Renormalization-group free-energy resonance ($m{\lambda }_t+n{\lambda }_h=D$)

An analytic source monomial $g_{mn}{t}^m{h}^n$ in an additive renormalization-group transformation of free-energy density is resonant when its scaling degree equals the spatial dimension. The analytic background coefficient would require division by $1-b^{m{\lambda }_t+n{\lambda }_h-D}=0$. On a branch with $t\ne 0$ and ${\lambda }_t>0$, a particular nonanalytic background is $-g_{mn}{t}^m{h}^n\log |t|/({\lambda }_t\log b)$: substituting $t^{\prime }=b^{{\lambda }_t}t$, $h^{\prime }=b^{{\lambda }_h}h$ verifies the additive recursion exactly. Thus logarithms can be required even when the eliminated-mode source is analytic. This is distinct from marginal interaction running, which can create additional logarithmic corrections.