DISCRETE GAUGE SYMMETRIES AND THE ORIGIN OF BARYON AND LEPTON NUMBER CONSERVATION IN SUPERSYMMETRIC VERSIONS OF THE STANDARD MODEL

In the supersymmetric standard model operators of dimension 4 and 5 generically violate B and L number. One usually assumes the presence of some discrete symmetry ("matter parities") in order to forbid dangerous operators which may lead otherwise to unacceptable violations of B and L. We give a general classification of such discrete Z, symmetries (and R-symmetries) and show that the number of independent possibilities is substantially reduced by equivalences. We argue that normal discrete symmetries may be expected to be violated by quantum gravity effects and hence are not enough to inhibit nucleon decay. On the other hand, gauge (either discrete or continuous) symmetries are stable under quantum gravity effects and we discuss how such symmetries may eliminate the dangerous B- or L-vioiating operators. We find that the massless fermion content of models with discrete "gauge" symmetries is strongly constrained by the cancellation of "discrete gauge anomalies". We show that there are two preferred Z(N) symmetries which are discrete anomaly free with the minimal light matter content. One of them is the standard R-parity whereas the other is a unique Z3 symmetry allowing for lepton number violation. We argue that from the point of view of arranging for proton stability without fine-tuning the second option should be preferred. The differences in the phenomenology of the various sypersymmetric models dictated by the different symmetries are discussed.