Note on Cyclic Sum and Combination Sum of Color-ordered Gluon Amplitudes

TL;DR

This paper investigates the boundary behavior of cyclic and combined permutation sums of gluon amplitudes under BCFW shifts, using BCJ and KK relations, advancing understanding of amplitude symmetries.

Contribution

It proves the large-z behavior of cyclic and combined sums of gluon amplitudes under BCFW deformation, employing BCJ and KK relations for the first time in this context.

Findings

Proved boundary behavior of cyclic sums with nonadjacent BCFW shifts.

Simplified proof for cyclic sums with adjacent BCFW shifts using KK relations.

Introduced a new observation for partial-ordered permutation sums.

Abstract

Continuing our previous study \cite{Du:2011se} of permutation sum of color ordered tree amplitudes of gluons, in this note, we prove the large-$z$ behavior of their cyclic sum and the combination of cyclic and permutation sums under BCFW deformation. Unlike the permutation sum, the study of cyclic sum and the combination of cyclic and permutation sums is much more difficult. By using the generalized Bern-Carrasco-Johansson (BCJ) relation, we have proved the boundary behavior of cyclic sum with nonadjacent BCFW deformation. The proof of cyclic sum with adjacent BCFW deformation is a little bit simpler, where only Kleiss-Kuijf (KK) relations are needed. Finally we have presented a new observation for partial-ordered permutation sum and applied it to prove the boundary behavior of combination sum with cyclic and permutation.