Color-kinematics duality for QCD amplitudes

Abstract

We show that color-kinematics duality is present in tree-level amplitudes of quantum chromodynamics with massive flavored quarks. Starting with the color structure of QCD, we work out a new color decomposition for n-point tree amplitudes in a reduced basis of primitive amplitudes. These primitives, with k quark-antiquark pairs and (n − 2k) gluons, are taken in the (n − 2)!/k! Melia basis, and are independent under the color-algebra Kleiss-Kuijf relations. This generalizes the color decomposition of Del Duca, Dixon, and Maltoni to an arbitrary number of quarks. The color coefficients in the new decomposition are given by compact expressions valid for arbitrary gauge group and representation. Considering the kinematic structure, we show through explicit calculations that color-kinematics duality holds for amplitudes with general configurations of gluons and massive quarks. The new (massive) amplitude relations that follow from the duality can be mapped to a well-defined subset of the familiar BCJ relations for gluons. They restrict the amplitude basis further down to (n − 3)!(2k − 2)/k! primitives, for two or more quark lines. We give a decomposition of the full amplitude in that basis. The presented results provide strong evidence that QCD obeys the color-kinematics duality, at least at tree level. The results are also applicable to supersymmetric and D-dimensional extensions of QCD.

Proof of the fundamental BCJ relations for QCD amplitudes

“…We will use this freedom to fix a particle at position 1 and a second particle at position n. Since we also assume that there is at least one gluon, let us label this gluon by 2 g . In [14] Johansson and Ochirov conjecture that which follows from eq. (2.6) by momentum conservation.…”

Manifesting enhanced cancellations in supergravity: integrands versus integrals

“…In this construction the duality (2.2) needs to be manifest in only one of the two gauge theories [25,49]. This construction also extends to cases where the gauge theory includes fundamental-representation matter particles [50,51].…”

BCJ numerators from reduced Pfaffian

“…More generally, once we have two gauge theories (with matter interactions or supersymmetry extension) that both can satisfy the color-kinematic duality, double copy constructions will always lead to gravity scattering amplitudes [7,[18][19][20][21][22][23].…”

“…For a given tree, we replace the end points of each path by k, and the internal vertices by k µ k ν . For example, we have: 18) where V 1 = 1, and V 2 can be either the root 1, or one of the φ(i)'s. 9 Finally, n NLSM (1, β β β, n) is obtained by summing up all these contributing trees.…”