I will discuss general results on the UV and IR asymptotics
of 4D unitary quantum field theory. The main tool is a generalization
of the Komargodski-Schwimmer proof for the $a$-theorem. I will 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. I will 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.
