Abstract
Inspired by the idea of viewing amplitudes in mathcal{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 mathcal{N}=4 SYM, which can also be thought of as “bosonizing” superamplitudes in non-chiral superspace. Remarkably all tree-level amplitudes in mathcal{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-shell-diagram 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 mathcal{N}=4 SYM.
Highlights
Theories in general dimension, the natural kinematic space is the space of Mandelstam variables, and differential forms there, dubbed “scattering forms”, have very different meaning
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
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
Summary
We will refer to the configuration space for n massless momenta as Γn, where both λ’s and λ’s form 2 × n matrices, and these two matrices are subject to momentum conservation: n. For a massless particle with helicity ±1, it is natural to dress it with h This already allows one to combine different helicity amplitudes with gluons or photons, into a 2n-form with no little-group weight. Note that the 4-form for a pair of fermions have the same unit with the form for two spin-one particles In this way, we can combine different amplitudes in massless QCD/QED, at least for theories with no more than four flavors of fermions, e.g. Fntr,QeeCD = Fntregeluons + Fqtqre+e(n−2) g + · · · , which contains the form for n-gluon tree amplitudes in YangMills theory, that with (n−2) gluons and a pair of quarks qq, etc. This is certainly not the case for a general differential form on Γn, i.e. it can contain contractions like λ dλ or [λdλ], the 2n form we just defined is a special form on Γn, where we can safely distinguish between spinor indices of λ, λand those of dλ, dλ
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