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)
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]