Asymptotic of Lorentzian Polyhedra Propagator

A certain operator $\T=\int_{\SL}\dd g\, Y^{\dagger}gY$ can be found in various Lorentzian EPRL calculations. The properties of this operator has been studied here in large $j$ limit. The leading order of $\T$ is proportional to the identity operator. Knowing the operator $\T$ one can renormalize spin-foam's edge self-energy by computing the amplitude of sum of a series of edges with increasing number of vertices and bubbles. This amplitude is calculated and is shown to be convergent. Moreover some technical tools useful in Lorentzian Spin-Foam calculation has been developed.