Naturalness and a line of fixed points in a 4 d O ( N ) scalar field model in the large - N limit

We try to use scale-invariance and the large-N limit to construct a non-trivial 4d O(N) scalar field model with good UV behavior and naturally light scalar excitations. The physical principle is to fix the interactions (not assumed polynomial) at each order in 1/N by requiring the effective action for arbitrary background fields to be scale-invariant, after including quantum effects. We find a line of non-trivial fixed-points in the large N limit, parameterized by a dimensionless coupling. Since part of the interaction potential is canceled by quantum effects, it is not of direct physical interest. Nevertheless, it grows logarithmically slower than a quartic potential for large fields and N . The finite and scale-free effective action for arbitrary backgrounds is obtained in an expansion around constant backgrounds. A relevant mass deformation is considered. The line of fixed points makes it natural to set the mass to zero. Doing so leads to scaling symmetry. The model has phases where O(N) invariance is either unbroken or spontaneously broken by the scalar vev. 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 ensure cancelation of divergences and maintenance of a line of fixed points for constant backgrounds. PACS: 11.10.Gh, 11.10.Hi, 11.15.Pg, 11.30.Qc, 14.80.Cp, 05.10.Cc.