Note on Bonus Relations for N=8 Supergravity Tree Amplitudes

TL;DR

This paper demonstrates how bonus relations can simplify the calculation of all tree-level amplitudes in N=8 supergravity, reducing the complexity from (n-2)! to (n-3)! permutations, with explicit examples and a general pattern.

Contribution

It introduces a novel application of bonus relations to simplify supergravity tree amplitudes, reducing permutation sums and providing explicit calculations and a general pattern.

Findings

Simplified supergravity amplitude formulas from (n-2)! to (n-3)! permutations.

Explicit calculations for NMHV and N^2MHV amplitudes.

Identified a general pattern of bonus coefficients for all tree amplitudes.

Abstract

We study the application of non-trivial relations between gravity tree amplitudes, the bonus relations, to all tree-level amplitudes in N=8 supergravity. We show that the relations can be used to simplify explicit formulae of supergravity tree amplitudes, by reducing the known form as a sum of (n-2)! permutations obtained by solving on-shell recursion relations, to a new form as a (n-3)!-permutation sum. We demonstrate the simplification by explicit calculations of the next-to-maximally helicity violating (NMHV) and next-to-next-to-maximally helicity violating (N^2MHV) amplitudes, and provide a general pattern of bonus coefficients for all tree-level amplitudes.