Anomalous dimensions and RG flow - Polchinski 1984

Question

While defining a renormalizable field theory in the UV, we can assume it to have a simple form, i.e only containing a few operators (the relevant and marginal ones).

Now, as we start flowing along the RG flow with change in energy scale, new operators will be introduced but in the IR the irrelevant operators do not go to zero but become functions of the relevant and marginal operators

Naively, I used to think that the irrelevant operators go to zero as we flow to the IR, but it doesn't. They just become functions of the relevant and marginal operators.

Can someone please explain the above point more clearly and intuitively.

Also, we can compute anomalous dimensions of various operators in perturbation theory, and is there some example which we can construct where the object is classically irrelevant, but the addition of due to anomalous dimensions make it relevant for the theory. Can somebody give an example of the above behaviour?

Answer 1

I don't know off the bat an example for your second question but your first one requires an extended discussion of the Wilsonian RG and what traditional renormalization or removal of UV cutoffs means in that context which I gave here:

What is the Wilsonian definition of renormalizability?

The quick answer is that relevant and irrelevant operators is a local notion in theory space which is relative to the fixed point you are close to. Usually the fixed point which serves as a reference is the trivial Gaussian one (say $x^{\ast}$ in the picture below). In 3d or in dimension $4-\epsilon$ the $\phi^4$ direction is relevant with respect to the Gaussian fixed point but becomes irrelevant with respect to the infrared WF fixed point. What I explained in my answer linked to above is that UV renormalization essentially is an explicit form of the (un)stable manifold theorem (or center-unstable if you include marginal and not just relevant directions): you are parametrizing the unstable manifold ($W_{\rm u}^{\rm loc}$ in the picture below) of the Gaussian fixed point (the set of QFTs which flow from that fixed point in the UV) by the linear space of unstable directions of the differential/linearization of the RG at the Gaussian fixed point. This linear space ($E^{+}$ in the picture below) is also the tangent space of the unstable manifold. Theory space is the cartesian product of this tangent space and the tangent space ($E^{-}$ below) of the stable manifold (if the fixed point is hyperbolic, again I am being cavalier with marginal directions). So the unstable manifold is the graph of a function which tells you the irrelevant coordinates in terms of the relevant (plus marginal) ones.