Charged Lepton Spectrum Approximation in a Three Body Nucleon Decay

Only phase space is typically used to obtain final state particle spectra in rare decay searches, which is a crude approximation in the case of three body processes. We will demonstrate how both dynamics and phase space can be approximately accounted for, in processes such as nucleon decays $p \rightarrow e^+ \bar{\nu} \nu$ or $p \rightarrow \mu^+ \bar{\nu} \nu$ originating from Grand Unification models, using general effective Fermi theory formalism of electroweak muon decay $\mu \rightarrow e^+ \bar{\nu} \nu$. This approach allows for a more precise and only weakly model dependent approximation of final particle spectra for these and similar decays, which may improve rare process searches in current and near-future experiments.