Ground-state energy of the spinor Bose-Einstein condensate

It is known that for a weakly interacting Bose-Einstein condensate (BEC), the assumption of a two-body [Formula Presented] interaction described by a constant coupling strength gives rise to a divergent ground-state energy. A similar divergence occurs in the spinor condensate in which the spin-spin interaction is included in addition to the repulsive [Formula Presented] interaction. In this paper, we examine, in the standard Bogoliubov approximation, the ground-state energy of a homogeneous spinor BEC with hyperfine spin [Formula Presented] The renormalized coupling constants are calculated and expressed in terms of the bare ones using the standard second-order perturbation method. With these renormalized coupling constants, we show that the ultraviolet divergence of the ground-state energy can be exactly eliminated.