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Nonrelativistic Effective Field Theory for Axions

Axions can be described by a relativistic field theory with a real scalar field $ϕ$ whose self-interaction potential is a periodic function of $ϕ$. Low-energy axions, such as those produced in the early universe by the vacuum misalignment mechanism, can be described more simply by a nonrelativistic effective field theory with a complex scalar field $ψ$ whose effective potential is a function of $ψ^*ψ$. We determine the coefficients in the expansion of the effective potential to fifth order in $ψ^*ψ$ by matching low-energy axion scattering amplitudes. In order to describe a Bose-Einstein condensate of axions that is too dense to expand the effective potential in powers of $ψ^*ψ$, we develop a sequence of systematically improvable approximations to the effective potential that include terms of all orders in $ψ^*ψ$.