  We perform a global renormalization group study of O(N) symmetric Wess-Zumino
theories and their phases in three euclidean dimensions. At infinite N the
theory is solved exactly. The phases and phase transitions are worked out for
finite and infinite short-distance cutoffs. A distinctive new feature arises at
strong coupling, where the effective superfield potential becomes multi-valued,
signalled by divergences in the fermion-boson interaction. Our findings resolve
the long-standing puzzle about the occurrence of degenerate O(N) symmetric
phases. At finite N, we find a strongly-coupled fixed point in the local
potential approximation and explain its impact on the phase transition. We also
examine the possibility for a supersymmetric Bardeen-Moshe-Bander phenomenon,
and relate our findings with the spontaneous breaking of supersymmetry in other
models.
