Finite quartic graph and superficial scaling conventions

Let $D\in {\mathbb{Z}}_{\ge 1}$ be the number of Euclidean momentum coordinates. Use the finite undirected multigraph convention of Degree in a multigraph, indegree and outdegree in a digraph, and their underlying connectivity: loops are allowed, distinct edges with the same endpoints are parallel, and connectedness here means that the underlying simple graph obtained by deleting loops and identifying each parallel class is connected.

A finite quartic graph is data $\mathrm{\Gamma }=(\mathcal{V},{\mathcal{E}}_{\mathrm{i}\mathrm{n}\mathrm{t}},\eta ,{\mathcal{H}}_{\mathrm{e}\mathrm{x}\mathrm{t}},(S_v{)}_{v\in \mathcal{V}},a)$ with the following properties. The vertex set $\mathcal{V}$ is finite and nonempty, the internal-edge set ${\mathcal{E}}_{\mathrm{i}\mathrm{n}\mathrm{t}}$ and external-half-edge set ${\mathcal{H}}_{\mathrm{e}\mathrm{x}\mathrm{t}}$ are finite, and $\eta :{\mathcal{E}}_{\mathrm{i}\mathrm{n}\mathrm{t}}\times \{0,1\}\to \mathcal{V}$ assigns the two (unoriented, but separately labelled) ends of each internal edge to vertices. Its endpoint set is $\{\eta (e,0),\eta (e,1)\}$, a singleton for a loop; distinct elements of ${\mathcal{E}}_{\mathrm{i}\mathrm{n}\mathrm{t}}$ with the same unordered endpoints remain distinct parallel edges. The internal multigraph $(\mathcal{V},{\mathcal{E}}_{\mathrm{i}\mathrm{n}\mathrm{t}},\eta )$ is connected in the sense just specified.

For every $v\in \mathcal{V}$, the incidence-slot set $S_v$ has exactly four elements. Set $\mathcal{S}:={⨆}_{v\in \mathcal{V}}(\{v\}\times S_v)$. The attachment map

$a:({\mathcal{E}}_{\mathrm{i}\mathrm{n}\mathrm{t}}\times \{0,1\})\bigsqcup {\mathcal{H}}_{\mathrm{e}\mathrm{x}\mathrm{t}}⟶\mathcal{S}$

is a bijection, and for each internal end $(e,j)$ its slot has vertex coordinate $\eta (e,j)$. Thus the two ends of a loop occupy distinct slots. Each external half-edge $h\in {\mathcal{H}}_{\mathrm{e}\mathrm{x}\mathrm{t}}$ is attached by $a(h)$ to exactly one vertex slot and has no second endpoint. In particular, each vertex has exactly four slots and every slot is used exactly once by an internal-edge end or an external half-edge. Write $V:=\mathrm{\mid }\mathcal{V}\mathrm{\mid }$, $I:=\mathrm{\mid }{\mathcal{E}}_{\mathrm{i}\mathrm{n}\mathrm{t}}\mathrm{\mid }$, and $E:=\mathrm{\mid }{\mathcal{H}}_{\mathrm{e}\mathrm{x}\mathrm{t}}\mathrm{\mid }$.

Define the cycle rank and loop count by $L:=I-V+1$. For this mathematical power-counting convention, a single formal loop-momentum integration has assigned degree $+D$: for $\mathrm{\ell }\in {\mathbb{R}}^D$, Lebesgue measure $d^D\mathrm{\ell }$ acquires the factor ${\lambda }^D$ under $\mathrm{\ell }\mapsto \lambda \mathrm{\ell }$ for $\lambda >0$. The total loop contribution is defined to be $DL$, the cycle rank multiplied by this one-loop degree; no momentum routing or actual graph integral is part of this definition. Assign degree $-2$ to each internal scalar-propagator factor and degree $0$ to each quartic-vertex factor. For $q=(q_1,\dots ,q_D)\in {\mathbb{R}}^D\setminus \{0\}$ use the Euclidean norm $\mathrm{\mid }q{\mathrm{\mid }}^2:={\sum }_{j=1}^D{q}_j^2$. For example, if $m\ge 0$, then $(\mathrm{\mid }q{\mathrm{\mid }}^2+m^2{)}^{-1}$ has assigned ultraviolet degree $-2$, since

${\lim }_{\lambda \to \mathrm{\infty }}{\lambda }^2(\mathrm{\mid }\lambda q{\mathrm{\mid }}^2+m^2{)}^{-1}=\mathrm{\mid }q{\mathrm{\mid }}^{-2}\mathrm{.}$

The superficial degree of $\mathrm{\Gamma }$ under this assignment is

${\omega }_D(\mathrm{\Gamma }):=DL-2I\mathrm{.}$

By convention, ${\omega }_D(\mathrm{\Gamma })=0$ is called a logarithmic superficial degree. This is finite graph bookkeeping under the stated assignment. It does not by itself establish convergence or divergence of an actual graph integral; proper subgraphs, numerator cancellations, exceptional infrared data, and distributional singularities require separate hypotheses and analysis.