The tree-level point-like weak decay rate for a lepton into three light or nearly…

PHYSICS · Subjective · Fermi University Physics · Particle Physics & Quantum Fields · Particles, interactions and experiment · Particles, interactions and experiment

Symmetry and Flavour, Ch From amplitude to rate: Lorentz-invariant phase space, widths and cross sections

Problem

The tree-level point-like weak decay rate for a lepton into three light or nearly massless products scales as Γ∝GF2m5\Gamma\propto G_F^2 m^5. The measured branching fraction of the τ−\tau^- into the muonic channel is B(τ−→μ−νˉμντ)=17.39 %\mathcal{B}(\tau^-\to\mu^-\bar\nu_\mu\nu_\tau) = 17.39\,\% (PDG 2024). Using the measured muon lifetime τμ=2.1970 μs\tau_\mu = 2.1970\,\mu\mathrm{s}, the masses mμ=105.658 MeVm_\mu = 105.658\,\mathrm{MeV} and mτ=1776.86 MeVm_\tau = 1776.86\,\mathrm{MeV}, and the fifth-power mass scaling of the decay rate, predict the τ\tau lifetime. Compare with the measured value ττ=2.903×10−13 s\tau_\tau = 2.903\times10^{-13}\,\mathrm{s}.

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