Leptonic Decay of Muon and Tau

Question

So the question asks:

(a) Draw the Feynman diagrams showing the dominant leptonic decay mode of the $\mu^-$ and the $\tau^-$.

(b) Assuming $m_\tau \gg m_\mu \gg m_e$, estimate the ratio of the rate of the two decay modes.

Part (a) is simple.

Part (b), I'm not sure what I am meant to do. I know the decay widths for the decay modes are $$ \Gamma(\mu^- \to e^-+\bar{\nu}_e+\nu_\mu) = \frac{G_F^eG_F^\mu m_\mu^5}{192 \pi^3}, $$ $$ \Gamma(\tau^- \to e^-+\bar{\nu}_e+\nu_\tau) = \frac{G_F^eG_F^\tau m_\tau^5}{192 \pi^3}, $$ $$ \Gamma(\tau^- \to \mu^-+\bar{\nu}_\mu+\nu_\tau) = \frac{G_F^\mu G_F^\tau m_\tau^5}{192 \pi^3} $$

and by lepton universality, $$ G_F^e=G_F^\mu = G_F^\tau. $$

I'm not sure how to proceed from here. Is it simply finding the branching ratios and dividing one by the other?

Answer 1

You don't need to worry about branching ratios. The two decay modes being referred to are the dominant leptonic decay mode of the muon and the dominant leptonic decay mode for the tau (not "tauon"). You have those decay widths already, so you can just take their ratio. Most of the factors will cancel out, leaving just a power of the muon-tau mass ratio, indicating the heavier tau has faster leptonic decays.

(Of course, what makes real tau decays really interesting is that the tau is heavy enough to have lots of hadronic decay modes, but those are obviously much more complicated.)