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Splitting a scalar field into slow and fast components $\varphi =\phi +\eta$, the Wick contractions in $\lambda ⟨(\phi +\eta {)}^6⟩$ generate $15\lambda ⟨{\eta }^2⟩{\phi }^4$. A vanishing bare quartic coupling is therefore not generally preserved by renormalization-group flow.

The relevance of a perturbation is determined by the eigenvalue of its dimensionless coupling under linearized renormalization-group flow. Positive, negative and zero eigenvalues define relevant, irrelevant and marginal perturbations, respectively.

A renormalization-group transformation integrates out selected short-distance degrees of freedom and restores chosen cutoff and field-normalization conventions. It defines a map $R_b$ on couplings at a length factor $b$. Its discrete Jacobian matrix multipliers differ from eigenvalues of the generator of a continuous renormalization-group flow; the relation for a scaling direction is $\lambda =b^y$.

A smooth-cutoff scalar propagator is the inverse of a positive regulated quadratic kernel. It agrees with the unregulated scalar propagator at low momentum and decreases rapidly at high momentum. Smooth suppression is not identical to vanishing support. Its cutoff derivative is the line weight in an exact renormalization-group flow.

Universality of critical phenomena is the agreement of critical exponents or normalized scaling forms between microscopically different systems. Dimension, symmetry and interaction range help determine the universality class. A renormalization-group flow can explain this agreement by showing that different microscopic models approach the same fixed-point description, while irrelevant differences fade at large scales.

A Wilsonian effective action is obtained by integrating out field modes above a chosen momentum scale while retaining lower-momentum modes as backgrounds. Its local couplings change with the scale according to a renormalization-group flow.