  We present the theoretical underpinnings of scale without conformal
invariance in quantum field theory. In light of our results the gradient-flow
interpretation of renormalization-group (RG) flow is challenged, due to deep
connections between scale-invariant theories and recurrent behaviors in the RG.
We show that, on scale-invariant trajectories, there is a redefinition of the
dilatation current that leads to generators of dilatations that generate
dilatations. Finally, we develop a systematic algorithm for the search of
scale-invariant trajectories in perturbation theory.
