Multi-Loop Positivity of the Planar ${\cal N}=4$ SYM Six-Point Amplitude

TL;DR

This paper investigates the positivity properties of the six-point NMHV ratio function in planar ${ m f N}=4$ SYM theory, providing evidence that it remains positive through multiple loops and exploring implications for underlying geometric structures.

Contribution

It demonstrates the positivity of the six-point NMHV ratio function across various loops and kinematic regions, suggesting a deeper geometric and structural property of the amplitude.

Findings

Ratio function is positive at one, two, and five loops.

Evidence of monotonicity in a radial direction.

BDS-like normalized MHV amplitude remains positive through five loops.

Abstract

We study the six-point NMHV ratio function in planar ${\cal N}=4$ SYM theory in the context of positive geometry. The Amplituhedron construction of the integrand for the amplitudes provides a kinematical region in which the integrand was observed to be positive. It is natural to conjecture that this property survives integration, i.e. that the final result for the ratio function is also positive in this region. Establishing such a result would imply that preserving positivity is a surprising property of the Minkowski contour of integration and it might indicate some deeper underlying structure. We find that the ratio function is positive everywhere we have tested it, including analytic results for special kinematical regions at one and two loops, as well as robust numerical evidence through five loops. There is also evidence for not just positivity, but monotonicity in a "radial" direction. We also investigate positivity of the MHV six-gluon amplitude. While the remainder function ceases to be positive at four loops, the BDS-like normalized MHV amplitude appears to be positive through five loops.