Bulk amplitude and degree of divergence in 4d spin foams

TL;DR

This paper analyzes the divergence properties of 4d spin foam models, deriving formulas for bulk amplitudes based on graph topology, and identifies melonic graphs as the most divergent in physically relevant regimes.

Contribution

It introduces a new formula for bulk amplitudes of 4d spin foams based on graph properties and extends divergence analysis to arbitrary connected 2-complexes.

Findings

Derived a simple expression for divergence degree using combinatorial and topological invariants.

Identified melonic graphs as the most divergent configurations in the physical regime.

Revealed phase transitions indicated by different behaviors across parameter space.

Abstract

We study the 4-d holomorphic Spin Foam amplitude on arbitrary connected 2-complexes and degrees of divergence. With recently developed tools and truncation scheme, we derive a formula for a certain class of graphs, which allows us to write down the value of bulk amplitudes simply based on graph properties. We then generalize the result to arbitrary connected 2-complexes and extract a simple expression for the degree of divergence only in terms of combinatorial properties and topological invariants. The distinct behaviors of the model in different regions of parameter space signal phase transitions. In the regime which is of physical interest for recovering diffeomorphsim symmetry in the continuum limit, the most divergent configurations are melonic graphs. We end with a discussion of physical implications.