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Positivity, Grassmannian Geometry and Simplex-like Structures of Scattering Amplitudes

This article revisits and elaborates the significant role of positive geometry of momentum twistor Grassmannian for planar N=4 SYM scattering amplitudes. First we establish the fundamentals of positive Grassmannian geometry for tree amplitudes. Then we formulate this subject around these four major facets: 1. Deriving the tree and 1-loop BCFW recursion relations solely from positivity, after introducing the simple building blocks called positive components. 2. Applying Grassmannian geometry and Pluecker coordinates to determine the signs in N$^2$MHV homological identities. Most of them in fact reflect the secret incarnation of the simple 6-term NMHV identity. 3. Deriving the stacking positivity relation, which is powerful for parameterizing the matrix representative in terms of positive variables, of a given Grassmannian geometric configuration. 4. Introducing a highly refined formalism for tree BCFW recursion relation, which reveals its two-fold simplex-like structures. First, the BCFW contour in terms of (reduced) Grassmannian geometry representatives is dissected into a triangle-shape sum, as only a small fraction of the sum needs to be explicitly identified. Second, this fraction can be further dissected, according to different growing modes and parameters. The growing modes possess the shapes of solid simplices of various dimensions, with which infinite number of BCFW cells can be entirely captured by the characteristic objects called fully-spanning cells. We find that for a given k, beyond n=4k+1 there is no more new fully-spanning cell, which signifies the essential termination of the recursive growth of BCFW cells. As n increases beyond the termination point, the BCFW contour simply replicates itself according to the simplex-like patterns, which enables us to master all BCFW cells once for all without actually identifying most of them.