  The plethora of increasingly precise experiments which hunt for axion-like
particles (ALPs), as well as their widely different energy reach, call for the
theoretical understanding of ALP couplings at loop-level. We derive the
one-loop contributions to ALP-SM effective couplings, including finite
corrections. The complete leading-order -- dimension five -- effective linear
Lagrangian is considered. The ALP is left off-shell, which is of particular
impact on LHC and accelerator searches of ALP couplings to $\gamma\gamma$,
$ZZ$, $Z\gamma$, $WW$, gluons and fermions. All results are obtained in the
covariant $R_\xi$ gauge. A few phenomenological consequences are also explored
as illustration, with flavour diagonal channels in the case of fermions: in
particular, we explore constraints on the coupling of the ALP to top quarks,
that can be extracted from LHC data, from astrophysical sources and from Dark
Matter direct detection experiments such as PandaX, LUX and XENON1T.
Furthermore, we clarify the relation between alternative ALP bases, the role of
gauge anomalous couplings and their interface with chirality-conserving and
chirality-flip fermion interactions, and we briefly discuss renormalization
group aspects.
