I am currently studying SuperSymmetry and I have reached a problem which I have not found an answer to.
I clearly understand how the Goldstone theorem works for the boson case (without any susy) however, the way I have seen the theorem, is by working with the SSB matrix, and see what generators are broken or not.
Now, I am learning SUSY by studying notes that follow Wess and Bagers book which state that you have SuperSymmetric Spontaneous breaking when your energy is positive (which I understand why), as therefore you have to see if there are solutions for whom the scalar potential vanishes or not.
In a usual Chiral case only, you need the Spinor fields to have zero mass in order to break susy which again I understand.
My problem comes when we take a supergauge invariant vector multiplet case where in order to break susy we need to enter a Fayet-Iliopoulos term.
Now, the book only argues that because δλ will be non-zero (where λ will be the spinor field in this case), λ will be our goldstino fermion now.
I dont really understand, why having a non-zero variation of the vacuum state of λ, will result to λ being massless and therefore connecting to when a generator breaks or not.