Abstract
We show that at N=∞ and below its upper critical dimension, d<d_{up}, the critical and tetracritical behaviors of the O(N) models are associated with the same renormalization group fixed point (FP) potential. Only their derivatives make them different with the subtleties that taking their N→∞ limit and deriving them do not commute and that two relevant eigenperturbations show singularities. This invalidates both the ε-and the 1/N-expansions. We also show how the Bardeen-Moshe-Bander line of tetracritical FPs at N=∞ and d=d_{up} can be understood from a finite-N analysis.
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