Large N limit of irreducible tensor models: O(N) rank-3 tensors with mixed permutation symmetry

It has recently been proven that in rank three tensor models, the antisymmetric and symmetric traceless sectors both support a large N expansion dominated by melon diagrams [1]. We show how to extend these results to the last irreducible O(N) tensor representation available in this context, which carries a two-dimensional representation of the symmetric group S3. Along the way, we emphasize the role of the irreducibility condition: it prevents the generation of vector modes which are not compatible with the large N scaling of the tensor interaction. This example supports the conjecture that a melonic large N limit should exist more generally for higher rank tensor models, provided that they are appropriately restricted to an irreducible subspace.