Proof of the fundamental BCJ relations for QCD amplitudes

TL;DR

This paper proves the conjectured fundamental BCJ relations for QCD amplitudes, extending known results from pure gluonic and supersymmetric theories to include massive and massless quarks.

Contribution

It provides a rigorous proof that the fundamental BCJ relations hold for QCD amplitudes, including cases with massive quarks.

Findings

BCJ relations hold for massless and massive quarks in QCD

The proof extends the validity of BCJ relations beyond pure gluonic cases

Supports the universality of BCJ relations in gauge theories

Abstract

The fundamental BCJ-relation is a linear relation between primitive tree amplitudes with different cyclic orderings. The cyclic orderings differ by the insertion place of one gluon. The coefficients of the fundamental BCJ-relation are linear in the Lorentz invariants $2 p_i p_j$. The BCJ-relations are well established for pure gluonic amplitudes as well as for amplitudes in ${\mathcal N}=4$ super-Yang-Mills theory. Recently, it has been conjectured that the BCJ-relations hold also for QCD amplitudes. In this paper we give a proof of this conjecture. The proof is valid for massless and massive quarks.