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

TL;DR

This paper proves the fundamental Bern-Carrasco-Johansson (BCJ) relations for tree-level amplitudes in N=4 SYM using the connected formulation, simplifying understanding of amplitude identities.

Contribution

It provides a straightforward proof of BCJ relations within the connected formulation, linking different amplitude representations.

Findings

BCJ relations are proven in the connected formulation.

Connected formula is effective for studying amplitude relations.

Simplifies the understanding of amplitude identities in N=4 SYM.

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.