Kerr black holes with synchronised scalar hair and boson stars in the Einstein-Friedberg-Lee-Sirlin model

We study a new family of self-gravitating axially-symmetric, rotating boson stars in the two-component Friedberg-Lee-Sirlin model minimally coupled to Einstein gravity in four spacetime dimensions. In the flat space limit the boson stars tend to the corresponding Q-balls. Subject to the usual synchronization condition, the model admits spinning hairy black hole solutions with two different types of scalar hair. We here investigate parity-even and parity-odd boson stars and their associated hairy black holes. We explore the domain of existence of the solutions and address some of their physical properties. The solutions exhibit close similarity to the corresponding boson stars and Kerr black holes with synchronised scalar hair in the $O(3)$-sigma model coupled to Einstein gravity and to the corresponding solutions in the Einstein-Klein-Gordon theory with a complex scalar field, where the latter are recovered in a limit.