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 $\Gamma$, we associate to each vertex a position $x_v \in \mathbb R$ and to each edge $e$ the combination $s_e = a_e^{-\frac 12} \left( x^+_e - x^-_e \right)$, where $x^\pm_e$ are the positions of the two end vertices of $e$ and $a_e$ a Schwinger parameter. The $``$topological propagator$"$ $P_e = e^{-s_e^2}$d$s_e$ includes a part proportional to d$x_v$ and a part proportional to d$a_e$. Integrating the product of all $P_e$ over positions produces a differential form $\alpha_\Gamma$ in the variables $a_e$. We prove that $\alpha_\Gamma \wedge \alpha_\Gamma=0$.
- Publication:
-
arXiv e-prints
- Pub Date:
- August 2024
- DOI:
- 10.48550/arXiv.2408.03192
- arXiv:
- arXiv:2408.03192
- Bibcode:
- 2024arXiv240803192B
- Keywords:
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- Mathematical Physics;
- High Energy Physics - Theory;
- Mathematics - Combinatorics;
- 81T18;
- 81Q30;
- 05C31;
- 18G85
- E-Print:
- 42 pages