Beta functions and relevant/irrelevant operators

[eherrtelle59]

Ok, I'm having some conceptual difficulty here. When discussing beta functions and the relation how these differential equations flow, I still don't quite get the difference between relevant vs. marginally relevant and irrelevant vs. marginally irrelevant.

For instance, take the β function with coupling g_s

$\frac{dg^2_s}{d\ln M} = -\frac{14}{16\pi^2}g^4_s$

The solution is $\frac{1}{g^2_s}=\frac{14}{16\pi^2} \ln(M/M')$
such that the theory diverges at M'. The theory's obviously asymptotically free, as when the scale M grows, the coupling g_s decreases.

So, since the beta function is negative, I know this is either irrelevant or marginally irrelevant. What's the difference?

 

[eherrtelle59]

Actually, I'm wrong above.

At lower and lower energy scales M, g becomes larger and larger and therefore relevant. Why is it marginally relevant instead of relevant?

 

[eherrtelle59]

In case I'm being to obscure above, let's just work with QED vs. QCD.

How do you know these theories are marginally (ir)relevant as opposed to (ir)relevant?

Thanks