Limit Cycles and Conformal Invariance

TL;DR

This paper demonstrates the existence of conformal field theories with nonzero beta functions that exhibit limit cycles, challenging the traditional view that zero beta functions are necessary for conformality, and explores their properties and implications.

Contribution

It provides the first explicit examples of cyclic CFTs with nonzero beta functions and analyzes their properties using perturbation theory and Weyl consistency conditions.

Findings

CFTs can have nonzero beta functions on limit cycles.

S function vanishes at fixed points and matches the generator Q on cycles.

Unitarity and scale invariance imply conformal invariance in 4D perturbation theory.

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.