 Solution

Write the leptonic partial decay width asHere $G_F$ is defined from muon decay, so the coupling denominator fixes the convention. Measure ${\Gamma }_{\ell }=B_{\ell }/{\tau }_{\tau }$. The ratio $B_{\mu }/B_e$ tests $|g_{\mu }/g_e{|}^2$ after the known phase space and radiative corrections. Comparing $B_e/{\tau }_{\tau }$ with the muon lifetime and the fifth power of the mass ratio tests $|g_{\tau }/g_{\mu }{|}^2$. Hadronic comparisons such as tau-to-pion versus pion-to-muon decay supply additional universality tests with their own hadronic and electromagnetic corrections.