# Momentum space measure -- change of variables

## Main Question or Discussion Point

I'm working through an article called "Cosmic abundances of stable particles -- improved analysis" (link -- viewable only in Firefox afaik), the result of which, equation (3.8), is cited quite a lot. I'm more interested in how they arrived there.
Particularly, how come momentum space measure, $\mathrm{d}^3\mathbf{p}_1\mathrm{d}^3\mathbf{p}_2$, suddenly becomes $4\pi p_1\mathrm{d} E_1 4\pi p_2\mathrm{d} E_2 \frac{1}{2}p_1 p_2 \mathrm{d}\cos\theta$, where $\theta=\angle(\mathbf{p}_1, \mathbf{p}_2)$ and $p_i=|\mathbf{p}_i|$ (equation (3.2))? The dimensions don't match at all.. The closest I can come up with is $\mathrm{d}^3\mathbf{p}_i = p_i^2 \sin\theta_i\mathrm{d}\theta_i\mathrm{d}\phi_i\mathrm{d}p_i \hat{\mathbf{p}}_i = [\mbox{integrate over angles}] = 4\pi p_i^2 \mathrm{d}p_i \hat{\mathbf{p}}_i$, which after substituting with $p_i \mathrm{d}p_i=E_i \mathrm{d}E_i$ (because $p_i^2 = E_i^2 - m_i^2$) gives $\mathrm{d}^3\mathbf{p}_i = 4\pi p_i E_i \mathrm{d}E_i \hat{\mathbf{p}}_i$. So I think it should be something like $\mathrm{d}^3\mathbf{p}_1\mathrm{d}^3\mathbf{p}_2 = 4\pi E_1 \mathrm{d}E_1 4\pi E_2 \mathrm{d}E_2 p_1 p_2 \cos\theta=16\pi^2 E_1 E_2 p_1 p_2\cos\theta\mathrm{d}E_1\mathrm{d}E_2$. With this I'm off by one diffenetial and 1/2...
It also bugs me that they make change of variables, $E_+ = E_1 + E_2$, $E_- = E_1 - E_2$, $s = (E_1 + E_2, \mathbf{p}_1 + \mathbf{p}_2)^2$, which again leads to a peculiar result, $\mathrm{d}^3 p_1\mathrm{d}^3p_2 = 2\pi^2 E_1 E_2 \mathrm{d}E_+\mathrm{d}E_-\mathrm{d}s$ (equation 3.4). I mean, doing this rigorously doesn't work out, unless you assume that $E_+ = E_+(E_2)$, $E_- = E_-(E_1)$ and $s = s(p_1 p_2 \cos\theta)$ or something..

Any ideas what went wrong in (3.2) and (3.4)?
(Sorry for the question being too technical; if you feel this thread doesn't belong to this forum, move it away)
Oh, and happy new year ..

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edit: ok, I have figured out that $\mathrm{d}^3\mathbf{p}_2 = 2\pi p_2^2 \mathrm{d}p_2 \mathrm{d}\cos\theta$ and evidently equation (3.2) is indeed missing a factor $p_1 p_2$. Still, the change of variables into $E_\pm$ and $s$ puzzles me.

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