Notes on scattering amplitudes as differential forms

Abstract

Inspired by the idea of viewing amplitudes in N = 4 SYM as differential forms on momentum twistor space, we introduce differential forms on the space of spinor variables, which combine helicity amplitudes in any four-dimensional gauge theory as a single object. In this note we focus on such differential forms in N = 4 SYM, which can also be thought of as "bosonizing" superamplitudes in non-chiral superspace. Remarkably all tree-level amplitudes in N = 4 SYM combine to a d log form in spinor variables, which is given by pushforward of canonical forms of Grassmannian cells. The tree forms can also be obtained using BCFW or inverse-soft construction, and we present all-multiplicity expression for MHV and NMHV forms to illustrate their simplicity. Similarly all-loop planar integrands can be naturally written as d log forms in the Grassmannian/on-shelldiagram picture, and we expect the same to hold beyond the planar limit. Just as the form in momentum twistor space reveals underlying positive geometry of the amplituhedron, the form in terms of spinor variables strongly suggests an "amplituhedron in momentum space". We initiate the study of its geometry by connecting it to the moduli space of Witten's twistor-string theory, which provides a pushforward formula for tree forms in N = 4 SYM.

The momentum amplituhedron

“…The push-forward of the Grassmannian top form through (2.17) is therefore: where q,q are defined in (2.9). This formula agrees with the result found in [12]. This calculation can be easily generalized to any MHV amplitude.…”

“…Therefore, we are interested in the following sequences of brackets: We want to show that the number of sign flips equals k − 2 in the sequence (2.23) and k in the sequence (2.24). This corresponds to the condition (N,Ñ ) = (k − 2, k) in the conjecture in [12]. It is easy to see that the number of sign flips in the sequence (2.24) is k since the formula in (2.18) forỸ is the definition of the ordinary amplituhedron [6] with m = 2 and k = k .…”

“…One can notice that the canonical variables are just an uplift of the formula found in [12] 4 . The push-forward is computed as After using our procedure (2.37) for extracting the amplitude, we find that the expression (3.17) reduces to the formula found in [12]. While for the denominator it can be easily seen, the numerator requires a more careful analysis.…”

“…This result agrees with the one we get for the ordinary amplituhedron A (2) n,n−2 . Let us notice that, when Ω n,k is written explicitly as a logarithmic form (3.8), it can be easily compared with results in [12]: it is sufficient to project through a fixed Y , which results in removing all Y -dependence, and to consider the differentials to act on λ. Finally, the volume function for MHV n amplitudes is…”

“…In order to achieve our goal, we will first introduce its bosonized version: (Λ i ,Λ i ). By assuming that the external kinematic data Λ andΛ satisfy particular positivity conditions, we will reproduce the sign flips postulated in [12]. Additionally, further constraints entangling Λ andΛ will enforce positivity of Mandelstam variables.…”