  We present a relativistic effective field theory for the interaction between
acoustic and gapped phonons in the limit of a small gap. We show that, while
the former are the Goldstone modes associated with the spontaneous breaking of
spacetime symmetries, the latter are pseudo-Goldstones associated with some
(small) explicit breaking. We hence dub them "pseudo-acoustic" phonons. In this
first investigation, we build our effective theory for the cases of one and two
spatial dimensions, two atomic species, and assuming large distance isotropy.
As an illustrative example, we show how the theory can be applied to compute
the total lifetime of both acoustic and pseudo-acoustic phonons. This
construction can find applications that range from the physics of bilayer
graphene to sub-GeV dark matter detectors.
