RG limit cycles and unconventional fixed points in perturbative QFT

Abstract

We study quantum field theories with sextic interactions in 3-ϵ dimensions, where the scalar fields φab form irreducible representations under the O(N)2 or O(N) global symmetry group. We calculate the beta functions up to four-loop order and find the renormalization group (RG) fixed points. In an example of large N equivalence, the parent O(N)2 theory and its antisymmetric projection exhibit identical large N beta functions that possess real fixed points. However, for projection to the symmetric traceless representation of O(N), the large N equivalence is violated by the appearance of an additional double-trace operator not inherited from the parent theory. Among the large N fixed points of this daughter theory we find complex conformal field theories. The symmetric traceless O(N) model also exhibits very interesting phenomena when it is analytically continued to small noninteger values of N. Here we find unconventional fixed points, which we call "spooky."They are located at real values of the coupling constants gi, but two eigenvalues of the Jacobian matrix ∂βi/∂gj are complex. When these complex conjugate eigenvalues cross the imaginary axis, a Hopf bifurcation occurs, giving rise to RG limit cycles. This crossing occurs for Ncrit≈4.475, and for a small range of N above this value we find RG flows that lead to limit cycles.