  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.
