Gaussian free-energy logarithm at effective dimension two ($F_S\propto t\log t$)

For a quadratic determinant in two effective momentum dimensions, differentiating the free-energy density with respect to the squared mass $R$ gives a term proportional to $\log ({\Lambda }^2/R)$. Integration gives $F_S\propto R\log R$ after subtracting analytic terms. Its thermal power index is still 1, but a strictly homogeneous pure-power expression misses this logarithm. It is a Gaussian determinant resonance, distinct from interaction-generated logarithms at an upper critical dimension.