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Let $f=-V^{-1}\log \mathcal{Z}$ be the dimensionless free-energy density. The transformed volume is $V^{\prime }=V/B$. Including the normalization from eliminated modes giveswhere $g_b$ is an additive shell contribution. After subtraction of regular backgrounds, the leading singular part obeys the scaling hypothesis for critical phenomenaChoose $b=t^{-1/2}$ for the high-temperature side $t>0$. Then the nominal homogeneous form isThe volume exponent is $D_{\mathrm{eff}}$, so inserting $d$ in the isotropic hyperscaling formula would give the wrong answer.

There are two qualifications to interpreting this as the total free energy. Completing the Gaussian square gives the source contribution exactly as $-h^2/(2t)$, and the zero-field determinant contributesRescaling $q_{\parallel }\sim t^{1/2}$ and $q_{\perp }\sim t^{1/4}$ gives the power $t^{(d+1)/4}$, while $2\Delta -1=(d+1)/4$ makes the source term consistent with it. Cutoff-dependent analytic terms need not obey the homogeneous law. Furthermore Gaussian free-energy logarithms at integer singular powers replace the pure zero-field power by $t^n\log t$ when $(d+1)/4=n$ is a positive integer, including $d=3,7$. This is the usual logarithmic qualification of the nominal exponents. The quadratic model also has no stable negative-$t$ phase unless a stabilizing interaction is added.