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I am wondering if there are any (insightful) examples for models which exhibit asymptotic freedom both in the UV and the IR. I know it sounds odd, but if anyone has come across something like that, please share your understanding :)

Thanks a lot in advance.

Hamurabi
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    Presumably short of a free theory :) – Qmechanic Jan 11 '20 at 11:18
  • That is of course one example, haha! But I was rather looking for examples where in the classical action one does have an interaction term. – Hamurabi Jan 11 '20 at 12:17
  • Can you be more precise? You want decreasing coupling with energy at both ends of the energy range, but not in-between, and no limit cycles? – Cosmas Zachos Jan 11 '20 at 15:11
  • The theory I am considering is interacting and lives in a compact domain. Studying the beta functions in both regimes shows that the theory exhibits both in IR and UV the Gaussian fixed point. I have not yet studied the intermediate regime, but was simply wondering if people here have come across a similar situation, perhaps even for a theory on a non-compact domain. – Hamurabi Jan 11 '20 at 18:30
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    Well, if you could only somehow reverse the flows of Russian doll models ... – Cosmas Zachos Jan 11 '20 at 20:54
  • Could you elaborate on that point just a little bit please? Thanks! – Hamurabi Jan 11 '20 at 23:48
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    Another example is where different degrees of freedom are "free" at the UV and IR fixed points respectively. For example, you can flow from the free boson CFT to the free fermion (Ising) CFT in $(1+1)$d (this is the RG flow of $\phi^4$ theory), but the bosons and fermions are mutually nonlocal, so this probably isn't what you have in mind. – Seth Whitsitt Jan 12 '20 at 00:33
  • Well, the model I look at is based on a scalar field theory and the interaction is in fact non-local. If you could provide a reference to your comment, this would be great! – Hamurabi Jan 14 '20 at 08:26

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Here is an example of a theory which is free in both the IR and the UV, but importantly the particles which are free in the UV are not the same particles which are free in the IR. It is scalar $\phi^4$ theory in two dimensions: $$ \mathcal{S} = \int d^2 x \left[ \left( \partial_{\mu} \phi \right)^2 + m^2 \phi^2 + \lambda \phi^4 \right]. $$ Here, I'm writing this as a 2D Euclidean field theory, but you are free to Wick rotate a dimension and interpret this as a (1+1)-dimensional relativistic quantum field theory (it has no effect on the RG flow).

The precise definition of the above theory depends on how we choose to regulate it, but no matter what choice we make, the RG flow qualitatively looks like this (picture credit: https://arxiv.org/abs/1811.03182):

Wilson-Fisher RG flow

The arrows point in the direction of RG flow to the infrared. The precise shape and details of this RG flow will depend on your regularization scheme, but you will always have some line in $m^2$-$\lambda$ space which runs from the point $G$, representing the free massless boson CFT, and $WF$, which is known as the Wilson-Fisher CFT. It is along this line that I claim we have free theories in the UV and the IR.

In the UV this is obvious - we just flow to the massless free boson. But in the IR this is very nontrivial - we actually flow to a theory of a massless fermion. The fermionic degrees of freedom are extremely nonlocal with respect to the field $\phi$, but nonetheless there is ample evidence for this due to the special properties of 2D conformal field theory (CFT). In particular, the space of unitary 2D CFTs is highly constrained due to the 2D conformal group being infinite-dimensional (and all scale-invariant 2D QFTs are automatically CFTs). Therefore, given that a theory has conformal invariance, we can very often identify its properties in detail based on what other symmetries or properties the theory has.

In the present case, a rather direct line of reasoning (provided you know the "basics" of 2D CFTs) comes from Zamolodchikov's paper Conformal symmetry and multicritical points in two-dimensional quantum field theory. In this paper, he argues that scalar theories with interaction terms of the form $\phi^{2(p-1)}$ correspond to the so-called Virasoro minimal models with central charge $c = 1 - 6/p(p+1)$. For our case ($p = 3$), this corresponds to the $c = 1/2$ minimal model. But one can show that the theory of a free massless fermion also has $c = 1/2$, and furthermore there are arguments (from modular invariance) that there is only one consistent manifestation of the $c=1/2$ CFT, so these two theories are really identical. So the fixed point $WF$ is the theory of a free massless fermion.

I should mention that there are many other arguments that the IR fixed point is described by free fermions. The usual argument is that this theory should describe the phase transition of the 2D Ising model, and there are many derivations which show that this transition is described by free fermions in the IR. But since you are interested in QFTs which are defined in the UV and the IR I thought taking the CFT approach would be more cautious.

Seth Whitsitt
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