Abstract
Superfluid dark matter postulates that the centers of galaxies contain
superfluid condensates. An important quantity regarding these superfluids is
their chemical potential $ $. 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, $ $ is sometimes introduced by shifting the phonon field
by $ 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.
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