Connecting infinity to soft factors

A bstract 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 [1]. 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 𝒪( z −2 ) behavior while explicitly showing the source of the mysterious cancellation among Feynman diagrams that tames the behavior from the 𝒪( z n −5 ) of individual Feynman diagrams down to the 𝒪( z −2 ) of the amplitude.