Combinatorial proof of a non-renormalization theorem

Abstract

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 x_v ∈ ℝ and to each edge e the combination \( {s}_e={a}_e^{-\frac{1}{2}}\left({x}_e^{+}-{x}_e^{-}\right) \), where \( {x}_e^{\pm } \) are the positions of the two end vertices of e, and a_e is a Schwinger parameter. The “topological propagator” \( {P}_e={e}^{-{s}_e^2}{\textrm{d}s}_e \) includes a part proportional to dx_v and a part proportional to da_e. Integrating the product of all P_e over positions produces a differential form α_Γ in the variables a_e. We derive an explicit combinatorial formula for α_Γ, and we prove that α_Γ ∧ α_Γ = 0 for all graphs except for trees.