Positive Amplitudes In The Amplituhedron

TL;DR

The paper proposes that the amplitude form $\Omega$ is positive inside the amplituhedron for all $n,k,L$, suggesting a dual amplituhedron and a widened geometric picture. It develops a zero-surface construction where the numerator vanishes on all spurious boundaries, yielding invariant, globally positive expressions in several concrete geometries and supported by extensive numerical checks. The work further argues that positive representations, including local and potentially dual formulations, illuminate the positivity of not only the integrand but also the logarithm of the MHV amplitude and the ratio function, hinting at deep geometric constraints on planar $\mathcal{N}=4$ SYM amplitudes. Together, these results motivate a broader program to understand positive geometries and dual descriptions that make positivity manifest beyond triangulations.

Abstract

The all-loop integrand for scattering amplitudes in planar N = 4 SYM is determined by an "amplitude form" with logarithmic singularities on the boundary of the amplituhedron. In this note we provide strong evidence for a new striking property of the superamplitude, which we conjecture to be true to all loop orders: the amplitude form is positive when evaluated inside the amplituhedron. The statement is sensibly formulated thanks to the natural "bosonization" of the superamplitude associated with the amplituhedron geometry. However this positivity is not manifest in any of the current approaches to scattering amplitudes, and in particular not in the cellulations of the amplituhedron related to on-shell diagrams and the positive grassmannian. The surprising positivity of the form suggests the existence of a "dual amplituhedron" formulation where this feature would be made obvious. We also suggest that the positivity is associated with an extended picture of amplituhedron geometry, with the amplituhedron sitting inside a co-dimension one surface separating "legal" and "illegal" local singularities of the amplitude. We illustrate this in several simple examples, obtaining new expressions for amplitudes not associated with any triangulations, but following in a more invariant manner from a global view of the positive geometry.