 Solution

In the Wilsonian effective action picture, integrate out field modes in a high-momentum shell and encode their effects in the action for the retained modes. Rescale lengths and fields to compare the resulting theory at the original cutoff. The couplings then follow a renormalization-group flow, while low-energy predictions are preserved. In general all local interactions allowed by the symmetries are generated, even if only a few appear in the initial action.

Near a renormalization-group fixed point, a perturbation $u\int d^{d}x\mathcal{O}(x)$ with scaling dimension $\Delta$ has linearized eigenvalue $y=d-\Delta$. Under coarse graining by $b>1$, its dimensionless coupling scales as $u^{\prime }\simeq b^{y}u$. A relevant operator has $y>0$, so its perturbation grows toward long distances; an irrelevant operator has $y<0$ and decreases; a marginal operator has $y=0$ and needs nonlinear flow to determine its behavior. Marginality at linear order need not mean exact scale independence. Interactions can be marginally relevant or marginally irrelevant operators.