Abstract
The scattering amplitudes of planar $$ \mathcal{N} $$ = 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 nonplanar sector of $$ \mathcal{N} $$ = 4 super-Yang-Mills theory.
Highlights
In this paper, we investigate how some of these properties carry over to the nonplanar sector
The dual formulation of planar N = 4 super-Yang-Mills scattering amplitudes using onshell diagrams and the positive Grassmannian makes manifest that the integrand has only logarithmic singularities, and can be written in a dlog form
We gave additional nontrivial evidence for the conjecture that only logarithmic singularities appear in nonplanar amplitudes [52, 53], which is another characteristic feature of planar amplitudes resulting from the amplituhedron construction
Summary
We summarize known properties of planar amplitudes in N = 4 SYM theory that we wish to carry beyond the planar limit to amplitudes of the full theory We emphasize those features associated with the amplituhedron construction. The variables are associated with the faces of each diagram, are globally defined for all diagrams, and allow us to define a unique integrand by appropriately symmetrizing over the faces [31] With these variables, we can sum all diagrams under one integration symbol and write an L-loop amplitude as. The integrand form dI for the n-point amplitude is a unique rational function with many extraordinary properties that we will review . Effective ways of constructing the integrand are unitarity cut methods [62,63,64] or BCFW recursion relations [31, 65]
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