# On the exactness of soft theorems

## TL;DR

This paper investigates the nature of soft theorems in massless theories, distinguishing between those protected by symmetries and those affected by divergences, with evidence from string theory amplitudes.

## Contribution

It clarifies the distinction between exact symmetry-based soft theorems and those modified by divergences, using current algebra and string amplitude analysis.

## Key findings

- Type (II) soft theorems are exact and symmetry-protected.
- Type (I) soft theorems are broken by divergences.
- String amplitudes confirm the validity of symmetry-based soft theorems up to O(α'^6).

## Abstract

Soft behaviours of S-matrix for massless theories reflect the underlying symmetry principle that enforces its masslessness. As an expansion in soft momenta, sub-leading soft theorems can arise either due to (I) unique structure of the fundamental vertex or (II) presence of enhanced broken-symmetries. While the former is expected to be modified by infrared or ultraviolet divergences, the latter should remain exact to all orders in perturbation theory. Using current algebra, we clarify such distinction for spontaneously broken (super) Poincar\'e and (super) conformal symmetry. We compute the UV divergences of DBI, conformal DBI, and A-V theory to verify the exactness of type (II) soft theorems, while type (I) are shown to be broken and the soft-modifying higher-dimensional operators are identified. As further evidence for the exactness of type (II) soft theorems, we consider the alpha' expansion of both super and bosonic open strings amplitudes, and verify the validity of the translation symmetry breaking soft-theorems up to O(alpha'^6). Thus the massless S-matrix of string theory "knows" about the presence of D-branes.