Why Might the Standard Large N Analysis Fail in the O (N) Model: The Role of Cusps in 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= its standard implementation misses some fixed points of the renormalization group in all dimensions smaller than four. These new fixed points show singularities under the form of cusps at N= in their effective potential that become a boundary layer at finite N. We show that they have a physical impact on the multicritical physics of the O(N) model at finite N. We also show that the mechanism at play holds also for the O(N) - O(2) model and is thus probably generic.