Decay ratio

1. The problem statement, all variables and given/known data

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

2. Relevant equations

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

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

3. The attempt at a solution

[tex]\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}[/tex]

[tex]\frac{dn}{dE} = \frac{dn}{dp} \frac{dp}{dE}[/tex]

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