= Solution
<Critical exponents> describe leading power-law singularities as the <reduced temperature> $t=(T-T_c)/T_c$ and the <conjugate field> tend to zero. Their amplitudes depend on microscopic details, while the exponents are characteristic of a <universality class>. Define $m_0\sim(-t)^\beta$ on a selected ordered branch, $\chi\sim|t|^{-\gamma}$ at zero field, $C_{V,s}\sim|t|^{-\alpha}$ for the singular heat capacity, and $m(t=0,h)\sim\operatorname{sgn}(h)|h|^{1/\delta}$. Also define $\xi\sim|t|^{-\nu}$ and $G(r,t=0)\sim r^{-(D-2+\eta)}$. In these statements $\beta$ is the <order-parameter critical exponent>, not inverse temperature.
Assume $r=r_t t$ with $r_t>0$ and a nonzero positive quartic coefficient $u$. Minimizing the quartic potential gives $m_0=\sqrt{-r/u}$, hence $\beta=1/2$. Differentiating its <equation of state> gives
$$
\chi=(r+3um^2)^{-1}=\begin{cases}r^{-1},&t>0,\\(-2r)^{-1},&t<0\text{ on a selected phase},\end{cases}
$$
so $\gamma=1$ with different amplitudes. At $r=0$, $h=um^3$, giving $\delta=3$. The minimized potential is zero above the transition and $-r^2/(4u)$ below, so its second temperature derivative has a finite jump: $\alpha=0$.
The quadratic fluctuation kernel around the stable uniform minimum is $\kappa p^2+V''(m_0)$, so $\xi^2=\kappa/V''(m_0)$. Since $V''$ is $r$ above and $-2r$ below, $\nu=1/2$. Its critical momentum dependence is $p^{-2}$, giving $\eta=0$. Thus the ordinary <mean-field critical exponents> are
$$
\boxed{\alpha=0,\quad\beta=\tfrac12,\quad\gamma=1,\quad\delta=3,\quad\nu=\tfrac12,\quad\eta=0.}
$$
These are predictions of the <Landau approximation>, whose validity is constrained by the <Ginzburg criterion>; they need not be the exponents of the fluctuating theory below its <upper critical dimension>.
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