Color-factor symmetry and BCJ relations for QCD amplitudes

Robert W Brown,Stephen G Naculich

doi:10.1007/jhep11(2016)060

Abstract

Tree-level n-point gauge-theory amplitudes with n − 2k gluons and k pairs of (massless or massive) particles in the fundamental (or other) representation of the gauge group are invariant under a set of symmetries that act as momentum-dependent shifts on the color factors in the cubic decomposition of the amplitude. These symmetries lead to gauge-invariant constraints on the kinematic numerators. They also directly imply the BCJ relations among the Melia-basis primitive amplitudes previously obtained by Johansson and Ochirov.

Highlights

JHEP11(2016)060 established in ref. [9], states that the amplitude is invariant under certain momentumdependent shifts of the color factors
Johansson and Ochirov found that the BCJ relations satisfied by the Melia primitives are directly tied to the presence of external gluons in the amplitude
We describe the cubic vertex expansion for An,k that was introduced in ref. [9], and show that the color-factor symmetry leads to gauge-invariant constraints on the kinematic numerators

Summary

Proper decompositions of gauge-theory amplitudes

We describe proper decompositions of gauge-theory amplitudes with particles in the adjoint and fundamental representations; that is, decompositions in terms of an independent set of color factors and gauge-invariant primitive amplitudes. The Melia basis is the set of color-ordered amplitudes A(1, γ(3), · · · , γ(n), 2), where γ is any permutation of {3, · · · , n} such that the set of k − 1 ψand k − 1 ψ in γ form a Dyck word of length 2k − 2. Having chosen the color-ordered amplitudes in the Melia basis A(1, γ, 2) to be the primitive amplitudes in a proper decomposition (2.3), one may ask what is the corresponding set of independent color factors C1γ2? Having defined the primitive amplitudes and corresponding color factors, we can write the proper (Melia-Johansson-Ochirov) decomposition of the amplitude An,k as. This expression is equivalent to eq (2.1), as was explicitly verified for n ≤ 8 (and for n = 9, k = 4) in ref.

BCJ relations for QCD amplitudes

Color-factor shifts

BCJ relations from color-factor symmetry

Conclusions