  In this work we study a Spin Foam model for 4d Riemannian gravity, and
propose a new way of imposing the simplicity constraints that uses the recently
developed holomorphic representation. Using the power of the holomorphic
integration techniques, and with the introduction of two new tools: the
homogeneity map and the loop identity, for the first time we give the analytic
expressions for the behaviour of the Spin Foam amplitudes under 4-dimensional
Pachner moves. It turns out that this behaviour is controlled by an insertion
of nonlocal mixing operators. In the case of the 5-1 move, the expression
governing the change of the amplitude can be interpreted as a vertex
renormalisation equation. We find a natural truncation scheme that allows us to
get an invariance up to an overall factor for the 4-2 and 5-1 moves, but not
for the 3-3 move. The study of the divergences shows that there is a range of
parameter space for which the 4-2 move is finite while the 5-1 move diverges.
This opens up the possibility to recover diffeomorphism invariance in the
continuum limit of Spin Foam models for 4D Quantum Gravity.
