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\author{ {\fontspec{Trebuchet MS}Marcin Chrz\k{a}szcz} (CERN, IFJ PAN)}
\institute{UZH}
\title[Anomalies in Physics]{Anomalies in 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}{0.9\textwidth}
			\flushright\fontspec{Trebuchet MS}\bfseries \Huge {Anomalies in Physics or Physics of anomalies?}
		\end{column}
		\begin{column}{0.2\textwidth}
		  %\includegraphics[width=\textwidth]{SHiP-2}
		\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]{cern}{~}{~} 
\includegraphics[height=1.1cm]{ifj.png}
\end{column}
\end{columns}

\vspace{1em}
%		\footnotesize\textcolor{gray}{With N. Serra, B. Storaci\\Thanks to the theory support from M. Shaposhnikov, D. Gorbunov}\normalsize\\
\vspace{0.5em}
	\textcolor{normal text.fg!50!Comment}{IFJ FCAL workshop \\May 10, 2018}
\end{center}
\end{frame}
}


\begin{frame}{Outline}

	\begin{minipage}{\textwidth}
{~}\\

\begin{enumerate}
\item Why flavour is important.
\item The flavour anomalies:
\begin{itemize}
\item $R(\PDstar)$
\item $R_K$ and $R_{\PKstar}$
\item $P_5^{\prime}$
\end{itemize}
\item Global fits results.
\item Conclusions.
\end{enumerate}


	\end{minipage}
		\vspace*{2.cm}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Why flavour physics
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\begin{center}
\begin{Huge}
Why Flavour is important?
\end{Huge}
\end{center}




\end{frame}



\begin{frame}{A lesson from history - GIM mechanism}
	\begin{minipage}{\textwidth}

\begin{center}
\includegraphics[width=0.62\textwidth]{images/GIM2.png}
\end{center}
\begin{columns}
\column{0.7\textwidth}
\begin{itemize}
\begin{footnotesize}


\item Cabibbo angle was successful in explaining dozens of decay rates in the 1960s.
\item There was, however, one that was not observed by experiments: $\PKzero \to \Pmuon \APmuon$.
\item Glashow, Iliopoulos, Maiani (GIM) mechanism was proposed in the 1970 to fix this problem. The mechanism required the existence of a $4^{th}$ quark.
\item At that point most of the people were skeptical about that. Fortunately in 1974 the discovery of the $\PJpsi$ meson silenced the skeptics.
\end{footnotesize}
\end{itemize}
\column{0.3\textwidth}
\begin{center}
\includegraphics[width=0.95\textwidth]{images/GIM3.png}\\
\includegraphics[width=0.7\textwidth]{images/604.jpg}\\{~}\\{~}
\end{center}
\end{columns}



\end{minipage}

		\vspace*{2.1cm}
\end{frame}

\begin{frame}{A lesson from history - CKM matrix}
	\begin{minipage}{\textwidth}

\begin{center}
{~}\\{~}\\
\includegraphics[width=0.5\textwidth]{images/CKMmatrix.png}

\end{center}
\begin{columns}
\column{0.6\textwidth}
\begin{itemize}
\begin{small}




\item Similarly, CP violation was discovered in 1960s in the neutral kaons decays.
\item $2 \times 2$ Cabbibo matrix could not allow for any CP violation.
\item For CP violation to be possible one needs at least a $3 \times 3$ unitary matrix \\ $\looparrowright$ Cabibbo-Kobayashi-Maskawa matrix (1973).
\item It predicts existence of $\Pbottom$ (1977) and $\Ptop$ (1995) quarks.
\end{small}

\end{itemize}
\column{0.4\textwidth}
\begin{center}
{~}
%\includegraphics[height=2cm]{images/CP.png}\\
\includegraphics[width=0.96\textwidth]{bottom.jpg}

\end{center}
\end{columns}



\end{minipage}

		\vspace*{2.1cm}
\end{frame}


\begin{frame}{A lesson from history - Weak neutral current}
	\begin{minipage}{\textwidth}

\begin{center}
\includegraphics[height=3cm]{images/weakcurr.png}{~}
\includegraphics[height=3cm]{images/weakcurr2.png}
\end{center}

\begin{columns}
\column{0.6\textwidth}
\begin{itemize}
\begin{small}


\item Weak neutral currents were first introduced in 1958 by Buldman.
\item Later on they were naturally incorporated into unification of weak and electromagnetic interactions.
\item 't Hooft proved that the GWS models was renormalizable.
\item Everything was there on theory side, only missing piece was the experiment, till 1973.
\end{small}

\end{itemize}
\column{0.4\textwidth}
\begin{center}
{~}
%\includegraphics[height=2cm]{images/CP.png}\\
\includegraphics[width=0.85\textwidth]{images/bubblecern.png}
\end{center}
\end{columns}



\end{minipage}

		\vspace*{2.1cm}
\end{frame}




\iffalse



\begin{frame}
\begin{center}
\begin{Huge}
LHCb detector
\end{Huge}
\end{center}




\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% DETECTOR
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\only<1>{\frametitle{LHCb detector - tracking}
\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 ($\delta p/p \sim 0.4 - 0.6\%$) and inv. mass resolution.\\
$\Rightarrow$ Low combinatorial background.

\end{itemize}


}

\only<2>{\frametitle{LHCb detector - particle identification}
\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}{Modern challenges: loops come in to the game}
	\begin{minipage}{\textwidth}
\begin{columns}

\column{0.5\textwidth}
\begin{itemize}
\item Standard Model contributions suppressed or absent:
\begin{itemize}
\item Flavour Changing Neutral Currents.
\item CP violation
\item Lepton Flavour/Number or Lepton Universality violation.
\end{itemize}
\item In general can probe physics beyond General  Purpose Detectors reach.
\end{itemize}
\column{0.5\textwidth}
\includegraphics[width=0.99\textwidth]{{images/TauLFV_UL_2014001_averaged}.png}


\end{columns}
\begin{center}
\includegraphics[width=0.75\textwidth]{images/Bsmumu.png}
\includegraphics[width=0.20\textwidth]{{images/bsmumu_SM}.png}
\end{center}
\end{minipage}

		\vspace*{2.1cm}
\end{frame}
\fi





\begin{frame}\frametitle{Modern Flavour Physics}
\begin{center}
\only<1>{
\includegraphics[width=1.05\textwidth]{1.png}
}
\only<2>{
\includegraphics[width=1.05\textwidth]{2.png}

}
\end{center}




\end{frame}


\begin{frame}\frametitle{Why semi-leptonic decays?}
\begin{large}
\ARROW A decay is semi-leptonic if its products are part leptons
and part hadrons.
\end{large}

\begin{center}
\includegraphics[width=0.99\textwidth]{3.png}
\end{center}

\begin{large}
\ARROW These decays can be factorised into the weak and
strong parts, greatly simplifying theoretical calculations.
\end{large}


\end{frame}



\begin{frame}\frametitle{Types of semi-leptonic decays}

\begin{center}
\includegraphics[width=1.05\textwidth]{4.png}
\end{center}


\end{frame}


\begin{frame}\frametitle{Anomalies}
\only<1>{
\begin{center}
\includegraphics[width=1.05\textwidth]{6.png}
\end{center}
}
\only<2>{
\ARROW Today I will talk about three anomalies in $\PB$
decays:
\begin{itemize}
\item $R(\PDstar)$
\item $R_{\PK/\PKstar}$
\item $P^{\prime}_5$
\end{itemize}

}

\end{frame}



\begin{frame}

\begin{center}

\begin{Huge}
Anomaly 1\\
\begin{align*}
R(\PDstar) = \frac{\Br (\PB \to \PDstar \tau \nu )}{\Br(\PB \to \PDstar \mu \nu)} 
\end{align*}
\end{Huge}


\end{center}


\end{frame}

\begin{frame}\frametitle{$R(\PDstar) $}

\begin{large}

\ARROW Large rate of charged current decays allow for measurement
in semi-tauonic decays
\end{large}

\begin{columns}
\column{0.5\textwidth}
\begin{align*}
R(\PDstar) = \frac{\Br (\PB \to \PDstar \tau \nu )}{\Br(\PB \to \PDstar \mu \nu)} 
\end{align*}
\column{0.5\textwidth}
\begin{large}
\ARROW Form ratio of decays with different lepton generations.\\
\ARROW Cancel QCD uncertainties.
\end{large}

\end{columns}
~\\
\begin{large}

\ARROW $R(\PDstar)$ is sensitive to the NP with strong 3rd generation couplings.
\end{large}

\includegraphics[width=0.8\textwidth]{images/7.png}


\end{frame}

\begin{frame}\frametitle{The Rule of three}

\begin{center}

\includegraphics[width=1.\textwidth]{images/8.png}

\end{center}


\end{frame}


\begin{frame}\frametitle{Experimental challenges}
\begin{large}
\ARROW With the $\tau \to \mu \nu \nu$ decay we are missing 3 neutrinos!\\
\ARROW No sharp peak in any distributions.\\
\end{large}
{~}\\
\begin{columns}
\column{0.5\textwidth}
\begin{large}
\ARROW At B-factories, can control this
using tagging technique.\\
\end{large}


\includegraphics[width=0.9\textwidth]{9.png}

\column{0.5\textwidth}


\includegraphics[width=0.9\textwidth]{10.png}\\
\begin{large}
\ARROW More difficult at LHCb, compensate
using large boost (flight information)
and huge B production\\
\end{large}
\end{columns}

\end{frame}

\begin{frame}\frametitle{Signal fits}
\ARROW Three main backgrounds:
\begin{itemize}
\item $\PB \to \PDstar \ell \nu$.
\item $\PB \to \PD^{\ast \ast} \ell \nu$.
\item $\PB \to \PD \PDstar X$
\end{itemize}
\begin{center}
\includegraphics[width=0.9\textwidth]{12.png}
\end{center}
\ARROW Fit variables which discriminate between the signal and background modes.

\end{frame}

\begin{frame}\frametitle{Results}
\ARROW All experiments see an access w.r.t. to SM prediction:
\begin{center}
\includegraphics[angle=-90,width=0.5\textwidth]{rdrds_fpcp2017.pdf}
\end{center}
\ARROW Theoretical uncertainties negligible.\\
\ARROW The ball is on the experimental side.

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%





\begin{frame}\frametitle{Introduction to anomaly 2 \& 3}                                                                          

\begin{columns}                                                                                                     
\column{4in}                                                                                                        
\begin{itemize}                                                                                                     
\item The SM allows only the charged interactions to change flavour.                                                 
\begin{itemize}                                                                                                     
\item Other interactions are flavour conserving.                                                                    
\end{itemize}                                                                                                       
\item One can escape this constraint and produce $\Pbottom \to \Pstrange$ and $\Pbottom \to \Pdown$ at loop level.   
\begin{itemize}                                                                                                     
\item These 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]{lupa.png}                                                                          
\includegraphics[scale=0.3]{example.png}                                                                       
\end{center}                                                                                                        
\column{1.5in}                                                                                                      
\includegraphics[width=0.61\textwidth]{couplings.png}                                                          
\end{columns}                                                                                                       
                                                                                                                    
\end{frame}                                                                                                         


\begin{frame}{Analysis of Rare decays}
\begin{footnotesize}

%{\Large Since a long time ago...} \\ \medskip 
%\hspace*{1.4cm}$\Rightarrow$ $b \to s \gamma$ and $b \to s \ell\ell $ {\bf Flavour Changing Neutral Currents} have been used as {\bf \cred Our  Portal} \\ to explore the fundamental theory beyond SM. \\ 
%\medskip
%\medskip
%\hfill....... with not much success till 2013.\hspace*{1cm}
%\bigskip

Analysis of FCNC in a model-independent approach, effective Hamiltonian:
\vspace*{-0.1cm}
\begin{columns}
\begin{column}{1cm}
~
\end{column}
\begin{column}{8cm}
\begin{equation*}
b\to s\gamma(^*): {\mathcal H}^{SM}_{\Delta F=1} \propto
  \sum_{i=1}^{10} V_{ts}^* V_{tb} {\cgreen \C{i}} \alert{  {\cal O}_i} + \ldots
\end{equation*}

\vspace{-0.2cm}

\begin{itemize}
\item $\alert{ {\cal O}_7} = \frac{e}{16 \pi^2}m_b\, 
        (\bar s\sigma^{\mu\nu} P_R b) F_{\mu\nu}\,$ %\quad [real or soft photon]
\item $\alert{ {\cal O}_9}=\frac{e^2}{16 \pi^2}(\bar{s}\gamma_\mu P_L b)\  (\bar\ell\gamma_\mu\ell)$
 %\quad [$b\to s\mu\mu$ via $Z$/hard $\gamma$]
\item $\alert{ {\cal O}_{10}}=\frac{e^2}{16 \pi^2}(\bar{s}\gamma_\mu P_L b) \  (\bar\ell\gamma_\mu\gamma_5\ell)$, ...
%\quad [$b\to s\mu\mu$ via $Z$]
\end{itemize}
\end{column}
\begin{column}{5.5cm}
\includegraphics[width=3.5cm]{images/qum1.png} 
%\includegraphics[width=3cm]{bsll.pdf}
\end{column}
\end{columns}

%\hspace*{5cm} with no clear success yet...
%\bigskip


%\centerline{{\bf Goal}: \underline{Decode the short distance physics to find a smoking gun of BSM}\hspace*{2cm}}


\bigskip
\hspace*{0.0cm} $\bullet$ {\bf SM} Wilson coefficients up to NNLO + e.m. corrections at $\mu_{ref}=4.8$ GeV [{\cgreen Misiak et al.}]: $${\cal C}_7^{\rm SM}=-0.29,\, {\cal C}_9^{\rm SM}=4.1,\, {\cal C}_{10}^{\rm SM}=-4.3$$
%BUT, like in the film there is always the good, the bad and the ugly.
\bigskip
$\bullet$ {\bf NP} changes short distance ${\cal C}_i-{\cal C}_i^{\rm SM}={\cal C}_i^{\rm NP}$ and induce new operators, like ${\cal O}^\prime_{7,9,10}={\cal O}_{7,9,10}\,\, (P_L \leftrightarrow P_R)$ ... also scalars, pseudoescalar, tensor operators...%\bigskip


\end{footnotesize}

\end{frame}


\begin{frame}

\begin{center}

\begin{Huge}
Anomaly 2\\
\begin{align*}
R_{\PK/\PKstar} = \frac{\Br(\PB \to \PK/\PKstar \mu \mu)}{\Br(\PB \to \PK/\PKstar e e)}
\end{align*}
\end{Huge}


\end{center}


\end{frame}



\begin{frame}\frametitle{Measurement at LHCb}

\ARROW Most precise measurements performed at LHCb.\\
\ARROW Main challenge is due to electron Bremsstrahlung.\\
\begin{center}
\includegraphics[width=0.99\textwidth]{13.png}
\end{center}
\ARROW To protect ourself from electron reconstruction issue we use double ratio:
\begin{align*}
R_K  = \frac{ \Br(\PB \to \PK \mu \mu ) \times \Br(\PB \to \PK \PJpsi(\to e e))   }{ \Br(\PB \to \PK e e ) \times \Br(\PB \to \PK \PJpsi(\to \mu \mu)) }
\end{align*}


\end{frame}


\begin{frame}\frametitle{Result}

\begin{center}
\begin{large}
\begin{align*}
R_K = 0.745^{+0.090}_{-0.074} ({\rm stat.}) \pm  0.036 ({\rm syst})
\end{align*}
\end{large}

\includegraphics[width=0.7\textwidth]{images/RK.png}\\

\end{center}
\ARROW $2.6~\sigma$ away from SM prediction.
\end{frame}

\begin{frame}\frametitle{The continuation - $R_{\PKstar}$}
\ARROW The neutral continuation of the $R_K$ measurement is to measure its partner:
\begin{center}
\begin{align*}
R_{\PKstar} = \frac{\Br(\PB \to \PKstar \mu \mu)}{\Br(\PB \to \PKstar e e)}
\end{align*}
\end{center}


\begin{columns}                                                                                                                                                                                                                                
\column{0.4\textwidth}                                                                                                                                                                                                                         
\ARROW Measurement performed in two $q^2$ bins. \\                                                                                                                                                                                             
\ARROW Normalized in double ratio to $\PB \to \PKstar \PJpsi$.\\                                                                                                                                                                               
\includegraphics[width=0.95\textwidth]{images/plot.png}                                                                                                                                                                                        
                                                                                                                                                                                                                                               
                                                                                                                                                                                                                                               
\column{0.6\textwidth}                                                                                                                                                                                                                         
\begin{center}                                                                                                                                                                                                                                 
\includegraphics[width=0.95\textwidth]{Fig10a.pdf}
\end{center}                                                                                                                                                                                                                                   
                                                                                                                                                                                                                                               
\end{columns}                                                                                                                                                                                                                                  
                    


\end{frame}




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Branching fraction measurements of $\PB \to \PKstar^{\pm} \Pmu \Pmu$}
{~}
\includegraphics[width=0.5\textwidth]{images/ksmumu_BF.png}
\includegraphics[width=0.5\textwidth]{images/kmumu_BF.png}

\begin{center}
\begin{columns}

\column{0.4\textwidth}
\begin{itemize}
\item Despite large theoretical errors the results are consistently smaller than SM prediction.
\end{itemize}
\column{0.6\textwidth}
\includegraphics[width=0.87\textwidth]{images/bukst_BF.png}


\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[width=0.65\textwidth]{images/bs2phipi.png}\\
	\end{center}

\begin{itemize}
\item Recent LHCb measurement [JHEPP09 (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.
\end{itemize}


\end{minipage}
		\vspace*{2.1cm}
\end{frame}







\begin{frame}

\begin{center}

\begin{Huge}
Anomaly 3\\
\begin{align*}
P_5^{\prime} = \sqrt{2} \frac{\Re (\apeL \apaL^{\ast} - \apeR \apaR^{\ast} )}{\sqrt{ |\aze|^2 ( |\ape|^2 + |\aze|^2 )}}
\end{align*}
\end{Huge}


\end{center}


\end{frame}


%\azeLR
%\apaLR
%\apeLR

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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 $\PKstar$ ($\overline{\PKstar}$) rest frame and the direction of the $\PKstar$ ($\overline{\PKstar}$) 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=0.95\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[
J_{1s} \sin^2\theta_K + J_{1c} \cos^2\theta_K + (J_{2s} \sin^2\theta_K + J_{2c} \cos^2\theta_K) \cos 2\theta_l\nn\\[1.5mm]
&&\hspace{-2.7cm}+ J_3 \sin^2\theta_K \sin^2\theta_l \cos 2\phi + J_4 \sin 2\theta_K \sin 2\theta_l \cos\phi  + J_5 \sin 2\theta_K \sin\theta_l \cos\phi \nn\\[1.5mm]
&&\hspace{-2.7cm}+ (J_{6s} \sin^2\theta_K +  {J_{6c} \cos^2\theta_K})  \cos\theta_l
+ J_7 \sin 2\theta_K \sin\theta_l \sin\phi  + J_8 \sin 2\theta_K \sin 2\theta_l \sin\phi \nn\\[1.5mm]
&&\hspace{-2.7cm}+ 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.

}

\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:
{\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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\begin{center}
\begin{Huge}
LHCb measurement of $\PBd \to \PKstar \Pmu \Pmu$
\end{Huge}
\end{center}




\end{frame}
\iffalse

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{LHCbs $\PBzero \to \PKstar \Pmuon \APmuon$, Selection}
{~}
	\begin{minipage}{\textwidth}
\begin{footnotesize}
\begin{columns}
\column{0.2in}
{~}
\column{2in}
\ARROW Trigger
\begin{itemize}
\item Muon trigger.
\item Topological trigger.
\end{itemize}
\ARROW Good modelling with MC. \\
\ARROW Selection:
\begin{itemize}
\item As loose as possible.
\item Based on the $\PBzero$ vertex quality, impact parameters, loose Particle identification for the hadrons.
\item The variables were chosen in a way we are sure the are correctly modelled in MC.
\end{itemize}
\column{2.8in}

\includegraphics[angle=-90,width=0.75\textwidth]{{images/tistos_L0Muon_pt}.pdf}\\
\includegraphics[angle=-90,width=0.75\textwidth]{{images/tistos_L0Muon_costhetal}.pdf}
\end{columns}



\end{footnotesize}



\end{minipage}
		\vspace*{2.1cm}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Peaking backgrounds}
{~}
	\begin{minipage}{\textwidth}
\begin{footnotesize}
\ARROW A number of peaking backgrounds that can mistaken as your signal.\\
\ARROW There where a specially designed vetoes to fight each of them.
\begin{center}
\begin{tiny}
\hspace{-1cm}\begin{tabular}{ r | c c | c c }                                                                       
  \hline                                                                                                                
  & \multicolumn{2}{c|}{after preselection, before vetoes} & \multicolumn{2}{c }{after vetoes and selection}\\        
  Channel          & Estimated events & \% signal   & Estimated events & \% signal \\                     
  \hline                                                                                                                
  \hline                                                                                                                
  $\PLambda_b \to \PLambda^{\ast}(1520)^{0} \mu\mu$      &$ (1.0\pm0.5)\times10^3 $&$ 19\pm8 $&$ 51\pm25 $&$ 1.0\pm0.4$\\                    
   $\PLambda_b \to {\rm p } \PK \mu\mu$        &$ (1.0\pm0.5)\times10^2 $&$ 1.9\pm0.8 $&$ 5.7\pm2.8 $&$ 0.11\pm0.05$ \\           
 $\PBd \to \PKplus \mu \mu$        &$ 28\pm7 $&$ 0.55\pm0.06 $&$ 1.6\pm0.5 $&$ 0.031\pm0.006$\\                      
$\PBs \to \Pphi \mu \mu$      &$ (3.2\pm1.3)\times10^2 $&$ 6.2\pm2.1 $&$ 17\pm7 $&$ 0.33\pm0.12$\\           
signal swaps    &$ (3.6\pm0.9)\times10^2 $&$ 6.9\pm0.6 $&$ 33\pm9 $&$ 0.64\pm0.06$ \\         
$\PBd \to \PKstar \PJpsi$ swaps &$ (1.3\pm0.4)\times10^2 $&$ 2.6\pm0.4 $&$ 2.7\pm2.8 $&$ 0.05\pm0.05$ \\  
             
   \hline                                                                                                               
 \end{tabular}                                                                                                          
\end{tiny}
\includegraphics[angle=-90,width=0.49\textwidth]{{images/h_Bd_Kstmm_vetoes}.pdf}               
\includegraphics[angle=-90,width=0.49\textwidth]{{h_Lb_L1520mm_vetoes}.pdf}        

\end{center}




\end{footnotesize}



\end{minipage}
		\vspace*{2.1cm}
\end{frame}
\fi


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Multivariate simulation}
{~}
	\begin{minipage}{\textwidth}
\begin{columns}

\column{0.5\textwidth}
\begin{itemize}
\begin{footnotesize}
\item PID, kinematics and isolation variables used in a Boosted Decision Tree (BDT) to discriminate signal and background.
\item BDT with k-Folding technique.
\item Completely data driven. 
\end{footnotesize}
\end{itemize}
\begin{center}
\includegraphics[width=0.70\textwidth]{images/Chopping_Distrib.pdf}
\end{center}

\column{0.5\textwidth}

\includegraphics[angle=-90,width=0.82\textwidth]{images/Fig1.pdf} \\
\includegraphics[width=0.88\textwidth]{images/fold.png}

\end{columns}



\end{minipage}
		\vspace*{2.1cm}
\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Multivariate simulation, efficiency}
{~}

	\begin{minipage}{\textwidth}
	\begin{footnotesize}

\ARROW BDT was also checked in order not to bias our angular distribution:
\begin{center}
\includegraphics[angle=-90,width=0.8\textwidth]{images/BDT_Eff_Comp.pdf}
\end{center}
\ARROW The BDT has small impact on our angular observables. We will correct for these effects later on.

\end{footnotesize}
\end{minipage}
		\vspace*{2.1cm}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\iffalse

\begin{frame}{Mass modelling}
{~}
\begin{minipage}{\textwidth}
\begin{tiny}
\begin{columns}
\column{0.1in}
{~}
\column{2.5in}
\ARROW The signal is modelled by a sum of two Crystal-Ball functions with common mean.\\
\ARROW The background is a single exponential.\\
\ARROW The base parameters are obtained from the proxy channel: $\PBd \to \PJpsi (\mu\mu) \PKstar$.\\
\ARROW All the parameters are fixed in the signal pdf.\\
\ARROW Scaling factors for resolution are determined from MC.\\
\ARROW In fitting the rare mode only the signal, background yield and the slope of the exponential is left floating.\\
\begin{center}
\includegraphics[angle=-90,width=0.9\textwidth]{images/msignal.pdf}\\

\end{center}
\ARROW We found $624\pm30$ candidates in the most interesting $\left[1.1,6.0\right]~\GeV^2/c^4$ region \\ and $2398 \pm 57$ in the full range $\left[ 1.1, 19.\right]~\GeV^2/c^4$. 
\column{2.5in}
\includegraphics[angle=-90,width=0.95\textwidth]{{images/FitJpsiKstar_withBDT_withoutPartially}.pdf}\\
\includegraphics[angle=-90,width=0.95\textwidth]{{images/Scaling_factor}.pdf}\\
\ARROW The S-wave fraction is extracted using a \texttt{LASS} model.
\end{columns}

\end{tiny}
\end{minipage}
		\vspace*{2.1cm}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Monte Carlo corrections}
{~}
	\begin{minipage}{\textwidth}
\begin{footnotesize}
\ARROW No Monte Carlo simulation is perfect! One needs to correct for remaining differences.\\
\ARROW We reweighted our $\PBd \to \PKstar \mu \mu$ Monte Carlo accordingly to differences between the $\PBd \to \PKstar \PJpsi$ in data (Splot) and Monte Carlo.
\only<1>{
\begin{center}
\includegraphics[angle=-90,width=0.38\textwidth]{images/pt.pdf} 
\includegraphics[angle=-90,width=0.38\textwidth]{images/vertex.pdf}\\
\includegraphics[angle=-90,width=0.38\textwidth]{images/nTracks.pdf} 
\end{center}
}
\only<2>{
\begin{center}
\includegraphics[angle=-90,width=0.38\textwidth]{images/eta_logy.pdf}
\includegraphics[angle=-90,width=0.38\textwidth]{images/B0_p.pdf} \\
\includegraphics[angle=-90,width=0.38\textwidth]{{images/bdt_data_mc_nominalMkpi}.pdf}
\end{center}
}

\end{footnotesize}
\end{minipage}
		\vspace*{2.1cm}
\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\fi



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Detector acceptance}
{~}
	\begin{minipage}{\textwidth}
\begin{footnotesize}

\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} 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 The coefficients were determined using Method of Moments, with a huge simulation sample.
\item The simulation was done assuming a flat phase space and reweighing the $q^2$ distribution to make is flat.
\item To make this work the $q^2$ distribution needs to be reweighted to be flat.
\end{itemize}
%\includegraphics[width=0.75\textwidth]{images/q2PHSP.png}



\column{0.4\textwidth}
\only<1>{

\includegraphics[width=0.99\textwidth]{images/det.png}
}
\end{columns}


\end{footnotesize}
\end{minipage}
		\vspace*{2.1cm}
\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Control channel}
{~}
	\begin{minipage}{\textwidth}
\begin{footnotesize}
\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}
\end{footnotesize}
\begin{center}
\includegraphics[angle=-90,width=0.4\textwidth]{images/mlogjpsi.pdf}
\includegraphics[angle=-90,width=0.4\textwidth]{images/mkpijpsi.pdf}\\
\includegraphics[width=0.99\textwidth]{images/angles3.png}
\end{center}


\end{minipage}
		\vspace*{2.1cm}
\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{The columns of New Physics}
{~}
	\begin{minipage}{\textwidth}
	\begin{center}
	\includegraphics[width=0.94\textwidth]{images/columns.png}

	\end{center}


\end{minipage}
		\vspace*{2.1cm}
\end{frame}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{The columns of New Physics}
{~}
	\begin{minipage}{\textwidth}
\begin{enumerate}
\item Maximum likelihood fit:
\begin{itemize}
\item The most standard way of obtaining the parameters.
\item Suffers from convergence problems, under coverages, etc. in low statistics.
\end{itemize} 
\item Method of moments:
\begin{itemize}
\item Less precise then the likelihood estimator ($10-15\%$ larger uncertainties).
\item Does not suffer from the problems of likelihood fit.
\end{itemize}
\item Amplitude fit:
\begin{itemize}
\item Incorporates all the physical symmetries inside the amplitudes! The most precise estimator.
\item Has theoretical assumptions inside!
\end{itemize}
\end{enumerate}

\end{minipage}
		\vspace*{2.1cm}
\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Maximum likelihood fit - Results}
{~}
\begin{footnotesize}

	\begin{minipage}{\textwidth}
\ARROW In the maximum likelihood fit one could weight the events accordingly to the $\dfrac{1}{\varepsilon(\cos \thetal, \cos \thetak, \phi, q^2)}$\\
\ARROW Better alternative is to put the efficiency into the maximum likelihood fit itself:
\begin{align*}
\mathcal{L}=\prod_{i=1}^N \epsilon_i(\Omega_i, q_i^2) \mathcal{P}(\Omega_i, q_i^2) / \int \epsilon(\Omega, q^2) \mathcal{P}(\Omega, q^2) d\Omega dq^2
\end{align*}
\ARROW Only the relative weights matters!\\
\ARROW The Procedure was commissioned with TOY MC study.\\
\ARROW Use Feldmann-Cousins to determine the uncertainties. \\
\ARROW Angular background component is modelled with $2^{\rm nd }$ order Chebyshev polynomials, which was tested on the side-bands.\\
\ARROW S-wave component treated as nuisance parameter.\\
\includegraphics[angle=-90,width=0.33\textwidth]{{images/FC_Afb3}.pdf}
\includegraphics[angle=-90,width=0.33\textwidth]{{images/FC_P11}.pdf}
\includegraphics[angle=-90,width=0.33\textwidth]{{images/FC_P57}.pdf}

\end{minipage}
\end{footnotesize}
		\vspace*{2.1cm}
\end{frame}
\iffalse
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Maximum likelihood fit - Results}
{~}
	\begin{minipage}{\textwidth}
\begin{center}
\only<1>{
\includegraphics[angle=-90,width=0.49\textwidth]{images/FLPad.pdf}
\includegraphics[angle=-90,width=0.49\textwidth]{images/S3Pad.pdf}\\
\includegraphics[angle=-90,width=0.49\textwidth]{images/S4Pad.pdf}
\includegraphics[angle=-90,width=0.49\textwidth]{images/S5Pad.pdf}
}
\only<2>{
\includegraphics[angle=-90,width=0.49\textwidth]{images/AFBPad.pdf}
\includegraphics[angle=-90,width=0.49\textwidth]{images/S7Pad.pdf}\\
\includegraphics[angle=-90,width=0.49\textwidth]{images/S8Pad.pdf}
\includegraphics[angle=-90,width=0.49\textwidth]{images/S9Pad.pdf}
}


\end{center}	
	

\end{minipage}
		\vspace*{2.1cm}
\end{frame}





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Maximum likelihood fit - Results}
	\begin{minipage}{\textwidth}
	\begin{center}
	\includegraphics[angle=-90,width=0.65\textwidth]{images/P5pPadOverlay.pdf}\\
	\end{center}

\begin{itemize}
\item Tension with $3~\invfb$ gets confirmed!
\item two bins both deviate by $2.8~\sigma$ from SM prediction.
\item Result compatible with previous result.
\end{itemize}


\end{minipage}
		\vspace*{2.1cm}
\end{frame}




\fi




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Method of moments}
{~}
\begin{footnotesize}
	\begin{minipage}{\textwidth}
\ARROW See {\color{blue}{\href{http://arxiv.org/abs/1503.04100}{Phys.Rev.D91(2015)114012}}}, F.Beaujean , M.Chrzaszcz, N.Serra, D. van Dyk for details.\\
\ARROW The idea behind Method of Moments is simple: Use orthogonality of spherical harmonics, $f_j(\overrightarrow{\Omega})$ to solve for coefficients within a $q^2$ bin:
\begin{align*}
\int f_i(\overrightarrow{\Omega})  f_j(\overrightarrow{\Omega}) = \delta_{ij}
\end{align*}
\begin{align*}
M_i = \int \left( \dfrac{1}{d(\Gamma+ \bar{\Gamma})/dq^2} \right) \dfrac{d^3(\Gamma+\bar{\Gamma})}{d \overrightarrow{\Omega}} f_i(\overrightarrow{\Omega})d \Omega
\end{align*}
\ARROW Don’t have true angular distribution but we ''sample'' it with our data.\\
\ARROW Therefore: $\int \to \sum$ and $M_i \to \widehat{M}_i$
\begin{align*}
\hat{M}_i=\dfrac{1}{\sum_e \omega_e} \sum_e \omega_e f_i(\overrightarrow{\Omega}_e)
\end{align*}
\ARROW The weight $\omega$ accounts for the efficiency. Again the normalization of weights does not matter.


\end{minipage}
\end{footnotesize}
		\vspace*{2.1cm}
\end{frame}



\begin{frame}{Method of moments - results}
{~}
\begin{footnotesize}
	\begin{minipage}{\textwidth}
	
\only<3>
{
\ARROW Method of Moments allowed us to measure for the first time a new observable:
}	
	
\begin{center}
\only<1>{
\includegraphics[angle=-90,width=0.49\textwidth]{images/compare_FLPad.pdf}
\includegraphics[angle=-90,width=0.49\textwidth]{images/compare_S3Pad.pdf}\\
\includegraphics[angle=-90,width=0.49\textwidth]{images/compare_S4Pad.pdf}
\includegraphics[angle=-90,width=0.49\textwidth]{images/compare_S5Pad.pdf}
}
\only<2>{
\includegraphics[angle=-90,width=0.49\textwidth]{images/compare_AFBPad.pdf}
\includegraphics[angle=-90,width=0.49\textwidth]{images/compare_S7Pad.pdf}\\
\includegraphics[angle=-90,width=0.49\textwidth]{images/compare_S8Pad.pdf}
\includegraphics[angle=-90,width=0.49\textwidth]{images/compare_S9Pad.pdf}
}
\only<3>{
\includegraphics[angle=-90,width=0.75\textwidth]{images/S6cPad.pdf}
}

\end{center}	
	
\end{minipage}
\end{footnotesize}
		\vspace*{2.1cm}
\end{frame}







\begin{frame}{Compatibility with SM}
{~}

	\begin{minipage}{\textwidth}

\begin{columns}
\column{0.1in}
{~}
\column{2in}
\ARROW Use \texttt{EOS} software package to test compatibility with SM.\\
\ARROW Perform the $\chi^2$ fit to the measured:
\begin{center}
\begin{align*}
F_L, A_{FB}, S_{3,..., 9} .
\end{align*}
\end{center}
\ARROW Float a vector coupling: $\Re(C_9)$.\\
\ARROW Best fit is found to be $3.4~\sigma$ away from the SM.


\column{3in}
\begin{align*}
\Delta \Re (C_9) \equiv \Re(C_9)^{{\rm fit}} - \Re(C_9)^{{\rm SM}} = -1.03
\end{align*}
\includegraphics[angle=-90,width=0.95\textwidth]{images/wilsonchi2.pdf}
\end{columns}



\end{minipage}

		\vspace*{2.1cm}
\end{frame}



\begin{frame}\frametitle{Global picture of $P_5^{\prime}$}

\begin{columns}

\column{0.4\textwidth}
\only<1>{{\color{gray}{\ARROW 2013 LHCb: \href{https://arxiv.org/pdf/1308.1707.pdf}{arXiv::1308.1707}}}\\}
\only<1>{\ARROW 2015 LHCb: \href{https://arxiv.org/abs/1512.04442}{arXiv::1512.0444}\\}
\only<1>{{\color{red}{\ARROW 2016 Belle: \href{https://arxiv.org/abs/1604.04042}{arXiv::1604.04042}}}\\}
\only<1>{\ARROW 2017: {\color{blue}{\href{https://atlas.web.cern.ch/Atlas/GROUPS/PHYSICS/CONFNOTES/ATLAS-CONF-2017-023/}{ATLAS-CONF-2017-023}}} $(20.5~\rm fb^{-1})$ and {\color{OliveGreen}{\href{http://cds.cern.ch/record/2256738?ln=en}{CMS-PAS-BPH-15-008}}} $(20.8~\rm fb^{-1})$} 



\column{0.6\textwidth}
\only<1>{
\ARROW Theory: 
~~DHMV: \href{https://arxiv.org/abs/1407.8526}{arXiv::1407.8526}
~~ASZB: \href{https://arxiv.org/abs/1411.3161}{arXiv::1411.3161}
}
%\includegraphics[width=0.95\textwidth]{images/P5p1.png}
\only<1>{
\includegraphics[angle=-90,width=0.9\textwidth]{images/P5p.pdf}
}


\end{columns}
\only<1>{


}

\end{frame}



\iffalse

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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 [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 $\PLambdab \to \PLambda \Pmu \Pmu$}
{~}
	\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}
\begin{itemize}
\item If $\PZprime$ is responsible for the $P'_5$ anomaly, does it couple equally to all flavours?
\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}{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 $0.0004 < q^2 <1~\GeV^2$.
	\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{equation}
  \label{eq:physPars}
  \begin{split}
  \FL &=\frac{|A_0|^2}{|A_0|^2+|A_{||}|^2 + |A_\perp|^2}\\
  \ATD &= \frac{|A_\perp|^2-|A_{||}|^2}{|A_\perp|^2+|A_{||}|^2}\\
  \ATRe &= \frac{2\Real(A_{||L}A^*_{\perp L} + A_{||R}A^*_{\perp R})}{|A_{||}|^2 + |A_\perp|^2}\\
  \ATIm &= \frac{2\Imag(A_{||L}A^*_{\perp L} + A_{||R}A^*_{\perp R})}{|A_{||}|^2 + |A_\perp|^2},
\end{split}\nonumber
\end{equation}

\end{minipage}
}
\only<2>{
\begin{center}
\includegraphics[width=0.5\textwidth]{images/Kstee.png}\\
\end{center}
\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}

}
		\vspace*{2.1cm}
\end{frame}

\fi

\begin{frame}
\begin{center}
\begin{Huge}
Global fit to $\Pbeauty \to \Pstrange \ell \ell$ measurements
\end{Huge}
\end{center}




\end{frame}



\begin{frame}{Link the observables}
\begin{footnotesize}

\ARROW Fits prepare by \texttt{S. Descotes-Genon, L. Hofer, J. Matias, J. Virto}, \href{http://arxiv.org/abs/1510.04239}{\color{blue}{arXiv::1510.04239}}

\begin{itemize}

\item Inclusive

\begin{itemize}
\item $B\to X_s\gamma$ {\color{gray}($BR$)
.......................................................... } {\color{red} $\C7^{(\prime)}$}%, {\color{brown} $\C{\rm had}$}
\item $B\to X_s\ell^+\ell^-$ {\color{gray}($dBR/dq^2$)
............................................ } {\color{red} $\C7^{(\prime)}$, $\C9^{(\prime)}$, $\C{10}^{(\prime)}$}%, {\color{brown} $\C{\rm had}$}
\end{itemize}

\item Exclusive leptonic

\begin{itemize}
\item $B_s\to \ell^+\ell^-$ {\color{gray}($BR$)
........................................................ } {\color{red} $\C{10}^{(\prime)}$}
\end{itemize}

\item Exclusive radiative/semileptonic
\begin{itemize}
\item $B\to K^*\gamma$ {\color{gray}($BR$, $S$, $A_I$)
................................................ } {\color{red} $\C7^{(\prime)}$}%, {\color{brown} $\C{\rm had}$}

\item $B\to K\ell^+\ell^-$ {\color{gray}($dBR/dq^2$)
.............................................. } {\color{red} $\C7^{(\prime)}$, $\C9^{(\prime)}$, $\C{10}^{(\prime)}$}%, {\color{brown} $\C{\rm had}$}

\item $\bf \color{Red} B\to K^*\ell^+\ell^-$ {\color{gray}($dBR/dq^2$, {\bf Optimized Angular Obs.})
.. } {\color{red} $\C7^{(\prime)}$, $\C9^{(\prime)}$, $\C{10}^{(\prime)}$}%, {\color{brown} $\C{\rm had}$}

\item $B_s\to \phi \ell^+\ell^-$ {\color{gray}($dBR/dq^2$, Angular Observables)
.............. } {\color{red} $\C7^{(\prime)}$, $\C9^{(\prime)}$, $\C{10}^{(\prime)}$}%, {\color{brown} $\C{\rm had}$}

\item $\Lambda_b\to \Lambda\ell^+\ell^-$ {\color{gray}(None so far)}
\item etc.
\end{itemize}



\end{itemize}
\end{footnotesize}

\end{frame}

\frame{ \frametitle{Statistic details}

\begin{footnotesize}

\ARROW Frequentist approach:
\medskip

$$\chi^2(C_i) = [O_\text{exp}- O_\text{th}(C_i)]_j \, [Cov^{-1}]_{jk}\, [O_\text{exp}- O_\text{th}(C_i)]_k$$


\begin{itemize}
\item $\bf Cov = Cov^\text{exp} + Cov^\text{th}$. We have $Cov^\text{exp}$ for the first time 
\item Calculate $Cov^\text{th}$: correlated multigaussian scan over all nuisance parameters
\item $Cov^\text{th}$ depends on $C_i$: Must check this dependence\\[5mm]
\end{itemize}
For the Fit:
\begin{itemize}
\item Minimise $\chi^2 \to \chi^2_\text{min} = \chi^2(C_i^0)\quad$ (Best Fit Point = $C_i^0$)  
\item Confidence level regions: $\chi^2(C_i) - \chi^2_\text{min} < \Delta\chi_{\sigma,n}$
%\item Compute pulls by inversion of the above formula
\end{itemize}
\medskip
\ARROW The results from 1D scans:{~}\\{~}\\
\begin{tiny}
\begin{tabular}{crccc}
%\toprule[1.6pt]
Coefficient ${\cal C}_i^{NP}={\cal C}_i-{\cal C}_i^{SM}$ & Best fit & 1$\sigma$ & 3$\sigma$ & Pull$_{\rm SM}$ \\   \hspace{10mm} \\[5mm] 
% \midrule 
  $\bf\cred\C9^{\rm NP}$ & $ -1.09 $ & $ [-1.29,-0.87] $ & $ [-1.67,-0.39] $ &  $\,\,\,\,\,\,\bf 4.5 
 \cred \Leftarrow$  \hspace{5mm}  \\[3mm] 
  $\C9^{\rm NP}=-\C{10}^{\rm NP}$ & $ -0.68 $ & $ [-0.85,-0.50] $ & $ [-1.22,-0.18] $ & \bf \quad  4.2 
   $\cred\Leftarrow$  \hspace{5mm}  \\[3mm] 
   $\C9^{\rm NP}=-\C{9'}^{\rm NP}$ & $ -1.06 $ & $ [-1.25,-0.86] $ & $ [-1.60,-0.40] $ &  \quad \quad \quad \,\,\quad 4.8
     $\cred\Leftarrow$ (no $R_K$)\hspace{5mm}  \\[3mm] 

 \hspace{5mm}  \\[3mm] 
%  \bottomrule[1.6pt] 
\end{tabular}
\end{tiny}
\end{footnotesize}

}





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Theory implications}
{~}
	\begin{minipage}{\textwidth}

\begin{itemize}
\item The data can be explained by modifying the $C_9$ Wilson coefficient.
\item Overall there is around $4.5~\sigma$ discrepancy wrt. SM.
\end{itemize}
\includegraphics[width=0.9\textwidth]{images/C9.png}




\end{minipage}
		\vspace*{2.1cm}
\end{frame}



\begin{frame}{2D scans}
{~}
\begin{footnotesize}

	\begin{minipage}{\textwidth}
\begin{columns}
\begin{column}{0.5cm}


\end{column}
\begin{column}{17cm}

\renewcommand{\arraystretch}{1.4}
 \setlength{\tabcolsep}{13pt}
\begin{tabular}{cccr}
\hline
Coefficient & Best Fit Point & Pull$_{\rm SM}$  \\  \hline

  $(\C7^{\rm NP},\C9^{\rm NP})$ &  $(-0.00,-1.07)$  &  {\bf 4.1}  \hspace{5mm} \\ 
 $(\C9^{\rm NP},\C{10}^{\rm NP})$ &  $(-1.08,0.33)$  &  {\bf 4.3}  \hspace{5mm} \\ 
 $(\C9^{\rm NP},\C{7'}^{\rm NP})$ &  $(-1.09,0.02)$  &  {\bf 4.2}  \hspace{5mm} \\ 
 $(\C9^{\rm NP},\C{9'}^{\rm NP})$ &  $(-1.12,0.77)$  &  {\bf 4.5}  \hspace{5mm} \\ 
 $(\C9^{\rm NP},\C{10'}^{\rm NP})$ &  $(-1.17,-0.35)$  & {\bf  4.5}  \hspace{5mm} \\ 
 $(\C{9}^{\rm NP}=-\C{9'}^{\rm NP},\C{10}^{\rm NP}=\C{10'}^{\rm NP})$ &  $(-1.15,0.34)$  & \!\!\!\!\!\!\!\!\!\!\! {\bf 4.7} \\ 
 $(\C{9}^{\rm NP}=-\C{9'}^{\rm NP},\C{10}^{\rm NP}=-\C{10'}^{\rm NP})$ &  $(-1.06,0.06)$  &  {\bf 4.4}  \hspace{5mm} \\ 
 $(\C{9}^{\rm NP}=\C{9'}^{\rm NP},\C{10}^{\rm NP}=\C{10'}^{\rm NP})$ &  $(-0.64,-0.21)$  &  3.9  \hspace{5mm} \\ 
 $(\C{9}^{\rm NP}=-\C{10}^{\rm NP},\C{9'}^{\rm NP}=\C{10'}^{\rm NP})$ &  $(-0.72,0.29)$  &  3.8  \hspace{5mm} \\ 
% $(\C{9}^{\rm NP}=-\C{10}^{\rm NP},\C{9'}^{\rm NP}=-\C{10'}^{\rm NP})$ &  $(-0.66,0.03)$  &  2.0 & 23.0  
 %$(\C{9}^{\rm NP}=-\C{10}^{\rm NP},\C{9'}^{\rm NP}=-\C{10'}^{\rm NP})$ &  $(-0.69,0.05)$  &  1.9 & 22.0 \hspace{5mm} \\ 

\end{tabular}
\end{column}
\end{columns}


\medskip
\begin{itemize}
\item $C_9^{NP}$ always play a dominant role
\item All 2D scenarios above 4$\sigma$ are quite indistinguishable. 

\end{itemize}



\end{minipage}

\end{footnotesize}
		\vspace*{2.1cm}
\end{frame}

%%%%%%%%%%%%%%%%%



\begin{frame}{$B\to K^\ast \ell\ell\ $ Amplitudes}

\small

\mbox{
\includegraphics[width=3cm,height=2cm]{bsg0.jpg}\hspace{5mm}
\includegraphics[width=3cm,height=2cm]{bsg1.jpg}\hspace{5mm}
\includegraphics[width=3cm,height=2.4cm]{bsg2.jpg}
}

\vspace{3mm}

\mbox{
\hspace{-10mm}
\colorbox{llgray}{
\hspace{1mm}
$\displaystyle
A_\lambda^{L,R} = N_\lambda\ \bigg\{
(C_9 \mp C_{10}) {\blue \F_\lambda(q^2)}
+\frac{2m_b M_B}{q^2} \bigg[ C_7 {\blue \F_\lambda^{T}(q^2)}
- 16\pi^2 \frac{M_B}{m_b} {\red \H_\lambda(q^2)} \bigg]
\bigg\}
$
\hspace{2mm}
}
}

\vspace{5mm}
{\small 

\hspace{-8mm} \btr {\brown Local (Form Factors) :} \hspace{2mm} {\blue $ \F_\lambda^{(T)}(q^2) = \av{\bar M_\lambda(k)| \,\bar s\, \Gamma_\lambda^{(T)}\, b\, | \bar{ B}(k+q)}$}
\\[5mm]

\mbox{
\hspace{-9mm} 
\btr {\brown Non-Local :} \hspace{0mm} {\red $\displaystyle  \H_\lambda(q^2) = i \,{\cal P}_\mu^\lambda \int d^4 x\ e^{i q\cdot x}\,
\av{\bar{M}_\lambda(k)|
T\big\{ {\cal J}_{\rm em}^\mu(x), \C{i} \, \mathcal{O}(0) \big\} | \bar{B}(q+k)}$}
}

\vspace{3mm}

\hspace{-8mm} 
\btr CKM structure : \hspace{2mm} $\displaystyle \H_\lambda = {\color{gray}- \frac{\lambda_u}{\lambda_t} \H_\lambda^{(u)}} - \frac{\lambda_c}{\lambda_t} \H_\lambda^{(c)}$ \hspace{5mm} $\Rightarrow\ \mathcal{O} \sim (\bar{ c} b)(\bar{ s} c)$

}

\end{frame}





\begin{frame}{Analytic structure of $\H_\lambda(q^2)$}

\Cite{Bobeth, Chrzaszcz, van Dyk, Virto 1707.07305}
\vspace{3mm}


Neglecting OZI- and CKM-suppressed contributions :

\begin{center}
\includegraphics[width=7.5cm]{Analyticq2.png}
\end{center}


$\displaystyle { \hat{\mathcal{H}}_\lambda(q^2)} = (q^2 - M_{J/\psi}^2)(q^2 -M_{\psi(2S)}^2) \,{ {\mathcal{H}}_\lambda(q^2)} \quad $ has no poles.

\end{frame}



\begin{frame}{Accessing $q^2 > 0$ : $\ z\ $ expansion}

\small
\Cite{Bobeth, Chrzaszcz, van Dyk, Virto 1707.07305}
\vspace{2mm}

\btr Conformal mapping : \hspace{5mm} $q^2 \mapsto \ z\,(q^2) = \frac{\sqrt{t_+ - q^2} - \sqrt{t_+ - t_0}}{\sqrt{t_+ - q^2} + \sqrt{t_+ - t_0}}$

\mbox{
\hspace{-10mm}
\raisebox{8mm}{\includegraphics[width=5.6cm]{Analyticq22.png}}
\hspace{1mm}
\includegraphics[width=6.5cm]{Analyticz.png}
}

\vspace{-7mm}

\btr  ${\red \hat \H_\lambda (q^2(z))}$ is {\bf analytic in $|z|<1$}\\[3mm]

\btr Taylor expand $\red \hat{\H}_\lambda(z)$ around $z=0$.\\[3mm]

\btr  Expansion needed for $|z| < 0.52\ $   ( $-7\,\GeV^2 \leq q^2 \leq 14 \GeV^2$ )


\end{frame}




\begin{frame}{Accessing $q^2 > 0$ : $\ z\ $ expansion}
\small
\Cite{Bobeth, Chrzaszcz, van Dyk, Virto 1707.07305}

\vspace{3mm}


\hspace{-5mm} {\bf \brown Some details for actual parametrisation :}

\mbox{\hspace{-5mm} \btr Try to capture most features of the expansion (better convergence)}

\mbox{\hspace{-5mm} \btr Parametrize the ratios $\H_\lambda(q^2)/\F_\lambda(q^2)$ instead}

\mbox{\hspace{-5mm} \btr The poles should not modify the asymptotic behaviour at $|q^2|\to \infty$}

\begin{eqnarray}
\H_\lambda(z) &=&
\frac{1-z\, z^*_{J/\psi}}{z-z_{J/\psi}} \frac{1-z\,z^*_{\psi(2S)}}{z-z_{\psi(2S)}} \ \hat\H_\lambda(z)
\nonumber\\[2mm]
%
\hat\H_\lambda(z) &=& \Big[ \sum_{k=0}^K \alpha_k^{(\lambda)} z^{k} \Big] \F_\lambda(z)
\nonumber
\end{eqnarray}


where $\alpha^{(\lambda)}_k$ are complex coefficients, and the expansion is truncated after the term $z^{K}$.
We will take $K=2$ ({\brown 16} real parameters).


\end{frame}




\begin{frame}{Experimental constraints on $\ z\ $ parametrisation }
\small

\vspace{-1mm}
\hspace{-5mm}
\Cite{Bobeth, Chrzaszcz, van Dyk, Virto 1707.07305}
\vspace{1mm}

\hspace{-5mm} {\bf \brown Experimental constraints :}

\mbox{\hspace{-5mm} \btr The residues of the poles are given by $B\to K^* \psi_n$ :}

$$
\H_\lambda(q^2 \to M_{\psi_n}^2) \sim
\frac{M_{\psi_n} f^{\,*}_{\psi_n} \A^{\psi_n}_\lambda}{M_B^2 (q^2 - M_{\psi_n}^2)} + \cdots
$$

\mbox{\hspace{-5mm} \btr Angular analyses \Cite{Belle, Babar, LHCb} determine : }

$$
|r_\perp^{\psi_n}|,\,
|r_\|^{\psi_n}|,\,
|r_0^{\psi_n}|,\,
\arg\{r_\perp^{\psi_n} r_{0}^{\psi_n*}\},\,
\arg\{r_\|^{\psi_n} r_{0}^{\psi_n*}\},
$$

where $\quad \displaystyle r_\lambda^{\psi_n} \equiv \operatorname*{Res}_{q^2\to M^2_{\psi_n}} \frac{\H_\lambda(q^2)}{\F_\lambda(q^2)}
\sim
\frac{M_{\psi_n} f^*_{\psi_n} \A^{\psi_n}_\lambda}{M_B^2\, \F_\lambda(M_{\psi_n}^2)}$\\[3mm]


\mbox{\hspace{-5mm} \btr We produce correlated pseudo-observables from a fit (5+5).}



\end{frame}




\begin{frame}{Prior Fit to $\ z\ $ parametrisation }
\small

\vspace{-1mm}
\hspace{-5mm}
\vspace{1mm}

\hspace{-5mm} {\bf \brown (Prior) Fit to Experimental and theoretical pseudo-observables :}



\begin{table}[b]
%    \resizebox{.85\textwidth}{!}{%
\centering
\renewcommand{\arraystretch}{1.5}
\renewcommand{\tabcolsep}{3.1mm}
\begin{tabular}{@{}crrr@{}}
\hline
$k$  &  0\hspace{7mm} & 1\hspace{7mm} & 2\hspace{7mm} \\
\hline
%re perp
${\rm Re}[\alpha_{k}^{(\perp)}]$ & $-0.06 \pm 0.21$  & $-6.77 \pm 0.27$ & $18.96 \pm 0.59$ \\
%re para  
${\rm Re}[\alpha_{k}^{(\parallel)}]$ & $-0.35 \pm 0.62$  & $-3.13 \pm 0.41$ & $12.20 \pm 1.34$ \\
%re long  
${\rm Re}[\alpha_{k}^{(0)}]$ & $0.05 \pm 1.52$  & $17.26 \pm 1.64$ & --  \\
%im perp  
${\rm Im}[\alpha_{k}^{(\perp)}]$ & $-0.21 \pm 2.25$  &  $1.17 \pm 3.58$   & $-0.08 \pm 2.24$ \\
%im para  
${\rm Im}[\alpha_{k}^{(\parallel)}]$ & $-0.04 \pm 3.67$  & $-2.14 \pm 2.46$   &  $6.03 \pm 2.50$ \\
%im long  
${\rm Im}[\alpha_{k}^{(0)}]$  & $-0.05 \pm 4.99$  &  $4.29 \pm 3.14$ & --  \\
\hline        
\end{tabular}
%    }
\caption{Mean values and standard deviations (in units of $10^{-4}$)
of the prior PDF for the parameters $\alpha_k^{(\lambda)}$.}
\label{alphak}
\end{table}

\end{frame}

\begin{frame}{New Physics Analysis   }
\small

\vspace{-1mm}
\hspace{-5mm}
\vspace{1mm}

\hspace{-5mm}  {\bf \brown SM predictions and Fit including $B\to K^* \mu^+\mu^-$ data and $\C{9}^{\rm NP}$  :}\\[4mm]

\mbox{
\hspace{-10mm}
\includegraphics[width=12cm]{NPFit.png}
}

The NP hypothesis with {\red $\C{9}^{\bf NP}\sim -1$} is favored strongly in the global fit

\end{frame}






%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Scale of NP?}
{~}
	\begin{minipage}{\textwidth}
\includegraphics[width=0.99\textwidth]{15.png}

\ARROW Stolen from M. Nardecchia
\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}{Amplitudes method}
{~}
\begin{footnotesize}
	\begin{minipage}{\textwidth}

\ARROW Fit for amplitudes as (continuous) functions of $q^2$ in the region: $q^2 \in \left[ 1.1. 6.0 \right]~\GeV^2/c^4$.\\
\ARROW Needs some Ansatz:
\begin{align*}
A(q^2) = \alpha + \beta q^2+ \dfrac{\gamma}{q^2}
\end{align*}
\ARROW The assumption is tested extensively with toys.\\
\ARROW Set of 3 complex parameters $ \alpha, \beta, \gamma $ per vector amplitude:\begin{itemize}
\item {\color{Magenta}{$L, ~R$}}, {\color{Cerulean}{$0 ,~\| ,~\bot$}}, {\color{PineGreen}{$\Re ,~\Im$}} $\rightarrowtail$~~ $3 \times {\color{PineGreen}{2}} \times {\color{Cerulean}{3}} \times {\color{Magenta}{2}} = 36$  DoF.
\item Scalar amplitudes: $+4$ DoF.
\item Symmetries of the amplitudes reduces the total budget to: $28$.
\end{itemize}
\ARROW The technique is described in \href{http://arxiv.org/pdf/1504.00574v2.pdf}{\color{blue}{JHEP06(2015)084}}.\\
\ARROW Allows to build the observables as continuous functions of $q^2$:
\begin{itemize}
\item At current point the method is limited by statistics.
\item In the future the power of this method will increase.
\end{itemize}
\ARROW Allows to measure the zero-crossing points for free and with smaller errors than previous methods.
\end{minipage}
\end{footnotesize}
		\vspace*{2.1cm}
\end{frame}



\begin{frame}{Amplitudes - results}
{~}
\begin{footnotesize}
	\begin{minipage}{\textwidth}
\begin{center}
\begin{columns}
\column{0.45\textwidth}
\includegraphics[angle=-90,width=0.95\textwidth]{{images/amplitudes_AFBOverlay}.pdf}\\
\includegraphics[angle=-90,width=0.95\textwidth]{{images/amplitudes_S4Overlay}.pdf}

\column{0.45\textwidth}
\includegraphics[angle=-90,width=0.95\textwidth]{{images/amplitudes_S5Overlay}.pdf}\\
{~}\\{~}\\{~}\\{~}\\
\begin{large}
Zero crossing points:
\end{large}
\begin{align*}
q_0(S_4) & <2.65 & {\rm{~at~}} & 95\% ~CL \\
q_0(S_5) & \in \left[ 2.49,3.95 \right] & {\rm{~at~}} & 68\% ~CL \\
q_0(A_{FB}) & \in \left[ 3.40, 4.87 \right] & {\rm{~at~}} & 68\% ~CL 
\end{align*}


\end{columns}
\end{center}

\end{minipage}
\end{footnotesize}
		\vspace*{2.1cm}
\end{frame}



\backupend

\end{document}
