The muon lifetime is

PHYSICS · Multi-part · Fermi University Physics · Particle Physics & Quantum Fields · Electroweak theory and the Standard Model · Electroweak theory and the Standard Model

Symmetry and Flavour, Ch Quark mixing: the Cabibbo angle, the GIM mechanism and the CKM matrix

Problem

The muon lifetime is τμ=2.1969×10−6 s\tau_\mu = 2.1969\times10^{-6}\ \mathrm{s}, with mμ=105.658 MeVm_\mu = 105.658\ \mathrm{MeV}. The τ\tau lepton has mτ=1776.86 MeVm_\tau = 1776.86\ \mathrm{MeV} and a measured branching fraction B(τ→μνˉμντ)=17.39%\mathcal{B}(\tau \to \mu\bar\nu_\mu\nu_\tau) = 17.39\%. Assume the τ\tau and μ\mu couple to the WW with the same strength (charged-current universality), so that Γ(τ→μνˉν)=0.973 Γ(μ→eνˉν) (mτ/mμ)5\Gamma(\tau \to \mu\bar\nu\nu) = 0.973\,\Gamma(\mu \to e\bar\nu\nu)\,(m_\tau/m_\mu)^5, where 0.9730.973 is the reduction in phase space caused by the final muon's mass (the electron's mass in muon decay changes nothing at this precision). ((a)) Predict the τ\tau lifetime, ττ=τμ(mμ/mτ)5 B(τ→μνˉν)/0.973\tau_\tau = \tau_\mu(m_\mu/m_\tau)^5\,\mathcal{B}(\tau\to\mu\bar\nu\nu)/0.973. ((b)) The measured τ\tau lifetime is 290.3 fs290.3\ \mathrm{fs}. Taking the width to scale as the square of the τ\tau coupling relative to the muon coupling, find gτ/gμg_\tau/g_\mu.

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