In a supersymmetric gauge theory with a U(1) factor, we can add a Fayet-Illiopoulos term LFI=eξD to the lagrangian, where D is the auxilliary field for the U(1) gauge field, and ξ is a parameter with dimensions of mass-squared.
a) Explain why adding this term preserves supersymmetry. Explain why the corresponding gauge field cannot be nonabelian.
b) Add this term to the SQED lagrangian that you found in problem 94.4, and eliminate the auxilliary fields.
c) Minimize the resulting potential. Show that supersymmetry is spontaneously broken if ξ is in a certain range.