  We revisit certain subtleties of renormalization that arise when one derives
a low-energy effective action by integrating out the heavy fields of a more
complete theory. Usually these subtleties are circumvented by matching some
physical observables, such as scattering amplitudes, but a more involved
procedure is required if one is interested in deriving the effective theory to
all orders in the light fields (but still to fixed order in the derivative
expansion). As a concrete example, we study the $U(1)$ Goldstone low-energy
effective theory that describes the spontaneously broken phase of a $\phi^4$
theory for a complex scalar. Working to lowest order in the derivative
expansion, but to all orders in the Goldstones, we integrate out the radial
mode at one loop and express the low-energy effective action in terms of the
renormalized couplings of the UV completion. This yields the one-loop equation
of state for the superfluid phase of (complex) $\phi^4$. We perform the same
analysis for a renormalizable scalar $SO(N)$ theory at finite chemical
potential, integrating out the gapped Goldstones as well, and confirm that the
effective theory for the gapless Goldstone exhibits no obvious sign of the
original $SO(N)$ symmetry.
