= Solution
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 $|\phi_j|=|\widetilde\phi_j|$, whereas <F-flatness> gives $\widetilde\phi_j\phi_j=0$; together they force both fields to vanish. Thus only the <Coulomb branch> remains,
$$
\boxed{\phi_i=\widetilde\phi_i=0,\qquad x\in\mathbb C},
$$
of complex dimension one. The points $x=m_i$ are special because the corresponding charged multiplets become massless there.
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