# Why does the integral for a “generic one-loop snail diagram” in scalar QFT blow up?

I am embarrassed to ask this but I can't figure it out. Say we are in scalar QFT and we have a Feynman diagram like the following (straight from my course notes)

Apparently this integral blows up for $large$ $q$ and that's why we need to do something like introduce a momentum cutoff. My misunderstanding here is that I see this integral blowing up for $small$ q and so clearly I don't understand this integral at all. Can somebody please be kind and help me out? I know this is probably a dumb question but I really need to understand this and I'm not getting very far on my own.

It blows up for large $q$ because of the integral measure, $\operatorname{d}^4q$. One way of writing that after Wick rotation is $$\Omega_4 q^3 \operatorname{d}q,$$ which grows as $q$ for $q\gg m$. Note that $\Omega_4$ is the solid angle in $4$-dimensional space.
That comes from the fact that in hyperspherical coordinates the volume element is given by $$\operatorname{d}^4q = q^3 \sin^2(\phi_1)\,\sin(\phi_2) \operatorname{d}q \operatorname{d}\phi_1 \operatorname{d}\phi_2 \operatorname{d}\phi_3.$$ Doing the angular integrals yields $\Omega_4$ because the integrand is invariant under $4$-d rotations.