Super-renormalizable theory and $\beta$-function 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. 
 A: In a qft, it may be possible to redefine other parameters than coupling to absorb the infinities coming from higher order corrections. In this way, coupling constant does not get renormalized and hence the beta function vanishes. It is a possibility in super-renormalizable theory as fewer diagrams are divergent and the condition may be satisfied.
As an example for the Sine- Gordon model, the action is
$$\mathcal{S}(\theta)=\int d^2x [\frac{1}{2}(\partial_\mu\theta(x))^2-\frac{m^2}{k^2}cosk\theta(x)]$$
Redefining, $\theta=k\theta$ gives
$$\mathcal{S}(\theta)=\frac{1}{t}\int d^2x [\frac{1}{2}(\partial_\mu\theta(x))^2-m^2cos\theta(x)]$$ with $t=k^2$.
Perturbative expansion in the power of k only modifies the $cos\theta$ term as a self interaction and the divergences arising can be absorbed by a redefinition of m. In this way, coupling constant does not get renormalized and hence beta function vanishes.
This property is not true in general as the vanishing of beta function to all orders implies a finite theory ($\mathcal{N}=4$ SYM) which is a result need to be obtained from a non-perturbative analysis unless it is trivially true as in the former case. Most qft exist perturbatively and the existence of fixed points is not known non-perturbatively. A super-renormalizable theory does not have a vanishing beta function generally as can be seen from $\phi^3$ theory beta function which in d dimension reads, 
$\beta(g)=(d/2-3)g-\frac{3g^3}{256\pi^3}+O(g^5)$ ( Collins "Renormalization," eqn. 7.3.7)
$\phi^3$ theory is super-renormalizable for $d<6$ but $\beta$ function is not zero. It however shows asymptotic freedom which is a property of super-renormalizable theories ( I am not aware of the proof though).
