 Solution

The upper critical dimension is the dimension at which the leading nonlinear interaction is marginal under Gaussian scale transformations. Normalize the gradient term and rescale $x=b{x}^{\prime }$ so that $\varphi (x)=b^{-(D-2)/2}{\varphi }^{\prime }(x^{\prime })$. A coupling $g_{2p}\int {\varphi }^{2p}{d}^{D}x$ then scales byThusBelow this dimension, the interaction is a relevant operator at the Gaussian fixed point, so neglecting long-wavelength fluctuations is not self-consistent near criticality. A direct Ginzburg criterion gives the same result: the fluctuation in a correlation volume scales as $⟨(\delta \varphi {)}^2{⟩}_{\xi }\sim {\xi }^{2-D}$. For ordinary ordering $M^2\sim r/u\sim {\xi }^{-2}/u$, their ratio is proportional to $u{\xi }^{4-D}$. Along a tricritical trajectory $M^2\sim (|r|/v{)}^{1/2}\sim {\xi }^{-1}/\sqrt{v}$, giving ratio proportional to $\sqrt{v}{\xi }^{3-D}$. These ratios grow without bound below the corresponding upper critical dimension.

At $D=D_c$, the interaction is marginal. Mean-field powers can survive, but acquire multiplicative logarithmic corrections; it is too strong to say that all mean-field exponent values necessarily fail at equality. Above $D_c$, fluctuations do not change the leading mean-field critical exponents, although a dangerously irrelevant coupling can invalidate naive hyperscaling.