# 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?

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.