= Quartic stabilization at a double no-scale root
{title2=$V\sim(\operatorname{Re}S-s_0)^4$}
For the <cubic-matter no-scale vacuum with an exponential dilaton>, put $x=S+\bar S$ and $A=ae^{-\alpha S}$. At $C=0$, zero energy requires $G(S)=b+(1+\alpha x)A=0$. On a fixed-imaginary-part slice, $s=\operatorname{Re}S$ 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 $-2\alpha^2A$, so $G=-\alpha^2A(s-s_0)^2+O((s-s_0)^3)$. Therefore $V=|G|^2/(xy^3)$ 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.
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