# Super-renormalizable theory and $\beta$-function

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There is the statement that $\beta$-function vanishes for super-renormalizable theories. In $D=2$, scalar field has mass dimension zero. So any polynomial interaction is super-renormalizable. Then shouldn't all of them have vanishing $\beta$-functions? But there are many theories (e.g, sine-Gordon) in $2D$ which have nontrivial $\beta$-function. I must be missing something very basic here.

This post imported from StackExchange Physics at 2016-09-10 11:18 (UTC), posted by SE-user Physics Moron
retagged Sep 10, 2016

There is the statement that β-function vanishes for super-renormalizable theories.

I'm rather skeptical about the statement. What's the context? Can you provide a source or an argument?

Sine-Gordon has a non-polynomial interaction, hence is not covered by your argument as it stands, independent of whether the ingredients of the argument are valid.

@JiaYiyang First line of Page 770 of this book by Zinn-Justin (4th edition) : http://www.amazon.in/Quantum-Critical-Phenomena-International-Monographs/dp/0198509235 ;
The statement reads : "The theory is super-renormalizable and thus the β-function vanishes."

That would make all super-renormalizable theories scale (and possibly conformal) invariant theories. Is this true?

Yes. That's my confusion. What would be the statement?

As I understand it, super-renormalizable interactions are those with positive mass dimension, that also behave as relevant operators that lead away from a fixed point when following the RG flow towards lower energy scales (?). So to me super-renormalizable theories seem to be rather not scale invariant and I therefore dont see why their $\beta-$ functions should vanish ...

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