Fundamental BCJ Relation in N=4 SYM From The Connected Formulation

TL;DR

The paper shows that the fundamental BCJ relation in ${\cal N}=4$ SYM can be proven directly within the RSVW connected formulation, exploiting its permutation invariance and the MHV-like factor that encodes color ordering. By extracting soft-like factors and applying the eikonal identity, the authors demonstrate that the BCJ sum vanishes on the delta-function support, and that every RSVW residue individually satisfies KK and BCJ. This residue-level validity implies the full connected amplitude obeys these amplitude relations, highlighting the connected formulation as a natural framework for manifesting BCJ/KK structures and offering a bridge to gravitational relations via KLT. The work emphasizes that each solution in the RSVW construction carries the same global amplitude identities, reinforcing the deep connection between permutation invariance and color-kinematics duality.

Abstract

Tree-level amplitudes in N=4 SYM can be decomposed into partial or color-ordered amplitudes. Identities relating various partial amplitudes have been known since the 80's. They are Kleiss-Kuijf (KK) identities. In 2008, Bern, Carrasco and Johansson (BCJ) introduced a new set of identities which reduce the number of independent partial amplitudes to (n-3)!. In recent years, several formulations for partial amplitudes have been discovered and shown to be equivalent to each other. These can be thought of as simple dualities in the sense that different formulations make manifest different properties of the same object; the amplitude. One such formulation is the ACCK Grassmannian formulation which makes Yangian invariance of individual partial amplitudes manifest. A different formulation is the so-called connected formula introduced by Witten in twistor space and formulated in momentum space by Roiban, Spradlin and Volovich. It has been argued that the connected formula is ideal for studying properties which are related to the full amplitude, such as the KK relations, and not to particular partial amplitudes, like Yangian invariance. Following this logic, it is very natural to expect that the BCJ identities should have a very simple proof in the connected formulation. In this short note we show that this is indeed the case.