Generating Tree Amplitudes in N=4 SYM and N=8 SG

TL;DR

This paper develops generating functions for n-point tree amplitudes in N=4 super Yang-Mills and N=8 supergravity, enabling efficient calculations and exploring symmetry structures and recursion relations.

Contribution

It introduces new generating functions for MHV and NMHV amplitudes, and analyzes their properties and implications for recursion and symmetry in supergravity and gauge theories.

Findings

Generated functions obey SUSY Ward identities.

MHV generating function simplifies loop amplitude calculations.

NMHV recursion relations are valid for certain supergravity amplitudes.

Abstract

We study n-point tree amplitudes of N=4 super Yang-Mills theory and N=8 supergravity for general configurations of external particles of the two theories. We construct generating functions for n-point MHV and NMHV amplitudes with general external states. Amplitudes derived from them obey SUSY Ward identities, and the generating functions characterize and count amplitudes in the MHV and NMHV sectors. The MHV generating function provides an efficient way to perform the intermediate state helicity sums required to obtain loop amplitudes from trees. The NMHV generating functions rely on the MHV-vertex expansion obtained from recursion relations associated with a 3-line shift of external momenta involving a reference spinor |X]. The recursion relations remain valid for a subset of N=8 supergravity amplitudes which do not vanish asymptotically for all |X]. The MHV-vertex expansion of the n-graviton NMHV amplitude for n=5,6,...,11 is independent of |X] and exhibits the asymptotic behavior z^{n-12}. This presages difficulties for n > 12. Generating functions show how the symmetries of supergravity can be implemented in the quadratic map between supergravity and gauge theory embodied in the KLT and other similar relations between amplitudes in the two theories.