 Scalar graph power-counting identity

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.