quantum field theory - Help understanding loops with negative superficial degree of divergence

Consider

$$\int \frac{d^4k}{(2\pi)^4} \frac{1}{k^2}\frac{1}{k^2}\frac{1}{k^2}.\tag{1}$$

We can Wick rotate $k_0 \to i k_0$:

$$i \int \frac{d^4k_E}{(2\pi)^4} \frac{1}{k_E^2}\frac{1}{k_E^2}\frac{1}{k_E^2}\tag{2}$$ and switch to spherical coordinates

$$= 2\pi^2i \int \frac{dk_E}{(2\pi)^4} \frac{k_E^3}{k_E^6} \tag{3}$$ $$= 2\pi^2i \int \frac{dk_E}{(2\pi)^4} \frac{1}{k_E^3}\tag{4}$$

which doesn't converge according to Mathematica. I'm basically confused because $\int dk \frac{1}{k^n}$ doesn't converge for any $n$ using Mathematica.

I may be missing an obvious point. Why isn't this finite?

Does it change significantly if the denominator involves a mass?

Answer

The integral is infra-red divergent. Power counting is about ultra-violet divergences. As you can check for yourself, the singular behaviour is in the lower limit: $$\int_\epsilon^\infty \frac{1}{x^n}\mathrm dx=(1-n)^{-1}x^{1-n}\bigg|^\infty_\epsilon\sim\epsilon^{1-n}$$ which blows up if $n>1$.

The superficial degree of divergence tells you that the integral is superficially convergent as far as the $k\to\infty$ integration region is concerned. It tells nothing about possible singularities at finite values of $k$. In other words, the degree of divergence is all about the UV, but it knows nothing about the IR. In this example, the singularity is of the second type.

To regulate this integral you have to introduce a non-zero mass in the propagators, or use the dimensional regularisation trick $$\int x^n\mathrm dx\equiv 0\quad \forall n$$ which is known as Veltman's formula. It can be argued that this formula is consistent as far as perturbation theory is concerned. Pragmatically speaking, this can be accomplished by introducing a non-zero mass and taking the massless limit carefully. In more formal terms, the consistency of the prescription can be proven by means of analytical continuation to complex $n$ (cf. 1.1666512).

For more details, see Massless integrals in dim-reg and Divergence of the tree level scattering amplitude in quantum field theory. The latter contains a list of useful references on IR divergences, their physical meaning and regularisation techniques.