How can I understand the reaction rate of WIMP dark matter intuitively? I'm reading a cosmology textbook(sorry it's Japanese) and stacked at statement that


*

*If the mass of WIMP is enough smaller than 100GeV, the reaction rate is represented as $\langle \sigma_a |\mathbf{\upsilon}|\rangle \simeq \frac{c}{\hbar^4} G_F^2 m^2$

*If the mass of WIMP is enough larger than 100GeV, the reaction rate is $\langle \sigma_a |\mathbf{\upsilon}|\rangle \simeq \frac{c}{\hbar^4} G_F^2 \frac{m_w^4}{m^2}$
I'm not familiar with particle theory so not sure how the mass dependences appear:
$m^2$ v.s. $\frac{m_w^4}{m^2}$.
My naive guess is the difference comes from the loop effect(radiative correction) of WIMP/W-boson particles, but I don't know how it affects in details.
Is there any intuitive way to understand the behavior of the mass dimension?
 A: Sounds like the wings of a Breit-Wigner distribution for virtual weak boson production by the dark matter, with a peak at $m^2 \approx m_W^2$ and a width related to $\Gamma^2 \approx 1/G_F$, and the energy scale set by the mass of the WIMP.
Dimensional analysis: ${G_F}{(\hbar c)^{-3}} m c^2$ has units of $\rm GeV^{-1}$. You've got that squared, multiplied by $(\hbar c)^2$ to give units of area, multiplied again by $c$ so that both sides have units of volume per unit time. To convert to a proper rate you have to multiply again by the number density of dark matter particles, but $\sigma v$ is where the new-particle physics is.
Using $\hbar = c = 1$ units, the Breit-Wigner distribution would have a complete expression like
$$
\left<\sigma |v| \right> \propto 
\frac
{k}
{\left( m^2 - m_W^2 \right)^2 + m_W^2/G_F}
$$
where the normalization $k$ depends on $m_W$ and $G_F$ in a messy way, with dimensions of $\rm GeV^{+2}$.
I had thought I would do some binomial expansion in the limits of $m \ll m_W$ and $m_W \ll m$, but that's not working for me today. But the Lorentzian has a going-up side and a going-down side, and $m^2$ and $m^{-2}$ are the simplest nonlinear going-up and going-down functions of $m$, and you can hide a lot of handwaving by writing $\simeq$ and making the units right.
