RG limit cycles and unconventional fixed points in perturbative QFT

Abstract

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 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 ${g}^{i}$, but two eigenvalues of the Jacobian matrix $\ensuremath{\partial}{\ensuremath{\beta}}^{i}/\ensuremath{\partial}{g}^{j}$ 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 ${N}_{\text{crit}}\ensuremath{\approx}4.475$, and for a small range of $N$ above this value we find RG flows that lead to limit cycles.

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