The shift-invariant orders of an ALP

TL;DR

This paper develops a framework to quantify and identify the breaking of axion shift symmetries through flavor-invariants in Yukawa couplings, providing explicit conditions and analyzing their behavior across energy scales.

Contribution

It introduces flavor-invariant order parameters for axion shift symmetry breaking and applies them to both high-energy and low-energy effective theories, linking UV models to observable effects.

Findings

Explicit conditions for exact axion shift symmetry in Yukawa couplings.

Identification of flavor-invariants as order parameters for symmetry breaking.

Analysis of CP-parities indicating mostly CP-odd shift-breaking invariants.

Abstract

It is generally believed that global symmetries, in particular axion shift symmetries, can only be approximate. This motivates us to quantify the breaking of the shift invariance that characterizes the couplings of an axion-like particle (ALP), and to identify proper order parameters associated to this breaking. Focusing on the flavorful effective Yukawa couplings to Standard Model fermions, we work out explicit conditions for them to maintain an exact axion shift symmetry. Those conditions are given in terms of Jarlskog-like flavor-invariants and can be directly evaluated from the values of the different Yukawa couplings. Therefore, they represent order parameters for the breaking of the axion shift symmetry. We illustrate this construction by matching the axion EFT to UV models, and by showing that the renormalization group running closes on those shift-breaking flavor-invariants, as it should on any complete set of order parameters. Furthermore, the study of the invariants' CP-parities indicate that all but one are CP-odd, hence the assumption of CP conservation suffices to cancel all but one sources of shift-breaking in the theory. We also investigate similar conditions in the low-energy EFT below the electroweak scale, and comment on relations inherited from a UV completion which realizes the electroweak symmetry linearly. Finally, we discuss the order parameter associated to the non-perturbative shift-breaking induced by the axion-gluons coupling, which is also flavorful.