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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.

Thanks

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SUSY is spontaneously broken if SUSY variation of a vacuum state of any field $A$ is non-zero: $$ \delta_\epsilon \langle A\rangle\neq 0. $$ But the only fields that can have non-zero SUSY variation of their vacuum states are fermions since $$ \delta_\epsilon\langle\psi\rangle\sim\epsilon\langle F\rangle\\ \delta_\epsilon \langle\lambda\rangle\sim\epsilon\langle D\rangle. $$ Here $\psi$ is a fermion from a chiral multiplet, and $\lambda$ is a gaugino; $F$ and $D$ are the corresponding auxiliary fields. Thus, if $\langle F\rangle $ and $\langle D\rangle $ are zero then the vacuum is supersymmetric, if either of them is non-zero then SUSY is broken.

Existence of a massless Goldstone field follows from the Goldstone theorem. Since the broken generator is fermionic in our case, the goldstone field is also a (Majorana or Weyl) fermion -- a goldstino.

Considering a theory of a single U(1) gauge multiplet, if SUSY is spontaneously broken Goldstone says there must be a massless goldstino. But the only fermion is the gaugino $\lambda$, so it will play the role of the goldstino.

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