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Won't it be correct to define a CFT as a QFT such that the beta-function of all the couplings vanish?

But couldn't it be possible that the beta-function of a dimensionful coupling vanishes but it does so at a non-zero value of it - then the scale invariance is not generated though the renormalization flow is stopped? Is this possible?

(..it is obviously true that a theory with no intrinsic scale or dimensionfull parameter can still not be a CFT - like a marginal deformation of a CFT may not keep it a CFT and then this deformation parameter has to flow to a fixed point for a new CFT to be produced at that fixed point value of the marginal coupling..)

user6818
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1 Answers1

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Your definition is quite good and works almost always. I'm quite sure it is rigorously true in 2D. You'll actually find it in some lecture notes. Remember that a theory is conformal if the trace of the stress tensor vanishes: $T \equiv T_\mu^{\mu} = 0.$ Indeed there is a folk theorem that states that

$T = \sum \beta_I \mathcal{O}^I$

where the sum runs over those operators $O^I$ in the theory with their beta functions $\beta_I$ (up to terms generating the conformal anomaly in curved space).

However, this is not completely true, and there are important classes of counterexamples where additional terms appear. Recently, these examples have led to some confusion in the literature (in the search for scale but not conformally invariant theories). All of this is well understood now and a good starting point for your studies would be 1204.5221 [hep-th].

Edit: don't forget that operator dimensions aren't protected and change under the RG flow.

Vibert
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