The SAGEX Review on Scattering Amplitudes, Chapter 7: Positive Geometry of Scattering Amplitudes

Abstract

Scattering amplitudes are both a wonderful playground to discover novel ideas in Quantum Field Theory and simultaneously of immense phenomenological importance to make precision predictions for e.g.~particle collider observables and more recently also for gravitational wave signals. In this review chapter, we give an overview of some of the exciting recent progress on reformulating QFT in terms of mathematical, geometric quantities, such as polytopes, associahedra, Grassmanians, and the amplituhedron. In this novel approach, standard notions of locality and unitarity are derived concepts rather than fundamental ingredients in the construction which might give us a handle on a number of open questions in QFT that have evaded an answer for decades. We first give a basic summary of positive geometry, before discussing the associahedron -- one of the simplest physically relevant geometric examples -- and its relation to tree-level scattering amplitudes in bi-adjoint ${\varphi }^3$ theory. Our second example is the amplituhedron construction for scattering amplitudes in planar maximally supersymmetric Yang-Mills theory.

35 pages, see also the overview article arXiv:2203.13011; updated references to match journal article

References in corpus (13)

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• Gluon scattering amplitudes at strong coupling
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• BFKL Pomeron, Reggeized gluons and Bern-Dixon-Smirnov amplitudes
• On the Singularity Structure of Maximally Supersymmetric Scattering Amplitudes
• Stratifying On-Shell Cluster Varieties: the Geometry of Non-Planar On-Shell Diagrams
• Matching polytopes, toric geometry, and the non-negative part of the Grassmannian
• All-loop cuts from the Amplituhedron
• Weights and recursion relations for ${\varphi }^p$ tree amplitudes from the positive geometry
• Momentum Amplituhedron meets Kinematic Associahedron

Cited by in corpus (8)

• The SAGEX Review on Scattering Amplitudes
• Integrated negative geometries in ABJM
• The Loop Momentum Amplituhedron
• Internal boundaries of the loop amplituhedron
• Non-planar BCFW Grassmannian Geometries
• Poles At Infinity in On-shell Diagrams
• Large $N$ analytical functional bootstrap I: 1D CFTs and total positivity
• BCFW like recursion for Deformed Associahedron