RG limit cycles and unconventional fixed points in perturbative QFT

Abstract

We study quantum field theories with sextic interactions in 3 − 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 fixed points. In an example of large N equivalence, the parent O(N )2 theory and its anti-symmetric projection exhibit identical large N beta functions which 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 CFTs. The symmetric traceless O(N ) model also exhibits very interesting phenomena when it is analytically continued to small non-integer 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 which lead to limit cycles.