Abstract
In this paper we provide detailed proofs for some of the uniqueness results presented in ref. [1]. 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.
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