No-scale vacuum with a cubic matter superpotential ($K=-3\log (T+\bar{T})+|C{|}^2,W=C^3+B$)

The no-scale supergravity identity cancels the negative gravitino contribution, leaving $V=e^{|C{|}^2}|3C^2+\bar{C}(C^3+B){|}^2/(T+\bar{T}{)}^3$. Zero-energy minima occur at $C=0$ and, for $B\ne 0$, three phase-related nonzero solutions with $|B|=|C|(3+|C{|}^2)$. Their $T$ supergravity auxiliary field is nonzero if $B\ne 0$, while both real components of $T$ are flat on each vacuum branch. For $B=0$, the $C=0$ branch is instead supersymmetric and has zero gravitino mass from a superpotential.