Towards the graviton from spinfoams: Higher order corrections in the 3D toy model

We consider the recent calculation by Rovelli of the graviton propagator in the spinfoam formalism. Within the 3D toy model introduced by Speziale, we test how the spinfoam formalism can be used to construct the perturbative expansion of graviton amplitudes. Although the 3D graviton is a pure gauge, one can choose to work in a gauge where it is not zero and thus reproduce the structure of the 4D perturbative calculations. We compute explicitly the next-to-leading and next-to-next-to-leading orders, corresponding to one-loop and two-loop corrections. We show that while the first arises entirely from the expansion of the Regge action around the flat background, the latter receives contributions from the microscopic, non-Regge-like, quantum geometry. Surprisingly, this new contribution reduces the magnitude of the next-to-next-to-leading order. It thus appears that the spinfoam formalism is likely to substantially modify the conventional perturbative expansion at higher orders. This result supports the interest in this approach. We then address a number of open issues in the rest of the paper. First, we discuss the boundary state ansatz, which is a key ingredient in the whole construction. We propose a way to enhance the ansatz in order to make the edge lengths and dihedral angles conjugate variables in a mathematically well-defined way. Second, we show that the leading order is stable against different choices of the face weights of the spinfoam model; the next-to-leading order, on the other hand, is changed in a simple way, and we show that the topological face weight minimizes it. Finally, we extend the leading-order result to the case of a regular, but not equilateral, tetrahedron.