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 Γ, we associate to each vertex a position xv ∈ ℝ and to each edge e the combination se=ae−12xe+−xe−$$ {s}_e={a}_e^{-\frac{1}{2}}\left({x}_e^{+}-{x}_e^{-}\right) $$, where xe±$$ {x}_e^{\pm } $$ are the positions of the two end vertices of e, and ae is a Schwinger parameter. The “topological propagator” Pe=e−se2dse$$ {P}_e={e}^{-{s}_e^2}{\textrm{d}s}_e $$ includes a part proportional to dxv and a part proportional to dae. Integrating the product of all Pe over positions produces a differential form αΓ in the variables ae. We derive an explicit combinatorial formula for αΓ, and we prove that αΓ ∧ αΓ = 0 for all graphs except for trees.