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\author{ {\fontspec{Trebuchet MS}Marcin Chrz\k{a}szcz} (Universit\"{a}t Z\"{u}rich, IFJ PAN)}
\institute{UZH, IFJ PAN}
\title[Quark flavour anomalies of the SM]{Quark flavour anomalies of the SM}
\date{25 September 2014}
\begin{document}
\tikzstyle{every picture}+=[remember picture]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{
\setbeamertemplate{sidebar right}{\llap{\includegraphics[width=\paperwidth,height=\paperheight]{bubble2}}}
\begin{frame}[c]%{\phantom{title page}}
\begin{center}
\begin{center}
\begin{columns}
\begin{column}{0.75\textwidth}
\flushright\fontspec{Trebuchet MS}\bfseries \LARGE {Quark flavour\\ anomalies of the SM}
\end{column}
\begin{column}{0.02\textwidth}
{~}
\end{column}
\begin{column}{0.23\textwidth}
% \hspace*{-1.cm}
\vspace*{-3mm}
\includegraphics[width=0.6\textwidth]{lhcb-logo}
\end{column}
\end{columns}
\end{center}
\quad
\vspace{3em}
\begin{columns}
\begin{column}{0.44\textwidth}
\flushright \vspace{-1.8em} {\fontspec{Trebuchet MS} \Large Marcin Chrząszcz\\\vspace{-0.1em}\small \href{mailto:mchrzasz@cern.ch}{mchrzasz@cern.ch}}
\end{column}
\begin{column}{0.53\textwidth}
\includegraphics[height=1.3cm]{uzh-transp}{~}{~}
\includegraphics[height=1.1cm]{ifj.png}
\end{column}
\end{columns}
\vspace{1em}
\footnotesize\textcolor{gray}{on behalf of the LHCb collaboration,\\ Universit\"{a}t Z\"{u}rich, \\ Institute of Nuclear Physics, Polish Academy of Science}\normalsize\\
\vspace{0.5em}
\textcolor{normal text.fg!50!Comment}{Quark Confinement and Hadron Spectrum,\\ Thessaloniki, 28 August - 3 September 2016}
\end{center}
\end{frame}
}
\iffalse
\begin{frame}{Outline}
\begin{minipage}{\textwidth}
\begin{enumerate}
\item Why flavour is important.
\item $\Pbeauty \to \Pstrange \ell \ell$ theory in a nutshell.
\item LHCb measurements of $\Pbeauty \to \Pstrange \ell \ell$.
\item Global fit to $\Pbeauty \to \Pstrange \ell \ell$ measurements.
\item Conclusions.
\end{enumerate}
\end{minipage}
\vspace*{2.cm}
\end{frame}
\fi
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}\frametitle{Why electroweak penguin decays?}
\begin{columns}
\column{4in}
\begin{itemize}
\item In SM allows only the charged interactions to change flavour.
\begin{itemize}
\item Other interactions are flavour conserving.
\end{itemize}
\item One can escape this constrain and produce $\Pbottom \to \Pstrange$ and $\Pbottom \to \Pdown$ at loop level.
\begin{itemize}
\item This kind of processes are suppressed in SM $\to$~Rare decays.
\item New Physics can enter in the loops.
\end{itemize}
\end{itemize}
\begin{center}
\includegraphics[scale=0.3]{images/lupa.png}
\includegraphics[scale=0.3]{images/example.png}
\end{center}
\column{1.5in}
\includegraphics[width=0.61\textwidth]{images/couplings.png}
\end{columns}
\end{frame}
\begin{frame}\frametitle{Tools}
\begin{itemize}
\item \textbf{Operator Product Expansion and Effective Field Theory}
\end{itemize}
\begin{columns}
\column{0.1in}{~}
\column{1.2in}
\begin{small}
\begin{align*}
H_{eff} = - \dfrac{4G_f}{\sqrt{2}} V V^{\prime \ast}\ \sum_i \left[\underbrace{C_i(\mu)O_i(\mu)}_\text{left-handed} +\underbrace{C'_i(\mu)O'_i(\mu)}_\text{right-handed}\right],
\end{align*}
\end{small}
\column{1.8in}
\begin{tiny}
\begin{description}
\item[i=1,2] Tree
\item[i=3-6,8] Gluon penguin
\item[i=7] Photon penguin
\item[i=9.10] EW penguin
\item[i=S] Scalar penguin
\item[i=P] Pseudoscalar penguin
\end{description}
\end{tiny}
\end{columns}
where $C_i$ are the Wilson coefficients and $O_i$ are the corresponding effective operators.
\begin{center}
\includegraphics[width=0.85\textwidth,height=3cm]{images/all.png}
\end{center}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\only<1>{\frametitle{LHCb detector - tracking}
\begin{columns}
\column{3in}
\includegraphics[width=0.9\textwidth]{images/1050px-Lhcbview.jpg}
\column{2in}
\includegraphics[width=0.95\textwidth]{images/sketch.png}
\end{columns}
\begin{itemize}
\item Excellent Impact Parameter (IP) resolution ($20~\rm \mu m$).\\
$\Rightarrow$ Identify secondary vertices from heavy flavour decays
\item Proper time resolution $\sim~40~\rm fs$.\\
$\Rightarrow$ Good separation of primary and secondary vertices.
\item Excellent momentum ($\delta p/p \sim 0.5 - 1.0\%$) and inv. mass resolution.\\
$\Rightarrow$ Low combinatorial background.
\end{itemize}
}
\only<2>{\frametitle{LHCb detector - PID}
\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}
}
\textref{Int. J. Mod. Phys. A30 (2015) 1530022}
\vspace*{2.1cm}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Recent measurements of $\Pbeauty \to \Pstrange \Plepton \Plepton$}
{~}
\only<1>{
\begin{minipage}{\textwidth}
\begin{columns}
\column{0.5\textwidth}
$\color{JungleGreen}{\Rrightarrow}$ {\color{WildStrawberry}{Branching fractions:}}
\begin{description}
\item [$\PB \to \PK \Pmuon \APmuon$] {~}{~} \textt{\begin{tiny}1606.04731\end{tiny}
%\item [$\PB^{0} \to \PKstar \Pmuon \APmuon$] {~}{~}CMS, Jul 15
\item [$\PBs \to \Pphi \Pmuon \APmuon$] {~}{~}{~}\begin{tiny}JHEP 09 (2015) 179\end{tiny}
\item [$\PB^{\pm} \to \Ppi^{\pm} \Pmuon \APmuon$] {~}\begin{tiny}JHEP 12 (2012) 125\end{tiny}
\item [$\PLambdab \to \PLambda \Pmuon \APmuon$] {~}{~}{~}{~}\begin{tiny}JHEP 06 (2015) 115\end{tiny}
\item [$\PB \to\Pmuon \APmuon$] {~}{~}{~}{~}{~}\begin{tiny}Nature 15\end{tiny}}
\end{description}
$\color{JungleGreen}{\Rrightarrow}$ {\color{WildStrawberry}{CP asymmetry:}}
\begin{description}
\item [$\PB^{\pm} \to \Ppi^{\pm} \Pmuon \APmuon$] {~}{~}\begin{tiny}JHEP 10 (2015) 034\end{tiny}
\end{description}
$\color{JungleGreen}{\Rrightarrow}$ {\color{WildStrawberry}{Isospin asymmetry:}}
\begin{description}
\item [$\PB \to \PK \Pmuon \APmuon$] {~}{~}{~}{~}{~}\begin{tiny}JHEP 06 (2014) 133\end{tiny}
\end{description}
\column{0.5\textwidth}
$\color{JungleGreen}{\Rrightarrow}$ {\color{WildStrawberry}{Lepton Universality:}}
\begin{description}
\item [$\PB^{\pm} \to \PK^{\pm} \Plepton \APlepton$] {~}{~}\begin{tiny}PRL 113, (2014)\end{tiny}
\end{description}
$\color{JungleGreen}{\Rrightarrow}$ {\color{WildStrawberry}{Angular:}}
\begin{description}
\item [$\PB^{0} \to \PK^{\ast} \Plepton \APlepton$] {~}{~}{~}\begin{tiny}JHEP 02 (2016) 104\end{tiny}
\item [$\PB^{0,\pm} \to \PK^{\ast,\pm} \Plepton \APlepton$] \begin{tiny}PRD 86 032012\end{tiny}
\item [$\PBs \to \Pphi \Pmu \Pmu$] {~}{~}{~}\begin{tiny}JHEP 09 (2015) 179\end{tiny}
\item [$\PLambdab \to \PLambda \Pmuon \APmuon$] {~}{~}\begin{tiny}JHEP 06 (2015) 115\end{tiny}
\end{description}
\end{columns}
\end{minipage}
}
\only<2>{
\begin{minipage}{\textwidth}
\begin{columns}
\column{0.5\textwidth}
$\color{JungleGreen}{\Rrightarrow}$ {\color{WildStrawberry}{Branching fractions:}}
\begin{description}
\item [{\color{red}{$\PB \to \PK \Pmuon \APmuon$}}] {~}{~}{\color{red}{\begin{tiny}1606.04731\end{tiny}}}
%\item [$\PB^{0} \to \PKstar \Pmuon \APmuon$] {~}{~}CMS, Jul 15
\item [{\color{red}{$\PBs \to \Pphi \Pmuon \APmuon$}}] {~}{~}{~}{\color{red}{\begin{tiny}JHEP 09 (2015) 179\end{tiny}}}
\item [$\PB^{\pm} \to \Ppi^{\pm} \Pmuon \APmuon$] {~}\begin{tiny}JHEP 12 (2012) 125\end{tiny}
\item [$\PLambdab \to \PLambda \Pmuon \APmuon$] {~}{~}{~}{~}\begin{tiny}JHEP 06 (2015) 115\end{tiny}
\item [{\color{red}{$\PB \to\Pmuon \APmuon$}}] {~}{~}{~}{~}{~}{\color{red}{\begin{tiny}Nature 15\end{tiny}}}
\end{description}
$\color{JungleGreen}{\Rrightarrow}$ {\color{WildStrawberry}{CP asymmetry:}}
\begin{description}
\item [$\PB^{\pm} \to \Ppi^{\pm} \Pmuon \APmuon$] {~}{~}\begin{tiny}JHEP 10 (2015) 034\end{tiny}
\end{description}
$\color{JungleGreen}{\Rrightarrow}$ {\color{WildStrawberry}{Isospin asymmetry:}}
\begin{description}
\item [$\PB \to \PK \Pmuon \APmuon$] {~}{~}{~}{~}{~}\begin{tiny}JHEP 06 (2014) 133\end{tiny}
\end{description}
\column{0.5\textwidth}
$\color{JungleGreen}{\Rrightarrow}$ {\color{WildStrawberry}{Lepton Universality:}}
\begin{description}
\item [{\color{red}{$\PB^{\pm} \to \PK^{\pm} \Plepton \APlepton$}}] {~}{~}{\color{red}{\begin{tiny}PRL 113, (2014)\end{tiny}}}
\end{description}
$\color{JungleGreen}{\Rrightarrow}$ {\color{WildStrawberry}{Angular:}}
\begin{description}
\item [{\color{red}{$\PB^{0} \to \PK^{\ast} \Plepton \APlepton$}}] {~}{~}{~}\begin{tiny}PRD 86 032012\end{tiny}
\item [{\color{red}{$\PB^{0,\pm} \to \PK^{\ast,\pm} \Plepton \APlepton$}}] {\color{red}{\begin{tiny}JHEP 09 (2015) 179\end{tiny}}}
\item [$\PBs \to \Pphi \Plepton \APlepton$] {~}{~}{~}\begin{tiny}JHEP 09 (2015) 179\end{tiny}
\item [{\color{red}{$\PLambdab \to \PLambda \Pmuon \APmuon$}}] {~}{~}{\color{red}{\begin{tiny}JHEP 06 (2015) 115\end{tiny}}}
\end{description}
\begin{alertblock}{}
$>2~\sigma$ deviations from SM
\end{alertblock}
\end{columns}
\end{minipage}
}
\vspace*{2.1cm}
\end{frame}
\iffalse
\fi
\begin{frame}{Observables in $\PB \to \PKstar \Pmu \Pmu$}
{~}
\begin{minipage}{\textwidth}
\ARROW The kinematics of $\PBzero \to \PKstar \Pmuon \APmuon$ decay is described by three angles $\thetal$, $\thetak$, $\phi$ and invariant mass of the dimuon system ($q^2)$.\\
\ARROW The angular distribution can be written as:
\begin{tiny}
\begin{align*}
\left.\frac{1}{{\rm d}(\Gamma+\bar{\Gamma})/{\rm d}q^2}\frac{{\rm d}(\Gamma+\bar{\Gamma})}{{\rm dcos}\thetal\,{\rm dcos}\thetak\,{\rm d}\phi}\right|_{\rm P} =
\tfrac{9}{32\pi}\bigl[
&\tfrac{3}{4} (1-{F_{\rm L}})\sin^2\thetak \label{eq:pdfpwave}\\[-0.75em]
&+ {F_{\rm L}}\cos^2\thetak + \tfrac{1}{4}(1-{F_{\rm L}})\sin^2\thetak\cos 2\thetal\nonumber\\
&- {F_{\rm L}} \cos^2\thetak\cos 2\thetal + {S_3}\sin^2\thetak \sin^2\thetal \cos 2\phi\nonumber\\
&+ {S_4} \sin 2\thetak \sin 2\thetal \cos\phi + {S_5}\sin 2\thetak \sin \thetal \cos \phi\nonumber\\
&+ \tfrac{4}{3} {A_{\rm FB}} \sin^2\thetak \cos\thetal + {S_7} \sin 2\thetak \sin\thetal \sin\phi\nonumber\\
&+ {S_8} \sin 2\thetak \sin 2\thetal \sin\phi + {S_9}\sin^2\thetak \sin^2\thetal \sin 2\phi \nonumber
\bigr].
%\end{split}
%\bigr],\
\end{align*}
\end{tiny}
\ARROW This equation is valid in the SM for massless leptons!
\end{minipage}
\vspace*{2.1cm}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Link to effective operators}
{~}
\begin{minipage}{\textwidth}
$\color{JungleGreen}{\Rrightarrow}$ The observables ${\color{red}{S_i}}$ are bilinear combinations of transversity amplitudes: $\apeLR,~\apaLR,~\azeLR $. \\
$\color{JungleGreen}{\Rrightarrow}$ So here is where the magic happens. At leading order the amplitudes can be written as:
{\tiny{
\eqa{
\apeLR &=&\sqrt{2} N m_B(1- \hat s)\bigg[ (\Ceff9 + \Cpeff9) \mp (\C{10} + \Cp{10})
+\frac{2\hat{m}_b}{\hat s} (\Ceff7 + \Cpeff7) \bigg]\xi_{\bot}(E_{K^*}) \nn \\[2mm]
\apaLR &=& -\sqrt{2} N m_B (1-\hat s)\bigg[(\Ceff9 - \Cpeff9) \mp (\C{10} - \Cp{10})
+\frac{2\hat{m}_b}{\hat s}(\Ceff7 - \Cpeff7) \bigg] \xi_{\bot}(E_{K^*}) \nn \\[2mm]
\azeLR &=& -\frac{N m_B (1-\hat s)^2}{2 \hat{m}_{K^*} \sqrt{\hat s}} \bigg[ (\Ceff9 - \Cpeff9) \mp (\C{10} - \Cp{10}) + 2\hat{m}_b (\Ceff7 - \Cpeff7) \bigg]\xi_{\|}(E_{K^*}),
\label{LargeRecoilAs}\nonumber}
}}
where $\hat s = q^2 /m_B^2$, $\hat{m}_i = m_i/m_B$. The $\xi_{\|,\bot }$ are the soft form factors. \\
\pause
$\color{JungleGreen}{\Rrightarrow}$ Now we can construct observables that cancel the $\xi$ soft form factors at leading order:
\eq{P_5^{\prime} = \dfrac{S_5+\bar{S}_5}{2\sqrt{F_L (1-F}}\nonumber
}
\end{minipage}
\vspace*{2.1cm}
\end{frame}
% symmetries
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{$\PBzero \to \PKstar \Pmuon \APmuon$, Selection}
{~}
\begin{minipage}{\textwidth}
\begin{columns}
\column{0.55\textwidth}
\begin{itemize}
\item PID, kinematics and isolation variables used in a Boosted Decision Tree (BDT) to reject background.
\item Reject the regions of $\PJpsi$ and $\Ppsi(2S)$.
\item Specific vetos for backgrounds: $\PLambdab \to \Pproton \PK \Pmu \Pmu$, $\PBs \to \Pphi \Pmu \Pmu$, etc.
\item Using k-Fold technique and signal proxy $\PB \to \PJpsi \PKstar$ for training the BDT.
\item Improved selection allowed for finer binning than the $1\invfb$ analysis.
\end{itemize}
\column{0.45\textwidth}
\includegraphics[angle=-90,width=0.88\textwidth]{images/Fig1.pdf} \\
\includegraphics[width=0.88\textwidth]{images/fold.png}
\end{columns}
\end{minipage}
\textref{JHEP 02 (2016) 104}
\vspace*{2.1cm}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{$\PBzero \to \PKstar \Pmuon \APmuon$, Selection}
{~}
\begin{minipage}{\textwidth}
\begin{itemize}
\item Signal modelled by a sum of two Crystal-Ball functions.
\item Shape is defined using $\PB \to \PJpsi \PKstar$ and corrected for $q^2$ dependency.
\item Combinatorial background modelled by exponent.
\end{itemize}
\begin{columns}
\column{0.5\textwidth}
\begin{itemize}
\item $\PK \Ppi$ system:
\begin{itemize}
\item Beside the $\PKstar$ resonance there might might a tail from other higher mass states.
\item We modelled it in the analysis.
\item Reduced the systematic compared to previous analysis.
\end{itemize}
\end{itemize}
\column{0.5\textwidth}
\includegraphics[width=0.88\textwidth]{images/pbkg}
\end{columns}
\begin{large}
\begin{itemize}
\item In total we found $2398\pm57$ candidates in the $(0.1,19)~\GeV^2$ $q^2$ region.
\item $624 \pm 30$ candidates in the theoretically the most interesting $(1.1-6.0)~\GeV^2/c^4$ region.
\end{itemize}
\end{large}
\end{minipage}
\textref{JHEP 02 (2016) 104}
\vspace*{2.1cm}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Detector acceptance}
{~}
\begin{minipage}{\textwidth}
\begin{columns}
\column{0.6\textwidth}
\begin{itemize}
\item Detector distorts our angular distribution.
\item We need to model this effect.
\item 4D function is used:
\begin{align*}
\epsilon (\cos \thetal, \cos \thetak, \phi, q^2) = \\\sum_{ijkl} c_{ijkl} P_i(\cos \thetal) P_j(\cos \thetak ) P_k(\phi) P_l(q^2),
\end{align*}
where $P_i$ is the Legendre polynomial of order $i$.
\item We use up to $4^{th}, 5^{th}, 6^{th}, 5^{th}$ order for the $\cos \thetal, \cos \thetak, \phi, q^2$.
\item $600$ terms in total!
\end{itemize}
\column{0.4\textwidth}
\includegraphics[width=0.99\textwidth]{images/det.png}
\end{columns}
\end{minipage}
\textref{JHEP 02 (2016) 104}
\vspace*{2.1cm}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Control channel}
{~}
\begin{minipage}{\textwidth}
\begin{itemize}
\item We tested our unfolding procedure on $\PB \to \PJpsi \PKstar$.
\item The result is in perfect agreement with other experiments and our different analysis of this decay.
\end{itemize}
\begin{columns}
\column{0.5\textwidth}
\includegraphics[width=0.95\textwidth]{images/mlogjpsi.png}
\column{0.5\textwidth}
\includegraphics[width=0.95\textwidth]{images/mkpijpsi.png}
\end{columns}
\begin{columns}
\column{0.33\textwidth}
\includegraphics[width=0.91\textwidth]{images/costhetakjpsi.png}
\column{0.33\textwidth}
\includegraphics[width=0.91\textwidth]{images/costhetaljpsi.png}
\column{0.33\textwidth}
\includegraphics[width=0.91\textwidth]{images/phijpsi.png}
\end{columns}
%\includegraphics[width=0.99\textwidth]{images/angles2.png}
\end{minipage}
\textref{JHEP 02 (2016) 104}
\vspace*{2.1cm}
\end{frame}
\begin{frame}{Method of moments}
{~}
\begin{minipage}{\textwidth}
\begin{itemize}
\item Use orthogonality of spherical harmonics, $f_j(\cos \thetal, \cos \thetak, \phi)$:
\begin{equation*}
\int f_i(\cos \thetal, \cos \thetak, \phi) \cdot f_j(\cos \thetal, \cos \thetak, \phi) = \delta_{ij}
\end{equation*}
\begin{equation*}
M_i = \int \frac{1}{{\rm d}(\Gamma+\bar{\Gamma})/{\rm d}q^2}\frac{{\rm d}(\Gamma+\bar{\Gamma})}{{\rm dcos}\thetal\,{\rm dcos}\thetak\,{\rm d}\phi} f_i(\cos \thetal, \cos \thetak, \phi)
\end{equation*}
\item Don’t have true angular distribution but we ''sample'' it with our data.
\item Therefore: $\int \rightarrow \sum$ and $M_i \rightarrow \widehat{M}_i$
%\begin{equation*}
%M_i = \int \frac{1}{{\rm d}(\Gamma+\bar{\Gamma})/{\rm d}q^2}\frac{{\rm d}(\Gamma+\bar{\Gamma})}{{\rm dcos}\thetal\,{\rm dcos}\thetak\,{\rm d}\phi} f_i(\cos \thetal, \cos \thetak, \phi)
%\end{equation*}
\item Acceptance corrections is included by:
\begin{equation*}
\widehat{M}_i = \dfrac{1}{\sum_e w_e} \sum w_e f_i(\cos \thetal, \cos \thetak, \phi)
\end{equation*}
\item The weight $w_e$ accounts for the efficiency from previous slide.
\end{itemize}
\end{minipage}
\textref{Phys. Rev. D 91, 114012 (2015)}
\vspace*{2.1cm}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
\begin{frame}\frametitle{$\PBzero \rightarrow \PK^{\ast} \Pmu \Pmu$ results}
\begin{minipage}{\textwidth}
\begin{columns}
\column{2.5in}
\includegraphics[angle=-90,width=0.95\textwidth]{images/compare_FLPad.pdf}\\
\includegraphics[angle=-90,width=0.95\textwidth]{images/compare_S3Pad.pdf}
\column{2.5in}
\includegraphics[angle=-90,width=0.95\textwidth]{images/compare_S4Pad.pdf}\\
\includegraphics[angle=-90,width=0.95\textwidth]{images/compare_S5Pad.pdf}
\end{columns}
\end{minipage}
\textref{JHEP 02 (2016) 104}
\vspace*{2.1cm}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
\begin{frame}\frametitle{$\PBzero \rightarrow \PK^{\ast} \Pmu \Pmu$ results}
\begin{minipage}{\textwidth}
\begin{columns}
\column{2.5in}
\includegraphics[angle=-90,width=0.95\textwidth]{images/compare_AFBPad.pdf}\\
\includegraphics[angle=-90,width=0.95\textwidth]{images/compare_S7Pad.pdf}
\column{2.5in}
\includegraphics[angle=-90,width=0.95\textwidth]{images/compare_S8Pad.pdf}\\
\includegraphics[angle=-90,width=0.95\textwidth]{images/compare_S9Pad.pdf}
\end{columns}
\end{minipage}
\textref{JHEP 02 (2016) 104}
\vspace*{2.1cm}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Results in $\PB \to \PKstar \Pmu \Pmu$}
{~}
\begin{minipage}{\textwidth}
\begin{center}
\includegraphics[angle=-90,width=0.65\textwidth]{images/compare_P5pPad.pdf}\\
\end{center}
\begin{itemize}
\item Tension gets confirmed!
\item The two bins deviate by $2.8$~and~$3.0~\sigma$ from SM prediction.
\item Result compatible with previous result.
\end{itemize}
\end{minipage}
\textref{JHEP 02 (2016) 104}
\vspace*{2.1cm}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Results in $\PB \to \PKstar \Pmu \Pmu$}
{~}
\begin{minipage}{\textwidth}
\begin{itemize}
\item Thanks to Method of Moments there was the possibility to measure a new observable $S_{6c}$.
\end{itemize}
\begin{center}
\includegraphics[angle=-90,width=0.65\textwidth]{images/S6cPad.pdf}\\
\end{center}
\begin{itemize}
\item Measurement is consistent with the SM prediction.
\end{itemize}
\end{minipage}
\textref{JHEP 02 (2016) 104}
\vspace*{2.1cm}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{BF measurements of $\PB \to \PKstar^{0,\pm} \Pmu \Pmu$}
{~}
\includegraphics[angle=-90,width=0.5\textwidth]{images/ksmumu_BF.pdf}
\includegraphics[angle=-90,width=0.5\textwidth]{images/kmumu_BF.pdf}
\begin{center}
\begin{columns}
\column{0.5\textwidth}
\begin{itemize}
\item Despite large theoretical errors the results are consistently smaller than SM prediction.
%\item \href{http://dx.doi.org/10.1007/JHEP06(2014)133}{{\color{blue}{JHEP 06 (2014) 133}}}
\end{itemize}
\column{0.5\textwidth}
\includegraphics[angle=-90,width=0.850\textwidth]{images/bukst_BF.pdf}
\end{columns}
\end{center}
\textref{JHEP 06 (2014) 133}
\vspace*{2.1cm}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{BF measurements of $\PBs \to \Pphi \Pmu \Pmu$}
{~}
\begin{minipage}{\textwidth}
\begin{center}
\includegraphics[height=4cm]{images/bs2phipi.png}
\includegraphics[height=4cm]{images/BsSel.png}
\end{center}
\begin{itemize}
\item Suppressed by $\frac{f_s}{f_d}$.
\item Cleaner because of narrow $\Pphi$ resonance.
\item $3.3~\sigma$ deviation in SM in the $1-6\GeV^2/c^4$ bin.
\item Angular part in agreement with SM ($S_5$ is not accessible).
\end{itemize}
\end{minipage}
\textref{JHEP09 (2015) 179}
\vspace*{2.1cm}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Measurements of $\PLambdab \to \PLambda \Pmu \Pmu$}
{~}
\begin{minipage}{\textwidth}
\includegraphics[width=0.32\textwidth]{images/Lb_BR.png} \includegraphics[width=0.32\textwidth]{images/Lblow.png}
\includegraphics[width=0.32\textwidth]{images/Lbhigh.png}
\begin{small}
\begin{itemize}
\item In total $\sim 300$ candidates in data set.
\item Decay not present in the low $q^2$.
\item For the bins in which we have $>3~\sigma$ significance the forward backward asymmetry is measured for the hadronic and leptonic system.
\end{itemize}
\end{small}
\vspace{-0.5cm}
\begin{center}
\includegraphics[width=0.7\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}
\textref{JHEP 06 (2015) 115}
\vspace*{2.1cm}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Lepton universality test}
{~}
\begin{minipage}{\textwidth}
\begin{columns}
\column{3.0in}
\begin{itemize}
%\includegraphics[width=0.99\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=1.05\textwidth]{images/uni2.png} \\
\includegraphics[width=0.99\textwidth]{images/RK.png}\\
%\includegraphics[width=0.99\textwidth]{{images/rk_}.png}\\
\end{columns}
\includegraphics[width=0.85\textwidth]{{images/rk_}.png}\\
\end{minipage}
\textref{Phys. Rev. Lett. 113, 151601 (2014)}
\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 $0.0004 < q^2 <1~\GeV^2/c^4$.
\item Electrons channels are extremely challenging experimentally:
\begin{itemize}
\item Bremsstrahlung.
\item Trigger efficiencies.
\end{itemize}
\item Determine the angular observables: $\FL$, $\ATD$, $\ATRe$, $\ATIm$:
\end{itemize}
\begin{itemize}
\item Results in full agreement with the SM.
\item Similar strength on $C_7$ Wilson coefficient as from $\Pbeauty \to \Pstrange \Pphoton$ decays.
\end{itemize}
\begin{center}
\includegraphics[width=0.9\textwidth]{images/Kstee2.png}
\end{center}
}
\end{minipage}
\textref{JHEP 04 (2015) 064}
\vspace*{2.1cm}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Theory implications}
{~}
\begin{minipage}{\textwidth}
\begin{itemize}
\item A fit prepared by \texttt{S. Descotes-Genon, L. Hofer, J. Matias, J. Virto}.
\item Took into the fit:
\begin{itemize}
\item $\mathcal{B} ( \PB \to X_s \Pphoton) = (3.36 \pm 0.23) \times 10^{-4} $, {\color{blue}{ Misiak et. al. PRL 114, 221801 (2015)}}
\item $\mathcal{B} ( \PB \to\Pmu \Pmu)$, theory: {\color{blue}{ Bobeth PRD 89, (2014)}}, experiment: LHCb+CMS average (2015)
\item $\mathcal{B} ( \PB \to X_s \Pmu \Pmu$), {\color{blue}{ Huber et al Nucl Phys B802, 2008}}
\item $\mathcal{B} ( \PB \to \PK \Pmu \Pmu$), {\color{blue}{ Bouchard et al JHEP11 (2011) 122}}
\item $\PB_{(s)} \to \PKstar(\Pphi) \Pmu \Pmu$,{\color{blue}{ Horgan et al PRL 112, (2014)}}
\item $\PB \to \PK \Pe \Pe$, $\PB \to \PKstar \Pe \Pe$ and $R_k$.
\end{itemize}
%\item Overall there is $>4~\sigma$ discrepancy wrt. SM.
\end{itemize}
\end{minipage}
\textref{JHEP 06 (2016) 092}
\vspace*{2.1cm}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Theory implications}
{~}
\begin{minipage}{\textwidth}
\begin{itemize}
\item A fit prepared by \texttt{S. Descotes-Genon, L. Hofer, J. Matias, J. Virto}.
\item The data can be explained by modifying the $C_9$ Wilson coefficient.
\item Overall there is $>4~\sigma$ discrepancy wrt. SM prediction.
\end{itemize}
\includegraphics[width=0.9\textwidth]{images/C9.png}
\end{minipage}
\textref{JHEP 06 (2016) 092}
\vspace*{2.1cm}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{If not NP?}
{~}
\begin{minipage}{\textwidth}
\begin{itemize}
\item We are not there yet!
\item There might be something not taken into account in the theory.
\item Resonances ($\PJpsi$, $\Ppsi(2S)$) tails can mimic NP effects.
\item There might be some non factorizable QCD corrections.\\
'' However, the central value of this effect would have to be significantly larger than expected on the basis of existing estimates'' \texttt{D.Straub}, \href{http://arxiv.org/abs/1503.06199}{ {\color{blue}{arXiv:1503.06199}}}
.
\end{itemize}
\only<1>{
\includegraphics[width=0.6\textwidth]{images/charmloop.png}
}
\only<2>{
\begin{center}
\includegraphics[width=0.6\textwidth]{images/charmloop2.png}
\end{center}
}
\end{minipage}
\vspace*{2.1cm}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{There is more!}
{~}
\begin{minipage}{\textwidth}
\begin{itemize}
\item There is one other Lepton Universality Violation decay recently measured by LHCb.
\item $R(\PDstar)=\dfrac{\mathcal{B}(\PB \to \PDstar \Ptau \Pnu)}{\mathcal{B}(\PB \to \PDstar \Pmu \Pnu)}$
\item Clean SM prediction: $R(\PDstar)=0.252(3)$, {\color{blue}{PRD 85 094025 (2012)}}
\item LHCb result: $R(\PDstar)= 0.336 \pm 0.027 \pm 0.030$
\item HFAG average: $R(\PDstar)=0.322 \pm 0.022$
\item $4.0~\sigma$ discrepancy wrt. SM.
\end{itemize}
\begin{center}
\includegraphics[width=0.4\textwidth]{images/RDstar.png}
\end{center}
\end{minipage}
\vspace*{2.1cm}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Conclusions}
{~}
\begin{minipage}{\textwidth}
\begin{itemize}
\item Clear tensions wrt. SM predictions!
\item Measurements cluster in the same direction.
\item We are not opening the champagne yet!
\item Still need improvement both on theory and experimental side.
\item Time will tell if this is QCD+fluctuations or new Physics:
\end{itemize}
\pause
''... when you have eliminated all the\\
Standard Model explanations, whatever remains,\\
however improbable, must be New Physics.''\\
Prof. Joaquim Matias
\end{minipage}
\vspace*{2.1cm}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
{~}
\begin{minipage}{\textwidth}
\begin{center}
\begin{LARGE}
Thank you for the attention!
\end{LARGE}
\includegraphics[width=0.8\textwidth]{images/Joke.jpg}
\end{center}
\end{minipage}
\vspace*{2.1cm}
\end{frame}
\backupbegin
\begin{frame}\frametitle{Backup}
\end{frame}
\begin{frame}{Theory implications}
{~}
\begin{minipage}{\textwidth}
\includegraphics[height=0.9\textheight]{images/table.png}
\end{minipage}
\vspace*{2.1cm}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{If not NP?}
{~}
\begin{minipage}{\textwidth}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{itemize}
\item How about our clean $P_i$ observables?
\item The QCD cancel as mentioned only at leading order.
\item Comparison to normal observables with the optimised ones.
\end{itemize}
\includegraphics[width=0.9\textwidth]{images/C9_S_P.png}
\end{minipage}
\vspace*{2.1cm}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Transversity amplitudes }
{~}
\begin{minipage}{\textwidth}
$\color{JungleGreen}{\Rrightarrow}$ One can link the angular observables to transversity amplitudes
{\tiny{
\eqa{
J_{1s} & = & \frac{(2+\beta_\ell^2)}{4} \left[|\apeL|^2 + |\apaL|^2 +|\apeR|^2 + |\apaR|^2 \right]
+ \frac{4 m_\ell^2}{q^2} \re\left(\apeL\apeR^* + \apaL\apaR^*\right)\,,\nn\\[1mm]
%
J_{1c} & = & |\azeL|^2 +|\azeR|^2 + \frac{4m_\ell^2}{q^2} \left[|A_t|^2 + 2\re(\azeL^{}\azeR^*) \right] + \beta_\ell^2\, |A_S|^2 \,,\nn\\[1mm]
%
J_{2s} & = & \frac{ \beta_\ell^2}{4}\left[ |\apeL|^2+ |\apaL|^2 + |\apeR|^2+ |\apaR|^2\right],
\hspace{0.92cm} J_{2c} = - \beta_\ell^2\left[|\azeL|^2 + |\azeR|^2 \right]\,,\nn\\[1mm]
%
J_3 & = & \frac{1}{2}\beta_\ell^2\left[ |\apeL|^2 - |\apaL|^2 + |\apeR|^2 - |\apaR|^2\right],
\qquad J_4 = \frac{1}{\sqrt{2}}\beta_\ell^2\left[\re (\azeL\apaL^* + \azeR\apaR^* )\right],\nn \\[1mm]
%
J_5 & = & \sqrt{2}\beta_\ell\,\Big[\re(\azeL\apeL^* - \azeR\apeR^* ) - \frac{m_\ell}{\sqrt{q^2}}\,
\re(\apaL A_S^*+ \apaR^* A_S) \Big]\,,\nn\\[1mm]
%
J_{6s} & = & 2\beta_\ell\left[\re (\apaL\apeL^* - \apaR\apeR^*) \right]\,,
\hspace{2.25cm} J_{6c} = 4\beta_\ell\, \frac{m_\ell}{\sqrt{q^2}}\, \re (\azeL A_S^*+ \azeR^* A_S)\,,\nn\\[1mm]
%
J_7 & = & \sqrt{2} \beta_\ell\, \Big[\im (\azeL\apaL^* - \azeR\apaR^* ) +
\frac{m_\ell}{\sqrt{q^2}}\, \im (\apeL A_S^* - \apeR^* A_S)) \Big]\,,\nn\\[1mm]
%
J_8 & = & \frac{1}{\sqrt{2}}\beta_\ell^2\left[\im(\azeL\apeL^* + \azeR\apeR^*)\right]\,,
%
\hspace{1.9cm} J_9 = \beta_\ell^2\left[\im (\apaL^{*}\apeL + \apaR^{*}\apeR)\right] \,,
\label{Js}\nonumber}
}}
\end{minipage}
\vspace*{2.1cm}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Link to effective operators}
{~}
\begin{minipage}{\textwidth}
$\color{JungleGreen}{\Rrightarrow}$ So here is where the magic happens. At leading order the amplitudes can be written as (soft form factors):
{\tiny{
\eqa{
\apeLR &=&\sqrt{2} N m_B(1- \hat s)\bigg[ (\Ceff9 + \Cpeff9) \mp (\C{10} + \Cp{10})
+\frac{2\hat{m}_b}{\hat s} (\Ceff7 + \Cpeff7) \bigg]\xi_{\bot}(E_{K^*}) \nn \\[2mm]
\apaLR &=& -\sqrt{2} N m_B (1-\hat s)\bigg[(\Ceff9 - \Cpeff9) \mp (\C{10} - \Cp{10})
+\frac{2\hat{m}_b}{\hat s}(\Ceff7 - \Cpeff7) \bigg] \xi_{\bot}(E_{K^*}) \nn \\[2mm]
\azeLR &=& -\frac{N m_B (1-\hat s)^2}{2 \hat{m}_{K^*} \sqrt{\hat s}} \bigg[ (\Ceff9 - \Cpeff9) \mp (\C{10} - \Cp{10}) + 2\hat{m}_b (\Ceff7 - \Cpeff7) \bigg]\xi_{\|}(E_{K^*}),
\label{LargeRecoilAs}\nonumber}
}}
where $\hat s = q^2 /m_B^2$, $\hat{m}_i = m_i/m_B$. The $\xi_{\|,\bot }$ are the form factors. \\
\pause
$\color{JungleGreen}{\Rrightarrow}$ Now we can construct observables that cancel the $\xi$ form factors at leading order:
\eq{P_5^{\prime} = \dfrac{J_5+\bar{J}_5}{2\sqrt{-(J_2^c+\bar{J}_2^c)(J_2^s+\bar{J}_2^s)} }\nonumber
}
\end{minipage}
\vspace*{2.1cm}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{$\PBzero \to \PKstar \Pmuon \APmuon$ kinematics}
{~}
\begin{minipage}{\textwidth}
$\color{JungleGreen}{\Rrightarrow}$ The kinematics of $\PBzero \to \PKstar \Pmuon \APmuon$ decay is described by three angles $\thetal$, $\thetak$, $\phi$ and invariant mass of the dimuon system ($q^2)$.
\only<1>{
\begin{columns}
\column{0.5\textwidth}
$\color{JungleGreen}{\Rrightarrow}$ $\cos \thetak$: the angle between the direction of the kaon in the $\PKstar$ ($\overline{\PK}^{\ast}$) rest frame and the direction of the $\PKstar$ ($\overline{\PK}^{\ast}$) in the $\PBzero$ ($\APBzero$) rest frame.\\
$\color{JungleGreen}{\Rrightarrow}$ $\cos \thetal$: the angle between the direction of the $\Pmuon$ ($\APmuon$) in the dimuon rest frame and the direction of the dimuon in the $\PBzero$ ($\APBzero$) rest frame.\\
$\color{JungleGreen}{\Rrightarrow}$ $\phi$: the angle between the plane containing the $\Pmuon$ and $\APmuon$ and the plane containing the kaon and pion from the $\PKstar$.
\column{0.5\textwidth}
\includegraphics[width=0.99\textwidth]{images/angles.png}
\end{columns}
}
\only<2>{
{\tiny{
\eqa{\label{dist}
\frac{d^4\Gamma}{dq^2\,d\!\cos\theta_K\,d\!\cos\theta_l\,d\phi}&=&\frac9{32\pi} \bigg[
{\color{red}{J_{1s}}} \sin^2\theta_K + {\color{red}{J_{1c}}} \cos^2\theta_K + ({\color{red}{J_{2s} }}\sin^2\theta_K + {\color{red}{J_{2c}}} \cos^2\theta_K) \cos 2\theta_l\nn\\[1.5mm]
&&\hspace{-2.7cm}+ {\color{red}{J_3}} \sin^2\theta_K \sin^2\theta_l \cos 2\phi + {\color{red}{J_4}} \sin 2\theta_K \sin 2\theta_l \cos\phi + {\color{red}{J_5}} \sin 2\theta_K \sin\theta_l \cos\phi \nn\\[1.5mm]
&&\hspace{-2.7cm}+ ({\color{red}{J_{6s}}} \sin^2\theta_K + {\color{red}{{J_{6c}}}} \cos^2\theta_K) \cos\theta_l
+ {\color{red}{J_7}} \sin 2\theta_K \sin\theta_l \sin\phi + {\color{red}{J_8}} \sin 2\theta_K \sin 2\theta_l \sin\phi \nn\\[1.5mm]
&&\hspace{-2.7cm}+ {\color{red}{J_9}} \sin^2\theta_K \sin^2\theta_l \sin 2\phi \bigg]\,,
\nonumber}
}}\\{~}\\
$\color{JungleGreen}{\Rrightarrow}$ This is the most general expression of this kind of decay.\\
$\color{JungleGreen}{\Rrightarrow}$ The $CP$ averaged angular observables are defined:\\
\eq{
S_i = \dfrac{J_i+ \bar{J}_i}{(d \Gamma + d \bar{\Gamma})/dq^2}\nonumber
}
}
\end{minipage}
\vspace*{2.1cm}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Link to effective operators}
{~}
\begin{minipage}{\textwidth}
$\color{JungleGreen}{\Rrightarrow}$ The observables ${\color{red}{J_i}}$ are bilinear combinations of transversity amplitudes: $\apeLR,~\apaLR,~\azeLR $. \\
$\color{JungleGreen}{\Rrightarrow}$ So here is where the magic happens. At leading order the amplitudes can be written as:
{\tiny{
\eqa{
\apeLR &=&\sqrt{2} N m_B(1- \hat s)\bigg[ (\Ceff9 + \Cpeff9) \mp (\C{10} + \Cp{10})
+\frac{2\hat{m}_b}{\hat s} (\Ceff7 + \Cpeff7) \bigg]\xi_{\bot}(E_{K^*}) \nn \\[2mm]
\apaLR &=& -\sqrt{2} N m_B (1-\hat s)\bigg[(\Ceff9 - \Cpeff9) \mp (\C{10} - \Cp{10})
+\frac{2\hat{m}_b}{\hat s}(\Ceff7 - \Cpeff7) \bigg] \xi_{\bot}(E_{K^*}) \nn \\[2mm]
\azeLR &=& -\frac{N m_B (1-\hat s)^2}{2 \hat{m}_{K^*} \sqrt{\hat s}} \bigg[ (\Ceff9 - \Cpeff9) \mp (\C{10} - \Cp{10}) + 2\hat{m}_b (\Ceff7 - \Cpeff7) \bigg]\xi_{\|}(E_{K^*}),
\label{LargeRecoilAs}\nonumber}
}}
where $\hat s = q^2 /m_B^2$, $\hat{m}_i = m_i/m_B$. The $\xi_{\|,\bot }$ are the soft form factors. \\
\pause
$\color{JungleGreen}{\Rrightarrow}$ Now we can construct observables that cancel the $\xi$ soft form factors at leading order:
\eq{P_5^{\prime} = \dfrac{J_5+\bar{J}_5}{2\sqrt{-(J_2^c+\bar{J}_2^c)(J_2^s+\bar{J}_2^s)} }\nonumber
}
\end{minipage}
\vspace*{2.1cm}
\end{frame}
% symmetries
\begin{frame}{Symmetries in $\PB \to \PKstar \Pmu \Pmu$}
{~}
\begin{minipage}{\textwidth}
$\color{JungleGreen}{\Rrightarrow}$ We have 12 angular coefficients ($S_i$).\\
$\color{JungleGreen}{\Rrightarrow}$ There exist 4 symmetry transformations that leave the angular distributions unchanged:
\begin{tiny}
\eq{
n_\|=\binom{A_\|^L}{A_\|^{R*}}\ ,\quad
n_\bot=\binom{A_\bot^L}{-A_\bot^{R*}}\ ,\quad
n_0=\binom{A_0^L}{A_0^{R*}}\ .\nonumber
}
\end{tiny}
\begin{tiny}
\eq{
n_i^{'} = U n_i=
\left[
\begin{array}{ll}
e^{i\phi_L} & 0 \\
0 & e^{-i \phi_R}
\end{array}
\right]
\left[
\begin{array}{rr}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\end{array}
\right]
\left[
\begin{array}{rr}
\cosh i \tilde{\theta} & -\sinh i \tilde{\theta} \\
- \sinh i \tilde{\theta} & \cosh i \tilde{\theta}
\end{array}
\right]
n_i \,.
\label{symmassless}\nonumber}
\end{tiny}
$\color{JungleGreen}{\Rrightarrow}$ Using this symmetries one can show that there are 8 independent observables. The pdf can be written as:
\begin{tiny}
\begin{align*}
\left.\frac{1}{{\rm d}(\Gamma+\bar{\Gamma})/{\rm d}q^2}\frac{{\rm d}(\Gamma+\bar{\Gamma})}{{\rm dcos}\thetal\,{\rm dcos}\thetak\,{\rm d}\phi}\right|_{\rm P} =
\tfrac{9}{32\pi}\bigl[
&\tfrac{3}{4} (1-{F_{\rm L}})\sin^2\thetak \label{eq:pdfpwave}\\[-0.75em]
&+ {F_{\rm L}}\cos^2\thetak + \tfrac{1}{4}(1-{F_{\rm L}})\sin^2\thetak\cos 2\thetal\nonumber\\
&- {F_{\rm L}} \cos^2\thetak\cos 2\thetal + {S_3}\sin^2\thetak \sin^2\thetal \cos 2\phi\nonumber\\
&+ {S_4} \sin 2\thetak \sin 2\thetal \cos\phi + {S_5}\sin 2\thetak \sin \thetal \cos \phi\nonumber\\
&+ \tfrac{4}{3} {A_{\rm FB}} \sin^2\thetak \cos\thetal + {S_7} \sin 2\thetak \sin\thetal \sin\phi\nonumber\\
&+ {S_8} \sin 2\thetak \sin 2\thetal \sin\phi + {S_9}\sin^2\thetak \sin^2\thetal \sin 2\phi \nonumber
\bigr].
%\end{split}
%\bigr],\
\end{align*}
\end{tiny}
\end{minipage}
\vspace*{2.1cm}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\backupend
\end{document}
mchrzasz