Naturalness via scale invariance and non-trivial UV fixed points in a 4d O(N) scalar field model in the large-N limit

We try to use scale-invariance and the 1/N expansion to construct a non-trivial 4d O(N) scalar field model with controlled UV behavior and naturally light scalar excitations. The principle is to fix interactions at each order in 1/N by requiring the effective action for arbitrary background fields to be scale-invariant. We find a line of non-trivial UV fixed-points in the large-N limit, parameterized by a dimensionless coupling. Nether action nor measure is scale invariant, but the effective action is. Scale invariance makes it natural to set a mass deformation to zero. The model has phases where O(N) invariance is unbroken or spontaneously broken. Masses of the lightest excitations above the unbroken vacuum are found. Slowly varying quantum fluctuations are incorporated at order 1/N. We find the 1/N correction to the potential, beta function of mass and anomalous dimensions of fields that preserve a line of fixed points for constant backgrounds.