Amplitudes and Spinor-Helicity in Six Dimensions
Clifford Cheung, Donal O'Connell
TL;DR
The paper develops a six-dimensional spinor-helicity formalism by solving the Dirac equation in SO(6) and organizing states under the SU(2)×SU(2) little group, yielding compact expressions for three-, four-, and five-point tree amplitudes in Yang-Mills theory and their gravity counterparts via KLT. It then introduces a covariant BCFW framework suited to higher dimensions, enabling efficient computation of higher-point amplitudes in 6D and, upon dimensional reduction, in four dimensions. The gravity amplitudes emerge in remarkably simple, gauge-invariant forms (notably the 4-point amplitude) and the method extends naturally toward higher dimensions and potentially to the (2,0) theory, with appendices detailing the Clifford algebra and technical rearrangements. Overall, the work showcases how six-dimensional spinor-helicity variables encode physical states and simplify on-shell S-matrix structures, with clear paths to further dimensional generalizations and supersymmetric extensions.
Abstract
The spinor-helicity formalism has become an invaluable tool for understanding the S-matrix of massless particles in four dimensions. In this paper we construct a spinor-helicity formalism in six dimensions, and apply it to derive compact expressions for the three, four and five point tree amplitudes of Yang-Mills theory. Using the KLT relations, it is a straightforward process to obtain amplitudes in linearized gravity from these Yang-Mills amplitudes; we demonstrate this by writing down the gravitational three and four point amplitudes. Because there is no conserved helicity in six dimensions, these amplitudes describe the scattering of all possible polarization states (as well as Kaluza-Klein excitations) in four dimensions upon dimensional reduction. We also briefly discuss a convenient formulation of the BCFW recursion relations in higher dimensions.