Evidence for a Nonplanar Amplituhedron

TL;DR

This work advances the program of extending the amplituhedron paradigm beyond planar ${\cal N}=4$ SYM by constructing pure integrand bases for nonplanar amplitudes that exhibit logarithmic singularities and no poles at infinity. It shows, through explicit two-loop four-point, three-loop four-point, and two-loop five-point examples, that the amplitudes can be represented as sums over diagrammatic pure integrands with unit leading singularities, with all coefficients fixed by homogeneous zero conditions and unitarity cuts up to an overall normalization. The results provide concrete nonplanar evidence for an amplituhedron-like geometric organization, suggesting that nonplanar amplitudes may be determined entirely by zero conditions in analogy with the planar case. If a nonplanar amplituhedron exists, it would illuminate a deeper geometric structure underlying gauge theories and could impact computations in QCD and gravity via color-kinematics duality and the double-copy construction.

Abstract

The scattering amplitudes of planar N = 4 super-Yang-Mills exhibit a number of remarkable analytic structures, including dual conformal symmetry and logarithmic singularities of integrands. The amplituhedron is a geometric construction of the integrand that incorporates these structures. This geometric construction further implies the amplitude is fully specified by constraining it to vanish on spurious residues. By writing the amplitude in a dlog basis, we provide nontrivial evidence that these analytic properties and "zero conditions" carry over into the nonplanar sector. This suggests that the concept of the amplituhedron can be extended to the the nonplanar sector of N = 4 super-Yang-Mills theory.

Figures (9)

• Figure 1: Sample on-shell diagrams. The black and white dots respectively represent MHV and $\overline{\text{MHV}}$ three-point amplitudes. Black lines are on-shell particles.
• Figure 2: A simple amplituhedron example Arkani-Hamed:2014dca. The area of the pentagon formed by the black solid line is the amplituhedron. The points $X_{i,i+2}$ define zeros of a numerator, so the conic given by the outer (blue) solid lines connecting the points represents the numerator.
• Figure 3: The planar two-loop four-point amplitude can be represented in terms of double-box diagrams.
• Figure 4: Example of a nonplanar on-shell diagram.
• Figure 5: The integrals appearing in the two-loop four-point amplitude of ${\cal N} = 4$ SYM theory.