= Solution
\b[Landau description.] Introduce a macroscopic <order parameter> and expand an appropriate <free energy> analytically in that parameter, retaining terms permitted by the physical symmetries. Stable equilibrium minimizes this <free energy> globally; a <local minimum> with larger <free energy> is metastable. Coefficient changes with <temperature> can exchange the preferred minima. A continuous emergence from zero can produce a continuous transition, while competing separated minima can produce a first-order transition. Stability requires a positive highest retained even-power term. This local mean-field description neglects spatial fluctuations and <nucleation> dynamics unless gradient and dynamical terms are added; near a <critical point> those omissions can matter.
For this <temperature-dependent cubic-quartic Landau free energy>, abbreviate $B=a/(12\pi)>0$ and $A(T)=c(T^2-T_0^2)$, so $F=A\phi^2/2-BT\phi^3+\lambda\phi^4/4$ and $\gamma=2B^2/(\lambda c)$.
\b[Local behavior at the origin.] To determine <constrained order-parameter boundary stability>, note that $F'(0)=0$ and $F''(0)=c(T^2-T_0^2)$, the origin is a <local minimum> for $T>T_0$ and a <local maximum> for $0\leq T<T_0$. At $T=T_0$, the quadratic term vanishes, but $F=-BT_0\phi^3+O(\phi^4)<0$ for small positive $\phi$. Therefore
$$
\boxed{\phi=0\text{ is locally stable for }T>T_0,\quad
\text{a one-sided <local maximum> for }0\leq T\leq T_0.}
$$
The equality case uses the stipulated domain $\phi\geq0$. On an unrestricted real line, the nonzero cubic term would instead make it a stationary inflection point.
\b[Global stability in both inline cases.] For $\phi>0$, complete the square in $F/\phi^2$:
$$
\frac{F}{\phi^2}
=\frac{\lambda}{4}\left(\phi-\frac{2BT}{\lambda}\right)^2
+\frac c2\left[(1-\gamma)T^2-T_0^2\right].
$$
The quartic coefficient guarantees a bounded <global minimum>. A negative value exists precisely when the minimum of this quadratic factor is negative. Hence case a, $\gamma>1$, has a nonzero globally stable phase with $F<0$ for every physical <temperature>. Case b, $0<\gamma<1$, has such a phase exactly for
$$
\boxed{T<T^*=\frac{T_0}{\sqrt{1-\gamma}}}.
$$
At $T=0$, the square's centre is at the boundary, but its negative constant still gives $F<0$ for small positive $\phi$, so this endpoint is included.
At $T=T^*$,
$$
F(\phi,T^*)=\frac{\lambda}{4}\phi^2(\phi-\phi^*)^2,\qquad
\boxed{\phi^*=\frac{2BT^*}{\lambda}=\frac{aT^*}{6\pi\lambda}}.
$$
The two global minima $\phi=0,\phi^*$ have equal <free energy>, and $F''(\phi^*)=\lambda(\phi^*)^2/2>0$. The <coexistence barrier of a cubic-quartic Landau transition> has its maximum at $\phi^*/2$, with barrier height $\lambda(\phi^*)^4/64$.
For locating the extrema away from coexistence,
$$
\phi_\pm(T)=\frac{3BT\pm\sqrt{9B^2T^2-4\lambda c(T^2-T_0^2)}}{2\lambda}.
$$
Retain only real positive roots; when both are positive, $\phi_-$ is a <local maximum> and $\phi_+$ a <local minimum>. For $T<T_0$, only $\phi_+$ is positive. If $\gamma<8/9$, the pair first appears on cooling at $T_{\rm sp}=T_0/\sqrt{1-9\gamma/8}>T^*$; if $8/9\leq\gamma<1$, a higher-free-energy nonzero minimum already exists at arbitrarily high <temperature>. These local-stability limits must not be mistaken for the global coexistence condition. The boundary $\gamma=1$ also favors a nonzero minimum at every finite <temperature>, but has no finite $T^*$.
\Image[/past-exam-of-the-thermal-and-statistical-physics-course-of-the-university-of-cambridge/2015/paper-1-landau-potentials.png]
{title=Original cubic-quartic free-energy curves above, at and below the relevant stability regimes}
\b[Cooling and transition <temperature>.] For $\gamma<1$, equilibrium starts at the zero minimum at high <temperature> and switches to the nonzero minimum at $T^*$. The jump is finite, $0\to\phi^*>0$, and there is a barrier between the coexisting states. This is a first-order transition. The <latent heat of a temperature-dependent cubic-quartic Landau transition> follows from differentiating the minimized <free energy>: the heat released on cooling is
$$
L=T^*(S_0-S_+)
=c(1-\gamma)(T^*)^2(\phi^*)^2
=cT_0^2(\phi^*)^2>0.
$$
If $F$ is an <energy> density rather than total <energy>, $L$ is likewise a density. An actual system can supercool below $T^*$ before <nucleation>, so $T^*$ is the equilibrium transition <temperature>.
For <thermal symmetry nonrestoration in a cubic-quartic model>, with $\gamma>1$, the equilibrium <global minimum> is nonzero already at arbitrarily high <temperature>. Following it does not force a transition. A system initially prepared in the <metastable> zero minimum can still leave it by <nucleation> while $T>T_0$; that minimum loses local stability at $T_0$. This is the sense in which a change is possible rather than inevitable. \b[There is no equilibrium coexistence temperature for $\gamma>1$ in this model.] A kinetic transition can depend on preparation and <nucleation> dynamics; $T_0$ is the limiting loss of local stability of the zero state, not a second coexistence temperature. Continuing $T^*$ to an imaginary value does not define a physical transition <temperature>.
\b[Mass window in the specified <electroweak phase transition> model.] For the <mass window in a cubic electroweak transition model>, put $x=m_H^2/m_0^2>0$. The supplied ratios imply
$$
\gamma=\frac1{72\pi^2x(A+x)},\qquad
\frac{\phi^*}{T^*}=\frac1{6\pi x}.
$$
The two strict conditions therefore require
$$
x(A+x)>\frac1{72\pi^2},
\qquad
x<\frac1{6\pi}.
$$
Since $x(A+x)$ is increasing for $x>0$, the finite window is
$$
\boxed{\frac{-A+\sqrt{A^2+1/(18\pi^2)}}2
<x<\frac1{6\pi}}.
$$
Using the specified $A$ and $m_0$ gives
$$
0.00147900<x<0.05305165,\qquad
\boxed{8.61\ \mathrm u<m_H<51.57\ \mathrm u}.
$$
At $m_H=134$ u, $x=0.3581801$, $\gamma\simeq0.0030033<1$ but $\phi^*/T^*\simeq0.1481<1$. Thus \b[134 u lies outside the required window], failing the order-parameter-strength condition. This does not remove the <polynomial> model's finite first-order jump; it fails the additional requirement $\phi^*>T^*$. The result is a bound within the specified simplified model, not a model-independent Higgs-mass bound.
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