Diagrammatic proof of the BCFW recursion relation for gluon amplitudes in QCD

TL;DR

This paper provides a diagrammatic proof of the tree-level BCFW recursion for gluon amplitudes in QCD by linking BCFW decompositions to Feynman diagrams in a carefully chosen gauge. It develops key kinematical identities for shifted momenta, classifies diagrams by the number of main-line propagators, and shows that sums over all cuts of hatted diagrams reproduce the full color-ordered amplitude through cancellations dictated by Yang-Mills vertex structure and gauge invariance. The result clarifies why BCFW recursions reorganize rather than generate new physics, and highlights the essential role of YM vertices—a feature absent in scalar theories—while providing a framework for understanding cancellations at the diagram level. Overall, the work gives a transparent, gauge-friendly diagrammatic underpinning for BCFW, with implications for generality and potential extensions.

Abstract

We present a proof of the Britto-Cachazo-Feng-Witten tree-level recursion relation for gluon amplitudes in QCD, based on a direct equivalence between BCFW decompositions and Feynman diagrams. We demonstrate that this equivalence can be made explicit when working in a convenient gauge. We exhibit that gauge invariance and the particular structure of Yang-Mills vertices guarantees the validity of the BCFW construction.