= Solution
Use the <connected correlation function> $G(x)=\langle\phi(x)\phi(0)\rangle-\langle\phi\rangle^2$, so that the constant ordered contribution is removed. Away from a <thermodynamic critical point> its long-distance decay defines the <correlation length>: $-\lim_{R\to\infty}R^{-1}\log|G(R)|=\xi^{-1}$ when this limit exists. In a massive rotation-invariant phase the <exponential decay> is typically multiplied by an algebraic factor.
Expand the <Landau-Ginzburg theory> free energy about a stable uniform minimum $M$, writing $\phi=M+\delta\phi$. To quadratic order,
$$
\mathcal F_2=\frac12\int d^Dx\{\kappa|\nabla\delta\phi|^2+c(\delta\phi)^2\},\qquad c=V''(M)=r+3uM^2+5vM^4>0.
$$
integration of the <Gaussian field theory> gives the Fourier-space <connected correlation function> $\widetilde G(q)=k_BT/(c+\kappa q^2)$. Thus the <Ornstein--Zernike correlation function> has
$$
\boxed{\xi=\sqrt{\kappa/c}.}
$$
For an ordinary quartic transition, $M=0$ above the transition and $M^2=-r/u$ below, so $c=r$ and $c=-2r$, respectively. The <correlation length> diverges as $|T-T_c|^{-1/2}$ on both sides, with different amplitudes. At criticality the mass vanishes and the Gaussian <correlation function> is proportional to $R^{-(D-2)}$ for $D>2$. Interactions can change this power to $R^{-(D-2+\eta)}$.
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