Three Applications of a Bonus Relation for Gravity Amplitudes

TL;DR

This paper explores bonus relations in gravity amplitudes, demonstrating their utility in simplifying formulas for MHV amplitudes and providing new proofs and connections to the BGK conjecture.

Contribution

The paper applies bonus relations to relate different amplitude formulas, offering new proofs and confirming the BGK formula's recursion compatibility.

Findings

Proved a formula by Elvang and Freedman.

Derived a new formula based on Bedford et al.

Provided an alternative proof of Mason and Skinner's formula.

Abstract

Arkani-Hamed et. al. have recently shown that all tree-level scattering amplitudes in maximal supergravity exhibit exceptionally soft behavior when two supermomenta are taken to infinity in a particular complex direction, and that this behavior implies new non-trivial relations amongst amplitudes in addition to the well-known on-shell recursion relations. We consider the application of these new bonus relations to MHV amplitudes, showing that they can be used quite generally to relate (n-2)!-term formulas typically obtained from recursion relations to (n-3)!-term formulas related to the original BGK conjecture. Specifically we provide (1) a direct proof of a formula presented by Elvang and Freedman, (2) a new formula based on one due to Bedford et. al., and (3) an alternate proof of a formula recently obtained by Mason and Skinner. Our results also provide the first direct proof that the conjectured BGK formula, only very recently proven via completely different methods, satisfies the on-shell recursion.