Classical nonrelativistic effective field theory and the role of gravitational interactions

Coherent oscillation of axions or axionlike particles may give rise to long-lived clumps, called axion stars, because of the attractive gravitational force or its self-interaction. Such a kind of configuration has been extensively studied in the context of oscillons without the effect of gravity, and its stability can be understood by an approximate conservation of particle number in a nonrelativistic effective field theory (EFT). We extend this analysis to the case with gravity to clarify the EFT expansion scheme in terms of gradient energy and Newton's constant. Our EFT is useful to calculate the axion star configuration and its classical lifetime without any ad hoc assumption. In addition, we derive a simple stability condition against perturbations in the case of self-gravitating objects. Finally, we discuss the consistency of other nonrelativistic effective field theories proposed in the literature.