On the soft limit of closed string amplitudes with massive states

Abstract

We extend our analysis of the soft behavior of string amplitudes with massive insertions to closed strings at tree level (sphere). Relying on our previous results for open strings on the disk and on KLT formulae we check universality of the soft behavior for gravitons to sub-leading order for superstring amplitudes and show how this gets modified for bosonic strings. At sub-sub-leading order we argue in favor of universality for superstrings on the basis of OPE of the vertex operators and gauge invariance for the soft graviton. The results are illustrated by explicit examples of 4-point amplitudes with one massive insertion in any dimension, including D=4, where use of the helicity spinor formalism drastically simplifies the expressions. As a by-product of our analysis we confirm that the ‘single valued projection’ holds for massive amplitudes, too. We briefly comment on the soft behavior of the anti-symmetric tensor and on loop corrections.