# Quantum field theories with asymptotic freedom

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.

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## 2 Answers

Yes, such examples are generic in 2+1 and 1+1D.

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2
Could you elaborate on this? You only answer the first part of the question. – Manishearth♦ Dec 17 '12 at 8:50

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

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Is it artificial? Since there is no true ground state for $\phi^3$ thoery. The strong coupling property at low energy may indicates the collapse of perturbation theory. – Tengen Jan 24 at 8:23
You are right, the theory is unstable. – Frederic Brünner Jan 24 at 10:21
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