I was recently learning about re-normalization in quantum field theory (in particular I was looking at the re-normalization 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 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' re-normalizable 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 re-normalization 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?