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For a connected scalar field theory graph without derivative vertices, each independent loop momentum supplies $d$ powers and each internal scalar propagator removes two. Thus $D=dL-2I$. The graph identities $L=I-{\sum }_n{V}_n+1$ and $2I+E={\sum }_{n}n{V}_n$ give the displayed formula. The mass dimension of the field is $(d-2)/2$ and that of $g_n$ is $n-d(n/2-1)$. Negative-dimensional couplings allow the superficial degree to grow with vertex count. This is overall power counting in quantum field theory; subgraphs can diverge even when the whole graph has negative degree.

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 $q^{5-2r}$.

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 $g$.

A translation-invariant connected generating functional with a constant external source is extensive in spacetime volume ${\mathcal{V}}_d$. Its total logarithm is dimensionless, while its density has mass dimension $d$. Consequently its dimensionless representation at fixed physical volume must retain ${\mu }^d{\mathcal{V}}_d$. For a free massive scalar with source coupling ${\mu }^{-ϵ/2}h\varphi$, completing the square gives $F={\mathcal{V}}_d{\mu }^{-ϵ}{h}^2/(2m^2)$. This already disproves a literal rewriting of the full uniform-source logarithm using only $\lambda ,m^2/{\mu }^2,h/{\mu }^3$ at fixed volume. In infinite volume the total functional is generally infinite; use the density, a source of compact support, or an explicit volume regulator. After writing the density as ${\mu }^{d}f$, its renormalization group equation has the engineering term $+df$. Ultraviolet subtraction alone does not establish infrared finiteness for a massless constant source.

A Wilson coefficient weights a local operator in an effective field theory. For an operator of mass dimension $d$ in four spacetime dimensions, a dimensionless coefficient is conventionally accompanied by ${\Lambda }^{4-d}$. The coefficient encodes short-distance dynamics and can be a matrix in flavour space.