A conformal fixed point is defined by

$$\beta(g)=0$$

We hence know that couplings, masses and dimensions of operators do not flow in the effective Lagrangian when we change the renormalization scale $\mu$.

My question then is, can we take $\mu$ to infinity and retain finite results for all separated correlation functions at this point? Intuitively that seems sensible to me, but perhaps there is a technical obstruction.

If no, could you provide me with an example where separated correlation functions diverge in a CFT?

This question is related, but doesn't seem to answer my specific question.

**First Edit**

The answer is definitely no. Here is an example of a UV divergent correlation function in planar $\mathcal{N}=4$ SYM, a theory which is conformal at all values of the coupling!

But I still don't understand where this divergence comes from! Surely if $\beta=0$ then nothing flows in the effective Lagrangian, so there can't be a divergence? Why is this simplistic viewpoint erroneous?

**Second Edit**

Actually I've changed my mind. My example above had a divergence before renormalization. But after renormalization it's fine! How come you need to renormalize, even though the coupling constants and anomalous dimensions are fixed? Well, you still have the wavefunction renormalization $Z$ which is unaffected by $\beta(g)=0$. So a CFT can have UV divergences which must be cancelled in $Z$ so that the correlation functions are finite.

To quote from that paper

The UV divergences, on the other hand, require renormalisation. In $\mathcal{N} = 4$ SYM theory, the appropriate combinations of the self-energies of the elementary fields and the one-particle-irreducible (1PI) corrections to the elementary vertices are UV finite, ensuring the vanishing of the $\beta$-function. The only sources for UV divergences are the insertions of composite operators as external states, which hence need to be renormalised.

If anyone disagrees with my reasoning let me know, otherwise I'll write this up into a Q and A!

This post imported from StackExchange Physics at 2015-05-24 09:00 (UTC), posted by SE-user Edward Hughes