Why is the decay rate ratio squared not fourth power of coupling constant?

[vertices]

Please take a look at the file "Feynman.pdf". The ratio rate for the two decays is shown in the second file, "rate.pdf".

I understand it takes that form because the usW vertex is Carribbo supressed relative to the udW vertex. My concern is that the ratio is only squared.

Rate is proportional to the amplitude *squared*. And the amplitude is proportional to the square of the coupling constant (multiplied by the propagator). So the transition rate must be proportional to the fourth power of the coupling constant, right?

Is the expression in rate.pdf wrong?

 

[pam]

The W-mu-nu coupling is the same for each so the amplitude is proportional to the first power of each g.
However, I would also expect a strong dependence on the mass ratio k/pi because the muon
has to have positive helicity for this decay.

 

[Vanadium 50]

Additionally, the ratio can't be right because it completely ignores the W propagator: there's 10 times as much Q in the kaon decay.

 

[vertices]

Thanks for your replies pam and Vanadium.