Testing the nature of neutrinos from four-body τ decays

This paper is designed to discuss four-body lepton number violating tau decay. We study the processes $\tau^+ \to e^+ e^+ \pi^- \nu_\tau$ and $\tau^+ \to e^+ e^+ \pi^- \nu_e$ to determine the nature of neutrino. The first process violates lepton number by two units which can only happen through an internal Majorana. The second one conserves lepton number but violates lepton flavor which can take place with both Majorana neutrino and Dirac neutrino. We calculate their branching ratio $Br$ and differential branching ratio $dBr/dE_\pi$ to distinguish Majorana neutrino vs. Dirac neutrino. We also propose the possibility of experiment to perform this detection.