Scalar engineering dimensions and tree-level dilation

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.