Estimating K$^+$ Decay Ratios Without Mass Dependence

[GoodKopp]

Homework Statement

Give a simple estimate of the ratio of decay rates for $$K^+ \rightarrow e^+ \nu_e / K^+ \rightarrow \mu^+ \nu_\mu.$$

Homework Equations

Fermi's Golden Rule for decay width $$\Gamma = \hbar W = 2\pi (dn/dE_f)|M_{if}|^2.$$

For comparison, from the PDG, $$\Gamma / \Gamma_i$$ are $$1.55 \pm 0.07 \times 10^{-5}$$ and $$63.44 \pm 0.14$$ for $$K^+ \rightarrow e^+ \nu_e$$ and $$K^+ \rightarrow \mu^+ \nu_\mu$$, respectively.

The Attempt at a Solution

$$\frac{\Gamma (K^+ \rightarrow e^+ \nu_e)}{\Gamma (K^+ \rightarrow \mu^+ \nu_\mu)} = \frac{\frac{dn}{dE} e^+ \nu_e}{\frac{dn}{dE} \mu^+ \nu_\mu}$$

$$\frac{dn}{dE} = \frac{dn}{dp} \frac{dp}{dE}$$

Am I going about this the right way? Where do I go from here?

 

[pam]

You have to include a very strong dependence on the e or mu mass.
This comes from the weak interaction.
The same factor comes up in pi decay.