= Solution
The negative <muon> emits a virtual $W^-$ and becomes a <muon neutrino>; the boson creates an <electron> and an <electron antineutrino>:
$$
\mu^-\to\nu_\mu+W^{-*}\to\nu_\mu+e^-+\bar\nu_e.
$$
Both vertices are <weak charged currents>.
\Image[/past-exam-of-the-particle-and-nuclear-physics-course-of-the-university-of-cambridge/2016/paper-1-muon-decay.png]
{title=<muon> decay through a virtual charged weak boson}
{description=Original tree diagram. Arrows indicate fermion-number flow, including the reversed arrow on the outgoing <electron antineutrino>.}
Typical transfers satisfy $|q^2|\lesssim m_\mu^2\ll M_W^2$, so the propagator factor $(M_W^2-q^2)^{-1}$ expands as $M_W^{-2}$ plus small corrections. Contracting this short-distance exchange produces the local <Fermi interaction>, with strength contained in $G_F$. This effective description does not assume that the real W <mass> vanishes. The same treatment applies to leptonic tau decays with the relevant <masses> changed.<Review of Particle Physics by Particle Data Group (2022)>[Muon Decay Parameters, §57.1 introduction and §57.2]{ref}
In GeV units, $m_\mu=0.106\,\mathrm{GeV}$ and $\hbar=6.582\times10^{-25}\,\mathrm{GeV\,s}$. Thus
$$
\Gamma_\mu=\frac{(1.166\times10^{-5})^2(0.106)^5}{192\pi^3}\,\mathrm{GeV}
=3.0562\times10^{-19}\,\mathrm{GeV},
$$
$$
\boxed{\tau_\mu=\frac{\hbar}{\Gamma_\mu}=2.154\times10^{-6}\,\mathrm s.}
$$
The modest difference from a precision experimental lifetime is expected from the rounded <mass> and omitted radiative and final-electron-mass corrections.
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