Past exam of the particle and nuclear physics course of the University of Cambridge / 2016 / Paper 1 / 3 / c / Solution

The negative muon emits a virtual $W^{-}$ and becomes a muon neutrino; the boson creates an electron and an electron antineutrino:Both vertices are weak charged currents.

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.^[ref]

In GeV units, $m_{\mu }=0.106\mathrm{GeV}$ and $\hslash =6.582\times 10^{-25}\mathrm{GeV}\mathrm{s}$. ThusThe modest difference from a precision experimental lifetime is expected from the rounded mass and omitted radiative and final-electron-mass corrections.