On Tree Amplitudes in Gauge Theory and Gravity

TL;DR

This paper provides a physical understanding of the conditions under which BCFW recursion relations are valid for computing tree amplitudes in gauge theories and gravity, revealing symmetry properties at infinite momentum.

Contribution

It introduces a background field approach and symmetry analysis to explain amplitude behavior at infinity, enabling the use of BCFW recursion in broader contexts.

Findings

Certain gauge and gravity amplitudes vanish at infinity

BCFW recursion applies to all gluon and graviton tree amplitudes in any dimension

Enhanced spin symmetries govern amplitude behavior at large momentum

Abstract

The BCFW recursion relations provide a powerful way to compute tree amplitudes in gauge theories and gravity, but only hold if some amplitudes vanish when two of the momenta are taken to infinity in a particular complex direction. This is a very surprising property, since individual Feynman diagrams all diverge at infinite momentum. In this paper we give a simple physical understanding of amplitudes in this limit, which corresponds to a hard particle with (complex) light-like momentum moving in a soft background, and can be conveniently studied using the background field method exploiting background light-cone gauge. An important role is played by enhanced spin symmetries at infinite momentum--a single copy of a "Lorentz" group for gauge theory and two copies for gravity--which together with Ward identities give a systematic expansion for amplitudes at large momentum. We use this to study tree amplitudes in a wide variety of theories, and in particular demonstrate that certain pure gauge and gravity amplitudes do vanish at infinity. Thus the BCFW recursion relations can be used to compute completely general gluon and graviton tree amplitudes in any number of dimensions. We briefly comment on the implications of these results for computing massive 4D amplitudes by KK reduction, as well understanding the unexpected cancelations that have recently been found in loop-level gravity amplitudes.