= Solution
Use the usual homogeneous, local, analytic <Landau-Ginzburg theory>; <rotational symmetry> and <Z2 symmetry> alone do not specify locality, analyticity, or the signs of the coefficients. The <local derivative expansion> contains every rotational scalar with an even total number of fields:
$$
\begin{aligned}
F[\phi]&=\int d^dx\left[V(\phi)+\frac12 K(\phi)(\nabla\phi)^2+c_1(\nabla^2\phi)^2+c_2\big((\nabla\phi)^2\big)^2+\cdots\right],\\
V(\phi)&=f_0+\frac r2\phi^2+\frac u4\phi^4+\frac v6\phi^6+\cdots,\qquad K(\phi)=\gamma+\gamma_2\phi^2+\cdots.
\end{aligned}
$$
The omitted terms include all allowed higher field powers and contracted spatial derivatives. Terms differing by a total derivative are equivalent for bulk behavior with suitable boundary conditions. The displayed <gradient energy> assumes spatial homogeneity. If only rotations about an origin are imposed, rotationally invariant position-dependent coefficients are also possible.
In the <mean-field approximation>, take a spatially uniform <order parameter> and minimize its <Landau free energy>. For $\gamma>0$, $u>0$ and $r=a(T-T_c)$ with $a>0$, keeping the quadratic and quartic terms gives
$$
\phi_0=0\quad(r\geq0),\qquad \phi_0=\pm\sqrt{-r/u}\quad(r<0).
$$
The single minimum splits into two minima as the <temperature> passes below the <critical temperature>. Choosing one minimum gives <spontaneous symmetry breaking> of the <Z2 symmetry> and a <continuous phase transition>. The <order parameter> is the expectation in a selected ordered state; averaging equally over both states gives zero. A <Z2 symmetry> does not guarantee a <continuous phase transition>: a negative quartic coefficient stabilized by a positive sextic coefficient can instead give a <first-order phase transition>.
Relaxing the <Z2 symmetry> permits odd terms. A nonzero linear source generally rounds this ordinary <continuous phase transition>. A cubic term can instead give a <first-order phase transition>: for $V(\phi)=r\phi^2/2+w\phi^3/3+u\phi^4/4$, $u>0$ and $w\ne0$, equal minima occur at
$$
\phi_*= -\frac{2w}{3u},\qquad r_* =\frac{2w^2}{9u}>0.
$$
The <order parameter> jumps from zero to $\phi_*$. A <continuous phase transition> without an exact <Z2 symmetry> is still possible with additional tuning; odd terms do not logically exclude it.
For the <multicritical even Landau potential>, set $r=\mu^2=a(T-T_c)$ and assume an integer $n\geq2$, $\lambda_n>0$ and $\gamma>0$. For $n>2$, all lower even interactions must be absent or tuned to zero to obtain these <mean-field critical exponents>. The stationary equation is
$$
0=r\phi_0+2n\lambda_n\phi_0^{2n-1},\qquad |\phi_0|^{2n-2}=\frac{-r}{2n\lambda_n}\quad(r<0).
$$
Its nonzero solutions are minima because $V''(\phi_0)=-(2n-2)r>0$. Thus the <order-parameter critical exponent> is
$$
\boxed{\beta_{\mathrm{MF}}=\frac{1}{2(n-1)}}.
$$
Substituting into the minimum <free energy> density gives
$$
f_{\min}=-\frac{n-1}{2n}(2n\lambda_n)^{-1/(n-1)}(-r)^{n/(n-1)}.
$$
The singular <heat capacity> per unit volume is $c_s=-T\,\partial_T^2 f_{\min}$. Smooth variation of the other coefficients does not change the leading power. Consequently
$$
c_s\propto (T_c-T)^{n/(n-1)-2},\qquad \boxed{\alpha_{\mathrm{MF}}=\frac{n-2}{n-1}}.
$$
For $n=2$, this means a finite <heat capacity> jump, with $\alpha_{\mathrm{MF}}=0$; for $n=3$, the <tricritical point> has $\beta_{\mathrm{MF}}=1/4$ and $\alpha_{\mathrm{MF}}=1/2$. The case $n=1$ would contain only a quadratic potential and would not stabilize an ordered phase.
The <Gaussian fluctuation correction near a critical point> is more singular than the <mean-field approximation> contribution when
$$
2-\frac d2>\frac{n-2}{n-1}\quad\Longleftrightarrow\quad \boxed{d<d_c=\frac{2n}{n-1}}.
$$
This is the <upper critical dimension of an even scalar interaction>. Equivalently, its coupling has <engineering dimension> $2n-(n-1)d$, positive below $d_c$. Dominant fluctuations signal breakdown of the <mean-field approximation>; the Gaussian expression is not then a calculation of the exact interacting <heat-capacity critical exponent>. At $d=d_c$, equality of the powers allows logarithmic corrections.
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