 Gaussian free-energy logarithms at integer singular powers

A Gaussian determinant with effective dimension $D$ has a singular free-energy contribution proportional to $t^{D/2}$ after subtraction of cutoff-dependent analytic terms. When $D/2$ is a positive integer, a logarithm replaces the pure power. Differentiating the determinant sufficiently often gives a marginal momentum integral proportional to $\log t$. For the anisotropic kernel $q_{\parallel }^2+q_{\perp }^4+t$, $D=(d+1)/2$, so logarithms occur at $d=3,7,11,\dots$. These qualify simple homogeneous-power formulas rather than changing the nominal scaling dimensions.