= 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 $u\int d^dx\,\mathcal O(x)$ with scaling dimension $\Delta$ has linearized eigenvalue $y=d-\Delta$. Under coarse graining by $b>1$, its dimensionless coupling scales as $u'\simeq b^yu$. 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. \b[Marginality at linear order need not mean exact scale independence]. Interactions can be marginally relevant or <marginally irrelevant operators>.
Back to article page
