Connecting Infinity to Soft Factors

TL;DR

This paper explores a deformation of graviton scattering amplitudes connecting hard and soft limits, providing new methods to compute large $z$ behavior and simplifying amplitude calculations using soft limits and CHY-like techniques.

Contribution

It introduces a deformation framework linking hard and soft graviton limits, enabling direct computation of amplitude behavior and simplifying complex calculations with new soft-limit techniques.

Findings

Reproduces known large $z$ behavior for gravity amplitudes.

Derives simple formulas for NMHV and N$^2$MHV amplitudes.

Supports results with CHY-like computational methods.

Abstract

In this note we study tree-level scattering amplitudes of gravitons under a natural deformation which in the large $z$ limit can be interpreted either as a $k$-hard-particle limit or as a $(n-k)$-soft-particle limit. When $k=2$ this becomes the standard BCFW deformation while for $k=3$ it leads to the Risager deformation. The hard- to soft-limit map we define motivates a way of computing the leading order behavior of amplitudes for large $z$ directly from soft limits. We check the proposal by applying the $k=3$ and $k=4$ versions to NMHV and N$^2$MHV gravity amplitudes respectively. The former reproduces in a few lines the result recently obtained by using CHY-like techniques in \cite{BCL}. The N$^2$MHV formula is also remarkably simple and we give support for it using a CHY-like computation. In the $k=2$ case applied to any gravity amplitude, the multiple soft-limit analysis reproduces the correct ${\cal O}(z^{-2})$ behavior while explicitly showing the source of the mysterious cancellation among Feynman diagrams that tames the behavior from the ${\cal O}(z^{n-5})$ of individual Feynman diagrams down to the ${\cal O}(z^{-2})$ of the amplitude.