# Decay ratio

1. May 7, 2008

### GoodKopp

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

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

2. Relevant 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.

3. The attempt at a solution

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

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