Note on the non-adjacent BCFW deformations

TL;DR

This paper proves the consistency and bonus relations for non-adjacent BCFW deformations using both field theory and S matrix analysis, enhancing understanding of amplitude relations in gauge theories.

Contribution

It provides new proofs of the consistency and bonus relations for non-adjacent BCFW deformations, including an alternative S matrix approach in $ ext{N}=4$ SYM.

Findings

Proof of the consistency condition via Kleiss-Kuijf relation

Proof of the bonus relation via BCJ relation

Alternative S matrix proof of bonus relation in $ ext{N}=4$ SYM

Abstract

As the first part, we provide a proof of the consistency condition by Kleiss-Kuijf relation and the bonus relation by BCJ relation for the non-adjacent BCFW deformations. On the other hand, rather than appealing to field theory argument, we provide an alternative proof of bonus relation for the non-adjacent BCFW deformations by a purely S matrix analysis in the context of $\mathcal{N}=4$ SYM theory.