 Cubic-matter no-scale vacuum with an exponential dilaton

For $W=C^3+A+b$, $A=a{e}^{-\alpha S}$, the potential is $V=|C^3+b+(1+\alpha s)A{|}^2/(s{t}^3)+3|C{|}^4/(s{t}^2)$. A finite zero-energy vacuum requires $C=0$ and $b+(1+\alpha s)A=0$. With $a,b\ne 0$, such a solution exists exactly when $0<|b/a|\le 2e^{-1/2}$: the magnitude equation is $|b/a|=(1+x)e^{-x/2}$, $x=\alpha s>0$, whose maximum occurs at $x=1$. At that vacuum $F^S=F^C=0$ but $F^T=e^{K/2}t\bar{W}\ne 0$. Both components of $T$ are flat, while $C$ is quartically lifted. If no root exists, the positive potential falls toward zero as $t\to \infty$ and has no finite minimum. If $a=b=0$, the $C=0$ family is supersymmetric instead.