= Solution
<Critical exponents> describe leading singular powers near a <continuous phase transition>. Using the <reduced temperature> $t=(T-T_c)/T_c$, the standard definitions include
$$
M(t,0)\sim(-t)^\beta\quad(t<0),\qquad
\chi(t,0)\sim |t|^{-\gamma},\qquad
C_{\mathrm{sing}}(t)\sim |t|^{-\alpha},\qquad
\xi(t)\sim |t|^{-\nu}.
$$
These are the <order-parameter critical exponent>, <magnetic-susceptibility critical exponent>, <heat-capacity critical exponent> and <correlation-length critical exponent>. The <critical-isotherm exponent> is defined by $M(0,H)\sim\operatorname{sgn}(H)|H|^{1/\delta}$. The <anomalous dimension> $\eta$ gives the critical long-distance <connected correlation function>, $G_c(x)\sim |x|^{-(d-2+\eta)}$, when an ordinary scale-invariant critical point exists.
In the <Landau approximation>, minimize the <Landau free energy> to obtain the <equation of state>, differentiate it with respect to the <field conjugate to an order parameter> to obtain <magnetic susceptibility>, and differentiate the minimized <free energy> with respect to temperature to obtain <heat capacity>. These operations determine $\beta,\delta,\gamma,\alpha$, respectively. Expanding the <Landau-Ginzburg theory> about its uniform minimum gives the quadratic fluctuation kernel $Kq^2+V''(M)$, whose <Fourier transform> determines the <correlation length> and the <anomalous dimension>. At the ordinary quartic critical point this gives $\xi^2=K/V''(M)\propto |t|^{-1}$ and $\eta=0$.
For $u>0$, $r=0$ gives $H=uM^3$ and hence $\delta=3$; the equilibrium <free energy> $-r^2/(4u)$ gives a finite <heat capacity> discontinuity, conventionally $\alpha=0$. Together with the explicit calculation below, the ordinary <mean-field critical exponents> are
$$
\boxed{\alpha=0,\quad\beta=\frac12,\quad\gamma=1,\quad
\delta=3,\quad\nu=\frac12,\quad\eta=0.}
$$
These describe neglect of fluctuation corrections, rather than the exact exponents of every low-dimensional <Landau-Ginzburg theory>.
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