Cubic-matter no-scale vacuum with an exponential dilaton ($V_{\min }=0,F^T\ne 0\text{ on the nontrivial finite branch}$)

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.

For the cubic-matter no-scale vacuum with an exponential dilaton, put $x=S+\bar{S}$ and $A=a{e}^{-\alpha S}$. At $C=0$, zero energy requires $G(S)=b+(1+\alpha x)A=0$. On a fixed-imaginary-part slice, $s=\mathrm{Re}S$ gives $dG/ds=\alpha (1-2\alpha s)A$. At the double root $s_0=1/(2\alpha )$, the first derivative vanishes but the second is $-2{\alpha }^{2}A$, so $G=-{\alpha }^{2}A(s-s_0{)}^2+O((s-s_0{)}^3)$. Therefore $V=|G{|}^2/(x{y}^3)$ begins at fourth order. This real scalar has zero quadratic mass but is not an exact flat direction of a scalar potential. The imaginary part of $S$ is still lifted quadratically, and the matter field $C$ also has a quartic leading potential. Exact flat directions instead come from the complex volume modulus $T$ along the zero-energy locus.