Questions on RG flows, CFTs, and UV and IR theories

Question

In the space of field theories, Conformal field theories are fixed points in the RG flow. However, a lot of literature on CFT usually talks about a QFT being the RG flow between two CFTs: one UV and the other IR. However, I do not understand what UV and IR CFTs are. Suppose there are only 2 fixed points, with characteristic lengths $R_c$ of $0$ and $\infty$, the trajectory from $R_c = \infty$ to $R_c = 0$ is a trajectory through QFTs. We have the following relationship on the trajectory,

$$ Z^{-1/2}(L)\langle \phi(x_1/L)...\phi(x_n/L) \rangle_{A'} = \langle\phi(x_1)...\phi(x_n) \rangle_A$$

where, $RG_L[A] = A'$. But I don't understand if the fixed point at $L = \infty$ would be considered UV or the other way around and why.

Clearly, there are QFTs that do not lie on this trajectory or any trajectory between 2 fixed points. In this case why do people associate CFTs with the UV and IR limits of QFTs?

Answer 1

• The theory with characteristic length $R_\text{c}=0$ is the UV CFT, since it is probing things at the smallest possible length scale, i.e. at the largest possible energy. In other words, its characteristic energy scale is $\Lambda_{\text{UV}}=\infty$. Similarly the $R_\text{c}=\infty$ is the IR CFT, with $\Lambda_{\text{IR}}=0$.

• The non-trivial statement here is the existence of a UV CFT. If there exists such a theory, there exists an IR CFT too. It might be trivial, but it's there. Namely, just turn on a relevant deformation in your UV CFT and start flowing down. In the deep IR there are the following possible scenarios:

  1. The theory is gapless. There are excitations which you can reach without spending energy. It is described either by a free or by an interacting CFT, depending on the details.
  2. The theory is trivially gapped. There is a unique ground state, and then an energy gap. If you go below that energy gap, the theory is empty. It is the trivial theory containing only the operator $\mathbf{1}$ corresponding to the unique vacuum state.
  3. The theory is topological. It is a theory of multiple degenerate vacua. Its spectrum is a set of topological operators corresponding to the various vacua.

  Hence, starting from a UV CFT, you will eventually reach an IR CFT, passing through QFTs in the middle. This is the way to understand QFTs as intermediate points between CFTs. However, if you just start with a QFT, at some energy scale $0<\Lambda<\infty$, you can only flow down, to its IR CFT. There is no unambiguous way to run up the RG flow, making sure that you remain with a local theory (and hence a QFT). Turning on an irrelevant deformation, usually results in non-local theories (cf. $\text{T}\overline{\text{T}}$-deformed QFTs, and related stories).