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QCD is the best-known example of theories with negtive beta function, i.e., coupling constant decreases when increasing energy scale. I have two questions about it:

(1) Are there other theories with this property? (non-Abelian gauge theory, principal chiral field, non-linear sigma model, Kondo effect, and ???)

(2) Are there any simple (maybe deep) reason why these theories are different from others? It seems that the non-linear constraint of the non-linear sigma model (and principal chiral model) is important, but I have no idea how to generalize this argument to other theories.

Tengen
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2 Answers2

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As gih correctly pointed out, there are a lot of QFTs in low dimensions which are asymptotically free.

In 3d, for example:

  1. Scalar $\phi^4$ field theory in 3d is asymptotically free. Renormalization flow increases the quartic coupling, taking you away from the free theory.
  2. Likewise, the 3d Gross-Neveu model (Dirac fermions with a 4-fermion interaction) is also asymptotically free. The 4-fermion interaction is relevant in 3d (rather than non-renormalizable, as it is in 4d).
  3. Plain old 3d Yang-Mills gauge theory in 3d is asymptotically free. (Even better, you can set things up so that the YM kinetic term itself is irrelevant, giving you a topological theory at low energies.)

It's not hard to combine these basic models to get more complicated but still asymptotically free models.

To address the 2nd part of your question: There's probably not anything model-specific going on here. It's just that the basic ingredients in these lower dimensional theories are better behaved.

user1504
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One example would be $\varphi^3$ theory, which is treated extensively in Srednicki.

Frederic Brünner
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