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.

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