= Sextic scalar field theory
{title2=$\phi^6$}
A sextic scalar field theory includes a term proportional to $\phi^6$ in its scalar potential. For a real field with reflection symmetry, a useful model is
$$
V(\phi)=\frac r2\phi^2+\frac u4\phi^4+\frac v6\phi^6,\qquad v>0.
$$
The positive sextic coefficient stabilizes large field values even when the quartic coefficient $u$ is negative. This permits competition between a minimum at zero and symmetry-related nonzero minima, making the model useful for <first-order phase transitions> and a <tricritical point>.
At $r=u=0$ in a mean-field treatment, the sextic term is the leading restoring nonlinearity. Gradient terms determine interfaces between different minima. The classical stability statement is distinct from quantum power counting: in three spacetime dimensions a canonically normalized $\phi^6$ interaction is marginal by engineering dimension, whereas in four it has a negative-dimension coupling.
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