Diverging Integral: 4 Powers of q in Numerator, 2 in Denominator

[Elwin.Martin]

Quick question:
What does it mean that this has "four powers of q in the numerator and two in the denominator"? Apparently, this diverges quadratically at large q and has an infrared divergence as q→0 (I'm not concerned about the second one all that much though).
I mean, simply looking at a comparison of powers, since the integration is over all of q, it's feels like they get this result by just saying we have 4/2=2 on top. . .but that's not really legitimate.

g\int \frac{d^4 q}{\left(2\pi\right)^4}\frac{1}{q^2-m^2}

Thanks for any help! I don't doubt the divergence, but I'm just not sure what is meant by it ^^; in one dimension, I know that it blows up at q=±m, I'm just not sure what implications this has on the divergence in d⁴q

 

[torquil]

The integrand depends only on |q|. Therefere, imagine the integration domain to be a 4-ball of radius Q. You can use 4d spherical coordinates and perform all angular integrals. You are left with an integral over the radial coordinate |q|. It will have some factors of |q| in the numerator and denominator of its integrand. Then analyse the degree of divergence by checking how the result depends on Q approaches infinity. Similarly can be done e.g for integral divergence caused by integrand divergencies such as what happens at q^2 = m^2