Although bosons in the same state are indistinguishable, it is possible to construct non-trivial configurations with bosons in different excited states. A system of bosons in different states that only interact with each other gravitationally can be described by the following Lagrangian density.
In the simplest case of a multi-state boson, one has the ground state and the first excited state. Such configurations are stable if the number of particles in the ground state is larger than the number of particles in the excited state [25, 7]
Similar results were found in the Newtonian limit , however, with a slightly higher stability limit . This work stresses that combining several excited states makes it possible to obtain flatter rotation curves than only with ground state, producing better models for galactic dark matter halos (see also discussion of boson stars as an explanation of dark matter in Section 5.3).
Ref.  considers two scalar fields describing boson stars that are phase shifted in time with respect to each other, studying the dynamics numerically. In particular, one can consider multiple scalar fields with an explicit interaction (beyond just gravity) between them, say . Refs. [37, 38] construct such solutions, considering the individual particle-like configurations for each complex field as interacting with each other.
Living Rev. Relativity 15, (2012), 6
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