Combinatorial proof of a Non-Renormalization Theorem

We provide a direct combinatorial proof of a Feynman graph identity which implies a wide generalization of a formality theorem by Kontsevich. For a Feynman graph $\mathrm{\Gamma }$, we associate to each vertex a position $x_v\in \mathbb{R}$ and to each edge $e$ the combination $s_e=a_e^{-\frac{1}{2}}(x_e^{+}-x_e^{-})$, where $x_e^{\pm }$ are the positions of the two end vertices of $e$, and $a_e$ is a Schwinger parameter. The “topological propagator” $P_e=e^{-s_e^2}\text{d}{s}_e$ includes a part proportional to $\text{d}{x}_v$ and a part proportional to $\text{d}{a}_e$. Integrating the product of all $P_e$ over positions produces a differential form ${\alpha }_{\mathrm{\Gamma }}$ in the variables $a_e$. We derive an explicit combinatorial formula for ${\alpha }_{\mathrm{\Gamma }}$, and we prove that ${\alpha }_{\mathrm{\Gamma }}\wedge {\alpha }_{\mathrm{\Gamma }}=0$.