Abstract
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.
Highlights
Dimensions, in a regime where perturbation theory is applicable
We have shown that the Komargodski-Schwimmer proof of the weak version of the ctheorem includes the more general case that a renormalization group flow goes from a fixed point or cycle to another fixed point or cycle
Regarding the strong version of the c-theorem, proven in perturbation theory by Jack and Osborn, we pointed out that the quantity that plays the role of c is Bb (defined in (3.19)) which is closely related to the a-anomaly; these quantities agree at fixed points and on cycles, but are not generally the same
Summary
We review the derivation of the Weyl consistency conditions of JO. The method uses as a starting point the expressions of Weyl invariance used by KS and by LPR. It is from counterterms that do not vanish for spacetime-independent coupling constants that the βa,b,c-anomalies arise It is convenient, in order to keep track of curvature-dependent terms, to do this in a more general background metric,. This is the well-known trace anomaly, accounting for the effects of curved background and spacetime-dependent coupling constants This equation is not quite correct in the most generality: there are two terms missing on the right-hand side. When the kinetic terms of the Lagrangian exhibit a continuous symmetry the current associated with this symmetry is a dimension-three operator and a new type of counterterm is required in the presence of spacetime-dependent couplings, that is, a counterterm proportional to the product of the current and the derivative of a coupling The remaining terms in (2.6) can be found
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