For the bounty please verify the following reasoning
[copied from comment below]
Ah right, so the idea is that overall observable quantities must be independent of the renormalization scale. But at each order in perturbation theory the result can depend on renormalization scale, right? And they do so in exactly the right way to invalidate the use of a perturbation series when external momenta are of the wrong order. This is because the external momenta get caught up in loop integrals due to momenta conservation at vertices and the renormalization group equation comes along and means the right things cancel.
People say that perturbation theory breaks down when the couplings run to high values. But given that this running depends on an arbitrary mass scale, how is this argument logical?!
A Longer Exposition (TL;DR)
It's obvious that Feynman diagram techniques work best when the coupling constant is small, since then you can neglect higher order terms. But various sources (e.g. Peskin and Schroeder) make claims about running couplings that seem incomplete to me.
I often read sentences like
- if the renormalized coupling is small at low energy, then Feynman diagrams are good in that region
- for asymptotically free theories, Feynman diagrams are good for high energy calculations
I understand exactly why these are true. But surely something much more general is true, namely
- if the coupling constant is small at any renormalization scale the Feynman diagrams are good for calculations
My reasoning is as follows. Observable quantities are completely independent of the running of the coupling, so you can just choose your scale appropriately so that the coupling is small for the expansion. Hey presto, you can use Feynman diagrams.
Is this true? I strongly expect the answer to be no but I can't convince myself why. Here's my attempt at a self-rebuttal.
- my argument above is incorrect, because I've assumed there's only a single coupling. In reality there's contributions from "irrelevant" terms at higher energies whose couplings can't be fixed from low energy observations.
This leads me to hypothesise that the following is true (would you agree?)
- if the coupling constant is small at any renormalization scale above the scale of your observations then Feynman diagrams are good
This seems more plausible to me, but it would mean that Feynman diagrams would be good for low energy strong interaction processes for example. This feels wrong is a sense because the renormalized coupling is large there. But then again the renormalization scale is arbitrary, so maybe that doesn't matter.
Many thanks in advance!