Gondolo-Gelmini Change of Variables

Question

In the article Cosmic abundances of stable particles: Improved analysis, P. Gondolo and G. Gelmini, Nucl. Phys. B 360 (1991), p. 145-179, they convert $rm{d}^3p_1rm{d}^3p_2=2pi^2p_1p_2rm{d}E_1rm{d}E_2rm{d}cos{theta}$ (eq.3.2) into  $rm{d}^3p_1rm{d}^3p_2=2pi E_1E_2rm{d}E_+rm{d}E_-rm{d}s$ (eq. 3.4) with the following change of variables:
$$E_+=E_1+E_2, quad E_-=E_1-E_2, quad s=2m^2+2E_1E_2-2p_1p_2cos{theta}, quad rm{(eq. 3.3)}$$
When I try to derive the second expresion of $rm{d}^3p_1rm{d}^3p_2$ using these new variables $E_+, E_-, s$, I always get second order diferentials that in theory should vanish. I don't know how one could get $rm{d}^3p_1rm{d}^3p_2=2pi E_1E_2rm{d}E_+rm{d}E_-rm{d}s$.
They also define new limits of integration due to this change of variables: ${ E_1>m, E_2>m,  |cos{theta}|leq 1 }$ changes to ${ sgeq 4m^2, E_+geq sqrt{s},  |E_-|leqsqrt{1-4m^2/s}sqrt{E_+^2-s} }$ (eq. 3.5). I get the first two, but I don't know how to compute $E_-$. The limit cases give some idea on how the expresion should be, but this is not enough to know the exact expresion, something like:
$$rm{If} quad E_+^2=s implies E_-=0$$
$$rm{If} quad s=4m^2 implies E_-=0$$
$$rm{Then (?):} quad |E_-| leq sqrt{1-frac{4m^2}{s}}sqrt{E_+^2-s}$$

Adri Escañuela · Accepted Answer

Since nobody has answered and I figured it out some time ago, I'll respond to my own post. You have to use the Jacobian to change variables:
$$    J_{ij}=frac{partial y_i}{partial x_j} =
    begin{pmatrix}
    frac{partial E_1}{partial E_+} & frac{partial E_1}{partial E_+} & frac{partial E_1}{partial s}
    frac{partial E_2}{partial E_+} & frac{partial E_2}{partial E_-} & frac{partial E_2}{partial s}
    frac{partial cos{theta}}{partial E_+} & frac{partial cos{theta}}{partial E_-} & frac{partial cos{theta}}{partial s}
    end{pmatrix}^{-1} =
    begin{pmatrix}
    1 & 1 & 2E_2
    1 & -1 & 2E_1
    0 & 0 & -2 p_1 p_2
    end{pmatrix}^{-1}.$$
Knowing that the determinant of the inverse matrix is the inverse of the determinant of the original matrix, we get $det{J}=(4p_1 p_2)^{-1}$.
$$
    mathrm{d}^3p_1 mathrm{d}^3p_2 = left(4pi p_1 mathrm{d}E_1right) left(4pi p_2 mathrm{d}E_2right) left(frac{1}{2}mathrm{d}cos{theta}right) = 8pi^2 p_1 p_2 frac{1}{4p_1 p_2}mathrm{d}E_+ mathrm{d}E_- mathrm{d}s = 2pi^2mathrm{d}E_+ mathrm{d}E_- mathrm{d}s.
$$
I guess the integrating limits logic is good enough. If anybody has something else to add there, please point it out.

Gondolo-Gelmini Change of Variables

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