Calculating a cross section

  • Thread starter Davide82
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Hi.

I am looking for some help calculating a cross section to derive the frequency of reaction in the early universe.

The reaction taking place is:
[tex]\nu_e + n \longleftrightarrow p + e^-[/tex]

After some calculation I came here:

[tex]
n_\nu \langle \sigma v \rangle = \frac {(2\pi)} {8 m^2} \int \frac {d \vec p_\nu} {(2\pi)^3 \, 2 p_\nu} \int \frac {d \vec p_e} {(2\pi)^3 \, 2 p_e} \cdot e^{-(p_\nu)/(k T)} \cdot \delta (p_\nu + m_n - m_p -p_e) \, |\mathcal{M} |^2
[/tex]

[tex]n_\nu \text{ is the numerical density of neutrinos}[/tex]

[tex]|\mathcal{M}|^2 = 32 \, G_\mathrm{F}^2 (1 + 3 \, g_\mathrm{A}^2) \, m_p^2 \, p_\nu \, p_e[/tex]

I know the passages up until here are pretty much correct.

But I can't get to the known result:
[tex]n_\nu \langle \sigma v \rangle = \frac {255} {2 \tau_\mathrm n x^5} (12+6x+x^2)[/tex]

[tex]\tau_\mathrm n \text{ is the neutron's mean lifetime}[/tex]
[tex]x = \frac {\Delta m}{T}[/tex]
 

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