Higher Order Corrections to the Effective Field Theory of Low-energy Axions

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.