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% particles                                                               
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% useful decays                                                            
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\def\BuToKmm    {\decay{\Bu}{\Kp\mumu}}                                    
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%% some other decays                                                       
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\author{ {\fontspec{Trebuchet MS}Marcin Chrz\k{a}szcz} (CERN, IFJ PAN)}
\institute{UZH}
\title[Quo Vadis flavor anomalies?]{Quo Vadis flavor anomalies?}
\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 {Quo Vadis flavor 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}{The Future of Particle Physics: A Quest for Guiding Principles \\October 2, 2018}
\end{center}
\end{frame}
}


\begin{frame}{Outline}

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

\begin{enumerate}
\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}






\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>{
\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}

\iffalse

\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=0.6\textwidth]{rdrds_winter2017.pdf}
\end{center}
\ARROW Theoretical uncertainties negligible.\\
\ARROW The ball is on the experimental side.

\end{frame}


\fi
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%





\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}

\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 ($q^2)$:
{\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}
}}




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


\fi

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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}{Compatibility with SM}
{~}

	\begin{minipage}{\textwidth}

\begin{columns}
\column{0.1in}
{~}
\column{2in}
\includegraphics[angle=-90,width=0.95\textwidth]{images/PBasis_P5pPad.pdf}

\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: $\Re(C_9)$.\\


\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} \\

\ARROW Best fit is found to be $3.4~\sigma$ away from the SM.

\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}

}


\frame{ \frametitle{Where to look for more?}

\begin{footnotesize}
\ARROW There are couple of models that can accommodate these (see next talk).\\
\ARROW Usual models need high mass particle outside the reach of LHC.\\

\begin{exampleblock}{My opinion}
Before moving to high energy frontier to look for something we should explore more precisely the electroweak sector. 

We can get more clues about the underlying physics
\end{exampleblock}

\begin{center}
\includegraphics[width=0.5\textwidth]{images/download.pdf}
\end{center}




\end{footnotesize}

}










\backupbegin

\begin{frame}\frametitle{Backup}


\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}



\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}
