\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}{9}{section.2}
\contentsline {subsection}{\numberline {2.1}Neutrino Oscillation}{9}{subsection.2.1}
\contentsline {subsection}{\numberline {2.2}New physics}{10}{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}{14}{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}{29}{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}{32}{section.6}
\contentsline {subsection}{\numberline {6.1}General information}{32}{subsection.6.1}
\contentsline {subsection}{\numberline {6.2}Preprocessing}{32}{subsection.6.2}
\contentsline {subsubsection}{\numberline {6.2.1}Dataset 1}{32}{subsubsection.6.2.1}
\contentsline {subsubsection}{\numberline {6.2.2}Dataset 2}{33}{subsubsection.6.2.2}
\contentsline {section}{\numberline {7}RNN's used}{34}{section.7}
\contentsline {subsection}{\numberline {7.1}RNN for track prediction}{34}{subsection.7.1}
\contentsline {subsection}{\numberline {7.2}RNN for classification of tracks}{35}{subsection.7.2}
\contentsline {section}{\numberline {8}Analysis}{38}{section.8}
\contentsline {subsection}{\numberline {8.1}Best $\chi ^2$}{38}{subsection.8.1}
\contentsline {subsection}{\numberline {8.2}RNN classifier with RNN track prediction input}{38}{subsection.8.2}
\contentsline {subsection}{\numberline {8.3}XGBoost}{40}{subsection.8.3}
\contentsline {subsection}{\numberline {8.4}Comparison in performance of the RNN and XGBoost}{42}{subsection.8.4}
\contentsline {section}{\numberline {9}Results}{43}{section.9}
\contentsline {subsection}{\numberline {9.1}Results}{43}{subsection.9.1}
\contentsline {subsection}{\numberline {9.2}Outlook and potential}{43}{subsection.9.2}
\contentsline {section}{\numberline {10}Acknowledgements}{44}{section.10}