\section{Mu3e experiment}

\subsection{Requirements}

The ultimate goal of this experiment is to observe a $\mu \rightarrow eee$ event. As we strive for a sensitivity of $10^{-16}$ , we should be able to observe this process if its branching ratio would be higher than our sensitivity. Otherwise we want to exclude a branching ratio $>10^{-16}$ with a $90\%$ certainty.\\
To get to this sensitivity, more than $5.5 \cdot 10^{16}$ muon decays have to be observed. To reach this goal within one year, a muon stopping rate of $2 \cdot 10^9 Hz$ in combination with a high geometrical acceptance as well as a high efficiency of the experiment is required.

\subsection{Phase I}

Phase I of the experiment serves as an exploratory phase to gain more experience with the new technology and validate the experimental concept. At the same time it already strives to produce competitive measurements with a sensitivity of $10^{-15}$. \footnote{Current experiments are in the $10^{-12}$ sensitivity range} This will be done, by making use of the already existing muon beams at PSI with around $1$-$1.5\cdot10^{8}Hz$ of muons on target. The lowered sensitivity also allows for some cross-checks as the restrictions on the system are much more relaxed than in phase II.

\subsection{Phase II}

Phase II strives to reach the maximum sensitivity of $10^{-16}$. To achieve this in a reasonable time a new beamline will be used which delivers more than $2\cdot10^{9}Hz$ of muons.

\subsection{Experimental setup}
\label{exp_setup}
The detector is of cylindrical shape around the beam. It has a total length of around $2m$ and is situated inside a $1T$ solenoid magnet with $1m$ of inner radius and a total length of $2.5m$. This form was chosen to cover as much phase space as possible. For an unknown decay such $\mu \rightarrow eee$, it crucial to have a high order of acceptance in all regions of phase space. There are only two kind of tracks that get lost. The first one are up- and downstream tracks and the second one are low transverse momenta tracks (no transversing of enough detector planes to be reconstructed).\\

\begin{figure}[H]
\begin{center}
\begin{subfigure}{0.8\textwidth}
\includegraphics[width=1\textwidth]{img/setup-Ia.png}
\caption{Setup of the detector in the first part of phase I}
\label{setup_Ia}
\end{subfigure}
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=0.8\textwidth]{img/tracks-phase_I.png}
\caption{Tracks in the detector in the first part of phase I}
\label{tracks_Ia}
\end{subfigure}
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=0.8\textwidth]{img/tracks-phase_II.png}
\caption{Tracks in the detector in the second part of phase I and Phase II}
\label{tracks_Ib,_II}
\end{subfigure}
\begin{subfigure}{1\textwidth}
\includegraphics[width=1\textwidth]{img/setup-Ib.png}
\caption{Setup of the detector in the second part of phase I}
\label{setup_Ib}
\end{subfigure}
\begin{subfigure}{1\textwidth}
\includegraphics[width=1\textwidth]{img/setup-II.png}
\caption{Setup of the detector in phase II}
\label{setup_II}
\end{subfigure}
\end{center}
\end{figure}\newpage

As seen in figure \ref{setup_II}, the final version of the detector can be divided into 5 separate parts in the longitudinal direction. There is the central part with the target, two inner silicon pixel layers, a fibre tracker and two outer silicon layers. The forward and backward parts, called recurl stations, consist only of a tile timing detector surrounded by two silicon recurl layers. A big advantage pf this layout is that even a partially constructed detector (gradally over phase I to phase II parts get added) can give us competitive measurements.\\\

The target itself is a big surfaced double cone with a surface length of $10cm$ and a width of $2cm$. The target was chosen specifically to be of this shape to facilitate separating tracks coming from different muons and hereby also helping to reduce accidental background.\\\
The two inner detector layers, also called vertex layers, span a length $12cm$. The innermost layer consists of 12 tiles while the outer vertex layer consists of 18 tiles. The tiles are each of $1cm$ width, with the inner layer having an average radius of $1.9cm$, respectively $2.9cm$ \cite{augustin2017mupix}, \cite{philipp2015hv}, \cite{augustin2015mupix}. They are supported by two half cylinder made up of $25\mu m$ thin Kapton foil mounted on plastic. The detector layers itself are $50\mu m$ thin and cooled by gaseous helium. The vertex detectors are read out at a rate of $20MHz$, giving us a time resolution of $20ns$.\\
After the vertex layers the particles pass through the fibre tracker (see Figure \ref{tracks_Ib,_II}, \ref{setup_II}). It is positioned around $6cm$ away from the center. Its main job is to provide accurate timing information for the outgoing electrons and positrons. It consist of three to five layers, each consisting of $36cm$ long and $250\mu m$ thick scintillating fibres with fast silicon photomultipliers at the end. They provide us a timing information of less than a $1ns$.\\
Next the outgoing particles encounter the outer silicon pixel detectors. They are mounted just after the fibre detector with average radii of $7.6cm$ and $8.9cm$. The inner layer has 24 and the outer has 28 tiles of $1cm$ length. The active area itself has a length of $36cm$. Similarly to the vertex detectors, they are mounted on $25\mu m$ thin Kapton foil with plastic ends.\\
The stations beam up- and downwards only consist of the outer pixel detector layers as well as a timing detector. While the silicon detector are the same as in the central station, the timing tracker was chosen to be much thicker than the fibre detector in the central station. It consists of scintillating tiles with dimensions of $7.5 \cross 7.5 \cross 5 mm^3$. They provide an even better time resolution than the fibre tracker in the center. Incoming particles are supposed to be stopped here. The outer stations are mainly used to determine the momenta of the outgoing particles and have an active length of $36cm$ and a radius of around $6cm$.

\subsection{The problem of low longitudinal momentum recurlers}

As explained in section \ref{exp_setup}, the outgoing particles are supposed to recurl back into the outer stations of the detector to enable a precise measurement of the momentum. A problem arises if the particles have almost no momentum in the beam direction. Then they can recurl back into the central station and cause additional hits there. As the the central station is designed to let particles easily pass through, they can recurl inside the central station many more times without getting stopped. As we have a $20ns$ time window for the readout of the pixel detectors, we need a very reliable way to identify and reconstruct these tracks as recurling particles as otherwise they look exactly like newly produced particles coming from our target. As one can imagine this influences the precision of our measurements by a big margin. So finding a way to identify these low beam direction momentum particles consistently is of great importance as it is crucial for the experiment to reduce the background as much as possible.\\\\

There is already an existing software to reconstruct particle tracks. However it struggles to find the right tracks for a lot of the particles recurling back into the center station.\\
These recurlers will typically leave eight hits or more, four (one on each silicon pixel detector layer) when initially leaving the detector  and another four when initially falling back in. It is possible for these recurlers to produce even more hits when leaving the detector again but for this thesis we will be only focusing on these 8 hit tracks.\\
The current reconstruction algorithm works by fitting helix paths with a $\chi^2$ method onto the 8 hits.\\
However experience has shown that often the fit with the lowest $\chi^2$ isn't necessarily the right track. If we increase the $\chi^2$ limit value to some arbitrary limit, we get a selection of several possible tracks per particle. Without any additional tools however, it is impossible to figure out if the right track is in the selection\footnote{\alignLongunderstack{\text{Based on detector efficiency it is possible for a particle to leave less}\\ \text{than 8 tracks and therefore not be reconstructed by the algorithm}}} and if yes which one of them correct track is.\\ 

\begin{figure}[H]
\begin{center}
\includegraphics[width=0.8\textwidth]{img/tracks_in_det_xy.png}
\caption{Particle recurling back into the center station}
\label{recurler}
\end{center}
\end{figure}