Muon decay calculation

  1. ChrisVer

    ChrisVer 2,271
    Gold Member

    Please, I'd need some help. Although I am not sure if this is again the correct thread, but since it concerns muon decay I bring it here. So...

    I am trying to find out why the differential below, in spherical cordinates becomes:
    [itex]d^{3}p_{\bar{v_{e}}}=-\frac{E_{\bar{v_{e}}} E_{v_{μ}}}{E_{e}} dE_{\bar{v_{e}}} dE_{v_{μ}} dφ (0)[/itex]

    I already have derived the equation:
    [itex] E_{v_{μ}}^{2}= E_{\bar{v_{e}}}^{2}+E_{e}^{2}+2E_{\bar{v_{e}}}E_{e}cosθ (1)[/itex]
    I also have the conservation of energy due to delta function:
    [itex] E_{v_{μ}}= m_{μ}-E_{\bar{v_{e}}}-E_{e} (2)[/itex]

    I stop in a very bad position not knowing how to continue:
    [itex]d^{3}p_{\bar{v_{e}}}= p_{\bar{v_{e}}}^{2} dp_{\bar{v_{e}}} dcosθ dφ=E_{\bar{v_{e}}}^{2} dE_{\bar{v_{e}}} dcosθ dφ [/itex]
    How would you recommend I continue? I would try to differentiate the [itex](1)[/itex] but it has also cosθ and generally a mess is happening. I also could try to differentiate [itex](2)[/itex] but I would get weird results not coinciding with [itex](0)[/itex]
    Any suggestion?
    (the mass of muon only exists, in the game, so the electron and neutrinos' masses are neglected, and thus their energies are equal to their momentum's magnitudes)
     
    Last edited: Dec 17, 2013
  2. jcsd
  3. Meir Achuz

    Meir Achuz 2,058
    Science Advisor
    Homework Helper
    Gold Member

    Start with
    [tex]d^3p_1d^3p_2d^3p_e\delta^4()/E_1E_2E_e[/tex].
    [tex]\rightarrow d^3p_1d^3p_2\delta(E_1+E_2+E_e-M)/E_1E_2E_e[/tex],
    [tex]\rightarrow 8\pi^2p_1dE_1p_2dE_2d\cos(\theta)\delta(E_1+E_2+E_e-M)/E_e[/tex],
    with [itex]E_e=\sqrt{m^2+p^2_1+p^2_2+2p_1p_2\cos(\theta)}[/itex].
    The delta function integration over d\theta gives
    [tex]8\pi^2dE_1dE_2.[/tex]
     
    Last edited: Dec 26, 2013
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