Gaussian specific-heat infrared threshold ($\int d^{D}p(p^2+t{)}^{-2}$)

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.