# Is there a "direct" way to see why the irreversibility of the RG flow is needed for the the enhancement of scale invariance to conformal invariance?

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In this paper, the author investigates under what conditions scale invariance of a quantum field theory is enhanced to the full conformal symmetry by making use of a holographic argument.

The argument outlined in the introduction contains that the scale invariance of a quantum field theory can only be enhanced to conformal symmetry, if the RG flow is irreversible, which is also in agreement with the c-theorem that roughly speaking says that the number of relevant degrees of freedom can not increase in the course of the RG flow (some kind of a second law of thermodynamics ...?). This means there can for example be no cyclic (limit point) behavior of the RG flow. By holographic considerations the author relates a cyclic behavior of the RG flow on the field theory side to inconistencies in the quantum gravity on the gravity side, which means that for certain systems scale invariance alone can not exist.

I do not well understand, why the RG flow has to be irreversible for the enhancement of scale invariance to conformal invariance in the first place. Is there a "direct" why how one could (and should ...) see this by only looking at the quantum field theory without making use of holographic arguments?

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