Possible large-N fixed points and naturalness for O(N) scalar fields

We try to use scale invariance and the large-N limit to find a non-trivial 4D O(N) scalar field model with controlled UV behavior and naturally light scalar excitations. The principle is to fix interactions by requiring the effective action for spacetime dependent background fields to be finite and scale invariant when regulators are removed. We find a line of non-trivial UV fixed points in the large-N limit, parameterized by a dimensionless coupling. They reduce to classical λϕ⁴ theory when ℏ → 0. For ℏ ≠ 0, neither 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. We derive a nonlinear equation for oscillations about the broken vacuum. The interaction potential is shown to have a locality property at large N. In 3D, our construction reduces to the line of large-N fixed points in |ϕ|⁶ theory.