\section{Tests with large statistics}\label{valid}

In order to validate the \fcncfitter framework  and the functionality of the angular acceptance correction, tests with large statistical samples are performed. First, the simulation samples introduced in \refSec{AnaIntro-MC} are fitted. As the Monte Carlo simulation uses a form-factor model \texttt{BTOSLLBALL}~\cite{FIT-btosllBall}, the extraction of the initial angular moments \allAng  very complicated. Therefore, instead of extracting the angular moments from the \texttt{BTOSLLBALL} model itself, a \emph{generator-level} simulation is studied in order to obtain the values of the angular parameters used at generation. Generator level sample is free of any acceptance, reconstruction or selection effects. The agreement of the angular moments of the full Monte Carlo simulation and the generator level simulation is a crucial validation of the angular acceptance correction procedure.

The full simulation sample is also used to validate the folding procedure described in \refSec{ANA_folding}. The large statistics allows for a full angular fit. The angular observables obtained by the full angular fit are compared to the results of the folded fit.

The last step of the validation is the fit to the reference \BuToKstJpsi channel. The statistical power of the reference channel allows to test the functionality of the \fcncfitter framework also on the data with present background contribution. The fit is validated by comparing the measured angular moments to previous measurements. Moreover, the \BuToKstJpsi decay data sample is also used to further validate the folding method.


\subsection{Fit to the simulation sample}\label{valid}

The simulation sample used for the validation is treated the same way as the real data: the events pass the \lhcb acceptance and the full selection. The angular acceptance correction is applied. The fit is performed using the same \qsq binning as in the data sample. There is no \swave pollution present in the simulation sample: only the \pwave is fitted. The fit projections are shown in \refFig{FIT-MC-sigFit}. The fit converges and describes the data very well.

%
%\input{Chapters/ParameterMeasurement/MC_MassFit_Jpsi_Run1}
%\input{Chapters/ParameterMeasurement/MC_MassFit_Jpsi_Run2}

\begin{figure}[hbt!]
	\centering
	\includegraphics[width=0.32\textwidth]{FCNC/MCfit/Signal/ctk_MC_SignalFit_5BINS_bin0_SimultaneousFit_Run12_AllPDFs.eps}
	\includegraphics[width=0.32\textwidth]{FCNC/MCfit/Signal/ctl_MC_SignalFit_5BINS_bin0_SimultaneousFit_Run12_AllPDFs.eps}
	\includegraphics[width=0.32\textwidth]{FCNC/MCfit/Signal/phi_MC_SignalFit_5BINS_bin0_SimultaneousFit_Run12_AllPDFs.eps}\\ \vspace{-3pt}
	
	\includegraphics[width=0.32\textwidth]{FCNC/MCfit/Signal/ctk_MC_SignalFit_5BINS_bin1_SimultaneousFit_Run12_AllPDFs.eps}
	\includegraphics[width=0.32\textwidth]{FCNC/MCfit/Signal/ctl_MC_SignalFit_5BINS_bin1_SimultaneousFit_Run12_AllPDFs.eps}
	\includegraphics[width=0.32\textwidth]{FCNC/MCfit/Signal/phi_MC_SignalFit_5BINS_bin1_SimultaneousFit_Run12_AllPDFs.eps}\\ \vspace{-3pt}
	
	\includegraphics[width=0.32\textwidth]{FCNC/MCfit/Signal/ctk_MC_SignalFit_5BINS_bin2_SimultaneousFit_Run12_AllPDFs.eps}
	\includegraphics[width=0.32\textwidth]{FCNC/MCfit/Signal/ctl_MC_SignalFit_5BINS_bin2_SimultaneousFit_Run12_AllPDFs.eps}
	\includegraphics[width=0.32\textwidth]{FCNC/MCfit/Signal/phi_MC_SignalFit_5BINS_bin2_SimultaneousFit_Run12_AllPDFs.eps}\\ \vspace{-3pt}
	
	\includegraphics[width=0.32\textwidth]{FCNC/MCfit/Signal/ctk_MC_SignalFit_5BINS_bin3_SimultaneousFit_Run12_AllPDFs.eps}
	\includegraphics[width=0.32\textwidth]{FCNC/MCfit/Signal/ctl_MC_SignalFit_5BINS_bin3_SimultaneousFit_Run12_AllPDFs.eps}
	\includegraphics[width=0.32\textwidth]{FCNC/MCfit/Signal/phi_MC_SignalFit_5BINS_bin3_SimultaneousFit_Run12_AllPDFs.eps}\\ \vspace{-3pt}
	
	\includegraphics[width=0.32\textwidth]{FCNC/MCfit/Signal/ctk_MC_SignalFit_5BINS_bin4_SimultaneousFit_Run12_AllPDFs.eps}
	\includegraphics[width=0.32\textwidth]{FCNC/MCfit/Signal/ctl_MC_SignalFit_5BINS_bin4_SimultaneousFit_Run12_AllPDFs.eps}
	\includegraphics[width=0.32\textwidth]{FCNC/MCfit/Signal/phi_MC_SignalFit_5BINS_bin4_SimultaneousFit_Run12_AllPDFs.eps}\\ \vspace{-3pt}
		
	\captionof{figure}[Projections of the fit to the simulated \BuToKstmm decay sample.]{Projections of the fit to the simulated \BuToKstmm decay sample. All events are weighted according to the acceptance correction function. The black markers represent the data, blue area represents the fit. Each figure represents one \qsq interval (\qsq range of the interval is denoted in the figures in \gevgev). On the left, \ctk projecitons are shown, in the middle \ctl projections and on the right $\phi$ projecitions are shown.}\label{fig:FIT-MC-sigFit}
\end{figure}

\clearpage

\subsection{Generator level simulation fit}\label{valid-genMC}

In order to extract the angular parameters used for the \lhcb Monte Carlo simulation, an independent generator level sample of 200\,000 \Bu mesons decaying at rest to $\Kstarp_{[\Kp\piz]}\mumu$ have been generated. In \refFig{FIT-GenLvl-vs-sigMC}, the measured values of \pwave angular moments are shown. 
%For readers comfort, the larger bin of $[1.1\gevgev-6\gevgev]$ is removed.
The agreement between the \lhcb simulation and the generator level event simulation is very good. The difference between them is below three standard deviations in all bins and all variables, showing the functionality of the angular acceptance corrections in all \qsq regions.

\begin{figure}[hbt!]\vspace{-10pt}
	\centering
	\includegraphics[width=0.40\textwidth]{FCNC/MCfit/GenLvl_vs_MC_Fl.eps}
	\includegraphics[width=0.40\textwidth]{FCNC/MCfit/GenLvl_vs_MC_S3.eps}\\
	\includegraphics[width=0.40\textwidth]{FCNC/MCfit/GenLvl_vs_MC_S4.eps}
	\includegraphics[width=0.40\textwidth]{FCNC/MCfit/GenLvl_vs_MC_S5.eps}\\
	\includegraphics[width=0.40\textwidth]{FCNC/MCfit/GenLvl_vs_MC_Afb.eps}
	\includegraphics[width=0.40\textwidth]{FCNC/MCfit/GenLvl_vs_MC_S7.eps}\\
	\includegraphics[width=0.40\textwidth]{FCNC/MCfit/GenLvl_vs_MC_S8.eps}
	\includegraphics[width=0.40\textwidth]{FCNC/MCfit/GenLvl_vs_MC_S9.eps}\\
	\captionof{figure}[Fit to the generator level compared to the fit to the full simulation.]{Fit to the generator level simulation compared to the fit to the \lhcb simulation results. The brown stripes represent the resonant \qsq regions. These regions are excluded from the fit. The red boxes represent the difference between the processed \lhcb simulation and the generator level simulation in terms of standard deviations, \stdev. The fitted values are in agreement, proving the functionality of the angular acceptance correction in all \qsq regions.}\label{fig:FIT-GenLvl-vs-sigMC}
\end{figure}

%\input{Chapters/ParameterMeasurement/GenLvl_Vs_Davids}
%\input{Chapters/ParameterMeasurement/GenLvlMC_5BINS_bin0_Run12}
%\input{Chapters/ParameterMeasurement/GenLvlMC_5BINS_bin1_Run12}
%\input{Chapters/ParameterMeasurement/GenLvlMC_5BINS_bin2_Run12}
%\input{Chapters/ParameterMeasurement/GenLvlMC_5BINS_bin3_Run12}
%\input{Chapters/ParameterMeasurement/GenLvlMC_5BINS_bin4_Run12}
%\input{Chapters/ParameterMeasurement/GenLvlMC_8BINS_bin0_Run12}
%\input{Chapters/ParameterMeasurement/GenLvlMC_8BINS_bin1_Run12}
%\input{Chapters/ParameterMeasurement/GenLvlMC_8BINS_bin2_Run12}
%\input{Chapters/ParameterMeasurement/GenLvlMC_8BINS_bin3_Run12}
%\input{Chapters/ParameterMeasurement/GenLvlMC_8BINS_bin4_Run12}
%\input{Chapters/ParameterMeasurement/GenLvlMC_8BINS_bin5_Run12}
%\input{Chapters/ParameterMeasurement/GenLvlMC_8BINS_bin6_Run12}
%\input{Chapters/ParameterMeasurement/GenLvlMC_8BINS_bin7_Run12}


\subsection{Validation of the folding method}\label{valid-folding}

In order to validate the \fcncfitter framework's folding method classes, a check using the \lhcb simulation sample of \BuToKstmm is performed. The sample is fitted using the full angular description as well as using all five folding techniques described in \refSec{ANA_folding}. The results of the full fit are compared to the fit results using the five folding methods. As shown in \refFig{FIT-Fold-comparison}, there is a  perfect agreement between all five folding methods and the full angular fit in all \qsq bins and all angular observables. 

%-----------------------------------------
\begin{figure}[hbt!]\vspace{-10pt}
	\centering
	\includegraphics[width=0.40\textwidth]{FCNC/MCfit/Folding_Fl_new.pdf}
	\includegraphics[width=0.40\textwidth]{FCNC/MCfit/Folding_S3.pdf}\\
	\includegraphics[width=0.40\textwidth]{FCNC/MCfit/Folding_S4.eps}
	\includegraphics[width=0.40\textwidth]{FCNC/MCfit/Folding_S5.eps}\\
	\includegraphics[width=0.40\textwidth]{FCNC/MCfit/Folding_Afb.eps}
	\includegraphics[width=0.40\textwidth]{FCNC/MCfit/Folding_S7.eps}\\
	\includegraphics[width=0.40\textwidth]{FCNC/MCfit/Folding_S8.eps}
	\includegraphics[width=0.40\textwidth]{FCNC/MCfit/Folding_S9.eps}\\
	\captionof{figure}[Full angular fit compared to fits using angular folding method.]{Full angular fit compared to fits using angular folding method as indicated in each figure by 'Fld'. The brown stripes represent the resonance \qsq regions. These regions are excluded from the fit. The red boxes represent the difference between the processed \lhcb simulation and the generator level simulation in terms of standard deviations \stdev. The results from the full angular fit and the fits using the folding methods are in perfect agreement in all \qsq regions. This proves the functionality of the angular acceptance correction in the folded fits.}\label{fig:FIT-Fold-comparison}
\end{figure}
%-----------------------------------------


%\input{Chapters/ParameterMeasurement/MC_Signal_5BINS_bin0_folding0_Run12}
%\input{Chapters/ParameterMeasurement/MC_Signal_5BINS_bin0_folding1_Run12}
%\input{Chapters/ParameterMeasurement/MC_Signal_5BINS_bin0_folding2_Run12}
%\input{Chapters/ParameterMeasurement/MC_Signal_5BINS_bin0_folding3_Run12}
%\input{Chapters/ParameterMeasurement/MC_Signal_5BINS_bin0_folding4_Run12}
%\input{Chapters/ParameterMeasurement/MC_Signal_5BINS_bin1_folding0_Run12}
%\input{Chapters/ParameterMeasurement/MC_Signal_5BINS_bin1_folding1_Run12}
%\input{Chapters/ParameterMeasurement/MC_Signal_5BINS_bin1_folding2_Run12}
%\input{Chapters/ParameterMeasurement/MC_Signal_5BINS_bin1_folding3_Run12}
%\input{Chapters/ParameterMeasurement/MC_Signal_5BINS_bin1_folding4_Run12}
%\input{Chapters/ParameterMeasurement/MC_Signal_5BINS_bin2_folding0_Run12}
%\input{Chapters/ParameterMeasurement/MC_Signal_5BINS_bin2_folding1_Run12}
%\input{Chapters/ParameterMeasurement/MC_Signal_5BINS_bin2_folding2_Run12}
%\input{Chapters/ParameterMeasurement/MC_Signal_5BINS_bin2_folding3_Run12}
%\input{Chapters/ParameterMeasurement/MC_Signal_5BINS_bin2_folding4_Run12}
%\input{Chapters/ParameterMeasurement/MC_Signal_5BINS_bin3_folding0_Run12}
%\input{Chapters/ParameterMeasurement/MC_Signal_5BINS_bin3_folding1_Run12}
%\input{Chapters/ParameterMeasurement/MC_Signal_5BINS_bin3_folding2_Run12}
%\input{Chapters/ParameterMeasurement/MC_Signal_5BINS_bin3_folding3_Run12}
%\input{Chapters/ParameterMeasurement/MC_Signal_5BINS_bin3_folding4_Run12}
%\input{Chapters/ParameterMeasurement/MC_Signal_5BINS_bin4_folding0_Run12}
%\input{Chapters/ParameterMeasurement/MC_Signal_5BINS_bin4_folding1_Run12}
%\input{Chapters/ParameterMeasurement/MC_Signal_5BINS_bin4_folding2_Run12}
%\input{Chapters/ParameterMeasurement/MC_Signal_5BINS_bin4_folding3_Run12}
%\input{Chapters/ParameterMeasurement/MC_Signal_5BINS_bin4_folding4_Run12}

\include{./Chapters/Validation/refFit}

\clearpage



