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Not only must supersymmetry be broken, it must be broken in a way that doesn't lead to phenomenological disasters. In the MSSM, $D$-term breaking with a Fayet-Iliopoulos term achieves neither: it doesn't break supersymmetry but it does lead to phenomenological disasters! Consider again your potential $$ \begin{align} V&=\sum_i |m_i|^2 |\phi_i|^2 +1/2 ...

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