Hidden zeros are equivalent to enhanced ultraviolet scaling and lead to unique amplitudes in Tr($\phi^3$) theory

TL;DR

This paper links hidden amplitude zeros to enhanced ultraviolet scaling in certain theories, proving their equivalence and demonstrating how they uniquely determine amplitudes in Tr(φ^3) and related models.

Contribution

It establishes the equivalence between hidden amplitude zeros and subset enhanced scaling, leading to new insights into amplitude uniqueness in scalar and gauge theories.

Findings

Hidden zeros correspond to subset enhanced scaling under BCFW shifts.

Tr(φ^3) amplitudes are uniquely fixed by zeros and propagator structure.

Zeros, combined with BCJ duality, constrain gluon polarization structures.

Abstract

We investigate the hidden amplitude zeros discovered by Arkani-Hamed et al., which describe a non-trivial 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 $X_{ij}{=}(p_i{+}p_{i+1}{+}\ldots{+}p_{j-1})^2$. This directly applies to Tr($\phi^3$), non-linear sigma model, 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($\phi^3$) amplitudes are uniquely fixed by the zeros, up to an overall normalization, when assuming an ordered and local propagator structure and trivial numerators. In the case of 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 $\lfloor n/2\rfloor$ 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.