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Beta functions and relevant/irrelevant operators

  1. May 12, 2012 #1
    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

    [itex]\frac{dg^2_s}{d\ln M} = -\frac{14}{16\pi^2}g^4_s[/itex]

    The solution is [itex]\frac{1}{g^2_s}=\frac{14}{16\pi^2} \ln(M/M')[/itex]
    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?
  2. jcsd
  3. May 12, 2012 #2
    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?
  4. May 12, 2012 #3
    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?

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