PhD-Kopecna-Renata/Chapters/AnalysisTheory/anaTheroy.tex

\section[Theoretical introduction to \texorpdfstring{${\BuToKstmm}$}{BuToKstmm}]
{Theoretical introduction to \texorpdfstring{$\BuToKstmmBF$}{BuToKstmm}}\label{sec:ANA_Theo}

\vspace{-5pt}
The theory of \bsll transitions is generally introduced in \refSec{SM_bsll}. In this section, this general introduction is broadened using the specific example of \BuToKstmm decay. Note that throughout the thesis the inclusion of opposite charged decays (\ie\xspace \BuToKstmmCH) is implied. The vector meson \Kstarp decays further in a \Kp\piz pair. The \Kstarp meson considered in this works is the $\Kstarp(892)$ resonance with natural width of $50.8\pm0.8\mev$~\cite{PDG}, spin 1, and parity -1.

%The \BuToKstmm decay is a FCNC transition. Therefore it can only occur through loop diagrams in the SM. The lowest SM diagrams that contribute to the decay are two penguin diagrams exchanging a photon or a \Z boson and a box diagram exchanging two $\W$ bosons (see \refFig{penguin_bsll}). The amplitudes can be expressed in terms of Wilson coefficients \C7, \C9 and \C10. 

The decay of \BuToKstKppizmm was first observed by the BaBar collaboration~\cite{ANA-KstMuMuDecayEvidence}.  As the final state of the decay contains neutral particles, which are challenging to reconstruct, the angular analysis including this decay has been so far only performed by BaBar~\cite{ANA-BaBarAngDist} and Belle~\cite{ANA-BelleAngDist}. %Recently, studies of \BuToKstKspimm have been carried out by the  LHCb~\cite{ANA-LHCb-angular4}, and CMS~\cite{ANA-CMS-angular} collaborations.

The \BuToKstmm decay is a FCNC transition. Therefore it can only occur through loop diagrams in the SM. The lowest SM diagrams that contribute to the decay are two penguin diagrams exchanging a photon or a \Z boson and a box diagram exchanging two $\W$ bosons, as shown in \refFig{penguin_bsll}). % The amplitudes can be expressed in terms of Wilson coefficients \C7, \C9 and \C10.
Looking back at \refSec{SM_bsll}, the contribution to the effective hamiltonian is polluted by QCD contributions (see \refFig{bsll_meson}). One way to validate the form-factor corrections to the decay is to change the \emph{spectator} quark in the decay: swapping the \uquark quark and \dquark quark changes the decay from \BuToKstmm to \BdToKstmm.  Hence, it is important to study the \BuToKstmm decay and compare the results to previous extensive measurements of \BdToKstmm decay.


\subsection{Decay topology}\label{sec:ANA_Theo_Topo}

Due to the spin structure of the decay \BuToKstmm, the differential decay rate can be fully expressed using only four variables: the dimuon invariant mass squared \qsq and the three angles defined by the direction of flight of the decay products: \thetak, \thetal and $\phi$. These angles are shown in \refFig{anglesB+}. Denoting the normalized vector of a particle X in the rest frame of Y, $\hat{p}_X^{(Y)}$, the angles can be defined as in \refEq{anglesB+}:
This definition is compatible with previous \lhcb measurements~\cite{ANA-LHCb-angular3, ANA-LHCb-angular4, ANA-LHCb-angular1,ANA-LHCb-angular2}.
%
\begin{align} \label{eq:anglesB+}\begin{split}
\cos\thetal =& \left(\hat{p}_{\mupm}^{(\mumu)}\right) \cdot  \left(\hat{p}_{\mumu}^{(\Bpm)}\right)
= \left(\hat{p}_{\mupm}^{(\mumu)}\right) \cdot  \left(-\hat{p}^{\mumu}_{(\Bpm)}\right)\,,\\
%
\cos\thetak =& \left(\hat{p}_{\Kpm}^{(\Kstarpm)}\right) \cdot
\left(\hat{p}_{\Kstarpm}^{(\Bpm)}\right) = \left(\hat{p}_{\Kpm}^{(\Kstarpm)}\right)  \cdot
\left(-\hat{p}^{\Kstarpm}_{(\Bpm)}\right)\,,\\
%
\cos\phi =& \left[\left(\hat{p}_{\mupm}^{(\Bpm)}\right) \times \left(\hat{p}_{\mump}^{(\Bpm)}\right)\right]
\cdot  \left[\left(\hat{p}_{\Kpm}^{(\Bpm)}\right) \times \left(\hat{p}_{\piz}^{(\Bpm)}\right)\right]\,,\\
%
\sin\phi =& \left[\left(\hat{p}_{\mupm}^{(\Bpm)}\right) \times \left(\hat{p}_{\mump}^{(\Bpm)}\right)\right]
\times  \left[\left(\hat{p}_{\Kpm}^{(\Bpm)}\right) \times \left(\hat{p}_{\piz}^{(\Bpm)}\right)\right]
\cdot  \left(\hat{p}_{\Kstarpm}^{(\Bpm)}\right)\,.\\
%
\end{split} \end{align}
%

\begin{figure}[htb!]  \centering  
	%\hspace{-20pt}
	\includegraphics[width=0.65\textwidth]{./AnalysisTheory/anglesB+.pdf} 
	\captionof{figure}[Definition of the angles in the \BuToKstmm decay.]
	{
		Definition of the angles in the \BuToKstmm decay. The angle \thetal is 
		defined as the angle between the \mup flight direction in the \mup\mun rest frame and the flight direction of the \mup\mun pair in the \Bu meson rest frame. Similarly, \thetak is defined as the angle between the \Kp in the rest frame of \Kstarp and the flight direction of \Kstarp in the 
		\Bu meson rest frame. Finaly, the angle $\phi$ is the angle between the normal vector of the \Kp\piz system and the normal vector of the \mup\mun system. \label{fig:anglesB+}
	}
\end{figure}

\subsection{Differential decay rate}\label{sec:ANA_Theo_BR}

As mentioned in the previous section, the differential decay rate of \BuToKstmm can be fully expressed using only four variables: the dimuon invariant mass squared  \qsq and the angles \thetak, \thetal and $\phi$. The decay rate then takes the form of
%
\begin{equation}\label{eq:decay_rate}
    \frac{\deriv^4\Gamma}{\deriv\cos\thetak\deriv\cos\thetak\deriv\phi\deriv q^2} = \frac{9}{32\pi}\sum_i{ J_i\left(q^2\right)f_i\left(\cos\thetal,\cos\thetak,\phi\right)}\,,\\
\end{equation}
%
where $f_i$ are angular terms only depending on the three decay angles. They are given purely by the spin structure of the decay. The coefficients $J_i$ are angular observables depending on the dimuon invariant mass squared \qsq. They 'control' how much the different $f_i$ terms contribute to the differential decay rate. For the full form, see \refApp{ANA-Theo}, \refEq{decay_rate_full}. The $J_i$ coefficients are directly connected to the Wilson coefficients listed in \refTab{SM_Wilson_sensitivity}. For the exact relation formulas, see~Ref.\,\cite{ANA-wilsonJi}.


Similarly, the decay rate of \BuToKstmmCH can be expressed in terms of $\bar{J_i}$. Given the CP asymmetry of this decay in the Standard Model is negligibly small, it is useful to define a set of \emph{CP averaged} angular observables $S_i$ instead of having a set of $J_i$ and $\bar{J_i}$:
%
\begin{equation}\label{eq:Si_definition}
    S_i = \frac{J_i+\bar{J_i}}{\Gamma+\bar{\Gamma}}\,.\\
\end{equation}


The dependence of the decay rate on the dimuon invariant mass squared \qsq is sketched in \refFig{q2_theory}. At $q^2\approx0$ the dominating operator is \Ope7. Then, between 1\gevgev$\lesssim\qsq\lesssim8\gevgev$, the interference of \Ope7 and \Ope9 plays a role. In the region of charm resonances, the decay is dominated by tree-level diagrams. Above the resonances, $q^2\gtrsim15\gevgev$, the operators \Ope9 and \Ope10 dominate. The observables are measured in bins of \qsq. In each bin, the \qsq-averaged observables are defined as
%
\begin{equation}\label{eq:S_i}
\langle S_i\rangle \left(q_{min},q_{max}\right)
= \frac{\int_{q_{min}}^{q_{max}}\deriv q^2
	\left(J_i + \bar{J_i}\right)}{\int_{q_{min}}^{q_{max}}\deriv q^2 \frac{ \deriv\left(\Gamma + \bar{\Gamma}\right)}{q^2}}\,.
\end{equation}

Following \refEq{decay_rate}, the available $S_i$ parameters are $S_{1s,6s}$ and $S_{3,4,5,7,8,9}$. This basis is also convenient from the experimental point of view: as the \BuToKstmm decay is a rare decay, measuring the $S_i$ rather than the $J_i$ and the $\bar{J}_i$ observables effectively doubles the signal yield. The $S_i$ observables are (linearly) connected to two historical observables: the forward-backward asymmetry of the \mumu pair, \AFB, and longitudinal polarization of \Kstar, \FL:
%
\begin{align}\label{eq:FLAFB_definition}\begin{split}
F_L &= 1 - \frac{4}{3} S_{1s}\,,\\
A_ {FB} &= \frac{3}{4} S_{6s}\\
\end{split}\end{align} 

As mentioned in \refSec{SM_bsll}, there is a non-negligible form-factor contribution to the decay rate. The influence of form-factor uncertainties can be transformed in a way that the theoretical uncertainties mostly cancel when studying a \emph{single} parameter. The uncertainties are then shifted to other observables. Taking into account all correlations between the angular moments, this basis does not bring any advantage. In the scope of this work the main advantage of this basis is the possibility of a direct comparison to previous \lhcb measurements and measurements of the angular observables performed by other experiments. The basis is expressed as a set of \Pprime{i} observables and \FL:
%
%\begin{align}\label{eq:P'_definition}\begin{split}
%P'_1 & = \frac{S_3}{1-F_L}\,,\\
%P'_2 & = \frac{S_{6s}}{1-F_L}\,,\\
%P'_3 & = \frac{S_9}{1-F_L}\,,\\
%P^{'}_4 & = \frac{S_4}{\sqrt{F_L\left(1-F_L\right)}}\,,\\
%P^{'}_5 & = \frac{S_5}{\sqrt{F_L\left(1-F_L\right)}}\,,\\
%P^{'}_6 & = \frac{S_7}{\sqrt{F_L\left(1-F_L\right)}}\,,\\
%P^{'}_8 & = \frac{S_8}{\sqrt{F_L\left(1-F_L\right)}}\\
%\end{split}\end{align} 
% 
\begin{align}\label{eq:P'_definition}\begin{split}
 P'_1  = \frac{S_3}{1-F_L}\,, \qquad\qquad\qquad P^{'}_4 & = \frac{S_4}{\sqrt{F_L\left(1-F_L\right)}}\,,\\ 
 P'_2  = \frac{S_{6s}}{1-F_L}\,, \qquad\qquad\qquad P^{'}_5 & = \frac{S_5}{\sqrt{F_L\left(1-F_L\right)}}\,,\\
 P'_3  = \frac{S_9}{1-F_L}\,, \qquad\qquad\qquad P^{'}_6 & = \frac{S_7}{\sqrt{F_L\left(1-F_L\right)}}\,,\\
  P^{'}_8 & = \frac{S_8}{\sqrt{F_L\left(1-F_L\right)}}\\
\end{split}\end{align} 

 
\subsection{\swave contribution}\label{sec:ANA_Theo_SWave}

The decay rate, as described by \refEq{decay_rate_full}, takes into account only the decay \BuToKstmm, where the \Kstar decays via \KstToKpPi. This is referred to as the \pwave. 
 However, in the measurement one has to consider the possible contributions from other higher \Kstarp resonances \eg $K^{*+}(1430)_0$ (\swave). As the \Kstarp resonance is very wide in mass, it is very difficult to distinguish the \swave component from the \pwave in the selection process. By considering only the events with the reconstructed $\Kstarp(892)$ mass being close to the $\Kstarp(892)$  rest mass ($|m_{\Kstar}^{reco}-m_{\Kstar}|<100\mev$), the \swave contribution is suppressed, but not fully eliminated.
  The \swave component has a different angular structure and therefore pollutes the angle \angles distributions.
The decay rate of the \swave is
\begin{equation}\label{eq:decay_rate_S}
\left.
\frac{{\rm d}(\Gamma+\bar{\Gamma})}{{\rm dcos}\theta_L\,{\rm dcos}\theta_K\,{\rm d}\phi}
\right |_{\rm S} 
=\frac{3}{16\pi}{F_S}\sin^2\theta_L\,.
\end{equation}
However, both \pwave and \swave amplitudes are complex. The addition of them results in interference term: % Both the \swave and interference term need to be added to the full decay rate.
%
\begin{equation}\label{eq:decay_rate_PS}
\left.
\frac{{\rm d}(\Gamma+\bar{\Gamma})}{{\rm dcos}\theta_L\,{\rm dcos}\theta_K\,{\rm d}\phi}
\right |_{\rm PS} 
 =\frac{3}{16\pi}\sum_1^5 S_{Si} f_{S_i}(\theta_L,\theta_K\,\phi)\,. 
\end{equation}
%
The full expression is given in \refEq{decay_rate_PS_full}. For the full angular description of the \BuToKstmm decay, both the \swave and the interference terms have to be added to \refEq{decay_rate}.




%In this work, due to limited statistical power of the selected dataset, the \swave contribution is fixed according to previous measurement in~Ref.\,\cite{ANA-LHCb-angular4}. \todoAddCitation{Check}
  
\subsection{Angular observables}\label{sec:ANA_Theo_Angular}

Using  \pwave term from \refEq{decay_rate}, \swave term from 
\refEq{decay_rate_S}, and their interference term \refEq{decay_rate_PS} it is possible to construct the full angular description of \BuToKstmm. The full procedure is described in \refApp{ANA-Theo}. The full decay description is then given by \refEq{decay_rate_final}.
%
In total, there are eight moments related to the \pwave contribution and six moments related to the \swave contribution and its interference with the \pwave. The observables are for readers convenience shown in red. Each of the observable is measured in bins of \qsq. In order to measure all these variables, the statistical power of the measured sample is required to be rather large.


\vspace{-20pt}
\begin{equation} \label{eq:decay_rate_final} 
\begin{aligned}
	\left.
	\frac{{\rm d}(\Gamma+\bar{\Gamma})}{{\rm dcos}\theta_L\,{\rm dcos}\theta_K\,{\rm d}\phi}
	\right |_{\rm S+P} 
	=& \left.\frac{{\rm d}(\Gamma+\bar{\Gamma})}{{\rm dcos}\theta_L\,{\rm dcos}\theta_K\,{\rm d}\phi}
	\right |_{\rm P} 
	+&& \left.\frac{{\rm d}(\Gamma+\bar{\Gamma})}{{\rm dcos}\theta_L\,{\rm dcos}\theta_K\,{\rm d}\phi}
	\right |_{\rm S} 
	+ \left.\frac{{\rm d}(\Gamma+\bar{\Gamma})}{{\rm dcos}\theta_L\,{\rm dcos}\theta_K\,{\rm d}\phi}
	\right |_{\rm PS}\\
	=(1-\textcolor{red}{F_S})\frac{9}{32\pi}\Bigl[&\tfrac{3}{4}(1-\textcolor{red}{F_{L}})\sin^2\theta_K &&+ \textcolor{red}{F_{L}}\cos^2\theta_K \\
	+& \tfrac{1}{4}(1-\textcolor{red}{F_{L}})\sin^2\theta_K\cos 2\theta_L &&- \textcolor{red}{F_{L}} \cos^2\theta_K\cos 2\theta_L \\
	+& \textcolor{red}{S_3}\sin^2\theta_K \sin^2\theta_L \cos 2\phi &&+ \textcolor{red}{S_4} \sin 2\theta_K \sin 2\theta_L \cos\phi \\
	+& \textcolor{red}{S_5}\sin 2\theta_K \sin \theta_L \cos \phi &&+ \tfrac{3}{4}\textcolor{red}{A_{FB}} \in^2\theta_K \cos\theta_L \\
	+& \textcolor{red}{S_7} \sin 2\theta_K \sin\theta_L \sin\phi &&+ \textcolor{red}{S_8} \sin 2\theta_K \sin 2\theta_L \sin\phi\\
	+& \textcolor{red}{S_9}\sin^2\theta_K \sin^2\theta_L \sin 2\phi 
	\Bigr]\\
	+\frac{3}{16\pi}\Bigl[&\textcolor{red}{F_S}\sin^2\theta_L &&+\textcolor{red}{S_{S1}}\sin^2\theta_L\cos\theta_K \\
	+& \textcolor{red}{S_{S2}}\sin2\theta_L \sin \theta_K \cos\phi &&+ \textcolor{red}{S_{S3}}\sin\theta_L \sin \theta_K \cos\phi \\
	+& \textcolor{red}{S_{S4}}\sin\theta_L \sin \theta_K \sin\phi &&+ \textcolor{red}{S_{S5}}\sin2\theta_L \sin \theta_K \sin\phi
	\Bigr].
\end{aligned}\end{equation}


\subsection{Folding of angles}\label{sec:ANA_folding}

In total, there are 14 angular observables for each \qsq bin. The full description also requires the parametrization of the mass distributions of the \Bu and \Kstar mesons, adding more free parameters to the decay description. Moreover, the background contribution needs to be modeled in all dimensions.  The expected signal yield in this rare decay is in the order of less than a hundred events in each \qsq bin.  This can result in a great instability in the data fit.

In order to improve the stability, a dedicated procedure is implemented. The \emph{folding} of angles is an angular transformation exploiting the symmetry of the angular functions in \refEq{decay_rate_final}. An example is a tranformation of $\phi\to\phi+\pi$ for signal candidates with $\phi < 0$. This results in canceling out the terms dependent on $\cos\phi$ and $\sin\phi$ and leaves the \refEq{decay_rate_final} only with observables $F_L$, $S_3$, \AFB, $S_9$ (and $F_S$ and $S_{S1}$). This method has been successfully applied in previous measurements, see \eg~Ref.\,\cite{ANA-LHCb_P5_AnaNote}.

Using a total of five foldings, all observables of interest are accessible. They are listed in \refEq{foldings}. This procedure greatly increases the stability of the data fit as most of the observables are canceled out. The price to pay is the loss of information about correlations between the observables.
\clearpage 

%
\setstretch{1.0}
\begin{equation}\label{eq:foldings}
\begin{aligned}
{\rm \textbf{folding 0:}}\\
\phi 	&\;\to		\;\phi+\pi 		&&\;{\rm for}\; 	\phi < 0\\[10pt]
{\rm \textbf{folding 1:}}\\
\phi 	&\;\to		\;-\phi  		&&\;{\rm for}\; 	\phi < 0\\
\phi		&\;\to		\pi-\phi 		&&\;{\rm for}\;	\ctl < 0\\
\ctl		&\;\to		\;-\ctl			&&\;{\rm for}\;	\ctl < 0\\[10pt]
{\rm \textbf{folding 2:}}\\
\phi 	&\;\to		\;-\phi  		&&\;{\rm for}\; 	\phi < 0\\
\ctl		&\;\to		\;-\ctl			&&\;{\rm for}\;	\ctl < 0\\[10pt]
{\rm \textbf{folding 3:}}\\
\ctl		&\;\to		\;-\ctl			&&\;{\rm for}\;	\ctl < 0\\
\phi		&\;\to		\;\pi-\phi		&&\;{\rm for}\;	\phi > \sfrac{\pi}{2}\\
\phi		&\;\to		\;-\pi-\phi		&&\;{\rm for}\;	\phi < -\sfrac{\pi}{2}\\[10pt]
{\rm \textbf{folding 4:}}\\
\ctk		&\;\to		\;-\ctk	&&\;{\rm for}\;	\ctl < 0\\
\ctl		&\;\to		\;-\ctl			&&\;{\rm for}\;	\ctl < 0\\
\phi		&\;\to		\;\pi-\phi		&&\;{\rm for}\;	\phi > \sfrac{\pi}{2}\\
\phi		&\;\to		\;-\pi-\phi		&&\;{\rm for}\;	\phi < -\sfrac{\pi}{2}\\
\end{aligned}\end{equation}\setstretch{1.25}


A tabular overview of the sensitivity of the angular folding to \pwave angular moments is presented in \refTab{folding}. Using all the five angular foldings gives access to all eight \pwave angular moments. The \swave angular moments sensitivity is shown in \refTab{folding-s}.

\begin{table}[hbt!] \centering
	\begin{tabular}{crccccc}\hline
		observable	&moment	&0	&1	&2	&3	&4\\		\hline\hline
		$S_{1s}$	&$\cos^2\theta_K$							&\checkmark	&\checkmark	&\checkmark	&\checkmark	&\checkmark	\\
		$S_{3}$		&$\sin^2\theta_K \sin^2\theta_L \cos 2\phi$	&\checkmark	&\checkmark	&\checkmark	&\checkmark	&\checkmark	\\
		$S_{4}$		&$\sin 2\theta_K \sin 2\theta_L \cos\phi$	&		-	&\checkmark	&	-		&	-		&	-		\\
		$S_{5}$		&$\sin 2\theta_K \sin \theta_L \cos \phi$	&	-		&	-		&\checkmark	&	-		&	-		\\
		$S_{6s}$	&$\sin^2\theta_K \cos\theta_L$				&\checkmark	&	-		&	-		&	-		&	-		\\
		$S_{7}$		&$\sin 2\theta_K \sin\theta_L \sin\phi$		&	-		&	-		&	-		&\checkmark	&	-		\\
		$S_{8}$		&$\sin 2\theta_K \sin 2\theta_L \sin\phi$	&	-		&	-		&	-		&	-		&\checkmark	\\
		$S_{9}$		&$\sin^2\theta_K \sin^2\theta_L \sin 2\phi$	&\checkmark	&	-		&	-		&	-		&	-		\\
		\hline
	\end{tabular}	
	\captionof{table}[Angular folding sensitivity to \pwave angular moments.]{
		Angular folding sensitivity to \pwave angular moments. \label{tab:folding}
	}
\end{table}


\begin{table}[hbt!] \centering
	\begin{tabular}{crccccc}\hline
observable	&moment			  &0	&1	&2	&3	&4\\		\hline\hline
$F_{S}$	    &$\sin^2\theta_L$ 						&\checkmark	&\checkmark	&\checkmark	&\checkmark	&\checkmark	\\
$S_{S1}$	&$\ctk\sin^2\theta_L$					&\checkmark	&\checkmark	&\checkmark	&\checkmark	&    -	\\
$S_{S2}$	&$\sin\theta_K \sin2\theta_L \cos\phi$	&		-	&\checkmark	&	-		&	-		&	-		\\
$S_{S3}$	&$\sin\theta_K \sin\theta_L \cos\phi$	&	-		&	-		&\checkmark	&	-		&	-		\\
$S_{S4}$	&$\sin\theta_K \cos\theta_L \sin\phi$	&  	-		&	-		&	-		&	\checkmark		&	\checkmark		\\
$S_{S5}$	&$\sin\theta_K \sin2\theta_L \sin\phi$	&	-		&	-		&	-	& - &	-		\\
		\hline
	\end{tabular}	
	\captionof{table}[Angular folding sensitivity to \swave angular moments.]{
		Angular folding sensitivity to \swave angular moments. \label{tab:folding-s}
	}
\end{table}

\subsection{Previous measurements}\label{sec:ANA_previous}

Experimentally, there are two main ways of studying  the \bsll transitions: measurements of branching ratios, and of angular observables\footnote{\bsll transitions are also an important tool in studying lepton flavor universality. However, this is beyond the scope of this work.}. First measurements of branching fractions were agreeing with the SM predictions~\cite{ANA-BR-Belle,ANA-BR-CDF,ANA-BR-LHCb}, as the statistical power of the measurements did not allow for precision measurements. However, with more available data first discrepancies started to appear, such as in the  differential branching fractions measurement of $\Bu\to K^{(*)}\mup\mun$ decays~\cite{SM-LHCb_BR} or in the  $\Bs\to\phi\mup\mun$ branching fraction measurement~\cite{ANA-BR-BPHI-LHCb}.

% The results from this study are shown in \refFig{BR_LHCb}. A clear discrepancy between data and SM calculation appears in different decay channels. Similarly a discrepancy of over 3~standard deviations $\sigma$ appeared . All these measurements suggest branching-ratios below the SM predictions. It is interesting to note here that with increasing statistics in the experiments the discrepancy would remain around 3\stdev due to large (hadronic) theory uncertainties. 
%
%\begin{figure}[htb!]  \centering  
%    \includegraphics[width=0.32\textwidth]{./AnalysisTheory/kmumu_BF.pdf}
%    \includegraphics[width=0.32\textwidth]{./AnalysisTheory/ksmumu_BF.pdf}
%    \includegraphics[width=0.32\textwidth]{./AnalysisTheory/bukst_BF.pdf}
%    \captionof{figure}[ Differnetal branching fraction results for the \Bu\to\Kp\mup\mun, \Bd\to\Kz\mup\mun and \BdToKstmm decays.]
%    {
%        Differential branching fraction measurement of the \Bu\to\Kp\mup\mun, \Bd\to\Kz\mup\mun and \BuToKstmm decays. The uncertainties shown on the data points are the quadratic sum
%        of the statistical and systematic uncertainties. The shaded regions illustrate the theoretical
%        predictions and their uncertainties from LCSR and lattice QCD calculations. In \qsq below \jpsi, measured BR are consistently below SM prediction value. Taken from~Ref.\,\cite{SM-LHCb_BR}. \label{fig:BR_LHCb}
%    }
%\end{figure}

A discrepancy between a measurement and the SM predictions of angular observables appeared already in 2013, when \lhcb analyzed 1\invfb of data in the decay of \BdToKstmm~\cite{ANA-LHCb-angular1}. One out of 24 measurements (four \Pprime{} parameters in six bins of \qsq) is 3.7\stdev away from the SM prediction. The parameter is \Pprime{5}. If there is a New Physics contribution in the Wilson Coefficients \C9 and \C10, it would show first in the \Pprime{5}: this discrepancy sparked a lot of interested.

Since then, many similar measurements have been performed~\cite{ANA-LHCb-angular3, ANA-LHCb-angular2,ANA-Belle_P5,ANA-CMS_P5,ANA-ATLAS_P5}. These measurements are summarized in \refFig{P5prime_All}. The latest \lhcb result using the \BdToKstmm decay~\cite{ANA-LHCb-angular3} is yet not present in the figure. In this last measurement, the \Pprime{5} discrepancy in low \qsq increased from 2.4\stdev in~Ref.\,\cite{ANA-LHCb-angular2} to 2.8\stdev. %On the other hand, the angular analysis of \Bs\to$\phi$\mumu measured the angular observables in agreement with the Standard Model predictions~\cite{ANA-BR-BPHI-LHCb2}. 

\begin{figure}[htb!]  \centering  
    \includegraphics[width=0.65\textwidth]{./AnalysisTheory/P5prime_all.pdf}
    \captionof{figure}[Measurements of the parameter \Pprime{5} compared to theory predictions.]
    {Measurements of the optimized angular observable  \Pprime{5} in bins of \qsq. The shaded areas represent charmonium resonances that are dominated by tree-level diagrams.  Experimental results are taken from~Ref.\,\cite{ANA-LHCb-angular2,ANA-Belle_P5,ANA-CMS_P5,ANA-ATLAS_P5}, theory predictions are taken from~Ref.\,\cite{ANA-P5-theo1,ANA-P5-theo2,ANA-P5-theo3,ANA-P5-theo4}. \label{fig:P5prime_All}
    }
\end{figure}

Moreover, the first angular study of \BuToKstKspimm at \lhcb~\cite{ANA-LHCb-angular4} has been recently published. The \Pprime{5} measured in eight \qsq bins is shown in \refFig{P5prime_David}. A global evaluation of the result in terms of the real part of the Wilson coefficient \C9 prefers a shift of Re(\C9 )=-1.9 from the Standard Model value with a significance of 3.1 standard deviations.

\begin{figure}[htb!]  \centering  
	\includegraphics[width=0.65\textwidth]{./AnalysisTheory/P5_David.pdf}
	\captionof{figure}[Measurement of the parameter \Pprime{5} in the \BuToKstKspimm decay compared to theory predictions.]
	{Measurements of the optimized angular observable \Pprime{5} in bins of \qsq from  \lhcb in decay of \BuToKstKspimm. The shaded areas represent charmonium resonances that are dominated by tree-level diagrams and $\phi$ pollution in the region around 1\gevgev.  Experimental results are taken from~Ref.\,\cite{ANA-LHCb-angular4}, theory predictions are obtained from~Ref.\,\cite{ANA-P5-theo3,ANA-P5-theo5} using the \flavio package~\cite{ANA-flavio}. \label{fig:P5prime_David}
	}
\end{figure}

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It is clear there are numerous independent measurements in tension with the Standard Model in the order of 2-3\stdev. Similar tensions are also observed in the lepton flavor universality tests~\cite{ANA-LFU-1,ANA-LFU-2,ANA-LFU-3}. Moreover, there are similar tensions outside of the \bsll scope, such as the $R_{D}$ and $R_{D^*}$ measurement~\cite{ANA-LHCb-RD} or the $R_\jpsi$ measurement~\cite{ANA-LHCb-Rjpsi}. They all are consistent with each other, painting a picture of possible New Physics contribution. Specifically, New Physics contribution to the Wilson coefficients \C9 and \C10 may contribute to the anomalies~\cite{ANA-wilsonNP}. An example of a global fit to all these measurements is shown in \refFig{ANA-Moriond}. Further measurements, such as the work presented here, and improved theory calculations will cast light on these tensions in the near future. \vspace{\baselineskip} 

\begin{figure}[htb!]  \centering  
	\includegraphics[width=0.65\textwidth]{AnalysisIntro/C9-C10_allExperiments.pdf} 
	\captionof{figure}[Constraints to the NP contribution to Wilson coefficients $C_9$ and $C_{10}$.]
	{Constraints to the New Physics contribution to Wilson coefficients $C_9$ and $C_{10}$ taken from ~\cite{ANA-Moriond2017}. All other Wilson coefficients are assumed to have Standard Model values. The bands represent the constrains from \B\to\Kstar\mup\mun and \Bs\to$\phi$\mup\mun measurements performed by listed collaborations, the countours represent one standard deviation $\sigma$. Branchingratio only measurements are shown as the yellow band. The global fit of these results is represented in red with the one, two and three $\sigma$ contours. In the case of no New Physics contribution, the $C_9^{NP}$ and $C_{10}^{NP}$ are equal to zero. Note that the global fit is however dominated by the \lhcb results and \cms measurements are compatible with the Standard Model. For the details about the global fit procedure, see~Ref.\,\cite{ANA-Moriond2017}. \label{fig:ANA-Moriond}
	}
\end{figure}



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