  We study the system of multi-body interacting bosons on a two dimensional
optical lattice and analyze the formation of bound bosonic pairs in the context
of the Bose-Hubbard model. Assuming a repulsive two-body interaction we obtain
the signatures of pair formation in the regions between the Mott insulator
lobes of the phase diagram for different choices of higher order local
interactions. Considering the most general Bose-Hubbard model involving local
multi-body interactions we investigate the ground state properties utilizing
the cluster mean-field theory approach and further confirm the results by means
of sophisticated infinite Projected Entangled Pair States calculations. By
using various order parameters, we show that the choice of higher-order
interaction can lead to pair superfluid phase in the system between two
different Mott lobes. We also analyze the effect of temperature and
density-dependent tunneling to establish the stability of the PSF phase.
