  We study a paradigmatic model in field theory where a global $U(1)$ and scale
symmetries are jointly and spontaneously broken. At zero density the model has
a non-compact flat direction, which at finite density needs to be slightly
lifted. The resulting low-energy spectrum is composed by a standard gapless
$U(1)$ Nambu-Goldstone mode and a light dilaton whose gap is determined by the
chemical potential and corrected by the couplings. Even though $U(1)$ and scale
symmetries commute, there is a mixing between the $U(1)$ Nambu-Goldstone and
the dilaton that is crucial to recover the expected dynamics of a conformal
fluid and leads to a phonon propagating at the speed of sound. The results rely
solely on an accurate study of the Ward-Takahashi identities and are checked
against standard fluctuation computations. We extend our results to a boosted
superfluid, and comment the relevance of our findings to condensed matter
applications.
