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

Abstract

We study the six-point NMHV ratio function in planar $\mathcal{N}=4$ 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.

45 pages, 7 figures, 4 ancillary files. v2: typos corrected; version to appear in JHEP

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• Classical Polylogarithms for Amplitudes and Wilson Loops
• Hexagon Wilson loop = six-gluon MHV amplitude
• The Two-Loop Six-Gluon MHV Amplitude in Maximally Supersymmetric Yang-Mills Theory
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• Symbology for elliptic multiple polylogarithms and the symbol prime
• Bootstrapping elliptic Feynman integrals using Schubert analysis
• Intersection theory rules symbology
• Two-Loop Five-Point Two-Mass Planar Integrals and Double Lagrangian Insertions in a Wilson Loop
• An Infinite Family of Elliptic Ladder Integrals
• Feynman Integrals from Positivity Constraints
• Poles At Infinity in On-shell Diagrams
• A Two-Loop Four-Point Form Factor at Function Level