Two-charge scalar gauge potential
= Two-charge scalar gauge potential
{title2=$V\supset M^2|\psi-b\phi^2|^2$}
With scalar charges $e$ and $2e$, both $\psi$ and $\phi^2$ transform with phase $e^{2i\alpha}$. Therefore $M^2|\psi-b\phi^2|^2$ is a gauge-invariant potential term. It contains a nonzero direct cubic interaction $-M^2(b\psi^*\phi^2+\mathrm{h.c.})$. Adding positive quadratic and quartic modulus terms gives a real bounded-below potential. All expanded interactions have field degree at most four.