Sextic scalar field theory (${\varphi }^6$)

A sextic scalar field theory includes a term proportional to ${\varphi }^6$ in its scalar potential. For a real field with reflection symmetry, a useful model isThe 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 ${\varphi }^6$ interaction is marginal by engineering dimension, whereas in four it has a negative-dimension coupling.