 Solution

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.