\subsection[Fit to the reference channel \texorpdfstring{ $\BuToKstJpsi$}{.}]{Fit to the \texorpdfstring{\BuToKstJpsiBF}{.}}\label{valid-reference}

As the statistical power of the rare \BuToKstmm channel is highly limited, tests are performed using the \BuToKstJpsi data sample. Candidates with a dimuon invariant mass squared between 8.68 and 10.09 \gevgev are considered. The data sample is split between the \runI and \runII samples. The angular parameters are shared between both samples. The two samples are fitted simultaneously in four dimensions of the \Bu meson reconstructed mass \mBu and the angles \angles exploiting the maximum-likelihood method. The parameter $F_S$ is extracted from a two-dimensional fit to the \Bu meson mass and the \Kstarp mass. The statistical power of this sample is large enough to test the functionality of the \fcncfitter framework as well as all corrections applied to the data. 

The projection of the full fit is presented in \refFig{MainFit-Ref}. The results of this fit are compared to previous measurements by \babar studying both decays of \Bu and \Bd mesons to \Kstar\mup\mun~\cite{FIT-BaBar}, \belle focusing on the \BuToKstJpsi decay~\cite{FIT-Belle}, and \lhcb measurements of the \BdToJPsiKst decay~\cite{FIT-LHCb-Jpsi} and of the \BuToKstmm decay with \KstToKsPi~\cite{ANA-LHCb-angular4}, where the fit to \BuToKstJpsi is also performed as an important check of the fitter framework. The results of the full angular fit are shown in \refTab{FIT-otherMeasurements-Ref}. The statistical uncertainty is approximated by the \hesse calculation (see \refSec{toy-ref}) and scaled according to the pull widths listed in \refTab{toys-Ref-pull}. For readers convenience, the difference between this measurement and the measurement listed in \refTab{FIT-otherMeasurements-Ref} are shown in terms of standard deviations in \refTab{FIT-otherMeasurements-Ref-sigma}.

The measured values from the full angular fit agree very well with the other measurements. The only exception is the parameter $S_4$ that is consistently below the values measured by other experiments. This can  be caused by the complicated modeling of the angular acceptance: the parameter $S_4$ is very sensitive to the symmetry of the \ctl distribution. However, the angular acceptance description does not always prefer a symmetry in \ctl, see \refApp{AngCorr}. A dedicated test by varying the order of the polynomial used to correct the angular acceptance in \ctl can be done. 


\begin{figure}[hbt!]
	\centering
	\includegraphics[width=0.35\textwidth]{FCNC/MainFit/ctk_JpsiFit_1BIN_bin0_SimultaneousFit_Run12_AllPDFs.eps}\hspace{-15pt}
	\includegraphics[width=0.35\textwidth]{FCNC/MainFit/ctl_JpsiFit_1BIN_bin0_SimultaneousFit_Run12_AllPDFs.eps}\hspace{-15pt}
	\includegraphics[width=0.35\textwidth]{FCNC/MainFit/phi_JpsiFit_1BIN_bin0_SimultaneousFit_Run12_AllPDFs.eps}
	\captionof{figure}[Full angular fit to the reference channel \BuToKstJpsi.]{Full angular fit to the reference channel \BuToKstJpsi. The black markers represent the data. The black line shows the full fit model. The blue space represents the signal contribution. From left to right, \ctk, \ctl and $\phi$ projecitions are shown. Red surface represents the background contribution. The green dashed line shows only the \pwave component, the orange dotted line represents the \swave components and the pink dot-and-dash line depicts the interference between the \pwave and the \swave.}\label{fig:MainFit-Ref}
\end{figure}


\begin{table}[hbt!]\small \centering
\begin{tabular}{l|lllll}
			& this fit 			 & \lhcb~\Bu 		& Belle (\Bu)      & BaBar ($\B^+$+\Bd)& \lhcb~\Bd \\\hline
$F_{L}$  & 0.563$\pm$0.014  & \cellOneSigma 0.572$\pm$0.005    & \cellTwoSigmas 0.604$\pm$0.015  & \cellOneSigma 0.556$\pm$0.009   & \cellOneSigma 0.572$\pm$0.008\\
$S_{3}$  & 0.014$\pm$0.011  & \cellTwoSigmas -0.002$\pm$0.007  & \cellTwoSigmas -0.018$\pm$0.017 & \cellOneSigma 0.011$\pm$0.011   & \cellTwoSigmas -0.013$\pm$0.010\\
$S_{4}$  & -0.211$\pm$0.014 &\cellThreeSigmas -0.246$\pm$0.008 &\cellThreeSigmas -0.255$\pm$0.010& \cellTwoSigmas -0.237$\pm$0.007 &\cellThreeSigmas -0.250$\pm$0.006\\
$S_{5}$  & -0.013$\pm$0.015 & \cellOneSigma -0.003$\pm$0.008   & \cellOneSigma 0.000$\pm$0.000   & \cellOneSigma 0.000$\pm$0.000   & \cellOneSigma 0.000$\pm$0.000\\
$A_{FB}$ & 0.002$\pm$0.007  & \cellOneSigma -0.002$\pm$0.005   & \cellOneSigma 0.000$\pm$0.000   & \cellOneSigma 0.000$\pm$0.000   & \cellOneSigma 0.000$\pm$0.000\\
$S_{7}$  & 0.002$\pm$0.014  & \cellOneSigma -0.001$\pm$0.008   & \cellOneSigma 0.000$\pm$0.000   & \cellOneSigma 0.000$\pm$0.000   & \cellOneSigma 0.000$\pm$0.000\\
$S_{8}$  & -0.062$\pm$0.015 & \cellOneSigma -0.063$\pm$0.008   & \cellTwoSigmas -0.037$\pm$0.018 & \cellOneSigma -0.058$\pm$0.015  & \cellOneSigma -0.048$\pm$0.007\\
$S_{9}$  & -0.074$\pm$0.011 & \cellOneSigma -0.084$\pm$0.007   & \cellTwoSigmas -0.041$\pm$0.016 & \cellTwoSigmas -0.095$\pm$0.014 & \cellOneSigma -0.084$\pm$0.006\\
\end{tabular}\captionof{table}[Full angular fit to \BuToKstJpsi compared to previous measurements.]{Comparison of the full angular fit to reference channel \BuToKstJpsi to previously done measurements by \babar, \belle and two \lhcb measurements ~\cite{FIT-BaBar, FIT-Belle,FIT-LHCb-Jpsi,ANA-LHCb-angular4}, focusing on different combinations of \Bu and \Bd meson decays, as indicated. The measurements are published in the form of polarization amplitudes. The amplitudes are converted into the $S_i$ angular observables and the uncertanities are propagated to the basis using 100\,000 randomly generated samples. The full angular fit results are in agreement with the previously published measurements. The statistical uncertainty of this result is scaled according to \refTab{toys-Ref-pull}, as the pseudoexperiment studies showed an overestimation of the statistical uncertainities (for the details see \refSec{toys}). Dark green represents an agreement better than one standard deviation, lime represents an agreement better than two standard deviations and yellow represents an agreement better than three standard deviations.}\label{tab:FIT-otherMeasurements-Ref}
\end{table}

\begin{table}[hbt!]\small \centering
	\begin{tabular}{l|cccc}
		&  \lhcb \Bu & Belle (\Bu)  & BaBar ($B^++$\Bd) & \lhcb \Bd \\\hline
$F_{L}$  &  -0.61 			 &  -2.00   		& \phantom{-}0.42  & -0.56\\
$S_{3}$  &  \phantom{-}1.23  &  \phantom{-}1.58 & \phantom{-}0.19  & \phantom{-}1.82\\
$S_{4}$  & \phantom{-}2.17 	 & \phantom{-}2.56  &  \phantom{-}1.66 &  \phantom{-}2.56\\
$S_{5}$  &  -0.59  		     &  -0.87   		&  -0.87 		   & -0.87\\
$A_{FB}$ &  \phantom{-}0.46  &  \phantom{-}0.29 &  \phantom{-}0.29 &  \phantom{-}0.29\\
$S_{7}$  &  \phantom{-}0.19  &  \phantom{-}0.14 &  \phantom{-}0.14 &  \phantom{-}0.14\\
$S_{8}$  &  \phantom{-}0.06  &  -1.07 			& -0.19  		   & -0.85\\
$S_{9}$  &  \phantom{-}0.77  &  -1.70 			& \phantom{-}1.18  &  \phantom{-}0.80\\
	\end{tabular}\captionof{table}[Full angular fit to \BuToKstJpsi compared to previous measurements in terms of standard deviations.]{The difference between the full angular fit to reference channel \BuToKstJpsi and the previously done measurements by \babar, \belle and two \lhcb measurements ~\cite{ANA-LHCb-angular4,FIT-BaBar,FIT-Belle,FIT-LHCb-Jpsi} in terms of the standard deviations. The measurements are published in the form of polrization amplitudes. The amplitudes are converted into the $S_i$ angular observables and the uncertanities are propagated to the basis using 100\,000 randomly generated samples. The full angular fit results are in agreement with the previously published measurements.}\label{tab:FIT-otherMeasurements-Ref-sigma}
\end{table}



Moreover, the reference channel \BuToKstJpsi is used to further test the angular folding method. The values obtained using the full angular fit are compared to the values obtained via folding methods 0-4 listed in \refTab{FIT-fld-Ref}. The agreement between the results is almost perfect. The projections of these fits are shown in \refFig{MainFit-Ref-fld}.


\begin{table}[hbt!] \small \centering
\begin{tabular}{l|cccccc}
			 & Full angular & Folding  0 	  & Folding 1 		& Folding 2 	   & Folding 3		 & Folding 4 \\\hline
$F_{L}$  &\phantom{-}0.563$\pm$0.014  &\phantom{-}0.563$\pm$0.012 &\phantom{-}0.565$\pm$0.011 &\phantom{-}0.564$\pm$0.011  &\phantom{-}0.564$\pm$0.012 &\phantom{-}0.564$\pm$0.019\\
$S_{3}$  &\phantom{-}0.014$\pm$0.011  &\phantom{-}0.015$\pm$0.005 &\phantom{-}0.015$\pm$0.005 &\phantom{-}0.016$\pm$0.005  &\phantom{-}0.016$\pm$0.005 &\phantom{-}0.016$\pm$0.005\\
$S_{4}$  & -0.211$\pm$0.014 &    ---          & -0.218$\pm$0.008&    ---           &    ---       	 &    ---      \\
$S_{5}$  & -0.013$\pm$0.015 &    ---          &    ---      	& -0.012$\pm$0.007 &    ---       	 &    ---      \\
$A_{FB}$ &\phantom{-}0.002$\pm$0.007  &\phantom{-}0.001$\pm$0.004 &    ---      	&    ---       	   &    ---       	 &    ---      \\
$S_{7}$  &\phantom{-}0.002$\pm$0.014  &    ---          &    ---       	&    ---       	   &\phantom{-}0.002$\pm$0.007 &    ---      \\
$S_{8}$  & -0.062$\pm$0.015 &    ---          &    ---       	&    ---           &    ---          & -0.069$\pm$0.010\\
$S_{9}$  & -0.074$\pm$0.011 & -0.074$\pm$0.005&    ---       	&    ---           &    ---          &    ---      \\
\end{tabular}\captionof{table}[Full angular fit to \BuToKstJpsi comparison to fits via the angular folding method.]{Comparison of the full angular fit results to reference channel \BuToKstJpsi to the fits using angular folding method. The results are in perfect agreement. This proves the functionality of the folding methods.}\label{tab:FIT-fld-Ref}
\end{table}

\clearpage
%\thispagestyle{empty} 
\begin{figure}[hbt!] \vspace{-10pt}
	\centering
	\includegraphics[width=0.32\textwidth]{FCNC/MainFit/ctk_JpsiFit_1BIN_bin0_SimultaneousFit_folding0_Run12_AllPDFs.eps}
	\includegraphics[width=0.32\textwidth]{FCNC/MainFit/ctl_JpsiFit_1BIN_bin0_SimultaneousFit_folding0_Run12_AllPDFs.eps}
	\includegraphics[width=0.32\textwidth]{FCNC/MainFit/phi_JpsiFit_1BIN_bin0_SimultaneousFit_folding0_Run12_AllPDFs.eps}\\ \vspace{-3pt}
	\includegraphics[width=0.32\textwidth]{FCNC/MainFit/ctk_JpsiFit_1BIN_bin0_SimultaneousFit_folding1_Run12_AllPDFs.eps}
	\includegraphics[width=0.32\textwidth]{FCNC/MainFit/ctl_JpsiFit_1BIN_bin0_SimultaneousFit_folding1_Run12_AllPDFs.eps}
	\includegraphics[width=0.32\textwidth]{FCNC/MainFit/phi_JpsiFit_1BIN_bin0_SimultaneousFit_folding1_Run12_AllPDFs.eps}\\\vspace{-3pt}
	\includegraphics[width=0.32\textwidth]{FCNC/MainFit/ctk_JpsiFit_1BIN_bin0_SimultaneousFit_folding2_Run12_AllPDFs.eps}
	\includegraphics[width=0.32\textwidth]{FCNC/MainFit/ctl_JpsiFit_1BIN_bin0_SimultaneousFit_folding2_Run12_AllPDFs.eps}
	\includegraphics[width=0.32\textwidth]{FCNC/MainFit/phi_JpsiFit_1BIN_bin0_SimultaneousFit_folding2_Run12_AllPDFs.eps}\\\vspace{-3pt}
	\includegraphics[width=0.32\textwidth]{FCNC/MainFit/ctk_JpsiFit_1BIN_bin0_SimultaneousFit_folding3_Run12_AllPDFs.eps}
	\includegraphics[width=0.32\textwidth]{FCNC/MainFit/ctl_JpsiFit_1BIN_bin0_SimultaneousFit_folding3_Run12_AllPDFs.eps}
	\includegraphics[width=0.32\textwidth]{FCNC/MainFit/phi_JpsiFit_1BIN_bin0_SimultaneousFit_folding3_Run12_AllPDFs.eps}\\\vspace{-3pt}
	\includegraphics[width=0.32\textwidth]{FCNC/MainFit/ctk_JpsiFit_1BIN_bin0_SimultaneousFit_folding4_Run12_AllPDFs.eps}
	\includegraphics[width=0.32\textwidth]{FCNC/MainFit/ctl_JpsiFit_1BIN_bin0_SimultaneousFit_folding4_Run12_AllPDFs.eps}
	\includegraphics[width=0.32\textwidth]{FCNC/MainFit/phi_JpsiFit_1BIN_bin0_SimultaneousFit_folding4_Run12_AllPDFs.eps}\\\vspace{-3pt}
	\captionof{figure}[Full angular fit to the reference channel via the angular folding method.]{Full angular fit to the reference channel \BuToKstJpsi for the five folding methods. The black markers represent the data, the blue space represents the signal contribution. On the left, \ctk projeciton is shown, in the middle \ctl projection and on the right $\phi$ projecition is shown. Red surface represents the background contribution. The green dashed line shows only the \pwave component, the orange dotted line represents the \swave components and the pink dot-and-dash line depicts the interference between the \pwave and the \swave.}\label{fig:MainFit-Ref-fld}
\end{figure}
\begin{textblock*}{23cm}(1.13\textwidth,4.7cm) % {block width} (coords)

	\rotatebox{-90}{\centering Folding 0 \hspace{1.85cm} Folding 1 \hspace{1.85cm} Folding 2 \hspace{1.85cm} Folding 3 \hspace{1.85cm} Folding 4}
\end{textblock*}

\clearpage    


