A fixed point can hide another one: the nonperturbative behavior of the tetracritical fixed point of the O($N$) models at large $N$

We show that at $N=\mathrm{\infty }$ and below its upper critical dimension, $d<d_{\mathrm{u}\mathrm{p}}$, 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 $N\to \mathrm{\infty }$ limit and deriving them do not commute and that two relevant eigenperturbations show singularities. This invalidates both the $\epsilon -$ and the $1\mathrm{/}N-$ expansions. We also show how the Bardeen-Moshe-Bander line of tetracritical FPs at $N=\mathrm{\infty }$ and $d=d_{\mathrm{u}\mathrm{p}}$ can be understood from a finite-$N$ analysis.