Complete LQG propagator. II. Asymptotic behavior of the vertex

In a previous article we have shown that there are difficulties in obtaining the correct graviton propagator from the loop-quantum-gravity dynamics defined by the Barrett-Crane vertex amplitude. Here we show that a vertex amplitude that depends nontrivially on the intertwiners can yield the correct propagator. We give an explicit example of asymptotic behavior of a vertex amplitude that gives the correct full graviton propagator in the large distance limit.

Figure 1

The five classes of pairings: from the upper left: (10), $(5,5)$, $(7,3)$, $(6,4)$, and $(4,3,3)$.