= Finite zero-energy vacuum for a single exponential superpotential
For $K=-\log(S+\bar S)-3\log(T+\bar T)$ and $W=ae^{-\alpha S}+b$ with $\alpha>0$, the potential is $|b+[1+\alpha(S+\bar S)]ae^{-\alpha S}|^2/[(S+\bar S)(T+\bar T)^3]$. With nonzero $a,b$, a finite zero-energy vacuum exists exactly when $0<|b/a|\le2e^{-1/2}$. To prove this, set $x=\alpha\operatorname{Re}S>0$; the magnitude equation is $|b/a|=(1+2x)e^{-x}$, whose derivative changes sign at $x=1/2$ and whose maximum is $2e^{-1/2}$. The phase fixes the imaginary part of $S$. At a zero-energy solution the $T$ <auxiliary field> remains nonzero, so <supersymmetry> is broken and both real components of $T$ are flat. If $a=b=0$, the family is instead supersymmetric.
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