Why does a perturbative theory imply coupling must be increasing or decreasing?

Question

I am reading some of Schwartz's quantum field theory book, and I have come across a passage I find confusing. It states that

When the coupling is small, the theory is perturbative, and then the coupling must either increase or decrease with scale. (p442, 1st edition)

I think I am confusing what may be two different definitions of perturbative. My understanding of perturbative was that the coupling is small enough to justify a perturbation expansion. But I do not see how this concept of perturbative implies anything about how the coupling should change with scale, more specifically how a small coupling implies that the first derivative of the coupling with respect to the scale should be of only one sign. Thanks!

Answer 1

I'd agree that the sentence is not correct (in a sense that the author should't have written "must"). There are actually three possibilities:

First:

If the coupling increases with $\mu$, as in QED, it goes to zero at long distances.

Second:

If it decreases with $\mu$, as in QCD, it goes to zero at sort distances...

And the third possibility:

The third possibility in a perturbative theory is that $\beta(\alpha) = 0$ exactly, in which case the theory is scale-invariant.

Also I'd agree that it has nothing to do with perturbativity at all. Since even if the theory is non-perturbative...

Even if the theory is non-perturbative, one can still define a coupling through the value of the Green's function.

BTW -- all the quotes above are from the same paragraph where your sentence is coming from. So I'd say that that sentence is quite unfortunate.

Edit: As for the questions in comments -- I'll add just another quote:

With multiple couplings there are other possibilities for solutions to the RGEs. For example, one can imagine a situation in which couplings circle around each other. It is certainly easy to write down coupled differential equations with bizarre solutions...

So, yes, there are possibilities for more complex RG behaviors. I guess what author wanted to do -- is to give an overview that goes from simple examples to more complex ones. Starting with perturbative single-coupling QFT with $\beta \ne 0$ and then adding more complex possibilities. And I'd agree that the whole paragraph is not very good at conveying this idea.

Answer 2

Perturbation theory and the fact that couplings run in most quantum field theories are not actually directly related to each other. As long as the coupling is small enough, one can use perturbation theory. The smaller the better of course.

The coupling runs, because the strength of the interaction differs at different energy scale. This shows up when one computes higher orders in the perturbative expansion. From the results of the higher order calculations one can compute something called the beta-function and from this one can compute how the coupling runs. For more detail on how this works, one can read up on renormalization group analysis.

It may happen (as it does in quantum chromodynamics or QCD) that the coupling becomes so strong at a particular energy scale that perturbation theory breaks down. The perturbative expansion does not work in this situation, because higher orders become as significant as, if not more significant than, the lower orders. In such situations, one needs to resort to so-called non-pertubative methods (such as Schwinger-Dyson equations).

Answer 3

Your understanding of perturbation theory is fine. Matt's statement is a little bit sloppy.

Much of his book is on how one calculates the renormalization group trajectories, that is the variation of the coupling with energy/scale, in perturbation theory, the easiest method to achieve that in QFT. (But you see in that article this is not the only method. Notably, block-spinning techniques in lattice models achieve this nonperturbatively.)

Perturbation theory around a fixed point, a small g_c, allows determination of the β-function, i.e. the evolution of the coupling with scale, as a power series in g: so, away from g_c, the coupling may increase, decrease, or not budge! (This last contingency is actually exhibited by the celebrated N=4 supersymmetric Yang-Mills theory.)

Worse yet, in non-relativistic systems you might even find limit cycles, that is the coupling may increase, and then decrease with scale, etc. But these are rare recondite technical models, see e.g. Curtright, Jin & Zachos, 2012, rarely encountered in "real life".

In most QFT applications in HEP and condensed matter, by far!, evolution with scale/energy off a fixed point is monotonic, as he summarizes, and the coupling either decreases with energy, as in QCD, or else it increases with energy, as in QED, or stays put as in the N=4 super-YM mentioned.