RG limit cycles and unconventional fixed points in perturbative QFT

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