\section{Introduction to the analysis}\label{sec:AnaIntro}

The aim of this thesis is to study angular parameters \Si{i} and \Pprime{i} in the decay of \BuToKstmm where \KstToKpPi. This chapter gives an overview of the analysis procedure.  The biggest obstacle in this analysis is the presence of a neutral pion in the final state. Therefore, the reconstruction of the neutral pion is discussed in detail. Then, as discussed thoroughly in \refSec{SM_bsll}, the split of data into different classes based on the dimuon invariant mass squared \qsq is explained. Lastly, the simulation samples used in this analysis are listed.


\subsection{Analysis strategy}\label{sec:AnaIntro-strategy}

The analysis uses data from the full \runI and \runII data taking periods. During this time, the \lhcb experiment collected a dataset corresponding to an integrated luminosity of 9\invfb. The integrated luminosity $\int$\lum, the beam energy \Ebeam and the center-of-mass energy \sqs for each data-taking year is given in \refTab{ANA-dataset}. 

In the first step of this analysis the recorded data is filtered to select only events containing the signal decay. The decay of \BuToKstmm is a rare decay with a branching fraction  $\BR_{\BuToKstmm}=(9.6\pm1.0)\times10^{-7}$~\cite{PDG}. In order to get the full branching fraction of the \BuToKstmmFull decay, the branching fraction is multiplied by the branching fraction of $\BR_{\KstToKpPi}=0.5$~\cite{PDG}. This leads to very strict requirements on data selection: the background rejection needs to be as high as possible while keeping high signal selection efficiency. 

At first, the data is selected centrally: events have to pass the trigger (online) selection and then the centrally-processed selection called \emph{stripping}. After this, specific preselection cuts are applied.
%
\begin{table}[hbt!]
	\begin{center}\begin{tabular}{cccc} %\hline
			year &$\int$\lum\,[fb\textsuperscript{-1}] &\Ebeam\,[TeV] & \sqs\,[TeV]\\
			\hline
			2011 	&1.11	&3.5 	&7.0\\
			2012 	&2.08	&4.0 	&8.0\\
			2015 	&0.33	&6.5 	&13.0\\
			2016 	&1.67	&6.5 	&13.0\\
			2017 	&1.71	&6.5 	&13.0\\
			2018 	&2.19	&6.5 	&13.0\\
			%\hline
	\end{tabular}\end{center}
	\captionof{table}[Data recording conditions for \lhcb in the years 2011-2018.]{
		Data recording conditions for \lhcb in the years 2011-2018. For each year, the recorded integrated luminosity $\int$\lum, beam energy \Ebeam and center-of-mass energy $\sqs$ are given.  \label{tab:ANA-dataset}
	}
\end{table} %\vspace{\baselineskip}
% 
The remaining background is studied and identified with the help of multi-variate classifiers. The selection is validated using the \BuToKstJpsi decay. 
This \emph{reference} decay has the same final-state particles as the \emph{signal} decay \BuToKstmm and very similar kinematics.  Moreover, due to the branching ratio of \BR(\BuToKstJpsi, \JpsiTomm)$=(1.43\times10^{-3})\times(5.96\times10^{-2})=8.52\times10^{-5}$~\cite{PDG}, the reference channel is $\sim$ 200 times more abundant than the signal channel.





After choosing the optimal selection criteria, the angular acceptance correction is applied. The \lhcb acceptance covers only the forward region. Moreover, all the subdetectors cover different regions in phase-space. For the accurate measurement of angles, angular acceptance corrections are crucial. 

Lastly, the measured angle distributions are fitted in order to extract the \Si{i} parameters. As one can see from \refEq{decay_rate_final}, the distribution is not trivial and the fit requires careful approach, especially given the limited size of the sample. The fit is done in the reconstructed mass of the \Bu meson \BMass, the reconstructed mass of the \Kstarp meson \KstarMass and \ctk, \ctl and $\phi$ dimensions. The reference channel is used to validate the fitter framework. A pseudoexperiment study is performed in order to examine the possible sensitivity of this measurement. 


\subsection{The neutral pion reconstruction}\label{sec:AnaIntro-piz}

It is mentioned in \refSec{ANA_Theo} that the available measurements of the \BuToKstmm decay are not as extensive as the measurements of the \BdToKstmm decay. The main reason is the intrinsic property of all high-energy detectors: the reconstruction of neutral particles is non-trivial and therefore not very effective. As \Kstarz decays to \Kp\pim, the final state contains four charged particles. The detection of those particles is rather simple\footnote{\Kstarz of course also decays to \Kz\piz, which is extremely difficult to reconstruct. Therefore it is typically omitted from the measurements.}. In the case of \Kstarp meson, it either decays into \Kz\pip or \Kp\piz. For \lhcb, the only relevant \Kz meson is \KS meson, as \KL meson is stopped in the \hcal without leaving any signal in the tracking detectors\footnote{Assuming a boost of $\gamma=20$ $(v\simeq0.999c)$, a free-flying \KL would decay after traveling 300\m.}. \KS mesons on the other hand decay fast enough into a \pip\pim pair, so the \lhcb tracking system  is able to register both charged pions\footnote{Once again assuming $\gamma=20$, \KS decays after 53\cm. This means depending on the \KS boost, it either decays inside or outside of \velo, leading to more complications in analyzing this subdecay.}. 

This analysis focuses on the decay channel where \KstToKpPi. The \Kp meson is detected by the tracking detectors. The \piz meson typically decays into a \g\g pair (branching ratio of \pizTogg is $\simeq$ 98.8\%~\cite{PDG}). Both photons are registered by the electromagnetic calorimeter. Due to the finite granularity of \ecal, the two photons can either be registered by one or two \ecal cells. A sketch of this is shown in \refFig{piz-principle}. 
%
\begin{wrapfigure}[11]{r}{0.48\textwidth}  \vspace{-0pt}
	\includegraphics[width=0.18\textwidth]{./AnalysisIntro/Resolved.png}\hspace{15pt}%\\ \vspace{30pt}
	\includegraphics[width=0.18\textwidth]{./AnalysisIntro/Merged.png}    
	\captionof{figure}[Neutral pion reconstruction illustration.]
	{ Illustration of {\resolved} (left) and \merged (right) \piz mesons reconstruction in the \ecal cells. \label{fig:piz-principle}
	}
\end{wrapfigure}
%
The neutral pions reconstructed from photons hitting one \ecal cell are called \emph{merged} pions, the ones reconstructed from two cells \emph{resolved} pions. 


For the reconstruction of the \piz meson, proper settings of the electromagnetic calorimeter are essential. This is done in three steps: initial adjustment of \ecal energy scale, energy flow calibration and fine calibration of the \ecal cells~\cite{ANA-CALO}. The methods used for calibration are essentially the same in \runI and \runII. The main changes in \runII are the full automation of the calibration process and skipping the intermediate step. 

The initial adjustment of the energy scale is done by adjusting the photo-multipliers' (PMTs) gain using the \ecal's LED monitoring system. A LED light is attached to PMTs generating a known signal. The voltage of the PMTs is adjusted to match the measured and the known signal. This adjustment leads to a precision of the cell-to-cell inter-calibration of 10\%. The reason for this uncertainty is the dispersion in the photoelectron yields and the accuracy of the light yield determination. The LED-based calibration is preformed approximately once a week.

Then, the energy flow calibration is performed. This is done in order to smooth the fluctuations in the flux among neighboring cells due to initial miscalibrations. The method is rather simple: one exploits the symmetry of the energy flow of the calorimeter surface~\cite{ANA-CALO-flow}. Simulations with known mis-calibration showed that the flux adjustments improves the calibration by a factor of $\sim3$, assuming an initial precision of the calibration of 10\%.

Aging negatively affects the \ecal's performance and has to be accounted for. This can be nicely seen from the time variation (decrease) of the reconstructed \piz mass presented in \refFig{piz-calibration}. To account for this effect, fine calibration exploiting the \piz mass is performed. The mean $\pizMass^{reco}$ is obtained from \g\g pairs from minimum-bias events\footnote{Minimum-bias events are events with at least one charged track in the VELO detector or the downstream tracking system.} with low multiplicity to remove possible pile-up events. The photons are reconstructed using $3\times3$ clusters with single photon signals, where the cell with the highest energy deposit is called seed. The seeds are then corrected to match the nominal \piz mass. The effect of this correction is depicted in \refFig{piz-calibration}. This calibration is performed every LHC-runnning month.
%
\begin{figure}[hbt!] \centering
    \includegraphics[width=0.43\textwidth]{./AnalysisIntro/Fig14a.pdf}\hspace{10pt}    
    \includegraphics[width=0.46\textwidth]{./AnalysisIntro/PizCalib.pdf}    
    \captionof{figure}[\ecal calibration using the mass of \piz meson.]
    {On the left, fitted neutral pion mass \pizMass as a function of run number (time) using 2011 data. The \piz mass is 135\mev~\cite{PDG}. The clear decrease in the \pizMass value is due to the ECAL ageing. On the right, invariant mass distribution for \pizTogg candidates used for the fine calibration. The red curve corresponds to the distribution before applying the fine calibration, the blue curve is the final \pizMass distribution. Values in the boxes are the mean and width of the signal peak distribution in\mev before (red box) and after (blue box) applying the \pizMass calibration. Taken from~Ref.\,\cite{ANA-piz-reco2}.
    \label{fig:piz-calibration}
    }
\end{figure}


For this analysis, only resolved \piz mesons are used. Merged \piz mesons tend to have higher momenta (as the higher boost results in more collimated photons). In this work, where the \piz mesons come from a \Kstarp meson, the statistical contribution of these events is low. The \piz mesons originating from the \Kstar have typically transverse momentum of a few gigaelectronvolts. In \refFig{piz-eff} left the higher abundance of resolved \piz mesons at lower momentum is shown. As merged and resolved \piz require their own careful approach, merged \piz mesons are not included in the analysis. 

\begin{figure} [hbt!]\centering
    \includegraphics[width=0.45\textwidth]{./AnalysisIntro/piz_resolved_merged2.png}
    \includegraphics[width=0.45\textwidth]{./AnalysisIntro/piz_eff2.png}    
    \captionof{figure}[Transverse momentum distribution and reconstruction efficiency of \piz.]
    {On the left, transverse momentum distributions of \merged (red) and \resolved (blue) \piz in the \lhcb acceptance originating from \Bd\to\pip\pim\piz decay.         
        On the right, the overall \merged (dashed line) and \resolved (full line) \piz efficiency (number of identified \piz / number of \piz in detector acceptance with $\pt^{\piz}>200$\mev). The black points represent the overall efficiency for both \resolved and \merged neutral pions. Taken from~Ref.\,\cite{ANA-piz-reco2}.\label{fig:piz-eff}
    }
\end{figure}

The mass resolution of a decay including a \piz meson can be seen in \refFig{piz-resolution}. In this case, the \Dz\to\Km\pip\piz candidate mass is reconstructed using resolved and merged pions. It is clearly visible that the mass resolution of the \Dz meson candidate is better for resolved \piz meson.

Moreover, the \ecal resolution effects come into play. The resolution of \ecal is $\sigma/E = 0.1/\sqrt{E}\oplus0.01$, which is a very good resolution for a sampling calorimeter.  The advantage is that the resolution \emph{decreases} with \emph{increasing} deposited energy. However, for low-energy photons this does not bring any asset.

To summarize, we expect the neutral pion mass peak to be wide and to be affected by the combinatorial background coming from the \ecal cells. As we focus on resolved \piz, combinatorial background contributions from \g\g combinations are expected. On the other hand, the usage of resolved \piz improved the particle-identification as we have information from two cells: the probability of misidentifying a random photon as a \piz meson is lower. The maximal efficiency to reconstruct resolved $\piz$ mesons is $\sim40$\% at low $\pt^\piz$. 
 
%
\begin{figure}[hbt!]  \centering
	\includegraphics[width=0.45\textwidth]{./AnalysisIntro/fig27a.pdf}  \hspace{15pt}
	\includegraphics[width=0.45\textwidth]{./AnalysisIntro/fig27b.pdf}   
	\captionof{figure}[Mass distribution of the reconstructed \Dz\to\Km\pip\piz candidates.]
	{Mass distribution of the reconstructed \Dz\to\Km\pip\piz candidates with resolved \piz (left) and merged \piz (right) obtained from the 2011 data sample. The blue curve corresponds to a fit. The signal component of the fit function (red dashed line) and the background (green dash-dotted line) contributions are shown. One can easily see the mass resolution of the \Dz candidate is much worse for merged \piz. Taken from~Ref.\,\cite{ANA-piz-reco2}.\label{fig:piz-resolution}
	}
\end{figure}

\vspace{-\baselineskip}
\subsection{Binning in the dimuon invariant mass}\label{sec:AnaIntro-qsq}

It is explained in \refSec{SM_bsll} that different physics processes dominate in different \qsq (dimuon invariant mass squared) regions. Therefore, a veto in the resonance regions is applied to eliminate \BuToKstJpsi and \BuToKstPsi decay contributions. Moreover, an additional veto to eliminate the rare \BuToKstPhi decays is introduced. All three resonances \jpsi, \psitwos and $\phi$ are indistinguishable from the signal as they  have a very short decay time and therefore they are not displaced enough from the \Bu vertex. 

The range of the measured \qsq is limited by the rest mass of the two muons value and by the difference of the \Bu and \Kstarp mass squared $(\mBu - \mKstarp)^2=19.25\gevgev$. However, as the \lhcb acceptance at very high \qsq is low, the upper limit in this measurement is set to 18\gevgev.

In this work, \qsq is segmented into four signal regions and the three vetoed regions.  As a result of the challenging reconstruction there is a smaller statistical significance in the signal yield compared to the previous analyses~\cite{ANA-LHCb-angular3,ANA-LHCb-angular4,ANA-LHCb-angular1,ANA-LHCb-angular2} and the \qsq segmentation is therefore coarser. These signal regions are larger than the ones chosen in previous analyses. The \qsq bins including the vetoed regions are listed in \refTab{q2-binning}. Furthermore, a wide bin $[1.1,6.0]$ is added on top of the four \qsq bins. 
The wide bin is the theoretically favored region to observe New Physics effects and it has been included also in the previous measurements~\cite{ANA-LHCb-angular3,ANA-LHCb-angular4,ANA-LHCb-angular1,ANA-LHCb-angular2}.

\begin{table}[hbt!]
	\centering
	\begin{tabular}{clc}
			bin		&\qsq [\gevgev] &veto\\
			\hline
			1		&[0.1,~ 0.98]		&\\
					&[0.98, 1.1]	&\Pphi\\
			1		&[1.1,~ 4.0]		&\\						
			2		&[4.0,~ 8.0]		&\\
					&[8.0,~ 11.0]	&\jpsi\\
			3		&[11.0, 12.5]	&\\
					&[12.5, 15.0]	&\psitwos\\
			4		&[15.0, 18.0]	&\\
			5		&[1.1,~ 6.0]	&\\
	\end{tabular}
	\captionof{table}[The binning scheme of the dimuon invariant mass squared \qsq.]
	{The binning scheme of the dimuon invariant mass squared \qsq in the angular analysis including the vetoed regions of resonances decaying to \mumu pair. In the first bin, the \Pphi resonance is removed. \label{tab:q2-binning}}
\end{table}


\subsection{Simulation samples}\label{sec:AnaIntro-MC}
 
In order to study the effects of the detector response and to identify possible backgrounds, several Monte Carlo simulation samples are used in the analysis. The exhaustive list of the MC samples is presented in \refTab{ANA-MCsamples}. The two main samples consist of the signal decay \BuToKstmm and the reference decay \BuToKstJpsi.

In order to study the acceptance of the detector, a PHase SPace (PHSP) Monte Carlo sample is used. This sample neglects the spin structure of the decay, reflecting only the kinematic properties of the decay. This effectively means the distributions of the angles \angles are flat. Moreover, an additional requirement is imposed on the sample: the dimuon invariant mass squared \qsq distribution is generated to be flat. The sample is used to understand the angular acceptance in four dimensions of \angles and \qsq. 

Lastly, for the background investigation, an inclusive sample of \BuToXJpsi is used, where $X$ stands for any particle that a \Bu can decay into additionally to the \jpsi meson. This is particularly useful for identifying pollutions from other decays.


\begin{table}[!htb]
	\begin{center}\begin{tabular}{ccx{3cm}} %\hline
			MC decay, type	& Year & Number of generated events per polarity  \\  \hline \hline	            
			\BuToKstmm  	& 2011 & 50\,000\\
			Signal channel  & 2012 & 50\,000\\
							& 2015 & 100\,000\\
							& 2016 & 100\,000\\
							& 2017 & 115\,000\\
							& 2018 & 120\,000\\ \hline
			\BuToKstJpsi	& 2011 & 100\,000\\
		Reference channel	& 2012 & 100\,000\\
							& 2015 & 100\,000\\
							& 2016 & 100\,000\\ \hline
			\BuToKstmm  	& 2011 & 85\,000\\
			Phase space		& 2012 & 225\,000\\
							& 2015 & 95\,000\\
							& 2016 & 260\,000\\
							& 2017 & 240\,000\\
							& 2018 & 290\,000\\ \hline
			\BuToXJpsi  	& 2011 & 250\,000\\
			Inclusive sample& 2012 & 250\,000\\			
							& 2016 & 500\,000 			
			%\hline
	\end{tabular}\end{center}	
	\captionof{table}[Monte Carlo simulation samples used in this work.]{
		Monte Carlo simulation samples used in this work. The bending magnet polarity is regurarly flipped during the data taking (see \refSec{det_tracking_vertexing}). Therefore, two samples, one for each polarity configuration, are produced.  \label{tab:ANA-MCsamples}
	}
\end{table}

%
%\begin{table}[!htb]
%    \begin{center}\begin{tabular}{ccccc} %\hline
%            MC decay, type	& Year & SimVer & Number of generated events (Down+up)  \\  \hline \hline	            
%			\BuToKstmm  	& 2011 & Sim09a & 507551  + 502787 \\
%			Signal channel  & 2012 & Sim09a & 514015  + 500458 \\
%			            	& 2015 & Sim09i & 1033424 + 1027977 \\
%			            	& 2016 & Sim09i & 1045028 + 1013635 \\
%			            	& 2017 & Sim09e & 1151738 + 1153816 \\
%			            	& 2018 & Sim09h & 1235528 + 1196153 \\ \hline
%			\BuToKstJpsi	& 2011 & Sim09a & 1011831 + 1007920 \\
%		Reference channel	& 2012 & Sim09a & 1003888 + 1000278 \\
%							& 2015 & Sim09e & 1007712 + 1009484 \\
%							& 2016 & Sim09e & 1010609 + 1000281 \\ \hline
%			\BuToKstmm  	& 2011 & Sim09f & 85736   + 86004 \\
%			 Phase space	& 2012 & Sim09f & 226933  + 226430 \\
%							& 2015 & Sim09h & 95323   + 94980 \\
%							& 2016 & Sim09f & 264917  + 267674 \\
%							& 2017 & Sim09f & 246457  + 242673 \\
%							& 2018 & Sim09f & 291850  + 299252 \\ \hline
%			\BuToXJpsi  	& 2011 & Sim08c & 2508491 + 2514495 \\
%			Inclusive sample& 2012 & Sim08a & 2504990 + 2535488 \\			
%							& 2016 & Sim09b & 5090001 + 6055765\\
%									          			            
%            %\hline
%    \end{tabular}\end{center}
%    
%    \captionof{table}[Monte Carlo simulaiton samples used in this work.]{
%        Monte Carlo simulaiton samples used in this work.  \label{tab:ANA-MCsamples}
%    }
%\end{table}

\clearpage    