\babel@toc {english}{} \contentsline {section}{\numberline {1}Standard Model}{4}{section.1} \contentsline {subsection}{\numberline {1.1}Elementary particles and forces}{4}{subsection.1.1} \contentsline {subsection}{\numberline {1.2}Interaction rules}{7}{subsection.1.2} \contentsline {section}{\numberline {2}Physics beyond the SM}{8}{section.2} \contentsline {subsection}{\numberline {2.1}Neutrino Oscillation}{8}{subsection.2.1} \contentsline {subsection}{\numberline {2.2}New physics}{9}{subsection.2.2} \contentsline {section}{\numberline {3}$\mu \rightarrow eee$ decay}{12}{section.3} \contentsline {subsection}{\numberline {3.1}Kinematics}{12}{subsection.3.1} \contentsline {subsection}{\numberline {3.2}Background events}{12}{subsection.3.2} \contentsline {subsubsection}{\numberline {3.2.1}Internal conversions}{12}{subsubsection.3.2.1} \contentsline {subsubsection}{\numberline {3.2.2}Michel decay}{13}{subsubsection.3.2.2} \contentsline {subsubsection}{\numberline {3.2.3}Radiative muon decay}{13}{subsubsection.3.2.3} \contentsline {subsubsection}{\numberline {3.2.4}BhaBha scattering}{13}{subsubsection.3.2.4} \contentsline {subsubsection}{\numberline {3.2.5}Pion decays}{13}{subsubsection.3.2.5} \contentsline {subsubsection}{\numberline {3.2.6}Analysis of the background}{14}{subsubsection.3.2.6} \contentsline {section}{\numberline {4}Mu3e experiment}{15}{section.4} \contentsline {subsection}{\numberline {4.1}Requirements}{15}{subsection.4.1} \contentsline {subsection}{\numberline {4.2}Phase I}{15}{subsection.4.2} \contentsline {subsection}{\numberline {4.3}Phase II}{15}{subsection.4.3} \contentsline {subsection}{\numberline {4.4}Experimental setup}{15}{subsection.4.4} \contentsline {subsection}{\numberline {4.5}The problem of low longitudinal momentum recurlers}{18}{subsection.4.5} \contentsline {section}{\numberline {5}Machine learning}{21}{section.5} \contentsline {subsection}{\numberline {5.1}Introduction}{21}{subsection.5.1} \contentsline {subsection}{\numberline {5.2}Artificial neural networks}{21}{subsection.5.2} \contentsline {subsubsection}{\numberline {5.2.1}General concepts}{21}{subsubsection.5.2.1} \contentsline {subsubsection}{\numberline {5.2.2}Activation functions}{23}{subsubsection.5.2.2} \contentsline {subsubsection}{\numberline {5.2.3}Concepts of training}{24}{subsubsection.5.2.3} \contentsline {subsubsection}{\numberline {5.2.4}Loss functions}{24}{subsubsection.5.2.4} \contentsline {subsubsection}{\numberline {5.2.5}Stochastic gradient descent}{25}{subsubsection.5.2.5} \contentsline {subsubsection}{\numberline {5.2.6}Stochastic gradient descent with Momentum}{25}{subsubsection.5.2.6} \contentsline {subsubsection}{\numberline {5.2.7}RMSProp}{25}{subsubsection.5.2.7} \contentsline {subsubsection}{\numberline {5.2.8}Adam}{26}{subsubsection.5.2.8} \contentsline {subsubsection}{\numberline {5.2.9}Decaying learning rate}{27}{subsubsection.5.2.9} \contentsline {subsubsection}{\numberline {5.2.10}Batch normalisation}{27}{subsubsection.5.2.10} \contentsline {subsection}{\numberline {5.3}Recurrent Neural Networks}{27}{subsection.5.3} \contentsline {subsubsection}{\numberline {5.3.1}General concepts}{27}{subsubsection.5.3.1} \contentsline {subsubsection}{\numberline {5.3.2}Most common architectures}{28}{subsubsection.5.3.2} \contentsline {subsubsection}{\numberline {5.3.3}Cell types}{29}{subsubsection.5.3.3} \contentsline {subsection}{\numberline {5.4}XGBoost}{30}{subsection.5.4} \contentsline {section}{\numberline {6}Data}{31}{section.6} \contentsline {subsection}{\numberline {6.1}General information}{31}{subsection.6.1} \contentsline {subsection}{\numberline {6.2}Preprocessing}{31}{subsection.6.2} \contentsline {subsubsection}{\numberline {6.2.1}Dataset 1}{31}{subsubsection.6.2.1} \contentsline {subsubsection}{\numberline {6.2.2}Dataset 2}{32}{subsubsection.6.2.2} \contentsline {section}{\numberline {7}RNN's used}{33}{section.7} \contentsline {subsection}{\numberline {7.1}RNN for track prediction}{33}{subsection.7.1} \contentsline {subsection}{\numberline {7.2}RNN for classification of tracks}{34}{subsection.7.2} \contentsline {section}{\numberline {8}Results}{37}{section.8} \contentsline {subsection}{\numberline {8.1}Best $\chi ^2$}{37}{subsection.8.1} \contentsline {subsection}{\numberline {8.2}RNN classifier with RNN track prediction input}{37}{subsection.8.2} \contentsline {subsection}{\numberline {8.3}XGBoost}{37}{subsection.8.3} \contentsline {subsection}{\numberline {8.4}Comparison in performance of the RNN and XGBoost}{39}{subsection.8.4} \contentsline {subsection}{\numberline {8.5}Outlook}{39}{subsection.8.5} \contentsline {section}{\numberline {9}Acknowledgements}{40}{section.9}