Scale without Conformal Invariance: Theoretical Foundations

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.