RG Limit Cycles and Unconventional Fixed Points in Perturbative QFT

We study quantum field theories with sextic interactions in $3-\epsilon$ dimensions, where the scalar fields $\phi^{ab}$ 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 $g^i$, but two eigenvalues of the Jacobian matrix $\partial \beta^i/\partial g^j$ 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 $N_{\rm crit} \approx 4.475$, and for a small range of $N$ above this value we find RG flows which lead to limit cycles.