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\title{Development of a pattern recognition algorithm for reconstructing multi-particle events in the Mu2e detector}




\author{
  H.~Kitagawa\from{ins:x}\thanks{Corresponding author: \texttt{hussain.kitagawa@phd.unipi.it}}, 
  S.~Donati\from{ins:x}, 
  \atque 
  P.~Murat\from{ins:y}
}

\instlist{\inst{ins:x} Dipartimento di Fisica, Universit\`a di Pisa - Pisa, Italy
  \inst{ins:y} Fermi National Accelerator Laboratory - Batavia, USA}
%% When only one author is present, please do not use the command \from{} near the author name.





\begin{document}
\pagestyle{plain}

\maketitle

\begin{abstract}
Mu2e is an upcoming experiment at Fermilab and its main goal is to search for the Charged Lepton Flavor Violation (CLFV) in the coherent transition of a muon into an electron on an Al target. In Mu2e, multi-particle events can occur simultaneously within the same time region and it is crucial to accurately identify each particle track, including signals, to improve the robustness of track-finding methods and enhance reconstruction efficiency. $\bar{p}$ annihilation is one of the background events and produces multiple particles that can mimic signal events. Additionally, photons from radiative pion captures can produce a $\gamma \rightarrow e^{+} e^{-}$ pair, which can be used to calibrate the Mu2e momentum scale and the resolution. The Mu2e track reconstruction sequence begins by grouping hits produced in the tracker based on time and z coordinate information, called TimeCluster, and selected hits are processed to reconstruct helices and determine their momentum. The current pattern recognition algorithms identify a single helix per TimeCluster for single-track events. A new pattern recognition algorithm is being developed to reconstruct multi-particle events and its features for finding multiple tracks and the current evaluation results are reported.
\end{abstract}



\section{Mu2e experiment}
The Mu2e experiment at Fermilab will search for the CLFV process in the coherent transition of a muon into an electron on an Al target~\cite{bartoszek2015mu2etechnicaldesignreport}. The muon apparatus consists of three main parts: the production solenoid~(PS), transport solenoid~(TS), and detector solenoid~(DS). A schematic view of the beamline and detector components is shown in Figure~\ref{fig:mu2e_detector}. The proton pulsed beam with the kinetic energy of 8 GeV is delivered to the PS and produces secondary particles. The S-shaped TS efficiently transfers charged particles to the DS under the magnetic field. Muons are primarily produced from pion decay and stop in the stopping target. A high-intensity low-momentum pulsed negative muon beam is selected thanks to the TS geometry and collimator system and transported inside the DS. Charged particles emitted from the stopping target will undergo helical motion in the solenoidal magnetic field in the DS. The Mu2e experiment uses aluminum (Z = 13) as a muon-stopping target material. The straw-tube tracker is located in the DS under 1 T magnetic field region to reconstruct the trajectories and momenta of particles. An array of 18 layers (referred to as stations) of gas-filled straw drift tubes aligned transversely to the beam axis as shown in Figure~\ref{fig:mu2e_strawtube_tracker}. The fiducial area of the tracker extends from about $40 < R < 70$ cm to have good acceptance for conversion electrons and reduce background hits with low momenta. The detecting sensitivity of muon-to-electron conversion, $R_{\mu e}$ is obtained by taking the ratio of the muon-to-electron conversion process to the total muon capture process. The current experimental upper limit for this process is $R_{\mu e} < 7\times10^{-13}~(90\%~\rm{CL})$ in Au target by the SINDRUM II experiment~\cite{SINDRUM-2}. For Run 1, Mu2e is expected to achieve a sensitivity $R_{\mu e} < 6.2\times10^{-16}~(90\%\rm{CL})$~\cite{mu2ecollaboration2022mu2e}, which is an improvement of a factor $\times 1000$ compared to 
SINDRUM II.



\begin{figure}[htbp]
    \begin{center}
        \includegraphics[width=0.70\linewidth]{images/mu2e_esperiment.eps}
    \end{center}
    \caption{\centering Schematic view of the Mu2e experiment~\cite{mu2ecollaboration2022mu2e}.}
    \label{fig:mu2e_detector}
\end{figure}


\begin{figure}[htbp]
  \begin{minipage}[b]{0.49\linewidth}
    \centering
    \includegraphics[width=0.9\columnwidth]{images/mu2e_tracker_02.eps}
    %\subcaption{ }
  \end{minipage}
  \begin{minipage}[b]{0.49\linewidth}
    \centering
    \includegraphics[width=0.9\columnwidth]{images/mu2e_tracker_00.eps}
    %\subcaption{ }
  \end{minipage}
  \caption{Mu2e straw-tube tracker with the entire 18 stations (left) and a single station with 12 panels in XY view (right). Each panel is rotated by $\theta = 30\degree$ and covers the trajectories of high momentum particles~\cite{bartoszek2015mu2etechnicaldesignreport} \label{fig:mu2e_strawtube_tracker}.}
\end{figure}



\section{Development of a pattern recognition algorithm}
The Mu2e track reconstruction algorithm involves various pattern recognition modules to construct helices from particle hits in the tracker and an assumption of the particle charge and direction of motion. The current reconstruction algorithm scans overall hits within the same time region (TimeCluster) in the XY plane of the tracker to identify hits that form a helix, which is then used for further steps to reconstruct a trajectory. While the pattern recognition algorithm is well-tuned for reconstructing conversion electron-like particles, further improvements will be crucial for real data analysis and $\phi Z$ clustering is one of the prospective methods to enhance robustness. The new algorithm will classify hits in the $\phi Z$ plane by taking advantage of the spiral trajectory of particles and the geometrical structure of the Mu2e tracker. The spiral trajectory becomes linear in the $\phi Z$ plane, which provides more explicit information for pattern recognition and potentially enhances track reconstruction efficiency. This track finding method could also improve reconstructing multi-track events within a single TimeCluster, which are prevalent in $\bar{p}$ annihilation background events. Reconstructing multi-track events will also be crucial to calibrate the Mu2e momentum scale using photons from radiative pion capture~(RPC) on hydrogen ($\pi^- p \rightarrow n \gamma$)~\cite{Mu2ePhysics50071}. The RPC photon converts in the detector material and produces a $\gamma \rightarrow e^+ e^-$ pair, with each particle carrying a momentum of 65~MeV/$c$. This work aims to enhance the capability and performance of Mu2e track reconstruction by introducing a new track-finding method which utilizes the unique geometry of the Mu2e tracker.


\section{Multi-track events}
\subsection{Antiproton-induced backgrounds}
Antiprotons are generated in the production target, and due to their negative charge and similar momentum range~($<100$ MeV/$c$) to muons, they can be transported to the stopping target by passing through the TS. Once antiprotons come to rest and annihilate in the stopping target, they produce neutral and charged pions, which decay into photons and muons, which further convert and decay into electrons. In order to limit the number of antiprotons reaching the stopping target, an absorber is mounted upstream of the TS. Figure~\ref{fig:5-pbar-phiz-view} shows the $\bar{p}$ annihilation events with the Mu2e tracker's geometric view.

\begin{figure}[htbp]
  \centering
  \includegraphics[width=0.9\textwidth]{images/phiz_xy_TZ_pbar.eps}
  \caption{Event display of $\bar{p}$ annihilation event showing three particles (a muon~(green) and two pions~(pink)) each producing a track. The left figure shows XY view, the middle figure shows $\phi Z$ view, and the right figure shows TZ view.}
  \label{fig:5-pbar-phiz-view}
\end{figure}

\subsection{Photons produced in the RPC on hydrogen}
Photons from RPC on hydrogen can be used to calibrate the Mu2e momentum scale and measure the momentum resolution. A $\pi^{-}$ stopped in hydrogen is captured by the hydrogen atom and undergoes the following interactions: $\pi^- p \rightarrow n \pi^{0}$, $\pi^- p \rightarrow n \gamma$. The branching ratio of $\pi^- p \rightarrow n \gamma$ is approximately 40\% and the energy spectrum of the emitted photons is monochromatic, peaking at 129.4 MeV~\cite{PhysRevC.5.1867}. RPC photon produces a $\gamma \rightarrow e^{+} e^{-}$ pair (65~MeV/$c$ for both) in the detector and induces enough hits in the tracker, which allows reconstruction of tracks and estimates the momentum of the converted photon. %The pion degrader installed in the Mu2e experiment employs hydrogen-containing compounds, polyethylene, with a thin converter foil, where conversion photons are emitted at angles close to $90\degree$ with respect to the DS axis with a clear signal. 
Figure~\ref{fig:photon_rpc} shows an electron and a positron from $\gamma \rightarrow e^{+} e^{-}$ conversion with 2 tracks in XY view.

\begin{figure}[htbp]
  \centering
  \includegraphics[width=0.7\textwidth]{images/photon_rpc_0.eps}
  \caption{\centering Event display of $\gamma \rightarrow e^+ e^-$ in XY view~\cite{PhysRevC.5.1867}.}
  \label{fig:photon_rpc}
\end{figure}



\section{Helix candidate search}
\subsection{Segment identification in the $\phi Z$ plane}
The algorithm processes a list of hits from a single TimeCluster and classifies them into subgroups, called segments, in accordance with their slope values in the $\phi Z$ plane. The reconstruction workflow proceeds as follows: (a) select 3 consecutive stations, each with at least 1 hit and require a total of at least 5 hits; (b) compute the slope value $d\phi/dz$ and calculate the residual $\Delta\phi$ between a hit and the slope. If $\Delta\phi \leq 0.4$, then save that hit as a candidate for the segment; (c) extend the slope to neighboring stations and continue until a hit is found in consecutive stations. If $\Delta\phi \leq 0.4$, then save that hit as a candidate; (d) if more than 2 segment candidates are found, calculate the fraction of overlapping hits between them. If the fraction of shared hits between 2 segments is more than 70\%, one of the segments is removed.










\subsection{Segment finding in multi-track events}
$\bar{p}$ annihilation events generate multiple particles at the stopping target within the same time range, which are classified into a single cluster. This characteristic feature of $\bar{p}$ annihilation complicates distinguishing two or more track events and separates corresponding hits from each other. The event in Figure~\ref{fig:5-pbar-phiz-view} shows three distinct tracks produced by three particles (such as muons and pions) within the same time window. While the tracks overlap in the XY plane, they exhibit distinct slope structures in the $\phi Z$ plane. Figure~\ref{fig:7-PhiZSeedFinder-result} shows segment candidates identified by the algorithm: two of these segments intersect at $Z \sim 600$~mm, with a positive slope associated with the pion and a negative slope associated with the muon.

\begin{figure}[h]
    \begin{center}
        \includegraphics[width=0.9\linewidth]{images/7-pbar-phiz-view-segment-1.eps}
    \end{center}
    \caption{\centering Segments identified in the $\phi Z$ view of a $\bar{p}$ annihilation event.}
    \label{fig:7-PhiZSeedFinder-result}
\end{figure}






\subsection{Helix reconstruction and validation}
Segments identified are grouped into a single candidate and passed to the next stage to search for a helix. Hits within the segment are used to reconstruct the helix by performing a circle-fit. The helix search is tested with $\bar{p}$ annihilation events with a single track to evaluate segment finding performance in a simplified case. The quality of the reconstructed helix is assessed by comparing the difference between the MC radius and the fit radius, as shown in Figure~\ref{fig:cicle_fit}. The circle-fitting process iteratively selects the best hits until the conditions $\chi^2/ \rm{ndf} < 5$ and $N_{hits} \geq 10$ are satisfied. The distribution of $R_{MC} - R_{Fit}$ yields a mean value of 11.4 mm. The segment search can accidentally collect nearby hits belonging to other particles, especially those positioned far away in the XY plane, which tends to result in a large radius in the circle-fit. To improve the helix quality, a cleanup process is required to remove surrounding hits belonging to other particles.

\begin{figure}[htbp]
  \centering
  \includegraphics[width=0.5\textwidth]{images/circle_fit.eps}
  \caption{The difference between the radius of the MC particle trajectory (truth level) and the reconstructed helix.}
  \label{fig:cicle_fit}
\end{figure}



\section{Summary}
The development of the new pattern recognition algorithm aims to enhance track reconstruction performance and robustness by introducing segment searches in the $\phi Z$ plane. Particles traversing the tracker exhibit linear trajectories in the $\phi Z$ plane, and the algorithm is designed to help identify these tracks, potentially improving reconstruction efficiency for events with different topologies. The helix search was tested with $\bar{p}$ events to quantify the performance of the circle-fit. The fitting results show cases where the fitted circles yield larger radii than the MC truth values, implying that a cleanup process for hits in the segment candidates is necessary to improve helix quality. A strategy to enhance the purity of hits in the segment candidates is under development and will be integrated into the track reconstruction sequence.




%\acknowledgments
%The author acknowledge XXX, YYY.
\begin{thebibliography}{0}

\bibitem{bartoszek2015mu2etechnicaldesignreport} \BY{Bartoszek~L. \etal}
  \TITLE{Mu2e Technical Design Report},
  arXiv preprint \texttt{arXiv:1501.05241}, 2015, \texttt{physics.ins-det}.

\bibitem{SINDRUM-2} \BY{Bertl~W., Engfer~R., Hermes~E.~A., Kurz~G., Kozlowski~T., Kuth~J., Otter~G., Rosenbaum~F., Ryskulov~N.~M., van der Schaaf~A., Wintz~P., Zychor~I. \atque The SINDRUM II Collaboration}
  \TITLE{A search for $\mu$ - e conversion in muonic gold},
  \IN{Eur. Phys. J. C}{47}{2006}{337--346}, Issue 2.

\bibitem{mu2ecollaboration2022mu2e} \BY{Mu2e Collaboration}
  \TITLE{Mu2e Run I Sensitivity Projections for the Neutrinoless $\mu^- \to e^-$ Conversion Search in Aluminum},
  arXiv preprint \texttt{arXiv:2210.11380}, 2022, \texttt{hep-ex}.

\bibitem{Mu2ePhysics50071} \BY{Ciampa~K., Miller~J. \atque Murat~P.}
  \TITLE{On a Possibility of the Mu2e Momentum Scale Calibration at Full Field 1. Proof of Principle},
  tech. report MU2E/PHYSICS/50071, version 1.02, Mu2e Collaboration, October 7, 2024.
  Note: Boston University (BU), Fermilab (FNAL).  

\bibitem{PhysRevC.5.1867} \BY{Bistirlich~J.~A., Crowe~K.~M., Parsons~A.~S.~L., Skarek~P. \atque Tru\"ol~P.}
  \IN{Phys. Rev. C}{5}{1972}{1867--1883}, Issue 6.


\end{thebibliography}

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