Degree of divergence of a Feynman diagram

Question

I am studying the degrees of divergence of Feynman diagrams. I feel that I miss something but I don't really understand what. Please apologize if this question is silly. Anyway.

As an introduction to systematics of renormalization, Peskin and Schroeder (chapter 10) introduce the counting of ultraviolet divergences. They say that an expression for any diagram typically looks like

$$\sim \int \dfrac{d^4 k_1 d^4 k_2 \dots d^4 k_L}{(\bar{k}_i-m) \dots k^2_j \dots }$$ where $\bar{k}$ is $k$ slash (i.e. $\gamma^{\mu} k_{\mu}$). The number of $d^4 k$ will depend on the number of loops, while the number of $(\bar{k}-m)$ and $k^2$ in the denominator will depend on the number of fermion/boson propagators. Ok.

They then say :

For each loop there is a potentially divergent 4-momentum integral, but each propagator aids the convergence of this integral by putting one or two powers of momentum into the denominator.

I don't understand that. To be more clear, I'll develop a little bit more...

They then introduce the superficial degree of divergence $D$ as $$ D \equiv (\text{power of } k \text{ in numerator}) - (\text{power of } k \text{ in denominator}) $$

and they say that we can expect a diagram to have a divergence proportional to $\Lambda^D$ ($\Lambda$ is a cut-off) when $D>0$, proportional to $\log(\Lambda)$ when $D=0$ and no divergence when $D<0$.

I know that it's a tool and not always exact and so on to evaluate the divergence of a Feynman diagram. That's not the point. What I don't get is why is a diagram with $D<0$ not divergent ? Integrals like

$$ \int \dfrac{d^4 k}{k^7} $$ don't converge, so the associated diagram should be divergent too... Or is it because we are only dealing with UV divergences, and that diagrams with $D<0$ diverge because of IR divergences ?

Answer 1

You hit upon the answer yourself: they are not talking about IR divergences. The renormalisation program is about absorbing unknown (not necessarily divergent) short distance physics into effective local operators. The beauty of this is we do not need a full understanding or rigorous formulation of the short distance physics if we are only interested on much larger distance scales. So you play this game of absorbing UV corrections into counter-terms. All the associated power counting is there to organise the UV corrections into a form that can be absorbed this way, order by order in perturbation theory.

IR divergences are a different beast. They come about because you have massless particles, so it is impossible in practice to distinguish a state with or without "soft" photons (or whatever massless things are in your theory). The way to handle this is to regulate in the IR, carefully compute observable quantities (inclusive cross sections) with wavepackets and so on. It is enough in principle to simply put the system in a box the size of the universe. Anything physical should not depend on the regulator in the end. One thing you definitely can't do is absorb IR divergences - which represent long range effects - into effective local counter-terms. You can integrate out a massless field, but you end up with a non-local effective theory that is a mess to deal with.