Positive Amplitudes In The Amplituhedron

TL;DR

This paper provides evidence that the amplitude form in the amplituhedron is positive inside its geometry, suggesting a new dual formulation and a deeper understanding of scattering amplitudes in planar N=4 SYM.

Contribution

It introduces the conjecture that the superamplitude form is positive within the amplituhedron, a property not evident in current methods, and proposes a potential dual geometric framework.

Findings

Amplitude form is positive inside the amplituhedron.

Positivity is not manifest in existing approaches.

New amplitude expressions derived from global geometric perspective.

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.