\subsection{Signal estimation}\label{sec:sel-SignalEstimation}

In order to select the most events with the least background, the cut value on the MLP response is optimized. As the figure of merit, the significance $\mathcal{S}$ defined in \refEq{significance} is chosen. Therefore, it is crucial to know the number of expected signal and background candidates in the data sample. Optimizing the MLP response cut using the \BuToKstmm data sample could bias our result. Therefore, the reference decay 
\mbox{\BuToKstJpsi} is used to extrapolate the expected number of signal candidates.

Let the number of \Bu mesons decaying into $\Kstarp \mumu$ be $N_\mumu$. The number of all \Bu meson decays is denoted $N_{all}$. Then, branching ratio $BR_\mumu$ can be defined as
%
\begin{equation}\label{eq:BR_definition}
BR_\mumu = N_\mumu / N_{all}\,.
\end{equation}

 It is not possible to directly measure $N_\mumu$. Due to limited efficiency of the detector, $\varepsilon_{\mumu}$, the measured yield is $Y_\mumu = N_\mumu \times \varepsilon_{\mumu}$ events.
 
  The branching ratio definition holds also for $\Bu$ decaying into $\Kstarp \jpsi$. The branching ratio, $BR_\jpsi$, is the ratio of number of \BuToKstJpsi decay events, $N_\jpsi$,  relative to all decays of the \Bu meson, $N_{all}$. Let the measured yield of \BuToKstJpsi events be $Y_\jpsi$ and the efficiency of detecting the decay events $\varepsilon_{\jpsi}$. Putting this together, the following formula gives the relation between the yield of \BuToKstmm and \BuToKstJpsi:
%
\begin{equation}\label{eq:YieldEstimation1}
 N_{all}  
 = \frac{N_{\mumu}}{BR_\mumu} 
 =\frac{Y_\mumu}{BR_\mumu \times \varepsilon_\mumu}  = \frac{N_{\jpsi}}{BR_\jpsi} 
 = \frac{Y_\jpsi}{BR_\jpsi \times \varepsilon_\jpsi}  \,.
\end{equation}

However, in this measurement only \BuToKstJpsi decays, where \jpsi\to\mumu, are considered. Therefore, the branching ratio of  \JpsiTomm, denoted $ BR_{\jpsi\to\mumu}$, has to be added to \refEq{YieldEstimation1}. The efficiency of detecting \BuToKstJpsi, \JpsiTomm is denoted $\varepsilon_{\jpsi,\jpsi\to\mumu}$. The formula used for estimation of the signal yield $Y_\mumu$ then becomes:
%
\begin{equation}\label{eq:YieldEstimation2}
Y_\mumu = \frac{\varepsilon_\mumu}{\varepsilon_{\jpsi,\jpsi\to\mumu}}\times\frac{BR_\mumu}{BR_\jpsi \times BR_{\jpsi\to\mumu}}\times {Y_{\jpsi,\jpsi\to\mumu}}\,.
\end{equation}
%
The used values of branching fractions, taken from~Ref.\,\cite{PDG}, are presented in \refTab{BR}. 

\begin{table}[hbt!]
	\centering
	\begin{tabular}{l|c}
		$BR_{\BuToKstmm}$    & 8.668$\times10^{-7}$ \\ \hline
		$BR_{\BuToKstJpsi}$  & 1.43$\times10^{-3}$ \\ \hline
		$BR_{\jpsi\to\mumu}$ & 5.961$\times10^{-2}$                     
	 \end{tabular}
	\captionof{table}[Branching ratios of \BuToKstmm, \BuToKstJpsi and $\jpsi\to\mumu$.]{Branching ratios of \BuToKstmm, \BuToKstJpsi and $\jpsi\to\mumu$ decays used for estimating the signal yields. The values are taken from the \flavio package~\cite{ANA-flavio}. \label{tab:BR}}
\end{table}

In order to estimate the yield of \BuToKstmm, the total efficiency of both the \BuToKstmm and the \BuToKstJpsi decay selection is needed. The total efficiency depends on many factors: the detector acceptance (acc), reconstruction (rec), tracking (tr), selection (sel) and MLP efficiencies, as expressed in~\refEq{TotEfficiency}\vspace{-0.25\baselineskip}
%
\begin{equation}\label{eq:TotEfficiency}
\varepsilon_{tot} =\varepsilon_{acc} \times  \varepsilon_{rec} \times \varepsilon_{tr} \times\varepsilon_{sel} \times \varepsilon_{MLP}\,.
\end{equation}
%
However, for estimating the signal yield $Y_\mumu$, only the ratio of the signal and  the reference efficiency $\varepsilon_\mumu/\varepsilon_{\jpsi}$ is needed. A lot of effects cancel out in this ratio. For practical reasons, the total efficiency is then evaluated in three steps: the acceptance efficiency $\varepsilon_{acc}$, the reconstruction, tracking, and selection efficiency $\varepsilon_{rec+tr+sel}$ and the efficiency of the multi-variate selection $\varepsilon_{MLP}$. The total efficiency then becomes \vspace{-0.25\baselineskip}
%
\begin{equation}\label{eq:EffEfficiency}
\varepsilon_{tot} =\varepsilon_{acc} \times  \varepsilon_{rec+tr+sel} \times \varepsilon_{MLP}\,.
\end{equation}
The evaluation of this efficiency is described in the following section.


\input{Chapters/EventSelection/Efficiency}
