Why Might the Standard Large $N$ Analysis Fail in the O($N$) Model: The Role of Cusps in the Fixed Point Potentials

The large $N$ expansion plays a fundamental role in quantum and statistical field theory. We show on the example of the O$(N)$ model that at $N=\mathrm{\infty }$, its traditional implementation misses in all dimensions below four some fixed points of the renormalization group. These new fixed points show singularities at $N=\mathrm{\infty }$ in their effective potential that become a boundary layer at finite $N$. We show that they have a physical impact on the multicritcal physics of the $O(N$) model at finite $N$. We also show that the mechanism at play holds also for the O($N$)$\otimes$O(2) model and is thus probably generic.