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The superficial degree of divergence is the power of a common large-momentum scale obtained by scaling all independent internal momenta together, before cancellations or subtraction of divergent subdiagrams. In a scalar field theory with a standard quadratic kinetic term and non-derivative interactions, each loop momentum integral supplies $d$ powers of momentum and each internal propagator supplies minus two. ThusFor a connected Feynman diagram, counting vertex half-edges and applying the stated identity for the loop order givesIt follows that $D=(d-2)L-2V+2$. Meanwhile, ${\sum }_n(n-4)V_n=2I+E-4V=2L-2V+E-2$. Substitution provesThis power counting in quantum field theory concerns the overall scaling: even a Feynman diagram with $D<0$ can contain divergent subdiagrams, and cancellations may improve convergence when $D\ge 0$.

The canonical mass dimensions are $[\varphi ]=(d-2)/2$ and $[g_n]=d-n(d-2)/2$. A finite polynomial scalar field theory has power-counting renormalizability when its interactions have nonnegative mass dimensions and the allowed local counterterms form a finite set closed under subtraction of divergent subdiagrams. For $d>2$, non-derivative interactions therefore satisfy $n\le 2d/(d-2)$. Equivalently,so adding vertices of such interactions cannot force an indefinitely increasing number of external legs or derivatives in divergent counterterms. In dimensions $d\le 2$ a fixed finite polynomial still has this favorable power counting in quantum field theory; the displayed upper bound on $n$ is intended only for $d>2$.

For six-dimensional cubic scalar field theory, $3V=2I+E$ and $L=I-V+1$ yieldThe coupling $g$ has zero mass dimension. The possible overall divergences have $E\le 3$: vacuum energy, a linear tadpole diagram term, a mass term, a kinetic term, and a cubic interaction. These are finitely many local counterterms. Divergent subdiagrams of higher-point Feynman diagrams are removed by the same counterterms, establishing perturbative renormalizability of cubic scalar theory in six dimensions. A real cubic potential is unbounded below, so this claim is one of perturbative quantum field theory, not a claim that the polynomial defines a stable nonperturbative vacuum.

After Wick rotation, use the Euclidean action with positive kinetic and mass terms and interaction $g{\varphi }^3/3!$. The Euclidean momentum-space Feynman rules are: an internal scalar propagator $1/(k^2+m^2)$, a vertex $-g$, momentum conservation at each vertex, integration $d^{6}k/(2\pi {)}^6$ for each independent loop, and division by the Feynman-diagram symmetry factor. Strip the overall momentum-conserving delta function and external propagators when computing an amputated Green's function.

The one-particle-irreducible two-point vertex at one loop is the bubble:

Interchanging its two internal lines is an automorphism, giving Feynman-diagram symmetry factor two. In the normalization of the printed Minkowski amputated Green's function, the two vertices contribute $(-ig{)}^2$, the two internal propagators contribute $(-i{)}^2$, and Wick rotation supplies the overall $i$ already factored out there. Thus the Euclidean bubble contribution isFor this self-energy calculation we impose a vanishing one-point function, so tadpole diagram insertions are canceled by the linear counterterm. Without that convention a connected but one-particle-reducible Feynman diagram with a tadpole insertion is also possible: both external legs attach to one vertex, a zero-momentum line joins it to a second vertex, and that second vertex has a self-loop. Its amputated contribution is $g^2/(2m^2)\int d^{6}k/[(2\pi {)}^6(k^2+m^2)]$, independent of $p$. It is not the bubble drawn above; keeping it without a tadpole subtraction would add this constant to the full amputated answer.

To find the bubble's local divergence, put $K=k^2+m^2$ and expand at large $|k|$:Odd terms vanish under the angular integral. Rotational symmetry gives $⟨(k\cdot p{)}^2⟩=k^2{p}^2/6$. The terms through order $p^2$ are thereforeand the $p^2$ coefficient approaches $-1/(3k^6)$. Terms of order $p^4$ and higher are integrable at infinity in six dimensions; subtracting the two displayed terms leaves an ultraviolet-convergent integral. Hence the divergent part is a local polynomial $A{p}^2+B$, with momentum-independent coefficients. For example, when $m^2>0$ its logarithmically divergent coefficient isChanging the reference scale in this logarithm only changes a finite local term.

At zero external momentum the spherical ultraviolet cutoff and $\mathrm{Vol}(S^5)={\pi }^3$ giveWith $u=k^2$, integrate $u^2/(u+m^2{)}^2=1-2m^2/(u+m^2)+m^4/(u+m^2{)}^2$. This gives the momentum-cutoff two-point function in six-dimensional cubic scalar theoryIn particular, the divergent part of this constant iswhile the last term in $B_{\Lambda }$ tends to a finite constant. For $m=0$ the zero-momentum integral gives $B_{\Lambda }=g^2{\Lambda }^2/(256{\pi }^3)$ directly. The finite parts of a local subtraction can depend on the regularization prescription; the conclusion about mass and kinetic counterterms does not.

At fixed $m>0$ and $\Lambda >0$, the radial integrands in the scalar shell integral in six dimensions behave as $q^{5-2r}$ in the ultraviolet. ThusConsequently the proper box quartic vertex is ultraviolet finite. The reducible triangle insertion contains the logarithmic cubic-vertex divergence. The internal bubble contains the mass divergence; its nonzero-momentum expansion additionally has a logarithmic kinetic-term divergence. An unsubtracted tadpole attachment has the quartic one-point divergence and is canceled by tadpole subtraction, or incorporated consistently in the chosen stationary background.

These divergent lower-point subgraphs require counterterms, not a new independent divergent ${\varphi }^4$ coupling. More generally, cubic graph identities give the superficial degree of divergenceVacuum, one-, two-, and three-point functions need vacuum-energy, linear, mass, wave-function renormalization and cubic-coupling counterterms. Proper higher-point functions have negative superficial degree and are finite after subtraction of divergent subgraphs. Hence the perturbative renormalizability of cubic scalar theory in six dimensions permits a continuum perturbative expansion with finitely many renormalized parameters; induced irrelevant higher-field and derivative interactions are finite predictions at finite scale, not an infinite list of ultraviolet parameters.

This perturbative statement does not establish a nonperturbative real Euclidean measure. For any nonzero real $g$, $m^2{\varphi }^2/2+g{\varphi }^3/6$ tends to $-\infty$ in one field direction, so even its constant-field integral is divergent. Thus $\boxed{\text{perturbatively renormalizable, with no convergent positive real Euclidean cubic measure}.}$ Stability or a specified analytic continuation would be additional input beyond these loop calculations.