Why is vanishing beta function associated with scale-invariance? Coupling constants have change rate of zero at some scale, but how is that related to scale-invariance?

Association of vanishing beta function with scale-invariance also indicates that a theory that is not scale-invariant at some scale cannot have vanishing beta function. Would that be correct?

  • $\begingroup$ You have effective scale invariance at a zero of β, but not far from it. The invariance is w.r.t. small/infinitesimal rescaling, not large ones. A theory with vanishing β everywhere would have no need for it, and would be globally scale invariant. Are you concerned about N=4 super-YM? $\endgroup$ – Cosmas Zachos Oct 23 '18 at 21:45
  • $\begingroup$ Look at what happens for positive and negative beta. (keeping in mind what Cosmas said about being infinitesimal as well) $\endgroup$ – Diego Oct 23 '18 at 23:53

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