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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 $p \rightarrow e^+ \barν ν$ or $p \rightarrow μ^+ \barν ν$ originating from Grand Unification models, using general effective Fermi theory formalism of electroweak muon decay $μ\rightarrow e^+ \barν ν$. 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.