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 = infinity 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 = infinity 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) circle times O(2) model and is thus probably generic.