= Gaussian specific-heat infrared threshold
{c}
{title2=$\int d^Dp\,(p^2+t)^{-2}$}
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>.
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