section I analyze these two situations in the context of the solutions of section \[Sol\].

Critical Dynamics of the fields and the theory of topological defect formation at second order transitions
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In the vicinity of a symmetry breaking second order transition the physical mass squared of the fields will approach zero, with a universal critical exponent $\nu$, $$\begin{aligned}
m^2(T) \sim m^2 \vert {T - T_c \over T_c} \vert^\nu, \nonumber\end{aligned}$$ where $T_c$ is the critical temperature. This can be seen already at first order in perturbation theory, albeit only with the mean-field value of $\nu=1/2$. In 3D we have $$\begin{aligned}
m^2(T) \sim  -m^2 + 3 \lambda \int^\Lambda {d^3 k \over (2 \pi)^3} {T \over k^2  } = 
-m^2 + {3 \lambda \over 2 \pi^2} \Lambda T.
\label{tadpole}\end{aligned}$$ $\Lambda$ is the ultraviolet cutoff which in many cases has physical meaning because generally scalar field theories are effective low energy models. An example is the scale of separation between the true quantum behavior of excitations at large $k^2$ and the effective classical dynamics for the long wave-length modes which we describe stochastically. In this case, at high temperature and small coupling $\lambda$, $\Lambda \simeq  T$.

The interesting fact about $m^2(T) \rightarrow 0$ 