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Supersymmetry non-breaking $iff$ no "Goldstone fermion"?

Physics Asked on January 12, 2021

In Supersymmetry and Morse Theory (1982) by E Witten,

Concern whether the supersymmetry is broken by checking whether
$$
Q | 0 rangle=0
$$

exists or not — Witten said:

SUSY breaking: A solution may be shown not to exist by calculating a reliable, positive lower bound to the energy eigenvalues.

SUSY non-breaking: It may be shown that a solution does exist by showing that the theory has a mass gap so that there is no potential "Goldstone fermion".

my question is that why SUSY non-breaking has something to do with

  • no potential "Goldstone fermion"?
  • theory has a mass gap?

Are these if and only if conditions?

$$
text{Supersymmetry non-breaking} iff text{mass gap and no “Goldstone fermion”?}
$$

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