8 cluding both the repulsive and attractive interspecies in- teraction. We used the many-body version of Rayleigh- Schr¨ odinger perturbation theory in the kinetic energy term with respect to the ground state of the system when the ki- netic energy term is absent, and calculate ground state ener- gies needed to carry out our analysis. This technique was previously used to discuss the phase diagram of the single- species BH model [19–21, 23], extended BH model [24], and of the hardcore BH model with a superlattice [25], and its results showed an excellent agreement with Monte Carlo sim- ulations [23, 25]. Motivated by the success of this technique with these models, here we generalized it to the two-species BH model, hoping to develop an analytical approach which could be as accurate as the numerical ones. sions. Since the available Monte Carlo calculations [9, 10] do not provide the Mott insulator to superfluid transition phas e boundary in the experimentally more relevant chemical po- tential versus hopping plane, we could not compare our results with them. This comparison is highly desirable to judge the accuracy of our strong-coupling expansion results. A possible direction to extend this work is to consider the limit where hopping of one-species is much larger than the other. In this limit, the two-species BH model reduces to the Bose-Bose version of the Falicov-Kimball model [28], the Fermi-Fermi version of which has been widely discussed in the condensed-matter literature and the Fermi-Bose version has just been studied [29]. It is known for such models that there is a tendency towards both phase separation and density wave order [30], which requires a new calculation partially similar to that of Ref. [24]. One can also examine how the momentum distribution changes with the hopping in the insu- lating phases [23, 31], which has direct relevance to ultracold atomic experiments. We derived analytical expressions for the phase boundary between the incompressible Mott insulator and the compress- ible superfluid phase up to third order in the hoppings. We als o proposed a chemical potential extrapolation technique based on the scaling theory to extrapolate our third-order power se- ries expansion into a functional form that is appropriate for the Mott lobes. In particular, when the interspecies interaction is sufficiently large and attractive, we found a re-entrant qua n- tum phase transition from paired superfluid (superfluidity o f composite bosons, i.e. Bose-Bose pairs) to Mott insulator and again to a paired superfluid in all one, two and three dimen- The author thanks Anzi Hu, L. Mathey and J. K. 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