 Solution

For $ϵ>0$, the quartic interaction has a nonzero limiting value ${\lambda }_{\ast }=O(ϵ)$ instead of disappearing under coarse-graining. Long-wavelength thermal fluctuations are therefore interacting, so the fluctuation-free Landau approximation cannot give the generic critical powers. For example, its ordinary correlation-length critical exponent is $1/2$, whereas the relevant thermal eigenvalue just derived givesThe tadpole is momentum independent, so the anomalous dimension begins at $O({ϵ}^2)$. Combining this with the scaling relations yields $\gamma =1+ϵ/6+O({ϵ}^2)$, $\beta =1/2-ϵ/6+O({ϵ}^2)$, and $\alpha =ϵ/6+O({ϵ}^2)$, exhibiting the deviation from ordinary mean-field indices. These are controlled statements for small positive $ϵ$, not exact numerical exponents at arbitrary dimension. The Landau-Ginzburg theory itself is not invalid: integrating its fluctuations is precisely how the Wilson-Fisher fixed point was obtained. The invalid step is replacing that interacting long-distance theory by a uniform saddle and ignoring its fluctuation corrections.