Possible large-Nfixed points and naturalness forO(N) scalar fields

Abstract

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 space-time 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 λφ 4 theory whenh → 0. Forh = 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 non-linear 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.We investigate the naturalness concept of 't Hooft [1] applied to 4-dimensional O(N ) scalar fields. By this dogma, if there is a scalar particle very light compared to the microscopic scale at which the model is superseded, it must be for a good reason e.g., a symmetry. We observe that a non-trivial fixed-point in scalar field theory would be enough to make small masses natural. For, setting masses to zero would restore symmetry under rescaling. We try to realize this scenario by developing an idea of Rajeev [2] to find a fixed-point in the large-N limit. Background and motivationsMany discussions of 4d QFT begin with massive λφ 4 theory, but it most likely does not have a non-trivial continuum limit [3,4,5,6]. So we wish to know if there is a non-trivial 4d scalar field theory. Another motivation concerns the UV and naturalness problems in the scalar sector of the standard model of particle physics. The importance of QFTs with controlled UV behavior is well-known: Yang-Mills theories, with a gaussian UV fixed-point provide our best models for strong and weak interactions. In equilibrium statistical mechanics of magnets, the gaussian fixed-point (GFP) controls high energy behavior while the lower energy dynamics is governed by a crossover to the non-trivial Wilson-Fisher fixed-point (WFP) [7]. However, the situation in 4d massive λφ 4 theory, the simplest (but unconfirmed) model for W ± ,Z masses, is less satisfactory. λφ 4 is based on the gaussian IR fixed-point, but doesn't flow to any fixed-point in the UV. Perturbatively, interactions become strong (Landau pole) at energies of O(m exp [16π 2 /3λ]), where m, λ are the parameters of the model in the IR. This is in contrast with asymptotically free theories or theories based on an interacting UV fixed-point which might (in principle) be valid up to a higher energy. Numerical [3,4] and analytic [6,5] calculations suggest that without a UV cutoff, λφ 4 theory is 'trivial' 1 . Unfortunately, the non-trivial WFP in φ 4 theory in 4-ǫ dime...