= Solution
When $u_0=0$, the field theory is a <Gaussian field theory>, so integrating out short-wavelength modes can be done exactly rather than merely at a saddle. Introduce an ultraviolet momentum cutoff $\Lambda$ and a scale factor $b>1$, then split $\phi=\phi_<+\phi_>$ into $|q|<\Lambda/b$ and $\Lambda/b<|q|<\Lambda$. Orthogonality of <Fourier modes> eliminates cross terms in the quadratic Hamiltonian. A constant source couples only to the zero-momentum mode and therefore does not couple to $\phi_>$.
Before rescaling, the <Gaussian functional integral> factorizes:
$$
Z[h]=Z_>(r_0)\int\mathcal D\phi_<\,e^{-H_<[\phi_<,h]},
$$
where $H_<$ has the original $r_0$ and $h$, restricted to low momenta. Up to a field-independent normalization, the eliminated modes contribute
$$
\Delta F_>=\frac12\operatorname{Tr}_>\log(-\nabla^2+r_0).
$$
The determinant affects the additive <free energy>, but generates neither a mass correction nor a source correction, since no interaction mixes low and high modes. This remains true beyond the <Landau approximation>: it is an exact integration of a <Gaussian field theory>, not neglect of the high-mode fluctuations.
To restore the original cutoff, use $x=bx'$, $q'=bq$, and choose the field rescaling
$$
\phi'(x')=b^{(D-2)/2}\phi_<(bx').
$$
Substitution into the low-mode Hamiltonian gives the <Gaussian momentum-shell scaling>
$$
H_< =\int d^Dx'\left[
\frac12(\nabla'\phi')^2+\frac12b^2r_0\phi'^2
-b^{(D+2)/2}h\phi'\right].
$$
The <gradient> coefficient is unchanged: the volume factor is $b^D$, the two derivatives contribute $b^{-2}$, and the two fields contribute $b^{2-D}$. The mass term gains $b^2$, while the source term gains $b^{(D+2)/2}$. Thus
$$
\boxed{r_0'=b^2r_0,\qquad h'=b^{(D+2)/2}h,\qquad u_0'=0.}
$$
For a nonconstant source the corresponding formula is $h'(x')=b^{(D+2)/2}h(bx')$, after restricting its coupling to retained modes. The constant-source case avoids that extra source filtering.
Writing $b=e^\ell$, the exact Gaussian <renormalization-group flow> has
$$
\boxed{\frac{dr_0}{d\ell}=2r_0,\qquad
\frac{dh}{d\ell}=\frac{D+2}{2}h.}
$$
Both perturbations are relevant at the massless zero-source <Gaussian fixed point>, with $y_t=2$ and $y_h=(D+2)/2$. There is no anomalous field rescaling here, so $\eta=0$ and the <correlation-length critical exponent> is again $\nu=1/y_t=1/2$. The coefficient changes arise from rescaling, not from an interaction-induced shift in $r_0$ during shell integration.
A stable real <Gaussian functional integral> requires $r_0>0$, or an infrared prescription that treats the zero mode at the massless point. With $u_0=0$ and $r_0<0$, the field energy is unbounded below; the <Gaussian field theory> alone cannot describe a stable ordered phase. The scaling laws are therefore interpreted about the <Gaussian fixed point> from the stable side, with finite-volume zero-mode regularization when needed.
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