  Axions and axion-like particles (ALPs) are ubiquitous in popular attempts to
solve supercalifragilisticexpialidocious puzzles of Nature. A widespread and
vivid experimental programme spanning a vast range of mass scales and decades
of couplings strives to find evidence for these elusive but theoretically
well-motivated particles. In the absence of clear guiding principle, effective
field theories (EFTs) prove to be an efficient tool in this experimental quest.
Hilbert series technologies are a privileged instrument of the EFT toolbox to
enumerate and classify operators. In this work, we compute explicitly the
Hilbert series capturing the interactions of a generic ALP to the Standard
Model particles above and below the electroweak symmetry scale, which allow us
to build bases of operators up to dimension 8. In particular, we revealed a
remarkable structure of the Hilbert series that isolates the shift-symmetry
breaking and preserving interactions. In addition, with the Hilbert series
method, we enumerate the sources of CP violation in terms of CP-even, CP-odd
and CP-violating operators. Furthermore, we provide an ancillary file of the
Hilbert series up to dimension 15 to supplement our findings, which can be used
for further analysis and exploration.
