  Multi-critical point principle (MPP) 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
multi-critical 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
$m^2$ 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
multi-critical points. In this paper, we study the multi-critical points of a
general scalar field $\phi$ 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 $\phi$. 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 still can be
widely applicable to many models of the Coleman-Weinberg mechanism and its
generalizations. After classifying the multi-critical points at low-energy
scales, we will briefly mention the possibility of criticalities at high-energy
scales and their implications for cosmology.
