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Muon decay calculation 
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#1
Dec1713, 02:52 PM

P: 872

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) 


#2
Dec2613, 07:53 AM

Sci Advisor
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P: 2,010

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_eM)/E_1E_2E_e[/tex], [tex]\rightarrow 8\pi^2p_1dE_1p_2dE_2d\cos(\theta)\delta(E_1+E_2+E_eM)/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] 


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