The role of the chemical potential in coupling superfluid dark matter to baryons

Superfluid dark matter postulates that the centers of galaxies contain superfluid condensates. An important quantity regarding these superfluids is their chemical potential $ \mu $. Here, we discuss two issues related to this chemical potential. First, there is no exactly conserved quantity associated with this chemical potential due to the symmetry-breaking baryon-phonon coupling. Second, $ \mu $ is sometimes introduced by shifting the phonon field by $ \mu \cdot t $ which -- again due to the symmetry-breaking baryon-phonon coupling -- introduces an explicit time dependence in the Lagrangian. We investigate under which conditions introducing a chemical potential is nevertheless justified and show how to correctly introduce it when these conditions are met. We further propose a model that recovers superfluid dark matter's zero-temperature equations of motion including a chemical potential even if the aforementioned conditions for justifying a chemical potential are not met.