Multi-graviton theory, a latticized dimension and the cosmological constant

Beginning with the Pauli–Fierz theory, we construct a model for multi-graviton theory. Couplings between gravitons belonging to nearest-neighbour ʹtheory spacesʹ lead to a discrete mass spectrum. Our model coincides with the Kaluza–Klein theory whose fifth dimension is latticized. We evaluate one-loop vacuum energy in models with a circular latticized extra dimension as well as with compact continuous dimensions. We find that the vacuum energy can take a positive value, if the dimension of the continuous spacetime is 6, 10, .... Moreover, since the amount of vacuum energy can be an arbitrary small value depending on the choice of parameters in the model, our models are useful for explaining the small positive dark energy in the present universe.