Proof of the fundamental BCJ relations for QCD amplitudes

TL;DR

This work proves the fundamental BCJ relations for primitive tree amplitudes in QCD, extending their validity from pure gluon and ${\mathcal N}=4$ SYM cases to configurations with massless and massive quarks. The authors employ a three-particle BCFW shift for external legs 1, 2$_{g}$, and $n$, define a deformation-dependent function $I_n(z)$, and analyze its large-$z$ behavior to exclude infinity contributions. The proof proceeds by induction on the number of external legs, using contour integration to relate $I_n(0)$ to lower-point BCJ relations and momentum-conservation identities, with detailed treatment of massive spinors, light-like decompositions, and reference-spinor choices. The result significantly broadens the applicability of BCJ relations in QCD, enabling systematic relations among primitive amplitudes with both massless and massive quarks and potentially simplifying amplitude computations in phenomenological contexts.

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.

Figures (1)

• Figure 1: Diagrams showing how the two outermost soft currents $J^{\mathrm{soft}}_a$ and $J^{\mathrm{soft}}_{a+1}$ may couple to the hard particle $\hat{2}$. There may be further soft currents attached to the hard lines below the dashed line.