The kinematic algebra from the self-dual sector

Abstract

We identify a diffeomorphism Lie algebra in the self-dual sector of Yang-Mills theory, and show that it determines the kinematic numerators of tree-level MHV amplitudes in the full theory. These amplitudes can be computed off-shell from Feynman diagrams with only cubic vertices, which are dressed with the structure constants of both the Yang-Mills colour algebra and the diffeomorphism algebra. Therefore, the latter algebra is the dual of the colour algebra, in the sense suggested by the work of Bern, Carrasco and Johansson. We further study perturbative gravity, both in the self-dual and in the MHV sectors, finding that the kinematic numerators of the theory are the BCJ squares of the Yang-Mills numerators.

Algebras for amplitudes

“…This duality hints at a Lie algebra structure underlying the kinematic dependence of the amplitudes, akin to the Yang-Mills Lie algebra. Such a kinematic Lie algebra was indeed recently found for MHV amplitudes [12], where it was shown to be inherited from…”

“…This already hints at a possible special role played by three-vertices. An underlying three-vertex structure for n i was found in [12] for MHV amplitudes. That structure is inherited from the self-dual sector of the gauge theory, which will be reviewed below.…”

“…As emphasized in [12], this three-point function is the product of the structure constants of two Lie algebras: obviously one of these factors is f abc , the structure constants of the gauge group of the theory, while the other factor is…”

“…In the case of the more general diffeomorphism algebra, such combinations lead to a singular matrix θ IJ for n ≥ 8 in four dimensions, if we make a random choice Ω µν for each element of the basis Θ I . However, this choice of auxiliary algebra allows for a special treatment of MHV amplitudes, different from the procedure in [12]. Consider the case where the structure constant is…”

“…dimensional vector space. In [12], an explicit color-kinematics dual representation of MHV amplitudes was found in light-cone gauge by using one of the negative helicity particles to define the light-cone direction. In the expression above, we treat all the external particles democratically.…”