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Textbooks often mention that if we perturb a UV theory in the "direction" of a relevant operator, the fixed point theory one eventually ends up with, under RG flow, is/can be a CFT. Now, this answer gives an example of a CFT for which a Lagrangian description is not possible. On the other hand, when we do RG transformations, we typically calculate the flow of various coupling parameters as a function of some UV cutoff (in the Wilsonian picture) or some typical energy scale (in the continuum RG picture). But if we cannot write the fixed point of this flow (the above counter-example CFT, for example) within a Lagrangian description, in what sense do we call this theory a legitimate fixed point of the map that sends one set of couplings to another, i.e. the RG map?

I seem to be missing the connection between how we understand typical quantum field theories within the RG philosophy and the way we describe a CFT in terms of the so-called "CFT data". So, how should I reconcile these points of view?

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The answer to this question is broadly presented in Cardy's book, titled "Scaling and Renormalization in Statistical Physics". The relevant sections are 3.7, 3.8 and 5.2.

The basic idea is that we start with the partition function of a fixed point theory $Z^*$ and perturb it by a linear combination of scaling fields, $$Z = Z^* \exp{\left(-\sum_i g_i \sum_r a^{x_i} \phi_i(r)\right)},$$ where $a$ is a microscopic length scale that serves as a UV cutoff, and $\{\phi_i\}$ are the scaling fields. Now we can expand this exponential in powers of $\{g_i\}$; note that we are only interested in perturbative stability to deformations of the fixed point theory, so this makes sense. Upon such an expansion we find, $$Z=Z^*\left [1- \sum_i g_i \int \frac{d^d r}{a^{d-x}}\langle \phi(r) \rangle + \sum_{i,j}\frac{g_i g_j}{2} \int \frac{d^d r_1 d^d r_2}{a^{2d-x_i-x_j}} \langle \phi_i(r_1) \phi_j(r_2) \rangle +\cdots \right],$$ where the expectation values are to be computed in the fixed point theory.

Now the RG scaling can be implemented on these correlation functions. The behavior of the expansion under a scaling transformation leads to RG equations for the coupling constants $\{g_i\}$, which in turn tells us about the RG relevance/irrelevance of the corresponding scaling fields. An important piece of information in taking this through is the operator product expansion (OPE) of the scaling fields, which is part of the data of the fixed point theory we started with.

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