Abstract
We elaborate the two-fold simplex-like structures of tree amplitudes in planar maximally supersymmetric Yang-Mills (N=4 SYM), through its connection to a mathematical structure known as the positive Grassmannian. Exploiting the reduced Grassmannian geometry and the matrix form of on-shell recursion relation in terms of super momentum twistors, we manifest that tree amplitudes can be remarkably refined via the essential building blocks named as fully-spanning cells. For a fixed number of negative helicities, an amplitude can be entirely captured by finite, compact information of the relevant fully-spanning cells up to an arbitrarily large number of external particles.
Highlights
In recent years, enormous progress on scattering amplitudes has been made using various modern approaches beyond Feynman diagrams
Amplitudes of N 1⁄4 4 SYM in the planar limit are most understood due to its unmatched symmetries. At both tree and loop levels, dual superconformal invariance manifested by momentum twistors [5], greatly facilitates the calculation of amplitudes and loop integrands in planar N 1⁄4 4 SYM [6]. This is realized by the momentum twistor version of Britto-Cachazo-Feng-Witten (BCFW) recursion relation [7,8], which constructs amplitudes solely from on-shell subamplitudes, eliminating gauge redundancy as well as unphysical internal particles
Another unanticipated magic, namely the positive Grassmannian together with on-shell diagrams and decorated permutations [9,10,11], provides new insights into the on-shell construction of amplitudes. This is mostly achieved in the space of massless spinors, while transforming its entire machinery into momentum twistor space brings extra complexity [12], since each momentum twistor is not characterized by the momentum of its literally corresponding particle, but a kinematic mixture of numerous adjacent particles
Summary
Enormous progress on scattering amplitudes has been made using various modern approaches beyond Feynman diagrams (see e.g., [1,2,3,4] for reviews). At both tree and loop levels, dual superconformal invariance manifested by (super) momentum twistors [5], greatly facilitates the calculation of amplitudes and loop integrands in planar N 1⁄4 4 SYM [6] This is realized by the momentum twistor version of Britto-Cachazo-Feng-Witten (BCFW) recursion relation [7,8], which constructs amplitudes solely from on-shell subamplitudes, eliminating gauge redundancy as well as unphysical internal particles. Back to planar N 1⁄4 4 SYM, to enhance the advantage brought by positive Grassmannian, we introduce another interesting excursion which brings even more insights and richer structures of amplitudes [19], at tree level for the moment It is a purely geometric approach working in momentum twistor space without referring to on-shell diagrams and decorated permutations, through establishing the exact correspondence between Grassmannian geometric configurations and Yangian invariants generated by recursion. For a fixed number of negative helicities, an amplitude can be entirely captured by finite characteristic objects called fully spanning cells up to an arbitrarily large number of external particles
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