Past exam of the particle physics course of the University of Oxford / 2016 / 5 / 1 / Solution

The local Fermi interaction gives a three-body leptonic decay width of the format tree level with negligible neutrino masses and universal weak coupling constant values. The fifth power follows from the dimension of $G_F$ and the three-body phase space; $f$ results from integrating the V-A interaction matrix element over the reduced phase space when the charged daughter has nonzero mass. It is more than a purely geometric volume correction because the spin-dependent matrix element is included.

As $z\to 0$, $f(z)\to 1$; increasing daughter mass reduces the available energy and suppresses the rate. The largest ratio is $m_{\mu }/m_{\tau }$, so the $\tau \to \mu$ correction is visibly below one. The electron is extremely light relative to either parent. Using the lepton masses givesThese reproduce the supplied factors to their stated accuracy. Radiative and finite-W boson corrections are separate small corrections when a precision lepton universality test is performed.