 Perturbative renormalizability of cubic scalar theory in six dimensions

For a cubic graph with $E$ external legs, $3V=2I+E$ and $L=I-V+1$. Its superficial degree of divergence is therefore $6L-2I=6-2E$. Divergences in vacuum, one-, two-, and three-point functions can be absorbed into vacuum energy, a linear term, mass, wave-function renormalization, and cubic-coupling counterterms. Higher-point proper diagrams are superficially convergent after subtraction of divergent subgraphs. This establishes perturbative renormalizability, without asserting a nonperturbative positive measure for the unstable real cubic potential.