Gaussian critical exponent ($\alpha =2-d/2,\Delta =(d+2)/4$)

For quadratic dispersion $p^2+t$ with no interacting anomalous dimension, the thermal and uniform-source scaling eigenvalues are $y_t=2$ and $y_h=(d+2)/2$. The generic homogeneous singular free-energy density has $2-\alpha =d/2$ and gap exponent $\Delta =(d+2)/4$. These Gaussian-model values below the upper critical dimension are not the exponents of an interacting scalar critical point.

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.

The Gaussian momentum-shell scaling has thermal eigenvalue $2$ and source eigenvalue $(D+2)/2$. The formal homogeneous magnetization index is therefore $[D-(D+2)/2]/2=(D-2)/4$. A free quadratic Hamiltonian alone does not possess a stable negative-mass ordered phase, so this index is not a claim of spontaneous magnetization in an unstabilized Gaussian integral. In $2<D<4$, a quartic interaction is relevant at the Gaussian fixed point and changes the generic scalar critical behavior.

Two thermal derivatives of a regulated Gaussian determinant produce this integral. For $0<D<4$, rescaling $p=\sqrt{t}q$ yields a divergent power $t^{D/2-2}$ and heat-capacity critical exponent $\alpha =(4-D)/2$. At four dimensions the singularity is logarithmic, with power exponent zero. Above four the leading heat capacity is a finite cutoff-dependent background; its leading bounded-power convention has exponent zero, although the background-subtracted singular Gaussian contribution retains the negative index $2-D/2$. In a stable quartic theory the mean-field ordered free energy proportional to $-t^2/u$ instead produces a heat-capacity jump. This is why a dangerously irrelevant coupling qualifies naive hyperscaling relation.