Real-slice diagnosis of chiral-scalar vacua

A chiral superfield has a complex scalar. Minimizing its potential only on the real axis can miss supersymmetric vacua. For example $W^{\prime }(\phi )=1+{\phi }^2$ has positive potential $(1+x^2{)}^2$ for real $x$, yet its complex vacua $\phi =\pm i$ have zero energy. Supersymmetry breaking must therefore be tested in the full complex field space, not inferred from a positive minimum on an arbitrarily chosen real slice.