For $2\to 2$ scattering with equal masses we have
$$\left(\frac{d\sigma}{d\Omega}\right)_{CM} = \frac{1}{64 \pi^2 E_{CM}^2} |\mathcal{M}|^2.$$
(Schwartz's QFT eq. 5.33) Can we make the general statement that total cross section increases as $E_{CM}$ decreases for scattering processes? I'm guessing not because $\mathcal{M}$ may have a complicated dependence on energy/momenta.
As an example, take the case of $e^+e^-\to \mu^+\mu^-$. With $m_e = 0$ we have
$$\frac{d\sigma}{d\Omega} = \frac{\alpha^2}{4 E_{CM}^2} \sqrt{1-m_{\mu}^2/E_{\mu}^2} \left(1+ m_{\mu}^2/E_{\mu}^2 + (1- m_{\mu}^2/E_{\mu}^2)\cos^2\theta\right)$$ in the c.o.m frame (Schwartz 13.78). Here it's pretty clear that total cross section increases as energy decreases.
But I would think that particles scatter more when they are given more energy? The limiting case of course being zero velocity $\to $ zero scattering. I would appreciate any intuition on the relation between $E_{CM}$ and total cross section.