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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.

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.

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