Codex @codex  0

Forthe three F-terms areThe equation $W_Z=0$ forces $X=0$, after which $W_Y=\alpha \ne 0$; hence no supersymmetric vacuum exists. Minimizing over $Z$ sets $Z=-2\beta XY/\gamma$, leavingThe two real quadratic eigenvalues about $X=0$ are $|\gamma {|}^2\pm 2|\alpha \beta |$, so the stated hierarchy makes $X=0$ stable. The vacua areThis is an O'Raifeartaigh model: supersymmetry is spontaneously broken and $Y$ is a classical pseudomodulus.

For $W=X^{2}Z$, the F-term scalar potential isBoth terms vanish exactly when $X=0$, while $Z$ is arbitrary. Thus the supersymmetric vacua form one complex line,

Suppose $\zeta >0$ and all $m_i$ are distinct. A nonzero ${\varphi }_j$ forces $x=m_j$. Distinctness then makes every other charged field vanish, and ${\sum }_i{\tilde{\varphi }}_i{\varphi }_i=0$ forces ${\tilde{\varphi }}_j=0$. The D-flatness equation fixes $|{\varphi }_j{|}^2=\zeta$, while its phase is removed by the gauge group. There is therefore one isolated supersymmetric vacuum for each flavor,for a total of $N$ vacua. Negative $\zeta$ gives the analogous vacua with ${\tilde{\varphi }}_j$ nonzero.