# Subleading Soft Graviton Theorem for Loop Amplitudes

## TL;DR

This paper proves the subleading soft graviton theorem to all orders in perturbation theory within superstring field theories, establishing key properties of quantum gravity amplitudes involving soft gravitons.

## Contribution

It provides a general proof of the subleading soft graviton theorem for loop amplitudes in superstring theories, valid for arbitrary external states and soft gravitons.

## Key findings

- Proof of subleading soft graviton theorem to all orders in superstring theories.
- Validation of leading soft graviton theorem for multiple external states and gravitons.
- Results applicable to any finite quantum gravity theory respecting coordinate invariance.

## Abstract

Superstring field theory gives expressions for heterotic and type II string loop amplitudes that are free from ultraviolet and infrared divergences when the number of non-compact space-time dimensions is five or more. We prove the subleading soft graviton theorem in these theories to all orders in perturbation theory for S-matrix elements of arbitrary number of finite energy external states but only one external soft graviton. We also prove the leading soft graviton theorem for arbitrary number of finite energy external states and arbitrary number of soft gravitons. Since our analysis is based on general properties of one particle irreducible effective action, the results are valid in any theory of quantum gravity that gives finite result for the S-matrix order by order in perturbation theory without violating general coordinate invariance.