Because the axion is a pseudo-scalar Boson, as the interaction between axions and matter, spin-dependent interactions have been mainly discussed up to the present.
Spin-independent terms, which break the axion shift symmetry, are considered to be small, being suppressed by the axion mass.
However, we have found that couplings of 4-point interactions between nucleons and axions are not negligibly small.
The term is of particular importance for the axion dark matter, which has a long Compton length, because the scattering amplitude is enhanced by the coherent effect.
We discuss the consequences and the possible application of the interactions.