# Renormalization of phi cubed theory

I was recently learning about renormalization in quantum field theory (in particular I was looking at the renormalization of phi to the fourth theory). The superficial degrees of divergence of a Feynman diagram for a theory involving one scalar field with a self interaction term that goes like $$gϕ^n$$ is given by the following formula: $$D = 4 - [g]V - E$$ where $$[g]$$ is the mass dimension of the coupling constant $$g$$ (which in our case is equal to $$4-n$$, where $$n$$f is the exponent of the interaction term), $$V$$ is the number of vertices in the diagram, and $$E$$ is the number of external lines in the Feynman diagram. This works out to be $$D = 4 -V(4-n) - E$$.

Now, with $$ϕ^3$$ theory the superficial degree of divergence $$D = 4 - V - E$$. Any diagrams where $$D$$ is greater than or equal to $$0$$ have UV divergences that need to absorbed into some bare parameter in the Lagrangian. There are only a few diagrams that I can find in $$ϕ^3$$ theory with $$D$$ greater than or equal to zero (to one loop order). Naively I'd expect that $$ϕ^3$$ theory should be 'more' renormalizable than $$ϕ^4$$ theory, since for $$ϕ^4$$ theory $$D = 4 - E$$, whereas for $$ϕ^3$$ theory $$D = 4 - V - E$$ (it get's lower for each additional vertex). But this is probably not the case since I can't seem to find anyone anywhere talking about the renormalization of $$ϕ^3$$ theory in 3+1 dimensions (in 5+1 dimensions I believe I've seen someone tackle it). The only divergent diagrams that I can find to one loop order are some vacuum diagrams with no external legs and the one loop correction to the $$ϕ$$-propagator ($$D = 4 - 2 - 2 = 0$$, so a log divergence). So am I just missing something here? Can $$ϕ^3$$ theory not be re-normalized in 3+1 dimensions?

$\phi^3$ theory in 4 dimensions is what is called a super-renormalizable theory. This means that as yourself pointed out that the superficial degrees of divergence decrease at higher orders. The reason that people rarely speak of it is that $\phi^4$ theory is arguably more interesting.
A feature of super-renormalizable theories is that they are quite often very sensative to UV physics. A nice way to see this would be to add a heavy fermion with mass $M$ and coupling $g' \phi \bar{\Psi}\Psi$ to the spectrum. You would find a vertex correction to the $\phi^3$ coupling, g, of the form $$\delta g \sim M g'^3 \left( \mathcal{O}(1)+logs \right)$$ Which looks very strange if we try to take the $M\rightarrow \infty$ limit.