Constructive Quantum Field Theory II pp 331-331 | Cite as

# Supersymmetry Breaking in Wess-Zumino Models

## Abstract

A combination of index theory [1,2] and multiphase analysis [3,4] should lead to an understanding of the vacuum structure of the weakly coupled two-dimensional Wess-Zumino model in infinite volume. The supersymmetric interaction is |*V*′(*φ*)|^{2} + *V″*(*φ*)\(\bar{\psi }\) *ψ*, where for the *N =* 1 model *φ*,*ψ* are real, while for the *N =* 2 model *φ*,*ψ* are ocmplex. We take *V* to be a polynomial of degree *n*¡ and scale it as *V*(Ф) → λ^{−2} *V*(λФ) with λ small. If *V*′has *n*−1 distinct zeros, then |*V*′|^{2} has *n*−1 minima at zero. At small λ multiphase expansion should be possible, leading as in [4] to a construction of the stable vacua. The index theorems proven in [5] should help determine the number of stable vacua. For example, in the *N =* 2 model the periodic partition function is equal to *n*−1, so the free energy must be zero and so supersymmetry is unbroken. A careful analysis of the expansion should lead to the further conclusion that all *n*−1 possible vacua are stable thermodynamically [3]. These conclusions must be modified in the *N =* 1 case because the Pffaffian in the measure of alternates in sign from one minimum to the next.

## References

- 1.Jaffe, A., Lesniewski, A., these proceedings.Google Scholar
- 2.Witten, E.: Nucl. Phys.
*B202*, 253 (1982).MathSciNetADSCrossRefGoogle Scholar - 3.Borgs, C., Imbrie, J.Z.: Multiphase Analysis in Quantum Field Theory, in preparation.Google Scholar
- 4.
- 5.Jaffe, A., Lesniewski, A.. Weitzman, J.: Commun Math. Phys.
*112*,75 (1987). The Loop Space*S*′→R and Supersymmetric Quantum Fields, to appear.Google Scholar