Estimating size of loop integrals

[RedX]

Is there a way to measure how large a numerical value a loop integral will give?

For example, take this integral over loop momenta k:

$$\frac{1}{k^2+m^2} \frac{1}{(k+p)^2+m^2}$$

How does it compare to setting p=0:

$$\frac{1}{k^2+m^2} \frac{1}{(k)^2+m^2}$$

or to a double integral over loop momenta k and q:

$$\frac{1}{k^2+m^2} \frac{1}{q^2+m^2}\frac{1}{(k+q)^2+m^2}$$

All I know how to do is to Wick rotate, regulate, renormalize, and I also know that $$\frac{1}{k^2+m^2}$$ should really be: $$\frac{1}{k^2+m^2-i\epsilon}$$ so that the integral over the energy component will not blow up - any blow up will be in the 3-momentum component.

But I'm not sure what's really going on with all these integrals. Some of them are infinite and you have to renormalize, and you're left with a finite part, but how to estimate the magnitude of the finite part?

 

[DarMM]

Well you can remove the divergent part via renormalization and then control the remaining finite part with analytic estimates.

State of the art in this regard gets a good (though you will need to be comfortable with Combinatorics, Graph Theory and Analysis) in Vincent Rivasseau's "From perturbative to constructive Renormalization"