The multicritical point principle as the origin of classical conformality and its generalizations

Abstract

Abstract The multicritical point principle is one of the interesting theoretical possibilities that can explain the fine-tuning problems of the universe. It simply claims that “the coupling constants of a theory are tuned to one of the multicritical points, where some of the extrema of the effective potential are degenerate.” One of the simplest examples is the vanishing of the second derivative of the effective potential around a minimum. This corresponds to the so-called classical conformality, because it implies that the renormalized mass m2 vanishes. More generally, the form of the effective potential of a model depends on several coupling constants, and we should sweep them to find all the multicritical points. We study the multicritical points of a general scalar field ϕ at one-loop level under the circumstance that the vacuum expectation values of the other fields are all zero. For simplicity, we also assume that the other fields are either massless or so heavy that they do not contribute to the low-energy effective potential of ϕ. This assumption makes our discussion very simple because the resultant one-loop effective potential is parametrized by only four effective couplings. Although our analysis is not completely general because of the assumption, it can still be widely applicable to many models of the Coleman–Weinberg mechanism and its generalizations. After classifying the multicritical points at low-energy scales, we will briefly mention the possibility of criticalities at high-energy scales and their implications for cosmology.