  Coherent oscillation of axions or axion-like 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 non-relativistic effective field theory (EFT). We extend this
analysis to the case with gravity to discuss the longevity of axion stars and
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 small perturbations. Finally, we discuss the
consistency of other non-relativistic effective field theories proposed in the
literature.
