= Renormalization-group free-energy resonance
{title2=$m\lambda_t+n\lambda_h=D$}
An analytic source monomial $g_{mn}t^mh^n$ 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 $t\ne0$ and $\lambda_t>0$, a particular nonanalytic background is $-g_{mn}t^mh^n\log|t|/(\lambda_t\log b)$: substituting $t'=b^{\lambda_t}t$, $h'=b^{\lambda_h}h$ 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.
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