Breaking of scale invariance in π⁶ theory: Tricriticality and critical end-points

We show how the breaking of scale invariance, recently observed in the large-N limit of π₃⁶ by Bardeen, Moshe and Bander, manifests itself in dimensional regularization. In the process we trace the full phase diagram of the system below d = 3, where tricriticality is always replaced by critical end-points. The appearance of a non-trivial β-function is discussed in the context of perturbation theory, though the effect is non-perturbative. In three dimensions the effective potential is flat in the broken phase, the vacuum energy is thus zero in both phases.