\section{Mass fit}

\subsection{Yield estimation}

To know the order of magnitude of how many events are expected to be found in the fit, two different estimations are calculated. The first one is a standalone estimation, which involves the theoretical predictions of the branching fraction \BF(\Btokpipiee). The number of events that are in our sample can be estimated
\begin{equation}
	n_{events} = \int\lum\ dt \cdot \sigma_{\bquark\bquarkbar} \cdot \etot/\epsilon_{geo} \cdot f(\Bu) \cdot 2 \cdot \BR(\decay{\B}{\kaon_1\epem}),
\end{equation}
with an integrated luminosity $\intlum{1.11\invfb}$, a \bbbar production cross section in the accepted $\eta$ region of $\sigma_{\bquark\bquarkbar} = 72.0 \pm 0.3 \stat\pm 6.8\syst\mub$\cite{Aaij:2016avz}, an efficiency of $\etot/\epsilon_{geo} = 0.377\%$ as the geometric efficiency is already taken into account in the production cross section $\sigma_{\bquark\bquarkbar}$, the hadronisation factor of $f(\Bu) = 0.377\pm0.005\%$, which is obtained using the methods described in \cite{LHCb-PAPER-2015-019} with the $f_s/f_d$ ratio from \cite{LHCb-PAPER-2012-037} under the assumption that $f_u \approx f_d$, and the branching ratio $\BR(\decay{\B}{\kaon_1\epem}) = (2.7_{-1.2-0.3}^{+1.5+0.0}) \times 10^{-6}$.
This estimation yields $n_{events} \approx 602 \pm 341$ where the total uncertainty is dominated by the statistical uncertainty on $\BR(\decay{\B}{\kaon_1\epem})$.

The second estimation uses the already measured ratio of the $\decay{\Bz}{\Kstarz\ellell}$ decays with \lepton equal to either \electron or \muon. Together with the $\BR(\decay{\Bu}{\Kp\pip\pim\mup\mun})$, we can estimate the yield for our mode. As several factors are the same between our mode and the \mup\mun final state, only the yield, the different integrated luminosities and parts of the efficiency have to be taken into account. The estimated number of events is given by
\begin{equation}
	n_{events} = \dfrac{\BR(\BdKstee)}{\BR(\BdToKstmm)} \cdot n_{events} (\decay{\Bu}{\Kp\pip\pim\mup\mun}) \cdot \dfrac{\etot^{\epem}}{\etot^{\mup\mun}} \cdot \dfrac{\lum^{\epem}}{\lum^{\mup\mun}}
\end{equation}
with the ratio $R_\Kstarz = \dfrac{\BR(\BdToKstmm)}{\BR(\BdKstee)} = 0.69_{-0.07}^{+0.11}\stat\pm0.05\syst$\cite{Aaij:2017vbb}, the number of events obtained from the fit to the \mup\mun final state $n_{events}^{\mup\mun} = 144.80_{-14.31}^{+14.89}$, the efficiency of the \mup\mun final state $\etot^{\mup\mun}/\epsilon_{geo} = 1.062$\cite{Aaij:2014kwa} and the efficiency for our mode obtained in Sec. \ref{sec:selection:efficiency} $\etot^{\epem}/\epsilon_{geo} = 0.377$, both without the geometric acceptance, the integrated luminosity for the \epem mode $\lum^{\epem} = 1.11\invfb$ and for the \mup\mun mode $\lum^{\mup\mun} = 3.19\invfb$. This yields an estimated number of events of $n_{events} \approx 19.0_{-2.8}^{+3.6}$.

Both estimations do not agree well with each other. Comparing with the measured branch ratio and the predictions of \decay{\Bu}{\Kp\pip\pim\mup\mun}, the measured one is lower by a factor of about six, a similar deviation is expected here. Also the uncertainties on the predicted branching ratio of \Btokpipiee is comparably large. This considerations favour the second estimation, which still would yield enough events for an observation, at least if the data taken in 2012 is used as well.

\subsection{Fits}
To determine the number of events, a fit to the vertex constrained $\B$ invariant mass is performed. From the \root software package, the \roofit library with python bindings is used.

The probability density function (pdf) for the fit is constructed using a linear combination of an exponential pdf as background shape and a double crystal-ball (CB) function\footnote{A CB function is a Gaussian distribution with exponential tails.} for the signal shape\cite{Skwarnicki:1986xj}. A double CB function is a linear combination of two CB functions, the ratio of the two normalisations is a fit parameter. An extended unbinned maximum likelihood fit is performed, leaving the number of the background and signal events as free parameters to the fit.

Four fits are performed in total to fix certain parameters and to correct for simulation differences. First, the fit is performed on the \jpsi samples and the ratio between the MC and data mean is taken to correct the mean obtained in the non-resonant mode with that factor.

\begin{enumerate}
	\item Fit to the \B invariant mass with vertex and \jpsi mass constrained of \Btojpsikpipiee MC as shown in Fig. \ref{fig:mass_fit:jpsiee_mc}
	\begin{itemize}
		\item Fit without background.
		\item All parameters are floating freely including the ration between the two CB functions.
	\end{itemize}
	\begin{figure}[tb]
		\centering
		\includegraphics[width=0.7\linewidth]{figs/mass-fit/jpsiee_mc.pdf}
		\caption{Fit to \Btojpsikpipiee MC}
		\label{fig:mass_fit:jpsiee_mc}
	\end{figure}

	\item Fit to the \B invariant mass with vertex and \jpsi mass constrained of \Btojpsikpipiee data as shown in Fig. \ref{fig:mass_fit:jpsiee_real}
	\begin{itemize}
		\item Fit with background.
		\item Exponential tail parameters and fraction are fixed from MC fit.
		\item Free parameters are the mean, width and the scaling.
	\end{itemize}
	\begin{figure}[tb]
	\centering
	\includegraphics[width=0.7\linewidth]{figs/mass-fit/jpsiee_real.pdf}
	\caption{Fit to \Btojpsikpipiee data}
	\label{fig:mass_fit:jpsiee_real}
	\end{figure}
	
	\item Fit to the \B invariant mass with vertex constrained of \Btokpipiee MC as shown in Fig. \ref{fig:mass_fit:k1ee_mc}
	\begin{itemize}
		\item Fit without background.
		\item All parameters are floating freely.
	\end{itemize}
	\begin{figure}[tb]
		\centering
	\includegraphics[width=0.7\linewidth]{figs/mass-fit/k1ee_mc.pdf}
	\caption{Fit to \Btokpipiee MC}
	\label{fig:mass_fit:k1ee_mc}
	\end{figure}

	\item Blind-fit to the \B invariant mass with vertex constrained of \Btokpipiee data blinding the region $5100-5380\mev$ around the \B mass of $5279\mev$ as shown in Fig. \ref{fig:mass_fit:k1ee_real}.
	\begin{itemize}
		\item Fit with background
		\item All signal parameters are fixed from the previous MC fit. The mean is corrected by the ratio of the mean between the fits to the MC and data of the \jpsi.
	\end{itemize}
	\begin{figure}[tb]
		\centering
	\includegraphics[width=0.7\linewidth]{figs/mass-fit/k1ee_real_blind.pdf}
	\caption{Fit to \Btokpipiee data with the region $5100-5380\mev$ blinded.}
	\label{fig:mass_fit:k1ee_real}
	\end{figure}
\end{enumerate}

For an unblinding of the fit, a clean signal region is required. As can be seen in Fig. \ref{fig:mass_fit:k1ee_real} at the lower bound of the blinded region, there seems to be a peak, most probably originating from physical background and reaching into our signal region. This background would bias our yield and has to be further investigated before an unblinding of the fit is possible.

