1. What is the definition of dangerously irrelevant renormalization-group (RG) fixed point?

  2. What are some examples of dangerously irrelevant RG fixed points?

  3. Do we also have the use of dangerously relevant RG fixed points? Do we also have the use of dangerously marginal RG fixed points? What are the examples?

  4. The term "irrelevant, marginal, relevant" are for the IR (infra-red) low energy long distant fixed-point perspective, commonly used in the low energy physics and in condensed matter physics. In high energy and particle physics, people also consider from the UV perspective, thus "irrelevant, marginal, relevant" become "non-renormalizable, renormalizable, super-renormalizable" at the UV (ultra-violet) high energy short distant perspective. So do we also have concepts of: $$ \text{dangerously irrelevant} \leftrightarrow \text{dangerously non-renormalizable}, $$ $$ \text{dangerously marginal} \leftrightarrow \text{dangerously renormalizable}, $$ $$ \text{dangerously relevant} \leftrightarrow \text{dangerously super-renormalizable}, $$

See also a possible example here: Are irrelevant terms in the Kahler potential always irrelevant, even at strong coupling?. Partial answers and partial comments are all welcome. You need not to provide the full answer to write an answer.

  • 3
    $\begingroup$ For first subquestion (v1), see Wikipedia. $\endgroup$ – Qmechanic Nov 14 '16 at 19:23
  • $\begingroup$ For 2. : an example is the quartic interaction in the ϕ4-theory for dimensions larger than four dimensions. The reason is that the scaling functions carry a singular dependence on the irrelevant coupling. One cannot simply take the irrelevant coupling to vanish in the effective treatment. Consequently, I don't think there is something is dangerously relevant couplings as relevant couplings are always taken into account. $\endgroup$ – Anne O'Nyme Jun 17 '17 at 10:55
  • $\begingroup$ To add, I think, the terminology refers to an operator, not a fixed point. The idea being an operator that is irrelevant by power counting, but is important for stabilizing the correct ground state. $\endgroup$ – vik May 19 at 19:55

I just saw this interesting discussion. Agree that adjective "dangerously" only goes with "irrelevant operators" either marginal or non-marginal ones. They refer to operators which are irrelevant from a RG point of view but relevant from the point of view of symmetry breaking etc. They can play a role in tri-critical phenomena etc.


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