Large field excursions and approximate discrete symmetries from a clockwork axion

We present a renormalizable theory of scalars in which the low-energy effective theory contains a pseudo-Goldstone boson with a compact field space of 2 pi F and an approximate discrete shift symmetry Z(Q) with Q >> 1, yet the number of fields in the theory goes as log Q. Such a model can serve as a UV completion to models of relaxions and is a new source of exponential scale separation in field theory. While the model is local in "theory space," it appears not to have a continuum generalization (i.e., it cannot be a deconstructed extra dimension). Our framework shows that super-Planckian field excursions can be mimicked while sticking to renormalizable four-dimensional quantum field theory. We show that a supersymmetric extension is straightforwardly obtained, and we illustrate possible UV completions based on a compact extra dimension, where all global symmetries arise accidentally as a consequence of gauge invariance and five-dimensional locality.