  Using the Feynman-Dyson diagram technique, we study nonlinear
polariton-polariton scattering in a two-dimensional micropillar-based optical
superlattice with hexagonal symmetry. We demonstrate that both the emerging
polariton chirality and the loop Feynman diagrams up to infinite order should
be strictly accounted for in the evaluation of the self-energy of the system.
Further, we explicitly show that in such a design the time of polariton
scattering towards the Dirac points can be drastically decreased which can be
used, for instance, in engineering novel classes of polariton lasers with
substantially reduced thresholds.
