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Speed of an antimuon in a pi meson decay

  1. Nov 12, 2015 #1
    1. The problem statement, all variables and given/known data

    I'm trying to confirm the speed of an antimuon in the [itex] \pi^+ \rightarrow \mu^+ \nu_{\mu} [/itex] decay through the laws of conservation but it doesn't add up.

    2. Relevant equations

    1.Energy-momentum relation:

    [tex]E^2 = (pc)^2 + (mc^2)^2 [/tex]

    2. Rest masses:

    [tex]m_{\pi} = 139.6 \ \frac{MeV}{c^2}[/tex]
    [tex]m_{\mu} = 105.7 \ \frac{MeV}{c^2}[/tex]
    [tex]m_{\nu} \approx 0 \frac{MeV}{c^2}[/tex]

    3. Relativistic kinetic energy formula:

    [tex]E_k =m_{\mu}c^2 \left( \frac{1}{\sqrt{1 - \frac{v_{\mu}^2}{c^2}}} - 1 \right)[/tex]

    3. The attempt at a solution

    By the way, the pi meson decays at rest, so [itex]p_{\pi}=0[/itex].

    I'm considering the difference of mass, before and after the decay, as pure kinetic energy, so around [itex](m_{\pi} - m_{\mu})c^2 = 33.9 MeV[/itex].

    [tex]m_{\mu}c^2 \left( \frac{1}{\sqrt{1 - \frac{v_{\mu}^2}{c^2}}} - 1 \right) = 33.9 \ MeV [/tex]

    Carrying this out yields [itex]v_{\mu}=0.65c[/itex] when in fact it should be [itex]0.27c[/itex].

    What am I doing wrong?
     
  2. jcsd
  3. Nov 13, 2015 #2

    fzero

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    Science Advisor
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    In order for momentum to be conserved, the neutrino must have momentum, so not all of that energy is available to the muon.
     
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