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Decay ratio

  1. May 7, 2008 #1
    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?
    Last edited: May 7, 2008
  2. jcsd
  3. May 8, 2008 #2


    User Avatar

    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.
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