Uniqueness from gauge invariance and the Adler zero

TL;DR

This paper proves the uniqueness of certain tree-level scattering amplitudes in gauge theories and gravity based on gauge invariance and soft limits, highlighting the fundamental constraints that determine these amplitudes.

Contribution

It provides detailed proofs that gauge invariance and soft limit conditions uniquely determine tree-level amplitudes in Yang-Mills, gravity, NLSM, and DBI theories, with new evidence for broader conjectures.

Findings

Yang-Mills and gravity amplitudes are fixed by gauge invariance with minimal assumptions.

Scalar and DBI amplitudes are determined by locality and the Adler zero condition.

Evidence suggests gauge invariance alone may imply locality and unitarity, leading to non-unique solutions.

Abstract

In this paper we provide detailed proofs for some of the uniqueness results presented in arXiv:1612.02797. We show that: (1) Yang-Mills and General Relativity tree-level amplitudes are completely determined by gauge invariance in $n-1$ particles, with minimal assumptions on the singularity structure, (2) scalar non-linear sigma model and Dirac-Born-Infeld tree-level amplitudes are fixed by imposing full locality and the Adler zero condition (vanishing in the single soft limit) on $n-1$ particles. We complete the proofs by showing uniqueness order by order in the single soft expansion for Yang-Mills and General Relativity, and the double soft expansion for NLSM and DBI. We further present evidence for a greater conjecture regarding Yang-Mills amplitudes, that a maximally constrained gauge invariance alone leads to both locality and unitarity, without any assumptions on the existence of singularities. In this case the solution is not unique, but a linear combination of amplitude numerators.