Minimal constrained superfields and the Fayet-Iliopoulos model

We show how the necessary constraints to project out all the components of a chiral superfield except for some scalar degrees of freedom originate from simple operators in the microscopic theory. This is in particular useful in constructing the simplest models of a goldstone boson/inflaton; or extracting the Standard Model Higgs doublet from a supersymmetric electroweak sector. We use the Fayet-Iliopoulos model as an example of the origin for the supersymmetry breaking. We consider the regime where both gauge symmetry and supersymmetry are spontaneously broken, leaving (in the decoupling limit) the goldstino as the only light mode in this sector. We show in three different ways, both in components and in superspace language, how the nilpotent goldstino superfield emerges. We then use it to write different effective operators and extract some of the consequences for the low energy spectrum.