= Scalar graph power-counting identity
{title2=$D=d-(d/2-1)E-\sum_n[n-d(n/2-1)]V_n$}
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 $L=I-\sum_nV_n+1$ and $2I+E=\sum_n nV_n$ 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.
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