Super-renormalizable theory and $\beta$-function

Question

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

Comments

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? 

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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.

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@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."

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

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

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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 ...

Answer 1

I still haven't got hold of Zinn-Justin's book mentioned in the comment, but here is a plausible/consistent interpretation of what that statement might mean. Generally one has to distinguish two kinds of RG flows and thus two kinds of beta functions, one is about pushing the UV cutoff to infinity and defining a QFT (i.e. UV completion), the other is about coarse graining the theory to gain IR information (Wilson-Kandanoff-Fisher).  The former slides a unphysical cutoff scale $\Lambda$ and the latter slides a physical probing/experimental scale $\mu$, and the beta functions are respectively denoted by $\beta_\text{UV}(g_\text{bare}(\Lambda))$ and $\beta_\text{Wilson}(g_\text{ren}(\mu))$.

The two are perturbatively very similar when one has dimensionless couplings ( see the "Update" section of this answer. In fact Wilson and Kogut only called the latter kind "RG trajectories", and the former "canonical curves"), but no a priori relation exists for general couplings.

For super-renomalizable theories, UV-finiteness require $g_\text{bare}(\Lambda)=g_0+o(1)$ (small o notation, with respect to $\Lambda$, and $g_0$ is independent of $\Lambda$), so the beta function in this sense must be 0 near $\Lambda=\infty$, and it might be this beta function that Zinn-Justin was talking about. On the other hand, since the coupling is dimensionful, we can almost be sure that the theory is not scale invariant hence the beta function in the sense of Wilson is not 0.