Hidden Zeros Are Equivalent to Enhanced Ultraviolet Scaling, and Lead to Unique Amplitudes in Tr(ϕ^{3}) Theory

We investigate the hidden amplitude zeros which describe a nontrivial vanishing of scattering amplitudes on special external kinematics. We first prove that every type of hidden zero is equivalent to what we call a "subset" enhanced scaling under Britto-Cachazo-Feng-Witten shifts for any rational function built from planar Lorentz invariants X_{ij}=(p_{i}+p_{i+1}+⋯+p_{j-1})^{2}. This directly applies to Tr(ϕ^{3}), nonlinear sigma models, or Yang-Mills-scalar amplitudes, revealing a novel type of enhanced UV scaling in these theories. We also use this observation to prove the conjecture that Tr(ϕ^{3}) amplitudes are uniquely fixed by the zeros, up to an overall normalization, when assuming an ordered and local propagator structure and trivial numerators. In this context, unitarity (residue factorization) may be viewed as a consequence of the zeros. For Yang-Mills theory, we conjecture the zeros, combined with the Bern-Carrasco-Johansson color-kinematic duality in the form of amplitude relations, uniquely fix the ⌊n/2⌋ distinct polarization structures of n-point gluon amplitudes. Our approach opens a new avenue for understanding previous similar uniqueness results, and also extending them beyond tree level for the first time.