Instability of fermionic zero modes and dynamical supersymmetry violation

We investigate the Dirac operator of massless and partially massless supersymmetric theories in four dimensions. We find that for a very general class of theories, all zero modes disappear from the spectrum under a generic variation of the scalar background field. The reason is the impossibility of maintaining a normalizable asymptotic behavior both at the origin and at infinity. As an application, we prove that the vacuum energy density of a large class of supersymmetric theories is strictly negative, a phenomenon which signifies a dynamical violation of the SUSY algebra. Certain implications for the anomaly equation as well as for completely massive theories are also discussed. (orig.)