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For $\zeta =0$ and distinct masses, any nonzero charged field can occur only at one $x=m_j$. At such a point, D-flatness gives $|{\varphi }_j|=|{\tilde{\varphi }}_j|$, whereas F-flatness gives ${\tilde{\varphi }}_j{\varphi }_j=0$; together they force both fields to vanish. Thus only the Coulomb branch remains,of complex dimension one. The points $x=m_i$ are special because the corresponding charged multiplets become massless there.

When every $m_i$ and $\zeta$ vanishes, the Coulomb branch again has ${\varphi }_i={\tilde{\varphi }}_i=0$ and arbitrary $x$. A Higgs branch also occurs at $x=0$: it is the zero-level quotient ofby U(1), and has complex dimension $2N-2$ for $N>1$. The two branches meet at the origin. For $N=1$, the F- and D-flat equations force both charged fields to vanish, so the Higgs branch collapses to that intersection point.