Shift-Invariant Orders of an Axionlike Particle

It is generally believed that global symmetries, in particular, axion shift symmetries, can only be approximate. This motivates one to quantify the breaking of the shift invariance that characterizes the flavorful effective couplings of an axionlike particle to standard model fermions. We identify a minimal set of Jarlskog-like flavor invariants that vanish if and only if the axion is shift symmetric. Therefore, they represent order parameters for the breaking of the axion shift symmetry. We illustrate properties of the invariants by matching to a UV model, studying the CP transformation of the invariants, calculating their renormalization group evolution, and investigating similar conditions in the low-energy effective field theory below the electroweak scale. Finally, we discuss the order parameter associated to the nonperturbative shift-breaking induced by the axion-gluons coupling, which is also flavorful.