  We study the possible IR and UV asymptotics of 4D Lorentz invariant unitary
quantum field theory. Our main tool is a generalization of the
Komargodski-Schwimmer proof for the $a$-theorem. We use this to rule out a
large class of renormalization group flows that do not asymptote to conformal
field theories in the UV and IR. We show that if the IR (UV) asymptotics is
described by perturbation theory, all beta functions must vanish faster than
$(1/|\ln\mu|)^{1/2}$ as $\mu \to 0$ ($\mu \to \infty$). This implies that the
only possible asymptotics within perturbation theory is conformal field theory.
In particular, it rules out perturbative theories with scale but not conformal
invariance, which are equivalent to theories with renormalization group
pseudocycles. Our arguments hold even for theories with gravitational
anomalies. We also give a non-perturbative argument that excludes theories with
scale but not conformal invariance. This argument holds for theories in which
the stress-energy tensor is sufficiently nontrivial in a technical sense that
we make precise.
