= 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.
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