Limit Cycles and Conformal Invariance

TL;DR

The paper demonstrates that four-dimensional unitary quantum field theories can host conformal field theories on RG limit cycles with nonzero beta functions. Using Weyl consistency (Osborn) and the S-function, it shows $S$ vanishes at fixed points and equals the cycle generator $Q$ on cycles, with a third-order perturbative check validating $S=Q$. It proves that, perturbatively, unitarity and scale invariance enforce full conformal invariance in $d=4$, and it extends the strong version of the $c$-theorem to flows that end on or start from cycles via a monotonically decreasing quantity $\widetilde{B}_b$. The work also clarifies the structure of trace anomalies in the presence of spacetime-dependent couplings, introduces cyclic CFTs, analyzes their operator spectra, and discusses implications for the $a$-theorem and RG irreversibility in the presence of cycles.

Abstract

There is a widely held belief that conformal field theories (CFTs) require zero beta functions. Nevertheless, the work of Jack and Osborn implies that the beta functions are not actually the quantites that decide conformality, but until recently no such behavior had been exhibited. Our recent work has led to the discovery of CFTs with nonzero beta functions, more precisely CFTs that live on recurrent trajectories, e.g., limit cycles, of the beta-function vector field. To demonstrate this we study the S function of Jack and Osborn. We use Weyl consistency conditions to show that it vanishes at fixed points and agrees with the generator Q of limit cycles on them. Moreover, we compute S to third order in perturbation theory, and explicitly verify that it agrees with our previous determinations of Q. A byproduct of our analysis is that, in perturbation theory, unitarity and scale invariance imply conformal invariance in four-dimensional quantum field theories. Finally, we study some properties of these new, "cyclic" CFTs, and point out that the a-theorem still governs the asymptotic behavior of renormalization-group flows.

Figures (6)

• Figure 1: The contour of integration for $\int\frac{ds}{s^3}\,\mathcal{A}_\text{fwd}(s)$.
• Figure 2: Diagrams that contribute to $N_{a|ij}$ at one loop and their corresponding counterterms.
• Figure 3: Feynman diagrams that contribute to $N_I$ at two loops.
• Figure 4: Three-loop diagrams that contribute to $N_I$ not symmetric under $a\leftrightarrow b$, and thus leading to contributions to $S$ at three loops.
• Figure :