Momentum Space Measure Change of Variables: Exploring Cosmic Abundances

[guest1234]

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, [itex]\mathrm{d}^3\mathbf{p}_1\mathrm{d}^3\mathbf{p}_2[/itex], suddenly becomes [itex]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[/itex], where [itex]\theta=\angle(\mathbf{p}_1, \mathbf{p}_2)[/itex] and [itex]p_i=|\mathbf{p}_i|[/itex] (equation (3.2))? The dimensions don't match at all.. The closest I can come up with is [itex]\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 [/itex], which after substituting with [itex]p_i \mathrm{d}p_i=E_i \mathrm{d}E_i[/itex] (because [itex]p_i^2 = E_i^2 - m_i^2[/itex]) gives [itex]\mathrm{d}^3\mathbf{p}_i = 4\pi p_i E_i \mathrm{d}E_i \hat{\mathbf{p}}_i [/itex]. So I think it should be something like [itex]\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[/itex]. With this I'm off by one diffenetial and 1/2...
It also bugs me that they make change of variables, [itex]E_+ = E_1 + E_2[/itex], [itex]E_- = E_1 - E_2[/itex], [itex]s = (E_1 + E_2, \mathbf{p}_1 + \mathbf{p}_2)^2[/itex], which again leads to a peculiar result, [itex]\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[/itex] (equation 3.4). I mean, doing this rigorously doesn't work out, unless you assume that [itex]E_+ = E_+(E_2)[/itex], [itex]E_- = E_-(E_1)[/itex] and [itex]s = s(p_1 p_2 \cos\theta)[/itex] 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 [itex]\mathrm{d}^3\mathbf{p}_2 = 2\pi p_2^2 \mathrm{d}p_2 \mathrm{d}\cos\theta[/itex] and evidently equation (3.2) is indeed missing a factor [itex]p_1 p_2[/itex]. Still, the change of variables into [itex]E_\pm[/itex] and [itex]s[/itex] puzzles me.

 

[Ambitwistor]

Thank you for bringing up this interesting question. I have read through the article and have some thoughts on the issues you have raised.

Firstly, regarding equation (3.2), I agree with your analysis that there seems to be a missing factor of p_1 p_2. This could be a typo or an oversight by the authors. I suggest reaching out to them for clarification on this matter.

As for the change of variables in equations (3.4) and (3.8), I believe the authors are using a relativistic approach where they are considering the energy and momentum of particles as components of a four-vector. In this approach, the variables E_+ and E_- represent the total energy and the difference in energy between the two particles, respectively. Similarly, s represents the square of the total four-momentum of the two particles. This approach may not be immediately intuitive, but it is commonly used in particle physics and cosmology.

I would also like to point out that the dimensions in equations (3.4) and (3.8) do match, as the factor of 4\pi comes from the integration over angles, which is accounted for in the \mathrm{d}\cos\theta term.

I hope this helps clarify some of your concerns. If you have any further questions or would like to discuss this further, please do not hesitate to reach out.

Happy new year to you as well.