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I have a few questions related to irrelevant couplings in the Wilsonian approach to the renormalization group (RG).

What is so great about RG theory is that one can trade the 'real physics', the one valid at any energies or scales, for an effective description at the scale of interests. In particular, it is sufficient to consider an action including

$$S= (\text{kinetic term}) \,+ \,(\text{relevant interactions})\,, $$

while all irrelevant interactions can be set to zero.

  • I have difficulties accepting that last point as the irrelevant quantities only vanish at the fixed point. Why can you simply neglect these irrelevant couplings altogether?

  • Why is an RG-flow so 'boring', i.e. one never encounters limit points, bifurcations, etc....? Does there exist "physical", in contrast to toy examples, showing a limit cycle behavior? If so, what is the role of the irrelevant operators? Surely these cannot be taken to vanish?

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There's really two questions here.

1) Why can we ignore the irrelevant interactions?

In general, you can't. If you define your theory with a finite cutoff -- for instance, using a lattice -- you can have non-negligible interactions which are nonetheless classified as 'irrelevant'. These interactions only become negligible as the lattice shrinks to nothing, i.e., as you approach the IR fixed point.

2) Why is RG flow boring?

In 2d continuum theories, it's because of the c-theorem. In almost all 2d QFTs (IIRC, you need a stress-energy tensor), you can define a quantity, the coefficient $c$ of the conformal anomaly, which always decreases under renormalization flow. This makes limit cycles impossible; you can never get back to where you started.

In 4d, an analogous theorem (a "physics theorem", but the proof will carry over to any rigorous setting mathematicians cook up) was proven in 2011 by Komargodski & Schwimmer.

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