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

TL;DR

This paper examines the role and proper implementation of the chemical potential in superfluid dark matter models, especially considering symmetry-breaking effects and their impact on the equations of motion.

Contribution

It clarifies when and how to correctly introduce the chemical potential in superfluid dark matter, proposing a model that accounts for symmetry-breaking effects.

Findings

Chemical potential's validity depends on symmetry considerations.

Proper introduction of chemical potential requires specific conditions.

A new model recovers superfluid dark matter equations with chemical potential even when conditions are not met.

Abstract

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.