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.

Let $D\in {\mathbb{Z}}_{\ge 1}$ and let $\mathrm{\Gamma }$ be a finite quartic graph in the sense of Finite quartic graph and superficial scaling conventions. With $V$ vertices, $I$ internal edges, $E$ external half-edges, and loop count $L$, the incidence and loop-count identities are $4V=2I+E$ and $L=I-V+1$. Its superficial degree under the stated Euclidean scalar power-counting convention is ${\omega }_D(\mathrm{\Gamma })=DL-2I=D+(D-4)V-\frac{D-2}{2}E$. In $D=4$ this reduces to ${\omega }_4(\mathrm{\Gamma })=4-E$. A zero value is a logarithmic superficial degree by the convention in the definition; these identities do not assert convergence or divergence of an actual graph integral.

Let $D\in {\mathbb{Z}}_{\ge 1}$, let $\varphi \in C_c^{\mathrm{\infty }}({\mathbb{R}}^D;\mathbb{R})$ (the smooth real-valued functions with compact support), let $r\ge 3$ be an integer, and let $g_r\in \mathbb{R}\setminus \{0\}$. The Euclidean gradient is ${\mathrm{\nabla }}_E\varphi :=({\mathrm{\partial }}_1\varphi ,\dots ,{\mathrm{\partial }}_D\varphi )$ and its squared norm is $\mathrm{\mid }{\mathrm{\nabla }}_E\varphi {\mathrm{\mid }}^2:={\sum }_{j=1}^D({\mathrm{\partial }}_j\varphi {)}^2$. With Lebesgue measure $d^{D}x$ on ${\mathbb{R}}^D$, define the real-valued functionals

$S_{\mathrm{k}\mathrm{i}\mathrm{n}}(\varphi ):=\frac{1}{2}{\int }_{{\mathbb{R}}^D}\mathrm{\mid }{\mathrm{\nabla }}_E\varphi (x){\mathrm{\mid }}^2{d}^{D}x,S_r(\varphi ):=g_r{\int }_{{\mathbb{R}}^D}\varphi (x{)}^r{d}^{D}x\mathrm{.}$

They are finite because $\varphi$ and its derivatives have compact support. Use the following formal natural-unit mass-degree convention: set $\mathrm{\hslash }=c=1$, assign degree $-1$ to each coordinate $x^{\mu }$ ($1\le \mu \le D$), degree $1$ to each partial derivative, and degree $-D$ to $d^{D}x$; degrees add under products. Normalize both displayed functionals as dimensionless action terms of degree $0$. The symbols $[\varphi ]$ and $[g_r]$ denote their formal mass degrees in this convention; these are the engineering dimensions. They are $[\varphi ]=(D-2)\mathrm{/}2$ and $[g_r]=D-r(D-2)\mathrm{/}2$. Under the dilation ${\varphi }_b(x)=b^{-(D-2)\mathrm{/}2}\varphi (x\mathrm{/}b)$ for $b>0$, $S_{\mathrm{k}\mathrm{i}\mathrm{n}}({\varphi }_b)=S_{\mathrm{k}\mathrm{i}\mathrm{n}}(\varphi )$ and $S_r({\varphi }_b)=b^{D-r(D-2)\mathrm{/}2}{S}_r(\varphi )$. Call the monomial relevant, marginal, or irrelevant at tree level according as this dilation exponent is positive, zero, or negative. Thus a quartic monomial has exponent $4-D$: it is relevant for $D<4$, marginal by engineering dimension for $D=4$, and irrelevant for $D>4$. These labels concern only this classical engineering-dimension convention; no quantum correction, fixed point, or renormalization-group flow is asserted.