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At a critical point, a local field operator $\mathcal{O}$ has scaling dimension ${\Delta }_{\mathcal{O}}$ if $\mathcal{O}(x)\mapsto b^{-{\Delta }_{\mathcal{O}}}\mathcal{O}(x/b)$ under coarse-graining by $b$. The coupling $u$ in $\int d^{d}xu\mathcal{O}$ has renormalization-group eigenvalueA perturbation is a relevant operator when $y_u>0$, an irrelevant operator when $y_u<0$, and a marginal operator when $y_u=0$; nonlinear terms in its renormalization-group beta function decide the fate of a marginal coupling. Irrelevant microscopic interactions decay under coarse-graining, so many distinct systems approach the same fixed point and share one universality class.

Near criticality the singular free energy density is of order one per correlation volume:The heat capacity contains two derivatives with respect to temperature, soConsequently the hyperscaling relation is

Introduce the renormalization-group beta functions and field anomalous dimensionIndependence of physics from the arbitrary sliding scale gives the functional Callan-Symanzik equationup to the equivalent sign convention obtained by defining ${\gamma }_{\varphi }$ with a minus sign. For an operator $O_i$ of dimension $d_i$ containing $n_i$ fields, differentiating its coefficient shows the canonical and wave-function pieceswhere ${\beta }_i^{\mathrm{vertex}}$ contains mixing among local field operators and genuine corrections at higher loop order. Thus $d_i<4$, $d_i=4$, and $d_i>4$ canonically produce relevant operators, marginal operators, and irrelevant operators, respectively, before anomalous corrections.