  Dark matter (DM) can be composed of a collection of axions, or axion-like
particles (ALPs), whose existence is due to the spontaneous breaking of the
Peccei-Quinn $U(1)$ symmetry which is the most compelling solution of the
strong $CP$-problem of Quantum Chromodynamics (QCD). Axions must be spin-$0$
particles with very small masses and extremely weak interactions with
themselves as well as with the particles that constitute the Standard Model. In
general, the physics of axions is detailed by a quantum field theory of a real
scalar field, $\phi$. Nevertheless, it is more convenient to implement a
non-relativistic effective field theory with a complex scalar field, $\psi$, to
characterize the mentioned axions in the low-energy regime. A possible
application of this equivalent description is to study the collapse of cold
dark matter into more complex structures. There have been a few derivations of
effective Lagrangians for the complex field $\psi$; resulting to be all
equivalent after a nonlocal-space transformation between $\phi$ and $\psi$ was
found, and some other corrections were introduced. Our contribution herein is
to further provide higher order corrections, in particular, we compute the
effective field theory Lagrangian up to order $(\psi^\ast\psi)^5$,
incorporating also the fast-oscillating field fluctuations into the dominant
slowly-varying non-relativistic field.
