Goldstino zero mode from vacuum stationarity ($W_{ij}\overline{W_j}=0$)

Stationarity of a canonical global F-term scalar potential gives ${\partial }_{i}V={\sum }_j{W}_{ij}\overline{W_j}=0$. If some $F_j=-\overline{W_j}$ is nonzero, this vector is in the kernel of the chiral-superfield fermion mass matrix. It supplies the massless goldstino direction. If every $F_j$ vanishes, this argument gives no broken-symmetry mode.