# The B-field soft theorem and its unification with the graviton and dilaton

## TL;DR

This paper formulates a unified soft theorem for gravitons, dilatons, and B-fields in string theories, filling a gap in understanding the soft behavior of the Kalb-Ramond B-field at tree level.

## Contribution

It introduces a new soft theorem for the B-field, unifies it with graviton and dilaton soft theorems, and demonstrates gauge-invariant amplitude decompositions at all orders.

## Key findings

- Soft theorem fixed by on-shell gauge invariance at subleading order.
- Explicit verification in bosonic and Type II superstring theories.
- Identification of gauge-invariant terms at higher orders that do not follow the soft theorem.

## Abstract

In theories of Einstein gravity coupled with a dilaton and a two-form, a soft theorem for the two-form, known as the Kalb-Ramond B-field, has so far been missing. In this work we fill the gap, and in turn formulate a unified soft theorem valid for gravitons, dilatons and B-fields in any tree-level scattering amplitude involving the three massless states. The new soft theorem is fixed by means of on-shell gauge invariance and enters at the subleading order of the graviton's soft theorem. In contrast to the subsubleading soft behavior of gravitons and dilatons, we show that the soft behavior of B-fields at this order cannot be fully fixed by gauge invariance. Nevertheless, we show that it is possible to establish a gauge invariant decomposition of the amplitudes to any order in the soft expansion. We check explicitly the new soft theorem in the bosonic string and in Type II superstring theories, and furthermore demonstrate that, at the next order in the soft expansion, totally gauge invariant terms appear in both string theories which cannot be factorized into a soft theorem.