Multiloop Soft Theorem for Gravitons and Dilatons in the Bosonic String

TL;DR

This paper extends soft theorems for gravitons and dilatons to multiloop amplitudes in bosonic string theory with compactified dimensions, showing that soft behaviors are preserved at one loop for D>4, with specific infrared divergence considerations.

Contribution

It demonstrates that soft theorems for gravitons and dilatons hold at multiloop level in bosonic string theory with compactification, revealing the role of infrared divergences.

Findings

Soft theorems are valid at one loop for D>4.

Infrared divergences are independent of external particles for D>4.

Additional subsubleading terms vanish at one loop for the dilaton.

Abstract

We construct, in the closed bosonic string, the multiloop amplitude involving $N$ tachyons and one massless particle with $26 -D$ compactified directions, and we show that at least for $D>4$, the soft behaviors of the graviton and dilaton satisfy the same soft theorems as at the tree level, up to one additional term at the subsubleading order, which can only contribute to the dilaton soft behavior and which we show is zero at least at one loop. This is possible, since the infrared divergences due to the non-vanishing tachyon and dilaton tadpoles do not depend on the number of external particles and are therefore the same both in the amplitude with the soft particle and in the amplitude without the soft particle. Therefore this leaves unchanged the soft operator acting on the amplitude without the soft particle. The additional infrared divergence appearing for $D \leq 4$ depend on the number of external legs and must be understood on their own.