Six-dimensional cubic scalar field theory ($\mathcal{L}=\frac{1}{2}(\partial \varphi {)}^2+\frac{1}{2}{m}^2{\varphi }^2+g{\varphi }^3/3!$)

A canonically normalized real scalar field in six dimensions has mass dimension two. The cubic coupling is classically a marginal coupling, whereas a local quartic coupling has mass dimension minus two. This theory is perturbatively renormalizable by power counting in quantum field theory, but a real cubic Euclidean potential is unbounded below. Its expansion about a massive Gaussian theory is therefore formal perturbation theory, not a convergent positive functional integral at nonzero real $g$.

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.

In a six-dimensional cubic scalar field theory, three labelled box Feynman diagrams give the one-loop local quartic vertex at zero external momentum. With an effective-action term $g_4{\varphi }^4/4!$, their contribution is $\delta g_4=-3g^4{I}_4$. To check its sign and multiplicity, expand the one-loop scalar effective action as $\frac{1}{2}\mathrm{Tr}\log (D+g\varphi )$, where $D=-{\partial }^2+m^2$. Its fourth-order term is $-g^4\mathrm{Tr}(D^{-1}\varphi {)}^4/8$, giving the stated coefficient. The associated amputated connected diagram insertion has the opposite sign.

For $m>0$ and $0<\Lambda <{\Lambda }_0$, defineThe radial formula uses the area ${\pi }^3$ of the unit five-sphere. Its mass dimension is $6-2r$. As ${\Lambda }_0\to \infty$, $I_1$ diverges quartically, $I_2$ quadratically, $I_3$ logarithmically, and $I_r$ is ultraviolet finite for $r\ge 4$ at fixed $\Lambda ,m$. This follows directly by comparison with $q^{5-2r}$.