On the internal inconsistency of the Wess-Zumino model

TL;DR

This work probes the internal consistency of the supersymmetric Wess-Zumino model by combining three assumptions: that counterterms mirror perturbative structure, that the spectrum is positive-definite (no ghosts), and that canonical commutation relations survive renormalization. Using canonical quantization, Schwinger equations, and Källén-Lehmann spectral representations, it derives stringent constraints on the wave-function renormalization factor $Z(\infty,...)$ and the propagator structure. It shows that either $Z(\infty,...) = 0$ (infinite renormalization) leads to vanishing scalar propagators and an empty spectrum, or $Z(\infty,...) \neq 0$ (finite renormalization) yields contradictions with UV fixed-point behavior via the Pohlmeyer theorem; in both branches, the theory becomes inconsistent and likely contains negative-norm states (Landau poles). The results thus challenge the self-consistency and unitarity of the WZ model under the stated assumptions, aligning with prior suggestions of ghost-like pathology in non-asymptotically free SUSY theories.

Abstract

We prove the internal inconsistency of the supersymmetric Wess-Zumino model. Our proof is based on three assumptions. The first assumption is that in the full theory the structure of counter temcs coincides with the structure of the counter terms in the perturbation theory. The second assumption is the positivity of norm states - no ghosts in the spectrum of the model. The third assumption is that the canonical commutation relations among generalized coordinates and momenta are valid in renormalized theory. The obtained results mean that there are negative norm states in the spectrum of the WZ model.