  We study how the coupling between axion-like particles (ALPs) and matter can
be obtained at the level of on-shell scattering amplitudes. We identify three
conditions that allow us to compute amplitudes that correspond to
shift-symmetric Lagrangians, at the level of operators with dimension 5 or
higher, and we discuss how they relate and extend the Adler's zero condition.
These conditions are necessary to reduce the number of coefficients consistent
with the little-group scaling to the one expected from the Lagrangian approach.
We also show how our formalism easily explains that the dimension-5 interaction
involving one ALP and two massless spin-1 bosons receive corrections from
higher order operators only when the ALP has a non-vanishing mass. As a direct
application of our results, we perform a phenomenological study of the
inelastic scattering $\ell^+\ell^- \to \phi h$ (with $\ell^\pm$ two charged
leptons, $\phi$ the ALP and $h$ the Higgs boson) for which, as a result of the
structure of the 3-point and 4-point amplitudes, dimension-7 operators can
dominate over the dimension-5 ones well before the energy reaches the cutoff of
the theory.
