= D-term vacuum branches of a cubic superpotential
{c}
{title2=$W=\lambda\phi_0\phi_+\phi_-$}
For canonical <Kähler potential>, set $a=|\phi_0|^2$, $b=|\phi_+|^2$, $c=|\phi_-|^2$. The supplied <supergravity F-term potential> is $|\lambda|^2e^{\kappa^2(a+b+c)}[ab+ac+bc+3\kappa^2abc+\kappa^4abc(a+b+c)]$. Adding $g(b-c-\zeta)^2$ with $g>0$, $\lambda\ne0$ and $\zeta>0$ gives a <supersymmetric vacuum> at $a=c=0$, $b=\zeta$. The branch $b=c=0$ has positive energy $g\zeta^2$, <D-term> breaking and a <flat direction of a scalar potential> in $\phi_0$. Its charged squared masses are $|\lambda|^2ae^{\kappa^2a}\mp2g\zeta$, so it is a <local minimum> valley only when both are positive. For negative $\zeta$, exchange the charged fields. These are branches rather than two isolated minima; a conventional constant <Fayet–Iliopoulos term> additionally needs a consistent <supergravity> gauge completion.
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