Boson stars in $f(T)$ extended theory of gravity

Spherically symmetric configurations of the non-interacting massive complex scalar field, representing non-rotating boson stars, are considered within the framework of the modified torsion based $f(T)$ gravity, with $f(T) = T + \alpha \, T^2/2$. We find that with sufficiently large negative value of $\alpha$ the mass of the boson stars can be made arbitrarily large. This is in contrast to General Relativity where an upper bound, $M_{max} \sim M_{Planck}^2/m$, to the mass of the boson stars built from the non-interacting scalar field exists and where the masses of boson stars in the astrophysical regime can be obtained only with the introduction of the scalar field self-interaction. With sufficiently large negative $\alpha$ we also find negative gravitational binding energy for all masses, which can be seen as an indication of the stability of such configurations. In its positive regime, $\alpha$ can not be made arbitrarily large as a phase transition in the stress--energy components of the $f(T)$-fluid develops. This phenomenon has already been reported to occur in polytropic stars constructed within the $f(T)$ gravity theory.