Supersymmetry breaking and the Goldstino

The positive supercharge anticommutator implies that a normalizable zero-energy vacuum is annihilated by the supercharges in the rigid theory. For canonical matter and gauge kinetic terms, nonzero vacuum F or D expectations make a fermion transformation contain an inhomogeneous constant. This diagnoses spontaneous breaking when the action itself is invariant. Explicitly adding unrelated scalar and fermion mass terms is a different operation: the algebra need not remain a symmetry of that action. Mass splitting alone does not identify which mechanism produced it.

Absence of a finite solution to the F equations also needs interpretation. A potential might approach zero only at infinite field values, leaving a runaway with no normalizable vacuum rather than a stable broken ground state. A zero Witten index is similarly inconclusive without spectral and boundary information. These are practical restrictions on using an algebraic criterion. We will instead inspect a model with a positive-energy classical minimum family and nonnegative transverse curvatures in a stated parameter range, before discussing its remaining quantum stability question.

If m is nonzero, setting the second numbered field derivative to zero forces phi one to zero. The X derivative then remains f and cannot vanish when f is nonzero. Thus no finite supersymmetric F-flat configuration exists. For $m^2>hf$, the classical minimum has phi one and phi two zero, while X is arbitrary. The energy density is f squared. Its positivity follows from the complete potential, not from assigning an arbitrary additive constant to a nongravitational scalar potential independently of the supercharge algebra.

At X equal to zero, write phi one as a plus i b divided by square root two. The two real scalar curvatures are $m^2+hf$ and $m^2-hf$, while the phi two scalars have mass squared m squared. Both transverse phi-one curvatures are positive in the stated range. The complex X scalar remains classically flat and is called a pseudomodulus. This flatness is not a Goldstone theorem and can be lifted by quantum corrections to the effective action. A complete vacuum analysis must check that lifting rather than calling every flat scalar protected.

In this minimum family only F X is nonzero, so the fermion in X shifts under supersymmetry by a constant proportional to f. It is the Goldstino direction and is massless in the rigid theory under the usual assumptions for a genuine spontaneously broken vacuum. At X equal to zero the other two Weyl fermions form a massive pair through the m term. The Goldstino's protection comes from the broken conserved supercharge; the X scalar's classical flatness has a different origin and need not survive radiative effects.

Coupling gravity changes the discussion. Local supersymmetry allows the Goldstino to be absorbed into the gravitino through the super-Higgs mechanism, and supergravity's scalar potential includes terms absent from the positive rigid sum. One cannot infer the cosmological constant from the simple rigid example without its gravitational completion. Likewise, a proof of a Goldstino in a toy action does not establish an experimentally observed particle. The calculation demonstrates how symmetry, auxiliary order parameters, stability, and low-energy modes fit together within a specified theoretical framework.