
\date{25 September 2014}
\begin{document}
\tikzstyle{every picture}+=[remember picture]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{
\setbeamertemplate{sidebar right}{\llap{\includegraphics[width=\paperwidth,height=\paperheight]{bubble2}}}
\begin{frame}[c]%{\phantom{title page}}
\begin{center}
\begin{center}
\begin{columns}
\begin{column}{0.75\textwidth}
\flushright\fontspec{Trebuchet MS}\bfseries \LARGE {Anomalies in Flavour physics}
\end{column}
\begin{column}{0.02\textwidth}
{~}
\end{column}
\begin{column}{0.23\textwidth}
% \hspace*{-1.cm}
\vspace*{-3mm}
\includegraphics[width=0.6\textwidth]{lhcb-logo}
\end{column}
\end{columns}
\end{center}
\quad
\vspace{3em}
\begin{columns}
\begin{column}{0.44\textwidth}
\flushright \vspace{-1.8em} {\fontspec{Trebuchet MS} \Large Marcin Chrząszcz\\\vspace{-0.1em}\small \href{mailto:mchrzasz@cern.ch}{mchrzasz@cern.ch}}
\end{column}
\begin{column}{0.53\textwidth}
\includegraphics[height=1.3cm]{uzh-transp}{~}{~}
\includegraphics[height=1.1cm]{ifj.png}
\end{column}
\end{columns}
\vspace{1em}
\footnotesize\textcolor{gray}{on behalf of the LHCb collaboration,\\ Universit\"{a}t Z\"{u}rich, \\ Institute of Nuclear Physics, Polish Academy of Science}\normalsize\\
\vspace{0.5em}
\textcolor{normal text.fg!50!Comment}{Miami, 16-22 December 2015}
\end{center}
\end{frame}
}
\iffalse
\begin{frame}{Outline}
\begin{minipage}{\textwidth}
\begin{enumerate}
\item Why flavour is important.
\item $\Pbeauty \to \Pstrange \ell \ell$ theory in a nutshell.
\item LHCb measurements of $\Pbeauty \to \Pstrange \ell \ell$.
\item Global fit to $\Pbeauty \to \Pstrange \ell \ell$ measurements.
\item Conclusions.
\end{enumerate}
\end{minipage}
\vspace*{2.cm}
\end{frame}
\fi
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}\frametitle{Why rare decays?}
\begin{columns}
\column{4in}
\begin{itemize}
\item In SM allows only the charged interactions to change flavour.
\begin{itemize}
\item Other interactions are flavour conserving.
\end{itemize}
\item One can escape this constrain and produce $\Pbottom \to \Pstrange$ and $\Pbottom \to \Pdown$ at loop level.
\begin{itemize}
\item This kind of processes are suppressed in SM $\to$~Rare decays.
\item New Physics can enter in the loops.
\end{itemize}
\end{itemize}
\begin{center}
\includegraphics[scale=0.3]{images/lupa.png}
\includegraphics[scale=0.3]{images/example.png}
\end{center}
\column{1.5in}
\includegraphics[width=0.61\textwidth]{images/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{1.2in}
\begin{small}
\begin{align*}
H_{eff} = - \dfrac{4G_f}{\sqrt{2}} V V^{\prime \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*}
\end{small}
\column{1.8in}
\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}
where $C_i$ are the Wilson coefficients and $O_i$ are the corresponding effective operators.
\begin{center}
\includegraphics[width=0.85\textwidth,height=3cm]{images/all.png}
\end{center}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\only<1>{\frametitle{LHCb detector - tracking, \href{http://arxiv.org/abs/1412.6352}{Int. J. Mod. Phys. A 30}}
\begin{columns}
\column{3in}
\includegraphics[width=0.9\textwidth]{images/1050px-Lhcbview.jpg}
\column{2in}
\includegraphics[width=0.95\textwidth]{images/sketch.png}
\end{columns}
\begin{itemize}
\item Excellent Impact Parameter (IP) resolution ($20~\rm \mu m$).\\
$\Rightarrow$ Identify secondary vertices from heavy flavour decays
\item Proper time resolution $\sim~40~\rm fs$.\\
$\Rightarrow$ Good separation of primary and secondary vertices.
\item Excellent momentum ($\delta p/p \sim 0.5 - 1.0\%$) and inv. mass resolution.\\
$\Rightarrow$ Low combinatorial background.
\end{itemize}
}
\only<2>{\frametitle{LHCb detector - PID, \href{http://arxiv.org/abs/1412.6352}{Int. J. Mod. Phys. A 30}}
\begin{columns}
\column{3in}
\includegraphics[width=0.9\textwidth]{images/1050px-Lhcbview.jpg}
\column{2in}
\includegraphics[width=0.95\textwidth]{images/cher.png}
\end{columns}
\begin{itemize}
\item Excellent Muon identification $\epsilon_{\mu \to \mu} \sim 97\%$, $\epsilon_{\pi \to \mu} \sim 1-3\%$
\item Good $\PK-\Ppi$ separation via RICH detectors, $\epsilon_{\PK \to \PK} \sim 95\%$, $\epsilon_{\Ppi \to \PK} \sim 5\%$.\\
$\Rightarrow$ Reject peaking backgrounds.
\item High trigger efficiencies, low momentum thresholds.
Muons: $p_T > 1.76 \GeV$ at L0, $p_T > 1.0 \GeV$ at HLT1,\\
$B \to \PJpsi X $: Trigger $\sim 90\%$.
\end{itemize}
}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Recent measurements of $\Pbeauty \to \Pstrange \Plepton \Plepton$}
{~}
\only<1>{
\begin{minipage}{\textwidth}
\begin{columns}
\column{0.5\textwidth}
$\color{JungleGreen}{\Rrightarrow}$ {\color{WildStrawberry}{Branching fractions:}}
\begin{description}
\item [$\PB \to \PK \Pmuon \APmuon$] {~}{~}LHCb, Mar 14
\item [$\PB^{0} \to \PKstar \Pmuon \APmuon$] {~}{~}CMS, Jul 15
\item [$\PBs \to \Pphi \Pmuon \APmuon$] {~}{~}{~}LHCb, Jun 15
\item [$\PB^{\pm} \to \Ppi^{\pm} \Pmuon \APmuon$] {~}LHCb, Sep 15
\item [$\PLambdab \to \PLambda \Pmuon \APmuon$] {~}{~}{~}{~}LHCb, Mar 15
\item [$\PB \to\Pmuon \APmuon$] {~}{~}{~}{~}{~}CMS+LHCb, Jun 15
\end{description}
$\color{JungleGreen}{\Rrightarrow}$ {\color{WildStrawberry}{CP asymmetry:}}
\begin{description}
\item [$\PB^{\pm} \to \Ppi^{\pm} \Pmuon \APmuon$] {~}{~}LHCb, Sep 15
\end{description}
$\color{JungleGreen}{\Rrightarrow}$ {\color{WildStrawberry}{Isospin asymmetry:}}
\begin{description}
\item [$\PB \to \PK \Pmuon \APmuon$] {~}{~}{~}{~}{~}LHCb, Mar 14
\end{description}
\column{0.5\textwidth}
$\color{JungleGreen}{\Rrightarrow}$ {\color{WildStrawberry}{Lepton Universality:}}
\begin{description}
\item [$\PB^{\pm} \to \PK^{\pm} \Plepton \APlepton$] {~}{~}LHCb, Jun 14
\end{description}
$\color{JungleGreen}{\Rrightarrow}$ {\color{WildStrawberry}{Angular:}}
\begin{description}
\item [$\PB^{0} \to \PK^{\ast} \Plepton \APlepton$] {~}{~}{~}LHCb, Dec 15
\item [$\PB^{0,\pm} \to \PK^{\ast,\pm} \Plepton \APlepton$] BaBar, Aug 15
\item [$\PBs \to \Pphi \Pmu \Pmu$] {~}{~}{~}LHCb, Jun 15
\item [$\PLambdab \to \PLambda \Pmuon \APmuon$] {~}{~}LHCb, Mar 15
\end{description}
\end{columns}
\end{minipage}
}
\only<2>{
\begin{minipage}{\textwidth}
\begin{columns}
\column{0.5\textwidth}
$\color{JungleGreen}{\Rrightarrow}$ {\color{WildStrawberry}{Branching fractions:}}
\begin{description}
\item [{\color{red}{$\PB \to \PK \Pmuon \APmuon$}}] {~}{~}{\color{red}{LHCb, Mar 14}}
\item [$\PB^{0} \to \PKstar \Pmuon \APmuon$] {~}{~}CMS, Jul 15
\item [{\color{red}{$\PBs \to \Pphi \Pmuon \APmuon$}}] {~}{~}{~}{\color{red}{LHCb, Jun 15}}
\item [$\PB^{\pm} \to \Ppi^{\pm} \Pmuon \APmuon$] {~}LHCb, Sep 15
\item [$\PLambdab \to \PLambda \Pmuon \APmuon$] {~}{~}{~}{~}LHCb, Mar 15
\item [{\color{red}{$\PB \to\Pmuon \APmuon$}}] {~}{~}{~}{~}{~}{\color{red}{CMS+LHCb, Jun 15}}
\end{description}
$\color{JungleGreen}{\Rrightarrow}$ {\color{WildStrawberry}{CP asymmetry:}}
\begin{description}
\item [$\PB^{\pm} \to \Ppi^{\pm} \Pmuon \APmuon$] {~}{~}LHCb, Sep 15
\end{description}
$\color{JungleGreen}{\Rrightarrow}$ {\color{WildStrawberry}{Isospin asymmetry:}}
\begin{description}
\item [$\PB \to \PK \Pmuon \APmuon$] {~}{~}{~}{~}{~}LHCb, Mar 14
\end{description}
\column{0.5\textwidth}
$\color{JungleGreen}{\Rrightarrow}$ {\color{WildStrawberry}{Lepton Universality:}}
\begin{description}
\item [{\color{red}{$\PB^{\pm} \to \PK^{\pm} \Plepton \APlepton$}}] {~}{~}{\color{red}{LHCb, Jun 14}}
\end{description}
$\color{JungleGreen}{\Rrightarrow}$ {\color{WildStrawberry}{Angular:}}
\begin{description}
\item [{\color{red}{$\PB^{0} \to \PK^{\ast} \Plepton \APlepton$}}] {~}{~}{~}LHCb, Dec 15
\item [{\color{red}{$\PB^{\pm} \to \PK^{\ast,\pm} \Plepton \APlepton$}}] {\color{red}{BaBar, Aug 15}}
\item [$\PBs \to \Pphi \Plepton \APlepton$] {~}{~}{~}LHCb, Jun 15
\item [{\color{red}{$\PLambdab \to \PLambda \Pmuon \APmuon$}}] {~}{~}{\color{red}{LHCb, Mar 15}}
\end{description}
\begin{alertblock}{}
$>2~\sigma$ deviations from SM
\end{alertblock}
\end{columns}
\end{minipage}
}
\vspace*{2.1cm}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{$\PBzero \to \PKstar \Pmuon \APmuon$ kinematics}
{~}
\begin{minipage}{\textwidth}
$\color{JungleGreen}{\Rrightarrow}$ The kinematics of $\PBzero \to \PKstar \Pmuon \APmuon$ decay is described by three angles $\thetal$, $\thetak$, $\phi$ and invariant mass of the dimuon system ($q^2)$.
\only<1>{
\begin{columns}
\column{0.5\textwidth}
$\color{JungleGreen}{\Rrightarrow}$ $\cos \thetak$: the angle between the direction of the kaon in the $\PKstar$ ($\overline{\PK}^{\ast}$) rest frame and the direction of the $\PKstar$ ($\overline{\PK}^{\ast}$) in the $\PBzero$ ($\APBzero$) rest frame.\\
$\color{JungleGreen}{\Rrightarrow}$ $\cos \thetal$: the angle between the direction of the $\Pmuon$ ($\APmuon$) in the dimuon rest frame and the direction of the dimuon in the $\PBzero$ ($\APBzero$) rest frame.\\
$\color{JungleGreen}{\Rrightarrow}$ $\phi$: the angle between the plane containing the $\Pmuon$ and $\APmuon$ and the plane containing the kaon and pion from the $\PKstar$.
\column{0.5\textwidth}
\includegraphics[width=0.99\textwidth]{images/angles.png}
\end{columns}
}
\only<2>{
{\tiny{
\eqa{\label{dist}
\frac{d^4\Gamma}{dq^2\,d\!\cos\theta_K\,d\!\cos\theta_l\,d\phi}&=&\frac9{32\pi} \bigg[
{\color{red}{J_{1s}}} \sin^2\theta_K + {\color{red}{J_{1c}}} \cos^2\theta_K + ({\color{red}{J_{2s} }}\sin^2\theta_K + {\color{red}{J_{2c}}} \cos^2\theta_K) \cos 2\theta_l\nn\\[1.5mm]
&&\hspace{-2.7cm}+ {\color{red}{J_3}} \sin^2\theta_K \sin^2\theta_l \cos 2\phi + {\color{red}{J_4}} \sin 2\theta_K \sin 2\theta_l \cos\phi + {\color{red}{J_5}} \sin 2\theta_K \sin\theta_l \cos\phi \nn\\[1.5mm]
&&\hspace{-2.7cm}+ ({\color{red}{J_{6s}}} \sin^2\theta_K + {\color{red}{{J_{6c}}}} \cos^2\theta_K) \cos\theta_l
+ {\color{red}{J_7}} \sin 2\theta_K \sin\theta_l \sin\phi + {\color{red}{J_8}} \sin 2\theta_K \sin 2\theta_l \sin\phi \nn\\[1.5mm]
&&\hspace{-2.7cm}+ {\color{red}{J_9}} \sin^2\theta_K \sin^2\theta_l \sin 2\phi \bigg]\,,
\nonumber}
}}\\{~}\\
$\color{JungleGreen}{\Rrightarrow}$ This is the most general expression of this kind of decay.\\
$\color{JungleGreen}{\Rrightarrow}$ The $CP$ averaged angular observables are defined:\\
\eq{
S_i = \dfrac{J_i+ \bar{J}_i}{(d \Gamma + d \bar{\Gamma})/dq^2}\nonumber
}
}
\end{minipage}
\vspace*{2.1cm}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Link to effective operators}
{~}
\begin{minipage}{\textwidth}
$\color{JungleGreen}{\Rrightarrow}$ The observables ${\color{red}{J_i}}$ are bilinear combinations of transversity amplitudes: $\apeLR,~\apaLR,~\azeLR $. \\
$\color{JungleGreen}{\Rrightarrow}$ So here is where the magic happens. At leading order the amplitudes can be written as:
{\tiny{
\eqa{
\apeLR &=&\sqrt{2} N m_B(1- \hat s)\bigg[ (\Ceff9 + \Cpeff9) \mp (\C{10} + \Cp{10})
+\frac{2\hat{m}_b}{\hat s} (\Ceff7 + \Cpeff7) \bigg]\xi_{\bot}(E_{K^*}) \nn \\[2mm]
\apaLR &=& -\sqrt{2} N m_B (1-\hat s)\bigg[(\Ceff9 - \Cpeff9) \mp (\C{10} - \Cp{10})
+\frac{2\hat{m}_b}{\hat s}(\Ceff7 - \Cpeff7) \bigg] \xi_{\bot}(E_{K^*}) \nn \\[2mm]
\azeLR &=& -\frac{N m_B (1-\hat s)^2}{2 \hat{m}_{K^*} \sqrt{\hat s}} \bigg[ (\Ceff9 - \Cpeff9) \mp (\C{10} - \Cp{10}) + 2\hat{m}_b (\Ceff7 - \Cpeff7) \bigg]\xi_{\|}(E_{K^*}),
\label{LargeRecoilAs}\nonumber}
}}
where $\hat s = q^2 /m_B^2$, $\hat{m}_i = m_i/m_B$. The $\xi_{\|,\bot }$ are the soft form factors. \\
\pause
$\color{JungleGreen}{\Rrightarrow}$ Now we can construct observables that cancel the $\xi$ soft form factors at leading order:
\eq{P_5^{\prime} = \dfrac{J_5+\bar{J}_5}{2\sqrt{-(J_2^c+\bar{J}_2^c)(J_2^s+\bar{J}_2^s)} }\nonumber
}
\end{minipage}
\vspace*{2.1cm}
\end{frame}
% symmetries
\begin{frame}{Symmetries in $\PB \to \PKstar \Pmu \Pmu$}
{~}
\begin{minipage}{\textwidth}
$\color{JungleGreen}{\Rrightarrow}$ We have 12 angular coefficients ($S_i$).\\
$\color{JungleGreen}{\Rrightarrow}$ There exists 4 symmetry transformations that leave the angular distributions non changed:
\begin{tiny}
\eq{
n_\|=\binom{A_\|^L}{A_\|^{R*}}\ ,\quad
n_\bot=\binom{A_\bot^L}{-A_\bot^{R*}}\ ,\quad
n_0=\binom{A_0^L}{A_0^{R*}}\ .\nonumber
}
\end{tiny}
\begin{tiny}
\eq{
n_i^{'} = U n_i=
\left[
\begin{array}{ll}
e^{i\phi_L} & 0 \\
0 & e^{-i \phi_R}
\end{array}
\right]
\left[
\begin{array}{rr}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\end{array}
\right]
\left[
\begin{array}{rr}
\cosh i \tilde{\theta} & -\sinh i \tilde{\theta} \\
- \sinh i \tilde{\theta} & \cosh i \tilde{\theta}
\end{array}
\right]
n_i \,.
\label{symmassless}\nonumber}
\end{tiny}
$\color{JungleGreen}{\Rrightarrow}$ Using this symmetries one can show that there are 8 independent observables. The pdf can be wrote as:
\begin{tiny}
\begin{align*}
\left.\frac{1}{{\rm d}(\Gamma+\bar{\Gamma})/{\rm d}q^2}\frac{{\rm d}(\Gamma+\bar{\Gamma})}{{\rm dcos}\thetal\,{\rm dcos}\thetak\,{\rm d}\phi}\right|_{\rm P} =
\tfrac{9}{32\pi}\bigl[
&\tfrac{3}{4} (1-{F_{\rm L}})\sin^2\thetak \label{eq:pdfpwave}\\[-0.75em]
&+ {F_{\rm L}}\cos^2\thetak + \tfrac{1}{4}(1-{F_{\rm L}})\sin^2\thetak\cos 2\thetal\nonumber\\
&- {F_{\rm L}} \cos^2\thetak\cos 2\thetal + {S_3}\sin^2\thetak \sin^2\thetal \cos 2\phi\nonumber\\
&+ {S_4} \sin 2\thetak \sin 2\thetal \cos\phi + {S_5}\sin 2\thetak \sin \thetal \cos \phi\nonumber\\
&+ \tfrac{4}{3} {A_{\rm FB}} \sin^2\thetak \cos\thetal + {S_7} \sin 2\thetak \sin\thetal \sin\phi\nonumber\\
&+ {S_8} \sin 2\thetak \sin 2\thetal \sin\phi + {S_9}\sin^2\thetak \sin^2\thetal \sin 2\phi \nonumber
\bigr].
%\end{split}
%\bigr],\
\end{align*}
\end{tiny}
\end{minipage}
\vspace*{2.1cm}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{LHCb update of the $\PBzero \to \PKstar \Pmuon \APmuon$, Selection}
{~}
\begin{minipage}{\textwidth}
\begin{columns}
\column{0.55\textwidth}
\begin{itemize}
\item \href{http://arxiv.org/abs/1512.04442}{{\color{blue}{arXiv:1512.04442}}}
\item PID, kinematics and isolation variables used in a Boosted Decision Tree (BDT) to reject background.
\item Reject the regions of $\PJpsi$ and $\Ppsi(2S)$.
\item Specific vetos for backgrounds: $\PLambdab \to \Pproton \PK \Pmu \Pmu$, $\PBs \to \Pphi \Pmu \Pmu$, etc.
\item Using k-Fold technique and signal proxy $\PB \to \PJpsi \PKstar$ for training the BDT.
\item Improved selection allowed for finer binning than the $1\invfb$ analysis.
\end{itemize}
\column{0.45\textwidth}
\includegraphics[angle=-90,width=0.88\textwidth]{images/Fig1.pdf} \\
\includegraphics[width=0.88\textwidth]{images/fold.png}
\end{columns}
\end{minipage}
\vspace*{2.1cm}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{LHCb update of the $\PBzero \to \PKstar \Pmuon \APmuon$, \href{http://arxiv.org/abs/1512.04442}{arXiv:1512.04442}}
{~}
\begin{minipage}{\textwidth}
\begin{itemize}
\item Signal modelled by a sum of two Crystal-Ball functions.
\item Shape is defined using $\PB \to \PJpsi \PKstar$ and corrected for $q^2$ dependency.
\item Combinatorial background modelled by exponent.
\end{itemize}
\begin{columns}
\column{0.5\textwidth}
\begin{itemize}
\item $\PK \Ppi$ system:
\begin{itemize}
\item Beside the $\PKstar$ resonance there might might a tail from other higher mass states.
\item We modelled it in analysis.
\item Reduced the systematic compared to previous analysis.
\end{itemize}
\end{itemize}
\column{0.5\textwidth}
\includegraphics[width=0.88\textwidth]{images/pbkg}
\end{columns}
\begin{large}
\begin{itemize}
\item In total we found $2398\pm57$ candidates in the $(0.1,19)~\GeV^2$ $q^2$ region.
\item $624 \pm 30$ candidates in the theoretically the most interesting $(1.1-6.0)~\GeV^2$ region.
\end{itemize}
\end{large}
\end{minipage}
\vspace*{2.1cm}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Detector acceptance, \href{http://arxiv.org/abs/1512.04442}{arXiv:1512.04442}}
{~}
\begin{minipage}{\textwidth}
\begin{columns}
\column{0.6\textwidth}
\begin{itemize}
\item Detector distorts our angular distribution.
\item We need to model this effect.
\item 4D function is used:
\begin{align*}
\epsilon (\cos \thetal, \cos \thetak, \phi, q^2) = \\\sum_{ijkl} c_{ijkl} P_i(\cos \thetal) P_j(\cos \thetak ) P_k(\phi) P_l(q^2),
\end{align*}
where $P_i$ is the Legendre polynomial of order $i$.
\item We use up to $4^{th}, 5^{th}, 6^{th}, 5^{th}$ order for the $\cos \thetal, \cos \thetak, \phi, q^2$.
\item $600$ terms in total!
\end{itemize}
\column{0.4\textwidth}
\includegraphics[width=0.99\textwidth]{images/det.png}
\end{columns}
\end{minipage}
\vspace*{2.1cm}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Moments
\begin{frame}{Method of moments}
{~}
\begin{minipage}{\textwidth}
\begin{itemize}
\item \href{http://arxiv.org/abs/1503.04100}{{\color{blue}{Phys. Rev. D 91, 114012 (2015)}}}
\item Use orthogonality of spherical harmonics, $f_j(\cos \thetal, \cos \thetak, \phi)$:
\begin{equation*}
\int f_i(\cos \thetal, \cos \thetak, \phi) \cdot f_j(\cos \thetal, \cos \thetak, \phi) = \delta_{ij}
\end{equation*}
\begin{equation*}
M_i = \int \frac{1}{{\rm d}(\Gamma+\bar{\Gamma})/{\rm d}q^2}\frac{{\rm d}(\Gamma+\bar{\Gamma})}{{\rm dcos}\thetal\,{\rm dcos}\thetak\,{\rm d}\phi} f_i(\cos \thetal, \cos \thetak, \phi)
\end{equation*}
\item Don’t have true angular distribution but we ''sample'' it with our data.
\item Therefore: $\int \rightarrow \sum$ and $M_i \rightarrow \widehat{M}_i$
%\begin{equation*}
%M_i = \int \frac{1}{{\rm d}(\Gamma+\bar{\Gamma})/{\rm d}q^2}\frac{{\rm d}(\Gamma+\bar{\Gamma})}{{\rm dcos}\thetal\,{\rm dcos}\thetak\,{\rm d}\phi} f_i(\cos \thetal, \cos \thetak, \phi)
%\end{equation*}
\item Acceptance corrections is included by:
\begin{equation*}
\widehat{M}_i = \dfrac{1}{\sum_e w_e} \sum w_e f_i(\cos \thetal, \cos \thetak, \phi)
\end{equation*}
\item The weight $w_e$ accounts for the efficiency from previous slide.
\end{itemize}
\end{minipage}
\vspace*{2.1cm}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Control channel, \href{http://arxiv.org/abs/1512.04442}{arXiv:1512.04442}}
{~}
\begin{minipage}{\textwidth}
\begin{itemize}
\item We tested our unfolding procedure on $\PB \to \PJpsi \PKstar$.
\item The result is in perfect agreement with other experiments and our different analysis of this decay.
\end{itemize}
\begin{columns}
\column{0.5\textwidth}
\includegraphics[width=0.95\textwidth]{images/mlogjpsi.png}
\column{0.5\textwidth}
\includegraphics[width=0.95\textwidth]{images/mkpijpsi.png}
\end{columns}
\begin{columns}
\column{0.33\textwidth}
\includegraphics[width=0.95\textwidth]{images/costhetakjpsi.png}
\column{0.33\textwidth}
\includegraphics[width=0.95\textwidth]{images/costhetaljpsi.png}
\column{0.33\textwidth}
\includegraphics[width=0.95\textwidth]{images/phijpsi.png}
\end{columns}
%\includegraphics[width=0.99\textwidth]{images/angles2.png}
\end{minipage}
\vspace*{2.1cm}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
\begin{frame}\frametitle{$\PBzero \rightarrow \PK^{\ast} \Pmu \Pmu$ results, \href{http://arxiv.org/abs/1512.04442}{arXiv:1512.04442}}
\begin{minipage}{\textwidth}
\begin{columns}
\column{2.5in}
\includegraphics[angle=-90,width=0.95\textwidth]{images/compare_FLPad.pdf}\\
\includegraphics[angle=-90,width=0.95\textwidth]{images/compare_S3Pad.pdf}
\column{2.5in}
\includegraphics[angle=-90,width=0.95\textwidth]{images/compare_S4Pad.pdf}\\
\includegraphics[angle=-90,width=0.95\textwidth]{images/compare_S5Pad.pdf}
\end{columns}
\end{minipage}
\vspace*{2.1cm}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
\begin{frame}\frametitle{$\PBzero \rightarrow \PK^{\ast} \Pmu \Pmu$ results, \href{http://arxiv.org/abs/1512.04442}{arXiv:1512.04442}}
\begin{minipage}{\textwidth}
\begin{columns}
\column{2.5in}
\includegraphics[angle=-90,width=0.95\textwidth]{images/compare_AFBPad.pdf}\\
\includegraphics[angle=-90,width=0.95\textwidth]{images/compare_S7Pad.pdf}
\column{2.5in}
\includegraphics[angle=-90,width=0.95\textwidth]{images/compare_S8Pad.pdf}\\
\includegraphics[angle=-90,width=0.95\textwidth]{images/compare_S9Pad.pdf}
\end{columns}
\end{minipage}
\vspace*{2.1cm}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Results in $\PB \to \PKstar \Pmu \Pmu$, \href{http://arxiv.org/abs/1512.04442}{arXiv:1512.04442}}
{~}
\begin{minipage}{\textwidth}
\begin{center}
\includegraphics[angle=-90,width=0.65\textwidth]{images/compare_P5pPad.pdf}\\
\end{center}
\begin{itemize}
\item Tension gets confirmed!
\item The two bins deviate by $2.8$~and~$3.0~\sigma$ from SM prediction.
\item Result compatible with previous result.
\end{itemize}
\end{minipage}
\vspace*{2.1cm}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Results in $\PB \to \PKstar \Pmu \Pmu$, \href{http://arxiv.org/abs/1512.04442}{arXiv:1512.04442}}
{~}
\begin{minipage}{\textwidth}
\begin{itemize}
\item Thanks to Method of Moments there was the possibility to measure a new observable $S_{6c}$.
\end{itemize}
\begin{center}
\includegraphics[angle=-90,width=0.65\textwidth]{images/S6cPad.pdf}\\
\end{center}
\begin{itemize}
\item Measurement is consistent with the SM prediction.
\end{itemize}
\end{minipage}
\vspace*{2.1cm}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Branching fraction measurements of $\PB \to \PKstar^{\pm} \Pmu \Pmu$}
{~}
\includegraphics[angle=-90,width=0.5\textwidth]{images/ksmumu_BF.pdf}
\includegraphics[angle=-90,width=0.5\textwidth]{images/kmumu_BF.pdf}
\begin{center}
\begin{columns}
\column{0.5\textwidth}
\begin{itemize}
\item Despite large theoretical errors the results are consistently smaller than SM prediction.
\item \href{http://dx.doi.org/10.1007/JHEP06(2014)133}{{\color{blue}{JHEP 06 (2014) 133}}}
\end{itemize}
\column{0.5\textwidth}
\includegraphics[angle=-90,width=0.850\textwidth]{images/bukst_BF.pdf}
\end{columns}
\end{center}
\vspace*{2.1cm}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Branching fraction measurements of $\PBs \to \Pphi \Pmu \Pmu$}
{~}
\begin{minipage}{\textwidth}
\begin{center}
\includegraphics[height=4cm]{images/bs2phipi.png}
\includegraphics[height=4cm]{images/BsSel.png}
\end{center}
\begin{itemize}
\item Recent LHCb measurement, \href{https://cds.cern.ch/record/2029820/files/JHEP09-179.pdf}{{\color{blue}{JHEP09 (2015) 179}}}.
\item Suppressed by $\frac{f_s}{f_d}$.
\item Cleaner because of narrow $\Pphi$ resonance.
\item $3.3~\sigma$ deviation in SM in the $1-6\GeV^2$ bin.
\item Angular part in agreement with SM ($S_5$ is not accessible).
\end{itemize}
\end{minipage}
\vspace*{2.1cm}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Branching fraction measurements of $\PLambdab \to \PLambda \Pmu \Pmu$}
{~}
\begin{minipage}{\textwidth}
\begin{center}
\only<1>{
\includegraphics[width=0.65\textwidth]{images/Lb_BR.png}
}
\only<2>{
\includegraphics[width=0.45\textwidth]{images/Lblow.png}
\includegraphics[width=0.45\textwidth]{images/Lbhigh.png}
}
\end{center}
\begin{itemize}
\item This years LHCb measurement \href{http://arxiv.org/abs/1503.07138}{{\color{blue}{JHEP 06 (2015) 115}}}.
\item In total $\sim 300$ candidates in data set.
\item Decay not present in the low $q^2$.
\end{itemize}
\end{minipage}
\vspace*{2.1cm}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Angular analysis of $\PLambdab \to \PLambda \Pmu \Pmu$, \href{http://arxiv.org/abs/1503.07138}{{JHEP 06 (2015) 115}}}
{~}
\begin{minipage}{\textwidth}
\begin{itemize}
\item For the bins in which we have $>3~\sigma$ significance the forward backward asymmetry for the hadronic and leptonic system.
\end{itemize}
\begin{center}
\includegraphics[width=0.9\textwidth]{{images/AFB_Lb}.png}
\end{center}
\begin{itemize}
\item $A_{FB}^H$ is in good agreement with SM.
\item $A_{FB}^{\ell}$ always in above SM prediction.
\end{itemize}
\end{minipage}
\vspace*{2.1cm}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Lepton universality test}
{~}
\begin{minipage}{\textwidth}
\begin{columns}
\column{3.0in}
\includegraphics[width=0.9\textwidth]{images/uni2.png}
\begin{itemize}
%\includegraphics[width=0.9\textwidth]{images/uni2.png}
\item Challenging analysis due to bremsstrahlung.
\item Migration of events modeled by MC.
\item Correct for bremsstrahlung.
\item Take double ratio with $\PBplus \to \PJpsi \PKplus$ to cancel systematics.
\item In $3\invfb$, 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]{images/RK.png}\\
\begin{itemize}
\item \href{http://arxiv.org/abs/1406.6482}{{\color{blue}{Phys. Rev. Lett. 113, 151601 (2014)}}}.
\end{itemize}
\end{columns}
\end{minipage}
\vspace*{2.1cm}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Angular analysis of $\PBzero \to \PKstar \Pe \Pe$}
{~}
\only<1>{
\begin{minipage}{\textwidth}
\begin{itemize}
\item With the full data set ($3\invfb$) we performed angular analysis in $0.0004 < q^2 <1~\GeV^2$, \href{http://arxiv.org/abs/1503.07138}{{\color{blue}{JHEP 04 (2015) 064}}}.
\item Electrons channels are extremely challenging experimentally:
\begin{itemize}
\item Bremsstrahlung.
\item Trigger efficiencies.
\end{itemize}
\item Determine the angular observables: $\FL$, $\ATD$, $\ATRe$, $\ATIm$:
\end{itemize}
\begin{itemize}
\item Results in full agreement with the SM.
\item Similar strength on $C_7$ Wilson coefficient as from $\Pbeauty \to \Pstrange \Pphoton$ decays.
\end{itemize}
\begin{center}
\includegraphics[width=0.9\textwidth]{images/Kstee2.png}
\end{center}
}
\end{minipage}
\vspace*{2.1cm}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Theory implications}
{~}
\begin{minipage}{\textwidth}
\begin{itemize}
\item A preliminary fit prepared by \texttt{S. Descotes-Genon, L. Hofer, J. Matias, J. Virto}, presented in \href{http://arxiv.org/abs/1510.04239}{\color{blue}{arXiv::1510.04239}}
\item Took into the fit:
\begin{itemize}
\item $\mathcal{B} ( \PB \to X_s \Pphoton) = (3.36 \pm 0.23) \times 10^{-4} $, Misiak et. al. 2015.
\item $\mathcal{B} ( \PB \to\Pmu \Pmu)$, theory: Bobeth et al 2013, experiment: LHCb+CMS average (2015)
\item $\mathcal{B} ( \PB \to X_s \Pmu \Pmu$), Huber et al 2015
\item $\mathcal{B} ( \PB \to \PK \Pmu \Pmu$),Bouchard et al 2013, 2015
\item $\PB_{(s)} \to \PKstar(\Pphi) \Pmu \Pmu$, Horgan et al 2013
\item $\PB \to \PK \Pe \Pe$, $\PB \to \PKstar \Pe \Pe$ and $R_k$.
\end{itemize}
%\item Overall there is $>4~\sigma$ discrepancy wrt. SM.
\end{itemize}
\end{minipage}
\vspace*{2.1cm}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Theory implications}
{~}
\begin{minipage}{\textwidth}
\begin{itemize}
\item A preliminary fit prepared by \texttt{S. Descotes-Genon, L. Hofer, J. Matias, J. Virto}, presented in \href{http://arxiv.org/abs/1510.04239}{\color{blue}{arXiv::1510.04239}}
\item The data can be explained by modifying the $C_9$ Wilson coefficient.
\item Overall there is $>4~\sigma$ discrepancy wrt. SM prediction.
\end{itemize}
\includegraphics[width=0.9\textwidth]{images/C9.png}
\end{minipage}
\vspace*{2.1cm}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{If not NP?}
{~}
\begin{minipage}{\textwidth}
\begin{itemize}
\item We are not there yet!
\item There might be something not taken into account in the theory.
\item Resonances ($\PJpsi$, $\Ppsi(2S)$) tails can mimic NP effects.
\item There might be some non factorizable QCD corrections.\\
'' However, the central value of this effect would have to be significantly larger than expected on the basis of existing estimates'' \texttt{D.Straub}, \href{http://arxiv.org/abs/1503.06199}{ {\color{blue}{arXiv::1503.06199}}}
.
\end{itemize}
\only<1>{
\includegraphics[width=0.75\textwidth]{images/charmloop.png}
\begin{flushright}
Courtesy of T.Blake
\end{flushright}
}
\only<2>{
\begin{center}
\includegraphics[width=0.6\textwidth]{images/charmloop2.png}
\end{center}
}
\end{minipage}
\vspace*{2.1cm}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{There is more!}
{~}
\begin{minipage}{\textwidth}
\begin{itemize}
\item There is one other LUV decay recently measured by LHCb.
\item $R(\PDstar)=\dfrac{\mathcal{B}(\PB \to \PDstar \Ptau \Pnu)}{\mathcal{B}(\PB \to \PDstar \Pmu \Pnu)}$
\item Clean SM prediction: $R(\PDstar)=0.252(3)$, PRD 85 094025 (2012)
\item LHCb result: $R(\PDstar)= 0.336 \pm 0.027 \pm 0.030$, HFAG average: $R(\PDstar)=0.322 \pm 0.022$
\item $3.9~\sigma$ discrepancy wrt. SM prediction
\end{itemize}
\begin{center}
\includegraphics[width=0.52\textwidth]{images/RDstar.png}
\end{center}
\end{minipage}
\vspace*{2.1cm}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Conclusions}
{~}
\begin{minipage}{\textwidth}
\begin{itemize}
\item Clear tensions wrt. SM predictions!
\item Measurements cluster in the same direction.
\item We are not opening the champagne yet!
\item Still need improvement both on theory and experimental side.
\item Time will tell if this is QCD+fluctuations or new Physics:
\end{itemize}
\pause
''... when you have eliminated all the\\
Standard Model explanations, whatever remains,\\
however improbable, must be New Physics.''\\
prof. Joaquim Matias
\end{minipage}
\vspace*{2.1cm}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
{~}
\begin{minipage}{\textwidth}
\begin{center}
\begin{LARGE}
Thank you for the attention!
\end{LARGE}
\includegraphics[width=0.8\textwidth]{images/Joke.jpg}
\end{center}
\end{minipage}
\vspace*{2.1cm}
\end{frame}
\backupbegin
\begin{frame}\frametitle{Backup}
\end{frame}
\begin{frame}{Theory implications}
{~}
\begin{minipage}{\textwidth}
\includegraphics[height=0.9\textheight]{images/table.png}
\end{minipage}
\vspace*{2.1cm}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{If not NP?}
{~}
\begin{minipage}{\textwidth}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{itemize}
\item How about our clean $P_i$ observables?
\item The QCD cancel as mentioned only at leading order.
\item Comparison to normal observables with the optimised ones.
\end{itemize}
\includegraphics[width=0.9\textwidth]{images/C9_S_P.png}
\end{minipage}
\vspace*{2.1cm}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Transversity amplitudes }
{~}
\begin{minipage}{\textwidth}
$\color{JungleGreen}{\Rrightarrow}$ One can link the angular observables to transversity amplitudes
{\tiny{
\eqa{
J_{1s} & = & \frac{(2+\beta_\ell^2)}{4} \left[|\apeL|^2 + |\apaL|^2 +|\apeR|^2 + |\apaR|^2 \right]
+ \frac{4 m_\ell^2}{q^2} \re\left(\apeL\apeR^* + \apaL\apaR^*\right)\,,\nn\\[1mm]
%
J_{1c} & = & |\azeL|^2 +|\azeR|^2 + \frac{4m_\ell^2}{q^2} \left[|A_t|^2 + 2\re(\azeL^{}\azeR^*) \right] + \beta_\ell^2\, |A_S|^2 \,,\nn\\[1mm]
%
J_{2s} & = & \frac{ \beta_\ell^2}{4}\left[ |\apeL|^2+ |\apaL|^2 + |\apeR|^2+ |\apaR|^2\right],
\hspace{0.92cm} J_{2c} = - \beta_\ell^2\left[|\azeL|^2 + |\azeR|^2 \right]\,,\nn\\[1mm]
%
J_3 & = & \frac{1}{2}\beta_\ell^2\left[ |\apeL|^2 - |\apaL|^2 + |\apeR|^2 - |\apaR|^2\right],
\qquad J_4 = \frac{1}{\sqrt{2}}\beta_\ell^2\left[\re (\azeL\apaL^* + \azeR\apaR^* )\right],\nn \\[1mm]
%
J_5 & = & \sqrt{2}\beta_\ell\,\Big[\re(\azeL\apeL^* - \azeR\apeR^* ) - \frac{m_\ell}{\sqrt{q^2}}\,
\re(\apaL A_S^*+ \apaR^* A_S) \Big]\,,\nn\\[1mm]
%
J_{6s} & = & 2\beta_\ell\left[\re (\apaL\apeL^* - \apaR\apeR^*) \right]\,,
\hspace{2.25cm} J_{6c} = 4\beta_\ell\, \frac{m_\ell}{\sqrt{q^2}}\, \re (\azeL A_S^*+ \azeR^* A_S)\,,\nn\\[1mm]
%
J_7 & = & \sqrt{2} \beta_\ell\, \Big[\im (\azeL\apaL^* - \azeR\apaR^* ) +
\frac{m_\ell}{\sqrt{q^2}}\, \im (\apeL A_S^* - \apeR^* A_S)) \Big]\,,\nn\\[1mm]
%
J_8 & = & \frac{1}{\sqrt{2}}\beta_\ell^2\left[\im(\azeL\apeL^* + \azeR\apeR^*)\right]\,,
%
\hspace{1.9cm} J_9 = \beta_\ell^2\left[\im (\apaL^{*}\apeL + \apaR^{*}\apeR)\right] \,,
\label{Js}\nonumber}
}}
\end{minipage}
\vspace*{2.1cm}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Link to effective operators}
{~}
\begin{minipage}{\textwidth}
$\color{JungleGreen}{\Rrightarrow}$ So here is where the magic happens. At leading order the amplitudes can be written as (soft form factors):
{\tiny{
\eqa{
\apeLR &=&\sqrt{2} N m_B(1- \hat s)\bigg[ (\Ceff9 + \Cpeff9) \mp (\C{10} + \Cp{10})
+\frac{2\hat{m}_b}{\hat s} (\Ceff7 + \Cpeff7) \bigg]\xi_{\bot}(E_{K^*}) \nn \\[2mm]
\apaLR &=& -\sqrt{2} N m_B (1-\hat s)\bigg[(\Ceff9 - \Cpeff9) \mp (\C{10} - \Cp{10})
+\frac{2\hat{m}_b}{\hat s}(\Ceff7 - \Cpeff7) \bigg] \xi_{\bot}(E_{K^*}) \nn \\[2mm]
\azeLR &=& -\frac{N m_B (1-\hat s)^2}{2 \hat{m}_{K^*} \sqrt{\hat s}} \bigg[ (\Ceff9 - \Cpeff9) \mp (\C{10} - \Cp{10}) + 2\hat{m}_b (\Ceff7 - \Cpeff7) \bigg]\xi_{\|}(E_{K^*}),
\label{LargeRecoilAs}\nonumber}
}}
where $\hat s = q^2 /m_B^2$, $\hat{m}_i = m_i/m_B$. The $\xi_{\|,\bot }$ are the form factors. \\
\pause
$\color{JungleGreen}{\Rrightarrow}$ Now we can construct observables that cancel the $\xi$ form factors at leading order:
\eq{P_5^{\prime} = \dfrac{J_5+\bar{J}_5}{2\sqrt{-(J_2^c+\bar{J}_2^c)(J_2^s+\bar{J}_2^s)} }\nonumber
}
\end{minipage}
\vspace*{2.1cm}
\end{frame}
\backupend
\end{document}
mchrzasz