On the soft limit of closed string amplitudes with massive states

TL;DR

This paper investigates the universal properties of soft graviton emissions in closed string amplitudes with massive states at tree level, extending known results from open strings and analyzing modifications across different string theories.

Contribution

It extends the analysis of soft limits to closed strings with massive states, confirming universality at sub-leading order for superstrings and exploring modifications for bosonic strings.

Findings

Universality of soft graviton behavior at sub-leading order in superstring amplitudes.

Modification of soft behavior in bosonic string amplitudes.

Confirmation of single-valued projection for massive string amplitudes.

Abstract

We extend our analysis of the soft behaviour 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 behaviour 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 favour 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 behaviour of the anti-symmetric tensor and on loop corrections.