Uniqueness from gauge invariance and the Adler zero
Laurentiu Rodina
TL;DR
This work delivers detailed proofs that Yang-Mills and General Relativity tree amplitudes are uniquely fixed by gauge invariance given a minimal singularity structure, and that scalar theories (NLSM and DBI) are fixed by Adler zero together with locality. The authors develop constrained gauge invariance and a soft-limit framework to order-by-order demonstrate that locality and unitarity emerge from these symmetry principles, without assuming them a priori. They extend the analysis to general and nonlocal singularities, proving a sharp uniqueness result for the canonical cases and proposing broader conjectures about amplitude polynomials and numerators. The results have implications for formalisms like BCFW, BCJ, and CHY by clarifying the extent to which gauge symmetry alone constrains scattering amplitudes and the role of singularity structure in determining locality and unitarity.
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.