TITLE: Primitive log-divergent graphs and convergence of Feynman amplitudes
QUESTION [5 upvotes]: To a connected graph $G$, quantum field theory attaches the integral
$$
I_G=\int_{\sigma} \frac{\Omega_G}{\Psi_G^2}
$$ where $N_G$ is the number of edges of the graph, $\sigma$ is the simplex of points $(x_1: \ldots : x_{N_G}) \in \mathbb{P}^{N_G-1}(\mathbb{R})$ such that $x_i \geq 0$ for all $i$, $\Omega_G$ is the volume form
$$
\Omega_G=\sum (-1)^i x_i dx_1 \wedge \cdots \widehat{dx_i} \cdots \wedge x_{N_G}
$$ and $\Psi_G$ is the first Symanzik polynomial, defined by
$$
\Psi_G=\sum_{T \subseteq G} \prod_{e \notin T} x_e, 
$$ where the sum runs over spanning trees. 
The integral is in general divergent, but converges for a class of graphs called primitive log-divergent. These are those for which $N_G=2b_1(G)$ (first Betti number) and $N_\gamma>2b_1(\gamma)$ for any proper subgraph $\gamma$. I have found this statement in several places but no proof. 
Does anybody know how to prove the convergence of $I_G$ in the primitive log-divergent case?
Is it an if and only if?
Any intuition about what the condition means?

REPLY [4 votes]: A convergence proof ("if and only if") follows proposition 5.2 of On Motives Associated to Graph Polynomials (2005).