# Calculating a cross section

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:
$$\nu_e + n \longleftrightarrow p + e^-$$

After some calculation I came here:

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

$$n_\nu \text{ is the numerical density of neutrinos}$$

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

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

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

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