Limit cycles and conformal invariance

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.

Comments on scale and conformal invariance

“…While in d = 2 spacetime dimensions scale invariance together with unitarity implies conformal invariance [1,2], 1 the problem has remained open in higher dimensions. Evidence that the conjecture may hold in four dimensions was presented in [4] while candidate counterexamples [5][6][7] were shown to actually be CFTs in [8,9]. As in the two-dimensional case, there is a connection between RG-flow and properties of the Wess-Zumino action [10,11] (see also [12,13]), the a-theorem, [14,15], and the scale vs conformal invariance problem.…”

A functional RG equation for the c-function

“…6 The c-function and Newton's constant 24 6.1 Relation between c k and β G k 24 6.2 Minimally coupled scalar 24 6.3 Self-interacting scalar 26 …”

The local RG equation and chiral anomalies

“…This feature corresponds to a scheme dependent artifact, in which the RG flow consists of a flavor rotation induced by wavefunction renormalization. An extreme example of this artifact was demonstrated in [8], where conformal theories with RG induced flavor rotations, or cyclic flows, were found.…”