β_{i} = Λ \frac{d g _{i}}{d Λ}, β_{m^{2}} = Λ \frac{d m ^{2}}{d Λ}, γ_{ϕ} = \frac{1}{2} Λ \frac{d l o g Z _{Λ}}{d Λ} .

Independence of physics from the arbitrary sliding scale gives the functional Callan-Symanzik equation

[Λ \partial_{Λ} + β_{m^{2}} \partial_{m^{2}} + i \sum β_{i} \partial_{g_{i}} - γ_{ϕ} \int d^{4} x ϕ (x) \frac{δ}{δ ϕ ( x )}] S_{Λ}^{eff} = 0,

up to the equivalent sign convention obtained by defining

γ_{ϕ}

with a minus sign. For an operator

O_{i}

d_{i}

containing

n_{i}

fields, differentiating its coefficient shows the canonical and wave-function pieces

β_{i} = (d_{i} - 4 + n_{i} γ_{ϕ}) g_{i} + β_{i}^{vertex},

where

β_{i}^{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.

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