Solving Muon Decay Calc: Need Help!

[ChrisVer]

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:
$d^{3}p_{\bar{v_{e}}}=-\frac{E_{\bar{v_{e}}} E_{v_{μ}}}{E_{e}} dE_{\bar{v_{e}}} dE_{v_{μ}} dφ (0)$

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

I stop in a very bad position not knowing how to continue:
$d^{3}p_{\bar{v_{e}}}= p_{\bar{v_{e}}}^{2} dp_{\bar{v_{e}}} dcosθ dφ=E_{\bar{v_{e}}}^{2} dE_{\bar{v_{e}}} dcosθ dφ$
How would you recommend I continue? I would try to differentiate the $(1)$ but it has also cosθ and generally a mess is happening. I also could try to differentiate $(2)$ but I would get weird results not coinciding with $(0)$
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)

 

[Meir Achuz]

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