Scattering Amplitudes and the Positive Grassmannian

By N. Arkani-Hamed, J. Bourjaily, F. Cachazo, A. Goncharov, Alexander Postnikov, J. Trnka. 2012

378 citations recorded by the source corpus.

We establish a direct connection between scattering amplitudes in planar four-dimensional theories and a remarkable mathematical structure known as the positive Grassmannian. The central physical idea is to focus on on-shell diagrams as objects of fundamental importance to scattering amplitudes. We show that the all-loop integrand in N=4 SYM is naturally represented in this way. On-shell diagrams in this theory are intimately tied to a variety of mathematical objects, ranging from a new graphical representation of permutations to a beautiful stratification of the Grassmannian G(k,n) which generalizes the notion of a simplex in projective space. All physically important operations involving on-shell diagrams map to canonical operations on permutations; in particular, BCFW deformations correspond to adjacent transpositions. Each cell of the positive Grassmannian is naturally endowed with positive coordinates and an invariant measure which determines the on-shell function associated with the diagram. This understanding allows us to classify and compute all on-shell diagrams, and give a geometric understanding for all the non-trivial relations among them. Yangian invariance of scattering amplitudes is transparently represented by diffeomorphisms of G(k,n) which preserve the positive structure. Scattering amplitudes in (1+1)-dimensional integrable systems and the ABJM theory in (2+1) dimensions can both be understood as special cases of these ideas. On-shell diagrams in theories with less (or no) supersymmetry are associated with exactly the same structures in the Grassmannian, but with a measure deformed by a factor encoding ultraviolet singularities. The Grassmannian representation of on-shell processes also gives a new understanding of the all-loop integrand for scattering amplitudes, presenting all integrands in a novel dLog form which directly reflects the underlying positive structure.

Subfields: Geometry and Topology, Nuclear and High Energy Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/28bbe358c4037427b1f168929bfca36866a3be67

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Citations in the corpus

Space-time S-matrix and flux tube S-matrix II. Extracting and matching data (2013, background): “We shall make extensive use, in particular, of the work [7] which builds on the recent developments about the all-loop integrand [8,9] and where all the one-loop amplitudes were given in manifestly conformally invariant notation.”
Space-time S-matrix and flux-tube S-matrix III. The two-particle contributions (2014, background): “It seems also propitious to look for connections between the OPE program and recent perturbative approaches such as the Integrand [42], the Positive Grassmanian [43], or the Amplituhedron [44].”
Lectures on differential equations for Feynman integrals (2014, background): “See also the discussion in [60] regarding the expected transcendental weight properties of the integrated result.”
General Argyres-Douglas theory (2012, background): “ach AD theory, its cluster coordinates correspond to a planar bipartite graph on a disk. Such planar bipartite graph has been used extensively in studying the scattering amplitude of N = 4 SYM theory [65]. This same exact mathematical structure appears in two seemingly completely diﬀerent physical s”